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# The role of self-similarity in singularities of PDE’s
Jens Eggers∗ and Marco A. Fontelos† ∗School of Mathematics, University of
Bristol, University Walk,
Bristol BS8 1TW, United Kingdom
† Instituto de Ciencias Matemáticas, (ICMAT, CSIC-UAM-UCM-UC3M),
C/ Serrano 123, 28006 Madrid, Spain
###### Abstract
We survey rigorous, formal, and numerical results on the formation of point-
like singularities (or blow-up) for a wide range of evolution equations. We
use a similarity transformation of the original equation with respect to the
blow-up point, such that self-similar behaviour is mapped to the fixed point
of a dynamical system. We point out that analysing the dynamics close to the
fixed point is a useful way of characterising the singularity, in that the
dynamics frequently reduces to very few dimensions. As far as we are aware,
examples from the literature either correspond to stable fixed points, low-
dimensional centre-manifold dynamics, limit cycles, or travelling waves. For
each “class” of singularity, we give detailed examples.
## 1 Introduction
Non-linear partial differential equations (PDE’s) are distinguished by the
fact that, starting from smooth initial data, they can develop a singularity
in finite time [1, 2, 3, 4]. 111Of course, there are also many examples of
nonlinear PDE’s for which global existence can be established! Very often,
such a singularity corresponds to a physical event, such as the solution (e.g.
a physical flow field) changing topology, and/or the emergence of a new
(singular) structure, such as a tip, cusp, sheet, or jet. On the other hand, a
singularity can also imply that some essential physics is missing from the
equation in question, which should thus be supplemented with additional terms.
(Even in the latter case, the singularity may still be indicative of a real
physical event).
Consider for example the physical case shown in Fig. 1, which we will treat in
section 4 below. Shown is a snapshot of one viscous fluid dripping into
another fluid, close to the point where a drop of the inner fluid pinches off.
This process is driven by surface tension, which tries to minimise the surface
area between the two fluids. At a particular point $x_{0},t_{0}$ in space and
time, the local radius $h(x,t)$ of the fluid neck goes to zero; this point is
a singularity of the underlying equation of motion. Since the drop breaks into
two pieces, there is no way the problem can be continued without generalising
the formulation to one that includes topological changes. However, in this
review we adopt a broader view of what constitutes a singularity. We consider
it as such whenever there is a loss of regularity, which implies that there is
a length scale which goes to zero. This is the situation under which one
expects self-similar behaviour, which is our guiding principle.
Figure 1: A drop of Glycerin dripping through Polydimethylsiloxane near pinch-
off [5]. The nozzle diameter is $0.48$ cm, the viscosity ratio is
$\lambda=0.95$.
A fascinating aspect of the study of singularities is that they describe a
great variety of phenomena which appear in the natural sciences and beyond
[3]. Some examples of such singular events occur in free-surface flows [6],
turbulence and Euler dynamics (singularities of vortex tubes [7, 8] and sheets
[9]), elasticity [10], Bose-Einstein condensates [11], non-linear wave physics
[12], bacterial growth [13, 14], black-hole cosmology [15, 16], and financial
markets [17].
In this paper we consider evolution equations
$h_{t}=F[h],$ (1.1)
where $F[h]$ represents some (nonlinear) differential or integral operator. We
will also discuss cases where $h$ is a vector, and thus (1.1) is a system of
equations. Furthermore, the spatial variable $x$ may also have several
dimensions, and thus potentially different scaling in different coordinate
directions. We will cite some examples below, but few of the higher-
dimensional cases have so far been analysed in detail. For the purpose of the
following discussion, let us suppose that both $x$ and $h$ are scalar
quantities, and that the singularity occurs at a single point in space and
time $x_{0},t_{0}$. If $t^{\prime}=t_{0}-t$ and $x^{\prime}=x-x_{0}$, we are
looking for local solutions of (1.1) which have the structure
$h(x,t)=t^{\prime\alpha}H(x^{\prime}/t^{\prime\beta}),$ (1.2)
with appropriately chosen values of the exponents $\alpha,\beta$. Note that
later the prime is also used to indicate a derivative. However, this will
always be with respect to a spatial variable like $x,z$, or the similarity
variable $\xi$, hence confusion should not arise.
Giga and Kohn [18, 19] proposed to introduce self-similar variables
$\tau=-\ln(t^{\prime})$ and $\xi=x^{\prime}/t^{\prime\beta}$ to study the
asymptotics of blow up. Namely, putting
$h(x,t)=t^{\prime\alpha}H(\xi,\tau),$ (1.3)
(1.1) is turned into the “dynamical system”
$H_{\tau}=G[H]\equiv\alpha H-\beta\xi H_{\xi}+F[H].$ (1.4)
By virtue of (1.4), solutions to the original PDE (1.1) for given initial data
can be viewed as orbits in some infinite dimensional phase phase, for
instance, $L^{2}$. To understand the blow-up of (1.1), Giga and Kohn proposed
to study the long-time behaviour of the dynamical system (1.4). Thus in
particular, one is interested in the attractors of (1.4) ($\omega$-limit sets
in the notation which is customary in the context of partial differential
equations, see [20] and references therein). If (1.2) is indeed a solution of
(1.1), the right hand side of (1.4) is independent of $\tau$, and self-similar
solutions of the form (1.2) are fixed points of (1.4), which we will denote by
$\overline{H}(\xi)$. By studying the dynamics close to the fixed point, we
find that the dynamical system (1.4) frequently reduces to very few
dimensions. Thus on one hand one obtains detailed information on the behaviour
of the original problem (1.1) near blowup. On the other hand, one also gains a
fruitful means of classifying, or at least characterising singularities.
The most basic linear stability analysis of this self-similar solution
consists in linearising around the fixed point according to
$H=\overline{H}(\xi)+\epsilon P(\xi,\tau),$ (1.5)
which gives
$P_{\tau}={\cal L}P,$ (1.6)
where ${\cal L}\equiv{\cal L}(\overline{H})$ depends on the fixed point
solution $\overline{H}$. To solve (1.6), we write $P$ as a superposition of
eigenfunctions $P_{j}$ of the operator ${\cal L}$:
$P(\xi)=\sum_{j=1}^{\infty}a_{j}(\tau)P_{j}(\xi),$ (1.7)
where $\nu_{j}$ is the eigenvalue:
${\cal L}P_{j}=\nu_{j}P_{j}.$ (1.8)
In the cases we know, the spectrum turns out to be discrete. For evolution
PDE’s involving second order elliptic differential operators, such as
semilinear parabolic equations, mean curvature or Ricci flows, the
discreteness of the spectrum of the linearisation about the fixed point is a
direct consequence of Sturm-Liouville theory [21, 22]. This theory establishes
that, under quite general conditions on the coefficients of a second order
linear differential operator and the boundary conditions, its spectrum is
discrete and the corresponding eigenfunctions form a complete set in a
suitably weighed $L^{2}$ space. Some explicit examples are presented in
subsection 3.1.1. For general linear operators such a theory is not available,
and one has to study the spectrum case by case.
Now the solution of (1.6) corresponding to $P_{j}$ is
$P=e^{\nu_{j}\tau}P_{j},$ (1.9)
and all eigenvalues need to be negative for the similarity solution to be
stable. In that case, convergence to the fixed point is exponential, or
algebraic in the original time variable $t^{\prime}$. Soon the solution has
effectively reached the fixed point, and there is very little change in the
self-similar behaviour. If one or several of the eigenvalues around the fixed
point vanish, the approach to the fixed point is slow, and the dynamics is
effectively described by a dynamical system whose dimension corresponds to the
number of vanishing eigenvalues. The same holds true if the attractor has few
dimensions (such as a limit cycle or a low-dimensional chaotic attractor).
Thus although singular behaviour is in principle a problem to be solved in
infinite dimensions, in practise it typically reduces to a dynamical problem
of few dimensions. In this review we analyse singularities from the point of
view of the slow dynamics contained in (1.4), to obtain an overview and
tentative classification of possible scaling behaviours. We also emphasise the
physical significance of these different types of behaviours.
The perspective described above suggests a close relationship to the
description of scaling phenomena by means of the renormalisation group,
developed in the context of critical phenomena [23, 24]; we will continue to
point out similarities, but we are not aware that a classification similar to
ours has been achieved using the language of the renormalisation group. For a
computational perspective on analysing (1.4) in terms of its slow dynamics,
see [25]. Finally, another approach sometimes associated with the
classification of singularities is catastrophe theory [26]. However, as far as
we are aware catastrophe theory only yields useful results if the problem can
be mapped onto a low-dimensional geometrical problem, which can in turn be
rephrased in terms of normal forms of polynomials. This has been shown to be
the case for wave problems such as shock formation and wave breaking [27], as
well as singularities of the eikonal equation [28] and related problems [29].
In this paper we discuss the following cases:
1. (I)
Stable fixed points (section 2)
In this case the fixed point is approached exponentially in the logarithmic
variable $\tau$, so the dynamics is described by the self-similar law (1.2).
This pure power-law behaviour is also known as type-I self-similarity [30].
2. (II)
Centre manifold (section 3)
Here one or more of the eigenvalues around the fixed point are zero. As a
result, the approach to the fixed point is only algebraic, leading to
logarithmic corrections to scaling. This is called type-II self-similarity
[30]; it characterises cases where the blow-up rate is different from what is
expected on the basis of a solution of the type (1.2).
3. (III)
Travelling waves (section 4)
Solutions of (1.1) converge to $h=t^{\prime\alpha}\phi(\xi+c\tau)$, which is a
travelling wave solution of (1.4) with propagation velocity $c$.
4. (IV)
Limit cycles (section 5)
Solutions have the form
$h={t^{\prime}}^{\alpha}\mathbf{\psi}\left[\xi,\tau\right]$ with $\psi$ being
a periodic function of period $T$ in $\tau$. This is known as “discrete self-
similarity” [15, 31], since at times $\tau_{n}=\tau_{0}+nT$, n integer, the
solution looks like a self-similar one.
5. (V)
Strange attractors (section 6)
The dynamics on scale $\tau$ are described by a nonlinear (low-dimensional)
dynamical system, such as the Lorenz equation.
6. (VI)
Multiple singularities (section 7)
Blow-up may occur at several points $(x_{0},t_{0})$ (or indeed in any set of
positive measure), in which case the description (1.4) is not useful. We also
describe cases where (1.2) still applies, and blow-up occurs at a single
point, but the underlying dynamics is really one of two singularities which
merge at the singular time.
Equation | Type | Dynamics | Section
---|---|---|---
Free surface flow
$h_{t}+\nabla\cdot(h^{n}\nabla\triangle h)\pm\nabla(h^{p}\nabla h)=0$ | I,II | stable ? | 2.1.1
$(h^{2})_{t}+(h^{2}u)_{x}=0$ | I | |
$\rho(u_{t}+uu_{x})=(h^{2}u_{x})_{x}/h^{2}-(h^{-1})_{x}$ | | stable | 2.1.1
$h_{t}=\left[h\kappa_{x}/(1+h_{x}^{2})^{1/2}\right]_{x},$ | I | |
$\kappa=1/(h(1+h_{x}^{2})^{1/2})-h_{xx}/(1+h_{x}^{2})^{3/2}$ | | stable | 2.1
$h_{t}+(hu)_{x}=0,\quad u_{t}+uu_{x}=h_{xxx}$ | I | stable | 2.4.2
$\int\frac{\ddot{a}(\xi,t)d\xi}{\sqrt{(x-\xi)^{2}+a(x,t)}}=\frac{\dot{a}^{2}}{2a}$ | II | $v_{\tau}=-v^{3}$ | 3.2.1
$u(x)=\frac{1}{4}\int\frac{h_{z}(z)}{\sqrt{h^{2}(z)+(x-z)^{2}}}dz$ | | |
$(h^{2})_{t}+(h^{2}u)_{x}=0$ | III | stable | 4
Geometric evolution equations
$h_{t}=h_{zz}/(1+h_{z}^{2})-1/h$ | II | $u_{\tau}=-u^{2}$ | 3.1.1
$\psi_{t}=\psi_{ss}-(n-1)(1-\psi_{s}^{2})/\psi$ | II | $u_{\tau}=-u^{2}$ | 3.1.1
Reaction-diffusion equations
$u_{t}-\triangle u=f(u)$ | II | $u_{\tau}=-u^{2}$ | 3.1.2
$u_{t}-\nabla\cdot(|u|^{m}\nabla u)=u^{p}$ | II | unknown | 3.1.2
$\rho_{t}+\nabla\cdot(\rho\nabla S-\nabla\rho)=0,\quad\rho=-\triangle S$ | II | $u_{\tau}=-u^{3}$ | 3.2.2
Nonlinear dispersive equations
$u_{t}+uu_{x}=0$ | I | stable | 2.4
$i\psi_{t}+\triangle\psi+|\psi|^{p}\psi=0$ | I,II | $u_{\tau}=-u^{2}/v$ |
| | $v_{\tau}=-uv$ | 3.3
$u_{t}+u^{p}u_{x}+u_{xxx}=0$ | II | unknown | 3.3.1
$u_{t}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}$ | I | unknown | 3.3.1
$u_{t}=2fv,\quad v_{t}=-2fu,\quad f_{t}=f^{2}$ | IV | circle | 5
Choptuik equations | I, IV | limit cycle | 5
$u_{tt}=u_{xx}+|u|^{p}u$ | I,II | unknown | 7.2
Fluid equations
$u_{t}+(u\cdot\nabla)u=-\nabla p+\triangle u,\quad\nabla\cdot u=0$ | I, IV ? | unknown | 2.2
$u_{t}+(u\cdot\nabla)u=-\nabla p,\quad\nabla\cdot u=0$ | I, IV ? | unknown | 2.2
$u_{t}+uu_{x}+vu_{y}=-p_{x}+u_{yy},\quad u_{x}+v_{y}=0$ | I | stable | 2.2
Table 1: A summary of PDE’s discussed in this paper. The first column gives
the PDE in question, the second the type of dynamics near the fixed point
according to the classification enumerated above. In the case of attracting
fixed-point dynamics, it is classed as “stable”, otherwise the equation
governing the slow dynamics is given.
This paper’s aim is to assemble the body of knowledge on singularities of
equations of the type (1.1) that is available in both the mathematical and the
applied community, and to categorise it according to the types given above. In
addition to rigorous results we pay particular attention to various
phenomenological aspects of singularities which are often crucial for their
appearance in an experiment or a numerical simulation. For example, what are
the observable implications of the convergence onto the self-similar form
(1.2) being slow? In most cases, we rely on known examples from the
literature, but the problem is almost always reformulated to conform with the
formulation advocated above. However, some examples are entirely new, which we
will indicate as appropriate. For each of the above categories, we will
present at least one example in greater detail, so the analysis can be
followed explicitely. A concise overview of the equations presented in this
review is given in Table 1.
## 2 Stable fixed points
A sub-classification into self-similarity of the first and second kind has
been expounded in [32, 33, 34, 35]. Self-similar solutions are of the first
kind if (1.2) only solves (1.1) for one set of exponents $\alpha,\beta$; their
values are fixed by either dimensional analysis or symmetry, and are thus
rational. Solutions are of the second kind if solutions (1.2) exist locally
for a continuous set of exponents $\alpha,\beta$; however, in general these
solutions are inconsistent with the boundary or initial conditions. Imposing
these conditions leads to a non-linear eigenvalue problem, whose solution
yields irrational exponents in general.
### 2.1 Self-similarity of the first kind
Figure 2: SEM images illustrating the pinch-off of a row of rectangular
troughs in silicon (top) [36]. The bottom picture shows the same sample after
10 minutes of annealing at $1100^{\circ}$C. The troughs have pinched off to
form a row of almost spherical voids. The dynamics is driven by surface
diffusion.
Our example, exhibiting self-similarity of the first kind [35], is that of a
solid surface evolving under the action of surface diffusion. Namely, atoms
migrate along the surface driven by gradients of chemical potential, see
Fig.2. The resulting equations in the axisymmetric case, where the free
surface is described by the local neck radius $h(x,t)$, are [37]:
$h_{t}=\frac{1}{h}\left[\frac{h}{(1+h_{x}^{2})^{1/2}}\kappa_{x}\right]_{x},$
(2.1)
where
$\kappa=\frac{1}{h(1+h_{x}^{2})^{1/2}}-\frac{h_{xx}}{(1+h_{x}^{2})^{3/2}}$
(2.2)
is the mean curvature. In (2.1),(2.2), all lengths have been made
dimensionless using an outer length scale $R$ (such as the initial neck
radius), and the time scale $R^{4}/D_{4}$, where $D_{4}$ is a forth-order
diffusion constant.
Physically, it is important to point out that (2.1) describes the evolution of
the free surface at elevated temperatures, above the so-called roughening
transition. This implies that the solid surface is smooth and does not exhibit
facets, coming from the underlying crystal structure. Above the roughening
transition, a continuum description is still possible [38]. The study of these
models has lead to a number of interesting similarity solutions describing
singular behaviour of the surface, such as grooves [39] or mounds [40, 41].
Figure 3: The approach to the self-similar profile for equation (2.1). The
dashed line is the stable similarity solution $H(\xi)$ as found from (2.4).
The full lines are rescaled profiles found from the original dynamics (2.1) at
$h_{m}=10^{-1},10^{-2}$, and $h_{m}=10^{-3}$, respectively. As the singularity
is approached, they converge rapidly onto the similarity solution (2.3).
At a time $t^{\prime}\ll 1$ away from breakup, dimensional analysis implies
that $\ell=t^{\prime 1/4}$ is a local length scale. This suggests the
similarity form
$h(x,t)=t^{\prime 1/4}H(x^{\prime}/t^{\prime 1/4}),$ (2.3)
and thus the exponents $\alpha,\beta$ of (1.2) are fixed by dimensional
analysis, which is typical for self-similarity of the first kind. Of course,
the result (2.3) also follows when directly searching for a solution of (2.1)
in the form of (1.2). In other cases, a unique set of local scaling exponents
is determined by symmetry [42]. The similarity form of the PDE becomes
$-\frac{1}{4}(H-\xi
H_{\xi})=\frac{1}{H}\left[\frac{H}{(1+H_{\xi}^{2})^{1/2}}\kappa_{\xi}\right]_{\xi},\quad\xi=\frac{x^{\prime}}{t^{\prime
1/4}}$ (2.4)
where $\kappa$ is the mean curvature of $H$.
Solutions of (2.4) have been studied extensively in [43]. To ensure matching
to a time-independent outer solution, the leading order time dependence must
drop out from (2.3), implying that
$H(\xi)\sim c|\xi|,\quad\xi\rightarrow\pm\infty;$ (2.5)
the general form of this matching condition for self-similar solutions of the
form (1.2) is
$H(\xi)\sim c|\xi|^{\frac{\alpha}{\beta}},\quad\xi\rightarrow\pm\infty.$ (2.6)
All solutions of the similarity equation (2.1), and which obey the growth
condition (2.5) are symmetric, and form a discretely infinite set [43],
similar to a number of other problems discussed below. The series of
similarity solutions is conveniently ordered by descending values of the
minimum, see table 2. Only the lowest order solution $H_{0}(\xi)$ is stable,
and is shown in Fig. 3; we return to the issue of stability in section 2.5
below. The fact that permissible similarity solutions form a discrete set
implies a great deal of “universality” in the way pinching can occur. It means
that the local solution is independent of the outer solution, and rather that
the former imposes constraints on the latter; in particular, the prefactor $c$
in (2.5) must be determined as part of the solution (see Table 2).
i | $H_{i}(0)$ | $c_{i}$ |
---|---|---|---
0 | 0.701595 | 1.03714 |
1 | 0.636461 | 0.29866 |
2 | 0.456842 | 0.18384 |
3 | 0.404477 | 0.13489 |
4 | 0.355884 | 0.10730 |
5 | 0.326889 | 0.08942 |
Table 2: A series of similarity solutions of (2.4) as given in [43]. The
higher-order solutions become successively thinner and flatter.
#### 2.1.1 Thin films and thin jets
A further class of solutions displaying self-similarity of the first kind is
the generalised long-wave thin-film equation
$h_{t}+\nabla\cdot(h^{n}\nabla\Delta h-Bh^{m}\nabla h)=0\ ,\ n>0.$ (2.7)
The mass flux in this equation has two contributions: the first is due to
surface tension, and the second is due to an external potential. When $n=m=3$,
then $z=h(\mathbf{x},t)$ represents the height of a film or a drop of viscous
fluid over a flat surface, located at $z=0$; the external potential is
gravity. If $B$ is negative, (2.7) describes a film that is hanging from a
ceiling. Regardless of the sign of $B$, there is no singularity in this case
[44]. The case $n=1$ and $B=0$ corresponds to flow between two solid plates,
to which we return in section 7.1 below.
Solutions to (2.7) are said to develop point singularities if $h$ goes to zero
in finite time. This happens if one incorporates van der Waals forces, which
at leading order implies $n=3$ and $m=-1$ with $B<0$. In [45], [46] (see also
the review [47], where further full numerical simulations and mathematical
theory are reported) the existence of radially symmetric self-similar
touchdown solutions of the form
$h(r,t)=t^{\prime\frac{1}{5}}H(\xi),\quad\xi=r/t^{\prime\frac{2}{5}}$ (2.8)
is shown numerically in this case. Self-similar solutions that touch down
along a line exist as well, but they are unstable. A proof of formation of
singularities in this context has been provided by Chou and Kwong [48].
A related set of equations are those for thin films and jets, but which are
isolated instead of being in contact with a solid. Problems of this sort
furnish many examples of type-I scaling, as reviewed from a physical
perspective in [49]. If the motion is no longer dampened by the presence of a
solid, inertia often has to be taken into account. This means that a separate
equation for the velocity is needed, which is essentially the Navier-Stokes
equation below, but often simplified by a reduction to a single dimension.
Thus one has solutions of the form
$h(x,t)=t^{\prime\alpha}H(\xi),\quad u(x,t)=t^{\prime\beta-1}U(\xi),$ (2.9)
where $\xi=x^{\prime}/t^{\prime\beta}$. If $\alpha>\beta$ the profile is
slender, and the dynamics is well described in a shallow-water theory. In this
case the equations for an axisymmetric jet with surface tension become
$\partial_{t}h^{2}+\partial_{x}(uh^{2})=0$ (2.10)
and
$\rho(\partial_{t}u+u\partial_{x}u)=-(\gamma/\rho)\partial_{x}(1/h)+3\nu\frac{\partial_{x}(\partial_{x}uh^{2})}{h^{2}}.$
(2.11)
The system (2.10),(2.11) is interesting because it exhibits different scaling
behaviours depending on the balance between the three different terms in
(2.11) [42]. This is an illustration of the principle of dominant balance,
which is of great practical importance in practise, where it is a priori not
known which physical effect will be dominant. In the case of (2.11), these are
the forces of inertia on the left, surface tension (first term on the right),
and viscosity (second term on the right). Pinching is driven by surface
tension, so it must always be part of the balance. Three different possible
balances remain [42]:
(i) In the first case [50], all forces in (2.11) are balanced as the
singularity is approached. The exponents $\alpha=1,\beta=1/2$ in (2.9) follow
directly from this condition. As shown in [51], there is a discretely infinite
sequence of self-similar profiles $H(\xi),U(\xi)$ corresponding to this
balance. Numerical evidence strongly suggests that only the first profile,
corresponding to the thickest thread, is stable [6]. All the other profiles
are unstable, and thus cannot be observed. We will revisit this general
scenario again below, when we study the stability of fixed points more
generally.
(ii) The second possibility corresponds to a balance between surface tension
and viscous forces, thus putting $\rho=0$ in (2.11). Physically, this occurs
if the fluid is very viscous [52]. In section 2.4.1 below we will describe the
pinching solution corresponding to this case in more detail, as an example of
self-similarity of the second kind. The exponent $\alpha=1$ is fixed by the
balance, but $\beta$ is fixed only by an integrability condition. This once
more results in an infinite sequence of solutions, ordered by the value of
$\beta$. Again, only one profile, which has the largest value of
$\beta=0.17487$ is stable. This time, this corresponds to the smallest value
of the minimum radius $R_{0}$, or the thinnest thread, as opposed to thickest
thread in the case of the inertial-surface tension-viscous balance.
If one inserts this viscous solution into the original equation (2.11), one
finds that in the limit $t^{\prime}\rightarrow 0$, the inertial term on the
left grows faster than the two terms on the right. This means that regardless
how large the viscosity, eventually all three terms become of the same order,
and one observes a crossover to the inertial-surface tension-viscous
similarity solution described above, which is characterised by another set of
scaling exponents and similarity profiles. In particular, the surface tension-
viscous solution is symmetric about the pinch point, whereas the solution
containing inertia is highly asymmetric [53]. We remark that crossover between
different similarity solutions may also occur by another mechanism, not
directly related to the dominant balance between different terms in the
equation (cf. section 7.1).
Equations (2.10),(2.11) correspond to a viscous liquid, surrounded by a gas,
which is not dynamically active. The case of an external viscous fluid is
considered in detail in section 4 below. The case of no internal fluid is
special, in that the dynamics decouples completely into one for independent
slices [54]. As a result, there is no universal profile associated with the
breakup of a bubble in a viscous environment, but rather it is determined by
the initial conditions.
(iii) At very low viscosity ($\nu\approx 0$ in (2.11)), the relevant balance
is one where inertia is balanced by surface tension, so one might want to set
$\nu=0$ in (2.11), as done originally in [55]. However, the resulting
equations do not lead to a selection of the values of the scaling exponents
$\alpha,\beta$; instead, there is a continuum of solutions [56], parameterised
by the value of $\alpha$, each with a continuum of possible similarity
profiles. In fact, for vanishing viscosity (2.10),(2.11) does not go toward a
pinching solution, but the slope of the interface steepens, and one finds a
shock solution [57], similar to the generic scenario described in section 2.4
below.
It was however shown numerically in [58, 59], and investigated in more detail
in [60], that pinch-off of an inviscid fluid is well described by a solution
of the full three-dimensional, axisymmetric potential flow equations. This is
thus an example of a similarity solution of higher order in the independent
variable, but both coordinate directions scale in the same way. The scaling
exponents in (2.9) are $\alpha=\beta=2/3$ in this case, which violates the
assumption $\alpha>\beta$ for the validity of the shallow water equations
(2.10),(2.11). In addition, we note that the similarity profile can no longer
even be written as a graph as assumed in (2.9), but turn over, as first
observed experimentally in [61]. It is not known whether there also exists a
sequence of similarity solutions, as in the case of the other balances. The
case of no internal fluid is again very special, and leads to type-II scaling.
It is considered in section 3.2.1 below.
Finally, variations of (2.10),(2.11) have been investigated in [62]. Breakup
was considered in arbitrary dimensions $d$ (yet retaining axisymmetry) and
with the pressure term $1/h$ replaced by an arbitrary power law $1/h^{p}$.
After introducing a new variable $1/h^{p}$, there remains a single parameter
$r=(d-1)/p$, which can formally be varied continuously. For all values of $r$,
discrete sequences of type-I solutions are obtained. For $r>1/2$, profiles are
asymmetric, while below that value they are symmetric. At the critical value,
both types of solutions coexist. Another interesting feature of the limit
$r=1/2$ is that the viscous term becomes subdominant at leading order.
However, similar to the case $d=3,p=1$ mentioned above, no selection takes
place in the absence of the viscous term. Nevertheless, the solutions selected
by the presence of the viscous term are very close to an appropriately chosen
member of the family of inviscid solutions.
### 2.2 Singularities in Euler and Navier-Stokes equations
One of the most important open problems, both in physics and mathematics, is
the existence of singularities in the equations of fluid mechanics: Euler and
Navier-Stokes equations in three space dimensions. The Navier-Stokes equations
represent the evolution of a viscous incompressible fluid and are of the form
$\mathbf{u}_{t}+\mathbf{u}\cdot\nabla\mathbf{u}=-\nabla
p+Re^{-1}\Delta\mathbf{u},\qquad\nabla\cdot\mathbf{u}=0,$ (2.12)
where $\mathbf{u}$ represents the velocity field, $p$ the pressure in the
fluid and Re is a dimensionless parameter called Reynolds number. Formally, by
making Re$\rightarrow\infty$, the term involving $\Delta\mathbf{u}$ vanishes
and we arrive at the Euler system, that models the evolution of the velocity
and pressure fields of an inviscid incompressible fluid:
$\mathbf{u}_{t}+\mathbf{u}\cdot\nabla\mathbf{u}=-\nabla
p,\qquad\nabla\cdot\mathbf{u}=0.$ (2.13)
We exclude from our discussion certain “exact” blow-up solutions of the Euler
equations [63], which have the defect that the velocity goes to infinity
uniformly in space; in other words, they lack the crucial mechanism of
focusing. Formally, they are of course similarity solutions of (2.13), but
with spatial exponent $\alpha=0$.
As we mentioned above, the existence of singular solutions is unknown.
Nevertheless, some scenarios have been excluded. For the Navier-Stokes
equations, there exists no nontrivial self-similar solution of the first kind
$\mathbf{u}(\mathbf{x},t)=t^{\prime-1/2}\mathbf{U}\left(\mathbf{\xi}\right),\quad\mathbf{\xi}={\bf
x}^{\prime}/t^{\prime 1/2}$ (2.14)
in $L^{2}(\mathbb{R}^{3})$. This was proved by Necas, Ruzicka and Sverak [64].
However, this does not exclude the formation of a singularity in a localised
region: the matching condition (2.6) for this case implies $|{\bf
U}|\propto|\xi|^{-1}$ as $|\xi|\rightarrow\infty$, which is not in $L^{2}$.
Therefore, the theorem [64] does not apply.
A possible self-similar solution consisting of two skewed vortex-pairs has
been proposed by Moffatt in [7] in the spirit of the scenario suggested by the
numerical simulations of Pelz [65], of the implosion of six vortex pairs in a
configuration with cubic symmetry. More recent numerical experiment by Hou and
Li [66] seem to indicate that, although the velocity field may grow to very
large values, singularities in the above mentioned scenarios saturate
eventually and the solutions remain smooth. It has been argued in [67] that no
self-similar solutions for Euler system should exist and that the ”limit-
cycle” scenario described in section 5 could apply.
Under certain circumstances, such as special symmetry conditions or
appropriate asymptotic limits, the Navier-Stokes and Euler systems may
simplify and give rise to models for which the question of existence of
singular solutions is somewhat simpler to analyse. This is the case for the
Prandtl boundary-layer equations for the 2-D evolution of the velocity field
$(u,v)$ in $y\geq 0$:
$u_{t}+uu_{x}+vu_{y}=-p_{x}+u_{yy},\quad u_{x}+v_{y}=0$ (2.15)
with boundary conditions $u=v=0$; $p$ is a given pressure field and the
behaviour of the velocity field at infinity is prescribed. Equation (2.15)
describes the asymptotic limit of the Navier-Stokes equation near a solid body
in the limit of large Reynolds numbers $Re$. The variable $x$ measures the
arclength along the body, and $Re^{1/2}y$ is the distance from the body.
Historically, a lot of attention was focused on the stationary version of
(2.15), considering it as an evolution equation in $x$. At some position
$x_{s}$ along the body, the so-called Goldstein singularity
$v\propto(x_{s}-x)^{-1/2}$ is encountered [68], which signals separation of
the flow from the body. However, in reality the outer flow changes as a result
of the appearance of a stagnation point, and one has to consider the
interaction between the boundary layer and the outer flow [69].
It is thus conceptually simpler to consider the case of unsteady boundary
layer separation, which is described by the first singularity of (2.15) at
time $t_{0}$. The formation of singularities of (2.15) in finite time was
proved by E and Engquist [70]. It was first found numerically by van Dommelen
and Shen [71], and its analytical structure was investigated in [72], using
Lagrangian variables, which follow fluid particles as they separate from the
surface (see also [73]). In the original Eulerian variables, the self-similar
structure is [74, 75]
$u=-u_{0}+t^{\prime
1/2}\phi_{0}^{1/2}U(\xi,\eta),\quad\xi=\frac{x^{\prime}-u_{0}t^{\prime}}{t^{\prime
3/2}\phi_{0}^{1/2}},\;\eta=\frac{y\phi_{0}^{1/4}}{t^{\prime 1/4}\Lambda},$
(2.16)
where $u_{0},\phi_{0}$, and $\Lambda$ are constants which depend on the
problem, while $U$ is universal and can be given in terms of elliptic
integrals. Note that the exponents for $u$ and $x$ are the generic exponents
for a developing shock (see section 2.4 below), while the similarity exponent
in the $y$-direction is different from the scaling for two-dimensional
breaking waves [27]. We stress that the appearance of a singularity in (2.15)
does not mean that the full 2D Navier-Stokes equation has developed a
singularity. Instead, lower order terms in the asymptotic expansion that lead
to (2.15) become important close to the singularity.
In relation with singularities in fluid mechanics, we can mention briefly a
few important problems involving models or suitable approximations to the
original Euler and Navier-Stokes systems. One concerns weak solutions to the
Euler system for which the vorticity ($\omega=\nabla\times\mathbf{u}$) is
concentrated in curves or surfaces. This is the case of the so called vortex
filaments and sheets in which the vorticity remains concentrated for all
times, in absence of viscosity. A useful way to represent the vortex sheet,
when it evolves in 2D, is by assuming the location of its points
$(x(\alpha,t),y(\alpha,t))$ as complex numbers
$z(\alpha,t)=x(\alpha,t)+iy(\alpha,t)$. Then, the evolution of $z(\alpha,t)$
is given by the so-called Birkhoff-Rott equation [76]:
$z_{t}^{\ast}(\alpha,t)=\frac{1}{2\pi
i}PV\int_{-\infty}^{\infty}\frac{\gamma(z(\alpha^{\prime},t),t)}{z(\alpha,t)-z(\alpha^{\prime},t)}z_{\alpha}(\alpha^{\prime},t)d\alpha^{\prime}\
,$ (2.17)
where $z^{\ast}$ stands for the complex conjugate of $z$. The principal value
is denoted by PV, and $\gamma$ is the vortex strength and is such that
$d\Gamma=\gamma(z(\alpha,t),t)z_{\alpha}(\alpha,t)d\alpha$ is constant along
particle paths of the flow. The question then is whether or not these
geometrical objects will remain smooth at all times or develop singularities
in finite time. In the case of vortex sheets, singularities are known to
develop in the form of a divergence of the curvature at some point. These are
called Moore’s singularities after their observation and description by D. W.
Moore [77]. A mathematical proof of existence of these singularities is
provided by Caflisch and Orellana in [78]. These singularities exhibit self-
similarity of the first kind as shown, for instance, in [79]: if one defines
the inclination angle $\theta(s,t)$ in terms of the arclength parameter $s$ as
such that $z_{s}=e^{i\theta}$, then the curvature is given by
$\kappa=\theta_{s}$ and may blow-up in the self-similar form (up to
multiplicative constants):
$\kappa(s,t^{\prime})=\frac{1}{t^{\prime\delta}}g(\eta),\quad\eta=s^{\prime}/t^{\prime}\
,\ \ 0<\delta<1,$ (2.18)
where
$g(\eta)=\frac{1}{(1+\eta^{2})^{\frac{\delta}{2}}}\sin(\delta\arctan\eta)\ .$
(2.19)
Interestingly, numerical simulations and Moore’s original observations suggest
that, although singular solutions with any $\delta$ are possible, that the
solution with $\delta=\frac{1}{2}$ is preferred. Thus the generically observed
geometry near the singularity is of the form $y=\left|x\right|^{\frac{3}{2}}$,
including the case of 3D simulations. This poses an interesting ”selection
problem” for the $\frac{3}{2}$ power which has not received a definitive
answer so far.
Another type of solution of (2.17) has the from of a double-branched spiral
vortex sheet [80]. The explicit form is
$z(\beta,t)=\left\\{\begin{array}[]{l}t^{\prime q}\beta^{\nu}\quad\beta>0\\\
t^{\prime q}|\beta|^{\nu}\quad\beta<0\quad\;,\end{array}\right.$ (2.20)
where the two cases correspond to the two branches of the spiral. The
parameter $\beta$ is related to integration variable $\alpha$ of (2.17) by
$d\beta=z_{\alpha}d\alpha$. The exponents are of the form $\nu=1/2+ib$ and
$q=1/2+i\mu b$, corresponding to a vortex of radius $r=t^{\prime 1/2}$
collapsing in finite time. However, in this case the vortex sheet strength is
found to increase exponentially at infinity [80].
Vortex filaments result as the limit of a vortex tube when the thickness tends
to zero. The fluid flow around a vortex filament is frequently approximated by
a truncation of the Biot-Savart integral for the velocity in terms of the
vorticity. This leads to a geometric evolution equation for the filament (see
[81], chapter 7, and references therein) that can be transformed, via Hasimoto
transformation, into the cubic Nonlinear-Schrödinger in 1D. This fact allowed
Gutierrez, Rivas and Vega to construct exact self-similar solutions for
infinite vortex filaments [82]. One can also consider the vorticity
concentrated in a region separating two fluids of different density and in the
presence of gravitational forces. This is the case of the surface water waves
system for which the existence of singularities is open [83].
A different approach in the study of singularities for Euler and Navier-Stokes
equations in three space dimensions relies on the development of models that
share some of the essential mathematical difficulties of the original systems,
but in a lower space dimension. This is the case of the surface quasi-
geostrophic equation popularised by Constantin, Majda and Tabak [84]:
$\displaystyle\theta_{t}+{\bf v}\cdot\nabla\theta=0,$ (2.21a)
$\displaystyle{\bf v}=\nabla^{\bot}\psi,\quad\theta=-(-\triangle)^{1/2}\psi,$
(2.21b)
to be solved in $d=2$. This system of equations describes the convection of an
active scalar $\theta$, representing the temperature, with a velocity field
which is an integral operator of the scalar itself. Nevertheless, the mere
existence of singular solutions to this equation in the form of blow-up for
the gradient of $\theta$ is still an open problem. One-dimensional analogues
of this problem, representing the convection of a scalar with a velocity
field, which is the Hilbert transform of the scalar itself do have
singularities in the form of cusps, as proved in [85], [86]. The structure of
such singularities has been described in [87] and they are, in fact, of the
type described in the next section, that is of the second kind.
### 2.3 Self-similarity of the second kind
In the example of the previous subsection, the exponents can be determined by
dimensional analysis, or from considerations of symmetry, and therefore assume
rational values. In many other problems, however, the scaling behaviour
depends on external parameters, set for example by the initial conditions. In
that case, the scaling exponent can assume any value. Often, this value is
fixed by a compatability condition, resulting in an irrational answer. We will
call this situation self-similarity of the second kind [32, 35]. Since it is
relatively rare that results are tractable analytically, we mention two simple
examples for which this is possible, although they do not come from time-
dependent problems.
The first example is that of viscous flow near a solid corner of opening angle
$2\alpha$ [88]. For analogues of this problem in elasticity, see [89, 90] as
well as the discussion in [35]. This flow is described by a Stokes’ equation,
whose solution near the corner is expected to be
$\psi=r^{\lambda}f_{\lambda}(\theta).$ (2.21v)
If one of the boundaries is moving, scaling is of the first kind, and
$\lambda=2$ (the so-called Taylor scraper [91]). However, if the flow is
driven by two-dimensional stirring at a distance from the corner, $\lambda$ is
determined by the transcendental equation
$\sin 2(\lambda-1)\alpha=-(\lambda-1)\sin 2\alpha.$ (2.21w)
If $2\alpha<146^{\circ}$, (2.21w) admits complex solutions, which correspond
to an infinite sequence of progressively smaller corner eddies. Since
$\lambda$ is complex, The strength of the eddies decreases as one comes closer
to the corner.
The second example consists in calculating the electric field between two non-
conducting spheres, where an external electric field is applied in the
direction of the symmetry plane [92]. In this case the electric potential
between the spheres is proportional to $(\rho/(Rh))^{\sqrt{2}-1}$, where
$\rho$ is the radial distance from the symmetry axis, $R$ the sphere radius,
and $h$ the distance between the spheres. Thus in accordance with the the
general ideas of self-similarity of the second kind, the singular behaviour is
not controlled by the local quantity $\rho/h$, but the “outer” parameter $R$
comes into play as well. We now explain two analytically tractable dynamical
examples of self-similarity of the second kind.
### 2.4 Breaking waves in conservation laws
Figure 4: Fringe pattern showing the steepening of a wave in a gas, leading to
the formation of a shock, which is travelling from left to right [93]. The
vertical position of a given fringe is proportional to the density at that
point. In the last picture a jump of seven fringes occurs.
We only consider the simplest model for the formation of a shock wave in gas
dynamics, which is Burger’s equation
$u_{t}+uu_{x}=0.$ (2.21x)
It is generally believed that any system of conservation laws that exhibits
blow up will locally behave like (2.21x) [94]. For example, Fig. 4 shows the
steepening of a density wave in a gas, leading to a jump of the density in the
picture on the right. In the words of [93]: “We conclude that an infinite
slope in the theoretical solution corresponds to a shock in real life”. As
throughout this review, we only consider the dynamics up to the singularity.
Which structure emerges after the singularity depends on the regularisation
used, as the continuation to times after the singularity is not unique [95,
96]. If the regularisation is diffusive, a shock wave forms [97]; if it is a
third derivative, one finds a KDV soliton. Finally, regularisation by higher-
order nonlinearities has been considered in [27] as a model of wave breaking.
It is well known [98] that (2.21x) can be solved exactly using the method of
characteristics. This method consists in noting that the velocity remains
constant along the characteristic curve
$z=u_{0}(x)t+x,$ (2.21y)
where $u_{0}(x)=u(x,0)$ is the initial condition. Thus
$u(z,t)=u_{0}(x)$ (2.21z)
is an exact solution to (2.21x), given implicitly.
It is geometrically obvious that whenever $u_{0}(x)$ has a negative slope,
characteristics will cross in finite time and produce a discontinuity of the
solution. This happens when $\partial z/\partial x=0$, which will occur for
the first time at the singularity time
$t_{0}=\min\left\\{-\frac{1}{\partial_{x}u_{0}(x)}\right\\},$ (2.21aa)
at a spatial position $x=x_{m}$. This means a singularity will first form at
$x_{0}=x_{m}-\frac{u_{0}(x_{m})}{\partial_{x}u_{0}(x_{m})}.$ (2.21ab)
Since (2.21x) is invariant under any shift in velocity, we can assume without
loss of generality that $u_{0}(x_{m})=0$, and thus that $x_{0}=x_{m}$. This
means the velocity is zero at the singularity. We now analyse the formation of
the singularity using the local coordinates $x^{\prime},t^{\prime}$. In [27],
this was done by expanding the initial condition $u_{0}$ in $x^{\prime}$, and
using (2.21z), using ideas from catastrophe theory [26]. Here instead we use
the similarity ideas developed in this paper.
The local behaviour of (2.21x) near $t_{0}$ can be obtained using the scaling
$u(x,t)=t^{\prime\alpha}U\left(x^{\prime}/t^{\prime\alpha+1}\right),$ (2.21ac)
which solves (2.21x). The similarity equation becomes
$-\alpha U+(1+\alpha)\xi U_{\xi}+UU_{\xi}=0,$ (2.21ad)
with implicit solution
$\xi=-U-CU^{1+1/\alpha}.$ (2.21ae)
The special case $\alpha=0$ has the solution $U=-\xi$, which is inconsistent
with the matching condition (2.6), and thus has to be discarded.
We are thus left with a continuum of possible scaling exponents $\alpha>0$, as
is typical for self-similarity of the second kind. A discretely infinite
sequence of exponents $\alpha_{n}$ is however selected by the requirement that
(2.21ae) defines a smooth function for all $\xi$. Namely, one must have
$1+1/\alpha$ odd, or
$\alpha_{i}=\frac{1}{2i+2},\quad i=0,1,2\dots,$ (2.21af)
and we denote the corresponding similarity profile by $U_{i}$. The constant
$C$ in (2.21ae) must be positive, but is otherwise arbitrary. It is set by the
initial conditions, which is another hallmark of self-similarity of the second
kind. However, $C$ can be normalised to 1 by rescaling $x$ and $U$. We will
see in section 2.5 that the solution with $\alpha_{0}$,
$u(x,t)=t^{\prime 1/2}U_{0}\left(x^{\prime}/t^{\prime 3/2}\right),$ (2.21ag)
is the only stable one, all higher-order solutions are unstable.
It is interesting to look at some possible exceptions to the form of blow-up
given above, suggested by [94]:
$u_{t}+uu_{x}=u^{\sigma}.$ (2.21ah)
This equation is also solved easily using characteristics. For $\sigma\leq 2$
the blow-up is alway of the form (2.21ag), for $\sigma>2$ two different
behaviours are possible. For small initial data $u_{0}(x)$, a singularity
still forms like (2.21ag), but in addition $u$ may also go to infinity.
However, there is a boundary between the two behaviours [94], where the slope
blows up at the same time that $u$ goes to infinity. For this case, one
expects all terms in (2.21ah) to be of the same order, giving
$u(x,t)=t^{\prime\frac{1}{1-\sigma}}U\left(\xi)\right),\quad\xi=x^{\prime}/t^{\prime\frac{\sigma-2}{\sigma-1}},$
(2.21ai)
with similarity equation
$\frac{U}{1-\sigma}+\frac{\sigma-2}{\sigma-1}\xi U_{\xi}=U^{\sigma}-UU_{\xi}.$
(2.21aj)
Figure 5: The similarity solution (2.21ak) for $\sigma=4$.
The solution to (2.21aj) that has the right decay at infinity is
$\xi=-\frac{1}{(\sigma-2)U^{\sigma-2}}\pm
C\frac{\left(1-(\sigma-1)U^{\sigma-1}\right)^{\frac{\sigma-2}{\sigma-1}}}{U^{\sigma-2}},$
(2.21ak)
where $C>0$ is an arbitrary constant. The + and - signs describe the solution
to the right and left of
$\xi^{*}=-(\sigma-1)^{\frac{\sigma-2}{\sigma-1}}/(\sigma-2)$, respectively.
The special case $\sigma=4$ is shown in Fig. 5. The similarity solution
(2.21ak) is not smooth at its maximum; rather, its first derivative behaves
like $U_{\xi}\propto(\xi-\xi^{*})^{1/(\sigma-2)}$. This can be understood from
the exact solution; in order for blow-up to occur at the same time that a
shock is formed, the initial profile must already have a maximum with the same
regularity as (2.21ak). Thus, the situation leading to (2.21ai) is a very
special one, requiring very peculiar initial conditions.
#### 2.4.1 Viscous pinch-off
As explained in section 2.1.1, the pinch-off of a very viscous fluid is
described by (2.10), (2.11), with $\rho=0$, but only for finite range of
scales. The equations can be simplified considerably by introducing Lagrangian
variables, i.e. writing all profiles as a function of a particle label $s$.
This means the particle is at position $z(s,t)$ at time $t$, and $z_{t}(s,t)$
is the velocity at time $t$. The jet profile can be obtained from
$z_{s}=1/h^{2}(s,t)$, and (2.11) becomes
$h_{t}(s,t)=\frac{1}{6}\left(1+\frac{C(t)}{h(s,t)}\right).$ (2.21al)
The typical velocity scale is $\gamma/\eta$, where $\gamma$ is the surface
tension and $\eta$ is the viscosity; (2.21al) has been made dimensionless
accordingly. The time-dependent constant of integration $C(t)$ has to be
determined self-consistently. Note that the self-similar form (1.2) is a
solution of (2.21al) for $\alpha=1$, and any value of $\beta$; the exponent
$\beta$ will be determined by the consistency condition (2.21as) below.
Figure 6: A drop of viscous fluid falling from a pipette 1 mm in diameter
[99]. Note the long neck.
Since $\alpha=1$, a scaling solution of (2.21al) has the form
$h^{-2}(s,t)=t^{\prime-2}f\left(\xi\right),\quad\mbox{with}\quad\xi=s^{\prime}/t^{\prime\gamma}$
(2.21am)
and
$C(t)=-C_{0}t^{\prime}\ .$ (2.21an)
The relationship with the exponent $\beta$ defined in (2.9) is simply
$\beta=\gamma-2$, as found from passing from Lagrangian to Eulerian variables.
Inserting (2.21am),(2.21an) into (2.21al) we obtain
$\frac{1}{\sqrt{f}}+3\left(\frac{2}{f}+\frac{\gamma\xi
f_{\xi}}{f^{2}}\right)=C_{0},$ (2.21ao)
where $C_{0}$ is a constant. Imposing symmetry and regularity of $f$, we
expand $f(\xi)$ in the form
$f_{i}(\xi)=R_{0}^{-2}+\xi^{2i+2}+O(\xi^{2i+4})\ ,\ i=0,1,2,\dots$ (2.21ap)
where we have normalised the coefficient of $\xi^{2i+2}$ to one. This is
consistent, since any solution of (2.21al) is only determined up to a scale
factor. Instead, the axial scale is fixed by the initial conditions. The
parameter $R_{0}$ is the rescaled minimum of the profile:
$h_{m}=R_{0}t^{\prime}$. Inserting (2.21ap) into (2.21ao), at order
$\xi^{2i+2}$ one obtains
$R_{0}=\frac{1}{12(\overline{\gamma}-1)},\quad
C_{0}=\frac{1}{24}\frac{2\overline{\gamma}-1}{(\overline{\gamma}-1)^{2}}$
(2.21aq)
where we have put $\overline{\gamma}=(i+1)\gamma$.
Each choice of $i$ corresponds to one member in an infinite sequence of
similarity solutions. Equation (2.21ao) can easily be integrated in terms of
$\ln\xi$ and $y=\sqrt{f}$:
$\int\frac{dy}{\left(\left(1+6R_{0}\right)y^{3}-y^{2}-6R_{0}y\right)}=\frac{1}{6R_{0}\gamma}\ln\xi+\widetilde{C}=\frac{1}{6R_{0}\overline{\gamma}}\ln\xi^{i+1}+\widetilde{C},$
with $\widetilde{C}$ an arbitrary constant. Computing the integral above we
obtain
$y^{-\overline{\gamma}}\left(\left(2\overline{\gamma}-1\right)y+1\right)^{\overline{\gamma}-\frac{1}{2}}\left(1-y\right)^{\frac{1}{2}}=\xi^{i+1},$
(2.21ar)
which is an implicit equation for the i-th similarity profile $y\equiv
y_{i}(\xi)=\sqrt{f_{i}(\xi)}$.
The value of the velocity $U_{\infty}$ at infinity must be a constant to be
consistent with boundary conditions. It can be found by integrating
$z_{ts}=(h^{-2})_{t}=t^{\prime-3}(2f+\gamma\xi f_{\xi})$ from zero to
infinity:
$U_{\infty}=\int_{0}^{\infty}z_{ts}ds=\frac{t^{\prime\gamma-3}}{3}\int_{0}^{\infty}\left(\left(\frac{1}{24}\frac{2\overline{\gamma}-1}{(\overline{\gamma}-1)^{2}}\right)f^{2}-f^{\frac{3}{2}}\right)d\xi=0,$
(2.21as)
where we have used (2.21ao). The above condition $U_{\infty}=0$, which ensures
that $U_{\infty}$ does not diverge as $t^{\prime}\rightarrow 0$, is the
equation which determines the exponent $\gamma$. Taking the derivative of
(2.21ar) we obtain
$(i+1)\xi^{i}\frac{d\xi}{dy}=\frac{d}{dy}\left(y^{-\overline{\gamma}}\left(\left(2\overline{\gamma}-1\right)y+1\right)^{\overline{\gamma}-\frac{1}{2}}\left(1-y\right)^{\frac{1}{2}}\right)=$
$=-y^{-\overline{\gamma}-1}\left(2y\overline{\gamma}-y+1\right)^{\overline{\gamma}-\frac{3}{2}}\frac{\overline{\gamma}}{\sqrt{\left(1-y\right)}}$
which can be used to transform the integral in (2.21as) to the variable $y$:
$\displaystyle
K_{i}(\gamma)\equiv\frac{3U_{\infty}}{(12(\overline{\gamma}-1))^{3}}=\frac{\overline{\gamma}}{i+1}\int_{0}^{1}\left(\left(\frac{1}{2}\frac{2\overline{\gamma}-1}{\overline{\gamma}-1}\right)y^{4}-y^{3}\right)\cdot$
$\displaystyle\left(y^{-\frac{i+1+\overline{\gamma}}{i+1}}\left(\left(2\overline{\gamma}-1\right)y+1\right)^{-\frac{1}{2}\frac{2i-2\overline{\gamma}+3}{i+1}}\left(1-y\right)^{-\frac{1}{2}\frac{2i+1}{i+1}}\right)dy=0.$
(2.21at)
The function $K_{i}(\gamma)$ may be written explicitly as
$\displaystyle
K_{i}(\gamma)=\gamma\frac{\Gamma\left(4-\gamma\right)\Gamma\left(\frac{1}{2i+2}\right)}{\Gamma\left(4-\gamma+\frac{1}{2i+2}\right)}\left(\frac{1}{2}\frac{(2i+2)\gamma-1}{(i+1)\gamma-1}\right)\cdot$
$\displaystyle
F\left(\frac{2i+3}{2i+2}-\gamma,4-\gamma;4-\gamma+\frac{1}{2i+2};1-(2i+2)\gamma\right)-\gamma\frac{\Gamma\left(3-\gamma\right)\Gamma\left(\frac{1}{2i+2}\right)}{\Gamma\left(3-\gamma+\frac{1}{2i+2}\right)}\cdot$
$\displaystyle
F\left(\frac{2i+3}{2i+2}-\gamma,3-\gamma;3-\gamma+\frac{1}{2i+2};1-(2i+2)\gamma\right),$
(2.21au)
where $F(a,b;c,z)$ is the hypergeometric function [100]. Roots of $\gamma_{i}$
are given in Table 3.
To summarise, each exponent $\gamma_{i}$ corresponds to a new member
$f_{i}(\xi)$ of an infinite hierarchy of similarity profiles, to be found from
(2.21ar). If one converts the Lagrangian variables back to the original
spatial variables, one obtains
$h(x,t)=t^{\prime}\phi^{(n)}_{St}\left(x^{\prime}/t^{\prime\gamma-2}\right).$
(2.21av)
Thus for $t^{\prime}\rightarrow 0$ the typical radial scale $t^{\prime}$ of
the generic $i=0$ solution rapidly becomes smaller than the axial scale
$t^{\prime 0.175}$ (cf. Table 3). This explains the long necks seen in Fig. 6.
i | $\gamma_{i}$ | $R_{0}$
---|---|---
0 | 2.1748 | 0.0709
1 | 2.0454 | 0.0797
2 | 2.0194 | 0.0817
3 | 2.0105 | 0.0825
4 | 2.0065 | 0.0828
5 | 2.0044 | 0.0832
Table 3: A list of exponents, found from $K_{i}(\gamma)=0$ using MAPLE, with
$K_{i}$ given by (2.21au). The number $2i+2$ gives the smallest non-vanishing
power in a series expansion of the corresponding similarity solution around
the origin. Only the solution with $i=0$ is stable. The rescaled minimum
radius is found from (2.21aq).
#### 2.4.2 More examples
Other recent examples for scaling of the second kind have been observed for
the breakup of a two-dimensional sheet with surface tension. In a shallow-
water approximation, which is justified for a description of breakup, the
equations read [101]
$h_{t}+(hu)_{x}=0,\quad u_{t}+uu_{x}=h_{xxx}$ (2.21aw)
after appropriate rescaling. Local similarity solutions can be found in the
form
$h(x,t)=t^{\prime 4\beta-2}H(\eta),\quad u(x,t)=t^{\prime\beta-1}U(\eta),$
(2.21ax)
where $\eta=x^{\prime}/t^{\prime\beta}$. The exponent $\beta$ is not
determined by dimensional analysis. Instead, it must be found from a
solvability condition on the nonlinear system of equations for the similarity
functions $H,U$.
The result of the numerical calculation is [101] $\beta=0.6869\pm 0.0003$,
which is curiously close to $\beta=2/3$, which is the value that had been
conjectured earlier [102], but contains a small correction. The value
$\beta=2/3$ comes out if both length scales in the longitudinal and
transversal directions are assumed to be the same, implying that
$4\beta-2=\beta$. This is a natural expectation for problems governed by
Laplace’s equation, such as inviscid, irrotational flow [59], and indeed is
observed for three-dimensional drop breakup [58, 59]. However, in present
case, even if the full two-dimensional irrotational flow equations are used,
$\beta\neq 2/3$.
Other physical problems which frequently involve anomalous scaling exponents
are strong explosions on one hand, and collapse of particles or gases into a
singular state on the other. These types of problems have been reviewed in
great detail in a number of textbooks and articles [32, 34, 33, 35], but
continue to attract a great deal of attention. As with many other singular
problems, the type of scaling depends on the details of the underlying
physics, and scaling of both the first and second kind is observed. For
example, the radius of a shock wave resulting from a strong explosion can be
calculated from dimensional analysis to be $r_{s}\propto t^{2/5}$ [103].
However, in the seemingly analogous case of a strong implosion, an anomalous
exponent is observed, which moreover depends on the parameters of the problem
[104, 98]. Cases were collapse and shock formation coincide were given by
[105] (similar to section 2.4 above). In a somewhat different context,
anomalous scaling is observed in model calculations for the collapse of self-
gravitating particles [106] and Bose-Einstein condensates [107]. It is
important to remember that these examples come from kinetic equations
describing the stochastic collision of waves or particles, and hence involving
nonlocal collision operators. However, the kinetic equations appear to be
closely related to certain PDE problems [108], which are analogous to other
evolution equations studied in this article.
### 2.5 Stability of fixed points
Self-similar solutions correspond to fixed points of the dynamical system
(1.4), whose stability we now investigate by linearising around the fixed
point. We explain the situation for the example of section 2.1 in more detail,
for which the transformation reads
$h(x,t)=t^{\prime 1/4}H(\xi,\tau),$ (2.21ay)
where $\tau=-\ln(t^{\prime})$. The similarity form of (2.1) becomes
$H_{\tau}=\frac{1}{4}(H-\xi
H_{\xi})+\frac{1}{H}\left[\frac{H}{(1+H_{\xi}^{2})^{1/2}}\kappa_{\xi}\right]_{\xi},$
(2.21az)
which reduces to (2.4) if the left hand side is set to zero. To assure
matching of (2.21az) to the outer solution, we have to require that (2.21ay)
is to leading order time-independent as $\xi$ is large, which leads to the
boundary condition
$H_{\tau}-(H-\xi H_{\xi})/4\rightarrow
0\quad\mbox{for}\quad|\xi|\rightarrow\infty.$ (2.21ba)
This is the natural extension of (2.5) to the time-dependent case.
Next we linearise around any one of the similarity solutions
$\overline{H}(\xi)=H_{i}(\xi)$ listed in Table 2, as described in the
Introduction. The stability is controlled by eigenvalues of the eigenvalue
equation (1.8). Inserting the eigensolution (1.9) into (2.21ba) one finds that
$P_{j}$ must grow at infinity like
$P_{j}(\xi)\propto\xi^{1-4\nu_{j}}.$ (2.21bb)
Similarly, the growth condition for the general case of a similarity solution
of the form (1.2) is
$P_{j}(\xi)\propto\xi^{\frac{\alpha-\nu_{j}}{\beta}}.$ (2.21bc)
If the similarity solution $\overline{H}(\xi)$ is to be stable, the real part
of the eigenvalues of ${\cal L}$ must be negative. However, there are always
two positive eigenvalues, which are related to the invariance of the equation
of motion (2.1) under translations in space and time, as noted by [109, 110].
Namely, for any $\epsilon$, the translated similarity solution
$h^{(\epsilon)}(x,t)=t^{\prime
1/4}\overline{H}(\frac{x^{\prime}+\epsilon}{t^{\prime 1/4}})$ (2.21bd)
is an equally good self-similar solution of (2.1), and thus of (2.21az). In
particular, we can expand (2.21bd) to lowest order in $\epsilon$, and find
that
$H^{(\epsilon)}(\xi,\tau)=\overline{H}(\xi)+\epsilon
e^{\beta\tau}\overline{H}_{\xi}(\xi)+O(\epsilon^{2}),$ (2.21be)
where the linear term is a solution of (1.6).
Thus
$\left(e^{\beta\tau}\overline{H}_{\xi}\right)_{\tau}=e^{\beta\tau}\beta\overline{H}_{\xi}=e^{\beta\tau}{\cal
L}\overline{H}_{\xi}.$ (2.21bf)
But this means that $\nu_{x}=\beta\equiv 1/4$ is an eigenvalue of ${\cal L}$
with eigenfunction $\overline{H}_{\xi}(\xi)$. Similarly, considering the
transformation $t\rightarrow t+\epsilon$, one finds a second positive
eigenvalue $\nu_{t}=1$, with eigenfunction $\xi\overline{H}_{\xi}$. However,
these two positive eigenvalues do not correspond to instability. Instead, the
meaning of these eigenvalues is that upon perturbing the similarity solution,
the singularity time as well as the position of the singularity will change.
Thus if the coordinate system is not adjusted accordingly, it looks as if the
solution would flow away from the fixed point. If, on the other hand, the
solution is represented relative to the perturbed values of $x_{0}$ and
$t_{0}$, the eigenvalues $\nu_{x}$ and $\nu_{t}$ will not appear.
The eigenvalue problem (1.8) was studied numerically in [43]. It was found
that each similarity solution $\overline{H}_{i}$ has exactly $2i$ positive
real eigenvalues, disregarding $\nu_{x},\nu_{t}$. The result is that the
linearisation around the “ground state” solution $\overline{H}_{0}$ only has
negative eigenvalues while all the other solutions have at least one other
positive eigenvalue. This means that $\overline{H}_{0}$ is the only similarity
solution that can be observed, all other solutions are unstable. Close to the
fixed point, the approach to $\overline{H}_{0}$ will be dominated by the
largest negative eigenvalue $\nu_{1}$:
$h(x,t)=t^{\prime 1/4}\left[\overline{H}(\xi)+\epsilon
t^{\prime-\nu_{1}}P_{1}(\xi)\right].$ (2.21bg)
For large arguments, the point $\xi_{cr}$ where the correction becomes
comparable to the similarity solution is $\xi\sim\epsilon
t^{\prime-\nu_{1}}\xi^{1-4\nu_{1}}$, and thus $\xi_{cr}\sim t^{\prime-1/4}$.
This means that the region of validity of $\overline{H}(\xi)$ expands in
similarity variables, and is constant in real space. This rapid convergence is
reflected by the numerical results reported in Fig. 3. More formally, one can
say that for any $\epsilon$ there is a $\delta$ such that
$\left|h(x,t)-t^{\prime 1/4}\overline{H}(\xi)\right|\leq\epsilon$ (2.21bh)
if $|x^{\prime}|\leq\delta$ uniformly as $t^{\prime}\rightarrow 0$.
We suspect that the situation described above is more general: the ground
state is stable, while each following profile has a number of additional
eigenvalues. In the case of the sequence of profiles $\overline{H}_{i}$ of
(2.4), two new positive eigenvalues appear for each new profile, corresponding
to a symmetric and an antisymmetric eigenfunction. Below we give two more
examples of the same scenario, for which we are able to give a simple
geometrical interpretation for the appearance of two additional positive
eigenvalues at each stage of the hierarchy of similarity solutions. The
simplest case is that of shock wave formation (cf. section 2.4), for which
everything can be worked out analytically.
The dynamical system corresponding to the self-similar solution (2.21ac) is
$U_{\tau}-\alpha U+\left(1+\alpha\right)\xi U_{\xi}+UU_{\xi}=0,$ (2.21bi)
and so the eigenvalue equation for perturbations $P$ around the base profile
$\overline{U}_{i}$ becomes
$(\alpha_{i}-\nu)P-(1+\alpha_{i})\xi
P_{\xi}-P(\overline{U}_{i})_{\xi}-P_{\xi}\overline{U}_{i}=0,\quad i=0,1,\dots$
(2.21bj)
Here $\overline{U}_{i}$ is the ith similarity function defined by (2.21ae) for
the exponents $\alpha_{i}$ as given by (2.21af).
The eigenvalue equation (2.21bj) is solved easily by transforming from the
variable $\xi$ to the variable $\overline{U}$, using (2.21ae):
$P\left[(\alpha_{i}-\nu)(1+(2i+3)\overline{U}_{i}^{2i+2})+1\right]=\frac{\partial
P}{\partial\overline{U}}\left[\alpha_{i}\overline{U}_{i}+(1+\alpha_{i})\overline{U}_{i}^{2i+3}\right],$
(2.21bk)
with solution
$P=\frac{\overline{U}_{i}^{3+2i-2\nu(i+1)}}{1+(2i+3)\overline{U}_{i}^{2i+2}}.$
(2.21bl)
The exponent $3+2i-2\nu(i+1)$ must be an integer for (2.21bl) to be regular at
the origin, so the eigenvalues are
$\nu_{j}=\frac{2i+4-j}{2i+2},\quad j=1,2,\dots$ (2.21bm)
As usual, the eigensolutions are alternating between even and odd. However, we
are interested in the first instance, given by (2.21aa), at which a shock
forms. This implies that the second derivative of the profile must vanish at
the location of the shock, and the amplitude of the $j=3$ perturbation must be
exactly zero.
Thus for $i=0$ the remaining eigenvalues are $\nu=3/2,1,0,-1/2,\dots$; the
first two are the eigenvalues $\nu_{x}=\beta=1+\alpha$ and $\nu_{t}=1$ found
above. The vanishing eigenvalue occurs because there is a family of solutions
parameterised by the coefficient $C$ in (2.21ae). All the other eigenvalues
are negative, which shows that the similarity solution (2.21ag) is stable. In
the same vein, for $\alpha_{1}=1/4$ there are two more positive exponents:
$\nu=5/4,1,1/2,1/4$, so the solution must be unstable. The same is of course
true for all higher order solutions. Thus in conclusion the ground state
solution $\overline{U}_{0}$ given by (2.21ag) is the only observable form of
shock formation. The same conclusion was reached in [27] by a stability
analysis based on catastrophe theory.
The sequence of profiles for viscous pinch-off, found in section 2.3, suggests
a simple mechanism for the fact that two new unstable directions appear with
each new similarity profile of higher order. In fact, the argument is
strikingly similar to that given for shock formation. Differentiating (2.21al)
with respect to $s$ one finds that a local minimum point $s_{min}$ remains a
minimum. Thus the local time evolution of the profile can be written as
$h(s,t)=h_{m}+\sum_{j=2}^{\infty}B_{j}(t)s^{\prime j}.$ (2.21bn)
For generic initial data $B_{2}(0)\neq 0$, so there is no reason why $B_{2}$
should vanish at the singular time, which means that the self-similar solution
$f_{0}$ will develop, which has a quadratic minimum. This situation is
structurally stable, so one expects the eigenvalues of the linearisation to be
negative. If however the coefficients $B_{j}(0)$ are zero for $j=2,\dots
2n-1$, they will remain zero for all times. Namely, if the first $k$
$s$-derivatives of $h$ vanish, one has
$\partial_{s}^{j}h_{t}=-\frac{C\partial_{s}^{j}h}{h^{2}},\quad j=1,\dots,k,$
(2.21bo)
so the first $k$ derivatives will remain zero. Thus to find the similarity
profile with $i=1$, one needs $B_{2}(0)=B_{3}(0)=0$ as an initial condition.
This is a non-generic situation, and a slight perturbation will make $B_{2}$
and $B_{3}$ nonzero. In other words, there are two unstable directions, which
take the solution away from $f_{1}(\xi)$, as defined by (2.21ap). In the
general case, the linearisation around $f_{i}(\xi)$ will have $2i$ positive
eigenvalues (apart from the trivial ones). Extensive numerical simulations of
drop pinch-off in the inertial-surface tension-viscous regime (cf. section
2.1.1) suggests that the the hierarchy of similarity solutions again has
similar properties in this case as well, although stability has not been
studied theoretically. The ground-state profile is stable, while all the
others are unstable [42]. Even when using a higher-order similarity solution
as an initial condition, it is immediately destabilised, and converges onto
the ground state solution [51].
## 3 Centre manifold
In section 2 we described the generic situation that the behaviour of a
similarity solution is determined by the linearisation around it. In the case
of a stable fixed point, convergence is exponentially fast, and the observed
behaviour is essentially that of the fixed point. In this section, we describe
a variety different cases where the the dynamics is slow. In all cases we are
able to associate this slow dynamics with a fixed point in the appropriate
variable(s), around which the eigenvalues vanish. Instead, higher-order non-
linear terms have to be taken into account, and the slow approach to the fixed
point is determined by a low-dimensional dynamical system.
We consider essentially two different cases:
* (a)
The dynamical system (1.4) possesses a fixed point $H_{0}(\xi)$, which has a
vanishing eigenvalue, with corresponding eigenfunction $\psi(\xi)$. The
dynamics in the slow direction $\psi$ is described by a nonlinear equation for
the amplitude $a(\tau)$, which varies on a logarithmic time scale:
$h=t^{\prime\alpha}\left[H_{0}(\xi)+a(\tau)\psi(\xi)\right],\quad\xi=x^{\prime}/t^{\prime\beta}.$
(2.21a)
* (b)
The dynamical system does not possess a fixed point, but has a solution of a
slightly more general form:
$h=h_{0}(\tau)H(\xi),\quad\xi=x^{\prime}/W(\tau),$ (2.21b)
where $h_{0}$ and $W$ are not necessarily power laws. To expand about a fixed
point, we define the generalised exponents
$\alpha=-\partial_{\tau}h_{0}/h_{0},\quad\beta=-\partial_{\tau}W/W$ (2.21c)
which now depend on time. In the case of a type-I similarity solution, this
reduces to the usual definition of the exponent. In the cases considered
below, one derives a finite dimensional dynamical system for the exponents
$\alpha,\beta$ (potentially including other, similarly defined scale factors).
Once more, the exponents vary on a logarithmic time scale, which can be
understood from the fact that the dynamical system possesses a fixed point
with vanishing eigenvalues.
Zero eigenvalues can also be associated to symmetries of the singularity, like
rotational or translational symmetries, which lead to the existence of a
continuum of similarity solutions. Another example, which concerns the
dynamics inside the singular object itself, is wave steepening as described by
(2.21ae) above. As seen from (2.21bm), there indeed is a vanishing eigenvalue
associated with this continuum of solutions. Below we will not be concerned
with this case, but only consider approach to the singularity starting from
nonsingular solutions.
### 3.1 Quadratic non-linearity: geometric evolution and reaction-diffusion
equations
The appearance of this type of nonlinearity is characteristic for various
nonlinear parabolic equations and systems. The blow-up behaviour is
characterised by the presence of logarithmic corrections in the similarity
profiles.
#### 3.1.1 Geometric evolution equations: Mean curvature and Ricci flows
Axisymmetric motion by mean curvature in three spatial dimensions is described
by the equation
$h_{t}=\left(\frac{h_{xx}}{1+h_{x}^{2}}-\frac{1}{h}\right),$ (2.21d)
where $h(x,t)$ is the radius of the moving free surface. A very good physical
realization of (2.21d) is the melting and freezing of a 3He crystal, driven by
surface tension [111], see Fig. 7. As before, the time scale $t$ has been
chosen such that the diffusion constant, which sets the rate of motion, is
normalised to one. A possible boundary condition for the problem is that
$h(0,t)=h(L,t)=R$, where $R$ is some prescribed radius. For certain initial
conditions $h(x,0)\equiv h_{0}(x)$ the interface will become singular at some
time $t_{0}$, at which $h(x_{0},t_{0})=0$ and the curvature blows up. The
moment of blow-up is shown in panel h of Fig. 7, for example.
Figure 7: Nine images (of width 3.5 mm) showing how a 3He crystal “flows” down
from the upper part of a cryogenic cell into its lower part [112]. The
recording takes a few minutes, the temperature is 0.32 K. 11 mK. The crystal
first “drips” down, so that a crystalline “drop” forms at the bottom (a to c);
then a second drop appears (d) and comes into contact with the first one (e);
coalescence is observed (f) and subsequently breakup occurs (h).
Inserting the self-similar solution (1.2) into (2.21d), one finds a balance
for $\alpha=\beta=1/2$. The corresponding similarity equation is
$-\frac{\phi}{2}+\xi\frac{\phi_{\xi}}{2}=\left(\frac{\phi_{\xi\xi}}{1+\phi_{\xi}^{2}}-\frac{1}{\phi}\right),\quad\xi=\frac{x^{\prime}}{t^{\prime
1/2}}.$ (2.21e)
One solution of (2.21e) is the constant solution $\phi(\xi)=\sqrt{2}$. Another
potential solution is one that grows linearly at infinity, to ensure matching
onto a time-independent outer solution. However, it can be shown that no
solution to (2.21e), which also grows linearly at infinity, exists [113, 114].
Our analysis below follows the rigorous work in [30], demonstrating type-II
self-similarity. In addition, we now show how the description of the dynamical
system can be carried out to arbitrary order.
The relevant solution is thus the constant solution, but which of course does
not match onto a time-independent outer solution. We thus write the solution
as
$h(x,t)=t^{\prime 1/2}\left[\sqrt{2}+g(\xi,\tau)\right],$ (2.21f)
with $\tau=-\ln(t^{\prime})$ as usual. The equation for $g$ is then
$g_{\tau}=g-\frac{\xi
g_{\xi}}{2}+\frac{g_{\xi\xi}}{1+g_{\xi}^{2}}-\frac{g^{2}}{2^{3/2}+2g},$
(2.21g)
which we solve by expanding into eigenfunctions of the linear part of the
operator
${\cal L}g=g-\xi g_{\xi}/2+g_{\xi\xi}.$ (2.21h)
It is easily confirmed that
${\cal L}H_{2i}(\xi/2)=\nu_{i}H_{2i}(\xi/2),\quad i=0,1,\dots,$ (2.21i)
where $H_{n}$ is the n-th Hermite polynomial [100]:
$H_{n}(y)=(-1)^{n}e^{y^{2}}\frac{d^{n}}{dy^{n}}e^{-y^{2}},$ (2.21j)
and $\nu_{i}=1-i$. Thus the first eigenvalue is $\nu_{0}=1$, which corresponds
to the positive eigenvalue $\nu_{t}$ coming from the arbitrary choice of
$t_{0}$. The other positive eigenvalue eigenvalue $\nu_{x}$ does not appear,
since we have chosen to look at symmetric solutions, breaking translational
invariance. However, the largest non-trivial eigenvalue $\nu_{1}$ is zero, and
the linear part of (2.21g) becomes
$\frac{\partial a_{i}}{\partial\tau}=(1-i)a_{i},\quad i=0,1,\dots.$ (2.21k)
Thus all perturbations with $i>1$ decay, but to investigate the approach of
the cylindrical solution, one must include nonlinear terms in the equation for
$a_{1}$.
If we write
$g(\xi,\tau)=\sum_{i=1}^{\infty}a_{i}(\tau)H_{2i}(\xi/2),$ (2.21l)
the equation for $a_{1}$ becomes
$\frac{da_{1}}{d\tau}=-2^{3/2}a_{1}^{2}+O(a_{1}a_{j}),$ (2.21m)
whose solution is
$a_{1}=1/(2^{3/2}\tau).$ (2.21n)
Thus instead of the expected exponential convergence onto the fixed point, the
approach is only algebraic. Since all other eigenvalues are negative, the
$\tau$-dependence of the $a_{i}$ is slaved by the dynamics of $a_{1}$. Namely,
as we will see below, $a_{j}=O(\tau^{-j})$, so corrections to (2.21m) are of
higher order. To summarise, the leading-order behaviour of (2.21d) is given by
$h(x,t)=t^{\prime 1/2}\left[\sqrt{2}+a_{1}(\tau)H_{2}(\xi)\right],$ (2.21o)
as was proven by [30].
Now we compute the specific form of the higher-order corrections to (2.21o),
which have not been worked out explicitly before. If one linearises around
(2.21n), putting $a_{1}=a_{1}^{(0)}+\epsilon_{1}$, one finds
$\frac{d\epsilon_{1}}{d\tau}=-\frac{2}{\tau}\epsilon_{1}+\mbox{other terms}.$
(2.21p)
This means that the coefficient $A$ of $\epsilon_{1}=A/\tau^{2}$ remains
undetermined, and a simple expansion of $a_{i}$ in powers of $\tau^{-1}$
yields an indeterminate system. Instead, at quadratic order, a term of the
form $\epsilon_{1}=A\ln\tau/\tau^{2}$ is needed. Fortunately, this is the only
place in the system of nonlinear equations for $a_{i}$ where such an
indeterminacy occurs. Thus all logarithmic dependencies can be traced, leading
to the general ansatz
$a_{i}^{(n)}=\frac{\delta_{i}}{\tau^{i}}+\sum_{k=i+1}^{n}\sum_{l=0}^{k-i}\frac{(\ln\tau)^{l}}{\tau^{k}}\delta_{lki},$
(2.21q)
where $\delta_{i}$ and $\delta_{lki}$ are coefficients to be determined. The
index $n$ is the order of the truncation.
The coefficients can now be found recursively by considering terms of
successively higher order in $\tau^{-1}$ in the first equation:
$\displaystyle\frac{da_{1}}{d\tau}=-2^{3/2}a_{1}^{2}-24\sqrt{2}a_{1}a_{2}+22a_{1}^{3}-$
$\displaystyle 272\sqrt{2}a_{1}^{4}-191\sqrt{2}a_{2}^{2}+192a_{1}^{2}a_{2}$
(2.21ra)
$\displaystyle\frac{da_{2}}{d\tau}=-a_{2}-\sqrt{2}/4a_{1}^{2}+6a_{1}^{3}-8\sqrt{2}a_{1}a_{2}.$
(2.21rb)
The next two orders will involve the next coefficient $a_{3}$. From (2.21ra)
and (2.21rb), one first finds $\delta_{121}$ and $\delta_{2}$, by considering
$O(\tau^{-3})$ and $O(\tau^{-2})$, respectively. Then, at order
$O(\tau^{-(n+1)})$ in the first equation, where $n=3$, one finds all remaining
coefficients $\delta_{lki}$ in the expansion (2.21q) up to $k=n$. At each
order in $\tau^{-1}$, there is of course a series expansion in $\ln\tau$ which
determines all the coefficients.
We constructed a MAPLE program to compute all the coefficients up to
arbitrarily high order (10th, say). Up to third order in $\tau^{-1}$ the
result is:
$\displaystyle
a_{1}=1/4\,{\frac{\sqrt{2}}{\tau}}+{\frac{17}{16}}\,{\frac{\ln\left(\tau\right)\sqrt{2}}{{\tau}^{2}}}-{\frac{73}{16}}\,{\frac{\sqrt{2}}{{\tau}^{3}}}+$
$\displaystyle{\frac{867}{128}}\,{\frac{\ln\left(\tau\right)\sqrt{2}}{{\tau}^{3}}}-{\frac{289}{128}}\,{\frac{\left(\ln\left(\tau\right)\right)^{2}\sqrt{2}}{{\tau}^{3}}}$
(2.21rsa) $\displaystyle
a_{2}=-1/32\,{\frac{\sqrt{2}}{{\tau}^{2}}}+{\frac{5}{16}}\,{\frac{\sqrt{2}}{{\tau}^{3}}}-{\frac{17}{64}}\,{\frac{\ln\left(\tau\right)\sqrt{2}}{{\tau}^{3}}},$
(2.21rsb)
and thus $h(x,t)$ becomes
$h(x,t)=t^{\prime
1/2}\left[\sqrt{2}+a_{1}(\tau)\left(-2+\xi^{2}\right)+a_{2}(\tau)\left(12-12\xi^{2}+\xi^{4}\right)\right],$
(2.21rst)
from which one of course immediately finds the minimum. To second order, the
result is
$h_{m}=(2t^{\prime})^{1/2}\left[1-\frac{1}{2\tau}-\frac{3+17\ln\tau}{8\tau^{2}}\right].$
(2.21rsu)
Figure 8: A plot of
$\left[h_{m}/\sqrt{2t^{\prime}}-1+1/(2\tau)\right]\tau^{2}$ (dashed line) and
$\tau_{0}/2-(3+17\ln(\tau+\tau_{0})/8)$ (full line) with $\tau_{0}=4.56$.
First, the presence of logarithms implies that there is some dependence on
initial conditions built into the description. The reason is that the argument
inside the logarithm needs to be non-dimensionalised using some “external”
time scale. More formally, any change in time scale $\tilde{t}=t/t_{0}$ leads
to an identical equation if also lengths are rescaled according to
$\tilde{h}=h/\sqrt{t_{0}}$. This leaves the prefactor in (2.21rsu) invariant,
but adds an arbitrary constant $\tau_{0}$ to $\tau$. This is illustrated by
comparing to a numerical simulation of the mean curvature equation (2.21d)
close to the point of breakup, see Fig. 8. Namely, we subtract the analytical
result (2.21rsu) from the numerical solution $h_{m}/(2\sqrt{t^{\prime}})$ and
multiply by $\tau^{2}$. As seen in Fig.8, the remainder is varying slowly over
12 decades in $t^{\prime}$. If the constant $\tau_{0}$ is adjusted, this small
variation is seen to be consistent with the logarithmic dependence predicted
by (2.21rsu).
The second important point is that convergence in space is no longer uniform
as implied by (2.21bh) for the case of type I self-similarity. Namely, to
leading order the pinching solution is a cylinder. For this to be a good
approximation, one has to require that the correction is small:
$\xi^{2}/\tau\ll 1$. Thus corrections become important beyond
$\xi_{cr}\sim\tau$, which, in view of the logarithmic growth of $\tau$,
implies convergence in a constant region in similarity variables only. As
shown in [111], the slow convergence toward the self-similar behaviour has
important consequences for a comparison to experimental data.
Mean curvature flow is also an example of a broader class of problems called
generically ”geometric evolution equations”. These are evolution equations
intended to gain topological insight by flowing geometrical objects (such as
metric or curvature) towards easily recognisable objects such as constant or
positive curvature manifolds. The most remarkable example is the so called
Ricci flow, introduced in [115], which is the essential tool in the recent
proof of the geometrisation conjecture (including Poincaré’s conjecture as a
consequence) by Grigori Perelman.
Namely, Poincaré’s conjecture states that every simply connected closed
3-manifold is homeomorphic to the 3-sphere. Being homeomorphic means that both
are topologically equivalent and can be transformed one into the other through
continuous mappings. Such mappings can be obtained from the flow associated to
an evolutionary PDE involving fundamental geometrical properties of the
manifold. Thurston’s geometrisation conjecture is a generalisation of
Poincaré’s conjecture to general 3-manifolds and states that compact
3-manifolds can be decomposed into submanifolds that have basic geometric
structures.
Perelman sketched a proof of the full geometrisation conjecture in 2003 using
Ricci flow with surgery [116]. Starting with an initial 3-manifold, one
deforms it in time according to the solutions of the Ricci flow PDE (2.21rsv)
we consider below. Since the flow is continuous, the different manifolds
obtained during the evolution will be homeomorphic to the initial one. The
problem is in the fact that Ricci flow develops singularities in finite time,
one of which we describe below. One would like to get over this difficulty by
devising a mechanism of continuation of solutions beyond the singularity,
making sure that such a mechanism controls the topological changes leading to
a decomposition into submanifolds, whose structure is given by Thurston’s
geometrisation conjecture. Perelman obtained essential information on how
singularities are like, essentially three dimensional cylinders made out of
spheres stretched out along a line, so that he could develop the correct
continuation (also called “surgery”) procedure and continue the flow up to a
final stage consisting of the elementary geometrical objects in Thurston’s
conjecture.
Ricci flow is defined by the equation
$\frac{\partial g_{ij}}{\partial t}=-2R_{ij}$ (2.21rsv)
for a Riemannian metric $g_{ij}$, where $R_{ij}$ is the Ricci curvature
tensor. The Ricci tensor involves second derivatives of the curvature and
terms that are quadratic in the curvature. Hence, there is the potential for
singularity formation and singularities are, in fact, formed. As Perelman
poses it, the most natural way to form a singularity in finite time is by
pinching an almost round cylindrical neck. The structure of this kind of
singularity has been studied in [117]. By writing the metric of a
$(n+1)$-dimensional cylinder as
$g=ds^{2}+\psi^{2}g_{can}\ ,$ (2.21rsw)
where $g_{can}$ is the canonical metric of radius one in the $n-$sphere
$S^{n}$, $\psi(s,t)$ is the radius of the hypersurface
$\left\\{s\right\\}\times S^{n}$ at time $t$ and $s$ is the arclength
parameter of the generatrix of the cylinder.
The equation for $\psi$ then becomes
$\psi_{t}=\psi_{ss}-\frac{(n-1)(1-\psi_{s}^{2})}{\psi}.$ (2.21rsx)
In [117] it is shown that for $n>1$ the solution close to the singularity
admits a representation that resembles the one obtained for mean curvature
flow:
$\psi(s,t)=\frac{1}{2^{\frac{1}{2}}(n-1)^{\frac{1}{2}}t^{\prime
1/2}}u(\xi,\tau),\qquad\xi=s/t^{\prime 1/2}.$ (2.21rsy)
Namely, (2.21rsx) admits a constant solution $u(\xi,\tau)=1$, and the
linearisation around it gives the same linear operator (2.21h) as for mean
curvature flow. Thus a pinching solution behaves as
$u(\xi,\tau)=1+a(\tau)H_{2}(\xi/2)+o(\tau^{-1}),$ (2.21rsz)
where the equation for $a$ is $a_{\tau}=-8a^{2}$, with solution $a=1/(8\tau)$.
#### 3.1.2 Reaction-diffusion equations
The semilinear parabolic equation
$u_{t}-\Delta u-\left|u\right|^{p-1}u=0$ (2.21rsaa)
is again closely related to the mean curvature flow problem (2.21d). Namely,
disregarding the higher order term in $h_{x}$, (2.21d) becomes
$h_{t}=h_{xx}-\frac{1}{h}.$ (2.21rsab)
Putting $u=1/h$ one finds
$u_{t}=u_{xx}+u^{3}-2u_{x}^{2}/u,$ (2.21rsac)
which is (2.21rsaa) in one space dimension and $p=3$, once more neglecting
higher-order non-linearities. As before, (2.21rsaa) has the exact blow-up
solution
$u=(p-1)^{\frac{1}{1-p}}t^{\prime-\frac{1}{p-1}}.$ (2.21rsad)
If $1<p<p_{c}=\frac{d+2}{d-2}$, where $d$ is the space dimension, then there
are no other self-similar solutions to (2.21rsaa) [18], and blow-up is of the
form (2.21rsad) (see [118], [119] and [120] for a recent review). As in the
case of mean curvature flow, corrections to (2.21rsad) are described by a
slowly varying amplitude $a$:
$u=t^{\prime
1/(p-1)}(p-1)^{\frac{1}{1-p}}\left[1-aH_{2}(\xi/2)+O(1/\tau^{2})\right],\quad\xi=x^{\prime}/t^{\prime
1/2},$ (2.21rsae)
where $a$ obeys the equation
$a_{\tau}=-4pa^{2}.$ (2.21rsaf)
This result holds in 1 space dimension. In higher dimensions, one has to
replace $x$ by the distance to the blow-up set.
This covers all range of exponents (larger than one, because otherwise there
is no blow-up) in dimensions $1$ and $2$. The situation if $p>p_{c}$ is not so
clear: if $p>1+\frac{2}{d}$ then there are solutions that blow-up and ”small”
solutions that do not blow-up. Nevertheless, the construction of solutions as
perturbations of constant self-similar solutions holds for any $d$ and any
$p>1$. A simple generalisation of (2.21rsaa) results from considering a
nonlinear diffusion operator,
$u_{t}-\nabla\cdot(|u|^{m}\nabla u)=u^{p}$ (2.21rsag)
and now the blow-up character depends on the two parameters m and p, see
[121].
### 3.2 Cubic non-linearity: Cavity breakup and Chemotaxis
More complex logarithmic corrections are possible if the linearisation around
the fixed point leads to a zero eigenvalue and cubic nonlinearities.
#### 3.2.1 Cavity break-up
As shown in [122], the equation for a slender cavity or bubble is
$\int_{-L}^{L}\frac{\ddot{a}(\xi,t)d\xi}{\sqrt{(x-\xi)^{2}+a(x,t)}}=\frac{\dot{a}^{2}}{2a},$
(2.21rsah)
where $a(x,t)\equiv h^{2}(x,t)$ and $h(x,t)$ is the radius of the bubble. Dots
denote derivatives with respect to time $t$. The length $L$ measures the total
size of the bubble. If for the moment one disregards boundary conditions and
looks for solutions to (2.21rsah) of cylindrical form, $a(x,t)=a_{0}(t)$, one
can do the integral to find
$\ddot{a}_{0}\ln\left(\frac{4L^{2}}{a_{0}}\right)=\frac{\dot{a}_{0}^{2}}{2a_{0}}.$
(2.21rsai)
It is easy to show that an an asymptotic solution of (2.21rsai) is given by
$a_{0}\propto\frac{t^{\prime}}{\tau^{1/2}},$ (2.21rsaj)
corresponding to a power law with a small logarithmic correction. Indeed,
initial theories of bubble pinch-off [123, 124] treated the case of an
approximately cylindrical cavity, which leads to the radial exponent
$\alpha=1/2$, with logarithmic corrections.
Figure 9: The pinch-off of an air bubble in water [125]. An initially smooth
shape develops a localised pinch-point.
However both experiment [125] and simulation [122] show that the cylindrical
solution is unstable; rather, the pinch region is rather localised, see Fig.
9. Therefore, it is not enough to treat the width of the cavity as a constant
$L$; the width $W$ is itself a time-dependent quantity. In [122] we show that
to leading order the time evolution of the integral equation (2.21rsah) can be
reduced to a set of ordinary differential equations for the minimum $a_{0}$ of
$a(x,t)$, as well as its curvature $a_{0}^{\prime\prime}$.
Figure 10: A comparison of the exponent $\alpha$ between full numerical
simulations of bubble pinch-off (solid line) and the leading order asymptotic
theory (2.21rsaq) (dashed line).
Namely, the integral in (2.21rsah) is dominated by a local contribution from
the pinch region. To estimate this contribution, it is sufficient to expand
the profile around the minimum at $z=0$:
$a(x,t)=a_{0}+(a^{\prime\prime}_{0}/2)z^{2}+O(z^{4})$. As in previous
theories, the integral depends logarithmically on $a$, but the axial length
scale is provided by the inverse curvature
$W\equiv(2a_{0}/a^{\prime\prime}_{0})^{1/2}$. Thus evaluating (2.21rsah) at
the minimum, one obtains [122] to leading order
$\ddot{a}_{0}\ln(4W^{2}/a_{0})=\dot{a}_{0}^{2}/(2a_{0}),$ (2.21rsak)
which is a coupled equation for $a_{0}$ and $W$. Thus, a second equation is
needed to close the system, which is obtained by evaluating the the second
derivative of (2.21rsah) at the pinch point:
$\ddot{a}^{\prime\prime}_{0}\ln\left(\frac{8}{e^{3}a^{\prime\prime}_{0}}\right)-2\frac{\ddot{a}_{0}a^{\prime\prime}_{0}}{a_{0}}=\frac{\dot{a}_{0}\dot{a}_{0}^{\prime\prime}}{a_{0}}-\frac{\dot{a}_{0}^{2}a_{0}^{\prime\prime}}{2a_{0}^{2}}.$
(2.21rsal)
The two coupled equations (2.21rsak),(2.21rsal) are most easily recast in
terms of the time-dependent exponents
$2\alpha\equiv-\partial_{\tau}a_{0}/a_{0},\quad
2\delta\equiv-\partial_{\tau}a^{\prime\prime}_{0}/a^{\prime\prime}_{0},$
(2.21rsam)
where $\beta=\alpha-\delta$, so $\alpha,\beta$ are generalisations of the
usual exponents in (1.2). The exponent $\delta$ characterises the time
dependence of the aspect ratio $W$. Returning to the collapse (2.21rsai)
predicted for a constant solution, one finds that $\alpha=1/2$ and $\delta=0$.
In the spirit of the the previous subsection, this is the fixed point
corresponding to the cylindrical solution. Now we expand the values of
$\alpha$ and $\delta$ around their expected asymptotic values $1/2$ and $0$:
$\alpha=1/2+u(\tau),\quad\delta=v(\tau).$ (2.21rsan)
and put $w(\tau)=1/\ln(a^{\prime\prime}_{0})$.
To leading order, the resulting equations are
$u_{\tau}=u+w/4,\quad v_{\tau}=-v-w/4,\quad w_{\tau}=2vw^{2}.$ (2.21rsao)
The linearisation around the fixed point thus has the eigenvalues $0$ and
$-1$, in addition to the eigenvalue $1$ coming from time translation. As
before, the vanishing eigenvalue is the origin of the slow approach to the
fixed point observed for the present problem. The derivatives $u_{\tau}$ and
$v_{\tau}$ are of lower order in the first two equations of (2.21rsao), and
thus to leading order $u=v$ and $v=-w/4$. Using this, the last equation of
(2.21rsao) can be simplified to
$w_{\tau}=-w^{3}/2.$ (2.21rsap)
Equation (2.21rsap) is analogous to (2.21m), but has a degeneracy of third
order, rather than second order. Equation (2.21rsap) yields, in an expansion
for small $\delta$ [122],
$\alpha=1/2+\frac{1}{4\sqrt{\tau}}+O(\tau),\quad\delta=\frac{1}{4\sqrt{\tau}}+O(\tau^{-3/2}).$
(2.21rsaq)
Thus the exponents converge toward their asymptotic values $\alpha=\beta=1/2$
only very slowly, as illustrated in Fig. 10. This explains why typical
experimental values are found in the range $\alpha\approx 0.54-0.58$ [125],
and why there is a weak dependence on initial conditions [126].
#### 3.2.2 Keller-Segel model for chemotaxis
This model describes the aggregation of microorganisms driven by chemotactic
stimuli. The problem has biological meaning in 2 space dimensions. If we
describe the density of individuals by $u(x,t)$ and the concentration of the
chemotactic agent by $v(x,t)$, then the Keller-Segel system reads
$\displaystyle u_{t}$ $\displaystyle=$ $\displaystyle\Delta
u-\chi\nabla\cdot(u\nabla v),$ (2.21rsara) $\displaystyle\Gamma v_{t}$
$\displaystyle=$ $\displaystyle\Delta v+(u-1),$ (2.21rsarb)
where $\Gamma$ and $\chi$ are positive constants. In [13, 127] it was shown
that for radially symmetric solutions of (2.21rsara),(2.21rsarb) singularities
are such that to leading order $u$ blows up in the form of a delta function.
The profile close to the singularity is self-similar and of the form
$u(r,t)=\frac{1}{R^{2}(t)}U\left(\frac{r}{R(t)}\right),$ (2.21rsaras)
where
$R(t)=Ce^{-\frac{1}{2}\tau-\frac{\sqrt{2}}{2}\tau^{\frac{1}{2}}-\frac{1}{4}\ln\tau+\frac{1}{4}\frac{\ln\tau}{\sqrt{\tau}}}(1+o(1))$
(2.21rsarat)
and
$U(\xi)=\frac{8}{\chi(1+\xi^{2})}.$ (2.21rsarau)
The result comes from a careful matched asymptotics analysis that, in our
notation, amounts to introducing the time-dependent exponent
$\gamma=-\partial_{\tau}R/R,$ (2.21rsarav)
which has the fixed point $\gamma=1/2$. Corrections are of the form
$\gamma=\frac{1}{2}+\frac{\alpha}{2}\left(\alpha-\alpha^{2}+1\right),$
(2.21rsaraw)
where $\alpha$ is controlled by a third-order non-linearity, as in the bubble
problem:
$\alpha_{\tau}=-\alpha^{3}(1-\alpha+o(\alpha)).$ (2.21rsarax)
### 3.3 Beyond all orders: The nonlinear Schrödinger equation
The cubic nonlinear Schrödinger equation
$i\varphi_{t}+\Delta\varphi+\left|\varphi\right|^{2}\varphi=0\,$ (2.21rsaray)
appears in the description of beam focusing in a nonlinear optical medium, for
which the space dimension is $d=2$. Equation (2.21rsaray) belongs to the more
general family of nonlinear Schrödinger equations of the form
$i\varphi_{t}+\Delta\varphi+\left|\varphi\right|^{p}\varphi=0,$ (2.21rsaraz)
and in any dimension $d$. Of particular interest, from the point of view of
singularities, is the critical case $p=4/d$. In this case, singularities with
slowly converging similarity exponents appear due to the presence of zero
eigenvalues. We will describe this situation below, based on the formal
construction of Zakharov [128], later proved rigorously by Galina Perelman
[129]. At the moment, the explicit construction has only been given for $d=1$,
that is, for the quintic Schrödinger equation. The same blow-up estimates have
been shown to hold for any space dimension $d<6$ by Merle and Raphaël [130],
[131], without making use of Zakharov’s [128] formal construction. Merle and
Raphaël also show that the stable solutions to be described below are in fact
global attractors.
In the critical case (2.21rsaraz) becomes in d=1:
$i\varphi_{t}+\varphi_{xx}+\left|\varphi\right|^{4}\varphi=0.$ (2.21rsarba)
This equation has explicit self-similar solutions (in the sense that rescaling
$x\rightarrow\lambda x$, $t\rightarrow\lambda^{2}t$,
$\varphi\rightarrow\lambda^{\frac{1}{2}}\varphi$ leaves the solutions
unchanged except for the trivial phase factor $e^{-2i\mu_{0}\ln\lambda}$) of
the form
$\varphi(x,t)=e^{i\mu_{0}\tau}e^{-\frac{\xi^{2}}{8}i}\frac{1}{t^{\prime\frac{1}{4}}}\varphi_{0}(\xi),\quad\xi=x^{\prime}/t^{\prime
1/2}.$ (2.21rsarbb)
The function $\varphi_{0}(\xi)$ solves
$-\varphi_{0,\xi\xi}+\varphi_{0}-\left|\varphi_{0}\right|^{4}\varphi_{0}=0,$
(2.21rsarbc)
and is given explicitly by
$\varphi_{0}(\xi)=\frac{(3\mu_{0})^{\frac{1}{4}}}{\cosh^{\frac{1}{2}}(2\sqrt{\mu_{0}}\xi)}.$
(2.21rsarbd)
We seek solutions of (2.21rsarba) using a generalisation of (2.21rsarbb),
which allow for a variation of the phase factors, and the amplitude to be
different from a power law:
$\varphi(x,t)=e^{i\mu(t)-i\beta(t)z^{2}/4}\lambda^{\frac{1}{2}}(t)\varphi_{a}(z),$
(2.21rsarbe)
where $z=\lambda(t)x$ and $\varphi_{a}$ satisfies
$-\varphi_{a,\xi\xi}+\varphi_{a}-\frac{1}{4}az^{2}\varphi_{a}-\left|\varphi_{a}\right|^{4}\varphi_{a}=0.$
(2.21rsarbf)
When $h$ $(=\sqrt{a})$ is constant, (2.21rsarbe) is a solution of (2.21rsarba)
if $(\mu,\lambda,\beta)$ satisfy
$\displaystyle\mu_{t}$ $\displaystyle=$ $\displaystyle\lambda^{2}$
(2.21rsarbga) $\displaystyle\lambda^{-3}\lambda_{t}$ $\displaystyle=$
$\displaystyle\beta$ (2.21rsarbgb)
$\displaystyle\beta_{t}+\lambda^{2}\beta^{2}$ $\displaystyle=$
$\displaystyle\lambda^{2}h^{2}.$ (2.21rsarbgc)
Notice that the equation for $\mu$ is uncoupled, so we only need to solve the
equations for $(\lambda,\beta)$ simultaneously and then integrate the equation
for $\mu$. It is interesting for the following that, in addition to the
solutions for constant $a$, one can let $a$ vary slowly in time. The resulting
system for $(\lambda,\beta,h)$ is
$\displaystyle\lambda^{-3}\lambda_{t}$ $\displaystyle=$ $\displaystyle\beta$
(2.21rsarbgbha) $\displaystyle\beta_{t}+\lambda^{2}\beta^{2}$ $\displaystyle=$
$\displaystyle\lambda^{2}h^{2}$ (2.21rsarbgbhb) $\displaystyle h_{t}$
$\displaystyle=$ $\displaystyle-c\lambda^{2}e^{-S_{0}/h}/h.$ (2.21rsarbgbhc)
Note the appearance of the factor $e^{-S_{0}/h}$ in the last equation, which
comes from a semiclassical limit of a linear Schrödinger equation with
appropriate potential (see [129]), and
$S_{0}=\int_{0}^{2}\sqrt{1-s^{2}/4}ds=\frac{\pi}{2}.$ (2.21rsarbgbhbi)
$S_{0}$ is an It follows from the presence of this factor that the non-
linearity is beyond all orders, smaller than any given power, in contrast to
the examples given above.
As in section 3.2.1, we rewrite the equations in terms of similarity
exponents,
$\alpha=-\frac{\lambda_{\tau}}{\lambda},\ \gamma=-\frac{\beta_{\tau}}{\beta},\
\delta=-\frac{h_{\tau}}{h}$ (2.21rsarbgbhbj)
to obtain the system:
$\displaystyle\alpha_{\tau}$ $\displaystyle=$
$\displaystyle-(1+2\alpha+\gamma)\alpha$ (2.21rsarbgbhbka)
$\displaystyle\gamma_{\tau}$ $\displaystyle=$
$\displaystyle(1+2\alpha+\gamma)\alpha-(\gamma+\alpha)(1+2\alpha+2\delta-\gamma)$
(2.21rsarbgbhbkb) $\displaystyle\delta_{\tau}$ $\displaystyle=$
$\displaystyle(-1-2\alpha+2\delta)\delta-\delta^{2}\frac{S_{0}}{h}$
(2.21rsarbgbhbkc) $\displaystyle h_{\tau}$ $\displaystyle=$
$\displaystyle-\delta h.$ (2.21rsarbgbhbkd)
The advantage of this formulation is that the exponents have fixed points.
There are two families of equilibrium points for
(2.21rsarbgbhbka)-(2.21rsarbgbhbkd):
1. (1)
$\alpha=-\frac{1}{2},\ \gamma=0\ ,\delta=0,\ h$ arbitrary positive or zero.
2. (2)
$\alpha=-1,\ \gamma=1\ ,\delta=0,\ h$ arbitrary positive or zero.
We first investigate case (1) by writing
$\alpha=-\frac{1}{2}+\alpha_{1},\ \gamma=\gamma_{1},\ \delta=\delta_{1},\
h=h_{1}.$ (2.21rsarbgbhbkbl)
The final fixed point corresponding to the singularity is going to be
$\alpha_{1}=\gamma_{1}=\delta_{1}=h_{1}=0$. However, there are also
equilibrium points for any $h>0$, in which case the linearisation reads:
$\displaystyle\alpha_{1,\tau}$ $\displaystyle=$
$\displaystyle\alpha_{1}+\frac{1}{2}\gamma_{1}$ (2.21rsarbgbhbkbma)
$\displaystyle\gamma_{1,\tau}$ $\displaystyle=$
$\displaystyle-\gamma_{1}+\delta_{1}$ (2.21rsarbgbhbkbmb)
$\displaystyle\delta_{1,\tau}$ $\displaystyle=$ $\displaystyle
2\delta_{1}^{2}-2\alpha_{1}\delta_{1}-\delta_{1}^{2}\frac{S_{0}}{h}.$
(2.21rsarbgbhbkbmc)
This system has the matrix
$A=\left(\begin{array}[]{ccc}1&\frac{1}{2}&0\\\ 0&-1&1\\\
0&0&0\end{array}\right),$
whose eigenvalues are: $1,0$, and $-1$. The vanishing eigenvalue corresponds
to the line of equilibrium points for $h>0$, the positive eigenvalue to the
direction of instability generated by a change in blow-up time. The
eigenvector corresponding to the negative eigenvalue gives the direction of
the stable manifold.
At the point $h=0$, there is an additional vanishing eigenvalue, and the
equations become:
$\displaystyle\alpha_{1,\tau^{\prime}}$ $\displaystyle=$
$\displaystyle(\alpha_{1}+\frac{1}{2}\gamma_{1})h_{1}$ (2.21rsarbgbhbkbmbna)
$\displaystyle\gamma_{1,\tau^{\prime}}$ $\displaystyle=$
$\displaystyle(-\gamma_{1}+\delta_{1})h_{1}$ (2.21rsarbgbhbkbmbnb)
$\displaystyle\delta_{1,\tau^{\prime}}$ $\displaystyle=$
$\displaystyle(2\delta_{1}^{2}-2\alpha_{1}\delta_{1})h_{1}-\delta_{1}^{2}S_{0}$
(2.21rsarbgbhbkbmbnc) $\displaystyle h_{1,\tau^{\prime}}$ $\displaystyle=$
$\displaystyle-\delta_{1}h_{1}^{2},$ (2.21rsarbgbhbkbmbnd)
where $d\tau^{\prime}=d\tau/h_{1}$. The first two equations reduce to leading
order to $\gamma_{1}=\delta_{1}h_{1}$ and $\alpha_{1}=-\delta_{1}h_{1}^{2}/2$,
while the last two equations reduce to the nonlinear system:
$\delta_{1,\tau^{\prime}}=-\delta_{1}^{2}S_{0},\quad
h_{1,\tau^{\prime}}=-\delta_{1}h_{1}^{2},\quad\tau_{\tau^{\prime}}=h_{1}.$
(2.21rsarbgbhbkbmbnbo)
In the original $\tau$-variable, the dynamical system is
$\delta_{1,\tau}=-\delta_{1}^{2}S_{0}/h_{1}\quad
h_{1,\tau^{\prime}}=-\delta_{1}h_{1},$ (2.21rsarbgbhbkbmbnbp)
which controls the approach to the fixed point. The system
(2.21rsarbgbhbkbmbnbp) is two-dimensional, corresponding to the two vanishing
eigenvalues.
Integrating the first equation of (2.21rsarbgbhbkbmbnbo) one gets
$\delta_{1}\sim 1/(S_{0}\tau^{\prime})$, and thus using the second equation
$h_{1}\sim S_{0}/\ln\tau^{\prime}$. From the last equation one obtains to
leading order $\tau^{\prime}\sim\tau\ln\tau/S_{0}$, so that
$h_{1}\sim\frac{S_{0}}{\ln\tau}\ ,\ \delta_{1}\sim\frac{1}{\tau\ln\tau}.$
(2.21rsarbgbhbkbmbnbq)
Thus we can conclude that
$\alpha(\tau)\simeq\frac{1}{2}-\frac{1}{2\tau\ln\tau},\quad\gamma(\tau)\simeq\frac{1}{\tau\ln\tau},\quad\delta(\tau)\simeq\frac{1}{\tau\ln\tau}.$
(2.21rsarbgbhbkbmbnbr)
In this fashion, one can construct a singular solution such that
$\displaystyle\varphi(x,t)=e^{-i\tau\ln\tau-i\frac{1}{t^{\prime}}x^{2}/4}\frac{(\ln\tau)^{\frac{1}{4}}}{t^{\prime\frac{1}{4}}}\varphi_{h^{2}\tau}\left(\frac{(\ln\tau)^{\frac{1}{2}}}{t^{\prime\frac{1}{2}}}x\right)$
$\displaystyle\sim
e^{-i\tau\ln\tau}\frac{(\ln\tau)^{\frac{1}{4}}}{t^{\prime\frac{1}{4}}}\varphi_{0}\left(\frac{(\ln\tau)^{\frac{1}{2}}}{t^{\prime\frac{1}{2}}}x\right)$
(2.21rsarbgbhbkbmbnbs)
Note the remarkable smallness of this correction to the “natural” scaling
exponent of $t^{\prime 1/4}$, which enters only as the logarithm of
logarithmic time $\tau$.
The fixed points (2) can be analysed in a similar fashion. The linearisation
leads to
$\displaystyle\alpha_{1,\tau}$ $\displaystyle=$ $\displaystyle
2\alpha_{1}+\gamma_{1}$ (2.21rsarbgbhbkbmbnbta) $\displaystyle\gamma_{1,\tau}$
$\displaystyle=$ $\displaystyle\gamma_{1}$ (2.21rsarbgbhbkbmbnbtb)
$\displaystyle\delta_{1,\tau}$ $\displaystyle=$ $\displaystyle\delta_{1}.$
(2.21rsarbgbhbkbmbnbtc)
All eigenvalues are positive, so one cannot expect these equilibrium points to
be stable.
One may also consider the blow-up of vortex solutions to both critical and
supercritical solutions to nonlinear Schrödinger equation in 2D. These are a
subset of the general solutions to NLSE that present a phase singularity at a
given point. The singularities appear in the form of collapse of rings at that
point. Both the existence of such solutions and their stability have been
considered recently in [132, 133].
#### 3.3.1 Other nonlinear dispersive equations
The nonlinear Schrödinger equation belongs to the broader class of nonlinear
dispersive equations, for which many questions concerning existence and
qualitative properties of singular solutions are still open. Nevertheless,
there have been recent developments that we describe next.
The Korteweg-de Vries (KdV) equation
$u_{t}+(u_{xx}+u^{2})_{x}=0$ (2.21rsarbgbhbkbmbnbtbu)
describes the propagation of waves with large wave-length in a dispersive
medium. For example, this is the case of water waves in the shallow water
approximation, where $u$ represents the height of the wave. In the case of an
arbitrary exponent of the nonlinearity, (2.21rsarbgbhbkbmbnbtbu) becomes the
generalised Korteweg de Vries equation:
$u_{t}+(u_{xx}+u^{p})_{x}=0\ ,\ p>1.$ (2.21rsarbgbhbkbmbnbtbv)
Based on numerical simulations, [134] conjectured the existence of singular
solutions of (2.21rsarbgbhbkbmbnbtbv) with type-I self-similarity if $p\geq
5$. In [135], [136] it was shown that in the critical case $p=5$ solutions may
blow-up both in finite and in infinite time. Lower bounds on the blow-up rate
were obtained, but they exclude blow-up in the self-similar manner proposed by
[134].
The Camassa-Holm equation
$u_{t}-u_{xxt}+3u_{x}u=2u_{x}u_{xx}+u_{xxx}u$ (2.21rsarbgbhbkbmbnbtbw)
also represents unidirectional propagation of surface waves on a shallow layer
of water. It’s main advantage with respect to KdV is the existence of
singularities representing breaking waves [137]. The structure of these
singularities in terms of similarity variables has not been addressed to our
knowledge.
## 4 Travelling wave
The pinching of a liquid thread in the presence of an external fluid is
described by the Stokes equation [138]. For simplicity, we consider the case
that the viscosity $\eta$ of the fluid in the drop and that of the external
fluid are the same. An experimental photograph of this situation is shown in
Fig. 1. To further simplify the problem, we make the assumption (the full
problem is completely analogous) that the fluid thread is slender. Then the
equations given in [5] simplify to
$h_{t}=-v_{x}h/2-vh_{x},$ (2.21rsarbgbhbkbmbnbta)
where
$v=\frac{1}{4}\int_{x_{-}}^{x_{+}}\left(\frac{h^{2}(y)}{\sqrt{h^{2}(y)+(x-y)^{2}}}\right)_{y}\kappa\;dy,$
(2.21rsarbgbhbkbmbnbtb)
and the mean curvature is given by (2.2). Here we have written the velocity in
units of the capillary speed $v_{\eta}=\gamma/\eta$. The limits of integration
$x_{-}$ and $x_{+}$ are for example the positions of the plates which hold a
liquid bridge [139].
Dimensionally, one would once more expect a local solution of the form
$h(x,t)=t^{\prime}H\left(\frac{x^{\prime}}{t^{\prime}}\right),$
and $H(\xi)$ has to be a linear function at infinity to match to a time-
independent outer solution. In similarity variables, (2.21rsarbgbhbkbmbnbtb)
has the form
$V(\xi)=\frac{1}{4}\int^{x_{b/t^{\prime}}}_{-x_{b}/t^{\prime}}\left(\frac{H^{2}(\eta)}{\sqrt{H^{2}(\eta)+(\xi-\eta)^{2}}})\right)_{\eta}\kappa\;d\eta.$
(2.21rsarbgbhbkbmbnbtc)
We have chosen $x_{b}$ as a real-space variable close to the pinch-point, such
that the similarity description is valid in $[-x_{b},x_{b}]$. But if $H$ is
linear, the integral in (2.21rsarbgbhbkbmbnbtc) diverges like $b\ln
t^{\prime}$, where
$b=-\frac{1}{4}\left[\frac{H_{+}}{1+H_{+}^{2}}+\frac{H_{-}}{1+H_{-}^{2}}\right].$
(2.21rsarbgbhbkbmbnbtd)
Here $H_{+}$ and $H_{-}$ are the slopes of the similarity profile at
$\pm\infty$. But this means that a simple “fixed point” solution (4) is
impossible.
However by subtracting the singularity as $t^{\prime}\rightarrow 0$, one can
define a self-similar velocity profile according to
$V^{\rm{(fin)}}(\xi)=\lim_{\Lambda\to\infty}\frac{1}{4}\int^{\Lambda}_{-\Lambda}\left(\frac{H^{2}(\eta)}{\sqrt{H^{2}(\eta)+(\xi-\eta)^{2}}}\right)_{\eta}\kappa\;d\eta+b\ln\Lambda,$
(2.21rsarbgbhbkbmbnbte)
where now
$V(\xi)=V^{\rm{(fin)}}(\xi)-b\tau,$ (2.21rsarbgbhbkbmbnbtf)
and an arbitrary constant has been absorbed into $V^{\rm{(fin)}}$. In terms of
$V^{\rm{(fin)}}$, and putting
$h(x,t)=t^{\prime}H\left(\xi,\tau\right),$
the dynamical system for $H$ becomes
$H_{\tau}=H-\left(\xi+V^{\rm(fin)}\right)H_{\xi}-HV^{\rm(fin)}_{\xi}/2+b\tau
H_{\xi}.$
This equation has a solution in the form of a travelling wave:
$H(\xi,\tau)=\overline{H}(\zeta),\quad
V^{\rm(fin)}(\xi,\tau)=\overline{V}(\zeta),\quad\mbox{where}\quad\zeta=\xi-b\tau.$
(2.21rsarbgbhbkbmbnbtg)
The profiles $\overline{H},\overline{V}$ of the travelling wave obey the
equation
$\overline{H}-(\zeta+\overline{V})\overline{H}_{\zeta}=\overline{H}\;\overline{V}_{\zeta}/2.$
The numerical solution of the integro-differential equation (4) gives
$h_{\min}=a_{\rm{out}}v_{\eta}t^{\prime},\quad\mbox{where}\quad
a_{\rm{out}}=0.033.$ (2.21rsarbgbhbkbmbnbth)
The slope of the solution away from the pinch-point are given by
$H_{+}=6.6\quad\mbox{and}\quad H_{-}=-0.074,$ (2.21rsarbgbhbkbmbnbti)
which means the solution is very asymmetric, as confirmed directly from Fig.
1. These results are reasonably close to the exact result, based on a full
solution of the Stokes equation [5]; in particular, the normalised minimum
radius is $a_{\rm{out}}=0.0335$ for the full problem.
## 5 Limit cycles
An example for this kind of blow-up was introduced into the literature in [15]
in the context of cosmology. There is considerable numerical evidence [140]
that discrete self-similarity occurs at the mass threshold for the formation
of a black hole. The same type of self-similarity has also been proposed for
singularities of the Euler equation [141, 67], the porous medium equation
driven by buoyancy [141], and for a variety of other phenomena [142]. A
reformulation of the original cosmological problem leads to the following
system:
$\displaystyle f_{x}=\frac{(a^{2}-1)f}{x},$ (2.21rsarbgbhbkbmbnbtaa)
$\displaystyle(a^{-2})_{x}=\frac{1-(1+U^{2}+V^{2})/a^{2}}{x},$
(2.21rsarbgbhbkbmbnbtab)
$\displaystyle(a^{-2})_{t}=\left[\frac{(f+x)U^{2}-(f-x)V^{2}}{x}+1\right]/a^{2}-1,$
(2.21rsarbgbhbkbmbnbtac) $\displaystyle
U_{x}=\frac{f[(1-a^{2})U+V]-xU_{t}}{x(f+x)},$ (2.21rsarbgbhbkbmbnbtad)
$\displaystyle V_{x}=\frac{f[(1-a^{2})U+V]+xV_{t}}{x(f-x)}.$
(2.21rsarbgbhbkbmbnbtae)
In [16], the self-similar description corresponding to the system
(2.21rsarbgbhbkbmbnbtaa)-(2.21rsarbgbhbkbmbnbtae) was solved using formal
asymptotics and numerical shooting procedures. This leads to the solutions
observed in [15]. We now propose another system, which shares some of the
structure of (2.21rsarbgbhbkbmbnbtaa)-(2.21rsarbgbhbkbmbnbtae), but which we
are able to solve analytically:
$\displaystyle u_{t}(x,t)=2f(x,t)v(x,t),$ (2.21rsarbgbhbkbmbnbtaba)
$\displaystyle v_{t}(x,t)=-2f(x,t)u(x,t),$ (2.21rsarbgbhbkbmbnbtabb)
$\displaystyle f_{t}(x,t)=f^{2}(x,t).$ (2.21rsarbgbhbkbmbnbtabc)
The system (2.21rsarbgbhbkbmbnbtaba)-(2.21rsarbgbhbkbmbnbtabc) is driven by
the simplest type of blow-up equation (2.21rsarbgbhbkbmbnbtabc), and can be
solved using characteristics. However, in the spirit of this review, we
transform to similarity variables according to:
$\displaystyle u=U(\xi,\tau)$ (2.21rsarbgbhbkbmbnbtabca) $\displaystyle
v=V(\xi,\tau)$ (2.21rsarbgbhbkbmbnbtabcb) $\displaystyle
f=t^{\prime-1}F(\xi,\tau)$ (2.21rsarbgbhbkbmbnbtabcc)
It is seen directly from (2.21rsarbgbhbkbmbnbtabc) that $f$ first blows up at
a local maximum $f_{max}>0$. Near a maximum, the horizontal scale is the
square root of the vertical scale $t^{\prime}$, and thus we must have
$\xi=x^{\prime}/t^{\prime 1/2}$. With that, the similarity equations become
$\displaystyle U_{\tau}=-\xi U_{\xi}/2+FV$ (2.21rsarbgbhbkbmbnbtabcda)
$\displaystyle V_{\tau}=-\xi V_{\xi}/2-FU$ (2.21rsarbgbhbkbmbnbtabcdb)
$\displaystyle F_{\tau}=-F-\xi F_{\xi}/2+F^{2}.$ (2.21rsarbgbhbkbmbnbtabcdc)
The fixed point solution of the last equation is
$F=\frac{1}{1+c\xi^{2}},$ (2.21rsarbgbhbkbmbnbtabcde)
where $c>0$ is a constant. The equations for $U,V$ are solved by the ansatz
$U=U_{0}\sin\left(C(\xi)+\tau\right),\quad
V=U_{0}\cos\left(C(\xi)+\tau\right),\quad$ (2.21rsarbgbhbkbmbnbtabcdf)
and for the function $C(\xi)$ one finds
$\xi C^{\prime}(\xi)/2=F-1,$ (2.21rsarbgbhbkbmbnbtabcdg)
with solution $C(\xi)=-\ln(1+c\xi^{2})$. Thus (a single component of) the
singular solution is indeed of the general form
$U=\psi(\phi(\xi)+\tau),$ (2.21rsarbgbhbkbmbnbtabcdh)
where $\psi$ is periodic in $\tau$. This is a particularly simple version of
discretely self-similar behaviour, i.e. when $T$ is the period of $\psi$, the
same self-similar picture is obtained for $\tau=\tau_{0}+nT$.
## 6 Strange attractors and exotic behaviour
In connection to limit cycles and in the context of singularities in
relativity, a few interesting situations have been found numerically quite
recently. One of them is the existence of Hopf bifurcations where a self-
similar solution (a stable fixed point) is transformed into a discrete self-
similar solution (limit cycle) as a certain parameter varies (see [143]).
Other kinds of bifurcations, for example of the Shilnikov type, are found as
well [144]. Before coming to simple explicit examples, we mention that
possible complex dynamics in $\tau$ has long been suggested for simplified
versions of the inviscid Euler equations [145, 146, 141]. For a critical
discussion of this work, see [147, 81].
The problems considered in these papers were the 2D axisymmetric Euler
equations with swirl, which produces a centripetal force. In the limit that
the rotation is confined to a small annulus, the direction of acceleration is
locally uniform, and the equation reduces to that of 2D Boussinesq convection,
where the centripetal force is replaced by a “gravity” force. Another related
model is 2D porous medium convection, for which the equation reads
$\frac{\partial T}{\partial t}+\left(T{\bf e}_{y}-\nabla\phi\right)\cdot\nabla
T=0,$ (2.21rsarbgbhbkbmbnbtabcda)
where ${\bf v}=T{\bf e}_{y}-\nabla\phi$ plays the role of the velocity field
and $T$ is the temperature. The potential $\phi$ follows from the constraint
of incompressibility, which gives $\triangle\phi=T_{y}$. Simulations provide
evidence of a self-similar dynamics of the form [141]
$T=t^{\prime\eta}M({\bf x}^{\prime}/t^{\prime 1+\eta},\tau),$
(2.21rsarbgbhbkbmbnbtabcdb)
where $\eta$ is approximately 0.1 and $M$ is a function that is slowly varying
with $\tau$.
Depending on the model, both periodic behaviour as well as more complicated,
chaotic motion has been observed in numerical simulations. Oscillations of
temperature in $\tau$ are motivated by the observation that a sharp, curved
interface (i.e. the transition region between a rising “bubble” of hot fluid
and its surroundings) becomes unstable and rolls up. However, owing to
incompressibility, the sheet is also stretched, which stabilises the
interface, leading to an eventual decrease in gradients. Locality suggests
that this process could repeat itself periodically on smaller and smaller
scales [141]. However, simulations of the Euler equation have also shown
examples of a more complicated dependence on $\tau$, which might be chaotic
behaviour [145]. We also mention that corresponding chaotic behaviour has been
proposed for the description of spin glasses in the theory of critical
phenomena [148]. We now give some explicit examples of chaos in the
description of a singularity.
In section 3.1.1 we treated a system of an infinite number of ordinary
differential equations for the coefficients of the expansion of an arbitrary
perturbation to an explicit solution. Such high-dimensional systems in
principle allow for a rich variety of dynamical behaviours, including those
found in classical finite dimensional dynamical systems, such as chaos.
Consider for instance an equation for the perturbation $g$ (the analogue of
(2.21g)) of the form
$g_{\tau}=\mathit{L}g+F(g,g),$ (2.21rsarbgbhbkbmbnbtabcdc)
where $\mathit{L}g$ is a linear operator. Assuming an appropriate non-linear
structure for the function $F$, an arbitrary nonlinear (chaotic) dynamics can
be added.
To give an explicit example of a system of PDE’s exhibiting chaotic dynamics,
consider the structure of the example given in section 5. It can be
generalised to produce any low-dimensional dynamics near the singularity, as
follows by considering the system
(2.21rsarbgbhbkbmbnbtaba)-(2.21rsarbgbhbkbmbnbtabc)
$\displaystyle u^{(i)}_{t}(x,t)=2fF_{i}(\\{u^{(i)}\\}),\quad i=1,\dots,n,$
(2.21rsarbgbhbkbmbnbtabcdda) $\displaystyle f_{t}(x,t)=f^{2}(x,t).$
(2.21rsarbgbhbkbmbnbtabcddb)
Using the ansatz analogous to (2.21rsarbgbhbkbmbnbtabcdf):
$u^{(i)}=U^{(i)}\left(C(\xi)+\tau,\xi\right),$ (2.21rsarbgbhbkbmbnbtabcdde)
and choosing $C(\xi)=-\ln(1+c\xi^{2})$, one obtains the system
$U^{(i)}_{\tau}=F_{i}\left\\{U^{(i)}\right\\}.$ (2.21rsarbgbhbkbmbnbtabcddf)
To be specific, we consider $n=3$ and
$F_{1}=\sigma(u^{(2)}-u^{(1)}),\quad F_{2}=\rho
u^{(1)}-u^{(2)}-u^{(1)}u^{(3)},\quad F_{3}=u^{(1)}u^{(2)}-\beta u^{(3)},$
(2.21rsarbgbhbkbmbnbtabcddg)
so that (2.21rsarbgbhbkbmbnbtabcddf) becomes the Lorenz system [149]. As
before, for $t^{\prime}\rightarrow 0$, the variable $\tau$ goes to infinity,
and near the singularity one is exploring the long-time behaviour of the
dynamical system (2.21rsarbgbhbkbmbnbtabcdde). In the case of
(2.21rsarbgbhbkbmbnbtabcddg), and for sufficiently large $\rho$, the resulting
dynamics will be chaotic. Specifically, taking $\sigma=10$, $\rho=28$, and
$\beta=8/3$, as done by Lorenz [150], the maximal Lyapunov exponent is
$0.906$. The initial conditions with which (2.21rsarbgbhbkbmbnbtabcdde) is to
be solved depend on $\xi$. Thus the chaotic dynamics will follow a completely
different trajectory for each space point. As a result, it will be very
difficult to detect self-similar behaviour of this type as such, even if data
arbitrarily close to the singularity time is taken. If for example a rescaled
spatial picture is observed at constant intervals of logarithmic time $\tau$,
the spatial structure of the singularity will appear to be very different.
However, as pointed out in [145], chaotic motion is characterised by unstable
periodic orbits, for which one could search numerically.
## 7 Multiple singularities
The singularities described so far occur at a single point $x_{0}$ at a given
time $t_{0}$. This need not be the case, but blow-up may instead occur on sets
of varying complexity, including sets of finite measure. We begin with a case
where singularity formation involves two different points in space.
### 7.1 Hele-Shaw equation
A particularly rich singularity structure is found for a special case of (2.7)
in one space dimension with $n=1$. Dropping the second term on the right,
which will typically be small, one arrives at
$h_{t}+(hh_{xxx})_{x}=0.$ (2.21rsarbgbhbkbmbnbtabcdda)
This is a simplified model for a neck of liquid of width $h$ confined between
two parallel plates, a so-called Hele-Shaw cell. which is a simplified model
for the free surface in a so-called Hele-Shaw cell [151]. Breakup of a fluid
neck inside the cell corresponds to $h$ going to zero in finite time.
Singular solutions displaying type-I self-similarity would be of the form
$h(x,t)=t^{\prime\alpha}H(x^{\prime}/t^{\prime(\alpha+1)/4}),$
(2.21rsarbgbhbkbmbnbtabcddb)
but are never observed. Instead, several types of pinch solutions different
from (2.21rsarbgbhbkbmbnbtabcddb) have been found for
(2.21rsarbgbhbkbmbnbtabcdda) using a combination of numerics and asymptotic
arguments [152, 102, 153]. On one hand, singularities exhibit type-II self-
similarity. On the other hand, the simple structure
(2.21rsarbgbhbkbmbnbtabcddb) is broken by the fact that the location of the
pinch point is moving in space. The root for this behaviour lies in the fact
that two singularities are interacting over a distance much larger than their
own spatial extend. Below we report on three different kinds of singularities
whose existence has been confirmed by numerical simulation of
(2.21rsarbgbhbkbmbnbtabcdda).
The first kind of singularity was called the imploding singularity in [153],
since it consists of two self-similar solutions which form mirror images, and
which collide at the singular time. Locally, the solution can be written
$h(x,t)=t^{\prime 6}H((x^{\prime}+at^{\prime})/t^{\prime 3}),$
(2.21rsarbgbhbkbmbnbtabcddc)
where $-a$ is the constant speed of the singular point. Note that the scaling
exponents do not agree with (2.21rsarbgbhbkbmbnbtabcddb). The reason is that
the singularity is moving, so $h$ is the solution of
$hh_{xxx}=J(t^{\prime})\equiv t^{\prime 3},$ (2.21rsarbgbhbkbmbnbtabcddd)
where $J$ is determined by matching to an outer region. The similarity profile
$H$ is a solution of the equation $HH^{\prime\prime\prime}=1$, with boundary
conditions
$H(\eta)\propto\eta^{2}/2,\;\eta\rightarrow-\infty;\quad
H(\eta)\propto\sqrt{8/3}(A-\eta)^{3/2},\;\eta\rightarrow\infty.$
(2.21rsarbgbhbkbmbnbtabcdde)
Figure 11: A simulation of (2.21rsarbgbhbkbmbnbtabcdda) with spatially
periodic boundary conditions and initial condition
(2.21rsarbgbhbkbmbnbtabcddf), with $w=0.02$ and $\delta=0.1$.
One might wonder whether this behaviour is generic, in the sense that it might
depend on the initial conditions being exactly symmetric around the eventual
point of blow up. The simulation of (2.21rsarbgbhbkbmbnbtabcdda) shown in Fig.
11 shows that this is not the case. The initial condition is
$h(x,0)=1-(1-w)\left[\frac{3}{2}\cos\pi x-\frac{6}{10}\cos 2\pi
x+\frac{1}{10}\cos 3\pi x(1+\delta\sin 2\pi x)\right],$
(2.21rsarbgbhbkbmbnbtabcddf)
which for $\delta=0$ reduces to the symmetric initial condition considered by
[153]. The type of singularity that is observed (or no singularity at all)
depends on the parameter $w$. The simulation shown in Fig. 11 shows that even
at finite $\delta$ (non-symmetric initial conditions) the final collapse is
described by a symmetric solution.
Figure 12: Same as Fig.11, but both parameters $w=0.07$ and $\delta=0.01$.
The second kind is the exploding singularity [153], since now the two self-
similar solutions are moving apart, cf. Fig.12. This time even a very small
asymmetry ($\delta=1/100$) makes one pinching event “win” over the other.
However, this does not affect the asymptotics described briefly below.
Locally, the solution can be written
$h(x,t)=\delta^{2}(t^{\prime})H((x^{\prime}-at^{\prime})/\delta(t^{\prime})),$
(2.21rsarbgbhbkbmbnbtabcddg)
with $\delta=t^{\prime}/ln(t^{\prime})$, which is similar to examples
considered in section 3. However, an additional complication consists in the
fact that the singularity is moving, so there is a coupling to the parabolic
region between the two pinch-points. This matching is unaffected by the fact
that in the simulation shown in Fig. 12 one side of the solution touches down
first. In [153], a possible generalisation is also conjectured, which has the
form
$h(x,t)=\delta^{2}(t^{\prime})H((x^{\prime}-at^{\prime\frac{r-1}{2}})/\delta(t^{\prime})),$
(2.21rsarbgbhbkbmbnbtabcddh)
and $\delta=t^{\prime\frac{r-1}{2}}/\ln t^{\prime}$. In principle, any value
of $r$ is possible, but numerical evidence has been found for $r\approx 3$
(above) and $r\approx 5/2$ only.
Finally, a third type is the symmetric singularity of [153], which does not
move. In that case, the structure of the solution is
$h(x,t)=h_{0}(t^{\prime})H((x^{\prime}/\delta(t^{\prime})),$
(2.21rsarbgbhbkbmbnbtabcddi)
with $h_{0}=\delta^{2}P(\ln\delta)$, where $P$ is a polynomial. The time
dependence of $\delta$ is not reported. Evidently, many aspects of the
exploding and of the symmetric singularity remain to be confirmed and/or to be
worked out in more detail.
The most intriguing feature of the Hele-Shaw equation
(2.21rsarbgbhbkbmbnbtabcdda) is that several types of stable singularities
have been observed for the same equation. Within a one-parameter family of
smooth initial conditions, all three types of singularities can be realized as
$h\rightarrow 0$. Each type is observed over an interval of the parameter $w$.
Near the boundary of the intervals, a very interesting crossover phenomenon
occurs: the solution is seen to follow one type of singularity at first (the
exploding singularity, say), and then crosses over to a solution of another
singularity (the imploding singularity). The dynamics of each singularity can
be followed numerically over many decades in $t^{\prime}$. By tuning $w$, the
crossover can be made to occur at arbitrarily small values of $h$.
The switch in behaviour is driven by the slow dynamics of scaling regions
exterior to (2.21rsarbgbhbkbmbnbtabcddc) or (2.21rsarbgbhbkbmbnbtabcddg). It
is a signature of the very long-ranged interactions (both in real space as
well as in scale), that exist in (2.21rsarbgbhbkbmbnbtabcdda). Thus an outside
development can trigger a change of behaviour that is taking place on the
local scale of the singularity. To mention another example, applying different
boundary conditions for the pressure at the outside of the cell can change the
singular behaviour completely [154]. This makes the crossover behaviour of
(2.21rsarbgbhbkbmbnbtabcdda) very different from that observed for drop pinch-
off (cf. (2.10),(2.11)), which is driven by a change in the dominant balance
between different terms in (2.11).
### 7.2 Semilinear wave equation
It appears that the Hele-Shaw equation is not an isolated example, but rather
is representative of a more general phenomenon. Namely, another example of a
potentially complex singularity structure is the semilinear wave equation
$u_{tt}-\Delta u=|u|^{p-1}u,\ p>1.$ (2.21rsarbgbhbkbmbnbtabcddj)
It has trivial singular solutions of the form
$u(x,t)=b_{0}(T-t)^{-\frac{2}{p-1}},$ (2.21rsarbgbhbkbmbnbtabcddk)
with $b_{0}=\left[\frac{2(p+1)}{(p-1)^{2}}\right]^{\frac{1}{p-1}}$.
Nevertheless, the existence of different self-similar solutions is known in a
few particular cases, like the case $p\geq 7$, where $p$ is an odd integer
(see [155]) or in space dimension $d=1$ (see [156]).
The character of the blow-up is controlled by the blow-up curve $T(x)$, which
is the locus where the equation first blows up at a given point in space. It
has been shown for $d=1$ [157] that there exists a set of characteristic
points, where the blow-up curve locally coincides with the characteristics of
(2.21rsarbgbhbkbmbnbtabcddj). The set of non-characteristic points $I_{0}$ is
open, and $T$ is $C^{1}$ on $I_{0}$. Recently, it has been shown [158] that
the blow-up at characteristic points is of type II. Even more intriguingly, it
appears [158] that the structure of blow-up at these points is such that the
singularity results from the collision of two peaks at the blow-up point, very
similar to the observation shown in Fig. 11.
### 7.3 More complicated sets
In the Hele-Shaw equation of the previous subsection, different parts of the
solution, characterised by different scaling laws, interacted with each other.
In the generic case, however, finally blow-up only occurred at a single point
in space. An example where singularities may even occur on sets of finite
measure is given by reaction-diffusion equations of the family
$u_{t}-\Delta u=u^{p}-b\left|\nabla u\right|^{q}\quad\mbox{for}\quad
x\in\Omega.$ (2.21rsarbgbhbkbmbnbtabcddl)
where $\Omega$ is any bounded, open set in dimension $d$. Depending on the
values of $p>1$ and $q>1$ singularities of (2.21rsarbgbhbkbmbnbtabcddl) may be
regional ($u$ blows up in subsets of $\Omega$ of finite measure), or even
global (the solution blows-up in the whole domain); see for instance [159] and
references therein.
Singularities may even happen in sets of fractional Hausdorff dimension, i.e.,
fractals. This is the case of the inviscid one-dimensional system for jet
breakup (cf. [160]) and might be case of the Navier-Stokes system in three
dimensions, where the dimension of the singular set at the time of first blow-
up is at most $1$ (cf. [161]). This connects to the second issue we did not
address here. It is the nature of the singular sets both in space and time,
i.e. including possible continuation of solutions after the singularity. In
some instances, existence of global in time (for all $0\leq t<\infty$)
solutions to nonlinear problems can be established in a weak sense. For
example, this has been achieved for systems like the Navier Stokes equations
[162], reaction-diffusion equations [163], and hyperbolic systems of
conservation laws [96]. Weak solutions allow for singularities to develop both
in space and time. In the case of the three-dimensional Navier-Stokes system,
the impossibility of singularities ”moving” in time, that is of curves
$\mathbf{x}=\mathbf{\varphi}(t)$ within the singular set is well-known [161].
Hence, provided certain kinds of singularities do not persist in time, the
question is how to continue the solutions after a singularity has developed.
A first version of this paper was an outgrowth of discussions between the
authors and R. Deegan, preparing a workshop on singularities at the Isaac
Newton Institute, Cambridge. The present version was written during the
programme: “Singularities in mechanics: formation, propagation and microscopic
description”, organised with C. Josserand and L. Saint-Raymond, which took
place between January and April 2008 at the Institut Henri Poincaré in Paris.
We are grateful to all participants for their input, in particular C. Bardos,
M. Brenner, M. Escobedo, F. Merle, H. K. Moffatt, Y. Pomeau, A. Pumir, J.
Rauch, S. Rica, L. Vega, T. Witten, and S. Wu. We also thank J. M. Martin-
Garcia and J. J. L. Velazquez for fruitful discussions and for providing us
with valuable references.
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|
arxiv-papers
| 2008-12-07T11:39:58 |
2024-09-04T02:48:59.272938
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jens Eggers and Marco A. Fontelos",
"submitter": "Jens Eggers",
"url": "https://arxiv.org/abs/0812.1339"
}
|
0812.1415
|
arxiv-papers
| 2008-12-08T03:36:11 |
2024-09-04T02:48:59.291700
|
{
"license": "Public Domain",
"authors": "Ester Aliu (for the VERITAS Collaboration)",
"submitter": "Ester Aliu",
"url": "https://arxiv.org/abs/0812.1415"
}
|
|
0812.1503
|
# Trans-Coordinate Physics
Richard Mould111Department of Physics and Astronomy, State University of New
York, Stony Brook, New York 11794-3800; richard.mould@stonybrook.edu;
http://ms.cc.sunysb.edu/~rmould
###### Abstract
Standard practice attempts to remove coordinate influence in physics through
the use of invariant equations. Trans-coordinate physics proceeds differently
by not introducing space-time coordinates in the first place. Differentials
taken from a novel limiting process are defined for a particle’s wave
function, allowing the particle’s dynamic principle to operate ‘locally’
without the use of coordinates. These differentials replace the covariant
differentials of Riemannian geometry. With coordinates out of the way
‘regional conservation principles’ and the ‘Einstein field equation’ are no
longer fundamentally defined; although they are constructible along with
coordinate systems so they continue to be analytically useful. Gravity is
solely described in terms of gravitons and quantized geodesics and curvatures.
Keywords: covariance, invariance, geometry, metric spaces, state reduction;
03.65.a, 03.65.Ta, 04.20.Cv
## Introduction
James Clerk Maxwell was the first to use space-time coordinate systems in the
way they are used in contemporary physics. They play a role in his formulation
of electromagnetic field theory that makes them virtually indispensable.
Einstein embraced Maxwell’s methodology but devoted himself to eliminating the
influence of coordinates because they have nothing to do with physics.
However, the influence of coordinates is not eliminated by relativistic
invariance as will be evident below where these space-time representations are
removed _entirely_ from physics.
Trans-coordinate physics proceeds on the assumption that space-time
coordinates should not be introduced at any level. As a practical matter, and
for many analytic reasons, coordinates are very useful and probably always
will be. But if nature does not use numerical labeling for event
identification and/or analytic convenience, and if we are interested in the
most fundamental way of thinking about nature, then we should avoid space-time
coordinates from the beginning.
Without coordinates the domain of relativity lies solely in the properties of
the embedding metric space, and the domain of quantum mechanics resides in
properties of local wave functions that are assigned to particles. These two
domains overlap ‘locally’ where Lorentz invariant quantum mechanics is
assumed. Photons in the space ‘between’ massive particles have a reduced
function and definition.
As a result, the variables of a particle’s wave packet are wholly contained
inside the packet and are coordinate independent. They move with a particle’s
wave function in the embedding metric space, but they do not locate it in that
space. No particle has a _net velocity_ or _kinetic energy_ when considered in
isolation, for these quantities require a coordinate framework for their
definition. This alone reveals the radical nature of removing coordinates
_entirely_ from physics, and the inadequacy of general relativistic invariance
for that purpose.
Another consequence of this program is that energy and momentum are not
propagated through the empty space between particles. Although particle
energy, momentum, and angular momentum are conserved in local interactions, we
say that nature does not provide for the exchange of energy and momentum
between separated particles. We are the ones who arrange these transfers
through our introduction of regional coordinates that we use to give ourselves
the big picture. It facilitates analysis. The organizing power of coordinates
and an opportune distribution of matter in space and time often allows us to
find a system of coordinates that supports regional conservation; however, we
can also find coordinates that do not support conservation. Therefore,
regional conservation is coordinate dependent. It is not an invariant idea. It
follows from a favorable construction on our part rather than from something
intrinsic to the system222A region surrounded by flat space will not conserve
energy and momentum if no coordinates are chosen in the region, or if certain
discontinuous coordinates are chosen in the region. Here again conservation
depends on a coordinate choice or on the choice of a transformation group..
General relativity is a product of energy-momentum conservation that relies on
regional coordinates for its meaning. It therefore joins regional conservation
principles as something coming from coordinate construction rather than
something fundamental. It is found for instance that while the metric tensor
$g_{\mu\nu}$ can be defined at any event inside the wave packet of a massive
particle, there is no trans-coordinate continuous function $g_{\mu\nu}$
associated with it. That is, a continuous metric tensor is not _physically_
defined. Therefore derivatives of $g_{\mu\nu}$ at an event are not physically
defined. General relativity suffers accordingly. The separation we establish
between quantum mechanics and general relativity avoids a clash of these
mismatched disciplines [G. ’t Hooft, (2008); J. Maldacena, (2005)], and weighs
in favor of quantum mechanics.
And finally, a new definition of state is proposed in this paper. In the
absence of regional coordinates there is no common time for two or more
particles, so a state definition is proposed that spans the no-mans-land
between particles. It is shown in another paper how to write the Hamiltonian
for a system of separated particles of this kind [R. A. Mould, (2008)]. The
new definition of state and the Hamiltonian that applies to it imposes a
consistent framework on a system of trans-coordinate particles.
If an atom emits a photon, then the system’s energy and momentum will be
locally conserved. If that photon is not subsequently detected in another part
of the universe it will essentially disappear from the system because a photon
in isolated flight is energetically invisible. This does not violate
conservation principles because those principles are satisfied at the emission
site.
If the photon is detected somewhere else, then the energy and momentum at the
detector site will also be conserved. The difficulty is that the energy
emitted by the atom and the energy received by the detector might not be the
same, so there is no general basis for claiming that energy conservation holds
for the entire two-site system. That’s because nature, we say, does not care
about conservation over more than one interaction. It cares only about
_conservation at individual interactions_. However, regional coordinates often
make it possible to compose energy differences of this kind in such a way as
to validate regional conservation; and hence the great advantage of regional
coordinates. They give us a useful analytic tool and a satisfying big picture
as well as (sometimes) regional conservation laws. But these laws are not
fundamental. They are only products of a fortunate coordinate construction.
This treatment is primarily concerned with electromagnetic interactions.
## Partition Lines
In Minkowski space one must choose a single world line to define the future
time cone of an event a. If there is a non-zero mass particle present in the
space it should be possible to choose a unique world line at each location
inside the particle’s wave packet that is specific to the particle at that
location. That world line corresponds to the direction of _square modular
flow_ at that event. The collection of these world lines over the particle’s
wave packet can be thought of as the _streamlines_ of its square modular flow
in space and time. They will be called _partition lines_. We also define
_perpendiculars_ that are space-like lines drawn through each event
perpendicular to the local partition line. We will first develop the
properties of partition lines in a 1 + 1 space, and then in 2 + 1 and 3 + 1
spaces.
Figure 1 is a 1 + 1 Mnkowski surface with light paths given by $45^{\circ}$
dashed lines. Partition lines of an imagined particle wave packet are
represented in the figure by the five slightly curved and more-or-less
vertical lines. They tell us that the wave packet moves to the left with ever
decreasing velocity and that it spreads out as it goes. This description is
not trans-coordinate because it is specific to the Lorentz frame in the
diagram; but these lines provide a scaffold on which it is possible to hang a
trans-coordinate wave function.
Figure 1: Partition Lines in a Minkowski Space
Partition lines pass through every part of the particle’s wave packet and do
not cross one another. They are not defined outside of a wave packet. Just as
the space is initially given to us in the form of a metric background, any
particle is initially given in the form of partition lines with the above
characteristics. The interpretation of these lines is given in the next
paragraph where values are assigned to them in a way that reflects the
intended _given conditions_. These conditions are not ‘initial’ in the usual
temporal sense, but are rather ‘given’ over the space-time region of interest.
Let the third partition line from the left (i.e., the middle line in Fig. 1)
portion off 1/2 of the packet, so half of the particle lies to the left of an
event such as a in the figure. That is, there is a 0.5 probability that the
particle will be found on the perpendicular extending to the left of a. This
statement is assumed to have objective invariant meaning. Of course, the other
half of the particle lies to the right of event a on the perpendicular through
a. The middle partition line is made up of all the events in the wave packet
that satisfy this condition, so they together constitute a continuous line to
which we assign the value of 1/2. There is a 0.5 probability that the particle
will be found _somewhere_ on the left side of this line when the included
events are all those on both sides of the line.
In a similar way we suppose that the second partition line in Fig. 1 portions
off, say, 1/4 of the packet on the perpendicular to the left of an event b,
and that the first line portions off 1/100 of the particle or some other
diminished amount. We further assume that the fifth line goes out to 99/100 of
the particle packet, so the entire particle is represented by streamlines that
split the particle into objectively defined fractional parts.
When a wave function is finally assigned we will show that its total square
modulus remains ‘constant in time’ between any two partition lines in 1 + 1
space, and is similarly confined in higher dimensions.
## Neighborhoods
Every event inside the wave packet has a unique time direction defined for it
by the partition line passing through the event. This allows us to define
unique _inertial_ neighborhoods associated with each event.
Figure 2: Establishing neighborhoods
Consider a flat space inside the wave packet of a massive particle, and assign
a Minkowski metric that is intrinsic to that space. Beginning with an event a
in Fig. 2a, proceed up the particle’s partition line through a by an amount
$-\Delta$ which is the magnitude of the invariant interval from event a to an
event b. This interval ab is negative and identifies the chosen time axis
inside the particle packet at event a. Then find event $\textbf{b}^{\prime}$
by proceeding down the partition line the same invariant interval $-\Delta$ .
Construct a backward time cone with b at its vertex and a forward time cone
with $\textbf{b}^{\prime}$ at its vertex and identify the intersection events
c and $\textbf{c}^{\prime}$. Since these events are embedded in a flat space,
the positive space-like interval $\textbf{cc}^{\prime}$ will pass through
event a and will be bisected by it with
$\textbf{ca}=\textbf{ac}^{\prime}=\textbf{cc}^{\prime}/2=\Delta>0$
For any $\Delta$, all of the events included in the intersection of the light
cones of b and b′ are defined to be a _neighborhood_ of event a. The events
along the line cc′ are defined to be a _spatial neighborhood_ of a. The limit
as $\Delta$ goes to zero is identical with the limit of small neighborhoods
around a.
## Curved Space
The above considerations for a ‘flat’ space also apply _locally_ in any curved
space, so we let the conditions in Fig. 2a be generally valid in the limit as
$\Delta\rightarrow 0$. Figure 2b shows the resulting Minkowski diagram in the
local inertial system with $\hat{x}$ and $\hat{t}$ as the space and time unit
vectors in the directions $\textbf{ac}^{\prime}$ and ab respectively.
The unit of these vector directions is given by $\sqrt{\Delta}$ in meters,
although we have not established coordinates in those units along those
directions. Specifically, we have not established a unique numerical value
attached to an event a or a distant zero-point for that value; so the
development so far is consistent with the trans-coordinate (or coordinate-
less) aims of this paper.
The unit vectors at event a will be referred to as the _local grid_ at event
a, where the time direction is always along the partition line going through
a. These definitions have nothing to do with the curvature of the space in the
wave packet at or beyond the immediate vicinity of a. Every event inside a
particle packet has a similar local grid. The local grids of other events in
the neighborhood of event a will be continuous with the local grid at a in
this 1+1 space, but not for higher dimensions as we will see.
## The Wave Function
We specify the quantum mechanical wave function at each event a in a particle
wave packet over the space-time region of interest
$\varphi(\textbf{a})$ (1)
which is identified in the manner of Euclid’s geometry since there are no
coordinate numbers involved. There are four auxilary conditions on this
function.
First: The function $\varphi(\textbf{a})$ is a complex number given at event a
that is continuous with all of its neighbors. The units of $\varphi$ are
$m^{-1/2}$ in this 1 + 1 space.
Second: Partial derivatives of $\varphi(\textbf{a})$ are defined in the limit
of small neighborhoods around a (i.e., for small values of $\Delta$).
$\displaystyle\partial\varphi(\textbf{a})/\partial x$ $\displaystyle=$
$\displaystyle\lim_{\Delta\rightarrow
0}\frac{\varphi(\textbf{c}^{\prime})-\varphi(\textbf{c})}{2\sqrt{\Delta}}$ (2)
$\displaystyle\partial\varphi(\textbf{a})/\partial t$ $\displaystyle=$
$\displaystyle\lim_{\Delta\rightarrow
0}\frac{\varphi(\textbf{b})-\varphi(\textbf{b}^{\prime})}{2\sqrt{\Delta}}$
The second spatial derivative is then
$\partial^{2}\varphi(\textbf{a})/\partial x^{2}=\lim_{\Delta\rightarrow
0}\frac{\partial\varphi(\textbf{c}^{\prime})/\partial
x-\partial\varphi(\textbf{c})/\partial x}{2\sqrt{\Delta}}$
Notice that we have defined derivatives in the directions $\hat{x}$ and
$\hat{t}$ without using coordinates to ‘locate’ or numerically ‘identify’
events along either of those directions. Only $\Delta$ _intervals_ between
events along the time line are taken from the invariant metric space.
Third: The value of $\varphi$ at event a is related to its neighbors through
the _dynamic principle_. This principle determines how $\varphi(\textbf{a})$
evolves relative to its own time against the metric background, and how it
relates spatially to its immediate neighbors.
Fourth: The objective fraction of the particle found between the partition
line through event c in Fig. 2a and a partition line through event
$\textbf{c}^{\prime}$ is equal to $f_{cc^{\prime}}$. In the limit as
$\textbf{cc}^{\prime}$ = $2\Delta$ goes to zero the fraction of the particle
between differentially close partition lines goes to $df$. Normalization of
$\varphi(\textbf{a})$ is stictly ’local’ and requires
$\varphi^{*}(\textbf{a})\varphi(\textbf{a})=\lim_{\Delta\rightarrow\
0}\frac{f_{cc^{\prime}}}{2\Delta}$ (3)
It follows that
$\varphi^{*}(\textbf{a})\varphi(\textbf{a})=\varphi^{*}(\textbf{b})\varphi(\textbf{b})=\varphi^{*}(\textbf{b}^{\prime})\varphi(\textbf{b}^{\prime})$
because the fractional difference between any two the partition lines is the
same over any perpendicular. Therefore, the square modular flow will be
_constant in time_ between any two partition lines as previously claimed.
These four auxiliary conditions must be satisfied when taken together with the
initally given partition lines, but there is no guarantee that there exists a
wave function that qualifies. _Finding a solution_ therefore consists of
varying the partition lines (i.e., the given conditions) until a wave function
exists that satisfies these conditions.
The choice of a world line based on partition lines is not a coordinate
choice, nor is the limiting procedure that follows. So these definitions are
not just coordinate invariant, they are fully _coordinate free_. They allow us
to find physically creditable derivatives of any continuous function in a way
that is independent of the curvature of the surrounding space, and to found a
physics on that basis.
## One Particle
Partition lines do not extend beyond the particle, so in the absence of
‘external’ coordinates that do extend beyond the particle (in an otherwise
empty space) there is no basis for claiming that the particle has a _net
velocity, kinetic energy, or net momentum_. This will be true of both zero and
non-zero mass particles. It is a consequence of a trans-coordinate physics
that particles take on these dynamic properties only in interaction with other
particles.
A massive particle has an ‘internal’ energy defined at each event in its wave
packet, but since that may differ from one event to another there is no single
internal energy representing the particle as a whole. Similarly, each part of
the particle’s wave packet follows its own world line, so the there is no
single world line for the particle as a whole as shown in Fig. 1. It is our
claim that nature attends to the particle as a whole by dealing separately
with each part. One exception is that the particle as a whole does produce a
gravitational disturbance in the background invariant metric that has its
origin in the regional distribution of the particle’s internal mass/energy.
## Two Particles
Figure 3 shows the partition lines of two separated massive particles where
each has its own definition of a grid that is different from the other
particle. It is a consequence of the trans-coordinate picture that these
particles in isolation will seem to have nothing to do with one another.
However, the positional relationship of one to the other is objectively
defined in the metric space in the background of both. Every event in the wave
packet of each particle has a definite location in the metric space, and that
fixes the positional relationship of each part of each particle with other
parts of itself and with other particles. In addition, each massive particle
produces a gravitational disturbance that has an invariant influence on the
other. That influence is a function of the relative velocity between the two,
even though kinetic energy is not defined for either one. Kinetic energy is a
coordinate-based idea as has been said, whereas metrical positions and
gravitational disturbances in the metric are invariant. We assume that the
latter are based solely on graviton activity.
Figure 3: Two particles and a photon
## A Radiation Photon
The pack of four lines that rise along the light line in Fig. 3 are intended
to be the partition lines of a radiation photon that has a group velocity
equal to the velocity of light. Photons can have partition lines as do massive
particles. They separate the photon into its fractional parts, which is a
separation by phase differences. The photon in Fig. 3 is confined to the
packet that is distributed over the perpendicular (dashed) light path $l$.
Normally in physics we do not hesitate to use coordinates in empty space, so a
photon by itself will be given a period and wavelength relative to that
coordinate frame, and hence an energy and momentum. But if coordinates in
empty space have no legitimate place in physics, than like any other particle
a photon by itself will lack translational variables (e.g., energy and
momentum); and since it has no internal energy (i.e., rest mass/energy), the
gravitational perturbation of its light line will be zero. There is no photon
mass/energy to perturb it. It should also be clear from the diagram in Fig. 3
that the photon bundle has _no definable_ wavelength or frequency at event k.
Vacuum fluctuations exist in the ‘empty’ space between massive particles and
their polarizing effects are physically significant. But if vacuum fluctuation
particles are not themselves polarized they will not interact with a passing
photon (resulting in a scattering of the photon). So the photon cannot use
these particle grids to define its period and wavelength. Fluctuation
particles do not contribute in any other way to the discussion, so their
presence is ignored.
## Information Transfer
It is the photon’s phases that affect a transfer of energy and momentum from
one particle to another. This is shown in Fig. 4 where two particles are
narrowly defined to be moving over world lines $w_{1}$ and $w_{2}$. The two
dashed lines represent the partition lines of a passing photon with ‘relative’
phase differences given by $\delta\pi$. If the photon wave is a superposition
of two different frequencies 1 and 2, then
$\delta\pi=\delta\pi_{1}+\delta\pi_{2}$.
Figure 4: Two particles and a photon
A photon interacting with the first particle at event $\bf{a}$ will have a
local energy and momentum given by $e_{\gamma}(\bf{a})$,
$p_{\gamma}(\textbf{a}),$ and as it interacts with the second particle at
event $\bf{b}$ it will have a local energy and momentum given by
$e_{\gamma}(\bf{b})$, $p_{\gamma}(\textbf{b})$. These quantities are related
through the phase relationships that are transmitted between particles, and
are articulated in the local grid of the interacting particle.
$\displaystyle a\hskip 2.84544ptphoton\hskip 2.84544ptat\hskip 2.84544ptevent$
$\displaystyle\textbf{a}:$ $\displaystyle
e_{\gamma}(\textbf{a})=\hbar\Sigma_{i}\omega_{i}(\textbf{a})\hskip
14.22636ptp_{\gamma}(\textbf{a})=\hbar\Sigma_{i}k_{i}(\textbf{a})$ (4)
$\displaystyle a\hskip 2.84544ptphoton\hskip 2.84544ptat\hskip 2.84544ptevent$
$\displaystyle\textbf{b}:$ $\displaystyle
e_{\gamma}(\textbf{b})=\hbar\Sigma_{i}\omega_{i}(\textbf{b})\hskip
14.22636ptp_{\gamma}(\textbf{b})=\hbar\Sigma_{i}k_{i}(\textbf{b})$
where $\omega_{i}(\bf{a})$ = $\partial_{t}\pi_{i}(\bf{a})$ and $k_{i}(\bf{a})$
= $\partial_{x}\pi_{i}(\bf{a})$. These derivatives refer to the local grid of
each event in each particle, and are defined like those in Eq. 2.
## Electromagnetic Variables
The parallel lines passing by event k in Fig. 3 are lines of constant
‘relative’ phase of the photon. Differential phase changes $\delta\pi$ over a
light line like $l$ are preserved across the length of the photon wave packet.
However, since the photon in flight between two particles does not have its
own local grid, components cannot be defined for the electromagnetic field any
more than can for energy and momentum.
In empty space the _electromagnetic potential_ of a radiation photon is
normally given by a fourvector $A^{\mu}(\textbf{a})$, where the d’Alembertian
operating on $A^{\mu}(\textbf{a})$ is equal to zero. However, trans-coordinate
physics cannot use the d’Alembertian in empty space although the photon’s
behavior there is lawful – it follows a dynamic principle of some kind. Where
a grid exists we can give analytic expression to the dynamic principle; but
where there is no grid we must settle for another kind of description. All we
can do in this case is notice the physical manifestations of the dynamic
principle, and there are just four in 3 + 1 space. First, different relative
phases appear on different parallel layers along a light line as in Figs. 3
and 4. There is a definite phase relationship between any two of these layers.
Second, the probability that a photon goes into a particular solid angle from
an emission site a depends on the distribution given by an atomic decay at a,
or by the interaction of $A_{\mu}(\textbf{a})$ with the current
$j(\textbf{a})$ at that site. The only mid-flight indication of the strength
of a signal in a given solid angle is the probability of a photon emission in
that direction. Third, we say that the magnitude of $A_{\mu}$ arriving at a
material target is _determined by_ that probability – rather than probability
determined by magnitude. In the case of a single photon (or for any definite
number of photons) the components of $A_{\mu}$ at a material destination are
indeterminate, and the magnitude of the transmission diminishes with square
distance from the source by virtue of the constancy of photon number in a
solid angle. The fourth property provides for Huygens’ wavelets. So far we
have considered a photon as moving undeflected in an outward direction from a
source along a light cone. We now say that an event such as k in Fig. 3 acts
as a point source of radiation is all directions. The wavelet from k has the
same (relative) phase as event k, and it reradiates the “probability
intensity” at k uniformly in all directions with a velocity $c$. Two wavelets
that arrive at a third event m have a definite phase difference that produces
interference there.
Notice that a Huygens’ electromagnetic wavelet is a ‘scalar’ like the primary
wave that gives rise to it. The vector nature of an EM wave does not appear
until it interacts with matter, and only then when an indefinite number of
photons are phased in such a way as to make that happen.
## Photon Scattering
If a photon scatters at an event a inside the wave packet of a particle, the
grid for that purpose will be the particle’s grid at a. There will be no
quantum jump or wave collapse in a scattering of this kind. Instead, some
fraction of the particle $p$ and photon $\gamma$ will evolve continuously into
a scattered wave that consists of a correlated particle $p^{\prime}$ and a
photon $\gamma^{\prime}$. Energy and momentum will be defined for each of the
four particles $p$, $p^{\prime}$, $\gamma$, and $\gamma^{\prime}$ that are
mapped together on that common grid of $p$ at event a, and the dynamic
principles of these particles (plus their interaction) will insure that total
conservation applies to all four. Each component of the scattered wave of
$p^{\prime}$ will also have a grid that is well defined at event a, and is a
Lorentz transformation away from the grid of $p$. Energy and momentum will be
conserved on the grid of each component of $p^{\prime}$. The velocity of any
component of $p^{\prime}$ relative to $p$ is not explicitly given in the
trans-coordinate case; however, it is implicit in the Lorentz transformation
that is required to go from the locally evolving grid of $p$ to that of
$p^{\prime}$.
## Virtual Photons
So far we have talked about _radiation_ photons that travel at the velocity of
light. _Virtual_ photons (in a Coulomb field) do not bundle themselves into
wave packets, so they do not have a ‘group’ velocity that requires the
identification of a world line over which the group travels. It makes no sense
to say that they travel over light lines. It may therefore be possible to give
the virtual photon a local grid in the same way that we created a grid for
particles with non-zero mass. Its vector nature would then be more evident.
However, we choose not to do that. It is unnecessary and would put the virtual
photon grid in competition with the particle grid during an interaction
between the two. That would necessitate a choice between one or the other in
any case; so _all_ photons will be considered gridless in this treatment –
just like radiation photons. They all lack internal energies. They also lack
translational variables such as energy and momentum when in transit between
particles; and they acquire these values only when they overlap the charged
particles with which they interact. We say in effect that there is no
fundamental difference between ‘near’ field photons and ‘far’ field photons in
an electromagnetic disturbance.
## Gravity
If a photon in transit (radiation or virtual) has no frequency or
translational energy $h\nu$, it will not have a weight in the presence of a
gravitating body or create a curvature in the surrounding metric space.
However, massive objects having rest energy _do_ create curvatures in their
vicinity in which _light line geodesics_ are well defined. We claim that
radiation photons follow these geodesics without themselves contributing to
the curvature of space. Although photons in transit are massless and hence
weightless, they nonetheless behave as though they are attracted to
gravitational masses.
This does not mean that current photon trajectories are in error, or that
particle masses have to be adjusted. The mass of an electron found from the
oil drop experiment is currently assumed to include the mass of the
accompanying electromagnetic field. From a trans-coordinate point of view the
electric field surrounding a charged particle is not defined, so this
experiment reveals the ‘bare’ mass of the electron. The mass of the Sun
obtained from the period of a planet is normally assumed to include the mass
of the radiation field surrounding the sun. From a trans-coordinate point of
view the radiation field is not defined, so this calculation reveals the
‘bare’ mass of the sun – that is, the total number of each kind of solar
particle times its mass. These changes will not result in observational
anomalies in particle theory or astronomy, for we have no way to separately
weigh the electromagnetic field of a charged particle, or to count the number
of particles in the Sun.
## Binding Energy
Even in coordinate language we are able to give up the idea of electromagnetic
field energy, so the binding energy of particles in a nucleus can be
considered a property of the particles themselves. Imagine two positive
particles of rest mass $m_{0}$ that approach one another in the center-of-mass
system with kinetic energy $T$. The momentum of one of these particles
decreases as a result of virtual photon exchange; however, its energy will not
change. A virtual photon leaving one particle will carry away a certain amount
of energy, but that energy is restored in equal amount by the virtual photon
that is received from the other particle. This means that the net energy of
the advancing particle will be unchanged during the trip. When the particle
reaches the point at which it has lost all of its kinetic energy and has
combined with the other particle due to nuclear forces, we would say that the
initial kinetic energy of one of them has become its binding energy $BE$,
where
$E=BE+m_{0}c^{2}=T+m_{0}c^{2}$
As the particle moves inward its energy square $E^{2}=P^{2}+m_{0}^{2}c^{4}$
remains constant while $P^{2}$ goes to zero. Therefore $E^{2}$ becomes
identified with an increased mass $M^{2}c^{4}$ giving $E=Mc^{2}=BE+m_{0}$.
Then
$Binding\hskip 5.69046ptenergy=Mc^{2}-m_{0}c^{2}$
In relativity theory a particle’s (relativistic) mass is a function of kinetic
energy. We can also say it is a function of an interaction with other
particles, thereby avoiding any notion of ‘field’ energy.
These ideas are peculiar to the center-of-mass coordinate system but are not
correct from a trans-coordinate point of view. Fundamentally there is no
energy associated with the particle as a whole. There is only the time
derivative of $\varphi$ at each separate event inside the particle’s wave
packet. There is also no kinetic energy of the particle or binding energy of a
captured particle. The ‘correct’ trans-coordinate account of a coulomb
interaction is given below.
## Virtual Interaction
The virtual (Coulomb) interaction cannot be thought of as a single virtual
photon interacting with a single charged particle because that is not
energetically possible. However, the interaction is _continuous_ like Compton
scattering; so in spite of the fact that the theory is based on photons the
interaction does not manifest itself as discrete quantum jumps. A particle in
a Coulomb interaction is therefore continuously receiving and transmitting
equal amounts of energy, which means that it undergoes a change of momentum
with no change of energy. The resulting behavior of the charged particle is
given by a continuum of particle grids along its partition line that are
related by infinitesimal Lorentz transformations. Energy and momentum are
conserved on any one of these grids. Since each particle is well localized in
the background metric space, predictable continuous transformations of the
world line of each event in the packet are _all that is necessary_ to
determine the packet’s complete behavior. Nature is not concerned with the
coordinate-based energies of the previous section – and does not need to be.
## Regional Coordinates and Conservation
Trans-coordinate physics does not provide for energy and momentum conservation
in the region between particles. We cannot assign frequency or wavelength to a
radiation photon in an otherwise empty space as we have seen, so we cannot say
that it carries energy $h\nu$ or momentum $h/\lambda$ from one part of space
to another. Also, a massive particle has no velocity or acceleration when it
is considered in isolation. It moves into its future time cone over the
invariant metric background following its dynamic principle, but that path
does not break down into spatial and temporal directions relative to which the
wave packet can be said to be moving with a kinetic energy or velocity $v$. So
it cannot be said to carry a net momentum $mv$.
Regional conservation of these quantities is therefore related to the
possibility of system-wide coordinates that _we_ construct. Having done that
we can define a metric tensor throughout the region. That is, from the
background invariant metric it is generally possible to find the continuous
metric tensor $g_{\mu\nu}$ that goes with the chosen coordinates. If that
tensor is time independent then _energy_ will be conserved in the region
covered by those coordinates. If it is independent of a spatial coordinate
such as $x$, then _momentum in the $x$-direction_ will be conserved in the
region covered by the coordinates. If the metric is symmetric about some axis
(in 3 + 1 space) then _angular momentum_ will be conserved about that axis [R.
A. Mould, (2002)]. It is therefore useful for us to construct system-wide
coordinates in order to take advantage of these regional conservation
principles. It is important to remember however that we do this, not nature.
Nature has no need to analyze as we do over extended regions. For the most
part it only _performs_ on a local platform.
If there is a difference in energy between $e_{\gamma}(\textbf{a})$ and
$e_{\gamma}(\textbf{b})$ in Eq. 4, it is possible that the photon in Fig. 4 is
_Doppler shifted_ because of a relative velocity between the two particles, or
that particle #2 is at a different _gravitational potential_ than particle #1.
When a coordinate system is chosen the velocity of one particle is decided
relative to the other particle, and only then will the extent of the Doppler
influence be determined. Only then will it be clear how the organizing power
of a coordinate system makes use of gravity to explain the non-Doppler
difference between $e_{\gamma}(\textbf{a})$ and $e_{\gamma}(\textbf{b})$.
## Trans-Coordinate Tensors
Every event in a massive particle wave packet has a grid associated with it.
In 3 + 1 space the spatial part is a three dimensional grid. When this is
combined with the metric background the metric tensor $g_{\mu\nu}$ is
determined at each event. One can therefore raise and lower indices of vectors
inside the wave packet in the trans-coordinate case.
However, we do not assign derivatives to $g_{\mu\nu}$ because it is not a
uniquely continuous function. For a given $g_{\mu\nu}(\textbf{a})$ at event a
there are an infinite number of ways that a continuous $g_{\mu\nu}$ field
‘might’ be applied in the region around a, corresponding to the infinite
number coordinate systems that ‘might’ be employed in that region. But if we
do not attach physical significance to coordinates, then physical significance
cannot be attached to a continuous metric tensor. Derivatives of that tensor
are therefore not defined in trans-coordinate physics. This applies to the
derivatives in Eq. 2 as well as to the covariant derivatives of Riemannian
geometry. Therefore, Christoffel symbols are not defined in trans-coordinate
physics.
It follows that the Riemann and Ricci tensors and the field equation of
general relativity are also not fundamentally defined. Like energy and
momentum conservation from which it is derived, the gravitational field
equation is a regional creation of ours that is analytically useful and that
gives us a satisfying big picture – but that is all. Of course the ‘curvature’
is objectively defined everywhere because it follows directly from the
invariant metric background in which everything is embedded.
We can be guided by our experience with general relativity when choosing the
most useful coordinate system in a given region of interest. A metric tensor
can then be defined; and from the symmetry of its components, energy,
momentum, and angular momentum conservation can be established over the
region. However, there is no assurance that one can always find an agreeable
system, for general relativity does not guarantee that the chosen coordinates
will conserve energy, momentum, or angular momentum without introducing
special pseudo-tensors that are devised for that purpose [L. Landou and E.
Lifshitz, (1971)].
## Gravitons
If general relativity is not fundamental then gravitons must be the exclusive
cause of gravitational effects. The geodesics that result from graviton
interactions between massive particles are not the smooth curves of general
relativity, but are quantized by discrete graviton interactions. The wider
effect of gravitons is to bend the background metric space between geodesics.
Their influence will spread through the invariant metric space; and as a
result, the curvature produced by gravitons will follow the average curvature
of general relativity except that it will have the jagged edge of
quantization. General Relativity is therefore a science that only approximates
the underlying reality. It is a science we initiate when we introduce the
coordinates that permit the definition of metric tensor derivatives and allow
the formulation of Einstein s field equation.
## Internal Coordinates
In additional to regional coordinates that cover the space between particles,
we want to give ourselves an _internal picture_ of the particle. We want the
wave function $\varphi{(\bf{a)}}$ in Eq. 1 in a form that permits analysis. To
do this starting at event a, integrate the minus square root of the metric
along the partition line going through a and assign a time coordinate $t_{a}$
with an origin at a. Then integrate the square root of the metric over the
perpendicular going through event a and assign a space coordinate $x_{a}$ with
an origin at a. The coordinates $x$ and $t$ may be extended over the entire
object yielding a wave function that can be written in the conventional way
$\varphi(x,t)$. These internal coordinates will have the same status as
external coordinates. They are only created by us for the purpose of analysis.
With internal coordinates we can integrate across one of the perpendiculars to
find the _width_ of the wave packet. It should also be possible to integrate
the square modulus over a perpendicular to find the _total normalization_.
That total will be equal to 1.0 if $df$ is equal to the fraction of the
particle sandwiched between two differentially close partition lines as
claimed. We can also use internal coordinates to give expression to the
internal variables of a particle, such as its total internal energy and net
momentum.
## Three and Four Dimensions
Imagine that a particle’s wave packet occupies the two-dimensional area shown
on the space-like surface in Fig. 5. The surface is divided into a patchwork
of squares, each of which is made to contain a given fraction of the particle,
like 1/100th of the particle.
Figure 5: Two dimensional scaffold
Each of these squares has four distinguishable crossing points or corners. A
similar two-dimensional scaffold is constructed on all of the space-like
surfaces through which the particle passes in time, thereby creating a
continuous 2 + 1 scaffold. Each of the enclosed areas generated in this way is
required to contain $1/100$ of the particle, and its corners will constitute
the partition lines of the particle. As in the 1 + 1 case, these lines may be
thought of as streamlines of the square modular flow of the particle through
time. In the limit as this fraction goes to zero, partition lines pass through
each event on the space-like surface in the figure and they do not cross one
another.
It is possible to find the direction of the partition line through an event a
without having to erect a system-wide scaffolding like that of Fig. 5. Any
small neighborhood of a has a probability that the particle will be found
within it; and that probability will be ‘minimal’ when the partition line
going through a coincides with the preferred direction of time for that
neighborhood.
Space-time directions are chosen for a given partition line in a way that is
similar to the procedure in Fig. 2. Starting with an event a in Fig. 6a, move
up its partition line a metrical distance $-\Delta$ to event b. Then find
$\textbf{b}^{\prime}$ by proceeding down the partition line the same invariant
interval $-\Delta$. Construct a backward time cone with b at its vertex and a
forward time cone with $\textbf{b}^{\prime}$ at its vertex and identify the
closed two-dimensional loop intersection shown in Fig. 6a in the limit as
$\Delta$ goes to zero. In the local inertial system, two perpendicular unit
vectors $\hat{x}$ and $\hat{y}$ are chosen along the radius of the circle of
radius $\Delta$ that spans the spatial part of the local grid at event a. For
any $\Delta$, choose a space-like line beginning at a that is aligned with
$\hat{x}$ and extends to the circumference of the circle in Fig. 6a. It
intercepts the circle at the event we call $\textbf{c}^{\prime}$ in Eq. 2. The
space-like line that begins with -$\hat{x}$ intercepts the circle at the event
we call c in Eq. 2. These space-like lines do not have to be ‘straight’, so
long as they are initially aligned with the unit vector and intercept the
circle in only one place.
Figure 6: Establishing space-like unit vectors
The spatial grids of nearby events such as a and $\textbf{a}^{\prime}$ in Fig.
6b do not have to line up in any particular way. Even if they are in each
other’s spatial neighborhood for some value of $\Delta$, $\hat{x}$ and
$\hat{x}^{\prime}$ will generally point in different directions.
In 3 + 1 space the intersection of a backward and forward time cone will
produce a spherical surface like the one pictured in Fig. 6c. In this case
choose four mutually perpendicular unit vectors $\hat{x}$, $\hat{y}$,
$\hat{z}$, and $\hat{t}$ to form the local grid at event a. As before, the
orientation of the spatial part of these grids is of no importance. They may
be arbitrarily directed because their only purpose is to locally define all
three spatial derivatives of the function $\varphi$. That function is
continuous throughout the wave packet in any direction; therefore, it does not
matter which grid orientation is chosen at any event for the purpose of
specifying the function and its derivatives there. The Dirac solution has four
components $\varphi_{\mu}$ where each satisfies all of the above conditions in
the 3 + 1 directions.
Since every event on the surface of the sphere in Fig. 6c locates a partition
line, the event a is enclosed by a sphere with a differential volume $d\Omega$
that contains a differential fraction $df$ of the entire particle, where
$\varphi^{*}(\textbf{a})\varphi(\textbf{a})d\Omega=df$
which normalizes the 3 + 1 wave function.
## Applying the Dynamic Principle (3 + 1)
The third condition on a wave function $\varphi(\textbf{a})$ in Eq. 1 requires
that the dynamic principle applies throughout the space. This can be done in
the 3 + 1 space of an event a by using the grid defined in Fig. 6c. Since we
can do this at any event and for any orientation of the grid, we state the
more general form of the third condition:
> _The wave function $\varphi(\textbf{a})$ of a particle at any event a is
> subject to a dynamic principle that is applied locally to any four mutually
> perpendicular space-time directions centered at a, where time is directed
> along the partition line through a. This principle determines how
> $\varphi(\textbf{a})$ evolves relative to its own time against the metric
> background, and how it relates spatially to its immediate neighbors._.
The continuity condition applies to the function $\varphi$ along any finite
segment of line emanating from any event.
## Atoms and Solids
Consider how all this might apply to a hydrogen atom. Each massive particle
carries a local grid that is independently defined at each event in its wave
packet. This insures separate normalization at each event for each particle.
The proton and electron grids may overlap but they need not be aligned because
the particles do not directly interact. They are connected through the Coulomb
field by virtual photons that carry no grid of their own. There are two
interactions, one involving a virtual photon and the event grids of the
proton, and one involving a virtual photon and the event grids of the
electron. These are described in the section “Virtual Interaction”.
In the non-relativistic case both particles can be covered by a single _common
inertial frame_ in which the total energy and momentum is conserved. It does
no harm and it facilitates analysis to imagine that each grid in the system is
aligned with this common coordinate frame. The time $t$ assigned to each
proton grid and the time $t^{\prime}$ assigned to each electron grid are then
set equal to each other and to the time of the common inertial frame. The
retarded interaction $j_{\mu}A^{\mu}$ at each end of the interaction will then
give the Coulomb intensity of $(e^{2}/4\pi r)\delta(t-t^{\prime})$ where $r$
is the distance between the particles in the common frame [R. P. Feynman et
al., (1995)]. Relativistic corrections to this occur when the spatial
components of the current fourvectors are taken into account.
The above inertial system is one that we impose on the atom. By itself, the
system operates on the basis of individual event grids alone. A photon passing
over the atom will interact with each separate event in the proton wave
function throughout its volume, and with each separate event of the electron
throughout its volume. Energy and momentum conservation is required at each
site, but the system will not support conservation unless the interaction
Hamiltonian [R. A. Mould, (2008)] includes the entire system in a ‘single’
interaction. It is the interaction Hamiltonian that makes the difference
between particles in a single interaction that conserves energy and momentum,
and particles in separate interactions that may or may not conserve these
quantities. In the atomic case the dynamic principle for the entire atom
provides the unity that can give rise to a quantum jump that carries the
product $pe\gamma$ of the proton, the electron, and the photon, into a new
product $p^{\prime}e^{\prime}\gamma^{\prime}$, conserving energy and momentum
in the process.
In the case of macroscopic crystals, metals, and other stationary solid forms
in a flat space, each event in each particle wave packet has its own space-
time grid and is separately normalized. However, they are all interactively
aligned to such an extent that we can usually impose a single common
coordinate system. We require the coordinates of this system to co-move with
the average density of matter in the solid. If that system has the right
symmetry properties it will insure macroscopic energy, momentum, and angular
momentum conservation.
## Containers
Figure 7: Particle in container
Let the central region of the hollow spherical container in Fig. 7 be a
general relativistic space of unknown curvature. The center of the sphere is
initially empty (suppressing vacuum fluctuations). A massive object leaves
event a and at some later time arrives at event b. At each event along the way
it is propelled by its dynamic principle into its forward time cone; and since
the resulting path of the packet cannot be broken down into spatial and
temporal parts, its velocity, energy, momentum, and distance traveled on that
path are not determined. The particle will have ‘internal’ energy and momentum
that are derived from interal coordinates, but these will not be its
‘translational’ energy and momentum in the usual sense going from a to b. A
radiation photon will not even have these internal properties over its path;
for it will only acquire the energy and momentum in Eq. 4 when it encounters a
particle in the container wall.
We can certainly construct a common coordinate system over this system,
extending the co-moving coordinates of the solid into the center of the
sphere. We will then know how far the object goes and its velocity along the
way. If the metric of that system is time independent, then total energy will
be conserved throughout the trip from event a to event b. Although we can
usually cover the system with extended coordinates and a metric, there is no
guarantee that resulting system will conserve total energy and momentum
without introducing the pseudo-potentials of L. Landou and E. Lifshitz,
(1971).
## A Gaseous System
The introduction of many gas particles in the space of Fig. 7 does not change
anything of substance. Molecular collisions occurring on the inside surface of
the container and between molecules are distinct physical events. But we still
do not have a natural basis for ascribing a numerical distance between any of
these collisions or the molecular velocities between them.
Molecular collisions are here assumed to be electromagnetic in nature. Parts
of the colliding molecules may or may not overlap, but they each (i.e., the
internal parts of each) maintain their separate grids for the purpose of
normalization. These grids do not compete with one another during a collision
because the interaction between them is conducted through virtual photons, and
these are declared to be gridless.
## States
In coordinate physics we normally define a physical ‘state’ across a
horizontal plane at some given time. This definition identifies an origin of
coordinates relative to which the system’s particles are located at that time.
That scheme will not work in the trans-coordinate case because the “same time”
for separated particles is undefined. Indeed, the time of a single particle at
a single location is undefined. The meaning of _state_ must therefore be
revised.
The state of a system of three particles is now given by
$\Psi(\textbf{a},\textbf{b},\textbf{c})=\phi_{1}(\textbf{a})\phi_{2}(\textbf{b})\phi_{3}(\textbf{c})$
where a, b, and c are events anywhere within each of the given wave functions,
subject only to the constraint that each event has a _space-like_ relationship
to the others. Each of these three functions is defined relative to its own
local grid and is related to its time-like successors through its dynamic
principle. These events are connected by the space-like line in Fig. 8,
thereby defining the state $\Psi$ of the particles that are specified along
their separate world lines $w_{1}$, $w_{2}$, and $w_{3}$.
Figure 8: New state definition
A _successor state_ can be written
$\Psi^{\prime}(\textbf{a}^{\prime},\textbf{b}^{\prime},\textbf{c}^{\prime})=\phi_{1}(\textbf{a}^{\prime})\phi_{2}(\textbf{b}^{\prime})\phi_{3}(\textbf{c}^{\prime})$
(5)
where events $\textbf{a}^{\prime}$, $\textbf{b}^{\prime}$, and
$\textbf{c}^{\prime}$ in the new state must also have space-like relationships
to each other; and in addition, they are required to be in the forward time
cones of events a, b, and c respectively. These events lie along a space-like
line in Fig. 8 giving the state function $\Psi^{\prime}$. Equation 5 does not
say that each event has advanced by the same amount of time. It says only that
each particle has advanced continuously along its own world line (i.e., along
its own partition line) under its own dynamic principle, and has reached the
designated ‘primed’ events.
We might also let $\textbf{b}^{\prime\prime}$ replace event
$\textbf{b}^{\prime}$, where $\textbf{b}^{\prime\prime}$ has space-like
relationships to $\textbf{a}^{\prime}$ and $\textbf{c}^{\prime}$ and is in the
forward time cone of event b. The resulting state
$\Psi^{\prime\prime}(\textbf{a}^{\prime},\textbf{b}^{\prime\prime},\textbf{c}^{\prime})$
is not the same as
$\Psi^{\prime}(\textbf{a}^{\prime},\textbf{b}^{\prime},\textbf{c}^{\prime})$,
but it is just as much a successor of the initial state
$\Psi(\textbf{a},\textbf{b},\textbf{c})$. Also, $\Psi^{\prime\prime}$ is a
successor of $\Psi^{\prime}$ because $\textbf{b}^{\prime\prime}$ is a
successor of $\textbf{b}^{\prime}$. This definition of state is far more
general than the coordinate based (planar) definition, giving us an important
degree of flexibility as will be demonstrated below and in another paper [R.
A. Mould, (2008)]. The Hamiltonian for this kind of state can be defined in
such a way as to establish the _conservation of probability current_ flow, as
is also shown in this reference.
## An Application
Consider the case of an atom emitting a photon that is captured by a distant
detector. The initial spontaneous decay of the atom can be written in the form
$\varphi=(a_{1}+a_{0}\gamma)D$ (6)
where $a_{1}$ is the initial state of the atom, $a_{0}$ is its ground state,
$\gamma$ is the emitted photon, and $D$ is a distant detector that is not
involved in the decay. At this point we do not specify specific events or use
the new definition of state. In response to the dynamic principle, the
probability current flows from the first component in Eq. 6 to the second
component inside the bracket, so the first component decreases in time and the
second component increases in such a way as to conserve square modulus as
shown in Ref. 3. At some moment of time a stochastic choice occurs and the
state undergoes a quantum jump from $\varphi$ to $\varphi^{\prime}$ conserving
energy and momentum and giving
$\varphi^{\prime}=a_{0}\gamma D$ (7)
that describes the state of the system during the time the photon is in flight
from the atom to the detector. When the photon interacts with the detector the
equation of state becomes
$\varphi^{\prime\prime}=a_{0}(\gamma D+D^{\prime\prime})$ (8)
where $D^{\prime\prime}$ is the detector after capture. The atom $a_{0}$ is
not a participant in this interaction. Again, probability current flows from
$\gamma D$ to $D^{\prime\prime}$ and this, we assume, results in another
stochastic hit conserving energy and momentum and yielding
$\varphi^{\prime\prime\prime}=a_{0}D^{\prime\prime}$
When the _new definition_ of state is applied to this case Eq. 6 is written
$\varphi(\textbf{a},\textbf{c})=[a_{1}(\textbf{a})+a_{0}(\textbf{a})\gamma(\textbf{a})]D(\textbf{c})$
(9)
where the atom and the photon overlap at event a. The photon uses the grid of
the atom at event a to evaluate its frequency and wavelength, whereas the
detector uses its own grid. Nonetheless, the dynamic principle in the form of
the Hamiltonian defined in Ref. 3 applies to this interaction equation that is
local to event a.
Equation 7 for the proton in flight is then
$\varphi^{\prime}(\textbf{a},\textbf{k},\textbf{c})=a_{0}(\textbf{a})\gamma(\textbf{k})D(\textbf{c})$
(10)
where the energy of the atom and the detector are given by their time
derivatives at events a and c, but there is no energy associated with the
independent photon in this equation. The function $\gamma(\textbf{k})$ is of
the form exp$[i\theta(\textbf{k})$] where k is the event appearing in Fig. 3,
so frequency and wavelength are not given. The photon’s Hamiltonian applied to
this equation equals zero. Equation 9 applies so long as the photon is located
on a definite partition line of the atom; but the moment the photon event
appears apart from the atom, Eq. 10 will apply.
Equation 8 using the above state definition is
$\varphi^{\prime\prime}(\textbf{a},\textbf{c})=a_{0}(\textbf{a})[\gamma(\textbf{c})D(\textbf{c})+D^{\prime\prime}(\textbf{c})]$
where the photon overlaps the detector at event c. In this case the photon
uses the grid of the detector at event c to evaluate its frequency and
wavelength, and the energy of the atom is given by its time derivative on the
grid of event a. Here again the dynamic principle applies to this interaction
equation that is local to event c.
Actually the atom should be written as a product of the proton $p$ and the
electron $e$ giving $a=pe$. In the parts of the atom where the proton and the
electron _do not_ overlap, Eq. 9 could be written as either
$\varphi^{\prime\prime}(\textbf{a},\textbf{b},\textbf{c})=[p_{1}(\textbf{a})e_{1}(\textbf{b})+p_{0}(\textbf{a})e_{0}(\textbf{b})\gamma(\textbf{a})]D(\textbf{c})$
or
$\varphi^{\prime\prime}(\textbf{a},\textbf{b},\textbf{c})=[p_{1}(\textbf{a})e_{1}(\textbf{b})+p_{0}(\textbf{a})e_{0}(\textbf{b})\gamma(\textbf{b})]D(\textbf{c})$
Both equations are correct. They both describe the interaction of the photon
on different grids associated with different parts of the atom, where the
dynamic principle applies in each case.
Equations of this kind are used more extensively in Ref. 3, and the rules that
govern them are given in the Appendix of that reference.
## Unifying Features
The most important non-local unifying feature of a trans-coordinate system is
the _invariant metric space_ in which everything is embedded. Another
important unifying feature is the _dynamic principle_ applied to each particle
by itself and to any system of particles as a whole.
_Non-local correlations_ are another unifying features of the functions
generated by the dynamic principle. These qualify the location of one particle
relative to the location of another particle; so the equation of state of two
particles $p_{1}$ and $p_{2}$ is written
$\Phi=p_{1}p_{2}(\textbf{a},\textbf{b})$, rather than
$\Phi=p_{1}(\textbf{a})p_{2}(\textbf{b})$. These particles have their separate
grids as always, to which the dynamic principle separately applies as always.
The difference is that the range of b depends on the value of a and visa
versa, and their joint values determine $\Phi$. This function is local to both
events a and b, so it is a bi-local function.
The fourth unifying feature is the _collapse of the wave function_ over finite
regions of space.
## Modified Hellwig-Kraus Collapse
A local quantum mechanical measurement can have regional consequences through
the collapse of a wave function. The question is: How can that superluminal
influence be invariantly transmitted over a relativistic metric space?
Hellwig and Kraus answered this question by saying that the collapse takes
place across the surface of the backward time cone of the triggering event [K.
E. Hellwig and K. Kraus, (1970)]. The Hellwig-Kraus collapse has been
criticized because it appears to result in causal loops [Y. Aharonov and D. Z.
Albert, (1981)], but the situation changes dramatically with the new trans-
coordinate definition of state. We keep the idea that the influence of a
collapse is communicated along the backward time cone; however, the state of
the system that survives a collapse (i.e., the finally realized eigenstate) is
not defined along a “simultaneous” surface. The increased flexibility of the
new state definition allows the remaining (uncollapsed) state to retain its
original relationship with the event that initiates the collapse. When this
program is consistently carried out causal loops are eliminated, even in a
system of two correlated particles. I will not elaborate on this idea in this
paper but it is demonstrated in detail in [R. A. Mould, (2008)].
## Another Approach
Invariance under coordinate transformation is not discussed at any length in
this paper because coordinates are not introduced in the first place; but it
should be noted that the idea of coordinate invariance is limited. General
relativity is not truly independent of coordinates because it does not include
_all possible_ coordinates in its transformation group. It does not include
‘discontinuous’ coordinate systems, many of which are capable of uniquely
identifying all the events in a space-time continuum – as is claimed to be the
purpose of a space-time coordinate system. For example, imagine Minkowski
coordinates in which the number 1.0 is added to all irrational numbers but not
to rational numbers. This system is perfectly capable of systematically and
uniquely identifying all of the events in a space-time continuum, but it is
thoroughly discontinuous in a way that prevents it from being useful to
general relativity. It only takes one example of unfit coordinates to
disqualify invariance as a fundamental requirement in physics, and there are
many discontinuous coordinates like this one. Of course one can always reject
coordinates that don’t work in the desired way on the basis of the fact that
they don’t work in the desired way. But that avoids the issue. The point is
that the influence of unnatural identification labels cannot be eliminated
from physics through an invariance principle that affects only a sub-set of
unnatural identification labels. Another approach is indicated.
## References
* Y. Aharonov and D. Z. Albert, (1981) Aharonov, Y., Albert, D. Z. (1981) _Phys. Rev. D_ 24, 359
* R. P. Feynman et al., (1995) Feynman, R. P., Morinigo, F. B., Wagner, W. G. (1995) _Feynman Lectures on Gravitation_ , B. Hatfield. ed., Addison-Westley, New York, 33
* K. E. Hellwig and K. Kraus, (1970) Hellwig, K. E., Kraus, K. (1970) _Phys. Rev. D_ 1, 566
* L. Landou and E. Lifshitz, (1971) Landou, L. and Lifshitz, E. _The Classical Theory of Fields_ , Pergamon Press, New York, (1971) p. 316
* J. Maldacena, (2005) Maldacena, J. (2005) ”The Illusion of Gravity”, _Sci. Am._ Nov, 56
* R. A. Mould, (2002) Mould, R. A. (2002) _Basic Relatvity_ , Springer,New York )2002) Eq. 8.66
* R. A. Mould, (2008) Mould, R. A. (2008) “Trans-Coordinate States”, arXiv:0812.1937
* G. ’t Hooft, (2008) ’t Hooft, G. (2008) “A Grand View of Physics”, _Int’l J. Mod. Phys._ A23 3755, sect 3; arXiv:0707.4572
|
arxiv-papers
| 2008-12-08T15:16:08 |
2024-09-04T02:48:59.297867
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Richard A. Mould",
"submitter": "Richard A. Mould",
"url": "https://arxiv.org/abs/0812.1503"
}
|
0812.1600
|
# $K$-trivials are $\operatorname{NCR}$
Antonio Montalbán Department of Mathematics
University of Chicago
5734 S. University ave.
Chicago, IL 60637, USA antonio@math.uchicago.edu and Theodore A. Slaman
Department of Mathematics, University of California, Berkeley Berkeley, CA
94720-3840 USA slaman@math.berkeley.edu
## 1\. Introduction
In RS (07, 08), Reimann and Slaman raise the question “For which infinite
binary sequences $X$ do there exist continuous probability measures $\mu$ such
that $X$ is effectively random relative to $\mu$?”
### 1.1. Randomness relative to continuous measures
We begin by reviewing the basic definitions needed to precisely formulate this
question.
###### Notation 1.1.
* •
For $\sigma\in 2^{<\omega}$, $[\sigma]$ is the basic open subset of
$2^{\omega}$ consisting of those $X$’s which extend $\sigma$. Similarly, for
$W$ a subset of $2^{<\omega}$, let $[W]$ be the open set given by the union of
the basic open sets $[\sigma]$ such that $\sigma\in W$.
* •
For $U\subseteq 2^{\omega}$, $\lambda(U)$ denotes the measure of $U$ under the
uniform distribution. Thus, $\lambda([\sigma])$ is $1/2^{\ell}$, where $\ell$
is the length of $\sigma$.
###### Definition 1.2.
A representation $m$ of a probability measure $\mu$ on $2^{\omega}$ provides,
for each $\sigma\in 2^{<\omega}$, a sequence of intervals with rational
endpoints, each interval containing $\mu([\sigma])$, and with lengths
converging monotonically to 0.
###### Definition 1.3.
Suppose $Z\in 2^{\omega}$. A _test relative to $Z$_, or _$Z$ -test_, is a set
$W\subseteq\omega\times 2^{<\omega}$ which is recursively enumerable in $Z$.
For $X\in 2^{\omega}$, $X$ _passes_ a test $W$ if and only if there is an $n$
such that $X\not\in[W_{n}]$.
###### Definition 1.4.
Suppose that $m$ represents the measure $\mu$ on $2^{\omega}$ and that $W$ is
an $m$-test.
* •
$W$ is _correct for $\mu$_ if and only if for all $n$, $\sum_{\sigma\in
W_{n}}\mu([\sigma])\leq 2^{-n}.$
* •
$W$ is _Solovay-correct for $\mu$_ if and only if
$\sum_{n\in\omega}\mu([W_{n}])<\infty$.
###### Definition 1.5.
$X\in 2^{\omega}$ is _$1$ -random relative to a representation $m$ of $\mu$_
if and only if $X$ passes every $m$-test which is correct for $\mu$. When $m$
is understood, we say that $X$ is 1-random relative to $\mu$.
By an argument of Solovay, see Nie (09), $X$ is $1$-random relative to a
representation $m$ of $\mu$ if an only if for every $m$-test which is Solovay-
correct for $\mu$, there are infinitely many $n$ such that $X\not\in[W_{n}]$.
###### Definition 1.6.
$X\in\operatorname{NCR}_{1}$ if and only if there is no representation $m$ of
a continuous measure $\mu$ such that $X$ is 1-random relative to the
representation $m$ of $\mu$.
In RS (08), Reimann and Slaman show that if $X$ is not hyperarithmetic, then
there is a continuous measure $\mu$ such that $X$ is 1-random relative to
$\mu$. Conversely, Kjøs-Hanssen and Montalbán, see Mon (05), have shown that
if $X$ is an element of a countable $\Pi^{0}_{1}$-class, then there is no
continuous measure for which $X$ is 1-random. As the Turing degrees of the
elements of countable $\Pi^{0}_{1}$-classes are cofinal in the Turing degrees
of the hyperarithmetic sets, the smallest ideal in the Turing degrees that
contains the degrees represented in $\operatorname{NCR}_{1}$ is exactly the
Turing degrees of the hyperarithmetic sets.
In RSte , Reimann and Slaman pose the problem to find a natural
$\Pi^{1}_{1}$-norm for $\operatorname{NCR}_{1}$ and to understand its
connection with the natural norm mapping a hyperarithmetic set $X$ to the
ordinal at which $X$ is first constructed. As of the writing of this paper,
this problem is open in general, but completed in RSte for
$X\in\Delta^{0}_{2}$.
Suppose that $X\in\Delta^{0}_{2}$ and that for all $n$,
$X(n)=\lim_{t\to\infty}X_{t}(n)$, where $X_{t}(n)$ is a computable function of
$n$ and $t$. Let $g_{X}$ be the convergence function for this approximation,
that is for all $n$, $g_{X}(n)$ is the least $s$ such that for all $t\geq s$
and all $m\leq n$, $X_{t}(m)=X(m)$. Let $f_{X}$ be function obtained by
iterated application of $g_{X}$: $f_{X}(0)=g_{X}(0)$ and
$f_{X}(n+1)=g_{X}(f_{X}(n))$.
For a representation $m$ of a continuous measure $\mu$, the granularity
function $s_{m}$ maps $n\in\omega$ to the least $\ell$ found in the
representation of $\mu$ by $m$ such that for all $\sigma$ of length $\ell$,
$\mu([\sigma])<1/2^{n}$. Note that, $s_{m}$ is well-defined by the compactness
of $2^{\omega}$.
###### Theorem 1.7 (Reimann and Slaman RSte ).
If $X$ is 1-random relative the representation $m$ of $\mu$, then the
granularity function $s_{m}$ for $\mu$ is eventually bounded by $f_{X}$.
Thus, there is a continuous measure relative to which $X$ is 1-random if and
only if there is a continuous measure whose granularity is eventually bounded
by $f_{X}$. The latter condition is arithmetic, again by a compactness
argument.
### 1.2. $K$-triviality
$K$-triviality is a property of sequences which characterizes another aspect
of their being far from random. We briefly review this notion and the results
surrounding it. A full treatment is given in Nies Nie (09).
For $\sigma\in 2^{<\omega}$, let $K(\sigma)$ denote the prefix-free Kolmogorov
complexity of $\sigma$. Intuitively, given a universal computable $U$ with
domain an antichain in $2^{<\omega}$, $K(\sigma)$ is length of the shortest
$\tau$ such that $U(\tau)=\sigma$. Similarly, for $X\in 2^{\omega}$, let
$K^{X}(\sigma)$ denote the prefix-free Kolmogorov complexity of $\sigma$
relative to $X$. That is, $K^{X}$ is determined by a function universal among
those computable relative to $X$.
###### Definition 1.8.
A sequence $X\in 2^{\omega}$ is _$K$ -trivial_ if and only if there is a
constant $k$ such that for every $\ell$, $K(X\restriction\ell)\leq
K(0^{\ell})+k$, where $0^{\ell}$ is the sequence of $0$’s of length $\ell$.
By early results of Chaitin and Solovay and later results of Nies and others,
there are a variety of equivalents to $K$-triviality and a variety of
properties of the $K$-trivial sets. For example, $X$ is $K$-trivial if and and
only if for every sequence $R$, $R$ is 1-random for $\lambda$ if and only if
$R$ is 1-random for $\lambda$ relative to $X$.
In the next section, we will apply the following.
###### Theorem 1.9 (Nies Nie (09), strengthening Chaitin Cha (76)).
If $X$ is $K$-trivial, then there is a computably enumerable and $K$-trivial
set which computes $X$.
The following theorem follows from the work of Nies and others Nie (09). Some
versions of this property have been used by Kučera extensively, e.g. in Kuč
(85).
###### Theorem 1.10.
Suppose $X$ is $K$-trivial and $\\{U_{e}^{X}:e\in\omega\\}$ a uniformly
$\Sigma^{0,X}_{1}$ family of sets. Then, there is a computable function $g$
and a $\Sigma^{0}_{1}$ set $V$ of measure less than 1 such for every $e$, if
$\lambda(U_{e}^{Z})<2^{-g(e)}$ for every oracle $Z$, then $U_{e}^{X}\subseteq
V$.
###### Proof.
(George Barmpalias) Let $\big{(}(E_{i}^{e})\big{)}_{e\in\mathbb{N}}$ be a
uniform sequence of all oracle Martin-Löf tests. A standard construction of a
universal oracle Martin-Löf test $(T_{i})$ (e.g. see Nie (09)) gives a
recursive function $f$ such that $\forall Z\subseteq\omega\
(E_{f(i,e)}^{e,Z}\subseteq T_{i}^{Z})$ for all $e,i\in\mathbb{N}$. Let
$T:=T_{2}$ and $f(e):=f(2,e)$ for all $e\in\mathbb{N}$, so that
$\mu(T^{Y})\leq 2^{-2}$ for all $Y\in 2^{\omega}$ and $E_{f(e)}^{e}\subseteq
T$ for all $e\in\mathbb{N}$. In KH (07) it was shown that $X$ is $K$-trivial
iff for some member $T$ of a universal oracle Martin-Löf test, there is a
$\Sigma^{0}_{1}$ class $V$ with $T^{X}\subseteq V$ and $\mu(V)<1$.
Now given a uniform enumeration $(U_{e})$ of oracle $\Sigma^{0}_{1}$ classes
we have the following property of $T$:
> There is a recursive function $g$ such that for each $e$,
> either $\exists Z\subseteq\omega\ (\mu(U_{e}^{Z})\geq 2^{-g(e)-1})$, or
> $\forall Z\subseteq\omega\ (U_{e}^{Z}\subseteq T^{Z})$.
To see why this is true, note that every $U_{e}$ can be effectively mapped to
the oracle Martin-Löf test $(M_{i})$ where $M_{i}^{Z}=U_{e}^{Z}[s_{i}]$ and
$s_{i}$ is the largest stage such that $\mu(U_{e}^{Z}[s_{i}])<2^{-i-1}$ (which
could be infinity). Effectively in $e$ we can get an index $n$ of $(M_{i})$.
It follows that if $\mu(U^{Z}_{e})<2^{-f(n)-1}$ for all $Z$, then
$U_{e}^{X}=M_{f(n)}^{X}=E^{n,X}_{f(n)}\subseteq T^{X}\subseteq V$. So
$g(e)=f(n)+1$ is as wanted. ∎
### 1.3. $X$ is $K$-trivial implies $X\in\operatorname{NCR}_{1}$
Intuitively, $X\in\operatorname{NCR}_{1}$ asserts that $X$ is not effectively
random relative to any continuous measure and $X$ is $K$-trivial asserts that
relativizing to $X$ does change the evaluation of randomness relative to the
uniform distribution. In the next section, we connect the two notions by
showing that if $X$ is $K$-trivial then $X\in\operatorname{NCR}_{1}$.
## 2\. The Main Theorem
###### Theorem 2.1.
Every $K$-trivial set belongs to $\operatorname{NCR}_{1}$.
###### Proof.
Let $Y$ be $K$-trivial and let $\mu$ be a continuous measure with
representation $m$; we want to show $Y$ is not $\mu$-random. By Theorem 1.9,
let $X$ be a computably enumerable $K$-trivial sequence that computes $Y$. Let
$f$ be the iterated convergence function as defined above for the computable
approximation to $Y$ given by approximating $X$’s computation of $Y$. Since
$X$ is computably enumerable, $X$ can compute the convergence function for its
own enumeration and hence $f$ is computable from $X$.
Let $s_{m}$ be the granularity function for $\mu$ as represented by $m$. By
Theorem 1.7, $f$ eventually dominates $s_{m}$. By changing finitely many
values of $f$, we may assume that $f$ dominates $s_{m}$ everywhere. So, we
have that for every $n$
$\mu([Y\mathop{\upharpoonright}f(n)])\leq 2^{-n}.$
Further, we may assume that $f$ can be obtained as the limit of a computable
function $f(n,s)$ such that for all $s$, $f(n-1,s)\leq f(n,s)\leq f(n,s+1)$.
We will build an $m$-test $\\{S_{i}:i\in\omega\\}$ which is Solovay-correct
for $\mu$ and which $Y$ does not pass, thereby concluding that $Y$ is not
$\mu$-random. That is, we plan to build $\\{S_{i}:i\in\omega\\}$ to be a
uniformly $\Sigma^{0,m}_{1}$ sequence of sets such that
$\sum_{i\in\omega}\mu(S_{i})$ is bounded and such that there are co-finitely
$i$ for which $Y\in[S_{i}]$. Our construction will not be uniform.
$X$’s $K$-triviality is exploited in the form of Theorem 1.10. Let $V$ and $g$
be given by Theorem 1.10 where $\\{U_{e}^{X}:e\in\omega\\}$ is a listing of
all $\Sigma^{0,X}_{1}$ sets. We will build an oracle $\Sigma^{0}_{1}$ class
$U$ along the construction. We use the recursion theorem to assume that in
advance we know an index $e$ such that $U=U_{e}$. During the construction we
will make sure that for every oracle $Z$, $\lambda(U^{Z})<2^{-g(e)}$. Theorem
1.10 then implies that $U^{X}\subseteq V$ where $V$ is a $\Sigma^{0}_{1}$
class of measure less than 1. To simplify our notation, let $a$ denote $g(e)$.
Furthermore, assume $a$ is large enough so that $\lambda(V)+2^{-a}<1$.
We use the approximation to $X$ as a computably enumerable set to enumerate
approximations to initial segments of $Y$ into the sets $S_{i}$; we rely on
the $K$-triviality of $X$ to keep the total $\mu$-measure of the $S_{i}$’s
bounded.
For each $n>a$ we have a requirement $R_{n}$ whose task is to enumerate
$Y\mathop{\upharpoonright}f(n)$ into $S_{n}$. Let
$y_{n,s}=Y_{s}\mathop{\upharpoonright}f(n,s)$ the stage $s$ approximation to
$Y\mathop{\upharpoonright}f(n)$. Let $x_{n,s}$ be the initial segment of
$X_{s}$ necessary to compute $y_{n,s}$ and $f(n,s)$. So, if $y_{n,s+1}\neq
y_{n,s}$, it is because $x_{n,s+1}\neq x_{n,s}$. In this case, $x_{n,s+1}$ is
not only different than $x_{n,s}$, but also incomparable. At stage $s$,
$R_{n}$ would like to enumerate $y_{n,s}$ into $S_{n}$, but before doing that
it will ask for confirmation using the fact that $U^{X}\subseteq V$. Since we
are constrained to keep $\lambda(U^{X})$ less than or equal to $2^{-a}$, we
will restrict $R_{n}$ to enumerate at most $2^{-n}$ measure into $U^{X}$. The
reason why we need a bit of security before enumerating a string in $S_{n}$ is
that we have to ensure that $\sum_{i}\mu(S_{i})$ is bounded. For this purpose,
we will only enumerate mass into $S_{n}$ when we see an equivalent mass going
into $V$.
Action of requirement $R_{n}$:
1. (1)
The first time after $R_{n}$ is initialized, $R_{n}$ chooses a clopen subset
of $2^{\omega}$, $\sigma_{n}$, of $m$-measure $2^{-n}$, that is disjoint form
$V_{s}$ and $U_{s}^{X_{s}}$. Note that since $V$ and $U^{X_{s}}$ have measure
less than $\lambda(V)+2^{-a}<1$, we can always find such a clopen set.
Furthermore we can chose $\sigma_{n}$ to be different from the $\sigma_{i}$
chosen by other requirements $R_{i}$, $i>a$. We note the value of $\sigma_{n}$
might change if $R_{n}$ is initialized.
2. (2)
To confirm $x_{n,s}$, requirement $R_{n}$ enumerates $\sigma_{n}$ into
$U^{x_{n,s}}$. Requirement $R_{n}$ will not be allowed to enumerate anything
else into $U^{X_{s}}$ unless $X_{s}$ changes below $x_{n,s}$. This way $R_{n}$
is always responsible for at most $2^{-n}$ measure enumerated in $U^{X_{s}}$.
3. (3)
Then, we wait until a stage $t>s$ such that
1. (a)
either $x_{n,s}\not\subseteq x_{n,t}$ (as strings),
2. (b)
or $\sigma_{n}\subseteq V_{t}$.
Observe that if $x_{n,s}$ is actually an initial segment of $X$, then we will
have $\sigma_{n}\subseteq U^{X}\subseteq V$. So, we will eventually find such
a stage $t$.
* •
In Case 3(a), we start over with $R_{n}$. Note that in this case $\sigma_{n}$
has come out of $U^{X_{t}}$, and hence $R_{n}$ is responsible for no measure
inside $U^{X_{t}}$ at stage $t$.
* •
In Case 3(b), if $\mu([y_{n,t}])\leq 2^{-n}$, enumerate $y_{n,t}$ into
$S_{n}$. (Recall that we are allowed to use the representation of $\mu$ as an
oracle when enumerating $S_{n}$.)
Since we only enumerate $y_{n,t}$ of $\mu$-measure less than $2^{-n}$ when
$\sigma_{n}$ is enumerated in $V$, we have that
$\sum_{i}\mu(S_{i})\leq\lambda(V)<1.$
It is not hard to check that
$\lambda(U^{X})\leq\sum_{n=a+1}^{\infty}2^{-n}=2^{-a}$, so we actually have
that $U^{X}\subseteq V$. Also notice that once $x_{n,s}$ is a initial segment
of $X$, we will eventually enumerate $\sigma_{n}$ into $V$ and an initial
segment of $Y$ into $S_{n}$. ∎
## References
* Cha [76] Gregory J. Chaitin. Information-theoretic characterizations of recursive infinite strings. Theoret. Comput. Sci., 2(1):45–48, 1976.
* KH [07] Bjørn Kjos-Hanssen. Low for random reals and positive-measure domination. Proc. Amer. Math. Soc., 135(11):3703–3709 (electronic), 2007.
* Kuč [85] Antonín Kučera. Measure, $\Pi^{0}_{1}$-classes and complete extensions of ${\rm PA}$. In Recursion theory week (Oberwolfach, 1984), volume 1141 of Lecture Notes in Math., pages 245–259. Springer, Berlin, 1985.
* Mon [05] Antonio Montalbán. Beyond the Arithmetic. PhD thesis, Cornell University, 2005.
* Nie [09] André Nies. Computability and randomness. to appear, 2009.
* RS [07] Jan Reimann and Theodore A. Slaman. Probability measures and effective randomness. preprint, 2007.
* RS [08] Jan Reimann and Theodore A. Slaman. Measures and their random reals. preprint, 2008.
* [8] Jan Reimann and Theodore A. Slaman. The structure of the never continuously random sequences. in preparation, no date.
|
arxiv-papers
| 2008-12-09T01:04:51 |
2024-09-04T02:48:59.310367
|
{
"license": "Public Domain",
"authors": "Antonio Montalban, Theodore A. Slaman",
"submitter": "Theodore A. Slaman",
"url": "https://arxiv.org/abs/0812.1600"
}
|
0812.1710
|
# LABORATORI NAZIONALI DI FRASCATI SIS-Pubblicazioni
LNF–08/27(P) November 29, 2008
SENSITIVITY AND ENVIRONMENTAL RESPONSE OF THE CMS RPC GAS GAIN MONITORING
SYSTEM
L. Benussi1, S. Bianco1, S. Colafranceschi${}^{1},2,3$, F. L. Fabbri1, M.
Giardoni1
B. Ortenzi1, A. Paolozzi,${}^{1},2$ L. Passamonti1, D. Pierluigi1
B. Ponzio1, A. Russo1, A. Colaleo4, F. Loddo4, M. Maggi4
A. Ranieri4, M. Abbrescia${}^{4},5$, G. Iaselli${}^{4},5$, B.
Marangelli${}^{4},5$, S. Natali${}^{4},5$
S. Nuzzo${}^{4},5$, G.Pugliese${}^{4},5$, F. Romano${}^{4},5$, G.
Roselli${}^{4},5$, R. Trentadue${}^{4},5$
S. Tupputi${}^{4},5$, R. Guida3, G. Polese${}^{3},6$, A. Sharma3, A.
Cimmino${}^{7},8$
D. Lomidze8, D. Paolucci8, P. Piccolo8, P. Baesso9, M. Necchi9
D. Pagano9, S. P. Ratti9, P. Vitulo9, C. Viviani9
1) INFN Laboratori Nazionali di Frascati, Via E. Fermi 40, I-00044 Frascati,
Italy.
2) Università degli Studi di Roma ”La Sapienza”, Piazzale A. Moro.
3) CERN CH-1211 Genéve 23 F-01631 Switzerland.
4) INFN Sezione di Bari, Via Amendola, 173I-70126 Bari, Italy.
5) Dipartimento Interateneo di Fisica, Via Amendola, 173I-70126 Bari, Italy.
6) Lappeenranta University of Technology, P.O. Box 20 FI-538 1 Lappeenranta,
Finland.
7) INFN Sezione di Napoli, Complesso Universitario di Monte Sant’Angelo,
edificio 6, 80126 Napoli, Italy.
8) Università di Napoli Federico II, Complesso Universitario di Monte
Sant’Angelo, edificio 6, 80126 Napoli, Italy.
9) INFN Sezione di Pavia, Via Bassi 6, 27100 Pavia, Italy and Università degli
studi di Pavia, Via Bassi 6, 27100 Pavia, Italy.
Results from the gas gain monitoring (GGM) system for the muon detector using
RPC in the CMS experiment at the LHC is presented. The system is designed to
provide fast and accurate determination of any shift in the working point of
the chambers due to gas mixture changes.
PACS: 07.77.Ka; 95.55.Vj; 29.40.Cs
Presented by S. Colafranceschi at the IEEE 08 - 23 October 2008, Dresden,
Germany
## 1 Introduction
Resistive Plate Chambers (RPC) detectors are widely used in HEP experiments
for muon detection and triggering at high-energy, high-luminosity hadron
colliders, in astroparticle physics experiments for the detection of extended
air showers, as well as in medical and imaging applications. At the LHC, muon
systems of the CMS experiment rely on Drift Tubes (DT), Cathode Strip Chambers
(CSC) and RPCs for their muon trigger system, with a total gas volume of about
50 m3. Utmost attention has to be paid to the possible presence of gas
contaminants which degrade the chamber performance. The gas gain monitoring
(GGM) system monitors the gas quality online and is based on small RPC
detectors. The working point - gain and efficiency - is continuously monitored
along with environmental parameters, such as temperature, pressure and
humidity, which are important for the operation of the muon detector system.
Design parameters, construction, prototyping and preliminary commissioning
results of the CMS RPC Gas Gain Monitoring (GGM) system have been presented
previously [1],[7]. In this paper, results on the response of the GGM
detectors to environmental changes are presented.
The CMS RPCs are bakelite-based double-gap RPC with strip readout (for
construction details see [2] and reference therein) operated with 96.2% C2H2F4
\- 3.5% Iso-C4H10 \- 0.3% SF6 gas mixture humidified at about 40%. The large
volume of the whole CMS RPC system and the cost of gas used make mandatory the
operation of RPC in a closed-loop gas system (for a complete description see
[3]), in which the gas fluxing the gaps is reused after being purified by a
set of filters[4].
The operation of the CMS RPC system is strictly correlated to the ratio
between the gas mixture components, and to the presence of pollution due to
contaminants that can be be produced inside the gaps during discharges (i.e.
HF produced by SF6 or C2H2F4 molecular break-up and further fluorine
recombination), accumulated in the closed-loop or by pollution that can be
present in the gas piping system (tubes, valves, filters, bubblers, etc.) and
flushed into the gaps by the gas flow. The monitoring of the presence of these
contaminants, as well as the gas mixture stability, is therefore mandatory to
avoid RPC damage and to ensure their correct functionality.
A monitoring system of the RPC working point due to changes of gas composition
and pollution must provide a faster and sensitive response than the CMS RPC
system itself in order to avoid irreversible damage of the whole system. Such
a Gas Gain Monitoring system monitors efficiency and signal charge
continuously by means of a cosmic ray telescope based on RPC detectors. In the
following will be briefly described the final setup of the GGM system, and the
first results obtained during its commissioning at the ISR test area (CERN).
## 2 The Gas Gain Monitoring System
The GGM system is composed by the same type of RPC used in the CMS detector
but of smaller size (2mm Bakelite gaps, 50$\times$50 cm2). Twelve gaps are
arranged in a stack located in the CMS gas area (SGX5 building) in the
surface, close to CMS assembly hall (LHC-P5). The choice to install the system
in the surface instead of underground allows one to profit from maximum cosmic
muon rates. In order to ensure a fast response to working point shifts with a
precision of 1%, $10^{4}$ events are are required, corresponding to about 30
minutes exposure time on surface, to be compared with a 100-fold lesser rate
underground. The trigger is provided by four out of twelve gaps of the stack,
while the remaining eight gaps are used to monitor the working point
stability.
The eight gaps are arranged in three sub-system: one sub-system (two gaps) is
fluxed with the fresh CMS mixture and its output sent to vent. The second sub-
system (three gaps) is fluxed with CMS gas coming from the closed-loop gas
system and extracted before the gas purifiers, while the third sub-system
(three gaps) is operated with CMS gas extracted from the closed-loop extracted
after the gas filters. The basic idea is to compare the operation of the three
sub-systems and, if some changes are observed, to send a warning to the
experiment. In this way, the gas going to and coming from the CMS RPC detector
is monitored by using the two gaps fluxed with the fresh mixture as reference
gaps. This setup will ensure that pressure, temperature and humidity changes
affecting the gaps behavior do cancel out by comparing the response of the
three sub-system operating in the same ambient condition.
The monitoring is performed by measuring the charge distributions of each
chamber. The eight gaps are operated at different high voltages, fixed for
each chamber, in order to monitor the total range of operating modes of the
gaps. The operation mode of the RPC changes as a function of the voltage
applied. A fraction of the eight gaps will work in pure avalanche mode, while
the remaining will be operated in avalanche+streamer mode. Comparison of
signal charge distributions and the ratio of the avalanche to streamer
components of the ADC provides a monitoring of the stability of working point
for changes due to gas mixture variations.
Details on the construction of GGM can be found in [7]. Each chamber of the
GGM system consists of a single gap with double sided pad read-out: two copper
pads are glued on the two opposite external side of the gap. The signal is
read-out by a transformer based circuit A3 (Fig.1). The circuit allows to
algebraically subtract the two signal, which have opposite polarities, and to
obtain an output signal with subtraction of the coherent noise, with an
improvement by about a factor 4 of the signal to noise ratio. The output
signals from circuit A3 are sent to a CAEN V965 ADC [6] for charge analysis.
Figure 1: The electric scheme of the read-out circuit providing the algebraic
sum of the two pad signal (PAD + and PAD -).
A typical ADC distribution of a GGM gap is shown in fig.2 for two different
effective operating voltage, defined as the high voltage set on the HV power
supply corrected for the local atmospheric pressure and temperature. Fig.2 a)
corresponding to HVeff=9.9kV shows a clean avalanche peak well separated from
the pedestal. Fig.2 b) shows the charge distribution at HVeff=10.7kV with two
signal regions corresponding to the avalanche and to avalanche+streamer mode.
Figure 2: Typical ADC charge distributions of one GGM chamber at two operating
voltages. Distribution (a) correspond to HVeff = 9.9kV while distribution (b)
to HVeff=10.7kV. In (b) is clearly visible the streamer peak around 1900 ADC
channels. The events on the left of the vertical line (1450 ADC channels in
this case) are assumed to be pure avalanche events.
Figure 3: Efficiency plot (full dots) of GGM chambers as a function of HVeff.
The efficiency is defined as the ratio between the number of ADC entries above
3$\sigma_{ped}$ and the number of acquired triggers. Open dot plots correspond
to the streamer fraction of the chamber signal as a function of HVeff.
Fig.3 shows the GGMS single gap efficiency (full dots), and the ratio between
the avalanche and the streamer component (open circles), as a function of the
effective high voltage. Each point corresponds to a total of 10000 entries in
the full ADC spectrum. The efficiency is defined as the ratio between the
number of triggers divided by the number of events above 3$\sigma_{ped}$ over
ADC pedestal, where $\sigma_{ped}$ is the pedestal width. The avalanche to
streamer ratio is defined by counting the number of entries in the avalanche
(below the ADC threshold (fig.2 b) and above the pedestal region) and dividing
it by the number of streamer events above the avalanche threshold. Both
efficiency and avalanche plateau are in good agreement with previous results
[5].
In order to determine the sensitivity of GGM gaps to working point shifts, the
avalanche to streamer transition was studied by two methods, the charge method
and the efficiency method. In the charge method, the mean value of the ADC
charge distribution in the whole ADC range is studied as a function of HVeff
(fig.4). Each point corresponds to 10000 events in the whole ADC spectrum. In
the plot three working point regions are identified
1. 1.
inefficiency (HV${}_{eff}<$ 9.7 kV);
2. 2.
avalanche (9.7 kV $<$ HV${}_{eff}<$ 10.6 kV;
3. 3.
avalanche+streamer mode (HV${}_{eff}>$ 10.6 kV).
The best sensitivity to working point shifts is achieved in the
avalanche+streamer region, estimated to be about 25 ADC ch/10 V or 1.2pC/10V.
Figure 4: Avarege avalanche charge of the eight monitor chamber signal as a
function of HVeff. The slope is about 25 ADC ch/10 or 1.2pC/10V. Each point
corresponds to 10000 triggers.
In the efficiency method, the ADC avalanche event yield is studied as a
function of HVeff (5). The avalanche signal increases by increasing the HV
applied to the gap, until it reaches a maximum value after which the streamer
component starts to increase. The 9.0kV-10.0kV shows a sensitivity to work
point changes of approximately 1.3/%/10V.
Figure 5: Streamer and avalanche yields as a function of HVeff. Each point
corresponds to 10000 collected triggers. The solid line has a slope of
approximately 130 events/10 V corresponding to a sensitivity of 1.3%/10V.
## 3 Response of GGM to environmental effects
The workpoint of GGM is affected by environment. However, all environmental
effects cancel out thanks to redundancy of the system. Each environmental
effect not connected with a modification of gas mixture will be cancelled out
by a comparison between different RPC chamber flown with the same gas, which
are affected by the same environmental parameters.
An example of such cancellation is shown in Fig. 6, where the average charge
distribution (black dots) is plotted across a changeover of gas bottles. Data
show a sudden increase in the average charge distribution which may
interpreted as a shift of working point due to changes in gas mixture
composition. By weighing the average charge with a correction factor linearly
depending on atmosferic pressure, however, no significant increase is left in
the distribution of corrected average charge (green dots) which may signal an
anomalous shift due to gas mixture. The cancellation algorithm is applied by
correcting variables withing gaps belonging to the same subdetector. Fig. 7
shows the average charge for two chambers working in different regimes at
different voltages. The average charge of both chambers is completely
correlated, and very well correlated to the atmosferic pressure variations.
Figure 6: Average charge and pressure-corrected charge
Figure 7: Average charge of two chambers at different voltages as influenced
by pressure
## 4 Conclusions
Results from the Gas Gain Monitoring System for the CMS RPC Detector have been
reported on. The purpose of GGM is to monitor any shift of the working point
of the CMS RPC detector. The GGM is being commissioned at CERN and is planned
to start operation by the end of 2008. Preliminary results show good
sensitivity to working point changes. The system redundancy allows for
effectively cancelling out the environmental effects. Further tests are in
progress to determine the sensitivity to gas variations.
## References
* [1] M. Abbrescia et al., “Gas analysis and monitoring systems for the RPC detector of CMS at LHC”, presented by S.Bianco at the IEEE 2006, San Diego (USA), arXiv:physics/0701014.
* [2] R. Adolphi et al. [CMS Collaboration], JINST 3 (2008) S08004
* [3] R. Guida et al., “ The CMS RPC gas system and the Closed Loop recirculation system”, presented at the RPC Conference, Mumbai, 2008.
* [4] G. Saviano et al., “Material, Filter and Gas analysis for the CMS RPC detector in the Closed Loop test setup”, presented at the RPC Conference, Mumbai, 2008 to appear on Journal of Instrumentation
* [5] M. Abbrescia et al., “Cosmic ray tests of double-gap resistive plate chambers for the CMS experiment”, Nucl. Instrum. Meth. A 550, 116 (2005).
* [6] C.A.E.N. Costruzioni Apparecchiature Elettroniche Nucleari S.p.A. Via Vetraia 11 - 55049 Viareggio (Italy).
* [7] L. Benussi et al., “The CMS RPC Gas Gain Monitoring System: an Overview and Preliminary Result”, presented in Mumbai08 to appear on Nuclear Instruments and Methods arXiv:0812.1108 [physics.ins-det].
|
arxiv-papers
| 2008-12-09T14:59:19 |
2024-09-04T02:48:59.319642
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. Benussi (1), S. Bianco (1), S. Colafranceschi (1,2,3), F. L. Fabbri\n (1), M. Giardoni (1), B. Ortenzi (1), A. Paolozzi (1,2), L. Passamonti (1),\n D. Pierluigi (1), B. Ponzio (1), A. Russo (1), A. Colaleo (4), F. Loddo (4),\n M. Maggi (4), A. Ranieri (4), M. Abbrescia (4,5), G. Iaselli (4,5), B.\n Marangelli (4,5), S. Natali (4,5), S. Nuzzo (4,5), G. Pugliese (4,5), F.\n Romano (4,5), G. Roselli (4,5), R. Trentadue (4,5), S. Tupputi (4,5), R.\n Guida (3), G. Polese (3,6), A. Sharma (3), A. Cimmino (7,8), D. Lomidze (8),\n D. Paolucci (8), P. Piccolo (8), P. Baesso (9), M. Necchi (9), D. Pagano (9),\n S. P. Ratti (9), P. Vitulo (9), C. Viviani (9), ((1) INFN Laboratori\n Nazionali di Frascati, (2) Universit\\`a degli Studi di Roma \"La Sapienza\",\n (3) CERN, (4) INFN Sezione di Bari, (5) Dipartimento Interateneo di Fisica di\n Bari, (6) Lappeenranta University of Technology, (7) INFN Sezione di Napoli,\n (8) Universit\\`a di Napoli Federico II, (9) INFN Sezione di Pavia.)",
"submitter": "Stefano Colafranceschi",
"url": "https://arxiv.org/abs/0812.1710"
}
|
0812.1790
|
# M-brane singularity formation
J.Eggers1 and J.Hoppe2 1School of Mathematics, University of Bristol,
University Walk, Bristol BS8 1TW, United Kingdom
2Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm,
Sweden
###### Abstract
We derive self-similar string solutions in a graph representation, near the
point of singularity formation, which can be shown to extend to point-like
singularities on M-branes, as well as to the radially symmetric case.
## I Introduction
For more than 40 years, Barbashov and Chernikov (1966, 1967, 1968) (see also
Whitham (1974)), various ways of solving the non-linear equation
$\ddot{z}(1+z^{\prime
2})-z^{\prime\prime}(1-\dot{z}^{2})=2\dot{z}z^{\prime}\dot{z}^{\prime}$ (1)
are known. Recent work on higher dimensional time-like zero mean curvature
hypersurfaces includes Christodoulou (2007); Milbredt (2008); Hoppe (2008);
Bellettini et al. (2008) Here we show that (1) can develop singularities in
finite time, starting from finite initial data. The structure of these
singularities is described by the self-similar ansatz
$z(t,x)=z_{0}-\hat{t}+\hat{t}^{\alpha}h\left(\frac{x}{\hat{t}^{\beta}}\right)+\dots$
(2)
where $\hat{t}:=t_{0}-t\rightarrow 0$ (the dots are indicating lower order
terms). Inserting (2) into (1) one finds the above ansatz to be consistent
provided $\beta=(1+\alpha)/2>1$, and
$h^{\prime\prime}\left(2\alpha
h-\frac{(\alpha+1)^{2}}{4}\xi^{2}\right)=(\alpha-1)\left[h^{\prime 2}+\alpha
h-\frac{3}{4}(\alpha+1)\xi h^{\prime}\right]\;,$ (3)
for (2) to be an asymptotic solution of (1). For consistency with a finite
outer solution of (1), the profile $h$ must satisfy
$h(\xi)\propto
A_{\pm}\xi^{\frac{2\alpha}{\alpha+1}}\quad\mbox{for}\quad\xi\rightarrow\pm\infty$
(4)
(for a general discussion of matching self-similar solutions to the exterior
see Eggers and Fontelos (2008)).
The ansatz (2) is formally consistent for a continuum of similarity exponents
$\alpha\geq 1$ and for any solution of the similarity equation (3). However,
by considering the regularity of solutions of (3) in the origin $\xi=0$ the
similarity exponent must be one of the sequence
$\alpha=\alpha_{n}=\frac{n+1}{n},\quad n\in\mathbb{N},$ (5)
certainly if $h(0)=0=h^{\prime}(0)$, and presumably in general (i.e. all
relevant solutions of (3)). Of this infinite sequence, we believe that only
$\alpha=2$ is realized for generic initial data; indeed, in this case
$\xi=\zeta+c\zeta^{3}/3,\quad h(\xi)=\zeta^{2}/2+c\zeta^{4}/4,$ (6)
which we will deduce from a parametric string solution corresponding to (1).
The importance of the similarity solution (2) lies in the fact that it can be
generalized to arbitrary dimensions, in particular to membranes. We find that
the same type of singular solution is observed in any dimension, even having
the same spatial structure (6).
## II The similarity equation
A way of satisfying (3) is to demand
$L^{2}:=h^{\prime 2}+2\alpha h-(\alpha+1)\xi h^{\prime}=0.$ (7)
(differentiating e.g. $(1+\alpha)\xi=h^{\prime}+2\alpha h/h^{\prime}$ one can
eliminate $h^{\prime\prime}$, reducing (3) to an identity, as long as
$h^{\prime}\neq 1$).
The transformation
$h(\xi)=\xi^{2}g(\xi)=\xi^{2}\left(\frac{(1+\alpha)^{2}}{8\alpha}-\frac{v^{2}}{2\alpha}\right)$
(8)
yields
$-\frac{d\xi}{\xi}=\frac{vdv}{v^{2}\pm\alpha
v+(\alpha^{2}-1)/4}=\frac{1}{2}\left(\frac{\alpha+1}{v\mp\frac{\alpha+1}{2}}-\frac{\alpha-1}{v\mp\frac{\alpha-1}{2}}\right)dv,$
(9)
i.e. (choosing the upper sign)
$\frac{\left|v-(\alpha+1)/2\right|^{\alpha+1}}{\left|v-(\alpha-1)/2\right|^{\alpha-1}}=\frac{E}{\xi^{2}}.$
(10)
This yields solutions $v\in[(\alpha-1)/2,(\alpha+1)/2)$,
$\displaystyle
v\approx\frac{\alpha-1}{2}+\left(\frac{\xi^{2}}{E}\right)^{\frac{1}{\alpha-1}}+\dots\quad\mbox{as}\quad\xi\rightarrow
0$ (11) $\displaystyle
v\approx\frac{\alpha+1}{2}-\left(\frac{\xi^{2}}{E}\right)^{\frac{1}{\alpha+1}}+\dots\quad\mbox{as}\quad\xi\rightarrow\pm\infty,$
i.e.
$\displaystyle h(\xi)\geq 0,h(0)=0,$ (12) $\displaystyle
h(\xi)\propto\xi^{2}/2\quad\mbox{as}\quad\xi\rightarrow 0$ $\displaystyle
h(\xi)\propto\frac{1+\alpha}{2\alpha}\xi^{\frac{2\alpha}{1+\alpha}}\quad\mbox{as}\quad\xi\rightarrow\pm\infty.$
Note that these solutions are consistent with the growth conditions (4).
To solve the second order equation (3) we note that
$\tilde{h}(\xi):=c\;h(\xi/\sqrt{c})$ solves (3), if $h$ does, and that
$\frac{h^{\prime}}{\xi}-\frac{2h}{\xi^{2}}=\frac{1}{\alpha}f\left(\sqrt{\frac{(\alpha+1)^{2}}{4}-2\alpha\frac{h(\xi)}{\xi^{2}}}\right)\quad\equiv\left(\frac{1}{\alpha}f(v)\right)$
(13)
reduces (3) to
$-\left(v^{2}-\frac{(\alpha+1)^{2}}{4}\right)\left(v^{2}-\frac{(\alpha-1)^{2}}{4}\right)=f\left(\alpha
vf^{\prime}-(\alpha-1)f-(\alpha+2)v^{2}+(\alpha^{2}-1)(\alpha-2)/4\right),$
(14)
and
$\frac{d\xi}{\xi}=-\frac{vdv}{f(v)}=\frac{\alpha dg}{f}.$ (15)
The growth condition (4) implies that $h$ grows less than quadratically at
infinity. Thus we deduce from (13) that $f$ vanishes at $(\alpha+1)/2$.
Furthermore, from a direct calculation using the growth exponent (4) we find
the first derivative, yielding the initial conditions
$f\left(\frac{\alpha+1}{2}\right)=0,\quad
f^{\prime}\left(\frac{\alpha+1}{2}\right)=1.$ (16)
Using (16), (14) yields a polynomial solution
$f(v)=\left(v-\frac{(\alpha+1)}{2}\right)\left(v-\frac{(\alpha-1)}{2}\right)=v^{2}-\alpha
v+\frac{\alpha^{2}-1}{4},$ (17)
(i.e.) (10), which corresponds to the first order equation (7), but also an
infinity of other solutions (a Taylor expansion around
$v_{\infty}=(\alpha+1)/2$ shows that (14) leaves
$f^{\prime\prime}((\alpha+1)/2)$ undetermined, when (16) holds). We note that
(14) also has the solution $f_{-}(v)=f(-v)$, and for the special case
$\alpha=2$ another pair of polynomial solutions,
$\tilde{f}(v)=\left(v+3/2\right)\left(v-1/2\right)=v^{2}+v-3/4,$ (18)
and $\tilde{f}_{-}(v)=\tilde{f}(-v)$. While the asymptotic behavior following
from (18) is in disagreement with (4), integration methods similar to those
used by Abel Abel (1881) perhaps permit a complete reduction of (14) to
quadratures.
In any case, (14) can be simplified in various ways. For $\alpha=2$, e.g. it
reduces to
$yy^{\prime}=y-\frac{1}{4v^{5/2}}(v^{2}-9/4)(v^{2}-1/4)$ (19)
via
$f(v)=\sqrt{v}y(4v^{3/2}/3).$ (20)
The solution (17), which is consistent with the growth condition (4), is
equivalent to the solution (10) of (7) given before. If one investigates the
behavior of the solution in the origin (either using (10) directly or by
series expansion of (7)), one finds that only for $\alpha=\alpha_{n}$ (cf.
(5)) a smooth solution is possible. Thus the first consistent solution is
found for $n=1$ or $\alpha=2$. Higher order solutions $n=2,3,\dots$ are also
possible in principle. They have the property that apart from
$f^{\prime\prime}(0)$, the first non-vanishing derivative is $f^{(2n+2)}(0)$.
However, we believe that they correspond to non-generic initial conditions,
whose derivatives have corresponding properties of vanishing up to a certain
order. To demonstrate this point, one would have to perform a stability
analysis of the corresponding solution Eggers and Fontelos (2008). In the
string picture discussed below this can be shown explicitly, as higher order
solutions correspond to non-generic initial data.
## III Higher dimensions
The solutions of (7), found to govern singularities of (1), also apply to
higher dimensions. The reason is that the left hand side of (7) is the leading
order term of
${\cal L}^{2}=1-\dot{z}^{2}+z^{\prime 2}.$ (21)
In other words, the asymptotic singular solutions discussed above have ${\cal
L}^{2}=0+$ lower order. In fact, differentiating (21) with respect to $t$ and
$x$ one easily shows that ${\cal L}^{2}=0$ provides solutions of (1). In
higher dimensions, differentiating $1-z^{\alpha}z_{\alpha}=0$ gives
$z^{\alpha}z_{\alpha\beta}=0$, and hence
$(1-z_{\alpha}z^{\alpha})\square z+z^{\beta}z^{\alpha}z_{\alpha\beta}=0.$ (22)
Thus solutions of ${\cal L}^{2}=0$ also solve the M-brane equation (22) in
arbitrary dimensions.
For the special case of radially symmetric membranes:
$\ddot{z}(1+z^{\prime
2})-z^{\prime\prime}(1-\dot{z}^{2})-2\dot{z}z^{\prime}\dot{z}^{\prime}=\frac{z^{\prime}}{r}\left(1-\dot{z}^{2}+z^{\prime
2}\right)\equiv\frac{z^{\prime}}{r}{\cal L}^{2}.$ (23)
Insert the radial version of the ansatz (2),
$z(t,r)=-\hat{t}+\hat{t}^{\alpha}h\left(\frac{r-r_{0}}{\hat{t}^{\beta}}\right)+\dots,$
(24)
into (23). If $r_{0}\neq 0$, the entire right hand side of (23) is of lower
order in $\hat{t}$, and the similarity equation (3) remains the same.
Geometrically, this corresponds to the singularity forming along a circular
ridge of radius $r_{0}$.
If on the other hand $r_{0}=0$, i.e. the singularity forms along the axis, the
right hand side is of the same order, and the similarity equation becomes
$h^{\prime\prime}\left(2\alpha
h-\frac{(\alpha+1)^{2}}{4}\xi^{2}\right)+(1-\alpha)\left[h^{\prime 2}+\alpha
h-\frac{3}{4}(\alpha+1)\xi
h^{\prime}\right]=-\frac{h^{\prime}}{\xi}\left[h^{\prime 2}+2\alpha
h-(\alpha+1)\xi h^{\prime}\right].$ (25)
This equation can in principle have solutions different from (3). For
solutions of (7), however, the expression in angular brackets in (25)
vanishes, hence solutions of (7) also solve (25). Thus (24),(6) describe a
point-like singularity on a membrane. These observations straightforwardly
generalize to higher M-branes, $M>2$.
## IV Parametric string solution
Let us now compare our findings with the solution of closed bosonic string
motions given by equation (50) of Hoppe (1995). (note that the definitions of
$f$ and $g$ are changed by $\pi/4$, and that the constant $\lambda$ is chosen
to be 1):
$\dot{{\bf x}}(t,\varphi)=\sin(f-g)\left(\begin{array}[]{c}-\sin(f+g)\\\
\cos(f+g)\end{array}\right)$ (26) ${\bf
x}^{\prime}(t,\varphi)=\cos(f-g)\left(\begin{array}[]{c}\cos(f+g)\\\
\sin(f+g)\end{array}\right),$ (27)
where $f=f(\varphi+t)$ and $g=g(\varphi-t)$. From (27) one finds the curvature
$k(t,\varphi)=\frac{f^{\prime}+g^{\prime}}{\cos(f-g)}.$ (28)
The hodograph transformation
$\displaystyle(t,\varphi)\rightarrow t=x^{0},\quad x=x^{1}(t,\varphi),$ (29)
$\displaystyle x^{2}(t,\varphi)=z(t,x^{1}(t,\varphi)),$
implying $\dot{z}=\dot{y}-\dot{x}y^{\prime}/x^{\prime}$,
$z^{\prime}=y^{\prime}/x^{\prime}$, $(\partial\phi/\partial
x^{0}=-\dot{x}/x^{\prime},(\partial\phi/\partial x^{1}=1/x^{\prime})$ permits
to go between the parametric string picture (26)-(28) and the graph
description (1). In particular,
$1-\dot{z}^{2}+z^{\prime 2}=\left(\frac{\cos(f-g)}{\cos(f+g)}\right)^{2}$ (30)
is manifestly non-negative in the parametric string-description, while for
solutions of (1) one has to demand it explicitly - leading e.g. to the
exclusion of solutions with $h^{\prime}(0)=0$, $h(0)<0$.
Let us give an explicit example of $\mathbb{M}_{2}\subset\mathbb{R}^{1,2}$
being at $t=0$ a regular graph, while for $t=1$ a curvature singularity has
developed. Let $\mathbb{M}_{2}$ be described by ${\bf x}(\varphi,t)$, as
defined by (26),(27), with $\varphi\in\mathbb{R}$, $t\geq 0$. Let
$\displaystyle f(w)=\left\\{\begin{array}[]{ll}\arctan w&\mbox{for}\quad
w\geq\epsilon\\\ \chi_{\epsilon}(w)\arctan w&\mbox{for}\quad 0\leq
w\leq\epsilon<0\\\ 0&\mbox{for}\quad w\leq 0\end{array}\right.,$ (34)
where $\chi_{\epsilon}(w\geq\epsilon)=1$, $\chi_{\epsilon}(w\leq 0)=0$, and
$\chi_{\epsilon}(0<w<0)$ such that $f^{\prime}(w)\geq 0$. We also assume that
$g(w)=-f(-w)$. A simple calculation then shows that for
$\varphi\in[-t+\epsilon,t-\epsilon]$ one obtains (${\bf x}_{0}(t=0,u=0)=0$)
$\displaystyle x(\varphi,t)=-\varphi+\arctan(\varphi+t)+\arctan(\varphi-t)$
(35) $\displaystyle
y(\varphi,t)=\ln\sqrt{\left(1+(\varphi+t)^{2}\right)\left(1+(\varphi-t)^{2}\right)}$
$\displaystyle
k(\varphi,t)=\frac{2}{\sqrt{\left(1+(\varphi+t)^{2}\right)\left(1+(\varphi-t)^{2}\right)}}\frac{\varphi^{2}+1+t^{2}}{\varphi^{2}+1-t^{2}}$
Note that for $t>1$ this is no longer a graph.
Figure 1: The formation of a swallowtail, as described by (40). Shown is a
smooth minimum, ($\hat{t}>0$), a minimum with a 4/3 singularity ($\hat{t}=0$),
and a swallowtail or double cusp ($\hat{t}<0$).
The example (34) underlies a general structure that can be uncovered by a
local expansion of the functions $f$ and $g$ around the singularity. Namely,
as seen from (28), the singularity occurs when $f-g$ is a multiple of $\pi/2$;
for simplicity, we also assume that $g(w)=-f(-w)$ as before.
Then a local Taylor expansion yields
$f(\zeta)=\pi/4+a(\zeta-\zeta_{0})-b(\zeta-\zeta_{0})^{2}+O(\zeta-\zeta_{0})^{3},$
(36)
so together with the symmetry requirement we find
$f-g=\pi/2+2a(t-\zeta_{0})-2b\varphi^{2}+(t-\zeta_{0})^{2},$ (37)
where we neglected higher-order terms in the expansion, which will turn out to
be irrelevant for the singularity formation.
From this expression it is clear that $\zeta_{0}$ has to be identified with
the singular time $t_{0}$ and $a>0$ for the solution to be regular for
$t<t_{0}$. Similarly, one must have $b>0$ (otherwise $f-g$ would be $\pi/2$ at
an earlier time), and the singularity occurs at $\varphi=0$. Thus to leading
order we have
$f-g\approx\pi/2-2a\hat{t}-2b\varphi^{2},\quad f+g\approx 2a\varphi,$ (38)
from which we get
$x^{\prime}=2a\hat{t}+2b\varphi^{2},\quad
y^{\prime}=4a^{2}\hat{t}\varphi+4ab\varphi^{3}.$ (39)
Integrating this expression, using the integrability condition (27), gives
$x=\hat{t}\varphi+c\varphi^{3}/3,\quad
y=-\hat{t}+\hat{t}\varphi^{2}/2+c\varphi^{4}/4,$ (40)
where we have used a rescaling of the parameter $\varphi$.
The curve described by (40) is shown in Fig. 1, but disregarding the spatial
translation of $z$ by the term $-\hat{t}$. In catastrophe theory, this is
known as the swallowtail Arnold (1984). For $\hat{t}>0$ the curve is smooth,
while for $\hat{t}=0$ a rather mild singularity develops; at the origin,
$y\propto x^{4/3}$. After the singularity ($\hat{t}<0$) the curve self-
intersects. The solution (40) leads directly to the similarity form (2), if
one notices that $\varphi$ is of order $\hat{t}^{1/2}$ for the terms in (40)
to balance. Thus, using the notation of (2), and putting
$\zeta=\varphi/\hat{t}^{1/2}$, one finds $\alpha=2,\beta=3/2$ for the
exponents, and (6) for the similarity function. The crucial point is that
although (6) came out of an expansion, all higher order terms are subdominant
in the limit $\hat{t}\rightarrow 0$. Thus (6) is in fact an exact solution of
(3) with $\alpha=2$. Moreover, it is even a solution of (7), and precisely of
the form (10).
In the case of non-generic initial conditions other solutions are possible.
For example, instead of (36)
$f(\zeta)=\pi/4+a(\zeta-\zeta_{0})-b(\zeta-\zeta_{0})^{2n},$ (41)
where $n\in\mathbb{N}$ but $n>1$. Only even powers $2n$ are allowed, otherwise
the singularity occurs for all $\varphi$ at the same time, i.e. it is no
longer point-like. If the leading order term is not linear but itself of
higher order, the resulting similarity profile becomes singular at the origin,
cf. 51. Repeating the above calculation along the same lines, we find
$x^{\prime}=2a\hat{t}+2b\varphi^{2n},\quad
y^{\prime}=4a^{2}\hat{t}\varphi+4ab\varphi^{2n+1},$ (42)
which is equivalent to the symmetric shape function
$\displaystyle\xi=\zeta+2d(n+1)\zeta^{2n+1}/(2n+1)$ (43) $\displaystyle
h=\zeta^{2}/2+d\zeta^{2n+2}.$
The corresponding similarity exponent is $\alpha=\alpha_{n}$, as given by (5).
We thus retrieve the exact same solutions identified by our previous analysis,
based on a similarity description.
*
## Appendix A Additional solutions of the similarity equation
It is possible to construct many more solutions to the similarity equation
(7), which are all defined on the real line, but which we reject since they
either contradict (4) or are not smooth. The simplest case is
$h(\xi)=\frac{\xi^{2}}{2},$ (44)
which is a solution for any $\alpha$, but evidently does not satisfy the
matching condition (4).
Recall that for $\alpha=\alpha_{n}$, (10) furnishes smooth solutions on the
real line. On the other hand, while for $\alpha=3$ the second derivative of
the resulting solution is well-defined, the third derivative is discontinuous.
Namely, for $E=4$ (e.g.) one finds that for $\xi>0$,
$\displaystyle h(\xi)=-\frac{2}{3}\left(\xi+2(1+\xi)-2(1+\xi)^{3/2}\right)$
(45) $\displaystyle h^{\prime}(\xi)=2\left(\sqrt{1+\xi}-1\right)>0$
$\displaystyle h^{\prime\prime}(\xi)=1/\sqrt{1+\xi},$
so that
$\displaystyle
h^{\prime\prime\prime}(\xi)=\left\\{\begin{array}[]{ll}-(1+\xi)^{-3/2}/2&\mbox{for}\quad\xi>0\\\
(1+\xi)^{-3/2}/2&\mbox{for}\quad\xi<0.\\\ \end{array}\right.$ (48)
Other solutions whose scaling exponent is not from the set (5), but which have
well-defined second derivatives, can be found from the parametric string
solution as described in section IV. If the expansion of $f$ does not start
with a linear term as in (36), but at higher order, e.g.
$f(\zeta)=\pi/4+(\zeta-\zeta_{0})^{3}/2-b(\zeta-\zeta_{0})^{4},$ (49)
one finds instead of (39)
$x^{\prime}=3\hat{t}\varphi^{2}+2b\varphi^{4},\quad
y^{\prime}=3\hat{t}\varphi^{5}+2b\varphi^{7}.$ (50)
Integrating (50), the result once more conforms with (2), with a similarity
exponent of $\alpha=4$, and the similarity function has the parametric form
$\displaystyle\xi=\zeta^{3}+2b\zeta^{5}/5$ (51) $\displaystyle
h=\zeta^{6}/2+b\zeta^{8}/4.$
It is confirmed easily that (51) solves (7) with $\alpha=4$, but the third
derivative of $h(\xi)$ is singular at the origin.
###### Acknowledgements.
J.H. would like to thank P.T. Allen and J. Isenberg for a discussion and
correspondence, as well as the Swedish Research Council and the Marie Curie
Training Network ENIGMA (contract MRNT-CT-2004-5652) for financial support.
## References
* Barbashov and Chernikov (1966) B. M. Barbashov and N. A. Chernikov, Sov. Phys. JETP 23, 861 (1966).
* Barbashov and Chernikov (1967) B. M. Barbashov and N. A. Chernikov, Sov. Phys. JETP 24, 437 (1967).
* Barbashov and Chernikov (1968) B. M. Barbashov and N. A. Chernikov, Sov. Phys. JETP 27, 971 (1968).
* Whitham (1974) G. B. Whitham, _Linear and Nonlinear Waves_ (John Wiley & Sons, 1974).
* Christodoulou (2007) D. Christodoulou, _The Formation of Shocks in 3-dimensional Fluids_ (EMS Monographs in Mathematics, 2007).
* Milbredt (2008) O. Milbredt, _The Cauchy problem for membranes_ , Ph.D. Thesis, FU Berlin (2008).
* Hoppe (2008) J. Hoppe, _Non-linear realization of Poincaré invariance in the graph-representation of extremal hypersurfaces_ (2008), URL http://arxiv:0806.0656.
* Bellettini et al. (2008) G. Bellettini, M. Novaga, and G. Orlandi, _Time-like Lorentzian minimal submanifolds as singular limits of nonlinear wave equations_ (2008).
* Eggers and Fontelos (2008) J. Eggers and M. A. Fontelos, _The role of self-similarity in singularities of partial differential equations_ (2008), URL http://arXiv:0812.1339.
* Abel (1881) N. H. Abel, _Oeuvres II_ (Imprimerie de Grøndahl & Søn, Christiania, 1881).
* Hoppe (1995) J. Hoppe, _Conservation laws and formation of singularities in relativistic theories of extended objects_ (1995), URL http://arxiv:hep-th/9503069.
* Arnold (1984) V. I. Arnold, _Catastrophe Theory_ (Springer, 1984).
|
arxiv-papers
| 2008-12-09T20:42:22 |
2024-09-04T02:48:59.325866
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jens Eggers and Jens Hoppe",
"submitter": "Jens Eggers",
"url": "https://arxiv.org/abs/0812.1790"
}
|
0812.2032
|
# Spatial Resolution Enhancement in Quantum Imaging beyond the Diffraction
Limit Using Entangled Photon-Number State
Jianming Wen,111Electronic address: jianm1@umbc.edu Morton H. Rubin, and
Yanhua Shih Physics Department, University of Maryland, Baltimore County,
Baltimore, Maryland 21250, USA
###### Abstract
In this paper we study the resolution of images illuminated by sources
composed of $N+1$ photons in which one non-degenerate photon is entangled with
$N$ degenerate photons. The $N$ degenerate photons illuminate an object and
are collected by an $N$ photon detector. The signal from the $N$ photon
detector is measured in coincidence with the non-degenerate photon giving rise
to a ghost image. We discuss the case of three photons in various
configurations and generalize to $N+1$. Using the Rayleigh criterion, we find
that the system may give an improvement in resolution by a factor of $N$
compared to using a classical source. For the case that the $N$-photon number
detector is a point detector, a coherent image is obtained. If the $N$-photon
detector is a bucket detector, the image is incoherent. The visibility of the
image in both cases is $1$. In the opposite case in which the non-degenerate
photon is scattered by the object, then, using an $N$-photon point detector
may reduce the Airy disk by a factor of $N$.
###### pacs:
42.50.Dv, 42.30.Kq, 42.50.St, 07.07.Df
## I Introduction
Diffraction puts a limit on the the resolution of optical devices. According
to the Rayleigh criterion rayleigh ; goodman , the ability to resolve two
point sources is limited by the wavelength of the light. The Rayleigh or
diffraction limit is not an absolute limit and proposals to exceed it have
been known for a long time goodman . Recently, new proposals to improve
resolution beyond the Rayleigh limit have been made based on the use of
entangled sources and new measurement techniques. Improving the resolving
power of optical systems beyond the diffraction limit not only is of interest
to the fundamental research, but also holds promise applications in remote
sensing and quantum sensors.
Classical imaging can be thought of as a single photon process in the sense
that the light detected is composed of photons each of which illuminates the
object, consequently, the image can be constructed one photon at time. What we
mean by referring to this as classical is that the source of the light may be
described by a density matrix with a positive P-function pos1 ; pos2 . In this
sense the Rayleigh limit may be thought of as a single photon limit. Recall
that ideal imaging is a process in which there is a point-to-point mapping of
the object to a unique image plane. Diffraction causes each point of the
object to be mapped onto a disk, the Airy disk, in the image plane.
One of the new approaches to improving resolution is based on using non-
classical light sources. Quantum ghost imaging todd ; streklov ; rubin ;
imaging ; shih ; milenaL is a process that uses two-photon entanglement. The
unique features of this process are that entanglement allows only one photon
to illuminate the object while the second photon does not. All the photons
that illuminates the object are detected in a single (bucket) detector that
does not resolve the image. The point detectors that detect the second photon
must lie in a specific plane. This plane is called the image plane although
there is no image in that plane; the image is formed in the correlation
measurement of entangled photons. The image is constructed one pair at a time.
The resolution of this system has recently been discussed rubin2008 ; milena .
Losses in this system affect the counting rate but not the quality of the
image.
A second approach using non-classical source is based on entangled photon-
number states dowling , e.g., N00N state. When the number of entangled photons
exceeds two there are many possible imaging schemes that can be envisioned and
so the analysis of these cases is still being carried out. This
interferometric approach achieves a sub-wavelength spatial resolution by a
factor $N$ and requires an $N$-photon absorption process. Another quantum
source used to study imaging is to generate squeezed states squeezed . The
image can be reconstructed through the homodyne detection homodyne . However,
both of these techniques are severely limited by the loss of photons.
A second class of approaches to improving resolution uses classical light
sources. One method uses classical light with measurements based on
correlations similar to ghost imaging and the Hanbury-Brown and Twiss
experiment h-t ; texts . This method has the advantage of being more robust
with respect to losses thermal ; thermal2 . Another approach is to build an
interferometric lithography with use of classical coherent state coherent ;
coherent2 , which has similar setup to the case using entangled photon-number
states.
In this paper we will consider improving spatial resolution beyond the
Rayleigh diffraction limit using quantum imaging with an entangled photon-
number state $|1,N\rangle$. In our imaging scheme by sending the $N$
degenerate photons to the object while keeping the non-degenerate photon and
imaging lens in the laboratory, a factor of $N$ improvement can be achieved in
spatial resolution enhancement compared to classical optics. The assumptions
required for the enhancement by a factor of $N$ are that the $N$ photons sent
to the object scatter off the same point and are detected by either an
$N$-photon number detector or a bucket detector. This sub-Rayleigh imaging
resolution may have important applications in such as improving sensitivities
of classical sensors and remote sensing. We emphasize that it is the quantum
nature of the state that offers such sub-wavelength resolving power with high
visibility. However, the system is very sensitive to loss. While we give
general results, our main concern will be with the case in which the object is
far from the source and the detectors and optics are close to the source. A
different but related approach to the one discussed here is given in
giovannetti .
We organize the paper as follows. We will discuss our imaging scheme with
entangled photon-number state $|1,2\rangle$ in some detail in Sec. II. In
previous work wen1 ; wen2 we have shown that imaging occurs in correlation
measurement, as in the ghost imaging case. Here we will show that under
certain stringent conditions, the resolution can be improved by a factor of
$2$ compared to classical optics. In Sec. III we generalize the scheme to the
$|1,N\rangle$ case and show that resolution improvement by a factor of $N$ can
be obtained. In Sec. IV some discussions will be addressed on other
experimental configurations. Finally we will draw our conclusions in Sec. V.
In an appendix we discuss the meaning of the approximation that the $N$
photons illuminate the same point on the object.
## II Three-Photon Optics
We start with three photons because this is the easiest case to investigate
the various configurations. Throughout the paper we shall assume that the
source of the three photons is a pure state and that the three-photon counting
rate for three point detectors is give by
$R_{cc}=\frac{1}{T^{2}}\int_{0}^{T}dt_{1}\int_{0}^{T}dt_{2}\int_{0}^{T}dt_{3}|\Psi(1,2,3)|^{2},$
(1)
where the three-photon amplitude is determined by matrix element between the
vacuum state and the three-photon state $|\psi\rangle$
$\Psi(1,2,3)=\langle 0|E^{(+)}_{1}E^{(+)}_{2}E^{(+)}_{3}|\psi\rangle,$ (2)
and
$E^{(+)}_{j}(\vec{\rho}_{j},z_{j},t_{j})=\int{d}\omega_{j}\int{d^{2}}\alpha_{j}E_{j}f_{j}(\omega_{j})e^{-i\omega_{j}t_{j}}g_{j}(\vec{\alpha}_{j},\omega_{j};\vec{\rho}_{j},z_{j})a(\vec{\alpha}_{j},\omega_{j}),$
(3)
where $E_{j}=\sqrt{\hbar\omega_{j}/2\epsilon_{0}}$, $\vec{\alpha}_{j}$ is the
transverse wave vector, and $a(\vec{\alpha}_{j},\omega_{j})$ is a photon
annihilation operator at the output surface of the source,
$[a(\vec{\alpha},\omega),a^{\dagger}(\vec{\alpha}\prime,\omega\prime)]=\delta(\vec{\alpha}-\vec{\alpha}\prime)\delta(\omega-\omega\prime).$
(4)
The function $f_{j}(\omega)$ is a narrow bandwidth filter function which is
assumed to be peaked at $\Omega_{j}$. The function $g_{j}$ is the Green’s
function goodman ; rubin that describes the propagation of each mode from the
output surface of the source to the $j$th detector at the transverse
coordinate $\vec{\rho}_{j}$, at the distance from the output surface of the
crystal to the plane of the detector, $z_{j}$. $\Psi$ is referred to as the
three-photon amplitude (or three-photon wavefunction).
We start with the case in which the source produces three-photon entangled
states with a pair of degenerate photons, that is
$\psi\rightarrow\psi_{{1,2}}$
$|\psi_{1,2}\rangle=\int{d}\omega_{1}{d}\omega_{2}\int{d^{2}}\alpha_{1}d^{2}\alpha_{2}\delta(2\omega_{1}+\omega_{2}-\Omega)\delta(2\vec{\alpha}_{1}+\vec{\alpha}_{2})a^{\dagger}(\vec{\alpha}_{2},\omega_{2})\big{[}a^{\dagger}(\vec{\alpha}_{1},\omega_{1})\big{]}^{2}|0\rangle,$
(5)
where $\Omega$ is a constant, $\omega_{1,2}$ and $\vec{\alpha}_{1,2}$ are the
frequencies and transverse wave vectors of the degenerate and non-degenerate
photons, respectively. The $\delta$-functions indicate that the source is
assumed to produce three-photon states with perfect phase matching. We assume
the paraxial approximation holds and that the temporal and transverse behavior
of the waves factor. The frequency correlation determines the three-photon
temporal properties. The transverse momentum correlation determines the
spatial properties of entangled photons. It is this wave-vector correlation
that we are going to concentrate on. As discussed in wen1 , several imaging
schemes can be implemented with this three-photon source. To demonstrate
spatial resolution enhancement beyond the Rayleigh diffraction limit, consider
the experimental setup shown in Fig. 1. It will be shown that for this
configuration the spatial resolving power is improved by a factor of 2,
provided the degenerate photons illuminate the same point on the object and
are detected by a two photon detector.
Figure 1: (color online) Schematic of quantum imaging with a three-photon
entangled state $|1,2\rangle$. $d_{1}$ is the distance from the output surface
of the source to the object. $L_{1}$ is the distance from the object to a
2-photon detector, D1. $d_{2}$ is the distance from the output surface of the
source to the imaging lens with focal length $f$ and $L_{2}$ is the length
from the imaging lens to a single-photon detector D2, which scans coming
signal photons in its transverse plane. “C.C.” represents the joint-detection
measurement.
As depicted in Fig. 1, two degenerate photons with wavelength $\lambda_{1}$
are sent to a two-photon detector (D1) after illuminating an object, and the
non-degenerate photon with wavelength $\lambda_{2}$ propagates to a single-
photon detector (D2) after an imaging lens with focal length $f$. The three-
photon amplitude (2) for detectors D1 and D2, located at
$(z_{1},\vec{\rho}_{1})$ and $(z_{2},\vec{\rho}_{2})$, now is
$\Psi\rightarrow\Psi_{1,2}=\langle
0|E^{(+)}_{2}(\vec{\rho}_{2},z_{2},t_{2})\big{[}E^{(+)}_{1}(\vec{\rho}_{1},z_{1},t_{1})\big{]}^{2}|\psi_{1,2}\rangle,$
(6)
Following the treatments in rubin ; goodman ; wen1 , we evaluate the Green’s
functions $g_{1}(\vec{\alpha}_{1},\omega_{2};\vec{\rho}_{1},z_{1})$ and
$g_{2}(\vec{\alpha}_{2},\omega_{2};\vec{\rho}_{2},z_{2})$ for the experimental
setup of Fig. 1 assuming that the narrow bandwidth filter allows us to make
the assumption that $\omega_{j}=\Omega_{j}+\nu_{j}$ where
$|\nu_{j}|\ll\Omega_{j}$ and $2\Omega_{1}+\Omega_{2}=\Omega$.
In the paraxial approximation it is convenient to write
$g_{j}(\vec{\alpha}_{j},\omega_{j};\vec{\rho}_{j},z_{j})=\frac{\omega_{j}e^{i\omega_{j}z_{j}/c}}{i2\pi
cL_{j}d_{j}}\chi_{j}(\vec{\alpha}_{j},\omega_{j};\vec{\rho}_{j},z_{j}),$ (7)
then
$\displaystyle\chi_{1}(\vec{\alpha}_{1},\Omega_{1};\vec{\rho}_{1},z_{1})$
$\displaystyle=$ $\displaystyle
e^{-i\frac{d_{1}|\vec{\alpha}_{1}|^{2}}{2K_{1}}}\int{d^{2}}\rho_{o}A(\vec{\rho}_{o})e^{i\frac{K_{1}|\vec{\rho}_{o}|^{2}}{2L_{1}}}e^{-i\frac{K_{1}\vec{\rho}_{1}\cdot\vec{\rho}_{o}}{L_{1}}}e^{i\vec{\alpha}_{1}\cdot\vec{\rho}_{o}},$
(8) $\displaystyle\chi_{2}(\vec{\alpha}_{2},\Omega_{2};\vec{\rho}_{2},z_{2})$
$\displaystyle=$ $\displaystyle
e^{-i\frac{d_{2}|\vec{\alpha}_{2}|^{2}}{2K_{2}}}\int{d^{2}}\rho_{l}e^{i\frac{K_{2}|\vec{\rho}_{l}|^{2}}{2}(\frac{1}{L_{2}}-\frac{1}{f})}e^{i(\vec{\alpha}_{2}-\frac{K_{2}}{L_{2}}\vec{\rho}_{2})\cdot\vec{\rho}_{l}},$
(9)
where we replace $\omega_{j}$ by $\Omega_{j}$ in $\chi_{j}$,
$K_{j}=\Omega_{j}/c=2\pi/\lambda_{j}$, $z_{1}=d_{1}+L_{1}$, and
$z_{2}=d_{2}+L_{2}$, respectively. In Eqs. (8) and (9), $A(\vec{\rho}_{o})$ is
the aperture function of the object, and $\vec{\rho}_{o}$ and $\vec{\rho}_{l}$
are two-dimensional vectors defined, respectively, on the object and the
imaging lens planes. With use of Eqs. (3) and (5), the three-photon amplitude
(6) becomes
$\Psi_{1,2}=e^{i(2\Omega_{1}\tau_{1}+\Omega_{2}\tau_{2})}\Phi_{1,2},$ (10)
where $\tau_{j}=t_{j}-z_{j}/c$ and
$\displaystyle\Phi_{1,2}=\int{d}\nu_{1}d\nu_{2}\delta(2\nu_{1}+\nu_{2})e^{i(2\nu_{1}\tau_{1}+\nu_{2}\tau_{2})}f_{1}(\Omega_{1}+\nu_{1})^{2}f_{2}(\Omega_{2}+\nu_{2})B_{1,2}.$
(11)
where
$\displaystyle B_{1,2}$ $\displaystyle=$ $\displaystyle
B_{0}\int{d^{2}}\rho_{o}A(\vec{\rho}_{o})e^{i\frac{K_{1}|\vec{\rho}_{o}|^{2}}{2L_{1}}}e^{-i\frac{K_{1}\vec{\rho}_{1}\cdot\vec{\rho}_{o}}{L_{1}}}\int{d^{2}}\rho^{\prime}_{o}A(\vec{\rho}^{\prime}_{o})e^{i\frac{K_{1}|\vec{\rho}^{\prime}_{o}|^{2}}{2L_{1}}}e^{-i\frac{K_{1}\vec{\rho}_{1}\cdot\vec{\rho}^{\prime}_{o}}{L_{1}}}\int{d^{2}}\rho_{l}e^{i\frac{K_{2}|\vec{\rho}_{l}|^{2}}{2}(\frac{1}{L_{2}}-\frac{1}{f})}e^{-i\frac{K_{2}}{L_{2}}\vec{\rho}_{2}\cdot\vec{\rho}_{l}}$
(12)
$\displaystyle\times\int{d^{2}}\alpha_{1}e^{-i|\vec{\alpha}_{1}|^{2}(\frac{d_{1}}{K_{1}}+\frac{2d_{2}}{K_{2}})}e^{-i\vec{\alpha}_{1}\cdot(2\vec{\rho}_{l}-\vec{\rho}_{o}-\vec{\rho}^{\prime}_{o})},$
where we collect all the slowly varying quantities into the constant $B_{0}$.
To proceed the discussion, in the following we will consider two different
detection schemes. One uses a point two-photon detector for two degenerate
photons after the object and the other has a two-photon bucket detector.
### II.1 Point Two-Photon Detector Scheme
In this detection scheme, a point two-photon detector is necessary to retrieve
the information of degenerate photons scattered off the same point in the
object. We therefore make the key assumption that the detector D1 is only
sensitive to the signals from the same point in the object, i.e.,
$\delta(\vec{\rho}_{o}-\vec{\rho}^{\prime}_{o})$ [The validity of this
assumption is addressed in the Appendix]. With this assumption, Eq. (12)
becomes
$\displaystyle B_{1,2}$ $\displaystyle=$ $\displaystyle
B_{0}\int{d^{2}}\rho_{o}A^{2}(\vec{\rho}_{o})e^{i\frac{K_{1}|\vec{\rho}_{o}|^{2}}{L_{1}}}e^{-i\frac{2K_{1}\vec{\rho}_{1}\cdot\vec{\rho}_{o}}{L_{1}}}\int{d^{2}}\rho_{l}e^{i\frac{K_{2}|\vec{\rho}_{l}|^{2}}{2}(\frac{1}{L_{2}}-\frac{1}{f})}e^{-i\frac{K_{2}}{L_{2}}\vec{\rho}_{2}\cdot\vec{\rho}_{l}}$
(13)
$\displaystyle\times\int{d^{2}}\alpha_{1}e^{-i|\vec{\alpha}_{1}|^{2}(\frac{d_{1}}{K_{1}}+\frac{2d_{1}}{K_{2}})}e^{-2i\vec{\alpha}_{1}\cdot(\vec{\rho}_{l}-\vec{\rho}_{o})}.$
Completing the integration on the transverse mode $\vec{\alpha}_{1}$ in Eq.
(13) gives
$\displaystyle B_{1,2}$ $\displaystyle=$ $\displaystyle
B_{0}\int{d^{2}}\rho_{o}A^{2}(\vec{\rho}_{o})e^{iK_{1}|\vec{\rho}_{o}|^{2}[\frac{1}{L_{1}}+\frac{1}{d_{1}+(2\lambda_{2}/\lambda_{1})d_{2}}]}e^{-i\frac{2K_{1}\vec{\rho}_{1}\cdot\vec{\rho}_{o}}{L_{1}}}$
(14)
$\displaystyle\times\int{d^{2}}\rho_{l}e^{i\frac{K_{2}|\vec{\rho}_{l}|^{2}}{2}[\frac{1}{L_{2}}+\frac{1}{d_{2}+(\lambda_{1}/2\lambda_{2})d_{1}}-\frac{1}{f}]}e^{-iK_{2}\vec{\rho}_{l}\cdot[\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{\rho}_{o}}{d_{2}+(\lambda_{1}/2\lambda_{2})d_{1}}]}.$
By imposing the Gaussian thin-lens imaging condition in Eq. (14)
$\displaystyle\frac{1}{f}=\frac{1}{L_{2}}+\frac{1}{d_{2}+(\lambda_{1}/2\lambda_{2})d_{1}},$
(15)
the transverse part of the three-photon amplitude reduces to
$\displaystyle B_{1,2}$ $\displaystyle=$ $\displaystyle
B_{0}\int{d^{2}}\rho_{o}A^{2}(\vec{\rho}_{o})e^{iK_{1}|\vec{\rho}_{o}|^{2}[\frac{1}{L_{1}}+\frac{1}{d_{1}+(2\lambda_{2}/\lambda_{1})d_{2}}]}e^{-i\frac{2K_{1}\vec{\rho}_{1}\cdot\vec{\rho}_{o}}{L_{1}}}\mathbf{somb}\bigg{(}\frac{2\pi{R}}{\lambda_{2}[d_{2}+(\lambda_{1}/2\lambda_{2})d_{1}]}\bigg{|}\vec{\rho}_{o}+\frac{\vec{\rho}_{2}}{m}\bigg{|}\bigg{)},$
(16)
where $R$ is the radius of the imaging lens,
$R/[d_{2}+(\lambda_{1}/2\lambda_{2})d_{1}]$ may be thought of as the numerical
aperture of the imaging system, and
$m=L_{2}/[d_{2}+(\lambda_{1}/2\lambda_{2})d_{1}]$ is the magnification factor.
In Eq. (16) the Airy disk is determined, as usual, by
$\mathbf{somb}(x)=2J_{1}(x)/x$, where $J_{1}(x)$ is the first-order Bessel
function.
Before proceeding with the discussion of resolution, let us look at the
physics behind Eqs. (15) and (16). Equation (15) defines the image plane where
the ideal the point-to-point mapping of the object plane occurs. The unique
point-to-point correlation between the object and the imaging planes is the
result of the transverse wavenumber correlation and the fact that we have
assumed that the degenerate photons illuminate the same object point. Let us
make a comparison with the two-photon and three-photon geometrical optics
rubin ; wen1 ; rubin2008 . In the Gauss thin lens equation the distance
between the imaging lens and the object planes,
$d_{2}+(\lambda_{1}/2\lambda_{2})d_{1}$ is similar to the form that appears in
the non-degenerate two-photon case except for the factor of 2. This factor 2
comes from the degeneracy of the pair of photons that illuminate the object.
As we will show below, this factor of 2 is the source of the improved spatial
resolution. Equation (16) implies that a coherent and inverted image magnified
by a factor of m is produced in the plane of $D_{2}$. Of course, there really
is no such image and the true image is nonlocal. The point-spread function in
Eq. (16) is generally determined by both wavelengths of the degenerate and
non-degenerate photons.
To examine the resolution using the Rayleigh criterion, we consider an object
consisting of two point scatters, one located at the origin and the other at
the point $\vec{a}$ in the object plane,
$A(\vec{\rho}_{o})^{2}=A_{0}^{2}\delta(\vec{\rho}_{o})+A_{\vec{a}}^{2}\delta(\vec{\rho}_{o}-\vec{a}).$
(17)
By substituting Eq. (17) into (16) we obtain
$\displaystyle
B_{1,2}=B_{0}\bigg{(}{A}^{2}_{0}\mathbf{somb}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}\bigg{|}\bigg{)}+e^{i\varphi_{2}}A_{\vec{a}}^{2}\mathbf{somb}\bigg{[}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{a}}{d_{2}+(\lambda_{1}/2\lambda_{2})d_{1}}\bigg{|}\bigg{]}\bigg{)},$
(18)
where the phase
$\varphi_{2}=K_{1}\bigg{[}|\vec{a}|^{2}\bigg{(}\frac{1}{L_{1}}+\frac{1}{d_{1}+d_{2}(2\lambda_{2}/\lambda_{1})}\bigg{)}-\frac{\vec{a}\cdot(\vec{\rho}_{1}+\vec{\rho}^{\prime}_{1})}{L_{1}}\bigg{]}$
(19)
indicates that the image is coherent. For a point $2$-photon detector, we
require $\vec{\rho}_{1}=\vec{\rho}^{\prime}_{1}$ in Eq. (19). As is well-known
goodman for coherent imaging the Rayleigh criterion is not the best choice
for characterizing the resolution, however, it is indicative of the resolution
that can be attained and it is convenient. For a circular aperture, the radius
of the Airy disk, $\xi$, is determined by the point-spread function, which is
$\xi=0.61\frac{\lambda_{2}L_{2}}{R}.$ (20)
Note that the radius of the Airy disk is proportional to the wavelength of the
non-degenerate photon. This is the standard result as obtained in classical
optics. Using the Rayleigh criterion, the image of the second term in Eq. (18)
is taken to lie on the edge of the Airy disk of the first term, therefore,
$a_{\mathrm{m}}=0.61\frac{\lambda_{2}}{R}\bigg{(}d_{2}+\frac{\lambda_{1}}{2\lambda_{2}}d_{1}\bigg{)}.$
(21)
We see from Eq. (21) that the resolution depends on the wavelengths of the
degenerate and the non-degenerate photons. In the case that $d_{1}\gg{d}_{2}$,
so that $d_{2}+(\lambda_{1}/2\lambda_{2})d_{1}$, is approximately
$(\lambda_{1}/2\lambda_{2})d_{1}$. In this case Eq. (15) implies that
$L_{2}\approx f$ and the radius of the Airy disk approaches to
$1.22\lambda_{2}f/R$, and
$a_{\mathrm{m}}=0.61\frac{\lambda_{1}d_{1}/2}{R}.$ (22)
Equation (22) shows a gain in spatial resolution of a factor of 2 compared to
classical optics. Furthermore, there is no background term which is
characteristic of the quantum case.
### II.2 Bucket Detector Scheme
If the two-photon detector is replaced by a bucket detector and the two
degenerate photons are collected by two single-photon detection events,
located at $(L_{1},\vec{\rho}_{1})$ and $(L_{1},\vec{\rho}^{\prime}_{1})$, in
the bucket, Eq. (12) becomes
$\displaystyle B_{1,2}$ $\displaystyle=$ $\displaystyle
B_{0}\int{d^{2}}\rho_{o}A(\vec{\rho}_{o})e^{i\frac{K_{1}|\vec{\rho}_{o}|^{2}}{2L_{1}}}e^{-i\frac{K_{1}\vec{\rho}_{1}\cdot\vec{\rho}_{o}}{L_{1}}}\int{d^{2}}\rho^{\prime}_{o}A(\vec{\rho}^{\prime}_{o})e^{i\frac{K_{1}|\vec{\rho}^{\prime}_{o}|^{2}}{2L_{1}}}e^{-i\frac{K_{1}\vec{\rho}^{\prime}_{1}\cdot\vec{\rho}^{\prime}_{o}}{L_{1}}}\int{d^{2}}\rho_{l}e^{i\frac{K_{2}|\vec{\rho}_{l}|^{2}}{2}(\frac{1}{L_{2}}-\frac{1}{f})}e^{i\frac{K_{2}}{L_{2}}\vec{\rho}_{2}\cdot\vec{\rho}_{l}}$
(23)
$\displaystyle\times\int{d^{2}}\alpha_{1}e^{-i|\vec{\alpha}_{1}|^{2}(\frac{d_{1}}{K_{1}}+\frac{2d_{2}}{K_{2}})}e^{-i\vec{\alpha}_{1}\cdot(2\vec{\rho}_{l}-\vec{\rho}_{o}-\vec{\rho}^{\prime}_{o})}.$
Under the assumption that the two degenerate photons are scattered off the
same point in the object, Eq. (23) takes the similar form as Eq. (13), except
that the second phase term in the first integrand of (13) is replaced by
$\mathrm{exp}\big{[}-i\frac{K_{1}(\vec{\rho}_{1}+\vec{\rho}^{\prime}_{1})\cdot\vec{\rho}_{o}}{L_{1}}\big{]}$.
It is easy to show that the Gaussian thin-lens equation takes the same form as
Eq. (15). By performing the same analysis as done in Sec. IIA on the resolving
two spatially close point scatters, the three-photon amplitude (18) now is
$\displaystyle
B_{1,2}=B_{0}\bigg{(}A_{0}^{2}\mathbf{somb}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}\bigg{|}\bigg{)}+e^{i\varphi_{2}}A_{\vec{a}}^{2}\mathbf{somb}\bigg{[}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{a}}{d_{2}+(\lambda_{1}/2\lambda_{2})d_{1}}\bigg{|}\bigg{]}\bigg{)}.$
(24)
Since the bucket detector gives no position information, we must square the
amplitude and integrating over the bucket detector,
$I=\int{d^{2}}\rho_{1}\int{d^{2}}\rho_{1}^{\prime}|B_{1,2}|^{2}=s_{b}^{2}|B_{0}|^{2}\bigg{(}|A_{0}|^{4}\mathbf{somb}^{2}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}\bigg{|}\bigg{)}+|A_{\vec{a}}|^{4}\mathbf{somb}^{2}\bigg{[}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{a}}{d_{2}+(\lambda_{1}/2\lambda_{2})d_{1}}\bigg{|}\bigg{]}\bigg{)}$
(25)
where $s_{b}$ is the area of the bucket detector. It is easy to see that the
spatial resolution improvement is the same as in Sec. IIA, the difference is
that now we get an incoherent image. The advantage is that a two photon bucket
detector should be easier to construct than a point two photon detector.
## III $N+1$ Photon Optics
Figure 2: (color online) Generalization of quantum imaging with $N+1$
entangled photons in state $|1,N\rangle$. For notations please refer to Fig. 1
except that here D1 is an $N$-photon detector. The image is formed in the
coincidence measurement and is not localized at either detector.
In Sec. II, we have shown that with the entangled photon-number state
$|1,2\rangle$, the ability to resolve two point sources in the object can be
improved by a factor of 2 by sending two degenerate photons to the object
while keeping the non-degenerate photon and imaging lens in the laboratory. In
this section, we are going to generalize the experimental configuration (Fig.
1) with use of the entangled state of $|1,N\rangle$, as described in Fig. 2.
For simplicity, we first address the case shown in Fig. 2 where the $N$
degenerate photons traverse to the $N$-photon detector, D1, after the object
and the non-degenerate photon propagates to the single-photon detector, D2.
The assumption required for the enhancement by a factor of $N$ are that the
$N$ photons sent to the object scatter off the same point and are detected by
the $N$-photon detector, D1.
The $N+1$ photons are assumed to be in a non-normalized pure state
$\displaystyle|\psi_{1,N}\rangle=\int{d}\omega_{1}{d}\omega_{2}\int{d^{2}}\alpha_{1}d^{2}\alpha_{2}\delta(N\omega_{1}+\omega_{2}-\Omega)\delta(N\vec{\alpha}_{1}+\vec{\alpha}_{2})a^{\dagger}_{\vec{k}_{2}}\big{(}a^{\dagger}_{\vec{k}_{1}}\big{)}^{N}|0\rangle.$
(26)
Again the $\delta$-functions in Eq. (26) indicate perfect phase matching. The
$N+1$-photon coincidence counting rate is defined as
$\displaystyle
R_{cc}=\frac{1}{T}\int^{T}_{0}dt_{1}\int^{T}_{0}dt_{2}\cdots\int^{T}_{0}dt_{N+1}|\Psi_{1,N}(1,2,\cdots,N+1)|^{2},$
(27)
where $\Psi_{1,N}$ is referred to as the $N+1$-photon amplitude. That is
$\displaystyle\Psi_{1,N}(1,2,\cdots,N+1)$ $\displaystyle=$
$\displaystyle\langle
0|E^{(+)}_{1}E^{(+)}_{2}\cdots{E}^{(+)}_{N+1}|\psi_{1,N}\rangle$ (28)
$\displaystyle=$ $\displaystyle\langle
0|E^{(+)}_{2}(\vec{\rho}_{2},z_{2},t_{2})[E^{(+)}_{1}(\vec{\rho}_{1},z_{1},t_{1})]^{N}|\psi_{1,N}\rangle.$
Following the procedure done for the $|1,2\rangle$ case, we calculate the
transverse part of the $N+1$-photon amplitude $\Psi_{1,N}$ (28) as
$\displaystyle\Psi_{1,N}$ $\displaystyle=$ $\displaystyle
e^{i(N\Omega_{1}\tau_{1}+\Omega_{2}\tau_{2})}\Phi_{1,N}(\tau_{1},\tau_{2})B_{1,N}$
$\displaystyle B_{1,N}$ $\displaystyle=$ $\displaystyle
B_{0}\underbrace{\int{d^{2}}\rho_{o}A(\vec{\rho}_{o})e^{i\frac{K_{1}|\vec{\rho}_{o}|^{2}}{2L_{1}}}e^{-i\frac{K_{1}\vec{\rho}_{1}\cdot\vec{\rho}_{o}}{L_{1}}}\cdots\int{d^{2}}\rho^{\prime}_{o}A(\vec{\rho}^{\prime}_{o})e^{i\frac{K_{1}|\vec{\rho}^{\prime}_{o}|^{2}}{2L_{1}}}e^{-i\frac{K_{1}\vec{\rho}_{1}\cdot\vec{\rho}^{\prime}_{o}}{L_{1}}}}_{\mathrm{N\;fold}}\int{d^{2}}\rho_{l}e^{i\frac{K_{2}|\vec{\rho}_{l}|^{2}}{2}(\frac{1}{L_{2}}-\frac{1}{f})}e^{-i\frac{K_{2}}{L_{2}}\vec{\rho}_{2}\cdot\vec{\rho}_{l}}$
(29)
$\displaystyle\times\int{d^{2}}\alpha_{1}e^{-i\frac{N^{2}|\vec{\alpha}_{1}|^{2}}{2}(\frac{d_{1}}{NK_{1}}+\frac{d_{2}}{K_{2}})}e^{-i\vec{\alpha}_{1}\cdot(N\vec{\rho}_{l}-\underbrace{\vec{\rho}_{o}-\cdots-\vec{\rho}^{\prime}_{o}}_{\mathrm{N}})}.$
Here $\Phi_{1,N}(\tau_{1},\tau_{2})$ describes the temporal behavior of
entangled three photons. By applying the same argument that the $N$-photon
detector D1 only receives the signals from the same spatial point in the
object, Eq. (29) can be further simplified as
$\displaystyle B_{1,N}$ $\displaystyle=$ $\displaystyle
B_{0}\int{d^{2}}\rho_{o}A^{N}(\vec{\rho}_{o})e^{i\frac{NK_{1}|\vec{\rho}_{o}|^{2}}{2L_{1}}}e^{-i\frac{NK_{1}\vec{\rho}_{1}\cdot\vec{\rho}_{o}}{L_{1}}}\int{d^{2}}\rho_{l}e^{i\frac{K_{2}|\vec{\rho}_{l}|^{2}}{2}(\frac{1}{L_{2}}-\frac{1}{f})}e^{-i\frac{K_{2}}{L_{2}}\vec{\rho}_{2}\cdot\vec{\rho}_{l}}$
(30)
$\displaystyle\times\int{d^{2}}\alpha_{1}e^{-i\frac{N^{2}|\vec{\alpha}_{1}|^{2}}{2}(\frac{d_{1}}{NK_{1}}+\frac{d_{2}}{K_{2}})}e^{-Ni\vec{\alpha}_{1}\cdot(\vec{\rho}_{l}-\vec{\rho}_{o})}.$
Performing the integration on the transverse mode $\vec{\alpha}_{1}$ in Eq.
(30) gives
$\displaystyle B_{1,N}$ $\displaystyle=$ $\displaystyle
B_{0}\int{d^{2}}\rho_{o}A^{N}(\vec{\rho}_{o})e^{i\frac{NK_{1}|\vec{\rho}_{o}|^{2}}{2}[\frac{1}{L_{1}}+\frac{1}{d_{1}+(N\lambda_{2}/\lambda_{1})d_{2}}]}e^{-i\frac{NK_{1}\vec{\rho}_{1}\cdot\vec{\rho}_{o}}{L_{1}}}$
(31)
$\displaystyle\times\int{d^{2}}\rho_{l}e^{i\frac{K_{2}|\vec{\rho}_{l}|^{2}}{2}[\frac{1}{L_{2}}+\frac{1}{d_{2}+(\lambda_{1}/N\lambda_{2})d_{1}}-\frac{1}{f}]}e^{-iK_{2}\vec{\rho}_{l}\cdot[\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{\rho}_{o}}{d_{2}+(\lambda_{1}/N\lambda_{2})d_{1}}]},$
where, again, we have assumed multimode generation in the process. Applying
the Gaussian thin-lens imaging condition
$\displaystyle\frac{1}{f}=\frac{1}{L_{2}}+\frac{1}{d_{2}+(\lambda_{1}/N\lambda_{2})d_{1}},$
(32)
the transverse part of the $N+1$-photon amplitude (31) between detectors D1
and D2 now becomes
$\displaystyle B_{1,N}$ $\displaystyle=$ $\displaystyle
B_{0}\int{d^{2}}\rho_{o}A^{N}(\vec{\rho}_{o})e^{i\frac{NK_{1}|\vec{\rho}_{o}|^{2}}{2}[\frac{1}{L_{1}}+\frac{1}{d_{1}+(N\lambda_{2}/\lambda_{1})d_{2}}]}e^{-i\frac{NK_{1}\vec{\rho}_{1}\cdot\vec{\rho}_{o}}{L_{1}}}\mathbf{somb}\bigg{[}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{\rho}_{o}}{d_{2}+(\lambda_{1}/N\lambda_{2})d_{1}}\bigg{|}\bigg{]}.$
(33)
As expected, Eqs. (32) and (33) have the similar forms as Eqs. (15) and (16)
for the $|1,2\rangle$ case. The unique point-to-point relationship between the
object and the imaging planes is enforced by the Gaussian thin-lens equation
(32). The coherent and inverted image is demagnified by a factor of
$L_{2}/[d_{2}+d_{1}(\lambda_{1}/N\lambda_{2})]$. The spatial resolution is
determined by the width of the point-spread function in Eq. (33). Note that a
factor of $N$ appears in the distance between the imaging lens and the object
planes, $d_{2}+d_{1}(\lambda_{1}/N\lambda_{2})$. We emphasize again that the
image is nonlocal and exists in the coincidence events.
To study the spatial resolution, we again consider the object represented by
Eq. (17). Plugging Eq. (17) into (33) yields
$\displaystyle
B_{1,N}=B_{0}\bigg{(}A_{0}^{N}\mathbf{somb}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}\bigg{|}\bigg{)}+e^{i\varphi_{N}}A_{\vec{a}}^{N}\mathbf{somb}\bigg{[}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{a}}{d_{2}+(\lambda_{1}/N\lambda_{2})d_{1}}\bigg{|}\bigg{]}\bigg{)}.$
(34)
For $N$ single photon detectors located at
$\vec{\rho}_{1}^{(1)},\cdots,\vec{\rho}_{1}^{(N)}$ the phase is given by
$\varphi_{N}=K_{1}\bigg{[}\frac{N|\vec{a}|^{2}}{2}\bigg{(}\frac{1}{L_{1}}+\frac{1}{d_{1}+d_{2}(N\lambda_{2}/\lambda_{1})}\bigg{)}-\frac{\vec{a}\cdot(\overbrace{\vec{\rho}_{1}^{(1)}+\vec{\rho}^{(2)}_{1}+\cdots}^{\mathrm{N}})}{L_{1}}\bigg{]}$
(35)
For a point $N$-photon number detector, we require
$\vec{\rho}_{1}^{(1)}=\vec{\rho}^{(2)}_{1}=\cdots$ and a coherent imaging is
achievable in this case. The first term on the right-hand side in Eq. (18)
gives the radius of the Airy disk, which is the same as the $|1,2\rangle$
case, see Eq. (20). Applying the Rayleigh criterion, the minimum resolvable
distance between two points in the transverse plane now is
$\displaystyle
a_{\mathrm{m}}=0.61\frac{\lambda_{2}}{R}\bigg{(}d_{2}+\frac{\lambda_{1}}{N\lambda_{2}}d_{1}\bigg{)}.$
(36)
For the case of $N=2$, Eq. (36) reduces to Eq. (21). In the case that
$d_{1}\gg{d_{2}}$, this becomes
$\displaystyle a_{\mathrm{m}}=0.61\frac{\lambda_{1}d_{1}}{NR}.$ (37)
As expected, Eq. (37) shows a gain in sub-Rayleigh resolution by a factor of
$N$ with respect to what one would obtain in classical optics. We therefore
conclude that in the proposed imaging protocol, the spatial resolving power
can be improved by a factor of $N$ with use of the entangled photon-number
state $|1,N\rangle$. Furthermore, because we are using an entangled state with
a specific type of detector, the image has high contrast because of the lack
of background noise.
By following the analysis in Sec. IIB, we can show that by replacing the
$N$-photon detector with an $N$-photon bucket detector, we get an incoherent
image but the sub-Rayleigh imaging process is not changed.
## IV Discussions and other Configurations
In the previous two sections, we have analyzed a novel ghost imaging by
sending $N$ degenerate photons to the object while keeping the non-degenerate
photon and imaging lens in the lab. We find that if the distance between the
object plane and the output surface of the source is much greater than the
distance between the imaging lens and the single-photon detector planes, we
can gain spatial resolution improvement in the object by a factor of $N$
compared to classical optics. In the cases that we have discussed in this
paper, this enhancement beyond the Rayleigh criterion is due to the quantum
nature of the entangled photon-number state. The assumptions required for such
an enhancement are that the $N$ degenerate photons sent to the object scatter
off the same point and are detected by either an $N$-photon number detector or
a bucket detector. An $N$-photon bucket detector is much easier to realize
than an $N$-photon point detector. Such a bucket detector could be an array of
single photon point detectors which only sent a signal to the coincidence
circuit if exactly $N$ of them fired.
Figure 3: (color online) Other schematics of quantum ghost imaging with three
entangled photons in state $|1,2\rangle$. (a) Both the imaging lens and the
object are inserted in the non-degenerate photon channel. (b) The imaging lens
is placed in the degenerate photon pathway while the object is in the non-
degenerate optical pathway.
Besides the favorable configuration discussed above, one may wonder what
happens if we switch the $N$ degenerate photons to detector $D_{1}$ and the
non-degenerate photon to $D_{2}$ after an imaging lens and an object? Do we
gain any spatial resolution improvement? To answer the questions, let us look
at the $|1,2\rangle$ case as illustrated in Fig. 3(a). Following the
treatments in Sec. IIA, after some algebra we find that the transverse part of
the three-photon amplitude (6) is
$\displaystyle B_{1,2}$ $\displaystyle=$ $\displaystyle
B_{0}\int{d^{2}}\rho_{o}A(\vec{\rho}_{o})e^{i\frac{K_{2}|\vec{\rho}_{o}|^{2}}{2}(\frac{1}{L_{2}}+\frac{1}{d^{\prime}_{2}})}e^{-i\frac{K_{2}\vec{\rho}_{2}\cdot\vec{\rho}_{o}}{L_{2}}}$
(38)
$\displaystyle\times\int{d^{2}}\rho_{l}e^{i\frac{K_{2}|\vec{\rho}_{l}|^{2}}{2}[\frac{1}{d^{\prime}_{2}}+\frac{1}{d_{2}+(\lambda_{1}/2\lambda_{2})L_{1}}-\frac{1}{f}]}e^{-iK_{2}\vec{\rho}_{l}\cdot[\frac{\vec{\rho}_{o}}{d^{\prime}_{2}}+\frac{\vec{\rho}_{1}}{d_{2}+(\lambda_{1}/2\lambda_{2})L_{1}}]}.$
In the derivation of Eq. (38), the Green’s functions associated with each beam
give
$\displaystyle\chi_{1}(\vec{\alpha}_{1},\Omega_{1};\vec{\rho}_{1},L_{1})$
$\displaystyle=$ $\displaystyle
e^{-i\frac{L_{1}|\vec{\alpha}_{1}|^{2}}{2K_{1}}}e^{i\vec{\rho}_{1}\cdot\vec{\alpha}_{1}},$
$\displaystyle\chi_{2}(\vec{\alpha}_{2},\Omega_{2};\vec{\rho}_{2},z_{2})$
$\displaystyle=$ $\displaystyle
e^{-i\frac{d_{2}|\vec{\alpha}_{2}|^{2}}{2K_{2}}}\int{d^{2}}\rho_{o}A(\vec{\rho}_{o})e^{i\frac{K_{2}|\vec{\rho}_{o}|^{2}}{2}(\frac{1}{L_{2}}+\frac{1}{d^{\prime}_{2}})}e^{-i\frac{K_{2}\vec{\rho}_{2}\cdot\vec{\rho}_{o}}{L_{2}}}\int{d^{2}}\rho_{l}e^{i\frac{K_{2}|\vec{\rho}_{l}|^{2}}{2}(\frac{1}{d^{\prime}_{2}}-\frac{1}{f})}e^{i\vec{\rho}_{l}\cdot(\vec{\alpha}_{2}-\frac{K_{2}\vec{\rho}_{o}}{d^{\prime}_{2}})}.$
Applying the Gaussian thin-lens imaging condition
$\displaystyle\frac{1}{d^{\prime}_{2}}+\frac{1}{d_{2}+(\lambda_{1}/2\lambda_{2})L_{1}}=\frac{1}{f},$
(39)
the transverse spatial part of the three-photon amplitude (38) reduces to
$\displaystyle
B_{1,2}=B_{0}\int{d^{2}}\rho_{o}A(\vec{\rho}_{o})e^{i\frac{K_{2}|\vec{\rho}_{o}|^{2}}{2}(\frac{1}{L_{2}}+\frac{1}{d^{\prime}_{2}})}e^{-i\frac{K_{2}\vec{\rho}_{2}\cdot\vec{\rho}_{o}}{L_{2}}}\mathbf{somb}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{o}}{d^{\prime}_{2}}+\frac{\vec{\rho}_{1}}{d_{2}+(\lambda_{1}/2\lambda_{2})L_{1}}\bigg{|}\bigg{)}.$
(40)
From this we see that the magnification is
$m=[d_{2}+(\lambda_{1}/2\lambda_{2})L_{1}]/d^{\prime}_{2}$. Comparing Eqs.
(39) and (40) with Eqs. (15) and (16), we see that the distances between the
object and the thin lens and between the thin lens and the imaging plane are
interchanged. Since the degenerate photons are measured at the imaging plane
in the setup of Fig. 3(a), the requirement of a point $N$-photon detector
cannot be relaxed.
Computing the spatial resolution as in Sec. II we have
$\displaystyle
B_{1,2}=B_{0}\bigg{[}A_{0}\mathbf{somb}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{1}}{d_{2}+(\lambda_{1}/2\lambda_{2})L_{1}}\bigg{|}\bigg{)}+e^{i\varphi^{\prime}}A_{\vec{a}}\mathbf{somb}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{a}}{d^{\prime}_{2}}+\frac{\vec{\rho}_{1}}{d_{2}+(\lambda_{1}/2\lambda_{2})L_{1}}\bigg{|}\bigg{)}\bigg{]},$
(41)
where
$\varphi^{\prime}=K_{2}\big{[}\frac{|\vec{a}|^{2}}{2}((\frac{1}{L_{2}}+\frac{1}{d^{\prime}_{2}})-\frac{\vec{\rho}_{2}\cdot\vec{a}}{L_{2}}\big{]}.$
The radius of the Airy disk is
$\displaystyle\xi=0.61\frac{\lambda_{2}}{R}\bigg{(}\frac{\lambda_{1}}{2\lambda_{2}}L_{1}+d_{2}\bigg{)}.$
(42)
If $L_{1}\gg{d_{2}}$,
$\xi\rightarrow\frac{0.61L_{1}}{R}(\frac{\lambda_{1}}{2})$, so that the width
of the point-spread function shrinks to one half its value compared to the
classical cases. Applying the Rayleigh criterion to see the minimum resolvable
distance between two point sources in the object. From the second term of Eq.
(41) the minimum distance turns out to be
$\displaystyle a_{\mathrm{min}}=0.61\frac{d^{\prime}_{2}\lambda_{2}}{R},$ (43)
which only is a function of the wavelength of the non-degenerate photon;
therefore, no spatial resolution improvement can be achieved compared to
classical optics.
Finally, we consider the configuration shown in Fig. 3(b) which was analyzed
in wen1 where it was shown that no well-defined images could be obtained.
It is straightforward to generalize the above two configurations with use of
the $|1,N\rangle$ state. By replacing the source state by the state
$|1,N\rangle$ in Fig. 3(a), it can be shown that the radius of the Airy disk
becomes
$\displaystyle\xi=0.61\frac{\lambda_{2}}{R}\bigg{(}\frac{\lambda_{1}}{N\lambda_{2}}L_{1}+d_{2}\bigg{)}.$
(44)
If $L_{1}\gg{d_{2}}$,
$\xi\rightarrow\frac{0.61L_{1}}{R}(\frac{\lambda_{1}}{N})$, so the Airy disk
shrinks to one $N$th of its radius compared to classical optics. However, if
$L_{1}\ll{d_{2}}$, Eq. (44) gives the same result as in classical optics.
Replacing the source with photon state $|1,N\rangle$ in Fig. 3(b), the above
conclusion is still valid. The analysis has been presented in wen2 and we
will not repeat here.
## V Conclusions
In summary, we have proposed a quantum-imaging scheme to improve the spatial
resolution in the object beyond the Rayleigh diffraction limit by using an
entangled photon-number state $|1,N\rangle$. We have shown that by sending the
$N$ degenerate photons to the object, keeping the non-degenerate photon and
imaging lens in the lab, and using a resolving $N$-photon detector or a bucket
detector, a factor of $N$ can be achieved in spatial resolution enhancement
using the Rayleigh criterion. The image is nonlocal and the quantum nature of
the state leads to the sub-Rayleigh imaging resolution with high contrast. We
have also shown that by sending the $N$ degenerate photons freely to a point
$N$-photon detector while propagating the non-degenerate photon through the
imaging lens and the object, the Airy disk in the imaging can be shrunk by a
factor of $N$ under certain conditions. However, it may be possible to show
that a similar effect can occur using non-entangled sources. In the language
of quantum information, the non-degenerate photon may be thought of as an
ancilla onto which the information about the object is transferred for
measurement. Our imaging protocol may be of importance in many applications
such as imaging, sensors, and telescopy.
## VI Acknowledgement
This work was supported in part by U.S. ARO MURI Grant W911NF-05-1-0197 and by
Northrop Grumman Corporation through the Air Force Research Laboratory under
contract FA8750-07-C-0201 as part of DARPA’s Quantum Sensors Program.
## Appendix A Validity of the Assumption Made in Eq. (13)
In going from Eq. (12) to Eq. (13), we have made an assumption that requires
the detector D1 is only sensitive to the scattered photons from the same
spatial point in the object. This allowed us to collapse the $N$ integrations
over the object into a single integral. In this Appendix, we give an example
of how this assumption may be satisfied for multi-photon scattering off the
target. Our example assumes that each point of the object transmits or
scatters the light with a random phase which satisfies Gaussian statistics.
The result is that the visibility decreases.
We start with the case of $2+1$ photons. From Eq. (12) the integration over
the transverse vector $\vec{\alpha}_{1}$, which gives
$\displaystyle B_{1,2}$ $\displaystyle=$ $\displaystyle
B_{0}\int{d}^{2}\rho_{o}A(\vec{\rho}_{o})e^{i\frac{K_{1}|\vec{\rho}_{o}|^{2}}{4}[\frac{2}{L_{1}}+\frac{1}{d_{1}+(2\lambda_{2}/\lambda_{1})d_{2}}]}e^{-i\frac{K_{1}\vec{\rho}_{1,1}\cdot\vec{\rho}_{o}}{L_{1}}}e^{i\phi(\vec{\rho}_{o})}\int{d}^{2}\rho^{\prime}_{o}A(\vec{\rho}^{\prime}_{o})e^{i\frac{K_{1}|\vec{\rho}^{\prime}_{o}|^{2}}{4}[\frac{2}{L_{1}}+\frac{1}{d_{1}+(2\lambda_{2}/\lambda_{1})d_{2}}]}$
(45)
$\displaystyle\times{e}^{-i\frac{K_{1}\vec{\rho}_{1,2}\cdot\vec{\rho}^{\prime}_{o}}{L_{1}}}e^{i\phi(\vec{\rho}^{\prime}_{o})}e^{i\frac{K_{1}\vec{\rho}_{o}\cdot\vec{\rho}^{\prime}_{o}}{2[d_{1}+(2\lambda_{2}/\lambda_{1})d_{2}]}}\int{d}^{2}\rho_{l}e^{i\frac{K_{2}|\vec{\rho}_{l}|^{2}}{2}[\frac{1}{L_{2}}+\frac{1}{d_{2}+(\lambda_{1}/2\lambda_{2})d_{1}}-\frac{1}{f}]}e^{-iK_{2}\vec{\rho}_{l}\cdot[\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{\rho}_{o}+\vec{\rho}^{\prime}_{o}}{2d_{2}+(\lambda_{1}/\lambda_{2})d_{1}}]},$
where $\vec{\rho}_{1,j}$ is a point at which a photon is detected on the
bucket detector, each point of the amplitude has a random phase associated
with its transmission amplitude and, as usual, all the slowly varying terms
have been grouped into $B_{0}$. Using the the Gaussian thin-lens imaging
condition (15) gives
$\displaystyle B_{1,2}$ $\displaystyle=$ $\displaystyle
B_{0}\int{d}^{2}\rho_{o}A(\vec{\rho}_{o})e^{i\frac{K_{1}|\vec{\rho}_{o}|^{2}}{4}[\frac{2}{L_{1}}+\frac{1}{d_{1}+(2\lambda_{2}/\lambda_{1})d_{2}}]}e^{-i\frac{K_{1}\vec{\rho}_{1,1}\cdot\vec{\rho}_{o}}{L_{1}}}e^{i\phi(\vec{\rho}_{o})}\int{d}^{2}\rho^{\prime}_{o}A(\vec{\rho}^{\prime}_{o})e^{i\frac{K_{1}|\vec{\rho}^{\prime}_{o}|^{2}}{4}[\frac{2}{L_{1}}+\frac{1}{d_{1}+(2\lambda_{2}/\lambda_{1})d_{2}}]}$
(46)
$\displaystyle\times{e}^{-i\frac{K_{1}\vec{\rho}_{1,2}\cdot\vec{\rho}^{\prime}_{o}}{L_{1}}}e^{i\phi(\vec{\rho}^{\prime}_{o})}e^{i\frac{K_{1}\vec{\rho}_{o}\cdot\vec{\rho}^{\prime}_{o}}{2[d_{1}+(2\lambda_{2}/\lambda_{1})d_{2}]}}\mathbf{somb}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{\rho}_{o}+\vec{\rho}^{\prime}_{o}}{2d_{2}+(\lambda_{1}/\lambda_{2})d_{1}}\bigg{|}\bigg{)}.$
Generalizing to the case of $N+1$, using the Gaussian thin-lens equation (32)
$\displaystyle B_{1,N}$ $\displaystyle=$ $\displaystyle
B_{0}\int{d}^{2}\rho_{o,1}A(\vec{\rho}_{o,1})e^{i\frac{K_{1}|\vec{\rho}_{o,1}|^{2}}{2L_{1}}}e^{-i\frac{K_{1}\vec{\rho}_{1,1}\cdot\vec{\rho}_{o,1}}{L_{1}}}e^{i\phi(\vec{\rho}_{o,1})}\cdots\int{d}^{2}\rho_{o,N}A(\vec{\rho}_{o,N})e^{i\frac{K_{1}|\vec{\rho}_{o,N}|^{2}}{2L_{1}}}e^{-i\frac{K_{1}\vec{\rho}_{1,N}\cdot\vec{\rho}_{o,N}}{L_{1}}}$
(47)
$\displaystyle\times{e}^{i\phi(\vec{\rho}_{o,N})}e^{i\frac{K_{1}|\vec{\rho}_{+}|^{2}}{2[d_{1}+(\lambda_{2}/N\lambda_{1})d_{2}]}}\mathbf{somb}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{\rho}_{+}}{d_{2}+(\lambda_{1}/N\lambda_{2})d_{1}}\bigg{|}\bigg{)},$
where $\vec{\rho}_{+}=\frac{1}{N}\sum_{j=1}^{N}\vec{\rho}_{o,j}$.
To compute the counting rate we first calculate the magnitude square of the
amplitude averaged over the random phases. Starting with the $N=2$ case and
assuming that the ensemble average, $\langle\cdots\rangle$, over those phases
satisfies Gaussian statistics so that
$\displaystyle\langle{e}^{i[\phi(\vec{\rho}_{o})+\phi(\vec{\rho}^{\prime}_{o})-\phi(\vec{\rho}^{\prime\prime}_{o})-\phi(\vec{\rho}^{\prime\prime\prime}_{o})]}\rangle=\delta(\vec{\rho}_{o}-\vec{\rho}^{\prime\prime}_{o})\delta(\vec{\rho}^{\prime}_{o}-\vec{\rho}^{\prime\prime\prime}_{o})+\delta(\vec{\rho}_{o}-\vec{\rho}^{\prime\prime\prime}_{o})\delta(\vec{\rho}^{\prime}_{o}-\vec{\rho}^{\prime\prime}_{o}),$
(48)
We have assumed that the correlation length of the random phase is
sufficiently small so that the Gaussian distribution can be approximated by
delta functions. We find
$\displaystyle\langle{B}^{*}_{1,2}B_{1,2}\rangle=|B_{0}|^{2}\int{d}^{2}\rho_{o}\int{d}^{2}\rho^{\prime}_{o}|A(\vec{\rho}_{o})A(\vec{\rho}^{\prime}_{o})|^{2}\mathbf{somb}^{2}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{\rho}_{o}+\vec{\rho}^{\prime}_{o}}{2d_{2}+(\lambda_{1}/\lambda_{2})d_{1}}\bigg{|}\bigg{)}\bigg{[}1+e^{-i\frac{K_{1}(\vec{\rho}_{1}-\vec{\rho}^{\prime}_{1})\cdot(\vec{\rho}_{o}-\vec{\rho}^{\prime}_{o})}{L_{1}}}\bigg{]}.$
(49)
When we integrate over the bucket detector, the first term will be a constant
while the second term will give us a delta function in $\vec{\rho}_{o}$ times
the area of the bucket detector, $s_{b}$. Equation (49) reduces to
$\displaystyle\int{d}^{2}\rho_{1,1}\int{d}^{2}\rho_{1,2}\langle|B_{1,2}|^{2}\rangle=C+|B_{0}|^{2}s_{b}^{2}(\frac{L_{1}\lambda_{1}}{2\pi{s_{b}}})\int{d}^{2}\rho_{o}|A(\vec{\rho}_{o})|^{4}\mathbf{somb}^{2}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{\rho}_{o}}{d_{2}+(\lambda_{1}/2\lambda_{2})d_{1}}\bigg{|}\bigg{)}.$
(50)
First note that for the second term is similar to Eq. (25), the difference
being the term in parenthesis which is the ratio of effect of diffraction to
the area of the bucket detector, it is essentially the inverse of the Fresnel
number. Computing the constant, $C$, is generally difficult and depends in
detail on the geometry of the object, we can obtain an upper bound on $C$
quite easily,
$|C|\leq{s}_{b}^{2}|B_{0}|^{2}\bigg{|}\int{d}^{2}\rho_{o}|A(\vec{\rho}_{o})|^{2}\bigg{|}^{2},$
(51)
consequently, the visibility will be much less than for the ideal case
discussed above. From Eq. (50) the second term is proportional to
$L_{1}\lambda_{1}$ which implies that as this product increases the visibility
increases, however, recall for the case of sensors $L_{1}\simeq d_{1}$, so as
this term increases the minimum resolvable distance also increases.
The generalization to the case of $N+1$ photons is straightforward. The
ensemble phase average now becomes
$\displaystyle\left\langle\mathrm{exp}\bigg{[}i\bigg{(}\sum_{j=1}^{N}\phi(\vec{\rho}_{oj})-\sum_{j=1}^{N}\phi(\vec{\rho}^{\prime}_{oj})\bigg{)}\bigg{]}\right\rangle=\sum_{P_{N}}\prod_{r=1}^{N}\delta(\vec{\rho}_{o,r}-\vec{\rho}^{\prime}_{o,P_{N}(r)}),$
(52)
where the $N$ degenerate transmitted or reflected photons acquire random
phases $\phi(\vec{\rho}_{o,j})$ and $P_{N}$ is the set of permutations of the
numbers $(1,\cdots,N)$. In Eq. (52) there are $N!$ terms. We can show that
$\displaystyle\langle|B_{1,N}|^{2}\rangle$ $\displaystyle=$
$\displaystyle|B_{0}|^{2}\int{d}^{2}\rho_{o,1}\cdots\int{d}^{2}\rho_{o,N}|A(\vec{\rho}_{o,1})\cdots{A}(\vec{\rho}_{o,N})|^{2}\mathbf{somb}^{2}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{\rho}_{+}}{N[d_{2}+(\lambda_{1}/N\lambda_{2})d_{1}]}\bigg{|}\bigg{)}$
(53)
$\displaystyle\times\sum_{P_{N}}e^{-i\frac{K_{1}}{L_{1}}\sum_{r=1}^{N}{\vec{\rho}_{1,r}\cdot(\vec{\rho}_{o,r}-\vec{\rho}_{o,P_{N}(r)})}}.$
When we integrate over the bucket detector, we get a complicated result. Two
terms are simple, the identity permutation gives a constant and the single
cycle subgroup give an incoherent image with a resolution that depends on
$\lambda_{1}/N$. These are the only terms for $N=2$. The remaining terms will
lead to terms which are essentially constant. For $N=3$ we get
$\displaystyle\int{d}^{2}\rho_{1,1}\int{d}^{2}\rho_{1,2}\int{d}^{2}\rho_{1,3}\langle|B_{1,3}|^{2}\rangle=C+3|B_{0}|^{2}s^{3}_{b}\bigg{(}\frac{L_{1}\lambda_{1}}{2\pi{s}_{b}}\bigg{)}\int{d}^{2}\rho_{+}\int{d}^{2}\zeta|A(\vec{\rho}_{+}-2\vec{\zeta})|^{2}|A(\vec{\rho}_{+}+\zeta)|^{4}$
(54)
$\displaystyle\times\mathbf{somb}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{\rho}_{+}}{3[d_{2}+(\lambda_{1}/3\lambda_{2})d_{1}]}\bigg{|}\bigg{)}+|B_{0}|^{2}s^{3}_{b}\bigg{(}\frac{L_{1}\lambda_{1}}{2\pi{s}_{b}}\bigg{)}^{2}\int{d}^{2}\rho_{o,1}|A(\vec{\rho}_{o,1})|^{6}$
$\displaystyle\times\mathbf{somb}^{2}\bigg{(}\frac{2\pi{R}}{\lambda_{2}}\bigg{|}\frac{\vec{\rho}_{2}}{L_{2}}+\frac{\vec{\rho}_{o,1}}{d_{2}+(\lambda_{1}/N\lambda_{2})d_{1}}\bigg{|}\bigg{)}.$
From Eq. (54) the second term shows explicitly how the general terms will lead
to a complicated average over the illuminated area of the object. This result
shows that the image will have very poor visibility for large $N$, it is not
certain whether there might be arrangement of detectors for the $N$ photons
which will give better results.
## References
* (1) It should be noted that there are different, related, meanings of the Rayleigh criterion. The one used here refers to the ability to resolve two point sources in the object. The other is the minimum angle between plane waves falling on an aperture that can be resolved and can be interpreted as resolving two point sources at infinity.
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* (12) M. D’Angelo, A. Valencia, M. H. Rubin, and Y.-H. Shih, Phys. Rev. A 72, 013810 (2005).
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* (19) D. Zhang, Y.-H. Zhai, L.-A. Wu, and X.-H. Chen, Opt. Lett. 30, 2354 (2005); Y. Bai and S. Han, Phys. Rev. A 76, 043828 (2007); Y. J. Cai and S. Y. Zhu, Phys. Rev. E 71, 056607 (2005); A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, Phys. Rev. A 70, 013802 (2004); R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, Phys. Rev. Lett. 92, 033601 (2004).
* (20) S. J. Bentley and R. W. Boyd, Opt. Express 12, 5735 (2004); A. Pe’er, B. Dayan, M. Vucelja, Y. Silberberg, and A. A. Friesem, ibid. 12, 6600 (2004); P. R. Hemmer, A. Muthukrishnan, M. O. Scully, and M. S. Zubairy, Phys. Rev. Lett. 96, 163603 (2006).
* (21) M. Kiffner, J. Evers, and M. S. Zubairy, Phys. Rev. Lett. 100, 073602 (2008).
* (22) V. Giovannetti, S. Lloyd, L. Maccone, and J. Shapiro, arXiv:0804.2875v1 [quant-ph].
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|
arxiv-papers
| 2008-12-10T21:16:00 |
2024-09-04T02:48:59.336413
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianming Wen, Morton H. Rubin, and Yanhua Shih",
"submitter": "Jianming Wen",
"url": "https://arxiv.org/abs/0812.2032"
}
|
0812.2309
|
# Classification of Cell Images Using MPEG-7-influenced Descriptors and
Support Vector Machines in Cell Morphology
Tobias Abenius
tobbe@tobbe.nu
###### Abstract
Counting and classifying blood cells is an important diagnostic tool in
medicine. Support Vector Machines are increasingly popular and efficient and
could replace artificial neural network systems. Here a method to classify
blood cells is proposed using SVM. A set of statistics on images are
implemented in C++. The MPEG-7 descriptors Scalable Color Descriptor, Color
Structure Descriptor, Color Layout Descriptor and Homogeneous Texture
Descriptor are extended in size and combined with textural features
corresponding to textural properties perceived visually by humans. From a set
of images of human blood cells these statistics are collected. A SVM is
implemented and trained to classify the cell images. The cell images come from
a CellaVision™ DM-96 machine which classify cells from images from microscopy.
The output images and classification of the CellaVision™ machine is taken as
ground truth, a truth that is 90-95% correct. The problem is divided in two —
the primary and the simplified. The primary problem is to classify the same
classes as the CellaVision™ machine. The simplified problem is to differ
between the five most common types of white blood cells. An encouraging result
is achieved in both cases — error rates of 10.8% and 3.1% — considering that
the SVM is misled by the errors in ground truth. Conclusion is that further
investigation of performance is worthwhile.
###### Sammanfattning
Att räkna och klassificera blodceller är ett viktigt diagnostiskt redskap inom
läkarvetenskapen. Support Vector Machines är effektiva, ökar i popularitet och
kan ersätta artificiella neurala nätverkssystem. Här föreslås en metod för att
klassificera blodceller m.h.a. SVM. En mängd statistika på bilder
implementeras i C++. De s.k. MPEG-7 descriptors Scalable Color Descriptor,
Color Structure Descriptor, Color Layout Descriptor och Homogeneous Texture
Descriptor utvidgas i storlek och kombineras med textur-mått motsvarande
textur-egenskaper som uppfattas visuellt av människor. Från en mängd bilder av
mänskliga blodceller samlas dessa mått. En SVM implementeras och tränas att
klassificera cellbilderna. Cellbilderna kommer från en CellaVision™ DM-96 som
klassificerar celler från mikroskoperade bilder. Bilderna och dess klasser
från en CellaVision™ DM-96-maskin tas som facit, ett facit som är 90-95%
korrekt. Problemet delas i två — det primära och det förenklade. Det primära
problemet är att skilja mellan de klasser som CellaVision™s maskin gör. Det
förenklade problemet är att skilja mellan de fem vanligaste typerna av vita
blodkroppar. Ett glädjande resultat uppnås i båda fallen — felfrekvenser om
10,8% och 3,1% — med tanke på att SVM missleddes av felen i det tagna facitet.
Slutsatsen är att vidare studier angående prestanda är lönsamma.
veelo companion subsection
firstpageempty firstpage Examensarbete för 30 hp
Institutionen för datavetenskap, Naturvetenskapliga fakulteten, Lunds
universitet
Thesis for a diploma in Computer Science, 30 ECTS credits
Department of Computer Science, Faculty of Science, Lund University
firstpage0.85 adjustwidth*-
adjustwidth*-
—
Klassificering av cellbilder med hjälp av MPEG-7-inspirerade mått och support
vector machines i cellmorfologi
to Britta,
to my family
subsection
###### Contents
1. Acknowledgments
2. 1 Introduction
3. 2 Background
1. 1 Support Vector Machines
1. 1.1 Supervised Learning
2. 1.2 Linear Learning Machines
3. 1.3 Maximum Margin Classifier
4. 1.4 Optimization Theory
5. 1.5 The Kernel Trick
6. 1.6 Gradient Ascent
7. 1.7 Multiclass SVM
2. 2 Features
1. 2.1 Scalable Color Descriptor
2. 2.2 Color Structure Descriptor
3. 2.3 Color Layout Descriptor
4. 2.4 Homogeneous Texture Descriptor
5. 2.5 Visual Texture Features
1. 2.5.1 Neighborhood Gray-Tone Difference Matrix
2. 2.5.2 Coarseness
3. 2.5.3 Contrast
4. 2.5.4 Busyness
5. 2.5.5 Complexity
6. 2.5.6 Texture strength
3. 3 Fast 2D Convolution
4. 4 Scaling data
4. 3 Material and Methods
1. 5 Material
2. 6 Implementation details
1. 6.1 Support Vector Machine
1. 6.1.1 A Stochastic Gradient Ascent Variant
2. 6.1.2 Multiclass SVM
2. 6.2 Features
1. 6.2.1 Scalable Color Descriptor
2. 6.2.2 Color Structure Descriptor
3. 6.2.3 Color Layout Descriptor
4. 6.2.4 Homogeneous Texture Descriptor
5. 6.2.5 Neighborhood Gray-Tone Difference Matrix
3. 6.3 Convolution
4. 6.4 Data View
5. 4 Experimental Setup and Results
1. 7 Experimental Setup
1. 7.1 Performance test method
2. 7.2 Description of the simplified problem
2. 8 Results
1. 8.1 Primary Problem
2. 8.2 Simplified Problem
6. 5 Discussion
7. 6 Software Usage
1. 6.A train – Train a model
2. 6.B cellfeatures – Generate examples from the cell database
3. 6.C jpeg_genfeature – Feature generation from images
4. 6.D predict – Predicting a set of features
5. 6.E extractcelltype – Extract a class of images from the cell database
6. 6.F extractcellid – Extract given instances from the cell database
7. 6.G extractcellinfo – Extract statistics of instances from the cell database
8. 6.H tolibsvm – Save cell features in libSVM format
###### List of Tables
1. 1.1 Abundance of different types of white blood cells (leukocytes) in healthy humans
2. 3.1 Cell types classified in the data set
3. 4.1 Cell types left in the simplified problem
4. 4.2 SVM cell classifier results for the primary problem
5. 4.3 Confusion Matrix for the primary problem
6. 4.4 SVM cell classifier results for the simplified problem
7. 4.5 Confusion matrix for the simplified problem
## Acknowledgments
First of all thanks to Doc. Christian Balkenius and Doc. Jacek Malec for
supervising my thesis. To Dr. Ferenc Belik for managing all practical details.
to Doc. B.S. Manjunath for inspiration and lending me figure 2.3. To Sebastian
Ganslandt for initial chats about support vector machines and thesis ideas. To
all others that made this work possible.
## 1 Introduction
After the introduction of MPEG-7 descriptors by the Movie Producers Expert
Group (MPEG) committeemanjunath2001ColorTextureDescriptors it is interesting
to see how these features perform in the field of machine learning. In this
thesis a subset of them will be tested on the problem of classifying different
cell types, i.e. cell morphology, by using Support Vector Machines.
In medicine, more specifically the fields of hematology and infectious
diseases, classifying different kinds of blood cells can be used as a tool in
diagnosis — by counting certain cells’ relative frequencies and comparing to
what is normal, conclusions can be made about possible diagnosis.
Type | Approx. Abundance
---|---
neutrophil granulocytes | 70%
eosinophil granulocytes | 1-6%
basophil granulocytes | 0.01-0.3%
lymphocyte | 20-40%
monocytes | 3-8%
Table 1.1: Abundance of different types of white blood cells (leukocytes) in
healthy humans
[Neutrophil Granulocyte, segmented (class 1)] [Neutrophil Granulocyte, band
(class 6)] [Eosinophil Granulocyte (class 2)] [Basophil Granulocyte (class 3)]
[Lymphocyte (class 4)] [Monocyte (class 5)]
Figure 1.1: Some typical images of common white blood cells
Classifying cells using microscopy is used to classify infectious diseases by
determining the relative amount of cells called neutrophils compared to the
amount of cells called lymphocytes. Typical relative frequencies of the cells
are found in table 1.1. Typical images of some common cells are found in
figure 1.1.
Another method used is flow cytometry where receptors on the cells are colored
and the different types of cells are counted. Flow cytometry uses a
complicated and expensive apparatus while microscopy is very cheap.
However, microscopy is personnel intensive, many cells are hard to classify
even for human experts, often several experts are needed to be certain. To be
able to classify cells, great efforts of training are required, even more, to
sustain competence, regular frequent work is required. This competence is
impossible to sustain at small clinics or in the countryside especially in
developing countries. Instead, samples have to be sent to hematology labs.
As processing power becomes cheaper and machine learning and computer vision
algorithms grow better, machines can help less experienced personnel or give
preliminary results while waiting for definite results.
The problem this thesis try to investigate is how well these different types
of white blood cells can be classified using a Support Vector Machine and a
set of measures on the images, called features.
There has been a lot of hype about Support Vector Machines since its
introduction in the 1990’s. SVM is applied within a broad range of fields,
from bioinformaticsLengauerBioinformaticsPredictionOfHIVCoreceptorUsage to
food engineeringDuMultiClassificationOfPizzaUsingComputerVisionAndSVM , iris
recognitionIrisRecognitionBasedOnScoreLevelFusionByUsingSVM , texture
classification and object
recognitionZhangLocalFeaturesAndKernelsForClassificationOfTextureAndObjectCategories
. It is now one of standard tools available for machine learning—A recent
search for “Support Vector Machine” (SVM) gave 6 394 articles compared to 17
893 for “Artificial Neural Network” (ANN) which has existed for much longer.
That is why my supervisor and I chose to work with SVM.
The SVM is trained with measures of the cell images, called features or
descriptors. These are values that describe the essence of an image. In this
thesis I will describe and implement a subset of the color and texture
descriptors found in the MPEG-7 standard with minor variance. I chose to work
with MPEG-7 as a guide because of the MPEG committee’s well known expertise.
The MPEG committee developed e.g. the audio compression techniques used in
MPEG-1 Layer 3 (MP3), the video compression used in e.g. DVDs (MPEG-4) and
MPEG-7. The committee consists of experts from a broad range of areas that
deal with digital information.urlMPEG7
MPEG-7 identify several descriptors which has proved useful in the Color and
Texture Core Experimentsmanjunath2001ColorTextureDescriptors while developing
of the standard. They have proved useful in image browsing, search and
retrievalmanjunath2000ATextureDescriptorForBrowsingAndSimilarityRetrieval as
well as in image
classificationSpyrouFuzzySVMForImageClassificationFusingMPEG7VisualDescriptors
. Color histogram based features has been successful both in image
retrievalSergyanColorHistogramFeaturesBasedImageClassificationInContentBasedImageRetrievalSystems
and image
classificationSergyanColorHistogramFeaturesBasedImageClassificationInContentBasedImageRetrievalSystems
, ChapelleSVMForHistogramBasedImageClassification ,
BarlaOldFashionedStateOfTheArtImageClassification systems. Texture features
like Gabor Wavelet Filter Bank used in MPEG-7 has been successfully applied to
irisIrisRecognitionBasedOnScoreLevelFusionByUsingSVM and facial
expressionBuciuICAAndGaborRepresentationForFacialExpressionRecognition
recognition.
## 2 Background
### 1 Support Vector Machines
In this section I will briefly introduce Support Vector Machines from a
theoretical perspective. Further introduction may be found in Bishop’s
book[Bishop, , chapters 6,7 and E]. If more substance is wanted I recommend
reading the whole book by Christianini and Shawe-TaylorNello . The very
thorough coverage of the topic by its original implementor Vapnik in his
bookVapnik , sometimes called the bible, was often an additional useful source
for me.
#### 1.1 Supervised Learning
Supervised learning is a kind of machine learning where the machine is fed
with examples, i.e. instances of data tied to their class. The machine is told
what class an instance belongs to.
The task that a learning machine performs is to recognize an element
$\boldsymbol{\mathbf{x}}\in\mathcal{X}$ as a member of a class — to classify
it. These classes are called destination values and I use the notation
$y\in\mathcal{Y}$. In the binary case for example $\mathcal{Y}=\\{-1,+1\\}$.
The task would then be to construct a function such that
$d(\boldsymbol{\mathbf{x}},\boldsymbol{\mathbf{\alpha}})=y$, given
$\boldsymbol{\mathbf{\alpha}}$ is the information the machine has previously
gathered during the training process. During training, the machine observes a
tuple of pairs
$\displaystyle
S=\big{(}(\boldsymbol{\mathbf{x}}_{1},y_{1}),\ldots,(\boldsymbol{\mathbf{x}}_{\ell},y_{\ell})\big{)}\subseteq(\mathcal{X}\times\mathcal{Y})^{\ell},$
which is called the training set, and produces parameters
$\boldsymbol{\mathbf{\alpha}}\in\mathbb{R}^{n}$ deduced from this
information.Nello
#### 1.2 Linear Learning Machines
Imagine the space $\mathcal{X}$ which has $n$ dimensions. To be able to
classify instances into the two classes labeled positive, $y=+1$, or negative,
$y=-1$, a hyperplane, i.e. an affine subspace of dimension $n-1$, must be
found that separates the instances of the respective classes from each other.
If such a hyperplane exists, the data is said to be linearly separable.
Imagine a two-dimensional coordinate system in which the instances are placed.
If a straight line can be placed between the two classes of instances, the
data is linearly separable. That straight line is a hyperplane of dimension 1.
The generalized hyperplane of dimension $n-1$ is defined by the equation
$\displaystyle\langle\boldsymbol{\mathbf{w}},\boldsymbol{\mathbf{x}}\rangle+b=0.$
The normal vector $\boldsymbol{\mathbf{w}}$ is orthogonal to the hyperplane
and the bias $b$ is the hyperplane’s offset from the origin.
Now consider the function
$\displaystyle
f(\boldsymbol{\mathbf{x}})=\langle\boldsymbol{\mathbf{w}},\boldsymbol{\mathbf{x}}\rangle+b=\sum_{i=1}^{n}w_{i}x_{i}+b$
(2.1)
Where: $\boldsymbol{\mathbf{x}}$ – instance
$\boldsymbol{\mathbf{w}}$ – coefficients learned
$b$ – system bias
It will tell whether an instance is above or below the hyperplane. This is
similar to linear regression in statistics.
A decision function for the binary classification case then becomes
$\displaystyle d(\boldsymbol{\mathbf{x}})$
$\displaystyle=\mathrm{sgn}(f(\boldsymbol{\mathbf{x}}))$
$\displaystyle\mathrm{sgn}(a)$ $\displaystyle=\begin{cases}-1,&\;a<0\\\
+1,&\;a\geq 0\end{cases}$
An example of an iterative algorithm that find the vector
$\boldsymbol{\mathbf{w}}$ from a set of
$\boldsymbol{\mathbf{x}}\in\mathcal{X}$ is Rosenblatt’s perceptron which was
the first and simplest type of an Artificial Neural Networks (ANN). It is
guaranteed to converge if the data is linearly separable. This criterion could
also be written
$\displaystyle\exists\boldsymbol{\mathbf{w}}\forall i:\gamma_{i}$
$\displaystyle=y_{i}(\langle\boldsymbol{\mathbf{w}},\boldsymbol{\mathbf{x}}_{i}\rangle+b)>0,$
$\displaystyle i\in[0,\ell),$
i.e. all instances are classified correctly. The quantity $\gamma_{i}$ is
called the margin as it specifies how far from the hyperplane an instance is.
If $\boldsymbol{\mathbf{w}}$ and $b$ are normalized, to
$\frac{\boldsymbol{\mathbf{w}}}{\lVert\boldsymbol{\mathbf{w}}\rVert}$ and
$\frac{b}{\lVert\boldsymbol{\mathbf{w}}\rVert}$, then the margin is called the
geometric margin which measures the euclidean distances of the points
$\boldsymbol{\mathbf{x}}$ to the hyperplane. The closest point, the
$\boldsymbol{\mathbf{x}}_{i}$ with minimal $\gamma_{i}$, define the margin of
a hyperplane which is a stripe of empty space where no instances are. If the
data is not linearly separable $\exists i:\gamma_{i}\leq 0$.Nello , Bishop
#### 1.3 Maximum Margin Classifier
The task of a maximum margin classifier is to maximize the margin which can be
motivated, using statistical learning theory, gives the least generalization
error.
The maximum margin solution, the optimal $\boldsymbol{\mathbf{w}}$ and $b$, is
found by solving
$\displaystyle\operatorname*{arg\
max}_{\boldsymbol{\mathbf{w}},b}\left\\{\min_{i}\frac{\gamma_{i}}{\lVert\boldsymbol{\mathbf{w}}\rVert}\right\\}=\operatorname*{arg\
max}_{\boldsymbol{\mathbf{w}},b}\left\\{\frac{1}{\lVert\boldsymbol{\mathbf{w}}\rVert}\min_{i}y_{i}(\langle\boldsymbol{\mathbf{w}},\boldsymbol{\mathbf{x}}_{i}\rangle+b)\right\\}$
To solve this first rescale
$\boldsymbol{\mathbf{w}}\rightarrow\kappa\boldsymbol{\mathbf{w}}$ and
$b\rightarrow\kappa b$. The distance to the hyperplane is still the same
$\min_{i}\gamma_{i}$. Then set
$\displaystyle\gamma_{j}=y_{j}(\langle\boldsymbol{\mathbf{w}},\boldsymbol{\mathbf{x}}_{j}\rangle+b=1$
for the point $\boldsymbol{\mathbf{x}}_{j}$ that is closest to the hyperplane.
All points will then have $\gamma_{i}\geq 1$ and since the minimum
$\gamma_{j}=1$ all that have to be done is to maximize
$\lVert\boldsymbol{\mathbf{w}}\rVert^{-1}$ or minimize
$\lVert\boldsymbol{\mathbf{w}}\rVert^{2}$. The problem that is left is to
$\begin{split}\text{find}&\quad\operatorname*{arg\
min}_{\boldsymbol{\mathbf{w}},b}\frac{\lVert\boldsymbol{\mathbf{w}}\rVert^{2}}{2},\\\
\text{subject to}&\quad\gamma_{i}\geq 1,\end{split}$ (2.2)
which is much easier. This problem is what is called a quadratic programming
problem and can be solved using the theory of optimization theory and Lagrange
Multipliers.Nello , Bishop
#### 1.4 Optimization Theory
The theory on Lagrangian multipliers states that to
$\begin{split}\text{optimize}\quad&f(\boldsymbol{\mathbf{x}})\\\ \text{subject
to}\quad&g(\boldsymbol{\mathbf{x}})\geq 0\\\ \end{split}$
one should optimize the Lagrangian function
$\begin{split}&L(\boldsymbol{\mathbf{x}},\alpha)=f(\boldsymbol{\mathbf{x}})+\alpha
g(\boldsymbol{\mathbf{x}})\\\ \text{subject
to}\quad&g(\boldsymbol{\mathbf{x}})\geq 0\\\ &\alpha\geq 0\\\ &\alpha
g(\boldsymbol{\mathbf{x}})=0.\end{split}$
These conditions are known as the Karush-Kuhn-Tucker(KKT) conditions. More
generally, to add more constraints $g_{j}(\boldsymbol{\mathbf{x}})$, replace
the $\alpha g(\boldsymbol{\mathbf{x}})$ with a linear combination of all
Lagrange multipliers $\alpha_{j}$ and their corresponding functions
$g_{j}(\boldsymbol{\mathbf{x}})$Bishop :
$\begin{split}\text{optimize}\quad\qquad\\!&L(\boldsymbol{\mathbf{x}},\\{\alpha_{j}\\})=f(\boldsymbol{\mathbf{x}})+\sum_{j=1}^{J}\alpha_{j}g_{j}(\boldsymbol{\mathbf{x}})\\\
\text{subject to}\quad\forall j:\ &g_{j}(\boldsymbol{\mathbf{x}})\geq 0\\\
&\alpha_{j}\geq 0\\\ &\alpha_{j}g_{j}(\boldsymbol{\mathbf{x}})=0.\end{split}$
In order to quickly find a solution to (2.2) it can now be rewritten as the
Lagrangian function
$\displaystyle
L(\boldsymbol{\mathbf{w}},b,\alpha)=\underbrace{\frac{1}{2}\lVert\boldsymbol{\mathbf{w}}\rVert^{2}}_{f(\boldsymbol{\mathbf{x}})}-\sum_{i=1}^{\ell}\alpha_{i}\underbrace{(y_{i}(\langle\boldsymbol{\mathbf{w}},\boldsymbol{\mathbf{x}}_{i}\rangle+b)-1)}_{g_{i}(\boldsymbol{\mathbf{x}})}.$
The constraint function is negative because we are minimizing wrt
$\lVert\boldsymbol{\mathbf{w}}\rVert$ and $b$ while maximizing wrt
$\boldsymbol{\mathbf{\alpha}}$. To finally arrive at what is called the dual
representation of the maximum margin problem the derivatives of $L$ wrt to
$\boldsymbol{\mathbf{w}}$ and $b$, are set to $0$. Maximizing this dual
representation,
$\begin{split}&W(\boldsymbol{\mathbf{\alpha}})={\tilde{L}}(\boldsymbol{\mathbf{\alpha}})=\sum_{i=1}^{\ell}\alpha_{i}-\frac{1}{2}\sum_{i=1}^{\ell}\sum_{j=1}^{\ell}\alpha_{i}\alpha_{j}y_{i}y_{j}\langle\boldsymbol{\mathbf{x}}_{i},\boldsymbol{\mathbf{x}}_{j}\rangle,\\\
\text{by finding}\quad&\boldsymbol{\mathbf{\alpha}},\\\ \text{subject
to}\quad&\forall i:\alpha_{i}\geq 0,\\\
&\sum_{i=1}^{\ell}\alpha_{i}y_{i}=0,\end{split}$ (2.3)
will construct the maximal margin classifier.Nello , Bishop , Vapnik
The instances that have a corresponding $\alpha_{i}>0$ are called support
vectors. That is because they lie on the margin. They are thus used in the
decision function.
Note how the input variables $\boldsymbol{\mathbf{x}}_{i}$ are only used in an
inner product which let the SVM avoid the curse of dimensionality caused by a
data set with instances of too high dimension.Nello
#### 1.5 The Kernel Trick
The Kernel Trick is used implicitly in Support Vector Machines but it has also
been tried out in e.g. RBF Networks, which is a kind of ANN.Bishop
The inner product used in the dual optimization problem can be a linear one.
Though it will not separate the instances fully when the dataset is not
linearly separable, data must be mapped to another space where it is.
A non-linear feature function $\phi(\boldsymbol{\mathbf{x}})$ can do such a
mapping. However, there is no need to know the feature function explicitly, it
is easier to define it implicitly via a Mercer Kernel.Nello
A complete, normed space with an inner product is called a Hilbert Space One
of the beauties of Hilbert spaces lies in that any given function in the
$L_{2}$ space could be approximated infinitely well in the
$\lVert\cdot\rVert_{2}$ and represented by an infinite linear combination of
some coefficients and some basis functions. An example of this is the Fourier
Series using Fourier coefficients and the Dirichlet Kernel Functions
$\\{e^{-ikx}\\}_{k}$.
A special kind of Hilbert spaces are the ones which are called Reproducing
Kernel Hilbert spaces. A function
$\langle\boldsymbol{\mathbf{x}}_{i},\boldsymbol{\mathbf{x}}_{j}\rangle=K(\boldsymbol{\mathbf{x}}_{i},\boldsymbol{\mathbf{x}}_{j})=\phi(\boldsymbol{\mathbf{x}}_{i})\phi(\boldsymbol{\mathbf{x}}_{j})$
is called a kernel when it satisfies the criteria in Mercer’s Theorem.
A Mercer kernel $K$ is defined as an inner product on elements of some space
$\mathcal{X}$.Nello An inner product is a function that is a positive-
definite sesqui-linear111anti-linear in the second argument and linear in the
first form. In the $\mathbb{R}$ case this becomes a function
$\displaystyle\langle\cdot,\cdot\rangle$
$\displaystyle:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$ such that
$\displaystyle K(\boldsymbol{\mathbf{x}},\boldsymbol{\mathbf{z}})$
$\displaystyle=\langle\boldsymbol{\mathbf{x}},\boldsymbol{\mathbf{z}}\rangle={\langle\boldsymbol{\mathbf{z}},\boldsymbol{\mathbf{x}}\rangle}={K(\boldsymbol{\mathbf{z}},\boldsymbol{\mathbf{x}})}$
(Symmetry) $\displaystyle
K(a\boldsymbol{\mathbf{x}}+b\boldsymbol{\mathbf{y}},c\boldsymbol{\mathbf{z}})$
$\displaystyle=ab{c}\big{(}K(\boldsymbol{\mathbf{x}},\boldsymbol{\mathbf{z}})+K(\boldsymbol{\mathbf{y}},\boldsymbol{\mathbf{z}})\big{)}$
(Bilinearity)
$\displaystyle\forall\boldsymbol{\mathbf{x}}:K(\boldsymbol{\mathbf{x}},\boldsymbol{\mathbf{x}})$
$\displaystyle\geq 0$ (Positivity) $\displaystyle
K(\boldsymbol{\mathbf{x}},\boldsymbol{\mathbf{x}})$
$\displaystyle=0\iff\boldsymbol{\mathbf{x}}=\boldsymbol{\mathbf{0}}$
(Definiteness) A Mercer kernel also have non-negative eigenvalues
$\lambda_{i}$ of the Gram matrix $\boldsymbol{\mathbf{G}}$ since it’s defined
as a Hermitian matrix $\displaystyle\forall i:\lambda_{i}$ $\displaystyle\geq
0|\boldsymbol{\mathbf{G}}$ (Positive semi-definite Gram matrix)
$\displaystyle\boldsymbol{\mathbf{G}}$
$\displaystyle=\Big{(}K\big{(}{\boldsymbol{\mathbf{x}}}_{i},{\boldsymbol{\mathbf{x}}}_{j}\big{)}\Big{)}_{i,j\in[1,\ell]^{2}}$
(2.4)
Note that the elements of the space $\mathcal{X}$ do not need to be real
vectors as they will be in this context, they could also be e.g. strings of
symbols as well. As soon as a symmetric sesqui-linear positive-definite
function could be defined on the elements of the space $\mathcal{X}$, the
space becomes an inner product space and the Support Vector Machine will do
its job.Nello
Here are some commonly used Mercer kernels defined on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$Nello , Bishop , Vapnik :
$\displaystyle\langle\boldsymbol{\mathbf{x}},\boldsymbol{\mathbf{y}}\rangle_{Linear}$
$\displaystyle={\boldsymbol{\mathbf{x}}}^{\mathsf{T}}\boldsymbol{\mathbf{y}}$
(Linear, dot product, kernel)
$\displaystyle\langle\boldsymbol{\mathbf{x}},\boldsymbol{\mathbf{y}}\rangle_{Poly}$
$\displaystyle=\Big{(}{\boldsymbol{\mathbf{x}}}^{\mathsf{T}}\boldsymbol{\mathbf{y}}+1\Big{)}^{d}$
(Complete Polynomial of degree $d$)
$\displaystyle\langle\boldsymbol{\mathbf{x}},\boldsymbol{\mathbf{y}}\rangle_{RBF}$
$\displaystyle=\exp\left(-\frac{1}{2\sigma^{2}}\lVert\boldsymbol{\mathbf{x}}-\boldsymbol{\mathbf{y}}\rVert\right)$
(Gaussian, Radial Basis Function)
$\displaystyle\langle\boldsymbol{\mathbf{x}},\boldsymbol{\mathbf{y}}\rangle_{MLP}$
$\displaystyle=\tanh({\boldsymbol{\mathbf{x}}}^{\mathsf{T}}\boldsymbol{\mathbf{y}}+b)$
(Multilayer perceptron, for some $b$) the norm used in RBF is usually the
euclidean distance, $p=2$ below $\displaystyle\lVert x-y\rVert_{L^{p}}$
$\displaystyle=\Big{(}\sum_{i}\lvert x_{i}-y_{i}\rvert^{p}\Big{)}^{1/p}$
($L^{p}$ distance)
#### 1.6 Gradient Ascent
An easy approach to find coefficients $\boldsymbol{\mathbf{\alpha}}$ is to
update them in the direction of the gradient of the objective function
$W(\boldsymbol{\mathbf{\alpha}})$,
$\displaystyle\frac{\partial
W(\boldsymbol{\mathbf{\alpha}})}{\partial\alpha_{i}}$
$\displaystyle=1-y_{i}\sum_{j=1}^{\ell}\alpha_{j}y_{j}\langle\boldsymbol{\mathbf{x}}_{i},\boldsymbol{\mathbf{x}}_{j}\rangle.$
To maximize the objective function $W(\boldsymbol{\mathbf{\alpha}})$ one could
just iterate $\displaystyle\alpha_{i}^{\prime}$
$\displaystyle\leftarrow\alpha_{i}+\eta\frac{\partial
W(\boldsymbol{\mathbf{\alpha}})}{\partial\alpha_{i}}.$
Where: $\eta$ – the learning rate
It is shown e.g. in Nello’s book that setting
$\eta=\frac{1}{K(\boldsymbol{\mathbf{x}}_{i},\boldsymbol{\mathbf{x}}_{j})}$
maximizes the gain if the $\alpha_{i}\in[0,C],C\in\mathbb{R}$ and that
convergence is guaranteed if the hyperplane exists.Nello
#### 1.7 Multiclass SVM
There are three major methods for training a set of classifiers to be able to
classify several classesHsuLinMultiClass , i.e. $|\mathcal{Y}|=k>2$.
In the one-against-the-rest method $k$ binary classifiers are created where
classifier $i\in[0,k)$ is told that all examples with class $i$ are positive
and the rest are negative. When predicting which class
$\boldsymbol{\mathbf{x}}$ belongs to all classifiers are tested and the one
which gave the highest certainty wins.
In the one-against-one method $k(k-1)/2$ binary classifiers are created such
that all 2-combinations of classes $i,j$ have a corresponding classifier.
$\displaystyle C^{n}_{2}={n\choose
2}=\frac{n!}{2!(n-2)!}=\frac{n(n-1)(n-2)!}{2(n-2)!}=\frac{n(n-1)}{2}$
The prediction is then done by voting, all binary classifiers vote on their
respective class $i$ or $j$. The class with the highest vote wins, this
approach is called the ”Max Wins” strategy.
Direct Acyclic Graph SVM (DAGSVM) is the third method. It uses the same
training method as one-against-one but a different decision mechanism. The
classifiers are placed in a rooted DAG with the classifiers as internal nodes
and the classes as leaves. Starting at the root a binary decision means move
either left or right. When a leaf is reached the decision is
done.HsuLinMultiClass
### 2 Features
Features, or descriptors, try to take useful information out of an image —
color distribution, measures on edges and texture properties. They capture
information in a more condensed and efficient way than by just using the color
values in each pixel.
These descriptors are also scale invariant — it does not matter which size the
images have. This is necessary as the images have different sizes.
Scalable Color Descriptor, Color Structure Descriptor and Color Layout
Descriptor are the three color descriptors that I describe below and that are
implemented in the project. After the description of those come descriptions
of two texture descriptors. One of them is similar to the Homogeneous Texture
Descriptor from MPEG-7. Another set of descriptors, named Visual Texture
Features, is from an article by Amadasum and King which describe computational
measures which approximate how humans perceive
texture.Amadasun1989TexturalFeaturesCorrespondingToTexturalProperties
#### 2.1 Scalable Color Descriptor
The HSV space is uniformly quantized into a 3D histogram of 256 bins. Hue is
divided into 16 levels, Saturation into 4 and Value into 4. In the MPEG-7
specification the $16\times 4\times 4=256$ bins are truncated to a 11-bit
integer mapped to a non-linear 4-bit representation and then encoded using a
Haar transform to drastically reduce space footprint. The scalability in this
descriptor comes from the ability to choose how many Haar coefficients to
store, see an article by Manjunath et al. for more
details.manjunath2001ColorTextureDescriptors
#### 2.2 Color Structure Descriptor
To express local color structure in an image this descriptor slides an
$8\times 8$-structuring element across the image counting in how many of these
elements each color exists. By this technique one can differ between the
images in figure 2.1.
This descriptor is scale invariant as the structuring elements spatial extent
scale with the image size. The structure element uses replacement sub-sampling
if the image is larger than $256\times 256$ pixels. If e.g. a $512\times 512$
image is processed every other row and column will represent the image and the
rest of the $2\times 2$ areas are thrown away. More generally
$\displaystyle p$
$\displaystyle=\max\\{0,\mathrm{round}(0.5\log_{2}(WH)-8)\\}$ (2.5)
$\displaystyle K$ $\displaystyle=2^{p},\,\,E=8K$ (2.6)
Where: $E\times E$ – spatial extent of the structuring element
$K$ – sub-sampling factor
Each bin in the generated histogram represents the number of occasions a
structuring element is found to contain the color associated with the bin.
[Blackness $\pi 2^{2}$] [Blackness $4\pi 1^{2}$]
Figure 2.1: These images contain the same amount of black and would yield an
identical color histogram but a different color structure descriptor.
#### 2.3 Color Layout Descriptor
This is kind of a low-pass filter capturing spatial information. Again it is
inspired by the MPEG-7 specification. The image is first divided in $8\times
8$ blocks. Then interpolation sub-sampling222the average of all pixels
involved in the block represent the whole block as opposed to replacement sub-
sampling where a single pixel represent the whole block is applied, i.e.
calculating the average color in each block, giving one representative color
for each block. A 2D discrete cosine transform (DCT-II) is performed on the
resulting $8\times 8$ matrix. Low-frequency coefficients are selected using
zigzag scanning order, see figure 2.2. In MPEG-7 the 6 first Y, the 3 first of
U and V coefficients are extracted.
$\displaystyle\begin{pmatrix}1&3&4&10&11\\\ 2&5&9&12&19\\\ 6&8&13&18&20\\\
7&14&17&21&24\\\ 15&16&22&23&25\\\ \end{pmatrix}$ Figure 2.2: Zigzag scan
order of a $5\times 5$ matrix
#### 2.4 Homogeneous Texture Descriptor
Gabor wavelets have proved to be the best set of features compared to pyramid-
structured wavelet transform (PWT), tree-structured wavelet transform (TWT)
and multi-resolution simultaneous autoregressive model (MR-SAR) based
descriptors.manjunath1996TextureFeaturesForBrowsingAndRetrievalOfImageData
They are used in the MPEG-7 Homogeneous Texture Descriptor (HTD).
Gabor wavelets are a family of modulated Gaussians, they form a complete basis
set implying that, any given function $f(\cdot,\cdot)$ can be expanded in
terms of these basis functions. However, as they are not orthonormal, there is
redundant information present in a set of coefficients. To decrease that
redundancy I follow the strategy used by Manjunath et al., that is aligning
the Gaussians such that their half-peaks meet like in figure
2.3.manjunath2000ATextureDescriptorForBrowsingAndSimilarityRetrieval
Figure 2.3: $U_{hi}=0.4,U_{lo}=0.05,S=5,K=6$
. To achieve this we first make a change of variables. The Gaussian is a
Gaussian in both frequency and space domains. The width of the Gaussian in the
frequency domain ($\sigma_{u},\sigma_{v}$) is inversely related to the
Gaussian in the space domain ($\sigma_{x},\sigma_{y}$). In other words, the
wider the Gaussian, the narrower its bandwidth.wikiGaussianFunction ,
Wallis_Linear_Models_Of_Simple_Cells_Mammal_Vision_Model
$\displaystyle\sigma_{x}$
$\displaystyle=\frac{1}{2\pi\sigma_{u}},\quad\\!\sigma_{y}=\frac{1}{2\pi\sigma_{v}}$
These parameters are needed for scaling
$\displaystyle a$
$\displaystyle=(U_{hi}/U_{lo})^{1/(S-1)},\quad\\!\sigma_{u}=\frac{(a-1)U_{hi}}{(a+1)\sqrt{2\ln
2}},$ $\displaystyle\sigma_{v}$
$\displaystyle=\tan\left(\frac{\pi}{2K}\right)\\!\left[U_{hi}-2\ln
2\left(\frac{\sigma^{2}_{u}}{U_{hi}}\right)\middle]\\!\middle[2\ln
2-\left(\frac{(2\ln
2)\sigma_{u}}{U_{hi}}\right)^{\\!\\!2\,}\right]^{\frac{1}{2}}$
Where: $U_{lo}\in\mathbb{R}$ – lower center frequency of interest
$U_{hi}\in\mathbb{R}$ – upper center frequency of interest
$m\in[0,S)\subset\mathbb{Z}^{+}$ – scale index
$S\subset\mathbb{N}$ – number of scales
$a>1\in\mathbb{R}$ – scale factor
For different orientations the image needs to be rotated before filtering and
scaling wrt $a$.
$\displaystyle x^{\prime}$ $\displaystyle=a^{-m}(x\cos\theta+y\sin\theta)$
$\displaystyle y^{\prime}$ $\displaystyle=a^{-m}(-x\sin\theta+y\cos\theta)$
$\displaystyle\theta$ $\displaystyle=n\pi/K$
Where: $n\in[0,K)\subset\mathbb{Z}^{+}$ – orientation index
$K\in\mathbb{N}$ – number of orientations
$\theta\in[0,\pi)$ – orientation angle
The generated filter bank are matrices that should be convoluted with the
image
$\displaystyle I^{\prime}=I*G$
Where: $*$ – the convolution operator
See section 3 for details about 2D convolution. In figure 3.1 images of the
Gabor wavelet filter bank kernels of different orientations are presented.
In MPEG-7, rotation invariance is achieved in this descriptor, by rotating the
features in the direction of the dominant direction.
#### 2.5 Visual Texture Features
The features described in the article by Amadasun and King are implemented.
These are features corresponding to properties of texture that humans can
perceive. In the article measures of coarseness, contrast, busyness,
complexity and strength are introduced and compared by rank with how humans
sensed ten natural textures from the widely used Brodatz’s album. I give here
a very brief overview of the proposed measures. They all use a column vector
called neighborhood gray-tone difference matrix
(NGTDM).Amadasun1989TexturalFeaturesCorrespondingToTexturalProperties
##### 2.5.1 Neighborhood Gray-Tone Difference Matrix
In a pixel $p$ with coordinates $\langle k,l\rangle$ neighborhood of size $d$,
i.e. of the square surrounding a pixel, but without the center pixel the mean
is calculated.
$\displaystyle\begin{split}\bar{A}_{p}=\bar{A}(k,l)=\frac{1}{W-1}\left[\sum_{m=-d}^{d}\sum_{n=-d}^{d}f(k+m,l+n)\right],\\\
\quad(m,n)\neq(0,0)\end{split}$ (2.7)
Where: $W=(2d+1)²$
The $i$th entry in the NGTDM is a sum of deviations from the mean of the
center pixel, only concerning those pixels in the image which do not lie in
the peripheral regions of width $d$.
$\displaystyle s(i)=\begin{cases}\displaystyle\sum_{p\in
N_{i}}\left|i-\bar{A}_{p}\right|,&\;\text{there is a pixel with gray-tone
}i\\\ 0,&\;\text{otherwise}\end{cases}$ (2.8)
Where: $N_{i}$ – the pixels with gray-tone $i$
$G_{h}$ – the largest gray-tone
The relative frequency, i.e. the probability of occurrence, of different gray-
tones is calculated as:
Amadasun1989TexturalFeaturesCorrespondingToTexturalProperties
$\displaystyle p_{i}$ $\displaystyle=\lvert N_{i}\rvert/n,$ $\displaystyle n$
$\displaystyle=(width-2d)(height-2d).$ (2.9)
Note that (2.9) allows a rectangular region of interest as opposed to the
square regions used in the article by Amadasun and King, and that $n$ replaces
$n^{2}$ in the formulas.
##### 2.5.2 Coarseness
Coarseness is a measure of how rough a surface is, e.g. how large particles it
is composed of.
$\displaystyle f_{cos}=\left[\epsilon+\sum_{i=0}^{G_{h}}p_{i}s(i)\right]^{-1}$
This is (inversely) a weighted sum of the deviations from the center pixels
wrt the surrounding pixels. The small value $\epsilon$ is to cope with
division by $0$.
##### 2.5.3 Contrast
High contrast means the intensity difference between neighboring regions is
large.
$\displaystyle f_{con}$
$\displaystyle=\left[\frac{1}{N_{g}(N_{g}-1)}\sum_{i=0}^{G_{h}}\sum_{j=0}^{G_{h}}p_{i}p_{j}(i-j)²\middle]\middle[\frac{1}{n}\sum_{i=0}^{G_{h}}s(i)\right]$
$\displaystyle N_{g}$ $\displaystyle=\sum_{i=0}^{G_{h}}Q_{i}$ $\displaystyle
Q_{i}$ $\displaystyle=\begin{cases}1,&\;\text{if }p_{i}\neq 0\\\
0,&\;\text{otherwise}\end{cases}$
Where: $N_{g}$ – the number of different gray-tones present in the image
The first factor is used to reflect the dynamic range of gray scale weighted
with the product of relative frequencies of the two gray-tone values under
consideration. The second factor increases with the amount of local variation
in intensity.
##### 2.5.4 Busyness
A busy texture is one where the spatial frequency of intensity changes are
high.
$\begin{split}f_{bus}=\left.\sum_{i=0}^{G_{h}}p_{i}s(i)\middle/\sum_{i=0}^{G_{h}}\sum_{j=i}^{G_{h}}ip_{i}-jp_{j}\right.\\!\\!,\\\
p_{i}\neq 0,\>p_{j}\neq 0\end{split}$
The numerator is a measure of the spatial rate of change in intensity,
inversely related to coarseness. The denominator is a summation of the
magnitude of differences between the different gray-tone values. This formula
differs slightly from the one described in the article by Amadasun and
KingAmadasun1989TexturalFeaturesCorrespondingToTexturalProperties — I’m
certain there’s a typo in that formula making it always zero.
##### 2.5.5 Complexity
Complexity means high information content. This could mean many primitives or
patches, especially if they have different average intensity.
$\displaystyle f_{com}=\sum_{i=0}^{G_{h}}\sum_{j=0}^{G_{h}}\left.\frac{\lvert
i-j\rvert}{n(p_{i}+p_{j})}\Big{(}p_{i}s(i)+p_{j}s(j)\Big{)}\right.$
An elaborate description of this formula (and the others in this section) are
found in the article by Amadasun and
KingAmadasun1989TexturalFeaturesCorrespondingToTexturalProperties .
##### 2.5.6 Texture strength
A strong texture is generally referred to as strong if its building blocks are
easily definable and clearly visible. Such texture tend to look attractive.
However a strong texture is difficult to define
conciselyAmadasun1989TexturalFeaturesCorrespondingToTexturalProperties . It is
defined as
$\displaystyle
f_{str}=\frac{\displaystyle\sum_{\begin{subarray}{c}i=0\end{subarray}}^{G_{h}}\sum_{\begin{subarray}{c}j=0\end{subarray}}^{G_{h}}\left(p_{i}+p_{j}\middle)\middle(i-j\right)^{2}}{\displaystyle\epsilon+\sum_{i=0}^{G_{h}}s(i)}\\!,\qquad{p_{i}\neq
0,\>p_{j}\neq 0}.$
Where the numerator is a factor stressing the differences between intensity
levels, and therefore may reflect intensity differences between adjacent
primitives. The probabilities $p_{.}$ tend to be high for large primitives.
The denominator would be small for coarse texture and high for busy or fine
textures considering the definition in (2.8).
### 3 Fast 2D Convolution
Two-dimensional discrete convolution in the spatial domain is defined as
$\displaystyle(f*g)[n]\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{m=-\infty}^{\infty}f[m]\cdot
g[n-m].$
By the Circular Convolution TheoremwikiCircularConvolution this can instead
be done in the frequency domain considering
$\displaystyle\mathcal{F}\\{f*g\\}=\mathcal{F}\\{f\\}\cdot\mathcal{F}\\{g\\}$
(2.10)
Where: $*$ – the convolution operator
$\mathcal{F}\\{\cdot\\}$ – the Fourier Transform (FT)
First apply FT to image and to convolution kernel, then multiply the two
matrices element-wise. To get the filtered image just apply inverse FT.
For this to work the kernel has to be placed in a matrix the same size as the
image, wrapped around the origin333origin aka DC component, zero frequency,
which in FFTW is at position $\langle 0,0\rangle$, like in figure 2.4. Also,
there are border cases in the image, it has to be padded with wraparound
pixels.Convolution2DNVIDIA
$\displaystyle\overbrace{\begin{pmatrix}11&12&13&14&15\\\ 21&22&23&24&25\\\
31&32&33&34&35\\\ 41&42&43&44&45\\\ 51&52&53&54&55\\\
\end{pmatrix}}^{\text{Example $5\times 5$
kernel}}\xrightarrow{\text{Layout}}\overbrace{\begin{pmatrix}33&34&35&0&\dots&0&31&32\\\
43&44&45&0&\dots&0&41&42\\\ 53&54&55&0&\dots&0&51&52\\\ 0&0&0&0&\dots&0&0&0\\\
[5]{8}\\\ 0&0&0&0&\dots&0&0&0\\\ 13&14&15&0&\dots&0&11&12\\\
23&24&25&0&\dots&0&21&22\end{pmatrix}}^{\text{Image height $\times$ Image
width matrix}}$ Figure 2.4: How to make sure the kernel wraps around the
origin in frequency space
### 4 Scaling data
Scaling is very important. If scaling is not applied to all features a feature
with a larger numeric range may dominate others with smaller numeric range.
$\displaystyle\displaystyle range$ $\displaystyle=\max_{i}x_{i}-\min_{i}x_{i}$
$\displaystyle midrange$
$\displaystyle=\left(\displaystyle\max_{i}x_{i}+\min_{i}x_{i}\middle)\right/\\!2$
$\displaystyle x^{\prime}_{i}$
$\displaystyle=\begin{cases}\displaystyle\frac{x_{i}-midrange}{range/2},&range\neq
0\\\ 0,&range=0\end{cases}$
Where: $x_{i}$ – feature value of example $i$
$i\in[0,\ell)\subset\mathbb{Z}^{+}$
## 3 Material and Methods
### 5 Material
Blood samples were taken from four individuals. The cells were photographed on
a CellaVision™ DM-96. The width of the images lies in the range $[119,267]$.
The height of the images lies in the range $[119,258]$. On average an image is
about $139\times 139$ pixels. This correspond to about $13.7$
$\umu\mathrm{m}$.
The cells are normal, e.g. there are no cancer cells or malaria infected
cells. There are very few (2) blast cells indicating the only possible cancer
type would be lymphoma, i.e. a cancer in the lymph nodes.
The cells were classified on the CellaVision™ DM-96 and its result was taken
as ground truth. The machine is 90% to 95% correct depending on the
individual. The cell types of the data set are given in table 3.1. Typical
relative frequencies of the cells are found in table 1.1. Typical images of
some common cells are found in figure 1.1.
Class No. | Class Name
---|---
1 | neutrophil granulocytes, segmented
6 | neutrophil granulocytes, band
2 | eosinophil granulocytes
3 | basophil granulocytes
4 | lymphocytes
7 | lymphocytes, variants
5 | monocytes
9 | myelocytes
10 | meta-myelocytes
11 | blast, immature cell
21 | artifacts
24 | broken cell
25 | thrombocytes (platelets)
29 | clots of thrombocytes
Table 3.1: Cell types classified in the data set
From the set of images of the cells a range of descriptors, or features, were
extracted. A set of features extracted from a single image, called instance or
example, is denoted $\boldsymbol{\mathbf{x}}$ and the space of all possible
features is denoted $\mathcal{X}$.
A Support Vector Machine (SVM) was trained using the set of features
described.
### 6 Implementation details
#### 6.1 Support Vector Machine
The SVM was written in C++ within the Boost C++ Libraries framework. The Gram
matrix $\boldsymbol{\mathbf{G}}$, defined in (2.4), the output of the kernel
function, is cached in memory to dramatically reduce running time.
##### 6.1.1 A Stochastic Gradient Ascent Variant
Stochastic gradient ascent differs from ordinary gradient ascent in that the
coefficients $\alpha_{i}$ updated are used right away, instead of in the next
iteration. In this project a variant of the stochastic gradient ascent method
of training a SVM were implemented.
The coefficients $\boldsymbol{\mathbf{\alpha}}_{KKT}$ that invalidate the
Karush-Kuhn-Tucker (KKT) conditions are selected first for update. They are
likely the ones that will affect the solution most rapid. When these satisfies
the KKT conditions, or when no progress has been made in some iterations, the
greater problem of updating all coefficients $\boldsymbol{\mathbf{\alpha}}$ is
considered.
##### 6.1.2 Multiclass SVM
I use the one-against-the-rest methodHsuLinMultiClass because it is the
simplest and it has similar precision to the latter twoVapnik ,
HsuLinMultiClass . The latter two are however faster to train because they can
train all the classifiers at once.Nello
#### 6.2 Features
##### 6.2.1 Scalable Color Descriptor
In MPEG-7 the 3D color histogram bins are reduced in size by truncation and
encoding (see 2.1). To release the SVM from this hassle it receives the values
as ordinary real values representing the relative frequency of color channel
values. The bounded time complexity to calculate this descriptor is
$O(3W\\!H)$.
##### 6.2.2 Color Structure Descriptor
This is implemented by calculating a histogram for each structuring element
and then summing over all structuring elements
$\displaystyle
h(m)=\sum_{i=1}^{\frac{W-8K}{K}}\sum_{j=1}^{\frac{H-8K}{K}}\min\\{1,h_{s_{i,j}}(c_{m})\\}$
(3.1)
Where: $m$ – bin index in the final histogram
$c_{m}$ – quantized color level
$h_{s_{i,j}}$ – histogram for structuring element $\langle i,j\rangle$
Calculating this descriptor is much more expensive than Scalable Color
Descriptor described in section 2.1, $O(\frac{(w-8k)(h-8k)}{k}8^{2})$ for each
channel, this is more than a 30-fold increase on a $640\times 480$ image
compared to the above.
##### 6.2.3 Color Layout Descriptor
The Discrete Cosine Transform of type DCT-II is calculated using the software
library FFTW3 (Fastest Fourier Transform in the West). The zigzag scanning
order described in figure 2.2 is implemented as an C++ STL iterator using the
simple algorithm presented in listing 1. A wider low pass band is used than in
MPEG-7. The 10 first Y (6 in MPEG-7), the 5 first of U and V (3) coefficients
are extracted.
⬇
x = 0; y = 0; forward = true;
value_type get_current() { return source(x,y); }
void next() {
if (forward)
if (y < length-1) {
y ++; x –;
if (x < 0) {
x = 0;
forward = false;
}
} else
if (y == length-1) {
x ++;
forward = false;
}
else
if (x < length-1) {
x ++; y –;
if (y < 0) {
y = 0;
forward = true;
}
} else
if (x == length-1) {
y ++;
forward = true;
}
}
Listing 1: Simplified source for the implemented zigzag order on a
length$\times$length square matrix
##### 6.2.4 Homogeneous Texture Descriptor
By symmetry the filter might as well be rotated instead of the image and since
that is more efficient that is what is done. The bandwidth $b$ is set to 1
octave by relation (3.2) and setting $\sigma=\sigma_{x}$
$\displaystyle\frac{\sigma}{\lambda}$
$\displaystyle=\frac{1}{\pi}\sqrt{\frac{\ln
2}{2}}\frac{2^{b}+1}{2^{b}-1}\approx 0.5622$ (3.2)
In MPEG-7 rotation invariance in this descriptor is achieved by rotating the
features in the direction of the dominant direction. This is not implemented
in this project.
In figure 3.1 images of the Gabor wavelet filter bank kernels of different
orientations are presented.
[$\theta=0°$] [$\theta=36°$] [$\theta=72°$]
[$\theta=108°$] [$\theta=144°$] [$\theta=180°$]
Figure 3.1: Gabor Filter bank at scale = $S-1$ at different orientations. Gray
areas are the ones with zero magnitude, darker is negative, lighter is
positive
##### 6.2.5 Neighborhood Gray-Tone Difference Matrix
The $\bar{A}$ used in the Neighborhood Gray-Tone Difference Matrix (2.7) can
be divided into subproblems which do not need to be calculated every time. By
keeping the center value $(m,n)=(0,0)$ in the sum (not writing out
normalization)
$\displaystyle A^{\prime}(k,l)$
$\displaystyle=\sum_{m=-d}^{d}\sum_{n=-d}^{d}f(k+m,l+n),$ it can also be
written as $\displaystyle A^{\prime}(k,l)$
$\displaystyle=\begin{cases}\begin{split}\underbrace{A^{\prime}(k,l-1)}_{\text{above}}+\phantom{\qquad\text{or
as }}\\\ \sum_{m=-d}^{d}f(k+m,l+d)-f(k+m,l-d-1)\qquad\text{or as
}\end{split}\\\ \begin{split}\underbrace{A^{\prime}(k-1,l)}_{\text{to the
left}}+\\\ \sum_{n=-d}^{d}f(k+d,l+n)-f(k-d-1,l+n).\end{split}\end{cases}$
Given the value above or the value to the left the others can be calculated
faster.
To find all $\bar{A}$ first fill in a table with all $A^{\prime}$, from left
to right, top-down. Then for all positions remove the center value and make
sure the accumulated value is correctly normalized. The time complexity is
thereby reduced from $O(d^{2})$ per pixel to $O(d)$ per pixel.
#### 6.3 Convolution
Using the method for convolution described in section 3 is much more efficient
than the naive approach of doing the calculations in the spatial domain. It
reduces the complexity from $O(K²)$ per pixel, where $K$ is the size of the
convolution kernel, to $O(\log N)$, where the image is $N\times N$ in size and
$N=2^{k},k\in\mathbb{Z}^{+}$. The last requirement make sure that the much
more efficient Fast Fourier Transform (FFT) can be used instead of a normal
Discrete Fourier Transform (DFT).
With the largest kernel used, $K²=91²=8281$, and a $1000\times 1000$ image,
$\log 1000\approx 6.9$, a thousandfold speed-up can be achieved.
These figures are however for FFT on matrices of size $N=2^{k}$. Padding to
the next larger 2-power is not implemented since the software library used for
FFT, called FFTW444Heavily used library with an impressing architecture, used
in e.g. Matlab (Fastest Fourier Transform in the West) supports other sizes
too and still provides great speed.
#### 6.4 Data View
The classifiers view data. Rather than giving them the data structure holding
data directly an abstraction was built named DataView. The abstraction was
realized in 11 classes which are found together with their base abstract class
in figure 3.2. The derived classes can all be used transparently releasing the
classifier and data set loader from the tasks of the views.
Figure 3.2: Abstract class (interface) to data views and their realizations
These three views below contain pointers to the real data.
* DataSetView
view of data represented by a DataSet instance
* ExampleView
view of data represented by a vector of Example instances
* ArrayView
view of data from an boost::Array, convenient for the unit tests concerning
views
The views below contain other views and just map their values. They are often
chained together to get the wanted view.
* DataViewScaled
view the features as if they were in the range $[-1,1]$, avoids feature-wise
bias, see section 4
* DataViewRange
selected only a subset of the examples, used in e.g. cross-validation
* DataViewConcat
view two views as if they were one, also used in cross-validation
* DataViewShuffle
shuffle the order of examples. It is of course not wanted to split an ordered
set and train on the first part and test on the other, a class may then be
present only in the latter
* DataViewClassMapLinear
if e.g. only classes $\\{0,3,42,\ldots\\}$ exists it is convenient if they can
be represented by $\\{0,1,\ldots\\}$
* DataViewClassMapBinary
one class given is said to be positive, all other is said to be negative. Used
in multiclass classifier
* DataViewClassJoin
join groups of classes into new classes
* DataViewClassRemove
view with a class removed
## 4 Experimental Setup and Results
### 7 Experimental Setup
The CellaVision™ DM-96 machine achieves an error rate of approx. 5-10%
depending on individual. Thus there are errors in the ground truth.
I have divided the problem in two parts.
* •
the primary problem — the SVM should classify all classes present in the data
set.
* •
the simplified problem — some classes are merged and others are removed.
#### 7.1 Performance test method
In both cases 2-fold cross-validation is used to test performance. This means
that two models will be trained. In the first, half the data set is the
training set and the other half is the test set. In the other, the roles of
the subsets are swapped. This way both halves will act as both training and
test sets.
#### 7.2 Description of the simplified problem
Class 1 and 6, Neutrophil granulocytes, segmented and band variants are merged
to form class 30. Even human experts have approx. 25% error rate on these. It
is often more a matter of opinion than of objective decision.
Class 4 and 7, Lymphocytes and their variants, are joined. The variants are
rather uncommon, there are only 8 instances in the dataset, compared to 160 of
Lymphocytes. Due to the skew distribution these are merged to form class 31.
The following classes are removed. Class 0 are unidentified objects, it is a
very heterogeneous group but there are only 6 of them. Class 21 are artifacts,
random garbage, they are removed. Class 24 are broken cells, there are only 7
of them. Class 25 and 29 are thrombocytes and clots of them, i.e. platelets.
Since they aren’t even white blood cells they are removed. Class 11, called
blast is a kind of immature cell which would be interesting to classify but
there are only two of them so they are removed as well. Class 9 and 10 are
myelocytes and meta-myelocytes, which are a development stage of different
granulocytes. There can be e.g. eosinophilic myelocytes and basophilic
myelocytes. In the dataset they are also too rare to train a general
classifier. There are only a total of 4 myelocytes in this heterogeneous
group. All classes that are left are presented again in table 4.1.
Class No. | Class Name
---|---
30 (1+6) | neutrophil granulocytes
2 | eosinophil granulocytes
3 | basophil granulocytes
31 (4+7) | lymphocytes and variants
5 | monocytes
Table 4.1: Cell types left in the simplified problem
### 8 Results
#### 8.1 Primary Problem
The error rate in the primary problem is 10.8%. The type of kernel function
that was the most successful was the Polynomial kernel. This is compared to
the slightly better result using libSVM, 9.6%. See table 4.2.
Most confusion occurs between classes 1 (segmented neutrophil granulocytes)
and 6 (band neutrophil granulocytes). Much confusion is also present when
recognizing class 3 (basophil granulocytes) — they are often (2 of their total
of 7) misclassified as class 1 (segmented neutrophil granulocytes), which is a
very large group.
#### 8.2 Simplified Problem
In the simplified problem the error rate is 3.1%. Also in this problem the
most successful kernel was the Polynomial kernel. This is compared to the
better result using libSVM, 2.3%. See table 4.4.
In the simplified problem most confusion (by number) occurs between class 5
(monocytes) and the new class 31 (lymphocytes). By percentage the largest
confusion occurs between class 30 (segmented and band neutrophil granulocytes)
and class 3 (basophil granulocytes). Class 3 have only 8 instances of which 3
were misclassified as 30.
Implemented SVM Results
---
Kernel Type | Error Rate (%) | Parameters
| Total | Max | Min |
RBF with $L^{2}$ norm | 11,5385 | 12,0192 | 11,0577 | $\sigma²=20$
RBF with $L^{2}$ norm | 11,5385 | 12,0192 | 11,0577 | $\sigma²=22$
Polynomial | 11,7788 | 12,5 | 11,0577 | $d=2$
Polynomial | 11,0577 | 11,5385 | 10,5769 | $d=3$
Polynomial | 11,5385 | 12,0192 | 11,0577 | $d=4$
Polynomial | 10,8173 | 11,0577 | 10,5769 | $d=5$
Polynomial | 11,2981 | 12,0192 | 10,5769 | $d=6$
libSVM Results
RBF | 9,5923 | | $C=512,\gamma^{-1}=8192$
Table 4.2: SVM cell classifier results for the primary problem Number of
Confusions
---
| Guessed Class |
Class | $(n)$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 11 | 21 | 24
0 | (4) | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
1 | (205) | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
2 | (14) | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
3 | (7) | $\cdot$ | 2 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
4 | (104) | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 2 | $\cdot$ | 1 | $\cdot$
5 | (32) | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 2 | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$
6 | (12) | 1 | 6 | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
7 | (6) | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
11 | (1) | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
21 | (31) | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
24 | (1) | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$
Table 4.3: Confusion Matrix for the primary problem Implemented SVM Results
---
Kernel Type | Error Rate (%) | Parameters
| Total | Max | Min
RBF | 3,64109 | 4,42708 | 2,86458 | $C=128,\sigma^{2}=16$
RBF | 3,25098 | 4,16667 | 2,34375 | $C=512,\sigma^{2}=128$
Polynomial | 3,12094 | 3,38542 | 2,864 | $d=3$
Polynomial | 3,51105 | 3,90625 | 3,125 | $d=5$
libSVM results
RBF | 2,470 | | $C=8,\gamma^{-1}=128$
Polynomial | 2,3407 | | $C=8,\gamma^{-1}=128,d=3$
Polynomial | 3,5111 | | $C=8,\gamma^{-1}=128,d=5$
Table 4.4: SVM cell classifier results for the simplified problem Number of
Confusions
---
| Guessed class |
Class | $(n)$ | 2 | 3 | 5 | 30 | 31
2 | (20) | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
3 | (8) | $\cdot$ | $\cdot$ | $\cdot$ | 3 | $\cdot$
5 | (56) | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 2
30 | (517) | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$
31 | (168) | $\cdot$ | $\cdot$ | 7 | $\cdot$ | $\cdot$
Table 4.5: Confusion matrix for the simplified problem
## 5 Discussion
The accuracy achieved in the primary problem was 89.2% and in the simplified
problem 96.9%. I regard these results as good when compared to CellaVision™
DM-96’s result of the primary problem, 90-95%. One has to consider that there
are errors in the ground truth misleading the SVM. Thus, it is uncertain
whether the results are better than the DM-96 or worse. Because the DM-96 has
an error rate of about 5-10% a 0% error rate in the primary problem would mean
something like 5-10% error, while a 5% error could possibly mean 0-15% error.
I conclude that using the combination of MPEG-7 descriptors and visual texture
features in combination with SVM to classify cells is good but need further
investigation to find out how good. A more comprehensive study could
investigate whether a set of SVM or ANN variants perform better on the set of
features implemented or on the set of features developed at CellaVision™.
I would like to stress that using a SVM instead of an Artificial Neural
Network as in the CellaVision™ DM-96 machine is more statistically rigor —
Confidence intervals of the classifier can be found, which to my knowledge is
impossible in ANNs. In medicine it is important to know the strength of the
method used.
It would be very interesting to test the features on the real training set
they have developed at CellaVision™. The company has a training set of
thousands of cells classified by field experts. Some cell images required five
experts to be certain of the cell type. Without the errors in the ground truth
the results could possibly compete with the CellaVision™machine.
The result of the primary problem states that the most confused instances are
those that are guessed to be a segmented neutrophil (class 1) but that are a
band neutrophil (class 6) in the ground truth. These two often look very
similar, humans often have different opinions about which class cells are.
Also the CellaVision™ DM-96 have problems with these classes indicating that
there are several errors in the ground truth. The errors in ground truth
probably mislead the SVM. There are only 12 cell images of class 6 of which
some have the wrong class and there are 205 images of class 1, of which not
all are truly class 1. This situation pushes bias to the larger class.
In the result of the simplified problem most confusion occur between the
monocytes (class 5) and the lymphocytes (class 31). This was expected as they
are very hard to classify for both humans and the CellaVision™ DM-96. Late in
writing this thesis I discovered that there is great discrepancy in size of
these two types of cells. The discrepancy indicates that the size of the cells
could be used as a feature too.
#### Practical Use
Even the simplified problem would give useful information when applied to
medicine. Standard measures used in diagnosis involve counting the total
number of white blood cells, leukocytes, determining the distribution of
lymphocytes and granulocytes and determining the number of monocytes.
Malaria infected and cancer cells look different compared to healthy blood
cells. It would be interesting to test the features on these kind of cells to
be able to classify them as well.
#### Runtime Performance
To increase cache performance in the Color Structure Descriptor (section 2.2)
it would be wise to first extract all sub-samples i.e. the representative
color for each $K\times K$ area as the other pixels aren’t used. They will
otherwise quickly fill up the cache during memory pre-fetch. Now, the sub-
samples are viewed using a sub sampling view present in Generic Image Library
(boost::gil), contributed to Boost by Adobe. The views in GIL are virtual,
meaning they only keep information about offset calculations — no data is
duplicated.
The 2D convolution was first done in the spatial domain but I soon realized it
was way to slow with my bigger Gabor filter kernels of which the largest are
$91²$ pixels big. Instead the calculations are done in the frequency domain
which is much faster, see sections 3 and 6.3.
To improve performance of the Gabor Wavelet Filter further the kernels should
of course be kept in memory when generating features of many images, however
they are not.
To improve SVM training performance the Gradient Ascent training algorithm
must be replaced or at least improved. The algorithm implemented divide the
problem into a subproblem where the coefficients violating the KKT conditions
are first optimized. This is a heuristic called chunking in the
literatureNello . By using this, fewer elements of the Gram matrix, and their
corresponding support vectors, need to be kept in memory. This is something I
don’t take advantage of because I had enough memory for my purposes. By
refining chunking into decomposition where a fixed size chunk is optimized,
more data points can be used and convergence speed is increased. The
Sequential Minimization Optimization (SMO) takes decomposition to the extreme
and optimizes only two coefficients at a time and can thereby make sure that
the KKT condition, $\sum_{i=1}^{\ell}\alpha_{i}y_{i}=0$, is always true.
LibSVM uses a variant of this approach and it offer great
performance.CC01aLIBSVM ,
LIBSVMDimensionalityReductionViaSparseSupportVectorMachines
#### Beyond Gabor Filters
If modeling human brains is the objective, considering other approaches than
the Gabor wavelet would be interesting. A type of neurons in the first visual
cortex, called simple cells, have been recorded from monkey and cat. The
recordings and the elaborate analytical discussion in an article by Wallis
show that both difference of Gaussian$\times$Gaussian (DoGG) and Cauchy
functions model cortical cells better than Gabor wavelets for the measured
parameters.Wallis_Linear_Models_Of_Simple_Cells_Mammal_Vision_Model In an
article by Ashour et al. three other types of transforms are suggested —
ridgelets, curvelets and
contourlets.Ashour2008SupervisedTextureClassificationUsingSeveralFeaturesExtractionTechniquesBasedOnAnnAndSvm
Perhaps they can show increased performance.
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http://en.wikipedia.org/w/index.php?title=Discrete_Fourier_transform&oldid=226961702
#Circular_convolution_theorem_and_cross-correlation_theorem, 2008. [Online;
accessed 21-July-2008].
* [23] Wikipedia. ”Gaussian function — Wikipedia, The Free Encyclopedia”.
http://en.wikipedia.org/w/index.php?title=Gaussian_function&oldid=219261026,
2008. [Online; accessed 24-June-2008].
* [24] P. Wu, B.S. Manjunath, S. Newsam, and H.D. Shin. ”A texture descriptor for browsing and similarity retrieval”. Signal Processing: Image Communication, 16(1-2):33–43, 2000.
* [25] J. Zhang, M. Marszalek, and S. Lazebnik. ”Local Features and Kernels for Classification of Texture and Object Categories: A Comprehensive Study”. International Journal of Computer Vision, 73(2):213–239, 2007.
## 6 Software Usage
The software produced in this project can be found at
* •
`http://tobbe.nu/pub/2008/cell.morph.mpeg7.svm/`
The software has only been tested on an Ubuntu Linux system. However, the
software is written in portable C99 C++ and should work on all *nix platforms
that can supply the dependencies, perhaps even under cygwin under MS Windows.
The dependencies are
* •
C99 compliant C++ compiler (GNU g++ tested)
* •
Boost C++ Libraries, http://www.boost.org/
* •
FFTW3 (Fastest Fourier Transform in the West 3), http://www.fftw.org/
* •
GSL (GNU Scientific Library), http://www.gnu.org/software/gsl/
* •
libjpeg
* •
libpng
Below is a brief overview on how to use the most important programs in the
software package. There are other programs in the package but they are mostly
related to testing.
## Appendix 6.A train – Train a model
This is the program where most processing is done. It can
* •
train a model from a dataset
* •
test a model with a dataset
* •
load and/or save a model from/to a file
* •
perform cross-validation
Here is the syntax of the program train
MAIN ::= (MAIN_HELP | MAIN_DO)
MAIN_HELP ::= ./train [-h]
MAIN_DO ::= ./train MODE DATASET
MODEL_PARAMS SAVE_MODEL
MODE ::= LOAD_MODEL XVALIDATION
LOAD_MODEL ::= -l MODEL.model
XVALIDATION ::= -f N_FOLDS
N_FOLDS ::= 1 | INTEGER
DATASET ::= -d INTEGER
MODEL_PARAMS ::= -k KERN -p KERN_PARAM
-C DOUBLE -g GAP_TOL -m TERM
KERN ::= KERN_LIST | KERN_TYPE
KERN_LIST ::= 0
KERN_TYPE ::= 1 | 2 | 3 | 4 | 5 | 6 | 7
KERN_PARAM ::= DOUBLE
GAP_TOL ::= DOUBLE
TERM ::= BITMASK
BITMASK ::= 1 | 2 | 3
SAVE_MODEL ::= -o MODEL.model
Both cross-validation and saving of a model can be performed at the same time
if wanted. However, this will mean that train will create one model for each
fold but it is just the last one that will be saved. If cross-validation is
not wanted pass one (-f 1) fold. The double precision floating point number
passed with -C is a number used in the classifier, it is related to the KKT
conditions. The gap tolerance is also a double precision floating point number
which is used as a convergence criterion. It is the allowed gap between the
primal and dual objective function, the feasibility gap, which should be a
small number. The default gap is set to $10^{-3}$. The m terminator is a bit-
mask which control when a classifier is considered optimal, i.e. when training
will stop. The feasibility gap constraint is not used if -m 2 is passed, i.e.
when the first bit (1) is zero. The primary training terminator bit is 2 which
means that all KKT conditions must be satisfied to terminate training. The
default of 3 means that both these conditions must be satisfied.
## Appendix 6.B cellfeatures – Generate examples from the cell database
To generate features from all pairs of (image,ground truth class) in the cell
database the program cellfeatures is used. The file cellfeatures.data is
backed up before writing the features generated to it. This file can be used
by the program train.
MAIN ::= (MAIN_HELP | MAIN_DO)
MAIN_HELP ::= ./cellfeatures
MAIN_DO ::= ./cellfeatures DB
## Appendix 6.C jpeg_genfeature – Feature generation from images
To generate a set of features from image(s) the program called jpeg_genfeature
is used. It generate a set of features that can be classified later with
predict.
MAIN ::= (MAIN_HELP | MAIN_DO)
MAIN_HELP ::= ./jpeg_genfeature -?
MAIN_DO ::= ./jpeg_genfeature CROPIMAGE* -o FEATURESET.feat
CROPIMAGE ::= -i IMAGE.jpeg [-x left -y top -w width -h height]
## Appendix 6.D predict – Predicting a set of features
To predict a set of features, generated by jpeg_genfeature, the program called
predict is used. It needs a previously trained model generated by train.
MAIN ::= (MAIN_HELP | MAIN_DO)
MAIN_DO ::= ./predict -l MODEL.model -f FEATURESET.feat
MAIN_HELP ::= ./predict -?
## Appendix 6.E extractcelltype – Extract a class of images from the cell
database
To extract a specific class (as classified by CellaVision™ DM-96) from the
cell database, the program called extractcelltype is used.
MAIN ::= (MAIN_HELP | MAIN_DO)
MAIN_HELP ::= ./extractcelltype
MAIN_DO ::= ./extractcelltype CLASS DB
CLASS ::= INTEGER
DB ::= ALLXMLFILES | (XMLFILE ’ ’)*
ALLXMLFILES ::= ’.’
## Appendix 6.F extractcellid – Extract given instances from the cell
database
To extract given instances from a list of id numbers, the program called
extractcellid is used.
MAIN ::= (MAIN_HELP | MAIN_DO)
MAIN_HELP ::= ./extractcellid
MAIN_DO ::= ./extractcellid IDLIST DB
IDLIST ::= (INTEGER ’ ’)* ’x’
## Appendix 6.G extractcellinfo – Extract statistics of instances from the
cell database
To extract statistics about size, resolution and number of instances of a
specific class or of all classes the program called extractcellinfo is used.
MAIN ::= (MAIN_HELP | MAIN_DO)
MAIN_HELP ::= ./extractcellinfo
MAIN_DO ::= ./extractcellinfo CLASS DB
CLASS ::= CLASS_ALL | CLASS
CLASS_ALL ::= ’-1’
## Appendix 6.H tolibsvm – Save cell features in libSVM format
This program load the features saved in cellfeatures.data and dump them in
libSVM format on standard output. It takes no parameters.
./tolibsvm > cellfeatures.data.libsvm
|
arxiv-papers
| 2008-12-12T08:27:02 |
2024-09-04T02:48:59.350196
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tobias Abenius",
"submitter": "Tobias Abenius",
"url": "https://arxiv.org/abs/0812.2309"
}
|
0812.2363
|
# On the apsidal motion of BP Vulpeculae
Csizmadia, Sz szilard.csizmadia@dlr.de Illés-Almár, E Borkovits, T
Institute of Planetary Research, German Aerospace Center, D-12489 Berlin,
Rutherfordstrasse 2., Germany Konkoly Observatory, H-1525 Budapest, P. O. Box
67., Hungary Baja Astronomical Observatory, H-6500 Baja, Szegedi út Kt. 766,
Hungary
###### Abstract
BP Vulpeculae is a bright eclipsing binary system showing apsidal motion. It
was found in an earlier study that it shows retrograde apsidal motion which
contradicts theory. In this paper we present the first $BV$ light curve of the
system and its light curve solution as well as seven new times of the minima
from the years 1959-1963. This way we could expanded the baseline of the
investigation to five decades. Based on this longer baseline we concluded that
the apsidal motion is prograde agreeing with the theoretical expectations and
its period is about 365 years and the determined internal structure constant
is close to the theoretically expected one.
###### keywords:
stars: binaries: eclipsing
###### PACS:
97.80.Hn , Eclipsing binaries
††journal: New Astronomy
## 1 Introduction
The eclipsing nature of the $10^{th}$ magnitude star BP Vulpeculae was
discovered by Illés-Almár (1960) and she gave a period value of $1.938$ days
which was slightly corrected by Huth (1965). Then the star was neglected until
the end of the 20th century when Lacy (1992) published $UBV$ colours of the
system at certain phases. Later Lacy et al. (2003) presented more than 5000
data points in one colour ($V$) and they have solved their light curve and
determined very accurately absolute dimensions of the system. This $V$ light
curve was again solved by a simple, but automatized code (Devor, 2005) and the
result was similar to that of Lacy et al. (2003).
The system has an age of about 1 Gyr consisting of A7m V + F2m V spectral type
components and is slightly eccentric ($e=0.0345$, Lacy et al., 2003). In
eccentric binary star systems the apsidal line is revolving which is called
apsidal motion and therefore the time difference between a primary minimum and
the subsequent secondary minimum is variable. The apsidal motion is usually
caused by two effects: the tidal forces of the components and the effects of
general relativity. In the case of a more or less ordinary eccentric binary,
i.e. no very close third companion in the system, and no extreme stellar
rotation, both phenomena predict that the apside shows a prograde motion. The
total apsidal motion is a sum of the classical and relativistic contributions.
It is worthy to mention that there are several systems, like DI Herculis or AS
Camelopardalis where the observed apsidal motions highly differ from the
theoretically predicted ones (see e.g. Claret, 1998, and references therein).
It seems that the mentioned other effects, like third bodies etc. cause these
pecularities (for an overview see Borkovits et al., 2007).
Regarding the case of BP Vul, Lacy (2003) established a very short apsidal
motion period ($77\pm 22$yr) and they found that it is a retrograde one. If BP
Vul really would have had a retrograde apsidal motion it would be a new
representative of the systems which confronts theory and requires further
study because a retrograde apsidal motion is in contradiction with theory. But
the observational window in the work of Lacy (2003) was about one decade only
therefore this rough estimation for the period of apsidal motion should be
refined. For this refinement we present $BV$ observations of BP Vul which were
obtained 45-49 years ago and were not published until now.
## 2 Observations
The $BV$ data points of BP Vul published here were obtained by one of us (E.
Illés-Almár) during the years 1959-1963. The observations were carried out by
the 60 cm Newtonian-telescope of the Konkoly Observatory, which is located at
Budapest and it was installed in 1926. The detector was an 1P21 RCA
photoelectric tube and $B$ and $V$ filters were used. The observations were
reduced via a standard way. These differential magnitude data (they had been
measured to a comparison star and transformed into standard system) are
published here (see Tables 1-2 which are available electronically via the
SIMBAD homepage111http://simbad.u-strasbg.fr/simbad/). The $B$ and $V$ curves
can be seen in Figure 1.
Using the method of Kwee & van Woerden (1956) from these observations we
determined seven new times of minima which are reported in Table 3. All the
minima times we used for the analysis can be found in that Table, too.
Table 3: Observed times of minima of BP Vul. Meaning of weights (W): 10:
CCD/photoelectric minimum, 2: plate minimum, 1: visual observation, 0: not
used for the calculation because it is an outlier.
Time of Min. | W | Type | Ref. | Time of Min. | W | Type | Ref.
---|---|---|---|---|---|---|---
(HJD-2 400 000) | | | | (HJD-2 400 000) | | |
21787.723 | 2 | p | 1 | 37898.427 | 2 | p | 2
22144.750 | 2 | p | 1 | 37933.365 | 2 | p | 2
22637.569 | 2 | p | 1 | 37935.294 | 2 | p | 2
23607.741 | 2 | p | 1 | 38001.256 | 2 | p | 2
25831.362 | 2 | p | 2 | 38255.437 | 2 | p | 2
26465.843 | 2 | p | 1 | 38288.425 | 10 | p | 3
26512.439 | 2 | p | 2 | 38323.326 | 2 | p | 2
26535.718 | 2 | p | 1 | 38614.433 | 2 | p | 2
26545.469 | 2 | p | 2 | 38938.446 | 1 | p | 4
26647.308 | 2 | s | 2 | 38938.452 | 1 | p | 4
26648.332 | 2 | p | 2 | 38938.453 | 1 | p | 4
26868.522 | 2 | s | 2 | 41082.509 | 1 | p | 5
26930.399 | 0 | p | 2 | 41107.549 | 0 | p | 5
27965.784 | 2 | p | 1 | 41115.516 | 1 | p | 5
28074.410 | 2 | p | 2 | 41965.387 | 1 | p | 6
28078.299 | 2 | p | 2 | 42386.456 | 2 | p | 1
28460.527 | 2 | p | 1 | 42666.772 | 2 | s | 1
29073.744 | 2 | p | 1 | 43317.780 | 2 | p | 1
29114.476 | 2 | p | 2 | 43423.529 | 2 | s | 1
29857.637 | 2 | p | 1 | 44757.540 | 1 | p | 7
31318.694 | 2 | p | 1 | 44875.875 | 2 | p | 1
33854.692 | 2 | p | 1 | 45114.551 | 1 | p | 8
34209.792 | 2 | p | 1 | 45504.566 | 1 | p | 9
34221.446 | 2 | p | 2 | 45541.452 | 1 | p | 10
34580.427 | 2 | p | 2 | 45611.295 | 1 | p | 11
35224.585 | 2 | p | 2 | 45611.302 | 1 | p | 11
35226.576 | 2 | p | 2 | 45636.490 | 2 | p | 1
35721.355 | 2 | p | 2 | 45785.882 | 2 | p | 1
36433.446 | 2 | p | 2 | 45855.778 | 2 | p | 1
36790.450 | 2 | p | 2 | 45931.461 | 1 | p | 12
36859.3445 | 10 | s | 3 | 45933.404 | 1 | p | 13
36860.331 | 10 | p | 2 | 46003.248 | 1 | p | 14
36860.3311 | 10 | p | 3 | 46290.424 | 1 | p | 15
37116.458 | 2 | p | 2 | 46321.464 | 1 | p | 15
37438.547 | 2 | p | 2 | 46356.387 | 1 | p | 15
37506.4677 | 10 | p | 3 | 46385.508 | 2 | p | 1
37543.3344 | 10 | p | 3 | 46534.864 | 2 | p | 1
37572.4389 | 10 | p | 3 | 46612.497 | 1 | p | 16
37642.260 | 2 | p | 2 | 46612.497 | 1 | p | 16
37867.3714 | 10 | p | 3 | 46612.500 | 1 | p | 16
1: Torres & Guilbault (2003); 2: Huth (1965); 3: Present paper; 4: BAV 7; 5:
BBSAG 30; 6: BBSAG 12; 7: BAV 34; 8: BBSAG 61; BBSAG 67 10: BAV 38; 11: BBSAG
69; 12: BAAVSS 61; 13: BBSAG 73; 14: BBSAG 74; 15: BBSAG 78; 16: BRNO 28
Table 3: (Continue.)
Time of Min. | W | Type | Ref. | Time of Min. | W | Type | Ref.
---|---|---|---|---|---|---|---
(HJD-2 400 000) | | | | (HJD-2 400 000) | | |
46612.501 | 1 | p | 16 | 49216.450 | 1 | p | 29
46612.502 | 1 | p | 16 | 49216.461 | 1 | p | 29
46612.503 | 1 | p | 16 | 49216.464 | 1 | p | 30
46612.504 | 1 | p | 16 | 49216.468 | 1 | p | 30
46612.506 | 1 | p | 16 | 49218.393 | 1 | p | 29
46612.507 | 1 | p | 16 | 49218.402 | 1 | p | 29
46612.509 | 1 | p | 16 | 49218.411 | 1 | p | 30
46614.441 | 1 | p | 16 | 49218.413 | 1 | p | 31
46614.445 | 1 | p | 16 | 49251.390 | 1 | p | 30
46614.447 | 1 | p | 16 | 49321.242 | 1 | p | 30
46614.447 | 1 | p | 16 | 49934.390 | 1 | p | 32
46614.448 | 1 | p | 16 | 49967.380 | 1 | p | 32
46614.452 | 1 | p | 16 | 50002.313 | 1 | p | 32
46678.476 | 1 | p | 16 | 50324.400 | 1 | p | 33
46678.478 | 1 | p | 16 | 50357.372 | 1 | p | 33
46709.547 | 2 | p | 1 | 50547.551 | 1 | p | 34
46973.428 | 1 | p | 17 | 50681.418 | 1 | p | 34
47026.696 | 2 | s | 1 | 50718.283 | 1 | p | 35
47039.370 | 1 | p | 18 | 50751.277 | 1 | p | 35
47064.618 | 2 | p | 1 | 51036.496 | 1 | p | 36
47361.497 | 1 | p | 19 | 51063.6717 | 10 | p | 37
47363.441 | 1 | p | 19 | 51128.645 | 10 | s | 37
47392.530 | 1 | p | 20 | 51129.646 | 10 | p | 37
47392.535 | 1 | p | 20 | 51327.564 | 1 | p | 38
47392.543 | 1 | p | 20 | 51364.416 | 1 | p | 39
47396.424 | 1 | p | 19 | 51397.4114 | 10 | p | 40
47431.331 | 1 | p | 21 | 51464.3104 | 10 | s | 40
47466.271 | 1 | p | 21 | 52031.90450 | 10 | p | 41
47788.3674 | 10 | p | 22 | 52064.89086 | 10 | p | 41
47790.313 | 1 | p | 23 | 52096.86757 | 10 | s | 41
47823.313 | 1 | p | 24 | 52098.80834 | 10 | s | 41
48112.405 | 1 | p | 25 | 52099.8166 | 10 | p | 41
48112.412 | 1 | p | 25 | 52101.75702 | 10 | p | 41
48147.327 | 1 | p | 25 | 52164.7794 | 10 | s | 41
48176.432 | 1 | p | 25 | 52165.78900 | 10 | p | 41
48533.457 | 1 | p | 26 | 52425.79570 | 10 | p | 42
48723.609 | 1 | p | 27 | 52487.88765 | 10 | p | 42
48859.422 | 1 | p | 28 | 52488.81917 | 10 | s | 42
17: BBSAG 84; 18: BBSAG 86; 19: BBSAG 89; 20: BRNO 30; 21: BBSAG 90; 22: BAV
56 23: BBSAG 92; 24: BBSAG 93; 25: BBSAG 96; 26: BBSAG 99; 27: BBSAG 101; 28:
BBSAG 102 29: BRNO 31; 30: BBSAG 105; 31: BBSAG 104; 32: BBSAG 110; 33: BBSAG
113; 34: BBSAG 115; 35: BBSAG 116 36: BBSAG 118; 37: Lacy et al. (1999); 38:
BBSAG 120; 39: BRNO 32; 40 Agerer et al. (2001); 41: Lacy et al. (2002); 42:
Lacy (2002)
Table 3: (Continue.)
Time of Min. | W | Type | Ref. | Time of Min. | W | Type | Ref.
---|---|---|---|---|---|---|---
(HJD-2 400 000) | | | | (HJD-2 400 000) | | |
52495.64880 | 10 | p | 42 | 53526.9042 | 10 | s | 47
52562.5517 | 10 | s | 42 | 53527.91289 | 10 | p | 47
52595.5379 | 10 | s | 42 | 53898.5192 | 10 | p | 48
52724.589 | 1 | p | 43 | 53933.4432 | 10 | p | 49
52782.8192 | 10 | p | 44 | 53987.7740 | 10 | p | 50
52814.7964 | 10 | s | 44 | 54026.5809 | 10 | p | 50
52817.74512 | 10 | p | 44 | 54325.3939 | 10 | p | 51
53169.8789 | 10 | s | 45 | 54388.4173 | 10 | s | 51
53186.4111 | 10 | p | 46 | | | |
43: Diethelm (2003); 44: Lacy (2003); 45: Lacy (2004); 46: Zejda (2004); 47:
Lacy (2006); 48: Hübscher et al. (2006); 49: Hübscher (2007); 50: Lacy (2007);
51: Hübscher et al. (2008)
## 3 Period analysis
The $O-C$ values were calculated with the following ephemeris:
$\mathrm{Min~{}I}=2\,436\,860.3311+1.9403494\times E$ (1)
where the initial epoch was our first primary minimum observation while the
period was taken from Lacy et al. (2003). The $O-C$ diagram is presented in
Figure 2. It is clear from Figure 2 that the time lag between the primary and
secondary minima has changed so the apsidal motion is clearly present. In the
following calculations CCD and photoelectric times of minima had weights of
10, plate minima had 2, visual observations had 1.
Using a second-order approximation in the eccentricity, the times of primary
and secondary minima will occur at the times given below:
${\mathrm{Min~{}I}}=T_{0}+EP_{\mathrm{s}}-\frac{eP_{\mathrm{a}}}{\pi}\cos\omega_{E}+\frac{3}{8}\frac{e^{2}P_{\mathrm{a}}}{\pi}\sin
2\omega_{E}+...$ (2)
${\mathrm{Min~{}II}}=T_{0}+EP_{\mathrm{s}}-\frac{P_{\mathrm{a}}}{2}+\frac{eP_{\mathrm{a}}}{\pi}\cos\omega_{E}+\frac{3}{8}\frac{e^{2}P_{\mathrm{a}}}{\pi}\sin
2\omega_{E}-...$ (3)
where $T_{0}$ are the epoch of a primary minimum, $P_{\mathrm{a}}$ is the
anomalistic period, $P_{\mathrm{s}}$ is the sidereal period, i.e.
$P_{\mathrm{s}}\approx P_{\mathrm{a}}(1-\omega^{\prime}/2\pi)$, $e$ is the
eccentricity, $\omega_{E}=\omega_{0}+\omega^{\prime}E$ where
$\omega^{\prime}=2\pi P/U$. $U$ is the apsidal motion period.
Applying these formulae we determined the apsidal motion period with the
upgraded version of LiteAM software developed by T. Borkovits (see e.g.
Borkovits et al., 2002). We found from the fitting of the $O-C$ curve that
$U/P=68700\pm 500$ and this means $U=365$ years, $T_{0}=51063.6537\pm
0.0001\mathrm{(}HJD)$ and $\omega_{0}=150^{\circ}\pm 5^{\circ}$. This latter
value is in good agreement with $\omega=154.7^{\circ}\pm 3.9^{\circ}$ found by
spectroscopic measurements (Lacy et al., 2003). The apsidal motion period
yields $\dot{\omega}\approx 1.0^{\circ}$/year.
Note that the $O-C$ curve is not well-covered yet. There is need for more
observations to determine an exact value of the apsidal motion period in BP
Vul – our value given above can be regarded as a first approximation. But more
interesting than the exact value of this period is that the new value yielded
a prograde motion of the semi-major axis instead of a retrograde one.
## 4 Light curve solution
For the light curve solution we used the Wilson-Devinney Code (Wilson, 1998).
The free parameters were the inclination, the dimensionless surface
potentials, argument of periastron and its time-derivative and the
luminosities of the components. Limb-darkeking coefficients were fixed and
these fixed values were interpolated ones from tables of van Hamme (1993).
Gravity darkening and reflexion coefficients were also fixed. Mass ratio,
surface temperatures of the components and eccentricity of the orbit were
fixed at the values given in Lacy et al. (2003). Then differential correction
analysis were carried out and the stopping criteria was that the change in the
parameters in the final step should be lower than its standard deviations.
Since BP Vul has a fast apsidal motion (see previous Section) we used time as
an independent variable during the modeling rather than phase. The result of
the light curve solution can be found in Table 4.
Comparing our results to the one of Lacy et al. (2003) we found a remarkably
excellent agreement in luminosity ratio, but other elements are slightly
different. However, the precision of our light curve does not reach the
precision of their one although we have colour information, too. Moreover,
they used the so-called EBOP code (Popper & Etzel, 1981) which has a slightly
different input physics. Since we solve these old light curves for the purpose
to determine the argument of periastron independently, these slight
differences do not destroy the validity of our light curve solution.
Thus we concentrate the position of the periastron hereafter. As one can see
from Table 4 $\omega=126^{\circ}\pm 5^{\circ}$ at epoch HJD
=$2\,436\,860.3311$ (our adopted epoch) which nearly corresponds 1959 October
18. From their spectroscopic measurements Lacy et al. (2003) had given
$\omega=154.7^{\circ}$ for epoch HJD $2\,451\,023.254$ which nearly
corresponds to 1998 July 28. It is easy to compute that these two measurements
yield $0.74^{\circ}$/yr apsidal motion or $U=486\pm 57$ years.
This value is more than the 365 years apsidal motion period – determined from
the $O-C$ analysis – by about 120 years. According to us this 33% difference
is not because the $O-C$ diagram is not well-covered and a new light curve
solution based on multi-colour observations are needed.
However, the fitted $\dot{\omega}$ gives $1.0^{\circ}$/yr which is fully in
agreement with the results of the $O-C$ analysis. Regarding the uncertainty in
the position of $\omega$ determined from these old light curves, one can
conclude that very likely the light curve solution gives $\dot{\omega}\approx
1.0^{\circ}$/yr.
Table 4: Light curve solution of BP Vulpeculae. Denotions have their usual meaning. Mode 0 of the Wilson-Devinney Code was used (see Wilson, 1998). $r_{1,2}$ were derived from $\Omega_{1,2}$ and $q$ by the WD-code itself. Quantity | | This paper | Lacy et al. (2003)
---|---|---|---
$i$ | adjusted | $86.64\pm 0.16$ | $87.67$
$L_{1}/L_{\mathrm{tot}}$ (B) | adjusted | $0.719\pm 0.02$ | -
$L_{1}/L_{\mathrm{tot}}$ (V) | adjusted | $0.696\pm 0.02$ | $0.718$
$\omega$ | adjusted | $126^{\circ}\pm 5^{\circ}$ | $154.7^{\circ}$
$\dot{\omega}$ | adjusted | $1.006^{\circ}\pm 0.002^{\circ}$ | -
$\Omega_{1}$ | adjusted | $7.04\pm 0.09$ | -
$\Omega_{2}$ | adjusted | $6.89\pm 0.07$ | -
$HJD0$ | fixed | 2 436 860.3311 | 2 451 023.254
mean $r_{1}$ | derived | $0.162\pm 0.027$ | 0.1931
mean $r_{2}$ | derived | $0.141\pm 0.031$ | 0.1552
$g_{1}$ | fixed | $1.0$ | -
$g_{2}$ | fixed | $1.0$ | -
$A_{1}$ | fixed | $1.0$ | -
$A_{2}$ | fixed | $1.0$ | -
$T_{1}$ | fixed | $7709$K | $7709$K
$T_{2}$ | fixed | $6823$K | $6823$K
$x_{1,\mathrm{bol}}$ | fixed | $0.538$ | -
$x_{2,\mathrm{bol}}$ | fixed | $0.467$ | -
$x_{1,\mathrm{B}}$ | fixed | $0.604$ | -
$x_{2,\mathrm{B}}$ | fixed | $0.621$ | -
$x_{1,\mathrm{V}}$ | fixed | $0.534$ | $0.50\pm 0.03$
$x_{2,\mathrm{V}}$ | fixed | $0.507$ | $0.56\pm 0.03$
$e$ | fixed | $0.0345$ | $0.0345$
$q$ | fixed | $0.811$ | $0.811$
## 5 The internal structure constant $k_{2}$
Our next calculations are based on Gimenez (1985). From the known
eccentricity, masses and period given in Lacy et al. (2003) one can calculate
that the relativistic contribution to the apsidal motion in BP Vulpeculae is
$7.529\cdot 10^{-4}$ degree/cycle. From the observed $U=365$ years see above
we found $\dot{\omega}_{\mathrm{obs}}=5.24\cdot 10^{-3}$ degree/cycle. So the
Newtonian term in the apsidal motion is
$\dot{\omega}_{N}=\dot{\omega}_{\mathrm{obs}}-\dot{\omega}_{\mathrm{rel}}=4.49\cdot
10^{-4}$ degree/cycle.
Using the well-known relationship
$k_{2,\mathrm{obs}}=\frac{1}{c_{21}+c_{22}}\frac{\dot{\omega}_{N}}{360^{\circ}}$
(4)
we found $\log k_{2,\mathrm{obs}}=-2.66\pm 0.08$. Here $c_{21}$ and $c_{22}$
are the functions of eccentricity, mass ratio and fractional radii of the
components and their precise form is given in Gimenez (1985).
Using the tables of Claret (2004) with X=0.70, Z=0.02, t=1 Gyr and with mixing
length parameter $\alpha=1.68$ and overshooting parameter $\alpha_{OV}=0.2$ we
could calculate $\log k_{2,\mathrm{theo}}=-2.47$. Regarding the uncertainties
in the determined apsidal motion period which appears in the determination of
$k_{2}$ internal structure constant we may conclude that the observed and
theoretically expected values are close to each other. Also note that we have
only one secondary minimum from the 1960s which is a key point in similar
calculations.
## 6 Summary
BP Vulpeculae is an eccentric eclipsing binary star showing the so-called
periastron-precession effect. Lacy et al. (2003) concluded that this effect
causes a retrograde motion of the semi-major axis and it has a period of
$77\pm 22$ years based on their about one decade long observational material.
However, retrograde motion contradicts theory. Their explanation was that a
possible third body in the system could perturb the orbit yielding the
observed peculiar periastron precession. Nevertheless no spectroscopic
evidence was found by them for such a third body.
BP Vul was observed at the Konkoly Observatory more than forty years before
the work of Lacy et al. (2003) by one of the authors of this study. This
observational material allowed us to determine the times of six primary and
one secondary minima. With these early observations the baseline could be
expanded to approximately five decades which was enough to refine the apsidal
motion period determined by Lacy et al. (2003). Our $O-C$ analysis based on
the extended time-line showed that an unseen, dark third body in the system
cannot be extracted from the presently available minima observations. All of
these makes very unlikely the presence of a third body with a mass and orbit
which would cause a peculiar periastron precession.
First time two-colour light curves were presented by us for BP Vul. The light
curve solution – using the Wilson-Devinney Code – yielded very similar results
comparing to Devor (2005)’s one and Lacy et al. (2003)’s one . In addition,
the $O-C$ analysis in this paper showed that the apsidal motion period in BP
Vul is prograde and it has a period of about 365 years – it is in agreement
with the value determined with less accuracy from the light curve. The
prograde motion means that BP Vul is not a representative of problematic cases
and it is in agreement with theoretical expectations. Nevertheless we
concluded that the negative apsidal motion rate determined by Lacy et al.
(2003) is only a consequence of their short observational window.
We also calculated the $k_{2}$ internal structure of BP Vul and found it being
close to the theoretical value. The slight difference should be refined in the
future with a better observed $O-C$ diagram without gaps. Therefore the minima
observations of BP Vulpeculae in the future are needed.
The comments on the first version of the manuscript by Drs J. Jurcsik and K.
Oláh is acknowledged.
Figure 1: The $B$ (top) and $V$ (bottom) light curves of BP Vul obtained in
the years 1959-1963. Bottom is the B curve while top curve is the V one which
is shifted by 0.3 magnitudes for the sake of clarity. Figure 2: The $O-C$
diagram of BP Vulpeculae. Filled and open squares represent primary and
secondary minima, respectively. Lines show the fits to the $O-C$ residuals for
the primary and secondary minima, respectively. The weights of the different
kind of minima can be found in the text.
## References
* Agerer et al. (2001) Agerer, F., Dahm, M., Hübscher, J., 2001, IBVS 5017
* Borkovits et al. (2002) Borkovits T., Csizmadia Sz., Hegedüs T., Bíró I. B., Sándor Zs., Opitz A., 2002, A&A 392, 895
* Borkovits et al. (2007) Borkovits T., Forgács-Dajka E., Regály Zs., 2007, A&A 473, 191
* Claret (1998) Claret, A., 1998, A&A, 330, 533
* Claret (2004) Claret, A., 2004, A&A, 424, 919
* Devor (2005) Devor, J., 2005, ApJ 628, 411
* Diethelm (2003) Diethelm, R., 2003, IBVS 5438 (BBSAG 129)
* Gimenez (1985) Gimenez, A., 1985, ApJ 297, 405
* Illés-Almár (1960) Illés-Almár, E., 1960, ATsir, 210, 21
* Huth (1965) Huth, H., 1965, IBVS 96
* Hübscher et al. (2006) Hübscher, J., Paschke, A., Walter, F., 2006, IBVS 5731 (BAV 178)
* Hübscher (2007) Hübscher, J., 2007, IBVS 5802 (BAV 186)
* Hübscher et al. (2008) Hübscher, J., Steinbach, H.-M., Walter, F., 2008, IBVS 5830 (BAV 193)
* Kwee & van Woerden (1956) Kwee, K.K., van Woerden, H., 1956, Bull. Astron. Inst. Neth., 12, 327
* Lacy (1992) Lacy, C. H., 1992, AJ 104, 801
* Lacy (2002) Lacy, C. H., 2002, IBVS 5357
* Lacy (2003) Lacy, C. H., 2003, IBVS 5487
* Lacy (2004) Lacy, C. H., 2004, IBVS 5557
* Lacy (2006) Lacy, C. H., 2006, IBVS 5670
* Lacy (2007) Lacy, C. H. S., 2007, IBVS 5764
* Lacy et al. (1999) Lacy, C. H., Marcrum, K., Ibanoglu, C., 1999, IBVS 4737
* Lacy et al. (2002) Lacy, C. H., Straughn, A., Denger, F., 2002, IBVS 5251
* Lacy et al. (2003) Lacy, C. H., Torres, G., Claret, A., Sabby, J. A., 2003, AJ 126, 1905
* Popper & Etzel (1981) Popper, D. M., Etzel, P. B., 1981, AJ 81, 102
* Torres & Guilbault (2003) Torres, G., Guilbault, P. R., 2003, IBVS 5421
* van Hamme (1993) van Hamme, W., 1993, AJ 106, 2096
* Wilson (1998) Wilson, R. E., 1998, Computing Observable Binary Stars (University of Florida)
* Zejda (2004) Zejda, M., 2004, IBVS 5583
|
arxiv-papers
| 2008-12-12T12:55:10 |
2024-09-04T02:48:59.364006
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sz. Csizmadia, E. Illes-Almar, T. Borkovits",
"submitter": "Szil\\'ard Csizmadia",
"url": "https://arxiv.org/abs/0812.2363"
}
|
0812.2483
|
# Statistical Properties of Gamma-Ray Burst Polarization
Kenji Toma11affiliation: Department of Astronomy and Astrophysics,
Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA
22affiliation: Division of Theoretical Astronomy, National Astronomical
Observatory of Japan (NAOJ), 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan ,
Takanori Sakamoto33affiliation: CRESST and NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA 44affiliation: Joint Center for Astrophysics,
University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD
21250, USA , Bing Zhang55affiliation: Department of Physics and Astronomy,
University of Nevada Las Vegas, Las Vegas, NV 89154, USA , Joanne E.
Hill33affiliation: CRESST and NASA Goddard Space Flight Center, Greenbelt, MD
20771, USA 66affiliation: Universities Space Research Association, 10211
Wincopin Circle, Suite 500, Columbia, MD, 21044-3432, USA , Mark L.
McConnell77affiliation: Space Science Center, University of New Hampshire,
Durham, NH 03824, USA , Peter F. Bloser77affiliation: Space Science Center,
University of New Hampshire, Durham, NH 03824, USA , Ryo
Yamazaki88affiliation: Department of Physical Science, Hiroshima University,
Higashi-Hiroshima, Hiroshima 739-8526, Japan , Kunihito Ioka99affiliation:
Theory Division, KEK (High Energy Accelerator Research Organization), 1-1 Oho,
Tsukuba 305-0801, Japan , and Takashi Nakamura1010affiliation: Department of
Physics, Kyoto University, Kyoto 606-8502, Japan toma@astro.psu.edu
###### Abstract
The emission mechanism and the origin and structure of magnetic fields in
gamma-ray burst (GRB) jets are among the most important open questions
concerning the nature of the central engine of GRBs. In spite of extensive
observational efforts, these questions remain to be answered and are difficult
or even impossible to infer with the spectral and lightcurve information
currently collected. Polarization measurements will lead to unambiguous
answers to several of these questions. Recent developments in X-ray and
$\gamma$-ray polarimetry techniques have demonstrated a significant increase
in sensitivity enabling several new mission concepts, e.g. POET (Polarimeters
for Energetic Transients), providing wide field of view and broadband
polarimetry measurements. If launched, missions of this kind would finally
provide definitive measurements of GRB polarizations. We perform Monte Carlo
simulations to derive the distribution of GRB polarizations in three emission
models; the synchrotron model with a globally ordered magnetic field (SO
model), the synchrotron model with a small-scale random magnetic field (SR
model), and the Compton drag model (CD model). The results show that POET, or
other polarimeters with similar capabilities, can constrain the GRB emission
models by using the statistical properties of GRB polarizations. In
particular, the ratio of the number of GRBs for which the polarization degrees
can be measured to the number of GRBs that are detected ($N_{m}/N_{d}$) and
the distributions of the polarization degrees ($\Pi$) can be used as the
criteria. If $N_{m}/N_{d}>30\%$ and $\Pi$ is clustered between 0.2 and 0.7,
the SO model will be favored. If instead $N_{m}/N_{d}<15\%$, then the SR or CD
model will be favored. If several events with $\Pi>0.8$ are observed, then the
CD model will be favored.
###### Subject headings:
gamma rays: bursts — magnetic fields — polarization — radiation mechanisms:
non-thermal
††slugcomment: accepted for publication in ApJ
## 1\. Introduction
Gamma-ray bursts (GRBs) are brief, intense flashes of $\gamma$-rays
originating at cosmological distances, and they are the most luminous objects
in the universe. They also have broadband afterglows long-lasting after the
$\gamma$-ray radiation has ceased. It has been established that the bursts and
afterglows are emitted from outflows moving towards us at highly relativistic
speeds (Taylor et al., 2004), and at least some GRBs are associated with the
collapse of massive stars (e.g., Hjorth et al., 2003; Stanek et al., 2003).
Observations suggest that the burst is produced by internal dissipation within
the relativistic jet that is launched from the center of the explosion, and
the afterglow is the synchrotron emission of electrons accelerated in a
collisionless shock driven by the interaction of the jet with the surrounding
medium (for recent reviews, Piran, 2005; Mészáros, 2006; Zhang, 2007).
In spite of extensive observational and theoretical efforts, several key
questions concerning the nature of the central engines of the relativistic
jets and the jets themselves remain poorly understood. In fact, some of these
questions are very difficult or even impossible to answer with the spectral
and lightcurve information currently collected. On the other hand,
polarization information, if retrieved, would lead to unambiguous answers to
these questions. In particular, polarimetric observations of GRBs can address
the following:
Magnetic composition of GRB jets – It is highly speculated that strong
magnetic fields are generated at the GRB central engine, and may play an
essential role in the launch of the relativistic jets. However, it is unclear
whether the burst emission region is penetrated by a globally structured,
dynamically important magnetic field, and whether the burst is due to shock
dissipation or magnetic reconnection (e.g., Spruit et al., 2001; Zhang &
Mészáros, 2002; Lyutikov et al., 2003).
Emission mechanisms of the bursts – The leading model for the emission
mechanism of the prompt burst emission is synchrotron emission from
relativistic electrons in a globally ordered magnetic field carried from the
central engine, or random magnetic fields generated in-situ in the shock
dissipation region (Rees & Mészáros, 1994). Other suggestions include Compton
drag of ambient soft photons (Shaviv & Dar, 1995; Eichler & Levinson, 2003;
Levinson & Eichler, 2004; Lazzati et al., 2004), synchrotron self-Compton
emission (Panaitescu & Meszaros, 2000), and the combination of a thermal
component from the photosphere and a non-thermal component (e.g., synchrotron)
(Ryde et al., 2006; Thompson et al., 2007; Ioka et al., 2007).
Geometric structure of GRB jets – Although it is generally believed that GRB
outflows are collimated, the distribution of the jet opening angles, the
observer’s viewing direction, and whether there are small-scale structures
within the global jet are not well understood (Zhang et al., 2004; Yamazaki et
al., 2004; Toma et al., 2005).
To date, robust positive detections of GRB polarization have been made only in
the optical band in the afterglow phase. Varying linear polarizations have
been observed in several optical afterglows several hours after the burst
trigger, with a level of $\sim 1-3\%$, which is consistent with the
synchrotron emission mechanism of GRB afterglow (for reviews, see Covino et
al., 2004; Lazzati, 2006). An upper limit $(<8\%)$ has been obtained for the
early $(t\sim 200~{}{\rm s})$ optical afterglow of GRB 060418 (Mundell et al.,
2007). Also for radio afterglows, we have several upper limits for the
polarization degree (Taylor et al., 2005; Granot & Taylor, 2005) (for some
implications, see Toma et al., 2008). As for the prompt burst emission, strong
linear polarization of the $\gamma$-ray emission at a level of $\Pi=80\pm
20\%$ was claimed for GRB 021206 based on an analysis of RHESSI data (Coburn &
Boggs, 2003), although this claim remains controversial because of large
systematic uncertainties (Rutledge & Fox, 2004; Wigger et al., 2004). Several
other reports of high levels of polarization in the prompt burst emission are
also statistically inconclusive (Willis et al., 2005; Kalemci et al., 2007;
McGlynn et al., 2007).
Recently, more sensitive observational techniques for X-ray and $\gamma$-ray
polarimetry have been developed, and there are several polarimeter mission
concepts. These include Polarimeters for Energetic Transients (POET, Hill et
al., 2008; Bloser et al., 2008), Polarimeter of Gamma-ray Observer (PoGO,
Mizuno et al., 2005), POLAR (Produit et al., 2005), Advanced Compton Telescope
(ACT, Boggs et al., 2006), Gravity and Extreme Magnetism (GEMS, Jahoda et al.,
2007) XPOL (Costa et al., 2007), Gamma-Ray Burst Investigation via Polarimetry
and Spectroscopy (GRIPS, Greiner et al., 2008), and so on.
Several of these missions, if launched, would provide definitive detections of
the burst polarizations and enable us to discuss the statistical properties of
the polarization degrees and polarization spectra. Although there are several
polarimetry mission concepts described in the literature, POET is the only one
to date that incorporates a broadband capability for measuring the prompt
emission from GRBs, and for this reason it provides a good case study for our
simulations. POET will make measurements with two different polarimeters, both
with wide fields of view. The Gamma-Ray Polarimeter Experiment (GRAPE; 60-500
keV) and the Low Energy Polarimeter (LEP; 2-15 keV) provide a broad energy
range for the observations. Suborbital versions of both POET instruments are
currently being prepared for flight within the next few years. GRAPE will fly
on a sub-orbital balloon in 2011, and the Gamma-Ray Burst Polarimeter (GRBP, a
smaller version of LEP) will fly on a sounding rocket.
Theoretically, it has been shown that similarly high levels of linear
polarization can be obtained in several GRB prompt emission models; the
synchrotron model with a globally ordered magnetic field, the synchrotron
model with a small-scale random magnetic field (Granot, 2003; Lyutikov et al.,
2003; Nakar et al., 2003), and the Compton drag model (Lazzati et al., 2004;
Eichler & Levinson, 2003; Levinson & Eichler, 2004; Shaviv & Dar, 1995). Thus
the detections of GRB prompt emission polarization would support these three
models. In this paper, we show that these models can be distinguished by their
statistical properties of observed polarizations. We performed detailed
calculations of the distribution of polarization degrees by including
realistic spectra of GRB prompt emission and assuming realistic distributions
of the physical parameters of GRB jets, and show that POET, or other
polarimeters with similar capabilities, can constrain the GRB emission models.
We use the limits of POET for GRB detection and polarization measurements as
realistic and fiducial limits. This paper is organized as follows. We first
introduce the POET mission concept in § 2. In § 3, we summarize the properties
of the observed linear polarization from uniform jets within the three
emission models. Based on these models, we perform Monte Carlo simulations of
observed linear polarizations and show how the statistical properties of
observed polarization may constrain GRB emission mechanisms in § 4. A summary
and discussion are given in § 5.
## 2\. Properties of POET satellite
POET (Polarimeters for Energetic Transients) is a Small Explorer (SMEX)
mission concept, that will provide highly sensitive polarimetric observations
of GRBs and can also make polarimetry measurements of solar flares, pulsars,
soft gamma-ray repeaters, and slow transients. The payload consists of two
wide field of view (FoV) instruments: a Low Energy Polarimeter (LEP) capable
of polarization measurements in the 2-15 keV energy range and a high energy
polarimeter (Gamma-Ray Polarimeter Experiment; GRAPE) that will measure
polarization in the 60-500 keV energy range. POET can measure GRB spectra from
2 keV up to 1 MeV. The POET spacecraft provides a zenith-pointed platform for
maximizing the exposure to deep space and spacecraft rotation provides a means
of effectively dealing with systematics in the polarization response. POET
provides sufficient sensitivity and sky coverage to detect up to 200 GRBs in a
two-year mission.
LEP and GRAPE determine polarization by measuring the number of events versus
the event azimuth angle (EAA) as projected onto the sky. This is referred to
as a modulation profile and represents a measure of the polarization magnitude
and direction of polarization for the incident beam. Depending on the type of
polarimeter, the EAA is either the direction of the ejected photoelectron
(LEP) or the direction of the scattered photon (GRAPE). The response of a
polarimeter to $100\%$ polarized photons can be quantified in terms of the
modulation factor, $\mu$, which is given by:
$\mu=\frac{C_{\rm max}-C_{\rm min}}{C_{\rm max}+C_{\rm min}}$ (1)
Where $C_{\rm max}$ and $C_{\rm min}$ are the maximum and minimum of the
modulation profile, respectively. The polarization fraction ($\Pi$) of the
incident flux is obtained by dividing the measured modulation by that expected
for $100\%$ polarized flux. The polarization angle ($\Phi_{0}$) corresponds
either to the maximum of the modulation profile (LEP) or the minimum of the
modulation profile (GRAPE). To extract these parameters from the data, the
modulation histograms are fit to the functional form:
$C(\Phi)=A+B\cos^{2}(\Phi-\Phi_{0})$ (2)
The sensitivity of a polarimeter is defined in terms of the minimum detectable
polarization (MDP), which refers to the minimum level of polarization that is
detectable with a given observation (or, equivalently, the apparent
polarization arising from statistical fluctuations in unpolarized data). The
precise value of the MDP will depend on the source parameters (fluence,
spectrum, etc.) and the polarimeter characteristics. At the $99\%$ confidence
level, the MDP can be expressed as,
$MDP=\frac{4.29}{\mu R_{s}}\sqrt{\frac{R_{s}+R_{b}}{t}},$ (3)
where $R_{s}$ is the observed source strength (${\rm cts}~{}{\rm s}^{-1}$),
$R_{b}$ is the total observed background rate (${\rm cts}~{}{\rm s}^{-1}$),
and $t$ is the observing time (s). The ultimate sensitivity, however, may not
be limited by statistics but by systematic errors created by false modulations
that arise from azimuthal asymmetries in the instrument.
Table 1 Instrument Parameters | GRAPE | LEP
---|---|---
Polarimetry | 60–500 keV | 2–15 keV
Detectors | BGO/plastic scintillator (62) | $Ne:CO_{2}:CH_{3}NO_{2}$ Gas (8)
Spectroscopy | 15 keV – 1 MeV | 2 – 15 keV
Detectors | NaI(TI) scintillator (2) | as above
Field-of-View | $\pm 60^{o}$ | $\pm 44^{o}$
At energies from $\sim 50$ keV up to several MeV, photon interactions are
dominated by Compton scattering. The operational concept for GRAPE is based on
the fact that, in Compton scattering, photons are preferentially scattered at
a right angle to the incident electric field vector (the polarization vector)
(Bloser et al., 2008, 2006; Jason et al., 2005). If the incident beam of
photons is polarized, the azimuthal distribution of scattered photons will be
asymmetric. The direction of the polarization vector is defined by the minimum
of the scatter angle distribution. The GRAPE performance characteristics are
shown in Table 1. The design of the GRAPE instrument is very modular, with 62
independent polarimeter modules and 2 spectroscopy modules. Each polarimeter
module incorporates an array of optically independent 5x5x50 $mm^{3}$ non-
hygroscopic scintillator elements aligned with and optically coupled to the
8x8 scintillation light sensors of a 64-channel MAPMT. Two types of
scintillators are employed. Low-Z plastic scintillator is used as an effective
medium for Compton scattering. High-Z inorganic scintillator (Bismuth
Germanate, BGO) is used as a calorimeter, for absorbing the full energy of the
scattered photon. The arrangement of scintillator elements within a module has
28 BGO calorimeter elements surrounding 32 plastic scintillator scattering
elements. Valid polarimeter events are those in which a photon Compton
scatters in one of the plastic elements and is subsequently absorbed in one of
the BGO elements. These events can be identified as a coincident detection
between one plastic scintillator element and one BGO calorimeter element. The
azimuthal scatter angle is determined for each valid event by the relative
locations of hit scintillator elements. It is not necessary to know where
within each element the interaction takes place (e.g., the depth of
interaction). It is sufficient to know only the lateral location of each
element to generate a histogram of photon scatter angles.
At energies below $\sim 50$ keV, the most sensitive technique for broadband
polarimetry is the photoelectric effect. The LEP measures the polarization of
incident photons with the innovative operation of a Time Projection Chamber
(TPC) (Black et al., 2007). The LEP polarimeter enclosure consists of four
dual-readout detector modules each with an isolated gas volume contained by a
Be X-ray window. Each detector module contains two 6 x 12 x 24 $cm^{3}$
(LxWxH) TPCs that share a single X-ray transparent drift electrode. Each TPC
is comprised of a micropattern proportional counter, consisting of a shared
drift electrode and a high-field gas electron multiplier (GEM) positioned 1 mm
from a strip readout plane. When an X-ray is absorbed in the gas between the
drift electrode and the GEM, a photoelectron is ejected in a preferential
direction with a $\cos^{2}\Phi$ distribution, where $\Phi$ is the azimuthal
angle measured from the X-ray polarization vector. As the photoelectron
travels through the gas it creates a path of ionization that drifts in a
moderate, uniform field to the GEM where an avalanche occurs. The charge
finally drifts to the strip detector where it is read out.
Figure 1.— Contour of MDPs (i.e., the minimum values of polarization that is
detectable at the 99% confidence level) of GRAPE (thick lines) and LEP (thin
lines) for variable $E_{p,{\rm obs}}$ and $F$ and for fixed
$\alpha=-0.2,\beta=1.2,T=20$ sec and the incident angle $30$ degree, where
$E_{p,{\rm obs}}$ is the observed photon energy at the spectral peak, $F$ is
the time-averaged flux in 2-400 keV, $\alpha$ and $\beta$ are the lower and
higher indices of the $F_{\nu}$ spectrum, respectively, and $T$ is the
duration of the prompt emission. The combination of GRAPE and LEP enable us to
measure polarizations with reasonable sensitivity in very wide energy range.
To estimate realistic MDP values for GRBs detected by GRAPE and LEP, we
perform an analytical calculation for LEP and a Monte Carlo simulation for
GRAPE using the current instrument configuration (Table 1). The input spectrum
in the calculation and the simulation is a typical GRB spectrum which can be
described as a smoothly broken power-law spectra characterized by photon
energy at the $\nu F_{\nu}$ spectral peak, $E_{p,{\rm obs}}$, and lower and
higher indices of the $F_{\nu}$ spectrum, $\alpha$ and $\beta$, respectively
(Band et al., 1993). (We treat the spectral indices of the specific energy
flux $F_{\nu}$, while Band et al. (1993) define $\alpha_{B}$ and $\beta_{B}$
as the indices of the photon number flux, i.e., $\alpha=-(\alpha_{B}+1)$ and
$\beta=-(\beta_{B}+1)$, since we will calculate the net polarizations by using
specific energy fluxes (equation (7)).) The various $E_{p,{\rm obs}}$ and
time-averaged flux in 2-400 keV, $F$, are investigated with fixed
$\alpha=-0.2,\beta=1.2$, and a burst duration of $T=20$ s. We also assume the
incident angle of bursts to be 30 degree off-axis. We interpret simulated
events with $\Pi>MDP$ as ’$\Pi$-measurable events’. Figure 1 shows the contour
of the MDP values in the $E_{p,{\rm obs}}-F$ plane for GRAPE and LEP. As can
be seen in the figure, with the combination of LEP and GRAPE, it is possible
to measure the polarization of GRBs with $E_{p,obs}$ ranging from a few keV to
MeV with reasonable sensitivity.
## 3\. Theoretical Models
We calculate the linear polarization for instantaneous emission from a thin
spherical shell moving radially outward with a bulk Lorentz factor $\gamma\gg
1$ and an opening angle $\theta_{j}$. The comoving-frame emissivity has the
functional form of $j^{\prime
I}_{\nu^{\prime}}=A_{0}f(\nu^{\prime})\delta(t^{\prime}-t^{\prime}_{0})\delta(r^{\prime}-r^{\prime}_{0})$,
where $A_{0}$ is the normalization which may depend on direction in the
comoving frame and other physical quantities of the shell and
$f(\nu^{\prime})$ represents the spectral shape. A prime represents the
physical quantities in the comoving frame. The delta functions describe the
instantaneous emission at $t=t_{0}$ and $r=r_{0}$. The normalization, $A_{0}$,
has units of ${\rm erg}~{}{\rm cm}^{-2}~{}{\rm str}^{-1}~{}{\rm Hz}^{-1}$.
Using the spherical coordinate system $(r,\theta,\phi)$ in the lab frame,
where $\theta=0$ is the line of sight, we obtain the spectral fluence (Granot
et al., 1999; Woods & Loeb, 1999; Ioka & Nakamura, 2001):
$I_{\nu}=\frac{1+z}{d_{L}^{2}}\int d\phi\int
d(\cos\theta)r_{0}^{2}\frac{A_{0}f(\nu^{\prime})}{\gamma^{2}(1-\beta\cos\theta)^{2}},$
(4)
where $z$ and $d_{L}$ are the redshift and the luminosity distance of the
source, respectively, and $\nu^{\prime}=(1+z)\nu\gamma(1-\beta\cos\theta)$.
The integration is performed within the jet cone, so that it depends on the
viewing angle $\theta_{v}$, i.e., the angle between the jet axis and the line
of sight. The corresponding Stokes parameters of the local emission (i.e., the
emission from a given point on the shell) are given by $j^{\prime
Q}_{\nu^{\prime}}=j^{\prime
I}_{\nu^{\prime}}\Pi^{\prime}_{0}\cos(2\chi^{\prime})$ and $j^{\prime
U}_{\nu^{\prime}}=j^{\prime
I}_{\nu^{\prime}}\Pi^{\prime}_{0}\sin(2\chi^{\prime})$, where
$\Pi^{\prime}_{0}$ and $\chi^{\prime}$ are the polarization degree and
position angle of the local emission measured in the comoving frame,
respectively. The Stokes parameters of the emission from the whole shell can
be obtained by integrating those of the local emission similarly to the
intensity $I_{\nu}$:
$\left\\{\begin{array}[]{c}Q_{\nu}\\\
U_{\nu}\end{array}\right\\}=\frac{1+z}{d_{L}^{2}}\int d\phi\int
d(\cos\theta)r_{0}^{2}\frac{A_{0}f(\nu^{\prime})}{\gamma^{2}(1-\beta\cos\theta)^{2}}\Pi_{0}\left\\{\begin{array}[]{c}\cos(2\chi)\\\
\sin(2\chi)\end{array}\right\\}.$ (5)
The polarization degree is Lorentz invariant, i.e.,
$\Pi^{\prime}_{0}=\Pi_{0}$. The position angle $\chi$ is calculated by taking
account of the Lorentz transformation of the electromagnetic waves, and it is
measured from a fixed direction, which we choose to be the direction from the
line of sight to the jet axis. Then by calculating
$\\{I,Q,U\\}=\int^{\nu_{2}}_{\nu_{1}}d\nu\\{I_{\nu},Q_{\nu},U_{\nu}\\}$, we
obtain the time-averaged linear polarization in the given wavebands
$[\nu_{1},\nu_{2}]$:
$\Pi=\frac{\sqrt{Q^{2}+U^{2}}}{I}.$ (6)
We consider synchrotron and Compton drag (CD) mechanisms for the GRB prompt
emission. In the synchrotron case, the magnetic field consists of a globally
ordered field, ${\mathbf{B}}_{\rm ord}$, and small-scale random field,
${\mathbf{B}}_{\rm rnd}$, i.e., ${\mathbf{B}}={\mathbf{B}}_{\rm
ord}+{\mathbf{B}}_{\rm rnd}$. The field ${\mathbf{B}}_{\rm ord}$ may originate
from the central engine, while ${\mathbf{B}}_{\rm rnd}$ may be produced in the
emission region itself. Here we consider two extreme cases; synchrotron model
with an ordered field (SO), in which $B^{2}_{\rm ord}\gg\langle B^{2}_{\rm
rnd}\rangle$, and a synchrotron model with a random field (SR), in which
$B^{2}_{\rm ord}\ll\langle B^{2}_{\rm rnd}\rangle$. For the SO model, in
particular, we assume a toroidal magnetic field. In the following sub-
sections, we describe $A_{0}$, $f(\nu^{\prime})$, $\Pi_{0}$, and $\chi$ as
functions of $(\theta,\phi)$ for each model, and calculate the linear
polarization for given parameters $\gamma$, $\theta_{j}$, $\theta_{v}$, and
$z$.
### 3.1. SO model: synchrotron with ordered field
The prompt emission of GRBs could be explained by synchrotron emission from
accelerated electrons that have a non-thermal energy spectra by some
dissipation process within the jet, e.g, internal shocks. Synchrotron emission
from the relativistically moving shell within a globally ordered magnetic
field results in a net observed linear polarization, reflecting the direction
of the field (Lyutikov et al., 2003; Granot, 2003; Nakar et al., 2003). Let us
assume that the jet is permeated by a toroidal field. This is a likely
configuration if a magnetic field is advected by the jet with a constant speed
from the central engine (e.g., Spruit et al., 2001; Fendt & Ouyed, 2004).
A general formula for calculating the observed linear polarization for
synchrotron emission from a uniform jet, in which the electrons have a single
power-law energy spectrum and an isotropic pitch angle distribution and the
magnetic field is ordered globally, is derived by Granot (2003) and Granot &
Taylor (2005). Here we adopt their formulation and extend it for the electrons
having a broken power-law energy spectrum in order to reproduce the typical
observed spectra of GRBs (Band et al., 1993). We adopt the following form for
the radiation spectrum: $f(\nu^{\prime})=\tilde{f}(x)$ where
$x=\nu^{\prime}/\nu^{\prime}_{0}$ and
$\tilde{f}(x)=\left\\{\begin{array}[]{lr}x^{-\alpha}e^{-x}&{\rm
for}~{}x\leq\beta-\alpha\\\
x^{-\beta}(\beta-\alpha)^{\beta-\alpha}e^{\alpha-\beta}&{\rm
for}~{}x\geq\beta-\alpha.\end{array}\right.$ (7)
where $\nu^{\prime}_{0}$, $\alpha$, and $\beta$ are the break frequency and
low-energy and high-energy spectral indices of the comoving spectrum,
respectively. 111 In our model the radiation spectrum (7) is thought to be
produced by the broken power-law energy spectrum of electrons:
$N(\gamma_{e})\propto\gamma_{e}^{-p_{1}}$ for $\gamma_{e}<\gamma_{0}$ and
$N(\gamma_{e})\propto\gamma_{e}^{-p_{2}}$ for $\gamma_{e}>\gamma_{0}$, where
$\alpha=(p_{1}-1)/2$ and $\beta=(p_{2}-1)/2$. This formulation also includes
the case of $p_{1}<1/3$, in which $\alpha=-1/3$,
$A_{0}\propto(\sin\theta^{\prime}_{B})^{2/3}$, and $\Pi_{0}^{\rm syn}=1/2$ for
$x\leq\beta-\alpha$ (Granot, 2003). If we assume that the energy spectrum of
the electrons and the strength of the magnetic field are uniform in the
emitting shell, then we may write
$A_{0}=(\sin\theta^{\prime}_{B})^{\alpha+1}$, where $\theta^{\prime}_{B}$ is
the angle between the direction of the emitted radiation and the local
direction of the magnetic field (Rybicki & Lightman, 1979). The local
polarization degree is given by:
$\Pi_{0}=\Pi^{\rm
syn}_{0}\equiv\left\\{\begin{array}[]{lr}(\alpha+1)/(\alpha+\frac{5}{3})&{\rm
for}~{}x\leq\beta-\alpha\\\ (\beta+1)/(\beta+\frac{5}{3})&{\rm
for}~{}x\geq\beta-\alpha.\end{array}\right.$ (8)
For a globally ordered magnetic field, the Faraday depolarization effect may
be strong within the emitting region (e.g., Toma et al., 2008; Matsumiya &
Ioka, 2003; Sagiv et al., 2004), but we neglect it here for simplicity. By
using a new variable $y\equiv(\gamma\theta)^{2}$, we obtain (see Appendix
A.1):
$\sin\theta^{\prime}_{B}=\left[\left(\frac{1-y}{1+y}\right)^{2}+\frac{4y}{(1+y)^{2}}\frac{(a-\cos\phi)^{2}}{1+a^{2}-2a\cos\phi}\right]^{1/2},$
(9)
$\chi=\phi+\arctan\left(\frac{1-y}{1+y}\frac{\sin\phi}{a-\cos\phi}\right),$
(10)
where $a=\theta/\theta_{v}$. Then the formulation of the net polarization
degree in the observed frequency region $[\nu_{1},\nu_{2}]$ becomes:
$\begin{array}[]{l}\Pi=\left|\int^{\nu_{2}}_{\nu_{1}}d\nu\int^{(1+q)^{2}y_{j}}_{0}\frac{dy}{(1+y)^{2}}\right.\\\
\left.\times\int^{\Delta\phi(y)}_{-\Delta\phi(y)}d\phi~{}\tilde{f}(x)(\sin\theta^{\prime}_{B})^{\alpha+1}\Pi^{\rm
syn}_{0}(x)\cos(2\chi)\right|\\\
\times\left[\int^{\nu_{2}}_{\nu_{1}}d\nu\int^{(1+q)^{2}y_{j}}_{0}\frac{dy}{(1+y)^{2}}\int^{\Delta\phi(y)}_{-\Delta\phi(y)}d\phi~{}\tilde{f}(x)(\sin\theta^{\prime}_{B})^{\alpha+1}\right]^{-1},\end{array}$
(11)
where $x=(1+z)\nu(1+y)/2\gamma\nu^{\prime}_{0}$, and
$q=\frac{\theta_{v}}{\theta_{j}},~{}~{}~{}y_{j}=(\gamma\theta_{j})^{2},$ (12)
$\Delta\phi(y)=\left\\{\begin{array}[]{lr}0&{\rm for}~{}q>1~{}{\rm
and}~{}y<(1-q)^{2}y_{j},\\\ \pi&{\rm for}~{}q<1~{}{\rm
and}~{}y<(1-q)^{2}y_{j},\\\
\cos^{-1}\left[\frac{(q^{2}-1)y_{j}+y}{2q\sqrt{y_{j}y}}\right]&{\rm
otherwise}.\end{array}\right.$ (13)
The polarization degree, $\Pi$, in the waveband [$\nu_{1},\nu_{2}$] can be
calculated if the geometrical parameters, $y_{j},q$, the spectral parameters,
$\gamma\nu^{\prime}_{0},\alpha,\beta$, and the redshift, $z$, are given.
Figure 2 shows the polarization degree in the $60-500$ keV band as a function
of $q$ for several values of $y_{j}$. The other parameters are
$\gamma\nu^{\prime}_{0}=350$ keV, $\alpha=-0.2$, $\beta=1.2$, and $z=1$. The
polarization degree is negligible for $q\approx 0$, because in this case the
local polarization vectors are axisymmetric around the line of sight, i.e.,
$\chi=\phi$ (see Appendix A.1), and the local polarizations are canceled out.
For $y_{j}>1$, a high level of polarization is obtained for $y_{j}^{-1/2}<q<1$
(i.e., $\gamma^{-1}<\theta_{v}<\theta_{j}$). In this case, only a fraction of
the emitting shell (i.e., $\theta<\gamma^{-1}$) is bright because of the
relativistic beaming effect, and the direction of the magnetic field is quite
ordered in the bright region. The contribution of the emission from high
latitude, $\theta>\gamma^{-1}$, is negligible, especially for $y_{j}\geq 100$,
so that the net polarization degree is determined only by the emission from
the bright region with $\theta<\gamma^{-1}$ and then it is nearly constant.
The results of our calculations for the case of $\alpha=\beta$ and $y_{j}\geq
100$ are consistent with the results of Granot (2003) and Lyutikov et al.
(2003). For $y_{j}<1$, a high level of polarization is obtained for $q\sim
1+y_{j}^{-1/2}$ (i.e., $\theta_{v}\sim\theta_{j}+\gamma^{-1}$). In this case,
the bright region on the emitting shell is small, also.
Figure 2.— Linear polarization degrees in the $60-500$ keV band as a function
of $q=\theta_{v}/\theta_{j}$, where $\theta_{v}$ is the viewing angle of the
observer and $\theta_{j}$ is the jet opening angle, for several values of
$y_{j}=(\gamma\theta_{j})^{2}$, calculated in the SO model (synchrotron model
with globally ordered magnetic field). The other parameters are
$\gamma\nu^{\prime}_{0}=350$ keV, $\alpha=-0.2,\beta=1.2$, and $z=1$.
The polarization is higher for softer spectra (i.e., larger $\alpha$ and
$\beta$). For example, for $y_{j}=100$, $\gamma\nu^{\prime}_{0}=350$ keV, and
$z=1$, the polarization degree at the plateau for $q<1$ is $\simeq 0.28$ for
$\alpha=-0.5$ and $\beta=0.9$, while it is $\simeq 0.52$ for $\alpha=0.4$ and
$\beta=1.8$. This is caused mainly by the dependence of the synchrotron
polarization on the spectral indices (equation 8). The maximum polarization
degree obtained in the SO model is $\simeq 0.8$ for $y_{j}\geq 0.01$,
$\alpha\leq 0.4$, and $\beta\leq 1.8.$
### 3.2. SR model: synchrotron with random field
If the magnetic field is produced at the shock itself within the jet, the
directions of the field would be random on a scale as small as the plasma skin
depth (Gruzinov & Waxman, 1999; Medvedev & Loeb, 1999). It is quite plausible
that the directions of the field are not completely random, but have symmetry
around the direction normal to the shock. The less isotropic the magnetic
field directions behind the shock, the higher the local polarization. We
consider the extreme case in which the field is random strictly within the
plane of the shock. In this model, the directions of the local polarization
vectors on the shell are axisymmetric around the line of sight (see below), so
that no net polarization remains if the visible region, $\theta<\gamma^{-1}$,
is wholly within the jet cone. However, if the observer views the jet from an
off-axis angle and the symmetry is broken a high level of polarization remains
(Waxman, 2003; Sari, 1999; Ghisellini & Lazzati, 1999).
Similarly to the SO model, we adopt the broken power-law form of the spectrum:
$f(\nu^{\prime})=\tilde{f}(x),$ where $x=\nu^{\prime}/\nu^{\prime}_{0}$ and
$\tilde{f}(x)$ is given by equation (7). We assume that the energy
distribution of the electrons and the strength of the magnetic field are
uniform in the emitting shell. The local Stokes parameters are given by
averaging them with respect to the magnetic field directions within the shock
plane (see Appendix A.2). Thus we may write
$A_{0}=\langle(\sin\theta^{\prime}_{B})^{\alpha+1}\rangle$, where
$\langle\rangle$ represents the average. The local polarization degree is
given by $\Pi_{0}=\Pi_{0}^{\rm
syn}\langle(\sin\theta^{\prime}_{B})^{\alpha+1}\cos(2\phi^{\prime}_{B})\rangle/\langle(\sin\theta^{\prime}_{B})^{\alpha+1}\rangle$,
where:
$\langle(\sin\theta^{\prime}_{B})^{\alpha+1}\rangle=\frac{1}{\pi}\int^{\pi}_{0}d\eta^{\prime}~{}\left[1-\frac{4y}{(1+y)^{2}}\cos^{2}\eta^{\prime}\right]^{(\alpha+1)/2},$
(14)
$\begin{array}[]{r}\langle(\sin\theta^{\prime}_{B})^{\alpha+1}\cos(2\phi^{\prime}_{B})\rangle=\frac{1}{\pi}\int^{\pi}_{0}d\eta^{\prime}~{}\left\\{\left[1-\frac{4y}{(1+y)^{2}}\cos^{2}\eta^{\prime}\right]^{(\alpha-1)/2}\right.\\\
\times\left.\left[\sin^{2}\eta^{\prime}-\left(\frac{1-y}{1+y}\right)^{2}\cos^{2}\eta^{\prime}\right]\right\\}.\end{array}$
(15)
The local polarization position angle measured in the lab frame is given by
$\chi=\phi$, therefore we obtain the formulation for the net polarization in
the observed frequency region $[\nu_{1},\nu_{2}]$:
$\begin{array}[]{l}\Pi=\left|\int^{\nu_{2}}_{\nu_{1}}d\nu\int^{(1+q)^{2}y_{j}}_{0}\frac{dy}{(1+y)^{2}}\tilde{f}(x)\Pi_{0}^{\rm
syn}(x)\right.\\\
\left.\times\langle(\sin\theta^{\prime}_{B})^{\alpha+1}\cos(2\phi^{\prime}_{B})\rangle\sin(2\Delta\phi(y))\right|\\\
\times\left[\int^{\nu_{2}}_{\nu_{1}}d\nu\int^{(1+q)^{2}y_{j}}_{0}\frac{dy}{(1+y)^{2}}\tilde{f}(x)\langle(\sin\theta^{\prime}_{B})^{\alpha+1}\rangle~{}2\Delta\phi(y)\right]^{-1},\end{array}$
(16)
where $q=\theta_{v}/\theta_{j}$, $y_{j}=(\gamma\theta_{j})^{2}$,
$x=(1+z)\nu(1+y)/2\gamma\nu^{\prime}_{0}$, and $\Pi_{0}^{\rm syn}$ and
$\Delta\phi(y)$ are given by equations (8) and (13), respectively.
Figure 3.— Same as Figure 2, but in the SR model (synchrotron model with
small-scale random magnetic field).
Figure 3 shows the polarization degree in the $60-500$ keV band as a function
of $q$ for several values of $y_{j}$. The other parameters are
$\gamma\nu^{\prime}_{0}=350$ keV, $\alpha=-0.2,\beta=1.2$, and $z=1$. The
results of our calculations for the case of $\alpha=\beta$ are consistent with
those of Granot (2003) and Nakar et al. (2003). A high level of polarization
is obtained for $q\sim 1+y_{j}^{-1/2}$ (i.e.,
$\theta_{v}\sim\theta_{j}+\gamma^{-1}$) for each value of $y_{j}$. Since the
local polarization vectors are axisymmetric around the line of sight, the
local polarizations are canceled out if the line of sight is within the jet
cone. If the jet is observed from an off-axis angle, the net polarization
remains. The local polarization degree is highest for emission where
$\theta=\gamma^{-1}$, so that the net polarization has a maximum value. The
maximum $\Pi$ is higher for smaller $y_{j}$, because the contribution of the
emission from high latitude points ($\theta>\gamma^{-1}$), with a low level of
local polarization, is smaller.
Similarly to the SO model, the polarization is higher for softer spectra,
mainly because of the dependence of the local polarization degree on frequency
(equation 8). For example, for $y_{j}=1$, $\gamma\nu^{\prime}_{0}=350$ keV,
and $z=1$, the maximum polarization is $\simeq 0.32$ for $\alpha=-0.5$ and
$\beta=0.9$, while it is $\simeq 0.49$ for $\alpha=0.4$ and $\beta=1.8$. For
$y_{j}\geq 0.01$, $\alpha\leq 0.4$, and $\beta\leq 1.8,$ the maximum
polarization degree in the SR model is $\simeq 0.8$.
### 3.3. CD model: Compton drag model
The prompt emission from GRBs could be produced by bulk inverse Comptonization
of soft photons from the relativistic jet (Lazzati et al., 2004; Eichler &
Levinson, 2003; Levinson & Eichler, 2004; Shaviv & Dar, 1995). The local
polarization position angles are symmetric around the line of sight, similarly
to the SR model. Therefore this model also requires an off-axis observation of
the jet to achieve a high level of polarization. However, the CD model is
different from the SR model in the fact that the CD model can in principle
achieve $\Pi\sim 1$ under the most optimistic geometric configurations,
whereas the maximum $\Pi$ is $\sim(\beta+1)/(\beta+\frac{5}{3})\sim 0.8$ in
the SR model.
We assume that the seed radiation is unpolarized and has a non-thermal,
isotropic spectrum, and the scattered radiation has the broken power-law
spectrum $f(\nu^{\prime})=\tilde{f}(x)$, where
$x=\nu^{\prime}/\nu^{\prime}_{0}$ and $\tilde{f}(x)$ is given by equation (7).
If the intensity of the seed radiation and the electron number density of the
shell are assumed to be uniform then we may write
$A_{0}=(1+\cos^{2}\theta^{\prime})/2$, and
$\Pi_{0}=(1-\cos^{2}\theta^{\prime})/(1+\cos^{2}\theta^{\prime})$ (Rybicki &
Lightman, 1979; Begelman & Sikora, 1987). The polarization vectors in the
comoving frame are perpendicular to both incident and scattering directions of
photons, so that we obtain $\chi=\phi+\frac{\pi}{2}$ in the lab frame.
Therefore we achieve the formulation for the net linear polarization in the
observed frequency region $[\nu_{1},\nu_{2}]$:
$\begin{array}[]{l}\Pi=\left|\int^{\nu_{2}}_{\nu_{1}}d\nu\int^{(1+q)^{2}y_{j}}_{0}\frac{dy}{(1+y)^{2}}\tilde{f}(x)\frac{2y}{(1+y)^{2}}\sin(2\Delta\phi(y))\right|\\\
\times\left[\int^{\nu_{2}}_{\nu_{1}}d\nu\int^{(1+q)^{2}y_{j}}_{0}\frac{dy}{(1+y)^{2}}\tilde{f}(x)\frac{1+y^{2}}{(1+y)^{2}}2\Delta\phi(y)\right]^{-1},\end{array}$
(17)
where $q=\theta_{v}/\theta_{j}$, $y_{j}=(\gamma\theta_{j})^{2}$,
$x=(1+z)\nu(1+y)/2\gamma\nu^{\prime}_{0}$, and $\Delta\phi(y)$ is given by
equation (13).
Figure 4 shows the polarization degree in the $60-500$ keV band as a function
of $q$ for several values of $y_{j}$. The other parameters are
$\gamma\nu^{\prime}_{0}=350$ keV, $\alpha=-0.2,\beta=1.2,$ and $z=1$. The
results of our calculations for the case of $\alpha=\beta$ are consistent with
those of Lazzati et al. (2004). The results are similar to those of the SR
model, but the polarization degree is higher than in the SR model.
Figure 4.— Same as Figure 2, but in the CD model (Compton drag model).
The polarization is higher for softer spectra, although the local polarization
degree is not dependent on the frequency in this model. For instance, for
$y_{j}=1$, $\gamma\nu^{\prime}_{0}=350$ keV, and $z=1$, the maximum
polarization is $\simeq 0.66$ for $\alpha=-0.5$ and $\beta=0.9$, while it is
$\simeq 0.71$ for $\alpha=0.4$ and $\beta=1.8$, but the variation is smaller
than for the synchrotron models (see § 3.1 and 3.2). This variation is caused
by the kinematic effect. The local polarization degree is a maximum for
$\theta=\gamma^{-1}$ (i.e., $\theta^{\prime}=\pi/2$). Thus the net
polarization is higher when the contribution of the emission from higher
latitude with $\theta>\gamma^{-1}$ is smaller. The high latitude emission is
dimmer as the radiation spectrum is softer. Therefore the net polarization is
higher when the spectrum is softer. This effect also arises in the SO and SR
models, although in those models the intrinsic dependence of polarization on
the spectrum (equation 8) is rather strong (see § 3.1 and 3.2). For $y_{j}\geq
0.01$, $\alpha\leq 0.4$, and $\beta\leq 1.8,$ the maximum polarization degree
for the CD model is $\simeq 1.0$.
## 4\. Statistical properties
In this section we show the results of our Monte Carlo simulation of the GRB
prompt emission polarization. First, in § 4.1, we give the values of the model
parameters so that the observed fluences and peak energies of simulated bursts
are consistent with the data obtained with the HETE-2 satellite. In § 4.2, we
examine the properties of the polarization distribution of bursts detectable
by the POET satellite, regardless of instrument MDP. Next, in § 4.3, we show
the distribution of polarizations that can be measured by POET, and discuss
how we may constrain the emission models.
### 4.1. Model parameters
We performed Monte Carlo simulations to obtain the distribution of the
observed spectral energies and fluences in the three emission models. Such
simulations have been developed to discuss the empirical correlation between
spectral peak energies in the cosmological rest frame and isotropic
$\gamma$-ray energies among GRBs and X-ray flashes in several models of
geometrical structure of GRB jets (Zhang et al., 2004; Yamazaki et al., 2004;
Dai & Zhang, 2005; Toma et al., 2005; Donaghy, 2006). We generated 10,000 GRB
jets with Lorentz factor, $\gamma$, and opening angle, $\theta_{j}$, and a
random viewing angle for each jet according to the probability distribution of
$\sin\theta_{v}d\theta_{v}d\phi$ with $\theta_{v}<0.22$ rad. 222 We confirmed
that the bursts with $\theta_{v}\geq 0.22$ rad in our simulation are not
detected by HETE-2 or POET with the parameters we adopt in this paper. We can
therefore discuss the distribution of several quantities of the detectable
bursts and the event rate ratio of bursts for which polarizations can be
measured to the detectable bursts without considering the bursts with
$\theta_{v}\geq 0.22$ rad. For each burst generated we calculate the $\nu
I_{\nu}$ spectrum to obtain the spectral peak energy, $E_{p,{\rm obs}}$, and
the fluence, $I$, in the $2-400$ keV range by using equation (4). Since
$E_{p,{\rm obs}}$’s and $I$’s calculated for each $q=\theta_{v}/\theta_{j}$ in
the three models are different only by factors less than 2, $E_{p,{\rm
obs}}$’s and $I$’s of the simulated bursts may be calculated using just one
model, for which we chose the CD model.
Figure 5.— The $E_{p,{\rm obs}}-F$ diagram calculated in our Monte Carlo
simulation. The simulated events that can be detected by WXM on HETE-2 are
represented by dots. They are compared with the HETE-2 data (points with
errorbars) (Sakamoto et al., 2005).
The distributions of $\gamma$ and $\theta_{j}$ for GRB jets are highly
uncertain. We make a simple assumption for the distribution and in § 4.3 we
perform some simulations for different assumptions. We fix $\gamma=100$. We
assume the distribution of $\theta_{j}$ as
$f(\theta_{j})d\theta_{j}\propto\left\\{\begin{array}[]{lr}\theta_{j}^{q_{1}}d\theta_{j}&{\rm
for}~{}0.001\leq\theta_{j}\leq 0.02\\\ \theta_{j}^{q_{2}}d\theta_{j}&{\rm
for}~{}0.02\leq\theta_{j}\leq 0.2,\end{array}\right.$ (18)
where $q_{1}=0.5$ and $q_{2}=-2.0$. The value of $q_{2}=-2$ is inferred from
the observations of the steepening breaks (i.e., jet breaks) of some optical
afterglows (Frail et al., 2001; Zeh et al., 2006) and from analysis of BATSE
data using some empirical relations (Yonetoku et al., 2005). There are several
suggestions of events with very small $\theta_{j}$ (e.g., Schady et al., 2007;
Racusin et al., 2008), although the value of $q_{1}$ is highly uncertain. The
spectral parameters $r_{0}^{2}A_{0},\gamma\nu^{\prime}_{0},\alpha,$ and
$\beta$ are assumed as follows. The first two parameters are given so that the
rest-frame spectral peak energies and isotropic $\gamma$-ray energies
calculated for a jet viewed with $\theta_{v}=0$ are consistent with those of
typical GRBs. Such an on-axis emission has approximately
$E_{p}=2\gamma\nu^{\prime}_{0}$ and $E_{\rm
iso}=16\pi^{2}r_{0}^{2}A_{0}\gamma\nu^{\prime}_{0}$. The parameters
$r_{0}^{2}A_{0}$ and $\gamma\nu^{\prime}_{0}$ are given through the empirical
relations $E_{\rm iso}\theta_{j}^{2}/2=10^{51}\xi_{1}$ erg and
$E_{p}=80\xi_{2}(E_{\rm iso}/10^{52}~{}{\rm erg})^{1/2}$ keV (e.g., Frail et
al., 2001; Amati et al., 2002). We assume that the coefficients $\xi_{1}$ and
$\xi_{2}$ obey the log-normal distribution (Ioka & Nakamura, 2002) with
averages of $1$ and logarithmic variances of $0.3$ and $0.15$, respectively.
The last two parameters are fixed by $\alpha=-0.2$ and $\beta=1.2$, which are
typical values for GRB prompt emission (Preece et al., 2000; Sakamoto et al.,
2005). The distribution of the source redshift, $z$, is assumed to be in
proportional to the cosmic star formation rate. We adopt the model SF2 in
Porciani & Madau (2001), i.e., the comoving GRB rate density is assumed to be
proportional to
$R(z)=\frac{\exp(3.4z)}{\exp(3.4z)+22}\frac{\sqrt{\Omega_{M}(1+z)^{3}+\Omega_{\Lambda}}}{(1+z)^{3/2}}.$
(19)
We take the standard cosmological parameters of $H_{0}=70~{}{\rm km}~{}{\rm
s}^{-1}~{}{\rm Mpc}^{-1},\Omega_{M}=0.3,$ and $\Omega_{\Lambda}=0.7$.
Figure 5 shows the results of $E_{p,{\rm obs}}$ and time-averaged flux, $F$.
The time-averaged flux is calculated by $F=I/T$, where $T$ is the duration of
a burst. We fix $T=20$ s, which is a typical value for long GRBs (e.g.,
Sakamoto et al., 2005). We show only the simulated bursts that have fluxes
above the detectable limit of the HETE-2 satellite. They are consistent with
the data obtained by HETE-2 (Sakamoto et al., 2005). The scatter of the
simulated bursts is due both to the scatter of the assumed jet parameters and
the viewing angle effect (Yamazaki et al., 2004; Donaghy, 2006). 333 Yamazaki
et al. (2004) showed a deviation from the Amati relation ($E_{p}\propto E_{\rm
iso}^{1/2}$) for $E_{p}<10$ keV in the uniform jet model, but the $E_{p,{\rm
obs}}-F$ diagram we derive is still consistent with the observed dataset.
### 4.2. Properties of polarization distribution
We calculated the linear polarization, $\Pi$, by using equations (11), (16),
and (17) to obtain the polarization distribution of the simulated bursts that
can be detected by GRAPE and LEP. The detection limits of GRAPE and LEP are
set to be the MDP contours of 1.0. (see Figure 1) 444 The detection limits of
GRAPE and LEP for signal-to-noise ratio $>5$ are similar but not identical to
the MDP contours of 1.0. Thus our setting for the detection limits is just for
simplicity. . Figures 6 and 7 show the $E_{p,{\rm obs}}-\Pi$ diagrams of all
the simulated bursts that can be detected by GRAPE and LEP, respectively, in
the SO (red open circles), SR (green filled circles), and CD (blue plus signs)
models. In the SO model, most of the detectable bursts have $0.3<\Pi<0.5$ in
the GRAPE band (60-500 keV), while they have $0.2<\Pi<0.3$ in the LEP band
(2-15 keV). In the SR and CD models, most of the detectable bursts have
$\Pi<0.1$ in both GRAPE and LEP bands. The events with $\Pi\geq 0.1$ are
distributed uniformly with $\Pi<0.4$ and $\Pi<0.9$ for the SR and CD models,
respectively.
Figure 6.— $E_{p,{\rm obs}}-\Pi$ diagrams for the simulated events that can
be detected by GRAPE in the SO (red open circles), SR (green filled circles),
and CD (blue plus signs) models. The adopted parameters are as follows. The
fixed parameters are $\gamma=100,q_{1}=0.5,q_{2}=-2.0,\alpha=-0.2,$
$\beta=1.2,$ and $T=20$ s. The distribution of the source redshift $z$ is
assumed to be in proportional to the cosmic star formation rate. The
parameters $r_{0}^{2}A_{0}$ and $\gamma\nu^{\prime}_{0}$ are distributed so
that the simulated $E_{p,{\rm obs}}-F$ diagram is consistent with the observed
data (see Figure 5). See text for the cases of the spectral indices
distributed realistically, for $-0.5<\alpha<0.4$ and $0.9<\beta<1.8$. Figure
7.— Same as Figure 7, but for LEP.
This result can be roughly explained by the polarizations calculated as
functions of $y_{j}$ and $q=\theta_{v}/\theta_{j}$ for
$\gamma\nu^{\prime}_{0}=350$ keV and $z=1$ (see Figures 2, 3, 4) and the
distribution of $\theta_{j}$ and $q$ for the detectable bursts in this
simulation, shown in Figure 8. The detectable events are dominated by the
events with $q<1$, since events with $q<1$ are much brighter than those with
$q>1$ because of the relativistic beaming effect. For events with $q>1$,
narrower jets are easier to detect since they have intrinsically higher
emissivities by our assumption. Most of the detectable events have $q<1$ and
$\theta_{j}>0.02$ (i.e., $y_{j}>4$). For these events the SO model gives
$0.3<\Pi<0.5$ in most cases, while the SR and CD models give $\Pi<0.1$, for
the GRAPE band as shown in Figures 2, 3, and 4. The remaining detectable
events mainly have $q>1$ and $\theta_{j}>0.005$ (i.e., $y_{j}>0.25$). These
events have $\Pi<0.6$ in the SO model, $\Pi<0.5$ in the SR model, and
$\Pi<0.9$ in the CD model, for the GRAPE band as shown in Figures 2, 3, and 4.
The results for the LEP band can be explained similarly.
In all the three models, the results show $\Pi(60-500~{}{\rm
keV})>\Pi(2-15~{}{\rm keV})$ for almost all the detectable bursts with
$\Pi>0.1$. This is due to the fact that typically the contribution of the
high-energy photons with spectral index $\beta$ is larger in the GRAPE band
than in the LEP band. The emission with softer spectrum has higher
polarization because of the intrinsic property of the synchrotron polarization
(equation 8) for the SO and SR models and the kinematic effect for the CD
model (see § 3.3), respectively.
In the SO model, the polarization of GRBs with $q<1$ is higher for lower
$E_{p,{\rm obs}}$ for the GRAPE band. This is because the contribution from
high-energy photons, with energy spectral index $\beta$, is larger. In the SR
and CD models, the higher $\Pi$ GRBs can be obtained for smaller $\theta_{j}$.
The maximum $\Pi$ is obtained for $\theta_{j}\simeq 0.002$.
Figure 8.— Distribution of $\theta_{j}$ and $q=\theta_{v}/\theta_{j}$ of the
detectable bursts by GRAPE in the model described by Figure 6.
### 4.3. Cumulative distribution of measurable polarizations
We obtain the distribution of polarization that can be measured, by using the
MDP values we derived for $\alpha=-0.2$, $\beta=1.2$, and $T=20$ s (see § 2).
We interpret the simulated events with $\Pi>MDP$ as ‘$\Pi$-measurable events’.
Figure 9 shows the cumulative distribution of $\Pi$ that can be measured by
GRAPE and LEP in the SO, SR, and CD models. We have set the number of
detectable events $N_{d}=200$. Since the polarization in the LEP band is lower
than in the GRAPE band for almost all the cases as discussed in § 4.2, the
number of events for which polarization can be measured by LEP is smaller than
for GRAPE. In the SO model, the number of $\Pi$-measurable bursts is
$N_{m}>60$, and the cumulative distribution of measurable $\Pi$ varies rapidly
at $0.3<\Pi<0.4$ for the GRAPE band. In the SR model, $N_{m}<10$, and the
maximum polarization is $\Pi_{\rm max}<0.4$. In the CD model, $N_{m}<30$, and
$\Pi_{\rm max}<0.8$.
To investigate general properties of the cumulative distribution that do not
depend on the model parameters, we performed simulations for other values of
$\gamma,$ $q_{1},$ and $q_{2}$, the Lorentz factor of the jets and the power-
law indices of the distribution of the opening angles of the jets,
respectively. We refer to the parameters adopted for the above simulation as
‘typical’ parameters. We now consider a range of parameters: $\gamma\geq 100$,
$q_{1}\geq 0.5,$ and $q_{2}\geq-3.0$, which are quite reasonable for GRBs
(e.g., Lithwick & Sari, 2001; Yonetoku et al., 2005). Within these parameter
ranges we obtain the lower (upper) limit of $N_{m}/N_{d}$ for the SO model
(the SR/CD models).
Figure 10 shows the results for $\gamma=300$ and the same ‘typical’ values for
the other parameters. The number $N_{m}$ is larger in the SO model and smaller
in the SR and CD models than the case of $\gamma=100$. As $\gamma$ is larger,
the beaming effect is stronger and the ratio of the bursts with $q<1$ for
detectable bursts is larger. Thus the number of bursts with a high degree of
polarization is larger in the SO model and smaller in the SR and CD models.
Figure 11 shows the results for $q_{1}=1.0$ and the same ‘typical’ values for
the other parameters. Since the ratio of the number of the bursts with smaller
$y_{j}$ to that of detectable bursts is smaller, $N_{m}$ is slightly smaller
than that for the ‘typical’ parameters in the SR and CD models. Figure 12
shows the results for $q_{2}=-3.0$ and the ‘typical’ values for the other
parameters. In this case $N_{m}$ is slightly larger than that for the
‘typical’ parameters in the SR and CD models. The number $N_{m}$ in the SO
model is similar for Figure 9, 11, and Figure 12 in the GRAPE band. To
summarize, for the parameters $\gamma\geq 100$, $q_{1}\geq 0.5$,
$q_{2}\geq-3.0$, $\alpha=-0.2$ and $\beta=1.2$, we can say that
$N_{m}/N_{d}>30\%$ for GRAPE and the cumulative distribution of measurable
$\Pi$ varies rapidly from $0.3<\Pi<0.4$ in the SO model. For the SR model,
$N_{m}/N_{d}<5\%$ for GRAPE, with a maximum polarization $\Pi_{\rm max}<0.4$.
For the CD model, $N_{m}/N_{d}<15\%$ for GRAPE, and $\Pi_{\rm max}<0.8$.
Since the dependence of the polarization degree on the spectral indices is
relatively large in the SO and SR models, we should take account of the
distribution of $\alpha$ and $\beta$. The observed spectral parameters
$\alpha$ and $\beta$ are distributed roughly as $-0.5<\alpha<0.4$ and
$0.9<\beta<1.8$ (Preece et al., 2000; Sakamoto et al., 2005). Within these
ranges of $\alpha$ and $\beta$, the polarization degree for $y_{j}>10$, $q<1$,
and $50<E_{p,{\rm obs}}<10^{3}$ keV is $0.2<\Pi<0.7$ in the SO model. Thus the
measurable polarizations are clustered at $0.2<\Pi<0.7$. The maximum
polarization obtained in the SO model for $y_{j}\geq 0.01,\alpha\leq 0.4,$ and
$\beta\leq 1.8$ is $\simeq 0.8$ (see § 3.1). In this case $N_{m}/N_{d}$ will
be larger than 30%. In the CD model, the result will not be significantly
different from the case of fixed $\alpha$ and $\beta$. In the SR model, the
polarization degree does not exceed those calculated in the CD model, and thus
$N_{m}/N_{d}<15\%$. The maximum polarization obtained in the SR model for
$y_{j}\geq 0.01,\alpha\leq 0.4,$ and $\beta\leq 1.8$ is $\simeq 0.8$ (see §
3.2).
In conclusion, we can constrain the emission mechanism of GRBs by using the
cumulative distribution obtained by GRAPE. If $N_{m}/N_{d}>30\%$, the SR and
CD models may be ruled out, and in this case if the measured polarizations are
clustered at $0.2<\Pi<0.7$, the SO model will be favored. If
$N_{m}/N_{d}<15\%$, the SO model may be ruled out, but we cannot distinguish
between the SR and CD models with different distributions of $y_{j}$,
$\alpha$, and $\beta$. If several bursts with $\Pi>0.8$ are detected, however,
the CD model which includes adequate number of small $y_{j}$ bursts will be
favored.
## 5\. Summary and discussion
Recently there has been an increasing interest in the measurement of X-ray and
$\gamma$-ray polarization, and the observational techniques can now achieve
significant sensitivity in the relevant energy bands. Several polarimetry
mission concepts, such as POET, are being planned. The POET concept has two
polarimeters, GRAPE (60-500 keV) and LEP (2-15 keV) both of which have wide
fields of view. If launched, missions of this type would provide the first
definitive detection of the polarization of GRB prompt emission. This would
enable the discussion of the statistical properties of the polarization degree
and polarization spectra, which will give us diagnostic information on the
emission mechanism of GRBs and the nature of the GRB jets that cannot be
obtained from current spectra and lightcurve observations. We have performed
Monte Carlo simulations of the linear polarization from GRB jets for three
major emission models: synchrotron model with globally ordered magnetic field
(SO model), synchrotron model with small-scale random magnetic field (SR
model), and Compton drag model (CD model). We assumed that the physical
quantities for the emission of the jets are uniform on the emitting surface
and that the jets have sharp edges. Our jet angle distribution allows the
detections of GRBs with very small opening angles (i.e., smaller than 1
degree) as suggested by several Swift bursts (Schady et al., 2007; Racusin et
al., 2008). We have shown that the POET mission or other polarimeters with
similar capabilities, i.e., broadband spectral capabilities for the
determination of $E_{p,{\rm obs}}$ and sensitive broadband polarimetric
capabilities to minimize MDP, can constrain the emission models of GRBs.
Furthermore, these simulations indicate that an increase in the LEP effective
area would be beneficial to compensate for the lower expected polarization at
lower energies.
As shown in Figures 2, 3, and 4, the SR and CD models require off-axis
observations of the jets to achieve a high level of polarization, while the SO
model does not. In this sense the SR and CD models are categorized as
geometric models, and the SO model as an intrinsic model (Waxman, 2003;
Lazzati, 2006). The distribution of observed polarizations obtained by our
simulations show that the geometric SR/CD models will be ruled out if the
number ratio of the $\Pi$-measurable bursts to detected bursts is larger than
$30\%$, and in this case the SO model will be favored if the measurable
polarizations are clustered at $0.2<\Pi<0.7$. If the number ratio is smaller
than $15\%$, the SO model may be ruled out, but we cannot distinguish between
the SR and CD models with different distributions of
$y_{j}=(\gamma\theta_{j})^{2}$, $\alpha$, and $\beta$, where $\gamma$ and
$\theta_{j}$ are the bulk Lorentz factor and the opening angle of the GRB jet,
respectively, and $\alpha$ and $\beta$ are lower and higher indices of the
energy spectrum. However, if several bursts with $\Pi>0.8$ are detected, the
CD model which includes an adequate number of small $y_{j}$ bursts will be
favored.
If the cumulative distribution of the measurable polarizations favors the SO
model, the globally ordered magnetic field would be advected from the central
engine. If we understand the strength of the magnetic field in the emitting
region from the luminosity and the spectrum of the emission, we can constrain
the strength of the field at the central engine. If the geometric SR/CD models
are favored from the observations, it will be established, independently of
the afterglow observations, that GRB outflows are not spherical but highly
collimated. If the CD model is favored by the observations, we may constrain
the distribution of the parameter $y_{j}=(\gamma\theta_{j})^{2}$ of GRB jets.
The CD model needs a dense optical/UV photon field interacting within the
relativistic jets (Lazzati et al., 2000; Eichler & Levinson, 2003).
We have made some simplifications in our simulations, and there are some
caveats. We have assumed that the jets are uniform on the emitting surfaces
and have sharp edges. To compare the simulations and the observations further,
more sophisticated modeling is required (e.g., Zhang et al., 2004; Toma et
al., 2005).
We have interpreted bursts as a simple combination of pulses, without taking
account of the temporal variation of the Lorentz factor $\gamma$ of the jet.
If this is accounted for, each pulse may have different
$y_{j}=(\gamma\theta_{j})^{2}$ but the same $q=\theta_{v}/\theta_{j}$. We
should then average the polarization with respect to fluence of each pulse
having different $y_{j}$ (Granot, 2003; Nakar et al., 2003). However, in the
SO model, the cumulative distribution of measurable $\Pi$ will not be changed
significantly as long as $y_{j}>10$, because $\Pi$ is clustered into a small
range for $q<1$ and $y_{j}>10$. To average the polarization in the case of
$q>1$, the relation between the luminosity and the Lorentz factor for each
pulse is required to predict the polarization distribution.
For the SR model we have assumed that the directions of the magnetic field are
confined within the shock plane. They may be more isotropic in reality, in
which case the polarization degree in the SR model will be reduced.
In the synchrotron model with a combination of the globally ordered magnetic
field and the locally random field, $\mathbf{B}=\mathbf{B}_{\rm
ord}+\mathbf{B}_{\rm rnd}$, the linear polarization can be calculated by
$\Pi=(Q_{\rm ord}+Q_{\rm rnd})/(I_{\rm ord}+I_{\rm rnd})\approx(\Pi_{\rm
ord}+\eta\Pi_{\rm rnd})/(1+\eta)$, where $\\{I,Q\\}_{\rm ord}$ and
$\\{I,Q\\}_{\rm rnd}$ are the Stokes parameters from the ordered and random
fields, respectively. $\Pi_{\rm ord}$ and $\Pi_{\rm rnd}$ are described by
equations (11) and (16), and $\eta\equiv(B_{\rm rnd}/B_{\rm ord})^{\alpha+1}$.
This model will reduce the number ratio of $\Pi$-measurable bursts to detected
bursts to less than 30% and the clustering of measurable polarizations will be
at $\Pi<0.7$.
This work is supported in part by the Grant-in-Aid from the Ministry of
Education, Culture, Sports, Science and Technology (MEXT) of Japan,
No.19047004 (RY, KI, and TN), No.18740153 (RY), No.18740147 (KI), and in part
by the Grant-in-Aid for the global COE program The Next Generation of Physics,
Spun from Universality and Emergence from the Ministry of Education, Culture,
Sports, Science and Technology (MEXT) of Japan. BZ acknowledges NASA
NNG05GB67G and NNX08AE57A (Nevada NASA EPSCoR program) and KT acknowledges
NASA NNX08AL40G for partial supports.
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Figure 9.— The cumulative distribution of $\Pi$ that can be measured by GRAPE
(left) and LEP (right) in the SO (solid), SR (dashed), and CD (dot-dashed)
models in which the number of detectable bursts is 200. The adopted parameters
are as follows. The fixed parameters are
$\gamma=100,q_{1}=0.5,q_{2}=-2.0,\alpha=-0.2,$ $\beta=1.2,$ and $T=20$ s. The
distribution of the source redshift $z$ is assumed to be in proportional to
the cosmic star formation rate. The parameters $r_{0}^{2}A_{0}$ and
$\gamma\nu^{\prime}_{0}$ are distributed so that the simulated $E_{p,{\rm
obs}}-F$ diagram is consistent with the observed date (see Figure 5). See text
for the cases of the spectral indices distributed realistically, for
$-0.5<\alpha<0.4$ and $0.9<\beta<1.8$.
Figure 10.— Same as Figure 9, but the Lorentz factor of the jets
$\gamma=300$.
Figure 11.— Same as Figure 9, but the lower power-law index of the
$\theta_{j}$ distribution $q_{1}=1.0$ (see equation 18).
Figure 12.— Same as Figure 9, but the higher power-law index of the
$\theta_{j}$ distribution $q_{2}=-3.0$ (see equation 18).
## Appendix A Some notes on synchrotron polarization
### A.1. The SO model: synchrotron with ordered field
We consider the synchrotron radiation from the shell moving radially outward
with a bulk Lorentz factor $\gamma\gg 1$, and the magnetic field in the shell
is globally ordered within the plane parallel to the shock plane. If the
matter of the shell expands with a constant speed, the strength of magnetic
field with radial direction scales as $R^{-2}$ while that with transverse
direction scales as $R^{-1}$. Thus the field advected with the shell is likely
to have the direction parallel to the shock plane.
We set the line of sight (i.e., the direction from the central engine to the
earth) in the lab frame to be $z$ axis, and the direction of the magnetic
field on a given point of the shell, projected onto the plane perpendicular to
$z$ axis, to be $\bar{x}$ axis. The given point can be described by spherical
coordinates $(\theta,\varphi)$. Then the components of the velocity vector of
the given point and the unit wave vector can be described by the right-handed
coordinate system $\bar{x}\bar{y}z$ as
$\mathbf{\beta}=(\beta\sin\theta\cos\varphi,\beta\sin\theta\sin\varphi,\beta\cos\theta)$
and $\hat{\mathbf{k}}=(0,0,1)$, respectively. The unit wave vector in the
comoving frame is
$\hat{\mathbf{k}^{\prime}}=\frac{1}{\gamma(1-\mathbf{\beta}\cdot\hat{\mathbf{k}})}\left[\hat{\mathbf{k}}+\mathbf{\beta}\left(\frac{\gamma^{2}}{\gamma+1}\mathbf{\beta}\cdot\hat{\mathbf{k}}-\gamma\right)\right].$
(A1)
Since the direction of the magnetic field in the comoving frame is
perpendicular to the velocity vector of the fluid,
$\hat{\mathbf{B}}^{\prime}=(\cos\theta/\sqrt{\cos^{2}\theta+\sin^{2}\theta\cos^{2}\varphi},0,-\sin\theta\cos\varphi/\sqrt{\cos^{2}\theta+\sin^{2}\theta\cos^{2}\varphi})$.
Then we may calculate
$\cos\theta^{\prime}_{B}=\hat{\mathbf{B}^{\prime}}\cdot\hat{\mathbf{k}^{\prime}}$,
and we obtain
$\sin\theta^{\prime}_{B}\approx\left(\frac{1-\gamma^{2}\theta^{2}}{1+\gamma^{2}\theta^{2}}\right)^{2}\cos^{2}\varphi+\sin^{2}\varphi$
(A2)
in the limit $\gamma\gg 1$.
The direction of the polarization vector of the synchrotron radiation is
calculated by
$\mathbf{e}^{\prime}\parallel\hat{\mathbf{B}^{\prime}}\times\hat{\mathbf{k}^{\prime}}$.
Then we obtain the direction of the polarization vector in the lab frame by
$\mathbf{e}=\gamma(1+\mathbf{\beta}\cdot\hat{\mathbf{k}^{\prime}})\mathbf{e}^{\prime}-(\mathbf{\beta}\cdot\mathbf{e}^{\prime})\left(\frac{\gamma^{2}}{\gamma+1}\mathbf{\beta}+\gamma\hat{\mathbf{k}^{\prime}}\right).$
(A3)
The results are $e_{z}=0$ and
$\tan\chi_{B}\equiv\frac{e_{y}}{e_{x}}=\tan\varphi-\frac{\beta-\cos\theta}{\beta\sin^{2}\theta}\frac{1}{\sin\varphi\cos\varphi}.$
(A4)
The angle $\chi_{B}$ is the polarization position angle measured from the axis
$\bar{x}$ (i.e., the direction of the local magnetic field). The above
equation can be rewritten as
$\chi_{B}\approx\varphi+\arctan[(1-\gamma^{2}\theta^{2})\cot\varphi/(1+\gamma^{2}\theta^{2})]$.
This result is consistent with that of Granot (2003).
Based on the above results, we consider the case that the magnetic field is
axisymmetric around the jet and has a toroidal configuration. We set the
direction from the line of sight to the jet axis to be $x$ axis. Below we will
rewrite the above results by using the azimuthal angle $\phi$ measured from
$x$ axis. In the coordinate system of $xyz$, the jet axis and the coordinates
of a given point on the shell are described as
$\mathbf{J}=(\sin\theta_{v},0,\cos\theta_{v})$, and
$\mathbf{R}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$, respectively.
The magnetic field at the given point is given by
$\hat{\mathbf{B}^{\prime}}=\mathbf{R}\times\mathbf{J}/|\mathbf{R}\times\mathbf{J}|$.
Let the unit vectors of the directions of $\mathbf{R}$ and
$\hat{\mathbf{B}^{\prime}}$ projected onto $xy$ plane be $\hat{\mathbf{r}}$
and $\hat{\mathbf{b}}$, and
$\cos\varphi=\hat{\mathbf{b}}\cdot\hat{\mathbf{r}}$. Then we obtain
$\cos^{2}\varphi\approx\frac{\sin^{2}\phi}{1+a^{2}-2a\cos\phi},$ (A5)
where $a\equiv\theta/\theta_{v}$. Equation (9) is given by inserting equation
(A5) into equation (A2). If we measure the position angle from the $x$ axis,
we obtain equation (10), i.e., $\chi=\chi_{B}-\varphi+\phi$. These results are
consistent with those of Granot & Taylor (2005).
### A.2. The SR model: synchrotron with random field
Here we consider that the directions of the magnetic fields are confined
within the plane parallel to the shock plane and that they are completely
random. This field configuration is possible if the field is generated by the
shock. In the comoving frame of the shell, we set the direction of
$\hat{\mathbf{k}^{\prime}}$ to be axis $3$, and set a right-handed coordinate
system $123$. Let the polar and azimuthal angles of
$\hat{\mathbf{B}^{\prime}}$ be $\theta^{\prime}_{B}$ and $\phi^{\prime}_{B}$,
respectively. In this coordinate system, the Stokes parameters of synchrotron
emissivity are given by
$j^{\prime Q}_{\nu^{\prime}}=-j^{\prime
I}_{\nu^{\prime}}\Pi_{0}\cos(2\phi^{\prime}_{B}),~{}~{}j^{\prime
U}_{\nu^{\prime}}=-j^{\prime
I}_{\nu^{\prime}}\Pi_{0}\sin(2\phi^{\prime}_{B}).$ (A6)
Next we set another right-handed coordinate system $xyz$ of which $z$ axis is
along the velocity vector of the fluid and $xz$ plane includes
$\mathbf{k}^{\prime}$. Then the angle between $\mathbf{k}^{\prime}$ and $z$
axis is $\theta^{\prime}$. Here the magnetic field $\mathbf{B}^{\prime}$ is
confined within $xy$ plane. Let the azimuthal angle of $\mathbf{B}^{\prime}$
be $\eta^{\prime}$, and we obtain the relations between the components of
$\mathbf{B}^{\prime}$ in the systems $123$ and $xyz$.
$\begin{array}[]{l}\sin\theta^{\prime}_{B}\sin\phi^{\prime}_{B}=\cos\theta^{\prime}\cos\eta^{\prime},\\\
\sin\theta^{\prime}_{B}\cos\phi^{\prime}_{B}=\sin\eta^{\prime},\\\
\cos\theta^{\prime}_{B}=\sin\theta^{\prime}\cos\eta^{\prime}.\end{array}$ (A7)
Then we obtain
$\begin{array}[]{l}\sin\theta^{\prime}_{B}=\left[1-\frac{4\gamma^{2}\theta^{2}}{(1+\gamma^{2}\theta^{2})^{2}}\cos^{2}\eta^{\prime}\right]^{1/2},\\\
\cos(2\phi^{\prime}_{B})=\frac{1}{\sin^{2}\theta^{\prime}_{B}}\left[\sin^{2}\eta^{\prime}-\left(\frac{1-\gamma^{2}\theta^{2}}{1+\gamma^{2}\theta^{2}}\right)^{2}\cos^{2}\eta^{\prime}\right].\end{array}$
(A8)
To obtain the polarization degree of synchrotron radiation from the random
field, we average the Stokes parameters with respect to $\eta^{\prime}$. This
leads to $\langle j^{\prime U}_{\nu^{\prime}}\rangle=0$. Then we can calculate
the polarization degree by $\Pi_{0}=\langle j^{\prime
Q}_{\nu^{\prime}}\rangle/\langle j^{\prime
I}_{\nu^{\prime}}\rangle=\Pi_{0}^{\rm
syn}\langle(\sin\theta^{\prime}_{B})^{\alpha+1}\cos(2\phi^{\prime}_{B})\rangle/\langle(\sin\theta^{\prime}_{B})^{\alpha+1}\rangle$,
and the polarization vector is along axis 1, i.e., the direction perpendicular
to $\mathbf{k}^{\prime}$ and within the plane including $\mathbf{k}^{\prime}$
and $\mathbf{\beta}$.
|
arxiv-papers
| 2008-12-12T21:11:20 |
2024-09-04T02:48:59.372983
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kenji Toma, Takanori Sakamoto, Bing Zhang, Joanne E. Hill, Mark L.\n McConnell, Peter F. Bloser, Ryo Yamazaki, Kunihito Ioka, Takashi Nakamura",
"submitter": "Kenji Toma Dr.",
"url": "https://arxiv.org/abs/0812.2483"
}
|
0812.2502
|
# Not each sequential effect algebra is sharply dominating††thanks: This
project is supported by Natural Science Found of China (10771191 and
10471124).
Shen Jun1,2, Wu Junde1 E-mail: wjd@zju.edu.cn
###### Abstract
Let $E$ be an effect algebra and $E_{S}$ be the set of all sharp elements of
$E$. $E$ is said to be sharply dominating if for each $a\in E$ there exists a
smallest element $\widehat{a}\in E_{s}$ such that $a\leq\widehat{a}$. In 2002,
Professors Gudder and Greechie proved that each $\sigma$-sequential effect
algebra is sharply dominating. In 2005, Professor Gudder presented 25 open
problems in International Journal of Theoretical Physics, Vol. 44, 2199-2205,
the 3th problem asked: Is each sequential effect algebra sharply dominating?
Now, we construct an example to answer the problem negatively.
1Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
2Department of Mathematics, Anhui Normal University, Wuhu 241003, P. R. China
Key Words. Sequential effect algebra, sharply dominating, sharp element.
Effect algebra is an important model for studying the unsharp quantum logic
(see [1]). In 2001, in order to study quantum measurement theory, Professor
Gudder began to consider the sequential product of two measurements $A$ and
$B$ (see [2]). In 2002, moreover, Professors Gudder and Greechie introduced
the abstract sequential effect algebra structure and studied its some
important properties. In particular, they proved that each $\sigma$-sequential
effect algebra is sharply dominating ([3, Theorem 6.3]). In 2005, Professor
Gudder presented 25 open problems in [4] to motive the study of sequential
effect algebra theory, the 3th problem asked: Is each sequential effect
algebra sharply dominating? Now, we construct an example to answer the problem
negatively.
First, we need the following basic definitions and results for effect algebras
and sequential effect algebras.
An effect algebra is a system $(E,0,1,\oplus)$, where 0 and 1 are distinct
elements of $E$ and $\oplus$ is a partial binary operation on $E$ satisfying
that [1]:
(EA1) If $a\oplus b$ is defined, then $b\oplus a$ is defined and $b\oplus
a=a\oplus b$.
(EA2) If $a\oplus(b\oplus c)$ is defined, then $(a\oplus b)\oplus c$ is
defined and
$(a\oplus b)\oplus c=a\oplus(b\oplus c).$
(EA3) For each $a\in E$, there exists a unique element $b\in E$ such that
$a\oplus b=1$.
(EA4) If $a\oplus 1$ is defined, then $a=0$.
In an effect algebra $(E,0,1,\oplus)$, if $a\oplus b$ is defined, we write
$a\bot b$. For each $a\in(E,0,1,\oplus)$, it follows from (EA3) that there
exists a unique element $b\in E$ such that $a\oplus b=1$, we denote $b$ by
$a^{\prime}$. Let $a,b\in(E,0,1,\oplus)$, if there exists a $c\in E$ such that
$a\bot c$ and $a\oplus c=b$, then we say that $a\leq b$. It follows from [1]
that $\leq$ is a partial order of $(E,0,1,\oplus)$ and satisfies that for each
$a\in E$, $0\leq a\leq 1$, $a\bot b$ if and only if $a\leq b^{\prime}$.
Let $(E,0,1,\oplus,\circ)$ be an effect algebra and $a\in E$. If $a\wedge
a^{\prime}=0$, then $a$ is said to be a sharp element of $E$. The set
$E_{S}=\\{x\in E|\ x\wedge x^{\prime}=0\\}$ is called the set of all sharp
elements of $E$ (see [5-6]). The effect algebra $(E,0,1,\oplus,\circ)$ is
called sharply dominating if for each $a\in E$ there exists a smallest sharp
element $\widehat{a}\in E_{s}$ such that $a\leq\widehat{a}$. That is, if $b\in
E_{s}$ satisfies $a\leq b$, then $\widehat{a}\leq b$. An important example of
sharply dominating effect algebras is the standard Hilbert space effect
algebra $\cal E(H)$ of positive linear operators on a complex Hilbert space
$\cal H$ with norm less than 1([5-6]). The sharply dominating effect algebras
have many nice properties, for example, recently, Riecanova and Wu showed that
sharply dominating Archimedean atomic lattice effect algebras can be
characterized by the property called basic decomposition of elements, etc (see
[5]).
A sequential effect algebra is an effect algebra $(E,0,1,\oplus)$ and another
binary operation $\circ$ defined on $(E,0,1,\oplus)$ satisfying [3]:
(SEA1) The map $b\mapsto a\circ b$ is additive for each $a\in E$, that is, if
$b\bot c$, then $a\circ b\bot a\circ c$ and $a\circ(b\oplus c)=a\circ b\oplus
a\circ c$.
(SEA2) $1\circ a=a$ for each $a\in E$.
(SEA3) If $a\circ b=0$, then $a\circ b=b\circ a$.
(SEA4) If $a\circ b=b\circ a$, then $a\circ b^{\prime}=b^{\prime}\circ a$ and
$a\circ(b\circ c)=(a\circ b)\circ c$ for each $c\in E$.
(SEA5) If $c\circ a=a\circ c$ and $c\circ b=b\circ c$, then $c\circ(a\circ
b)=(a\circ b)\circ c$ and $c\circ(a\oplus b)=(a\oplus b)\circ c$ whenever
$a\bot b$.
Let $(E,0,1,\oplus,\circ)$ be a sequential effect algebra. Then the operation
$\circ$ is said to be a sequential product on $(E,0,1,\oplus,\circ)$. If
$a,b\in(E,0,1,\oplus,\circ)$ and $a\circ b=b\circ a$, then $a$ and $b$ is said
to be sequentially independent and denoted by $a|b$ (see [2-3]). The
sequential effect algebra is an important and interesting mathematical model
for studying the quantum measurement theory [2-4, 7-8].
Let $(E,0,1,\oplus,\circ)$ be a sequential effect algebra. If $a\in E$, then
it follows from ([3, Lemma 3.2]) that $a$ is a sharp element of
$(E,0,1,\oplus,\circ)$ iff $a\circ a=a$.
A $\sigma$-effect algebra is an effect algebra $(E,0,1,\oplus)$ such that
$a_{1}\geq a_{2}\geq a_{3}\cdots$ implies that $\bigwedge a_{i}$ exists in
$E$. A $\sigma$-sequential effect algebra $(E,0,1,\oplus,\circ)$ is a
sequential effect algebra and is a $\sigma$-effect algebra satisfying [3]:
(1). If $a_{1}\geq a_{2}\geq a_{3}\cdots$, then $b\circ(\bigwedge
a_{i})=\bigwedge(b\circ a_{i})$ for each $b\in E$;
(2). If $a_{1}\geq a_{2}\geq a_{3}\cdots$ and $b|a_{i},i=1,2,\cdots$, then
$b|(\bigwedge a_{i})$.
It is known that ${\cal E(H)}$ is a $\sigma$-sequential effect algebra (see
[3]).
In 2002, Professors Gudder and Greechie proved the following important
conclusion ([3, Theorem 6.3]): Every $\sigma$-sequential effect algebra is
sharply dominating.
In 2005, by the motivation of the above result, Professor Gudder asked ([4,
Problem 3]): Is each sequential effect algebra sharply dominating?
Now, we construct a sequential effect algebra which is not sharply dominating,
thus, we answer the above problem negatively.
Let $E_{0}=\\{0,1,a_{n},b_{n},c_{\wedge,n},d_{\wedge,n}|\
n\in{\mathbf{N}}^{+},\wedge\in\Lambda\\}$, where $\mathbf{N}^{+}$ be the
positive integer set and $\Lambda$ be the set of all finite nonempty subsets
of $\mathbf{N}^{+}$. First, we define a partial binary operation $\oplus$ on
$E_{0}$ as following (when we write $x\oplus y=z$, we always mean that
$x\oplus y=z=y\oplus x$):
For each $x\in E_{0}$, $0\oplus x=x$,
$a_{n}\oplus a_{m}=a_{n+m}$,
For $n<m$, $a_{n}\oplus b_{m}=b_{m-n}$, $a_{n}\oplus b_{n}=1$,
$a_{n}\oplus c_{\wedge,m}=c_{\wedge,n+m}$,
For $n<m$, $a_{n}\oplus d_{\wedge,m}=d_{\wedge,m-n}$,
For $\wedge\cap I=\emptyset$, $c_{\wedge,n}\oplus c_{I,m}=c_{\wedge\cup
I,m+n-1}$,
For $\wedge\subset I\ and\ n\leq m$, $c_{\wedge,n}\oplus
d_{I,m}=d_{I\backslash\wedge,m-n+1}(when\ \wedge\neq I)\ or\ b_{m-n}(when\
\wedge=I\ and\ n<m)\ or\ 1(when\ \wedge=I\ and\ n=m)$.
No other $\oplus$ operation is defined.
Next, we define a binary operation $\circ$ on $E_{0}$ as following (when we
write $x\circ y=z$, we always mean that $x\circ y=z=y\circ x$):
For each $x\in E_{0}$, $0\circ x=0$, $1\circ x=x$,
$a_{n}\circ a_{m}=0$, $a_{n}\circ b_{m}=a_{n}$, $b_{n}\circ b_{m}=b_{m+n}$,
$a_{n}\circ c_{\wedge,m}=0$, $c_{\wedge,n}\circ b_{m}=c_{\wedge,n}$,
$a_{n}\circ d_{\wedge,m}=a_{n}$, $b_{n}\circ d_{\wedge,m}=d_{\wedge,m+n}$,
$d_{\wedge,n}\circ d_{I,m}=d_{\wedge\cup I,n+m-1}$,
$c_{\wedge,n}\circ c_{I,m}=c_{\wedge\cap I,1}(when\ \wedge\cap
I\neq\emptyset)\ or\ 0(when\ \wedge\cap I=\emptyset)$,
$c_{\wedge,n}\circ d_{I,m}=c_{\wedge\backslash I,n}(when\ \wedge\backslash
I\neq\emptyset)\ or\ a_{n-1}(when\ \wedge\backslash I=\emptyset\ and\ n>1)\
or\ 0(when\ \wedge\backslash I=\emptyset\ and\ n=1)$.
Proposition 1. $(E_{0},0,1,\oplus,\circ)$ is a sequential effect algebra.
Proof. First we verify that $(E_{0},0,1,\oplus)$ is an effect algebra.
(EA1) and (EA4) are trivial.
We verify (EA2), we omit the trivial cases about 0,1:
$a_{n}\oplus(a_{m}\oplus a_{k})=(a_{n}\oplus a_{m})\oplus a_{k}=a_{k+m+n}$.
$a_{n}\oplus(a_{m}\oplus c_{\wedge,k})=(a_{n}\oplus a_{m})\oplus
c_{\wedge,k}=c_{\wedge,k+m+n}$.
Each $a_{n}\oplus(a_{m}\oplus b_{k})$ or $(a_{n}\oplus a_{m})\oplus b_{k}$ is
defined iff $n+m\leq k$, $a_{n}\oplus(a_{m}\oplus b_{k})=(a_{n}\oplus
a_{m})\oplus b_{k}=b_{k-m-n}(when\ m+n<k)\ or\ 1(when\ m+n=k)$.
Each $a_{n}\oplus(a_{m}\oplus d_{\wedge,k})$ or $(a_{n}\oplus a_{m})\oplus
d_{\wedge,k}$ is defined iff $n+m<k$, $a_{n}\oplus(a_{m}\oplus
d_{\wedge,k})=(a_{n}\oplus a_{m})\oplus d_{\wedge,k}=d_{\wedge,k-m-n}$.
Each $a_{n}\oplus(c_{\wedge,m}\oplus d_{I,k})$ or $(a_{n}\oplus
c_{\wedge,m})\oplus d_{I,k}$ or $(a_{n}\oplus d_{I,k})\oplus c_{\wedge,m}$ is
defined iff $\wedge\subset I\ and\ n+m\leq k$, $a_{n}\oplus(c_{\wedge,m}\oplus
d_{I,k})=(a_{n}\oplus c_{\wedge,m})\oplus d_{I,k}=(a_{n}\oplus d_{I,k})\oplus
c_{\wedge,m}=d_{I\backslash\wedge,k-m-n+1}(when\ \wedge\neq I)\ or\
b_{k-m-n}(when\ \wedge=I\ and\ m+n<k)\ or\ 1(when\ \wedge=I\ and\ m+n=k)$.
Each $a_{n}\oplus(c_{\wedge,m}\oplus c_{I,k})$ or $(a_{n}\oplus
c_{\wedge,m})\oplus c_{I,k}$ is defined iff $\wedge\cap I=\emptyset$,
$a_{n}\oplus(c_{\wedge,m}\oplus c_{I,k})=(a_{n}\oplus c_{\wedge,m})\oplus
c_{I,k}=c_{\wedge\cup I,n+m+k-1}$.
Each $c_{\wedge,n}\oplus(c_{I,m}\oplus c_{Y,k})$ or $(c_{\wedge,n}\oplus
c_{Y,k})\oplus c_{I,m}$ is defined iff $\wedge\cap I\ and\ \wedge\cap Y\ and\
Y\cap I\ are\ all\ \emptyset$, $c_{\wedge,n}\oplus(c_{I,m}\oplus
c_{Y,k})=(c_{\wedge,n}\oplus c_{Y,k})\oplus c_{I,m}=c_{\wedge\cup I\cup
Y,n+m+k-2}$.
Each $c_{\wedge,n}\oplus(c_{I,m}\oplus d_{Y,k})$ or $(c_{\wedge,n}\oplus
c_{I,m})\oplus d_{Y,k}$ is defined iff $\wedge\cap I=\emptyset\ and\
\wedge\cup I\subset Y\ and\ n+m\leq k+1$, $c_{\wedge,n}\oplus(c_{I,m}\oplus
d_{Y,k})=(c_{\wedge,n}\oplus c_{I,m})\oplus d_{Y,k}=d_{Y\backslash(\wedge\cup
I),k-m-n+2}(when\ \wedge\cup I\neq Y)\ or\ b_{k-n-m+1}(when\ \wedge\cup I=Y\
and\ m+n<k+1)\ or\ 1(when\ \wedge\cup I=Y\ and\ m+n=k+1)$.
Thus, (EA2) is proved. We verify (EA3):
$a_{n}\oplus b_{n}=1$, $c_{\wedge,n}\oplus d_{\wedge,n}=1$.
So $(E,0,1,\oplus)$ is an effect algebra.
We now verify that $(E,0,1,\oplus,\circ)$ is a sequential effect algebra.
(SEA2) and (SEA3) and (SEA5) are trivial.
We verify (SEA1), we omit the trivial cases about 0,1:
$a_{n}\circ(a_{m}\oplus a_{k})=a_{n}\circ a_{m}\oplus a_{n}\circ a_{k}=0$,
$b_{n}\circ(a_{m}\oplus a_{k})=b_{n}\circ a_{m}\oplus b_{n}\circ
a_{k}=a_{m+k}$,
$c_{\wedge,n}\circ(a_{m}\oplus a_{k})=c_{\wedge,n}\circ a_{m}\oplus
c_{\wedge,n}\circ a_{k}=0$,
$d_{\wedge,n}\circ(a_{m}\oplus a_{k})=d_{\wedge,n}\circ a_{m}\oplus
d_{\wedge,n}\circ a_{k}=a_{m+k}$.
$a_{n}\circ(a_{m}\oplus c_{\wedge,k})=a_{n}\circ a_{m}\oplus a_{n}\circ
c_{\wedge,k}=0$,
$b_{n}\circ(a_{m}\oplus c_{\wedge,k})=b_{n}\circ a_{m}\oplus b_{n}\circ
c_{\wedge,k}=c_{\wedge,m+k}$,
$c_{I,n}\circ(a_{m}\oplus c_{\wedge,k})=c_{I,n}\circ a_{m}\oplus c_{I,n}\circ
c_{\wedge,k}=c_{\wedge\cap I,1}(when\ \wedge\cap I\neq\emptyset)\ or\ 0(when\
\wedge\cap I=\emptyset)$,
$d_{I,n}\circ(a_{m}\oplus c_{\wedge,k})=d_{I,n}\circ a_{m}\oplus d_{I,n}\circ
c_{\wedge,k}=c_{\wedge\backslash I,m+k}(when\ \wedge\backslash
I\neq\emptyset)\ or\ a_{m+k-1}(when\ \wedge\backslash I=\emptyset)$.
For $m<k$,
$a_{n}\circ(a_{m}\oplus d_{\wedge,k})=a_{n}\circ a_{m}\oplus a_{n}\circ
d_{\wedge,k}=a_{n}$,
$b_{n}\circ(a_{m}\oplus d_{\wedge,k})=b_{n}\circ a_{m}\oplus b_{n}\circ
d_{\wedge,k}=d_{\wedge,n+k-m}$,
$c_{I,n}\circ(a_{m}\oplus d_{\wedge,k})=c_{I,n}\circ a_{m}\oplus c_{I,n}\circ
d_{\wedge,k}=c_{I\backslash\wedge,n}(when\ I\backslash\wedge\neq\emptyset)\
or\ a_{n-1}(when\ I\backslash\wedge=\emptyset\ and\ n>1)\ or\ 0(when\
I\backslash\wedge=\emptyset\ and\ n=1)$,
$d_{I,n}\circ(a_{m}\oplus d_{\wedge,k})=d_{I,n}\circ a_{m}\oplus d_{I,n}\circ
d_{\wedge,k}=d_{\wedge\cup I,n+k-m-1}$.
For $m\leq k$,
$a_{n}\circ(a_{m}\oplus b_{k})=a_{n}\circ a_{m}\oplus a_{n}\circ b_{k}=a_{n}$,
$b_{n}\circ(a_{m}\oplus b_{k})=b_{n}\circ a_{m}\oplus b_{n}\circ
b_{k}=b_{n+k-m}$,
$c_{\wedge,n}\circ(a_{m}\oplus b_{k})=c_{\wedge,n}\circ a_{m}\oplus
c_{\wedge,n}\circ b_{k}=c_{\wedge,n}$,
$d_{\wedge,n}\circ(a_{m}\oplus b_{k})=d_{\wedge,n}\circ a_{m}\oplus
d_{\wedge,n}\circ b_{k}=d_{\wedge,n+k-m}$.
For $\wedge\cap I=\emptyset$,
$a_{n}\circ(c_{\wedge,m}\oplus c_{I,k})=a_{n}\circ c_{\wedge,m}\oplus
a_{n}\circ c_{I,k}=0$,
$b_{n}\circ(c_{\wedge,m}\oplus c_{I,k})=b_{n}\circ c_{\wedge,m}\oplus
b_{n}\circ c_{I,k}=c_{\wedge\cup I,m+k-1}$,
$c_{Y,n}\circ(c_{\wedge,m}\oplus c_{I,k})=c_{Y,n}\circ c_{\wedge,m}\oplus
c_{Y,n}\circ c_{I,k}=c_{Y\cap(\wedge\cup I),1}(when\ Y\cap(\wedge\cup
I)\neq\emptyset)\ or\ 0(when\ Y\cap(\wedge\cup I)=\emptyset)$,
$d_{Y,n}\circ(c_{\wedge,m}\oplus c_{I,k})=d_{Y,n}\circ c_{\wedge,m}\oplus
d_{Y,n}\circ c_{I,k}=c_{(\wedge\cup I)\backslash Y,m+k-1}(when\ (\wedge\cup
I)\backslash Y\neq\emptyset)\ or\ a_{m+k-2}(when\ (\wedge\cup I)\backslash
Y=\emptyset\ and\ m+k>2)\ or\ 0(when\ (\wedge\cup I)\backslash Y=\emptyset\
and\ m+k=2)$.
For $\wedge\subset I\ and\ m\leq k$,
$a_{n}\circ(c_{\wedge,m}\oplus d_{I,k})=a_{n}\circ c_{\wedge,m}\oplus
a_{n}\circ d_{I,k}=a_{n}$,
$b_{n}\circ(c_{\wedge,m}\oplus d_{I,k})=b_{n}\circ c_{\wedge,m}\oplus
b_{n}\circ d_{I,k}=d_{I\backslash\wedge,n+k-m+1}(when\ \wedge\neq I)\ or\
b_{n+k-m}(when\ \wedge=I)$,
$c_{Y,n}\circ(c_{\wedge,m}\oplus d_{I,k})=c_{Y,n}\circ c_{\wedge,m}\oplus
c_{Y,n}\circ d_{I,k}=c_{Y\backslash(I\backslash\wedge),n}(when\
Y\backslash(I\backslash\wedge)\neq\emptyset)\ or\ a_{n-1}(when\
Y\backslash(I\backslash\wedge)=\emptyset\ and\ n>1)\ or\ 0(when\
Y\backslash(I\backslash\wedge)=\emptyset\ and\ n=1)$,
$d_{Y,n}\circ(c_{\wedge,m}\oplus d_{I,k})=d_{Y,n}\circ c_{\wedge,m}\oplus
d_{Y,n}\circ d_{I,k}=d_{Y\cup(I\backslash\wedge),n+k-m}$.
Thus, (SEA1) is proved. We verify (SEA4), we omit the trivial cases about 0,1:
$a_{n}\circ(a_{m}\circ a_{k})=(a_{n}\circ a_{m})\circ a_{k}=0$,
$a_{n}\circ(a_{m}\circ b_{k})=b_{k}\circ(a_{n}\circ
a_{m})=a_{m}\circ(a_{n}\circ b_{k})=0$,
$a_{n}\circ(a_{m}\circ c_{\wedge,k})=c_{\wedge,k}\circ(a_{n}\circ
a_{m})=a_{m}\circ(a_{n}\circ c_{\wedge,k})=0$,
$a_{n}\circ(a_{m}\circ d_{\wedge,k})=d_{\wedge,k}\circ(a_{n}\circ
a_{m})=a_{m}\circ(a_{n}\circ d_{\wedge,k})=0$,
$a_{n}\circ(b_{m}\circ b_{k})=b_{k}\circ(a_{n}\circ
b_{m})=b_{m}\circ(a_{n}\circ b_{k})=a_{n}$,
$a_{n}\circ(b_{m}\circ c_{\wedge,k})=c_{\wedge,k}\circ(a_{n}\circ
b_{m})=b_{m}\circ(a_{n}\circ c_{\wedge,k})=0$,
$a_{n}\circ(b_{m}\circ d_{\wedge,k})=d_{\wedge,k}\circ(a_{n}\circ
b_{m})=b_{m}\circ(a_{n}\circ d_{\wedge,k})=a_{n}$,
$a_{n}\circ(c_{I,m}\circ c_{\wedge,k})=c_{\wedge,k}\circ(a_{n}\circ
c_{I,m})=c_{I,m}\circ(a_{n}\circ c_{\wedge,k})=0$,
$a_{n}\circ(c_{I,m}\circ d_{\wedge,k})=d_{\wedge,k}\circ(a_{n}\circ
c_{I,m})=c_{I,m}\circ(a_{n}\circ d_{\wedge,k})=0$,
$a_{n}\circ(d_{I,m}\circ d_{\wedge,k})=d_{\wedge,k}\circ(a_{n}\circ
d_{I,m})=d_{I,m}\circ(a_{n}\circ d_{\wedge,k})=a_{n}$,
$b_{n}\circ(b_{m}\circ b_{k})=b_{k}\circ(b_{n}\circ b_{m})=b_{m+n+k}$,
$b_{n}\circ(b_{m}\circ c_{\wedge,k})=c_{\wedge,k}\circ(b_{n}\circ
b_{m})=b_{m}\circ(b_{n}\circ c_{\wedge,k})=c_{\wedge,k}$,
$b_{n}\circ(b_{m}\circ d_{\wedge,k})=d_{\wedge,k}\circ(b_{n}\circ
b_{m})=b_{m}\circ(b_{n}\circ d_{\wedge,k})=d_{\wedge,n+m+k}$,
$b_{n}\circ(c_{I,m}\circ c_{\wedge,k})=c_{\wedge,k}\circ(b_{n}\circ
c_{I,m})=c_{I,m}\circ(b_{n}\circ c_{\wedge,k})=c_{I\cap\wedge,1}(when\
I\cap\wedge\neq\emptyset)\ or\ 0(when\ I\cap\wedge=\emptyset)$,
$b_{n}\circ(c_{I,m}\circ d_{\wedge,k})=d_{\wedge,k}\circ(b_{n}\circ
c_{I,m})=c_{I,m}\circ(b_{n}\circ d_{\wedge,k})=c_{I\backslash\wedge,m}(when\
I\backslash\wedge\neq\emptyset)\ or\ a_{m-1}(when\
I\backslash\wedge=\emptyset\ and\ m>1)\ or\ 0(when\
I\backslash\wedge=\emptyset\ and\ m=1)$,
$b_{n}\circ(d_{I,m}\circ d_{\wedge,k})=d_{\wedge,k}\circ(b_{n}\circ
d_{I,m})=d_{I,m}\circ(b_{n}\circ d_{\wedge,k})=d_{I\cup\wedge,n+m+k-1}$,
$c_{Y,n}\circ(c_{I,m}\circ c_{\wedge,k})=c_{\wedge,k}\circ(c_{Y,n}\circ
c_{I,m})=c_{Y\cap I\cap\wedge,1}(when\ Y\cap I\cap\wedge\neq\emptyset)\ or\
0(when\ Y\cap I\cap\wedge=\emptyset)$,
$c_{Y,n}\circ(c_{I,m}\circ d_{\wedge,k})=d_{\wedge,k}\circ(c_{Y,n}\circ
c_{I,m})=c_{I,m}\circ(c_{Y,n}\circ d_{\wedge,k})=c_{(Y\cap
I)\backslash\wedge,1}(when\ (Y\cap I)\backslash\wedge\neq\emptyset)\ or\
0(when\ (Y\cap I)\backslash\wedge=\emptyset)$,
$c_{Y,n}\circ(d_{I,m}\circ d_{\wedge,k})=d_{\wedge,k}\circ(c_{Y,n}\circ
d_{I,m})=d_{I,m}\circ(c_{Y,n}\circ d_{\wedge,k})=c_{Y\backslash(\wedge\cup
I),n}(when\ Y\backslash(\wedge\cup I)\neq\emptyset)\ or\ a_{n-1}(when\
Y\backslash(\wedge\cup I)=\emptyset\ and\ n>1)\ or\ 0(when\
Y\backslash(\wedge\cup I)=\emptyset\ and\ n=1)$,
$d_{Y,n}\circ(d_{I,m}\circ d_{\wedge,k})=d_{\wedge,k}\circ(d_{Y,n}\circ
d_{I,m})=d_{\wedge\cup I\cup Y,n+m+k-2}$.
(SEA4) is proved and so $(E_{0},0,1,\oplus,\circ)$ is a sequential effect
algebra.
Our main result is:
Theorem 1. Not each sequential effect algebra is sharply dominating.
Proof. In fact, in the sequential effect algebra $(E_{0},0,1,\oplus,\circ)$,
its all sharp elements is the set $E_{s}=\\{0,1,c_{\wedge,1},d_{\wedge,1}|\
\wedge\in\Lambda$, where $\Lambda$ is the set of all finite nonempty subsets
of $\mathbf{N}^{+}\\}$. Note that when $\wedge_{1}\subset\wedge_{2}$ and
$\wedge_{1}\neq\wedge_{2}$, $c_{\wedge_{1},1}\oplus
c_{\wedge_{2}\backslash\wedge_{1},1}=c_{\wedge_{2},1}$,
$d_{\wedge_{2},1}\oplus
c_{\wedge_{2}\backslash\wedge_{1},1}=d_{\wedge_{1},1}$, so
$c_{\wedge_{1},1}<c_{\wedge_{2},1}$, $d_{\wedge_{2},1}<d_{\wedge_{1},1}$. For
each finite subset $\wedge$ of ${\mathbf{N}}^{+}$, $a_{1}\oplus
d_{\wedge,2}=d_{\wedge,1}$, so $a_{1}<d_{\wedge,1}$, and there is no
comparison relation between $a_{1}$ and $c_{\wedge,1}$. So the set of elements
in $E_{s}$ larger than $a_{1}$ is $A=\\{1,d_{\wedge,1}|\ \wedge\in\Lambda\\}$,
nevertheless, there is no smallest element in $A$. Thus,
$(E_{0},0,1,\oplus,\circ)$ is not sharply dominating and the theorem is
proved.
Moreover, we show that the sequential effect algebra
$(E_{0},0,1,\oplus,\circ)$ in Proposition 1 is not even a $\sigma$-effect
algebra. At first, we need the following theorem:
Theorem 2. Let $(E,0,1,\oplus,\circ)$ be a sequential effect algebra, $I$ be
an index set, $\\{a_{\alpha}\\}_{\alpha\in I}\subset E_{s}$.
(1) If $\bigwedge\limits_{\alpha\in I}a_{\alpha}$ exists, then
$\bigwedge\limits_{\alpha\in I}a_{\alpha}\in E_{s}$;
(2) if $\bigvee\limits_{\alpha\in I}a_{\alpha}$ exists, then
$\bigvee\limits_{\alpha\in I}a_{\alpha}\in E_{s}$.
Proof. Just the same as the proof of [3] corollary 4.3.
Proposition 2. $(E_{0},0,1,\oplus)$ is not a $\sigma$-effect algebra.
Proof. Let $\\{\wedge_{i}\\}_{i\in{\mathbf{N}}^{+}}$ be a strictly increasing
sequence of finite nonempty subsets of $\mathbf{N}^{+}$. We note from the
proof of Theorem 1 that
$\\{d_{\wedge_{i},1}|\ i\in{\mathbf{N}}^{+}\\}\subset E_{s}$ and satisfying
$d_{\wedge_{1},1}>d_{\wedge_{2},1}>\cdots>d_{\wedge_{n},1}>\cdots\ .$
If $(E_{0},0,1,\oplus)$ is a $\sigma$-effect algebra, then
$\bigwedge\limits_{i\in{\mathbf{N}}^{+}}d_{\wedge_{i},1}$ will exist, and it
follows from Theorem 2 that
$\bigwedge\limits_{i\in{\mathbf{N}}^{+}}d_{\wedge_{i},1}\in E_{s}$.
By the proof of Theorem 1 again, we have $a_{1}<d_{\wedge,1}$, so
$a_{1}\leq\bigwedge\limits_{i\in{\mathbf{N}}^{+}}d_{\wedge_{i},1}$. Note that
there is no comparison relation between $a_{1}$ and $c_{\wedge,1}$ (proof of
Theorem 1), so $\bigwedge\limits_{i\in{\mathbf{N}}^{+}}d_{\wedge_{i},1}$ is
not $c_{\wedge,1}$. Also, it is obvious that
$\bigwedge\limits_{i\in{\mathbf{N}}^{+}}d_{\wedge_{i},1}$ is not 0 or 1.
It follows from above and $E_{s}=\\{0,1,c_{\wedge,1},d_{\wedge,1}|\
\wedge\in\Lambda,$ where $\Lambda$ is the set of all finite nonempty subsets
of $\mathbf{N}^{+}\\}$ that there exists some $\wedge_{0}$ such that
$d_{\wedge_{0},1}=\bigwedge\limits_{i\in{\mathbf{N}}^{+}}d_{\wedge_{i},1}$.
But then we will have $d_{\wedge_{0},1}\leq d_{\wedge_{i},1}$ and
$\wedge_{0}\supset\wedge_{i}$ for all $i\in{\mathbf{N}}^{+}$, which is
impossible since $\wedge_{0}$ is a finite subset of $\mathbf{N}^{+}$.
Acknowledgement
The authors wish to express their thanks to the referee for his valuable
comments and suggestions.
References
[1]. Foulis, D J, Bennett, M K. Effect algebras and unsharp quantum logics.
Found Phys 24 (1994), 1331-1352.
[2]. Gudder, S, Nagy, G. Sequential quantum measurements. J. Math. Phys.
42(2001), 5212-5222.
[3]. Gudder, S, Greechie, R. Sequential products on effect algebras. Rep.
Math. Phys. 49(2002), 87-111.
[4]. Gudder, S. Open problems for sequential effect algebras. Inter. J.
Theory. Phys. 44 (2005), 2219-2230.
[5]. Gudder, S. Sharply dominating effect algebras. Tatra Mt. Math. Publ.,
15(1998), 23-30.
[6] Riecanova, Z, Wu Junde. States on sharply dominating effect algebras.
Science in China A: Mathematics, 51(2008), 907-914.
[7]. Gheondea, A, Gudder, S. Sequential product of quantum effects. Proc.
Amer. Math. Soc. 132 (2004), 503-512.
[8]. Gudder, S, Latr moli re, F. Characterization of the sequential product on
quantum effects. J. Math. Phys. 49 (2008), 052106-052112.
|
arxiv-papers
| 2008-12-12T23:09:02 |
2024-09-04T02:48:59.383983
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shen Jun, Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0812.2502"
}
|
0812.2581
|
# Improved spacecraft radio science using an on-board atomic clock:
application to gravitational wave searches
Massimo Tinto Massimo.Tinto@jpl.nasa.gov Jet Propulsion Laboratory,
California Institute of Technology, Pasadena, CA 91109 George J. Dick
George.J.Dick@jpl.nasa.gov Jet Propulsion Laboratory, California Institute of
Technology, Pasadena, CA 91109 John D. Prestage John.D.Prestage@jpl.nasa.gov
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA
91109 J.W. Armstrong John.W.Armstrong@jpl.nasa.gov Jet Propulsion
Laboratory, California Institute of Technology, Pasadena, CA 91109
###### Abstract
Recent advances in space-qualified atomic clocks (low-mass, low power-
consumption, frequency stability comparable to that of ground-based clocks)
can enable interplanetary spacecraft radio science experiments at
unprecedented Doppler sensitivities. The addition of an on-board digital
receiver would allow the up- and down-link Doppler frequencies to be measured
separately. Such separate, high-quality measurements allow optimal data
combinations that suppress the currently-leading noise sources: phase
scintillation noise from the Earth’s atmosphere and Doppler noise caused by
mechanical vibrations of the ground antenna. Here we provide a general
expression for the optimal combination of ground and on-board Doppler data and
compute the sensitivity such a system would have to low-frequency
gravitational waves (GWs). Assuming a plasma scintillation noise calibration
comparable to that already demonstrated with the multi-link CASSINI radio
system, the space-clock/digital-receiver instrumentation enhancements would
give GW strain sensitivity of $2.0\times 10^{-17}$ for randomly polarized,
monochromatic GW signals over a two-decade ($\sim 0.0001-0.01$ Hz) region of
the low-frequency band. This is about an order of magnitude better than
currently achieved with traditional two-way coherent Doppler experiments. The
utility of optimally combining simultaneous up- and down-link observations is
not limited to GW searches. The Doppler tracking technique discussed here
could be performed at minimal incremental cost to also improve other radio
science experiments (i.e. tests of relativistic gravity, planetary and
satellite gravity field measurements, atmospheric and ring occultations) on
future interplanetary missions.
###### pacs:
04.80.Nn, 95.55.Ym, 07.60.Ly
## I Introduction
Measurements of the relative velocity between the Earth and an interplanetary
spacecraft, by means of coherent microwave tracking, have allowed studies of
solar system bodies kliore_etal2004 , tests of relativistic gravity Vessot1993
; bertotti_etal2003 , searches for low-frequency gravitational radiation
EW1975 ; Tinto1996 ; Armstrong2006 , and other science objectives
kliore_etal2004 . In the frequency band ($10^{-6}-10^{-2}$) Hz, typical deep
space tracks are limited by phase scintillation caused by random refractivity
variations in the solar wind, and the ionosphere asmar_etal2005 . The most
sensitive deep-space Doppler observations to date, however, calibrate and
largely remove these noises bertotti_etal1993 ; Tinto2002 ; bertotti_etal2003
; armstrong_etal2003 and are then limited by antenna mechanical noise
(unmodeled motion of the phase center of the ground antenna) and residual
post-calibration tropospheric scintillation (i.e. Doppler fluctuations caused
by refractive index fluctuations in the Earth’s atmosphere) Tinto1996 ;
asmar_etal2005 . The most sensitive observations hit the limit identified by
these noise sources with an Allan standard deviation of about $3\times
10^{-15}$ for integration times of a few thousand seconds. Improved
sensitivity would benefit the science disciplines listed above, but antenna
mechanical noise, in particular, has seemed irreducible at reasonable cost
since it would require a large, moving, steel structure much more rigid than
that of the current ground tracking stations.
Several ideas have been proposed to reduce the antenna mechanical noise or the
tropospheric noise, or both Armstrong_Estabrook_etal . Those ideas do not
involve modifications to the spacecraft, but rather use simultaneous tracking
on the ground with appropriate linear combinations of those data to synthesize
an observable with the ground/tropospheric noises of the better of the two
receiving stations. If, however, additional microwave instrumentation on the
spacecraft is considered, such as a space-borne, high-stability frequency
standard Prestage_Weaver2007 ; Dick1990 and a space-qualified digital
receiver Tyler_etal_2008 ; RASSI_2008 , then an alternative method for noise
suppression is possible. This involves properly combining the two one-way
(spacecraft to Earth and simultaneous Earth to spacecraft) Doppler data taken
onboard and on the ground in such a way to maximize the signal-to-noise ratio
of the observed physical observable.111It should be emphasized that the
tropospheric, antenna mechanical and ionospheric noise suppression can also be
accomplished by combining the two-way coherent Doppler data with the one-way
Doppler measurement performed at the ground Tinto1996 ; Piran_etal1986 . We
have however analyzed the configuration with the two one-way measurements for
reasons of symmetry and simplicity. Multiple Doppler observations with
different noises, or different transfer functions to the same noises, are
clearly useful in identifying noise sources and minimizing (and in some cases
canceling) their effects on the final observable. In particular, the use of
multiple radio links (some of which driven by a high-quality space-borne
frequency standard) was pioneered by R. Vessot in the Gravity Probe A sub-
orbital experiment Vessot1970 . Because of mass and power constraints,
however, no very high quality frequency standards have yet been flown on deep
space probes.
Recent advances in clock technology indicate that a new era of space-
qualified, highly stable frequency standards has started Prestage_Weaver2007 ,
which will result into significantly improved Doppler radio science
experiments. A summary of this paper is given below.
In Section II we present a brief overview of the theory underlying Doppler
tracking experiments relying only on the two one-way Doppler data measured
onboard and on the ground. Although this experimental configuration has been
discussed in previous publications Vessot1970 ; Piran_etal1986 , it has been
shown only relatively recently Tinto1996 ; Tinto2002 how to fully take
advantage of it for improving the precision of Doppler tracking radio science
experiments. A brief description of an onboard atomic clock and a digital
receiver is given in the Appendix, where an account of all the one-sided power
spectral densities of the noises affecting the Doppler data is also presented.
The main advantage of spacecraft Doppler tracking experiments relying on the
two one-way Doppler data, over those based on two-way coherent measurements,
is in their ability of exactly canceling the frequency fluctuations due to the
Earth atmosphere and ionosphere, and the mechanical vibrations of the ground
antenna, presently the main noise sources of Doppler tracking experiments
Tinto1996 . This is possible because there exists a unique linear combination
of the properly time-shifted one-way measurements that does exactly that
Tinto1996 . Depending on the specific radio science experiment performed with
this technique, it is actually possible to combine optimally the two one-way
Doppler measurements to maximize the signal-to-noise ratio (SNR) of the
experiment. After deriving the expression of the optimal SNR in Section III,
we apply it to searches for gravitational waves. Under the assumption of
calibrating the frequency fluctuations induced by the interplanetary plasma,
we find that a Doppler broad-band sensitivity of $2.0\times 10^{-17}$ to
randomly polarized monochromatic signals uniformly distributed over the sky
can be achieved. This is about one order of magnitude better than that
obtainable with two-way coherent Doppler tracking experiments. Narrow-band
searches at frequencies where the transfer function of the onboard clock
reaches sharp nulls (i.e. the “xylophone” frequencies) Tinto1996 further
enhance the strain sensitivity of these experiments to about $7.0\times
10^{-18}$.
In Section IV we finally present our comments and conclusions, and emphasize
that the Doppler tracking technique discussed in this article can be performed
at minimal additional cost by forthcoming interplanetary missions.
## II The One-Way Doppler Tracking Observables
In Doppler tracking experiments aimed at detecting low-frequency (milliHertz)
gravitational radiation, a distant interplanetary spacecraft is monitored from
Earth through a radio link, with the Earth and the spacecraft acting as free-
falling test particles 222Spacecraft Doppler GW searches piggy-back on
interplanetary probes used primarily for other (e.g., solar system) science
goals. Doppler tracking is the current generation GW detector in the low-
frequency band. A much more sensitive, dedicated GW mission, LISA, is
currently in the design and technology development stage and could launch
sometimes in the next decade.. A radio signal of nominal frequency $\nu_{0}$
is transmitted to the spacecraft, and coherently transponded back to Earth
where the received signal is compared to a signal referenced to a highly
stable clock (typically a hydrogen-maser). Relative frequency changes
$\Delta\nu/\nu_{0}$, as functions of time, are measured. When a gravitational
wave crossing the solar system propagates through the radio link, it causes
small perturbations in $\Delta\nu/\nu_{0}$, which are replicated three times
in the Doppler data with maximum spacing given by the two-way light
propagation time between the Earth and the spacecraft EW1975 .
An alternative way of performing Doppler tracking searches for gravitational
radiation was suggested in Vessot_Levine1978 ; Piran_etal1986 ; Tinto1996 . By
adding a highly-stable clock and a digital receiver to the spacecraft radio
instrumentation (Figure 1), two one-way Doppler time series can be recorded
simultaneously at the ground station and on board the spacecraft.
Figure 1: Block diagram of the radio hardware at the ground antenna of the
NASA Deep Space Network (DSN) and on board the spacecraft (S/C), which allows
the acquisition and recording of the two Doppler data $E(t)$, $S(t)$. A
description of each individual block in this diagram is provided in the
Appendix.
If we introduce a set of Cartesian orthogonal coordinates ($X,Y,Z$) in which
the wave is propagating along the $Z$-axis and ($X,Y$) are two orthogonal axes
in the plane of the wave (see Figure 2), then the two one-way relative
frequency fluctuations at time $t$ can be written in the following form after
first-order Doppler and other systematic Doppler effects are modeled out from
the data 333The one-way Doppler data measured onboard can be digitally
recorded, time tagged, and telemetered back to Earth in real time or at a
later time during the mission.Tinto1996
Figure 2: A radio signal of nominal frequency $\nu_{0}$ is transmitted to a
spacecraft and simultaneously another radio signal from the spacecraft and
referenced to the onboard clock is transmitted to the ground. The
gravitational wave train propagates along the $Z$ direction, and the cosine of
the angle between its direction of propagation and the radio beam is denoted
by $\mu$. See text for a complete description.
$\displaystyle\left(\frac{\Delta\nu(t)}{\nu_{0}}\right)_{E}\equiv E(t)$
$\displaystyle=$ $\displaystyle\frac{1-\mu}{2}\ \left[h(t-(1+\mu)L)\ -\
h(t)\right]\ +\ C_{S}(t-L)\ -\ C_{E}(t)$ (1) $\displaystyle+$ $\displaystyle
T(t)\ +\ B(t-L)\ +\ A_{S}(t-L)\ +\ EL_{E}(t)\ +\ P_{E}(t)\ ,$
$\displaystyle\left(\frac{\Delta\nu(t)}{\nu_{0}}\right)_{S}\equiv S(t)$
$\displaystyle=$ $\displaystyle\frac{1+\mu}{2}\ \left[h(t-L)\ -\ h(t-\mu
L)\right]\ +\ C_{E}(t-L)\ -\ C_{S}(t)$ (2) $\displaystyle+$ $\displaystyle
T(t-L)\ +\ B(t)\ +\ A_{E}(t-L)\ +\ EL_{S}(t)\ +\ P_{S}(t)\ ,$
where $h(t)$ is equal to
$h(t)=h_{+}(t)\cos(2\phi)+h_{\times}(t)\sin(2\phi)\ .$ (3)
Here $h_{+}(t)$, $h_{\times}(t)$ are the wave’s two amplitudes with respect to
the ($X,Y$) axis, ($\theta,\phi$) are the polar angles describing the location
of the spacecraft with respect to the ($X,Y,Z$) coordinates, $\mu$ is equal to
$\cos\theta$, and $L$ is the distance to the spacecraft (units in which the
speed of light $c=1$)
In Eqs. (1, 2) we have assumed the Earth and the on board clocks to be
perfectly synchronized. Although this condition is impossible to achieve in
practice, it has been previously shown by one of us Tinto2002 that the
accuracy required for successfully implementing a noise cancellation scheme
similar to the one discussed in this paper requires a clock synchronization
accuracy of about $0.5\ s$, which is easy to achieve.
In Eqs. (1,2) we have denoted by $C_{E}(t)$, $C_{S}(t)$ the random processes
associated with the frequency fluctuations of the clock on Earth and onboard
respectively, $B(t)$ the joint effect of the noise from buffeting of the probe
by non gravitational forces and from the antenna of the spacecraft, $T(t)$ the
joint frequency fluctuations due to the troposphere, ionosphere and ground
antenna, $A_{E}(t)$ the noise of the radio transmitter on the ground,
$A_{S}(t)$ the noise of the radio transmitter on board, $EL_{E}(t)$,
$EL_{S}(t)$, the noise from the electronics on the ground and onboard
respectively, and $P_{E}(t)$, $P_{S}(t)$ the fluctuations on the two links due
to the interplanetary plasma. Since the frequency fluctuations induced by the
plasma are, to first order, inversely proportional to the square of the radio
frequency, by using high frequency radio signals or by monitoring two
different radio frequencies transmitted to and from the spacecraft, this noise
source can be suppressed to very low levels or entirely removed from the data
respectively bertotti_etal1993 . In what follows we will assume dual frequency
to be used, and disregard the noise effects of the plasma fluctuations in our
analysis.
From Eqs. (1,2) we deduce that gravitational wave pulses of duration longer
than the one-light-time $L$ give a Doppler response that, to first order,
tends to zero. The tracking system essentially acts as a pass-band device, in
which the low-frequency limit $f_{l}$ is roughly equal to $(L)^{-1}$ Hz, and
the high-frequency limit $f_{H}$ is set by the thermal noise in the receiver.
Since the clocks and some electronic components are most stable at integration
times around $1000$ seconds, Doppler tracking experiments are performed when
the distance to the spacecraft is of the order of a few astronomical units.
This sets the value of $f_{l}$ for a typical experiment to about $10^{-4}$ Hz,
while the thermal noise gives an $f_{H}$ of about $10^{-2}$ Hz.
It is important to note the characteristic time signatures of the clock
noises, $C_{E}(t)$ and $C_{S}(t)$, of the probe antenna and spacecraft
buffeting noise $B(t)$, of the troposphere, ionosphere, and ground antenna
noise $T(t)$, and the transmitters $A_{E}(t)$, $A_{S}(t)$. The time signature
of the two clock noises, for instance, can be understood by observing that the
frequency of the signal received at the ground station at time $t$ contains
fluctuations from the onboard clock that were transmitted $L$ seconds earlier
and the noise from the ground clock enters with a negative sign at time $t$
due to the heterodyne nature of the Doppler measurement. The time signature of
the noises $T$, $B(t)$, $A_{E}(t)$, and $A_{S}(t)$ in Eq. (1,2) can be
understood through similar considerations.
Since the major noise source affecting the two one-way measurements is
represented by the fluctuations induced by the Earth troposphere and the
mechanical vibrations of the ground station, it has been emphasized Tinto1996
that there exists a combination of the two Doppler data that cancels these
noises. It is easy to see from inspection of Eqs. (1,2) that such a
combination is equal to
$x(t)\equiv S(t)-E(t-L)\ .$ (4)
After substituting into Eq. (4) the expressions for $E(t)$, $S(t)$ given in
Eqs. (1,2) we get
$\displaystyle x(t)$ $\displaystyle=$ $\displaystyle h(t-L)-\frac{1+\mu}{2}\
h(t-\mu L)-\frac{1-\mu}{2}\ h(t-2L-\mu L)$ (5) $\displaystyle+\
2C_{E}(t-L)-[C_{S}(t)+C_{S}(t-2L)]+[B(t)-B(t-2L)]$ $\displaystyle+\
A_{E}(t-L)-A_{S}(t-2L)+EL_{S}(t)-EL_{E}(t-L)\ .$
From Eq. (5) we may notice that the spacecraft buffeting noise, $B$, does not
cancel exactly and it gets suppressed by its transfer function to the $x$
combination at frequencies smaller than the inverse of the round-trip-light-
time.
## III Gravitational Wave Sensitivities
This section describes the derivation of the sensitivity, which is defined on
average over the sky, to be equal to the strength of a sinusoidal
gravitational wave required to achieve a signal-to-noise ratio of $1$ in a
forty-day integration time. Note that the sensitivity is therefore a function
of Fourier frequency, $f$. We have chosen the integration time to be equal to
forty days since this was the tracking time of the CASSINI gravitational wave
experiments Armstrong2006 . Sensitivity is essentially the noise-to-signal
ratio and it will be computed for both the new data combination $x$ as well as
for the traditional two-way coherent tracking measurement, $y$, for comparison
reasons. For convenience we provide below the expression of the two-way
Doppler response, $y(t)$, which will be used later on for estimating its
sensitivity
$\displaystyle y(t)$ $\displaystyle=$ $\displaystyle-\frac{(1-\mu)}{2}\ h(t)\
-\ \mu\ h(t-(1+\mu)L)+\frac{(1+\mu)}{2}\ h(t-2L)$ (6) $\displaystyle+$
$\displaystyle C_{E}(t-2L)\ -\ C_{E}(t)\ +\ 2B(t-L)\ +\ T(t-2L)\ +\ T(t)$
$\displaystyle+$ $\displaystyle A_{E}(t-2L)\ +\ A_{S}(t-L)\ \ +\ TR(t-L)\ +\
EL_{E_{2}}(t)\ ,$
where $TR$ is the random process associated with the relative frequency
fluctuations due to the onboard microwave transponder. For more details we
refer the reader to Armstrong1987 .
### III.1 Signal Averaged Power
The averaged signal power in the combination $x$, estimated at an arbitrary
Fourier frequency $f$, is computed by (i) taking the Fourier transform of the
signal entering into the combination $x$ and its modulus-squared, and (ii) by
integrating the resulting expression over an ensemble of sinusoidal signals
uniformly distributed over the celestial sphere. Such a calculation is long
but straightforward, and the resulting expression, $S_{x_{h}}$, is equal to
$S_{x_{h}}=S_{h}\ \left[\frac{4}{3}+\frac{2}{3}\ \cos^{2}(2\pi
fL)-\frac{\sin^{2}(2\pi fL)}{2(\pi fL)^{2}}\right]\ ,$ (7)
where $S_{h}$ is the gravitational wave signal one-sided power spectral
density.
The calculation of the averaged signal power of the observable $y$ can
similarly be carried through, resulting into the following expression
$S_{y_{h}}=S_{h}\ \frac{8\pi^{3}f^{3}L^{3}+2\sin(4\pi fL)-\frac{2}{3}\pi fL\
(3+4\pi^{2}f^{2}L^{2})\cos(4\pi fL)-6\pi fL}{8\pi^{3}f^{3}L^{3}}$ (8)
Figure 3 shows the two “transfer functions” of the signal one-sided power
spectral densities, i.e. $q_{x}\equiv S_{x_{h}}/S_{h}$ and $q_{y}\equiv
S_{y_{h}}/S_{h}$. The transfer function $q_{x}$ is slightly larger than
$q_{y}$ in the region $[5\times 10^{-4}-1]$ Hz, indicating that a constructing
interference of the signal with itself is taking place in this part of
frequency band. On the other hand, in the low-part of the frequency band the
combination $x$ shows a coupling to gravitational radiation that is weaker
than that of $y$. This is to be expected, as $x$ is the difference of the two
one-way measurements, which in the “long-wavelength” limit become equal to
each other.
Figure 3: Averaged power transfer functions of the Doppler responses $x$ and
$y$ to an ensemble of sinusoidal signals randomly polarized and uniformly
distributed over the celestial sphere. The x-transfer function shows
constructing interference at frequencies that are integer multiples of the
inverse of the round-trip-light-time, $2L$, taken here to be equal to $5500$
seconds. The coupling of the $x$ data combination to such a stochastic
ensemble of gravitational radiation is slightly stronger than that of the two-
way $y$ response at frequencies larger than $5.0\times 10^{-4}$ Hz. At lower
frequencies the transfer function of the $x$ combination decays more rapidly
than that of $y$ as a consequence of being the difference of the two one-way
measurements.
### III.2 Noise Spectra
To compute the sensitivity of the combination $x$, and compare it against that
of the two-way Doppler measurement, $y$, we need the one-sided power spectral
densities of the main noise sources and their transfer functions to the
observables $x$ and $y$. If we assume all the noise sources to be
uncorrelated, from equations (5, 6) we can derive the following expressions
for the two noise spectra $S_{x_{n}}$ and $S_{y_{n}}$
$\displaystyle S_{x_{n}}$ $\displaystyle=$ $\displaystyle
4S_{C_{E}}+4S_{C_{S}}\cos^{2}(2\pi fL)+4S_{B}\sin^{2}(2\pi
fL)+S_{A_{E}}+S_{A_{S}}+S_{EL_{E}}+S_{EL_{S}}\ ,$ (9) $\displaystyle
S_{y_{n}}$ $\displaystyle=$ $\displaystyle 4S_{C_{E}}\sin^{2}(2\pi
fL)+4S_{T}\cos^{2}(2\pi
fL)+4S_{B}+S_{A_{E}}+S_{A_{S}}+S_{EL_{E}}+S_{EL_{S}}+S_{TR}\ ,$ (10)
where the meaning of the various terms appearing into Eqs. (9,10) is self-
explanatory. We provide in the Appendix the expressions for the various noise
spectra corresponding to a gravitational wave search performed with a
spacecraft out to a distance of $5.5$ AU from Earth (as was during the first
gravitational wave experiment with the CASSINI spacecraft), and equipped with
an onboard microwave instrumentation similar to the one flown on CASSINI.
The gravitational wave sensitivity is the wave amplitude required to achieve a
signal-to-noise ratio of $1$, and it can be computed as a function of Fourier
frequency using the following expression Armstrong2006
$\Sigma_{z}(f)\equiv\sqrt{\frac{S_{z_{n}}(f)\ B}{q_{z}(f)}}\ ,$ (11)
where $z$ means either $x$ or $y$, and $B$ is the frequency bandwidth
corresponding to a $40$ days integration time, the duration of the
gravitational wave experiments performed with the CASSINI spacecraft
Armstrong2006 .
Figure 4: Sensitivities of the $x$ (solid-line) and $y$ (dash-line) Doppler
responses to a randomly polarized and uniformly distributed stochastic
ensemble of sinusoidal gravitational wave signals. The sensitivity is
expressed as a function of the frequency and represents the equivalent
sinusoidal strain required to produce a signal-to-noise ration of $1$. The
spacecraft is assumed to be out to a distance of $5.5$ AU, and eighty percent
tropospheric noise calibration is applied to the two-way Doppler data $y$
(dash-line). The two sensitivity curves reflect the noise spectral levels and
shapes (given in the Appendix), their transfer functions to the observables
$x$ and $y$ (Eqs. 9, 10), and the gravitational wave transfer functions shown
in Fig. 3)
The main difference between the two observables $x$ and $y$ is of course in
the absence in the $x$ combination of the joint disturbances from the
troposphere and the mechanical vibration of the ground antenna (the random
process denoted $T$ in Eq.(6)) and the presence (in $x$) of the spacecraft
clock noise process. As onboard and ground microwave instrumentation have in
recent years reached unprecedented frequency stabilities, $T$ has indeed
become the major sensitivity limitation of two-way Doppler tracking searches
for gravitational radiation Armstrong2006 . Since the effects from the
troposphere can be mitigated by relying on simultaneous measurements performed
by a radiometer located in the proximity of the tracking station, our
sensitivity analysis reflects the assumption of being able to calibrate out
eighty percent of the tropospheric effects from the $y$ observable (as
demonstrated by the CASSINI experiments). Figure 4 shows the estimated
sensitivity of the observable $x$ (continuous-line) formed out of the two one-
way measurements, and compares it against that of the two-way measurement in
which eighty percent of the tropospheric effects are calibrated out (dash-
line). The $x$ combination displays the best sensitivity in the frequency band
$10^{-4}-10^{-1}$ Hz, which is of most interest to gravitational wave search
experiments. At higher frequencies, between about $10^{-1}-3.0\times 10^{-1}$
Hz, effects related to the locking of the atomic clock to its local oscillator
introduce a small degradation in the $x$ sensitivity over that of the $y$
data. Also, at frequencies lower than $3.0\times 10^{-5}$ Hz, the $x$
combination shows a sensitivity worse than that of $y$ due to the cancellation
of the signal in this low-frequency region.
As the two one-way Doppler measurements can be combined to synthesize the two-
way measurement $y$ Piran_etal1986 ; Tinto1996 , one could argue that at these
frequencies one could of course rely on the synthesized $y$ data to take
advantage, if needed, of its better sensitivity in these two regions of the
accessible frequency band. This observation suggests that it must be possible
to identify a combination of the two one-way Doppler data that maximizes the
sensitivity to gravitational waves in the entire band of interest.
### III.3 Optimal Sensitivity
In order to derive the combination of the two one-way Doppler data that
achieves optimal sensitivity, let us consider the following linear combination
$\eta(f)$ of the Fourier transforms of $E(t)$ and $S(t)$
$\eta(f)\equiv a_{1}(f,{\vec{\lambda}})\ {\widetilde{E}}(f)\ +\
a_{2}(f,{\vec{\lambda}})\ {\widetilde{S}}(f)\ ,$ (12)
where the $\\{a_{i}(f,\vec{\lambda})\\}_{i=1,2}$ are arbitrary complex
functions of the Fourier frequency $f$, and of a vector $\vec{\lambda}$
containing parameters characterizing the signal and the noises affecting the
two Doppler data. For a given choice of the two functions
$\\{a_{i}\\}_{i=1,2}$, $\eta$ gives a specific Doppler data combination, and
our goal is therefore that of identifying, for a given signal, the two
functions $\\{a_{i}\\}_{i=1,2}$ that maximize the signal-to-noise ratio
Helstrom68 , $SNR_{\eta}^{2}$, of the combination $\eta$
$SNR_{\eta}^{2}=\int_{f_{l}}^{f_{u}}\frac{|a_{1}\ {\widetilde{E}_{s}}+a_{2}\
{\widetilde{S}_{s}}|^{2}}{{\langle|a_{1}\ {\widetilde{E}_{n}}+a_{2}\
{\widetilde{S}_{n}}|^{2}\rangle}}\ df\ .$ (13)
In equation (13) the subscripts $s$ and $n$ refer to the signal and the noise
parts of (${\widetilde{E}},{\widetilde{S}}$) respectively, the angle brackets
represent noise ensemble averages, and the interval of integration
($f_{l},f_{u}$) corresponds to the accessible frequency band.
The $SNR_{\eta}^{2}$ can be regarded as a functional over the space of the two
complex functions $\\{a_{i}\\}_{i=1,2}$, and their expressions that maximize
it can of course be derived by solving the associated set of Euler-Lagrange
equations. The derivation of the expression of the optimal SNR,
${SNR_{\eta}^{2}}_{\rm opt}$, is long but straightforward, and it is equal to
(see PTLA02 for details)
${SNR_{\eta}^{2}}_{\rm opt.}=\int_{f_{l}}^{f_{u}}{\bf z}^{(s)*}_{i}\ ({\bf
C}^{-1})_{ij}\ {\bf z}^{(s)}_{j}\ df\ .$ (14)
In Eq. (14) the convention of sum over repeated indices is assumed, ${\bf
z}^{(s)}$ is the vector of the signals,
(${\widetilde{E_{s}}},{\widetilde{S_{s}}}$), and ${\bf C}$ is the hermitian,
non-singular, correlation matrix of the vector random process ${\bf
z}_{n}\equiv({\widetilde{E_{n}}},{\widetilde{S_{n}}})$
$({\bf C})_{rt}\equiv\langle{\bf z}^{(n)}_{r}{\bf z}^{(n)*}_{t}\rangle\ .$
(15)
Eq. (14) can now be used for estimating the sensitivity to an ensemble of
sinusoidal gravitational wave signals randomly polarized and uniformly
distributed over the celestial sphere. Figure 5 shows the estimated optimal
sensitivity (solid-line) obtained by relying on the two one-way measurements,
and again that of a two-way coherent tracking experiment (dash-line), in which
all the parameters characterizing the experiment are as in Figure (4). Note
how the sensitivity of the optimal combination is now consistently below that
of the two-way measurement, and it coincides with that of the $x$ combination
in most of the accessible frequency band.
Figure 5: Sensitivity curves of the optimal combination (solid-line) and the
two-way Doppler tracking data (dash-line). The noise parameters are equal to
those used in figure 4. Note how the sensitivity of the optimal combination is
now consistently below that of the two-way measurement.
## IV Conclusions
We have discussed a method for significantly enhancing the sensitivity of
Doppler tracking experiments aimed at the detection of gravitational waves.
The main result of our analysis shows that by adding an atomic frequency
standard and a digital receiver on board the spacecraft we can achieve a
broad-band sensitivity of $2.0\times 10^{-17}$ in the milliHertz band. This
sensitivity figure is obtained by completely removing the frequency
fluctuations due to the interplanetary plasma. Our method relies on a properly
chosen linear combination of the one-way Doppler data recorded on board with
the data measured on the ground. It allows us to optimally suppress the
frequency fluctuations due to the troposphere, ionosphere, and antenna
mechanical and, for a spacecraft that is tracked for $40$ days out to $5.5$
AU, to reach a sensitivity that is about one order of magnitude better than
that achievable by a state-of-the art two-way Doppler tracking search.
The expression of the optimal combination of the two one-way Doppler data can
be used in all the classic tests of relativistic theory of gravity in which
one-way and two-way spacecraft Doppler measurements are used as primary data
sets. We will analyze the implications of the sensitivity improvements that
this technique will provide for direct measurements of the gravitational red-
shift, the second-order relativistic Doppler effect predicted by the theory of
special relativity, searches for possible anisotropy in the velocity of light,
measurements of the parameterized post-Newtonian parameters, and measurements
of the deflection and time delay by the Sun in radio signals. This research is
in progress, and will be the subject of a forthcoming investigation.
## Acknowledgements
It is a pleasure to that Frank B. Estabrook for his constant encouragement
during the development of this work. This research was performed at the Jet
Propulsion Laboratory, California Institute of Technology, under contract with
the National Aeronautics and Space Administration.(c) 2008 California
Institute of Technology. Government sponsorship acknowledged.
## Appendix
### IV.1 Noise sources and their spectra
This appendix provides a general description of the radio hardware needed for
implementing the technique described in the main body of this paper, the
corresponding one-sided power spectral densities of the frequency fluctuations
introduced by these subsystems into the observables $E(t)$ and $S(t)$, and
discusses the frequency fluctuations due to the Earth atmosphere. For a more
comprehensive analysis on the radio hardware the reader is referred to
Tinto1996 ; Tinto2000 ; Armstrong2006 ; Prestage_Weaver2007 ; Dick1990 ;
RASSI_2008 , while a review on the propagation noises is given in
Armstrong2006
The ground master clock and the frequency & timing distribution represent the
overall contribution of the reference clock itself and the cabling system that
takes the signal generated by the master clock to the antenna. This can be
located several kilometers away from the site of the clock, implying that the
need of a highly-stable cabling system is required. It has been shown at JPL
that optical-fiber cables would not degrade significantly the frequency
stability of the signal generated by the master clock. The corresponding one-
sided power spectral density of the frequency fluctuations, introduced by
these two noise sources, is equal to Tinto2000
$S_{C_{E}}(f)=6.2\times[10^{-28}f+10^{-33}f^{-1}+10^{-30}]+1.3\times
10^{-28}f^{2}\ \ {\rm Hz}^{-1}\ .$ (16)
The Ground and onboard Transmitters include all the frequency multipliers that
are needed to generate the desired frequency of the transmitted radio signal,
starting from the frequency provided by the clocks. It also accounts for the
radio amplifier, and the extra phase delay changes occurring between the
amplifier and the feed cone of the antennas. The noise due to the amplifiers
is the dominant one, and it has been characterized in Tinto1996 . The one-
sided power spectral densities of the frequency fluctuations are given by
$S_{A_{E}}(f)+S_{A_{S}}(f)=2.3\times 10^{-28}+4.0\times 10^{-25}f\ \ {\rm
Hz}^{-1}\ ,$ (17)
The noises introduced by the Receiver at the ground station can be modeled as
white phase fluctuations. The contribution to the overall noise budget from
the receiver chain on the ground can be repartitioned into thermal noise
(finiteness of the signal-to-noise ratio) and fluctuations introduced into the
signal as it propagates through the cables and waveguides running from the
feed of the antenna to the actual receiver. The effects of the latter noise
source is nowadays minimized with the use of beam waveguide (BWG) antennas.
These new antennas have become operational in the year 2004 at the NASA Deep
Space Network three sites: in North America (Goldstone, California), Europe
(Madrid, Spain), and Australia (Canberra).
Under the assumption of relying on a $34$ meter diameter beam waveguide
antenna for receiving a coherent Ka-Band ($32$ GHz) signal transmitted by a
spacecraft out a distance of $5.5$ AU, a ground system noise temperature of
about $70$ degrees Kelvin, an onboard Ka-Band amplifier of $10$ W, and a
spacecraft High Gain Antenna (HGA) with a diameter of about $4$ meters, we
find the following one-sided power spectral density of the frequency
fluctuations at Ka-Band Armstrong2006
$S_{EL_{E}}(f)=6.3\times 10^{-27}\ f^{2}\ {\rm Hz}^{-1}\ .$ (18)
The buffeting of the spacecraft will introduce unwanted frequency fluctuations
in the one-way Doppler observable. Estimates of its magnitude have been given
in Riley_etal1990 , and the one-sided power spectral density of the frequency
fluctuations is given by the following expression
$S_{B}(f)=5.0\ \times 10^{-42}\ f^{-3}\ +\ 1.0\ \times 10^{-31}\ {\rm
Hz}^{-1}\ .$ (19)
The noise introduced into the Doppler observable $y$ by the Earth Atmosphere
and ionosphere, and by the scintillation of the interplanetary plasma, have
been studied extensively in the literature AWE1979 ; Armstrong2006 ; Keihm1995
.
The scintillation introduced into the Doppler observables by the Atmosphere
are independent of the microwave frequency at which the spacecraft is tracked.
Since gravitational wave searches are performed in a band whose upper
frequency cutoff is smaller than $1$ Hz (thermal noise at higher frequencies
becomes unacceptably large), the one-sided power spectral density of the noise
due to the atmosphere can be written as follows Linfield1998
$\displaystyle S_{T}(f)$ $\displaystyle=$ $\displaystyle 2.8\ \times 10^{-28}\
f^{-2/5}\ \ {\rm Hz}^{-1}\ \ \ \ \ \ \ 10^{-5}\leq f\leq 10^{-2}\ \ {\rm Hz}$
(20) $\displaystyle=$ $\displaystyle 2.2\ \times 10^{-30}\ f^{-3}\ \ \ \ {\rm
Hz}^{-1}\ \ \ \ \ \ \ 10^{-2}\leq f\leq 1\ \ \ \ \ \ \ {\rm Hz}\ .$
The first term in the equation above accounts for the remaining effect of the
atmosphere after eighty percent calibration is applied to the data with the
use of a water vapor radiometer Armstrong2006 , while the second term accounts
for the effect of aperture averaging, which causes a reduction in delay
fluctuations on time scales less than the antenna wind speed crossing time
($1$ to $10$ seconds) Linfield1998 .
The Transponder entering into the $y$ measurement is responsible for keeping
the phase coherence between the incoming and outgoing radio signals on the
spacecraft. Its performance depends on the accuracy of tracking of the up-link
signal by the phase locked loop, and the noise floor and non-linearities of
its electronic components Riley_etal1990 . Frequency stability measurements of
the Ka-Band ($32$ GHz) transponder flown onboard the CASSINI mission have
resulted in the following one-sided power spectral density of the relative
frequency fluctuations
$S_{TR}(f)=1.6\times 10^{-26}\ f\ \ {\rm Hz}^{-1}\ .$ (21)
The onboard clock provides the frequency and timing reference for the onboard
radio instrumentation, and identifies the stability of the microwave signal
transmitted to the ground. The space-qualified clock presently under
realization at the Jet Propulsion Laboratory relies on a combined “interplay”
between a local quartz oscillator and a trapped Hg ions clock. The frequency
of the hyperfine transition made by the Hg ions is used for monitoring and
correcting the frequency of the quartz oscillator. This steering process takes
place on a typical time scale of about $10$ seconds, making then possible over
longer time scale to significantly improve the stability of the resulting
combined instrument. A frequency stability comparable to that of the Deep
Space Network ground clocks has already been demonstrated with a prototype,
showing an Allan standard deviation of a few parts in $10^{-15}$ at an
integration time of a few thousand secondsPrestage_Weaver2007 ; Dick1990 . The
corresponding one-sided power spectral density of the relative frequency
fluctuations of such a clock is given by the following expression
$\displaystyle S_{C_{S}}$ $\displaystyle=$ $\displaystyle 5.0\times 10^{-27}\
\ \ \ {\rm Hz}^{-1}\ \ \ \ \ \ \ \ \ \ \ 10^{-5}\leq f\leq 2.0\times 10^{-2}\
\ \ \ \ {\rm Hz}$ (22) $\displaystyle=$ $\displaystyle 2.5\times 10^{-25}f\ \
{\rm Hz}^{-1}\ \ \ \ \ \ 2.0\times 10^{-2}\leq f\leq 2.0\times 10^{-1}\ \ \
{\rm Hz}$ $\displaystyle=$ $\displaystyle 10^{-26}f^{-1}\ \ \ \ \ \ \ {\rm
Hz}^{-1}\ \ \ \ \ \ 2.0\times 10^{-1}\leq f\leq 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
{\rm Hz}$
The onboard digital receiver used in this method measures the amplitude and
phase of the uplink signal and telemeters that information to the ground.
The frequency fluctuations of the receiver chain on board the spacecraft is
estimated to be entirely due to thermal noise because of a simpler cabling
system, and its performance being essentially identical to that of the ground
receiver. By assuming again a $34$ meter diameter beam waveguide antenna
transmitting with an $800$ W Ka-Band ($32$ GHz) amplifier to a spacecraft out
to a distance of $5.5$ AU, equipped with a $4$ meter diameter (HGA), and a
system noise temperature of $400$ Kelvin, we find the following one-sided
power spectral density of the frequency fluctuations of the onboard
electronics noise at Ka-Band Armstrong2006
$S_{EL_{S}}(f)=7.2\times 10^{-28}\ f^{2}\ {\rm Hz}^{-1}\ .$ (23)
## References
## References
* (1) A. Kliore, J.D. Anderson, J.W. Armstrong, S.W. Asmar, C.L. Hamilton, N.J. Rappaport, H.D. Wahlquist, R. Ambrosini, F.M. Flasar, R.G. French, L. Iess, E.A. Marouf, & A.F. Nagy, Space Sci. Rev., 115, 1–70, (2004).
* (2) R.F.C. Vessot. In: Proceedings of the XXVIIIth RENCONTRE DE MORIOND, eds. J. Tran Thanh Van, T. Damour, E. Hinds, and J. Wilkerson (Editions FRONTIERS), p.471-489, (1993).
* (3) B. Bertotti, L. Iess, & P. Tortora, Nature, 425, 374–376, (2003).
* (4) F.B. Estabrook and H.D. Wahlquist, Gen. Relativ. Gravit. 6, 439 (1975).
* (5) J.W. Armstrong, Living Reviews in Relativity 9, 1, (2006).
* (6) M. Tinto, Phys. Rev. D 53, 5354, (1996).
* (7) S.W. Asmar, J.W. Armstrong, L. Iess, & P. Tortora, Radio Sci., 40, (2005).
* (8) M. Tinto, Radio Sci. 37, 3, 1045, doi:10.1029/2001RS002473, (2002).
* (9) B. Bertotti, G. Comoretto, & L. Iess, Astron. Astrophys., 269, 608–616, (1993).
* (10) J.W. Armstrong, L. Iess, P. Tortora, & B. Bertotti, Astrophys. J., 599, 806–813, (2003).
* (11) J.A. Barnes, A.R. Chi, L.S. Cutler, D.J. Healey, D.B. Leeson, T.E. McGuniga, J.A. Mullen, W.L. Smith, R.L. Sydnor, R.F.C. Vessot, & G.M.R. Winkler, IEEE Trans. Instrum. Meas., 20, 105–120, (1971).
* (12) J.W. Armstrong, F.B. Estabrook, S.W. Asmar, L. Iess, & P. Tortora, Radio Sci., 43, RS3010, doi:10.1029/2007RS003766 28 June 2008.
* (13) J.D. Prestage, & G.L. Weaver. In: Proceedings of the IEEE, 95, 11, 2235-2247 (2007).
* (14) G.J. Dick, J.D. Prestage, C.A. Greenhall, & L. Maleki. In: Proceedings of the 22nd annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting, The U.S. Naval Observatory publication. Available at: http://tycho.usno.navy.mil/ptti/1990/Vol
* (15) G. L. Tyler , I. R. Linscott, M. K. Bird, D. P. Hinson, D. F. Strobel, M. P tzold, M. E. Summers and K. Sivaramakrishnan, Space Sci. Rev., DOI 10.1007/s1214-007-9302-3, (2008).
* (16) The Radio Atmospheric Sounding and Scattering Instrument (RASSI), JPL proposal for constructing a flight radio science digital receiver. JPL internal publication (2008).
* (17) D. Kleppner, R.F.C. Vessot, & N.F. Ramsey, Astrophys. Space Sci., 6, 13-32, (1970).
* (18) R.F.C. Vessot, & M.W. Levine. In: A closeup of the Sun, Eds. M. Neugebauer, & R.W. Davis. JPL Publication 78-80, NASA (1978).
* (19) T. Piran, E. Reiter, W.G. Unruh, and R.F.C. Vessot, Phys. Rev. D, 34, 984, (1986).
* (20) J.W. Armstrong, R. Woo, and F.B. Estabrook, Ap.J., 230,570 (1979).
* (21) J.W. Armstrong, In: Proceedings of the NATO Advanced Research Workshop: Gravitational Wave Data Analysis, St. Nichols, Cardiff, Wales, 6 – 9 July 1987, NATO ASI Series C, vol. 253, 153–172, (Kluwer, Dordrecht, Netherlands; Boston, U.S.A., 1989), ed. B.F. Schutz.
* (22) M. Tinto, Open Loop Radio Science, Jet Propulsion Laboratory Publication - DSMS Telecommunications Link Design Handbook, 810-005, Rev. E, (2000). (http://eis.jpl.nasa.gov/deepspace/dsndocs/810-005/)
* (23) A.L. Riley, D. Antsos, J.W. Armstrong, P. Kinman, H.D. Wahlquist, B. Bertotti, G. Comoretto, B. Pernice, G. Carnicella, & R. Giordani, Cassini Ka-Band Precision Doppler and Enhanced Telecommunications System Study, Jet Propulsion Laboratory Report, Pasadena, California, January 22, 1990.
* (24) S.J. Keihm, TDA Progress Report, 42 - 122, 1-11, August 15, (1995).
* (25) R.P. Linfield, Radio Science, 33, 5 1353-1359, (1998).
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|
arxiv-papers
| 2008-12-13T20:33:33 |
2024-09-04T02:48:59.391129
|
{
"license": "Public Domain",
"authors": "Massimo Tinto, George J. Dick, John D. Prestage, and J.W. Armstrong",
"submitter": "Massimo Tinto",
"url": "https://arxiv.org/abs/0812.2581"
}
|
0812.2696
|
# Emergent Electroweak Gravity
Bob McElrath bob.mcelrath@cern.ch CERN theory group, Geneva 23, CH 1211,
Switzerland
###### Abstract
We show that any massive cosmological relic particle with small self-
interactions is a super-fluid today, due to the broadening of its wave packet,
and lack of any elastic scattering. The WIMP dark matter picture is only
consistent its mass $M\gg M_{\rm Pl}$ in order to maintain classicality. The
dynamics of a super-fluid are given by the excitation spectrum of bound state
quasi-particles, rather than the center of mass motion of constituent
particles. If this relic is a fermion with a repulsive interaction mediated by
a heavy boson, such as neutrinos interacting via the $Z^{0}$, the condensate
has the same quantum numbers as the vierbein of General Relativity. Because
there exists an enhanced global symmetry $SO(3,1)_{space}\times
SO(3,1)_{spin}$ among the fermion’s self-interactions broken only by it’s
kinetic term, the long wavelength fluctuation around this condensate is a
Goldstone graviton. A gravitational theory exists in the low energy limit of
the Standard Model’s Electroweak sector below the weak scale, with a strength
that is parametrically similar to $G_{N}$.
## I Introduction
In the early universe, relics including photons, neutrinos and dark matter
evolve out of thermal equilibrium as their interaction strength becomes small
at low temperature in a process known as “freeze-out”. This calculation is
essentially classical, assuming particles are point-like and using the
Boltzmann equation griest_cosmic_1987 ; srednicki_calculations_1988 .
After freeze-out the number density of particles is fixed, and the temperature
just evolves with Hubble expansion. Their time evolution is given only by the
free particle kinetic term. It is usually assumed that the interaction
strength is so weak that it can be neglected and that particles remain
localized point particles forever. The free particle Hamiltonian propagates
particles and also broadens their wave packets, described by their uncertainty
$\Delta x$. This is due to the fact that the localization of particles causes
them to not be an eigenstate of the Hamiltonian if they are massive.
There are two limits of interest for the particle uncertainty $\Delta x$
relative to the number density $n$. The classical gas limit is $\Delta x\ll
n^{-1/3}$. Elastic scattering collisions and the Boltzmann equation describe
this system. The opposite limit, $\Delta x\gg n^{-1/3}$ is a quantum liquid.
Because particles have wave function overlap with their neighbors, one must
take into account collective effects due to contact interactions. If there
exists an attractive interaction in any partial wave, then the vacuum energy
can be lowered by forming bound state quasi-particles. The system will undergo
a phase transition to a super-fluid described by quasi-particles.
If the system contains global symmetries that are broken when the system
becomes a super-fluid, then Goldstone bosons will emerge. As these are
massless, their dynamics are extremely important.
The idea of gravity emerging from spinors is not new and fairly obvious, as
one can construct a spin-2 particle as the direct product of spinors
ohanian_gravitons_1969 ; kraus_photons_2002 . However no workable theory has
been yet constructed. The first idea of this type is due to Bjorken
bjorken_dynamical_1963 , who attempted to formulate the photon and graviton as
a composite state. The most recent attempt and the most successful is due to
Hebecker and Wetterich hebecker_spinor_2003 ; wetterich_gravity_2003 . Their
theory can be regarded as a reformulation of gravity in terms of spinors, but
they give no dynamics for the spinors which would lead to such a theory. This
line of research was largely killed by the paper of Weinberg and Witten
Weinberg:1980kq , which showed that a spin-2 particle could not couple to a
covariant conserved current. Two ways out of this theorem are to quantize
geometry (the approach of string theory), or to abandon diffeomorphism
invariance as an exact symmetry. Sakharov originally suggested that the
graviton could be emergent, and in such theories, diffeomorphism invariance
can only be approximate Sakharov:1967pk .
## II Quantum Liquid Transition
The quantum liquid regime for a system occurs when the position uncertainty
$\Delta x$ is larger than the inter-particle spacing
$\Delta x\gg n^{-1/3}.$ (1)
In this limit the system is not classical, and the condition of scattering
theory that the impact parameter $b\gg\Delta x$ cannot be satisfied (often
known as the “well-localized” assumption).
Particles in the classical gas limit will eventually time-evolve into a
quantum liquid in the absence of interactions. The expansion of a free
particle wave packet in time is
$\Delta x(t)^{2}=\Delta x_{0}^{2}+\Delta v^{2}t^{2}.$ (2)
This can be intuitively understood because different momentum components may
move with different velocities. The wave number at $p+\Delta p$ has a velocity
$(p+\Delta p)/E$ while the wave number at $p-\Delta p$ has a smaller velocity
$(p-\Delta p)/E$ and these two wave numbers will separate in space as they
propagate if $E>p$.
The condition for the time-independent super-fluid transition can be derived
by neglecting the second term of Eq. 2. In the non-relativistic limit one
arrives at
$T<\frac{\lambda^{2}n^{2/3}}{3mk_{B}}.$ (3)
The cross-section does not enter into this calculation, and the uncertainty
$\Delta x_{0}$ is assumed to be proportional to the thermal de Broglie
wavelength, $\Delta x_{0}=1/\Delta p=\lambda/p=\lambda/\sqrt{3mkT}$, where
$\lambda$ is an $\mathcal{O}(1)$ parameter reflecting how “localized” the
state is. This temperature may be further suppressed by elastic collisions,
which must occur frequently enough to keep particles localized to their
thermal de Broglie wavelength, but not so often that they destroy the
condensate.
In the relativistic case, we also use Eq. 2, however the velocity uncertainty
for relativistic states is
$\Delta v=\frac{\Delta p}{E}(1-v^{2})$ (4)
where $v=p/E$. This correctly reflects the relativistic limit, $v\to c$;
massless wave packets do not broaden as each wave number propagates with the
same velocity, $v=c$.
The relevant time scale for wave packet broadening is the mean time between
collisions $\tau=1/\sigma nv$ in terms of the cross section $\sigma$ since the
uncertainty of a wave packet $\Delta x_{0}$ is set by the 3-momentum of an
elastic scattering collision. The condition for a quantum liquid is then
$\frac{1}{p^{2}}+\frac{(1-v^{2})^{2}}{\sigma^{2}n^{2}}>\frac{1}{\lambda^{2}n^{2/3}}.$
(5)
In the limit that the first term on the left side is small compared to the
second (e.g. for decoupled relics), the quantum liquid condition is:
$\sigma<\frac{\lambda(1-v^{2})}{n^{2/3}}.$ (6)
Thus, for any decoupled cosmological relic, it becomes a quantum liquid when
its cross section is approximately less than the square of the inter-particle
separation. This occurs faster for non-relativistic relics $v\to 0$ than
relativistic ones $v\to 1$, and can be delayed if collisions are “well-
localized” relative to the inter-particle separation ($\lambda\to 0$).
This condition (Eq.6) is extremely well satisfied for massive neutrinos and
Weakly Interacting Massive Particle (WIMP) dark matter, so that today, WIMPs
and at least two neutrino mass eigenstates are definitely quantum liquids.
An important implication of this result is that non-relativistic relics such
as WIMP dark matter must be treated as quantum liquids. The phenomena
currently attributed to dark matter can only be achieved by a classical gas of
particles which must satisfy $\Delta x(t)\ll n(t)^{-1/3}$ One can see that
under virtually any assumptions about Hubble expansion and decoupling, these
theories are only consistent if $M\gg M_{\rm Pl}$. Such a heavy object is very
unlikely to be consistently described as a single quantum particle.
If attractive contact interactions exist, the system will make a phase
transition to a super-fluid in exactly the same way as a BCS superconductor or
3He. For WIMP dark matter, the required contact interaction occurs by
integrating out any heavy particles which couple to the WIMP to give a 4-point
operator. Collisions are so rare that they can’t break up the collective
excitations of the super-fluid, and the relevant condensation criterion is not
given by the thermal wavelength (Eq. 3) but rather the time-expanded wave
packet as in Eq. 6. In the next section we show that an attractive interaction
always exists among fermions, though it may be in a higher partial wave.
## III The Kohn-Luttinger Effect
Beyond wave-function overlap, a necessary condition for a super-fluid state is
the existence of a ground state with lower energy than the original vacuum
Lagrangian. In the case of an attractive 4-fermion interaction, there
obviously exists a lower energy ground state where the fermions bind into
$s$-wave quasi-particles. For WIMP dark matter theories this is a possibility.
For the Standard Model (SM), neutrino self-interactions are repulsive
caldi_cosmological_1999 . However Kohn and Luttinger showed that even a
repulsive fermionic quantum liquid cannot behave as a classical gas. The
reason is that at one loop, 4-point interactions induce a singularity at the
Fermi surface that is attractive PhysRevLett.15.524 ; KaganChubakov.47.525 ;
efremov-2000-90 . Since higher partial wave interactions are exponentially
suppressed relative to the $s$-wave, and this correction scales only as
$\ell^{-4}$, in terms of the partial wave number $\ell$. For some large $\ell$
this correction dominates. For cosmological relics this occurs already in the
$p$ wave.
The relevant correction comes from an exchange (box) diagram and its
contribution to the BCS potential $V(x)$ in the $\ell$th partial wave is
$\delta
V_{\ell}=(-1)^{\ell+1}\frac{mp_{F}}{4\pi^{2}}\frac{|V(\cos\theta=-1)|^{2}}{\ell^{4}}$
(7)
where $p_{F}=(3\pi n)^{1/3}$ and $V(\cos\theta)$ is the tree-level potential
evaluated on the Fermi surface. This is attractive for odd $\ell$, The
relevant infrared divergence occurs for $\cos\theta=-1$ and corresponds to an
exchange of the propagating neutrino with a background neutrino. The
divergence occurs at $2p_{F}$ because it occurs in the internal loops, which
contain two fermion propagators, both of which must lie on the Fermi surface.
This potential is parametrically $\mathcal{O}(p_{F}^{2}G_{F}^{2})$. Therefore
this condensation is a much more important effect than scattering, which is
associated with the mean free path and is $\mathcal{O}(p_{F}^{5}G_{F}^{2})$.
Note that $\delta V_{1}$ is also parametrically the same order as Newton’s
constant $G_{N}$.
Therefore, an attractive self-interaction always exists in a neutrino or
fermionic WIMP fluid, regardless of the sign of the fundamental interaction.
If the mass is sufficiently small so that the conditions of the previous
section are also satisfied, then such a cosmological relic is a super-fluid
today. The two heavier neutrino species and WIMP dark matter are super-fluids
today. Lighter species such the lightest neutrino (if sufficiently light)
would require an early-universe analysis to determine if the conditions of the
previous section can be satisfied.
## IV Condensate Quantum Numbers
A condensate will break Lorentz invariance, but if the underlying theory is
invariant, we can classify the condensates by their Lorentz representation. A
Weyl fermion condenses as
$(\frac{1}{2},0)\otimes(\frac{1}{2},0)=(0,0)\oplus(1,0)$ according to its
representation under the spin Lorentz group. A $p$-wave condensate must
contain a derivative, giving
$\displaystyle A_{\mu}(x,y)$ $\displaystyle=$
$\displaystyle\frac{i}{2}(\tilde{\partial}_{\mu}\chi\epsilon\xi-\chi\epsilon\tilde{\partial}_{\mu}\xi);$
(8) $\displaystyle E^{a}_{\mu}(x,y)$ $\displaystyle=$
$\displaystyle\frac{i}{2}(\tilde{\partial}_{\mu}\chi^{\dagger}\overline{\sigma}^{a}\xi-\chi^{\dagger}\overline{\sigma}^{a}\tilde{\partial}_{\mu}\xi),$
(9)
where $\tilde{\partial}_{\mu}$ represents the deviation in momentum from the
Fermi surface, $p_{0}=0$, $|\vec{p}|=2p_{F}$, and we abbreviate $\chi=\chi(x)$
and $\xi=\chi(y)$. In condensed matter nomenclature, these excitations are
“zero-sound”.
The four-point operator for these two condensates is the same since they are
related by a Fierz transformation, therefore we may write it as
$-\frac{g_{Z}^{4}mp_{F}}{4\pi^{2}M_{Z}^{4}}\int_{xy}\left[(1-\eta_{\nu})E^{a\dagger}_{\mu}E_{a}^{\mu}+\eta_{\nu}A_{\mu}^{\dagger}A^{\mu}\right],$
(10)
where
$\eta_{\nu}=\frac{n_{\nu}-n_{\overline{\nu}}}{n_{\nu}+n_{\overline{\nu}}}$
(11)
is the asymmetry between neutrinos and anti-neutrinos. After the phase
transition (Eq.6) has occurred, the original Fermi gas is described by
momentum distribution functions for $A_{\mu}$ and $E^{a}_{\mu}$, rather than
original one for free fermions.
The condensate $E^{a}_{\mu}$ contains both particles and antiparticles, while
$A_{\mu}$ contains only particles (or antiparticles). Therefore, $A_{\mu}$
only condenses among the unpaired particles that don’t have an antiparticle
partner. The Cosmic Neutrino Background (CNB) is expected to contain very
nearly equal numbers of neutrinos and anti-neutrinos. The asymmetry
$\eta_{\nu}$ is proportional to the baryon to photon ratio, $\eta_{b}\sim
6\times 10^{-10}$. Therefore $E^{a}_{\mu}$ is the dominant condensate and the
dynamics of $A_{\mu}$ are sub-leading so we will neglect them. A right-handed
neutrino state (if they are Dirac) has interactions that are much weaker than
the left-handed state, and can be ignored. Likewise, repulsive Majorana dark
matter such as a bino is usually not assumed to have any matter/antimatter
asymmetry and again can be treated as a single Weyl spinor super-fluid which
condenses into $E^{a}_{\mu}$.
## V Lorentz Breaking
The condensation of $A_{\mu}$ and $E^{a}_{\mu}$ breaks Poincaré invariance,
since both fields have Lorentz indices, and the neutrinos should have a
spatially varying density distribution. This symmetry breaking is dynamical
and spontaneous, due to the condensation of a physical background; the SM is
Poincaré and Lorentz invariant. As a consequence of the symmetry breaking,
both have corresponding Goldstone bosons, which are long wavelength
fluctuations about the expectation values for $A_{\mu}$ and $E^{a}_{\mu}$.
Neutrino self-interactions are mediated by the $Z^{0}$ boson. In the Feynman
gauge we may write the tree level effective 4-point operator as
$-\frac{g_{Z}^{2}}{2M_{Z}^{2}}\int_{xy}\left\\{\chi^{\dagger}\overline{\sigma}^{a}\chi\xi^{\dagger}\overline{\sigma}_{a}\xi\right\\}.$
(12)
This interaction has the enhanced symmetry $SO(3,1)\times SO(3,1)$. The only
term that breaks this enhanced symmetry is the fermion’s kinetic term, which
ties together a derivative and a gamma or sigma matrix of the spin Lorentz
group:
$i\int_{x}\chi^{\dagger}\overline{\sigma}^{\mu}\partial_{\mu}\chi=\int_{xy}E^{a}_{\mu}\delta_{a}^{\mu}\delta^{4}(x-y).$
(13)
However this term is a tadpole for the condensate $E^{a}_{\mu}$. As such, when
$E^{a}_{\mu}$ condenses, the field must be shifted
$E^{a}_{\mu}\to\tilde{E}^{a}_{\mu}+\delta^{a}_{\mu}\delta^{4}(x-y)$ to remove
this tadpole, and $\tilde{E}^{a}_{\mu}$ is the order parameter of the
$SO(3,1)\times SO(3,1)$ symmetry breaking. In the limit that
$\tilde{E}^{a}_{\mu}\to 0$, the effective action has this enhanced symmetry
(and the fermion has no kinetic energy).
A free fermion $\psi(x)$ transforms with two Lorentz symmetries. The first is
defined on the coordinates of space-time, with the generators
$L_{\mu\nu}=i(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu}).$ (14)
Under this symmetry $\psi$ transforms as a scalar. The second Lorentz symmetry
is defined with the generators
$S_{ab}=\frac{i}{2}(\gamma_{a}\gamma_{b}-\gamma_{b}\gamma_{a}),$ (15)
under which $\psi$ transforms in the $1/2$ (spinor) representation. Normally
we consider these to be two different representations of the same $SO(3,1)$
Lorentz symmetry. The SM Lagrangian is not symmetric under both groups
separately. We write Greek indices for the space-time Lorentz group, and Roman
indices for the spinor Lorentz group to indicate the difference. Since both
groups contain the Minkowski metric $\eta_{\mu\nu}$ and $\eta_{ab}$, we will
use this to raise and lower indices. We can define the mixed generators
$M_{\mu\nu}=L_{\mu\nu}+S_{ab}e^{a}_{\mu}e^{b}_{\nu};\qquad
N_{\mu\nu}=L_{\mu\nu}-S_{ab}e^{a}_{\mu}e^{b}_{\nu}$ (16)
where $e^{a}_{\mu}=\langle\tilde{E}^{a}_{\mu}\rangle\simeq\delta^{a}_{\mu}$.
The new operator $N_{\mu\nu}$ is the broken generator, and corresponds for a
massless fermion to local violations of being in a helicity eigenstate. A
plane wave could be a helicity eigenstate, but a localized state is not an
energy or momentum eigenstate, and therefore is also cannot be a helicity
eigenstate unless it is completely delocalized. Thus $e^{a}_{\mu}$ is the
order parameter of the $SO(3,1)\times SO(3,1)\to SO(3,1)$ symmetry breaking.
By Goldstone’s theorem, a vacuum expectation value for $\tilde{E}^{a}_{\mu}$
not only breaks this symmetry but also generates Goldstone bosons from the
broken symmetry generators. Here care must be taken because the number of
Goldstones is not the same as the number of broken generators, because the
broken symmetry is a space-time symmetry Goldstone:1961eq ; Goldstone:1962es ;
Low:2001bw .
The Goldstones carry a representation of the unbroken group $M_{\mu\nu}$. The
field $\tilde{E}^{a}_{\mu}$ however carries an index of both the original
groups. The propagating Goldstone is
$g_{\mu\nu}=\tilde{E}^{a}_{\mu}\tilde{E}^{b}_{\nu}\eta_{ab}$ (17)
which we identify as spin-2 graviton under $M_{\mu\nu}$. This should be
familiar from the Palatini formalism for quantizing gravity, if we identify
$\tilde{E}^{a}_{\mu}$ as the vierbein (tetrad).
The gravitational theory arising here does not conflict with the Weinberg-
Witten Theorem because of the presence of a physical background, and
consequently this emergent gravitational theory isn’t diffeomorphism invariant
Weinberg:1980kq . There are many ways to see this, but in particular, the
Lorentz symmetry is not exact in the gravitational theory, spatial variations
of $p_{F}$ lead to a spatially varying interaction strength (Eq.10), and the
emergent vierbein (Eq.9) is nonlocal.
From here one can almost directly follow the program of “Spinor Gravity”
hebecker_spinor_2003 ; wetterich_gravity_2003 , with the exception that due to
the Lorentz symmetry breaking, we have the metric $\eta_{\mu\nu}$ with which
to tie up spacetime indices, which gives rise to a spin connection which was
absent in “Spinor Gravity”. The existence of $\eta_{\mu\nu}$ implies more
invariants as well.
## VI Conclusions
We have shown that massive cosmological relics are not classical gasses. If
they have attractive interactions or are fermions, they instead are a super-
fluid. This implies that WIMP dark matter scenarios are inconsistent: WIMPs
cannot both be decoupled and localized for the age of the universe.
Cosmic background neutrinos must exist. They are a super-fluid, and their
self-interactions are a gravitational theory. These dynamics arise in the SM,
which is a renormalizable quantum field theory. We suggest that this may
actually be the gravity that we observe.
## VII Acknowledgements
We thank Bruce Campbell, Steve Carlip, Jessica De Haene, Francois Gelis,
Patrick Huber, Nemanja Kaloper, Alessio Notari, Thomas Schwetz, Steve Sekula,
Aleksi Vuorinen, Edward Witten, and Jure Zupan for useful comments.
## References
* [1] Kim Griest and David Seckel. Cosmic asymmetry, neutrinos and the sun. Nuclear Physics B, 283:681–705, 1987.
* [2] Mark Srednicki, Richard Watkins, and Keith A. Olive. Calculations of relic densities in the early universe. Nuclear Physics B, 310:693–713, December 1988.
* [3] Per Kraus and E. T Tomboulis. Photons and gravitons as goldstone bosons, and the cosmological constant. hep-th/0203221, March 2002. Phys.Rev. D66 (2002) 045015.
* [4] Hans C. Ohanian. Gravitons as goldstone bosons. Physical Review, 184:1305, 1969.
* [5] J. D. Bjorken. A dynamical origin for the electromagnetic field. Annals of Physics, 24:174–187, October 1963.
* [6] A. Hebecker and C. Wetterich. Spinor gravity. hep-th/0307109, July 2003. Phys.Lett. B574 (2003) 269-275.
* [7] C. Wetterich. Gravity from spinors. hep-th/0307145, July 2003. Phys.Rev. D70 (2004) 105004.
* [8] Steven Weinberg and Edward Witten. Limits on Massless Particles. Phys. Lett., B96:59, 1980.
* [9] A. D. Sakharov. Vacuum quantum fluctuations in curved space and the theory of gravitation. Sov. Phys. Dokl., 12:1040–1041, 1968.
* [10] D. G Caldi and Alan Chodos. Cosmological neutrino condensates. hep-ph/9903416, March 1999.
* [11] D. V. Efremov, M. S. Mar’enko, M. A. Baranov, and M. Yu Kagan. Superfluid transition temperature in a fermi gas with repulsion. higher orders perturbation theory corrections. SOV.PHYS.JETP, 90:861, 2000.
* [12] M. Yu. Kagan and A. V. Chubukov. Possibility of a superfluid transition in a slightly nonideal fermi gas with repulsion. Pis’ma Zh. Eksp. Teor. Fiz., 47(10):525–528, May 1988.
* [13] W. Kohn and J. M. Luttinger. New mechanism for superconductivity. Phys. Rev. Lett., 15(12):524–526, Sep 1965.
* [14] J. Goldstone. Field Theories with Superconductor Solutions. Nuovo Cim., 19:154–164, 1961.
* [15] Jeffrey Goldstone, Abdus Salam, and Steven Weinberg. Broken Symmetries. Phys. Rev., 127:965–970, 1962.
* [16] Ian Low and Aneesh V. Manohar. Spontaneously broken spacetime symmetries and Goldstone’s theorem. Phys. Rev. Lett., 88:101602, 2002.
|
arxiv-papers
| 2008-12-15T15:37:28 |
2024-09-04T02:48:59.401147
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bob McElrath",
"submitter": "Bob McElrath",
"url": "https://arxiv.org/abs/0812.2696"
}
|
0812.2713
|
# Applying Bayesian Neural Network to Determine Neutrino Incoming Direction in
Reactor Neutrino Experiments and Supernova Explosion Location by Scintillator
Detectors
Weiwei Xua Ye Xua Corresponding author, e-mail address: xuye76@nankai.edu.cn
Yixiong Menga Bin Wua
###### Abstract
In the paper, it is discussed by using Monte-Carlo simulation that the
Bayesian Neural Network(BNN) is applied to determine neutrino incoming
direction in reactor neutrino experiments and supernova explosion location by
scintillator detectors. As a result, compared to the method in Ref.[1], the
uncertainty on the measurement of the neutrino direction using BNN is
significantly improved. The uncertainty on the measurement of the reactor
neutrino direction is about 1.0∘ at the 68.3% C.L., and the one in the case of
supernova neutrino is about 0.6∘ at the 68.3% C.L.. Compared to the method in
Ref.[1], the uncertainty attainable by using BNN reduces by a factor of about
20. And compared to the Super-Kamiokande experiment(SK), it reduces by a
factor of about 8.
###### keywords:
Bayesian neural network, neutrino incoming direction, reactor neutrino,
supernova neutrino
aDepartment of Physics, Nankai University, Tianjin 300071, The People’s
Republic of China
PACS numbers: 07.05.Mh, 29.85.Fj, 14.60.Pq, 95.85.Ry
## 1 Introduction
The location of a $\nu$ source is very important to study galactic supernova
explosion. The determination of neutrino incoming direction can be used to
locate a supernova, especially, if the supernova is not optically visible. The
method based on the inverse $\beta$ decay, $\bar{\nu_{e}}+p\rightarrow
e^{+}+n$, has been discussed in the Ref.[1]. The method can be applied to
determine a reactor neutrino direction and a supernova neutrino direction. But
the uncertainty of location of the $\nu$ source attainable by using the method
is not small enough and almost 2 times as large as that in the Super-
Kamiokande experiment(SK). So we try to apply the Bayesian neural
network(BNN)[2] to locate $\nu$ sources in order to decrease the uncertainty
on the measurement of the neutrino incoming direction.
BNN is an algorithm of the neural networks trained by Bayesian statistics. It
is not only a non-linear function as neural networks, but also controls model
complexity. So its flexibility makes it possible to discover more general
relationships in data than the traditional statistical methods and its
preferring simple models make it possible to solve the over-fitting problem
better than the general neural networks[3]. BNN has been used to particle
identification and event reconstruction in the experiments of the high energy
physics, such as Ref.[4, 5, 6, 7].
In this paper, it is discussed by using Monte-Carlo simulation that the method
of BNN is applied to determine neutrino incoming direction in reactor neutrino
experiments and supernova explosion location by scintillator detectors.
## 2 Regression with BNN[2, 6]
The idea of BNN is to regard the process of training a neural network as a
Bayesian inference. Bayes’ theorem is used to assign a posterior density to
each point, $\bar{\theta}$, in the parameter space of the neural networks.
Each point $\bar{\theta}$ denotes a neural network. In the method of BNN, one
performs a weighted average over all points in the parameter space of the
neural network, that is, all neural networks. The methods make use of training
data {($x_{1}$,$t_{1}$), ($x_{2}$,$t_{2}$),…,($x_{n}$,$t_{n}$)}, where $t_{i}$
is the known target value associated with data $x_{i}$, which has $P$
components if there are $P$ input values in the regression. That is the set of
data $x=$($x_{1}$,$x_{2}$,…,$x_{n}$) which corresponds to the set of target
$t=$($t_{1}$,$t_{2}$,…,$t_{n}$). The posterior density assigned to the point
$\bar{\theta}$, that is, to a neural network, is given by Bayes’ theorem
$p\left(\bar{\theta}\mid
x,t\right)=\frac{\mathit{p\left(x,t\mid\bar{\theta}\right)p\left(\bar{\theta}\right)}}{p\left(x,t\right)}=\frac{p\left(t\mid
x,\bar{\theta}\right)p\left(x\mid\bar{\theta}\right)p\left(\bar{\theta}\right)}{p\left(t\mid
x\right)p\left(x\right)}=\frac{\mathit{p\left(t\mid
x,\bar{\theta}\right)p\left(\bar{\theta}\right)}}{p\left(t\mid x\right)}$ (1)
where data $x$ do not depend on $\bar{\theta}$, so
$p\left(x\mid\theta\right)=p\left(x\right)$. We need the likelihood
$p\left(t\mid x,\bar{\theta}\right)$ and the prior density
$p\left(\bar{\theta}\right)$, in order to assign the posterior density
$p\left(\bar{\theta}\mid x,t\right)$to a neural network defined by the point
$\bar{\theta}$. $p\left(t\mid x\right)$ is called evidence and plays the role
of a normalizing constant, so we ignore the evidence. That is,
$Posterior\propto Likelihood\times Prior$ (2)
We consider a class of neural networks defined by the function
$y\left(x,\bar{\theta}\right)=b+{\textstyle{\displaystyle\sum_{j=1}^{H}v_{j}sin\left(a_{j}+\sum_{i=1}^{P}u_{ij}x_{i}\right)}}$
(3)
The neural networks have $P$ inputs, a single hidden layer of $H$ hidden nodes
and one output. In the particular BNN described here, each neural network has
the same structure. The parameter $u_{ij}$ and $v_{j}$ are called the weights
and $a_{j}$ and $b$ are called the biases. Both sets of parameters are
generally referred to collectively as the weights of the BNN, $\bar{\theta}$.
$y\left(x,\bar{\theta}\right)$ is the predicted target value. We assume that
the noise on target values can be modeled by the Gaussian distribution. So the
likelihood of $n$ training events is
$p\left(t\mid
x,\bar{\theta}\right)=\prod_{i=1}^{n}exp[-((t_{i}-y\left(x_{i},\bar{\theta}\right))^{2}/2\sigma^{2}]=exp[-\sum_{i=1}^{n}(t_{i}-y\left(x_{i},\bar{\theta}\right)/2\sigma^{2})]$
(4)
where $t_{i}$ is the target value, and $\sigma$ is the standard deviation of
the noise. It has been assumed that the events are independent with each
other. Then, the likelihood of the predicted target value is computed by Eq.
(4).
We get the likelihood, meanwhile we need the prior to compute the posterior
density. But the choice of prior is not obvious. However, experience suggests
a reasonable class is the priors of Gaussian class centered at zero, which
prefers smaller rather than larger weights, because smaller weights yield
smoother fits to data . In the paper, a Gaussian prior is specified for each
weight using the BNN package of Radford Neal111R. M. Neal, _Software for
Flexible Bayesian Modeling and Markov Chain Sampling_ ,
http://www.cs.utoronto.ca/~radford/fbm.software.html. However, the variance
for weights belonging to a given group(either input-to-hidden
weights($u_{ij}$), hidden -biases($a_{j}$), hidden-to-output weights($v_{j}$)
or output-biases($b$)) is chosen to be the same: $\sigma_{u}^{2}$,
$\sigma_{a}^{2}$, $\sigma_{v}^{2}$, $\sigma_{b}^{2}$, respectively. However,
since we don’t know, a priori, what these variances should be, their values
are allowed to vary over a large range, while favoring small variances. This
is done by assigning each variance a gamma prior
$p\left(z\right)=\left(\frac{\alpha}{\mu}\right)^{\alpha}\frac{z^{\alpha-1}e^{-z\frac{\alpha}{\mu}}}{\Gamma\left(\alpha\right)}$
(5)
where $z=\sigma^{-2}$, and with the mean $\mu$ and shape parameter $\alpha$
set to some fixed plausible values. The gamma prior is referred to as a
hyperprior and the parameter of the hyperprior is called a hyperparameter.
Then, the posterior density, $p\left(\bar{\theta}\mid x,t\right)$, is gotten
according to Eqs. (2),(4) and the prior of Gaussian distribution. Given an
event with data $x^{\prime}$, an estimate of the target value is given by the
weighted average
$\bar{y}\left(x^{\prime}|x,t\right)=\int
y\left(x^{\prime},\bar{\theta}\right)p\left(\bar{\theta}\mid
x,t\right)d\bar{\theta}$ (6)
Currently, the only way to perform the high dimensional integral in Eq. (6) is
to sample the density $p\left(\bar{\theta}\mid x,t\right)$ with the Markov
Chain Monte Carlo (MCMC) method[2, 8, 9, 10]. In the MCMC method, one steps
through the $\bar{\theta}$ parameter space in such a way that points are
visited with a probability proportional to the posterior density,
$p\left(\bar{\theta}\mid x,t\right)$. Points where $p\left(\bar{\theta}\mid
x,t\right)$ is large will be visited more often than points where
$p\left(\bar{\theta}\mid x,t\right)$ is small.
Eq. (6) approximates the integral using the average
$\bar{y}\left(x^{\prime}\mid
x,t\right)\approx\frac{1}{L}\sum_{i=1}^{L}y\left(x^{\prime},\bar{\theta_{i}}\right)$
(7)
where $L$ is the number of points $\bar{\theta}$ sampled from
$p\left(\bar{\theta}\mid x,t\right)$. Each point $\bar{\theta}$ corresponds to
a different neural network with the same structure. So the average is an
average over neural networks, and is closer to the real value of
$\bar{y}\left(x^{\prime}\mid x,t\right)$, when $L$ is sufficiently large.
## 3 Toy Detector and Simulation[5]
In the paper, a toy detector is designed to simulate the central detector in
the reactor neutrino experiment, such as Daya Bay experiment[11] and Double
CHOOZ experiment[12], with CERN GEANT4 package[13]. The toy detector consists
of three regions, and they are the Gd-doped liquid scintillator(Gd-LS from now
on), the normal liquid scintillator(LS from now on) and the oil buffer,
respectively. The toy detector of cylindrical shape like the detector modules
of Daya Bay experiment and Double CHOOZ experiment is designed in the paper.
The diameter of the Gd-LS region is 2.4 meter, and its height is 2.6 meter.
The thickness of the LS region is 0.35 meter, and the thickness of the oil
part is 0.40 meter. In the paper, the Gd-LS and LS are the same as the
scintillator adopted by the proposal of the CHOOZ experiment[14]. The 8-inch
photomultiplier tubes (PMT from now on) are mounted on the inside the oil
region of the detector. A total of 366 PMTs are arranged in 8 rings of 30 PMTs
on the lateral surface of the oil region, and in 5 rings of 24, 18, 12, 6, 3
PMTs on the top and bottom caps.
The response of the neutrino and background events deposited in the toy
detector is simulated with GEANT4. Although the physical properties of the
scintillator and the oil (their optical attenuation length, refractive index
and so on) are wave-length dependent, only averages[14] (such as the optical
attenuation length of Gd-LS with a uniform value is 8 meter and the one of LS
is 20 meter) are used in the detector simulation. The program couldn’t
simulate the real detector response, but this won’t affect the result of the
comparison between the BNN and the method in the Ref.[1].
## 4 Event Reconstruction[5]
The task of the event reconstruction in the reactor neutrino experiments is to
reconstruct the energy and the vertex of a signal. The maximum likelihood
method (MLD) is a standard algorithm of the event reconstruction in the
reactor neutrino experiments. The likelihood is defined as the joint Poisson
probability of observing a measured distribution of photoelectrons over the
all PMTs for given ($E,\overrightarrow{x}$) coordinates in the detector. The
Ref.[15] for the work of the CHOOZ experiment shows the method of the
reconstruction in detail.
In the paper, the event reconstruction with the MLD are performed in the
similar way with the CHOOZ experiment[15], but the detector is different from
the detector of the CHOOZ experiment, so compared to Ref.[15], there are some
different points in the paper:
(1) The detector in the paper consists of three regions, so the path length
from a signal vertex to the PMTs consist of three parts, and they are the path
length in Gd-LS region, the one in LS region, and the one in oil region,
respectively.
(2) Considered that not all PMTs in the detector can receive photoelectrons
when a electron is deposited in the detector, the $\chi^{2}$ equation is
modified in the paper and different from the one in the CHOOZ experiment, that
is, $\chi^{2}=\sum_{N_{j}=0}\bar{N_{j}}+\sum_{N_{j}\neq
0}(\bar{N}_{j}-N_{j}+N_{j}log(\frac{N_{j}}{\bar{N_{j}}}))$, where $N_{j}$ is
the number of photoelectrons received by the j-th PMT and $\bar{N_{j}}$ is the
expected one for the j-th PMT[15].
(3) $c_{E}\times N_{total}$ and the coordinates of the charge center of
gravity for the all visible photoelectrons from a signal are regarded as the
starting values for the fit parameters($E,\overrightarrow{x}$), where
$N_{total}$ is the total numbers of the visible photoelectrons from a signal
and $c_{E}$ is the proportionality constant of the energy $E$, that is,
$E=c_{E}\times N_{total}$. $c_{E}$ is obtained through fitting $N_{total}$’s
of the 1 MeV electron events, and is $\frac{1}{235/MeV}$ in the paper.
## 5 Monte-Carlo Sample
### 5.1 Monte-Carlo Sample for Reactor Neutrinos
According to the anti-neutrino interaction in the detector of the reactor
neutrino experiments[16], the neutrino events from the random direction and
the particular direction, (0.433,0.75,-0.5), are generated uniformly
throughout GD-LS region of the toy detector. Fig. 1 shows the four important
physics quantities of the Monte-Carlo reactor neutrino events and they are
$E_{e^{+}},E_{n},$$\Delta$$t_{e^{+}n}$$,d_{e^{+}n}$, respectively. The
selections of the neutrino events are as follows:
(1) Positron energy: 1.3 MeV < $E_{e^{+}}$ < 8 MeV;
(2) Neutron energy: 6 MeV < $E_{n}$ < 10 MeV;
(3) Neutron delay: 2 $\mu$s < $\Delta$$t_{e^{+}n}$ < 100 $\mu$s;
(4) Relative positron-neutron distance: $d_{e^{+}n}$ < 100 cm.
10000 events from the random directions and 5000 events from (0.433,0.75,-0.5)
are selected according to the above criteria, respectively. The events from
the random direction are regarded as the training sample of BNN, and the
events from (0.433,0.75,-0.5) are regarded as the test sample of BNN.
### 5.2 Monte-Carlo sample for Supernova Neutrinos
The neutrino events for the random direction and the particular direction,
(0.354,0.612,-0.707), are generated uniformly throughout GD-LS region of a
liquid scintillator detector with the same geometry and the same target as the
toy detector in the sec. 3, according to the following supernova
$\bar{\nu_{e}}$ energy distribution[1, 17]:
$\frac{dN}{dE}=C\frac{E^{2}}{1+e^{E/T}}$ (8)
with $T=3.3MeV$ and the supernova is considered to be at $10Kpc$. The number
of the fixed direction neutrino events, for a supernova at $10Kpc$, could be
detected in a liquid scintillator experiment with mass equal to that of SK[1].
The events from the random direction are regarded as the training sample of
BNN, and the events from (0.354,0.612,-0.707) are regarded as the test sample
of BNN. Fig. 2 shows the four important physics quantities of the Monte-Carlo
supernova neutrino events and they are
$E_{e^{+}},E_{n},$$\Delta$$t_{e^{+}n}$$,d_{e^{+}n}$, respectively.
## 6 Location of the neutrino source using the method in the Ref.[1]
The inverse-$\beta$ decay can be used to locate the neutrino source in
scintillator detector experiments. The method is based on the neutron boost in
the forward direction. And neutron retains a memory of the neutrino source
direction. The unit vector $\hat{X}_{e^{+}n}$, having its origin at the
positron reconstructed position and pointing to the captured neutron position,
is defined for each neutrino event. The distribution of the projection of this
vector along the known neutrino direction is forward peaked , but its r.m.s.
value is not far from that of a flat distribution($\sigma_{flat}=1/\sqrt{3}$).
$\vec{p}$ is defined as the average of vectors $\hat{X}_{e^{+}n}$, that is
$\vec{p}=\frac{1}{N}\sum\hat{X}_{e^{+}n}$ (9)
The measured neutrino direction is the direction of $\vec{p}$.
The neutrino direction lies along the z axis is assumed to evaluate the
uncertainty in the direction of $\vec{p}$. From the central limit theorem
$\vec{p}$ follows that the distribution of the three components is Gaussian
with $\sigma=1/\sqrt{3N}$ centered at (0,0,$|\vec{p}|$). Therefore, the
uncertainty on the measurement of the neutrino direction can be given as the
cone around $\vec{p}$ which contains 68.3% of the integral of this
distribution.
## 7 Location of the neutrino source using BNN
In the paper, the x,y,z components of the neutrino incoming direction are
predicted by the three BNNs, respectively. The BNNs have the input layer of 6
inputs, the single hidden layer of 15 nodes and the output layer of a output.
Here we will explain the case of predicting the x component of the neutrino
incoming direction in detail:
(1) The data format for the training sample is
$d_{i},f_{i},E_{e^{+}},E_{n},$$\Delta$$t_{e^{+}n}$$,d_{e^{+}n},t_{i}$ (i=x),
where $d_{i}$ is the difference of $v_{i}$ and $n_{i}$ (i=x). $v_{i}$(i=x) is
the x components of the $\hat{X}_{e^{+}n}$ in the section 6. $n_{i}$(i=x) is
the x component of the known neutrino incoming direction ($\vec{n}$).
$f_{i}$(i=x) is the x component of the reconstructed positron position.
$d_{i},f_{i},E_{e^{+}},E_{n},$$\Delta$$t_{e^{+}n}$$,d_{e^{+}n}$ are used as
inputs to a BNN, and $t_{i}$ is the known target. The target can be obtained
by Eq. 10. That is
$t_{i}=\frac{1}{1+exp(0.5v_{i}/n_{i})}(i=x).$ (10)
where
(2) The inputs of the test sample are similar with that of the train sample,
but the $d_{i}$(i=x) is different from that of the training sample. The
$\vec{p}$ obtained by the method in the section 6 is substituted for the known
neutrino incoming direction in the process of computing $d_{i}$(i=x). The
$tp_{i}$(i=x) is the output of the BNN, that is, it is the predicted value
using the BNN. We make use of the $tp_{i}$ value to compute the x component of
neutrino incoming direction via the following equation(In fact, Eq. 11 is the
inverse-function of Eq. 10.):
$m_{i}=\frac{0.5v_{i}}{ln(1/tp_{i}-1)}(i=x),$ (11)
where $v_{i}$(i=x) is the x component of the $\hat{X}_{e^{+}n}$. $m_{i}$(i=x)
is just the x component of the direction vector ($\vec{m}$) predicted by the
BNN.
A Markov chain of neural networks is generated using the BNN package of
Radford Neal, with the training sample, in the process of predicting the x
component of neutrino incoming direction by using the BNN. One thousand
iterations, of twenty MCMC steps each, are used in the paper. The neural
network parameters are stored after each iteration, since the correlation
between adjacent steps is very high. That is, the points in neural network
parameter space are saved to lessen the correlation after twenty steps. It is
also necessary to discard the initial part of the Markov chain because the
correlation between the initial point of the chain and the points of the part
is very high. The initial three hundred iterations are discarded in the paper.
Certainly, the y,z components of the $\vec{m}$ are obtained in the same
method, if only i=y,z, respectively. Here $\vec{L}$ is defined as the unit
vector of the $\vec{m}$ predicted by the BNNs for each event in the test
sample. We can also define the direction $\vec{q}$ as the average of the unit
direction vectors predicted by the BNNs in the same way as the section 6. That
is
$\vec{q}=\frac{1}{N}\sum\vec{L}.$ (12)
The $\vec{q}$ is just the neutrino incoming direction predicted by the BNNs.
The uncertainty in this value is evaluated in the same method as the section
6. We can know the r.m.s. value of the distribution of the projection of the
unit direction vectors predicted by the BNNs in the same method as the
Ref.[1]. From the central limit theorem $\vec{q}$ follows that the
distributions of its three components are Gaussian with
$\sigma=r.m.s./\sqrt{N}$ centered at (0,0,$|\vec{q}|$). Therefore, the
uncertainty on the measurement of the neutrino direction can be given as the
cone around $\vec{q}$ which contains 68.3% of the integral of this
distribution.
## 8 Results
Fig. 3 shows the distributions of the projections of the $\hat{X}_{e^{+}n}$ in
the sec. 6 and the $\vec{L}$ predicted by the method of BNN along the reactor
neutrino incoming direction. The r.m.s. attainable by using BNN is only about
0.41, and less than that attainable by using the method in the Ref.[1]. The
results of the determination of the reactor neutrino incoming direction using
the method in the Ref.[1] and the method of BNN are shown in Table 1. The
uncertainty attainable by using the method in the Ref.[1] is 21.1∘,and the one
attainable by using BNN is 1.0∘. Fig. 4 shows the distributions of the
projections of the $\hat{X}_{e^{+}n}$ in the sec. 6 and the $\vec{L}$
predicted by the method of BNN along the supernova neutrino incoming
direction. The r.m.s. attainable by using BNN is also about 0.35. The results
of the determination of the supernova neutrino incoming direction using the
method in the Ref.[1] and the method of BNN are shown in Table 2. The
uncertainty attainable by using the method in the Ref.[1] is 10.7∘, and the
one attainable by using BNN is 0.6∘.
So compared to the method in Ref.[1], the uncertainty attainable by using BNN
is significantly improved and reduces by a factor of about 20 (21∘ compared to
1∘ in the case of reactor neutrinos and 11∘ compared to 0.6∘ in the case of
supernova neutrinos). And compared to SK, it reduces by a factor of about 8
(5∘ compared to 0.6∘). Why such good results can be obtained with BNN? First,
neutrino directions obtained with the method in the Ref.[1] are used as inputs
to BNN, that is such good results obtained with BNN is on the base of the
results of the method in the Ref.[1]; Second, BNN can extract some unknown
information from its inputs and discover more general relationships in data
than traditional statistical methods; Third, the over-fitting problem can be
solved by using Bayesian methods to control model complexity. So results
obtained with BNN can be much better than that of the method in the Ref.[1].
In a word, the method of BNN can be well applied to determine neutrino
incoming direction in reactor neutrino experiments and supernova explosion
location by scintillator detectors.
## 9 Acknowledgements
This work is supported by the National Natural Science Foundation of China
(NSFC) under the contract No. 10605014.
## References
* [1] M. Apollonio et al., Physical Review D61, 012001 (1999)
* [2] R. M. Neal, _Bayesian Learning of Neural Networks_. New York: Springer-Verlag, 1996
* [3] R. Beale and T. Jackson, _Neural Computing: An Introduction_ , New York: Adam Hilger, 1991
* [4] Y. Xu, J. Hou and K. E. Zhu, Chinese Physics C (HEP&NP), 32(3), 201-204 (2008)
* [5] Y. Xu, W. W. Xu, Y. X. Meng, and W. Xu, Nuclear Instruments and Methods in Physics Rearch A592, 451-455 (2008), arXiv: 0712.4042
* [6] P. C. Bhat and H. B. Prosper _Beyesian Neural Networks_. In: L. Lyons and M. K. Unel ed. _Proceedings of Statistical Problems in Particle Physics, Astrophysics and Cosmology, Oxford, UK 12-15, September 2005_. London: Imperial college Press. 2006. 151-154
* [7] Y. Xu, Y. X. meng, and W. W. Xu, Journal of Instrumentation 3, P08005 (2008), arXiv: 0808.0240
* [8] S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth, Physics Letters, B195, 216-222 (1987)
* [9] M. Creutz and A. Gocksch, Physical Review Letters, 1989 63, 9-12
* [10] P. B. Mackenzie, Physics Letters, B226, 369-371 (1989)
* [11] Daya Bay Collaboration, _Daya Bay Proposal: A Precision Measurement of the Neutrino Mixing Angle $\theta_{13}$ Using Reactor Antineutrino At Daya Bay_, arXiv: hep-ex/0701029
* [12] F. Ardellier et al., _Double Chooz: A Search for the Neutrino Mixing Angle $\theta_{13}$_, arXiv: hep-ex/0606025
* [13] Geant4 Reference Manual, vers. 9.0 (2007)
* [14] _The CHOOZ Experiment Proposal (1993)_ , available at the WWW site http://duphy4.physics.drexel.edu/chooz_pub/
* [15] M. Apollonio et al., European Physical Journal C27, 331 (2003)
* [16] Y. X. Sun, J. Cao, and K. J. Luk, et al., HEP & NP, 29(6), 543-548 (2005)
* [17] S. A. Bludman and P. J. Schinder, Astrophysical Journal 326, 265 (1988)
Table 1: Measurement of reactor neutrino direction | |$\vec{p}$| or |$\vec{q}$| | $\phi$ | $\theta$ | uncertainty
---|---|---|---|---
known neutrino incoming direction | – | 60∘ | 120∘ | –
Direction determined by the method in Ref.[1] | 0.033 | 42.5∘ | 111.4∘ | 21.1∘
Direction determined by BNN | 0.708 | 56.7∘ | 118.9∘ | 1.0∘
Table 2: Measurement of supernova neutrino direction | |$\vec{p}$| or |$\vec{q}$| | $\phi$ | $\theta$ | uncertainty
---|---|---|---|---
known neutrino incoming direction | – | 60∘ | 135∘ | –
Direction determined by the method in Ref.[1] | 0.066 | 61.0∘ | 149.2∘ | 10.7∘
Direction determined by BNN | 0.727 | 55.8∘ | 138.5∘ | 0.6∘
Figure 1: The reactor neutrino events for the Monte-Carlo simulation of the
toy detector are uniformly generated throughout Gd-LS region. (a) is the
distribution of the positron energy; (b) is the distribution of the energy of
the neutron captured by Gd; (c) is the distribution of the distance between
the positron and neutron positions; (d) is the distribution of the delay time
of the neutron signal. Figure 2: The supernova neutrino events for the Monte-
Carlo simulation of a liquid scintillator detector with the same geometry and
the same target as the toy detector in the sec. 3 are uniformly generated
throughout Gd-LS region. (a) is the distribution of the positron energy; (b)
is the distribution of the energy of the neutron captured by Gd; (c) is the
distribution of the distance between the positron and neutron positions; (d)
is the distribution of the delay time of the neutron signal. Figure 3: The
distributions of the projections of the $\hat{X}_{e^{+}n}$ in the sec. 6 and
the $\vec{L}$ predicted by the method of BNN along the reactor neutrino
incoming direction.
Figure 4: The distributions of the projections of the $\hat{X}_{e^{+}n}$ in
the sec. 6 and the $\vec{L}$ predicted by the method of BNN along the
supernova neutrino incoming direction.
|
arxiv-papers
| 2008-12-15T02:30:09 |
2024-09-04T02:48:59.407459
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Weiwei Xu, Ye Xu, Yixiong Meng, Bin Wu",
"submitter": "Ye Xu",
"url": "https://arxiv.org/abs/0812.2713"
}
|
0812.2840
|
# Comprehensive Characterization of InGaAs/InP Avalanche Photodiodes at 1550
nm with an Active Quenching ASIC
Jun Zhang, Rob Thew, Jean-Daniel Gautier, Nicolas Gisin, and Hugo Zbinden
Manuscript received. This work was supported by the Swiss NCCR Quantum
Photonics and the Swiss CTI.J. Zhang, R. Thew, J.-D. Gautier, N. Gisin, and H.
Zbinden are with the Group of Applied Physics, University of Geneva, 1211
Geneva 4, Switzerland e-mail: (Jun.Zhang@unige.ch).
###### Abstract
We present an active quenching application specific integrated circuit (ASIC),
for use in conjunction with InGaAs/InP avalanche photodiodes (APDs), for 1550
nm single-photon detection. To evaluate its performance, we first compare its
operation with that of standard quenching electronics. We then test 4
InGaAs/InP APDs using the ASIC, operating both in the free-running and gated
modes, to study more general behavior. We investigate not only the standard
parameters under different working conditions but also parameters such as
charge persistence and quenching time. We also use the multiple trapping model
to account for the afterpulsing behavior in the gated mode, and further
propose a model to take account of the afterpulsing effects in the free-
running mode. Our results clearly indicate that the performance of APDs with
an on-chip quenching circuit significantly surpasses the conventional
quenching electronics, and makes them suitable for practical applications,
e.g., quantum cryptography.
###### Index Terms:
Avalanche photodiodes (APDs), SPAD, single-photon detection, telecom
wavelengths, ASIC, quantum cryptography.
## I Introduction
Single-photon detectors are the key components in numerous photonics-related
applications such as quantum cryptography [1], optical time domain
reflectometry [2, 3] and integrated circuit testing [4]. We can classify
single-photon detection into four classes: photomultiplier tubes [5];
semiconductor APDs [6, 7]; superconducting detectors [8]; and novel proposals
such as using a single-electron transistor consisting of a semiconductor
quantum dot [9]. In the telecommunication regime (1550 nm), InGaAs/InP APDs
are currently the best choice for practical applications such as quantum
cryptography [1] due to their favorable characteristics such as cost, size and
robust operation with only thermo-electric cooling required.
To detect single photons, APDs must work in the so-called Geiger mode in which
an inverse bias voltage ($V_{bv}$), exceeding the breakdown voltage
($V_{br}$), is applied, such that even a single photoexcited carrier
(electron-hole pair) can create a persistent avalanche and a subsequent
macroscopic current pulse due to the process of impact ionization. After the
avalanche, a passive or active quenching circuit [6], is used to reduce
$V_{bv}$ down to below $V_{br}$, output a synchronized pulse and reset the APD
for detecting the next photon.
InGaAs/InP APDs are currently fabricated with separate absorption, charge and
multiplication layers [7] to ensure the lattice matching and preserve a low
electric field in the InGaAs absorption layer with a narrower bandgap
($E_{g}=0.75$ eV for In0.53Ga0.47As), minimizing the induced leakage currents,
while a high electric field in the InP multiplication layer, enhancing the
impact ionization effect. The middle charge layer can efficiently control the
electric field profiles of the absorption and multiplication layers. The
parameters of APDs are affected by many factors such as the crystalline
quality of semiconductor device, imperfections of design and fabrication,
quenching circuit, and operational conditions. Therefore, actual performance
of these APDs is always compromised and optimized for different applications.
In the past decade efforts have been made to characterize and further improve
APD performance on the single-photon level at 1550 nm [10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22]. Recently, integration of the quenching
electronics for InGaAs/InP APDs to an ASIC [23, 24] has been implemented. The
measured results on some key parameters of APDs demonstrate active quenching
ASICs can efficiently improve the noise-efficiency performance, and it has
been shown that these APDs can work in a free-running mode [24]. However, full
characterization of APDs with the ASIC is still necessary to better understand
the improvements they provided. In this paper, we fully test 4 InGaAs/InP APDs
at 1550 nm with an active quenching ASIC operating both in the free-running
and gated modes, and compare the improvements with conventional electronics.
## II The setup and the principle of the ASIC
The schematic setup for testing APDs is shown in Fig. 1. A digital delay pulse
generator (DG 535, Stanford Research Systems Inc.) provides synchronous
signals for the whole system. One of its periodic outputs drives a 1550 nm
laser diode (LD) to produce short optical pulses with $\sim 200$ ps FWHM. The
optical pulses are split into two parts by a 10/90 asymmetric fiber
beamsplitter (BS). 90% of the signal is monitored by a power meter (IQ 1100,
EXFO Co.) to regulate the precise variable attenuator (Var. ATT, IQ 3100, EXFO
Co.) in real-time and stabilize the intensity of the output from the
attenuator that goes to the pigtailed APD. The pins of the APD and ASIC are
soldered together on a small printed circuit board, while the body of APD is
fixed on the top of 4-layer thermoelectric cooler and actively stabilized with
a closed-loop control.
Figure 1: The experimental setup.
The schematic diagram of the ASIC, fabricated with a $0.8\,\mu$m complementary
metal oxide semiconductor (CMOS) process, is shown in the gray area of Fig. 1.
The amplitude of the gate signals from the complex programmable logic device
(CPLD) is first converted to the power supply voltage VDD ($+5$ V) of the
chip. Two pulses are then generated to control the PMOS and NMOS switches
respectively which have extremely fast rise and fall times. There is a very
short delay between the two control pulses to avoid the simultaneous
conduction of the two switches. The timing is such that at the beginning of
the gate the PMOS switch is closed while the NMOS switch is opened, to charge
the voltage at the quenching point (Q) up to VDD, and then the PMOS switch is
reopened. The total voltage difference between cathode and anode of the APD is
$V_{bv}=$VDD$+|V_{DC}|$, exceeding $V_{br}$ for Geiger mode operation. The
NMOS switch remains open until the end of the gate if no avalanche happens, or
until the active quenching after a triggered avalanche. During the avalanche
process, current across the APD rapidly increases and results in an increasing
voltage drop across the resistor $R_{q}$. The comparator and the following
circuit quickly detects the the voltage drop at Q and immediately informs the
buffer to close the NMOS switch to drop the voltage at Q to zero, and also
generate a synchronous detection output to the CPLD. Normally the detector
output maintains the high level until the falling edge of the next gate.
Actually, when a detection is registered the CPLD inserts a short reset pulse
after the gate, otherwise the CPLD does nothing. In the free-running mode, the
gates from the CPLD are not used and VDD is applied to the cathode of the APD
until an avalanche is excited. Further technical description about the ASIC
can be found in Ref. [23].
## III Performance tests of APDs
We have tested 4 commercial APDs: #1 (JDSU0131T1897); #2 (JDSU0122E1711); and
#3 (Epitaxx9951E9559) from JDSU; as well as #4 (PLI-DOI61910-040W059-076) from
Princeton Lightwave, Inc., and compared the different performance
characterizations of these APDs with the ASIC quenching system.
### III-A Integrated versus conventional quenching electronics
Firstly, we perform the key parameter measurements on the same (#3) APD using
the new (ASIC) and old (conventional non-integrated circuit) [11, 14]
quenching electronics under the same settings (T$=223$ K ). Fig. 3 shows the
comparison results for dark count ($P_{DC}$ per ns) vs single-photon detection
efficiency ($P_{DE}$) probabilities, afterpulse probability ($P_{AP}$) and
jitter, respectively. Using the double-gate method [14] (we discuss this in
the latter section) as shown in Fig. 2, these parameters can be related to:
$P_{DC}=\frac{C_{DC}}{f\tau_{AB}},\,\,P_{AP}=\frac{C_{AP}}{C_{DE}\tau_{CD}},\,\,P_{DE}=\frac{1}{\mu}\ln\frac{1-\frac{C_{DC}}{f}}{1-\frac{C_{DE}}{f}},$
(1)
where $C_{DC}$ ($C_{AP}$, $C_{DE}$) is the observed dark count (afterpulse,
detection) rate, $\tau_{AB}$ ($\tau_{CD}$) is the effective width of detection
(afterpulse) gate in ns and $\mu$ is the mean photon number per optical pulse
with repetition frequency of $f$. During the experiment, the conditions are
$f=10$ kHz, $\tau_{AB}=\tau_{CD}=100$ ns and $\mu=1$, and these are fixed
unless specifically mentioned in this paper.
Figure 2: The timing diagram for the afterpulse measurements using the double-
gate method.
The three curves manifestly exhibit the performance improvements provided by
the new quenching electronics. The improvement of a factor of 3 in the
$P_{DC}$-$P_{DE}$ performance for #3 APD shown in Fig. 3a is better than
expected. As we know, due to the ASIC the size of the electronics are greatly
decreased and the electronic cables and the lengths of wires are reduced. This
brings a lot of benefits such as superior signal integrity, minimized
parasitic capacitance and reducing fake avalanche signals due to signal
reflections or electronic noise. We also observe $P_{DC}$-$P_{DE}$ performance
improvements on other APDs, for instance, for #2 APD shown in Fig. 3a the
ratio is always about 1 (no improvement) when $P_{DE}<13\%$ and slowly
increases to about 2 when $P_{DE}\sim 25\%$. The $P_{DC}$-$P_{DE}$ performance
improvement ratio strongly depends on the APD devices and operational
conditions. Although the reasons of the significant improvement for #3 APD are
not clear yet, one possibility could be different gate heights and
discrimination approaches between the two quenching systems, as it is the
noise that is improved here, for a given excess bias voltage.
a)
b)
c)
Figure 3: a) Dark counts per ns ($P_{DC}$) versus detection efficiency
($P_{DE}$). b) Afterpulse probability per ns ($P_{AP}$) versus deadtime
($\tau_{d}$.) c) Time jitter versus $P_{DE}$.
We see, in Fig. 3b, a significant improvement in the $P_{AP}$ between the two
cases as expected. The $P_{AP}$ is generally proportional to the total number
of carriers generated during an avalanche and hence motivates small and
rapidly quenched avalanches. The results here clearly illustrate the circuit
response and quenching time of the new system for the avalanche discrimination
are faster than the old system. We will come back to this in more details in
the following sections.
Timing jitter (time resolution) is another key parameter. It is defined as the
temporal uncertainty of detection output for an avalanche with fixed arrival
time of photons. Time jitter strongly depends on device fabrication and
$P_{DE}$, corresponding to excess bias ($V_{eb}$) on the APD. Larger $V_{eb}$
can generate higher electric fields, which will shorten the trapping time of
the carriers in the absorption and grading layers, and also the buildup time
of avalanche, hence reducing the jitter. To measure this we use a time-
correlated single photon counting (TCSPC) board (SPC-130, Becker & Hickl GmbH)
with a time resolution of $6$ ps FWHM and minimum time slot of $815$ fs, to
measure the jitter properties. A synchronized signal from a pulse generator is
used as the TCSPC’s “stop” while the detection output signal is used as
“start”. The measured jitter is the overall jitter of the system, including
the jitter ($<60$ ps) and width ($\sim 200$ ps) of arrived optical pulses, the
APD’s intrinsic jitter owing to the stochastic process of carrier dynamics, as
well as from the associated electronics. The jitter performance is shown in
Fig. 3c and we only see a minor improvement when $P_{DE}<10~{}\%$. We expect
the electronic jitter to be slightly better as the ASIC can efficiently reduce
the propagation time and jitter of the signals. At higher $P_{DE}$ we don’t
observe the improvement and the negligible difference between the two cases is
due to contributions from the associated external electronics, e.g., CPLD and
discriminator that are used with the new system but not the old one. However,
varying degrees of improvement have been observed on other APDs even at higher
$P_{DE}$.
### III-B $P_{DC}$, $P_{DE}$ and thermal activation energy
In order to illustrate the universal improvements afforded by this new
quenching system, we use the new system operating in the gated mode to repeat
the measurement on different APDs and temperature settings, as shown in Fig. 4
and Fig. 5.
Figure 4: $P_{DC}$ versus $P_{DE}$ of 4 APDs.
Figure 5: $P_{DC}$ versus $P_{DE}$ of #2 APD at different T.
The $P_{DC}$-$P_{DE}$ behavior of #1, #2 and #4 APDs are very similar, with
$P_{DC}\sim 1.6\times 10^{-6}$ ns-1 and $P_{DE}=10\%$ at $223$ K, as shown in
Fig. 4, but much better than #3 APD. Fig. 5 shows the $P_{DC}$ behavior of #2
APD from $210$ K to $238$ K, and we see a reduction in $P_{DC}$ to $4.5\times
10^{-7}$ ns-1 for the same $P_{DE}$.
The origin of the dark counts is mainly due to the defect concentration in the
semiconductor device. There are two main mechanisms for the generation of dark
carriers: thermal generation; and tunneling generation. The thermal generation
means that a carrier is transferred from the valence band to the conduction
band either directly or via the midgap defects, owing to the thermal
excitation. Tunneling generation means that a carrier tunnels between the two
bands, or it is trapped by a defect first and then tunnels to the conduction
band, which is also called trap-assisted tunneling (TAT) [21, 22].
Combinations of the two mechanisms are normally not taken into account. The
simulations for $1.06\,\mu$m InGaAsP/InP APDs performed by Donnelly _et al._
[25] show that TAT in the multiplication layer dominates the $P_{DC}$ at low
temperature, while at high temperature the two mechanisms compete with each
other. Unfortunately, the dark count model for $1550$ nm InGaAs/InP APD is
more complicated than this, though one can investigate the so-called thermal
activation energy ($E_{a}$) to identify the dominant mechanism [19, 20, 21].
Theoretically, the relationship between $P_{DC}$, $E_{a}$ and temperature (T)
can be expressed as [20]
$P_{DC}\propto T^{2}e^{-\frac{E_{a}(T)}{kT}},$ (2)
where $k$ is the Boltzmann constant and $E_{a}$(T) is a function of
temperature with slow variations. In Fig. 6, four curves of
$log(P_{DC}/T^{2})$ versus $1/k$T for #1 APD with different $V_{eb}$ values
are plotted. We evaluate the difference of $E_{a}$ values for two small
temperature ranges ($216$ K $\sim 223$ K and $233$ K $\sim 238$ K). The
fitting values are displayed in Fig. 6. The results clearly show that
generally higher temperatures induce larger $E_{a}$ values and suggest that
the thermal generation mechanism around $238$ K dominates $P_{DC}$ while the
TAT mechanism is more significant around $216$ K, see also ref. [26].
Figure 6: Plot of $P_{DC}/T^{2}$ as a function of $1/k$T for #1 APD.
### III-C Afterpulsing
During an avalanche process, due to a photon detection, dark count effects, or
afterpulsing itself, a carrier can be trapped by a defect in the
multiplication layer. This carrier may excite another avalanche - an
afterpulse, during subsequent gates. This process severely limits the APD
performance for high frequency operation due to the need to apply long,
typically $\sim$ 10 $\mu$s, deadtimes where the APD is inactive. There are two
methods to measure the $P_{AP}$ behavior. The first approach measures the
total noise behavior as a function of $\tau_{d}$. When $\tau_{d}$ is large
enough, say, $100~{}\mu$s, the $P_{AP}$ is negligible and the measured noise
is primarily due to dark counts. After subtracting $P_{DC}$, the quantity of
noise left can be attributed to afterpulsing. This method was used in Ref [24]
but, while straightforward, generally overestimates $P_{AP}$.
The other approach, the double-gate method [14], as used in our setup is
illustrated in Fig. 2. If there is a click during the detection (AB) gate, the
CPLD will also generate an afterpulse (CD) gate after AB’s reset pulse with a
delay of $\tau_{d}$ to the falling edge of the AB gate. This corresponds to
the deadtime. The CPLD also generates a reset pulse for the CD gate only when
an afterpulse detection is registered during this gate. This method directly
measures $P_{AP}$.
Assuming a Poisson distribution, $P_{AP}$ can be expressed as
$P_{AP}=(1-e^{-R_{AP}(\tau_{d})\eta_{av}\tau_{CD}})/\tau_{CD},$ (3)
where $R_{AP}(\tau_{d})$ is the detrapping rate at time $\tau_{d}$ and
$\eta_{av}$ is the avalanche probability. We use a multiple trapping model
(multiple detrapping times) to describe $R_{AP}(\tau_{d})$ [19, 20],
$\displaystyle R_{AP}(\tau_{d})=\sum_{i}\frac{N_{i}}{\Delta
t_{i}}e^{-\tau_{d}/\Delta t_{i}},$ (4)
where $N_{i}$ is the number of trapped carriers at the end of the detection
gate with a detrapping time constant of $\Delta t_{i}$. There are single
trapping models that use a single detrapping time constant $\Delta t$ but in
many cases this does not correspond to the measured results. The multiple
trapping model effectively fits the measured results but some physical
questions remain, e.g., why only 2 detrapping time parameters are needed for
modeling one APD while 3 parameters are required for another _etc_. In fact,
quantitive description and modeling for $P_{AP}$ behavior is still an
intractable problem.
Figure 7: $P_{AP}$ versus $\tau_{d}$.“end”, “mid” and “front” mean that
incident photons appear in the end, middle and front of AB gates,
respectively. The minimum $\tau_{d}$ is always $800$ ns in Fig. 7-9.
To illustrate the suitability for free-running operations we look at the
$P_{AP}$ as we make our detection gates longer. The results for #1 APD are
plotted in Fig. 7 and fitted using the multiple trapping model. $\tau_{CD}$ is
fixed at $100$ ns while $\tau_{AB}$ and the photon’s arrival positions are
altered. If the active quenching was slow then the arrival position, or time,
of the photon’s appearance in the $AB$ gate would be reflected in the $P_{AP}$
behavior. A photon creating an avalanche at the start of a long gate would
generate more carriers, increasing the chances for subsequent afterpulses,
than in the short gate regime or if the photon arrived at the end of a gate.
The overlapping curves show that the $P_{AP}$ behavior doesn’t change for long
gates, nor is it dependent on the arrival time, and hence shouldn’t change
when we move to a free-running regime.
Figure 8: $P_{AP}$ versus $\tau_{d}$ for #1 (a) and #2 (b) APDs at different T. TABLE I: The detrapping time parameters of fitting curves in Fig. 8. APD | T(K) | $\Delta t_{1}$(ns) | $\Delta t_{2}$(ns) | $\Delta t_{3}$(ns)
---|---|---|---|---
#1 | 216 | 1135 | 5645 |
#1 | 238-223 | 860 | 4385 |
#2 | 210 | 615 | 2560 | 10135
#2 | 238-223 | 1020 | 2165 | 5075
We finally study the temperature dependence of afterpulse. The experimental
results and fitting curves are shown in Fig. 8 and the fitting parameters are
listed in Table I. When the temperature is varied from $238$ K to $223$ K the
$P_{AP}$ behavior is almost identical due to the close trap lifetime
parameters, but when the temperature is at $216$ K (#1 APD) or $210$ K (#2
APD), there is a distinct increase for the $P_{AP}$. The detrapping lifetime
can be modeled as [27]
$\Delta t\propto e^{\frac{E_{ta}}{kT}}/{T^{2}},$ (5)
where $E_{ta}$ is the trapping activation energy. This formula means that
lower temperatures cause larger $\Delta t$ for traps, corresponding to larger
$P_{AP}$.
Moreover, when $\tau_{d}\lesssim 10\,\mu$s, the $P_{AP}$ of #2 APD at $210$ K,
in Fig. 8, is less than at other temperatures, but the $P_{AP}$ of #1 APD at
$216$ K is not. According to the fitting results at $210$ K, there is a trap
type with a fast detrapping lifetime of $615$ ns in #2 APD, which causes rapid
detrapping at small $\tau_{d}$, but when $\tau_{d}$ becomes large, the effect
of this trap type is gradually diminished while the other trap types with
$2560$ ns and $10135$ ns lifetimes start to dominate the detrapping process.
Unfortunately, this kind of fast detrapping time is too short and/or too weak
to be measured at the other three temperatures and for #1 APD.
Figure 9: $P_{AP}$ versus $\tau_{d}$ for #4 APD at different T.
In order to validate the above phenomena, we perform the measurements of
$P_{AP}$ behaviors of #4 APD from another manufacturer, whose results are
shown in Fig. 9. The $P_{AP}$ increases from $233$ K to $210$ K while the
cross point appears between $210$ K to $203$ K, which agrees well with our
explanation for the different $P_{AP}$ behaviors. We believe that the $P_{AP}$
models so far are not perfect and further investigations, including effective
models and experiments, are still needed.
### III-D Free-running mode
Free-running operation is very important for many applications such as
asynchronous and CW photon counting and quantum cryptography [1] _etc_. Due to
the lower noise characteristics of InGaAs/InP APDs that use this active
quenching ASIC, some of us have recently been able to show that this is now
also possible for APDs in the telecom regime.
Figure 10: Plot of the detection and noise rates as a function of deadtime for
#2 APD at $V_{DC}=48.62$ V, $N=10$ KHz with CW photons and T$=210$ K,
operating in the free-running mode.
As in the gated regime, the operation in the free-running mode depends on the
parameters of $V_{DC}$, $\tau_{d}$ and T. However, unlike the gated mode, the
afterpulse parameter in the free-running mode is more difficult to evaluate.
As we said, with respect to Fig. 7, the $P_{AP}$ does not depend on the width
of the gate, which is applicable for the free-running mode. Indeed, it may not
be obvious how the afterpulse probablity evolves when the gate is open for
such long times, though it would appear that at worst, the probability
continues to decrease over the period of detection. Nonetheless, we have
previously seen that for short deadtimes the afterpulsing dominates [24]. As
we have now been able to use the double-gate method to characterize the
afterpulsing, in the gated regime, we can use a simple model to describe the
detection and noise rates for the free-running mode,
$\displaystyle R=\eta N(1-\eta N\tau_{d})(1+\overline{P_{AP}}),$ (6)
with $\eta=1-e^{-\mu P_{DE}}(1-P_{DC})$, considering the Poisson distribution.
$P_{DE}$ and $P_{DC}$ are the detection efficiency and dark count probability,
and $N$ is the input photon number. The term of $(1-\eta N\tau_{d})$ is for
deadtime correction. If we put $\mu$ = 0, we recover the noise rate.
$\overline{P_{AP}}$ is the total afterpulsing contribution at $\tau_{d}$,
calculated from integrating over the gated afterpulse probability from
$\tau_{d}$ to infinity (in practice 100 $\mu$s is sufficient). Fig. 10 shows
the experimental rates as well as the results of our model as a function of
$\tau_{d}$.
It is clear that a more complicated model is warranted. However, the physics
of these limitations is clear. In the small $\tau_{d}$ region we underestimate
the rates as we do not take account of cascaded afterpulses, i.e., higher
order effects. The more interesting region, from 20 - 40 $\mu$s, we are
overestimating due to the difficulty in defining an appropriate integration
range, which will also change as a function of the photon flux, the intrinsic
detection efficiency and the deadtime. Importantly, we can also conclude that
for small $\tau_{d}$, if $N$ increases, then the $\overline{P_{AP}}$ value
will decrease, since photon clicks will increase while the multiple
afterpulsing effects will be relatively less likely.
Our model makes a first attempt to both understand the afterpulsing and to
develop a reliable technique for determining the detector’s characteristics,
without resorting to complicated techniques in a double-gate regime, there is
still some way to go. Although the apparent need for large $\tau_{d}$ that, in
turn, limits the maximum count rate, this is highly dependent on the photon
flux to be detected and free-running APDs are certainly highly advantageous
for applications with low to moderate count rates.
### III-E Charge persistence
Charge persistence is not normally a problem for synchronous detectors as the
photons arrive during the gate. However, what happens if a photon arrives
before the gate is applied, as is possible in the free-running mode, before
the APD is activated after a deadtime?
When the detector is “off”, i.e., at $V_{bv}$ below $V_{br}$, with only a few
volts so that primary dark carriers can still be generated and multiplied by
the average dc gain but with a small probability. When the gate pulse arrives
some of the carriers that remained in the multiplication layer can induce
avalanches [28]. This is called “charge persistence” (CP), or sometimes
referred as the “twilight effect” [29]. Similarly, when the CP carriers are
released before the gate pulses with the time difference less than the
effective transit time, they can also create afterpulses [28]. Now let us
consider another case, where photons always appear before the gate. Based on
the same principle, in this case the number of dark CP carriers will be
increased and the CP effect will be expanded.
We experimentally test this effect and the results are shown in Fig. 11. By
varying the time difference between the arrival times of gates (Td) and
photons (Tp), we observe the changes of the normalized (for $\mu$) noise per
gate, for #4 APD. The two almost identical behaviors show that the CP effect
is proportional to photon numbers and, per photon, can generate noise of about
$10\%$ of the dark count level with the time difference less than $1$ ns. When
the time difference is larger than $\sim 5$ ns, the CP effect is negligible
due to the characteristic exponential decrease. Moreover, through using TCSPC,
we also observe the detection events at the beginning of the gate are more
than those at other regions. The CP effect will cause nonnegligible noise in
the case of high frequency gating or asynchronous high flux detection.
Figure 11: The noise, including CP and dark counts per gate, normalized by
$\mu$, as a function of time difference between detection gate (Td) and
photons (Tp). The horizontal line is the dark count level. The results are
tested using #4 APD at $P_{DE}$=10%.
### III-F Quenching time
Quantifying the quenching time, including the circuit reaction time and gate
closing time as shown in Fig. 12, of an avalanche is very important to
understand the avalanche dynamics of APDs. Although an active quenching ASIC
should have a faster quenching time than conventional electronics this has not
previously been measured. More generally, these results are also pertinent for
rapid gating schemes that use very short gates and hence terminate avalanches
very quickly.
Figure 12: The principle of measuring the quenching time.
Figure 13: The count rates of detections and afterpulses with $\tau_{d}=$5
$\mu$s versus the delay of detection gate (Td). Points and lines are
experimental values and theoretical S fits, respectively. The results are
tested with #1 APD at T$=223$ K and $P_{DE}$=10%.
The principle for measuring the quenching time is to compare the count rate
behaviors for detections and afterpulses, see Fig. 12, using the double-gate
measurement electronics. The total number of carriers, during an avalanche,
should be proportional to the excess bias on APD and the excess bias duration,
or the integral of excess bias over the quenching time. Now we consider the
case where photons arrive at the end of the detection gates, by delaying
photons. From phase 1 to phase 2 in Fig. 12, the count rates of detection and
afterpulse are both almost constant, while from phase 2 to phase 3 the
detection rate is still constant but the afterpulse rate decreases first due
to the decrease of the integral. The time difference between the two phases
can be regarded as the reaction time, to detect the onset of the avalanche and
send the signal to close the NMOS switch. After phase 3, both of the rates
drastically decrease until the end of the closing time.
Fig. 13 shows the results of these measurements on #1 APD. From the slope of
the detection rate, we can obtain the closing time of the gate, which is only
around $1$ ns. Although it is very hard to determine a precise value of the
reaction time from the fitting results, the slight shift between the detection
and afterpulse rates indicates that the reaction time is much less than the
closing time.
## IV Conclusion
In summary, we have fully characterized an active quenching ASIC and compared
its operation with a conventional electronic circuit. To show the improvements
are universal we also characterized and compared 4 different InGaAs/InP APDs.
The APDs operating in the gated mode exhibit substantial performance
improvements compared with the conventional quenching electronics and allow
for free-running operation. We also extract thermal activation energies to
identify the dominant mechanism of dark counts, and by employing the multiple
detrapping model in the gated mode and proposed model in the free-running mode
the afterpulse behaviors are well illustrated. Moreover, we have characterized
the charge persistence and quenching time. The advantages of low afterpulsing
and noise in both theses regimes are mostly attributed to the state-of-the-art
ASIC.
## Acknowledgment
The authors would like to thank Dr. A. Rochas and Dr. M. A. Itzler for useful
discussions.
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|
arxiv-papers
| 2008-12-15T15:51:19 |
2024-09-04T02:48:59.416306
|
{
"license": "Public Domain",
"authors": "Jun Zhang, Rob Thew, Jean-Daniel Gautier, Nicolas Gisin, and Hugo\n Zbinden",
"submitter": "Jun Zhang",
"url": "https://arxiv.org/abs/0812.2840"
}
|
0812.2994
|
# On topological charged braneworld black holes
Ahmad Sheykhi 1,2111sheykhi@mail.uk.ac.ir and Bin Wang 3222wangb@fudan.edu.cn
1Department of Physics, Shahid Bahonar University, P.O. Box 76175, Kerman,
Iran
2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),
Maragha, Iran
3 Department of Physics, Fudan University, Shanghai 200433, China
###### Abstract
We study a class of topological black hole solutions in RSII braneworld
scenario in the presence of a localized Maxwell field on the brane. Such a
black hole can carry two types of charge, one arising from the extra
dimension, the tidal charge, and the other one from a localized gauge field
confined to the brane. We find that the localized charge on the brane modifies
the bulk geometry and in particular the bulk Weyl tensor. The bulk geometry
does not depend on different topologies of the horizons. We present the
temperature and entropy expressions associated with the event horizon of the
braneworld black hole and by using the first law of black hole thermodynamics
we calculate the mass of the black hole.
In the past years there has been a lot of interest in the braneworld scenario,
based on the assumption that all gauge fields in standard model of particle
physics are confined on a $3$-brane, playing the role of our $4$-dimensional
universe, embedded in a higher dimensional spacetime, while the gravitational
field, in contrast, is usually considered to live in the whole spacetime. The
first picture appeared in braneworld scenarios was the second Randall-Sundrum
model (RSII) in which, our universe observed as a positive tension $3$-brane
embedded in a $5$-dimensional anti de-Sitter bulk RS . In this model the
localization of gravity happens on the brane due to the negative bulk
cosmological constant and the cross over between $4$-dimensional and
$5$-dimensional gravity is set by the anti de-Sitter radius. Within the
context of the RSII scenario, it is important that the induced metric on the
brane is, in the low energy regime, the solution predicted by standard general
relativity in four dimensions. Otherwise the usual astrophysical properties of
black holes and stars would not be recovered. Therefore it is natural to
assume the formation of black hole in the braneworld due to gravitational
collapse of matter trapped on the brane. In fact, the construction and study
of black hole solutions on the brane has been one of the most important and
intriguing challenge in braneworld physics. There are several reasons why this
problem is so challenging. First, the effective gravitational field equation
on the brane is not the usual Einstein one but contains higher correction
terms due to the nonlocal bulk effects on the brane and therefore is more
complicated compared with the usual gravitational field equations. Second,
even one finds the solution of the effective gravitational field equations on
the brane, one can not regard it as a braneworld black hole solution. One can
just consider this solution as an initial data for the evolution of the brane
into the bulk. The first attack on this problem was done by Chamblin, Hawking
and Reall who investigated the gravitational collapse of uncharged, non-
rotating matter in RSII braneworld model Cham1 . They showed that a static
uncharged black hole on the brane is described by a “black cigar” solution in
five dimensions. If this cigar extends all the way down to the anti-de Sitter
horizon, then we recover the metric for a black string in anti-de Sitter
spacetime. However, such a black string is unstable near the anti-de Sitter
horizon Gre1 ; Gre2 . An exact braneworld black hole solution satisfies a
closed system of effective gravitational field equations on the brane,
describing an uncharged black hole in the RSII scenario was obtained in Dad .
By using the braneworld gravitational field equations derived in Shi , it was
shown that a Reissner-Nordstrom geometry could arise on the brane provided
that the bulk Weyl tensor takes a particular form. The solution in Dad
carries a “tidal charge”, arising from the projection of the bulk free
gravitational field effects onto the brane. However, it was argued in Cham2
that although the solution in Dad was claimed to describe an uncharged black
hole, one can not regard it as a braneworld black hole solution. One can just
consider this solution as an initial data for the evolution of the brane into
the bulk. Until this evolution is performed and boundary conditions in the
bulk are imposed, it is not clear what this solution represents. For example,
it might give rise to some pathology such as a naked curvature singularity.
Therefore the main problem remains in the braneworld black hole physics is to
study the effect of the braneworld black hole on the bulk geometry, and in
particular the nature of the off-brane horizon structure. Indeed, the
analytical solution for the bulk spacetime has not been found until now. The
numerical calculations on the bulk geometry in the case of charged and
uncharged braneworld black holes have been investigated in Cham2 and Shibata
, respectively. Other attempts on the study of braneworld black holes and
their physical properties have been carried out in Cas1 ; Cas2 ; Bro ; Gre3 ;
Ali ; kof ; Bwang ; Bwang2 ; Yosh ; Yosh2 .
The purpose of the present Letter is to tackle the first problem mentioned
above in the braneworld black hole physics. We will consider the Maxwell gauge
fields confined onto the brane. Employing a simple strategy, we solve
gravitational field equations on the brane and obtain the charged topological
braneworld black hole solutions. Our solution is the generalization of Cham2
to different horizon topologies. We also present the temperature and entropy
expressions associated with the event horizon of the braneworld black hole and
calculate the mass of the black hole by using the first law of black hole
thermodynamics. Since the flux lines of gauge fields can pierce the horizon
only when they intersect the brane, our bulk theory is the same as that of the
uncharged case and one might expect that the “black cigar” solution still
describes the bulk containing the charged braneworld black hole. Here we will
not repeat the discussion on the bulk metric, since we see that the bulk
geometry does not depend on different topologies of the horizons, thus our
bulk metric is the same as that discussed in Cham2 for the spherically
symmetric braneworld black hole.
We start with the effective field equations on a 3-brane embedded in the
5-dimensional anti de-Sitter spacetime with $\mathbb{Z}_{2}$ symmetry
expressed as Shi
$\displaystyle G_{\mu\nu}=-\Lambda g_{\mu\nu}+8\pi
GT_{\mu\nu}+\kappa_{5}^{4}\pi_{\mu\nu}-E_{\mu\nu},$ (1)
where
$\displaystyle G$ $\displaystyle=$
$\displaystyle\frac{\kappa^{4}_{5}}{48\pi}\lambda,\hskip
19.91684pt\Lambda=\frac{\kappa_{5}^{2}}{2}\Bigl{(}\Lambda_{5}+\frac{\kappa_{5}^{2}}{6}\lambda^{2}\Bigr{)}.$
(2)
Here $\kappa_{5}$ and $\Lambda_{5}$ are, respectively, the five-dimensional
gravity coupling constant and cosmological constant. The factor $\Lambda$ is
the effective cosmological constant on the brane, $\lambda$ is the brane
tension, and $T_{\mu\nu}$ is the stress energy tensor confined onto the brane,
so $T_{AB}\,n^{B}=0$, where $n^{A}$ is the unit normal to the brane. The first
correction term relative to Einstein’s gravity is the inclusion of a quadratic
term $\pi_{\mu\nu}$ in the stress-energy tensor, arising from the extrinsic
curvature term in the projected Einstein tensor, and is given by
$\pi_{\mu\nu}=\frac{1}{12}TT_{\mu\nu}-\frac{1}{4}T_{\mu\alpha}T_{\nu}^{\
\alpha}{}+\frac{1}{8}\,g_{\mu\nu}\left(T_{\alpha\beta}T^{\alpha\beta}-\frac{1}{3}T^{2}\right)\,.$
(4)
The second correction term, ${E}_{\mu\nu}$, is the projection of the five-
dimensional bulk Weyl tensor onto the brane, which is defined as
$E_{\mu\nu}={}^{(5)}C_{\mu\alpha\nu\beta}n^{\alpha}n^{\beta}$ and encompasses
the nonlocal bulk effect. The only general known property of this nonlocal
term is that it is traceless, namely ${E}^{\mu}{}_{\mu}=0$. Using the
traceless property of the projected Weyl tensor, Eq. (1) can be simplified
into
$R=4\Lambda-8\pi
G\,T-\frac{\kappa_{5}^{4}}{4}\left(T_{\alpha\beta}T^{\alpha\beta}-\frac{1}{3}T^{2}\right)\,.$
(5)
We would like to find the topological black hole solutions of the field
equations (1). We assume the induced metric on the brane in the form
$ds^{2}=-f(r)dt^{2}+{dr^{2}\over f(r)}+r^{2}d\Omega_{k}^{2},$ (6)
where $d\Omega_{k}^{2}$ is the line element of a two-dimensional hypersurface
$\Sigma$ with constant curvature,
$d\Omega_{k}^{2}=\left\\{\begin{array}[]{ll}$$d\theta^{2}+\sin^{2}\theta
d\phi^{2}$$,\quad\quad\\!\\!{\rm for}\quad$$k=1$$,&\\\
$$d\theta^{2}+\theta^{2}d\phi^{2}$$,\quad\quad\quad{\rm for}\quad$$k=0$$,&\\\
$$d\theta^{2}+\sinh^{2}\theta d\phi^{2}$$,\quad{\rm
for}\quad$$k=-1$$.&\end{array}\right.$ (7)
For $k=1$, the topology of the event horizon is the two-sphere $S^{2}$, and
the spacetime has the topology $R^{2}\times S^{2}$. For $k=0$, the topology of
the event horizon is a torus and the spacetime has the topology $R^{2}\times
T^{2}$. For $k=-1$, the surface $\Sigma$ is a two-dimensional hypersurface
$H^{2}$ with constant negative curvature. In this case the topology of
spacetime is $R^{2}\times H^{2}$. It is not necessary to take the exact metric
describing a topological braneworld black hole in the form (6). In general one
may expect that $g_{rr}\neq-{g_{tt}}^{-1}$. But, it is well known that the
induced metric describing a charged black hole should be close to Reissner-
Nordstrom metric, so our ansatz for the braneworld black hole metric is a good
guess Cham2 .
Assuming the localized gauge field on the brane is the Maxwell field with
action
$S=-\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}F_{\mu\nu}F^{\mu\nu}.$ (8)
The corresponding localized energy-momentum tensor on the brane can be written
as
$T_{\mu\nu}=\frac{1}{4\pi
G}\left(F_{\mu\rho}F_{\nu}\,^{\rho}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\right).$
(9)
which is traceless, satisfying $T=T_{\mu}^{\ \mu}=0$. We also assume that
there is a localized static point charge on the brane which produces an
electric field
$F_{tr}=\frac{q}{r^{2}},$ (10)
where $q$ is the charge parameter. Using metric (6), the electric field (10)
and Eq. (9) for the total energy-momentum tensor localized on the brane, one
can show that Eq. (5) has a solution of the form
$f(r)=k-{\frac{2m}{r}}-\frac{\Lambda}{3}{r}^{2}+{\frac{\beta+q^{2}}{{r}^{2}}}+{\frac{1}{240}}\,{\frac{{\kappa_{5}}^{4}{q}^{4}}{{r}^{6}}},$
(11)
where $m$ and $\beta$ are arbitrary integration constants and we have assumed
$4\pi G=1$, for simplicity. Although in Dad , $\beta>0$ has been interpreted
as a tidal charge associated with the bulk Weyl tensor, in the presence of
localized charge on the brane, it is quite possible to take $\beta<0$ as
pointed out in Cham2 . Indeed, the projected Weyl tensor, transmits the tidal
charge stresses from the bulk to the brane. One may also interpret $\beta$ as
a five-dimensional mass parameter Cham2 . The horizons can be found by solving
Eq. $f(r)=0$. This equation cannot be solved analytically except for $q=0$.
The event horizon of the charged braneworld black hole locates at $r_{+}$
where $r=r_{+}$ is the largest root of equation $f(r)=0$. Inserting solution
(11) into field equations (1), we obtain the components of the five-
dimensional bulk Weyl tensor. The result is
$E^{t}_{\ t}=E^{r}_{\ r}=-E^{i}_{\
i}=\frac{\beta}{r^{4}}+\frac{1}{24}\frac{\kappa_{5}^{4}q^{4}}{r^{8}},$ (12)
where $i=1,2$. Clearly the traceless nature of the Weyl tensor is obeyed. Eqs.
(1) with solutions (11) and (12) form a closed system of equations on the
brane.
Some discussions on our solution are needed. In the special case $k=1$ and
$\Lambda=0$, $q=0$, our solution (11) reduces to the uncharged braneworld
black hole solution found in Dad . In the case $k=1$ and $\Lambda=0$, our
solution (11) reduces to the charged black hole solution presented in Cham2 .
With the presence of the charge on the brane, the bulk geometry has to change,
since now $T_{\mu\nu}\neq 0$. In other words, the localized charge on the
brane will induce changes in the bulk geometry and therefore modifies the bulk
Weyl tensor. This property keeps for different topologies of the horizon.
Further from Eq. (12) we see that the horizon topology of the braneworld black
hole does not affect the bulk geometry and therefore the bulk Weyl tensor is
independent of the constant curvature $k$.
In the following we are going to calculate the conserved and thermodynamic
quantities of the braneworld black hole. We will adopt a simple strategy based
on the profound connection between gravity and thermodynamics which has
recently been revealed in various gravity theories Jac -Sheywang , showing the
deep correspondence between the gravitational equation describing the gravity
in the bulk and the first law of thermodynamics on the apparent horizon. This
connection sheds the light on holography since the gravitation equations
persist the information in the bulk while the first law of thermodynamics on
the apparent horizon contains the information on the boundary. Besides, this
connection was shown as a useful tool to extract the entropy of the
braneworld. In the general case, gravity on the brane does not obey the
Einstein theory and the usual area formula for the black hole entropy does not
hold on the brane. The relation between the braneworld black hole horizon
entropy and its geometry is not known. It was argued in Shey1 ; Shey2 that
the entropy associated with the apparent horizon on the brane can be extracted
from the obtained gravity and thermodynamics correspondence. The entropy and
temperature associated with the apparent horizon of the FRW universe on the
brane, in the RSII braneworld model, are found in Cai4 ; Shey1 with form
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{2\pi\ell}{G_{5}}{\displaystyle\int^{\tilde{r}_{A}}_{0}\frac{\tilde{r}_{A}^{2}}{\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}}d\tilde{r}_{A}}=\frac{2\pi{\tilde{r}_{A}}^{3}}{3G_{5}}\times{}_{2}F_{1}\left(\frac{3}{2},\frac{1}{2},\frac{5}{2},-\frac{{\tilde{r}_{A}}^{2}}{\ell^{2}}\right),$
(13) $\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi{\tilde{r}_{A}}},$ (14)
where $\tilde{r}_{A}$ is the apparent horizon radius and $\ell$ is the AdS
radius of the bulk spacetime which is related to the bulk cosmological
constant. Here ${}_{2}F_{1}(a,b,c,z)$ is a hypergeometric function and
$G_{5}=\kappa_{5}^{2}/8\pi$ is the gravitational constant in five dimensions.
Recently, we have shown that the extracted apparent horizon entropy, given in
Eq. (13), satisfies the generalized second law of thermodynamics Sheywang .
The satisfaction of the generalized second law of thermodynamics further
supports that the entropy (13) is a reasonable thermodynamical entropy
describing the brane.
Now we suppose that the temperature and entropy formula (13) and (14) also
hold on the event horizon of the black hole on the brane. Replacing the
apparent horizon radius $\tilde{r}_{A}$ by the black hole horizon radius
$r_{+}$, we have the temperature and entropy on the event horizon of the
braneworld black hole
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{2\pi\ell}{G_{5}}{\displaystyle\int^{r_{+}}_{0}\frac{r_{+}^{2}}{\sqrt{r_{+}^{2}+\ell^{2}}}dr_{+}}=\frac{2\pi{r_{+}}^{3}}{3G_{5}}\times{}_{2}F_{1}\left(\frac{3}{2},\frac{1}{2},\frac{5}{2},-\frac{r_{+}^{2}}{\ell^{2}}\right),$
(15) $\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi r_{+}}.$
(16)
Eq. (16) is exactly the Hawking temperature on the event horizon. The validity
of (15) to describe the event horizon entropy of the braneworld black hole can
be justified by considering its limiting case with $\tilde{r}_{+}\ll\ell$.
Physically this limit means that the size of extra dimension is very large if
compared with the black hole event horizon radius. In this limit Eq. (15)
reduces to the five-dimensional area formula for the black hole entropy
$S=2\Omega_{3}{\tilde{r}_{+}}^{3}/4G_{5}$, where $\Omega_{3}=4\pi/3$ is the
volume of a unit sphere. The factor $2$ comes from the $\mathbb{Z}_{2}$
symmetry in the bulk. This is an expected result since in this regime the anti
de-Sitter bulk reduces to the Minkowski spacetime. And due to the absence of
the negative cosmological constant in the Minkowski bulk, no localization of
gravity happens on the brane. Thus the gravity on the brane is still five-
dimensional and the entropy formula on the black hole event horizon obeys the
five-dimensional area formula Shey1 .
Adopting the first law of black hole thermodynamics on the event horizon
$r_{+}$ and considering that the electric charge of black hole does not affect
its mass, we just need to discuss the uncharged case with the first law
$dM=TdS.$ (17)
Integrating (17) and inserting (15) and (16), we obtain the mass of the
braneworld black hole
$M=\frac{\ell}{G_{5}}\int^{r_{+}}_{0}\frac{r_{+}dr_{+}}{\sqrt{r_{+}^{2}+\ell^{2}}}=\frac{\ell}{G_{5}}\left(\sqrt{r_{+}^{2}+\ell^{2}}-\ell\right).$
(18)
It is interesting to see that in the limiting case $\tilde{r}_{+}\ll\ell$, the
mass formula (18) reduces to
$M=\frac{r_{+}^{2}}{2G_{5}},$ (19)
which is exactly the mass of the five-dimensional black hole in Einstein
gravity.
In conclusion, we have obtained a class of topological black hole solutions in
RSII braneworld scenario in the presence of a localized Maxwell field on the
brane. We have shown that the localized charge on the brane modifies the bulk
geometry and in particular the bulk Weyl tensor. The horizon topology of the
braneworld black holes does not affect the geometry of extra dimension. We
presented the temperature and entropy expressions associated with the event
horizon of the braneworld black hole. We also obtained the mass of the
braneworld black holes through the use of the first law of black hole
thermodynamics.
We would like to mention here that in this Letter we have not studied fully
the effect of the braneworld black hole on the bulk geometry, and in
particular the nature of the off-brane horizon structure. This has been done
for solutions which reduce to the Schwarzschild black hole on the brane Cham1
. We have adopted a different approach: instead of starting from an induced
metric on the brane, we have solved the closed system of the effective field
equations for the induced metric on the brane in RSII model, and found a class
of topological braneworld black holes. Therefore the main problem remains to
find the exact bulk metric that describes a topological braneworld black hole.
This was solved for uncharged black holes in three dimensions Emp .
Unfortunately, the higher dimensional generalization of this metric is still
not known. In general the bulk spacetime may be given, by solving the full
five-dimensional equations, and the geometry of the embedded brane is then
deduced. Due to the complexity of the five-dimensional equations, one may
follow the strategy outlined in this Letter, by considering the intrinsic
geometry on the brane, which encompasses the imprint from the bulk, and
consequently evolve the metric off the brane. However, in this Letter we did
not study the effects of the braneworld black hole on the bulk geometry, and
in particular the nature of the topological horizon structure in the bulk.
Indeed, determining the bulk geometry is an extremely difficult task which
needs numerical calculations, so it was not explored here.
###### Acknowledgements.
This work has been supported financially by Research Institute for Astronomy
and Astrophysics of Maragha, Iran. The work of B. W. was support in part by
NNSF of China, Shanghai Education Commission and Shanghai Science and
Technology Commission.
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|
arxiv-papers
| 2008-12-16T08:06:03 |
2024-09-04T02:48:59.427295
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi and Bin Wang",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/0812.2994"
}
|
0812.3013
|
# Wigner function for twisted photons
I. Rigas Departamento de Óptica, Facultad de Física, Universidad Complutense,
28040 Madrid, Spain L. L. Sánchez Soto Departamento de Óptica, Facultad de
Física, Universidad Complutense, 28040 Madrid, Spain A. B. Klimov
Departmento de Física, Universidad de Guadalajara, 44420 Guadalajara, Jalisco,
Mexico J. Řeháček Department of Optics, Palacky University, 17\. listopadu
50, 772 00 Olomouc, Czech Republic Z Hradil Department of Optics, Palacky
University, 17. listopadu 50, 772 00 Olomouc, Czech Republic
###### Abstract
A comprehensive theory of the Weyl-Wigner formalism for the canonical pair
angle-angular momentum is presented, with special emphasis in the implications
of rotational periodicity and angular-momentum discreteness.
###### pacs:
03.65.Wj, 03.75.Lm, 42.50.Dv
## I Introduction
A quantum system has a dynamical symmetry group $G$ if its Hamiltonian is a
function of the generators of $G$. In this case, the Hilbert space of the
system splits into a direct sum invariant subspaces (carriers of the
irreducible representations of $G$) and the discussion of any physical
property can be restricted to one of these subspaces Barut and Ra̧czka (1987).
The existence of such a symmetry also allows for the explicit construction of
a phase space for the system as the coadjoint orbit associated with an
irreducible representation of $G$ Kostant (1970); Kirillov (1976.) (in fact,
it turns out to be a symplectic manifold). In consequence, to every operator
on Hilbert space we can associate a function on phase space, opening the way
to formally representing quantum mechanics as a statistical theory on
classical phase space. Various aspects of this formalism for basic quantum
systems have been developed by a number of authors Weyl (1950); Wigner (1932);
Moyal (1949); Stratonovich (1956); Agarwal and Wolf (1970); Berezin (1975);
Agarwal (1981); Bertrand and Bertrand (1987); Varilly and Gracia-Bondía
(1989); Atakishiyev et al. (1998); Brif and Mann (1998); Benedict and Czirják
(1999).
There are, however, important differences with respect to a classical
description. They come from the noncommuting nature of conjugate quantities,
which precludes their simultaneous precise measurement and, therefore, imposes
a fundamental limit to the accuracy with which we can determine a point in
phase space. As a distinctive consequence of this, there is no unique rule by
which we can associate a classical phase-space variable to a quantum operator
and depending on the operator ordering, various functions can be defined. For
example, the quantum state (i.e., the density matrix) of the system can be
mapped onto a whole family of functions parametrized by a number $s$; the
values $+1$, 0, and $-1$ corresponding to the Husimi $Q$, the Wigner $W$, and
the Glauber-Sudarshan $P$ functions, respectively. These phase-space functions
are known as quasiprobability distributions, as in quantum mechanics they play
a role similar to that of genuine probability distributions in classical
statistical mechanics (for reviews, see Refs. Balazs and Jennings (1984);
Hillery et al. (1984); Lee (1995); Jr. (1996)).
Apart from the description of the harmonic oscillator (for which $G$ is the
Heisenberg-Weyl group and the corresponding phase space is the plane
$\mathbb{R}^{2}$), this formalism has also been successfully applied to spin-
like systems (or qubits in the modern parlance of quantum information), for
which $G$ is the group SU(2) and the phase space is the two-dimensional Bloch
sphere. However, one can rightly argue that this Wigner function, although
describing a discrete system, is not defined in a discrete phase space. In
fact, the growing interest in quantum information has fueled the search for
discrete phase-space counterparts of the Wigner function (see Ref. Klimov et
al. (2008) for a complete and up-to-date review). The main advantage of such a
representation consists in that even states from different irreducible
representations can be pictured on the same phase space, which is basically a
direct product of two-dimensional discrete tori.
There is still another “mixed” canonical pair: angle and angular momentum.
Now, the symmetry group $G$ is noncompact and can be taken as the two-
dimensional Euclidean group E(2), whereas the associated phase space is the
discrete cylinder $\mathbb{Z}\times\mathcal{S}_{1}$ ($\mathcal{S}_{1}$ denotes
here the unit circle), since one of the variables is continuous and the other
is discrete. Several interesting properties of a number of systems, such as
molecular rotations, electron wave packets, Hall fluids, and light fields, to
cite only a few examples, can be described in terms of this symmetry group
Rigas et al. (2008). In quantum optics, it is the basic tool to deal with the
orbital angular momentum of the so-called twisted photons Molina-Terriza et
al. (2007); Franke-Arnold et al. (2008), which have been proposed for
applications in quantum experiments Vaziri et al. (2002).
The construction of a proper Wigner function for this case is still under
discussion. Although some interesting attempts have been published Mukunda
(1979); Bizarro (1994); Mukunda et al. (2005), they seem of difficult
application to practical problems. Quite interesting group-theoretical
approaches to this problem can be also found in Refs. Nieto et al. (1998);
Plebański et al. (2000). In this paper, we approach this interesting problem
from the perspective of finite-dimensional systems and construct a bona fide
Wigner function that fulfills all the reasonable requirements and is easy to
handle and to interpret. We also discuss its applications to some relevant
quantum states.
## II Wigner function for position-momentum
In this section we briefly recall the relevant structures needed to set up the
Wigner function for Cartesian quantum mechanics. This is to facilitate
comparison with the angular case later on. For simplicity, we choose one
degree of freedom only, so the associated phase space is the plane
$\mathbb{R}^{2}$.
The canonical Heisenberg commutation relations between Hermitian coordinate
and momentum operators $\hat{q}$ and $\hat{p}$ are (in units $\hbar=1$)
$[\hat{q},\hat{p}]=i\,,$ (1)
so that they are the generators of the Heisenberg-Weyl algebra. In the unitary
Weyl form this is expressed as
$\hat{U}(p)\hat{V}(q)=\hat{V}(q)\hat{U}(p)\,e^{iqp}\,,$ (2)
where
$\hat{V}(q)=\exp(-iq\hat{p})\,,\qquad\hat{U}(p)=\exp(ip\hat{q})\,,$ (3)
are the generators of translations in position and momentum, respectively. In
the Cartesian case, these exponentials can be entangled to define a
displacement operator
$\hat{D}(q,p)=\hat{U}(p)\hat{V}(q)e^{-iqp/2}=\exp[i(p\hat{q}-q\hat{p})]\,,$
(4)
with the parameters $(q,p)$ labelling phase-space points. However, this cannot
be done for other canonical pairs, as we shall see.
The displacement operators form a complete trace-orthonormal set (in the
continuum sense) in the space of operators acting on $\mathcal{H}$ (the
Hilbert space of square integrable functions on $\mathbb{R}$):
$\mathop{\mathrm{Tr}}\nolimits[\hat{D}(q,p)\,\hat{D}^{\dagger}(q^{\prime},p^{\prime})]=2\pi\delta(q-q^{\prime})\delta(p-p^{\prime})\,.$
(5)
Note that $\hat{D}^{\dagger}(q,p)=\hat{D}(-q,-p)$, while
$\hat{D}(0,0)=\hat{\openone}$.
The mapping of the density matrix $\hat{\varrho}$ into a Wigner function
defined on $\mathbb{R}^{2}$ is established in a canonical way:
$\displaystyle
W(q,p)=\mathop{\mathrm{Tr}}\nolimits[\hat{\varrho}\,\hat{w}(q,p)]\,,$ (6)
$\displaystyle\hat{\varrho}=\displaystyle\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}\hat{w}(q,p)W(q,p)\,dqdp\,,$
where the (Hermitian) Wigner kernel $\hat{w}$ (a particular instance of a
Stratonovitch-Weyl quantizer) is the double Fourier transform of the
displacement operator:
$\hat{w}(q,p)=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}\exp[-i(pq^{\prime}-qp^{\prime})]\hat{D}(q^{\prime},p^{\prime})\,dq^{\prime}dp^{\prime}\,.$
(7)
One can immediately check that the Wigner kernels are also a complete trace-
orthonormal set. Furthermore, they transform properly under displacements
$\hat{w}(q,p)=\hat{D}(q,p)\,\hat{w}(0,0)\,\hat{D}^{\dagger}(q,p)\,,$ (8)
where
$\hat{w}(0,0)=\int_{\mathbb{R}^{2}}\hat{D}(q,p)\,dqdp=2\hat{P}\,,$ (9)
and $\hat{P}$ is the parity operator.
The Wigner function in (6) fulfills all the basic properties required for any
good probabilistic description. First, due to the Hermiticity of
$\hat{w}(q,p)$, it is real for Hermitian operators. Second, on integrating
$W(q,p)$ over one variable, the probability distribution of the conjugate
variable is reproduced
$\int_{\mathbb{R}}W(q,p)\,dp=\langle
q|\hat{\varrho}|q\rangle\,,\quad\int_{\mathbb{R}}W(q,p)\,dq=\langle
p|\hat{\varrho}|p\rangle\,.$ (10)
Third, $W(q,p)$ is covariant, which means that for the displaced state
$\hat{\varrho}^{\prime}=\hat{D}(q_{0},p_{0})\,\hat{\varrho}\,\hat{D}^{\dagger}(q_{0},p_{0})$,
one has
$W_{\hat{\varrho}^{\prime}}(q,p)=W_{\hat{\varrho}}(q-q_{0},p-p_{0})\,,$ (11)
so that the Wigner function follows displacements rigidly without changing its
form, reflecting the fact that physics should not depend on a certain choice
of the origin.
Finally, the overlap of two density operators is proportional to the integral
of the associated Wigner functions:
$\mathop{\mathrm{Tr}}\nolimits(\hat{\varrho}_{1}\,\hat{\varrho}_{2})\propto\int_{\mathbb{R}^{2}}W_{1}(q,p)W_{2}(q,p)\,dqdp\,.$
(12)
This property (often known as traciality) offers practical advantages, since
it allows one to predict the statistics of any outcome, once the Wigner
function of the measured state is known.
## III Wigner function for discrete systems
Many quantum systems can be appropriately described in a finite-dimensional
Hilbert space. The previous standard approach can be extended to these
discrete systems, since they do have a dynamical symmetry group. However, in a
continuous Wigner function for these systems, there is a lot of information
redundancy. The goal of this section is to carry out a non-redundant discrete
phase-space analysis for this case.
Let us consider a system living in a Hilbert space $\mathcal{H}_{d}$, of
dimension $d$ (a qudit). It is useful to choose a computational basis
$|n\rangle$ ($n=0,\ldots,d-1$) in $\mathcal{H}_{d}$ and introduce the basic
operators Schwinger (1960)
$\hat{X}|n\rangle=|n+1\rangle\,,\qquad\hat{Z}|n\rangle=\omega(n)|n\rangle\,,$
(13)
where addition and multiplication must be understood modulo $d$ and, for
simplicity, we use the notation
$\omega(m)\equiv\omega^{m}=\exp(i2\pi m/d)\,,$ (14)
$\omega=\exp(i2\pi/d)$ being a $d$th root of the unity. The operators
$\hat{X}$ and $\hat{Z}$ generate a group under multiplication known as the
generalized Pauli group Nielsen and Chuang (2000) and obey
$\hat{Z}\hat{X}=\omega\,\hat{X}\hat{Z}\,,$ (15)
which is the finite-dimensional version of the Weyl form (2) of the
commutation relations.
The monomials $\\{\hat{Z}^{k}\hat{X}^{l}\\}$ ($k,l=0,1,\ldots,d-1$) form a
basis in the space of all the operators acting in $\mathcal{H}_{d}$ Klimov et
al. (2005). It seems then natural to introduce the unitary displacement
operators
$\hat{D}(k,l)=e^{i\phi(k,l)}\hat{Z}^{k}\hat{X}^{l}\,,$ (16)
where $\phi(k,l)$ is a phase. The unitarity condition imposes that
$\phi(k,l)+\phi(-k,-l)=-\frac{2\pi}{d}kl\,.$ (17)
Different choices have been analyzed in the literature Vourdas (2007); one of
special relevance is
$\phi(k,l)=\frac{2\pi}{d}2^{-1}\,kl\,,$ (18)
where $2^{-1}$ is the multiplicative inverse of 2 in $\mathbb{Z}_{d}$ when d
is prime and $2^{-1}=1/2$ for nonprime dimensions.
In this way, we have got a discrete phase space of the system as a $d\times d$
grid of points, in a such a way that the coordinate of each point $(k,l)$
define powers of $Z$ (“position”) and $X$ (“momentum”) and the whole phase
space is isomorphic to a discrete two-dimensional torus.
The following mapping from the Hilbert space into the discrete phase space
[equivalent to (II)]
$\displaystyle
W(k,l)=\mathop{\mathrm{Tr}}\nolimits[\hat{\varrho}\,\hat{w}(k,l)]\,,$ (19)
$\displaystyle\hat{\varrho}=\displaystyle\frac{1}{d^{2}}\sum_{k,l}\hat{w}(k,l)W(k,l)\,,$
is established in terms of the following (Hermitian) Wigner kernel
$\hat{w}(k,l)=\frac{1}{d^{2}}\sum_{m,n}\omega(kn-lm)\,\hat{D}(m,n)\,,$ (20)
which is normalized, satisfies the overlap condition
$\mathop{\mathrm{Tr}}\nolimits[\hat{w}(k,l)\hat{w}(k^{\prime},l^{\prime})]=d\,\delta_{k,k^{\prime}}\,\delta_{l,l^{\prime}}\,,$
(21)
and it is explicitly covariant:
$\hat{w}(k,l)=\hat{D}(k,l)\,\hat{w}(0,0)\,\hat{D}^{\dagger}(k,l)\,,$ (22)
where
$\hat{w}(0,0)=\frac{1}{d^{2}}\sum_{k,l}\hat{D}(k,l)\,.$ (23)
It is interesting to note that the phase (18) for prime dimensions leads to
$\hat{w}(0,0)=\hat{P}$, $\hat{P}$ being the parity operator. In view of these
properties, one can easily conclude that the corresponding Wigner function
$W(k,l)$ fulfills properties fully analogous as those for the continuous
harmonic oscillator.
## IV Wigner function for angle-angular momentum
In this section, we consider the conjugate pair angle and angular momentum. To
avoid the difficulties linked with periodicity, the simplest solution Louisell
(1963); Mackey (1963); Carruthers and Nieto (1968) is to adopt two angular
coordinates, such as, e.g., cosine and sine, we shall denote by $\hat{C}$ and
$\hat{S}$ to make no further assumptions about the angle itself. One can
concisely condense all this information using the complex exponential of the
angle $\hat{E}=\hat{C}+i\hat{S}$, which satisfies the commutation relation
$[\hat{E},\hat{L}]=\hat{E}\,,$ (24)
or, equivalently,
$\displaystyle[\hat{C},\hat{L}]=i\hat{S},\qquad[\hat{S},\hat{L}]=-i\hat{C}\,,$
(25) $\displaystyle[\hat{C},\hat{S}]=0\,.$
In mathematical terms, this defines the Lie algebra of the two-dimensional
Euclidean group E(2). Note also, that from the Baker-Campbell-Hausdorff
formula, one gets
$e^{-i\phi\hat{L}}\hat{E}=e^{i\phi}\,\hat{E}e^{-i\phi\hat{L}}\,,$ (26)
which is the unitary Weyl form of (24).
The action of $\hat{E}$ on the angular momentum basis is
$\hat{E}|\ell\rangle=|\ell-1\rangle\,,$ (27)
and, since the integer $\ell$ runs from $-\infty$ to $+\infty$, $\hat{E}$ is a
unitary operator whose normalized eigenvectors
$|\phi\rangle=\frac{1}{\sqrt{2\pi}}\sum_{\ell\in\mathbb{Z}}e^{i\ell\phi}|\ell\rangle\,,$
(28)
form a complete basis
$\langle\phi|\phi^{\prime}\rangle=\sum_{\ell\in\mathbb{Z}}\delta(\phi-\phi^{\prime}-2\ell\pi)=\delta_{2\pi}(\phi-\phi^{\prime})\,,$
(29)
where $\delta_{2\pi}$ represents the periodic delta function (or Dirac comb)
of period $2\pi$.
As anticipated in the Introduction, the phase space is now the semi-discrete
cylinder $\mathbb{Z}\times\mathcal{S}_{1}$. Following the ideas of Sec. III, a
displacement operator can be introduced as
$\hat{D}(\ell,\phi)=e^{i\alpha(\ell,\phi)}\,\hat{E}^{-\ell}e^{-i\phi\hat{L}}\,,$
(30)
where $\alpha(\ell,\phi)$ is a phase to be specified. Note that here there is
no possibility to rewrite Eq. (30) as an entangled exponential, since the
action of the operator to be exponentiated would not be well defined. The
requirement of unitarity imposes now
$\alpha(\ell,\phi)+\alpha(-\ell,-\phi)=\ell\phi\,.$ (31)
As desired, the displacement operators form a complete trace-orthonormal set:
$\mathop{\mathrm{Tr}}\nolimits[\hat{D}(\ell,\phi)\hat{D}^{\dagger}(\ell^{\prime},\phi^{\prime})]=2\pi\,\delta_{\ell,\ell^{\prime}}\,\delta_{2\pi}(\phi-\phi^{\prime})\,,$
(32)
whose resemblance with relation (5) is evident.
We can introduce then the canonical mapping
$\displaystyle
W(\ell,\phi)=\mathop{\mathrm{Tr}}\nolimits[\hat{\varrho}\,\hat{w}(\ell,\phi)]\,,$
(33)
$\displaystyle\displaystyle\hat{\varrho}=\frac{1}{(2\pi)^{2}}\,\sum_{{\ell}\in\mathbb{Z}}\int_{2\pi}\hat{w}(\ell,\phi)W(\ell,\phi)\,d\phi\,,$
where the Wigner kernel $\hat{w}$ is defined, in close analogy to the previous
cases, as
$\hat{w}(\ell,\phi)=\frac{1}{(2\pi)^{2}}\sum_{{\ell^{\prime}}\in\mathbb{Z}}\int_{2\pi}\exp[-i(\ell^{\prime}\phi-\ell\phi^{\prime})]\hat{D}(\ell^{\prime},\phi^{\prime})\,d\phi^{\prime}\,.$
(34)
The set of Wigner kernels constitutes a complete orthogonal Hermitian operator
basis. In addition, they are explicitly covariant:
$\hat{w}(\ell,\phi)=\hat{D}(\ell,\phi)\,\hat{w}(0,0)\,\hat{D}^{\dagger}(\ell,\phi)\,,$
(35)
with
$\hat{w}(0,0)=\frac{1}{(2\pi)^{2}}\sum_{{\ell}\in\mathbb{Z}}\int_{2\pi}\hat{D}(\ell,\phi)\,d\phi\,,$
(36)
although the interpretation of $\hat{w}(0,0)$ as the parity on the cylinder is
problematic.
All these properties automatically guarantee that we have indeed a well-
behaved Wigner function for this canonical pair.
## V Examples
To work out the explicit form of the Wigner function for a given state, one
first needs to specify the phase $\alpha(\ell,\phi)$ in Eq. (31). For
convenience, in this paper the choice
$\alpha(\ell,\phi)=-\ell\phi/2$ (37)
shall be used, as it is linear in both arguments, and it appears to be the
simplest function fulfilling the unitarity condition and the periodicity in
$\phi$ Rigas et al. (2008).
In this case, the Wigner kernel (34) becomes
$\displaystyle\hat{w}(\ell,\phi)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{(2\pi)^{2}}\sum_{{\ell^{\prime},\ell^{\prime\prime}}\in\mathbb{Z}}\int_{2\pi}e^{i\ell^{\prime}\phi^{\prime}/2}\,e^{-i\ell^{\prime\prime}\phi^{\prime}}$
(38) $\displaystyle\times$ $\displaystyle\displaystyle
e^{i(\ell\phi^{\prime}-\ell^{\prime}\phi)}|\ell^{\prime\prime}\rangle\langle\ell^{\prime\prime}-\ell^{\prime}|\,d\phi^{\prime}\,.$
After some manipulations, we obtain
$\displaystyle\hat{w}(\ell,\phi)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{2\pi}\sum_{{\ell^{\prime}}\in\mathbb{Z}}e^{-2i\ell^{\prime}\phi}|\ell+\ell^{\prime}\rangle\langle\ell-\ell^{\prime}|$
(39) $\displaystyle+$
$\displaystyle\displaystyle\frac{1}{2\pi^{2}}\sum_{{\ell^{\prime},\ell^{\prime\prime}}\in\mathbb{Z}}\frac{(-1)^{\ell^{\prime\prime}}}{\ell^{\prime\prime}+1/2}e^{-(2\ell^{\prime}+1)i\phi}$
$\displaystyle\times$
$\displaystyle|\ell+\ell^{\prime\prime}+\ell^{\prime}+1\rangle\langle\ell+\ell^{\prime\prime}-\ell^{\prime}|\,,$
which coincides with the kernel derived by Plebanski and coworkers Plebański
et al. (2000) in the context of deformation quantization.
Note that (39) splits into “even” and “odd” parts, depending on whether the
matrix elements
$\varrho_{\ell\ell^{\prime}}=\langle\ell|\hat{\varrho}|\ell^{\prime}\rangle$
have $\ell\pm\ell^{\prime}$ even (first sum) or odd (second sum).
For an angular momentum eigenstate $|\ell_{0}\rangle$, one immediately gets
$W_{|\ell_{0}\rangle}(\ell,\phi)=\frac{1}{2\pi}\delta_{\ell,\ell_{0}}\,,$ (40)
which is quite reasonable in this case: it is flat in $\phi$ and the integral
over the whole phase space gives the unity, reflecting the normalization of
$|\ell_{0}\rangle$.
For an angle eigenstate $|\phi_{0}\rangle$, one has
$W_{|\phi_{0}\rangle}(\ell,\phi)=\frac{1}{2\pi}\,\delta_{2\pi}(\phi-\phi_{0})\,.$
(41)
Now, the Wigner function is flat in the conjugate variable $\ell$, and thus,
the integral over the whole phase space diverges, which is a consequence of
the fact that the state $|\phi_{0}\rangle$ is not normalizable.
The coherent states $|\ell_{0},\phi_{0}\rangle$ (parametrized by points on the
cylinder) introduced in Ref. Kowalski et al. (1996) (see also Refs. González
and del Olmo (1998); Kastrup (2006) for a detailed discussion of the
properties of these relevant states) are characterized by
$\displaystyle\langle\ell|\ell_{0},\phi_{0}\rangle$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{\sqrt{\vartheta_{3}\left(0\big{|}\frac{1}{e}\right)}}e^{-i\ell\phi_{0}}\,e^{-(\ell-\ell_{0})^{2}/2}\,,$
$\displaystyle\langle\phi|\ell_{0},\phi_{0}\rangle$ $\displaystyle=$
$\displaystyle\displaystyle\frac{e^{i\ell_{0}(\phi-\phi_{0})}}{\sqrt{\vartheta_{3}\left(0\big{|}\frac{1}{e}\right)}}\vartheta_{3}\left(\frac{\phi-\phi_{0}}{2}\Big{|}\frac{1}{e^{2}}\right),$
where $\vartheta_{3}$ denotes the third Jacobi theta function Mumford (1983).
The Wigner function for the state $|\ell_{0},\phi_{0}\rangle$ splits as
$W_{|\ell_{0},\phi_{0}\rangle}(\ell,\phi)=W^{(+)}_{|\ell_{0},\phi_{0}\rangle}(\ell,\phi)+W^{(-)}_{|\ell_{0},\phi_{0}\rangle}(\ell,\phi)\,.$
(43)
The “even” part turns out to be
$W^{(+)}_{|\ell_{0},\phi_{0}\rangle}(\ell,\phi)=\frac{1}{2\pi\vartheta_{3}\left(0\big{|}\frac{1}{e}\right)}e^{-(\ell-\ell_{0})^{2}}\vartheta_{3}\left(\phi-\phi_{0}\Big{|}\frac{1}{e}\right)\,.$
(44)
This seems a sensible result, since it is a discrete Gaussian in the variable
$\ell$, and for the continuous angle $\phi$ it is a Jacobi theta function,
which plays the role of the Gaussian for circular statistics Řeháček et al.
(2008). However, the “odd” contribution spoils this simple picture:
$\displaystyle W^{(-)}_{|\ell_{0},\phi_{0}\rangle}(\ell,\phi)$
$\displaystyle=$
$\displaystyle\frac{e^{i(\phi-\phi_{0})-1/2}}{2\pi^{2}\vartheta_{3}\left(0\big{|}\frac{1}{e}\right)}\vartheta_{3}\left(\phi-\phi_{0}+i/2\Big{|}\frac{1}{e}\right)$
(45) $\displaystyle\times$
$\displaystyle\sum_{{\ell^{\prime}}\in\mathbb{Z}}(-1)^{\ell^{\prime}-\ell+\ell_{0}}\frac{e^{-\ell^{\prime}{}^{2}-\ell^{\prime}}}{\ell^{\prime}+\ell_{0}-\ell+1/2}\,.$
Figure 1: Plot of the Wigner function for a coherent state with $\ell_{0}=0$
and $\phi_{0}=0$. The cylinder extends vertically from $\ell=-4$ to $\ell=+4$.
The two corresponding marginal distributions are shown.
In Fig. 1, the Wigner function for the coherent state
$|\ell_{0}=0,\phi_{0}=0\rangle$ is plotted on the discrete cylinder. A
pronounced peak at $\phi=0$ for $\ell=0$ and slightly smaller ones for
$\ell=\pm 1$ can be observed. The associated marginal distributions [obtained
from Eq. (43) by integrating over $\phi$ or by summing over $\ell$,
respectively] are also plotted. They are strictly positive, as correspond to
true probability distributions.
Figure 2: Unwrapped plot of the Wigner function for a coherent state with
$\ell_{0}=0$ and $\phi_{0}=0$. The plane extends from $\ell=-4$ to $\ell=+4$
and from $\phi=-\pi$ to $\phi=\pi$.
For quantitative comparisons, however, sometimes it may be convenient to “cut”
this cylindrical plot along a line $\phi$=constant and unwrap it. This is
shown in Fig. 2. Here, the range of $\ell$ is from -4 to 4, while the angle is
plotted between $-\pi$ to $\pi$.
A closer look at these figures reveals also a remarkable fact: for values
close to $\phi=\pm\pi$ and $\ell=\pm 1$, the Wigner function takes negative
values. Actually, a numeric analysis suggests the existence of negativities
close to $\phi=\pm\pi$ for any odd value of $\ell$.
Figure 3: Plot and marginal distributions of the Wigner function for an even
superposition $|\ell_{1}+_{\theta}\ell_{2}\rangle$ with $\ell_{1,2}=\pm 3$ for
$\ell=-4$ to $\ell=+4$. Figure 4: Plot and marginal distributions of the
Wigner function for an even superposition $|\ell_{1}+_{\theta}\ell_{2}\rangle$
with $\ell_{1}=4,\ell_{2}=-3$ for $\ell=-4$ to $\ell=+5$.
As our last example, we address the superposition
$|\Psi\rangle=\frac{1}{\sqrt{2}}(|\ell_{1}\rangle+e^{i\phi_{0}}|\ell_{2}\rangle)$
(46)
of two angular-momentum eigenstates with a relative phase $e^{i\phi_{0}}$. The
analysis can be carried out for the superposition of any number of
eigenstates, but (46) is enough to display the relevant features.
The Wigner function splits again; now the “even” part reads as
$\displaystyle W_{|\Psi\rangle}^{(+)}(\ell,\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\\{\delta_{\ell,\ell_{1}}+\delta_{\ell,\ell_{2}}$
(47) $\displaystyle+$ $\displaystyle
2\delta_{\ell_{1}+\ell_{2},2\ell}\,\cos[\phi_{0}+(\ell_{2}-\ell_{1})\phi]\\}\,.$
For the “odd” part, the diagonal contributions vanish, and one has
$\displaystyle W_{|\Psi\rangle}^{(-)}(\ell,\phi)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{\pi^{2}}\cos[\phi_{0}+(\ell_{2}-\ell_{1})\phi]$
(48) $\displaystyle\times$
$\displaystyle\displaystyle\frac{(-1)^{\ell+(\ell_{1}+\ell_{2}-1)/2}}{\ell_{1}+\ell_{2}-2\ell}\delta_{\ell_{1}+\ell_{2}=\mathrm{odd}}\,,$
where $\delta_{\ell_{1}+\ell_{2}=\mathrm{odd}}$ indicates that the sum is
nonzero only when $\ell_{1}+\ell_{2}$ is odd.
In consequence, when $|\ell_{1}-\ell_{2}|$ is odd, the interference term
contains contributions for any $\ell$, damped as $1/\ell$. When
$|\ell_{1}-\ell_{2}|$ is an even number, the contribution (48) vanishes and we
have three contributions: two flat slices coming from the states
$|\ell_{1}\rangle$ and $|\ell_{2}\rangle$ and an interference term located at
$\ell=(\ell_{1}+\ell_{2})/2$.
These features are illustrated in Figs. 3 and 4. The state $|\Psi\rangle$ is
plotted for $\ell_{2}=-3$ and $\ell_{1}=3$ and (Fig. 3) and $\ell_{2}=-3$ and
$\ell_{1}=4$ (Fig. 4). Changing the relative phase $\phi_{0}$ results in a
global rotation of the cylinder. In can be observed in Fig. 4 that the two
rings at $\ell=-3$ and $\ell=4$ (as opposed to the rings at $\ell=\pm 3$ in
Fig. 3), are not flat in $\phi$, but show a weak dependence on the angle due
to the odd contributions added to the flat Kronecker deltas.
## VI Concluding remarks
In summary, we have carried out a full program for a complete phase-space
description in terms a Wigner function for the canonical pair angle-angular
momentum. An experimental demonstration in terms of optical beams is presently
underway in our laboratory.
###### Acknowledgements.
We acknowledge discussions with Hubert de Guise, Jose Gracia-Bondia, Hans
Kastrup, Jakub Rembielinski and Krzysztof Kowalski. This work was supported by
the Czech Ministry of Education, Project MSM6198959213, the Czech Grant
Agency, Grant 202/06/0307, the Spanish Research Directorate, Grant
FIS2005-06714, and the Mexican CONACYT, Grant 45705.
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|
arxiv-papers
| 2008-12-16T09:52:30 |
2024-09-04T02:48:59.433408
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "I. Rigas, L. L. Sanchez-Soto, A. B. Klimov, J. Rehacek and Z. Hradil",
"submitter": "Luis L. Sanchez. Soto",
"url": "https://arxiv.org/abs/0812.3013"
}
|
0812.3236
|
# On the Siegel-Weil Theorem for Loop Groups (I)
Howard Garland and Yongchang Zhu The second author’s research is supported by
Hong Kong Research Grant Council earmark grant number HKUST 604507
## 1 Introduction
The notion of an snt-module and our extension (1.14) of the Siegel-Weil
Theorem (also see Theorem 8.1, §8) grew out of our work on the Siegel-Weil
Theorem for arithmetic quotients of loop groups, which we prove in Part II of
this paper ([3]). In fact (1.14) is a vital step in our proof of the Siegel-
Weil theorem in the loop case, and as far as we know, it also seems to be a
new result for automorphic forms on certain, finite-dimensional, non-reductive
groups. At the same time, the theory of automorphic forms on arithmetic
quotients of loop groups, specifically a loop version of Godement’s criterion
(Theorem 5.3, Part II [3]) is used to prove the convergence theorem (Theorem
3.3) we need for the Eisenstein series ${\rm Et}$ defined in (3.10). At the
moment, even though the statement of Theorem 3.3 only involves finite-
dimensional groups, the proof using loop groups is the only one we have!
To state our main result of this part, we first recall the Siegel-Weil theorem
proved in [7].
Let $F$ be a number field, ${\bf A}$ be the ring of adeles of $F$, and $M$ be
a symplectic space over $F$ with symplectic pairing $\langle,\rangle$. Let
$(V,(\,,\,))$ be a finite dimensional vector space over $F$ with the non-
degenerate, symmetric, bilinear form $(\,,\,)$. The space $M\otimes_{F}V$ has
the symplectic form given by
$\langle u_{1}\otimes v_{1},u_{2}\otimes v_{2}\rangle=\langle
u_{1},u_{2}\rangle(v_{1},v_{2}).$
The group $Sp_{M}$ and the orthogonal group $G$ of $V$ form a dual pair in the
symplectic group $Sp_{2N}$ of $M\otimes_{F}V$ (where $2N={\rm dim}\,M\,{\rm
dim}\,V$). Let
$M=M_{-}\oplus M_{+}$ (1.1)
be a direct sum of Lagrangian subspaces of $M$, then
$M\otimes V=M_{-}\otimes V\oplus M_{+}\otimes V$
is a direct sum of Lagrangian subspaces of $M\otimes V$. The Hilbert space
$L^{2}((M_{-}\otimes V)_{\bf A})$ is an irreducible unitary representation of
the metaplectic group $\widehat{Sp}_{2N}({\bf A})$, which is called the Weil
representation. The dense subspace ${\cal S}((M_{-}\otimes V)_{\bf A}))$,
formed by the Schwartz-Bruhat functions on $(M_{-}\otimes V)_{\bf A}$, is
invariant under the action of $\widehat{Sp}_{2N}({\bf A})$. The theta
functional
$\theta:{\cal S}((M_{-}\otimes V)_{\bf A})\to{\mathbb{C}}$
given by
$\theta(\phi)=\sum_{r\in M_{-}\otimes V}\phi(r)$
is $Sp_{2N}(F)$-invariant.
The group $\widehat{Sp}_{2N}({\bf A})$ contains the commuting pair of groups
$\widehat{Sp}_{M}({\bf A})$ and $G({\bf A})$, where $\widehat{Sp}_{M}({\bf
A})$ is the preimage of $Sp_{M}({\bf A})$ in $\widehat{Sp}_{2N}({\bf A})$.
For a given $\phi\in{\cal S}((M_{-}\otimes V)_{\bf A})$, there are two simple
ways to construct a function on $Sp_{M}(F)\backslash\widehat{Sp}_{M}({\bf
A})$: one is
${\rm I}(\phi,g)\stackrel{{\scriptstyle\rm def}}{{=}}\int_{G(F)\backslash
G({\bf A})}\theta(\pi(g,h)\phi)dh,\,\,\,\,\,\,g\in\widehat{Sp}_{M}({\bf A}),$
(1.2)
where $\pi$ is the Weil representation and $dh$ is the Haar measure on $G({\bf
A})$ such that the volume of $G(F)\backslash G({\bf A})$ is $1$. For the
second way, we first consider the function $g\mapsto(\pi(g)\phi)(0)$, which is
$P({\bf A})$-invariant (where $P$ is the Siegel parabolic subgroup of $Sp_{M}$
that fixes $M_{+}$). We then form the Eisenstein series
${\rm E}(\phi,g)\stackrel{{\scriptstyle\rm def}}{{=}}\sum_{r\in P(F)\backslash
Sp_{M}(F)}(rg\cdot\phi)(0).$ (1.3)
This gives a function on $Sp_{M}(F)\backslash\widehat{Sp}_{M}({\bf A})$. When
${\rm dim}\,V>{\rm dim}\,M+2$, both (1.2) and (1.3) converge, and the the
Siegel-Weil formula asserts that the above two constructions are equal, i.e.
${\rm I}(\phi,g)={\rm E}({\phi},g).$ (1.4)
Weil [7] proved such an identity in more generality for dual pairs constructed
from semisimple algebras with involutions. Generalizations of this formula to
non-convergent cases can be found in [5].
Put ${\rm I}({\phi})\stackrel{{\scriptstyle\rm def}}{{=}}{\rm I}({\phi},e)$
and ${\rm E}({\phi})\stackrel{{\scriptstyle\rm def}}{{=}}{\rm E}({\phi},e)$,
we have
${\rm I}({\phi})={\rm E}(\phi)$ (1.5)
Since ${\rm I}({\phi},g)={\rm I}({g\cdot\phi})$ and ${\rm E}(\phi,g)={\rm
E}({g\cdot\phi})$, we see that (1.4) and (1.5) are equivalent.
Since the cosets $P(F)\backslash Sp_{M}(F)$ are in one-to-one correspondence
with the set $Gr(M)$ of Lagrangian subspaces of $M$, via the map $Pg\mapsto
M_{+}g$ (we assume the symplectic group acts on $M$ from the right). The
Eisenstein series (1.3) can also be written as a summation over $Gr(M)$ as
follows. Let $\pi_{-}:M\to M_{-}$ denote the projection with respect to the
decomposition (1.1). For $r\in P(F)\backslash Sp_{M}(F)$, let $U=M_{+}r$ .
Then the symplectic pairing $\pi_{-}(U)\times M_{+}\to F$ factors through a
non-degenerate pairing $\pi_{-}(U)\times M_{+}/(M_{+}\cap U)\to F$. For each
$v\in\pi_{-}(U)$, let $\tilde{v}\in U$ be a lifting of $v$, write
$\tilde{v}=\tilde{v}_{-}+\tilde{v}_{+}$ according the decomposition (1.1),
then the map
$\rho:\pi_{-}(U)\to M_{+}/(M_{+}\cap
U),\,\,\,\,v\mapsto\tilde{v}_{+}+M_{+}\cap U$ (1.6)
is well-defined. One can prove that
$\pi(r)\phi(0)=\int_{(\pi_{-}(U)\otimes V)_{\bf A}}\psi(\frac{1}{2}\langle
x,\rho x\rangle)\phi(x)dx\stackrel{{\scriptstyle\rm def}}{{=}}E(\phi,U),$
(1.7)
where $\psi$ is the additive character of ${\bf A}/F$ used in the definition
of the Weil representation, and the measure on the right hand side is the Haar
measure on $(\pi_{-}(U)\otimes V)_{\bf A}$ normalized by the condition that
the covolume of the lattice $\pi_{-}(U)\otimes V$ is $1$. So the Eisenstein
series (1.3) can be written as
${\rm E}(\phi)=\sum_{U\in Gr(M)}E(\phi,U).$
In our generalization of the Siegel-Weil formula (1.5), we assume the
symplectic space $M$ has an additional structure, which we call an snt-module.
The groups involved are no longer reductive groups (examples:
$Sp_{n}(F[t]/(t^{k}))$ and $G(F[t]/(t^{k}))$) . To state our generalization,
we define
###### Definition 1.1
By a symplectic, nilpotent $t$-module (=snt-module) $M$, we mean an
$F[t]$-module which is finite-dimensional when considered as a vector space
over $F,$ and which is equipped a symplectic form $\langle\,,\,\rangle$ such
that the following conditions are satisfied
(i) there exists a positive integer $N>0$ such that
$t^{N}\cdot\xi=0,\,\,\,{\rm for\,all}\,\,\xi\in M.$
(ii). The operator $t$ is self-dual, i.e.
$\langle t\xi,\eta\rangle=\langle\xi,t\eta\rangle$ (1.8)
Since $t^{N}=0$ on $M$, we may regard $M$ as an $F[[t]]$-module. We give a
simple example of an snt-module. Consider $F[[t]]$-module
$H_{k}=F[[t]]/(t^{k})\oplus F[[t]]/(t^{k}),$ (1.9)
with a symplectic form $\langle,\rangle$ defined by the conditions
(i). Each of the two summands is isotropic.
(ii).
$\langle(t^{i},0),(0,t^{j})\rangle=\left\\{\begin{array}[]{c}0,\;i+j\neq
k-1\\\ 1,\;i+j=k-1\end{array}\right.,$ where $i,j=0,1,....,k-1.$
We shall prove later (see Lemma 2.2) that every snt-module is a direct sum of
the above examples. For a given snt-module $M$, $g\in GL(M)$ is called an snt-
module automorphism if $g$ preserves both the $F[[t]]$-module structure and
the symplectic structure. We denote by
$Sp(M,t)$
the group of all snt-automorphisms on $M$.
For a given snt-module $M$ and a space $V$ with non-degenerate, bilinear,
symmetric form $(\,,\,)$, then the space
$M\otimes_{F}V=M\otimes_{F[[t]]}V[[t]]$ (1.10)
has a natural snt-module structure, where $V[[t]]=V\otimes_{F}F[[t]]$. And the
symplectic form is defined using the first tensor product:
$\langle x_{1}\otimes v_{1},x_{2}\otimes v_{2}\rangle=\langle
x_{1},x_{2}\rangle(v_{1},v_{2})$
and the $F[[t]]$-module structure is defined using the second tensor product.
Let $Sp_{2N}$ ($2N={\rm dim}\,M{\rm dim}\,V$) denote the symplectic group of
the symplectic space $M\otimes_{F}V$, the group $Sp(M,t)$ is a subgroup of
$Sp_{2N}$. The orthogonal group $G(F)$ acts on $M\otimes_{F}V$ preserving the
snt-module structure. But a larger group $G(F[[t]])$ acts on (1.10): for
$x\otimes v\in M\otimes_{F[[t]]}V[[t]]$, $g\in G(F[[t]])$,
$(x\otimes v)\cdot g=x\otimes(v\cdot g).$
It is easy to see that the action preserves both the symplectic structure and
the $F[[t]]$-module structure, so we also have a group morphism
$G(F[[t]])\to Sp_{2N}.$
We denote $G^{q}(F[[t]])$ the image. The subgroups $Sp(M,t)$ and
$G^{q}(F[[t]])$ in $Sp(M\otimes V,t)$ obviously commute. Our generalization of
the Siegel-Weil formula is concerned with the commuting pair
($Sp(M,t),G^{q}(F[[t]])$), which are not reductive groups in general. Let
$Gr(M,t)$
denote the set of all $F[[t]]$-stable Lagrangian subspaces of $M$, so
$Gr(M,t)\subset Gr(M)$.
We take a direct sum decomposition $M=M_{-}\oplus M_{+}$ such that $M_{-}\in
Gr(M),M_{+}\in Gr(M,t)$. As before $L^{2}((M_{-}\otimes V)_{\bf A})$ is a
representation of ${\widehat{S}p}_{2N}({\bf A})$, with the theta functional
$\theta:{\cal S}((M_{-}\otimes V)_{\bf
A})\to{\mathbb{C}},\,\,\,\phi\mapsto\theta(\phi)=\sum_{r\in M_{-}\otimes
V}\phi(r).$
For a subspace $U\in Gr(M)$, let $E(\phi,U)$ be as in (1.7), and we define
${\rm Et}(\phi)\stackrel{{\scriptstyle\rm def}}{{=}}\sum_{W\in
Gr(M,t)}E(\phi,W).$ (1.11)
And we define
${\rm It}(\phi)\stackrel{{\scriptstyle\rm
def}}{{=}}\int_{G^{q}(F[[t]])\backslash G^{q}({\bf
A}[[t]])}\theta(h\cdot\phi)dh,$ (1.12)
where $dh$ denotes the Haar measure on $G^{q}({\bf A}[[t]])$ such that the
volume of $G^{q}(F[[t]])\backslash G^{q}({\bf A}[[t]])$ is $1$.
By Lemma 2.2, $M$ is isomorphic to a direct sum of $n$ copies of $H_{k}$’s:
$M\cong H_{k_{1}}\oplus\dots\oplus H_{k_{n}}$ (1.13)
assume ${\rm dim}\,V>6n+2$, and the quadratic form $(\,)$ on $V$ is
$F$-anisotropic or ${\rm dim}V-r>\frac{1}{2}{\rm dim}\,M+1$, where $r$ is the
dimension of a maximal isotropic subspace of $V$, then we have the following
generalization of the Siegel-Weil formula (Theorem 7.3, Theorem 8.1) :
${\rm Et}(\phi)={\rm It}(\phi).$ (1.14)
The condition ${\rm dim}\,V>6n+2$ is for the convergence of ${\rm Et}(\phi)$,
and the condition that $(V,(,))$ is $F$-anisotropic or ${\rm
dim}\,V-r>\frac{1}{2}{\rm dim}\,M+1$ is for convergence of ${\rm It}(\phi)$.
This formula reduces to the classical formula (1.5) when
$k_{1}=\dots=k_{n}=1$.
In general, the $Sp(M,t)$-action on $Gr(M,t)$ is not transitive, but there are
only finitely many orbits, so the sum ${\rm Et}(\phi)=\sum E(U,\phi)$ is a sum
of Eisenstein series induced from several ”parabolic” subgroups (rather than
only one), and each corresponds to a $Sp(M,t)$-orbit in $Gr(M,t)$.
In the case that $M$ is a direct sum of $n$-copies of the snt-module $H_{k}$,
then $Sp(M,t)=Sp_{2n}(F[t]/(t^{k}))$ and $G^{q}(F[[t]])=G(F[t]/(t^{k}))$. The
formula (1.14) means that the Siegel-Weil formula holds for symplectic and
orthogonal groups over $F[t]/(t^{k})$. It is implies that the Siegel-Weil
formula holds for symplectic and orthogonal groups over $F[t]/(p(t))$ for
arbitrary polynomial $p(t)$.
We now give an explicit example of our formula. Let $F={\mathbb{Q}}$, and the
snt - module $M$ be
$M={\mathbb{Q}}[t]/(t^{2})\oplus{\mathbb{Q}}/(t^{2})$ (1.15)
with the snt-module structure given as in (1.9). And we take a positive
definite even unimodular lattice $L$ of rank $N$ with the bilinear form given
by $(,)$, and let $V={\mathbb{Q}}L$. It is well-known that $N$ is divisible by
$8$. Let $L_{1},\dots,L_{g}$ be the list of the positive definite even
unimodular lattices of rank $N$ (up to isomorphism), let $()_{j}$ denote the
the pairing of $L_{j}$, and $|{\rm Aut}_{j}|$ be the order of automorphism
group of $L_{j}$. We denote $1\oplus 0$ and $0\oplus 1$ in (1.15) by $e_{1}$
and $e_{2}$ respectively, then $e_{1},te_{1},e_{2},te_{2}$ is a
${\mathbb{Q}}$-basis of $M$. Let $M_{-}={\mathbb{Q}}e_{1}+{\mathbb{Q}}te_{1}$
and $M_{+}={\mathbb{Q}}te_{2}+{\mathbb{Q}}e_{2}$. It is clear that $M_{-}$ and
$M_{+}$ are Lagrangian subspaces of $M$ and
$M=M_{-}\oplus M_{+}.$
We have the Weil representation of $Sp_{4N}({\bf A})$ on
${\cal S}((M_{-}\otimes V)_{\bf A}).$
Take $\phi=\Pi_{v}\phi_{v}\in{\cal S}((M_{-}\otimes V)_{\bf A})$ as follows:
for a finite place $p$ of ${\mathbb{Q}}$, $\phi_{p}$ is the characteristic
function of $e_{1}\otimes L_{{\bf Z}_{p}}+te_{1}\otimes L_{{\bf Z}_{p}}$
(where ${\bf Z}_{p}$ denotes the ring of $p$-adic integers and $L_{{\bf
Z}_{p}}=L\otimes{\bf Z}_{p}$); for the real place $\infty$ of ${\mathbb{Q}}$,
$\phi_{\infty}(e_{1}\otimes v_{1}+te_{1}\otimes v_{2})=e^{\pi
i\tau_{1}(v_{1},v_{1})+\pi i\frac{-1}{\tau_{2}}(v_{2},v_{2})},$
where $\tau_{1},\tau_{2}$ are complex numbers in the upper half plane. With
the above choice of $\phi$, our new Siegel-Weil formula (1.14) becomes
$\displaystyle 1+\frac{1}{2}\sum_{a\geq 1,b\in{\bf
Z}:(a,b)=1}\,\sum_{m,n\in{\bf
Z}:(m,n)=1}(am^{2}\tau_{1}+an^{2}\tau_{2}+b)^{-\frac{N}{2}}$ (1.16)
$\displaystyle=$ $\displaystyle C\sum_{j=1}^{g}\frac{1}{|{\rm
Aut}_{j}|}\sum_{u,v\in L_{j},u,v\,{\rm colinear}}e^{\pi i\tau_{1}(u,u)+\pi
i\tau_{2}(v,v)}$
where $(m,n)=1$ ( $(a,b)=1$) means $m,n$ (resp. $a,b$) are relatively prime,
and $u,v$ colinear means the ${\mathbb{Q}}$-span of $u,v$ is at most
$1$-dimensional. And the constant $C$ is given by (1.18) below.
We compare (1.16) with the classical Siegel-Weil formula
$\frac{1}{2}\sum_{m,n\in{\bf
Z},(m,n)=1}(m\tau+n)^{-\frac{N}{2}}=C\sum_{j=1}^{g}\frac{1}{|{\rm
Aut}_{j}|}\sum_{u\in L_{j}}e^{\pi i\tau(u,u)_{j}}$ (1.17)
This formula expresses the $\frac{N}{2}$-th Eisenstein series of
$SL(2,{\mathbb{Z}})$ as a sum of Theta series. The constant term in
$q$-expansion of both sides gives the density formula
$1=C\sum_{j=1}^{g}\frac{1}{|{\rm Aut}_{j}|},$ (1.18)
which determines $C$.
If we take more general test function $\phi$, our Siegel-Weil formula (1.14)
is
$\displaystyle 1+\frac{1}{2}\sum_{a\geq 1,b\in{\bf
Z}:(a,b)=1}\,\sum_{m,n\in{\bf
Z}:(m,n)=1}(am^{2}\tau_{11}+2amn\tau_{12}+an^{2}\tau_{22}+b)^{-\frac{N}{2}}$
(1.19) $\displaystyle=$ $\displaystyle C\sum_{j=1}^{g}\frac{1}{|{\rm
Aut}_{j}|}\sum_{u,v\in L_{j},u,v\,{\rm colinear}}e^{\pi
i\left(\tau_{11}(u,u)+2\tau_{12}(u,v)+\tau_{22}(v,v)\right)},$
where $C$ is determined by (1.18).
The paper is organized as follows: In §2 we study the structure of an snt-
module $M$ (Lemma 2.2) and the structure of the corresponding group $Sp(M,t)$
of snt-automorphisms of $M$ (Cor. 2.6). In §3 we recall basic facts about the
Weil representation and study the Eisenstein series ${\rm Et}(\phi)$ for the
Weil representation associated to an snt-module. In particular, we state the
convergence theorem (Theorem 3.3) for such Eisenstein series. As we already
stated, the proof depends on the convergence theorem for Eisenstein series on
loop groups, and will be given in Part II of this paper ([3]). We also state a
consequence, Theorem 3.4, of Theorem 3.3 together with Prop. 1 and Prop. 2 of
[7].
In §4 we extend a result in Weil [7] and obtain Theorem 4.7, which identifies
certain abstract measures associated with the map $T_{W}$ (see (3.13) for the
definition of $T_{W})$ with certain gauge measures in the sense of [7]. In §5
we obtain Theorems 5.4 and 5.8. In particular, we identify the space of
$G(F[[t]])$ \- orbits in $M_{\\_}\otimes V$ with the set of pairs $(W,i)$ with
$W\in Gr(M_{\\_},t)$ (the $t$ \- Grassmannian), $i\in S_{t}^{2}(W)$ such that
$U(i)_{F}$ is non-empty ( $U(i)_{F}$ is defined in §5, just before the
statement of Theorem 5.8).
In §6 we discuss $\theta$ \- series and finally, we prove Theorems 7.3 and
8.1, our versions of the Siegel-Weil theorem for snt-modules.
## 2 The structure of symplectic, nilpotent $t$-modules.
In this section, we prove some results about the structure of symplectic,
nilpotent $t$-modules (snt-modules) defined in Section 1 (Definition 1.1).
###### Lemma 2.1
Let $M$ be an snt-module. For all $\xi\in M$, $k\in\mathbb{Z}_{>0}$, we have
$\langle\xi,t^{k}\xi\rangle=0.$
Proof. We have for all $\xi\in M$, $k\in\mathbb{Z}_{\geq 0},$
$\langle\xi,t^{k}\cdot\xi\rangle=-\langle t^{k}\cdot\xi,\xi\rangle,$
since $\langle\cdot,\cdot\rangle$ is skew symmetric. On the other hand
$\langle\xi,t^{k}\cdot\xi\rangle=\langle t^{k}\cdot\xi,\xi\rangle$
by (1.8). Hence
$\langle\xi,t^{k}\cdot\xi\rangle=0,$
$\Box$
###### Lemma 2.2
Every snt-module $M$ is isomorphic to a direct sum
$M\cong H_{k_{1}}\oplus....\oplus H_{k_{n}}$
$k_{1}\geq k_{2}\geq....\geq k_{n}$. Where $H_{k}$ is given as (1.9).
Moreover, $n$ and the $k_{i}$ are uniquely determined by $M$.
Proof. The uniqueness is clear from the theory of elementary divisors. For
$\xi\in M$, we define the order of $\xi$ to be the smallest positive integer
$k,$ such that $t^{k}\cdot\xi=0.$ Pick $\xi\in M$ of maximal order $(N$ say).
Then the vectors
$\xi,t\cdot\xi,....,t^{N-1}\cdot\xi$
are linearly independent over $F.$ To see this, we consider $F[[t]]$-submodule
$F[[t]]\xi$. Since $F[[t]]$ is a PID, $F[[t]]\xi$ is isomorphic to
$F[[t]]/(t^{k})$. It is clear that $k=N$.
Since $\langle\,,\,\rangle$ is non-degenerate, we can find $\eta\in M$, so
that
$\langle t^{N-1}\cdot\xi,\eta\rangle=1,\,\,\,\,\langle
t^{j}\cdot\xi,\eta\rangle=0,\,\,\,j=0,1,...,N-2$
But then,
$\langle\xi,t^{N-1}\cdot\eta\rangle=\langle t^{N-1}\cdot\xi,\eta\rangle=1$
Hence
$t^{N-1}\cdot\eta\neq 0,$
and so
$\eta,t\cdot\eta,....,t^{N-1}\cdot\eta$
are linearly independent (by the above argument) and
$t^{N}\cdot\eta=0$
(since $N$ was assumed the maximal order of any element of $M$). But then if
$C_{1}=F\text{-span of }\xi,t\cdot\xi,....,t^{N-1}\cdot\xi,$
$C_{2}=F\text{-span of }\eta,t\cdot\eta,....,t^{N-1}\cdot\eta,$
we see that
$H=C_{1}\oplus C_{2}$
with symplectic structure given by the restriction of $\langle,\rangle$ on $M$
is isomorphic to the snt-module $H_{N}$, with $C_{1},$ $C_{2}$ corresponding
to the two direct summands $F[[t]]/(t^{N})$. Since $\langle,\rangle$
restricted to $H$ is non-degenerate, $M$ decomposes as a direct sum of
$snt$-modules
$M\cong H\oplus H^{\bot},$
and applying the induction hypothesis to $H^{\bot},$ we obtain the lemma.
$\Box$
For an snt-module $M$, so $M$ is in particular a symplectic space. As in
Section 1, we let $Gr(M)$ denote the set of all Lagrangian subspaces, and
$Gr(M,t)$ denote the set of Lagrangian subspaces which are stable under the
action of $F[[t]]$, so $Gr(M,t)\subset Gr(M)$. We call elements in $Gr(M,t)$
$t$-Lagrangian subspaces. We have:
###### Lemma 2.3
If $U\subset M$ is a $F[[t]]$-stable, isotropic and it is not properly
contained in any other $F[[t]]$-stable, isotropic subspaces, then $U$ is
$t$-Lagrangian, i.e. $U\in Gr(M,t)$.
Proof. Assume $U$ is not maximal isotropic. Then there exists $v\in M$,
$v\notin U,$ such that
$\langle v,u\rangle=0,\,\,\text{ all }\,u\in U.$
But then for all $u\in U,$ $i\in\mathbb{Z}_{\geq 0}$
$\langle t^{i}\cdot v,u\rangle=\langle v,t^{i}\cdot u\rangle=0,$
since $t^{i}\cdot u\in U.$
Hence, the space $Span_{F[[t]]}\\{v,U\\}$ is $F[[t]]$-stable, isotropic and
contains $M$ properly. This contradicts the maximality of $M$. $\Box$
As in Section 1, we let $Sp(M,t)$ be the subgroup of all $\sigma\in Sp(M)$
such that $\sigma$ is an $F[[t]]$-module automorphism. For a finite
dimensional $F[[t]]$-module $U$, a subset of non-zero elements
$e_{1},\dots,e_{n}$ is called a quasi-basis of $U$ if every element in $U$ can
be written as an $F[[t]]$-linear combination of $e_{i}$’s and
$a_{1}e_{1}+\dots+a_{n}e_{n}=0$ implies that all $a_{i}e_{i}=0$. For example,
for
$U=F[[t]]/(t^{k_{1}})\oplus\dots\oplus F[[t]]/(t^{k_{n}}),$
the set (n elements) $(1,0,\dots,0),(0,1,\dots,0),\dots(0,\dots,1)$ is a
quasi-basis. If $U$ has a quasi-basis consisting of $n$-elements, then the
$F$-vector space $\bar{U}\stackrel{{\scriptstyle\rm def}}{{=}}U/tU$ is
$n$-dimensional. If we have two such $F[[t]]$-modules $U_{1}$ and $U_{2}$, and
$T:U_{1}\to U_{2}$ is a morphism, then $T$ induces a $F$-linear map
$\bar{T}:{\bar{U}}_{1}\to{\bar{U}}_{2}$.
An snt-module $M$ with decomposition as Lemma 2.2 is called homogeneous if
$k_{1}=k_{2}=....=k_{n}=k$. In this case the group $Sp(M,t)$ is determined by
###### Lemma 2.4
Let $M$ be a homogeneous snt-module as in (1.13) with $k_{1}=\dots=k_{n}=k$.
Then $Sp(M,t)$ is isomorphic to $Sp_{2n}(F[[t]]/(t^{k}))$. In particular, its
reduction ${\rm mod}\,t$ defines a surjective group homomorphism
$\pi_{0}:Sp(M,t)\rightarrow Sp_{2n}(F),$
Proof. We first consider the free $F[[t]]/(t^{k})$-module
$(F[[t]]/(t^{k}))^{2n}$. It has as standard $F[[t]]/(t^{k})$-valued symplectic
form $\langle\,,\,\rangle^{\wedge}$. We define an $F$-valued symplectic form
$\langle\,,\,\rangle$ on $(F[[t]]/(t^{k}))^{2n}$ by
$\langle a,b\rangle={\rm coefficient\,\,of\,\,}\,\,t^{k-1}\,\,{\rm
in}\,\,\langle a,b\rangle^{\wedge}.$
With this $\langle\,,\,\rangle$, $(F[[t]]/(t^{k}))^{2n}$ is an snt-module,
which is clearly isomorphic to $M$ in the lemma. It follows from the
construction that $Sp_{2n}(F[[t]]/(t^{k}))$ preserves the snt-module
structure, so we have
$Sp_{2r}(F[[t]]/(t^{k}))\subset Sp(M,t).$
For the converse inclusion is also clear. $\Box$
For a homogeneous snt-module $M$ as in Lemma 2.4 , $\bar{M}=M/tM$ has a
symplectic structure defined as follows, if $\bar{a},\bar{b}\in\bar{M}$, let
$a,b\in M$ be their liftings, then
$\langle\bar{a},\bar{b}\rangle\stackrel{{\scriptstyle\rm def}}{{=}}\langle
a,t^{k-1}b\rangle.$ (2.1)
Now we turn to the case of general (possibly non-homogeneous) snt-modules $M$,
with direct sum decomposition as in Lemma 2.2. For fixed $k$, we let
$M(k)=\oplus_{k_{i}=k}H_{k_{i}},$
so
$M=M(l_{1})\oplus...\oplus M(l_{s}),\,\,\,\,\,l_{1}>l_{2}>....>l_{s},$ (2.2)
with each $M(l_{i})$ a homogeneous snt-submodule. The $M(l_{i})$’s are
mutually orthogonal with respect to $\langle,\rangle$.
Now if $\sigma\in Sp(M,t),$ then $\sigma$ (acting on the right, recall) has a
block decomposition with respect to (2.2),
$\sigma=\left[\begin{array}[]{c}\sigma_{1}^{1}....\sigma_{s}^{1}\\\
........\\\ \sigma_{1}^{s}....\sigma_{s}^{s}\end{array}\right]$ (2.3)
where $\sigma_{j}^{i}:M(l_{i})\rightarrow M(l_{j}),$ and for
$\xi=(\xi_{1},....,\xi_{s})\in M$ $(\xi_{i}\in M(l_{i}))$
$\xi\sigma=(\xi_{1},....,\xi_{s})\left[\begin{array}[]{c}\sigma_{1}^{1}....\sigma_{s}^{1}\\\
........\\\ \sigma_{1}^{s}....\sigma_{s}^{s}\end{array}\right].$
We have also the decomposition ${\bar{M}}=M/tM$ induced from the decomposition
(2.2),
${\bar{M}}={\bar{M}}(l_{1})\oplus...\oplus{\bar{M}}(l_{s}),\,\,\,\,l_{1}>l_{2}>....>l_{s},$
(2.4)
So $\bar{\sigma}:{\bar{X}}\to{\bar{X}}$ has a block decomposition:
$\bar{\sigma}=\left[\begin{array}[]{c}\bar{\sigma}_{1}^{1}....\bar{\sigma}_{s}^{1}\\\
........\\\ \bar{\sigma}_{1}^{s}....\bar{\sigma}_{s}^{s}\end{array}\right]$
(2.5)
###### Lemma 2.5
The matrix $\bar{\sigma}$ is block-upper triangular, i.e.
$\bar{\sigma}_{i}^{j}=0\,\,{\rm for}\,\,i<j$ (2.6)
and the diagonal block $\bar{\sigma}_{i}^{i}$ is in $Sp(\bar{X}(l_{i}))$
(recall $\bar{X}(l_{i})$ has the symplectic structure defined by (2.1)).
Proof. For $i<j$, $v_{j}\in M(l_{j})$, we have $t^{l_{j}}v_{j}=0$ and
$\sigma_{i}^{j}$ is $t$-linear, this implies that
$t^{i}(v_{j}\sigma_{i}^{j})=0$, then
$v_{j}\sigma_{i}^{j}\in t^{l_{i}-l_{j}}M(l_{i})$ (2.7)
This implies that $\bar{\sigma}_{i}^{j}=0$. For $a,b\in M(l_{i})$, we have
$\langle t^{l_{i}-1}a,b\rangle=\langle
t^{l_{i}-1}a\sigma,b\sigma\rangle=\sum_{j=1}^{s}\langle
t^{l_{i}-1}a\sigma_{j}^{i},b\sigma_{j}^{i}\rangle,$ (2.8)
if $j<i$, by (2.7) we have
$t^{l_{i}-1}a\sigma_{j}^{i}\in
t^{l_{i}-1}t^{l_{j}-l_{i}}M(l_{j})=t^{l_{j}-1}M(l_{j})$
and
$b\sigma_{j}^{i}\in t^{l_{j}-l_{i}}M(l_{j})$
it implies that
$\langle t^{l_{i}-1}a\sigma_{j}^{i},b\sigma_{j}^{i}\rangle=0.$
For $j>i$, then $l_{i}-1\geq l_{j}$, we have
$t^{l_{i}-1}a\sigma_{j}^{i}\in t^{l_{i}-1}M(l_{j})=0.$
So (2.8) gives that
$\langle t^{l_{i}-1}a,b\rangle=\langle
t^{l_{i}-1}a\sigma_{i}^{i},b\sigma_{i}^{i}\rangle,$
by the definition (2.1) of the symplectic form on ${\bar{M}}(l_{i})$, we prove
that $\bar{\sigma}_{i}^{i}$ is a symplectic isomorphism of ${\bar{M}}(l_{i})$.
$\Box$
###### Corollary 2.6
Let $M$ be an snt-module as in (2.2), then $Sp(M,t)$ is the semi-direct
product
$Sp(M,t)=N\ltimes H,$
where $N$ is the unipotent radical of $Sp(M,t)$ and
$H\cong\Pi_{i=1}^{s}Sp_{2r_{i}}(F),$
where $r_{i}$ is the number $H_{l_{i}}$’s in the decomposition of $M(l_{i})$.
Next we discuss the classification of $t$-Lagrangian subspaces. First for the
snt-module (1.9), let $e_{1},e_{2}$ denote $(1,0),(0,1)$ respectively. For
each $0\leq i\leq k-1$, let $L_{i}$ be the $F$-subspace with basis
$t^{i}e_{1},t^{i+1}e_{1},\dots,t^{k-1}e_{1},t^{k-i}e_{2},t^{k-i+1}e_{2},\dots,t^{k-1}e_{2}.$
It is clear that $L_{i}$ is a $t$-Lagrangian subspace. For an snt-module $M$
with decomposition as in Lemma 2.2, let $L_{i_{j}}\subset H_{k_{j}}$ be the
subspace described as above, it is clear that
$L_{i_{1}}\oplus\dots\oplus L_{i_{n}}$ (2.9)
is an $t$-Lagrangian subspace of $M$. We have
###### Proposition 2.7
Let $M$ be a snt-module with decomposition as in Lemma 2.2, then every
$t$-Lagrangian subspace can be tranformed by some $g\in Sp(M,t)$ to an
$t$-Lagrangian subspace as in (2.9).
This proposition will not be used later, we skip its proof.
## 3 The Weil representation and $t$-Eisenstein series.
In this section, we first recall some basic facts about the Weil
representation associated to a symplectic space over $F$, then we study the
Eisenstein series ${\rm Et}(\phi)$ (1.11) for the Weil representations
associated to an snt-module.
We shall fix a non-trivial additive character $\psi:{\bf A}\to S^{1}$ that is
trivial on $F$. Let $F^{2N}$ be the standard symplectic space over $F$, and
$C=C_{-}\oplus C_{+}$ be a direct sum into Lagrangian subspaces; then a two-
fold cover, denoted by $\widehat{Sp}_{2N}({\bf A})$, of the adelic group
$Sp_{2N}({\bf A})$ acts on $L^{2}(C_{-,{\bf A}})$. The subspace ${\cal
S}(C_{-,{\bf A}})$, formed by the Schwartz-Bruhat functions is invariant under
this action. We recall now the action formula. For $g\in Sp_{2N}({\bf A})$,
let
$\left[\begin{array}[]{cc}\alpha_{g}&\beta_{g}\\\
\gamma_{g}&\delta_{g}\end{array}\right]$ (3.1)
be the block decomposition of $g$ with respect to the decomposition
${\bf A}^{2N}=C_{-,{\bf A}}\oplus C_{+,{\bf A}}.$
In this paper, we always assume the action of $Sp_{2N}$ on $F^{2N}$ (as well
as other symplectic group actions on symplectic spaces) is from the right. So
$\gamma_{g}$ in (3.1) is a map from $C_{+}\to C_{-}$. Let
$\tilde{g}\in\widehat{Sp}_{2N}({\bf A})$ be a lifting of $g$. For
$\phi\in{\cal S}(C_{-,{\bf A}})$, $({\tilde{g}}\cdot\phi)(x)$ equals to
$\lambda\int_{{\rm
Im}\,\gamma_{g}}S_{g}(x+x^{*})\phi(x\alpha_{g}+x^{*}\gamma_{g})d(x^{*}\gamma_{g}),$
(3.2)
where $\lambda\in{\mathbb{C}}^{*}$ is a certain scalar depending only on
$\tilde{g}$, $d(x^{*}\gamma_{g})$ is a Haar measure on ${\rm Im}\,\gamma_{g}$
and
$S_{g}(x+x^{*})=\psi\left(\frac{1}{2}\langle
x\alpha_{g},x\beta_{g}\rangle+\frac{1}{2}\langle
x^{*}\gamma_{g},x^{*}\delta_{g}\rangle+\langle
x^{*}\gamma_{g},x\beta_{g}\rangle\right);$
it is easy to see that $\langle x^{*}\gamma_{g},x^{*}\delta_{g}\rangle$
depends only on $x^{*}\gamma_{g}$ (not on the choice of $x^{*}$), therefore
$S_{g}(x+x^{*})$ is a function of $x$ and $x^{*}\gamma_{g}$. Since $\tilde{g}$
is unitary, $\lambda$ can be determined up to a factor in $S^{1}$.
Let $P$ be the subgroup of $Sp_{2N}$ that consists of elements that maps
$C_{+}$ to itself. An element $g$ is in $P({\bf A})$ iff $\gamma_{g}=0$. Then
there is a lifting $P({\bf A})\subset\widehat{Sp}_{2N}({\bf A})$ so that for
$g\in P({\bf A})$ and $\phi\in{\cal S}(C_{-,{\bf A}})$,
$(g\cdot\phi)(x)=|det(\alpha_{g})|_{\bf
A}^{\frac{1}{2}}\,\psi(\frac{1}{2}\langle
x\alpha_{g},x\beta_{g}\rangle)\phi(x\alpha_{g}),$ (3.3)
where the factor $|det(\alpha_{g})|_{\bf A}^{\frac{1}{2}}$ guarantees the
unitarity of the operator $g$. There is also a lifting
$Sp_{2N}(F)\subset\widehat{Sp}_{2N}({\bf A})$
such that theta functional
$\theta:{\cal S}(C_{-,{\bf A}})\to{\mathbb{C}}$
given by
$\theta(\phi)=\sum_{r\in C_{-}}\phi(r)$
is invariant under $Sp_{2N}(F)$. The action of $Sp_{2N}(F)$ is given by (3.2)
with $\lambda=1$ and the Haar measure is given by the condition that the
covolume of $({\rm Im}\,\gamma_{g})(F)$ is $1$.
For a given snt-module $M$ with
$M=H_{k_{1}}\oplus\dots\oplus H_{k_{n}}$ (3.4)
with $H_{k_{i}}$ is as in (1.9), and a space $V$ with non-degenerate,
bilinear, symmetric form $(\,,\,)$. Let $G$ denote the orthogonal group of
$V$. Recall that
$M\otimes_{F}V=M\otimes_{F[[t]]}V[[t]]$ (3.5)
has a natural snt-module structure (Section 1). Let $Sp_{2N}$ (where $2N={\rm
dim}\,M{\rm dim}\,V$) denote the symplectic group of the symplectic space
$M\otimes_{F}V$. The group $Sp(M,t)$ is a subgroup of $Sp_{2N}$. The group
$G(F[[t]])$ acts on $M\otimes_{F[[t]]}V[[t]]$ in the second factor, so we have
a group morphism
$G(F[[t]])\to Sp(M\otimes V,t).$
Suppose $l={\rm max}(k_{1},\dots,k_{n})$; then the image $G^{q}(F[[t]])$ of
the above homomorphism is isomorphic to $G(F[[t]]/(t^{l}))$. We have a
commuting pair ($Sp(M,t),G(F[[t]]/(t^{l}))$) in $Sp(M\otimes_{F}V,t)$.
Suppose we have a direct sum decomposition
$M=M_{-}\oplus M_{+}$ (3.6)
such that $M_{+},M_{-}\in Gr(M,t)$. We put
$X\stackrel{{\scriptstyle\rm def}}{{=}}M_{-}\otimes V.$
The space $L^{2}(X_{\bf A})$ is a representation of metaplectic group
${\widehat{S}p}_{2N}({\bf A})$, and we have the theta functional
$\theta:{\cal S}(X_{\bf
A})\to{\mathbb{C}},\,\,\,\phi\mapsto\theta(\phi)=\sum_{r\in X}\phi(r).$ (3.7)
Recall the Eisenstein series (1.3), (1.7) for $\phi\in{\cal S}(X_{\bf A})$ is
given by
${\rm E}(\phi)=\sum_{U\in Gr(M)}E(\phi,U)=\sum_{U\in
Gr(M)}\int_{(\pi_{-}(U)\otimes V)_{\bf A}}\psi(\frac{1}{2}\langle x,\rho
x\rangle)\phi(x)dx.$
Let $\pi_{-}:M\to M_{-}$ be the projection map with respect to (3.6). It gives
a map
$Gr(M)\to Gr(M_{-}):\,\,\,\,\,U\mapsto\pi_{-}(U).$ (3.8)
We wish to describe the inverse image of a given $W\in Gr(M_{-})$. Let
$W^{\bot}=\\{x\in M_{+}\,|\,\langle W,x\rangle=0\\}.$
If $U\in Gr(M)$ satisfying $\pi_{-}(U)=W$, then it is easy to see that
$W^{\bot}=M_{+}\cap U.$
The symplectic pairing $W\times M_{+}\to F$ factors through a non-degenerate
pairing
$\langle\,\,\rangle:W\times M_{+}/W^{\bot}\to F.$
We may identify $W^{*}$ with $M_{+}/W^{\bot}$ using this pairing. Recall we
have the map
$\rho_{U}:W\to M_{+}/W^{\bot}$
as defined in (1.6). It is easy to prove that $\rho_{U}$ is self-dual. We have
###### Lemma 3.1
For a given $W\in Gr(M_{-})$, the map
$U\mapsto\rho_{U}$
is a bijection from the set of $U\in Gr(M)$ such that $\pi_{-}(U)=W$ to the
set self-dual linear maps from $W$ to $W^{*}=M_{+}/W^{\bot}$
Proof. It is clear that the map $U\mapsto\rho_{U}$ is one-to-one. If
$\rho:W\to W^{*}=M_{+}/W^{\bot}$ is self-dual, then
$U\stackrel{{\scriptstyle\rm def}}{{=}}\\{w+\rho(w)+W^{\bot}\,|\,w\in W\\}$
(3.9)
is a Lagrangian subspace of $M$, and $\rho_{U}=\rho$. So the map in the lemma
is also onto. $\Box$
Recall the $t$-Eisenstein series defined in (1.11) is a sub-series of ${\rm
E}(\phi)$ given by
${\rm Et}(\phi)=\sum_{U\in Gr(X,t)}E(\phi,U).$ (3.10)
Since $M_{-}$ and $M_{+}$ are $F[[t]]$-submodules of $M$, the projection map
$\pi_{-}:M\to M_{-}$
is an $F[[t]]$-module homomorphism. For each $U\in Gr(M,t)$, $\pi_{-}(U)$ is
an $F[[t]]$-submodule of $M_{-}$. Denote $Gr(M_{-},t)$ the set of
$F[[t]]$-submodules of $M_{-}$, so we have map
$P:Gr(M,t)\to Gr(M_{-},t):\,\,\,U\mapsto\pi_{-}(U).$ (3.11)
For a $W\in Gr(M_{-},t)$, we wish to describe the inverse image
$P_{W}\stackrel{{\scriptstyle\rm def}}{{=}}P^{-1}(W).$
For any $U\in P_{W}$, i.e. $U\in Gr(M,t)$ and $\pi_{-}(U)=W$, we have the
linear map
$\rho_{U}:W\to M_{+}/W^{\bot},$
as defined in (1.6). As before $\rho_{U}$ is self-dual. We now prove
$\rho_{U}$ is $F[t]$-linear. If $w\in W$, since $\pi(U)=W$, there is
$w^{\prime}\in W_{+}$ such that $w+w^{\prime}\in U$. By our definition of
$\rho_{U}$, $\rho_{U}(w)=w^{\prime}\,\,\,mod(W^{\bot})$. Since $U$ is
$t$-Lagrangian, $tw+tw^{\prime}\in U$, this implies
$\rho_{U}(tw)=tw^{\prime}\,\,\,mod(W^{\bot})$.
###### Lemma 3.2
For each $U\in P_{W}$,
$\rho_{U}:W\to M_{+}/W^{\bot}=M_{+}/(M_{+}\cap U)$
is $F[[t]]$-linear and self-dual. Conversely for each $\rho:W\to
M_{+}/W^{\bot}$ that is $F[[t]]$-linear and self-dual, there is a unique $U\in
P_{W}$ such that $\rho_{U}=\rho$. Therefore $U\mapsto\rho_{U}$ is a bijection
from $P_{W}$ to $F_{W}$, the space of all $\rho:W\to U_{+}/W^{\bot}$ that is
self-dual and $F[[t]]$-linear .
This lemma is an $t$-analog of Lemma 3.1. Suppose $\rho$ is $F[[t]]$-linear
and self-dual, then $U$ given as (3.9) is the unique $t$-Lagrangian subspace
such that $\rho_{U}=\rho$.
We set
${\rm Et}_{W}(\phi)=\sum_{U\in P_{W}}E(\phi,U),$
By Lemma 3.2, we have
${\rm Et}_{W}(\phi)=\sum_{\rho\in F_{W}}\int_{(W\otimes V)_{\bf
A}}\phi(x)\psi(\frac{1}{2}\langle x,\rho(x)\rangle)dx,$ (3.12)
where $dx$ denotes the Haar measure on $(W\otimes V)_{\bf A}$ such that the
covolume of $W\otimes V$ is $1$. We have
${\rm Et}(\phi)=\sum_{W\in Gr(M_{-},t)}{\rm Et}_{W}(\phi).$
For an $F[[t]]$-submodule $W\subset M_{-},$ we let
$S_{t}^{2}(W)\subset W\otimes_{F[[t]]}W$
denote the $F[[t]]$-submodule of symmetric tensors. We define
$T_{W}:W\otimes_{F}V=W\otimes_{F[[t]]}V[[t]]\rightarrow S_{t}^{2}(W)$
by
$T_{W}:\sum_{i=1}^{s}w_{i}\otimes
v_{i}\mapsto\sum_{i,j=1}^{s}(v_{i},v_{j})w_{i}\otimes w_{j},$ (3.13)
where $w_{i}\in W,$ $v_{i}\in V$, $i=1,....,s,$ and where $w_{i}\otimes w_{j}$
denotes the tensor product of $w_{i},$ $w_{j}$ in $S_{t}^{2}(W).$ Of course
$T_{W}$ can be extended to an adelic map
$T_{W}:(W\otimes_{F}V)_{{\bf A}}\rightarrow S_{t}^{2}(W)_{{\bf A}},$
and for $r\in S_{t}^{2}(W)_{\bf A},$ we set
$\mathcal{U}_{r}=\\{x\in(W\otimes_{F}V)_{{\bf A}}\,|\,T_{W}(x)=r\\}.$
We now consider ${\rm Et}_{W}(\phi)$ as defined in (3.12). In that expression
we consider $\rho\in F_{W}$. We have a pairing
$S_{t}^{2}(W)\times F_{W}\to F,\,\,\,\,\,\,(\sum w_{i}\otimes
u_{i},\rho)=\sum\langle w_{i},\rho(u_{i})\rangle,$ (3.14)
which is clearly non-degenerate. And we have
$\langle w,\rho(w)\rangle=(T_{W}(w),\rho),\,\,\,\,\,w\in(W\otimes V)_{\bf A}.$
We may then rewrite the right hand side of (3.12) as
$\sum_{\rho\in F_{W}}\int_{(W\otimes V)_{{\bf
A}}}\phi(x)\psi(\frac{1}{2}(T_{W}(x),\rho))dx.$ (3.15)
The following result is a corollary of convergence of Eisenstein series on
loop groups, it will be proved in part II.
###### Theorem 3.3
Let $M$ be an snt-module as in (3.4), where $H_{k}$ is as in (1.9). If ${\rm
dim}V>6n+2$, then the series (3.10) converges absolutely and the convergence
is uniform for $\phi$ varying over a compact subsets in ${\cal S}(X_{\bf A})$.
It follows that the series (3.12)= (3.15) converges absolutely and the
convergence is uniform for $\phi$ varying over a compact subset of ${\cal
S}((W\otimes V)_{\bf A})$.
We can apply Proposition 1 and Proposition 2 of [7]. Using Weil’s notation as
in [7]:
$X=(W\otimes V)_{\bf A},\,\,\,\,\,G=S_{t}^{2}(W)_{\bf
A},\,\,\,\,\,\Gamma=S_{t}^{2}(W),\,\,\,\,\,f=T_{W}.$ (3.16)
and $G^{*}=(F_{W})_{\bf A}$, $\Gamma_{*}=F_{W}$, where we regard
$G^{*}=(F_{W})_{\bf A}$ as Pontryagin dual of $G$ by the pairing
$S_{t}^{2}(W)_{\bf A}\times(F_{W})_{\bf A}\to
S^{1},\,\,\,\,\,\,\\{a,\rho\\}=\psi(\frac{1}{2}(a,\rho)).$
We have
$F_{\phi}^{*}(g^{*})=\int_{(W\otimes V)_{{\bf
A}}}\phi(x)\psi(\frac{1}{2}\\{T_{W}(x),g^{*}\\})dx=\int_{X}\phi(x)\\{f(x),g^{*}\\}dx.$
Theorem 3.3 implies that the condition of Proposition 2 in [7] is satisfied;
that is
$\sum_{r^{*}\in\Gamma_{*}}|F_{\Phi}^{*}(g^{*}+\gamma^{*})|$
converges and the convergence is uniform as $(\phi,g^{*})$ varies over a
compact subset of ${\cal S}(X)\times G^{*}$. By using of Proposition 1 and
Proposition 2 [7], we obtain
###### Theorem 3.4
Suppose ${\rm dim}V>6n+2$. To every $r\in S_{t}^{2}(W)_{\bf A},$ there
corresponds a unique positive measure $\mu_{r}$ on $(W\otimes V)_{\bf A}$
whose support is contained in $\mathcal{U}_{r},$ so that for every function
$\phi$ on $(W\otimes V)_{\bf A}$ which is continuous with compact support, the
function $F_{\phi}(r)=\int\phi d\mu_{r}$ is continuous and satisfies
$\int F_{\phi}(r)dr=\int\phi(x)dx$
where $dr,$ $dx$ are fixed Haar measures on $S_{t}^{2}(W)_{\bf A},$
$(W\otimes_{F}V)_{\bf A},$ respectively. Moreover, the $\mu_{r}$’s are
tempered measures and for $\phi\in{\cal S}((W\otimes V)_{\bf A}),$ $F_{\phi}$
is continuous, is an element of $L^{1}(S_{t}^{2}(W)_{\bf A})$, and is the
Fourier transform of the function $F_{\phi}^{\ast}(\cdot)$ on
$S_{t}^{2}(W^{\ast})_{\bf A}$ given by
$F_{\phi}^{\ast}({\rho})=\int_{(W\otimes V)_{\bf
A}}\phi(x)\psi(\frac{1}{2}(T_{W}(x),{\rho}))dx.$
Finally,
${\rm Et}_{W}(\phi)=\sum_{r\in S_{t}^{2}(W)}\int_{(W\otimes V)_{\bf
A}}{\phi}d\mu_{r},$ (3.17)
the series on the right being absolutely convergent.
Since the convergence of the right hand side of (3.17) is uniform as $\phi$
varies on a compact subset of ${\cal S}((W\otimes V)_{\bf A})$, ${\rm Et}_{W}$
is a tempered measure on $(W\otimes V)_{\bf A}$. The formula (3.17) can be
restated as:
###### Corollary 3.5
We have the identity of the tempered distributions:
${\rm Et}_{W}=\sum_{r\in S_{t}^{2}(W)}\mu_{r}.$ (3.18)
## 4 An extension of Weil’s abstract lemma
In this section we study the measure $d\mu_{r}$ in Theorem 3.4. We begin with
a statement of Proposition 1 in [7] (page 6)
###### Lemma 4.1
Let $X$ and $G$ be two locally compact, abelian groups with fixed Haar
measures $dx$, $dg$ respectively. Let
$f:X\rightarrow G$
be a continuous map such that
1. (A)
For any $\Phi\in\mathcal{S}(X),$ the function $F_{\Phi}^{\ast}$ on $G^{\ast}$
defined by
$F_{\Phi}^{\ast}(g^{\ast})=\int_{X}\Phi(x)\\{f(x),g^{\ast}\\}dx,$ (4.1)
(where $dx$ $\\{\,,\,\\}$ denotes the pairing between $G$ and $G^{\ast})$ is
integrable on $G^{\ast}$ and the integral $\int|F_{\Phi}^{\ast}|dg^{\ast}$
(where $(dg^{\ast}$ is Haar measure on $G^{\ast})$ dual to $dg$) converges
uniformly on every compact subset of $\mathcal{S}(X).$
Then one can find a uniquely determined family of positive measures
$\\{\mu_{g}\\}_{g\in G},$ on $X,$ where support($\mu_{g})\subseteq
f^{-1}(\\{g\\}),$ and so that for every continuous function with compact
support $\Phi$ on $X,$ the function $F_{\Phi}$ on $G$ defined by
$F_{\Phi}(g)=\int\Phi d\mu_{g},$ (4.2)
is continuous and satisfies
$\int F_{\Phi}dg=\int\Phi dx.$ (4.3)
Moreover, the measures $\mu_{g}$ are tempered measures and for
$\Phi\in\mathcal{S}(X),$ $F_{\Phi}$ is continuous, belongs to $L^{1}(G),$
satisfies (4.3), and is the Fourier transform of $F_{\Phi}^{\ast}.$
We call $(X,G,f)$ as in Lemma 4.1, an admissible triple.
Let $M,M_{-},M_{+},V$ be as in Section 3. And as Section 3, $W$ denotes a
$F[[t]]$-submodule of $M_{-}$. For a fixed place $v$ of $F$, we set
$X_{v}=(W\otimes V)_{F_{v}},\,\,\,\,\,\,\,\,G_{v}=S_{t}^{2}(W)_{F_{v}},$
and let $T_{v}:X_{v}\to G_{v}$ be the $F_{v}$-linear extension of $T_{W}$
defined in (3.13). The dual group $G_{v}^{*}$ of $G_{v}$ is identified with
$(F_{W})_{F_{v}}$.
###### Lemma 4.2
If ${\rm dim}\,V>6n+2$, then above triple $(X_{v},G_{v},T_{v})$ is an
admissible triple, equivalently,
$F_{\Phi}^{\ast}(g^{\ast})=\int_{X}\Phi(x)\\{f(x),g^{\ast}\\}dx,$
satisfies condition (A) in Lemma 4.1.
This lemma is an analog of Proposition 5 in [7] (page 45). We expect that the
condition ${\rm dim}\,V>6n+2$ can be replaced by the weaker condition ${\rm
dim}\,V>6n+1$. For our purpose, the condition in the lemma is enough.
Proof. For simplicity, we write $X,G,T$ for $X_{v},G_{v},T_{v}$. Let $X_{\bf
A}=(W\otimes V)_{{\bf A}}$, $G_{\bf A}=S_{t}^{2}(W)_{{\bf A}}$, and $T_{\bf
A}:X_{\bf A}\to G_{\bf A}$ be $T_{W}\otimes{\bf A}$ then we have
$X_{\bf A}=X\times X^{c},\,\,\,\,\,\,\,G_{\bf A}=G\times G^{c}$
where $X^{c}$ is the restricted product of $(W\otimes V)_{F_{w}}$’s for $w\neq
v$, $G^{c}$ is the restricted product of $S_{t}^{2}(W)_{F_{w}}$’s for $w\neq
v$. And $T_{\bf A}=T\times T_{c}$, where $T_{c}:X^{c}\to G^{c}$ is defined
similarly as $T$. Let $C$ be a compact subset of ${\cal S}(X)$. We choose a
function $\phi_{0}\in{\cal S}(X^{c})$ such that
$\int_{X^{c}}\phi_{0}(x_{c})dx_{c}\neq 0.$
Then $F^{*}_{\phi_{0}}(g_{c}^{*})$ given by
$F^{*}_{\phi_{0}}(g_{c}^{*})=\int_{X^{c}}\phi_{0}(x_{c})\\{T_{c}(x_{c}),g_{c}^{*}\\}dx_{c},$
satisfies that $F^{*}_{\phi_{0}}(0)\neq 0$. Since
$F^{*}_{\phi_{0}}(g_{c}^{*})$ is continuous, we have
$\int_{G^{c}}|F_{\phi_{0}}(g_{c}^{*})|dg_{c}^{*}=M\neq 0.$
For each function $\phi(x)\in{\cal S}(X)$ , $\phi(x)\phi_{0}(x_{c})$ is in
${\cal S}(X_{\bf A})$. Since $C$ is a compact subset of ${\cal S}(X)$,
$C\phi_{0}$ is a compact subset of ${\cal S}(X_{\bf A})$. By Theorem 3.3, and
the Proposition 2 in [7], we know that $X_{\bf A},G_{\bf A},T_{\bf A}$ is an
admissible triple. We consider the function
$\int_{X}\int_{X^{c}}\phi(x)\phi_{0}(x_{c})\\{T(x),g^{*}\\}\\{T_{c}(x_{c}),g_{c}^{*}\\}dxdx_{c}=F_{\phi}(g^{*})F_{\phi_{0}}(g_{c}^{*}).$
Since $C\phi_{0}$ is compact, the integral
$\int_{G^{*}\times
G_{c}^{*}}|F_{\phi}(g^{*})F_{\phi_{0}}(g_{c}^{*})|dg^{*}dg_{c}^{*}$
converges uniformly as $\phi$ varies over $C$. By the Fubuni theorem, we have
$\displaystyle\int_{G^{*}\times
G_{c}^{*}}|F_{\phi}(g^{*})F_{\phi_{0}}(g_{c}^{*})|dg^{*}dg_{c}^{*}$
$\displaystyle=\int_{G^{*}}|F_{\phi}(g^{*})|dg^{*}\int_{G_{c}^{*}}|F_{\phi_{0}}(g_{c}^{*})|d\bar{g}_{c}^{*}$
$\displaystyle=M\int_{G^{*}}|F_{\phi}(g^{*})|dg^{*}.$
This implies that
$\int_{G^{*}}|F_{\phi}(g^{*})|dg^{*}$
converges uniformly as $\phi$ varies on $C$. $\Box$.
Since $W$ is a finite dimensional over $F$ and $t^{N}W=0$ for $N$ large, $W$
is isomorphic to
$F[t]/(t^{k_{1}})e_{1}\oplus\dots\oplus F[t]/(t^{k_{m}})e_{m}$
as a $F[[t]]$-module, where $e_{1},\dots,e_{m}$ is a quasi-basis of $W$. Let
$\bar{W}$ denote the quotient $W/tW$, we have
$\bar{W}\cong F\bar{e}_{1}\oplus\dots\oplus F\bar{e}_{m},$
where $\bar{e}_{i}$ is the projection of $e_{i}$. Let $\bar{G}_{v}$ denote
$S^{2}(\bar{W}_{F_{v}})$, where $S^{2}(\bar{W}_{F_{v}})$ is the subspace of
the symmetric tensors in $\bar{W}_{F_{v}}\otimes\bar{W}_{F_{v}}$. For
simplicity, we shall write $G,X,\bar{G}$ for $G_{v},X_{v},\bar{G}_{v}$. We
have $\bar{T}$ given by
$\bar{T}:\bar{X}\stackrel{{\scriptstyle\rm def}}{{=}}(\bar{W}\otimes
V)_{v}\to\bar{G}\stackrel{{\scriptstyle\rm
def}}{{=}}S^{2}(W_{v}),\,\,\,\,\,\,\,\sum_{i}u_{i}\otimes
v_{i}\mapsto\sum_{i,j}(v_{i},v_{j})u_{i}\otimes u_{j}.$
The condition ${\rm dim}V>6n+2$ implies in particular ${\rm dim}V>6n+2\geq
6m+2$; this implies that the condition for Proposition 5 in [7] (page 45) is
satisfied, so $(\bar{X},\bar{G},\bar{T})$ is an admissible triple. The
canonical map $W\to\bar{W}$ induces surjective linear maps
$\pi_{X}:X\to\bar{X},\,\,\,\,\,\,\,\pi_{G}:G\to\bar{G}.$
We have the commutative diagram
$\begin{array}[]{ccc}X&\overset{\pi_{X}}{\longrightarrow}&\bar{X}\\\
\downarrow T&&\downarrow\bar{T}\\\
G&\overset{\pi_{G}}{\longrightarrow}&\bar{G}\end{array}$ (4.4)
Let $f:X\to\bar{G}$ denote $\pi_{G}\circ T=\bar{T}\circ\pi_{X}$.
###### Lemma 4.3
For $x\in X$, the following conditions are equivalent
(1). $T$ is submersive at $x$.
(2). $f$ is submersive at $x$.
(3). $\bar{T}$ is submersive at $\pi_{X}(x)$.
Proof. Since $\pi_{X}$ is linear and surjective, it is submmersive at every
point. It follows that (2) and (3) are equivalent. Since $\pi_{G}$ is linear
and surjective, it is submmersive at every point, it follows that (1) implies
(3). The fact that (3) implies (1) follows directly from Lemma 5.7 in Section
5.
###### Lemma 4.4
The map $f:X\to\bar{G}$ satisfies the condition (A) in Lemma 4.1, so
$(X,\bar{G},f)$ is an admissible triple.
Proof. We use the following diagram to prove the lemma:
$\begin{array}[]{ccc}X&\overset{\pi_{X}}{\longrightarrow}&\bar{X}\\\
&f\searrow&\downarrow\bar{T}\\\ &&\bar{G}\end{array}$
Let $K$ denote the kernel of $\pi_{X}$. We have a map from ${\cal
S}(X)\to{\cal S}(\bar{X})$ given by
$\Phi\mapsto\bar{\Phi}(\bar{x})=\int_{K}\Phi(k+\bar{x})dk,$ (4.5)
where $k$ denotes the Haar measure on $K$. It is clear that this map is
continuous. Consider
$F_{\Phi}^{\ast}({\bar{g}}^{\ast})=\int_{X}\Phi(x)<f(x),{\bar{g}}^{*}>dx$
In the right hand side, we integrate over $K$ first, and notice that
$<f(x),{\bar{g}}^{*}>=<\bar{T}(\bar{x}),{\bar{g}}^{*}>$
where $\bar{x}=\pi_{X}(x)$, we get
$F_{\Phi}^{\ast}({\bar{g}}^{\ast})=\int_{\bar{X}}\bar{\Phi}(\bar{x})<\bar{T}(\bar{x}),{\bar{g}}^{\ast}>d\bar{x},$
since $(\bar{X},\bar{G},\bar{T})$ is an admissible triple, the right hand side
is in $L^{1}(\bar{G})$. And if $\Phi$ runs through a compact subset of ${\cal
S}(X)$, then $\bar{\Phi}$ which is related to $\Phi$ by (4.5) runs through a
corresponding compact subset of ${\cal S}(\bar{X})$, so the integral
$\int|F_{\Phi}^{\ast}({\bar{g}}^{\ast})|d\bar{g}^{*}$
converges uniformly. This proves the lemma. $\Box$
By Lemma 4.1, we have a family of measures
$\\{\mu_{\bar{g}}\\}_{\bar{g}\in\bar{G}}$ on $X$, with
${\rm support}(\mu_{\bar{g}})\subset f^{-1}(\bar{g})$, such that for every
$\Phi\in C_{c}(X)$, $\int\Phi d\mu_{\bar{g}}$ is continuous function of
$\bar{g}$ and
$\int_{\bar{G}}\left(\int\Phi d\mu_{\bar{g}}\right)d\bar{g}=\int_{X}\Phi dx.$
(4.6)
On the other hand, apply Lemma 4.1 to the admissible triple
$(\bar{G},\bar{X},\bar{T})$, we have a family of measures
$\\{\mu_{\bar{g}}^{\bar{T}}\\}_{\bar{g}\in\bar{G}}$ on $\bar{X}$, with ${\rm
support}(\mu_{\bar{g}}^{\bar{T}})\subset\bar{T}^{-1}(\bar{g})$, and
$\int_{\bar{G}}\left(\int\bar{\Phi}d\mu_{\bar{g}}^{\bar{T}}\right)d\bar{g}=\int_{\bar{X}}\bar{\Phi}d\bar{x}.$
(4.7)
Suppose that $\Phi$ and $\bar{\Phi}$ are related by (4.5) and the Haar
measures $dg,d\bar{g},dk$ are compatible so that the right hand sides of (4.6)
and (4.7) are equal. We then have
$\int_{\bar{G}}\left(\int\Phi
d\mu_{\bar{g}}\right)d\bar{g}=\int_{\bar{G}}\left(\int\bar{\Phi}d\mu_{\bar{g}}^{\bar{T}}\right)d\bar{g}$
(4.8)
We claim that the truth of (4.8) for all $\Phi\in C_{c}(X)$ implies that
$\int\Phi d\mu_{\bar{g}}=\int\bar{\Phi}d\mu_{\bar{g}}^{\bar{T}}.$ (4.9)
To prove this, take arbitrary $h(\bar{g})\in C_{c}(\bar{G})$, let
$f^{*}h=h\circ f$, replace $\Phi$ in (4.8) by $f^{*}h\Phi$. We get
$\int_{\bar{G}}h(\bar{g})\left(\int\Phi
d\mu_{\bar{g}}\right)d\bar{g}=\int_{\bar{G}}h(\bar{g})\left(\int\bar{\Phi}d\mu_{\bar{g}}^{\bar{T}}\right)d\bar{g}.$
The above is true for all $h(\bar{g})\in C_{c}(\bar{G})$, and $\int\Phi
d\mu_{\bar{g}}$, $\int\bar{\Phi}d\mu_{\bar{g}}^{\bar{T}}$ are continuous
functions of $\bar{g}$, so we have (4.9). We rewrite (4.9) as
$\int\Phi
d\mu_{\bar{g}}=\int(\int_{K}\Phi(k+\bar{x})dk)d\mu_{\bar{g}}^{\bar{T}}$ (4.10)
Recall Lemma 17 [7] (page 52), the support of $\mu_{\bar{g}}^{\bar{T}}$ is on
the $\bar{T}^{-1}(\bar{g})_{\rm re}$ (the regular points (= submerssive
points) in $\bar{T}^{-1}(\bar{g})$). By (4.10), the support of $\mu_{\bar{g}}$
is in $\pi_{X}^{-1}\bar{T}^{-1}(\bar{g})_{\rm re}$, which is precisely the set
of the regular points in $f^{-1}(\bar{g})$ by Lemma 4.3. We have proved
###### Lemma 4.5
The measure $\mu_{\bar{g}}$ is supported on $f^{-1}(\bar{g})_{\rm re}$, the
subset of regular points of $f^{-1}(\bar{g})$.
We consider the diagram
$\begin{array}[]{ccc}X&&\\\ \downarrow T&\searrow f&\\\
G&\overset{\pi_{G}}{\longrightarrow}&\bar{G}\end{array}$
For the admissible triple $(X,G,T)$, Lemma 4.1 implies that we have a family
of measures $\mu_{g}$ ($g\in G$) supported on $T^{-1}(g)$ such that for
$\Phi\in C_{c}(X)$, $F_{\Phi}(g)\stackrel{{\scriptstyle\rm def}}{{=}}\int\Phi
d\mu_{g}\in C(G)$ , we have
$\int_{G}F_{\Phi}(g)dg=\int_{X}\Phi(x)dx$ (4.11)
We take a subspace of $G$ that maps isomorphically onto $\bar{G}$, we denote
this space by $\bar{G}$, so we have the identification $G=K\times\bar{G}$,
where $K$ is the kernal of $\pi_{G}$. Since $\Phi$ has compact support,
$F_{\Phi}$ has compact support, and it is continuous, so we have
$\bar{g}\to\int_{K}F_{\Phi}(\bar{g}+k)dk$
is in $C(\bar{G})$. The left hand side of (4.11) can be written as
$\int_{\bar{G}}\int_{K}F_{\Phi}(\bar{g}+k)dkd\bar{g},$
so we have
$\int_{\bar{G}}\int_{K}F_{\Phi}(\bar{g}+k)dkd\bar{g}=\int_{X}\Phi(x)dx$ (4.12)
On the other hand, use the triple $(X,\bar{G},f)$, we have by (4.6
$\int_{\bar{G}}(\int\Phi d\mu_{\bar{g}})d\bar{g}=\int_{X}\Phi(x)dx$ (4.13)
Comparing (4.12) and (4.13), we get
$\int_{\bar{G}}\int_{K}F_{\Phi}(\bar{g}+k)dkd\bar{g}=\int_{\bar{G}}(\int\Phi
d\mu_{\bar{g}})d\bar{g}.$ (4.14)
Take an arbitrary $h\in C_{c}(\bar{G})$, let $f^{*}h=h\circ f\in C(X)$, and
replacing $\Phi$ in (4.14) by $f^{*}h\Phi$, we get
$\int_{\bar{G}}h(\bar{g})\int_{K}F_{\Phi}(\bar{g}+k)dkd\bar{g}=\int_{\bar{G}}h(\bar{g})(\int\Phi
d\mu_{\bar{g}})d\bar{g}.$
This is true for arbitrary $h\in C_{c}(\bar{G})$, and both
$\int_{K}F_{\Phi}(\bar{g}+k)dk$ and $\int\Phi d\mu_{\bar{g}}$ are continuous
functions, so we have
$\int_{K}F_{\Phi}(\bar{g}+k)dk=\int\Phi d\mu_{\bar{g}}.$
But the measure $\mu_{\bar{g}}$ is supported on $f^{-1}(\bar{g})_{\rm re}$, so
it is a gauge measure (see section 5 of [7] for the definition of ”gauge”
measure),
$f^{-1}(\bar{g})=T^{-1}(\pi_{G}^{-1}(\bar{g}))=T_{X}^{-1}(g+K)$
Note that
$f^{-1}(\bar{g})_{\rm re}=\cup_{k\in K}T^{-1}(g+k)_{\rm re}$
For each given $\bar{g}$, $T^{-1}(\bar{g}+k)_{\rm re}$ is non-singular
subvariety of $X$, we have a gauge form $d\delta_{k}$ on it, we have
$\int\Phi d\mu_{\bar{g}}=\int_{K}\int_{T^{-1}(\bar{g}+k)}\Phi d\delta_{k}dk$
we obtain
$\int_{K}F_{\Phi}(\bar{g}+k)dk=\int_{K}\int{T^{-1}(\bar{g}+k)}\Phi(x)d\delta_{k}dk.$
(4.15)
The above holds for arbitrary $\Phi\in C_{c}(X)$, use the same method we used
to deduce (4.9) from (4.8), we deduce from (4.15) that
$F_{\Phi}(\bar{g}+k)=\int{T^{-1}(\bar{g}+k)_{\rm re}}\Phi d\delta_{k}$
So we have proved
###### Lemma 4.6
The measure $\mu_{g,v}$ for the triple $(X_{v},G_{v},T_{v})$ is supported on
$T_{v}^{-1}(g)_{\rm re}$ and is the gauge measure.
Lets recall the meaning of ”gauge” measure ( [7], section 5). In the situation
as Lemma 4.6. We first take an invariant top form $\eta$ on $G$ and an
invariant top form $\omega$ on $X$. Let $X^{\prime}$ be the open set of $X$
that consists of all the points where $T$ is submersive. Near each point $x\in
X^{\prime}$, there is a form $\theta_{x}$ such that $\theta\wedge
T^{*}\eta=\omega$. For each $y\in G$, the local forms $\theta$, restrict to
$T^{-1}(y)_{\rm re}=T^{-1}(y)\cap X^{\prime}$, give a top form $\theta_{y}$ on
$T^{-1}(y)_{\rm re}$, which defines a measure which is equal to $\mu$ in Lemma
4.6.
Now we consider the global situation. Let $X,G,T_{W}$ as in (3.16). For each
$r\in S^{2}_{t}(W)$, we consider the inverse image $T_{W}^{-1}(r)$. Notice
that $T$ is submersive at a generic point, the space $X^{\prime}$ formed by
the points at which $f$ is submersive is $F$-open in $X$. We take a top form
$\eta$ over $G$ and a top form $\omega$ on $X$, we assume that the Tamagawa
measures on $G({\bf A})$ ($X_{\bf A}$ resp.) with respect to $\eta$ (resp
$\omega$ ) are the Haar measure normalized by the condition that the covolume
of $G(F)$ ($X(F)$) is $1$. The space $X^{\prime}$ can be covered by $F$-open
subsets $U_{\lambda}$ such that, there is a form $\theta_{\lambda}$ rational
over $F$ satisfying $\theta\wedge f^{*}=\omega$. For each $i\in G(F)$, then
the $\theta$’s restrict on $f^{-1}(i)\cap X^{\prime}$ to get a top form
$\theta_{i}$ on $f^{-1}(i)\cap X^{\prime}$. By Lemma 4.6, $d\mu_{i,v}$ is
given by $|\theta_{i}|_{v}$. Using a similar argument as in [7], Section 42,
we can prove that $1$ is a system of convergence factor of $|\theta_{i}|_{\bf
A}$. And we have
###### Theorem 4.7
For each $r\in S_{t}^{2}(W)$, the measure $\mu_{r}$ in (3.17) is supported in
$T_{W}^{-1}(r)_{\rm re}$ and it is the same as $|\theta_{r}|_{\bf A}$, the
measure define by the form $\theta_{r}$.
## 5 Classification of orbits of orthogonal groups
As in Section 3, we denote $M$ an snt-module with decomposition $M=M_{-}\oplus
M_{+}$ into $t$-Lagrangian subspaces and $V$ a finite dimensional vector space
over $F$ with a non-degenerate, bilinear symmetric form $(,)$. The orthogonal
group $G(F[[t]])$ acts on $M\otimes V$, leaving the subspace $M_{-}\otimes V$
invariant. The purpose of this section to give a complete set of invariants of
$G(F[[t]])$-orbits in $M_{-}\otimes V$ (see Theorem 5.4 below).
###### Definition 5.1
Let $W$ be a finitely generated $F[[t]]$-module. A submodule $L\subset W$ is
called a primitive submodule if one of the following equivalent conditions is
satisfied:
(1). there is a complement $F[[t]]$-submodule $L^{\prime}$, i.e. $W=L\oplus
L^{\prime}$.
(2). the natural map $L/tL\to W/tW$ induced form the embedding
$L\hookrightarrow M$ is injective.
Examples. (1) Let $W=F[[t]]/(t^{k_{1}})\oplus\dots\oplus F[[t]]/(t^{k_{m}})$,
For any $l\leq m$, $L=F[[t]]/(t^{k_{1}})\oplus\dots\oplus F[[t]]/(t^{k_{l}})$
is a primitive submodule of $M$.
(2). Let $F[[t]]^{m}$ be a free $F[[t]]$-module of rank $m$, then $F[[t]]^{l}$
(consists of elements with last $(m-l)$-components $0$ is a primitive
submodule. (3). $\\{0\\}$ is a primitive submodule for any $M$.
Let $W$ be a finitely generated $F[[t]]$-module, $e_{1},\dots,e_{m}$ is called
a quasi-basis of $W$ if every $e_{i}\neq 0$, every element $x\in W$ can be
written as a $F[[t]]$-linear combination of $e_{1},\dots,e_{m}$, and
$a_{1}e_{1}+\dots+a_{m}e_{m}=0$ ($a_{i}\in F[[t]]$) implies that all
$a_{i}e_{i}=0$. If $W$ is a finite dimensional $F[[t]]$-module, the ”quasi-
basis” defined above is the same notion as defined in Section 2. If $W$ is a
free $F[[t]]$-module, then a quasi-basis of $W$ is the same as a basis of $W$.
In the example (1) above, there is a quasi-basis of $m$ elements. It is clear
that $e_{1},\dots,e_{m}$ is a quasi-basis of $W$ if and only if the images of
$e_{1},\dots,e_{m}$ in $W/tW$ form an $F$-basis of vector space $W/tW$.
Therefore any two quasi-bases have the same number of elements. The
cardinality of a quasi-basis is called the rank of $W$.
Every
$x=\sum_{i}u_{i}\otimes v_{i}\in M_{-}\otimes V=M_{-}\otimes_{F[[t]]}V[[t]]$
gives rise to an $F[[t]]$-linear map
$f_{x}:V[[t]]\to M_{-},\,\,\,\,\,\,f_{x}(v)=\sum_{i}(v_{i},v)u_{i}.$ (5.1)
We denote by ${\rm Im}\,f_{x}$ the image of $f_{x}$, which is an
$F[[t]]$-submodule of $M_{-}$. Let $e_{1},\dots,e_{m}$ be a quasi-basis of
${\rm Im}\,f_{x}$, then
${\rm Im}\,f_{x}\cong F[[t]]/(t^{k_{1}})e_{1}\oplus\dots\oplus
F[[t]]/(t^{k_{m}})e_{m},$ (5.2)
where $k_{i}$ is the smallest positive integer such that $t^{k_{i}}e_{i}=0$.
###### Lemma 5.2
Let $e_{1},\dots,e_{m}$ be a quasi-basis of ${\rm Im}\,f_{x}$, and suppose
$v_{1},\dots,v_{m}\in V[[t]]$ satisfy $f_{x}(v_{i})=e_{i}$ ($i=1,\dots,m)$,
then ${\rm Span}_{F[[t]]}\\{v_{1},\dots,v_{m}\\}$ is a primitive submodule of
$V[[t]]$ and $v_{1},\dots,v_{m}$ is a basis of ${\rm
Span}_{F[[t]]}\\{v_{1},\dots,v_{m}\\}$.
Proof. Set $L={\rm Span}_{F[[t]]}\\{v_{1},\dots,v_{m}\\}$, then
$f_{x}|_{L}:L\to{\rm Im}\,f_{x}$ induces a linear isomorphism:
$\bar{f_{x}}:L/tL\to{\rm Im}\,f_{x}/t({\rm Im}\,f_{x}).$
This implies in particular, ${\rm dim}\,L/tL=m$, so the map $L/tL\to
V[[t]]/tV[[t]]$ induced from $L\subset V[[t]]$ is injective, so $L$ is
primitive submodule of $V[[t]]$. The other conclusions are clear. $\Box$
The bilinear form $(,)$ on $V$ can be extended to a $F[[t]]$-valued bilinear
form on $V[[t]]=V\otimes_{F}F[[t]]$. This $F[[t]]$-valued bilinear form on
$V[[t]]$ is non-degenerate. It is easy to prove
###### Lemma 5.3
Let $e_{1},\dots,e_{m}$ be a quasi-basis of ${\rm Im}\,f_{x}$, then there are
elements $w_{1},\dots,w_{m}\in V[[t]]$ such that
(1) ${\rm Span}_{F[[t]]}\\{w_{1},\dots,w_{m}\\}$ is a primitive submodule of
$V[[t]]$ and $w_{1},\dots,w_{m}$ is basis of ${\rm
Span}_{F[[t]]}\\{w_{1},\dots,w_{m}\\}$.
(2)
$x=e_{1}\otimes w_{1}+\dots+e_{m}\otimes w_{m}.$
Proof. Choose $v_{i}\in V[[t]]$ ( $i=1,\dots,m$) such that
$f_{x}(v_{i})=e_{i}$. By Lemma 5.2, ${\rm
Span}_{F[[t]]}\\{v_{1},\dots,v_{m}\\}$ is a primitive submodule of $V[[t]]$
and $v_{1},\dots,v_{m}$ is a basis. It is clear that
$V[[t]]={\rm Span}_{F[[t]]}\\{v_{1},\dots,v_{m}\\}\oplus ker(f_{x}).$
Let $v_{m+1},\dots,v_{N}$ be a basis of $ker(f_{x})$. Then $v_{1},\dots,v_{N}$
is basis of $V[[t]]$. Now let $w_{1},\dots,w_{N}$ be the dual basis if
$v_{1},\dots,v_{N}$, i.e., $(v_{i},w_{j})=\delta_{ij}$. This is clear that
$v=\sum_{i=1}^{m}e_{i}\otimes w_{i}$. $\Box$
From Lemma 5.3, we see that $x\in{\rm Im}\,f_{x}\otimes_{F[[t]]}V[[t]]$. We
define a map
$T:{\rm Im}\,f_{x}\otimes_{F[[t]]}V[[t]]\to S_{t}^{2}({\rm
Im}\,f_{x}),\,\,\,\,\,\,\sum_{i}^{N}u_{i}\otimes
v_{i}\mapsto\sum_{i,j=1}^{N}(v_{i},v_{j})u_{i}\otimes u_{j}.$
Where $S_{t}^{2}({\rm Im}\,f_{x})$ denote the subspace of symmetric tensors in
${\rm Im}\,f_{x}\otimes_{F[[t]]}{\rm Im}\,f_{x}$. We remark that though ${\rm
Im}\,f_{x}$ is a submodule of $M_{-}$, but the natural map $S_{t}^{2}({\rm
Im}\,f_{x})\to S_{t}^{2}(M_{-})$ is in general not an embedding.
###### Theorem 5.4
Two elements $x,y\in M_{-}\otimes V$ are in the same $G(F[[t]])$-orbit iff
${\rm Im}\,f_{x}={\rm Im}\,f_{y}$ and $T(x)=T(y)$ .
We need some preparations for proving the theorem.
We recall a special case of Witt’s theorem (see [4] ):
###### Theorem 5.5
If $L_{1},$ $L_{2}\subseteq V[[t]]$ are two primitive submodules of $V[[t]]$
and if $\sigma:L_{1}\rightarrow L_{2}$ is an isometry, then $\sigma$ can be
extended to an isometry in $G(F[[t]])$.
It is clear that if $x,y$ are in the same $G(F[[t]])$-orbit, then ${\rm
Im}\,f_{x}={\rm Im}\,f_{y}$ and $T(x)=T(y).$ Conversely, if ${\rm
Im}\,f_{x}={\rm Im}\,f_{y}\stackrel{{\scriptstyle\rm def}}{{=}}W$ and
$T(x)=T(y)$. Let $e_{1},....,e_{m}$ be a quasi-basis for $W$, and $W$ be as
(5.2). We may assume that $k_{1}\geq k_{2}\geq\dots\geq k_{m}$. Then
$S_{t}^{2}(W)$ has a quasi-basis $e_{ij}\stackrel{{\scriptstyle\rm
def}}{{=}}e_{i}\otimes e_{j}+e_{j}\otimes e_{i}$ ($1\leq i\leq j\leq m$), and
$S_{t}^{2}(W)=\sum_{1\leq i\leq j\leq m}F[[t]]/(t^{k_{j}})e_{ij}.$
By lemma 5.3, we may write
$x=e_{1}\otimes a_{1}+\dots+e_{m}\otimes a_{m},\,\,\,\,\,\,y=e_{1}\otimes
b_{1}+\dots+e_{m}\otimes b_{m}$
with $\\{a_{1},\dots,a_{m}\\}$ and $\\{b_{1},\dots,b_{m}\\}$ satisfy condition
(1) in Lemma 5.3. Then $T(x)=T(y)$ implies that
$(a_{i},a_{j})=(b_{i},b_{j})\,\,\,\,\,{\rm mod}\,\,t^{{\rm
min}(k_{i},k_{j})}.$ (5.3)
###### Lemma 5.6
Let $L_{1}$ and $L_{2}$ be two primitive submodules of $V[[t]]$ with bases
$a_{1},\dots,a_{m}$ and $b_{1},\dots,b_{m}$. Let $1\geq k_{1}\geq\dots\geq
k_{m}$. If (5.3) holds, then the set $b_{1},\dots,b_{m}$ can be altered to
another set ${\tilde{b}}_{1},\dots,{\tilde{b}}_{m}$ such that
${\tilde{b}}_{i}=b_{i}\,\,\,\,\,\,{\rm mod}\,t^{k_{i}},\,\,\,\,{\rm
for}\,1\leq i\leq m,$ (5.4)
and
$({\tilde{b}}_{i},{\tilde{b}}_{i})=(a_{i},a_{j})\,\,\,\,\,\,{\rm for}\,1\leq
i,j\leq m,$ (5.5)
and the $F[[t]]$-span of ${\tilde{b}}_{1},\dots,{\tilde{b}}_{m}$ is a
primitive submodule of $V[[t]]$ with ${\tilde{b}}_{1},\dots,{\tilde{b}}_{m}$
as a basis.
Suppose the truth of Lemma 5.6, then Theorem 5.4 can be proved as follows. The
equation (5.4) implies that
$y=\sum e_{i}\otimes b_{i}=\sum e_{i}\otimes\tilde{b}_{i}.$
The equation (5.5) implies that the map $\sigma:L_{1}\rightarrow L_{2}$ give
by $a_{i}\mapsto{\tilde{b}}_{i}$ is an isometry, by Theorem 5.5, $\sigma$ can
be extended to $g\in G(F[[t]])$. then
$y\cdot g=(\sum e_{i}\otimes\tilde{b}_{i})\cdot g=\sum
e_{i}\otimes\sigma(\tilde{b}_{i})=\sum e_{i}\otimes{a}_{i}=x.$
It remains to prove Lemma 5.6.
Proof of Lemma 5.6. We use induction on $m$. For case $m=1$, we first take
$c\in V[[t]]$ such that $(b_{1},c)=1$, we want to find
${\tilde{b}}_{1}=b_{1}+t^{k_{1}}h(t)c$ where
$h(t)=h_{0}+h_{1}t+h_{2}t^{2}+\dots\in F[[t]]$ such that We have
$(a_{1},a_{1})=({\tilde{b}}_{1},{\tilde{b}}_{1})$
which is equivalent to
$(a_{1},a_{1})-(b_{1},b_{1})=2t^{k_{1}}h(t)+t^{2k_{1}}h(t)^{2}(c,c).$ (5.6)
Since $(a_{1},a_{1})=(b_{1},b_{1})\,\,{\rm mod}\,t^{k_{1}}$, we see that 5.6
holds mod $t^{k_{1}}$ for arbitrary $h(t)$. Compare the coefficient of
$t^{k_{1}}$, we solve for $h_{0}$, after $h_{0}$, we compare coefficient of
$t^{k_{1}+1}$, we solve $h_{1}$. It is clear that the similar process can be
continued to solve all $h_{i}$. Assume the Lemma holds for $m-1$, so we can
find ${\tilde{b}}_{1},\dots,{\tilde{b}}_{m-1}$ such that
${\tilde{b}}_{i}\cong b_{i}\,\,\,\,\,\,{\rm mod}\,t^{k_{i}},\,\,\,\,{\rm
for}\,1\leq i\leq m-1,$ (5.7)
and
$({\tilde{b}}_{i},{\tilde{b}}_{i})=(a_{i},a_{j})\,\,\,\,\,\,{\rm for}\,1\leq
i,j\leq m-1.$ (5.8)
We may assume $b_{i}={\tilde{b}}_{i}$ for $i=1,\dots,m-1$. Since $(\,)$ on
$V[[t]]$ is non-degenerate, we can find $c_{1},c_{2},\dots,c_{m}\in V[[t]]$
such that
$(b_{i},c_{j})=\delta_{i,j}.$ (5.9)
We want to find $h_{1}(t),\dots,h_{m}(t)\in F[[t]]$ such that
${\tilde{b}}_{m}=b_{m}+t^{k_{m}}\left(h_{1}(t)c_{1}+\dots+h_{m}(t)c_{m}\right)$
satisfies the following $m$ equations
$({\tilde{b}}_{i},{\tilde{b}}_{m})=(a_{i},a_{m}),\,\,\,\,\,\,\,i=1,\dots,m$
(5.10)
Let $h_{i}(t)=\sum_{s=0}^{\infty}h_{i,s}t^{s}$. The equations (5.10) already
hold mod $t^{k_{m}}$. Compare the coefficient of $t^{k_{m}}$ of (5.10), we get
a linear system with $m$-variables $h_{1,0},\dots,h_{m,0}$ and $m$ equations,
this system has non-zero determinant, thanks to (5.9), we can solve for
$h_{1,0},\dots,h_{m,0}$. Then we compare coefficient of $t^{k_{m}+1}$ of
(5.10), we get a linear system with $m$ equations and $m$ variables
$h_{1,1},\dots,h_{m,1}$, and again because of (5.9), the system has a
solution. It is clear that this process can be continued to solve for all
$h_{i,s}$’s. $\Box$
###### Lemma 5.7
Let $v$ be a place of $F$, $W$ be a $F[[t]]$-submodule of $M_{-}$, let
$W_{v}=W\otimes F_{v},V_{v}=V\otimes F_{v}$. The map $T:W_{v}\otimes V_{v}\to
S_{t}^{2}(W_{v})$ given by
$T(\sum_{i}u_{i}\otimes v_{i})=\sum_{i,j}(v_{i},v_{j})u_{i}\otimes u_{j}$
is submersive at $x_{0}\in W_{v}\otimes V_{v}$ iff ${\rm Im}\,f_{x_{0}}=W_{v}$
Proof. For simplicity, we denote $W_{v},V_{v},F_{v}$ by $W,V,F$ respectively.
Let $a_{1},\dots,a_{m}$ be a quasi-basis of the $F[[t]]$-module ${\rm
Im}\,f_{x_{0}}$. By Lemma 5.3, $x_{0}$ can be written as
$x_{0}=\sum_{i}a_{i}\otimes b_{i}=a_{1}\otimes b_{1}+\dots+a_{m}\otimes b_{m}$
where $b_{1},\dots,b_{m}$ is a basis of $Span\\{b_{1},\dots,b_{m}\\}$ and
$Span\\{b_{1},\dots,b_{m}\\}$ is a primitive $F[[t]]$-submodule of $V[[t]]$.
We first find a formula for the tangent map
$dT_{x_{0}}:T_{x_{0}}=W\otimes_{F[[t]]}V[[t]]\to T_{y_{0}}=S^{2}_{t}(W)$
where $y_{0}=T(x_{0})$. Take a line $x(\epsilon)=\sum_{i}a_{i}\otimes
b_{i}+\epsilon\sum_{j}{u_{j}\otimes v_{j}}$ passing through $x_{0}$ in the
direction $\sum_{j}{u_{j}\otimes v_{j}}$, Then
$T(x(\epsilon))=\sum(b_{i},b_{j})a_{i}\otimes
a_{j}+\epsilon\sum(b_{i},v_{j})(a_{i}\otimes u_{j}+u_{j}\otimes
a_{i})+\epsilon^{2}\sum(v_{i},v_{j})u_{i}\otimes u_{j}.$
So we have
$dT_{x_{0}}(\sum_{j}{u_{j}\otimes v_{j}})=\sum(b_{i},v_{j})(a_{i}\otimes
u_{j}+u_{j}\otimes a_{i}).$ (5.11)
From this formula, we see that $dT_{x_{0}}$ is $F[[t]]$-linear. If ${\rm
Im}\,f_{x_{0}}\not=W$, then ${\rm Im}\,dT_{x_{0}}\subset{\rm
Im}\,f_{x^{0}}\otimes W+{\rm Im}\,f_{x^{0}}\otimes W$, $dT_{x_{0}}$ is not
surjective, i.e. $T$ is not submersive at $x_{0}$. If ${\rm Im}\,f_{x^{0}}=W$,
since $(\,)$ on $V[[t]]$ is non-degenerate, for each $1\leq k\leq m$, we can
find $v\in V[[t]]$ such that $(b_{i},v)=\delta_{i,k}$, then
$dT_{x_{0}}(a_{l}\otimes v)=a_{l}\otimes a_{k}+a_{k}\otimes a_{l}$. So all
$a_{l}\otimes a_{k}+a_{k}\otimes a_{l}$ are in ${\rm Im}\,f_{x_{0}}$. So
$dT_{x_{0}}$ is surjective and $T$ is submersive at $x_{0}$. $\Box$
For an $F[[t]]$-submodule $W$ of $M_{-}$, $T_{W}:W\otimes V\to S_{t}^{2}(W)$
as in Section 3. For $i\in S_{t}^{2}(W)$, We denote by $U(i)$ the variety of
elements in $T_{W}^{-1}(i)$ where $T_{W}$ is submerssive. Of course the set of
$F$-points $U(i)_{F}$ may be empty. By Lemma 5.7, Theorem 5.4 can be
reformulated as
###### Theorem 5.8
The $G(F[[t]])$-orbits in $M_{-}\otimes V$ are in one-to-one correspondence
with the set of pairs $(W,i)$ with $W\in Gr(M_{-},t)$, $i\in S_{t}^{2}(W)$
such that $U(i)_{F}$ is not empty. The correspondence is the following, for
$x\in M_{-}\otimes V$, its orbit corresponds to the pair $(W,i)$, where
$W={\rm Im}\,f_{x}$, and $i=T_{W}(x)$.
## 6 Theta series.
We continue with the snt-module $M$ with decomposition $M=M_{-}\oplus M_{+}$
and $V$ a finite dimensional vector space with a non-degenerate, bilinear
symmetric form $(\,,\,)$ as in Section 3. For each $\phi\in{\cal S}(X_{\bf
A})={\cal S}((M_{-}\otimes V)_{\bf A})$, the theta functional $\theta(\phi)$
is defined by (3.7). Recall $G^{q}({\bf A}[[t]])$ acts on ${\cal S}(X_{\bf
A})$, the action formula is given as follows. An element $g\in G^{q}({\bf
A}[[t]])$ has the block decomposition
$\left[\begin{array}[]{cc}\alpha_{g}&\beta_{g}\\\
\gamma_{g}&\delta_{g}\end{array}\right]$
with respect the decomposition
$(M\otimes V)_{\bf A}=(M_{-}\otimes V)_{\bf A}\oplus(M_{+}\otimes V)_{\bf A}.$
Since $G^{q}({\bf A}[[t]])$ preserves $(M_{\pm}\otimes V)_{\bf A}$, so we have
$\beta_{g}=0$ and $\gamma_{g}=0$ . Then the action of $g$ on ${\cal
S}((M_{-}\otimes V)_{\bf A})$ is given by
$(g\cdot\phi)(x)=\phi(x\alpha_{g})=\phi(xg)$ (6.1)
If $g\in G^{q}(F[[t]])$, it is clear that
$\theta(g\cdot\phi)=\theta(\phi).$
So $\theta(g\cdot\phi)$ is a continuous function on $G^{q}(F[[t]])\backslash
G^{q}({\bf A}[[t]])$. Let $dg$ be the Haar measure on $G^{q}({\bf A}[[t]])$
such that the volume of $G^{q}(F[[t]])\backslash G^{q}({\bf A}[[t]])$ is $1$.
###### Lemma 6.1
If $(V,(\,,\,))$ is anisotropic over $F$ or ${\rm dim}V-r>\frac{1}{2}{\rm
dim}\,M+1$, where $r$ is the dimension of a maximal isotropic subspace of $V$,
then the integral
${\rm It}(\phi)\stackrel{{\scriptstyle\rm
def}}{{=}}\int_{G^{q}(F[[t]])\backslash G^{q}({\bf
A}[[t]])}\theta(g\cdot\phi)dg$ (6.2)
converges
Proof. If $(V,(\,,\,))$ is anisotropic over $F$, then $G^{q}(F[[t]])\backslash
G^{q}({\bf A}[[t]])$ is compact, so (6.2) converges. Let $U$ be the unipotent
radical of $G^{q}$, then $G^{q}=G\ltimes U$. Let $D$ be a compact fundamental
domain of $U(F)\backslash U({\bf A})$. Then
$(\ref{4.5})=\int_{G(F)\backslash G^{q}(A)}\int_{a\in D}\sum_{r\in
M_{-}\otimes V}\phi(rag)dadg=\int_{G(F)\backslash
G^{q}(A)}\theta(g\cdot\bar{\phi})dg,$ (6.3)
where $\bar{\phi}(x)=\int_{D}\phi(xa)da$. Since $D$ is compact,
$\bar{\phi}\in{\cal S}((M_{-}\otimes V)_{\bf A})$. By the convergence
criterion in Proposition 8 [7], the right hand side of (6.3) converges under
the condition ${\rm dim}V-r>\frac{1}{2}{\rm dim}\,M+1$. $\Box$
We can write the integral (6.2) as a sum of orbital integrals. Let ${\cal O}$
be a set of representatives of $G(F[[t]])$-orbit in $M_{-}\otimes V$. For each
$\xi\in{\cal O}$, let $G_{\xi}$ denote its isotropy subgroup in
$G^{q}(F[[t]])$. Then the integral (6.2) can be written as
$\int_{G^{q}(F[[t]])\backslash G^{q}({\bf A}[[t]])}\sum_{\xi\in{\cal
O}}\sum_{\tau\in G_{\xi}\backslash G^{q}(F[[t]])}\phi(\xi\tau g)dg,$ (6.4)
which can be further written as
$\sum_{\xi\in\mathcal{O}}vol(G_{\xi}\backslash G_{\xi,{\bf
A}})\int_{G_{\xi,{\bf A}}\backslash G^{q}({\bf A}[[t]])}\phi(\xi g)dg$ (6.5)
and we have thereby expressed ${\rm It}(\phi)$ as
${\rm It}(\phi)=\sum_{\xi\in\mathcal{O}}vol(G_{\xi}\backslash G_{\xi,{\bf
A}})\int_{G_{\xi,{\bf A}}\backslash G^{q}({\bf A}[[t]])}\phi(\xi g)dg.$ (6.6)
## 7 Siegel-Weil formula
We assume in this section $M$ and $V$ satisfies the conditions that ${\rm
dim}\,V>6n+2$, where $n$ is the number of $H_{k}$’s in the decomposition of
$M$ as in (3.4) and $V$ satisfies the conditions in Lemma 6.1. By Theorem 3.3,
the $t$-Eisenstein series $\phi\in{\cal S}((M_{-}\otimes V)_{\bf
A})\mapsto{\rm Et}(\phi)$ is a tempered distribution on $(M_{-}\otimes V)_{\bf
A}$. And for each $W\in Gr(M_{-},t)$, we have a tempered distribution
$\phi\mapsto{\rm Et}_{W}(\phi)$, given by (3.12). We have
${\rm Et}=\sum_{W\in Gr(M_{-},t)}{\rm Et}_{W}.$
By Theorem 4.7,
${\rm Et}_{W}=\sum_{i\in{\rm St}^{2}(W)}\mu_{i}.$
We denote $\mu_{i}$ by ${\rm Et}_{W,i}$. Therefore we have
${\rm Et}=\sum_{W\in Gr(M_{-},t)}\sum_{i\in{\rm St}^{2}(W)}\mu_{W,i}.$ (7.1)
Moreover the measure ${\rm Et}_{W,i}=\mu_{i}$ is the gauge measure as
described in Theorem 4.7, which implies in particular $\mu_{i}$ is $0$ if
$U(i)_{\bf A}$ in empty.
On the other hand,
$\phi\in{\cal S}((M_{-}\otimes V)_{\bf A})\mapsto{\rm It}(\phi)$
given in (6.2) is a tempered distribution and it has a decomposition given by
(6.6). By Theorem 5.8, each orbit corresponds uniquely to a pair $(W,i)$ where
$W\in Gr(U_{-},t)$ and $i\in{\rm St}^{2}(W)$. So we may write
${\rm It}=\sum_{W,i}{\rm I}_{W,i}$
where
${\rm I}_{W,i}(\phi)=vol(G_{\xi}\backslash G_{\xi,{\bf A}})\int_{G_{\xi,{\bf
A}}\backslash G^{q}({\bf A}[[t]])}\phi(\xi g)dg.$
where $(W,i)$ corresponds to the orbit containing $\xi$, i.e., ${\rm
Im}\,f_{\xi}=W$ and $T_{W}(\xi)=i$. If $U(i)$ is empty, $(W,i)$ doesn’t
corresponds to any orbit, in this case we define
${\rm I}_{W,i}\stackrel{{\scriptstyle\rm def}}{{=}}0.$
We shall prove that ${\rm Et}={\rm It}$, and actually we shall prove more:
${\rm Et}_{W,i}={\rm It}_{W,i}$ for any pair $(W,i)$ . We use the induction on
${\rm dim}\,M$. The case ${\dim}\,M=2$ is the classical result in [7].
Our proof is entirely parallel to that of [7]. To start with, we introduce
some notations. Let $\pi:\widehat{Sp}_{2N}({\bf A})\to Sp_{2N}({\bf A})$
denote the double cover (recall $2N={\rm dim}\,M{\rm dim}\,V$). Let
$\widehat{Sp}(M,t)_{\bf A}$ denote $\pi^{-1}({Sp}(M,t)_{\bf A})$. We let
$M(k)=\oplus_{k_{i}=k}H_{k_{i}},$
so
$M=M(l_{1})\oplus\dots\oplus M(l_{s}),\,\,\,\,\,l_{1}>\dots>l_{s}.$
Recall Corollary 2.6, $Sp(M,t)=N\ltimes H$, where $N$ is the unipotent
radical, and
$H=\Pi_{i=1}^{s}Sp_{2r_{i}}(F),$
where $r_{i}$ is the number $H_{k_{i}}$’s in the decomposition of $M_{l_{i}}$.
Since $M$ has decomposition (3.4), $M_{-}$ has decomposition
$M_{-}=F[t]/(t^{k_{1}})\oplus\dots\oplus F[t]/(t^{k_{n}}).$ (7.2)
Let $T$ be a maximal torus of $H$, we may take $T$ such that $T$ preserves
$M_{\pm}$ and preserves each component in (7.2). Then $T={\rm G}_{\rm m}^{n}$,
where the $i$-th $G_{m}$ acts on the $i$-th component in (7.2). We have
$T_{\bf A}=I_{F}^{n}$, where $I_{F}$ denotes the idele group of $F$. Let
$T_{\bf A}^{\prime}$ be the subset of $T_{\bf A}$ formed by
$T_{\bf A}^{\prime}=\\{(t_{1},\dots,t_{n})\,|\,|t_{1}|_{\bf
A}\geq\dots\geq|t_{r_{1}}|_{\bf A}\geq 1,\dots,|t_{n-r_{s}+1}1|_{\bf
A}\geq\dots\geq|t_{n}|_{\bf A}\geq 1\\}.$
Since $N$ is unipotent, the space $N(F)\backslash N({\bf A})$ is compact. By
the reduction theory for the semi-simple group $H$, there exists a compact
$C\subset Sp(M,t)_{\bf A}$ such that
$Sp(M,t)_{\bf A}=Sp(M,t)_{F}T_{\bf A}^{\prime}C.$ (7.3)
Since $T$ preserves $M_{+}$, we may regard $T_{\bf A}$ as a subgroup of
$\widehat{Sp}(M,t)_{\bf A}$, the decomposition (7.3) implies that
${\widehat{S}p}(M,t)_{\bf A}=Sp(M,t)_{F}T_{\bf A}^{\prime}C.$ (7.4)
for some compact subset $C\subset{\widehat{S}p}(M,t)_{\bf A}$.
As in [7], for each $\tau\in{\mathbb{R}}_{>0}$, we let $a_{\tau}\in I_{F}$
denote the idele such that $(a_{\tau})_{v}=\tau$ for each infinite place $v$
and $(a_{\tau})_{v}=1$ for each finite place. We let $\Theta(T)$ denote the
set of all $(a_{\tau_{1}},\dots,a_{\tau_{n}})$, and set
$\Theta(T)^{\prime}=\Theta(T)\cap T_{\bf A}^{\prime}$.
###### Lemma 7.1
If $\hat{E}$ is a positive tempered measure on $X_{\bf A}=(M_{-}\otimes
V)_{\bf A}$, and is a sum of positive measures $\hat{\mu}_{i}$ supported on
$U(i)_{\bf A}$ ($i\in S_{t}^{2}(M_{-})$ ), and is $T_{F}$-invariant, and there
is a place $v$ of $F$ and a subgroup $G_{v}^{\prime}$ of $G^{q}(F_{v}[[t]])$
that acts transitively on $U(i)_{v}$ such that $\hat{E}$ is invariant under
$G_{v}^{\prime}$. Then the function $S\mapsto\hat{E}(S\phi)$ is bounded on
$T^{\prime}_{\bf A}$, uniformly for $\phi$ in a compact subset in ${\cal
S}(X_{\bf A})$.
This lemma is a generalization of Lemma 23 in [7]. Our proof below closely
follows that of [7].
Proof. Let $e_{1},\dots,e_{n}$ be a quasi-basis of $M_{-}$:
$M_{-}=F[t]/(t^{k_{1}})e_{1}\oplus\dots\oplus F[t]/(t^{k_{n}})e_{n}.$
Then $e_{i}\otimes e_{j}+e_{j}\otimes e_{i}$ is a quasi-basis of ${\rm
St}^{2}(M_{-})$. For each $\alpha\in\\{0,1,\dots,n\\}$, let
$S_{t}^{2}(M_{-})^{(\alpha)}$ be the set that consists of elements
$\sum_{i,j>\alpha}k_{ij}(e_{i}\otimes e_{j}+e_{j}\otimes e_{i})$
such that
$k_{\alpha+1,j}(e_{\alpha+1}\otimes e_{j}+e_{j}\otimes e_{\alpha+1})\neq 0$
for at least one $j\geq{\alpha+1}$. We set
$S_{t}^{2}(M_{-})^{(n)}=\\{0\\}$
by convention. It is clear that $S_{t}^{2}(M_{-})$ is a disjoint union of
$S_{t}^{2}(M_{-})^{(\alpha)}$. Let $\hat{E}_{\alpha}$ be the sum of
$\hat{\mu}_{i}$ for $i\in S_{t}^{2}(M_{-})^{(\alpha)}$. It is clear that
$\hat{E}=\hat{E}_{0}+\hat{E}_{1}+\dots+\hat{E}_{n}$
and $\hat{E}_{\alpha}$ satisfies all the conditions in the lemma. it is enough
to prove the result for each $\hat{E}_{\alpha}$.
Now we fix $0\leq\alpha\leq n$. Let $q$ be the constant which is $1$ if $v$ is
infinite and is equal to the cardinality of the residue field if $v$ is a
finite place. As in [7], there is a compact subset $C\subset I_{F}$ (where
$I_{F}$ is the idele group of $F$), such that every $t\in I_{F}$ with
$1\leq|t|\leq S$ (where $S$ is a fixed constant ) can be written as $rc$ with
$r\in F$, $c\in C$. We denote $C^{n}$ the subset of $T_{\bf A}$ formed by
elements $(c_{1},\dots,c_{n})$ with all $c_{i}\in C$, and let
$\Theta^{\prime}_{\alpha}=\\{(a_{\tau_{1}},\dots,a_{\tau_{n}})\,|\,\tau_{1}=\dots=\tau_{\alpha+1}\geq\dots\geq\tau_{n}\geq
1\\}.$
We will apply Lemma 6 of [7] to the space $X_{F}$. We consider $X_{F}$ as a
product space $\Pi_{i=1}^{n-\alpha}X_{F}^{(i)}$, where $X_{F}^{(1)}$ is the
$F[[t]]$-submodule generated by $e_{1}\otimes V,\dots,e_{\alpha+1}\otimes V$,
and for $n-\alpha\geq i\geq 2$, $X_{F}^{(i)}$ is $F[[t]]e_{i+\alpha}$,
$Y_{F}\subset M_{-}\otimes_{F[[t]]}M_{-}$ is the submodule over $F[[t]]$
spanned by $e_{i}\otimes e_{j}$ with $j\leq\alpha+1$ . And $p:X_{F}\to Y_{F}$
is given by
$p(\sum_{i=1}^{j=1}e_{i}\otimes
v_{i})=\sum_{i=1}^{n}\sum_{j=1}^{\alpha+1}(v_{i},v_{j})e_{i}\otimes e_{j}.$
Apply Lemma 6 of [7], there is $\phi_{0}\in S(X_{\bf A})$ such that
$|((\theta c)\cdot\phi)(x)|\leq\phi_{0}(x)$ (7.5)
for all $x\in X_{\bf A}$ with $T(x)\in S_{t}(M_{-})^{(\alpha)}$, all
$\theta\in\Theta^{\prime}_{\alpha}$, $c\in C^{n}$, and $\phi\in C_{0}$.
It can be proved ([7] page 71) that each element $t=(t_{1},\dots,t_{n})\in
T^{\prime}_{\bf A}$ can be written as
$t=ry\theta c$
where $r\in T_{F}$, $y=(y_{1},\dots,y_{n})\in T_{v}$ with all $|y_{i}|_{v}\geq
1$ and $y_{\alpha+1}=\dots=y_{n}=1$, $\theta\in\Theta^{\prime}_{\alpha}$ and
$c\in C^{n}$. Since $\hat{E}_{\alpha}$ is invariant under $T_{F}$, we have
$|\hat{E}_{\alpha}(t\cdot\phi)|\leq\hat{E}_{\alpha}(y\cdot\phi_{0}).$
We may assume that $\phi_{0}=\phi_{v}\phi^{\prime}$ where $\phi_{v}\in{\cal
S}(X_{v})$, $\phi^{\prime}\in{\cal S}(\Pi^{\prime}_{w\neq v}X_{w})$. By Lemma
22 [7], we have
$\hat{E}_{\alpha}(y\cdot\phi_{0})=\sum_{i\in
St^{2}(M_{-})^{(\alpha)}}c_{i}(\phi^{\prime})\int_{U(i)_{v}}y\cdot\phi_{v}|\theta_{i}|_{v}.$
From this, we obtain that
$\hat{E}_{\alpha}(y\cdot\phi_{0})=\Pi_{i=1}^{n}|y_{i}|_{v}^{k_{1}+\dots+k_{i-1}+(n-m/2-i+2)k_{i}}\hat{E}_{\alpha}(\phi_{0})$
since $k_{i}\leq k_{j}$ for $i\leq j$, and $m>6n+2$, we see that the exponent
of $|y_{i}|_{v}$ is $\leq 0$, and since $|y_{i}|_{v}\geq 1$, so have
$\hat{E}_{\alpha}(y\cdot\phi_{0})\leq\hat{E}_{\alpha}(\phi_{0}).$
This proves the lemma. $\Box$
The following theorem is a generalization of Theorem 4 in [7].
###### Theorem 7.2
Suppose ${\rm dim}\,V>6n+2$. If there is a place $v$ of $F$ such that
$U(0)_{v}$ is not empty, and a subgroup $G_{v}^{\prime}$ of
$G^{q}(F_{v}[[t]])$ acts transitively on $U(i)_{v}$ for every $i\in
S_{t}^{2}(M_{-})$. And if $E^{\prime}$ is a positive tempered measure on
$X_{\bf A}$ invariant under $Sp(M,t)$ and $G_{v}^{\prime}$ and
$E^{\prime}-{\rm Et}$ is supported on the union of $U(i)_{\bf A}$ for $i\in
S_{t}^{2}(M_{-})$ . Then $E^{\prime}={\rm Et}$.
Sketch of Proof. Using Lemma 7.1 and 7.4, it is easy to see that the function
$\widehat{Sp}(M,t)_{\bf A}\to{\mathbb{C}}$ given by $S\mapsto(E^{\prime}-{\rm
Et})(S\phi)$ is bounded on $\widehat{Sp}(M,t)_{\bf A}$ uniformly for $\phi$ in
every compact subset of ${\cal S}(X_{\bf A})$. The remainder of the proof is
similar to that of Theorem 4 in [7]. $\Box$
Now we can prove the main theorem of this work:
###### Theorem 7.3
Suppose ${\rm dim}\,V>6n+2$ and $V$ is anisotropic or ${\rm
dim}V-r>\frac{1}{2}{\rm dim}\,M+1$, where $r$ is the dimension of a maximal
isotropic subspace of $V$, then
${\rm Et}={\rm It}.$
The condition on $V$ in the Theorem is for the convergence of ${\rm It}$ (see
6.1). The proof uses the induction on ${\rm dim}\,M$. The induction assumption
implies that $E^{\prime}={\rm It}$ satisfies the conditions of Theorem 7.2,
therefore ${\rm Et}={\rm It}$.
## 8 Corollaries of Siegel-Weil formula
In this section we prove a slightly more general form of the Siegel-Weil
formula (Theorem 8.1) for snt-modules that will be used in part II. Let $M,V$
be as in Section 3. Let
$M=M_{-}\oplus M_{+}$ (8.1)
be a direct sum such that $M_{+}\in Gr(M,t)$, $M_{-}\in Gr(M)$ but not
necessarily in $Gr(M,t)$. Let $X=M_{-}\otimes V$. The space $L^{2}(X_{\bf A})$
is a representation of the metaplectic group ${\widehat{S}p}_{2N}({\bf A})$
($2N={\rm dim}M{\rm dim}V$) with the usual theta functional
$\theta:{\cal S}(X_{\bf
A})\to{\mathbb{C}},\,\,\,\phi\mapsto\theta(\phi)=\sum_{r\in X}\phi(r).$
Recall the Eisenstein series (1.3), (1.7) for $\phi\in{\cal S}(X_{\bf A})$ is
given by
${\rm E}(\phi)=\sum_{U\in Gr(M)}E(\phi,U)=\sum_{U\in
Gr(M)}\int_{(\pi_{-}(U)\otimes V)_{\bf A}}\psi(\frac{1}{2}\langle x,\rho
x\rangle)\phi(x)dx.$
The $t$-Eisenstein series defined in (1.11) is a subseries given by
${\rm Et}(\phi)=\sum_{U\in Gr(X,t)}E(\phi,U).$ (8.2)
Though $M_{-}$ is not an $F[[t]]$-submodule of $M$, it has $F[[t]]$-module
structure via the isomorphism $M_{-}=M/M_{+}$. In the case that $M_{-}$ is an
$F[[t]]$-submodule, the two $F[[t]]$-module structures on $U_{-}$ clearly
coincide. The projection map
$\pi_{-}:M\to M/M_{+}=M_{-}$
is an $F[[t]]$-module homomorphism. For each $U\in Gr(M,t)$, $\pi_{-}(U)$ is
an $F[[t]]$-submodule of $M_{-}$. Denote $Gr(M_{-},t)$ the set of
$F[[t]]$-submodules of $M_{-}$, so we have a map
$P:Gr(M,t)\to Gr(M_{-},t):\,\,\,U\mapsto\pi_{-}(U).$ (8.3)
For $W\in Gr(M_{-},t)$, we set
${\rm Et}_{W}(\phi)=\sum_{U\in Gr(M,t):\pi_{-}(U)=W}E(\phi,U).$
We recall the action formula of $G^{q}({\bf A}[[t]])$ on ${\cal
S}((M_{-}\otimes V)_{\bf A})$. An element $g\in G^{q}({\bf A}[[t]])$ has the
block decomposition
$\left[\begin{array}[]{cc}\alpha_{g}&\beta_{g}\\\
\gamma_{g}&\delta_{g}\end{array}\right]$
with respect the decomposition
$X_{\bf A}=(M_{-}\otimes V)_{\bf A}\oplus(M_{+}\otimes V)_{\bf A}.$
Since $G^{q}({\bf A}[[t]])$ preserves $(M_{+}\otimes V)_{\bf A}$, so we have
$\gamma_{g}=0$. Since $M_{-}$ is not an $F[[t]]$-submodule in general,
$\beta_{g}$ may not be $0$. Then the action of $g$ on ${\cal S}((X\otimes
V)_{\bf A})$ is given by
$(g\cdot\phi)(x)=\psi(\frac{1}{2}\langle
x\alpha_{g},x\beta_{g}\rangle)\phi(x\alpha_{g})$ (8.4)
If $g\in G^{q}(F[[t]])$ and $\xi\in V$, then
$\frac{1}{2}\langle\xi\alpha_{g},\xi\beta_{g}\rangle\in F,$
and so $\psi(\frac{1}{2}\langle\xi\alpha_{g},\xi\beta_{g}\rangle)=1$ and
$(g\cdot\phi)(\xi)=\phi(\xi\alpha_{g}).$
Therefore, we have
$\theta(g\cdot\phi)=\theta(\phi),\,\,\,\,\,{\rm for}\,\,g\in G^{q}(F[[t]]).$
(8.5)
The function $\theta(g\cdot\phi)$ (as a function on $G^{q}({\bf A}[[t]])$ is
actually a function on $G^{q}(F[[t]])\backslash G^{q}({\bf A}[[t]])$. Assume
$(V,(\,,\,))$ satisfies the conditions in Lemma 6.1, then we can form the
convergent integral
${\rm It}(\phi)\stackrel{{\scriptstyle\rm
def}}{{=}}\int_{G^{q}(F[[t]])\backslash G^{q}({\bf
A}[[t]])}\theta(g\cdot\phi)dg,$ (8.6)
the convergence can be proved in the same way as Lemma 6.1.
We wish to write the integral (8.6) as a sum of orbital integrals. We
introduce the set
$\Omega=S^{1}\times(M_{-}\otimes V)_{\bf A},$
on which $G^{q}({\bf A}[[t]])$ acts as
$(s,x)\cdot g=(s\psi(\frac{1}{2}\langle
x\alpha_{g},x\beta_{g}\rangle),x\alpha_{g}).$ (8.7)
One can then check directly that (8.7) does define a group action; i.e., that
$((s,x)g_{1})g_{2}=(s,x)g_{1}g_{2},\;\,\,\,\,g_{1},g_{2}\in G^{q}({\bf
A}[[t]]).$
On the other hand, if
$\phi\in{\cal S}((U_{-}\otimes V)_{\bf A}),$
we can extend $\phi$ to $\Omega$ by
$\varphi(s,x)=s\varphi(x),\;\,\,\,\,x\in(U_{-}\otimes V)_{{\bf
A}},\,\,\,\,s\in S^{1}.$
And note that we can rewrite the integral (8.6) as
$\int_{G^{q}(F[[t]])\backslash G^{q}({\bf A}[[t]])}\left(\sum_{\xi\in
U_{-}\otimes V}\phi((1,\xi)\cdot g)\right)dg.$ (8.8)
We note that the subset
$(1,U_{-}\otimes V)\subset\Omega$
is invariant under $G^{q}(F[[t]]).$ We let $\mathcal{O}\subset(1,U_{-}\otimes
V)$ be a family of orbit representatives for the action of $G^{q}(F[[t]])$. Of
course $\mathcal{O}$ can be identified with a subset of $U_{-}\otimes_{F}V$
$((1,\xi)\mapsto\xi,$ $\xi\in U_{-}\otimes V),$ and as such, it is a family of
orbit representatives for the action
$\xi\longmapsto\xi\alpha_{g},\;\,\,\,\,g\in G^{q}(F[[t]])$
of $G^{q}(F[[t]])$ on $U_{-}\otimes V$. We may rewrite (8.8) as
$\int_{G^{q}(F[[t]])\backslash G^{q}({\bf
A}[[t]])}\sum_{\xi\in\mathcal{O}}\sum_{\tau\in G_{\xi}\backslash
G^{q}(F[[t]])}\varphi((1,\xi)\tau g)dg.$ (8.9)
Using the fact that $G_{\xi,{\bf A}}$ is unimodular, we can prove (8.9) is
equal to
${\rm It}(\phi)=\sum_{\xi\in\mathcal{O}}vol(G_{\xi}\backslash G_{\xi,{\bf
A}})\int_{G_{\xi,{\bf A}}\backslash G^{q}({\bf
A}[[t]])}\phi((1,\xi)g_{2})dg_{2},$ (8.10)
by a similar argument as in [7] (page 16). We have thereby expressed the ${\rm
It}(\phi)$ as a sum of orbital integrals. For each orbit ${\cal O}$, we have a
corresponding $W\in Gr(M_{-},t)$ (see Theorem 5.8). Let ${\rm It}_{W}(\phi)$
be the subseries of (8.10) that is over all $\xi$ corresponding to $W$. Then
we have
${\rm It}(\phi)=\sum_{W\in Gr(M_{-},t)}{\rm It}_{W}(\phi).$
###### Theorem 8.1
Suppose ${\rm dim}\,V>6n+2$ and $(V,(\,,\,))$ satisfies the condition in Lemma
6.1. Let $M=M_{-}\oplus M_{+}$ be a decomposition such that $M_{-}\in Gr(M)$
and $M_{+}\in Gr(M,t)$. Put $X=M_{-}\otimes V$. We can define tempered
distributions ${\rm Et}$ and ${\rm It}$ on $X_{\bf A}$ as in (8.2) and (8.6).
Then
${\rm Et}={\rm It}.$ (8.11)
And for each $W\in Gr(M_{-},t)$, we have
${\rm Et}_{W}={\rm It}_{W}.$ (8.12)
Notice that we didn’t assume $M_{-}\in Gr(M,t)$, but this theorem can be
reduced to Theorem 7.3. Let
$M=\bar{M}_{-}\oplus\bar{M}_{+}$
be a decomposition of $t$-Lagrangian subspaces. Both $L^{2}((M_{-}\otimes
V)_{\bf A})$ and $L^{2}((\bar{M}_{-}\otimes V)_{\bf A})$ are models of the
Weil representation of $\widehat{Sp}_{2N}({\bf A})$. We shall define an
intertwining operator
$T:L^{2}((M_{-}\otimes V)_{\bf A})\to L^{2}((\bar{M}_{-}\otimes V)_{\bf A})$
such that $T$ sends ${\cal S}((M_{-}\otimes V)_{\bf A})$ to ${\cal
S}((\bar{M}_{-}\otimes V)_{\bf A})$, and
${\rm Et}(\phi)={\rm Et}(T\phi),\,\,\,\,\,\,\,{\rm It}(\phi)={\rm It}(T\phi).$
By Theorem 7.3, ${\rm Et}(T\phi)={\rm It}(T\phi)$, so we have , ${\rm
Et}(\phi)={\rm It}(\phi)$. The more detailed proof follows.
Proof. Recall for the symplectic space $M\otimes V$, we have the associated
Heisenberg group
$H=(M\otimes V)_{\bf A}\times S^{1}$
with the product given by
$(a_{1},s_{1})(a_{2},s_{2})=(a_{1}+a_{2},s_{1}s_{2}\psi(\frac{1}{2}\langle
a_{1},a_{2}\rangle).$
Since $M_{+}$ is a Lagrangian subspace, $A\stackrel{{\scriptstyle\rm
def}}{{=}}(M_{+}\otimes V)_{\bf A}\times S^{1}$ is a maximal abelian subgroup
of $H$. And
$\chi:(M_{+}\otimes V)_{\bf A}\times S^{1}\to S^{1},\,\,\,\,\,(a,s)\mapsto s$
is a $1$-dimensional representation. The induced representation
$Ind_{A}^{H}\chi$
consists of functions $f$ on $H$ such that $f(ax)=\chi(a)f(x)$ for all $a\in
A$ and the restriction $f$ on $(M_{-}\otimes V)_{\bf A}$ is in
$L^{2}((M_{-}\otimes V)_{\bf A})$. Let $M=\bar{M}_{-}\oplus\bar{M}_{+}$ be a
direct sum of $t$-Lagrangian subspaces. Similarly, for the maximal abelian
subgroup $(\bar{M}_{-}\otimes V)_{\bf A}\times S^{1}$, and the character
$\bar{\chi}:(\bar{M}_{-}\otimes V)_{\bf A}\times S^{1}\to
S^{1},\,\,\,\,\,(\bar{a},s)\mapsto s$
we have the induced representation
$Ind_{\bar{A}}^{H}\bar{\chi}.$
The representation space $Ind_{A}^{H}\chi$ ( $Ind_{\bar{A}}^{H}\bar{\chi}$,
resp.) can be identified with $L^{2}((M_{-}\otimes V)_{\bf A})$ (
$L^{2}((\bar{M}_{-}\otimes V)_{\bf A})$, resp.) by the restricting a function
on $H$ to $(\bar{M}_{-}\otimes V)_{\bf A}$ ($(\bar{M}_{-}\otimes V)_{\bf A}$,
reps.). The space of smooth vectors are ${\cal S}(M_{-}\otimes V)_{\bf A})$
and ${\cal S}((\bar{M}_{-}\otimes V)_{\bf A})$, respectively. Let $h\in Sp(M)$
be an element such that
$M_{-}h=\bar{M}_{-},\,\,\,\,\,\,\,\,M_{+}h=\bar{M}_{+}.$
We define
$T:{\cal S}(M_{-}\otimes V)_{\bf A})\to{\cal S}((\bar{M}_{-}\otimes V)_{\bf
A})$
by
$(Tf)(x)=(h\cdot f)(xh^{-1}).$
It is easy to check that $T$ is an isomorphism of $H$-representations. There
are two actions of $\widehat{{Sp}_{2N}}({\bf A})$ on ${\cal
S}((\bar{M}_{-}\otimes V)_{\bf A})$ (or on $L^{2}({\bar{M}}_{-}\otimes V)_{\bf
A})$): the Weil representation action which we denote by $\pi(g)$, and
$\pi^{\prime}(g)=T\pi(g)T^{-1}$ (where $\pi(g)$ denote the Weil representation
action on ${\cal S}({M}_{-}\otimes V)_{\bf A})$. They both satisfy that, for
every $\alpha\in H:$
$\pi(g^{-1})\pi(\alpha)\pi(g)=\pi(\alpha\cdot g)$
$\pi^{\prime}(g^{-1})\pi(\alpha)\pi^{\prime}(g)=\pi(\alpha\cdot g)$
So $\pi^{\prime}(g)=c_{g}\pi(g)$ for some scalar $c_{g}$. It is clear that
$c_{g_{1}g_{2}}=c_{g_{1}}c_{g_{2}}$. Since $Sp_{2N}(F)$ is a perfect group, we
have $c_{g}=1$ for all $g\in Sp_{2N}(F)$. This implies that $c_{g}=1$ for all
$g\in\widehat{{Sp}_{2N}}({\bf A})$. It then follows that $T$ is an isomorphism
of the Weil representations. We define as usual the theta function functionals
$\theta:{\cal S}((M_{-}\otimes V)_{\bf
A})\to{\mathbb{C}},\,\,\,\,\,\,\,\bar{\theta}:{\cal S}((\bar{M}_{-}\otimes
V)_{\bf A})\to{\mathbb{C}}.$
We have
$\bar{\theta}(Tf)=\sum_{r\in\bar{M}_{-}}(Tf)(r)=\sum_{r\in\bar{M}_{-}}(h\cdot
f)(rh^{-1})=\sum_{r\in M_{-}}(h\cdot f)(r)=\theta(h\cdot f)=\theta(f).$
And
$\displaystyle{\rm It}(Tf)$ $\displaystyle=$
$\displaystyle\int_{G^{q}(F[[t]]\backslash G^{q}({\bf A}[[t]]}\theta(g\cdot
Tf)dg$ $\displaystyle=$ $\displaystyle\int_{G^{q}(F[[t]]\backslash G^{q}({\bf
A}[[t]]}\bar{\theta}(T(g\cdot f))dg$ $\displaystyle=$
$\displaystyle\int_{G^{q}(F[[t]]\backslash G^{q}({\bf A}[[t]]}\theta((g\cdot
f))dg$ $\displaystyle=$ $\displaystyle{\rm It}(f).$
Let $P\subset Sp(M)$ be the parabolic subgroup that consists of all $g\in
Sp(M)$ such that $M_{+}g=M_{+}$, similarly, let $\bar{P}\subset Sp(M)$ be the
parabolic subgroup that consists of all $g\in Sp(M)$ such that
$\bar{M}_{+}g=\bar{M}_{+}$. And let $S$ denote the set of all $g\in
P\backslash Sp(M)$ such that $M_{+}g$ is a $t$-Lagrangian subspace. Similarly
let $\bar{S}$ denote the set of all $g\in\bar{P}\backslash Sp(M)$ such that
$\bar{M}_{+}g$ is a $t$-Lagrangian subspace. Since $M_{+}h=\bar{M}_{+}$, the
map $L_{h}:\bar{S}\to S,g\mapsto hg$ is a bijection.
$\displaystyle{\rm Et}(Tf)$ $\displaystyle=$
$\displaystyle\sum_{g\in\bar{S}}(g\cdot Tf)(0)$ $\displaystyle=$
$\displaystyle\sum_{g\in\bar{S}}(Tgf)(0)$ $\displaystyle=$
$\displaystyle\sum_{g\in\bar{S}}(hgf)(0)$ $\displaystyle=$
$\displaystyle\sum_{g\in S}(gf)(0)$ $\displaystyle=$ $\displaystyle{\rm
Et}(f).$
This proves ${\rm Et}(\phi)={\rm It}(\phi)$. To prove ${\rm Et}_{W}(\phi)={\rm
It}_{W}(\phi)$ for every $W\in Gr(M_{-},t)$, we first note for every $W\in
Gr(M_{-},t)$, $W+M_{+}$ is an $F[[t]]$-submodule of $M$. Let
$W^{\bot}=\\{v\in M_{+}\,|\,\langle v,W\rangle=0\\}.$
It is easy to see that $W^{\bot}$ is the radical of the restriction of
$\langle\,,\,\rangle$ on $W+M_{+}$. Therefore
$M_{W}\stackrel{{\scriptstyle\rm def}}{{=}}(W+M_{+})/W^{\bot}$
is an snt-module. And
$M_{W}=W\oplus M_{+}/W^{\bot}$
is a decomposition into Lagrangian subspaces, and $M_{+}/W^{\bot}\in
Gr(M_{W},t)$. We consider ${\cal S}((W\otimes V)_{\bf A})$, and apply (8.11)
to the pair $(Sp(M_{W},t),G)$, we have
$\sum_{W^{\prime}\in Gr(M_{-},t):W^{\prime}\subset W}{\rm
Et}_{W^{\prime}}(\phi)=\sum_{W^{\prime}\in Gr(M_{-},t):W^{\prime}\subset
W}{\rm It}_{W^{\prime}}(\phi).$
The above equality holds for all $W\in Gr(M_{-},t)$, which implies ${\rm
Et}_{W}(\phi)={\rm It}_{W}(\phi)$. $\Box$
## References
* [1] H. Garland, Certain Eisenstein series on loop groups: convergence and the constant term, Algebraic Groups and Arithmetic Tata Inst. Fund. Res. Mumbai (2004), 275-319.
* [2] H. Garland, Absolute convergence of Eisenstein series on loop groups, Duke Math J. Vol. 135, No. 2 (2006), 203-260.
* [3] H. Garland, Y.Zhu, On the Siegel-Weil theorem for loop groups (II), preprint, 2008.
* [4] D.G. James, On Witt’s theorem for unimodular, quadratic forms II, Pacific Jour. of Math. 33, 1970.
* [5] S.S. Kudla, S. Rallis, On the Weil-Siegel formula, J. reine angew. Math. 387 (1988), 1-68.
* [6] A. Weil, Sur certaines groupes d’oprateurs unitaries, Acta Math. 111 (1964), 143-211.
* [7] A. Weil, Sur la formule de Siegel dans la theorie des groupes calssiques, Acta Math. 113 (1965), 1-88.
* [8] Y. Zhu, Theta functions ans Weil representations of loop symplectic groups, Duke Math. J, vol 143 (2208), no. 1, 17-39.
Dept of Math, Yale University, New Haven, CT 06520-8283.
hgarland@math.yale.edu
Dept of Math, Hong Kong University of Science and Tecgnology. mazhu@ust.hk
|
arxiv-papers
| 2008-12-17T10:00:51 |
2024-09-04T02:48:59.445327
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Howard Garland, Yongchang Zhu",
"submitter": "Yongchang Zhu",
"url": "https://arxiv.org/abs/0812.3236"
}
|
0812.3665
|
# Grid Diagrams, Braids, and Contact Geometry
Lenhard Ng and Dylan Thurston Mathematics Department, Duke University,
Durham, NC 27708 ng@math.duke.edu Mathematics Department, Barnard College,
Columbia University, New York, NY 10027 dpt@math.columbia.edu
###### Abstract.
We use grid diagrams to present a unified picture of braids, Legendrian knots,
and transverse knots.
## 1\. Introduction
Grid diagrams, also known in the literature as arc presentations, are a
convenient combinatorial tool for studying knots and links in
$\mathbb{R}^{3}$. Although grid diagrams (or equivalent structures) have been
studied for over a century ([Bru, Cro, Dyn]), they have recently regained
prominence due to their role in the combinatorial formulation of knot Floer
homology ([MOS, MOST]).
It has been known for some time that grid diagrams are closely related to
contact geometry as well as to braid theory. Our purpose here is to indicate
the extent to which the relationships are similar. Indeed, braids, like the
Legendrian and transverse knots in contact geometry, can be viewed as certain
equivalence classes of grid diagrams, and we will see that the various
equivalences fit into one single description. Furthermore, this description is
compatible with the various maps between these objects, like the transverse
knot constructed from a braid. Much of the picture we will present has
previously appeared, but we believe that the full picture (especially the part
concerning braids) is new.
###### Definition 1.
A grid diagram with grid number $n$ is an $n\times n$ square grid with $n$
$X$’s and $n$ $O$’s placed in distinct squares, such that each row and each
column contains exactly one $X$ and one $O$.
We will employ the word “knot” throughout as shorthand for “oriented knot or
oriented link”. Then any grid diagram yields a diagram of a knot in a standard
way: connect $O$ to $X$ in each row, connect $X$ to $O$ in each column, and
have the vertical line segments pass over the horizontal ones (Figure 1). In
addition, one can associate to any grid diagram not only a topological knot
but also a braid, a Legendrian knot, and a transverse knot. We will use the
following notation:
$\displaystyle\mathcal{G}$ $\displaystyle=\\{\text{grid diagrams}\\}$
$\displaystyle\mathcal{K}$ $\displaystyle=\\{\text{isotopy classes of
topological knots}\\}$ $\displaystyle\mathcal{B}$
$\displaystyle=\\{\text{isotopy classes of braids modulo conjugation and
exchange}\\}$ $\displaystyle\mathcal{L}$ $\displaystyle=\\{\text{Legendrian
isotopy classes of Legendrian knots}\\}$ $\displaystyle\mathcal{T}$
$\displaystyle=\\{\text{transverse isotopy classes of transverse knots}\\}.$
(For definitions, see Section 2.)
In Section 2, we will review maps between these various sets that fit together
into the following commutative diagram:
(1)
$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{K}.}$
Here the map from $\mathcal{G}$ to $\mathcal{K}$ is as described above. For
the other maps, see also [Ben, Cro, Dyn, KN, MM, OST].
\begin{picture}(11144.0,3349.0)(1179.0,-3533.0)\end{picture}
Figure 1. A grid diagram and corresponding knot diagram and Legendrian front.
In [Cro] (see also [Dyn]), Cromwell provides a list of alterations of grid
diagrams that do not change topological knot type, the grid-diagram equivalent
of Reidemeister moves for knot diagrams. These are collectively known as
Cromwell moves and consist of translations, commutations, and
stabilizations/destabilizations. The last we distinguish into four types,
X:NW, X:NE, X:SW, and X:SE, following [OST].
###### Proposition 1 (Cromwell [Cro]).
The map $\mathcal{G}\rightarrow\mathcal{K}$ sending grid diagrams to
topological knots induces a bijection
$\mathcal{K}\longleftrightarrow\mathcal{G}/(\text{translation, commutation,
(de)stabilization}).$
We will see that the maps from $\mathcal{G}$ to $\mathcal{B}$, $\mathcal{L}$,
and $\mathcal{T}$ can be similarly understood. More precisely, we have the
following result.
###### Proposition 2.
Let $\tilde{\mathcal{G}}$ denote the quotient set
$\mathcal{G}/(\text{translation, commutation})$. The maps
$\mathcal{G}\rightarrow\mathcal{B}$, $\mathcal{G}\rightarrow\mathcal{L}$, and
$\mathcal{G}\rightarrow\mathcal{T}$ induce bijections
$\displaystyle\mathcal{B}$
$\displaystyle\longleftrightarrow\tilde{\mathcal{G}}/(\text{\text{\it{X:NE}},\text{\it{X:SE}}{}
(de)stabilization})$ $\displaystyle\mathcal{L}$
$\displaystyle\longleftrightarrow\tilde{\mathcal{G}}/(\text{\text{\it{X:NE}},\text{\it{X:SW}}{}
(de)stabilization})$ $\displaystyle\mathcal{T}$
$\displaystyle\longleftrightarrow\tilde{\mathcal{G}}/(\text{\text{\it{X:NE}},\text{\it{X:SW}},\text{\it{X:SE}}{}
(de)stabilization}).$
It follows from this result that the maps between
$\mathcal{B},\mathcal{L},\mathcal{T},\mathcal{K}$ can also be interpreted in
terms of grid diagrams. For instance, the map $\mathcal{B}\to\mathcal{T}$ is
the quotient
$\tilde{\mathcal{G}}/(\text{\text{\it{X:NE}},\text{\it{X:SE}}{}
(de)stabilization})\longrightarrow\tilde{\mathcal{G}}/(\text{\text{\it{X:NE}},\text{\it{X:SW}},\text{\it{X:SE}}{}
(de)stabilization}).$
Similarly, the maps $\mathcal{B}\to\mathcal{K}$, $\mathcal{L}\to\mathcal{T}$,
$\mathcal{L}\to\mathcal{K}$, $\mathcal{T}\to\mathcal{K}$, in terms of grid
diagrams, are quotients by various (de)stabilizations.
Legendrian knotstransverse knotstopological knotsSESWNWNE
Figure 2. Quotienting $\tilde{\mathcal{G}}$, the set of grid-diagram orbits
under translation and commutation, by various combinations of $X$
(de)stabilizations yields equivalence classes of braids and various types of
knots.
Proposition 2 is summarized diagrammatically in Figure 2. The bijections in
Proposition 2 involving $\mathcal{L}$ and $\mathcal{T}$ have already been
established in [OST]; the new content in this note is the bijection involving
$\mathcal{B}$.
We note that stabilization operations on braids and Legendrian and transverse
knots can be expressed in terms of Cromwell moves. More precisely, we have the
following.
###### Proposition 3.
Under the identifications of Proposition 2, we have
positive braid stabilization
$\displaystyle\longleftrightarrow\text{\text{\it{X:SW}}{} stabilization}$
negative braid stabilization
$\displaystyle\longleftrightarrow\text{\text{\it{X:NW}}{} stabilization}$
positive Legendrian stabilization
$\displaystyle\longleftrightarrow\text{\text{\it{X:NW}}{} stabilization}$
negative Legendrian stabilization
$\displaystyle\longleftrightarrow\text{\text{\it{X:SE}}{} stabilization}$
transverse stabilization
$\displaystyle\longleftrightarrow\text{\text{\it{X:NW}}{} stabilization}.$
Proposition 3 follows from an inspection of the effect of the various $X$
stabilizations on the corresponding braid or Legendrian or transverse knot.
See also the table at the end of Section 2.4.
Propositions 2 and 3 give an alternate proof via grid diagrams of the
following result.
###### Proposition 4 (Transverse Markov Theorem [OSh, Wr]).
Two braids represent isotopic transverse knots if and only if they are related
by a sequence of conjugations and positive braid stabilizations and
destabilizations.
In the usual formulation of Proposition 4, the map from braids to transverse
knots uses a contact-geometric construction of Bennequin [Ben] (cf. Section
2.4), rather than the map we use here; see [KN] for a proof that the two maps
coincide.
In Section 2, we recall the various relevant constructions and discuss the
effects of grid-diagram symmetries on the maps in Formula (1). We prove our
main result, Proposition 2, in Section 3.
## 2\. Definitions and Maps
### 2.1. Grid diagrams
The Cromwell moves on grid diagrams, translation, commutation, and
stabilization/destabilization, are illustrated in Figure 3 and defined below.
From that figure it is clear that each Cromwell move preserves the topological
type of the corresponding knot.
Translation views a grid diagram as lying on a torus by identifying opposite
ends of the grid, and changes the diagram by translation in the torus. Any
translation is a composition of some number of vertical translations, which
move the top row of the diagram to the bottom or vice versa, and horizontal
translations, which move the leftmost column of the diagram to the rightmost
or vice versa.
\begin{picture}(11766.0,11744.0)(1607.0,-11033.0)\end{picture}
Figure 3. Illustration of a sequence of Cromwell moves. In succession: X:SE
destabilization; horizontal commutation; vertical torus translation; vertical
commutation; horizontal torus translation; O:SW stabilization. The highlighted
sections of each diagram indicate the portion that changes under the following
move.
Commutation interchanges two adjacent rows (vertical commutation) or two
adjacent columns (horizontal commutation). These adjacent rows or columns are
required to be disjoint or nested in the following sense. For rows, the four
$X$’s and $O$’s in the adjacent rows must lie in distinct columns, and the
horizontal line segments connecting $O$ and $X$ in each row must be either
disjoint or nested (one contained in the other) when projected to a single
horizontal line; there is an obvious analogous condition for columns.
An $X$ (resp. $O$) _destabilization_ replaces a $2\times 2$ subgrid containing
two $X$’s and one $O$ (resp. two $O$’s and one $X$) with a single square
containing an $X$ (resp. $O$), eliminating one row and one column in the
process. _Stabilization_ is the inverse of destabilization. Each
(de)stabilization is identified by its type, $X$ or $O$, along with the corner
in the $2\times 2$ subgrid not occupied by a symbol. This yields eight
possibilities: X:NW, X:NE, X:SW, X:SE, O:NW, O:NE, O:SW, O:SE. It is easy to
check that any O:NW (resp. O:NE, O:SW, O:SE) (de)stabilization can be
expressed as a composition of translations, commutations, and one X:SE (resp.
XSW, X:NE, X:NW) (de)stabilization. Thus we restrict our set of Cromwell moves
to include only $X$ (de)stabilizations.
###### Remark 5.
By the argument of [OST, Lemma 4.3], we can instead drop torus translations
and keep matching $O$ (de)stabilizations to yield alternate definitions for
topological, Legendrian, and transverse knots in terms of grid diagrams. In
particular, X:NE, X:SW, O:SW, and O:NE (de)stabilizations, combined with
commutations, generate all torus translations. The same argument can also be
adapted for braids: that is, $\mathcal{B}$ is also $\mathcal{G}$ modulo
commutation and X:NE, X:SE, O:NW, and O:SW (de)stabilization, as follows.
Sequences of moves similar to those from [OST, Lemma 4.3] show that any
horizontal torus translation can be achieved by these moves, as can any
vertical torus translation where the $O$ appears to the left of the $X$. But
any vertical torus translation can be put into the correct position by
horizontal torus translations.
### 2.2. Braids
As usual, a braid of braid index $n$ is an element of the group
$\mathcal{B}_{n}$ generated by $\sigma_{1},\dots,\sigma_{n-1}$ with relations
$\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}$ for
$1\leq i\leq n-2$ and $\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}$ for
$|i-j|\geq 2$. Note the natural inclusion
$\mathcal{B}_{n}\subset\mathcal{B}_{n+1}$ sending $\sigma_{i}$ to itself for
$i\leq n-1$. The relevant moves to consider on braids are:
* •
braid conjugation: $B\mapsto B^{\prime}B(B^{\prime})^{-1}$ for
$B,B^{\prime}\in\mathcal{B}_{n}$;
* •
exchange move [BM]: $B_{1}\sigma_{n-1}B_{2}\sigma_{n-1}^{-1}\mapsto
B_{1}\sigma_{n-1}^{-1}B_{2}\sigma_{n-1}$ on $\mathcal{B}_{n}$, where
$B_{1},B_{2}\in\mathcal{B}_{n-1}\subset\mathcal{B}_{n}$;
* •
braid stabilization: either positive braid stabilization
$(B\in\mathcal{B}_{n})\mapsto(B\sigma_{n}\in\mathcal{B}_{n+1})$ or negative
braid stabilization
$(B\in\mathcal{B}_{n})\mapsto(B\sigma_{n}^{-1}\in\mathcal{B}_{n+1})$; and
* •
braid destabilization: the inverse of braid stabilization.
In fact, by an observation of Birman and Wrinkle [BW], an exchange move can be
expressed as a combination of one positive stabilization, one positive
destabilization, and a number of conjugations. (Here the positive
stabilization and positive destabilization can equally well be replaced by a
negative stabilization and negative destabilization.) For reference, we
include the calculation here.
$\displaystyle B_{1}\sigma_{n-1}B_{2}\sigma_{n-1}^{-1}$
$\displaystyle\stackrel{{\scriptstyle\text{conj}}}{{\longmapsto}}\sigma_{n-1}B_{1}\sigma_{n-1}B_{2}\sigma_{n-1}^{-2}\stackrel{{\scriptstyle+\text{
stab}}}{{\longmapsto}}\sigma_{n-1}B_{1}\sigma_{n-1}B_{2}\sigma_{n-1}^{-2}\sigma_{n}$
$\displaystyle\stackrel{{\scriptstyle\text{conj}}}{{\longmapsto}}B_{1}\sigma_{n-1}B_{2}\sigma_{n-1}^{-2}\sigma_{n}\sigma_{n-1}=B_{1}\sigma_{n-1}B_{2}\sigma_{n}\sigma_{n-1}\sigma_{n}^{-2}$
$\displaystyle\stackrel{{\scriptstyle\text{conj}}}{{\longmapsto}}\sigma_{n}^{-2}B_{1}\sigma_{n-1}\sigma_{n}B_{2}\sigma_{n-1}=B_{1}\sigma_{n-1}\sigma_{n}\sigma_{n-1}^{-2}B_{2}\sigma_{n-1}$
$\displaystyle\stackrel{{\scriptstyle\text{conj}}}{{\longmapsto}}\sigma_{n-1}^{-2}B_{2}\sigma_{n-1}B_{1}\sigma_{n-1}\sigma_{n}\stackrel{{\scriptstyle+\text{
destab}}}{{\longmapsto}}\sigma_{n-1}^{-2}B_{2}\sigma_{n-1}B_{1}\sigma_{n-1}$
$\displaystyle\stackrel{{\scriptstyle\text{conj}}}{{\longmapsto}}B_{1}\sigma_{n-1}^{-1}B_{2}\sigma_{n-1}.$
We will depict braids horizontally from left to right, with strands numbered
from top to bottom; for instance, $\sigma_{1}$ interchanges the top two
strands, with the top strand passing over the other as we move from left to
right.
### 2.3. Legendrian and transverse knots
We give a quick description of Legendrian and transverse knots, which occur
naturally in contact geometry; see, e.g., [Et] for more details. A Legendrian
knot is a knot in $\mathbb{R}^{3}$ along which the standard contact form
$dz-y\,dx$ vanishes everywhere; a transverse knot is a knot in
$\mathbb{R}^{3}$ along which $dz-y\,dx>0$ everywhere. (Note for the condition
$dz-y\,dx>0$ that the knot is oriented.) We consider Legendrian (resp.
transverse) knots up to Legendrian isotopy (resp. transverse isotopy), which
is simply isotopy through Legendrian (resp. transverse) knots.
One convenient way to depict a Legendrian knot is through its front
projection, or projection in the $xz$ plane. A generic front projection has
three features: it has no vertical tangencies; it is immersed except at cusp
singularities; and at all crossings, the strand of larger slope passes
underneath the strand of smaller slope. Any front with these features
corresponds to a Legendrian knot, with the $y$ coordinate given by $y=dz/dx$.
The knot diagram corresponding to any grid diagram can be viewed as the front
projection of a Legendrian knot by rotating it $45^{\circ}$ counterclockwise
and smoothing out the corners, creating cusps where necessary; see Figure 1
for an example. This yields a map $\mathcal{G}\to\mathcal{L}$ from grid
diagrams to isotopy classes of Legendrian knots. Note that our convention
differs from the convention of [OST]: the convention there is to reverse all
crossings in the grid diagram and then rotate $45^{\circ}$ clockwise. See also
Section 2.5.
In [OST], it is verified that changing a grid diagram by translation,
commutation, or (in our convention) X:SW, X:NE (de)stabilization does not
change the isotopy class of the corresponding Legendrian knot. Changing by
X:NW (resp. X:SE) stabilization does change the Legendrian knot type, by
positive Legendrian stabilization (resp. negative Legendrian stabilization).
Legendrian stabilizations can be described in the front projection as adding a
zigzag, as shown in Figure 4.
+-
Figure 4. Positive and negative Legendrian stabilizations of the front
projection of a Legendrian knot.
Any Legendrian knot is isotopic to one obtained from some grid diagram. It is
shown in [OST] that the set of equivalence classes of Legendrian knots under
Legendrian isotopy corresponds precisely to grid diagrams modulo translation,
commutation, and X:NE, X:SW (de)stabilization, as presented in Proposition 2.
A Legendrian knot can be $C^{0}$ perturbed to a transverse knot, its positive
transverse pushoff. The resulting map $\mathcal{L}\to\mathcal{T}$ is not
injective; negative Legendrian stabilization does not change the transverse
isotopy type of the positive transverse pushoff. It is a standard fact in
contact geometry [EFM] that this gives a bijection
$\mathcal{T}\longleftrightarrow\mathcal{L}/(\text{negative Legendrian
stabilization}).$
Since negative Legendrian stabilization corresponds to an X:SE Cromwell move,
the characterization in Proposition 2 of $\mathcal{T}$ as a quotient of
$\mathcal{G}$ holds. Note that positive Legendrian stabilization becomes the
“transverse stabilization” operation on transverse knots.
### 2.4. Maps between
$\mathcal{G},\mathcal{B},\mathcal{L},\mathcal{T},\mathcal{K}$
Here we collect the constructions of the maps in Formula (1). It suffices to
define $\mathcal{G}\to\mathcal{L}$, $\mathcal{G}\to\mathcal{B}$,
$\mathcal{L}\to\mathcal{T}$, $\mathcal{B}\to\mathcal{T}$, and
$\mathcal{T}\to\mathcal{K}$, since the other maps follow by composition. We
note that the commutativity of the square
$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}}$
was established in [KN], and in fact our description of the maps is
essentially identical to the one given there. The maps
$\mathcal{G}\to\mathcal{L}$ and $\mathcal{L}\to\mathcal{T}$ have already been
discussed; since the map $\mathcal{T}\to\mathcal{K}$ is obvious, we are left
with $\mathcal{G}\to\mathcal{B}$ and $\mathcal{B}\to\mathcal{T}$.
We begin with the map $\mathcal{G}\to\mathcal{B}$, as described in [Cro, Dyn];
this is also called a “flip” in [MM]. Any braid in $B_{n}$ can be viewed as a
braid diagram: a tangle diagram of $n$ strands in the strip
$[0,1]\times\mathbb{R}$, oriented so that the orientation points rightward at
all points, with some collection of $n$ distinct points
$x_{1},\dots,x_{n}\in\mathbb{R}$ for which the braid intersects
$\\{0\\}\times\mathbb{R}$ and $\\{1\\}\times\mathbb{R}$ in
$\\{(0,x_{1}),\dots,(0,x_{n})\\}$ and $\\{(1,x_{1}),\dots,(1,x_{n})\\}$
respectively. Define a rectilinear braid diagram (cf. “braided rectangular
diagram” [MM]) to be a tangle diagram in $[0,1]\times\mathbb{R}$ with the same
boundary conditions as a braid diagram, but consisting exclusively of
horizontal and vertical line segments, satisfying the following properties:
* •
vertical segments always pass over horizontal segments;
* •
each strand can be oriented so that every horizontal segment is oriented
rightwards.
Any rectilinear braid diagram can be perturbed into a standard braid diagram
by perturbing vertical segments slightly to point rightwards, as in Figure 5.
\begin{picture}(13994.0,3194.0)(1179.0,-3533.0)\end{picture}
Figure 5. Braid version (left) of the grid diagram in Figure 1. Omitting the
$X$’s and $O$’s produces a rectilinear braid diagram, which can be perturbed
to become a braid, in this case
$\sigma_{2}^{-1}\sigma_{1}\sigma_{2}^{2}\sigma_{1}^{2}\in\mathcal{B}_{3}$.
Now given a grid diagram, one obtains a knot diagram as usual by drawing
horizontal and vertical lines. Turn this into a rectilinear braid diagram by
replacing any horizontal line oriented leftwards from $O$ to $X$ by two
horizontal lines, one pointing rightwards from the $O$, one pointing
rightwards to the $X$, and have these new horizontal lines pass under all
vertical line segments as usual. The rectilinear braid diagram corresponds to
a braid as described above. This produces the desired map
$\mathcal{G}\to\mathcal{B}$.
It remains to define the map $\mathcal{B}\to\mathcal{T}$. The original
contact-geometric definition from [Ben] is as follows. Identify ends of $B$ to
obtain a knot or link in the solid torus $S^{1}\times D^{2}$. View the solid
torus as a small (framed) tubular neighborhood of the standard transverse
unknot in $\mathbb{R}^{3}$ with self-linking number $-1$. Then $B$ becomes a
transverse knot in a neighborhood of the transverse unknot.
There is also a combinatorial description for the map
$\mathcal{B}\to\mathcal{T}$, which we now describe. (This description is
proven to coincide with the contact-geometric description in [KN]; see also
[MM, OSh]). Create a front by replacing each braid crossing as shown in Figure
6 and joining corresponding braid ends. (Joining ends introduces $2n$ cusps
for a braid with $n$ strands; see Figure 6.) This construction produces a
Legendrian knot from any braid.
B
Figure 6. A Legendrian front for a braid $B$.
It is an easy exercise in Legendrian Reidemeister moves to show that changing
the braid by isotopy changes the Legendrian knot by isotopy and negative
Legendrian (de)stabilization; the stabilization is needed when one introduces
cancelling terms $\sigma_{i}\sigma_{i}^{-1}$ or $\sigma_{i}^{-1}\sigma_{i}$ in
the braid. Similarly, a conjugation or exchange move on a braid produces a
Legendrian isotopy of the Legendrian knot. See Figure 7 for the exchange move.
$B_{1}$$B_{2}$$B_{1}$$B_{2}$$B_{1}$$B_{2}$$B_{1}$$B_{2}$
Figure 7. A braid exchange move produces a Legendrian-isotopic front.
Equality denotes Legendrian isotopy.
The map $\mathcal{B}\to\mathcal{T}$ is now given as follows: given a braid,
the corresponding Legendrian front is well-defined up to isotopy and negative
Legendrian stabilization, and hence its positive transverse pushoff is well-
defined. This transverse knot (equivalently, the class of the Legendrian knot
modulo negative Legendrian (de)stabilization) is unchanged by braid
conjugation and exchange.
Table 1 has a summary of the effect of the Cromwell moves on grid diagrams
correspond to changes in the associated braid, Legendrian knot, and transverse
knot. The braid column is verified in Section 3, while the Legendrian and
transverse columns were established in [OST]. For completeness, the table
includes $O$ as well as $X$ stabilizations.
Grid diagram | Braid | Legendrian knot | Transverse knot
---|---|---|---
torus translation | conjugation | Legendrian isotopy | transverse isotopy
vertical commutation | unchanged | Legendrian isotopy | transverse isotopy
horizontal commutation | conj, exchange | Legendrian isotopy | transverse isotopy
X:NE, O:SW stab | unchanged | Legendrian isotopy | transverse isotopy
X:SW, O:NE stab | conj, $+$ braid stab | Legendrian isotopy | transverse isotopy
X:SE, O:NW stab | unchanged | $-$ Legendrian stab | transverse isotopy
X:NW, O:SE stab | conj, $-$ braid stab | $+$ Legendrian stab | transverse stab
Table 1. The effect of Cromwell moves on associated topological structures.
### 2.5. Symmetries and conventions
Here we discuss various symmetries of grid diagrams and how they relate the
conventions for the maps in Formula (1) to other, sometimes conflicting,
conventions in the literature. In this section, we will denote the maps
$\mathcal{G}\to\mathcal{L}$, $\mathcal{G}\to\mathcal{T}$,
$\mathcal{G}\to\mathcal{B}$ described in Section 2.4 by $G\mapsto L(G)$,
$G\mapsto T(G)$, $G\mapsto B_{\shortrightarrow}(G)$, respectively.
Consider the symmetries $S_{1}$, $S_{2}$, $S_{3}$, and $S_{4}$ of grid
diagrams defined as follows:
* •
$S_{1}$ rotates the grid diagram $180^{\circ}$;
* •
$S_{2}$ reflects the diagram about the NE-SW diagonal and interchanges $X$’s
and $O$’s;
* •
$S_{3}$ reflects the diagram across the horizontal axis; and
* •
$S_{4}$ rotates the grid diagram $180^{\circ}$ and interchanges $X$’s and
$O$’s.
Both $S_{1}$ and $S_{2}$ preserve topological knot type, while $S_{3}$
produces the topological mirror knot $m(K)$ (with reversed orientation on
$\mathbb{R}^{3}$), and $S_{4}$ produces the inverse (i.e., orientation-
reversed) knot $-K$.
The symmetries descend to the quotient $\tilde{\mathcal{G}}$ of grid diagrams
by translation and commutation. On $\tilde{\mathcal{G}}$, it is readily
checked that the symmetries permute the four $X$ stabilizations as shown in
Table 2. We will use this information to examine the effect of the symmetries
on Legendrian and transverse knots and braids, as shown in the table and
explained below.
$\displaystyle\begin{array}[]{@{}c*{4}{r@{}>{{}}c<{{}}@{}l}c@{}}\hline\cr\hline\cr\text{Symmetry}&\lx@intercol\hfil\text{Knot}\hfil\lx@intercol&\lx@intercol\hfil\text{Braid}\hfil\lx@intercol&\lx@intercol\hfil\text{Legendrian}\hfil\lx@intercol&\lx@intercol\hfil\text{Transverse}\hfil\lx@intercol&X\text{
stabilizations}\\\ \hline\cr
S_{1}&K&\mapsto&K&B_{\shortrightarrow}&\mapsto&B_{\shortleftarrow}&\hskip
3.00003ptL&\mapsto&\mu(L)&&\clap{\text{---}}&&\vbox{\lx@xy@svg{\hbox{\raise
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S_{2}&K&\mapsto&K&B_{\shortrightarrow}&\mapsto&B_{\shortuparrow}&L&\mapsto&L&T&\mapsto&T&\vbox{\lx@xy@svg{\hbox{\raise
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27.10411pt\raise-25.74994pt\hbox{\hbox{\kern 0.0pt\raise
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S_{3}&K&\mapsto&m(K)&B_{\shortrightarrow}&\mapsto&m(B_{\shortrightarrow})&&\clap{\text{---}}&&&\clap{\text{---}}&&\vbox{\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
12.97916pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\raise
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S_{4}&K&\mapsto&-K&B_{\shortrightarrow}&\mapsto&-B_{\shortrightarrow}&L&\mapsto&-\mu(L)&T&\mapsto&-\mu(T)&\vbox{\lx@xy@svg{\hbox{\raise
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\hline\cr\hline\cr\end{array}$
Table 2. The effect of symmetries of a grid diagram on associated topological
structures.
Since $S_{1}$ and $S_{2}$ send X:NE, X:SW stabilizations to themselves or each
other, Proposition 2 implies that these symmetries descend to maps on
$\mathcal{L}$. Indeed, it can be shown (see, e.g., [OST, Lemma 4.6]) that
$S_{2}$ does not change Legendrian isotopy type: $L\circ S_{2}(G)=L(G)$. It
follows also that $T\circ S_{2}(G)=T(G)$. On the other hand, we have $L\circ
S_{1}(G)=\mu(L(G))$, where $\mu:\thinspace\mathcal{L}\to\mathcal{L}$ is the
Legendrian mirror operation, which reflects Legendrian front diagrams in the
horizontal axis [FT, OST]. In general, the two maps lead to two distinct
Legendrian knots [Ng]; note that Legendrian “mirroring” preserves topological
type. We remark that $S_{3}$ does not descend to a map on $\mathcal{L}$ (there
is no Legendrian version of the topological mirror construction), and
Legendrian mirrors do not descend to the transverse category.
The map $S_{4}$ on Legendrian knots produces the orientation reverse of the
Legendrian mirror: $L\mapsto-\mu(L)$. This operation descends to (oriented)
transverse knots, in an operation that could be called the transverse mirror.
We next consider braids. Given a grid diagram, there are four equally valid
ways to obtain a map $\mathcal{G}\to\mathcal{B}$ that preserves topological
knot type. One can require that the braid goes from left to right, as we do in
Section 2.4, but one could instead require that the braid go from bottom to
top, right to left, or top to bottom. We write the resulting maps as $G\mapsto
B_{\shortrightarrow}(G)$, $G\mapsto B_{\shortuparrow}(G)$, $G\mapsto
B_{\shortleftarrow}(G)$, and $G\mapsto B_{\shortdownarrow}(G)$, respectively.
In general, these maps lead to four distinct braids, related by
$B_{\shortrightarrow}\circ S_{1}(G)=B_{\shortleftarrow}(G)\hskip
25.83325ptB_{\shortrightarrow}\circ S_{2}(G)=B_{\shortuparrow}(G)\hskip
25.83325ptB_{\shortrightarrow}\circ S_{1}\circ
S_{2}(G)=B_{\shortdownarrow}(G).$
As noted in [KN], it follows from $L\circ S_{2}(G)=L(G)$ that the braids
$B_{\shortrightarrow}(G)$ and $B_{\shortuparrow}(G)$ represent the same
element of $\mathcal{T}$ even though they usually differ in $\mathcal{B}$, and
the same is true of the pair $B_{\shortleftarrow}(G)$ and
$B_{\shortdownarrow}(G)$. In addition, if we define operations $B\mapsto m(B)$
and $B\mapsto-B$ on braids, where $m(B)$ replaces every letter in $B$ by its
inverse and $-B$ is the braid word $B$ read backwards, then
$B_{\shortrightarrow}\circ S_{3}(G)=m(B_{\shortrightarrow}(G))$ and
$B_{\shortrightarrow}\circ S_{4}(G)=-B_{\shortrightarrow}(G)$.
All symmetries of the NW-NE-SE-SW square are generated by $S_{1},S_{2},S_{3}$.
The following generalization of Proposition 2 is an immediate consequence of
the symmetries and Proposition 2.
###### Corollary 6.
We have bijections
$\displaystyle\tilde{\mathcal{G}}/(\text{\it{X:NE}},\text{\it{X:SE}})$
$\displaystyle\stackrel{{\scriptstyle
B_{\rightarrow}}}{{\longrightarrow}}\mathcal{B}$
$\displaystyle\tilde{\mathcal{G}}/(\text{\it{X:SW}},\text{\it{X:SE}})$
$\displaystyle\stackrel{{\scriptstyle
B_{\uparrow}}}{{\longrightarrow}}\mathcal{B}$
$\displaystyle\tilde{\mathcal{G}}/(\text{\it{X:NW}},\text{\it{X:SW}})$
$\displaystyle\stackrel{{\scriptstyle
B_{\leftarrow}}}{{\longrightarrow}}\mathcal{B}$
$\displaystyle\tilde{\mathcal{G}}/(\text{\it{X:NW}},\text{\it{X:NE}})$
$\displaystyle\stackrel{{\scriptstyle
B_{\downarrow}}}{{\longrightarrow}}\mathcal{B}$
$\displaystyle\tilde{\mathcal{G}}/(\text{\it{X:NE}},\text{\it{X:SW}})$
$\displaystyle\stackrel{{\scriptstyle L}}{{\longrightarrow}}\mathcal{L}$
$\displaystyle\tilde{\mathcal{G}}/(\text{\it{X:NW}},\text{\it{X:SE}})$
$\displaystyle\stackrel{{\scriptstyle L\circ
S_{3}}}{{\longrightarrow}}\mathcal{L}$
$\displaystyle\tilde{\mathcal{G}}/(\text{\it{X:NE}},\text{\it{X:SW}},\text{\it{X:SE}})$
$\displaystyle\stackrel{{\scriptstyle T}}{{\longrightarrow}}\mathcal{T}$
$\displaystyle\tilde{\mathcal{G}}/(\text{\it{X:NW}},\text{\it{X:NE}},\text{\it{X:SW}})$
$\displaystyle\stackrel{{\scriptstyle T\circ
S_{1}}}{{\longrightarrow}}\mathcal{T}$
$\displaystyle\tilde{\mathcal{G}}/(\text{\it{X:NW}},\text{\it{X:SW}},\text{\it{X:SE}})$
$\displaystyle\stackrel{{\scriptstyle T\circ
S_{3}}}{{\longrightarrow}}\mathcal{T}$
$\displaystyle\tilde{\mathcal{G}}/(\text{\it{X:NW}},\text{\it{X:NE}},\text{\it{X:SE}})$
$\displaystyle\stackrel{{\scriptstyle T\circ S_{3}\circ
S_{2}}}{{\longrightarrow}}\mathcal{T}$
where $L$, $T$ are induced from the maps $\mathcal{G}\to\mathcal{L}$,
$\mathcal{G}\to\mathcal{T}$ described in Section 2.4.
Note that three of the bijections in Proposition 6 involve $S_{3}$ and thus
topological mirroring.
We now discuss the conventions used in Section 2.4 in light of symmetries of
grid diagrams. Our conventions are chosen to make the maps in Formula (1)
always preserve topological knot type. This involves making several choices:
* •
vertical over horizontal line segments in grid diagrams (vs. horizontal over
vertical), and Legendrian fronts obtained by $45^{\circ}$ counterclockwise
rotation (vs. clockwise);
* •
transverse knots given by positive pushoffs of Legendrian knots (vs.
negative);
* •
braids going from left to right (vs. bottom to top, right to left, top to
bottom).
These choices largely agree with the standard conventions in the literature
[Cro, Dyn, EFM, Et, MOS, MOST]. One can obtain different conventions from ours
by applying grid-diagram symmetries. For braids, this is discussed above,
while for transverse knots, positive pushoffs become negative pushoffs by
applying the symmetry $S_{1}$: negative pushoffs are transversely isotopic
under X:NW,X:NE,X:SW (de)stabilization.
For the knot Floer homology invariant introduced in [OST] and subsequently
used in [KN, NOT], a slightly different set of conventions is useful. Here an
element $\lambda^{+}$ of combinatorial knot Floer homology $\mathit{HK}^{-}$
is associated to any grid diagram, and $\lambda^{+}$ is shown to be invariant
under translation, commutation, and X:NW,X:SW,X:SE (de)stabilization. (Another
element $\lambda^{-}$ is also considered in [OST]; in our notation,
$\lambda^{-}=\lambda^{+}\circ S_{1}$.) If we apply symmetry $S_{2}\circ S_{3}$
to a grid diagram $G$ before calculating $\lambda^{+}$, then $\lambda^{+}$
becomes an invariant of the transverse knot $T(G)$.
In [KN, NOT, OST], the map $\mathcal{G}\to\mathcal{L}$ is thus given by
$G\mapsto(L\circ S_{2}\circ S_{3})(G)$ rather than $G\mapsto L(G)$. More
explicitly, given a grid diagram, one can use the horizontal-over-vertical
convention and $45^{\circ}$ clockwise rotation to obtain a Legendrian front,
as is done in these papers. (In particular, to translate from our conventions
to those of [KN], first apply $S_{2}\circ S_{3}$ to all grid diagrams.) Note
that due to the presence of $S_{3}$, $\lambda^{+}$ becomes an element of
$\mathit{HK}^{-}$ of the topological mirror of the transverse knot.
## 3\. Proof of Proposition 2
Let $B(G)$ ($=B_{\rightarrow}(G)$ from Section 2.5) denote the braid
associated to a grid diagram $G$ as described in Section 2. Proposition 2 (or,
more precisely, the braid statement of Proposition 2) is a direct consequence
of the following stronger result.
###### Proposition 7.
Let $G$ be a grid diagram.
1. (1)
Changing $G$ by torus translation or X:NE,X:SE (de)stabilization changes
$B(G)$ by conjugation.
2. (2)
Changing $G$ by commutation changes $B(G)$ by a combination of conjugation and
exchange moves.
3. (3)
The map $G\mapsto B(G)$ induces a bijection between
$\mathcal{G}$/(translation, commutation, X:NE, X:SE (de)stabilization) and
$\mathcal{B}$/(conjugation, exchange).
###### Proof.
We first check claims (1) and (2). A quick inspection of braid diagrams
reveals that changing a grid diagram $G$ by horizontal commutation or by X:NE
or X:SE stabilization does not change the braid isotopy type of $B(G)$.
Changing $G$ by horizontal torus translation changes $B(G)$ by conjugation;
some portion of the beginning of $B(G)$ is moved to the end, or vice versa.
See Figure 8.
Next we claim that changing $G$ by vertical torus translation also changes
$B(G)$ by conjugation. Indeed, consider moving the topmost column of $G$ to
the bottom. By conjugating by a horizontal torus translation if necessary, we
may assume that in the relevant row, the $O$ lies to the left of the $X$. Then
moving the column keeps the braid unchanged; see Figure 8 again.
\begin{picture}(15794.0,3794.0)(-21.0,-3833.0)\end{picture}
Figure 8. The effect on $B(G)$ of changing $G$ by horizontal (left) and
vertical (right) torus translation. The bold $X$ and $O$ represent the
column/row being moved.
Finally, we claim that changing $G$ by a vertical commutation changes $B(G)$
by conjugation and/or exchange. Indeed, by conjugating with an appropriate
torus translation if necessary, we may assume the following: the two relevant
rows are the bottom two rows in the grid diagram; the $X$ and $O$ in the
bottom row both lie to the right of the $X$ and $O$ in the row above it; and
the bottom right corner of the grid diagram is occupied by an $X$ or $O$. If
$X$ lies to the left of $O$ in both rows, then the commutation changes $B(G)$
by exchange; otherwise, it does not change $B(G)$. See Figure 9.
$B_{1}$$B_{2}$$B_{2}$$B_{1}$
Figure 9. The effect on $B(G)$ of changing $G$ by horizontal commutation. In
three cases, $B(G)$ is unchanged. In the other case (upper left), the
$n$-strand braid $B(G)$ changes from $B_{1}\sigma_{n-1}^{-1}B_{2}\sigma_{n-1}$
to $B_{1}\sigma_{n-1}B_{2}\sigma_{n-1}^{-1}$, an exchange move.
We now establish claim (3). From claims (1) and (2), the map in (3) is well-
defined. To prove bijectivity, we construct an inverse. Any braid $B$ can be
given a rectilinear braid diagram by replacing each crossing by an appropriate
rectilinear version; see Figure 10.
\begin{picture}(8444.0,1454.0)(1179.0,-1763.0)\end{picture}
Figure 10. Turning a braid diagram into a rectilinear braid diagram.
Perturb the resulting rectilinear diagram slightly to another rectilinear
diagram for which no vertical line segments have the same $x$-coordinate
(i.e., are collinear), and no horizontal line segments have the same
$y$-coordinate except for those that are identified when the ends of the braid
are identified. The perturbed diagram is oriented (from left to right), and
each corner can be assigned an $X$ or $O$ in the usual way. The collection of
$X$’s and $O$’s forms a grid diagram $G(B)$, and by construction we have
$B=B(G(B))$.
$B_{2}$$B_{1}$$B_{1}$$B_{2}$$B_{1}$$B_{1}$$B_{2}$$B_{2}$
Figure 11. Accomplishing an exchange move through a sequence of commutation
and (de)stabilization moves. The first arrow is given by commutations, one
X:NE destabilization, and one X:SE destabilization; the second is a horizontal
commutation; the third is commutations, one X:NE stabilization, and one X:SE
destabilization. See also Figure 12 for the moves corresponding to the first
and third arrows.
\begin{picture}(12944.0,5519.0)(1179.0,-5708.0)\end{picture}
Figure 12. Detail of local moves in the first step of Figure 11. A vertical
commutation move is followed by X:NE and X:SE destabilization.
Note that $G(B)$ depends on the choice of perturbation from rectilinear braid
diagram to grid diagram, but a different perturbation simply changes $G(B)$ by
commutation. In fact, up to commutation and X:SW,X:SE (de)stabilization,
$G(B)$ is well-defined for an isotopy class of braids $B$. This fact is
readily established by examining how $G(B)$ changes when the braid word for
$B$ changes by one of the relations
$\sigma_{i}\sigma_{i}^{-1}=\sigma_{i}^{-1}\sigma_{i}=1$,
$\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}$ for $|i-j|\geq 2$, and
$\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}$. See
[Cro] for details.
In addition, changing $B$ by conjugation changes $G(B)$ by horizontal torus
translation, while changing $B$ by an exchange move changes $G(B)$ by a
combination of horizontal commutations and X:NE,X:SE (de)stabilizations; see
Figures 11 and 12. Thus $B$ induces a map from $\mathcal{B}$/(conjugation,
exchange) to $\mathcal{G}$/(translation, commutation, X:NE, X:SE
(de)stabilization).
If we consider $G$ and $B$ as maps between $\mathcal{G}$/(translation,
commutation, X:NE, X:SE (de)stabilization) and $\mathcal{B}$/(conjugation,
exchange), then as noted earlier, $B\circ G$ is the identity, and one readily
checks that $G\circ B$ is the identity as well. Claim (3) follows, and the
proof of Proposition 7 is complete. ∎
## Acknowledgments
LLN thanks the participants of the conference “Knots in Washington XXVI” for
useful comments on a preliminary version of the results presented here. DPT
thanks Ciprian Manolescu, Peter Ozsváth, and Zoltán Szabó for helpful
conversations. LLN was supported by NSF grant DMS-0706777; DPT was supported
by a Sloan Research Fellowship.
## References
* [Ben] D. Bennequin, Entrelacements et équations de Pfaff, Astérisque 107–108 (1983), 87–161.
* [BM] J. S. Birman and W. M. Menasco, Studying links via closed braids IV: Composite links and split links, Invent. Math. 102 (1990), no. 1, 115–139.
* [BW] J. S. Birman and N. C. Wrinkle, On transversally simple knots, J. Differential Geom. 55 (2000), no. 2, 325–354; arXiv:math.GT/9910170.
* [Bru] H. Brunn, Über verknotete Kurven, Verhandlungen des Internationalen Math. Kongresses (Zürich 1897), 256–259, 1898.
* [Cro] P. R. Cromwell, Embedding knots and links in an open book I: Basic properties, Topology Appl. 64 (1995), no. 1, 37–58.
* [Dyn] I. A. Dynnikov, Arc-presentations of links: monotonic simplification, Fund. Math. 190 (2006), 29–76; arXiv:math.GT/0208153.
* [EFM] J. Epstein, D. Fuchs, and M. Meyer, Chekanov–Eliashberg invariants and transverse approximations of Legendrian knots, Pacific J. Math. 201 (2001), no. 1, 89–106.
* [Et] J. B. Etnyre, Legendrian and transversal knots, in Handbook of knot theory, 105–185, Elsevier B. V., Amsterdam, 2005; arXiv:math.SG/0306256.
* [FT] D. Fuchs and S. Tabachnikov, Invariants of Legendrian and transverse knots in the standard contact space, Topology 36 (2007), no. 5, 1025–1053.
* [KN] T. Khandhawit and L. Ng, A family of transversely nonsimple knots, arXiv:0806.1887.
* [MM] H. Matsuda and W. Menasco, On rectangular diagrams, Legendrian knots and transverse knots, arXiv:0708.2406.
* [MOS] C. Manolescu, P. Ozsváth, and S. Sarkar, A combinatorial description of knot Floer homology, arXiv:math/0607691.
* [MOST] C. Manolescu, P. Ozsváth, Z. Szabó, and D. Thurston, On combinatorial link Floer homology, Geom. Topol. 11 (2007), 2339–2412; arXiv:math/0610559.
* [Ng] L. Ng, Computable Legendrian invariants, Topology 42 (2003), no. 1, 55–82; arXiv:math.GT/0011265.
* [NOT] L. Ng, P. Ozsváth, and D. Thurston, Transverse knots distinguished by knot Floer homology, J. Symplectic Geom., to appear; arXiv:math/0703446.
* [OSh] S. Yu. Orevkov and V. V. Shevchishin, Markov theorem for transversal links, J. Knot Theory Ramifications 12 (2003), no. 7, 905–913; arXiv:math.GT/0112207.
* [OST] P. S. Ozsváth, Z. Szabó, and D. P. Thurston, Legendrian knots, transverse knots and combinatorial Floer homology, arXiv:math/0611841.
* [Wr] N. C. Wrinkle, The Markov Theorem for transverse knots, arXiv:math.GT/0202055.
|
arxiv-papers
| 2008-12-19T17:44:44 |
2024-09-04T02:48:59.466858
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lenhard Ng and Dylan Thurston",
"submitter": "Dylan Thurston",
"url": "https://arxiv.org/abs/0812.3665"
}
|
0812.3685
|
# Adiabatic dynamics of a quantum critical system coupled to an environment:
Scaling and kinetic equation approaches
Dario Patanè MATIS CNR-INFM $\&$ Dipartimento di Metodologie Fisiche e
Chimiche (DMFCI), Università di Catania, viale A. Doria 6, 95125 Catania,
Italy Departamento de F sica de Materiales, Universitad Complutense, $28040$
Madrid, Spain Alessandro Silva The Abdus Salam International Centre for
Theoretical Physics, Strada Costiera $11$, $34100$ Trieste, Italy Luigi Amico
MATIS CNR-INFM $\&$ Dipartimento di Metodologie Fisiche e Chimiche (DMFCI),
Università di Catania, viale A. Doria 6, 95125 Catania, Italy Departemento de
F sica de Materiales, Universitad Complutense, $28040$ Madrid, Spain Rosario
Fazio NEST-CNR-INFM $\&$ Scuola Normale Superiore, Piazza dei Cavalieri 7,
I-56126 Pisa, Italy Giuseppe E. Santoro International School for Advanced
Studies (SISSA), Via Beirut $2-4$, $34014$ Trieste, Italy CNR-INFM Democritos
National Simulation Center, Via Beirut $2-4$, $34014$ Trieste, Italy The
Abdus Salam International Centre for Theoretical Physics, Strada Costiera
$11$, $34100$ Trieste, Italy
###### Abstract
We study the dynamics of open quantum many-body systems driven across a
critical point by quenching an Hamiltonian parameter at a certain velocity.
General scaling laws are derived for the density of excitations and energy
produced during the quench as a function of quench velocity and bath
temperature. The scaling laws and their regimes of validity are verified for
the XY spin chain locally coupled to bosonic baths. A detailed derivation and
analysis of the kinetic equation of the problem is presented.
## I Introduction
A series of beautiful experiments on the dynamics of cold atomic gases Greiner
; Kinoshita ; Sadler spurred renewed interest in the study of non-equilibrium
quantum many-body systems. On the theoretical side these experiments triggered
an intense investigation mostly devoted to the simplest paradigm of
nonequilibrium quantum dynamics: the controlled variation in time of one of
the system parameters (quantum quenches). In the case of sudden quenches,
where the driving parameter is changed on a time scale much shorter than
typical time scales of the system, a number of important issues have been
addressed. We mention, for example, the study of the signatures of
universality in the quench dynamics of quantum critical systems Sengupta , the
presence of thermalization in integrable vs. nonintegrable systems Rigol , as
well as the description of generic nonequilibrium quenches using thermodynamic
variables Polkovnikov08-2 and their statistics Silva08 .
In this paper we will focus on the opposite case in which the control
parameter is varied adiabatically, a case which becomes particularly
interesting when a critical point is crossed during the adiabatic evolution.
Because of the vanishing of the energy gap at criticality, the system is
unable to follow adiabatically the driving remaining in its equilibrium/ground
state when passing through the quantum critical point the system will not be
able no matter how slow is the quench. The study of these deviation from the
adiabatic dynamics is a problem which is very important in a number of
different branches of physics ranging from the defect formation in the early
universe KZ ; KibbleReview to adiabatic quantum computation farhi01 or
quantum annealingsantoro02 ; santoro06 . Depending on the context, the loss of
adiabaticity has been characterized by the excess energy at the end of the
quench, by the density of defects (if the final state was a fully ordered
system), or by the fidelity of the time evolved state with the ground state of
the Hamiltonian at the end of the quench.
The scaling of the density of excitations generated during the dynamics as a
function of the velocity of the quench was first predicted in Ref. zurek05, ;
polkovnikov05, for a quantum critical system. The mechanism behind the
generation of excitations/defects is similar to so-called Kibble-Zurek (KZ)
mechanism KZ first proposed for classical phase transitions. Following these
initial works a number of specific models were scrutinizedDziarmaga05 ;
Damski05 ; Schutzhold06 ; Cherng06 ; Damski07 ; Cucchietti07 ; Cincio07 ;
Caneva07 ; Sengupta08 ; Polkovnikov08-1 ; Caneva08 ; Deng08 ; Sen08 ;
Pellegrini08 ; Divakaran08 , thereby confirming the general picture.
All the works mentioned previously assumed unitary Hamiltonian dynamics. We
know, however, that understanding the effect of the external environment on
the adiabatic dynamics is of paramount importance for several reasons. In the
case of adiabatic quantum computation, decoherence is a fundamental limiting
factor to the ability of implementing quantum algorithms. Furthermore, an
experimental verification of the KZ scaling in a quantum phase transition can
only occur through the detection of this effect at low temperatures, i.e. when
the quantum critical system is in contact with a thermal bath. Despite its
importance the adiabatic dynamics of open critical systems is a much less
studied problem. The effect of classical and quantum noise acting uniformly on
a quantum Ising chain was considered in Ref.Fubini07, and Ref.Mostame07,
respectively. Numerical simulations for a model of local noise on a disordered
Ising model were performed in Ref.Amin08, . Moreover the effect a static spin
bath locally coupled to an ordered Ising model is studied in Ref.Cincio08, .
In a recent Letter PatanePRL , we have addressed the universality of the
production of defects in the adiabatic dynamics in the presence of an
environment by generalizing the scaling theory to open critical system and by
formulating a quantum kinetic equation approach for the adiabatic dynamics
across the quantum critical region. We found that, at weak coupling and for
not too slow quenches the density of excitations is universal also in the
presence of an external bath. In this paper we extend the results presented in
Ref. PatanePRL, and provide a detailed derivation of the kinetic equations
and of the scaling approach.
The paper is organized as follows. We first derive qualitatively the scaling
laws obeyed by both the density of defects and of energy generated in a quench
for a generic open quantum critical system (Sec.II). We then address a
specific one-dimensional model possessing a quantum critical point: the XY
spin chain in transverse magnetic field. To model a thermal reservoir we
couple the system to a set of bosonic degrees of freedom, as in the spin-boson
model. Baths are chosen with power-law spectral density and are locally
coupled to strings of neighboring spins. The model, a generalization of the
one studied in Ref.PatanePRL, , is discussed in Sec.III. For this model, we
derive a kinetic equation within the Keldysh technique (Sec. IV and Appendix
A1-A2) which allows us to compute the density of defects. In Sec. V (and
Appendix B) we discuss the spectrum of relaxation times needed for a
comparison with the scaling approach. The density of defects and of excitation
generated in a quench, and a comparison with the scaling laws is presented in
Sec. VI. Finally in Sec.VII we summarize our conclusions.
## II Scaling analysis
In this section we discuss the scaling laws obeyed by the density of
excitations PatanePRL and by the energy density following a linear quench of
a control parameter $h$ from an initial value $h_{i}$ to a final one $h_{f}$,
through a second-order quantum critical point at $h_{c}$. The system, during
the whole dynamics, is kept in contact with a bath at temperature $T$. In the
$h-T$ plane the adiabatic quench is described by the horizontal line shown in
Fig.1. For adiabatic quenches occurring at zero temperature, the system stops
following the external drive adiabatically and can be considered as frozen
around the quantum critical point. This happens roughly when the time it takes
to reach and cross the quantum critical point becomes comparable to the
internal time scale (the inverse gap $\Delta(h)\simeq|h-h_{c}|^{-\nu z}$). The
determination of this crossover point is the fundamental ingredient which
leads to the scaling in the case of unitary evolution zurek05 ; polkovnikov05
. In the case of a finite temperature quench there is a new important
timescale which enters the problem, the time at which the system enters (and
eventually leaves) the quantum critical region (see Fig.1). Initially the
system is in equilibrium with the bath at a low temperature
($T\ll\Delta(h_{i})$) and the behavior is, to a large extent, as in the zero-
temperature case. In the quantum critical region SachdevBOOK , characterized
by the crossover temperature $T\sim|h-h_{c}|^{\nu z}$, the gap is much smaller
than the temperature itself. One can therefore expect that during the interval
the system spends in the quantum critical region a number of excitations will
be produced by the presence of the environment. Interestingly also this
contribution to the defect production obeys a scaling law footnote1 .
Figure 1: A sketch of the finite temperature crossover phase-diagram close to
the quantum critical point. Crossover lines $T\sim|h-h_{c}|^{\nu z}$
separating the semiclassical regions from the quantum critical region are
shown. The latter is traversed by the system during the quench in a time
$t_{QC}$.
We now proceed with the derivation of the scaling laws. In the rest of the
paper we will consider the density $\mathcal{E}$ and the energy density ${E}$
of excitations, defined respectively:
$\displaystyle\mathcal{E}$ $\displaystyle=$
$\displaystyle\int\frac{d^{d}k}{\left(2\pi\right)^{d}}\mathcal{P}_{k}\;,$ (1)
$\displaystyle{E}$ $\displaystyle=$
$\displaystyle\int\frac{d^{d}k}{\left(2\pi\right)^{d}}E_{k}\mathcal{P}_{k}\;,$
(2)
where $\mathcal{P}_{k}$ is the population of the excitation with quantum
number $k$, $E_{k}$ the energy spectrum at $h_{f}$, and $d$ is the
dimensionality of the quantum system.
The first assumption we make consists in separating the density of
excitations/energy at the end of the quench in the sum of two (coherent and
incoherent) contributions:
$\displaystyle\mathcal{E}$ $\displaystyle\simeq$
$\displaystyle\mathcal{E}_{coh}+\mathcal{E}_{inc}\;,$ (3) $\displaystyle{E}$
$\displaystyle\simeq$ $\displaystyle{E}_{coh}+{E}_{inc}\;.$ (4)
In the previous equation, ${E}_{coh}$ ($\mathcal{E}_{coh}$) is the density of
energy (excitations) of the system produced coherently in the absence of the
bath; the incoherent contribution ${E}_{inc}$ ($\mathcal{E}_{inc}$) arises
instead from the bath/system interaction. The separation of a coherent and an
incoherent contribution, eqs. (3) and (4), requires weak coupling $\alpha$
between the system and the bath. Evidence for the validity of this assumption
will be shown below (see Eqs.(10) and (11)).
In the absence of an environment, the density of excitations was shown to obey
the KZ scaling zurek05 ; polkovnikov05
$\mathcal{E}_{coh}=\mathcal{E}_{KZ}\propto v^{d\nu/(z\nu+1)}\;.$ (5)
In order to obtain a similar relation for the energy density in Eq. (30)
additional information on $E_{k}$ at $h_{f}$ is needed. Thus, the scaling of
this quantity depends on the details of the system at the end of the quench. A
simple scaling law can be obtained only in specific situations, e.g. for
quenches halted at the critical point $h_{f}=h_{c}$, where $E_{k}\propto
k^{z}$. By using techniques similar to those employed in Ref. polkovnikov05,
one obtains
${E}_{coh}\propto v^{\nu(d+z)/(z\nu+1)}\;.$ (6)
Let us now derive a scaling law for the incoherent contributions
$\mathcal{E}_{inc}$ and ${E}_{inc}$. To this end it is convenient to divide
the quench in three steps (see Fig. 1): initially the system is in the so-
called low-temperature region at $T\ll\Delta$. Here the relatively high energy
gap suppresses thermal excitation and the system remains in the ground state.
Close to the critical point the system passes through the quantum critical
region: thermal excitations are unavoidably created because of the relatively
high temperature $T\gg\Delta$. As we shall see below, the density of
excitations generated in this region are universal functions on the velocity
of the quench and on the temperature as long as only the low-energy details of
the system spectrum matter. On the contrary, the bath-induced relaxation
occurring once the system leaves the quantum critical region, entering the
other semiclassical region ($T\ll\Delta$), depends on the details of the
energy spectrum; hence the relaxation towards an asymptotic thermal state at
temperature $T$ is not expected to be universal if the final $h_{f}$ is far
off the critical point $h_{c}$. In our analysis below we will neglect the
effects of this non-universal relaxation. We are therefore assuming that the
time elapsed between the moment when the critical region is left and when the
quench is stopped (and the measurement of the density of excitations/energy is
made) is short as compared to the typical relaxation times in the
semiclassical region. In Section VI we will further comment on this point for
the specific case of the quantum XY chain, showing that the scenario just
depicted holds for a wide range of $h_{f}$ and $v$. The dynamics of the
probability to excite the model $k$ $\mathcal{P}_{k}$ can be described, inside
the quantum critical region, in terms of a phenomenological rate equation:
$\frac{d}{dt}\mathcal{P}_{k}=-\frac{1}{\tau}\left[\mathcal{P}_{k}-\mathcal{P}_{k}^{th}\left(h_{c}\right)\right]\;,$
(7)
where $\mathcal{P}_{k}^{th}(h_{c})$ is the critical thermal equilibrium
distribution and $\tau^{-1}$ is the relaxation rate, $\tau^{-1}\propto\alpha
T^{\theta}$. As shown in Section V and appendix C, $\theta$ can be related to
characteristics of the bath and to the critical indices of the phase
transition (see Eq. (56)). From the relation $T\sim\Delta\sim|h-h_{c}|^{\nu
z}$ we deduce the time spent inside the quantum critical region is
$t_{QC}=2T^{1/\nu z}v^{-1}\;\;.$
A direct integration of Eq. (7) gives for the thermal excitation created in
the quantum critical region
$\mathcal{P}_{k}\sim(1-e^{t_{QC}/\tau})\mathcal{P}_{k}^{th}\left(h_{c}\right)$.
Finally, integrating the latter over all k-modes we get:
$\displaystyle\mathcal{E}_{inc}$ $\displaystyle\propto$
$\displaystyle\left(1-e^{t_{QC}/\tau}\right)\int\\!dE\,E^{d/z-1}\,\mathcal{P}_{k}^{th}(h_{c})\;,$
(8)
where we used the scaling of the excitation energy $E\propto k^{z}$. For the
density of energy, a similar relation holds in the case of quenches halted at
the critical point
$\displaystyle{E}_{inc}$ $\displaystyle\propto$
$\displaystyle\left(1-e^{t_{QC}/(2\tau)}\right)\int\\!dE\,E^{d/z}\,\mathcal{P}_{k}^{th}(h_{c})\;,$
(9)
where $t_{QC}/2$ is due to the fact that in this case only half of the quantum
critical region is crossed. Finally, since the thermal distribution is a
function of $E/T$, a simple change of variable gives the required results
$\displaystyle\mathcal{E}_{inc}$ $\displaystyle\propto$
$\displaystyle\alpha\,v^{-1}\,T^{\theta+\frac{d\nu+1}{\nu z}}\;,$ (10)
$\displaystyle{E}_{inc}$ $\displaystyle\propto$
$\displaystyle\alpha\,v^{-1}\,T^{\theta+\frac{(d+z)\nu+1}{\nu z}}\;,$ (11)
that are valid in the limit $T^{1/\nu z}\ll v\tau$. Eqs. (10) and (11)
together with Eqs. (5) and (6) give, through the assumptions Eqs. (3) and (4),
the general scaling-law for the quench dynamics of open systems. The different
scaling of the two contributions with respect to the velocity $v$ implies that
for slow quenches $v<v_{cross}$ the incoherent mechanism of excitation
dominates over the coherent one, and viceversa for $v>v_{cross}$. The
crossover velocity can be deduced by equating
$\mathcal{E}_{inc}\simeq\mathcal{E}_{coh}$ and ${E}_{inc}\simeq{E}_{coh}$
yielding:
$\displaystyle v_{cross}^{\mathcal{E}}$ $\displaystyle\propto$
$\displaystyle\alpha^{\frac{\nu
z+1}{\nu(z+d)+1}}\,T^{\left(1+\frac{(\theta-1)\nu
z}{\nu(z+d)+1}\right)\left(1+\frac{1}{\nu z}\right)}\;,$ (12) $\displaystyle
v_{cross}^{E}$ $\displaystyle\propto$ $\displaystyle\alpha^{\frac{\nu
z+1}{\nu(2z+d)+1}}\,T^{\left(1+\frac{(\theta-1)\nu
z}{\nu(2z+d)+1}\right)\left(1+\frac{1}{\nu z}\right)}\;.$ (13)
## III Quantum XY model with thermal reservoir
The scaling laws derived above will be tested against a specific model: an XY-
chain coupled to a set of bosonic baths. The Hamiltonian of the XY chain is
defined as
$H_{S}=-\frac{1}{2}\\!\sum_{j}^{N}\\!\left(\frac{1+\gamma}{2}\sigma_{j}^{x}\sigma_{j+1}^{x}+\frac{1-\gamma}{2}\sigma_{j}^{y}\sigma_{j+1}^{y}\\!+\\!h\sigma_{j}^{z}\right)\;.$
(14)
Here $N$ is the number of sites ($\sigma^{x,\ y,\ z}$ are Pauli matrices).
Each spin is coupled to its neighbors d by anisotropic Ising-like interaction
and subject to a transverse magnetic field $h$ (the couplings are expressed in
terms of the exchange energy). In the thermodynamic limit
$N\rightarrow\infty$, a quantum phase transition at $h_{c}=1$ separates a
paramagnetic phase for $h>1$ from a ferromagnetic phase ($h<1$) where the
$Z_{2}$ symmetry is spontaneously broken and a magnetic order along $\vec{x}$
appears, $\left\langle\sigma^{x}\right\rangle\neq 0$.
The spin Hamiltonian Eq.i (14) can be diagonalized by using the Jordan-Wigner
transformation pfeuty to map the spins into spinless fermions $c_{j}$ and
thus obtain in momentum space (after a projection in a definite parity
subspace)
$\displaystyle H_{S}$ $\displaystyle=$
$\displaystyle\sum_{k>0}\Psi_{k}^{\dagger}\hat{\mathcal{H}}_{k}\Psi_{k}$
$\displaystyle\hat{\mathcal{H}}_{k}$ $\displaystyle=$ $\displaystyle-(\cos
k+h)\,\hat{\tau}_{z}+\gamma\sin k\,\hat{\tau}_{y}\;,$ (15)
where
$\Psi_{k}^{\dagger}=\left(\begin{array}[]{cc}c_{k}^{\dagger}&c_{-k}\end{array}\right)$
are Nambu spinors and $\hat{\tau}$ are Pauli matrices in Nambu space. Finally,
a Bogoliubov rotation diagonalizes the Hamiltonian:
$H_{S}=\sum_{k>0}\Lambda_{k}(\eta_{k}^{\dagger}\eta_{k}-\eta_{-k}\eta_{-k}^{\dagger})$,
where
$\Lambda_{k}=\sqrt{(\cos k+h)^{2}+(\gamma\sin k)^{2}}\;,$ (16)
is the quasi-particle dispersion. At $h=h_{c}$ the spectrum becomes gapless
with a linear dispersion relation $\Lambda_{k}\propto\pi-k$; accordingly, the
critical indexes of the model are $\nu=z=1$.
The spins are also locally coupled to a set of $N/l$ bosonic baths
$H_{int}=-\frac{1}{2}\sum_{j=0}^{N/l-1}\left(\sum_{r=0}^{l-1}\sigma_{jl+r}^{z}\right)X_{j}$
(17)
where $X_{j}=\sum_{\beta}\lambda_{\beta}(b_{\beta,j}^{\dagger}+b_{-\beta,j})$
and $b_{\beta,j}^{\dagger}$ ($b_{\beta,j}$) are the creation (annihilation)
operators of the j-th bosonic bath. As a result of the coupling in Eq. (17)
each baths correlated in a string of $l$ adjacent spins. The model presented
above generalizes the one considered in Ref. PatanePRL, , where the spins were
individually coupled ($l=1$) to Ohmic baths ($s=1$).
The total Hamiltonian reads:
$H=H_{S}+H_{int}+H_{B}$ (18)
where $H_{B}=\\!\sum_{j,\beta}\\!\omega_{\beta}b_{\beta j}^{\dagger}b_{\beta
j}$. The spectral density of the baths
$J(\omega)=\sum_{\beta}\lambda_{\beta}^{2}\delta(\omega-\omega_{\beta})$ is
$J(\omega)=2\alpha\omega^{s}e^{-\omega/\omega_{c}}\theta(\omega)$ (19)
where $\alpha$ is the system/bath coupling, $\omega_{c}$ is a high energy
cutoff and $\theta(s)$ is the step function WeissBOOK .
Figure 2: A cartoon of the spins-baths coupling (17) for the $l=3$, where
each bath is coupled to three spins.
In the situation we are interested, the system is initialized in its ground
state at large $h$. The coupling of the spin to the bath through $\sigma^{z}$
preserves parity symmetry (see Eq. (20)). Therefore, once the system has
initially a specific parity, it will remain in the corresponding sector for
the entire evolution. Throughout this paper we consider $N$ even: in this case
the ground state has always an even number of fermions $c_{k}$ and we are thus
allowed to select the even parity sector and neglect the odd one.
In momentum space
$b_{\beta,q}=\frac{1}{\sqrt{N/l}}\sum_{j=0}^{N/l-1}\exp(-iqj)\,b_{\beta,j}$
with $q=\frac{2m\pi}{N/l}$ and after the Jordan-Wigner transformation we get
$H_{int}=-\frac{1}{\sqrt{N}}\sum_{k}\sum_{q}F(q)\Psi_{k}^{\dagger}\hat{\tau}^{z}\Psi_{k+\frac{q}{l}}X_{q}$
(20)
where $F(q)=1/\sqrt{l}\sum_{r=0}^{l-1}\exp(-irq/l)$. For $l=1$ each spin
interact with a different bath and according to Eq. (20) all $k$ modes are
coupled (i.e., transitions $k\leftrightarrow k^{\prime}$, $\forall
k,k^{\prime}$ are induced). In the opposite case of $l=N$ (just one bath for
the whole system) no transition between different $k$ is allowed. In the
intermediate case each mode interacts with the other modes in an interval of
width $2\pi l/N$.
It is important to notice that correlations between baths over a finite
distance would not change qualitatively our picture as long as we focus on the
critical properties of the model. Indeed, near criticality the divergence of
the correlation length makes the details of the bath correlations over
microscopic distances unimportant. For the same reason, as long as one is
interested in the low-$T$ properties of the bath, the specific value of $l$ is
not relevant provided $l/N\rightarrow 0$ in the thermodynamic limit.
Specifically, for all values of $l$ such that
$T\ll\left(\frac{1}{l}\right)^{z}$ the same dissipative dynamics is obtained,
since transitions with large $\Delta k$ (and hence large energy) are thermally
suppressed (we used $E\propto k^{z}$ at a fixed $T$). In this regime,
therefore, the system cannot resolve the microscopic details of different
system-bath couplings (i.e., whether $l=1,2,3,\dots$). In the following, we
thus focus on the case $l=1$: specific high-temperature and non-critical
behaviors for different $l$ could be easily investigated within the same
scheme considered below. Only if $l=N$ the dynamics of the system changes
qualitatively.
## IV Kinetic equation
In this section we derive a kinetic equation for the Green’s function of the
Jordan-Wigner fermions within the Keldysh formalism. In terms of this Green’s
function we will then calculate both the excitation and the energy densities,
Eq. (1) and (2). Our analysis in terms of a kinetic equation will provide
support for the scaling laws obtained above, Eqs. (10)-(11), while allowing us
to study the non-universal dynamics beyond the limit of applicability of the
scaling approach.
The fermionic Keldysh Green’s function is a matrix in Nambu space defined by
$[G_{k}(t_{1},t_{2})]_{i,j}\equiv-i\left\langle\mathcal{T}_{\gamma}\,\Psi_{ki}^{\phantom{\dagger}}(t_{1})\Psi_{kj}^{\dagger}(t_{2})\right\rangle$
(21)
(see appendix A1 for notations) where $\gamma$ is the Keldysh contour. In the
following we will neglect the initial correlations between system and bath
Rammer86 . Hence $\gamma$ consists of just a forward and backward branch on
the real time axis. Below we will sketch of the main steps of the derivation:
the remaining details can be found in Appendix A1-A2.
The starting point of our derivation is the Dyson’s equation in its integro-
differential form
$\displaystyle\left[i\partial_{t_{1}}-\mathcal{\hat{H}}_{k}(t_{1})\right]G_{k}(t_{1},t_{2})$
$\displaystyle=$ $\displaystyle\delta(t_{1}-t_{2})$ $\displaystyle+$
$\displaystyle\int_{\gamma}\\!d\bar{t}\,\Sigma_{k}(t_{1},\bar{t})\,G_{k}(\bar{t},t_{2})$
and an analogous one obtained by differentiation with respect $t_{2}$. Here
$\Sigma_{k}$ is the self-energy associated to the interaction of the system
with the bath. In order to compute the energy and excitations density we need
to find the equal-time statistical Green’s functions. The latter are defined
as
$\displaystyle[G_{k}^{<}(t_{1},t_{2})]_{i,j}$ $\displaystyle\doteq$
$\displaystyle
i\langle\Psi_{k,j}^{\dagger}(t_{2})\Psi_{k,i}^{\phantom{\dagger}}(t_{1})\rangle$
(23) $\displaystyle{}[G_{k}^{>}(t_{1},t_{2})]_{i,j}$ $\displaystyle\doteq$
$\displaystyle-i\langle\Psi_{ki}^{\phantom{\dagger}}(t_{1})\Psi_{kj}^{\dagger}(t_{2})\rangle$
(24)
An equation for these correlators can be obtained from Eq. (IV) by using
standard techniques HaugBOOK (see Appendix A1 for details). For the equal-
time Green’s function $G_{k}^{<}(t,t)$ we obtain:
$\displaystyle i\partial_{t}G_{k}^{<}$ $\displaystyle=$
$\displaystyle\left[\mathcal{\hat{H}}_{k},G_{k}^{<}\right]+$
$\displaystyle\Sigma_{k}^{>}\cdot G_{k}^{<}-\Sigma_{k}^{<}\cdot
G_{k}^{>}+G_{k}^{<}\cdot\Sigma_{k}^{>}-G_{k}^{>}\cdot\Sigma_{k}^{<}\;,$
where the dots indicate the convolution:
$\Sigma_{k}^{>}\cdot
G_{k}^{<}\doteq\int_{0}^{t}d\bar{t}\,\Sigma_{k}^{>}(t,\bar{t})G_{k}^{<}(\bar{t},t)\;.$
In order to proceed with the solution of Eq. (IV) it is now important to
discuss the approximations we make for the self-energy. Let us first notice
that long-time correlations induced by the bath may change the universality
class of the transition, by renormalizing the low energy spectrum of the
system Werner07 . As previously mentioned, we will not consider this case
here. Therefore, we assume that the bosons have a non-zero inverse lifetime
$\Gamma\ll T$ which provides a natural cutoff-time for the bath correlation
functions. Within this assumption it is now possible to describe the kinetics
of the system using a Markov approximation together with a self-consistent
Born approximation. The latter is justified for weak system/bath coupling
($\alpha\ll 1$) and is represented diagrammatically in Fig. 3-(a).
a)b)
Figure 3: Lowest-order diagrams contributing to the self-consistent Born
approximation: dashed lines correspond to the non-interacting bath Green’s
function $g$, while solid lines to the interacting system Green function $G$.
a) corresponds to Eq. (46), while b) to Eq. (47).
We will neglect the tadpole diagram (b), which represents just a small shift
of the energy levels.
By going to the interaction picture
$\displaystyle\tilde{G}_{k}(t_{1},t_{2})$ $\displaystyle\doteq$
$\displaystyle\mathcal{\hat{U}}_{k}^{\dagger}(t_{1})G_{k}(t_{1},t_{2})\hat{\mathcal{U}}_{k}^{\phantom{\dagger}}(t_{2})\;,$
it is now evident that, within our assumptions, the evolution of
$\tilde{G}_{k}$ can be considered slow as compared to that of the bath
correlators appearing in the self-energies. Using this separation of time
scales it is possible to implement the Markov approximation and transform the
general integro-differential kinetic equation into a simple differential
equation (see Appendix A1-A2). We then obtain, in the case in which each spin
is coupled to its own bath ($l=1$), the kinetic equation
$\displaystyle\partial_{t}G_{k}^{<}$ $\displaystyle+$ $\displaystyle
i\left[\mathcal{H}_{k},\,G_{k}^{<}\right]=$ (26)
$\displaystyle\frac{1}{N}\sum_{q}\tau^{z}({\bf
1}+iG_{q}^{<})\hat{D}_{qk}G_{k}^{<}$ $\displaystyle+$
$\displaystyle\tau^{z}G_{q}^{<}\hat{D}_{kq}^{\dagger}({\bf 1}+iG_{k}^{<})+{\it
H.c.}$
where
$\hat{D}_{qk}=i\int_{0}^{\infty}\\!ds\,g^{>}(s)\,{\hat{\mathcal{U}}}_{q}^{\dagger}(t,t-s)\,\hat{\tau}^{z}\,{\hat{\mathcal{U}}}_{k}(t,t-s)\;,$
(27)
$g^{>}(t)=-i\left\langle X_{q}(t)X_{q}(0)\right\rangle$, and
${\hat{\mathcal{U}}}_{k}(t_{0},t)$ is the evolution operator satisfying
$i\partial_{t}{\mathcal{\hat{U}}}_{k}={\hat{\mathcal{H}}}_{k}(t){\hat{\mathcal{U}}}_{k}$.
The left-hand-side of Eq. (26) represents the free evolution term, while the
right-hand-side describes the scattering between the $k$ modes mediated by the
bath degrees of freedom. Notice that the number of equations scales linearly
with the system size $N$, in contrast to conventional systems of master
equations whose number scales exponentially with $N$ as a result of the fact
that the full density matrix (i.e. all $m-$points Green functions) is
considered. The fact that we are considering only the two-point Green’s
function self consistently using the Born approximation is responsible for the
non-linear nature of Eq. (26), in contrast to the linearity of the master
equation.
In the eigenbasis of the Hamiltonian $\mathcal{\hat{H}}_{k}$ the Green’s
function can be parameterized as
$-iG_{k}^{<}=\left(\begin{array}[]{cc}\mathcal{P}_{k}&\mathcal{C}_{k}\\\
\mathcal{C}_{k}^{*}&1-\mathcal{P}_{k}\end{array}\right)$ (28)
where $\mathcal{P}_{k}=\langle\eta_{k}^{\dagger}\eta_{k}\rangle$ is the
population of the excited mode $k$ and
$\mathcal{C}_{k}=\left\langle\eta_{-k}\eta_{k}\right\rangle$ can be regarded
as a “coherence” term CohenBOOK . In the static case, where the evolution
operator is $\hat{\mathcal{U}}_{k}=\exp(-i\mathcal{\hat{H}}_{k}t)$, the
stationary solution of the kinetic equation (26) is correctly the thermal
equilibrium one: $\mathcal{C}_{k}=\mathcal{C}_{k}^{th}=0$ and
$\mathcal{P}_{k}=\mathcal{P}_{k}^{th}=(e^{\Lambda_{k}/k_{B}T}+1)^{-1}$ (the
Fermi function).
Finally, once the solution of the kinetic equation (26) is obtained, the
density of excitations and energy produced during the quench can be expressed
as
$\displaystyle\mathcal{E}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{k>0}\mathcal{P}_{k}$ (29) $\displaystyle{E}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{k>0}\Lambda_{k}\mathcal{P}_{k}$ (30)
We conclude this section by commenting on a useful approximation to evaluate
numerically the kernel of $\hat{D}_{qk}$, Eq. (27), discussed in full detail
in Appendix A2. It consists in approximating the evolution operator
${\hat{\mathcal{U}}}_{k}$ appearing in $\hat{D}_{qk}$ with
${\hat{\mathcal{U}}}_{k}(t,t-s)\simeq\exp\left(i\hat{\mathcal{H}}_{k}(t)s\right)$,
thus obtaining
$\hat{D}_{qk}\simeq
i\int_{0}^{\infty}\\!ds\,g^{>}(s)\exp\left(-i\hat{\mathcal{H}}_{k}(t)s\right)\hat{\tau}^{z}\exp\left(i\hat{\mathcal{H}}_{k}(t)s\right)\;.$
(31)
This is again consistent with the separation of time scales mentioned above,
and in particular with the Markov approximation. Indeed, while the exact
relaxation rate matrix (27) depends on the velocity of the quench, if the
quench is slow on the time scale characteristic of the bath, the correlation
function $g^{>}(s)$ can be seen as strongly peaked at $s=0$. Hence the system
can be considered “frozen” at the instantaneous value of $h(t)$ and,
consistently, its evolution operator is the exponential of the Hamiltonian.
## V Relaxation time
In order to make further progress in understanding the quench dynamics of the
system we will first extract from the kinetic equation the characteristic
relaxation time for the populations of the excitations $\mathcal{P}_{k}$ (see
Eq.(28)) as a function of the magnetic field $h$ and the temperature $T$. For
this purpose, it is sufficient to consider only the diagonal elements of Eq.
(28). This is equivalent to the so called “secular approximation” for the
master equation CohenBOOK , which is valid for weak couplings ($\alpha\ll 1$
in the present case). For generic $N$, we deal with a set of ${N}/{2}$
equations (only ${N}/{2}$ modes are independent) of the form:
$\frac{d}{dt}\mathcal{P}_{k}=a_{k}+\sum_{q}b_{kq}\mathcal{P}_{q}+\sum_{q}c_{kq}\mathcal{P}_{k}\mathcal{P}_{q}\;.$
The asymptotic relaxation can be studied by linearizing the previous set of
equations near the thermal equilibrium fixed point. We obtain, with the vector
notation
$\delta\mathcal{\underline{P}}=\left(\delta\mathcal{P}_{1},\delta\mathcal{P}_{2},\dots,\delta\mathcal{P}_{N/2}\right)^{\rm
tr}$ where $\delta\mathcal{P}_{k}=\mathcal{P}_{k}-\mathcal{P}_{k}^{th}$:
$\frac{d}{dt}\delta\mathcal{\underline{P}}=-\mathcal{R\
}\delta\mathcal{\underline{P}}$ (32)
where non-linear terms in $\delta\mathcal{P}_{k}$ have been neglected. The
diagonal and off-diagonal elements of the $N/2\times N/2$ matrix $\mathcal{R}$
are:
$\displaystyle\mathcal{R}_{kk}$ $\displaystyle=$
$\displaystyle\frac{2}{N}\sum_{q>0,q\neq
k}\left[\mathcal{G}_{q}^{th}\left(1-\cos(\theta_{k}+\theta_{q})\right)g[-\Lambda_{k}-\Lambda_{q}]+\mathcal{G}_{q}^{th}\left(1+\cos(\theta_{k}+\theta_{q})\right)g[\Lambda_{k}-\Lambda_{q}]\right.$
$\displaystyle\hskip
42.67912pt\left.+\mathcal{P}_{q}^{th}\left(1-\cos(\theta_{k}+\theta_{q})\right)g[\Lambda_{k}+\Lambda_{q}]+\mathcal{P}_{q}^{th}\left(1+\cos(\theta_{k}+\theta_{q})\right)g[-\Lambda_{k}+\Lambda_{q}]\right]$
$\displaystyle+$ $\displaystyle\frac{2}{N}\
4\sin^{2}\theta_{k}\left(\mathcal{G}_{k}^{th}g[-2\Lambda_{k}]+\mathcal{P}_{k}^{th}g[2\Lambda_{k}]\right)$
$\displaystyle\mathcal{R}_{kq}$ $\displaystyle=$
$\displaystyle\frac{2}{N}\left[-\mathcal{G}_{k}^{th}\left(1+\cos(\theta_{k}+\theta_{q})\right)g[-\Lambda_{k}+\Lambda_{q}]+\mathcal{G}_{q}^{th}\left(1-\cos(\theta_{k}+\theta_{q})\right)g[-\Lambda_{k}-\Lambda_{q}]\right.$
$\displaystyle\hskip
14.22636pt\left.-\mathcal{P}_{q}^{th}\left(1+\cos(\theta_{k}+\theta_{q})\right)g[\Lambda_{k}-\Lambda_{q}]+\mathcal{P}_{q}^{th}\left(1-\cos(\theta_{k}+\theta_{q})\right)g[\Lambda_{k}+\Lambda_{q}]\right]$
where $g[E]$ is the Laplace transform of bath correlator (Appendix A2:
Approximation for the kinetic equation matrices $\hat{D}$),
$\mathcal{G}_{k}^{th}$ is the thermal equilibrium value of the population of
the ground-state of mode $k$, $\mathcal{G}_{k}^{th}=1-\mathcal{P}_{k}^{th}$,
and
$\theta_{k}=\arccos-\frac{(\cos k+h)}{\Lambda_{k}}\;.$ (35)
The eigenvalues of $\mathcal{R}$, $\\{\lambda_{i}\\}$, are the characteristic
relaxation rates of the long-time dynamics. Hence the solution of Eq. (32) for
each population would be a linear combination containing all the
characteristic relaxation times:
$\delta\mathcal{P}_{k}=\sum_{j}r_{kj}e^{-\lambda_{j}t}$
At long times $t\gg\left(\min_{j}\lambda_{j}\right)^{-1}\doteq\tau$ all modes
relax with the same relaxation time $\tau$. In the following we first analyze
the longest relaxation time $\tau$, extending the results presented in Ref.
PatanePRL, ; we then study the structure of the entire spectrum of relaxation
times.
Figure 4: Relaxation rate $1/\tau$ as a function of $T$ and $h$ for $N=400$
(here $\gamma=1$,and $s=1$).
In Fig. 4 we show the general behavior of $\tau$ in the finite temperature
phase-diagram, calculated by numerically diagonalizing the matrix
$\mathcal{R}$. As $T\rightarrow 0$, $\tau$ diverges and close to the critical
point two different behaviors are found in the semiclassical regions and in
the quantum critical region (see also Fig.1):
$\tau^{-1}\propto\begin{cases}T^{1+s}&T\gg\Delta\\\
e^{-\Delta/T}&T\ll\Delta\end{cases}$ (36)
These relations extend the results obtained for the relaxation rate in Ref.
PatanePRL, to the generic case of non-Ohmic baths and give the exponent
$\theta$ as $\theta=1+s$.
An analytic expression for the power-law scaling inside the quantum critical
region can be obtained by approximating the smallest eigenvalue of
$\mathcal{R}$ with the smallest diagonal element. This is justified by the
fact that the off-diagonal elements are of the order $O(1/N)$ (see Eq. (V)).
For $h=h_{c}=1$, considering the gapless mode $k=\pi$, we have from Eq. (V) in
the continuum limit:
$\displaystyle\tau_{diag}^{-1}$ $\displaystyle\doteq\mathcal{R}_{\pi\pi}$
$\displaystyle=\frac{2}{\pi}\int_{0}^{\pi}\\!dq\,\left(\mathcal{G}_{q}^{th}g[-\Lambda_{q}]+\mathcal{P}_{q}^{th}g[\Lambda_{q}]\right)$
(37)
$\displaystyle=4\alpha\int_{0}^{\pi}\\!dq\,\frac{\Lambda_{q}^{s}}{\sinh\left(\Lambda_{q}/T\right)}$
$\displaystyle\simeq
8\alpha\,(1-2^{-1-s})\,\Gamma(1+s)\,\zeta(1+s)\left(\gamma/T\right)^{-1-s}$
where $\Gamma$ and $\zeta$ are the Gamma and the zeta functions, and we used
the critical dispersion relation $\Lambda_{q}\simeq\gamma(\pi-q)$ obtained by
linearizing Eq. (16) around the gapless point $k=\pi$ (we extended the
integration to $-\infty$ since at low-temperature only the low-energy modes
contribute to the integral). Fig. 5 demonstrates that the analytical
expression in Eq. (37) agrees very well with the numerical solution (obtained
by diagonalizing $\mathcal{R}$), especially at low temperature.
$s=1$$s=3$$s=6$
Figure 5: Relaxation rate $1/\tau$ as a function of $T$ as obtained from the
exact diagonalization of $\mathcal{R}$ (symbols) and the approximation in Eq.
(37) (dashed lines). Upper panel: $s=1$, $3$, $6$, with $\gamma=1$; lower
panel: $s=1$ with different $\gamma=0.3$, $0.5$, $1$ from top to bottom.
As we have shown in Fig. 5, inside the quantum critical region the exponent
$\theta$ is universal within the range of anisotropy $0<\gamma\leq 1$ where
the system belongs to the Ising universality class. This suggests a relation
between $\theta$ and the critical indexes of the quantum phase transition.
Indeed, it can be shown, within the Fermi golden rule (see Appendix B), that
for a generic system coupled to a bosonic bath the following expression holds
inside the quantum critical region:
$\tau^{-1}\propto T^{s+d/z}\;.$ (38)
An important feature of the relaxation dynamics can be extracted by analyzing
the spectrum of the eigenvalues $\\{\lambda_{j}\\}$ of $\mathcal{R}$. In Fig.
6 the $\lambda_{j}$’s are shown for some values of temperature and magnetic
field. We find that in the semiclassical regions $T\ll\Delta$ the smallest
eigenvalue of $\mathcal{R}$ is separated from the rest of the spectrum by a
gap (even in the $N\rightarrow\infty$ limit). On the contrary, inside the
quantum critical region such eigenvalue merges with the rest of the spectrum.
This can be quantified by the relative gap of the spectrum of relaxation
times, that is identified by
$(\lambda_{2}^{-1}-\lambda_{1}^{-1})/\lambda_{1}^{-1}$, being $\lambda_{1,2}$
the lowest eigenvalues of $\mathcal{R}$ (see Fig. 7). Such result indicates
that while the exponential divergence of the relaxation time
$\tau\propto\exp\left\\{\Delta/T\right\\}$ in the semiclassical regions is due
to an isolated eigenvalue, the long-time behavior in the quantum critical
region is, instead, built up by a continuum of eigenvalues contributing to the
$\tau^{-1}\propto T^{2}$ scaling.
$h=0.8$
$h=1$
Figure 6: Spectrum of the eigenvalues of $\mathcal{R}$ for the Ising model
($\gamma=1$) with Ohmic baths ($s=1$), for $N=100$. The values of temperature
$T=0.1$ and magnetic field $h=0.8$, or $1$ are chosen to belong, respectively,
to the semiclassical and the quantum critical region.
$T$$h$
Figure 7: Ising model ($\gamma=1$) with Ohmic baths ($s=1$). Relative gap
between the two longest relaxation times:
$(\lambda_{2}^{-1}-\lambda_{1}^{-1})/\lambda_{1}^{-1}$ ($\lambda_{1,2}$ being
the two lowest eigenvalue of $\mathcal{R})$; crossover lines
$T=T_{cross}=|h-1|$ are plotted for comparison.
## VI Adiabatic quenches
Equipped with the kinetic equation and the knowledge of the scaling of the
relaxation times, we now analyze the quench dynamics of the model in Eq. (18))
by solving the kinetic equation (26) numerically. The system is initialized at
$h_{i}\gg h_{c}$ in equilibrium with the bath at a fixed temperature
$T\ll\Delta(h_{i})$, and the transverse field is then ramped linearly
$h(t)=h_{i}-vt$ down to a final value $h_{f}$ (the bath temperature is kept
fixed).
In Fig. 8 we plot the density of excitations as a function of the quench
velocity for different system sizes (a similar behavior is obtained for the
density of energy). Additionally we considered separately the coherent
($\mathcal{E}_{coh}$) and incoherent contribution ($\mathcal{E}_{inc}$) to the
final density of excitations. The first one is obtained by integrating the
kinetic equation for $\alpha=0$, i.e. no coupling with the bath. The
incoherent term is due to thermal excitations created by the bath and it is
obtained by integrating the kinetic equation and ignoring the unitary
evolution term $i[\hat{\mathcal{H}}_{k},\hat{G}^{<}_{k}]$, responsible for the
coherent excitation process.
$\mathcal{E}$$v$
Figure 8: Density of excitation $\mathcal{E}$ (circles) Vs quench velocity
for different system sizes $N=26,\leavevmode\nobreak\ 50,\leavevmode\nobreak\
100,\leavevmode\nobreak\ 200,\leavevmode\nobreak\ 400,\leavevmode\nobreak\
800$ from bottom to top, according to the arrow; the points corresponding to
$N=800$ and $N=400$ are indistinguishable. Parameters are set $\alpha=0.01$,
$T=0.1$ $\gamma=1$ and $s=1$ and the quench is halted at $h_{f}=0.8$. Dotted
line is the coherent contribution $\mathcal{E}_{coh}$ obtained for $\alpha=0$
and stars represent the incoherent contribution $\mathcal{E}_{inc}$ due to
thermal excitation (see text); both curves refere to $N=800$, even if for
$\mathcal{E}_{inc}$ the same curve is obtained already at $N\sim 30$.
In order to understand the two excitation mechanisms we analyse directly the
dynamics of the populations $\mathcal{P}_{k}$. From the results shown in Fig 9
(left) it emerges that excitations are generated close to the critical point
and when the system is driven in the semiclassical region (and $T\ll\Delta$)
they are relaxed out by the bath. The density of excitation generated is the
sum of the the incoherent and coherent contribution, thus proving the validity
of the Ansazt (3) and (4) (see Fig. 9 (right)).
$\mathcal{P}_{k}$$h(t)$$k/\pi$
Figure 9: Populations of the excited states $\mathcal{P}_{k}$ for $N=400$,
$v=0.0017$ and the same values of Fig. 8. Left: dynamics of two low energy
modes $k_{1}\sim\pi$ (dashed line) and $k_{2}\sim 0.9\pi$ (dotted-dashed line)
obtained by solving the kinetic equation (instantaneous thermal equilibrium
values are plotted as reference as dotted lines). The Inset shows the energy
levels near the critical point and the scale of temperature; marked levels of
the excited band refer to $k_{1}$ and $k_{2}$. The energy gap closes at
$k=\pi$. Right: distribution of $\mathcal{P}_{k}$ as a function of the mode k
at $h(t)=1$. Stars represent the excitations created incoherently and
triangles are the coherent excitations produced in the case of no coupling to
the bath. The two excitation mechanisms act on different energy scales, being
lowest energy modes coherently populated and the highest one thermally
excited.
For the XY model the integration of Eqs. (8) and (9) can be performed
explicitly by using the critical spectrum $\Lambda_{k}\sim\gamma(\pi-k)$. We
obtain:
$\displaystyle\mathcal{E}_{inc}$ $\displaystyle=$ $\displaystyle\frac{\log
2}{2\pi\gamma}T\left(1-e^{-2T/(\tau v)}\right)$ (39) $\displaystyle{E}_{inc}$
$\displaystyle=$ $\displaystyle\frac{\pi}{24\gamma}T^{2}\left(1-e^{-T/(\tau
v)}\right)$ (40)
where the latter holds for quenches halted at $h_{f}=h_{c}$. In the previous
formulas, the expression derived for $\tau$ in Eq. (37) can be used to get a
fully analytical expression. Expanding the exponentials in Eqs. (39) and (40)
we obtain
$\displaystyle\mathcal{E}_{inc}$ $\displaystyle\simeq$
$\displaystyle\frac{8\log 2}{\pi}\leavevmode\nobreak\
\varphi(s)\leavevmode\nobreak\ \alpha\gamma^{-2-s}\leavevmode\nobreak\
v^{-1}T^{3+s}$ (41) $\displaystyle{E}_{inc}$ $\displaystyle\simeq$
$\displaystyle\frac{\pi}{3}\leavevmode\nobreak\ \varphi(s)\leavevmode\nobreak\
\alpha\gamma^{-2-s}\leavevmode\nobreak\ v^{-1}T^{4+s}$ (42)
where $\varphi(s)=(1-2^{-1-s})\Gamma(1+s)\zeta(1+s)$. The previous relations
are consistent with Eqs.(10) and (11) with $\theta=1+s$ (see Eq. (38)).
$\mathcal{E}$${E}$$v$
Figure 10: Density of energy (lowest panel) and of excitations versus quench
velocity $v$ for $h_{f}=0.8$, and $1$. Parameters are set to $\gamma=0.7$,
$s=1.5$, $\alpha=0.01$, and $T=0.15$, or $0.1$ (upper and lower curves of each
panel). Circles are obtained by solving the kinetic equation; dotted lines are
the coherent contributions $\mathcal{E}_{coh}$ and ${E}_{coh}$ evaluated by
solving the kinetic equation for $\alpha=0$; a fit gives correctly
$\mathcal{E}_{coh}\propto\sqrt{v}$ and ${E}_{coh}\propto v$ consistently with
the KZ scaling-law for excitations (5) and the modified scaling we derived for
the energy density (6). Solid and dashed lines are Eqs. (3) and (4) using for
the incoherent contributions the expressions (39) and (40) and their
linearized forms (41) and (42), respectively.
$vE_{inc}$$v^{E}_{cross}$$v\mathcal{E}_{inc}$$v^{\mathcal{E}}_{cross}$$T$$T$
Figure 11: System and bath parameters are fixed as in of Fig. 10. Upper
panel: Data collapse of $\mathcal{E}_{inc}$ and $E_{inc}$ obtained from the
kinetic equation; data refer to $10^{-3}\lesssim v\lesssim 10^{-2}$ (data
relative to $h_{f}=1$ for $\mathcal{E}_{inc}$ are rescaled by a factor $2$,
since in this case only half quantum critical region is crossed). Lower panel:
scaling of $v_{cross}$ is obtained equating
$\mathcal{E}_{inc}=\mathcal{E}_{coh}$ and analogously for $E$. The fits
confirm the scaling predicted by (10), (11) and (12), (13), that, for the
specific case of $s=1.5$ considered, are shown in their corresponding plots.
$\mathcal{E}$$v$
Figure 12: Density of excitations versus quench velocity $v$ for $h_{f}=0.8,\
0.6,\ 0.4,\ 0.2,\ 0$. Parameters are set to $\gamma=1$, $s=1$, $\alpha=0.01$
and $T=0.1$. Upper solid line is the scaling-law (3) using the expression
(39). Decreasing the value of $h_{f}$, the crossing of the semiclassical
region, after the critical point, becomes more relevant at low $v$ and scaling
no longer holds strictly.
In Fig. 10 the density of excitations and energy obtained from the solution of
kinetic equation is compared with the scaling-law derived in Sec. II using the
specific expressions, Eqs. (39) and (40), derived above for the XY model. The
scaling-laws are found in good agreement with the numerical data. The results
shown in Fig. 11 further confirm the scaling as a function of the temperature
and the relations (12), (13) for the crossover velocity.
Finally we comment on the role of the final value of magnetic field $h_{f}$ at
which the quench is halted. The agreement with the scaling Ansatz becomes
worse for decreasing $h_{f}$ (see Fig. 12). This is due to the non-critical
relaxation induced by the bath when the system crosses the semiclassical
region after the critical point (see Fig. 1 and Fig. 9 left). At low $v$ the
time spent therein at a relative low-temperature $T\ll\Delta$ is so long that
the bath is able to relax the excitations created close to the critical point.
## VII Conclusions
We have studied the dynamics of a quantum critical system coupled to a thermal
reservoir and subject to an adiabatic quench across its quantum critical
point. We considered the regime of weak coupling, low-temperature and slow
quench velocity.
The bath has two effects on the system: the first one is to create excitations
inside the quantum critical region and the second one is to trigger the
relaxation of the excitations created close to the critical point when the
system is driven in the semiclassical region (see Fig. 1). While the first
mechanism is universal, being entirely ruled by the critical properties of the
low-energy spectrum, the latter depends on the details of the system far-off
the critical point. Hence, as far as the evolution is halted close to the
critical point and the non-critical relaxation mechanism is negligible,
universal scaling behavior is recovered. We derived scaling-laws for the
density of energy produced by the quench at finite temperature extending the
previous results obtained for the density of excitations in Ref. PatanePRL, .
To check the validity of the scaling-laws, we considered the specific case of
the quantum XY model (14) coupled locally to a set of bosonic baths, Eq. (17)
(see Fig. 2). In order to study the dynamics we derived a kinetic equation,
within the Keldysh formalism. A detailed analysis of the characteristic
relaxation time obtained from the kinetic equation was given in Sec. V. An
analytic expression for the critical relaxation time was obtained in Eq. (37)
and verified in Fig. 5. As shown in Appendix B, the scaling of the latter as a
function of the temperature is related to the critical exponents of the model
(see Eq. (56)). Finally, we considered the quench dynamics. The kinetic
equations derived allow us to study the dissipative dynamics also beyond the
universal regime. We checked the scaling-laws derived and their range of
validity in Figs. 10 and 12.
We remark that the method described here to obtain a kinetic equation for the
XY model, may be extended to describe the dissipative dynamics of other models
that can be mapped into fermionic degrees of freedom, like other spin chains
and ladders or certain $2d$ models of the Kitaev-type.
## Acknowledgments
We acknowledge F. Guinea, V. Kravtsov, A. Polkovnikov, A.J. Leggett, R.
Raimondi and F. Sols, T. Caneva, G. Carleo and M. Schirò for fruitful
discussions. D.P. acknowledges the ISTANS (grant 1758) program of ESF for
financial support.
## Appendix A1: kinetic equation
Here we present a detailed derivation of the kinetic equation. Apart from the
Keldysh (21), lesser (23) and greater (24) Green’s function we need also the
retarded and the advanced ones, and also the bath Green functions:
$\displaystyle G_{kij}^{a(r)}(t_{1},t_{2})$ $\displaystyle\doteq$
$\displaystyle(-)i\theta(t_{2(1)}-t_{1(2)})\left\langle\left\\{\Psi_{ki}(t_{1}),\
\Psi_{kj}^{\dagger}(t_{2})\right\\}\right\rangle$ $\displaystyle
g_{q}(t_{1},t_{2})$ $\displaystyle\doteq$
$\displaystyle-i\left\langle\mathcal{T}_{\gamma}\
X_{q}(t_{1})X_{q}(t_{2})\right\rangle$ $\displaystyle
g_{q}^{<(>)}(t_{1},t_{2})$ $\displaystyle\doteq$
$\displaystyle(-)i\left\langle\ X_{q}(t_{2(1)})X_{q}(t_{1(2)})\right\rangle$
$\displaystyle g_{q}^{a(r)}(t_{1},t_{2})$ $\displaystyle\doteq$
$\displaystyle(-)i\theta(t_{2(1)}-t_{1(2)})\left\langle\left[X_{q}(t_{1}),\
X_{q}(t_{2})\right]\right\rangle$
where we used commutators (anticommutators) for the retarded and advanced bath
(system) Green’s functions. The starting point is the Dyson’s equation (IV).
In order to obtain from the Dyson’s equation (IV) an equation for the lesser
and greater Green’s function we use the Keldysh book-keeping for a generic
convolution
$C(t_{1},t_{2})\doteq\int_{\gamma}d\bar{t}A(t_{1},\bar{t})B(\bar{t},t_{2})$ is
$C^{r(a)}(t_{1},t_{2})=\int_{0}^{t}d\bar{t}A^{r(a)}(t_{1},\bar{t})B^{r(a)}(\bar{t},t_{2})$
and
$C^{<(>)}(t_{1},t_{2})=\int_{0}^{t}dt_{1}A^{r}(t_{1},\bar{t})B^{<(>)}(\bar{t},t_{2})+A^{<(>)}(t_{1},\bar{t})B^{a}(\bar{t},t_{2})$
vanLeeuwen05 . Using the previous formulas we rewrite the Dyson’s equations
as:
$\displaystyle i\partial_{t_{1}}G_{k}^{<}(t_{1},t_{2})$ $\displaystyle=$
$\displaystyle\mathcal{H}_{k}(t_{1})G_{k}(t_{1},t_{2})+$
$\displaystyle\int_{0}^{t}d\bar{t}\
\Sigma_{k}^{r}(t_{1},\bar{t})G_{k}^{<}(\bar{t},t_{2})+\Sigma_{k}^{<}(t_{1},\bar{t})G_{k}^{a}(\bar{t},t_{2})$
and an analogous equation for $\partial_{t_{2}}$. We are interested in the
_equal-time_ Green’s function and hence we perform a change of variables:
$\displaystyle t$ $\displaystyle=$ $\displaystyle\frac{t_{1}+t_{2}}{2}$
$\displaystyle\delta t$ $\displaystyle=$ $\displaystyle t_{1}-t_{2}$
whose Jacobian is simply $\partial_{t}=\partial_{t_{1}}+\partial_{t_{2}}$ and
$\partial_{\delta t}=\frac{1}{2}(\partial_{t_{1}}-\partial_{t_{2}})$. At equal
time ($\delta t=0$), for the lesser Green’s function
$G_{k}^{<}=G_{k}^{<}(t_{1},t_{1})=G_{k}^{<}(t)$ we get:
$\displaystyle i\partial_{t}G_{k}^{<}$ $\displaystyle=$
$\displaystyle[\mathcal{H}_{k}(t),\ G_{k}]+$ $\displaystyle\Sigma_{k}^{r}\cdot
G_{k}^{<}+\Sigma_{k}^{<}\cdot
G_{k}^{a}-G_{k}^{r}\cdot\Sigma_{k}^{<}-G_{k}^{<}\cdot\Sigma_{k}^{a}$
where the dot indicates the convolution $\Sigma_{k}^{r}\cdot
G_{k}^{<}\doteq\int_{0}^{t}d\bar{t}\,\Sigma_{k}^{r}(t,\bar{t})G_{k}^{<}(\bar{t},t)$.
Now we use the relations:
$\displaystyle G^{r}(t_{1},t_{2})$ $\displaystyle=$
$\displaystyle\theta(t_{1}-t_{2})\left(G^{>}(t_{1},t_{2})-G^{<}(t_{1},t_{2})\right)$
$\displaystyle G^{a}(t_{1},t_{2})$ $\displaystyle=$
$\displaystyle\theta(t_{2}-t_{1})\left(G^{<}(t_{1},t_{2})-G^{>}(t_{1},t_{2})\right)$
and similar relations that hold also for the $\Sigma^{r,a}$ (see vanLeeuwen05
):
$\displaystyle\Sigma^{r(a)}(t_{1},t_{2})$ $\displaystyle=$
$\displaystyle\Sigma^{\delta}\delta(t_{1},t_{2})+$
$\displaystyle\theta(t_{1(2)}-t_{2(1)})\left(\Sigma^{>(<)}(t_{1},t_{2})-\Sigma^{<(>)}(t_{1},t_{2})\right)$
where we can neglect the term $\Sigma^{\delta}$ that only renormalizes the
Hamiltonian and is not relevant in our case (see Eq. (48) below). At equal
times we get:
$\displaystyle i\partial_{t}G_{k}^{<}$ $\displaystyle=$
$\displaystyle\left[\mathcal{H}_{k},G_{k}^{<}\right]+$
$\displaystyle\Sigma_{k}^{>}\cdot G_{k}^{<}-\Sigma_{k}^{<}\cdot
G_{k}^{>}+G_{k}^{<}\cdot\Sigma_{k}^{>}-G_{k}^{>}\cdot\Sigma_{k}^{<}$
We now perform a Markov approximation. This will transform the integro-
differential kinetic equation into a differential equation. Let us define the
interaction picture for a general function:
$\displaystyle\tilde{O}_{k}(t_{1},t_{2})$ $\displaystyle\doteq$
$\displaystyle\mathcal{U}_{k}^{\dagger}(t_{1})O_{k}(t_{1},t_{2})\mathcal{U}_{k}(t_{2})$
where $\mathcal{U}_{k}$ is the _free_ evolution matrix for the system obeying
$i\,\mathcal{\dot{U}}_{k}=\mathcal{H}_{k}\mathcal{U}_{k}$. Such transformation
gauges away the free evolution and the new Green’s function $\tilde{G}_{k}$
dynamics is solely governed by the self energy:
$i\partial_{t}\tilde{G}_{k}^{<}=\tilde{\Sigma}_{k}^{>}\cdot\tilde{G}_{k}^{<}-\tilde{\Sigma}_{k}^{<}\cdot\tilde{G}_{k}^{>}+\tilde{G}_{k}^{<}\cdot\tilde{\Sigma}_{k}^{>}-\tilde{G}_{k}^{>}\cdot\tilde{\Sigma}_{k}^{<}\;.$
Since the self-energy carries the “small” perturbative coupling parameter the
evolution of $\tilde{G}_{k}$ can be regarded as “slow” with respect to the
time scale of the self energy, that is the same as that of the bath. In fact
$\Sigma$ contains the bath-correlation function $g(t_{1},t_{2})$ (see Eq.
(48)) that is strongly peaked at $t_{1}\simeq t_{2}$ because of the assumption
of a cutoff-time for the bosonic modes (see Sec. IV). Thus we can take
$\tilde{G}$ out of the convolutions:
$\displaystyle i\partial_{t}\tilde{G}_{k}^{<}$ $\displaystyle\eqsim$
$\displaystyle\left(\int_{0}^{t}d\bar{t}\tilde{\Sigma}_{k}^{>}(t,\bar{t})\right)\tilde{G}_{k}^{<}-\left(\int_{0}^{t}d\bar{t}\tilde{\Sigma}_{k}^{<}(t,\bar{t})\right)\tilde{G}_{k}^{>}$
(45)
$\displaystyle+\tilde{G}_{k}^{<}\left(\int_{0}^{t}d\bar{t}\tilde{\Sigma}_{k}^{>}(\bar{t},t)\right)-\tilde{G}_{k}^{>}\left(\int_{0}^{t}d\bar{t}\tilde{\Sigma}_{k}^{<}(\bar{t},t)\right)$
Eq. (45) is quite general and it is based solely on the assumption of
Markovian baths. We now use the explicit form of the self-energy for the
coupling system-bath (20) with $l=1$, that within the self-consistent Born
approximation reads:
$\displaystyle\Sigma_{k}(t_{1},t_{2})$ $\displaystyle=$
$\displaystyle\frac{i}{N}\sum_{q}\>g_{k-q}(t_{1},t_{2})\tau^{z}G_{q}(t_{1},t_{2})\tau^{z}$
(46) $\displaystyle=$
$\displaystyle\frac{i}{N}g(t_{1},t_{2})\tau^{z}\sum_{q}G_{q}(t_{1},t_{2})\tau^{z}$
where $g_{q}(t_{1},t_{2})=-i\left\langle\mathcal{T}\
X_{q}(t_{1})X_{q}(t_{2})\right\rangle$ is the non-interacting bath Keldysh
Green’s function that does not explicitly depend on the moment $q$ (since all
baths have the same spectral function). In Eq. (46) we neglected the polaronic
shift contribution (corresponding to the tadpole diagram, Fig. 3b)
$\Sigma_{k}^{\delta}(t_{1},t_{2})=-\frac{i}{N}\delta(t_{1,}t_{2})\tau^{z}\int_{\gamma}d\bar{t}\,g(t_{1},\bar{t})\sum_{q}\textrm{Tr}[\tau^{z}G_{q}(\bar{t},\bar{t})]$
(47)
In fact, being such term proportional to a $\delta(t_{1,}t_{2})$, it has only
the irrelevant effect of renormalizing the Hamiltonian (see Sec. IV). Using
again the Keldysh book-keeping HaugBOOK ; vanLeeuwen05 , we obtain from Eq.
(46), for the lesser and greater self energy
$\displaystyle\Sigma_{k}^{\lessgtr}(t_{1},t_{2})$ $\displaystyle=$
$\displaystyle\frac{i}{N}g^{\lessgtr}(t_{1},t_{2})\tau^{z}\sum_{q}G_{q}^{\lessgtr}(t_{1},t_{2})\tau^{z}$
(48)
(notice that $g^{>}(t_{1},t_{2})=-g^{<}(t_{1},t_{2})^{*}$). Evaluating
explicitly the self-energy kernels we obtain:
$\displaystyle\int_{0}^{t}d\bar{t}\tilde{\Sigma}_{k}^{>}(t,\bar{t})$
$\displaystyle=$ $\displaystyle\frac{i}{N}\sum_{q}\int_{0}^{t}d\bar{t}\
g^{>}(t,\bar{t})\mathcal{U}_{k}^{\dagger}(t)\tau^{z}G_{q}^{>}(t,\bar{t})\tau^{z}\mathcal{U}_{k}(\bar{t})$
$\displaystyle=$ $\displaystyle\frac{i}{N}\sum_{q}\int_{0}^{t}d\bar{t\
}g^{>}(t-\bar{t})\mathcal{U}_{k}^{\dagger}(t)\tau^{z}\mathcal{U}_{q}(t)\tilde{G}_{q}^{>}(t,\bar{t})\mathcal{U}_{q}^{\dagger}(\bar{t})\tau^{z}\mathcal{U}_{k}(\bar{t})$
$\displaystyle\simeq$
$\displaystyle\frac{i}{N}\sum_{q}\mathcal{U}_{k}^{\dagger}(t)\tau^{z}\mathcal{U}_{q}(t)\tilde{G}_{q}^{>}(t,t)\left(\int_{0}^{\infty}\\!ds\
g^{>}(s)\mathcal{U}_{q}^{\dagger}(t-s)\tau^{z}\mathcal{U}_{k}(t-s)\right)$
$\displaystyle\int_{0}^{t}d\bar{t}\tilde{\Sigma}_{k}^{>}(\bar{t},t)$
$\displaystyle=$ $\displaystyle\frac{i}{N}\sum_{q}\int_{0}^{t}d\bar{t}\
g^{>}(\bar{t},t)\mathcal{U}_{k}^{\dagger}(\bar{t})\tau^{z}G_{q}^{>}(\bar{t},t)\tau^{z}\mathcal{U}_{k}(t)$
$\displaystyle=$ $\displaystyle\frac{i}{N}\sum_{q}\int_{0}^{t}d\bar{t}\
g^{>}(\bar{t}-t)\mathcal{U}_{k}^{\dagger}(\bar{t})\tau^{z}\mathcal{U}_{q}(\bar{t})\tilde{G}_{q}^{>}(\bar{t},t)\mathcal{U}_{q}^{\dagger}(t)\tau^{z}\mathcal{U}_{k}(t)$
$\displaystyle\simeq$
$\displaystyle\frac{i}{N}\sum_{q}\left(\int_{0}^{\infty}\\!ds\
g^{>}(-s)_{q}\mathcal{U}_{k}^{\dagger}(t-s)\tau^{z}\mathcal{U}_{q}(t-s)\right)\tilde{G}_{q}^{>}(t,t)\mathcal{U}_{q}^{\dagger}(t)\tau^{z}\mathcal{U}_{k}(t)$
where $\simeq$ refers again to the Markov approximation. For the greater
kernels simply interchange “$<$” with ‘$>$”. Finally, in the Schrödinger
picture, using the relation $G^{>}=-i{\bf 1}+G^{<}$, we obtain Eq. (26).
## Appendix A2: Approximation for the kinetic equation matrices $\hat{D}$
In this appendix we comment on the validity of the approximation (31) for the
matrices (27) appearing in the kinetic equation. To calculate $\hat{D}$
exactly, we need to know the evolution operator $\mathcal{\hat{U}}_{k}$,
solution of the differential equation
$i\,\mathcal{\dot{\hat{U}}}_{k}=\mathcal{\hat{H}}_{k}\mathcal{\hat{U}}_{k}$.
This can be obtained exactly by mapping the dynamics of a generic mode $k$
into a Landau-Zener two-level system dynamics Dziarmaga05 ; Fubini07
$\mathcal{\hat{H}}^{LZ}\equiv
h^{LZ}(t)\hat{\tau}^{z}+\Delta^{LZ}\hat{\tau}^{x}$ (49)
with $\Delta_{k}^{LZ}=\gamma\sin k$ and $h_{k}^{LZ}=-vt$ that can be obtained
from (15) by a simple change of the variable $t$. Using the solution of the
Landau-Zener problem for a quench that starts at $t=-\infty$ we have for the
matrix elements of $\mathcal{U}^{LZ}(-\infty,t)$,
$\mathcal{U}^{LZ}(-\infty,t)=\left(\begin{array}[]{cc}\mathcal{U}_{11}^{LZ}(t)&-\mathcal{U}_{21}^{LZ}(t)^{*}\\\
\mathcal{U}_{21}^{LZ}(t)&\mathcal{U}_{11}^{LZ}(t)^{*}\end{array}\right)$
the following results:
$\displaystyle\mathcal{U}_{11}^{LZ}(t)$ $\displaystyle=$ $\displaystyle
e^{i\frac{\pi}{4}}\exp\left\\{-\frac{\pi\left(\Delta^{LZ}\right)^{2}}{8v}\right\\}\mathcal{D}_{-p}\left((-1+i)\sqrt{v}\
t\right)$ $\displaystyle\mathcal{U}_{21}^{LZ}(t)$ $\displaystyle=$
$\displaystyle\frac{\Delta^{LZ}}{\sqrt{2v}}\exp\left\\{-\frac{\pi\left(\Delta^{LZ}\right)^{2}}{8v}\right\\}\mathcal{D}_{-p-1}\left((-1+i)\sqrt{v}\
t\right)$
where $p=-i\left(\Delta^{LZ}\right)^{2}/2v$ and $\mathcal{D}_{p}$ are
parabolic cylinder functions. Finally the evolution operator from generic
$\bar{t}$ to $t$ can be obtained using the simple property:
$\displaystyle\mathcal{U}^{LZ}(\bar{t},t)$ $\displaystyle=$
$\displaystyle\mathcal{U}^{LZ}(-\infty,t)\mathcal{U}^{LZ}(\bar{t},-\infty)$
$\displaystyle=$
$\displaystyle\mathcal{U}^{LZ}(-\infty,t)\mathcal{U}^{LZ\dagger}(-\infty,\bar{t})$
The second ingredient we need in order to calculate $\hat{D}$ is the bath
thermal equilibrium correlation function $g^{>}$. From its definition:
$\displaystyle g^{>}(t)$ $\displaystyle\doteq$ $\displaystyle-i\left\langle
X(t)X^{\dagger}(0)\right\rangle$ $\displaystyle=$
$\displaystyle-i\sum_{\beta}\lambda_{\beta}^{2}\left(e^{-i\omega_{\beta}t}\left\langle
b_{-\beta}b_{-\beta}^{\dagger}\right\rangle+e^{i\omega_{\beta}t}\left\langle
b_{\beta}^{\dagger}b_{\beta}\right\rangle\right)$ $\displaystyle=$
$\displaystyle-i\int_{0}^{\infty}\\!d\omega\,J(\omega)[e^{-i\omega
t}(1+n_{B}(\omega))+e^{i\omega t}n_{B}(\omega)]$
where $n_{B}\equiv 1/(e^{\omega/T}-1)$ is the Bose function and we used the
definition (19) of spectral function for the bath
$J(\omega)=\sum_{\beta}\lambda_{\beta}^{2}\delta(\omega-\omega_{\beta})$. The
correlation function can be written explicitly as:
$\displaystyle g^{>}(t)$ $\displaystyle=$
$\displaystyle-i\int_{0}^{\infty}\\!d\omega\,J(\omega)\left(\coth(\frac{\omega}{2T})\cos(\omega\tau)-i\sin(\omega\tau)\right)$
$\displaystyle=$ $\displaystyle-2i\alpha T^{s+1}\Gamma(s+1)\times$
$\displaystyle\left(\zeta(s+1,\,T\frac{1+\frac{\omega_{c}}{T}-i\omega_{c}\tau}{\omega_{c}})+\zeta(s+1,\,T\frac{1+i\omega_{c}\tau}{\omega_{c}})\right)$
where $\Gamma$ is the Gamma function and
$\zeta(z,u)\equiv\sum_{n=0}^{\infty}\frac{1}{(n+u)^{z}},\;u\neq 0,-1,-2\dots$.
We are now able to calculate explicitly the matrix $\hat{D}$ and check the
validity of the approximation in (31). As stated in Sec. IV, the approximation
consists in considering the instantaneous transition rates induced by the
bath, independent on the velocity of the quench. This is ultimately justified
by the assumption of “fast” and memoryless Markovian baths. Hence, for slow
quenches when the typical time of the quench is much larger than the typical
bath time-scale we expect that the magnetic field can be regarded as not
evolving for the bath. Within the approximation in Eq. (31) we can perform
explicitly the integration for the matrix elements of $\hat{D}_{qk}$ over
time, giving the Laplace transform of the bath Green function. We are
interested only in the real part of the latter
$\displaystyle g[E]$ $\displaystyle\doteq$
$\displaystyle\Re[\int_{0}^{\infty}ig^{>}(t)e^{iEt}]$ $\displaystyle=$
$\displaystyle\pi\left(J(E)-J(-E)\right)\frac{\exp\left(\beta
E\right)}{\exp\left(\beta E\right)-1}$
since the imaginary part gives a renormalization contribution that is
negligible in the weak coupling limit $\alpha\rightarrow 0$ CohenBOOK . In the
basis of the eigenvectors of $\mathcal{\hat{H}}_{k}$ we obtain:
$\hat{D}_{qk}^{approx}=\frac{1}{2}\left(\begin{array}[]{cc}\cos^{++}g^{+-}+\cos^{+-}g^{++}&i\left(\sin^{+-}g^{-+}+\sin^{++}g^{--}\right)\\\
-i\left(\sin^{+-}g^{+-}+\sin^{++}g^{++}\right)&\cos^{-+}g^{--}-\cos^{++}g^{-+}\end{array}\right)\;.$
(51)
where we defined
$\displaystyle\cos^{\pm\pm}$ $\displaystyle\doteq$
$\displaystyle\pm\cos\theta_{k}\pm\cos\theta_{q}$ $\displaystyle\sin^{\pm\pm}$
$\displaystyle\doteq$ $\displaystyle\pm\sin\theta_{k}\pm\sin\theta_{q}$
$\displaystyle g^{\pm\pm}$ $\displaystyle\doteq$ $\displaystyle
g[\pm\Lambda_{k}\pm\Lambda_{q}]$
The $\sin$ and $\cos$ are geometric factors specific of the system operator
that couples to the bath (in our case $\sigma^{z}$), while the Laplace
transform of the bath $g[.]$ (see Appendix A2: Approximation for the kinetic
equation matrices $\hat{D}$) carries information about the relaxation rates
between the different energy levels, and depends explicitly on the temperature
and on the nature of the baths (i.e., its spectral function). For simplicity
we consider the equal indexes $\hat{D}_{kk}$ matrices. The same results hold
also for unequal indexes since the integral of the matrix elements have the
same structure in both cases. In Fig. 13 we compare the matrix elements
obtained using the exact evolution operator with the ones given by Eq. (51).
The agreement is good, validating the approximation. Deviations appear only in
the limit of fast quenches $\sqrt{v}\gg T$, and in such regime the bath has a
less relevant effect on the dynamics because of the short interaction time
during the quench. Besides that, deviations appear far from the critical point
(corresponding to $h^{LZ}\simeq 0$), i.e., far from the most relevant part of
the quench according to Secs. II and VI.
Figure 13: Matrix elements of $\hat{D}_{kk}$ as a function of the rescaled
field $h^{LZ}$ (49); $h^{LZ}=0$ correspond to the critical point $h\simeq 1$
for the relevant low energy modes. Lower panel: diagonal elements
$i\left(D_{kk}\right)_{21}$ (up) and $-i\left(D_{kk}\right)_{12}$ (down);
upper panel shows the difference
$\left(D_{kk}\right)_{11}-\left(D_{kk}\right)_{22}$. Plots refer to
$T/\Delta^{LZ}=5$; continuous line is the approximation (51) (which is
independent on $v$) and symbols are the exact value for $\sqrt{v}/T=0.1,\ 1,\
5$; deviations from the approximation are appreciable only for the last value
of $v$.
## Appendix B: Fermi golden rule for the relaxation time
In this section we derive an expression for the critical relaxation time using
the Fermi golden rule for a generic system interacting with a bosonic bath.
Let us assume the system-bath interaction Hamiltonian to have the form
$H_{int}=AZ$ where $A$ and $Z$ are system and bath operator respectively.
Consider a quench of the system from zero temperature to a certain finite $T$.
The transition rate for the process of thermalization in presence of the
reservoir $\rho_{B}^{th}\otimes(|GS\rangle\langle
GS|)_{S}\rightarrow\rho_{B}^{th}\otimes\rho_{S}^{th}$ (where $B$ and $S$ refer
to system and bath, respectively) is:
$\displaystyle\frac{1}{\tau}$ $\displaystyle=$ $\displaystyle
2\pi\sum_{f,i,k}\delta(E_{f}+E_{k}-E_{i}-E_{GS})P_{B}^{th}(E_{i}/T)$ (52)
$\displaystyle\times P_{S}^{th}(E_{k}/T)|\left\langle k,\
f\right|H_{int}\left|GS,\ i\right\rangle|^{2}$
where $i$, $f$ and $k$ address the bath eigenvalues and the final state of the
system respectively; $P_{S(B)}^{th}$ are thermal weights. We rewrite the
$\delta$-function as
$\frac{1}{2\pi}\int_{-\infty}^{\infty}dte^{-i(E_{f}-E_{i})t}e^{-i(E_{k}-E_{GS})t}$.
Summing over $f$ and $i$ we get the bath correlation function
$z(t)=\left\langle Z(t)Z(0)\right\rangle$:
$\frac{1}{\tau}=\sum_{k}\int_{-\infty}^{\infty}dte^{-i(E_{k}-E_{GS})t}z(t)P_{S}^{th}(E_{k}/T)|\left\langle
k\right|A\left|GS\right\rangle|^{2}$ (53)
The time integral gives the Fourier transform of the bath correlation
function, that we parametrize as
$z[E]=J(E)f(E/T)\;.$ (54)
For instance, for a bosonic bath with spectral function $J(E)\propto E^{s}$ we
have
$z[E]=\begin{cases}J(E)\,(1+n_{B}(E/T))&E>0\\\
J(|E|)\,n_{B}(|E|/T)&E<0\end{cases}\;.$
By integrating over the k-modes (setting $E_{GS}=0$), using the critical
density of states $\rho(E)\propto E^{d/z-1}$, we get:
$\displaystyle\frac{1}{\tau}$ $\displaystyle\propto$
$\displaystyle\int\\!dE\,\rho(E)\,z[E]\,P_{S}^{th}(E/T)\,|A_{GS}(E)|^{2}\;,$
where $A_{GS}(E)=\left\langle k(E)\right|A\left|GS\right\rangle$. Now,
assuming that the low-energy modes (that are the relevant ones at low-
temperature) are coupled uniformly by the bath $A_{GS}(E)\simeq A_{GS}(0)$ we
obtain
$\frac{1}{\tau}\propto\int\\!dE\,E^{d/z-1}\ J(E)\,f(E/T)\,P_{S}^{th}(E/T)$
(55)
and finally, using for the spectral density $J(E)\propto E^{s}$ and performing
a change of variable to $x=E/T$ we obtain
$\tau^{-1}\propto T^{s+d/z}\;.$ (56)
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* (35) S. Sachdev, “Quantum Phase Transitions” (Cambridge University-Press, Cambridge 1999).
* (36) Notice that depending on the characteristics of the bath, the system-bath coupling may lead to a change of universality class of the transition as a result of bath induced long-range correlations effects. We will not address these issues here, which deserve a careful separate study. Therefore we will assume the bath correlations to be short ranged in time PatanePRL .
* (37) P. Pfeuty, Ann. Phys. (N.Y.), Ann. Phys. 57, 79-90 (1970).
* (38) U. Weiss, Quantum Dissipative Systems (World Scientific, 1992).
* (39) P. Werner et al, Phys. Rev. Lett. 94 047201 (2005).
* (40) H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors (Springer, Berlin, 1996).
* (41) J. Rammer and H. Smith, Rev. Mod. Phys., 58, 323(1986).
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* (43) L. Landau, Physics of the Soviet Union 2: 46-51, (1932); C. Zener, Proceedings of the Royal Society of London, Series A 137 (6): 692-702, (1932).
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|
arxiv-papers
| 2008-12-19T00:03:34 |
2024-09-04T02:48:59.475590
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dario Patan\\`e, Alessandro Silva, Luigi Amico, Rosario Fazio, Giuseppe\n E. Santoro",
"submitter": "Dario Patan\\`e",
"url": "https://arxiv.org/abs/0812.3685"
}
|
0812.3725
|
# Three Views of a Secret in Relativistic Thermodynamics
Tadas K. Nakamura CFAAS, Fukui Prefectural University tadas@fpu.ac.jp
###### Abstract
It has been shown three different views in relativistic thermodynamics can be
derived from the basic formulation proposed by van Kampen and Israel. The way
to decompose energy-momentum into the reversible and irreversible parts is not
uniquely determined, and different choices result in different views. The
effect of difference in the definition of a finite volume is also considered.
relativity, thermodynamics, Lorentz transform
There has been long controversy about the relativistic thermodynamics. A
number of theories were proposed in 1960s, and the discussion seems to have
arrived at a vague general agreement that each theory is consistent in its own
framework by early 1970s Yuen (1970). However, papers has been still published
long after that, even until today (e.g., de Parga et al. (2005); Requardt ),
proposing new formulations which are allegedly better than others.
Roughly speaking there are three different views on the relativistic
thermodynamics, which are characterized by the difference in the temperature
$T$ of a moving body in the following:
$\begin{cases}\textnormal{I)}&T=T_{0}\gamma^{-1}\,,\\\
\textnormal{II)}&T=T_{0}\gamma\,,\\\ \textnormal{III)}&T=T_{0}\,.\end{cases}$
(1)
where $T_{0}$ is the temperature measured in a frame comoving with the body,
$\gamma$ is the Lorentz factor defined as $\gamma=1/\sqrt{1-v^{2}}$ with the
speed of the body $v$ relative to the rest frame (we use the unit of $c=1$).
Most of theories proposed so far can be categorized into one of the above
three. Papers published right after the establishment of special relativity
(e.g., Einstein (1907); Planck (1908)) are based on View I. Theories with View
II were extensively investigated in the middle of 1960s (e.g, Gamba (1965);
Kibble (1966)) stimulated by the papers by Ott (1963) and Arzelies (1965). A
little later a theory in View III was proposed by Lansberg (1966).
There is another theory by van Kampen (1968) in View III; his stand point is
quite different from other theories in Views I, II, and III. He treats the
three components of the velocity as thermodynamical parameters in addition to
the temperature. The van Kampen’s theory was later refined by Israel (1976)
into a more transparent form.
The author of the present paper strongly believes this van Kampen-Israel
theory is the one in _the book_ , i.e., the very fundamental one based on
which other formulations can be derived. The purpose of the present paper is
to show the above three views can be actually derived from the van Kampen-
Israel theory. The great advantage of the van Kampen-Israel theory is in the
point that it does not need the concept of heat or work. The second law can be
expressed with well defined mechanical quantities such as the energy-momentum
or four velocity.
The difference of the above three views mainly comes from the difference in
the definition of heat. Non-relativistic thermodynamics decomposes the energy
increase $\Delta E$ into two parts, heat $\Delta Q$ and work $\Delta W$
namely, as $\Delta E=\Delta Q+\Delta W$. Since the energy is one component of
the energy-momentum four vector, the decomposition should be expressed in the
form of four vectors in the relativistic thermodynamics:
$\Delta\bar{G}=\Delta\bar{Q}+\Delta\bar{W}\,,$ (2)
where $\Delta\bar{G}$ is the change of the energy-momentum, and its reversible
and irreversible parts are expressed as $\Delta\bar{W}$ and $\Delta\bar{Q}$;
we denote a four vector as a whole by a bar (e.g., $\bar{G}$) and its each
component by indices (e.g., $G^{\mu}$) in this paper.
Most of theories in Views I and II determine the temperature from the
following entropy expression.
$\Delta S=\frac{\Delta Q}{T}\,,$ (3)
where $\Delta Q$ is the temporal component of $\Delta\bar{Q}$. However, there
is ambiguity in the decomposition in (2) and the heat is not uniquely
determined as we will see in the present paper. Views I and II define the heat
as $\Delta Q=\Delta Q_{0}\gamma^{-1}$ and $\Delta Q=\Delta Q_{0}\gamma$
($\Delta Q_{0}$ is the heat measured in the comoving frame) respectively, and
both definitions are consistent as the temporal component of an irreversible
energy-momentum change. Consequently two different temperatures (Views I and
II) are derived from (1) since the entropy is supposed to be Lorentz
invariant. View III tries to accommodate both somehow.
The author considers this difference of $\Delta\bar{Q}$ is the main reason for
the confusion in relativistic thermodynamics. However, the definition of a
finite volume may also make the problem complicated. This point has been known
since the very early years of relativity (Fermi (1923)), and must have been
well recognized during the controversy in 1960s (Gamba (1965); Kibble (1966);
Yuen (1970)). However, curiously enough, its importance is not well understood
and a number of erroneous statements on this point are found in papers since
1960s to this date.
We will see in the present paper these confusions can be cleared by the
covariant expression of the van Kampen-Israel theory. For this purpose it is
convenient to define the volume as a four vector (Nakamura (2006)):
$V^{\eta}(\bar{w})=\frac{w^{\eta}V_{0}}{w^{\mu}u_{\mu}}\,.$ (4)
This four vector represents a space-like volume orthogonal to the unit vector
$w^{\eta}$; in other words, this vector defines the volume viewed in a
reference frame with the four velocity $\bar{w}$.
The van Kampen-Israel theory defines the entropy change of a matter with a
finite volume as
$\Delta S=\beta_{0}u_{\mu}V^{\nu}\Delta T_{\nu}^{\mu}-\beta_{0}u_{\mu}P\Delta
V^{\mu}\,,$ (5)
where $P$, $T_{\nu}^{\mu}$ are the pressure and energy momentum tensor
respectively, and $\beta_{0}=1/T_{0}$ is the inverse temperature measured in
the rest frame.
When we define the total energy-momentum four vector as
$G^{\mu}(\bar{w})=V^{\nu}(\bar{w})\,T_{\nu}^{\mu}$ then (5) can be expressed
as
$\Delta S=\beta_{0}u_{\mu}[\Delta G^{\mu}(\bar{w})-P\Delta
V^{\mu}(\bar{w})]\,.$ (6)
Both $\Delta\bar{G}$ and $\Delta\bar{V}$ depends on $\bar{w}$, i.e., the
direction of the volume in the Minkowski space, however, the dependence is
canceled out by taking inner product with $u_{\mu}$ and $\Delta S$ becomes
invariant.
The heat/work is defined as an irreversible/reversible part of the energy-
momentum $\Delta\bar{G}$ in (2), which means
$\beta_{0}u_{\mu}\Delta Q^{\mu}>0\,,\,\,\,\beta_{0}u_{\mu}[\Delta
W^{\mu}-P\Delta V^{\mu}]=0\,.$ (7)
Obviously the above conditions cannot determine the heat and work uniquely;
when we define new values of the heat and work by
$\Delta\bar{Q}^{\prime}=\Delta\bar{Q}+\bar{A}$ and
$\Delta\bar{W}^{\prime}=\Delta\bar{W}-\bar{A}$ with an arbitrary four vector
$\bar{A}$ that satisfies $u_{\mu}A^{\mu}=0$, (7) holds for the new values
$\Delta\bar{Q}^{\prime}$ and $\Delta\bar{W}^{\prime}$. This ambiguity causes
the difference in (1) as we will see in the following.
Suppose a matter moving in the $x$ direction with a four velocity
$(u_{t},u_{x},0,0)$. Then energy momentum tensor may be written in the rest
frame as
$T_{\mu\nu}=\left(\begin{array}[]{cc}u_{t}^{2}\varepsilon_{0}+u_{x}^{2}P&u_{t}u_{x}(\varepsilon_{0}+P)\\\
u_{t}u_{x}(\varepsilon_{0}+P)&u_{x}^{2}\varepsilon_{0}+u_{t}^{2}P\end{array}\right)\,,$
with $\varepsilon_{0}$ being the energy density measured in the comoving
frame. We ignore the dimension in $y$ and $z$ direction for simplicity. Note
that $\varepsilon_{0}$ and $P$ do not depend on $t$ or $x$ because the matter
is in the equilibrium state.
We introduce a parameter $\theta=\tanh^{-1}(w^{x}/w^{t})$ to define the volume
in (4). Then the total energy-momentum can be expressed as a function of
$\theta$ in the following:
$\bar{G}(\theta)=\left(\begin{array}[]{c}E(\theta)\\\
G(\theta)\end{array}\right)=\left(\begin{array}[]{c}E_{0}\cosh\alpha+{\displaystyle
PV_{0}\sinh\alpha\tanh(\theta-\alpha)}\\\ E_{0}\sinh\alpha+{\displaystyle
PV_{0}\cosh\alpha\tanh(\theta-\alpha)}\end{array}\right)$ (8)
where $E_{0}=\varepsilon_{0}V_{0}$ is the total energy measured in the
comoving frame, and the velocity of the matter is parametrized by
$\alpha=\tanh^{-1}(u^{x}/u^{t})$ instead of $\bar{u}$.
We need another parameter to fix the ambiguity of heat in (2). Let us
introduce a parameter $\phi$ such that
$\Delta\bar{Q}=\left(\begin{array}[]{c}\Delta Q\\\ \Delta
Q\tanh\phi\end{array}\right)$
to this end. This parameter $\phi$ specifies the frame in which the heat is
purely timelike, in other words, the frame in which the heat looks “heat” only
without momentum. The rest frame and comoving frame are represented by
$\phi=0$ and $\phi=\alpha$ respectively.
The work $\Delta\bar{W}$ is then calculated as
$\Delta\bar{W}=\left(\begin{array}[]{c}\Delta
E_{0}\cosh\alpha+{\displaystyle\Delta(PV_{0})\sinh\alpha\tanh(\theta-\alpha)-\Delta
Q}\\\ \Delta
E_{0}\sinh\alpha+{\displaystyle\Delta(PV_{0})\cosh\alpha\tanh(\theta-\alpha)}-\Delta
Q\tanh\phi\end{array}\right)\,.$
Since the work $\Delta\bar{W}$ must satisfy (7), the heat $\Delta Q$ is
uniquely determined when $\phi$ is given:
$\Delta Q(\phi)=\frac{\cosh\phi}{\cosh(\phi-\alpha)}\,\Delta Q_{0}\,,$ (9)
where $\Delta Q_{0}=\Delta E_{0}-P\Delta V_{0}$.
Various formulations can be derived by expressing $\Delta Q(\phi)$ in (9) with
the energy-momentum $\bar{G}(\theta)$ in (8) by choosing different $\phi$ and
$\theta$. Any value of $\phi$ and $\theta$ can determine the relativistic
thermodynamical equation in general, however, the value of the rest frame or
the comoving frame ($0$ or $\alpha$) are practically preferable choices. In
the following we examine three typical choices in Views I, II, and III.
Typical theories choose the same value for $\phi$ and $\theta$ ($\phi=\theta$)
because they consider the heat exchange and volume change in the same frame
111This is typical, but not always. For example, the result by Kibble (1966)
can be derived with $\phi=\alpha$ and $\theta=0$.. For example, Ott (1963)
assumes a Carnot cycle in the comoving frame; the steps in the cycle,
including the heat exchange and volume change, take place in the moving frame.
Then (9) can be cast in the form of
$Q(\theta)=\frac{\cosh\theta}{\cosh(\theta-\alpha)}\Delta E_{0}-P\Delta
V^{t}(\theta)\,.$
The temporal component $\Delta V^{t}(\theta)$ is regarded as the volume change
in these theories, and denoted simply by $\Delta V$ and little attention has
been paid to its dependence on $\theta$. Then the above equation can be
regarded as to correspond to the definition of heat $\Delta Q=\Delta E-P\Delta
V$ in non-relativistic thermodynamics. The coefficient of the $\Delta E_{0}$
term determines the transformation rule of the heat, and consequently, that of
the temperature.
View I is typically derived with $\phi=\theta=0$, which means both the heat
and volume are defined in the rest frame. This choice gives $\Delta Q=\Delta
Q_{0}\gamma^{-1}$ and thus $T=T_{0}\gamma^{-1}$ because of (3). The
calculation of heat is obtained by subtracting $\gamma^{-1}u_{x}\Delta G$ from
the energy, regarding this as the work to cause acceleration:
$\Delta Q(0)=E(0)-\gamma^{-1}u_{x}\Delta G(0)-P\Delta V(0)\,,$ (10)
where we write $\Delta V=\Delta V^{t}$.
The typical choice of View II is $\phi=\theta=\alpha$, resulting
$T=T_{0}\gamma$. The heat and the volume are defined in the comoving frame,
and the expression in the rest frame is a result of their Lorentz transform.
The calculation of $\Delta Q$ is straightforward:
$\Delta Q(\alpha)=\Delta E(\alpha)-P\Delta V(\alpha)\,.$ (11)
Lansberg (1966) considered the temperature must be a Lorentz-invariant as in
View III from the symmetry. When two identical systems are moving relative to
each other, there is no reason for one system to have a temperature higher
than the others’. This argument can be represented by choosing
$\phi=\frac{1}{2}\alpha$ in (9) to treat the rest frame and the comoving frame
symmetrically. It should be noted that his actual calculation is more
complicated, but we do not examine its details here.
In the present paper we have successfully derived three different views of
relativistic thermodynamics from one basic formulation proposed by van Kampen
(1968) and Israel (1976). The difference comes from two factors, the
definitions of the heart and volume namely, which are represented by the two
parameters $\phi$ and $\theta$ here. The papers published so far on this topic
are so numerous that it is not practical to check all of them. However, the
author believes all the formulations can be derived from the van Kampen-Israel
theory as long as they are not wrong.
## References
* Yuen (1970) C. K. Yuen, Amer. J. Phys. 38, 246 (1970).
* (2) M. Requardt, eprint arXiv:0801.2639.
* de Parga et al. (2005) G. A. de Parga, Lopéz-Carrera, and F. Anulo-Brown, J. Math. Phys. 38, 2821 (2005).
* Einstein (1907) A. Einstein, Jb. Radioaktivitat 4, 411 (1907).
* Planck (1908) M. Planck, Ann. d. Phys. 76, 1 (1908).
* Gamba (1965) A. Gamba, Nuovo Cimento 37, 1792 (1965).
* Kibble (1966) T. W. B. Kibble, Nuovo Cimento 41B, 167 (1966).
* Ott (1963) H. Ott, Z. Physik 175, 70 (1963).
* Arzelies (1965) H. Arzelies, Nuovo Cimento 35, 792 (1965).
* Lansberg (1966) P. T. Lansberg, Nature 212, 571 (1966).
* van Kampen (1968) N. G. van Kampen, Phys. Rev. 173, 295 (1968).
* Israel (1976) W. Israel, Ann. Phys. 106, 310 (1976).
* Fermi (1923) E. Fermi, Nuovo Cimenmto 25, 159 (1923).
* Nakamura (2006) T. K. Nakamura, Phys. Lett. A 352, 175 (2006), eprint arXiv:physics/0505004.
|
arxiv-papers
| 2008-12-19T08:53:12 |
2024-09-04T02:48:59.485837
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tadas K. Nakamura",
"submitter": "Tadas Nakamura",
"url": "https://arxiv.org/abs/0812.3725"
}
|
0812.3755
|
# Modification of Heisenberg uncertainty relations in
non-commutative Snyder space-time geometry
Marco Valerio Battisti battisti@icra.it ICRA - International Center for
Relativistic Astrophysics Dipartimento di Fisica (G9), Università di Roma
“Sapienza” P.le A. Moro 5, 00185 Rome, Italy Stjepan Meljanac
meljanac@irb.hr Rudjer Bovskovic Institute, Bijenivcka c.54, HR-10002 Zagreb,
Croatia
###### Abstract
We show that the Euclidean Snyder non-commutative space implies infinitely
many different physical predictions. The distinct frameworks are specified by
generalized uncertainty relations underlying deformed Heisenberg algebras.
Considering the one-dimensional case in the minisuperspace arena, the bouncing
Universe dynamics of loop quantum cosmology can be recovered.
###### pacs:
04.60.Bc; 02.40.Gh; 11.10.Nx
## I Introduction
Non-commutative geometries are widely considered as plausible candidates for
describing physics at the Planck scale noncom and have natural connections
with string theory SW . Moreover some of these models can be related to the
intuitions of doubly special relativity (DSR) AMS where another invariant
scale (apart from the speed of light) is introduced ab initio in the theory.
Interest in DSR is also increased because such a framework can be regarded as
a semi-classical limit of quantum gravity (see RovSmo and references
therein).
In this paper the Snyder proposal Sny of a non-commutative space-time is
analyzed from a physical point of view. This model can be understood by means
of the projective geometry approach to the de Sitter space of momenta with two
universal constants and is relevant since it can be related to some of DSR
models Kov . Furthermore, it has some motivations from loop quantum gravity LO
and two-time physics tt .
The starting point of our analysis is the requirement that the only deformed
commutator in the Euclidean Snyder framework is one between the coordinates.
This way, the translation group is not deformed and the rotational symmetry is
preserved. We then show that, infinitely many commutators between the non-
commutative coordinates and momenta are possible, such that in all the cases
the algebra closes. This way, infinitely many different physical predictions
of the Snyder space are allowed. These are summarized in the deformed
symplectic geometry and in the generalized uncertainty principle at classical
and quantum level, respectively. The physical interesting framework of a
deformed quantum cosmology is also analyzed. Here we deal with a one-
dimensional system and our picture is almost uniquely fixed. We show that this
framework naturally leads to the non-singular (bouncing) Friedmann dynamics
obtained in recent issues of loop quantum cosmology (LQC) bloop .
The paper is organized as follows. In Section II the algebraic structure of
the Euclidean Snyder space is analyzed. Section III is devoted to discuss the
physical implications of this framework. Concluding remarks follow. Over the
paper we adopt units such that $\hbar=c=1$.
## II Realizations of Snyder space
The algebraic structure of the non-commutative Snyder space is analyzed in
this Section. All possible realizations of this space, the general form of the
uncertainty principle and the required hermiticity conditions are showed. The
known algebras are then recovered as particular cases of our construction.
Realizations. Let us start by considering a $n$-dimensional non-commutative
(deformed) Euclidean space such that the commutator between the coordinates
has the non-trivial structure ($\\{i,j,...\\}\in\\{1,...,n\\}$)
$[\tilde{x}_{i},\tilde{x}_{j}]=\kappa M_{ij}\,,$ (1)
where with $\tilde{x}_{i}$ we refer to the non-commutative coordinates and
$\kappa\in\mathbb{R}$ is the deformation parameter with dimension of a squared
length. We then demand that the rotation generators
$M_{ij}=-M_{ji}=i(x_{i}p_{j}-x_{j}p_{i})$ satisfy the ordinary $SO(n)$ algebra
$[M_{ij},M_{kl}]=\delta_{jk}M_{il}-\delta_{ik}M_{jl}-\delta_{jl}M_{ik}+\delta_{il}M_{jk}$
(2)
and that the translation group is not deformed, i.e. $[p_{i},p_{j}]=0$. In
order to preserve the rotational symmetry the commutators between $M_{ij}$ and
the coordinates $\tilde{x}_{i}$, as well as between $M_{ij}$ and $p_{k}$, have
to be undeformed. Therefore, we assume that the relations
$\displaystyle[M_{ij},\tilde{x}_{k}]$ $\displaystyle=$
$\displaystyle\tilde{x}_{i}\delta_{jk}-\tilde{x}_{j}\delta_{ik},$ (3)
$\displaystyle[M_{ij},p_{k}]$ $\displaystyle=$ $\displaystyle
p_{i}\delta_{jk}-p_{j}\delta_{ik}$
hold. This way we deal with the (Euclidean) Snyder space Sny . The above
relations however do not uniquely fix the commutators between $\tilde{x}_{i}$
and $p_{j}$. In particular, there are infinitely many of such commutators
which are all compatible (in the sense that the algebra closes in virtue of
the Jacobi identities) with the above natural requirements.
This feature can be understood by analyzing the realizations Mel ; Luk ; Gosh
of such a non-commutative space. The concept of realization was developed in a
series of papers Mel (for a similar approach in the $\kappa$-deformed space-
time see Luk and a related analysis in the context of DSR can be found in
Gosh ). A realization of the Snyder algebra (1) is defined as a rescaling of
the non-commutative coordinates $\tilde{x}_{i}$ in terms of the ordinary phase
space variables ($x_{i},p_{j}$). The most general $SO(n)$ covariant
realization for $\tilde{x}_{i}$ is given by
$\tilde{x}_{i}=x_{i}\varphi_{1}(\mu,\nu)+\kappa(x_{j}p_{j})p_{i}\varphi_{2}(\mu,\nu),$
(4)
where the convention $a_{i}b_{i}=\sum_{i}a_{i}b_{i}$ is adopted and
$\varphi_{1}$ and $\varphi_{2}$ are two arbitrary finite functions depending
on the dimensionless quantities $\mu=\kappa p^{2}$ and $\nu=\kappa m^{2}$. In
particular, the second quantity accounts for a mass-like term $m^{2}$ which
can be positive, negative or zero. In order to recover the ordinary Heisenberg
algebra, suitable boundary conditions on these functions have to be imposed.
We have to demand that, in the $\kappa\rightarrow 0$ ($\mu,\nu\rightarrow 0$)
limit, $\varphi_{1}(0,0)=1$.
The realization above is, of course, not completely arbitrary since it depends
on the adopted algebraic structure. In particular, the two functions
$\varphi_{1}$ and $\varphi_{2}$ are constrained by the relations (1) and (3).
Inserting the formula (4) into the non-commutative coordinate commutator (1),
the first restriction we obtain reads
$2\left(\varphi_{1}^{\prime}\varphi_{1}+\mu\varphi_{1}^{\prime}\varphi_{2}\right)-\varphi_{1}\varphi_{2}+1=0,$
(5)
where $\varphi_{1}^{\prime}=\partial\varphi_{1}/\partial\mu$. The other
condition on $\varphi_{1}$ and $\varphi_{2}$ arises after considering the
realization (4) into the commutator $[M_{ij},\tilde{x}_{k}]$ in (3). Such
second constraint can be written as
$\left(x_{l}[M_{ij},p_{l}]p_{k}+[M_{ij},x_{l}]p_{l}p_{k}\right)\varphi_{2}=0$
(6)
and is immediate to verify that the argument in the brackets identically
vanishes. Therefore, only one condition on $\varphi_{1}$ and $\varphi_{2}$
appears. As a matter of fact, given any function $\varphi_{1}(\mu,\nu)$
satisfying the boundary condition $\varphi_{1}(0,0)=1$, the function
$\varphi_{2}(\mu,\nu)$ is uniquely determined by the equation (5) and reads
$\varphi_{2}=(1+2\varphi_{1}^{\prime}\varphi_{1})/(\varphi_{1}-2\mu\varphi_{1}^{\prime})$.
In other words, there are infinitely many ways to express, via $\varphi_{1}$,
the non-commutative coordinates (1) in terms of the ordinary ones without
deforming either the rotation and the translation groups.
It is worth noting that: (i) The realizations (4) have sense if there exist
the inverse transformation $x_{i}=\tilde{x}_{j}(\varphi^{-1})_{ji}$ and the
necessary and sufficient condition is $\det|\delta_{ij}\varphi_{1}+\kappa
p_{i}p_{j}\varphi_{2}|>0$. If we deal with a $n\geq 2$ dimensional space, such
a condition reads $\varphi^{n-1}_{1}(\varphi_{1}+\mu\varphi_{2})>0$, i.e.
$\varphi_{1}>0$ and $\varphi_{1}+\mu\varphi_{2}>0$. (ii) Our analysis can be
straightforward generalized to a Snyder Minkowskian space-time. In this case,
all the relations above hold as soon as the following replacements are taken
into account ($\\{\alpha,\beta\\}\in\\{0,...,n\\}$): the $SO(n)$ generators
are substituted by the Lorentz generators $L_{\alpha\beta}$ and
$(\tilde{x}_{\alpha},p_{\alpha})$ now transform as four-vectors under Lorentz
algebra which indices are raised and lowered by the Minkowski metric
$\eta_{\alpha\beta}$, i.e $p^{2}=\eta^{\alpha\beta}p_{\alpha}p_{\beta}$ is
Lorentz invariant.
Uncertainty relations. In order to complete the analysis of the deformed
algebra we need to analyze the $(\tilde{x}-p)$ commutation relation. This way,
the general form of the uncertainty principle, and thus the physical
consequences of the model, can be discussed. The commutator between
$\tilde{x}_{i}$ and $p_{j}$ arises from the realization (4) and reads
$[\tilde{x}_{i},p_{j}]=i\left(\delta_{ij}\varphi_{1}+\kappa
p_{i}p_{j}\varphi_{2}\right).$ (7)
Of course, the ordinary one is recovered in the $\kappa\rightarrow 0$ limit.
From the commutator above we can immediately obtain the generalized
uncertainty principle underlying the Snyder non-commutative space, i.e.
$\Delta\tilde{x}_{i}\Delta
p_{j}\geq\frac{1}{2}|\delta_{ij}\langle\varphi_{1}\rangle+\kappa\langle
p_{i}p_{j}\varphi_{2}\rangle|.$ (8)
Three remarks are in order. (i) The algebra we obtain can be regarded as a
deformed Heisenberg algebra. More precisely, the deformation of the only
commutator between the spatial coordinates as in (1) leads to infinitely many
realizations of the algebra, and thus generalized uncertainty relations (8),
all consistent with the assumptions underlying the model. (ii) Unless
$\varphi_{2}=0$ no compatible observables exist. These are coupled with each
other and an exactly simultaneous measurable couple $(\tilde{x}_{i},p_{j})$ is
not longer allowed. A measure of the $i$-component of the (non-commutative)
position will always affect a measure of the $j(\neq i)$-component of the
momentum by an uncertainty $\Delta p_{j}\gtrsim|\kappa\langle
p_{i}p_{j}\varphi_{2}\rangle|/\Delta\tilde{x}_{i}$. (iii) For any fixed
$\varphi_{1}$ the non-commutative framework is unique, but we can realize the
commutator (7) in terms of any commutative coordinates $x_{i}^{\prime}$ and
corresponding canonical momenta $p_{i}^{\prime}$ satisfying
$[x_{i}^{\prime},p_{j}^{\prime}]=i\delta_{ij}$. Of course all these
descriptions lead to the same physical consequences.
Hermiticity conditions. The non-commutative coordinates $\tilde{x}_{i}$ have
to be hermitian operators in any given realization. All the commutators given
above are invariant under the formal anti-linear involution “${\dagger}$”
$\tilde{x}_{i}^{\dagger}=\tilde{x}_{i},\quad p_{i}^{\dagger}=p_{i},\quad
M_{ij}^{\dagger}=-M_{ij}\,,$ (9)
where the order of elements is inverted under the involution. However, the
realization (4) in general is not hermitian. The hermiticity condition can be
immediately implemented as soon as the expression
$\tilde{x}_{i}=\frac{1}{2}\left(x_{i}\varphi_{1}+\kappa(x_{j}p_{j})p_{i}\varphi_{2}+\varphi_{1}^{\dagger}x_{i}^{\dagger}+\kappa\varphi_{2}^{\dagger}p_{i}^{\dagger}(x_{j}p_{j})^{\dagger}\right)$
(10)
is taken into account. However, the physical result do not depend on the
choice of the representation as long as exist a smooth limit
$\tilde{x}_{i}\rightarrow x_{i}$ as $\kappa\rightarrow 0$. Therefore, we can
restrict our attention to non-hermitian realization only.
Recovering the know realizations. The non-commutative Snyder space has been
analyzed in literature from different points of view Kov ; LO ; tt (see also
GB ), but only two particular realizations of its algebra are known: the
Snyder Sny and the Maggiore Mag ones. The original realization of Snyder, in
particular, expressed through the commutator between $\tilde{x}$ and $p$,
reads
$[\tilde{x}_{i},p_{j}]=i\left(\delta_{ij}+\kappa p_{i}p_{j}\right).$ (11)
It is not difficult to see that this is a particular case of our realization
(4) as soon as $\varphi_{1}=1$. From this condition, the function
$\varphi_{2}$ is fixed by (5) as $\varphi_{2}=1$ and the above commutation
relation is recovered. The condition on the inverse mapping implies that
$p^{2}>-1/\kappa$. On the other hand, the Maggiore algebra
$[\tilde{x}_{i},p_{j}]=i\delta_{ij}\sqrt{1-\kappa(p^{2}+m^{2})},$ (12)
can be regarded as the particular case of (4) when the condition
$\varphi_{2}=0$ is taken into account. But this requirement alone is not
enough. In fact, from the constraint (5), the function $\varphi_{1}$ is not
uniquely fixed but reads $\varphi_{1}=\sqrt{1-\mu+f(\nu)}$, where $f(\nu)$ is
a generic function of $\nu$ (the inverse mapping condition entails
$p^{2}<(1+f)/\kappa$). Only in the specific case $f(\nu)=-\nu$ the commutator
(12) is recovered. Finally, we note that, the deformed algebra proposed by
Kempf et al. in Kem can be regarded as a particular case of (12) as $|\mu|\ll
1$ and $m=0$, i.e. $[\tilde{x}_{i},p_{j}]=i\delta_{ij}(1+\beta p^{2})$ where
$\beta=-\kappa/2$ with $\kappa<0$. In the one-dimensional framework (see
below), this algebra is the same of the Snyder one (11).
## III Physical implications
As understood, the physical consequences of a non-commutative space geometry
are deeply and completely new scenarios are opened at both classical and
quantum levels. Two physically relevant frameworks are analyzed in this
Section: a generic mechanical system and the so-called quantum cosmological
arena.
Mechanical system. Let us start by considering a mechanical system, i.e. a
model with a finite number of degrees of freedom described by a Hamiltonian
$H=H(\tilde{x},p)$. At classical level the deformations induced on the system
appear as soon as the (classical) limit
$-i[\cdot,\cdot]\rightarrow\\{\cdot,\cdot\\}$ is taken into account. In doing
this the deformation parameter $\kappa$ is regarded as an independent constant
with respect to $\hbar$. Modifications of a non-commutative framework on the
classical dynamics are then summarized in the deformed Poisson brackets
$\\{F,G\\}=\left(\frac{\partial F}{\partial\tilde{x}_{i}}\frac{\partial
G}{\partial p_{j}}-\frac{\partial F}{\partial p_{i}}\frac{\partial
G}{\partial\tilde{x}_{j}}\right)\\{\tilde{x}_{i},p_{j}\\}+\frac{\partial
F}{\partial\tilde{x}_{i}}\frac{\partial
G}{\partial\tilde{x}_{j}}\\{\tilde{x}_{i},\tilde{x}_{j}\\}$ (13)
between any phase space functions. This symplectic structure is not fixed but
depends on the two functions $\varphi_{1}$ and $\varphi_{2}$ constrained by
(5) and $\varphi_{1}(0,0)=1$. From the relation above, the equations of motion
of a mechanical system are deformed as
$\displaystyle\dot{\tilde{x}}_{i}$ $\displaystyle=$
$\displaystyle\\{\tilde{x}_{i},H\\}=\frac{\partial H}{\partial
p_{j}}\left(\delta_{ij}\varphi_{1}+\kappa
p_{i}p_{j}\varphi_{2}\right)+\frac{\kappa}{i}\frac{\partial
H}{\partial\tilde{x}_{j}}M_{ij},$ $\displaystyle\dot{p}_{i}$ $\displaystyle=$
$\displaystyle\\{p_{i},H\\}=-\frac{\partial
H}{\partial\tilde{x}_{j}}\left(\delta_{ij}\varphi_{1}+\kappa
p_{i}p_{j}\varphi_{2}\right).$ (14)
When the deformation parameter vanishes ($\kappa\rightarrow 0$) the ordinary
Hamilton equations are recovered. At quantum level our picture implies either
modifications of the Ehrenfest theorem through (III), either deformations of
the canonical quantization prescription via the commutator (7). As we said,
this commutator is not fixed at all by the assumptions described above and for
any choice of the realization (4) of the non-commutative coordinates, the
corresponding Hilbert spaces are thus unitarily inequivalent. Each quantum
framework (Hilbert space) corresponds to a specific choice of the realization
(4). We also stress that given an eigenvalue problem
$\hat{H}(\tilde{x},p)\psi(x)=E\psi(x)$, the wave function $\psi$ and the
spectrum $E$ depend on $\varphi_{1}$.
This is not surprising since the deformation of the canonical commutation
relations can be viewed, from the realization (4), as an algebra homomorphism
which is a non-canonical transformation. In particular, it can not be
implemented at quantum level as an unitary transformation. From this point of
view, the set of predictions of any deformed Heisenberg algebra can not be
matched by the set of predictions arising from another one, i.e. neither by
the set of prediction of the ordinary framework (the $\kappa\rightarrow 0$
limit). New features are then introduced, for any fixed $\varphi_{1}$, at both
classical and quantum level. This way, a Snyder structure (1) in which the
translation and rotation groups are undeformed, leads to infinitely many
different physical predictions through (4).
A notable problem to be considered is the harmonic oscillator with non-
commutative quadratic potential, i.e. $H=p^{2}/2m+m\omega^{2}\tilde{x}^{2}/2$.
In the one-dimensional case the symmetry group is trivial ($SO(1)=\text{Id}$)
and the most general realization is given by $\tilde{x}=x\sqrt{1-\mu+f(\nu)}$.
Considering the representation of this algebra (we take $f=0$) in the momentum
space, the deformed stationary Schrödinger equation for this model is given by
the so-called Mathieu equation and the energy spectrum appears to be modified
as
$E_{n}=\omega(2n+1)/2-\omega\kappa(2n^{2}+2n+1)/8d^{2}+\mathcal{O}(\kappa^{2}/d^{4})$
where $d=1/\sqrt{m\omega}$ is the characteristic length scale (for more
details see Bat ). We note that, if $\kappa>0$ this is the spectrum obtained
in polymer (loop) quantum mechanics pol , while if $\kappa<0$ this result
resembles the one recovered in DJM03 .
Quantum cosmology. An interesting quantum mechanical arena to test such a
framework is the so-called minisuperspace reduction of the dynamics, i.e.
quantum cosmology. As well-know Wald , by imposing symmetries on the spatial
Cauchy surfaces which fill the space-time manifold, a considerable
simplification on the gravitational theory occurs. In particular, by requiring
the spatial homogeneity the phase space of general relativity reduces to six
dimensions. The system is described by three degrees of freedom, i.e. the
three scalar factors of the Bianchi models. Furthermore, by imposing also the
spatial isotropy, we deal with one-dimensional mechanical systems. These are
the Friedmann-Robertson-Walker (FRW) models which describe the observed
Universe and on which the standard model of cosmology is based.
In order to discuss the implications of the Snyder algebra on the FRW
Universes we consider the one-dimensional case of the scheme analyzed above.
If we assume the minisuperspace as Snyder-deformed, then the isotropic scale
factor $a$ (namely the radius of the Universe) and its conjugate momentum $p$
satisfy the commutation relation $[a,p]=i\sqrt{1-\mu+f(\nu)}$. It is worth
stressing that, when $\kappa>0$ (taking $f=0$) a natural cut-off on the
momentum arises, i.e. $|p|<\sqrt{1/\kappa}$, while as $\kappa<0$ the
uncertainty relation (8) predicts a minimal observable length
$\Delta{\tilde{x}}_{\text{min}}=\sqrt{-\kappa}$. Moreover, at the first order
in $\kappa$, the string theory result String
$\Delta{\tilde{x}}\gtrsim(1/\Delta p+l_{s}^{2}\Delta p)$, in which the string
length $l_{s}$ can be identify with $\sqrt{-\kappa/2}$, is recovered.
Following Bat is possible to show that the effective Friedmann equation of
Snyder-deformed flat FRW cosmological model becomes
$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G}{3}\rho\left(1-\frac{\rho}{\rho_{c}}+f(\nu)\right),$ (15)
where $G$ is the gravitational constant, $\rho=\rho(a)$ denotes a generic
matter energy density and $\rho_{c}=(2\pi G/3\kappa)\rho_{P}$ is the critical
energy density ($\rho_{P}$ being the Planck one). When the limit
$\kappa\rightarrow 0$ is taken into account, the critical energy density
diverges (the function $f(\nu)$ disappears) leading to the ordinary dynamics.
It is worth noting that, if $f(\nu)=0$ and $\kappa>0$, the equation (15)
resembles exactly the effective bouncing Friedmann equation of LQC bloop .
Such a dynamics is singularity-free since, when $\rho$ reaches the critical
energy density, $\dot{a}$ vanishes and the Universe experiences a (big)-bounce
instead of the classical big-bang. On the other hand, if $f(\nu)=0$ and
$\kappa<0$, the effective braneworlds dynamics is recovered Roy .
Summarizing, the non-commutative Snyder minisuperspace framework can clarify
similarities and differences between different quantum gravity theories. Other
comparisons between deformed and loop-polymer quantum cosmology, in view of
discussing the fate of the cosmological singularity at quantum level, were
performed considering the flat FRW model filled with a massless scalar field
BM07a and the Taub cosmological model BM07b . Such investigations deserve
interest either in clarifying the role of loop quantization techniques in
cosmology, either in establishing a phenomenological contact with some
frameworks relevant in a flat space-time limit of quantum gravity.
## IV Concluding remarks
In this paper we have shown how there are infinitely many realizations of the
Snyder algebra, equations (1-3), implying different commutation relations
between the non-commutative coordinates $\tilde{x}$ and momenta $p$, i.e. we
deal with deformed Heisenberg algebras. These depend on an arbitrary function
$\varphi_{1}(\mu,\nu)$ such that $\varphi_{1}(0,0)=1$ ensuring the correctness
of the picture. Therefore, different non-commutative spaces, described by
distinct commutations relations (7), imply different (unitarily inequivalent)
physical consequences. On the other hand, in the one-dimensional case the
commutator between $\tilde{x}$ and $p$ is fixed (up to a function of the mass-
like term) and, when implemented in the minisuperspace dynamics, the loop as
well as the braneworlds cosmological evolutions are recovered.
Acknowledgments. We thank Daniel Meljanac for comments. M.V.B. thanks S.M. for
the warm hospitality in Zagreb during which this paper was carried out. This
work was supported in part by Ministry of Science and Technology of the
Republic of Croatia under contract No. 098-0000000-2865.
## References
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|
arxiv-papers
| 2008-12-19T11:26:03 |
2024-09-04T02:48:59.491292
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marco Valerio Battisti and Stjepan Meljanac",
"submitter": "Marco Valerio Battisti",
"url": "https://arxiv.org/abs/0812.3755"
}
|
0812.3789
|
# Optimizing Nuclear Reaction Analysis (NRA) using Bayesian Experimental
Design
U. von Toussaint, T. Schwarz-Selinger, and S. Gori Max-Planck-Institut für
Plasmaphysik, EURATOM Association, Boltzmannstr. 2, 85748 Garching, Germany
###### Abstract
Nuclear Reaction Analysis with 3He holds the promise to measure Deuterium
depth profiles up to large depths. However, the extraction of the depth
profile from the measured data is an ill-posed inversion problem. Here we
demonstrate how Bayesian Experimental Design can be used to optimize the
number of measurements as well as the measurement energies to maximize the
information gain. Comparison of the inversion properties of the optimized
design with standard settings reveals huge possible gains. Application of the
posterior sampling method allows to optimize the experimental settings
interactively during the measurement process.
###### pacs:
02.50.Le, 02.50.Tt, 25..55.-e, 29.85.Fj
## I Introduction
The rising price for oil has recently shifted the focus to other possible
sources of energy, preferably without adverse effects to the environment. One
of the methods presently being developed is nuclear magnetic fusion. The
objective of fusion research is to harness the energy provided by the fusion
of hydrogen isotopes. In the fusion experiment ITER, presently under
construction in Cadarache, France, the necessary data to design and operate
anelectricity-producing plant shall be gained. ITER is a tokamak, an
intermittent operating device in which strong magnetic fields confine a torus-
shaped plasma.
Figure 1: In-vessel view of ITER. The surface of the main chamber is
Beryllium, the material of the divertor (in lower part of the vacuum vessel)
is carbon and tungsten. Source:Iter08 , published with permission of ITER.
Since the confinement is not perfect (and must not be) there are always
interactions between the plasma and the plasma-facing (wall) components (PFCs)
which have to be taken into account. One of the key aspects in the licensing
process of ITER is a strict upper limit of the total amount of radioactive
tritium accumulated in the vessel walls, which is presently at 700g
tritiumIter08 . The prediction of the amount of retained tritium is
complicated by the material choice of ITER (Fig.1): The main vessel walls are
Beryllium, the strike-points are made of carbon (CFC) and the other parts of
the divertor are tungsten. During the operation of ITER the interaction of the
plasma and high energy 14MeV-neutrons with the vessel walls will lead to
erosion, redeposition, material mixing and alloy formation. Since even the
hydrogen retention properties of pure materials are still subject to current
research, a significant amount of additional experimental data is required to
develop and calibrate the theoretical models which will be needed to process
the huge number of material combinations created in ITER.
However, even the first step - measuring hydrogen depth profiles in material
composites - is challenging for many reasons; here we will mention only two:
a) Hydrogen and its isotopes are very volatile, which can easily distort
measurements of depth profiles and b) Hydrogen is usually the main component
of the residual gas in vacuum chambers which precludes the use of many well-
established analysis methods.
One method which holds great promise to overcome these difficulties is the
Nuclear Reaction Analysis (NRA) of deuterium using 3He as probing particle. It
is a specific and sensitive method, and has a sufficient analysis depth.
However every data point takes about 30min to measure and the extraction of
the concentration depth profile is an ill-posedinversion problem requiring the
deconvolution of the measured data vector, here even more challenging than in
Rutherford BackscatteringToussaint2000 . Therefore the experimental setup (ie
the choice of the analysis energies) should provide a maximum of information.
So far the most common choice of the 3He energies for the measurements was
simply equidistant. Using Bayesian Experimental Design the performance of the
method can be improved considerably (in some cases up to orders of magnitude)
and quantitative measures can bederived about the expected utility of further
(time consuming) measurements. sectionNuclear Reaction Analysis The basic
principle of Nuclear Reaction Analysis is straightforward: The sample is
subjected to an energetic ion beam (here 3He) with initial energy
${E}_{i}^{0}$ and incoming angle $\phi$, which reacts predominately with the
species of interest (Deuterium) and the products of the reaction are measured
under a specified angle $\theta$. Given the total number of impinging ions Ni,
the energy dependent cross-section of the reaction $\sigma\left(E\right)$, the
efficency of the detection and the geometry of the set-up $\mu$ the measured
total signal counts $d_{i}$ depend (in the limit of small concentrations)
linearly on the concentration profile $c\left(x\right)$ of the species in the
depth $x$:
$d_{i}=d\left(E_{i}^{0}\right)=\mu\mathrm{N}_{i}\int_{0}^{x\left(E_{i}^{0}\right)}\\!dx\,\sigma\left(E\right)c\left(x\right)=\mu\mathrm{N}_{i}\int_{0}^{x\left(E_{i}^{0}\right)}\\!dx\,\sigma\left(E\left(x,E_{i}^{0}\right)\right)c\left(x\right)+\epsilon_{i},$
(1)
where $\epsilon_{i}\sim{\phantom{a}}N\left(0,\sigma_{i}\right)$ represents
normal distributed noise. Repeated measurements with different initial
energies of 3He provide increasing information about the Deuterium depth
profile. The question addressed in the following is: Given a set of already
measured data $d\left(E_{i}^{0}\right)$ which measurement energy should be
chosen next?
To evaluate Eq. 1 we first need to specify the cross section
$\sigma\left(E\right)$ and the energy $E\left(x\right)$ of the incident
particle on its path through the sample.
### I.1 Cross-Section
The relevant cross-section for the reaction D+${}^{3}\mathrm{He}\rightarrow
p+{}^{4}\mathrm{He}$\+ 18.352 MeV (in standard notation written as
$\mathrm{D}\left({}^{3}\mathrm{He},p\right){}^{4}\mathrm{He}$) has been
(re-)measured recently Alimov05 in the range of 550 keV to 6MeV and the
obtained cross-section values have been given in tabular form. Using the same
method as Alimov05 we added several cross-section measurements at energies
below 690keV and fitted both data sets taking into account also earlier
measurements Moeller80 ; Bosch92
$\sigma\left(E\left[MeV\right]\right)=829.98*\frac{E^{2.83962}\left(0.270713*e^{-2.2158E}+0.0182765\right)}{E^{3.47626}+0.270713*e^{-1.17229E}-0.00123669}\left[mb\right].$
(2)
using the Levenberg-Marquardt algorithm minimizing the $\chi^{2}$-misfit with
the variance set to $d_{i}$.
Figure 2: a) Differential cross-section of the nuclear reaction D(3He,p)4He in
the laboratory system with a reaction energy of Q=18.352 MeV (left). b) On the
right hand side the typical results for an NRA measurement are shown
(simulated data of a tungsten sample with an exponentially decaying D
concentration profile (cf. Eq.4). The uncertainties due to the counting
statistics are usually dominated by the uncertainty of the analysis current.
The cross-section is plotted in Fig.2a and shows a broad maximum around 630
keV and is above 3 MeV nearly constant at 8 mb/sr up to 6 MeV (above that
there are no data available). The reaction energy is very high (Q=18.352 MeV)
and most of the energy is transferred to the resulting proton. This leads to a
very good S/N-ratio of the measurement because other particles can easily be
separated by energy.
### I.2 Energy Loss
The energy loss of the impinging 3He-ion in the sample is determined by the
stopping power $S(E)$ of the sample
$\frac{dE}{dx}=-S(E),$ (3)
which can be solved to get the depth dependent energy $E_{i}(x)$ for different
initial energies ${E}_{i}^{0}$. Parameterizations and tables of $S$ for
different elements are given in Tesmer95 . Since the amount of hydrogen in the
sample is usually well below $1\%$ (with the exception of a very thin surface
layer), the influence of the hydrogen concentration on the stopping power can
be neglected in most cases.
### I.3 Simulation of Mock Data
To simulate mock data for typical accelerator parameters a tungsten sample
$\left(\rho=19.3\mathrm{g/cm}^{3}\right)$ with a (high) surface concentration
of 12$\%$ Deuterium, followed by an exponentially decaying Deuterium
concentration down to a constant background level, described by
$c(x)=a_{0}*\exp\left(-\frac{x}{a_{1}}\right)+a_{2}=0.1\exp\left(-\frac{x}{2.5*10^{18}\frac{\mathrm{at}}{\mathrm{cm}^{2}}}\right)+0.02$
(4)
has been used 111Generally ion-beam analysis methods are sensitive to the
areal density of the species $\left(\mathrm{at/cm^{2}}\right)$ which can be
converted into a depth scale if the density of the material is known. In
tungsten $a_{1}=2.5*10^{18}\frac{\mathrm{at}}{\mathrm{cm}^{2}}$ correspond to
$a_{1}=$395nm.. The corresponding mock data for a set of initial energies
E0={500, 700, 1000, 1300, 1600, 2000, 2500, 3000}keV is shown in Fig. 2b. The
variations in the detected yields display the interplay of the increasing
range of the ions with increasing energy and the reduced cross-section at
higher energies modulated with the decreasing Deuterium concentration at
larger depths. The increase of the signal by raising the initial energy from
2500 keV to 3000 keV is already caused by the constant Deuterium background of
2%. The accelerator time which would be needed to obtain the 8 data points is
around one working day taking into account the necessary interleaved
calibration measurements:The ion bombardment causes an energy and depth
dependent loss of Deuterium. Commonly a first order correction is applied by
normalizing the yields with respect to the yields obtained from repeated
calibration measurements using the same (typically low, e.g. 690keV) initial
energy222Remark: This approach of taking into account the Deuterium loss,
although in widespread use, will almost always introduce a systematic bias in
the derived concentration profile: The loss of Deuterium near the surface is
used to correct the signal resulting from the overall Deuterium concentration
profile, where the losses are usually different..
The uncertainty of the detector is given by Poisson-statistics. However,
fluctuations in the beam current measurements are very often the dominating
factor, affecting the pre-factor $\mathrm{N}_{i}$ in Eq.1. An accuracy of up
to 3$\%$ can be achieved (e.g. by using the number of Rutherford-scattered 3He
ions on a thin gold-coating on top of the sample as reference). The error of
the renormalization procedure is harder to quantify. For simplicity we will
use $\sigma_{i}=\mathrm{max}\left(5\%d_{i},\sqrt{d_{i}}\right)$ as uncertainty
of the data in the following, acknowledging that there is room forimprovement.
## II Bayesian Experimental Design
Bayesian Experimental Design (BED) offers the tempting possibility to actively
select (and optimize) the experimental parameters for the next measurement(s)
based on objective criteria. Especially if measurements are expensive or time
consuming (like in the case of energy changes of an accelerator) it is a huge
advantage to know where to look next, so as to learn as much as possible. The
problem of experimental design has already been studied by Lindley back in
1956 Lindley56 in a Bayesian setting and Fedorov published his influential
book in 1972 Fedorov72 \- but the limitations in computational power limited
the application of experimental design almost always to simple (linear)
problems. This situation changed in the recent years and consequently there is
a renewed interest to apply BED also to (non-linear) real-world problems (see
e.g. Loredo03 and references therein or e.g. Preuss08 ; Dreier07 ; Fischer04
; Caticha00 . Not surprising the interest is biggest in branches of physics
where the experimental possibilities are severely restricted: Astronomy,
Fusion research,… Given the excellent account of BED in Loredo03 we only
summarize the key principles: In a first step an appropriate utility function
U has to be agreed upon: It describes the value which we assign to the
measurement results of an experiment and may include parameters like costs of
an experiment, duration, parameter uncertainty etc. Several utility functions
are considered in Chaloner95 . With focus on parameter estimation it was
proposed Lindley56 ; Bernardo79 to use the Kullback-Leibler divergence (KLd)
between the posterior and the prior distributions as utility function. The KLd
for a new datum D is given by
$U_{KL}\left(D,\underline{d},\eta\right)=\int\\!d\underline{\alpha}\;p\left(\underline{\alpha}|D,\underline{d},\eta\right)\log\left[\frac{p\left(\underline{\alpha}|D,\underline{d},\eta\right)}{p\left(\underline{\alpha}|\underline{d},\eta\right)}\right].$
(5)
Next we try to identify the action $\eta$ which maximizes the expected
utility. ’Expected’ utility because we have to account for the prediction
uncertainty for $D$. To compute the expected utility (EU) we have to average
over the new datum D weighted by the marginal likelihood for the new datum
given the observation of the old data $\underline{d}$
$\displaystyle EU\left(\underline{d},\eta\right)$ $\displaystyle=$
$\displaystyle\int\\!dD\;p\left(D|\underline{d},\eta\right)\cdot
U_{KL}\left(D,\underline{d},\eta\right)$ (6) $\displaystyle=$
$\displaystyle\int\\!dD\;p\left(D|\underline{d},\eta\right)\int\\!d\underline{\alpha}\;p\left(\underline{\alpha}|D,\underline{d},\eta\right)\log\left[\frac{p\left(\underline{\alpha}|D,\underline{d},\eta\right)}{p\left(\underline{\alpha}|\underline{d},\eta\right)}\right]$
$\displaystyle=$
$\displaystyle\int\\!dD\;p\left(D|\underline{d},\eta\right)\int\\!d\underline{\alpha}\;\frac{p\left(D|\underline{\alpha},\underline{d},\eta\right)p\left(\underline{\alpha}|\underline{d},\eta\right)}{p\left(D|\underline{d},\eta\right)}\log\left[\frac{p\left(D|\alpha,\underline{d},\eta\right)p\left(\alpha|\underline{d},\eta\right)}{p\left(\underline{\alpha}|\underline{d},\eta\right)p\left(D|\underline{d},\eta\right)}\right]$
$\displaystyle=$
$\displaystyle\int\\!dD\;\int\\!d\underline{\alpha}\;p\left(D|\underline{\alpha},\underline{d},\eta\right)p\left(\underline{\alpha}|\underline{d},\eta\right)\log\left[\frac{p\left(D|\underline{\alpha},\underline{d},\eta\right)}{p\left(D|\underline{d},\eta\right)}\right]$
$\displaystyle=$
$\displaystyle\int\\!dD\;\int\\!d\underline{\alpha}\;p\left(D|\underline{\alpha},\underline{d},\eta\right)p\left(\underline{\alpha}|\underline{d},\right)\log\left[\frac{p\left(D|\underline{\alpha},\underline{d},\eta\right)}{\int\\!d\underline{\alpha}\;p\left(D|\underline{\alpha},\underline{d},\eta\right)p\left(\underline{\alpha}|\underline{d}\right)}\right]$
where we dropped the $\eta-$dependence of the posterior of
$\underline{\alpha}$ in the last line, since our knowledge about
$\underline{\alpha}$ is not influenced by a possible future action. Closer
inspection of Eq. 6 reveals that only two different probability distributions
are required to compute the expected utility: the posterior distribution of
$\underline{\alpha}$ given the old data $\underline{d}$,
$p\left(\underline{\alpha}|\underline{d}\right)$ and the likelihood of the new
datum $D$ based on the previous measurements,
$p\left(D|\underline{\alpha},\underline{d},\eta\right)$.
### II.1 The Linear Design
Assuming that the concentration profile $c(x)$ depends linearly on the
concentrations $c_{i}\left(x_{i}\right),i=1..q$ at a given set of $q$ support
points $\underline{x}$ then Eq.1 can be recast in the following form
$\underline{d}=\underline{f}+\underline{\epsilon}=\underline{\underline{M}}\,\underline{c}+\underline{\epsilon},$
(7)
where the data vector $\underline{d}$ is of size $p$, the matrix
$\underline{\underline{M}}$ is a $p\mathrm{x}q-$matrix and the parameter-
vector $\underline{c}$ has $q$ components. However, the requirement of
linearity applies only to the concentration parameter vector
${\underline{c}}$, the functional form of the concentration may be much more
complex, e.g. $c(x)=c_{1}*\left(x-x_{3}\right)^{4}+c_{2}*\sqrt{|x-x_{1}|}$,
although almost always $c(x)$ is chosen to be constant between the different
support points: $c(x)=c_{i},\forall x\in\left[x_{i},x_{i+1}\right]$ or as
linear interpolation between the support points. The noise vector
$\underline{\epsilon}$ is normally distributed
$\underline{\epsilon}\sim{\phantom{a}}N\left(0,\underline{\underline{\Sigma^{-1}}}\right)$,
where $\underline{\underline{\Sigma}}$ is a diagonal matrix with the entries
$\underline{\underline{\Sigma}}_{ii}=1/\sigma_{i}^{2}$. Every row of
$\underline{\underline{M}}_{j}$ is given by the solution of Eq. 1 for a
specified initial energy $E_{j}^{0}$,
${\underline{m}\left(E_{j}^{0}\right)}^{T}$. The consideration of the
uncertainties in the entries of the matrix due to energy straggling of the
impinging particles is beyond the scope of the present paper, but see e.g.
Mayer08 .
With a Gaussian likelihood for the existing data and a flat prior for the
parameters the posterior of the concentration vector reads
$\displaystyle p\left(\underline{c}|\underline{d},\eta\right)\propto
p\left(\underline{d}|\underline{c},\eta\right)$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\exp\left(-\frac{1}{2}\left(\underline{d}-\underline{\underline{M}}\,\underline{c}\right)^{T}\underline{\underline{\Sigma}}\left(\underline{d}-\underline{\underline{M}}\,\underline{c}\right)\right)$
(8) $\displaystyle=$
$\displaystyle\sqrt{\frac{\det{\underline{\underline{A}}}}{\left(2\pi\right)^{q}}}\exp\left(-\frac{1}{2}\left(\underline{c}-\underline{c_{0}}\right)^{T}\underline{\underline{A}}\left(\underline{c}-\underline{c_{0}}\right)\right)$
with
$\underline{\underline{A}}={\underline{\underline{M}}}^{T}\underline{\underline{\Sigma}}\,\underline{\underline{M}}\;\;\;\;\mathrm{and}\;\;\;\;\underline{c_{0}}={\underline{\underline{A}}}^{-1}{\underline{\underline{M}}}^{T}\underline{\underline{\Sigma}}\,\underline{d}.$
(9)
The posterior distribution of $\underline{c}$ including the new data point $D$
with its uncertainty $\sigma$,
$p\left(\underline{c}|D,\underline{d},\eta\right)$ can similarly be cast in a
Gaussian form. Therefore Eq. 6 can be solved analytically Fedorov72 and
yields a simple closed form for the exponential utility Dreier07 ; Preuss08 :
$\mathrm{EU}\left(\underline{d},\eta\right)=\frac{1}{2}\left(\log\left(1+\mathrm{G}\right)-r\right)$
(10)
with
$G=\frac{\underline{m}\left(\eta\right)^{T}\underline{\underline{A}}^{-1}\underline{m}\left(\eta\right)}{\sigma^{2}}.$
(11)
If $p\left(D|\underline{c},\eta\right)$ is Gaussian then $r=0$. The variation
of the EU depends on the vector $\underline{m}\left(\eta\right)$ which in turn
is uniquely determined by the choice of the initial energy $E_{p+1}^{0}$. The
optimum (maximum of the EU) can be found by a simple 1-D scan of the energy.
The sequential design approach in action is displayed in Fig 3. Starting from
the surface the concentration at increasingly larger depth intervals is of
interest. For this example the chosen depths are 0 nm, 80 nm, 240 nm, 470 nm
and 950 nm. After initial measurements at 400 keV, 700 keV and 3000 keV
(representing the lower and upper limit of the useful energy range for the
measurements and one calibration measurement) the best energy for the next
measurement has to be determined. The EU for this first cycle has a maximum at
1250keV (solid line). After a measurement with this energy the EU for the next
measurement has its maximum at 960keV and about twice the EU than before.
This, on the first glance, surprising increase of the EU can be made
transparent: With 5 unknowns and 5 (informative) measurements the solution
space of this linear problem no longer covers a sub-manifold of the parameter
space: It ’collapses’ and the volume of the ’occupied’ parameter space starts
to be determined by the measurement uncertainties. Therefore the 5-th
measurement has a very high EU. In the following cycle(s) the amplitude of the
EU is much lower since the subsequent measurements now gradually shrink the
’volume’ of the parameter posterior distribution. As long as the EU is above
the intended threshold for new measurements (which depends on the addressed
physical problem) further measurements are indicated.
How much better is the BED derived experiment compared to an experiment with
the same number but equidistant chosen initial energies? The entropy of the
parameter posterior distribution would be the obvious quantity to compare.
However, for the time being, many scientists are not happy with this measure
and prefer a more familiar measure, e.g. the condition number. The condition
number of the (pseudo-)inverse of $\underline{\underline{M}}$ is often used to
characterize linear least squares problems numrecipes and is a measure how
strongly uncertainties in the data vector $\underline{d}$ may be amplified by
multiplication with the pseudo-inverse matrix. Using this measure the BED
optimized setting surpasses the equidistant experiment by a factor of more
than 100 (!).
Figure 3: Expected Utility for subsequent measurements. All measurements are
performed with the initial energy $E^{0}$ suggested by the maximum of the EU
in the corresponding cycle (indicated by marks on the energy axis).
### II.2 Non-linear Design
The analytical solution in the preceeding case was possible because several
approximations have been applied: The integration range of the integration
over the predicted datum (a positive quantity) had to be changed from
$\int_{0}^{\infty}\\!dD$ to $\int_{-\infty}^{\infty}\\!dD$. Given the actual
number of counts and the uncertainties this can easily be justified.
Unfortunately, a similar change of the integration limits had to be applied
also in the parameter integration (from $\int_{0}^{1}\\!d\underline{c}$ to
$\int_{-\infty}^{+\infty}\\!d\underline{c}$) and here it definitely affects
the results. The analysis could be repeated substituting the analytical
integration by the numerical counterparts (e.g. using codes like VEGAS
numrecipes or MCMC approaches). Furthermore, the uncertainty of the predicted
datum D is not constant but proportional to the signal $\sigma_{D}\propto D$
and therefore also the integrations over the data space have to be done
numerically. Under those circumstances there is no difference in the
computation to a non-linear experimental design problem. Additionally it
turned out that the actual quantity of interest is the decay length of the
hydrogen depth profile and that quite accurate data for the surface hydrogen
concentration are available (additionally measuring the ${}^{4}\mathrm{He}$ of
the $\mathrm{D}\left({}^{3}\mathrm{He},p\right){}^{4}\mathrm{He}$ reaction).
Therefore the optimal energy settings for the estimation of the parameters
$a_{1}$ and $a_{2}$ of concentration profiles of the functional form of Eq. 4
have to be computed. However, in non-linear experimental design the measured
data influences the EU (in contrast to the linear case: the maximum of the EU
is independent of the actually measured data, cf. Eq. 10) and this poses a
practical problem: The next accelerator energy has to be determined after the
previous measurement. And longer computation times to optimize the EU, causing
delays, are not tolerable.
Here the posterior sampling approach, suggested in Loredo03 , proved very
valuable. It turned out that sets of posterior samples
$p\left(a_{1i},a_{2i}|\underline{d},\eta\right),i=1..N$ drawn from
$p\left(a_{1},a_{2}|\underline{d},\eta\right)$ could be generated quite
efficiently (partly due to the low dimensionality of the parameter space).
With that sample (typically of size 1000) the denominator of the logarithm in
Eq. 6 is given by a simple summation
$\int\\!d\underline{\alpha}\,p\left(D|\underline{\alpha},\underline{d},\eta\right)p\left(\underline{\alpha}|\underline{d}\right)\approx\sum_{i=1}^{N}p\left(D|\underline{\alpha_{i}},\underline{d},\eta\right).$
(12)
The biggest saving comes from the fact that the posterior sample is
independent from the actual value of $D$ and of the design action $\eta$: all
computations are reduced to repeated evaluations of the likelihood, which can
efficiently be vectorized. Finding the best energy is a matter of less than 5
minutes(!) on contemporary hardware (Linux-PC, 2GHz).
In Fig. 4 three cycles of the non-linear BED are shown: After a first
measurement at 500keV the posterior distribution of
$\left\\{a_{1},a_{2}\right\\}$ is visualized in the upper left graph by the
posterior sample. The single measurement does not allow to distinguish between
a large decay constant $a_{1}$ and low constant offset $a_{2}$ or vice versa.
The EU, plotted in the upper right graph, favors now a measurement at the
other end of the energy range (the maximum of the utility function is
encircled). After a measurement with 3MeV ${}^{3}\mathrm{He}$ the ’area’ of
the posterior distribution is significantly reduced (middle row, left graph):
The background concentration is below 3% but the decay length is still quite
undetermined. The EU has a maximum at 1500 keV, still with a pretty high EU.
Performing a measurement with 1500keV localizes the posterior distribution
around the true (but unknown value of $a_{1}=395$nm and $a_{2}=0.02$). The
next measurement should be performed at 1200keV but the EU is significantly
lower than before: subsequent measurements are predominantly improving the
statistics: a second measurement at 3 MeV provides nearly the same
information.
Figure 4: Three cycles of the Experimental Design process: On the left hand
side 1000 samples drawn from the posterior distribution
$p\left(a_{1},a_{2}|\underline{d},\eta\right)$ are displayed. On the right
hand side the EU is plotted and the maximum is indicated by a circle. The
corresponding abscissa value is the suggested next measurement energy.
Performingthat measurement yields the posterior distribution given in the next
row.
## III Conclusion and Outlook
The concept of Bayesian Experimental Design allows to objectively optimize
experimental designs. Here we presented two different approaches to optimize
NRA depth profiling: First in a linear setting, allowing an analytical
solution and straightforward parametric studies. Second, a time-critical non-
linear experimental design problem which could be tackled using posterior
sampling. Both optimization procedures may considerably increase the accuracy
of the derived depth profiles compared to the present approach and at the same
time reduce the overall measurement time by signaling a diminishing utility of
further measurements. With the posterior sampling approach many sequential
measurements can now be optimized on the fly: This opens up the door for a
wealth of new applications of BED in the field of ion beam
analysisToussaint2000 as well as in other physical areas Loredo03 ; Preuss08
; Toussaint2006
## References
* (1) ITER Organization, http://www.iter.org, (2008).
* (2) U. von Toussaint and R. Fischer and K. Krieger and V. Dose, Depth Profile Determination with Confidence Intervals from Rutherford Backscattering Data, New Journal of Physics 1, 11 (1999).
* (3) V. Kh. Alimov and M. Mayer and J. Roth, Differential cross-section of the D$\left({}^{3}\mathrm{He},p\right){}^{4}\mathrm{He}$ nuclear reaction and depth profiling of deuterium up to large depths, Nucl. Instr. Meth. B 234, 169-175 (2005).
* (4) W. Möller and F. Besenbacher, A note on the 3He+D nuclear reaction cross section, Nucl. Instr. and Meth. 168(1), 111-114 (1980).
* (5) H.-S. Bosch and G. M. Hale, Improved formulas for fusion cross-sections and thermal reactivities, Nucl. Fusion 32, 611-632 (1992).
* (6) V. Kh. Alimov et al, Deuterium retention in tungsten exposed to low-energy, high-flux clean and carbon-seeded deuterium plasmas, J. Nucl. Mat. 375, 192-201 (2008).
* (7) J. R. Tesmer and M. Nastasi and J.C. Barbour and C. J. Maggiore and J. W. Mayer, Handbook of Modern Ion Beam Analysis, Materials Research Society, Pittsburgh, PA, USA (1995).
* (8) M. Clyde and P. Müller and G. Parmigiani, Exploring expected utility surfaces by markov chains, Source: http://ftp.stat.duke.edu/WorkingPapers/95-39.ps (1995).
* (9) D. MacKay, Information-based objective functions for active data selection, Neural Computation 4(4), 590-604 (1992).
* (10) D. V. Lindley, On the measure of information provided by an experiment, Ann. Stat 27, 986-1005 (1956).
* (11) V. V. Fedorov, Theory of Optimal Experiments, Academic, New York (1972).
* (12) T. J. Loredo, ’Bayesian Adaptive Exploration’ in Bayesian Inference and Maximum Entropy Methods in Science and Engineering, edited by G. Erickson and Y. Zhai, AIP, Melville, NY, vol. Conf. Proc 707, 330-346 (2003).
* (13) R. Fischer, ’Bayesian Experimental Design - Studies for Fusion Diagnostics’ in Bayesian Inference and Maximum Entropy Methods in Science and Engineering, edited by R. Fischer, R. Preuss and U. von Toussaint, AIP, Melville, NY, vol. Conf. Proc 735, 76-83 (2004).
* (14) P. Riegler and N. Caticha, ’MaxEnt queries and sequential sampling’ in Bayesian Inference and Maximum Entropy Methods in Science and Engineering, edited by A. Mohammad-Djafari, AIP, Melville, vol. Conf. Proc 568, 270-279 (2001).
* (15) H. Dreier, Bayesian Experimental Design: Applications in Nuclear Fusion, PhD-thesis, IPP-Report 13/8, Max-Planck-Institut für Plasmaphysik (2007).
* (16) R. Preuss and H. Dreier and A. Dinklage and V. Dose, Data adaptive control parameter estimation for scaling laws for magnetic fusion devices, EPL 81(5), 55001 (2008).
* (17) K. Chaloner and I. Verdinelli, Bayesian experimental design: A review, Stat. Sci. 10, 273-304 (1995).
* (18) J. M. Bernardo, Expected Information as Expected Utility, Ann. Stat. 7, 686-690 (1979).
* (19) M. Mayer, E. Gauthier, K. Sugiyama, and U. von Toussaint, Quantitative depth profiling of Deuterium up to very large depths, to be submitted.
* (20) W. H. Press and S. A. Teukolsky and W. T. Vetterling and B. P. Flannery, Numerical Recipes in Fortran 77, Oxford Science Publications, Cambridge University Press, 2nd edition (1992).
* (21) U. von Toussaint and V. Dose, Bayesian Analysis in surface physics, Applied Physics A 82, 403-413 (2006).
|
arxiv-papers
| 2008-12-19T13:53:15 |
2024-09-04T02:48:59.498068
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "U. von Toussaint, T. Schwarz-Selinger, S. Gori",
"submitter": "Udo V. Toussaint",
"url": "https://arxiv.org/abs/0812.3789"
}
|
0812.3897
|
# Origin of the Ising Ferrimagnetism and Spin-Charge Coupling in LuFe2O4
H. J. Xiang National Renewable Energy Laboratory, Golden, Colorado 80401, USA
E. J. Kan Department of Chemistry, North Carolina State University, Raleigh,
North Carolina 27695-8204, USA Su-Huai Wei National Renewable Energy
Laboratory, Golden, Colorado 80401, USA M.-H. Whangbo Department of
Chemistry, North Carolina State University, Raleigh, North Carolina
27695-8204, USA Jinlong Yang Hefei National Laboratory for Physical Sciences
at Microscale, University of Science and Technology of China, Hefei, Anhui
230026, P. R. China
###### Abstract
The spin ordering and spin-charge coupling in LuFe2O4 were investigated on the
basis of density functional calculations and Monte Carlo simulations. The 2:1
ferrimagnetism arises from the strong antiferromagnetic intra-sheet Fe3+-Fe3+
and Fe3+-Fe2+ as well as some substantial antiferromagnetic Fe2+-Fe3+ inter-
sheet spin exchange interactions. The giant magnetocapacitance at room
temperature and the enhanced electric polarization at 240 K of LuFe2O4 are
explained by the strong spin-charge coupling.
###### pacs:
75.80.+q,71.20.-b,77.80.-e,64.60.De
Recently, multiferroics Kimura2003 ; Hur2004 ; Ikeda2005 ; Xu2008 ;
Subramanian2006 ; Zhang2008 ; Xiang2007A ; Xiang2007B ; Xiang2008 ; Angst2008
have attracted much attention because of their potential applications in novel
magnetoelectric and magneto-optical devices. Among the newly discovered
multiferroics, LuFe2O4 is particularly interesting due to its large
ferroelectric (FE) polarization Ikeda2005 and giant magnetocapacitance at
room temperature Subramanian2006 . In the high-temperature crystal structure
of LuFe2O4 with space group R3̄m, layers of composition Fe2O4 alternate with
layers of Lu3+ ions, such that there are three Fe2O4 layers per unit cell.
Each Fe2O4 layer is made up of two triangular sheets (hereafter, T-sheets) of
corner-sharing FeO5 trigonal bipyramids (Fig. 1). Below 320 K ($T_{CO}$)
LuFe2O4 undergoes a three-dimensional (3D) charge ordering (CO) (2Fe2.5+
$\Rightarrow$ Fe2+ \+ Fe3+) with the $\sqrt{3}\times\sqrt{3}$ superstructure
in each T-sheet; in each Fe2O4 layer, one T-sheet has the honeycomb network of
Fe2+ ions with a Fe3+ ion at the center of each Fe2+ hexagon (hereafter, the
type A T-sheet), while the other T-sheet has an opposite arrangement of the
Fe2+ and Fe3+ ions (hereafter the type B T-sheet).
LuFe2O4, with the novel CO-driven “electronic ferroelectricity”, Ikeda2005
presents several fundamental questions. First, LuFe2O4 shows strong Ising
behavior with the easy axis along $c$ Iida1993 ; Wu2008 . The spin anisotropy
of the non-CO state is understandable because the spin down electron of the
Fe2.5+ ion partially occupies the degenerate ($d_{x^{2}-y^{2}}$,$d_{xy}$)
orbitals Xiang2007A ; Dai2005 . However, the Ising behavior below $T_{CO}$ is
puzzling because the insulating $\sqrt{3}\times\sqrt{3}$ CO breaks the 3-fold
rotational symmetry hence lifting the degeneracy of the
($d_{x^{2}-y^{2}}$,$d_{xy}$) orbitals Xiang2007A . Second, LuFe2O4 undergoes a
ferrimagnetic spin ordering below 240 K ($T_{N}$) Iida1993 ; Tanaka1989 ;
Siratori1992 ; Christianson2008 . A number of experimental studies found this
spin ordering to be two-dimensional (2D) in nature Iida1993 ; Tanaka1989 ;
Funagashi1984 . In contrast, a recent neutron diffraction study observed a
finite spin correlation along $c$ and suggested a 3D spin structure without
considering CO Christianson2008 . The Mössbauer Tanaka1989 and neutron
diffraction Siratori1992 studies led to a detailed ferrimagnetic structure of
LuFe2O4, in which the majority spin lattice consists of all Fe2+ ions plus
one-third of the total Fe3+ ions while the minority spin sublattice consists
of the remaining Fe3+ ions. This 2:1 ferrimagnetic order was suggested to
originate from weak ferromagnetic (FM) interactions between the next-nearest
neighbor (NNN) Fe sites in the triangular antiferromagnetic (AFM) Ising
lattice Iida1993 . However, using the spin exchange parameters estimated from
the energy parameters of LaFeO3, Naka et al. Naka2008 predicted quite a
different spin structure that includes some Fe sites without unique spin
direction. Therefore, the detailed ferrimagnetic structure and its origin
remain unclear. Third, LuFe2O4 exhibits a giant magnetodielectric response at
room temperature Subramanian2006 , and a room-temperature dynamic
magnetoelectric coupling was also reported Park2007 . Furthermore, the FE
polarization of LuFe2O4 was found to increase around $T_{N}$ Ikeda2005 . These
observations suggest the occurrence of coupling between the CO and magnetism.
The understanding of the spin-charge coupling is crucial for future
magnetodielectric applications of LuFe2O4.
In this Letter, we explore these isuues on the basis of first principles
density functional calculations for the first time. A large spin anisotropy is
found along the $c$ direction due mainly to the Fe2+ ions of the B-sheet, the
spin ground state of the $\sqrt{3}\times\sqrt{3}$ CO state has the 2:1
ferrimagnetic spin arrangement proposed by Siratori et al. Siratori1992 , and
there occurs strong spin-charge coupling in LuFe2O4.
Our density functional theory calculations employed the frozen-core projector
augmented wave method PAW encoded in the Vienna ab initio simulation package
VASP , and the generalized-gradient approximation (GGA) Perdew1996 . To
properly describe the strong electron correlation in the 3d transition-metal
oxide, the GGA plus on-site repulsion U method (GGA+U) Liechtenstein1995 was
employed with the effective $U$ value ($U_{eff}=U-J$ with $J=0$) of 4.61 eV
Xiang2007A . It is known experimentally Iida1993 ; Tanaka1989 ; Funagashi1984
that the interlayer magnetic interactions in LuFe2O4 are weak, which is
understandable due to its layered structure. In this work, therefore, we focus
on the 2D spin ordering within a single Fe2O4 layer. For the
$\sqrt{3}\times\sqrt{3}$ CO state of LuFe2O4, the FE ordering of the Fe2O4
layers will be assumed.
We first examine the magnetic anisotropy of the Fe ions by performing GGA+U
calculations, with spin-orbit coupling (SOC) included, for the FM state of
LuFe2O4 with the $\sqrt{3}\times\sqrt{3}$ CO. As shown in Fig. 1(a), there are
two kinds of Fe2+ ions and two kinds of Fe3+ ions in the
$\sqrt{3}\times\sqrt{3}$ CO state. We label the Fe2+ and Fe3+ ions of the type
A T-sheet as 2A and 3A, respectively, and those of the type B T-sheet as 2B
and 3B, respectively. In our GGA+U+SOC calculations with spins pointing along
several different directions, all Fe2+ and Fe3+ spins are kept in the same
direction. Our calculations show that the easy axis is along the $c$
direction, as experimentally observed Iida1993 ; Wu2008 ; the
$\parallel$c-spin orientation is more stable than the $\perp$c-spin
orientation by 1.5 meV per formula unit (FU). The orbital moments of 2A, 2B,
3A, and 3B for the $\parallel$c-spin orientation are 0.101, 0.156, 0.031 and
0.035, respectively, which are greater than those for the $\perp$c-spin
orientation by 0.019, 0.062, 0.015, and 0.018 $\mu_{B}$, respectively. As
expected, the Fe3+ ($d^{5}$) ions have a very small anisotropy, However, two
kinds of the Fe2+ ions also have different degree of spin anisotropy. The spin
down electron of the 2B Fe2+ ion occupies the ($d_{x^{2}-y^{2}}$,$d_{xy}$)
manifold Xiang2007A , therefore the 2B Fe2+ ion has the largest spin
anisotropy along $c$. Our calculations indicate a non-negligible orbital
contribution to the total magnetization, in agreement with the X-ray magnetic
circular dichroism result Wu2008 .
To determine the magnetic ground state of LuFe2O4 in the
$\sqrt{3}\times\sqrt{3}$ CO state, we extract its spin exchange parameters by
mapping the energy differences between ordered spin states obtained from GGA+U
calculations onto the corresponding energy differences obtained from the Ising
Hamiltonian whangbo2003 :
$H=\sum_{i,j}J_{ij}S_{iz}S_{jz},$ (1)
where the energy is expressed with respect to the spin disorder (paramagnetic)
state, $J_{ij}$ is the spin exchange parameter between the spin sites $i$ and
$j$, and $S_{iz}$ is the spin component along the $c$ direction ($|S_{z}|=2$
and $2.5$ for Fe2+ and Fe3+ ions, respectively). We consider all 15 possible
superexchange (SE) interactions and all 19 super-superexchange (SSE)
interactions with the O…O distance less than 3.2 Å. The intra- and inter-sheet
interactions within each Fe2O4 layer as well as the SSE interactions between
adjacent Fe2O4 layers are taken into account. To evaluate these 34 spin
exchange parameters reliably, we considered 111 different ordered spin states
leading to 110 energy differences. The 34 spin exchange parameters were
determined by performing a linear least-square fitting analysis. The SSE
interactions are generally much weaker than the SE interactions with the
magnitude of all SSE interactions less than 1.4 meV. The calculated SE
parameters are reported in Table 1. All intra-sheet SE interactions are AFM,
and the strongest interactions ($\sim 7.3$ meV) occurs between the 3B Fe3+
ions because of the large energy gain of the AFM configuration and almost zero
FM coupling. The inter-sheet SE interactions are weaker than the the intra-
sheet SE interactions, and are mostly AFM.
With the calculated spin exchange parameters, one can identify the spin ground
state of the CO state. The Metropolis Monte Carlo simulation of the Ising
model is performed to search for the ground state. Simulations with supercells
of several different sizes show that the spin ground state has the magnetic
structure shown in Fig. 2(a), which has the same cell as the
$\sqrt{3}\times\sqrt{3}$ CO structure. In this state, all Fe2+ ions contribute
to the majority spin, and the Fe3+ ions are antiferromagnetically coupled to
the Fe2+ ions in the type A T-sheet. In the honeycomb lattice of the type B
T-sheet, the Fe3+ spins are antiferromagnetically coupled. Thus, the spin
ground state is ferrimagnetic, as experimentally observed Iida1993 . This 2:1
ferrimagnetic structure is the same as the magnetic structure proposed by
Siratori et al. Siratori1992 , and differs from the structure proposed by Naka
et al. Naka2008 .
The observed ferrimagnetic ordering can be readily explained in terms of the
calculated exchange parameters. In the honeycomb network of the type B
T-sheet, the nearest-neighbor (NN) 3B ions are antiferromagnetically coupled
since their SE interaction is strongly AFM. In the type A T-sheet, the SE
interactions between the 2A ions are AFM, and so are those between the 2A and
3A ions, which leads to spin frustration. As a consequence, two possible spin
arrangements compete with each other in the type A T-sheet; the first is the
state in which the coupling between the NN 2A ions are AFM with the spin
direction of the 3A ion undetermined, and the second is the state in which all
2A ions are antiferromagnetically coupled to the 3A ions. The energies of
these two states (considering only the SE interaction) are
$E_{1}=-4(J_{2A1,2A2}+J_{2A1,2A4})$ per 3A ion, and
$E_{2}=-10(J_{3A1,2A1}+J_{3A1,2A2}+J_{3A1,2A3})+4(J_{2A1,2A2}+J_{2A1,2A4})$
per 3A ion, respectively. Due to the relatively strong AFM interactions
between the 3A and 2A ions (See Table 1) and the large spin of the 3A ions,
the second state has a lower energy, i.e., $E_{2}$ $<$ $E_{1}$. Without loss
of generality, we can assume the 2A (3A) ions constitute the majority
(minority) spin in the second state. Now, we examine the spin orientation of
the Fe2+ ions in the type B T-sheet. The intra-sheet interactions of the 2B
ion with 3B ions vanish due to the AFM ordering of the 3B ions. As for the
inter-sheet interactions involving the 2B ions, the dominant one is the AFM
interaction of the 2B ion with the 3A ion ($J_{3A1-2B1}$ in Table 1).
Consequently, we obtain the ferrimangetic ground state shown in Fig. 2(a), in
which the spin of the 2B ion contributes to the majority spin of the Fe2O4
layer. For the stability of the ferrimangetic ground state, the inter-sheet
interaction is essential. This was neglected in the model Hamiltonian study of
Naka et al. Naka2008 . The ferrimangetic state is not due to the FM
interactions between NNN Fe ions of the T-sheet because they must be
vanishingly weak and mostly AFM.
The electronic structure of the ferrimangetic state calculated for the
$\sqrt{3}\times\sqrt{3}$ CO structure of LuFe2O4 is shown in Fig. 3. Also
shown is the electronic structure calculated for the FM state. Both states are
semiconducting, and the highest occupied (HO) and the lowest unoccupied (LU)
levels of both states come from the spin-up Fe2+ and Fe3+ ions, respectively
Xiang2007A . In addition, the band dispersion from $\Gamma$ to A is rather
small, indicating a very weak interlayer interaction. However, there are some
important differences. First, the ferrimangetic state has a larger band gap
(1.68 eV) than does the FM state (0.77 eV). This is consistent with the
stability of the ferrimangetic state. Second, the FM state has an indirect
band gap with the HO and LU levels located at K and $\Gamma$, respectively. In
the ferrimangetic state, however, the LU level has the highest energy at
$\Gamma$ and the band dispersions of the HO and LU levels are almost flat from
M to K. This difference comes from the orbital interaction between the spin
down ($d_{x^{2}-y^{2}}$,$d_{xy}$) levels of the spin up Fe3+ and Fe2+ ions.
To probe the presence of spin-charge coupling in LuFe2O4, it is necessary to
consider the spin ordering in a CO state other than the
$\sqrt{3}\times\sqrt{3}$ CO state. The previous electrostatic calculations
Xiang2007A ; Naka2008 showed that the chain CO, in which one-dimensional (1D)
chains of Fe2+ ions alternate with 1D chains of Fe3+ ions in each T-sheet
[Fig. 2(b)], is only slightly less stable than the $\sqrt{3}\times\sqrt{3}$
CO, and has no FE polarization. We extract exchange parameters by mapping
analysis as described above. It is found that the intra-sheet SE between the
Fe3+ ions is the strongest ($J=6.7$ meV) as in the $\sqrt{3}\times\sqrt{3}$ CO
case. All intra-sheet SE’s are AFM with $J$(Fe3+-Fe3+) $>$ $J$(Fe2+-Fe3+) $>$
$J$(Fe2+-Fe2+). The inter-sheet SE between the Fe3+ ions is very weak
($|J|<0.3$ meV), and that between the Fe2+ and Fe3+ ions is FM with $J=-1.4$
meV. Interestingly, the inter-sheet SE between the Fe2+ ions is rather
strongly AFM ($J=6.3$ meV). Monte Carlo simulations using these spin exchange
parameters indicate that the spin state shown in Fig. 2(b) is the spin ground
state. In this spin ordering, the spins within each chain of Fe2+ ions or Fe3+
ions are antiferromagnetically coupled. The NN chains of Fe2+ ions belonging
to different T-sheets are coupled antiferromagnetically, whereas the
corresponding chains of Fe3+ are almost decoupled.
The above results show that the spin ordering of the chain CO state is
dramatically different from that of the $\sqrt{3}\times\sqrt{3}$ CO state. The
most important difference is that the total spin moments are 2.33 $\mu_{B}$/FU
for the $\sqrt{3}\times\sqrt{3}$ CO, but 0 $\mu_{B}$/FU for the chain CO. This
evidences a strong spin-charge coupling in LuFe2O4. The external magnetic
field will have different effects on the two CO states due to the the Zeeman
effect. It is expected that the magnetic field will further stabilize the
ferrimagnetic $\sqrt{3}\times\sqrt{3}$ CO state. Consequently, an external
magnetic field will reduce the extent of charge fluctuation and hence decrease
the dielectric constant. This supports our explanation for the giant
magnetocapacitance effect of LuFe2O4 at room temperature Xiang2007A .
Without considering the inter-sheet interactions, Naka et al. Naka2008
suggested that the degeneracy of the spin ground state of the
$\sqrt{3}\times\sqrt{3}$ CO state is of the order O($2^{N/3}$)( N is the
number of the spin sites), which is much larger than the spin degeneracy
[O($2^{\sqrt{N}}$)] of the chain CO state. Thus, they proposed that spin
frustration induces reinforcement of the polar $\sqrt{3}\times\sqrt{3}$ CO by
a gain of spin entropy. However, our calculations show that there are
substantial inter-sheet spin exchange interactions between the 2B1 and 3A1
ions, which would remove the macroscopic degeneracy of the spin ground state
of the $\sqrt{3}\times\sqrt{3}$ CO state. The macroscopic degeneracy still
persists for the chain CO state. Thus, our work provides a picture opposite to
what Naka et al. proposed. Furthermore, we find that the
$\sqrt{3}\times\sqrt{3}$ CO state is more favorable for the spin ordering than
is the chain CO state; with respect to the paramagnetic state, the spin ground
state is lower in energy by $-78$ meV/FU for the $\sqrt{3}\times\sqrt{3}$ CO,
but by $-57$ meV/FU for the chain CO. The model of Naka et al. Naka2008
predicts that the polar $\sqrt{3}\times\sqrt{3}$ CO state is destabilized and
the electric polarization is reduced by the magnetic field, since it will lift
the macroscopic spin degeneracy. In contrast, our work predicts that the
magnetic field stabilizes the ferrimagnetic $\sqrt{3}\times\sqrt{3}$ CO state
due to the Zeeman effect, and provides an explanation for why the electric
polarization increases when the temperature is lowered below the Neel
temperature Ikeda2005 , because the charge fluctuation has an onset well below
$T_{CO}$ Xu2008 .
In summary, our first principles results explain the experimentally observed
Ising ferrimagnetism, and manifest the spin-charge coupling and
magnetoelectric effect in LuFe2O4.
Work at NREL was supported by the U.S. Department of Energy, under Contract
No. DE-AC36-08GO28308, and work at NCSU by the U. S. Department of Energy,
under Grant DE-FG02-86ER45259.
## References
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* (3) N. Ikeda, H. Ohsumi, K. Ohwada, K. Ishii, T. Inami, K. Kakurai, Y. Murakami, K. Yoshii, S. Mori, Y. Horibe, and H. Kitô, Nature (London) 436, 1136 (2005).
* (4) M. A. Subramanian, T. He, J. Chen, N. S. Rogado, T. G. Calvarese, and A. W. Sleight, Adv. Mater. 18, 1737 (2006).
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* (8) X. S. Xu, M. Angst, T. V. Brinzari, R. P. Hermann, J. L. Musfeldt, A. D. Christianson, D. Mandrus, B. C. Sales, S. McGill, J.-W. Kim, and Z. Islam, Phys. Rev. Lett. 101, 227602 (2008).
* (9) H. J. Xiang and M.-H. Whangbo, Phys. Rev. Lett. 99, 257203 (2007).
* (10) H. J. Xiang, S.-H. Wei, M.-H. Whangbo, and J. L. F. Da Silva, Phys. Rev. Lett. 101, 037209 (2008).
* (11) J. Iida, M. Tanaka, Y. Nakagawa, S. Funahashi, N. Kimizuka, and S. Takekawa, J. Phys. Soc. Jpn. 62, 1723 (1993).
* (12) W. Wu, V. Kiryukhin, H.-J. Noh, K.-T. Ko, J.-H. Park, W. Ratcliff II, P. A. Sharma, N. Harrison, Y. J. Choi, Y. Horibe, S. Lee, S. Park, H. T. Yi, C. L. Zhang, and S.-W. Cheong, Phys. Rev. Lett. 101, 137203 (2008).
* (13) D. Dai and M.-H. Whangbo, Inorg. Chem. 44, 4407 (2005).
* (14) M. Tanaka, H. Iwasaki, K. Siratori, and I. Shindo, J. Phys. Soc. Jpn. 58, 1433 (1989).
* (15) K. Siratori, S. Funahashi, J. Iida, and M. Tanaka, Proc. 6th Intern. Conf. Ferrites, Tokyo and Kyoto, Japan, 1992, p. 703.
* (16) A. D. Christianson, M. D. Lumsden, M. Angst, Z. Yamani, W. Tian, R. Jin, E. A. Payzant, S. E. Nagler, B. C. Sales, and D. Mandrus, Phys. Rev. Lett. 100, 107601 (2008).
* (17) S. Funahashi, J. Akimitsu, K. Siratori, N. Kimizuka, M. Tanaka, and H. Fujishita, J. Phys. Soc. Jpn. 53, 2688 (1984).
* (18) M. Naka, A. Nagano, and S. Ishihara, Phys. Rev. B 77, 224441 (2008); A. Nagano, M. Naka, J. Nasu, and S. Ishihara, Phys. Rev. Lett. 99, 217202 (2007).
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* (20) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994); G. Kresse and D. Joubert, ibid 59, 1758 (1999).
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* (24) M.-H. Whangbo, H.-J. Koo and D. Dai, J. Solid State Chem. 176, 417 (2003).
Table 1: Calculated superexchange parameters (in meV) in the $\sqrt{3}\times\sqrt{3}$ CO state of LuFe2O4 (For the spin sites of the 2A, 3A, 2B and 3B ions,see Fig. 1 ) A-A | $J_{3A1,2A1}$ | $J_{3A1,2A2}$ | $J_{3A1,2A3}$ | $J_{2A1,2A2}$ | $J_{2A1,2A4}$
---|---|---|---|---|---
| 3.2 | 4.0 | 4.7 | 1.9 | 3.6
B-B | $J_{3B1,3B2}$ | $J_{3B1,3B4}$ | $J_{2B1,3B1}$ | $J_{2B1,3B2}$ | $J_{2B1,3B3}$
| 7.0 | 7.6 | 1.5 | 2.8 | 1.3
A-B | $J_{3A1,3B1}$ | $J_{3A1,2B1}$ | $J_{2A1,2B1}$ | $J_{2A1,3B2}$ | $J_{2A1,3B3}$
| 2.0 | 1.9 | $\sim 0$ | $-0.6$ | 1.2
Figure 1: (Color online) Schematic representation of the
$\sqrt{3}\times\sqrt{3}$ CO structure. Large, medium, and small circles
represent the Fe2+, Fe3+, and O2- ions, respectively. The type A (type B)
T-sheet has the honeycomb network of Fe2+ (Fe3+) ions with a Fe3+ (Fe2+) ion
at the center of each hexagon. 2A and 3A (2B and 3B) refer to the Fe2+ and
Fe3+ ions of the type A (type B) T-sheet, respetively. The region enclosed by
dashed lines indicates the unit cell of the CO structure. There is a mirror
plane of symmetry, which is parallel to the $c$ axis and crosses the 3A1 and
2B1 sites. The inset shows an isolated FeO5 trigonal bipyramid. Figure 2:
(Color online) Schematic representations of (a) the spin ground state of the
$\sqrt{3}\times\sqrt{3}$ CO structure and (b) one of the macroscopic spin
ground states of the chain CO structure. The arrows denote the spin
directions. The region enclosed by the dashed lines on the bottom T-sheet
indicates the magnetic unit cell of the spin structure. Figure 3: (Color
online) Band structures calculated for (a) the FM state and (b) the
ferrimagnetic state of the $\sqrt{3}\times\sqrt{3}$ CO structure of LuFe2O4.
The solid and dashed lines represent the up-spin and down-spin bands,
respectively. The $\sqrt{3}\times\sqrt{3}\times 1$ hexagonal cell is used in
the calculations.
|
arxiv-papers
| 2008-12-19T21:24:43 |
2024-09-04T02:48:59.506700
|
{
"license": "Public Domain",
"authors": "H. J. Xiang, E. J. Kan, Su-Huai Wei, M.-H. Whangbo, and Jinlong Yang",
"submitter": "H. J. Xiang",
"url": "https://arxiv.org/abs/0812.3897"
}
|
0812.3907
|
# Microfabricated Chip Traps for Ions111This is a chapter from the forthcoming
book “Atom Chips” edited by J. Reichel and V. Vuletic (to be published by
WILEY-VCH).
J. M. Amini, J. Britton, D. Leibfried, and D. J. Wineland
Time and Frequency Division
National Institute of Standards and Technology
Boulder, CO 80305
(December 17, 2008)
###### Contents
1. 1 Introduction
2. 2 Radio-frequency (rf) ion traps
1. 2.1 Motion of ions in a spatially inhomogeneous rf field
2. 2.2 Electrode geometries for linear quadrupole traps
3. 3 Design considerations for Paul traps
1. 3.1 Doppler cooling
2. 3.2 Micromotion
3. 3.3 Exposed dielectrics
4. 3.4 Loading ions
5. 3.5 Electrical connections
6. 3.6 Motional heating
4. 4 Measuring heating rates
5. 5 Multiple trapping zones
6. 6 Trap modeling
1. 6.1 Modeling 3D geometries
2. 6.2 Analytic solutions for surface electrode traps
7. 7 Trap examples
8. 8 Future
9. 9 Acknowledgments
## 1 Introduction
Most chapters of this monograph focus on trapping and manipulating neutral
atoms with magnetic and optical fields. In this chapter, we discuss the
trapping of atomic ions. This is of current high interest because individual
ions can be the physical representations of qubits for quantum information
processing [1]. For recent reviews see [2, 3]. The goals are similar to those
of neutral atom traps in that we wish to create microfabricated structures to
trap, transport, and arrange ions in an array. Microfabrication holds the
promise of forming large arrays of traps that would allow the scaling of
current quantum information processing capabilities to the level needed to
implement useful algorithms [4, 5, 6, 7].
There are two primary types of ion traps used in low energy atomic physics:
Penning traps and Paul traps. In a Penning trap, charged particles are trapped
by a combination of static electric and magnetic fields [8, 9]. In a Paul
trap, a spatially varying sinusoidally oscillating electric field, typically
in the radio-frequency (rf) domain, confines atomic or molecular ions in space
[10]. In this review only the Paul type will be considered.
Neutral atom traps operate by a coupling between external trapping fields and
atoms’ electric or magnetic moments. Trap depths of a few kelvins are common.
In ion traps, an ion is trapped by a coupling between the applied electric
trapping fields and the atom’s net (or overall) charge. Typical ion trap
depths are 1 eV. This coupling does not depend on the ion’s internal
electronic state, leaving it largely unperturbed.
We begin this chapter with an introduction to the dynamics of ions confined in
Paul traps based on the pseudopotential approximation. Subsequent topics
include numeric and analytic models for various Paul trap geometries, a list
of considerations for practical trap design and finally an overview of
microfabricated trapping structures. A discussion of future directions
concludes this chapter.
## 2 Radio-frequency (rf) ion traps
In this section we discuss the equations of motion of a charged particle in a
spatially inhomogeneous radio-frequency (rf) field based on the
pseudopotential approximation model. We then present examples of suitable
electrode geometries.
### 2.1 Motion of ions in a spatially inhomogeneous rf field
Most schemes for quantum information processing with trapped ions are based on
a linear rf trap shown schematically in fig. 1a. This trap is essentially a
linear quadrupole mass filter [10] with its ends plugged by static potentials
[11]. The radial confinement (the $x$-$y$ plane in fig. 1a) is provided by an
rf potential applied to two of the electrodes with the other electrodes held
at rf ground. In this linear geometry, the rf potential cannot generate full
3D confinement, so static potentials $V_{1}$ and $V_{2}$ applied to control
electrodes provide axial ($z$ axis) confinement. We will assume the axial
trapping fields are relatively weak so that the accompanying static radial
fields do not significantly perturb the radial trapping.
Applying a potential of $V_{0}\cos(\Omega_{\rm rf}t)$ to the rf electrodes
while grounding the other electrodes ($V_{1}=V_{2}=0$), the rf potential near
the geometric center of the four rods takes the form
$\Phi\approx\frac{1}{2}V_{0}\cos(\Omega_{\rm
rf}t)(1+\frac{x^{2}-y^{2}}{R^{2}}),$ (1)
where $R$ is a distance scale that is approximately the distance from the trap
axis to the nearest surface of the electrodes [10, 4, 11]. The resulting
electric field is shown in fig. 1b. There is a field null at the trap center;
the field magnitude increases linearly with distance from the center.
We can think of the rf electric field as analogous to the electric field from
the trapping laser in an optical dipole trap [4, 12]. For a neutral atom, the
laser’s electric field induces a dipole moment. If the electric field is
inhomogeneous, the force on the dipole, averaged over one cycle of the
radiation, can give a trapping force. For detunings red of the atom’s resonant
frequency $\omega_{0}$, the resulting potential is a minimum at high fields,
while for detunings blue of $\omega_{0}$ it is a minimum at low fields. An
ion, however, is a free particle in the absence of a trapping field and its
eigenfrequency is zero. The rf trapping potential is therefore analogous to a
blue detuned light field and the ion seeks the position of lowest intensity.
In the case of eq. (1), that corresponds to $x=y=0$.
Figure 1: (a) Schematic drawing of the electrodes for a linear Paul trap. A
common rf potential $V_{0}\cos(\Omega_{\rm rf}t)$ is applied to the two
continuous electrodes, as indicated. The other electrodes are held at rf
ground through capacitors (not shown) connected to ground. In (b), we show the
radial ($x$-$y$) instantaneous electric fields from the applied rf potential.
Contours of the pseudopotential due to this rf-field are shown in (c). A
static trapping potential is created along the z-axis by applying a positive
potential $V_{1}>V_{2}$ (for positive ions) to the outer segments relative to
the center segments.
The motion for an ion placed in this field is commonly treated in one of two
ways: as an exact solution of the Mathieu differential equation or as an
approximate solution of a static effective potential called the
‘pseudopotential’. The Mathieu solutions provide insights on trap stability
and high frequency motion; the pseudopotential approximation is more
straightforward and is convenient for the analysis of trap designs.
We define the pseudopotential that governs the secular motion as follows [13].
The motion of an ion in the rf field is a combination of fast ‘micromotion’ at
the rf frequency on top of a slower ‘secular’ motion. For a particle of charge
$q$ and mass $m$ in a uniform electric field $E=E_{0}\cos(\Omega_{\rm rf}t)$,
the ion motion (neglecting a drift term) takes the form
$x(t)=-x_{\mu m}\cos(\Omega_{\rm rf}t),$ (2)
where $x_{\mu m}=qE_{0}/(m\Omega_{\rm rf}^{2})$ is the amplitude of what we
will call micromotion. If the rf field amplitude has a spatial dependence
$E_{0}(x)$ along the $x$ direction, there is a nonzero net force on the ion
when we average over an rf cycle:
$F_{\rm
net}=\left<qE(x)\right>\approx-\frac{1}{2}q\left.\frac{dE_{0}(x)}{dx}\right|_{x\rightarrow
x_{s}}x_{\mu m}=-\frac{q^{2}}{4m\Omega_{\rm
rf}^{2}}\left.\frac{dE_{0}^{2}(x)}{dx}\right|_{x\rightarrow
x_{s}}=-\frac{d}{dx}(q\Phi_{\rm pp}),$ (3)
where $x$ is evaluated at what we designate as the secular position $x_{s}$,
and the pseudopotential $\Phi_{\rm pp}$ is defined by
$\Phi_{\rm pp}(x_{s})\equiv\frac{1}{4}\frac{qE_{0}^{2}(x_{s})}{m\Omega_{\rm
rf}^{2}}.$ (4)
We have made the approximation that the solution in eq. (2) holds over an rf
cycle and have dropped terms of higher order in the Taylor expansion of
$E_{0}(x)$ around $x_{s}$. For regions near the center of the trapping
potential, these approximations hold. In three dimensions, we make the
substitution
$E_{0}^{2}\rightarrow|E|^{2}=E_{0,x}^{2}+E_{0,y}^{2}+E_{0,z}^{2}$. Note that
the pseudopotential depends on the magnitude of the electric field, not its
direction.
For the quadrupole field given in eq. (1), the pseudopotential is that of a 2D
harmonic potential (see fig. 1c):
$q\Phi_{\rm pp}=\frac{1}{2}m\omega_{r}^{2}(x^{2}+y^{2}),$ (5)
where $\omega_{r}\simeq qV_{0}/(\sqrt{2}m\Omega R^{2})$ is the resonant
frequency. As an example, for ${}^{24}Mg^{+}$ in a Paul trap with $V_{0}=50$
V, $\Omega_{\rm rf}/2\pi=100$ MHz and $R=50$ µm, which are typical parameters
for a microfabricated trap, the radial oscillation frequency is
$\omega_{r}/2\pi=14$ MHz.
The rf pseudopotential provides confinement of the ion in the radial ($x$-$y$)
plane. Axial trapping is obtained by the addition of the static control
potentials $V_{1}$ and $V_{2}$, as shown in fig. 1a.
For $\omega_{z}\ll\omega_{r}$, multiple ions trapped in the same potential
well will form a linear ‘crystal’ along the trap axis due to a balance between
the axial trapping potential and the ions’ mutual Coulomb repulsion. The
inter-ion spacing is determined by the axial frequency ($\omega_{z}$). The
characteristic length scale of ion-ion spacing is
$s=\left(\frac{q^{2}}{4\pi\epsilon_{0}m\omega_{z}^{2}}\right)^{1/3}.$ (6)
For a three-ion crystal the adjacent separation of the ions is
$s_{3}=(5/4)^{1/3}s$ [4]. For example, $s_{3}=5.3$ µm for
$\,{}^{24}\text{Mg}^{+}$ and $\omega_{z}/2\pi=1.0$ MHz. For multiple ions in a
linear Paul trap, $\omega_{z}$ is the frequency of the lowest vibrational mode
(the center of mass mode) along the trap axis.
A single ion’s radial motion in the potential given by eq. (5) can be
decomposed into uncoupled harmonic motion in the $x$ and $y$ directions, both
with the same trap frequency $\omega_{r}$. Because the potential is
cylindrically symmetric about $z$, we could choose the decomposition about any
two orthogonal directions, called the principle axes. We will see in section
3.1 when discussing Doppler cooling that we need to break this cylindrical
symmetry by the application of static electric fields. In that case, the
choice of the principle axes becomes fixed with corresponding radial trapping
frequencies $\omega_{1}$ and $\omega_{2}$, one for each principle axis.
### 2.2 Electrode geometries for linear quadrupole traps
Designs for miniaturized ion traps conserve the basic features of the Paul
trap shown in fig. 1. Figure 2 shows a few geometries that have been
experimentally realized. All these geometries generate a radial quadratic
potential near the trap axis, though the extent of deviations from the ideal
quadrupole potential away from the axis will depend on the design.
In one particular geometry, the electrodes all lie in a single plane, as shown
in fig. 2d with the ion suspended above the plane [14, 15, 16, 17, 18, 19,
20]. Trapping in such surface electrode (SE) traps is possible over a wide
range of geometries, albeit with $1/6$ to $1/3$ the motional frequencies and
$1/30$ to $1/200$ the trap depth of more conventional quadrupolar geometries
at comparable rf potentials and ion-electrode distances [14].
Advantages of the SE trap geometry over the other geometries shown in fig. 2
include easier fabrication and the possibility of integrating control
electronics on the same trap wafer [6]. A SE trap at cryogenic temperature was
demonstrated at MIT in 2008 [19].
Figure 2: Examples of microfabricated trap structures: (a) two wafers
mechanically clamped over a spacer [21, 22, 23, 24, 25], (b) two layers of
electrodes fabricated onto a single wafer [26], (c) three wafers clamped with
spacers (not shown) [27], and (d) surface electrode construction [14, 15, 16,
17, 18, 19, 20].
Research on SE trap designs is ongoing and holds promise to yield complex
geometries that would be difficult to realize in non-surface electrode
designs.
## 3 Design considerations for Paul traps
In this section, we will discuss the requirements that need to be addressed
when designing a practical ion trap.
### 3.1 Doppler cooling
For Doppler laser cooling of an ion in a trap, only a single laser beam is
needed; trap strengths far exceed the laser beam radiation pressure. The
cooling is offset by heating from photon recoil. Therefore, to cool in all
directions, the Doppler cooling beam k-vector must have a component along all
three principal axes of the trap [28]. This also implies that the trap
frequencies are not degenerate, otherwise one principal axis could be chosen
normal to the laser beam’s k-vector.
Meeting the first condition is usually straightforward for non-SE type traps,
where access for the laser beam is fairly open (see fig. 3). For SE traps,
where laser beams are typically constrained to run parallel to the chip
surface, care has to be taken in designing the trap so that neither radial
principle axis is perpendicular to the trap surface. Alternately, for SE
traps, we could bring the Doppler laser beam at an angle to the surface but
the beam would have to strike the surface. This can cause problems with
scattered light affecting detection of the ion and with charging of exposed
dielectrics (see section 3.3).
Figure 3: Doppler cooling with a single laser beam. The dashed lines are
equipotential curves for the pseudopotential. The overlap with the axial
direction ($z$) is fairly straightforward, as in (a), but care has to be taken
that the orientation of the two radial modes $\omega_{1}$ and $\omega_{2}$
does not place one of the mode axes perpendicular to the laser beam, as shown
in (b). For efficient cooling, the axes must be at an angle with respect to
the laser beam k-vector (c).
If any two trap frequencies are degenerate, then the trap axes in the plane
containing those modes are not well defined and the motion in a direction
perpendicular to the Doppler laser beam k-vector will not be cooled and will
be heated due to photon recoil. The axial trap frequency can be set
independently of the radial frequencies and can be chosen to prevent a
degeneracy with either of the radial modes. However, the two radial modes
could still be degenerate. There are several ways to break this degeneracy,
but usually the axial trapping potential is sufficient. When we apply an axial
trapping potential, Laplace’s equation forces us to have a radial component to
the electric field. In general, this radial field is not cylindrically
symmetric about $z$ and will distort the net trapping potential, as shown in
fig. 4, thereby lifting the degeneracy of the radial frequencies. If this is
not sufficient, offsetting all the control electrodes by a common potential
with respect to the rf electrodes will result in a static field that has the
same spatial dependence (that is the same function of $x$ and $y$) as the
field generated by the rf electrodes. This field, shown in fig. 1b, can be
used to split the radial frequencies. We will refer to the axes’ orientation
resulting from the offset of all control electrodes as the ‘intrinsic’ trap
axes since it does not depend on the segmentation of the control electrodes,
but only on the overall geometry of the rf and control electrodes. The static
axial potential might or might not define trap axes aligned with the intrinsic
axes, but, overall, the control electrodes and axial potential can be
configured to prevent either radial modes from being normal to the surface in
an SE trap. Furthermore, in some cases additional control electrodes are
designed into the trap to lift the degeneracy independent of both the axial
potential and the intrinsic axes.
Figure 4: The degeneracy in the radial trap modes can be lifted by the radial
component of the static axial confinement field. In (a), the quadrupole field
is shown overlaid on the cylindrically symmetric pseudopotential. The radial
component of the electric field (b) deforms the net potential seen by the ion
(c), breaking the cylindrical symmetry.
### 3.2 Micromotion
If the pseudopotential at the equilibrium position of a trapped ion is
nonzero, then the ion motion will include a persistent micromotion component
at frequency $\Omega_{\rm rf}$. There are two mechanisms that can generate a
nonzero equilibrium pseudopotential. As the trapping structures become more
complicated and the symmetry of the simple Paul trap in fig. 1 is broken,
there can be a component of the rf field in the axial direction at the
pseudopotential minimum; that is, the pseudopotential minimum need not be a
pseudopotential zero. Since this effect is caused by the geometry of the trap,
we refer to the resulting micromotion as ‘intrinsic’ micromotion [29].
Secondly, if there is a static electric field at the pseudopotential zero, the
equilibrium position of an ion will be shifted away from the pseudopotential
minimum. Because shim potentials can be applied to the control electrodes to
null these fields [29], the micromotion due to this mechanism is called
‘excess’ micromotion.
Both intrinsic and excess micromotion can cause problems with the laser-ion
interactions, such as Doppler cooling, ion fluorescence, and Raman transitions
[4, 29]. An ion with micromotion experiences a frequency-modulated laser field
due to the Doppler shift. In the rest frame of the ion, this modulation
introduces sidebands to the laser frequency (as seen by the ion) at integer
multiples of $\Omega_{\rm rf}$ and reduces the laser beam’s intensity at the
carrier frequency, as shown in fig. 5. The strength of these sidebands is
parametrized by the modulation index $\beta$, given by
$\beta=\frac{2\pi x_{\mu m}}{\lambda}\cos\theta$ (7)
where $x_{\mu m}$ is the micromotion amplitude, $\lambda$ is the laser
wavelength, and $\theta$ is the angle the laser beam k-vector makes with the
micromotion. For laser beams tuned near resonance, ion fluorescence becomes
weaker and can disappear entirely. As another example, when $\beta=1.43$, the
carrier and first micromotion sideband have equal strength. For $\beta<1$, the
fractional loss of on-resonance fluorescence is approximately $\beta^{2}/2$.
As a rule of thumb, we aim for $\beta<0.25$, which corresponds to a drop of
less than five percent in on-resonant fluorescence.
Figure 5: In the rest frame of the ion, micromotion induces sidebands of a
probing laser. Here, the monochromatic laser spectrum has been convoluted with
the atomic linewidth.
For a given static electric field $E_{\rm dc}$ in the radial plane, an ion’s
radial displacement $x_{d}$ from the trap center and the resulting excess
micromotion amplitude $x_{\mu m}$ are
$x_{d}=\frac{qE_{\rm dc}}{m\omega_{r}^{2}},\\\ x_{\mu
m}\simeq\sqrt{2}\frac{w_{r}}{\Omega_{\rm rf}}x_{d},$ (8)
where $\omega_{r}$ is the radial trapping frequency.
Assume ${}^{24}Mg^{+}$, $\Omega_{\rm rf}/2\pi=100$ MHz and
$\omega_{r}/2\pi=10$ MHz. A typical SE trap with $R\sim 50$ µm and an excess
potential of 1 V on a control electrode will produce a radial electric field
at the ion of $\sim 500$ V/m. The resulting displacement is $x_{d}=500$ nm and
the corresponding micromotion amplitude is $x_{\mu m}=70$ nm. This results in
a laser modulation index of $\beta=1.14$.
Stray electric fields can be nulled if the control electrode geometry permits
application of independent compensation fields along each radial principle
axis. For the Paul trap in fig. 1, a common potential applied to the control
electrodes can only generate a field at the trap center that is along the
diagonal connecting the electrodes. We can compensate for other directions by
applying, for example, a static potential offset to one of the rf electrodes
or by adding extra compensation electrodes.
There are several experimental approaches to detecting and minimizing excess
micromotion [29]. One technique uses the dependence of the fluorescence from a
cooling laser beam on the micromotion modulation index. The micromotion can be
minimized by maximizing the fluorescence when the laser is near resonance and
minimizing the fluorescence when tuned to the rf sidebands.
Intrinsic micromotion can also be caused by an rf phase difference $\phi_{\rm
rf}$ between the two rf electrodes. A phase difference can arise due to a path
length difference or a differential capacitive coupling to ground for the
leads supplying the electrodes with rf potential [29, 20]. We aim for
$\beta<0.25$ (see section 3.2) for typical parameters, which requires
$\phi_{\rm rf}<0.5^{\circ}$.
### 3.3 Exposed dielectrics
Exposed dielectric surfaces near the trapping region can pose a problem due to
charging of these surfaces and resulting stray electric fields. Charging can
be caused by photo-emission by the probe laser or from electron sources such
as those used for loading ions into the traps. Depending on the resistivity of
the dielectric, these charges can remain on the surfaces for minutes or
longer, requiring time-dependent micromotion nulling or waiting a sufficient
time for the charge to dissipate.
Surface electrode traps can be particularly prone to this problem. The
metallic trapping electrodes are often supported by an insulating substrate
and the spaces between the electrodes expose the substrate. The effect of
charging these regions can be mitigated by increasing the ratio of electrode
conductor thickness to the inter-electrode spacing.
Figure 6 illustrates a model for estimating how thick electrodes can suppress
the field from a strip of exposed substrate charged to a potential $V_{s}$.
The sidewalls are assumed conducting and grounded. Along the midpoint of the
trench the potential drops exponentially with height [30]. Using this solution
to relate $V_{s}$ to the potential at the top of the trench, and employing the
techniques described in section 6.2 to relate the surface potential to a field
at the ions, we obtain an approximate expression for the field seen by the
ion:
$|E|\simeq\frac{aV}{\pi R^{2}}\times\left\\{\begin{array}[]{ll}1,&t=0\\\
\frac{4}{\pi}e^{-\pi t/a},&t\geq\frac{a}{\pi},\\\ \end{array}\right.$ (9)
where $a$ is the width of the exposed strip of substrate, $t$ is the electrode
thickness, $R$ is the distance from the trap surface to the ion, and we have
assumed $R\gg a$. Thus, the effect of the stray charges drops off rapidly with
the ratio of electrode thickness to gap spacing.
Figure 6: Model used for estimating the effect of stray charging. We assume
that $R\gg a$ and $t\geq a/\pi$.
### 3.4 Loading ions
Ions are loaded into traps by ionizing neutral atoms as they pass through the
trapping region. The neutral atoms are usually supplied by a heated oven but
can also come from background vapor in the vacuum or laser ablation of a
sample.
It is necessary that the neutral atom flux reach the trapping region but not
deposit on insulating spacers, which might cause shorting between adjacent
trap electrodes. In practice, this is accomplished by careful shielding and,
in some SE traps, undercutting of electrodes to form a shadow mask (see fig.
18). Alternately, for SE traps, a hole machined through the substrate can be
used to direct neutral flux from an oven on the back side of the wafer to a
small region of the trap, preventing coating of the surface. This is called
backside loading and has been demonstrated in several traps (see section 7).
### 3.5 Electrical connections
The control potentials and rf trapping potentials are delivered to the trap
electrodes by wiring that includes conducting traces on the trap substrate.
Care is needed to avoid several pitfalls.
The high-voltage rf potential is typically produced with resonant rf
transformers [31, 32, 33]. Rf losses in a microtrap’s electrodes or insulating
substrate can degrade the resonator (loaded) quality factor ($Q_{L}$) and can
cause ohmic heating of the microtrap itself. This can be be mitigated by use
of low-loss insulators (for example, quartz or alumina) and decreasing the
capacitive coupling of the rf electrodes to ground through the insulators.
Typical rf parameters are $\Omega_{\rm rf}/{2\pi}=10$ to 100 MHz, $V_{\rm
rf}\simeq 100~{}V$ and $Q_{L}\simeq 200$.
The rf electrodes have a small capacitive coupling $C_{s}$ to each control
electrode (typically less than $0.1$ pF), which can result in rf potential on
the control electrodes. This rf potential needs to be shunted to ground by a
capacitor $C_{f}$ as shown in fig. 7. A low-pass RC filter (typically
$R=1~{}k\Omega$ and $C_{f}=1~{}nF$) on each control electrode is used to
filter noise introduced by the externally-applied control electrode
potentials. The impedance of the lines between the control electrodes and
$C_{f}$ should be low or the rf shunting to ground will be compromised. Proper
grounding, shielding and filtering of the electronics supplying the control
electrode potentials are also important to suppress pickup and ground loops
(which can cause motional heating; see section 3.6).
Figure 7: Figure showing typical filtering and grounding of a trap control
electrode. Inside the vacuum system are low pass RC filters which reduce noise
from the control potential source and provide low impedance shorts to ground
for the rf coupled to the control electrodes by stray capacitances $C_{\rm
s}\ll C_{\rm f}$. The RC filters typically lie inside the vacuum system,
within 2 cm of the trap electrodes. The control potential is referenced to the
trap rf ground and is supplied over a properly shielded wire.
### 3.6 Motional heating
Doppler and Raman cooling can place a trapped ion’s harmonic motion into the
ground state with high probability [4, 34, 35, 36]. If we are to use the
internal states of an ion to store information, we must turn off the cooling
laser beams during that period. Unfortunately, the ions do not remain in the
motional ground state and this heating can reduce the fidelity of operations
performed with the ions. One source of heating comes from laser interactions
used to manipulate the electronic states [37]. Another source is ambient
electric fields that have a frequency component at the ion’s motional
frequencies. We expect such fields from the Johnson noise on the electrodes
[4, 38, 39, 40], but the heating rates observed experimentally are typically
several orders of magnitude larger than the Johnson noise can account for.
Currently, the source of this anomalous heating is not explained, but recent
experiments [39, 19] indicate it is thermally activated and consistent with
patches of fluctuating potentials with a size scale smaller than the ion-
electrode spacing [38].
The spectral density of electric field fluctuations $S_{E}$ at the ion’s
position inferred from ion heating measurements in a number of traps is
plotted versus the minimum ion-electrode separation $R$ in fig. 8. The
dependence of $S_{E}$ on $R$ and on the trap frequency $\omega$ follows a
roughly $R^{-\alpha}\omega^{-\beta}$ scaling, where $\alpha\approx 3.5$[38,
39] and $\beta\approx 0.8$ to 1.4 [38, 39, 16, 19]. In addition to being too
small to account for these measured heating rates, Johnson noise scales as
$R^{-2}$ [4, 38]. One candidate mechanism that does scale as $R^{-4}$ is noise
caused by small fluctuating patch potentials on the electrode surfaces [38].
The potentials on these patches fluctuate at megahertz frequencies and
generate a corresponding fluctuating electric field at the ion’s equilibrium
position. This field can lead to heating of the ion [41, 42, 43, 38, 39, 40].
In the context of ion quantum information processing, microtraps are
advantageous because quantum logic gate speeds and ion packing densities
increase as the trap size decreases [4, 5, 44, 6]. However, these gains are at
odds with the highly unfavorable dependence of motional heating on ion-
electrode distance. For example, extrapolating from the room temperature
heating results of [16], a $R=10$ µm trap might exceed $10^{6}$ quanta per
second. Heating between gate operations can also be problematic because hot
ions require more time to recool to the motional ground state.
Figure 8: Spectral density of electric-field fluctuations inferred from
observed ion motional heating rates. Data points show heating measurements in
ion traps observed in different ion species by several research groups [34,
35, 45, 46, 38, 47, 21, 48, 39, 49, 26, 50, 19, 20, 25]. Unless specified, the
data was taken with the trap at room temperature. The dashed line shows a
$R^{-4}$ trend for ion heating vs ion-electrode separation $R$.
## 4 Measuring heating rates
Heating rates have often been measured by observing an ion’s energy increase
after cooling to the motional ground state, a relatively complicated and
technically challenging undertaking [34, 35]. This section outlines a method
to measure ion motional heating with a single low power laser beam [50, 51,
23]. Near resonance, an atom’s fluorescence rate is influenced by its motion
due to the Doppler effect. This can be exploited in the following way:
1. 1.
Cool a trapped ion to its Doppler limit.
2. 2.
Let it remain in the dark for some time. Ambient electric fields couple to the
ion’s motion and heat it.
3. 3.
Turn on the Doppler cooling laser and measure the ion’s time-resolved
fluorescence, as shown in fig. 9.
4. 4.
A fit to a theoretical model of the ion fluorescence rate versus time (during
recooling) [51] gives an estimate of the ion’s temperature at the end of step
2.
The theoretical model in [51] explored cooling of hot ions where the average
modulus of the Doppler shift is on the order of, or greater than, the cooling
transition line width $\Gamma$. The model is a one-dimensional semiclassical
theory of Doppler cooling in the weak binding limit where
$\omega_{z}ll\Gamma$. It is assumed that hot ions undergo harmonic
oscillations with amplitudes corresponding to the Maxwell-Boltzman energy
distribution when averaged over many experiments.
As a one-dimensional (1D) model, only a single motional mode is assumed to be
hot. Since the electric field spectral density $S_{E}$ at the ion is observed
to scale approximately as $S_{E}\propto\omega^{-\beta}$, where $\beta\approx
0.8$ to 1.4 [38, 39, 16, 19], the heating is effectively 1D if
$\omega_{z}<<\omega_{x},\omega_{y}$. This is also important experimentally
because efficient Doppler cooling requires laser beam overlap with all modes
simultaneously: a change in ion fluorescence can arise from heating of any
mode.
Heating rates measured with the recooling technique were found to be in
reasonable agreement with rates measured starting from the ground state and
allowing heating to only a few average motional quanta [34, 35]. In these
comparisons, heating seems to be approximately linear from the ground state to
at least 10000 motional quanta. The disadvantage of the recooling technique is
that for small heating rates, the duration of step 2 can become quite long.
Figure 9: Plot showing normalized fluorescence rate $dN/dt$ during Doppler
cooling of a hot ion versus the time the cooling laser is turned off (dark
time). The experimental data averaged over many experiments is fit to the 1D
model [51] briefly discussed in the text. The fit has a single free parameter:
the ion’s temperature at the outset of cooling. The error bars are based on
counting statistics. Data taken on ${}^{25}Mg^{+}$ with a dark time of 25 s
[51].
## 5 Multiple trapping zones
Much of the emphasis in the recent generation of ion traps is towards traps
that can store ions in multiple trapping zones and can transport ions between
the zones.
We can modify the basic Paul trap in fig. 1 to support multiple zones and ion
transport by dividing the control electrodes into a series of segments as
shown in fig. 10a. By applying appropriate potentials [21, 52, 22, 53, 54, 55,
56, 23] to these segments, an axial harmonic well can be moved along the
length of the trap carrying ions along with it (fig. 10b). In the adiabatic
limit (with respect to $\omega_{z}^{-1}$), ions have been transported a
distance of 1.2 mm in 50 µs with undetectable heating or internal-state
decoherence [21].
Figure 10: (a) Example of a multizone trap. By applying appropriate waveforms
to the segmented control electrodes, ions can be (b) shuttled from zone to
zone or (c) pairs of ions can be merged into a single zone or split into
separate zones.
As an example relevant to quantum information processing, we need to be able
to take pairs of ions in a single zone (for example, zone 2 in fig. 10c) and
separate them into independent zones (one ion in zone 1 and a second in zone
3) without excessive heating. Likewise, we need to reverse this process and
combine the ions into a single well. Separating and recombining are more
difficult tasks than ion transport; the theory is discussed in [53] and
experimentally demonstrated in [21, 52]. The basis for these potentials is the
quadratic and quartic terms of the axial potential. Proper design of the trap
electrodes can increase the strength of the quartic term and facilitate faster
ion separation and merging with less heating. Groups of two and three ions
have been separated while heating the center of mass mode to less than 10
quanta and the higher order modes to less than 2 quanta [52].
The segmented Paul trap in fig. 10 forms a linear series of trapping zones,
but other geometries are desirable. Of particular interest are junctions with
linear trapping regions extending from each leg. Specific junction geometries
are discussed in section 7. The broad goal is to create large interconnected
trapping structures that can store, transport and reorder ions so that any two
ions can be brought together in a common zone [4, 5].
## 6 Trap modeling
Calculation of trap depth, secular frequencies, and transport and separation
waveforms requires detailed knowledge of the potential and electric fields
near the trap axis. In the pseudopotential approximation, the general time-
dependent problem is simplified to a slowly varying electrostatic one. For
simple four-rod type traps, good trap design is not difficult using numerical
simulation owing to their symmetry. However, SE trap design is more
complicated since the potential may have large anharmonic terms and highly
asymmetric designs are common. Fortunately, for certain SE trap geometries,
analytic solutions exist. These closed-form expressions permit efficient
parametric optimization of electrode geometries not practical by numerical
methods. In this section, we will first discuss the full 3D calculations and
then introduce the analytic solutions.
### 6.1 Modeling 3D geometries
There are several numerical methods for solving the general electrostatic
problem. In our trap simulations, we use the boundary element method
implemented in a commercial software package. In contrast to the finite
element method, the solutions from the boundary element method are in
principle differentiable to all orders. A simulation consists of calculating
the potential due to each control electrode when that electrode is set to a
fixed non-zero potential and all others are grounded. The solution for an
arbitrary set of potentials on the control electrodes is then a linear
combination of these particular solutions. Similarly, the pseudopotential is
obtained by scaling the field calculated for a finite potential on the rf
electrodes and ground on the control electrodes and then squaring the field
according to eq. (4).
### 6.2 Analytic solutions for surface electrode traps
Numerical calculations work for any electrode geometry, but they are are slow
and not well suited to automatic optimization of SE trap electrode shapes. For
the special case of SE traps, an analytic solution exists subject to a few
realistic geometric constraints. Electrodes are modeled as a collection of
separately biased regions embedded in an infinite ground plane (see fig. 11)
without gaps between the electrodes. The electric field that would be observed
from a biased region is proportional to the magnetic field produced by a
current flowing along its perimeter [57]. The problem is then reduced from
solving Laplace’s equation to integrating a Biot-Savart type integral around
the patch boundary. Furthermore, for patches that have boundaries composed of
straight line segments, the integrals have analytic solutions. The application
of this technique to SE traps is given in [58].
Figure 11: Surface electrode trap composed of two rf electrodes embedded in a
ground plane (four-wire trap) (a). The field lines from the Biot-Savart type
integral are shown in (b).
The main shortcoming of this method is the requirement that there be no gaps
between the electrodes. Typical SE trap fabrication techniques produce 1 to 5
µm gaps which can only be accounted for at the level important to ion dynamics
by full numerical simulations.
Fields for arbitrarily shaped patches can be calculated using this Biot-Savart
technique, but for simplicity we restrict ourselves to strips that extend to
infinity in the $z$-direction of fig. 11. For this particular case, we can
also derive potentials from the calculated fields. A strip extending from
$x=a$ to $x=b$ with $a<b$ held at potential $U_{s}$ leads to a spatial
potential
$\Phi_{s}(a,b)=\frac{U_{s}}{\pi}\times\left\\{\begin{array}[]{ll}\tan^{-1}\left(\frac{x-a}{y}\right)-\tan^{-1}\left(\frac{x-b}{y}\right),&-\infty<a<b<\infty\\\
\frac{\pi}{2}-\tan^{-1}\left(\frac{x-b}{y}\right),&a=-\infty\\\
\frac{\pi}{2}+\tan^{-1}\left(\frac{x-a}{y}\right).&b=\infty\\\
\end{array}\right.$ (10)
The potentials of multiple, non-overlapping strips can then be summed for more
complex structures.
Two basic SE trap geometries are the ‘four-wire’ trap and the ‘five-wire’
trap. An example four-wire trap consists of an rf electrode from $x=-d$ to
$x=0$ and another semi-infinite rf electrode from $x=d$ to $x=\infty$ (see
fig. 11a and fig. 16). An example five-wire trap consists of two symmetric rf
electrodes from $x=-3/2d$ to $x=-1/2d$ and $x=1/2d$ to $x=3/2d$. Their
respective potentials are given by
$\Phi_{\rm 4w}=\Phi_{s}(-d,0)+\Phi_{s}(d,\infty);~{}~{}\Phi_{\rm
5w}=\Phi_{s}\left(-\frac{3d}{2},-\frac{d}{2}\right)+\Phi_{s}\left(\frac{d}{2},\frac{3d}{2}\right).$
(11)
From the electric fields and eq. (4) we can derive the pseudopotential. Note
that the potential minima coincide with the points of zero electric field that
lie in the line of symmetry around $x=0$ at $y_{4w}=d$ and
$y_{5w}=\sqrt{3}d/2$, respectively. For an ion of mass $m$ and charge $q$, the
trap frequencies along the two degenerate radial directions are
$\omega_{\rm 4w}=\frac{qU_{s}}{\sqrt{2}m\pi\Omega_{\rm
rf}d^{2}};~{}~{}\omega_{\rm
5w}=\sqrt{\frac{2}{3}}\frac{qU_{s}}{m\pi\Omega_{\rm rf}d^{2}},$ (12)
where $\Omega_{\rm rf}$ is the rf-drive frequency. Figure 12 shows the general
shape of the pseudopotential well along the $y$-axis at $x=0$ for the four-
wire trap (for the five-wire trap the potential looks very similar).
Figure 12: Analytic pseudopotential of the four-wire trap along $y$ at $x=0$.
The trapping zero is at $y=d$; the maximum defining the well depth is at
$s_{\rm 4w}=d~{}\sqrt{2+\sqrt{5}}$
The potential is zero at $y=d$ where the ion is trapped, then rises to a
maximum and finally asymptotically drops towards zero for
$y\rightarrow\infty$. The positions of the maxima are at
$s_{\rm 4w}=d~{}\sqrt{2+\sqrt{5}};~{}~{}s_{\rm 5w}=d~{}\sqrt{3/4+\sqrt{3}},$
(13)
and the pseudopotential well depth (in eV) is
$W_{\rm 4w}=\left(\frac{qU_{s}^{2}}{4m\Omega_{\rm
rf}^{2}}\right)\frac{2}{\pi^{2}d^{2}(11+5\sqrt{5})};~{}~{}W_{\rm
5w}=\left(\frac{qU_{s}^{2}}{4m\Omega_{\rm
rf}^{2}}\right)\frac{1}{\pi^{2}d^{2}(7+4\sqrt{3})}.$ (14)
To get an idea of practical parameters, we can calculate the radial frequency
and pseudopotential well depth of a four-wire trap with a geometry similar to
the trap described in [16]. For $\Omega_{\rm rf}/2\pi=87$ MHz, $U_{s}=103.2$
V, $d=$ 40 µm and $m$ the mass of a 24Mg+ ion, we get $\omega_{\rm
4w}/2\pi=$16.9 MHz and $W_{\rm 4w}=$203 meV.
## 7 Trap examples
Having covered the general principles for Paul trap designs, we now give
specific examples of microfabricated ion traps. A number of fabrication
techniques have been used for micro-traps, starting with assembling multiple
wafers to form a traditional Paul trap type design [21, 22, 27, 23, 24, 20,
25]. Recently, trap fabrication has been extended to monolithic designs using
substrate materials such as Si, GaAs, quartz, and printed circuit board [15,
16, 17, 26, 59, 18, 19, 20]. The fabrication process includes such
microfabrication standards as photolithography, metalization, and chemical
vapor deposition as well as other less used techniques such as laser
machining.
The microfabricated equivalent to the prototypical four-rod Paul trap can use
two insulating substrates patterned with electrodes that are then clamped or
bonded together with an insulating spacer. This approach has been implemented
in a number of traps [21, 52, 22, 20, 24, 25] using two substrates, as shown
in fig. 13a. Alternatively, it is possible to build this structure into a
single monolithic device [26], as indicated schematically in fig. 14a.
Reference [27] describes a three-wafer trap design like that shown in fig. 13b
incorporating a ‘T’ shaped junction. At NIST, a two-layer trap with an ‘X’
junction has recently been implemented [25] and is shown in fig. 15. Such two-
dimensional geometries will be important in order to combine arbitrarily
selected qubits from an array together in the same trap zone.
Figure 13: Multiwafer traps can be formed by mechanically clamping or bonding
multiple substrates to form (a) a four-rod quadrupolar Paul trap type
structure or (b) a modified Paul trap using a three-layer structure [27]. The
segmentation of the control electrodes on the bottom substrate is similar to
that of the top substrate. Figure 14: (a) Four-rod Paul trap realized by
successively deposited layers of GaAs and AlGaAs on a GaAs wafer [26]. In (b),
conducting gold strips deposited on two glass substrates and alternately
driven at opposite phases of an rf source (phases denoted by ‘+’ and ‘-’)
generates a trapping volume between the substrates [60]. Static potentials at
the edges of the trap along the $z$ axis, applied with electrodes that are not
shown, confine the ions to the central region of the trap. Figure 15: Example
of a two-wafer trap with an ‘X’ junction [25]. The trap electrodes are
fabricated with evaporated and electroplated gold that is deposited on laser-
machined alumina substrates.
Another approach demonstrated recently used two patterned substrates, without
slots, that are mounted with the conducting layers facing each other [60] (see
fig. 14b). The array of conducting gold electrode strips is driven with rf
that alternates between a phase of $0\,^{\circ}$ and $180\,^{\circ}$ from one
strip to the next. This creates a pseudopotential that is near zero for much
of the space between the wafers but which rises sharply near the substrates.
When combined with static potentials at the edges of the wafers, this trap
generates a near field-free region bounded by ‘hard’ potential walls (fig.
14b). Arrays of cylindrical Paul type traps have been microfabricated on
silicon for use as mass spectrometers [59].
Surface electrode (SE) traps have the benefit of using standard
microfabrication methods where layers of metal and insulator are deposited on
the surface of the wafer without the need for milling of the substrate itself.
There are two general versions of the surface trap electrode geometry, as
described in section 6.2 and shown in fig. 16. The four-wire geometry has the
intrinsic trap axes rotated at 45∘ to the substrate plane, which allows for
efficient laser cooling of the ion. The five-wire geometry has one intrinsic
trap axis perpendicular to the surface, which can make that axis difficult to
Doppler cool (see section 3.1). To enable Doppler cooling, additional control
electrodes can be added to the design to rotate the trap axes away from the
intrinsic direction. Alternately, a hybrid between the four- and five-wire
designs where the rf strips are of unequal widths (an ‘asymmetric’ five-wire
trap) will rotate the intrinsic axes and enable cooling.
Figure 16: (a) Four-wire SE trap geometry and (b) symmetric five-wire SE trap
geometry. In practice, the symmetric five-wire geometry is typically not used
because of the difficulty of cooling the vertical motion of the trapped ions.
Surface electrode traps are relatively new and only a few designs have been
demonstrated [15, 16, 17, 18, 19]. An SE trap was first demonstrated with
charged polystyrene balls using standard PC board fabrication techniques [15].
The first SE trap for atomic ions was constructed on a fused quartz substrate
with electroplated gold electrodes [16, 20]. In addition, meander-line
resistors were fabricated on the chip as part of the control electrode
filtering. Surface-mount capacitors were gap welded to the chip to complete
the filters (see section 3.5). The fabrication process sequence is shown in
fig. 18. The bonding pads and the thin meander-line resistors were formed by
liftoff of evaporated gold. Charging of the exposed substrate between the
electrodes was a concern, so the trap electrodes were made of 6 µm thick
electroplated gold with 8 µm gaps so as to shield the ion somewhat from the
charges on the quartz surface.
Figure 17: An example of a four-wire SE trap constructed of electroplated gold
on a quartz substrate [16]. Figure 18: Fabrication steps for the example SE
trap in fig. 17 [16]. The copper seed layer could not be used under the
meander line resistors because the final step of etching the seed layer would
fully undercut the narrow meander pattern.
A similar design was built by a group at MIT for low-temperature testing using
1 µm evaporated silver on quartz [19]. They reported a strong dependence of
the anomalous ion heating on temperature (see section 3.6).
The construction of the traps in [16] and [19] was based on adding conducting
layers to an insulating substrate. An alternate fabrication method used boron-
doped Si wafers anodically bonded to a glass substrate [17] and boron-doped
silicon-on-insulator (SOI) wafers [20]. In both cases trenches were etched
through the silicon layer to the glass or embedded insulating layer to define
the trap electrodes. The SOI design demonstrated multiple trapping zones in a
SE trap and backside loading of ions.
Surface electrode traps allow for complex arrangements of trapping zones, but
making electrical connections to these electrodes quickly becomes intractable
as the complexity grows. This problem can be addressed by incorporating
multiple conducting layers into the design with only the field from the top
layer affecting the ion [6, 61]. An example of such a multilayer trap
fabricated on an amorphous quartz substrate at NIST is shown in fig. 19. The
metal layers are separated by chemical vapor deposited (CVD) silicon dioxide
and connections between metal layers are made by vias that are plasma etched
through the oxide, as shown in fig. 20. The fabrication process for the
surface gold layer is similar to the electroplating shown in fig. 18.
Figure 19: Multilayer, multi-zone, linear SE trap mounted in its carrier and
an enlargement of the active region [61]. Figure 20: Fabrication of an
asymmetric five-wire multilayer SE trap. A CVD oxide insulates the surface
electrodes from the second layer of interconnects. Plasma etched holes in the
insulated layer connect the two conducting layers.
In the last three years, microfabricated traps have also been produced by
Sandia National Laboratory (contact: M. Blaine, SNL) and Lucent Technologies
(contact: R. Slusher, Georgia Tech Research Institute) and distributed to
several ion trap groups in the framework of a "trap foundry" initiated by DTO
(now IARPA). Several groups have seen trapping in the Lucent trap, a 17-zone
SE trap. The Sandia trap, a 5-zone planar trap where the ions reside in-plane
with the electrodes, has also been used to trap ions in two laboratories.
## 8 Future
As ion traps become smaller, trap complexity increases and features such as
junctions promise to expand the capabilities of such traps. The two
experimentally demonstrated atomic ion traps with junctions (see [27] and fig.
15) are based on multilayer designs. The slots and difficulty of alignment and
bonding in multiwafer traps make it difficult to scale such structures.
Figure 21a shows an example design of a ‘Y’ version of an SE trap junction.
The shape of the rf junction is an example of the optimization that is
possible with SE traps because of the efficient methods described in section
6.2 to calculate the fields. The electrode geometry has been optomized to
generate a pseudopotential that has minimal axial ‘bumps’ so that rf
micromotion during ion transport will be minimized (see section 3.2).
Components such as this ‘Y’ could then be assembled into larger structures as
shown in fig. 21b. Surface electrode traps fabricated using standard recipes
in a foundry and using standard patterns may eventually make ion traps more
accessible to research groups that do not have the resources needed to develop
their own.
Figure 21: (a) Example of a SE trap ‘Y’ junction and (b) a prototype design
using multiple ‘Y’ junctions to link experimental regions and loading zones.
With increased trap complexity, several other issues arise. One of these is
the question of how to package traps and provide all the electrical
connections needed to operate them. Another issue is that of corresponding
complexity of the lasers used in manipulating the ions. Beyond cooling, state
preparation, and detection, lasers are needed to coherently manipulate the
internal states of the ions and couple pairs or groups of ions. Multiplexing
sets of lasers to address multiple trapping zones for parallel processing will
be difficult. Alternatives to laser optical field state manipulation have been
proposed [62, 63, 64, 65, 66] where magnetic structures, both active wire
loops and passive magnetic layers, replace laser beams. If proven to be
viable, this would transfer much of the experimental complexity from large
laser systems to electronic packages, which can be more reliably engineered
and should be scalable [6].
## 9 Acknowledgments
Work supported by the NIST Quantum Information Program and IARPA. This
manuscript is a publication of NIST and is not subject to U.S. copyright.
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## Index
* analytic solutions §6.2
* anomalous heating §3.6
* degenerate frequencies §3.1
* dielectric charging §3.3
* Doppler cooling §3.1
* filters §3.5
* five-wire trap §6.2
* four-wire trap §6.2
* heating §3.6, §4
* intrinsic axes §3.1
* ion trap §1, §5
* ions
* Doppler cooling §3.1
* loading §3.4
* motional heating §3.6
* spacing §2.1
* trapping §2
* Johnson noise §3.6
* junction §7, §8
* Mathieu equation §2.1
* micromotion §2.1, §3.2
* modeling §6
* Paul trap §1
* geometries §2.2
* ion motion §2.1
* Penning trap §1
* pseudopotential §2.1
* radial frequency §2.1
* recooling §4
* resonator §3.5
* rf shunt §3.5
* rf trap §2
* secular motion §2.1
* separation §5
* sidebands §3.2
* surface electrode (SE) trap §2.2, §6.2, §7
* transport §5
* zones §5
|
arxiv-papers
| 2008-12-19T22:11:34 |
2024-09-04T02:48:59.513376
|
{
"license": "Public Domain",
"authors": "J. M. Amini, J. Britton, D. Leibfried, D. J. Wineland",
"submitter": "Jason Amini",
"url": "https://arxiv.org/abs/0812.3907"
}
|
0812.4044
|
# The Offset Tree for Learning with Partial Labels
Alina Beygelzimer beygel@us.ibm.com
IBM Research John Langford jl@yahoo-inc.com
Yahoo! Research
###### Abstract
We present an algorithm, called the $\operatorname{Offset\ Tree}$, for
learning to make decisions in situations where the payoff of only one choice
is observed, rather than all choices. The algorithm reduces this setting to
binary classification, allowing one to reuse of any existing, fully supervised
binary classification algorithm in this partial information setting. We show
that the Offset Tree is an optimal reduction to binary classification. In
particular, it has regret at most $(k-1)$ times the regret of the binary
classifier it uses (where $k$ is the number of choices), and no reduction to
binary classification can do better. This reduction is also computationally
optimal, both at training and test time, requiring just $O(\log_{2}k)$ work to
train on an example or make a prediction.
Experiments with the $\operatorname{Offset\ Tree}$ show that it generally
performs better than several alternative approaches.
Keywords: Supervised learning, active learning Bandits, Reinforcement
Learning, Interactive Learning.
## 1 Introduction
This paper is about learning to make decisions in partial feedback settings
where the payoff of only one choice is observed rather than all choices.
As an example, consider an internet site recommending ads or other content
based on such observable quantities as user history and search engine queries,
which are unique or nearly unique for every decision. After the ad is
displayed, a user either clicks on it or not. This type of feedback differs
critically from the standard supervised learning setting since we don’t
observe whether or not the user would have clicked had a different ad beed
displayed instead.
In an online version of the problem, a policy chooses which ads to display and
uses the observed feedback to improve its future ad choices. A good solution
to this problem must explore different choices and properly exploit the
feedback.
The problem faced by an internet site, however, is more complex. They have
observed many interactions historically, and would like to exploit them in
forming an initial policy, which may then be improved by further online
exploration. Since exploration decisions have already been made, online
solutions are not applicable. To properly use the data, we need _non-
interactive_ methods for learning with partial feedback.
This paper is about constructing a family of algorithms for non-interactive
learning in such partial feedback settings. Since any non-interactive solution
can be composed with an exploration policy to form an algorithm for the online
learning setting, the algorithm proposed here can also be used online. Indeed,
some of our experiments are done in an online setting.
### Problem Definition
Here is a formal description of non-interactive data generation:
1. 1.
Some unknown distribution $D$ generates a feature vector $x$ and a vector
$\vec{r}=(r_{1},r_{2},...,r_{k})$, where $r_{i}\in[0,1]$ is the reward of the
$i$-th action, $i\in\\{1,\ldots,k\\}$. Only $x$ is revealed to the learner.
2. 2.
An existing policy chooses an action $a\in\\{1,\ldots,k\\}$.
3. 3.
The reward $r_{a}$ is revealed.
The goal is to learn a policy $\pi:X\rightarrow\\{1,\ldots,k\\}$ for choosing
action $a$ given $x$, with the goal of maximizing the expected reward with
respect to $D$, given by
$\eta(\pi,D)=\mathbf{E}_{(x,\vec{r})\sim D}\left[r_{\pi(x)}\right].$
We call this a _partial label problem_ (defined by) $D$.
### Existing Approaches
Probably the simplest approach is to regress on the reward $r_{a}$ given $x$
and $a$, and then choose according to the largest predicted reward given a new
$x$. This approach reduces the partial label problem to a standard regression
problem.
A key technique for analyzing such a reduction is _regret analysis_ , which
bounds the “regret” of the resulting policy in terms of the regressor’s
“regret” on the problem of predicting $r_{a}$ given $x$ and $a$. Here _regret_
is the difference between the largest reward that can be achieved on the
problem and the reward achieved by the predictor; or—defined in terms of
losses—the difference between the incurred loss and the smallest achievable
loss. One analyzes excess loss (i.e., regret) instead of absolute loss so that
the bounds apply to inherently noisy problems. It turns out that the simple
approach above has regret that scales with the square root of the regressor’s
regret (see section 6 for a proof). Recalling that the latter is upper bounded
by 1, this is undesirable.
Another natural approach is to use the technique in Zadrozny . Given a
distribution $p(a)$ over the actions given $x$, the idea is to transform each
partial label example $\left(x,a,r_{a},p(a)\right)$ into an importance
weighted multiclass example $\left(x,a,r_{a}/p(a)\right)$, where $r_{a}/p(a)$
is the cost of not predicting label $a$ on input $x$. These examples are then
fed into any importance weighted multiclass classification algorithm, with the
output classifier used to make future predictions. Section 6 shows that when
$p(a)$ is uniform, the resulting regret on the original partial label problem
is bounded by $k$ times the importance weighted multiclass regret, where $k$
is the number of choices. The importance weighted multiclass classification
problem can, in turn, be reduced to binary classification, but all known
conversions yield worse bounds than the approach presented in this paper.
### Results
We propose the $\operatorname{Offset\ Tree}$ algorithm for reducing the
partial label problem to binary classification, allowing one to reuse any
existing, fully supervised binary classification algorithm for the partial
label problem.
The $\operatorname{Offset\ Tree}$ uses the following trick, which is easiest
to understand in the case of $k=2$ choices (covered in section 3). When the
observed reward $r_{a}$ of choice $a$ is low, we essentially pretend that the
other choice $a^{\prime}$ was chosen and a different reward
$r_{a^{\prime}}^{\prime}$ was observed. Precisely how this is done and why, is
driven by the regret analysis. This basic trick is composable in a binary tree
structure for $k>2$, as described in section 4.
The $\operatorname{Offset\ Tree}$ achieves computational efficiency in two
ways: First, it improves the dependence on $k$ from $O(k)$ to $O(\log_{2}k)$.
It is also an oracle algorithm, which implies that it can use the implicit
optimization in existing learning algorithms rather than a brute-force
enumeration over policies, as in the Exp4 algorithm EXP4 . We prove that the
$\operatorname{Offset\ Tree}$ policy regret is bounded by $k-1$ times the
regret of the binary classifier in solving the induced binary problems.
Section 5 shows that no reduction can provide a better guarantee, giving the
first nontrivial lower bound for learning reductions. Since the bound is tight
and has a dependence on $k$, it shows that the partial label problem is
inherently different from standard fully supervised learning problems like
$k$-class classification.
Section 6 analyzes several alternative approaches. An empirical comparison of
these approaches is given in section 7.
### Related Work
The problem considered here is a non-interactive version of the contextual
bandit problem (see Auer ; EXP4 ; EMM ; Robbins ; Woodruff for background on
the bandit problem). The interactive version has been analyzed under various
additional assumptions Banditron ; Kulkarni ; Epoch-Greedy ; hierarchy_bandit
; Associate ; WKP , including payoffs as a linear function of the side
information ABL ; Auer . The Exp4 algorithm EXP4 has a nice assumption-free
analysis. However, it is intractable when the number of policies we want to
compete with is large. It also relies on careful control of the action
choosing distribution, and thus cannot be applied to historical data, i.e.,
non-interactively.
Sample complexity results for policy evaluation in reinforcement learning RL
and contextual bandits Epoch-Greedy show that Empirical Risk Minimization
type algorithms can find a good policy in a non-interactive setting. The
results here are mostly orthogonal to these results, although we do show in
section A that a constant factor improvement in sample complexity is possible
using the offset trick.
The Banditron algorithm Banditron deals with a similar setting but does not
address several concerns that the $\operatorname{Offset\ Tree}$ addresses: (1)
the Banditron requires an interactive setting; (2) it deals with a
specialization of our setting where the reward for one choice is $1$, and $0$
for all other choices; (3) its analysis is further specialized to the case
where linear separators with a small hinge loss exist; (4) it requires
exponentially in $k$ more computation; (5) the Banditron is not an oracle
algorithm, so it is unclear, for example, how to compose it with a decision
tree bias.
Transformations from partial label problems to fully supervised problems can
be thought of as learning methods for dealing with sample selection bias bias
, which is heavily studied in Economics and Statistics.
## 2 Basic Definitions
This section reviews several basic learning problems and the
$\operatorname{Costing}$ method costing used in the construction.
A _$k$ -class classification_ problem is defined by a distribution $Q$ over
$X\times Y$, where $X$ is an arbitrary feature space and $Y$ is a label space
with $|Y|=k$. The goal is to learn a classifier $c:X\rightarrow Y$ minimizing
the _error rate_ on $Q$,
$e(c,Q)=\mathbf{Pr}_{(x,y)\sim Q}[{c(x)\neq y}]=\mathbf{E}_{(x,y)\sim
Q}[\,\mathbf{1}({c(x)\neq y})],$
given training examples of the form $(x,y)\in X\times Y$. Here
$\mathbf{1}(\cdot)$ is the indicator function which evaluates to 1 when its
argument is true, and to 0 otherwise.
Importance weighted classification is a generalization where some errors are
more costly than others. Formally, an _importance weighted classification_
problem is defined by a distribution $P$ over $X\times Y\times[\,0,\infty)$.
Given training examples of the form $(x,y,w)\in X\times Y\times[\,0,\infty)$,
where $w$ is the cost associated with mislabeling $x$, the goal is to learn a
classifier $c:X\rightarrow Y$ minimizing the _importance weighted loss_ on
$P$, $\mathbf{E}_{(x,y,w)\sim P}[w\cdot\mathbf{1}({c(x)\neq y})]$.
A folk theorem costing says that for any importance weighted distribution
$P$, there exists a constant $\overline{w}=\mathbf{E}_{(x,y,w)\sim P}[w]$ such
that for any classifier $c:X\rightarrow Y$,
$\mathbf{E}_{(x,y)\sim Q}[\,\mathbf{1}(c(x)\neq
y)]=\frac{1}{\overline{w}}\mathbf{E}_{(x,y,w)\sim P}[w\cdot\mathbf{1}(c(x)\neq
y)],$
where $Q$ is the distribution over $X\times Y$ defined by
$Q(x,y,w)=\frac{w}{\overline{w}}P(x,y,w),$
marginalized over $w$. In other words, choosing $c$ to minimize the error rate
under $Q$ is equivalent to choosing $c$ to minimize the importance weighted
loss under $P$.
The $\operatorname{Costing}$ method costing can be used to resample the
training set drawn from $P$ using rejection sampling on the importance weights
(an example with weight $w$ is accepted with probability proportional to $w$),
so that the resampled set is effectively drawn from $Q$. Then, any binary
classification algorithm can be run on the resampled set to optimize the
importance weighted loss on $P$.
$\operatorname{Costing}$ runs a base classification algorithm on multiple
draws of the resampled set, and averages over the learned classifiers when
making importance weighted predictions (see costing for details). To simplify
the analysis, we do not actually have to consider separate classifiers. We can
simply augment the feature space with the index of the resampled set and then
learn a single classifier on the union of all resampled data. The implication
of this observation is that we can view $\operatorname{Costing}$ as a machine
that maps importance weighted examples to unweighted examples. We use this
method in Algorithms 1 and 2 below.
## 3 The Binary Case
This section deals with the special case of $k=2$ actions. We state the
algorithm, prove the regret bound (which is later used for the general $k$
case), and state a sample complexity bound. For simplicity, we let the two
action choices in this section be $1$ and $-1$.
### 3.1 The Algorithm
The $\operatorname{Binary\ Offset}$ algorithm is a reduction from the 2-class
partial label problem to binary classification. The reduction operates per
example, implying that it can be used either online or offline. We state it
here for the offline case. The algorithm reduces the original problem to
binary importance weighted classification, which is then reduced to binary
classification using the $\operatorname{Costing}$ method described above. A
base binary classification algorithm $\operatorname{Learner}$ is used as a
subroutine.
The key trick appears inside the loop in Algorithm 1, where importance
weighted binary examples are formed. The offset of $1/2$ changes the range of
importances, effectively reducing the variance of the induced problem. This
trick is driven by the regret analysis in section 3.2.
set $S^{\prime}=\emptyset$
for __each $(x,a,r_{a},p(a))\in S$__ do Form an importance weighted example
$\displaystyle\quad(x,y,w)=\left(x,\mbox{sign}\left(a\left(r_{a}-1/2\right)\right),\frac{1}{p(a)}\left|r_{a}-1/2\right|\right).$
Add $(x,y,w)$ to $S^{\prime}$.
return $\operatorname{Learner}(\operatorname{Costing}(S^{\prime}))$.
Algorithm 1 $\operatorname{Binary\ Offset}$ (binary classification algorithm
$\operatorname{Learner}$, 2-class partial label dataset $S$)
### 3.2 Regret Analysis
This section proves a regret transform theorem for the $\operatorname{Binary\
Offset}$ reduction. Informally, _regret_ measures how well a predictor
performs compared to the best possible predictor on the same problem. A
_regret transform_ shows how the regret of a base classifier on the induced
(binary classification) problem controls the regret of the resulting policy on
the original (partial label) problem. Thus a regret transform bounds only
excess loss due to suboptimal prediction.
$\operatorname{Binary\ Offset}$ transforms partial label examples into binary
examples. This process implicitly transforms the distribution $D$ defining the
partial label problem into a distribution $Q_{D}$ over binary examples, via a
distribution over importance weighted binary examples. Note that even though
the latter distribution depends on both $D$ and the action-choosing
distribution $p$, the induced binary distribution $Q_{D}$ depends only on $D$.
Indeed, the probability of label 1 given $x$ and $\vec{r}$, according to
$Q_{D}$, is
$\displaystyle\mathbf{E}_{a\sim p}$
$\displaystyle\left[\frac{|r_{a}-{1/2}|}{p(a)}\cdot\mathbf{1}\left(a(r_{a}-\frac{1}{2})>0\right)\right]$
$\displaystyle=\mathbf{1}(r_{1}>\frac{1}{2})\left|r_{1}-\frac{1}{2}\right|+\mathbf{1}(r_{-1}<\frac{1}{2})\left|r_{-1}-\frac{1}{2}\right|,$
independent of $p$.
The _binary regret_ of a classifier $c:X\rightarrow\\{-1,1\\}$ on $Q_{D}$ is
given by
$\operatorname{reg}_{e}(c,{Q}_{D})=e(c,{Q}_{D})-\min_{c^{\prime}}e(c^{\prime},{Q}_{D}),$
where the min is over all classifiers $c^{\prime}:X\rightarrow\\{1,-1\\}$. The
_importance weighted regret_ is definited similarly with respect to the
importance weighted loss.
For the $k=2$ partial label case, the policy that a classifier $c$ induces is
simply the classifier. The regret of policy $c$ is defined as
$\operatorname{reg}_{\eta}(c,D)=\max_{c^{\prime}}\eta(c^{\prime},D)-\eta(c,D),$
where
$\eta(c,D)=\mathbf{E}_{(x,\vec{r})\sim D}\left[r_{c(x)}\right],$
is the value of the policy.
The theorem below states that the policy regret is bounded by the binary
regret. We find it surprising because strictly less information is available
than in binary classification. Note that the lower bound in section 5 implies
that no reduction can do better. Redoing the proof with the offset set to $0$
rather than $1/2$ also reveals that $2\operatorname{reg}_{e}(c,Q_{D})$ bounds
the policy regret, implying that the offset trick gives a factor of 2
improvement in the bound.
Finally, note that the theorem is quantified over all classifiers, which
includes the classifier returned by $\operatorname{Learner}$ in the last line
of the algorithm.
###### Theorem 3.1
_( $\operatorname{Binary\ Offset}$ Regret)_ For all $2$-class partial label
problems $D$ and all binary classifiers $c$,
$\displaystyle\operatorname{reg}_{\eta}(c,D)\leq\operatorname{reg}_{e}(c,Q_{D}).$
Furthermore, there exists $D$ such that for all values $v\in[0,1]$ there
exists $c$ such that
$v=\operatorname{reg}_{\eta}(c,D)=\operatorname{reg}_{e}(c,Q_{D})$ (i.e. the
bound is tight).
Proof We first bound the partial label regret of $c$ in terms of importance
weighted regret, and then apply known results to relate the importance
weighted regret to binary regret.
Conditioned on a particular value of $x$, we either make a mistake or we do
not. If no mistake is made, then the regrets of both sides are $0$, and the
claim holds trivially. Assume that a mistake is made. Without loss of
generality, $r_{1}>r_{-1}$ and label $-1$ is chosen. The expected importance
weight of label $-1$ is given by
$\displaystyle\mathbf{E}_{\vec{r}\sim D|x}\,$ $\displaystyle\mathbf{E}_{a\sim
p(a)}\left[\frac{1}{p(a)}\left|r_{a}-1/2\right|\cdot\mathbf{1}\left({a(r_{a}-1/2)<0}\right)\right]$
$\displaystyle=\mathbf{E}_{\vec{r}\sim
D|x}\left[\left(\frac{1}{2}-r_{1}\right)_{+}+\left(r_{-1}-\frac{1}{2}\right)_{+}\right]$
where we use the operator $(Z)_{+}=Z\cdot\mathbf{1}\left({Z>0}\right)$. The
difference in expected importance weights between label $1$ and label $-1$ is
$\displaystyle\mathbf{E}_{\vec{r}\sim D|x}$
$\displaystyle\left[\left(\frac{1}{2}-r_{-1}\right)_{+}+\left(r_{1}-\frac{1}{2}\right)_{+}\right]$
$\displaystyle-\mathbf{E}_{\vec{r}\sim
D|x}\left[\left(\frac{1}{2}-r_{1}\right)_{+}+\left(r_{-1}-\frac{1}{2}\right)_{+}\right]$
$\displaystyle=\mathbf{E}_{\vec{r}\sim
D|x}\left[\left(\frac{1}{2}-r_{-1}\right)+\left(r_{1}-\frac{1}{2}\right)\right]$
$\displaystyle=\mathbf{E}_{\vec{r}\sim
D|x}[r_{1}-r_{-1}]=\operatorname{reg}_{\eta}(c,D|x).$
This shows that the importance weighted regret of the binary classifier is the
policy regret. The folk theorem from section 2 (see costing ) says that the
importance weighted regret is bounded by the binary regret, times the expected
importance. The latter is $\mathbf{E}_{\vec{r}\sim D|x}\,\mathbf{E}_{a\sim
p(a)}\left[\frac{1}{p(a)}|r_{a}-1/2|\right]=\mathbf{E}_{\vec{r}\sim
D|x}\left[\,|r_{1}-1/2|+|r_{-1}-1/2|\,\right]\leq 1,$ since both $r_{1}$ and
$r_{-1}$ are bounded by 1. This proves the first part of the theorem.
For the second part, notice that the proof of the first part can be made an
equality by having a reward vector $(0,1)$ for each $x$ always, and letting
the classifier predict label $1$ with probability $(1-v)$ over the draw of
$x$.
## 4 The Offset Tree Reduction
In this section we deal with the case of large $k$.
### 4.1 The Offset Tree Algorithm
The technique in the previous section can be applied repeatedly using a tree
structure to give an algorithm for general $k$. Consider a maximally balanced
binary tree on the set of $k$ choices, conditioned on a given observation $x$.
Every internal node in the tree is associated with a classification problem of
predicting which of its two inputs has the larger expected reward. At each
node, the same offsetting technique is used as in the binary case described in
section 3.
For an internal node $v$, let $\Gamma(T_{v})$ denote the set of leaves in the
subtree $T_{v}$ rooted at $v$. Every input to a node is either a leaf or a
winning choice from another internal node closer to the leaves.
Fix a binary tree $T$ over the choices
for __each internal node $v$ in order from leaves to root__ do Set
$S_{v}=\emptyset$
for __each $(x,a,r_{a},p(a))\in S$ such that $a\in\Gamma(T_{v})$ and all nodes
on the path $v\leadsto a$ predict $a$ on $x$__ do Let $a^{\prime}$ be the
other choice at $v$ and y=1(a’ comes from the left subtree of $v$)
if $r_{a}<1/2$,
add $(x,y,\dfrac{p(a)+p(a^{\prime})}{p(a^{\prime})}(1/2-r_{a}))$ to $S_{v}$
else add $(x,1-y,\dfrac{p(a)+p(a^{\prime})}{p(a)}(r_{a}-1/2))$
Let $c_{v}=\operatorname{Learner}(\textrm{Costing}(S_{v}))$
return $c=\\{c_{v}\\}$
Algorithm 2 $\operatorname{Offset\ Tree}$ (binary classification algorithm
$\operatorname{Learner}$, partial label dataset $S$)
The training algorithm, $\operatorname{Offset\ Tree}$, is given in Algorithm
2. The testing algorithm defining the predictor is given in Algorithm 3.
return unique action $a$ for which every classifier $c_{v}$ from $a$ to root
prefers $a$.
Algorithm 3 Offset Test (classifiers $\\{c_{v}\\}$, unlabeled example $x$)
### 4.2 The Offset Tree Regret Theorem
The theorem below gives an extension of Theorem 3.1 for general $k$. For the
analysis, we use a simple trick which allows us to consider only a single
induced binary problem, and thus a single binary classifier $c$. The trick is
to add the node index as an additional feature into each importance weighted
binary example created algorithm 2, and then train based upon the union of all
the training sets.
As in section 3, the reduction transforms a partial label distribution $D$
into a distribution $Q_{D}$ over binary examples. To draw from $Q_{D}$, we
draw $(x,\vec{r})$ from $D$, an action $a$ from the action-choosing
distribution $p$, and apply algorithm 2 to transform $(x,\vec{r},a,p(a))$ into
a set of binary examples (up to one for each level in the tree) from which we
draw uniformly at random. Note that $Q_{D}$ is independent of $p$, as
explained in the beginning of section 3.
Denote the policy induced by the Offset-Test algorithm using classifier $c$ by
$\pi_{c}$. For the following theorem, the definitions of regret are from
section 3.
###### Theorem 4.1
_( $\operatorname{Offset\ Tree}$ Regret)_ For all $k$-class partial label
problems $D$, for all binary classifiers $c$,
$\displaystyle\operatorname{reg}_{\eta}(\pi_{c},D)$
$\displaystyle\leq\operatorname{reg}_{e}(c,Q_{D})\cdot\mathbf{E}_{(x,\vec{r})\sim
D}\hskip-7.22743pt\sum_{v(a,a^{\prime})\in
T}\hskip-12.28577pt\big{[}\,|r_{a}-\frac{1}{2}|+|r_{a^{\prime}}-\frac{1}{2}|\,\big{]}$
$\displaystyle\leq(k-1)\operatorname{reg}_{e}(c,Q_{D}),$
where $v(a,a^{\prime})$ ranges over the $(k-1)$ internal nodes in $T$, and $a$
and $a^{\prime}$ are its inputs determined by $c$’s predictions.
Note: Section 5 shows that no reduction can give a better regret transform
theorem. With a little bit of side information, however, we can do better: The
offset minimizing the regret bound turns out to be the median value of the
reward given $x$. Thus, it is generally best to pair choices which tend to
have similar rewards. Note that the algorithm need not know how well $c$
performs on $Q_{D}$.
The proof below can be reworked with the offset set to $0$, resulting in a
regret bound which is a factor of $2$ worse.
Proof We fix $x$, taking the expectation over the draw of $x$ at the end. The
first step is to show that the partial label regret is bounded by the sum of
the importance weighted regrets over the binary prediction problems in the
tree. We then apply the costing analysis costing to bound this sum in terms
of the binary regret.
The proof of the first step is by induction on the nodes in the tree. We want
to show that the sum of the importance weighted regrets of the nodes in any
subtree bounds the regret of the output choice for the subtree. The hypothesis
trivially holds for one-node trees.
Consider a node $u$ making an importance weighted decision between choices $a$
and $a^{\prime}$. The expected importance of choice $a$ is given by
$\displaystyle\mathbf{E}_{\vec{r}\sim D|x}$
$\displaystyle\left[p(a)\frac{p(a)+p(a^{\prime})}{p(a)}(r_{a}-1/2)_{+}\right.$
$\displaystyle\left.+p(a^{\prime})\frac{p(a)+p(a^{\prime})}{p(a^{\prime})}(1/2-r_{a^{\prime}})_{+}\right]$
$\displaystyle=\mathbf{E}_{\vec{r}\sim
D|x}[(r_{a}-1/2)_{+}+(1/2-r_{a^{\prime}})_{+}].$
It is important to note that, by construction, only two actions can generate
examples for a given internal node. Without loss of generality, assume that
$a^{\prime}$ has the larger expected reward. The expected importance weighted
binary regret $\operatorname{wreg}_{u}$ of the classifier’s decision is either
$0$ if it predicts $a^{\prime}$, or
$\displaystyle\operatorname{wreg}_{u}=$ $\displaystyle\mathbf{E}_{\vec{r}\sim
D|x}\left[\left(r_{a}^{\prime}-1/2\right)_{+}+\left(1/2-r_{a}\right)_{+}\right]$
$\displaystyle-\mathbf{E}_{\vec{r}\sim
D|x}\left[\left(r_{a}-1/2\right)_{+}+\left(1/2-r_{a^{\prime}}\right)_{+}\right]$
$\displaystyle=$ $\displaystyle\mathbf{E}_{\vec{r}\sim
D|x}[1/2-r_{a}+r_{a^{\prime}}-1/2]=\mathbf{E}_{\vec{r}\sim
D|x}[r_{a^{\prime}}-r_{a}]$
if the classifier predicts $a$.
Let $T_{v}$ be the subtree rooted at node $v$, and let $a$ be the choice
output by $T_{v}$ on $x$. If the best choice in $\Gamma(T_{v})$ comes from the
subtree $L$ producing $a$, the policy regret of $T_{v}$ is given by
$\displaystyle\operatorname{Reg}({T_{v}})$
$\displaystyle=\max_{y\in\Gamma(L)}\mathbf{E}_{\vec{r}\sim
D|x}[r_{y}]-\mathbf{E}_{\vec{r}\sim D|x}[r_{a}]$
$\displaystyle=\operatorname{Reg}({L})\leq\sum_{u\in
L}\operatorname{wreg}_{u}\leq\sum_{u\in{T_{v}}}\operatorname{wreg}_{u}.$
If on the other hand the best choice comes from the other subtree $R$, we have
$\displaystyle\operatorname{Reg}({T_{v}})$
$\displaystyle=\max_{y\in\Gamma(R)}\mathbf{E}_{\vec{r}\sim
D|x}[r_{y}]-\mathbf{E}_{\vec{r}\sim D|x}[r_{a}]$
$\displaystyle=\operatorname{Reg}({R})+\mathbf{E}_{\vec{r}\sim
D|x}[r_{a^{\prime}}]-\mathbf{E}_{\vec{r}\sim D|x}[r_{a}]$
$\displaystyle\leq\sum_{u\in
R}\operatorname{wreg}_{u}+\operatorname{wreg}_{v}\leq\sum_{u\in{T_{v}}}\operatorname{wreg}_{u},$
proving the induction.
The induction hypothesis applied to $T$ tells us that
$\operatorname{Reg}(T)\leq\sum_{v\in T}\operatorname{wreg}_{v}$. According to
the Costing theorem discussed in section 2, the importance weighted regret is
bounded by the unweighted regret on the resampled distribution, times the
expected importance. The expected importance of deciding between actions $a$
and $a^{\prime}$ is
$\displaystyle\mathbf{E}_{\vec{r}\sim
D|x}\left[p(a)\frac{1}{p(a)}|r_{a}-1/2|+p(a^{\prime})\frac{1}{p(a^{\prime})}|r_{a^{\prime}}-1/2|\right]\leq
1$
since all rewards are between 0 and 1. Noting that
$\operatorname{Reg}(T)=\operatorname{reg}_{\eta}(\pi_{c},D\,|\,x)$, we thus
have
$\operatorname{reg}_{\eta}(\pi_{c},D\,|\,x)\leq(k-1)\operatorname{reg}_{e}(c,Q_{D}\,|\,x),$
completing the proof for any $x$. Taking the expectation over $x$ finishes the
proof.
The setting above is akin to Boosting adaboost : At each round $t$, a booster
creates an input distribution $D_{t}$ and calls an oracle learning algorithm
to obtain a classifier with some error $\epsilon_{t}$ on $D_{t}$. The
distribution $D_{t}$ depends on the classifiers returned by the oracle in
previous rounds. The accuracy of the final classifier is analyzed in terms of
$\epsilon_{t}$’s. The binary problems induced at internal nodes of an offset
tree depend, similarly, on the classifiers closer to the leaves. The
performance of the resulting partial label policy is analyzed in terms of the
oracle’s performance on these problems. (Notice that Theorem 4.1 makes no
assumptions on the error rates on the binary problems; in particular, it
doesn’t require them to be bounded away from $1/2$.)
For the analysis, we use the simple trick from the beginning of this
subsection to consider only a single binary classifier. The theorem is
quantified over all classifiers, and thus it holds for the classifier returned
by the algorithm. In practice, one can either call the oracle multiple times
to learn a separate classifier for each node (as we do in our experiments), or
use iterative techniques for dealing with the fact that the classifiers are
dependent on other classifiers closer to the leaves.
## 5 A Lower Bound
This section shows that no method for reducing the partial label setting to
binary classification can do better. First we formalize a learning reduction
which relies upon a binary classification oracle. The lower bound we prove
below holds for all such learning reductions.
###### Definition 5.1
_(Binary Classification Oracle) A binary classification oracle $O$ is a
(stateful) program that supports two kinds of queries:_
1. 1.
Advice. An advice query $O(x,y)$ consists of a single example $(x,y)$, where
$x$ is a feature vector and $y\in\\{1,-1\\}$ is a binary label. An advice
query is equivalent to presenting the oracle with a training example, and has
no return value.
2. 2.
Predict. A predict query $O(x)$ is made with a feature vector $x$. The return
value is a binary label.
All learning reductions work on a per-example basis, and that is the
representation we work with here.
###### Definition 5.2
_(Learning Reduction) A learning reduction is a pair of algorithms $R$ and
$R^{-1}$._
1. 1.
The algorithm $R$ takes a partially labeled example $(x,a,r_{a},p(a))$ and a
binary classification oracle $O$ as input, and forms a (possibly dependent)
sequence of advice queries.
2. 2.
The algorithm $R^{-1}$ takes an unlabeled example $x$ and a binary
classification oracle $O$ as input. It asks a (possibly dependent) sequence of
predict queries, and makes a prediction dependent only on the oracle’s
predictions. The oracle’s predictions may be adversarial (and are assumed so
by the analysis).
We are now ready to state the lower bound.
###### Theorem 5.1
For all reductions $(R,R^{-1})$, there exists a partial label problem $D$ and
an oracle $O$ such that
$\operatorname{reg}_{\eta}(R^{-1}(O),D)\geq(k-1)\operatorname{reg}_{e}(O,R(D)),$
where $R(D)$ is the binary distribution induced by $R$ on $D$, and $R^{-1}(O)$
is the policy resulting from $R^{-1}$ using $O$.
Proof The proof is by construction. We choose $D$ to be uniform over $k$
examples, with example $i$ having 1 in its $i$-th component of the reward
vector, and zeros elsewhere. The corresponding feature vector consists of the
binary representation of the index with reward 1. Let the action-choosing
distribution be uniform.
The reduction $R$ produces some simulatable sequence of advice calls when the
observed reward is 0. The oracle ignores all advice calls from $R$ and chooses
to answer all queries with zero error rate according to this sequence.
There are two cases: Either $R$ observes $0$ reward (with probability
$(k-1)/k$) or it observes reward $1$ (with probability $1/k$). In the first
case, the oracle has $0$ error rate (and, hence $0$ regret). In the second
case, it has error rate (and regret) of at most $1$. Thus the expected error
rate of the oracle on $R(D)$ is at most $1/k$.
The inverse reduction $R^{-1}$ has access to only the unlabeled example $x$
and the oracle $O$. Since the oracle’s answers are independent of the draw
from $D$, the output action has reward $0$ with probability $(k-1)/k$ and
reward $1$ with probability $1/k$, implying a regret of $(k-1)/k$ with respect
to the best policy. This is a factor of $k-1$ greater than the regret of the
oracle, proving the lower bound.
## 6 Analysis of Simple Reductions
This section analyzes two simple approaches for reducing partial label
problems to basic supervised learning problems. These approaches have been
discussed previously, but the analysis is new.
### 6.1 The Regression Approach
The most obvious approach is to regress on the value of a choice as in
Algorithm 4, and then use the argmax classifier as in Algorithm 5. Instead of
learning a single regressor, we can learn a separate regressor for each
choice.
Let $S^{\prime}=\emptyset$
for _each $(x,a,r_{a})\in S$_ do Add $((x,a),r_{a})$ to $S^{\prime}$.
return $f=\textrm{Regress}(S^{\prime})$.
Algorithm 4 Partial-Regression (regression algorithm Regress, partial label
dataset $S$)
return $\arg\max_{a}f(x,a)$
Algorithm 5 Argmax (regressor $f$, unlabeled example $x$)
The squared error of a regressor $f:X\rightarrow\mathbb{R}$ on a distribution
$P$ over $X\times\mathbb{R}$ is denoted by
$\ell_{r}(f,P)=\mathbf{E}_{(x,y)\sim P}(f(x)-y)^{2}.$
The corresponding regret is given by
$\operatorname{reg}_{r}(f,P)=\ell_{r}(f,P)-\min_{f^{\prime}}\ell_{r}(f^{\prime},P)$.
The following theorem relates the regret of the resulting predictor to that of
the learned regressor.
###### Theorem 6.1
For all $k$-class partial label problems $D$ and all squared-error regressors
$f$,
$\operatorname{reg}_{\eta}(\pi_{f},D)\leq\sqrt{2k\operatorname{reg}_{r}(f,P_{D})},$
where $P_{D}$ is the regression distribution induced by Algorithm 4 on $D$,
and $\pi_{f}$ is the argmax policy based on $f$. Furthermore, there exist $D$
and $h$ such that the bound is tight.
The theorem has a square root, which is undesirable, because the theorem is
vacuous when the right hand side is greater than 1.
Proof Let $\pi_{f}$ choose some action $a$ with true value
$v_{a}=\mathbf{E}_{(x,\vec{r})\sim D}[r_{a}]$. Some other action $a^{*}$ may
have a larger expected reward $v_{a^{*}}>v_{a}$. The squared error regret
suffered by $f$ on $a$ is $\mathbf{E}_{(x,\vec{r})\sim
D}[(r_{a}-v_{a})^{2}-(r_{a}-f(x,a))^{2}]=(v_{a}-f(x,a))^{2}$. Similarly for
$a^{*}$, we have regret $\left(v_{a^{*}}-f(x,a^{*})\right)^{2}$. In order for
$a$ to be chosen over $a^{*}$, we must have $f(x,a)\geq f(x,a^{*})$. Convexity
of the two regrets implies that the minima is reached when
$f(x,a)=f(x,a^{*})=\frac{v_{a}+v_{a^{*}}}{2}$, where the regret for each of
the two choices is $\left(\frac{v_{a^{*}}-v_{a}}{2}\right)^{2}$. The regressor
need not suffer any regret on the other $k-2$ arms. Thus with average regret
$\frac{\left(v_{a^{*}}-v_{a}\right)^{2}}{2k}$ a regret of $v_{a^{*}}-v_{a}$
can be induced, completing the proof of the first part. For the second part,
note that an adversary can play the optimal strategy outlined above achieving
the bound precisely.
### 6.2 Importance Weighted Classification
Zadrozny Zadrozny noted that the partial label problem could be reduced to
importance weighted multiclass classification. After Algorithm 6 creates
importance weighted multiclass examples, the weights are stripped using
Costing (the rejection sampling on the weights discussed in Section 2), and
then the resulting multiclass distribution is converted into a binary
distribution using, for example, the all-pairs reduction all-pairs ). The last
step is done to get a comparable analysis.
Let $S^{\prime}=\emptyset$
for __each $(x,a,p(a),r_{a})\in S$__ do Add $(x,a,\frac{r_{a}}{p(a)})$ to
$S^{\prime}$.
return $\textrm{All-Pairs-
Train}\,(\textrm{Learn},\textrm{Costing}(S^{\prime}))$
Algorithm 6 IWC-Train (binary classification algorithm Learn, partial label
dataset $S$)
All-Pairs-Train uses a given binary learning algorithm Learn to distinguish
each pair of classes in the multiclass distribution created by Costing. The
learned classifier $c$ predicts, given $x$ and a distinct pair of classes
$(i,j)$, whether class $i$ is more likely than $j$ given $x$. At test time, we
make a choice using All-Pairs-Test, which takes $c$ and an unlabeled example
$x$, and returns the class that wins the most pairwise comparisons on $x$,
according to $c$.
return $\textrm{All-Pairs-Test}(c,x)$.
Algorithm 7 IWC-Test (binary classifier $c$, unlabeled example $x$)
A basic theorem applies to this approach.
###### Theorem 6.2
For all $k$-class partial label problems $D$ and all binary classifiers $c$,
__
$\displaystyle\operatorname{reg}_{\eta}(\pi_{c},D)$
$\displaystyle\leq\operatorname{reg}_{e}(c,Q_{D})(k-1)\mathbf{E}_{(x,\vec{c})\sim
D}\sum_{a}(1-c_{a})$ $\displaystyle\leq\operatorname{reg}_{e}(c,Q_{D})(k-1)k,$
where $\pi_{c}$ is the _IWC-Test_ policy based on $c$ and $Q_{D}$ is the
binary distribution induced by _IWC-Train_ on $D$.
Proof The proof first bounds the policy regret in terms of the importance
weighted multiclass regret. Then, we apply known results for the other
reductions to relate the policy regret to binary classification regret.
Fix a particular $x$. The policy regret of choosing action $a$ over the best
action $a^{*}$ is $\mathbf{E}_{r\sim D|x}[r_{a^{*}}]-\mathbf{E}_{r\sim
D|x}[r_{a}]$. The importance weighted multiclass loss of action $a$ is
$\mathbf{E}_{r\sim D|x}\sum_{a^{\prime}\neq
a}\frac{p(a^{\prime})r_{a^{\prime}}}{p(a^{\prime})}=\mathbf{E}_{r\sim
D|x}\sum_{a^{\prime}\neq a}r_{a^{\prime}}$
since the loss is proportional to $\frac{1}{p(a^{\prime})}r_{a^{\prime}}$ with
probability $p(a^{\prime})$. This implies the importance weighted regret of
$\mathbf{E}_{r\sim D|x}\sum_{a^{\prime}\neq a}r_{a^{\prime}}-\mathbf{E}_{r\sim
D|x}\sum_{a^{\prime}\neq a^{*}}r_{a^{\prime}}=\mathbf{E}_{r\sim
D|x}[r_{a^{*}}-r_{a}],$
which is the same as the policy regret.
The importance weighted regret is bounded by the unweighted regret, times the
expected importance (see costing ), which in turn is bounded by $k$.
Multiclass regret on $k$ classes is bounded by binary regret times $k-1$ using
the all-pairs reduction all-pairs , which completes the proof.
Relative to the $\operatorname{Offset\ Tree}$, this theorem has an undesirable
extra factor of $k$ in the regret bound. While this factor is due to the all-
pairs reduction being a weak regret transform, we are aware of no alternative
approach for reducing multiclass to binary classification that in composition
can yield the same regret transform as the $\operatorname{Offset\ Tree}$.
## 7 Experimental Results
| Properties | | Single regressor | $k$ regressors |
---|---|---|---|---|---
Dataset | $k$ | $m$ | Weighting | M5P | REPTree | M5P | REPTree | Offset Tree
ecoli | 8 | 336 | 0.3120 | 0.5663 | 0.3376 | 0.3752 | 0.3811 | 0.2311
flare | 7 | 1388 | 0.1565 | 0.1570 | 0.1685 | 0.1570 | 0.1592 | 0.1506
glass | 6 | 214 | 0.5938 | 0.6662 | 0.5846 | 0.5800 | 0.6077 | 0.5000
letter | 25 | 20000 | 0.3546 | 0.6974 | 0.5491 | 0.4456 | 0.5352 | 0.3790
lymph | 4 | 148 | 0.2953 | 0.5267 | 0.4622 | 0.3422 | 0.3400 | 0.3114
optdigits | 10 | 5620 | 0.1682 | 0.5426 | 0.4108 | 0.1948 | 0.2956 | 0.1649
page-blocks | 5 | 5473 | 0.0407 | 0.0590 | 0.0451 | 0.0571 | 0.0465 | 0.0488
pendigits | 10 | 10992 | 0.1029 | 0.2492 | 0.1840 | 0.1408 | 0.1774 | 0.0976
satimage | 6 | 6435 | 0.1703 | 0.2027 | 0.1968 | 0.1787 | 0.1878 | 0.1853
soybean | 19 | 683 | 0.6533 | 0.8824 | 0.7327 | 0.7688 | 0.7473 | 0.5971
vehicle | 4 | 846 | 0.3719 | 0.6142 | 0.5665 | 0.3886 | 0.4114 | 0.3743
vowel | 11 | 990 | 0.6403 | 0.9034 | 0.8919 | 0.7440 | 0.8198 | 0.6501
yeast | 10 | 1484 | 0.5406 | 0.6626 | 0.5679 | 0.5406 | 0.5697 | 0.4904
Table 1: Dataset-specific test error rates (see section 7.1). Here $k$ is the
number of choices and $m$ is the number of examples
We conduct two sets of experiments. The first set compares the Offset Tree
with the two approaches from section 6. The second compares with the Banditron
Banditron on the dataset used in that paper.
### 7.1 Comparisons with Reductions
Ideally, this comparison would be with a data source in the partial label
setting. Unfortunately, data of this sort is rarely available publicly, so we
used a number of publicly available multiclass datasets UCI and allowed
queries for the reward ($1$ or $0$ for correct or wrong) of only one value per
example.
Figure 1: Error rates (in %) of $\operatorname{Offset\ Tree}$ versus the
regression approach using two different base regression algorithms (left) and
$\operatorname{Offset\ Tree}$ versus Importance Sampling (right) on several
different datasets using decision trees as a base classifier learner.
For all datasets, we report the average result over 10 random splits (fixed
for all methods), with $2/3$ of the dataset used for training and $1/3$ for
testing. Figure 1 shows the error rates (in %) of the $\operatorname{Offset\
Tree}$ plotted against the error rates of the regression (left) and the
importance weighting (right). Decision trees (J48 in Weka weka ) were used as
a base binary learning algorithm for both the $\operatorname{Offset\ Tree}$
and the importance weighting. For the regression approach, we learned a
separate regressor for each of the $k$ choices. (A single regressor trained by
adding the choice as an additional feature performed worse.) M5P and REPTree,
both available in Weka weka , were used as base regression algorithms.
The $\operatorname{Offset\ Tree}$ clearly outperforms regression, in some
cases considerably. The advantage over importance weighting is moderate: Often
the performance is similar and occasionally it is substantially better.
We did not perform any parameter tuning because we expect that practitioners
encountering partial label problems may not have the expertise or time for
such optimization. All datasets tested are included. Note that although some
error rates appear large, we are choosing among $k$ alternatives and thus an
error rate of less than $1-1/k$ gives an advantage over random guessing.
Dataset-specific test error rates are reported in Table 1.
### 7.2 Comparison with the Banditron Algorithm
The Banditron Banditron is an algorithm for the special case of the problem
where one of the rewards is $1$ and the rest are $0$. The sample complexity
guarantees provided for it are particularly good when the correct choice is
separated by a multiclass margin from the other classes.
We chose the Binary Perceptron as a base classification algorithm since it is
the closest fully supervised learning algorithm to the Banditron. Exploration
was done according to Epoch-Greedy Epoch-Greedy instead of Epsilon-Greedy (as
in the Banditron), motivated by the observation that the optimal rate of
exploration should decay over time. The Banditron was tested on one dataset, a
4-class specialization of the Reuters RCV1 dataset consisting of 673,768
examples. We use precisely the same dataset, made available by the authors of
Banditron .
Since the Banditron analysis suggests the realizable case, and the dataset
tested on is nearly perfectly separable, we also specialized the
$\operatorname{Offset\ Tree}$ for the realizable case. In particular, in the
realizable case we can freely learn from every observation implying it is
unnecessary to importance weight by $1/p(a)$. We also specialize Epoch-Greedy
to this case by using a realizable bound, resulting in a probability of
exploration that decays as $1/t^{1/2}$ rather than $1/t^{1/3}$.
The algorithms are compared according to their error rate. For the Banditron,
the error rate after one pass on the dataset was $16.3\%$. For the realizable
$\operatorname{Offset\ Tree}$ method above, the error rate was $10.72\%$. For
the fully agnostic version of the $\operatorname{Offset\ Tree}$, the error
rate was $18.6\%$. These results suggest there is some tradeoff between being
optimal when there is arbitrary noise, and performance when there is no or
very little noise. In the no-noise situation, the realizable
$\operatorname{Offset\ Tree}$ performs substantially superior to the
Banditron.
## 8 Discussion
We have analyzed the tractability of learning when only one outcome from a set
of $k$ alternatives is known, in the reductions setting. The
$\operatorname{Offset\ Tree}$ approach has a worst-case dependence on $k-1$
(Theorem 4.1), and no other reduction approach can provide a better guarantee
(Section 5). Furthermore, with an $O(\log k)$ computation, the
$\operatorname{Offset\ Tree}$ is qualitatively more efficient than all other
known algorithms, the best of which are $O(k)$. Experimental results suggest
that this approach is empirically promising.
The algorithms presented here show how to learn from one step of exploration.
By aggregating information over multiple steps, we can learn good policies
using binary classification methods. A straightforward extension of this
method to deeper time horizons $T$ is not compelling as $k-1$ is replaced by
$k^{T}$ in the regret bounds. Due to the lower bound proved here, it appears
that further progress on the multi-step problem in this framework must come
with additional assumptions.
## 9 Acknowledgements
We would like to thank Tong Zhang, Alex Strehl, and Sham Kakade for helpful
discussions. We would also like to thank Shai Shalev-Shwartz for providing
data and helping setup a clean comparison with the Banditron.
## References
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## A Sample Complexity Bound
This section proves a simple sample complexity bound on the performance of
$\operatorname{Binary\ Offset}$. For ease of comparison with existing results,
we specialize the problem set to partial label _binary classification_
problems where one label has reward $1$ and the other label has reward $0$.
Note that this is not equivalent to assuming realizability: Conditioned on
$x$, any distribution over reward vectors $(0,1)$ and $(1,0)$ is allowed.
Comparing the bound with standard results in binary classification (see, for
example, Tutorial ), shows that the bounds are identical, while eliminating
the offset trick weakens the performance by a factor of roughly 2.
When a sample set is used as a distribution, we mean the uniform distribution
over the sample set (i.e., an empirical average).
###### Theorem A.1
_( $\operatorname{Binary\ Offset}$ Sample Complexity)_ Let the action choosing
distribution be uniform. For all partial label binary classification problems
$D$ and all sets of binary classifiers $C$, after observing a set $S$ of $m$
examples drawn independently from $D$, with probability at least $1-\delta$,
$\displaystyle|\eta(c,D)-\eta(c,S)|\leq\sqrt{\frac{\ln|C|+\ln(2/\delta)}{2m}}$
holds simultaneously for all classifiers $c\in C$. Furthermore, if the offset
is set to $0$, then
$\displaystyle|\eta(c,D)-\eta(c,S)|\leq\sqrt{\frac{\ln|C|+\ln(3/\delta)}{m-2\sqrt{m\ln(3/\delta)}}}.$
Proof First note that for partial label binary classification problems, the
$\operatorname{Binary\ Offset}$ reduction recovers the correct label. Since
all importance weights are $1$, no examples are lost in converting from
importance weighted classification to binary classification. Consequently, the
Occam’s Razor bound on the deviations of error rates implies that, with
probability $1-\delta$, for all classifiers $c\in C$,
$|e(c,Q_{D})-e(c,Q_{D})|\leq\sqrt{(\ln|C|+\ln(2/\delta))/{2m}}$, where the
induced distribution $Q_{D}$ is $D$ with the two reward vectors encoded as
binary labels. Observing that $e(c,Q_{D})=\eta(c,D)$ finishes the first half
of the proof.
For the second half, notice that rejection sampling reduces the number of
examples by a factor of two in expectation; and with probability at least
$1-\delta/3$, this number is at least $m/2-\sqrt{m\ln(3/\delta)}$. Applying
the Occam’s Razor bound with probability of failure $2\delta/3$, gives
$|e(c,Q_{D})-e(c,Q_{D})|\leq\sqrt{\frac{\ln|C|+\ln(3/\delta)}{m-2\sqrt{m\ln(3/\delta)}}}.$
Taking the union bound over the two failure modes proves that the above
inequality holds with probability $1-\delta$. Observing the equivalence
$e(c,Q_{D})=\eta(c,D)$ gives us the final result.
The sample complexity bound provides a stronger (absolute) guarantee, but it
requires samples to be independent and identically distributed. The regret
bound, on the other hand, provides a relative assumption-free guarantee, and
thus applies always.
|
arxiv-papers
| 2008-12-21T17:45:27 |
2024-09-04T02:48:59.526970
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Alina Beygelzimer and John Langford",
"submitter": "John Langford",
"url": "https://arxiv.org/abs/0812.4044"
}
|
0812.4454
|
# Two and One-dimensional Honeycomb Structure of Boron Nitride
M. Topsakal UNAM-Institute of Materials Science and Nanotechnology, Bilkent
University, Ankara 06800, Turkey E. Aktürk UNAM-Institute of Materials
Science and Nanotechnology, Bilkent University, Ankara 06800, Turkey S.
Ciraci ciraci@fen.bilkent.edu.tr UNAM-Institute of Materials Science and
Nanotechnology, Bilkent University, Ankara 06800, Turkey Department of
Physics, Bilkent University, Ankara 06800, Turkey
###### Abstract
This paper presents a systematic study of two and one dimensional honeycomb
structure of boron nitride (BN) using first-principles plane wave method. Two-
dimensional (2D) graphene like BN is a wide band gap semiconductor with ionic
bonding. Phonon dispersion curves demonstrate the stability of 2D BN flakes.
Quasi 1D armchair BN nanoribbon are nonmagnetic semiconductors with edge
states. Upon passivation of B and N with hydrogen atoms these edge states
disappear and band gap increases. Bare zigzag BN nanoribbons are metallic, but
become a ferromagnetic semiconductor when their both edges are passivated with
hydrogen. However, their magnetic ground state, electronic band structure and
band gap are found to be strongly dependent on whether B- or N-edge of the
ribbon is saturated with hydrogen. Vacancy defects in armchair and zigzag
nanoribbons affects also magnetic state and electronic structure. In order to
reveal dimensionality effects these properties are contrasted with those of
various 3D BN crystals and 1D BN atomic chain.
###### pacs:
73.22.-f, 75.75.+a, 63.22.-m
## I introduction
Synthesis of a single atomic plane of graphite, i.e. Graphene with covalently
bonded honeycomb lattice has been a breakthrough for several reasons novo ;
zhang ; berger . Firstly, electrons behaving as if massless Dirac Fermions
have made the observation of several relativistic effects possible. Secondly,
stable graphene has disproved previous theories, which were concluded that
two-dimensional structures cannot be stable. Graphene displaying exceptional
properties, such as high mobility even at room temperature, ambipolar effect,
Klein tunelling, anomalous quantum hall effect etc. seems to offer novel
applications in various fields graphene_applications1 . Not only 2D graphene,
but also its quasi 1D forms, such as armchair and zigzag nanoribbons have
shown novel electronic and magnetic properties graphene_applications2 ;
graphene_applications3 ; graphene_applications4 , which can lead to important
applications in nanotechnology. As a result, 2D honeycomb structures derived
from Group IV elements and Group III-V and II-VI compounds are currently
generating significant interest owing to their unique properties.
Boron-Nitride (BN) in ionic honeycomb lattice which is the Group III-V
analogue of graphene have also been produced having desired insulator
characteristics bn-synthesis . Nanosheets bn-nanosheets1 ; bn-nanosheets2 ,
nanocones bn-nanocones , nanotubes bn-nanotubes , nanohorns bn-nanohorns ,
nanorods bn-nanorods and nanowires bn-nanowires of BN have already been
synthesized and these systems might hold promise for novel technological
applications. Among all these different structures, BN nanoribbons, where the
charge carriers are confined in two dimension and free to move in third
direction, are particularly important due to their well defined geometry and
possible ease of manipulation.
BN nanoribbons posses different electronic and magnetic properties depending
on their size and edge termination. Recently, the variation of band gaps of BN
nanoribbons with their widths and Stark effect due to applied electric field
have been studied Guo ; louie-nanoletter . Recently the magnetic properties of
zigzag BN nanoribbons have been investigated barone . Half-metallic properties
have been revealed from these studies which might be important for spintronic
applications. Production of graphene nanoribbons as small as 10 nm in width
has been achieved dai1 ; dai2 and similar techniques are expected to be
developed for BN nanoribbons.
A thorough understanding of 2D BN honeycomb structure and their various
nanoribbons is important for further study of this graphene like compounds. BN
by itself provides with very interesting chemical and physical properties,
which may lead to important applications. In this paper, we present a detailed
ab-initio study of electronic, magnetic and elastic properties of 2D (graphene
like) BN and bare and hydrogen passivated, quasi 1D BN nanoribbons (BNNRs). We
also investigated the effect of the vacancy defects on these properties. To
reveal the dimensionality effects we include also a short discussion regarding
3D BN bulk crystals and 1D BN atomic chains. We found that 2D BN is a
nonmagnetic, wide band gap semiconductor. The ionic bonding due to significant
amount of charge from B to N atom opens a gap and hence dominates electronic
structure. Calculated phonon dispersion curves provide a clear evidence that
2D BN flakes is stable. The armchair and zigzag nanoribbons of BN display even
more interesting electronic and magnetic properties. Bare and hydrogen
passivated armchair BN nanoribbons (A-BNNR) are nonmagnetic wide band gap
semiconductor. The value of band gap of A-BNNR having width $w>10$ Å is
practically independent from the width of nanoribbons. While the bare zigzag
BN nanoribbons (Z-BNNR) are ferromagnetic metal, they become nonmagnetic
semiconductor upon the passivation of both edges. We found that 2D BN and its
nanoribbons have properties, which are complementary to graphene.
## II Model and Methodology
We have performed first-principles plane wave calculations within density
functional theory (DFT) using PAW potentials paw . The exchange correlation
potential has been approximated by generalized gradient approximation (GGA)
using PW91 pw91 functional both for spin-polarized and spin-unpolarized
cases. All structures have been treated within supercell geometry using the
periodic boundary conditions. A plane-wave basis set with kinetic energy
cutoff of 500 eV has been used. In the self-consistent potential and total
energy calculations the Brillouin zone (BZ) is sampled by special k-points.
The numbers of these k-points are (15x15x15) for bulk BN, (25x25x1) for 2D BN
and (25x1x1) for nanoribbons, respectively, and are scaled according to the
size of superlattices. All atomic positions and lattice constants are
optimized by using the conjugate gradient method, where the total energy and
atomic forces are minimized. The convergence for energy is chosen as 10-5 eV
between two steps, and the maximum Hellmann-Feynman forces acting on each atom
is less than 0.02 eV /Å upon ionic relaxation. A large spacing ($\sim$ 10 Å)
between monolayers has been taken to prevent interactions between them. The
pseudopotentials having 3 and 5 valence electrons for the B (B: $2s^{2}$
$2p^{1}$) and N ions (N: $2s^{2}$ $2p^{3}$) were used. Numerical calculations
have been performed by using VASP package vasp1 ; vasp2 . The phonon
dispersion curves are calculated within density functional perturbation theory
(DFPT) using plane wave methods as implemented in PWSCF software pwscf .
## III 3D BN Crystals and 1D atomic chain
In this section, we present our theoretical calculations on 3D bulk BN
crystals and truly 1D BN atomic chain. Earlier these 3D bulk crystals
BNcrystal1 ; BNcrystal2 ; BNcrystal3 ; BNcrystal4 ; BNcrystal5 and 1D atomic
chains BNchain have been studied theoretically by using different methods.
Our purpose in including these crystals of BN in different dimensionalities is
to contrast their properties with those of 2D and quasi 1D honeycomb
structures of BN and also reveal dimensionality effects.
### III.1 3D Bulk BN crystals
Three dimensional bulk crystals include hexagonal-layered BN (h-BN), wurtzite
BN (wz-BN) and zincblende BN (zb-BN) structures. Their atomic configurations
and primitive unit cells are described in Fig. 1. By using the expression,
$E_{C}={E[BN]}-E[B]-E[N]$ (1)
where E[BN] is the total energy per B-N pair of the optimized structure of BN
crystal; E[B] and E[N] are the total energies of free B and N atoms; we
calculated the equilibrium cohesive energies of h-BN, wz-BN and zb-BN crystals
as -17.65, -17.45 and -17.49 eV per B-N pair, respectively. Accordingly, h-BN,
which is the analogue of graphite, is the most energetic bulk structure. On
the other hand, the cubic BN structure is known to be the second hardest
material of all.
The lattice constants of the optimized structures in equilibrium are $a=2.511$
Å, $c/a=2.66$ and the distance between the nearest B and N atoms is $d=1.450$
Åfor h-BN layered crystal. For wz-BN, optimized values of $a$, $c/a$ and $d$
are calculated to be 2.542 Å, 1.64, and 1.561 Å, respectively. The zincblende
structure has lattice constant $a=2.561$ Å and $d=1.568$ Å. All our results
related with the structural parameters are in good agreement with the
experimental and theoretical values BNcrystal1 ; BNcrystal2 ; BNcrystal3 ;
BNcrystal4 ; bulk-references within the average error of $\sim 1\%$.
The calculated electronic band structure, total and partial (or orbital
projected) density of states (DOS) of 3D crystals are presented in Fig. 1.
These h-BN, wz-BN and zb-BN crystals are indirect band gap semiconductors with
calculated band gaps being $E_{G}$=4.47, 5.72, and 4.50 eV, respectively. The
calculated values of $E_{G}$ differ from the earlier ones depending on the
method used DFTgap . For h-BN having 2D BN atomic layers in the (x,y)-plane.
The band structure is composed from the band structures of these individual
atomic layers with hexagonal symmetry, which are slightly split due to weak
coupling between them. Highest valence band has N-$p_{z}$ character; the
states of the lowest conduction band is formed from B-$p_{z}$ orbitals (the z
direction corresponds to “c” in Fig. 1 ). Overall features of the total
density of states (TDOS) are similar for three 3D crystal structures. Valence
band consists of two parts separated by a wide intra band gap. The lower part
at $\sim$ -20 eV is projected mainly to N-$s$ and partly to N-$p$ and B-$s$
orbitals. The upper part is due to mainly N-$p$ and partly B-$p$ orbitals and
has similarities in both zb-BN and wz-BN crystals. As for the lower part of
the conduction band it is derived mainly from B-$p$ orbitals. The differences
of three 3D crystals are pronounced in the lower part of the conduction band.
Figure 1: (Color online) Optimized atomic structure, energy bands, total
(TDOS) and orbital projected density of states (PDOS) of various 3D crystals
of BN. (a) Hexagonal (h-BN) whose B(N) atoms are on top of the N(B) atoms in
the consecutive layer; (b) wurtzite (wz-BN); and (c) zincblende (zb-BN)
crystals. Dark-green and light-gray balls represent B and N atoms,
respectively. The band gaps between conduction and valence bands are
highlighted. The orbital character of states are indicated for the conduction
and valence band edges. The zero of energy is set to the Fermi energy EF. All
structures are fully optimized.
We calculate the amount of charge on constituent B and N atoms in 3D crystals
by performing the Löwdin lowdin analysis in terms of the projection of plane-
waves into atomic orbitals. By subtracting the valencies of free B and N atoms
from the calculated charge values on the same atoms in 3D crystals we obtain
the charge transfer, $\Delta Q$. The calculated values of $\Delta Q$ for h-BN,
wz-BN and zb-BN are 0.416, 0.342, 0.334 electrons, respectively. The fact that
$\Delta Q$ of zb-BN and wz-BN have almost equal values, but $\Delta Q$ of h-BN
crystal is significantly larger related to the shorter B-N bond length in h-BN
crystal.
### III.2 1D BN Atomic chain
BN forms stable segments of linear atomic chain BNchain like carbon tongay .
This situation is in contrast to second and third row elements (such as Si and
Ge) and III-V compounds and metals (such as Al, Au etc) which can form stable
zigzag chain structures instead of linear chain structures. Our results on
optimized chain structure yield the cohesive energy $E_{C}$=16.04 eV per B-N
pair, the B-N distance $d=$1.307 Å, the indirect band gap E${}_{G}=$3.99 eV
and charge transfer from B to N, $\Delta Q=$0.511 electrons. Hence the double
bond between B and N is ionic.
## IV 2D Honeycomb Structure of BN
Having discussed the overall structural and elastic properties of 3D and 1D
BN, we now consider 2D BN with hexagonal symmetry. The atomic structure of 2D
BN is similar to the honeycomb structure of graphene, except that the
constituent atoms of the former are from III and V columns of the Periodic
Table. Normally, the bond between nearest B and N atoms is formed from the
bonding combination of B-$sp^{2}$ and N-$sp^{2}$ orbitals. However, owing to
the electronegativity difference between B and N atoms electrons are
transferred from B to N. As a result, in contrast to purely covalent bond in
graphene the bonding between B and N gains an ionic character. The charge
transfer from B to N dominates several properties of 2D BN including the
opening of the band gap. In this respect the BN honeycomb structure is
complementary to graphene.
### IV.1 Charge density analysis and electronic structure
The atomic structure, atomic charge, charge transfer from B to N and the
electronic structure of 2D BN are presented in Fig. 2. Contour plots of total
charge indicates high density around N atoms. The difference charge density is
calculated by subtracting charge densities of free B and N atoms from the
charge density of 2D BN, i.e. $\Delta\rho=\rho_{BN}-\rho_{B}-\rho_{N}$. High
density contour plots around N atoms protruding towards the B-N bonds indicate
charge transfer from B to N atoms. This way the B-N bonds achieve an ionic
character. The amount of transfer of charge is calculated by Löwdin analysis
to be $\Delta Q$=0.429 electrons. Interestingly, $\Delta Q$ is slightly larger
than that calculated for h-BN, but significant larger than those calculated
for wz-BN and zb-BN crystals.
2D BN is a semiconductor. Calculated electronic energy bands are similar to
those calculated for h-BN crystal. The $\pi$\- and $\pi^{*}$\- bands of
graphene which cross at the K- and K∗-points of the BZ open a gap in 2D BN as
a bonding and antibonding combination of N-$p_{z}$ and B-$p_{z}$ orbitals. The
contribution of N-$p_{z}$ is pronounced for the filled band at the edge of
valence band. The calculated band gap is indirect and $E_{G}=$4.64 eV. TDOS
and partial density of states show also similarity to those of h-BN layered
crystal presented in Fig. 1.
Figure 2: (Color online) (a) Primitive unit cell of the honeycomb structure of
2D BN together with Bravais lattice vectors. Calculated total charge density
$\rho_{BN}$ and difference charge density $\Delta\rho$, are also shown in the
same panel. (b) Calculated electronic structure of 2D BN honeycomb crystal
together with total, TDOS and partial density of states, PDOS on B and N
atoms. The orbital character of the states are also indicated.
### IV.2 Phonon spectrum
Even if the structure optimization resulting in the honeycomb structure in
Fig. 2 can be taken as an indication for the stability of 2D BN, calculation
of phonon dispersion curves through the diagonalization of dynamical matrix
provides a more stringent test for stability. One of acoustical branches for
$\Gamma$ to $K$ curves taking negative value even at a small region of BZ
indicates the instability of the structure. There have been a number of
experimental Rokuta and theoretical studies of phonon spectrum of 2D Wirtz
and 3D honeycomb BN Kern ; Yu ; Serrano ; Solozhenkan ; Miyamoto . Here, the
phonon dispersion curves of h-BN, 2D BN and 1D BN chain and density of states
together with the infrared (IR) and Raman (R) active modes of 2D BN and h-BN
at $\Gamma$-point have been calculated by using density functional
perturbation theory (DFPT) as implemented in PWSCF software pwscf . For the
DFPT phonon calculation of bulk h-BN, we used a four atom primitive cell,
which yield 12 phonon branches at the center of BZ in Fig. 3 (a). The symmetry
point group is calculated as D6h (space group P6/mmm). The irreducible
representations at $\Gamma$ is 2 E2g\+ 2 B2g+2 A2u+2 E1u. While the modes E1u
and E2g are doubly degenerate, B2g and A2u are non degenerate. The modes E1u
and A2u are IR active, the E2g is Raman active. B2g is an inactive mode. Our
results are in agreement with previously calculated and experimental data, but
differ slightly from those of Serrano et al.Serrano . While present GGA
calculations predict B2g mode as an inactive mode, LDA calculations by Serrano
et al. found B1g as an inactive mode. Most of the phonon bands of h-BN are
degenerate. This indicates that the coupling between BN layers in h-BN is
weak. However, it is well known that the BN is polar material with long range
dipole-dipole interaction. This gives rise to the splitting between
longitudinal optical (LO) and transverse optical (TO) mode at $\Gamma$ point.
The lowest transverse acoustical mode has parabolic dispersion as k
$\rightarrow$ 0 owing to rapidly decaying interatomic forces for transversal
displacements decay . Another feature is the overlap of the lowest transversal
optical mode with the acoustical modes.
In Fig. 3 (b) we show the phonon dispersion curve of BN atomic chain. Two TA
modes have low frequency and get very small but negative values near the zone
center. This indicates structural instabilty as $\lambda\rightarrow\infty$.
However, the linear segments of BN atomic chain can be stable. Similar to
h-BN, the doubly degenerate TO branch overlaps with the LA branch.
For 2D BN honeycomb structure, the unit cell consists of two atoms.
Accordingly, there are three acoustical and three optical branches in Fig. 3
(c). The symmetry point group is D3h (space group (P-62m)). Optical phonon
modes at the $\Gamma$-point is given by A${}^{{}^{\prime\prime}}_{2}$+2
E${}^{{}^{\prime}}$. The mode A${}^{{}^{\prime\prime}}_{2}$ is IR active and
the E${}^{{}^{\prime}}$ mode is both IR and Raman active. The similarity
between calculated phonon dispersion curves of h-BN and 2D-BN is remarkable.
We also calculate the phonon dispersion curves of 2D BN honeycomb structure by
using PAW potentials paw as implemented in VASP vasp1 for further checking
of the results of our phonon calculation. Force constants are determined from
the $(8\times 8\times 1)$ supercells. The phonon modes were calculated by
using the direct method as implemented in the PHON alfe software. The
calculated phonon frequencies are almost identical with those calculated by
DFPT method. In Fig. 3 (d), we present the phonon density of states calculated
for 2D BN honeycomb structure. Note that both calculations yield that TA (or
ZA) mode displaying parabolic dispersion gets negative frequencies as
$k\rightarrow 0$. similar to BN atomic chains, this indicates structural
instability as $\lambda\rightarrow\infty$. Accordingly, finite size of 2D BN
flakes are expected to have stable structure.
Figure 3: (Color online) Calculated phonon frequencies versus k-vectors. (a)
h-BN crystal. (b) 1D BN atomic chain. (c) 2D BN honeycomb structure. Phonon
modes calculated by force constant direct method are shown by the blue-dashed
curve. (d) Density of phonon frequencies (DOS) for the 2D BN honeycomb
structure.
## V 1D BN Nanoribbons
Similar to graphene graphene-nanoribbons , two unique orientation in 2D BN
yield nanoribbons with uniform edges: These are armchair (A-BNNR) and zigzag
(Z-BNNR) nanoribbons. The profile of the atomic configuration at both edges of
the nanoribbon determines their electronic and magnetic properties. The
properties can be modified by the passivation of dangling bond of edge atoms
by hydrogen. Because of their interesting electronic and spintronic
properties, BN nanoribbons are attractive nanostructures for various device
applications. Electronic properties of BN nanoribbons have been investigated
in recent papers Guo ; louie-nanoletter ; barone . Present study is
complementary to previous studies.
### V.1 Electronic structure
Here we present the results of our study on the electronic and magnetic
properties of bare and hydrogen passivated A-BNNR and Z-BNNRs. Bare and
hydrogen passivation A-BNNR are wide band gap semiconductors. Similarly,
hydrogen passivated Z-BNNRs are also semiconductor. The band gaps of these BN
nanoribbons depend on the width of the nanoribbons $w$ or the numbers of BN
pairs, $n$ in the primitive unit cell. The variation of the band gap EG as a
function of $n$ is given in Fig. 4. Normally, the properties of nanoribbons
approaches to those of 2D honeycomb structure as the width
$n\rightarrow\infty$. However, due to the localized edge states the band gap
of Z-BNNR approaches to a gap smaller than that of 2D BN honeycomb structure
louie-nanoletter . For narrow ($n<8$) bare and hydrogen passivated A-BNNRs the
band gaps vary with $n$, but they are practically unaltered for $n>8$. For
$n>8$ the band gap of bare A-BNNR is 0.4 eV smaller than that of hydrogen
passivated A-BNNR. The band gap of hydrogen passivated Z-BNNR is 4.5 eV for
$n=3$, but decrease to 3.8 eV for $n=16$. However, its variation with $n$ is
not monotonic for $5<n<13$, it rather display family dependent oscillatory
variation with changes as large as 0.4 eV between two consecutive values of
$n$. On the other hand, bare Z-BNNRs are found to be metallic.
Figure 4: (Color online). Energy band gap versus the width of the nanoribbons
given in terms of the number of B-N atom pairs in the primitive unit cell,
$n$. Bare armchair nanoribbons A-BNNR, hydrogen passivated A-BNNR, and
hydrogen passivated zigzag nanoribbons Z-BNNR. Dotted line indicates the bulk
band gap of 2D-BN. Figure 5: (Color online) (a) Energy band structure of bare
armchair nanoribbon A-BNNR having $n=12$ B-N pairs in the primitive unit cell.
At the right hand side of bands, the schematic description of atomic structure
with primitive unit cell delineated by dotted lines and isosurface charge
distribution of specific states are shown. (b) Same as (a) but the dangling
bonds at both edges are passivated by hydrogen atoms.
The atomic and electronic structure of bare and hydrogen passivated A-BNNR are
described in Fig. 5 for $n=12$. The atoms at the edges of the bare A-BNNR are
reconstructed; while one edge atom, B is lowering, adjacent edge atom, N is
raised. Two bands of edge states occur below the conduction band edge. These
bands are normally degenerate for large $n$, but split around the center of BZ
due to their coupling. The bands of edge states occur $\sim$-1 eV below the
top of the valance band edge. Normal states, on the other hand, have charge
distributed uniformly in the ribbon. Because of the edge states the band gap
is indirect and is $\sim$4.22 eV wide. Upon passivation of the dangling bonds
of B and N atoms situated at the edges with hydrogen atoms, these edge state
bands are discarded from the band gap and reconstruction of edge atoms
disappear. At the end, the band gap of H-passivated A-BNNR becomes direct and
increases by $\sim$0.3 eV.
Figure 6: (Color online) Top panels: Atomic structures of zigzag nanoribbons
(Z-BNNR). The primitive unit cell has $n=6$ B-N pairs delineated by dotted
lines. The unit cell is doubled due to antiferromagnetic interaction between
adjacent N atoms. Middle panels: Isosurface plots of difference charge density
between up spin and down spin states,
$\Delta\rho=\rho(\uparrow)-\rho(\downarrow)$. Bottom panels: Energy band
structure with dotted (blue) and solid (red) lines showing spin up and spin
down states, respectively. (a) Bare Z-BNNR; (b) B-side free, but N-side is
passivated by hydrogen atoms; (c) N-side free, but B-side is saturated by
hydrogen atoms; (d) Both sides are saturated by hydrogen atoms. The bands in
(a), (b), and (c) are calculated using double cell.
The electronic and magnetic states of Z-BNNR depend on whether their edges are
passivated with hydrogen atoms. While a bare Z-BNNR is magnetic and metallic,
it becomes nonmagnetic and a wide band gap semiconductor upon the passivation
of B and N atoms at both edges. Moreover, its electronic and magnetic
properties depend on whether only B- or N-side is passivated with hydrogen
atoms. Accordingly, Z-BNNRs provide us for several alternatives for different
electronic and magnetic properties barone . However, different magnetic states
corresponding to different edge configuration, namely bare or hydrogen
passivated, are very sensitive to the parameters of calculation. In Fig. 6 we
present the calculated electronic structures of a Z-BNNR with $n$=6 B-N pairs
in a primitive unit cell for four different cases. These are both side free,
only N-side is passivated with hydrogen, only B-side is passivated with
hydrogen and both edges are passivated with hydrogen.
Bare Z-BNNR having both edges are free display different magnetic states
(magnetic order), which are close in energy. Moreover, the ordering of these
magnetic states with respect to their energy is sensitive to the criterion of
energy convergence. To ensure the antiferromagnetic (AFM) order at edges, we
considered double cells. The possible magnetic states are spin-up, spin-down
for adjacent B atoms at one side and spin-up, spin-up for the adjacent N atoms
at the other side; namely $\uparrow\downarrow$ / $\uparrow\uparrow$ spin
configuration. Other possible spin configurations are $\uparrow\uparrow$ /
$\downarrow\downarrow$; $\uparrow\uparrow$ / $\uparrow\uparrow$ ;
$\uparrow\uparrow$ / $\uparrow\downarrow$; $\uparrow\downarrow$ /
$\uparrow\uparrow$. We found that the spin configuration, $\uparrow\downarrow$
/ $\uparrow\uparrow$ for Z-BNNR having 12 B-N pairs in double unit cell
corresponds to the ground state. The other excited configurations,
$\uparrow\uparrow$ / $\downarrow\downarrow$; $\uparrow\uparrow$ /
$\uparrow\uparrow$ ; $\uparrow\uparrow$ / $\uparrow\downarrow$;
$\uparrow\downarrow$ / $\uparrow\downarrow$, have 6,7,35,131 meV higher
energies than ground state. The ordering of these configuration is slightly
different from that reported earlier barone . Nevertheless, the difference
between the earlier and present ground state energies are within the accuracy
limits of DFT calculations. The ground state spin configuration
$\uparrow\downarrow$ / $\uparrow\uparrow$ of the bare Z-BNNR having both edges
free is found to be ferrimagnetic metal with $\mu=1.77$ $\mu_{B}$ per double
cell. Whereas the excited magnetic state with configuration $\uparrow\uparrow$
/ $\downarrow\downarrow$ is half metallic.
In Fig. 6 (b), Z-BNNR with N-edge passivated with hydrogen atoms is an AFM
semiconductor. The AFM edge state is localized at the B-side. When only the
B-side is passivated with hydrogen atoms the magnetic edge state is, this
time, localized at the N-side of the ribbon. As seen in Fig. 6 (c) the ground
state of Z-BNNR is ferromagnetic with $\mu$ =2 $\mu_{B}$ per double cell. Our
calculations suggest that the nearest neighbor N-N interaction is
ferromagnetic, the B-B interaction is antiferromagnetic. Finally, the Z-BNNR
becomes non magnetic, when the atoms at both edges are passivated with
hydrogen atoms. Earlier, Hwan and Louie louie-nanoletter studied hydrogen
passivated A-BNNRs and Z-BNNRs with widths up to 10 nm. Our results for
hydrogen passivated nanoribbons are in good agreement with their results,
except that our results for zigzag ribbons obtained using GGA as well as LDA
exhibit family dependent oscillations for $5<n<13$.
### V.2 Elastic properties
The elastic properties of BNNRs are examined through the variation of the
total energy $E_{T}$ with respect to the applied uniaxial strain
$\epsilon=\Delta c/c$, $c$ being the lattice constant along the nanoribbon
axis. Owing to ambiguities in defining the cross section of the ribbon one
cannot determine the Young’s modulus rigorously. Instead we calculate
$\kappa=\partial^{2}E_{T}/\partial c^{2}$ from the variation of $E_{T}$ to
specify the elastic properties of quasi 1D nanoribbons. In Fig. 7 (a) we show
the variation of the total energy $E_{T}$ versus $\epsilon$. In order to lift
the constraints imposed by periodic boundary conditions, calculations are
performed for a supercell comprising five primitive unit cells having lattice
constant $c_{s}=5c$. For $\epsilon<0.10$, the variation of $E_{T}(\epsilon)$
is parabolic, and hence $\kappa$ is independent of $\epsilon$. For
$\epsilon>10$ $E_{T}(\epsilon)$ curve deviates from parabola and becomes
anharmonic. For higher values of strain in the plastic region, the ribbon
undergoes structural transformation. For example, such a transformation
occurred at $\epsilon=0.24$ with a sudden change in $E_{T}(\epsilon)$ curve.
The corresponding structure is illustrated as inset. The lattice constant
$c_{s}$ increased from the initial value 21.5 Å to 27.4 Åcorresponding to
$\epsilon=0.27$.
Figure 7: (Color online) (a) Variation of total energy of hydrogen saturated
A-BNNR with strain , $\epsilon$ is shown by dashed curve with large black dots
indicating the calculated data points ($c_{s}=5c$ and $n$ = 9). Harmonic,
anharmonic and plastic regions are distinguished. The harmonic part is fitted
to a parabola presented by red-solid curve. Atomic structure shown by filled,
empty and very small empty circles represent B, N, and H atoms. Supercell
comprising five primitive unit cells are shown in the harmonic and plastic
regions. (b) Variation of $\kappa=\partial^{2}E_{T}/\partial c^{2}$ versus
ribbon width $n$ calculated for bare and hydrogen passivated A-BNNR.
In Fig. 7 (b) $\kappa$ versus the width of the ribbon in terms of the number
of B-N pair in the primitive unit cell $n$ is plotted for bare and hydrogen
passivated A-BNNR. $\kappa(n)$ shows an approximately linear variation
indicating that the force constant is directly proportional to the width of
the ribbon. One also sees that the strength of the ribbon increases upon
passivation with hydrogen.
The behavior of bare and hydrogen passivated Z-BNNR under uniaxial tensile
stress is similar to that of A-BNNR. In Fig. 8 three regions, namely elastic
harmonic, elastic-anharmonic and plastic regions are seen. The sudden change
in the $E_{T}(\epsilon)$ curve at $\epsilon\sim 0.23$ indicates a structural
phase transformation, where the lattice constant $c_{s}$ elongates from the
initial $\epsilon=0$ value of 19.8 Å to 25.7 Å corresponding to
$\epsilon=0.3$. The structure of hydrogen passivated Z-BNNR before and after
the structural transformation are shown as inset. Variation of $\kappa$ versus
the ribbon width $n$ is calculated for bare and hydrogen passivated Z-BNNR
show an overall linear behavior as presented in Fig. 7 (b).
Figure 8: (Color online) Variation of total energy of hydrogen saturated
Z-BNNR shown by dashed curve with large black dots indicating the calculated
data points. Harmonic, anharmonic and plastic regions are distinguished. The
harmonic part is fitted to a parabola presented by a red solid curve. Atomic
structure of the ribbon in a supercell comprising eight unit cells ($c_{s}=8c$
and $n$ = 6) are shown before and after structural transformation as inset.
### V.3 Vacancy and antisite defects
It is known that the vacancy defect in 2D graphene esquinazi ; Iijima ; yazyev
; guinea ; brey2 and graphene nanoribbons brey1 ; delik give rise to crucial
changes in the electronic and magnetic structure. According to Lieb’s theorem
lieb , the net magnetic moment per cell is determined with the difference in
the number of atoms belonging to different sublattices, i.e.
$\mu=(N_{B}-N_{N})\mu_{B}$. While DFT calculations on vacancies in 2D graphene
and armchair graphene nanoribbons confirmed Lieb’s theorem, results are
diversified for vacancies in zigzag graphene nanoribbons brey1 ; delik .
Therefore, the effect of vacancy defects on the properties of BNNRs is of
interest.
Earlier activation energies and reaction paths for diffusion and nucleation
mono and divacancy in h-BN layers have been investigated by using density
functional tight-binding method zobelli . The formation energies were
calculated to be 11.22 eV and 8.91 eV, for a B- and N-vacancy, respectively.
The possible magnetism induced by nonmagnetic impurities and vacancy defects
in a BN sheet have been investigated from the first-principles. The magnetic
moment associated by nonmagnetic atoms substituting B or N has been calculated
to be $1\mu_{B}$ liu . Based on first-principles calculations, the magnetic
moment of a N-vacancy in a 2D BN sheet has been predicted to be 1 $\mu_{B}$.
In the case of a B-vacancy, three neighboring N atoms are displaced further
apart from each other and the net magnetic moment is predicted to be 3
$\mu_{B}$ si . Another calculation of defects in a BN monolayer found that
three dangling bonds associated with a B-vacancy lead to total spin S=3/2, i.e
3 $\mu_{B}$ azeveda .
Figure 9: (Color online) Relaxed atomic structures and corresponding energy
bands of hydrogen passivated A-BNNR with $n=12$ having a point defect located
periodically in every four primitive cell. Blue filled, empty and small
circles represent B, N, and H atoms, respectively. Blue-dark and yellow-light
isosurface plots are for spin-up and spin-down states. (a) Single B-vacancy;
(b) single N vacancy; (c) B-N divacancy; (d) Antisite defect.
The effects of vacancies of BN nanoribbons have not been treated yet. Here we
investigated the effect of B-, N-, B+N-divacancy and B+N-anti site on the
electronic and magnetic properties of A- and Z-BNNR. Within periodic boundary
conditions, a vacancy defect in an A-BNNR of width $n=12$ is repeated in every
5th primitive unit cell to yield minute defect-defect coupling. As shown in
Fig. 9 (a) A-BNNR with B-vacancy becomes ferromagnetic with a net magnetic
moment of $\mu=1$ $\mu_{B}$ per unit cell. Similarly, a N vacancy gives rise
to a net magnetic moment of $\mu=1$ $\mu_{B}$ per unit cell. A-BNNR having
either periodic B+N-divacancy or anti site defect for every five unit cell
remain nonmagnetic. The calculated values of magnetic moments are in
compliance with Lieb’s theorem. We found that the structural relaxation is
crucial to obtain correct values of magnetic moments. In particular, initially
we calculated $\mu=3$ $\mu_{B}$ for relaxed structure of the B-vacancy.
However, the neighboring N atoms distorted slightly from their equilibrium,
the structure is relaxed further and had lowered the total energy. As a
result, the magnetic moment was calculated as $\mu=1$ $\mu_{B}$. The energy
band structures in Fig. 9 (a)-(d) are calculated for periodic vacancy defects
repeating in every four primitive cell. The Fermi levels are assigned
according to the occupancy of vacancy states. We note that the empty state
associated with the B-vacancy in Fig. 9 is hole like. The states associated
with the N-vacancy occur near the edge of the conduction band are donor like.
Figure 10: (Color online) Relaxed atomic structures of hydrogen passivated
Z-BNNR with $n=6$ having a vacancy defect located periodically in every eight
primitive cell ($c_{s}\approx 8c$ and $n$ = 6). Filled, empty and small
circles represent B, N, and H atoms, respectively. Blue and pink isosurface
plots are spin up and spin down states, respectively. (a) Single B-vacancy;
(b) single N vacancy; (c) B+N divacancy; (d) anti site defect.
The situation with Z-BNNR is similar to that in A-BNNR discussed above, since
hydrogen passivated Z-BNNR is nonmagnetic as A-BNNR. A periodic B- or
N-vacancy repeated in every eight unit cell of hydrogen saturated Z-BNNR with
$n$=6 has a net magnetic moment of $\mu=1$ $\mu_{B}$ per supercell. Whereas
Z-BNNR passivated with hydrogen atoms at both edges and having either periodic
B+N-divacancy or anti site defect repeating in every eight unit cell is
nonmagnetic. The type of the periodic vacancy defect modifies the band gap of
Z-BNNR from 4.2 eV to 2.21 eV for divacancy, but to 2.8 eV for anti site. The
calculated magnetic moment of hydrogen passivated Z-BNNR are in agreement with
Lieb’s theorem. We note that in zigzag graphene nanoribbons magnetic edge
states survive even after hydrogen passivation, and interact with the magnetic
moments of vacancies delik . This interaction causes deviation from the
prediction of Lieb’s theorem.
## VI Discussion and Conclusions
In various allotropic forms of BN the dimensionality play a crucial role. For
the sake of comparison, we present the calculated values of BN for different
allotropic forms of BN in different dimensionalities. One sees that the B-N
double bond of 1D BN atomic chain is shortest and is 1.31 Å. The s$p^{2}$ bond
of h-BN and 2D BN has intermediate value of 1.45 Å. Therefore, h-BN can
considered to be quasi two dimensional. Three dimensional wurtzite and
zincblende BN crystal have s$p^{3}$-bonding with $d=1.56$ Å, which is largest
among the allotropic forms studied here. According to GGA results the cohesive
energy of 2D BN is 3 meV larger than that of h-BN. This is due to fact that
the GGA calculation cannot account the van der Waals interaction between
atomic layers of h-BN. However, the calculations using LDA, where the van der
Waals interactions are better accounted, yield the cohesive energy of h-BN is
$\sim$57 meV larger than that of 2D BN as one expects. The charge transfer
$\Delta Q$ from B to N atom increases with decreasing dimensionality. This due
to fact that $d$ decreases with decreasing dimensionality. As for the
coordination number increases with increasing dimension.
In 2D BN honeycomb structures and in its zigzag and armchair nanoribbons, the
B-N bond formed from the bonding s$p^{2}$ hybrid orbitals from B and N atoms
is essential. Owing to the transfer of charge from B to N the B-N bond
acquires an ionic character, which underlies the semiconducting properties
with wide band gap.
Table 1: Values of the bond length $d$ in Å, cohesive energy $E_{C}$ in eV per B-N pair, band gap EG in eV, charge transfer from B to N ($\Delta Q$) in electrons and lattice constants (a,c) in Åcalculated for various allotropic forms of BN in different dimensionality. | $d$ | $E_{C}$ | EG | $\Delta$ Q | Lattice
---|---|---|---|---|---
1D Chain | 1.307 | -16.04 | 3.99 | 0.511 | a=2.614
2D BN | 1.452 | -17.65 | 4.64 | 0.429 | a=2.511
h-BN | 1.450 | -17.65 | 4.47 | 0.416 | a=2.511 , c/a=2.66
Wurtzite | 1.561 | -17.45 | 5.726 | 0.342 | a=2.542 , c/a=1.63
Zincblende | 1.568 | -17.49 | 4.50 | 0.334 | a=2.561
Bare armchair nanoribbon of 2D BN is again a nonmagnetic wide band gap
semiconductor, the band gap of which is practically unaltered with width
$n>8$. Upon passivation with hydrogen band gap of the ribbon increase by 0.3
eV. As for zigzag nanoribbons, they provide a number of interesting
properties. When its both edges are bare, it is ferromagnetic metal. When its
N-edge is passivated with hydrogen, it becomes an antiferromagnetic
semiconductor. In the reverse case, namely when B-side is passivated, it
becomes a ferromagnetic semiconductor. When both edges are passivated, it
becomes a nonmagnetic, wide band gap semiconductor. The band gap as well as
the magnetic state of a ribbon can be modified by periodic vacancy defects.
Finally, BN nanoribbons have been found to be strong, quasi one dimensional
and stable structures. They can sustain up to high strains, and they stretch
in the plastic region with structural transformations.
Briefly, the calculated electronic, magnetic and mechanical properties of 2D
BN honeycomb structure and its nanoribbons present interesting but some
differences from graphene. In this respect BN honeycomb structure and its
nanoribbons are complimentary to graphene. The properties of 2D BN honeycomb
structure can be changed upon functionalization with foreign atoms.
Interesting quantum structures (such as single and series quantum dots,
resonant tunneling double barriers and multiple quantum well structures) based
on heterostructures and core shell structures of lattice matched graphene and
BN can be formed, since the band gap of BN is much larger than that of
graphene.
###### Acknowledgements.
Part of the computational resources have been provided by UYBHM at Istanbul
Technical University through grant Grant No. 2-024-2007. E. Aktürk gratefully
acknowledgments the receipt of a BIDEB Postdoctoral Fellowship from TUBITAK.
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|
arxiv-papers
| 2008-12-23T21:19:10 |
2024-09-04T02:48:59.545101
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Topsakal, E. Akturk and S. Ciraci",
"submitter": "Ethem Akturk",
"url": "https://arxiv.org/abs/0812.4454"
}
|
0812.4586
|
# Detecting barrier to cross-jet Lagrangian transport and its destruction in a
meandering flow
M.V. Budyansky, M.Yu. Uleysky, and S.V. Prants Pacific Oceanological
Institute
of the Russian Academy of Sciences, 43 Baltiiskaya st., 690041 Vladivostok,
Russia
###### Abstract
Cross-jet transport of passive scalars in a kinematic model of the meandering
laminar two-dimensional incompressible flow which is known to produce chaotic
mixing is studied. We develop a method for detecting barriers to cross-jet
transport in the phase space which is a physical space for our model. Using
tools from theory of nontwist maps, we construct a central invariant curve and
compute its characteristics that may serve good indicators of the existence of
a central transport barrier, its strength, and topology. Computing fractal
dimension, length, and winding number of that curve in the parameter space, we
study in detail change of its geometry and its destruction that are caused by
local bifurcations and a global bifurcation known as reconnection of
separatrices of resonances. Scenarios of reconnection are different for odd
and even resonances. The central invariant curves with rational and irrational
(noble) values of winding numbers are arranged into hierarchical series which
are described in terms of continued fractions. Destruction of central
transport barrier is illustrated for two ways in the parameter space: when
moving along resonant bifurcation curves with rational values of the winding
number and along curves with noble (irrational) values.
###### pacs:
05.45.-a,05.60.Cd,47.52.+j
## I Introduction
A meandering jet is a fundamental structure in laboratory and geophysical
fluid flows. Strong oceanic and atmospheric jet currents separate water and
air masses with distinct physical properties. For example, the Gulf Stream
separates the colder and fresher slope ocean waters from the salty and warmer
Sargasso sea ones. Recently, there has been much interest in applying ideas
and methods from dynamical systems theory to study mixing and transport in
meandering jets. In steady horizontal velocity fields, water (air) parcels
move along streamlines in a regular way. When the velocity field changes in
time, the motion becomes much more complicated even if the change is periodic.
The phenomenon of chaotic advection of passive particles in
(quasi)periodically-disturbed fluid flows has been studied theoretically and
experimentally A65 ; A84 ; A02 ; Ottino .
In the context of dynamical systems theory, chaotic advection is Hamiltonian
chaos in two-dimensional incompressible flows (for a review of Hamiltonian
chaos see, for example, AKN85 ; LL84 ; Ott ; Z05 ). The coordinates $x$ and
$y$ of a passive particle on the horizontal plane satisfy to simple Lagrangian
equations of motion
$\frac{dx}{dt}=u(x,y,t)=-\frac{\partial\Psi}{\partial
y},\quad\frac{dy}{dt}=v(x,y,t)=\frac{\partial\Psi}{\partial x},$ (1)
where $u$ and $v$ are zonal and meridional velocities of the particle, and the
stream function $\Psi$ plays the role of a Hamiltonian. The phase space of the
dynamical system (1) with one and half degrees of freedom is a configuration
space for passive particles.
A number of simple kinematic and dynamically consistent model stream functions
have been proposed to study large-scale chaotic mixing and transport in
geophysical meandering jet flows S92 ; DW96 ; Miller97 ; YPJ02 ; Chaos06 ;
UlBP07 ; PK06 ; Book08 . Deterministic models do not pretend to quantify
transport fluxes in real oceanic and atmospheric currents but they are useful
to reveal large-scale space-time structures that specify qualitatively mixing
and transport of water and air masses. Whether or not the jet provides an
effective barrier to meridional or cross-jet transport, under which conditions
the barrier becomes permeable and to which extent, these are crucial questions
in physical oceanography and physics of the atmosphere. The problem must be
treated from different points of view. In the straightforward numerical
approach based on full-physics nonlinear models, the velocity field is
generated as an outcome of a basin circulation model and a flux across the jet
(if any) can be estimated integrating a large number of tracers. The kinematic
and linear dynamically consistent models are less realistic, but they allows
to identify and analyze different factors which could enhance or suppress the
cross-barrier transport.
As to meandering currents, both the approaches have been applied to study
cross-jet Lagrangian transport. A simple kinematic model with the basic
streamfunction in the form (2) has been shown to reproduce some features of
the large-scale Lagrangian dynamics of the Gulf Stream water masses B89 . The
phase portrait of Eqs. (1) with the meandering Bickley jet (2) is plotted in
Fig. 1 (a) in the frame moving with the meander phase velocity. Time
dependence of the meander amplitude or the introduction of a secondary
meander, superimposed on the basic flow, may break the boundaries between
distinct regions in Fig. 1 (a) producing chaotic mixing and transport between
them S92 ; Meyers94 ; DW96 ; Chaos06 ; UlBP07 ; JPA08 . The numerical
calculations, based on computing the Melnikov function Melnik , have shown
that transport across the jet was much weaker than that between the jet ($J$),
the circulation cells ($C$), and the peripheral currents ($P$) in Fig. 1 (a),
i. e. the perturbation mixes the water along each side of the jet more
efficiently than across the jet core S92 . An attempt to analytically predict
the parameter values for the destruction of the transport barrier was made in
Ref. Meyers94 using the heuristic Chirikov criterion for overlapping
resonances Chir79 . A technique, based on computing the finite-scale Lyapunov
exponent, as a function of initial position of tracers, has been found useful
in Ref. BLR01 to detect the presence of cross-jet barriers in the kinematic
model (2). An analysis of cross-jet transport, based on lobe dynamics, has
been applied in RWig06 to describe how particles can cross the jet from the
north to the south and vice versa.
The study of cross-jet transport has been motivated also by a series of
laboratory experiments SMS89 ; BMS91 ; SHS93 on Rossby waves propagating
along an azimuthal jet in a rapidly rotating tank. This flow can be modeled in
the linear approximation of the corresponding fluid equations DM93 by a
stream function which is a superposition of a Bickley jet and two neutral
modes (Rossby waves). The destruction of a barrier to cross-jet transport has
been studied analytically by using the Chirikov criterion in the pendulum
approximation and numerically by using Poincaré sections DM93 . It was shown
that one needs very large values of the perturbation amplitudes to break the
barrier. The analytic model proposed in DM93 has been used recently to study
Lagrangian dynamics of atmospheric zonal jets and the permeability of the
stratospheric polar vortex. Poincaré sections and finite-time Lyapunov
exponents revealed a robust transport barrier which can be broken either due
to large perturbation amplitudes of the Rossby waves or as a result of an
increase of their phase velocities Rypina . A comparison of properties of
cross-jet transport in ad hoc kinematic and dynamically consistent models of
atmospheric zonal jets has been done recently in Ref. P.H.Haynes .
Being motivated by Lagrangian observations of the oceanic currents, cross-jet
transport and mixing have been studied in numerical models of meandering jets
Miller97 ; YPJ02 ; YPJ04 . It has been shown both in barotropic and baroclinic
nonlinear numerical models, where the meander amplitude can not be made
arbitrary large, that cross-jet chaotic transport, resulting from the
meandering motions, are maximized at a subsurface level. Since the undisturbed
velocity is weaker at deeper levels, the corresponding separatrices are closer
to the jet core. Therefore, separatrix reconnection should occur below some
critical depth and transport across the jet should be facilitated.
Independent on the work on cross-jet transport in the geophysical community,
there have been a number of theoretical and numerical investigations of
chaotic transport in so-called area-preserving nontwist maps HH84 ; W88 ; DM93
; PhysD96 ; Aizawa ; PhysD97 ; Shinohara97 ; Wurm05 ; Wurm04 . We mention
specially the early study of different reconnection scenario HH84 ; W88 and
the first systematic study of cross-jet transport in nontwist maps DM93 ;
PhysD96 ; PhysD97 . These maps locally violate the twist condition, a map
analogue of the non-degeneracy condition for Hamiltonian systems. Nontwist
maps are of interest because many important mathematical results, including
KAM and Aubry-Mather theory, depend on the twist condition. Apart from their
mathematical importance, nontwist maps are of a physical interest because they
are able to model transition to global chaos, the term meaning in the
mathematical community a cross-jet transport. Nontwist maps allow to study
different scenarios for this transition: reconnection of separatrices,
meandering and breakup of invariant tori, and others.
The onset of global chaos in oscillatory Hamiltonian systems, where the
eigenfrequency possesses a local extremum as a function of energy, has been
studied analytically and numerically in Refs. SMSoskin ; S.M.S . In such
systems with two or more separatrices, global chaos may occur at unusually
small magnitudes of perturbation due to overlap in the phase space between
resonances of the same order and their overlap in energy with chaotic layers
of the corresponding unperturbed separatrices.
In the present paper we develop a method for detecting a barrier to cross-jet
Lagrangian transport (or global chaos in a more general context), apply it to
the kinematic model of a meandering jet flow, study changes in its topology
under varying the perturbation parameters, and scenarios of its destruction.
In Sec. II we briefly introduce a model streamfunction which is known to
produce chaotic advection S92 ; Chaos06 ; RWig06 and compute the
amplitude–frequency $\varepsilon$–$\nu$ diagram demonstrating the parameter
range for which cross-jet transport exists. Based on the symmetry of the flow,
we propose in Sec. IIIa a numerical method to identify a central invariant
curve (CIC) which is a diagnostic means to detect the process of destruction
of a central transport barrier (CTB). The CIC is constructed by successive
iterations of so-called indicator points Aizawa . Computing the fractal
dimension of a set of iterations of those points at different values of the
parameters, we identify whether the CIC and CTB are broken or not. In Sec.
IIIb we study possible geometries of the CIC that may change dramatically with
varying $\varepsilon$ and $\nu$. Before the total destruction, the CIC
experiences a number of local bifurcations becoming a complicated meandering
curve whose properties can be specified by its length and the winding number
$w$. A structure of the set of CICs is revealed in a continued fraction
representation of their winding numbers. The CICs with rational $w$ are
arranged in hierarchical series connected with the corresponding resonances.
Whereas, the CICs with noble numbers form their own series. Destruction of CTB
is studied in Sec. IV for two ways in the parameter space. When moving along a
so-called resonant bifurcation curve with a rational value of $w$, one
specifies the values of $\varepsilon$ and $\nu$ for which the CIC is broken
but CTB remains. In contrary to that, when moving along any curve with noble
value of $w$, a CIC exists providing CTB. The process of CTB destruction in
both the cases is illustrated in Sec. IV.
## II The amplitude–frequency diagram for cross-jet transport in the model
flow
We take the Bickley jet with a running wave imposed as a kinematic model of a
meandering shear flow in the ocean. The respective normalized stream function
in the frame moving with the phase velocity of the meander has the following
form Chaos06 :
$\Psi=-\tanh{\left(\frac{y-A\cos
x}{L_{\text{jet}}\sqrt{1+A^{2}\sin^{2}x}}\right)}+Cy,$ (2)
where the jet’s width $L_{\text{jet}}$, meander’s amplitude $A$ and its phase
velocity $C$ are the control parameters. The phase portrait of the advection
equations (1) with the streamfunction (2), shown in Fig. 1 (a), consists of
three different regions: the central eastward jet $J$, chains of the northern
and southern circulation cells $C$ and the peripheral westward currents $P$.
The flow is steady in the moving frame of reference, and passive particles
follow the streamlines. In Fig. 1 (b) we plot a frequency map $f(x_{0},y_{0})$
that shows by nuances of the grey color the value of the frequency $f$ of
particles with initial positions ($x_{0},y_{0}$) in the unperturbed system.
The maximal value of the frequency, $f_{\text{max}}=1.278$, have the particles
moving in the central jet.
As a perturbation, we take the simple periodic modulation of the meander’s
amplitude
$A=A_{0}+\varepsilon\cos\nu t.$ (3)
Under the perturbation, the separatrices, connecting saddle points, are
destroyed and transformed into stochastic layers. The strength of chaos
depends strongly on both the perturbation parameters, the perturbation
amplitude $\varepsilon$ and frequency $\nu$. In the model used the normalized
control parameters are connected with the dimensional ones as follows Chaos06
: $A=ak$, $C=c/u_{\text{m}}\lambda k$, and $L_{\text{jet}}=\lambda k$, where
$a,k$, and $c$ are amplitude, wave number, and phase velocity of a meander,
respectively, $\lambda$ and $u_{\text{m}}$ are characteristic width and
maximal zonal velocity in the jet on the surface. All these parameters change
in a wide range in the Gulf Stream B89 ; S92 ; Meyers94 : $\lambda\simeq
40\div 100$ km, $a\simeq 50\div 60$ km, $2\pi/k\simeq 200\div 400$ km,
$c\simeq 0.1\div 0.5$ m/sec, $u_{\text{m}}\simeq 1\div 1.5$ m/sec. So, we get
$L_{\text{jet}}\simeq 0.1\div 3$, $A\simeq 0.7\div 2$, and $C\simeq 0.02\div
0.3$. Being motivated by mixing and transport in the Gulf Stream, we took the
following normalized values of the control parameters that will be used in all
our numerical experiments: $A_{0}=0.785,C=0.1168$ and $L_{\text{jet}}=0.628$.
The equations of motion (1) with the stream function (2) and the perturbation
(3) have the symmetry
$\hat{S}:\left\\{\begin{aligned} x^{\prime}&=\pi+x,\\\
y^{\prime}&=-y\end{aligned}\right.$ (4)
and the time reversal symmetry
$\hat{I}_{0}:\left\\{\begin{aligned} x^{\prime}&=-x,\\\
y^{\prime}&=y.\end{aligned}\right.$ (5)
The symmetries (4) and (5) are involutions, i. e. $\hat{S}^{2}=1$ and
$\hat{I}_{0}^{2}=1$. Due to the symmetry $\hat{S}$, motion can be considered
on the cylinder with $0\leq x\leq 2\pi$. 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 should
be stressed that the phase space in two-dimensional incompressible flows is a
configuration space for advected particles.
Figure 1: (a) Phase portrait of the unperturbed flow in the frame moving with
the meander’s phase velocity. Streamlines in the circulation cells ($C$), jet
($J$), and peripheral currents ($P$) are shown. (b) Frequency map represents
by color values of the frequency $f$ of particles with initial positions
($x_{0},y_{0}$) advected by the unperturbed flow.
The following numerical procedure has been applied to establish the fact of
cross-jet transport in the kinematic model of the meandering jet current. The
advection equations (1) for given values of the perturbation amplitude
$\varepsilon$ and frequency $\nu$ and with twenty particles, released nearby
the northern saddle point, have been integrated up to the time instant when
one of the particles was detected to cross the straight line $y=y_{s}$ passing
through the southern saddle. If after the time $T_{\text{max}}=1000\times
2\pi/\nu$ none of the particles crosses the line $y=y_{s}$, we assume that for
given values of the parameters there is no cross-jet transport. The
$\varepsilon$–$\nu$ diagram in Fig. 2 shows the values of the parameters for
which cross-jet chaotic transport exists (white-color zones). There are a
number of the frequency values for which transport occurs at surprisingly
small values of the perturbation amplitude $\varepsilon$. The absolute minimal
value of the perturbation amplitude, at which the cross-jet chaotic transport
occurs, $\varepsilon_{\text{min}}=0.0218\approx A_{0}/36$, corresponds to the
frequency $\nu=1.165$ which is close to the natural frequencies of the
particles moving in the central jet.
Figure 2: Amplitude–frequency $\varepsilon$–$\nu$ diagram showing the
parameter values for which cross-jet chaotic transport exists (white zones) or
not (black zones).
## III Topology of a barrier to cross-jet transport and central invariant
curve
The amplitude–frequency diagram is useful to detect cross-jet transport but
its computation is a time consuming procedure. Moreover, it says nothing about
the properties of barrier to transport and mechanism of its destruction. A
fractal-like boundary between the colors in Fig. 2 reflects an intermittency
in appearance and destruction of the cross-jet barrier when varying
$\varepsilon$ and $\nu$. Further insight into topology of the barrier could be
obtained if one would be able to find an indicator of cross-jet transport, i.
e. an object in the phase space whose form contains an information about
permeability of the barrier.
### III.1 Detecting the central invariant curve
First of all, we need to give definitions of some basic structures specifying
a cross-jet barrier and its destruction. The central transport barrier (CTB)
is defined as a strip between the southern and northern unperturbed
separatrices confined by marginal northern and southern ballistic trajectories
(excluding orbits of ballistic resonances in the stochastic layer). All the
trajectories inside the CTB are ballistic, some of them are regular and the
other ones are chaotic. The amplitude–frequency diagram in Fig. 2 demonstrates
clearly destruction of the CTB at some values of the perturbation parameters
$\varepsilon$ and $\nu$ and onset of cross-jet transport.
Our Hamiltonian flow with the streamfunction (2) is degenerate, i. e. it
violates the non-degeneracy condition, $\partial f/\partial I\neq 0$, for some
values of the natural frequency of passive particles $f$ and their actions $I$
in the unperturbed system. Physically it means that the zonal velocity profile
$u(y)$ has a maximum. In theory of nontwist maps the curve, for which the
twist condition (analogue of the non-degeneracy condition) is violated, is
called a nonmonotonic curve Wurm05 . In our model flow (2) it is some value of
unperturbed streamfunction along which the frequency $f$ is maximal (see Fig.
1 (b)).
Instead of integrating the advection equations (1), we integrate the
corresponding Poincaré map, an orbit of which is defined as a set of points
$\\{(x_{i},y_{i})\\}_{i=-\infty}^{\infty}$ on the phase plane such that
$\hat{G}_{T}(x_{i},y_{i})=(x_{i+1},y_{i+1})$, where $\hat{G}_{t}$ is an
evolution operator on a time interval $t$ and $T\equiv 2\pi/\nu$ is the period
of perturbation. Operator $\hat{G}_{T}$ can be factorized as a product of two
involutions $\hat{G}_{T}=\hat{I}_{1}\hat{I}_{0}$, where
$\hat{I}_{1}=\hat{G}_{T}\hat{I}_{0}$ is also a time reversal simmetry PhysD96
.
A periodic orbit of period $nT$ ($n=1,2,\dots$) is an orbit such that
$(x_{i+n},y_{i+n})=(x_{i}+2\pi m,y_{i})$, $\forall i$, where $m$ is an
integer. An invariant curve is a curve invariant under the map. The
nonmonotonic curve is not an invariant curve under a perturbation. The winding
(or rotation) number $w$ of an orbit is defined as the limit
$w=\lim\limits_{i\to\infty}[(x_{i}-x_{0})/(2\pi i)]$, when it exists. The
winding number is a ratio between the frequency of perturbation $\nu$ and the
natural frequency $f$. Periodic orbits have rational winding numbers $w=m/n$.
It simply means that a ballistic passive particle in the flow flies $m$ frames
before returning to its initial position $x_{0}$ (modulo $2\pi$) after $n$
periods of perturbation. Winding numbers of quasiperiodic orbits are
irrational.
Now we are ready to introduce the important notion of a central invariant
curve (CIC). We define CIC as an curve which is invariant under the operators
$\hat{S}$ and $\hat{G}_{T}$. It can be shown, that two curves invariant under
$\hat{S}$ have at least two common points. The curves, which are invariant
under $\hat{G}_{T}$, cannot intersect each other. So, the CIC is a unique
curve. Following to Ref. Shinohara97 , one can show that the CIC corresponds
to a local extremum on the winding number profile with an irrational value of
$w$. Such curves are called <<shearless curves>> in theory of nontwist maps
Wurm05 . The significance of a shearless curve is that it acts as a barrier to
global transport in the phase space of a nontwist map. The violation of the
twist condition leads to existence of more than one orbit with the same
winding number arising in pairs on both sides of the shearless curve. Those
pairs of orbits can collide and annihilate at certain parameter values. The
collision of the orbits involves in phenomenon, which was called as
reconnection of invariant manifolds of the corresponding hyperbolic orbits
HH84 .
The CIC should not be thought as the last cross-jet barrier curve in the CTB
in the sense that it breaks down under increasing the perturbation amplitude
in the last turn. Sometimes it is the case, but sometimes it is not.
Nevertheless, the CIC serves a good indicator of the strength of the CTB and
its topology.
The CIC can be constructed by successive iterations of so-called indicator
points Aizawa . In our model flow (2) with the symmetries (4) and (5),
indicator points are the points ($x_{j}^{(k)}$, $y_{j}^{(k)}$), $k=1,2$, which
are solutions of the equations
$\hat{I}_{0}(x_{j}^{(1)},y_{j}^{(1)})=\hat{S}(x_{j}^{(1)},y_{j}^{(1)}),$ (6)
or
$\hat{I}_{1}(x_{j}^{(2)},y_{j}^{(2)})=\hat{S}(x_{j}^{(2)},y_{j}^{(2)}),$ (7)
where index $j$ numerates the points. The equation (6) gives a pair of
indicator points: ($x_{1}^{(1)}=\pi/2$, $y_{1}^{(1)}=0$) and
($x_{2}^{(1)}=3\pi/2$, $y_{2}^{(1)}=0$). Instead of solving Eq. (7) we solve
the equivalent equation
$\hat{G}_{T}(x,y)=\hat{I}_{0}\hat{S}(x,y)\equiv(\pi-x,-y).$ (8)
If some ($x$, $y$) is a solution of (8), then $\hat{I}_{0}(x$, $y)$ is a
solution of (7). The equation (8) cannot be solved analitically, so we apply
the numerical method based on computing a minimum of the function
$r(x,y)=||\hat{G}_{T}(x,y)-(\pi-x,-y)||$, where $||\cdot||$ is a norm on the
cylinder. Since $r(x,y)\geq 0$ for any $(x,y)$, the points with $r(x,y)=0$ are
minima of the function $r(x,y)$. Thus, solution of Eq. (8) reduces to
searching for a local minimum of the function $r(x,y)$ with the additional
condition $r(x,y)=0$ at the point of the minimum. There are a number of
numerical methods for doing that job. We prefer to use the downhill simplex
method. In our problem the function $r(x,y)$ always has two minima, i. e. two
indicator points transforming to each other under action of the operator
$\hat{S}$.
Next, we study iterations, i. e. Poincaré mapping of one of the indicator
points $(x_{0},y_{0})$. If the iterations
$(x_{i},y_{i})=\hat{G}_{T}^{i}(x_{0},y_{0})$ are confined between invariant
curves in a bounded region, the following three cases are possible in
dependence on the dimension $d$ of the set $(x_{i},y_{i})$:
1) The iterations lie on a curve on the phase plane with $d=1$ which is a CIC.
2) The iterations is an organized set of points with $d=0$. It means that they
constitute either a central periodic orbit or a central almost periodic orbit,
an orbit that could not form a smooth curve on the phase plane for a limited
integration time.
3) The iterations form a central stochastic layer with $d=2$.
If the iterations are not confined by any invariant curves in a bounded
region, i. e. they occupy all the accessible phase plane to the south and
north from the central jet, then there exists global chaotic transport. Thus,
the type of motion of indicator points provides an indicator of global chaos
and the absence of barriers to cross-jet transport.
Possible topologies of the corresponding CTB at the fixed frequency $\nu=1.2$
and with increasing values of the perturbation amplitude $\varepsilon$ are
illustrated in Fig. 3 plotting iterations of the indicator points computed by
the above-mentioned method. The panel (a) illustrates the case when those
iterations form a CIC. Another typical situation is shown in Fig. 3 (b) where
the iterations fall in small segments filling at $\tau\to\infty$ a continuous
curve which is a central almost periodic orbit. If the iterations fill up not
a curve but a bounded region between invariant curves, then there appears a
central stochastic layer preventing cross-jet transport (Fig. 3 (c)). When
iterations of the indicator points occupy a region that is not confined by any
invariant curves, it means destruction of the CTB and onset of global chaos,
i. e., chaos in a large region of the phase space accompanied by cross-jet
transport.
The indicator points have been found with the help of the above-mentioned
numerical procedure, and their iterations have been computed in the following
range of the control parameters: $\nu\in[0.95:1.5]$ and
$\varepsilon\in[0.01:1]$. We assume that the iterations are bounded, if their
coordinates do not cross the unperturbed separatrices after $5\times 10^{4}$
iterations. The dimension $d$ of the set of those iterations is computed by
the box-counting method, where the value of $d$ for the box size
$e_{k}=(1/2)^{k}$ is defined as
$d_{k}=\log_{2}\frac{N_{k+1}}{N_{k}},$ (9)
where $N_{k}$ is a number of boxes of the size $e_{k}$ containing set points.
The dimension $d_{k}$ goes to zero with decreasing $e_{k}$, and one cannot
distinguish in this limit between the central almost periodic orbit and the
central stochastic layer at large $k$. Comparing the values of $d_{k}$ at
different values of $k$, we were able to find the empirical value $k=4$ which
is enough to make the difference.
The results of computation of the dimension $d_{4}(\varepsilon,\nu)$ for a set
of iterations of the indicator points are shown in the bird-wing diagram in
Fig. 4. That one and the other bird-wing diagrams in the parameter space show
the properties of CTB and CIC in the range of comparatively small values of
the perturbation amplitude ($0.01\leq\varepsilon\leq 0.1$) and the frequency
($1.15\leq\nu\leq 1.5$) corresponding to particles moving in the central jet
(Fig. 1 (b)). White color corresponds to the regime of global chaotic
transport with unbounded motion of iterations of the indicator points.
Otherwise, the CTB exists but its topology is different. Grey color means that
there exists a CIC with $0.95\leq d_{4}\leq 1.05$ in the corresponding range
of the parameters. White rectangles, which are hardly visible in the main
panel (see their magnification on the inset of the figure), means existence of
a central almost periodic orbit with $d_{4}<0.95$ and black strips — a central
stochastic layer with $d_{4}>1.05$.
### III.2 Geometry of the central invariant curve and its bifurcations
To quantify complexity of the CIC we define its length $L$ as a sum of the
distances between the iterations of the indicator points $(x_{i},y_{i})$
ordered on the phase plane in the following way:
1. 1.
The first step. A point $B_{0}$, belonging to a set of iterations of the
Poincaré map $(x_{i},y_{i})$, is marked.
2. 2.
The $(j+1)$-th step. We find and mark among all the unmarked points that one,
$B_{j+1}$, which minimizes the Euclidean distance $D_{j}=D(B_{j},B_{j+1})$
between $B_{j}$ and $B_{j+1}$.
3. 3.
The procedure is repeated unless all the points will be marked.
Figure 3: Poincaré mapping of indicator points. In the first three panels the
orbit of these points is bounded and there exists a central transport barrier
(CTB) with (a) CIC $(\varepsilon=0.01,\nu=1.2)$, (b) central almost periodic
orbit $(\varepsilon=0.011997277,\nu=1.2)$, and (c) central stochastic layer at
$(\varepsilon=0.01177,\nu=1.2)$ with the inset demonstrating a magnification
of a small region. (d) Destruction of CTB and onset of global chaotic
transport as a result of unbounded iterations of indicator points
$(\varepsilon=0.041,\nu=1.2)$.
Figure 4: Bird-wing diagram of the box-counting dimension
$d_{4}(\varepsilon,\nu)$. White color: regime with global chaotic transport
with unbounded motion of indicator points (see Fig. 3 (d)). If the motion of
indicator points is bounded, then there exists a CTB but its topology may
differ. Grey color ($0.95\leq d_{4}\leq 1.05$): regime with a CIC (see Fig. 3
(a)). Small white regions which are hardly visible inside the grey ‘‘wing’’
($d_{4}<0.95$): regime with a central almost periodic orbit (see Fig. 3 (b)).
Black color ($d_{4}>1.05$): regime with a central stochastic layer (see Fig. 3
(c)). Inset shows magnification of a small region in the parameter space with
visible white and black regions.
As an output we have an ordered set of points $B_{j}$ constituting a CIC. The
accuracy is controlled by the quantity $\max{D_{j}}$. Large values of this
quantity mean that the points are ordered in a wrong way or a set of points is
chaotic. To increase the number of points we use in addition to the original
points $(x_{i},y_{i})$ their ‘‘images’’ $(x_{i}+\pi,-y_{i})$ as well. To
minimize the computation time the points are sorted in accordance with their
$x$ coordinates.
Figure 5 illustrates metamorphosis of the CIC as the perturbation amplitude
increases. We start with the CIC, shown in Fig. 5 (a), which we call a
nonmeandering CIC. At the critical value $\varepsilon\approx 0.011758$,
invariant manifolds of hyperbolic orbits of two chains of the 1:1 resonance
islands on both sides of the CIC connect, and after that the CIC becomes a
meandering curve of the first order (Fig. 5 (b)) and period $T$. The period of
CIC’s meandering is simply a period of nearby main islands Shinohara97 ;
Simo98 . At the next critical value $\varepsilon=0.01178721$, reconnection of
invariant manifolds of secondary resonance islands takes place. The
corresponding second-order meandering CIC with period $79T$ is shown in Fig. 5
(c). Highly meandering CICs of higher orders appear with further increasing
the perturbation amplitude (Fig. 5 (d)).
Figure 5: Metamorphosis of the central invariant curve (CIC). (a)
Nonmeandering CIC ($\nu=1.2$, $\varepsilon=0.01174929$). (b) Meandering CIC of
the first order and period $T$ ($\nu=1.2$, $\varepsilon=0.01178721$). (c)
Meandering CIC of the second order and period $79T$ ($\nu=1.2$,
$\varepsilon=0.01179027$). (d) Meandering CIC of a higher order ($\nu=1.2$,
$\varepsilon=0.01179339$).
Some smooth invariant curves inside the CTB break down under the perturbation
(3), and chains of ballistic resonance islands appear at their place. Those
islands appear in pairs to the north and south from a CIC due to the flow
symmetries (4) and (5) (see Fig. 5 (a) and (b)). Geometry of the CIC, size and
number of the islands, and topology of their invariant manifolds change with
variation of the perturbation amplitude $\varepsilon$ and frequency $\nu$ in a
very complicated way.
In Fig. 6 we plot in the parameter space the values of the CIC length $L$
coding it by nuances of the grey color. White color corresponds to those
values of the parameters $\varepsilon$ and $\nu$ for which cross-jet transport
exists due to destruction of the CTB. Black color codes the regime with a
broken CIC but a remaining CTB that prevents cross-jet transport
($d_{4}>1.05$). Dotted and dashed lines on the plot are the resonant
bifurcation curves along which the CIC winding number $w$ is rational. The
$m/n$ resonant bifurcation curve is the set of values of the control
parameters for which a reconnection of invariant manifolds of the $n:m$
resonances takes place. The dotted lines correspond to even resonances with
$w=~{}(2k~{}-~{}1)~{}/~{}2k$ and the dashed lines are odd resonances with
$w=2k/(2k+1)$, $k=1,2,\dots$. All those curves end up in the dips of the bird-
wing diagram.
Figure 6: Bird-wing diagram showing the length $L$ of the CIC by nuances of
the grey color in the parameter space $(\varepsilon,\nu)$. White zone: regime
with a broken CTB and cross-jet transport. Black color: regime with a broken
CIC but a remaining CTB preventing cross-jet transport. Resonant bifurcation
curves, along which the CIC winding numbers $w$ are rational, end up in the
dips of ‘‘the wing’’. Dotted and dashed lines correspond to even and odd
resonances, respectively.
Figure 7: Dependence of the length of CIC $L$ on the perturbation amplitude at
the fixed frequency $\nu=1.2$. (a) General view of the dependence
$L(\varepsilon)$ in the range of interest $\varepsilon=[0.01175:0.012]$.
Vertical dotted lines correspond to rational values of the CIC winding number
$w$. The resonance $1/1$ appears at $\varepsilon=0.11756$. The arrangement of
the spikes is explained in the text. (b) Magnification of one of the wide
spikes in panel (a). Solid line is a winding number profile $w(\varepsilon)$
with the value $w=62/63$ shown by the dashed line.
In order to analyze a fractal-like boundary of the bird-wing diagram in Fig.
6, we cross it horizontally at the frequency $\nu=1.2$ and consider the plot
$L(\varepsilon)$ in the range of interest of $\varepsilon$ (Fig. 7 (a)). The
perturbation frequency $\nu=1.2$ is close to the maximal frequency
$f_{\text{max}}$ of particles in the middle of the jet in the unperturbed flow
(see Fig. 1 (b)). The plot $L(\varepsilon)$ consists of a number of spikes
with different height and width. In the range of small values of the
perturbation amplitude ($\varepsilon<0.011756$), the length of the CIC is
approximately the same $L\approx 7.35$ (a small fragment of the function
$L(\varepsilon)$ is shown in Fig. 7 (a) just to the left from the vertical
line $1/1$). In that range, the CIC is a nonmeandering curve (see Fig. 5 (a))
surrounded by smooth invariant curves and $1:1$ resonance islands with a
heteroclinic topology. The size of those islands is comparable with the frame
size. The width of the CTB, filled by invariant curves around the CIC,
decreases with increasing the perturbation amplitude $\varepsilon$.
Figure 8: Poincaré sections at $\nu=1.2$ and increasing values of
$\varepsilon$ in the range corresponding to the wide spike with $w=62/63$ in
Fig. 7 (b). (a) Meandering CIC surrounded by meandering invariant curves
$(\varepsilon=0.011931)$. (b) CIC meandering between the odd $63:62$ islands
born as a result of a saddle-center bifurcation
($\varepsilon_{\text{sc}}=0.011934)$. (c) CIC destruction due to connection of
invariant manifolds of the $63:62$ islands $(\varepsilon=0.01193511)$. A
narrow stochastic layer appears at the place of the CIC. (d) CIC appears again
$(\varepsilon=0.0119352)$.
The CIC winding number $w$ changes under a variation of the perturbation
amplitude. At $\varepsilon\simeq 0.011756$, invariant manifolds of the 1:1
resonance connect, and a central stochastic layer appears at the place of the
CIC. This layer exists up to $\varepsilon\simeq 0.011785$ (see a random set of
points in Fig. 7 (a) in that range of $\varepsilon$). At
$\varepsilon>0.011785$, the CIC appears again. Now it is a meandering curve of
the first order (see Fig. 5 (b)) whose length is larger due to reconnection of
the 1:1 resonance islands. As $\varepsilon$ increases further, the CIC length
$L$ changes in a wide range. Smooth fragments with approximately the same
value of $L\approx 20$ alternate with spikes of different height and width.
The spikes are condensed, when approaching to the value $w=1/1$, and overlap
in the range $\varepsilon\simeq[0.011756:0.011785]$.
Figure 9: Poincaré sections at $\varepsilon=0.015$ and increasing values of
$\nu$. (a) CIC between islands of the even 2:1 resonance
($\nu=2.51<2f_{\text{max}}=2.556$). (b) Reconnection of invariant manifolds of
that resonance and a formation of a vortex pair with a narrow stochastic layer
shown by bold curves ($\nu=2.55$). (c) The vortex size decreases with
increasing $\nu$ ($\nu=2.555$). (d) Past some critical value of $\nu$, the
vortex pair disappears and CIC appears again ($\nu=2.56$).
The arrangement of the spikes in Fig. 7 (a) can be explained using a
representation of rational numbers by continued fractions. A continued
fraction is the expression
$c=[a_{0};a_{1},a_{2},a_{3},\dots]=a_{0}+\frac{1}{a_{1}+\dfrac{1}{a_{2}+\dfrac{1}{a_{3}+\cdots}}},$
(10)
where $a_{0}$ is an integer number and the other $a_{n}$ are natural numbers.
Any rational (irrational) number can be represented by a continued fraction
with a finite (infinite) number of elements. The spikes in Fig. 7 (a) are
arranged in convergent series in such a way that each spike in a series
generates a series of spikes of the next order. For example, the series of the
integer $n:1$ resonance has the winding numbers equal to $1/n$ or $[0;n]$ in
the continued-fraction representation. Each spike in that series generates a
series of resonance spikes of the next order converging to the parent spike.
Winding numbers of those resonances are $[0;n,i]$, $i=2,3,4\dots$ (at $i=1$,
one gets a spike in the main series because of the identity
$[a_{0};a_{1},\dots,a_{n},1]\equiv[a_{0};a_{1},\dots,a_{n}+1]$). The spikes
with $[0;1,i]=i/(i+1)$ converge to the spike of the $1:1$ resonance. That is
clearly seen in Fig. 7 (a). The direction of convergence of the spikes in a
series alternate with the series order: the winding number increases in the
series of the first order, decreases in the series of the second order, and
increases again in the series of the third order. That is why a chaotic region
in Fig. 7 (a) is situated to the right from the $1:1$ resonance, i. e. in the
range of smaller values of $w$. Whereas, it is to the left for the series of
the second order, i. e. in the range of larger values of $w$. There also
exists an additional hierarchical structure with fractional $1:n$ resonances,
whose frequencies are below the $1:1$ resonance frequency, and a series with
resonances corresponding to the spikes below $[0;1,i,(1)]$, for example, a
clearly visible series of spikes below $[0;1,4,(1)]$ converging to the spike
$5/6$ in the $L$ diagram (see Fig. 6).
Unfortunately, we could not identify series of the third and a higher order
because numerical errors in identifying the winding numbers are greater than
the distance between the spikes of higher-order series. Unresolved regions on
the plot $L(\varepsilon)$ in Fig. 7 (a) appear because of a decrease of the
distance between the spikes in the same series with increasing series number,
a process resembling Chirikov’s overlapping of resonances.
A magnification of one of the wide spikes is shown in Fig. 7 (b). We plot the
winding number profile $w(\varepsilon)$ together with the function
$L(\varepsilon)$ for the spike. To illustrate what happens with the CIC and
its surrounding with increasing $\varepsilon$ we plot the corresponding
Poincaré section in Fig. 8. In the range $\varepsilon\simeq[0.0119:0.01192]$
the lengths of the CIC and surrounding invariant curves increase slowly due to
small changes in their geometry (Fig. 8 (a)). After a saddle-center
bifurcation at $\varepsilon_{\text{sc}}\simeq 0.011934$, there appear two
chains of homoclinically connected $63:62$ islands separated by a meandering
CIC (Fig. 8 (b)). The amplitude of the CIC meanders increases with further
increasing $\varepsilon$ in the range $\varepsilon\simeq[0.011934:0.011935]$.
In that range the CIC disappears and appears again in a random-like manner
(see the corresponding fragment on the plot $L(\varepsilon)$ in Fig. 7 (b))
due to overlapping of higher-order resonances and reconnection of their
invariant manifolds. The example of such a reconnection for the $63:62$
resonance at $\varepsilon=0.01193511$ is shown in Fig. 8 (c) where a
stochastic layer appears at the place of the CIC. As $\varepsilon$ increases
further, the CIC appears again but with a smaller number of meanders (Fig. 8
(d)). Animation of the corresponding patterns is available at
http://dynalab.poi.dvo.ru/papers/cic.avi.
The other wide spikes in the plot $L(\varepsilon)$ with a similar structure
are caused by another odd resonances between the external perturbation and
particle’s motion along the CIC. Under a CIC resonance with the winding number
$w=m/n$, we mean reconnection of invariant manifolds of the resonance $n:m$
and onset of a local stochastic layer. The narrow spikes, situated between the
wide ones in Fig. 7 (a), correspond to reconnection of even resonances. They
are hardly resolved on the plot. Even resonances of higher orders have a
smaller effect on CIC geometry then odd resonances. As an example, we
illustrate in Fig. 9 metamorphosis of the CIC with the winding number $w=1/2$.
The perturbation amplitude is fixed at a rather small value
$\varepsilon=0.015$ and the frequency increases in the range
$2.51<\nu<2.556=2f_{\text{max}}$. At $\nu=2.51$ there are islands of the even
$2:1$ resonance separated by a CIC (Fig. 9 (a)). At some critical value of
$\nu$ invariant manifolds of the $2:1$ resonance connect and the islands form
a tight vortex-pair structure surrounded by a narrow stochastic layer (see
Fig. 9 (b) at $\nu=2.55$). The size of the pair decreases gradually with
further increasing $\nu$, and the corresponding hyperbolic orbits approach
each other (see Fig. 9 (c) at $\nu=2.555$). At some critical value of $\nu$,
hyperbolic and elliptic orbits of the resonance collide and annihilate, and
CIC appears again (see Fig. 9 (d) at $\nu=2.56$). Vortex pairs of the other
even resonances are formed in a similar way. The higher is the order of the
resonance, the smaller is the vortex size.
We conclude this section by computing winding numbers $w$ of the CIC in the
parameter space. The result is shown in the bird-wing diagram in Fig. 10. The
CIC does not exist in the white region where the CTB is broken and cross-jet
transport takes place. The curves, which end up on the tips of the ‘‘feathers
of the wing’’, have winding numbers $w$ with the following continued-fraction
representation: $[a_{0};a_{1},\dots,a_{n},(1)]$. These are so-called noble
numbers which are known to be the numbers that cannot be approximated by
continued-fraction sequences to better accuracy than the so-called Diophantine
condition (see, for example, Almeida ). The CICs with noble winding numbers
are in a sense the most structurally robust invariant curves, i. e. they may
survive under a comparatively large perturbation preventing cross-jet
transport. The noble curves are arranged in series like the resonant
bifurcation curves with rational winding numbers which end up in the dips of
‘‘the wing’’ in the bird-wing diagram in Fig. 6. For example, the noble series
$[0;1,i,(1)]$ in Fig. 10 corresponds to the resonance series $[0;1,i]$ in Fig.
6. In the $w$ diagram we show a few representatives of the noble series
$[0;1,i,(1)]$ and series of the next order $[0;1,i,j,(1)]$ (see Fig. 10 with
$j=2,3$).
Figure 10: Bird-wing diagram $w(\varepsilon,\nu)$ in the parameter space
showing values of the winding number $w$ of the CIC by nuances of the grey
color. White zone: regime with broken CTB and cross-jet transport. The curves
with irrational winding numbers end up on the tips of the ‘‘feathers of the
wing’’ (some of them are marked by the corresponding noble numbers), whereas
the curves with rational winding numbers (shown in Fig. 6) end up in the dips
of ‘‘the wing’’.
## IV Breakdown of central transport barrier
Figure 11: Destruction of central transport barrier upon moving in the
parameter space along a resonant bifurcation curve with the rational winding
number $w=8/9$. (a) Narrow stochastic layer on the Poincaré section is
confined between invariant curves which provide a transport barrier. The
perturbation parameters ($\varepsilon=0.04889$, $\nu=1.31625$) are chosen on
the curve with $w=8/9$ nearby its right edge (see Fig. 6). (b) Onset of cross-
jet transport at the values of parameters ($\varepsilon=0.054$, $\nu=1.285$)
chosen in the white zone of that dip.
Figure 12: Destruction of central transport barrier upon moving in the
parameter space along a curve with the noble value of the CIC’s winding number
$w=[0;1,5,1,(1)]$. When approaching a tip of the corresponding ‘‘feather of
the wing’’ in Fig. 10, one observes on the Poincaré section a decrease in the
width of transport barrier with a CIC (bold curves) inside. (a)
$\varepsilon=0.07017$, $\nu=1.367875$. (b) $\varepsilon=0.07416$,
$\nu=1.350375$. (c) $\varepsilon=0.0796067$, $\nu=1.325875$. (d) Onset of
cross-jet transport at the values of parameters ($\varepsilon=0.08$,
$\nu=1.3142$) chosen beyond a tip of that ‘‘feather’’ in the white zone in
Fig. 10.
We have studied in the preceding section properties of the CIC which has been
shown to be a diagnostic means to characterize CTB and its destruction. CTB
separates water masses to the south and the north from the central jet and
prevents their mixing. It is not a homogeneous jet-like layer but consists of
chains of ballistic islands, narrow stochastic layers, and meandering
invariant curves of different orders and periods (including a CIC) to be
confined by invariant curves from the south and the north. Those curves break
down one after another when increasing the perturbation amplitude
$\varepsilon$, producing stochastic layers at their place on both sides of the
central jet, until the stochastic layers merge with one another and with
stochastic layers around the southern and northern circulation cells producing
a global stochastic layer and onset of cross-jet transport.
Upon moving along any resonant bifurcation curve with a rational value of the
winding number $w$ in the bird-wing diagram in Fig. 6, we have those values of
the perturbation amplitude $\varepsilon$ and frequency $\nu$ at which the
corresponding CIC is broken due to reconnection of invariant manifolds. It
does not mean that CTB is broken as well. That is the case only if we are at
the dips of the ‘‘wing’’. The process of CTB destruction for this type of
movement in the parameter space is illustrated in Fig. 11. We fix a point
($\varepsilon=0.04889$, $\nu=1.31625$) on the resonant bifurcation curve with
$w=8/9$ nearby its right edge in Fig. 6 and plot the corresponding Poincaré
section. A narrow stochastic layer, confined between invariant curves
providing a transport barrier, appears on the Poincaré section in panel (a) at
the place of a broken CIC. The barrier will be broken if one would choose the
values of parameters in the white zone in Fig. 6. Merging of southern and
northern stochastic layers and onset of cross-jet transport are shown in panel
(b) at $\varepsilon=0.054$, $\nu=1.285$.
Upon moving along any curve with a noble value of the winding number $w$ in
the bird-wing diagram in Fig. 10, we have those values of the perturbation
amplitude $\varepsilon$ and frequency $\nu$ at which a CIC with the
corresponding noble number exists. The process of CTB destruction for the
motion in the parameter space along the noble curve with $w=[0;1,5,1,(1)]$ is
illustrated in Fig. 12. When moving to the tip of the corresponding ‘‘feather
of the wing’’ in Fig. 10, one observes progressive destruction of invariant
curves and decrease of the width of the transport barrier (panels (a) and (b))
unless a single CIC remains as the last barrier to cross-jet transport (panel
(c)). Onset of cross-jet transport (panel (d)) happens at the values of
parameters chosen beyond a tip of that ‘‘feather’’ in the white zone in
Fig.10.
Upon moving along any resonant bifurcation curve to the corresponding dip of
the bird-wing diagram in Fig. 6, we find cross-jet transport at smaller values
of the perturbation amplitude as compared to the case with irrational winding
numbers because in order to provide cross-jet transport in the first case it
is enough to destruct all the KAM curves. Whereas, CICs with irrational and
especially noble values of the winding number may deform in a complicated way
but still survive under increasing $\varepsilon$ up to comparatively large
values.
## V Conclusion
Being motivated by the problem of cross-jet transport in geophysical flows in
the ocean and atmosphere, we have studied in detail topology of a central
transport barrier (CTB) and its destruction in a simple kinematic model of a
meandering current with chaotic advection of passive particles (Fig. 1) that
belong to the class of non-degeneracy Hamiltonian systems. Direct computation
of the amplitude–frequency diagram (Fig. 2) demonstrated onset of cross-jet
transport at surprisingly small values of the perturbation amplitude
$\varepsilon$ provided that the perturbation frequency $\nu$ was sufficiently
large. As an indicator of the strength of the CTB and its topology, we used a
central invariant curve (CIC) which was constructed by iterating indicator
points by a numerical procedure borrowed from theory of nontwist maps. The CTB
has been shown to exist provided a set of the iterations was bounded.
Otherwise, cross-jet transport has been observed (Fig. 3). The results were
presented as a diagram of the box-counting dimension of those sets of
iterations in Fig. 4.
Geometry of the CIC has been shown to be highly sensitive to small variations
in the parameters near a fractal-like boundary of the diagram (Fig. 5).
Quantifying complexity of the CIC’s form by its length $L$, we computed the
corresponding $L$ diagram looking like a bird wing with a fractal-like
boundary (Fig. 6). Resonant bifurcation curves with rational winding numbers
$m/n$ end up in the dips of the boundary. Along those curves in the parameter
space, invariant manifolds of the corresponding $n:m$ resonances connect
providing a destruction of the CIC. Scenarios of the reconnection are
different for odd (Fig. 8) and even (Fig. 9) resonances. Animation of the
process at http://dynalab.poi.dvo.ru/papers/cic.avi provides a visual
demonstration of complexity of topology of the CTB and its destruction.
Computing the winding number $w$ of the CIC, we have got an information about
those values of the perturbation parameters at which the CTB is strong or
weak. Using representation of the values of winding numbers by continued
fractions, we were able to order spikes with rational values of $w$ in Fig. 7
into hierarchical series of the corresponding CIC resonances. The curve, which
end up on the tips of ‘‘feathers of the wing’’ in the winding-number diagram
in Fig. 10, have noble winding numbers which are so irrational that the
corresponding CICs break down in the last turn when varying the perturbation
parameters. The noble curves have been found to be arranged in series like the
resonant bifurcation curves with rational values of $w$. Destruction of CTB is
illustrated for two ways in the parameter space: upon moving along resonant
bifurcation curves with rational values of $w$ (Fig. 11) and along curves with
noble values of $w$ (Fig. 12).
In conclusion we address two points that may be important in possible
aplications of the results abtained. Molecular diffusion in laboratory
experiment and turbulent diffusion in geophysical flows are expected to wash
out ideal fractal-like structures caused by chaotic advection after a
characteristic time scale. The question is what is this scale. As to molecular
diffusion in the ocean, the diffusion time-scale $L^{2}/D$ is very large since
the diffusion coefficient is of the order of $D\simeq 10^{-5}$ cm2/sec and $L$
is of a kilometer scale. The scale of molecular diffusion in laboratory tanks
is, of course, much smaller. However, some fractal-like structures have been
observed in real laboratory experiments (see, for example, Ref. SKG96 and the
book Book08 for a recent review of experiments). Modelling of a combined
effect of chaotic advection and turbulent diffusion in the ocean is a hard
problem deserving a special consideration. Any kind of diffusion is expected
to intensify cross-jet transport.
The advantage of the kinematic approach is its ability to identify different
factors that may enhance or suppress cross-jet transport. However, the results
obtained with our simplified kinematic model should be taken with caution to
describe cross-jet transport in real geophysical flows. In any kinematic model
the velocity field is postulated based on known features of the current while
in dynamic models it should obey dynamical equations following from the
conservation of potential vorticity P87 ; P91 . It is very difficult to
formulate an analytic and dynamically consistent model with chaotic advection
(for a discussion and examples of such models see DM93 ; P.H.Haynes ; PK06 ;
KSD08 ). One approach is to seek solutions of the fluid dynamics equations
that are self-consistent to linear order Lipps . Some aspects of cross-jet
transport in a linearized model with a zonal Bickley jet current and two
Rossby waves have been studied in Refs. DM93 ; PL95 ; Rypina . Potential
vorticity is not exactly conserved within the linear approximation but models
that are self-consistent to linear order provide a compromise between the
self-consistency demands and the fruitfulness of Hamiltonian models.
The question, how predictions of kinematic models in destruction of barriers
to cross-jet transport carry over to more realistic dynamical models, remains
open. We plan in the future to apply the methods developed in the present
paper to a dynamically consistent model of a meandering current with Rossby
waves.
## Acknowledgments
The work was supported partially by the Program ‘‘Fundamental Problems of
Nonlinear Dynamics’’ of the Russian Academy of Sciences and by the Russian
Foundation for Basic Research (project no. 09-05-98520).
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|
arxiv-papers
| 2008-12-25T06:37:47 |
2024-09-04T02:48:59.555805
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.V. Budyansky, M.Yu. Uleysky, and S.V. Prants",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/0812.4586"
}
|
0812.4600
|
# $R$-symmetric Gauge Mediation With Fayet-Iliopoulos Term
Mingxing Luo and Sibo Zheng
Zhejiang Institute of Modern Physics, Department of Physics,
Zhejiang University, Hangzhou 310027, P. R. China.
E-mail luo@zimp.zju.edu.cn sibozheng.zju@gmail.com
###### Abstract:
We have studied $R$-symmetrc gauge mediation models with Fayet-Iliopoulos
terms. We give a concrete example of hidden sector with an $U(1)_{H}$ gauge
theory and a Fayet-Iliopoulos term, which can induce distinctive soft terms in
the visible sector, and help solving fine tuning problems in models of
$R$-symmetric gauge mediation.
Fayet-Iliopoulos Term, Supersymmetric Standard Model, Seiberg Duality
## 1 Introduction
Of all the candidates to stabilize the hierarchy between the weak and Plank
scales, supersymmetry seems to be the most plausible and predictive at LHC.
Supersymmetry is broken in some hidden sector and mediated to the visible
sector via gravity or gauge interactions [2, 3]. Such mechanisms induce the
necessary soft terms in supersymmetric Standard Model (SSM) for which to be
phenomenoloically viable. In models with direct gauge mediation, a number of
hidden models have been successfully constructed [21]. There exist now a
general framework to calculate the soft masses [17].
Recently, a new class of gauge mediation models has been proposed [4, 5], in
which $R$ symmetry is retained in this class of models. In usual gauge
mediation models, $R$ symmetry has to be spontaneously broken in order to
generate Marojana gaugino massses. By adding suitable $C$-parity chiral fields
in adjoint representations, the gauginos can acquire Dirac masses.
Phenomenologies are rather distinctive in these kind of models [5].
Unfortunately, some of the usual flavor problems persist in these models and
fine tunings are needed.
In this paper, we will first have a close look at hidden sectors with
$R$-symmetry and SUSY-breaking. It has been long understood that these two
issues are closely connected to each other [16]. The $R$-symmetry can be
broken at the tree level. If it is not broken at the tree level, it can still
be spontaneously broken due to quantum corrections in general O
${}^{{}^{\prime}}$Raifeartaigh (OR) model, provided that the $R$-charges of
superfields in the superpotential take values different from 0 and 2 [20].
This implies that the general form of superpotential is determined as
$W=X^{n}f_{n}(\Phi)$, where $X,\Phi$ are $R$-charges 2 and 0 respectively,
$f_{n}(\Phi)$ are polynomial of $\Phi$ constrained only by the renormalization
of the theory. In section II, we will deform these models by including a
Fayet-Iliopoulos (FI) term. And one sees that the $R$-symmetry can still be
preserved.
In section III, we will construct a hidden sector with a FI term to realize
the scenario of $R$-symmetric gauge mediation. Embedded into SUSY theories via
an $U(1)$ group, the phenomenological implications of the FI term have been
extensively discussed in gauge and anomaly mediation111Applications in other
fields, such as extra dimensions, cosmology and string theory, are beyond
present discussions.. In gauge mediation scheme, an $U(1)$ FI term can be used
to spontaneously break supersymmetry, while $R$-symmetry is usually unbroken
as proposed in [16]. In $N=2$ models, this can naturally generate Dirac
gaugino masses[10]. In [11], an FI term is employed for $N=1$ models with
Majorana gaugino masses, in which $R$-symmetry is spontaneously broken by
gaugino condensation in a strong-coupled Yang-Mills hidden sector. FI terms
can also be applied in the construction of SUSY grand unification theories
(GUT). If the role of messenger sector are replaced by an $U(1)$ with FI term,
there will be less modifications to RG running of SM gauge couplings. A GUT
can be relatively easier realized [12]. Moreover, doublet-triplet splitting
problem in high rank $SU(n)$ SUSY theories that include $SU(5)$ GUT can be
solved by introducing a new mass scale carried by FI term [13]. In anomaly
mediation scheme, introducing a FI term can help to solve the tachyonic
slepton problem, or even accommodate neutrino masses when suitable $U(1)$
charges are chosen [14, 15].
In section IV, we calculate the soft masses, and $\mu/B\mu$ terms. The model
preserves most distinctive features of $R$-symmetric gauge mediation. However,
we can allievate the fine tunings and the $\mu/B\mu$ problem. To summarize,
the model has the following features:
* •
$R$-symmetry is conserved in SUSY-breaking and visible sectors.
* •
There are no $A$ terms, and adjoint fields $\Phi_{r}$ (r=1,2,3) that combined
with gauginos have to be added to construct Dirac gauginos.
* •
There is an unbroken $U(1)_{H}$ gauge theory in the hidden sector. The SUSY-
breaking effects contain the contributions of $F$\- and $D$-terms at meantime.
* •
The negative sfermions masses squared coming from the $D$-term may solve the
fine tuning problems in $R$-symmetric gauge mediation with only $F$-term
induced visible effects [4]. The Dirac gaugino masses can be heavier than the
sfermions masses.
* •
The little hierarchy between $\mu$ and $B\mu$ terms can also be obtained by
adjusting the $D$-term.
We expect that some of phenomenologies in the visible sector are model
dependent. It would be interesting to develop a general framework to
distinguish the general characters in $R$-symmetric gauge mediation, as done
in [17]. On the other hand, the $\mu/B\mu$ problem should also be discussed in
depth to find a more economical mechanism. Finally, we conclude in section V
with discussions.
## 2 SUSY-breaking Sectors with $R$-symmetry
According to [20], the $R$-symmetry is maintained only if that the R charges
of superfields in the superpotential are either 0 or 2. Otherwise, radiative
corrections will spontaneously break the $R$-symmetry even if it is conserved
at tree level. This implies that the general form of superpotential with
$R$-symmetry reads,
$\displaystyle W=X^{n}f_{n}(\Phi_{m}),$ (2.1)
where $R(X^{n})=2$ and $R(\Phi_{m})=0$, $n,m$ denote different superfields of
$X$ and $\Phi$, respectively, repeated $n$ implies summation. $f_{n}(\Phi)$
are $n$ polynomial of $\Phi$ constrained only by the renormalization of the
theory. The supersymmetry preserving vacua is given by vanishing F terms:
$\displaystyle
F_{n}=f_{n}(\Phi_{m})=0,~{}~{}~{}~{}~{}F_{m}=X^{n}\partial_{m}f_{n}(\Phi_{m})=0$
(2.2)
$F_{m}=0$ can easily be satisfied by simply taking all $X^{n}=0$. In cases
with $n>m$, generally $F_{n}=0$ can not be simultaneously satisfied, thus
leading to supersymmetry spontaneously broken. The $R$-symmetry is preserved
at the origin of the moduli space. In cases with $n\leq m$, there are two
possibilities. In a generally renormalizable theory, $f_{n}(\Phi_{m})$ always
assumes the following form,
$\displaystyle
f_{n}(\Phi_{m})=k_{n}+M_{n}^{m}\Phi_{m}+\lambda_{n}^{mm^{{}^{\prime}}}\Phi_{m}\Phi_{m^{{}^{\prime}}}$
(2.3)
If all $k_{n}=0$, we expect the supersymmetry and $R$ symmetry are both
unbroken at the origin of the moduli space. On the other hand, if some
$k_{n}\neq 0$ as in the ordinary OR model, it is then possible to obtain SUSY-
breaking models with $R$ symmetry. For example, the ISS model constructed in
[19] belongs to this type.
If there is an $U(1)_{H}$ gauge interaction in the hidden sector, one can also
have a FI term in principle. This yields the mechanism of $D$-term scenario,
in addition to the $F$-term scenario, to induce SUSY-breaking. Generically, a
hidden sector with a $U(1)_{H}$ gauge can have the following potential,
$\displaystyle
V=F_{m}^{2}+F_{n}^{2}+\frac{D^{2}}{2g^{2}},~{}~{}~{}~{}D=g^{2}\left(\xi+g^{2}q_{i}(\mid\Phi_{i}\mid^{2}-\mid\Phi_{i}^{\prime}\mid^{2})\right)$
(2.4)
where $g$ is the $U(1)_{H}$ gauge coupling, $q_{i}$ are the $U(1)_{H}$ charges
of $\Phi_{i}$. Without $k_{n}$ terms, one obviously cannot have $F=0$ and
$D=0$ in the same time. The supersymmetry is thus broken spontaneously. In the
specific case that the absolute minimum of the potential is at $\Phi=0$, one
would have $V=g^{2}\xi^{2}/2$ and the SUSY-breaking comes only from the
$D$-term.
In the case that there are non-zero $k_{n}$’s, the SUSY-breaking will come
both from $F$\- and $D$-term, if we can arrange the parameters in the model
such that the absolute minimum of the potential is still at $\Phi=0$. In this
case, the $R$ symmetry is unbroken with a minima
$V=k^{2}_{1}+\frac{1}{2}g^{2}\xi^{2}$. As we will see, the $R$-symmetric gauge
mediation with such a hidden sector has distinctive phenomenologies compared
with only $F$ term induced SUSY-breaking [4].
For simplicity, we have assumed that the Kahler potential is canonical, i.e,
$K=X_{n}^{\dagger}X_{n}+\Phi_{m}^{\dagger}\Phi_{m}$, In principle, it can be
modified by the radiative corrections which can in turn induce a non-trivial
metric of moduli space.
## 3 A concrete example of hidden sector
Consider a hidden sector with the gauge groups of $SU(5)\times U(1)$, two
types of chiral superfields $Q$ in the fundamental representation, and two
types of chiral superfields $\bar{Q}$ in the anti-fundamental representation,
as shown in the table. The global flavor symmetry is $SU(6)$. The $U(1)_{H}$
charges of $Q$’s and $\bar{Q}$’s are either $+1$ or $-1$, so the model is
anomaly free.
| $SU(5)$ | $U(1)_{H}$
---|---|---
$Q_{i}$ | $\square$ | $+1$
$\bar{Q}_{i}$ | $\bar{\square}$ | $-1$
$Q_{6}$ | $\square$ | $0$
$\bar{Q}_{6}$ | $\bar{\square}$ | $0$
Table 1: $Q_{i}$ denote the first five chiral superfields. Note that the gauge
groups in the hidden sector are the simplest extension of that in [4], which
are well motivated and can be easily constructed in intersecting branes
models.
In the infrared, the strong coupling $SU(5)$ theory can be described by the
dual magnetic theory of ISS-type [19], with a superpotneial
$W_{mag}=\lambda\bar{q}\mathcal{M}q-f^{2}Tr\mathcal{M}$ (3.1)
where $q=(\varphi,\psi)$ are the dual quarks and $\mathcal{M}$ the mesons,
$\mathcal{M}=\left(\begin{array}[]{cc}\omega M&kN\\\ k\bar{N}&k^{\prime}Y\\\
\end{array}\right)$ (3.2)
In detail, the superpotential of our theory can be written as222We can also
begin with a hidden sector with superpotential $W_{mag}$ and $D$-term, and
consider the $SU(5)\times U(1)_{H}$ theory as one ultraviolet completion. For
instance, abelian SUSY theory with a set of chiral superfields including
singlets is another candidate.,
$\displaystyle{}W_{mag}$ $\displaystyle=$ $\displaystyle X^{n}f_{n}(\Phi_{m})$
$\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle-f^{2}\omega+\lambda\varphi\bar{\varphi},$ $\displaystyle f_{2}$
$\displaystyle=$ $\displaystyle\lambda k\bar{\varphi}\psi,$ $\displaystyle
f_{3}$ $\displaystyle=$ $\displaystyle\lambda k\varphi\bar{\psi},$
$\displaystyle f_{4}$ $\displaystyle=$ $\displaystyle-f^{2}+\lambda
k^{\prime}\bar{\psi}\psi,$ (3.3)
where $X_{n}$ denote the mesons singlets $(M,N,\bar{N},Y)$ of $R$-charge 2,
the rest of $R$-charge 0. When the $U(1)_{H}$ coupling constant $g\rightarrow
0$, the theory returns to the usual Seiberg duality [18] and corresponding ISS
model.
Here is the rational for such an assumption. In the standard model the strong
$SU(3)_{c}$ quark theory can be well described by the dual effective theory of
composite mesons and baryons at low energy, which are not invalidated by extra
electroweak interactions. To reach the macroscopic superpotential (3), we have
assumed that the $U(1)_{H}$ gauge theory does not spoil the validity of
Seiberg duality with small enough coupling constant $g$. There are
spontaneously broken $N=2$ dual theories with FI terms [9], and $N=1$
dualities which are not spoiled by deformations of IR irrelevant couplings
[7]. There are also Seiberg dualities with non-simple Lie groups. For example,
there are $N=1$ theories with two gauge groups [8]. Similar assumption has
been applied to study other topics in earlier works [6].
We now read out the $U(1)_{H}$ charges of the dual mesons and quarks. The
singlet mesons $M,~{}Y$ are also $U(1)_{H}$ singlets . The $N,~{}\bar{N}$
mesons are not $U(1)_{H}$ singlets, but of $U(1)_{H}$ charges $+1,~{}-1$,
respectively. The dual quarks $q$ carry the same $U(1)_{H}$ charges as those
of $Q$. In summary, all dual fields are $U(1)_{H}$ singlets except
$\varphi,\bar{\varphi},N,\bar{N}$.
It can be shown that the supersymmetry is broken in our model. The absolute
minimum of $V$ is located at the origin of muduli space with
$\displaystyle{}F_{Tr~{}M}=\omega
f^{2},~{}~{}~{}~{}~{}\bar{\psi}\psi=\upsilon^{2}=f^{2}/(\lambda k^{\prime})$
(3.4)
Since $\psi,~{}\bar{\psi}$ have no $U(1)_{H}$ charges, their nonzero vacuum
expectation values do not break the $U(1)_{H}$ gauge symmetry
spontaneously333Recently, it is proposed that unbroken weak coupling
$U(1)_{H}$ theory can work as a model of dark matter [23].. However, the
global $SU(6)$ symmetry is spontaneously broken to $SU(5)$. The corresponding
Nambu-Goldstone (NG) bosons acquire significant masses due to interactions
with the messenger sector [4].
To mediate the SUSY-breaking to the visible sector, we gauge the remaining
global $SU(5)$ flavor symmetry. The $\phi$ and $N$ fields will serve as the
messengers. The scalar mass matrix for messengers is given by,
$\displaystyle\left(\begin{array}[]{cccc}\varphi^{*}&\bar{\varphi}&N^{*}&\bar{N}\\\
\end{array}\right)\left(\begin{array}[]{cccc}M^{2}+D&-zM^{2}&0&0\\\
-zM^{2}&M^{2}-D&0&0\\\ 0&0&M^{2}+D&0\\\ 0&0&0&M^{2}-D\\\
\end{array}\right)\left(\begin{array}[]{c}\varphi\\\ \bar{\varphi}^{*}\\\ N\\\
\bar{N}^{*}\\\ \end{array}\right)$ (3.14)
where $M=\sqrt{\lambda\omega/z}f$ and $z=\omega k^{\prime}/k^{2}$. From Eq.
(3.14) we see that the eigenstates of the upper $2\times 2$ block of scalar
mass matrix are $\phi_{\pm}=(\bar{\varphi}^{*}\pm\varphi)/\sqrt{2}$ of
eigenvalues,
$\displaystyle{}\phi:~{}~{}m_{\pm}^{2}=M^{2}(1\pm\tilde{z}),~{}~{}~{}~{}~{}~{}\tilde{z}=\sqrt{z^{2}+x^{2}}$
(3.15)
where $x=D/M^{2}$. In order to avoid tachyons in the spectrum, we need to
impose $x^{2}<(1-z^{2})$. The eigenvalues of $N,\bar{N}$ are given by,
$\displaystyle{}N:~{}~{}m_{\pm}^{2}=M^{2}(1\pm x)$ (3.16)
The fermion mass matrix for messengers $\varphi,N$ is off-diagonal,
$\displaystyle\left(\begin{array}[]{cc}\varphi&N\\\
\end{array}\right)\left(\begin{array}[]{cc}0&Me^{i\theta}\\\
Me^{-i\theta}&0\\\ \end{array}\right)\left(\begin{array}[]{c}\bar{\varphi}\\\
\bar{N}\\\ \end{array}\right)$ (3.22)
They are all degenerate at $M$. The spectra for NG particles and the remaining
messengers $X,\psi,\bar{\psi}$ are the same as those given in [4]. Compared
with the messenger spectra in [4], the masses of scalar messengers $\phi$ and
$N$ are modified by the $D$-term. The sfermions masses squared and Dirac
gaugino masses in the visible sector will be induced by the mass splitting of
$\phi$ and $N$ in the loop(s).
## 4 Soft terms and fine tunings
We now analyze the soft terms in the visible sector. Before going to the
details, we outline the main characteristics of $R$-symmetric gauge mediation
with $D$-term,
* •
The sfermion masses are decreased by negative $D$-term induced contribution.
When $\sqrt{D}\sim M\sim 10^{3}$TeV, the sfemions are still of order
$\mathcal{O}(1)$TeV. On the other hand, the Dirac gaugino masses increase by
positive $D$-term induced contribution. Without $D$-term effects the sfermions
masses are usually heavier than gauginos masses. However, this can easily be
reversed with the $D$-term. This reversion of the gaugino and scalar mass
ratio helps to evade constraints on flavor structures.
* •
As shown in [4], fine tunings are needed to have viable diagonal and off-
diagonal coefficients $c_{D}$ and $c_{OD}$ for the sfermion mass matrix. An
adjustable $D$-term helps to obtain reasonable relations $c_{D}\sim c_{OD}\sim
1$.
* •
$B\mu$ and $\mu$ terms receive the nonzero and zero contributions from
$D$-term respectively, which can be used to adjust the small hierarchy between
$\mu$ and $B\mu$.
* •
Gravitino is the lightest superparticle (LSP) of a mass in the order of $eV$,
which is determined by $m_{3/2}\sim(D+f^{2})/M_{Pl}$ in supergravity.
The sfermion masses squared receive contributions from the ultraviolet (UV)
and infrared (IR) physics. The IR contribution is due to the mass splitting of
the messengers induced by the SUSY breaking in the hidden sector, starting
from two-loops [22]. The UV contribution can be written generically,
$\displaystyle\int
d^{4}\theta\frac{c_{ij}}{\Lambda^{2}}(\Xi\Xi^{\dagger})Q^{\dagger}_{i}Q_{j}$
(4.1)
where $\Xi=<TrM>=\theta^{2}\omega f^{2}$. In total, we have
$\displaystyle{}(\tilde{m}^{2})_{ij}$ $\displaystyle=$ $\displaystyle
c_{ij}(\tilde{m}_{UV}^{2})_{ij}-(\tilde{m}_{IR}^{2})_{ij},$
$\displaystyle(\tilde{m}_{IR}^{2})_{ij}$ $\displaystyle=$
$\displaystyle\delta_{ij}\frac{g_{s}^{4}M^{2}}{(16\pi^{2})^{2}}J(x,\tilde{z}),$
$\displaystyle\tilde{m}_{UV}$ $\displaystyle=$
$\displaystyle\left(\frac{z}{\lambda}\right)\left(\frac{M}{\Lambda}\right)M$
(4.2)
where
$\displaystyle{}J(x,\tilde{z})=\frac{7}{9}(x^{4}+\tilde{z}^{4})+\frac{38}{75}(x^{5}+\tilde{z}^{5})+\mathcal{O}\left(x^{6},\tilde{z}^{6}\right)$
(4.3)
$c_{ij}$ are the coefficients appearing in the UV operator. The UV operators
responsible for Dirac gaugino masses are,
$\displaystyle\int
d^{2}\theta\frac{\bar{D}^{2}D_{\alpha}(\Xi\Xi^{\dagger})}{\Lambda^{3}}Tr\left(W^{\alpha}\Phi\right)$
(4.4)
where $W^{\prime},W$ refer to the $U(1)_{H}$ and SSM spinor superfields
respectively. The IR contribution to gaugino masses is again due to the mass
splitting of the messengers, starting from one-loop. Since the masses of the
scalars $N,\bar{N}$ in the loop are shifted, gaugino masses are changed, in
comparison with those in [4]. Explicitly, we have
$\displaystyle{}m_{1/2}$ $\displaystyle=$ $\displaystyle m_{IR}+m_{UV},$
$\displaystyle m_{IR}$ $\displaystyle=$
$\displaystyle\frac{g_{s}y}{16\pi^{2}}Mcos\left(\frac{\theta}{\upsilon}\right)Q(x,\tilde{z})$
(4.5)
where
$\displaystyle{}m_{UV}\simeq\left(\frac{\tilde{m}_{UV}}{\Lambda}\right)\tilde{m}_{UV}$
(4.6)
and $\theta$ is the NG bosons, The function $Q(x,\tilde{z})$ is defined by ,
$\displaystyle Q(x,\tilde{z})$ $\displaystyle=$
$\displaystyle\frac{1}{\tilde{z}}\left((1+\tilde{z})\log(1+\tilde{z})-(1-\tilde{z})\log(1-\tilde{z})-2\tilde{z}\right)+(\tilde{z}\rightarrow
x)$ (4.7)
The coefficients $c_{ij}$ can also be written as [4],
$\displaystyle
c_{D}=\frac{\tilde{m}^{2}+\tilde{m}^{2}_{IR}}{\tilde{m}^{2}_{UV}},~{}~{}~{}~{}~{}~{}~{}c_{OD}=\delta\left(\frac{\tilde{m}^{2}}{\tilde{m}^{2}_{UV}}\right)$
(4.8)
which refer to the non-perturbative behavior of the hidden sector, which
arises when the RG scale near the Landau pole $\Lambda$ in the direct gauge
mediation444 The messengers introduced to mediate the SUSY-breaking effects
substantially modify the slopes of gauge coupling $\alpha_{s}$ running, lead
to the divergence of $\alpha_{s}$ at $\Lambda$.. In principle, $c_{ij}$ are of
the order $\mathcal{O}(1)$ and cannot be calculated in perturbative method.
However, they are constrained by ratio of gaugino mass over sfermion mass to
avoid flavor problems. In the ISS model, $c_{D}$ should be smaller than
$10^{-2}$ to be phenomenlogically viable. This implies that the non-
perturbative physics of the hidden sector is seriously constrained and this is
the origin of fine tuning.
In models with $D$-term breaking, $c_{D}$ can be in the neighborhood of unity.
Take $M\sim 10^{3}$TeV and $\Lambda\sim 10^{4}$TeV for illustrations, in which
$\alpha_{3}(M)\sim 0.15$. We start with the following parameter space,
$\displaystyle{}z=0.1~{}~{}~{}and~{}~{}~{}~{}\lambda=1,$ (4.9)
which can yield the typical mass relations,
$\displaystyle{}\tilde{m}\sim\tilde{m}_{UV}\sim\tilde{m}_{IR}\sim
10^{-3}M,~{}~{}~{}~{}m_{1/2}\sim m_{IR}\sim 10^{-2}M$ (4.10)
Explicit calculations show that $c_{D}\leq 10$ and $c_{OD}\leq 1$ are
phenomenlogically viable. Shown in Figure. 1 are the masses of sfermions and
gauginos for two typical regions of $c_{ij}$. Clearly, for the given parameter
space (4.9), there are no fine tunings and no extra Landau pole coming from
large Yukawa coupling $\lambda$.
When going up along the positive direction of $z$, one has to increase $x\sim
1$ and (or) $\lambda$ in order to restore the typical mass relations (4.10).
However, there is the upper bound on $x\sim(1-z)$, as one wants to avoid
tachyons. Roughly $z$ cannot be greater than $0.5$.
Figure 1: The massses of sfermion (dashed line) and gaugino (solid line) (TeV
scale) as functions of $x$ parameter. Left figure is for the region of small
$c_{D}=0.1$, where a large mass ratio can be obtained near $x\sim 0.16$ ($y=4$
and $M=3\times 10^{3}$ TeV). Right figure is for the region of large
$c_{D}=8$, where a large mass ratio can be obtained near $x\sim 0.69$ ($y=3$
and $M=10^{3}$ TeV).
Finally, we address the $\mu/B\mu$ term in $R$-symmetric gauge mediation.
Generally, there are three mechanisms to generate $\mu/B\mu$ term of weak
scale in gauge mediation without $R$-symmetry. One is by introducing a gauge
singlet $X$, usually dubbed as NMSSM [24]. Another is by introducing massive
vector-like pairs. Thirdly, one takes the conformal sequestering into account
[25]. In $R$-symmetric gauge mediated theories, the unbroken $R$-symmetry
severely restricts the $\mu$ term, while $B\mu$ of weak scale can be generated
from either the $F$\- or $D$-terms. An economical scheme to the $\mu$ term is
to introduce two extra $SU(2)$ doublets $R_{u,d}$ of $R$-charge 2 [4]. The UV
operators corresponding to $\mu$ and $B\mu$ are
$\displaystyle{}\int d^{4}\theta
c_{F}\frac{\Xi^{\dagger}}{\Lambda}\left(H_{u}R_{d}+H_{d}R_{u}\right)$ (4.11)
and,
$\displaystyle{}\int d^{4}\theta
c^{\prime}_{F}\frac{\Xi\Xi^{\dagger}}{\Lambda^{2}}H_{u}H_{d}+\int d^{2}\theta
c^{\prime}_{D}\frac{W^{\prime}_{\alpha}W^{\prime\alpha}}{M^{2}}H_{u}H_{d}$
(4.12)
respectively. Here the second term comes from the unbroken $U(1)_{H}$ gauge
theory. They are both at the weak scale if $\sqrt{D}\sim M\sim 10^{3}$TeV
$\displaystyle\mu\sim c_{F}m_{UV},~{}~{}~{}~{}B\mu\sim
c^{\prime}_{F}m_{UV}^{2}+c^{\prime}_{D}\frac{D^{2}}{M^{2}}$ (4.13)
Note that the simplest way to generate $\mu$ term is to include a $R$-charge
zero spurion field of nonzero $F$-term in the messenger sector. According to
discussions in section 2, this can be realized. For instance, the Wess-Zumino
model with only chiral superfields is a feasible candidate. It would be
interesting to develop a scheme to dicuss the general $R$-symmetric gauge
mediation as done in ordinary gauge mediation [17], to see which of the
features outlined above are general and which are model dependent.
## 5 Conclusions
In this paper, we have discussed the possibilities to obtain SUSY-breaking
hidden sectors with $R$-symmetry, with an extra $U(1)_{H}$ sector and a FI
term. A concrete example of hidden sector is constructed. In this particular
model, we find that in the visible sector, the ratio of Dirac gaugino mass
over sfermion mass substantially increases compared with those with only
$F$-term [4]. The $\mu$ and $B\mu$ terms receive zero and nonzero
contribution, respectively. These help evading the fine tunings in
$R$-symmetric gauge mediation with interesting flavor phenomenologies.
As we point out in section 2, there are other candidates as hidden sectors in
$R$-symmetric gauge mediation. It is worth developing a general framework to
see which characters are model independent, especially the generations of
$\mu/B\mu$ terms, which closely connect with some important topics such as
electro-weak symmetry breaking and the dark matter model of supersymmetric
neutralino.
## Acknowledgement
This work is supported in part by the National Science Foundation of China
(10425525).
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|
arxiv-papers
| 2008-12-25T11:37:41 |
2024-09-04T02:48:59.565821
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mingxing Luo and Sibo Zheng",
"submitter": "Sibo Zheng",
"url": "https://arxiv.org/abs/0812.4600"
}
|
0812.4639
|
# Sensitivity to neutrino mixing parameters with atmospheric neutrinos
Abhijit Samanta 111E-mail address: abhijit@hri.res.in Harish-Chandra Research
Institute, Chhatnag Road, Jhusi, Allahabad 211 019, India
###### Abstract
We have analyzed the atmospheric neutrino data to study the octant of
$\theta_{23}$ and the precision of the oscillation parameters for a large Iron
CALorimeter (ICAL) detector. The ICAL being a tracking detector has the
ability to measure the energy and the direction of the muon with high
resolution. From bending of the track in magnetic field it can also
distinguish its charge. We have generated events by Nuance and then considered
only the muons (directly measurable quantities) produced in charge current
interactions in our analysis. This encounters the main problem of wide
resolutions of energy and baseline. The energy-angle correlated two
dimensional resolution functions are used to migrate the energy and the zenith
angle of the neutrino to those of the muon. A new type of binning has been
introduced to get better reflection of the oscillation pattern in chi-square
analysis. Then the marginalization of the $\chi^{2}$ over all parameters has
been carried out for neutrinos and anti-neutrinos separately. We find that the
measurement of $\theta_{13}$ is possible at a significant precision with
atmospheric neutrinos. The precisions of $\Delta m_{32}^{2}$ and
$\sin^{2}\theta_{23}$ are found $\sim$ 8% and 38%, respectively, at 90% CL.
The discrimination of the octant as well as the deviation from maximal mixing
of atmospheric neutrinos are also possible for some combinations of
($\theta_{23},~{}\theta_{13}$). We also discuss the impact of the events at
near horizon on the precision studies.
###### pacs:
14.60.Pq
## I Introduction
Recent discovery of the neutrino mass has opened up a new window into physics
beyond the standard model. Aside from this fact, the two surprising sets of
results Fogli:2008ig , i) the extremely small masses of neutrinos (very
different from quark sector) and ii) a dramatically different mixing pattern
from quarks, indicate a new direction of this field. The first one may be the
hint of a new symmetry such as $B-L$ at high scales so that one can use a
mechanism like seesaw to resolve the puzzle of the smallness of the masses. On
the other hand, the second one poses a much more challenging problem. One can
expect a new symmetry for leptons as well as for quarks to solve this problem.
Currently, there are many theoretical ideas. For example, the $\mu-\tau$
symmetry Mohapatra:2005yu is invoked to explain the maximal mixing. However,
if this $\mu-\tau$ symmetry exists, it leads to $\delta_{CP}=0$ and
$\theta_{13}=0$. It should be broken if there appears a nonzero $\theta_{13}$
and CP violation. In that case the octant (the sign of
$\theta_{23}-45^{\circ}$) and the nonzero value of $\theta_{13}$ emerges other
new possibilities. It is also expected that neutrino theories may have
implications on the very fascinating fields like observed matter-antimatter
asymmetry of the universe, grand unification, supersymmetry, extra dimensions,
etcMohapatra:2005wg .
Active endeavors are under way to launch the era of precision experiments with
a thrust to uncover the underlying principle that gives neutrino masses and
their mixing. This is one of the most promising ways to explore physics beyond
the standard model. In the standard oscillation picture there are six
parameters. The present 1$\sigma$, 2$\sigma$ and 3$\sigma$ confidence level
(CL) ranges from global $3\nu$ oscillation analysis (2008) Fogli:2008ig are
very exciting (see table 1). Recently, new bounds,
$\theta_{13}=-0.07^{+0.18}_{-0.11}$ and the asymmetry
$\theta_{23}-\pi/4=0.03^{+0.09}_{-0.15}$ at 90% CL have been shown in
Escamilla:2008vq ; Roa:2009wp from an analysis considering all present
neutrino data. The CP-violating phase $\delta_{CP}$ is still unconstrained.
Parameter | $\Delta m_{21}^{2}/10^{-5}\mathrm{\ eV}^{2}$ | $\sin^{2}\theta_{12}$ | $\sin^{2}\theta_{13}$ | $\sin^{2}\theta_{23}$ | $|\Delta m_{31}^{2}|/10^{-3}\mathrm{\ eV}^{2}$
---|---|---|---|---|---
Best fit | 7.67 | 0.312 | 0.016 | 0.466 | 2.39
$1\sigma$ range | 7.48 – 7.83 | 0.294 – 0.331 | 0.006 – 0.026 | 0.408 – 0.539 | 2.31 – 2.50
$2\sigma$ range | 7.31 – 8.01 | 0.278 – 0.352 | $<0.036$ | 0.366 – 0.602 | 2.19 – 2.66
$3\sigma$ range | 7.14 – 8.19 | 0.263 – 0.375 | $<0.046$ | 0.331 – 0.644 | 2.06 – 2.81
Table 1: Global 3$\nu$ oscillation analysis (2008)
Despite of these spectacular achievements, a lot of things are still missing.
Tremendous efforts are underway to determine the mass ordering (sign of
$\Delta m_{32}^{2}$), the values of $\theta_{13}$ and $\delta_{CP}$, and to
discriminate the octant degeneracy of $\theta_{23}$ in future experiments. We
define $\Delta m^{2}_{32}=m_{3}^{2}-m_{2}^{2}$. There are many ongoing and
planned experiments: UNO Jung:1999jq , T2K Itow:2001ee , NOvA Ayres:2004js ,
Hyper-Kamiokande Nakamura:2003hk , INO Arumugam:2005nt and many others. The
main characteristic feature of a magnetized Iron CALorimeter (ICAL) detector
proposed at India-based Neutrino Observatory (INO) is that it has the
capability to detect $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ separately, which
measures directly the matter effect.
Unlike a fixed baseline neutrino beam experiment, the atmospheric neutrino
flux covers a wide range of baseline (a few km – 12900 km) and energy (sub GeV
– a few hundred GeV). On the other hand, it is not known well and there are
huge uncertainties in its estimation. It is also a very rapidly falling
function of energy. So, the extraction of the results from the experimental
data is very complicated.
The deviation from maximal mixing and the discrimination of octant degeneracy
of $\theta_{23}$ have been studied in Choubey:2005zy ; Indumathi:2006gr with
atmospheric neutrinos for a large magnetized ICAL detector. However, the
results have been obtained without marginalization and assuming the Gaussian
resolution functions of fixed widths for whole range of energy and zenith
angle. The energy range for the atmospheric neutrinos is very wide. The
resolutions are changed significantly over its range and are very different
for neutrinos and anti-neutrinos. Moreover, the energy resolutions appear to
be non-Gaussian due to some unmeasurable product particles like neutral
hadrons in neutrino interaction even if one considers all visible hadrons.
For a given neutrino energy and direction, there is a distribution in the
reconstructed energy and direction. Again, a particular reconstructed energy
and direction can come from a wide range of true neutrino energy and
direction. So, it is not possible to convert a distribution in reconstructed
energy and direction obtained from an experiment to a distribution in actual
neutrino energy and direction. This restricts the binning of the data for chi-
square analysis only in experimentally measured energy and direction. On the
other hand, the actual resolution functions have no regular pattern and
significantly deviate from the Gaussian nature even if we consider the visible
hadrons. Again, the width changes with neutrino energy. For a simplistic
analysis, if one considers a Gaussian resolution with a width that gives equal
space under the surface of resolution function, the correct theoretical data
smearing this approximated Gaussian resolution function can not be obtained
for chi-square analysis. As a consequence, the best-fits and the contours of
oscillation parameters will differ largely from the true values. In
literature, there are many analyses where both the theoretical as well as the
experimental data are obtained by smearing the Gaussian resolution functions.
For an example, see ref. Indumathi:2006gr . However, the result changes very
rapidly with change of the width of the resolution. So, realistic estimation
of the capability of an experiment can be done only by an analysis with
experimentally measurable quantities and exact resolution functions.
Till now, the precision studies with atmospheric neutrinos have mainly carried
out for water Cherenkov detector, a non-magnetized detector. It is very
important to see the capability of a large magnetized detector. We have
studied the neutrino oscillation considering neutrinos and anti-neutrinos
separately in the chi-square analysis. Here, we consider the muons (directly
measurable quantities at ICAL) produced by the charge current interactions. We
generate events by Nuance-v3 Casper:2002sd . The two dimensional energy-angle
correlated resolution functions are used to migrate the energy and the zenith
angle of the neutrino to the energy and the zenith angle of the muon.
The above method has been introduced in Samanta:2006sj and later used in
Samanta:2008ag . The goal of the previous work Samanta:2008ag was only to
compare the allowed parameter space of oscillation parameters obtained from
different types of binning. The considered systematic uncertainties were very
much different from the present systematic uncertainties. The purpose of this
work is to study the following.
We consider whole data set in previous studies. But in reality, the horizontal
events cannot be detected when the iron slabs are stacked horizontally. In
this paper, we have studied the impact of these events in determining the
precision of the parameters with and without considering a rejection criteria
for the horizontal events. This is very crucial to determine whether
horizontal stacking of iron plates is better than the vertical stacking or
not.
As discussed in Samanta:2008ag , the binning of the data neither in $\log
E-\cos\theta_{\rm zenith}$ nor in $\log E-\log L$ is the optimum. In this
paper, we have optimized the binning in $L$. These are equal binned grids in
$\log E-L^{0.4}$ plane, which can capture the oscillation behavior for all $L$
and $E$ in a better way in the chi-square analysis. Again, the number of bins
in both axes need optimization between resolutions and statistics. However, it
should be noted here that if the statistics is huge for whole range of $E$ and
$L$, one can solve this problem by making the bin size very small and then the
type of binning will not play any crucial role. However, the type of binning
is very crucial when the analyses is in experimentally measurable energy and
directions. Here the statistics over measured energy and direction is
redistributed notably from the true neutrino energy and direction.
Finally, we have made a detailed study on the sensitivity of a magnetized ICAL
detector in determining the precision of $\Delta m_{32}^{2}$ and $\theta_{13}$
as well as in discriminating the octant ambiguity of $\theta_{23}$. We find
the sensitivities of the parameters in two dimensional parameter space after
marginalization over whole allowed ranges of the parameters. The absolute
bounds of each parameter are also studied.
## II Atmospheric neutrino flux and events
The atmospheric neutrinos are produced by the interactions of the cosmic rays
mainly with nucleuses of molecules in the earth’s atmosphere. The knowledge of
primary spectrum of the cosmic rays has been improved from the observations by
BESSMaeno:2000qx and AMSAlcaraz:2000ss . However, large regions of parameter
space have not been explored and they are interpolated or extrapolated from
the measured flux. The difficulties and the uncertainties in the calculation
of the neutrino flux depend on the neutrino energy. The low energy fluxes have
been known quite well. The cosmic ray fluxes ($<$ 10 GeV) are modulated by the
solar activity and the geomagnetic field through a rigidity (momentum/charge)
cutoff. At the higher neutrino energy ($>$ 100 GeV), the solar activity and
the rigidity cutoff are irrelevantHonda:2004yz . There is 10% agreement among
the calculations for neutrino energy below 10 GeV because different hadronic
interaction models are used in the calculations and because the uncertainty in
the cosmic ray flux measurement is 5% for the cosmic ray energy below 100 GeV
Honda:2004yz . In our simulation, we have used a typical Honda flux calculated
in 3-dimensional schemeHonda:2004yz .
The interactions of neutrinos with the detector material are simulated using
the Monte Carlo model Nuance (version-3)Casper:2002sd . Here, the charged
current (CC) and neutral current (NC) interactions are considered for
(quasi-)elastic, resonance, coherent, diffractive, and deep inelastic
scattering processes.
Figure 1: The oscillogram of $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\mu}$
oscillation probability in $E-\cos\theta_{\rm zenith}$ plane for
$\theta_{23}=40^{\circ}$ (left column) and $50^{\circ}$ (right column) with
$\theta_{13}=5^{\circ}$ (lower row) and $7.5^{\circ}$ (upper row). We choose
$\Delta m_{32}^{2}=-2.5\times 10^{-3}$eV2 and $\delta_{CP}=0$.
## III Oscillation of atmospheric neutrinos
The present atmospheric neutrino data are well explained by two flavor
oscillation Ashie:2005ik ; Ashie:2004mr . However, one expects a considerable
$\nu_{\mu}\rightarrow\nu_{e}$ oscillation of atmospheric neutrinos in 3-flavor
framework if $\theta_{13}$ is nonzero. To understand the analytical solution
one may adopt the so called “one mass scale dominance” (OMSD) frame work:
$|\Delta m_{21}^{2}|<<|m_{3}^{2}-m_{1,2}^{2}|$. Then the oscillation
probabilities can be expressed as:
$\displaystyle\mbox{P}_{\mu e}$ $\displaystyle=$
$\displaystyle\mbox{P}_{e\mu}$ $\displaystyle=$
$\displaystyle\sin^{2}\theta_{23}\sin^{2}2\theta_{13}\sin^{2}\left(\frac{1.27\Delta
m_{31}^{2}L}{E}\right);$ $\displaystyle\mbox{P}_{\mu\mu}$ $\displaystyle=$
$\displaystyle 1$ (1)
$\displaystyle-4\cos^{2}\theta_{13}\sin^{2}\theta_{23}(1-\cos^{2}\theta_{13}\sin^{2}\theta_{23})$
$\displaystyle\times\sin^{2}\left(\frac{1.27\Delta m_{31}^{2}L}{E}\right).$
These oscillation probabilities are derived for vacuum. Since the oscillation
involves electron neutrino, it will be modulated by the matter effect
Mikheev:1986gs ; Wolfenstein:1977ue . Then,
$\displaystyle\mbox{P}^{m}_{\mu e}$ $\displaystyle=$
$\displaystyle\mbox{P}^{m}_{e\mu}$ $\displaystyle=$
$\displaystyle\sin^{2}\theta_{23}\sin^{2}2\theta_{13}^{m}\sin^{2}\left(\frac{1.27\Delta(m_{31}^{2})^{m}L}{E}\right).$
(2)
Here, $E$, $L$ and $\Delta m^{2}_{31}$ are in GeV, km and eV2, respectively.
$\displaystyle P^{m}_{\mu\mu}$ $\displaystyle=$
$\displaystyle{1-\cos^{2}\theta^{m}_{13}\;{\sin^{2}2\theta_{23}}}$ (3)
$\displaystyle\times\sin^{2}\left[1.27\;\left(\frac{{(\Delta
m_{31}^{2})}+A+{(\Delta m_{31}^{2})^{m}}}{2}\right)\;\frac{L}{E}\right]$
$\displaystyle~{}-~{}\sin^{2}\theta^{m}_{13}\;\sin^{2}2\theta_{23}$
$\displaystyle\times\sin^{2}\left[1.27\;\left(\frac{{(\Delta
m_{31}^{2})}+A-{(\Delta m_{31}^{2})^{m}}}{2}\right)\;\frac{L}{E}\right]$
$\displaystyle~{}-~{}{{\sin^{4}\theta_{23}}}\;{{\sin^{2}2\theta^{m}_{13}\;\sin^{2}\left[1.27\;{(\Delta
m_{31}^{2})^{m}}\;\frac{L}{E}\right].}}$
The mass squared difference ${{{(\Delta m_{31}^{2})^{m}}}}$ and mixing angle
${{\sin^{2}2\theta_{13}^{m}}}$ in matter are related to their vacuum values by
$\displaystyle{(\Delta m_{31}^{2})^{m}}=\sqrt{({(\Delta m_{31}^{2})}\cos
2\theta_{13}-A)^{2}+({(\Delta m_{31}^{2})}\sin 2\theta_{13})^{2}},$
$\displaystyle sin2\theta^{m}_{13}=\frac{{{(\Delta m_{31}^{2})}\sin
2\theta_{13}}}{\sqrt{({(\Delta m_{31}^{2})}\cos 2\theta_{13}-A)^{2}+({(\Delta
m_{31}^{2})}\sin 2\theta_{13})^{2},}}$ (4)
where, $A=2\sqrt{2}G_{F}N_{e}E$, $G_{F}$ is the Fermi constant, $N_{e}$ is the
electron density of the medium and $E$ is neutrino energy Giunti:1997fx . The
matter potential term $A$ has the same absolute value, but opposite sign for
neutrino and anti-neutrino. The superscript ‘m’ denotes effective parameters
in matter. Due to this matter effect, the Mikheyev-Smirnov-Wolfenstein (MSW)
resonance occurs in $\mbox{P}(\nu_{\mu}\rightarrow\nu_{e})$ or
$\mbox{P}(\nu_{e}\rightarrow\nu_{\mu})$. It happens for Normal Hierarchy (NH)
with neutrinos and for Inverted Hierarchy (IH) with anti-neutrinos. It can be
understood from Eq. 3 and 4 that a resonance in above oscillation
probabilities will occur for neutrinos (anti-neutrinos) with NH (IH) when
$\sin^{2}2\theta_{13}^{m}\rightarrow 1~{}~{}~{}~{}{\rm or,}~{}~{}A=\Delta
m^{2}_{31}\cos 2\theta_{13}.$ (5)
Then the resonance energy can be expressed as
$\displaystyle E=\left[\frac{1}{2\times 0.76\times
10^{-4}Y_{e}}\right]\left[\frac{|\Delta m_{31}^{2}|}{\rm eV^{2}}\cos
2\theta_{13}\right]\left[\frac{\rm gm/cc}{\rho}\right].$ (6)
The resonance energy corresponding to a baseline can be seen in Samanta:2006sj
.
The oscillogram of muon survival probability is demonstrated in Fig. 1 for
$\theta_{13}=5^{\circ}$ and $7.5^{\circ}$ with $\theta_{23}=40^{\circ}$ and
$50^{\circ}$, respectively. Here, we show the resonance ranges for the
neutrinos passing through the core of the earth (with $E\approx 3-6$ GeV) and
the mantle of the earth (with $E\approx 5-10$ GeV). We also see a difference
for $\theta_{23}=40^{\circ}$ and $50^{\circ}$ due to the $\sin^{4}\theta_{23}$
term (Eq. 3), which dominates over the other terms due to the matter effect.
Figure 2: The $\nu_{\mu}\rightarrow\nu_{\mu}$ oscillation probability in
vacuum. We choose $\Delta m_{32}^{2}=-2.5\times 10^{-3}$eV2,
$\theta_{23}=45^{\circ}$ and $\theta_{13}=0^{\circ}$.
## IV Binning of the events
For binning of the data in $E$, we need to consider the following facts. I)
The atmospheric neutrino flux falls very rapidly with increase in energy. II)
Again, the wide resolutions of $E$ and $L$ between true neutrinos and
reconstructed neutrinos smear the oscillation effect to a significant extent.
The wide resolutions arise mainly due to the interaction kinematics. This huge
uncertainty in reconstructed neutrino momentum is due to the un-observable
product particles and slightly due to the un-measurable momentum of recoiled
nucleus when $E\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$<$}\hss}{\lower
2.15277pt\hbox{$\sim$}}}1$ GeV. For this reason, the energy resolutions
deviate largely from the Gaussian nature. These are strongly neutrino energy
dependent. At low energy ($E\mathrel{\hbox to0.0pt{\raise
2.15277pt\hbox{$<$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}1.5$ GeV) the quasi-
elastic process dominates and the muon carries almost whole energy of the
neutrino. The energy resolution is very good here. With increase in energy,
the width of the resolution increases significantly as the deep inelastic
event dominates as well as the flux also falls very rapidly. This is one of
the main problems in the atmospheric neutrino experiments. III) There is also
an important characteristics of the oscillation probability when both $L$ and
$E$ are varied simultaneously. We explain it here for
$\nu_{\mu}\leftrightarrow\nu_{\mu}$ oscillation in vacuum, which is a
sinusoidal function of $L/E$. If we plot it in $L-E$ plane (see Fig. 2), it is
seen that the distance between two consecutive peaks of oscillation in $E$ for
a fixed $L$ increases very rapidly with $E$. These three points suggest
increase in bin size with increase in $E$. We choose equal bin size in $\log
E$.
Again, the distance between two consecutive peaks of oscillation in $L$ for a
fixed $E$ increases rapidly as we go to lower values of $L$. When this
distance is very small compared to the resolution width of $L$, the
oscillation effect is averaged out. Only when the distance is large, it
contributes to oscillation measurements. To get the reflection of this
oscillation pattern in $\chi^{2}$, we need decreasing bin size of $L$ with
decreasing of its value. This has been studied in detail for three common
choices of binning of the data in Samanta:2008ag , and it has been found that
neither $\log E-\cos\theta_{\rm zenith}$ nor in $\log E-\log L$ is optimum. In
this work we optimize the binning the data in the grids of $\log E-L^{0.4}$
plane.
The number of bins used for this analysis is discussed later in the section
VI. Here, it should be noted that one cannot make the bin size arbitrarily
small. The number of event in a bin may be a fraction of 1 in theoretical data
for chi-square analysis, but the number of event in experimental data is
either zero or integer number greater or equal to 1. Obviously, no chi-square
method will work if many of the bins have number of event equal to zero or
just equal to 1. However, the number of events per bin $\geq 1$ is not also
sufficient. We have checked that one needs number of event per bin at least
$>4$ to obtain $\chi^{2}$/d.o.f$\approx 1.$ This indicates the optimization of
bin size with statistics.
Figure 3: The variation of the half width at half maxima with $E_{\mu}$ for
the distribution of ${\theta_{\nu}}^{\rm zenith}-{\theta_{\mu}}^{\rm zenith}$
at horizon. The distribution is obtained for each $E_{\mu}$ bin from 500 years
un-oscillated atmospheric data of 1 Mton ICAL.
## V Selection of events
The up going and down going events are mixed at the near horizon due the
uncertainty in scattering angle between neutrino and muon. The up going
neutrinos get oscillated and down going neutrinos remain almost un-oscillated
due to the short distance from the source to the detector. When the iron
plates of ICAL detector are placed horizontally, all these events cannot be
detected. The high energy events will have normally small scattering angle,
but very long tracks in the detector. So, they may be detected. If we plot the
distribution of the difference in zenith angles between neutrino and the
corresponding muon for a fixed energy, it gives a Gaussian plot. The half
width at the half maxima of this distribution as a function of muon energy is
shown in fig 3. We put a selection criteria that the events for a given muon
energy having the difference $|90^{\circ}-\theta_{\rm zenith}|$ within the
above half width are rejected. Here we expect roughly that these events cannot
be detected in case of real experiments. The precisions determined with and
without this cut is discussed later.
## VI The $\chi^{2}$
The number of events falls very rapidly with increase in energy and there is a
very small statistics at the high energy. However, the contributions to the
sensitivities of the oscillation parameters is significant from these high
energy events. For the low statistics at the high energy, the $\chi^{2}$ is
calculated according to the Poisson probability distribution defined by the
expression:
$\displaystyle\chi^{2}$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{n_{L},n_{E}}\left[2\left\\{N^{p}_{ij}\left(1+\sum_{k=1}^{n_{s}}f^{k}_{ij}\cdot\xi^{k}\right)-N^{o}_{ij}\right\\}\right.$
(7)
$\displaystyle\left.-2N^{o}_{ij}\ln\left(\frac{N^{p}_{ij}\left(1+\sum_{k=1}^{n_{s}}f^{k}_{ij}\cdot\xi_{k}\right)}{N^{o}_{ij}}\right)\right]\
$ $\displaystyle+\sum_{k=1}^{n_{s}}{\xi_{k}}^{2}$
Here, $N^{o}_{ij}$ is the number of observed events generated by Nuance for a
given set of oscillation parameters with an exposure of 1 Mton.year of ICAL
and $N^{p}_{ij}$ is the number of predicted events (discussed later). These
are obtained in a 2-dimensional grids in the plane of $\log E-L^{0.4}$. The
term $f^{k}_{ij}$ is the systematic uncertainty of $N^{p}_{ij}$ due to the
$k$th uncertainty (discussed later) and ${\xi_{k}}$ is the pull variable for
the $k$th systematic uncertainty. We use total number of $\log E$ bins $n_{E}$
= 35 (0.8 $-$ 40 GeV) and the number of $L^{0.4}$ bins as a function of the
energy. We consider $n_{L}=2\times 25,~{}2\times 27,~{}2\times 29,~{}2\times
31,$ and $~{}2\times 33$ for $E=0.8-1,~{}1-2,~{}2-3,~{}3-4,~{}{\rm and}~{}>4$
GeV, respectively. For the down-going events, the binning is done by replacing
‘$L^{0.4}$’ by $`-L^{0.4}$’. The factor ‘2’ is taken to consider both up and
down going cases. For the up going neutrino, $L$ is the distance traveled by
the neutrino from the source at the atmosphere to the detector in the
underground. In case of the down going neutrino, the $L$ is the ‘mirror $L$’
which is the same $L$ if the neutrino comes from exactly opposite direction.
The table of Honda flux is given in 20 $\cos\theta_{\rm zenith}$ bins and 101
$E$ bins ($0.1-10^{4}$ GeV). It should be noted that we first re-binned the
data into 300 $\cos\theta_{\rm zenith}$ bins and 200 $\log E$ bins ($0.8-40$
GeV) to get the oscillation pattern accurately. This large number of
$\cos\theta_{\rm zenith}$ bins also help in proper re-binning of the data into
$L^{0.4}$ bins.
### VI.1 Migration from neutrino to muon
To generate the theoretical data for the chi-square analysis, we first
generate 500 years un-oscillated data for 1 Mton detector by Nuance. From this
data we find the energy-angle correlated resolutions (see Figs. 4) in 35
$E^{\nu}$ bins (in log scale for the range of $0.8-40$ GeV) and 17
$\cos\theta_{\rm zenith}^{\nu}$ bins (for the range $-1$ to $+1$). For a given
$\log E_{\nu}$ bin, we calculate the efficiency of having $E_{\mu}\geq$ 0.8
GeV (threshold of the detector). For each set of oscillation parameters, we
integrate the oscillated atmospheric neutrino flux folding the cross section,
the exposure time, the target mass, the efficiency and the resolution function
to obtain the predicted data. We use the CC cross section of Nuance-v3
Casper:2002sd and the Honda flux in 3-dimensional scheme Honda:2004yz . This
method has been discussed in detail in Samanta:2006sj , but the number of bins
and resolution functions have been changed here.
One can do this directly by generating 500 Mton.year data (to ensure that the
statistical error is negligible) for each set of oscillation parameters and
then reducing it to 1Mton.year equivalent data, which would be the more
straight forward method. The marginalization study with this method is almost
an un-doable job in a normal CPU. However, an exactly equivalent result is
obtained here using the energy-angle correlated resolution function.
We have done this study for ideal muon detector. From GEANT simulation of ICAL
detector it is seen that the energy resolution of muon varies 4–10% depending
on the direction and energy. Since the iron plates are stacked horizontally,
the resolution will be better for vertical events than the slanted events. The
angular resolution varies from 4–12% for the considered range of energy and
zenith angle. Here, the thickness of iron plates are considered to be 6 cm.
From Fig. 4 it is clear that these are negligible compared to the resolutions
obtained from kinematics of scattering processes.
The addition of the hadron energy to the muon energy of an event, which might
improve the reconstructed neutrino energy resolution, is not considered here
for conservative estimation of the sensitivity. It would be realistic in case
of GEANT-based studies since the number of hits produced by the hadron shower
strongly depends on the thickness of iron layers. However, ICAL can also
detect the neutral current events. Though it is expected that these events
will not have any directional information; energy dependency of the
oscillation, averaged over all directions can also contribute to the total
$\chi^{2}$ in the sensitivity studies separately.
### VI.2 Systematic uncertainties
The atmospheric neutrino flux is not known precisely, there are huge
uncertainties in its estimation. We may divide them into two categories: I)
overall uncertainties (which are independent of energy and zenith angle), and
II) tilt uncertainties (which are dependent of energy and/or zenith angle). We
consider the following types of uncertainties.
The energy dependent uncertainty, which arises due to the uncertainty in
spectral indices, can be expressed as
$\Phi_{\delta_{E}}(E)=\Phi_{0}(E)\left(\frac{E}{E_{0}}\right)^{\delta_{E}}\approx\Phi_{0}(E)\left[1+\delta_{E}\log_{10}\frac{E}{E_{0}}\right].$
(8)
Similarly, the vertical/horizontal flux uncertainty as a function of zenith
angle can be expressed as
$\Phi_{\delta_{z}}(\cos\theta_{z})\approx\Phi_{0}(\cos\theta_{z})\left[1+\delta_{z}(|\cos\theta_{z}|-0.5)\right].$
(9)
Next, we consider the overall flux normalization uncertainty $\delta_{f_{N}}$,
and the overall neutrino cross section uncertainty $\delta_{\sigma}$.
For $E<1$ GeV we consider $\delta_{E}=5\%$ and $E_{0}=1$ GeV and for $E>10$
GeV, $\delta_{E}=5\%$ and $E_{0}=10$ GeV. We take $\delta_{f_{N}}=10\%$,
$\delta_{\sigma}=15\%$. We consider $\delta_{z}=4\%$ which leads to 2%
vertical/horizontal flux uncertainty. We derived these uncertainties from
Honda:2006qj .
For each set of oscillation parameters, we calculate the $\chi^{2}$ in two
stages. First we used $\xi_{k}$ such that
$\frac{\delta\chi^{2}}{\delta\xi_{k}}=0$, which can be obtained solving the
equations Fogli:2002pt . Then we calculate the final $\chi^{2}$ with these
$\xi_{k}$ values. Finally, we minimize the $\chi^{2}$ with respect to all
oscillation parameters 222Here, we consider all uncertainties as a function of
reconstructed neutrino energy and zenith angle. Here we assumed that the tilt
uncertainties will not be changed too much due to reconstruction. However, on
the other hand if any tilt uncertainties arises in reconstructed neutrino
events from the reconstruction method or the kinematics of the scattering, it
is then accommodated in $\chi^{2}$..
Figure 4: The sample energy-angle correlated resolution plots for neutrino
(left column) and anti-neutrino (right column) for the bins of
$E_{\nu}=0.85-0.98$ GeV with $\cos\theta_{\rm zenith}=-0.40$ to $-0.20$ (upper
row) and $E_{\nu}=6.84-7.86$ GeV with $\cos\theta_{\rm zenith}=0$ to $0.20$
(lower row). The data is obtained from the simulation of 500 MTon.year
exposure of ICAL considering no oscillation.
## VII Marginalization and Results
A global scan of $\chi^{2}$ is carried out over the oscillation parameters
$\Delta m_{32}^{2},~{}\theta_{23}$, $\theta_{13}$ and $\delta_{CP}$ with
neutrinos and anti-neutrinos separately. We have chosen the range of $|\Delta
m_{32}^{2}|=2.0-3.0\times 10^{-3}$eV2, $\theta_{23}=38^{\circ}-52^{\circ}$,
$\theta_{13}=0^{\circ}-12.5^{\circ}$ and $\delta_{CP}=0^{\circ}-360^{\circ}$.
The 2-dimensional 68%, 90%, 99% confidence level allowed parameter spaces
(APSs) are obtained by considering $\chi^{2}=\chi^{2}_{\rm
min}+2.48,~{}4.83,~{}9.43$, respectively. For every set of data we have
checked that chi-square/d.o.f remains $\mathrel{\hbox to0.0pt{\raise
2.15277pt\hbox{$<$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}1.1$ at its minimum
value. We obtain the APS in $|\Delta m_{32}^{2}|-\theta_{23}$ and $|\Delta
m_{32}^{2}|-\theta_{13}$ plane. We set the input of $|\Delta
m_{32}^{2}|=2.5\times 10^{-3}$eV2 and $\delta_{CP}=0$.
It is important to note here that the statistics changes significantly over
$L-E$ plane with the change of oscillation parameters. Moreover, the fluxes
and the resolutions are very different at different $L-E$ zones. The upper and
lower bounds of an oscillation parameter depends significantly on the
statistics as well as on the resolutions of the specific zones in $L-E$ plane.
The binning of the data, which captures the oscillation patten also plays the
vital role.
However, for some sets of input parameters the chi-square remains almost flat
over a significant range of a parameter and then changes rapidly. It happens
due to the fact that I) the change of oscillation probability is
insignificant, and/or II) the above change is significant, but it is eaten by
the systematic uncertainties in chi-square analysis. In this circumstances,
the best-fit values may change significantly from the input values. This is a
very common feature in analyses with generating events by Monte Carlo method.
But, in methods without Monte Carlo, the number of events are determined with
an accuracy of a fraction of 1 and then best-fit values is always close to the
input values.
In some cases, the deviations of the best-fit values are large. This is
happened due to the following reasons. Here, we have just folded the total
charge current cross section of all processes to find the number events for a
particular neutrino energy to generate the theoretical data. We see
significant fluctuations more than 1 $\sigma$ in number of events between
“theoretical data” and “experimental data” in some particular energy bins for
a given set of oscillation parameters (see Fig. 5). This happens mainly at the
neutrino energy $\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$<$}\hss}{\lower
2.15277pt\hbox{$\sim$}}}3$ GeV, where the resonances occur. Here, the neutrino
cross sections depend on the type of nucleus. The generation of events is very
complicated here and it depends on the models. These all are not considered in
the same way as in Nuance in generation of theoretical data, which causes
energy dependent systematic uncertainty. However, this has no regular pattern.
In our analysis we consider only the over all uncertainty in the cross
section. These energy dependent uncertainties have not been considered in our
analysis. When $\theta_{23}$ deviates from $\pi/4$, the difference between
peak and dip decreases and the fluctuations becomes relatively prominent.
Again, when $\theta_{13}$ becomes large, the periodic pattern of oscillation
is lost due to matter effect. We have checked that the fluctuations are larger
for $\theta_{23}=50^{\circ}$ and $\theta_{13}=7.5^{\circ}$ than
$\theta_{23}=45^{\circ}$ and $\theta_{13}=0^{\circ}$. In this region of
oscillation parameters, significant deviations of best-fit values of
oscillation parameters from their true values are obtained.
The variation of $\Delta\chi^{2}[=\chi^{2}-\chi^{2}_{\rm min}]$ with each of
$\theta_{23},~{}\theta_{13}$ and $\Delta m^{2}_{32}$, are shown in Fig. 6, 7,
8, and 9. These are after marginalization over all the oscillation parameters
except one, with which it varies. We present the cases for inputs
$\theta_{13}=0^{\circ}$,and $7.5^{\circ}$ with
$\theta_{23}=40^{\circ}~{},45^{\circ}$, and $50^{\circ},$ respectively.
Figure 5: The typical distribution of events with $E_{\nu}$ keeping
$\cos\theta^{\nu}_{\rm zenith}$ fixed at $\approx-0.367$ and with
$\cos\theta^{\nu}_{\rm zenith}$ keeping $E_{\nu}$ fixed at $\approx 2.24$ GeV.
We set $\Delta m^{2}_{32}=-2.5\times 10^{-3}$eV2, $\theta_{23}=45^{\circ}$,
$\theta_{13}=0^{\circ}$ and $\delta_{CP}=0^{\circ}$.
Figure 6: The variation of $\Delta\chi^{2}=(\chi^{2}-\chi^{2}_{\rm min})$ with
$\theta_{23}$ for input value of $\theta_{23}=40^{\circ}$, $45^{\circ}$ and
$50^{\circ}$ with $\theta_{13}=0^{\circ}$ and $\theta_{13}=7.5^{\circ}$,
respectively. Here, we consider both neutrinos and anti-neutrinos together.
The type of input hierarchy is inverted. Figure 7: The variation of
$\Delta\chi^{2}=(\chi^{2}-\chi^{2}_{\rm min})$ with $\theta_{13}$ for input
value of $\theta_{13}=0^{\circ}$, and $\theta_{23}=45^{\circ}$ considering
neutrino, anti-neutrino and both types of neutrinos, respectively.
Figure 8: The same as Fig. 6, but with $\theta_{13}$.
Figure 9: The same as Fig. 6, but with $\Delta m^{2}_{32}$.
### VII.1 Sensitivity to $\theta_{23}$ and its octant discrimination
As the present experiments indicate that the value of $\theta_{13}$ is small
compared to $\theta_{23}$, the atmospheric neutrino oscillation is mainly
governed by two flavor oscillation
$\nu_{\mu}~{}(\bar{\nu}_{\mu})~{}\leftrightarrow\nu_{\tau}~{}(\bar{\nu}_{\tau})$.
This constrains $\sin^{2}2\theta_{23}$ and $|\Delta m_{32}^{2}|$.
From Fig. 10, we see that the deviation from the maximal mixing between 2 and
3 flavor eigen states can be observed. However, a degeneracy in $\theta_{23}$
arises in case of $\theta_{13}=0$, whether it is larger or smaller than
$45^{\circ}$. But, when the matter effect comes into the play, a resonance
occurs in
$\nu_{\mu}~{}(\bar{\nu}_{\mu})~{}\leftrightarrow\nu_{e}~{}(\bar{\nu}_{e})$
oscillation and it leads to a large effective value of $\theta_{13}$ (see Eq.
4). This helps to dominate the $\sin^{4}\theta_{23}$ term in Eq. 3 and breaks
the $\theta_{23}$ degeneracy in its measurement. Since the atmospheric
neutrinos cover a large region of $E-L$ plane, it can observe the matter
resonance and has an ability to discriminate the octant degeneracy. In Fig. 6,
the variations of $\Delta\chi^{2}=(\chi^{2}-\chi^{2}_{\rm min})$ with
$\theta_{23}$ are shown for input values of
$\theta_{23}=40^{\circ},~{}45^{\circ}$ and $50^{\circ}$ with
$\theta_{13}=0^{\circ}$ and $7.5^{\circ}$, respectively. We see that with
increase in $\theta_{13}$, the matter effect not only discriminates the
octant, but increases the precision also.
In Fig. 10 we see that for $\theta_{13}=7.5^{\circ}$ the octant discrimination
is possible for input of $\theta_{23}=40^{\circ}$ and $50^{\circ}$ with IH.
But it is not possible for NH. Normally, the flux of $\nu_{\mu}$ is higher
than $\bar{\nu}_{\mu}$. In case of IH (NH), $\bar{\nu}_{\mu}$ ($\nu_{\mu}$) is
suppressed. The statistics remains high for IH compared to NH, which leads
better octant discrimination possibility for IH.
### VII.2 Sensitivity to $\theta_{13}$
The effect of $\theta_{13}$ in oscillation probability does not appear
dominantly neither in atmospheric nor in solar neutrino oscillation, but as a
subleading in both oscillations. In case of atmospheric neutrino, its effect
is seen at a) $E\sim$ 1 GeV for propagation of neutrinos through vacuum as
well as through matter (no matter resonance), and b) $E\approx 2-10$ GeV for
propagation only through matter (matter resonance). The matter effect enhances
the difference in oscillation probabilities between two $\theta_{13}$ values
for neutrinos with NH and for anti-neutrinos with IH (see Eq. 4). In Fig. 7 we
show the cases a) and b) considering neutrinos and anti-neutrinos separately.
We find that the effect of case a) is negligible.
We have plotted the APS in $\theta_{13}-|\Delta m_{32}^{2}|$ plane in Fig 11
for $\theta_{13}=0^{\circ},~{}5^{\circ},$ and $7.5^{\circ}$ with
$\theta_{23}=40^{\circ}$, $45^{\circ}$ and $50^{\circ}$, respectively. We find
that the matter effect significantly constrains $\theta_{13}$ over the present
limit. Though the matter effect acts either on neutrinos or on anti-neutrinos
depending on the type of the hierarchy, but we have checked that it improves
when we consider both neutrinos and anti-neutrinos. The sensitivity of
$\theta_{13}$ is not generally expected to be improved for the case of
analysis with neutrinos and anti-neutrinos in together. However, this happens
here due to the marginalization which restricts $\theta_{23}$ more tightly for
the case of $\nu$ and $\bar{\nu}$ in together than either with $\nu$ or
$\bar{\nu}$ and indirectly constrains $\theta_{13}$. It is also seen that the
APS is strongly dependent on the input of $\theta_{23}$ and a better
constraint is obtained for $\theta_{23}>45^{\circ}$. However, it is notable
here that the uncertainty is very high and the best-fit values deviate largely
from its input values for nonzero $\theta_{13}$ inputs due to the reasons
discussed at the beginning of this section.
### VII.3 Sensitivity to $\Delta m^{2}_{32}$
We show the constraint on $|\Delta m^{2}_{32}|$ in Fig. 10 and 11. We see that
the precision is little better when $\theta_{13}=0.$ The reason behind this is
that a regular oscillation pattern with periodic rise and fall is observed
when $\theta_{13}=0$.
It is seen that the APS is larger for NH than IH. The matter effect does not
act on neutrino for IH and anti-neutrinos for NH with an addition to the fact
that the flux is higher for neutrino than anti-neutrino. As discussed above,
the APS is more restricted when there is no matter effect. Here, for input
with IH the number of neutrino events is high and they do not have any matter
effect. This leads to smaller APS for IH compared to NH for large values of
$\theta_{13}$.
### VII.4 Effect of events at near horizon on precision measurements
For a given set of input parameters, if we compare the APSs with zenith angle
cut (discussed in section V) with those without any cut, we find no
significant differences. As a demonstrating example, we have shown the APSs in
Fig. 12 without imposing any zenith angle cut for a given set of oscillation
parameters. One can find the corresponding plots with zenith angle cut in
Figs. 10 and 11.
From the study of this paper, we can conclude that the events at near horizon
cannot contribute significantly in precision measurements. The fact is that
the $L$ resolution is very poor here. A little change in zenith angle at near
horizon changes $L$ values drastically. Again, the discrimination of up and
down going events are not possible. So, the oscillation effect is almost
smeared out by the resolutions. From the $L/E$ dip considering the $L$ and $E$
values of neutrinos, one can expect a large contribution in precision from
these events. But, in practical situation, there is no appreciable improvement
after addition of these events.
The vertical (horizontal) stacking of iron plates will be able to detect the
horizontal (vertical) events. So, from this study one can conclude that
horizontal stacking is expected to give better precision than the vertical
stacking.
### VII.5 Precision of the parameters
For a quantitative assessment of the result, we define the precision of a
parameter $t$ as:
$P=2\left(\frac{t^{\rm max}-t^{\rm min}}{t^{\rm max}+t^{\rm min}}\right).$
(10)
We find that the precisions are strongly dependent on the set of input
parameters. We obtained the precision of $|\Delta m_{32}^{2}|\approx
6.4\%,~{}8.8\%$ and $12\%$ at 68%, 90% and 99% CL, respectively and the
precision of $\sin^{2}\theta_{23}\approx 31\%$, 38%, and 41% at 68%, 90% and
99% CL, respectively for the input of $\theta_{23}=45^{\circ}$ and
$\theta_{13}=0^{\circ}$.
The oscillation dip moves towards the lower $L/E$ values as $|\Delta
m_{32}^{2}|$ increases. The statistics also decreases at the lower $L/E$
region. So, the precision is expected to be weaker as the input of $|\Delta
m_{32}^{2}|$ increases.
A comparison of the precisions of $\Delta m^{2}_{32}$ and
$\sin^{2}\theta_{23}$ among different future baseline experiments is made in
Huber:2004dv . The variation of the precisions with the change of input
parameters are also presented there. We compare our results with 5 years run
of T2K, which is the best in determining precision of atmospheric oscillation
parameters in the list in Huber:2004dv . The precision of $\Delta m^{2}_{32}$
is almost same with T2K ($\approx 12\%$) and precision of
$\sin^{2}\theta_{23}$ from ICAL is 41% while from T2K is 46%. Here we present
the results for 10 years run of 100 kTon ICAL detector. From this work it is
also seen that atmospheric neutrinos at ICAL detector are in very good
position to discriminate octant of $\theta_{23}$. The main advantage here is
that atmospheric neutrinos are natural sources and the cost goes only to build
and run the detector.
Figure 10: The 68%, 90%, 99% CL allowed regions in $\theta_{23}-|\Delta
m_{32}^{2}|$ plane for the input of $\theta_{23}=40^{\circ}$ (first row),
$45^{\circ}$ (second row), $50^{\circ}$ (third row) with
$\theta_{13}=0^{\circ}$ (first column), $5^{\circ}$ (second column),
$7.5^{\circ}$ (third column) with IH and $7.5^{\circ}$ (fourth column) with
NH.
Figure 11: The same as Fig. 10, but in $\theta_{13}-|\Delta m_{32}^{2}|$
plane.
Figure 12: The allowed regions without any zenith angle cut for the events at
the horizon.
## VIII Conclusion
We have studied the precisions of the oscillation parameters from atmospheric
neutrino oscillation experiment at the large magnetized ICAL detector
generating events by Nuance and considering only the muons produced by the
charge current interactions. The distance between two consecutive peaks of
oscillation in $E$ for fixed $L$ increases as one goes from higher $L$ values
to its lower values. This indicates the need of finer binning at lower $L$
values in $\chi^{2}$ analysis. We optimize the binning of the data in the
grids of $\log E-L^{0.4}$ plane. We find that the impact of the events at near
horizon on the precision measurements is very negligible due to poor $L$
resolution.
From the marginalized $\chi^{2}$ study separately for neutrinos and anti-
neutrinos, we find that the measurement of $\theta_{13}$ is possible at a
considerable precision with atmospheric neutrinos. The precision of
$\theta_{13}$ depends crucially on its input value. For $\theta_{13}=0$, we
find its upper bound $\approx 4^{\circ},~{}6^{\circ}$ and $9^{\circ}$ at 68%,
90% and 99% CL, respectively. The both lower and upper bounds of $\theta_{13}$
are also possible for some combinations of ($\theta_{23},\theta_{13}$) and it
happens mainly for $\theta_{23}\mathrel{\hbox to0.0pt{\raise
2.15277pt\hbox{$>$}\hss}{\lower 2.15277pt\hbox{$\sim$}}}45^{\circ}$.
The precision of $|\Delta m_{32}^{2}|$ and $\theta_{23}$ can also be very high
and the determination of octant of $\theta_{23}$ is possible for some
combinations of ($\theta_{23},~{}\theta_{13}$).
It should also be noted here that in $\chi^{2}$ analysis the theoretical data
and the experimental data are not generated in the same way. The different
models of neutrino interactions generate energy dependent systematic
uncertainties at some energies. These are not included in this analysis. This
causes sometimes large deviation of the best-fit values of the oscillation
parameters from the input values.
Acknowledgments: This research has been supported by funds from Neutrino
Physics projects at HRI. The use of excellent cluster computational facility
installed from this project is gratefully acknowledged. A part of the
computation was also carried out in HRI general cluster facility.
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|
arxiv-papers
| 2008-12-30T09:40:55 |
2024-09-04T02:48:59.573124
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Abhijit Samanta",
"submitter": "Abhijit Samanta",
"url": "https://arxiv.org/abs/0812.4639"
}
|
0812.4640
|
HRI-P-08-12-004
A comparison of the sensitivities of the parameters
with atmospheric neutrinos for different analysis methods
Abhijit Samanta 111E-mail address: abhijit@hri.res.in
Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211 019,
India
###### Abstract
In the atmospheric neutrino experiments the primary problems are the huge
uncertainties of flux, very rapid fall of flux with increase of energy, the
energy dependent wide resolutions of energy and zenith angle between true
neutrinos and reconstructed neutrinos. These all in together make the choice
of binning of the data for chi-square analysis complicated. The large iron
calorimeter has the ability to measure the energy and the direction of the
muon with high resolution. From the bending of the track in the magnetic field
it can also distinguish its charge. We have analyzed the atmospheric neutrino
oscillation generating events by Nuance and then considering the muons
produced in the charge current interactions as the reconstructed neutrinos.
This practically takes into account the major problem of wide resolutions. We
have binned the data in three ways: i) in the grids of $\log E-\log L$ plane,
ii) in the grids of $\log E-\cos\theta_{\rm zenith}$ plane, and iii) in the
bins of $\log(L/E)$. We have performed a marginalized $\chi^{2}$ study over
$\Delta m_{32}^{2},~{}\theta_{13}$ and $\theta_{23}$ for neutrinos and anti-
neutrinos separately for each method and finally compared the results.
PACS 14.60.Pq
Keywords: neutrino oscillation, atmospheric neutrino, INO
## 1 Introduction
The atmospheric neutrino anomaly was first observed by IMB in 1986 and then
confirmed by Kamiokande in 1988 [1, 2]. Finally, the neutrino oscillation was
discovered in 1998 with atmospheric neutrino experiment [3]. The atmospheric
neutrinos are produced by the interactions of the cosmic rays with the
atmosphere. At the neutrino energies above a few GeV, the effect of geo-
magnetic field on the cosmic rays is negligible and then the atmospheric
neutrino flux can be predicted to be up-down symmetric. The flight lengths for
up and down going neutrinos are very different. The atmospheric neutrino
experiments exploit these features to study the neutrino oscillation.
The atmospheric neutrinos are also equally important in the precision era of
neutrino physics. The main thrust is now on the precise measurements of
oscillation parameters. This helps to identify the right track to understand
the underlying principle that gives the neutrino masses and their mixing. In
the recent years, the studies of neutrinos has become a popular tool to probe
the physics beyond the standard model. In the standard oscillation picture,
there are six parameters. The present 1$\sigma$, 2$\sigma$ and 3$\sigma$
confidence level ranges from global $3\nu$ oscillation analysis (2008) 222The
CP-violating phase $\delta_{CP}$ is still unconstrained. [4] are tabulated in
table 1.
Parameter | $\Delta m_{21}^{2}/10^{-5}\mathrm{\ eV}^{2}$ | $\sin^{2}\theta_{12}$ | $\sin^{2}\theta_{13}$ | $\sin^{2}\theta_{23}$ | $|\Delta m_{31}^{2}|/10^{-3}\mathrm{\ eV}^{2}$
---|---|---|---|---|---
Best fit | 7.67 | 0.312 | 0.016 | 0.466 | 2.39
$1\sigma$ range | 7.48 – 7.83 | 0.294 – 0.331 | 0.006 – 0.026 | 0.408 – 0.539 | 2.31 – 2.50
$2\sigma$ range | 7.31 – 8.01 | 0.278 – 0.352 | $<0.036$ | 0.366 – 0.602 | 2.19 – 2.66
$3\sigma$ range | 7.14 – 8.19 | 0.263 – 0.375 | $<0.046$ | 0.331 – 0.644 | 2.06 – 2.81
Table 1: Global 3$\nu$ oscillation analysis (2008)
This spectacular achievement is very stimulating to uncover the facts which
are still missing. To determine the mass ordering (sign of $\Delta
m_{32}^{2}$) 333$\Delta m^{2}_{32}=m_{3}^{2}-m_{2}^{2}$., the values of
$\theta_{13}$ and $\delta_{CP}$ with good precision, the octant of
$\theta_{23}$ with atmospheric neutrinos as well as neutrinos from artificial
beams, there are many ongoing and planned experiments: INO [5], UNO [6], T2K
[7], NOvA [8], Hyper-Kamiokande [9] and many others. In the current few years,
a large fraction of effort in particle physics research has gone to study the
physics potential of these detectors [10]. The current research activity [11,
12, 13, 14, 15, 16, 17, 18, 19, 20] shows the uniqueness in physics potential
of the large magnetized Iron CALorimeter (ICAL) detector at the India-based
Neutrino Observatory (INO). It should be noted here that its position at
PUSHEP has a special feature. It gives the magic baseline from CERN for beam
experiments, which provides the oscillation probabilities relatively
insensitive to the yet unconstrained CP phase compared to all other baselines
and permits to make the precise measurements of the masses and their mixing
avoiding the degeneracy issues [16]. On the other hand, ICAL can detect
$\nu_{\mu}$ and $\bar{\nu}_{\mu}$ separately using the magnetic field for
charge current events. The oscillation study with atmospheric neutrinos is the
primary goal of ICAL at INO. Before going into the detailed techniques of the
analysis methods, we will first discuss the basic nature of atmospheric
neutrino oscillation and the detection characteristics of ICAL detector.
### 1.1 The atmospheric neutrino oscillation and the ICAL detector
The present atmospheric neutrino data from the pioneering Super Kamiokande
(SK) experiment are well explained by two flavor oscillation [21, 22].
However, one expects the reflection of $\nu_{\mu}\rightarrow\nu_{e}$
oscillation in data for standard 3-flavor framework in the data if
$\theta_{13}$ is nonzero. Neglecting the $\Delta m_{21}^{2}$ term the
oscillation probability can be expressed as:
$\displaystyle\mbox{P}(\nu_{\mu}\rightarrow\nu_{e})$ $\displaystyle=$
$\displaystyle\mbox{P}(\nu_{e}\rightarrow\nu_{\mu})$ $\displaystyle=$
$\displaystyle\sin^{2}\theta_{23}\sin^{2}2\theta_{13}\sin^{2}\left(\frac{1.27\Delta
m^{2}L}{E}\right)$ $\displaystyle\mbox{P}(\nu_{\mu}\rightarrow\nu_{\mu})$
$\displaystyle=$ $\displaystyle 1$ (1)
$\displaystyle-4\cos^{2}\theta_{13}\sin^{2}\theta_{23}(1-\cos^{2}\theta_{13}\sin^{2}\theta_{23})$
$\displaystyle\times\sin^{2}\left(\frac{1.27\Delta m^{2}L}{E}\right)$
These oscillation probabilities are derived for vacuum. Since it involves
electron neutrino, the oscillation will be modulated by the matter effect [23,
24]. Then,
$\displaystyle\mbox{P}(\nu_{\mu}\rightarrow\nu_{e})$ $\displaystyle=$
$\displaystyle\mbox{P}(\nu_{e}\rightarrow\nu_{\mu})$ (2) $\displaystyle=$
$\displaystyle\sin^{2}\theta_{23}\sin^{2}2\theta_{13}^{M}\sin^{2}\left(\frac{1.27\Delta{m^{2}}_{M}L}{E}\right).$
The symbol ‘M’ denotes effective parameters in matter. The effective mixing
angle is
$\displaystyle\sin^{2}2\theta_{13}^{M}$ $\displaystyle=$
$\displaystyle\frac{\sin^{2}2\theta_{13}}{(\cos 2\theta_{13}-A_{CC}/\Delta
m^{2})^{2}+\sin^{2}2\theta_{13}}$ (3)
and
$\displaystyle\Delta{m^{2}}_{M}$ $\displaystyle=$
$\displaystyle\sqrt{\left(\Delta m^{2}\cos
2\theta_{13}-A_{CC}\right)^{2}+\left(\Delta m^{2}\sin
2\theta_{13}\right)^{2}}$ (4)
with
$\displaystyle A_{CC}$ $\displaystyle=$ $\displaystyle 2\sqrt{2}G_{F}N_{e}E,$
(5)
where $G_{F}$ is the Fermi constant, $N_{e}$ is the electron density of the
medium and $E$ is neutrino energy [25]. The matter potential term $A_{CC}$ has
the same absolute value, but opposite sign for neutrinos and anti-neutrinos.
The Mikheyev-Smirnov-Wolfenstein (MSW) resonance occurs when neutrino passes
through the matter (see eq. 3). It happens for Normal Hierarchy (NH) with
neutrinos and for Inverted Hierarchy (IH) with anti-neutrinos. The resonance
energy corresponding to a baseline can be seen in [26].
The muon neutrino (anti-neutrino) produces $\mu^{-}$ ($\mu^{+}$) in Charge
Current (CC) weak interactions. The magnetized ICAL can distinguish $\mu^{+}$
and $\mu^{-}$ with the magnetic field. The energy ($E$) and zenith angle
($\theta_{\rm zenith}$) or baseline ($L$) resolutions of the muons are very
high at ICAL [5]. The hadron energy can also be measured at ICAL. However, its
resolution is very poor and strongly depends on thickness of the iron layers.
The atmospheric neutrinos are expected to be very useful in precision studies
for its very wide energy range (MeV $-$ few hundred GeV) and wide baseline
range (few km $-12950$ km). It gives both neutrino and anti-neutrino, which
behave oppositely with matter. This helps to detect the sign( $\Delta
m_{32}^{2}$), the value of $\theta_{13}$ as well as the octant of
$\theta_{23}$. One can exploit this feature to measure the precision of these
parameters and the mass ordering at the magnetized ICAL detector at INO. It
should be noted here that the non-magnetized detectors, like water Cherenkov
detector, can also contribute in this study since the cross section, the
$y(=(E_{\nu}-E_{\rm lepton})/E_{\nu})$ dependence of the cross section are
different for $\nu$ and $\bar{\nu}$. The water detectors may also be able to
distinguish statistically $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ due to different
capture rates and lifetimes of the charged muons in water.
However, one of the crucial problems in neutrino physics experiments is the
wide resolutions of $E$ and $L$ between true neutrinos and reconstructed
neutrinos, which smears the oscillation effect to some significant extent.
This arises mainly due to interaction kinematics. The un-observable product
particles, un-measurable momentum of recoiled nucleus are the main sources of
this huge uncertainty in reconstructed neutrino momentum. These are strongly
neutrino energy dependent.
Due to the above complications, the method of extraction of the results from
the data is not straightforward. The results depend crucially on the way of
the analysis and particularly on the type of binning of the data. This fact is
well-known from the analysis of atmospheric neutrino data of SK experiment
[21, 22]. In this paper we consider the reconstructed energy and the direction
of an event only from the muon generating it by the neutrino event generator
Nuance-v3[27]. The addition of hadrons to the muon, which might increase the
reconstructed neutrino energy resolution, is not considered here for
conservative estimation of the sensitivity. It would be realistic in case of
GEANT-based studies since the number of hits produced by the hadron shower
strongly depends on the iron thickness. However, INO can also detect the
neutral current events. Though it is expected that these will not have any
directional information, the energy dependency of the averaged oscillation
over all directions can also contribute to the total $\chi^{2}$ separately in
the sensitivity studies. Here we have studied the atmospheric neutrino
oscillation by binning the data in different ways and finally compared the
results. These are discussed in the next sections.
## 2 The $\chi^{2}$ analysis
Now we will describe a general expression for $\chi^{2}$, the method for
generation of the theoretical data, the estimated systematic uncertainties,
and finally the ways of binning of the data. The number of events falls very
rapidly with the increase of energy and the statistics is very poor at high
energy. However, the contribution to the sensitivities of the oscillation
parameters is significant from these high energy events. To incorporate these
events at high energy, the $\chi^{2}$ value is calculated according to Poisson
probability distribution. For all types of binning, we define a general
expression of $\chi^{2}$ as
$\displaystyle\chi^{2}$ $\displaystyle=$
$\displaystyle\sum_{I=1}^{N}\left[2\left\\{N^{p}_{I}-N^{o}_{I}\right\\}-2N^{o}_{I}\ln\left(\frac{N^{p}_{I}}{N^{o}_{I}}\right)\right]+\sum_{k=1}^{n_{s}}{\xi_{k}}^{2}$
(6)
with
$\displaystyle N^{p}_{I}$ $\displaystyle=$ $\displaystyle\sum_{i,j=n_{c}^{\rm
low},n_{E}^{\rm low}}^{n_{c}^{\rm high},n_{E}^{\rm
high}}N^{p}_{ij}\left(1+\sum_{k=1}^{n_{s}}f^{k}_{ij}\cdot\xi^{k}\right),$
$\displaystyle{\rm and}$ (7) $\displaystyle N^{o}_{I}$ $\displaystyle=$
$\displaystyle\sum_{i,j=n_{c}^{\rm low},n_{E}^{\rm low}}^{n_{c}^{\rm
high},n_{E}^{\rm high}}N^{o}_{ij}$ (8)
The $N^{o}_{ij}$ ($N^{p}_{ij}$) is considered as the number of observed
(predicted) events in the $ij$th grid in the plane of $\log E-\cos\theta_{\rm
zenith}$. Here we consider the data for 1 Mton.year exposure of the detector.
The $f^{k}_{ij}$ is the systematic error of $N^{p}_{ij}$ due to the $k$th
uncertainty. The ${\xi_{k}}$ is the pull variable for the $k$th systematic
error. We consider $n_{s}=5$. Here we have considered 30 bins of $\log E$ and
300 bins of $\cos\theta_{\rm zenith}$ for both $N^{p}_{ij}$ and $N^{o}_{ij}$.
However, it should be noted here that in calculation of the oscillated flux we
consider 200 bins of $\log E$ and 300 bins of $\cos\theta_{\rm zenith}$ to
find the accurate oscillation pattern. We consider the $E$ range $0.8-50$ GeV
and $\cos\theta_{\rm zenith}$ range $-1$ to $+1$. It should be noted here that
the energy and angular resolutions between the muons and the neutrinos of the
events differ significantly for neutrinos and anti-neutrinos due to their
different ways of interactions.
To generate the theoretical data $N^{p}_{ij}$ for the chi-square analysis, we
first generate 500 years un-oscillated data for 1 Mton detector. From this
data we find the energy-angle correlated resolutions (see figs. 1) in 30 bins
of energy (in log scale) and 10 bins of cosine of zenith angle ($-1$ to $+1$).
For a given $E_{\nu}$, we calculate the efficiency of having $E_{\mu}\geq$ 0.8
GeV (threshold of the detector). For each set of oscillation parameters, we
integrate the oscillated atmospheric neutrino flux folding the total CC cross
section, the exposure time, the target mass, the efficiency and the resolution
function to obtain the predicted data in the reconstructed $\log
E-\cos\theta_{\rm zenith}$ grid 444One can do this in an another way. This is
generating the theoretical data directly for each set of oscillation
parameters. To ensure that the statistical error is negligible, one needs
first to generate a huge number of events. For example, one may generate
events for 500 Mton.year exposure of the detector for each set of oscillation
parameters. Then to obtain the theoretical data, one needs to normalize the
data to 1 Mton.year exposure of the detector dividing the events of each
energy and zenith angle bin by 500 since the experimental data is considered
for 1 Mton.year exposure. This would be the more straightforward method. But
the marginalization study with this method is almost an undoable job in normal
CPU. However, an exactly equivalent result is obtained here using the energy-
angle correlated resolution function.. We use the CC cross section of
Nuance-v3 [27] and the Honda flux of 3-dimensional scheme [28].
The atmospheric neutrino flux is not known precisely. There are huge
uncertainties in its estimation. We may divide them into two categories: I)
overall uncertainties (which are flat with respect to energy and zenith
angle), and II) tilt uncertainties (which are function of energy and/or zenith
angle). These have been estimated as the following [21]:
1. 1.
The energy dependence uncertainty which arises due to the uncertainty in
spectral indices, can be expressed as:
$\Phi_{\delta_{E}}(E)=\Phi_{0}(E)\left(\frac{E}{E_{0}}\right)^{\delta_{E}}\approx\Phi_{0}(E)\left[1+\delta_{E}\log_{10}\frac{E}{E_{0}}\right].$
(9)
The uncertainty of ${\delta_{E}}=$5% and $E_{0}=2$ GeV is considered.
2. 2.
Again, the flux uncertainty as a function of zenith angle can be expressed as
$\Phi_{\delta_{z}}(\cos\theta_{z})\approx\Phi_{0}(\cos\theta_{z})\left[1+\delta_{z}|\cos\theta_{z}|\right].$
(10)
The uncertainty of $\delta_{z}$ is considered to be $2\%$.
3. 3.
A flux normalization uncertainty of 20%.
4. 4.
An over all uncertainty of 10% in neutrino cross section.
5. 5.
An overall 5% uncertainty for this analysis.
We consider three types of binning:
* •
Type I: The events are binned in the grid of $\log E-\log L$ plane. We use
total number of $\log E$ bins $n_{E}$ = 30 (0.8 $-$ 50 GeV) and the number of
$\log L$ bins as a function of of the energy. We consider $n_{L}=2\times
14,~{}2\times 18,~{}2\times 22,~{}2\times 26,$ and $~{}2\times 30$ for
$E=0.8-1.2,~{}1.2-2.4,~{}2.4-3.6,~{}3.6-4.8,~{}{\rm and}~{}>4.8$ GeV,
respectively. For the down-going events the binning is done by replacing
‘$\log L$’ by $`-\log L$’. The factor ‘2’ is to consider both up and down
going cases.
* •
Type II: The events are binned in the grid of $\log E-\cos\theta_{\rm zenith}$
plane with exactly in the same fashion of type I. The only difference is that
the binning is done in $\cos\theta_{\rm zenith}$ instead of $\log L$.
* •
Type III: The events are binned in 100 $\log(L/E)$ bins and replacing
‘$\log(L/E)$’ by ‘$-\log(L/E)$’ for down-going events.
For the up-going neutrino, $L$ is the distance traveled by the neutrino from
the detector to the source at the atmosphere. In case of down-going neutrinos
the distance traveled from the source to detector is negligible for getting an
appreciable oscillation. However, these events help to minimize the systematic
uncertainties when considered in the $\chi^{2}$ analysis. The flux for a fixed
$E$ is strongly dependent on the zenith angle. So, for the down-going
neutrinos, we mapped the zenith angle into $L$ considering the mirror $L$.
This is the same $L$ if the neutrino comes from exactly opposite direction. It
should be noted here that the angular error makes a much smaller error to $L$
when the tracks are near vertical. It increases gradually when the tracks are
slanted and very rapidly when they are near horizontal.
For each set of oscillation parameters we calculate the $\chi^{2}$ in two
stages. First we used $\xi_{k}$ such that
$\frac{\delta\chi^{2}}{\delta\xi_{k}}=0$, which can be obtained solving the
equations [29]. Then we calculate the final $\chi^{2}$ with these $\xi_{k}$
values. Finally, we find the minimum from these $\chi^{2}$ with respect to all
oscillation parameters 555Here we consider all uncertainties as a function of
reconstructed neutrino energy and direction. We assumed that the tilt
uncertainties will not be changed too much due to reconstruction. However, on
the other hand, if any tilt uncertainty arises in reconstructed neutrino
events from the reconstruction method or kinematics of scattering, these are
then accommodated in $\chi^{2}$. We first incorporate all uncertainties in
$\log E-\cos\theta_{\rm zenith}$ bins. Then we re-bin the data in the form
what we want, e.g.; $\log E-\log L$ bins. It should be noted that we first
binned the data into a large number of $\cos\theta_{\rm zenith}$ bins compared
to number of $L$ bins to get proper binning in $\log L$..
Figure 1: The sample energy-angle correlated resolution plots for neutrino
(left column) and anti-neutrino (right column) for the bins of
$E_{\nu}=0.85-0.98$ GeV with $\cos\theta_{\rm zenith}=-0.40$ to $-0.20$ (upper
row) and $E_{\nu}=6.84-7.86$ GeV and $\cos\theta_{\rm zenith}=0$ to $0.20$
(lower row). The data are obtained from the simulation of 500 MTon.year
exposure of ICAL considering no oscillation. Figure 2: The oscillation
probability of $\nu_{\mu}\rightarrow\nu_{\mu}$. We choose $\Delta
m_{32}^{2}=-2.5\times 10^{-3}$eV2, $\theta_{23}=45^{\circ}$ and
$\theta_{13}=0^{\circ}$. Figure 3: The typical distribution of
$\Delta\chi^{2}$ with $\Delta m^{2}_{32}$. We choose the input of $\Delta
m_{32}^{2}=+2.5\times 10^{-3}$eV2, $\theta_{23}=42^{\circ}$ and
$\theta_{13}=7.5^{\circ}$.
## 3 Result
In this section, we first discuss the results qualitatively in a very general
way for all analysis techniques. Then we compare the results for different
techniques. In all cases a global scan is carried out over the three
oscillation parameters $\Delta m_{32}^{2},~{}\theta_{23}$ and $\theta_{13}$
for both normal and inverted hierarchies with neutrinos and anti-neutrinos
separately. We have considered the range of $\Delta m_{32}^{2}=2.0-3.0\times
10^{-3}$eV2, $\theta_{23}=37^{\circ}-54^{\circ}$, and
$\theta_{13}=0^{\circ}-12.5^{\circ}$. We have fixed other parameters $\Delta
m_{21}^{2}$ and $\theta_{12}$ at their best-fit values and $\delta_{\rm
CP}=0$. The 2-dimensional 68%, 90%, 99% confidence level allowed parameter
spaces (APSs) are obtained by considering $\chi^{2}=\chi^{2}_{\rm
min}+2.48,~{}4.83,~{}9.43$. To obtain the APS in $\theta_{13}-\Delta
m_{32}^{2}$ ($\Delta m_{32}^{2}-\theta_{23}$) plane, we marginalize the
$\chi^{2}$ over $\theta_{23}~{}(\theta_{13})$ over its whole range.
The experiment indicates that the value of $\theta_{13}$ is very small
compared to $\theta_{23}$ [30]. So, the atmospheric neutrino oscillation is
mainly governed by two flavor oscillation
$\nu_{\mu}~{}(\bar{\nu}_{\mu})~{}\leftrightarrow\nu_{\tau}~{}(\bar{\nu}_{\tau})$.
This constrains $\sin^{2}2\theta_{23}$ and $|\Delta m_{32}^{2}|$. Now, there
appears a degeneracy in $\theta_{23}$ whether it is larger or smaller than
$45^{\circ}$ due to the $\sin^{2}2\theta_{23}$ dependence of oscillation
probability. However, when the matter effect comes into the play, the
effective value of $\theta_{13}$ becomes large and a resonance occurs in
$\nu_{\mu}~{}(\bar{\nu}_{\mu})~{}\leftrightarrow\nu_{e}~{}(\bar{\nu}_{e})$
oscillation. This breaks the above $\theta_{23}$ degeneracy.
The difference in oscillation probability between two $\theta_{13}$ values for
neutrinos with NH and for anti-neutrinos with IH becomes significant when
matter effect comes in the picture (see eq. 3). We have plotted the APS in
$\theta_{13}-\Delta m_{32}^{2}$ plane considering both neutrinos and anti-
neutrinos (i.e. with $\chi^{2}_{\rm
total}=\chi^{2}_{\nu}+\chi^{2}_{\bar{\nu}}$) for different sets of input
parameters at 68%, 90% and 99% CL in fig 4, 5 and 6, respectively for each
type of binning of the data. We see that the matter effect significantly
constrains $\theta_{13}$ over the present limit, which is a very stimulating
result for atmospheric neutrino oscillation analysis.
Again, for the APS in $\Delta m^{2}_{32}-\theta_{23}$ plane, $\theta_{13}$ is
marginalized over the present allowed range. The APSs are shown in fig. 7, 8
and 9 at 68%, 90% and 99% CL, respectively for each type of binning of the
data. If the value of $\theta_{13}$ is nonzero, the matter effect plays a role
in determination of the octant of $\theta_{23}$ as discussed previously and
also constrains the $\theta_{23}$ range (compare its range for zero and non-
zero values of $\theta_{13}$). We find that for some combinations of
($\theta_{13},~{}\theta_{23}$), the octant determination is possible.
Now we will compare the APSs coming from different analysis method. From the
APSs it is clear that the $L/E$ analysis gives very poor results compared to
the other two methods. It happens due to the mixing of events from different
$E$ and $L$ resolutions since the resolution widths are strongly energy
dependent. It should be noted here that we have not used any selection
criteria for the events, which might improve the results.
Now we will compare the positive and the negative sides of the rest two
methods. We find a relatively stronger upper bound of $\Delta m^{2}_{32}$ in
case of binning in the grids of $\log E-\log L$ plane than the other case.
This is very important since it comes from the events with high $E$ and low
$L$ values. The $L$ resolution is very poor at low $L$ and the statistics is
low at high $E$. However, a stronger bound is obtained for this special type
of binning. We will explain it with the oscillation probability in vacuum,
which is a sinusoidal function of $L/E$. For a fixed $L$, the distance between
two consecutive peaks in $E$ increases rapidly with $E$. Again, if we compare
the distances between two consecutive peaks in $E$ for two fixed values of
$L$, it is larger for smaller $L$ value. Therefore, as one goes to smaller $L$
values, this distance in $E$ increases rapidly. So, these two consecutive
peaks of the oscillation in $E$ can be resolved with much better resolution as
one goes gradually from larger $L$ values to lower $L$ values. This is
pictorially illustrated in fig. 2. To get the reflection of this fact in
$\chi^{2}$, the finer binning at lower $L$ is essential. Though the angular
resolution is worsened at lower $L$, but the rapid increase of $E$ resolution
between two peaks wins the competition here. This is the main advantage of
this type of binning. So, we binned the data in a two dimensional grids of
$\log L-\log E$ plane 666This captures the oscillation effect well in
$\chi^{2}$ analysis without mixing events from different $E$ and $L$
resolutions.. In type II this behavior is not taken into account in the
binning of the data. However, there is a disadvantage in type I that the bin
size at high $L$ values is very large compared to type II, which gives weaker
lower bound on $\Delta m_{32}^{2}$. So, the combination of type I and II (type
I at the lower range of $L$ and type II for the rest) is a better choice than
the individual cases. However, this is not studied in this paper, but is
reflected when we compare two results. This is also demonstrated in terms of
$\Delta\chi^{2}$ for a typical set of parameters in fig. 3. It should be noted
here that the contrast between two methods would be prominent when the number
of bins in $L$ or $\cos\theta_{\rm zenith}$ will be relatively lowered than
that used in this paper.
For a quantitative assessment of the result, we define the precision $P$ of a
parameter $t$ as:
$P=2\left(\frac{t^{\rm max}-t^{\rm min}}{t^{\rm max}+t^{\rm min}}\right)$ (11)
We see, one can achieve the precision of $\Delta m_{32}^{2}\approx$
$4.8-7.5\%~{}(5.4-8.0\%),~{}6.9-10.9\%~{}(8.0-12.6\%)~{}{\rm
and}~{}9.6-15.7\%~{}(10.9-17.6\%)$, at 68%, 90% and 99% CL, respectively in
type I (II) method. For the input with bi-maximal mixing of $\theta_{23}$, we
find its precision in terms of $\sin^{2}\theta_{23}\approx$
$14.3-31.8\%~{}(16.9-36.9\%),~{}21.6-36.8\%~{}(22.4-41.7\%),{~{}\rm
and}~{}28.5-42.1\%~{}(27.9-45.9\%)$ at 68%, 90% and 99% CL, respectively in
type I (II) method. The precision of $\theta_{13}$ is strongly dependent on
its input value. For $\theta_{13}=0$, we find its upper bound $\approx
6.4^{\circ}~{}(8.0^{\circ}),~{}8.0^{\circ}~{}(9.5^{\circ})$ and
$10.1^{\circ}~{}(11.5^{\circ})$ at 68%, 90% and 99% CL, respectively in type I
(II) methods. The both lower and upper bounds are also possible for some
combinations of ($\theta_{23},\theta_{13}$) and it happens mainly for
$\theta_{23}\mathrel{\hbox to0.0pt{\raise 2.15277pt\hbox{$>$}\hss}{\lower
2.15277pt\hbox{$\sim$}}}45^{\circ}$.
## 4 Conclusion
In this paper we have binned the atmospheric data in three ways: i) in the
grids of $\log E-\log L$ plane, ii) in the grids of $\log E-\cos\theta_{\rm
zenith}$ plane, and iii) in the bins of $\log(L/E)$. We have performed a
marginalized $\chi^{2}$ study over $\Delta m_{32}^{2},~{}\theta_{13}$ and
$\theta_{23}$ for neutrinos and anti-neutrinos separately for each method.
Finally, we find that in spite of very poor resolutions at low $L$, which is
the main problem as $\Delta m_{32}^{2}$ goes to the upper range, one can
obtain a relatively stronger upper bound in case of binning in $\log E-\log L$
plane compared to the binning in $\log E-\cos\theta_{\rm zenith}$ plane.
However, it is also found from both analysis that considerable precisions of
$\theta_{13}$ and $\Delta m_{32}^{2}$ can be achieved and the octant
discrimination can also be possible for some combinations of
($\theta_{23},\theta_{13}$).
Figure 4: The 68% CL allowed regions in $\theta_{13}-\Delta m_{32}^{2}$ plane
for type I (top) type II (middle) and type III (bottom) binning of the data
with the input of
$\theta_{23}=40^{\circ},42^{\circ},45^{\circ},48^{\circ},50^{\circ}$ with
$\theta_{13}=0^{\circ}$ (first column), $5^{\circ}$ (second column),
$7.5^{\circ}$ (third column) from neutrinos with NH and $7.5^{\circ}$ (fourth
column) from anti-neutrinos with IH.
Figure 5: The same plots of fig. 4 but with 90% CL.
Figure 6: The same plots of fig. 4 but with 99% CL.
Figure 7: The 68% CL allowed regions in $\theta_{23}-\Delta m_{32}^{2}$ plane
for type I (top) type II (middle) and type III (bottom) binning of the data
with the input of
$\theta_{23}=40^{\circ},42^{\circ},45^{\circ},48^{\circ},50^{\circ}$ with
$\theta_{13}=0^{\circ}$ (first column), $5^{\circ}$ (second column),
$7.5^{\circ}$ (third column) from neutrinos with NH and $7.5^{\circ}$ (fourth
column) from anti-neutrinos with IH.
Figure 8: The same plots of fig. 7 but with 90% CL.
Figure 9: The same plots of fig. 7 but with 99% CL.
Acknowledgments: I would like to acknowledge the excellent cluster
computational facility of HRI, which makes the work possible. The research has
been supported by the funds from Neutrino Physics project at HRI.
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|
arxiv-papers
| 2008-12-30T10:17:04 |
2024-09-04T02:48:59.581518
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Abhijit Samanta",
"submitter": "Abhijit Samanta",
"url": "https://arxiv.org/abs/0812.4640"
}
|
0812.4717
|
# Thermalization dynamics close to a quantum phase transition
Dario Patanè MATIS CNR-INFM $\&$ Dipartimento di Metodologie Fisiche e
Chimiche (DMFCI), Università di Catania, viale A. Doria 6, $I-95125$ Catania,
Italy Departamento de Física de Materiales, Universitad Complutense de
Madrid, $E-28040$ Madrid, Spain Alessandro Silva The Abdus Salam
International Centre for Theoretical Physics, Strada Costiera $11$, $I-34100$
Trieste, Italy Fernando Sols Departamento de Física de Materiales,
Universitad Complutense de Madrid, $E-28040$ Madrid, Spain Luigi Amico MATIS
CNR-INFM $\&$ Dipartimento di Metodologie Fisiche e Chimiche (DMFCI),
Università di Catania, viale A. Doria 6, $I-95125$ Catania, Italy
Departamento de Física de Materiales, Universitad Complutense de Madrid,
$E-28040$ Madrid, Spain
###### Abstract
We investigate the dissipative dynamics of a quantum critical system in
contact with a thermal bath. In analogy with the standard protocol employed to
analyze aging, we study the response of a system to a sudden change of the
bath temperature. The specific example of the XY model in a transverse
magnetic field whose spins are locally coupled to a set of bosonic baths is
considered. The peculiar nature of the dynamics is encoded in the correlations
developing out of equilibrium. By means of a kinetic equation we analyze the
spin-spin correlations and block correlations. We identify some universal
features in the out-of-equilibrium dynamics. Two distinct regimes,
characterized by different time and length scales, emerge. During the initial
transient the dynamics is characterized by the same critical exponents as
those of the equilibrium quantum phase transition and resembles the dynamics
of thermal phase transitions. At long times equilibrium is reached through the
propagation along the chain of a thermal front in a manner similar to the
classical Glauber dynamics.
_Introduction._ Understanding the out-of-equilibrium dynamics of quantum many-
body systems is a central issue of modern condensed matter physics from both a
fundamental and an applicative point of view. Theoretical interest on these
problems traces back to the studies of irreversibility in non equilibrium
thermodynamics. In quantum systems the interplay between phase coherence,
strong interactions, and low dimensionality may result in surprising dynamical
behaviors NONthermalization . Remarkably, this kind of issues can be explored
experimentally at the quantum level by realizing highly controllable quantum
many-particle systems. In this sense, cold atoms in optical lattices are the
paradigmatic example of an interacting system where the interaction strength
and the geometrical settings can be fine tuned. The engineered Hamiltonians
can mimic condensed matter systems Lewenstein07 and also provide feasible
tools to investigate many interesting issues in non-equilibrium statistical
mechanics. coldatoms . Similarly, arrays of coupled microcavities have been
shown to have the potential to act as simulators of quantum many-body
dynamics, with characteristics complementary to those of optical lattices
Hartmann06 .
In this context it is desirable to consider simple but illustrative enough
situations. An important problem that has been studied is the response of a
system to a _strong_ perturbation driving it out of equilibrium.
Paradigmatically, this occurs when a given system is forced out of equilibrium
by a sudden change of a control parameter (quantum quench) quantum_quenches .
The richness and complexity of this problem is ultimately related to the
nonlocal character of the correlations developing the system temporal
evolution. An elegant picture of the phenomenon has been provided by Calabrese
and Cardy Calabrese07 , according to which correlations among spatially
distant parts of the system are established because, from each point, pairs of
entangled quasiparticles are emitted and propagate ballistically in opposite
directions with a characteristic velocity. This picture, which allows to
describe the dynamics of correlation functions and block entropy, has been
tested in different contexts (see Calabrese08 and references therein).
The above results concern the idealized situation of the unitary dynamics of a
closed system. A central problem then, is to understand the influence of an
external environment on the dynamics. This issue, originally explored in the
context of mean-field spin glasses Cugliandolo98 ; Kennett01 , has very
recently obtained some attention in the context of the adiabatic dynamics of
open quantum critical systems Fubini07 ; Mostame07 ; Amin08 ; PatanePRL ;
Cincio08 .
In this Letter we study the problem of a _thermal quench_ , i.e. the response
of the system to a sudden change of the temperature keeping the Hamiltonian
parameters fixed. This is the purely dissipative analog of a quantum quench
thermalquench . When the temperature of the bath in contact with the system is
changed, the system is forced to relax towards the new thermal equilibrium
state. We will be focusing on such a thermalization process close to a quantum
phase transition. Some key questions we wish to address are: How is
thermalization achieved microscopically? What are the characteristic time and
length scales emerging from the out-of equilibrium dynamics? And, ultimately:
What are the dynamical signatures of the quantum phase transition?
We address the questions above for a benchmark class of systems with a quantum
critical point, namely, the quantum XY model. To model a thermal reservoir the
system is coupled to a bosonic bath. As in the unitary case (conservative
system), we will focus on the study of spin-spin correlation functions. A
detailed study of the dynamics of the system will allow us to understand how
not only quasi-equilibrium but also far-from-equilibrium dynamics reveals
signatures of criticality.
_Model._ We consider a chain of $N$ spins with an anisotropic Ising
interaction and subject to a transverse magnetic field:
$H_{S}=-\frac{\mathcal{J}}{2}\\!\sum_{j}^{N}\\!\left(\frac{1+\gamma}{2}\sigma_{j}^{x}\sigma_{j+1}^{x}+\frac{1-\gamma}{2}\sigma_{j}^{y}\sigma_{j+1}^{y}\\!+\\!h\sigma_{j}^{z}\right)\,$
(1)
where $\sigma^{x,y,z}$ are Pauli matrices. We fix the energy scale
$\mathcal{J}=1$ and consider $h>0$. In the anisotropic case $0<\gamma\leq 1$,
the magnetic field $h$ induces a phase transition at $h_{c}=1$ that separates
a paramagnetic phase at $h>1$ from a ferromagnetic ordered phase with
$\left\langle\sigma^{x}\right\rangle\neq 0$; such a phase transition belongs
to the Ising universality class with critical indexes $\nu=z=1$. The
Hamiltonian can be diagonalized in momentum space, in terms of Jordan-Wigner
fermions $c_{k}$, as
$\sum_{k>0}\Psi_{k}^{\dagger}\mathcal{\hat{H}}_{k}\Psi_{k}$, where
$\Psi_{k}^{\dagger}=\left(c_{k}^{\dagger},\ c_{-k}\right)$,
$\mathcal{\hat{H}}_{k}=-(\cos k+h)\hat{\tau}^{z}+\sin k\ \hat{\tau}^{y}$,
where $\hat{\tau}^{\alpha}$ are Pauli matrices. The $T=0$ quantum phase
transition leaves an imprint at low temperatures, leading, close to the
quantum critical point, to a crossover at temperatures $T\sim\Delta$ with
$\Delta\equiv|h-h_{c}|$ the energy gap. In particular for $T\ll\Delta$ the
spin-spin correlation function is factorized into a quantum and a thermal term
that can be described semiclassically in terms of quasiparticle excitations,
while in the quantum critical region ($T\gg\Delta$) quasiparticle excitations
no longer exist SachdevBOOK .
To model a thermal reservoir we consider a set of bosonic baths coupled
locally to each spin PatanePRL , such that the global Hamiltonian reads
$H=H_{S}+\sum_{j}^{N}\\!X_{j}\sigma_{j}^{z}+H_{B}\;.$ (2)
where $X_{j}=\sum_{\beta}\lambda_{\beta}(b_{\beta,j}^{\dagger}+b_{\beta,j})$
and $H_{B}=\\!\sum_{j,\beta}\\!\omega_{\beta}b_{\beta j}^{\dagger}b_{\beta
j}$. The system-bath coupling is chosen to have power law spectral densities
$\sum_{\beta}\lambda_{\beta}^{2}\delta(\omega-\omega_{\beta})=2\alpha\omega^{s}\exp(-\omega/\omega_{c})$
WeissBOOK . The system-bath coupling we are considering, Eq. (2), breaks the
integrability of the model, inducing transitions between all energy levels,
and thus complete relaxation.
The quantum quench dynamics for the closed XY model was studied in quenchXY .
It is customary to consider the physical system initially uncorrelated, e.g.
by applying a strong magnetic field $h$. After a quench of the magnetic field,
correlations between parts of the system will start to develop because of the
dynamics induced by the new Hamiltonian. Analogously, in the case of thermal
quenches, we consider the system to be initially prepared in equilibrium with
the bath at a very high temperature, again with no correlations because the
density matrix of the system is proportional to the identity. After a quench
of the temperature of the bath at $t=0$, the system is forced out of
equilibrium and eventually reaches a new stationary thermal state. In the
following we investigate how such correlations develop and how thermal
equilibrium is eventually approached. All the results shown in the figures
below refer to the Ising model ($\gamma=1$) coupled to Ohmic baths ($s=1$).
However, the results stated in the text refer to the general case
$0<\gamma\leq 1$, $s>0$. We discuss the spin-spin correlation function and
later we consider the quantum mutual information.
$\log t/\alpha$$\log C_{zz}$$R$
Figure 1: Correlation function $C_{zz}$ for a quench from $T=\infty$ to
$T=0.1$ at $h=1$; red thick line on the right at large $t$ is the thermal
equilibrium $C_{zz}(R)$. During the initial transient ($t\ll\alpha$),
$C_{zz}\propto t^{2}$ for all $R$, as marked by the fits on the left.
_Spin-spin correlations._ We consider the equal-time connected correlation
function
$C_{zz}(t,\
R)=\left\langle\sigma_{j}^{z}(t)\sigma_{j+R}^{z}(t)\right\rangle-\left\langle\sigma_{j}^{z}(t)\right\rangle\left\langle\sigma_{j+R}^{z}(t)\right\rangle\,.$
(3)
In the case of thermal quenches the dynamics is purely dissipative. Since for
weak coupling $\alpha\ll 1$ the dynamics of populations and coherences
decouple, if the system starts in a mixed state no coherences will develop
after the quench (this is consistent with the so called “secular
approximation” CohenBOOK ). Therefore in this limit at each time the system is
approximately in a statistical mixture of the Hamiltonian eigenvectors, i.e. a
gaussian state. Hence, by exploiting this the correlation function can be
expressed as $C_{zz}=\left|\frac{4}{N}\sum_{k>0}\langle
c_{k}c_{-k}\rangle\sin(kR)\right|^{2}-\left(\frac{4}{N}\sum_{k>0}\cos(kR)\langle
c_{k}^{\dagger}c_{k}\rangle\right)^{2}$. In order to evaluate the two point
fermionic correlators we use the kinetic equation derived in PatanePRL ;
PatanePRL_long within the weak coupling and Markov approximations. From the
analysis of our results (Fig. 1) two regimes can be outlined: right after the
quench, for $t\ll\alpha$, correlations increase as $C_{zz}\propto t^{2}$,
while in the opposite limit, at times $t\gg\alpha$, the system is close to
thermal equilibrium. During the initial transient, $C_{zz}$ reaches for far
distant spins ($R\gg 1$) values greater than those of thermal equilibrium,
$C_{zz}(R)>C_{zz}^{\mathrm{th}}(R)\propto\exp(-R/\xi)$, with $\xi$ the thermal
correlation length SachdevBOOK . Thus, the crossover to the long-time regime
displays a non-monotonous behavior as a function of time, so that that
$C_{zz}$ increases up to a maximum value and then relaxes to the thermal
equilibrium value (Fig. 1).
$|C_{zz}|$$\partial_{L}\mathcal{I}$$\partial_{L}\mathcal{I}$$C_{zz}$$R$$L$
Figure 2: Initial transient: snapshots of $C_{zz}$ (left) and
$\partial_{L}\mathcal{I}$ (right) at a fixed $t/\alpha\ll 1$ after a quench
from $T=\infty$ to $T=0.1$ and for $h=0.8,\ 0.9,\ 0.95,\ 0.975,\ 1$ (from
bottom to top). The spikes relative to $C_{zz}$ in the left panel mark the
distances $\xi_{t}(h)$ at which $C_{zz}$ changes sign. Dashed lines are
plotted for comparison. _Inset:_ $\xi_{t}$ as a function of $|h-h_{c}|$; for
$\partial_{L}\mathcal{I}$, $\xi_{t}$ is calculated as the maximum of
$\partial_{L}^{2}\mathcal{I}$ which marks the crossover between $L^{-1}$ and
$L^{-4}$ scaling.
Let us analyze first the initial transient. We observe that, for noncritical
values of the magnetic field ($h\neq 1$), $C_{zz}$ changes its sign at a
certain distance $\xi_{t}(h)$ such that $C_{zz}\lessgtr 0$ for
$R\gtrless\xi_{t}$ (see Fig. 2). That distance marks the crossover between two
power-law behaviors with different exponent, respectively $R^{-4}$ and
$R^{-2}$, and close to the critical point it diverges as
$\xi_{t}\propto|h-h_{c}|^{-1}.$ (4)
Collecting all the above results, we find that the long $R$ behavior close to
the critical point is described by
$C_{zz}\propto t^{2}\begin{cases}R^{-2}&\ R\ll\xi_{t}\\\ R^{-4}&\
R\gg\xi_{t}\end{cases}\ \ \ $ (5)
At equilibrium, close to the phase transition, the correlation length $\xi$
marks the crossover between the exponential decay for $R\gg\xi$ and the
critical power-law for $R\ll\xi$. Similarly, in the present non-equilibrium
case $\xi_{t}$ can be interpreted as an effective crossover scale between two
power-law regimes. Eqs. (4) and (5) are independent of the specific value of
the final temperature at which the system is quenched and of the specific
exponent $s$ of the bath spectral function. Moreover, they are robust within
the range $0<\gamma\leq 1$ in which the system belongs to the Ising
universality class. Remarkably, although in this regime the system is far from
equilibrium, Eqs. (4) and (5) are characterized by the equilibrium critical
indexes: $\xi\propto|h-h_{c}|^{-1}$ and $C_{zz}\propto R^{-2},\ R\ll\xi$.
$C_{zz}$$R$$\downarrow$$R_{{\rm th}}$$t/\alpha$
Figure 3: Thermalization of the spin-spin correlation function $C_{zz}$ after
a quench from $T=\infty$ to $T=0.1$ at $h=1$. Left: snapshots of $C_{zz}(R)$
at $t/\alpha=10,\ 20,\ 30,\ 40$ from top to bottom; thick red line is the
(exponential) thermal equilibrium $C_{zz}^{\mathrm{th}}$. At a given time
after the quench $C_{zz}$ is thermalized up to a distance $R_{\mathrm{th}}$
that increases with time. Right: corresponding time dependence of
$R_{\mathrm{th}}$; the linear fit gives $v_{\mathrm{th}}\simeq 0.32$.
We now focus on the long time regime. The analysis of Fig. 3 indicates that,
at a given time, the correlation function is thermalized up to a distance
$R_{\mathrm{th}}$. This _thermal front_ exists because the long-distance
correlations are dominated by the slowly relaxing low-energy modes. The front
is found to propagate ballistically with a speed $v_{\mathrm{th}}$ that is a
function of $T$ and $h$. In particular, as shown in Fig. 4, the velocity
scales as
$v_{\mathrm{th}}\propto\begin{cases}T^{s}&T\gg\Delta\\\
e^{-\Delta/T}&T\ll\Delta\end{cases}$ (6)
where $\Delta=|h-1|$ is the energy gap of the XY model.
$v_{{\rm th}}$$1/T$$C_{zz}$$\partial_{L}\mathcal{I}$
Figure 4: Thermal front velocity $v_{\mathrm{th}}$ as a function of $1/T$.
From top to bottom $h=1,\ 0.9,\ 1.2$ (so that $\Delta=0,\ 0.1,\ 0.2$). Lines
are the fit $\ Ta(1+b\frac{\Delta}{T})\ e^{-\Delta/T}$ with $a=3.3,\ b=0.9$
for the specific case $\gamma=1,\ s=1$.
_Block correlations._ We now analyse how correlations between a block of spins
and the rest of the chain develop after a thermal quench. In order to quantify
such a correlation, we use a tool originally developed in the context of
quantum information theory. For a certain bipartition of the system into two
blocks of $L$ and $N-L$ spins, the mutual information is defined as
$\mathcal{I}(L)=S(\rho_{L})+S(\rho_{N-L})-S(\rho_{N})\;,$ (7)
where $S(\rho)=-\mathrm{Tr}\left(\rho\log\rho\right)$ and $\rho_{N}$ is the
density matrix of the entire system. The mutual information measures the
correlations between the two blocks of $L$ and $N-L$ spins Wolf08 . For the XY
model $\mathcal{I}(L)$ is known to diverge logarithmically as a function of
$L$ at the critical point, while it saturates for noncritical values. In the
following we concentrate on the derivative $\partial_{L}\mathcal{I}$, which
measures the sensitivity of the correlations to the block size. It is useful
to study $\partial_{L}\mathcal{I}$ because, at equilibrium, it shows features
similar to the spin-spin correlation function: it scales as
$\partial_{L}\mathcal{I}\propto L^{-1}$ at the critical point, while it decays
exponentially for noncritical values.
Equation (7) can be computed in terms of two-point fermionic correlators
(obtained by solving the kinetic equation) using the results of Ref. Peschel03
. We study for the mutual information the same setting of thermal quenches
investigated above for the spin-spin correlation function. Remarkably the
scenario emerging is very similar to that depicted in Fig. 1. There are two
regimes: an initial transient governed by
$\partial_{L}\mathcal{I}\propto t^{2}\begin{cases}L^{-1}&\ L\ll\xi_{t}\\\
L^{-4}&\ L\gg\xi_{t}\end{cases}\ \ \ $ (8)
with the same correlation length (4) (see Fig. 2). In the quasi-equilibrium
regime at long times, $\partial_{L}\mathcal{I}$ exhibits a thermal front
propagation similar to that shown in Fig. 3 and with the same velocity found
for $C_{zz}$ (see Fig. 4).
_Conclusions._ We have analyzed the dynamics of spin-spin and block
correlation functions following a sudden cooling of the bath coupled to a
quantum system. For both quantities we find that the dynamics displays two
regimes: at short times the correlations develop according to (5) and (8),
while at long times a well defined thermal front propagates along the chain
with velocity (6), the latter being sensitive to the critical properties of
the system. We remark that the system does not exhibit aging because it is
quenched away from the critical point. Nevertheless in its early stages
relaxation does show critical features analogously to those of thermal phase
transitions. In particular, for systems quenched at the critical temperature,
it is known that equal-time two-point correlation function scales, during the
initial transient, as a power law both in time $\propto t^{a}$ (here $a=2$)
and in space $R^{-d+2-\eta}$ (in our case $R^{-2}$) thermalquench . Besides,
we point out that the scaling of the thermalization velocity
$v_{\mathrm{th}}\propto\exp(-\Delta/T)$, which we find in the semiclassical
regions $T\ll\Delta$, holds also in the classical Ising model within the
Glauber dynamics Glauber . This similarity can be ascribed to the fact that
the system-bath coupling generates incoherent relaxation without conserving
the order parameter, as happens in the phenomenological Glauber model for the
dynamics of the classical spins.
_Acknowledgments._ We thank R. Fazio, G.E. Santoro and P. Calabrese for
discussions and comments. D.P. acknowledges the ESF (INSTANS) for financial
support. L.A. and F.S. acknowledge support from MEC (FIS2007-65723).
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* (15) Such a problem has been extensively studied for classical systems, especially in connection with aging. See e.g P. Calabrese and A. Gambassi, J. Phys. A 38 (2005) R133; L. Cugliandolo, in Slow relaxations and non-equilibrium dynamics in condensed matter, Les Houches, Session LXXVII, J. L. Barrat, M. Feigelman, J. Kurchan, J. Dalibard Edts, Springer-EDP Sciences (2003).
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|
arxiv-papers
| 2008-12-26T22:19:41 |
2024-09-04T02:48:59.589434
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dario Patan\\`e, Alessandro Silva, Fernando Sols, Luigi Amico",
"submitter": "Dario Patan\\`e",
"url": "https://arxiv.org/abs/0812.4717"
}
|
0812.4794
|
# Propagation of extremely short electromagnetic pulses in a doubly-resonant
medium
Y. Frenkela, I. Gabitovb, A. Maimistovc, and V. Roytburda
aDepartment of Mathematical Sciences, Rensselaer Polytechnic
Institute, Troy, NY 12180-3590; bDepartment of Mathematics,
The University of Arizona, Tucson, AZ 85721-0089;
cDepartment of Solid State Physics, Moscow Engineering
Physics Institute, Kashirskoe sh. 31, Moscow, 115409 Russia
###### Abstract
Propagation of extremely short electromagnetic pulses in a homogeneous doubly-
resonant medium is considered in the framework of the total Maxwell-Duffing-
Lorentz model, where the Duffing oscillators (anharmonic oscillators with
cubic nonlinearities) represent the dielectric response of the medium, and the
Lorentz harmonic oscillators represent the magnetic response. The wave
propagation is governed by the one-dimensional Maxwell equations.
It is shown that the model possesses a one-parameter family of traveling-wave
solutions with the structure of single or multiple humps. Solutions are
parametrized by the velocity of propagation. The spectrum of possible
velocities is shown to be continuous on a small interval at the lower end of
the spectrum; elsewhere the velocities form a discrete set. A correlation
between the number of humps and the velocity is established. The traveling-
wave solutions are found to be stable with respect to weak perturbations.
Numerical simulations demonstrate that the traveling-wave pulses collide in a
nearly elastic fashion.
## 1 Introduction
The recent demonstration of artificial materials (metamaterials) with the left
oriented triplet of electric $\vec{E}$, magnetic $\vec{H}$ and wave vector
$\vec{k}$ of electromagnetic field SSS01 , Valent08 , SCCYSDK05 , ZPMOB05
stimulated studies of nonlinear optical phenomena in such materials ASBZ04 ,
ZSK03 , LT05 , AMSB05 , SZK05 , PS06 , MG07 . Nonlinear dynamics of extremely
short optical pulses in left-handed materials was the subject of particular
interest in several recent papers LT05 , SAMABCSB05 , BVKR05 , GILMSS06 . The
first experimental realization of the left-handed property based on the
resonant response of the artificial material to both electric and magnetic
fields was described in SSS01 . To mention just one of the latest experimental
achievements, Valentine et al Valent08 were able to observe the negative
refractive index in the balk material in the optical range. A theoretical
description of the electromagnetic wave interaction with such double resonance
materials (DRM) was considered in ZH01 , PHRS98 , PHRS99 , PSS02 , MS02 .
Presence of two frequency intervals with different orientation of
$(\vec{E},\vec{H},\vec{k})$ triplets is a characteristic feature of such
materials.
Most of the studies of electromagnetic pulse propagation in DRM has been
conducted in the slowly varying envelope approximation. On the other hand,
there is a broad area of nonlinear optical phenomena taking place in the limit
of extremely short pulses, when the slowly varying envelope approximation is
not valid BK00 . The case of extremely short electromagnetic pulses offers a
new type of nonlinear interaction, when different frequency components of
electromagnetic pulses have different orientations of the
$(\vec{E},\vec{H},\vec{k})$ triplets.
The design of currently available DRM is based upon the use of embedded
metallic structures whose size is on the same order as the spatial size of an
extremely short electromagnetic pulse. Therefore a theoretical and numerical
investigation of the currently existing DRM would require 3D computer
simulation on Maxwell’s equations that takes into account the strong
inhomogeneity of composite materials. Recently, there have been introduced
some qualitatively different approaches to design of DRM, including the use of
multilevel atoms TM06 , Krowne08a , Krowne08b ; the latter gives rise to a
spatially homogeneous medium. Possibilities of experimental realizations of
such an approach were recently discussed in Yelin07 , Yelin08 . As a first
step in the theoretical investigation of electrodynamics of homogeneous DRM in
this paper we study a simple model of a homogeneous doubly-resonant medium.
Even under such simplification, dynamics of extremely short pulses turn out to
be quite complex.
## 2 Basic equations
The system of equations that describe interaction of coherent light with a
medium consisting of molecules (considered as harmonic oscillators) is known
as the Maxwell-Lorentz model AE75 . In this work we use a version of the
Maxwell-Lorentz system that is extended to account for simultaneous magnetic
and electric resonances, with the magnetic susceptibility being linear, while
the electric polarization being nonlinear. Consider the general form Maxwell’s
equations:
$\displaystyle~{}\nabla\times\vec{E}=-c^{-1}\vec{B}_{t},~{}~{}\;\nabla\times\vec{H}=-c^{-1}\vec{D}_{t}$
(2.1)
$\displaystyle~{}\vec{B}=\vec{H}+4\pi\vec{M},~{}~{}\;\vec{D}=\vec{E}+4\pi\vec{P}$
For simplicity, we consider transverse electromagnetic plane waves propagating
along the $z$-axis with the electric field $\vec{E}=(E(z,t),0,0)$ and the
magnetic field $\vec{B}=(0,B(z,t),0).$ Then the Maxwell equations transform to
the scalar form:
$\displaystyle~{}\frac{\partial E}{\partial z}+\frac{1}{c}\frac{\partial
B}{\partial t}=0,~{}~{}\frac{\partial H}{\partial z}+\frac{1}{c}\frac{\partial
D}{\partial t}=0$ (2.2) $\displaystyle~{}B=H+4\pi M,~{}~{}~{}\;D=E+4\pi P$
(2.3)
which leads to
$E_{z}+c^{-1}H_{t}=-4\pi c^{-1}M_{t},~{}~{}\;H_{z}+c^{-1}E_{t}=-4\pi
c^{-1}P_{t}$ (2.4)
The system (2.4) must be closed by two additional equations describing the
interaction of the electric and magnetic fields with the DR medium. As usual,
it is convenient to avoid the $4\pi$-factors by changing the units for $M$ and
$P$: $\tilde{M}=4\pi M,$ $\tilde{P}=4\pi P.$ In the sequel we drop the tildes
over $M$ and $P.$
Assume that the medium polarization is defined by the plasma oscillation of
electron density,
$P_{tt}=\omega_{p}^{2}E$
Here $\omega_{p}$ is an effective parameter characterizing polarizability of
the medium; in the case of metallic nanostructures it would be the effective
plasma frequency. To account for the dimensional quantization due to the
confinement of the plasma in nanostructures one should include the additional
term $\omega_{D}^{2}P$, where $\omega_{D}$ is the frequency of dimensional
quantization. We take into account nonlinearity in the lowest order of $P$,
which is $P^{3}$. A more accurate analysis, based on a quantum mechanical
approach R97 and experimental measurements DBNS04 confirms validity of this
assumption. Therefore we consider the modeling equation for the medium
polarization dynamics in the following form
$P_{tt}+\omega_{D}^{2}P+\kappa P^{3}=\omega_{p}^{2}E$ (2.5)
where $\kappa$ is a constant of anharmonisity. To account for magnetic
resonances we use the standard model ZH01
$M_{tt}+\omega_{T}^{2}M=-\beta H_{tt}$ (2.6)
Here $\beta$ is a constant characterizing magnetization.
We represent equations (2.4), (2.5) and (2.6) in a dimensionless form by
introducing $\tau=t/\tau_{0}$ ($\tau_{0}=1/\omega_{p}$ is the characteristic
time), $\eta=z/z_{0}$ ($z_{0}=c\tau_{0}$ is the characteristic distance),
$q=P/P_{0}$ ($P_{0}=\omega_{p}/\sqrt{\kappa}$ is the maximal achievable medium
polarization). It is convenient to normalize remaining variables as follows:
$m=M/P_{0}$, $e=E/P_{0}$, $h=H/P_{0}$. The system of dimensionless equations
then takes the following form:
$\displaystyle~{}h_{\tau}+e_{\eta}=-m_{\tau},$
$\displaystyle~{}e_{\tau}+h_{\eta}=-q_{\tau},$
$\displaystyle~{}q_{\tau\tau}+\omega_{1}^{2}q+\gamma q^{3}=e$ (2.7)
$\displaystyle~{}m_{\tau\tau}+\omega_{2}^{2}m=-\beta h_{\tau\tau},$
where $\gamma=\kappa/\left(\left|\kappa\right|\omega_{p}^{2}\right)$,
$\omega_{1}=\omega_{D}/\omega_{p}$, $\omega_{2}=\omega_{T}/\omega_{p}$.
The system possesses the following conserved quantity:
$\displaystyle\frac{1}{2}\frac{\partial}{\partial\tau}\int\left[\beta\omega_{2}^{2}\left(e^{2}+\omega_{1}^{2}q^{2}+\frac{\gamma}{2}q^{4}\right)+\beta\omega_{2}^{2}\left(h+m\right)^{2}+\omega_{2}^{2}\left(1-\beta\right)m^{2}\right.$
(2.8)
$\displaystyle\left.+\beta\omega_{2}^{2}\left(q_{\tau}\right)^{2}+\left[m_{\tau}+\beta
h_{\tau}\right]^{2}\right]d\eta=0$
which is positive-definite for $\beta<1.$ For the traveling-wave solutions the
conservation relation (2.8) yields conservation of electromagnetic energy
$\frac{1}{2}\int\left(e^{2}+h^{2}\right)d\eta=\mathrm{const}$
(see frenkel for details). A natural question arises is whether the system in
(2.7) possesses any solitary-wave solutions. We address this issue in the
following section.
## 3 Solitary wave solutions
Consider a traveling wave solution of (2.7), i.e., a solution that is a
function of the variable $\zeta=\tau-\eta/V.$ Then the PDEs in (2.7) become
ODEs, and one obtains the following system:
$\displaystyle h^{\prime}-e^{\prime}/V$ $\displaystyle=-m^{\prime}$ (3.9)
$\displaystyle e^{\prime}-h^{\prime}/V$ $\displaystyle=-q^{\prime}$ (3.10)
$\displaystyle q^{\prime\prime}+\omega_{1}^{2}q+\gamma q^{3}$
$\displaystyle=e$ (3.11) $\displaystyle m^{\prime\prime}+\omega_{2}^{2}m$
$\displaystyle=-\beta h^{\prime\prime}$ (3.12)
Upon the integration of equations (3.9) and (3.10) once, we get the algebraic
conservation relations
$\displaystyle Vh-e$ $\displaystyle=-mV+R$ $\displaystyle-h+eV$
$\displaystyle=-qV+S$
We are interested in a traveling-wave solution on the zero background, hence
$h=m=q=e=0$ at $\pm\infty;$ therefore the constants of integration $R=S=0.$
This yields the following expressions for $h$ and $e$
$\displaystyle h$ $\displaystyle=a_{1}m+a_{2}q$ (3.13) $\displaystyle e$
$\displaystyle=a_{2}m+a_{1}q$ (3.14)
where
$\displaystyle a_{1}=V^{2}\left(1-V^{2}\right)^{-1},\quad
a_{2}=V\left(1-V^{2}\right)^{-1}$ (3.15)
We insert expressions (3.13) and (3.14) for $h$ and $e$ into the equations
(3.11) and (3.12) for $q$ and $m$ and obtain the following system of second
order equations:
$\displaystyle
q^{\prime\prime}+\left(\omega_{1}^{2}-a_{1}\right)q-a_{2}m+\gamma q^{3}$
$\displaystyle=0$ $\displaystyle\beta a_{2}q^{\prime\prime}+\left(1+\beta
a_{1}\right)m^{\prime\prime}+\omega_{2}^{2}m$ $\displaystyle=0$
This system can be diagonalized with respect to the second derivatives
$\displaystyle Q^{\prime\prime}+A_{11}Q+A_{12}M+\gamma Q^{3}$
$\displaystyle=0$ (3.16) $\displaystyle M^{\prime\prime}+A_{21}Q+A_{22}M$
$\displaystyle=0$
by the means of the transformation
$\left[\begin{array}[c]{l}q\\\
m\end{array}\right]=\left[\begin{array}[c]{ll}1&0\\\ \dfrac{-\beta
a_{2}}{1+\beta a_{1}}&\dfrac{\omega_{2}\sqrt{\beta}}{1+\beta
a_{1}}\end{array}\right]\left[\begin{array}[c]{l}Q\\\ M\end{array}\right]$
The matrix $A$ in (3.16) is symmetric $A_{12}=A_{21}:$
$A=\left[\begin{array}[c]{ll}\omega_{1}^{2}-a_{1}+\dfrac{\beta
a_{2}^{2}}{1+\beta a_{1}}&-\dfrac{a_{2}\omega_{2}\sqrt{\beta}}{1+\beta
a_{1}}\\\ -\dfrac{a_{2}\omega_{2}\sqrt{\beta}}{1+\beta
a_{1}}&\dfrac{\omega_{2}^{2}}{1+\beta a_{1}}\end{array}\right]$ (3.17)
Instead of the second order system (3.16) we will consider the following
equivalent $4\times 4$ first order system
$\frac{d}{d\zeta}\left[\begin{array}[c]{c}Q\\\ M\\\ Q_{1}\\\
M_{1}\end{array}\right]=\left[\begin{array}[c]{cccc}0&0&1&0\\\ 0&0&0&1\\\
-A_{11}&-A_{12}&0&0\\\
-A_{21}&-A_{22}&0&0\end{array}\right]\left[\begin{array}[c]{c}Q\\\ M\\\
Q_{1}\\\ M_{1}\end{array}\right]-\left[\begin{array}[c]{c}0\\\ 0\\\ \gamma
Q^{3}\\\ 0\end{array}\right]$ (3.18)
Obviously $[0,0,0,0]$ (the zero background) is the only equilibrium solution
(the critical point) of the system. The pulse solutions are the trajectories
of the system (3.18) that start and end at the equilibrium (homoclinic
orbits). Thus, the investigation of solitary pulses is mathematically
equivalent to studying homoclinic solutions.
Figure 1: The left figure shows the $E=0$ cross-section of the potential
energy landscape $U(Q,M)=0$. The Newtonian particle trajectory corresponds to
a one-hump solution presented in the right figure.
## 4 Structure of solitary waves
To investigate the structure of homoclinic solutions, we linearize the system
in (3.18) near the critical point $Q=M=Q_{1}=M_{1}=0$:
$\frac{d}{d\zeta}\left[\begin{array}[c]{c}Q\\\ M\\\ Q_{1}\\\
M_{1}\end{array}\right]=\left[\begin{array}[c]{cccc}0&0&1&0\\\ 0&0&0&1\\\
-A_{11}&-A_{12}&0&0\\\
-A_{21}&-A_{22}&0&0\end{array}\right]\left[\begin{array}[c]{c}Q\\\ M\\\
Q_{1}\\\ M_{1}\end{array}\right]:=\tilde{A}\left[\begin{array}[c]{c}Q\\\ M\\\
Q_{1}\\\ M_{1}\end{array}\right]$ (4.19)
The characteristic equation of the matrix $\tilde{A}$ on the right-hand side
is given by
$p^{4}+\left(A_{11}+A_{22}\right)p^{2}+A_{11}A_{22}-A_{21}A_{12}=0$
Therefore, the values of $p^{2}$ coincide with the eigenvalues of the matrix
$-A.$ It is easy to see that
$\det A=\dfrac{\left(\omega_{1}^{2}-a_{1}\right)\omega_{2}^{2}}{1+\beta
a_{1}}$ (4.20)
Thus, the condition
$\omega_{1}^{2}-a_{1}<0$ (4.21)
makes $\det A<0,$ causing $A$ to have eigenvalues of opposite signs, which is
a necessary condition for the existence of homoclinic orbits. Indeed, in the
case of $A$ having eigenvalues of the opposite signs, the $4\times 4$ matrix
$\tilde{A}$ has two pure imaginary eigenvalues (square roots of the negative
eigenvalue of $-A$) and one negative, and one positive eigenvalues. Therefore
the nonlinear system has one-dimensional stable and unstable manifolds, and a
two-dimensional center manifold (corresponding to the imaginary eigenvalues).
Figure 2: The Newtonian particle trajectory (left) corresponds to the two-hump
solitary wave (right).
It was first noticed in GILMSS06 that the nonlinear system in (3.18) has
Hamiltonian structure. If the kinetic, $E,$ and potential, $U,$ energies and
the Hamiltonian, $H,$ are introduce as follows
$\displaystyle E$ $\displaystyle=\frac{1}{2}\left(Q_{1}^{2}+M_{1}^{2}\right),$
(4.22) $\displaystyle U$
$\displaystyle=A_{12}QM+\frac{1}{2}\left(A_{11}Q^{2}+A_{22}M^{2}\right)+\frac{\gamma}{4}Q^{4},$
(4.23) $\displaystyle H$ $\displaystyle=E+U$ (4.24)
then the system (3.18) takes the form
$\displaystyle\partial_{\zeta}Q_{1}$ $\displaystyle=-\partial H/\partial
Q,\quad\partial_{\zeta}M_{1}=-\partial H/\partial M,$
$\displaystyle\partial_{\zeta}Q$ $\displaystyle=\partial H/\partial
Q_{1},\quad\partial_{\zeta}M=\partial H/\partial M_{1}.$
Since the Hamiltonian is a conserved quantity, $\partial_{\zeta}H=0,$ any
trajectory issued from the critical point $\left[0,0,0,0\right]$ stays on the
zero energy level surface $H=0$ for all time. Note that the surface $H=0$ is a
3D manifold in $\mathbb{R}^{4}.$ The intersection of this 3D hypersurface with
the hyperplanes $Q_{1}=0$ and $M_{1}=0$ is a curve $\Gamma$ in the $QM$-plane
$U(Q,M)=A_{12}QM+\frac{1}{2}\left(A_{11}Q^{2}+A_{22}M^{2}\right)+\frac{\gamma}{4}Q^{4}=0$
(4.25)
(the figure-eight shaped curve on the left in Fig. 1 and 2). If for a given
$V$ there exits a homoclinic trajectory of (3.18), then on this trajectory
$E+U=0$ and since $E\geq 0$, necessarily $U\leq 0$. At the extrema of $U$ its
gradient is zero:
$\frac{\partial U}{\partial M}=A_{22}M+A_{12}Q=0,\quad\frac{\partial
U}{\partial Q}=A_{12}M+A_{11}Q+\gamma Q^{3}=0$
By eliminating $M$ from the equations above, we obtain the cubic equation
$-\frac{A_{12}^{2}}{A_{22}}Q+A_{11}Q+\gamma Q^{3}=0$
whose roots are easily found:
$Q=0,\quad Q=\pm\sqrt{\frac{-\det A}{A_{22}}}=\pm\sqrt{a_{1}-\omega_{1}^{2}}.$
Thus, $\nabla U=0$ at the points
$\left(0,~{}0\right),~{}~{}\left(\pm\sqrt{a_{1}-\omega_{1}^{2}},~{}\pm\frac{a_{2}}{\omega_{2}}\sqrt{\beta\left(a_{1}-\omega_{1}^{2}\right)}\right)$
which are real if
$a_{1}=\frac{V^{2}}{1-V^{2}}>\omega_{1}^{2}$
thus producing the figure eight level curves. We already encountered this
inequality above, see (4.21). After some algebra it can be rewritten as the
following constraints on the traveling wave velocity:
$V_{0}<V<1,\quad V_{0}=\sqrt{\omega_{1}^{2}/(1+\omega_{1}^{2})}$ (4.26)
Thus, a possible velocity of the propagating pulse is bounded below.
## 5 Numerical study of solitary waves
The nonlinear system (3.18) has the time reversal symmetry; therefore, if
$[Q,M,Q_{1},$ $M_{1}](t):=\mathbf{u}(t)$ is a homoclinic orbit,
$[Q,M,-Q_{1},-M_{1}](-t)$ also is (recall that $Q_{1}$ and $M_{1}$ are time
derivatives of $Q$ and $M$).
(a) Number of solitons per bin: 100 bins is depicted.
(b) The hump distribution: number of humps vs. the velocity.
Figure 3: Statistics of solitary wave solutions.
A priori it is not clear why any homoclinic solution would possess this
symmetry, and it is quite likely that there exist non-symmetric homoclinic
orbits; we plan to investigate them elsewhere. The characteristic property of
a time reversal orbit is that at the symmetry point $Q_{1}=M_{1}=0,$ and
consequently the kinetic energy $E$ must be zero; i.e., the symmetry point
lies on the curve $\Gamma,$ see (4.25). Moreover at the symmetry point the
trajectory is orthogonal to $\Gamma$ (for an illustration, see the $QM$
diagrams on the left of Fig. 1 and Fig. 2).
Figure 4: Solitary waves examples $V=0.84322$ and $V=0.91461$. The left and
right figures illustrate four-hump and eight-hump solitary wave solutions.
Our search algorithm for finding solitary-wave solutions is based on the
following minimization idea. If for a given value of the propagation velocity
$V$ there exists a homoclinic orbit with the time-reversal symmetry, then at
some point both the kinetic and potential energies are zero. The algorithm
takes the initial condition $\mathbf{u}_{0}$ from a domain $S$ on the zero-
energy surface, near the critical point $(0,0,0,0)$ and in the direction close
to that of the unstable eigenvector of the linearized problem. Then the
following optimization problem is posed: Determine
$\Phi(V)=\min_{\mathbf{u}_{0}\in
S}\min_{\zeta_{0}<\zeta<\zeta_{0}+\tau_{0}}E[\mathbf{u}(\zeta|\mathbf{u}_{0})]$
(5.27)
Figure 5: The energy per hump vs. the velocity
where $E$ is the kinetic energy, $\mathbf{u}(\zeta|\mathbf{u}_{0})$ is the
solution of (3.18) with the initial condition
$\mathbf{u}(0|\mathbf{u}_{0})=\mathbf{u}_{0},$ recall that
$\mathbf{u}=[Q,M,Q_{1},M_{1}].$ The parameter $\tau_{0}$ is the expected
”width” of the pulse. Since $E=0$ at $\zeta=0$ we take $\zeta>\zeta_{0}$ to
obtain a nontrivial solution for the energy minimization problem. Computation
of any particular value of
$\min_{\zeta_{0}<\zeta<\zeta_{0}+\tau_{0}}E[\mathbf{u}(\zeta|\mathbf{u}_{0})]$
involves a numerical solution of the nonlinear system of ODEs.
Figure 6: Stable propagation of a eight-hump solitary wave; $V=0.822$.
The search of the optimal initial datum is stochastic and is organized via a
version of simulated annealing simannealing . On each step the initial datum
is obtained by sampling a random distribution with the density determined by
the results of the previous step (see frenkel , yfvr09 for more detail). If
$\Phi(V)=0$ then there exists a homoclinic solutions with velocity $V.$
When the kinetic energy possesses several local minima along the trajectory
the corresponding solitary wave has the multi-hump structure. Figures 1 and 2
illustrate this phenomenon. The figure-eight shaped curves on the left
correspond to the $E=0$ cross-section of the potential energy landscapes; the
curves inside the domains represent the Newtonian particle trajectories in the
$QM$ configuration space. The graphs on the right show the profiles of the
corresponding solitary wave solutions. Fig. 1 illustrates a typical one-hump
solution. In contrast, the trajectory shown in Fig. 2 has a point of the
nearest approach to the boundary where the kinetic energy attains a local
minimum. The resulting solution has a two-hump structure. Multi-hump solutions
correspond to more complicated trajectories. Each of these trajectories has
the return point at which it has the normal incidence with the $E=0$ contour.
Figure 7: The initial solitary wave pulse with a perturbation added (left);
Evolution of this pulse governed by the PDEs (right).
For the fixed set of physical parameter values, the shape of the potential
energy landscape is controlled by the pulse velocity $V$ via the coefficients
$a_{1}$ and $a_{2}$ in (3.15). We investigated numerically the set
$\mathfrak{V}$ of values of $V$ which give rise to homoclinic orbits; in some
sense one might think of these $V$s as the ”spectrum” of the problem. For
numerous applications with soliton-like solutions the velocity value is known
to change continually (a continuous spectrum). However, for the Maxwell-
Duffing model under consideration our numerical investigation demonstrates
that the spectrum $\mathfrak{V}$ contains both an interval of a continuous
spectrum and a discrete subset of parameter values $V$ for which a wave
solution exists. One of the principal issues is to understand the
correspondence between types of solitary wave solutions and values of
$V\in\mathfrak{V}$.
We first investigated numerically the distribution density of the values of
$V$, which give rise to homoclinic orbits. For all numerical computations of
this section we adopted the following values of the nondimensional physical
parameters:
$\omega_{1}=1,\;\omega_{2}=5,\;\gamma=0.01,\;\beta=0.5$ (5.28)
For $\omega_{1}=1$ the allowable range of values of $V$ from (4.26) is given
by $1/\sqrt{2}<V<1.$ The plot in Fig. 3(a) illustrates the density
distribution of $V\in\mathfrak{V}$ on the interval $[0.73,0.95]$. The search
algorithm tested potential values of $V$ on the grid $\delta V=10^{-4}.$ The
plot depicts the number of “successful” homoclinic orbits per velocity
interval $\Delta V$ (a “bin”); in this particular case the value has been
chosen as $\Delta V=0.002$.
Figure 8: Solitary wave collisions: two-hump solitons with $V=0.9$ and
$V=-0.9$ (left); an eight-hump soliton with $V=0.89$ and a phase-inverted
soliton with $V=-0.75$ (right).
Our numerical computations show that on a rather small interval
$\mathfrak{V}_{c}=[0.73,0.7642]$ at the low end of the spectrum, every attempt
of computing a homoclinic orbit was successful (20 orbits per bin). These
results stay consistent with the refinement of the computational grid size
$\delta V.$
All the solitary wave solutions in $\mathfrak{V}_{c}$ are of the one-hump
variety; note, however that the single-hump solitons are not exclusively
confined to the lower end of $\mathfrak{V}$. Elsewhere the spectrum density is
very low, and the solitons are mostly of a multi-hump kind. Somewhat
arbitrarily, we define a hump as a local maximum of the electric field $e,$
which is at least 50% of the global maximum.
Next we studied the distribution of the different type of solitary wave
solutions on the interval of velocities $[0.73,0.95]$. The figure (Fig. 3(b))
gives a very clear idea of the placement of solitons according to the number
of humps, which ranges from one to ten. Some typical soliton profiles for
four- and eight-hump solutions with $V=0.84322$ and $V=0.91416$ respectively
are collected in Fig. 4.
(a)
Figure 9: a) Initial pulses for propagation study; b) Energy dissipation for
the given initial pulses. Larger Gaussian waves quickly shed energy while
breaking up into near-solitary waves. The near solitary-waves slowly lose
energy while converging to solitary waves.
Different types of solitary wave solutions have different energy values.
Because of the multi-hump nature of these solutions it is convenient to
introduce the energy of the electromagnetic field per one hump. We analyzed
dependence of the electromagnetic field energy $\mathcal{E}$ per one hump
versus velocity of solitary wave, see Fig. 5. Here $\mathcal{E}$ is defined as
$\mathcal{E}=\frac{1}{2N}\int_{-\infty}^{\infty}\left[e^{2}(t,x)+h^{2}(t,x)\right]dt,$
where $N$ is the number of humps. As follows from this figure, in the log-log
coordinates the energy increase is very well approximated by a linear
function. The least square fit of the data from this figure shows that the
energy increases approximately as a polynomial of fifth degree in $V.$
## 6 Formation, stability, and interaction of solitary waves: computer
simulations
In this section through direct numerical simulations we study evolution of
waves as well as wave interactions. We consider formation of solitary wave
solutions from arbitrary initial-boundary condition, stability of traveling
waves under small perturbations and stability under strong perturbations due
to wave collisions. In all numerical simulations of this section we use the
same values of physical parameters (5.28) as in Sec. 5.
Numerically we solve the signaling problem for (2.7). In other words we give
boundary conditions on either one or both ends of the spatial interval
$(0,L)$; as initial conditions we assign zero values for all the variables,
which corresponds to propagation in a quiescent medium. For solving the
initial-boundary value problem for the system in (2.7) we devised a simple
fractional step numerical method.
Because the first two equations in (2.7) are hyperbolic PDEs while the rest
are ODEs the choice of the fractional steps is extremely natural: on the first
half-step we propagate the PDE part of the governing equations, and on the
second half-step we march according to the system of ODEs. The resulting ODE
system is solved by using the midpoint rule, while the PDEs are solved by the
explicit McCormack method laveque . The midpoint rule and the McCormack method
are both second order accurate. To increase the accuracy of the fractional
step method we utilize the Strang split approximation strang , which results
in the second order convergence of the final numerical scheme.
For many of the solitary-wave solutions discussed in Sec. 5 we ran direct
numerical simulations on the model with these solitary waves as input pulses.
All the waves tested, even the ones of a rather intricate shape, propagate
with constant speed and without any shape distortion. See, for example, Fig. 6
where propagation of an eight-hump soliton is depicted.
Figure 10: Evolution of the $15\exp(-0.1\tau^{2})$ Gaussian. A “sharp”
Gaussian quickly evolves into a near-solitary wave, leaving some disturbance
in the wake. The speed of the near-solitary wave is significantly higher than
the speed of propagation of the radiation; thus the wave quickly leaves the
disturbance behind
This suggests that the solitary waves are (nonlinearly) stable with respect to
numerical perturbations. We remark that although because of the scale of Fig.
6, the pulses appear rather singular, they are in fact completely smooth and
numerically resolved. The numerical resolution of this computation is $8$ mesh
intervals per unit length, which provides about 60 mesh points per each hump
of the traveling wave. Similarly, fine computational meshes are employed in
all the simulations below.
The issue of stability can be addressed analytically by studying the
linearization of the system of partial differential equations (2.7) about
arbitrary traveling wave solutions and analyzing the corresponding linear
evolution operator. Our analysis showed that this operator is skew-Hermitian
in ${L}_{2}$ with the appropriate norm. Therefore the spectrum of the
evolution operator is pure imaginary and the traveling wave solutions are
neutrally linearly stable (see frenkel for detail).
Figure 11: Evolution of the $15\exp(-0.05\tau^{2})$ Gaussian. A medium size
Gaussian evolves into two waves. The velocity of the smaller wave is on the
order of the velocity of radiation.
To further elucidate the issue of stability we consider stability with respect
to a finite-size perturbation in the initial wave shape. This situation is
illustrated in Fig. 7. To the two-hump numerical soliton we add a rather
substantial perturbation and employ the thus obtained functions as boundary
data for the system of partial differential equations (2.7). As the result of
evolution, the solution relaxes to the solitary wave shape followed by a low-
amplitude “continuous radiation”.
Stability with respect to strong perturbations due to collision of two
traveling wave solutions is illustrated in Fig. 8. We take two solitary waves
obtained by numerical solutions of ODEs and use these solutions as the
boundary conditions for the PDEs. The left part of the Fig. 8 shows collision
of two-hump solitary wave solutions. The right part of the figure shows
collision of eight-hump and one-hump solitary waves. In both cases collision
of solitary waves leads to formation of the steady state solutions. The
collisions are followed by emission of a small amplitude continuous radiation
and a residual phase shift.
A soliton nature of solutions of (2.7) is further confirmed by the set of
numerical simulations we present next. We consider propagation of solutions
with the pulses in Fig. 9 given as a series of boundary conditions at the
$x=0$ boundary. The soliton of velocity $V=0.75$ (see Fig. 1) propagates in a
stable fashion, while its least-squares approximation by a Gaussian
$9.65\exp{-0.1t^{2}}$ approaches the soliton shape after shedding a small
amount of residual continuous radiation. These time evolutions are not
included for space saving (the energy dissipation curves for these cases show
conservation of electromagnetic energy, see Fig. 9(a)).
r[ht]
Figure 12: Evolution of the $15\exp(-0.01\tau^{2})$ Gaussian. Large Gaussian
quickly breaks up into four near-solitary waves, leaving some disturbance in
the wake. The waves become more separated over the time, since the near
solitary waves with higher amplitude have higher velocities.
In the next three figures (Fig. 10 -12) we present evolutions of the larger
Gaussian pulses from Fig. 9. Evolution of the sharpest Gaussian
($\sigma=\sqrt{5})$ is displayed in Fig. 10. Very fast the solution forms a
solitary wave that moves with constant velocity with no shape change. It is
followed by low magnitude oscillations whose leading edge also moves with
constant speed. During the evolution, the oscillatory part disperses more and
more. This part of the solutions appears to be of a nonlinear nature; it will
be studied separately. We note that although because of the scale of the
figure, the pulse appears very sharp, it is in fact completely smooth with
“width” about $20$ and about $150$ computational mesh points within the pulse.
The evolution of a wider Gaussian, $\sigma=\sqrt{10}$ (see Fig. 11) is similar
with a very interesting distinction. Now the leading soliton is trailed by a
slower low amplitude soliton. The latter is followed by low amplitude
oscillations that again lag behind and disperse. The waves become more
separated over time because the solitary waves with higher amplitude have
higher velocities. Finally, the widest Gaussian, $\sigma=5\sqrt{2},$ develops
into a train of four solitons, see Fig. 12.
To characterize the energy exchange between the propagating pulse and the
medium, in Fig. 9(a) we present plots of the total electromagnetic energy as a
function of time for all the input profiles from Fig. 9. For the soliton
solution there is a dynamic equilibrium between the energy stored in the
medium and the electromagnetic energy of the pulse. In case of the input
impulse being not a soliton, the balance between the medium and the pulse is
violated, which leads to the dissipation of the electromagnetic energy into
the medium.
## 7 Concluding Remarks
In this paper we considered propagation of extremely short pulses in a
nonlinear medium, which is characterized by both electric and magnetic
resonance responses. Interaction of the electromagnetic field with the medium
was described in the framework of the Maxwell-Duffing model. In particular we
employed the classical Maxwell-Lorenz model for describing the magnetic
resonance, ZH01 . For describing the interaction of the electric field
component with the medium we used a generalized Maxwell-Lorentz model which
takes into account cubic anharmonism of the polarization response (i.e., the
Maxwell-Duffing system). Our findings demonstrate that the model supports a
wide array of traveling-wave solutions. We investigated the structure and
properties of these solutions through a combination of analysis and numerical
modeling. We determined that the family of traveling-wave solutions is
parameterized by one parameter, which is the velocity of a steady wave
solution, normalized by the speed of light in vacuum. The spectrum
$\mathfrak{V}$ contains both an interval of a continuous spectrum and a
discrete subset of parameter values for which a traveling-wave solution
exists. Computer modeling demonstrated a multi-hump structure of these
solutions. Their multi-hump nature suggests to characterize solitary wave
solutions by a number of humps (types). All types are determined by not
overlapping sets of velocities.
Direct numerical simulations showed that solitary-wave solutions are
dynamically stable. This dynamical stability is consistent with the analysis
of the system linearized about solitary wave solutions frenkel . Stability of
these solutions with respect to strong perturbations was studied by means of
solitary wave collisions. Computer simulations indicated nearly elastic nature
of scattering followed by a residual excessive radiation and a phase shift. In
addition to traveling-wave solutions, numerical simulations demonstrated
presence of another type of nonlinear oscillatory solutions with extended
tail.
## Acknowledgment
Frenkel’s work was partially supported by the NSF EMSW21-RTG Grant No.
DMS-0636358. Part of this work is based on his Ph.D. thesis frenkel . This
work was partially supported by NSF (grant DMS-0509589), ARO-MURI award
50342-PH-MUR, the State of Arizona (Proposition 301), and by the Russian
Foundation for Basic Research through grant 06-02-16406. Roytburd’s work was
partially supported by the National Science Foundation, while working at the
Foundation. Part of his work was performed during a sabbatical leave at the
Lawrence Berkeley National Laboratory. The authors would like to thank M.
Stepanov for the enlightening discussions and for the valuable help in
preparation of this manuscript.
## References
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* [3] V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, Opt. Lett. 30, 3356-3358, 2005
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* [5] V. M. Agranovich, Y.R. Shen, R.H. Baughman, A. A. Zakhidov, Phys. Rev. B 69, 165112, 2004
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* [9] I.V. Shadrivov , A.A. Zharov, Yu.S. Kivshar, J.Opt.Soc.Amer. B. 23, (2006) 529-534.
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* [11] A.I. Maimistov, I.R. Gabitov, Eur. Phys. J. Special Topics ”Nonlinear waves in complex systems: energy flow and geometry” 147(1), 265-286, 2007 (Springer, 2007)
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|
arxiv-papers
| 2008-12-28T05:37:43 |
2024-09-04T02:48:59.597031
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y. Frenkel, I. Gabitov, A. Maimistov, and V. Roytburd",
"submitter": "Ildar Gabitov",
"url": "https://arxiv.org/abs/0812.4794"
}
|
0812.4862
|
# Bistable states of quantum dot array junctions for high-density memory
David M.-T. Kuo1 and Yia-Chung Chang2,3 Department of Electrical Engineering,
National Central University,
Chung-Li, Taiwan 320, Republic of China 2Research Center for Applied
Sciences, Academia Sinica, Taipei, Taiwan 115, R.O.C. 3Department of Physics
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
###### Abstract
We demonstrate that two-dimensional (2D) arrays of coupled quantum dots (QDs)
with six-fold degenerate p orbitals can display bistable states, suitable for
application in high-density memory device with low power consumption. Due to
the inter-dot coupling of $p_{x}$ and $p_{y}$ orbitals in these QD arrays, two
dimensional conduction bands can be formed in the x-y plane, while the $p_{z}$
orbitals remain localized in the x-y plane such that the inter-dot coupling
between them can be neglected. We model such systems by taking into account
the on-site repulsive interactions between electrons in $p_{z}$ orbitals and
the coupling of the localized $p_{z}$ orbitals with the 2D conduction bands
formed by $p_{x}$ and $p_{y}$ orbitals. The Green’s function method within an
extended Anderson model is used to calculate the tunneling current through the
QDs. We find that bistable tunneling current can exist for such systems due to
the interplay of the on-site Coulomb interactions (U) between the $p_{z}$
orbitals and the delocalized nature of conduction band states derived from the
hybridization of $p_{x}$/$p_{y}$ orbitals. This bistable current is not
sensitive to the detailed band structure of the two dimensional band, but
depends critically on the strength of $U$ and the ratio of the left and right
tunneling rates. The behavior of the electrical bistability can be sustained
when the 2D QD array reduces to a one-dimensional QD array, indicating the
feasibility for high-density packing of these bistable nanoscale structures.
Intrinsic hysteresis in DC current-voltage characteristics is one of the most
intriguing problems for resonant tunneling diodes (RTDs)1,2. Such a
bistability has an important application in memory devices3,4. Whether this
phenomenon exists in nanoscale devices such as single-electron transistors
(SETs) and single molecular transistors (SMTs) has been theoretically
investigated in refs [5-7]. Alexandrov and coworkers5 pointed out that the
tunneling current through a highly degenerate states of a single QD
(molecular) can lead to a switching effect only in the case of attractive
electron Coulomb interactions, which is mediated by electron-phonon
interaction. On the basis of Hartree approximation and polaron effect Galperin
et al proposed that the hysteresis of I-V characteristics can be observed in a
single molecular junction with effective attractive electron Coulomb
interaction.6 Recently, Magna and Deretzis showed hysteresis feature of
tunneling current in a polaron model beyond the Hartree approximation.7
Although previous theoretical studies predicted the existence of hysteresis in
a QD (or molecular) junction,5-7 such a phenomenon still lacks conclusive
experimental support. Moon et al. have experimentally examined the tunneling
current through a carbon nanotube QD, which exhibits a periodic oscillatory
behavior with respect to the applied gate voltage arising from the eightfold
degenerate state.8 In addition, Liljeroth et al. have reported a periodic
oscillatory differential conductance as a result of tunneling current through
a single spherical PbSe QD with a sixfold degenerate state9. These two
experiments did not exhibit the bistable tunneling current. Their results
indicate that electron-phonon interactions in nanotube QDs or PbS QDs are not
sufficient to yield the strongly attractive Coulomb interactions needed for
observing the bistability. Thus it remains questionable whether a single QD
junction can display the bistable memory effect.
Recently, it was demonstrated that semiconductor quantum dot arrays (QDAs) can
be chemically fabricated to form a superlattice.10-13 Via nanoscale
manipulation, experimentalists can now control the lattice constant and QD
size to tune charges of QDA in the Coulomb blockade regime or semiconducting
regime.10,14 Consequently, QDA is not only a good physical system for
investigating strongly correlated problem but also a promising integrated
electronic device.10,15 Although many theoretical efforts have been devoted to
the charge transport through a single QD,16,17 not many studies are on the
tunneling current through a QDA junction.18 In this letter we illustrate that
a new mechanism exists in a QDA junction involving degenerate p-like orbitals
which can lead to bistable tunneling current, making it a good candidate for
high density storage device.
Figure. 1 illustrates the system of a QD array embedded in an insulator
connected with metallic electrodes. The system can be described by the
Anderson Hamiltonian, $H=H_{0}+H_{T}+H_{d}$. The
$H_{0}=\sum_{k,\sigma,\beta}\epsilon_{k}a^{\dagger}_{k,\sigma,\beta}a_{k,\sigma,\beta}$
describes the electronic states in the metallic leads.Here
$a^{\dagger}_{k,\sigma,\beta}$ ($a_{k,\sigma,\beta}$) creates (destroys) an
electron of momentum $k$ and spin $\sigma$ with energy $\epsilon_{k}$ in the
$\beta$ metallic electrode. The $H_{T}$ term describes the coupling between
the electrodes and the $p_{z}$ orbitals of the QD array.
$H_{T}=\sum_{k,\sigma,\beta,\ell}V_{k,\beta,\ell}a^{\dagger}_{k,\sigma,\beta}d_{\ell,\sigma}\\\
+\sum_{k,\sigma,\beta,\ell}V^{*}_{k,\beta,\ell}d^{\dagger}_{\ell,\sigma}a_{k,\sigma,\beta},$
(1)
where $V_{k,\beta,\ell}$ describes the coupling between the band states in the
electrodes and the localized $p_{z}$ states. Here we assume that the coupling
between the electrodes and the $p_{x}/p_{y}$ orbitals of the QD array is
negligible since the $p_{x}/p_{y}$ orbitals are much more localized along the
$z$ axis than the $p_{z}$ orbitals. At last, the $H_{d}$ term describes
electronic states and their interactions in the QD array.
$\displaystyle H_{d}$ $\displaystyle=$
$\displaystyle\sum_{\ell,\sigma}E_{p_{z}}d^{\dagger}_{\ell,\sigma}d_{\ell,\sigma}+\sum_{p,\lambda}(\epsilon_{p,\lambda}+U(N_{c}-N_{\lambda}))c^{\dagger}_{p,\lambda}c_{p,\lambda}$
$\displaystyle+$
$\displaystyle\sum_{\ell,p,\sigma}(v_{p,\ell}c^{\dagger}_{p,\sigma}d_{\ell,\sigma}+h.c)+\sum_{\ell,\sigma}U_{\ell}d^{\dagger}_{\ell,\sigma}d_{\ell,\sigma}d^{\dagger}_{\ell,-\sigma}d_{\ell,-\sigma}$
$\displaystyle+$
$\displaystyle\frac{U_{dc}}{N}\sum_{\ell,p,p^{\prime},\sigma}c^{\dagger}_{p,\lambda}c_{p^{\prime},\lambda}e^{i({\bf
p-p}^{\prime})\cdot{\bf
R}_{\ell}}d^{\dagger}_{\ell,\sigma,\lambda}d_{\ell,\sigma}.$
$d^{\dagger}_{\ell,\sigma}$ ($d_{\ell,\sigma}$) creates (destroys) an electron
in the $p_{z}$ orbital (with energy $E_{p_{z}}=E_{a}$) of the QD at site
$\ell$. The second term in Eq. (2) describes the conduction bands of QD array
arising from the $p_{x}$ and $p_{y}$ orbitals. $\lambda$ labels the conduction
bands (including spin). $U$ denotes the on-site Coulomb interaction between
two electrons in the $p_{x}$ and $p_{y}$ orbitals. Note that if we ignore the
quadrupole and higher-order terms in the expansion of $1/r_{12}$, then the
Coulomb repulsion integrals between two electrons in any of the three
degenerate p-like orbitals are the same. $N_{\lambda}$ is the occupation
number per unit cell for the $\lambda$-th conduction band, and
$N_{c}=\sum_{\lambda}N_{\lambda}$ is the total occupation number per unit cell
for the conduction bands. A mean-field theory (which is justified for extended
states) has been applied to the 2D conduction bands to obtain the second term
in the above equation. The third term in Eq. (1) describes the hopping
coupling between the $p_{z}$ orbital and the $p_{x}/p_{y}$ orbitals within the
tight-binding model. The last two terms in Eq. (2) involve $U_{\ell}=U$, and
$U_{dc}$, which denote the on-site repulsive Coulomb energy in the $p_{z}$
orbital, and electron Coulomb interactions between the $p_{z}$ and
$p_{x}/p_{y}$ orbitals. $N$ denotes the number of QDs in the matrix. Here, we
focus on the $p_{z}$ orbital rather than the ground state orbital, even though
its wave function is more localized than that of $p_{z}$, since in the range
of applied bias considered, the QD ground state energy level is deeply below
the Fermi levels of both electrodes and the electron tunneling through the QD
ground state is blockaded. Consequently, carriers in the QD ground states only
lead a constant-shift to all the p orbitals. It is worth noting that Eq. (2)
is similar to the so-called the extended Falicov-Kimball model, which has been
used extensively to study the semiconductor-metal transition in a solid
consisting of localized orbitals and delocalized orbitals.19-21
Using Keldysh Green’s function technique22,23, the tunneling current through
the $\ell$th QD can be expressed by
$J_{\ell,\sigma}=\frac{-e}{\hbar}\int\frac{d\epsilon}{\pi}[f_{L}(\epsilon)-f_{R}(\epsilon)]\frac{\Gamma_{\ell,L}\Gamma_{\ell,R}}{\Gamma_{\ell,L}+\Gamma_{\ell,R}}ImG^{r}_{\ell,\ell}(\epsilon),$
(3)
where $f_{L}=f(\epsilon-\mu_{L})$ and $f_{R}=f(\epsilon-\mu_{R})$ are the
Fermi distribution functions for the left and right electrodes, respectively.
The chemical potential difference between these two electrodes is related to
the applied bias by $\mu_{L}-\mu_{R}=eV_{a}$. $\Gamma_{\ell,L}(\epsilon)$ and
$\Gamma_{\ell,R}(\epsilon)$ [$\Gamma_{\ell,\beta}=2\pi\sum_{{\bf
k}}|V_{\ell,\beta,{\bf k}}|^{2}\delta(\epsilon-\epsilon_{{\bf k}})]$ denote
the tunneling rates from the $p_{z}$ orbitals to the electrodes. Notations $e$
and $\hbar$ denote the electron charge and Plank’s constant. In the wide-band
limit, these tunneling rates are approximately energy-independent. Therefore,
the calculation of tunneling current is entirely determined by the spectral
function $A=ImG^{r}_{\ell,\ell}(\epsilon)$, which is the imaginary part of the
retarded Green’s function $G^{r}_{\ell,\ell}(\epsilon)$.
Using the equation of motion for $G^{r}_{\ell,\ell}$, we obtain
$\displaystyle(\epsilon-E_{0}+i\Gamma)G^{r}_{i,j}(\epsilon)$ $\displaystyle=$
$\displaystyle\delta_{i,j}+U<n_{i,-\sigma}d_{i,\sigma}d^{\dagger}_{j,\sigma}>$
$\displaystyle+$
$\displaystyle\sum_{p}v_{i,p}G^{r}_{p,j}+\sum_{p^{{}^{\prime\prime}},p^{{}^{\prime}},\sigma}g_{p^{{}^{\prime\prime}},p^{{}^{\prime}}}<c^{\dagger}_{p^{{}^{\prime\prime}},\sigma^{\prime}}c_{p^{{}^{\prime}},\sigma^{\prime}}d_{i,\sigma}d^{\dagger}_{j,\sigma}>,$
$\displaystyle(\epsilon-E_{0}+i\Gamma)G^{r}_{i,p}(\epsilon)$ $\displaystyle=$
$\displaystyle U<n_{i,-\sigma}d_{i,\sigma}c^{\dagger}_{p,\sigma}>$
$\displaystyle+$
$\displaystyle\sum_{p^{\prime}}v_{i,p^{\prime}}G^{r}_{p^{\prime},p}+\sum_{p^{{}^{\prime\prime}},p^{{}^{\prime}},\sigma}g_{p^{{}^{\prime\prime}},p^{{}^{\prime}}}<c^{\dagger}_{p^{{}^{\prime\prime}},\sigma^{\prime}}c_{p^{{}^{\prime}},\sigma^{\prime}}d_{i,\sigma}c^{\dagger}_{p,\sigma}>,$
$\displaystyle(\epsilon-\epsilon_{p^{\prime},\lambda}-U(N_{c}-N_{\lambda}))G^{r}_{p^{\prime},p}(\epsilon)$
$\displaystyle=$
$\displaystyle\delta_{p^{\prime},p}+v_{p^{\prime},i}G^{r}_{i,p}+\sum_{i,p^{{}^{\prime\prime}},\sigma}g_{p^{{}^{\prime\prime}},p^{{}^{\prime}}}<(n_{i,\uparrow}+n_{i,\downarrow})d_{p^{{}^{\prime\prime}},\sigma}c^{\dagger}_{p,\sigma}>,$
and
$\displaystyle(\epsilon-\epsilon_{p^{\prime},\lambda}-U(N_{c}-N_{\lambda}))G^{r}_{p^{\prime},j}(\epsilon)$
$\displaystyle=$ $\displaystyle
v_{p^{\prime},i}G^{r}_{i,p}+\sum_{i,p^{{}^{\prime\prime}},\sigma}g_{p^{{}^{\prime\prime}},p^{{}^{\prime}}}<(n_{i,\uparrow}+n_{i,\downarrow})d_{p^{{}^{\prime\prime}},\sigma}d^{\dagger}_{j,\sigma}>.$
Here, $\Gamma=(\Gamma_{\ell,L}+\Gamma_{\ell,R})/2$ and
$g_{p,p^{{}^{\prime}}}=\frac{U_{dc}}{N}e^{i({\bf p-p}^{\prime})\cdot{\bf
R}_{i}}$. In Eqs. (4)-(7), we have introduced four one-particle Green’s
functions $G^{r}_{i,j}(\epsilon)=<d_{i,\sigma}d^{\dagger}_{j,\sigma}>$,
$G^{r}_{i,p}(\epsilon)=<d_{i,\sigma}c^{\dagger}_{p,\sigma}>$,
$G^{r}_{p^{\prime},p}(\epsilon)=<c_{p^{\prime},\sigma}c^{\dagger}_{p,\sigma}>$
and
$G^{r}_{p^{\prime},j}(\epsilon)=<c_{p^{\prime},\sigma}d^{\dagger}_{j,\sigma}>$.
These four single-particle Green’s function are coupled with two-particle
Green’s functions via $U$ and $U_{dc}$. The equation of motion for the two-
particle Green’s function (defined as
$<n_{i,-\sigma}d_{i,\sigma}d^{\dagger}_{j,\sigma}>$,
$<n_{i,-\sigma}d_{i,\sigma}c^{\dagger}_{p,\sigma}>$,
$<c^{\dagger}_{p^{{}^{\prime\prime}},\sigma^{\prime}}c_{p^{{}^{\prime}},\sigma^{\prime}}d_{i,\sigma}d^{\dagger}_{j,\sigma}>$,
$<c^{\dagger}_{p^{{}^{\prime\prime}},\sigma^{\prime}}c_{p^{{}^{\prime}},\sigma^{\prime}}d_{i,\sigma}c^{\dagger}_{p,\sigma}>$,
$<(n_{i,\uparrow}+n_{i,\downarrow})d_{p^{{}^{\prime\prime}},\sigma}c^{\dagger}_{p,\sigma}>$,
and
$<(n_{i,\uparrow}+n_{i,\downarrow})d_{p^{{}^{\prime\prime}},\sigma}d^{\dagger}_{j,\sigma}>$)
are coupled to the three-particle Green’s functions. In order to terminate the
heirachy of the equation of motions, we use the Hartree-Fock approximation
method19-21 to decouple terms involving the $U_{dc}$ factor. Meanwhile in the
derivation for $<n_{i,-\sigma}d_{i,\sigma}d^{\dagger}_{j,\sigma}>$ and
$<n_{i,-\sigma}d_{i,\sigma}c^{\dagger}_{p,\sigma}>$, the treatment for
coupling terms between localized states and the electrodes (or 2-D conduction
band) is employed in the scheme considered in our previous method, which is
valid for the Coulomb blockade regime.16,17 Solving Eqs. (4)-(7), we obtain
$\displaystyle
G^{r}_{pp^{\prime}\lambda}(\epsilon)=\frac{\delta_{p,p^{\prime}}}{\epsilon-\epsilon_{p}-\Delta_{\lambda}}+\frac{v^{2}G^{r}_{\ell,\ell}(\omega)}{(\epsilon-\epsilon_{p}-\Delta_{\lambda})(\epsilon-\epsilon_{p^{{}^{\prime}}}-\Delta_{\lambda})},$
(8)
where
$\Delta_{\lambda}=U_{dc}(N_{d,\sigma}+N_{d,-\sigma})+U(N_{c}-N_{\lambda})$ and
$\displaystyle G^{r}_{\ell,\ell}(\epsilon)$ $\displaystyle=$
$\displaystyle\frac{1-N_{d,-\sigma}}{\epsilon-
E_{0}-\Delta_{c}-(\Gamma_{b}(\epsilon)-i\Gamma)}$ $\displaystyle+$
$\displaystyle\frac{N_{d,-\sigma}}{\epsilon-
E_{0}-U-\Delta_{c}-(\Gamma_{b}(\epsilon)-i\Gamma)}.$
The retarded Green’s function $G^{r}_{\ell,\ell}(\epsilon)$ has the self-
energies $\Delta_{c}=U_{dc}N_{c}$ and
$\Gamma_{b}(\epsilon)=\frac{1}{N}\sum_{p}\frac{v^{2}}{\epsilon-\epsilon_{p}-\Delta_{d}+i\delta}$
($\delta$ is a positive infinitesimal number), which results from the
interaction between the localized states and conduction band. $N_{d}$ is the
occupation number of $p_{z}$ orbital in each unit cell. The second term in Eq.
(8) describes the scattering amplitude of the conduction electron due to
interaction with the $p_{z}$ orbitals. This term would be important for
studying the charge transport through the $p_{x}$ and $p_{y}$ orbitals in the
x-y plane . In this study we focus on the longitudinal transport (along
z-axis) rather than transverse transport (in the x-y plane), thus this term
can be ignored.
To reveal the tunneling current behavior, the occupation number $N_{d,\sigma}$
determining the probability amplitude of resonant channels
$\epsilon=E_{0}+\Delta_{c}+(\Gamma_{b}-i\Gamma)$ and
$\epsilon=E_{0}+U+\Delta_{c}+(\Gamma_{b}-i\Gamma)$ is solved by the equation
$N_{d,\sigma}=-\int\frac{d\epsilon}{\pi}\frac{\Gamma_{L}f_{L}(\epsilon)+\Gamma_{R}f_{R}(\epsilon)}{\Gamma_{L}+\Gamma_{R}}ImG^{r}_{\ell,\ell}(\epsilon).$
(10)
As for $N_{c}$, we have
$N_{c}=-\sum_{p,\lambda}\int\frac{d\epsilon}{\pi}\frac{\Gamma_{L,c}f_{L}(\epsilon)+\Gamma_{R,c}f_{R}(\epsilon)}{\Gamma_{L,c}+\Gamma_{R,c}}Im{\cal
G}^{r}_{p\lambda,p\lambda}(\epsilon)/N,$ (11)
where ${\cal
G}^{r}_{p\lambda,p\lambda}(\epsilon)=1/(\epsilon-\epsilon_{p,\lambda}-U(N_{c}-N_{\lambda})-2U_{dc}N_{d}+i(\Gamma_{L,c}+\Gamma_{R,c})/2)$.
As mentioned above, the coupling between the electrodes and the $p_{x}/p_{y}$
orbitals of QD array is negligible. (i.e
$\Gamma_{c}=(\Gamma_{L,c}+\Gamma_{R,c})/2$, where $\Gamma_{L,c}(\Gamma_{R,c})$
denotes the tunneling between the left (right) electrode and $p_{x}/p_{y}$
orbitals is small), therefore, $Im{\cal
G}^{r}_{p\lambda,p\lambda}(\epsilon)\approx\pi\delta(\epsilon-\epsilon_{p,\lambda}-U(N_{c}-N_{\lambda})-2U_{dc}N_{d})$.
The range of applied bias considered here would not be enough to overcome the
charging energy of $U+\Delta_{c}$, therefore, the second term in Eq. (9) can
be ignored and we have
$G^{r}_{\ell,\ell}(\epsilon)=(1-N_{d,-\sigma})/(\epsilon-
E_{0}-\Delta_{c}-\Gamma_{b}+i\Gamma)$ in which
$\Gamma_{b}(\epsilon)=-i\Gamma_{0}$ (ignoring the small real part). The
occupation number at zero temperature is calculated by
$\displaystyle N_{d,\sigma}$ $\displaystyle=$
$\displaystyle\frac{(1-N_{d,-\sigma})}{\pi}\frac{\Gamma_{L}}{\Gamma_{L}+\Gamma_{R}}$
$\displaystyle\int_{-\infty}^{E_{F}+eV_{a}}d\epsilon\frac{\Gamma_{0}+\Gamma}{(\epsilon-
E_{0}-\alpha eV_{a}-\Delta_{c})^{2}+(\Gamma_{0}+\Gamma)^{2}}$
or
$\displaystyle\frac{\Gamma_{L}+\Gamma_{R}}{\Gamma_{L}}\pi N_{d}/(1-N_{d})$
$\displaystyle=$ $\displaystyle\cot^{-1}(\frac{E_{F}+eV_{a}-E_{0}-\alpha
eV_{a}-U_{cd}N_{c}}{\Gamma_{0}+\Gamma}),$
in which $\alpha eV_{a}$ term arises from the applied bias crossing QDA and
$\alpha$ is a dimensionless scaling factor determined by the QDA location, and
$N_{\lambda}=\frac{\Gamma_{L,c}}{\Gamma_{L,c}+\Gamma_{R,c}}\int_{-\infty}^{E_{F}+eV_{a}}d\epsilon
D_{\lambda}(\epsilon-2U_{dc}N_{d}-U(N_{c}-N_{\lambda})),$ (14)
where $D_{\lambda}(\epsilon)=\sum_{p}\delta(\epsilon-\epsilon_{p,\lambda})/N$
denotes the density of states per unit cell of the $\lambda$-th conduction
band. Due to the fact that the $p_{z}$ energy level is always above the Fermi
energy of right electrodes (in the range of bias considered), we can ignore
the electron injection from the right electrode in Eqs. (12) and (14). We
first consider the simple case in which $N_{x}=N_{y}$ (valid for a square
lattice) and we approximate the density of states by a square pulse function
$D_{x}(\epsilon)=D_{y}(\epsilon)=1/W\mbox{ for }E_{b}<\epsilon<E_{b}+W,$ (15)
where $E_{b}$ denotes the bottom of the conduction band and $W$ is the band
width. Such an approximation allows Eq. (11) to have a simple analytic
solution of the form
$N_{\lambda}=b-cN_{d}$ (16)
with $b=[E_{f}+(1-\alpha)eV_{a}-E_{b}]/(\gamma W+3U)$ and $c=2U_{dc}/(\gamma
W+3U)$, where $\gamma=(\Gamma_{L,c}+\Gamma_{R,c})/\Gamma_{L,c}$. Substituting
this into Eq. (13) allows a simple transcendental equation, which can be
solved numerically. The equation allows a maximum of three roots, out of which
only two are stable roots.
We can also solve two coupled transcendental equations as given in Eqs. (13)
and (14) numerically for a more realistic density states, which is derived for
a 2D tight-binding model. We consider a tight-binding model for $p_{x}$ and
$p_{y}$ orbitals arranged on a rectangular lattice with lattice constants $a$
and $b$. Figure 2 illustrates the rectangular lattice. The band structure for
the $p_{x}$ band is given by
$E_{x}({\bf k})=E_{p}-2v_{l}\cos(k_{x}a)-2v_{t}\cos(k_{y}b),$ (17)
where $v_{l}$ denotes the $(pp\sigma)$ interaction and $v_{t}$ denotes the
$(pp\pi)$ interaction.24 For the $p_{y}$ band, we have
$E_{y}({\bf
k})=E_{p}-2v^{\prime}_{l}\cos(k_{y}b)-2v^{\prime}_{t}\cos(k_{x}a).$ (18)
The density of states per unit cell form the $p_{x}$ band is given by (if
$v_{l}>v_{t}$)
$D_{x}(\epsilon)=\left\\{\begin{array}[]{lll}\frac{1}{\pi^{2}}\int_{0}^{\pi}d\eta[(2v_{l})^{2}-(2v_{l}+2v_{t}(1-\cos\eta)-\tilde{\epsilon})^{2}]^{-1/2}\theta(\tilde{\epsilon}-2v_{t}(1-\cos\eta))\;\mbox{
for }0<\tilde{\epsilon}<4v_{t}\\\
\frac{1}{\pi^{2}}\int_{0}^{\pi}d\eta[(2v_{t})^{2}-(\tilde{\epsilon}-2v_{t}-2v_{l}(1-\cos\eta))^{2}]^{-1/2}\theta(\tilde{\epsilon}-2v_{l}(1-\cos\eta))\;\mbox{
for }4v_{t}<\tilde{\epsilon}<4v_{l}\\\
\frac{1}{\pi^{2}}\int_{0}^{\pi}d\eta[(2v_{l})^{2}-(2v_{l}+2v_{t}(1+\cos\eta)-\bar{\epsilon})^{2}]^{-1/2}\theta(\bar{\epsilon}-2v_{t}(1+\cos\eta))\;\mbox{
for }0<\bar{\epsilon}<4v_{t}\end{array}\right.,$ (19)
where $\tilde{\epsilon}=\epsilon-E_{p}+2v_{l}+2v_{t}$ and
$\bar{\epsilon}=E_{p}+2v_{l}+2v_{t}-\epsilon$. If $v_{l}<v_{t}$, then the
roles of $v_{l}$ and $v_{t}$ should be exchanged in the above expression. The
DOS described by Eq. (19) contains the Van Hove singularities. Similar
expression ($D_{y}(\epsilon)$) holds for the $p_{y}$ band with the hopping
parameters $v_{l}$ and $v_{t}=v^{{}^{\prime}}_{t}$ replaced by
$v^{\prime}_{l}=v_{l}$ and $v^{\prime}_{t}$. By varying these hopping
parameters (for instance, fix lattice constant a and tune b), we can study the
behavior of bistable tunneling current for systems between the 1D and 2D
limits.
We numerically solve the coupled nonlinear Eqs. (13) and (14) for
$U_{dc}=U=50meV$, $\Gamma_{L}=1~{}meV(\Gamma_{L,c}=\Gamma_{L}/10)$, and
$\Gamma_{R}=1meV(\Gamma_{R,c}=\Gamma_{R}/10)$. Throughout the paper, we shall
use $T=0K$, $v^{\prime}_{t}=5meV$, $v_{l}=20meV$, $\alpha=0.5$, and
$E_{F}+V_{0}=E_{p}$, where $V_{0}$ is a reference bias for $V_{a}$. For
simplicity, $v_{0}=0$. The tight-binding parameters are assumed to scale
according to the $1/R^{2}$ rule24, where $R$ is the separation between two
QDs. Thus, we have $v^{\prime}_{l}=v_{l}(a/b)^{2}$ and
$v_{t}=v^{{}^{\prime}}_{t}(a/b)^{2}$. The occupation number $N_{d}$ as a
function of the applied bias at zero temperature for the square lattice case
($a=b$) is shown as solid line in Fig. 3. The result remains very similar if
we use the constant DOS approximation as described in Eqs. (15) and (16) with
the same bandwidth, $W=4(v_{l}+v_{t})$. Although there are Van Hove
singularities in tight binding DOS, the structure of bias-dependent occupation
number does not exhibit an anormal feature. This is because $p_{z}$ orbital is
correlated with $p_{x}$ and $p_{y}$ via $N_{c}$, which is related to the
integral over the DOS.
We see that the occupation number, $N_{d}$ has bistable roots. Although, the
QDA crystal structure reported in references[10,13] has a triangle lattice,
the results of Fig. 3 indicate that the hysteresis behavior will not depend on
the detailed band structure. Once the occupation numbers are solved, the
tunneling current can be obtained by
$J=\frac{e}{\hbar}\Gamma_{R}N_{d}=J_{0}N_{d}$, which is valid for zero
temperature and when the carrier injection from the right lead can be ignored.
Consequently, $N_{d}$ via the applied bias directly shows the tunneling
current characteristics. The roots for the turn-on and turn-off processes in
Fig. 3 are determined by selecting the root closest to the root corresponding
to the previous value of $V_{a}$ when multiple roots are allowed.
It is crucial to clarify how the system selects the high conductivity state
(larger $N_{d}$) or the low conductivity state (smaller $N_{d}$) as the
applied bias is turned on or off. In Fig. 4a, we plot the bistable current for
various strengths of $U_{dc}$. Curves 1, 2, and 3 denote, respectively,
$U$=50, 30, and 20 $~{}meV$. For smaller Coulomb interactions, the bistable
current vanishes. The critical Coulomb interactions to maintain the bistable
current depend on the physical parameters such a bandwidth of 2-D conduction
band, tunneling rates between dot array and electrodes, the broadening
$\Gamma_{0}$, and temperatures. From the application of memory devices, larger
$U$ (smaller dot size) is favored because the bistability behavior will be
more robust against the increase of temperature and broadening. Although Fig.
(4a) exhibits the bistable current, it does not show any negative differential
conductivity (NDC), unlike the bistable current in quantum well systems, which
is typically associated with NDC.1,2 Fig. 4b shows the behavior of $N_{c}$ for
the same set of on-site Coulomb interaction strengths as in Fig. 4a. It is
useful to understand the behavior of $N_{c}$ for clarifying the bistable
mechanism of $N_{d}$, which is described below.
The physical mechanism for this bistablity behavior can be explained as
follows. As we gradually increase the bias from below the resonance level
$E_{p}$, the allowed solution to the occupation number, $N_{d}$ remains small
during the ”turn-on” process, as a result of inter-level Coulomb blockade
(i.e. the $p_{z}$ level is pushed up by the amount $N_{c}U_{dc}$, where the
2-D conduction band occupation number $N_{c}$ is appreciable). Once the
applied bias reaches a critical value (the inter-level Coulomb blockade is
overcome), $N_{d}$ increases quickly to a value around 1/3. Now, charges
accumulate in the localized $p_{z}$ orbitals, which leads to an increase of
the self-energy of the conduction band states by $2U_{dc}N_{d}$, and hence
reduces $N_{c}$ to a much smaller value (from $N_{c2}$ to $N_{c3}$ as
illustrated in Fig. 4b). This in turn causes the $p_{z}$ energy level to
decease during the ”turn-off” process while $N_{d}$ maintains around 1/3 to
keep the transcendental equation self-consistent. When the applied bias
continues to decrease to a critical value, $N_{c}$ is switched from $N_{c4}$
to $N_{c1}$ (see Fig. 4b), and $N_{d}$ quickly goes back to the lower value
which becomes the only allowed self-consistent solution to the transcendental
equation. The results of Fig. 4 demonstrate that the bistable tunneling
current arises from the on-site repulsive Coulomb interactions $U_{dc}$. In
the polaron model adopted in Refs. 6 and 7, the bistable mechanism arises from
an attractive potential of
$-\frac{2\lambda_{p}^{2}}{W_{0}}d^{\dagger}_{\ell,\sigma}d_{\ell,\sigma}d^{\dagger}_{\ell,-\sigma}d_{\ell,-\sigma}$,where
$\lambda_{p}$ and $W_{0}$ are the electron-phonon interaction strength and the
phonon frequency. This two particle interaction term should be corrected as
$(U-\frac{2\lambda_{p}^{2}}{W_{0}})d^{\dagger}_{\ell,\sigma}d_{\ell,\sigma}d^{\dagger}_{\ell,-\sigma}d_{\ell,-\sigma}$
when the intralevel Coulomb interaction is included in a single QD or
molecule. Due to the large repulsive intralevel Coulomb interaction U, a net
attractive electron-electron interactions mediated by phonon is difficult to
achieve, which may explain why a bistable tunneling current through a single
QD junction has not been observed.
To realize nanoscale memory structures, we need to examine whether this
bistable current exists in one dimensional array. Figure. 5 show tunneling
current through $p_{z}$ orbital for different ratios of $b/a$. Other
parameters are the same as those for Fig. 3. From two dimensional array to
quasi-one dimensional array, the bistability behavior is sustained (although
somewhat weaker in the 1D limit). When $b/a$ is greater than 3, (results not
shown here) the bistable current behaves essentially the same as in the
$b/a=3$ case. This indicates that we already reached the 1D limit for
$b/a\approx 3$, and the bistable current still exists. If we assume $a=3nm$,
$b=9nm$, and 50 coupled QDs along the chain in the $x$ direction are needed to
establish a band-like behavior, then the density of the memory device is
around $1/(1350nm^{2})\approx 0.5TB/in^{2}$. The operating voltage needed is
around 100mV, which indicates very low power consumption. Because this is a
quantum device, the switching time is expected to be comparable to the
tunneling rate, which is on the order of 1 THz.
Finally, we show in Fig. 6 the tunneling current for various tunneling rate
ratios $\Gamma_{L}/\Gamma_{R}$ for the $b/a=2$ case. It is seen that the
bistability disappears for $\Gamma_{L}=0.1$ meV and $\Gamma_{R}=1~{}$meV
(shell-tunneling condition). In this case, charges are unable to accumulate in
the $p_{z}$ orbitals. Consequently, the effect of $U_{dc}$ is suppressed. On
the other hand, for $\Gamma_{L}=1$ meV and $\Gamma_{R}=0.1$meV (shell-filling
condition), the effect of $U_{dc}$ is enhanced. This leads to the wider
voltage range of bistable current. From results of Fig. 6, we can control the
bistable current by adjusting the tunneling rate ratio.
In conclusion, we have illustrated a novel mechanism for generating the
bistable tunneling by using a junction involving a 2D periodic array of QDs
with six-fold degenerate $p$-like states, which consist of localized $p_{z}$
orbitals interacting with non-localized $p_{x}$ and $p_{y}$ orbitals via the
on-site Coulomb interaction. Due to the interplay of Coulomb blockade effect
for the localized state and the self-energy correction to the 2D conduction
bands formed by the $p_{x}$ and $p_{y}$ orbitals, a bistable tunneling current
with well defined hysteresis behavior can be achieved. This bistable current
is not sensitive to the details of the band structure, but sensitive to the
charging energy and the ratio of incoming to outgoing tunneling rate. Such a
hysteresis behavior arises from a collective effect, not observable in a
single QD. It is shown that this hysteresis behavior can also exist in one
dimensional QDA and very high density integrated memory circuits can be
realized.
This work was supported by the National Science Council of the Republic of
China under Contract Nos. NSC 97-2112-M-008-017-MY2, 96-2120-M-008-001, and
95-2112-M-001-068-MY3.
## References
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* (4) (4) Tseng, R. J.;Huang, J. ;Ouyang, J.;Kaner, R. B.;Yang, Y. Nano Lett 2005, 5, 1077.
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* (8) (8) Moon, S.;Song, W. ;Lee, J. S. ;Kim, N. ;Kim, J. ;Lee, S. G.;M. Choi, S. Phys. Rev. Lett 2007, 99, 176804.
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* (25)
Figure Captions
Fig. 1. Quantum dot array is embedded into an insulated matrix sandwiched
between two metallic electrodes.
Fig. 2. Two band ($p_{x}$ and $p_{y}$) rectangular lattice with lattice
constants $a$ and $b$. $v_{l}(v^{{}^{\prime}}_{t})$ and
$v_{t}(v^{{}^{\prime}}_{l})$ denote, respectively, the electron hopping
strength of $p_{x}(p_{y})$ orbital in $a$ and $b$.
Fig. 3. Occupation number as a function of applied bias for $U_{dc}=U=50meV$,
$\Gamma_{L}=1~{}meV(\Gamma_{L,c}=\Gamma_{L}/10)$, and
$\Gamma_{R}=1~{}meV(\Gamma_{R,c}=\Gamma_{R}/10)$. Solid line and dashed line
denote, respectively, tight binding DOS and constant DOS.
Fig. (4a) Bistable current as a function of applied bias for variation
strengths $U$ at zero temperature. Fig. (4b) shows $N_{c}$ for different $U$.
Other parameters are the same those of Fig. 3. Note that tunneling current is
in units of $J_{0}=e\Gamma_{R}/\hbar$, which depends on the tunneling rate of
$\Gamma_{R}$.
Fig. 5. Bistable current as a function of applied bias for $b/a=$ 1, 2, and 3.
Other parameters are the same those for Fig. 3.
Fig. 6. Bistable current as a function of applied bias for the various ratios
of $\Gamma_{L}/\Gamma_{R}$ and $b/a=3$. Other parameters are the same as those
for Fig. 3.
|
arxiv-papers
| 2008-12-29T01:32:48 |
2024-09-04T02:48:59.607714
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "David M.-T. Kuo and Yia-Chung Chang",
"submitter": "Mingting Kuo david",
"url": "https://arxiv.org/abs/0812.4862"
}
|
0812.4955
|
# Quantum Crooks fluctuation theorem and quantum Jarzynski equality in the
presence of a reservoir
H. T. Quan Theoretical Division, MS B213, Los Alamos National Laboratory, Los
Alamos, NM, 87545, U.S.A. H. Dong Institute of Theoretical Physics, Chinese
Academy of Sciences, Beijing, 100190, P.R. China
###### Abstract
We consider the quantum mechanical generalization of Crooks Fluctuation
Theorem and Jarzynski Equality for an open quantum system. The explicit
expression for microscopic work for an arbitrary prescribed protocol is
obtained, and the relation between quantum Crooks Fluctuation Theorem, quantum
Jarzynski Equality and their classical counterparts are clarified. Numerical
simulations based on a two-level toy model are used to demonstrate the
validity of the quantum version of the two theorems beyond linear response
theory regime.
###### pacs:
05.70.Ln, 05.40.-a
## I Introduction:
Nonequilibrium thermodynamics has been an intriguing research subject for more
than one hundred years nonequilibrium . Yet our understanding about
nonequilibrium thermodynamic phenomena, especially about those far-from-
equilibrium regime (beyond the linear response regime), remains very limited.
In the past fifteen years, there are several significant breakthroughs in this
field, such as Evans-Searls Fluctuation Theorem evans , Jarzynski Equality
(JE) JE , and Crooks Fluctuation Theorem (Crooks FT) crooks . These new
theorems not only have important applications in nanotechnology and
biophysics, such as extracting equilibrium information from nonequilibrium
measurements, but also shed new light on some fundamental problems, such as
improving our understanding of how the thermodynamic reversibility arise from
the underlying time reversible dynamics.
Since the seminal work by Jarzynski and Crooks a dozen of years ago, the
studies of nonequilibrium thermodynamics in small system attract numerous
attention followup , and the validity and universality of these two theorems
in classical systems has been extensively studied not only by numerical
studies classicalnumerical , but also by experimental exploration
classicalexperiment in single RNA molecules, and for both deterministic and
stochastic processes. For quantum systems, possible quantum extension of
Crooks FT and JE have also been reported quantumJE . Nevertheless, we notice
that almost all of these reports about quantum extension of Crooks FT focus on
isolated quantum systems isolatedcrooks , and the explicit expression of
microscopic work, and their distributions in the presence of a heat bath are
not extensively studied. In addition, the relationship between classical and
quantum Crooks FT is not addressed adequately so far. As a result, the
experimental studies of quantum Crooks FT and JE are not explored (an
exception is the experimental scheme of quantum JE of isolated system based on
trapped ions schmidt ).
Figure 1: (Color Online) Trajectories of a quantum system in a nonequilibrium
process. Similar to Ref. crooks each step (from $t_{n}$ to $t_{n+1}$) is
divided into two substeps: the controlling substep of time $\tau_{Q}^{n}$, in
which the energy spectrum (black solid line) of the system change with time,
and the relaxation substep of time $\tau_{R}^{n}$ in which the energy spectrum
(black dashed line) remains unchanged. In the controlling substep (solid line)
work is done, but there is no heat exchange; While in the relaxation substep,
there is heat exchange between the system and the heat bath, but there is no
work done. Blue trajectory corresponds to fast controlling protocol, during
which there are usually interstate excitations in the controlling substep. Red
trajectory corresponds to slow (quantum adiabatic) controlling protocol, and
the system remains in its instantaneous eigenstate in the controlling substep.
Red trajectory is the counterpart of classical case.
In this paper, we will give a detailed proof of the validity of quantum Crooks
FT and quantum JE for an open quantum system based on the explicit expression
of microscopic work and their corresponding probability distributions for an
arbitrary prescribed controlling protocol. We also clarify the relation
between quantum Crooks FT, quantum JE and their classical counterparts. In the
last part of the paper, the studies based on a two-level system are given as
an illustration to demonstrate our central idea.
## II Notations and assumptions:
Crooks FT crooks is firstly derived in classical systems in a microscopically
reversible Markovian stochastic process. In the proof of a classical Crooks
FT, a key technique is to separate work steps from heat steps. In the
following discussion of quantum extension of Crooks FT and JE, we will employ
the same technique as that used in Ref. crooks to separate the controlling
process into two substeps: controlling substep and relaxation substep (see
Fig. 1). The controlling substep proceeds so quickly in comparison with the
thermalization process of the system that we can ignore the influence of the
heat bath during the controlling substep. So there is only work done in the
controlling substep. In the relaxation substep, on the other hand, there is
only heat exchange.
Having clarified the main strategy (separating work substep from heat
substep), let us come to the details of the notations and assumptions. We
employ the same notations and assumptions as that in Ref. crooks to prove the
quantum Crooks FT. In Ref. crooks the author assumes discrete time and
discrete phase space. Here, the discrete energy spectrum in a quantum system
in place of the discrete phase space of a classical system arises naturally.
We also assume discrete time $t_{0}$, $t_{1}$, $t_{2}$, $t_{3}$, $\cdots$,
$t_{N}$ for the quantum system (see Fig. 1). The parameter $\lambda(t)$ is
controlled according to an arbitrary prescribed protocol
$\lambda(t_{0})=\lambda_{A}$, $\lambda(t_{1})=\lambda_{1}$,
$\lambda(t_{2})=\lambda_{2}$, $\cdots$, $\lambda(t_{N})=\lambda_{B}$, where
$A$ and $B$ depict the initial and final points of the process. Every step
$t_{n}\rightarrow t_{n+1}$ is seperated into controlling substep of time
$\tau_{Q}^{i}$ and relaxation substep of time time $\tau_{R}^{i}$,
$t_{i+1}=t_{i}+\tau_{Q}^{i}+\tau_{R}^{i}$ (see Fig. 1). If we use
$\left|i_{n},\lambda_{m}\right\rangle$ and $E(i_{n},\lambda_{m})$ to depict
the $i_{n}$-th instantaneous eigenstate and eigenenergy of the system
Hamiltonian $H(\lambda_{m})$, we can rewrite the trajectory $A\rightarrow B$
of Ref. crooks in the following way
$\begin{split}\left|i_{0},\lambda_{0}\right\rangle\rightarrow\left|i_{0},\lambda_{1}\right\rangle\underrightarrow{\lambda_{1}}\left|i_{1},\lambda_{1}\right\rangle\rightarrow\left|i_{1},\lambda_{2}\right\rangle\underrightarrow{\lambda_{2}}\left|i_{2},\lambda_{2}\right\rangle\\\
\rightarrow\cdots\rightarrow\left|i_{N-1},\lambda_{N-1}\right\rangle\rightarrow\left|i_{N-1},\lambda_{N}\right\rangle\underrightarrow{\lambda_{N}}\left|i_{N},\lambda_{N}\right\rangle.\end{split}$
(1)
In the classical case, the system remains in its $i_{n}$-th state of the
discrete phase space during the controlling substep. Analogously, in quantum
systems, this process corresponds to the quantum adiabatic regime, i.e., the
system remains in its $i_{n}$-th eigenstate of the instantaneous Hamiltonian
when we control the parameter $\lambda(t)$ of the Hamiltonian $H[\lambda(t)]$
so slowly that the quantum adiabatic conditions are satisfied, and the above
trajectories (1) can be achieved (red trajectory of Fig. 1). However, if we
control the parameter of the Hamiltonian very quickly in the controlling
substep, and then the quantum adiabatic conditions are not satisfied, the
trajectory $A\rightarrow B$ in general should be written as (see blue
trajectory of Fig. 1)
$\begin{split}\left|i_{0},\lambda_{0}\right\rangle\rightarrow\left|i_{0}^{\prime},\lambda_{1}\right\rangle\underrightarrow{\lambda_{1}}\left|i_{1},\lambda_{1}\right\rangle\rightarrow\left|i_{1}^{\prime},\lambda_{2}\right\rangle\underrightarrow{\lambda_{2}}\left|i_{2},\lambda_{2}\right\rangle\\\
\rightarrow\cdots\rightarrow\left|i_{N-1},\lambda_{N}\right\rangle\rightarrow\left|i_{N-1}^{\prime},\lambda_{N}\right\rangle\underrightarrow{\lambda_{N}}\left|i_{N},\lambda_{N}\right\rangle.\end{split}$
(2)
The main difference of the above two kinds of trajectories (1) and (2) is that
after the controlling substep the system may not be in the same eigenstate as
that before the controlling, i.e., $i_{n}\neq i^{\prime}_{n}$. The internal
excitation
$\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle$
is due to randomness caused by quantum non-adiabatic transition and has no
classical counterpart. Actually this difference of trajectories (1) and (2)
highlights the main difference between the quantum and classical Crooks FT.
For a quantum system, the microscopic work done in every controlling substep
is equal to the difference of the energy before and after the controlling
substep: $W_{n}=E(i_{n}^{\prime},\lambda_{n+1})-E(i_{n},\lambda_{n})$, and the
heat exchanged with the heat bath is equal to the difference of the energy of
the system before and after the relaxation substep
$Q_{n}=E(i_{n},\lambda_{n})-E(i_{n-1}^{\prime},\lambda_{n})$. For the
trajectory (2) as a whole, we must make $2N$ times quantum measurements to
confirm the microscopic work done and heat exchanged with the heat bath.
Similar to the classical case, the total work $W$ performed on the system, and
the total heat $Q$ exchanged with the heat bath are given by the summation of
work and heat in every step,
$W=\sum_{n=0}^{N-1}\left[E(i_{n}^{\prime},\lambda_{n+1})-E(i_{n},\lambda_{n})\right],Q=\sum_{n=0}^{N}\left[E(i_{n},\lambda_{n})-E(i_{n-1}^{\prime},\lambda_{n})\right]$,
and the total change in energy is $\Delta
E=Q+W=E(i_{N},\lambda_{N})-E(i_{0},\lambda_{0})$. Note that the work and heat
depend on the trajectory, but the energy change depends only on the initial
and final energy, and does not depend on the trajectory.
Similar to the classical case crooks we assume the trajectory (2) to be
Markovian, and the forward process starts from the thermal equilibrium
distribution $P(\left|i_{0},\lambda_{0}\right\rangle)=e^{-\beta
E(i_{0},\lambda_{0})}/(\sum_{i}e^{-\beta E(i,\lambda_{0})})$. The joint
probability for a given trajectory (2) can be expressed as
$\begin{split}P_{F}(A\rightarrow
B)=&P(\left|i_{0},\lambda_{0}\right\rangle)\prod_{n=0}^{N-1}P_{F}(\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)\\\
&\times
P_{F}(\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle\rightarrow\left|i_{n+1},\lambda_{n+1}\right\rangle).\end{split}$
(3)
It can be seen that the above probability (3) of a trajectory for a quantum
case is different from the classical case crooks by the extra term
$P(\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)$
arising from randomness due to quantum non-adiabatic transition. When the
quantum adiabatic conditions are satisfied,
$P(\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)=\delta_{i_{n},i_{n}^{\prime}}$,
we regain the probability of a trajectory in classical systems crooks . We
will see later that the quantum Crooks FT and quantum JE in the quantum
adiabatic regime are the counterpart of classical Crooks FT and classical JE.
To prove the quantum Crooks FT, we also need to consider the time-reversed
trajectory reverse of the original trajectory (2). The time-reversed
trajectory corresponding to the forward time trajectory $A\leftarrow B$ in Eq.
(2) can be written as
$\begin{split}\Theta\left|i_{0},\lambda_{0}\right\rangle\leftarrow\Theta\left|i_{0}^{\prime},\lambda_{1}\right\rangle\underleftarrow{\lambda_{1}}\Theta\left|i_{1},\lambda_{1}\right\rangle\leftarrow\Theta\left|i_{1}^{\prime},\lambda_{2}\right\rangle\underleftarrow{\lambda_{2}}\\\
\cdots\leftarrow\Theta\left|i_{N-1},\lambda_{N}\right\rangle\leftarrow\Theta\left|i_{N-1}^{\prime},\lambda_{N}\right\rangle\underleftarrow{\lambda_{N}}\Theta\left|i_{N},\lambda_{N}\right\rangle\end{split}$
(4)
where
$\Theta\left|i_{n},\lambda_{n}\right\rangle=\left|i_{n},\lambda_{n}\right\rangle^{\ast}$
is the microscopic state in the time-reversed trajectory sakurai . The
sequence in which states are visited is reversed, as is the order in which
$\lambda$ is changed. The work done $W$, the heat exchange $Q$ with the heat
bath, the change of the internal energy $\Delta E$, and the change of free
energy $\Delta F$ for the reversed time direction are the negative value of
that of the forward time trajectory. The joint probability for time reversed
trajectory $A\leftarrow B$ can be expressed as
$\begin{split}P_{R}(A\leftarrow
B)=&\prod_{n=0}^{N-1}P_{R}(\Theta\left|i_{n},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)\\\
&\times
P_{R}(\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle\leftarrow\Theta\left|i_{n+1},\lambda_{n+1}\right\rangle)\\\
&\times P(\Theta\left|i_{N},\lambda_{N}\right\rangle),\end{split}$ (5)
where $P(\Theta\left|i_{N},\lambda_{N}\right\rangle)=e^{-\beta
E(i_{N},\lambda_{N})}/\sum_{i}e^{-\beta E(i,\lambda_{N})})$ is the initial
thermal distribution for the time-reversed trajectory. Also there is en extra
term
$P_{R}(\Theta\left|i_{n},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)$
arising due to the randomness caused by quantum non-adiabatic transition in
comparison with the classical case.
## III Proof of quantum crooks FT and quantum JE
As we have mentioned before, in a trajectory every step consists of two
substeps, the controlling substep (not necessarily to be quantum adiabatic)
and the relaxation substep. The relaxation substeps are assumed to be
microscopically reversible, and therefore obey the detailed balance crooks ;
chandler for all fixed value of the external control parameter $\lambda$
$\frac{P_{F}(\left|i_{n-1}^{\prime},\lambda_{n}\right\rangle\rightarrow\left|i_{n},\lambda_{n}\right\rangle)}{P_{R}(\Theta\left|i_{n-1}^{\prime},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n},\lambda_{n}\right\rangle)}=\frac{e^{-\beta
E(i_{n},\lambda_{n})}}{e^{-\beta E(i_{n-1}^{\prime},\lambda_{n})}}.$ (6)
To compare the ratio of the probabilities of forward (3) and time-reversed (5)
trajectories, we also need to know the ratio of the probabilities in the
controlling substep. In the following we will focus on the study of
controlling substep and its time reversal. As we mentioned before, during the
controlling substep, the system can be regarded as an isolated quantum system
and the evolution is completely determined by a time-dependent Hamiltonian
$H[\lambda(t)]$. For example, when the controlling parameter $\lambda$ is
changed from $\lambda_{n}$ to $\lambda_{n+1}$, the probability of the
transition from a microscopic state $\left|i_{n},\lambda_{n}\right\rangle$ to
another microscopic state $\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle$
can be expressed as
$P_{F}(\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)=|\left\langle
i_{n}^{\prime},\lambda_{n+1}\right|U\left|i_{n},\lambda_{n}\right\rangle|^{2}$
(7)
where $U=\mathrm{T}\exp\\{-i\int_{t_{0}}^{t_{1}}H[\lambda(t)]dt\\}$ is the
unitary matrix describing the evolution of the isolated quantum system in the
controlling substep, and $\mathrm{T}$ is the time-ordered operator. Similarly,
in the time-reversed trajectory the excitation probability from the
microscopic state $\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle$ to
another microscopic state $\Theta\left|i_{n},\lambda_{n}\right\rangle$ in the
time reversed trajectory can be expressed as theta
$\begin{split}P_{R}(\Theta&\left|i_{n},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)\\\
&=|\left(\left\langle
i_{n},\lambda_{n}\right|\overleftarrow{\Theta}\right)\Theta
U\overleftarrow{\Theta}\left(\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle\right)|^{2},\end{split}$
(8)
where $\Theta
U\overleftarrow{\Theta}=\mathrm{T}\exp\\{-i\int_{t_{0}}^{t_{1}}H[\lambda(t_{0}+t_{1}-t)]dt\\}=(U^{\dagger})^{\ast}=U^{T}$
is the time-reversed unitary matrix. Because of the property of the time-
reversed transformation
$\Theta\left|i_{n},\lambda_{n}\right\rangle=\left|i_{n},\lambda_{n}\right\rangle^{\ast}$,
and the property of the Hermitian conjugate matrix,
$(\left\langle
i_{n},\lambda_{n}\right|)^{\ast}U^{T}(\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)^{\ast}\equiv\left\langle
i_{n}^{\prime},\lambda_{n+1}\right|U\left|i_{n},\lambda_{n}\right\rangle$ (9)
it is not difficult to prove that
$\frac{P_{F}(\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)}{P_{R}(\Theta\left|i_{n},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)}\equiv
1.$ (10)
Based on the above two results (6), (10) and Eqs. (3) and (5), we reproduce
the Crooks FT for a quantum mechanical system
$\frac{P_{F}(A\rightarrow B)}{P_{R}(A\leftarrow B)}=e^{\beta(W-\Delta F)}.$
(11)
From Eq. (11) we group all those trajectories with the same amount of
microscopic work, and obtain
$\frac{P_{F}(W|_{a})}{P_{R}(-W|_{-a})}=e^{\beta(a-\Delta F)}.$ (12)
Eq. (12) is the Crooks FT. Similar to the derivation in Ref. crooks , we
obtain the JE for a quantum open system straightforwardly $\left\langle
e^{-\beta W}\right\rangle=e^{-\beta\Delta F}$ from $\int P_{R}(-W|_{-a})da=1$.
Here, we would like to emphasize that though quantum generalization of Crooks
FT and JE have been reported in some previous work, the explicit consideration
of the influence of the heat bath, i.e., the explicit expression of
microscopic work in the presence of a heat bath has not been reported before.
Also the relation between quantum and classical trajectories are not addressed
clearly. Hence our quantum mechanical extensions of Crooks FT and JE are
highly nontrivial.
## IV Illustration of quantum Crooks FT and quantum JE in a two-level system
Figure 2: (Color Online) Microscopic work distribution $P_{F}(W)$ of forward
trajectories (solid lines), and the negative reverse work distribution
$P_{R}(-W)$ of their corresponding time-reversed trajectories (dashed lines).
The probabilities have been normalized. Here we fix $\Delta(t_{0})$ and
$\Delta(t_{N})$. Different distributions represent different controlling time
(the more steps, the longer control time). The controlling steps are chosen to
be $N=5$ (red $\bullet$), $N=10$ (blue $\bigtriangleup$), $N=15$ (green
$\square$), and $N=20$ (black $\bigcirc$). It can be seen that the work
distributions for both forward and reversed trajectories are not Gaussian.
Moreover, with the decrease of the controlling speed, the fluctuation of the
distributions decreases, and the difference between the work distribution of
the forward and time-reversed trajectories becomes less obvious. The
corresponding forward and negative reverse work distribution cross at
$W=\Delta F$, and this is a direct consequence of the quantum Crooks FT. The
free energy difference $\Delta F$ ia marked by the red vertical dash-dotted
line.
Having generalized the Crooks FT and JE to quantum systems in the presence of
a heat bath. In the following, we use the studies based on a two-level system
quan08 as an illustration to demonstrate our main idea. The Hamiltonian of
the two-level system is $H=\Delta(t)\left(\sigma_{z}+1\right)/2$, where
$\Delta(t)$ is the parameter of the Hamiltonian, and $\sigma_{z}$ is Pauli
matrix. The initial and final value of the parameter are
$\Delta_{A}=\Delta(t_{0})$ and $\Delta_{B}=\Delta(t_{N})$ respectively. The
controlling scheme is the same as that in Ref. quan08 : We divide the whole
process into N even steps. Hence the parameter in the $n$th step is
$\Delta(t_{n})=\Delta(t_{0})+n\Delta$, $n=1$, $2$, $\cdots$, $N$, where
$\Delta=(\Delta_{B}-\Delta_{A})/N$ is the change of the parameter in every
step. Every step consists of two substeps: the controlling substep, in which
we change the parameter from $\Delta(t_{n})$ to
$\Delta_{n+1}=\Delta(t_{n})+\Delta$, and the relaxation substep. For
simplicity, we consider the case where the system reaches thermal equilibrium
with the heat bath in every relaxation substep. Hence, the probability for the
forward and reverse relaxation substep can be expressed as
$P_{F}(\left|i_{n-1}^{\prime},\lambda_{n}\right\rangle\rightarrow\left|i_{n},\lambda_{n}\right\rangle)=e^{-\beta
E(i_{n},\lambda_{n})}/(\sum_{i}e^{-\beta E(i,\lambda_{n})})$, and
$P_{R}(\Theta\left|i_{n-1}^{\prime},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n},\lambda_{n}\right\rangle)=e^{-\beta
E(i_{n-1}^{\prime},\lambda_{n})}/(\sum_{i}e^{-\beta E(i,\lambda_{n})})$. Also
we assume the quantum adiabatic conditions are satisfied in every controlling
substep. That is
$P_{F}(\left|i_{n},\lambda_{n}\right\rangle\rightarrow\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)=\delta_{i_{n},i_{n}^{\prime}}$,
and
$P_{R}(\Theta\left|i_{n},\lambda_{n}\right\rangle\leftarrow\Theta\left|i_{n}^{\prime},\lambda_{n+1}\right\rangle)=\delta_{i_{n},i_{n}^{\prime}}$.
Based on these assumptions, the microscopic work distribution for the forward
trajectories can be obtained quan08
$P_{F}(W|_{k\Delta})=P^{F}_{e}\prod_{l=0}^{N-k-1}\frac{e^{\beta\Delta_{B}}-e^{\beta(\Delta_{A}+l\Delta)}}{e^{\beta(l+1)\Delta}-1},$
(13)
where
$P^{F}_{e}=\prod_{j=1}^{N}\frac{e^{-\beta[\Delta_{A}+(j-1)\Delta]}}{1+e^{-\beta[\Delta_{A}+(j-1)\Delta]}},k=0,1,2,\cdots,N.$
(14)
Similarly, the microscopic work distribution for the time-reversed trajectory
can be expressed as
$P_{R}(-W|_{-k\Delta})=P^{R}_{e}\prod_{l=0}^{N-k-1}\frac{e^{\beta\Delta}[e^{\beta\Delta_{B}}-e^{\beta(\Delta_{A}+l\Delta)}]}{e^{\beta(l+1)\Delta}-1},$
(15)
where
$P^{R}_{e}=\prod_{j=1}^{N}\frac{e^{-\beta[\Delta_{B}-(j-1)\Delta]}}{1+e^{-\beta[\Delta_{B}-(j-1)\Delta]}},k=0,1,2,\cdots,N.$
(16)
We plot the above distributions (13) and (14) of microscopic work in Fig. 2.
Here the probability distribution in the excited state are
$P_{e}(\Delta_{A})=e^{-\beta\Delta_{A}}/(1+e^{-\beta\Delta_{A}})=1/3$, and
$P_{e}(\Delta_{B})=e^{-\beta\Delta_{B}}/(1+e^{-\beta\Delta_{B}})=1/5$. The
free energy difference is $\Delta
F_{AB}=\left[\ln(1+1/2)-\ln(1+1/4)\right]k_{B}T\approx 0.263\ln 2k_{B}T$. It
can be seen (see Fig. 2) that the corresponding forward and negative reverse
work distributions cross at $W=\Delta F$, no matter what the controlling
protocol is, and this result is a direct consequence of Crooks FT. It should
be pointed out that the work distributions (13) and (15) are non-Gaussian
quan08 . Hence, the processes discussed here are beyond the linear response
regime. Yet we will see both Crooks FT and JE holds. We also plot the
logarithm of the ratio of the forward and negative reverse work distribution
(See Fig. 3(a)). It can be seen that all data collapse onto the same straight
line. In addition, the slope of the line is equal to unit, and the line cross
the horizontal axis at $W=0.263\ln 2k_{B}T=\Delta F_{AB}$. Thus our numerical
simulation confirms the validity of quantum Crooks FT when the process is
beyond the linear response regime. We also plot the logarithm of the exponent
averaged work $\ln\left\langle e^{-\beta W}\right\rangle$ and averaged work
$\left\langle W\right\rangle$ of the forward process (see Fig. 3(b)) to test
the validity of quantum JE. It can be seen that the averaged work is greater
than the free energy difference $\left\langle W\right\rangle\geqslant\Delta
F$, while the logarithm of the exponent averaged work is identical to the
difference of the free energy $\ln\left\langle e^{-\beta
W}\right\rangle\equiv\Delta F\approx 0.1823k_{B}T$ no matter what the
controlling protocol is. Hence, Fig. 3(b) verifies quantum JE when the process
is beyond the linear response regime.
Figure 3: (Color Online) (a) The logarithm of the probabilities of forward and
time-reversed trajectories as a function of work. It can be seen that all data
of different work and different control protocols ($N=5$ (red $\bullet$),
$N=10$ (blue $\bigtriangleup$), $N=15$ (green $\square$), and $N=20$ (black
$\bigcirc$) ) collapse onto the same straight line. The slop of the line is
equal to unity, and the line cross the horizontal axes at $W=\Delta F$. Thus
the numerical result verifies the quantum Crooks FT
$\ln\left[P_{F}(W|_{a})/P_{R}(-W|_{-a})\right]=\beta(a-\Delta F)$. (b) The
averaged work VS. the logarithm of averaged exponent work for different
control protocols. It can be seen that the averaged work $\left\langle
W\right\rangle$ (red $\bigcirc$) is always greater than the difference of free
energy $\Delta F_{AB}$ and differ from one control protocol to another, while
the logarithm of the exponentially averaged work $\ln\left\langle\exp[-\beta
W]\right\rangle$ (blue $\square$) is always equivalent to the difference of
free energy irrespective of the control protocols. Thus the numerical result
verifies the JE $\ln{\left\langle\exp[-\beta W]\right\rangle}\equiv\Delta F$.
## V Conclusion and remarks
In this paper, we explicitly consider the quantum Crooks FT and quantum JE in
the presence of an external heat bath. Our proof includes the proof of
classical Crooks FT as a special case. When the quantum adiabatic conditions
are satisfied, we reproduce the result of Crooks FT and JE for classical
systems. Our work indicates that in quantum systems, the probabilities (Eqs.
(3) and (5)) comes from the quantum non-adiabatic transition and statistical
mechanical randomness, while in classical system, the randomness only comes
from the later case. We use the two-level system as an illustration to
demonstrate the validity of quantum Crooks FT and quantum JE beyond the linear
response regime.
Before concluding the paper, we would like to mention the following points.
First, though the quantum non-adiabatic transition is introduced into the
controlling substep, this substep is time reversal symmetric. I. e., all the
time asymmetry is due the relaxation substep (statistical mechanical
randomness), rather than the controlling substep (quantum non-adiabatic
transition). This is the same as the classical case. Second, when we change
the Hamiltonian slowly, we reproduce the proof of Crooks for classical
systems. In this sense, we say that our proof includes the classical Crooks FT
and classical JE as a special case. Third, for classical system, the Crooks FT
and JE have been experimentally verified classicalexperiment . However, for a
quantum mechanical system, the experimental exploration on Crooks FT and JE
has not been reported (an exception is schmidt ). This perhaps is mainly due
to the fact that microscopic work in a quantum mechanical system is not a well
defined observable hanggi . There is no well defined pressure or force for a
quantum system quan0811 . Hence, we cannot follow the way that we do in
classical system to measure the force and make the integral of the force by
the extension. On the contrary, we will have to introduce quantum measurement
processes to confirm the initial and final energy of the system and then
calculate the microscopic work done from the difference of the initial and
final energy difference mukemal . Fourth, though the numerical simulations
consider only the special cases: 1) the system reach thermal equilibrium with
the heat bath in every relaxation substep, and 2) the quantum adiabatic
conditions are satisfied in every controlling substep, the quantum Crooks FT
and quantum JE are not constrained in these special cases. Finally, our
numerical simulations based on a two-level system can possibly be testified by
employing Josephson junction charge qubit chargequbit . Discussion about
employing Josephson Junction qubit to test the quantum Crooks FT and quantum
JE will be given later.
## VI acknowledgments
HTQ thanks Wojciech H. Zurek, G. Crooks and Rishi Sharma for stimulating
discussions and gratefully acknowledges the support of the U.S. Department of
Energy through the LANL/LDRD Program for this work.
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* (11) For classical systems, if the forward process is described by a trajectory in the phase space $(\vec{p}_{0},\vec{q}_{0})\rightarrow(\vec{p}_{1},\vec{q}_{1})$ as the Hamiltonian is changed from $H(\lambda_{0})$ to $H(\lambda_{1})$. The time-reversed trajectory is $(-\vec{p}_{1},\vec{q}_{1})\rightarrow(-\vec{p}_{0},\vec{q}_{0})$ as the Hamiltonian is changed from $H(\lambda_{1})$ to $H(\lambda_{0})$. For quantum systems, if the forward trajectory is $\left|\psi(t_{0})\right\rangle\rightarrow\left|\psi(t_{1})\right\rangle$ as the Hamiltonian is changed from $H(\lambda_{0})$ to $H(\lambda_{1})$, the time-reversed trajectory is $\Theta\left|\psi(t_{1})\right\rangle\rightarrow\Theta\left|\psi(t_{0})\right\rangle$ when the Hamiltonian is changed from $H(\lambda_{1})$ to $H(\lambda_{0})$ sakurai .
* (12) J. J. Sakurai, _Modern Quantum Mechanics_ (Revised Edition), (Reading, Addison-Wesley, 1994).
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|
arxiv-papers
| 2008-12-29T19:09:48 |
2024-09-04T02:48:59.617256
|
{
"license": "Public Domain",
"authors": "H. T. Quan, and H. Dong",
"submitter": "Haitao Quan",
"url": "https://arxiv.org/abs/0812.4955"
}
|
0812.4985
|
# On the Capacity of Partially
Cognitive Radios
G. Chung, S. Sridharan, and S. Vishwanath Wireless Networking and
Communication Group
University of Texas at Austin
Austin, TX 78712, USA
Email: {gchung,sridhara,sriram}@ece.utexas.edu C. S. Hwang Communication
Lab., SAIT
Samsung Electronics Co. Ltd.
Yongin, Korea
Email: cshwang@samsung.com
###### Abstract
This paper considers the problem of cognitive radios with partial-message
information. Here, an interference channel setting is considered where one
transmitter (the “cognitive” one) knows the message of the other (“legitimate”
user) partially. An outer bound on the capacity region of this channel is
found for the “weak” interference case (where the interference from the
cognitive transmitter to the legitimate receiver is weak). This outer bound is
shown for both the discrete-memoryless and the Gaussian channel cases. An
achievable region is subsequently determined for a mixed interference Gaussian
cognitive radio channel, where the interference from the legitimate
transmitter to the cognitive receiver is “strong”. It is shown that, for a
class of mixed Gaussian cognitive radio channels, portions of the outer bound
are achievable thus resulting in a characterization of a part of this
channel’s capacity region.
Note that results in this paper specialize to the case of the weak/mixed
interference channel and the cognitive radio channel with full-message
information. 111This work is supported by a grant from Samsung Advanced
Institute of Technology.
## I Introduction
A cognitive radio is one that possesses information that allows it to tailor
its transmission to maximize network throughput while meeting constraints
imposed on it [1]. There are multiple notions of cognition in literature [1],
[2] with an increasingly popular strategy known as overlay cognition, where
both the cognitive and the legitimate users transmit their own messages in the
same sub-band simultaneously, as in [3]. In this setting, the cognitive
transmitter has access to (limited) information about the legitimate user so
as to mitigate network interference and thus increase overall throughput.
In previous work, the class of interference channels with degraded message
sets has been considered [6], where the cognitive user has access to the
entire message of the legitimate user. Examples of this setting include [7],
where the authors determine the capacity region of this channel for both the
case of “weak” interference and for a class of “strong” interference channels.
However, the paper’s assumption of perfect and complete message information
should be relaxed in order to apply the ideas and concepts to more general
classes of cognitive radio channels.
This paper considers a cognitive radio channel model where the cognitive radio
is not fully cognitive of the other transmitter’s message set. In this
setting, the cognitive radio has access only to a portion of the message. Note
that as this portion varies from nothing to everything, it includes the
interference channel (IFC) in literature [8], [9], [10], and IFC with fully-
degraded message set [7] as special cases. This channel is referred to as an
interference channel with a partially cognitive transmitter. Note that this
channel model is motivated by practical constraints, where the cognitive
transmitter is only able to garner limited information about the legitimate
transmitter’s message.
The interference channel with a partially cognitive transmitter has already
been studied in [4], with a specific focus on strong interference settings.
This paper focuses on the weak and mixed interference settings. Specifically,
we derive an outer bound on the capacity region of this channel for both the
discrete memoryless and Gaussian cases when the interference from the
cognitive transmitter to the legitimate receiver is “weak”. Subsequently, we
show for the Gaussian case that Gaussian distributions satisfying the
constraints on the inputs/auxiliary random variables which makes the outer
bound extreme exist. Finally, for a special class of mixed interference
channels (where the interference from the cognitive transmitter to the
legitimate receiver is “weak” and that from the legitimate transmitter to the
cognitive receiver is “strong”), we show that a portion of the capacity region
can be characterized, i.e., a non-trivial subset of the outer bound is
achievable.
This paper is organized as follows. The next section details the system model
and notations used in the paper. In Section III, we describe an outer bound on
the partially cognitive radio channel for the discrete memoryless case and for
the Gaussian channel. In Section IV, we describe an achievable region for the
Gaussian partially cognitive radio channel. In Section V, we derive channel
conditions under which the achievable region is optimal. We conclude in
Section IV.
## II System Model and Preliminaries
The notation used in this paper is based largely on that of [7]. Random
variables (RVs) are denoted by capital letters, and their realizations using
the corresponding lower case letters. $X_{m}^{n}$ denotes the random vector
$(X_{m},...,X_{n})$, $X^{n}$ denotes the random vector $(X_{1},...,X_{n})$,
and $X^{n\backslash m}$ denotes the random vector
$(X_{1},...,X_{m-1},X_{m+1},...,X_{n})$. Also, for any set $S$, $\overline{S}$
denotes the convex hull of $S$, and $\widetilde{S}$ means the complementary
set of $S$. Finally, the notation $X\Rightarrow Y\Rightarrow Z$ is used to
denote that $X$ and $Z$ are conditionally independent given $Y$.
### II-A Discrete Memoryless Partially Cognitive Radio Channels
A two user interference channel as in Fig. 1 is a quintuple
$(\mathcal{X}_{1},\mathcal{X}_{2},\mathcal{Y}_{1},\mathcal{Y}_{2},p)$, where
$\mathcal{X}_{1},\mathcal{X}_{2}$ are two input alphabet sets;
$\mathcal{Y}_{1},\mathcal{Y}_{2}$ are two output alphabet sets;
$p(y_{1},y_{2}|x_{1},x_{2})$ is a transition probability. Since we confine
channel to be memoryless, the transition probability of $y_{1}^{n},y_{2}^{n}$
given $x_{1}^{n},x_{2}^{n}$ is
$\displaystyle
p(y_{1}^{n},y_{2}^{n}|x_{1}^{n},x_{2}^{n})=\displaystyle\prod_{i=1}^{n}p(y_{1,i},y_{2,i}|x_{1,i},x_{2,i})$
Figure 1: The discrete memoryless partially cognitive radio model
This channel model is similar to that of an interference channel with the
difference being the message sets at each transmitter. Transmitter 1 is the
legitimate user, who communicates messages from the sets
$W_{0}\in\\{1,...,M_{0}\\}$ and $W_{1}\in\\{1,...,M_{1}\\}$ to Receiver 1, the
legitimate receiver. Transmitter 2, the cognitive transmitter communicates
messages $W_{2}\in\\{1,...,M_{2}\\}$ to Receiver 2, the cognitive receiver.
The unique feature of this channel is that the realization of $W_{0}$ is known
to both Transmitters 1 and 2, which allows for partial unidirectional
cooperation between the transmitters. An
$(R_{0},R_{1},R_{2},n,P_{e,0},P_{e,1},P_{e,2})$ code is any code with the rate
vector $(R_{0},R_{1},R_{2})$ and block size $n$, where
$R_{t}\triangleq\log(M_{t})/n$ bits per usage for $t=0,1,2$. As discussed
above, $W_{0}$, and $W_{1}$ are the messages from Receiver 1 which must be
decoded with (average) probabilities of error of at most $P_{e,0},P_{e,1}$
respectively, and $W_{2}$ must be retrieved at Receiver 2 while suffering an
error probability of no more than $P_{e,2}$. Rate pair $(R_{0},R_{1},R_{2})$
is said to be achievable if the error probabilities $P_{e,t}$ for $t=0,1,2$
can be made arbitrarily small as the block size $n$ grows. The capacity region
of the interference channel with partially cognitive transmitter is the
closure of the set of all achievable rate pairs $(R_{0},R_{1},R_{2})$. The
main goal of the users, legitimate and cognitive, is to maximize in general
the ${\mu}_{0}R_{0}+{\mu}_{1}R_{1}+{\mu}_{2}R_{2}$ for some non-negative
number ${\mu}_{0},{\mu}_{1}$, and ${\mu}_{2}$. We also have a restriction on
the pair $(R_{0},R_{1})$, such that $R_{1}\geq\mu R_{0}$ for some positive
number $\mu$. This restriction is to ensure that optimization of
$({\mu}_{0},{\mu}_{1},{\mu}_{2})$ in order to maximize
${\mu}_{0}R_{0}+{\mu}_{1}R_{1}+{\mu}_{2}R_{2}$ does not drive $R_{1}$ to zero,
which results in a fully cognitive solution.
### II-B Gaussian Partially Cognitive Radio Channel
In the Gaussian IFC, input and output alphabets are the reals $\mathbb{R}$,
and outputs are the linear combination of the inputs and additive white
Gaussian noise. A Gaussian IFC model in Fig 2. is characterized mathematically
as follows:
$\displaystyle\ Y_{1}$ $\displaystyle=X_{1}+bX_{2}+Z_{1}$ $\displaystyle
Y_{2}$ $\displaystyle=aX_{1}+X_{2}+Z_{2},$ (1)
where $a$ and $b$ are real numbers and $Z_{1}$ and $Z_{2}$ are independent,
zero-mean, unit-variance Gaussian random variables. Further, each transmitter
has a power constraint
$\displaystyle\frac{1}{n}\displaystyle\sum_{i=1}^{n}{\mathbb{E}}[X_{t,i}^{2}]\leq
P_{t},t=1,2.$
Figure 2: The Gaussian partially cognitive radio channel
This concludes our description of the models considered in this paper. The
next section describes the outer bound on the capacity region for these
channels under “weak” interference.
## III The Outer Bound region
### III-A Discrete Memoryless Partially Cognitive Radio Channels
For a discrete memoryless channel, under the condition
$\displaystyle X_{2}|X_{1}\Rightarrow Y_{1}|X_{1}\Rightarrow Y_{2}|X_{1},$ (2)
we say that the legitimate receiver is observing weak interference. For the
Gaussian case, the weak interference constraint can be interpreted as the
requirement of $b<1$ in (1). First, we reproduce a useful lemma from [5].
###### Lemma 1 ([5])
The following forms a Markov chain for the partially cognitive radio channel:
$\displaystyle(W_{0},W_{t})\Rightarrow(W_{0},X_{t})\Rightarrow Y_{t}$ (3)
where $t=1,2$.
We present the outer bound in the following:
###### Theorem 1
The convex closure of the following inequalities defines an outer bound on the
capacity region of “weak” partially cognitive radio channels:
$\displaystyle R_{0}$ $\displaystyle\leq I(U,X_{1};Y_{1}|V)$ (4)
$\displaystyle R_{0}+R_{1}$ $\displaystyle\leq I(U,X_{1};Y_{1})$ (5)
$\displaystyle R_{2}$ $\displaystyle\leq I(X_{2};Y_{2}|U,X_{1})$ (6)
$\displaystyle R_{1}$ $\displaystyle\geq\mu R_{0}$ (7)
for any $p(u,v))p(x_{1}|u,v)p(x_{2}|u)$ such that:
1\. $V$ and $X_{2}$ are independent
2\. $X_{1}$ is a function of $U$ and $V$
3\. $(U,V)\Rightarrow(X_{1},X_{2})\Rightarrow(Y_{1},Y_{2})$.
Proof: First we prove the outer bound for $R_{0}$ in (5) and $R_{2}$ in (7).
We have
$\displaystyle nR_{0}$ $\displaystyle=H(W_{0}|W_{1})$ $\displaystyle\leq
I(W_{0};Y_{1}^{n}|W_{1})+n\epsilon_{0}$
$\displaystyle=\displaystyle\sum_{i=1}^{n}[H(Y_{1,i}|Y_{1}^{i-1},W_{1})-H(Y_{1,i}|Y_{1}^{i-1},W_{0},W_{1})]+n\epsilon_{0}$
$\displaystyle\leq\begin{array}[]{l}\displaystyle\sum_{i=1}^{n}[H(Y_{1,i}|W_{1})-H(Y_{1,i}|Y_{1}^{i-1},X_{1}^{n\backslash
i},W_{0},W_{1},X_{1,i})]\\\ +n\epsilon_{0}\end{array}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}\begin{array}[]{l}\displaystyle\sum_{i=1}^{n}[H(Y_{1,i}|W_{1})-H(Y_{1,i}|Y_{2}^{i-1},X_{1}^{n\backslash
i},W_{0},W_{1},X_{1,i})]\\\ +n\epsilon_{0}\end{array}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\displaystyle\sum_{i=1}^{n}[H(Y_{1,i}|V_{i})-H(Y_{1,i}|U_{i},V_{i},X_{1,i})]+n\epsilon_{0}$
$\displaystyle=\displaystyle\sum_{i=1}^{n}I(U_{i},X_{1,i};Y_{1,i}|V_{i})+n\epsilon_{0}$
where $(a)$ results from the conditional Markov chain
$Y_{2,i}|X_{1}^{n}\Rightarrow Y_{1}^{i-1}|X_{1}^{n}\Rightarrow
Y_{2}^{i-1}|X_{1}^{n}$, which can be derived from the Markov chain for the
weak interference channel, $X_{2}\Rightarrow Y_{1}\Rightarrow Y_{2}$, given
$X_{1}$ in (3) as in the proof of Lemma 3.6 in [7]. $(b)$ results from
identifying auxiliaries $U_{i}=(Y_{2}^{i-1},X_{1}^{n\backslash i},W_{0})$ and
$V_{i}={W_{1}}$. For $R_{2}$,
$\displaystyle nR_{2}$ $\displaystyle=H(W_{2}|W_{0})$ $\displaystyle\leq
I(W_{2};Y_{2}^{n}|W_{0})+n\epsilon_{2}$ $\displaystyle\leq
I(W_{2};Y_{2}^{n},X_{1}^{n}|W_{0})+n\epsilon_{2}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}I(W_{2};Y_{2}^{n}|X_{1}^{n},W_{0})+n\epsilon_{2}$
$\displaystyle=H(Y_{2}^{n}|X_{1}^{n},W_{0})-H(Y_{2}^{n}|X_{1}^{n},W_{0},W_{2})+n\epsilon_{2}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}H(Y_{2}^{n}|X_{1}^{n},W_{0})-H(Y_{2}^{n}|X_{1}^{n},W_{0},X_{2}^{n})+n\epsilon_{2}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{\leq}}\displaystyle\sum_{i=1}^{n}[H(Y_{2,i}|U_{i},X_{1,i})-H(Y_{2,i}|U_{i},X_{1,i},X_{2,i})]+n\epsilon_{2}$
$\displaystyle=\displaystyle\sum_{i=1}^{n}I(X_{2,i};Y_{2,i}|U_{i},X_{1,i})+n\epsilon_{2}$
where $(a)$ is due to the independence of $W_{2}$ and $X_{1}^{n}$, $(b)$ is
from Lemma 1, and $(c)$ comes from the same definition above of
$U_{i}={Y_{2}^{i-1},X_{1}^{n\backslash i},W_{0}}$. Next, we prove the outer
bound for the sum rate $R_{0}+R_{1}$ in (6). We have
$\displaystyle n(R_{0}+R_{1})$ $\displaystyle=H(W_{0},W_{1})$
$\displaystyle\leq I(W_{0},W_{1};Y_{1}^{n})+n\epsilon_{1}$
$\displaystyle=H(Y_{1}^{n})-H(Y_{1}^{n}|W_{0},W_{1})+n\epsilon_{1}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}H(Y_{1}^{n})-H(Y_{1}^{n}|W_{0},X_{1}^{n})+n\epsilon_{1}$
$\displaystyle=\begin{array}[]{l}\displaystyle\sum_{i=1}^{n}\left[\begin{array}[]{l}H(Y_{1,i}|Y_{1}^{i-1})\\\
-H(Y_{1,i}|Y_{1}^{i-1},X_{1}^{n\backslash
i},W_{0},X_{1,i})\end{array}\right]\\\ +n\epsilon_{1}\end{array}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\begin{array}[]{l}\displaystyle\sum_{i=1}^{n}\left[\begin{array}[]{l}H(Y_{1,i}|Y_{1}^{i-1})\\\
-H(Y_{1,i}|Y_{2}^{i-1},X_{1}^{n\backslash
i},W_{0},X_{1,i})\end{array}\right]\\\ +n\epsilon_{1}\end{array}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{\leq}}\displaystyle\sum_{i=1}^{n}[H(Y_{1,i})-H(Y_{1,i}|U_{i},X_{1,i})]+n\epsilon_{1}$
$\displaystyle=\displaystyle\sum_{i=1}^{n}I(U_{i},X_{1,i};Y_{1,i})+n\epsilon_{1}$
$(a)$ results from (Lemma 1), $(b)$ is due to the conditional Markov chain
$Y_{2,i}|X_{1}^{n}\Rightarrow Y_{1}^{i-1}|X_{1}^{n}\Rightarrow
Y_{2}^{i-1}|X_{1}^{n}$, and $(c)$ follows from the definition above of
$U_{i}={Y_{2}^{i-1},X_{1}^{n\backslash i},W_{0}}$. Note that the choice of
auxiliary random variables automatically satisfies the constraints imposed on
them in Theorem 1. Finally, (8) comes from the restriction on the
$(R_{0},R_{1})$, which is described in the section II.A.
### III-B Gaussian Partially Cognitive Radio Channel
First, note that similar proof will ensure the outer bound for the rate region
defined in Theorem 1 to be valid for the Gaussian partially cognitive radio
channel. The main details of proof are omitted here. Next, we establish three
lemmas that will be essential in proving the optimality of a jointly Gaussian
input distribution for the region defined in Theorem 1.
###### Lemma 2 (Lemma 1 in [11])
Let $X_{1},X_{2},...,X_{k}$ be arbitrarily distributed zero-mean random
variables with covariance matrix $K$. Let $S$ be any subset of
$\\{1,2,...,k\\}$ and $\widetilde{S}$ be its complement. Then
$\displaystyle h(X_{S}|X_{\widetilde{S}})\leq
h(X_{S}^{*}|X_{\widetilde{S}}^{*}),$ (8)
where $X_{1}^{*},X_{2}^{*},...,X_{k}^{*}\sim N(0,K)$.
###### Lemma 3
Let $X_{1},X_{2},V$ be an arbitrarily distributed zero-mean random variables
with covariance matrix $K$, where $X_{2}$ and $V$ is independent of each
other. Let $X_{1}^{*},X_{2}^{*},V^{*}$ be the zero mean Gaussian distributed
random variables with the same covariance matrix as $X_{1},X_{2},V$. Then,
$\displaystyle{\mathbb{E}}[X_{1}X_{2}]={\mathbb{E}}[X_{1}^{*}X_{2}^{*}|V^{*}]$
(9)
Proof: Without loss of generality $X_{1}^{*}$ can be written as
$X_{1}^{*}=W^{*}+cV^{*}$, where $W^{*}$ is the zero mean Gaussian random
variable independent of $V^{*}$. Then
$\displaystyle{\mathbb{E}}[X_{1}X_{2}]$
$\displaystyle={\mathbb{E}}[X_{1}^{*}X_{2}^{*}]$
$\displaystyle={\mathbb{E}}[{\mathbb{E}}[X_{1}^{*}X_{2}^{*}|V^{*}]]$
$\displaystyle={\mathbb{E}}[{\mathbb{E}}[(W^{*}+cV^{*})X_{2}^{*}|V^{*}]]$
$\displaystyle={\mathbb{E}}[{\mathbb{E}}[W^{*}X_{2}^{*}|V^{*}]]+c{\mathbb{E}}[{\mathbb{E}}[V^{*}X_{2}^{*}|V^{*}]]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}{\mathbb{E}}[X_{1}^{*}X_{2}^{*}|V^{*}]+c{\mathbb{E}}[V^{*}{\mathbb{E}}[X_{2}^{*}]]$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}{\mathbb{E}}[X_{1}^{*}X_{2}^{*}|V^{*}]$
where $(a)$ results from the independence of $X_{2}^{*}$ and $V^{*}$. And,
$(b)$ results from the fact that $X_{2}^{*}$ is zero mean.
###### Lemma 4
${\mathbb{E}}[X_{1}^{*}X_{2}^{*}|V^{*}]\leq({\mathbb{E}}[(X_{1}^{*})^{2}|V^{*}])^{\frac{1}{2}}({\mathbb{E}}[({\mathbb{E}}[X_{2}^{*}|X_{1}^{*}])^{2}])^{\frac{1}{2}}$
Proof: Note that
$\displaystyle{\mathbb{E}}[X_{1}^{*}X_{2}^{*}|V^{*}]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}{\mathbb{E}}[{\mathbb{E}}[X_{1}^{*}X_{2}^{*}|V^{*},X_{1}^{*}]]$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}{\mathbb{E}}[X_{1}^{*}{\mathbb{E}}[X_{2}^{*}|V^{*},X_{1}^{*}]|V^{*}]$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{\leq}}({\mathbb{E}}[(X_{1}^{*})^{2}|V^{*}])^{\frac{1}{2}}({\mathbb{E}}[({\mathbb{E}}[X_{2}^{*}|V^{*},X_{1}^{*}])^{2}])^{\frac{1}{2}}$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{\leq}}({\mathbb{E}}[(X_{1}^{*})^{2}|V^{*}])^{\frac{1}{2}}({\mathbb{E}}[({\mathbb{E}}[X_{2}^{*}|X_{1}^{*}])^{2}])^{\frac{1}{2}}$
where $(a)$ comes from the law of iterated expectations, $(b)$ from the
independence of $X_{2}^{*}$ and $V^{*}$, $(c)$ from the Cauchy-Schwartz
inequality, and $(d)$ from the fact that entropy can only be reduced by
conditioning.
###### Definition 1
Define the rate region $\mathcal{R}_{out}^{\alpha,\beta}$ to be the convex
hull of all rate pairs $(R_{0},R_{1},R_{2})$ satisfying
$\begin{array}[]{l}R_{0}\leq\frac{1}{2}\log\left(1+\frac{\beta
P_{1}+b^{2}(1-\alpha)P_{2}+2b\sqrt{(\beta(1-\alpha)P_{1}P_{2})}}{(1+b^{2}\alpha
P_{2})}\right)\\\
R_{0}+R_{1}\leq\frac{1}{2}\log\left(1+\frac{P_{1}+b^{2}(1-\alpha)P_{2}+2b\sqrt{(\beta(1-\alpha)P_{1}P_{2})}}{(1+b^{2}\alpha
P_{2})}\right)\\\ R_{2}\leq\log(\alpha P_{2}+1)\\\ R_{1}\geq\mu
R_{0}\end{array}$ (10)
for some $\alpha\in[0,1]$ and $\beta\in[0,1]$
###### Definition 2
Define the rate region $\mathcal{R}_{out}$ to be convex hull of the union of
rate region $\mathcal{R}_{out}^{\alpha,\beta}$:
$\mathcal{R}_{out}\triangleq\overline{\bigcup_{0\leq\alpha,\beta\leq
1}\mathcal{R}_{out}^{\alpha,\beta}}.$ (11)
We denote $\mathcal{C}$ to be the capacity region of the Gaussian weak
partially cognitive radio channel. An outer bound for $\mathcal{C}$ is
obtained as follows.
###### Theorem 2
$\mathcal{R}_{out}$ is an outer bound of the capacity region for the Gaussian
weak partially cognitive radio channel:
$\mathcal{C}\subset\mathcal{R}_{out}.$
Proof: We start from the rate region in Theorem 1.
$\displaystyle\ R_{0}$ $\displaystyle\leq
I(U,X_{1};Y_{1}|V)=h(Y_{1}|V)-h(Y_{1}|V,U,X_{1})$
$\displaystyle=h(Y_{1}|V)-h(Y_{1}|U,X_{1})$ (12) $\displaystyle R_{0}+R_{1}$
$\displaystyle\leq I(U,X_{1};Y_{1})=h(Y_{1})-h(Y_{1}|U,X_{1})$ (13)
$\displaystyle R_{2}$ $\displaystyle\leq
I(X_{2};Y_{2}|U,X_{1})=h(Y_{2}|U,X_{1})-h(N_{2})$ (14)
(III-B) follows from the Markov chain, $V\Rightarrow(U,X_{1})\Rightarrow
Y_{1}$. First, we set
$\displaystyle\ h(Y_{2}|U,X_{1})=\frac{1}{2}\log(2\pi e(1+\alpha P_{2}))$ (15)
without loss of generality for some $\alpha\in[0,1]$. Note that
$\displaystyle Y_{1}=b(X_{2}+Z_{1})+X_{1}+Z^{\prime}$ $\displaystyle
h(Y_{1}|U,X_{1})=h(b(X_{2}+Z_{1})+Z^{\prime}|U,X_{1}),$ (16)
where $b<1$ because legitimate receiver faces a weak interference, and
$Z^{\prime}$ is a Gaussian distributed random variable with variance
$1-b^{2}$. By entropy power inequality (EPI)[14], we have,
$\displaystyle 2^{2h(Y_{1}|U,X_{1})}$ $\displaystyle\geq
2^{2h(bY_{2}|U,X_{1})}+2^{2h(Z^{\prime})}.$
$\displaystyle=b^{2}2^{2h(Y_{2}|U,X_{1})}+2\pi e(1-b^{2})$ $\displaystyle=2\pi
e(1+b^{2}\alpha P_{2}),$
which yields
$\displaystyle h(Y_{1}|U,X_{1})\geq\frac{1}{2}\log(2\pi e(1+b^{2}\alpha
P_{2})).$ (17)
Next, we need to bound $h(Y_{1})$ and $h(Y_{1}|V)$. Note that by setting
$h(Y_{2}|U,X_{1})=\frac{1}{2}\log(2\pi e(1+\alpha P_{2}))$ we have the
following result.
$\displaystyle\ h(Y_{2}|U,X_{1})$ $\displaystyle\leq h(X_{2}+Z_{2}|X_{1})$
$\displaystyle\leq h(X_{2}^{*}+Z_{2}|X_{1}^{*})$
$\displaystyle=\frac{1}{2}\log(2\pi e(1+\mathop{\rm
Var}(X_{2}^{*}|X_{1}^{*}))),$ (18)
where $\mathop{\rm Var}(\cdot|\cdot)$ denotes the conditional covariance.
Combining (15) with (III-B), we obtain the bound
$\displaystyle\ \mathop{\rm Var}(X_{2}^{*}|X_{1}^{*})\geq\alpha P_{2}.$ (19)
Also,
$\displaystyle\ \mathop{\rm
Var}(X_{2}^{*}|X_{1}^{*})={\mathbb{E}}[{(X_{2}^{*})}^{2}]-{\mathbb{E}}[({\mathbb{E}}[X_{2}^{*}|X_{1}^{*}])^{2}].$
(20)
From (19) and (20), we obtain,
$\displaystyle{\mathbb{E}}[({\mathbb{E}}[X_{2}^{*}|X_{1}^{*}])^{2}]\leq(1-\alpha)P_{2}.$
(21)
Again, we set ${\mathbb{E}}[(X_{1}^{*})^{2}|V^{*}]=\beta P_{1}$ for some
$\beta\in[0,1]$ without loss of generality. Now combining Lemma 3, Lemma 4,
and the above results,
$\displaystyle{\mathbb{E}}[X_{1}X_{2}]\leq\sqrt{(\beta
P_{1})}\sqrt{(1-\alpha)P_{2}}.$ (22)
Therefore, we obtain the bound for $h(Y_{1})$ as
$\displaystyle h(Y_{1})$ $\displaystyle\leq\frac{1}{2}\log\left(2\pi
e\left(\begin{array}[]{l}1+\mathop{\rm Var}(X_{1})+b^{2}\mathop{\rm
Var}(X_{2})\\\ +2b{\mathbb{E}}[X_{1}X_{2}]\end{array}\right)\right)$ (25)
$\displaystyle\leq\frac{1}{2}\log\left(2\pi
e\left(\begin{array}[]{l}1+P_{1}+b^{2}P_{2}\\\
+2b\sqrt{\beta(1-\alpha)P_{1}P_{2}}\end{array}\right)\right)$ (28)
For $h(Y_{1}|V)$, note that $(Y_{1}^{*},V^{*})$ has the same covariance matrix
as $(Y_{1},V)$ if $Y_{1}=X_{1}^{*}+bX_{2}^{*}$. Also, $Y_{1}$ is a mean zero
Gaussian distributed random variable. Thus,
$\displaystyle h(Y_{1}|V)\leq$ $\displaystyle h(Y_{1}^{*}|V^{*})$
$\displaystyle=$ $\displaystyle h(X_{1}^{*}+bX_{2}^{*}+Z_{1}|V^{*})$
$\displaystyle=$ $\displaystyle\frac{1}{2}\log\left(2\pi
e\left(\begin{array}[]{l}1+\mathop{\rm Var}(X_{1}^{*}|V^{*})\\\
+b^{2}\mathop{\rm Var}(X_{2}^{*}|V^{*})\\\
+2b{\mathbb{E}}[X_{1}^{*}X_{2}^{*}|V^{*}]\end{array}\right)\right)$ (32)
$\displaystyle\leq$ $\displaystyle\frac{1}{2}\log\left(2\pi
e\left(\begin{array}[]{l}1+\beta P_{1}+b^{2}P_{2}\\\
+2b\sqrt{(\beta(1-\alpha)P_{1}P_{2})}\end{array}\right)\right),$ (35)
which gives the desired outer bound for the capacity region.
## IV Achievable Region for the Gaussian Channel
In this section, we describe an achievable region for the Gaussian channel
model described in (II-B). In deriving the achievable region, we combine
superposition coding and dirty paper coding [13]. The legitimate transmitter
encodes messages $W_{0}$ and $W_{1}$ using Gaussian codebooks and superimposes
them to form its final codeword. The cognitive transmitter allocates a portion
of the power in communicating message $W_{0}$ to the legitimate receiver. The
remaining power is used in encoding its own message $W_{2}$ using dirty paper
coding treating the codewords (from $W_{0}$) as non-causally known
interference. Then the following two definitions and theorem present the
achievable region for the Gaussian partially cognitive radio channel.
###### Definition 3
Define the rate region $\mathcal{R}_{i}^{\alpha,\beta}$ to be the convex hull
of all rate pairs $(R_{0},R_{1},R_{2})$ satisfying
$\begin{array}[]{l}R_{0}\leq\frac{1}{2}\log\left(1+\frac{\beta
P_{1}+b^{2}(1-\alpha)P_{2}+2b\sqrt{\beta(1-\alpha)P_{1}P_{2}}}{1+b^{2}\alpha
P_{2}}\right)\\\ R_{1}\leq\frac{1}{2}\log\left(1+\frac{(1-\beta)P_{1}}{1+\beta
P_{1}+b^{2}P_{2}+2b\sqrt{\beta(1-\alpha)P_{1}P_{2}}}\right)\\\
R_{1}\leq\frac{1}{2}\log\left(1+\frac{a^{2}(1-\beta)P_{1}}{1+a^{2}\beta
P_{1}+P_{2}+2a\sqrt{\beta(1-\alpha)P_{1}P_{2}}}\right)\\\
R_{2}\leq\frac{1}{2}\log(1+\alpha P_{2})\end{array}$ (36)
for some $\alpha\in[0,1]$ and $\beta\in[0,1]$.
###### Definition 4
Define the rate region $\mathcal{R}_{i}$ to be convex hull of the union of
rate region $\mathcal{R}_{i}^{\alpha,\beta}$:
$\mathcal{R}_{i}\triangleq\overline{\bigcup_{0\leq\alpha,\beta\leq
1}\mathcal{R}_{i}^{\alpha,\beta}}.$ (37)
###### Theorem 3
For the Gaussian channel with partially cognitive radio as described in
(II-B), the region described by
$\mathcal{R}_{in}=\left\\{(R_{0},R_{1},R_{2})\in\mathcal{R}_{i}:R_{1}\geq\mu
R_{0}\right\\}$ (38)
is achievable.
Proof: In proving the theorem, we use an encoding strategy that combines
superposition coding and dirty paper coding. We first describe the encoding
strategy at the two transmitters.
Encoding Strategy at legitimate transmitter: For every message
$W_{0}\in\\{1,\ldots,M_{0}\\}$, the legitimate transmitter generates a
codeword $X_{10}^{n}(W_{0})$ from the distribution
$p(X_{10}^{n})=\Pi_{i=1}^{n}p(X_{10}(i))$ and
$X_{10}(i)\sim\mathcal{N}(0,\beta P_{1})$ for some $0\leq\beta\leq 1$. For
every message $W_{1}\in\\{1,\ldots,M_{1}\\}$, the legitimate transmitter
generates a codeword $X_{11}^{n}(W_{1})$ from the distribution
$p(X_{11}^{n})=\Pi_{i=1}^{n}p(X_{11}(i))$ and
$X_{11}(i)\sim\mathcal{N}(0,(1-\beta)P_{1})$. The legitimate transmitter then
superimposes these codewords to form the net codeword $X_{1}^{n}$ as
$X_{1}^{n}=X_{10}^{n}+X_{11}^{n}.$
Encoding strategy at cognitive transmitter: The cognitive transmitter
allocates a portion of its power in communicating the message $W_{0}$ to the
legitimate receiver. For message $W_{0}$, the cognitive transmitter generates
a codeword $X_{20}^{n}(W_{0})$ as follows:
$X_{20}^{n}(W_{0})=\sqrt{\frac{(1-\alpha)P_{2}}{\beta
P_{1}}}X_{10}^{n}(W_{0}).$
That is, the cognitive transmitter uses the same codeword for encoding message
$W_{0}$ as used by the legitimate transmitter except that it is scaled to
power $(1-\alpha)P_{2}$ for some $0\leq\alpha\leq 1$. Next, the cognitive
transmitter encodes message $W_{2}$ to codeword $X_{22}^{n}$. The codeword is
generated using dirty paper coding treating $aX_{10}^{n}+X_{20}^{n}$ as non-
causally known interference. A characteristic feature of Costa’s dirty paper
coding is that the codeword $X_{22}^{n}$ is independent of the interference
$X_{20}^{n}+aX_{10}^{n}$, and is distributed as
$p(X_{22}^{n})=\Pi_{i=1}^{n}p(X_{22}(i))$ and
$X_{22}(i)\sim\mathcal{N}(0,\alpha P_{2})$. The cognitive transmitter
superimposes the two codewords $X_{20}^{n}$ and $X_{22}^{n}$ to form its net
codeword $X_{2}^{n}$. That is,
$X_{2}^{n}=X_{20}^{n}+X_{22}^{n}.$
Next, we describe the decoding strategy and the rate constraints associated at
the two receivers.
Decoding strategy at legitimate receiver: The legitimate receiver obtains the
signal
$Y_{1}^{n}=X_{10}^{n}+X_{11}^{n}+bX_{20}^{n}+bX_{22}^{n}+Z_{1}^{n}.$
The receiver first decodes message $W_{1}$ treating $X_{10}^{n},X_{20}^{n}$
and $X_{22}^{n}$ as Gaussian noise. After decoding message $W_{1}$, the
receiver decodes message $W_{0}$ by treating $X_{22}^{n}$ as Gaussian noise
after canceling out $X_{11}^{n}$. In the first stage, the receiver can decode
message $W_{1}$ successfully if
$R_{1}\leq\frac{1}{2}\log\left(1+\frac{(1-\beta)P_{1}}{1+\beta
P_{1}+b^{2}P_{2}+2b\sqrt{\beta(1-\alpha)P_{1}P_{2}}}\right).$ (39)
Similarly, the receiver can decode message $W_{0}$ successfully if
$R_{0}\leq\frac{1}{2}\log\left(1+\frac{\beta
P_{1}+b^{2}(1-\alpha)P_{2}+2b\sqrt{\beta(1-\alpha)P_{1}P_{2}}}{1+b^{2}\alpha
P_{2}}\right).$ (40)
Decoding strategy at cognitive receiver: The cognitive receiver obtains the
signal
$Y_{2}^{n}=aX_{10}^{n}+aX_{11}^{n}+X_{20}^{n}+X_{22}^{n}+Z_{2}^{n}.$
Similar to the legitimate receiver, the cognitive receiver first decodes
message $W_{1}$ treating $X_{10}^{n},X_{20}^{n}$ and $X_{22}^{n}$ as Gaussian
noise. The receiver can decode message $W_{1}$ successfully if
$R_{1}\leq\frac{1}{2}\log\left(1+\frac{a^{2}(1-\beta)P_{1}}{1+a^{2}\beta
P_{1}+P_{2}+2a\sqrt{\beta(1-\alpha)P_{1}P_{2}}}\right).$ (41)
After decoding message $W_{1}$, the cognitive receiver decodes message $W_{2}$
using Costa’s dirty paper decoding. In decoding message $W_{2}$, the cognitive
receiver sees only $Z_{2}^{n}$ as Gaussian noise. $X_{10}^{n}$ and
$X_{20}^{n}$ do not appear as noise as they were canceled out at the encoder
side using Costa’s dirty paper coding. Hence, the receiver can decode message
$W_{2}$ successfully if
$R_{2}\leq\frac{1}{2}\log(1+\alpha P_{2}).$ (42)
Hence, the region described by $\mathcal{R}_{in}$ in (38) is achievable in the
Gaussian partially cognitive radio channel. This completes the proof of
Theorem 3.
###### Remark 1
It should be noted here that the cognitive receiver first cancels the
interference due to message $W_{1}$ before decoding message $W_{2}$. This
places a constraint on rate $R_{1}$ given by (41). Ideally, we would want the
constraint on $R_{1}$ given by (39) to be more binding than the constraint on
$R_{1}$ given by (41). This is possible if
$\displaystyle\frac{a^{2}(1-\beta)P_{1}}{1+a^{2}\beta
P_{1}+P_{2}+2a\sqrt{\beta(1-\alpha)P_{1}P_{2}}}$
$\displaystyle\geq\frac{(1-\beta)P_{1}}{1+\beta
P_{1}+b^{2}P_{2}+2b\sqrt{\beta(1-\alpha)P_{1}P_{2}}}.$ (43)
## V Conditions of Optimality of Achievable Region
In this section, we compare the achievable region and the outer bound and
derive conditions when the two meet. We say that the achievable region
described in Section IV is $(\mu_{0},\mu_{1},\mu_{2})$ optimal if
$\displaystyle\max_{(R_{0},R_{1},R_{2})\in\mathcal{R}_{in}}\mu_{0}R_{0}+\mu_{1}R_{1}+\mu_{2}R_{2}$
$\displaystyle=\max_{(R_{0},R_{1},R_{2})\in\mathcal{R}_{out}}\mu_{0}R_{0}+\mu_{1}R_{1}+\mu_{2}R_{2}$
(44)
Let $(R_{0}^{o},R_{1}^{o},R_{2}^{o})$ be $(\mu_{0},\mu_{1},\mu_{2})$ optimal
with respect to the outer bound. That is,
$(R_{0}^{o},R_{1}^{o},R_{2}^{o})=\mathop{\rm
arg\,max}_{(R_{0},R_{1},R_{2})\in\mathcal{R}_{out}}\mu_{0}R_{0}+\mu_{1}R_{1}+\mu_{2}R_{2}.$
(45)
Let $(\alpha^{o},\beta^{o})$ be the optimal power splits at the two
transmitters that maximizes the $(\mu_{0},\mu_{1},\mu_{2})$ sum rate with
respect to the outer bound. That is,
$(\alpha^{o},\beta^{o})=\mathop{\rm arg\,max}_{0\leq\alpha,\beta\leq
1}\max_{(R_{0},R_{1},R_{2})\in\mathcal{R}_{out}^{\alpha,\beta}}\mu_{0}R_{0}+\mu_{1}R_{1}+\mu_{2}R_{2}.$
(46)
Then, we have the following lemma.
###### Lemma 5
$\beta^{o}=1\ $ for all $\ (\mu_{0},\mu_{1},\mu_{2})$.
The proof of the lemma follows from the observation that
$\mathcal{R}_{out}^{\alpha,1}\supseteq\mathcal{R}_{out}^{\alpha,\beta}$ for
all $0\leq\beta\leq 1$. We next look at the conditions when the achievable
region meets the outer bound.
We first consider the case $\mu_{0}\geq\mu_{1}$. Then, we have
$\begin{array}[]{l}R_{0}^{o}+R_{1}^{o}=\frac{1}{2}\log\left(1+\frac{P_{1}+b^{2}(1-\alpha^{o})P_{2}+2b\sqrt{\beta^{o}(1-\alpha^{o})P_{1}P_{2}}}{1+b^{2}\alpha^{o}P_{2}}\right),\vspace{0.1cm}\\\
R_{1}^{o}=\mu R_{0}^{o}\vspace{0.1cm}\\\
R_{2}^{o}=\frac{1}{2}\log(1+\alpha^{o}P_{2}).\end{array}$ (47)
The conditions for optimality are then given by the following lemma.
###### Lemma 6
If the following two conditions are satisfied
$\displaystyle\frac{a^{2}}{1+a^{2}\beta^{o}P_{1}+P_{2}+2a\sqrt{(1-\alpha^{o})P_{1}P_{2}}}$
$\displaystyle\geq\frac{1}{1+\beta^{o}P_{1}+b^{2}P_{2}+2b\sqrt{\beta^{o}(1-\alpha^{o})P_{1}P_{2}}},$
(48)
$\begin{array}[]{l}\log\left(1+\frac{P_{1}}{1+b^{2}P_{2}}\right)\geq\mu\log\left(1+\frac{b^{2}(1-\alpha^{o})P_{2}}{1+b^{2}\alpha^{o}P_{2}}\right)\end{array},$
(49)
then the achievable region is $(\mu_{0},\mu_{1},\mu_{2})$ sum optimal for
$\mu_{0}\geq\mu_{1}$.
Proof: The proof of the lemma is fairly simple and we briefly explain the two
conditions.
The first condition comes from ensuring that constraint on $R_{1}$ in the
achievable region due to decoding message $m_{1}$ at the legitimate receiver
is more binding than the constraint due to decoding message $m_{1}$ at the
cognitive receiver.
The second condition comes in ensuring that the point which maximizes the
$(\mu_{0},\mu_{1},\mu_{2})$ sum in $\mathcal{R}_{out}^{\alpha^{0},1}$ is also
achievable. The main details of the proof are omitted here.
Next, we consider the case $\mu_{0}<\mu_{1}$. In this case,
$R_{0}^{o},R_{1}^{o}$ and $R_{2}^{o}$ are given by
$\begin{array}[]{l}R_{0}^{o}=0\\\
R_{1}^{o}=\frac{1}{2}\log\left(1+\frac{P_{1}+b^{2}(1-\alpha^{o})P_{2}+2b\sqrt{(1-\alpha^{o})P_{1}P_{2}}}{1+b^{2}\alpha^{o}P_{2}}\right)\\\
R_{2}^{o}=\frac{1}{2}\log(1+\alpha^{o}P_{2}).\end{array}$ (50)
The condition of optimality when $\mu_{0}<\mu_{1}$ is given by the following
lemma.
###### Lemma 7
When $\mu_{0}<\mu_{1}$, if we have $\alpha^{o}=1$, then the achievable region
is $(\mu_{0},\mu_{1},\mu_{2})$ sum optimal.
The proof of the lemma follows from the argument that if $\alpha^{o}=1$, then
the corresponding point $(R_{0}^{o},R_{1}^{o},R_{2}^{o})$ is also achievable
by substituting $\alpha=1$ and $\beta=0$.
## VI Conclusions
In this paper, we investigated the capacity region of interference channel
with partially cognitive radios. For the general discrete memoryless IFC
setting, we obtained the outer bound for the capacity region when the
legitimate receiver observes the weak interference. And, for a mixed
interference Gaussian channel, we showed that the portions of the outer bound
can be achieved.
## VII Acknowledgment
We thank Ivana Maric for useful discussions and comments.
## References
* [1] J.Mitola, “Cognitive Radio,” Ph.D. dissertation, Royal Institute of Technology (KTH),Stockholm, Sweden, 2000.
* [2] S. Haykin, Cognitive radio: brain-empowered wireless communications, IEEE J. Sel. Areas in Commun., vol. 23, pp. 201-220, Feb. 2005.
* [3] N. Devroye, P. Mitran and V. Tarokh, “Achievable Rates in Cognitive Rado Channels,” IEEE Trans. Inform. Theory, vol. 52, pp. 1813-1827, May 2006.
* [4] I. Maric, A. Goldsmith, G. Kramer, S. Shamai (Shitz), “On the Capacity of Interference Channel with a Partially-Cognitive Traansmitter,”IEEE Trans. Inform. Theory.
* [5] I. Maric, R. Yates, “The Strong Interference Channel with Common Information,” Allerton Conf. Communications, Monticello, Il, Sep. 2005.
* [6] I. Maric, R. Yates, G. Kramer, “The strong interference channel with unidirectional cooperation,” presented at the Information Theory and Applications (ITA) Inaugural Workshop, Feb. 2006.
* [7] W. Wu, S. Vishwanath and A. Arapostathis, “On the Capacity of Interference Channel with degraded Message Sets,”IEEE Trans. Inform. Theory.
* [8] T. S. Han and K. Kobayashi, “A new achievable rate region for the interference channel,”IEEE Trans. Inform. Theory, vol. 27, pp. 49-60, Jan. 1981.
* [9] H. Sato, “Two-user communication channels,” IEEE Trans. Inform. Theory, vol. 23, pp. 295-304, May 1977.
* [10] A. B. Carleial, “Outer bounds on the capacity of interference channels,” IEEE Trans. Inform. Theory, vol. 29, pp. 602-606, Jul. 1983. vol. IT-24, pp. 60.70, Jan. 1978.
* [11] J. A. Thomas, “Feedback can at most double Gaussian multiple access channel capacity,” IEEE Trans. Inform. Theory, vol. 33, pp. 711-716, Sep. 1987.
* [12] H. Weingarten, Y. Steinberg and S. Shamai (Shitz), “The Capacity region of the Gaussian MIMO broadcast channel,” IEEE Trans. Inform. Theory, vol. 52, pp. 3936-3964, Sep. 2006.
* [13] M. Costa, “Writing on dirty paper,” IEEE Trans. Inform. Theory, vol. 29, pp. 439-441, May 1983.
* [14] T. M Cover and J.A. Thomas, Elements of information theory, ser. Wiley Series in Telecommunications. New York: John Wiley $\&$ Sons Inc., 1991, a Wiley-Interscience Publication.
|
arxiv-papers
| 2008-12-29T23:04:08 |
2024-09-04T02:48:59.625000
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. Chung, S. Sridharan, S. Vishwanath, C. S. Hwang",
"submitter": "Goochul Chung",
"url": "https://arxiv.org/abs/0812.4985"
}
|
0812.5080
|
# BPS ansatzes as electric form-factors.
L. D. Lantsman.
18109, Rostock, Germany; Mecklenburger Allee, 7
llantsman@freenet.de
Tel. (049)-0381-7990724.
###### Abstract
We argue that BPS ansatzes, entering manifestly vacuum BPS monopole solutions
to equations of motion in the (Minkowskian) non-Abelian Higgs model play the
role of some electric form-factors and that this implies (soft) violating the
CP-invariance of the mentioned model, similar to taking place in the Euclidian
Yang-Mills (YM) theory with instantons, generating the $\theta$-term in the
appropriate effective Hamiltonian.
PACS: 14.80.Bn, 14.80.Hv.
Keywords: Non-Abelian Theory, BPS Monopole, Minkowski Space, Instanton.
###### Contents
1. 1 Introduction.
2. 2 BPS ansatzes as electric form-factors.
3. 3 Higgs BPS ansatzes and CP violating.
4. 4 Discussion.
## 1 Introduction.
The (without of quarks) non-Abelian YM theory involving vacuum BPS monopole
solutions in their Higgs and gauge sectors (as a result of the spontaneous
breakdown the initial $SU(2)$ gauge symmetry group to its $U(1)$ subgroup)
occupy a special position among another such theories with monopoles. This is
associated with manifest superfluid properties of the former model.
To elucidate this our assertion, let us at first write down explicitly the
action functional for the (Minkowskian) Yang-Mills-Higgs (YMH) model. It can
be represented as [1, 2, 3, 4, 5]
$S=-\frac{1}{4g^{2}}\int d^{4}xF_{\mu\nu}^{b}F_{b}^{\mu\nu}+\frac{1}{2}\int
d^{4}x(D_{\mu}\phi,D^{\mu}\phi)-\frac{\lambda}{4}\int
d^{4}x\left[(\phi^{b})^{2}-\frac{m^{2}}{\lambda}\right]^{2},$ (1.1)
with
$D_{\mu}\phi=\partial^{\mu}\phi+g[A^{\mu},\phi]$
being the covariant derivative an $g$ being the YM coupling constant.
The action functional (1.1) results the equations of motion [1]
$(D_{\nu}F^{\mu\nu})_{a}=-g\epsilon_{abc}\phi^{b}(D_{\mu}\phi)^{c},$ (1.2)
$(D^{\mu}D_{\mu}\phi)_{a}=-\lambda\phi_{a}({\vec{\phi}}\cdot{\vec{\phi}}-a^{2});\quad
a^{2}=m^{2}/\lambda.$ (1.3)
It turns out that going over to the limit
$\lambda\to 0,~{}~{}~{}~{}~{}~{}m\to
0:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{\epsilon}\equiv\frac{gm}{\sqrt{\lambda}}=ga\not=0;$
(1.4)
in Eq. (1.1) just induces the (topologically degenerated) vacuum BPS monopole
solutions in the Higgs and YM sectors of the model (1.1).
Historically, the idea to go over to the limit (1.4) in the YMH model is
originated from the works [6], and from this time it was refer to as the
Bogomol’nyi-Prasad-Sommerfeld (BPS) limit. Later, in the papers [3, 4] the BPS
limit was rearranged to the look (1.4), implicating the YM coupling constant
$g$.
It is remarkable that the ratio $a=m/\sqrt{\lambda}$ (having the mass
dimension) can take arbitrary values in the limit (1.4) and that the variable
$\epsilon\to 0$ of the length dimension is introduced therein. We shall make
sure soon that $\epsilon$ plays the role of the size parameter characterized
the core of a BPS monopole.
Vacuum BPS monopole solutions can be derived in the limit (1.4) at evaluating
the lowest bound of the energy for the given YMH configuration (often referred
to as the Bogomol nyi bound in the modern literature) [3, 4]:
$E_{\rm min}=4\pi{\bf m}\frac{a}{g}$ (1.5)
(where $\bf m$ denotes the magnetic charge).
As a result, one arrives at the so-called Bogomol nyi equation [3, 4, 7]
${\bf B}(\Phi)=\pm D\Phi$ (1.6)
relating the vacuum “magnetic” field ${\bf B}$ to the vacuum Higgs
configuration in the shape of a BPS monopole. The presence of two opposite
signs in the Bogomol nyi equation (1.6) corresponds to two opposite signs of
magnetic charges in nature.
The explicit way deriving the Bogomol nyi equation (1.6) and evaluating the
Bogomol nyi bound (1.5) was stated, for instance, in the monograph [8] (see
ibid §$\Phi$11). In particular, in the monograph [8] vacuum BPS monopole
solutions to the Bogomol nyi equation (1.6), arising in the Higgs and gauge
sectors of the YMH model [6], were written down. In the series of papers [3,
4] these solutions were reproduced with the only modification that the
effective Higgs mass $a=m/\sqrt{\lambda}$ (utilized in Ref. [8]) was replaced
with the parameter $\epsilon^{-1}$:
$\Phi^{a}_{(0)}(t,{\bf
x})=\frac{x^{a}}{gr}f_{0}^{BPS}(r)~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}f_{0}^{BPS}(r)=\left[\frac{1}{\epsilon\tanh(r/\epsilon)}-\frac{1}{r}\right],$
(1.7) $A^{a}_{i(0)}(t,{\bf x})\equiv\Phi^{aBPS}_{i}({\bf
x})=\epsilon_{iak}\frac{x^{k}}{gr^{2}}f^{BPS}_{1}(r),~{}~{}~{}~{}~{}~{}~{}~{}f^{BPS}_{1}(r)=\left[1-\frac{r}{\epsilon\sinh(r/\epsilon)}\right].$
(1.8)
Indeed, the BPS monopoles (1.7), (1.8) are topologically trivial fields 111The
topological degeneration of vacuum BPS monopole data (1.7), (1.8) can be carry
out by means of “large” gauge transformations (in the terminology [9])
proposed in the work [10]: ${\Phi_{i}}^{a(n)}:=v^{(n)}({\bf
x})[{\Phi_{i}}^{a(0)}+\partial_{i}]v^{(n)}({\bf x})^{-1},\quad v^{(n)}({\bf
x})=\exp[n\hat{\Phi}_{0}({\bf x})];$ $\Phi_{(n)a}=v^{(n)}({\bf
x})\Phi_{(0)a}v^{(n)}({\bf x})^{-1};\quad n\in{\bf Z}$ (latter Eq. was derived
in the paper [11]). Here
${\hat{\Phi}}_{0}(r)=-i\pi\frac{\tau^{a}x_{a}}{r}f_{01}^{BPS}(r);\quad
f_{01}^{BPS}(r)=[\frac{1}{\tanh(r/\epsilon)}-\frac{\epsilon}{r}]=f_{1}^{BPS}(r)/\epsilon;$
with $\tau^{a}$ ($a=1,2,3$) being the Pauli matrices. The exponential
multipliers $v^{(n)}({\bf x})$ were referred to as Gribov topological
multipliers in Ref. [10] while the value ${\hat{\Phi}}_{0}(r)$ as the Gribov
phase. It can be argued that specified in this way topologically degenerated
YM vacuum BPS monopole data ${\Phi_{i}}^{a(n)}$ satisfy the Coulomb gauge
$D^{i}{\Phi_{i}}^{a(n)}=0$. And moreover, YM vacuum BPS monopole data
${\Phi_{i}}^{a(n)}$ also turn out to be gauge invariant, i.e. physical,
functionals of YM fields. In particular, this is correctly for topologically
trivial YM BPS monopoles (1.8) [3, 4], and this indicates transparently the
purely physical nature of these monopoles. Also topologically trivial Higgs
BPS monopole modes (1.7) and their Gribov topological copies $\Phi^{(n)a}$
[11] prove to be manifestly gauge invariant, side by side with YM BPS
monopoles. Mention that the topologically degenerated YM vacuum BPS monopoles
${\Phi_{i}}^{a(n)}$ are patterns of topological Dirac variables, gauge
invariant and transverse (in the sense satisfying the Lorentz covariant
Coulomb gauge $D^{i}{\Phi_{i}}^{a(n)}=0$). The important point here that
topological Dirac variables $\Phi^{(n)a}$ are got indeed as solutions to the
YM Gauss law constraint [10] $\frac{\delta W}{\delta
A^{a}_{0}}=0\Longleftrightarrow[D^{2}(A)]^{ac}A_{0c}=D^{ac}_{i}(A)\partial_{0}A_{c}^{i}.$
As to Higgs BPS monopole modes $\Phi^{(n)a}$, the appropriate current
$\rho^{H}\sim ig\Phi D\Phi$ decouples from the YM Gauss law constraint in the
first order of the perturbation theory by the YM coupling constant $g$. The
detailed analysis of topological Dirac variables (including the answer why
Higgs BPS monopole modes $\Phi^{(n)a}$ disappear from the YM Gauss law
constraint in the first order of the perturbation theory) was performed in the
works [3, 4, 10, 12, 13], and we recommend these to our readers for studying
the matter. .
Now let us discuss the behaviour of the BPS anzatses $f_{0}^{BPS}(r)$ and
$f^{BPS}_{1}(r)$ at the origin of coordinates and at the spatial infinity.
Direct checking shows that [13]
$f_{0}^{BPS}(0)=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}f_{0}^{BPS}(\infty)=1;$
(1.9)
$f_{1}^{BPS}(0)=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}f_{1}^{BPS}(\infty)=1.$
(1.10)
Then the YM BPS monopoles (1.8) (with their Gribov copies ${\Phi_{i}}^{a(n)}$)
display an alike good behaviour (disappearing) at the origin of coordinates
and at the spatial infinity ($r\to\infty$). The other thing the behaviour of
Higgs vacuum BPS monopole modes $\Phi_{(n)a}$. These diverge at the origin of
coordinates, as it follows from (1.7) and (1.9).
The intersesting and important feature of YM BPS monopoles (1.8) is that they
merge (because of Eq. (1.10)) with Wu-Yang monopoles $\Phi^{Wa}_{i}$ [14]
222Remember that Wu-Yang monopoles $\Phi^{Wa}_{i}$ are solutions to the
classical equation of motion [3, 4, 10]
$D^{ab}_{k}(\Phi_{i}^{W})F^{bk}_{a}(\Phi_{i}^{W})=0\Longrightarrow\frac{d^{2}f}{dr^{2}}+\frac{f(f^{2}-1)}{r^{2}}=0$
in the “pure” YM theory (with absent Higgs and fermionic modes) corresponding
to the exact $SU(2)$ gauge group. One can distinguish three solutions to this
equation: $f_{1}^{PT}=0,\quad f_{1}^{W}=\pm 1\quad(r\neq 0).$ The first,
trivial, solution $f_{1}^{PT}=0$ corresponds to the naive unstable
perturbation theory, involving the asymptotic freedom formula [15, 16]. They
are just the Wu-Yang monopoles [14] with topological charges $\pm 1$,
respectively. .
The Bogomol nyi equation (1.6) can be treated as a potentiality condition for
the BPS monopole vacuum. A brief argumentation in favour of this statement was
advanced in the recent paper [5].
Really, mathematically, any potentiality condition may be written down as
${\rm rot}~{}{\rm grad}~{}{\Phi}=0$ (1.11)
for a scalar field $\Phi$. Thus any potential field may be represented as
${\rm grad}~{}{\Phi}$.
In the Minkowskian YMH theory involving BPS monopole solutions there exists
always such scalar fields. There are just Higgs vacuum BPS monopole modes
(1.7) (with their Gribov topological copies $\Phi_{(n)a}$ [11]).
Then it is easy to guess that the Bogomol’nyi equation (1.6), having the look
(1.11), can be treated as the potentiality condition for the Minkowskian YMH
vacuum involving vacuum BPS monopole solutions. It is so due to the Bianchi
identity $DB=0$ 333This becomes more transporent upon representing the
Bogomol’nyi equation (1.6) in the tensor shape [8]
$\frac{1}{2g}\epsilon^{ijk}F_{jk}^{a}=\nabla^{i}\Phi^{a}.$ Then due to the
Bianchi identity $\epsilon^{ijk}\nabla_{i}F_{jk}^{b}=0,$ the Bogomol’nyi
equation (1.6) results $D^{2}\Phi\sim{\rm rot}{\bf B}=0$ (at neglecting the
items in $DB$ directly proportional to $g$ and $g^{2}$). .
Indeed, there can be drawn a highly transparent parallel between the
Minkowskian YMH vacuum involving vacuum BPS monopole solutions and a liquid
helium II specimen described in the Bogolubov-Landau model [17].
In the latter case, the potential motion is proper to the superfluid component
in this liquid helium specimen.
The superfluid motion in a liquid helium II is the motion without a friction
between the superfluid component and the walls of the vessel where a liquid
helium specimen is contained.
Thus the viscosity of the superfluid component in a helium II is equal to
zero, and vortices (involving ${\rm rot}~{}{\bf v}\neq 0$) are absent in the
superfluid component of a helium.
As L. D. Landau showed [17], at velocities of the liquid exceeding a critical
velocity $v_{0}={\rm min}~{}(\epsilon/p)$ for the ratio of the energy
$\epsilon$ and momentum $p$ for quantum excitations spectrum in the liquid
helium II, the dissipation of the liquid helium energy occurs via arising
excitation quanta with momenta $\bf p$ directed antiparallel to the velocity
vector $\bf v$. Such dissipation of the liquid helium energy becomes
advantageous [18] just at
$\epsilon+{\bf p~{}v}<0\Longrightarrow\epsilon-p~{}v<0.$
From the above reasoning concerning properties of potential motions, it
becomes obvious that the vector ${\bf v}_{0}$ of the critical velocity for the
superfluid potential motion possesses the zero curl: ${\rm rot}~{}{\bf
v}_{0}=0$.
In this case, according to (1.11), the critical velocity ${\bf v}_{0}$ of the
superfluid potential motion in a liquid helium specimen may be represented
[19] as
${\bf v}_{0}=\frac{\hbar}{m}\nabla\Phi(t,{\bf r}),$ (1.12)
where $m$ is the mass of a helium atom and $\Phi(t,{\bf r})$ is the phase of
the complex-value helium Bose condensate wave function $\Xi(t,{\bf r})\in C$.
Thus the similar look for the vacuum ”magnetic” field $\bf B$ in the
Minkowskian Higgs model involving BPS monopole solutions, generating by the
Bogomol’nyi equation (1.6), and for the critical velocity ${\bf v}_{0}$ of the
superfluid motion in a liquid helium II, given by Eq. (1.12), testifies in
favour of the potential motions occurring therein.
In this case, drawing a highly transparent parallel between the Minkowskian
YMH vacuum involving BPS monopole solutions and a liquid helium II specimen
described in the Bogolubov-Landau model [17], we can also conclude about
manifest superfluid properties of the Minkowskian YMH vacuum involving BPS
monopoles.
As in the Bogolubov-Landau model [17] of liquid helium II, the ground cause of
the superfluid properties of the Minkowskian YMH vacuum with BPS monopoles
roots in long-range correlations of local excitations [20].
While in the Bogolubov-Landau model [17] of liquid helium II this comes to
repulsion forces between helium atoms as the cause of superfluidity effects,
in the Minkowskian YMH vacuum involving BPS monopole solutions, the cause of
the superfluidity taking place is in the strong YMH coupling $g$ (entering
effectively the appropriate action functional (1.1)).
The principal thing in alike superfluid effects occurring in a liquid helium
II specimen as well as in the Minkowskian YMH vacuum involving BPS monopoles
is that these both physical systems are non-ideal gases.
In ideal gases no superfluidity phenomena are possible.
There can be demonstrated [21] that in ideal gases a deal of particles is
accumulated on the zero energy quantum level at temperatures $T<T_{0}$;
herewith the temperature $T_{0}$
$kT_{0}=\frac{1}{(2.61)^{2/3}}\frac{h^{2}}{2\pi m}(\frac{N}{V})^{2/3}$ (1.13)
(with $k$ and $h$ being, respectively, the Boltzmann and Planck constants; $N$
being the complete number of particles; $V$ being the volume occupied by the
ideal Bose gas; $m$ being the mass of a particle) is called the condensation
temperature, while the above deal of particles is called the Bose condensate.
The just described superfluidity is absent in another Minkowskian Higgs models
with monopoles: for instance, in the ’t Hooft-Polyakov model [22, 23]. This
can be argued, repeating the arguments [5, 24], by disapearing the covariant
derivative $D^{i}\phi_{a}$ of a Higgs ’t Hooft-Polyakov monopole mode
$\phi_{a}$ at the spatial infinity.
In this case, asymptotically (at $r\to\infty$),
${\bf B}^{a}_{i}D^{i}\phi_{a}={\partial}_{i}({\bf B}^{i}_{a}\phi_{a})=0,$
(1.14)
because of the Bianchi identity $DB=0$ and the remark that ${\bf
B}^{i}_{a}\phi_{a}$ is a $U(1)\subset SU(2)$ scalar; thus one can replace the
covariant derivative $D$ with the partial one, $\partial$, for ${\bf
B}^{i}_{a}\phi_{a}$.
In turn, the complete energy of the YMH configuration may be represented as
[8, 24]
$E_{\rm compl}=\int d^{3}x~{}[\frac{1}{2}(D\phi_{a}\pm{\bf
B}_{a})^{2}+\frac{\lambda}{4}((\phi^{a})^{2}-a^{2})]+\frac{4\pi}{g^{2}}M_{W}.$
(1.15)
The last item in Eq. (1.15) involves the mass $M_{W}$ of the $W$-boson.
Such look of $E_{\rm compl}$ originates from the paper [22] devoted to the ’t
Hooft-Polyakov model.
The connection between the energy integral $E_{\rm compl}$ and the general
action functional (1.1) [3, 4] of the Minkowskian Higgs model is given by the
identity [24]
$(D\phi_{a})^{2}+{\bf B}_{a}^{2}=(D\phi_{a}\pm{\bf B}_{a})^{2}\mp 2{\bf
B}_{a}D\phi_{a}.$ (1.16)
Herewith the last item on the right-hand side of (1.16) vanishes at the
spatial infinity, as we have noted above. Just from Eq. (1.15) one can read
the Bogomol’nyi equation in the shape (1.6).
In the ’t Hooft-Polyakov model [22, 23] the Bogomol’nyi equation (1.6)
determines the Bogomol’nyi bound [24]
$M_{\rm mon}=\frac{4\pi}{g^{2}}M_{W}$ (1.17)
for the complete energy $E_{\rm compl}$, (1.15), of the YMH configuration at
going over to the BPS limit (1.4) [6].
Then the asymptotic $D_{i}\phi^{a}\to 0$ as $r\to\infty$ for ’t Hooft-Polyakov
monopoles [22, 23] forces to vanish identically the first item under the
integral sign in $E_{\rm compl}$ ($|{\bf B}|=0$).
In the light of the said above it becomes obvious that the vacuum ”magnetic”
field $\bf B$, playing the role of the (critical) velocity for the superfluid
motion in the Minkowskian non-Abelian vacuum with BPS monopoles, actually
approaches zero in the ’t Hooft-Polyakov model [22, 23], involving the
$D_{i}\phi^{a}\to 0$ as $r\to\infty$ asymptotic for Higgs monopoles.
The principal goal of the present note is to show that BPS ansatzes
$f^{BPS}_{1}(r)$, $f^{BPS}_{0}(r)$, one encounter in the Higgs BPS monopole
model, can serve as electric form-factors therein. Unlike this, electric form-
factors become trivial in the ’t Hooft-Polyakov theory [22, 23]. Grounding
this fact will be the topic of Section 2.
In Section 3 we show that presence of BPS ansatzes in the considered model
implies violating the CP invariance. It is the effect similar to that taking
place in the instanton models [1, 2, 8, 25], generating the $\theta$-items in
the appropriate effective Lagrangians. This effect violating the CP invariance
by the $\theta$-dependence of the instanton models was analyzed in the paper
[26], and we reconstruct partially the arguments [26] in Section 3.
## 2 BPS ansatzes as electric form-factors.
The starting point of our discussion will be the well known Dirac quantization
condition [27] for the electric and magnetic charges presented in a closed
system of quantum fields.
In a simple case when a quantum object is isolated from another, the Dirac
quantization condition [27] acquires the look [1, 2, 8]
$\frac{q{\bf m}}{4\pi}=\frac{1}{2}n;\quad n\in{\bf Z},$ (2.1)
where $q$ and ${\bf m}$ are, respectively, the electric and magnetic charges
of the considered object (in the system of units in which $\hbar=c=1$) 444In
some sources (for instance, [2, 8]) Eq. (2.1) is given in the slightly
modified air $q{\bf m}=\frac{1}{2}n.$ Going over from Eq. (2.1) to the latter
Eq. can be achieved [24] at setting ${\bf m}=4\pi/q$. The origin of this in
the Laplace equation [2] $\nabla\cdot{\bf B}=4\pi{\bf m}\delta^{3}(r)$ for the
point magnetic charge ${\bf m}$ creating the radial magnetic field ${\bf B}$,
resulting the total magnetic flux $\Phi=4\pi r^{2}B=4\pi{\bf m}$ through a
sphere with its centre in the origin of coordinates. Just this provides (see
§10.3 in [2]) the change $\Delta\alpha|_{\pi}=\frac{q}{\hbar c}\oint{\bf
A}\cdot{\bf dl}|_{\pi}=\frac{q}{\hbar c}\int{\rm rot}{\bf A}\cdot{\bf
dS}|_{\pi}=\frac{q}{\hbar c}\int{\bf B}\cdot{\bf dS}|_{\pi}=\frac{q}{\hbar
c}\Phi(r,\theta)|_{\theta=\pi}=\frac{q}{\hbar c}4\pi{\bf m}=2\pi n$ of the
dyon’s wave function ($\psi\equiv|\psi|e^{i\alpha}=|\psi|\exp[(-iq/\hbar
c){\bf A}\cdot{\bf r}]$) phase $\alpha$ at $\theta=\pi$ (the flux
$\Phi(r,\pi)$ is just the maximal possible flux, spreeded to the whole
sphere). . Each such quantum object possessing the electric and magnetic
charges simultaneously is referred to as a dyon in modern physical literature
555Formally, a particle possessing the zero electric and magnetic charges also
relates to the class of dyons. And moreover, if ${\bf m}=0$, Eq. (2.1) is
satisfied at arbitrary values of the electric charge $q$..
When a system of quantum fields consists of two dyons, Eq. (2.1) can be
generalized to Eq. [24, 26]
$q_{1}{\bf m}_{2}\pm q_{2}{\bf m}_{1}=2\pi n;\quad n\in{\bf Z};$ (2.2)
Eq. (2.2) was derived for the first time by Zwanziger and Schwinger [28]. The
reasoning for deriving this Eq. is [26] the classical formula for the angular
momentum of an electromagnetic field. The angular momentum in an
electromagnetic background of a two-particle system can be calculated easily.
It has the magnitude
$(e_{1}{\bf m}_{2}-e_{2}{\bf m}_{1})/4\pi c,$
that takes integer or half-integer values, as it is expected in quantum
mechanics, only if
$(e_{1}{\bf m}_{2}-e_{2}{\bf m}_{1})/\hbar c=2\pi n$
(with setting $\hbar=c=1$). Going over to the sign $+$ in (2.2) from the $-$
one is reduced simply to replacing ${\bf m}\leftrightarrow-{\bf m}$.
In Ref. [24] it was given the in definite sense generalization of Eq. (2.2) to
the case of an arbitrary gauge group $SU(N)$:
$\sum\limits_{i=1}^{N-1}e_{i}{\bf m}_{i}=2\pi n.$
Herewith it is easy to see that Eq. (2.2) (with the $+$ sign) is the
particular case of the latter relation for the gauge group $SU(2)$.
Let us now calculate, following [1], the total momentum
${\bf J}={\bf L}+{\bf T}$ (2.3)
of a particle in a magnetic monopole background. It involves its spatial
angular momentum ${\bf L}$ (including its ”ordinary” spin) and the generator
$\bf T$ of the internal (for instance, the gauge $U(1)$) symmetry.
On the other hand, ${\bf L}={\bf r}\times{\bf p}$, with $\bf p$ being the
canonical momentum
$p_{i}=mv_{i}+g({\bf A}_{i}\cdot{\bf T})$ (2.4)
for a (vacuum) ”gauge” monopole solution involving the mass $m$.
In the particular case of ’ t Hooft-Polyakov monopole solutions [22, 23], when
YM potentials have the look [2]
$A_{i}^{a}=\epsilon_{iab}\frac{r^{b}}{gr^{2}},$ (2.5)
Eq. (2.4) can be rewritten as
${\bf p}=m{\dot{\bf r}}+\frac{1}{r}{\bf n}\times{\bf T}.$ (2.6)
Then
${\bf J}={\bf r}\times{\bf p}+{\bf T}={\bf r}\times m{\dot{\bf r}}+{\bf
n}\times{\bf n}\times{\bf T}+{\bf T}=$ ${\bf r}\times m{\dot{\bf r}}+({\bf
n}\cdot{\bf T}){\bf n}.$ (2.7)
Now we must recall that in the ’ t Hooft-Polyakov model [22, 23] the radial
”magnetic” field $\bf B$ is given by Eq. [2] 666Note that Eqs. (2.5) and (2.8)
correspond to the ”standard” normalization YMH Lagrangian density [2]
involving the coefficient $-1/4$ in front of $F_{\mu\nu}^{2}$.
$F^{ij}=-\frac{1}{gr^{3}}\epsilon^{ijk}r_{k};\quad{\bf
B}_{k}\equiv-F^{ij}\epsilon_{ijk}.$ (2.8)
From the general reasoning [1] about magnetic monopoles, the equation of
motion for an electric charged particle in its field is read as
$m\ddot{\bf r}=e\dot{\bf r}\times{\bf B}.$ (2.9)
It is just the Lorenz force acting onto this particle in the magnetic monopole
background.
In this case the rate of change of the particle angular momentum $\bf L$ is
$\frac{d}{dt}({\bf r}\times m\dot{\bf r})={\bf r}\times m\ddot{\bf
r}=\frac{2\pi n}{\nu}{\bf r}\times(\dot{\bf r}\times{\bf
r})=\frac{d}{dt}(\frac{2\pi n}{\nu}{\bf n}),$ (2.10)
where [8] $\nu$ is the (minimal) positive number for which the condition
$\exp(\nu h)=1$ (with $h\equiv h(\Phi)\equiv\Phi/a$ being the generator of the
residual $U(1)$ gauge group in the quested YMH model) is satisfied 777Defining
$\nu$ in this way, one can argue [8] that ${\bf
m}(\Phi,A)=C~{}\zeta(\Phi,A),\quad\zeta(\Phi,A)\in{\bf Z}$ for the given
monopole YMH configuration $(\Phi,A)$ with $C=\nu/4\pi$..
Then, issuing from the above discussed Dirac quantization condition
$q{\bf m}=\frac{1}{2}n$ (2.11)
and the normalizations [8]
$q=\frac{2\pi n}{\nu}g,$ (2.12) ${\bf m}=\frac{\nu}{4\pi
g}~{}\zeta;\quad\zeta\in{\bf Z};$ (2.13)
for the electric and magnetic charges, respectively (thus the Dirac
quantization condition (2.11) is satisfied automatically), we just arrive to
Eq. (2.10) (with $q$ given in (2.12) and appropriate cancelling the YM
coupling constant $g$, which enters the relation for $\bf B$).
The said suggests the formal possibility to introduct the total momentum (2.3)
[1] of an electric charged particle in the magnetic monopole background in
such a wise that it is conserved:
${\bf J}={\bf r}\times m{\dot{\bf r}}-\frac{2\pi n}{\nu}{\bf n}={\bf r}\times
m{\dot{\bf r}}-(q/g){\bf n}.$ (2.14)
Comparing then the expressions (2.7) and (2.14) for the total momentum $\bf J$
of an electric charged particle in the magnetic monopole background got in the
’ t Hooft-Polyakov model [22, 23], one can conclude that
$(e/g)=-{\bf n}\cdot{\bf T}$ (2.15)
if $q=e$ ($e$ is the elementary charge).
At the particular choice $\bf n$ to be the $z$-direction, ${\bf n}\cdot{\bf
T}=T_{3}$. On the other hand, the value $2\pi/\nu$ can be normalized as
$2\pi/\nu=1$ (at considering [8] the $U(1)$ group space as the circle $S^{1}$
of the unit radius).
Because of (2.12), we can conclude that the isospin operator $\bf T$ ($T_{3}$)
is topologically degenerated. Geometrically, such topological degeneration of
the isospin operator $\bf T$ means extracting (”large” and ”small”) gauge
orbits in the $U(1)$ group space.
Specifying [1] the electric charge operator $Q_{U(1)}=eT_{3}$, we see
additionally that $Q_{U(1)}$ takes integers multipliers of $e$ (at setting
$2\pi/\nu=1$) in the presence of ’ t Hooft-Polyakov monopole modes 888This
effect was noted, for example, in Ref. [24] with that important correction
that besides $Q_{U(1)}=eT_{3}=ne$ charged states, $({n}+1/2)e$ states are also
possible, with $q=e/2$ being the minimal charge corresponding to ${\bf
T}=1/2$. For trivial topologies $n=0$, the minimal charge $q=e/2$ corresponds
to a fermionic field $\psi$ in the $\displaystyle
I=\frac{1}{2}:\quad\psi={\psi_{1}\choose\psi_{2}}$ representation of $SU(2)$
(if an YMH model is in question). .
The presence of YM BPS ansatz (1.8) in the YMH model with (vacuum) BPS
monopole solutions changes the computations [1] regarding the isospin operator
$\bf T$ and the total momentum $\bf J$.
So instead of (2.7), then it should be written down
${\bf J}={\bf r}\times m{\dot{\bf r}}+f_{1}^{BPS}({\bf n}\cdot{\bf T}){\bf
n}+{\bf T}(1-f_{1}^{BPS}).$ (2.16)
The third item appearing in (2.16) corresponds to the expression
${\bf p}=m{\dot{\bf r}}+(\frac{1}{r}f_{1}^{BPS}){\bf n}\times{\bf T}$
for the momentum $\bf p$ of a particle in the YM BPS monopole background.
Indeed, the presence of the YM BPS ansatz (1.8) complicates to a considerable
extent the computations comparing to those (2.10)- (2.14) [1] in the ’ t
Hooft-Polyakov monopole model [22, 23]. It is associated, for instance, with
the more complicated expression for $\bf B$ (see e.g. [7]) in the BPS monopole
theory.
But, in spite of these difficulties, one can conclude, issuing from (2.16),
that $f_{1}^{BPS}$ really plays the role of an electric form-factor in the BPS
monopole YMH model. It is because the second item in (2.16) can be represented
as
$f_{1}^{BPS}T_{3}{\bf n},$
that implies, as it is easy to understand, the replacement $e\leftrightarrow
f_{1}^{BPS}e$ in Eq. (2.16). And this is just equivalent to the role of the YM
BPS ansatz $f_{1}^{BPS}$, (1.8), as an electric form-factor screening the
(elementary) charge $e$.
The similar role of an electric form-factor is played also by the Higgs BPS
ansatz $f_{0}^{BPS}$, (1.7). The physical consequence of this is screening
effect for electric charges in the Higgs phase [24] of the YMH model,
additional to that rendering in the Higgs phase by the in average electrically
neutral Higgs Bose condensate.
## 3 Higgs BPS ansatzes and CP violating.
In this section, repeating the arguments [26], we shall attempt to demonstrate
that the presence of YM BPS ansatz $f_{1}^{BPS}$, (1.8), in the (Minkowskian)
YMH BPS monopole theory violates manifestly the CP invariance of that theory.
In the previous section we have discussed the Dirac-Zwanziger-Schwinger
quantization condition (2.2) [26]. It turns out that this condition says
something important about the difference between electric charges of two
magnetic monopoles.
Given, for example, two monopoles of minimum allowed charge $2\pi/e$ and of
electric charges $q$ and $q^{{}^{\prime}}$, one finds
$e_{1}{\bf m}_{2}-e_{2}{\bf m}_{1}=2\pi(q-q^{{}^{\prime}})/e,$ (3.1)
so that the Dirac-Zwanziger-Schwinger quantization condition (2.2) gives
$q-q^{{}^{\prime}}=ne.$ (3.2)
Thus the difference $q-q^{{}^{\prime}}$ must be an integer multiple of $e$.
But as it was noted in [28], there is no restriction onto $q$ and
$q^{{}^{\prime}}$ separately.
If, however, the Dirac-Zwanziger-Schwinger quantization condition (2.2) is
suplemented by the CP conservation, the allowed values of the electric charge
of an magnetic monopole are also quantized.
In fact, although the electric charge is odd under CP, the magnetic charge is
even (this is because electric and magnetic fields are transformed oppositely
under parity). Applied to a monopole of the charges $(q,2\pi/e)$, a CP
transformation gives the monopole of the charges $(-q,2\pi/e)$. For these two
particles
$e_{1}{\bf m}_{2}-e_{2}{\bf m}_{1}=4\pi q/e$ (3.3)
and is a multiple of $2\pi$ only if
$q=ne\quad{\rm or}\quad q=(n+1/2)e.$ (3.4)
Thus at assuming the CP conservation, monopoles can have integer or half-
integer electric charges (as multiples of $e$). And moreover, if monopoles of
integer charges exist, monopoles of half-integer charges do not exist and
vice-versa.
Appart from the CP conservation, there are no reasoning for satisfying the
claim (3.4). In nature the CP invariance is violated, but weakly. One can thus
suspect that monopoles possess almost (half)integer electric charges. The
deviation of monopoles from such charges would be proportional to the strenght
of CP violating [26].
The one source CP violating is the instanton YM model [25], resulting [1] the
effective Lagrangian
${\cal L}_{\rm eff}\equiv{\cal L}+\Delta{\cal L}={\cal
L}+\frac{g^{2}\theta}{16\pi^{2}}~{}{\rm
tr}~{}(F_{\mu\nu}^{a}\tilde{F}^{\mu\nu});\quad\tilde{F}_{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta};$
(3.5)
involving the quasimomentum $\theta\in[-\pi,\pi]$ 999Indeed, as it was argued
in the papers [11, 20, 29] (see also [30]), $\theta$ is a complex parameter.
This can be seen at performing the quantization procedure for the instanton YM
model [25]. The latter one is reduced to solving the system of equations [11,
20, 29, 30] ${\hat{H}}(i\delta/\delta
A,A)\Psi_{\epsilon}[A]=\epsilon\Psi_{\epsilon}[A],$
$\nabla_{i}^{ab}(A)(\frac{\delta}{i\delta
A_{ib}}\Psi_{\epsilon}[A])=0;\quad\nabla_{i}^{ab}(A)=\delta^{ab}\partial_{i}-g\epsilon^{abc}A_{ic};$
$T_{1}\Psi_{\epsilon}[A]=e^{i1\cdot\theta}\Psi_{\epsilon}[A].$ for the wave
function $\Psi_{\epsilon}[A]$, the quantum analogue of an instanton [25]
possessing the energy $\epsilon$. The first equation in this system is the
Schr$\rm\ddot{o}$dinger equation for the YM Hamiltonian ${\hat{H}}=\int
d^{3}x\frac{1}{2}[E^{2}+B^{2}];\quad E=\frac{\delta}{i\delta A}.$ The second
one expresses the normalization assumed in the instanton YM model [25] at
which the electric field $E$ is transverse [11]:
$\nabla_{i}E^{i}\Psi_{\epsilon}[A]=0.$ At last, the third equation implicates
the raising operator [1] $T_{1}|n>=|n+1>$ (with the winding number 1 as its
eigenvalue). In the terminology [9], the transverse electric field $E$ remains
invariant with respect to all the “large” transformations $T_{1}$, while the
instanton wavefunction $\Psi_{\epsilon}[A]$ is manifestly covariant with
respect to these transformations. The raising operator $T_{1}$ can be
represented explicitly in the shape [20]
$T_{1}=\exp(\frac{d}{dX[A]})=\exp\left\\{\left[\int
d^{3}xB^{2}\frac{g^{2}}{16\pi^{2}}\right]^{-1}\int
d^{3}xB_{i}^{a}\frac{\delta}{\delta A_{i}^{a}}\right\\},$ with $X[A]$ being
the YM winding number functional (its look is well known and we will not cite
it here). The above system of equations gives a correct definition (cf. [31])
of the $\theta$-vacuum as a pseudomomentum operator possessing the common
system of eigenfunctions $\\{\Psi_{\epsilon}[A]\\}$ with the momentum operator
$\nabla_{i}^{ab}(A)$. Indeed, one encounters the following problem with this
definition [20, 31] of the $\theta$-vacuum. The thing is that the operators
$\hat{H}$ and $T_{1}$ don’t commute (as it was argued in [20]) unless
$\epsilon=0$, i.e. the have no common eigenfunction $\Psi_{\epsilon}[A]$ at
$\epsilon\neq 0$ (on the contrary, the Hamilton operator $\hat{H}$ commutes
with the momentum $\nabla_{i}^{ab}(A)$). It is obvious [20, 29, 30] that at
setting $\epsilon=0$, the definition [20, 31] of the $\theta$-vacuum remains
valid for imaginary as well as for real values of $\theta$. This allows to
represent $\theta$ as a complex number [30] $\theta=\theta_{1}+i\theta_{2}$.
In particular, at purely imaginary values of $\theta=\theta_{2}$ and
$\epsilon=0$, there exists the plane wave $\Psi_{0}[A]=\exp\\{i(2\pi
k+\theta_{2})X[A]\\}\equiv\exp\\{iP_{\cal N}X[A]\\}\quad(k\in{\bf Z})$ which
satisfies formally our definition [20, 31] of the $\theta$-vacuum at $P_{\cal
N}=2\pi k+\theta_{2}=2\pi k\pm i8\pi^{2}/g^{2}.$ Here the real part of
$P_{\cal N}$, $\theta_{1}$, runs formally over the discrete set $2\pi k$,
while its imaginary part $\theta_{2}=8\pi^{2}/g^{2}$ is continuous. But this
plane wave functional diverges manifestly at the $-$ sign before
$8i\pi^{2}/g^{2}$ (this fact was pointed out, for instance, in the papers [4,
30]). Simultaneously, one can constract (repeating the arguments [20, 29]; see
also [30]) the family of purely real solutions for the topological momentum
$P_{\cal N}$ satisfying the Schr$\rm\ddot{o}$dinger equation at $\epsilon=0$
being parallel (via the $\theta$) the eigenvalue of the raising operator
$T_{1}$: $P_{\cal N}^{\rm R}=2\pi k\pm 8\pi^{2}/g^{2}.$ Issuing from this
Eq., one can equate $\theta_{1}=\pm 8\pi^{2}/g^{2}$ and assume $\theta_{1}$ to
vary in the interval $[-\pi,\pi]$; it is just that real $\theta$-angle
considered in modern gauge physics (see, for instance, [1]). On the other
hand, the presence of imaginary (i.e. space-like) momentum modes
$8i\pi^{2}/g^{2}$ in the $P_{\cal N}$ spectrum (with the appropriate
“diverged” plane waves) gives it impossibly to give the correct probability
description of the (topologically degenerated) $\theta$-vacuum sinse the
Hilbert space $\Psi_{0}^{(n)}[A]$ of its states becomes non-separable in this
case. The remarkable property of the real part $P_{\cal N}$ spectrum is
following [20, 30]. It is obvious that such real topological momentum $P_{\cal
N}^{\rm R}$ vanishes (i.e. the instanton YM configuration stops) in the limit
$g\to\infty$ for the YM coupling constant $g$. It is just the infrared QCD
confinement limit as it is understood in modern physic. In the terminology
[20] this case is referred to as the infrared catastrophe. Note that purely
imaginary (and thus space-like and unphysical) values $P_{\cal N}=\pm
8\pi^{2}i/g^{2}$ (at $k=0$) have no relation to the infrared catastrophe.
The results we have demonstrated now can be treated [20] as the presence of
unphysical solutions to the Schrödinger equation at the application of
ordinary quantization methods to a topologically nontrivial theory. In the
paper [11] this statement was referred to as the so-called no-go theorem. .
The name “quasimomentum” for $\theta$ has the following origin. The thing is
that even real values of the topological momentum $P_{\cal N}$ have rather a
fictive nature. This is so since the $\theta$-term in the instanton YM
effective Lagrangian (3.5) does not alter [8] the YM equations of motions
$D_{\mu}F^{\mu\nu}=0.$
It is obvious that the first degree of the quasimomentum $\theta$ in the
instanton YM effective Lagrangian (3.5) inplies its manifest P (and CP because
of this) covariance.
The question about the “separate” C-covariance of the $\theta$-item in (3.5)
is more delicate. To understand which a “delicacy” is concealed here let us
now resort to the arguments [26].
In this paper the effect influence the (real) $\theta$-angle upon the dyon
charge was investigated.
To determine the concrete effect CP violation of the dyon charge by
$\theta$-angle (latter proves to be conserved if $\theta=0$; we have made sure
in this above), one must apply a semi-clasical analysis. For instance, the
author [26] utilize the simple semi-clasical analysis has been performed in
the work [32].
In this work a semi-clasical quantizaation of clasical dyonic solutions. In a
gauge in which fields disappear at the (spatial) infinity, clasical dyonic
solutions are periodic in time. The semi-clasical quantizaation condition
comes to the claim [26, 32] that $S+ET$ (the action during the time period $T$
plus the energy times the time) should be an integer multiple of $2\pi$.
Let $I$ being the action per unit time. Then the above claim “that $S+ET$
should be an integer multiple of $2\pi$” is reduced to the relation
$T(I+E)=2\pi n\quad(n\in{\bf Z}).$
The clasical period $T$ and the “abbreviated action” [26] $I+E$ (at $T=1$)
were caslculated in Ref. [32] in the absence of the CP violation. It was found
that
$I+E=cq^{2};$ (3.6) $T=\frac{2\pi}{ec}~{}~{}\frac{1}{q},$ (3.7)
where $q$ is the charge of the dyon an $c$ is a constant 101010It is not a
large problem to calculate this constant, but a simple arguments shows that
the same constant $c$ appears in Eqs. (3.6) and (3.7)..
The condition $T(I+E)=2\pi n$ now gives
$q=ne,$ (3.8)
so that dyons possess integer charges as one expects in the absence of the CP
violation.
Let us assume now that $\theta\neq 0$ and let us repeat the above
calculations. At $\theta\neq 0$ the equations of motion are unchanged [8, 30],
and there no change in the period $T$ or the energy $E$. However, there is an
extra contribution to the action $I$ from the extra item $\Delta{\cal L}$ in
the effective instanton YM Lagrangian (3.5) 111111In the light of our above
conjecture [20, 29, 30] that $\theta=\theta_{1}+i\theta_{2}$, the expression
below, recast to the look
$E=cq^{2}-I+ceq\frac{\theta}{2\pi}=cq^{2}-I+ceq\frac{\theta_{1}+i\theta_{2}}{2\pi},$
shows transparently that the “complex” $\theta$-vacuum is not stationar. Due
to the ordinary quantum mechanic canons, the life time of a $\theta$-vacuum
state with the energy $E$ (indeed, we should set $E=0$ sinse only such
$\theta$-vacuum states are compatible with the definition, us given above, of
this vacuum) is $\tau=\frac{2\pi\hbar}{\theta_{2}}$ (appropriately, its line
width is $\theta_{2}/2\pi$). This means that the $\theta$-vacuum state with
the energy $E=0$ decays into (two) states: say, 1 and 2, for which
$E_{1}+E_{2}=0$. But these $\theta$-vacuum states involving nonzero energies
$E_{1}$ and $E_{2}$ are badly specified. As we have emphasized above,
repeating the arguments [20], these values of energy cannot be simultaneously
common eigenvalues of the YM Hamilton operator $\hat{H}$ and the raising
operator $T_{1}$. In this is the next in turn contradiction about the YM
instanton model. And moreover, the life time $\tau$ of a $\theta$-vacuum state
with the energy $E=0$ is a finite number as $\theta_{2}\neq 0$. Thus the
$E=0\to E_{1}+E_{2}$ vacuum decay occurs in the finite time $\tau$. This
implies once again bad specifying the quantum states $|1>$ and $|2>$
corresponding to the energies $E_{1}$ and $E_{2}$ as those not referring to
the time infinities. As it was discussed in Ref. [30] (repeating the said in
the monograph [33]), in that case $\tau\neq\pm\infty$ it is impssible to
describe correctly Feynman diagrams referring to the above $E=0\to
E_{1}+E_{2}$ vacuum decay since only at the claim $\tau\to\pm\infty$ the
interaction representation of the system of quantum fields, set with the aid
of the appropriate scattering matrix $S$, is true. If $\tau\neq\pm\infty$, one
cannot pick out (as a consequence of the Haag theorem [33]) a correct Fock
representation for interacting (quantum) fields. :
$I+E=cq^{2}+ceq\frac{\theta}{2\pi}.$ (3.9)
The semi-classical (Bohr-Sommerfeld) quantizaation of $T(I+E)$ then gives
$q=ne-\theta e/2\pi.$ (3.10)
So the allowed values of magnetic monopoles (if latter ones are contemplated
in the model: for instance if Higgs modes are present in such model, violating
the initial $SU(2)$ gauge symmetry) and are not integer if $\theta\neq 0$. In
particular, there does not exist an electrically neutral magnetic monopole in
that case.
If no magnetic and electric charges are presented in the gauge theory (i.e.
there are no magnetic monopoles in that theory): for instance in the “pure” YM
theory without Higgs modes (the instanton model is just such a case), one
would set $q=0$ in (3.10) (this does not contradict the Dirac quantization
condition (2.1) since one deals with the uncertainty $0\cdot 0$ in that case).
Then
$n=[\theta/2\pi],$ (3.11)
i.e. the integer part of the number $\theta/2\pi$. It is, in fact, the
ordinary connection between the $\theta$-angle and the integer topological
number $n$ in the instanton model.
Now we should recall that in the absence of magnetic monopoles the
$\theta$-angle results no “true” motions in the YM theories. Instead of this,
the $\theta$-dependense in a YM model comes to purely tunnelling effects
connected with instantons [1, 25] (for the topological number $n=1$ such
effects of the order $\exp(1/\alpha)$; $\alpha\equiv g^{2}/4\pi$).
But if magnetic monopoles (for instance, of the ’t Hooft-Polyakov type [22,
23]) are incorporated in the YM theory, in the monopole sector of that theory
there are classically allowed motions, dyons, with nonzero
$n\sim\int d^{4}xF_{\mu\nu}\tilde{F}^{\mu\nu}\in{\bf Z}.$
As a result, the $\theta$-dependense in the monopole sector is something
another than that connected with instantons: in particular, it is of the
leading order rather different from $\exp(1/\alpha)$.
What happens, asks the author [26], in the YM theories in which CP is violated
by another mechanism than $\theta$?
The fact that at $\theta=0$ the dyons have integer charges is associated with
the fact that $I+E$ in Eq. (3.6) is quadric in $q$, with no linear term. A
linear term as in Eq. (3.9) leads to noninteger charges: the “non-integrality”
is directly proportional to the coeficient of the linear term.
CP forbids such linear term sinse $q$ is odd under CP. If CP is violated,
regardless the violation mechanism, a linear term can be present.
Even if a linear term is absent on the classical level (for instance, if only
couplings to fermions violate CP, since fermions do not enter classical
solutions), it can be present on the quantum level due to loop corrections.
Roughly speaking, one should recalculate $I+E$ from the quantum effective
action (rather from the classical action). If CP is violated, loop corrections
to the effective action would induce a term linear in $q$ in the effective
$I+E$ and therefore to cause monopole charges (if exist) to be not quite
integer.
The same effect CP violating occurs, as we have demonstrated in the previous
section, in the YMH model involving BPS magnetic monopoles (without any
$\theta$-dependense). The crucial point here is the presence of the BPS ansatz
$f_{1}^{BPS}$, (1.8), in that model (as it can be seen from (2.16)). But these
BPS magnetic monopole solutions induce (as it can be demonstrated) rather tree
than loop Feynman diagrams.
## 4 Discussion.
The Dirac fundamental quantization [34] of the YMH model involving vacuum BPS
monopole solutions (coming, as we have explained above, to solving the Gauss
law constraint in terms of topological Dirac variables) implies some refining
us said in the previous section about the CP violation in the presence of
those vacuum BPS monopole solutions.
As it was discussed in the recent paper [30], resolving the YM Gauss law
constraint
$\frac{\delta W}{\delta
A^{a}_{0}}=0\Longleftrightarrow[D^{2}(A)]^{ac}A_{0c}=D^{ac}_{i}(A)\partial_{0}A_{c}^{i}.$
(4.1)
in terms of topological Dirac variables
$\hat{A}_{i}^{(n)}(t,{\bf x})=\hat{\Phi}_{i}^{(n)}({\bf
x})+{\hat{\bar{A}}}_{i}^{(n)}(t,{\bf x});\quad n\in{\bf
Z};\quad\hat{A}_{\mu}=g\frac{A_{\mu}^{a}\tau_{a}}{2i\hbar c}$ (4.2)
(here ${\hat{\bar{A}}}_{i}^{(n)}(t,{\bf x})$ are perturbation excitations,
multipoles, over the BPS monopole vacuum); satisfying the Coulomb gauge
$D^{i}{\hat{A}_{i}}^{a(n)}=0$, results the homogeneous equation
$[D^{2}(A)]^{ac}\hat{A}_{0c}=0$ (4.3)
for temporal components $\hat{A}_{0c}$ of YM fields.
As it was shown in Ref. [20], the solutions to the “homogeneous” YM Gauss law
constraint (4.3) can be found in the shape
$A_{0}^{c}(t,{\bf x})={\dot{N}}(t)\Phi_{(0)}^{c}({\bf x})\equiv Z^{c},$ (4.4)
implicating the topological dynamical variable $\dot{N}(t)$ and Higgs
(topologically trivial) vacuum Higgs BPS monopole modes $\Phi_{0}^{a}({\bf
x})$.
Issuing from zero mode solutions [20] $Z^{a}$ to the YM Gauss law constraint
(4.1), it is easy to write down $F_{i0}^{a}$ components of the YM tension
tensor, taking the shape of so-called vacuum ”electric” monopoles [3, 4]
$F^{a}_{i0}={\dot{N}}(t)D^{ac}_{i}(\Phi_{k}^{(0)})\Phi_{0c}({\bf x}).$
In turn, vacuum ”electric” monopoles $F^{a}_{i0}$ generate the action
functional
$W_{N}=\int d^{4}x\frac{1}{2}(F_{0i}^{c})^{2}=\int dt\frac{{\dot{N}}^{2}I}{2}$
(4.5)
involving the rotary momentum [12]
$I=\int_{V}d^{3}x(D_{i}^{ac}(\Phi_{k}^{0})\Phi_{0c})^{2}=\frac{4\pi^{2}\epsilon}{\alpha_{s}}=\frac{4\pi^{2}}{\alpha_{s}^{2}}\frac{1}{V<B^{2}>}.$
(4.6)
This action functional (describing the solid rotations of the YMH vacuum
suffered the Dirac fundamental quantization) is manifestly Poincare (P)
invariant sinse it can be recast to the look
$W_{N}=\int dt\frac{P_{N}^{2}(t)}{2I};\quad P_{N}={\dot{N}}I.$ (4.7)
Herewith
$P_{N}={\dot{N}}I=2\pi k+\theta;\quad\theta\in[-\pi,\pi];$ (4.8)
is the momentum corresponding to the abovementioned solid rotations of the YMH
vacuum. As it can be seen from (4.8), the $P_{N}$ spectrum is purely real and
thus rotary trajectories for the YMH vacuum suffered the Dirac fundamental
quantization belong to the physical ones.
It is just the momentum spectrum free from imaginary, i.e. tachionic, modes,
unlike the instanton case, us discussed in Section 3.
Besides that the action functional (4.7) is manifestly P-invariant, it is also
C-invariant. To understand this fact, we should compare Eq. (4.7) (it is a
definite functional of the YMH BPS monopole vacuum rotary energy
$P_{N}^{2}(t)/2I$) with Eq. (3.9) [26].
In the latter case the appropriate energy $I+E$ associated with the instanton
$\theta$ vacuum proves to be linear (i.e. odd) by $e$ and $\theta$. This just
implies CP violating.
The former case is rather different. As it follows from (4.6), (4.7), the
action functional (4.7) is directly proportional to the vacuum ”electric”
field (”electric” monopole) $F^{a}_{i0}$, induced, in turn, by a Higgs vacuum
BPS monopole mode $\Phi_{0c}({\bf x})$.
As it was shown in Ref. [12] (see also [30]),
$F^{a}_{i0}\equiv
E_{i}^{a}=\dot{N}(t)~{}(D_{i}(\Phi_{k}^{(0)})~{}\Phi_{(0)})^{a}=P_{N}\frac{\alpha_{s}}{4\pi^{2}\epsilon}B_{i}^{a}(\Phi_{(0)})=(2\pi
k+\theta)\frac{\alpha_{s}}{4\pi^{2}\epsilon}B_{i}^{a}(\Phi_{(0)}).$ (4.9)
Latter relations were got with the aid of the Bogomol’nyi equation (1.6).
Sinse the YM coupling constant $\alpha_{s}\equiv g^{2}/4\pi(\hbar c)^{2}$
enters Eq. (4.9), it is already squared by $g$, while the rotary momentum $I$,
(4.6), is of the order $g^{4}$.
But at accepting the normalization (2.12) [8] for the electric charges $q$
(referring to the Higgs Bose condensate), the vacuum ”electric” field
$F^{a}_{i0}$ will be of the order $q^{-2}$ (respectively, the YMH BPS monopole
vacuum rotary energy $P_{N}^{2}(t)/2I$ will be of the order $q^{-4}$). Thus
the YMH model with vacuum BPS monopoles quantized by Dirac is C-invariant as
that possessing the action functional (4.7) even by the total electric charge
$Q=\sum_{i}q_{i}$ of (topologically degenerated) Higgs vacuum BPS monopole
modes $\Phi_{c(n)}({\bf x})$ [11].
This us demonstrated CP conservation is the next in turn remarkable feature of
that model.
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|
arxiv-papers
| 2008-12-30T15:15:18 |
2024-09-04T02:48:59.636712
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Leonid Lantsman",
"submitter": "Leonid Lantsman",
"url": "https://arxiv.org/abs/0812.5080"
}
|
0812.5108
|
# Asymptotically FRW black holes
J.T. Firouzjaee Department of Physics, Sharif University of Technology,
Tehran, Iran firouzjaee@physics.sharif.edu Reza Mansouri Department of
Physics, Sharif University of Technology, Tehran, Iran and
School of Astronomy, Institute for Research in Fundamental Sciences (IPM),
Tehran, Iran mansouri@ipm.ir
###### Abstract
Special solutions of the LTB family representing collapsing over-dense regions
extending asymptotically to an expanding closed, open, or flat FRW model are
found. These solutions may be considered as representing dynamical mass
condensations leading to black holes immersed in a FRW universe. We study the
dynamics of the collapsing region, and its density profile. The question of
the strength of the central singularity and its nakedness, as well as the
existence of an apparent horizon and an event horizon is dealt with in detail,
shedding light to the notion of cosmological black holes. Differences to the
Schwarzschild black hole are addressed.
###### pacs:
95.30.Sf,98.80.-k, 98.62.Js, 98.65.-r
## I introduction
Let us use the term cosmological black hole for any solution of Einstein
equations representing a collapsing overdensity region in a cosmological
background leading to an infinite density at its center sultana . There have
been different attempts to construct solutions of Einstein equations
representing such a collapsing central mass. Gluing of a Schwarzschild
manifold to an expanding FRW manifold is one of the first attempts to
construct such a cosmological black hole, as done first by Einstein and Straus
Einstein Straus . Depending of the way the model is constructed, one is led to
un-physical behavior of the trajectories Baker .
Models not based on a cut-and-paste technology is much more interesting giving
more information on the behavior of the mass condensation within a FRW
universe model. The first attempt is due to McVittie McVittie introducing a
spacetime metric that represents a point mass embedded in a Friedmann-
Robertson-Walker (FRW) universe. There have been many other attempts to
construct cosmological black holes such as Nolan interior solution nolan m ,
and Sultana-Dyer solutionsultana , each of them contrasting some of the
features one expect from theory or observation.
The interest for cosmological black holes in the past has been mainly from the
theoretical side to understand concepts like black hole, singularity, horizon,
and thermodynamics of black holes haw-bh . Indeed, the conventional definition
of black holes implies an asymptotically flat space-time and a global
definition of the event horizon. In practice, however, the universe is not
asymptotically flat. The need for local definition of black holes and their
horizons has led to concepts such as Hayward’s trapping horizon Hayward94 ,
Ashtekar’s isolated horizon ashtekar99 , Ashtekar and Krishnan’s dynamical
horizon ashtekar02 , and Booth and Fairhurst’s slowly evolving horizon booth04
.
There are cases where both apparent and event horizon maybe possibly defined.
For example, for dynamical black holes one may define the event horizon as the
very last ray to reach future null infinity or the light ray that divides
those observer who cannot escape the future singularity from those that can
kra-hel-BH . Eardley proposed the conjecture that in such cases trapped
surfaces can be deformed to get arbitrarily close to the event horizon
Eardley98 . Numerical evidence was provided in Krishnan05 and later proved
analytically for the Vaidya metric Ben-Dov .
The precision cosmology has opened a new arena for questions like cosmological
black holes and their behavior. New observation of our galactic center allow
to resolve phenomena near the Schwarzschild horizon of the central black hole
doeleman . It is therefore desirable to have black hole models embedded in
cosmological environment to see if there may be considerable differences to
the familiar Schwarzschild black hole. There have been also increasing
interest in the gravitational lensing by a cosmological mass condensation such
as a cluster of galaxies in a cosmological background. The simplest cases are
the Kottler and the Einstein-Straus model rindler . The more complex situation
is lensing by a mass condensation within a dynamical background.
Now, a widely used metric to describe the gravitational collapse of a
spherically symmetric dust cloud is the so-called Tolman-Bondi-Lemai tre (LTB)
metric LTB . These models have been extensively studied for the validity of
the cosmic censorship conjecture cencorship ,joshi and joshi-cell . In
particular, we know already initial condition that, depending upon the
initial conditions defined in terms of the initial density and velocity
profiles from which the collapse develops, the central shell-focusing
singularity at $r=0$ can be either a black hole or a locally or globally naked
singularity. We may note however, that in all these papers a compact LTB
region is glued to the Schwarzschild metric or the FRW outer universe mansouri
. Therefore, the results have to be taken cautiously: any principally existent
event horizon is cut off by the outer static or homogeneous space-time. The
statement may still be correct that in a dynamic spacetime the cosmic
censorship hypotheses is valid, as discussed in wald98 . It is also possible
to glue two different LTB metrics to study the structure formation out of an
initial mass condensation or the formation of a galaxy with a central black
hole kra-hel-sf and kra-hel-BH . Here again the structure of the metric
outside the mass condensation is defined by hand to match with a specific
galaxy or cluster feature. Faraoni et al have tried to change McVitte metric
so that it resemble a collapsing mass condensation. Their solution, however,
represents a singularity within a horizon embedded in a universe filled with a
non-perfect fluid where the change of the mass is not because of the in-
falling matter but the heat flow faraoni2 . This metric gives us no clue
whatsoever about the dynamics of a possible collapsed mass condensation.
Harada et al, being interested in the behavior of primordial black holes
within cosmological models with a varying gravitational constant, use a LTB
solution to study the evolution of a background scalar field when a black hole
forms from the collapse of dust in a flat Friedmann universe probing the
gravitational memory harada .
Our goal is to look for a model of a cosmological black hole, i.e. a mass
condensation leading to a singularity within a FRW universe universe. In this
paper we propose the models for closed-, open-, and flat FRW universe studying
their density profiles, singularities, and horizon behaviors. There are many
nontrivial questions to be answered before understanding in detail the
differences of these cosmological black holes to the familiar Schwarzschild
ones, which are beyond the scope of this paper and are to be dealt with in
future publications.
The question of singularities and the definition of a black hole in such a
dynamical environment has been subject of different studies in the last 15
years. We review very shortly different definitions of horizons in section II
as a reference to the properties of model solutions we propose. Some initial
attempts to model black holes within a FRW universe is introduced in section
III. Section IV is devoted to the LTB metric as the generic solution
representing a spherically symmetric ideal fluid. Section V is devoted to
different models of cosmological black holes, their dynamics, density profile,
apparent and event horizons,and singularities. The question of strength and
the nakedness of singularities are dealt with in section VI. We then conclude
in section VII. Throughout the paper we assume $8\pi G=c=1$.
## II local definitions of black holes
Standard definition of black holes haw-bh needs some global assumptions such
as regular predictability and asymptotic flatness. In the cosmological context
concepts of asymptotic flatness and regular predictability have no
application. This has already been noticed by Demianski and Lasota na73
stressing the fact that in the cosmological context the standard global
definition of black holes using event horizons may not be used any more.
Tipler tipler77 also present a definition of black hole in non asymptotically
flat space time, but these definition did not have comprehensive property of
black hole such as thermodynamic laws. In the last decade the interest in a
local definition of black holes has led to four different concepts based
primarily on the concept of the apparent horizon.
Let us start by assuming a spacelike two surface $S$ with two normal null
vectors $\ell^{a}$ and $n^{a}$ on it.The corresponding expansions are then
defined as $\theta_{(\ell)}$,$\theta_{(n)}$.
Definition 1 Hayward94 . A _trapping horizon_ $H$ is a hypersurface in a
4-dimensional spacetime that is foliated by 2-surfaces such that
$\theta_{(\ell)}\mid_{H}=0$, $\theta_{(n)}\mid_{H}\neq 0$, and
$\pounds_{n}\theta_{(\ell)}\mid_{H}\neq 0$. A trapping horizon is called
_outer_ if $\pounds_{n}\theta_{(\ell)}\mid_{H}<0$, _inner_ if
$\pounds_{n}\theta_{(\ell)}\mid_{H}>0$, _future_ if $\theta_{(n)}\mid_{H}<0$,
and _past_ if $\theta_{(n)}\mid_{H}>0$. The most relevant case in the context
of black holes is the _future outer trapping horizon_(FOTH).
Definition 2 ashtekar99 . A _weakly isolated horizon_ is a three-surface H
such that :
1\. H is null;
2\. The expansion $\theta_{(\ell)}\mid_{H}=0$ where $\ell^{a}$, being null and
normal to the foliations $S$ of $H$;
3.$-T^{b}_{a}\ell^{a}$ is future directed and causal;
4\. $\pounds_{\ell}\omega_{a}=0$, where
$\omega_{a}=-n_{b}\nabla_{\underleftarrow{a}}\ell^{b}$, and the arrow
indicates a pull-back to H.
Weakly isolated horizon is a useful term to be used for characterization of
black holes not interacting with their surroundings, and corresponds to
isolated equilibrium states in thermodynamics. These definition do not apply
to cosmological mass condensations because of their dynamical behavior.
Definition3 ashtekar02 . A _marginally trapped tube_ T (MTT) is a hypersurface
in a 4-dimensional spacetime that is foliated by two-surfaces $S$, called
_marginally trapped surfaces_ , such that $\theta_{(n)}|_{T}<0$ and
$\theta_{(\ell)}|_{T}=0$. MTTs have no restriction on their signature, which
is allowed to vary over the hypersurface. This is a generalization of the
familiar concept of the apparent horizon ashtekar02 . If a MTT is everywhere
spacelike it is referred to as a _dynamical horizon_. If it is everywhere
timelike it is called a timelike membrane (TLM). In case it is everywhere null
and non-expanding then we have an isolated horizon. The apparent horizons
evolving in the our proposed models will not be everywhere spacelike and will
have a complex behavior.
Irrespective of different concepts related to the apparent horizon we may
still compromise on a definition of event horizon differing principally from
the apparent horizon. We follow the definition of kra-hel-BH as the very last
ray to reach future null infinity or the light ray that divides those observer
who cannot escape the future singularity from those that can. We will see in
the next sections that cosmological black holes may have distinct apparent and
even horizons, in contrast to the Schwarzschild black hole.
## III Existing metrics representing over-densities within a cosmological
background and their deficiencies
### III.1 McVittie s solutions
In 1933, McVittie McVittie found an exact solution of Einstein s equations
for a perfect fluid mimicking a black hole embedded in a cosmological
background. McVittie s solutions can be written in the form
$ds^{2}=-(\frac{1-\frac{M}{2N}}{1+\frac{M}{2N}})^{2}dt^{2}+e^{\beta(t)}(1+\frac{M}{2N})^{4}(dr^{2}+h^{2}d\Omega^{2}),$
(1)
where $M=me^{\beta(t)/2}$ and $m$ is a constant. Functions $h(r)$ and $N(r)$
depend on a constant $k$, and are given, respectively, by
$\mbox{$h(r)=$}~{}\begin{cases}&\sinh(r)~{}~{}~{}k=1{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
&r~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}k=0\\\ &\sin(r)~{}~{}~{}~{}k=-1\end{cases}\\\
\mbox{$N(r)=$}~{}\begin{cases}&2\sinh(r/2)~{}~{}~{}k=1{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}\\\
&r~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}k=0\\\ &2\sin(r/2)~{}~{}~{}~{}k=-1\end{cases}$
(2)
This metric represents a point mass embedded into an isotropic universe. It
possesses a curvature singularity at proper radius $R=2m$, in contrast to the
Schwarzschild metric where there is a coordinate singularity. It has been
shown that this singularity is space-like and weaknolan . The interpretation
of the metric in the region $R<2m$ is also not clear nolan . Therefore, the
McVittie’s metric is not a suitable solution of Einstein equations to
represent the collapse of a spherical mass distribution with over-density
within a cosmological setting.
### III.2 Sultana-Dyer solution
Recently Sultana and Dyer sultana found an exact solution representing a
primordial cosmological black hole. It describes an expanding event horizon in
the asymptotic background of the Einstein-de Sitter universe. The black hole
is primordial in the sense that it forms ab initio with the big bang
singularity and therefore does not represent the gravitational collapse of a
matter distribution.
This metric is given by
$ds^{2}=t^{4}[(1-\frac{2m}{r})dt^{2}-\frac{4m}{r}dtdr-(1-\frac{2m}{r})dr^{2}-r^{2}d\Omega^{2}].$
(3)
Though the metric has the same causal characteristics as the Schwarzschild
spacetime, there are significant differences for timelike geodesics. In
particular an increase in the perihelion precession and the non-existence of
circular timelike orbits should be mentioned. The matter content is described
by a non-comoving two-fluid source, one of which is a dust and the other is a
null fluid. At late times the dust becomes superluminal near horizon violating
the energy condition.
## IV Introducing realistic Models of Cosmological Mass Condensation
There maybe different ways of constructing solutions of Einstein equations
representing a collapsing mass concentration in a FRW background, as the
preceding sections show. We choose the direct way of a cosmological spherical
symmetric isotropic solution, and look for an overdensity mass distribution
within the model universe undergoing a collapse to see if and how a
singularity representing a black hole emerges. To begin with, we choose a so-
called flat LTB metric. This is the simplest spherically symmetric solution of
Einstein equations representing an inhomogeneous dust distribution LTB .
### IV.1 LTB metric
The LTB metric may be written in synchronous coordinates as
$ds^{2}=dt^{2}-\frac{R^{\prime 2}}{1+f(r)}dr^{2}-R(t,r)^{2}d\Omega^{2}.$ (4)
It represents a pressure-less perfect fluid satisfying
$\displaystyle\rho(r,t)=\frac{2M^{\prime}(r)}{R^{2}R^{\prime}},\hskip
22.76228pt\dot{R}^{2}=f+\frac{2M}{R}.$ (5)
Here dot and prime denote partial derivatives with respect to the parameters
$t$ and $r$ respectively. The angular distance $R$, depending on the value of
$f$, is given by
$\displaystyle R=-\frac{M}{f}(1-\cos\eta(r,t)),$ $\displaystyle\hskip
22.76228pt\eta-\sin\eta=\frac{(-f)^{3/2}}{M}(t-t_{n}(r)),$ (6)
$\displaystyle\dot{R}=(-f)^{1/2}\frac{sin(\eta)}{1-cos\eta},$ (7)
for $f<0$, and
$R=(\frac{9}{4}M)^{\frac{1}{3}}(t-t_{n})^{\frac{2}{3}},$ (8)
for $f=0$, and
$\displaystyle R=\frac{M}{f}(\cosh\eta(r,t)-1),$ $\displaystyle\hskip
22.76228pt\sinh\eta-\eta=\frac{f^{3/2}}{M}(t-t_{n}(r)),$ (9)
for $f>0$.
The metric is covariant under the rescaling $r\rightarrow\tilde{r}(r)$.
Therefore, one can fix one of the three free parameter of the metric, i.e.
$t_{n}(r)$, $f(r)$, and $M(r)$. The function $M(r)$ corresponds to the Misner-
Sharp mass in general relativity, as shown in the general case of spherically
symmetric solutions of Einstein equationsmis-sharp .
There are two generic singularities of this metric: the shell focusing
singularity at $R(t,r)=0$, and the shell crossing one at $R^{\prime}(t,r)=0$.
To get rid of the complexity of the shell focusing singularity, corresponding
to a non-simultaneous big bang singularity, we will assume $t_{n}(r)=0$. This
will enable us to concentrate on the behavior of the collapse of an
overdensity region in an expanding universe without interfering with the
complexity of the inherent bang singularity of the metric.
Now, an expanding universe means generally $\dot{R}>0$. However, in a region
around the center it may happen that $\dot{R}<0$, corresponding to the
collapsing region. It is then easy to show that in this collapsing region
$\theta_{(\ell)}\propto(1-\frac{\sqrt{\frac{2M}{R}+f}}{\sqrt{1+f}})$,
$\theta_{(n)}\propto(-1-\frac{\sqrt{\frac{2M}{R}+f}}{\sqrt{1+f}})<0$.
Therefore, $R=2M$, is obviously a _marginally trapped tube_ , as defined in
section 2, representing an apparent horizon according to the familiar
definitionshaw-bh ; ashtekar02 . It will turn out that this apparent horizon
is not always spacelike and can have a complicated behavior for different $r$,
as was first seen in booth-mtt .
### IV.2 Behavior of the curvature function $f(r)$
Now, we are interested in an expanding universe, meaning generally
$\dot{R}>0$. However, in a region around the center we expect to have a late
time behavior $\dot{R}<0$ corresponding to the collapse phase of the
overdensity region. From equations (IV.1), (8), and (IV.1) we infer that to
have a collapsing region one has to ask for $f(r)<0$ in that region. In
contrast, the universe outside the collapsing region being expanding leave us
to choose $f(r)>0$, $f(r)=0$, or $f(r)<0$ depending on the model. We may have
an asymptotically flat FRW universe, however, with $f(r)>0$ or $f(r)<0$
tending to zero for large $r$.
Now, we have still to make a choice for $f(r)$ at the center $r=0$. Expecting
the mass $M$ to be zero at $r=0$ to avoid a central singularity, we see from
(IV.1) and (IV.1) that $\frac{f^{3/2}}{M}|_{r=0}=const$, or $f(r=0)=0$ origin
con . This give us different possibilities of the function $f(r)$ to behave as
shown in Fig.1.
Figure 1: Different behaviors of the curvature function $f(r)$.
## V Construction of Models
We have now the necessary prerequisites to construct our models of mass
condensation immersed in FRW models leading to singularities and representing
cosmological black holes. Our cosmological black hole solutions evolve from
mass condensations within closed, open, or flat FRW universes, leading to
singularities having different horizons, and providing us examples of
collapsed regions behaving differently to known Schwarzschild ones.
### V.1 Example I: $f<0$: asymptotically closed LTB metric
As mentioned before, we are free to choose one of the three parameters of the
LTB metric. Assuming a negative $f(r)$, we may choose $r$ such that
$f(r)=-M(r)/r$ cencorship . Now, let us choose the mass function $M$ such that
$M(r)=2^{3}a^{2}r^{3}\frac{\alpha+r^{3}}{1+r^{3}},$
where $a$ and $\alpha$ are constants to be defined properly. We then obtain
from (IV.1)
$\displaystyle R=r(1-cos\eta(r,t))$ $\displaystyle\hskip 22.76228pt\eta-
sin(\eta)=\sqrt{\frac{7.2+r^{3}}{1+r^{3}}}2^{3/2}at.$ (10)
We are free to fix $a$ and $\alpha$ such that for the present time, $t_{0}$,
the region around the center of the overdensity, $r=0$, is collapsing while
far from the center the universe expands. Note that in contrast to the
familiar FRW universe, where the scale factor as a function of time, $t$, is
an explicit function having a straightforward behavior. In the LTB case,
$R(t)$ playing the role of the scale factor is an implicit function of time
and comoving coordinate $r$ given by (IV.1). We now fix $a$ and $\alpha$ such
that $r=0$ corresponds to $\eta=\frac{3\pi}{2}$, and $r\gg 1$ corresponds to
$\eta=\frac{5\pi}{6}$ (Fig. 2). We then find $a\simeq\frac{0.75}{t_{0}}$ with
$t_{0}$ being the present time, and $\alpha\simeq 7$.
Now, the expansion phase of the model is given by $\dot{R}$ (7). We then see
from (7) that the region around $r=0$, corresponding to
$\eta\sim\frac{3\pi}{2}$, is always collapsing for any time $t$, while the
regions far from the center, $r>>0$, at the present time, corresponding to
$\eta\sim\frac{5\pi}{6}$, are expanding. Note that this bound LTB model,
similar to the closed FRW one, has a maximum comoving radius corresponding to
$f(r)=-1$.
Figure 2: Evolution of the Cauchy surfaces.
The density evolution and the causal structure of the model is shown in Fig.
3. We see clearly how the central overdensity region collapses to a
singularity at $r=0$, while the universe is expanding. Note also how the slope
of outgoing null geodesics tend to infinity in the vicinity of the
singularity, i.e. $R^{\prime}\rightarrow+\infty$ at $R=0$.
Figure 3: The case of the asymptotically closed universe: in the central
region the density increases with time indefinitely while far from the center
the density is decreasing with time. The apparent horizon and the trapped
region is shown in the lower diagram.
### V.2 Example II: $f<0$, $\lim_{r\rightarrow\infty}f(r)\rightarrow 0$;
asymptotically flat LTB metric 1
Our favorite choice is a solution representing a collapsing overdensity region
at the center and a flat FRW far from the overdensity region. Of course the
overdensity region may take part in the expansion of the universe at early
times but gradually reversing the expansion and start collapsing. To achieve
this, we require $f(r)<0$ and $f(r)\rightarrow 0$ when $r\rightarrow\infty$.
This choice give us trivially $M(0)=0$.
Let us now make the ansatz $f(r)=-re^{-r}$ leading to
$M(r)=\frac{1}{a}r^{3/2}(1+r^{3/2}),$
where $a$ is a constant having the dimension $[a]=[L]^{-2}$. We fix $a$ by
$at_{0}=3\pi/2$. Similar to our previous model I, this value of $a$
corresponds to the collapsing mass condensation around $r=0$ starting in the
expanding phase of the bound LTB model.
Equation (IV.1), (8) then leads to
$\displaystyle R=\frac{\sqrt{r}(1+r^{3/2})}{ae^{-r}}(1-\cos\eta(r,t)),$
$\displaystyle\hskip 22.76228pt\eta-
sin(\eta)=\frac{e^{-\frac{3}{2}r}}{(1+r^{3/2})}at.$ (11)
We have plotted the density evolution and casual structure of this model in
Fig.4.
Figure 4: The density profile for the cosmic black hole within a closed but
asymptotically flat universe. The causal structure is shown below. Note the
behavior of the event horizon for arbitrary large but finite $t$.
As a result of $R^{\prime}\rightarrow+\infty$ near the singularity, the slope
of the outgoing null geodesics becomes infinite at the central singularity.
Again we see clearly how the collapse of the central region and the evolution
of the apparent horizon separates the overdense region from the expanding
universe.
The negativity of the curvature function $f(r)$ means that, although the
universe is asymptotically flat, waiting enough, every slice $r=constant$ will
collapse to the central region. We may , however, define an event horizon
according to the definition of section 2 for any large but finite time, as
shown in Fig.4.
### V.3 Example III: $f>0$, $f(r)\rightarrow 0$ when $r\rightarrow\infty$;
asymptotically flat LTB metric 2
What would happen if we choose the curvature function $f(r)$ such that it
tends to zero for large $r$ while it is positive? We still have a model which
tends to a flat FRW at large distances from the center, but having a density
less than the critical one.
Let us make the ansatz $f(r)=-r(e^{-r}-\frac{1}{r^{n}+c})$ with $n=2$ and
$c=20000$, leading to
$M(r)=\frac{1}{a}r^{3/2}(1+r^{3/2}),$
where $a$ is a constant having the dimension $[a]=[L]^{-2}$. We fix $a$ by
requiring $at_{0}=3\pi/2$. Equation (IV.1), (8) then leads to
$\displaystyle
R=\frac{\sqrt{r}(1+r^{3/2})}{a(e^{-r}-\frac{1}{r^{2}+20000})}(1-\cos\eta(r,t)),$
$\displaystyle\hskip 22.76228pt\eta-
sin\eta=\frac{(e^{-r}-\frac{1}{r^{2}+20000})^{1.5}}{(1+r^{3/2})}at.$ (12)
and for $f>0$ region,
$\displaystyle
R=\frac{\sqrt{r}(1+r^{3/2})}{a(\frac{1}{r^{2}+20000}-e^{-r})}(\cosh\eta(r,t)-1),$
$\displaystyle\hskip 22.76228pt\eta-
sinh\eta=\frac{(\frac{1}{r^{2}+20000}-e^{-r})^{1.5}}{(1+r^{3/2})}at.$ (13)
The solution is continuous at $r=1$, as can be checked by evaluating
$\dot{R}$, $R^{\prime}$, $\dot{R}^{\prime}$, $\ddot{R}$, and
$\ddot{R}^{\prime}$ at $r=1$(see the appendix).
We have plotted the density evolution and casual structure of this model in
Fig.5.
Figure 5: Evolution of the cosmic black hole within an open but
asymptotically flat universe is similar to the closed case. The causal
structure, however, is significantly different, as seen from the lower
diagram. Result of the numerical calculation of the locations of the event
horizon, apparent horizon and the singularity is also shown.
The term $\frac{1}{r^{n}+c}$ is responsible for $f(r)$ being positive and
tending to zero for large $r$ given $n\geq 2$ and $c>>1$. Let us check if this
may cause shell crossing in the region where $f^{\prime}(r)<0$ while $f>0$.
Using (IV.1) we obtain
$\frac{R^{\prime}}{R}=\frac{M^{\prime}}{M}(1-\phi_{4})+\frac{f^{\prime}}{f}(\frac{3}{2}\phi_{4}-1),$
(14)
where
$\frac{2}{3}\leq\phi_{4}=\frac{sinh\eta(sinh\eta-\eta)}{(cosh\eta-1)^{2}}\leq
1$. The condition for no shell crossing singularity is then $\frac{M\mid
f^{\prime}\mid}{fM^{\prime}}<\frac{1-\phi_{4}}{\frac{3}{2}\phi_{4}-1}$. For
$\phi_{4}\sim 1$, corresponding to $\eta>>1$ or $t>>1$ the inequality breaks
down leading to a shell crossing singularity. The shell crossing, however, can
be shifted to arbitrary large $t$ by choosing $f^{\prime}<<1$ corresponding to
$n>>1$ and $c>>1$ hell-shell . Therefore, for the model we are proposing the
shell crossing will happen out of the range of applicability of it.
As a result of $R^{\prime}\rightarrow+\infty$ near the singularity, the slope
of the outgoing null geodesics become infinite at the central singularity.
Again we see clearly how the collapse of the central region and the evolution
of the apparent horizon separates the overdense region from the expanding
universe. There is an event horizon defined by the very last ray to reach
future null infinity and separates those observer who can not scape the future
singularity from those that can. A fixed $r=r_{0}$ value, being the non-
trivial root of $f(r)=0$, divides the absolute collapsing region from the
absolute expanding region. We may be living in a region inside the event
horizon but outside the apparent one without noticing it soon!
This solution represents a collapsing mass within an asymptotically flat FRW
universe. The collapsed region is dynamical in the sense that its mass is not
constant. In fact the rate of change of the Misner-Sharp energy is given by
$\frac{dM(r)}{dt}|_{R=const}=\frac{dM(r)}{dr}\frac{dr}{dt}|_{R=const}>0$
because $\frac{dM(r)}{dr}>0$, $R^{\prime}dr+\dot{R}dt=0$, $R^{\prime}>0$, and
$\dot{R}<0$ for collapsing region, so $\frac{dr}{dt}|_{R=const}>0$. Therefore,
it is clear that concepts such as isolated horizon and slowly evolving horizon
do not apply to this case.
### V.4 Example IV: $f>0$: asymptotically open FRW metric
Now we look for a solution which goes to an open FRW metric at distances far
from the center. At the same time one should take care of the conditions
$M(0)=0$ and $\frac{f(0)^{3/2}}{M(0)}\neq\infty$. Let us choose
$f(r)=-r(1-r),$
and
$M(r)=\frac{1}{a}r^{3/2}(1+r^{3/2}),$
where $a$ is a constant, which may be fixed by assuming $r=0$ at the present
time $t_{0}$ corresponding to $\eta=\frac{3\pi}{2}$. This leads to
$at_{0}=3\pi/2+1$ . We then obtain from (IV.1)
$\displaystyle R=\frac{\sqrt{r}(1+r^{3/2})}{a(1-r)}(1-\cos\eta(r,t)),$
$\displaystyle\hskip 22.76228pt\eta-
sin(\eta)=\frac{(1-r)^{3/2}}{(1+r^{3/2})}at,$ (15)
for $r<1$, and
$\displaystyle R=\frac{\sqrt{r}(1+r^{3/2})}{a(r-1)}(cosh\eta(r,t)-1)$
$\displaystyle\hskip
22.76228ptsinh\eta-\eta=\frac{(r-1)^{3/2}}{(1+r^{3/2})}at,$ (16)
for $r>1$. The solution is again continuous at $r=1$, as can be checked by
evaluating $\dot{R}$, $R^{\prime}$, $\dot{R}^{\prime}$, $\ddot{R}$, and
$\ddot{R}^{\prime}$ at $r=1$(see the appendix).
The resulting density profile and the causal structure is plotted in Fig.6.
Obviously a singularity at the origin forms gradually while the universe is
expanding. The causal structure is also similar to the open but asymptotically
flat case.
Figure 6: The case of asymptotically open model: the density profile is
similar to the previous open but asymptotically flat case, except for the less
mass concentrated in the central region. The causal structure is also similar.
The locations of the event horizon, apparent horizon and the singularity are
also calculated numerically.The separation between the singularity and the
apparent horizon is not clear here due to the scale chosen.
This solution represents a collapsing mass within an open FRW universe. The
collapsed region is again dynamical in the sense that its mass is not
constant, and the rate of change of the Misner-Sharp energy is given by the
same amount as the previous model. Therefore, concepts of isolated horizon and
slowly evolving horizon do not apply to this case.
## VI Characteristics of singularities of proposed models
We have avoided in the models proposed the shell crossing singularities except
example III with a late time shell crossing singularity.
The shell focusing singularities, however, are unavoidable and in fact it is
what we are looking for to study characteristics of cosmological black holes.
An important aspect of such a singularity is its gravitational strength
strength , which is an important differentiating feature of black hole.
### VI.1 Strength of the shell focusing singularities
Heuristically, a singularity is termed gravitationally strong, or simply
strong, if it destroys by crushing or stretching any object which falls into
it. The prototype of such a singularity is the Schwarzschild one: a radially
infalling object is infinitely stretched in the radial direction and crushed
in the tangential directions, with the net result of crushing to zero volume.
Otherwise a singularity is termed weak where no object falling into it is
destroyed. To check the strength of singularities of our models we use the
criteria defined by Clarke strength .
Let $k^{\mu}$ be the tangent vector to the ingoing null geodesic, and
$\lambda$ the corresponding affine parameter being zero at the center.
$R_{\mu\nu}$ being the Ricci tensor, the singularity is said to be strong if
$\Psi=lim_{r\rightarrow 0}\lambda^{2}k^{\mu}k^{\nu}R_{\mu\nu}\neq 0.$ (17)
For a general LTB metric one obtains easily
$k^{\mu}k^{\nu}R_{\mu\nu}=2(k^{t})^{2}\frac{M^{\prime}}{R^{2}R^{\prime}}$. For
the three interesting cases of cosmological black holes in flat and open LTB
models we have done the calculation along the lines of the joshi using
appropriate coordinates near singularity. For our cases (V.2-V.3-V.4) we
obtain after some calculation $\Psi=0$ for $r\rightarrow 0$. Therefore, shell
focusing singularities occurring in the center of the models we are proposing
are week. This is in contrast to the Schwarzschild singularity which is a
strong one. We leave it to future studies if this weakness is generic of any
cosmological black holes.
### VI.2 Nakedness of singularities
We know already from Oppenheimer-Snyder collapse of a homogeneous dust
distribution how the shells become singular at the same time, and thus none of
them crosses. In the case of spherically symmetric inhomogeneous matter
configurations, however, the proper time of collapse depends on the comoving
radius $r$. Thus the piling up of neighboring matter shells at finite proper
radius can occur, thereby producing two-dimensional caustics where the energy
density and some curvature components diverge. These singularities can be
locally naked, but they are gravitationally weak cencorship ; nolan-sc-w ,
i.e. curvature invariants and tidal forces remain finite. It has also been
shown that analytic continuations of the metric, in a distributional sense,
can always be found in the neighborhood of the singularity shell-c .
Models proposed in this paper are, however, free from shell crossing
singularities. The shell crossing singularity of example III at late times
does not influence the following argumentation. Conditions for the absence of
shell crossing singularities have been studied in detail in hell-shell . In
our case these conditions are equivalent to $M^{\prime}(r)>0$ and
$R^{\prime}>0$, which are satisfied by the models discussed above. We may then
conclude that
$\frac{\frac{dt}{dr}|_{AH}}{\frac{dt}{dr}|_{null}}=\left(1-\frac{2M^{\prime}}{R^{\prime}}\right)<1.$
(18)
Therefore, the condition for the apparent horizon $R=2M$ to be spacelike is,
i.e. $-1<\frac{\frac{dt}{dr}|_{AH}}{\frac{dt}{dr}|_{null}}<1$, leads to the
condition $R^{\prime}-M^{\prime}>0$, which is not everywhere satisfied in our
model. As a result we notice that apparent horizons of the models proposed
here are not spacelike everywhere. Such a behavior has already been discussed
inbooth-mtt .
The case of shell focusing singularities is, however, a different one.
Irrespective of the behavior of the apparent and event horizons, it is then a
relevant question if the shell focusing singularity could be a naked one. We
notice that the slope of the outgoing null geodesics at the singularity are
greater than the slope of the singularity itself. Therefore, the singularity
is spacelike and no timelike or null geodesic can come out of the singularity.
We then conclude that the singularities we are facing can not be naked.
## VII Discussion and conclusions
Unlike models discussed so far in the literature, we have constructed models
of mass condensation within the FRW universe leading to cosmological black
holes without having the usual pathologies we know from other models: the
cosmic fluid is dust and ideal producing a singularity at the center in the
course of time. The central singularity is spacelike and not naked. In the
case of flat or open universe models the singularity is weak and has distinct
apparent and event horizons. The apparent horizons are not everywhere
spacelike, to be compared with the Schwarzschild one which is null everywhere.
This has already been noticed in a general context by booth-mtt . While the
apparent horizon is defined by the surfaces $R=2M$, similar to the
Schwarzschild horizon, the even horizon is further away. Models we have
proposed show that one has to expect new effects while considering dynamical
cosmic black holes. The simple Schwarzschild static model may not reflect all
the phenomena one may expect in observational cosmology, and the black hole
thermodynamics. Even the simple concept of mass is not a trivial one in such a
dynamical environment. the answer to these questions are beyond the scope of
these paper and will be deal with in future publications.
## Appendix A
The curvature function $f(r)$ has a zero point where it changes sign for
models III and IV, corresponding to two different solutions. Therefore, we
have to take care of joining two solutions across the hypersurface defined by
$f(r)=0$ to be continuous. This is done by looking at the metric functions and
their derivatives to be continuous.
Let us first look at the model IV. There we have to look at the metric
function $R$ and its derivatives, $R$, $R^{\prime}$, $\dot{R}$, $\ddot{R}$ and
$\ddot{R}^{\prime}$, at the point $r=1$ where $f$ vanishes. From the following
relations derived from the Einstein equations (5)
$\ddot{R}=-\frac{M}{R^{2}},$ (19)
$\dot{R}^{\prime}=\frac{M^{\prime}}{R\dot{R}}-\frac{MR^{\prime}}{\dot{R}R^{2}}+\frac{f^{\prime}}{2\dot{R}},$
(20)
and
$\ddot{R}^{\prime}=-\frac{M^{\prime}}{R^{2}}+\frac{2MR^{\prime}}{R^{3}},$ (21)
we infer that these second derivatives relevant for the Einstein equations to
be continuous on the hypersurface $f(r)=0$ are continuous if the $f$, $R$,
$R^{\prime}$, $\dot{R}$, $M^{\prime}$, and $M$ are continuous. Now, because of
the continuity of $f$, $M^{\prime}$, and $M$, we just have to prove the
continuity of $R$, $\dot{R}$, and $R^{\prime}$.
Let us look first at $R$ and its derivative $R^{\prime}$. In the case of $r<1$
we have
$\displaystyle R=\frac{a(r)}{1-r}(1-cos\eta),$ $\displaystyle\eta-
sin\eta=\frac{(1-r)^{1.5}}{b(r)}t,$ (22)
where $a(r)=\sqrt{r}+r^{2}$, $b(r)=1+r^{1.5}$, and $a(1)=2$,
$a^{\prime}(1)=2.5$, $b(1)=2$, $b^{\prime}(1)=1.5$, and
$\dot{R}=\frac{a\sqrt{1-r}}{b}\frac{sin\eta}{1-cos\eta}.$ (23) $\displaystyle
R^{\prime}=\frac{a^{\prime}(1-r)+a}{(1-r)^{2}}(1-cos\eta)-\frac{a}{1-r}$
$\displaystyle\frac{sin\eta}{1-cos\eta}\frac{1.5(1-r)^{0.5}b+b^{\prime}(1-r)^{1.5}}{b^{2}}t.$
(24)
Defining $1-r=x$, we have
$\eta-sin\eta=\frac{\eta^{3}}{6}-O(\eta^{5})=\frac{x^{3}}{2}t.$ (25)
Therefore, to first order in $\eta$ we have $\eta=\sqrt[3]{3t}\sqrt{x}$. Now
taking the limit $x\rightarrow 0^{-}$ we obtain
$\displaystyle\lim_{x\rightarrow 0^{-}}R(x)=\lim_{x\rightarrow
0^{-}}\frac{2}{x}(1-cos\eta)=\lim_{x\rightarrow
0^{-}}(\frac{\eta^{2}}{x}-O(\eta^{4})/x)$
$\displaystyle=(3t)^{2/3}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(26)
which is a well defined quantity.
In the case of $r>1$ we have
$\displaystyle R=\frac{a(r)}{r-1}(cosh\eta-1),$ $\displaystyle
sinh\eta-\eta=\frac{(r-1)^{1.5}}{b(r)}t,$ (27)
$\dot{R}=\frac{a\sqrt{r-1}}{b}\frac{sinh\eta}{cosh\eta-1},$ (28)
and
$\displaystyle
R^{\prime}=\frac{a^{\prime}(r-1)-a}{(r-1)^{2}}(cosh\eta-1)+\frac{a}{r-1}$
$\displaystyle\frac{sinh\eta}{cosh\eta-1}\frac{1.5(r-1)^{0.5}b-b^{\prime}(r-1)^{1.5}}{b^{2}}t.$
(29)
Now, defining $r-1=x$, and noting that
$sinh\eta-\eta=\frac{\eta^{3}}{6}+O(\eta^{5})=\frac{x^{3}}{2}t,$ (30)
we obtain to first order of $\eta$ the relation $\eta=\sqrt[3]{3t}\sqrt{x}$.
Therefore,
$\displaystyle\lim_{x\rightarrow 0^{+}}R(x)=\lim_{x\rightarrow
0^{+}}\frac{2}{x}(cosh\eta-1)=\lim_{x\rightarrow
0^{+}}\frac{\eta^{2}}{x}+O(\eta^{4})/x$
$\displaystyle=(3t)^{2/3}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}.$
(31)
Therefore the continuity of $R$ across $r=1$ is established.
Similar calculation for the first derivatives shows the continuity of
$R^{\prime}$ and $\dot{R}$ having well defined values on both sides of the
$r=r_{0}$ hypersurface:
$R^{\prime}(1)=2.5(3t)^{2/3}-\frac{3}{4(3t)^{1/3}}t,$ (32)
and
$\dot{R}(1)=\frac{2}{(3t)^{1/3}}.$ (33)
The case of model III is similar except for the hypersurface defined by
$g(r)=\frac{f(r)}{r}=e^{-r}-\frac{1}{r^{2}+20000}=0$ with the root of
$e^{-r_{0}}-\frac{1}{r_{0}^{2}+20000}=0$ being at a point $r=r_{0}$ different
from $r=1$. It is easy to see that $g(r)$ is an analytic function at
$r=r_{0}$, and can be approximated by $g(r)\approx
g^{\prime}(r_{0})(r-r_{0})+\frac{g^{\prime\prime}(r_{0})}{2}(r-r_{0})^{2}+...$.
Similar calculations verify the continuity of the metric function $R$ and its
relevant derivatives across the hypersurface $r=r_{0}$.
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|
arxiv-papers
| 2008-12-30T19:58:20 |
2024-09-04T02:48:59.646955
|
{
"license": "Public Domain",
"authors": "J.T. Firouzjaee, Reza Mansouri",
"submitter": "Javad Taghizadeh firouzjaee",
"url": "https://arxiv.org/abs/0812.5108"
}
|
0901.0015
|
# Maximum Entropy on Compact Groups
Peter Harremoës Centrum Wiskunde & Informatica, Science Park 123, 1098 GB
Amsterdam, Noord-Holland, The Netherlands
E-mail: P.Harremoes@cwi.nl
###### Abstract
On a compact group the Haar probability measure plays the role of uniform
distribution. The entropy and rate distortion theory for this uniform
distribution is studied. New results and simplified proofs on convergence of
convolutions on compact groups are presented and they can be formulated as
entropy increases to its maximum. Information theoretic techniques and Markov
chains play a crucial role. The convergence results are also formulated via
rate distortion functions. The rate of convergence is shown to be exponential.
###### keywords:
Compact group; Convolution; Haar measure; Information divergence; Maximum
entropy; Rate distortion function; Rate of convergence; Symmetry.
94A34,60B15
## 1 Introduction
It is a well-known and celebrated result that the uniform distribution on a
finite set can be characterized as having maximal entropy. Jaynes used this
idea as a foundation of statistical mechanics [1], and the Maximum Entropy
Principle has become a popular principle for statistical inference [2, 3, 4,
5, 6, 7, 8]. Often it is used as a method to get prior distributions. On a
finite set, for any distributions $P$ we have $H(P)=H(U)-D(P\|U)$ where $H$ is
the Shannon entropy, $D$ is information divergence, and $U$ is the uniform
distribution. Thus, maximizing $H(P)$ is equivalent to minimizing $D(P\|U)$.
Minimization of information divergence can be justified by the conditional
limit theorem by Csiszár [9, Theorem 4]. So if we have a good reason to use
the uniform distribution as prior distribution we automatically get a
justification of the Maximum Entropy Principle. The conditional limit theorem
cannot justify the use of the uniform distribution itself, so we need
something else. Here we shall focus on symmetry.
###### Example 1.
A die has six sides that can be permuted via rotations of the die. We note
that not all permutations can be realized as rotations and not all rotations
will give permutations. Let $G$ be the group of permutations that can be
realized as rotations. We shall consider $G$ as the symmetry group of the die
and observe that the uniform distribution on the six sides is the only
distribution that is invariant under the action of the symmetry group $G.$
###### Example 2.
$G=\mathbb{R}/2\pi\mathbb{Z}$ is a commutative group that can be identified
with the group $SO\left(2\right)$ of rotations in 2 dimensions. This is the
simplest example of a group that is compact but not finite.
For an object with symmetries the symmetry group defines a group action on the
object, and any group action on an object defines a symmetry group of the
object. A special case of a group action of the group $G$ is left translation
of the elements in $G$. Instead of studying distributions on objects with
symmetries, in this paper we shall focus on distributions on the symmetry
groups themselves. It is no serious restriction because a distribution on the
symmetry group of an object will induce a distribution on the object itself.
Convergence of convolutions of probability measures were studied by Stromberg
[10] who proved weak convergence of convolutions of probability measures. An
information theoretic approach was introduced by Csiszár [11]. Classical
methods involving characteristic functions have been used to give conditions
for uniform convergence of the densities of convolutions [12]. See [13] for a
review of the subject and further references.
Finally it is shown that convergence in information divergence corresponds to
uniform convergence of the rate distortion function and that weak convergence
corresponds to pointwise convergence of the rate distortion function. In this
paper we shall mainly consider convolutions as Markov chains. This will give
us a tool, which allows us to prove convergence of iid. convolutions, and the
rate of convergence is proved to be exponential.
The rest of the paper is organized as follows. In Section 2 we establish a
number of simple results on distortion functions on compact set. These results
will be used in Section 4. In Section 3 we define the uniform distribution on
a compact group as the uniquely determined Haar probability measures. In
Section 4 it is shown that the uniform distribution is the maximum entropy
distribution on a compact group in the sense that it maximizes the rate
distortion function at any positive distortion level. Convergence of
convolutions of a distribution to the uniform distribution is established in
Section 5 using Markov chain techniques, and the rate of convergence is
discussed in Section 6. The group $SO\left(2\right)$ is used as our running
example. We finish with a short discussion.
## 2 Distortion on compact groups
Let $G$ be a compact group where $\ast$ denotes the composition. The neutral
element will be denoted $e$ and the inverse of the element $g$ will be denoted
$g^{-1}$.
We shall start with some general comments on distortion functions on compact
sets. Assume that the group both plays the role as source alphabet and
reproduction alphabet. A _distortion function_ $d:G\times
G\rightarrow\mathbb{R}$ is given and we will assume that
$d\left(x,y\right)\geq 0$ with equality if and only if $x=y.$ We will also
assume that the distortion function is continuous.
###### Example 3.
As distortion function on $SO\left(2\right)$ we use the squared Euclidean
distance between the corresponding points on the unit circle, i.e.
$\displaystyle d\left(x,y\right)$ $\displaystyle=$ $\displaystyle
4\sin^{2}\left(\frac{x-y}{2}\right)$ $\displaystyle=$ $\displaystyle
2-2\cos\left(x-y\right).$
This illustrated in Figure 1.
Figure 1: Squared Euclidean distance between the rotation angles $x$ and $y.$
The distortion function might be a metric but even if the distortion function
is not a metric, the relation between the distortion function and the topology
is the same as if it was a metric. One way of constructing a distortion
function on a group is to use the squared Hilbert-Smidt norm in a unitary
representation of the group.
###### Theorem 4.
If $C$ is a compact set and $d:C\times C\rightarrow\mathbb{R}$ is a non-
negative continuous distortion function such that $d\left(x,y\right)=0$ if and
only if $x=y,$ then the topology on $C$ is generated by the distortion balls
$\left\\{{x\in C\mid d\left(x,y\right)<r}\right\\}$ where $y\in C$ and $r>0.$
###### Proof.
We have to prove that a subset $B\subseteq C$ is open if and only if for any
$y\in B$ there exists a ball that is a subset of $B$ and contains $y$. Assume
that $B\subset C$ is open and that $y\in B.$ Then $\complement B$ compact.
Hence, the function $x\rightarrow d\left(x,y\right)$ has a minimum $r$ on
$\complement B$ and $r$ must be positive because $r=d\left(x,y\right)=0$ would
imply that $x=y\in B.$ Therefore $\left\\{{x\in C\mid
d\left(x,y\right)<r}\right\\}\subseteq B.$
Continuity of $d$ implies that the balls $\left\\{{x\in C\mid
d\left(x,y\right)<r}\right\\}$ are open. If any point in $B$ is contained in
an open ball, then $B$ is a union of open set and open. ∎
The following theorem may be considered as a kind of uniform continuity of the
distortion function or as a substitute for the triangular inequality when $d$
is not a metric.
###### Lemma 5.
If $C$ is a compact set and $d:C\times C\rightarrow\mathbb{R}$ is a non-
negative continuous distortion function such that $d\left(x,y\right)=0$ if and
only if $x=y$, then there exists a continuous function $f_{1}$ satisfying
$f_{1}\left(0\right)=0$ such that
$\left|d\left(x,y\right)-d\left(z,y\right)\right|\leq
f_{1}\left(d\left(z,y\right)\right)\text{ for }x,y,z\in C.$ (1)
###### Proof.
Assume that the theorem does not hold. Then there exists $\epsilon>0$ and a
net $\left(x_{\lambda},y_{\lambda},z_{\lambda}\right)_{\lambda\in\Lambda}$
such that
$d\left(x_{\lambda},y_{\lambda}\right)-d\left(z_{\lambda},y_{\lambda}\right)>\epsilon$
and $d\left(z_{\lambda},y_{\lambda}\right)\rightarrow 0.$ A net in a compact
set has a convergent subnet so without loss of generality we may assume that
the net $\left(x_{\lambda},y_{\lambda},z_{\lambda}\right)_{\lambda\in\Lambda}$
converges to some triple $\left(x_{\infty},y_{\infty},z_{\infty}\right).$ By
continuity of the distortion function we get
$d\left(x_{\infty},y_{\infty}\right)-d\left(z_{\infty},y_{\infty}\right)\geq\epsilon$
and $d\left(z_{\infty},y_{\infty}\right)=0,$ which implies
$z_{\infty}=y_{\infty}$ and we have a contradiction. ∎
We note that if a distortion function satisfies (1) then it defines a topology
in which the distortion balls are open.
In order to define the weak topology on probability distributions we extend
the distortion function from $C\times C$ to $M_{+}^{1}\left(C\right)\times
M_{+}^{1}\left(C\right)$ via
$d\left(P,Q\right)=\inf E\left[\ d\left(X,Y\right)\right],$
where $X$ and $Y$ are random variables with values in $C$ and the infimum is
taken all joint distributions on $\left(X,Y\right)$ such that the marginal
distribution of $X$ is $P$ and the marginal distribution of $Y$ is $Q.$ The
distortion function is continuous so $\left(x,y\right)\rightarrow
d\left(x,y\right)$ has a maximum that we denote $d_{\max}.$
###### Theorem 6.
If $G$ is a compact set and $d:C\times C\rightarrow\mathbb{R}$ is a non-
negative continuous distortion function such that $d\left(x,y\right)=0$ if and
only if $x=y$, then
$\left|d\left(P,Q\right)-d\left(S,Q\right)\right|\leq
f_{2}\left(d\left(S,P\right)\right)\text{ for }P,Q,S\in
M_{+}^{1}\left(C\right)$
for some continuous function $f_{2}$ satisfying $f_{2}\left(0\right)=0.$
###### Proof.
According to Lemma 5 there exists a function $f_{1}$ satisfying (1). We use
that
$\displaystyle E\left[\left|d\left(X,Y\right)-d\left(Z,Y\right)\right|\right]$
$\displaystyle\leq E\left[f_{1}\left(d\left(Z,X\right)\right)\right]$
$\displaystyle=E\left[f_{1}\left(d\left(Z,X\right)\right)\mid
d\left(Z,X\right)\leq\delta\right]\cdot
P\left(d\left(Z,X\right)\leq\delta\right)$
$\displaystyle+E\left[f_{1}\left(d\left(Z,X\right)\right)\mid
d\left(Z,X\right)>\delta\right]\cdot P\left(d\left(Z,X\right)>\delta\right)$
$\displaystyle\leq f_{1}\left(\delta\right)\cdot
1+f_{1}\left(d_{\max}\right)\cdot\frac{E\left[d\left(Z,X\right)\right]}{\delta}$
$\displaystyle\leq
f_{1}\left(\delta\right)+f_{1}\left(d_{\max}\right)\cdot\frac{d\left(S,P\right)}{\delta}.$
This hold for all $\delta>0$ and in particular for
$\delta=\left(d\left(S,P\right)\right)^{1/2}$, which proves the theorem. ∎
The theorem can be used to construct the _weak topology_ on
$M_{+}^{1}\left(C\right)$ with
$\left\\{P\in M_{+}^{1}\left(C\right)\mid d\left(P,Q\right)<r\right\\},$
$P\in M_{+}^{1}\left(C\right),r>0$ as open balls that generate the topology.
We note without proof that this definition is equivalent with the quite
different definition of weak topology that one will find in most textbooks.
For a group $G$ we assume that the distortion function is _right invariant_ in
the sense that for all $x,y,z\in G$ a distortion function $d$ satisfies
$d\left(x\ast z,y\ast z\right)=d\left(x,y\right).$
A right invariant distortion function satisfies
$d\left(x,y\right)=d\left(x\ast y^{-1},e\right)$, so right invariant
continuous distortion functions of a group can be constructed from non-
negative functions with a minimum in $e$.
## 3 The Haar measure
We use $\ast$ to denote convolution of probability measures on $G.$ For $g\in
G$ we shall use $g\ast P$ to denote the $g$-translation of the measure $P$ or,
equivalently, the convolution with a measure concentrated in $g$. The $n$-fold
convolution of a distribution $P$ with itself will be denoted $P^{\ast n}.$
For random variables with values in $G$ one can formulate an analog of the
central limit theorem. We recall some facts about probability measures on
compact groups and their _Haar measures_.
###### Definition 7.
Let $G$ be a group. A measure $P$ is said to be a _left Haar measure_ if
$g\ast P=P$ for any $g\in G$. Similarly, $P$ is said to be a _right Haar
measure_ if $P\ast g=P$ for any $g\in G.$ A measure is said to be a _Haar
measure_ if it is both a left Haar measure and a right Haar measure.
###### Example 8.
The uniform distribution on $SO\left(2\right)$ or $\mathbb{R}/2\pi Z$ has
density $1/2\pi$ with respect to the Lebesgue measure on
$\left[0;2\pi\right[.$ The function
$f\left(x\right)=1+\sum_{n=1}^{\infty}a_{n}\cos\left(n\left(x+\phi_{n}\right)\right)$
(2)
is a density on a probability distribution $P$ on $SO\left(2\right)$ if the
Fourier coefficients $a_{n}$ are sufficiently small so that $f$ is non-
negative. A sufficient condition for $f$ to be non-negative is that
$\sum_{n=1}^{\infty}\left|a_{n}\right|\leq 1.$
Translation by $y$ gives a distribution with density
$f\left(x-y\right)=1+\sum_{n=1}^{\infty}a_{n}\cos\left(n\left(x-y+\phi_{n}\right)\right).$
The distribution $P$ is invariant if and only if $f$ is $1$ or, equivalently,
all Fourier coefficients $\left(a_{n}\right)_{n\in\mathbb{N}}$ are $0.$
A measure $P$ on $G$ is said to have _full support_ if the support of $P$ is
$G,$ i.e. $P\left(A\right)>0$ for any non-empty open set $A\subseteq G.$ The
following theorem is well-known [14, 15, 16].
###### Theorem 9.
Let $U$ be a probability measure on the compact group $G.$ Then the following
four conditions are equivalent.
* •
$U$ is a left Haar measure.
* •
$U$ is a right Haar measure.
* •
$U$ has full support and is idempotent in the sense that $U\ast U=U.$
* •
There exists a probability measure $P$ on $G$ with full support such that
$P\ast U=U.$
* •
There exists a probability measure $P$ on $G$ with full support such that
$U\ast P=U.$
In particular a Haar probability measure is unique.
In [14, 15, 16] one can find the proof that any locally compact group has a
Haar measure. The unique Haar probability measure on a compact group will be
called the _uniform distribution_ and denoted $U.$
For probability measures $P$ and $Q$ the _information divergence from_ $P$
_to_ $Q$ is defined by
$D\left(P\|Q\right)=\left\\{\begin{array}[]{cc}\int\log\frac{dP}{dQ}~{}dP,&\text{if
}P\ll Q;\\\ \infty,&\text{otherwise.}\end{array}\right.$
We shall often calculate the divergence from a distribution to the uniform
distribution $U,$ and introduce the notation
$D\left(P\right)=D\left(P\|U\right).$
For a random variable $X$ with values in $G$ we will sometimes write
$D\left(X\|U\right)$ instead of $D\left(P\|U\right)$ when $X$ has distribution
$P.$
###### Example 10.
The distribution $P$ with density $f$ given by (2) has
$\displaystyle D\left(P\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}f\left(x\right)\log\left(f\left(x\right)\right)~{}dx$
$\displaystyle\approx$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}f\left(x\right)\left(f\left(x\right)-1\right)~{}dx$
$\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{n=1}^{\infty}a_{n}^{2}.$
Let $G$ be a compact group with uniform distribution $U$ and let $F$ be a
closed subgroup of $G.$ Then the subgroup has a Haar probability measure
$U_{F}$ and
$D\left(U_{F}\right)=\log\left(\left[G:F\right]\right)$ (3)
where $\left[G:F\right]$ denotes the index of $F$ in $G.$ In particular
$D\left(U_{F}\right)$ is finite if and only if $\left[G:F\right]$ is finite.
## 4 The rate distortion theory
We will develop aspects of the rate distortion theory of a compact group $G.$
Let $P$ be a probability measure on $G.$ We observe that compactness of $G$
implies that a covering of $G$ by distortion balls of radius $\delta>0$
contains a finite covering. If $k$ is the number of balls in a finite covering
then $R_{P}\left(\delta\right)\leq\log\left(k\right)$ where $R_{P}$ is the
rate distortion function of the probability measure $P.$ In particular the
rate distortion function is upper bounded. The entropy of a probability
distribution $P$ is given by $H\left(P\right)=R_{P}\left(0\right)$. If the
group is finite then the uniform distribution maximizes the Shannon entropy
$R_{P}\left(0\right)$ but if the group is not finite then in principle there
is no entropy maximizer. As we shall see the uniform distribution still plays
the role of entropy maximizer in the sense that the uniform distribution
maximize the value $R_{P}\left(\delta\right)$ of the rate distortion function
for any positive distortion level $\delta>0$. The rate distortion function
$R_{P}$ can be studied using its convex conjugate $R_{P}^{\ast}$ given by
$R_{P}^{\ast}\left(\beta\right)=\sup_{\delta}\beta\cdot\delta-
R_{P}\left(\delta\right).$
The rate distortion function is then recovered by the formula
$R_{P}\left(\delta\right)=\sup_{\beta}\beta\cdot\delta-
R_{P}^{\ast}\left(\beta\right).$
The techniques are pretty standard [17].
###### Theorem 11.
The rate distortion function of the uniform distribution is given by
$R_{U}^{\ast}\left(\beta\right)=\log\left(Z\left(\beta\right)\right)$
where $Z$ is the partition function defined by
$Z\left(\beta\right)=\int_{G}\exp\left(\beta\cdot
d\left(g,e\right)\right)~{}dUg.$
The rate distortion function of an arbitrary distribution $P$ satisfies
$R_{U}-D\left(P\|U\right)\leq R_{P}\leq R_{U}.$ (4)
###### Proof.
First we prove a Shannon type lower bound on the rate distortion function of
an arbitrary distribution $P$ on the group. Let $X$ be a random variable with
values in $G$ and distribution $P$, and let $\hat{X}$ be a random variable
coupled with $X$ such that the mean distortion
$E\left[d\left(X,\hat{X}\right)\right]$ equals $\delta$. Then
$\displaystyle I\left(X;\hat{X}\right)$
$\displaystyle=D\left(X\|U\mid\hat{X}\right)-D\left(X\|U\right)$ (5)
$\displaystyle=D\left(X\ast\hat{X}^{-1}\|U\mid\hat{X}\right)-D\left(X\|U\right)$
(6) $\displaystyle\geq D\left(X\ast\hat{X}^{-1}\|U\right)-D\left(X\|U\right).$
(7)
Now,
$E\left[d\left(X,\hat{X}\right)\right]=E\left[d\left(X\ast\hat{X}^{-1},e\right)\right]$
and
$D\left(X\ast\hat{X}^{-1}\|U\right)\geq D\left(P_{\beta}\|U\right)$
where $P_{\beta}$ is the distribution that maximizes divergence under the
constraint $E\left[d\left(Y,e\right)\right]=\delta$ when $Y$ has distribution
$P_{\beta}.$ The distribution $P_{\beta}$ is given by the density
$\frac{dP_{\beta}}{dU}\left(g\right)=\frac{\exp\left(\beta\cdot
d\left(g,e\right)\right)}{Z\left(\beta\right)}.$
where $\beta$ is determined by the condition
$\delta=Z^{\prime}\left(\beta\right)/Z\left(\beta\right).$
If $P$ is uniform then a joint distribution is obtained by choosing $\hat{X}$
uniformly distributed, and choosing $Y$ distributed according to $P_{\beta}$
and independent of $\hat{X}.$ Then $X=Y\ast\hat{X}$ is distributed according
to $P_{\beta}\ast U=U$, and we have equality in (7). Hence the rate determined
the lower bound (7) is achievable for the uniform distribution, which prove
the first part of the theorem, and the left inequality in (4).
The joint distribution on $\left(X,\hat{X}\right)$ that achieved the rate
distortion function when $X$ has a uniform distribution, defines a Markov
kernel $\Psi:X\rightarrow\hat{X}$ that is invariant under translations in the
group. For any distribution $P$ the joint distribution on
$\left(X,\hat{X}\right)$ determined by $P$ and $\Psi$ gives an achievable pair
of distortion, and rate that is on the rate distortion curve of the uniform
distribution. This proves the right inequality in Equation (4). ∎
###### Example 12.
For the group $SO\left(2\right)$ the rate distortion function can be
parametrized using the modified Bessel functions $I_{j},j\in\mathbb{N}_{0}$.
The partition function is given by
$\displaystyle Z\left(\beta\right)$
$\displaystyle=\int_{G}\exp\left(\beta\cdot d\left(g,e\right)\right)~{}dUg$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\exp\left(\beta\cdot\left(2-2\cos
x\right)\right)~{}dx$
$\displaystyle=\exp\left(2\beta\right)\cdot\frac{1}{\pi}\int_{0}^{\pi}\exp\left(-2\beta\cdot\cos
x\right)~{}dx$ $\displaystyle=\exp\left(2\beta\right)\cdot
I_{0}\left(-2\beta\right).$
Hence $R_{U}^{\ast}\left(\beta\right)=$
$\log\left(Z\left(\beta\right)\right)=2\beta+\log\left(I_{0}\left(-2\beta\right)\right)$.
The distortion $\delta$ corresponding to $\beta$ is given by
$\delta=2-2\frac{I_{1}\left(-2\beta\right)}{I_{0}\left(-2\beta\right)}$
and the corresponding rate is
$\displaystyle R_{U}\left(\delta\right)$ $\displaystyle=$
$\displaystyle\beta\cdot\delta-\left(2\beta+\log\left(I_{0}\left(-2\beta\right)\right)\right)$
$\displaystyle=$ $\displaystyle-\beta\cdot
2\frac{I_{1}\left(-2\beta\right)}{I_{0}\left(-2\beta\right)}-\log\left(I_{0}\left(-2\beta\right)\right).$
These joint values of distortion and rate can be plotted with $\beta$ as
parameter as illustrated in Figure 2.
Figure 2: The rate distortion region of the uniform distribution on
$SO\left(2\right)$ is shaded. The rate distortion function is the lower
bounding curve. In the figure the rate is measured in nats. The critical
distortion $d_{crit}$ equals 2, and the dashed line indicates $d_{\max}=4.$
The minimal rate of the uniform distribution is achieved when $X$ and
$\hat{X}$ are independent. In this case the distortion is
$E\left[d\left(X,\hat{X}\right)\right]=\int_{G}d\left(x,e\right)~{}dPx.$ This
distortion level will be called the critical distortion and will be denoted
$d_{crit}.$ On the interval $\left]0;d_{crit}\right]$ the rate distortion
function is decreasing and the distortion rate function is the inverse
$R_{P}^{-1}$ of the rate distortion function $R_{P}$ on this interval. The
distortion rate function satisfies:
###### Theorem 13.
The distortion rate function of an arbitrary distribution $P$ satisfies
$R_{U}^{-1}\left(\delta\right)-f_{2}\left(d\left(P,U\right)\right)\leq
R_{P}^{-1}\left(\delta\right)\leq R_{U}^{-1}\left(\delta\right)~{}\text{for
}\delta\leq d_{crit}$ (8)
for some increasing continuous function $f_{2}$ satisfying
$f_{2}\left(0\right)=0.$
###### Proof.
The right hand side follows because $R_{U}$ is decreasing in the interval
$\left[0;d_{crit}\right]$ Let $X$ be a random variable with distribution $P$
and let $Y$ be a random variable coupled with $X.$ Let $Z$ be a random
variable coupled with $X$ such that
$E\left[d\left(X,Z\right)\right]=d\left(P,U\right).$ The couplings between $X$
and $Y$, and between $X$ and $Z$ can be extended to a joint distribution on
$X,Y$ and $Z$ such that $Y$ and $Z$ are independent given $X.$ For this joint
distribution we have
$I\left(Z;Y\right)\leq I\left(X,Y\right)$
and
$\left|E\left[d\left(Z,Y\right)\right]-E\left[d\left(X,Y\right)\right]\right|\leq
f_{2}\left(d\left(P,U\right)\right).$
We have to prove that
$E\left[d\left(X,Y\right)\right]\geq
R_{U}^{-1}\left(I\left(X,Y\right)\right)-f_{2}\left(d\left(P,U\right)\right)$
but $I\left(Z;Y\right)\leq I\left(X,Y\right)$ so it is sufficient to prove
that
$E\left[d\left(X,Y\right)\right]\geq
R_{U}^{-1}\left(I\left(Z,Y\right)\right)-f_{2}\left(d\left(P,U\right)\right)$
and this follows because $E\left[d\left(Z,Y\right)\right]\geq
R_{U}^{-1}\left(I\left(Z,Y\right)\right).$ ∎
## 5 Convergence of convolutions
We shall prove that under certain conditions the $n$-fold convolutions
$P^{\ast n}$ converge to the uniform distribution.
###### Example 14.
The function
$f\left(x\right)=1+\sum_{n=1}^{\infty}a_{n}\cos\left(n\left(x+\phi_{n}\right)\right)$
is a density on a probability distribution $P$ on $G$ if the Fourier
coefficients $a_{n}$ are sufficiently small. If $\left(a_{n}\right)$ and
$\left(b_{n}\right)$ are Fourier coefficients of $P$ and $Q$ then the
convolution has density
$\frac{1}{2\pi}\int_{0}^{2\pi}\left(1+\sum_{n=1}^{\infty}a_{n}\cos
n\left(x-y+\phi_{n}\right)\right)\left(1+\sum_{n=1}^{\infty}b_{n}\cos
n\left(y+\psi_{n}\right)\right)~{}dy\\\
=1+\frac{1}{2\pi}\sum_{n=1}^{\infty}\int_{0}^{2\pi}a_{n}b_{n}\cos
n\left(x-y+\phi_{n}\right)\cos n\left(y+\psi_{n}\right)~{}dy\\\
=1+\frac{1}{2\pi}\sum_{n=1}^{\infty}\int_{0}^{2\pi}a_{n}b_{n}\cos\left(n\left(x+\phi_{n}+\psi_{n}\right)-ny\right)\cos\left(ny\right)~{}dy\\\
=1+\frac{1}{2\pi}\sum_{n=1}^{\infty}\int_{0}^{2\pi}a_{n}b_{n}\left(\begin{array}[]{c}\cos
n\left(x+\phi_{n}+\psi_{n}\right)\cos\left(ny\right)\\\
+\sin\left(n\left(x+\phi_{n}+\psi_{n}\right)\right)\sin\left(ny\right)\end{array}\right)\cos\left(ny\right)~{}dy\\\
=1+\sum_{n=1}^{\infty}\frac{a_{n}b_{n}\cos\left(n\left(x+\phi_{n}+\psi_{n}\right)\right)}{2\pi}\int_{0}^{2\pi}\cos^{2}\left(ny\right)~{}dy\\\
=1+\sum_{n=1}^{\infty}\frac{a_{n}b_{n}\cos\left(n\left(x+\phi_{n}+\psi_{n}\right)\right)}{2}.$
Therefore the $n$-fold convolution has density
$1+\sum_{k=1}^{\infty}\frac{a_{k}^{n}\cos\left(k\left(x+n\phi_{k}\right)\right)}{2^{n-1}}=1+\sum_{k=1}^{\infty}\left(\frac{a_{k}}{2}\right)^{n}2\cos\left(k\left(x+n\phi_{k}\right)\right).$
Therefore each of the Fourier coefficients is exponentially decreasing.
Clearly, if $P$ is uniform on a proper subgroup then convergence does not
hold. In several papers on this topic [18, 13, and references therein] it is
claimed and “proved” that if convergence does not hold then the support of $P$
is contained in the coset of a proper normal subgroup. The proofs therefore
contain errors that seem to have been copied from paper to paper. To avoid
this problem and make this paper more self-contained we shall reformulate and
reprove some already known theorems.
In the theory of finite Markov chains is well-known that there exists an
invariant probability measure. Certain Markov chains exhibits periodic
behavior where a certain distribution is repeated after a number of
transitions. All distributions in such a cycle will lie at a fixed distance
from any (fixed) measure, where the distance is given by information
divergence or total variation (or any other Csiszár $f$-divergence). It is
also well-known that finite Markov chains without periodic behavior are
convergent. In general a Markov chain will converge to a “cyclic” behavior as
stated in the following theorem [19].
###### Theorem 15.
Let $\Phi$ be a transition operator on a state space $A$ with an invariant
probability measure $Q_{in}.$ If $D\left(S\parallel Q\right)<\infty$ then
there exists a probability measure $P^{\ast}$ such that
$D\left(\Phi^{n}S\parallel\Phi^{n}Q\right)\rightarrow 0$ and
$D\left(\Phi^{n}Q\parallel Q_{in}\right)$ is constant.
We shall also use the following proposition that has a purely computational
proof [20].
###### Proposition 16.
Let $P_{x},x\in X$ be distributions and let $Q$ be a probability distribution
on $X.$ Then
$\int D\left(P_{x}\parallel Q\right)~{}dQx=D\left(\int P_{x}dQx\parallel
Q\right)+\int D\left(P_{x}\parallel\int P_{t}~{}dQt\right)~{}dQx.$
We denote the set of probability measures on $G$ by $M_{+}^{1}\left(G\right)$.
###### Theorem 17.
Let $P$ be a distribution on a compact group $G$ and assume that the support
of $P$ is not contained in any nontrivial coset of a subgroup of $G.$ Then, if
$D\left(S\|U\right)$ is finite then $D\left(P^{\ast n}\ast
S\|U\right)\rightarrow 0$ for $n\rightarrow\infty.$
###### Proof.
Let $\Psi:G\rightarrow M_{+}^{1}\left(G\right)$ denote the Markov kernel
$\Psi\left(g\right)=P\ast g.$ Then $P^{\ast n}\ast S=\Psi^{n}\left(P\ast
S\right).$ Thus there exists a probability measure $Q$ on $G$ such that
$D\left(\Psi^{n}\left(P\right)\|\Psi^{n}\left(Q\right)\right)\rightarrow 0$
for $n\rightarrow\infty$ and such that $D\left(\Psi^{n}\left(Q\right)\right)$
is constant. We shall prove that $Q=U.$
First we note that
$\displaystyle D\left(Q\right)$ $\displaystyle=D\left(P\ast Q\right)$
$\displaystyle=\int_{G}\left(D\left(g\ast Q\right)-D\left(g\ast Q\|P\ast
Q\right)\right)~{}dPg$ $\displaystyle=D\left(Q\right)-\int_{G}D\left(g\ast
Q\|P\ast Q\right)~{}dPg\ .$
Therefore $g\ast Q=P\ast Q$ for $P$ almost every $g\in G.$ Thus there exists
at least one $g_{0}\in G$ such that $g_{0}\ast Q=P\ast Q.$ Then
$Q=\tilde{P}\ast Q$ where $\tilde{P}=g_{0}^{-1}\ast P.$
Let $\tilde{\Psi}:G\rightarrow M_{+}^{1}\left(G\right)$ denote the Markov
kernel $g\rightarrow\tilde{P}\ast g.$ Put
$P_{n}=\frac{1}{n}\sum_{i=1}^{n}\tilde{P}^{\ast
i}=\frac{1}{n}\sum_{i=1}^{n}\tilde{\Psi}^{i-1}\left(\tilde{P}\right).$
According to [19] this ergodic mean will converge to a distribution $T$ such
that $\tilde{\Psi}\left(T\right)=T$ so that $T\ast\tilde{P}=T.$ Hence we also
have that $T\ast T=T,$ i.e. $T$ is idempotent and therefore supported by a
subgroup of $G$. We know that $\tilde{P}$ is not contained in any nontrivial
subgroup of $G$ so the support of $T$ must be $G$. We also get $Q=T\ast Q,$
which together with Theorem 9 implies that $Q=U.$ ∎
by choosing $S=P$ we get the following corollary.
###### Corollary 18.
Let $P$ be a probability measure on the compact group $G$ with Haar
probability measure $U$. Assume that the support of $P$ is not contained in
any coset of a proper subgroup of $G$ and $D\left(P\|U\right)$ is finite. Then
$D\left(P^{\ast n}\|U\right)\rightarrow 0$ for $n\rightarrow\infty$.
Corollary 18 together with Theorem 11 implies the following result.
###### Corollary 19.
Let $P$ be a probability measure on the compact group $G$ with Haar
probability measure $U$. Assume that the support of $P$ is not contained in
any coset of a proper subgroup of $G$ and $D\left(P\|U\right)$ is finite. Then
the rate distortion function of $P^{\ast n}$ converges uniformly to the rate
distortion function of the uniform distribution.
We also get weak versions of these results.
###### Corollary 20.
Let $P$ be a probability measure on the compact group $G$ with Haar
probability measure $U.$ Assume that the support of $P$ is not contained in
any coset of a proper subgroup of $G.$ Then $P^{\ast n}$ converges to $U$ in
the weak topology, i.e. $d\left(P^{\ast n},U\right)\rightarrow 0$ for
$n\rightarrow\infty.$
###### Proof.
If we take $S=P_{\beta}$ then $D\left(P_{\beta}\right)$ is finite and
$D\left(P^{\ast n}\ast P_{\beta}\|U\right)\rightarrow 0$ for
$n\rightarrow\infty$. We have
$\displaystyle d\left(P^{\ast n}\ast P_{\beta},U\right)$ $\displaystyle\leq$
$\displaystyle d_{\max}\left\|P^{\ast n}\ast P_{\beta}-U\right\|$
$\displaystyle\leq$ $\displaystyle d_{\max}\left(2D\left(P^{\ast n}\ast
P_{\beta}\|U\right)\right)^{1/2}$
implying that $d\left(P^{\ast n}\ast P_{\beta},U\right)\rightarrow 0$ for
$n\rightarrow\infty$. Now
$\displaystyle\left|d\left(P^{\ast n},U\right)-d\left(P^{\ast n}\ast
P_{\beta},U\right)\right|$ $\displaystyle\leq$ $\displaystyle
f_{2}\left(d\left(P^{\ast n}\ast P_{\beta},P^{\ast n}\right)\right)$
$\displaystyle\leq$ $\displaystyle
f_{2}\left(d\left(P_{\beta},e\right)\right).$
Therefore $\lim_{n\rightarrow\infty}\sup d\left(P^{\ast n},U\right)\leq
f_{2}\left(d\left(P_{\beta},e\right)\right)$ for all $\beta$, which implies
that
$\lim_{n\rightarrow\infty}\sup d\left(P^{\ast n},U\right)=0.\qed$
###### Corollary 21.
Let $P$ be a probability measure on the compact group $G$ with Haar
probability measure $U.$ Assume that the support of $P$ is not contained in
any coset of a proper subgroup of $G$ and $D\left(P\|U\right)$ is finite. Then
$R_{P^{\ast n}}$ converges to $R_{U}$ pointwise on the interval
$\left]0;d_{\max}\right[$ for $n\rightarrow\infty.$
###### Proof.
Corollary 20 together with Theorem 13 implies uniform convergence of the
distortion rate function for distortion less than $d_{crit}$. This implies
pointwise convergence of the rate distortion function on
$\left]0;d_{crit}\right[$ because rate distortion functions are convex
functions. The same argument works in the interval
$\left]d_{crit};d_{\max}\right[.$ Pointwise convergence in $d_{crit}$ must
also hold because of continuity. ∎
## 6 Rate of convergence
Normally the rate of convergence will be exponential. If the density is lower
bounded this is well-known. We bring a simplified proof of this.
###### Lemma 22.
Let $P$ be a probability distribution on the compact group $G$ with Haar
probability measure $U.$ If $dP/dU\geq c>0$ and $D\left(P\right)$ is finite,
then
$D\left(P^{{}^{n}}\right)\leq\left(1-c\right)^{n-1}D\left(P\right).$
###### Proof.
First we write
$P=\left(1-c\right)\cdot S+c\cdot U$
where $S$ denotes the probability measure
$S=\frac{P-cU}{1-c}.$
For any distribution $Q$ on $G$ we have
$\displaystyle D\left(Q\ast P\right)$
$\displaystyle=D\left(\left(1-c\right)\cdot Q\ast S+c\cdot Q\ast U\right)$
$\displaystyle\leq\left(1-c\right)\cdot D\left(Q\ast S\right)+c\cdot
D\left(Q\ast U\right)$ $\displaystyle\leq\left(1-c\right)\cdot
D\left(Q\right)+c\cdot D\left(U\right)$ $\displaystyle=\left(1-c\right)\cdot
D\left(Q\right).$
Here we have used convexity of divergence. ∎
If a distribution $P$ has support in a proper subgroup $F$ then
$\displaystyle D\left(P\right)$ $\displaystyle\geq D\left(U_{F}\right)$
$\displaystyle=\log\left(\left[G:F\right]\right)$
$\displaystyle\geq\log\left(2\right)=\text{1 bit}.$
Therefore $D\left(P\right)<1$ bit implies that $P$ cannot be supported by a
proper subgroup, but it implies more.
###### Proposition 23.
If $P$ is a distribution on the compact group $G$ and $D\left(P\right)<1$ bit
then $\frac{d\left(P\ast P\right)}{dU}$ is lower bounded by a positive
constant.
###### Proof.
The condition $D\left(P\right)<1$ bit implies that
$U\left\\{\frac{dP}{dU}>0\right\\}>1/2.$ Hence there exists $\varepsilon>0$
such that $U\left\\{\frac{dP}{dU}>\varepsilon\right\\}>1/2.$ We have
$\displaystyle\frac{d\left(P\ast P\right)}{dU}\left(y\right)$
$\displaystyle=\int_{G}\frac{dP}{dU}\left(x\right)\cdot\frac{dP}{dU}\left(y-x\right)~{}dUx$
$\displaystyle\geq\int_{\left\\{\frac{dP}{dU}>\varepsilon\right\\}}\varepsilon\cdot\frac{dP}{dU}\left(y-x\right)~{}dUx$
$\displaystyle\geq\varepsilon\cdot\int_{\left\\{\frac{dP}{dU}\left(x\right)>\varepsilon\right\\}\cap\left\\{\frac{dP}{dU}\left(y-x\right)>\varepsilon\right\\}}\varepsilon~{}dUx$
$\displaystyle=\varepsilon^{2}\cdot
U\left(\left\\{\frac{dP}{dU}\left(x\right)>\varepsilon\right\\}\cap\left\\{\frac{dP}{dU}\left(y-x\right)>\varepsilon\right\\}\right).$
Using the inclusion-exclusion inequalities we get
$U\left(\left\\{\frac{dP}{dU}\left(x\right)>\varepsilon\right\\}\cap\left\\{\frac{dP}{dU}\left(y-x\right)>\varepsilon\right\\}\right)\\\
=U\left\\{\frac{dP}{dU}\left(x\right)>\varepsilon\right\\}+U\left\\{\frac{dP}{dU}\left(y-x\right)>\varepsilon\right\\}-U\left(\left\\{\frac{dP}{dU}\left(x\right)>\varepsilon\right\\}\cup\left\\{\frac{dP}{dU}\left(y-x\right)>\varepsilon\right\\}\right)\\\
\geq 2\cdot U\left\\{\frac{dP}{dU}\left(x\right)>\varepsilon\right\\}-1.$
Hence
$\frac{d\left(P\ast P\right)}{dU}\left(y\right)\geq
2\varepsilon^{2}\left(U\left\\{\frac{dP}{dU}\left(x\right)>\varepsilon\right\\}-1/2\right)$
for all $y\in G.$ ∎
Combining Theorem 17, Lemma 22, and Proposition 23 we get the following
result.
###### Theorem 24.
Let $P$ be a probability measure on a compact group $G$ with Haar probability
measure $U.$ If the support of $P$ is not contained in any coset of a proper
subgroup of $G$ and $D\left(P\right\|U)$ is finite then the rate of
convergence of $D\left(P^{\ast n}\right\|U)$ to zero is exponential.
As a corollary we get the following result that was first proved by Kloss [21]
for total variation.
###### Corollary 25.
Let $P$ be a probability measure on the compact group $G$ with Haar
probability measure $U.$ If the support of $P$ is not contained in any coset
of a proper subgroup of $G$ and $D\left(P\|U\right)$ is finite then $P^{\ast
n}$ converges to $U$ in variation and the rate of convergence is exponential.
###### Proof.
This follows directly from Pinsker’s inequality [22, 23]
$\frac{1}{2}\left\|P^{\ast n}-U\right\|^{2}\leq D\left(P^{\ast
n}\|U\right).\qed$
###### Corollary 26.
Let $P$ be a probability measure on the compact group $G$ with Haar
probability measure $U.$ If the support of $P$ is not contained in any coset
of a proper subgroup of $G$ and $D\left(P\|U\right)$ is finite, then the
density
$\frac{dP^{\ast n}}{dU}$
converges to 1 point wise almost surely for $n$ tending to infinity.
###### Proof.
The variation norm can be written as
$\left\|P^{\ast n}-U\right\|=\int_{G}\left|\frac{dP^{\ast
n}}{dU}-1\right|~{}dU.$
Thus
$U\left(\left|\frac{dP^{\ast
n}}{dU}-1\right|\geq\varepsilon\right)\leq\frac{\left\|P^{\ast
n}-U\right\|}{\varepsilon}.$
The result follows by the exponential rate of convergence of $P^{\ast n}$ to
$U$ in total variation combined with the Borel-Cantelli Lemma. ∎
## 7 Discussion
In this paper we have assumed the existence of the Haar measure by referring
to the literature. With the Haar measure we have then proved convergence of
convolutions using Markov chain techniques. The Markov chain approach can also
be used to prove the existence of the Haar measure by simply referring to the
fact that a homogenous Markov chain on a compact set has an invariant
distribution. The problem about this approach is that the proof that a Markov
chain on a compact set has an invariant distribution is not easier than the
proof of the existence of the Haar measure and is less known.
We have shown that the Haar probability measure maximizes the rate distortion
function at any distortion level. The normal proofs of the existence of the
Haar measure use a kind of covering argument that is very close to the
techniques found in rate distortion technique. There is a chance that one can
get an information theoretic proof of the existence of the Haar measure. It
seems obvious to use concavity arguments as one would do for Shannon entropy
but, as proved by Ahlswede [24], the rate distortion function at a given
distortion level is not a concave function of the underlying distribution, so
some more refined technique is needed.
As noted in the introduction for any algebraic structure $A$ the group
$Aut\left(A\right)$ can be considered as symmetry group, it it has a compact
subgroup for which the results of this paper applies. It would be interesting
to extend the information theoretic approach to the algebraic object $A$
itself, but in general there is no known equivalent to the Haar measure for
other algebraic structures. Algebraic structures are used extensively in
channel coding theory and cryptography so although the theory may become more
involved extensions of the result presented in this paper are definitely
worthwhile.
## Acknowledgement
The author want to thank Ioannis Kontoyiannis for stimulating discussions.
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|
arxiv-papers
| 2008-12-30T21:09:28 |
2024-09-04T02:48:59.658558
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Peter Harremoes",
"submitter": "Peter Harremo\\\"es",
"url": "https://arxiv.org/abs/0901.0015"
}
|
0901.0129
|
Also at ]the A.F. Ioffe Institute, St. Petersburg, Russia Present address:
]Department of Physics, Bose Institute, 93/1, A.P.C. Road, Kolkata 700 009,
India
# Rigorous treatment of electrostatics for spatially varying dielectrics
based on energy minimization
O. I. Obolensky [ T. P. Doerr R. Ray [ Yi-Kuo Yu yyu@ncbi.nlm.nih.gov
National Center for Biotechnology Information, National Library of Medicine,
National Institutes of Health, Bethesda, MD 20894
(August 27, 2024)
###### Abstract
A novel energy minimization formulation of electrostatics that allows
computation of the electrostatic energy and forces to any desired accuracy in
a system with arbitrary dielectric properties is presented. An integral
equation for the scalar charge density is derived from an energy functional of
the polarization vector field. This energy functional represents the true
energy of the system even in non-equilibrium states. Arbitrary accuracy is
achieved by solving the integral equation for the charge density via a series
expansion in terms of the equation’s kernel, which depends only on the
geometry of the dielectrics. The streamlined formalism operates with volume
charge distributions only, not resorting to introducing surface charges by
hand. Therefore, it can be applied to any spatial variation of the dielectric
susceptibility, which is of particular importance in applications to
biomolecular systems. The simplicity of application of the formalism to real
problems is shown with analytical and numerical examples.
electrostatics, energy functional, integral equations, bio-macromolecules,
protein-solvent interactions
###### pacs:
03.50.De, 41.20-Cv, 87.10.Tf, 87.15.hg, 87.15.kr, 02.30.Rz
## I Introduction
Molecular dynamics (MD) simulations of solute-solvent systems in chemistry and
biology require accurate computation of electrostatic forces in order to
obtain meaningful results. For practical purposes, computational efficiency is
also essential, and various formulations exist that strive to achieve a
balance between these two requirements. The explicit solvent methods simulate
behaviour of each single solvent molecule which may be prohibitively expensive
for a system of reasonable dimensions. In addition to having high
computational costs, explicit solvent methods are usually tailored for
reproducing one of the many physical properties of the solvent and therefore
may not be well suited for a general description of solute-solvent systems
(see Wallqvist ; Tirado-Rives ; Guillot for reviews and performance
analyses).
The alternative approach is to treat the solvent as a dielectric continuum,
and the solute as a different dielectric object in the solvent. The dielectric
properties of the solvent and the solute usually serve as parameters of the
model. In the literature this scheme is known as the implicit or continuum
solvent method (for reviews see BrooksIII ; Honig ; Tomasi1 ; Cramer ;
Bashford ). Computations based on these methods are inherently faster while
comparable in accuracy with those using explicit methods, at least in the
situations when interactions between solute and solvent molecules can be
neglected. For reasons of computational efficiency, many of the implemented
implicit solvent methods make use of assumptions which prevent improvement in
accuracy even as computational resources increase. The so-called generalized
Born model is a good example of such uncontrolled approximations (see Doerr
for a discussion).
To achieve controllable accuracy, we have recently proposed a novel scheme
Doerr based on determining surface charges satisfying the displacement field
boundary condition. With this scheme, one can achieve any level of accuracy
permitted by the available computing power, while remaining computationally
more efficient than explicit solvent methods. The main idea is to treat the
induced surface charges at the boundaries as the variables to be solved for.
This makes the potential, expressed directly in terms of the induced surface
charge density, continuous at the boundary. Therefore only the displacement
field boundary condition remains, and it leads to a set of algebraic equations
for the surface charge densities. The potential is obtained at no additional
cost.
One of the seeming oversimplifications in the implicit solvent methods is the
assumption of a sharp boundary between the solute and solvent. It is known,
for example, that the solute (e.g., proteins) may strongly interact with the
surrounding solvent molecules producing the so-called hydration layer(s)
Bagchi . To determine electrostatic forces acting on a protein coated with
such hydration layers, one needs to find induced charges in a spatially
varying dielectric medium. In this paper we develop a rigorous framework,
based on functional minimization, for handling spatially varying dielectrics.
Functional variation is a powerful approach in modern physics. Despite common
use in quantum electrodynamics, variational techniques in classical
electrostatics are relatively rare and focus mainly on boundary value problems
for linear dielectrics Schwinger2 . It has long been a textbook fact that the
true electrostatic potential minimizes the system’s energy for a given
configuration of charges Jackson . A suitable energy functional can be
constructed in general for any system of continuous media including systems
with inhomogeneous and nonlinear dielectric properties. For instance, free
energy functionals became an important tool in description of electrolyte
solutions within the mean-field (Poisson-Boltzmann) approach (see a recent
paper McCammon and references therein).
From our viewpoint the electrostatic potential is not the best choice for a
minimization variable as it contains information about both the cause and
effect, i.e. the source and induced charge densities. Moreover constitutive
relations must be assumed (as in Allen ). And finally this approach depends on
prior knowledge of the Green’s function with boundary conditions suitable for
the given problem. In contrast, we use the polarization as the fundamental
function as was proposed by Marcus over fifty years ago Marcus , albeit with a
different functional. The constitutive relations are then obtained as a result
of minimization of the energy functional. The only boundary condition needed
is that the potential goes to zero sufficiently rapidly (like inverse of the
distance) at large distances.
In Marcus’s formulation Marcus , the electric field and electric polarization
were strongly motivated as the vectors defining the electric state of the
system. This formulation was aimed at the processes (charge transfer chemical
reactions) which happen on a much shorter time scale than the molecular
rearrangement in response to the changing electric field. Marcus attempted to
deal with this problem by dividing the polarization into a fast reacting part
that is proportional to the local electric field and slowly reacting part that
is not a function of the local electric field. As a result, the free energy
functional derived by Marcus contains several electric fields and
polarizations of various origins.111If there is a true separation of time
scales between various portions of the electrical response, these excess
fields should be eliminated by a proper classification of the charges in the
system: charges that respond rapidly and whose redistribution is a function of
the local electric field contribute to the polarization, while charges that
respond slowly are part of the so-called free charge distribution. However, if
one were allowed to combine the induced charge due to fast-responding
polarization with the frozen free charges, Marcus’s functional becomes
identical to ours.
A physically sound free energy functional was proposed by Felderhof Felderhof
in the context of a discussion of thermal fluctuations of the polarization and
magnetization in dielectric magnetic media. However, a free energy of this
type seems not to have been adopted for calculation of electrostatic fields
until recently. An example of numerical implementation using Felderhof’s
scheme can be found in Levy . There, the polarization vector field was
expanded in a plane waves basis set. The energy functional is then an ordinary
function of the expansion coefficients which, in turn, become the variational
parameters of a standard multidimensional optimization problem. Fast Fourier
transforms were used to go from the real space to the reciprocal space
representations.
An approach close to Felderhof’s scheme was also taken in Attard where a
thermodynamic functional was constructed with the polarization as the
independent function. However, the techniques used there are suitable for
systems with sharp boundaries only (the susceptibility is not considered to be
a function of coordinates, but is rather treated as a piecewise constant).
In this paper we construct an energy minimization scheme suitable for a
rigorous treatment of systems with spatially varying dielectric functions, be
they linear or nonlinear. In the case of linear dielectrics, our functional is
equivalent to that proposed by Felderhof Felderhof . In section II we give the
details of the formulation and describe a systematic protocol for obtaining
the total charge density. To show the versatility of the scheme we apply it in
section III to systems with sharp boundaries for which the exact solutions (or
the exact equations governing the exact solutions) are known. In section IV we
present numerical results for the case of two interacting dielectric charged
spheres (solutes) placed in a dielectric solvent. We discuss the differences
in force and energy between the situations with sharp and smooth boundaries.
Finally we conclude with a discussion assessing the usefulness of the method.
Electrostatic CGS units are used throughout.
## II Fundamental Formulation
Polarization is the response of a dielectric medium to an applied electric
field. The phenomenon is usually visualized as the appearance of an induced
dipole moment due to a small shift in the relative positions of the positive
and negative charge centers at the atomic scale Feynman . The shift may be
either translational or rotational or both, depending on the quantum
mechanical and electromagnetic interactions at the atomic level. The applied
electric fields must be weak enough not to split the atoms or molecules into
their constituents. The system is in a state of equilibrium under the external
electromagnetic and the intrinsic restoring forces.
Quantitatively, polarization ${{\rm\bf P}}({{\rm\bf r}})$ is the density of
induced dipole moment at location ${{\rm\bf r}}$. This density in classical
electrodynamics is defined through averaging of dipole moments of constituent
atoms/molecules in a small volume centered around ${{\rm\bf r}}$. The amount
of polarization depends on the applied force and the susceptibility of the
medium to such forces. Determination of the susceptibility of the medium (or
rather the intrinsic restoring force in the medium) is the subject of quantum
mechanics rather than classical electrodynamics. Polarization is thus a
classical/macroscopic variable summarizing quantum mechanical effects at the
atomic/microscopic level. Therefore, we choose the polarization vector field
${{\rm\bf P}}({{\rm\bf r}})$ and electric field ${{\rm\bf E}}({{\rm\bf r}})$,
in contrast to the more commonly used pair ${{\rm\bf E}}({{\rm\bf r}})$ and
${{\rm\bf D}}({{\rm\bf r}})$, as our fundamental variables. This choice
provides a simpler connection to the parameters determined in microscopic
physics.
We express the energy as a functional $U[{{\rm\bf P}}]$
$U[{{\rm\bf P}}]=U_{\rm C}[{{\rm\bf P}}]+W[{{\rm\bf P}}],$ (1)
where $U_{\rm C}[{{\rm\bf P}}]$ is the electrostatic energy of interaction of
all charges present in the system, and $W[{{\rm\bf P}}]$ is the energy
required to create the given polarization vector field ${{\rm\bf P}}({{\rm\bf
r}})$.
From simple considerations it can be shown Feynman ; Landau that the
variation of polarization in the vicinity of a point is equivalent to the
presence of an induced charge density ${\rho_{\rm i}}({{\rm\bf
r}})=-\nabla\cdot{{\rm\bf P}}({{\rm\bf r}})$. Therefore, the total charge
density ${\rho_{\rm t}}({{\rm\bf r}})$ in the medium is a sum of the free
charge density ${\rho_{\rm f}}({{\rm\bf r}})$ and ${\rho_{\rm i}}({{\rm\bf
r}})$:
${\rho_{\rm t}}({\rm\bf r})={\rho_{\rm f}}({\rm\bf r})+{\rho_{\rm i}}({\rm\bf
r}).$ (2)
Then222When there is no possibility of confusion, we do not specify the
variable for the operator $\nabla$; otherwise, we indicate the variable by a
subscript.
$U_{\rm C}[{{\rm\bf P}}]=\frac{1}{2}\int\left[{\rho_{\rm f}}({{\rm\bf
r}})-\nabla\cdot{{\rm\bf P}}({{\rm\bf r}})\right]\frac{1}{|{{\rm\bf
r}}-{{\rm\bf r}}^{\prime}|}\left[{\rho_{\rm f}}({{\rm\bf
r}}^{\prime})-\nabla\cdot{{\rm\bf P}}({{\rm\bf r}}^{\prime})\right]d{{\rm\bf
r}}d{{\rm\bf r}}^{\prime}.$ (3)
Note first that we do not include any separate term for induced surface
charges as was done in some of the earlier formulations of functional
minimization Marcus ; Attard . The volume charge density is the most general
form of charge density possible. Secondly, (3) is the Coulomb energy in vacuum
and hence quite fundamental as opposed to the form with the dielectric
constant of the material in the denominator used in some of the earlier works
Attard .
The work functional $W[{{\rm\bf P}}]$ should contain the intrinsic self
interaction of the polarization vector field. Here we consider only local
contact terms for the intrinsic interactions. Noting that the energy
functional is a scalar and assuming ${{\rm\bf P}}\leftrightarrow-{{\rm\bf P}}$
symmetry, one can write the general work functional $W[{{\rm\bf P}}]$ as a
polynomial expansion in even powers of ${{\rm\bf P}}$ (or the components
$P_{i}$). Thus we may write,
$W[{{\rm\bf P}}]=\frac{1}{2}\int\left[P_{i}\,\left(\frac{1}{\chi({{\rm\bf
r}})}\right)_{ij}\,P_{j}+P_{i}P_{j}\,\left(\frac{1}{\mu({{\rm\bf
r}})}\right)_{ijkl}\,P_{k}P_{l}+\cdots\right]d{{\rm\bf r}},$ (4)
where the interaction tensors $1/\chi$, $1/\mu$, etc. describe the linear and
nonlinear dielectric properties of the media, isotropic or anisotropic
(summation over repeated indices is assumed). The effective dielectric
properties of the medium at the macroscopic level are now contained in these
quantities.
We emphasize that $U[{{\rm\bf P}}]$ is the actual energy functional unlike
various other functionals proposed in the literature Jackson ; Allen ;
McCammon ; Briggs which yield the energy or free energy of the system only at
equilibrium. The equilibrium distribution of polarization (as well as induced
charge distribution) can be obtained by minimizing this energy functional with
respect to the polarization. For any given external charge distribution and
spatially varying dielectric susceptibilities one can obtain the solution
analytically or numerically.
We may truncate the series in (4) at an order suitable for the problem at
hand. For example, if the field is very weak we can retain only the quadratic
term which corresponds to the case of linear dielectrics (isotropy is also
assumed for the sake of simplicity of presentation):
$U[{{\rm\bf P}}]=U_{C}[{{\rm\bf P}}]+\frac{1}{2}\int\frac{{{\rm\bf
P}}({{\rm\bf r}})\cdot{{\rm\bf P}}({{\rm\bf r}})}{\chi({{\rm\bf r}})}d{{\rm\bf
r}}.$ (5)
Performing a functional variation with respect to the polarization vector
${{\rm\bf P}}$, we arrive at an integro-differential equation defining the
equilibrium polarization
$\frac{{{\rm\bf P}}({{\rm\bf r}})}{\chi({{\rm\bf r}})}+\nabla_{{\rm\bf
r}}\int\frac{{\rho_{\rm f}}({{\rm\bf r}}^{\prime})-\nabla\cdot{{\rm\bf
P}}({{\rm\bf r}}^{\prime})}{|{{\rm\bf r}}-{{\rm\bf r}}^{\prime}|}d{{\rm\bf
r}}^{\prime}=0$ (6)
which implies
${{\rm\bf P}}({{\rm\bf r}})=\chi({{\rm\bf r}})\int\left[{\rho_{\rm
f}}({{\rm\bf r}}^{\prime})-\nabla\cdot{{\rm\bf P}}({{\rm\bf
r}}^{\prime})\right]\frac{{{\rm\bf r}}-{{\rm\bf r}}^{\prime}}{|{{\rm\bf
r}}-{{\rm\bf r}}^{\prime}|^{3}}d{{\rm\bf r}}^{\prime}=\chi({{\rm\bf
r}}){{\rm\bf E}}({{\rm\bf r}})\;.$ (7)
Thus the constitutive relation for a linear dielectric is obtained as a result
of functional minimization, with the expansion coefficient $\chi({{\rm\bf
r}})$ turning out to be the dielectric susceptibility. Inserting the
equilibrium polarization (7) in (5) results in the well known expression for
the total energy of the system:
$U=\frac{1}{2}\int{\rho_{\rm f}}({{\rm\bf r}})\frac{1}{|{{\rm\bf r}}-{{\rm\bf
r}}^{\prime}|}\left[{\rho_{\rm f}}({{\rm\bf r}}^{\prime})-\nabla\cdot{{\rm\bf
P}}({{\rm\bf r}}^{\prime})\right]d{{\rm\bf r}}d{{\rm\bf r}}^{\prime}.$ (8)
Keeping two (or more) terms in the series (4) introduces nonlinearity into the
problem. The energy functional in this case is given by
$U[{{\rm\bf P}}]=U_{C}[{{\rm\bf P}}]+\frac{1}{2}\int\frac{{{\rm\bf
P}}({{\rm\bf r}})\cdot{{\rm\bf P}}({{\rm\bf r}})}{\chi({{\rm\bf r}})}d{{\rm\bf
r}}+\frac{1}{2}\int\frac{\left[{{\rm\bf P}}({{\rm\bf r}})\cdot{{\rm\bf
P}}({{\rm\bf r}})\right]^{2}}{\mu({{\rm\bf r}})}d{{\rm\bf r}}\;.$ (9)
Performing a functional variation as above we now obtain
${{\rm\bf P}}({{\rm\bf r}})=\chi({{\rm\bf r}}){{\rm\bf E}}({{\rm\bf
r}})-2\frac{\chi({{\rm\bf r}})}{\mu({{\rm\bf r}})}\left[{{\rm\bf P}}({{\rm\bf
r}})\cdot{{\rm\bf P}}({{\rm\bf r}})\right]{{\rm\bf P}}({{\rm\bf r}})\;.$ (10)
Given that the first term on the right hand side is the dominant one, we can
obtain the solution via iteration. The first approximation would be the same
as the result for the linear dielectrics. Substituting it back into (10), we
obtain at the second order of approximation,
${{\rm\bf P}}({{\rm\bf r}})=\chi({{\rm\bf r}}){{\rm\bf E}}({{\rm\bf
r}})-2\frac{\chi^{4}({{\rm\bf r}})}{\mu({{\rm\bf r}})}\left[{{\rm\bf
E}}({{\rm\bf r}})\cdot{{\rm\bf E}}({{\rm\bf r}})\right]{{\rm\bf E}}({{\rm\bf
r}})\;.$ (11)
One can continue with this to obtain a series of terms with higher and higher
powers of $[{{\rm\bf E}}\cdot{{\rm\bf E}}]$. This gives the desired result for
nonlinear dielectrics. We should mention once more that this solution is true
for weak fields so that the higher order terms are successively weaker. To
ensure this condition we require $\mu({{\rm\bf r}})>>\chi^{3}({{\rm\bf r}})$
to be true to any order of approximation.
Let us now solve (7) for the case of linear dielectrics. We simplify the
analysis by choosing the (scalar) induced density ${\rho_{\rm
i}}=-\nabla\cdot{\bf P}$ as our variable.
Using the relation
$\nabla_{{\rm\bf r}}\cdot\left[{{{\rm\bf r}}-{{\rm\bf
r}}^{\prime}\over|{{\rm\bf r}}-{{\rm\bf
r}}^{\prime}|^{3}}\right]=4\pi\delta({{\rm\bf r}}-{{\rm\bf r}}^{\prime})\;,$
(12)
we obtain from (7)
$\nabla\cdot{\rm\bf P}({\rm\bf r})=\nabla\chi({\rm\bf r})\cdot\int{{\rm\bf
r}-{\rm\bf r}^{\prime}\over|{\rm\bf r}-{\rm\bf
r}^{\prime}|^{3}}\left[{\rho_{\rm f}}({\rm\bf r}^{\prime})-\nabla\cdot{\rm\bf
P}({\rm\bf r}^{\prime})\right]d{\rm\bf r}^{\prime}+4\pi\chi({\rm\bf
r})\left[{\rho_{\rm f}}({\rm\bf r})-\nabla\cdot{\rm\bf P}({\rm\bf r})\right]$
(13)
which implies
$\epsilon({\rm\bf r}){\rho_{\rm i}}({\rm\bf r})=-\nabla\chi({\rm\bf
r})\cdot\int\frac{{{\rm\bf r}}-{{\rm\bf r}^{\prime}}}{|{{\rm\bf r}}-{{\rm\bf
r}^{\prime}}|^{3}}\left[{\rho_{\rm f}}({\rm\bf r}^{\prime})+{\rho_{\rm
i}}({\rm\bf r}^{\prime})\right]d{\rm\bf r}^{\prime}-4\pi\chi({\rm\bf
r}){\rho_{\rm f}}({\rm\bf r})\;,$ (14)
where $\epsilon=1+4\pi\chi$. Equation (14) relates ${\rho_{\rm i}}$ and
${\rho_{\rm f}}$. We may rewrite this equation as
$\epsilon({\rm\bf r}){\rho_{\rm t}}({\rm\bf r})={\rho_{\rm f}}({\rm\bf
r})-\nabla\chi({\rm\bf r})\cdot\int\frac{{{\rm\bf r}}-{{\rm\bf
r}^{\prime}}}{|{{\rm\bf r}}-{{\rm\bf r}^{\prime}}|^{3}}{\rho_{\rm t}}({\rm\bf
r}^{\prime})d{\rm\bf r}^{\prime}$ (15)
or
${\rho_{\rm t}}({\rm\bf r})={{\rho_{\rm f}}({\rm\bf r})\over\epsilon({\rm\bf
r})}-{1\over\epsilon({\rm\bf r})}\nabla\chi({\rm\bf r})\cdot\int\frac{{{\rm\bf
r}}-{{\rm\bf r}^{\prime}}}{|{{\rm\bf r}}-{{\rm\bf r}^{\prime}}|^{3}}{\rho_{\rm
t}}({\rm\bf r}^{\prime})d{\rm\bf r}^{\prime}$ (16)
This integral equation is the most general equation for total charge density
in linear dielectric media. Note that it is a simple scalar equation for the
induced charge ${\rho_{\rm i}}$, as opposed to (7), a vector equation for the
polarization ${\bf P}$ whose numerical solution also requires calculation of
$\nabla\cdot{\bf P}$. Once (7) is solved for ${\rho_{\rm t}}$, the
polarization field is straightforwardly obtained by substituting ${\rho_{\rm
t}}$ for ${\rho_{\rm f}}-\nabla{\rm\bf P}$ in (7). The advantages of switching
to the induced charge persist even in the case of nonlinear dielectrics.
For a system with uniform susceptibility, we obtain the expected screening
${\rho_{\rm t}}({\rm\bf r})={{\rho_{\rm f}}({\rm\bf r})\over\epsilon}$, so
that ${\rho_{\rm i}}({\rm\bf r})=-(1-\frac{1}{\epsilon}){\rho_{\rm f}}({\rm\bf
r})$. The second term in (16) generates induced charges due to non-uniformity
of dielectric medium. In the case of a sharp boundary, the proper limit of
this term gives rise to surface charges. A planar interface example is
described in appendix A.
We may rewrite (16) in the form of an operator equation
$({\rm\bf I}+{\rm\bf C}){\rho_{\rm t}}=\frac{{\rho_{\rm f}}}{\epsilon},$ (17)
where the operators ${\rm\bf I}$ and ${\rm\bf C}$ are defined as
$\left[{\rm\bf I}\;h\right]({\rm\bf r})=\int\delta({\rm\bf r}-{\rm\bf
r}^{\prime})h({\rm\bf r}^{\prime})d{\rm\bf r}^{\prime}\;,$ (18)
$\left[{\rm\bf C}\;h\right]({\rm\bf r})=\int{\nabla\chi({\rm\bf
r})\over\epsilon({\rm\bf r})}\cdot{{\rm\bf r}-{\rm\bf r}^{\prime}\over|{\rm\bf
r}-{\rm\bf r}^{\prime}|^{3}}h({\rm\bf r}^{\prime})d{\rm\bf r}^{\prime}\;.$
(19)
We will frequently make use of the kernel of this operator defined as
${\rm\bf C}({\rm\bf r},{\rm\bf r}^{\prime})={\nabla\chi({\rm\bf
r})\over\epsilon({\rm\bf r})}\cdot{{\rm\bf r}-{\rm\bf r}^{\prime}\over|{\rm\bf
r}-{\rm\bf r}^{\prime}|^{3}}.$ (20)
Note that ${\rm\bf C}$ is completely determined by the geometry regardless of
the position of the source charge.
Using the formal inversion of ${\rm\bf I}+{\rm\bf C}$
$\left[{\rm\bf I}+{\rm\bf C}\right]^{-1}={\rm\bf I}-{\rm\bf C}+{\rm\bf
C}^{2}-{\rm\bf C}^{3}+\cdots\;,$ (21)
one may obtain the total charge density
${\rho_{\rm t}}=\left[{\rm\bf I}-{\rm\bf C}+{\rm\bf C}^{2}-{\rm\bf
C}^{3}+\cdots\right]{{\rho_{\rm f}}\over\epsilon}.$ (22)
If the off-diagonal part ${\rm\bf C}({\rm\bf r},{\rm\bf r}^{\prime})$ is small
compared to the diagonal delta function, series (21) converges quickly.
## III Three case studies
In this section we apply our energy minization method to three examples for
which the exact solutions or the equations governing the exact solutions are
known.
### III.1 A planar interface
Let $\chi$ depend only on one spatial variable $z$. For $z>a$,
$\chi={\chi_{1}}$, and for $z<-a$, $\chi={\chi_{2}}$. In the range $-a\leq
z\leq a$, $\chi$ is a smooth function of $z$. Then
$C({\rm\bf r},{\rm\bf
r}^{\prime})=\frac{\partial_{z}\chi}{\epsilon(z)}\hat{z}\cdot{{\rm\bf
r}-{\rm\bf r}^{\prime}\over|{\rm\bf r}-{\rm\bf
r}^{\prime}|^{3}}={\partial_{z}\chi\over\epsilon(z)}{z-z^{\prime}\over|{\rm\bf
r}-{\rm\bf r}^{\prime}|^{3}}.$ (23)
Let us put a free point charge $q$ at $z=d>a$ so that ${\rho_{\rm f}}({\rm\bf
r})=q\delta({\rm\bf r}-d\hat{z})$. The total charge density (22) becomes
$\displaystyle{\rho_{\rm t}}({\rm\bf r})$ $\displaystyle=$
$\displaystyle{q\over\epsilon_{1}}\delta({\rm\bf
r}-d\hat{z})-{\epsilon^{\prime}(z)\over
4\pi\epsilon(z)}\int{z-z^{\prime}\over|{\rm\bf r}-{\rm\bf
r}^{\prime}|^{3}}\delta({\rm\bf
r}^{\prime}-d\hat{z}){q\over\epsilon_{1}}d{\rm\bf r}^{\prime}$ (24)
$\displaystyle+{\epsilon^{\prime}(z)\over
4\pi\epsilon(z)}\int{z-z^{\prime}\over|{\rm\bf r}-{\rm\bf
r}^{\prime}|^{3}}{\epsilon^{\prime}(z^{\prime})\over
4\pi\epsilon(z^{\prime})}{z^{\prime}-z^{\prime\prime}\over|{\rm\bf
r}^{\prime}-{\rm\bf r}^{\prime\prime}|^{3}}\delta({\rm\bf
r}^{\prime\prime}-d\hat{z}){q\over\epsilon_{1}}d{\rm\bf r}^{\prime}d{\rm\bf
r}^{\prime\prime}+\cdots\;,$
where we have used $\epsilon=1+4\pi\chi$.
In the $a\to 0$ limit,
$\epsilon^{\prime}(z)=\delta(z)(\epsilon_{1}-\epsilon_{2})$, so
$\displaystyle{\rho_{\rm t}}({\rm\bf r})$ $\displaystyle=$
$\displaystyle{q\over\epsilon_{1}}\delta({\rm\bf
r}-d\hat{z})-{\epsilon_{1}-\epsilon_{2}\over
4\pi\epsilon(z=0)}\delta(z)\int{z-z^{\prime}\over|{\rm\bf r}-{\rm\bf
r}^{\prime}|^{3}}\delta({\rm\bf
r}^{\prime}-d\hat{z}){q\over\epsilon_{1}}d{\rm\bf r}^{\prime}$ (25)
$\displaystyle+\left({\epsilon_{1}-\epsilon_{2}\over
4\pi\epsilon(z=0)}\right)^{2}\delta(z)\int{z-z^{\prime}\over|{\rm\bf
r}-{\rm\bf
r}^{\prime}|^{3}}\delta(z^{\prime}){z^{\prime}-z^{\prime\prime}\over|{\rm\bf
r}^{\prime}-{\rm\bf r}^{\prime\prime}|^{3}}\delta({\rm\bf
r}^{\prime\prime}-d\hat{z}){q\over\epsilon_{1}}d{\rm\bf r}^{\prime}d{\rm\bf
r}^{\prime\prime}+\cdots\;.$
Note that each term from the second order on has a factor of $z\delta(z)$
which is zero for any $z$. We finally obtain
${\rho_{\rm t}}({\rm\bf r})={q\over\epsilon_{1}}\delta({\rm\bf
r}-d\hat{z})-{\epsilon_{1}-\epsilon_{2}\over
4\pi\epsilon(z=0)}\delta(z){z-d\over|{\rm\bf
r}-d\hat{z}|^{3}}{q\over\epsilon_{1}}\;.$ (26)
The surface charge density Jackson depending on the radial vector
$\boldsymbol{\rho}$ in the $x-y$ plane
$\sigma(\boldsymbol{\rho})={q\over
4\pi\epsilon_{1}}\frac{2(\epsilon_{1}-\epsilon_{2})}{(\epsilon_{1}+\epsilon_{2})}{d\over|\boldsymbol{\rho}-d\hat{z}|^{3}}\;,$
(27)
is then obtained by setting $\epsilon(z=0)=(\epsilon_{1}+\epsilon_{2})/2$. The
validity of using the average dielectric constant at the boundary is justified
by the following argument. Let there be a surface charge density $\sigma$ at
the boundary. It creates an electric field of magnitude $2\pi\sigma$ directed
along the normal vector to the surface. Assuming that there are no free
charges at the interface, the boundary condition requires that
$(E_{\perp}+2\pi\sigma)\epsilon_{1}=(E_{\perp}-2\pi\sigma)\epsilon_{2}$, where
$E_{\perp}$ is a normal component of electric field produced by sources other
than $\sigma$. Therefore,
$\sigma(\epsilon_{1}+\epsilon_{2})/2=(\epsilon_{2}-\epsilon_{1})E_{\perp}/4\pi$,
in agreement with setting $\epsilon(z=0)=(\epsilon_{1}+\epsilon_{2})/2$. In
Appendix A we present a thorough derivation of the $a\to 0$ limit, which
arrives at the same conclusion without invoking $\delta$-functions. It is
worthwhile to point out here that the surface charge density arises entirely
from the term containing the gradient of the susceptibility. Our formulation
is straightforward in this respect when contrasted with methods that first
neglect the gradient of $\chi$ and then introduce a surface charge density by
hand Attard .
### III.2 A point charge outside of a sphere
Consider a ball of radius $a_{1}$ centered at the origin and a point charge
$q$ located at point ${\rm\bf L}$, ${\rho_{\rm f}}({\rm\bf r})=q\delta({\rm\bf
r}-{\rm\bf L})$. In this subsection, we first obtain a set of equations for
the general case of spatially varying susceptibility, assuming only that it
changes in the radial direction. We then consider the case of a sharp boundary
and show that the simplified expressions for the induced density coincide with
the known results Menzel .
Let the susceptibility change in the radial direction from some value
$\chi_{1}$ inside the ball to another value ${\chi_{\rm o}}$ outside. Gradient
$\chi$ is then directed radially,
$\nabla\chi({\rm\bf r})=\frac{\partial\chi}{\partial
r}\hat{r}=\frac{\epsilon^{\prime}(r)}{4\pi}\hat{r},$ (28)
and we find for $C({\rm\bf r},{\rm\bf r}^{\prime})$
$C({\rm\bf r},{\rm\bf
r}^{\prime})=\frac{\epsilon^{\prime}(r)}{4\pi\epsilon(r)}\hat{r}\cdot{{\rm\bf
r}-{\rm\bf r}^{\prime}\over|{\rm\bf r}-{\rm\bf
r}^{\prime}|^{3}}=-\frac{\epsilon^{\prime}(r)}{4\pi\epsilon(r)}\partial_{r}{1\over|{\rm\bf
r}-{\rm\bf r}^{\prime}|}.$ (29)
Let us calculate
$\left[{\rm\bf C}\cdot{{\rho_{\rm f}}\over\epsilon}\right]({\rm\bf r})=\int
d{\rm\bf r}^{\prime}C({\rm\bf r},{\rm\bf r}^{\prime})\frac{{\rho_{\rm
f}}({\rm\bf r}^{\prime})}{\epsilon({\rm\bf
r}^{\prime})}=-\frac{q}{{\epsilon_{\rm
o}}}\frac{\epsilon^{\prime}(r)}{4\pi\epsilon(r)}\partial_{r}\frac{1}{|{\rm\bf
r}-{\rm\bf L}|}.$ (30)
Assuming, for simplicity, that the point charge is located far enough from the
ball, so that $\epsilon^{\prime}(r)\neq 0$ only where $r<L$ (a generalization
which would lift this condition is straightforward), we obtain the first order
approximation for the induced charge density,
${\rho_{\rm i}}^{(1)}({\rm\bf r})\equiv\left[-{\rm\bf C}\cdot{{\rho_{\rm
f}}\over\epsilon}\right]({\rm\bf
r})=\sum_{lm}\rho^{(1)}_{lm}(r)Y_{lm}(\hat{r})Y^{*}_{lm}(\hat{L}),$ (31)
where
$\rho^{(1)}_{lm}(r)=\frac{4\pi}{2l+1}\;\frac{q}{{\epsilon_{\rm
o}}}\frac{\epsilon^{\prime}(r)}{4\pi\epsilon(r)}\frac{lr^{l-1}}{L^{l+1}}$ (32)
and the expansion
$\frac{1}{|{\rm\bf r}_{1}-{\rm\bf
r}_{2}|}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\frac{4\pi}{2l+1}\frac{r_{<}^{l}}{r_{>}^{l+1}}Y_{lm}(\hat{r}_{<})Y^{*}_{lm}(\hat{r}_{>}),\quad
r_{<}\equiv{\rm min}(r_{1},r_{2}),\;r_{>}\equiv{\rm max}(r_{1},r_{2})$ (33)
was used. Note that any one of the spherical harmonics can bear the complex
conjugation sign.
The next order is obtained by applying the operator $(-{\rm\bf C})$ to
${\rho_{\rm i}}^{(1)}$:
${\rho_{\rm i}}^{(2)}({\rm\bf r})=\left[-{\rm\bf C}\cdot{\rho_{\rm
i}}^{(1)}\right]({\rm\bf r})=\sum_{lm}\left[\int d{\rm\bf
r}^{\prime}\left(-C({\rm\bf r},{\rm\bf
r}^{\prime})\right)\rho^{(1)}_{lm}(r^{\prime})Y_{lm}(\hat{r^{\prime}})\right]Y^{*}_{lm}(\hat{L}).$
(34)
The angular integration in (34) can be performed analytically using (29) and
(33):
$\displaystyle\int d{\rm\bf r}^{\prime}\left(-C({\rm\bf r},{\rm\bf
r}^{\prime})\right)\rho^{(1)}_{lm}(r^{\prime})Y_{lm}(\hat{r^{\prime}})=\frac{\epsilon^{\prime}(r)}{4\pi\epsilon(r)}\partial_{r}\int
d{\rm\bf r}^{\prime}\frac{1}{|{\rm\bf r}-{\rm\bf
r}^{\prime}|}\rho^{(1)}_{lm}(r^{\prime})Y_{lm}(\hat{r^{\prime}})$
$\displaystyle\quad=\frac{\epsilon^{\prime}(r)}{4\pi\epsilon(r)}\sum_{l^{\prime}m^{\prime}}\frac{4\pi}{2l^{\prime}+1}Y_{l^{\prime}m^{\prime}}(\hat{r})\left[\partial_{r}\int_{0}^{\infty}dr^{\prime}\frac{r_{<}^{l^{\prime}}}{r_{>}^{l^{\prime}+1}}\rho^{(1)}_{lm}(r^{\prime})\int
d\hat{r^{\prime}}Y^{*}_{l^{\prime}m^{\prime}}(\hat{r^{\prime}})Y_{lm}(\hat{r^{\prime}})\right].$
(35)
The orthogonality relation for the spherical harmonics,
$\int
d\hat{r^{\prime}}Y^{*}_{l^{\prime}m^{\prime}}(\hat{r^{\prime}})Y_{lm}(\hat{r^{\prime}})=\delta_{l^{\prime}l}\>\delta_{m^{\prime}m}$
(36)
removes the sum, so we obtain
$\displaystyle{\rho_{\rm i}}^{(2)}({\rm\bf r})$ $\displaystyle=$
$\displaystyle\sum_{lm}\rho^{(2)}_{lm}(r)Y_{lm}(\hat{r})Y^{*}_{lm}(\hat{L}),$
$\displaystyle\rho^{(2)}_{lm}(r)$ $\displaystyle=$
$\displaystyle\frac{4\pi}{2l+1}\frac{\epsilon^{\prime}(r)}{4\pi\epsilon(r)}\left[l\int_{r}^{\infty}\frac{r^{l-1}}{(r^{\prime})^{l-1}}\rho^{(1)}_{lm}(r^{\prime})dr^{\prime}-(l+1)\int_{0}^{r}\frac{(r^{\prime})^{l+2}}{r^{l+2}}\rho^{(1)}_{lm}(r^{\prime})dr^{\prime}\right].$
(37)
The same derivation leads us to a general recursive relation
$\displaystyle{\rho_{\rm i}}^{(n+1)}({\rm\bf r})$ $\displaystyle=$
$\displaystyle\sum_{lm}\rho^{(n+1)}_{lm}(r)Y_{lm}(\hat{r})Y^{*}_{lm}(\hat{L}),$
$\displaystyle\rho^{(n+1)}_{lm}(r)$ $\displaystyle=$
$\displaystyle\frac{4\pi}{2l+1}\frac{\epsilon^{\prime}(r)}{4\pi\epsilon(r)}\left[l\int_{r}^{\infty}\frac{r^{l-1}}{(r^{\prime})^{l-1}}\rho^{(n)}_{lm}(r^{\prime})dr^{\prime}-(l+1)\int_{0}^{r}\frac{(r^{\prime})^{l+2}}{r^{l+2}}\rho^{(n)}_{lm}(r^{\prime})dr^{\prime}\right].$
(38)
Therefore, using (22), we write the induced charge density for the general
case of a sphere with a radially varying susceptibility as
$\displaystyle{\rho_{\rm i}}({\rm\bf r})$ $\displaystyle=$
$\displaystyle\sum_{lm}\rho_{lm}(r)Y_{lm}(\hat{r})Y^{*}_{lm}(\hat{L}),$
$\displaystyle\rho_{lm}(r)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\rho^{(n)}_{lm}(r),$ (39)
where $\rho_{lm}^{(n)}(r)$ can be found via (32) and (38).
In the limit of a sharp boundary,
$\epsilon^{\prime}(r)=({\epsilon_{\rm o}}-\epsilon_{1})\delta(r-a_{1}),$ (40)
we immediately find that
$\rho^{(1)}_{lm}(r)=\frac{q}{{\epsilon_{\rm o}}}\frac{{\epsilon_{\rm
o}}-\epsilon_{1}}{4\pi\epsilon(a_{1})}\frac{la_{1}^{l-1}}{L^{l+1}}\delta(r-a_{1}),$
(41)
while the higher order contributions,
$\rho^{(n+1)}_{lm}(r)=\left(\frac{-1}{2}\right)^{n}\left(\frac{4\pi}{2l+1}\right)^{n+1}\left(\frac{{\epsilon_{\rm
o}}-\epsilon_{1}}{4\pi\epsilon(a_{1})}\right)^{n+1}\frac{la_{1}^{l-1}}{L^{l+1}}\delta(r-a_{1}),$
(42)
are found from (38) using the generalized definition of the Dirac
$\delta$-function,
$\int_{0}^{\infty}h(x)\delta(x)=\frac{1}{2}h(0).$ (43)
Finally, we sum all the contributions to obtain the total charge density:
$\displaystyle{\rho_{\rm t}}({\rm\bf r})$ $\displaystyle=$
$\displaystyle\left[\left({\rm\bf I}+\sum_{n=1}^{\infty}(-{\rm\bf
C})^{n}\right){\rho_{f}\over\epsilon}\right]({\rm\bf r})$ (44)
$\displaystyle=$ $\displaystyle{q\over{\epsilon_{\rm o}}}\delta({\rm\bf
r}-{\rm\bf L})+{q\over{\epsilon_{\rm o}}}\left({{\epsilon_{\rm
o}}-\epsilon_{1}\over
4\pi\epsilon(a_{1})}\right)\delta(r-a_{1})\sum_{lm}^{\infty}\frac{4\pi}{2l+1}{la_{1}^{l-1}\over
L^{l+1}}\times$
$\displaystyle\qquad\left[\sum_{n=1}^{\infty}\left(-{{\epsilon_{\rm
o}}-\epsilon_{1}\over 2\epsilon(a_{1})}{1\over 2l+1}\right)^{n-1}\right]\
Y_{lm}(\hat{r})Y^{*}_{lm}(\hat{L})$
The sum in square brackets is a geometric series with common factor less than
1 for all $l$. Substituting $\epsilon(a_{1})=({\epsilon_{\rm
o}}+\epsilon_{1})/2$ again, we derive
${\rho_{\rm t}}({\rm\bf r})={q\over{\epsilon_{\rm o}}}\delta({\rm\bf
r}-{\rm\bf L})+{q\over{\epsilon_{\rm o}}}({\epsilon_{\rm
o}}-\epsilon_{1})\delta(r-a_{1})\sum_{lm}^{\infty}{l\over\left[(l+1){\epsilon_{\rm
o}}+l\epsilon_{1}\right]}{a_{1}^{l-1}\over
L^{l+1}}Y_{lm}(\hat{r})Y^{*}_{lm}(\hat{L}).$ (45)
For the case in which the point charge is inside the ball, similar analysis
leads to
${\rho_{\rm t}}({\rm\bf r})={q\over\epsilon_{1}}\delta({\rm\bf r}-{\rm\bf
L})+{q\over\epsilon_{1}}({\epsilon_{\rm
o}}-\epsilon_{1})\delta(r-a_{1})\sum_{lm}^{\infty}{l+1\over\left[(l+1){\epsilon_{\rm
o}}+l\epsilon_{1}\right]}{L^{l}\over
a_{1}^{l+2}}Y_{lm}(\hat{r})Y^{*}_{lm}(\hat{L}),\qquad L<a_{1}$ (46)
Using the addition theorem for spherical harmonics,
$P_{l}(\hat{r}\cdot\hat{L})=\frac{4\pi}{2l+1}\sum_{m=-l}^{l}Y_{lm}(\hat{r})Y^{*}_{lm}(\hat{L}),$
(47)
and placing the point charge on the z-axis, ${\rm\bf L}=(0,0,L)$, one can
further simplify the derived equations:
${\rho_{\rm t}}({\rm\bf r})={q\over{\epsilon_{\rm o}}}\delta({\rm\bf
r}-L\hat{z})+{q\over{\epsilon_{\rm o}}}\frac{{\epsilon_{\rm
o}}-\epsilon_{1}}{4\pi}\delta(r-a_{1})\sum_{l}{l(2l+1)\over\left[(l+1){\epsilon_{\rm
o}}+l\epsilon_{1}\right]}{a_{1}^{l-1}\over L^{l+1}}P_{l}(\cos\theta),\qquad
L>a_{1},$ (48) ${\rho_{\rm t}}({\rm\bf r})={q\over\epsilon_{1}}\delta({\rm\bf
r}-L\hat{z})-{q\over\epsilon_{1}}\frac{{\epsilon_{\rm
o}}-\epsilon_{1}}{4\pi}\delta(r-a_{1})\sum_{l}{(l+1)(2l+1)\over\left[(l+1){\epsilon_{\rm
o}}+l\epsilon_{1}\right]}{L^{l}\over a_{1}^{l+2}}P_{l}(\cos\theta),\qquad
L<a_{1}.$ (49)
where $\theta$ is the polar angle of ${\rm\bf r}$. These expressions provide
the correct results for the surface charge densities which can be found in
Menzel .
### III.3 Multiple charges and multiple spheres
We now generalize to the situation of many point charges and many spheres. In
this case only the exact equation, not the exact solution, is known Doerr .
According to the linear superposition principle, the induced surface charge on
each sphere may be computed by using one free charge at a time and then adding
up the contributions.
Let us consider $N$ dielectric spheres of various radii and dielectric
constants immersed inside a dielectric medium of dielectric constant
${\epsilon_{\rm o}}$. The location of sphere $i$ is ${\rm\bf R}_{i}$, its
radius is $a_{i}$, and its interior has dielectric constant $\epsilon_{i}$. No
two spheres are in contact with one another. There are $K$ point charges
$q_{i}$ located at ${\rm\bf g}_{i}$ so that the free charge density reads
${\rho_{\rm f}}({\rm\bf r})=\sum_{i=1}^{K}q_{i}\delta({\rm\bf r}-{\rm\bf
g}_{i})$. We assume that the variation of susceptibility in the vicinity of
each sphere is radial with respect to the center of that sphere:
$\nabla\chi({\rm\bf
r})=\sum_{i=1}^{N}\frac{\partial\chi}{\partial{\tilde{r}}_{i}}{\hat{\tilde{r}}}_{i}\equiv\sum_{i=1}^{N}\frac{\epsilon^{\prime}({\tilde{r}}_{i})}{4\pi}{\hat{\tilde{r}}}_{i}\;.$
(50)
Here and throughout this section we use the tilde sign to denote radius
vectors centered at the corresponding spheres, ${\rm\bf r}={\rm\bf
R}_{i}+{\tilde{\rm\bf r}}_{i}$.
From (16) we have
${\rho_{\rm t}}({\rm\bf r})={{\rho_{\rm f}}({\rm\bf r})\over\epsilon({\rm\bf
r})}-\sum_{i}\frac{\epsilon^{\prime}({\tilde{r}}_{i})}{4\pi\epsilon({\tilde{r}}_{i})}\int{\hat{\tilde{r}}}_{i}\cdot{{\rm\bf
r}-{\rm\bf r}^{\prime}\over|{\rm\bf r}-{\rm\bf
r}^{\prime}|^{3}}\rho_{t}({\rm\bf r}^{\prime})d{\rm\bf
r}^{\prime}\equiv{{\rho_{\rm f}}({\rm\bf r})\over\epsilon({\rm\bf
r})}-\sum_{i}[{\rm\bf C}_{i}\rho_{t}]({\rm\bf r})$ (51)
where $\sum_{i}{\rm\bf C}_{i}$ plays the role of ${\rm\bf C}$ in (17).
Concentrating on the equation associated with a particular sphere $k$, we
decompose ${\rho_{\rm t}}({\rm\bf r})$ as
${\rho_{\rm t}}({\rm\bf r})=\rho_{k}({\rm\bf r})+{{\rho_{\rm f}}({\rm\bf
r})\over\epsilon({\rm\bf r})}+\sum_{j\neq k}\rho_{j}({\rm\bf r}),$ (52)
where $\rho_{i}({\rm\bf r})$ is the total charge density near the surface of
sphere $i$. Since we consider nonoverlapping spheres, ${\rm\bf C}_{i}{\rm\bf
C}_{j}=0$ for $i\neq j$. Therefore, when focusing on a spatial point near
sphere $k$, the only contribution to the overall charge density is
$\rho_{k}({\rm\bf r})$, so ${\rho_{\rm t}}({\rm\bf r})=\rho_{k}({\rm\bf r})$
for ${\rm\bf r}$ sufficiently close to sphere $k$. Then in vicinity of sphere
$k$ the charge density becomes
$\rho_{k}({\rm\bf
r})=-\frac{\epsilon^{\prime}({\tilde{r}}_{k})}{4\pi\epsilon({\tilde{r}}_{k})}\int{\hat{\tilde{r}}}_{k}\cdot{{\rm\bf
r}-{\rm\bf r}^{\prime}\over|{\rm\bf r}-{\rm\bf
r}^{\prime}|^{3}}\left[\frac{{\rho_{\rm f}}({\rm\bf
r}^{\prime})}{\epsilon({\rm\bf r}^{\prime})}+\rho_{k}({\rm\bf
r}^{\prime})+\sum_{j\neq k}\rho_{j}({\rm\bf r}^{\prime})\right],$ (53)
which may be expressed symbolically as
$\left[{\rm\bf I}+{\rm\bf C}_{k}\right]\rho_{k}=-{\rm\bf
C}_{k}\left({\rho_{f}\over\epsilon}+\sum_{j\neq k}\rho_{j}\right)$ (54)
with
$C_{k}({\rm\bf r},{\rm\bf
r}^{\prime})=\frac{\epsilon^{\prime}({\tilde{r}}_{k})}{4\pi\epsilon({\tilde{r}}_{k})}{\hat{\tilde{r}}}_{k}\cdot{{\rm\bf
r}-{\rm\bf r}^{\prime}\over|{\rm\bf r}-{\rm\bf r}^{\prime}|^{3}}\;.$ (55)
This implies a symbolic solution for $\rho_{k}$
$\rho_{k}=-\left[{\rm\bf I}-{\rm\bf C}_{k}+{\rm\bf C}_{k}^{2}-{\rm\bf
C}_{k}^{3}+\cdots\right]{\rm\bf C}_{k}\left({{\rho_{\rm
f}}\over\epsilon}+\sum_{j\neq k}\rho_{j}\right).$ (56)
Notice that the solution for the series acting on the free charges part will
be essentially the same as that for the one sphere problem dealt with in the
previous subsection. Let us consider ${\rm\bf C}_{k}\,\rho_{j\neq k}$.
${\rm\bf C}_{k}\,\rho_{j}={\epsilon^{\prime}({\tilde{r}}_{k})\over
4\pi\epsilon({\tilde{r}}_{k})}{\hat{\tilde{r}}}_{k}\cdot\int{{\rm\bf
r}-{\rm\bf r}^{\prime}\over|{\rm\bf r}-{\rm\bf
r}^{\prime}|^{3}}\rho_{j}({\rm\bf r}^{\prime})d{\rm\bf r}^{\prime}$ (57)
We switch to vectors centered on the corresponding spheres so that the final
expression is in terms of the local polar angle of ${\tilde{\rm\bf r}}_{k}$,
which allows easier manipulation later. In this notation,
$\displaystyle{\rm\bf C}_{k}\,\rho_{j}$ $\displaystyle=$
$\displaystyle{\epsilon^{\prime}({\tilde{r}}_{k})\over
4\pi\epsilon({\tilde{r}}_{k})}{\hat{r}_{k}}\cdot\int{{\tilde{\rm\bf
r}}_{k}-({\tilde{\rm\bf r}}^{\prime}_{j}-{\rm\bf L}_{j\to
k})\over|{\tilde{\rm\bf r}}_{k}-({\tilde{\rm\bf r}}^{\prime}_{j}-{\rm\bf
L}_{j\to k})|^{3}}\rho_{j}({\tilde{\rm\bf r}}^{\prime}_{j})d{\tilde{\rm\bf
r}}^{\prime}_{j}$ (58) $\displaystyle=$
$\displaystyle-{\epsilon^{\prime}({\tilde{r}}_{k})\over
4\pi\epsilon({\tilde{r}}_{k})}\;\partial_{r_{k}}\\!\\!\int{1\over|{\tilde{\rm\bf
r}}_{k}-({\tilde{\rm\bf r}}^{\prime}_{j}-{\rm\bf L}_{j\to
k})|}\rho_{j}({\tilde{\rm\bf r}}^{\prime}_{j})d{\tilde{\rm\bf
r}}^{\prime}_{j}$
where ${\rm\bf L}_{j\to k}\equiv{\rm\bf R}_{k}-{\rm\bf R}_{j}=-{\rm\bf
L}_{k\to j}$ represents the vector pointing from the center of sphere $j$ to
that of sphere $k$. Using the expansion (33), we obtain
${\rm\bf C}_{k}\,\rho_{j}({\rm\bf
r})=-{\epsilon^{\prime}({\tilde{r}}_{k})\over
4\pi\epsilon({\tilde{r}}_{k})}\sum_{lm}{4\pi l\over
2l+1}({\tilde{r}}_{k})^{l-1}Y_{lm}({\hat{\tilde{r}}}_{k})\int\frac{Y_{lm}^{*}({{\tilde{\rm\bf
r}}^{\prime}_{j}-{\rm\bf L}_{j\to k}\over|{\tilde{\rm\bf
r}}^{\prime}_{j}-{\rm\bf L}_{j\to k}|})}{|{\tilde{\rm\bf
r}}^{\prime}_{j}-{\rm\bf L}_{j\to k}|^{l+1}}\rho_{j}({\tilde{\rm\bf
r}}^{\prime}_{j})d{\tilde{\rm\bf r}}^{\prime}_{j}\;.$ (59)
The angular integral in the above equation was solved by Yu Yu and employed
in Doerr where $\rho_{j}\propto\delta({\tilde{r}}_{j}-a_{j})$. The process
for calculating ${\rm\bf C}_{k}^{n}\,\rho_{j}$ is not affected by the detailed
result of the integration. For now, it is sufficient to point out that the
integral gives rise to a geometrical factor with some factorials multiplied by
the multipole moment $Q^{j}_{lm}$ of the surface charge distribution of sphere
$j$. Denoting the integral by $\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to k})$,
$\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to
k})\equiv\int{Y_{lm}^{*}({{\tilde{\rm\bf r}}^{\prime}_{j}-{\rm\bf L}_{j\to
k}\over|{\tilde{\rm\bf r}}^{\prime}_{j}-{\rm\bf L}_{j\to
k}|})\over|{\tilde{\rm\bf r}}^{\prime}_{j}-{\rm\bf L}_{j\to
k}|^{l+1}}\rho_{j}({\tilde{\rm\bf r}}^{\prime}_{j})d{\tilde{\rm\bf
r}}^{\prime}_{j}\;,$ (60)
we may then write
${\rm\bf C}_{k}\,\rho_{j}({\rm\bf
r})=-{\epsilon^{\prime}({\tilde{r}}_{k})\over
4\pi\epsilon({\tilde{r}}_{k})}\sum_{lm}{4\pi\over
2l+1}l\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to
k}){\tilde{r}}_{k}^{l-1}Y_{lm}({\hat{\tilde{r}}}_{k})\;.$ (61)
For the case of sharp boundaries between the spheres and the external medium,
one then obtains
${\rm\bf C}_{k}\,\rho_{j}({\rm\bf r})=-{{\epsilon_{\rm o}}-\epsilon_{k}\over
4\pi\epsilon(a_{k})}\delta({\tilde{r}}_{k}-a_{k})\sum_{lm}{4\pi\over
2l+1}\left[la_{k}^{l-1}\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to
k})\right]Y_{lm}({\hat{\tilde{r}}}_{k})\;.$ (62)
Applying the ${\rm\bf C}_{k}$ operator once again and performing the
integration in the radial direction, we find
${\rm\bf C}_{k}^{2}\,\rho_{j}({\rm\bf r})=-\left({{\epsilon_{\rm
o}}-\epsilon_{k}\over
4\pi\epsilon(a_{k})}\right)^{2}\\!\\!{\delta({\tilde{r}}_{k}-a_{k})\over
2}\int{d{\hat{\tilde{r}}}^{\prime}_{k}\over|2-2{\hat{\tilde{r}}}_{k}\cdot{\hat{\tilde{r}}}^{\prime}_{k}|^{1/2}}\sum_{lm}{4\pi\over
2l+1}\left[la_{k}^{l-1}\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to
k})\right]Y_{lm}({\hat{\tilde{r}}}^{\prime}_{k})$ (63)
After performing the angular integration, ${\rm\bf
C}_{k}^{2}\,\rho_{j}({\rm\bf r})$ becomes
${\rm\bf C}_{k}^{2}\,\rho_{j}({\rm\bf r})=-\left({{\epsilon_{\rm
o}}-\epsilon_{k}\over
4\pi\epsilon(a_{k})}\right)\delta({\tilde{r}}_{k}-a_{k})\sum_{lm}\left({{\epsilon_{\rm
o}}-\epsilon_{k}\over 2\epsilon(a_{k})(2l+1)}\right){4\pi\over
2l+1}\left[la_{k}^{l-1}\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to
k})\right]Y_{lm}({\hat{\tilde{r}}}_{k})$ (64)
It is easy to see that this process continues and one ends up having
${\rm\bf C}_{k}^{n}\,\rho_{j}({\rm\bf r})=-\left({{\epsilon_{\rm
o}}-\epsilon_{k}\over
4\pi\epsilon(a_{k})}\right)\\!\\!\delta({\tilde{r}}_{k}-a_{k})\sum_{lm}\left({{\epsilon_{\rm
o}}-\epsilon_{k}\over
2\epsilon(a_{k})(2l+1)}\right)^{n-1}\\!\\!\\!\\!\\!\\!{4\pi\over
2l+1}\left[la_{k}^{l-1}\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to
k})\right]Y_{lm}({\hat{\tilde{r}}}_{k})$ (65)
and therefore
$\displaystyle\sum_{n=1}^{\infty}(-{\rm\bf C}_{k})^{n}\rho_{j}$
$\displaystyle=$ $\displaystyle-\left({{\epsilon_{\rm o}}-\epsilon_{k}\over
4\pi\epsilon(a_{k})}\right)\delta({\tilde{r}}_{k}-a_{k})\times$ (66)
$\displaystyle\times\sum_{lm}\left[\sum_{n=1}^{\infty}(-1)^{n}\\!\left({{\epsilon_{\rm
o}}-\epsilon_{k}\over
2\epsilon(a_{k})(2l+1)}\right)^{n-1}\\!\right]\\!\\!{4\pi\over
2l+1}\left[la_{k}^{l-1}\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to
k})\right]Y_{lm}({\hat{\tilde{r}}}_{k})$ $\displaystyle=$
$\displaystyle-\left({{\epsilon_{\rm o}}-\epsilon_{k}\over
4\pi}\right)\delta({\tilde{r}}_{k}-a_{k})\times$
$\displaystyle\times\sum_{lm}\left[{(2l+1)\over(l+1){\epsilon_{\rm
o}}+l\epsilon_{k}}\right]{4\pi\over
2l+1}\left[la_{k}^{l-1}\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to
k})\right]Y_{lm}({\hat{\tilde{r}}}_{k})\;,$
where $\epsilon(a_{k})=({\epsilon_{\rm o}}+\epsilon_{k})/2$ is used. We are
now in a position to write down the full solution using (45), (46), and (66).
Defining ${\cal I}_{k}\equiv\left\\{i\left|a_{k}>|{\rm\bf g}_{i}-{\rm\bf
R}_{k}|\right.\right\\}$ and ${\cal O}_{k}\equiv\left\\{i\left|a_{k}<|{\rm\bf
g}_{i}-{\rm\bf R}_{k}|\right.\right\\}$ to be the sets of charges inside and
outside sphere $k$, respectively, we find
$\displaystyle\rho_{k}({\tilde{\rm\bf r}}_{k})$ $\displaystyle=$
$\displaystyle-\sum_{{\cal I}_{k}}{q_{i}\over\epsilon_{k}}\left({\epsilon_{\rm
o}}-\epsilon_{k}\right)\delta({\tilde{r}}_{k}-a_{k})\sum_{lm}{(l+1)\over\left[(l+1){\epsilon_{\rm
o}}+l\epsilon_{k}\right]}{|{\rm\bf g}_{i}-{\rm\bf R}_{k}|^{l}\over
a_{k}^{l+2}}Y^{*}_{lm}\left(\frac{{\rm\bf g}_{i}-{\rm\bf R}_{k}}{|{\rm\bf
g}_{i}-{\rm\bf R}_{k}|}\right)Y_{lm}({\hat{\tilde{r}}}_{k})$ (67)
$\displaystyle+$ $\displaystyle\sum_{{\cal O}_{k}}{q_{i}\over\epsilon({\rm\bf
g}_{i})}\left({\epsilon_{\rm
o}}-\epsilon_{k}\right)\delta({\tilde{r}}_{k}-a_{k})\sum_{lm}{l\over\left[(l+1){\epsilon_{\rm
o}}+l\epsilon_{k}\right]}{a_{k}^{l-1}\over|{\rm\bf g}_{i}-{\rm\bf
R}_{k}|^{l+1}}Y^{*}_{lm}\left(\frac{{\rm\bf g}_{i}-{\rm\bf R}_{k}}{|{\rm\bf
g}_{i}-{\rm\bf R}_{k}|}\right)Y_{lm}({\hat{\tilde{r}}}_{k})$ $\displaystyle-$
$\displaystyle\sum_{j\neq k}\left({{\epsilon_{\rm o}}-\epsilon_{k}\over
4\pi}\right)\delta({\tilde{r}}_{k}-a_{k})\sum_{lm}\left[{(2l+1)\over(l+1){\epsilon_{\rm
o}}+l\epsilon_{k}}\right]{4\pi\over
2l+1}\left[la_{k}^{l-1}\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to
k})\right]Y_{lm}({\hat{\tilde{r}}}_{k})$
which, with appropriate rotations and taking a single point charge at the
center of each sphere, is equivalent to (11) in Doerr .
## IV Numerical case study
In this section we present results of numerical computations comparing the
force between two charged identical spheres with sharp boundaries to the force
between two charged identical spheres with smeared boundaries. For brevity,
the spheres with smeared boundaries will be called “fuzzy spheres” and the
spheres with sharp boundaries will be called “rigid spheres”. The dielectric
constant $\epsilon_{1}=4$ inside the spheres and ${\epsilon_{\rm o}}=80$
outside. For the fuzzy spheres there is an interface region $r_{0}-\delta
r<r<r_{0}+\delta r$ in which the dielectric constant changes smoothly from
$\epsilon_{1}$ to ${\epsilon_{\rm o}}$ in the radial direction (with respect
to the center of the corresponding sphere).
The simplest polynomial smoothly connecting $\epsilon_{1}$ and ${\epsilon_{\rm
o}}$, i.e., satisfying the conditions $\epsilon(r_{0}-\delta r)=\epsilon_{1}$,
$\epsilon(r_{0}+\delta r)={\epsilon_{\rm o}}$, $\epsilon^{\prime}(r_{0}-\delta
r)=\epsilon^{\prime}(r_{0}+\delta r)=0$, is cubic, so that the dielectric
constant can be defined around each sphere as
$\displaystyle\epsilon(r)$ $\displaystyle=$ $\displaystyle\epsilon_{1},\qquad
r<r_{0}-\delta r$ $\displaystyle\epsilon(r)$ $\displaystyle=$
$\displaystyle\left[\frac{\left(r-r_{0}\right)^{3}}{\delta
r^{3}}-3\frac{r-r_{0}}{\delta r}\right]\frac{\epsilon_{1}-{\epsilon_{\rm
o}}}{4}+\frac{\epsilon_{1}+{\epsilon_{\rm o}}}{2},\qquad r_{0}-\delta r\leq
r\leq r_{0}+\delta r$ $\displaystyle\epsilon(r)$ $\displaystyle=$
$\displaystyle{\epsilon_{\rm o}},\qquad r>r_{0}+\delta r.$ (68)
With a fifth order polynomial one can request additionally that
$\epsilon(r_{0}+\delta r_{\rm H})=\epsilon_{\rm H}$ and
$\epsilon^{\prime}(r_{0}+\delta r_{\rm H})=0$. Letting $\epsilon_{\rm H}=70$
and $\delta r_{\rm H}=0.5\delta r$ yields a non-monotonic profile, which may
be used to simulate the hydration layer phenomenon in bio-macromolecules and
clusters (see Fig. 1).
Figure 1: Radial dependence of (a) the dielectric constant $\epsilon(r/a)$
and (b) $\epsilon^{\prime}(r/a)/(4\pi\epsilon(r/a))$ for a monotonic step (red
broken line, Eq. (68)) and for a non-monotonic step simulating a hydration
layer (blue solid line). The dielectric constant changes smoothly from
$\epsilon_{1}=4$ inside the sphere to ${\epsilon_{\rm o}}=80$ outside. The
effective radii $r_{0}=1.13a$ and $r_{0}=1.17a$, respectively, are chosen so
that the Born solvation energy in each case is equal to that in the case of a
sharp boundary at radius $a$ (shown with dotted line). The half-width of the
steps $\delta r=0.2a$.
Let there be point charges $q_{1}$ and $q_{2}$ at the centers of spheres 1 and
2, respectively. The induced charge density is found for rigid spheres as the
self-consistent solution of (67) for $\rho_{1}({\tilde{r}}_{1})$ and
$\rho_{2}({\tilde{r}}_{2})$. Of course, (67) simplifies dramatically in the
case of two spheres and two free charges. For fuzzy spheres, one has to use a
continuous version of (67) in which summation over $n$ in (66) is carried out
numerically with the $n^{\rm th}$-order terms (65) calculated recursively via
numerical integration, analogously to the method for a point charge outside a
sphere, see (38), (42) and (44). Notice that the $l=0$ components of the
induced densities can only be produced by the free charge inside the
corresponding sphere. Notice also that the free charges in the centers of the
spheres induce only $l=0$, i.e. spherically-symmetric, components. For these
reasons it is convenient to distinguish the $l=0$ and $l\neq 0$ components of
the induced charge density.
In accordance with (8), the total energy of the system consists of the
following terms:
(i) interaction between the point charges (screened by $\epsilon_{1}$),
$\frac{q_{1}q_{2}}{\epsilon_{1}L},$ (69)
where $L$ is the length of the vector ${\rm\bf L}_{1\to 2}=-{\rm\bf L}_{2\to
1}$, connecting the centers of the two spheres,
(iia) interaction between each point charge and the $l=0$ component of the
induced charge in the interface region of the other sphere,
$\frac{1}{2}\left(q_{1}\int\frac{\left.\rho_{2}({\tilde{\rm\bf
r}}_{2})\right|_{l=0}}{|{\tilde{\rm\bf r}}_{2}+{\rm\bf L}_{1\to
2}|}\,d{\tilde{\rm\bf r}}_{2}+q_{2}\int\frac{\left.\rho_{1}({\tilde{\rm\bf
r}}_{1})\right|_{l=0}}{|{\tilde{\rm\bf r}}_{1}+{\rm\bf L}_{2\to
1}|}\,d{\tilde{\rm\bf r}}_{1}\right),$ (70)
(iib) interaction between each point charge and the $l\neq 0$ components of
the induced charge in the interface region of the other sphere,
$\frac{1}{2}\left(q_{1}\int\frac{\left.\rho_{2}({\tilde{\rm\bf
r}}_{2})\right|_{l\neq 0}}{|{\tilde{\rm\bf r}}_{2}+{\rm\bf L}_{1\to
2}|}\,d{\tilde{\rm\bf r}}_{2}+q_{2}\int\frac{\left.\rho_{1}({\tilde{\rm\bf
r}}_{1})\right|_{l\neq 0}}{|{\tilde{\rm\bf r}}_{1}+{\rm\bf L}_{2\to
1}|}\,d{\tilde{\rm\bf r}}_{1}\right),$ (71)
(iiia) interaction between each point charge and the $l=0$ component of the
induced charge in the interface region of the same sphere,
$\frac{1}{2}\left(q_{1}\int\frac{\left.\rho_{1}({\tilde{\rm\bf
r}}_{1})\right|_{l=0}}{{\tilde{r}}_{1}}\,d{\tilde{\rm\bf
r}}_{1}+q_{2}\int\frac{\left.\rho_{2}({\tilde{\rm\bf
r}}_{2})\right|_{l=0}}{{\tilde{r}}_{2}}\,d{\tilde{\rm\bf r}}_{2}\right),$ (72)
(iiib) interaction between each point charge and the $l\neq 0$ components of
the induced charge in the interface region of the same sphere,
$\frac{1}{2}\left(q_{1}\int\frac{\left.\rho_{1}({\tilde{\rm\bf
r}}_{1})\right|_{l\neq 0}}{{\tilde{r}}_{1}}\,d{\tilde{\rm\bf
r}}_{1}+q_{2}\int\frac{\left.\rho_{2}({\tilde{\rm\bf r}}_{2})\right|_{l\neq
0}}{{\tilde{r}}_{2}}\,d{\tilde{\rm\bf r}}_{2}\right),$ (73)
The sum of terms (i) and (iia) is equal to the energy of interaction of two
point charges in dielectric medium ${\epsilon_{\rm o}}$
$\frac{q_{1}q_{2}}{{\epsilon_{\rm o}}L}.$ (74)
This energy is the same for rigid and fuzzy spheres. In contrast, terms (iib)
are different for rigid and fuzzy spheres and are the main source of
differences in the forces in these two situations. Finally, terms (iiib) are
zero for the point charges located at the centers of the spheres, while terms
(iiia) are the Born solvation energy in this case.
Born solvation energies are quite different for rigid and fuzzy spheres, since
for fuzzy spheres the induced charge density tends to accumulate near the
inner boundary of the interface region. Indeed, the operator ${\rm\bf C}$ is
proportional to $\epsilon^{\prime}(r)/\epsilon(r)$ and $\epsilon(r_{0}-\delta
r)=\epsilon_{1}\ll\epsilon(r_{0}+\delta r)={\epsilon_{\rm o}}$. This asymmetry
is present at each order $n$ and is preserved after the summation over $n$.
Radial dependences of the $l=0$ components of the induced densities are
illustrated in Fig. 2. On the other hand, fuzzy and rigid spheres model the
same physical objects, so it is reasonable to assume that whatever profile of
the dielectric constant is chosen, the Born solvation energy should remain the
same. For this reason, we adjust the effective radius $r_{0}$ for each profile
of the dielectric constant so that the Born solvation energy is equal to that
of a rigid sphere of unit radius, see Fig. 1.
Figure 2: Radial dependence of the induced electric density
$\left.\rho(r/a)\right|_{l=0}$ for the monotonic (red broken line) and non-
monotonic (blue solid line) steps shown in Fig. 1. The density is normalized
by the value of the point charge in the center of the sphere. The inset
magnifies a small, oscillatory feature associated with the non-monotonic step.
In Fig. 3 we present the dependence of the interaction energy on distance for
a pair of rigid spheres and for two pairs of fuzzy spheres, with monotonic and
non-monotonic behaviour of the dielectric function in the interface region,
respectively. The energies are normalized to the energy of interaction of
point charges (74). The forces between two fuzzy spheres and between two rigid
spheres are shown in Fig. 4. The forces are normalized by the interaction
force between two point charges. We note that the seemingly weaker effect for
the fuzzy spheres with non-monotonic $\epsilon(r)$ dependence is due to the
fact that $\epsilon(r)$ changes faster near the inner surface of the interface
region to make room for the feature representing the hydration layer. This
makes the fuzzy spheres with non-monotonic $\epsilon(r)$ dependence
effectively more similar to rigid spheres for fixed $\delta r$ (compare the
charge density distributions in Fig. 2).
Figure 3: Energy of interaction between two spheres with sharp (thin line)
and smeared (thick lines) boundaries. The red broken thick line corresponds to
the case of the monotonic radial dependence of the dielectric constant, while
the blue solid thick line corresponds to the non-monotonic radial dependence
shown in Fig. 1. Free charges of the same sign are located at the centers of
the spheres. The energies are normalized by the Coulomb energy of these point
charges in the uniform dielectric medium ${\epsilon_{\rm o}}$. The vertical
dotted lines indicate the contact points. Figure 4: Interaction forces
between two spheres with sharp and smeared boundaries. The line
identifications are same as in Fig. 3.
For very thin interface regions ($\delta r\to 0$), the forces between two
rigid and two fuzzy spheres are equal, as expected. For fuzzy spheres with
moderate interface region widths, the repulsion increases with the width.
However, this trend quickly saturates (Fig. 5). Qualitatively, this saturation
can be explained by two opposing effects. The increase in the interface width
increases the size of the spheres thereby strengthening the repulsion. On the
other hand, the induced charge density tends to concentrate near the inner
surface of the interface which remains around $r=a$ to maintain constant Born
solvation energy. Therefore, the bulk of the induced charge on one sphere
becomes farther from that of the other sphere, hence weakening the repulsion.
Figure 5: Maximum difference in interaction forces between two spheres with
smeared and two spheres with sharp boundaries, occuring at the contact point
$2(r_{0}+\delta r)$, as a function of half width of the interface region
$\delta r$. The forces are normalized by the interaction force between two
spheres with sharp boundaries. The red broken line corresponds to the case of
the monotonic radial dependence of the dielectric constant, while the blue
solid line corresponds to the non-monotonic radial dependence shown in Fig. 1.
We finally note, that if the point charges are located away from the centers
of the spheres, the terms (iiib) depend on the relative position and
orientation of the spheres. In this case one can still define the Born
solvation energies as the sum of terms (iiia) and (iiib) at large separations,
but the terms (iiib) would contribute to the difference of interaction
forces/energies between the rigid and fuzzy spheres.
## V Conclusions
We have presented an energy minimization formulation of electrostatics that
allows computation of the electrostatic energy and forces to any desired
accuracy in a system with arbitrary dielectric properties. We have derived an
integral equation for the scalar charge density from an energy functional of
the polarization vector field. This energy functional represents the true
energy of the system even in non-equilibrium states. Arbitrary accuracy is
achieved by solving the integral equation for the charge density via a series
expansion in terms of the equation’s kernel, which depends only on the
geometry of the dielectrics. The streamlined formalism operates with volume
charge distributions only, not resorting to introducing surface charges by
hand as is done in various other studies of electrostatics via energy
minimization. Therefore, it can be applied to arbitrary spatial variations of
the dielectric susceptibility. The simplicity of application of the formalism
to real problems has been shown with three analytic examples and with a
numerical case study. We found that finite boundary widths introduce a
measurable correction to the interaction forces as compared to sharp boundary
case. For two charged identical spheres the correction is about 10%.
The formalism has various potential applications in modeling electrostatic
interactions between solvated molecules: it enables one to go beyond the
widely used simplification of atoms and molecules as dielectric balls immersed
in a dielectric solvent, as was first suggested by Born in the early twenties
of the last century Born . For example, the description of an aqueous solvent
as a continuous and homogeneous dielectric medium fails to account for the
strong dielectric response of water molecules around charges. Normally,
charged ions and surfaces give rise to hydration layers by orienting and
displacing surrounding water molecules. These hydration phenomena are very
important in many biological processes such as protein folding, protein
crystallization, and interactions between charged biopolymers inside the cell.
With our formalism one can now consider arbitrary structures for such
hydration layers and arrive at a possibly more realistic and reliable analysis
of the molecular mechanisms in bio-chemical interactions.
Applied to MD simulations, this formulation is still an implicit solvent
scheme, and the position-dependent susceptibility is therefore a model
parameter (indeed, the only one). To obtain an estimate of the macroscopic
dielectric susceptibility at the molecular level or at the intermolecular
boundaries one has to explore physics at the atomic level and introduce some
coarse graining. Given that the dielectric susceptibility is related to the
charge fluctuations as a response to external perturbations, one can estimate
susceptibilites through the study of linear/nonlinear response. For example,
the dielectric susceptibility can be related to the correlations of the net
system dipole moment and local polarization density Stern . A fully quantum
mechanical treatment of solvation of biological systems might be hindered by
limits of numerical accuracy Kohn.Nobel and will demand much more
computational power than currently available. We believe that quantum
mechanics, in particular, density functional theory, can in principle be used
to calculate the local dielectric susceptibility which in turn should be used
as input for the implicit solvent methods, such as the one described in this
paper.
## Acknowledgements
This research was supported by the Intramural Research Program of the NIH,
NLM. The computations were performed on the Biowulf Linux cluster at the
National Institutes of Health, Bethesda, MD (http://biowulf.nih.gov).
## Appendix A Sharp boundary limit in the planar interface problem
Let us demonstrate how a rigorous limiting procedure applied to (25) produces
correct expression for the surface charge density in the case of sharp planar
interface. The surface charge is found by integrating the charge density over
the range $-a\leq z\leq a$ in which $\chi$ changes from ${\chi_{2}}$ to
${\chi_{1}}$, and then taking the limit $a\rightarrow 0$.
We return to (24) and, making use of the azimuthal symmetry of the problem,
expand the kernels in terms of Bessel functions $J_{m}$ Jackson ,
$\displaystyle\frac{1}{|{\rm\bf r}-{\rm\bf r}^{\prime}|}$ $\displaystyle=$
$\displaystyle\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}e^{i\,m(\phi-\phi^{\prime})}\,J_{m}(k\rho)\,J_{m}(k\rho^{\prime})e^{-k(z_{>}-z_{<})}dk$
(75) $\displaystyle\frac{1}{|{\rm\bf r}-d\,\hat{z}|}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}J_{0}(k\rho)e^{-k(d-z)}dk.$ (76)
Here the position vectors ${\rm\bf r}$ and ${\rm\bf r}^{\prime}$ are
represented via the polar vectors $\boldsymbol{\rho}$ and
$\boldsymbol{\rho}^{\prime}$ in the $z=0$ plane, ${\rm\bf
r}=\boldsymbol{\rho}+z\hat{z}$ and ${\rm\bf
r}^{\prime}=\boldsymbol{\rho}^{\prime}+z^{\prime}\hat{z}$. The polar vectors
are in turn defined through their lengths $\rho=\sqrt{x^{2}+y^{2}}$ and
$\rho^{\prime}=\sqrt{x^{\prime 2}+y^{\prime 2}}$ and their azimuthal angles
$\phi$ and $\phi^{\prime}$. The notation $z_{>}$ ($z_{<}$) is used for the
greater (lesser) of the corresponding $z$ and $z^{\prime}$.
We now treat each of the terms in the expansion of (24) separately. The first
term is the screened point charge. All other terms form the induced charge
density at the interfacial region. The first contribution to the induced
charge density is given by
${\rho_{\rm i}}^{(1)}({\rm\bf
r})=-\frac{q}{\epsilon_{1}}\frac{\epsilon^{\prime}(z)}{4\pi\epsilon(z)}\frac{z-d}{|{\rm\bf
r}-d\,\hat{z}|^{3}}.$ (77)
The corresponding surface charge density is
${\sigma_{\rm
i}}^{(1)}(\boldsymbol{\rho})=-\frac{q}{\epsilon_{1}}\lim_{a\rightarrow
0}\int_{-a}^{a}\frac{\epsilon^{\prime}(z)}{4\pi\epsilon(z)}\left[\frac{-d}{|\boldsymbol{\rho}-d\,\hat{z}|^{3}}+{\cal
O}(z)\right]dz.$ (78)
All the ${\cal O}(z)$ terms vanish since for any bounded function $h(z)$
$\lim_{a\rightarrow 0}\int_{-a}^{a}z^{n}h(z)dz\leq\lim_{a\rightarrow
0}a^{n}\int_{-a}^{a}|h(z)|dz=0,\;\;\;\;\forall\;\;n>0.$ (79)
Thus,
${\sigma_{\rm
i}}^{(1)}(\boldsymbol{\rho})=\frac{q}{4\pi\epsilon_{1}}\frac{d}{|\boldsymbol{\rho}-d\,\hat{z}|^{3}}(f_{1}-f_{2}).$
(80)
Here we have used the notations $f(z)=\ln[\epsilon(z)]$,
$f_{1}=f(a)=\ln[\epsilon_{1}]$ and $f_{2}=f(-a)=\ln[\epsilon_{2}]$.
We can similarly evaluate all the other contributions to the induced surface
charge density. The second contribution to the induced charge density is
${\rho_{\rm i}}^{(2)}({\rm\bf
r})=\frac{q}{\epsilon_{1}}\frac{\epsilon^{\prime}(z)}{4\pi\epsilon(z)}\int\frac{z-z^{\prime}}{|{\rm\bf
r}-{\rm\bf
r}^{\prime}|^{3}}\frac{\epsilon^{\prime}(z^{\prime})}{4\pi\epsilon(z^{\prime})}\frac{z^{\prime}-d}{|{\rm\bf
r}^{\prime}-d\,\hat{z}|^{3}}\rho^{\prime}d\rho^{\prime}d\phi^{\prime}dz^{\prime}$
(81)
Using (75), (76), and the completeness relation for Bessel functions Jackson ,
$\int_{0}^{\infty}J_{m}(k\rho)J_{m}(k^{\prime}\rho)\rho
d\rho=\frac{1}{k}\delta(k-k^{\prime}),$ (82)
we obtain, after integration over $\phi^{\prime}$ and $\rho^{\prime}$,
$\displaystyle{\rho_{\rm i}}^{(2)}({\rm\bf r})$ $\displaystyle=$
$\displaystyle\frac{q}{\epsilon_{1}}\frac{\epsilon^{\prime}(z)}{4\pi\epsilon(z)}\frac{d}{dz}\int\frac{\epsilon^{\prime}(z^{\prime})}{4\pi\epsilon(z^{\prime})}\int_{0}^{\infty}J_{0}(k\rho)e^{-k(d-z^{\prime})}e^{-k(z_{>}-z_{<})}2\pi
dkdz^{\prime}$ (83) $\displaystyle=$
$\displaystyle\frac{q}{\epsilon_{1}}\frac{\epsilon^{\prime}(z)}{4\pi\epsilon(z)}\frac{1}{2}\int_{0}^{\infty}kdke^{-k(d-z)}J_{0}(k\rho)\times$
$\displaystyle\hskip
72.26999pt\left[\int_{z}^{a}\frac{\epsilon^{\prime}(z^{\prime})}{\epsilon(z^{\prime})}dz^{\prime}-\int_{-a}^{z}\frac{\epsilon^{\prime}(z^{\prime})}{\epsilon(z^{\prime})}e^{-2k(z-z^{\prime})}dz^{\prime}\right].$
The corresponding surface charge density is then
$\displaystyle{\sigma_{\rm i}}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{q}{\epsilon_{1}}\lim_{a\rightarrow
0}\int_{-a}^{a}dz\frac{\epsilon^{\prime}(z)}{4\pi\epsilon(z)}\frac{1}{2}\int_{0}^{\infty}kdke^{-k(d-z)}J_{0}(k\rho)\times$
(84) $\displaystyle\hskip
72.26999pt\left[\int_{z}^{a}\frac{\epsilon^{\prime}(z^{\prime})}{\epsilon(z^{\prime})}dz^{\prime}-\int_{-a}^{z}\frac{\epsilon^{\prime}(z^{\prime})}{\epsilon(z^{\prime})}e^{-2k(z-z^{\prime})}dz^{\prime}\right].$
Applying to (84) the same argument used in deriving (80),
${\sigma_{\rm
i}}^{(2)}=\frac{q}{\epsilon_{1}}\int_{0}^{\infty}kdke^{-kd}J_{0}(k\rho)\lim_{a\rightarrow
0}\int_{-a}^{a}dz\frac{\epsilon^{\prime}(z)}{4\pi\epsilon(z)}\frac{1}{2}\left[\int_{z}^{a}\frac{\epsilon^{\prime}(z^{\prime})}{\epsilon(z^{\prime})}dz^{\prime}-\int_{-a}^{z}\frac{\epsilon^{\prime}(z^{\prime})}{\epsilon(z^{\prime})}dz^{\prime}\right].$
(85)
The integral over $k$ is evaluated using (76) as
$\int kJ_{0}(k\rho)e^{-kd}dk=\left.\frac{d}{dz}\int
J_{0}(k\rho)e^{-k(d-z)}dk\right|_{z=0}=\left.\frac{d}{dz}\frac{1}{|{\rm\bf
r}-d\,\hat{z}|}\right|_{z=0}=\frac{d}{|\boldsymbol{\rho}-d\,\hat{z}|^{3}}.$
(86)
Then
${\sigma_{\rm
i}}^{(2)}=\frac{q}{\epsilon_{1}}\frac{d}{|\boldsymbol{\rho}-d\,\hat{z}|^{3}}\lim_{a\rightarrow
0}\int_{-a}^{a}dz\frac{\epsilon^{\prime}(z)}{4\pi\epsilon(z)}\left[\frac{1}{2}(f_{1}+f_{2})-f(z)\right].$
(87)
Finally, we obtain that ${\sigma_{\rm i}}^{(2)}=0$,
${\sigma_{\rm
i}}^{(2)}=\frac{q}{4\pi\epsilon_{1}}\frac{d}{|\boldsymbol{\rho}-d\,\hat{z}|^{3}}\int_{f_{2}}^{f_{1}}df\left[\frac{1}{2}(f_{1}+f_{2})-f(z)\right]=0.$
(88)
Analogously, the expressions for the induced surface charge densities up to
the fifth order are found to be
$\displaystyle{\sigma_{\rm i}}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{q}{4\pi\epsilon_{1}}\frac{d}{|\boldsymbol{\rho}-d\,\hat{z}|^{3}}(f_{1}-f_{2}),$
$\displaystyle{\sigma_{\rm i}}^{(2)}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle{\sigma_{\rm i}}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{q}{4\pi\epsilon_{1}}\frac{d}{|\boldsymbol{\rho}-d\,\hat{z}|^{3}}\frac{-1}{12}(f_{1}-f_{2})^{3},$
$\displaystyle{\sigma_{\rm i}}^{(4)}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle{\sigma_{\rm i}}^{(5)}$ $\displaystyle=$
$\displaystyle\frac{q}{4\pi\epsilon_{1}}\frac{d}{|\boldsymbol{\rho}-d\,\hat{z}|^{3}}\frac{1}{120}(f_{1}-f_{2})^{5}.$
(89)
In general, the surface charge density is of the form
$\displaystyle{\sigma_{\rm i}}^{(n)}(z)$ $\displaystyle=$
$\displaystyle-\frac{q}{\epsilon_{1}}\lim_{a\rightarrow
0}\int_{-a}^{a}dz\frac{\epsilon^{\prime}(z)}{4\pi\epsilon(z)}\frac{z-d}{|{\rm\bf
r}-d\,\hat{z}|^{3}}\times$ (90)
$\displaystyle\qquad\frac{1}{2}\left[\int_{z}^{a}\frac{\epsilon^{\prime}(z^{\prime})}{\epsilon(z^{\prime})}g^{(n-1)}(f(z^{\prime}))dz^{\prime}-\int_{-a}^{z}\frac{\epsilon^{\prime}(z^{\prime})}{\epsilon(z^{\prime})}g^{(n-1)}(f(z^{\prime}))dz^{\prime}\right]$
$\displaystyle=$ $\displaystyle-\frac{q}{\epsilon_{1}}\lim_{a\rightarrow
0}\frac{1}{2}\int_{-a}^{a}dz\frac{\epsilon^{\prime}(z)}{4\pi\epsilon(z)}\frac{z-d}{|{\rm\bf
r}-d\,\hat{z}|^{3}}g^{(n)}(f(z))$ $\displaystyle=$
$\displaystyle\frac{q}{4\pi\epsilon_{1}}\frac{d}{|\boldsymbol{\rho}-d\,\hat{z}|^{3}}\int_{f_{2}}^{f_{1}}g^{(n)}(f)df.$
The functions $g^{(n)}(f(z))$ up to $n=5$ are
$\displaystyle g^{(1)}(f(z))$ $\displaystyle=$ $\displaystyle 1$
$\displaystyle g^{(2)}(f(z))$ $\displaystyle=$
$\displaystyle-f(z)+\frac{1}{2}(f_{1}+f_{2})$ $\displaystyle g^{(3)}(f(z))$
$\displaystyle=$
$\displaystyle\frac{f^{2}(z)}{2}-\frac{1}{2}(f_{1}+f_{2})\,f(z)+\frac{1}{2}f_{1}f_{2}$
$\displaystyle g^{(4)}(f(z))$ $\displaystyle=$
$\displaystyle-\frac{f^{3}(z)}{6}+\frac{1}{4}(f_{1}+f_{2})\,f^{2}(z)-\frac{1}{2}f_{1}f_{2}f(z)-\frac{1}{24}(f_{1}+f_{2})(f_{1}^{2}-4f_{1}f_{2}+f_{2}^{2})$
$\displaystyle g^{(5)}(f(z))$ $\displaystyle=$
$\displaystyle\frac{f^{4}(z)}{24}-\frac{1}{12}(f_{1}+f_{2})\,f^{3}(z)+\frac{1}{4}f_{1}f_{2}f^{2}(z)+\frac{1}{24}(f_{1}+f_{2})(f_{1}^{2}-4f_{1}f_{2}+f_{2}^{2})f(z)$
(91) $\displaystyle\hskip
72.26999pt-\frac{1}{24}f_{1}f_{2}(f_{1}^{2}-3f_{1}f_{2}+f_{2}^{2}).$
We will show by induction that $g^{(n)}(f)$ is
$\displaystyle g^{(n)}(f)$ $\displaystyle=$
$\displaystyle(-1)^{n-1}\frac{1}{(n-1)!}f^{n-1}+\frac{1}{2}\left[C_{1}\,g^{(n-1)}(f)-\frac{1}{2!}C_{2}\,g^{(n-2)}(f)+\frac{1}{3!}C_{3}\,g^{(n-3)}(f)+\cdots\right.$
(92) $\displaystyle\hskip
108.405pt\left.+(-1)^{n-2}\frac{1}{(n-1)!}C_{n-1}\,g^{(1)}(f)\right]$
$\displaystyle=$
$\displaystyle\frac{(-1)^{n-1}f^{n-1}}{(n-1)!}+\frac{1}{2}\sum_{m=1}^{n-1}\frac{(-1)^{n-m-1}C_{n-m}}{(n-m)!}g^{(m)}(f),$
where the coefficients $C_{n}=f_{1}^{n}+f_{2}^{n}$. First, (92) can be
explicitly verified up to $n=5$ using (91). Second, we show that if this
expression holds for some integer $n$, then it also holds for $n+1$. From
Eq.(90) we can write,
$\displaystyle g^{(n+1)}(f(z))$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\int_{f(z)}^{f_{1}}g^{(n)}(f)df-\int_{f_{2}}^{f(z)}g^{(n)}(f)df\right]$
(93) $\displaystyle=$
$\displaystyle\frac{(-1)^{n-1}}{(n-1)!}\frac{1}{2}\left[\int_{f(z)}^{f_{1}}f^{n-1}df-\int_{f_{2}}^{f(z)}f^{n-1}df\right]$
$\displaystyle\hskip
36.135pt+\frac{1}{2}\sum_{m=1}^{n-1}\frac{(-1)^{n-m-1}C_{n-m}}{(n-m)!}\frac{1}{2}\left[\int_{f(z)}^{f_{1}}g^{(m)}(f)df-\int_{f_{2}}^{f(z)}g^{(m)}(f)df\right]$
$\displaystyle=$
$\displaystyle\frac{(-1)^{n}f^{n}}{n!}+\frac{1}{2}\frac{(-1)^{n-1}(f_{1}^{n}+f_{2}^{n})}{n!}+\frac{1}{2}\sum_{m=1}^{n-1}\frac{(-1)^{n-m-1}C_{n-m}}{(n-m)!}g^{(m+1)}(f)$
$\displaystyle=$
$\displaystyle\frac{(-1)^{(n+1)-1}f^{(n+1)-1}}{((n+1)-1)!}+\frac{1}{2}\sum_{m=1}^{(n+1)-1}\frac{(-1)^{(n+1)-m-1}C_{(n+1)-m}}{((n+1)-m)!}g^{(m)}(f)$
We thus proved that $g^{(n)}(f)$ is given by (92) for any given integer $n\geq
2$ with $g^{(1)}(f)=1$.
We now need to find the integral $\int{\sigma_{\rm i}}^{(n)}$ in (90). We will
show by induction that
$\int_{f_{2}}^{f_{1}}g^{(n)}(f)=-2\frac{E_{n}}{n!}u^{n},$ (94)
where $u=f_{1}-f_{2}$ and $E_{n}$ are the coefficients of the expansion
$\frac{2}{e^{u}+1}=\sum_{n=0}^{\infty}\frac{E_{n}}{n!}u^{n}.$ (95)
It is easy to see that $E_{0}=1$.
The base for the mathematical induction for (94) is easily established for the
first few terms using (91). Now we verify that (94) holds true for $n+1$ if it
is true for $n$. To do so, we integrate both sides of (93) and use the
assumption (94) to obtain
$\displaystyle\int_{f_{2}}^{f_{1}}g^{(n+1)}(f)$ $\displaystyle=$
$\displaystyle-\frac{(-1)^{(n+1)}}{(n+1)!}(f_{1}^{n+1}-f_{2}^{n+1})+\sum_{m=1}^{n}\frac{(-1)^{n+1-m}}{(n+1-m)!m!}C_{n+1-m}E_{m}u^{m}$
(96)
$\displaystyle=-2\frac{(-f_{1})^{n+1}}{(n+1)!}+\sum_{m=0}^{n}\frac{(-f_{1})^{n+1-m}}{(n+1-m)!}\frac{E_{m}u^{m}}{m!}+\sum_{m=0}^{n}\frac{(-f_{2})^{n+1-m}}{(n+1-m)!}\frac{E_{m}u^{m}}{m!}$
$\displaystyle=-2\frac{(-f_{1})^{n+1}}{(n+1)!}+\sum_{m=0}^{n+1}\left[\frac{(-f_{1})^{n+1-m}}{(n+1-m)!}+\frac{(-f_{2})^{n+1-m}}{(n+1-m)!}\right]\frac{E_{m}u^{m}}{m!}-2\frac{E_{n+1}u^{n+1}}{(n+1)!}.$
In the second step we have included an $m=0$ term in the summation and in the
third step we have added and subtracted an $m=n+1$ term. It can be easily
verified that the right hand side of (96) is the $s^{n+1}$ term of the
following expression.
$-2e^{-f_{1}s}+\left[e^{-f_{1}s}+e^{-f_{2}s}-2\right]\frac{2}{e^{us}+1}=-2\frac{2}{e^{us}+1}\\\
=-2\sum_{m=0}^{\infty}\frac{E_{m}u^{m}}{m!}s^{m}.$
This completes the proof.
Summing over all the terms, we have
$\sum_{n=1}^{\infty}\int_{f_{2}}^{f_{1}}g^{(n)}(f)=-2\sum_{n=0}^{\infty}\frac{E_{n}u^{n}}{n!}+2E_{0}=2\left(1-\frac{2}{e^{u}+1}\right)=\frac{2(\epsilon_{1}-\epsilon_{2})}{\epsilon_{1}+\epsilon_{2}}$
(97)
We note that the series converges for $|u|={\rm ln}{\epsilon_{\rm
o}}/\epsilon_{1}<\pi$. This means that if one medium is water (${\epsilon_{\rm
o}}\approx 80$) then for the other material the dielectric constant
$\epsilon_{1}>{\epsilon_{\rm o}}e^{-\pi}\approx 3.47$. However, using
techniques similar to Borel summation, one can show that the series can still
be summed to the correct final formula for larger values of $|u|$.
Finally the induced surface charge density becomes
${\sigma_{\rm
i}}(\boldsymbol{\rho})=\frac{q}{4\pi\epsilon_{1}}\frac{2(\epsilon_{1}-\epsilon_{2})}{\epsilon_{1}+\epsilon_{2}}\frac{d}{|\boldsymbol{\rho}-d\,\hat{z}|^{3}},$
(98)
which is identical to (27). Thus, we have rigorously justified using the
average dielectric constant $(\epsilon_{1}+\epsilon_{2})/2$ at the boundary.
## Appendix B Evaluation of $\Lambda$ for Spheres with Sharp Boundaries
To compute $\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to k})$, defined as
$\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to
k})\equiv\int{Y_{lm}^{*}({{\tilde{\rm\bf r}}_{j}-{\rm\bf L}_{j\to
k}\over|{\tilde{\rm\bf r}}_{j}-{\rm\bf L}_{j\to k}|})\over|{\tilde{\rm\bf
r}}_{j}-{\rm\bf L}_{j\to k}|^{l+1}}\rho_{j}({\tilde{\rm\bf
r}}_{j})d{\tilde{\rm\bf r}}_{j}\;,$ (99)
for the case of spheres with sharp boundaries, expand the charge density on
sphere $j$ as
$\rho_{j}({\tilde{\rm\bf
r}}_{j})=\delta({\tilde{r}}_{j}-a_{j})\sum_{l^{\prime},m^{\prime}}\sqrt{4\pi}\sigma^{j}_{l^{\prime}m^{\prime}}Y_{l^{\prime}m^{\prime}}({\hat{\tilde{r}}}_{j})$
(100)
to find
$\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to
k})=\sum_{l^{\prime},m^{\prime}}\sqrt{4\pi}\sigma^{j}_{l^{\prime}m^{\prime}}\int{Y_{lm}^{*}({{\tilde{\rm\bf
r}}_{j}-{\rm\bf L}_{j\to k}\over|{\tilde{\rm\bf r}}_{j}-{\rm\bf L}_{j\to
k}|})Y_{l^{\prime}m^{\prime}}({\hat{\tilde{r}}}_{j})\over L_{j\to
k}^{l+1}(1+t^{2}-2t\cos{\tilde{\theta}}_{j})^{(l+1)/2}}\delta({\tilde{r}}_{j}-a_{j})d{\tilde{\rm\bf
r}}_{j}\;,$ (101)
where use has been made of the geometrical fact that $|{\tilde{\rm\bf
r}}_{j}-{\rm\bf L}_{j\to k}|=L_{j\to
k}\sqrt{1+t^{2}-2t\cos{\tilde{\theta}}_{j}}$ with
$t\equiv{\tilde{r}}_{j}/L_{j\to k}$. The delta function renders the radial
integration trivial:
$\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to
k})=\sum_{l^{\prime},m^{\prime}}\sqrt{4\pi}a_{j}^{2}\sigma^{j}_{l^{\prime}m^{\prime}}\int{Y_{lm}^{*}(\vartheta,\varphi)Y_{l^{\prime}m^{\prime}}({\tilde{\theta}}_{j},{\tilde{\phi}}_{j})\over
L_{j\to
k}^{l+1}(1+t^{2}-2t\cos{\tilde{\theta}}_{j})^{(l+1)/2}}d(\cos{\tilde{\theta}}_{j})d{\tilde{\phi}}_{j}\;,$
(102)
where $\vartheta$ and $\varphi$ are the polar variables of $({\tilde{\rm\bf
r}}_{j}-{\rm\bf L}_{j\to k})/|{\tilde{\rm\bf r}}_{j}-{\rm\bf L}_{j\to k}|$ and
$t=a_{j}/L_{j\to k}$ now. All of the angular variables are measured with
respect to a coordinate system whose $z$ axis is parallel to ${\rm\bf L}_{j\to
k}$. The angles $\vartheta$ and $\varphi$ must be expressed as functions of
the integration variables ${\tilde{\theta}}_{j}$ and ${\tilde{\phi}}_{j}$:
$\displaystyle\cos\vartheta$ $\displaystyle=$
$\displaystyle(t\cos{\tilde{\theta}}_{j}-1)\over\sqrt{1+t^{2}-2t\cos{\tilde{\theta}}_{j}}$
(103) $\displaystyle\varphi$ $\displaystyle=$
$\displaystyle{\tilde{\phi}}_{j}\;.$ (104)
Since the definition of the spherical harmonics is
$Y_{lm}(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)!}}P_{lm}(\cos\theta)e^{im\phi}\;,$
(105)
$\Lambda$ is
$\displaystyle\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to k})$ $\displaystyle=$
$\displaystyle\sum_{l^{\prime},m^{\prime}}\frac{\sqrt{4\pi}a_{j}^{2}\sigma^{j}_{l^{\prime}m^{\prime}}}{L_{j\to
k}^{l+1}}\left[\frac{(2l+1)(l-m)!(2l^{\prime}+1)(l^{\prime}-m^{\prime})!}{4\pi(l+m)!4\pi(l^{\prime}+m^{\prime})!}\right]^{1/2}$
(106) $\displaystyle\times$
$\displaystyle\int\frac{P_{lm}(\frac{(t\cos{\tilde{\theta}}_{j}-1)}{\sqrt{1+t^{2}-2t\cos{\tilde{\theta}}_{j}}})P_{l^{\prime}m^{\prime}}(\cos{\tilde{\theta}}_{j})}{(1+t^{2}-2t\cos{\tilde{\theta}}_{j})^{(l+1)/2}}d(\cos{\tilde{\theta}}_{j})d{\tilde{\phi}}_{j}\;.$
(107)
The integration over ${\tilde{\phi}}_{j}$ produces $2\pi\delta_{mm^{\prime}}$.
The integration over $\cos{\tilde{\theta}}_{j}$ is then the integral calcuated
by YuYu . The final expression for $\Lambda$ is
$\Lambda^{j}_{lm}(a_{j},{\rm\bf L}_{j\to
k})=\sum_{l^{\prime}}\frac{Q^{j}_{l^{\prime}m}t^{l^{\prime}}(-1)^{l-m}(l+l^{\prime})!\sqrt{2l+1}}{L_{j\to
k}^{l+1}[4\pi(l+m)!(l^{\prime}+m)!(l-m)!(l^{\prime}-m)!(2l^{\prime}+1)]^{1/2}}\;,$
(108)
where $Q^{j}_{l^{\prime}m}\equiv 4\pi a_{j}^{2}\sigma^{j}_{l^{\prime}m}$.
## References
* (1) A. Wallqvist and R. D. Mountain, Reviews in Computational Chemistry 13, 183 (1999).
* (2) W. L. Jorgensen and J. Tirado-Rives, Proc. Natl. Acad. Sci. U.S.A. 102, 6665 (2005).
* (3) B. Guillot, J. Mol. Liq. 101, 219 (2002).
* (4) J. Chen, C. L. Brooks III, J. Khandogin, Current Opinion in Structural Biology 18, 140 (2008).
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* (8) D. Bashford and D. A. Case, Annu. Rev. Phys. Chem. 51, 129 (2000).
* (9) T. P. Doerr and Y.-K. Yu, Phys. Rev. E 373, 061902 (2006).
* (10) B. Bagchi, Chem. Rev. 105, 3197 (2005).
* (11) J. Schwinger, L. L. Deraad, K. A. Milton, W. Tsai and J. Norton, Classical Electrodynamics (Westview Press, 1998).
* (12) J. D. Jackson, Classical Electrodynamics 3rd Edn., Chapter 1, page 43, (John Wiley & Sons, Inc. 1999).
* (13) J. Che, J. Dzubiella, B. Li, and J. A. McCammon, J. Phys. Chem. B 112, 3058 (2008).
* (14) R. Allen, J-P Hansen and S. Melchionna, Phys. Chem. Chem. Phys. 3, 4177 (2001).
* (15) R. A. Marcus, J. Chem. Phys. 24, 979 (1956); ibid 24, 966 (1956).
* (16) B. U. Felderhof, J. Chem. Phys. 67, 493 (1977).
* (17) M. Marchi, D. Borgis, N. Levy and P. Ballone, J. Chem. Phys. 114, 4377 (2001). We believe that the citation supporting Eq. (2) (the energy functional) in this paper should only invoke Felderhof’s paper Felderhof (and not Marcus’s). N. Levy, D. Borgis and M. Marchi, Comp. Phys. Comm., 169, 69 (2005).
* (18) P. Attard, J. Chem. Phys. 119, 1365 (2003).
* (19) R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics II, Chapter 10, pages 10-2 (Addison-Wesley, 1964).
* (20) L. D. Landau, E. M. Lifschits, The course of theoretecal physics. Volume VIII: The electrodynamics of continuous media, 2 edition (Butterworth-Heinemann, 1984)
* (21) F. Fogolari and J. M. Briggs, Chem. Phys. Lett. 281, 135 (1997).
* (22) D. H. Menzel, Fundamental Formulas of Physics, (New York: Prentice-Hall, 1955)
* (23) Y.-K. Yu, Physica A 326, 522 (2003).
* (24) M. Born, Z. Phys. 1, 45 (1920).
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* (26) W. Kohn, Reviews of Modern Physics 71, 1253 (1999).
|
arxiv-papers
| 2008-12-31T19:03:34 |
2024-09-04T02:48:59.669299
|
{
"license": "Public Domain",
"authors": "O. I. Obolensky, T. P. Doerr, R. Ray, Yi-Kuo Yu",
"submitter": "Oleg Obolensky",
"url": "https://arxiv.org/abs/0901.0129"
}
|
0901.0287
|
Information flow in interaction networks II: channels, path lengths and
potentials
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
In our previous publication, a framework for information flow in interaction
networks based on random walks with damping was formulated with two
fundamental modes: emitting and absorbing. While many other network analysis
methods based on random walks or equivalent notions have been developed before
and after our earlier work, one can show that they can all be mapped to one of
the two modes. In addition to these two fundamental modes, a major strength of
our earlier formalism was its accommodation of context-specific directed
information flow that yielded plausible and meaningful biological
interpretation of protein functions and pathways. However, the directed flow
from origins to destinations was induced via a potential function that was
heuristic. Here, with a theoretically sound approach called the _channel mode_
, we extend our earlier work for directed information flow. This is achieved
by our newly constructed nonheuristic potential function that facilitates a
purely probabilistic interpretation of the channel mode. For each network
node, the channel mode combines the solutions of emitting and absorbing modes
in the same context, producing what we call a _channel tensor_. The entries of
the channel tensor at each node can be interpreted as the amount of flow
passing through that node from an origin to a destination. Similarly to our
earlier model, the channel mode encompasses damping as a free parameter that
controls the locality of information flow. Through examples involving the
yeast pheromone response pathway, we illustrate the versatility and stability
of our new framework.
## 1 Introduction
Biological pathways in protein interaction networks have been modelled (Tu _et
al._ , 2006; Stojmirović and Yu, 2007; Suthram _et al._ , 2008) as information
flow or equivalently random walks between pathway origins and destinations.
Ideally, the nodes visited by the flow should suggest a mechanism for the
pathway being investigated. For biological specificity of the results, it is
important that the flow is directed and localized, that is, the random walks
should follow more direct paths from origins to destinations, as opposed to
wandering around the whole network. Otherwise, if pathway origins and
destinations are distant, many proteins (particularly large network hubs)
unrelated to the pathway’s biological function may appear as significant. It
is therefore necessary to construct a model that is able to controllably pull
the information flow towards the pathway destinations.
In our earlier paper (Stojmirović and Yu, 2007), we developed a mathematical
framework that is capable of directing information flow in interaction
networks based on random walks. Via information damping/aging, our framework
naturally accommodates information loss/leakage that always occurs in all
networks. It requires no prior restriction to the sub-network of interest nor
it uses additional (and possibly noisy) information. The framework consisted
of two modes _absorbing_ and _emitting_. Given a set of information _sinks_ ,
the absorbing mode returns for any network node the likelihood of a random
walk starting at that node to terminate at sinks. The emitting mode returns
for each network node the expected number of visits to that node by a random
walk starting at information _sources_. The emitting mode can also be used to
model biological pathways: given sources and selected destinations
(pseudosinks), we introduced heuristic potential functions that adjust the
weights of network links to guide the information flow towards pseudosinks
(Stojmirović and Yu, 2007).
Although the introduction of potential to direct information flow is novel,
the concepts of diffusion and random walks have been extensively used for
analysis of protein interaction networks. Nabieva _et al._ (2005) introduced
an algorithm that used truncated diffusion from nodes in interactomes to
predict protein function. Tu _et al._ (2006) used simulations of random walks
to infer gene regulatory pathways, while Suthram _et al._ (2008) modelled the
interactome as an electrical network to interpret expression quantitative loci
(eQTLs). The latter two approaches are conceptually similar due to the
correspondence between random walks on (undirected) graphs and electrical
networks (Doyle and Snell, 1984). Missiuro _et al._ (2009) used the electrical
network approach to measure network centrality of each node in several
interactomes. Voevodski _et al._ (2009) proposed a spectral measure of
closeness between two proteins based on PageRank to discover functionally
related proteins. Most efforts in this direction – for example, the methods
proposed by Suthram _et al._ (2008), Missiuro _et al._ (2009) and Voevodski
_et al._ (2009) – can be mapped to our absorbing and emitting modes, without
potentials (see Section 2.3 for details).
While our earlier model provides very reasonable results on many examples from
yeast protein-protein interaction networks (Stojmirović and Yu, 2007), it also
has room for improvement. The potential functions were empirically chosen
since there was no theoretical foundation for the form they should take. In
addition, the choice of optimal potentials could be example-dependent, that
is, different potentials might be needed for different networks, sources and
pseudosinks. Consequently, the model values (visits) for each node can not be
directly interpreted but only in relation to each other. Furthermore, since
each choice of the origins and destinations results in a different network
graph, rapid computation at large-scale is hindered.
In this sequel, we present a major extension of our previous framework. By
appropriately combining the emitting and absorbing modes, we have devised a
new, _channel_ , mode that permits directed information flow with
probabilistic interpretation. The manuscript is structured as follows. Section
2 presents a succinct review of our previous work and shows how other proposed
methods can be mapped to its absorbing or emitting mode. Section 3 details our
extension. Section 4 discusses applications of the channel mode to protein
interaction networks using the yeast pheromone response pathway as an example.
Discussion and conclusions are in Section 5, with more technical details
provided in the Appendix.
## 2 Technical Background
### 2.1 Preliminaries
We will closely follow the notation from our earlier paper (Stojmirović and
Yu, 2007). 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)
We do not make distinction between a vertex $v\in V$ and its corresponding
state given by a particular ordering of vertices. Denote by $\mathbf{P}$ the
$n\times n$ matrix such that for all $i,j\in V$,
$P_{ij}=\frac{\alpha_{i}W_{ij}}{\sum_{k}W_{ik}},$ (2)
when $\sum_{k\in V}W_{ik}>0$ and $P_{ij}=0$ otherwise. Here
$\alpha_{i}\in(0,1]$ for all $i$.
When $\alpha_{i}=1$ for all $i$, the matrix $\mathbf{P}$ is a transition
matrix for a random walk or a Markov chain on $\Gamma$: for any pair of
vertices $i$ and $j$, $P_{ij}$ gives the transition probability from vertex
$i$ to vertex $j$ in one time step. In the general case, the node-specific
damping factors $\alpha_{i}$ model _dissipation_ of information: at each step
of the random walk there is some probability that the walk leaves the graph.
The value $\alpha_{i}$ measures the likelihood for the random walk leaving the
vertex $i$ to remain in the graph, or equivalently, the likelihood of
dissipation at $i$ is $1-\alpha_{i}$.
For this paper, it will be convenient to express dissipation in terms of a
uniform damping coefficient $\mu$, where
$\mu=\max_{i}\alpha_{i}.$ (3)
Let $a_{i}=\alpha_{i}/\mu$ and define the matrix $\mathbf{Q}$ by
$\mathbf{P}=\mu\mathbf{Q}$, that is,
$Q_{ij}=\frac{a_{i}W_{ij}}{\sum_{k}W_{ik}},$ (4)
for $i,j\in V$ by and $0<a_{i}\leq 1$. We will consider $\mu$ as a free
parameter in $(0,1]$ and the transition matrix $\mathbf{P}$ as dependent on
$\mu$.
### 2.2 Emitting and absorbing modes
We extract the properties of information flow through a given network by
examining the paths of discrete random walks. A random walker starts at an
originating node, chosen according to the application domain, and traverses
the network, visiting a node at each step. Each walk terminates at an explicit
_boundary_ vertex or due to dissipation, which is modeled as reaching an
implicit (out-of-network) boundary node.
We distinguish two types of boundary nodes: _sources_ and _sinks_. Sources
emit information, that is, serve as the origins of random walks. All
information entering a source from inside the network is dissipated, so a
walker is not allowed to visit the source more than once. Sinks absorb
information, serving as destinations of walks; information leaving each sink
is completely dissipated. The network graph together with a set of boundary
nodes and a vector of damping factors $\boldsymbol{\alpha}$ provides the
_context_ for the information flow being investigated.
The main variable of interest is the (averaged) number of times a vertex is
visited by a random walk given the context. Let $D$ denote the set of selected
boundary nodes, let $T=V\setminus D$ and let $m=\left|T\right|$. Assuming that
the first $n-m$ states correspond to vertices in $D$, we write the matrix
$\mathbf{P}$ in the canonical block form:
$\mathbf{P}=\left[\begin{array}[]{cc}\mathbf{P}_{DD}&\mathbf{P}_{DT}\\\
\mathbf{P}_{TD}&\mathbf{P}_{TT}\end{array}\right].$ (5)
Here $\mathbf{P}_{AB}$ denotes a matrix giving probabilities of moving from
$A$ to $B$ where $A,B$ stand for either $D$ or $T$. The states (vertices)
belonging to the set $T$ are called _transient_.
#### 2.2.1 Absorbing mode
Suppose that the boundary set $D$ consists only of sinks. Let $\mathbf{F}$
denote an $m\times(n-m)$ matrix such that $F_{ij}$ is the total probability
that the information originating at $i\in T$ is absorbed at $j\in D$. The
matrix $\mathbf{F}$ is found by solving the discrete Laplace equation
$(\mathbb{I}-\mathbf{P}_{TT})\mathbf{F}=\mathbf{P}_{TD},$ (6)
where $\mathbb{I}$ denotes the identity matrix. The matrix
$\Delta(\mathbf{P}_{TT})=\mathbb{I}-\mathbf{P}_{TT}$ is known as the discrete
Laplace operator of the matrix $\mathbf{P}_{TT}$. If
$\mathbb{I}-\mathbf{P}_{TT}$ is invertible, Equation (6) has a unique solution
$\mathbf{F}=\mathbf{G}\mathbf{P}_{TD},$ (7)
where $\mathbf{G}=(\mathbb{I}-\mathbf{P}_{TT})^{-1}$.
#### 2.2.2 Emitting mode
Now consider the dual problem where $D$ is a set of sources. Let $\mathbf{H}$
denote an $(n-m)\times m$ matrix such that $H_{ij}$ is the total expected
number of times the transient vertex $j$ is visited by a random walk emitted
from source $i$ (for all times). Again, $\mathbf{H}$ is found by solving the
discrete Laplace equation
$\mathbf{H}(\mathbb{I}-\mathbf{P}_{TT})=\mathbf{P}_{DT}.$ (8)
which, if $\mathbb{I}-\mathbf{P}_{TT}$ is invertible, has a unique solution
$\mathbf{H}=\mathbf{P}_{DT}\mathbf{G}.$ (9)
It is easy to show (Stojmirović and Yu, 2007) that the matrix
$\mathbf{G}=(\mathbb{I}-\mathbf{P}_{TT})^{-1}$, also known as the Green’s
function or the fundamental matrix of an absorbing Markov chain (Kemeny and
Snell, 1976), exists if every node can be connected to a boundary node or if
$\alpha_{i}<1$ for all $i$. 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$
(Kemeny and Snell, 1976). For any transient state $i$, the value
$T_{i}=\sum_{j\in T}G_{ij}$ (10)
gives the average length of a path traversed by a random walker starting at
$i$ before terminating (Kemeny and Snell, 1976). In this case, the walker is
allowed to revisit $i$ after leaving it $i$. In the Markov chain theory,
$T_{i}$ is also known as the average absorption time from $i$. For the
emitting mode, where the walker starts at $s\in S$ and cannot revisit it, it
can be shown that the average path length is
$T_{s}=1+\sum_{j\in T}H_{sj}$ (11)
### 2.3 Interpretations
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 $s$ deposited at the transient
vertex $j$. Furthermore, we can interpret $F_{ij}$ as the sum of probabilities
(weights) of the paths originating at the vertex $i\in T$ and terminating at
the vertex $j\in D$ while avoiding all other boundary nodes in the set $D$,
and $H_{ij}$ as the sum of probabilities (weights) of the paths originating at
the vertex $i\in D$ and terminating at the vertex $j\in T$, also avoiding all
other nodes in the set $D$. 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.
The entry $H_{ij}$ can alternatively be interpreted as equilibrium information
content at $j$ for information flow originating from $i$. In this case we
imagine the flow entering the network at node $i$ and leaving the network at
$i$ and any other node due to dissipation. The amount of inflow at $i$ is set
to $1$ and $H_{ij}$ denotes the steady state content at $j$. Hence, the
_equilibrium flow rate_ through an edge $(i,j)$ by the flow entering at $s\in
D$, denoted $\psi_{s}(i,j)$, is
$\psi_{s}(i,j)=H_{si}P_{ij}.$ (12)
#### 2.3.1 Electrical networks and heat conduction
A weighted undirected graph $\Gamma=(V,E,w)$ can be considered as an
electrical network with each edge weight $(i,j)$ being associated with
resistance $R_{ij}=1/W_{ij}$. Doyle and Snell (1984) have shown that voltages
and currents through nodes and edges can be interpreted in terms of random
walks with transition matrix $\mathbf{P}$ (where $\alpha_{i}=1$ for all $i\in
V$) and absorbing boundary. Let $\mathbf{f}$ denote the voltage vector over
all nodes and suppose that a unit voltage is applied between two nodes $a$ and
$b$, so that $f_{a}=1$ and $f_{b}=0$. Then, the solution for $\mathbf{f}$ over
$T=v\setminus\\{a,b\\}$ according to Kirchhoff’s laws is equivalent to the
$a$-th column of the absorbing mode matrix $\mathbf{F}$, that is,
$f_{i}=F_{ia}$. The current flowing through an edge $(i,j)$, which we denote
$I_{ij}$, is then given by
$I_{ij}=\frac{f_{i}-f_{j}}{R_{ij}}=(F_{ia}-F_{ja})W_{ij}.$ (13)
Therefore, modeling protein interaction networks as resistor networks is
equivalent to applying our absorbing mode without dissipation.
However, electrical network paradigm is only applicable to interaction
networks where all links can be modeled as undirected edges. This is the case
in (Missiuro _et al._ , 2009), where the authors only take physical
interactions between proteins as links in their networks. On the other hand,
the network constructed by Suthram _et al._ (2008) contained, in addition to
physical interactions, the transcription factor-to-gene interactions. These
interactions were modeled as directed edges and Suthram _et al._ (2008)
applied a heuristic approach to model the current flowing through them. In
contrast, our absorbing mode can be directly applied to directed networks,
although the columns of the matrix $\mathbf{F}$ cannot be interpreted as
voltages (Figure 1). We will show in 3.5 that, even in that case, $\mathbf{F}$
gives rise to potentials.
(a) |
---|---
|
(b) |
|
(c) |
|
Figure 1: Absorbing mode formalism can be extended beyond resistor networks.
Consider, for example, the directed graph shown in (a), where all edges,
directed and undirected have weight 1. This graph can be modeled as a resistor
network by treating all edges as undirected: (b). Applying a unit voltage at
node A and grounding at node B leads to the current flowing from A to B. The
voltage at each node is indicated by shading (dark means high voltage) while
the current at each edge is indicated by the thickness and the direction of
the arrow corresponding to that edge. The equivalents of voltage and current
can be obtained for the original graph using the absorbing mode with the same
boundary: (c). Note the qualitative difference between the results in (b) and
(c): the node shaped as square conducts significant current in (b) but is
totally isolated in (c).
Zhang _et al._ (2007) applied the same formalism without damping to social
networks as a recommendation model. They consider a graph $\Gamma$
corresponding to a social network, where items under consideration are mapped
to nodes, as a heat conduction medium and interpret $f$ as temperature. For
each recomendee, by setting his/her favorite items to ‘high-tempereature’ and
disliked items to ‘low-temperature’ and solving for $f$ over the remaining
nodes, they obtain the heat distribution over the entire $\Gamma$. The values
of $f$ can be used to recommend potential interesting items (other high
temperature nodes) to individuals.
#### 2.3.2 Topic-sensitive PageRank
Topic-sensitive PageRank was introduced by Haveliwala (2003) as a context
sensitive algorithm for web search and has been recently applied to protein
interaction networks by Voevodski _et al._ (2009). The PageRank vector
$\mathbf{p}$ is defined as the unique solution of the equation
$\mathbf{p}=\beta\mathbf{s}+(1-\beta)\mathbf{p}\mathbf{M},$ (14)
where $\mathbf{M}$ is the transition matrix for a graph (i.e. $\sum_{j\in
V}M_{ij}=1$), $0<\beta<1$ and $\mathbf{s}$ is a probability vector
($\sum_{j}s_{j}=1$). The vector $\mathbf{p}$ is interpreted as the steady
state for the random walk governed by $\mathbf{M}$, which at each step has
probability $\beta$ of restarting at a different node. The probability of
restarting at the node $j$ is $s_{j}$. Clearly, $\mathbf{p}$ can be written as
$\mathbf{p}=\beta\mathbf{s}(\mathbb{I}-(1-\beta)\mathbf{M})^{-1}.$ (15)
PageRank can be considered a special case of the emitting mode in the
following way. Add an additional vertex $v$ to the graph with no incoming
edges and with the weight of each outgoing edge $v\to i$ proportional to
$s_{i}$. Construct a matrix $\mathbf{P}$ using $\alpha_{i}=1-\beta$ for all
$i$ in the original graph and $\alpha_{v}=\beta$. Let $D=\\{v\\}$ be the
boundary set. Clearly, $(1-\beta)\mathbf{M}=\mathbf{P}_{TT}$ and
$\beta\mathbf{s}=\mathbf{P}_{DT}$, hence Equation (15) reduces to Equation
(8).
#### 2.3.3 Other methods based on random walks
Beyond the analysis of protein interaction networks, approaches based on
diffusion and random walks have received attention for a number of
applications. We will only mention here a few examples from machine learning
to illustrate the point.
A _kernel_ on a space $X$ is a symmetric positive (semi)definite map
$\kappa:X\times X\to\mathbb{R}$, which can be used to measure similarity
between two points in $X$. A kernel can naturally be treated as an inner
product on some feature space. Among other approaches, kernels are the
foundation of Support Vector Machines (SVMs), machine learning methods widely
used for classification and pattern recognition of data (Schoelkopf and Smola,
2002; Schölkopf _et al._ , 2004).
A variety of kernels were proposed to compare nodes in undirected graphs
(Fouss _et al._ , 2006), mostly derived from discrete Laplacians. Recall that
we called the matrix $\Delta(\mathbf{P}_{TT})=\mathbb{I}-\mathbf{P}_{TT}$ the
discrete Laplace operator of the matrix $\mathbf{P}_{TT}$. One can similarly
define the matrices $\Delta(\mathbf{W})=\mathbb{I}-W$ and
$\Delta(\mathbf{P})=\mathbb{I}-\mathbf{P}$, where $W$ is the adjacency matrix
and $\mathbf{P}$ is the transition matrix for a weighted undirected graph
$\Gamma$. Both $\Delta(\mathbf{W})$ and $\Delta(\mathbf{P})$ were sometimes
called the graph Laplacians for $\Gamma$.
Generally, the matrix $\Delta(\mathbf{W})$ need not be invertible (in
particular, $\Delta(\mathbf{P})$ is not invertible – see (Zhang _et al._ ,
2007)). Fouss _et al._ (2007) proposed using the Moore-Penrose pseudoinverse,
which generalizes a matrix inverse to matrices of less than full rank, of
$\Delta(\mathbf{W})$ as a kernel, with applications to collaborative
recommendation. The approach and the application domain of Fouss _et al._
(2007) are similar to that of Zhang _et al._ (2007).
The von Neumann diffusion kernel (Schoelkopf and Smola, 2002), proposed by
Katz (1953) has the form
$\kappa=\sum_{n=1}^{\infty}\beta^{n}[\mathbf{W}^{n}]=(\mathbb{I}-\beta\mathbf{W})^{-1}-\mathbb{I},$
(16)
where $\beta$ is a damping factor chosen so that
$(\mathbb{I}-\beta\mathbf{W})^{-1}$ exists. This approach is roughly similar
to ours where we compute $\mathbf{G}=(\mathbb{I}-\mu\mathbf{Q}_{TT})^{-1}$ in
that both $\kappa_{ij}$ and $G_{ij}$ include the sums of the weights for all
paths from $i$ to $j$. The main difference between the two approaches is that
the weight of each path of length $n$ included in $\kappa$ is the product of
weights of each link followed, while in our case it is the product of
probabilities and therefore has a probabilistic interpretation.
Exponential diffusion kernels, introduced by Kondor and Lafferty (2002), are
defined by
$\kappa=\sum_{n=0}^{\infty}\frac{\beta^{k}(-\Delta(\mathbf{W}))^{k}}{k!}=\exp(-\beta\Delta(\mathbf{W})),$
(17)
with a real parameter $\beta$. Diffusion kernels can be interpreted to model
continuous diffusion through graph, with infinitesimal time steps in contrast
to discrete-time diffusion implied by von Neumann diffusion kernel and other
similar random-walk based methods. Note that, since every kernel is required
to be symmetric, the above formalizations do not extend directly to directed
graphs.
## 3 Theory
Assume $V=S\sqcup T\sqcup K$, where the set $S$ denotes the sources, $K$
denotes the sinks and $T$ the transient nodes and write the matrix
$\mathbf{P}$ in the block form as
$\mathbf{P}=\left[\begin{array}[]{ccc}\mathbf{P}_{SS}&\mathbf{P}_{ST}&\mathbf{P}_{SK}\\\
\mathbf{P}_{TS}&\mathbf{P}_{TT}&\mathbf{P}_{TK}\\\
\mathbf{P}_{KS}&\mathbf{P}_{KT}&\mathbf{P}_{KK}\end{array}\right].$ (18)
Let us modify (add context to) the underlying graph $\Gamma$ so that the
random walk can only leave the sources and only enter the sinks. Furthermore,
no communication is allowed among sources or among sinks without going through
transient nodes. The modified transition matrix, denoted $\mathbf{\tilde{P}}$
has the form
$\mathbf{\tilde{P}}=\left[\begin{array}[]{ccc}\mathbf{0}&\mathbf{P}_{ST}&\mathbf{P}_{SK}\\\
\mathbf{0}&\mathbf{P}_{TT}&\mathbf{P}_{TK}\\\
\mathbf{0}&\mathbf{0}&\mathbf{0}\end{array}\right].$ (19)
Effectively, the flow moving through disallowed links in $\mathbf{P}$ is
dissipated in $\mathbf{\tilde{P}}$ instead.
Treating the vertices in $S$ and $T$ as transient for the absorbing mode in
2.2.1, we first derive the matrix $\mathbf{F}$ (of size $\left|S\cup
T\right|\times\left|K\right|$):
$\displaystyle\mathbf{F}$
$\displaystyle=\left(\mathbb{I}-\left[\begin{array}[]{cc}\mathbf{0}&\mathbf{P}_{ST}\\\
\mathbf{0}&\mathbf{P}_{TT}\end{array}\right]\right)^{-1}\left[\begin{array}[]{c}\mathbf{P}_{SK}\\\
\mathbf{P}_{TK}\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{cc}\mathbb{I}&\mathbf{P}_{ST}\mathbf{G}\\\
\mathbf{0}&\mathbf{G}\end{array}\right]\left[\begin{array}[]{c}\mathbf{P}_{SK}\\\
\mathbf{P}_{TK}\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{c}\mathbf{P}_{SK}+\mathbf{P}_{ST}\mathbf{G}\mathbf{P}_{TK}\\\
\mathbf{G}\mathbf{P}_{TK}\end{array}\right],$
where, as before, $\mathbf{G}=(\mathbb{I}-\mathbf{P}_{TT})^{-1}$.
Similarly, treating the vertices in $T$ and $K$ as transient for the emitting
mode in 2.2.2, we derive the matrix $\mathbf{H}$ (of size
$\left|S\right|\times\left|T\cup K\right|$):
$\displaystyle\mathbf{H}$
$\displaystyle=\left[\begin{array}[]{cc}\mathbf{P}_{ST}&\mathbf{P}_{SK}\end{array}\right]\left(\mathbb{I}-\left[\begin{array}[]{cc}\mathbf{P}_{TT}&\mathbf{P}_{TK}\\\
\mathbf{0}&\mathbf{0}\end{array}\right]\right)^{-1}$
$\displaystyle=\left[\begin{array}[]{cc}\mathbf{P}_{ST}&\mathbf{P}_{SK}\end{array}\right]\left[\begin{array}[]{cc}\mathbf{G}&\mathbf{G}\mathbf{P}_{TK}\\\
\mathbf{0}&\mathbf{\mathbb{I}}\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{cc}\mathbf{P}_{ST}\mathbf{G}&\mathbf{P}_{ST}\mathbf{G}\mathbf{P}_{TK}+\mathbf{P}_{SK}\end{array}\right].$
The entries of $\mathbf{F}$ and $\mathbf{H}$ are, as before, interpreted as
probabilities of absorption at sinks and average numbers of visits of walks
emitted from sources, respectively. Note that the same Green’s function,
$\mathbf{G}=(\mathbb{I}-\mathbf{P}_{TT})^{-1}$, needs to be computed for both
solutions. Also note that the ‘$S$’ rows of $\mathbf{F}$ form the transpose of
the ‘$K$’ columns of $\mathbf{H}$, that is, for all $s\in S$ and $k\in K$,
$F_{sk}=H_{sk}$.
The matrices $\mathbf{F}$ and $\mathbf{H}$ can be extended into the matrices
$\mathbf{\bar{F}}$ and $\mathbf{\bar{H}}$, of sizes $n\times\left|K\right|$
and $\left|S\right|\times n$, respectively (i.e. extended over the whole
graph) by setting $\bar{F}_{kk^{\prime}}=\delta_{kk^{\prime}}$ for
$k,k^{\prime}\in K$ and $\bar{H}_{ss^{\prime}}=\delta_{ss^{\prime}}$ for
$s,s^{\prime}\in S$. This is equivalent to setting the $K$ portion of
$\mathbf{\bar{F}}$ and $S$ portion of $\mathbf{\bar{H}}$ to appropriately
sized identity matrices:
$\displaystyle\mathbf{\bar{F}}$
$\displaystyle=\left[\begin{array}[]{ccc}\mathbf{P}_{SK}+\mathbf{P}_{ST}\mathbf{G}\mathbf{P}_{TK},&\mathbf{G}\mathbf{P}_{TK},&\mathbb{I}\end{array}\right]^{T}$
(21) $\displaystyle\mathbf{\bar{H}}$
$\displaystyle=\left[\begin{array}[]{ccc}\mathbb{I},&\mathbf{P}_{ST}\mathbf{G},&\mathbf{P}_{ST}\mathbf{G}\mathbf{P}_{TK}+\mathbf{P}_{SK}\end{array}\right]$
(23)
The matrices $\mathbf{\bar{F}}$ and $\mathbf{\bar{H}}$ contain explicit
boundary conditions with interpretations straightforwardly extended from
$\mathbf{{F}}$ and $\mathbf{{H}}$. Specifically,
$\bar{F}_{kk^{\prime}}=\delta_{kk^{\prime}}$ means that a random walk
originating from a sink cannot move anywhere else, while
$\bar{H}_{ss^{\prime}}=\delta_{ss^{\prime}}$ implies that a random walk
starting at a source will visit it exactly once and cannot return to it nor to
any other source.
###### Remark 3.1.
We explicitly assumed that a boundary node can either be a source or a sink.
Sometimes, it is desirable to examine flows that both start and terminate at
the same point. This case can be reduced to our assumption by introducing for
each source that is also a sink an additional node with all the edges of the
original node. The new enlarged graph will contain two ‘logical’ nodes for
each ‘physical’ source/sink node that plays a dual role and hence it will be
possible to have disjoint sets of sources and sinks on the boundary.
### 3.1 Channel tensor
Define the _channel tensor_ $\boldsymbol{\mathrm{\Phi}}\in V\otimes K\otimes
S^{*}$ by
$\varPhi_{i,k}^{s}=\bar{H}_{si}\bar{F}_{ik}.$ (24)
The entry $\varPhi_{i,k}^{s}$ gives the expected number of times a random walk
emerging from the source $s$ and terminating at the sink $k$ visits the vertex
$i$ (Lemma A.1). In particular, for all for all $s\in S$ and $k\in K$,
$\varPhi_{s,k}^{s}=\varPhi_{k,k}^{s}=F_{sk}=P_{sk}+[\mathbf{P}_{ST}\mathbf{G}\mathbf{P}_{TK}]_{sk}.$
(25)
Hence, the entries of $\boldsymbol{\mathrm{\Phi}}$ can be interpreted
similarly to the entries of $\mathbf{\bar{H}}$: as expected numbers of visits
to nodes in network by random walkers starting at a source node. While
$\bar{H}_{si}$ gives the total number of visits to $i$ by a random walker
starting at $s$, $\varPhi_{i,k}^{s}$ measures only those walkers that
ultimately reach the sink $k$. All other walkers, which either terminate due
to dissipation before reaching $k$, reach other sinks or reach any of the
sources, are not considered. Alternatively, $\varPhi_{i,k}^{s}$ measures the
amount of equilibrium flow through the node $i$ by a stream of particles
entering through $s$ and leaving from $k$. The corresponding equilibrium flow
through an edge $(i,j)$, denoted $\psi_{s,k(i,j)}$ is given by
$\psi_{s,k(i,j)}=\varPhi_{i,k}^{s}P_{ij}$.
Suppose $s$ and $k$ are connected through a directed path (equivalently
$F_{sk}>0$) and let $T_{sk}$ denote the expected length of the path traversed
by a walker starting at $s$ and terminating at $k$. Then, it can be shown
(Lemma C.1) that,
$T_{sk}=1+\sum_{i\in
T}\frac{\varPhi_{i,k}^{s}}{F_{sk}}=\frac{\mu}{F_{sk}}\frac{\partial
F_{sk}}{\partial\mu}.$ (26)
Other moments and cumulants of the distribution of lengths of paths traversed
by walkers starting at $s$ and terminating at $k$ can similarly be expressed
in terms of the Green’s function $\mathbf{G}$ or the derivatives of $F_{sk}$
with respect to $\mu$ (see Appendix C).
### 3.2 Normalized channel tensor
For brevity we will use a convention that when a set symbol replaces an
ordinary index, it means to sum over that entity index of the set in question.
For example, for any $i\in S\cup T$, $F_{iK}\equiv\sum_{k\in K}F_{ik}$ and for
all $s\in S$, $i\in V$, $\varPhi_{i,K}^{s}\equiv\sum_{k\in
K}\varPhi_{i,k}^{s}$.
For $s\in S$, $F_{sK}$ gives the probability (or expectation) of a random walk
emerging from the source $s$ reaching any of the sinks in $K$. Assuming
$F_{sK}>0$ for all $s\in S$, define the _normalized channel tensor_ ,
$\boldsymbol{\hat{\Phi}}\in V\otimes K\otimes S^{*}$ by
$\hat{\varPhi}_{i,k}^{s}=\frac{\varPhi_{i,k}^{s}}{F_{sK}}.$ (27)
The normalized channel tensor $\hat{\varPhi}_{i,k}^{s}$ gives the expectation
of the number of visits of $i$ by a random walk from $s$ to $k$, conditional
on the random walk being terminated at sinks only. It does not consider any of
the random walk paths that return to sources or terminate due to dissipation
at transient nodes.
### 3.3 Interpretations
Generally, the entries of $\boldsymbol{\mathrm{\Phi}}$ and
$\boldsymbol{\hat{\Phi}}$ can be interpreted in the same way as the entries of
$\mathbf{H}$ from the emitting mode. For practical applications, it is
sometimes desirable to reduce the amount of available information to a single
vector over $V$, which can be tabulated and graphically visualized using color
maps.
For a source $s\in S$, the _source specific content_ of a node $i\in V$ is
$\hat{\varPhi}_{i,K}^{s}$, the total number of visits to $i$ by a random
walker starting from $s$ and terminating at any of the sinks in $K$. Equations
(25-27) imply that for all $s\in S$,
$\hat{\varPhi}_{s,K}^{s}=\sum_{k\in K}\hat{\varPhi}_{k,k}^{s}=1,$ (28)
that is, the entire flow starting at $s$ and reaching one of the sinks is
normalized to unity. The _total content_ vector of $\boldsymbol{\hat{\Phi}}$,
denoted by $\hat{\boldsymbol{\tau}}$, sums all visits for each node:
$\hat{\tau}_{i}=\hat{\varPhi}_{i,K}^{S}.$ (29)
The concept of _destructive interference_ measures the overlap between visits
from different sources for each node. We define the interference vector
$\hat{\boldsymbol{\sigma}}$ over $V$ by
$\hat{\sigma}_{i}=\left|S\right|\min_{s\in S}\hat{\varPhi}_{i,K}^{s}.$ (30)
Hence, $\hat{\sigma_{i}}$ gives the total number of times the random walks
from all sources co-occur at each node (scaled by the number of sources). The
above formulas assume that each source emits the same amount of information.
If needed, $\hat{\varPhi}_{i,K}^{s}$ can be weighted by _source strength_
before evaluating total content or interference.
With damping factors less than unity, the random walks from sources to sinks
effectively visit a small portion of the entire underlying network.
Information Transduction Module or ITM is a notion that we coined to describe
the set of nodes most influenced by the flow. The nodes are ranked using their
values for the total content or interference and the most significant nodes
are selected. The number of selected nodes depends on the application-specific
considerations but we found that the _participation ratio_ $\pi$ (Stojmirović
and Yu, 2007) of the total content vector $\hat{\boldsymbol{\tau}}$ gives a
good estimate of the number of nodes whose relative amount of content is
significant. It is given by the formula
$\pi(\hat{\boldsymbol{\tau}})=\frac{\left(\sum_{i\in
V}\hat{\tau}_{i}\right)^{2}}{\sum_{j\in V}\hat{\tau}_{j}^{2}}.$ (31)
For undirected graphs, with a context consisting of a single source and a
single sink, the values of $\boldsymbol{\hat{\Phi}}$ are invariant under
interchange of sources and sinks (see Appendix B). In general, however,
reversing sources and sinks gives a different result, both due to asymmetry of
the weight matrix in directed graphs and because sources and sinks have
different roles if more than one of each are present: random walkers
originating from different sources can simultaneously visit a transient node
while a walk can terminate only at a single sink. Thus, the sinks split the
total information flow, that is, compete for it, while sources interfere,
either constructively or destructively.
### 3.4 Path lengths
Damping influences the normalized channel tensor differently from the non-
normalized one or the absorbing and emitting solutions. For the non-normalized
versions, damping factors control the amount of information reaching the
boundary and any intermediate points. In the normalized case, all “normalized”
information emitted from the sources reaches sinks (Equation (28)) and damping
controls a random walker’s average path length, which is always bounded below
by the length of the shortest path. Provided each source is connected to at
least one sink through a directed path, we have (Corollary C.3)
$T_{sK}=1+\sum_{i\in
T}\hat{\varPhi}_{i,K}^{s}=\frac{\mu}{F_{sK}}\frac{\partial
F_{sK}}{\partial\mu}.$ (32)
Small values of $\mu$ strongly favor the nodes on the shortest paths, while
large values allow random walks to take longer excursions and hence favor the
vertices with many connections. As $\mu\downarrow 0$, only the nodes at the
shortest path receive any flow and $T_{sK}\to\rho(s,K)$, the smallest distance
between $s$ and any sinks in $K$. Appendix C contains a more detailed analysis
of the role of damping with full statements and proofs.
As an interesting application of the $\mu$ dependence of $T_{sK}$ allows one
to determine the appropriate damping factor for a specified average path
length. From the results in Appendix C, it follows that $T_{sK}$ is a smooth
function of $\mu$, which is strictly increasing on $[0,1]$ ($\frac{\partial
T_{sK}}{\partial\mu}$ is positive). Therefore, the equation $T_{sK}(\mu)=x$
has a unique simple root for $\rho(s,K)\leq x\leq T_{sK}(1)$ and any root-
finding method can be used to find $\mu$ from $T_{sK}$. When there exist
multiple sources in a context, a desired (weighted) average of $T_{sK}$ over
all $s\in S$ can be set to obtain a global uniform damping factor $\mu$.
### 3.5 Potentials and normalized evolution operators
In our earlier paper (Stojmirović and Yu, 2007), we used a concept of a
_potential_ to redirect the flow towards desired destinations in the emitting
mode. To each node $j\in V$, we associated the value of the total potential
$\Theta(j)$ such that
$\Theta(j)=\sum_{k\in R}\theta_{k}(\rho(j,k)),$ (33)
where $R\subset T$ is the set of potential centers, $\rho(j,k)$ is the length
of the shortest path from $j$ to $k$, and $\theta_{k}$ is an increasing
function with a minimum at $k$. The exponential of the total potential was
then used to re-weight the weight of edges incoming to $j$ and form a new
matrix $\hat{\mathbf{W}}$:
$\hat{W}_{ij}=W_{ij}\exp(-\Theta(j)).$ (34)
The matrix $\hat{W}$ was then normalized to construct the transition matrix to
be used (after applying damping) for the emitting mode. It is possible to
express the application of the potential $\Theta$ as a direct transformation
of the transition matrix $\mathbf{P}$ (without dissipation included). Let
$f_{j}\equiv\exp(-\Theta(j))$ and let $\hat{\mathbf{P}}$ denote the new
transition matrix derived from $\hat{\mathbf{W}}$. Then, $\hat{\mathbf{P}}$
can be written as
$\hat{P}_{ij}=\frac{\hat{W}_{ij}}{\sum_{k\in
V}\hat{W}_{ik}}=c_{i}\frac{P_{ij}f_{j}}{f_{i}},$ (35)
where
$c_{i}=\frac{f_{i}\sum_{k\in V}W_{ik}}{\sum_{k\in V}W_{ik}f_{k}}.$ (36)
If $c_{i}=1$ for all $i$, we can express $\hat{\mathbf{P}}$ as a similarity
transformation of $\mathbf{P}$, where
$\hat{\mathbf{P}}=\mathbf{\Lambda}^{-1}\mathbf{P}\mathbf{\Lambda}$, where
$\Lambda_{ij}=\delta_{ij}f_{i}$. In general, this is not the case with the
heuristic potentials proposed in (Stojmirović and Yu, 2007). However, we will
now show (with proofs in Appendix D) that there exist a potential derived from
the matrix $\mathbf{F}$ that transforms the context specific matrix
$\mathbf{\tilde{P}}$ into a stochastic transition matrix over source and
transient nodes. The solution of the emitting mode using the new matrix
recovers the normalized channel tensor $\boldsymbol{\hat{\Phi}}$ and allows
additional generalizations.
Let $V_{K}=\\{i\in V:\bar{F}_{iK}>0\\}$ be the set of all nodes in $V$ that
are connected with any sink in $K$ by a directed path and denote by $S_{K}$
and $T_{K}$ the sets $S\cap V_{K}$ and $T\cap V_{K}$, respectively. Suppose
$0\leq\mu\leq 1$. For $i,j\in V_{K}$, define
$N_{ij}=\frac{\tilde{P}_{ij}f_{j}}{f_{i}},$ (37)
where $f_{k}>0$ are arbitrary for $k\in K$ and for $i\in S_{K}\cup T_{K}$
$f_{i}=\sum_{k\in K}\bar{F}_{ik}f_{k}.$ (38)
Since all transient nodes are assumed to be connected to a sink, the matrix
$\mathbf{N}$ is well defined for $0<\mu\leq 1$. It can be shown using parts of
Appendix C.2 that it is also well defined in the limit as $\mu\downarrow 0$.
Clearly, $N_{kj}=0$ for all $k\in K$ and $j\in V_{K}$. Over $S_{K}\cup T_{K}$,
the matrix $\mathbf{N}$ is stochastic (Proposition D.1), that is
$\sum_{j\in V_{K}}N_{ij}=1.$ (39)
Hence, $\mathbf{N}$ is an evolution operator for flow entering at sources and
terminating exclusively at a point in $K$. The dependence on $\mu$ is built in
the transition probabilities $N_{ij}$. Furthermore, Equation (38) is the only
way to construct a function over $V_{K}$ so that (37) gives a stochastic
transition matrix (Proposition D.1).
Denote by $\mathbf{G}(\mathbf{N})$, $\mathbf{\bar{F}}(\mathbf{N})$,
$\mathbf{\bar{H}}(\mathbf{N})$, $\boldsymbol{\mathrm{\Phi}}(\mathbf{N})$ the
quantities corresponding to $\mathbf{G}$, $\mathbf{F}$, $\mathbf{H}$ and
$\boldsymbol{\mathrm{\Phi}}$ respectively, when the transition matrix
$\tilde{\mathbf{P}}$ is replaced by $\mathbf{N}$. Since transformation (37) is
a similarity transformation from $\tilde{\mathbf{P}}$ to $\mathbf{N}$, the
following identities hold (Proposition D.2):
1. (i)
For all $i,j\in T_{K}$,
$\displaystyle[\mathbf{G}(\mathbf{N})]_{ij}=\frac{G_{ij}f_{j}}{f_{i}}$,
2. (ii)
For all $i\in V_{K}$ and $k\in K$,
$\displaystyle[\mathbf{\bar{F}}(\mathbf{N})]_{ik}=\frac{\bar{F}_{ik}f_{k}}{f_{i}}$,
3. (iii)
For all $s\in S_{K}$ and $i\in V_{K}$,
$\displaystyle[\mathbf{\bar{H}}(\mathbf{N})]_{si}=\frac{\bar{H}_{si}f_{i}}{f_{s}}$,
4. (iv)
For all $s\in S_{K}$, $i\in V_{K}$ and $k\in K$,
$\displaystyle[\boldsymbol{\mathrm{\Phi}}(\mathbf{N})]^{s}_{i,k}=\frac{\varPhi_{i,k}^{s}f_{k}}{f_{s}}$.
The special case where $f_{k}$’s are equal for all $k\in K$ results in
$[\mathbf{\bar{H}}(\mathbf{N})]_{si}=\hat{\varPhi}_{i,K}^{s}$ and
$[\boldsymbol{\mathrm{\Phi}}(\mathbf{N})]^{s}_{i,k}=\hat{\varPhi}_{i,k}^{s}$.
Hence, $\mathbf{N}$ in this case can be considered a ‘natural’ transition
operator for random walks or Markov chains that start at sources $S$ and
terminate at a point in $K$. The time evolution of such processes can be
followed by raising $\mathbf{N}$ to appropriate powers. As demonstrated in the
previous sections, the parameter $\mu$, which is implicit in $\mathbf{N}$,
controls the how fast the random walkers move towards their destinations.
Figure 2 shows a graphical example of the transformation of the operator
$\tilde{\mathbf{P}}$ into $\mathbf{N}$, which directs the flow towards the
sink.
(a) |
---|---
|
(b) |
|
(c) |
|
Figure 2: Transformation of the evolution operator using potentials. Part (a)
shows the directed graph from Figure 1 with transition probabilities indicated
by edge arrows. Nodes are shaded according to the potential associated with
the sink (octagon). Part (b) displays the normalized transition operator
$\mathbf{N}$ resulting from the application of the sink potential to the
context specific transition matrix (the single source is indicated as
hexagon). Part (c) shows the values of the normalized channel tensor as shades
and the directional flow through each edge as arrows. Comparison between (b)
and (c) shows that edges with large transition probabilities need not carry
significant flows.
In general, each value $f_{k}$ represents the _sink strength_ of the sink
$k\in K$. Equal sink strengths imply no prior preference for any sink while in
the case of unequal sink strengths the flow from sources towards sinks is
preferentially pulled towards sinks with larger strength. It is also possible
to exclude some sinks from consideration by setting their strength to $0$.
Since the scaling of $f_{k}$’s does not affect the transition matrix, they can
be considered as probabilities over $K$ and, in the Bayesian framework, as
priors. Indeed, the equation
$[\mathbf{\bar{F}}(\mathbf{N})]_{ik}=\frac{\bar{F}_{ik}f_{k}}{\sum_{k^{\prime}\in
K}\bar{F}_{ik^{\prime}}f_{k^{\prime}}}$ (40)
can be easily recognized as Bayes’ formula for posterior likelihood. Here
$\bar{F}_{ik}$ can be interpreted as the likelihood of a random walk from $i$
being absorbed at sink $k$, given that $k$ is absorbing; $f_{k}$ is the prior
probability that $k$ is absorbing; while $[\mathbf{\bar{F}}(\mathbf{N})]_{ik}$
is the likelihood that a walker starting at $i$ is absorbed at $k$, given that
it is absorbed at any of the ‘active’ sinks (i.e. sinks with $f_{k}>0$). This
suggests a use of the absorbing and channel modes as Bayesian inference
frameworks for forming and testing hypotheses. For example, sinks can be
associated with mutually exclusive hypotheses. The likelihood of each source
being associated with a hypothesis can then be evaluated using (40).
The matrix $\mathbf{N}$ can also be expressed in terms of potentials. Suppose
$f_{k}>0$ for each $k\in K$ and set the potential of each node $i\in V_{K}$ by
$\Theta(i)\equiv-\log\sum_{k\in K}F_{ik}f_{k}.$ (41)
Then, $\mathbf{N}$ can be written as
$N_{ij}=\tilde{P}_{ij}\exp\big{(}\Theta(i)-\Theta(j)\big{)},$ (42)
with the straightforward interpretation of the information flow moving from
high- to low- potential nodes. Unlike our earlier potential (34), which was
totally heuristic, this new potential is theoretically founded.
## 4 Applications to cellular networks
In the recent years, development of high-throughput genomic and proteomic
techniques resulted in proteome-wide interaction networks (interactomes) in a
number of model organisms (Ito _et al._ , 2001; Uetz _et al._ , 2000; Giot _et
al._ , 2003; Li _et al._ , 2004; Stelzl _et al._ , 2005; Rual _et al._ , 2005;
Ptacek _et al._ , 2005). Databases such as the BioGRID (Breitkreutz _et al._ ,
2008), IntAct (Kerrien _et al._ , 2007), DIP (Salwinski _et al._ , 2004) and
MINT (Chatr-Aryamontri _et al._ , 2007) have been established to collect and
curate sets of interactions from different experiments and make them publicly
available. Most databases contain physical binding interactions, while the
BioGRID additionally includes genetic interactions (such as synthetic
lethality) and biochemical interactions, which describe a biochemical effect
of one protein upon another.
A protein (or a protein state) is mapped to a node in a cellular protein
network. Hence, the solution of a channel mode context (as tensors
$\boldsymbol{\mathrm{\Phi}}$ and $\boldsymbol{\hat{\Phi}}$) will highlight an
ITM consisting of the proteins most visited by a directed flow from sources to
sinks, that is, the proteins lying on the most likely paths connecting sources
and sinks. Clearly, biological interpretations of the model results will
depend on the nature of interactions ascribed 6for links within the network
graphs: an ITM from a genetic or functional network should be interpreted
differently from an ITM from a physical network. We will mainly focus on the
physical networks where interactions correspond to binding between two
proteins (undirected) or a post-translational modification of one protein by
another (directed). Each step of a random walk in such a network is equivalent
to a physical event and dissipation naturally corresponds to protein
degradation by a protease and negative feedback mechanisms that limit
transmission of information. It is thus plausible that the information
channels obtained by solving the channel mode with suitable sources and sinks
may correspond to (portions of) actual signaling or gene regulation pathways.
However, it is important to note that the biological validity of a network
path is contingent upon the transitivity of biochemical effect along that path
as not all protein-protein interactions have the same downstream effect. Also,
even in the best case, the information obtained from a random walk models
would be primarily qualitative since cellular processes in general do not
correspond to linear models.
The simplest way to use the channel mode is for knowledge retrieval by
exploring large networks. In many model organisms, it is possible to construct
physical protein interaction networks that integrate proteome-wide data
collected from results of multiple experiments from different sources using a
variety of techniques. All three modes discussed in this paper, emitting,
absorbing and channel, can be used to explore network neighborhoods of
proteins of interest and learn more about their function(s). In particular,
given two proteins, one set as a source and the other as a sink, one may use
the channel mode to extract a sub-network containing only the proteins most
relevant to the possible functional relation between them. By using graphical
tools to visualize the sub-network and by examining the annotations for the
individual proteins within it, one can learn about their role within the cell
and hence understand the cellular context of the query proteins.
More complex settings of the channel mode can be used for hypothesis forming
and confirmation. For example, using destructive interference between flows
from multiple sources may reveal the points of crosstalk between different
biological pathways that can be selected for further experimental
investigation. Given one or more proteins of interest one can explore the
hypothesis about their function by using the property that sinks split the
flow. Set these proteins of interest as sources and set several sinks, each
associated with an a different biological role. After running a channel mode,
the sinks attracting most of the flow would point to the most likely cellular
role of the proteins, _given all alternatives_. Of course, if all alternatives
are biologically invalid, no valid functional inference can be made.
Since it is possible to arbitrarily specify sources and sinks and obtain model
results that may not correspond to any cellular role, it is desirable to be
able to check whether retrieved ITMs can be associated with any existing
annotation. A common way to do so is through enrichment analysis (Huang _et
al._ , 2009), which assigns terms from a controlled vocabulary such as Gene
Ontology (Ashburner _et al._ , 2000) or KEGG (Kanehisa _et al._ , 2010) to a
set of genes or proteins with weights. Each term from a controlled vocabulary
annotates one or more proteins and enrichment analysis aims to retrieve, by
statistical inference, those terms that best describe the set of submitted
proteins with weights. While many enrichment tools were developed for analysis
of microarrays (Huang _et al._ , 2009), we found that none of them are
entirely suitable for analyzing the results of our model. We have therefore
developed a novel tool, called _SaddleSum_ (Stojmirović and Yu, 2010), which
is based on asymptotic approximation of tail probabilities (Lugannani and
Rice, 1980). For each term, it computes the probability that a score greater
than or equal to the sum of weights, for all the proteins associated with that
term, can arise by chance. In the context of the channel mode, the quantities
that can serve as input to _SaddleSum_ are source specific content, total
content, and destructive interference.
### 4.1 Example: Yeast Pheromone Pathway
As an illustration, we will consider the mating pheromone response pathway in
Saccharomyces cerevisiae, the one of the best understood signalling pathways
in eukaryotes (Bardwell, 2005). The mating signal is transferred from a
membrane receptor to a transcription factor in nucleus, leading to
transcription of mating genes. We will only provide a very brief overview of
the pathway necessary for discussing our examples; more details are available
in the review by Bardwell (2005).
A mating pheromone binds the transmembrane G-protein coupled pheromone
receptors Ste2p/Ste3p. This leads to dissociation of Ste4p and Ste18p, the
membrane bound subunits of the G-protein complex, which also contains the
subunit Gpa1p. Ste4p then binds to the protein kinase Ste20p, which is
recruited to the membrane through Cdc42p, and the scaffold protein Ste5p. On
the scaffold, a MAPK (mitogen activated protein kinase) cascade occurs, where
each protein kinase in the cascade is activated by being phosphorylated by the
previous kinase and in turn activates the next protein. In this case, the
cascade goes Ste20p $\to$ Ste11p $\to$ Ste7p $\to$ Fus3p or Kss1p. The final
activated kinase Fus3p or Kss1p then migrates to the nucleus where it
phosphorylates the proteins Dig1p and Dig2p, the repressors of the Ste12p
transcription factor activity. The Ste12p transcription factor can then, in
coordination with other transcription factors such as Tec1p, promote the
transcription of the mating genes.
As a basis for the underlying network, we used all physical yeast protein-
protein interactions from the BioGRID-3.0.65 (Breitkreutz _et al._ , 2008). To
improve the fidelity of the network, we removed every interaction reported by
a single publication unless that publication described a low-throughput
experiment, which we assumed to be more targeted. We considered an experiment
low-throughput if it reported fewer than 300 interactions in total. We also
removed all interactions labelled with the ‘Affinity Capture-RNA’ experimental
system since they were not protein-to-protein. The physical binding
interactions were given a weight 1 in both directions while the interactions
labelled as ‘Biochemical Activity’ were interpreted as directional (bait $\to$
prey) and given a weight of 2. In cases where both physical and biochemical
interactions were reported, only biochemical were considered. Since it is
known (Steffen _et al._ , 2002) that proteins with a large number of non-
specific interaction partners might overtake the true signaling proteins in
the information flow modeling, we excluded a set of 165 nodes corresponding to
cytoskeleton proteins, histones and chaperones. We found that the excluded
nodes do not strongly affect the results of the particular examples presented
here. For each example we computed the normalized channel tensor summed over
all sinks, that is $\hat{\varPhi}_{i,K}^{s}$ in our notation.
(a) |
---|---
|
(b) |
|
Figure 3: ITMs for the MAPK cascade part of the yeast pheromone response
obtained by running the normalized channel mode with Ste20p as the source and
Ste12p as the sink ($\mu=0.85$). Grey shading of each node indicates its total
content (darker nodes represent more visits). The number of nodes shown is
determined by the participation ratio. Part (a) shows the result using the
network with ‘standard’ excluded nodes (histones, chaperones, cytoskeleton),
while (b) shows the result of additionally excluding the nodes for Slt2p and
Nup53p.
Fig. 3 focuses solely on the MAPK cascade portion of the pheromone pathway,
with Ste20p as a single source and Ste12p as a single sink. Selection of top
proteins by participation ratio (Fig. 3(a)) captures all important
participants of the cascade but emphasizes a ‘shortcut’ through Slt2p, which
is a MAP kinase involved in a different signalling pathway. Upon examination
of the reference (Zarzov _et al._ , 1996) used by the BioGRID to support the
Ste20p $\to$ Slt2p link, we discovered that it does not anywhere claim
existence of such interaction. Hence, we removed Slt2p from our network for
all subsequent queries and reran the query. In addition to the true pathway,
the new ITM (not shown) emphasized a path through Nup53p (a nuclear core
protein). We examined the publication (Lusk _et al._ , 2007) indicated by the
BioGRID to support the Ste20p $\to$ Nup53p link and found that while it is
true that Ste20p phosphorylates Nup53p _in vitro_ , another kinase was mainly
responsible for its phosphorylation _in vivo_. We therefore felt justified to
exclude Nup53p as well. The ITM resulting from the same query with Slt2p and
Nup53p additionally excluded is shown in Fig. 3(b). Enrichment analysis using
the GO database gives ‘receptor signaling protein serine/threonine kinase
activity’ as a top term under ‘Molecular Function’ and ‘filamentous growth’ as
a top term under ‘Biological Process’. Hence, the final ITM agrees well with
the canonical understanding of the MAPK cascade.
(a) | (b)
---|---
|
(c) | (d)
|
Figure 4: Yeast pheromone response ITMs obtained by running the normalized
channel mode with Ste2p and Cdc42p as the sources and Ste12p as the sink with
damping factors $\mu=0.85$ ((a) and (b)), $\mu=1$ (c) and $\mu=0.55$ (d). Part
(a) shows flow intensity from each source using a separate base color, while
(b), (c) and (d) show interference (darker nodes indicate stronger
interference). All images show the top 30 nodes in terms of the total content
for the case of $\mu=0.85$.
To obtain an ITM best describing the entire pheromone response pathway, it is
necessary to include two sources, the receptor Ste2p and the membrane-bound
protein Cdc42p (Fig. 4). Including only Ste2p is not sufficient because of the
shortcut through the link Gpa1p $\to$ Fus3p, which avoids the MAPK cascade.
Furthermore, inclusion of Cdc42p is biologically sensible because Cdc42p
activates Ste20p (Bardwell, 2005) and is hence necessary for the MAPK cascade.
Since the information flows from Ste2p and Cdc42p to Ste12p share some but
definitely not all paths in common (Fig. 4(a)), interference between them
(Fig. 4(b)), rather than total visits, is most appropriate to highlight the
most important proteins in the signalling pathway.
Figs. 4 (b,c and d) illustrate the effect of changing the damping factor
$\mu$. With $\mu=1$ (Fig. 4(c)) the flows from sources visit a much larger
portion of the network (the average path length
$\bar{T}_{sK}=\frac{1}{\left|S\right|}\sum_{s\in S}T_{sK}=194$) than with
$\mu=0.85$ (Fig. 4(b), $\bar{T}_{sK}=7.14$) or $\mu=0.55$ (Fig. 4(d),
$\bar{T}_{sK}=4.58$). The lower bound on path length is $3$, the shortest
distance from both sources to Ste12p. In terms of enrichment analysis with GO
(we provide full results in Appendix E), all three cases pick as significant
the terms related to cell growth but with different statistical significance.
In addition, both the $\mu=0.85$ and $\mu=1$ cases can be associated with
terms related to MAP kinase and signal transduction, while the $\mu=1$ case
alone produces terms related to ‘cell projection’ under ‘Cellular Component’.
Hence, in terms of biological interpretation, results for large $\mu$ tend to
give lower E-values but with lower specificity while small $\mu$ gives very
specific results but with less significant E-values. The $\mu$-dependence of
E-values for any given term is not surprising since different $\mu$s
correspond to different null models. Based on the images in Fig. 4, the
enrichment results as well as our experience in other model contexts, the
values of $\mu$ around 0.85, corresponding to a random walk visiting about
four more nodes than the minimum necessary to reach the sink, appear to give
the best results in emphasizing biologically relevant channels.
Figure 5: Alternative transcription factor targets of yeast pheromone response
pathway. ITM was obtained by running the normalized channel mode with Ste2p
and Cdc42p as the sources and the transcription factors Ste12p, Gal4p, Ino4p,
Ume6p, Yap1p and Rap1p as the sinks with damping factor $\mu=0.85$. Nodes are
shaded by interference. Most of the flow still reaches the proper target
Ste12p while the channels towards other sinks are weak.
The channel mode is relatively robust to addition of non-relevant sinks to its
contexts. In Fig. 5, we set as sinks Ste12p plus five additional transcription
factor proteins not known to be directly influenced by the pheromone response
pathway. As can be seen, the most visited nodes mostly belong to the channel
to Ste12p while the remaining sinks are linked to sources by weaker channels
(mostly not shown because the figure shows only the top 40 nodes). In this
case, Ste12p has $0.62$ total visits (out of $2$) with interference of $0.54$.
The remaining $1.38$ visits are distributed among the other five sinks.
Enrichment results are similar to those with additional sinks absent.
Figure 6: Reversal of sources and sinks for the yeast pheromone response
pathway. ITM was obtained by running the normalized channel mode with Ste2p
and Cdc42p as the sinks and Ste12p as the source ($\mu=0.85$). Nodes are
shaded by total content. The flow uses entirely different channels from Fig 4
and the MAPK cascade is missing.
Fig. 6 shows the effects of reversing sources and sinks. The resulting ITM
performs much worse in describing the pheromone pathway for both reasons
discussed in the last paragraph of 3.1. Firstly, the pheromone response
pathway is dominated by the MAPK phosphorylation cascade, which is in our case
modelled by directed links ‘towards’ Ste12p. Thus, the cascade cannot be seen
at all in the image. Secondly, since the sinks are competing, most of the
information emitted from Ste12p is captured by Cdc42p, leaving little for
Ste2p.
## 5 Discussion and Conclusion
We have described the channel mode for modeling context-specific information
flow in interaction networks. It supports discovery of the most likely
channels through networks between user-specified origins (sources) and
destinations (sinks) of information. The transition operator $\mathbf{N}$,
constructed by applying potentials centered on sinks to the original
transition operator, fully describes the dynamics of the flow within the
channels. The mathematical formulation of the channel mode is flexible and can
be easily modified for related cases. For example, it is possible to model the
flow through a sequence of ‘checkpoints’ by using destination from one context
as the origin for another.
Unlike other models based on random walks and/or electrical networks proposed
in the literature (Tu _et al._ , 2006; Suthram _et al._ , 2008; Missiuro _et
al._ , 2009; Voevodski _et al._ , 2009) that can be reduced to either emitting
or absorbing modes, our channel mode allows for “directed” information flow.
Furthermore, it can readily accommodate networks containing directed links and
multiple sources and sinks. Most importantly, in common with our original
framework (absorbing and emitting modes), the channel mode uses damping to
retain the information flow in the portion of the network most relevant to the
specified context and prevent visits to distant nodes. Damping is controlled
by a free parameter $\mu$ (or more generally, node specific parameters
$\alpha_{i}$), which in the case of the channel mode controls the amount of
path deviation from the shortest one. In statistical physics terms, this is
equivalent to using fugacity to control the number of particles of the system.
Computation of the model solution requires only a solution to a (sparse)
system of linear equations, without needing to simulate random walks as was
done in (Tu _et al._ , 2006). If computation of multiple contexts with the
same damping coefficients is required, it is possible to re-use the Green’s
function for one context to efficiently compute the Green’s function for
another (Appendix F)
Applied to physical protein interaction networks, the channel mode can be used
as a framework for knowledge retrieval through network exploration and
hypothesis formation and confirmation. The node weights obtained can be
interpreted directly as well as submitted to an enrichment tool for further
analysis. Note however that, in many cases, the annotation of a protein by a
term is directly tied to publications reporting its physical interactions.
As illustrated by our pheromone pathway example, the results of our model are
sensitive to ‘shortcuts’ between biologically unrelated protein nodes.
Therefore, to obtain valid conclusions from the ITMs retrieved, the underlying
interaction network must be constructed from high-quality data relevant to the
biological context under investigation. The nodes with many non-specific
interactions, as well as links that may not represent actual in vivo
interactions under the query context, should be removed from the network. The
damping factor $\mu$ also needs to be selected appropriately for the
biological context investigated and depending on whether the coverage (high
$\mu$) or the selectivity (low $\mu$) of the channel are desired more. The
results of enrichment analysis for the example shown in Fig. 4 indicate that
at least some interpretations are robust to the change of $\mu$.
We have already deployed a software implementation of our framework, called
ITM Probe, to the web for the use of biomedical researchers (Stojmirović and
Yu, 2009). In future, we plan to extend our information flow framework to a
platform for network-based context-specific qualitative analysis of cellular
process. The improved models will take into account additional biological
information, such as protein complex memberships, post-translational
modification states and abundances, possibly leading to non-linear transition
operators. Generally, while wishing to improve accuracy by incorporating more
specific information, we aim to preserve the simplicity of the present
framework.
## Acknowledgments
This work was supported by the Intramural Research Program of the National
Library of Medicine at the National Institutes of Health.
## Appendix
## Appendix A Channel tensor as expectation
###### Lemma A.1.
Let $Z^{s}_{i,k}$ be a random variable denoting the total number of times a
random walk starting at $s\in S$ and absorbed at $k\in K$ visits $i\in V$.
Then,
$\operatorname{\mathbb{E}}(Z^{s}_{i,k})=\varPhi_{i,k}^{s}.$ (43)
###### Proof.
Consider a path $x=x_{0},x_{1},x_{2}\ldots x_{\tau}$ from $s\in S$ to $k\in K$
of total length $\tau$ where $x_{0}=s$, $x_{\tau}=k$ and $x_{1},x_{2},\ldots
x_{\tau-1}\in T$. The total weight or probability associated with $x$ is
$\mathbb{P}(x)=P_{x_{0}x_{1}}P_{x_{1}x_{2}}\ldots P_{x_{\tau-1}x_{\tau}}$. For
any $i\in V$, let $X_{i}(x,t)=1$ if $x_{t}=i$ and $0$ otherwise. Then, the
total number of times $x$ visits $i$ is $N_{i}(x)=\sum_{t=0}^{\tau}X_{i}(x,t)$
and
$Z^{s}_{i,k}=\sum_{\tau=1}^{\infty}\sum_{x\in\mathcal{X}(\tau)}N_{i}(x),$
where $\mathcal{X}(\tau)$ denotes the set of all paths from $s$ to $k$ of
length $\tau$. Therefore,
$\displaystyle\operatorname{\mathbb{E}}(Z^{s}_{i,k})=\sum_{\tau=1}^{\infty}\sum_{x\in\mathcal{X}(\tau)}N_{i}(x)\mathbb{P}(x)$
$\displaystyle=\sum_{\tau=1}^{\infty}\sum_{x\in\mathcal{X}(\tau)}\sum_{t=0}^{\tau}X_{i}(x,t)\mathbb{P}(x)$
$\displaystyle=\sum_{\tau=1}^{\infty}\sum_{t=0}^{\tau}Y_{i}(t;\tau),$ (44)
where $Y_{i}(t;\tau)=\sum_{x\in\mathcal{X}(\tau)}X_{i}(x,t)\mathbb{P}(x)$.
There are three cases to consider depending on whether $i$ is a source, a sink
or a transient node.
If $i$ is a source, a path from $s$ can visit $i$ only if $i=s$ and $t=0$.
Therefore, $X_{i}(x,t)=\delta_{si}\delta_{t0}$ and hence
$Y_{i}(t;\tau)=\begin{cases}\delta_{si}P_{sk}&\text{if $t=0$ and $\tau=1$},\\\
\sum_{j,j^{\prime}\in
T}\delta_{si}P_{ij}\left[\mathbf{P}_{TT}^{\tau-2}\right]_{jj^{\prime}}P_{j^{\prime}k}&\text{if
$t=0$ and $\tau\geq 2$},\\\ 0&\text{otherwise}.\end{cases}$ (45)
Here $\left[\mathbf{P}_{TT}^{\tau-2}\right]_{jj^{\prime}}$ is exactly the
total weight of paths of length $\tau-2$ that start at $j\in T$, visit nodes
in $T$ and terminate at $j^{\prime}\in T$. Hence,
$\displaystyle\operatorname{\mathbb{E}}(Z^{s}_{i,k})$
$\displaystyle=\delta_{si}P_{ik}+\sum_{\tau=2}^{\infty}\sum_{j,j^{\prime}\in
T}\delta_{si}P_{ij}\left[\mathbf{P}_{TT}^{\tau-2}\right]_{jj^{\prime}}P_{j^{\prime}k}$
$\displaystyle=\delta_{si}\left[\mathbf{P}_{SK}\right]_{ik}+\delta_{si}\sum_{j,j^{\prime}\in
T}P_{ij}\sum_{n=0}^{\infty}\left[\mathbf{P}_{TT}^{n}\right]_{jj^{\prime}}P_{j^{\prime}k}$
$\displaystyle=\delta_{si}\left[\mathbf{P}_{SK}+\mathbf{P}_{ST}\mathbf{G}\mathbf{P}_{TK}\right]_{ik}$
$\displaystyle=\bar{H}_{si}\bar{F}_{ik}=\varPhi_{i,k}^{s}.$ (46)
Similarly, if $i$ is a sink, a walker from $s$ can visit $i$ and terminate at
$k$ only if $i=k$ and $0<t=\tau$. Thus, $X_{i}(x,t)=\delta_{ik}\delta_{t\tau}$
and
$Y_{i}(t;\tau)=\begin{cases}P_{si}\delta_{ik}&\text{if $t=\tau=1$},\\\
\sum_{j,j^{\prime}\in
T}P_{sj}\left[\mathbf{P}_{TT}^{\tau-2}\right]_{jj^{\prime}}P_{j^{\prime}i}\delta_{ik}&\text{if
$t=\tau\geq 2$},\\\ 0&\text{otherwise}.\end{cases}$ (47)
Therefore,
$\displaystyle\operatorname{\mathbb{E}}(Z^{s}_{i,k})$
$\displaystyle=P_{si}\delta_{ik}+\sum_{\tau=2}^{\infty}\sum_{j,j^{\prime}\in
T}P_{sj}\left[\mathbf{P}_{TT}^{\tau-2}\right]_{jj^{\prime}}P_{j^{\prime}i}\delta_{ik}$
$\displaystyle=\left[\mathbf{P}_{SK}\right]_{si}\delta_{ik}+\sum_{j,j^{\prime}\in
T}P_{sj}\sum_{n=0}^{\infty}\left[\mathbf{P}_{TT}^{n}\right]_{jj^{\prime}}P_{j^{\prime}i}\delta_{ik}$
$\displaystyle=\left[\mathbf{P}_{SK}+\mathbf{P}_{ST}\mathbf{G}\mathbf{P}_{TK}\right]_{si}\delta_{ik}$
$\displaystyle=\bar{H}_{si}\bar{F}_{ik}=\varPhi_{i,k}^{s}.$ (48)
Finally, suppose $i\in T$. In order to visit $i$ at time $t$ and terminate at
$k$ at time $\tau$, a path in $\mathcal{X}(\tau)$ must take one step to reach
$T$, spend there $t-1$ steps before arriving at $i$, then take another
$\tau-t-1$ steps in $T$ and an additional step to terminate at $k$.
Considering all possible paths that visit $i$ at time $t$, we have
$Y_{i}(t;\tau)=\begin{cases}\sum_{j,j^{\prime}\in
T}P_{sj}\left[\mathbf{P}_{TT}^{t-1}\right]_{ji}\left[\mathbf{P}_{TT}^{\tau-t-1}\right]_{ij^{\prime}}P_{j^{\prime}k}&\text{if
$1\leq t<\tau$},\\\ 0&\text{otherwise}.\end{cases}$ (49)
It follows that
$\displaystyle\operatorname{\mathbb{E}}(Z^{s}_{i,k})$
$\displaystyle=\sum_{\tau=2}^{\infty}\sum_{t=1}^{\tau-1}\sum_{j,j^{\prime}\in
T}P_{sj}\left[\mathbf{P}_{TT}^{t-1}\right]_{ji}\left[\mathbf{P}_{TT}^{\tau-t-1}\right]_{ij^{\prime}}P_{j^{\prime}k}$
$\displaystyle=\sum_{t=1}^{\infty}\sum_{\tau=t+1}^{\infty}\sum_{j,j^{\prime}\in
T}P_{sj}\left[\mathbf{P}_{TT}^{t-1}\right]_{ji}\left[\mathbf{P}_{TT}^{\tau-t-1}\right]_{ij^{\prime}}P_{j^{\prime}k}$
$\displaystyle=\sum_{j,j^{\prime}\in
T}P_{sj}\sum_{n=0}^{\infty}\left[\mathbf{P}_{TT}^{n}\right]_{ji}\sum_{m=0}^{\infty}\left[\mathbf{P}_{TT}^{m}\right]_{ij^{\prime}}P_{j^{\prime}k}$
$\displaystyle=\left[\mathbf{P}_{ST}\mathbf{G}\right]_{si}\left[\mathbf{G}\mathbf{P}_{TK}\right]_{ik}$
$\displaystyle=\bar{H}_{si}\bar{F}_{ik}=\varPhi_{i,k}^{s}.\qed$ (50)
## Appendix B Reversibility of sources and sinks
It is easy to see that in general, reversing sources and sinks produces
different values for the normalized channel tensor. One important exception,
however, is the case when the underlying graph is undirected and there is a
single source and a single sink.
###### Lemma B.1.
Let $\Gamma=(V,E,w)$ be an _undirected_ weighted graph with a weight matrix
$\mathbf{W}$ and transition matrix $\mathbf{P}$ as defined in (2), with
$\alpha_{i}\in[0,1]$ for all $i\in V$. Suppose $\Gamma$ is connected and let
$s,k\in V$. Denote by $\boldsymbol{\hat{\Phi}}$ the normalized channel tensor
over $\Gamma$ with $s$ as a single source and $k$ as a single sink, and denote
by $\boldsymbol{\hat{\Psi}}$ the normalized channel tensor over $\Gamma$ with
$k$ as a single source and $s$ as a single sink. Then, for all $i\in V$,
$\hat{\varPhi}_{i,k}^{s}=\hat{\Psi}_{i,s}^{k}.$ (51)
###### Proof.
Since $\Gamma$ is an undirected graph, it satisfies the detailed balance
equation
$\pi_{y}P_{xy}=\pi_{x}P_{yx}$ (52)
for all $x,y\in V$, where $\pi_{x}=\alpha_{x}/\sum_{z\in V}W_{xz}$. It
directly follows that
$\pi_{y}G_{xy}=\sum_{n=0}^{\infty}\pi_{y}[\mathbf{P}^{n}_{TT}]_{xy}=\sum_{n=0}^{\infty}\pi_{x}[\mathbf{P}^{n}_{TT}]_{yx}=\pi_{x}G_{yx}.$
(53)
For $i=s$ or $i=k$, one can immediately see that
$\hat{\varPhi}_{i,k}^{s}=1=\hat{\Psi}_{i,s}^{k}$. Observing that the transient
state is the same for both $\boldsymbol{\hat{\Phi}}$ and
$\boldsymbol{\hat{\Psi}}$, we have for each $i\in T$,
$\displaystyle\hat{\varPhi}_{i,k}^{s}$ $\displaystyle=\frac{\left(\sum_{j\in
T}P_{sj}G_{ji}\right)\left(\sum_{j^{\prime}\in
T}G_{ij^{\prime}}P_{j^{\prime}k}\right)}{P_{sk}+\sum_{j,j^{\prime}\in
T}P_{sj}G_{jj^{\prime}}P_{j^{\prime}k}}$ $\displaystyle=\frac{\left(\sum_{j\in
T}\frac{\pi_{s}}{\pi_{j}}P_{js}\frac{\pi_{j}}{\pi_{i}}G_{ij}\right)\left(\sum_{j^{\prime}\in
T}\frac{\pi_{i}}{\pi_{j^{\prime}}}G_{j^{\prime}i}\frac{\pi_{j^{\prime}}}{\pi_{k}}P_{kj^{\prime}}\right)}{\frac{\pi_{s}}{\pi_{k}}P_{ks}+\sum_{j,j^{\prime}\in
T}\frac{\pi_{s}}{\pi_{j}}P_{js}\frac{\pi_{j}}{\pi_{j^{\prime}}}G_{j^{\prime}j}\frac{\pi_{j^{\prime}}}{\pi_{k}}P_{kj^{\prime}}}$
$\displaystyle=\hat{\Psi}_{i,s}^{k}.$
∎
## Appendix C The role of the damping factor in the channel mode
Recall that $\mathbf{P}=\mu\mathbf{Q}$, where $\mu\in(0,1)$ is the uniform
damping factor and $\mathbf{Q}$ is given in (4). For this range of $\mu$, the
Green’s function
$\mathbf{G}=(\mathbb{I}-\mathbf{P}_{TT})^{-1}=\sum_{n=0}^{\infty}\mathbf{P}_{TT}^{n}=\sum_{n=0}^{\infty}\mathbf{Q}_{TT}^{n}\mu^{n}$
is well-defined (see (Stojmirović and Yu, 2007), Proposition 2.2) and hence
the solution matrices $\mathbf{\bar{F}}$ and $\mathbf{\bar{H}}$ from Equations
(21–23) are well defined and continuous as functions of $\mu$. As
$\mu\downarrow 0$, all the damping factors in $\boldsymbol{\alpha}$ uniformly
tend to $0$ and $\mathbf{P}\to\mathbf{0}$. However, we will show in C.2 that
the normalized channel tensor is well-defined in the limit as $\mu\to 0$
(provided it is well defined for other values of $\mu$).
At the other extreme, as $\mu\uparrow 1$ and $\mathbf{P}\to\mathbf{Q}$, the
Green’s function may not exist for every choice of boundary nodes, since the
spectral radius of $\mathbf{Q}_{TT}$ may be equal to $1$. If the vertex set is
restricted to $V(K)$, the set of all nodes connected through a directed path
to at least one sink, then by Proposition 2.1 of (Stojmirović and Yu, 2007),
the Green’s function is well-defined for $\mu=1$ as well. Also note that for a
channel tensor $\boldsymbol{\mathrm{\Phi}}$ to be non-trivial (i.e. non-zero
everywhere), it is also necessary that each source is connected to at least
one sink through a directed path, or equivalently, that $F_{sK}>0$ for all
$s\in S$.
### C.1 Path lengths
The damping parameter $\mu$ controls the distribution of lengths of the paths
(or the times) a random walk emitted from a source takes before being absorbed
at a sink.
For nodes $s\in S$ and $k\in K$, let $L_{sk}$ (more precisely, $L_{sk}(\mu)$)
denote the random variable giving the length of the path (or a number of
steps) taken by a random walk originating at $s$ and terminating at $k$. At
least one such path from $s$ to $k$ exists if and only if $F_{sk}>0$. The
underlying probability density $\mathbb{P}(L_{sk}=n)$ is given by
$\mathbb{P}(n)=\frac{1}{F_{sk}}\begin{cases}P_{sk}&\text{for $n=1$;}\\\
\left[\mathbf{P}_{ST}\mathbf{P}^{n-2}_{TT}\mathbf{P}_{TK}\right]_{sk}&\text{for
$n\geq 2$.}\end{cases}$ (54)
Let $M_{L_{sk}(\mu)}$ denote the moment generating function for $L_{sk}$ and
let $C_{L_{sk}(\mu)}\equiv\log M_{L_{sk}(\mu)}$ denote its cumulant generating
function. Let us write $F_{sk}$ as a function of $\mu$:
$\displaystyle F_{sk}(\mu)$
$\displaystyle=Q_{sk}\mu+\sum_{n=2}^{\infty}\left[\mathbf{Q}_{ST}\mathbf{Q}^{n-2}_{TT}\mathbf{Q}_{TK}\right]_{sk}\mu^{n},$
(55)
and observe that
$\displaystyle M_{L_{sk}(\mu)}(t)$
$\displaystyle=\sum_{n=0}^{\infty}\mathbb{P}(n)e^{nt}$
$\displaystyle=P_{sk}e^{t}+\sum_{n=2}^{\infty}\left[\mathbf{P}_{ST}\mathbf{P}^{n-2}_{TT}\mathbf{P}_{TK}\right]_{sk}e^{nt}$
$\displaystyle=Q_{sk}\mu
e^{t}+\sum_{n=2}^{\infty}\left[\mathbf{Q}_{ST}\mathbf{Q}^{n-2}_{TT}\mathbf{Q}_{TK}\right]_{sk}\mu^{n}e^{nt}$
$\displaystyle=F_{sk}(\mu e^{t}).$ (56)
Thus, all moments and cumulants of $L_{sk}$ can be expressed in terms of the
Green’s function $\mathbf{G}$ and its related quantities $\mathbf{F}$,
$\mathbf{H}$ and $\boldsymbol{\mathrm{\Phi}}$, both directly and in terms of
derivatives of their entires with respect to $\mu$. In particular,
$\displaystyle\operatorname{\mathbb{E}}(L_{sk})$
$\displaystyle=C^{\prime}_{L_{sk}(\mu)}(0)=\frac{\frac{\partial}{\partial
t}F_{sk}(\mu e^{t})}{F_{sk}(\mu e^{t})}\Big{|}_{t=0}=\frac{\mu
e^{t}F^{\prime}_{sk}(\mu e^{t})}{F_{sk}(\mu e^{t})}\Big{|}_{t=0}=\frac{\mu
F^{\prime}_{sk}(\mu)}{F_{sk}(\mu)}.$ (57)
Using the easily provable identity
$\sum_{n=0}^{\infty}(n+2)\mathbf{P}_{TT}^{n}=\mathbf{G}^{2}+\mathbf{G},$ (58)
we have
$\displaystyle F^{\prime}_{sk}(\mu)$
$\displaystyle=Q_{sk}+\sum_{n=2}^{\infty}\left[\mathbf{Q}_{ST}\mathbf{Q}^{n-2}_{TT}\mathbf{Q}_{TK}\right]_{sk}n\mu^{n-1}$
(59)
$\displaystyle=\frac{1}{\mu}\left(P_{sk}+\sum_{n=0}^{\infty}(n+2)\left[\mathbf{P}_{ST}\mathbf{P}^{n}_{TT}\mathbf{P}_{TK}\right]_{sk}\right)$
$\displaystyle=\frac{1}{\mu}\left(P_{sk}+\left[\mathbf{P}_{ST}(\mathbf{G}+\mathbf{G}^{2})\mathbf{P}_{TK}\right]_{sk}\right)$
$\displaystyle=\frac{1}{\mu}\left(F_{sk}+\left[\mathbf{P}_{ST}\mathbf{G}^{2}\mathbf{P}_{TK}\right]_{sk^{\prime}}\right).$
(60)
Therefore, by (57),
$\displaystyle\operatorname{\mathbb{E}}(L_{sk})$
$\displaystyle=1+\frac{\left[\mathbf{P}_{ST}\mathbf{G}^{2}\mathbf{P}_{TK}\right]_{sk}}{F_{sk}}$
(61) $\displaystyle=1+\sum_{i\in T}\frac{H_{si}F_{ik}}{F_{sk}}$
$\displaystyle=1+\sum_{i\in T}\frac{\varPhi_{i,k}^{s}}{F_{sk}},$ (62)
and we obtain the following
###### Lemma C.1.
Let $s\in S$, let $k\in K$ and let $\mu\in(0,1)$. Suppose $F_{sk}>0$. Then,
$T_{sk}=\operatorname{\mathbb{E}}(L_{sk})=1+\sum_{i\in
T}\frac{\varPhi_{i,k}^{s}}{F_{sk}}=\frac{\mu}{F_{sk}}\frac{\partial
F_{sk}}{\partial\mu}.$ (63)
Similarly,
$\displaystyle\operatorname{Var}(L_{sk})$
$\displaystyle=C^{\prime\prime}_{L_{sk}(\mu)}(0)$
$\displaystyle=\frac{\partial}{\partial t}\frac{\mu e^{t}F^{\prime}_{sk}(\mu
e^{t})}{F_{sk}(\mu e^{t})}\Big{|}_{t=0}$ $\displaystyle=\frac{\mu
e^{t}F^{\prime}_{sk}(\mu e^{t})+\mu^{2}e^{2t}F^{\prime\prime}_{sk}(\mu
e^{t})}{F_{sk}(\mu e^{t})}-\left(\frac{\mu e^{t}F^{\prime}_{sk}(\mu
e^{t})}{F_{sk}(\mu e^{t})}\right)^{2}\Big{|}_{t=0}$
$\displaystyle=\operatorname{\mathbb{E}}(L_{sk})+\frac{\mu^{2}F^{\prime\prime}_{sk}(\mu)}{F_{sk}(\mu)}-\operatorname{\mathbb{E}}^{2}(L_{sk}).$
(64)
Using another easily provable identity
$\sum_{n=0}^{\infty}(n+2)^{2}\mathbf{P}_{TT}^{n}=2\mathbf{G}^{3}+\mathbf{G}^{2}+\mathbf{G},$
(65)
and Equation (59), we have
$\displaystyle F^{\prime\prime}_{sk}(\mu)$
$\displaystyle=\sum_{n=2}^{\infty}\left[\mathbf{Q}_{ST}\mathbf{Q}^{n-2}_{TT}\mathbf{Q}_{TK}\right]_{sk}n(n-1)\mu^{n-2}$
$\displaystyle=\frac{1}{\mu^{2}}\sum_{n=0}^{\infty}(n+2)(n+1)\left[\mathbf{P}_{ST}\mathbf{P}^{n}_{TT}\mathbf{P}_{TK}\right]_{sk}$
$\displaystyle=\frac{2}{\mu^{2}}\left[\mathbf{P}_{ST}\mathbf{G}^{3}\mathbf{P}_{TK}\right]_{sk}.$
(66)
Hence, we obtain
###### Lemma C.2.
Let $s\in S$, let $k\in K$ and let $\mu\in(0,1)$. Suppose $F_{sk}>0$. Then,
$\operatorname{Var}(L_{sk})=\operatorname{\mathbb{E}}(L_{sk})+\frac{2\left[\mathbf{P}_{ST}\mathbf{G}^{3}\mathbf{P}_{TK}\right]_{sk}}{F_{sk}}-\operatorname{\mathbb{E}}^{2}(L_{sk}).$
(67)
Denote by $L_{sK}$ the random variable giving the length of the path (or the
number of steps) taken by a random walk originating at $s$ and terminating at
any sink in $K$. This random variable is well-defined provided $s$ is
connected with at least one $k\in K$ through a directed path, or equivalently,
if $\max_{k\in K}F_{sk}>0$. Let $\hat{K}(s)=\\{k\in K:F_{sk}>0\\}$. Then,
$L_{sK}$ can be expressed as a weighted sum of $L_{sk}$ over $k\in\hat{K}(s)$:
$L_{sK}=\sum_{k\in\hat{K}(s)}\frac{F_{sk}}{F_{sK}}L_{sk}.$ (68)
Here $F_{sk}/F_{sK}$ gives the conditional probability of a random walker from
$s$ reaching sink $k$, given that it reaches any of the sinks in $\hat{K}(s)$.
Through properties of mean, variance and the differential operator, this leads
to the following corollary.
###### Corollary C.3.
Let $s\in S$ and let $\mu\in(0,1)$. Suppose $\max_{k\in K}F_{sk}>0$. Then,
$T_{sK}=\operatorname{\mathbb{E}}(L_{sK})=1+\sum_{i\in
T}\hat{\varPhi}_{i,K}^{s}=\frac{\mu}{F_{sK}}\frac{\partial
F_{sK}}{\partial\mu}\\\ $ (69)
and,
$\operatorname{Var}(L_{sK})=\operatorname{\mathbb{E}}(L_{sK})+\frac{2\left[\mathbf{P}_{ST}\mathbf{G}^{3}\mathbf{P}_{TK}\right]_{sK}}{F_{sK}}-\sum_{k\in\hat{K}(s)}\frac{F_{sk}}{F_{sK}}\operatorname{\mathbb{E}}^{2}(L_{sk}).$
(70)
Since $\operatorname{\mathbb{E}}(L_{sk})$ and
$\operatorname{\mathbb{E}}(L_{sK})$ can be expressed in terms of sums and
products of entries of $\mathbf{G}$, they are continuous and increasing
functions of $\mu\in(0,1)$. The value of $\operatorname{\mathbb{E}}(L_{sK})$
is bounded from below: as $\mu\downarrow 0$, the variance of $L_{sK}$
vanishes, and, as will be shown in the remainder of this section, the average
path-length converges to the length of the shortest path from the source to
any of the sinks. If the graph nodes are restricted to $V(K)$, $\mathbf{G}$ is
well-defined for $\mu=1$ and $\operatorname{\mathbb{E}}(L_{sK})$ is bounded
and attains its maximum there. The value of the maximum varies depending on
the underlying network graph and the particular context.
### C.2 Large dissipation asymptotics
For all $i,j\in V$, let $\rho(i,j)$ denote the (unweighted) length of the
shortest directed path between $i$ and $j$. We allow $\rho(i,j)=\infty$ if
there exists no directed path between $i$ and $j$. It is well-known that
$\rho$ is a (not necessarily symmetric) distance that satisfies the triangle
inequality, that is, for all $i,j,k\in V$,
$\rho(i,j)+\rho(j,k)\geq\rho(i,k).$ (71)
For any source $s\in S$, recall that $\rho(s,K)=\min_{k\in K}\rho(s,k)$ and
let $K_{s}=\\{k\in K:\rho(s,k)=\rho(s,K)\\}$, the set of all the sinks closest
to $s$.
###### Theorem C.4.
Let $s\in S$, $i\in T$ and $k\in K$ such that $\rho(s,i)$ and $\rho(i,k)$ are
both finite. Then, if $k\in K_{s}$ and $i$ lies on the shortest path from $s$
to $k$,
$\lim_{\mu\downarrow
0}\hat{\varPhi}_{i,k}^{s}=\frac{\left[\mathbf{Q}_{ST}\mathbf{Q}^{\rho(s,i)-1}_{TT}\right]_{si}\left[\mathbf{Q}^{\rho(i,k)-1}_{TT}\mathbf{Q}_{TK}\right]_{ik}}{\sum_{k^{\prime}\in
K_{s}}\left[\mathbf{Q}_{ST}\mathbf{Q}^{\rho(s,k)-2}_{TT}\mathbf{Q}_{TK}\right]_{sk^{\prime}}}.$
(72)
Otherwise, $\lim_{\mu\downarrow 0}\hat{\varPhi}_{i,k}^{s}=0$.
###### Proof.
Let $s\in S$, $i\in T$ and $k\in K$. Since, $\rho(s,i)$ and $\rho(i,k)$ are
finite, it follows that $\rho(s,k)$ is also finite, that is, $k$ is reachable
from $s$ through $i$ and the normalized channel tensor
$\boldsymbol{\hat{\Phi}}$ is well defined for all $\mu\in(0,1)$. Recall that
$\hat{\varPhi}_{i,k}^{s}=\frac{\varPhi_{i,k}^{s}}{F_{sK}}=\frac{[\mathbf{P}_{ST}\mathbf{G}]_{si}[\mathbf{G}\mathbf{P}_{TK}]_{ik}}{\sum_{k^{\prime}\in
K}F_{sk^{\prime}}}$ (73)
where
$F_{sk^{\prime}}=[\mathbf{P}_{SK}+\mathbf{P}_{ST}\mathbf{G}\mathbf{P}_{TK}]_{sk^{\prime}}$.
Let $u,v\in T$ and let $d=\rho(u,v)$. It can be easily shown (see Lemma A.3
from (Stojmirović and Yu, 2007) for a partial proof) that
$\left[\mathbf{P}_{TT}^{n}\right]_{uv}=0$ for all $n<d$ and
$\left[\mathbf{P}_{TT}^{d}\right]_{uv}>0$. Therefore,
$G_{uv}=\sum_{n=d}^{\infty}\left[\mathbf{P}_{TT}^{n}\right]_{uv}=\sum_{n=d}^{\infty}\mu^{n}\left[\mathbf{Q}_{TT}^{n}\right]_{uv}=\mu^{d}\left[\mathbf{Q}^{d}_{TT}\right]_{uv}+O(\mu^{d+1})$
as $\mu\downarrow 0$. Hence,
$\displaystyle[\mathbf{P}_{ST}\mathbf{G}]_{si}$ $\displaystyle=\sum_{j\in
T}\mu^{\rho(j,i)+1}Q_{sj}\left[\mathbf{Q}^{\rho(j,i)}_{TT}\right]_{ji}+O(\mu^{\rho(j,i)+2})$
$\displaystyle=\mu^{\rho(s,i)}\left[\mathbf{Q}_{ST}\mathbf{Q}^{\rho(s,i)-1}_{TT}\right]_{si}+O(\mu^{\rho(s,i)+1}),$
(74) $\displaystyle[\mathbf{G}\mathbf{P}_{TK}]_{ik}$ $\displaystyle=\sum_{j\in
T}\mu^{\rho(i,j)+1}\left[\mathbf{Q}^{\rho(i,j)}_{TT}\right]_{ij}Q_{jk}+O(\mu^{\rho(i,j)+2})$
$\displaystyle=\mu^{\rho(i,k)}\left[\mathbf{Q}^{\rho(i,k)-1}_{TT}\mathbf{Q}_{TK}\right]_{ik}+O(\mu^{\rho(i,k)+1}).$
(75)
Let $\xi=\rho(s,k^{\prime\prime})$, where $k^{\prime\prime}\in K_{s}$. We will
consider the denominator of Equation (73) under two separate cases, $\xi=1$
and $\xi>1$.
If $\xi>1$, for all $k^{\prime}\in K$, the vertices $s$ and $k^{\prime}$ are
not adjacent and thus $P_{sk^{\prime}}=0$. Hence, since $s$ and $k^{\prime}$
are connected, there exist $j,j^{\prime}\in T$ such that
$\rho(s,k^{\prime})=\rho(s,j)+\rho(j,j^{\prime})+\rho(j^{\prime},k^{\prime})=\rho(j,j^{\prime})+2$,
implying
$\displaystyle[\mathbf{P}_{ST}\mathbf{G}\mathbf{P}_{TK}]_{sk^{\prime}}$
$\displaystyle=\sum_{j,j^{\prime}\in
T}\mu^{\rho(j,j^{\prime})+2}Q_{sj}\left[\mathbf{Q}^{\rho(j,j^{\prime})}_{TT}\right]_{jj^{\prime}}Q_{j^{\prime}k^{\prime}}+O(\mu^{\rho(j,j^{\prime})+3})$
$\displaystyle=\mu^{\rho(s,k^{\prime})}\left[\mathbf{Q}_{ST}\mathbf{Q}^{\rho(s,k^{\prime})-2}_{TT}\mathbf{Q}_{TK}\right]_{sk^{\prime}}+O(\mu^{\rho(s,k^{\prime})+1}).$
(76)
Similarly,
$\displaystyle F_{sK}$ $\displaystyle=\sum_{k^{\prime}\in
K_{s}}\mu^{\xi}\left[\mathbf{Q}_{ST}\mathbf{Q}^{\xi-2}_{TT}\mathbf{Q}_{TK}\right]_{sk^{\prime}}+O(\mu^{\xi+1}),$
(77)
and, as $\mu\downarrow 0$,
$\hat{\varPhi}_{i,k}^{s}\to\frac{\mu^{\rho(s,i)+\rho(i,k)}\left[\mathbf{Q}_{ST}\mathbf{Q}^{\rho(s,i)-1}_{TT}\right]_{si}\left[\mathbf{Q}^{\rho(i,k)-1}_{TT}\mathbf{Q}_{TK}\right]_{ik}}{\mu^{\xi}\sum_{k^{\prime}\in
K_{s}}\left[\mathbf{Q}_{ST}\mathbf{Q}^{\xi-2}_{TT}\mathbf{Q}_{TK}\right]_{sk^{\prime}}}$
(78)
By the triangle inequality and our assumptions on $s$, $i$ and $k$,
$\rho(s,i)+\rho(i,k)\geq\rho(s,k)\geq\xi.$ (79)
The first inequality becomes an equality if and only if $i$ lies on the
shortest path between $s$ and $k$ while the second is an equality if and only
if $k\in K_{s}$. Therefore, if the assumption of the theorem is satisfied, the
value of $\hat{\varPhi}_{i,k}^{s}$ converges to the value of the right hand
side of Equation (72), while otherwise $\lim_{\mu\downarrow
0}\hat{\varPhi}_{i,k}^{s}=0$.
On the other hand, if $\xi=1$, $F_{sK}\to\sum_{k^{\prime}\in K_{s}}\mu
Q_{sk^{\prime}}+O(\mu^{2})$ and therefore, since $\rho(s,i)+\rho(i,k)\geq 2$,
$\hat{\varPhi}_{i,k}^{s}\to 0$ as $\mu\downarrow 0$. ∎
We have therefore shown that, as $\mu\downarrow 0$, only the nodes associated
with the shortest path from each source to the sink(s) closest to it will have
positive values of the normalized channel tensor – all other entries will be
exactly $0$.
###### Corollary C.5.
Let $s\in S$ and suppose the normalized channel tensor
$\boldsymbol{\hat{\Phi}}$ is well defined for all $\mu\in(0,1)$. Then,
$\lim_{\mu\downarrow 0}\operatorname{\mathbb{E}}(L_{sK})=\rho(s,k),$ (80)
where $k\in K_{s}$.
###### Proof.
Let $s\in S$, let $k\in K_{s}$ and let $d=\rho(s,k)$. For $m=1,2\ldots d-1$,
let $\Pi_{s}(m)=\\{i\in T:\rho(s,i)=m\,\text{and}\,\rho(s,i)+\rho(i,k)=d\\}$.
The set $\Pi_{s}(m)$ consists of all transient nodes that are at the distance
$m$ from $s$ on a shortest path from $s$ to any of the sinks closest to $s$.
By Theorem C.4,
$\displaystyle\lim_{\mu\downarrow 0}\sum_{k^{\prime\prime}\in K}\sum_{i\in
T}\hat{\varPhi}_{i,k^{\prime\prime}}^{s}$
$\displaystyle=\sum_{k^{\prime\prime}\in
K_{s}}\sum_{m=1}^{d-1}\sum_{i\in\Pi_{s}(m)}\frac{\left[\mathbf{Q}_{ST}\mathbf{Q}^{m-1}_{TT}\right]_{si}\left[\mathbf{Q}^{d-m-1}_{TT}\mathbf{Q}_{TK}\right]_{ik^{\prime\prime}}}{\sum_{k^{\prime}\in
K_{s}}\left[\mathbf{Q}_{ST}\mathbf{Q}^{\rho(s,k)-2}_{TT}\mathbf{Q}_{TK}\right]_{sk^{\prime}}}$
$\displaystyle=\sum_{k^{\prime\prime}\in K_{s}}\sum_{m=1}^{d-1}\sum_{i\in
T}\frac{\left[\mathbf{Q}_{ST}\mathbf{Q}^{m-1}_{TT}\right]_{si}\left[\mathbf{Q}^{d-m-1}_{TT}\mathbf{Q}_{TK}\right]_{ik^{\prime\prime}}}{\sum_{k^{\prime}\in
K_{s}}\left[\mathbf{Q}_{ST}\mathbf{Q}^{d-2}_{TT}\mathbf{Q}_{TK}\right]_{sk^{\prime}}}$
$\displaystyle=\sum_{m=1}^{d-1}\frac{\sum_{k^{\prime\prime}\in
K_{s}}\left[\mathbf{Q}_{ST}\mathbf{Q}^{d-2}_{TT}\mathbf{Q}_{TK}\right]_{sk^{\prime\prime}}}{\sum_{k^{\prime}\in
K_{s}}\left[\mathbf{Q}_{ST}\mathbf{Q}^{d-2}_{TT}\mathbf{Q}_{TK}\right]_{sk^{\prime}}}$
$\displaystyle=d-1.$
Therefore, by Equation (69),
$\lim_{\mu\downarrow 0}\operatorname{\mathbb{E}}(L_{sK})=1+\lim_{\mu\downarrow
0}\sum_{k^{\prime}\in K}\sum_{i\in
T}\hat{\varPhi}_{i,k^{\prime}}^{s}=\rho(s,k),$
as required. ∎
## Appendix D Normalized evolution operator
In this appendix, we will prove the statements from 3.5. Recall that in 3.5,
we assumed $0\leq\mu\leq 1$ and defined the transition matrix $\mathbf{N}$
over $V_{K}=\\{i\in V:\bar{F}_{iK}>0\\}$ by
$N_{ij}=\frac{\tilde{P}_{ij}f_{j}}{f_{i}},$
where $f_{k}$ for $k\in K$ are assumed to be positive but otherwise arbitrary
and $f_{i}=\sum_{k\in K}\bar{F}_{ik}f_{k}$ for $i\in S_{K}\cup T_{K}$. Denote
by $\mathbf{G}(\mathbf{N})$, $\mathbf{\bar{F}}(\mathbf{N})$,
$\mathbf{\bar{H}}(\mathbf{N})$, $\boldsymbol{\mathrm{\Phi}}(\mathbf{N})$ the
quantities corresponding to $\mathbf{G}$, $\mathbf{F}$, $\mathbf{H}$ and
$\boldsymbol{\mathrm{\Phi}}$ respectively, when the transition matrix
$\mathbf{P}$ is replaced by $\mathbf{N}$. To make our arguments more concise
we will here additionally assume, without loss of generality, that every node
is connected to a sink via a directed path, that is, that $V_{K}=V$.
Note that $\mathbf{N}$ is indeed well defined in the limit as $\mu\downarrow
0$. For example, if $i,j\in T$, we have from (75)
$\displaystyle N_{ij}$
$\displaystyle=\frac{\tilde{P}_{ij}[\mathbf{G}\mathbf{P}_{TK}]_{jk}f_{k}}{\sum_{k^{\prime}\in
K}[\mathbf{G}\mathbf{P}_{TK}]_{ik^{\prime}}f_{k^{\prime}}}$
$\displaystyle\to\frac{\mu^{\rho(j,k)+1}Q_{ij}\left[\mathbf{Q}^{\rho(j,k)-1}_{TT}\mathbf{Q}_{TK}\right]_{jk}f_{k}}{\sum_{k^{\prime}\in
K}\mu^{\rho(i,k^{\prime})}\left[\mathbf{Q}^{\rho(i,k^{\prime})-1}_{TT}\mathbf{Q}_{TK}\right]_{ik^{\prime}}f_{k^{\prime}}}$
$\displaystyle=\frac{\mu^{\rho(j,k)+1}Q_{ij}\left[\mathbf{Q}^{\rho(j,k)-1}_{TT}\mathbf{Q}_{TK}\right]_{jk}f_{k}}{\mu^{\rho(i,K)}\sum_{k^{\prime}\in
K}\mu^{\rho(i,k^{\prime})-\rho(i,K)}\left[\mathbf{Q}^{\rho(i,k^{\prime})-1}_{TT}\mathbf{Q}_{TK}\right]_{ik^{\prime}}f_{k^{\prime}}}$
$\displaystyle=\begin{cases}0&\text{if $\rho(j,k)>\rho(i,K)-1$,}\\\
\frac{Q_{ij}\left[\mathbf{Q}^{\rho(i,K)-2}_{TT}\mathbf{Q}_{TK}\right]_{jk}f_{k}}{\sum_{k^{\prime}\in
K_{i}}\left[\mathbf{Q}^{\rho(i,K)-1}_{TT}\mathbf{Q}_{TK}\right]_{ik^{\prime}}f_{k^{\prime}}}&\text{if
$\rho(j,k)=\rho(i,K)-1$}.\end{cases}$ (81)
Other cases can also be easily shown using the results from Appendix C.2.
###### Proposition D.1.
Let $\mathbf{f}$ denote an arbitrary vector over $V$. Suppose $i\in S\cup T$.
Then,
$\sum_{j\in V}N_{ij}=1\iff f_{i}=\sum_{k\in K}\bar{F}_{ik}f_{k}.$ (82)
###### Proof.
Write the vector $\mathbf{f}$ as
$\mathbf{f}=[\mathbf{f}_{S},\mathbf{f}_{T},\mathbf{f}_{K}]^{T}$ and the matrix
$\bar{\mathbf{F}}$ as
$\bar{\mathbf{F}}=\left[\bar{\mathbf{F}}_{SK},\bar{\mathbf{F}}_{TK},\bar{\mathbf{F}}_{KK}\right]$,
where
$\bar{\mathbf{F}}_{SK}=\mathbf{P}_{ST}\mathbf{G}\mathbf{P}_{TK}+\mathbf{P}_{SK}$,
$\bar{\mathbf{F}}_{TK}=\mathbf{G}\mathbf{P}_{TK}$ and
$\bar{\mathbf{F}}_{KK}=\mathbb{I}$. The right equality from (82) can then be
written in the block matrix form as
$\left[\begin{array}[]{c}\mathbf{f}_{S}\\\
\mathbf{f}_{T}\end{array}\right]=\left[\begin{array}[]{c}\bar{\mathbf{F}}_{SK}\\\
\bar{\mathbf{F}}_{TK}\end{array}\right]\mathbf{f}_{K}.$ (83)
By definition of $\mathbf{N}$, our premise $\sum_{j\in V}N_{ij}=1$ is
equivalent to
$f_{i}=\sum_{j\in T}P_{ij}f_{j}+\sum_{j\in K}P_{ik}f_{k}.$ (84)
For $i\in T$, Equation (84) can be expressed in matrix form as
$\mathbf{f}_{T}=\mathbf{P}_{TT}\mathbf{f}_{T}+\mathbf{P}_{TK}\mathbf{f}_{K},$
(85)
that is,
$(\mathbb{I}-\mathbf{P}_{TT})\mathbf{f}_{T}=\mathbf{P}_{TK}\mathbf{f}_{K}.$
(86)
Since the matrix $\mathbb{I}-\mathbf{P}_{TT}$ is invertible by our assumption
of connectivity, this is further equivalent to
$\mathbf{f}_{T}=\mathbf{G}\mathbf{P}_{TK}\mathbf{f}_{K}=\bar{\mathbf{F}}_{TK}\mathbf{f}_{K}.$
(87)
For $i\in S$, Equation (84) can be written as
$\mathbf{f}_{S}=\mathbf{P}_{ST}\mathbf{f}_{T}+\mathbf{P}_{SK}\mathbf{f}_{K},$
(88)
which using (87) is equivalent to
$\mathbf{f}_{S}=\mathbf{P}_{ST}\mathbf{G}\mathbf{P}_{TK}\mathbf{f}_{K}+\mathbf{P}_{SK}\mathbf{f}_{K}=\bar{\mathbf{F}}_{SK}\mathbf{f}_{K},$
(89)
as required. ∎
###### Proposition D.2.
The following identities hold:
1. (i)
For all $i,j\in T$,
$\displaystyle[\mathbf{G}(\mathbf{N})]_{ij}=\frac{G_{ij}f_{j}}{f_{i}}$,
2. (ii)
For all $i\in V$ and $k\in K$,
$\displaystyle[\mathbf{\bar{F}}(\mathbf{N})]_{ik}=\frac{\bar{F}_{ik}f_{k}}{f_{i}}$,
3. (iii)
For all $s\in S$ and $i\in V$,
$\displaystyle[\mathbf{\bar{H}}(\mathbf{N})]_{si}=\frac{\bar{H}_{si}f_{i}}{f_{s}}$,
4. (iv)
For all $s\in S$, $i\in V$ and $k\in K$,
$\displaystyle[\boldsymbol{\mathrm{\Phi}}(\mathbf{N})]^{s}_{i,k}=\frac{\varPhi_{i,k}^{s}f_{k}}{f_{s}}$.
###### Proof.
All properties follow from the fact that the transformation from
$\tilde{\mathbf{P}}$ to $\mathbf{N}$ is a similarity transformation.
(i) Let $i,j\in T$. We have
$\displaystyle[\mathbf{G}(\mathbf{N})]_{ij}=\sum_{n=0}^{\infty}[\mathbf{N}_{TT}^{n}]_{ij}=\sum_{n=0}^{\infty}\frac{[\mathbf{P}_{TT}^{n}]_{ij}f_{j}}{f_{i}}=\frac{G_{ij}f_{j}}{f_{i}}.$
(ii) Let $k\in K$ and suppose $i\in K$. Then
$[\mathbf{\bar{F}}(\mathbf{N})]_{ik}=\delta_{ik}=\frac{\delta_{ik}f_{k}}{f_{i}}=\frac{\bar{F}_{ik}f_{k}}{f_{i}}$.
Now suppose $i\in T$. Then,
$\displaystyle[\mathbf{\bar{F}}(\mathbf{N})]_{ik}=[\mathbf{G}(\mathbf{N})\mathbf{N}_{TK}]_{ik}=\sum_{j\in
T}\frac{G_{ij}f_{j}}{f_{i}}\frac{P_{jk}f_{k}}{f_{j}}=\frac{\bar{F}_{ik}f_{k}}{f_{i}}.$
If $i\in S$, we have
$\displaystyle[\mathbf{\bar{F}}(\mathbf{N})]_{ik}=[\mathbf{N}_{SK}+\mathbf{N}_{ST}\mathbf{G}(\mathbf{N})\mathbf{N}_{TK}]_{ik}=\frac{P_{ik}f_{k}}{f_{i}}+\sum_{j\in
T}\sum_{l\in
T}\frac{P_{ij}f_{j}}{f_{i}}\frac{G_{jl}P_{lk}}{f_{j}}=\frac{\bar{F}_{ik}f_{k}}{f_{i}}.$
(iii) Let $s\in S$ and suppose $i\in S$. Then
$[\mathbf{\bar{H}}(\mathbf{N})]_{si}=\delta_{si}=\frac{\delta_{si}f_{i}}{f_{s}}=\frac{\bar{H}_{si}f_{i}}{f_{s}}$.
Now suppose $i\in K$. Then
$[\mathbf{\bar{H}}(\mathbf{N})]_{si}=[\mathbf{\bar{F}}(\mathbf{N})]_{si}=\frac{\bar{F}_{si}f_{i}}{f_{s}}=\frac{\bar{H}_{si}f_{i}}{f_{s}}$.
If $i\in T$,
$\displaystyle[\mathbf{\bar{H}}(\mathbf{N})]_{si}=[\mathbf{N}_{ST}\mathbf{G}(\mathbf{N})]_{si}=\sum_{j\in
T}\frac{P_{sj}f_{j}}{f_{s}}\frac{G_{ji}f_{i}}{f_{j}}=\frac{\bar{H}_{si}f_{i}}{f_{s}}.$
(iv) Let $s\in S$, $i\in V$ and $k\in K$. Then,
$\displaystyle[\boldsymbol{\mathrm{\Phi}}(\mathbf{N})]^{s}_{i,k}$
$\displaystyle=[\mathbf{\bar{H}}(\mathbf{N})]_{si}[\mathbf{\bar{F}}(\mathbf{N})]_{ik}=\frac{\bar{H}_{si}f_{i}}{f_{s}}\frac{\bar{F}_{ik}f_{k}}{f_{i}}=\varPhi_{i,k}^{s}\frac{f_{k}}{f_{s}}.$
∎
## Appendix E SaddleSum enrichment analysis results
Here we show the results of SaddleSum enrichment analysis for ITMs shown in
Fig. 4. The interference values of all nodes (not only those included in the
picture) were submitted to SaddleSum with an E-value cutoff of 0.01 to
retrieve significant terms. The terms database used was Gene Ontology.
### E.1 Fig. 4 (b), $\mu=0.85$
**** RESULTS ****
Database name GO: Saccharomyces cerevisiae
Total database terms 5687
Total database entities 6328
Submitted weights 3860
Valid submitted entity ids 3822
Minimum term size (weighted entities per term) 2
Used database terms 3871
Non-zero weight entities 3421
Unknown submitted entity ids 0
Duplicate submitted entity ids 0
Unresolvable (ignored) conflicting entity ids 0
Resolvable (accepted) conflicting entity ids 65
Entities without submitted weight 2506
E-value cutoff 1.00e-02
Effective database size 3.87e+03
Statistics Lugannani-Rice (sum of weights)
Discretized weights No
Top-ranked weights selected All
Minimum weight selected N/A
******** Molecular Function (3 significant terms) ********
Term ID Name Associ Score E-value
------------------------------------------------------------------------------------------
GO:0004707 MAP kinase activity 4 1.0718 1.69e-03
GO:0004702 receptor signaling protein serine/threon 11 1.1767 5.89e-03
GO:0005057 receptor signaling protein activity 12 1.1770 7.38e-03
******** Biological Process (25 significant terms) ********
Term ID Name Associ Score E-value
------------------------------------------------------------------------------------------
GO:0001403 invasive growth in response to glucose l 43 2.8283 4.02e-08
GO:0044182 filamentous growth of a population of un 64 2.9110 2.22e-07
GO:0070783 growth of unicellular organism as a thre 64 2.9110 2.22e-07
GO:0030447 filamentous growth 91 3.1452 2.90e-07
GO:0040007 growth 127 3.2711 1.15e-06
GO:0007124 pseudohyphal growth 53 2.5558 2.00e-06
GO:0016049 cell growth 66 2.6329 3.49e-06
GO:0008361 regulation of cell size 91 2.6920 1.45e-05
GO:0032535 regulation of cellular component size 93 2.6976 1.58e-05
GO:0090066 regulation of anatomical structure size 93 2.6976 1.58e-05
GO:0000750 pheromone-dependent signal transduction 25 1.8430 6.31e-05
GO:0032005 regulation of conjugation with cellular 25 1.8430 6.31e-05
GO:0019236 response to pheromone 73 2.3335 8.79e-05
GO:0007186 G-protein coupled receptor protein signa 31 1.8830 1.02e-04
GO:0031137 regulation of conjugation with cellular 29 1.8510 1.07e-04
GO:0043900 regulation of multi-organism process 29 1.8510 1.07e-04
GO:0046999 regulation of conjugation 29 1.8510 1.07e-04
GO:0007166 cell surface receptor linked signaling p 32 1.8833 1.16e-04
GO:0051704 multi-organism process 98 2.4439 1.77e-04
GO:0000746 conjugation 88 2.3403 2.28e-04
GO:0000749 response to pheromone involved in conjug 58 2.0239 4.23e-04
GO:0010033 response to organic substance 116 2.3968 6.75e-04
GO:0000747 conjugation with cellular fusion 84 2.0433 2.07e-03
GO:0019953 sexual reproduction 194 2.5998 3.44e-03
GO:0070887 cellular response to chemical stimulus 109 2.0933 5.04e-03
### E.2 Fig. 4 (c), $\mu=1.0$
**** RESULTS ****
Database name GO: Saccharomyces cerevisiae
Total database terms 5687
Total database entities 6328
Submitted weights 3860
Valid submitted entity ids 3822
Minimum term size (weighted entities per term) 2
Used database terms 3871
Non-zero weight entities 3422
Unknown submitted entity ids 0
Duplicate submitted entity ids 0
Unresolvable (ignored) conflicting entity ids 0
Resolvable (accepted) conflicting entity ids 65
Entities without submitted weight 2506
E-value cutoff 1.00e-02
Effective database size 3.87e+03
Statistics Lugannani-Rice (sum of weights)
Discretized weights No
Top-ranked weights selected All
Minimum weight selected N/A
******** Molecular Function (7 significant terms) ********
Term ID Name Associ Score E-value
------------------------------------------------------------------------------------------
GO:0005515 protein binding 440 8.6125 1.01e-04
GO:0004871 signal transducer activity 39 2.4375 1.36e-04
GO:0060089 molecular transducer activity 39 2.4375 1.36e-04
GO:0005488 binding 1103 16.2503 1.25e-03
GO:0004702 receptor signaling protein serine/threon 11 1.4392 1.66e-03
GO:0005057 receptor signaling protein activity 12 1.4433 2.34e-03
GO:0004707 MAP kinase activity 4 1.0216 7.06e-03
******** Cellular Component (7 significant terms) ********
Term ID Name Associ Score E-value
------------------------------------------------------------------------------------------
GO:0042995 cell projection 85 3.9866 7.46e-07
GO:0005937 mating projection 85 3.9866 7.46e-07
GO:0044463 cell projection part 80 3.6525 5.48e-06
GO:0043332 mating projection tip 76 3.5683 5.77e-06
GO:0030427 site of polarized growth 175 4.9448 6.44e-05
GO:0019897 extrinsic to plasma membrane 16 1.7951 2.01e-04
GO:0044459 plasma membrane part 49 2.2951 3.27e-03
******** Biological Process (51 significant terms) ********
Term ID Name Associ Score E-value
------------------------------------------------------------------------------------------
GO:0040007 growth 127 7.0735 3.17e-15
GO:0030447 filamentous growth 91 5.7178 5.09e-13
GO:0016049 cell growth 66 5.0323 1.16e-12
GO:0007165 signal transduction 227 8.1500 2.53e-12
GO:0023033 signaling pathway 234 8.2642 2.65e-12
GO:0023060 signal transmission 228 8.1544 2.79e-12
GO:0023046 signaling process 233 8.1910 4.03e-12
GO:0090066 regulation of anatomical structure size 93 5.4848 6.40e-12
GO:0032535 regulation of cellular component size 93 5.4848 6.40e-12
GO:0019236 response to pheromone 73 5.0215 6.71e-12
GO:0008361 regulation of cell size 91 5.4315 6.95e-12
GO:0007186 G-protein coupled receptor protein signa 31 3.8227 9.40e-12
GO:0032005 regulation of conjugation with cellular 25 3.6044 9.75e-12
GO:0000750 pheromone-dependent signal transduction 25 3.6044 9.75e-12
GO:0007166 cell surface receptor linked signaling p 32 3.8302 1.26e-11
GO:0023052 signaling 315 9.3581 1.44e-11
GO:0019953 sexual reproduction 194 7.2737 2.53e-11
GO:0031137 regulation of conjugation with cellular 29 3.6544 2.84e-11
GO:0043900 regulation of multi-organism process 29 3.6544 2.84e-11
GO:0046999 regulation of conjugation 29 3.6544 2.84e-11
GO:0000749 response to pheromone involved in conjug 58 4.4317 5.59e-11
GO:0001403 invasive growth in response to glucose l 43 3.9711 1.01e-10
GO:0051704 multi-organism process 98 5.2634 1.20e-10
GO:0000746 conjugation 88 5.0391 1.32e-10
GO:0044182 filamentous growth of a population of un 64 4.4501 1.95e-10
GO:0070783 growth of unicellular organism as a thre 64 4.4501 1.95e-10
GO:0000003 reproduction 287 8.4866 2.82e-10
GO:0010033 response to organic substance 116 5.4839 4.15e-10
GO:0007124 pseudohyphal growth 53 4.0334 7.92e-10
GO:0035556 intracellular signal transduction 112 5.3135 8.99e-10
GO:0000747 conjugation with cellular fusion 84 4.6377 2.21e-09
GO:0009966 regulation of signal transduction 65 4.0403 1.13e-08
GO:0023051 regulation of signaling process 65 4.0403 1.13e-08
GO:0070887 cellular response to chemical stimulus 109 4.9497 1.16e-08
GO:0010646 regulation of cell communication 73 4.1338 2.41e-08
GO:0048610 reproductive cellular process 157 5.6129 6.05e-08
GO:0022414 reproductive process 159 5.6190 7.49e-08
GO:0023034 intracellular signaling pathway 193 6.1789 8.00e-08
GO:0050794 regulation of cellular process 961 16.2476 8.21e-07
GO:0065008 regulation of biological quality 331 7.8986 1.04e-06
GO:0007154 cell communication 127 4.6615 1.65e-06
GO:0065009 regulation of molecular function 118 4.3229 6.69e-06
GO:0050789 regulation of biological process 1070 17.0825 8.02e-06
GO:0065007 biological regulation 1252 19.2448 9.39e-06
GO:0050790 regulation of catalytic activity 92 3.6138 4.70e-05
GO:0042221 response to chemical stimulus 320 6.7925 3.12e-04
GO:0007264 small GTPase mediated signal transductio 58 2.5248 2.03e-03
GO:0048284 organelle fusion 55 2.3440 5.73e-03
GO:0035466 regulation of signaling pathway 49 2.2229 6.06e-03
GO:0030010 establishment of cell polarity 78 2.7167 7.76e-03
GO:0051716 cellular response to stimulus 504 8.5748 8.08e-03
### E.3 Fig. 4 (d), $\mu=0.55$
**** RESULTS ****
Database name GO: Saccharomyces cerevisiae
Total database terms 5687
Total database entities 6328
Submitted weights 3860
Valid submitted entity ids 3822
Minimum term size (weighted entities per term) 2
Used database terms 3871
Non-zero weight entities 3421
Unknown submitted entity ids 0
Duplicate submitted entity ids 0
Unresolvable (ignored) conflicting entity ids 0
Resolvable (accepted) conflicting entity ids 65
Entities without submitted weight 2506
E-value cutoff 1.00e-02
Effective database size 3.87e+03
Statistics Lugannani-Rice (sum of weights)
Discretized weights No
Top-ranked weights selected All
Minimum weight selected N/A
******** Biological Process (5 significant terms) ********
Term ID Name Associ Score E-value
------------------------------------------------------------------------------------------
GO:0001403 invasive growth in response to glucose l 43 1.9837 8.15e-04
GO:0044182 filamentous growth of a population of un 64 1.9997 2.62e-03
GO:0070783 growth of unicellular organism as a thre 64 1.9997 2.62e-03
GO:0030447 filamentous growth 91 2.0688 5.19e-03
GO:0007124 pseudohyphal growth 53 1.7633 8.56e-03
## Appendix F Rapid Evaluation of Submatrix Inverses
Consider an invertible block matrix
$\mathbf{M}=\left[\begin{array}[]{cc}\mathbf{A}&\mathbf{B}\\\
\mathbf{C}&\mathbf{D}\end{array}\right]$, where $\mathbf{A}$ is a square
matrix. It is a well known result of linear algebra (see for example Press _et
al._ (2007), 2.7.4) that the inverse of $\mathbf{M}$ can be written as
$\mathbf{M}^{-1}=\left[\begin{array}[]{cc}\mathbf{A}^{-1}+\mathbf{A}^{-1}\mathbf{B}\mathbf{Q}^{-1}\mathbf{C}\mathbf{A}^{-1}&-\mathbf{A}^{-1}\mathbf{B}\mathbf{Q}^{-1}\\\
-\mathbf{Q}^{-1}\mathbf{C}\mathbf{A}^{-1}&\mathbf{Q}^{-1}\end{array}\right],$
(90)
where $\mathbf{Q}=\mathbf{D}-\mathbf{C}\mathbf{A}^{-1}\mathbf{B}$. Suppose we
are interested in computing matrices of the form $\mathbf{A}^{-1}\mathbf{U}$,
where $\mathbf{A}$ is very large and $\mathbf{U}$ is an arbitrary matrix with
appropriate number of rows. If it is necessary to perform a large number of
such computations with different square submatrices $\mathbf{A}$ (the matrix
$\mathbf{M}$ may be permuted in each case to reorder the indices), it could be
effective to precompute the matrix $\mathbf{M}^{-1}$ (or, computationally more
appropriately, its LU-decomposition) once and in each case extract the
required inverse $\mathbf{A}^{-1}$ through simple and relatively inexpensive
algebraic manipulations and permutations.
Indeed, write
$\mathbf{M}^{-1}=\left[\begin{array}[]{cc}\mathbf{X}&\mathbf{Y}\\\
\mathbf{Z}&\mathbf{W}\end{array}\right]$, with each of the blocks known and
with the block sizes the same as that in Equation (90). One observes that
$\mathbf{W}=\mathbf{Q}^{-1}$ and hence
$\mathbf{Y}\mathbf{W}^{-1}\mathbf{Z}=\mathbf{A}^{-1}\mathbf{B}\mathbf{Q}^{-1}\mathbf{C}\mathbf{A}^{-1}$.
Therefore,
$\mathbf{A}^{-1}=\mathbf{X}-\mathbf{Y}\mathbf{W}^{-1}\mathbf{Z},$ (91)
Since $\mathbf{W}$ is assumed to be much smaller in size than $\mathbf{A}$,
this gives rise to a rapid inverse formula with only index permutation needed.
This method was mentioned earlier in a similar context by Zhang _et al._
(2007).
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|
arxiv-papers
| 2009-01-02T21:45:34 |
2024-09-04T02:48:59.686925
|
{
"license": "Public Domain",
"authors": "Aleksandar Stojmirovi\\'c and Yi-Kuo Yu",
"submitter": "Aleksandar Stojmirovi\\'c",
"url": "https://arxiv.org/abs/0901.0287"
}
|
0901.0364
|
# Collins diffraction formula and the Wigner function in entangled state
representation
Hong-Yi Fan1,2 and Li-yun Hu${}^{1\text{*}}$
1Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China
2Department of Material Science and Engineering,
University of Science and Technology of China, Hefei, Anhui 230026, China
Corresponding author. E-mail address: hlyun2008@126.com.
###### Abstract
Based on the correspondence between Collins diffraction formula (optical
Fresnel transform) and the transformation matrix element of a three-parameters
two-mode squeezing operator in the entangled state representation (Opt. Lett.
31 (2006) 2622) we further explore the relationship between output field
intensity determined by the Collins formula and the input field’s probability
distribution along an infinitely thin phase space strip both in spacial domain
and frequency domain. The entangled Wigner function is introduced for
recapitulating the result.
OCIS codes: 070.2590, 270.6570
In a preceding Letter [1] we have reported that the Collins diffraction
formula in cylindrical coordinates is just the transformation matrix element
of a three-parameter ($k$ and $t$ are complex and satisfy the unimodularity
condition $kk^{\ast}-tt^{\ast}=1)$ two-mode squeezing operator [2, 3]
$F^{\left(t,k\right)}=\exp\left(\frac{t}{k^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}\right)\exp\left[\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}+1\right)\ln\left(k^{\ast}\right)^{-1}\right]\exp\left(-\frac{t^{\ast}}{k^{\ast}}a_{1}a_{2}\right),$
(1)
in the deduced entangled state representation $\left\langle
s,r^{\prime}\right|,$
$\displaystyle\phi_{s}\left(r^{\prime}\right)$ $\displaystyle\equiv$
$\displaystyle\left\langle
s,r^{\prime}\right|\left.\phi\right\rangle=\left\langle
s,r^{\prime}\right|F^{\left(t,k\right)}\left|\psi\right\rangle$ (2)
$\displaystyle=$
$\displaystyle\frac{\mathtt{i}^{s}}{2\mathtt{i}B}\int_{0}^{\infty}\mathtt{d}\left(r^{2}\right)\exp\left[\frac{\mathtt{i}}{2B}\left(Ar^{2}+Dr^{\prime
2}\right)\right]J_{s}\left(-\frac{rr^{\prime}}{B}\right)\psi_{s}\left(r\right),$
where $\psi_{s}$ and $\phi_{s}$ denote the incoming and output light,
respectively, $[a_{i},a_{j}^{\dagger}]=\delta_{i,j},$
$k=\frac{1}{2}\left[A+D-\mathtt{i}\left(B-C\right)\right],\text{\ \
}t=\frac{1}{2}\left[A-D+\mathtt{i}\left(B+C\right)\right],$ (3)
we see that the relation $kk^{\ast}-tt^{\ast}=1$ becomes $AD-BC=1$, $J_{s}$ is
the $s$th Bessel function, and
$\left\langle
s,r^{\prime}\right|=\frac{1}{2\pi}\int_{0}^{2\pi}\mathtt{d}\theta
e^{\mathtt{i}s\theta}\left\langle\eta=r^{\prime}e^{\mathtt{i}\theta}\right|,$
(4)
here $\left|\eta\right\rangle$ is the entangled states in two-mode Fock space
[4, 5, 6, 7] named after Einstein-Podolsky-Rosen (EPR)’s [8] concept of
quantum entanglement,
$\left|\eta\right\rangle=\exp\\{-\frac{1}{2}\left|\eta\right|^{2}+\eta
a_{1}^{\dagger}-\eta^{\ast}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger}\\}\left|00\right\rangle.$
(5)
Thus $\left\langle
s,r^{\prime}\right|F^{\left(t,k\right)}\left|\psi\right\rangle$ is the quantum
optics version of the Collins formula (generalized Hankel transformation). In
[9] we have also found
$\mathcal{K}^{\left(t,k\right)}\left(\eta^{\prime},\eta\right)=\frac{1}{\pi}\left\langle\eta^{\prime}\right|F^{\left(t,k\right)}\left|\eta\right\rangle=\frac{1}{2\mathtt{i}B\pi}\exp\left\\{\frac{\mathtt{i}}{2B}\left[A\left|\eta\right|^{2}-\left(\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}\right)+D\left|\eta^{\prime}\right|^{2}\right]\right\\}.$
(6)
Comparing with the integral kernel of usual Fresnel transform which describes
how a general beam $\psi\left(x^{\prime}\right),$ propagating through an
$\left(ABCD\right)$ optical paraxial system, becomes output field
$\phi\left(x\right)$[10, 11]
$\phi\left(x\right)=\int_{-\infty}^{\infty}\mathcal{K}\left(x,x^{\prime}\right)\psi\left(x^{\prime}\right)\mathtt{d}x^{\prime},$
(7)
where $AD-BC=1,$
$\mathcal{K}\left(x,x^{\prime}\right)=\frac{1}{\sqrt{2\pi\mathtt{i}B}}\exp\left[\frac{\mathtt{i}}{2B}\left(Ax^{\prime
2}-2x^{\prime}x+Dx^{2}\right)\right],$ (8)
we see that $\mathcal{K}^{\left(t,k\right)}\left(\eta^{\prime},\eta\right)$
can be considered as the integration kernel of 2-dimensional entangled optical
Fresnel transform,
$\Psi\left(\eta^{\prime}\right)=\int\mathcal{K}^{\left(t,k\right)}\left(\eta^{\prime},\eta\right)\Phi\left(\eta\right)\mathtt{d}^{2}\eta,$
(9)
in this sense $F^{\left(t,k\right)}$ can be named entangled Fresnel operator
(EFO), here
$\Phi\left(\eta\right)=\left\langle\eta\right|\left.\Phi\right\rangle,$
$\Psi\left(\eta^{\prime}\right)=\left\langle\eta^{\prime}\right|\left.\Psi\right\rangle,$
and we have used the completeness relation
$\int\frac{\mathtt{d}^{2}\eta}{\pi}\left|\eta\right\rangle\left\langle\eta\right|=1.$
Clearly, if the $\left[ABCD\right]$ system is changed to
$\left[D\left(-B\right)\left(-C\right)A\right]$ system, then Eq. (9) should
read
$\Psi\left(\eta^{\prime}\right)=\int\mathcal{K}_{2}^{\left(D,-B,-C\right)}\left(\eta^{\prime},\eta\right)\Phi\left(\eta\right)\mathtt{d}^{2}\eta,$
(10)
where $\mathcal{K}_{2}^{\left(D,-B,-C\right)}$ is
$\mathcal{K}_{2}^{\left(D,-B,-C\right)}\left(\eta^{\prime},\eta\right)=\frac{1}{-2\mathtt{i}B\pi}\exp\left\\{\frac{\mathtt{i}}{-2B}\left[D\left|\eta\right|^{2}-\left(\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}\right)+A\left|\eta^{\prime}\right|^{2}\right]\right\\}.$
(11)
On the other hand, signals or images in optical information theory may be
described directly or indirectly by the Wigner distribution function (WDF)
[12]. In one-dimensional (1D) case, the WDF of an optical signal field
$\psi\left(x\right)$ is defined as
$W_{\psi}(\nu,x)=\int_{-\infty}^{+\infty}\frac{\mathtt{d}u}{2\pi}e^{\mathtt{i}\nu
u}\psi^{\ast}\left(x+\frac{u}{2}\right)\psi\left(x-\frac{u}{2}\right).$ (12)
$W_{\psi}(\nu,x)$ involves both spatial distribution information and space-
frequency distribution information of the signal, $\nu$ is named space
frequency. Now, let us consider the entangled case. Like Eq. (12), it is
natural to introduce the 2-D complex Wigner transform as
$W\left(\sigma,\gamma\right)=\int\frac{\mathtt{d}^{2}\eta}{\pi^{3}}\psi\left(\sigma+\eta\right)\psi^{\ast}\left(\sigma-\eta\right)e^{\eta\gamma^{\ast}-\eta^{\ast}\gamma},$
(13)
where $\sigma,$ $\gamma,$ $\eta$ are all complex variables. To see its
physical meaning, using the integration formula of Dirac $\delta-$function, we
perform the following integration,
$\int\mathtt{d}^{2}\gamma
W\left(\sigma,\gamma\right)=\int\frac{\mathtt{d}^{2}\eta}{\pi}\psi\left(\sigma+\eta\right)\psi^{\ast}\left(\sigma-\eta\right)\delta\left(\eta\right)\delta\left(\eta^{\ast}\right)=\frac{1}{\pi}\left|\psi\left(\sigma\right)\right|^{2},$
(14)
which is just the probability distribution of the complex function
$\psi$($\sigma$). Further, let the ordinary Fourier transforms of
$\psi\left(\sigma\right)$ be $j\left(\zeta\right),$
$\psi\left(\sigma\right)=\int\frac{\mathtt{d}^{2}\zeta}{2\pi}j\left(-\zeta\right)e^{\left(\zeta^{\ast}\sigma-\zeta\sigma^{\ast}\right)/2},$
(15)
then substituting (15) into (13) leads to
$\displaystyle W\left(\sigma,\gamma\right)$ $\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\eta}{\pi^{3}}\frac{\mathtt{d}^{2}\zeta}{2\pi}\frac{\mathtt{d}^{2}\zeta^{{}^{\prime}}}{2\pi}j\left(-\zeta\right)j^{\ast}\left(-\zeta^{{}^{\prime}}\right)e^{\frac{\left(\zeta^{\ast}-\zeta^{/\ast}\right)\sigma-\left(\zeta-\zeta^{/}\right)\sigma^{\ast}}{2}}e^{\eta\left(\gamma^{\ast}+\frac{\zeta^{\ast}+\zeta^{/\ast}}{2}\right)-\eta^{\ast}\left(\gamma+\frac{\zeta+\zeta^{{}^{\prime}}}{2}\right)}$
(16) $\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\zeta}{\pi^{3}}j\left(-\zeta\right)j^{\ast}\left(2\gamma+\zeta\right)e^{\left(\zeta^{\ast}+\gamma^{\ast}\right)\sigma-\left(\zeta+\gamma\right)\sigma^{\ast}}=\int\frac{\mathtt{d}^{2}\zeta}{\pi^{3}}j\left(\gamma-\zeta\right)j^{\ast}\left(\gamma+\zeta\right)e^{\zeta^{\ast}\sigma-\zeta\sigma^{\ast}}.$
It then follows from (16) that
$\displaystyle\int\mathtt{d}^{2}\sigma W\left(\sigma,\gamma\right)$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\zeta}{\pi^{3}}j\left(\gamma-\zeta\right)j^{\ast}\left(\zeta+\gamma\right)\int\mathtt{d}^{2}\sigma
e^{\zeta^{\ast}\sigma-\zeta\sigma^{\ast}}$ (17) $\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\zeta}{\pi}j\left(\gamma-\zeta\right)j^{\ast}\left(\zeta+\gamma\right)\delta\left(\zeta\right)\delta\left(\zeta^{\ast}\right)=\frac{1}{\pi}\left|j\left(\gamma\right)\right|^{2},$
which is the probability distribution of the complex function
$j\left(\gamma\right)$. Thus our definition in (13) leads to two marginal
distributions in $\sigma$ and $\gamma$ phase space, respectively. Hence
$W\left(\sigma,\gamma\right)$ is indeed the correct complex 2-D Wigner
function (Wigner transform) of complex function $\psi\left(\sigma\right)$ or
$j\left(\gamma\right)$. If one wants to reconstruct the Wigner function by
using various probability distributions, obviously the “position density”
$\left|\left\langle\sigma\right|\left.\psi\right\rangle\right|^{2}$ and the
space-frequency density
$\left|\left\langle\gamma\right|\left.\psi\right\rangle\right|^{2}$ are not
enough, so we extend
$\delta\left(\eta\right)\delta\left(\eta^{\ast}\right)\equiv\delta\left(\eta_{1}\right)\delta\left(\eta_{2}\right)$
to
$\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right)\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)$
and generalize (14) to,
$R_{2}\left(\eta_{1},\eta_{2}\right)\equiv\pi\int\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right)\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)W\left(\sigma,\gamma\right)\mathtt{d}^{2}\sigma\mathtt{d}^{2}\gamma,$
(18)
$R_{2}\left(\eta_{1},\eta_{2}\right)$ is also a probability distribution along
an infinitely thin phase space strip denoted by the real parameters $B,D$,
which is a generalized entangled Radon transform [13, 14] of the two-mode
Wigner function (in the entangled form) [15, 16],
Then an interesting question naturally arises: what is the relation between
the generalized Fresnel transform and the WDF in entangled state
representation?
We begin with rewriting the 2-D WF (13) as
$\displaystyle W\left(\sigma,\gamma\right)$ $\displaystyle=$
$\displaystyle\int\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}\int\frac{\mathtt{d}^{2}\eta}{\pi^{3}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\delta^{(2)}\left(\sigma^{\prime}-\sigma-\eta\right)\delta^{(2)}\left(\sigma-\eta-\sigma^{\prime\prime}\right)e^{\eta\gamma^{\ast}-\eta^{\ast}\gamma}$
(19) $\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{\pi^{3}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\delta^{\left(2\right)}\left(2\sigma-\sigma^{\prime}-\sigma^{\prime\prime}\right)e^{\left(\sigma^{\prime}-\sigma\right)\gamma^{\ast}-\left(\sigma^{\prime}-\sigma\right)^{\ast}\gamma}.$
Substituting (19) into (18) we rewrite the Radon transform of
$W\left(\sigma,\gamma\right)$ as
($\mathtt{d}^{2}\sigma=\mathtt{d}\sigma_{1}\mathtt{d}\sigma_{2},$
$\mathtt{d}^{2}\gamma=\mathtt{d}\gamma_{1}\mathtt{d}\gamma_{2}$)
$\displaystyle R_{2}\left(\eta_{1},\eta_{2}\right)$ $\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{\pi^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\int\mathtt{d}^{2}\sigma\mathtt{d}^{2}\gamma\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)$
(20)
$\displaystyle\times\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right)\delta^{\left(2\right)}\left(2\sigma-\sigma^{\prime}-\sigma^{\prime\prime}\right)e^{\left(\sigma^{\prime}-\sigma\right)\gamma^{\ast}-\left(\sigma^{\prime}-\sigma\right)^{\ast}\gamma}$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{4\pi^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\int\mathtt{d}^{2}\gamma\delta\left(\eta_{2}-D\frac{\sigma_{2}^{\prime}+\sigma_{2}^{\prime\prime}}{2}+B\gamma_{1}\right)$
$\displaystyle\times\delta\left(\eta_{1}-D\frac{\sigma_{1}^{\prime}+\sigma_{1}^{\prime\prime}}{2}-B\gamma_{2}\right)\exp\left\\{\allowbreak
i\left[\left(\sigma_{2}^{\prime}-\sigma_{2}^{\prime\prime}\right)\gamma_{1}-\left(\sigma_{1}^{\prime}-\sigma_{1}^{\prime\prime}\right)\gamma_{2}\right]\right\\}$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{4B^{2}\pi^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)$
$\displaystyle\times\exp\left\\{\allowbreak\frac{i}{B}\left[\left(\sigma_{2}^{\prime}-\sigma_{2}^{\prime\prime}\right)\left(-\eta_{2}+D\frac{\sigma_{2}^{\prime}+\sigma_{2}^{\prime\prime}}{2}\right)-\left(\sigma_{1}^{\prime}-\sigma_{1}^{\prime\prime}\right)\left(\eta_{1}-D\frac{\sigma_{1}^{\prime}+\sigma_{1}^{\prime\prime}}{2}\right)\right]\right\\}$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{4B^{2}\pi^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\exp\left\\{\frac{i}{2B}\left[D\left(\left|\sigma^{\prime}\right|^{2}-\left|\sigma^{\prime\prime}\right|^{2}\right)-2\eta_{1}\left(\sigma_{1}^{\prime}-\allowbreak\sigma_{1}^{\prime\prime}\right)-2\eta_{2}\left(\sigma_{2}^{\prime}-\sigma_{2}^{\prime\prime}\right)\right]\right\\}.$
On the other hand, when the beam $\Phi\left(\eta\right)$ propagates through
the $\left[D\left(-B\right)\left(-C\right)A\right]$ optical system, according
to the Fresnel integration (10)-(11), we have
$\displaystyle\left|\Psi\left(\eta^{\prime}\right)\right|^{2}$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\eta}{\pi}\mathcal{K}_{2}^{\left(D,-B,-C\right)}\left(\eta^{\prime},\eta\right)\Phi\left(\eta\right)\int\frac{\mathtt{d}^{2}\eta^{\prime\prime}}{\pi}\mathcal{K}_{2}^{\ast\left(D,-B,-C\right)}\left(\eta^{\prime},\eta^{\prime\prime}\right)\Phi^{\ast}\left(\eta^{\prime\prime}\right)$
(21) $\displaystyle=$
$\displaystyle\frac{1}{4B^{2}}\int\frac{\mathtt{d}^{2}\eta}{\pi}\exp\left\\{\frac{\mathtt{i}}{2B}\left[-D\left|\eta\right|^{2}+\left(\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}\right)-A\left|\eta^{\prime}\right|^{2}\right]\right\\}\Phi\left(\eta\right)$
$\displaystyle\times\int\frac{\mathtt{d}^{2}\eta^{\prime\prime}}{\pi}\exp\left\\{\frac{\mathtt{i}}{2B}\left[D\left|\eta^{\prime\prime}\right|^{2}-\left(\eta^{\prime\prime\ast}\eta^{\prime}+\eta^{\prime\prime}\eta^{\prime\ast}\right)+A\left|\eta^{\prime}\right|^{2}\right]\right\\}\Phi^{\ast}\left(\eta^{\prime\prime}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4B^{2}\pi^{2}}\int\frac{\mathtt{d}^{2}\eta}{\pi}\Phi\left(\eta\right)\Phi^{\ast}\left(\eta^{\prime\prime}\right)\exp\left\\{\frac{\mathtt{i}}{2B}\left[D\left(\left|\eta^{\prime\prime}\right|^{2}-\left|\eta\right|^{2}\right)-2\eta_{1}^{\prime}\left(\eta_{1}^{\prime\prime}-\eta_{1}\right)-2\eta_{2}^{\prime}\left(\eta_{2}^{\prime\prime}-\eta_{2}\right)\right]\right\\},$
which is the same as $R_{2}\left(\eta_{1},\eta_{2}\right)$ in (20). So
combining (20), (10)-(11) and (21) we reach the conclusion
$\displaystyle\left|\frac{1}{-2\mathtt{i}B}\int\frac{\mathtt{d}^{2}\eta}{\pi}\exp\left\\{\frac{\mathtt{i}}{-2B}\left[D\left|\eta\right|^{2}-\left(\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}\right)+A\left|\eta^{\prime}\right|^{2}\right]\right\\}\Phi\left(\eta\right)\right|^{2}$
(22) $\displaystyle=$
$\displaystyle\pi\int\delta\left(\eta_{1}^{\prime}-D\sigma_{1}-B\gamma_{2}\right)\delta\left(\eta_{2}^{\prime}-D\sigma_{2}+B\gamma_{1}\right)W\left(\sigma,\gamma\right)\mathtt{d}^{2}\sigma\mathtt{d}^{2}\gamma,$
where $AD-BC=1$. The physical meaning of Eq. (22) is: when an input field
propagates through an optical $\left[D\left(-B\right)\left(-C\right)A\right]$
system, the energy density of the output field is equal to the Radon transform
of the two-mode entangled Wigner function of the input field. So far as our
knowledge is concerned, this conclusion seems new. Eq. (22) is the
relationship between the input amplitude and output one in spatial-domain.
Next we turn to the frequency domain.
If taking the matrix element of $F^{\left(t,k\right)}$ in the
$\left|\xi\right\rangle$ representation which is conjugate to
$\left|\eta\right\rangle$, where the overlap
$\left\langle\eta\right|\left.\xi\right\rangle$ is
$\left\langle\eta\right|\left.\xi\right\rangle=\frac{1}{2}\exp[(\xi\eta^{\ast}-\xi^{\ast}\eta)/2],$
we obtain the 2-dimensional GFT in its ‘frequency domain’, i.e.,
$\displaystyle\frac{1}{\pi}\left\langle\xi^{\prime}\right|F^{\left(t,k\right)}\left|\xi\right\rangle=\int\frac{\mathtt{d}^{2}\eta\mathtt{d}^{2}\eta^{\prime}}{\pi^{3}}\left\langle\xi^{\prime}\right|\left.\eta^{\prime}\right\rangle\left\langle\eta^{\prime}\right|F^{\left(t,k\right)}\left|\eta\right\rangle\left\langle\eta\right|\left.\xi\right\rangle$
(23) $\displaystyle=$
$\displaystyle\frac{1}{4}\int\frac{\mathtt{d}^{2}\eta\mathtt{d}^{2}\eta^{\prime}}{\pi^{2}}\exp\left(\frac{\xi^{\prime\ast}\eta^{\prime}-\xi^{\prime}\eta^{\prime\ast}+\xi\eta^{\ast}-\xi^{\ast}\eta}{2}\right)\mathcal{K}^{\left(t,k\right)}\left(\eta^{\prime},\eta\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2\mathtt{i}\left(-C\right)\pi}\exp\left[\frac{\mathtt{i}}{2\left(-C\right)}\left(D\left|\xi\right|^{2}+A\left|\xi^{\prime}\right|^{2}-\xi^{\prime\ast}\xi-\xi^{\prime}\xi^{\ast}\right)\right]\equiv\mathcal{K}_{2}^{N}\left(\xi^{\prime},\xi\right),$
where the superscript $N$ of $\mathcal{K}_{2}^{N}$ means that this transform
kernel corresponds to the parameter matrix $N=\left[D,-C,-B,A\right].$ Thus if
the $\left[D,-C,-B,A\right]$ system is changed to
$\tilde{N}=\left[A,C,B,D\right]$ system, the GFT in its ‘frequency domain’ is
given by
$\Psi\left(\xi^{\prime}\right)=\int\mathcal{K}_{2}^{\tilde{N}}\left(\xi^{\prime},\xi\right)\Phi\left(\xi\right)\mathtt{d}^{2}\xi,$
(24)
where $\mathcal{K}_{2}^{\tilde{N}}\left(\xi^{\prime},\xi\right)$ is
$\mathcal{K}_{2}^{N}\left(\xi^{\prime},\xi\right)=\frac{1}{2\mathtt{i}C\pi}\exp\left[\frac{\mathtt{i}}{2C}\left(A\left|\xi\right|^{2}+D\left|\xi^{\prime}\right|^{2}-\xi^{\prime\ast}\xi-\xi^{\prime}\xi^{\ast}\right)\right].$
(25)
It then follows from Eqs.(24) and (25) that
$\displaystyle\left|\Psi\left(\xi^{\prime}\right)\right|^{2}$ $\displaystyle=$
$\displaystyle\int\mathcal{K}_{2}^{\tilde{N}}\left(\xi^{\prime},\xi\right)\Phi\left(\xi\right)\mathtt{d}^{2}\xi\int\mathcal{K}_{2}^{\ast\tilde{N}}\left(\xi^{\prime},\xi^{\prime\prime}\right)\Phi^{\ast}\left(\xi^{\prime\prime}\right)\mathtt{d}^{2}\xi^{\prime\prime}$
(26) $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}C^{2}}\int\mathtt{d}^{2}\xi\mathtt{d}^{2}\xi^{\prime\prime}\Phi\left(\xi\right)\Phi^{\ast}\left(\xi^{\prime\prime}\right)$
$\displaystyle\times\exp\left\\{\frac{\mathtt{i}}{2C}\left[A\left(\left|\xi\right|^{2}-\left|\xi^{\prime\prime}\right|^{2}\right)+2\xi_{1}^{\prime}\left(\xi_{1}^{\prime\prime}-\xi_{1}\right)+2\xi_{2}^{\prime}\left(\xi_{2}^{\prime\prime}-\xi_{2}\right)\right]\right\\}.$
On the other hand, in similar to (18), we consider the integration transform,
$R_{2}\left(\xi_{1},\xi_{2}\right)=\pi\int\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right)W\left(\sigma,\gamma\right)\mathtt{d}^{2}\sigma\mathtt{d}^{2}\gamma,$
(27)
$R_{2}\left(\xi_{1},\xi_{2}\right)$ is also a probability distribution along
an infinitely thin phase space strip denoted by the real parameters $A,C$.
Substituting (19) into (27) yields
$\displaystyle R_{2}\left(\xi_{1},\xi_{2}\right)$ $\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{\pi^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\int\delta^{\left(2\right)}\left(2\sigma-\sigma^{\prime}-\sigma^{\prime\prime}\right)\mathtt{d}^{2}\sigma\mathtt{d}^{2}\gamma$
(28)
$\displaystyle\times\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right)\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)e^{\left(\sigma^{\prime}-\sigma\right)\gamma^{\ast}-\left(\sigma^{\prime}-\sigma\right)^{\ast}\gamma}$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{4\pi^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\int\mathtt{d}^{2}\gamma\delta\left(\xi_{2}-A\frac{\sigma_{2}^{\prime}+\sigma_{2}^{\prime\prime}}{2}+C\gamma_{1}\right)$
$\displaystyle\times\delta\left(\xi_{1}-A\frac{\sigma_{1}^{\prime}+\sigma_{1}^{\prime\prime}}{2}-C\gamma_{2}\right)\exp\left[\frac{\sigma^{\prime}-\sigma^{\prime\prime}}{2}\gamma^{\ast}-\frac{\sigma^{\prime\ast}-\sigma^{\prime\prime\ast}}{2}\gamma\right]$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{4\pi^{2}C^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)$
$\displaystyle\times\exp\left\\{\frac{\mathtt{i}}{2C}\left[A\left(\left|\sigma^{\prime}\right|^{2}-\left|\sigma^{\prime\prime}\right|^{2}\right)+2\xi_{1}\left(\sigma_{1}^{\prime\prime}-\sigma_{1}^{\prime}\right)+2\xi_{2}\left(\sigma_{2}^{\prime\prime}-\sigma_{2}^{\prime}\right)\right]\right\\},$
which is the same as $\left|\Psi\left(\xi^{\prime}\right)\right|^{2}$ in (26).
So combining (28), (24)-(25), and (26) we can draw the conclusion
$\displaystyle\left|\frac{1}{2\mathtt{i}C\pi}\int\exp\left[\frac{\mathtt{i}}{2C}\left(A\left|\xi\right|^{2}+D\left|\xi^{\prime}\right|^{2}-\xi^{\prime\ast}\xi-\xi^{\prime}\xi^{\ast}\right)\right]\Phi\left(\xi\right)\mathtt{d}^{2}\xi\right|^{2}$
(29) $\displaystyle=$
$\displaystyle\pi\int\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right)W\left(\sigma,\gamma\right)\mathtt{d}^{2}\sigma\mathtt{d}^{2}\gamma.$
This is the relationship between the output amplitude and input one’s
entangled Wigner function in ‘frequency domain’.
In sum, based on the correspondence between Collins diffraction formula
(optical Fresnel transform) and the transformation matrix element of a three-
parameters two-mode squeezing operator in the entangled state representation,
we have explored the relationship between output field intensity determined by
the Collins formula and the input field’s probability distribution along an
infinitely thin phase space strip. The entangled Wigner function is introduced
for recapitulating the result.
Work supported by the National Natural Science Foundation of China (Grant Nos
10775097 and 10874174).
## References
* [1] H.-Y. Fan and H.-L. Lu, “Collins diffraction formula studied in quantum optics,” Opt. Lett. 31, 2622-2624 (2006).
* [2] D. F. Walls, “Squeezed states of light,” Nature 324, 210 (1986).
* [3] H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226 (1976).
* [4] H. Y. Fan, H. R. Zaidi and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831 (1987).
* [5] H. Y. Fan, “Operator ordring in quantum optics theory and the development of Dirac ’s symbolic method,” J. Opt. B: Quantum Semiclassical Opt. 5, R147 (2003).
* [6] H. Y. Fan and J. R. Klauder, “Eigenvectors of two particles’ relative position and total momentum,” Phys. Rev. A 49, 704 (1994).
* [7] A. Wünsche, “About integration within ordered products in quantum optics,” J. Opt. B: Quantum Semiclassical Opt. 1, R11 (1999).
* [8] A. Einstein, B. Podolsky and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Phys. Rev. 47, 777 (1935).
* [9] H.-Y. Fan and H.-L. Lu, “2-mode Fresnel operator and entangled Fresnel transform”, Phys. Lett. A 334, 132 (2005).
* [10] D. F. V. James and G. S. Agarwal, “The Generalized Fresnel Transform and its Application to Optics,” Opt. Commun. 126, 207 (1996).
* [11] S. A. Collins, “Lens-system diffraction integral written in terms of matrix optic,” J. Opt. Soc. Am. A 60, 1168 (1970).
* [12] E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749 (1932).
* [13] J. Radon, “Uber die Bestimmung von Funktionen Durch Ihre Integralwerte Langs Gewisser Mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig Math. Phys. K1. 69, 262 (1917).
* [14] Y. Zhang, B. Guo, B. Dong and G. Yang, “Optical implementations of the Radon– Wigner display for one-dimensional signals,” Opt. Lett. 23, 1126 (1998).
* [15] Wolfgang P. Schleich, Quantum Optics in Phase Space, (Wiley-VCH, Berlin, 2001) and references therein.
* [16] H.-Y. Fan, “Time evolution of the Wigner function in the entangled-state representation,” Phys. Rev. A 65, 064102 (2002).
|
arxiv-papers
| 2009-01-04T08:00:19 |
2024-09-04T02:48:59.703742
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hong-yi Fan and Li-yun Hu",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/0901.0364"
}
|
0901.0457
|
# Out-of-plane impurities induced the deviation from the monotonic d-wave
superconducting gap in cuprate superconductors
Zhi Wang and Shiping Feng∗ Department of Physics, Beijing Normal University,
Beijing 100875, China
###### Abstract
The electronic structure of cuprate superconductors is studied within the
kinetic energy driven superconducting mechanism in the presence of out-of-
plane impurities. With increasing impurity concentration, although both
superconducting coherence peaks around the nodal and antinodal regions are
suppressed, the position of the leading-edge mid-point of the electron
spectrum around the nodal region remains at the same position, whereas around
the antinodal region it is shifted towards higher binding energies, this leads
to a strong deviation from the monotonic d-wave superconducting gap in the
out-of-plane impurity-controlled cuprate superconductors.
###### pacs:
74.62.Dh, 74.20.Rp, 74.25.Jb, 74.20.Mn
## I Introduction
The superconducting gap is a fundamental property of superconductors
schrieffer , and the nature of its anisotropy has played a crucial role in the
testing of the microscopic theory of superconductivity in cuprate
superconductors anderson . Experimentally, by virtue of systematic
measurements tsuei , particularly using the angle-resolved photoemission
spectroscopy (ARPES) technique shen , the d-wave nature of the superconducting
gap has been well established by now. In particular, this d-wave
superconducting symmetry remains one of the cornerstones of our understanding
of the physics in cuprate superconductors shen ; tsuei ; perali ; sangiovanni
; zhang . The early ARPES measurements on the cuprate superconductor
Bi2Sr2CaCu2O8+δ shi showed that in the real space the gap function and the
pairing force have a range of one lattice spacing, and then the
superconducting gap function is of the monotonic d-wave form $\Delta_{\bf
k}=\Delta[{\rm cos}k_{x}-{\rm cos}k_{y}]/2$. Later, the ARPES measurements on
the cuprate superconductor Bi2Sr2CaCu2O8+δ mesot indicated that the
superconducting gap significantly deviates from this monotonic d-wave form.
Furthermore, it was argued that this deviation should be attributed to an
increase of the electron correlation, which may increase the intensity of the
higher order of the harmonic component in the monotonic d-wave gap function
mesot . However, recent ARPES measurements kondo ; hashimoto on the cuprate
superconductors (Bi,Pb)2(Sr,La)2CuO6+δ and Bi2Sr1.6$Ln$0.4CuO6+δ ($Ln$-Bi2201)
with $Ln$=La, Nd, and Gd showed that a much stronger deviation from the
monotonic d-wave superconducting gap form is unlikely to be due to the strong
correlation effect.
The cuprate superconductors have a layered structure consisting of the two-
dimensional CuO2 layers separated by insulating layers bednorz ; kastner . The
single common feature is the presence of the CuO2 plane shen ; kastner , and
it seems evident that the unusual behaviors of cuprate superconductors are
dominated by this CuO2 plane anderson . It has been well established that the
Cu2+ ions exhibit an antiferromagnetic long-range order in the parent
compounds of cuprate superconductors, and superconductivity occurs when the
antiferromagnetic long-range order state is suppressed by doped charge
carriers kastner . Since these doped charge carriers are induced by the
replacement of ions by those with different valences or the addition of excess
oxygens in the block layer, therefore in principle, all cuprate
superconductors have naturally impurities (or disorder). However, for the
cuprate superconductors (Bi,Pb)2(Sr,La)2CuO6+δ and $Ln$-Bi2201, the mismatch
in the ionic radius between Bi and Pb or Sr and $Ln$ causes the out-of-plane
impurities eisaki , where the concentration of the out-of-plane impurities is
controlled by varying the radius of the Pb or $Ln$ ions, and then the
superconducting transition temperature $T_{c}$ is found to be decreasing with
increasing impurity concentration. These cuprate superconductors
(Bi,Pb)2(Sr,La)2CuO6+δ and $Ln$-Bi2201 are often referred to as the out-of-
plane impurity-controlled cuprate superconductors. Recently, the electronic
structure of the out-of-plane impurity-controlled cuprate superconductors and
the related superconducting gap function have been investigated experimentally
by using ARPES kondo ; hashimoto . It was shown that although the effect of
the out-of-plane impurity scattering around the antinodal region is much
stronger than that around the nodal region, both superconducting coherence
peaks around the nodal and antinodal regions are suppressed. Furthermore, the
magnitude of the deviation from the monotonic d-wave superconducting gap form
increases with increasing impurity concentration kondo ; hashimoto . The
appearance of the strong deviation from the monotonic d-wave superconducting
gap form observed recently in the out-of-plane impurity-controlled cuprate
superconductors (Bi,Pb)2(Sr,La)2CuO6+δ and $Ln$-Bi2201 is the most remarkable
effect kondo ; hashimoto , however, its full understanding is still a
challenging issue. To the best of our knowledge, this strong deviation from
the monotonic d-wave superconducting gap form in the out-of-plane impurity-
controlled cuprate superconductors has not been treated starting from a
microscopic superconducting theory yet.
In the absence of out-of-plane impurity scattering, the electronic structure
of cuprate superconductors in the superconducting state has been discussed guo
; feng within the framework of the kinetic energy driven superconductivity
feng1 , where the superconducting gap function has a monotonic d-wave form,
and the main features of the ARPES experiments shen on cuprate
superconductors have been reproduced. In this paper, we study the electronic
structure of the out-of-plane impurity-controlled cuprate superconductors in
the superconducting state and the related superconducting gap function along
with this line. We employ the $t$-$J$ model by considering the out-of-plane
impurity scattering, and then show explicitly that the strong deviation from
the monotonic d-wave superconducting gap form occurs due to the presence of
the impurity scattering. Although both sharp superconducting coherence peaks
around the nodal and the antinodal regions are suppressed, the effect of the
impurity scattering is stronger in the antinodal region than that in the nodal
region. Our results also show that the electronic structure of the out-of-
plane impurity-controlled cuprate superconductors in the superconducting state
can be understood within the framework of the kinetic energy driven
superconducting mechanism with the out-of-plane impurity scattering taken into
account.
This paper is organized as follows. In Sec. II we present the basic formalism
of the electronic structure calculation in the presence of the out-of-plane
impurities. Within this theoretical framework, we discuss the electronic
structure of the out-of-plane impurity-controlled cuprate superconductors in
the superconducting state and the related superconducting gap function in Sec.
III, where we show that the well pronounced deviation from the monotonic
d-wave superconducting gap form is mainly caused by the out-of-plane impurity
scattering. Finally, we give a summary in Sec. IV.
## II Formalism
It has been shown that the essential physics of cuprate superconductors is
properly accounted by the two-dimensional $t$-$J$ model on a square lattice
anderson ,
$\displaystyle H$ $\displaystyle=$
$\displaystyle-t\sum_{i\hat{\eta}\sigma}C^{\dagger}_{i\sigma}C_{i+\hat{\eta}\sigma}+t^{\prime}\sum_{i\hat{\tau}\sigma}C^{\dagger}_{i\sigma}C_{i+\hat{\tau}\sigma}+\mu\sum_{i\sigma}C^{\dagger}_{i\sigma}C_{i\sigma}$
(1) $\displaystyle+$ $\displaystyle J\sum_{i\hat{\eta}}{\bf S}_{i}\cdot{\bf
S}_{i+\hat{\eta}},$
acting on the Hilbert subspace with no doubly occupied site, i.e.,
$\sum_{\sigma}C^{\dagger}_{i\sigma}C_{i\sigma}\leq 1$, where
$\hat{\eta}=\pm\hat{x},\pm\hat{y}$, $\hat{\tau}=\pm\hat{x}\pm\hat{y}$,
$C^{\dagger}_{i\sigma}$ ($C_{i\sigma}$) is the creation (annihilation)
operator of an electron with spin $\sigma$, ${\bf
S}_{i}=(S^{x}_{i},S^{y}_{i},S^{z}_{i})$ are spin operators, and $\mu$ is the
chemical potential. To deal with the constraint of no double occupancy in
analytical calculations, the charge-spin separation fermion-spin theory feng2
has been developed, where the constrained electron operators $C_{i\uparrow}$
and $C_{i\downarrow}$ are decoupled as
$C_{i\uparrow}=h^{\dagger}_{i\uparrow}S^{-}_{i}$ and
$C_{i\downarrow}=h^{\dagger}_{i\downarrow}S^{+}_{i}$, respectively, here the
spinful fermion operator $h_{i\sigma}=e^{-i\Phi_{i\sigma}}h_{i}$ describes the
charge degree of freedom together with some effects of spin configuration
rearrangements due to the presence of the doped charge carrier itself, while
the spin operator $S_{i}$ describes the spin degree of freedom, then the
electron on-site local constraint for the single occupancy,
$\sum_{\sigma}C^{\dagger}_{i\sigma}C_{i\sigma}=S^{+}_{i}h_{i\uparrow}h^{\dagger}_{i\uparrow}S^{-}_{i}+S^{-}_{i}h_{i\downarrow}h^{\dagger}_{i\downarrow}S^{+}_{i}=h_{i}h^{\dagger}_{i}(S^{+}_{i}S^{-}_{i}+S^{-}_{i}S^{+}_{i})=1-h^{\dagger}_{i}h_{i}\leq
1$, is satisfied in analytical calculations. In particular, it has been shown
that under the decoupling scheme, this charge-spin separation fermion-spin
representation is a natural representation of the constrained electron defined
in the Hilbert subspace without double electron occupancy feng . Furthermore,
these charge carrier and spin are gauge invariant feng2 , and in this sense
they are real and can be interpreted as physical excitations laughlin . In
this charge-spin separation fermion-spin representation, the $t$-$J$ model (1)
can be expressed as,
$\displaystyle H$ $\displaystyle=$ $\displaystyle
t\sum_{i\hat{\eta}}(h^{\dagger}_{i+\hat{\eta}\uparrow}h_{i\uparrow}S^{+}_{i}S^{-}_{i+\hat{\eta}}+h^{\dagger}_{i+\hat{\eta}\downarrow}h_{i\downarrow}S^{-}_{i}S^{+}_{i+\hat{\eta}})$
(2) $\displaystyle-$ $\displaystyle
t^{\prime}\sum_{i\hat{\tau}}(h^{\dagger}_{i+\hat{\tau}\uparrow}h_{i\uparrow}S^{+}_{i}S^{-}_{i+\hat{\tau}}+h^{\dagger}_{i+\hat{\tau}\downarrow}h_{i\downarrow}S^{-}_{i}S^{+}_{i+\hat{\tau}})$
$\displaystyle-$
$\displaystyle\mu\sum_{i\sigma}h^{\dagger}_{i\sigma}h_{i\sigma}+J_{{\rm
eff}}\sum_{i\hat{\eta}}{\bf S}_{i}\cdot{\bf S}_{i+\hat{\eta}},$
with $J_{{\rm eff}}=(1-\delta)^{2}J$, and $\delta=\langle
h^{\dagger}_{i\sigma}h_{i\sigma}\rangle=\langle h^{\dagger}_{i}h_{i}\rangle$
being the charge carrier doping concentration. This $J_{{\rm eff}}$ is similar
to that obtained in Gutzwiller approach zhang . As an important consequence,
the kinetic energy term in the $t$-$J$ model has been transferred as the
interaction between charge carriers and spins, which reflects that even the
kinetic energy term in the $t$-$J$ Hamiltonian has a strong Coulombic
contribution due to the restriction of no double occupancy of a given site.
This interaction from the kinetic energy term is quite strong, and it has been
shown feng1 in terms of the Eliashberg’s strong coupling theory eliashberg
that in the case without an antiferromagnetic long-range order, this
interaction can induce a charge carrier pairing state (then the electron
Cooper pairing state) with d-wave symmetry by exchanging spin excitations in
the higher power of the charge carrier doping concentration $\delta$. In this
case, the electron Cooper pairs originating from the charge carrier pairing
state are due to the charge-spin recombination, and their condensation reveals
the d-wave superconducting ground-state. Furthermore, this d-wave
superconducting state is controlled by both the superconducting gap function
and the quasiparticle coherence, which leads to the fact that the maximal
superconducting transition temperature occurs around the optimal doping, and
then decreases in both underdoped and overdoped regimes feng1 . Moreover, it
has been shown guo ; feng that this superconducting state is the conventional
Bardeen-Cooper-Schrieffer (BCS) like schrieffer ; bcs with the d-wave
symmetry, so that the basic BCS formalism with the d-wave superconducting gap
function is still valid in quantitatively reproducing all main low energy
features of the ARPES experimental measurements on cuprate superconductors,
although the pairing mechanism is driven by the kinetic energy by exchanging
spin excitations, and other exotic magnetic scattering dai is beyond the BCS
formalism. Following previous discussions guo ; feng ; feng1 , the full charge
carrier Green’s function in the superconducting state with a monotonic d-wave
gap function can be obtained in the Nambu representation as wang ,
$\displaystyle\tilde{g}({\bf k},\omega)$ $\displaystyle=$ $\displaystyle
Z_{hF}{1\over\omega^{2}-E^{2}_{h{\bf
k}}}\left(\begin{array}[]{cc}{\omega+\bar{\xi}_{{\bf
k}}}&{\bar{\Delta}_{hZ}({\bf k})}\\\ {\bar{\Delta}_{hZ}({\bf
k})}&{\omega-\bar{\xi}_{{\bf k}}}\end{array}\right)$ (5) $\displaystyle=$
$\displaystyle Z_{hF}{\omega\tau_{0}+\bar{\Delta}_{hZ}({\bf
k})\tau_{1}+\bar{\xi}_{{\bf k}}\tau_{3}\over\omega^{2}-E^{2}_{h{\bf k}}},$ (6)
where $\tau_{0}$ is the unit matrix, $\tau_{1}$ and $\tau_{3}$ are the Pauli
matrices, the renormalized charge carrier excitation spectrum $\bar{\xi}_{{\bf
k}}=Z_{hF}\xi_{\bf k}$, with the mean-field charge carrier excitation spectrum
$\xi_{{\bf k}}=Zt\chi_{1}\gamma_{{\bf
k}}-Zt^{\prime}\chi_{2}\gamma^{\prime}_{{\bf k}}-\mu$, the spin correlation
functions $\chi_{1}=\langle S_{i}^{+}S_{i+\hat{\eta}}^{-}\rangle$ and
$\chi_{2}=\langle S_{i}^{+}S_{i+\hat{\tau}}^{-}\rangle$, $\gamma_{{\bf
k}}=(1/Z)\sum_{\hat{\eta}}e^{i{\bf k}\cdot\hat{\eta}}$, $\gamma^{\prime}_{{\bf
k}}=(1/Z)\sum_{\hat{\tau}}e^{i{\bf k}\cdot\hat{\tau}}$, $Z$ is the number of
the nearest neighbor or next nearest neighbor sites, the renormalized charge
carrier monotonic d-wave pair gap function $\bar{\Delta}_{hZ}({\bf
k})=Z_{hF}\bar{\Delta}_{h}({\bf k})$, where the effective charge carrier
monotonic d-wave pair gap function $\bar{\Delta}_{h}({\bf
k})=\bar{\Delta}_{h}\gamma^{(d)}_{{\bf k}}$ with $\gamma^{(d)}_{{\bf k}}=({\rm
cos}k_{x}-{\rm cos}k_{y})/2$, and the charge carrier quasiparticle spectrum
$E_{h{\bf k}}=\sqrt{\bar{\xi}^{2}_{{\bf k}}+\mid\bar{\Delta}_{hZ}({\bf
k})\mid^{2}}$. The charge carrier quasiparticle coherent weight $Z_{hF}$ and
effective charge carrier gap parameter $\bar{\Delta}_{h}$ are determined by
the following two equations guo ; feng ; feng1 ,
$\displaystyle 1$ $\displaystyle=$ $\displaystyle{1\over N^{3}}\sum_{{\bf
k,p,p^{\prime}}}\Lambda^{2}_{{\bf p+k}}\gamma^{(d)}_{{\bf
k-p^{\prime}+p}}\gamma^{(d)}_{{\bf k}}{Z^{2}_{hF}\over E_{h{\bf k}}}{B_{{\bf
p}}B_{{\bf p^{\prime}}}\over\omega_{{\bf p}}\omega_{{\bf
p^{\prime}}}}\left({F^{(1)}_{1}({\bf k,p,p^{\prime}})\over(\omega_{{\bf
p^{\prime}}}-\omega_{{\bf p}})^{2}-E^{2}_{h{\bf k}}}-{F^{(2)}_{1}({\bf
k,p,p^{\prime}})\over(\omega_{{\bf p^{\prime}}}+\omega_{{\bf
p}})^{2}-E^{2}_{h{\bf k}}}\right),$ (7a) $\displaystyle{1\over Z}_{hF}$
$\displaystyle=$ $\displaystyle 1+{1\over N^{2}}\sum_{{\bf
p,p^{\prime}}}\Lambda^{2}_{{\bf p}+{\bf k}_{0}}Z_{hF}{B_{{\bf p}}B_{{\bf
p^{\prime}}}\over 4\omega_{{\bf p}}\omega_{{\bf
p^{\prime}}}}\left({F^{(1)}_{2}({\bf p,p^{\prime}})\over(\omega_{{\bf
p}}-\omega_{{\bf p^{\prime}}}-E_{h{\bf
p-p^{\prime}+k_{0}}})^{2}}+{F^{(2)}_{2}({\bf p,p^{\prime}})\over(\omega_{{\bf
p}}-\omega_{{\bf p^{\prime}}}+E_{h{\bf p-p^{\prime}+k_{0}}})^{2}}\right.$ (7b)
$\displaystyle+$ $\displaystyle\left.{F^{(3)}_{2}({\bf
p,p^{\prime}})\over(\omega_{{\bf p}}+\omega_{{\bf p^{\prime}}}-E_{h{\bf
p-p^{\prime}+k_{0}}})^{2}}+{F^{(4)}_{2}({\bf p,p^{\prime}})\over(\omega_{{\bf
p}}+\omega_{{\bf p^{\prime}}}+E_{h{\bf p-p^{\prime}+k_{0}}})^{2}}\right),$
respectively, where ${\bf k}_{0}=[\pi,0]$, $\Lambda_{\bf k}=Zt\gamma_{\bf
k}-Zt^{\prime}\gamma^{\prime}_{\bf k}$, $B_{{\bf
p}}=2\lambda_{1}(A_{1}\gamma_{{\bf
p}}-A_{2})-\lambda_{2}(2\chi^{z}_{2}\gamma_{{\bf p}}^{\prime}-\chi_{2})$,
$\lambda_{1}=2ZJ_{{\rm eff}}$, $\lambda_{2}=4Z\phi_{2}t^{\prime}$,
$A_{1}=\epsilon\chi^{z}_{1}+\chi_{1}/2$,
$A_{2}=\chi^{z}_{1}+\epsilon\chi_{1}/2$, $\epsilon=1+2t\phi_{1}/J_{{\rm
eff}}$, the charge carrier’s particle-hole parameters $\phi_{1}=\langle
h^{\dagger}_{i\sigma}h_{i+\hat{\eta}\sigma}\rangle$ and $\phi_{2}=\langle
h^{\dagger}_{i\sigma}h_{i+\hat{\tau}\sigma}\rangle$, the spin correlation
functions $\chi^{z}_{1}=\langle S_{i}^{z}S_{i+\hat{\eta}}^{z}\rangle$ and
$\chi^{z}_{2}=\langle S_{i}^{z}S_{i+\hat{\tau}}^{z}\rangle$, $F^{(1)}_{1}({\bf
k,p,p^{\prime}})=(\omega_{{\bf p^{\prime}}}-\omega_{{\bf
p}})[n_{B}(\omega_{{\bf p}})-n_{B}(\omega_{{\bf
p^{\prime}}})][1-2n_{F}(E_{h{\bf k}})]+E_{h{\bf k}}[n_{B}(\omega_{{\bf
p^{\prime}}})n_{B}(-\omega_{{\bf p}})+n_{B}(\omega_{{\bf
p}})n_{B}(-\omega_{{\bf p^{\prime}}})]$, $F^{(2)}_{1}({\bf
k,p,p^{\prime}})=(\omega_{{\bf p^{\prime}}}+\omega_{{\bf
p}})[n_{B}(-\omega_{{\bf p^{\prime}}})-n_{B}(\omega_{{\bf
p}})][1-2n_{F}(E_{h{\bf k}})]+E_{h{\bf k}}[n_{B}(\omega_{{\bf
p^{\prime}}})n_{B}(\omega_{{\bf p}})+n_{B}(-\omega_{{\bf
p^{\prime}}})n_{B}(-\omega_{{\bf p}})]$, $F^{(1)}_{2}({\bf
p,p^{\prime}})=n_{F}(E_{h{\bf p-p^{\prime}+k_{0}}})[n_{B}(\omega_{{\bf
p^{\prime}}})-n_{B}(\omega_{{\bf p}})]-n_{B}(\omega_{{\bf
p}})n_{B}(-\omega_{{\bf p^{\prime}}})$, $F^{(2)}_{2}({\bf
p,p^{\prime}})=n_{F}(E_{h{\bf p-p^{\prime}+k_{0}}})[n_{B}(\omega_{{\bf
p}})-n_{B}(\omega_{{\bf p^{\prime}}})]-n_{B}(\omega_{{\bf
p^{\prime}}})n_{B}(-\omega_{{\bf p}})$, $F^{(3)}_{2}({\bf
p,p^{\prime}})=n_{F}(E_{h{\bf p-p^{\prime}+k_{0}}})[n_{B}(\omega_{{\bf
p^{\prime}}})-n_{B}(-\omega_{{\bf p}})]+n_{B}(\omega_{{\bf
p}})n_{B}(\omega_{{\bf p^{\prime}}})$, $F^{(4)}_{2}({\bf
p,p^{\prime}})=n_{F}(E_{h{\bf p-p^{\prime}+k_{0}}})[n_{B}(-\omega_{{\bf
p^{\prime}}})-n_{B}(\omega_{{\bf p}})]+n_{B}(-\omega_{{\bf
p}})n_{B}(-\omega_{{\bf p^{\prime}}})$, $n_{B}(\omega_{{\bf p}})$ and
$n_{F}(E_{h{\bf k}})$ are the boson and fermion distribution functions,
respectively, and the mean-field spin excitation spectrum,
$\displaystyle\omega^{2}_{{\bf p}}$ $\displaystyle=$
$\displaystyle\lambda_{1}^{2}\left[\left(A_{4}-\alpha\epsilon\chi^{z}_{1}\gamma_{{\bf
p}}-{1\over 2Z}\alpha\epsilon\chi_{1}\right)(1-\epsilon\gamma_{{\bf
p}})+{1\over 2}\epsilon\left(A_{3}-{1\over
2}\alpha\chi^{z}_{1}-\alpha\chi_{1}\gamma_{{\bf
p}}\right)(\epsilon-\gamma_{{\bf
p}})\right]+\lambda_{2}^{2}\left[\alpha\left(\chi^{z}_{2}\gamma_{{\bf
p}}^{\prime}-{3\over 2Z}\chi_{2}\right)\gamma_{{\bf p}}^{\prime}\right.$ (8)
$\displaystyle+$ $\displaystyle\left.{1\over 2}\left(A_{5}-{1\over
2}\alpha\chi^{z}_{2}\right)\right]+\lambda_{1}\lambda_{2}\left[\alpha\chi^{z}_{1}(1-\epsilon\gamma_{{\bf
p}})\gamma_{{\bf p}}^{\prime}+{1\over 2}\alpha(\chi_{1}\gamma_{{\bf
p}}^{\prime}-C_{3})(\epsilon-\gamma_{{\bf p}})+\alpha\gamma_{{\bf
p}}^{\prime}(C^{z}_{3}-\epsilon\chi^{z}_{2}\gamma_{{\bf p}})-{1\over
2}\alpha\epsilon(C_{3}-\chi_{2}\gamma_{{\bf p}})\right],~{}~{}~{}~{}~{}$
where $A_{3}=\alpha C_{1}+(1-\alpha)/(2Z)$, $A_{4}=\alpha
C^{z}_{1}+(1-\alpha)/(4Z)$, $A_{5}=\alpha C_{2}+(1-\alpha)/(2Z)$, and the spin
correlation functions
$C_{1}=(1/Z^{2})\sum_{\hat{\eta},\hat{\eta^{\prime}}}\langle
S_{i+\hat{\eta}}^{+}S_{i+\hat{\eta^{\prime}}}^{-}\rangle$,
$C^{z}_{1}=(1/Z^{2})\sum_{\hat{\eta},\hat{\eta^{\prime}}}\langle
S_{i+\hat{\eta}}^{z}S_{i+\hat{\eta^{\prime}}}^{z}\rangle$,
$C_{2}=(1/Z^{2})\sum_{\hat{\tau},\hat{\tau^{\prime}}}\langle
S_{i+\hat{\tau}}^{+}S_{i+\hat{\tau^{\prime}}}^{-}\rangle$,
$C_{3}=(1/Z)\sum_{\hat{\tau}}\langle
S_{i+\hat{\eta}}^{+}S_{i+\hat{\tau}}^{-}\rangle$, and
$C^{z}_{3}=(1/Z)\sum_{\hat{\tau}}\langle
S_{i+\hat{\eta}}^{z}S_{i+\hat{\tau}}^{z}\rangle$. In order to satisfy the sum
rule of the correlation function $\langle S^{+}_{i}S^{-}_{i}\rangle=1/2$ in
the case without the antiferromagnetic long-range order, an important
decoupling parameter $\alpha$ has been introduced in the above calculation guo
; feng ; feng1 , which can be regarded as the vertex correction. These two
equations (4a) and (4b) must be solved simultaneously with other self-
consistent equations, then all order parameters, the decoupling parameter
$\alpha$, and the chemical potential $\mu$ are determined by the self-
consistent calculation guo ; feng ; feng1 . In this sense, the calculations in
this kinetic energy driven superconductivity scheme are controllable without
using any adjustable parameters. We emphasize that the Green’s function (3) is
obtained within the kinetic energy driven superconducting mechanism, although
the similar phenomenological expression has been used to discuss the impurity
effect in cuprate superconductors haas ; graser .
With the charge carrier BCS formalism (3) under the kinetic energy driven
superconducting mechanism, we can now introduce the effect of impurity
scatterers into the electronic structure. In the presence of impurities, the
unperturbed charge carrier Green’s function in Eq. (3) is dressed by impurity
scattering wang ,
$\displaystyle\tilde{g}_{I}({\bf k},\omega)^{-1}=\tilde{g}({\bf
k},\omega)^{-1}-Z_{hF}^{-1}\tilde{\Sigma}({\bf k},\omega),$ (9)
with the self-energy function $\tilde{\Sigma}({\bf
k},\omega)=\sum_{\alpha}\Sigma_{\alpha}({\bf k},\omega)\tau_{\alpha}$. In this
case, the charge carrier Green’s function in Eq. (6) can be explicitly
rewritten as,
$\displaystyle\tilde{g}_{I}({\bf k},\omega)=\sum_{\alpha}g_{I\alpha}({\bf
k},\omega)\tau_{\alpha}=Z_{hF}{[\omega-\Sigma_{0}({\bf
k},\omega)]\tau_{0}+[\bar{\Delta}_{hZ}({\bf k})+\Sigma_{1}({\bf
k},\omega)]\tau_{1}+[\bar{\xi}_{{\bf k}}+\Sigma_{3}({\bf
k},\omega)]\tau_{3}\over[\omega-\Sigma_{0}({\bf
k},\omega)]^{2}-[\bar{\xi}_{{\bf k}}+\Sigma_{3}({\bf
k},\omega)]^{2}-[\bar{\Delta}_{hZ}({\bf k})+\Sigma_{1}({\bf k},\omega)]^{2}}.$
(10)
Based on this Green’s function (7), we wang have discussed the effect of the
extended impurity scatterers on the quasiparticle transport of cuprate
superconductors in the superconducting state within the nodal approximation of
the quasiparticle excitations and scattering processes, where the main effect
on the quasiparticle transport comes from the extended impurity forward (or
diagonal) scatterers, and therefore the component of the self-energy function
$\Sigma_{1}({\bf k},\omega)$ has been neglected, while the components of
$\Sigma_{0}({\bf k},\omega)$ and $\Sigma_{3}({\bf k},\omega)$ have been
treated within the framework of the T-matrix approximation. However, it has
been demonstrated that the superconducting transition temperature is
considerably affected by the out-of-plane impurity scattering in spite of a
relatively weak increase of the residual resistivity eisaki . This reflects
the fact that the superconducting pairing is very sensitive to the out-of-
plane impurity scattering, and then the effect of the out-of-plane impurity
scattering is always accompanied by breaking of the superconducting pairs. In
this case, the out-of-plane impurities can be described as the elastic off-
diagonal scatterers or pairing impurity scatterers. In particular, the
modulation of the out-of-plane impurity scattering potential observed in
scanning tunneling microscopy experiments pan has a characteristic wavelength
of a few lattice spacings, this may arise because the impurities give rise to
an atomic-scale modulation of the charge carrier pairing potential which
causes larger, coherence length size fluctuations in the out-of-plane impurity
scattering potential nunner . Furthermore, the crude effect of the order
parameter modulations on the quasiparticle scattering by allowing the order
parameter to be modulated on the four bonds around the impurity has been
estimated graser by adding the off-diagonal scattering potential,
$\displaystyle\hat{V}$ $\displaystyle=$ $\displaystyle\sum_{{\bf k},{\bf
k}^{\prime}}[V({\bf k})+V({\bf k}^{\prime})]\tau_{1}$ (11) $\displaystyle=$
$\displaystyle{1\over 2}V_{0}\sum_{{\bf k},{\bf k}^{\prime}}[({\rm
cos}k_{x}-{\rm cos}k_{y})+({\rm cos}k^{\prime}_{x}-{\rm
cos}k^{\prime}_{y})]\tau_{1},~{}~{}~{}$
to the phenomenological d-wave BCS Hamiltonian, then it was shown that the
scattering rate is largest at the antinode.
The exact form of the out-of-plane impurity scattering potential is very
important for a better understanding of the electronic structure of the out-
of-plane impurity-controlled cuprate superconductors. In the following
discussions, we determine the form of the out-of-plane impurity scattering
potential in terms of the calculation of Dyson’s equation. The potential which
scatters the electron is taken as summation of impurity potentials
$\tilde{V}=\sum_{l}V({\bf r}_{i}-{\bf R}_{l})$, where the summation is over
all impurity sites $l$, and then its Fourier transform is obtained kohn ;
mahan as $\tilde{V}({\bf q})=\rho_{i}V({\bf q})\rho({\bf q})$, where
$\displaystyle\rho({\bf q})$ $\displaystyle=$ $\displaystyle\sum_{{\bf
k}}h^{\dagger}_{{\bf k}+{\bf q}}h_{{\bf k}},$ (12) $\displaystyle\rho_{i}({\bf
q})$ $\displaystyle=$ $\displaystyle\sum_{l}e^{i{\bf q}\cdot{\bf R}_{l}},$
(13)
are the charge carrier density in the Nambu representation and the impurity
density, respectively. In the calculation of the self-energy function induced
by the impurity scattering, usually it is assumed that the impurities are
randomly located and that there is no correlation between their positions kohn
; mahan . In this case, the self-energy function can be obtained as
$\tilde{\Sigma}({\bf k},\omega)=\tilde{\Sigma}^{(1)}({\bf
k},\omega)+\tilde{\Sigma}^{(2)}({\bf k},\omega)$ within the Born
approximation, with the corresponding first-order and second-order self-energy
functions are evaluated as kohn ; mahan ,
$\displaystyle\tilde{\Sigma}^{(1)}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\rho_{i}\sum_{\bf k^{\prime}}\delta_{{\bf k^{\prime}}=0}V({\bf
k^{\prime}})=\rho_{i}V(0),$ (14a) $\displaystyle\tilde{\Sigma}^{(2)}({\bf
k},\omega)$ $\displaystyle=$ $\displaystyle\rho_{i}\sum_{{\bf k^{\prime}},{\bf
k^{\prime\prime}}}\delta_{{\bf k^{\prime}}+{\bf k^{\prime\prime}}=0}V({\bf
k^{\prime}})\tilde{g}_{I}({\bf k}+{\bf k^{\prime}},\omega)V({\bf
k^{\prime\prime}})$ (14b) $\displaystyle=$ $\displaystyle\rho_{i}\sum_{{\bf
k^{\prime}}}V({\bf k^{\prime}})\tilde{g}_{I}({\bf k}+{\bf
k^{\prime}},\omega)V(-{\bf k^{\prime}}),$
where $\rho_{i}$ is the impurity concentration. As we have mentioned above,
the out-of-plane impurities are the off-diagonal scatterers. Although their
scattering has a very weak effect on the residual resistivity for cuprate
superconductors, a heavy effect on the d-wave SC state is observed
experimentally eisaki . With these considerations, we introduce the following
out-of-plane impurity scattering potential,
$\displaystyle\tilde{V}=\sum_{{\bf k^{\prime}}}V({\bf
k^{\prime}})\tau_{1}=V_{0}\sum_{{\bf k^{\prime}}}[{\rm cos}k^{\prime}_{x}-{\rm
cos}k^{\prime}_{y}]\tau_{1}.$ (15)
In this case, $V(0)=V_{0}[{\rm cos}(0)-{\rm cos}(0)]=0$ (then
$\tilde{\Sigma}_{1}({\bf k},\omega)=0$), and $\tilde{\Sigma}({\bf
k},\omega)=\tilde{\Sigma}^{(2)}({\bf k},\omega)$. This form of the out-of-
plane impurity scattering potential in Eq. (12) is very similar to that in Eq.
(8) used in Ref. graser, , and the scattering rate is also largest at the
antinode. This is indeed confirmed by the quantitative characteristics
presented in the following section. With the help of the impurity scattering
potential in Eq. (12), the components of the charge carrier self-energy
function $\tilde{\Sigma}({\bf k},\omega)$ are obtained explicitly as,
$\displaystyle\Sigma_{0}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\rho_{i}{1\over N}\sum_{{\bf k^{\prime}}}|V({\bf
k^{\prime}})|^{2}{g}_{I0}({\bf k^{\prime}+k},\omega)$ (16a) $\displaystyle=$
$\displaystyle\rho_{i}{1\over N}\sum_{{\bf k^{\prime}}}|V({\bf
k^{\prime}-k})|^{2}{g}_{I0}({\bf k^{\prime}},\omega),$
$\displaystyle\Sigma_{3}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle-\rho_{i}{1\over N}\sum_{{\bf k^{\prime}}}|V({\bf
k^{\prime}})|^{2}{g}_{I3}({\bf k^{\prime}+k},\omega)$ (16b) $\displaystyle=$
$\displaystyle-\rho_{i}{1\over N}\sum_{{\bf k^{\prime}+k}}|V({\bf
k^{\prime}-k})|^{2}{g}_{I3}({\bf k^{\prime}},\omega),~{}~{}~{}~{}$
$\displaystyle\Sigma_{1}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\rho_{i}{1\over N}\sum_{{\bf k^{\prime}}}|V({\bf
k^{\prime}})|^{2}{g}_{I1}({\bf k^{\prime}+k},\omega)$ (16c) $\displaystyle=$
$\displaystyle\rho_{i}{1\over N}\sum_{{\bf k^{\prime}}}|V({\bf
k^{\prime}-k})|^{2}{g}_{I1}({\bf k^{\prime}},\omega).$
In the charge-spin separation fermion-spin theory feng2 , the electron
diagonal and off-diagonal Green’s functions are the convolutions of the spin
Green’s function guo ; feng ; feng1 $D^{-1}({\bf
p},\omega)=(\omega^{2}-\omega_{{\bf p}}^{2})/B_{{\bf p}}$ and the charge
carrier diagonal and off-diagonal Green’s functions in Eq. (7), respectively.
These convolutions reflect the charge-spin recombination anderson1 . Following
the previous discussions guo ; feng ; feng1 , we can obtain the electron
diagonal and off-diagonal Green’s functions in the present case. Then the
electron spectral function from the electron diagonal Green’s function is
found explicitly as,
$\displaystyle A({\bf k},\omega)$ $\displaystyle=$ $\displaystyle{1\over
N}\sum_{\bf p}{B_{\bf p}\over 2\omega_{\bf p}}\\{[n_{B}(\omega_{\bf
p})+n_{F}(\omega_{\bf p}-\omega)]A_{h}({\bf p}-{\bf k},\omega_{\bf p}-\omega)$
(17) $\displaystyle-$ $\displaystyle[n_{B}(-\omega_{\bf p})+n_{F}(-\omega_{\bf
p}-\omega)]A_{h}({\bf p}-{\bf k},-\omega_{\bf p}-\omega)\\},~{}~{}~{}~{}~{}$
where $A_{h}$ is the charge carrier spectral function, which can be expressed
as $A_{h}=-2{\rm Im}g^{dia}_{I}({\bf k},\omega)$, with $g^{dia}_{I}$ obtained
from Eq. (7) as,
$\displaystyle g^{dia}_{I}({\bf k},\omega)=Z_{hF}{\omega-\Sigma_{0}({\bf
k},\omega)+\bar{\xi}_{{\bf k}}+\Sigma_{3}({\bf
k},\omega)\over[\omega-\Sigma_{0}({\bf k},\omega)]^{2}-[\bar{\xi}_{{\bf
k}}+\Sigma_{3}({\bf k},\omega)]^{2}-[\bar{\Delta}_{hZ}({\bf
k})+\Sigma_{1}({\bf k},\omega)]^{2}}.$ (18)
## III Electronic structure of the out-of-plane impurity-controlled cuprate
superconductors
Experimentally, it has been shown that the average of the next-nearest
neighbor hopping $t^{\prime}$ is not appreciably affected by the out-of-plane
impurities hashimoto . In this case, the commonly used parameters in this
paper are chosen as $t/J=2.5$ and $t^{\prime}/t=0.3$. We are now ready to
discuss the electronic structure of the out-of-plane impurity-controlled
cuprate superconductors and the related superconducting gap. In cuprate
superconductors, the information revealed by ARPES experiments shen has shown
that around the nodal [$\pi/2,\pi/2$] and antinodal [$\pi,0$] points of the
Brillouin zone contain the essentials of the whole low energy quasiparticle
excitation spectrum. In this case, we have performed a calculation for the
electron spectral function $A({\bf k},\omega)$ in Eq. (14) at both nodal and
antinodal points. The results at (a) the nodal point and (b) the antinodal
point with the impurity concentration $\rho_{i}=0.001$ (solid line),
$\rho_{i}=0.002$ (dashed line), and $\rho_{i}=0.003$ (dotted line) under the
impurity scattering potential with $V_{0}=50J$ for the charge carrier doping
concentration $\delta=0.15$ are plotted in Fig. 1. For comparison, the
corresponding ARPES experimental results hashimoto for the out-of-plane
impurity-controlled cuprate superconductors $Ln$-Bi2201 in the superconducting
state are also presented in Fig. 1 (inset). Our results show that the
quasiparticle peak is strongly dependent on the impurity concentration, and
the peaks at both nodal and antinodal points are suppressed due to the
presence of impurity scattering. At the nodal point, there is a sharp
superconducting quasiparticle peak near the Fermi energy, however, although
the peak at the high impurity concentration is dramatically reduced compared
to that at the low impurity concentration, the position of the leading-edge
mid-point of the electron spectral function remains almost unchanged. In
particular, the position of the leading-edge mid-point of the electron
spectral function reaches the Fermi level, indicating that there is no
superconducting gap. On the other hand, the spectral intensity from the Fermi
energy down to $\sim-1.1J$ decreases as the impurity concentration increases
at the antinodal point, this is the same case as that at the nodal point.
However, the position of the leading-edge mid-point of the electron spectral
function is shifted towards higher binding energies with increasing impurity
concentration, this is in contrast with the behavior observed at the nodal
point, and indicates the presence of the superconducting gap. The present
results also show that the effect of the out-of-plane impurity scattering is
stronger at the antinodal point than at the nodal one, in qualitative
agreement with the experimental results kondo ; hashimoto .
Figure 1: The electron spectral function at (a) the nodal point and (b) the
antinodal point with $\rho_{i}=0.001$ (solid line), $\rho_{i}=0.002$ (dashed
line), and $\rho_{i}=0.003$ (dotted line) for $V_{0}=50J$ at $\delta=0.15$.
Inset: the corresponding experimental results taken from Ref. hashimoto, .
Figure 2: The superconducting gap as a function of the Fermi surface angle
$\phi$ with $\rho_{i}=0$ (dashed line) and $\rho_{i}=0.001$ (solid line) for
$V_{0}=50J$ at $\delta=0.15$. Inset: the corresponding experimental results
taken from Ref. kondo, .
The behavior of the electron spectrum in Fig. 1 indicates an enhancement of
the superconducting gap in the antinodal region by the impurity scattering. To
show this point clearly, we have calculated the electron spectral function
$A({\bf k},\omega)$ along the direction $[\pi,0]\rightarrow[\pi/2,\pi/2]$, and
then employed the shift of the leading-edge mid-point as a measure of the
magnitude of the superconducting gap at each momentum just as it has been done
in the experiments kondo ; hashimoto . The results for the extracted
superconducting gap as a function of the Fermi surface angle $\phi$, defined
in the inset, with the impurity concentration $\rho_{i}=0$ (dashed line) and
$\rho_{i}=0.001$ (solid line) under the impurity scattering potential with
$V_{0}=50J$ for the charge carrier doping concentration $\delta=0.15$ are
plotted in Fig. 2 in comparison with the corresponding ARPES experimental
results for the out-of-plane impurity-controlled cuprate superconductor
(Bi,Pb)2(Sr,La)2CuO6+δ in the superconducting state kondo (inset). It is
clearly shown that the superconducting gap $\Delta$ increases with the Fermi
surface angle decreasing from 45o (node) to 0o (antinode). Although the
superconducting gap in the presence of the impurity scattering is basically
consistent with the d-wave symmetry, it is obvious that there is a strong
deviation from the monotonic d-wave form around the antinodal region. In
particular, this strong deviation is mainly caused by a remarkable enhancement
of the superconducting gap value around the antinodal region, in qualitative
agreement with the experimental results kondo ; hashimoto . In other words,
the superconducting gap around the antinodal region is strongly enhanced by
the impurity scattering, whereas around the nodal region its value remains the
same. As a consequence, the well pronounced deviation from the monotonic
d-wave superconducting gap form in the out-of-plane impurity-controlled
cuprate superconductors is mainly caused by the effect of the out-of-plane
impurity scattering. This is also the reason why the superconducting gap
function for very high quality samples of the cuprate superconductor
La1-xSrxCuO4 has a monotonic d-wave form shi1 .
Figure 3: The superconducting gap as a function of $[{\rm cos}k_{x}-{\rm
cos}k_{y}]/2$ with $\rho_{i}=0001$ (solid line), $\rho_{i}=0.002$ (dashed
line), and $\rho_{i}=0.003$ (dotted line) for $V_{0}=50J$ at $\delta=0.15$.
Inset: the corresponding experimental results taken from Ref. hashimoto, .
For a better understanding of the impurity concentration dependence of the
deviation from the monotonic d-wave superconducting gap function, we have made
a series of calculations for the superconducting gap at different impurity
concentration levels, and the results of the superconducting gap as a function
of the monotonic d-wave function $[{\rm cos}k_{x}-{\rm cos}k_{y}]/2$ with the
impurity concentration $\rho_{i}=0001$ (solid line), $\rho_{i}=0.002$ (dashed
line), and $\rho_{i}=0.003$ (dotted line) under the impurity scattering
potential with $V_{0}=50J$ for the charge carrier doping concentration
$\delta=0.15$ are plotted in Fig. 3 in comparison with the corresponding ARPES
experimental results for the out-of-plane impurity-controlled cuprate
superconductors Ln-Bi2201 hashimoto (inset). Obviously, our results show that
the magnitude of the deviation from the monotonic d-wave superconducting gap
form around the antinodal region increases with increasing impurity
concentration, in qualitative agreement with the experimental results kondo ;
hashimoto . This strong out-of-plane impurity effect in the antinodal region
is also consistent with scanning tunneling spectroscopy results sugimoto ,
where the average of the superconducting gap size, which corresponds to the
antinodal superconducting gap in the ARPES spectra, increases with increasing
impurity concentration.
Within the framework of the kinetic energy driven superconducting mechanism
feng1 in the presence of the out-of-plane impurities, our present results
show that the out-of-plane impurity scattering potential (12) in which the
impurities modulate the pair interaction locally give qualitative agreement
with respect to the main features observed in the ARPES measurements on the
out-of-plane impurity-controlled cuprate superconductors in the
superconducting state. Although this out-of-plane impurity effect in cuprate
superconductors can also be discussed starting directly from a
phenomenological d-wave BCS formalism graser ; nunner , in this paper we are
primarily interested in exploring the general notion of the effects of the
out-of-plane impurity scatterers in the kinetic energy driven cuprate
superconductors in the superconducting state. The qualitative agreement
between the present theoretical results and ARPES experimental data also
indicates that the presence of the out-of-plane impurities has a crucial
impact on the electronic structure of cuprate superconductors. On the other
hand, we emphasize that the quasiparticle scattering rate in the antinodal
region is strongly increased by the impurity scattering potential (12), while
the nodal quasiparticles are very weakly scattered by the impurity scattering
potential (12), this is why the superconducting transition temperature is
considerably affected by the out-of-plane impurity scattering in spite of a
relatively weak increase of the residual resistivity eisaki , since the
transport properties are mainly governed by the quasiparticles in the nodal
region.
## IV Summary
In conclusion, we have shown very clearly in this paper that if the out-of-
plane impurity scattering is taken into account within the framework of the
kinetic energy driven d-wave superconductivity feng1 , the quasiparticle
spectrum of the $t$-$J$ model calculated based on the off-diagonal impurity
scattering potential (12) per se can correctly reproduce some main features
found in the ARPES measurements on the out-of-plane impurity-controlled
cuprate superconductors in the superconducting state kondo ; hashimoto . In
the presence of the out-of-plane impurities, although both sharp
superconducting coherence peaks around the nodal and antinodal regions are
suppressed, the effect of the impurity scattering is stronger in the antinodal
region than that in the nodal region, this leads to a strong deviation from
the monotonic d-wave superconducting gap form in the out-of-plane impurity-
controlled cuprate superconductors.
Finally, we have noted that within a phenomenological BCS approach, the
electron spectral properties of the underdoped cuprates as resulting from a
momentum dependent pseudogap in the normal state have been discussed
sangiovanni , where a normal state pseudogap function deviating from the
monotonic d-wave pseudogap form has been used to fit the ARPES experimental
data in the normal state. It has been shown kastner ; shen95 that there are
some subtle differences for different families of underdoped cuprates in the
normal state, and therefore it is possible that the pseudogap in the normal
state is effected by the impurity scattering as well.
###### Acknowledgements.
This work was supported by the National Natural Science Foundation of China
under Grant No. 10774015, and the funds from the Ministry of Science and
Technology of China under Grant Nos. 2006CB601002 and 2006CB921300.
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|
arxiv-papers
| 2009-01-05T10:28:16 |
2024-09-04T02:48:59.713333
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhi Wang and Shiping Feng",
"submitter": "Shiping Feng",
"url": "https://arxiv.org/abs/0901.0457"
}
|
0901.0522
|
KINETICS OF PARTON- ANTIPARTON PLASMA VACUUM CREATION IN THE TIME - DEPENDENT
CHROMO - ELECTRIC FIELDS OF ARBITRARY POLARIZATION
A.V. Filatov, S.A. Smolyansky, A.V. Tarakanov
Physical Department of Saratov State University, 410026, Saratov, Russia
E- mail: smol@sgu.ru
###### Abstract
The kinetic equation of non - Markovian type for description of the vacuum
creation of parton - antiparton pairs under action of a space homogeneous time
- dependent chromo - electric field of the arbitrary polarization is obtained
on the strict non - pertubative foundation in the framework of the oscillator
representation. A comparison of the effectiveness of vacuum creation with the
case of linear polarization one is fulfilled.
## 1 Introduction
The Schwinger effect [1] of the vacuum production of electron-positron pairs
(EPP’s) under the action of electromagnetic fields is one from a few QED
effects, that has not up to now an accurate experimental test. It is
stipulated by the huge electric fields $E\sim E_{c}=1,3\cdot 10^{16}V/cm$ for
the electrons that is necessary for observation of this effect in a constant
field. Such field strength is unachievable for static fields therefore main
attention was involved the theoretical study of pair creation by time-varying
electric fields ([2]-[6]). The detailed description was obtained for the case
of linearly polarized spatially homogeneous time dependent electric field. The
sufficiently strong electric field can be achieved nowadays with laser beams
only. The estimations made before ([2]-[8]) showed that pair creation by a
single laser pulse with $E\ll E_{c}$ could be hardly observed. The more
optimistic results have been obtained for a planning X-ray free electron
lasers ([9]-[11]) and for the counter-propagating laser beams of optical range
([12]-[14]).
In the present work we make step on the way of theoretical research of the
parton - antiparton vacuum creation in the nonstationary chromo - electric
field of arbitrary polarization. The corresponding kinetic equation (KE) will
be derived below on the strict non-perturbative dynamics basis. We will
restrict ourself here by consideration of the nonstationary Schwinger effect
in vacuum only leaving in a site the analysis of this effect in some plasma-
similar medium (see, e.g., [15]). We use the oscillator representation (OR)
for the construction of the kinetic theory (initially, this representation was
be suggested in the scalar QED [16]). OR leads in the shortest way to
quasipartical (QP) representation, in which all dynamical operators of
observable quantities have diagonal form [5]. On this basis, the Heisenberg-
like equations of motion for the creation and annihilation operators will be
obtained in the spinor QED (Sect.2) that corresponds to a large N in QCD. The
corresponding kinetic theory will be constructed in Sect. 3. The preliminary
communication about these results has made on the conference [17]. In general
case, the obtained KE of the non - Markovian type is rather complicated
because of spin effects. The case of rather weak of the chromo - electric
external field is considered in the Sect. 4. The short conclusions are
summarized in Sect. 5.
We use the metric $g^{\mu\nu}=diag(1,-1,-1,-1)$ and the natural units
$\hbar=c=1$.
## 2 Oscillator representation
Let us consider the QED system in the presence of an external quasi-classical
spatially homogeneous time-dependent electric field of arbitrary polarization
with the 4-potential (in the Hamilton gange) $A^{\mu}(t)=(0,\mathbf{A(t)})$
and the corresponding field strength $\mathbf{E}(t)=-\mathbf{\dot{A}}(t)$ (the
overdots denote the time derivative). Such a field can be considered either as
an external field, or as a result of the mean field approximation [18]. The
Lagrange function is
$\mathcal{L}=\frac{i}{2}\\{\overline{\psi}\gamma^{\mu}D_{\mu}\Psi-(D^{*}_{\mu}\overline{\psi})\gamma^{\mu}\Psi\\}-m\overline{\psi}\psi,$
(1)
where $D_{\mu}=\partial_{\mu}+ieA_{\mu}(t)$ and -e is the electron charge. The
equations of motion are
$\displaystyle(i\gamma^{\mu}D_{\mu}-m)\psi=0,$
$\displaystyle\overline{\psi}(i\gamma^{\mu}\overleftarrow{D}_{\mu}^{*}+m)=0,$
(2)
where $\bar{\psi}=\psi^{+}\gamma^{0}$. The fields $\psi$ and $\psi^{+}$
compose the pair of canonical conjugated variables. The corresponding
Hamiltonian is (k=1,2,3)
$H(t)=i\int d^{3}x\psi^{+}\dot{\psi}=\int
d^{3}x\bar{\psi}\\{-i\gamma^{k}D_{k}+m\\}\psi.$ (3)
In the considered case, the system is space homogeneous and nonstationary.
Therefore the transition in the Fock space can be realized on the basis
functions $\phi=\exp{(\pm i\mathbf{k}\mathbf{x})}$ and creation and
annihilation operators become the time dependent one, generally speaking.
Hence, we have the following decompositions of the field functions in the
discrete momentum space ($V=L^{3}$ and $p_{i}=(2\pi/L)n_{i}$ with an integer
$n_{i}$ for each $i=1,2,3$):
$\displaystyle\psi(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{V}}\sum_{\mathbf{k}}{\sum_{\alpha=1,2}}\left\\{e^{i\mathbf{k}\mathbf{x}}a_{\alpha}(\mathbf{k},t)u_{\alpha}(\mathbf{k},t)+e^{-i\mathbf{k}\mathbf{x}}b_{\alpha}^{+}(\mathbf{k},t)v_{\alpha}(\mathbf{k},t)\right\\},$
$\displaystyle\bar{\psi}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{V}}\sum_{\mathbf{k}}\sum_{\alpha=1,2}\\{e^{-i\mathbf{k}\mathbf{x}}a^{+}_{\alpha}(\mathbf{k},t)\bar{u}_{\alpha}(\mathbf{k},t)+e^{i\mathbf{k}\mathbf{x}}b_{\alpha}(\mathbf{k},t)\bar{v}_{\alpha}(\mathbf{k},t)\\}.$
(4)
The nearest aim is derivation of the equations of motion for the creation and
annihilation operators on the basis of the primary equations (2) and the use
of the free $u,v$-spinors as the basic functions with the natural substitution
of the canonical momentum with the corresponding kinematic one (that
corresponds to the basic OR idea). It is necessary to take into account, that
the electron and positron states are different by sings of the charges and
hence their kinematic momentum are $\mathbf{p}=\mathbf{k}-e\mathbf{A}$ for
electrons and $\mathbf{p}^{c}=\mathbf{k}+e\mathbf{A}$ for positrons. Thus, the
following ”free-like” equations for the spinors are postulated in OR:
$\displaystyle[{\gamma}p-m]u(\mathbf{k},t)=0,$
$\displaystyle{[{\gamma}p^{c}+m]}v(\mathbf{k},t)=0,$ (5)
where $p^{0}=\omega(\mathbf{p})=\sqrt{m^{2}+\mathbf{p}^{2}}$. These equations
have the orthogonal solutions which is convenient to normalize on unit [4, 19]
$\displaystyle{{u}}^{+}_{\alpha}(\mathbf{k},t){v_{\beta}}(-\mathbf{k},t)=0$ ,
$\displaystyle
u^{+}_{\alpha}(\mathbf{k},t)u_{\beta}(\mathbf{k},t)=v^{+}_{\alpha}(-\mathbf{k},t)v_{\beta}(-\mathbf{k},t)=\delta_{\alpha\beta}$
,
$\displaystyle{\bar{u}}_{\alpha}(\mathbf{k},t)u_{\beta}(\mathbf{k},t)=\frac{m}{\omega(\mathbf{k},t)}{\delta}_{\alpha\beta},\qquad{\bar{v}}_{\alpha}(\mathbf{k},t)v_{\beta}(\mathbf{k},t)=-\frac{m}{\omega(\mathbf{k},t)}{\delta}_{\alpha\beta}$
, (6)
The decompositions (2) and the relation (2) lead to the diagonal form of the
Hamiltonian (3) at once (before second quantization)
$H(t)=\sum_{\mathbf{k},\alpha}\omega(\mathbf{k},t)\left[a_{\alpha}^{+}(\mathbf{k},t)a_{\alpha}(\mathbf{k},t)-b_{\alpha}(-\mathbf{k},t)b^{+}_{\alpha}(-\mathbf{k},t)\right].$
(7)
Such form of the Hamiltonian is necessary for interpretation of the time
dependent operators $a^{+},a$ (and $b^{+},b$) as the operators of creation and
annihilation of quasi-particles (anti-quasi-particles). Thus, this way results
to QP representation at once.
Now, in order to get the equations of motion for the creation and annihilation
operators in the OR, let us substitute the decomposition (2) in the Eq.(2) and
use the relations (2). Then we obtain as the intermediate result the following
closed system of equations of motion in the matrix form:
$\displaystyle\dot{a}(\mathbf{k},t)+U_{(1)}(\mathbf{k},t)a(\mathbf{k},t)+U_{(2)}(\mathbf{k},t)b^{+}(-\mathbf{k},t)$
$\displaystyle=-i\omega(\mathbf{k},t)a(\mathbf{k},t),$
$\displaystyle\dot{b}(-\mathbf{k},t)-b(-\mathbf{k},t)V_{(2)}(\mathbf{k},t)+a^{+}(\mathbf{k},t)V^{+}_{(1)}(\mathbf{k},t)$
$\displaystyle=-i\omega(\mathbf{k},t)b(-\mathbf{k},t).$ (8)
The spinor constructions was introduced here
$\displaystyle U_{(1)}^{\alpha\beta}(\mathbf{k},t)$
$\displaystyle={u}^{+}_{\alpha}(\mathbf{k},t)\dot{u}_{\beta}(\mathbf{k},t),$
$\displaystyle U^{+}_{(1)}$ $\displaystyle=-U_{(1)},$ $\displaystyle
U_{(2)}^{\alpha\beta)}(\mathbf{k},t)$
$\displaystyle={u}^{+}_{\alpha}(\mathbf{k},t)\dot{v}_{\beta}(-\mathbf{k},t),$
$\displaystyle U^{+}_{(2)}$ $\displaystyle=-V_{(1)},$ $\displaystyle
V_{(2)}^{\alpha\beta}(\mathbf{k},t)$
$\displaystyle={v}^{+}_{\alpha}(-\mathbf{k},t)\dot{v}_{\beta}(-\mathbf{k},t),$
$\displaystyle V^{+}_{(2)}$ $\displaystyle=-V_{(2)}.$ (9)
The matrices $U_{(2)}$ and $V_{(1)}$ describe transitions between states with
the positive and negative energies and different spin while the matrixes
$U_{(1)}$ and $V_{(2)}$ show the spin rotations in the external field
$\mathbf{A}^{k}(t)$.
The equations (2) are compatible with the standard anti-commutation relations
because the matrix $U_{(1)}$ is anti-hermitian:
$\\{a_{\alpha}(\mathbf{k},t),a^{+}_{\beta}(\mathbf{k}^{\prime},t)\\}=\\{b_{\alpha}(\mathbf{k},t),b^{+}_{\beta}(\mathbf{k}^{\prime},t)\\}=\delta_{\mathbf{k}\mathbf{k}^{\prime}}\delta_{\alpha\beta}.$
(10)
Let us write the $u,v$-spinors in the explicit form using the corresponding
free spinors [20]:
$\displaystyle
u^{+}_{1}(\mathbf{k},t)=A(\mathbf{p})\begin{bmatrix}\omega_{+},0,p^{3},p_{-}\end{bmatrix},$
$\displaystyle
u^{+}_{2}(\mathbf{k},t)=A(\mathbf{p})\begin{bmatrix}0,\omega_{+},p_{+},-p^{3}\end{bmatrix},$
$\displaystyle
v^{+}_{1}(-\mathbf{k},t)=A(\mathbf{p})\begin{bmatrix}-p^{3},-p_{-},\omega_{+},0\end{bmatrix},$
$\displaystyle
v^{+}_{2}(-\mathbf{k},t)=A(\mathbf{p})\begin{bmatrix}-p_{+},p^{3},0,\omega_{+}\end{bmatrix},$
(11)
where $p_{\pm}=p^{1}\pm ip^{2}$, $\omega_{+}=\omega+m$ and
$A(\mathbf{p})=[2\omega\omega_{+}]^{-1/2}$. In this representation
$U_{(1)}=V_{(2)}$ and $U_{(2)}=-V_{(1)}$ so a sufficient set is
$\displaystyle U_{(1)}(\mathbf{k},t)$ $\displaystyle=i\omega
a[\mathbf{p}\mathbf{E}]\mathbf{{\boldsymbol{\sigma}}},$ $\displaystyle
U_{(2)}(\mathbf{k},t)$ $\displaystyle=\mathbf{q}\mathbf{\boldsymbol{\sigma}},$
(12)
where $\sigma^{k}$ are the Pauli matrices,
$\mathbf{q}=a[\mathbf{p}(\mathbf{p}\mathbf{E})-\mathbf{E}\omega\omega_{+}]$
and $a=e/2\omega^{2}\omega_{+}$.
The operator equations of motion (2) become more simple:
$\displaystyle\dot{a}(\mathbf{k},t)$
$\displaystyle=-U_{(1)}(\mathbf{k},t)a(\mathbf{k},t)-U_{(2)}b^{+}(-\mathbf{k},t)-i\omega(\mathbf{k},t)a(\mathbf{k},t),$
$\displaystyle\dot{b}(-\mathbf{k},t)$
$\displaystyle=b(-\mathbf{k},t)U_{(1)}(\mathbf{k},t)+a^{+}(\mathbf{k},t)U_{(2)}(\mathbf{k},t)-i\omega(\mathbf{k},t)b(-\mathbf{k},t).$
(13)
## 3 Kinetic equation (the general case)
In order to get KE for time dependent electric fields of arbitrary
polarization, let us introduce the one particle correlation functions of
electrons and positrons
$\displaystyle f_{\alpha\beta}(\mathbf{k},t)$
$\displaystyle=\,<a^{+}_{\beta}(\mathbf{k},t)a_{\alpha}(\mathbf{k},t)>,$
$\displaystyle{f}^{c}_{\alpha\beta}(\mathbf{k},t)$
$\displaystyle=\,<b_{\beta}(-\mathbf{k},t)b^{+}_{\alpha}(-\mathbf{k},t)>,$
(14)
where the averaging procedure is performed over the in-vacuum state [5]. The
diagonal parts of these correlators are connected with relations
$\sum\limits_{\mathbf{k},\alpha}\bigl{(}f_{\alpha\alpha}(\mathbf{k},t)+f^{c}_{\alpha\alpha}(\mathbf{k},t)\bigr{)}=Q,$
(15)
where $Q$ \- total electric charge of the system. Differentiation over time
leads to equations
$\displaystyle\dot{f}$
$\displaystyle=[f,U_{(1)}]-\bigl{(}U_{(2)}f^{(+)}+f^{(-)}U_{(2)}\bigr{)},$
$\displaystyle\dot{f}^{c}$
$\displaystyle=[f^{c},U_{(1)}]+\bigl{(}f^{(+)}U_{(2)}+U_{(2)}f^{(-)}\bigr{)},$
(16)
where the auxiliary correlation functions was introduced
$\displaystyle f^{(+)}_{\alpha\beta}(\mathbf{k},t)$
$\displaystyle=\,<a^{+}_{\beta}(\mathbf{k},t)b^{+}_{\alpha}(-\mathbf{k},t)>,$
$\displaystyle f_{\alpha\beta}^{(-)}(\mathbf{k},t)$
$\displaystyle=\,<b_{\beta}(-\mathbf{k},t)a_{\alpha}(\mathbf{k},t)>.$ (17)
The equations of motion for these functions can be obtained similarly:
$\displaystyle\dot{f}^{(+)}=[{f}^{(+)},U_{(1)}]+\bigl{(}U_{(2)}f-f^{c}U_{(2)}\bigr{)}+2i\omega
f^{(+)},$
$\displaystyle\dot{f}^{(-)}=[{f}^{(-)},U_{(1)}]+\bigl{(}fU_{(2)}-U_{(2)}f^{c}\bigr{)}-2i\omega
f^{(-)}$ (18)
with the connection $\stackrel{{\scriptstyle+}}{{f^{(+)}}}=f^{(-)}.$ In
general case, the Eqs.(3) and(18) represent the closed system of 16 ordinary
differential equations.
Accounting of charge symmetry (in consequence of that $f^{c}=1-f$ allows to
reduce this number up to 12. If to express the anomalous correlators (3) via
the original functions (3) with help of Eqs.(3), it can obtain the closed KE
in the integro-differential form [17]. Let us write this KE of non-Markovian
type in the following matrix form:
$\dot{f}(t)=[f(t),U_{(1)}]-U_{(2)}(t)S(t)\int\limits_{t_{0}}^{t}dt^{\prime}S^{+}(t^{\prime})[U_{(2)}(t^{\prime})f(t^{\prime})-{f}^{c}(t^{\prime})U_{(2)}(t^{\prime})]S(t^{\prime})S^{+}(t^{\prime})e^{2i\theta(t,t^{\prime})}\\\
-S(t)\int\limits_{t_{0}}^{t}dt^{\prime}S^{+}(t^{\prime})[f(t^{\prime})U_{(2)}(t^{\prime})-U_{(2)}(t^{\prime}){f}^{c}(t^{\prime})]S(t^{\prime})S^{+}(t^{\prime})U_{(2)}(t)e^{-2i\theta(t,t^{\prime})},$
(19)
where the evolution operator of the spin rotations $S(\mathbf{k},t)$ is
defined by equation
$\dot{S}=-U_{(1)}(t)S(t)$ (20)
with the initial condition $S(t_{0})=1$ ($t_{0}$ is some initial time) and
$\theta(t,t^{\prime})=\theta(t)-\theta(t^{\prime})$,
$\theta(t)=\int\limits_{t_{0}}^{t}dt^{\prime}\omega(\mathbf{k},t^{\prime}).$
(21)
In comparison with the KE for the known case of the linear polarized field
$\mathbf{A}(t)=\\{0,0,A^{3}(t)=A(t)\\},$ (22)
KE (19) has more complicated form because nontrivial spin effects. In general
case, KE (19) is not allow simplification because of $[U_{(1)},U_{(2)}]\neq
0$.
## 4 Perturbation theory
Let us write the source term (the right hand side) of KE (19) in the leading
(second) order of the perturbative theory with respect to weak external field,
$E_{m}/E_{c}\ll 1$. The adiabatic parameter [8] $\gamma=\frac{m\nu}{eE_{m}}$
is arbitrary (here $E_{m}$ is amplitude of external electric field, $\nu$ is
it characteristic frequency). In according to the relations (12), $U_{(1)}\sim
U_{(2)}\sim E_{m}$ in the leading approximation. Then in the leading order it
is necessary to put $S\to S_{0}=1$ according to Eq.(20).
We take into account also electroneutrality of the system and relation (10),
so $f^{c}=1-f$. In the considered leading approximation, the diagonal terms of
the correlation functions (12) if small in comparison with unit,
$f_{\alpha\alpha}$, and the non-diagonal terms $f_{\alpha\beta}\sim E^{2}$ for
$\alpha\neq\beta$, that allows to omit the corresponding contribution in the
source term
$\dot{f}(t)=\int\limits_{t_{0}}^{t}Sp\\{U_{(2)}(t)U_{(2)}(t^{\prime})\\}\cos{2\theta(t,t^{\prime})}.$
(23)
As it follows from Eq. (12) ($\omega_{+}=\omega_{0}$),
$\displaystyle
2Sp\\{U_{(2)}(t)U_{(2)}(t^{\prime})\\}=\frac{e^{2}}{2\omega^{2}\omega_{0}^{2}}\left\\{\mathbf{E}(t)\mathbf{E}(t^{\prime})\omega\omega_{0}-(\mathbf{E}(t)\mathbf{p})(\mathbf{E}(t^{\prime})\mathbf{p})\right\\}=\Phi(\mathbf{p}|t,t^{\prime}).$
(24)
If at the initial time before switch-on of an electric field the electrons and
positrons are absent, we can write the total density of quasiparticles
$n(t)=\frac{1}{4\pi^{3}}\int
d^{3}p\int\limits_{t_{0}}^{t}dt_{1}\int\limits_{t_{0}}^{t_{1}}dt_{2}\Phi(\mathbf{p}|t_{1},t_{2})\cos{[2\theta(t_{1},t_{2}]}.$
(25)
In the case of the linear polarization (22), from Eqs. (24) and (25) it
follows the well known result [13, 14]:
$n(t)=\frac{1}{4\pi^{3}}\int
d^{3}p\left|\int\limits_{t_{0}}^{t}dt^{\prime}\lambda(t^{\prime})\exp{(2i\theta(t,t^{\prime}))}\right|^{2},$
(26)
where $\lambda(\mathbf{p},t)=eE(t)\varepsilon_{\perp}/2\omega^{2}$ and
$\varepsilon_{\perp}^{2}=m_{2}+p_{\perp}^{2}$, $\mathbf{p}_{\perp}$ is the
transversal momentum relatively of the vector $\mathbf{E}(t)$.
The relations (25) and (26) are convenient for the numerical analysis, that is
planned to made in the following work.
## 5 Conclusion
Thus, it was shown that the oscillator representation may be used for the KE
derivation in the rather non-trivial case of the time-dependent chromo-
electric field of arbitrary polarization. The obtained KE’s can be used for
investigation of particle-antiparticle vacuum creation in strong laser fields
of optical and X-ray range as well as in the chromo-electric fields acting in
the pre-equilibrium stage of QGP evolution. Besides, the used method opens
prospects for further generalization ( e.g., the account of a constant
magnetic field).
## References
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|
arxiv-papers
| 2009-01-05T17:05:20 |
2024-09-04T02:48:59.722821
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.V. Filatov, S.A. Smolyansky, A.V. Tarakanov (Physical Department of\n Saratov State University)",
"submitter": "Alexander Tarakanov",
"url": "https://arxiv.org/abs/0901.0522"
}
|
0901.0537
|
Nonlinear Dimensionality Reduction Methods in Climate
Data Analysis
Ian Ross
Doctor of Philosophy
School of Geographical Sciences
ca. 80,000 words
October 2005
September 2008
Linear dimensionality reduction techniques, notably principal
component analysis, are widely used in climate data analysis as a
means to aid in the interpretation of datasets of high
dimensionality. These linear methods may not be appropriate for the
analysis of data arising from nonlinear processes occurring in the
climate system. Numerous techniques for nonlinear dimensionality
reduction have been developed recently that may provide a
potentially useful tool for the identification of low-dimensional
manifolds in climate data sets arising from nonlinear dynamics. In
this thesis I apply three such techniques to the study of El
Niño/Southern Oscillation variability in tropical Pacific sea
surface temperatures and thermocline depth, comparing observational
data with simulations from coupled atmosphere-ocean general
circulation models from the CMIP3 multi-model ensemble.
The three methods used here are a nonlinear principal component
analysis (NLPCA) approach based on neural networks, the Isomap
isometric mapping algorithm, and Hessian locally linear embedding.
I use these three methods to examine El Niño variability in the
different data sets and assess the suitability of these nonlinear
dimensionality reduction approaches for climate data analysis.
I conclude that although, for the application presented here,
analysis using NLPCA, Isomap and Hessian locally linear embedding
does not provide additional information beyond that already provided
by principal component analysis, these methods are effective tools
for exploratory data analysis.
I declare that the work in this dissertation was carried out in
accordance with the Regulations of the University of Bristol. The
material presented here is the result of my own independent research
performed at the University of Bristol, School of Geographical
Sciences, between and
, and no part of the dissertation has been submitted
for any other academic award. Sections of
Chapters <ref>, <ref> and <ref>
and all of Chapter <ref> have previously appeared as:
I. Ross, P. J. Valdes and S. Wiggins. ENSO dynamics in current
climate models: an investigation using nonlinear dimensionality
reduction. Nonlin. Processes Geophys., 15(2):339–363,
April 2008.
Any opinions expressed in this thesis are those of the author.
Many thanks to my supervisors, Paul Valdes and Steve Wiggins. Paul,
first of all, gave me a job and provided a nurturing and congenial
environment, in the form of the BRIDGE group. Paul's easy-going
leadership really set the tone for BRIDGE (“field trips” that
consist of a weekend camping and surfing in Devon, anyone?), which
ended up being a very productive arrangement for everyone concerned.
I've certainly enjoyed being part of that and the group will be
something I'll miss a lot when I leave Bristol.
As for Steve, during the three years of my Ph.D., we missed just a
handful of our weekly meetings due to his absence or other
engagements. For someone covering head of department
responsibilities while maintaining an active research programme,
that's an extraordinary level of commitment, one for which I am very
grateful. The only thing that (slightly) tempers this gratitude is
Steve's habit of sending emails in the dead of night with wads of
papers attached to them, all with the comment “You really should
know about this stuff...”. As a result of Steve's
“encouragement”, I've probably read about five times as much as I
otherwise would have done. I even enjoyed some of it.
Of the other BRIDGE-ites, special mention has to go to Rupes (for
always refusing to understand things in the most enlightening
fashion possible), Dan (“I trust David Blunkett!”), Gethin (a boy
from Wales more interested in computers than sheep) and Rachel (her
door is always open, she's always ready for a chat, and she lives at
the bottom of the steepest hill in Somerset). Also, apologies to
anyone who's had to give a group seminar with me in the front row
heckling (that's nearly everyone!).
This is a thesis about climate data analysis, so we need some
climate data. I've used data from the NCEP atmospheric and ocean
reanalyses, both truly excellent resources, I've used the NOAA ERSST
v2 data set, and I've used GCM simulations archived for the IPCC
Fourth Assessment Report. There's a blurb that goes with the IPCC
data: “I acknowledge the modelling groups, the Program for Climate
Model Diagnosis and Intercomparison (PCMDI) and the WCRP's Working
Group on Coupled Modelling (WGCM) for their roles in making
available the WCRP CMIP3 multi-model data set. Support for this
data set is provided by the Office of Science, U.S. Department of
Energy”. Those official words don't capture just how useful these
multi-model ensemble databases are and what a job it is to organise
them. All kudos to the people involved! On another official note,
I should mention that my Ph.D. work was funded by an e-Science
studentship from NERC, number NER/S/G/2005/13913.
Finally, of course, an enormous thank you to Rita. She lives with
me, shares her life with me, sometimes works with me, even puts up
with my “jokes”, and yet through all of this, she maintains the
sunniest of dispositions, the happiest of smiles. As anyone who
knows me will attest, this must mean that she is a very angel.
We've had a lot of fun over the last four and a half years,
including some things that were more “fun” than fun (the
completion of two Ph.D.s, broken collarbones, invisible fishbones,
immigration anxieties), but some that were absolute unalloyed
FUN (holidays in Ireland, Greece, even Austria, and every
everyday day). I am absolutely sure that we will have many years
more. Life is good, and the reason is Rita.
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|
arxiv-papers
| 2009-01-02T16:33:30 |
2024-09-04T02:48:59.729435
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ian Ross (University of Bristol)",
"submitter": "Ian Ross",
"url": "https://arxiv.org/abs/0901.0537"
}
|
0901.0739
|
FERMILAB-PUB-08-582-E
# Measurement of $\gamma+b+X$ and $\gamma+c+X$ production cross sections in
$p\bar{p}$ collisions at $\sqrt{s}=1.96~{}\text{TeV}$
V.M. Abazov36 B. Abbott75 M. Abolins65 B.S. Acharya29 M. Adams51 T. Adams49 E.
Aguilo6 M. Ahsan59 G.D. Alexeev36 G. Alkhazov40 A. Alton64,a G. Alverson63
G.A. Alves2 M. Anastasoaie35 L.S. Ancu35 T. Andeen53 B. Andrieu17 M.S.
Anzelc53 M. Aoki50 Y. Arnoud14 M. Arov60 M. Arthaud18 A. Askew49,b B. Åsman41
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Bagby50 B. Baldin50 D.V. Bandurin59 P. Banerjee29 S. Banerjee29 E. Barberis63
A.-F. Barfuss15 P. Bargassa80 P. Baringer58 J. Barreto2 J.F. Bartlett50 U.
Bassler18 D. Bauer43 S. Beale6 A. Bean58 M. Begalli3 M. Begel73 C. Belanger-
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1Universidad de Buenos Aires, Buenos Aires, Argentina 2LAFEX, Centro
Brasileiro de Pesquisas Físicas, Rio de Janeiro, Brazil 3Universidade do
Estado do Rio de Janeiro, Rio de Janeiro, Brazil 4Universidade Federal do
ABC, Santo André, Brazil 5Instituto de Física Teórica, Universidade Estadual
Paulista, São Paulo, Brazil 6University of Alberta, Edmonton, Alberta,
Canada, Simon Fraser University, Burnaby, British Columbia, Canada, York
University, Toronto, Ontario, Canada, and McGill University, Montreal, Quebec,
Canada 7University of Science and Technology of China, Hefei, People’s
Republic of China 8Universidad de los Andes, Bogotá, Colombia 9Center for
Particle Physics, Charles University, Prague, Czech Republic 10Czech
Technical University, Prague, Czech Republic 11Center for Particle Physics,
Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech
Republic 12Universidad San Francisco de Quito, Quito, Ecuador 13LPC,
Université Blaise Pascal, CNRS/IN2P3, Clermont, France 14LPSC, Université
Joseph Fourier Grenoble 1, CNRS/IN2P3, Institut National Polytechnique de
Grenoble, Grenoble, France 15CPPM, Aix-Marseille Université, CNRS/IN2P3,
Marseille, France 16LAL, Université Paris-Sud, IN2P3/CNRS, Orsay, France
17LPNHE, IN2P3/CNRS, Universités Paris VI and VII, Paris, France 18CEA, Irfu,
SPP, Saclay, France 19IPHC, Université Louis Pasteur, CNRS/IN2P3, Strasbourg,
France 20IPNL, Université Lyon 1, CNRS/IN2P3, Villeurbanne, France and
Université de Lyon, Lyon, France 21III. Physikalisches Institut A, RWTH
Aachen University, Aachen, Germany 22Physikalisches Institut, Universität
Bonn, Bonn, Germany 23Physikalisches Institut, Universität Freiburg,
Freiburg, Germany 24Institut für Physik, Universität Mainz, Mainz, Germany
25Ludwig-Maximilians-Universität München, München, Germany 26Fachbereich
Physik, University of Wuppertal, Wuppertal, Germany 27Panjab University,
Chandigarh, India 28Delhi University, Delhi, India 29Tata Institute of
Fundamental Research, Mumbai, India 30University College Dublin, Dublin,
Ireland 31Korea Detector Laboratory, Korea University, Seoul, Korea
32SungKyunKwan University, Suwon, Korea 33CINVESTAV, Mexico City, Mexico
34FOM-Institute NIKHEF and University of Amsterdam/NIKHEF, Amsterdam, The
Netherlands 35Radboud University Nijmegen/NIKHEF, Nijmegen, The Netherlands
36Joint Institute for Nuclear Research, Dubna, Russia 37Institute for
Theoretical and Experimental Physics, Moscow, Russia 38Moscow State
University, Moscow, Russia 39Institute for High Energy Physics, Protvino,
Russia 40Petersburg Nuclear Physics Institute, St. Petersburg, Russia 41Lund
University, Lund, Sweden, Royal Institute of Technology and Stockholm
University, Stockholm, Sweden, and Uppsala University, Uppsala, Sweden
42Lancaster University, Lancaster, United Kingdom 43Imperial College, London,
United Kingdom 44University of Manchester, Manchester, United Kingdom
45University of Arizona, Tucson, Arizona 85721, USA 46Lawrence Berkeley
National Laboratory and University of California, Berkeley, California 94720,
USA 47California State University, Fresno, California 93740, USA
48University of California, Riverside, California 92521, USA 49Florida State
University, Tallahassee, Florida 32306, USA 50Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA 51University of Illinois at Chicago,
Chicago, Illinois 60607, USA 52Northern Illinois University, DeKalb, Illinois
60115, USA 53Northwestern University, Evanston, Illinois 60208, USA
54Indiana University, Bloomington, Indiana 47405, USA 55University of Notre
Dame, Notre Dame, Indiana 46556, USA 56Purdue University Calumet, Hammond,
Indiana 46323, USA 57Iowa State University, Ames, Iowa 50011, USA
58University of Kansas, Lawrence, Kansas 66045, USA 59Kansas State
University, Manhattan, Kansas 66506, USA 60Louisiana Tech University, Ruston,
Louisiana 71272, USA 61University of Maryland, College Park, Maryland 20742,
USA 62Boston University, Boston, Massachusetts 02215, USA 63Northeastern
University, Boston, Massachusetts 02115, USA 64University of Michigan, Ann
Arbor, Michigan 48109, USA 65Michigan State University, East Lansing,
Michigan 48824, USA 66University of Mississippi, University, Mississippi
38677, USA 67University of Nebraska, Lincoln, Nebraska 68588, USA
68Princeton University, Princeton, New Jersey 08544, USA 69State University
of New York, Buffalo, New York 14260, USA 70Columbia University, New York,
New York 10027, USA 71University of Rochester, Rochester, New York 14627, USA
72State University of New York, Stony Brook, New York 11794, USA 73Brookhaven
National Laboratory, Upton, New York 11973, USA 74Langston University,
Langston, Oklahoma 73050, USA 75University of Oklahoma, Norman, Oklahoma
73019, USA 76Oklahoma State University, Stillwater, Oklahoma 74078, USA
77Brown University, Providence, Rhode Island 02912, USA 78University of
Texas, Arlington, Texas 76019, USA 79Southern Methodist University, Dallas,
Texas 75275, USA 80Rice University, Houston, Texas 77005, USA 81University
of Virginia, Charlottesville, Virginia 22901, USA 82University of Washington,
Seattle, Washington 98195, USA
(January 6, 2009)
###### Abstract
First measurements of the differential cross sections
$\mathrm{d}^{3}\sigma/(\mathrm{d}p_{T}^{\gamma}\mathrm{d}y^{\gamma}\mathrm{d}y^{\text{jet}})$
for the inclusive production of a photon in association with a heavy quark
($b$, $c$) jet are presented, covering photon transverse momenta
$30<p_{T}^{\gamma}<150$ GeV, photon rapidities $|y^{\gamma}|<1.0$, jet
rapidities $|y^{\text{jet}}|<0.8$, and jet transverse momenta $p_{T}^{\rm
jet}>15$ GeV. The results are based on an integrated luminosity of $1$ fb-1 in
$p\bar{p}$ collisions at $\sqrt{s}=1.96~{}\text{TeV}$ recorded with the D0
detector at the Fermilab Tevatron Collider. The results are compared with
next-to-leading order perturbative QCD predictions.
###### pacs:
13.85.Qk, 12.38.Qk
Photons ($\gamma$) produced in association with heavy quarks $Q$ ($\equiv c$
or $b$) in the final state of hadron-hadron interactions provide valuable
information about the parton distributions of the initial state hadrons
Aurenche et al. (1996); Pumplin et al. (2007). Such events are produced
primarily through the QCD Compton-like scattering process $gQ\to\gamma Q$,
which dominates up to photon transverse momenta ($p_{T}^{\gamma}$) of $\sim
90$ GeV for $\gamma+{c}+X$ and up to $\sim 120$ GeV for $\gamma+{b}+X$
production, but also through quark-antiquark annihilation $q\bar{q}\to\gamma
g\to\gamma Q\bar{Q}$. Consequently, $\gamma+Q+X$ production is sensitive to
the $b$, $c$, and gluon ($g$) densities within the colliding hadrons, and can
provide constraints on parton distribution functions (PDFs) that have
substantial uncertainties Wu-Ki et al. (2005); D. Stump et al. (2003). The
heavy quark and gluon content is an important aspect of QCD dynamics and of
the fundamental structure of the proton. In particular, many searches for new
physics, e.g. for certain Higgs boson production modes Brodsky et al. (2006);
He et al. (2006); K.A.Assamagan et al. (2003); Gluck et al. (2008), will
benefit from a more precise knowledge of the heavy quark and gluon content of
the proton.
This Letter presents the first measurements of the inclusive differential
cross sections
$\mathrm{d}^{3}\sigma/(\mathrm{d}p_{T}^{\gamma}\mathrm{d}y^{\gamma}\mathrm{d}y^{\text{jet}})$
for $\gamma+b+X$ and $\gamma+c+X$ production in $p\bar{p}$ collisions, where
$y^{\gamma}$ and $y^{\text{jet}}$ are the photon and jet rapidities rapidity .
The results are based on an integrated luminosity of 1.02$~{}\pm~{}0.06$ fb-1
Andeen et al. (1994) collected with the D0 detector Abazov et al. (2006) at
the Fermilab Tevatron Collider at $\sqrt{s}=1.96~{}\text{TeV}$. The highest
$p_{T}$ (leading) photon and jet are required to have $|y^{\gamma}|<1.0$ and
$|y^{\text{jet}}|<0.8$, and transverse momentum $30<p_{T}^{\gamma}<150$ GeV
and $p_{T}^{\rm jet}>15$ GeV. This selection allows one to probe PDFs in the
range of parton-momentum fractions $0.01\lesssim x\lesssim 0.3$, and hard
scatter scales of $9\times 10^{2}\lesssim
Q^{2}\equiv(p_{T}^{\gamma})^{2}\lesssim 2\times 10^{4}~{}\text{GeV}^{2}$.
Differential cross sections are presented for two regions of kinematics,
defined by $y^{\gamma}y^{\text{jet}}>0$ and $y^{\gamma}y^{\text{jet}}<0$.
These two regions provide greater sensitivity to the parton $x$ because they
probe different sets of $x_{1}$ and $x_{2}$ intervals, as discussed in Ref.
Abazov et al. (2008).
The triggers for this analysis identify clusters of large electromagnetic (EM)
energy, and are based on $p_{T}^{\gamma}$ and on the spatial distribution of
energy in the photon shower. The trigger efficiency is $\approx$96% for photon
candidates with $p_{T}^{\gamma}=30$ GeV and rises to nearly 100% for
$p_{T}^{\gamma}>40$ GeV.
To reconstruct photon candidates, towers Abazov et al. (2006) with large
depositions of energy are used as seeds to create clusters of energy in the EM
calorimeter in a cone of radius ${\cal R}=0.4$, where ${\cal
R}\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$ etaphi . Once an EM energy
cluster is formed, the final energy $E_{\text{EM}}$ is defined by a smaller
cone of ${\cal R}=0.2$. Photon candidates are required to be isolated within
the calorimeter, and must also have $>96$% of their energy in its EM section.
We require the sum of the total energy inside a cone of ${\cal R}=0.4$, after
the subtraction of $E_{\text{EM}}$, to be $<7$% of $E_{\text{EM}}$. We also
require the width of the energy-weighted shower in the most finely segmented
part of the EM calorimeter to be consistent with that expected for an
electromagnetic shower, and the probability for any track spatially matched to
the photon EM cluster to be $<$0.1%. Background from dijet events containing
$\pi^{0}$ and $\eta$ mesons that can mimic photon signatures is also rejected
using an artificial neural network for identifying photons ($\gamma$-ANN),
described in Ref. Abazov et al. (2008). The requirement that the $\gamma$-ANN
output be $>0.7$, combined with all other photon selection critera, reduces
the dijet event efficiency to 0.1–0.5%. We calculate photon detection
efficiencies using a Monte Carlo (MC) simulation. Signal events are generated
using pythia Sjöstrand et al. (2001) and processed through a geant-based Brun
and Carminati (2001) simulation of the detector geometry and response, and
reconstructed using the same software as for the data. The MC efficiencies are
calibrated to those in data using small correction factors measured in $Z\to
e^{+}e^{-}$ samples. The total efficiency of the above photon selection
criteria is 63–80%, depending on $p_{T}^{\gamma}$. The systematic
uncertainties on these values are 5%, and are mainly due to uncertainties in
the isolation, the track-match veto, and the $\gamma$-ANN requirements.
At least one jet must be present in each event. Jets are reconstructed using
the D0 Run II algorithm Zeppenfeld et al. (1994) with a radius of $0.5$. The
efficiency for a jet to be reconstructed and to satisfy the jet identification
criteria is 93%, 96.5%, and 94.5% for light ($u$, $d$, $s$ quark or $g$), $c$,
and $b$ jets at $p_{T}^{\gamma}=30~{}\text{GeV}$ and increases to $\approx
98$% at $p_{T}^{\gamma}=150$ GeV, independent of the jet flavor. The impact
from uncertainties on jet energy scale, jet energy resolution, and difference
in energy response between light and $b(c)$ jets is found to be between 8 %(6
%) and 2 %(2 %) for $p_{T}^{\rm jet}$ between 15 GeV and 150 GeV. The leading
jet is also required to have at least two associated tracks with $p_{T}>0.5$
GeV and the track leading in $p_{T}$ must have $p_{T}>1.0$ GeV, and each track
must have at least one hit in the silicon microstrip tracker. These criteria
ensure that the jet has sufficient information to be classified as a heavy-
flavor (HF) candidate. Light jets are suppressed using a dedicated artificial
neural network ($b$-ANN) c:bNN that exploits the longer lifetimes of heavy-
flavor hadrons relative to their lighter counterparts. The leading jet is
required to have a $b$-ANN output $>0.85$. Depending on $p_{T}^{\gamma}$, this
selection is 55–62% efficient for $\gamma+b$ jet, and 11–12% efficient for
$\gamma+c$ jet events, with 3–5% relative uncertainties on these values. Only
0.2–1% of light jets are misidentified as heavy-flavor jets.
A primary collision vertex with $\geq$3 tracks is required within 35 cm of the
center of the detector along the beam axis. The missing transverse momentum in
the event is required to be $<0.7p_{T}^{\gamma}$ so as to suppress background
from cosmic-ray muons and $W\to\ell\nu$ decays. Such a requirement is highly
efficient for signal, achieving an efficiency $\geq 96\%$ even for events with
semi-leptonic heavy-flavor quark decays.
About 13,000 events remain in the data sample after applying all selection
criteria. Background for photons, stemming mainly from dijet events in which
one jet is misidentified as a photon, is still present in this sample. To
estimate the photon purity, a template fitting technique is employed Barlow et
al. (1993). The $\gamma$-ANN distribution in data is fitted to a linear
combination of templates for photons and jets obtained from simulated
$\gamma~{}+$ jet and dijet samples, respectively. An independent fit is
performed in each $p_{T}^{\gamma}$ bin, yielding photon purities between 51%
and 93% for $30<p_{T}^{\gamma}<150~{}\text{GeV}$. The fractional contributions
of $b$ and $c$ jets are determined by fitting templates of $P_{\text{HF-
jet}}=-\ln\prod_{i}{P_{\rm track}^{i}}$ to the data, where $P_{\rm track}^{i}$
is the probability that a track originates from the primary vertex, based on
the significance of the track’s distance of closest approach to the primary
vertex. All tracks within the jet cone are used in the fit, except the one
with lowest value of $P_{\rm track}$. Jets from $b$ quarks usually have large
values of $P_{\text{HF-jet}}$, whereas light jets mostly have small values, as
their tracks originate from the primary vertex. Templates are used for the
shape information of the $P_{\text{HF-jet}}$ distributions. For $b$ and $c$
jets these are extracted from MC events whereas the light jet template is
taken from a data sample enriched in light jets, which is corrected for
contributions from $b$ and $c$ quarks.
Figure 1: Distribution of observed events for $P_{\text{HF-jet}}$ after all
selection criteria for the bin $50<p_{T}^{\gamma}<70$ GeV. The distributions
for the $b$, $c$, and light jet templates are shown normalized to their fitted
fraction. Error bars on the templates represent combined uncertainties from
statistics of the MC and the fitted jet flavor fractions, while the data
contain just statistical uncertainties. Fits in the other $p_{T}^{\gamma}$
bins are of similar quality.
The result of a maximum likelihood fit, normalized to the number of events in
data, is shown in Fig. 1 for $50<p_{T}^{\gamma}<70~{}\text{GeV}$. The
estimated fractions of $b$ and $c$ jets in all $p_{T}^{\gamma}$ bins vary
between 25–34% and 40–48%, respectively. The corresponding uncertainties range
between 7-24%, dominated at higher $p_{T}^{\gamma}$ by the limited data
statistics.
The differential cross sections are extracted in five bins of $p_{T}^{\gamma}$
and in the two regions of $y^{\gamma}y^{\text{jet}}$, and are all listed in
Table 1. The measured cross sections are corrected for the effect of finite
calorimeter energy resolution affecting $p_{T}^{\gamma}$ using the unfolding
procedure described in Ref. Abbott et al. (2001). Such corrections are 1–3%.
The measured differential cross sections are shown in Fig. 2 for
$\gamma+{b}+X$ and $\gamma+{c}+X$ production as a function of $p_{T}^{\gamma}$
for the jet and photon rapidity intervals in question. The cross sections fall
by more than three orders of magnitude in the range
$30<p_{T}^{\gamma}<150~{}\text{GeV}$. The statistical uncertainty on the
results ranges from 2% in the first $p_{T}^{\gamma}$ bin to $\approx 9\%$ in
the last bin, while the total systematic uncertainty varies between 15% and
28%. The main uncertainty at low $p_{T}^{\gamma}$ is due to the photon purity
(10.5%) and the heavy-flavor fraction fit (9%). At higher $p_{T}^{\gamma}$,
the uncertainty is dominated by the heavy-flavor fraction. Other significant
uncertainties result from the jet-selection efficiency (between 8% and 2%),
the photon selection efficiency (5%), and the luminosity (6.1%) Andeen et al.
(1994). Systematic uncertainties have a 60–68% correlation between adjacent
$p_{T}^{\gamma}$ bins for $30<p_{T}^{\gamma}<50$ GeV and 20–30% for
$p_{T}^{\gamma}>$70 GeV.
Figure 2: The $\gamma+{b}+X$ and $\gamma+{c}+X$ differential cross sections as a function of $p_{T}^{\gamma}$ in the two regions $y^{\gamma}y^{\text{jet}}>0$ and $y^{\gamma}y^{\text{jet}}<0$. The uncertainties on the data points include statistical and systematic contributions added in quadrature. The NLO pQCD predictions using cteq6.6M PDFs are indicated by the dotted lines. Figure 3: The data-to-theory ratio of cross sections as a function of $p_{T}^{\gamma}$ for $\gamma+{b}+X$ and $\gamma+{c}+X$ in the regions $y^{\gamma}y^{\text{jet}}>0$ and $y^{\gamma}y^{\text{jet}}<0$. The uncertainties on the data include both statistical (inner line) and full uncertainties (entire error bar). Also shown are the uncertainties on the theoretical pQCD scales and the cteq6.6M PDFs. The scale uncertainties are shown as dotted lines and the PDF uncertainties by the shaded regions. The ratio of the standard cteq6.6M prediction to two models of intrinsic charm is also shown. Table 1: The $\gamma+{b}+X$ and $\gamma+{c}+X$ cross sections in bins of $p_{T}^{\gamma}$ in the two regions $y^{\gamma}y^{\text{jet}}>0$ and $y^{\gamma}y^{\text{jet}}<0$ together with statistical, $\delta\sigma_{\text{stat}}$, and systematic, $\delta\sigma_{\text{syst}}$, uncertainties. The theory cross sections $\sigma_{\text{theory}}$ are taken from Ref. Stavreva et al. (2003). | | | $y^{\gamma}y^{\text{jet}}>0$ | | $y^{\gamma}y^{\text{jet}}<0$
---|---|---|---|---|---
| $p_{T}^{\gamma}$ bin | | $\langle p_{T}^{\gamma}\rangle$ | Cross section | $\delta\sigma_{\text{stat}}$ | $\delta\sigma_{\text{syst}}$ | $\sigma_{\text{theory}}$ | | $\langle p_{T}^{\gamma}\rangle$ | Cross section | $\delta\sigma_{\text{stat}}$ | $\delta\sigma_{\text{syst}}$ | $\sigma_{\text{theory}}$
| (GeV) | | (GeV) | (pb/GeV) | ($\%$) | ($\%$) | (pb/GeV) | | (GeV) | (pb/GeV) | ($\%$) | ($\%$) | (pb/GeV)
$\gamma+{b}+X$ | 30–40 | | 34.1 | 2.73$\times 10^{-1}$ | 1.5 | 18.5 | 2.96$\times 10^{-1}$ | | 34.1 | 2.23$\times 10^{-1}$ | 1.6 | 19.1 | 2.45$\times 10^{-1}$
| 40–50 | | 44.3 | 1.09$\times 10^{-1}$ | 2.5 | 15.5 | 9.31$\times 10^{-2}$ | | 44.2 | 9.53$\times 10^{-2}$ | 2.6 | 16.0 | 8.18$\times 10^{-2}$
| 50–70 | | 57.6 | 2.72$\times 10^{-2}$ | 3.3 | 15.2 | 2.66$\times 10^{-2}$ | | 57.4 | 2.67$\times 10^{-2}$ | 3.3 | 15.3 | 2.22$\times 10^{-2}$
| 70–90 | | 78.7 | 6.21$\times 10^{-3}$ | 6.6 | 20.8 | 6.39$\times 10^{-3}$ | | 78.3 | 6.10$\times 10^{-3}$ | 6.7 | 20.8 | 5.49$\times 10^{-3}$
| 90–150 | | 108.3 | 1.23$\times 10^{-3}$ | 8.2 | 26.2 | 1.11$\times 10^{-3}$ | | 110.0 | 1.09$\times 10^{-3}$ | 8.9 | 25.7 | 1.05$\times 10^{-3}$
$\gamma+{c}+X$ | 30–40 | | 34.1 | 1.90 | 1.5 | 18.1 | 2.02 | | 34.1 | 1.56 | 1.6 | 18.7 | 1.59
| 40–50 | | 44.3 | 5.14$\times 10^{-1}$ | 2.5 | 17.7 | 5.82$\times 10^{-1}$ | | 44.2 | 4.51$\times 10^{-1}$ | 2.6 | 18.1 | 4.56$\times 10^{-1}$
| 50–70 | | 57.6 | 1.53$\times 10^{-1}$ | 3.3 | 17.9 | 1.41$\times 10^{-1}$ | | 57.4 | 1.50$\times 10^{-1}$ | 3.3 | 18.0 | 1.10$\times 10^{-1}$
| 70–90 | | 78.7 | 4.45$\times 10^{-2}$ | 6.6 | 21.3 | 2.85$\times 10^{-2}$ | | 78.3 | 4.39$\times 10^{-2}$ | 6.7 | 21.3 | 2.22$\times 10^{-2}$
| 90–150 | | 108.3 | 9.63$\times 10^{-3}$ | 8.2 | 27.5 | 3.69$\times 10^{-3}$ | | 110.0 | 8.57$\times 10^{-3}$ | 8.9 | 27.0 | 3.28$\times 10^{-3}$
Next-to-leading order (NLO) perturbative QCD (pQCD) predictions, with the
renormalization scale $\mu_{R}$, factorization scale $\mu_{F}$, and
fragmentation scale $\mu_{f}$, all set to $p_{T}^{\gamma}$, are also given in
Table 1 and compared to data in Fig. 2. These predictions Stavreva et al.
(2003) are are based on techniques used to calculate the cross section
analytically Harris et al. (2002), and the ratios of the measured to the
predicted cross sections are shown in Fig. 3.
The uncertainty from the choice of the scale is estimated through a
simultaneous variation of all three scales by a factor of two, i.e., to
$\mu_{R,F,f}=0.5p_{T}^{\gamma}$ and $2p_{T}^{\gamma}$. The predictions utilize
cteq6.6M PDFs D. Stump et al. (2003), and are corrected for effects of parton-
to-hadron fragmentation. This correction for $b\,(c)$ jets varies from $7.5$%
($3$%) at $30<p_{T}^{\gamma}<40~{}\text{GeV}$ to 1% at
$90<p_{T}^{\gamma}<150~{}\text{GeV}$.
The pQCD prediction agrees with the measured cross sections for $\gamma+{b}+X$
production over the entire $p_{T}^{\gamma}$ range, and with $\gamma+{c}+X$
production for $p_{T}^{\gamma}<70$ GeV. For $p_{T}^{\gamma}>70$ GeV, the
measured $\gamma+{c}+X$ cross section is higher than the prediction by about
1.6–2.2 standard deviations (including only the experimental uncertainties)
with the difference increasing with growing $p_{T}^{\gamma}$.
Parameterizations for two models containing intrinsic charm (IC) have been
included in cteq6.6 Pumplin et al. (2007), and their ratios to the standard
cteq predictions are also shown in Fig. 3. Both non-perturbative models
predict a higher $\gamma+{c}+X$ cross section. In the case of the BHPS model
Pumplin et al. (2007) it grows with $p_{T}^{\gamma}$. The observed difference
may also be caused by an underestimated contribution from the $g\to Q\bar{Q}$
splitting in the annihilation process that dominates for
$p_{T}^{\gamma}>90~{}\text{GeV}$ Amsler et al. (2008).
In conclusion, we have performed the first measurement of the differential
cross section of inclusive photon production in association with heavy flavor
($b$ and $c$) jets at a $p\bar{p}$ collider. The results cover the range
$30<p_{T}^{\gamma}<150~{}\text{GeV}$, $|y^{\gamma}|<1.0$, and $|y^{\rm
jet}|<0.8$. The measured cross sections provide information about $b$, $c$,
and gluon PDFs for $0.01\lesssim x\lesssim 0.3$. NLO pQCD predictions using
cteq6.6M PDFs Stavreva et al. (2003) for $\gamma+{b}+X$ production agree with
the measurements over the entire $p_{T}^{\gamma}$ range. We observe
disagreement between theory and data for $\gamma+{c}+X$ production for
$p_{T}^{\gamma}>70$ GeV.
We are very grateful to the authors of the theoretical code, Tzvetalina
Stavreva and Jeff Owens, for providing predictions and for many fruitful
discussions. We thank the staffs at Fermilab and collaborating institutions,
and acknowledge support from the DOE and NSF (USA); CEA and CNRS/IN2P3
(France); FASI, Rosatom and RFBR (Russia); CNPq, FAPERJ, FAPESP and FUNDUNESP
(Brazil); DAE and DST (India); Colciencias (Colombia); CONACyT (Mexico); KRF
and KOSEF (Korea); CONICET and UBACyT (Argentina); FOM (The Netherlands); STFC
(United Kingdom); MSMT and GACR (Czech Republic); CRC Program, CFI, NSERC and
WestGrid Project (Canada); BMBF and DFG (Germany); SFI (Ireland); The Swedish
Research Council (Sweden); CAS and CNSF (China); and the Alexander von
Humboldt Foundation (Germany).
## References
* (1) Visitor from Augustana College, Sioux Falls, SD, USA.
* (2) Visitor from Rutgers University, Piscataway, NJ, USA.
* (3) Visitor from The University of Liverpool, Liverpool, UK.
* (4) Visitor from II. Physikalisches Institut, Georg-August-University, Göttingen, Germany.
* (5) Visitor from Centro de Investigacion en Computacion - IPN, Mexico City, Mexico.
* (6) Visitor from ECFM, Universidad Autonoma de Sinaloa, Culiacán, Mexico.
* (7) Visitor from Helsinki Institute of Physics, Helsinki, Finland.
* (8) Visitor from Universität Bern, Bern, Switzerland.
* (9) Visitor from Universität Zürich, Zürich, Switzerland.
* (10) Deceased.
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* (13) Pseudorapidity $\eta$ is defined as $\eta=-\ln[\tan(\theta/2)]$, where $\theta$ is the polar angle with respect to the proton beam direction, with origin at the center of the detector. $\phi$ is defined as the azimuthal angle in the plane transverse to the proton beam direction.
* Sjöstrand et al. (2001) T. Sjöstrand et al., Comput. Phys. Commun. 135, 238 (2001).
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|
arxiv-papers
| 2009-01-07T00:21:30 |
2024-09-04T02:48:59.746525
|
{
"license": "Public Domain",
"authors": "D0 Collaboration: V.M. Abazov, et al",
"submitter": "Dmitry Bandurin V",
"url": "https://arxiv.org/abs/0901.0739"
}
|
0901.0792
|
# Serendipity observations of far infrared cirrus emission in the Spitzer
Infrared Nearby Galaxies Survey:
Analysis of far-infrared correlations**affiliation: This work is based on
observations made with the _Spitzer Space Telescope_ , which is operated by
the Jet Propulsion Laboratory, California Institute of Technology, under a
contract with NASA.
Caroline Bot11affiliation: California Institute of Technology, Pasadena CA
91125, USA 55affiliation: Observatoire Astronomique de Strasbourg, 67000
Strasbourg, FRANCE and George Helou11affiliation: California Institute of
Technology, Pasadena CA 91125, USA and François Boulanger22affiliation:
Institut d’Astrophyisque Spatiale, 91405 Orsay, FRANCE and Guilaine
Lagache22affiliation: Institut d’Astrophyisque Spatiale, 91405 Orsay, FRANCE
and Marc-Antoine Miville-Deschenes22affiliation: Institut d’Astrophyisque
Spatiale, 91405 Orsay, FRANCE and Bruce Draine33affiliation: Princeton
University Observatory, Princeton, NJ08544, USA and Peter Martin44affiliation:
Canadian Institute for Theoretical Astrophysics, Toronto, Ontario, M5S 3H8,
Canada bot@astro.u-strasbg.fr
###### Abstract
We present an analysis of far-infrared dust emission from diffuse cirrus
clouds. This study is based on serendipitous observations at 160$\mu$m at high
galactic latitude with the Multiband Imaging Photometer (MIPS) onboard the
Spitzer Space Telescope by the Spitzer Infrared Nearby Galaxies Survey
(SINGS). These observations are complemented with IRIS data at 100 and
60$\mu$m and constitute one of the most sensitive and unbiased samples of far
infrared observations at small scale of diffuse interstellar clouds. Outside
regions dominated by the cosmic infrared background fluctuations, we observe a
substantial scatter in the 160/100 colors from cirrus emission. We compared
the 160/100 color variations to 60/100 colors in the same fields and find a
trend of decreasing 60/100 with increasing 160/100. This trend can not be
accounted for by current dust models by changing solely the interstellar
radiation field. It requires a significant change of dust properties such as
grain size distribution or emissivity or a mixing of clouds in different
physical conditions along the line of sight. These variations are important as
a potential confusing foreground for extragalactic studies.
ISM:clouds — infrared:ISM
## 1 Introduction
The InfraRed Astronomical Satellite (IRAS) showed for the first time that
extended infrared emission was present at high galactic latitude, far from
star forming regions (Low et al. 1984). In these diffuse regions, clouds are
optically thin to stellar radiation and the radiation field is relatively
uniform which results in very limited variations of dust equilibrium
temperature (Boulanger et al. 1996; Arendt et al. 1998; Lagache et al. 1998;
Schlegel et al. 1998). These high latitude cirrus also show a tight
correlation between their infrared emission (100$\mu$m to 1mm observed by
DIRBE and FIRAS) and the HI column density (Boulanger et al. 1996) and the
dust emission is well characterized with a constant dust emissivity per
hydrogen atom ($\tau/N_{H}=10^{-25}(\lambda/250)^{-2}cm^{2}$) close to the
value expected from models of interstellar dust grains (Draine & Lee 1984). At
shorter wavelength, the smaller dust grains emission is characterized by a
ratio of $I_{60\mu m}/I_{100\mu m}\sim 0.2$ (Laureijs et al. 1991; Abergel et
al. 1996; Boulanger et al. 2000). All in all, dust emission from local cirrus
is then seen as rather homogeneous and simply characterized on large scales.
However, little is known about the dust properties (e.g. optical properties
for absorption and emission, distribution,…) in these high latitude clouds, at
resolutions smaller than the DIRBE beam (0.7o).
Smaller scale analysis of infrared colors have been done on individual regions
and show clear variations of dust properties. Laureijs et al. (1996) and
Abergel et al. (1994) observed a decrease of $I_{60\mu m}/I_{100\mu m}$ toward
dense clouds. Bernard et al. (1999) studied the far infrared emission at the
arcminute scale in the Polaris flare with IRAS, ISOPHOT and PRONAOS (200 to
600$\mu$m) in a region where extended emission from cirrus is detected as well
as a denser structure. The spectrum of the extended cirrus indicates a low
dust temperature associated with a low 60/100 $\mu$m ratio. This was also
observed in the Polaris flare toward moderately dense regions ($A_{V}\sim 1$)
and in a denser filament in the Taurus complex (Cambrésy et al. 2001; Stepnik
et al. 2003). It might be explained by the formation of large dust aggregates
through the adhesion of small dust particles onto the surface of larger
grains, leading to a change of dust emissivity properties. In the dense
regions the very small grains seem to have disappeared almost completely.
However, all these observations were restricted to individual regions, most of
which are much denser than the diffuse local interstellar medium seen at high
galactic latitudes.
By comparing near-infrared extinction and extinction deduced from far-infrared
dust emission in the whole anticenter hemisphere, Cambrésy et al. (2005)
observed a discrepancy between the two quantities in regions above 1 mag. This
effect is also interpreted by a change of dust emissivity due to the presence
of fluffy grains and the grain-grain coagulation scenario was therefore
extended to larger regions.
Kiss et al. (2006) analyzed the far-infrared emission properties in a large
sample of interstellar clouds observed with ISOPHOT with respect to extinction
in regions of the order of 100 arcmin2. They find variations of the far
infrared dust emissivities in the coldest (12K$<T_{d}<$14K) and densest
regions that are consistent with a dust grain growth scenario. But they also
observe changes of the dust emissivities in the warmer regions
(14K$<T_{d}<$17.5K) and interpret them as an effect of mixing along the line
of sight of components with different temperatures or a change of the dust
grain size distribution. However, a fraction of their sample was chosen on the
basis of high brightnesses in the IRAS bands and could therefore be biased
toward regions with enhanced small grain emission.
Extinction measures toward high galactic latitude sightlines show a
substantial fraction of low $R_{V}=A_{V}/E(B-V)$ values, also indicative of
enhanced relative abundances of small grains. However, with the lack of longer
wavelength measurements at small scales, it is difficult to relate these
variations to possible changes in the dust grain properties.
The photometric data from the Spitzer Space Telescope enable us to have access
to sensitive observations up to 160$\mu$m. Among the large programs, the
Spitzer Infrared Nearby Galaxies Survey (SINGS) observed a sample of 75 nearby
galaxies in photometry with the IRAC and MIPS instruments. The fields observed
were chosen to be at high galactic latitude in order to limit the foreground
cirrus contamination in the study of the targeted galaxy. Because the region
observed was larger than the targeted galaxy, these serendipitous observations
are then ideal to study far infrared dust emission in a large sample of high
galactic latitude regions. We combine these new observations with IRIS data
(Miville-Deschênes & Lagache 2005) at 60 and 100$\mu$m, a reprocessing of IRAS
data including a better calibration of the infrared brightnesses and better
zodiacal light subtraction. The goal of this paper is to _study the infrared
colors of diffuse local dust emission on the scale of a few arcminutes._
## 2 The data
The Spitzer Infrared Nearby Galaxies Survey (SINGS Kennicutt et al. 2003)
observed in imagery with IRAC (Fazio et al. 2004) and MIPS (Rieke et al. 2004)
onboard Spitzer a sample of 75 nearby galaxies. While the IRAC images only
observed the galaxy itself, a significant part of the MIPS observations
(strips) encompass the surrounding sky. Since the SINGS observations were
chosen to be at high galactic latitude to limit the galactic foreground
contamination, the MIPS observations at 160$\mu$m provide a good opportunity
to study the low surface brightness diffuse infrared emission from high
galactic cirrus in a large number of fields at a resolution of $\sim 37"$.
These data are complemented with IRIS data (Miville-Deschênes & Lagache 2005)
at 100 and 60$\mu$m. The position of the fields on the sky are shown in Fig. 1
while their characteristics are summarized in Tab. 1.
Although SINGS observations were also done at 70$\mu$m with MIPS, the regions
observed are offset with respect to the galaxy targeted and only a small
fraction of the 70 and 160$\mu$m observations overlap outside the galaxy
itself, making them inappropriate for our galactic cirrus emission study.
$24\mu$m observations were also available, but they are dominated by point
sources emission as well as stronger zodiacal light. Once the point sources
removed and the regions at low ecliptic latitude are discarded, the 24$\mu$m
brightness have a low dynamic range in each field and no meaningful
correlation can be done with longer wavelength observations. This study was
therefore restricted to the comparison of 60, 100 and 160$\mu$m brightnesses.
The IRIS 25$\mu$m observations were however used together with longer
wavelength in order to remove point sources (like galaxies) more efficiently
in the observations.
### 2.1 Data treatment
The observations at 160$\mu$m were reduced using the GeRT
software111http://ssc.spitzer.caltech.edu/mips/gert/index.html on the raw MIPS
observations. Standard parameters were used for the reduction, but data where
flashes of the internal source led to a significant number of saturated pixels
which were removed. The removed data are most often positioned on the bright
center of the galaxy. Other saturated pixels removed from processing were due
to cosmic ray hits. These saturated flashes when not removed can bias the
sensitivity of the diffuse extended emission. This step in the reduction may
be not appropriate for the photometry of the galaxy, but significantly reduces
latents (stripes) in the outer regions we are interested in. Each region
targeted was observed twice. Discrepant fluxes at the same position between
the two observations are removed and the data are combined into a mosaic for
each region.
The MIPS 160$\mu$m maps and IRIS 60$\mu$m are convolved to the IRIS 100$\mu$m
resolution assuming gaussian beams with FWHM of 4.0’, 4.3’ and 37” for IRIS
60, 100$\mu$m and MIPS 160$\mu$m observations respectively.
For each MIPS strip, the galaxy and other point sources are detected in the
25, 60 and 100$\mu$m maps using the method described in Miville-Deschênes et
al. (2002). These point sources at the IRIS resolution (but with the MIPS
sampling) are then smoothed by a gaussian kernel with a full width half
maximum of $3\times 3$ pixels (at the MIPS original pixel size) to encompass
possible extended emission from these galaxies. The smoothed point sources are
then masked in all the maps. All maps are then projected on the IRIS grid to
avoid oversampling. The observation targeting the galaxy Holmberg IX was
removed from the sample since the emission in the whole strip is dominated by
the galaxy and its interaction features with nearby galaxies. We chose to
remove the observation containing the galaxy NGC3034 (M82) which was hampered
by saturation effects in the whole central region of the galaxy, affecting the
observation globally. The observations containing the galaxies NGC1266,
NGC2915 and M81 Dwarf B were also removed because the width of the region
observed were too narrow to be convolved meaningfully to the IRIS resolution.
We ended up with 70 fields of view222Although there are 75 galaxies in the
SINGS sample, some galaxies are in the same field of view: NGC3031 is in the
same observation as M81 dwarf B and NGC5195 was observed simultaneously with
NGC5194 at a resolution of 4.3’ observed at 60, 100 and 160$\mu$m. Due to
uncertainties in the zodiacal light subtraction at 60$\mu$m that can dominate
the flux at the low surface brightnesses we sample, we limited the sample at
60$\mu$m to the 9 observations at high ecliptic latitude ($|\beta|>15^{o}$).
A constant brightness of 0.78 MJy/sr is removed from the IRIS 100$\mu$m maps
to account for the cosmic infrared background (Lagache et al. 2000), i.e. the
emission from the distant unresolved galaxies (called hereafter CIB). The
exact level of CIB emission has not yet been established at 60$\mu$m and the
MIPS observations can have offsets in the calibration of the brightness that
are not physical. To overcome the uncertainties (physical or instrumental) on
the zero levels in the different maps, we hereafter perform the analysis of
the data through the use of correlations (c.f. §3.1).
The errors on the surface brightness are taken to be 0.03 and 0.06 MJy/sr at
60 and 100$\mu$m respectively (Miville-Deschênes & Lagache 2005). At
160$\mu$m, we take a quadratic combination of a constant sensitivity limit of
0.12 MJy/sr 333the sensitivity of the observations is computed for a 16s
integration time per pixel using the SENS-PET tool,
http://ssc.spitzer.caltech.edu/tools/senspet/ and is divided by $\sqrt{N}$
where $N=49$ is the number of MIPS 160$\mu$m PSF inside an IRIS PSF at
100$\mu$m and a 2% uncertainty on the brightness (due to the uncertainty on
the calibration factor from instrumental units to surface brightness
(Stansberry et al. 2007)).
### 2.2 Sample selection
In low surface brightness regions, the variations of the infrared emission in
the observations can come from cirrus emission, fluctuations in the cosmic
infrared background or from noise. Since we want to study the variations of
cirrus emission only, we want to select observations in the SINGS sample that
are dominated by the dust emission variations.
The different contributions (cosmic infrared background, cirrus emission) to
the infrared emission have different power spectra that can help to
disentangle them. In particular, the cirrus power spectrum normalization
depends on the mean surface brightness while the contribution from background
galaxies does not. This dependence can be translated into a relationship
between the mean brightness and the standard deviation square,
$\sigma_{cirrus}^{2}$, in a region and depends on the size of the region
(Miville-Deschênes et al. 2007). For each of the observed field of view, we
computed the standard deviation at 100$\mu$m and the mean brightness at
100$\mu$m (minus the average CIB contribution at this wavelength) and then
plot the $\sigma^{2}$–$<B>$ relationship observed at 100$\mu$m in our sample.
To model $\sigma_{cirrus}$, we use the relationship derived by Miville-
Deschênes et al. (2007) below 10 MJy/sr, for a maximum scale length of 50’
(dotted line in Fig 2). We observe that our observations are consistent with
the model, with a large scatter as in the original relationship. This
dispersion is likely enhanced due to the fact that our fields of view are
elongated and the size of the region mapped varies between field.The
contribution from the CIB fluctuations can be described by two terms: a
Poisson noise that represents the galaxies distributed homogeneously with
respect to the resolution and a component with correlated spatial variations
corresponding to the clustering of galaxies on large scales. The contribution
from the clustering of infrared galaxies is predicted by using the Lagache et
al. (2003) model for galaxy evolution, with a bias parameter from Lagache et
al. (2007). The contribution from the Poisson noise to the $\sigma^{2}$
observed at 100$\mu$m is taken to be that measured by Miville-Deschênes et al.
(2002) since we used the same point source detection method. However, compared
to their study, we removed point sources applying the detection scheme at all
wavelength (25, 60 and 100$\mu$m). This enables us to mask faint galaxies at
100$\mu$m more efficiently and the Poisson noise in our measurements could be
lower than their measurement. Because we want to select the fields with the
least contribution from other sources (CIB) than cirrus to the observed
variations, this choice is therefore conservative.
Combining all contributions (represented in Fig. 2) to the observed
variations, we determine that a cut at 2.5 MJy/sr corresponds to
$\sigma_{cirrus}/\sigma_{CIB}=1.5$ (so that the total infrared fluctuations
$\sigma_{tot}=\sqrt{\sigma_{cirrus}^{2}+\sigma_{CIB}^{2}}$ are less thatn 20%
larger than from cirrus fluctuations alone). The regions with a mean 100$\mu$m
brightness above this threshold will therefore be dominated by variations of
cirrus emission. In each field, we computed the mean brightness at 100$\mu$m
as well as the standard deviation at 60, 100 and 160$\mu$m (c.f. Tab.1). By
keeping only the fields above the 2.5 MJy/sr cut, the sub-sample we will study
in this paper is composed of 15 fields with 100 and 160$\mu$m brightnesses,
among which 9 can be studied as well at 60$\mu$m ($|\beta|>15^{o}$, see sec.
2.1).
Stellar reddenings obtained from the analysis of the Sloan Digital Sky Survey
data enable us to put an upper limit of 1.2 mag on the extinction in these
fields444Using $N(H)/A_{V}$ (Bohlin et al. 1978) and $B_{100}/N(H_{I})\approx
6.67\times 10^{-21}MJy/srcm^{2}$, this upper limit implies
$B_{100}<15.2MJy/sr$, which is fully consistent with the brightness observed
in our sample. This confirms that the variations in the infrared cirrus
emission studied in each region comes from diffuse clouds according to the van
Dishoeck & Black (1988) classification.
## 3 Results
### 3.1 Cirrus emission at 160 and 100$\mu$m
In each field of the selected sample, we plot the point to point correlation
between the brightnesses observed at 100 and 160$\mu$m (represented in Fig 3
and 4) and apply a linear fit taking into account the errors at both
wavelengths. This enables us to obtain for each field a slope corresponding to
the ratio $B_{160}/B_{100}$ unbiased by variations of the zero point level
(residuals from the zodiacal light subtraction, absolute value of the CIB).
The correlation coefficient and the slope derived in each region are
summarized in Tab. 2.
Large scale observations of high galactic latitude emission of cirrus with
COBE were well characterized by a modified black body with a dust temperature
of 17.5K and an emissivity index proportionnal to $\nu^{2}$ (Boulanger et al.
1996). Using this law, we estimate the large scale 160/100 color for cirrus to
be of $B_{160}/B_{100}=2.0$ (taking into account color corrections). This
ratio is represented in the correlation plots (Fig. 3 and 4) to guide the eye.
While some correlations between $B_{100}$ and $B_{160}$ are in agreement with
the $B_{160}/B_{100}=2.0$ obtained on large scales, clear deviations are also
seen (4 fields out of 15 have a slope discrepant at a 5-6$\sigma$ level with
respect to the value of 2.0. The most extreme case is the field of NGC2976,
with a fitted slope on the $B_{160}$ versus $B_{100}$ correlation that is
10$\sigma$ away from the 2.0 standard value).
In Fig. 5, we compare the obtained ratios $B_{160}/B_{100}$ to the mean
surface brightness at 100$\mu$m in each field (black points). We observe a
large dispersion in the 160/100 colors that can not be explained by the error
on the data or the fitting process. At 100 and 160$\mu$m, the interstellar
emission is dominated by the emission from big dust grains at thermal
equilibrium with the radiation field (Désert et al. 1990) and the
$B_{160}/B_{100}$ ratio is therefore related to that characteristic dust
temperature. Taking a standard emissivity of dust per hydrogen atom in Hi from
Boulanger et al. (1996) and an emissivity index of 2, the 160/100 color
variations we are probing can therefore be related for illustrative purposes
to temperatures between 15.7 and 18.9K for column densities ranging from
$N_{H}=3\times 10^{20}$ to $2\times 10^{21}$cm-2. We note that these
variations are consistent _on average_ with the large scale estimate (blue
solid line), confirming that the fields used in this study are sampling the
cirrus emission observed on large scale.
For a given grain size and composition, this characteristic temperature
depends on the local radiation field strength and spectrum which depends on
the presence and distance of nearby heating sources and on the extinction. In
the framework of this model, the presence of large variations in the
160/100$\mu$m ratio observed in our sample would suggests the presence of
large variations in the heating of grains at small scales (variations by a
factor of 3 of the intensity of the incident radiation field). This can be
surprising since at low FIR surface brightness and at high latitude the
interstellar radiation field might be expected to be homogeneous. We looked at
the far-infrared color temperature maps derived from DIRBE observations by
Lagache et al. (1998) and Schlegel et al. (1998). The regions we are studying
appear to be reasonably representative of the high latitude cirrus given the
small number statistics. For the sightlines covered by our sample, the FIR
color variations seen in the DIRBE data are compatible with the variations
that we observe. Our study is indeed more sensitive than previous observations
and therefore able to probe color variations smaller than the uncertainties in
the previous studies.
The shape of the optical spectrum heating the grains could also affect the
far-infrared colors: the radiation field could become gradually harder with
position off the galactic plane (Mattila 1980). Using the cirrus model from
Efstathiou & Rowan-Robinson (2003) with different stellar populations heating
the clouds, we checked that changes in the shape of the optical radiation
field is unlikely to affect the 160/100 and 60/100 colors of cirrus by more
than 20%.
The dust equilibrium temperature depends however also on the structure of the
grains. Grain aggregates for example cool more efficiently. The temperature
variations observed in the diffuse medium could therefore be either due to
variations of the intensity of the interstellar radiation field or to changes
in the grain structure.
We compared our findings with different studies of far infrared emission from
the literature: the quiescent high galactic latitude clouds from del Burgo et
al. (2003), the large sample from archival ISOPHOT data by Kiss et al. (2006),
the two regions in a high latitude cirrus MCLD 123.5+24.9 observed by Bernard
et al. (1999), and the quiescent filament in the Taurus molecular complex from
Stepnik et al. (2003). Because other observations were obtained with different
instruments, we have to interpolate the brightnesses at various far infrared
wavelengths to estimates at 100 and 160$\mu$m. To do so, we took the dust
temperatures determined in each study with a brightness at 100 or 200$\mu$m
and used a modified black body law with a spectral index. The power index is
either taken from the study itself (if it was computed) or is fixed to a
standard value of 2. For each region, we also compute the mean 100$\mu$m
brightness as observed by IRIS and subtract a mean CIB contribution as for our
observations. Despite large scatter, we observe a trend between $<B_{100}>$
and the 160/100 color that is consistent with the idea that denser regions are
colder. However, the effect of selection biases of these studies remains
unclear. The comparison of our results with that from the literature (Fig. 5)
shows that previous studies in the far-infrared have been targeting higher
$B_{160}/B_{100}$ and $<B_{100}>$, i.e. denser and colder clouds. Due to our
better sensitivity, our observations fill the gap at low $B_{100}$ and low
$B_{160}/B_{100}$.
For the first time, we observe interstellar dust emission at low surface
brightness in an unbiased way (in the observing strategy) with a high
sensitivity. These observations show that former studies on dust properties at
FIR wavelength at small scale, have been biased toward colder and denser
clouds. Our study shows that the 160/100 brightness ratios of high galactic
cirrus clouds at small scales are consistent on average with the observations
on large scales. However, these 160/100 colors show a wide dispersion that
could arise from variations in the heating of the clouds or from change of the
dust grain structure. In order to investigate further the origin of the
160/100 variations, we extend the comparison to the 60$\mu$m data.
### 3.2 Comparison to the 60$\mu$m data
To investigate the origin of the 160/100 color variations observed in the
diffuse cirrus, we compare the 160 and 100$\mu$m data to the 60$\mu$m
emission. The sample for this part of the study is however reduced to fields
with an ecliptic latitude above 15o in order to avoid artifacts due to the
uncertainties in the zodiacal light subtraction in the IRIS data. For each
field, we determine a 60/100 color by using the same correlation technique as
above. The correlations in each region are shown in Fig. 6 and the obtained
$B_{60}/B_{100}$ are summarized in Tab. 2.
Fig. 7 presents the $B_{60}/B_{100}$ ratio obtained in each field with respect
to the $B_{160}/B_{100}$ ratio. Here again, large variations are observed in
the 60/100 colors that can not be explained by the uncertainties in the data
or in the analysis. As for the 160/100 colors, the 60/100 brightness ratios
are consistent on average with the ”reference values” (the pink cross and the
blue star in the figure) obtained for high latitude emission on large scales
(Boulanger & Perault 1988; Boulanger et al. 1996; Arendt et al. 1998).
Furthermore, there is a trend of decreasing $B_{60}/B_{100}$ with increasing
$B_{160}/B_{100}$.
To try to interpret this trend, we used two models of the dust grain emission
at different interstellar radiation fields: the Draine & Li (2007) model for
the Milky Way 555We took the model with a PAH fraction q${}_{PAH}=4.58\%$ but
we checked that this parameter does not influence significantly the results of
this study and the ”DUSTEM” model (updated model based on Désert et al.
(1990)). The models take into account the shape of the IRAS and MIPS/Spitzer
filters, the color corrections. For both models, the abundances of different
grain types are kept constant. The tracks obtained are compared to the data in
Fig. 7. The comparison shows that, if the variations of colors are due to
variations in heating of the grains, this would imply large changes in the
interstellar radiation field at high galactic latitude (from $U\approx 0.3$ to
1). Furthermore, the trend observed between the 60/100 and 160/100 colors is
not well reproduced with the current models by changing the interstellar
radiation field alone. In that respect, the Draine & Li (2007) model is
however closer to the observed trend than the DUSTEM model (only 3 fields are
more than 3$\sigma$ deviant from the expected curve), but for
$B_{160}/B_{100}<2.1$ all observations show systematically higher
$B_{60}/B_{100}$ than predicted by both models, while for
$B_{160}/B_{100}>2.1$ all data points have systematically lower
$B_{60}/B_{100}$ values than expected from both models.
Current dust models might be missing an additional dust grain type. Such an
addition might reproduce all color variations while changing the interstellar
radiation field alone. Another way to interpret the observed trend is that the
variations in the dust emission spectrum reflect spatial changes in the grain
properties – composition, structure or size distribution.
The equilibrium temperatures of dust grains is expected to decrease for
increasing grain sizes and small grains ($\leq 0.01\mu$m) undergo temperature
excursions following single-photon heating that enhances the 60$\mu$m
emission. Thus regions with fewer small grains may have lower 60/100 ratios.
The observed trend between the 60/100 and 160/100 infrared colors could be
reproduced by changing the dust grains size distribution or composition. For
example, enhancing the amount of small grains in regions with higher
interstellar radiation field (i.e. higher temperatures) and reducing it at low
equilibrium temperatures could reproduce the observed variations.
In the same way, regions with enhanced populations of large grains may have
increased 160/100 ratios. In that case, reproducing the observations could be
obtained with only modest variations in the starlight heating rate and shifts
in the grain size distribution (fewer small grains and increased sizes for the
larger grains at low temperatures, more small grains and smaller sizes for the
big grains at higher dust temperatures).
The 60/100 colors we observe therefore suggest changes of the dust properties
(dust size distribution and/or composition) from one region to the other.
These changes are related to variations in the 160/100 brightness ratio.
Whether the 160/100 color variations require a change of the starlight
intensity heating the clouds or result from the change of dust properties
alone is unclear. The interpretation of the color variations and of the trend
between the 160/100 and 60/100 colors is discussed further in the next
section.
## 4 Discussion
Despite its rather constant color distribution on large scale, the far
infrared emission from diffuse cirrus at high galactic latitude is observed to
host large color variations on small scales. These variations seem related to
each other (the 60/100 color decreases as the 160/100 color increases). In
this section, we will first check that these variations come indeed from
cirrus emission and are not related to the galaxies targeted with the
observations. Second, we will discuss possible interpretations for these large
color variations and the trend between them.
### 4.1 Extended disks in galaxies
Because the MIPS observations were taken to observe nearby galaxies, it is
legitimate to ask whether the infrared emission that we observe could be
associated with these targets. In particular, Hi observations have shown that
gaseous disks can extend much farther than the optical diameters. Dust grains
could be present in these outer parts of the galaxies and bias our
measurements.
To avoid this extended emission from the galaxies, we were careful in masking
regions larger than the detected emission (c.f. Sec. 2.1). Some of the
galaxies in the observations used in this study have been observed in Hi
observations through The HI Nearby Galaxies Survey (THINGS, Walter et al.
2005). We checked that the Hi diameters reported for these galaxies are
smaller than the region masked for our study. We are therefore confident that
the variations observed in the infrared emission between fields do not come
from the targeted galaxies, but rather from diffuse cirrus emission.
### 4.2 Physical conditions in cirrus clouds
A possibility to interpret the variations of far-infrared colors at high
galactic latitude is that we are sampling clouds in different physical
conditions and/or composition (different heating of the clouds, different dust
size distribution, …). The color changes would then be due to mixing along the
sightline of these different components.
An unbiased survey of H2 absorption in high galactic latitude clouds by FUSE
(Gillmon et al. 2006) has been interpreted as showing that some clouds have
been compressed. The dynamical history leading to this compression may involve
shock waves or strong turbulence, which could also lead to changes in the
grain size distribution by shattering in grain-grain collisions, possibly
explaining the regions of higher than average 60/100 and lower than average
160/100 colors.
One tantalizing possibility, in terms of mixing, is the presence of
Intermediate or high velocity clouds (IVCs and HVCs) along the sightline.
These cloud falling onto our galaxy could have very different dust properties
(e.g. Miville-Deschênes et al. 2005) and would bias the measured infrared
emission from more local cirrus clouds. We checked for the presence of
intermediate velocity clouds in the LAB Hi survey spectra (Kalberla et al.
2005) in the direction of the fields in our sample. For about half of the
sample, there is a intermediate velocity component seen in Hi in the
sightline. Two sightlines also have a high velocity component. However, no
conclusive trend between the fraction of the Hi in the IVCs and/or HVCs and
the infrared colors could be seen. This may be due to the lack of resolution
of the Hi observations ($\sim 0.6^{o}$), to the difficulty to disentangle IVCs
and the Milky Way in some regions or to the small size of our sample.
### 4.3 Variations in grain properties
The grain size distribution is the result of processes such as sputtering,
shattering and coagulation, and sightline-to-sightline variations in the
wavelength-dependence of optical and ultraviolet extinction toward stars have
already demonstrated regional variations in the grain properties. Whether the
observed variations in emission can be fully explained by variations in the
size-distribution alone, or whether other properties (e.g., composition or
porosity) are also involved is uncertain.
In denser clouds, variations of infrared colors (Stepnik et al. 2003; Kiss et
al. 2006) have been interpreted with a grain coagulation scenario, combining
changes of the size distribution with changes of the dust emissivity
properties. The trend observed between the 60/100 and 160/100 colors would be
consistent with this idea. In this scenario, most of the 160/100 color
variations would then be due to the change of emissivity of dust grains (due
to changes of their structure), while the 60/100 colors would change with the
incorporation or release of small grains in large dust aggregates. We could be
witnessing variations of dust properties due to variations in turbulent
motions in the diffuse interstellar medium. Alternatively, dust grains in the
diffuse medium could retain for some time the aggregate structure they had
previously acquired in denser regions. So we could be seeing a sequence of
regions corresponding to increasing time since their release from high density
environments.
Such changes in the dust size distribution and structure of grains would imply
related variations of the UV-optical extinction curves at high galactic
latitude. An extinction and reddening study of stars at high galactic latitude
behind translucent clouds (Larson & Whittet 2005) shows variations in the
extinction curves obtained with respect to the average curve for the diffuse
interstellar medium. In particular, 48% of their sightlines have $R_{V}<2.8$,
much lower that the diffuse ISM average of $3.05$. Such values are indicative
of enhanced abundances of small grains, and these regions could have a higher
that average 60/100 color. To test if the high 60/100 colors and low $R_{V}$
values are connected, we computed the 60/100 colors from IRIS data in a
similar fashion to this study for a $\sim 1\times 1^{o}$ region around each
sightline of the Larson & Whittet (2005) sample. We do not observe any
correlation however between the $R_{V}$ and 60/100 color, nor between $A_{V}$
and the 60/100 color. Unfortunately, no longer wavelength observations exist
for these regions and it would be important to determine the 160/100 colors in
these regions as well and study their dependancy with extinction properties.
This result is however a concern for the coagulation scenario as an
interpretation of the 60/100 color variations we observe.
Interpreting the trend observed between the 60/100 and 160/100 colors with a
change of dust optical properties and dust size distribution remains
hypothetical without further observations. In particular, Hi observations of
these diffuse sightlines, with a high resolution (at least similar to the IRIS
one), will be needed to determine the emissivity of dust per hydrogen atom and
test if variations are observed and correlated with the infrared colors.
It is presently not possible to interpret further the far-infrared color
variations in terms of physical condition changes, grain size distribution,
grain properties, etc. A larger number of observations of high latitude cirrus
could help to probe the spatial variations. Hi studies of these regions at
high resolution would also be crucial to probe possible variations of dust
grain emissivities, or to check whether the velocity structure of the cirrus
correlates with the 60/100/160 colors. Finally, extinction curves on
sightlines where cirrus emission properties have been determined would be most
useful to see whether extinction properties would correlate with the FIR
colors.
## 5 Conclusion
We performed an unbiased study of dust emission from high galactic latitude
cirrus using serendipitous Spitzer MIPS observations at 160$\mu$m from the
SINGS survey, complemented by IRIS data at 60 and 100$\mu$m. After an
appropriate post-reduction of the data and a removal of the targeted galaxy, a
sub-sample is selected so that the variations of the cirrus emission dominate
over the CIB fluctuations in each field.
We observe 160/100 colors in our fields that are consistent on average with
large scale studies. However, strong variations are also observed from field
to field. This paper extends former studies on dust properties at high
galactic latitude to more diffuse, fainter and warmer clouds. The 60/100 color
is also observed to vary significantly in the sample and there is a trend of
decreasing 60/100 with increasing 160/100 ratios. This trend is not completely
reproduced by current models taking only into account variations of the
radiation field strength and requires changes in the dust properties,
composition or size distribution.
The exact origin of these variations remains unknown, but the variations of
the 60/100/160 colors may reflect variations of the grains size distribution,
of grain properties in addition to heating variations. However, we can not
completely rule out the possibility that our fields contain emission from
matter at different heights above the Galactic plane, the juxtaposition of
multiple components in the fields could be affecting the infrared color
estimates.
All in all, we observe unexpected variations of far-infrared colors in the
supposed ”homogeneous” cirrus at high galactic latitudes. These variations are
not yet understood and further studies will be needed to test their origin. In
particular, studies on a larger area of sky is needed to confirm the
significance of these variations and their spatial distribution on the sky
could give new clues on their origin.
These infrared color variations will most probably be linked with variations
of the infrared colors at longer wavelengths. They therefore represent an
important point to study for the Planck and Herschel missions. Such longer
wavelength observations will enable us to determine precisely the temperature
and spectral index of the dust, and their variations, in high latitude
regions.
We would like to thank our referee M. Rowan-Robinson for his useful and
interesting comments and A. Efstathiou for his help with their cirrus model.
We are also grateful to L. Cambrésy for useful discussions and work on upper
limit for extinction in our fields. Facilities: Spitzer.
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Figure 1: Position of the SINGS fields (red circles and blue squares) overlaid
on the dust column density maps from Schlegel et al. (1998) centered around
the north galactic pole (left panel) and the south galactic pole (right
panel). The blue squares correspond to the fields selected for this study (the
variation in the infrared emission is dominated by the cirrus component). A
grid of galactic coordinates is overlayed.
Figure 2: Variations of the square of the standard deviation (related to the
power spectrum of the signal) measured in each field with the mean brightness
at 100$\mu$m. The observed values are compared with models for the different
contributions: the infrared galaxies clustering (dotted-dashed line), the
poisson noise (dashed line) and the cirrus variation (dotted line). This
enables us to define a cut in 100$\mu$m brightness (the vertical black line)
above which the cirrus variations dominate over CIB fluctuations.
Figure 3: 160–100 scatterplots for all SINGS observations with $<B_{100}>\geq
2.45$ MJy/sr. In each plot, a canonical slope of 2.0 is represented by a
dashed line (corresponding to a temperature of 17.5K). A linear fit is
performed on the correlation and the best fit is represented with a solid
line. The value of the slope obtained is written in the legend.
Figure 4: idem as Fig. 3
Figure 5: Variations of the 160/100 surface brightness ratio with the mean
100$\mu$m surface brightness. The diamonds represent the SINGS observations.
The blue line denotes a typical ratio of 2.0.
Figure 6: 60–100 scatterplots for all SINGS observations with $<B_{100}>\geq
2.5$ MJy/sr. In each plot, a canonical slope of 0.2 is represented by a dashed
line. A linear fit is applied and the best fit is represented with a solid
line. The value of the slope obtained is written in the legend. The green
point and its bold uncertainties represent the value chosen to represent the
local background in the observation.
Figure 7: Color-color plot showing the evolution of the 160/100 and 60/100 brightness ratios with each other for all the the regions where $<B_{100}>\geq 2.5$ MJy/sr. Two reference values from the literature representing the colors measured on large scales are overplotted. The observed colors are compared to the trend obtained with two dust models: the Draine et al. (2007) model (dark solid line) and the DUSTEM model (blue dotted line). In both case, the incident interstellar radiation field intensity is the only parameter varying (the proportion of grain types are kept constant). Table 1: Characteristics of the observations: galactic and ecliptic coordinates, average 100$\mu$m cirrus brightnesses in each field as well as standard deviations at 60, 100 and 160$\mu$m field | (l,b) | ($\lambda$,$\beta$) | Area (deg2) | $\sigma_{60}$ | $<B_{100}>$ (MJy/sr) | $\sigma_{100}$ | $\sigma_{160}$
---|---|---|---|---|---|---|---
NGC0337 | ( 126.983 , -70.4576) | ( 10.0714 , -12.6317) | 0.048 | —- | 5.75 | 0.713 | 1.951
NGC0584 | ( 148.863 , -68.1957) | ( 17.8130 , -15.3840) | 0.100 | 0.057 | 2.38 | 0.177 | 0.387
NGC0628 | ( 138.043 , -46.3226) | ( 27.4637 , 5.05541) | 0.115 | —- | 3.09 | 0.287 | 0.809
NGC0855 | ( 143.962 , -32.1481) | ( 39.9931 , 13.3583) | 0.056 | —- | 3.33 | 0.111 | 0.399
NGC0925 | ( 144.527 , -25.8230) | ( 44.7625 , 17.7346) | 0.124 | 0.040 | 3.09 | 0.101 | 0.316
NGC1097 | ( 227.959 , -65.0380) | ( 26.0183 , -43.9238) | 0.113 | 0.037 | 1.08 | 0.112 | 0.250
NGC1291 | ( 247.760 , -57.5207) | ( 27.3597 , -55.9323) | 0.154 | 0.018 | 0.68 | 0.104 | 0.212
NGC1316 | ( 240.631 , -56.8186) | ( 32.3476 , -53.2688) | 0.197 | 0.042 | 1.00 | 0.093 | 0.189
NGC1377 | ( 212.367 , -52.3728) | ( 44.7413 , -38.7417) | 0.042 | 0.029 | 1.65 | 0.071 | 0.246
NGC0024 | ( 40.4333 , -80.1196) | ( 351.074 , -23.9783) | 0.099 | 0.043 | 1.14 | 0.073 | 0.130
NGC1404 | ( 237.002 , -53.9016) | ( 38.2431 , -52.8271) | 0.069 | 0.032 | 0.59 | 0.059 | 0.101
NGC1482 | ( 213.765 , -48.2291) | ( 50.1099 , -39.5384) | 0.039 | 0.040 | 2.44 | 0.087 | 0.212
NGC1512 | ( 248.603 , -48.4042) | ( 40.9466 , -61.8125) | 0.176 | 0.042 | 0.49 | 0.075 | 0.132
NGC1566 | ( 264.199 , -43.5133) | ( 32.0129 , -73.2076) | 0.116 | 0.025 | 0.39 | 0.081 | 0.124
NGC1705 | ( 260.913 , -38.8932) | ( 50.3989 , -74.3634) | 0.043 | 0.033 | 0.42 | 0.094 | 0.121
NGC2403 | ( 150.172 , 28.7424) | ( 102.746 , 43.5117) | 0.261 | 0.049 | 1.62 | 0.132 | 0.286
HolmbergII | ( 143.899 , 32.2721) | ( 106.073 , 49.5466) | 0.077 | 0.028 | 1.34 | 0.091 | 0.300
M81DwarfA | ( 143.536 , 32.5857) | ( 106.476 , 49.8996) | 0.041 | 0.043 | 1.00 | 0.121 | 0.248
DDO053 | ( 148.991 , 34.4963) | ( 110.352 , 45.7107) | 0.077 | 0.048 | 1.57 | 0.113 | 0.324
NGC2798 | ( 178.927 , 43.7838) | ( 127.750 , 25.1694) | 0.051 | 0.040 | 1.04 | 0.086 | 0.166
NGC2841 | ( 166.458 , 43.6210) | ( 124.989 , 33.8516) | 0.101 | 0.043 | 0.74 | 0.061 | 0.140
NGC2976 | ( 143.612 , 40.4772) | ( 118.782 , 50.4361) | 0.076 | 0.100 | 2.59 | 0.594 | 1.951
HolmbergI | ( 140.502 , 38.2525) | ( 115.197 , 52.8405) | 0.079 | 0.036 | 1.18 | 0.185 | 0.659
NGC3049 | ( 226.610 , 44.4312) | ( 146.934 , -2.98701) | 0.027 | —- | 2.20 | 0.223 | 0.504
NGC3190 | ( 211.891 , 54.4481) | ( 147.733 , 10.7909) | 0.090 | —- | 1.61 | 0.094 | 0.195
NGC3184 | ( 178.537 , 54.9707) | ( 139.768 , 28.3651) | 0.102 | 0.031 | 0.93 | 0.089 | 0.149
NGC3198 | ( 170.656 , 54.2275) | ( 138.036 , 32.7413) | 0.050 | 0.039 | 0.52 | 0.074 | 0.157
IC2574 | ( 139.983 , 43.1538) | ( 123.351 , 52.9795) | 0.156 | 0.036 | 1.38 | 0.205 | 0.509
NGC3265 | ( 200.457 , 58.5603) | ( 147.821 , 18.3074) | 0.029 | 0.030 | 1.30 | 0.055 | 0.146
MRK33 | ( 156.610 , 52.2144) | ( 135.166 , 41.0632) | 0.032 | 0.032 | 0.86 | 0.212 | 0.424
NGC3351 | ( 232.652 , 56.1499) | ( 157.328 , 3.64920) | 0.074 | —- | 1.95 | 0.108 | 0.210
NGC3521 | ( 254.316 , 52.8287) | ( 166.849 , -5.14601) | 0.149 | —- | 2.81 | 0.328 | 0.718
NGC3621 | ( 280.580 , 25.9208) | ( 184.582 , -34.1228) | 0.180 | 0.111 | 4.12 | 0.444 | 0.900
NGC3627 | ( 240.246 , 64.2844) | ( 165.038 , 8.27119) | 0.097 | —- | 1.70 | 0.179 | 0.248
NGC3773 | ( 249.132 , 66.7215) | ( 169.474 , 8.73315) | 0.034 | —- | 1.84 | 0.102 | 0.125
NGC3938 | ( 153.689 , 68.7273) | ( 156.788 , 39.2954) | 0.075 | 0.065 | 1.01 | 0.194 | 0.328
NGC4125 | ( 130.164 , 50.9404) | ( 139.741 , 57.1897) | 0.095 | 0.037 | 0.81 | 0.174 | 0.285
NGC4236 | ( 127.310 , 46.9854) | ( 134.438 , 60.6054) | 0.267 | 0.035 | 0.66 | 0.098 | 0.170
NGC4254 | ( 267.627 , 75.3684) | ( 177.739 , 15.3315) | 0.075 | 0.054 | 2.37 | 0.161 | 0.370
NGC4321 | ( 267.852 , 77.0715) | ( 178.035 , 17.0118) | 0.119 | 0.053 | 1.37 | 0.160 | 0.214
NGC4450 | ( 270.395 , 78.8333) | ( 178.801 , 18.7003) | 0.079 | 0.032 | 1.36 | 0.119 | 0.338
NGC4536 | ( 291.454 , 65.1324) | ( 186.331 , 5.66591) | 0.102 | —- | 1.64 | 0.090 | 0.239
NGC4552 | ( 285.274 , 74.9059) | ( 182.409 , 14.8782) | 0.106 | —- | 2.28 | 0.210 | 0.688
NGC4559 | ( 193.095 , 85.8409) | ( 175.296 , 29.2435) | 0.124 | 0.029 | 1.05 | 0.082 | 0.184
NGC4569 | ( 285.793 , 75.9750) | ( 182.385 , 15.9551) | 0.101 | 0.060 | 2.65 | 0.174 | 0.470
NGC4579 | ( 287.803 , 74.2939) | ( 183.194 , 14.3801) | 0.071 | —- | 2.37 | 0.126 | 0.311
NGC4594 | ( 297.305 , 51.1460) | ( 193.066 , -6.96614) | 0.151 | —- | 3.12 | 0.141 | 0.298
NGC4625 | ( 131.096 , 75.1008) | ( 168.734 , 41.5922) | 0.047 | 0.052 | 0.88 | 0.121 | 0.239
NGC4631 | ( 145.167 , 83.5710) | ( 174.202 , 33.9219) | 0.171 | 0.054 | 1.02 | 0.182 | 0.248
NGC4725 | ( 271.647 , 88.6791) | ( 179.905 , 28.4957) | 0.079 | 0.037 | 0.70 | 0.102 | 0.161
NGC4736 | ( 124.738 , 75.5220) | ( 170.796 , 42.2419) | 0.187 | 0.047 | 0.67 | 0.192 | 0.293
DDO154 | ( 110.504 , 89.5114) | ( 179.918 , 30.2879) | 0.042 | 0.043 | 0.57 | 0.085 | 0.137
NGC4826 | ( 311.183 , 85.0014) | ( 183.181 , 25.6681) | 0.125 | 0.059 | 1.89 | 0.182 | 0.239
DDO165 | ( 121.301 , 49.1180) | ( 142.645 , 62.9864) | 0.058 | 0.051 | 1.14 | 0.171 | 0.301
NGC5033 | ( 101.051 , 79.6584) | ( 178.930 , 40.1141) | 0.128 | 0.033 | 0.54 | 0.096 | 0.149
NGC5055 | ( 107.903 , 73.9458) | ( 175.505 , 45.4687) | 0.163 | 0.052 | 0.75 | 0.178 | 0.267
NGC5194 | ( 106.114 , 68.6881) | ( 174.458 , 50.7106) | 0.233 | 0.087 | 1.23 | 0.364 | 0.712
Tololo89 | ( 318.732 , 27.7794) | ( 219.272 , -19.6125) | 0.051 | 0.186 | 5.26 | 0.544 | 0.978
NGC5408 | ( 316.804 , 20.0386) | ( 223.038 , -26.7581) | 0.056 | 0.163 | 5.55 | 0.761 | 1.702
NGC5474 | ( 101.503 , 59.8880) | ( 174.929 , 59.7255) | 0.061 | 0.027 | 0.48 | 0.086 | 0.185
NGC5713 | ( 350.094 , 52.5981) | ( 217.013 , 14.3062) | 0.052 | —- | 2.08 | 0.070 | 0.187
NGC5866 | ( 92.3464 , 52.7620) | ( 186.246 , 66.8587) | 0.105 | 0.029 | 0.62 | 0.097 | 0.210
IC4710 | ( 327.879 , -22.1180) | ( 273.178 , -43.4420) | 0.034 | 0.119 | 5.69 | 0.426 | 1.007
NGC6822 | ( 24.6969 , -17.9857) | ( 294.785 , 6.07295) | 0.185 | —- | 7.77 | 0.745 | 1.705
NGC6946 | ( 95.3945 , 12.0399) | ( 356.855 , 72.2046) | 0.086 | 0.125 | 9.57 | 0.722 | 1.948
NGC7331 | ( 93.0851 , -20.9465) | ( 356.050 , 39.1427) | 0.126 | 0.121 | 3.91 | 0.429 | 0.761
NGC7552 | ( 347.564 , -64.7400) | ( 330.605 , -34.6836) | 0.046 | 0.060 | 0.81 | 0.128 | 0.173
NGC7793 | ( 5.28269 , -76.5898) | ( 344.625 , -29.1746) | 0.119 | 0.046 | 0.95 | 0.126 | 0.184
Table 2: Infrared colors in the observations, i.e. the 160/100 and 60/100 brightness ratios obtained from a fit of the correlations in each field field | 160/100 color | 60/100 color
---|---|---
NGC0337 | $2.3\pm 0.2$ | —
NGC0628 | $2.6\pm 0.1$ | —
NGC0855 | $2.7\pm 0.5$ | —
NGC0925 | $2.4\pm 0.3$ | $0.12\pm 0.03$
NGC2976 | $3.0\pm 0.1$ | $0.15\pm 0.01$
NGC3521 | $1.81\pm 0.07$ | —
NGC3621 | $1.86\pm 0.04$ | $0.225\pm 0.005$
NGC4569 | $2.0\pm 0.2$ | $0.31\pm 0.03$
NGC4594 | $2.3\pm 0.1$ | —
TOL89 | $1.7\pm 0.1$ | $0.32\pm 0.02$
NGC5408 | $2.22\pm 0.04$ | $0.19\pm 0.01$
IC4710 | $2.0\pm 0.1$ | $0.31\pm 0.01$
NGC6822 | $2.23\pm 0.04$ | —
NGC6946 | $2.5\pm 0.1$ | $0.16\pm 0.006$
NGC7331 | $1.75\pm 0.07$ | $0.238\pm 0.009$
|
arxiv-papers
| 2009-01-07T09:52:37 |
2024-09-04T02:48:59.754783
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Caroline Bot, George Helou, Francois Boulanger, Guilaine Lagache,\n Marc-Antoine Miville-Deschenes, Bruce Draine, Peter Martin",
"submitter": "Caroline Bot",
"url": "https://arxiv.org/abs/0901.0792"
}
|
0901.0812
|
# The relevance of random choice in tests of Bell inequalities with two atomic
qubits
Emilio Santos Departamento de Física. Universidad de Cantabria. Santander.
Spain
###### Abstract
It is pointed out that a loophole exists in experimental tests of Bell
inequality using atomic qubits, due to possible errors in the rotation angles
of the atomic states. A sufficient condition is derived for closing the
loophole.
PACS numbers: 03.65.Ud, 03.67.Mn, 37.10.Ty, 42.50.Xa
A recent experiment, by a group of Maryland, has measured the correlation
between the quantum states of two Yb+ ions separated by a distance of about 1
meter[1]. The authors claim that the experiment is relevant because it
violates a CHSH[2] (Bell) inequality, modulo the locality loophole, closing
the detection loophole. In my opinion that assertion does not make full
justice to the relevance of the experiment. The truth is that it is the first
experiment which has tested a genuine Bell inequality. Actually the results of
previous experiments, in particular those involving optical photon pairs[3],
did not test any genuine Bell inequality, that is an inequality which is a
necessary condition for the existence of local hidden variables (LHV) models.
The inequalities tested in those experiments should not be qualified as Bell´s
because their derivation involves additional assumptions. Consequently their
violation refutes only restricted families of LHV models, namely those
fulfilling the additional assumption. ( For details see[4].)
The aim of the present letter is to point out the existence of a loophole in
the Maryland experiment[1], or more generally in Bell tests with atomic
qubits, in addition to the locality loophole. Blocking that loophole will be
straightforward using random choice of the measurements, as is explained
below.
In general I will consider experiments where a pair of atoms (or ions) is
prepared in an entangled state. Then Alice performs a rotation of the state of
her atom by an angle $\theta_{a}$ and, after a short time, she may detect
fluorescence of the atom illuminated by an appropriate laser. Similarly Bob
performs a rotation of his atom by an angle $\theta_{b}$ and, after that, he
may detect fluorescence too. I shall label
$p_{++}\left(\theta_{a},\theta_{b}\right)$ the probability of coincidence
detection and $p_{--}\left(\theta_{a},\theta_{b}\right)$ the probability that
neither Alice nor Bob detect fluorescence. Similarly
$p_{-+}\left(\theta_{a},\theta_{b}\right)$ (
$p_{+-}\left(\theta_{a},\theta_{b}\right))$ will be the probability that only
Bob (Alice) detects fluorescence. In the Maryland experiment[1] (see their
eq.$\left(6\right))$, a function $E\left(\theta_{a},\theta_{b}\right)$ is
defined by
$E\left(\theta_{a},\theta_{b}\right)=p_{++}\left(\theta_{a},\theta_{b}\right)+p_{--}\left(\theta_{a},\theta_{b}\right)-p_{+-}\left(\theta_{a},\theta_{b}\right)-p_{-+}\left(\theta_{a},\theta_{b}\right).$
(1)
Then the authors define a parameter $S$ by
$S=\left|E\left(\theta_{a},\theta_{b}\right)+E\left(\theta_{a}^{\prime},\theta_{b}\right)\right|+\left|E\left(\theta_{a},\theta_{b}^{\prime}\right)-E\left(\theta_{a}^{\prime},\theta_{b}^{\prime}\right)\right|,$
(2)
and claim that the CHSH[2] inequality $S\leq 2$ is violated. The notation used
by the authors is, however, somewhat misleading. Instead of
eq.$\left(\ref{a7}\right)$ they write
$E\left(\theta_{a},\theta_{b}\right)=p\left(\theta_{a},\theta_{b}\right)+p\left(\theta_{a}+\pi,\theta_{b}+\pi\right)-p\left(\theta_{a},\theta_{b}+\pi\right)-p\left(\theta_{a}+\pi,\theta_{b}\right),$
(3)
where they label $p\left(\theta_{a},\theta_{b}\right)$ the quantity which I
have labeled $p_{++}\left(\theta_{a},\theta_{b}\right).$ Definition
eq.$\left(\ref{10}\right),$ in place of eq.$\left(\ref{a7}\right),$ rests upon
assuming the equalities
$\displaystyle p_{-+}\left(\theta_{a},\theta_{b}\right)$ $\displaystyle=$
$\displaystyle
p\left(\theta_{a}+\pi,\theta_{b}\right),p_{+-}\left(\theta_{a},\theta_{b}\right)=p\left(\theta_{a}+\pi,\theta_{b}\right),$
$\displaystyle p_{--}\left(\theta_{a},\theta_{b}\right)$ $\displaystyle=$
$\displaystyle p\left(\theta_{a}+\pi,\theta_{b}+\pi\right),$
which are true according to quantum mechanics, but may not be true in LHV
theories. In any case the authors measured
$E\left(\theta_{a},\theta_{b}\right)$ as defined in
eq.$\left(\ref{a7}\right)$[5].
In order to show that there is a loophole in the experiment, in addition to
the locality loophole, I begin remembering that, according to Bell[6], a LHV
model will contain a set of hidden variables, $\lambda,$ a positive normalized
density function, $\rho\left(\lambda\right),$ and two functions
$M_{a}\left(\lambda,\theta_{a}\right),$
$M_{b}\left(\lambda,\theta_{b}\right)$, $\theta_{a}$ and $\theta_{b}$ being
parameters which may be controlled by Alice and Bob respectively. The latter
functions fulfil
$M_{a}\left(\lambda,\theta_{a}\right),M_{b}\left(\lambda,\theta_{b}\right)\in\\{0,1\\}.$
(4)
In the Maryland experiment[1] the parameters $\theta_{a}$ and $\theta_{b}$ are
angles defining the quantum states of the two ions. The probability,
$p_{++}\left(\theta_{a},\theta_{b}\right),$ that the coincidence measurement
of two dichotomic variables, in two distant regions, gives a positive answer
for both variables should be obtained in the LHV model by means of the
integral
$p_{++}\left(\theta_{a},\theta_{b}\right)=\int\rho\left(\lambda\right)M_{a}\left(\lambda,\theta_{a}\right)M_{b}\left(\lambda,\theta_{b}\right)d\lambda.$
(5)
Similarly the probability, $p_{+-}\left(\theta_{a},\theta_{b}\right),$ that
Alice gets the answer “yes” and Bob the answer “no” is given by
$p_{+-}\left(\theta_{a},\theta_{b}\right)=\int\rho\left(\lambda\right)M_{a}\left(\lambda,\theta_{a}\right)\left[1-M_{b}\left(\lambda,\theta_{b}\right)\right]d\lambda,$
(6)
and analogous expressions for $p_{-+}$ and $p_{--}.$
A LHV model for an atomic experiment may be obtained by choosing
$\displaystyle\rho\left(\lambda\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi},\lambda\in\left[0,2\pi\right],\;M_{a}\left(\lambda,\theta_{a}\right)=\Theta\left(\frac{\pi}{2}-\left|\lambda-\theta_{a}\right|\right),$
$\displaystyle M_{b}\left(\lambda,\theta_{b}\right)$ $\displaystyle=$
$\displaystyle\Theta\left(\frac{\pi}{2}-\left|\lambda-\theta_{b}-\pi\right|\right),\mathop{\mathrm{m}od}\left(2\pi\right),$
(7)
where $\Theta\left(x\right)=1$ if $x>0$, $\Theta\left(x\right)=0$ if $x<0$. It
is easy to see, taking eqs.$\left(\ref{1}\right)$ and $\left(\ref{1a}\right)$
into account, that model predictions are (assuming
$\theta_{a},\theta_{b}\in\left[0,\pi\right])$
$\displaystyle p_{++}\left(\theta_{a},\theta_{b}\right)$ $\displaystyle=$
$\displaystyle
p_{--}\left(\theta_{a},\theta_{b}\right)=\frac{\left|\theta_{a}-\theta_{b}\right|}{2\pi},$
$\displaystyle p_{+-}\left(\theta_{a},\theta_{b}\right)$ $\displaystyle=$
$\displaystyle
p_{-+}\left(\theta_{a},\theta_{b}\right)=\frac{1}{2}-\frac{\left|\theta_{a}-\theta_{b}\right|}{2\pi}.$
(8)
Hence I get
$E\left(\theta_{a},\theta_{b}\right)=\frac{2}{\pi}\left|\theta_{a}-\theta_{b}\right|-1,$
(9)
and it is not difficult to show that, for any choice of the angles
$\theta_{a},\theta_{b},\theta_{a}^{\prime},\theta_{b}^{\prime},$ the model
predicts $S\leq 2$ with $S$ given by eq.$\left(\ref{7a}\right).$
Now let us assume that the experiment is performed so that Alice and Bob start
measuring the quantity $E\left(\theta_{a},\theta_{b}\right)$ in a sequence of
runs of the experiment. After that they measure
$E\left(\theta_{a},\theta_{b}^{\prime}\right)$ in another sequence, then they
measure $E\left(\theta_{a}^{\prime},\theta_{b}\right)$ and, finally, they
measure $E\left(\theta_{a}^{\prime},\theta_{b}^{\prime}\right).$ Let $\alpha$
be the error in the rotation performed by Bob on his atom in the first
sequence of runs, so that the rotation angle is $\theta_{b}+\alpha$ rather
than $\theta_{b}$ in the measurement of $E\left(\theta_{a},\theta_{b}\right).$
Similarly I shall assume that the rotation angles are
$\theta_{b}^{\prime}+\beta,\theta_{b}+\gamma$ and $\theta_{b}^{\prime}+\delta$
in the measurements of $E\left(\theta_{a},\theta_{b}^{\prime}\right)$,
$E\left(\theta_{a}^{\prime},\theta_{b}\right)$ and
$E\left(\theta_{a}^{\prime},\theta_{b}^{\prime}\right),$ respectively. For
simplicity I will assume that no error appears in Alice rotations. The errors
are considered small, specifically
$\left|\alpha\right|,\left|\beta\right|,\left|\gamma\right|,\left|\delta\right|<\pi/4.$
I shall prove that, taking into account the errors in the measurement of the
angles, the LHV model prediction for the parameter $S$,
eq.$\left(\ref{7a}\right)$ may apparently violate the CHSH[2] inequality
$S\leq 2.$ To do that let us choose, as in the Maryland experiment[1],
$\theta_{a}=\frac{\pi}{2},\theta_{b}=\frac{\pi}{4},\theta_{a}^{\prime}=0,\theta_{b}^{\prime}=\frac{3\pi}{4}.$
(10)
The values predicted by the LHV model for the relevant quantities are
$\displaystyle E\left(\theta_{a},\theta_{b}+\alpha\right)$ $\displaystyle=$
$\displaystyle-0.5-\frac{2\alpha}{\pi},\;E\left(\theta_{a},\theta_{b}^{\prime}+\beta\right)=-0.5+\frac{2\beta}{\pi},$
$\displaystyle E\left(\theta_{a}^{\prime},\theta_{b}+\gamma\right)$
$\displaystyle=$
$\displaystyle-0.5+\frac{2\gamma}{\pi},\;E\left(\theta_{a}^{\prime},\theta_{b}^{\prime}+\delta\right)=0.5+\frac{2\delta}{\pi}.$
(11)
Then the parameter actually measured in the experiment is
$S^{\prime}=\left|E\left(\theta_{a},\theta_{b}+\alpha\right)+E\left(\theta_{a}^{\prime},\theta_{b}+\gamma\right)\right|+\left|E\left(\theta_{a},\theta_{b}^{\prime}+\beta\right)-E\left(\theta_{a}^{\prime},\theta_{b}^{\prime}+\delta\right)\right|,$
(12)
and the LHV prediction for that parameter is
$S^{\prime}==2+\frac{2}{\pi}\left(\alpha-\beta-\gamma+\delta\right),$
which may violate the inequality $S^{\prime}\leq 2$ for some values of the
parameters $\alpha,\beta,\gamma$ and $\delta.$ In particular the results of
the Maryland experiment[1] are reproduced by choosing
$2\alpha/\pi=0.018,2\beta/\pi=-0.046,2\gamma/\pi=-0.081,2\delta/\pi=-0.073.$
The errors in the angles are of order 7o or less. It is plausible that errors
as high as these may appear in experiments with atomic qubits but not in
optical photon experiments. I stress that no violation of a Bell inequality by
a LHV model is produced. Actually the parameter $S^{\prime}$ of
eq.$\left(\ref{46}\right)$ is not a CHSH parameter as defined in
eq.$\left(\ref{7a}\right).$
In the following I shall prove that the loophole may be closed by random
choice of the angles to be measured. To begin with, it is easy to see that the
LHV model predictions do not violate the inequality $S^{\prime}\leq 2$ if the
error in the measurement, by Bob, of the angle $\theta_{b}$ is the same in all
measurements of that angle, and similarly for $\theta_{b}^{\prime}.$ In fact
the inequality is fulfilled if $\alpha=\beta$ and $\gamma=\delta,$ as may be
seen by looking at eq.$\left(\ref{46}\right).$ In the following I derive a
sufficient condition for the fulfillement of the inequality, $S^{\prime}\leq
2,$ for the actually measurable quantity $S^{\prime},$ by the predictions of
any LHV model.
Let us assume that there is a (normalized) probability distribution,
$f_{a}(x),$ for the errors when Alice rotates her atom by an angle
$\theta_{a}$ and another distribution, $f_{a}^{\prime}(y),$ when she rotates
her atom by an angle $\theta_{a}^{\prime}.$ Similarly I shall assume that
there are similar disitribuions $f_{b}(u)$ and $f_{b}^{\prime}(v)$ for the
errors in the rotations, by Bob, of the angles $\theta_{b}$ and
$\theta_{b}^{\prime}.$ I shall show that a sufficient condition for the
inequality $S^{\prime}\leq 2$ is that the distributions of errors, in the
rotations made by Alice, are the same independently of what rotation performs
Bob on the partner atom. And similarly for the rotations made by Bob. If this
is the case the predictions of any LHV model for the quantity $S^{\prime}$
will be obtained from probabilities defined as follows (compare with
eqs.$\left(\ref{1}\right)$ and $\left(\ref{1a}\right))$
$\displaystyle p_{++}\left(\theta_{a},\theta_{b}\right)$ $\displaystyle=$
$\displaystyle\int\rho\left(\lambda\right)M_{a}\left(\lambda,\theta_{a}+x\right)M_{b}\left(\lambda,\theta_{b}+u\right)d\lambda
f_{a}(x)dxf_{b}(u)du,$ (13) $\displaystyle
p_{+-}\left(\theta_{a},\theta_{b}\right)$ $\displaystyle=$
$\displaystyle\int\rho\left(\lambda\right)M_{a}\left(\lambda,\theta_{a}+x\right)\left[1-M_{b}\left(\lambda,\theta_{b}+u\right)\right]d\lambda
f_{a}(x)dxf_{b}(u)du,$
and similarly for the other quantities $p_{ij}$ with $i,j=+,-$. Now we may
define new quantities
$\displaystyle Q_{a}\left(\lambda,a\right)$ $\displaystyle=$
$\displaystyle\int M_{a}\left(\lambda,\theta_{a}+x\right)f_{a}(x)dx,$ (14)
$\displaystyle Q_{b}\left(\lambda,b\right)$ $\displaystyle=$
$\displaystyle\int M_{b}\left(\lambda,\theta_{b}+u\right)f_{b}(u)du,$
$\displaystyle Q_{a}\left(\lambda,a^{\prime}\right)$ $\displaystyle=$
$\displaystyle\int
M_{a}\left(\lambda,\theta_{a}^{\prime}+y\right)f_{a}^{\prime}(y)dy,$
$\displaystyle Q_{b}\left(\lambda,b^{\prime}\right)$ $\displaystyle=$
$\displaystyle\int
M_{b}\left(\lambda,\theta_{b}^{\prime}+v\right)f_{b}^{\prime}(v)dv,$
which fulfil the conditions (compare with eqs.$\left(\ref{1b}\right))$
$0\leq
Q_{a}\left(\lambda,a\right),Q_{a}\left(\lambda,a^{\prime}\right),Q_{b}\left(\lambda,b\right),Q_{b}\left(\lambda,b^{\prime}\right)\leq
1.$ (15)
The consequence is that we may obtain a new LHV model for the experiment with
the quantities $Q,$ eqs.$\left(\ref{40}\right),$ in place of the quantities
$M$, eqs.$\left(\ref{1b}\right).$ The existence of the model implies the
fulfillement of the inequality $S^{\prime}\leq 2.$
From our proof it is rather obvious that the essential condition required to
block the loophole is that the probability distribution of errors made by Bob
are independent of what rotation is performed by Alice in the partner atom,
and similarly the errors made by Alice should be independent of the rotation
performed by Bob. A simple method to insure that independence is that Alice
makes at random the choice whether to rotate her atom by the angle
$\theta_{a}$ or by the angle $\theta_{a}^{\prime},$ and similarly Bob. That
is, after every preparation of the entangled state of the atom, Alice should
make a random choice (with equal probabilities) between the rotation angles
$\theta_{a}$ and $\theta_{a}^{\prime}$ and similarly Bob should make a random
choice, independently of Alice, between $\theta_{b}$ and
$\theta_{b}^{\prime}.$
## References
* [1] D. N. Matsukevich, P. Maunz, D.L. Moehring, S. Olmschenk and C. Monroe, Phys. Rev. Lett. 100, 150404 (2008).
* [2] J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).
* [3] M. Genovese, Phys. Reports 413, 319 (2005).
* [4] E. Santos, Found. Phys. 34, 1643 (2004).
* [5] D. N. Matsukevich, private communication.
* [6] J. S. Bell, Physics 1, 195 (1964).
|
arxiv-papers
| 2009-01-07T12:17:59 |
2024-09-04T02:48:59.765050
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Emilio Santos",
"submitter": "Emilio Santos Corchero",
"url": "https://arxiv.org/abs/0901.0812"
}
|
0901.0955
|
# Four-state rock-paper-scissors games in constrained Newman-Watts networks
Guo-Yong Zhang Institute of Theoretical Physics, Lanzhou University, Lanzhou
730000, China Department of Computer Science, Hubei Normal University,
Huangshi 435002, China Yong Chen Corresponding author. Email:
ychen@lzu.edu.cn Institute of Theoretical Physics, Lanzhou University, Lanzhou
730000, China Wei-Kai Qi Institute of Theoretical Physics, Lanzhou
University, Lanzhou 730000, China Department of Industrial Systems and
Engineering, The Hong Kong polytechnic University, Hung Hom, Kowloon, Hong
Kong, China Shao-Meng Qing Institute of Theoretical Physics, Lanzhou
University, Lanzhou 730000, China
###### Abstract
We study the cyclic dominance of three species in two-dimensional constrained
Newman-Watts networks with a four-state variant of the rock-paper-scissors
game. By limiting the maximal connection distance $R_{max}$ in Newman-Watts
networks with the long-rang connection probability $p$, we depict more
realistically the stochastic interactions among species within ecosystems.
When we fix mobility and vary the value of $p$ or $R_{max}$, the Monte Carlo
simulations show that the spiral waves grow in size, and the system becomes
unstable and biodiversity is lost with increasing $p$ or $R_{max}$. These
results are similar to recent results of Reichenbach et al. [Nature (London)
448, 1046 (2007)], in which they increase the mobility only without including
long-range interactions. We compared extinctions with or without long-range
connections and computed spatial correlation functions and correlation length.
We conclude that long-range connections could improve the mobility of species,
drastically changing their crossover to extinction and making the system more
unstable.
###### pacs:
87.23.Cc, 89.75.Fb, 05.50.+q
††preprint: Phys. Rev. E 79, 062901 (2009).
The question of how biological diversity is maintained has initiated
increasingly more research from multiple angles in recent decades frean ; kerr
; reichenbach ; mobilia ; claussen ; murray . Mathematical modeling of
population dynamics is widely recognized as a useful tool in the study of many
interesting features of ecological systems. However, the enormous number of
interacting species found in the Earth’s ecosystems is a major challenge for
theoretical description. For this reason, researchers have built many
simplified models to describe the evolution of ecological systems over time
lotka ; volterra ; kerr ; durrett ; windus ; hastings . One of the simplest
cases is of three species that have relationships analogous to the game of
rock-paper-scissors (RPS), where rock smashes scissors, scissors cut paper,
and paper wraps rock. It is a well-studied model of population dynamics matti
; szabo1 ; szolnoki ; szabo2 ; efimov , and it can be classified in two ways:
three-state or four-state models, depending on whether we consider the empty
state or not. It is well known that such a cyclic dominance can lead to
nontrivial spatial patterns as well as coexistence of all three species.
Recently, Reichenbach and co-workers studied a stochastic spatial variant of
the RPS game reichenbach ; mobilia ; tobias . In their study, they run the
game with four states: the three original cyclically dominating states and a
fourth one that denotes empty space. In addition, they introduced a form of
mobility to mimic a central feature of real ecosystems: animal migration,
bacteria run and tumble. They found that mobility has a critical influence on
species diversity reichenbach ; mobilia ; tobias . When mobility exceeds a
certain value, biodiversity is jeopardized and lost. In contrast, below this
critical threshold value, spatial patterns can form and help enable stable
species diversity. We shall take this population model as a basis to construct
a new version of the three-species food chain in the constrained Newman-Watts
(NW) networks. In the model studied by Reichenbach and co-workers, they
consider mobile individuals of three species (referred to as $A$, $B$, and
$C$), arranged on a spatial lattice, where each individual can only interact
with its nearest neighbors. In this study, we introduce some stochastic long-
range interactions between elements of the lattice. The stochastic long-range
interactions occur when there exist long-range connections in the NW networks,
mimicking a more real ecosystem: e.g., birds can fly, so they can prey not
only near their nest but also at longer distances from the nest sabrina ,
pathogens disperse by air and water brown ; mccallum , biological invasions
related to human influence occur over long distances ruiz , and the long-range
dispersal of plant seeds is driven by large and migratory animals, ocean
currents and human transportation nathan . Of course, the long-range
interaction cannot be infinite, so we limit the distance of long-range
interactions to $R_{max}$. That is, the individuals are assigned an
interaction distance. For the sake of simplicity, we consider that the maximum
interaction distance $R_{max}$ and the long-range interaction probability $p$
are the same for all species. With Monte Carlo (MC) simulations we show that
the maximum interaction distance $R_{max}$ and the long-range interaction
probability $p$ play an important role in the coexistence of all three
species.
We consider the four-state RPS model which was described in detail in Refs.
reichenbach ; tobias ; matti . Here, we give a recapitulation:
$\displaystyle AB\stackrel{{\scriptstyle\sigma}}{{\rightarrow}}AE,\quad$
$\displaystyle BC\stackrel{{\scriptstyle\sigma}}{{\rightarrow}}BE,\quad$
$\displaystyle CA\stackrel{{\scriptstyle\sigma}}{{\rightarrow}}CE.$
$\displaystyle AE\stackrel{{\scriptstyle\mu}}{{\rightarrow}}AA,\quad$
$\displaystyle BE\stackrel{{\scriptstyle\mu}}{{\rightarrow}}BB,\quad$
$\displaystyle CE\stackrel{{\scriptstyle\mu}}{{\rightarrow}}CC.$ (1)
Here, $A$, $B$, and $C$ denote the three species which cyclically dominate
each other, and $E$ denotes an available empty space. An individual of species
$A$ can kill B, with successive production of E. Cyclic dominance occurs as A
can kill B, B preys on C, and C beats A in turn, closing the cycle. These
processes are called ‘selection’ and occur at a rate $\sigma$. To mimic a
finite carrying capacity, each species can reproduce only if an empty space is
available, as described by the reaction $AE\rightarrow AA$ and analogously for
$B$ and $C$. For all species, these reproduction events occur at a rate $\mu$.
In addition, to mimic the possibility of migration, one can amend the reaction
equations with an exchange reaction:
$\displaystyle XY\stackrel{{\scriptstyle\epsilon}}{{\rightarrow}}YX.$ (2)
where $X$ and $Y$ denote any state (including empty space) and $\epsilon$ is
the exchange rate. The mobility was defined as $m=2\epsilon N^{-1}$ in Ref.
reichenbach , where $N$ denotes the number of sites. From a dynamic viewpoint,
the RPS game can be described by the mean field rate equations tobias ; matti
,
$\displaystyle\partial_{t}a$ $\displaystyle=$ $\displaystyle
a[\mu(1-\rho)-\sigma c],$ $\displaystyle\partial_{t}b$ $\displaystyle=$
$\displaystyle a[\mu(1-\rho)-\sigma a],$ $\displaystyle\partial_{t}c$
$\displaystyle=$ $\displaystyle a[\mu(1-\rho)-\sigma b],$ (3)
where $a$, $b$, and $c$ are densities of the states $A$, $B$, and $C$,
respectively. That is,
$a=N_{a}/N,\quad b=N_{b}/N,\quad c=N_{c}/N,$ (4)
where $N_{a}$, $N_{b}$, and $N_{c}$ are the number of species of $A$, $B$, and
$C$, respectively. $\rho=a+b+c$ is the total density. These equations have a
reactive fixed point $a=b=c=\frac{\mu}{3\mu+\sigma}$, which is linearly
unstable tobias .
The mean field approach does not take into account the spatial structure and
assumes the system to be well mixed. Therefore, it can only serve as a rough
model for dynamic processes. Here, we consider the spatial version of the
above model in the complex NW network structure newman , and we use the Monte
Carlo simulation approach. The two-dimension (2D) NW network was constructed
as follows: (i) We first built a 2D $L\times L$ $(N=L^{2})$ regular square
lattice. So, the total number of connections is $2N$. (ii) Then, we randomly
chose two sites that have no direct connection. If the shortest path length
between the two sites was shorter than the maximal distance $R_{max}$, we
connected the sites; if not, we choose other two sites, until the number of
the long-range connections equaled $2pN$. Here, the shortest path length
refers to that we did not take into account the long-range connections.
Figure 1: (a) The structure of a constrained NW network. All the long-range
connections within the range $[1,R_{max}]$ and the range of $R_{max}$ is
$1\leq R_{max}\leq L$. (b) The characteristic path length as a function of the
long-range connection probability $p$. The maximal long-range connection
distance $R_{max}=L/4$.
The above procedure produces a constrained NW network structure as shown in
Fig. 1(a). In Fig. 1(b), we plot the characteristic path length as a function
of the long-range connection probability $p$. The characteristic path length
decreases with an increase in the long-range connection probability $p$. The
long-range connection probability $p$ is equivalent to the rewiring
probability in a Watts-Strogatz network but connections are added without
removing any of the original ones. So, the modified NW structure is
characterized by the probability $p$ and the maximal long-range connection
distance $R_{max}$. Once the network was built as described above, the
evolution of the system over time obeyed the following rules (modified from
Refs. reichenbach ; tobias ): (i) Consider mobile individuals of three species
(referred to as $A$, $B$, and $C$), scattered randomly on a square lattice as
in Fig. 1(a) with periodic boundary conditions. Every lattice site was
initially occupied by an individual of species $A$, $B$, or $C$, or left
empty. (ii) At each simulation step, a random individual was chosen to
interact with a randomly-chosen individual directly connected to it. A process
(selection, reproduction, or mobility) was chosen randomly with a probability
proportional to the rates, and the corresponding reaction is executed.
In the above process, $N=L^{2}$ simulation steps constitute one Monte Carlo
step (MCS). During one MCS all lattice sites had one chance to interact. Over
time, the spatial distributions of $A$, $B$, and $C$ species changed from one
MCS to another, providing the evolution of the system at the microscopic
level.
According to Ref. tobias , Eq. (3) could be cast into the form of the complex
Ginzburg-Landau equation (CGLE). In accordance with the known behaviors of the
CGLE, it was found that the spatial four-state model with diffusion leads to
the formation of spirals. The spirals’ wavelength $\lambda$ is proportional to
the square root of mobility reichenbach ; tobias . To investigate how the
long-range interaction probability $p$ and the maximal distance $R_{max}$
affect the behaviors of the four-state model, we ran MC simulations of this
model in the constrained NW network with periodic boundary conditions. All the
results that we present were obtained starting from a random initial
distribution of individuals and vacancies, and each site was in one of the
four possible states. The densities of $A$, $B$, and $C$ coincided with the
values of the unstable reactive fixed point of the rate equations (3). We
considered equal selection and reproduction rates, which were set
$\mu=\sigma=1$ tobias . So, all four states initially occurred with equal
probability $1/4$.
Figure 2: (Color online) Snapshots of the reactive steady state for $m=4\times
10^{-6}$, $\mu=\sigma=1$, and system size $L=1000$ ($\epsilon=2$). The long-
range connection probability $p=0.08$, and the maximal interaction distance
increases from $R_{max}=2$ to $120$.
In Fig. 2, we plot typical snapshots of the reactive steady states for various
values of the maximal interaction distance $R_{max}$. When $R_{max}$ is short,
long-range interactions have little effect, and all species coexist. The
pattern of spiral waves in Fig. 2(a) is similar to the case without long-range
interactions. With increasing $R_{max}$, the spiral waves grow in size and
eventually disappear for longer enough values of $R_{max}$. When the spiral
waves disappear, the system becomes a uniform state where only one species
exists and the others have died out. This process is similar to the result
from increased mobility $m$ in the lattice simulation without long-range
interactions in Ref. reichenbach . In addition, we computed the extinction
probability $P_{ext}$ that two species have gone extinct after $10000$ MCS
(see Fig. 3). Fig. 3 clearly shows that there exists a critical value
$R_{c}\approx 30$ in the process of phase transition from coexistence
($P_{ext}$ tends to zero) to extinction ($P_{ext}$ approaches $1$). Of course,
the critical value $R_{c}$ depends on the other parameters, such as the system
size, mobility, and so on.
Figure 3: Extinction probability as a function of the maximal long-range
interaction distance $R_{max}$. Parameters: $L=200$, $t=10,000$ MCS,
$\mu=\sigma=1$, and $m=1\times 10^{-4}$. As $R_{max}$ increases, the
transition from stable coexistence ($P_{ext}=0$) to extinction ($P_{ext}=1$)
sharpens at a critical value $R_{c}\approx 30$.
To investigate how the long-range connection probability $p$ affects the
coexistence, we fixed the maximal long-range interaction distance to
$R_{max}=10$ and varied the probability $p$. The simulation results are shown
in Fig. 4, and the dependence of the extinction probability $P_{ext}$ on $p$
is plotted in Fig. 5. It turns out that $p$ has effects similar to those of
$R_{max}$ on the extinction probability and spiral wave pattern. There also
exists a critical value $p_{c}\approx 0.06$ in the process of phase transition
from coexistence ($P_{ext}$ tends to zero) to extinction ($P_{ext}$ approaches
$1$). The critical value $p_{c}$ depends on the other parameters in the model,
as well.
Figure 4: (Color online) Snapshots of the reactive steady state for $m=1\times
10^{-4}$, $\mu=\sigma=1$, and system size $L=200$ ($\epsilon=2$). The fixed
maximal interaction distance $R_{max}=10$, and the long-range connection
probability increases from $p=0.0$ to $0.55$. Figure 5: The extinction
probability as a function of the long-range connection probability $p$.
Parameters: $L=100$, $t=10,000$ MCS, $\mu=\sigma=1$, and $m=2\times 10^{-5}$.
As $p$ increases, the transition from stable coexistence ($P_{ext}=0$) to
extinction ($P_{ext}=1$) sharpens at a critical value $p_{c}\approx 0.06$.
In Ref. reichenbach , the authors verified that the spiral wavelength
increases with individual mobility and that the wavelength is proportional to
$\sqrt{m}$. They found that there exists a critical mobility $M_{c}$. When
mobility is greater than $M_{c}$, the pattern outgrows the system size,
causing loss of biodiversity.
In this work, we obtain similar results in the case of fixed mobility and
variable long-range connection probability $p$ or variable maximal interaction
distance $R_{max}$. This means that increasing $p$ or $R_{max}$ is equivalent
to increasing the mobility. Although the long-range interaction does not
directly change the exchange rate $\epsilon$, it does change the spatial
structure and leads to faster interactions, particularly for exchange. So,
increasing $p$ or $R_{max}$ increases mobility $m$ indirectly.
In Fig. 6(a), we plot the dependence of $P_{ext}$ on mobility $m$ in the
presence of long-range interactions. With increasing mobility $m$, a sharpened
transition emerges at a critical value $M_{c}\approx 1.9\times 10^{-4}$, which
is smaller than the value $(4.5\pm 0.5)\times 10^{-4}$ provided in Ref.
reichenbach . In Fig. 6(b), we also compute $P_{ext}$ without long-range
interactions ($p=0$), holding the other parameters the same in Fig. 6(a). In
these conditions, it takes much longer ($t=10N$ MCS) to reach the critical
value $M_{c}\approx 4\times 10^{-4}$, which approximately coincides with the
value $(4.5\pm 0.5)\times 10^{-4}$. That is to say, in this case the system is
more stable than with long-range interactions.
Figure 6: The extinction probability as a function of mobility: (a) With long-
range connections, the transition from stable coexistence to extinction
sharpens at a critical mobility $M_{c}\approx 1.9\times 10^{-4}$,
$t=10,000MCS$. (b) Without long-range connections, $M_{c}\approx 4.0\times
10^{-4}$. Parameters: $\mu=\sigma=1$, $p=0.02$, $R_{max}=L/4$.
It is well known that long-range connections change spatial structure
dramatically. To learn more information about the effect of long-range
connections on the emerging spiral patterns, we computed the equal-time
correlation function $g_{AA}\left(|r-r^{\prime}|\right)$ at $r$ and
$r^{\prime}$ of species $A$ for the system’s steady state, which is defined in
Ref. tobias , as
$g_{AA}\left(|r-r^{\prime}|\right)=\left<a(r,t)a(r^{\prime},t))\right>-\left<a(r,t)\right>\left<a(r^{\prime},t)\right>,$
(5)
where $\left<\ldots\right>$ stands for an average over all histories.
Fig. 7(a) plots $g_{AA}$ obtained from MC simulations. When the separating
distance reaches zero, the correlation reaches its maximal value. With
increasing distance, the correlation decreases and the spatial oscillations
appear. This oscillation reflects the underlying spiraling spatial structures
where the three species alternate in turn. Furthermore, the correlation
functions could be characterized by their correlation length $l_{corr}$, which
is the length at which the correlations decay by a factor $1/e$ from their
maximal value. The value of $l_{corr}$ is obtained by fitting $g_{AA}(r)$ to
exponentials $e^{-r/l_{corr}}$. This value conveys information on the typical
size of the spirals tobias2 . In Fig. 7(b), we show the dependence of
$l_{corr}$ on the maximal long-range interaction distance $R_{max}$. The
results confirm the scaling relationship $l_{corr}\propto R_{max}$ for the
fixed long-range connection probability $p$. In addition, it can be observed
that the linear fit is less good when $R_{max}$ around 100. Through extra
numerical simulations on the correlation length, we found that when $R_{max}$
is around or above 100, the system would be at extinction in a high
probability. This could affect the correlation and correlation length.
Figure 7: (Color online) (a) The spatial correlation $g_{AA}(r)$ as a function
of $r$ in the reactive steady state. (b) The dependence of correlation length
$l_{corr}$ on the maximal long-range interaction distance $R_{max}$.
Correlation length is depicted as circle. The black line is the linear fit.
Parameters: $L=1000$, $m=1.2\times 10^{-5},t=6000MCS$, and $\mu=\sigma=1$
In summary, we studied the influence of random long-range connection on four-
state RPS games with NW networks based on extensive MC simulations. For a
fixed maximal interaction distance $R_{max}$, as the probability of long-range
connections $p$ increases, we observe that the spiral waves grow in size and
(for larger $p$) disappear. When the spiral waves disappear, the system
reaches a uniform state and biodiversity is lost. There exists a critical
value $p_{c}$ separating coexistence from extinction. Similar behaviors are
observed with increasing $R_{max}$ for a fixed $p$. To close more ecological
realistic model, we also consider the case that $XE->EX$ occur with a smaller
probability where $X$ and $E$ (empty place) are not neighboring sites. When
$p$ or $R_{max}$ increases, we observe that both the size of spiral waves grow
more slowly and the process of phase transition from coexistence to extinction
are slower than before.
We compared the critical value $M_{c}$ obtained in two cases: with and without
long-range connection. It is found that $M_{c}$ changes drastically and the
systems becomes more unstable if even a weak long-range connection is
presented. We conclude that long-range interactions could result in improved
mobility, and it has dramatic effects on species coexistence. This point is
also confirmed by the equal-time correlation functions for the system’s steady
state and by the correlation length for different $R_{max}$.
We are grateful to T. Reichenbach for helpful advice on simulation methods and
thank referees’ for their constructive suggestions. This work was supported by
the National Natural Science Foundation of China under Grant No. $10305005$.
## References
* (1) M. Frean and E. R. Abraham, Proc. R. Soc. Lond. B 268, 1323 (2001).
* (2) J. D. Murray, Mathematical Biology (Springer, New York, 2002), 3rd ed., Vols. I and II.
* (3) B. Kerr, M. A. Feldman, M. W. Feldman, and B. J. M. Bohannan, Nature 418, 171 (2002).
* (4) T. Reichenbach, M. Mobilia, and E. Frey, Nature 448, 1046 (2007).
* (5) T. Reichenbach and E. Frey, Phys. Rev. Lett. 101, 058102 (2008).
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* (8) V. Volterra, J. Cons. Int. Explor. Mer 3, 3 (1928).
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* (10) A. Windus and H. J. Jensen, J. Phys. A: Math. Gen. 40, 2287 (2007).
* (11) A. Hastings and T. Powell, Ecology 72, 896 (1991).
* (12) M. Peltomaki and M. Alava, Phys. Rev. E 78, 031906 (2008).
* (13) G. Szabó and T. Czárán, Phys. Rev. E 64, 042902 (2001).
* (14) A. Szolnoki and G. Szabó, Phys. Rev. E 70, 037102 (2004).
* (15) G. Szabó, A. Szolnoki, and R. Izsák, J. Phys. A: Math. Gen. 37, 2599 (2004).
* (16) A. Efimov, A. Shabunin, and A. Provata, Phys. Rev. E 78, 056201 (2008).
* (17) T. Reichenbach, M. Mobilia, and E. Frey, Phys. Rev. Lett. 99, 238105 (2007).
* (18) S. B. L. Araújo and M. A. M. de Aguiar, Phys. Rev. E 75, 061908 (2007).
* (19) J. K. M. Brown, M. S. Hovmoller, Science 297, 537 (2002).
* (20) H. McCallum, D. Harvell, A. Dobson, Ecol. Lett. 6, 1062 (2003).
* (21) G. M. Ruiz, T. K. Rawlings, F. C. Dobbs, A. Huq, R. Colwell, Nature 408, 49 (2000).
* (22) R. Nathan, F. M. Schurr, O. Spiegel, O. Steinitz, A. Trakhtenbrot and A. Tsoar, Trends Ecol. Evol. 23, 638 (2008).
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|
arxiv-papers
| 2009-01-08T00:58:07 |
2024-09-04T02:48:59.773174
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guo-Yong Zhang, Yong Chen, Wei-Kai Qi, and Shao-Meng Qin",
"submitter": "Yong Chen",
"url": "https://arxiv.org/abs/0901.0955"
}
|
0901.0966
|
ON TWO RESULTS OF MIXED MULTIPLICITIES
Le Van Dinh and Duong Quoc Viet
Department of Mathematics, Hanoi University of Education
136 Xuan Thuy Street, Hanoi, Vietnam
E-mail: duongquocviet@fmail.vnn.vn
This paper shows that the main result of Trung-Verma in 2007 [TV] only is an
immediate consequence of an improvement version of [Theorem 3.4, Vi1] in 2000.
Let $(A,\mathfrak{m})$ be a Noetherian local ring of Krull dimension $d=\dim
A>0$ with infinite residue field $k=A/\mathfrak{m}.$ Let $J$ be an
$\mathfrak{m}$-primary ideal and $I_{1},\ldots,I_{s}$ ideals in $A$ such that
their product $I=I_{1}\cdots I_{s}$ is non-nilpotent. It is known that the
Bhattacharya function $\ell_{A}\Big{(}\frac{J^{n}I_{1}^{n_{1}}\cdots
I_{s}^{n_{s}}}{J^{n+1}I_{1}^{n_{1}}\cdots I_{s}^{n_{s}}}\Big{)}$ is a
polynomial of degree $q-1$ for all sufficiently large $n,n_{1},\ldots,n_{s}$,
where $q=\dim A/0:I^{\infty}$ [Proposition 3.1, Vi1]. If the terms of total
degree $q-1$ in this polynomial have the form
$\sum_{k_{0}+k_{1}+\cdots+k_{s}=q-1}\frac{1}{k_{0}!k_{1}!\cdots
k_{s}!}e(J^{[k_{0}+1]},I_{1}^{[k_{1}]},\ldots,I_{s}^{[k_{s}]})n^{k_{0}}n_{1}^{k_{1}}\cdots
n_{s}^{k_{s}},$
then $e(J^{[k_{0}+1]},I_{1}^{[k_{1}]},\ldots,I_{s}^{[k_{s}]})$ are non-
negative integers and are called mixed multiplicities of the ideals
$J,I_{1},\ldots,I_{s}.$
The positivity and the relationship between mixed multiplicities and Hilbert-
Samuel multiplicities have attracted much attention.
Using different sequences, one transmuted mixed multiplicities into Hilbert-
Samuel multiplicities, for instance: in the case of $\mathfrak{m}$-primary
ideals, Risler-Teissier in 1973 [Te] by superficial sequences and Rees in 1984
[Re] by joint reductions; the case of arbitrary ideals, Viet in 2000 [Vi1] by
(FC)-sequences and Trung-Verma in 2007 [TV] by
$(\varepsilon_{1},\ldots,\varepsilon_{m})$-superficial sequences.
Definition 1 [see Definition, Vi1]. A element $x\in A$ is called an
(FC)-element of $A$ with respect to $(I_{1},\ldots,I_{s})$ if there exists
$i\in\\{1,2,\ldots,s\\}$ such that $x\in I_{i}$ and
* (FC1):
$(x)\cap I_{1}^{n_{1}}\cdots I_{i}^{n_{i}}\cdots
I_{s}^{n_{s}}=xI_{1}^{n_{1}}\cdots I_{i}^{n_{i}-1}\cdots I_{s}^{n_{s}}$ for
all large $n_{1},\ldots,n_{s}$.
* (FC2):
$x$ is a filter-regular element with respect to $I,$ i.e., $0:x\subseteq
0:I^{\infty}.$
* (FC3):
$\dim A/[(x):I^{\infty}]=\dim A/0:I^{\infty}-1.$
We call $x$ a weak-(FC)-element with respect to $(I_{1},\ldots,I_{s})$ if $x$
satisfies conditions (FC1) and (FC2). Let $x_{1},\ldots,x_{t}$ be a sequence
in $A$. For each $i=0,1,\ldots,t-1$, set $A_{i}=A/(x_{1},\ldots,x_{i})S$,
$\bar{I}_{j}=I_{j}[A/(x_{1},\ldots,x_{i})]$, $\bar{x}_{i+1}$ the image of
$x_{i+1}$ in $A_{i}$. Then $x_{1},\ldots,x_{t}$ is called an (FC)-sequence
(respectively, a weak-(FC)-sequence) of $A$ with respect to
$(I_{1},\ldots,I_{s})$ if $\bar{x}_{i+1}$ is an (FC)-element (respectively, a
weak-(FC)-element) of $A_{i}$ with respect to
$(\bar{I}_{1},\ldots,\bar{I}_{s})$ for all $i=1,\ldots,t-1$. 00footnotetext:
Mathematics Subject Classification (2000): Primary 13H15. Secondary 13D40,
14C17, 13C15.
KeyKey words and phrases: Mixed multiplicity, (FC)-sequence, superficial
sequence. Remark 2. Set $A^{*}=\dfrac{A}{0:I^{\infty}}$, $J^{*}=JA^{*}$,
${I_{i}}^{*}=I_{i}A^{*}$ for all $i=1,\ldots,s,$ $x\in I_{i}$ satisfies the
condition (FC2) and $x^{*}$ the image of $x$ in $A^{*}.$ Then the condition
(i) of (Definition in Sect. 3, [Vi1]) is
$(x^{*})\cap{I_{1}^{*}}^{n_{1}}\cdots{I_{i}^{*}}^{n_{i}}\cdots{I_{s}^{*}}^{n_{s}}=x^{*}{I_{1}^{*}}^{n_{1}}\cdots{I_{i}^{*}}^{n_{i}-1}\cdots{I_{s}^{*}}^{n_{s}}$
for all $n_{i}\geq n^{\prime}_{i}$ and all non-negative integers
$n_{1},\ldots,n_{i-1},n_{i+1},\ldots,n_{s}.$ Since $x$ is an $I$-filter-
regular element, $x^{*}$ is non-zero-divisor in $A^{*}.$ Hence
$\frac{x^{*}{{J^{*}}^{n_{0}}}{I_{1}^{*}}^{n_{1}}\cdots{I_{i}^{*}}^{n_{i}-1}\cdots{I_{s}^{*}}^{n_{s}}}{x^{*}{J^{*}}^{n_{0}+1}{I_{1}^{*}}^{n_{1}}\cdots{I_{i}^{*}}^{n_{i}-1}\cdots{I_{s}^{*}}^{n_{s}}}\cong\frac{{J^{*}}^{n_{0}}{I_{1}^{*}}^{n_{1}}\cdots{I_{i}^{*}}^{n_{i}-1}\cdots{I_{s}^{*}}^{n_{s}}}{{J^{*}}^{n_{0}+1}{I_{1}^{*}}^{n_{1}}\cdots{I_{i}^{*}}^{n_{i}-1}\cdots{I_{s}^{*}}^{n_{s}}}.$
Using this property, [Vi1] showed [Proposition 3.3,Vi1]. But in fact, by $x$
satisfies the condition (FC2),
$\lambda_{x}:I_{1}^{n_{1}}\cdots I_{i}^{n_{i}}\cdots
I_{s}^{n_{s}}\longrightarrow xI_{1}^{n_{1}}\cdots I_{i}^{n_{i}}\cdots
I_{s}^{n_{s}},\;\;y\mapsto xy$
is surjective and $\text{ker}\lambda_{x}=(0:x)\cap I_{1}^{n_{1}}\cdots
I_{i}^{n_{i}}\cdots I_{s}^{n_{s}}\subseteq(0:I^{\infty})\cap
I_{1}^{n_{1}}\cdots I_{i}^{n_{i}}\cdots I_{s}^{n_{s}}=0$ for all large
$n_{1},\ldots,n_{s}$ by Artin-Rees lemma . Therefore,
$I_{1}^{n_{1}}\cdots I_{i}^{n_{i}}\cdots I_{s}^{n_{s}}\cong
xI_{1}^{n_{1}}\cdots I_{i}^{n_{i}}\cdots I_{s}^{n_{s}}$
for all large $n_{1},\ldots,n_{s}.$ This follows that
$\frac{x\mathfrak{J}^{n_{0}}{I_{1}}^{n_{1}}\cdots{I_{i}}^{n_{i}-1}\cdots{I_{s}}^{n_{s}}}{x\mathfrak{J}^{n_{0}+1}{I_{1}}^{n_{1}}\cdots{I_{i}}^{n_{i}-1}\cdots{I_{s}}^{n_{s}}}\cong\frac{\mathfrak{J}^{n_{0}}{I_{1}}^{n_{1}}\cdots{I_{i}}^{n_{i}-1}\cdots{I_{s}}^{n_{s}}}{\mathfrak{J}^{n_{0}+1}{I_{1}}^{n_{1}}\cdots{I_{i}}^{n_{i}-1}\cdots{I_{s}}^{n_{s}}}$
for all large $n_{1},\ldots,n_{s}$ and for any ideal $\mathfrak{J}$ of $A.$
This proved that [Proposition 3.3,Vi1] and hence the results of [Vi1] and
[Vi2] are still true for the (FC)-sequences that is defined as in Definition
1.
In this context, Theorem 3.4 in [Vi1] is stated as follows: Theorem 3 [Theorem
3.4, Vi1]. Let $(A,\mathfrak{m})$ denote a Noetherian local ring with maximal
ideal $\mathfrak{m}$, infinite residue $k=A/\mathfrak{m},$ and an ideal
$\mathfrak{m}$-primary $J$, and $I_{1},\ldots,I_{s}$ ideals of $A$ such that
$I=I_{1}\cdots I_{s}$ is non nilpotent. Then the following statements hold.
1. (i)
$e(J^{[k_{0}+1]},I_{1}^{[k_{1}]},\ldots,I_{s}^{[k_{s}]},A)\not=0$ if and only
if there exists an (FC)-sequence $x_{1},\ldots,x_{t}$ $(t=k_{1}+\cdots+k_{s})$
with respect to $(J,I_{1},\ldots,I_{s})$ consisting of $k_{1}$ elements of
$I_{1}$, …, $k_{s}$ elements of $I_{s}.$
2. (ii)
Suppose that $e(J^{[k_{0}+1]},I_{1}^{[k_{1}]},\ldots,I_{s}^{[k_{s}]},A)\not=0$
and $x_{1},\ldots,x_{t}$ $(t=k_{1}+\cdots+k_{s})$ is an (FC)-sequence with
respect to $(J,I_{1},\ldots,I_{s})$ consisting of $k_{1}$ elements of $I_{1}$,
…, $k_{s}$ elements of $I_{s}$. Set
$\bar{A}=A/(x_{1},\ldots,x_{t}):I^{\infty}$. Then
$e(J^{[k_{0}+1]},I_{1}^{[k_{1}]},\ldots,I_{s}^{[k_{s}]},A)=e_{A}(J,\bar{A}).$
Note that Theorem 3 is an immediate cosequence of [Theorem 3.3, VTh] and the
filtration version of Theorem 3 is proved also in [DV]. Definition 4 [Sect.1,
TV]. Set $T=\bigoplus_{n_{1},\ldots,n_{s}\geqslant 0}\frac{I_{1}^{n_{1}}\cdots
I_{s}^{n_{s}}}{I_{1}^{n_{1}+1}\cdots I_{s}^{n_{s}+1}}.$ Let $\varepsilon$ be
an index with $1\leqslant\varepsilon\leqslant s.$ An element $x\in A$ is an
$\varepsilon$-superficial element for $I_{1},\ldots,I_{s}$ if $x\in
I_{\varepsilon}$ and the image $x^{*}$ of $x$ in $I_{\varepsilon}/I_{1}\cdots
I_{\varepsilon-1}I_{\varepsilon}^{2}I_{\varepsilon+1}\cdots I_{s}$ is a
filter-regular element in $T$, i.e., $(0:_{T}x^{*})_{(n_{1},\ldots,n_{s})}=0$
for $n_{1},\ldots,n_{s}\gg 0.$ Let $\varepsilon_{1},\ldots,\varepsilon_{m}$ be
a non-decreasing sequence of indices with $1\leqslant\varepsilon_{i}\leqslant
s.$ A sequence $x_{1},\ldots,x_{m}$ is an
$(\varepsilon_{1},\ldots,\varepsilon_{m})$-superficial sequence for
$I_{1},\ldots,I_{s}$ if for $i=1,\ldots,m$, $\bar{x}_{i}$ is an
$\varepsilon_{i}$-superficial element for $\bar{I}_{1},\ldots,\bar{I}_{s}$,
where $\bar{x}_{i},\bar{I}_{1},\ldots,\bar{I}_{s}$ are the images of
$x_{i},I_{1},\ldots,I_{s}$ in $A/(x_{1},\ldots,x_{i-1}).$
Theorem 5 [Theorem 1.4, TV]. Set $q=\dim A/0:I^{\infty}$. Let
$k_{0},k_{1},\ldots,k_{s}$ be non-negative integers such that
$k_{0}+k_{1}+\cdots+k_{s}=q-1.$ Assume that
$\varepsilon_{1},\ldots,\varepsilon_{m}$ $(m=k_{1}+\cdots+k_{s})$ is a non-
decreasing sequence of indices consisting of $k_{1}$ numbers $1,\ldots,$
$k_{s}$ numbers $s$. Let $Q$ be any ideal generated by an
$(\varepsilon_{1},\ldots,\varepsilon_{m})$-superficial sequence for
$J,I_{1},\ldots,I_{s}$. Then
$e(J^{[k_{0}+1]},I_{1}^{[k_{1}]},\ldots,I_{s}^{[k_{s}]})\neq 0$ if and only if
$\dim A/Q:I^{\infty}=k_{0}+1.$ In this case,
$e(J^{[k_{0}+1]},I_{1}^{[k_{1}]},\ldots,I_{s}^{[k_{s}]})=e\big{(}J,A/Q:I^{\infty}\big{)}.$
Then the relationship between
$(\varepsilon_{1},\ldots,\varepsilon_{m})$-superficial sequences and
weak-(FC)-sequences is given in [DV] by the following proposition. Proposition
6 [Proposition 4.3, DV]. Let $I_{1},\ldots,I_{s}$ be ideals in $A.$ Let $x\in
A$ be an $\varepsilon$-superficial element for $I_{1},\ldots,I_{s}.$ Then $x$
is a weak-(FC)-element with respect to $(I_{1},\ldots,I_{s}).$
Proof: Assume that $x$ is an $\varepsilon$-superficial element for
$I_{1},\ldots,I_{s}$. Without loss of generality, we may assume that
$\varepsilon=1.$ Then
$\big{(}I_{1}^{n_{1}+2}I_{2}^{n_{2}+1}\cdots I_{s}^{n_{s}+1}:x\big{)}\cap
I_{1}^{n_{1}}\cdots I_{s}^{n_{s}}=I_{1}^{n_{1}+1}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}$ (1)
for $n_{1},\ldots,n_{s}\gg 0.$ (1) implies
$\begin{split}\big{(}I_{1}^{n_{1}+2}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}:x\big{)}\cap I_{1}^{n_{1}}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}=I_{1}^{n_{1}+1}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}\end{split}$ (2)
for $n_{1},\ldots,n_{s}\gg 0$. We prove by induction on $k\geqslant 2$ that
$\begin{split}\big{(}I_{1}^{n_{1}+k}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}:x\big{)}\cap I_{1}^{n_{1}}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}=I_{1}^{n_{1}+k-1}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}\end{split}$ (3)
for $n_{1},\ldots,n_{s}\gg 0.$ The case $k=2$ follows from (2). Assume now
that
$\begin{split}\big{(}I_{1}^{n_{1}+k}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}:x\big{)}\cap I_{1}^{n_{1}}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}=I_{1}^{n_{1}+k-1}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}\end{split}$
for $n_{1},\ldots,n_{s}\gg 0.$ Then
$\displaystyle\big{(}$ $\displaystyle I_{1}^{n_{1}+k+1}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}:x\big{)}\cap I_{1}^{n_{1}}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}$ $\displaystyle=\big{(}I_{1}^{n_{1}+k+1}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}:x\big{)}\cap\big{(}I_{1}^{n_{1}+k}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}:x\big{)}\cap I_{1}^{n_{1}}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}$ $\displaystyle=\big{(}I_{1}^{n_{1}+k+1}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}:x\big{)}\cap I_{1}^{n_{1}+k-1}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}$ $\displaystyle=I_{1}^{n_{1}+k}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}$
for $n_{1},\ldots,n_{s}\gg 0.$ The last equality is derived from (2). Hence
the induction is complete and we get (3). It follows that for
$n_{1},\ldots,n_{s}\gg 0$,
$\displaystyle(0:x)$ $\displaystyle\cap I_{1}^{n_{1}}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}$ $\displaystyle=\Big{(}\bigcap_{k\geqslant
2}I_{1}^{n_{1}+k}I_{2}^{n_{2}+1}\cdots I_{s}^{n_{s}+1}:x\Big{)}\cap
I_{1}^{n_{1}}I_{2}^{n_{2}+1}\cdots I_{s}^{n_{s}+1}$
$\displaystyle=\Big{(}\bigcap_{k\geqslant
2}\big{(}I_{1}^{n_{1}+k}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}:x\big{)}\Big{)}\cap I_{1}^{n_{1}}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}$ $\displaystyle=\bigcap_{k\geqslant
2}\Big{(}\big{(}I_{1}^{n_{1}+k}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}:x\big{)}\cap I_{1}^{n_{1}}I_{2}^{n_{2}+1}\cdots
I_{s}^{n_{s}+1}\Big{)}$ $\displaystyle=\bigcap_{k\geqslant
2}I_{1}^{n_{1}+k-1}I_{2}^{n_{2}+1}\cdots I_{s}^{n_{s}+1}=0,$
that is, $(0:x)\cap I^{n}=0$ for $n\gg 0,$ here $I=I_{1}\cdots I_{s}.$ Hence
$0:x\subseteq 0:I^{\infty}.$ So $x$ is satisfies condition (FC2). Now we need
to prove that $I_{1}^{n_{1}}\cdots
I_{s}^{n_{s}}\cap(x)=xI_{1}^{n_{1}-1}I_{2}^{n_{2}}\cdots I_{s}^{n_{s}}$ for
$n_{1},\ldots,n_{s}\gg 0$. But this has from the proof of [Lemma 1.3, TV].
Hence $x$ is a weak-(FC)-element with respect to $(I_{1},\ldots,I_{s})$.
Remark 7. Return to Theorem 5, assume that $Q=(x_{1},\ldots,x_{m})$, where
$x_{1},\ldots,x_{m}$ is an
$(\varepsilon_{1},\ldots,\varepsilon_{m})$-superficial sequence for
$J,I_{1},\ldots,I_{s}$. As $x_{1},\ldots,x_{m}$ is a weak-(FC)-sequence with
respect to $(J,I_{1},\ldots,I_{s})$ by Proposition 6. Hence
$\dim A/Q:I^{\infty}\leqslant q-m=k_{0}+1$
with equality if and only if $x_{1},\ldots,x_{m}$ is an (FC)-sequence by
[Proposition 3.1(ii), Vi2]. This fact proved that Theorem 3 covers Theorem 5
that is the main result of Trung and Verma in [TV].
## References
* [DV] L. V. Dinh and D. Q. Viet, On Mixed multiplicities of good filtrations, preprint.
* [Re] D. Rees, Genaralizations of reductions and mixed multiplicities, J. London Math. Soc. 29 (1984), 397-414.
* [Te] B. Teissier, Cycles èvanescents, sections planes, et conditions de Whitney, Singularités à Cargése 1972, Astérisque 7-8 (1973), 285-362.
* [TV] N. V. Trung and J. Verma, Mixed multiplicities of ideals versus mixed volumes of polytopes, Trans. Amer. Math. Soc. 359(2007), 4711-4727.
* [Vi1] D. Q. Viet, Mixed multiplicities of arbitrary ideals in local rings, Comm. Algebra. 28(8)(2000), 3803-3821.
* [Vi2] D. Q. Viet, Sequences ditermining mixed multiplicities and reductions of ideals, Comm. Algebra. 31(10)(2003), 5047-5069.
* [VTh] D. Q. Viet and T. H. Thanh, Mixed multiplicities of multigraded algebras over noetherian local rings, preprint.
|
arxiv-papers
| 2009-01-08T03:19:29 |
2024-09-04T02:48:59.779100
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Le Van Dinh and Duong Quoc Viet",
"submitter": "Duong Quoc Viet",
"url": "https://arxiv.org/abs/0901.0966"
}
|
0901.0980
|
# Remarkable suppression of dc Josephson current on $d$-wave superconductor
junction
Takashi Hirai Nakatsugawa, gifu, 508-0101, Japan
###### Abstract
Josephson current in superconductor/insulator/superconductor junction is
studied theoretically. It is well known that when the zero-energy resonance
state exists both side of superconducting interface, the behaver of the
temperature dependence of the critical Josephson current is striking
enhancement at low temperature. On the other hand it is reported that if
$d+is$-wave exists at the interface, Josephson current is suppressed at low
temperature. In this paper, we discuss the existence of the imaginary part of
the pair potential at the interface and remarkable suppresses of dc Josephson
current on $d$-wave superconductor 110-junction.
###### pacs:
PACS numbers: 74.25.Fy, 74.45.+c, 74.50.+r
## I Introduction
In two decade, transport property of the unconventional superconducting
junctions is studied both theoretically and experimentally. In these
junctions, zero-energy resonance state (ZES) plays an important role. ZES2 ;
ZES3 It is well known that in the tunneling spectroscopy of the high-$T_{C}$
superconductor the zero-bias conductance peak ZBCP4 ; ZBCP5 ; ZBCP6 ; ZBCP7 ;
ZBCP8 ; ZBCP9 ; ZBCP10 ; ZBCP11 appears.
On the other hand Josephson current in superconductor / insulator /
superconductor junction is one of the characteristic phenomena. Anomalous
behaver is obtained on high-$T_{C}$ superconducting junction, $i.e.$ the
critical Josephson current enhances at low temperature when the lattice
orientation is $\alpha=\pi/4$ (110-junction) as the Fig. 1. This is caused by
the existence of the ZES formed at the interface. In previous papers we have
known a general formula as Furusaki-Tsukada formulation for dc Josephson
current, which include both macroscopic phase and ZES. This theory is based on
a microscopic calculation of the current represented in terms of the
coefficients of Andreev reflection. Andreev1 ; Andreev2 ; Andreev3
In this paper, we calculate and discuss dc Josephson current at 110-junction
in the $d$-component superconductor / insulator / $d$-wave superconductor
junction considering existence of $is$-wave state d+is and imaginary part of
the $d$-wave state. In these, we calculate spatial dependence of the pair
potential self-consistently.
Figure 1: A schematic of 110-junction of the wave function of $d$-wave
superconductors. The crystal orientation of right and left side of the
superconductor for junction are chosen as $\alpha=\pi/4$, respectively.
## II Formulation
In order to calculate Josephson current, we well know the Green’s function
method like this:
$\displaystyle I=\frac{e\hbar}{2im}\left(\frac{\partial}{\partial
x}-\frac{\partial}{\partial
x^{\prime}}\right)\mbox{Tr}G_{\omega_{m}}(x,x^{\prime}).$ (1)
And now, we use the quasi-classical method in this paper. First of all, Nanbu-
Gol’kov Green’s function is written as,
$\displaystyle
G(x,x^{\prime})=G_{++}(x,x^{\prime})e^{ik_{F}(x-x^{\prime})}+G_{--}(x,x^{\prime})e^{-ik_{F}(x-x^{\prime})}$
$\displaystyle+G_{+-}(x,x^{\prime})e^{ik_{F}(x+x^{\prime})}+G_{-+}(x,x^{\prime})e^{-ik_{F}(x+x^{\prime})}.$
(2)
Taking the differential into the equation, we obtain the following equation,
$\displaystyle I=\frac{e\hbar
k_{F}}{m}\mbox{Tr}\left(G_{++}(x,x^{\prime})e^{ik_{F}(x-x^{\prime})}\right.$
$\displaystyle\left.-G_{--}(x,x^{\prime})e^{-ik_{F}(x-x^{\prime})}\right)+O(1).$
(3)
The differential for $G_{\alpha\beta}(x,x^{\prime})$ is order $1$ ( $\ll
k_{F}$ ), so it’s ignored. The quantity $\alpha$, $\beta$ mean $\pm$. The
Green’s function $G_{\pm\mp}(x,x^{\prime})$ terms vanish in the differential.
We define the quasi-classical Green’s function. Green
$\displaystyle\hat{g}_{\alpha}=f_{1\alpha}\hat{\tau}_{1}+f_{2\alpha}\hat{\tau}_{2}+g_{\alpha}\hat{\tau}_{3}\mbox{
, }(\hat{g}_{\alpha})^{2}=\hat{1}$ (4)
Here $\hat{\tau}_{j}$($j=1,2,3$) are Pauri matrices and $\hat{1}$ is a unit
matrix. The quantities $f_{1\alpha},f_{2\alpha},g_{\alpha}$ obey the following
relations,
$\displaystyle f_{1\alpha}=$
$\displaystyle\alpha\left[iF_{\alpha}(x)+D_{\alpha}(x)\right]/\left[1-D_{\alpha}(x)F_{\alpha}(x)\right],$
(5) $\displaystyle f_{2\alpha}=$
$\displaystyle-\left[F_{\alpha}(x)-D_{\alpha}(x)\right]/\left[1-D_{\alpha}(x)F_{\alpha}(x)\right],$
(6) $\displaystyle g_{\alpha}=$
$\displaystyle\alpha\left[1+F_{\alpha}(x)D_{\alpha}(x)\right]/\left[1-D_{\alpha}(x)F_{\alpha}(x)\right].$
(7)
In these quasi-classical Green’s function, the quantity $D_{\alpha}(x)$ and
$F_{\alpha}(x)$ obey the Ricatti equations Ricatti
$\displaystyle\hbar|v_{F}|D_{\alpha}(x)=\alpha\left[2\omega_{m}D_{\alpha}(x)+\Delta(x,\theta)D_{\alpha}^{2}(x)\right.$
$\displaystyle\left.-\Delta^{*}(x,\theta)\right],$ (8)
$\displaystyle\hbar|v_{F}|F_{\alpha}(x)=\alpha\left[-2\omega_{m}F_{\alpha}(x)+\Delta^{*}(x,\theta)F_{\alpha}^{2}(x)\right.$
$\displaystyle\left.-\Delta(x,\theta)\right].$ (9)
The quantity $\theta$ is the angle between quasi-particle going through the
interface and $x$ direction, here $x$-axis is the vertical to the interface.
The boundary conditions at the interface are given by
$\displaystyle
F_{+L}=\frac{D_{-R}-RD_{+R}-(1-R)D_{-R}}{D_{-L}(RD_{-R}-D_{+R})+(1-R)D_{+R}D_{-R}},$
(10) $\displaystyle
F_{-L}=\frac{RD_{-R}-D_{+R}+(1-R)D_{+L}}{D_{+L}(D_{-R}-RD_{+R})-(1-R)D_{+R}D_{-R}},$
(11) $\displaystyle
F_{+R}=\frac{RD_{+L}-D_{-L}+(1-R)D_{-R}}{D_{-R}(D_{+L}-RD_{-L})-(1-R)D_{+L}D_{-L}},$
(12) $\displaystyle
F_{-R}=\frac{D_{+L}-RD_{-L}-(1-R)D_{+R}}{D_{+R}(RD_{+L}-D_{-L})+(1-R)D_{+L}D_{-L}},$
(13)
where we omit the index $(x=0)$. The quantity $R$ is $R=Z^{2}/(4+Z^{2})$ with
$Z=2mH/\hbar^{2}k_{F}$. Here the quantity $H$ is the height of the barrier
potential. Then we treat the insulator as the $\delta$-functional barrier
potential. The boundary condition for $D_{\alpha}(x)$ at $x=\pm\infty$ is
$\displaystyle
D_{\alpha}(\pm\infty)=\frac{\Delta^{*}(\pm\infty,\theta)}{\omega_{m}+\alpha\Omega_{\alpha}}.$
(14)
In these relation, we can write down Josephson current as following,
$\displaystyle I(\theta)=\frac{2e\hbar
k_{F}}{m}i\left(\left[g_{+}(x,\theta)\right]-\left[g_{-}(x,\theta)\right]\right).$
(15)
Josephson current in this formula is obtained by $x\rightarrow 0$
The spatial dependent pair potential is calculated as following
$\displaystyle\Delta(x,\theta)$ $\displaystyle=$ $\displaystyle\frac{2T}{\ln
T/T_{C}+\sum_{0\leq m}\frac{1}{m+1/2}}$ (16) $\displaystyle\times$
$\displaystyle\sum_{0\leq
m}\int_{-\pi/2}^{\pi/2}d\theta^{\prime}V(\theta,\theta^{\prime})f_{2+}$
where $V(\theta,\theta^{\prime})=2\sin 2\theta\sin 2\theta^{\prime}$ for
110-junction and $V(\theta,\theta^{\prime})=2\cos 2\theta\cos
2\theta^{\prime}$ for 100- junction, respectively for $d$-wave component, and
$V(\theta,\theta^{\prime})=1$ for $s$-wave component for both 110- and
100-junction case. In this equation, we can calculate spatial dependent of the
pair potential self-consistently (SCF).
Josephson current $I$ in these formula is obtained numerically solving Eq. II,
15, 16 under the boundary conditions Eq. 10,11,12,13,14.
Calculated result of Josephson current is normalized by normal conductance
$\sigma_{N}$,
$\displaystyle I(\eta)=\int_{-\pi/2}^{\pi/2}I(\theta)\cos\theta
d\theta/\sigma_{N},$ (17)
$\displaystyle\sigma_{N}=\int_{-\pi/2}^{\pi/2}\frac{4\cos\theta^{2}}{4\cos\theta^{2}+Z^{2}}\cos\theta
d\theta.$ (18)
Here, we define $\eta=\eta_{L}-\eta_{R}$, where $\eta_{L}$, $\eta_{R}$ is the
macroscopic phase of left and right side of the superconductors. In every
thing, we chose the temperature $T=0.05T_{C}$, where $T_{C}$ is the transition
temperature of superconductor. And the cutoff frequency $\omega_{D}$ is set to
be $\omega_{D}/2\pi T_{C}=1$ for summation of Matsubara frequency $m$.
## III Results
In this section, we show the calculated results of the superconducting
macroscopic phase $\eta$ dependence of the pair potential and Josephson
current. In all case we chose $T_{C_{s}}=0.2T_{C_{d}}$, where $T_{C_{s}}$ and
$T_{C_{d}}$ ($=T_{C}$) are the transition temperature of $s$-wave component
and $d$-wave component, respectively.
Figure 2: The $x$-dependence of the pair potential of the right and left side
of the superconductors at $\eta=0$. The a, b, c mean $Z=0$, $5$, $10$,
respectively. $\xi$ is the coherent length of the superconductor. Figure 3:
The $x$-dependence of the pair potential of the right and left side of the
superconductors at $\eta=\pi/2$. The a, b, c mean $Z=0$, $5$, $10$,
respectively. $\xi$ is the coherent length of the superconductor. Figure 4:
The $x$-dependence of the pair potential of the right and left side of the
superconductors at $\eta=\pi$. The a, b, c mean $Z=0$, $5$, $10$,
respectively. $\xi$ is the coherent length of the superconductor.
First of all, we show the $\eta$-dependence of the pair potential. The system
is 110-junction. In the Fig. 2, 3, 4, the solid line is real number, and doted
line is imaginary number, respectively. The $x$ axis is normalized by coherent
length $\xi$. For $\eta=0$, $Z$-dependences don’t appear in the pair potential
as Fig. 2. The reducing of the pair potential for $Z=0$ near the interface
receives spatial changing. In these, $is$ state exists near the interface.
This state doesn’t appear in the 100-junction’s case. When the quasi-particle
goes through the interface, it feels the opposite sign of the pair potential.
So the reducing occurs in spite of $Z=0$.
Second, let us show the $\eta=\pi/2$ case. For $Z=0$, $s$ and $is$ state not
exist. Since the pair potential contains the superconducting macroscopic phase
at the right side of the superconductor, in the Fig. 3, right side of the pair
potential of $d$-wave is imaginary number. Similarly $\eta=0$ case, since the
quasi-particle through the interface feels the different sign of the pair
potential, real number of $d$-wave (since it contains the macroscopic phase of
the superconductor, real number of $d$-wave appears as the imaginary number. )
at the right side of the pair potential is connected to the imaginary $d$-wave
at the left side of the pair potential.
For $Z=5$, pair potential behaves as Fig. 3 $b$. The existence of the barrier
potential affects the suppression of the right and left side of the pair
potential for both real and imaginary numbers near the interface. At the
region of the coherence length near the interface, imaginary parts of the
$d$-wave is enhanced for the right and left side of the superconductors.
When $Z=10$, $s$ and $is$ state appear for both side of the superconductors
near the interface. The existence of the $is$-wave is same reason as the
ordinary discussion for $is$-wave state at the edge of the $d$-wave
superconductor on $\alpha=\pi/4$ (110-junction).
Next, we show the $\eta=\pi$ case. The phase factor is $\exp(i\eta)=-1$, so
110-junction is same as in the 100-junction ($\alpha=0$). Therefore real part
of the $d$-wave is not spacial dependence in the $Z=0$ case, $i.e.$ pair
potential is constance for all region. $s$ and $is$-wave don’t appear and
$id$-wave doesn’t appear too.
When $Z=5$, since existence of the barrier potential makes the reflection of
the quasi-particle at the interface, $d$-wave factors are reduced. $Z=10$ case
is same as $Z=5$. The different point at the $Z=10$ is the existence of the
$is$ state at the interface.
Finally, we show the normalized dc Josephson current for 110-junction and
100-junction. For 110-junction, Josephson current is suppressed at the
$\eta=\pi/2\sim\pi$ region for $Z=10$, and it suppressed at all area for
$Z=5$. For $Z=15$, Josephson current behaves $\sin\eta$. On the other hand,
for 100-junction, Josephson current is not suppressed. And it is consistent
with the non-SCF calculation.
Comparing the 110-junction to 100-junction, the hight of the Josephson current
for 100-junction is higher than that for 110-junction. It is not consistent
with non-SCF calculations. This is our new dissolve.
## IV Summary
In this section, we summarize the obtained results. Now we have seen the
imaginary part of the pair potential exists at the $\eta\neq 0$, $\pi$ for
110-junction. That occurs by the existence of the macroscopic phase of the
superconductors. This results are different from the situation of the surface
of superconductor or junction between normal metal and superconductor. Since
the both $is$\- and $id$-wave state exist near the interface, Josephson
current is reduced on 110-junction. This results is not only by the $is$-wave
state but also by the existence of the imaginary part of the pair potential of
$d$-wave. This reducing is same as in the $s$-wave superconductor / $p$-wave-
superconductor / $s$-wave superconductor junction. In this junction similar
reducing occurs by the existence of another symmetry pair potential at the
junction. Yamashiro In this paper’s case $id$-wave component plays the
different symmetry for the $d$-wave component. On the other hand, Josephson
current is not reduced on 100-interface. These results are unusual. These
appear only in the SCF calculation. In the non-SCF calculation, these don’t
appear. These result for 110-junction is consistent with Ref. 10, where it’s a
high barrier limit case.
And adding one more thing, Josephson current disappears on $Z=0$ both for
110-junction and 100-junction. In the physical point of view, it is expected
that Josephson current only exists when insulating barrier or something
(normal metal or different type of superconductor) exist at the interface.
Therefore these results are valid physically.
In this paper, we discuss dc Josephson current for the 110-junction. Pair
potential has the imaginary part for $\eta\neq 0$, and Josephson current is
suppressed. This result appears only in the SCF calculation of the pair
potential.
Figure 5: Josephson current at the 110-junction (a) and 100-junction (b) for
$Z=5$, $10$ and $Z=15$.
###### Acknowledgements.
I greatly acknowledge useful comment with Y. Tanaka and N. Hayashi. I would
like to thank S. Kaya for giving me a calculating tool.
## References
* (1) J. Hara, and K. Nagai, Theor. Phys. 76, 1237 (1986).
* (2) C. R. Hu, Phys. Rev. Lett. 72, 1526 (1994).
* (3) S. Kashiwaya, Y. Tanaka, M. Koyanagi, H. Takashima and K. Kajimura, Phys. Rev. B 51, 1350 (1995).
* (4) L. Alff, H. Takashima, S. Kashiwaya, N. Terada, H. Ihara, Y. Tanaka, M. Koyanagi and K. Kajimura, Phys. Rev. B 55 (1997) R14757.
* (5) M. Covington, M. Aprili, E. Paraoanu, L. H. Greene, F. Xu, J. Zhu and C. A. Mirkin, Phys. Rev. Lett. 79, 277 (1997).
* (6) J. Y. T. Wei, N.-C. Yeh, D. F. Garrigus and M. Strasik, Phys. Rev. Lett. 81, 2542 (1998).
* (7) W. Wang, M. Yamazaki, K. Lee and I. Iguchi, Phys. Rev. B 60, 4272 (1999).
* (8) I. Iguchi, W. Wang, M. Yamazaki, Y. Tanaka and S. Kashiwaya, Phys. Rev. B 62, R6131 (2000).
* (9) Y. Tanaka and S. Kashiwaya, Phys. Rev. B 53, 9371 (1996).
* (10) Y. Tanaka and S. Kashiwaya, Phys. Rev. B 53, R11957 (1996); 56, 892 (1997); 58, R2948 (1998).
* (11) A. F. Andreev, Zh. Eksp. Theor. Fiz. 46, 1823 (1964).
* (12) A. Furusaki and M. Tsukada, Solid State Commum. 78, 299 (1991).
* (13) Y. Tanaka, Phys. Rev. Lett. 72, 3871 (1994).
* (14) M. Matsumoto and H. Shiba, J. Phys. Soc. Jpn, 64 ; 3384 (1995); 64, 4867 (1995).
* (15) K. Nagai(unpublished); M. Ashida, S. Aoyama, J. Hara and K. Nagai, Phys. Rev. B 40, 8673 (1989); Y. Nagato, K. Nagai and J. Hara, J. Low Temp. Phys. 93, 33 (1993); J. Kurkij$\ddot{\mbox{a}}$rvi and D. Rainer, in $HeliumThree$, edited by W. P. Halperin and L. P. Pitaevskii (Elsevier, Amsterdam, 1990); Y. Tanuma, Y. Tanaka and S. Kashiwaya, Phys. Rev. B 64, 214519 (2001)
* (16) M. Eschrig, Phys. Rev. B 61, 9061 (2000); A. Shelankov and M. Ozana, $ibid$, 61, 7077 (2000); N. Schopohi and K. Maki, $ibid$, 52, 490 (1995); C. Iniotakis, G. Graser, T. Dahm and N. Schopohi, $ibid$, 71, 214508 (2005).
* (17) M. Yamashiro, Y. Tanaka, N. Yoshida and S. Kashiwaya, J. Phys. Soc. Jpn, 68, 2019 (1999).
|
arxiv-papers
| 2009-01-08T06:55:58 |
2024-09-04T02:48:59.784125
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Takashi Hirai",
"submitter": "Takashi Hirai",
"url": "https://arxiv.org/abs/0901.0980"
}
|
0901.1023
|
Modified Friedmann model in Randers-Finsler space of approximate Berwald type
as a possible alternative to dark energy hypothesis
Zhe Chang 111changz@mail.ihep.ac.cn and Xin Li 222lixin@mail.ihep.ac.cn
Institute of High Energy Physics
Chinese Academy of Sciences
P. O. Box 918(4), 100049 Beijing, China
###### Abstract
Gravitational field equations in Randers-Finsler space of approximate Berwald
type are investigated. A modified Friedmann model is proposed. It is showed
that the accelerated expanding universe is guaranteed by a constrained
Randers-Finsler structure without invoking dark energy. The geodesic in
Randers-Finsler space is studied. The additional term in the geodesic equation
acts as repulsive force against the gravity.
PACS numbers: 02.40.-k, 04.50.Kd, 95.36.+x, 98.80.Jk
Einstein’s general relativity connects the Riemann geometry to gravitation. It
is the standard model of gravity. However, up to now, general relativity still
faces problems. One of them is that the flat rotation curves of spiral
galaxies violate the prediction of Einstein’s gravity. Another is related with
recent astronomical observations[1]. Our universe is acceleratedly expanding.
This result can not be obtained directly from Einstein’s gravity and his
cosmological principle.
The most widely adopted way to resolve these difficulties is the dark matter
and dark energy hypothesis. However, up to now, such things can not be
detected directly from observations. This situation causes that some
physicists imagine the dark matter and dark energy hypothesis possesses some
properties of the ether hypothesis at the early 20 century. It is reasonable
to test the connection between gravitation and new geometry. Modified
Einstein’s gravity may throw new light to the above problems. Models have been
built for alternative to the dark matter hypothesis. The famous one is the
modified Newtonian dynamics[2]. Models have also been built for alternative to
the dark energy hypothesis[3].
Finsler geometry, which takes Riemann geometry as its special case, is a good
candidate to solve the problems mentioned above. In our previous paper[4], a
modified Newton’s gravity was obtained as the weak field approximation of the
Einstein’s equation in Finsler space of Berwald type. We have shown that the
prediction of the modified Newton’s gravity is in good agreement with the
rotation curves of spiral galaxies without invoking of dark matter hypothesis.
In this Letter, we propose a modified Friedmann model in Randers-Finsler space
of approximate Berwald type for possible alternative to the dark energy
hypothesis.
It is well known that the violation of Lorentz symmetry is one of the origins
of new physics beyond Standard Model. An interesting case of Lorentz
violation, which was proposed by Cohen and Glashow[5], is the model of Very
Special Relativity (VSR) characterized by a reduced symmetry SIM(2). In fact,
Gibbons, Gomis and Pope[6] showed that the Finslerian line element
$ds=(\eta_{\mu\nu}dx^{\mu}dx^{\nu})^{(1-b)/2}(n_{\rho}dx^{\rho})^{b}$ is
invariant under the transformations of the group DISIM${}_{b}(2)$. Further
investigation of the VSR in Finsler cosmology was presented[7]. In
reference[8], we have used the similar method of Gibbons et al. to study the
Lorentz violations within the framework of Finsler geometry.
Randers space, as a special kind of Finsler space, was first proposed by G.
Randers[9]. Within the framework of Randers space, modified dispersion
relation has been discussed[8]. A generalized Friedmann-Robertson-Walker (FRW)
cosmology of Randers-Finsler geometry has been also suggested[10].
The gravity in Finsler space has been studied for a long time[11, 12, 13, 14].
The gravitational field equations derived from Riemannian osculating metric
were presented in [15]. The generalized FRW cosmology and the anisotropies of
the universe have been investigated for such a metric[7, 10]. However, their
gravitational field equations are not consistent with the Bianchi identity and
general covariance principle of Einstein. The gravitational field equations in
Berwald-Finsler space has been written down explicitly[16](the Greek indices
belong to {0, 1, 2, 3} and the Latin ones to {1, 2, 3}),
$\displaystyle\left[Ric_{\mu\nu}-\frac{1}{2}g_{\mu\nu}S\right]+\left\\{\frac{1}{2}B^{~{}\alpha}_{\alpha~{}\mu\nu}+B^{~{}\alpha}_{\mu~{}\nu\alpha}\right\\}=8\pi
GT_{\mu\nu}.$ (1)
Berwald space is just a bit more general than the Riemannian space. Given a
Berwald space, all its tangent spaces are linearly isometric to a common
Minkowski space[17]. This property of Berwald space is compatible with the
general covariance principle.
Before dealing with the gravitational field equations, first of all, we
introduce some basic notations of the Finsler geometry[18]. Denote by $T_{x}M$
the tangent space at $x\in M$, and by $TM$ the tangent bundle of $M$. Each
element of $TM$ has the form $(x,y)$, where $x\in M$ and $y\in T_{x}M$. The
natural projection $\pi:TM\rightarrow M$ is given by $\pi(x,y)\equiv x$. A
Finsler structure of $M$ is a function
$\displaystyle F:TM\rightarrow[0,\infty)$
with the following properties:
(i) Regularity: F is $C^{\infty}$ on the entire slit tangent bundle
$TM\backslash 0$.
(ii) Positive homogeneity : $F(x,\lambda y)=\lambda F(x,y)$ for all
$\lambda>0$.
(iii) Strong convexity: The $n\times n$ Hessian matrix
$\displaystyle g_{\mu\nu}\equiv\frac{\partial}{\partial
y^{\mu}}\frac{\partial}{\partial y^{\nu}}\left(\frac{1}{2}F^{2}\right)$
is positive-definite at every point of $TM\backslash 0$.
Throughout the Letter, the lowering and raising of indices are carried out by
the fundamental tensor $g_{\mu\nu}$ defined above, and its inverse
$g^{\mu\nu}$.
In Finsler manifold, there exists a unique linear connection - the Chern
connection[19]. It is torsion freeness and metric-compatibility,
$\displaystyle\Gamma^{\alpha}_{\mu\nu}=\gamma^{\alpha}_{\mu\nu}-g^{\alpha\lambda}\left(A_{\lambda\mu\beta}\frac{N^{\beta}_{\nu}}{F}-A_{\mu\nu\beta}\frac{N^{\beta}_{\lambda}}{F}+A_{\nu\lambda\beta}\frac{N^{\beta}_{\mu}}{F}\right),$
(2)
where $\gamma^{\alpha}_{\mu\nu}$ is the formal Christoffel symbols of the
second kind with the same form of Riemannian connection, $N^{\mu}_{\nu}$ is
defined as
$N^{\mu}_{\nu}\equiv\gamma^{\mu}_{\nu\alpha}y^{\alpha}-A^{\mu}_{\nu\lambda}\gamma^{\lambda}_{\alpha\beta}y^{\alpha}y^{\beta}$
and $A_{\lambda\mu\nu}\equiv\frac{F}{4}\frac{\partial}{\partial
y^{\lambda}}\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}(F^{2})$ is the Cartan tensor (regarded as a measurement of deviation
from the Riemannian Manifold).
The Randers metric is a Finsler structure $F$ on $TM$ with the form
$\displaystyle F(x,y)\equiv\alpha(x,y)+\beta(x,y)~{},$ (3)
where
$\displaystyle\alpha(x,y)$ $\displaystyle\equiv$
$\displaystyle\sqrt{\tilde{a}_{\mu\nu}(x)y^{\mu}y^{\nu}}$
$\displaystyle\beta(x,y)$ $\displaystyle\equiv$
$\displaystyle\tilde{b}_{\mu}(x)y^{\mu}.$ (4)
Here $\tilde{\alpha}$ is a Riemannian metric on the manifold $M$. In this
Letter, the indices decorated with a tilde are lowered and raised by
$\tilde{\alpha}_{\mu\nu}$ and its inverse matrix $\tilde{\alpha}^{\mu\nu}$. A
Finsler structure $F$ is said to be of Berwald type if the Chern connection
coefficients $\Gamma^{\alpha}_{\mu\nu}$ in natural coordinates have no $y$
dependence. Given a Randers space of Berwald type, Kikuchi[20] proved that
$\displaystyle\tilde{b}_{\mu|\nu}\equiv\frac{\partial\tilde{b}_{\mu}}{\partial
x^{\nu}}-\tilde{b}_{\kappa}\tilde{\gamma}^{\kappa}_{\mu\nu}=0,$ (5)
where $\tilde{\gamma}^{\kappa}_{\mu\nu}$ is the Christoffel symbols of
Riemannian metric $\tilde{\alpha}_{\mu\nu}$. In Randers space of Berwald type,
after some tedious calculations one obtains that
$\displaystyle\Gamma^{\kappa}_{\mu\nu}=\tilde{\gamma}^{\kappa}_{\mu\nu}.$ (6)
The curvature of Finsler space of Berwlad type is given as
$\displaystyle R^{~{}\lambda}_{\kappa~{}\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\partial\Gamma^{\lambda}_{\kappa\nu}}{\partial
x^{\mu}}-\frac{\partial\Gamma^{\lambda}_{\kappa\mu}}{\partial
x^{\nu}}+\Gamma^{\lambda}_{\alpha\mu}\Gamma^{\alpha}_{\kappa\nu}-\Gamma^{\lambda}_{\alpha\nu}\Gamma^{\alpha}_{\kappa\mu}.$
(7)
Thus, the curvature of Randers space of Berwald type can be simplified as
$\displaystyle R^{~{}\lambda}_{\kappa~{}\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\partial\tilde{\gamma}^{\lambda}_{\kappa\nu}}{\partial
x^{\mu}}-\frac{\partial\tilde{\gamma}^{\lambda}_{\kappa\mu}}{\partial
x^{\nu}}+\tilde{\gamma}^{\lambda}_{\alpha\mu}\tilde{\gamma}^{\alpha}_{\kappa\nu}-\tilde{\gamma}^{\lambda}_{\alpha\nu}\tilde{\gamma}^{\alpha}_{\kappa\mu}.$
(8)
This curvature is none other than the curvature of $\tilde{\alpha}$. The Ricci
tensor on Finsler manifold was first introduced by Akbar-Zadeh[21]. In Finsler
space of Berwald type, it reduces to
$\displaystyle
Ric_{\mu\nu}=\frac{1}{2}(R^{~{}\alpha}_{\mu~{}\alpha\nu}+R^{~{}\alpha}_{\nu~{}\alpha\mu}).$
(9)
It is manifestly symmetric and covariant. Apparently the Ricci tensor will
reduce to the Riemann-Ricci tensor if the Cartan tensor vanish identically.
The trace of the Ricci tensor gives the scalar curvature $S\equiv
g^{\mu\nu}Ric_{\mu\nu}$. In order to investigate the FRW cosmology, we set the
Riemannian metric $\tilde{\alpha}$ to be the Robertson-Walker one
$\displaystyle\tilde{a}_{\mu\nu}={\rm
diag}\left(1,-\frac{R^{2}(t)}{1-kr^{2}},-R^{2}(t)r^{2},-R^{2}(t)r^{2}\sin^{2}\theta\right),$
(10)
where $k=0,\pm 1$ for a flat, closed and hyperbolic geometry respectively.
Unfortunately, such a Randers space of Berwald type is just the Riemannian
space. That is the condition (5) only has solution $\tilde{b}=0$. Here, we set
$\tilde{b}_{\mu}=(\tilde{b}_{0},0,0,0)$ for satisfying the requirement that
the universe is homogenous and isotropic. If $\tilde{b}_{0}$ is sufficient
small, the space can be regarded as a Berwald space approximately. On such
approximation, we just neglect the term proportion to
$\int\frac{\partial\Gamma}{\partial y}dx$ in the field equation (1).
After some tedious but straightforward calculations, we obtain following
nonzero components of curvature in Randers space of approximate Berwald type
$\displaystyle Ric_{00}$ $\displaystyle=$
$\displaystyle-3\frac{\ddot{R}}{R}\tilde{a}_{00},$ $\displaystyle Ric_{ij}$
$\displaystyle=$
$\displaystyle-\left(\frac{\ddot{R}}{R}+2\frac{\dot{R}^{2}}{R^{2}}+\frac{2k}{R^{2}}\right)\tilde{a}_{ij},$
$\displaystyle S$ $\displaystyle=$
$\displaystyle-6\frac{\alpha}{F}\left(\frac{\ddot{R}}{R}+\frac{\dot{R}^{2}}{R^{2}}+\frac{k}{R^{2}}\right)$
(11)
$\displaystyle-3\frac{\ddot{R}}{R}\tilde{a}_{00}\frac{\alpha^{2}}{F^{2}}\left(\frac{\beta}{F}\tilde{a}_{00}\frac{y^{0}}{\alpha}\frac{y^{0}}{\alpha}-2\tilde{a}_{00}\frac{y^{0}}{\alpha}\tilde{b}^{0}\right)$
$\displaystyle-3\left(\frac{\ddot{R}}{R}+2\frac{\dot{R}^{2}}{R^{2}}+\frac{2k}{R^{2}}\right)\tilde{a}_{ij}\frac{\alpha^{2}}{F^{2}}\left(\frac{\beta}{F}\tilde{a}_{ij}\frac{y^{i}}{\alpha}\frac{y^{j}}{\alpha}\right).$
The terms $B^{~{}\alpha}_{\alpha~{}\mu\nu}$ and
$B^{~{}\alpha}_{\mu~{}\nu\alpha}$ vanish in Randers space of approximate
Berwald type, where
$\displaystyle
B_{\mu\nu\alpha\beta}=-A_{\mu\nu\lambda}R^{~{}\lambda}_{\theta~{}\alpha\beta}y^{\theta}/F.$
(12)
In the left side of the field equations, only symmetric part is left. Thus, we
should set the energy-momenta tensor as
$\displaystyle T^{\mu}_{\nu}={\rm diag}(\rho,-p,-p,-p),$ (13)
where $\rho\equiv\rho(x)$ and $p\equiv p(x)$ is the the energy density and
pressure of the cosmic fluid respectively. The $0-0$ component of the field
equations (1) gives the modified Friedmann equation
$\displaystyle\frac{\alpha}{F}\left(\frac{\dot{R}^{2}}{R^{2}}+\frac{k}{R^{2}}\right)-\frac{1}{2}\frac{\ddot{R}}{R}\tilde{a}_{00}\frac{\alpha^{2}}{F^{2}}\left(\frac{\beta}{F}\tilde{a}_{00}\frac{y^{0}}{\alpha}\frac{y^{0}}{\alpha}-2\tilde{a}_{00}\frac{y^{0}}{\alpha}\tilde{b}^{0}\right)\hskip
142.26378pt$
$\displaystyle+\frac{1}{2}\left(\frac{\ddot{R}}{R}+2\frac{\dot{R}^{2}}{R^{2}}+\frac{2k}{R^{2}}\right)\tilde{a}_{ij}\frac{\alpha^{2}}{F^{2}}\left(\frac{\beta}{F}\tilde{a}_{ij}\frac{y^{i}}{\alpha}\frac{y^{j}}{\alpha}\right)=\frac{8\pi
G}{3}\rho.$ (14)
By making use of the modified Friedmann equation (S0.Ex8) and omitting the
$O(b^{2})$ term, we obtain the $i-i$ component of the field equations (1)
$\displaystyle\frac{\alpha}{F}\frac{\ddot{R}}{R}\left(1+\frac{\alpha}{F}\left(\frac{\beta}{F}\tilde{a}_{00}\frac{y^{0}}{\alpha}\frac{y^{0}}{\alpha}-2\tilde{a}_{00}\frac{y^{0}}{\alpha}\tilde{b}^{0}\right)\right)=-\frac{4\pi
G}{3}(\rho+3p).$ (15)
From the equation (15), one can see clearly that the accelerated expanding
universe ($\ddot{R}>0$) is guaranteed by the constraint
$\displaystyle
1+\frac{\alpha}{F}\left(\frac{\beta}{F}\tilde{a}_{00}\frac{y^{0}}{\alpha}\frac{y^{0}}{\alpha}-2\tilde{a}_{00}\frac{y^{0}}{\alpha}\tilde{b}^{0}\right)<0,$
(16)
while the energy density and pressure of the cosmic fluid keep positive. Since
the Finsler structure $F$ and Riemannian length element $\alpha$ are positive,
a direct deduction from (16) is
$\displaystyle\tilde{b}_{0}$ $\displaystyle<$
$\displaystyle-\frac{1}{\frac{y^{0}}{\alpha}((\frac{y^{0}}{\alpha})^{2}-2)},$
(17) $\displaystyle\frac{y^{0}}{\alpha}$ $\displaystyle>$
$\displaystyle\sqrt{2}.$ (18)
The positive Finsler structure $F$ gives that $\tilde{b}_{0}\tilde{b}^{0}<1$.
So that the complete constraint on Randers-Finsler structure to support
accelerated expanding universe is
$\displaystyle-1<\tilde{b}_{0}<-\frac{1}{\frac{y^{0}}{\alpha}((\frac{y^{0}}{\alpha})^{2}-2)}.$
(19)
It means that a negative $\tilde{b}_{0}$ provides an effective repulsive force
in the course of universe expanding.
This fact also can be observed clearly from the geodesic with constant
Riemanian speed. Following the calculus of variations, one get the geodesic
equation of Finsler space[18]
$\displaystyle\frac{d^{2}\sigma^{\lambda}}{d\tau^{2}}+\gamma^{\lambda}_{\mu\nu}\frac{d\sigma^{\mu}}{d\tau}\frac{d\sigma^{\nu}}{d\tau}=\frac{d\sigma^{\mu}}{d\tau}\frac{d}{d\tau}\left(\log
F(\sigma,\frac{d\sigma}{d\tau})\right).$ (20)
Deducing from (20), we obtain the geodesic of Randers space with constant
Riemanian speed (namely, $\alpha(\frac{d\sigma}{d\tau})$ is constant)
$\displaystyle\frac{d^{2}\sigma^{\lambda}}{d\tau^{2}}+\tilde{\gamma}^{\lambda}_{\mu\nu}\frac{d\sigma^{\mu}}{d\tau}\frac{d\sigma^{\nu}}{d\tau}+\tilde{a}^{\lambda\mu}f_{\mu\nu}\alpha\left(\frac{d\sigma}{d\tau}\right)\frac{d\sigma^{\nu}}{d\tau}=0,$
(21)
where $f_{\mu\nu}\equiv\frac{\partial\tilde{b}_{\mu}}{\partial
x^{\nu}}-\frac{\partial\tilde{b}_{\nu}}{\partial x^{\mu}}$. The geodesic
equation of Randers space (21) has clearly physical meaning. The last term,
which is proportional to the asymmetrical term $f_{\mu\nu}$, acts as
electromagnetic force. The term $\tilde{b}_{\mu}$ can be regarded as the
electromagnetic potential. The negative $\tilde{b}_{0}$ means that
the“electromagnetic” force $f$ is repulsive and against the popular attractive
force.
Since the Finsler structure depends on both coordinates and velocities, it is
important to investigate the physical meaning of the velocity dependence. The
term $\frac{y^{0}}{\alpha}$[8] involved in (17) represents the energy–to–mass
ratio. The upper bound of the dimensionless parameter $\tilde{b}_{0}$ gives a
criteria that the repulsive effect equal to the attractive one. It means that
the universe is expanding with constant speed while $\tilde{b}_{0}$ equal to
its upper bound. The particle have enough energy to fight against the
attractive force while $\tilde{b}_{0}$ satisfies the constraint (17).
Acknowledgements
We would like to thank Prof. H. Y. Guo and C. G. Huang for useful discussions.
The work was supported by the NSF of China under Grant No. 10575106 and
10875129.
## References
* [1] A. G. Riess, et al., Astrophys J. 117, 707 (1999); S. Perlmutter, et al., Astrophys J. 517, 565 (1999); C. L. Bennett, et al., Astrophys J. 148 (Suppl), 1 (2003).
* [2] M. Milgrom, Astrophys. J. 270, 365 (1983).
* [3] S. Bludman, arXiv:astro-ph/0605198.
* [4] Z. Chang and X. Li, Phys. Lett. B 668, 453 (2008).
* [5] A. G. Cohen and S. L. Glashow, Phys. Rev. Lett. 97, 021601 (2006).
* [6] G. W. Gibbons, J. Gomis and C. N. Pope, Phys. Rev. D 76, 081701 (2007).
* [7] A. P. Kouretsis, M. Stathakopoulos and P. C. Stavrinos, arXiv:gr-qc/0810.3267.
* [8] Z. Chang and X. Li, Phys. Lett. B 663, 103 (2008).
* [9] G. Randers, Phys. Rev. 59, 195 (1941).
* [10] P. C. Stavrinos, A. P. Kouretsis and M. Stathakopoulos, arXiv:gr-qc/0612157.
* [11] Y. Takano, Lett. Nuovo Cimento 10, 747 (1974).
* [12] S. Ikeda, Ann. der Phys. 44, 558 (1987).
* [13] R. Tavakol and N. van den Bergh, Phys. Lett. A 112, 23 (1985).
* [14] G. Yu. Bogoslovsky, Phys. Part. Nucl. 24, 354 (1993).
* [15] G.S.Asanov, Finsler Geometry, Relativity and Gauge Theories, Reidel Pub.Com., Dordrecht, 1985.
* [16] X. Li and Z. Chang, arXiv: gr-qc/0711.1934.
* [17] Y. Ichijyō, Finsler manifolds modeled on a Minkowski space, J. Math. Kyoto Univ. 16-3, 639 (1976).
* [18] D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathmatics 200, Springer, New York, 2000.
* [19] S. S. Chern, Sci. Rep. Nat. Tsing Hua Univ. Ser. A 5, 95 (1948); or Selected Papers, vol. II, 194, Springer 1989.
* [20] S. Kikuchi, Tensor, N.S. 33, 242 (1979).
* [21] H. Akbar-Zadeh, Acad. Roy. Belg. Bull. Cl. Sci. (5) 74, 281 (1988).
|
arxiv-papers
| 2009-01-08T11:39:40 |
2024-09-04T02:48:59.790053
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhe Chang and Xin Li",
"submitter": "Xin Li",
"url": "https://arxiv.org/abs/0901.1023"
}
|
0901.1368
|
# Global well-posedness for a modified critical dissipative quasi-geostrophic
equation
Changxing Miao111Institute of Applied Physics and Computational Mathematics,
P.O. Box 8009, Beijing 100088, P.R. China. Email: miao_changxing@iapcm.ac.cn.
and Liutang Xue222 The Graduate School of China Academy of Engineering
Physics, P.O. Box 2101, Beijing 100088, P.R. China. Email: xue_lt@163.com.
###### Abstract
In this paper we consider the following modified quasi-geostrophic equation
$\partial_{t}\theta+u\cdot\nabla\theta+\nu|D|^{\alpha}\theta=0,\quad
u=|D|^{\alpha-1}\mathcal{R}^{\bot}\theta$
with $\nu>0$ and $\alpha\in]0,1[\,\cup\,]1,2[$. When $\alpha\in]0,1[$, the
equation was firstly introduced by Constantin, Iyer and Wu in [10]. Here, by
using the modulus of continuity method, when $\alpha\in]0,1[$ we prove the
global well-posedness of the system with the smooth initial data, and when
$\alpha\in]1,2[$ we show the similar global result under the condition that
the (scaling-invariant) $L^{\infty}$ norm of the initial data is small.
MSC(2000): 76U05, 76B03, 35Q35
Keywords: Modified quasi-Geostrophic equation, Modulus of continuity, Blow-up
criterion, Global well-posedness.
## 1 Introduction
In this paper we focus on the following modified 2D dissipative quasi-
geostrophic equation
$\begin{cases}\partial_{t}\theta+u\cdot\nabla\theta+\nu|D|^{\alpha}\theta=0\\\
u=|D|^{\alpha-1}\mathcal{R}^{\bot}\theta,\qquad\theta|_{t=0}=\theta_{0}(x)\end{cases}$
(1.1)
with $\nu>0$, $\alpha\in]0,1[\,\cup\,]1,2[$,
$|D|^{\beta}=(-\Delta)^{\frac{\beta}{2}}$ ($\beta=\alpha,\alpha-1$) is defined
via the Fourier transform
$\widehat{(|D|^{\beta}f)}(\zeta)=|\zeta|^{\beta}\hat{f}(\zeta)$ and
$\mathcal{R}^{\bot}\theta=(-\mathcal{R}_{2}\theta,\mathcal{R}_{1}\theta)=|D|^{-1}(\partial_{2}\theta,-\partial_{1}\theta)$
where $\mathcal{R}_{i}$($i=1,2$) are the usual Riesz transforms (cf. [15]).
When $\alpha=0$, this model describes the evolution of the vorticity of a two
dimensional damped inviscid incompressible fluid. The case of $\alpha=1$ just
is the critical dissipative quasi-geostrophic equation which arises in the
geostrophic study of rotating fluids. Although when $\alpha=2$ the flow term
in (1.1) vanishes, we can still view the model introduced in [16] as a
meaningful generalization of this endpoint case, where the model is derived
from the study of the full magnetohydrodynamic equations and the divergence-
free three-dimensional velocity $u$ satisfies $u=M[\theta]$ with $M$ a
nonlocal differential operator of order 1.
For convenience, we here recall the well-known 2D quasi-geostrophic equation
$(QG)_{\alpha}\quad\begin{cases}\partial_{t}\theta+u\cdot\nabla\theta+\nu|D|^{\alpha}\theta=0\\\
u=\mathcal{R}^{\bot}\theta,\qquad\theta(0,x)=\theta_{0}(x)\end{cases}$
where $\nu\geq 0$ and $0\leq\alpha\leq 2$. When $\nu>0$,
$\alpha\in]0,1[\,\cup\,]1,2[$, we observe that the system (1.1) is almost the
same with the quasi-geostrophic equation, and its only difference lies on
introducing an extra $|D|^{\alpha-1}$ in the definition of $u$. When
$\alpha\in]0,1[$, $|D|^{\alpha-1}$ is a negative derivative operator and
always plays a good role; while when $\alpha\in]1,2[$, $|D|^{\alpha-1}$ is a
positive derivative operator and always takes a bad part. Moreover,
corresponding to the dissipation operator $|D|^{\alpha}$, this additional
operator makes the equation $(QG)_{\alpha}$ be a new balanced state: the flow
term $u\cdot\nabla\theta$ scale the same way as the dissipative term
$|D|^{\alpha}\theta$, i.e., the equation (1.1) is scaling invariant under the
transformation
$\theta(t,x)\rightarrow\theta_{\lambda}(t,x):=\theta(\lambda^{\alpha}t,\lambda
x),\quad\mathrm{with}\quad\lambda>0.$
We note that in the sense of scaling invariance, the system (1.1) is similar
to the critical dissipative quasi-geostrophic equation.
Recently, when $\alpha\in]0,1[$, Constantin, Iyer and Wu in [10] introduced
this modified quasi-geostrophic equation and proved the global regularity of
Leray-Hopf weak solutions to the system with $L^{2}$ initial data. Basically,
they use the method from Caffarelli-Vasseur [3] which deal with the same issue
of 2D critical dissipative quasi-geostrophic equation $(QG)_{1}$. We also
remark that partially because of its simple form and its internal analogy with
the 3D Euler/Navier-Stokes equations, the quasi-geostrophic equation
$(QG)_{\alpha}$, especially the critical one $(QG)_{1}$, has been extensively
considered (see e.g. [1, 3, 6, 7, 8, 9, 11, 13, 17, 22] and reference
therein). While global existence of Navier-Stokes equations remains an
outstanding challenge in mathematical physics, the global issue of the 2D
critical dissipative quasi-geostrophic equation has been in a satisfactory
state. In [9] Constantin, Cordoba and Wu showed the global well-posedness of
the classical solution under the condition that the zero-dimensional
$L^{\infty}$ norm of the data is small. This smallness assumption was firstly
removed by Kiselev, Nazarov and Volberg in [17], where they obtained the
global well-posedness for the arbitrary periodic smooth initial data by using
a modulus of continuity method. Almost at the same time, Caffarelli and
Vasseur in [3] resolved the problem to establish the global regularity of weak
solutions associated with $L^{2}$ initial data by exploiting the De Giorgi
method. We also cite the work of Abidi-Hmidi [1] and Dong-Du [13], as extended
work of [17], in which the authors proved the global well-posedness with the
initial data belonging to the (critical) space $\dot{B}^{0}_{\infty,1}$ and
$H^{1}$ respectively without the additional periodic assumption.
The main goal in this paper is to prove the global well-posedness of the
smooth solutions for the system (1.1) with $\alpha\in]0,1[\,\cup\,]1,2[$. In
contrast with the work of [10], we here basically follow the pathway of [17]
to obtain the global results by constructing suitable moduli of continuity.
Precisely, we have
###### Theorem 1.1.
Let $\nu>0$, $\alpha\in]0,1[$ and $\theta_{0}\in H^{m}$, $m>2$, then there
exists a unique global solution
$\theta\in\mathcal{C}([0,\infty[;H^{m})\cap
L_{\mathrm{loc}}^{2}([0,\infty[;H^{m+\frac{\alpha}{2}})\cap\mathcal{C}^{\infty}(]0,\infty[\times\mathbb{R}^{2})$
to the modified quasi-geostrophic equation (1.1). Moreover, we get the uniform
bound of the Lipschitz norm
$\sup_{t\geq 0}\left\|\nabla\theta(t)\right\|_{L^{\infty}}\leq
C\left\|\nabla\theta_{0}\right\|_{L^{\infty}}e^{C\left\|\theta_{0}\right\|_{L^{\infty}}},$
where $C$ is an absolute constant depending only on $\alpha,\nu$.
###### Theorem 1.2.
Let $\nu>0$, $\alpha\in]1,2[$ and $\theta_{0}\in H^{m}$, $m>2$. Then an
absolute constant $c_{0}>0$ can be found such that if
$\left\|\theta_{0}\right\|_{L^{\infty}}\leq c_{0},$ (1.2)
there exists a unique global solution
$\theta\in\mathcal{C}([0,\infty[;H^{m})\cap
L_{\mathrm{loc}}^{2}([0,\infty[;H^{m+\frac{\alpha}{2}})\cap\mathcal{C}^{\infty}(]0,\infty[\times\mathbb{R}^{2})$
to the modified quasi-geostrophic equation (1.1). We also get $\sup_{t\geq
0}\left\|\nabla\theta(t)\right\|_{L^{\infty}}\leq
C\left\|\nabla\theta_{0}\right\|_{L^{\infty}}$.
The proof is divided into two parts. First through applying the classical
method, we obtain the local existence results (Proposition 4.1) and further
build the blowup criterion (Proposition 4.2). Then we adopt the nonlocal
maximum principle method of Kiselev-Nazarov-Volberg and finally manage to
remove all the possible breakdown scenarios by constructing suitable moduli of
continuity.
###### Remark 1.1.
The main new ingredients in the global existence part are two suitable moduli
of continuity, with their explicit formulae (5.2) and (5.3), which correspond
to the case $\alpha\in]0,1[$ and case $\alpha\in]1,2[$ respectively and are
extensions to the one in [17] with $\alpha=1$.
###### Remark 1.2.
The modulus of continuity (5.3) turns out to be a bounded one (i.e. as
$\xi\rightarrow\infty$, $\omega(\xi)<\infty$), and this is the only reason why
we introduce the smallness condition (1.2). In order to construct a more
efficient modulus of continuity, one has to truly improve the bound on the
positive term or the negative term in the case $\mathrm{II}.2$, and this does
not seem to be an easy task.
The paper is organized as follows. In Section 2, we present some preparatory
results. In Section 3, some facts about modulus of continuity are discussed.
In Section 4, we obtain the local results and establish blowup criterion.
Finally, we prove the global existence in Section 5.
## 2 Preliminaries
In this preparatory section, we present the definitions and some related
results of the Sobolev spaces and the Besov spaces, also we provide some
important estimates which will be used later.
We begin with introducing some notations.
$\diamond$ Throughout this paper $C$ stands for a constant which may be
different from line to line. We sometimes use $A\lesssim B$ instead of $A\leq
CB$, and use $A\lesssim_{\beta,\gamma\cdots}B$ instead of $A\leq
C(\beta,\gamma,\cdots)B$ with $C(\beta,\gamma,\cdots)$ a constant depending on
$\beta,\gamma,\cdots$. For $A\thickapprox B$ we mean $A\lesssim B\lesssim A$.
$\diamond$ Denote by $\mathcal{S}(\mathbb{R}^{n})$ the Schwartz space of
rapidly decreasing smooth functions, $\mathcal{S}^{\prime}(\mathbb{R}^{n})$
the space of tempered distributions,
$\mathcal{S}^{\prime}(\mathbb{R}^{n})/\mathcal{P}(\mathbb{R}^{n})$ the
quotient space of tempered distributions which modulo polynomials.
$\diamond$ $\mathcal{F}f$ or $\hat{f}$ denotes the Fourier transform, that is
$\mathcal{F}f(\zeta)=\hat{f}(\zeta)=\int_{\mathbb{R}^{n}}e^{-ix\cdot\zeta}f(x)\textrm{d}x,$
while $\mathcal{F}^{-1}f$ the inverse Fourier transform, namely,
$\mathcal{F}^{-1}f(x)=(2\pi)^{-n}\int_{\mathbb{R}^{n}}e^{ix\cdot\zeta}f(\zeta)\textrm{d}\zeta$.
Now we give the definition of $L^{2}$ based Sobolev space. For
$s\in\mathbb{R}$, the inhomogeneous Sobolev space
$H^{s}:=\Big{\\{}f\in\mathcal{S}^{\prime}(\mathbb{R}^{n});\left\|f\right\|^{2}_{H^{s}}:=\int_{\mathbb{R}^{n}}(1+|\zeta|^{2})^{s}|\hat{f}(\zeta)|^{2}\textrm{d}\zeta<\infty\Big{\\}}$
Also one can define the corresponding homogeneous space:
$\dot{H}^{s}:=\Big{\\{}f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})/\mathcal{P}(\mathbb{R}^{n});\left\|f\right\|^{2}_{\dot{H}^{s}}:=\int_{\mathbb{R}^{n}}|\zeta|^{2s}|\hat{f}(\zeta)|^{2}\textrm{d}\zeta<\infty\Big{\\}}$
The following calculus inequality is well-known(see [2])
###### Lemma 2.1.
$\forall m\in\mathbb{R}^{+}$, there exists a constant $c_{m}>0$ such that
$\left\|fg\right\|_{H^{m}}\leq
c_{m}\big{(}\left\|f\right\|_{L^{\infty}}\left\|g\right\|_{H^{m}}+\left\|f\right\|_{H^{m}}\left\|g\right\|_{L^{\infty}}\big{)}.$
(2.1)
To define Besov space we need the following dyadic unity partition (see e.g.
[5]). Choose two nonnegative radial functions $\chi$,
$\varphi\in\mathcal{D}(\mathbb{R}^{n})$ be supported respectively in the ball
$\\{\zeta\in\mathbb{R}^{n}:|\zeta|\leq\frac{4}{3}\\}$ and the shell
$\\{\zeta\in\mathbb{R}^{n}:\frac{3}{4}\leq|\zeta|\leq\frac{8}{3}\\}$ such that
$\chi(\zeta)+\sum_{j\geq
0}\varphi(2^{-j}\zeta)=1,\quad\forall\zeta\in\mathbb{R}^{n};\qquad\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\zeta)=1,\quad\forall\zeta\neq
0.$
For all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ we define the
nonhomogeneous Littlewood-Paley operators
$\Delta_{-1}f:=\chi(D)f;\;\;\Delta_{j}f:=\varphi(2^{-j}D)f,\;S_{j}f:=\sum_{-1\leq
k\leq j-1}\Delta_{k}f,\quad\forall j\in\mathbb{N},$
And the homogeneous Littlewood-Paley operators can be defined as follows
$\dot{\Delta}_{j}f:=\varphi(2^{-j}D)f;\;\dot{S}_{j}f:=\sum_{k\in\mathbb{Z},k\leq
j-1}\dot{\Delta}_{k},f\quad\forall j\in\mathbb{Z}.\quad$
Now we introduce the definition of Besov spaces . Let
$(p,r)\in[1,\infty]^{2}$, $s\in\mathbb{R}$, the nonhomogeneous Besov space
$B^{s}_{p,r}:=\Big{\\{}f\in\mathcal{S}^{\prime}(\mathbb{R}^{n});\left\|f\right\|_{B^{s}_{p,r}}:=\left\|2^{js}\left\|\Delta_{j}f\right\|_{L^{p}}\right\|_{\ell^{r}}<\infty\Big{\\}}$
and the homogeneous space
$\dot{B}^{s}_{p,r}:=\Big{\\{}f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})/\mathcal{P}(\mathbb{R}^{n});\left\|f\right\|_{\dot{B}^{s}_{p,r}}:=\left\|2^{js}\left\|\dot{\Delta}_{j}f\right\|_{L^{p}}\right\|_{\ell^{r}(\mathbb{Z})}<\infty\Big{\\}}.$
We point out that for all $s\in\mathbb{R}$, $B^{s}_{2,2}=H^{s}$ and
$\dot{B}^{s}_{2,2}=\dot{H}^{s}$.
The classical space-time Besov space $L^{\rho}([0,T],B^{s}_{p,r})$,
abbreviated by $L^{\rho}_{T}B^{s}_{p,r}$, is the set of tempered distribution
$f$ such that
$\left\|f\right\|_{L^{\rho}_{T}B^{s}_{p,r}}:=\left\|\left\|2^{js}\left\|\Delta_{j}f\right\|_{L^{p}}\right\|_{\ell^{r}}\right\|_{L^{\rho}([0,T])}<\infty.$
We can similarly extend to the homogeneous one
$L^{\rho}_{T}\dot{B}^{s}_{p,r}$.
Bernstein’s inequality is fundamental in the analysis involving Besov spaces
(see [5])
###### Lemma 2.2.
Let $f\in L^{a}$, $1\leq a\leq b\leq\infty$. Then for every
$(k,q)\in\mathbb{N}^{2}$ there exists a constant $C>0$ such that
$\sup_{|\alpha|=k}\left\|\partial^{\alpha}S_{q}f\right\|_{L^{b}}\leq
C2^{q(k+n(\frac{1}{a}-\frac{1}{b}))}\left\|f\right\|_{L^{a}},$
$C^{-1}2^{qk}\left\|f\right\|_{L^{a}}\leq\sup_{|\alpha|=k}\left\|\partial^{\alpha}\Delta_{q}f\right\|_{L^{a}}\leq
C2^{qk}\left\|f\right\|_{L^{a}}$
Finally we state an important maximum principle for the transport-diffusion
equation (cf. [11])
###### Proposition 2.3.
Let $u$ be a smooth divergence-free vector field and $f$ be a smooth function.
Assume that $\theta$ is the smooth solution of the equation
$\partial_{t}\theta+u\cdot\nabla\theta+\nu|D|^{\alpha}\theta=f,\quad\mathrm{div}u=0,$
with initial datum $\theta_{0}$ and $\nu\geq 0$, $0\leq\alpha\leq 2$, then for
every $p\in[1,\infty]$ we have
$\left\|\theta(t)\right\|_{L^{p}}\leq\left\|\theta_{0}\right\|_{L^{p}}+\int^{t}_{0}\left\|f(\tau)\right\|_{L^{p}}\,\textrm{d}\tau.$
(2.2)
## 3 Moduli of Continuity
In this section, we discuss the moduli of continuity which play a key role in
our global existence part.
We suppose that $\omega$ is a modulus of continuity, that is, a continuous,
increasing, concave function on $[0,\infty)$ such that $\omega(0)=0$. We say
that a function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ has modulus of
continuity if $|f(x)-f(y)|\leq\omega(|x-y|)$ for all $x,y\in\mathbb{R}^{n}$
and that $f$ has strict modulus of continuity if the inequality is strict for
$x\neq y$.
Next we introduce the pseudo-differential operators $\mathcal{R}_{\alpha,j}$
which may be termed as the modified Riesz transforms
###### Proposition 3.1.
Let $\alpha\in]0,2[$, $1\leq j\leq n$, $n\geq 2$, then for every
$f\in\mathcal{S}(\mathbb{R}^{n})$
$\mathcal{R}_{\alpha,j}f(x)=|D|^{\alpha-1}\mathcal{R}_{j}f(x)=c_{\alpha,n}\mathrm{p.v.}\int_{\mathbb{R}^{n}}\frac{y_{j}}{|y|^{n+\alpha}}f(x-y)\,\textrm{d}y,$
(3.1)
where $c_{\alpha,n}$ is the normalization constant such that
$\widehat{\mathcal{R}_{\alpha,j}f}(\zeta)=-i\frac{\zeta_{j}}{|\zeta|^{2-\alpha}}\hat{f}(\zeta).$
The proof is placed in the appendix. Also note that when $\alpha\in]0,1[$, we
do not need to introduce the principle value of integral expression in the
formula (3.1).
The pseudo-differential operators like the modified Riesz transforms do not
preserve the moduli of continuity generally but they do not destroy them too
much either. More precisely, we have
###### Lemma 3.2.
If the function $\theta$ has the modulus of continuity $\omega$, then
$u=(-\mathcal{R}_{\alpha,2}\theta,\mathcal{R}_{\alpha,1}\theta)$
($\alpha\in]0,2[$) has the modulus of continuity
$\Omega(\xi)=A_{\alpha}\bigg{(}\int^{\xi}_{0}\frac{\omega(\eta)}{\eta^{\alpha}}\textrm{d}\eta+\xi\int_{\xi}^{\infty}\frac{\omega(\eta)}{\eta^{1+\alpha}}\textrm{d}\eta\bigg{)}$
(3.2)
with some absolute constant $A_{\alpha}>0$ depending only on $\alpha$.
###### Proof.
The modified Riesz transforms are pseudo-differential operators with kernels
$K(x)=\frac{S(x^{\prime})}{|x|^{n-1+\alpha}}$ (in our special case, $n=2$ and
$S(x^{\prime})=\frac{x_{j}}{|x|},j=1,2$), where
$x^{\prime}=\frac{x}{|x|}\in\mathbb{S}^{n-1}$. The function $S\in
C^{1}(\mathbb{S}^{n-1})$ and
$\int_{\mathbb{S}^{n-1}}S(x^{\prime})d\sigma(x^{\prime})=0$. Assume that the
function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ has some modulus of
continuity $\omega$, that is $|f(x)-f(y)|\leq\omega(|x-y|)$ for all
$x,y\in\mathbb{R}^{n}$. Then take any $x,y$ with $|x-y|=\xi$, and consider the
difference
$\int K(x-t)f(t)\textrm{d}t-\int K(y-t)f(t)\textrm{d}t.$ (3.3)
First due to the canceling property of $S$ we have
$\bigg{|}\int_{|x-t|\leq
2\xi}K(x-t)f(t)\textrm{d}t\bigg{|}=\bigg{|}\int_{|x-t|\leq
2\xi}K(x-t)(f(t)-f(x))\textrm{d}t\bigg{|}\leq
C\int_{0}^{2\xi}\frac{\omega(r)}{r^{\alpha}}\textrm{d}r$
since $\omega$ is concave, we obtain
$\int_{0}^{2\xi}\frac{\omega(r)}{r^{\alpha}}\textrm{d}r\leq
2^{2-\alpha}\int_{0}^{\xi}\frac{\omega(r)}{r^{\alpha}}\textrm{d}r$ (3.4)
A similar estimate holds for the second integral in (3.3). Next, set
$z=\frac{x+y}{2}$, then
$\begin{split}&\bigg{|}\int_{|x-t|\geq
2\xi}K(x-t)f(t)\textrm{d}t-\int_{|y-t|\geq
2\xi}K(y-t)f(t)\textrm{d}t\bigg{|}\\\ &=\bigg{|}\int_{|x-t|\geq
2\xi}K(x-t)(f(t)-f(z))\textrm{d}t-\int_{|y-t|\geq
2\xi}K(y-t)(f(t)-f(z))\textrm{d}t\bigg{|}\\\ &\leq\int_{|z-t|\geq
3\xi}|K(x-t)-K(y-t)||f(t)-f(z)|\textrm{d}t\\\
&\quad+\int_{\frac{3\xi}{2}\leq|z-t|\leq
3\xi}(|K(x-t)|+|K(y-t)|)|f(t)-f(z)|\textrm{d}t\\\ &=I_{1}+I_{2}\end{split}$
To estimate the first integral, we use the smoothness condition of $S$ to get
$|K(x-t)-K(y-t)|\leq
C\frac{|x-y|}{|z-t|^{n+\alpha}}\quad\text{when}\,|z-t|\geq 3\xi$
thus
$I_{1}\leq
C\xi\int_{3\xi}^{\infty}\frac{\omega(r)}{r^{1+\alpha}}\textrm{d}r\leq
C3^{-\alpha}\xi\int_{\xi}^{\infty}\frac{\omega(3r)}{r^{1+\alpha}}\textrm{d}r\leq
C\xi\int_{\xi}^{\infty}\frac{\omega(r)}{r^{1+\alpha}}\textrm{d}r$
For the second integral, using the concavity of $\omega$ and (3.4), we have
$\begin{split}I_{2}\leq&2C\omega(3\xi)\xi^{1-\alpha}\int_{\xi\leq|x-t|\leq\frac{7}{2}\xi}\frac{1}{|x-t|^{n}}\textrm{d}t\\\
\leq&C\omega(\xi)\xi^{1-\alpha}\leq
C2^{\alpha}\int_{\xi}^{2\xi}\frac{\omega(r)}{r^{\alpha}}\textrm{d}r\leq
C\int_{0}^{\xi}\frac{\omega(r)}{r^{\alpha}}\textrm{d}r\end{split}$
∎
Now we consider the action of the fractional differential operators
$|D|^{\alpha}$($0<\alpha<2$) on the function having modulus of continuity.
Precisely,
###### Lemma 3.3.
If the function $\theta:\mathbb{R}^{2}\rightarrow\mathbb{R}$ has modulus of
continuity $\omega$, and especially satisfies
$\theta(x)-\theta(y)=\omega(\xi)$ at some $x,y\in\mathbb{R}^{2}$ with
$|x-y|=\xi>0$, then we have
$\begin{split}\bigl{[}(-|D|^{\alpha})\theta\bigr{]}(x)-\bigl{[}(-|D|^{\alpha})\theta\bigr{]}(y)\leq
B_{\alpha}&\int_{0}^{\frac{\xi}{2}}\frac{\omega(\xi+2\eta)+\omega(\xi-2\eta)-2\omega(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta\\\
+&B_{\alpha}\int_{\frac{\xi}{2}}^{\infty}\frac{\omega(2\eta+\xi)-\omega(2\eta-\xi)-2\omega(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta\end{split}$
(3.5)
where $B_{\alpha}>0$ is an absolute constant.
###### Remark 3.1.
In fact this result has occurred in [23], as a generalization of the one in
[17]. For convenience, we prove it again for the general $n$-dimensional case
and place the proof in the appendix. Also note that due to concavity of
$\omega$ both terms on the righthand side of (3.5) are strictly negative.
## 4 Local existence and Blowup criterion
Our purpose in this section is to prove the following local result
###### Proposition 4.1.
Let $\nu>0$, $0<\alpha<2$ and the initial data $\theta_{0}\in H^{m}$, $m>2$.
Then there exists a positive $T$ depending only on $\alpha$, $\nu$ and
$\left\|\theta_{0}\right\|_{H^{m}}$ such that the modified quasi-geostrophic
equation (1.1) generates a unique solution
$\theta\in\mathcal{C}([0,T],H^{m})\cap L^{2}([0,T],H^{m+\frac{\alpha}{2}})$.
Moreover we have $t^{\gamma}\theta\in L^{\infty}(]0,T],H^{m+\gamma\alpha})$
for all $\gamma\geq 0$, which implies
$\theta\in\mathcal{C}^{\infty}(]0,T]\times\mathbb{R}^{2})$.
We further obtain the following criterion for the breakdown of smooth
solutions
###### Proposition 4.2.
Let $T^{*}$ be the maximal existence time of $\theta$ in
$\mathcal{C}([0,T^{*}),H^{m})\cap L^{2}([0,T^{*}),H^{m+\frac{\alpha}{2}})$. If
$T^{*}<\infty$ then we necessarily have
$\int_{0}^{T^{*}}\left\|\nabla\theta(t,\cdot)\right\|_{L^{\infty}}^{\alpha}\textrm{d}t=\infty.$
(4.1)
The method of proof for the Proposition 4.1 is to regularize the equation
(1.1) by the standard Friedrich method, and then pass to the limit for the
regularization parameter.
Denote the frequency cutoff operator
$\mathcal{J}_{\epsilon}:L^{2}(\mathbb{R}^{2})\rightarrow
H^{m}(\mathbb{R}^{2})$, $\epsilon>0$, $m\geq 0$ by
$(\mathcal{J}_{\epsilon}f)(x)=\mathcal{F}^{-1}(\hat{f}(\cdot)1_{B_{1/\epsilon}}(\cdot))(x)=(2\pi)^{-2}\int_{\mathbb{R}^{2}}e^{ix\cdot\zeta}\hat{f}(\zeta)1_{\\{|\cdot|\leq\frac{1}{\epsilon}\\}}(\zeta)\mathrm{d}\zeta.$
The following properties of $\mathcal{J}_{\epsilon}$ are obvious.
###### Lemma 4.3.
Let $\mathcal{J}_{\epsilon}$ be the projection operator defined as above,
$m\in\mathbb{R}^{+}$, $k\in\mathbb{R}^{+}$, $\delta\in[0,m[$. Then
1. (i)
for all $f\in H^{m}$, $\lim_{\epsilon\rightarrow
0}\left\|\mathcal{J}_{\epsilon}f-f\right\|_{H^{m}}=0$.
2. (ii)
for all $f\in H^{m}$,
$|D|^{m}(\mathcal{J}_{\epsilon}f)=\mathcal{J}_{\epsilon}(|D|^{m}f)$ and
$\Delta_{j}(\mathcal{J}_{\epsilon}f)=\mathcal{J}_{\epsilon}(\Delta_{j}f)$.
3. (iii)
for all $f\in H^{m}$,
$\left\|\mathcal{J}_{\epsilon}f-f\right\|_{H^{m-\delta}}\lesssim\epsilon^{\delta}\left\|f\right\|_{H^{m}}$
and
$\left\|\mathcal{J}_{\epsilon}f\right\|_{H^{m+k}}\lesssim\frac{1}{\epsilon^{k}}\left\|f\right\|_{H^{m}}$.
Then we regularize the modified quasi-geostrophic equation (1.1) as follows
$\begin{cases}\begin{split}&\theta^{\epsilon}_{t}+\mathcal{J}_{\epsilon}\big{(}(\mathcal{J}_{\epsilon}u^{\epsilon})\cdot\nabla(\mathcal{J}_{\epsilon}\theta^{\epsilon})\big{)}+\nu\mathcal{J}_{\epsilon}|D|^{\alpha}\theta^{\epsilon}=0\\\
&u^{\epsilon}=|D|^{\alpha-1}\mathcal{R}^{\perp}\theta^{\epsilon},\quad\theta^{\epsilon}|_{t=0}=\mathcal{J}_{\epsilon}\theta_{0}.\end{split}\end{cases}$
(4.2)
For this approximate system (ODE), we can use the standard Cauchy-Lipschitz
argument combined with $L^{2}$ energy estimate to get
###### Proposition 4.4.
Let the initial data $\theta_{0}\in L^{2}$. Then for any $\epsilon>0$ there
exists a unique global solution
$\theta^{\epsilon}\in\mathcal{C}^{1}([0,\infty),H^{\infty})$ to the
regularized equation (4.2).
###### Remark 4.1.
From the proof we know
$\theta^{\epsilon}=\mathcal{J}_{\epsilon}\theta^{\epsilon}$, thus (4.2) will
be written as follows
$\begin{cases}\begin{split}&\theta^{\epsilon}_{t}+\mathcal{J}_{\epsilon}(u^{\epsilon}\cdot\nabla\theta^{\epsilon})+\nu|D|^{\alpha}\theta^{\epsilon}=0\\\
&u^{\epsilon}=|D|^{\alpha-1}\mathcal{R}^{\perp}\theta^{\epsilon},\quad\theta^{\epsilon}|_{t=0}=\mathcal{J}_{\epsilon}\theta_{0}.\end{split}\end{cases}$
(4.3)
In the sequel we shall instead consider this form.
Next, we prove the main result in this section.
###### Proof of Proposition 4.1.
Step 1: Uniform Bounds.
We claim that: the regularized solution
$\theta^{\epsilon}\in\mathcal{C}^{1}([0,\infty),H^{\infty})$ to equation (4.2)
satisfies
$\frac{d}{2dt}\left\|\theta^{\epsilon}\right\|_{B^{m}_{2,2}}^{2}+\frac{\nu}{2}\left\||D|^{\frac{\alpha}{2}}\theta^{\epsilon}\right\|_{B^{m}_{2,2}}^{2}\lesssim_{\nu,\alpha}\frac{1}{\nu}\left\|\nabla\theta^{\epsilon}\right\|_{L^{\infty}}^{\alpha}\left\|\theta^{\epsilon}\right\|_{L^{\infty}}^{2-\alpha}\left\|\theta^{\epsilon}\right\|_{B^{m}_{2,2}}^{2}+\left\|\theta^{\epsilon}\right\|_{L^{2}}^{2}\left\|\theta^{\epsilon}\right\|_{B^{m}_{2,2}}.$
(4.4)
Indeed, for every $q\in\mathbb{N}$, applying dyadic operator $\Delta_{q}$ to
both sides of regularized equation (4.3) yields
$\partial_{t}\Delta_{q}\theta^{\epsilon}+\mathcal{J}_{\epsilon}\big{(}(S_{q+1}u^{\epsilon})\cdot\nabla\Delta_{q}\theta^{\epsilon}\big{)}+\nu|D|^{\alpha}\Delta_{q}\theta^{\epsilon}=\mathcal{J}_{\epsilon}\big{(}F_{q}(u^{\epsilon},\theta^{\epsilon})\big{)},$
where
$F_{q}(u^{\epsilon},\theta^{\epsilon})=(S_{q+1}u^{\epsilon})\cdot\nabla\Delta_{q}\theta^{\epsilon}-\Delta_{q}(u^{\epsilon}\cdot\nabla\theta^{\epsilon}).$
Taking the $L^{2}$ inner product in the above equality with
$\Delta_{q}\theta^{\epsilon}$ and using the divergence free property, we have
$\begin{split}\frac{1}{2}\frac{d}{dt}\left\|\Delta_{q}\theta^{\epsilon}\right\|_{L^{2}}^{2}+\nu\left\||D|^{\frac{\alpha}{2}}\Delta_{q}\theta^{\epsilon}\right\|_{L^{2}}^{2}&\leq\Big{|}\int_{\mathbb{R}^{2}}\big{(}F_{q}(u^{\epsilon},\theta^{\epsilon})\big{)}(x)\mathcal{J}_{\epsilon}\Delta_{q}\theta^{\epsilon}(x)\mathrm{d}x\Big{|}\\\
&\leq
2^{-q\frac{\alpha}{2}}\left\|F_{q}(u^{\epsilon},\theta^{\epsilon})\right\|_{L^{2}}2^{q\frac{\alpha}{2}}\left\|\mathcal{J}_{\epsilon}\Delta_{q}\theta^{\epsilon}\right\|_{L^{2}}\\\
&\lesssim
2^{-q\frac{\alpha}{2}}\left\|F_{q}(u^{\epsilon},\theta^{\epsilon})\right\|_{L^{2}}\left\||D|^{\frac{\alpha}{2}}\Delta_{q}\theta^{\epsilon}\right\|_{L^{2}}.\end{split}$
Then by virtue of Young inequality, we deduce
$\frac{1}{2}\frac{d}{dt}\left\|\Delta_{q}\theta^{\epsilon}\right\|_{L^{2}}^{2}+\frac{\nu}{2}\left\||D|^{\frac{\alpha}{2}}\Delta_{q}\theta^{\epsilon}\right\|_{L^{2}}^{2}\leq\frac{C_{0}}{\nu}\Big{(}2^{-q\frac{\alpha}{2}}\left\|F_{q}(u^{\epsilon},\theta^{\epsilon})\right\|_{L^{2}}\Big{)}^{2}.$
(4.5)
From the inequality (6.2) in the appendix, we know that
$\begin{split}&2^{-q\frac{\alpha}{2}}\left\|F_{q}(u^{\epsilon},\theta^{\epsilon})\right\|_{L^{2}}\\\
\lesssim&\left\||D|^{1-\frac{\alpha}{2}}u^{\epsilon}\right\|_{L^{\infty}}\sum_{q^{\prime}\geq
q-4}2^{(q-q^{\prime})(1-\frac{\alpha}{2})}\left\|\Delta_{q^{\prime}}\theta^{\epsilon}\right\|_{L^{2}}+\left\||D|^{\frac{\alpha}{2}}\theta^{\epsilon}\right\|_{L^{\infty}}\sum_{|q^{\prime}-q|\leq
4}\left\|\Delta_{q^{\prime}}\theta^{\epsilon}\right\|_{L^{2}}\end{split}$
(4.6)
Also notice that for some number $K\in\mathbb{N}$
$\begin{split}\left\||D|^{1-\frac{\alpha}{2}}u^{\epsilon}\right\|_{L^{\infty}}+\left\||D|^{\frac{\alpha}{2}}\theta^{\epsilon}\right\|_{L^{\infty}}&\lesssim\left\||D|^{1-\frac{\alpha}{2}}|D|^{\alpha-1}\mathcal{R}^{\bot}\theta^{\epsilon}\right\|_{\dot{B}^{0}_{\infty,1}}+\left\||D|^{\frac{\alpha}{2}}\theta^{\epsilon}\right\|_{\dot{B}^{0}_{\infty,1}}\\\
&\lesssim\sum_{k=-\infty}^{K-1}2^{k\alpha/2}\left\|\dot{\Delta}_{k}\theta^{\epsilon}\right\|_{L^{\infty}}+\sum_{k=K}^{\infty}2^{-k(1-\frac{\alpha}{2})}\left\|\dot{\Delta}_{k}\nabla\theta^{\epsilon}\right\|_{L^{\infty}}\\\
&\lesssim
2^{K\alpha/2}\left\|\theta^{\epsilon}\right\|_{L^{\infty}}+2^{K(\frac{\alpha}{2}-1)}\left\|\nabla\theta^{\epsilon}\right\|_{L^{\infty}},\end{split}$
thus choosing $K$ satisfying
$\left\|\theta^{\epsilon}\right\|_{L^{\infty}}2^{K}\thickapprox\left\|\nabla\theta^{\epsilon}\right\|_{L^{\infty}}$,
we deduce
$\left\||D|^{1-\frac{\alpha}{2}}u^{\epsilon}\right\|_{L^{\infty}}+\left\||D|^{\frac{\alpha}{2}}\theta^{\epsilon}\right\|_{L^{\infty}}\lesssim\left\|\nabla\theta^{\epsilon}\right\|_{L^{\infty}}^{\frac{\alpha}{2}}\left\|\theta^{\epsilon}\right\|_{L^{\infty}}^{1-\frac{\alpha}{2}}.$
(4.7)
Plunging the above two estimates (4.7) and (4.6) into inequality (4.5), then
multiplying both sides by $2^{2qm}$ and summing up over $q\in\mathbb{N}$, we
obtain
$\frac{1}{2}\frac{d}{dt}\sum_{q\in\mathbb{N}}2^{2qm}\left\|\Delta_{q}\theta^{\epsilon}\right\|_{L^{2}}^{2}+\frac{\nu}{2}\sum_{q\in\mathbb{N}}2^{2qm}\left\||D|^{\frac{\alpha}{2}}\Delta_{q}\theta^{\epsilon}\right\|_{L^{2}}^{2}\lesssim\frac{1}{\nu}\left\|\nabla\theta^{\epsilon}\right\|_{L^{\infty}}^{\alpha}\left\|\theta^{\epsilon}\right\|_{L^{\infty}}^{2-\alpha}\left\|\theta^{\epsilon}\right\|_{B^{m}_{2,2}}^{2}.$
(4.8)
On the other hand, we apply the low frequency operator $\Delta_{-1}$ to the
regularized system (4.2) to get
$\partial_{t}\Delta_{-1}\theta^{\epsilon}+\nu|D|^{\alpha}\Delta_{-1}\theta^{\epsilon}=-\mathcal{J}_{\epsilon}\Delta_{-1}\big{(}u^{\epsilon}\cdot\nabla\theta^{\epsilon}\big{)}.$
Multiplying both sides by $\Delta_{-1}\theta^{\epsilon}$ and integrating in
the spatial variable, we obtain
$\begin{split}\frac{1}{2}\frac{d}{dt}\left\|\Delta_{-1}\theta^{\epsilon}\right\|_{L^{2}}^{2}+\nu\left\||D|^{\frac{\alpha}{2}}\Delta_{-1}\theta^{\epsilon}\right\|_{L^{2}}^{2}&\leq\Big{|}\int_{\mathbb{R}^{2}}\mathrm{div}\Delta_{-1}\big{(}u^{\epsilon}\,\theta^{\epsilon}\big{)}(x)\,\Delta_{-1}\mathcal{J}_{\epsilon}\theta^{\epsilon}(x)\mathrm{d}x\Big{|}\\\
&\lesssim\left\|u^{\epsilon}\right\|_{L^{\infty}}\left\|\theta^{\epsilon}\right\|_{L^{2}}^{2}.\end{split}$
We see that
$\begin{split}\left\|u^{\epsilon}\right\|_{L^{\infty}}&\leq\Big{(}\sum_{j\leq-1}+\sum_{j\geq
0}\Big{)}\left\|\dot{\Delta}_{j}|D|^{\alpha-1}\mathcal{R}^{\bot}\theta^{\epsilon}\right\|_{L^{\infty}}\\\
&\lesssim\sum_{j\leq-1}2^{j\alpha}\left\|\dot{\Delta}_{j}\theta^{\epsilon}\right\|_{L^{2}}+\sum_{j\geq
0}2^{j(\alpha-2)}\left\|\dot{\Delta}_{j}\nabla\theta^{\epsilon}\right\|_{L^{\infty}}\\\
&\lesssim\left\|\theta^{\epsilon}\right\|_{L^{2}}+\left\|\nabla\theta^{\epsilon}\right\|_{L^{\infty}},\end{split}$
(4.9)
thus we have
$\frac{1}{2}\frac{d}{dt}\left\|\Delta_{-1}\theta^{\epsilon}\right\|_{L^{2}}^{2}+\frac{\nu}{2}\left\||D|^{\frac{\alpha}{2}}\Delta_{-1}\theta^{\epsilon}\right\|_{L^{2}}^{2}\lesssim\left\|\theta^{\epsilon}\right\|_{B^{m}_{2,2}}\left\|\theta^{\epsilon}\right\|_{L^{2}}^{2}.$
(4.10)
Multiplying (4.10) by $2^{-2m}$ and combining it with (4.8) leads to (4.4).
Next, we prove that the solution family $(\theta^{\epsilon})$ is uniformly
bounded in $H^{m}$. Indeed, from estimate (4.4), Besov embedding and the fact
that
$\left\|\cdot\right\|^{2}_{B^{m}_{2,2}}/C_{0}\leq\left\|\cdot\right\|^{2}_{H^{m}}\leq
C_{0}\left\|\cdot\right\|^{2}_{B^{m}_{2,2}}$ with $C_{0}$ a universal number,
we have
$\begin{split}\frac{d}{dt}\Bigl{(}\left\|\theta^{\epsilon}(t)\right\|^{2}_{H^{m}}+\int_{0}^{t}\|\theta^{\epsilon}(\tau)\|^{2}_{H^{m+\frac{\alpha}{2}}}\mathrm{d}\tau\Bigr{)}&\leq
C\Big{(}\left\|\nabla\theta^{\epsilon}\right\|_{L^{\infty}}^{\alpha}\left\|\theta^{\epsilon}\right\|_{L^{\infty}}^{2-\alpha}\left\|\theta^{\epsilon}\right\|_{H^{m}}+\left\|\theta^{\epsilon}\right\|_{L^{2}}\Big{)}\left\|\theta^{\epsilon}\right\|_{H^{m}}^{2}\\\
&\leq
C_{1}(1+\left\|\theta^{\epsilon}(t)\right\|_{H^{m}}^{2})\left\|\theta^{\epsilon}(t)\right\|_{H^{m}}^{2},\end{split}$
(4.11)
where $C_{1}$ depends only on $m,\alpha,\nu$. Gronwall inequality yields that
$\sup_{0\leq t\leq
T}\left\|\theta^{\epsilon}\right\|_{H^{m}}^{2}+\|\theta^{\epsilon}\|_{L^{2}_{T}H^{m+\frac{\alpha}{2}}}^{2}\leq\frac{\left\|\theta_{0}\right\|_{H^{m}}^{2}}{(\|\theta_{0}\|_{H^{m}}^{2}+1)e^{-CT}-\|\theta_{0}\|_{H^{m}}^{2}}.$
(4.12)
Thus for some
$T<\frac{1}{C}\log(1+1/\|\theta_{0}\|_{H^{m}}^{2}),$
the family $(\theta^{\epsilon})$ is uniformly bounded in
$\mathcal{C}([0,T],H^{m})\cap L^{2}([0,T];H^{m+\frac{\alpha}{2}})$, $m>2$.
Step 2: Strong Convergence
We firstly claim that the solutions $(\theta^{\epsilon})$ to the approximate
equation (4.3) converge in $\mathcal{C}([0,T],L^{2}(\mathbb{R}^{2}))$. Indeed
for all $0<\tilde{\epsilon}<\epsilon$, we set that $\theta^{\epsilon}$ and
$\theta^{\tilde{\epsilon}}$ are two approximate solutions, then from a direct
calculation
$\begin{split}\big{(}\theta^{\epsilon}_{t}-\theta^{\tilde{\epsilon}}_{t},\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\big{)}=-\nu\big{(}|D|^{\alpha}\theta^{\epsilon}-|D|^{\alpha}\theta^{\tilde{\epsilon}},\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\big{)}-\Big{(}\big{(}\mathcal{J}_{\epsilon}(u^{\epsilon}\cdot\nabla\theta^{\epsilon})-\mathcal{J}_{\tilde{\epsilon}}(u^{\tilde{\epsilon}}\cdot\nabla\theta^{\tilde{\epsilon}})\big{)},\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\Big{)},\end{split}$
we have
$\begin{split}&\frac{1}{2}\frac{d}{dt}\left\|\theta^{\epsilon}(t)-\theta^{\tilde{\epsilon}}(t)\right\|_{L^{2}}^{2}+\nu\left\||D|^{\frac{\alpha}{2}}(\theta^{\epsilon}-\theta^{\tilde{\epsilon}})\right\|_{L^{2}}^{2}\\\
=&\Big{(}(\mathcal{J}_{\epsilon}-\mathcal{J}_{\tilde{\epsilon}})\big{(}u^{\epsilon}\cdot\nabla\theta^{\epsilon}\big{)},\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\Big{)}+\Big{(}\mathcal{J}_{\tilde{\epsilon}}\big{(}(u^{\epsilon}-u^{\tilde{\epsilon}})\cdot\nabla\theta^{\epsilon}\big{)},\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\Big{)}\\\
&+\Big{(}\mathcal{J}_{\tilde{\epsilon}}\big{(}u^{\tilde{\epsilon}}\cdot\nabla(\theta^{\epsilon}-\theta^{\tilde{\epsilon}})\big{)},\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\Big{)}\\\
:=&II_{1}+II_{2}+II_{3}.\end{split}$
We set $\delta_{0}:=\min\\{m-\alpha,1\\}$, then for $II_{1}$, by means of the
calculus inequality (2.1), divergence free condition and the following simple
inequality
$\left\|u^{\epsilon}\right\|_{H^{m-\alpha+1}}=\left\||D|^{\alpha-1}R^{\bot}\theta^{\epsilon}\right\|_{H^{m-\alpha+1}}\lesssim\left\|\theta^{\epsilon}\right\|_{H^{m}}\lesssim
M,$
we have
$\begin{split}|II_{1}|&\lesssim\epsilon^{\delta_{0}}\left\|u^{\epsilon}\theta^{\epsilon}\right\|_{H^{1+\delta_{0}}}\left\|\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\right\|_{L^{2}}\\\
&\lesssim\epsilon^{\delta_{0}}\big{(}\left\|u^{\epsilon}\right\|_{H^{1+\delta_{0}}}+\left\|\theta^{\epsilon}\right\|_{H^{1+\delta_{0}}}\big{)}\left\|\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\right\|_{L^{2}}\\\
&\lesssim_{M}\epsilon^{\delta_{0}}\left\|\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\right\|_{L^{2}}.\end{split}$
For $II_{2}$, we directly obtain
$\begin{split}|II_{2}|&\leq\left\|(u^{\epsilon}-u^{\tilde{\epsilon}})\cdot\nabla\theta^{\epsilon}\right\|_{\dot{H}^{-\frac{\alpha}{2}}}\left\||D|^{\frac{\alpha}{2}}(\theta^{\epsilon}-\theta^{\tilde{\epsilon}})\right\|_{L^{2}}\\\
&\leq
C_{\alpha}\left\||D|^{\alpha-1}\mathcal{R}^{\bot}(\theta^{\epsilon}-\theta^{\tilde{\epsilon}})\right\|_{\dot{H}^{1-\alpha}}^{2}\left\|\nabla\theta^{\epsilon}\right\|_{\dot{H}^{\frac{\alpha}{2}}}^{2}+\frac{\nu}{2}\left\||D|^{\frac{\alpha}{2}}(\theta^{\epsilon}-\theta^{\tilde{\epsilon}})\right\|_{L^{2}}^{2}\\\
&\leq
C_{M,\alpha}\left\|\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\right\|_{L^{2}}^{2}+\frac{\nu}{2}\left\||D|^{\frac{\alpha}{2}}(\theta^{\epsilon}-\theta^{\tilde{\epsilon}})\right\|_{L^{2}}^{2},\end{split}$
where in the second line we have used the classical product estimate (cf.
[14]) that for every $s,t<1$ and $s+t>0$,
$\left\|fg\right\|_{\dot{H}^{s+t-1}}\lesssim_{s,t}\left\|f\right\|_{\dot{H}^{s}}\left\|g\right\|_{\dot{H}^{t}}.$
For the last term, $II_{3}$, from the divergence free fact of
$u^{\tilde{\epsilon}}$ and
$\mathcal{J}_{\tilde{\epsilon}}\theta^{\epsilon}=\theta^{\epsilon}$ we get
$\begin{split}II_{3}=\Big{(}\big{(}u^{\tilde{\epsilon}}\cdot\nabla(\theta^{\epsilon}-\theta^{\tilde{\epsilon}})\big{)},\mathcal{J}_{\tilde{\epsilon}}(\theta^{\epsilon}-\theta^{\tilde{\epsilon}})\Big{)}=\frac{1}{2}\Big{(}u^{\tilde{\epsilon}},\nabla(\theta^{\epsilon}-\theta^{\tilde{\epsilon}})^{2}\Big{)}=0\end{split}$
Putting all these estimates together yields that for
$\delta_{0}=\min\\{m-\alpha,1\\}$
$\frac{1}{2}\frac{d}{dt}\left\|\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\right\|_{L^{2}}^{2}\lesssim_{M}\big{(}\epsilon^{\delta_{0}}+\left\|\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\right\|_{L^{2}}\big{)}\left\|\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\right\|_{L^{2}}.$
Furthermore
$\frac{d}{dt}\left\|\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\right\|_{L^{2}}\leq
C(M)\big{(}\epsilon^{\delta_{0}}+\left\|\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\right\|_{L^{2}}\big{)}.$
Thus the Grönwall inequality leads to the desired result:
$\begin{split}\sup_{0\leq t\leq
T}\left\|\theta^{\epsilon}-\theta^{\tilde{\epsilon}}\right\|_{L^{2}}&\leq\;e^{C(M)T}\big{(}\epsilon^{\delta_{0}}+\left\|\theta^{\epsilon}_{0}-\theta^{\tilde{\epsilon}}_{0}\right\|_{L^{2}}\big{)}\\\
&\lesssim_{T,\left\|\theta_{0}\right\|_{H^{m}}}a(\epsilon),\end{split}$ (4.13)
where
$a(\epsilon):=\epsilon^{\delta_{0}}+\left\|(Id-\mathcal{J}_{\epsilon})\theta_{0}\right\|_{L^{2}}$
satisfies that $a(\epsilon)\rightarrow 0$ as $\epsilon\rightarrow 0$.
From (4.13), we deduce that the solution family $(\theta^{\epsilon})$ is
Cauchy sequence in $\mathcal{C}([0,T],L^{2}(\mathbb{R}^{2}))$, so that it
converges strongly to a function
$\theta\in\mathcal{C}([0,T],L^{2}(\mathbb{R}^{2}))$. This result combined with
uniform bounds (4.12) and the interpolation inequality in Sobolev spaces gives
that for all $0\leq s<m$
$\begin{split}\sup_{0\leq t\leq
T}\left\|\theta^{\epsilon}-\theta\right\|_{H^{s}}&\leq C_{s}\sup_{0\leq t\leq
T}(\left\|\theta^{\epsilon}-\theta\right\|_{L^{2}}^{1-s/m}\left\|\theta^{\epsilon}-\theta\right\|_{H^{m}}^{s/m})\\\
&\lesssim_{s,T,\left\|\theta_{0}\right\|_{H^{m}}}a(\epsilon)^{1-s/m}.\end{split}$
Hence we obtain the strong convergence in
$\mathcal{C}([0,T],H^{s}(\mathbb{R}^{2}))$ for all $s<m$. With $2<s<m$, this
specially implies strong convergence in
$\mathcal{C}([0,T],\mathcal{C}^{1}(\mathbb{R}^{2}))$. Also from the equation
$\theta^{\epsilon}_{t}=-\nu|D|^{\alpha}\theta^{\epsilon}-\mathcal{J}_{\epsilon}(u^{\epsilon}\cdot\nabla\theta^{\epsilon}),$
we find that $\theta^{\epsilon}_{t}$ strongly converges to
$-\nu|D|^{\alpha}\theta-u\cdot\nabla\theta$ in
$\mathcal{C}([0,T],L^{2}(\mathbb{R}^{2}))$. Since
$\theta^{\epsilon}\rightarrow\theta$, the distribution limit of
$\theta^{\epsilon}_{t}$ has to be $\theta_{t}$. Thus
$\theta\in\mathcal{C}^{1}([0,T],L^{2}(\mathbb{R}^{2}))\cap\mathcal{C}([0,T],\mathcal{C}^{1}(\mathbb{R}^{2}))$
is a solution to the original equation (1.1). Using Fatou’s Lemma, from
(4.12), we also have $\theta\in L^{\infty}([0,T],H^{m}(\mathbb{R}^{2}))\cap
L^{2}([0,T],H^{m+\frac{\alpha}{2}}(\mathbb{R}^{2}))$.
Next, we show that $\theta\in\mathcal{C}([0,T],H^{m}(\mathbb{R}^{2}))$ indeed.
The proof is classical (cf. [18]). We first prove that
$\theta(t)\rightarrow\theta_{0}$ weakly in $H^{m}$ as $t\rightarrow 0$. Let
$\psi(x)\in\mathcal{C}^{\infty}_{0}(\mathbb{R}^{2})$, denote
$F^{\epsilon}(t,\psi):=(\theta^{\epsilon},\psi)=\int_{\mathbb{R}^{2}}\theta^{\epsilon}(t,x)\psi(x)\mathrm{d}x.$
Clearly $F^{\epsilon}(\cdot,\psi)\in\mathcal{C}([0,T])$. And by taking the
inner product of (4.3) with $\psi$, we get
$\frac{d}{dt}F^{\epsilon}(t,\psi)=-(u^{\epsilon}\theta^{\epsilon},\mathcal{J}_{\epsilon}\nabla\psi)-\nu(\theta^{\epsilon},|D|^{\alpha}\psi),$
thus for every $p\in]1,2]$
$\int_{0}^{T}|F^{\epsilon}_{t}|^{p}\mathrm{d}t\leq
T^{\frac{1}{p}-\frac{1}{2}}\|u^{\epsilon}\|_{L^{2}_{T}L^{2}}\|\theta^{\epsilon}\|_{L^{\infty}_{T}L^{2}}\|\psi\|_{H^{3}}+\nu\|\theta^{\epsilon}\|_{L^{p}_{T}L^{2}}\|\psi\|_{H^{\alpha}}.$
From the $L^{2}$ energy estimate
$\|\theta^{\epsilon}\|_{L^{\infty}_{T}L^{2}}^{2}+\|\theta^{\epsilon}\|_{L^{2}_{T}\dot{H}^{\alpha/2}}^{2}\leq\|\theta_{0}\|_{L^{2}}^{2}$,
we know
$\|F^{\epsilon}_{t}(\cdot,\psi)\|_{L^{p}([0,T])}\lesssim_{T,\|\psi\|_{H^{3}}}1$.
Hence by Arzela-Ascolli theorem, $\\{F^{\epsilon}(t,\psi)\\}_{\epsilon>0}$ is
compact in $\mathcal{C}([0,T])$, and we can choose a subsequence
$F^{\epsilon_{j}}(t,\psi)$ converging to a function
$F(t,\psi)\in\mathcal{C}([0,T])$ uniformly in $t$. In particular, from
$\theta^{\epsilon}\rightarrow\theta$ in $\mathcal{C}([0,T];L^{2})$, we can
further find a subsequence (still denote $F^{\epsilon_{j}}$) such that
$F(t,\psi)=(\theta(t),\psi)$ for all $t\in[0,T]$. Next, since
$C_{0}^{\infty}(\mathbb{R}^{2})$ is dense in the separable space
$H^{-m}(\mathbb{R}^{2})$ and $\|\theta^{\epsilon}(t)\|_{H^{m}}$ is uniformly
bounded in $[0,T]$, an appropriate subsequence $\epsilon_{j}$ can be picked
such that $F^{\epsilon_{j}}(t,\psi)$ converges to $F(t,\psi)$ uniformly in
$\epsilon$ for every $\psi\in H^{-m}$. Then for every $t>0$ and $\psi\in
H^{-m}$
$|(\theta(t)-\theta_{0},\psi)|\leq|(\theta(t)-\theta^{\epsilon_{j}}(t),\psi)|+|(\theta^{\epsilon_{j}}(t)-\theta^{\epsilon_{j}}_{0},\psi)|+|(\theta_{0}^{\epsilon_{j}}-\theta_{0},\psi)|.$
All the three terms in the RHS can be made small for sufficiently small
$\epsilon_{j}$ and $t$, thus $\theta(t)$ converges to $\theta_{0}$ weakly in
$H^{m}$ as $t\rightarrow 0$. So we have
$\|\theta_{0}\|_{H^{m}}\leq\liminf_{t\rightarrow 0}\|\theta(t)\|_{H^{m}}.$
(4.14)
Furthermore, from (4.11) we infer that for every $\epsilon>0$ the function
$\|\theta^{\epsilon}(t)\|_{H^{m}}^{2}$ is below the graph of the solution of
the equation
$\frac{d}{dt}y(t)=Cy(t)+Cy^{2}(t),\quad y(0)=\|\theta_{0}\|_{H^{m}}^{2}.$
By construction, the same holds for $\|\theta(t)\|_{H^{m}}^{2}$. Thus from the
continuity of $y(t)$, we find $\|\theta_{0}\|_{H^{m}}\geq\limsup_{t\rightarrow
0}\|\theta(t)\|_{H^{m}}$. Therefore $\|\theta_{0}\|_{H^{m}}=\lim_{t\rightarrow
0}\|\theta(t)\|_{H^{m}}$, and the conclusion follows from this fact combined
with the weak convergence.
Step 3: Uniqueness
Let $\theta^{1}$, $\theta^{2}\in L^{\infty}([0,T],H^{m}(\mathbb{R}^{2}))$ be
two smooth solutions to the modified quasi-geostrophic equation (1.1) with the
same initial data. Denote $u^{i}=|D|^{\alpha-1}R^{\bot}\theta^{i}$, $i=1,2$,
$\delta\theta=\theta^{1}-\theta^{2}$, $\delta u=u^{1}-u^{2}$, then we write
the difference equation as
$\partial_{t}\delta\theta+u^{1}\cdot\nabla\delta\theta+\nu|D|^{\alpha}\delta\theta=-\delta
u\cdot\nabla\theta^{2},\quad\delta\theta|_{t=0}=0$
We also use the $L^{2}$ energy method, and in a similar way as treating the
term $II_{3}$, we obtain
$\frac{d}{dt}\left\|\delta\theta\right\|_{L^{2}}\leq
C_{\alpha}\left\|\nabla\theta^{2}\right\|_{\dot{H}^{\frac{\alpha}{2}}}^{2}\left\|\delta\theta\right\|_{L^{2}}\leq
C_{\alpha}\left\|\theta^{2}\right\|_{H^{m}}^{2}\left\|\delta\theta\right\|_{L^{2}}.$
Thus the Grönwall inequality ensures $\delta\theta\equiv 0$, that is,
$\theta^{1}\equiv\theta^{2}$.
Step 4: Smoothing Effect
Precisely, we have that for all $\gamma\in\mathbb{R}^{+}$ and $t\in[0,T]$
$\left\|t^{\gamma}\theta(t)\right\|_{L^{\infty}_{T}H^{m+\gamma\alpha}}^{2}+\|t^{\gamma}\theta(t)\|_{L^{2}_{T}H^{m+\alpha/2+\gamma\alpha}}^{2}\leq
Ce^{C(\gamma+1)(T\left\|\theta\right\|_{L^{\infty}_{T}H^{m}}^{2}+T)}\left\|\theta_{0}\right\|_{H^{m}}^{2},$
(4.15)
where $C$ is an absolute constant depending only on $\alpha,\nu,m$. Notice
that $t^{\gamma}\theta$ ($\gamma>0$) satisfies
$\partial_{t}(t^{\gamma}\theta)+u\cdot\nabla(t^{\gamma}\theta)+\nu|D|^{\alpha}(t^{\gamma}\theta)=\gamma
t^{\gamma-1}\theta,\quad(t^{\gamma}\theta)|_{t=0}=0.$ (4.16)
which is a linear transport-diffusion equation with the velocity
$u=|D|^{\alpha-1}R^{\bot}\theta$, $\alpha\in]0,2[$. We first treat the case
$\gamma\in\mathbb{Z}^{+}$. For $\gamma=1$, in a similar way as obtaining
(4.4), and using the Sobolev embedding we infer
$\begin{split}\frac{d}{dt}\|t\theta(t)\|_{B^{m+\alpha}_{2,2}}^{2}+\|t\theta(t)\|_{B^{m+\frac{3}{2}\alpha}_{2,2}}^{2}&\lesssim(\|\nabla\theta(t)\|_{L^{\infty}}^{\alpha}\|\theta(t)\|_{L^{\infty}}^{2-\alpha}+\|\theta(t)\|_{L^{2}})\|t\theta(t)\|_{B^{m+\alpha}_{2,2}}^{2}+\|\theta(t)\|_{B^{m+\frac{\alpha}{2}}_{2,2}}^{2}\\\
&\lesssim(\|\theta(t)\|_{H^{m}}^{2}+1)\|t\theta(t)\|_{B^{m+\alpha}_{2,2}}^{2}+\|\theta(t)\|_{B^{m+\frac{\alpha}{2}}_{2,2}}^{2}.\end{split}$
Gronwall inequality yields that
$\begin{split}\|t\theta(t)\|_{B^{m+\alpha}_{2,2}}^{2}+\|t\theta(t)\|^{2}_{L^{2}_{T}B^{m+\frac{3}{2}\alpha}_{2,2}}\lesssim
e^{CT+CT\|\theta\|_{L^{\infty}_{T}H^{m}}^{2}}\int_{0}^{T}\|\theta(\tau)\|_{B^{m+\frac{\alpha}{2}}_{2,2}}^{2}\mathrm{d}\tau.\end{split}$
(4.17)
Meanwhile, similarly as obtaining (4.11), we get
$\|\theta(t)\|_{H^{m}}^{2}+\|\theta\|_{L^{2}_{T}H^{m+\frac{\alpha}{2}}}^{2}\leq\|\theta_{0}\|_{H^{m}}^{2}e^{CT+CT\|\theta\|_{L^{\infty}_{T}H^{m}}^{2}}.$
(4.18)
Thus (4.15) with $\gamma=1$ follows from (4.17) and (4.18) and the fact that
the space $B^{s}_{2,2}$ is equivalent with $H^{s}$, $s\in\mathbb{R}$. Now
suppose estimate (4.15) holds for $\gamma=N$, we shall consider the case
$N+1$. We use the equation (4.16) with $\gamma=N+1$. Similarly as above, and
observing that the constant $C$ in (4.17) is independent of $N$ if $\theta(t)$
is replaced by $t^{N}\theta(t)$ and $m$ by $m+N\alpha$, we have
$\begin{split}\|t^{N+1}\theta(t)\|_{H^{m+(N+1)\alpha}}^{2}+\|t^{N+1}\theta(t)\|_{L^{2}_{T}H^{m+(N+1)\alpha+\frac{\alpha}{2}}}^{2}&\lesssim
e^{CT+CT\|\theta\|_{L^{\infty}_{T}H^{m}}^{2}}\|t^{N}\theta(t)\|_{L^{2}_{T}H^{m+(N+\frac{1}{2})\alpha}}^{2}\\\
&\lesssim
e^{C(N+2)(T+T\|\theta\|_{L^{\infty}_{T}H^{m}}^{2})}\|\theta_{0}\|_{H^{m}}^{2}.\end{split}$
Thus the induction method ensures the estimate (4.15) for all
$\gamma\in\mathbb{Z}^{+}$. Also notice that for $\gamma=0$ the inequality
(4.15) is also satisfied. Hence we obtain estimate (4.15) for all
$\gamma\in\mathbb{N}$. For the general $\gamma\geq 0$, we set
$[\gamma]\leq\gamma<[\gamma]+1$, where $[\gamma]$ denotes the integer part of
$\gamma$, and use the interpolation inequality in Sobolev spaces to get
$\begin{split}\left\|t^{\gamma}\theta\right\|^{2}_{L^{\infty}_{T}H^{m+\gamma\alpha}}\leq&\|t^{[\gamma]}\theta\|_{L^{\infty}_{T}H^{m+[\gamma]\alpha}}^{2([\gamma]+1-\gamma)}\|t^{[\gamma]+1}\theta\|_{L^{\infty}_{T}H^{m+([\gamma]+1)\alpha}}^{2(\gamma-[\gamma])}\\\
\lesssim&e^{C(\gamma+1)(T+T\left\|\theta\right\|_{L^{\infty}_{T}H^{m}}^{2})}\left\|\theta_{0}\right\|_{H^{m}}^{2}.\end{split}$
Similar estimate holds for
$\|t^{\gamma}\theta\|^{2}_{L^{2}_{T}H^{m+(\gamma+\frac{1}{2})\alpha}}$.
Therefore, we conclude the Proposition 4.1. ∎
Now, we are devoted to building the blowup criterion.
###### Proof of Proposition 4.2.
We first note that the equation has a natural blowup criterion: if
$T^{*}<\infty$ then necessarily
$\left\|\theta\right\|_{L^{\infty}([0,T^{*}),H^{m})}+\left\|\theta\right\|_{L^{2}([0,T^{*}),H^{m+\frac{\alpha}{2}})}=\infty.$
Otherwise from the local result, the solution will continue over $T^{*}$.
In the same way as obtaining the estimate (4.4), we get the similar result for
the original equation
$\frac{1}{2}\frac{d}{dt}\left\|\theta(t)\right\|_{B^{m}_{2,2}}^{2}+\frac{\nu}{2}\left\|\theta(t)\right\|_{B^{m+\frac{\alpha}{2}}_{2,2}}^{2}\leq
C_{m,\alpha}\Big{(}\frac{1}{\nu}\left\|\nabla\theta\right\|_{L^{\infty}}^{\alpha}\left\|\theta\right\|_{L^{\infty}}^{2-\alpha}\left\|\theta\right\|_{B^{m}_{2,2}}^{2}+\left\|\theta\right\|_{L^{2}}^{2}\left\|\theta\right\|_{B^{m}_{2,2}}\Big{)}.$
Also due to the maximum principle Proposition 2.3, we have
$\frac{d}{dt}\Big{(}\left\|\theta(t)\right\|_{B^{m}_{2,2}}^{2}+\nu\int_{0}^{t}\left\|\theta(\tau)\right\|_{B^{m+\frac{\alpha}{2}}_{2,2}}^{2}\mathrm{d}\tau\Big{)}\lesssim_{\alpha,\nu,m}\big{(}\left\|\nabla\theta(t)\right\|_{L^{\infty}}^{\alpha}+1\big{)}\left\|\theta(t)\right\|_{B^{m}_{2,2}}^{2}.$
This together with the Grönwall inequality leads to
$\begin{split}\sup_{0\leq t\leq
T}\left\|\theta(t)\right\|_{H^{m}}^{2}+\left\|\theta\right\|_{L^{2}([0,T],H^{m+\frac{\alpha}{2}})}^{2}&\leq
C_{0}\sup_{0\leq t\leq
T}\left\|\theta(t)\right\|_{B^{m}_{2,2}}^{2}+C_{0}\left\|\theta\right\|_{L^{2}([0,T],B^{m+\frac{\alpha}{2}}_{2,2})}^{2}\\\
&\leq
C\exp\Big{\\{}CT+C\int_{0}^{T}\left\|\nabla\theta(t)\right\|_{L^{\infty}}^{\alpha}\textrm{d}t\Big{\\}}.\end{split}$
Further, if $T^{*}<\infty$ and the integral
$\int_{0}^{T^{*}}\left\|\nabla\theta(t)\right\|_{L^{\infty}}^{\alpha}\textrm{d}t<\infty$,
then from the above estimate we directly have
$\sup_{0\leq
t<T^{*}}\left\|\theta(t)\right\|_{H^{m}}+\left\|\theta\right\|_{L^{2}([0,T^{*}),H^{m+\frac{\alpha}{2}})}<\infty.$
Clearly this contradicts the upper natural blowup criterion. Thus, if
$T^{*}<\infty$, we necessarily have the equality
$\int_{0}^{T^{*}}\left\|\nabla\theta(t)\right\|_{L^{\infty}}^{\alpha}\textrm{d}t=\infty.$
∎
## 5 Global Existence
In this section, we use the modulus of continuity argument developed by
Kiselev, Nazarov and Volberg [17] to prove the global result, see also [1].
Throughout this section, we assume $T^{*}$ be the maximal existence time of
the solution in $\mathcal{C}([0,T^{*}),H^{m})\cap
L^{2}([0,T^{*}),H^{m+\frac{\alpha}{2}})$.
Let $\lambda>0$ be a real number which will be chosen later, then we define
the set
$\mathcal{I}:=\big{\\{}T\in[0,T^{*})|\forall t\in[0,T],\forall
x,y\in\mathbb{R}^{2},x\neq
y,|\theta(t,x)-\theta(t,y)|<\omega_{\lambda}(|x-y|)\big{\\}},$
where $\omega$ is a strict modulus of continuity also satisfying that
$\omega^{\prime}(0)<\infty$, $\lim_{\eta\searrow
0}\omega^{\prime\prime}(\eta)=-\infty$ and
$\omega_{\lambda}(|x-y|)=\omega(\lambda|x-y|).$
The explicit expression of $\omega$ will be shown later.
We first show that the set $\mathcal{I}$ is nonempty, that is, at least
$0\in\mathcal{I}$. The proof is almost the same with the one in [1] only by
setting $T_{1}$ there to be $0$. We omit it here and only note that to fit our
purpose $\lambda$ can be taken
$\lambda=\frac{\omega^{-1}(3\left\|\theta_{0}\right\|_{L^{\infty}})}{2\left\|\theta_{0}\right\|_{L^{\infty}}}\left\|\nabla\theta_{0}\right\|_{L^{\infty}}.$
(5.1)
Thus $\mathcal{I}$ is an interval of the form $[0,T_{*})$, where
$T_{*}:=\sup_{T\in\mathcal{I}}T$. We have three possibilities:
1. (a)
$T_{*}=T^{*}$
2. (b)
$T_{*}<T^{*}$ and $T_{*}\in\mathcal{I}$
3. (c)
$T_{*}<T^{*}$ and $T_{*}\notin\mathcal{I}$
For case (a), we necessarily have $T^{*}=\infty$, since the Lipschitz norm of
$\theta$ does not blow up from the definition of $\mathcal{I}$ which
contradicts with (4.1). This is our goal.
For case (b), we observe that this is just the case treated in [1] or [13]
showing that it is impossible. The proof only needs very small modification,
so we omit it either. We just point out in this case the smoothing effects
will be used, since we need the fact that
$\left\|\nabla^{2}\theta(T_{*})\right\|_{L^{\infty}}$ is finite.
Then our task is reduced to get rid of the case (c). We prove by
contradiction. If the case (c) is satisfied, then by the time continuity of
$\theta$, we necessarily get
$\sup_{x,y\in\mathbb{R}^{2},x\neq
y}\frac{|\theta(T_{*},x)-\theta(T_{*},y)|}{\omega_{\lambda}(|x-y|)}=1.$
We further have the following assertion (with its proof in the end of this
section).
###### Lemma 5.1.
If the above condition is assumed, there exists $x,y\in\mathbb{R}^{2}$, $x\neq
y$ such that
$\theta(T_{*},x)-\theta(T_{*},y)=\omega_{\lambda}(\xi),\quad\text{with}\quad\xi:=|x-y|.$
We shall show that this scenario can not happen, more precisely, we shall
prove
$f^{\prime}(T_{*})<0,\quad\textrm{with}\quad f(t):=\theta(t,x)-\theta(t,y).$
This is impossible because we necessarily have $f(t)\leq f(T_{*})$, for all
$0\leq t\leq T_{*}$ from the definition of $\mathcal{I}$.
We see that the modified quasi-geostrophic equation (1.1) can be defined in
the classical sense (from the smoothing effect), and thus
$\begin{split}f^{\prime}(T_{*})=&-\Big{[}(u\cdot\nabla\theta)(T_{*},x)-(u\cdot\nabla\theta)(T_{*},y)\Big{]}+\nu\Big{[}(-|D|^{\alpha}\theta)(T_{*},x)-(-|D|^{\alpha}\theta)(T_{*},y)\Big{]}\\\
:=&\,\mathcal{A}_{1}+\mathcal{A}_{2}\end{split}$
with
$u=|D|^{\alpha-1}\mathcal{R}^{\bot}\theta=\mathcal{R}^{\bot}_{\alpha}\theta:=(-\mathcal{R}_{\alpha,2}\theta,\mathcal{R}_{\alpha,1}\theta)$
where $\mathcal{R}_{\alpha,j}$ are the modified Riesz transforms introduced in
the section 3.
For the first term, $\mathcal{A}_{1}$, we find that
$(u\cdot\nabla)\theta(x)=\frac{d}{dh}\theta(x+hu)|_{h=0}$. Then due to the
fact that $\theta(T_{*},\cdot)$ also has the modulus of continuity, namely,
$|\theta(T_{*},x^{\prime})-\theta(T_{*},y^{\prime})|\leq\omega_{\lambda}(|x^{\prime}-y^{\prime}|)$
and Lemma 3.2 we have
$\theta\big{(}T_{*},x+hu(x)\big{)}-\theta\big{(}T_{*},y+hu(y)\big{)}\leq\omega_{\lambda}\big{(}|x-y|+h|u(x)-u(y)|\big{)}\leq\omega_{\lambda}\big{(}\xi+h\Omega_{\lambda}(\xi)\big{)},$
where $\Omega_{\lambda}(\xi)$ is defined from (3.2) in Lemma 3.2, i.e.
$\Omega_{\lambda}(\xi)=A\bigg{(}\int^{\xi}_{0}\frac{\omega_{\lambda}(\eta)}{\eta^{\alpha}}\textrm{d}\eta+\xi\int_{\xi}^{\infty}\frac{\omega_{\lambda}(\eta)}{\eta^{1+\alpha}}\textrm{d}\eta\bigg{)}=\lambda^{\alpha-1}\Omega(\lambda\xi).$
Since $\theta(T_{*},x)-\theta(T_{*},y)=\omega_{\lambda}(\xi)$, we have
$\begin{split}\lim_{h\rightarrow
0}&\frac{\big{\\{}\theta\big{(}T_{*},x+hu(x)\big{)}-\theta\big{(}T_{*},x\big{)}\big{\\}}-\big{\\{}\theta\big{(}T_{*},y+hu(y)\big{)}-\theta\big{(}T_{*},y\big{)}\big{\\}}}{h}\\\
&\leq\lim_{h\rightarrow
0}\frac{\omega_{\lambda}\big{(}\xi+h\Omega_{\lambda}(\xi)\big{)}-\omega_{\lambda}\big{(}\xi\big{)}}{h}\end{split}$
thus
$|\mathcal{A}_{1}|\leq\Omega_{\lambda}(\xi)\omega^{\prime}_{\lambda}(\xi)=\lambda^{\alpha}(\Omega\omega^{\prime})(\lambda\xi).$
For the second term, $\mathcal{A}_{2}$, we observe that this is just the
result of Lemma 3.3:
$\begin{split}\mathcal{A}_{2}\leq&\,\nu
B\int_{0}^{\frac{\xi}{2}}\frac{\omega_{\lambda}(\xi+2\eta)+\omega_{\lambda}(\xi-2\eta)-2\omega_{\lambda}(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta\\\
&+\nu
B\int_{\frac{\xi}{2}}^{\infty}\frac{\omega_{\lambda}(2\eta+\xi)-\omega_{\lambda}(2\eta-\xi)-2\omega_{\lambda}(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta\\\
\leq&\;\lambda^{\alpha}\Upsilon(\lambda\xi)\end{split}$
where
$\begin{split}\Upsilon(\xi):=&\,\nu
B\int_{0}^{\frac{\xi}{2}}\frac{\omega(\xi+2\eta)+\omega(\xi-2\eta)-2\omega(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta\\\
&+\nu
B\int_{\frac{\xi}{2}}^{\infty}\frac{\omega(2\eta+\xi)-\omega(2\eta-\xi)-2\omega(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta\end{split}$
Thus we obtain
$f^{\prime}(T_{*})\leq\lambda^{\alpha}\big{(}\Omega\omega^{\prime}+\Upsilon)(\lambda\xi\big{)}.$
Next we shall construct our special modulus of continuity in the spirit of
[17]. Choose two small positive numbers $0<\gamma<\delta<1$ and define the
continuous functions $\omega$ as follows that when $\alpha\in]0,1[$
$\mathrm{MOC}_{1}\;\begin{cases}\omega(\xi)=\xi-\xi^{1+\frac{\alpha}{2}}\quad&\text{if}\quad
0\leq\xi\leq\delta,\\\
\omega^{\prime}(\xi)=\frac{\gamma}{2(\xi+\xi^{\alpha})}\quad&\text{if}\quad\xi>\delta,\end{cases}$
(5.2)
and when $\alpha\in]1,2[$
$\mathrm{MOC}_{2}\;\begin{cases}\omega(\xi)=\xi-\xi^{1+r}\quad&\text{if}\quad
0\leq\xi\leq\delta,\\\
\omega^{\prime}(\xi)=\frac{\gamma}{4(\xi+\xi^{\alpha})}\quad&\text{if}\quad\xi>\delta,\end{cases}$
(5.3)
where when $\alpha\in]1,2[$ $\delta<\frac{1}{4}$ and $\delta^{r}=\frac{1}{2}$
(i.e. $r=\frac{\log 2}{\log(1/\delta)}\in]0,1/2[$). Note that, for small
$\delta$, the left derivative of $\omega$ at $\delta$ is about 1 (or at least
$\frac{1}{4}$), while the right derivative equals
$\frac{\gamma}{2(\delta+\delta^{\alpha})}(\textrm{or}\frac{\gamma}{4(\delta+\delta^{\alpha})})<\frac{1}{4}$.
So $\omega$ is concave if $\delta$ is small enough. Clearly in both cases,
$\omega(0)=0$, $\omega^{\prime}(0)=1$ and $\lim_{\eta\rightarrow
0+}\omega^{\prime\prime}(\eta)=-\infty$. Moreover, when $\alpha\in]0,1[$,
$\omega$ is unbounded (it has the logarithmic growth at infinity); while when
$\alpha\in]1,2[$, $\omega$ is unfortunately bounded (thus we have to a priori
assume that $\left\|\theta_{0}\right\|_{L^{\infty}}$ is small to give a
meaning of $\omega^{-1}(3\left\|\theta_{0}\right\|_{L^{\infty}})$; we also
note that this is the only point that the boundedness property of $\omega$ is
used).
Then our target is to show that, for these MOC $\omega$,
$\Omega(\xi)\omega^{\prime}(\xi)+\Upsilon(\xi)<0\quad\textrm{for
all}\quad\xi>0.$
More precisely, it reduces to proving the inequality
$\begin{split}A\bigg{[}\int^{\xi}_{0}\frac{\omega(\eta)}{\eta^{\alpha}}\textrm{d}\eta+&\xi\int_{\xi}^{\infty}\frac{\omega(\eta)}{\eta^{1+\alpha}}\textrm{d}\eta\bigg{]}\omega^{\prime}(\xi)+\nu
B\int_{0}^{\frac{\xi}{2}}\frac{\omega(\xi+2\eta)+\omega(\xi-2\eta)-2\omega(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta\\\
+&\nu
B\int_{\frac{\xi}{2}}^{\infty}\frac{\omega(2\eta+\xi)-\omega(2\eta-\xi)-2\omega(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta<0\quad\text{for
all}\quad\xi>0.\end{split}$
To check this, we first consider MOC1 and then MOC2. Case I: when
$\alpha\in]0,1[$
Case I.1: $\alpha\in]0,1[$ and $0<\xi\leq\delta$
Since $\frac{\omega(\eta)}{\eta}\leq\omega^{\prime}(0)=1$ for all $\eta>0$ and
$\eta\leq\eta^{\alpha}$ for $\eta\leq\delta<1$, we have
$\int_{0}^{\xi}\frac{\omega(\eta)}{\eta^{\alpha}}\textrm{d}\eta\leq\int_{0}^{\xi}\frac{\omega(\eta)}{\eta}\textrm{d}\eta\leq\xi,$
and
$\int_{\xi}^{\delta}\frac{\omega(\eta)}{\eta^{1+\alpha}}\textrm{d}\eta\leq\int_{\xi}^{\delta}\frac{1}{\eta^{\alpha}}\textrm{d}\eta=\frac{1}{1-\alpha}(\delta^{1-\alpha}-\xi^{1-\alpha})\leq\frac{1}{1-\alpha}.$
Further,
$\int_{\delta}^{\infty}\frac{\omega(\eta)}{\eta^{1+\alpha}}\textrm{d}\eta=\frac{1}{\alpha}\frac{\omega(\delta)}{\delta^{\alpha}}+\frac{1}{\alpha}\int_{\delta}^{\infty}\frac{\gamma}{2\eta^{\alpha}(\eta+\eta^{\alpha})}\textrm{d}\eta\leq\frac{1}{\alpha}+\frac{1}{\alpha^{2}}\frac{\gamma}{\delta^{\alpha}}\leq\frac{2}{\alpha},$
if $\gamma<\alpha\delta$. Obviously
$\omega^{\prime}(\xi)\leq\omega^{\prime}(0)=1$, so we get that the positive
part is bounded by $A\xi\frac{2}{\alpha(1-\alpha)}$.
For the negative part, we have
$\begin{split}\nu
B\int_{0}^{\frac{\xi}{2}}&\frac{\omega(\xi+2\eta)+\omega(\xi-2\eta)-2\omega(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta\leq\nu
B\int_{0}^{\frac{\xi}{2}}\frac{\omega^{\prime\prime}(\xi)2\eta^{2}}{\eta^{1+\alpha}}\textrm{d}\eta\\\
=&-\nu
B\frac{\alpha(2+\alpha)}{2^{1-\alpha}(2-\alpha)}\xi^{1-\frac{\alpha}{2}}\leq-\frac{\alpha}{2}\nu
B\xi^{1-\frac{\alpha}{2}}.\end{split}$
But, clearly $\xi\Big{(}A\frac{2}{\alpha(1-\alpha)}-\frac{\alpha}{2}\nu
B\xi^{-\frac{\alpha}{2}}\Big{)}<0$ on $(0,\delta]$ when $\delta$ is small
enough. Case I.2: $\alpha\in]0,1[$ and $\xi\geq\delta$
For $\eta\leq\delta<1$ we still use $\eta^{\alpha}\geq\eta$ and for
$\delta\leq\eta\leq\xi$ we use $\omega(\eta)\leq\omega(\xi)$, then
$\int_{0}^{\xi}\frac{\omega(\eta)}{\eta^{\alpha}}\textrm{d}\eta\leq\delta+\frac{\omega(\xi)}{1-\alpha}\Big{(}\xi^{1-\alpha}-\delta^{1-\alpha}\Big{)}\leq\omega(\xi)\Big{(}\frac{2}{\alpha}+\frac{\xi^{1-\alpha}}{1-\alpha}\Big{)},$
where the last inequality is due to
$\frac{\alpha}{2}\delta<\omega(\delta)\leq\omega(\xi)$ if $\delta$ is small
enough (i.e. $\delta<(1-\frac{\alpha}{2})^{2/\alpha}$). Also
$\int_{\xi}^{\infty}\frac{\omega(\eta)}{\eta^{1+\alpha}}\textrm{d}\eta=\frac{1}{\alpha}\frac{\omega(\xi)}{\xi^{\alpha}}+\frac{1}{\alpha}\int_{\xi}^{\infty}\frac{\gamma}{2\eta^{\alpha}(\eta+\eta^{\alpha})}\textrm{d}\eta\leq\frac{1}{\alpha}\frac{\omega(\xi)}{\xi^{\alpha}}+\frac{1}{\alpha^{2}}\frac{\gamma}{2}\frac{1}{\xi^{\alpha}}\leq\frac{2}{\alpha}\frac{\omega(\xi)}{\xi^{\alpha}}$
if $\gamma<\alpha^{2}\delta$ and $\delta$ is small enough. Thus the positive
term is bounded from above by
$A\omega(\xi)\bigg{(}\frac{2}{\alpha}+\Big{(}\frac{1}{1-\alpha}+\frac{2}{\alpha}\Big{)}\xi^{1-\alpha}\bigg{)}\omega^{\prime}(\xi)\leq
A\frac{\omega(\xi)}{\xi^{\alpha}}\frac{2}{\alpha(1-\alpha)}(\xi+\xi^{\alpha})\omega^{\prime}(\xi)\leq\frac{A\gamma}{\alpha(1-\alpha)}\frac{\omega(\xi)}{\xi^{\alpha}}.$
For the negative part, we first observe that for $\xi\geq\delta$,
$\omega(2\xi)=\omega(\xi)+\int_{\xi}^{2\xi}\omega^{\prime}(\eta)\textrm{d}\eta\leq\omega(\xi)+\frac{\log
2}{2}\gamma\leq\frac{3}{2}\omega(\xi),$
under the same assumptions on $\delta$ and $\gamma$ as above. Also, taking
advantage of the concavity we obtain
$\omega(2\eta+\xi)-\omega(2\eta-\xi)\leq\omega(2\xi)$ for all
$\eta\geq\frac{\xi}{2}$. Therefore
$\nu
B\int_{\frac{\xi}{2}}^{\infty}\frac{\omega(2\eta+\xi)-\omega(2\eta-\xi)-2\omega(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta\leq-\nu
B\frac{\omega(\xi)}{2}\int_{\frac{\xi}{2}}^{\infty}\frac{1}{\eta^{1+\alpha}}\textrm{d}\eta=-\nu
B\frac{2^{\alpha}}{2\alpha}\frac{\omega(\xi)}{\xi^{\alpha}}.$
But $\frac{\omega(\xi)}{\xi^{\alpha}}(\frac{A\gamma}{\alpha(1-\alpha)}-\nu
B\frac{2^{\alpha}}{2\alpha})<0$ if $\gamma$ is small enough (i.e.
$\gamma<\min\\{\alpha^{2}\delta,\frac{\nu(1-\alpha)B2^{\alpha}}{2A}\\}$).
Case II: when $\alpha\in]1,2[$
Case II.1: $\alpha\in]1,2[$ and $0<\xi\leq\delta$ Since
$\frac{\omega(\eta)}{\eta}\leq\omega^{\prime}(0)=1$ for all $\eta>0$, we have
$\int_{0}^{\xi}\frac{\omega(\eta)}{\eta^{\alpha}}\textrm{d}\eta\leq\int_{0}^{\xi}\frac{1}{\eta^{\alpha-1}}\textrm{d}\eta\leq\frac{1}{2-\alpha}\xi^{2-\alpha},$
and
$\int_{\xi}^{\delta}\frac{\omega(\eta)}{\eta^{1+\alpha}}\textrm{d}\eta\leq\int_{\xi}^{\delta}\frac{1}{\eta^{\alpha}}\textrm{d}\eta\leq\frac{1}{\alpha-1}\xi^{1-\alpha}.$
Further,
$\begin{split}\int_{\delta}^{\infty}\frac{\omega(\eta)}{\eta^{1+\alpha}}\textrm{d}\eta&=\frac{1}{\alpha}\frac{\omega(\delta)}{\delta^{\alpha}}+\frac{1}{\alpha}\int_{\delta}^{\infty}\frac{\gamma}{4\eta^{\alpha}(\eta+\eta^{\alpha})}\textrm{d}\eta\\\
&\leq\frac{1}{\alpha}\frac{1}{\delta^{\alpha-1}}+\frac{\gamma}{4\alpha^{2}}\frac{1}{\delta^{\alpha}}\leq
2\frac{1}{\delta^{\alpha-1}}\leq 2\xi^{1-\alpha}.\end{split}$
Obviously $\omega^{\prime}(\xi)\leq\omega^{\prime}(0)=1$, so we get that the
positive part is bounded by $A\xi^{2-\alpha}\frac{2}{(\alpha-1)(2-\alpha)}$.
For the negative part, we have
$\begin{split}\nu
B\int_{0}^{\frac{\xi}{2}}&\frac{\omega(\xi+2\eta)+\omega(\xi-2\eta)-2\omega(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta\leq\nu
B\int_{0}^{\frac{\xi}{2}}\frac{\omega^{\prime\prime}(\xi)2\eta^{2}}{\eta^{1+\alpha}}\textrm{d}\eta\\\
=&-\nu
B\frac{r(1+r)}{2^{2-\alpha}(2-\alpha)}\xi^{1-\alpha+r}\leq-\frac{r}{2(2-\alpha)}\nu
B\xi^{1-\alpha+r}.\end{split}$
But, clearly
$\xi^{2-\alpha}\Big{(}A\frac{2}{(\alpha-1)(2-\alpha)}-\frac{r}{2(2-\alpha)}\nu
B\xi^{-1+r}\Big{)}<0$ on $(0,\delta]$ when $\delta$ is small enough (i.e.
$\delta\log(1/\delta)<\frac{\nu(\log 2)(\alpha-1)B}{8A}$). Case II.2:
$\alpha\in]1,2[$ and $\xi\geq\delta$
For $0\leq\eta\leq\delta$ we still have $\omega(\eta)\leq\eta$ and for
$\delta\leq\eta\leq\xi$ we have $\omega(\eta)\leq\omega(\xi)$, then
$\int_{0}^{\xi}\frac{\omega(\eta)}{\eta^{\alpha}}\textrm{d}\eta\leq\frac{\delta^{2-\alpha}}{2-\alpha}+\frac{\omega(\xi)}{\alpha-1}\Big{(}\delta^{1-\alpha}-\xi^{1-\alpha}\Big{)}\leq\frac{2\delta^{1-\alpha}}{(\alpha-1)(2-\alpha)}\omega(\xi),$
where the last inequality is due to
$\frac{\delta}{2}=\omega(\delta)\leq\omega(\xi)$. Also
$\begin{split}\int_{\xi}^{\infty}\frac{\omega(\eta)}{\eta^{1+\alpha}}\textrm{d}\eta&=\frac{1}{\alpha}\frac{\omega(\xi)}{\xi^{\alpha}}+\frac{1}{\alpha}\int_{\xi}^{\infty}\frac{\gamma}{4\eta^{\alpha}(\eta+\eta^{\alpha})}\textrm{d}\eta\\\
&\leq\frac{1}{\alpha}\frac{\omega(\xi)}{\xi^{\alpha}}+\frac{\gamma}{4\alpha^{2}}\frac{1}{\xi^{\alpha}}\leq
2\frac{\omega(\xi)}{\xi^{\alpha}}.\end{split}$
Thus the positive term is bounded from above by
$A\omega(\xi)\bigg{(}\frac{2\delta^{1-\alpha}}{(\alpha-1)(2-\alpha)}+2\xi^{1-\alpha}\bigg{)}\omega^{\prime}(\xi)\leq\frac{A}{\delta^{\alpha-1}}\frac{\omega(\xi)}{\xi^{\alpha}}\frac{2(\xi+\xi^{\alpha})}{(\alpha-1)(2-\alpha)}\omega^{\prime}(\xi)\leq\frac{A\delta^{1-\alpha}\gamma}{2(\alpha-1)(2-\alpha)}\frac{\omega(\xi)}{\xi^{\alpha}}.$
For the negative part, we first observe that for $\xi\geq\delta$,
$\omega(2\xi)=\omega(\xi)+\int_{\xi}^{2\xi}\omega^{\prime}(\eta)\textrm{d}\eta\leq\omega(\xi)+\frac{(\log
2)\gamma}{4}\leq\frac{3}{2}\omega(\xi)$
under the same assumptions on $\delta$ and $\gamma$ as above. Also, taking
advantage of the concavity we obtain
$\omega(2\eta+\xi)-\omega(2\eta-\xi)\leq\omega(2\xi)$ for all
$\eta\geq\frac{\xi}{2}$. Therefore
$\nu
B\int_{\frac{\xi}{2}}^{\infty}\frac{\omega(2\eta+\xi)-\omega(2\eta-\xi)-2\omega(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta\leq-\nu
B\frac{\omega(\xi)}{2}\int_{\frac{\xi}{2}}^{\infty}\frac{1}{\eta^{1+\alpha}}\textrm{d}\eta=-\frac{2^{\alpha}\nu
B}{2\alpha}\frac{\omega(\xi)}{\xi^{\alpha}}.$
But
$\frac{\omega(\xi)}{\xi^{\alpha}}(\frac{A\gamma}{2\delta^{\alpha-1}(\alpha-1)(2-\alpha)}-\frac{\nu
B2^{\alpha}}{2\alpha})<0$ if $\gamma$ is small enough (i.e.
$\gamma<\min\\{\delta,\frac{2^{\alpha}(\alpha-1)(2-\alpha)\nu B}{\alpha
A}\delta^{\alpha-1}\\}$).
Therefore both cases yield $f^{\prime}(T_{*})<0$.
Now we discuss the smallness condition (1.2) based on the MOC2 (5.3). First
clearly $\omega(\delta)=\frac{\delta}{2}$, thus if
$\left\|\theta_{0}\right\|_{L^{\infty}}\leq\frac{\delta}{6}$, we have
$\omega^{-1}(3\left\|\theta_{0}\right\|_{L^{\infty}})\leq\delta$ and
$\frac{\omega^{-1}(3\left\|\theta_{0}\right\|_{L^{\infty}})}{2\left\|\theta_{0}\right\|_{L^{\infty}}}\leq\frac{3}{2}\frac{\omega^{-1}(\delta/2)}{\delta/2}=3$.
Second since
$\begin{split}\frac{\delta}{2}+\int_{\delta}^{\infty}\frac{\gamma}{4(\xi+\xi^{\alpha})}\mathrm{d}\xi&>\frac{\delta}{2}+\int_{\delta}^{1}\frac{\gamma}{8\xi}\mathrm{d}\xi+\int_{1}^{\infty}\frac{\gamma}{8\xi^{\alpha}}\mathrm{d}\xi\\\
&=\frac{\delta}{2}+\frac{\gamma}{8}\log(1/\delta)+\frac{\gamma}{8(\alpha-1)}\\\
&:=3c_{0}\end{split}$
where $\delta$ and $\gamma$ are arbitrary numbers satisfying
$0<\delta<\frac{1}{4},\;\delta\log(1/\delta)<\frac{\nu(\log
2)(\alpha-1)B}{8A};\quad
0<\gamma<\min\\{\delta,\frac{2^{\alpha}(\alpha-1)(2-\alpha)\nu B}{\alpha
A}\delta^{\alpha-1}\\}.$
Hence if $\left\|\theta_{0}\right\|_{L^{\infty}}\leq c_{0}$ ($c_{0}$ should be
chosen in a best way), we get
$\omega^{-1}(3\left\|\theta_{0}\right\|_{L^{\infty}})<\infty$ and thus all the
process above has no problem.
Finally, only case (a) occurs and we obtain $T^{*}=\infty$. Moreover
$\left\|\nabla\theta(t)\right\|_{L^{\infty}}<\lambda,\quad\forall
t\in[0,\infty)$
where the value of $\lambda$ is given by (5.1).
###### Proof of Lemma 5.1.
Set $C^{\prime}:=\omega^{-1}(3\left\|\theta_{0}\right\|_{L^{\infty}})$, then
from the maximum principle (2.2), we get
$\lambda|x-y|\geq
C^{\prime}\Rightarrow|\theta(T_{*},x)-\theta(T_{*},y)|<\frac{2}{3}\omega_{\lambda}(|x-y|).$
(5.4)
Since $\nabla\theta(t)\in\mathcal{C}([0,T^{*}),H^{m-1}(\mathbb{R}^{2}))$, then
for every $\epsilon>0$, there exists $R>0$ such that
$\left\|\nabla\theta(T_{*})\right\|_{L^{\infty}(\mathbb{R}^{2}\setminus
B_{R})}\leq
C_{0}\left\|\nabla\theta(T_{*})\right\|_{H^{m-1}(\mathbb{R}^{2}\setminus
B_{R})}\leq\epsilon,$
where $B_{R}$ is a ball centered at the origin with the radius $R$ and
$\mathbb{R}^{2}\setminus B_{R}$ is its complement. Thus for every $x,y$($x\neq
y$) satisfying that $\lambda|x-y|\leq C^{\prime}$ and $x$ or $y$ belongs to
$\mathbb{R}^{2}\setminus B_{R+C^{\prime}/\lambda}$, we get
$|\theta(T_{*},x)-\theta(T_{*},y)|\leq\left\|\nabla\theta(T_{*})\right\|_{L^{\infty}(\mathbb{R}^{2}\setminus
B_{R})}|x-y|\leq\epsilon|x-y|.$
Taking advantage of the following inequality from the concavity of $\omega$
$\frac{\omega(C^{\prime})}{C^{\prime}}\lambda|x-y|\leq\omega_{\lambda}(|x-y|),$
we can take $\epsilon$ small enough such that
$\epsilon<\frac{1}{2}\frac{\omega(C^{\prime})}{C^{\prime}}\lambda$ to obtain
$\lambda|x-y|\leq C^{\prime},\,x\,\mathrm{or}\,y\in\mathbb{R}^{2}\setminus
B_{R+\frac{C^{\prime}}{\lambda}};\Rightarrow|\theta(T_{*},x)-\theta(T_{*},y)|<\frac{1}{2}\omega_{\lambda}(|x-y|).$
(5.5)
Now it remains to consider the case when $x,y\in
B_{R+\frac{C^{\prime}}{\lambda}}$. From the smoothing effect we know
$\left\|\nabla^{2}\theta(T_{*})\right\|_{L^{\infty}}<\infty$, thus we have
(cf. [17])
$\left\|\nabla\theta(T_{*})\right\|_{L^{\infty}(B_{R+\frac{C^{\prime}}{\lambda}})}<\lambda\omega^{\prime}(0).$
Let $\delta^{\prime}\ll 1$ small enough, then we see
$\left\|\theta(T_{*})\right\|_{L^{\infty}(B_{R+\frac{C^{\prime}}{\lambda}})}<\lambda(1-\delta^{\prime})\frac{\omega(\delta^{\prime})}{\delta^{\prime}}.$
Thus for every $x,y$($x\neq y$) satisfying that
$\lambda|x-y|\leq\delta^{\prime}$ and both $x,y$ belongs to
$B_{R+C^{\prime}/\lambda}$, we have
$\begin{split}|\theta(T_{*},x)-\theta(T_{*},y)|&\leq\left\|\nabla\theta(T_{*})\right\|_{L^{\infty}(B_{R+\frac{C^{\prime}}{\lambda}})}|x-y|\\\
&<(1-\delta^{\prime})\frac{\omega(\delta^{\prime})}{\delta^{\prime}}\lambda|x-y|\leq(1-\delta^{\prime})\omega_{\lambda}(|x-y|).\end{split}$
(5.6)
We set
$\Omega:=\\{(x,y)\in\mathbb{R}^{2}\times\mathbb{R}^{2}:\max\\{|x|,|y|\\}\leq
R+\frac{C^{\prime}}{\lambda},\,|x-y|\geq\frac{\delta^{\prime}}{\lambda}\\},$
then from the above results we necessarily have
$1=\sup_{x\neq
y}\frac{|\theta(T_{*},x)-\theta(T_{*},y)|}{\omega_{\lambda}(|x-y|)}=\sup_{(x,y)\in\Omega}\frac{|\theta(T_{*},x)-\theta(T_{*},y)|}{\omega_{\lambda}(|x-y|)}.$
Thus the conclusion follows from the compactness of $\Omega$. ∎
## 6 Appendix
### 6.1 The formula for $\mathcal{R}_{\alpha,j}f$
###### Proof of Proposition 3.1.
The pseudo-differential operator $\mathcal{R}_{\alpha,j}$ ($\alpha\in]0,2[$)
is the composition of two operators $|D|^{\alpha-1}$ and $\mathcal{R}_{j}$,
which both are (constant coefficient) pseudo-differential operators, thus the
symbol of $\mathcal{R}_{\alpha,j}$ is $-i\zeta_{j}/|\zeta|^{2-\alpha}$. Now we
want to know the explicit formula of
$\mathcal{F}^{-1}(-i\zeta_{j}/|\zeta|^{2-\alpha})$.
From the equality in the distributional sense
$\frac{\partial}{\partial
x_{j}}|x|^{-(n+\alpha-2)}=-(n+\alpha-2)\mathrm{p.v.}\frac{x_{j}}{|x|^{n+\alpha}},$
and the known formula that for every $0<a<n$ (c.f. [15])
$(|x|^{-a})^{\wedge}(\zeta)=\frac{2^{n-a}\pi^{n/2}\Gamma(\frac{n-a}{2})}{\Gamma(\frac{a}{2})}|\zeta|^{-n+a},$
we directly have
$\begin{split}(\mathrm{p.v.}\frac{x_{j}}{|x|^{n+\alpha}})^{\wedge}(\zeta)&=-\frac{1}{n+\alpha-2}(\partial_{x_{j}}|x|^{-n-\alpha+2})^{\wedge}(\zeta)\\\
&=-\frac{i\zeta_{j}}{n+\alpha-2}(|x|^{-n-\alpha+2})^{\wedge}(\zeta)\\\
&=-\frac{i\zeta_{j}}{n+\alpha-2}\frac{2^{2-\alpha}\pi^{n/2}\Gamma(\frac{2-\alpha}{2})}{\Gamma(\frac{n+\alpha-2}{2})}|\zeta|^{\alpha-2}\\\
&=-i\frac{2^{1-\alpha}\pi^{n/2}\Gamma(\frac{2-\alpha}{2})}{\Gamma(\frac{n+\alpha}{2})}\cdot\frac{\zeta_{j}}{|\zeta|^{2-\alpha}}.\end{split}$
∎
### 6.2 A commutator estimate
The key to the proof of the uniform estimate is the following commutator
estimate
###### Lemma 6.1.
Let $v$ be a divergence free vector field over $\mathbb{R}^{n}$. For every
$q\in\mathbb{N}$, denote
$F_{q}(v,f):=S_{q+1}v\cdot\nabla\Delta_{q}f-\Delta_{q}(v\cdot\nabla f).$
Then for every $\beta\in]0,1[$, there exists a positive constant $C$ such that
$\begin{split}&2^{-q\beta}\left\|F_{q}(v,f)\right\|_{L^{2}}\\\
\leq&C\left\||D|^{1-\beta}v\right\|_{L^{\infty}}\Big{(}\sum_{q^{\prime}\leq
q+4}2^{q^{\prime}-q}\left\|\Delta_{q^{\prime}}f\right\|_{L^{2}}+\sum_{q^{\prime}\geq
q-4}2^{(q-q^{\prime})(1-\beta)}\left\|\Delta_{q^{\prime}}f\right\|_{L^{2}}\Big{)},\end{split}$
(6.1)
Especially, in the case $n=2$ and $v=|D|^{\alpha-1}\mathcal{R}^{\bot}f$
($\alpha\in]0,2[$), we further have for every
$\beta\in\big{]}\max\\{0,\alpha-1\\},1\big{[}$ and every $q\in\mathbb{N}$
$\begin{split}&2^{-q\beta}\left\|F_{q}(v,f)\right\|_{L^{2}}\\\
\leq&C\Big{(}\left\||D|^{1-\beta}v\right\|_{L^{\infty}}\sum_{q^{\prime}\geq
q-4}2^{(q-q^{\prime})(1-\beta)}\left\|\Delta_{q^{\prime}}f\right\|_{L^{2}}+\left\||D|^{\alpha-\beta}f\right\|_{L^{\infty}}\sum_{|q^{\prime}-q|\leq
4}\left\|\Delta_{q^{\prime}}f\right\|_{L^{2}}\Big{)}.\end{split}$ (6.2)
Moreover, when $\beta=0$, $\alpha\in]0,1[$, (6.1) and (6.2) hold if we replace
$\left\||D|^{1-\beta}v\right\|_{L^{\infty}}$ by $\left\|\nabla
v\right\|_{L^{\infty}}$; and when $\beta=1$, $\alpha=2$, then (6.1) and (6.2)
hold if we make such a modification
$\left\||D|^{1-\beta}v\right\|_{L^{\infty}}\rightarrow\left\|v\right\|_{B^{0}_{\infty,1}},\quad\left\||D|^{\alpha-\beta}f\right\|_{L^{2}}\rightarrow\left\|\nabla
f\right\|_{L^{\infty}}.$
###### Proof.
Using Bony decomposition, we decompose $F_{q}(v,f)$ into
$\sum_{i=1}^{6}F^{i}_{q}(v,f)$, where
$F_{q}^{1}(v,f)=(S_{q+1}v-v)\cdot\nabla\Delta_{q}f,\quad
F_{q}^{2}(v,f)=[\Delta_{-1}v,\Delta_{q}]\cdot\nabla f,$
$F_{q}^{3}(v,f)=\sum_{q^{\prime}\in\mathbb{N}}[S_{q^{\prime}-1}\widetilde{v},\Delta_{q}]\cdot\nabla\Delta_{q^{\prime}}f,\quad
F_{q}^{4}(v,f)=\sum_{q^{\prime}\geq-1}\Delta_{q^{\prime}}\widetilde{v}\cdot\nabla\Delta_{q}S_{q^{\prime}+2}f,$
$F_{q}^{5}(v,f)=-\sum_{q^{\prime}\in\mathbb{N}}\Delta_{q}\Big{(}\Delta_{q^{\prime}}\widetilde{v}\cdot\nabla
S_{q^{\prime}-1}f\Big{)},\quad
F_{q}^{6}(v,f)=-\sum_{q^{\prime}\geq-1}\mathrm{div}\Delta_{q}\Big{(}\Delta_{q^{\prime}}\widetilde{v}\sum_{i\in\\{\pm
1,0\\}}\Delta_{q^{\prime}+i}f\Big{)},$
where $[A,B]:=AB-BA$ denotes the commutator operator and
$\widetilde{v}:=v-\Delta_{-1}v$ denotes the high frequency part of $v$.
For $F_{q}^{1}$, from the divergence-free property of $v$ we directly obtain
that when $1-\beta>0$
$\begin{split}2^{-q\beta}\left\|F_{q}^{1}(v,f)\right\|_{L^{2}}&\lesssim\sum_{q^{\prime}\geq
q+1}2^{(1-\beta)(q-q^{\prime})}2^{q^{\prime}(1-\beta)}\left\|\Delta_{q^{\prime}}v\right\|_{L^{\infty}}\left\|\Delta_{q}f\right\|_{L^{2}}\\\
&\lesssim\left\||D|^{1-\beta}v\right\|_{L^{\infty}}\left\|\Delta_{q}f\right\|_{L^{2}}.\end{split}$
For $F_{q}^{2}$, since $F^{q}_{2}(v,f)=\sum_{|q^{\prime}-q|\leq
1}[\Delta_{-1}v,\Delta_{q}]\cdot\nabla\Delta_{q^{\prime}}f$, then from the
expression formula of $\Delta_{q}$ and mean value theorem, we get that when
$\beta>0$
$\begin{split}2^{-q\beta}\left\|F_{q}^{2}(v,f)\right\|_{L^{2}}&\lesssim
2^{-q\beta}2^{-q}\left\|\nabla\Delta_{-1}v\right\|_{L^{\infty}}\sum_{|q^{\prime}-q|\leq
1}2^{q^{\prime}}\left\|\Delta_{q^{\prime}}f\right\|_{L^{2}}\\\
&\lesssim\sum_{-\infty\leq
j\leq-1}2^{j\beta}\left\||D|^{1-\beta}\dot{\Delta}_{j}v\right\|_{L^{\infty}}\sum_{|q^{\prime}-q|\leq
1}\left\|\Delta_{q^{\prime}}f\right\|_{L^{2}}\\\
&\lesssim\left\||D|^{1-\beta}v\right\|_{L^{\infty}}\sum_{|q^{\prime}-q|\leq
1}\left\|\Delta_{q^{\prime}}f\right\|_{L^{2}}.\end{split}$
For $F_{q}^{3}$, similarly as estimating $F_{q}^{2}$, we infer
$\begin{split}2^{-q\beta}\left\|F_{q}^{3}(v,f)\right\|_{L^{2}}&\lesssim
2^{-q\beta}\sum_{|q^{\prime}-q|\leq 4}2^{-q}\left\|\nabla
S_{q^{\prime}-1}\widetilde{v}\right\|_{L^{\infty}}2^{q^{\prime}}\left\|\Delta_{q^{\prime}}f\right\|_{L^{2}}\\\
&\lesssim\sum_{|q^{\prime}-q|\leq 4}\sum_{q^{\prime\prime}\leq
q^{\prime}-2}2^{(q^{\prime\prime}-q^{\prime})\beta}\left\||D|^{1-\beta}\Delta_{q^{\prime\prime}}\widetilde{v}\right\|_{L^{\infty}}\left\|\Delta_{q^{\prime}}f\right\|_{L^{2}}\\\
&\lesssim\left\||D|^{1-\beta}v\right\|_{L^{\infty}}\sum_{|q^{\prime}-q|\leq
4}\left\|\Delta_{q^{\prime}}f\right\|_{L^{2}}.\end{split}$
For $F^{4}_{q}$ and $F^{5}_{q}$, from the spectral property and the fact
$2^{q^{\prime}(1-\beta)}\left\|\Delta_{q^{\prime}}\widetilde{v}\right\|_{L^{\infty}}\approx\left\|\Delta_{q^{\prime}}|D|^{1-\beta}\widetilde{v}\right\|_{L^{\infty}}$,
we have
$\begin{split}2^{-q\beta}\left\|F^{4}_{q}(v,f)\right\|_{L^{2}}\lesssim\sum_{q^{\prime}\geq
q-2}2^{(q-q^{\prime})(1-\beta)}2^{q^{\prime}(1-\beta)}\left\|\Delta_{q^{\prime}}\widetilde{v}\right\|_{L^{\infty}}\left\|\Delta_{q}f\right\|_{L^{2}}\lesssim\left\||D|^{1-\beta}v\right\|_{L^{\infty}}\left\|\Delta_{q}f\right\|_{L^{2}}.\end{split}$
$\begin{split}2^{-q\beta}\left\|F_{q}^{5}(v,f)\right\|_{L^{2}}&\lesssim
2^{-q\beta}\sum_{|q^{\prime}-q|\leq
4}2^{q^{\prime}}\left\|\Delta_{q^{\prime}}\widetilde{v}\right\|_{L^{\infty}}\sum_{q^{\prime\prime}\leq
q^{\prime}-2}2^{q^{\prime\prime}-q^{\prime}}\left\|\Delta_{q^{\prime\prime}}f\right\|_{L^{2}}\\\
&\lesssim\left\||D|^{1-\beta}v\right\|_{L^{\infty}}\sum_{q^{\prime\prime}\leq
q+2}2^{q^{\prime\prime}-q}\left\|\Delta_{q^{\prime\prime}}f\right\|_{L^{2}}.\end{split}$
Besides, for $F^{5}_{q}$ when $v=|D|^{\alpha-1}\mathcal{R}^{\bot}f$, we
alteratively have the following improvement that when $\beta>\alpha-1$
$\begin{split}2^{-q\beta}\left\|F_{q}^{5}(v,f)\right\|_{L^{2}}&\leq
2^{-q\beta}\sum_{|q^{\prime}-q|\leq
4}\left\|\Delta_{q^{\prime}}(Id-\Delta_{-1})|D|^{\alpha-1}\mathcal{R}^{\bot}f\right\|_{L^{2}}\left\|\nabla
S_{q^{\prime}-1}f\right\|_{L^{\infty}}\\\ &\lesssim\sum_{|q^{\prime}-q|\leq
4}\left\|\Delta_{q^{\prime}}f\right\|_{L^{2}}\sum_{-\infty\leq
q^{\prime\prime}\leq
q^{\prime}-2}2^{(\alpha-1-\beta)(q^{\prime}-q^{\prime\prime})}\left\||D|^{\alpha-\beta}\dot{\Delta}_{q^{\prime\prime}}f\right\|_{L^{\infty}}\\\
&\lesssim\left\||D|^{\alpha-\beta}f\right\|_{L^{\infty}}\sum_{|q^{\prime}-q|\leq
4}\left\|\Delta_{q^{\prime}}f\right\|_{L^{2}}.\end{split}$
Finally, for $F^{6}_{q}$ we easily have
$\begin{split}2^{-q\beta}\left\|F^{6}_{q}(v,f)\right\|_{L^{2}}&\lesssim\sum_{q^{\prime}\geq
q-3}2^{(q-q^{\prime})(1-\beta)}\,2^{q^{\prime}(1-\beta)}\left\|\Delta_{q^{\prime}}\widetilde{v}\right\|_{L^{\infty}}\sum_{i\in\\{\pm
1,0\\}}\left\|\Delta_{q^{\prime}+i}f\right\|_{L^{2}}\\\
&\lesssim\left\||D|^{1-\beta}v\right\|_{L^{\infty}}\sum_{q^{\prime}\geq
q-4}2^{(q-q^{\prime})(1-\beta)}\left\|\Delta_{q^{\prime}}f\right\|_{L^{2}}.\end{split}$
Combining the above estimates appropriately yields the inequalities (6.1) and
(6.2).
∎
### 6.3 Proof of Lemma 3.3
###### Proof.
We treat the general $n$-dimensional case. Let
$x=(x_{1},\tilde{x})=(x_{1},x_{2},\cdots,x_{n})$ and the Fourier variable
$\zeta=(\zeta_{1},\tilde{\zeta})=(\zeta_{1},\zeta_{2},\cdots,\zeta_{n})$.
First we observe that for every $\alpha\in]0,2[$ (cf. [4])
$(-|D|^{\alpha})\theta=\frac{d}{dh}e^{-h|D|^{\alpha}}\theta\Big{|}_{h=0}=\frac{d}{dh}\mathcal{P}^{\alpha}_{h,n}*\theta\Big{|}_{h=0}$
where
$\mathcal{P}^{\alpha}_{h,n}(x):=c^{\prime}_{n,\alpha}\frac{h}{(|x|^{2}+\alpha^{2}h^{\frac{2}{\alpha}})^{\frac{n+\alpha}{2}}}$
and $c^{\prime}_{n,\alpha}$ is the normalization constant such that
$\int\mathcal{P}^{\alpha}_{h,n}\textrm{d}x=1(=e^{-h|\zeta|^{\alpha}}|_{\zeta=0})$.
In the following we take $\mathcal{P}_{h,n}$ instead of
$\mathcal{P}^{\alpha}_{h,n}$ for brevity. Thus our task reduces to estimate
$(\mathcal{P}_{h,n}*\theta)(x)-(\mathcal{P}_{h,n}*\theta)(y).$
Due to the translation and rotation invariant properties, we may assume that
$x=(\frac{\xi}{2},0,\cdots,0)$ and $y=(-\frac{\xi}{2},0,\cdots,0)$. Then from
the symmetry and monotonicity of the kernel $\mathcal{P}_{h,n}$ and the fact
$\int_{\mathbb{R}^{n-1}}\mathcal{P}_{h,n}(x_{1},\tilde{x})\textrm{d}\tilde{x}=\mathcal{F}^{-1}(\widehat{\mathcal{P}_{h,n}}|_{\tilde{\zeta}=0})(x_{1})=\mathcal{F}^{-1}(e^{-h|\zeta_{1}|^{\alpha}})(x_{1})=\mathcal{P}_{h,1}(x_{1})$
we have
$\begin{split}&(\mathcal{P}_{h,n}*\theta)(x)-(\mathcal{P}_{h,n}*\theta)(y)\\\
&=\iint_{\mathbb{R}^{n}}\big{[}\mathcal{P}_{h,n}\bigl{(}\frac{\xi}{2}-\eta,-\tilde{\eta}\bigr{)}-\mathcal{P}_{h,n}\bigl{(}-\frac{\xi}{2}-\eta,-\tilde{\eta}\bigl{)}\big{]}\theta(\eta,\tilde{\eta})\textrm{d}\eta\textrm{d}\tilde{\eta}\\\
&=\int_{\mathbb{R}^{n-1}}\textrm{d}\tilde{\eta}\int_{0}^{\infty}\big{[}\mathcal{P}_{h,n}\bigl{(}\frac{\xi}{2}-\eta,\tilde{\eta}\bigr{)}-\mathcal{P}_{h,n}\bigl{(}-\frac{\xi}{2}-\eta,\tilde{\eta}\bigl{)}\big{]}\bigl{[}\theta(\eta,\tilde{\eta})-\theta(-\eta,\tilde{\eta})\bigr{]}\textrm{d}\eta\\\
&\leq\int_{\mathbb{R}^{n-1}}\textrm{d}\tilde{\eta}\int_{0}^{\infty}\big{[}\mathcal{P}_{h,n}\bigl{(}\frac{\xi}{2}-\eta,\tilde{\eta}\bigr{)}-\mathcal{P}_{h,n}\bigl{(}-\frac{\xi}{2}-\eta,\tilde{\eta}\bigl{)}\big{]}\omega(2\eta)\textrm{d}\eta\\\
&=\int_{0}^{\infty}\big{[}\mathcal{P}_{h,1}\bigl{(}\frac{\xi}{2}-\eta\bigr{)}-\mathcal{P}_{h,1}\bigl{(}-\frac{\xi}{2}-\eta\bigl{)}\big{]}\omega(2\eta)\textrm{d}\eta\\\
&=\int_{0}^{\frac{\xi}{2}}\mathcal{P}_{h,1}(\eta)\bigl{[}\omega(2\eta+\xi)+\omega(\xi-2\eta)\bigr{]}\textrm{d}\eta+\int_{\frac{\xi}{2}}^{\infty}\mathcal{P}_{h,1}(\eta)\bigl{[}\omega(2\eta+\xi)-\omega(2\eta-\xi))\bigr{]}\textrm{d}\eta\end{split}$
Because of
$\int_{0}^{\infty}\mathcal{P}_{h,1}(\eta)\textrm{d}\eta=\frac{1}{2}$, we have
the estimate of the difference
$\begin{split}(\mathcal{P}_{h,n}*\theta)(x)-&(\mathcal{P}_{h,n}*\theta)(y)-\omega(\xi)\\\
\leq&\int_{0}^{\frac{\xi}{2}}\mathcal{P}_{h,1}(\eta)\bigl{[}\omega(2\eta+\xi)+\omega(\xi-2\eta)-2\omega(\xi)\bigr{]}\textrm{d}\eta\\\
&+\int_{\frac{\xi}{2}}^{\infty}\mathcal{P}_{h,1}(\eta)\bigl{[}\omega(2\eta+\xi)-\omega(2\eta-\xi)-2\omega(\xi)\bigr{]}\textrm{d}\eta\end{split}$
Hence from the above estimates and the explicit formula of kernel
$\mathcal{P}_{h,1}$, we can conclude that
$\begin{split}&\bigl{[}(-|D|^{\alpha})\theta\bigr{]}(x)-\bigl{[}(-|D|^{\alpha})\theta\bigr{]}(y)\\\
&=\lim_{h\rightarrow
0}\frac{[(\mathcal{P}_{h,n}*\theta)(x)-\theta(x)]-[(\mathcal{P}_{h,n}*\theta)(y)-\theta(y)]}{h}\\\
&=\lim_{h\rightarrow
0}\frac{(\mathcal{P}_{h,n}*\theta)(x)-(\mathcal{P}_{h,n}*\theta)(y)-\omega(\xi)}{h}\\\
&\lesssim_{\alpha,n}\int_{0}^{\frac{\xi}{2}}\frac{\omega(\xi+2\eta)+\omega(\xi-2\eta)-2\omega(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta+\int_{\frac{\xi}{2}}^{\infty}\frac{\omega(2\eta+\xi)-\omega(2\eta-\xi)-2\omega(\xi)}{\eta^{1+\alpha}}\textrm{d}\eta\end{split}$
∎
Acknowledgments: The authors would like to thank Prof. P.Constantin for
helpful advice and discussion. They would also like to express their deep
gratitude to the anonymous referees for their kind suggestions. The authors
were partly supported by the NSF of China (No.10725102).
## References
* [1] H.Abidi, T.Hmidi, On the global wellposedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40(2008), 167-185.
* [2] A.L.Bertozzi and A.J.Majda, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, (2002).
* [3] L.Caffarelli and V.Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equations. Arxiv, math.AP/0608447, To appear in Annals of Math.
* [4] L. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian. Communications in PDE, 32, Issue 8(2007), 1245-1260
* [5] J-Y.Chemin, Perfect incompressible fluids, Clarendon press, Oxford, (1998).
* [6] Q.Chen, C.Miao and Z.Zhang, A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation. Comm. Math. Phys. 271(2007), 821-838
* [7] P.Constantin, A.J.Majda and E.Tabak, Formation of strong fronts in the $2$-D quasigeostrophic thermal active scalar. Nonlinearity 7(1994), 1495-1533.
* [8] P.Constantin and J.Wu, Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30(1999), 937-948.
* [9] P.Constantin, D.Cordoba and J.Wu, On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 50(2001), 97-107.
* [10] P.Constantin, G.Iyer and J.Wu, Global regularity for a modified critical dissipative quasi-geostrophic equation. Indiana Univ. Math. Journal, 57(2008), 2681-2692.
* [11] A. Córdoba and D. Córdoba, A maximum principle applied to the quasi-geostrophic equations. Comm. Math. Phys. 249(2004) 511-528
* [12] R. Danchin and M. Paicu. Global existence results for the anisotropic Boussinesq system in dimension two. Arxiv, math.AP/08094964.
* [13] H.Dong and D.Du, Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete Contin. Dyn. Syst. 21 (2008) no. 4, 1095-1101.
* [14] H. Dong, Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness. Discrete Contin. Dyn. Syst. 26 Issue 4(2010), 1197-1211.
* [15] J. Duoandikoetxea. Fourier analysis(Translated and revised by D. Cruz-Uribe). GSM 29, AMS, Providence, RI, 2001.
* [16] S. Friedlander and V. Vicol. Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics. Arxiv, math.AP/1007.1211v1.
* [17] A.Kiselev, F.Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167(2007), 445-453.
* [18] A. Kiselev, F. Nazarov and R. Shterenberg, On blow up and regularity in dissipative Burgers equation. Dynamics of PDE, 5(2008), 211-240.
* [19] A. Kiselev, Regularity and blow up for active scalars, Math. Model. Math. Phenom. 5 (2010), 225–255.
* [20] N. Ju, Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space. Commun. Math. Phys. 251(2004), 365-376.
* [21] R.May. Global well-posedness for a modified 2D dissipative quasi-geostrophic equation with initial data in the critical Sobolev space $H^{1}$, Arxiv, math.AP/0910.0998v1.
* [22] J.Wu, Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces. SIAM J. Math. Anal. 36(2004), 1014-1030.
* [23] X.Yu, Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation, J. Math. Anal. Appl. 339(2008), 359-371.
|
arxiv-papers
| 2009-01-10T10:34:56 |
2024-09-04T02:48:59.802591
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Changxing Miao and Liutang Xue",
"submitter": "Changxing Miao",
"url": "https://arxiv.org/abs/0901.1368"
}
|
0901.1393
|
# Two-particle Direct Photon-Jet Correlation Measurements in PHENIX
Justin Frantz for the PHENIX Collaboration
(Received: date / Revised version: date)
###### Abstract
Various 2-particle direct photon-hadron correlation yields in $p+p$ and
$Au+Au$ collisions at $\sqrt{s_{NN}}$ = 200 GeV are presented. The per-trigger
yield of direct photon hadron pairs from direct-photon-jet correlations is
obtained by a statistical subtraction of the decay photon pairs from inclusive
photon-hadron sample. The decay photon per-trigger yields are estimated from
the measured $\pi^{0}$-hadron by means of a Monte Carlo based calculation
which takes into account decay kinematics and detector response. Under the
assumption that the suppression is nearly $p_{T}$ independent using a specific
averaging scheme, we find a ratio of $Au+Au$ to $p+p$ per-trigger photon
yields $I_{AA}$ averaged over all available $p_{T}$ bins consistent with the
single particle suppression level $R_{AA}$, which can be interpreted as a
qualitative confirmation of the basic geometrical picture of jet suppression
at RHIC. The application of the event by event photon isolation cuts in $p+p$
results our highest precision measurement yet, and allows for precision
studies of the baseline fragmentation function $D(z)$ and also a determination
of the apparent intrinsic $k_{T}$, or non-zero transverse momentum of the
original collision partons. With a model dependent extraction method, the
average $\sqrt{<k_{T}^{2}>}$ at this $\sqrt{s}$ in $p+p$ is found to be
approximately 2.5-3 GeV, consistent with analysis of di-hadron (di-jet)
correlations [1]. Finally, we present a unique direct measurement of prompt
photons from jet fragmentation.
Hard Probes 2008 Conference Proceedings. June 9th, 2008. Illa da Toxa, Spain
Two-particle Direct Photon-Jet Correlation
Measurements in PHENIX
J. Frantza for the PHENIX Collaboration
a State University of New York (SUNY), Stony Brook, Stony Brook, NY, U.S.A
Contact e-mail: jfrantz@skipper.physics.sunysb.edu
## 1 Introduction
At the Relativistic Heavy Ion Collider (RHIC), experimental results from the
have established the formation of hot and dense matter of a fundamentally new
nature in Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV [2]. One of the most
important probes of this dense colored matter is energy loss by hard partons
leading to a suppression of normal jet production of hard (E $>\sim 1-2$ GeV)
particles. As a complement to di-hadron correlations, which can directly
access di-jet production and their structure [3], [4], direct photon-hadron
correlations can be used to study photon-jet production in the medium without
various biases and complications. This is due to the fact that since it lacks
color charge, the photon escapes the dense QCD matter without interacting. For
this reason, using high $p_{T}$ prompt photons, which have been demonstrated
to be unmodified when traversing the medium due to their lack of color charge
[5], as triggers in $\gamma$ \- hadron ($i.e.$ $\gamma$-jet) correlation
studies has long been considered to be a ”golden channel” to study energy
loss. When the photon trigger is a product of the dominant Leading Order QCD
Compton scattering process, the photon’s 4-momentum should be the same as the
opposite quark parton’s. Therefore the photon in its primordial state can be
used to directly measure the fragmentation functions of the opposite jet which
itself is modified by final state effects in the case of Au+Au collisions.
The above picture is based on a Leading Order (LO) approximation and
interpretation of events where there are always only two ”clean” outgoing
products, di-parton (_g, q_) or photon-parton for $\gamma$-jet, which are
perfectly balanced in momentum in the direction transverse to the incoming
colliding particles. In reality there are a number of complications that could
make this interpretation incorrect that need to be understood even in the
baseline elementary $p+p$ collisions. For example, Next to Leading Order
(NLO)/fragmentation contributions of hard prompt photons in di-jet events
obviously have different properties, in particular in $Au+Au$ are expected to
follow similar suppression effects as the di-jet events which makes making
quantifying the baseline probability for their appearance in the overall
$\gamma$-jet event samples crucial. Another example is the role of an apparent
intrinsic net transverse momentum $k_{T}$ of the incoming colliding parton-
parton system potentially due to initial state effects or non-perturbative or
higher order gluon radiation, which at the value discovered in dihadron
correlations [1] can considerably change the actual momentum transfer
($Q^{2}$) being sampled in the kinematic ranges currently being studied. Thus,
before studying $\gamma$-jet production A+A, one needs to understand these
intrinsic complications to the LO picture in $p+p$ to gauge their effect on
calculations and interpretations of energy loss observables.
This analysis was performed using approximately 950 million Au+Au minimum bias
events from the Run 4 data set and 471 million level-1 photon-triggered events
from the p+p Run 5 and Run 6 data sets. The Beam-Beam Counters (BBC) and Zero-
Degree Calorimeters (ZDC) are used to trigger the minimum bias data. These
detectors are also used to estimate the collision centrality. The p+p photon
trigger requires that a module in the Electromagnetic Calorimeter (EMC) fire
in coincidence with the BBC.
The PHENIX central arms, each covering the 0.7 units of pseudo-rapidity around
mid-rapidity and $90^{\circ}$ in azimuth, contain charge particle tracking
chambers and electromagnetic calorimetry. The EMC consists of two types of
detectors, six sectors of lead-scintillator sampling calorimeters and two of
lead-glass Cherenkov calorimeters. The fine segmentation of the EMC ($\sim
0.01\times 0.01$ for PbSc and $\sim 0.008\times 0.008$ for PbGl) allows for
the reconstruction of the $2\gamma\ $ $\pi^{0}$ and $\eta$ decays to $p_{T}$
$>20$ GeV.
Charged hadrons are detected using a drift chamber at a radial distance of 2.0
m and and a multi-wire proportional chambers (PC1) at a distance of 2.5 m. The
momentum resolution was determined to be $0.7\%\bigotimes 1.0\%p$ (GeV/c).
Secondary tracks from decays and conversions are suppressed by matching tracks
to hits in a second multi-wire chamber (PC3) and the EMC, both at distance of
$\sim 5.0$ m. Track projections to the EMC plane are used to veto charged
hadrons which shower in the EMC.
Two-particle correlations are performed by measuring the azimuthal angle
between photon triggers and charged hadrons. The correlation function
$C(\Delta\phi)\equiv\frac{N_{pairs}(\Delta\phi)}{N_{mixed}(\Delta\phi)}$
corrects for the limited acceptance of photon-hadrons pairs by dividing out
the mixed events distribution. The correlation function is decomposed via the
two-source model where the jet correlation is superimposed on an underlying
event which is modulated by elliptic flow. Hence, the jet function is
expressed as $J(\Delta\phi)\equiv C(\Delta\phi)-b_{0}(1+2\langle
v_{2}^{\gamma}v_{2}^{h}\rangle\cos{2\Delta\phi})$. The underlying event level,
$b_{0}$, is determined by the ZYAM procedure, described elsewhere [6].
We define a direct photon as any photon not from a decay process. It follows
that the per-trigger yield ($Y\equiv 1/N_{trigger}\ J(\Delta\phi)$) for direct
photons may be obtained by a statistical subtraction of the decay per-trigger
yield from the inclusive per-trigger yield according to:
$Y_{direct}=\frac{R_{\gamma}Y_{inclusive}-Y_{decay}}{R_{\gamma}-1}$ (1)
where $R_{\gamma}$$\equiv N_{inclusive}/N_{decay}$. $R_{\gamma}$ is determined
for Au+Au collisions in [5] and is derived from the $\pi^{0}$ [7] and direct
photons spectra in p+p [8]. The direct photon yields from the statistical
subtraction method do not, by definition, exclude photons from jet
fragmentation or medium induced photon production.
The decay photon per-trigger yields are determined from the parent ($\pi^{0}$
or $\eta$) per-trigger yields via a Monte Carlo procedure. A flat distribution
of parent mesons are decayed into the PHENIX aperture uniformly in the z and
phi directions. This determines the mapping of the parent to daughter $p_{T}$,
$\wp(p_{T}^{\pi^{0}}\rightarrow p_{T}^{\gamma})$, where $p_{T}^{\gamma}$ is
smeared by the detector resolution. In order to reproduce the correct
$p_{T}$dependence of the decay photon distribution W is applied as a weight
factor to the parent meson-hadron $p_{T}$distribution on a pair-by-pair basis.
The finite reconstruction efficiency of the parent mesons is corrected for
using the published PHENIX $p_{T}$ spectra [7], [9]. The decay per-trigger
yield from $\pi^{0}$’s can then be expressed in terms of the parent per-
trigger yield.
## 2 Constraining Direct Photon-Jet Production in Elementary $p+p$ Collisions
### 2.1 Isolation Cut Analysis
In addition to the above statistical subtraction method, standard photon
isolation cuts were applied event by event in a new analysis in order to
dramatically reduce the contamination of di-jet events with $\pi^{0}$ decay or
fragmentation photons in the $\gamma$-hadron event sample. To be considered
isolated, the sum of $p_{T}$from all tracks and and EMCal energies must be
$<0.1E_{\gamma}$ in a cone around the photon of size $\Delta
R=\sqrt{\Delta\phi^{2}+\Delta\eta^{2}}=0.5$. Statistical subtraction of the
remaining contribution for isolated $\pi^{0}$ production is achieved through a
modified version of the statistical method above where isolated $\pi^{0}$ are
used as input to the decay-photon calculation.
As in the statistical method the analysis is performed as a function of
$\Delta\phi$. We find good agreement with the statistical method results as
shown in figure 1. Since the statistical subtraction method includes, in
principle, the contribution from fragmentation photon triggers, the agreement
between the two methods places a constraint on the magnitude of such a
contribution.
Figure 1: For two trigger photon bins as indicated on the figure, the awayside
per-trigger yield vs. $\Delta\phi$ for the statistical and isolation cut
analysis. The two methods give consistent results with the isolation method
having much improved precision due to having only a small decay photon
background subtraction.
### 2.2 Per trigger yields and Awayside Jet Fragmentation Function Analysis
Results from the isolation cut analysis are shown in figure 2, plotted in
terms of the fragmentation variables $z_{T}=p_{T}^{assoc}/p_{T}^{\gamma}$ and
$x_{E}=z_{T}\cos{\Delta\phi}$ (taking only the component of the associated
hadron momentum that is in the same direction as the trigger photon). We
measure in 6 photon trigger $p_{T}$ bins covering the range 5-15 GeV/c. The
inverse slope parameter of exponential fits to the $x_{E}$ distribution with
$\pi^{0}$ and direct photon triggers are shown in figure 3. Comparisons with
theoretical calculations are currently underway.
Figure 2: Per trigger yields of direct photon-hadron pairs as a function of
$x_{E}/z_{T}$ for various selections of trigger $p_{T}$. The data have been
scaled by factors of 10 for visibility. Figure 3: The inverse slope parameter
of exponential fits to the $x_{E}$ distributions for $\pi^{0}$ direct photon
triggers.
### 2.3 Model Dependent Determination of $k_{T}$
Also in [1], a method for extracting the apparent intrinsic $k_{T}$ itself was
described and used to extract a value of $k_{T}$= $2.68\pm 0.35$ for
$\pi^{0}$-h correlations. Using isolated direct photon triggers, the method is
simplified and can be also be used to extract a $k_{T}$ value in this channel.
Please refer [1] for details of the method. The simplification occurs due to
the lack of an additional fragmentation variable on the nearside, i.e
$p_{Tt}\equiv\hat{p}_{Tt}\equiv p_{T\gamma}$. The measured experimental
variable $p_{out}$ whose distribution for various trigger $p_{T}$ bins is
shown in figure 4, is defined as $p_{Ta}\sin{\Delta\phi}$, or the associated
hadron’s transverse momentum component perpendicular to the trigger photon
direction, and is proportional to the $k_{T}$ on an event by event basis.
Figure 4: $p_{out}$ distributions for various values of trigger $p_{T}$. The
data have been scaled by factors of 10 for visibility.
By finding the average $p_{out}$ for each trigger photon $p_{T}$ bin as shown
in 5 $a)$ one can extract the quantity average $k_{T}$/$\hat{x}$ where
$\hat{x}$ is just the ratio of the true awayside jet momentum to that of the
direct photon trigger. This can be extracted from other measurements as in [1]
and eliminated, leaving a pure measurement of the $k_{T}$. For now, we simply
use the event generator PYTHIA $6.3$ to generate direct photon processes with
initial and final state radiation turned off, and phenomenological $k_{T}$
parameters as shown on 5 $b)$ to extract $\hat{x}$ and make a comparison to
PYTHIA, which shows that for PYTHIA-like distributions of $\hat{x}$ a $k_{T}$
parameter in the vicinity of 2.5-3 GeV gives values similar to the data, well-
consistent with the value of 2.68 found for di-hadron di-jet correlations.
This implies that the same $k_{T}$ bias effects in the di-hadron correlations
exist also in the direct photon-jet correlations. This makes the comparison
with the perturbative QCD calculations, which at only NLO should _not_ contain
such $k_{T}$ bias modifications even more important, as well as understanding
in energy loss models of $Au+Au$ how such a large value of imbalance between
the trigger photon and awayside jet might alter energy loss interpretations.
Figure 5: a) $\sqrt{<p_{out}^{2}>}$ vs trigger $p_{T}$ for direct photons b)
$k_{T}$/$\hat{x}$ vs trigger $p_{T}$ for direct photons along with
calculations from PYTHIA $6.3$.
### 2.4 Direct Measurement of Prompt Photons from Jet Fragmentation
PHENIX has made further strides towards understanding complications to the LO
intepretations of direct photon-jet correlations, in studying the contribution
of single prompt photons that occur in di-jet events, the so-called
fragmentation photons produced by NLO hard photon radiation or non-
perturbative fragmentation processes. A direct measurement technique discussed
in [10] has been used to measure the angular distribution of such photons with
respect to trigger _hadrons_ in events where high $p_{T}$ photons are tagged
to only be nearby such hard hadrons–an _anti_ -isolation selection.
Integrating this distribution one can find the fraction of such photons to
total hadron-correlated photons from all sources including decay, shown in
figure 6 which can possibly be related to the fraction of the total direct
photon production rate due to these fragmentation photons. However due to the
restricted kinematic region of the measurement, interpretations of this
fraction and the angular distribution itself need input from sophisticated
higher order pQCD calculations (which likely do not yet exist) and thus close
attention from the theory community. Nonetheless, the measurement is exciting
because it provides the first step towards making the measurement in the
$Au+Au$ environment where several interesting predictions of jet-medium
enhancement of the rate of production of such photons exist.
Figure 6: Fraction of hadron-photon that contain a prompt (fragmentation)
photon
## 3 Constraining Energy Loss theory in Au+Au with $\gamma$-jet
In previous conferences [11, 12] PHENIX has presented $Au+Au$ results of the
statistical subtraction method for direct photon-hadron correlation yields.
With the now statistically improved $p+p$ results (for this section, using the
pure statistical subtraction method in $p+p$ for comparison, combining Run6
and Run6 statistics) and an expanded $p_{T}$ range for the trigger photons in
$Au+Au$, we can now divide the per-trigger yields for $Au+Au$ and $p+p$in many
$p_{T}$ bins making the ratio $I_{AA}$ = $Y_{direct}^{A+A}/Y_{direct}^{p+p}$
for an awayside integration range of $2\pi/5$ radians around $\pi$. An example
is shown for the direct photon (trigger) $p_{T}$ bin 5-7 GeV/c in figure 7. It
is apparent that there are very large uncertainties, but most points have
positive yield in $Au+Au$ and a value of $I_{AA}$ between 0 and 1 which
indicates the basic expectation of suppression of the awayside jet.
Figure 7: Ratio $I_{AA}$ (see text) for the lowest trigger direct photon
$p_{T}$ bin.
We find that different sources of uncertainties dominate in different ranges
of trigger photon and associated hadron hadron $p_{T}$. As either $p_{T}$ is
increased the statistics in the raw correlation functions obviously decrease
as the production probabilities follow steeply falling spectra. However the
combinatoric background from pairs of hard particles and soft underlying or
otherwise uncorrelated particles is reduced dramatically as the $p_{T}$ of the
associated hadron is increased, having the effect of reducing the systematic
uncertainty from the underlying event subtraction. Further, as the $p_{T}$ of
the trigger photon is increased there is a larger fraction of direct photon-h
pairs in the inclusive photon-h pair sample due to the increasing direct
photon signal relative to the suppressed decay photon background, and thus the
higher trigger photon $p_{T}$ bins also have a reduction in the uncertainty
from the subtraction in equation 1. Because of these effects we find that the
total uncertainty remains more constant with increasing $p_{T}$ than the loss
in statistical precision that the falling production probability would
normally manifest. For this reason, integrating over large $p_{T}$ hadron or
trigger $p_{T}$ bins causes a loss of information since the steeply falling
production probability causes uncertainties to be dominated by the source
corresponding to the low end of the $p_{T}$ bin. Therefore in order to
effectively give the higher $p_{T}$ bins more weight, we take the plain
average of all $p_{T,\gamma},p_{T,h}$ bins. We call this average the Mean
Value $I_{AA}$ akin to performing the functional mean value:
$MeanValue\;\;I_{AA}=\frac{1}{{\Delta
p_{T}}}\int{I_{AA}(p_{T}^{assoc})dp_{T}}$ (2)
Under the further assumption that $I_{AA}$ remains constant with $p_{T}$ this
average indeed corresponds exactly to the functional mean value, and it is
found that this assumption is satisfied to what is likely a sufficient degree
(considering our large uncertainties which would dominate any such
uncertainties in the assumption) both in measurements of di-hadron
correlations [4] and [3] and in most theoretical predictions [13, 14] at
sufficiently high $p_{T}$ (associated $p_{T}$ $>\simeq$ GeV/c).
Figure 8: a) Mean value $I_{AA}$ for each trigger $p_{T}$ bin b) over all
$p_{T}$ bins for 0-20%, 20-40%, and 40-92% centrality bins.
Results of the Mean Value $I_{AA}$ are shown in figure 8. The transition of
different uncertainty sources from systematic to statistical is even more
apparent in figure 8 $a)$ for central events, and indeed the Mean Value
$I_{AA}$ seems to favor a constant value _vs._ trigger direct photon $p_{T}$
although the uncertainties are still much too large to rule out with any
significance rather large possible trigger photon $p_{T}$ dependence
scenarios. Still, again taking the average of _all_ $I_{AA}$ points together,
shown on figure 8 $b)$, now also for more peripheral bins, we find that the
Mean Value $I_{AA}$ for the central events is significantly positive at a two-
sigma level and consistent with the single particle suppression level
$R_{AA}$.
This confirms the basic geometrical picture of energy loss at RHIC [15],[16]
because vertices in $\gamma$-h correlations, and the trigger direct photon
themselves, come from the entire collision volume. Therefore the fraction of
$\gamma$-$h$ pairs that are observed without significant suppression, come
from the same geometrical region that surviving single high $p_{T}$ particles
do. Dividing this quantity by the unsuppressed direct photon yield is
analogous to dividing the surviving single particle production rate by the
expectation from $p+p$ multiplied over the entire production weighted
geometric volume of the initial distribution of all hard scattering production
points, as is done in the construction of $R_{AA}$Ṫhis means that the away-
side suppression in the $\gamma-jet$ channel should simply reflect the same
geometry as the single particle picture and, if geometry plays a dominant
role, give the same suppression level. Since this geometrical picture is
believed to be a ”surface bias” picture where the only di-jets initially
produced near the surface it could be a confirmation of the surface bias
picture, although further comparison to the di-hadron $I_{AA}$ comparing
$\gamma$-hadron and di-hadron correlations together in the same energy loss
framework, are necessary to make this statement, in order to rule out possible
effects in the di-hadron $I_{AA}$ that are believed _not_ to be from geometry.
Looking forward to the full release of our Run7 analysis, we also present a
first look in figure 9 at the lowest trigger $p_{T}$ bin using $\simeq$ 3.0
billion events, which entices us to look across several $p_{T}$ bins to look
for consistent behavior that may be consistent to jet-medium displaced peak as
seen already in di-hadron correlations, although at this point uncertainties
are too large to make a definitive statement with just a single $p_{T}$ bin
combination.
Figure 9: $\Delta\phi$ distribution of direct photon-hadron correlations from
Run7 data for 5-7 trigger direct photon bin.
## 4 Conclusions
Various 2-particle direct photon-hadron correlation yields in $p+p$ and
$Au+Au$ collisions at $\sqrt{s_{NN}}$ = 200 GeV have been presented. Under the
assumption that the suppression is nearly $p_{T}$ independent using a specific
averaging scheme, we find a ratio of $Au+Au$ to $p+p$ per-trigger photon
yields averaged over all available $p_{T}$ bins, $I_{AA}$, consistent with the
single particle suppression level $R_{AA}$, which can be interpreted as a
qualitative confirmation of the basic geometrical picture of jet suppression
at RHIC. To the extent that the prevailing geometric picture is believed to be
that of surface bias where only jets ejected near the surface dominately
contribute to yields, [15], [16], it may be said to confirm this picture,
however it should be noted that any geometric scenario would yield $I_{AA}$
$\simeq$ $R_{AA}$. Nonetheless future comparisons of $I_{AA}$ from
$\gamma$-jet to that of di-jets, may indeed yield some insight into the
details of possible surface bias models.
We have also presented the first event by event isolation cut 2-p correlation
results at RHIC. The application of the event by event photon isolation cuts
in $p+p$ results our highest precision measurement yet, and allows for
precision studies of the baseline fragmentation function $D(z)$, and well as a
variable $p_{out}$ which is proportional to the apparent intrinsic $k_{T}$, or
non-zero transverse momentum of the original collision partons. With a model
dependent extraction method, the average $\sqrt{<k_{T}^{2}>}$ at this
$\sqrt{s}$ in $p+p$ is found to be approximately 2.5-3 GeV, consistent with
analysis of di-hadron (di-jet) correlations [1]. The possible implications of
this and also the improved precision in the isolated yields warrant detailed
comparison with the baseline perturbative QCD (pQCD) calculations used in the
various models of jet energy loss. Finally, we have presented a unique direct
measurement of single prompt photons from jet fragmentation, of both angular
information and the fraction of these photons in the entire photon sample
(including decay photons) in the vicinity of the jet cone and for specific
$p_{T}$ cuts.
## References
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|
arxiv-papers
| 2009-01-10T22:48:47 |
2024-09-04T02:48:59.817413
|
{
"license": "Public Domain",
"authors": "Justin Frantz",
"submitter": "Justin Frantz",
"url": "https://arxiv.org/abs/0901.1393"
}
|
0901.1412
|
#
Understanding light scalar meson by color-magnetic wavefunction in QCD sum
rule
Yi Pang1 yipang@itp.ac.cn Mu-Lin Yan2 mlyan@ustc.edu.cn 1Kavli Institute for
Theoretical Physics China, Key Laboratory of Frontiers in Theoretical Physics,
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
P.R.China,
2Interdisciplinary Center of Theoretical Studies, USTC, Hefei, Anhui 230026,
P.R.China
###### Abstract
In this paper, we study the $0^{+}$ nonet mesons as tetraquark states with
interpolating currents induced from the color-magnetic wavefunction. This
wavefunction is the eigenfunction of effective color-magnetic Hamiltonian with
the lowest eigenvalue, meaning that the state depicted by this wavefunction is
the most stable one and is most probable to be observed in experiments. Our
approach can be recognized as determining interpolating currents dynamically.
We perform an OPE calculation up to dimension eight condensates and find that
the best QCD sum rule is achived when the current induced from the color-
magnetic wavefunction is a proper mixture of the tensor and pseudoscalar
diquark-antidiquark bound states. Compared with previous results, to
sigma(600) and kappa(800), our results appear better, due to larger pole
contribution. The direct instanton contribution are also considered, which
yields a consistent result with previous OPE results. Finally, we also discuss
the $\eta^{\prime}$ problem as a possible six-quark state.
###### pacs:
12.38.Cy, 12.38.Lg, 12.39.Mk, 12.40.Yx
††preprint: USTC-ICTS-09-01
## I Introduction
In past decades, the question how to validly interpret scalar mesons with
their mass below 1 GeV stimulated many discussions and controversies amsler .
In the naive constituent quark model, they are expected to be $SU(3)_{f}$
nonet consisting of a quark and an antiquark, with one unit of orbital
excitation for positive parity. However, due to the fact that the orbital
excitation contributes energy about 0.5 GeV, it is difficult to interpret
their light mass as well as their mass spectrum jaffe1 . Moreover,
$a_{0}(980)$ and $f_{0}(980)$ couple to $K\bar{K}$ channel strongly, which is
in contradictory to the prediction by naive $q\bar{q}$ mesons picture. This
situation very naturally leads to alternative interpretation about these
mesons, such as tetraquark states jaffe2 ; maiani ; brito ; wang ; zhu1 ; zhu2
; zhu3 ; zhu4 ; lee1 ; lee2 ; lee3 ; lee4 ; Kojo ; Matheus ; Zhang ; Latorre ,
which were put forward many years ago in jaffe3 ; jaffe4 . Recently, ’t Hooft
et. al. thooft and authors of schechter found out new evidence from the
instanton induced effective Lagrangian, implying that the predominant
component of light scalar meson is tetraquark.
In 1977, using the MIT bag model jaffe5 , Jaffe suggested the existence of a
light scalar nonet with masses below 1 GeV jaffe3 ; jaffe4 . This nonet is
composed by bound states of diquark and antidiquark. The dominant interaction
generating the bound state is from one-gluon exchange which induces the
following effective Hamiltonian
$H_{eff}=-\widetilde{C}\sum_{i\neq
j}(\lambda_{i}\cdot\lambda_{j})(\overrightarrow{\sigma}_{i}\cdot\overrightarrow{\sigma}_{j}),$
(1)
where $\widetilde{C}>0$ is the strength factor constant,
$\overrightarrow{\sigma}_{i}$ and $\lambda_{i}$ are $2\times 2$ Pauli matrices
and $3\times 3$ Gell-Mann color operators for the $i$th quark. This is a
simple generalization of the Breit spin-spin interaction to include a similar
color-color piece. It is also known as “color-magnetic” or “color-spin”
interaction of QCD, which was first discussed in the pioneering work of De
Rujula, Georgi, Glashow DGG . Hereafter, we will call the eigenstates of
$H_{eff}$ as color-magnetic eigenstates. The eigenfunctions and corresponding
eigenvalues of $H_{eff}$ for $q^{2}\bar{q}^{2}$ system (tetraquark) have been
presented in jaffe3 ; jaffe4 . In these work, the eigenstate with the largest
mass defect is
$|0^{+},\underline{9}\rangle=0.972\;|0^{+}\underline{9}[1]\rangle+0.233\;|0^{+}\underline{9}[405]\rangle,$
(2)
with
$H_{eff}|0^{+},\underline{9}\rangle=-43.36\widetilde{C}|0^{+},\underline{9}\rangle.$
(3)
where $0^{+}$ stands for the $J^{P}$, $\underline{9}$ denotes flavor
$SU(3)_{f}$-nonet, and $\underline{9}[1]$ ($\underline{9}[405]$) represents
the nonet belonging to $[1]$-representation ($[405]$-representation) of color-
spin $SU(6)_{CS}$. Explicitly, they are
$\displaystyle|0^{+}\underline{9}[1]\rangle$ $\displaystyle=$
$\displaystyle\sqrt{6\over
7}|(6,3)\underline{\bar{3}};\;(\bar{6},\bar{3})\underline{3};\;(1,1)\rangle+\sqrt{1\over
7}|(\bar{3},1)\underline{\bar{3}};\;(3,1)\underline{3};\;(1,1)\rangle,$ (4)
$\displaystyle|0^{+}\underline{9}[405]\rangle$ $\displaystyle=$
$\displaystyle\sqrt{1\over
7}|(6,3)\underline{\bar{3}};\;(\bar{6},\bar{3})\underline{3};\;(1,1)\rangle-\sqrt{6\over
7}|(\bar{3},1)\underline{\bar{3}};\;(3,1)\underline{3};\;(1,1)\rangle.$ (5)
In the right hand side of above equations, there is state of
$|(6,3)\underline{\bar{3}};\;(\bar{6},\bar{3})\underline{3};\;(1,1)\rangle$,
where $(6,3)\underline{\bar{3}}$ indicates that the diquark is in 6-dimension
symmetric representation of color $SU(3)_{C}$ with spin $S=1$ (so $2S+1=3$),
and in 3-dimension $\bar{3}$ representation of flavor $SU(3)_{f}$. While
$(\bar{6},\bar{3})\underline{3}$ means the antidiquark is in the conjugate
representation. And (1,1) means the bound state of diquark and antidiquark is
singlet both in color and spin. In the following, without ambiguity, the
diquark and antidiquark will be denoted according to their $SU(3)_{C}$
representations. For example, $\mathbf{6_{c}}$ diquark signifies the diquark’s
wavefunction is $(6,3)\underline{\bar{3}}$. Similarly,
$|(\bar{3},1)\underline{\bar{3}};\;(3,1)\underline{3};\;(1,1)\rangle$ is
comprised of spin-0 $\mathbf{\bar{3}_{c}}$ diquark and $\mathbf{3_{c}}$
antidiquark.
Basing on Eq. (3), Jaffe claimed that the scalar tetraquarks with masses below
1GeV exist and the color-spin part of their wavefunctions can be described by
$|0^{+}\underline{9}\rangle$. Utilizing the latest data, Jaffe’s statement
could be roughly checked for a visual comprehension. For instance, a data fit
of charmed baryons determines the constituent quark masses hogaasen1 ;
hogaasen2 ; dy :
$m^{c}_{u}\approx m^{c}_{d}\approx 360{\rm MeV}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ m^{c}_{s}\approx 540{\rm MeV},$ (6)
where $c$ is the abbreviation of “constituent”. The strength factor constants
related to the light quarks are
$\displaystyle\widetilde{C}$ $\displaystyle\approx$
$\displaystyle\widetilde{C}_{qq}\approx 20{\rm MeV},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\rm with}\;\;q\in\\{u,d\\},$ (7)
$\displaystyle(\widetilde{C}_{qs}$ $\displaystyle=$ $\displaystyle 15{\rm
MeV},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\widetilde{C}_{ss}=10{\rm MeV}).$
Then, if we assume $\sigma(600)$ as one member of $0^{+}$-tetraquark nonet,
the mass of $\sigma(600)$ could be roughly estimated:
$m_{\sigma}\approx\langle\sum_{i}m_{i}^{c}-\widetilde{C}\sum_{i\;j}(\lambda_{i}\cdot\lambda_{j})(\overrightarrow{\sigma}_{i}\cdot\overrightarrow{\sigma}_{j})\rangle_{\sigma}\approx
4\times 360{\rm MeV}-43.36\times 20{\rm MeV}\approx 573{\rm MeV}.$ (8)
Obviously, Jaffe’s claim is reasonable, and the underlying dynamical
consideration should be legitimate. Therefore, it is interesting to study
Jaffe’s tetraquark in the framework of QCD sum rule which relates the
nonperturbative aspects of QCD to the hadronic physics shifman ; reinders . In
other words, we will try to obtain a legitimate QCD sum rule for tetraquarks
in terms of their color-magnetic eigenfunctions.
QCD Sum Rule (SR) analysis for scalar nonet mesons as tetraquarks has been
widely discussed in the literature (e.g., see brito ; wang ; zhu1 ; zhu2 ;
zhu3 ; zhu4 ; lee1 ; lee2 ; lee3 ; lee4 ; Kojo ; Matheus ; Zhang ; Latorre ).
Since the correlator of tetraquark-type current operator for SR has higher
energy dimension than that of ordinary baryon-type one, the operator product
expansion (OPE) must be considered up to higher dimensional operators
(condensates) than ordinary baryons. Technically, it has been widely accepted
that the OPE contributions from condensates of dimensions higher than eight
are very small for tetraquarks lee2 . To single scalar tetraquark current, it
has been shown in lee1 that the contributions from the dimension eight
condensates are unexpectedly large and become dominant in the left hand sum
rule. What is worse, their negative contributions break down the physical
meaning of the left hand sum rule. In order to solve this problem, in lee2 ,
the authors demonstrated that the current including equal weight of scalar and
pseudoscalar diquark-antidiquarks leads to a strong cancelation of the
contributions from dimension eight operators in the OPE, and then gives a good
sum rule. In zhu2 , by assuming mixing of single tetraquark currents, the
authors performed a SR analysis for low-lying $0^{+}$-mesons as tetraquarks.
However, by now, all work on tetraquark SR has not considered a basic question
that whether the color-spin-flavor structures of the tetraquark-type currents
in SR are consistent with the color-magnetic hyperfine interaction mechanism
on tetraquarks. The aim of this paper is to pursue this question.
The key point of this paper is that we think the interpolating current used in
SR should inherit a color-spin-flavor structure from the color-magnetic
wavefunction. This means that we treat a current standing for linear
combination of $\mathbf{3_{c}}$-$\mathbf{\bar{3}_{c}}$ and
$\mathbf{6_{c}}$-$\mathbf{\bar{6}_{c}}$ tetraquarks as the SR interpolating
current. We emphasize that this combination or mixture of
$\mathbf{3_{c}}$-$\mathbf{\bar{3}_{c}}$ and
$\mathbf{6_{c}}$-$\mathbf{\bar{6}_{c}}$ tetraquarks is determined dynamically
by Eq. (3) without any additional ad hoc assumptions. Due to the non-
relativistic nature of color-magnetic interaction, it should be aware of that
the induced mixture is specific to energy scale around 1GeV, which is mass
scale of mesons we are interested in. In short, our method is based on the
well established concept that color-magnetic hyperfine interactions play a
crucial role in multiquark physics.
The strategy of the calculation is what follows. At the first step, we will
study the properties of the scalar tetraquark $SU(3)_{f}$-nonet as color-
magnetic eigenstate with the largest mass defect in QCD sum rule by OPE
expansion. With the method presented in Section 2, we construct interpolating
currents that can represent the color-magnetic structure of tetraquark. Then
utilizing these currents, and following the standard procedure for
tetraquark’s OPE calculations zhu1 ; zhu2 ; zhu3 ; zhu4 ; lee1 ; lee2 ; lee3 ;
lee4 , we obtain the contributions from the operators up to dimension eight.
Meanwhile, to achieve a reliable sum rule, we require that the pole
contributions should reach around $50\%$. Then we obtain $\sigma$ meson mass
$(600\pm 75)$MeV.
In addition, the instanton effects, in other words the topological
fluctuations of gluon fields, play an important role in the structure of QCD
vacuum schafer and spectroscopy of multiquark hadrons dorokhov ; schafer1 .
So they should be taken into account in the SR calculations. Combining the
contribution from OPE and instanton, we obtain $\sigma$ mass about 720 MeV
close to previous OPE results. At this stage, a complete sum rule description
of $0^{+}$ nonet meson has been obtained by us, including both the OPE and
instanton effects.
The paper is organized as follows. In Section II, we will deduce the
interpolating currents for $0^{+}$ tetraquarks from their color-magnetic
wavefunctions. In Section III, the analytic results of OPE calculation based
on previous currents will be presented, followed by the numerical results. In
Section IV, the single direct instanton contribution will be considered. In
Section V, we summarize the results briefly and make a speculation on the
extension of our method to study mesons with 6 quarks (Fermi-Yang meson). In
appendix, we will list some necessary formulas of spectral functions and
correlators.
## II Interpolating current for Jaffe tetraquark
Substituting Eqs. (4) and (5) into (2), we obtain the expression of the color-
magnetic wavefunction for Jaffe’s $0^{+}$ tetraquark nonet meson as follows
$|0^{+},9\rangle=0.988|(6,3)\underline{\overline{3}};(\overline{6},\overline{3})\underline{3};(1,1)\rangle+0.157|(\overline{3},1)\underline{\overline{3}};(3,1)\underline{3};(1,1)\rangle.$
(9)
The elements for $|0^{+},9\rangle$ are $\mathbf{6_{c}}$,
$\mathbf{\bar{3}_{c}}$ diquarks and $\mathbf{{\bar{6}}_{c}}$,
${\mathbf{3}_{c}}$ anti-diquarks. Generally, the composite operator for a
diquark with certain structure of color, flavor and spin is
$\sum_{\\{a\leftrightarrow b\\},\\{i\leftrightarrow
j\\}}(-1)^{P_{c}}(-1)^{P_{f}}q^{(i)\;T}_{a}C\Gamma q^{(j)}_{b},$ (10)
where $\\{a,b\\}$ and $\\{i,j\\}$ are color and flavor indices of quarks
respectively. Specifically,
$q^{(1)}_{a}=u_{a},\;q^{(2)}_{a}=d_{a},\;q^{(3)}_{a}=s_{a}$. $C$ is the charge
conjugation operator, and $\Gamma$ is Dirac matrix determined by the spin of
the system. $(-1)^{P_{c}}$ and $(-1)^{P_{f}}$ reflect the parities of the
diquark’s color and flavor wavefunctions respectively. As for wavefunctions
being symmetric in color or flavor, $P_{c}=0$, or $P_{f}=0$, and for anti-
symmetric ones, $P_{c}=1$ or $P_{f}=1$. Notation $\\{a\leftrightarrow
b\\},\\{i\leftrightarrow j\\}$ represent the color and flavor permutations
respectively. Since
$|(6,3)\underline{\overline{3}};(\overline{6},\overline{3})\underline{3};(1,1)\rangle$
signifies that the diquark and anti-diquark are symmetric in color and anti-
symmetric in flavor, the composite operator of $\mathbf{6_{c}}$ diquark can be
written as
$q^{(i)\;T}_{a}C\Gamma q^{(j)}_{b}-q^{(j)\;T}_{a}C\Gamma
q^{(i)}_{b}+q^{(i)\;T}_{b}C\Gamma q^{(j)}_{a}-q^{(j)\;T}_{b}C\Gamma
q^{(i)}_{a}.$ (11)
In the non-relativistic limit of diquark bispinor $q^{T}C\Gamma q$, spin-1
requires that
$\Gamma=\\{\sigma^{\mu\nu},\leavevmode\nobreak\
\gamma^{\mu},\leavevmode\nobreak\
\gamma^{\mu}\gamma^{5}\\}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
with}\;\;\sigma^{\mu\nu}={i\over
2}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}).$ (12)
Then, inserting (12) into (11), we obtain all possible composite operators for
$\mathbf{6_{c}}$ spin-1 diquark expressed as below,
$\displaystyle Q_{T}^{(ij)}(6)$ $\displaystyle\equiv$ $\displaystyle{1\over
2\sqrt{2}}(q^{(i)\;T}_{a}C\sigma^{\mu\nu}q^{(j)}_{b}-q^{(j)\;T}_{a}C\sigma^{\mu\nu}q^{(i)}_{b}+q^{(i)\;T}_{b}C\sigma^{\mu\nu}q^{(j)}_{a}-q^{(j)\;T}_{b}C\sigma^{\mu\nu}q^{(i)}_{a})$
(13) $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}(q^{(i)\;T}_{a}C\sigma^{\mu\nu}q^{(j)}_{b}-q^{(j)\;T}_{a}C\sigma^{\mu\nu}q^{(i)}_{b}),$
$\displaystyle Q_{A}^{(ij)}(6)$ $\displaystyle\equiv$ $\displaystyle{1\over
2\sqrt{2}}(q^{(i)\;T}_{a}C\gamma^{\mu}q^{(j)}_{b}-q^{(j)\;T}_{a}C\gamma^{\mu}q^{(i)}_{b}+q^{(i)\;T}_{b}C\gamma^{\mu}q^{(j)}_{a}-q^{(j)\;T}_{b}C\gamma^{\mu}q^{(i)}_{a})$
(14) $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}(q^{(i)\;T}_{a}C\gamma^{\mu}q^{(j)}_{b}-q^{(j)\;T}_{a}C\gamma^{\mu}q^{(i)}_{b}),$
$\displaystyle Q_{B}^{(ij)}(6)$ $\displaystyle\equiv$ $\displaystyle{1\over
2\sqrt{2}}(q^{(i)\;T}_{a}C\gamma^{\mu}\gamma^{5}q^{(j)}_{b}-q^{(j)\;T}_{a}C\gamma^{\mu}\gamma^{5}q^{(i)}_{b}+q^{(i)\;T}_{b}C\gamma^{\mu}\gamma^{5}q^{(j)}_{a}-q^{(j)\;T}_{b}C\gamma^{\mu}\gamma^{5}q^{(i)}_{a})$
(15) $\displaystyle=$ $\displaystyle 0.$
where $1/(2\sqrt{2})$ is a widely adopted normalization. Likewise, the
composite operators of $\mathbf{\bar{6}_{c}}$ spin-1 antidiquark are
$\displaystyle\overline{Q}_{T}^{(ij)}(6)$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}(\bar{q}^{(i)}_{a}\sigma_{\mu\nu}C\bar{q}^{(j)\;T}_{b}-\bar{q}^{(j)}_{a}\sigma_{\mu\nu}C\bar{q}^{(i)\;T}_{b}),$
(16) $\displaystyle\overline{Q}_{A}^{(ij)}(6)$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}(\bar{q}^{(i)}_{a}\gamma_{\mu}C\bar{q}^{(j)\;T}_{b}-\bar{q}^{(j)}_{a}\gamma_{\mu}C\bar{q}^{(i)\;T}_{b}),$
(17) $\displaystyle\overline{Q}_{B}^{(ij)}(6)$ $\displaystyle=$ $\displaystyle
0.$ (18)
Because
$|(\overline{3},1)\underline{\overline{3}};(3,1)\underline{3};(1,1)\rangle$
means that the diquark and antidiquark are anti-symmetric in color, spin and
flavor. The composite operators for $\mathbf{\bar{3}_{c}}$ spin-0 diquarks
belonging to representation $(\bar{3},1)\underline{\overline{3}}$ of
$SU(6)_{cs}\times SU(3)_{f}$ are the following ones,
$\displaystyle{Q}_{S}^{(ij)}(3)$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}({q}^{(i)\;T}_{a}C\gamma^{5}{q}^{(j)}_{b}-{q}^{(j)\;T}_{a}C\gamma^{5}{q}^{(i)}_{b}),$
(19) $\displaystyle{Q}_{P}^{(ij)}(3)$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}({q}^{(i)\;T}_{a}C{q}^{(j)}_{b}-{q}^{(j)\;T}_{a}C{q}^{(i)}_{b}).$
(20)
On the other hand, the composite operators of $\mathbf{{3}_{c}}$ spin-0
antidiquarks belonging to the conjugate representation are
$\displaystyle\overline{Q}_{S}^{(ij)}(3)$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}(\bar{q}^{(i)}_{a}\gamma^{5}C\bar{q}^{(j)\;T}_{b}-\bar{q}^{(j)}_{a}\gamma^{5}C\bar{q}^{(i)\;T}_{b}),$
(21) $\displaystyle\overline{Q}_{P}^{(ij)}(3)$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}(\bar{q}^{(i)}_{a}C\bar{q}^{(j)\;T}_{b}-\bar{q}^{(j)}_{a}C\bar{q}^{(i)\;T}_{b}).$
(22)
For the time being, we can express the composite operators related to
$|(6,3)\underline{\overline{3}};(\overline{6},\overline{3})\underline{3};(1,1)\rangle$
as
$\displaystyle T_{6}^{\\{ij\\}\\{lm\\}}$ $\displaystyle\equiv$ $\displaystyle
Q_{T}^{(ij)}(6)\overline{Q}_{T}^{(lm)}(6)$ (23) $\displaystyle=$
$\displaystyle
q^{(i)\;T}_{a}C\sigma^{\mu\nu}q^{(j)}_{b}\bar{q}^{(m)}_{a}\sigma_{\mu\nu}C\bar{q}^{(l)\;T}_{b}+q^{(i)\;T}_{b}C\sigma^{\mu\nu}q^{(j)}_{a}\bar{q}^{(l)}_{a}\sigma_{\mu\nu}C\bar{q}^{(m)\;T}_{b},$
$\displaystyle A_{6}^{\\{ij\\}\\{lm\\}}$ $\displaystyle\equiv$ $\displaystyle
Q_{A}^{(ij)}(6)\overline{Q}_{A}^{(lm)}(6)$ (24) $\displaystyle=$
$\displaystyle
q^{(i)\;T}_{a}C\gamma^{\mu}q^{(j)}_{b}\bar{q}^{(m)}_{a}\gamma_{\mu}C\bar{q}^{(l)\;T}_{b}+q^{(i)\;T}_{b}C\gamma^{\mu}q^{(j)}_{a}\bar{q}^{(l)}_{a}\gamma_{\mu}C\bar{q}^{(m)\;T}_{b},$
where $T$, $A$ represent “tensor” and “axial vector” respectively. These
notations lie with how the diquark and anti-diquark operators vary under
Lorentz transformation. In terms of Eqs. (19)-(22), the composite operators
corresponding to
$|(\bar{3},1)\underline{\overline{3}};(3,1)\underline{3};(1,1)\rangle$ are
$\displaystyle S_{3}^{\\{ij\\}\\{lm\\}}$ $\displaystyle\equiv$ $\displaystyle
Q_{S}^{(ij)}(3)\overline{Q}_{S}^{(lm)}(3)$ (25) $\displaystyle=$
$\displaystyle\epsilon_{abc}\epsilon_{ab^{\prime}c^{\prime}}q^{(i)\;T}_{b}C\gamma^{5}q^{(j)}_{c}\bar{q}^{(m)}_{b^{\prime}}\gamma^{5}C\bar{q}^{(l)\;T}_{c^{\prime}},$
$\displaystyle P_{3}^{\\{ij\\}\\{lm\\}}$ $\displaystyle\equiv$ $\displaystyle
Q_{P}^{(ij)}(3)\overline{Q}_{P}^{(lm)}(3)$ (26) $\displaystyle=$
$\displaystyle\epsilon_{abc}\epsilon_{ab^{\prime}c^{\prime}}q^{(i)\;T}_{b}Cq^{(j)}_{c}\bar{q}^{(m)}_{b^{\prime}}C\bar{q}^{(l)\;T}_{c^{\prime}},$
where $S$, $P$ stand for “scalar” and “pseudoscalar” respectively. Following
Jaffe, $\\{\sigma,\;f_{0},\;a_{+},\;\kappa\\}$ are assumed as
$0^{+}$-$SU(3)_{f}$ nonet tetraquarks. For $\sigma$, since its flavor content
is $\\{ud\\}\\{\bar{u}\bar{d}\\}$, by Eqs. (23) and (24), the operators
corresponding to
$|(6,3)\underline{\overline{3}};(\overline{6},\overline{3})\underline{3};(1,1)\rangle$
of $\sigma$ are
$\displaystyle T_{6}^{\sigma}$ $\displaystyle\equiv$ $\displaystyle
T_{6}^{\\{ud\\}\\{\bar{u}\bar{d}\\}}=u^{T}_{a}C\sigma^{\mu\nu}d_{b}\bar{d}_{a}\sigma_{\mu\nu}C\bar{u}^{T}_{b}+u^{T}_{b}C\sigma^{\mu\nu}d_{a}\bar{u}_{a}\sigma_{\mu\nu}C\bar{d}^{T}_{b},$
(27) $\displaystyle A_{6}^{\sigma}$ $\displaystyle\equiv$ $\displaystyle
A_{6}^{\\{ud\\}\\{\bar{u}\bar{d}\\}}=u^{T}_{a}C\gamma^{\mu}d_{b}\bar{d}_{a}\gamma_{\mu}C\bar{u}^{T}_{b}+u^{T}_{b}C\gamma^{\mu}d_{a}\bar{u}_{a}\gamma_{\mu}C\bar{d}^{T}_{b}.$
(28)
By Eqs. (25) and (26), the operators corresponding to
$|(\overline{3},1)\underline{\overline{3}};(3,1)\underline{3};(1,1)\rangle$ of
$\sigma$ are
$\displaystyle S_{3}^{\sigma}$ $\displaystyle\equiv$ $\displaystyle
S_{3}^{\\{ud\\}\\{\bar{u}\bar{d}\\}}=\epsilon_{abc}\epsilon_{ab^{{}^{\prime}}c^{{}^{\prime}}}u^{T}_{b}C\gamma^{5}d_{c}\bar{u}_{b^{{}^{\prime}}}\gamma^{5}C\bar{d}^{T}_{c^{{}^{\prime}}},$
(29) $\displaystyle P_{3}^{\sigma}$ $\displaystyle\equiv$ $\displaystyle
P_{3}^{\\{ud\\}\\{\bar{u}\bar{d}\\}}=\epsilon_{abc}\epsilon_{ab^{{}^{\prime}}c^{{}^{\prime}}}u^{T}_{b}Cd_{c}\bar{u}_{b^{{}^{\prime}}}C\bar{d}^{T}_{c^{{}^{\prime}}}.$
(30)
Similarly, for $f_{0}$, because of its flavor content
$\frac{1}{\sqrt{2}}(\\{us\\}\\{\bar{u}\bar{s}\\}+\\{ds\\}\\{\bar{d}\bar{s}\\})$,
the results are
$\displaystyle T_{6}^{f_{0}}\equiv
T_{6}^{\frac{1}{\sqrt{2}}(\\{us\\}\\{\bar{u}\bar{s}\\}+\\{ds\\}\\{\bar{d}\bar{s}\\})}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(u^{T}_{a}C\sigma^{\mu\nu}s_{b}\bar{s}_{a}\sigma_{\mu\nu}C\bar{u}^{T}_{b}+u^{T}_{b}C\sigma^{\mu\nu}s_{a}\bar{u}_{a}\sigma_{\mu\nu}C\bar{s}^{T}_{b})$
(31)
$\displaystyle+\frac{1}{\sqrt{2}}(d^{T}_{a}C\sigma^{\mu\nu}s_{b}\bar{s}_{a}\sigma_{\mu\nu}C\bar{d}^{T}_{b}+d^{T}_{b}C\sigma^{\mu\nu}s_{a}\bar{d}_{a}\sigma_{\mu\nu}C\bar{s}^{T}_{b}),$
$\displaystyle A_{6}^{f_{0}}\equiv
A_{6}^{\frac{1}{\sqrt{2}}(\\{us\\}\\{\bar{u}\bar{s}\\}+\\{ds\\}\\{\bar{d}\bar{s}\\})}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(u^{T}_{a}C\gamma^{\mu}s_{b}\bar{s}_{a}\gamma_{\mu}C\bar{u}^{T}_{b}+u^{T}_{b}C\gamma^{\mu}s_{a}\bar{u}_{a}\gamma_{\mu}C\bar{s}^{T}_{b})$
(32)
$\displaystyle\frac{1}{\sqrt{2}}(d^{T}_{a}C\gamma^{\mu}s_{b}\bar{s}_{a}\gamma_{\mu}C\bar{d}^{T}_{b}+d^{T}_{b}C\gamma^{\mu}s_{a}\bar{d}_{a}\gamma_{\mu}C\bar{s}^{T}_{b}),$
$\displaystyle S_{3}^{f_{0}}\equiv
S_{3}^{\frac{1}{\sqrt{2}}(\\{us\\}\\{\bar{u}\bar{s}\\}+\\{ds\\}\\{\bar{d}\bar{s}\\})}$
$\displaystyle=$
$\displaystyle\hskip-7.22743pt\frac{1}{\sqrt{2}}\epsilon_{abc}(\epsilon_{ab^{{}^{\prime}}c^{{}^{\prime}}}u^{T}_{b}C\gamma^{5}s_{c}\bar{u}_{b^{{}^{\prime}}}\gamma^{5}C\bar{s}^{T}_{c^{{}^{\prime}}}+\epsilon_{ab^{{}^{\prime}}c^{{}^{\prime}}}d^{T}_{b}C\gamma^{5}s_{c}\bar{d}_{b^{{}^{\prime}}}\gamma^{5}C\bar{s}^{T}_{c^{{}^{\prime}}}),$
(33) $\displaystyle P_{3}^{f_{0}}\equiv
S_{3}^{\frac{1}{\sqrt{2}}(\\{us\\}\\{\bar{u}\bar{s}\\}+\\{ds\\}\\{\bar{d}\bar{s}\\})}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\epsilon_{abc}(\epsilon_{ab^{{}^{\prime}}c^{{}^{\prime}}}u^{T}_{b}Cs_{c}\bar{u}_{b^{{}^{\prime}}}C\bar{s}^{T}_{c^{{}^{\prime}}}+\epsilon_{ab^{{}^{\prime}}c^{{}^{\prime}}}d^{T}_{b}Cs_{c}\bar{d}_{b^{{}^{\prime}}}C\bar{s}^{T}_{c^{{}^{\prime}}}).$
(34)
The results for $a_{+}\leavevmode\nobreak\ (\\{us\\}\\{\bar{d}\bar{s}\\})$,
$\kappa\leavevmode\nobreak\ (\\{ud\\}\\{\bar{d}\bar{s}\\})$ are the following
ones,
$\displaystyle T_{6}^{a_{+}}$ $\displaystyle\equiv$ $\displaystyle
T_{6}^{(\\{us\\}\\{\bar{d}\bar{s}\\})}=u^{T}_{a}C\sigma^{\mu\nu}s_{b}\bar{d}_{a}\sigma_{\mu\nu}C\bar{s}^{T}_{b}+u^{T}_{b}C\sigma^{\mu\nu}s_{a}\bar{d}_{a}\sigma_{\mu\nu}C\bar{s}^{T}_{b},$
(35) $\displaystyle A_{6}^{a_{+}}$ $\displaystyle\equiv$ $\displaystyle
A_{6}^{(\\{us\\}\\{\bar{d}\bar{s}\\})}=u^{T}_{a}C\gamma^{\mu}s_{b}\bar{d}_{a}\gamma_{\mu}C\bar{s}^{T}_{b}+u^{T}_{b}C\gamma^{\mu}s_{a}\bar{d}_{a}\gamma_{\mu}C\bar{s}^{T}_{b},$
(36) $\displaystyle S_{3}^{a_{+}}$ $\displaystyle\equiv$ $\displaystyle
S_{3}^{(\\{us\\}\\{\bar{d}\bar{s}\\})}=\epsilon_{abc}\epsilon_{ab^{{}^{\prime}}c^{{}^{\prime}}}u^{T}_{b}C\gamma^{5}s_{c}\bar{d}_{b^{{}^{\prime}}}\gamma^{5}C\bar{s}^{T}_{c^{{}^{\prime}}},$
(37) $\displaystyle P_{3}^{a_{+}}$ $\displaystyle\equiv$ $\displaystyle
P_{3}^{(\\{us\\}\\{\bar{d}\bar{s}\\})}=\epsilon_{abc}\epsilon_{ab^{{}^{\prime}}c^{{}^{\prime}}}u^{T}_{b}Cs_{c}\bar{d}_{b^{{}^{\prime}}}C\bar{s}^{T}_{c^{{}^{\prime}}}.$
(38) $\displaystyle{}T_{6}^{\kappa}$ $\displaystyle\equiv$ $\displaystyle
T_{6}^{(\\{ud\\}\\{\bar{d}\bar{s}\\})}=u^{T}_{a}C\sigma^{\mu\nu}d_{b}\bar{s}_{a}\sigma_{\mu\nu}C\bar{d}^{T}_{b}+u^{T}_{b}C\sigma^{\mu\nu}d_{a}\bar{s}_{a}\sigma_{\mu\nu}C\bar{d}^{T}_{b},$
(39) $\displaystyle A_{6}^{\kappa}$ $\displaystyle\equiv$ $\displaystyle
A_{6}^{(\\{ud\\}\\{\bar{d}\bar{s}\\})}=u^{T}_{a}C\gamma^{\mu}d_{b}\bar{s}_{a}\gamma_{\mu}C\bar{d}^{T}_{b}+u^{T}_{b}C\gamma^{\mu}d_{a}\bar{s}_{a}\gamma_{\mu}C\bar{d}^{T}_{b},$
(40) $\displaystyle S_{3}^{\kappa}$ $\displaystyle\equiv$ $\displaystyle
S_{3}^{(\\{ud\\}\\{\bar{d}\bar{s}\\})}=\epsilon_{abc}\epsilon_{ab^{{}^{\prime}}c^{{}^{\prime}}}u^{T}_{b}C\gamma^{5}d_{c}\bar{s}_{b^{{}^{\prime}}}\gamma^{5}C\bar{d}^{T}_{c^{{}^{\prime}}},$
(41) $\displaystyle P_{3}^{\kappa}$ $\displaystyle\equiv$ $\displaystyle
P_{3}^{(\\{ud\\}\\{\bar{d}\bar{s}\\})}=\epsilon_{abc}\epsilon_{ab^{{}^{\prime}}c^{{}^{\prime}}}u^{T}_{b}Cd_{c}\bar{s}_{b^{{}^{\prime}}}C\bar{d}^{T}_{c^{{}^{\prime}}}.$
(42)
Subsequently, from above results and basing on Eq. (9), we get the desired all
possible simplest interpolating currents for tetraquark $|0^{+},9\rangle$ as
follows
$J^{X}_{1}=\alpha T_{6}^{X}+\beta S_{3}^{X},$ $J^{X}_{2}=\alpha
T_{6}^{X}+\beta P_{3}^{X},$ $J^{X}_{3}=\alpha A_{6}^{X}+\beta S_{3}^{X},$
$J^{X}_{2}=\alpha A_{6}^{X}+\beta P_{3}^{X},$ (43)
where $X$ can signifies $\sigma,\kappa,a_{+}$ and $f_{0}$, with $\alpha=0.988$
and $\beta=0.157$. We notice that some indispensable contents of the best
mixed current in zhu2 disappear here. The reason is that they are forbidden
by requiring the wavefunction of diquark to be anti-symmetrized jaffe3 ;
jaffe4 .
## III QCD sum rule analysis without instanton contribution
### III.1 General formulas for QCD sum rule
In sum rule analysis, we usually consider two-point correlation functions:
$\Pi(q^{2})\equiv i\int d^{4}xe^{iqx}\langle 0|{\rm
T}J(x)J^{\dagger}(0)|0\rangle,$ (44)
where $J$ is an interpolating current for the tetraquark. We compute
$\Pi(q^{2})$ up to certain order in the expansion, which is matched with a
hadronic parametrization to extract information of hadron properties. At
hadron level, we express the correlation function in the form of dispersion
relation with a spectral function:
$\Pi(q^{2})=\int^{\infty}_{0}\frac{\rho(s)}{s-q^{2}-i\epsilon}ds,$ (45)
where
$\displaystyle\rho(s)$ $\displaystyle=$
$\displaystyle\pi\sum_{n}\delta(s-M_{n}^{2})\langle 0|J(x)|n\rangle\langle
n|J^{\dagger}(0)|0\rangle,$ (46) $\displaystyle=$ $\displaystyle 2\pi
f_{X}^{2}m_{X}^{8}\delta(s-M_{X}^{2})+\rm higher\leavevmode\nobreak\ states,$
with the convention
$\langle 0|J(x)|S_{i}\rangle=\sqrt{2}f_{i}m_{i}^{4}.$ (47)
The sum rule analysis is then performed after Borel transforming both sides of
Eqs. (44) and (45),
$\Pi^{(\rm
all)}(M_{B}^{2})=\mathcal{B}_{M_{B}^{2}}\Pi(q^{2})=\int^{\infty}_{0}e^{-s/M_{B}^{2}}\rho(s)ds.$
(48)
Usually, evaluating $\rho(s)$ by OPE or some other methods, then from Eq.
(48), one obtains the left hand sum rule (LHS). On the other hand, inserting
Eq. (46) into Eq. (48), one derives the right hand sum rule (RHS). By
definition,
$\Pi_{\rm RHS}(M_{B}^{2})=2\pi f_{X}^{2}m_{X}^{8}e^{-m_{X}^{2}/M_{B}^{2}}.$
(49)
The LHS and RHS are supposed to be equal, so we obtain
$\int^{S_{0}}_{0}e^{-s/M_{B}^{2}}\rho(s)ds=2\pi
f_{X}^{2}m_{X}^{8}e^{-m_{X}^{2}/M_{B}^{2}}.$ (50)
In above expressions, we have chosen a finite threshold $S_{0}$ to exclude the
contribution from the continuum. Differentiating Eq. (50) with respect to
$\frac{1}{M_{B}^{2}}$, and dividing it by Eq. (50), finally we obtain the
physical mass
$M_{X}^{2}=\frac{\int^{S_{0}}_{0}e^{-s/M_{B}^{2}}s\rho(s)ds}{\int^{S_{0}}_{0}e^{-s/M_{B}^{2}}\rho(s)ds}.$
(51)
In the following, we study both Eqs. (48) and (51) as functions of Borel mass
$M_{B}$ and threshold $S_{0}$.
### III.2 OPE calculation for $0^{+}$ nonet as Jaffe tetraquark
The $\sigma$-correlator can be expressed as follows:
$\displaystyle\Pi^{\sigma}(q^{2})$ $\displaystyle=$ $\displaystyle i\int
d^{4}xe^{iq\cdot x}\langle 0|TJ^{\sigma}(x)J^{\sigma{\dagger}}(0)|0\rangle$
(52) $\displaystyle=$ $\displaystyle\alpha^{2}\Pi^{\sigma{\rm
OPE}}_{A,A}+\beta^{2}\Pi^{\sigma{\rm OPE}}_{B,B}+\alpha\beta(\Pi^{\sigma{\rm
OPE}}_{A,B}+\Pi^{\sigma{\rm OPE}}_{B,A}).$
where $J^{\sigma}=\alpha A+\beta B$ represents any one of the four possible
currents in Eq. (43), $A$ represents the composite operator related to
$\mathbf{6_{c}}$-$\mathbf{\bar{6}_{c}}$, and $B$ is that associated with
$\mathbf{3_{c}}$-$\mathbf{\bar{3}_{c}}$. $\Pi_{A,B}$ is the correlator between
$A$-type content and $B$-type content. In this section, we will first compute
the spectral functions for the correlators through OPE expansion, then insert
these results into the Eq. (48) to obtain the Borel transformed correlators.
In the process of calculating OPE, we use the following propagators for quarks
zhu1 , which contain all the necessary terms for computing tetraquark spectral
functions.
$\displaystyle iS_{q}^{ab}(x)$ $\displaystyle\equiv$ $\displaystyle\langle
0|T[q_{a}(x)\bar{q}_{b}(0)]|0\rangle$ (53) $\displaystyle=$
$\displaystyle\frac{i\delta^{ab}}{2\pi^{2}x^{4}}\hat{x}+\frac{i}{32\pi^{2}}\frac{\lambda^{n}_{ab}}{2}\textsl{g}_{c}G^{n}_{\mu\nu}\frac{1}{x^{2}}(\sigma^{\mu\nu}\hat{x}+\hat{x}\sigma^{\mu\nu})-\frac{\delta^{ab}}{12}\langle\bar{q}q\rangle+\frac{\delta^{ab}x^{2}}{192}\langle\textsl{g}_{c}\bar{q}\sigma
Gq\rangle-\frac{\delta^{ab}m_{q}}{4\pi^{2}x^{2}}$
$\displaystyle+\frac{i\delta^{ab}m_{q}}{48}\langle\bar{q}q\rangle\hat{x}+\frac{i\delta^{ab}m_{q}^{2}}{8\pi^{2}x^{2}}\hat{x}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
with}\leavevmode\nobreak\ q\in\\{u,d\\}.$ $\displaystyle iS_{s}^{ab}(x)$
$\displaystyle\equiv$ $\displaystyle\langle
0|T[s_{a}(x)\bar{s}_{b}(0)]|0\rangle$ (54) $\displaystyle=$
$\displaystyle\frac{i\delta^{ab}}{2\pi^{2}x^{4}}\hat{x}+\frac{i}{32\pi^{2}}\frac{\lambda^{n}_{ab}}{2}\textsl{g}_{c}G^{n}_{\mu\nu}\frac{1}{x^{2}}(\sigma^{\mu\nu}\hat{x}+\hat{x}\sigma^{\mu\nu})-\frac{\delta^{ab}}{12}\langle\bar{s}s\rangle+\frac{\delta^{ab}x^{2}}{192}\langle\textsl{g}_{c}\bar{s}\sigma
Gs\rangle-\frac{\delta^{ab}m_{s}}{4\pi^{2}x^{2}}$
$\displaystyle+\frac{i\delta^{ab}m_{s}}{48}\langle\bar{s}s\rangle\hat{x}+\frac{i\delta^{ab}m_{s}^{2}}{8\pi^{2}x^{2}}\hat{x}.$
Actually, OPE computation for tetraquarks is rather long, but it can be
performed analytically. A convenient formulation for performing this
calculation has been presented in zhu1 ; zhu2 . The MATHMATICA with FEYNCALC
feynman may be helpful for computation. In the following, we use the
notations and formulations in zhu1 ; zhu2 . We have performed the OPE
calculation for spectral functions up to dimension eight, which is up to the
constant ($s^{0}$) term of $\rho(s)$. During the calculations, we have assumed
the vacuum is saturated for higher dimension operators, such as $\langle
0|\bar{q}q\bar{q}q|0\rangle\sim\langle 0|\bar{q}q|0\rangle^{2}$. After
finishing the OPE calculation, we obtain the following results for $\sigma$
meson,
$\displaystyle\rho^{\sigma\rm OPE}_{T,T}$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{1280}-\frac{m_{q}^{2}}{16\pi^{6}}s^{3}+(\frac{21m_{q}^{4}}{16\pi^{6}}+\frac{\langle\bar{q}q\rangle
m_{q}}{2\pi^{4}}+\frac{11\langle\textsl{g}^{2}GG\rangle}{768})s^{2}-(\frac{9m_{q}^{6}}{2\pi^{6}}+\frac{15\langle\bar{q}q\rangle
m_{q}^{3}}{\pi^{4}}+\frac{11\langle\textsl{g}^{2}GG\rangle
m_{q}^{2}}{64\pi^{6}})s$ (55)
$\displaystyle+(\frac{9m_{q}^{8}}{4\pi^{6}}+\frac{18\langle\bar{q}q\rangle
m_{q}^{5}}{\pi^{4}}+\frac{11\langle\textsl{g}^{2}GG\rangle
m_{q}^{4}}{64\pi^{6}}-\frac{3\langle\textsl{g}\bar{q}\sigma Gq\rangle
m_{q}^{3}}{\pi^{4}}+\frac{30\langle\bar{q}q\rangle^{2}m_{q}^{2}}{\pi^{2}}+\frac{11\langle\textsl{g}^{2}GG\rangle\langle\bar{q}q\rangle
m_{q}}{48\pi^{4}}),$ $\displaystyle\rho^{\sigma\rm OPE}_{S,S}$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{61440\pi^{6}}-\frac{m_{q}^{2}s^{3}}{1536\pi^{6}}+(\frac{3m_{q}^{4}}{256\pi^{6}}-\frac{m_{q}\langle\bar{q}q\rangle}{96\pi^{4}}+\frac{\langle\textsl{g}^{2}GG\rangle}{6144\pi^{6}})s^{2}-(\frac{3m_{q}^{6}}{64\pi^{6}}+\frac{\langle\textsl{g}^{2}GG\rangle
m_{q}^{2}}{1024\pi^{6}}+\frac{\langle\textsl{g}\bar{q}\sigma Gq\rangle
m_{q}}{32\pi^{4}}$ (56)
$\displaystyle-\frac{\langle\bar{q}q\rangle^{2}}{12\pi^{2}})s+(\frac{3m_{q}^{8}}{64\pi^{6}}+\frac{\langle\textsl{g}^{2}GG\rangle
m_{q}^{4}}{512\pi^{6}}+\frac{\langle\textsl{g}\bar{q}\sigma Gq\rangle
m_{q}^{3}}{16\pi^{4}}-\frac{m_{q}^{2}\langle\bar{q}q\rangle^{2}}{24\pi^{2}}-\frac{\langle\textsl{g}^{2}GG\rangle\langle\bar{q}q\rangle
m_{q}}{384\pi^{4}}+\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle\langle\bar{q}q\rangle}{12\pi^{2}}),$ $\displaystyle\rho^{\sigma\rm
OPE}_{T,S}$ $\displaystyle=$ $\displaystyle\rho^{\sigma\rm
OPE}_{S,T}=-\frac{\langle\textsl{g}^{2}GG\rangle}{1024\pi^{6}}s^{2}+\frac{3\langle\textsl{g}^{2}GG\rangle
m_{q}^{2}}{256\pi^{6}}s-(\frac{3\langle\textsl{g}^{2}GG\rangle
m_{q}^{4}}{256\pi^{6}}+\frac{\langle\textsl{g}^{2}GG\rangle\langle\bar{q}q\rangle
m_{q}}{64\pi^{4}}),$ (57) $\displaystyle\rho^{\sigma\rm OPE}_{P,P}$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{61440\pi^{6}}-\frac{m_{q}^{2}s^{3}}{512\pi^{6}}+(\frac{11m_{q}^{4}}{256\pi^{6}}+\frac{m_{q}\langle\bar{q}q\rangle}{32\pi^{4}}+\frac{\langle\textsl{g}^{2}GG\rangle}{6144\pi^{6}})s^{2}-(\frac{9m_{q}^{6}}{64\pi^{6}}+\frac{5\langle\bar{q}q\rangle
m_{q}^{3}}{8\pi^{4}}+\frac{3\langle\textsl{g}^{2}GG\rangle
m_{q}^{2}}{1024\pi^{6}}$ $\displaystyle-\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle
m_{q}}{32\pi^{4}}+\frac{\langle\bar{q}q\rangle^{2}}{12\pi^{2}})s+(\frac{3m_{q}^{8}}{64\pi^{6}}+\frac{3\langle\bar{q}q\rangle
m_{q}^{5}}{4\pi^{4}}+\frac{\langle\textsl{g}^{2}GG\rangle
m_{q}^{4}}{512\pi^{6}}-\frac{3\langle\textsl{g}\bar{q}\sigma Gq\rangle
m_{q}^{3}}{16\pi^{4}}+\frac{31m_{q}^{2}\langle\bar{q}q\rangle^{2}}{24\pi^{2}}$
(58) $\displaystyle+\frac{\langle\textsl{g}^{2}GG\rangle\langle\bar{q}q\rangle
m_{q}}{128\pi^{4}}-\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle\langle\bar{q}q\rangle}{12\pi^{2}}),$ $\displaystyle\rho^{\sigma\rm
OPE}_{T,P}$ $\displaystyle=$ $\displaystyle\rho^{\sigma\rm
OPE}_{P,T}=-\frac{\langle\textsl{g}^{2}GG\rangle}{512\pi^{6}}s^{2}+\frac{3\langle\textsl{g}^{2}GG\rangle
m_{q}^{2}}{128\pi^{6}}s-(\frac{3\langle\textsl{g}^{2}GG\rangle
m_{q}^{4}}{128\pi^{6}}+\frac{\langle\textsl{g}^{2}GG\rangle\langle\bar{q}q\rangle
m_{q}}{32\pi^{4}}),$ (59) $\displaystyle\rho^{\sigma\rm OPE}_{A,A}$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{7680\pi^{6}}-\frac{m_{q}^{2}s^{3}}{128\pi^{6}}+(\frac{5m_{q}^{4}}{32\pi^{6}}+\frac{\langle\textsl{g}^{2}GG\rangle}{3072\pi^{6}})s^{2}-(\frac{9m_{q}^{6}}{16\pi^{6}}+\frac{5\langle\bar{q}q\rangle
m_{q}^{3}}{4\pi^{4}}+\frac{9\langle\textsl{g}^{2}GG\rangle
m_{q}^{2}}{512\pi^{6}}+\frac{\langle\textsl{g}\bar{q}\sigma Gq\rangle
m_{q}}{8\pi^{4}}$ (60)
$\displaystyle+\frac{\langle\bar{q}q\rangle^{2}}{3\pi^{2}})s+(\frac{3m_{q}^{8}}{8\pi^{6}}+\frac{3\langle\bar{q}q\rangle
m_{q}^{5}}{2\pi^{4}}+\frac{5\langle\textsl{g}^{2}GG\rangle
m_{q}^{4}}{256\pi^{6}}+\frac{7m_{q}^{2}\langle\bar{q}q\rangle^{2}}{3\pi^{2}}+\frac{\langle\textsl{g}^{2}GG\rangle\langle\bar{q}q\rangle
m_{q}}{64\pi^{4}}+\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle\langle\bar{q}q\rangle}{3\pi^{2}}),$ $\displaystyle\rho^{\sigma\rm
OPE}_{A,S}$ $\displaystyle=$ $\displaystyle\rho^{\sigma\rm
OPE}_{S,A}=-\frac{3\langle\textsl{g}^{2}GG\rangle
m_{q}^{2}}{1024\pi^{6}}s+\frac{\langle\textsl{g}^{2}GG\rangle\langle\bar{q}q\rangle
m_{q}}{64\pi^{4}},$ (61) $\displaystyle\rho^{\sigma\rm OPE}_{A,P}$
$\displaystyle=$ $\displaystyle\rho^{\sigma\rm OPE}_{P,A}=0.$ (62)
In above equations, $\langle\bar{q}q\rangle$ is a dimension $d=3$ quark
condensate; $\langle\textsl{g}^{2}GG\rangle$ is a dimension $d=4$ gluon
condensate; $\langle\textsl{g}\bar{q}\sigma Gq\rangle$ is a dimension $d=5$
mixed condensate; the strong coupling constant takes its value at energy scale
about 1 GeV, that is the energy scale we are interested in. Long distance bulk
properties of physical vacuum are effectively parameterized in these vacuum
expectation values. At present, according to Eq. (43), we can make use of
above spectral functions to generate correlator of each kind interpolating
current belonging to $\sigma$. These correlators will be the starting point of
numerical calculation in the next section.
In order to prevent the long listing of formulas for spectral functions from
obscuring the conceptual content, we will put the necessary spectral functions
of $\kappa$, $a_{+}$ and $f_{0}$ into the appendix.
### III.3 Numerical analysis of QCD sum rule for OPE contribution
For numerical calculations, we use the following values of condensates
Yang:1993bp ; Narison:2002pw ; Gimenez:2005nt ; Jamin:2002ev ; Ioffe:2002be ;
Ovchinnikov:1988gk ; Yao:2006px :
$\displaystyle\langle\bar{q}q\rangle=-(0.240\mbox{ GeV})^{3}\,,$
$\displaystyle\langle\bar{s}s\rangle=-(0.8\pm 0.1)\times(0.240\mbox{
GeV})^{3}\,,$ $\displaystyle\langle g_{s}^{2}GG\rangle=(0.48\pm 0.14)\mbox{
GeV}^{4}\,,$ $\displaystyle m_{u}=m_{d}=m_{q}=0.1\times 2.4^{-3}\mbox{
GeV}\,,$ $\displaystyle m_{s}(1\mbox{ GeV})=125\pm 20\mbox{ MeV}\,,$ (63)
$\displaystyle\langle g_{s}\bar{q}\sigma
Gq\rangle=-M_{0}^{2}\times\langle\bar{q}q\rangle\,,$ $\displaystyle
M_{0}^{2}=(0.8\pm 0.2)\mbox{ GeV}^{2}\,.$
Figure 1 shows the LHS of four possible interpolating currents of the $\sigma$
meson, as a function of Borel mass squared, in the case of infinite threshold.
From the definition of Eq. (48), the LHS should be positive quantities.
However, in practical calculations, the positivity may not be necessarily
realized due to the insufficient convergence of OPE calculations. In our case,
from Figure. 1, we see that current $J_{1}^{\sigma}$ and current
$J_{2}^{\sigma}$ show better convergence than current $J_{3}^{\sigma}$ and
current $J_{4}^{\sigma}$.
Figure 1: LHS of four interpolating currents of $\sigma$ meson, as a functions
of Borel mass squared, with $s_{0}$=infinity, in units of $\rm GeV^{10}$.
To find the current with the best convergence, we have to refer to their Borel
transformed correlators in numerical expressions, which are:
$\displaystyle\Pi^{\sigma(\rm all)}_{1}$ $\displaystyle=$ $\displaystyle
1.9\times 10^{-5}M_{B}^{10}-1.9\times 10^{-8}M_{B}^{8}+9.5\times
10^{-6}M_{B}^{6}+3.7\times 10^{-8}M_{B}^{4}-8.5\times 10^{-8}M_{B}^{2},$
$\displaystyle\Pi^{\sigma(\rm all)}_{2}$ $\displaystyle=$ $\displaystyle
1.9\times 10^{-5}M_{B}^{10}-2.0\times 10^{-8}M_{B}^{8}+9.5\times
10^{-6}M_{B}^{6}-4.2\times 10^{-8}M_{B}^{4}-2.1\times 10^{-8}M_{B}^{2},$
$\displaystyle\Pi^{\sigma(\rm all)}_{3}$ $\displaystyle=$ $\displaystyle
3.2\times 10^{-6}M_{B}^{10}-2.5\times 10^{-9}M_{B}^{8}+1.6\times
10^{-6}M_{B}^{6}-6.2\times 10^{-6}M_{B}^{4}-5.1\times 10^{-6}M_{B}^{2},$
$\displaystyle\Pi^{\sigma(\rm all)}_{4}$ $\displaystyle=$ $\displaystyle
3.2\times 10^{-6}M_{B}^{10}-2.5\times 10^{-9}M_{B}^{8}+1.6\times
10^{-6}M_{B}^{6}+6.2\times 10^{-6}M_{B}^{4}-5.1\times 10^{-6}M_{B}^{2}.$ (64)
From these expressions, it is obvious that current $J_{2}^{\sigma}$ shows the
best convergence behavior, so we will utilize current $J_{2}^{\sigma}$ to
compute the physical mass of $\sigma$. We first choose an infinite threshold
to estimate the mass as the traditional sum rule has done reinders . In Figure
2, we exhibit the behavior of the mass of $\sigma$ meson as the function of
$M_{B}$ for infinite and finite $s_{0}$. In traditional sum rule, if the mass
as a function of $M_{B}$, has a wide minimum, then the minimum value of mass
function can be perceived as the real mass of the state. From Figure 2, we
observed that $M_{\sigma}$ as a function of $M_{B}^{2}$ indeed has a minimum
with $M_{\sigma(\rm min)}=0.59{\rm\leavevmode\nobreak\ GeV}$ at
$M_{B}^{2}=0.079{\rm\leavevmode\nobreak\ GeV}^{2}$. At this value of Borel
mass, the correlation function $\Pi^{\sigma(\rm all)}_{2}=3\times
10^{-9}\leavevmode\nobreak\ {\rm GeV}^{10}$, so the positivity of LHS is kept.
Although $M_{\sigma(\rm min)}$ is very close to the experimental center value
$\langle M_{\sigma}\rangle\sim 0.6\rm\leavevmode\nobreak\ GeV$, the minimum is
not wide enough as required. Therefore, to obtain an acceptable result, we
have to adopt finite thresholds scheme zhu1 ; zhu2 ; zhu3 ; zhu4 to repeat
the process of computing mass. The results for some values of threshold are
presented in the right part of Figure 2. We notice that when the mass becomes
weakly dependent on $M_{B}$, the value of mass is around 0.6 GeV. But we also
find that as the threshold increases, the mass will increase too. This may be
due to the fact that $\sigma$ is a broad resonance state. So there must be
some criteria to help us dictate which value of mass is the most believable
one. Combining the points of view adopted byzhu2 ; Kojo ; Matheus on judging
when an acceptable sum rule is arrived, we postulate the following criteria.
Figure 2: Mass of $\sigma$ is illustrated as function of Borel Mass squared.
The left figure is in the case of infinite threshold, while the right one is
in cases of finite thresholds. The results corresponding to $s_{0}$ =0.5, 0.6,
0.7 $\rm GeV^{2}$ are represented by a solid line, a dashed line and a dot-
dashed line respectively.
1\. The Borel transformed correlation function $\Pi(M_{B}^{2})$ should show a
good positivity for almost all values of Borel mass. This is usually related
the convergence of LHS.
2\. The physical mass should depend weakly on the value of Borel mass in a
wide region. In other words, there should be a Borel window.
3\. OPE convergence. This is a strong constraint to the lower bound of the
$M_{B}^{2}$ region. OPE series converge better for higher values of
$M_{B}^{2}$, so that requiring a good convergence sets a lower limit to
$M_{B}^{2}$. To current $J^{\sigma}_{2}$, we find such a lower limit of
$M_{B}^{2}$ in the following. We first rewrite the spectral function
corresponding to $J^{\sigma}_{2}$ as,
$\rho_{\sigma}^{(\rm{OPE})}=\Sigma_{n=0}^{4}c^{(8-2n)}s^{n}=\Sigma_{n=0}^{4}\rho^{n},$
(65)
where $c^{(8-2n)}$ denotes the operators of mass dimension $(8-2n)$,
$\rho^{n}\equiv c^{(8-2n)}s^{n}$. From Eqs. (55)-(62), we learn that terms
$\rho^{(3,4)}$ are perturbative contributions denoted as $\rho^{(pert)}$, in
other words, they do not contain condensate. Remaining terms represent
contributions from operators of dimension 4, 6 and 8. These terms are
dominated by condensates including the non-perturbative effect, denoted by
$\rho^{(2)}$, $\rho^{(1)}$, $\rho^{(0)}$ respectively. In Fig. 3, we present
the relative contribution of $\rho^{(2)}$, $\rho^{(1)}$, $\rho^{(0)}$ to the
total spectral function $\rho^{({\rm OPE})}_{\sigma}$.
Figure 3: convergence of OPE series of spectral function related to
current$J^{\sigma}_{2}$ for $s_{0}=0.6\rm{GeV}^{2}$.
The thick line denotes
[$\int^{0.6}_{0}(\rho^{(pert)}+\rho^{(2)})e^{-s/M_{B}^{2}}ds/\int^{0.6}_{0}\rho^{(\rm
OPE)}e^{-s/M_{B}^{2}}ds$], the dashed line signifies
[$\int^{0.6}_{0}(\rho^{(pert)}+\rho^{(2)}+\rho^{(1)})e^{-s/M_{B}^{2}}ds/\int^{0.6}_{0}\rho^{(\rm
OPE)}e^{-s/M_{B}^{2}}ds$], the dashed doted line represents
[$\int^{0.6}_{0}(\rho^{(pert)}+\rho^{(2)}+\rho^{(1)}+\rho^{(0)})e^{-s/M_{B}^{2}}ds/\int^{0.6}_{0}\rho^{(\rm
OPE)}e^{-s/M_{B}^{2}}ds$=1]. We see that, for $M_{B}^{2}>0.2\rm{GeV}^{2}$, the
addition of a subsequent term in expansion (65), brings the curve closer to an
asymptotic value (which is normalized to 1). Furthermore, the changes in this
curve become smaller with increasing dimension. Thus, for
$s_{0}=0.6\rm{GeV}^{2}$, the convergence is satisfied by
$M_{B}^{2}>0.2\rm{GeV}^{2}$. For $s_{0}=0.5,\leavevmode\nobreak\
0.7,\leavevmode\nobreak\ 0.8\rm{GeV}^{2}$, convergence limits
$M_{B}^{2}>0.2,\leavevmode\nobreak\ 0.3,\leavevmode\nobreak\ 0.4\rm{GeV}^{2}$,
respectively.
4\. For a given threshold, the pole contribution should be sufficient large.
By choosing suitable Borel mass, this can be satisfied. Since the Borel
transformation suppresses the contributions from $s_{0}>M_{B}^{2}$, small
value of $M_{B}^{2}$ are preferred to suppress the continuum contributions.
But $M_{B}^{2}$ cannot be arbitrarily small, or it will spoil previous three
requirements. To $\sigma$, we have found such optimal values of $M_{B}^{2}$
for different thresholds. We list the corresponding pole contributions in
Table I. The pole contribution is defined as
$\mbox{Pole
contribution}\equiv\frac{\int^{s_{0}}_{0}e^{-s/M_{B}^{2}}\rho(s)ds}{\int^{\infty}_{0}e^{-s/M_{B}^{2}}\rho(s)ds}\,.$
(66)
Table 1: Pole contributions of various threshold. $s_{0}\leavevmode\nobreak\ ({\rm GeV}^{2})$ | 0.5 | 0.6 | 0.7 | 0.8
---|---|---|---|---
$M_{B}^{2}\leavevmode\nobreak\ ({\rm GeV}^{2})$ | 0.2 | 0.2 | 0.3 | 0.4
Pole (%) | 40 | 52 | 35 | 25
$M_{\sigma}$ (GeV) | 0.6 | 0.6 | 0.7 | 0.75
From this table, we can extract following information that when threshold
changes from $0.5\leavevmode\nobreak\ {\rm GeV}^{2}$ to
$0.8\leavevmode\nobreak\ {\rm GeV}^{2}$, the pole contribution will vary from
40% to 25% correspondingly, but reaches its maximum 52% at $M_{B}^{2}$=0.2
${\rm GeV}^{2}$, when $s_{0}=0.6{\rm GeV}^{2}$. That the pole contribution
reaches 52% implies that a good sum rule has been obtained. We get
$m_{\sigma}=(600\pm 75)\rm{MeV},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \rm{with\leavevmode\nobreak\
Pole\leavevmode\nobreak\ contribution}(52\%),$ (67)
where $(\pm 75)$ MeV originates from the error of condensates (see Eq. III.3).
It is remarkable that the Pole contribution is larger than that given in zhu2
, where the Pole contribution is below 30%.
Applying the same analysis to meson $\kappa$, the LHS of four possible
interpolating currents of $\kappa$ can be found in Figure 4, with threshold
value $s_{0}$ being infinity.
Figure 4: LHS of $\kappa$ meson as functions of Borel mass squared with
$s_{0}$=infinity in units of $\rm GeV^{10}$.
The corresponding numerical expressions are listed below:
$\displaystyle\Pi^{\kappa(\rm all)}_{1}$ $\displaystyle=$ $\displaystyle
1.9\times 10^{-5}M_{B}^{10}-1.2\times 10^{-6}M_{B}^{8}+6.7\times
10^{-6}M_{B}^{6}-1.3\times 10^{-7}M_{B}^{4}-1.2\times 10^{-7}M_{B}^{2},$
$\displaystyle\Pi^{\kappa(\rm all)}_{2}$ $\displaystyle=$ $\displaystyle
1.9\times 10^{-5}M_{B}^{10}-1.2\times 10^{-6}M_{B}^{8}+6.7\times
10^{-6}M_{B}^{6}-1.9\times 10^{-7}M_{B}^{4}-5.7\times 10^{-8}M_{B}^{2},$
$\displaystyle\Pi^{\kappa(\rm all)}_{3}$ $\displaystyle=$ $\displaystyle
3.2\times 10^{-6}M_{B}^{10}-1.9\times 10^{-7}M_{B}^{8}+1.7\times
10^{-6}M_{B}^{6}-5.2\times 10^{-6}M_{B}^{4}-4.6\times 10^{-6}M_{B}^{2},$
$\displaystyle\Pi^{\kappa(\rm all)}_{4}$ $\displaystyle=$ $\displaystyle
3.2\times 10^{-6}M_{B}^{10}-1.9\times 10^{-7}M_{B}^{8}+1.7\times
10^{-6}M_{B}^{6}+5.2\times 10^{-6}M_{B}^{4}-4.6\times 10^{-6}M_{B}^{2}.$ (68)
From Figure 4 and above expressions, we notice that current $J_{2}^{\kappa}$,
which is a proper mixture between tensor and pseudoscalar contents, is the
best interpolating current. By setting the threshold to be infinity, we obtain
an estimation for the mass of $\kappa$. As shown in Figure 5, $M_{\kappa}$ as
a function of $M_{B}$ has a minimum with $M_{\kappa(\rm
min)}=0.90{\rm\leavevmode\nobreak\ GeV}$ at
$M_{B}^{2}=0.2{\rm\leavevmode\nobreak\ GeV}^{2}$. At this value of Borel mass,
the correlation function $\Pi^{\kappa(\rm all)}_{2}=1.6\times
10^{-7}\leavevmode\nobreak\ {\rm GeV}^{10}$ , the positivity of LHS is also
retained. But the minimum is still not wide enough, then the finite threshold
analysis should be performed. The results are shown in the right part of
Figure 5. At the Borel window, the mass of $\kappa$ is close to 0.8 GeV.
Figure 5: Mass of $\kappa$ is illustrated as function of Borel Mass squared.
The left figure is in the case of infinite threshold, while the right one is
in cases of finite thresholds. The results corresponding to $s_{0}$ =0.7, 0.8,
0.9 $\rm GeV^{2}$ are represented by a solid line, a dashed line and a dot-
dashed line respectively.
To find the best sum rule, following the previous criteria, we find that to
$\kappa$, the convergence limits $M_{B}^{2}>0.25\rm{GeV}^{2}$ for
$s_{0}=0.8,\leavevmode\nobreak\ 0.9\rm{GeV}^{2}$ and
$M_{B}^{2}>0.225,\leavevmode\nobreak\ 0.3\rm{GeV}^{2}$ for
$s_{0}=0.7,\leavevmode\nobreak\ 1.2\rm{GeV}^{2}$, respectively. For instance,
to $s_{0}=0.9\rm{GeV}^{2}$, the convergence of OPE series is shown in Fig. 6.
Figure 6: convergence of OPE series of spectral function related to
current$J^{\kappa}_{2}$ for $s_{0}=0.9\rm{GeV}^{2}$.
The pole contributions for several values of threshold are listed in Table II.
Table 2: Pole contributions of various threshold. $s_{0}\leavevmode\nobreak\ ({\rm GeV}^{2})$ | 0.7 | 0.8 | 0.9 | 1.2
---|---|---|---|---
$M_{B}^{2}\leavevmode\nobreak\ ({\rm GeV}^{2})$ | 0.225 | 0.25 | 0.25 | 0.5
Pole (%) | 43 | 47 | 56 | 27
$M_{\kappa}$ (GeV) | 0.75 | 0.8 | 0.82 | 0.95
When $s_{0}=0.9{\rm GeV}^{2}$, $M_{B}^{2}=0.25{\rm GeV}^{2}$, we get a pole
contribution 56%. Such a large pole contribution suggests that a good sum rule
has been obtained. We get the mass of $\kappa$,
$m_{\kappa}=(820\pm 80)\rm{MeV},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ with\leavevmode\nobreak\
Pole\leavevmode\nobreak\ contribution(56\%).$ (69)
This pole contribution is also larger than that given by zhu2 , where the pole
contribution approaches 45%.
Lastly, for $a_{+}$ and $f_{0}$ that are degenerate in OPE calculations, the
LHS of four possible interpolating currents are shown in Fig. 7, with
threshold value $s_{0}$ being infinity.
Figure 7: LHS of four interpolating currents of $a_{+}$ and $f_{0}$ as
functions of Borel mass squared with $s_{0}$=infinity in units of $\rm
GeV^{10}$.
Their numerical expressions are the following ones:
$\displaystyle\Pi^{a+,f0(\rm all)}_{1}$ $\displaystyle=$ $\displaystyle
1.9\times 10^{-5}M_{B}^{10}-2.3\times 10^{-6}M_{B}^{8}+4.0\times
10^{-6}M_{B}^{6}-5.8\times 10^{-8}M_{B}^{4}+7.2\times 10^{-7}M_{B}^{2},$
$\displaystyle\Pi^{a+,f0(\rm all)}_{2}$ $\displaystyle=$ $\displaystyle
1.9\times 10^{-5}M_{B}^{10}-2.3\times 10^{-6}M_{B}^{8}+4.0\times
10^{-6}M_{B}^{6}-1.1\times 10^{-8}M_{B}^{4}+7.7\times 10^{-7}M_{B}^{2},$
$\displaystyle\Pi^{a+,f0(\rm all)}_{3}$ $\displaystyle=$ $\displaystyle
3.2\times 10^{-6}M_{B}^{10}-3.7\times 10^{-7}M_{B}^{8}+1.8\times
10^{-6}M_{B}^{6}+4.2\times 10^{-6}M_{B}^{4}-4.1\times 10^{-6}M_{B}^{2},$
$\displaystyle\Pi^{a+,f0(\rm all)}_{4}$ $\displaystyle=$ $\displaystyle
3.2\times 10^{-6}M_{B}^{10}-3.7\times 10^{-7}M_{B}^{8}+1.8\times
10^{-6}M_{B}^{6}+4.2\times 10^{-6}M_{B}^{4}-4.0\times 10^{-6}M_{B}^{2}.$ (70)
From Fig. 7 and above expressions, current $J_{2}^{a_{+}}$ seems to be the
best one. But when applying the traditional sum rule method to estimate mass,
it turns out that there is no minimum as shown in Fig. 8. Furthermore, if we
choose certain threshold and Borel mass to reproduce the experimental center
value of the masses of $a_{+}$ and $f_{0}$, the pole contribution can only be
around 10%. This indicates that in contrast to the success of SR analysis of
$\sigma$ and $\kappa$, the SR fails to analyze $a_{+}$ and $f_{0}$, in terms
of the interpolating currents deduced from their wavefunctions as tetraquarks.
The reason is as follows. Jaffe’s wavefunctions are the eigenfunctions of
$H_{eff}$ in Eq. (1). However, $H_{eff}$ is only an approximate description of
color-magnetic interactions
$H_{CM}=-\sum_{i\;j}C_{ij}(\lambda_{i}\cdot\lambda_{j})(\overrightarrow{\sigma}_{i}\cdot\overrightarrow{\sigma}_{j})$
DGG ; hogaasen1 ; hogaasen2 ; dy . If the flavor $SU(3)_{f}$-symmetry is
exact, the interaction strengthes $C_{ij}$ are flavor-$(ij)$ independent,
i.e., $C_{ij}=C$, then $H_{CM}=H_{eff}$. But for real QCD, the constituent
mass $m^{c}_{u}\approx m^{c}_{d}$, while
$m^{c}_{s}>\hat{m^{c}}\equiv(m^{c}_{u}+m^{c}_{d})/2$. So $SU(3)_{f}$ must be
broken within order
$\mathcal{O}((m^{c}_{s}-\hat{m^{c}})/m^{c}_{s})\sim\mathcal{O}(0.3)$.
Therefore, both $H_{eff}$ and Jaffe’s wavefunction
$|0^{+},\underline{9}\rangle$ will suffer of this $SU(3)_{f}$ breaking effect.
In other words, $|0^{+},\underline{9}\rangle$ can only be thought of as the
leading term of the eigenfunction of $H_{CM}$, without considering the
correction from the next leading term caused by the strange quark content in
$0^{+}$-tetraquarks. In $\sigma(\\{ud\\}\\{\bar{u}\bar{d}\\})$, there is no
strange quark, so no such kind of corrections, hence
$|\sigma\rangle=|0^{+},\underline{9}\rangle_{\sigma}$ is suitable. In
$\kappa(\\{ud\\}\\{\bar{d}\bar{s}\\})$, there is one strange quark, its
correction is relatively small, and the wavefunction
$|0^{+},\underline{9}\rangle_{\kappa}$ may be still valid to some extent. This
is supported by numerical results. However, for
$f_{0}({1\over\sqrt{2}}(\\{us\\}\\{\bar{u}\bar{s}\\}+\\{ds\\}\\{\bar{d}\bar{s}\\}))$
or $a_{+}(\\{us\\}\\{\bar{d}\bar{s}\\})$, there are two strange quarks, the
$SU(3)_{f}$ breaking effects is doubled. To these cases, one cannot insist the
Jaffe’s wavefunctions $|f_{0}\rangle=|0^{+},\underline{9}\rangle_{f_{0}}$ and
$|a_{+}\rangle=|0^{+},\underline{9}\rangle_{a_{+}}$ be still good enough to
describe the non-perturbative QCD physics. Above all, we speculate that a
legitimate SR analysis for $f_{0}$ and $a_{+}$ should be based on the
tetraquark’s color-magnetic wavefunctions which are more precise, encoding the
$SU(3)_{f}$-symmetry breaking effects.
Figure 8: Mass of $a_{+}$ and $f_{0}$ as function of Borel mass $M_{B}$ with
$s_{0}$ being infinity.
## IV The direct instanton contribution to sum rule
### IV.1 Analytic results
In addition to the contribution of power type from the OPE expansion to the
QCD SR, there are exponential contributions coming from direct instanton
contributions. The direct instantion contributions originate from ’t Hooft’s
instanton induced interaction tHooft . If the physics considered is relevant
to two flavors, instanton effects induce a four-fermion interaction, as
illustrated in Fig. 9 (usually called two-body single instanton contribution
defined in lee2 ). In the framework of sum rule, this kind of instanton effect
can be encoded in the quark propagator. Now the quark propagator has two
terms,
$S^{q}_{ab}=S^{q({\rm st})}_{ab}+S^{q({\rm inst})}_{ab}.$ (71)
$S^{q({\rm st})}_{ab}$ corresponds to standard quark propagator (Eqs. (53) and
(54)) in Euclidean space, $S^{q({\rm inst})}_{ab}$ is related to instanton
contribution and can be calculated by using the following formula in Euclidean
space and regular gauge,
$S^{q({\rm
inst})}_{ab}=A_{q}(x,y)\gamma_{\mu}\gamma_{\nu}(1+\gamma_{5})(U\tau_{\mu}^{+}\tau_{\nu}^{-}U^{\dagger})_{ab},$
(72)
where
$A_{q}(x,y)=-i\frac{r^{2}}{16\pi^{2}m_{q}^{\ast}}\phi(x-z_{0})\phi(y-z_{0})$
(73)
and
$\phi(x-z_{0})=\frac{1}{[(x-z_{0})^{2}+r^{2}]^{3/2}}.$ (74)
Here $r$ stands for the instanton size, $z_{0}$ for the center of the
instanton. $U$ represents the color orientation matrix of the instanton in
$SU(3)_{c}$ and $\tau^{+,-}_{\mu,\nu}$ are $SU(2)_{c}$ matrices. The effective
mass of quark on the instanton vacuum is
$m_{q}^{\ast}=m_{q}-2\pi^{2}r^{2}_{c}\langle\bar{q}q\rangle/3$ with current
quark mass $m_{q}$, here $q\in\\{u,d,s\\}$. At the final stage, we multiply
the result by a factor of two to take into account the anti-instanton effect
and integrate over the color orientation and instanton size. When integrating
over the instanton size, Shuryak’s instanton liquid model schafer for QCD
vacuum with density $n_{r}=n_{eff}\delta(r-r_{c})$ has been used.
Figure 9: The leading direct instanton contribution to the correlator, where
“I” represents the instanton.
With the definition $Q^{2}=-q^{2}$, the direct instanton contributions to the
scalar nonet are listed below, corresponding to above two diagrams. Here, we
only exhibit the contributions to $\sigma$-correlator, and the reader can find
the results of other tetraquarks in appendix. We denote the total
contributions from intanton and anti-instanton by “inst”. Recalling that the
direct instanton contribution is possible only for different quark flavors, so
in case of $\sigma$, there is no direct three-body instanton contribution
(from instanton induced six-fermion interaction). But to $\kappa$, $a_{+}$,
$f_{0}$, three-body instanton contribution might be important. However, in
this paper, we only present the two-body instanton contributions for these
mesons, to capture the main physics.
$\displaystyle\Pi_{TT}^{\sigma(\rm inst)}$ $\displaystyle=$
$\displaystyle\frac{156n_{eff}r_{c}^{4}\langle\bar{q}q\rangle^{2}}{3\pi^{4}m_{q}^{\ast
2}}f_{0}(Q),$ (75) $\displaystyle\Pi_{SS}^{\sigma(\rm inst)}$ $\displaystyle=$
$\displaystyle\frac{32n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast
2}}f_{6}(Q)+\frac{19n_{eff}r_{c}^{4}\langle\bar{q}q\rangle^{2}}{18\pi^{4}m_{q}^{\ast
2}}f_{0}(Q),$ (76) $\displaystyle\Pi_{PP}^{\sigma(\rm inst)}$ $\displaystyle=$
$\displaystyle-\frac{32n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast
2}}f_{6}(Q)+\frac{19n_{eff}r_{c}^{4}\langle\bar{q}q\rangle^{2}}{18\pi^{4}m_{q}^{\ast
2}}f_{0}(Q),$ (77) $\displaystyle\Pi_{TS}^{\sigma(\rm inst)}$ $\displaystyle=$
$\displaystyle\Pi_{ST}^{\sigma(\rm
inst)}=\frac{2n_{eff}r_{c}^{4}\langle\bar{q}q\rangle^{2}}{\pi^{4}m_{q}^{\ast
2}}f_{0}(Q),$ (78) $\displaystyle\Pi_{TP}^{\sigma(\rm inst)}$ $\displaystyle=$
$\displaystyle\Pi_{PT}^{\sigma(\rm
inst)}=\frac{2n_{eff}r_{c}^{4}\langle\bar{q}q\rangle^{2}}{\pi^{4}m_{q}^{\ast
2}}f_{0}(Q),$ (79) $\displaystyle\Pi_{AA}^{\sigma(\rm inst)}$ $\displaystyle=$
$\displaystyle\frac{48n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast
2}}f_{6}(Q)+\frac{68n_{eff}r_{c}^{4}\langle\bar{q}q\rangle^{2}}{9\pi^{4}m_{q}^{\ast
2}}f_{0}(Q),$ (80) $\displaystyle\Pi_{AS}^{\sigma(\rm inst)}$ $\displaystyle=$
$\displaystyle\Pi_{SA}^{\sigma(\rm
inst)}=-\frac{20n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast 2}}f_{6}(Q),$ (81)
$\displaystyle\Pi_{AP}^{\sigma(\rm inst)}$ $\displaystyle=$
$\displaystyle\Pi_{PA}^{\sigma(\rm
inst)}=-\frac{20n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast 2}}f_{6}(Q).$ (82)
In above expressions,
$\displaystyle f_{6}(Q)$ $\displaystyle=$ $\displaystyle\int d^{4}z_{0}\int
d^{4}x\frac{e^{iq\cdot
x}}{x^{6}[z_{0}^{2}+r_{c}^{2}]^{3}[(x-z_{0})^{2}+r_{c}^{2}]^{3}},$
$\displaystyle f_{0}(Q)$ $\displaystyle=$ $\displaystyle\int d^{4}z_{0}\int
d^{4}x\frac{e^{iq\cdot
x}}{[z_{0}^{2}+r_{c}^{2}]^{3}[(x-z_{0})^{2}+r_{c}^{2}]^{3}}.$ (83)
The Borel transformation of $f_{6}(Q)$ and $f_{0}(Q)$ are:
$\displaystyle\hat{B}[f_{6}(Q)]$ $\displaystyle=$
$\displaystyle-\frac{\pi^{4}M^{12}_{B}}{2^{13}}\int^{1}_{0}dt\int^{1}_{0}dy\frac{e^{-M^{2}_{B}r_{c}^{2}/(4ty(1-y))}}{y^{2}(1-y)^{2}}(X^{2}+5X^{3}+10X^{4}$
$\displaystyle+10X^{5}+5X^{6}+X^{7}),$ $\displaystyle\hat{B}[f_{0}(Q)]$
$\displaystyle=$
$\displaystyle\frac{\pi^{4}M^{6}_{B}}{16}e^{-M^{2}_{B}r_{c}^{2}/2}(K_{0}(M^{2}_{B}r_{c}^{2}/2)+K_{1}(M^{2}_{B}r_{c}^{2}/2)),$
(84)
where we adopt the notations in paper lee2 , $X=(1-t)/t$ and $K_{n}(x)$ is the
McDonald function.
### IV.2 Numeric analysis of QCD sum rule with instanton effects
To evaluate the direct instanton effects quantitatively, we make use of the
following relation between the parameters of Shuryak instanton model schafer .
$\frac{n_{eff}}{m_{q}^{\ast
2}}=\frac{3}{4\pi^{2}r_{c}^{2}}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
q\in\\{u,\leavevmode\nobreak\ d\\},$ (85)
with
$r_{c}=1.6\leavevmode\nobreak\ \mbox{GeV}^{-1}.$ (86)
Considering the single instanton effects, the left hand sum rule becomes:
$\Pi_{\rm LHS}(Q^{2})=\Pi^{\rm OPE}(Q^{2})+\Pi^{\rm inst}(Q^{2}).$ (87)
After Borel transforming the both side of the QCD sum rule, we obtain the
following relation
$\mathcal{B}_{M_{B}^{2}}\Pi^{\rm OPE}(Q^{2})+\mathcal{B}_{M_{B}^{2}}\Pi^{\rm
inst}(Q^{2})=2\pi f_{X}^{2}m_{X}^{8}e^{-m_{X}^{2}/M_{B}^{2}}.$ (88)
In above expressions,
$\mathcal{B}_{M_{B}^{2}}\Pi^{{\rm
OPE}}(Q^{2})=\int^{S_{0}}_{0}e^{-s/M_{B}^{2}}\rho^{\rm OPE}(s)ds,$ (89)
where we have chosen a finite threshold to suppress the contribution from
continuum. Utilizing the results in previous sections, the left hand sum rule
can be performed for each possible interpolating current in (43) belonging to
a certain meson. Then we can make use of the best current to fit the right
hand sum rule to obtain the mass and residue. This approach was first
suggested by lee2 . In the following, for the sake of simplicity, we will only
present a detailed analysis for $\sigma$ meson. For other mesons, the results
are also exhibited.
In Fig. 10, we show the Borel transformed correlators $\Pi(M_{B}^{2})$,
including the instanton effects, at threshold value $s_{0}$=0.6
$\mbox{GeV}^{2}$. From the Figure, we see that the instanton contributions are
not always positive. To current $J_{1}^{\sigma}$, they provide little negative
contributions, and spoil the positivity of LHS obviously, when Borel mass is
small; to current $J_{3}^{\sigma}$ and $J_{4}^{\sigma}$, instanton effects
make the LHS rather negative, and this may be the usually called dangerous
instanton contribution to sum rule lee2 ; only to current $J_{2}^{\sigma}$,
the instanton effects improve the OPE calculation completely. This feature can
be seen more clearly, if we notice that in Eqs. (75)-(79):
$\displaystyle\Pi_{TT}^{\sigma(\rm inst)}$ $\displaystyle=$
$\displaystyle\frac{156n_{eff}r_{c}^{4}\langle\bar{q}q\rangle^{2}}{3\pi^{4}m_{q}^{\ast
2}}f_{0}(Q),$ $\displaystyle\Pi_{PP}^{\sigma(\rm inst)}$ $\displaystyle=$
$\displaystyle-\frac{32n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast
2}}f_{6}(Q)+\frac{19n_{eff}r_{c}^{4}\langle\bar{q}q\rangle^{2}}{18\pi^{4}m_{q}^{\ast
2}}f_{0}(Q),$ $\displaystyle\Pi_{TP}^{\sigma(\rm inst)}$ $\displaystyle=$
$\displaystyle\Pi_{PT}^{\sigma(\rm
inst)}=\frac{2n_{eff}r_{c}^{4}\langle\bar{q}q\rangle^{2}}{\pi^{4}m_{q}^{\ast
2}}f_{0}(Q).$ (90)
In above expressions, the coefficients of $f_{0}(Q)$ are positive, while the
coefficient of $f_{6}(Q)$ is negative. After Borel transformation, $f_{0}(Q)$
and $f_{6}(Q)$ are just as in Eq. (IV.1). Numerically, $\hat{B}[f_{0}(Q)]$ is
always positive, but $\hat{B}[f_{6}(Q)]$ is always negative, so totally, the
instanton contributions to the current $J_{2}^{\sigma}$ are positive. From
Fig. 10, it is clear that the instanton contributions improve the convergence
of current $J_{2}^{\sigma}$ when Borel mass is small.
Figure 10: LHS of $\sigma$ including the instanton effects of four
interpolating currents with $s_{0}$=0.6 $\mbox{GeV}^{2}$.
At this moment, we can use the numeric results associated with LHS of current
$J_{2}^{\sigma}$, at threshold value 0.6 $\mbox{GeV}^{2}$, to fit the RHS in
single resonance approximation that is just the Eq. (88), as illustrated in
Fig. 11. That choosing 0.6 $\mbox{GeV}^{2}$ as the value of threshold is
inspired by previous OPE results. The fitted mass and residues are listed in
Table 3:
Figure 11: The dashed and thick lines represent the left hand sum rule and
right hand sum rule, respectively. To RHS, the mass and residue are presented
in Table III.
From the table, we notice that after adding the instanton contribution, the
mass of $\sigma$ meson is still close to OPE result Eq.(67). Then the
instanton contribution is compatible with OPE results. It suggests that the
physical mass of $\sigma$ depends weakly on the choice of QCD vacuum.
Table 3: Fitted masses and residues in single resonance approximation $s_{0}({\rm GeV}^{2})$ | $M_{\sigma}$(GeV) | ${f}_{\sigma}(10^{-2}\rm GeV)$
---|---|---
0.6 | 0.72 | 0.94
1 | 0.73 | 0.93
In the case of $s_{0}=0.6\rm{GeV^{2}}$, considering a $(\pm 10\%)$ variation
of instanton size $r_{c}=1.6\pm 0.2$, we find a corresponding variation of
$m_{\sigma}=720^{-100}_{+\leavevmode\nobreak\ 60}\rm{MeV}$ and
$f_{\sigma}=0.94^{+\leavevmode\nobreak\ 0.4}_{-0.07}(10^{-2}\rm GeV)$. It
seems like that the change of physical quantity lies within an acceptable
range and the residue is more sensitive to the variation of instanton size
compared with the mass. In Kojo , the authors discussed the meaning of the
residue. In their notations, residue is defined as $\lambda^{2}=2\pi
f_{X}^{2}M_{X}^{8}$. So we obtain a residue $\lambda^{2}=4\times 10^{-5}{\rm
GeV^{10}}$, which is larger than $\lambda^{2}=2\times 10^{-6}{\rm GeV^{10}}$
presented in Kojo . According to the explanation of Kojo , large residue
signifies the interpolating current operators have enough overlaps to the
resonance states and the sum rule constructed with approximate OPE may contain
enough information for the resonance to be extracted. So in our case,
evaluating OPE up to dimension eight condensates seems reasonable.
Finally, in order to investigate further the widths of the $\sigma$ meson
states, it is necessary to find out three point correlation functions for
$\sigma\rightarrow\pi\pi$, which has got out of the scope of this paper.
As for other mesons, the current $J_{2}$ still shows the best performance. The
fitted masses and residues for $\kappa$, $a_{+}$ and $f_{0}$ are presented in
Table IV, V and VI in appendix , respectively.
## V Conclusion and Discussion
In this paper, we study the $0^{+}$ nonet mesons as tetraquark states with
interpolating currents induced from the color-magnetic wavefunction. This
wavefunction is the eigenfunction of the effective color-magnetic Hamiltonian
with the lowest eigenvalue, meaning that the state with this wavefunction is
the most stable one and is most probable to be observed in experiments. Our
approach can be recognized as constructing interpolating currents dynamically.
We find that based on a current which is a proper mixture of the tensor and
pseudoscalar contents, a good sum rule can be obtained. Our result can be
perceived as a direct support to multiquark scenario described by the color-
magnetic interaction, by means of QCD sum rule.
In the SR calculations performed in this paper, we have taken into account the
contributions from operators up to dimension $d=8$ in the OPE. The results of
SR analysis without instanton effects for $0^{+}$ meson nonet
$\\{\sigma,\;\kappa,\;f_{0},\;a_{+}\\}$ are :
1. 1.
$\sigma$: In the SR analysis , a good Borel stability turns out in the region
$M_{B}^{2}>0.2\leavevmode\nobreak\ {\rm GeV}^{2}$. Taking $M_{B}^{2}\approx
0.2\leavevmode\nobreak\ {\rm GeV}^{2}$ and the threshold $s_{0}\approx
0.6\leavevmode\nobreak\ {\rm GeV}^{2}$, the largest pole contribution is
$52\%$ implying that a good SR analysis is achieved. Where we extract the mass
of $\sigma$ $(600\pm 75)$ MeV.
2. 2.
$\kappa$: A good sum rule was found when $s_{0}=0.9\rm{GeV}^{2}$,
$M^{2}_{B}>0.25\rm{GeV}^{2}$. We obtain $\kappa$ mass $(820\pm 80)$MeV with
pole contribution approaching 56%.
3. 3.
$f_{0}\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ a_{+}$: to obtain a
mass about 1 GeV by choosing the threshold and Borel mass, the pole
contributions in SR are always around 10%. This indicates that the SR fails to
analyze $a_{+}$ and $f_{0}$ by using the interpolating currents deduced from
the wavefunctions. We guess the reason is that in
$f_{0}({1\over\sqrt{2}}(\\{us\\}\\{\bar{u}\bar{s}\\}+\\{ds\\}\\{\bar{d}\bar{s}\\}))$
or $a_{+}(\\{us\\}\\{\bar{d}\bar{s}\\})$, there are two strange quarks, so
$SU(3)_{f}$ breaking effects are too strong to be negligible. This causes the
Jaffe’s wavefunctions $|f_{0}\rangle=|0^{+},\underline{9}\rangle_{f_{0}}$ and
$|a_{+}\rangle=|0^{+},\underline{9}\rangle_{a_{+}}$ to miss some aspects of
the $f_{0}$\- and $a_{+}$-physics. We speculate that a legitimate SR analysis
for them should be based on the tetraquark color-magnetic wavefunctions
including the $SU(3)_{f}$-breaking effects due to $m_{s}^{c}>\hat{m}^{c}$.
Proceed stepwise, we consider the direct instanton contribution. To the
current $J_{2}$, the instanton effects are completely positive. Numerically,
this positive effects improve the small Borel mass behavior of the Borel
transformed correlator of current $J_{2}$. Meanwhile, adding instanton
effects, the LHS gives a result compatible with OPE results.
Finally, we go one step further and believe that the idea demonstrated in this
paper also applies to $0^{-}$-$q^{3}\bar{q}^{3}$ system. In DPY , the authors
have successfully extended Jaffe’s method from $q^{2}\bar{q}^{2}$ to
$q^{3}\bar{q}^{3}$ six-quark system (i.e., baryonium). One of the non-trivial
results in DPY for baryonium is the existance of a counterpart of $\sigma$.
We denote this state by $|0^{-},\underline{1}_{f}\rangle$. Corresponding to
Eq. (3) for tetraquark, DPY shows
$H_{eff}|0^{-},\underline{1}_{f}\rangle=-82.533\widetilde{C}|0^{-},\underline{1}_{f}\rangle.$
(91)
In baryonium contents, its color-spin-flavor wavefunction can be expressed as:
$\displaystyle|0^{-},\underline{1}_{f}\rangle$ $\displaystyle\equiv$
$\displaystyle|\mathbf{1},\mathbf{}1_{f}\otimes\mathbf{1}_{f}\rangle_{1}=0.591|(\mathbf{56}_{cs},\mathbf{10}_{c},\mathbf{4};\mathbf{1}_{f}),(\overline{\mathbf{56}}_{cs},\overline{\mathbf{10}}_{c},\mathbf{4};\mathbf{1}_{f}),\mathbf{1}_{c},\mathbf{1};\mathbf{1}_{f}\otimes\mathbf{1}_{f}\rangle$
(92)
$\displaystyle+0.807|(\mathbf{56}_{cs},\mathbf{8}_{c},\mathbf{2};\mathbf{1}_{f}),(\overline{\mathbf{56}}_{cs},\mathbf{8}_{c},\mathbf{2};\mathbf{1}_{f}),\mathbf{1}_{c},\mathbf{1};\mathbf{1}_{f}\otimes\mathbf{1}_{f}\rangle,$
where the notations in DPY have been used. Like $|\sigma\rangle$,
$|0^{-},\underline{1}_{f}\rangle$ has the largest mass defect among all the
baryoniums. This implies that $|0^{-},\underline{1}_{f}\rangle$, the lightest
baryonium meson, may represent a stable physical state. Like Eq. (8), the mass
of $|0^{-},\underline{1}_{f}\rangle$ can be estimated roughly in the naive
constituent quark model as follows
$\displaystyle m_{|0^{-},\underline{1}_{f}\rangle}$ $\displaystyle\approx$
$\displaystyle\langle\sum_{i}m_{i}^{c}-\widetilde{C}\sum_{i\;j}(\lambda_{i}\cdot\lambda_{j})(\overrightarrow{\sigma}_{i}\cdot\overrightarrow{\sigma}_{j})\rangle_{|0^{-},\underline{1}_{f}\rangle}$
(93) $\displaystyle\approx$ $\displaystyle(4\times 360{\rm MeV}+2\times
540{\rm MeV})-82.533\times\left({4\times 20{\rm MeV}+2\times 15{\rm MeV}\over
6}\right)$ $\displaystyle\approx$ $\displaystyle 1.007{\rm GeV}.$
We find that the mass of $|0^{-},\underline{1}_{f}\rangle$ is close to that of
$\eta^{\prime}(960)$ Yao:2006px . Furthermore, their quantum numbers are the
same. So in the multiquark picture, we might identify
$|0^{-},\underline{1}_{f}\rangle$ as $\eta^{\prime}(960)$, or perceive
$\eta^{\prime}(960)$ as a baryonium or a Fermi-Yang meson FY . Alternatively,
there may be a large weight baryonium component in $\eta^{\prime}(960)$.
Usually, in the $q\bar{q}$-picture, the mass of $\eta^{\prime}$ is attributed
to $U(1)_{A}$ anomaly with non-trivial $\theta$ vacuum in QCD tHooft .
However, that scenario has not excluded other schemes yet (e.g., see donoghue
). In our case, a further examination to the conjecture on $\eta^{\prime}$ in
non-perturbative QCD should be meaningful. Since we have already known the
color-magnetic wavefunction for $|0^{-},\underline{1}_{f}\rangle$, following
the method presented in this paper, a SR analysis is straightforward. The
result will be helpful to understand two interesting experimental measurements
that may reveal the baryonium content of $\eta^{\prime}$. Those experiments
are that:
i) to measure the anomalous enhancement near the mass threshold in the
$p\bar{p}$ invariant-mass spectrum from $J/\psi\rightarrow\gamma p\bar{p}$
reported by BES BES1 .
ii) to observe resonance $X(1835)$ in
$J/\psi\rightarrow\gamma\pi^{+}\pi^{-}\eta^{\prime}$ BES2 . In BES1 the data
fitting indicates that the enhancement is a S-wave Breit-Wigner resonance
$X(1835)$ 1 . It has been estimated that the decay branching fraction
$B(X\rightarrow p\bar{p})>4\%$ 2 . The decay mode of $X\rightarrow p\bar{p}$
is due to the tail effect of enhancement resonance of $X(1835)$ near the
threshold of process $J/\psi\rightarrow\gamma p\bar{p}$, therefore the fact of
$B(X\rightarrow p\bar{p})>4\%$ means the coupling between $X$ and $p\bar{p}$
is very very strong. The most natural interpretation to this fact is that
$X(1835)$ is simply a bound state of $p-\bar{p}$. Namely, $X(1835)$ is a
$q^{3}\bar{q}^{3}$-baryonium molecular state datta ; yan . In another hand,
the major decay mode for $X(1835)$ is
$X(1835)\rightarrow\pi^{+}\pi^{-}\eta^{\prime}$ observed by BES BES2 . It
indicates that $X(1835)$ is a molecular exciting state of meson
$\eta^{\prime}$ yan . Consequently, the quark component of $\eta^{\prime}$
should be same as $X(1835)$, i.e., $\eta^{\prime}$ would be a
$0^{-}$-baryonium meson, or a meson with large weight baryonium component. BES
observations BES1 ; BES2 provide evidence to this multiquark picture for
$\eta^{\prime}$ meson.
## ACKNOWLEDGEMENTS
We would like to thank R. L. Jaffe for helpful comments to this work and
information discussions. We also thank Gui-Jun Ding, Dao-Neng Gao, N. I.
Kochelev, Jia-Lun Ping for discussions and Yi Wang, Tower Wang for warm helps.
Especially, we are grateful to Shi-Lin Zhu for introducing useful OPE
calculation method to us. This work is partially supported by National Natural
Science Foundation of China under Grant Numbers 90403021, and by the Chinese
Science Academy Foundation under Grant Numbers KJCX-YW-N29.
## Appendix A
### A.1 Formulas of necessary spectral functions of $\kappa$, $a_{+}$ and
$f_{0}$
For $\kappa$ $(\\{ud\\}\\{\bar{s}\bar{d}\\})$, since the current mass $m_{s}$
is much bigger than $m_{u},m_{d}$, we can ignore terms proportional to
$m_{u},m_{d}$ when listing the necessary spectral functions. Having done this,
the length of formulas will be shortened, and the reader can have a clear
impression about the structure of spectral functions. We will do the same
thing for $a_{+}$ and $f_{0}$. However, in numerical calculations, the
contributions from the $u,d$ quark mass terms have been taken into account.
The spectral functions are the followings:
$\displaystyle\rho^{\kappa\rm OPE}_{T,T}$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{1280\pi^{6}}-\frac{m_{s}^{2}}{64\pi^{6}}s^{3}+(\frac{11\langle\textsl{g}^{2}GG\rangle}{768\pi^{6}}+\frac{m_{s}\langle\bar{s}s\rangle}{8\pi^{4}})s^{2}-\frac{11m_{s}^{2}\langle\textsl{g}^{2}GG\rangle}{256\pi^{6}}s+\frac{11m_{s}\langle\textsl{g}^{2}GG\rangle\langle\bar{s}s\rangle}{192\pi^{4}},$
(94) $\displaystyle\rho^{\kappa\rm OPE}_{S,S}$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{61440\pi^{6}}-\frac{{m_{s}}^{2}s^{3}}{3072\pi^{6}}+(\frac{\langle\textsl{g}^{2}GG\rangle}{6144\pi^{6}}-\frac{{m_{s}}\langle\bar{q}q\rangle}{192\pi^{4}}+\frac{{m_{s}}\langle\bar{s}s\rangle}{384\pi^{4}})s^{2}$
(95)
$\displaystyle+(-\frac{m_{s}^{2}\langle\textsl{g}^{2}GG\rangle}{2048\pi^{6}}-\frac{m_{s}\langle\textsl{g}\bar{q}\sigma
Gq\rangle}{128\pi^{4}}+\frac{\langle\bar{q}q\rangle^{2}}{24\pi^{2}}+\frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{24\pi^{2}})s$
$\displaystyle-\frac{m_{s}^{2}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}-\frac{m_{s}\langle\textsl{g}^{2}GG\rangle\langle\bar{q}q\rangle}{768\pi^{4}}+\frac{m_{s}\langle\textsl{g}^{2}GG\rangle\langle\bar{s}s\rangle}{1536\pi^{4}}+\frac{\langle\bar{q}q\rangle\langle\textsl{g}\bar{q}\sigma
Gq\rangle}{24\pi^{2}}+\frac{\langle\bar{s}s\rangle\langle\textsl{g}\bar{q}\sigma
Gq\rangle}{48\pi^{2}}+\frac{\langle\bar{q}q\rangle\langle\textsl{g}\bar{q}\sigma
Gq\rangle}{48\pi^{2}},$ $\displaystyle\rho^{\kappa\rm OPE}_{T,S}$
$\displaystyle=$ $\displaystyle\rho^{\kappa\rm
OPE}_{S,T}=-\frac{\langle\textsl{g}^{2}GG\rangle}{1024\pi^{6}}s^{2}+\frac{3\langle\textsl{g}^{2}GG\rangle
m_{s}^{2}}{1024\pi^{6}}s-\frac{\langle\textsl{g}^{2}GG\rangle\langle\bar{s}s\rangle
m_{s}}{256\pi^{4}},$ (96) $\displaystyle\rho^{\kappa\rm OPE}_{P,P}$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{61440\pi^{6}}-\frac{m_{s}^{2}}{3072\pi^{6}}s^{3}+(\frac{\langle\bar{q}q\rangle
m_{s}}{192\pi^{4}}+\frac{\langle\bar{s}s\rangle
m_{s}}{384\pi^{4}}+\frac{\langle\textsl{g}^{2}GG\rangle}{6144\pi^{6}})s^{2}-(\frac{\langle\bar{q}q\rangle^{2}}{24\pi^{2}}+\frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{24\pi^{2}}-\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle m_{s}}{128\pi^{4}}$ (97)
$\displaystyle+\frac{\langle\textsl{g}^{2}GG\rangle
m_{s}^{2}}{2048\pi^{6}})s+(\frac{m_{s}^{2}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}-\frac{\langle\textsl{g}\bar{s}\sigma
Gs\rangle\langle\bar{q}q\rangle}{48\pi^{2}}-\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle(\langle\bar{s}s\rangle+2\langle\bar{q}q\rangle)}{48\pi^{2}}+\frac{\langle\textsl{g}^{2}GG\rangle(\langle\bar{q}q\rangle+\langle\bar{s}s\rangle)m_{s}}{1536\pi^{4}}),$
$\displaystyle\rho^{\kappa\rm OPE}_{T,P}$ $\displaystyle=$
$\displaystyle\rho^{\kappa\rm
OPE}_{P,T}=-\frac{\langle\textsl{g}^{2}GG\rangle}{1024\pi^{6}}s^{2}+\frac{3\langle\textsl{g}^{2}GG\rangle
m_{s}^{2}}{1024\pi^{6}}s-\frac{\langle\textsl{g}^{2}GG\rangle\langle\bar{s}s\rangle
m_{s}}{256\pi^{4}},$ (98) $\displaystyle\rho^{\kappa\rm OPE}_{A,A}$
$\displaystyle=$
$\displaystyle\frac{s^{4}}{7680\pi^{6}}-\frac{m_{s}^{2}}{384\pi^{6}}s^{3}+(\frac{m_{s}(\langle\bar{s}s\rangle-\langle\bar{q}q\rangle)}{48\pi^{4}}+\frac{5\langle\textsl{g}^{2}GG\rangle}{3072\pi^{6}})s^{2}-(\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle m_{s}}{32\pi^{4}}+\frac{5\langle\textsl{g}^{2}GG\rangle
m_{s}^{2}}{1024\pi^{6}}-\frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{6\pi^{2}}$
(99)
$\displaystyle-\frac{\langle\bar{q}q\rangle\langle\bar{q}q\rangle}{6\pi^{2}})s+(\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle(2\langle\bar{q}q\rangle+\langle\bar{s}s\rangle)}{12\pi^{2}}-\frac{\langle\bar{q}q\rangle^{2}m_{s}^{2}}{3\pi^{2}}+\frac{m_{s}\langle\textsl{g}^{2}GG\rangle(5\langle\bar{s}s\rangle-2\langle\bar{q}q\rangle)}{768\pi^{4}}+\frac{\langle\textsl{g}\bar{s}\sigma
Gs\rangle\langle\bar{q}q\rangle}{12\pi^{2}}),$ $\displaystyle\rho^{\kappa\rm
OPE}_{A,S}$ $\displaystyle=$ $\displaystyle\rho^{\kappa\rm
OPE}_{S,A}=\frac{\langle\textsl{g}^{2}GG\rangle
m_{s}\langle\bar{q}q\rangle}{256\pi^{4}},$ (100) $\displaystyle\rho^{\kappa\rm
OPE}_{A,P}$ $\displaystyle=$ $\displaystyle\rho^{\kappa\rm
OPE}_{P,A}=0.\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
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\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (101)
For $a_{+}\leavevmode\nobreak\ (\\{us\\}\\{\bar{d}\bar{s}\\})$ and
$f_{0}\leavevmode\nobreak\
(\frac{1}{\sqrt{2}}(\\{su\\}\\{\bar{s}\bar{u}\\}+\\{sd\\}\\{\bar{s}\bar{d}\\})$,
we only list the spectral functions for $a_{+}$ below. This is because in the
widely adopted scheme Eq. (53), $u$ and $d$ quark take the same value of
current masses and condensates, which leads to a direct consequence that from
the OPE calculation of the correlators of currents, we can not discern $a_{+}$
and $f_{0}$. In other words, to each kind interpolating current in Eq. (43),
the correlators of $a_{+}$’s and the correlators of $f_{0}$’s take the same
expressions after completing the OPE calculation.
$\displaystyle\rho^{a_{+}\rm OPE}_{T,T}$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{1280}-\frac{m_{s}^{2}}{32\pi^{6}}s^{3}+(\frac{3m_{s}^{4}}{16\pi^{6}}+\frac{\langle\bar{s}s\rangle
m_{s}}{4\pi^{4}}+\frac{11\langle\textsl{g}^{2}GG\rangle}{768})s^{2}-(\frac{3\langle\bar{s}s\rangle
m_{s}^{3}}{2\pi^{4}}+\frac{11\langle\textsl{g}^{2}GG\rangle
m_{s}^{2}}{128\pi^{6}})s+(\frac{4\langle\bar{q}q\rangle^{2}m_{s}^{2}}{\pi^{2}}$
(102)
$\displaystyle+\frac{\langle\bar{s}s\rangle^{2}m_{s}^{2}}{\pi^{2}}+\frac{5\langle\textsl{g}^{2}GG\rangle
m_{s}^{4}}{128\pi^{6}}+\frac{11\langle\textsl{g}^{2}GG\rangle\langle\bar{s}s\rangle
m_{s}}{96\pi^{4}}),$ $\displaystyle\rho^{a_{+}\rm OPE}_{S,S}$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{61440\pi^{6}}-\frac{m_{s}^{2}}{1536\pi^{6}}s^{3}+(\frac{m_{s}^{4}}{256\pi^{6}}+\frac{m_{s}(\langle\bar{s}s\rangle-2\langle\bar{q}q\rangle)}{192\pi^{4}}+\frac{\langle\textsl{g}^{2}GG\rangle}{6144\pi^{6}})s^{2}-(\frac{m_{s}^{3}\langle\bar{s}s\rangle}{32\pi^{2}}-\frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{12\pi^{2}}-\frac{m_{s}^{3}\langle\bar{q}q\rangle}{16\pi^{2}}$
(103) $\displaystyle+\frac{\langle\textsl{g}^{2}GG\rangle
m_{s}^{2}}{1024\pi^{6}}+\frac{\langle\textsl{g}\bar{q}\sigma Gq\rangle
m_{s}}{64\pi^{4}})s+(\frac{m_{s}^{2}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}-\frac{m_{s}^{2}\langle\bar{s}s\rangle\langle\bar{q}q\rangle}{4\pi^{2}}+\frac{m_{s}^{2}\langle\bar{s}s\rangle^{2}}{48\pi^{2}}+\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle m_{s}^{3}}{32\pi^{4}}+\frac{\langle\textsl{g}\bar{s}\sigma
Gs\rangle\langle\bar{q}q\rangle}{24\pi^{2}}$
$\displaystyle+\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle\langle\bar{s}s\rangle}{24\pi^{2}}-\frac{\langle\textsl{g}^{2}GG\rangle\langle\bar{q}q\rangle
m_{s}}{384\pi^{4}}+\frac{\langle\textsl{g}^{2}GG\rangle\langle\bar{s}s\rangle
m_{s}}{768\pi^{4}}),$ $\displaystyle\rho^{a_{+}\rm OPE}_{T,S}$
$\displaystyle=$ $\displaystyle\rho^{a_{+}\rm
OPE}_{S,T}=-\frac{\langle\textsl{g}^{2}GG\rangle
s^{2}}{1024\pi^{6}}+\frac{3\langle\textsl{g}^{2}GG\rangle
m_{s}^{2}}{512\pi^{6}}s-(\frac{3\langle\textsl{g}^{2}GG\rangle
m_{s}^{4}}{1024\pi^{6}}+\frac{\langle\textsl{g}^{2}GG\rangle
m_{s}\langle\bar{s}s\rangle}{128\pi^{4}}),$ (104) $\displaystyle\rho^{a_{+}\rm
OPE}_{P,P}$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{61440\pi^{6}}-\frac{m_{s}^{2}}{1536\pi^{6}}s^{3}+(\frac{m_{s}^{4}}{256\pi^{6}}+\frac{m_{s}(\langle\bar{s}s\rangle+2\langle\bar{q}q\rangle)}{192\pi^{4}}+\frac{\langle\textsl{g}^{2}GG\rangle}{6144\pi^{6}})s^{2}-(\frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{12\pi^{2}}+\frac{m_{s}^{3}(\langle\bar{s}s\rangle+2\langle\bar{q}q\rangle)}{32\pi^{2}}$
(105) $\displaystyle+\frac{\langle\textsl{g}^{2}GG\rangle
m_{s}^{2}}{1024\pi^{6}}-\frac{\langle\textsl{g}\bar{q}\sigma Gq\rangle
m_{s}}{64\pi^{4}}-\frac{\langle\textsl{g}\bar{s}\sigma Gs\rangle
m_{q}}{64\pi^{4}})s+(\frac{3\langle\bar{s}s\rangle\langle\bar{q}q\rangle
m_{s}^{2}}{4\pi^{2}}+\frac{\langle\bar{q}q\rangle^{2}m_{s}^{2}}{12\pi^{2}}+\frac{m_{s}^{2}\langle\bar{s}s\rangle^{2}}{48\pi^{2}}-\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle m_{s}^{3}}{32\pi^{4}}$
$\displaystyle-\frac{\langle\textsl{g}\bar{s}\sigma
Gs\rangle\langle\bar{q}q\rangle}{24\pi^{2}}-\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle\langle\bar{s}s\rangle}{24\pi^{2}}+\frac{\langle\textsl{g}^{2}GG\rangle\langle\bar{q}q\rangle
m_{s}}{384\pi^{4}}+\frac{\langle\textsl{g}^{2}GG\rangle\langle\bar{s}s\rangle
m_{s}}{768\pi^{4}},$ $\displaystyle\rho^{a_{+}\rm OPE}_{T,P}$ $\displaystyle=$
$\displaystyle\rho^{a_{+}\rm OPE}_{P,T}=-\frac{\langle\textsl{g}^{2}GG\rangle
s^{2}}{1024\pi^{6}}+\frac{3\langle\textsl{g}^{2}GG\rangle
m_{s}^{2}}{512\pi^{6}}s-(\frac{3\langle\textsl{g}^{2}GG\rangle
m_{s}^{4}}{1024\pi^{6}}+\frac{\langle\textsl{g}^{2}GG\rangle
m_{s}\langle\bar{s}s\rangle}{128\pi^{4}}),$ (106) $\displaystyle\rho^{a_{+}\rm
OPE}_{A,A}$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{7680\pi^{6}}-\frac{m_{s}^{2}}{192\pi^{6}}s^{3}+(\frac{m_{s}^{4}}{32\pi^{6}}+\frac{(\langle\bar{s}s\rangle-\langle\bar{q}q\rangle)m_{s}}{24\pi^{4}}+\frac{5\langle\textsl{g}^{2}GG\rangle}{3072\pi^{6}})s^{2}-(\frac{5\langle\textsl{g}^{2}GG\rangle
m_{s}^{2}}{512\pi^{6}}+\frac{m_{s}^{3}(\langle\bar{s}s\rangle-\langle\bar{q}q\rangle)}{4\pi^{2}}$
(107)
$\displaystyle-\frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{3\pi^{2}}+\frac{5\langle\textsl{g}\bar{q}\sigma
Gq\rangle
m_{s}}{16\pi^{4}})s+(\frac{2m_{s}^{2}\langle\bar{q}q\rangle^{2}}{3\pi^{2}}-\frac{m_{s}^{2}\langle\bar{s}s\rangle\langle\bar{q}q\rangle}{\pi^{2}}+\frac{m_{s}^{2}\langle\bar{s}s\rangle^{2}}{6\pi^{2}}+\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle m_{s}^{3}}{8\pi^{4}}+\frac{\langle\textsl{g}\bar{s}\sigma
Gs\rangle\langle\bar{q}q\rangle}{6\pi^{2}}$
$\displaystyle+\frac{\langle\textsl{g}\bar{q}\sigma
Gq\rangle\langle\bar{s}s\rangle}{6\pi^{2}}-\frac{\langle\textsl{g}^{2}GG\rangle\langle\bar{q}q\rangle
m_{s}}{192\pi^{4}}+\frac{5\langle\textsl{g}^{2}GG\rangle
m_{s}^{4}}{1024\pi^{6}}+\frac{5\langle\textsl{g}^{2}GG\rangle\langle\bar{s}s\rangle
m_{s}}{384\pi^{4}}),$ $\displaystyle\rho^{a_{+}\rm OPE}_{A,S}$
$\displaystyle=$ $\displaystyle\rho^{a_{+}\rm
OPE}_{S,A}=-\frac{3\langle\textsl{g}^{2}GG\rangle
m_{s}^{2}}{4096\pi^{6}}s+(\frac{\langle\textsl{g}^{2}GG\rangle
m_{s}\langle\bar{s}s\rangle}{256\pi^{4}}+\frac{\langle\textsl{g}^{2}GG\rangle
m_{s}\langle\bar{q}q\rangle}{256\pi^{4}}),$ (108) $\displaystyle\rho^{a_{+}\rm
OPE}_{A,P}$ $\displaystyle=$ $\displaystyle\rho^{a_{+}\rm OPE}_{P,A}=0.$ (109)
To convince the reader that our calculations are reliable, we make a
comparison with the results of other authors. For example,
$\displaystyle\rho^{\kappa\rm OPE}_{T,T}$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{1280\pi^{6}}-\frac{m_{s}^{2}}{64\pi^{6}}s^{3}+(\frac{11\langle\textsl{g}^{2}GG\rangle}{768\pi^{6}}+\frac{m_{s}\langle\bar{s}s\rangle}{8\pi^{4}})s^{2}-\frac{11m_{s}^{2}\langle\textsl{g}^{2}GG\rangle}{256\pi^{6}}s+\frac{11m_{s}\langle\textsl{g}^{2}GG\rangle\langle\bar{s}s\rangle}{192\pi^{4}},$
$\displaystyle\rho^{\kappa\rm OPE}_{S,S}$ $\displaystyle=$
$\displaystyle\frac{s^{4}}{61440\pi^{6}}-\frac{{m_{s}}^{2}s^{3}}{3072\pi^{6}}+(\frac{\langle\textsl{g}^{2}GG\rangle}{6144\pi^{6}}-\frac{{m_{s}}\langle\bar{q}q\rangle}{192\pi^{4}}+\frac{{m_{s}}\langle\bar{s}s\rangle}{384\pi^{4}})s^{2}$
(111)
$\displaystyle+(-\frac{m_{s}^{2}\langle\textsl{g}^{2}GG\rangle}{2048\pi^{6}}-\frac{m_{s}\langle\textsl{g}\bar{q}\sigma
Gq\rangle}{128\pi^{4}}+\frac{\langle\bar{q}q\rangle^{2}}{24\pi^{2}}+\frac{\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{24\pi^{2}})s$
$\displaystyle-\frac{m_{s}^{2}\langle\bar{q}q\rangle^{2}}{12\pi^{2}}-\frac{m_{s}\langle\textsl{g}^{2}GG\rangle\langle\bar{q}q\rangle}{768\pi^{4}}+\frac{m_{s}\langle\textsl{g}^{2}GG\rangle\langle\bar{s}s\rangle}{1536\pi^{4}}+\frac{\langle\bar{q}q\rangle\langle\textsl{g}\bar{q}\sigma
Gq\rangle}{24\pi^{2}}+\frac{\langle\bar{s}s\rangle\langle\textsl{g}\bar{q}\sigma
Gq\rangle}{48\pi^{2}}+\frac{\langle\bar{q}q\rangle\langle\textsl{g}\bar{q}\sigma
Gq\rangle}{48\pi^{2}}.$
These are the expressions appearing in zhu2 .
### A.2 Instanton contribution to correlators of $\kappa$, $a_{+}$ and
$f_{0}$
We obtain the intanton contributions to $\kappa$ correlators as follows,
$\displaystyle\Pi_{TT}^{\kappa(\rm inst)}$ $\displaystyle=$
$\displaystyle(\frac{76n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{3\pi^{4}m_{q}^{\ast
2}}+\frac{144n_{eff}r_{c}^{4}\langle\bar{q}q\rangle^{2}}{3\pi^{4}m_{q}^{\ast}m_{s}^{\ast}})f_{0}(Q),$
(112) $\displaystyle\Pi_{SS}^{\kappa(\rm inst)}$ $\displaystyle=$
$\displaystyle(\frac{16n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}+\frac{16n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast
2}})f_{6}(Q)+(\frac{11n_{eff}r_{c}^{4}\langle\bar{q}q\rangle^{2}}{18\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}+\frac{19n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{36\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (113) $\displaystyle\Pi_{PP}^{\kappa(\rm inst)}$
$\displaystyle=$
$\displaystyle-(\frac{16n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}+\frac{16n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast
2}})f_{6}(Q)+(\frac{19n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{36\pi^{4}m_{q}^{\ast
2}}+\frac{11n_{eff}r_{c}^{4}\langle\bar{q}q\rangle^{2}}{18\pi^{4}m_{q}^{\ast}m_{s}^{\ast}})f_{0}(Q),$
$\displaystyle\Pi_{TS}^{\kappa(\rm inst)}$ $\displaystyle=$
$\displaystyle\Pi_{ST}^{\kappa(\rm
inst)}=\frac{n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{\pi^{4}m_{q}^{\ast
2}}f_{0}(Q),$ (114) $\displaystyle\Pi_{TP}^{\kappa(\rm inst)}$
$\displaystyle=$ $\displaystyle\Pi_{PT}^{\kappa(\rm
inst)}=\frac{n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{\pi^{4}m_{q}^{\ast
2}}f_{0}(Q),$ (115) $\displaystyle\Pi_{AA}^{\kappa(\rm inst)}$
$\displaystyle=$
$\displaystyle(\frac{24n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}+\frac{24n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast
2}})f_{6}(Q)+(\frac{37n_{eff}r_{c}^{4}\langle\bar{q}q\rangle^{2}}{6\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}+\frac{34n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{9\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (116) $\displaystyle\Pi_{AS}^{\kappa(\rm inst)}$
$\displaystyle=$ $\displaystyle\Pi_{SA}^{\kappa(\rm
inst)}=-(\frac{20n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}+\frac{10n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast
2}})f_{6}(Q),$ (117) $\displaystyle\Pi_{AP}^{\kappa(\rm inst)}$
$\displaystyle=$ $\displaystyle\Pi_{PA}^{\kappa(\rm
inst)}=-(\frac{20n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}+\frac{10n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast
2}})f_{6}(Q).$ (118)
The instanton contributions to $a_{+}$ are,
$\displaystyle\Pi_{TT}^{a_{+}(\rm inst)}$ $\displaystyle=$
$\displaystyle(\frac{152n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{3\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}+\frac{68n_{eff}r_{c}^{4}\langle\bar{s}s\rangle^{2}}{3\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (119) $\displaystyle\Pi_{SS}^{a_{+}(\rm inst)}$
$\displaystyle=$
$\displaystyle\frac{32n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}f_{6}(Q)+(\frac{19n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{18\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}+\frac{n_{eff}r_{c}^{4}\langle\bar{s}s\rangle^{2}}{12\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (120) $\displaystyle\Pi_{PP}^{a_{+}(\rm inst)}$
$\displaystyle=$
$\displaystyle-\frac{32n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}f_{6}(Q)+(\frac{19n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{18\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}+\frac{n_{eff}r_{c}^{4}\langle\bar{s}s\rangle^{2}}{12\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (121) $\displaystyle\Pi_{TS}^{a_{+}(\rm inst)}$
$\displaystyle=$ $\displaystyle\Pi_{ST}^{a_{+}(\rm
inst)}=(\frac{2n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}-\frac{n_{eff}r_{c}^{4}\langle\bar{s}s\rangle^{2}}{\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (122) $\displaystyle\Pi_{TP}^{a_{+}(\rm inst)}$
$\displaystyle=$ $\displaystyle\Pi_{PT}^{a_{+}(\rm
inst)}=(\frac{2n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}-\frac{n_{eff}r_{c}^{4}\langle\bar{s}s\rangle^{2}}{\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (123) $\displaystyle\Pi_{AA}^{a_{+}(\rm inst)}$
$\displaystyle=$
$\displaystyle\frac{48n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}f_{6}(Q)+(\frac{68n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{9\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}+\frac{43n_{eff}r_{c}^{4}\langle\bar{s}s\rangle^{2}}{18\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (124) $\displaystyle\Pi_{AS}^{a_{+}(\rm inst)}$
$\displaystyle=$ $\displaystyle\Pi_{SA}^{a+(\rm
inst)}=-(\frac{20n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}+\frac{10n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast
2}})f_{6}(Q),$ (125) $\displaystyle\Pi_{AP}^{a_{+}(\rm inst)}$
$\displaystyle=$ $\displaystyle\Pi_{PA}^{a_{+}(\rm
inst)}=-(\frac{20n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}+\frac{10n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast
2}})f_{6}(Q).$ (126)
The instanton contributions to $f_{0}$ are,
$\displaystyle\Pi_{TT}^{f_{0}(\rm inst)}$ $\displaystyle=$
$\displaystyle(\frac{152n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{3\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}-\frac{68n_{eff}r_{c}^{4}\langle\bar{s}s\rangle^{2}}{3\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (127) $\displaystyle\Pi_{SS}^{f_{0}(\rm inst)}$
$\displaystyle=$
$\displaystyle\frac{32n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}f_{6}(Q)+(\frac{19n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{18\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}-\frac{n_{eff}r_{c}^{4}\langle\bar{s}s\rangle^{2}}{12\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (128) $\displaystyle\Pi_{PP}^{f_{0}(\rm inst)}$
$\displaystyle=$
$\displaystyle-\frac{32n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}f_{6}(Q)+(\frac{19n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{18\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}-\frac{n_{eff}r_{c}^{4}\langle\bar{s}s\rangle^{2}}{12\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (129) $\displaystyle\Pi_{TS}^{f_{0}(\rm inst)}$
$\displaystyle=$ $\displaystyle\Pi_{ST}^{f_{0}(\rm
inst)}=(\frac{2n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}+\frac{n_{eff}r_{c}^{4}\langle\bar{s}s\rangle^{2}}{\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (130) $\displaystyle\Pi_{TP}^{f_{0}(\rm inst)}$
$\displaystyle=$ $\displaystyle\Pi_{PT}^{f_{0}(\rm
inst)}=(\frac{2n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}+\frac{n_{eff}r_{c}^{4}\langle\bar{s}s\rangle^{2}}{\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (131) $\displaystyle\Pi_{AA}^{f_{0}(\rm inst)}$
$\displaystyle=$
$\displaystyle\frac{48n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}f_{6}(Q)+(\frac{68n_{eff}r_{c}^{4}\langle\bar{q}q\rangle\langle\bar{s}s\rangle}{9\pi^{4}m_{q}^{\ast}m_{s}^{\ast}}-\frac{43n_{eff}r_{c}^{4}\langle\bar{s}s\rangle^{2}}{18\pi^{4}m_{q}^{\ast
2}})f_{0}(Q),$ (132) $\displaystyle\Pi_{AS}^{f_{0}(\rm inst)}$
$\displaystyle=$ $\displaystyle\Pi_{SA}^{f_{0}(\rm
inst)}=-(\frac{20n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}-\frac{10n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast
2}})f_{6}(Q),$ (133) $\displaystyle\Pi_{AP}^{f_{0}(\rm inst)}$
$\displaystyle=$ $\displaystyle\Pi_{PA}^{f_{0}(\rm
inst)}=-(\frac{20n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast}m_{s}^{\ast}}-\frac{10n_{eff}r_{c}^{4}}{\pi^{8}m_{q}^{\ast
2}})f_{6}(Q).$ (134)
To check our results, we take the $SU(3)_{f}$ limit, which is
$m^{\ast}_{q}=m^{\ast}_{s}$ and
$\langle\bar{q}q\rangle=\langle\bar{s}s\rangle$. In this limit, the instanton
contributions to $\kappa$ and $a_{+}$ are equal to each other. This is because
that they all belong to the octet representation of $SU(3)_{f}$. But this
check is not suitable for $f_{0}$, since it comes from ideally mixing of the
flavor singlet state with the isospin $I$=0 component of flavor octet state .
Table 4: Fitted masses and residues in single resonance approximation for $\kappa$ $s_{0}({\rm GeV}^{2})$ | $M_{\kappa}$(GeV) | ${f}_{\kappa}(10^{-2}\rm GeV)$
---|---|---
1 | 0.72 | 1.04
1.5 | 0.73 | 1.00
Table 5: Fitted masses and residues in single resonance approximation for
$a_{+}$
$s_{0}({\rm GeV}^{2})$ | $M_{a_{+}}$(GeV) | ${f}_{a_{+}}(10^{-2}\rm GeV)$
---|---|---
1 | 0.73 | 0.89
1.5 | 0.73 | 0.90
Table 6: Fitted masses and residues in single resonance approximation for
$f_{0}$
$s_{0}({\rm GeV}^{2})$ | $M_{f_{0}}$(GeV) | ${f}_{f_{0}}(10^{-2}\rm GeV)$
---|---|---
1 | 0.72 | 0.90
1.5 | 0.72 | 0.94
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|
arxiv-papers
| 2009-01-11T05:07:25 |
2024-09-04T02:48:59.824620
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yi Pang, Mu-Lin Yan",
"submitter": "Yi Pang",
"url": "https://arxiv.org/abs/0901.1412"
}
|
0901.1424
|
# Wigner functions of thermo number state, photon subtracted and added thermo
vacuum state at finite temperature††thanks: Project supported by the National
Natural Science Foundation of China (Grant Nos 10775097 and 10874174).
Li-yun Hu and Hong-yi Fan
Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China
Corresponding author. _E-mail address_ : hlyun2008@126.com (L-Y Hu).
###### Abstract
Based on Takahashi-Umezawa thermo field dynamics and the order-invariance of
Weyl ordered operators under similar transformations, we present a new
approach to deriving the exact Wigner functions of thermo number state, photon
subtracted and added thermo vacuum state. We find that these Wigner functions
are related to the Gaussian-Laguerre type functions of temperature, whose
statistical properties are then analysed.
## I Introduction
In recent years photon subtracted and added quantum states have been paid much
attention because these fields exhibit an abundant of nonclassical properties
and may give access to a complete engineering of quantum states and to
fundamental quantum phenomena [1-8]. However, all these discussions are
restricted to the case at zero point temperature. In fact, most systems are
not isolated, but are immersed in a “thermal reservoir”, excitation and de-
excitation processes of a system are influenced by its energy exchange with
reservoirs. In this work we study field properties by photon subtracting and
adding at finite temperature.
The Wigner function (WF) is a powerful tool to investigate the nonclassicality
of optical fields [9,10]. Its partial negativity implies the highly
nonclassical properties of quantum states and is often used to describe the
decoherence of quantum states [7,8,11,12]. In one dimensional case, the WF of
a density matrix $\rho$ is defined as
$\mathtt{Tr}\left[\rho\Delta(\alpha)\right],$ where $\Delta(\alpha)$ is the
single-mode Wigner operator, whose normally ordered form and Weyl ordered form
are given as [13-15], respectively,
$\Delta\left(\alpha\right)=\frac{1}{\pi}\colon
e^{-\left(q-Q\right)^{2}-\left(p-P\right)^{2}}\colon=\frac{1}{\pi}\colon
e^{-2\left(\alpha-a\right)\left(\alpha^{\ast}-a^{\dagger}\right)}\colon,$ (1)
and
$\Delta\left(\alpha\right)=\frac{1}{2}\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\alpha-a\right)\delta\left(\alpha^{\ast}-a^{\dagger}\right)\genfrac{}{}{0.0pt}{}{:}{:},$
(2)
where $\alpha=\left(q+\mathtt{i}p\right)/\sqrt{2},$
$a=\left(Q+\mathtt{i}P\right)/\sqrt{2}$, $\left[Q,P\right]=\mathtt{i},$
$\hbar=1;$ $a$ and $a^{\dagger}$ ($\left[a,a^{\dagger}\right]=1)$ are Bose
annihilation and creation operators, the symbols $\colon\colon$ and
$\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$ denote the normal
ordering and the Weyl ordering, respectively. Our main aim is to provide a new
and direct approach to deriving the WFs of quantum states at finite
temperature by using the order-invariance of Weyl ordered operators under
similar transformations [13-15], which means
$S\genfrac{}{}{0.0pt}{}{:}{:}\left(\circ\circ\circ\right)\genfrac{}{}{0.0pt}{}{:}{:}S^{-1}=\genfrac{}{}{0.0pt}{}{:}{:}S\left(\circ\circ\circ\right)S^{-1}\genfrac{}{}{0.0pt}{}{:}{:},$
(3)
as if the “fence” $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$did
not exist, so $S$ can pass through it. We also appeal to the Takahashi-Umezawa
thermo field dynamics (TFD) [16-18], we consider it convenient to obtaining
the explicit expressions of WFs.
## II Brief review of thermo state
The main point of TFD lies in converting the evaluation of ensemble average at
nonzero temperature into the equivalent expectation value with a pure state.
This worthwhile convenience is at the expense of introducing a fictitious
field (or a so-called tilde-conjugate field, denoted as operator
$\tilde{a}^{\dagger}$) in the extending Hilbert space $\tilde{H}$, thus the
original optical field state $\left|n\right\rangle$ in the Hilbert space
$\mathcal{H}$ is accompanied by a tilde state $\left|\tilde{n}\right\rangle$
in $\tilde{H}$. A similar rule holds for operators: every annihilation
operator $a$ acting on $\mathcal{H}$ has an image $\tilde{a}$ acting on
$\tilde{H}$. At finite temperature $T$ the thermal vacuum
$\left|0(\beta)\right\rangle$ is defined by the requirement that the vacuum
expectation value agrees with the statistical average [16-18], i.e.
$\left\langle A\right\rangle=\mathtt{Tr}\left(\rho_{c}A\right)=\left\langle
0(\beta)\right|A\left|0(\beta)\right\rangle=\mathtt{Tr}\left(Ae^{-\beta
H}\right)/\mathtt{Tr}\left(e^{-\beta H}\right),$ (4)
where $\beta=\frac{1}{kT},$ $k$ is the Boltzmann constant and $H$ is the
system’s Hamiltonian. For the ensemble of free bosons with Hamiltonian
$H_{0}=\omega a^{{\dagger}}a$, the thermal vacuum state
$\left|0(\beta)\right\rangle$ is
$\left|0(\beta)\right\rangle=\text{sech}\theta\exp\left[a^{\dagger}\tilde{a}^{\dagger}\tanh\theta\right]\left|0,\tilde{0}\right\rangle=S\left(\theta\right)\left|0,\tilde{0}\right\rangle,$
(5)
where $\left|0,\tilde{0}\right\rangle$ is annihilated by $a$ and $\tilde{a},$
$\left[\tilde{a},\tilde{a}^{\dagger}\right]=1,$ and
$S\left(\theta\right)\equiv\exp\left[\theta\left(a^{\dagger}\tilde{a}^{\dagger}-a\tilde{a}\right)\right],$
(6)
is the thermo squeezing operator which transforms the zero-temperature vacuum
$\left|0,\tilde{0}\right\rangle$ into the thermo vacuum state
$\left|0(\beta)\right\rangle,$ and $\theta$ is related to the Bose
distribution by
$\tanh\theta=\exp\left(-\frac{\omega}{2kT}\right),$ (7)
which is determined by comparing the Bose–Einstein distribution
$n_{c}=\left[\exp\left(\frac{\omega}{kT}\right)-1\right]^{-1}$ (8)
and
$\left\langle
0(\beta)\right|a^{\dagger}a\left|0(\beta)\right\rangle=\sinh^{2}\theta.$ (9)
In particular, when operator $A$ is the Wigner operator
$\Delta\left(\alpha\right)$ itself, it is easy to see that
$\displaystyle\mathtt{Tr}_{a}\left(\Delta\left(\alpha\right)e^{-\beta
H}\right)/\mathtt{Tr}_{a}\left(e^{-\beta H}\right)$ $\displaystyle=$
$\displaystyle\left\langle
0(\beta)\right|\Delta\left(\alpha\right)\left|0(\beta)\right\rangle$ (10)
$\displaystyle=$
$\displaystyle\mathtt{Tr}_{a,\tilde{a}}\left[\Delta\left(\alpha\right)\left|0(\beta)\right\rangle\left\langle
0(\beta)\right|\right],$
which is just the WF of thermo vacuum state. From Eq.(10) one can see that the
calculation of WF for thermo states is converted into the expectation value of
Wigner operator in themo vacuum state $\left|0(\beta)\right\rangle$
($\rho_{c}\rightarrow\left|0(\beta)\right\rangle\left\langle
0(\beta)\right|$), which is defined in the enlarged Fock space. This implies
that it is convenient to deriving some WFs of density operators at finite
temperature by doubly enlarging the original space.
## III Normally ordered form of
$S^{\dagger}\left(\theta\right)\Delta\left(\alpha\right)S\left(\theta\right)$
In order to deriving conveniently the WFs of density operators at finite
temperature, let’s first calculate the normally ordered form of
$S^{\dagger}\left(\theta\right)\Delta\left(\alpha\right)S\left(\theta\right).$
Recalling that for single-mode case the Weyl rule [13-15] is defined as
$\hat{H}\left(a,a^{{\dagger}}\right)=2\int\mathtt{d}^{2}\alpha
h\left(\alpha,\alpha^{\ast}\right)\Delta\left(\alpha\right),$ (11)
where $h\left(\alpha,\alpha^{\ast}\right)$ is the classical function
corresponding to operator $\hat{H}\left(a,a^{{\dagger}}\right).$ Eq.(11)
expresses the Weyl correspondence rule, using (2) it can be expressed as
$\displaystyle\hat{H}\left(a,a^{{\dagger}}\right)$ $\displaystyle=$
$\displaystyle\int\mathtt{d}^{2}\alpha
h\left(\alpha,\alpha^{\ast}\right)\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\alpha-a\right)\delta\left(\alpha^{\ast}-a^{\dagger}\right)\genfrac{}{}{0.0pt}{}{:}{:}$
(12) $\displaystyle=$
$\displaystyle\genfrac{}{}{0.0pt}{}{:}{:}h\left(a,a^{\dagger}\right)\genfrac{}{}{0.0pt}{}{:}{:},$
which means that Weyl ordered of operator
$\genfrac{}{}{0.0pt}{}{:}{:}h\left(a,a^{\dagger}\right)\genfrac{}{}{0.0pt}{}{:}{:}$,
whose Weyl correspondence is $h\left(\alpha,\alpha^{\ast}\right)$, can be
obtained by just respectively replacing $\alpha,\alpha^{\ast}$ in
$h\left(\alpha,\alpha^{\ast}\right)$ by $a$ and $a^{\dagger}$ without
disturbing the form of function $h$.
According to the Weyl ordering invariance under similar transformations [13]
and the following transform relation
$\displaystyle S^{\dagger}\left(\theta\right)aS\left(\theta\right)$
$\displaystyle=$ $\displaystyle a\cosh\theta+\tilde{a}^{\dagger}\sinh\theta,$
$\displaystyle S^{\dagger}\left(\theta\right)\tilde{a}S\left(\theta\right)$
$\displaystyle=$ $\displaystyle\tilde{a}\cosh\theta+a^{\dagger}\sinh\theta,$
(13)
it is easily seen
$\displaystyle
S^{\dagger}\left(\theta\right)\Delta\left(\alpha\right)S\left(\theta\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\alpha-a\cosh\theta-\tilde{a}^{\dagger}\sinh\theta\right)$
(14)
$\displaystyle\times\delta\left(\alpha^{\ast}-a^{{\dagger}}\cosh\theta-\tilde{a}\sinh\theta\right)\genfrac{}{}{0.0pt}{}{:}{:},$
which is just the Weyl ordering of
$S^{\dagger}\left(\theta\right)\Delta\left(\alpha\right)S\left(\theta\right)$
in the enlarged Fock space. Based on the Weyl rule, the classical
correspondence
$h\left(\beta,\beta^{\ast};\tilde{\beta},\tilde{\beta}^{\ast}\right)$ of the
operator
$S^{\dagger}\left(\theta\right)\Delta\left(\alpha\right)S\left(\theta\right)$
can be obtained by replacing ($a,a^{{\dagger}})$ and
($\tilde{a},\tilde{a}^{{\dagger}})$ with ($\beta,\beta^{\ast}$) and
($\tilde{\beta},\tilde{\beta}^{\ast})$, respectively, i.e.,
$\displaystyle
h\left(\beta,\beta^{\ast};\tilde{\beta},\tilde{\beta}^{\ast}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\delta\left(\alpha-\beta\cosh\theta-\tilde{\beta}^{\ast}\sinh\theta\right)$
(15)
$\displaystyle\times\delta\left(\alpha^{\ast}-\beta^{\ast}\cosh\theta-\tilde{\beta}\sinh\theta\right).$
It then follows from Eqs.(11) and (15) that
$S^{\dagger}\left(\theta\right)\Delta\left(\alpha\right)S\left(\theta\right)=4\int\mathtt{d}^{2}\beta\mathtt{d}^{2}\tilde{\beta}\Delta\left(\beta,\beta^{\ast};\tilde{\beta},\tilde{\beta}^{\ast}\right)h\left(\beta,\beta^{\ast};\tilde{\beta},\tilde{\beta}^{\ast}\right),$
(16)
where
$\Delta\left(\beta,\beta^{\ast};\tilde{\beta},\tilde{\beta}^{\ast}\right)$ is
the two-mode Wigner operator, whose normally ordering form is
$\Delta\left(\beta,\beta^{\ast};\tilde{\beta},\tilde{\beta}^{\ast}\right)=\frac{1}{\pi^{2}}\colon\exp\left[-2\left(a^{{\dagger}}-\beta^{\ast}\right)\left(a-\beta\right)-2\left(\tilde{a}^{{\dagger}}-\tilde{\beta}^{\ast}\right)\left(\tilde{a}-\tilde{\beta}\right)\right]\colon.$
(17)
On substituting Eq.(17) into Eq.(16) and using the integral formula [19]
$\int\frac{\mathtt{d}^{2}z}{\pi}e^{\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}}=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{
Re}\left(\zeta\right)<0,$ (18)
we can derive the normally ordered form of (16) as follows
$\displaystyle
S^{\dagger}\left(\theta\right)\Delta\left(\alpha\right)S\left(\theta\right)$
$\displaystyle=$ $\displaystyle
2\int\frac{\mathtt{d}^{2}\beta\mathtt{d}^{2}\tilde{\beta}}{\pi^{2}}\delta\left(\alpha-\beta\cosh\theta-\tilde{\beta}^{\ast}\sinh\theta\right)$
(19)
$\displaystyle\times\delta\left(\alpha^{\ast}-\beta^{\ast}\cosh\theta-\tilde{\beta}\sinh\theta\right)$
$\displaystyle\times\colon\exp\left[-2\left(a^{{\dagger}}-\beta^{\ast}\right)\left(a-\beta\right)-2\left(\tilde{a}^{{\dagger}}-\tilde{\beta}^{\ast}\right)\left(\tilde{a}-\tilde{\beta}\right)\right]\colon$
$\displaystyle=$
$\displaystyle\frac{\text{sech}2\theta}{\pi}e^{-2\left|\alpha\right|^{2}\text{sech}2\theta}\colon\exp\left\\{-\left(a\tilde{a}+a^{{\dagger}}\tilde{a}^{{\dagger}}\right)\tanh
2\theta\right.$
$\displaystyle+2\text{sech}2\theta\left[\sinh\theta\left(\allowbreak\alpha^{\ast}\tilde{a}^{{\dagger}}+\allowbreak\alpha\tilde{a}\right)+\cosh\theta\left(\alpha^{\ast}a+\alpha
a^{\dagger}\right)\right.$
$\displaystyle-\left(\tilde{a}^{{\dagger}}\tilde{a}\sinh^{2}\theta+a^{{\dagger}}a\cosh^{2}\theta\right)]\\}\colon,$
which is just the normally ordered form of (16). Eq.(19) directly leads to the
WF of the thermo vacuum state $\left|0(\beta)\right\rangle$,
$\displaystyle\left\langle
0(\beta)\right|\Delta\left(\alpha\right)\left|0(\beta)\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle
0,\tilde{0}\right|S^{\dagger}\left(\theta\right)\Delta\left(\alpha\right)S\left(\theta\right)\left|0,\tilde{0}\right\rangle=\frac{\text{sech}2\theta}{\pi}e^{-2\left|\alpha\right|^{2}\text{sech}2\theta}$
(20) $\displaystyle=$
$\displaystyle\frac{1-e^{-\beta\omega}}{\pi(1+e^{-\beta\omega})}e^{-2\left|\alpha\right|^{2}\frac{1-e^{-\beta\omega}}{1+e^{-\beta\omega}}}.$
## IV Wigner function of photon-subtracted thermo vacuum state
At finite temperature, the photon-subtracted thermo vacuum state can be
expressed as [20]
$\rho_{1}=C_{1}a^{n}\left|0(\beta)\right\rangle\left\langle
0(\beta)\right|a^{{\dagger}n},$ (21)
where $C_{1}$ is the normalized factor, defined by
$C_{1}^{-1}=\mathtt{Tr}\left[a^{n}S\left(\theta\right)\left|0,\tilde{0}\right\rangle\left\langle
0,\tilde{0}\right|S^{{\dagger}}\left(\theta\right)a^{{\dagger}n}\right],$ (22)
which can be calculated as follows. Using Eq.(5) and the binomial formula
$\sum_{l=0}^{\infty}\frac{\left(n+l\right)!}{n!l!}x^{l}=\left(1-x\right)^{-n-1},$
(23)
we have
$\displaystyle C_{1}^{-1}$ $\displaystyle=$ $\displaystyle\left\langle
0,\tilde{0}\right|S^{{\dagger}}\left(\theta\right)a^{{\dagger}n}a^{n}S\left(\theta\right)\left|0,\tilde{0}\right\rangle$
(24) $\displaystyle=$ $\displaystyle\text{sech}^{2}\theta\left\langle
0,\tilde{0}\right|e^{a\tilde{a}\tanh\theta}a^{{\dagger}n}a^{n}e^{a^{\dagger}\tilde{a}^{\dagger}\tanh\theta}\left|0,\tilde{0}\right\rangle$
$\displaystyle=$
$\displaystyle\text{sech}^{2}\theta\sum_{k,l=0}^{\infty}\tanh^{l+k}\theta\left\langle
k,\tilde{k}\right|a^{{\dagger}n}a^{n}\left|l,\tilde{l}\right\rangle$
$\displaystyle=$
$\displaystyle\text{sech}^{2}\theta\sum_{l=n}^{\infty}\frac{l!}{\left(l-n\right)!}\tanh^{2l}\theta=n!\sinh^{2n}\theta.$
By using Eqs. (21) and (19), we calculate the WF of photon-subtracted thermal
state $\rho_{1}$
$\displaystyle W_{1}\left(\alpha\right)$ $\displaystyle=$ $\displaystyle
C_{1}\left\langle
0,\tilde{0}\right|S^{{\dagger}}\left(\theta\right)a^{{\dagger}n}\Delta\left(\alpha\right)a^{n}S\left(\theta\right)\left|0,\tilde{0}\right\rangle$
(25) $\displaystyle=$ $\displaystyle\left\langle
0,\tilde{0}\right|\left[S^{{\dagger}}\left(\theta\right)a^{{\dagger}n}S\left(\theta\right)\right]S^{{\dagger}}\left(\theta\right)\Delta\left(\alpha\right)S\left(\theta\right)\left[S^{{\dagger}}\left(\theta\right)a^{n}S\left(\theta\right)\right]\left|0,\tilde{0}\right\rangle.$
Noticing Eq.(13) we see
$\displaystyle\left[S^{{\dagger}}\left(\theta\right)a^{n}S\left(\theta\right)\right]\left|0,\tilde{0}\right\rangle$
$\displaystyle=$
$\displaystyle\left(a\cosh\theta+\tilde{a}^{\dagger}\sinh\theta\right)^{n}\left|0,\tilde{0}\right\rangle$
(26) $\displaystyle=$
$\displaystyle\sqrt{n!}\sinh^{n}\theta\left|0,\tilde{n}\right\rangle,$
then substituting (26) into Eq.(25) and using Eq.(19) yields
$\displaystyle W_{1}\left(\alpha\right)$ $\displaystyle=$
$\displaystyle\frac{e^{-2\left|\alpha\right|^{2}\text{sech}2\theta}}{\pi\cosh
2\theta}\left\langle\tilde{n}\right|e^{\frac{2\sinh\theta}{\cosh
2\theta}\alpha^{\ast}\tilde{a}^{{\dagger}}}\left(\text{sech}2\theta\right)^{\tilde{a}^{{\dagger}}\tilde{a}}e^{\frac{2\sinh\theta}{\cosh
2\theta}\tilde{a}\alpha}\left|\tilde{n}\right\rangle$ (27) $\displaystyle=$
$\displaystyle\frac{e^{-2\left|\alpha\right|^{2}\text{sech}2\theta}}{\pi\cosh
2\theta}\sum_{k,l=0}^{n}\frac{\alpha^{\ast
k}\alpha^{l}}{k!l!}\left(\frac{2\sinh\theta}{\cosh
2\theta}\right)^{k+l}\left\langle\tilde{n}\right|\tilde{a}^{{\dagger}k}\left(\text{sech}2\theta\right)^{\tilde{a}^{{\dagger}}\tilde{a}}\tilde{a}^{l}\left|\tilde{n}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{e^{-2\left|\alpha\right|^{2}\text{sech}2\theta}}{\pi\cosh^{n+1}2\theta}\sum_{l=0}^{n}\frac{n!}{l!l!\left(n-l\right)!}\left(\frac{4\sinh^{2}\theta}{\cosh
2\theta}\left|\alpha\right|^{2}\right)^{l},$
where we have used the identity operator [21]
$\exp\left[\lambda\tilde{a}^{{\dagger}}\tilde{a}\right]=\colon\exp\left[\left(e^{\lambda}-1\right)\tilde{a}^{{\dagger}}\tilde{a}\right]\colon.$
(28)
Recalling that the definition of Laguerre polynomials [22],
$L_{n}(x)=\sum_{l=0}^{n}\frac{n!}{\left(l!\right)^{2}\left(n-l\right)!}(-x)^{l},$
(29)
Eq. (27) can be further put into the following neat form,
$W_{1}\left(\alpha\right)=\frac{e^{-2\left|\alpha\right|^{2}\text{sech}2\theta}}{\pi\cosh^{n+1}2\theta}L_{n}\left(-\frac{4\sinh^{2}\theta}{\cosh
2\theta}\left|\alpha\right|^{2}\right),$ (30)
which is just the WF of photon-subtracted thermo vacuum state, a Gaussian-
Laguerre type function of temperature, since
$\tanh\theta=\exp\left(-\frac{\omega}{2kT}\right)$. Due to $\cosh 2\theta>0$
and $L_{n}(-\frac{4\sinh^{2}\theta}{\cosh
2\theta}\left|\alpha\right|^{2})\geqslant 0$, for the photon-subtracted case,
$W_{1}\left(\alpha\right)$ has no chance to present the negative value in
phase space, which can be seen from Fig.1. On the other hand, the amplitude
value of WF in $\left(\left|\alpha\right|,\theta\right)$ space decreases with
the increasing temperature (corresponding to $\theta$). In appendix A, in
order to check the result in Eq. (30), we have derived the WF of photon-
subtracted thermo vacuum state by using the coherent state representation of
Wigner operator. Comparing with the result in Ref.[20], Eq.(30) seems _more
concise and convenient_ for further discussion.
Figure 1: Wigner function distributions of photon-subtracted thermo state in
($q,p$) phase space with (a) $n=1,\theta=0.2$, (b) $n=1,\theta=0.8$, (c)
$n=2,\theta=0.8$, and in $\left(\left|\alpha\right|,\theta\right)$ space with
(d) $n=1$.
## V Wigner function of photon-added thermo vacuum state
At finite temperature, the photon-added thermo vacuum state is expressed as
[23]
$\rho_{2}=C_{2}a^{{\dagger}n}\left|0(\beta)\right\rangle\left\langle
0(\beta)\right|a^{n}.$ (31)
By the same procedures as deriving Eqs. (22) and (26), we have
$C_{2}^{-1}=n!\cosh^{2n}\theta,$ (32)
and
$S^{{\dagger}}\left(\theta\right)a^{{\dagger}n}S\left(\theta\right)\left|0,\tilde{0}\right\rangle=\sqrt{n!}\cosh^{n}\theta\left|n,\tilde{0}\right\rangle.$
(33)
Uisng Eq.(32) and (33), the WF $W_{2}\left(\alpha\right)$ of $\rho_{2}\ $is
given by
$\displaystyle W_{2}\left(\alpha\right)$ $\displaystyle=$ $\displaystyle
C_{2}\left\langle
0,\tilde{0}\right|S^{{\dagger}}\left(\theta\right)a^{n}S\left(\theta\right)\left[S^{{\dagger}}\left(\theta\right)\Delta\left(\alpha\right)S\left(\theta\right)\right]S^{{\dagger}}\left(\theta\right)a^{{\dagger}n}S\left(\theta\right)\left|0,\tilde{0}\right\rangle$
(34) $\displaystyle=$ $\displaystyle\left\langle
n,\tilde{0}\right|S^{{\dagger}}\left(\theta\right)\Delta\left(\alpha\right)S\left(\theta\right)\left|n,\tilde{0}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{n}e^{-2\left|\alpha\right|^{2}\text{sech}2\theta}}{\pi\cosh^{n+1}2\theta}\sum_{l=0}^{n}\frac{n!}{l!l!\left(n-l\right)!}\left(-\frac{4\cosh^{2}\theta}{\cosh
2\theta}\left|\alpha\right|^{2}\right)^{l}$ $\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{n}e^{-2\left|\alpha\right|^{2}\text{sech}2\theta}}{\pi\cosh^{n+1}2\theta}L_{n}\left(\frac{4\cosh^{2}\theta}{\cosh
2\theta}\left|\alpha\right|^{2}\right),$
a Gaussian-Laguerre type function which may present negative region in phase
space (see Fig.2). In particular, when $n=1,$ Eq.(34) reduces to
$W_{2}\left(\alpha\right)=-\frac{e^{-2\left|\alpha\right|^{2}\text{sech}2\theta}}{\pi\cosh^{2}2\theta}\left(1-\frac{4\cosh^{2}\theta}{\cosh
2\theta}\left|\alpha\right|^{2}\right).$ (35)
In Fig. 2, the behaviour of WF distributions of photon-added thermo state are
plotted in ($q,p$) phase space and $\left(\left|\alpha\right|,\theta\right)$
space. From Fig.2, one can see clearly the modulation action of photon-added
number and temperature. The “oscillating frequency” of WF increases with the
increasing photon-added number; while the amplitude value of WF in
$\left(\left|\alpha\right|,\theta\right)$ space decreases with the increasing
temperature (corresponding to $\theta$), which indicates that the
nonclassicality is weakened at finite temperature.
Figure 2: Wigner function distributions of photon-added thermo state in
($q,p$) phase space with $\theta=0.2$ for (a) $n=1$, (b) $n=2,$ and in
$\left(\left|\alpha\right|,\theta\right)$ space with (c) $n=1$ and (d) $n=5$.
## VI Wigner function of thermo number state
At finite temperature, according to TFD, the number state
$\left|n\right\rangle$ is replaced by $\left|n,\tilde{n}\right\rangle,$ thus
the thermo number state (i.e., number states at finite temperature) is
$S\left(\theta\right)\left|n,\tilde{n}\right\rangle$ in the enlarged Fock
space. Using the un-normalized coherent state representation of number state,
$\left|n,\tilde{n}\right\rangle=\frac{1}{n!}\frac{d^{2n}}{dz^{n}d\tilde{z}^{n}}\left.\left|z,\tilde{z}\right\rangle\right|_{z=\tilde{z}=0},\text{\
}\left\langle z^{\prime}\right.\left|z\right\rangle=e^{z^{\prime\ast}z},$ (36)
where
$\left|z,\tilde{z}\right\rangle=\exp[za^{{\dagger}}+\tilde{z}\tilde{a}^{{\dagger}}]\left|0,\tilde{0}\right\rangle$
is the non-normalized two-mode coherent state, and employing Eq.(19), we
calculate the WF $W_{3}\left(\alpha\right)$ of thermo number state as
$\displaystyle W_{3}\left(\alpha\right)$ $\displaystyle=$
$\displaystyle\left\langle
n,\tilde{n}\right|S^{{\dagger}}\Delta\left(\alpha\right)S\left|n,\tilde{n}\right\rangle$
(37) $\displaystyle=$
$\displaystyle\frac{1}{n!^{2}}\frac{d^{2n}}{df^{n}dr^{n}}\frac{d^{2n}}{dz^{n}dt^{n}}\left\langle
f^{\ast},r^{\ast}\right|S^{{\dagger}}\Delta\left(\alpha\right)S\left.\left|z,t\right\rangle\right|_{f=r=z=t=0}$
$\displaystyle=$
$\displaystyle\mathcal{A}\frac{d^{2n}}{df^{n}dr^{n}}\frac{d^{2n}}{dz^{n}dt^{n}}\exp\left\\{-\left(tz+fr\right)\tanh
2\theta\right.$ $\displaystyle+\left.\left(\allowbreak rt-
fz\right)\text{sech}2\theta+zE^{\ast}+fE+r\allowbreak F^{\ast}+\allowbreak
tF\right\\}_{f=r=z=t=0},$
where we have set
$\mathcal{A=}\frac{e^{-2\left|\alpha\right|^{2}\text{sech}2\theta}}{\pi
n!^{2}\cosh 2\theta},\text{ }E=2\alpha\text{sech}2\theta\cosh\theta,\text{ \
}F=2\alpha\text{sech}2\theta\allowbreak\sinh\theta.$ (38)
Expanding the exponential term $\exp\left[\left(rt-\allowbreak
fz\right)\text{sech}2\theta\right]$ as series, we have
$\displaystyle W_{3}\left(\alpha\right)$ $\displaystyle=$
$\displaystyle\mathcal{A}\frac{d^{2n}}{df^{n}dz^{n}}\frac{d^{2n}}{dr^{n}dt^{n}}\exp\left[-\left(fr+tz\right)\tanh
2\theta\right]$ (39)
$\displaystyle\times\sum_{l,k=0}^{\infty}\frac{\left(-\allowbreak
1\right)^{k}\text{sech}^{l+k}2\theta}{l!k!}\left(rt\right)^{l}\left(\allowbreak
fz\right)^{k}\exp\left[zE^{\ast}+fE+tF\allowbreak+rF^{\ast}\right]_{z=t=f=r=0}$
$\displaystyle=$
$\displaystyle\mathcal{A}\sum_{l,k=0}^{\infty}\frac{\left(-\allowbreak
1\right)^{k}\text{sech}^{l+k}2\theta}{l!k!}\frac{\partial^{2l}}{\partial
F^{l}\allowbreak\partial F\allowbreak^{\ast l}}\frac{\partial^{2k}}{\partial
E^{k}\allowbreak\partial E\allowbreak^{\ast k}}$
$\displaystyle\times\frac{d^{2n}}{df^{n}dz^{n}}\frac{d^{2n}}{dr^{n}dt^{n}}\exp\left[-\left(fr+tz\right)\tanh
2\theta+fE\allowbreak+rF^{\ast}+zE^{\ast}+tF\right]_{z=t=f=r=0}.$
Then making the variable replacement for $f,r,t,z$ we can rewrite Eq.(39) as
$\displaystyle W_{3}\left(\alpha\right)$ $\displaystyle=$
$\displaystyle\mathcal{A}\tanh^{2n}2\theta\sum_{l,k=0}^{\infty}\frac{\left(-\allowbreak
1\right)^{k}\text{sech}^{l+k}2\theta}{l!k!}\frac{\partial^{2l}}{\partial
F^{l}\allowbreak\partial F\allowbreak^{\ast l}}\frac{\partial^{2k}}{\partial
E^{k}\allowbreak\partial E\allowbreak^{\ast k}}$ (40)
$\displaystyle\times\frac{d^{2n}}{df^{n}dr^{n}}\frac{d^{2n}}{dz^{n}dt^{n}}\exp\left[-fr+fE\allowbreak+\frac{rF^{\ast}}{\tanh
2\theta}-tz+zE^{\ast}+\frac{tF}{\tanh 2\theta}\right]_{z=t=f=r=0}$
$\displaystyle=$
$\displaystyle\mathcal{A}\tanh^{2n}2\theta\sum_{l,k=0}^{\infty}\frac{\left(-\allowbreak
1\right)^{k}\text{sech}^{l+k}2\theta}{l!k!}$
$\displaystyle\times\frac{\partial^{k+l}}{\partial E^{k}\partial
F\allowbreak^{\ast l}}\frac{\partial^{k+l}}{\partial E\allowbreak^{\ast
k}\allowbreak\partial F^{l}\allowbreak}H_{n,n}\left(E,\frac{F^{\ast}}{\tanh
2\theta}\right)H_{n,n}\left(E^{\ast},\frac{F}{\tanh 2\theta}\right).$
Noticing the formula
$\frac{\partial^{l+k}}{\partial\xi^{l}\partial\eta^{k}}H_{m,n}\left(\xi,\eta\right)=\frac{m!n!}{\left(m-l\right)!\left(n-k\right)!}H_{m-l,n-k}\left(\xi,\eta\right),$
(41)
we have
$\displaystyle W_{3}\left(\alpha\right)$ $\displaystyle=$
$\displaystyle\frac{n!^{2}e^{-2\left|\alpha\right|^{2}\text{sech}2\theta}}{\pi\cosh
2\theta}\sum_{l,k=0}^{n}\frac{\left(-\allowbreak
1\right)^{k}\text{sech}^{l+k}2\theta\tanh^{2\left(n-l\right)}2\theta}{l!k!\left[\left(n-l\right)!\left(n-k\right)!\right]^{2}}$
(42) $\displaystyle\times\left|H_{n-k,n-l}\left(E,\frac{F^{\ast}}{\tanh
2\theta}\right)\right|^{2}.$
From Eq.(42) one can see clearly that the WF of thermo number state is a real
number.
In particular, when $n=0$, noticing that
$\tanh\theta=e^{-\frac{1}{2}\omega\beta},$
$\cosh^{2}\theta=\frac{1}{1-e^{-\beta\omega}},\sinh^{2}\theta=\frac{e^{-\beta\omega}}{1-e^{-\beta\omega}},$
Eq.(42) reduces to the WF of thermo vacuum state $\left|0(\beta)\right\rangle$
in Eq.(20). On the other hand, when $T\rightarrow 0,$(i.e., finite temperature
case reduces to zero temperature case) $e^{-\beta\omega}\rightarrow
e^{-\infty}\rightarrow 0,$ $\sinh\theta\rightarrow 0,$ $\cosh\theta\rightarrow
1,$ $E\rightarrow 2\alpha,$ $\frac{F^{\ast}}{\tanh
2\theta}\rightarrow\alpha^{\ast},$ and noticing Eq.(29) and the definition of
two-variable Hermite polynomials [24,25],
$H_{m,n}\left(\xi,\kappa\right)=\sum_{l=0}^{\min(m,n)}\frac{m!n!\left(-1\right)^{l}\xi^{m-l}\kappa^{n-l}}{l!\left(n-l\right)!\left(m-l\right)!},$
(43)
which leads to
$H_{n-k,0}\left(2\alpha,\alpha^{\ast}\right)=\left(2\alpha\right)^{n-k},$ then
Eq.(42) becomes
$\displaystyle W_{3}\left(\alpha\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}e^{-2\left|\alpha\right|^{2}}\sum_{k=0}^{n}\frac{\left(-\allowbreak
1\right)^{k}n!}{k!\left[\left(n-k\right)!\right]^{2}}\left|H_{n-k,0}\left(2\alpha,\alpha^{\ast}\right)\right|^{2}$
(44) $\displaystyle=$ $\displaystyle\frac{\left(-\allowbreak
1\right)^{n}}{\pi}e^{-2\left|\alpha\right|^{2}}\sum_{k=0}^{n}\frac{n!}{k!\left[\left(n-k\right)!\right]^{2}}\left(-4\left|\alpha\right|^{2}\right)^{n-k}$
$\displaystyle=$ $\displaystyle\frac{\left(-\allowbreak
1\right)^{n}}{\pi}e^{-2\left|\alpha\right|^{2}}L_{n}(4\left|\alpha\right|^{2}),$
which is just the WF of number state $\left|n\right\rangle$ at zero
temperature.
In sum, by using TFD and Weyl ordered operators’ order-invariance under
similar transformations, we present a new approach to deriving the exact
expressions of Wigner functions for thermo number state, photon subtracted and
added thermo vacuum state. These WF are related to the Gaussian-Laguerre type
functions, which are easily to be further analysed. The affection of
temperature to nonclassical behaviour of the fields is manifestly shown. For
discussions about the decoherence at finite temperature, we refer to [30,31].
Appendix A Checking Eq.(30)
In fact, in original Fock space, the photon-subtracted thermo state is
expressed as [20]
$\rho_{1}=C_{1}\text{Tr}_{\tilde{a}}\left[a^{n}\left|0(\beta)\right\rangle\left\langle
0(\beta)\right|a^{{\dagger}n}\right]=C_{1}a^{n}\rho_{c}a^{{\dagger}n},$ (A1)
where $\rho_{c}$ is the thermo state
$\rho_{c}=\sum_{l=0}^{\infty}\frac{n_{c}^{l}}{\left(n_{c}+1\right)^{l+1}}\left|l\right\rangle\left\langle
l\right|=\frac{1}{n_{c}+1}e^{a^{{\dagger}}a\ln\frac{n_{c}}{n_{c}+1}},\text{
}n_{c}=\sinh^{2}\theta.$ (A2)
Using the the coherent state representation of Wigner operator [26],
$\Delta\left(\alpha\right)=e^{2\left|\alpha\right|^{2}}\int\frac{\mathtt{d}^{2}z}{\pi^{2}}\left|z\right\rangle\left\langle-z\right|\exp\left[-2\left(z\alpha^{\ast}-z^{\ast}\alpha\right)\right],$
(A3)
where $\left|z\right\rangle$ is the coherent state [27,28], we have
$\displaystyle W_{1}\left(\alpha\right)$
$\displaystyle=\text{Tr}\left(\Delta\left(\alpha\right)\rho_{1}\right)$
$\displaystyle=\frac{C_{1}e^{2\left|\alpha\right|^{2}}}{n_{c}+1}\int\frac{\mathtt{d}^{2}z}{\pi^{2}}\left\langle-z\right|a^{n}e^{a^{{\dagger}}a\ln\frac{n_{c}}{n_{c}+1}}a^{{\dagger}n}\left|z\right\rangle\exp\left[-2\left(z\alpha^{\ast}-z^{\ast}\alpha\right)\right].$
(A4)
Note that
$e^{a^{{\dagger}}a\ln\frac{n_{c}}{n_{c}+1}}a^{{\dagger}n}e^{-a^{{\dagger}}a\ln\frac{n_{c}}{n_{c}+1}}=\frac{n_{c}^{n}}{\left(n_{c}+1\right)^{n}}a^{{\dagger}n},$
(A5)
and
$e^{a^{{\dagger}}a\ln\frac{n_{c}}{n_{c}+1}}\left|z\right\rangle=e^{-\frac{2n_{c}+1}{2\left(n_{c}+1\right)^{2}}}\left|\frac{n_{c}z}{n_{c}+1}\right\rangle,$
(A6)
Eq.(A4) can be rewritten as
$\displaystyle W_{1}\left(\alpha\right)$
$\displaystyle=\frac{n_{c}^{n}C_{1}e^{2\left|\alpha\right|^{2}}}{\left(n_{c}+1\right)^{n+1}}\int\frac{d^{2}z}{\pi^{2}}\left\langle-z\right|a^{n}a^{{\dagger}n}\left|\frac{n_{c}z}{n_{c}+1}\right\rangle$
$\displaystyle\times\exp\left[-\frac{2n_{c}+1}{2\left(n_{c}+1\right)^{2}}\left|z\right|^{2}-2\left(z\alpha^{\ast}-z^{\ast}\alpha\right)\right].$
(A7)
Further using the operator identity [29]
$a^{n}a^{{\dagger}n}=\left(-1\right)^{n}\colon
H_{n,n}\left(ia^{{\dagger}},ia\right)\colon,$ (A8)
where $H_{m,n}\left(x,y\right)$ is the two-variable Hermite polynomials, whose
generating function is
$H_{m,n}\left(x,y\right)=\left.\frac{\partial^{m+n}}{\partial t^{m}\partial
t^{\prime
n}}\exp\left[-tt^{\prime}+tx+t^{\prime}y\right]\right|_{t=t^{\prime}=0},$ (A9)
we have
$\displaystyle W_{1}\left(\alpha\right)$
$\displaystyle=\frac{\left(-1\right)^{n}e^{2\left|\alpha\right|^{2}}}{n!\left(n_{c}+1\right)^{n+1}}\frac{\partial^{2n}}{\partial
t^{n}\partial\tau^{n}}e^{-t\tau}$
$\displaystyle\times\int\frac{\mathtt{d}^{2}z}{\pi^{2}}\exp\left\\{-\frac{2n_{c}+1}{n_{c}+1}\left|z\right|^{2}\right.\left.+\left(\frac{i\tau
n_{c}}{n_{c}+1}-2\alpha^{\ast}\right)z+\left(2\alpha-
it\right)z^{\ast}\right\\}_{t=\tau=0}$
$\displaystyle=\frac{\left(-1\right)^{n}}{n!\pi\left(n_{c}+1\right)^{n}}\frac{e^{2\left|\alpha\right|^{2}}}{2n_{c}+1}\frac{\partial^{2n}}{\partial
t^{n}\partial\tau^{n}}\exp\left[-t\tau\right]$
$\displaystyle\exp\left[\frac{n_{c}+1}{2n_{c}+1}\left(\frac{i\tau
n_{c}}{n_{c}+1}-2\alpha^{\ast}\right)\left(2\alpha-
it\right)\right]_{t=\tau=0}$
$\displaystyle=\frac{\left(-1\right)^{n}}{n!\pi\left(n_{c}+1\right)^{n}}\frac{e^{-\frac{2\left|\alpha\right|^{2}}{2n_{c}+1}}}{2n_{c}+1}\frac{\partial^{2n}}{\partial
t^{n}\partial\tau^{n}}\exp\left\\{-\frac{n_{c}+1}{2n_{c}+1}t\tau\right.$
$\displaystyle+\left.2i\alpha^{\ast}\frac{n_{c}+1}{2n_{c}+1}t+2i\alpha\frac{n_{c}}{2n_{c}+1}\tau\right\\}_{t=\tau=0}$
$\displaystyle=\frac{e^{-\frac{2\left|\alpha\right|^{2}}{2n_{c}+1}}}{\left(2n_{c}+1\right)^{n+1}}\frac{\left(-1\right)^{n}}{n!\pi}H_{n,n}\left(\frac{2in_{c}\alpha}{\sqrt{\left(2n_{c}+1\right)\left(n_{c}+1\right)}},2i\sqrt{\frac{n_{c}+1}{2n_{c}+1}}\alpha^{\ast}\right),$
(A10)
then using the relation
$\frac{\left(-1\right)^{n}}{n!}H_{n,n}\left(x,y\right)=L_{n}\left(xy\right),$
(A11)
and noticing that $n_{c}=\sinh^{2}\theta,$ $2n_{c}+1=\cosh 2\theta,$ Eq.(A10)
can be put into
$W_{1}\left(\alpha\right)=\frac{e^{-\frac{2\left|\alpha\right|^{2}}{2n_{c}+1}}}{\pi\left(2n_{c}+1\right)^{n+1}}L_{n}\left(-\frac{4n_{c}\left|\alpha\right|^{2}}{2n_{c}+1}\right),$
(A12)
which is just the Eq.(30). Thus we have checked the result using a new
appraoch.
## References
* (1) Parigi V, Zavatta A, Kim M S and Bellini M 2007 Science 317 1890
* (2) Boyd R W, Chan K W and O’Sullivan M N 2007 Science 317 1874
* (3) Wenger J, Tualle-Brouri R and Grangier P 2004 Phys. Rev. Lett. 92 153601
* (4) Zavatta A, Viciani S and Bellini M 2004 Science 306 660
* (5) Ourjoumtsev A, Dantan A, Tualle-Brouri R and Grangier P 2007 Phys. Rev. Lett. 98 030502
* (6) Ourjoumtsev A, Dantan A, Tualle-Brouri R and Grangier P 2006 Phys. Rev. Lett. 96 213601
* (7) Biswas A and Agarwal G S 2007 Phys. Rev. A 75 032104
* (8) Li-yun Hu and Hong-yi Fan 2008 J. Opt. Soc. Am. B 25 1955
* (9) Wigner E, 1932 Phys. Rev. 40 749
* (10) Wolfgang P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, Birlin, 2001
* (11) Kim M S and Bužek V, 1992 Phys. Rev. A 46 4239-4251.
* (12) Jeong H, Lund A P, and Ralph T C, 2005 Phys. Rev. A 72 013801; Jeong H, Lee J and Nha H, 2008 J. Opt. Soc. Am. B 25 1025
* (13) Hong-yi Fan, 1992 J. Phys. A 25 3443; Hong-yi Fan, 2008 Ann. Phys. 323 500
* (14) Hong-yi Fan, 1997 Mod. Phys. Lett. A 12 2325; 2000 Mod. Phys. Lett. A 15 2297
* (15) Hong-yi Fan and Yue Fan, 1998 Mod. Phys. Lett. A 13 433; 2002 Int. J. Mod. Phys. A 17 701
* (16) Y. Takahashi and Umezawa H, 1975 Collecive Phenomena 2 55
* (17) Memorial Issue for Umezawa H, 1996 Int. J. Mod. Phys. B 10 1695 memorial issue and references therein.
* (18) H. Umezawa, Advanced Field Theory – Micro, Macro, and Thermal Physics (AIP 1993).
* (19) R. R. Puri, Mathematical Methods of Quantum Optics (Springer-Verlag, Berlin, 2001), Appendix A.
* (20) Agarwal G S 1992 Phys. Rev. A 45 1787
* (21) Hong-yi Fan, et. al., 2006 Ann. Phys. 321 480
* (22) Magnus W et. al., Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd Ed., Springer Verlag. 1966
* (23) Agarwal G S and Tara K 1992 Phys. Rev. A 46 485
* (24) Wünsche A, 2001 J. Computational and Appl. Math. 133 665
* (25) Wünsche A, 2000 J . Phys. A: Math. and Gen. 33 1603
* (26) Hong-yi Fan, 1987 Phys. Lett. A 124 303
* (27) Glauber R J, 1963 Phys. Rev. 130 2529-2539; 1963 Phys. Rev. 131 2766-2788
* (28) Klauder J R and Skargerstam B S, Coherent States (World Scientific, Singapore, 1985).
* (29) Hong-yi Fan, 2004 Commun. Theor. Phys. 42 339
* (30) Hong-yi Fan and Li-yun Hu, 2008 Mod. Phys. Lett. B 22 2435; Hong-yi Fan and Hai-liang Lu, 2007 Mod. Phys. Lett. B 21 183
* (31) Hong-yi Fan and Li-yun Hu, 2008 Opt. Commun. 281 5571
|
arxiv-papers
| 2009-01-12T02:19:18 |
2024-09-04T02:48:59.835602
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-yun Hu and Hong-yi Fan",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/0901.1424"
}
|
0901.1497
|
# Improving Application of Bayesian Neural Networks to Discriminate Neutrino
Events from Backgrounds in Reactor Neutrino Experiments
Ye Xua Corresponding author, e-mail address: xuye76@nankai.edu.cn WeiWei Xua
YiXiong Menga Bin Wua
###### Abstract
The application of Bayesian Neural Networks(BNN) to discriminate neutrino
events from backgrounds in reactor neutrino experiments has been described in
Ref.[1]. In the paper, BNN are also used to identify neutrino events in
reactor neutrino experiments, but the numbers of photoelectrons received by
PMTs are used as inputs to BNN in the paper, not the reconstructed energy and
position of events. The samples of neutrino events and three major backgrounds
from the Monte-Carlo simulation of a toy detector are generated in the signal
region. Compared to the BNN method in Ref.[1], more 8He/9Li background and
uncorrelated background in the signal region can be rejected by the BNN method
in the paper, but more fast neutron background events in the signal region are
unidentified using the BNN method in the paper. The uncorrelated background to
signal ratio and the 8He/9Li background to signal ratio are significantly
improved using the BNN method in the paper in comparison with the BNN method
in Ref.[1]. But the fast neutron background to signal ratio in the signal
region is a bit larger than the one in Ref.[1].
###### keywords:
Bayesian neural networks, neutrino oscillation, identification
aDepartment of Physics, Nankai University, Tianjin 300071, China
PACS numbers: 07.05.Mh, 29.85.Fj, 14.60.Pq
## 1 Introduction
The main goals of reactor neutrino experiments are to detect
$\bar{\nu_{e}}\rightarrow\bar{\nu_{x}}$ oscillation and precisely measure the
mixing angle of neutrino oscillation $\theta_{13}$. The experiment is designed
to detect reactor $\bar{\nu_{e}}$’s via the inverse $\beta$-decay reaction
$\bar{\nu_{e}}+p\rightarrow e^{+}+n$.
The signature is a delayed coincidence between $e^{+}$ and the neutron
captured signals. In the paper, only three important sources of backgrounds
are taken into account and they are the uncorrelated background from natural
radioactivity and the correlated backgrounds from fast neutrons and 8He/9Li.
The backgrounds like the neutrino events consist of two signals, a fast signal
and a delay signal. It is vital to separate neutrino events from backgrounds
accurately in the reactor neutrino experiments. Bayesian neural networks
(BNN)[2] are algorithms of the neural networks trained by Bayesian statistics.
They are not only non-linear functions as neural networks, but also controls
model complexity. So their flexibility makes them possible to discover more
general relationships in data than traditional statistical methods and their
preferring simple models make them possible to solve the over-fitting problem
better than the general neural networks[3]. BNN have been used to particle
identification and event reconstruction in the experiments of the high energy
physics, such as Ref.[1, 4, 5, 6]. The application of BNN to discriminate
neutrino events from backgrounds in reactor neutrino experiments has been
described in Ref.[1]. In the paper, BNN are also used to identify neutrino
events in the signal region[1] in reactor neutrino experiments, but the
numbers of photoelectron received by PMTs are used as inputs to BNN, not the
reconstructed energy and position of events.
## 2 The Classification with BNN[1, 2, 6]
The idea of BNN is to regard the process of training a neural network as a
Bayesian inference. Bayes’ theorem is used to assign a posterior density to
each point, $\bar{\theta}$, in the parameter space of the neural networks.
Each point $\bar{\theta}$ denotes a neural network. In the methods of BNN, one
performs a weighted average over all points in the parameter space of the
neural network, that is, all neural networks. The methods are described in
detail in Ref.[1, 2, 6]. The posterior density assigned to the point
$\bar{\theta}$, that is, to a neural network, is given by Bayes’ theorem
$p\left(\bar{\theta}\mid x,t\right)\propto\mathit{p\left(t\mid
x,\bar{\theta}\right)p\left(\bar{\theta}\right)}$ (1)
Where $x$ is a set of input data which corresponds to a set of target $t$. The
likelihood $p\left(t\mid x,\bar{\theta}\right)$ can be obtained by using a
training sample. And a Gaussian prior is specified for each weight using the
Bayesian neural networks package of Radford Neal111R. M. Neal, _Software for
Flexible Bayesian Modeling and Markov Chain Sampling_ ,
http://www.cs.utoronto.ca/~radford/fbm.software.html. Given an event with data
$x^{\prime}$, an estimate of the probability that it belongs to the signal is
given by the weighted average
$\bar{y}\left(x^{\prime}|x,t\right)=\int
y\left(x^{\prime},\bar{\theta}\right)p\left(\bar{\theta}\mid
x,t\right)d\bar{\theta}$ (2)
Currently, the only way to perform the high dimensional integral in Eq. (2) is
to sample the density $p\left(\bar{\theta}\mid x,t\right)$ with Markov Chain
Marlo Carlo (MCMC) methods[2, 7, 8, 9]. In MCMC methods, one steps through the
$\bar{\theta}$ parameter space in such a way that points are visited with a
probability proportional to the posterior density, $p\left(\bar{\theta}\mid
x,t\right)$. Points where $p\left(\bar{\theta}\mid x,t\right)$ is large will
be visited more often than points where $p\left(\bar{\theta}\mid x,t\right)$
is small.
Eq. (2) approximates the integral using the average
$\bar{y}\left(x^{\prime}\mid
x,t\right)\approx\frac{1}{L}\sum_{i=1}^{L}y\left(x^{\prime},\bar{\theta_{i}}\right)$
(3)
where $L$ is the number of points $\bar{\theta}$ sampled from
$p\left(\bar{\theta}\mid x,t\right)$. Each point $\bar{\theta}$ corresponds to
a different neural network with the same structure. So the average is an
average over neural networks, and the probability of the data $x^{\prime}$
belongs to the signal. The average is closer to the real value of
$\bar{y}\left(x^{\prime}\mid x,t\right)$, when $L$ is sufficiently large.
## 3 Toy Detector and Monte-Carlo Simulation[5]
In the paper, a toy detector is used to simulate central detectors in the
reactor neutrino experiments, such as Daya Bay experiment[10] and Double Chooz
experiment[11], with CERN GEANT4 package[12]. The toy detector is the same as
Ref.[5]. A total of 366 PMTs are arranged in 8 rings of 30 PMTs on the lateral
surface of the oil region, and in 5 rings of 24, 18, 12, 6, 3 PMTs on the top
and bottom caps.
The responses of neutrino events and backgrounds deposited in the toy detector
are simulated with GEANT4. Although the physical properties of the
scintillator and the oil (their optical attenuation length, refractive index
and so on) are wave-length dependent, only averages[13] (such as the optical
attenuation length of Gd-LS with a uniform value is 8 meter and the one of LS
is 20 meter) are used in the detector simulation.
According to the anti-neutrino interaction in detectors of the reactor
neutrino experiments[14], a neutrino event is uniformly generated throughout
Gd-LS region (see Fig. 1). A uncorrelated background event is generated in
such a way that a $\gamma$ event generated on the base of the energy
distribute of the natural radioactivity in the proposal of the Day Bay
experiment[10] is regarded as the fast signal, a neutron event of the single
signal is regarded as the delay signal, its delay time is uniformly generated
from 2 $\mu$s to 100 $\mu$s and the positions of the fast signal and the delay
signal are uniformly generated throughout Gd-LS region. A fast neutron event
is uniformly generated throughout Gd-LS region and its energy are uniformly
generated from 0 MeV to 50 MeV, therein an event of two signals are regarded
as a fast neutron background event. Since the behaviors of 8He/9Li decay
events in detectors couldn’t be simulated by the Geant4 package, a 8He/9Li
event is generated in such a way that the neutron signal from a fast neutron
event is regarded as its delay signal, an electron event generated at the same
position as the fast neutron event on the base of the energy distribute of
8He/9Li events in the proposal of the Day Bay experiment[10] is regarded as
its fast signal in the paper.
Energies and positions of neutrino events and backgrounds are reconstructed by
the method in Ref.[5]. The signal region is determined by using the
reconstructed energies and positions, as well as the neutron delay
time(described in Ref.[1]).
## 4 Neutrino Discrimination with BNN
Choosing inputs to BNN is vital to identify neutrino events . The
reconstructed energies, the distance between reconstructed the positions of
neutron and positron and the neutron delay time were used as inputs to the BNN
method in Ref.[1], but the energies and the distance are both the
reconstructed physics variables, and they make BNN discriminations worse
because of their reconstruction uncertainties. So we try to use raw data as
inputs to BNN. Obviously, the numbers of photoelectrons received by 366 PMTs
are rawer than the reconstructed variables. An event consists of two signals
(a fast signal and a delay signal), so if the numbers of photoelectron
received by PMTs will be directly used as inputs to BNN, BNN will have 732
inputs at least. It will take too much time to run a BNN program in a general
computer because of such many inputs. The method of reducing inputs to BNN in
the paper is that the photoelectrons received by several neighboring PMTs are
added up. That is several neighboring PMTs incorporate a PMT patch. In the
paper, a PMT patch is a 3(azimuth direction)$\times$4(z direction) PMTs array
on the detector lateral surface or a 120∘ sector(including 21 PMTs) on the
detector top and bottom caps. The delay time between two signals is very
important to discriminate neutrino events from the uncorrelated background, so
the number of photoelectrons received by a patch is multiply by the delay
time, and the result is used as the inputs to all neural networks, which have
the same structure. Then all the networks have a input layer of 52 inputs, the
single hidden layer of fifteen nodes and a output layer of a single output
which is just the probability that an event belongs to the neutrino event.
Discriminating neutrino events from backgrounds is actually a binary response
problem, that is the target is ’1’ or ’0’. Neutrino events are labeled by t=1,
and background events are labeled by t=0. So the output of BNN has to be a
number between 0 and 1. If the output is less than 0.5, the event is regarded
as a background event, and If the output is larger than 0.5, the event is
regarded as a neutrino event.
A Markov chain of neural networks is generated using the Bayesian neural
networks package of Radford Neal, with a training sample consisting of
neutrino events and background events. One thousand iterations, of twenty MCMC
steps each, are used. The neural network parameters are stored after each
iteration, since the correlation between adjacent steps is very high. That is,
the points in neural network parameter space are saved to lessen the
correlation after twenty steps here. It is also necessary to discard the
initial part of the Markov chain because the correlation between the initial
point of the chain and the point of the part is very high. The initial three
hundred iterations are discarded here. It takes about 120 hours to run 1000
iterations on a computer with two 3.4GHz Intel Pentium D processors (only one
of which are used).
Neutrino identification efficiencies are defined by the ratios between the
number of the events in neutrino test sample regarded as neutrinos and the
number of neutrino test sample. Background identification efficiencies are
defined by the ratios between the numbers of the events in background test
samples regarded as background events and the numbers of background test
samples. The identification efficiencies are measured with the test sample
which is different from the training sample. Other 3000 events each of the
neutrino and the three backgrounds are used to test the identification
capability of the trained BNN. In the paper, BNN are trained by the different
training samples, which consist of neutrinos and three backgrounds at
different rates, since the different identification efficiencies are obtained
using those BNN.
## 5 Results and Discussion
As Tab. 1 shows, most neutrino events, uncorrelated background events and
8He/9Li background events in the signal region can be identified using the BNN
method in the paper, but only a small part of fast neutron background events
can be identified using the BNN method in the paper. Since most fast neutron
events can’t be discriminate from neutrino events using the BNN method in the
paper, neutrino discriminations are concerned with neutrino rates in training
samples, as well as ratios of neutrino events and fast neutron events in
training samples. The neutrino discrimination in the signal region increases
from 90.5% to 93.7% with the increase of the neutrino rate from 50.0% to 57.1%
in the training sample using the BNN method in the paper. And the neutrino
discrimination also increases from 90.5% to 94.1% with the increase of the
ratio of neutrino events and fast neutron events from 2:1 to 3:1 in the
training sample. The different background to signal ratios in the signal
region are obtained using the BNN trained by the training samples consisting
of neutrino events and background events at different rates in the reactor
neutrino experiments.
Neutrino events are discriminated from fast neutrons and 8He/9Li events via
their fast signals identification, that is positron signals from neutrino
events are separated from recoil proton signals from fast neutrons and
electron signals from 8He/9Li events. $\gamma$ signals induced by positrons
and recoil protons are closer to point sources, but $\gamma$ signals induced
by electrons are closer to line sources. There is an effect on the
distribution of photoelectrons over all the PMTs in the detector due to the
difference between a point source and a line source. The effect can be
extracted from the inputs by the BNN method in the paper. So neutrino events
can be better discriminated from 8He/9Li events using the BNN method in the
paper, but distinguishing between neutrinos events and fast neutrons becomes
worse using the BNN method in the paper.
The events in the signal region can be identified using BNN one by one, once
those BNN are trained by training samples. If the BNN method in the paper is
used to the reactor neutrino experiments, the background to signal ratios will
be changed. We only roughly estimate the changes here. We assume that the
uncorrelated background fraction in the signal region is $A/N$, the fast
neutrons background fraction in the signal region is $F/N$, and the 8He/9Li
background fraction in the signal region is $L/N$ in the reactor neutrino
experiments. Those background fractions are very low (for example, they are
<0.2%, 0.1%, 0.3% in one of the near detector claimed by the proposal of the
Daya Bay experiment[10], respectively). If neutrino events are discriminated
from background events using the BNN method in Ref.[1], the background to
signal ratios can reach 0.2*(A/N), 0.68*(F/N) and 0.66*(L/N), respectively. If
the efficiencies of the first column in Tab. 1 are use to the estimation, we
get the result of the identification using the BNN method in the paper:
Uncorrelated background/Signal=(A/N)*(1-0.983)/0.941=0.018*(A/N)
Fast neutrons Background/Signal=(F/N)*(1-0.293)/0.941=0.751*(F/N)
8He/9Li Background/Signal=(L/N)*(1-0.913)/0.941=0.092*(L/N)
As the above equations show, the uncorrelated background to signal ratio and
8He/9Li background to signal ratio in the signal region are significantly
improved using the BNN method in the paper in comparison with the BNN method
in Ref.[1]. And the fast neutron background to signal ratio is a bit larger
than the one in Ref.[1]. But the fast neutron fraction in the signal region is
lower than the ones of the uncorrelated background and 8He/9Li background, so
the total background to signal ratio using the BNN method in the paper is much
lower than the one in Ref.[1]. In a word, the BNN method in the paper can be
applied to discriminate neutrino events from background events better than the
BNN method in Ref.[1] and the method based on the cuts in reactor neutrino
experiments.
## 6 Acknowledgements
This work is supported in part by the National Natural Science Foundation of
China (NSFC) under the contract No. 10605014, the national undergraduate
innovative plan of China under the contract No.081005517 and the physical base
of Nankai University under the contract No. J0730315.
## References
* [1] Y. Xu, Y. X. Meng, and W. W. Xu, _Applying bayesian neural networks to separate neutrino events from backgrounds in reactor neutrino experiments_ , Journal of Instrumentation, 3, P08005 (2008), arXiv: 0808.0240
* [2] R. M. Neal, _Bayesian Learning of Neural Networks_. New York: Springer-Verlag, 1996
* [3] R. Beale and T. Jackson, _Neural Computing: An Introduction_ , New York: Adam Hilger, 1991
* [4] Y. Xu, J. Hou and K. E. Zhu, _Applying Bayesian neural networks to identify pion, kaon and proton in BESII_ , Chinese Physics C32, 201-204 (2008)
* [5] Y. Xu, W. W. Xu, Y. X. Meng, and W. Xu, _Applying Bayesian neural networks to event reconstruction in reactor neutrino experiments_ , Nuclear Instruments and Methods in Physics Rearch A592, 451-455 (2008), arXiv: 0712.4042
* [6] P. C. Bhat and H. B. Prosper _Beyesian Neural Networks_. In: L. Lyons and M. K. Unel ed. _Proceedings of Statistical Problems in Particle Physics, Astrophysics and Cosmology, Oxford, UK 12-15, September 2005_. London: Imperial college Press. 2006. 151-154
* [7] S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth, _Hybrid Monte Carlo_ , Physics Letters, B195, 216-222 (1987)
* [8] M. Creutz and A. Gocksch, _Higher-order hybrid Monte Carlo_ , Physical Review Letters, 1989 63, 9-12
* [9] P. B. Mackenzie, _An improved hybrid Monte Carlo method_ , Physics Letters, 1989 B226, 369-371
* [10] Daya Bay Collaboration, _Daya Bay Proposal: A Precision Measurement of the Neutrino Mixing Angle $\theta_{13}$ Using Reactor Antineutrino At Daya Bay_, arXiv: hep-ex/0701029
* [11] M. Goodman and T. Lasserre, _Double Chooz: A Search for the Neutrino Mixing Angle $\theta_{13}$_, arXiv: hep-ex/0606025
* [12] Geant4 Reference Manual, vers. 9.0 (2007)
* [13] _The CHOOZ Experiment Proposal (1993)_ , available at the WWW site http://duphy4.physics.drexel.edu/chooz_pub/
* [14] Y. X. Sun, J. Cao, and K. J. Luk, et al., _Baseline Optimization of Reactor Neutrino experiments_ , Chinese Physics C29, 543-548 (2005)
Table 1: The different identification efficiencies are obtained with the BNNs trained by the different training samples, which consist of the neutrino and three backgrounds at different rates. The term after $\pm$ is the statistical error of the identification efficiencies. The numbers of the train samples are 24000, respectively. The 3000 events each of the uncorrelated background, fast neutron and 8He/9Li are regarded as the test sample. neutrino rate (%) | 50.0 | 50.0 | 54.5 | 57.1
---|---|---|---|---
uncorrelated background rate (%) | 16.7 | 12.5 | 9.1 | 9.5
fast neutron rate (%) | 16.7 | 25.0 | 27.3 | 23.8
8He/9Li rate (%) | 16.7 | 12.5 | 9.1 | 9.5
neutrino eff.(%) | 94.1$\pm$0.43 | 90.5$\pm$0.54 | 92.6$\pm$0.48 | 93.7$\pm$0.44
uncorrelated background eff.(%) | 98.3$\pm$0.24 | 98.1$\pm$0.25 | 96.4$\pm$0.34 | 96.7$\pm$0.33
fast neutrons eff.(%) | 29.3$\pm$0.83 | 35.8$\pm$0.88 | 34.6$\pm$0.87 | 32.8$\pm$0.86
8He/9Li eff.(%) | 91.3$\pm$0.51 | 90.6$\pm$0.53 | 87.7$\pm$0.60 | 87.5$\pm$0.60
Figure 1: The neutrino events for the Monte-Carlo simulation of the toy
detector are uniformly generated throughout Gd-LS region. (a) is the
distribution of the positron energy; (b) is the distribution of the energy of
the neutron captured by Gd; (c) is the distribution of the distance between
the positron and neutron positions; (d) is the distribution of the delay time
of the neutron signal.
|
arxiv-papers
| 2009-01-12T02:40:37 |
2024-09-04T02:48:59.843198
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ye Xu, WeiWei Xu, YiXiong Meng and Bin Wu",
"submitter": "Ye Xu",
"url": "https://arxiv.org/abs/0901.1497"
}
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0901.1635
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# Epicyclic oscillations of non-slender fluid tori around Kerr black holes
Odele Straub1 and Eva Šrámková2 1Copernicus Astronomical Centre PAN, Bartycka
18, 00-716 Warsaw, Poland 2Department of Physics, Silesian University in
Opava, Bezručovo nám. 13, 746-01 Opava, Czech Republic odele@camk.edu.pl
sram_eva@centrum.cz
###### Abstract
Considering epicyclic oscillations of pressure-supported perfect fluid tori
orbiting Kerr black holes we examine non-geodesic (pressure) effects on the
epicyclic modes properties. Using a perturbation method we derive fully
general relativistic formulas for eigenfunctions and eigenfrequencies of the
radial and vertical epicyclic modes of a slightly non-slender, constant
specific angular momentum torus up to second-order accuracy with respect to
the torus thickness. The behaviour of the axisymmetric and lowest-order
($m=\pm 1$) non-axisymmetric epicyclic modes is investigated. For an arbitrary
black hole spin we find that, in comparison with the (axisymmetric) epicyclic
frequencies of free test particles, non-slender tori receive negative pressure
corrections and exhibit thus lower frequencies. Our findings are in
qualitative agreement with the results of a recent pseudo-Newtonian study of
analogous problem defined within the Paczyński-Wiita potential. Implications
of our results on the high-frequency QPO models dealing with epicyclic
oscillations are addressed.
###### pacs:
95.30.Lz, 95.30.Sf, 95.85.Nv, 97.60.Lf
††: Class. Quantum Grav.
,
## 1 Introduction
Oscillations of black hole accretion discs have been studied extensively in
various astrophysical contexts. Investigations of hydrodynamic oscillation
modes of geometrically thin accretion discs revealed three fundamental,
discoseismic classes of modes: acoustic pressure p-modes, gravity g-modes and
corrugation c-modes. Kato et al (1998); Wagoner (1999); Kato (2001a) and
Wagoner et al (2001) give comprehensive reviews on the subject of relativistic
’discoseismology’. Geometrically thick discs (tori) have been examined rather
less and mainly in matters of their stability (e.g. Papaloizou & Pringle,
1984; Kojima, 1986). Only recently oscillatory modes of fluid tori were
explored in more detail in several numerical studies (e.g. Zanotti et al,
2003; Rezzolla et al, 2003a; Rubio-Herrera and Lee, 2005a, b; Šrámková et al,
2007; Montero et al, 2007) and in a purely analytic work by Blaes, Arras and
Fragile (2006). The latter presents a thorough analysis of oscillatory modes
of relativistic slender tori.
The analysis of disc oscillation modes is motivated by observations of quasi-
periodic oscillations (QPOs) in the light curves of Galactic low-mass X-ray
binaries (see McClintock and Remillard, 2003; van der Klis, 2004, for
reviews). In particular the understanding of high frequency (HF) QPOs, which
are presumably a strong gravity phenomenon, would provide a deeper insight
into the innermost regions of accretion discs and the very nature of compact
objects. Black hole HF QPOs occur at frequencies that are constant in time and
characteristic for a particular source. If more than one HF QPO is detected in
a given system, the frequencies typically appear in ratios of small natural
numbers, whereas in most cases the ratio is close to 3:2 (Abramowicz and
Kluźniak, 2001; McClintock and Remillard, 2003).
Several models explain HF QPOs in terms of disc (or blob) oscillations. Stella
and Vietri (1998) for instance consider an orbiting hot spot and propose that
HF QPOs arise due to a modulation of the spot radiation by its precessional
motion. Pointing out the observed rational ratios of HF QPO pairs Kluźniak and
Abramowicz (2000) and Abramowicz and Kluźniak (2001) suggest an underlying
non-linear resonance between some modes of accretion disc oscillation. In the
resonance model the mode pair is commonly represented by the radial and
vertical epicyclic oscillation. The importance of epicyclic oscillations is
also stressed by Kato (2001b) who attributes the origin of HF QPOs to non-
axisymmetric g-mode oscillations. The corotation resonance in a disc which is
deformed by a warp would be responsible for the excitation of g-modes and
general relativistic effects trap them near the inner edge of the accretion
disc (Kato, 2003, 2004). HF QPOs could also result from acoustic p-mode
oscillations of a small accretion torus orbiting close to the black hole
(Rezzolla et al, 2003b). Recently Blaes, Arras and Fragile (2006) discussed
the possibility that the vertical epicyclic and the lowest-order (acoustic)
breathing mode of a relativistic slender torus might represent the two black
hole HF QPOs in 3:2 ratio.
The main interest of our work lies in the epicyclic modes of oscillations.
Most models that are dealing with them consider geodesic flows and are based
on free test particles. However, non-geodesic effects related to e.g. magnetic
fields, viscosity or pressure forces may play a certain role in this concern.
The aim of this paper is to investigate such non-geodesic effects on the two
epicyclic modes by means of a pressure-supported perfect fluid torus and to
find the consequential pressure corrections to the mode eigenfunctions and
eigenfrequencies.
For an infinitely slender torus, the frequencies of epicyclic oscillations are
consistent with the epicyclic frequencies of free test particles on a circular
orbit in the equatorial plane, where the torus pressure exhibits a maximum
value (Abramowicz et al, 2006). It has been shown that the epicyclic modes may
be retained also for thicker tori (Blaes et al, 2007). Numerical simulations
(Rezzolla et al, 2003a; Rubio-Herrera and Lee, 2005a, b; Šrámková et al, 2007)
as well as analytic calculations in pseudo-Newtonian approximation (Blaes et
al, 2007) showed that with growing torus thickness the (axisymmetric)
epicyclic frequencies decrease. What we have in mind here is a slightly non-
slender torus (with a radial extent that is very small in comparison to its
distance from the central object) in hydrostatic equilibrium in Kerr
spacetime. We perform a second-order perturbation analysis of the
eigenfunctions and eigenfrequencies of both modes with respect to the torus
thickness and derive exact analytic formulas for the pressure corrections.
Blaes et al (2007) have studied an analogous problem in the framework of
Newtonian physics using the pseudo-Newtonian potential of Paczyński and Wiita
(1980) to model the relevant general relativistic effects. Our work represents
a generalisation of their results into Kerr geometry. Like Blaes et al (2007)
we assume a non-self-gravitating, non-magnetic, stationary torus with a
constant specific angular momentum distribution. We neglect self-gravity
because the mass of a black hole exceeds the mass of a torus many times over.
Although effects of magnetic fields are neither negligible nor irrelevant to
our problem, we study here a purely hydrodynamical case. Our calculations are
only valid as long as magnetic fields are unimportant, i.e., as long as the
torus pressure dominates the magnetic pressure. The stability of these
hydrodynamic modes in presence of magnetohydrodynamical turbulence is an issue
that needs yet to be investigated. The above assumptions are supported by
numerical simulations. Proga & Begelman (2003) show for instance in an
inviscid hydrodynamical simulation that the inner axisymmetric accretion flow
settles into a pressure-rotation supported torus with constant specific
angular momentum. Once magnetic fields are introduced the angular momentum
distribution gets a different profile, torus-like configurations are, however,
still seen as ”inner tori” in global MHD simulations (e.g. Machida et al,
2006; Fragile et al, 2008). Although the described torus setup is not the most
likely to be found in nature, we think it is reasonable to assume such a
configuration as a first approximation.
In this work we focus on a mathematical description of the problem, whereas
the astrophysical applications shall be presented separately. The paper is
outlined in the following manner: In sections 2 and 3 we give a brief
introduction to the problem writing down the equations that describe the
relativistic equilibrium tori and the relativistic Papaloizou-Pringle
equation. Then, in section 4, we describe the perturbation method and derive
formulas for the radial and vertical epicyclic mode eigenfunctions and
eigenfrequencies. The results are presented for different values of the black
hole spin parameter in section 5 and discussed in section 6.
## 2 Equilibrium configuration
Consider an axisymmetric, non-self-gravitating perfect fluid torus in
hydrostatic equilibrium on the background of the Kerr geometry. The flow of
fluid is stationary and in a state of pure rotation. Generally, the line
element of a stationary, axially symmetric spacetime is given in Boyer-
Lindquist coordinates ($t,r,\theta,\phi$) by
$ds^{2}=g_{tt}dt^{2}+2g_{t\phi}dtd\phi+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}.$
(1)
We take the ($-+++$) signature and units where $c=G=M=1$. The explicit
expressions for the covariant and contravariant coefficients of the Kerr
metric then write
$\displaystyle g_{tt}=-\left(1-\frac{2r}{\Sigma}\right),\qquad$ $\displaystyle
g^{tt}=-\frac{\Xi}{\Sigma\Delta},$ (2) $\displaystyle
g_{t\phi}=-\frac{2ar}{\Sigma}\sin^{2}\theta,\qquad$ $\displaystyle
g^{t\phi}=-\frac{2ar}{\Sigma\Delta},$ $\displaystyle
g_{rr}=\frac{\Sigma}{\Delta},\qquad$ $\displaystyle
g^{rr}=\frac{\Delta}{\Sigma},$ $\displaystyle g_{\theta\theta}=\Sigma,\qquad$
$\displaystyle g^{\theta\theta}=\frac{1}{\Sigma},$ $\displaystyle
g_{\phi\phi}=\left(r^{2}+a^{2}+\frac{2a^{2}r}{\Sigma}\sin^{2}\theta\right)\sin^{2}\theta,\qquad$
$\displaystyle
g^{\phi\phi}=\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma\Delta\sin^{2}\theta},$
where $\Sigma\equiv r^{2}+a^{2}\cos^{2}\theta$, $\Delta\equiv r^{2}-2r+a^{2}$,
$\Xi\equiv(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta$ and $M$ is the mass and
$a$ the specific angular momentum (spin) of the black hole.
Because the flow is assumed to be purely azimuthal, the four-velocity has only
two non-zero components, $u^{\mu}=(u^{t},0,0,u^{\phi})$. One may derive the
fluid specific angular momentum $l$, angular velocity $\Omega$, specific
energy $\mathcal{E}$ and the contravariant $t$-component of the four-velocity,
often denoted as $A$, in the form
$\displaystyle l\equiv-$ $\displaystyle\frac{u_{\phi}}{u_{t}}$
$\displaystyle=-\frac{g_{t\phi}+\Omega g_{\phi\phi}}{g_{tt}+\Omega
g_{t\phi}},$ (3) $\displaystyle\Omega\equiv$
$\displaystyle\frac{u^{\phi}}{u^{t}}$
$\displaystyle=\frac{g^{t\phi}-lg^{\phi\phi}}{g^{tt}-lg^{t\phi}},$ (4)
$\displaystyle\mathcal{E}\equiv-$ $\displaystyle u_{t}$
$\displaystyle=(-g^{tt}+2lg^{t\phi}-l^{2}g^{\phi\phi})^{-1/2},$ (5)
$\displaystyle A\equiv$ $\displaystyle u^{t}$ $\displaystyle=(-g_{tt}-2\Omega
g_{t\phi}-\Omega^{2}g_{\phi\phi})^{-1/2}.$ (6)
The perfect fluid is characterised by the stress-energy tensor
$T^{\mu\nu}=(p+e)u^{\mu}u^{\nu}+pg^{\mu\nu}$. We restrict our consideration to
polytropic flows such that, measured in the fluid’s rest frame, pressure $p$,
internal energy density $e$ and rest mass density $\rho$ are related by
$p=K\rho^{(n+1)/n}$ and $e=np+\rho$, where $n$ is the polytropic index and $K$
the polytropic constant.
We assume the specific angular momentum to be constant throughout the torus,
i.e., $l(r,\theta)\equiv l_{0}=const$. Such a configuration is governed by the
relativistic Euler equation which may be written as
$-\frac{\partial_{\mu}p}{p+e}=\partial_{\mu}\left(\ln\mathcal{E}\right),\qquad\mu\in\\{r,\theta\\}.$
(7)
Introducing the enthalpy $H\equiv\int{dp/(p+e)}$, the integration of (7) leads
for a barotropic fluid, for which $p=p(e)$, to the following form of the
Bernoulli equation
$H+\ln\mathcal{E}=const.$ (8)
that determines the structure of the torus in the $r-\theta$ plane. The
subscript zero refers to the special location $r=r_{0}$ in the equatorial
plane where the pressure gradients vanish ($p$ has a maximal value) and the
fluid moves along a geodesic line.
For a small torus cross-section, when the adiabatic sound speed defined at the
pressure maximum $c^{2}_{s0}=(n+1)p_{0}/(n\rho_{0})$ satisfies
$c^{2}_{s0}<<c^{2}$, one may write $H\approx(n+1)p/\rho$ (Abramowicz et al,
2006). Following Abramowicz et al (2006) and Blaes et al (2007) we introduce
the function $f(r,\theta)$, which takes constant values at the isobaric and
isodensity surfaces, by
$\frac{p}{\rho}=\frac{p_{0}}{\rho_{0}}f(r,\theta).$ (9)
Form the Bernoulli equation (8) with the constant evaluated at the pressure
maximum $r_{0}$ and the above form of $H$ we get
$f=1-\frac{1}{nc_{s0}^{2}}\left(\ln\mathcal{E}-\ln\mathcal{E}_{0}\right).$
(10)
### 2.1 Epicyclic oscillations
It is advantageous to introduce the effective potential
$U=g^{tt}-2l_{0}g^{t\phi}+l_{0}^{2}g^{\phi\phi}$ (11)
that has its minimum at the torus pressure maximum $r_{0}$. A small
perturbation of a test particle orbiting on a geodesic line $r=r_{0}$ with
$l=l_{0}$ results in radial and vertical epicyclic oscillations around the
equilibrium point $r_{0}$ at a radial $\omega_{r0}$ and vertical
$\omega_{\theta 0}$ epicyclic frequency given by (e.g., Abramowicz et al
(2006))
$\omega_{r0}^{2}=\frac{1}{2}\left(\frac{\mathcal{E}^{2}}{A^{2}g_{rr}}\frac{\partial^{2}U}{\partial
r^{2}}\right)_{0}\qquad\textnormal{and}\qquad\omega_{\theta
0}^{2}=\frac{1}{2}\left(\frac{\mathcal{E}^{2}}{A^{2}g_{\theta\theta}}\frac{\partial^{2}U}{\partial\theta^{2}}\right)_{0}.$
(12)
In Kerr geometry (2) the above definitions lead to (Aliev and Galtsov, 1981;
Nowak and Lehr, 1998; Török and Stuchlík, 2005)
$\omega_{r0}^{2}=\Omega_{0}^{2}\left(1-\frac{6}{r_{0}}+\frac{8a}{r_{0}^{3/2}}-\frac{3a^{2}}{r_{0}^{2}}\right)\qquad\textnormal{and}\qquad\omega_{\theta
0}^{2}=\Omega_{0}^{2}\left(1-\frac{4a}{r_{0}^{3/2}}+\frac{3a^{2}}{r_{0}^{2}}\right),$
(13)
where $\Omega_{0}$ is the angular velocity at the pressure maximum $r_{0}$
that in Kerr geometry reads $\Omega_{0}=1/(r_{0}^{3/2}+a)$.
In order to investigate the behaviour of the equipotential function $f$ in
close vicinity of the equilibrium point $r_{0}$, Abramowicz et al (2006)
introduced local coordinates,
$x=\sqrt{g_{rr0}}\left(\frac{r-r_{0}}{r_{0}}\right)\qquad\textnormal{and}\qquad
y=\sqrt{g_{\theta\theta 0}}\left(\frac{\pi/2-\theta}{r_{0}}\right),$ (14)
satisfying $x=y=0$ at $r_{0}$. For small $x$ and $y$, and a constant specific
angular momentum torus, the equipotential function $f$ can be expressed as
$f=1-\frac{1}{\beta^{2}}\left(\bar{\omega}_{r0}^{2}x^{2}+\bar{\omega}_{\theta
0}^{2}y^{2}\right),$ (15)
where $\bar{\omega}_{r0}\equiv\omega_{r0}/\Omega_{0}$, $\bar{\omega}_{\theta
0}\equiv\omega_{\theta 0}/\Omega_{0}$, and $\beta$ is a dimensionless
parameter given by
$\beta^{2}\equiv\frac{2nc_{s0}^{2}}{r_{0}^{2}A_{0}^{2}\Omega_{0}^{2}}$ (16)
which determines the thickness of the torus.
Equation (15) describes the equipotential function in the vicinity of $r_{0}$
in terms of the test particle epicyclic frequencies and is congruent with the
formula derived in Newtonian theory (see equation (9) in Blaes et al (2007)).
In the slender torus limit $\beta\rightarrow 0$, the torus reduces to an
infinitesimally slender ring at the pressure maximum $r_{0}$. In a Newtonian
$1/r$ potential, $\omega_{r0}=\omega_{\theta 0}=\Omega_{0}$ and the slender
torus cross-section has a circular shape. In the general case, however,
$\omega_{r0}\neq\omega_{\theta 0}$ and the isobaric surfaces are ellipses with
semi-axes being in the ratio of the two epicyclic frequencies.
## 3 Perturbation equation
We consider small linear perturbations around the axisymmetric and stationary
torus equilibrium with azimuthal and time dependence in the form
$\propto\exp[\rmi(m\phi-\omega t)]$. The differential equation governing such
perturbations for constant specific angular momentum tori was in Newtonian
theory derived by Papaloizou & Pringle (1984) where it was expressed in terms
of a scalar variable $W$. Abramowicz et al (2006) recently derived the general
relativistic form of the Papaloizou-Pringle equation in terms of
$W=-\frac{\delta p}{A\rho(\omega-m\Omega)},$ (17)
which is related to the Eulerian perturbation in the four-velocity as
$\delta
u_{\mu}=\frac{\rmi\rho}{p+e}\partial_{\mu}W,\qquad\mu\in\\{r,\theta\\}.$ (18)
The relativistic Papaloizou-Pringle equation writes
$\displaystyle\frac{1}{(-g)^{1/2}}\left\\{\partial_{\mu}\left[(-g)^{1/2}g^{\mu\nu}f^{n}\partial_{\nu}W\right]\right\\}$
$\displaystyle-$ $\displaystyle\left(m^{2}g^{\phi\phi}-2m\omega
g^{t\phi}+\omega^{2}g^{tt}\right)f^{n}W$ (19)
$\displaystyle=-\frac{2n\mathcal{A}(\bar{\omega}-m\bar{\Omega})^{2}}{\beta^{2}r^{2}_{0}}f^{n-1}W$
where $\\{\mu,\nu\\}\in\\{r,\theta\\}$, $\mathcal{A}\equiv A^{2}/A^{2}_{0}$,
$\bar{\Omega}\equiv\Omega/\Omega_{0}$, $\bar{\omega}\equiv\omega/\Omega_{0}$
and $g$ denotes the determinant of the metric.
Following Blaes et al (2007) we write (19) as
$\hat{L}W=-2n\mathcal{A}\left(\bar{\omega}-m\bar{\Omega}\right)^{2}W,$ (20)
where $\hat{L}$ is a linear operator given by
$\displaystyle\hat{L}$ $\displaystyle=$
$\displaystyle[\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g})g^{\mu\nu}f\partial_{\nu}+\partial_{\mu}(g^{\mu\nu})f\partial_{\nu}+g^{\mu\nu}n\partial_{\mu}f\partial_{\nu}+g^{\mu\nu}f\partial^{2}_{\nu}$
(21) $\displaystyle-\left(m^{2}g^{\phi\phi}-2m\omega
g^{t\phi}+\omega^{2}g^{tt}\right)f]\beta^{2}r^{2}_{0}.$
## 4 Expanding the relativistic Papaloizou-Pringle equation about the slender
torus limit
We now use a perturbation method to derive the expressions for eigenfunctions
and eigenfrequencies of the epicyclic modes for thicker tori ($\beta>0$).
We start by transforming all variables to local coordinates, $\bar{x}=x/\beta$
and $\bar{y}=y/\beta$, measured from the equilibrium point. Then we expand
$\bar{\omega}$, $W$, $\mathcal{A}$, $\bar{\Omega}$, $f$ and $\hat{L}$ into a
power series in $\beta$ by writing
$Q=Q^{(0)}+Q^{(1)}\beta+Q^{(2)}\beta^{2}+\ldots\\\
\,\,\textnormal{where}\,\,\,\,Q\in\\{\bar{\omega},W,\mathcal{A},\bar{\Omega},f,\hat{L}\\}.$
(22)
Substituting all variables (22) into the perturbation equation (20) and
comparing terms of the same order in $\beta$ we obtain formulas for the
respective corrections to eigenfunctions and eigenfrequencies of the desired
modes.
### 4.1 Slender torus limit
In the slender torus limit ($\beta\rightarrow 0$) the relativistic Papaloizou-
Pringle equation (19) reduces to
$\displaystyle
f^{(0)}\left(\frac{\partial^{2}W^{(0)}}{\partial\bar{x}^{2}}+\frac{\partial^{2}W^{(0)}}{\partial\bar{y}^{2}}\right)+n\left(\frac{\partial
f^{(0)}}{\partial\bar{x}}\frac{\partial
W^{(0)}}{\partial\bar{x}}+\frac{\partial
f^{(0)}}{\partial\bar{y}}\frac{\partial W^{(0)}}{\partial\bar{y}}\right)=$
$\displaystyle-2n\mathcal{A}^{(0)}(\bar{\omega}^{(0)}-m\bar{\Omega}^{(0)})^{2}W^{(0)}.$
(23)
It represents the zeroth-order of (20) and may be written in operator form,
$\hat{L}^{(0)}W^{(0)}=-2n\mathcal{A}^{(0)}\sigma^{2}W^{(0)},$ (24)
where $\sigma\equiv\bar{\omega}^{(0)}-m\bar{\Omega}^{(0)}$ denotes the zeroth-
order eigenfrequency in the corotating frame scaled with the orbital velocity
$\Omega_{0}$. The encountered zeroth-order expansion terms are
$\displaystyle\mathcal{A}^{(0)}$ $\displaystyle=$ $\displaystyle 1,$ (25)
$\displaystyle\bar{\Omega}^{(0)}$ $\displaystyle=$ $\displaystyle 1,$ (26)
$\displaystyle f^{(0)}$ $\displaystyle=$ $\displaystyle
1-\bar{\omega}_{r0}^{2}\bar{x}^{2}-\bar{\omega}_{\theta 0}^{2}\bar{y}^{2},$
(27) $\displaystyle\hat{L}^{(0)}$ $\displaystyle=$ $\displaystyle
f^{(0)}\frac{\partial^{2}}{\partial\bar{x}^{2}}+f^{(0)}\frac{\partial^{2}}{\partial\bar{y}^{2}}+n\frac{\partial
f^{(0)}}{\partial\bar{x}}\frac{\partial}{\partial\bar{x}}+n\frac{\partial
f^{(0)}}{\partial\bar{y}}\frac{\partial}{\partial\bar{y}}.$ (28)
Equations (23) and (24) are of the same form as the Newtonian slender torus
limit of the Papaloizou-Pringle equation (Abramowicz et al, 2006; Blaes et al,
2007). They represent an eigenvalue equation for $\sigma$ with $\hat{L}^{(0)}$
being a self-adjoint operator with respect to the inner product
$\left\langle
W_{a}^{(0)}|W_{b}^{(0)}\right\rangle=\int\int\left(f^{(0)}\right)^{n-1}W_{a}^{(0)\ast}W_{b}^{(0)}d\bar{x}d\bar{y}=\delta_{ab},$
(29)
where the integrals are taken over the slender torus cross-section where
$f^{(0)}\geq 0$. This implies that the eigenvalues $\sigma$ are real numbers
and the zeroth-order eigenfunctions $W^{(0)}$ form a complete orthonormal set.
Therefore, any function defined over the torus cross-section may be expanded
in terms of $W^{(0)}$.
The explicit expressions for eigenfunctions and eigenfrequencies of the
complete set of normal modes of a constant specific angular momentum slender
torus in Newtonian potential, where $\bar{\omega}_{r0}=\bar{\omega}_{\theta
0}=1$, were given in Blaes (1985). Recently, Blaes, Arras and Fragile (2006)
derived the eigenfunctions and eigenfrequencies of the lowest-order modes of a
general relativistic slender torus, where
$\bar{\omega}_{r0}\neq\bar{\omega}_{\theta 0}$, for arbitrary specific angular
momentum distribution. The eigenfunctions and eigenfrequencies of the simplest
modes of the relativistic torus with constant specific angular momentum which
are relevant to our calculations are specified in table 1.
In the slender torus limit the two epicyclic modes correspond to $i=1$ and
$i=2$ from table 1. The corresponding eigenfunctions take the form
$W_{r}^{(0)}\equiv
W_{1}^{(0)}=a_{1}\bar{x}e^{\rmi(m\phi-\omega_{1}^{(0)}t)}\quad\textnormal{and}\quad
W_{\theta}^{(0)}\equiv
W_{2}^{(0)}=a_{2}\bar{y}e^{\rmi(m\phi-\omega_{2}^{(0)}t)}.$ (30)
They describe the global, incompressible modes in which the entire torus moves
in purely radial ($W_{r}^{(0)}$) or purely vertical ($W_{\theta}^{(0)}$)
direction at frequencies which are, in the corotating frame, consistent with
the epicyclic frequencies of free test particles,
$\omega_{r0}=\sigma_{1}\Omega_{0}=\omega_{1}^{(0)}-m\Omega^{(0)}$ and
$\omega_{\theta 0}=\sigma_{2}\Omega_{0}=\omega_{2}^{(0)}-m\Omega^{(0)}$.
### 4.2 Non-slender torus
#### 4.2.1 First-order corrections
Expanding the Papaloizou-Pringle equation (20) to first-order in $\beta$ we
arrive at 111We use the index $i$ to label the modes of interest. It takes
values $i=1$ for the radial or $i=2$ for the vertical epicyclic mode.
$\displaystyle\hat{L}^{(0)}W_{i}^{(1)}+\hat{L}^{(1)}W_{i}^{(0)}$
$\displaystyle=$ $\displaystyle
2n(2m\sigma_{i}\bar{\Omega}^{(1)}-\sigma_{i}^{2}\mathcal{A}^{(1)}-2\sigma_{i}\bar{\omega}_{i}^{(1)})W_{i}^{(0)}$
(31) $\displaystyle-2n\sigma_{i}^{2}W_{i}^{(1)}.$
The perturbed basis of eigenfunctions $W_{i}^{(1)}$ may now be expressed in
terms of the orthonormal zeroth-order basis as
$W_{i}^{(1)}=\sum_{j}{b_{ij}W_{j}^{(0)}}.$ (32)
The $b_{ij}$ coefficients may be determined by taking the inner product of
(31) with a zeroth-order eigenfunction $W_{k}^{(0)}$. If the subscript $k$
refers to a different mode from the one we are interested in (i.e., $k\neq i$)
we find
$b_{ij}=\frac{\left\langle
W_{j}^{(0)}|\hat{L}^{(1)}-4nm\sigma_{i}\bar{\Omega}^{(1)}+2n\sigma_{i}^{2}\mathcal{A}^{(1)}|W_{i}^{(0)}\right\rangle}{2n(\sigma_{j}^{2}-\sigma_{i}^{2})}.$
(33)
Whereas if $k$ refers to either of the two epicyclic modes ($k=i$) we obtain
the formula for the first-order correction to the radial ($i=1$) or vertical
($i=2$) eigenfrequency,
$\bar{\omega}_{i}^{(1)}=-\frac{\left\langle
W_{i}^{(0)}|\hat{L}^{(1)}-4nm\sigma_{i}\bar{\Omega}^{(1)}+2n\sigma_{i}^{2}\mathcal{A}^{(1)}|W_{i}^{(0)}\right\rangle}{4n\sigma_{i}}.$
(34)
The first-order expansion terms of $\mathcal{A}$, $\bar{\Omega}$, $f$ and
$\hat{L}$ have a ($\bar{x},\bar{y}$) dependence in the form
$\displaystyle\mathcal{A}^{(1)}$ $\displaystyle=\mathcal{A}_{11}\bar{x},$ (35)
$\displaystyle\bar{\Omega}^{(1)}$ $\displaystyle=\Omega_{11}\bar{x},$ (36)
$\displaystyle f^{(1)}$
$\displaystyle=f_{11}\bar{x}^{3}+f_{12}\bar{x}\bar{y}^{2},$ (37)
$\displaystyle\hat{L}^{(1)}$
$\displaystyle=\left(L_{101}\bar{x}+L_{102}\bar{x}^{3}+L_{103}\bar{x}\bar{y}^{2}\right)\partial_{\bar{x}}^{2}$
(38)
$\displaystyle+\left(L_{104}\bar{x}+L_{105}\bar{x}^{3}+L_{106}\bar{x}\bar{y}^{2}\right)\partial_{\bar{y}}^{2}$
$\displaystyle+\left(L_{107}+L_{108}\bar{x}^{2}+L_{109}\bar{y}^{2}\right)\partial_{\bar{x}}+L_{110}\bar{x}\bar{y}\partial_{\bar{y}},$
where the explicit forms of the coefficients
$\mathcal{A}_{11},\Omega_{11},f_{11},f_{12}$ and $L_{101}-L_{110}$ are
specified in the appendix.
Table 1: Eigenfunctions of the lowest-order modes of a general slender torus
with constant specific angular momentum (Blaes, Arras and Fragile, 2006). The
normalisation constants $a_{0}-a_{5}$ are given in table 3 and the
coefficients $w_{41}$, $w_{42}$, $w_{51}$, $w_{52}$ are specified in the
appendix.
* $i$ | $W_{i}^{(0)}$
---|---
0 | $a_{0}$
1 | $a_{1}\bar{x}$
2 | $a_{2}\bar{y}$
3 | $a_{3}\bar{x}\bar{y}$
4 | $a_{4}\left(1+w_{41}\bar{x}^{2}+w_{42}\bar{y}^{2}\right)$
5 | $a_{5}\left(1+w_{51}\bar{x}^{2}+w_{52}\bar{y}^{2}\right)$
Substituting (35)-(38) into (33) and considering the eigenmodes derived in
Blaes, Arras and Fragile (2006) we find for the radial mode ($i=1$) three non-
zero coefficients that correspond to the modes i = 0, 4 and 5 given in table
1. The resulting eigenfunction $W_{r}$ of the radial epicyclic mode for a
slightly non-slender torus in first-order accuracy reads
$W_{r}=a_{1}\bar{x}+\beta\left(C_{0}+C_{1}\bar{x}^{2}+C_{2}\bar{y}^{2}\right)+\mathcal{O}(\beta^{2}),$
(39)
where $C_{0}=a_{0}b_{10}+a_{4}b_{14}+a_{5}b_{15}$,
$C_{1}=a_{4}b_{14}w_{41}+a_{5}b_{15}w_{51}$ and
$C_{2}=a_{4}b_{14}w_{42}+a_{5}b_{15}w_{52}$. The normalisation constants
$a_{0}$, $a_{4}$, $a_{5}$ are listed in table 3 and the eigenfunction-related
terms $w_{41}$, $w_{42}$, $w_{51}$, $w_{52}$ are specified in the appendix.
The coefficients $b_{10}$, $b_{14}$, $b_{15}$, given by (33), take the
following forms
$\displaystyle b_{10}$ $\displaystyle=$
$\displaystyle-\frac{a_{0}a_{1}\pi}{4n^{2}(n+1)(\sigma_{0}^{2}-\sigma_{1}^{2})\bar{\omega}_{\theta
0}^{3}\bar{\omega}_{r0}^{3}}\left\\{L_{109}\bar{\omega}_{r0}^{2}+\left[L_{108}+2n(m-\bar{\omega}_{r0})\times\right.\right.$
(40)
$\displaystyle\left.\left.\times\left\\{\mathcal{A}_{11}(m-\bar{\omega}_{r0})-2m\Omega_{11}\right\\}+2L_{107}(n+1)\bar{\omega}_{r0}^{2}\right]\bar{\omega}_{\theta
0}^{2}\right\\},$ $\displaystyle b_{1q}$ $\displaystyle=$
$\displaystyle\frac{a_{1}a_{q}\pi}{8n^{2}(n+1)(n+2)(\sigma_{1}^{2}-\sigma_{q}^{2})\bar{\omega}_{\theta
0}^{5}\bar{\omega}_{r0}^{5}}\left[\\{L_{108}+2n\mathcal{A}_{11}(m-\bar{\omega}_{r0})^{2}\right.$
(41)
$\displaystyle\left.-4m^{2}n\Omega_{11}+4mn\Omega_{11}\bar{\omega}_{r0}\\}\times\\{[2(n+2)\bar{\omega}_{r0}^{2}+3W_{q1}]\bar{\omega}_{\theta
0}^{2}\right.$
$\displaystyle\left.+\bar{\omega}_{r0}^{2}W_{q2}\\}\bar{\omega}_{\theta
0}^{2}+L_{109}\\{[2(n+2)\bar{\omega}_{r0}^{2}+W_{q1}]\bar{\omega}_{\theta
0}^{2}+3\bar{\omega}_{r0}^{2}W_{q2}\\}\bar{\omega}_{r0}^{2}\right.$
$\displaystyle\left.+2L_{107}(n+2)\bar{\omega}_{\theta
0}^{2}\bar{\omega}_{r0}^{2}\\{[2(n+1)\bar{\omega}_{r0}^{2}+W_{q1}]\bar{\omega}_{\theta
0}^{2}+\bar{\omega}_{r0}^{2}W_{q2}\\}\right],$
where $q=4$ or $q=5$ in case of $b_{14}$ or $b_{15}$, respectively, and
$\sigma_{0}$, $\sigma_{1}$, $\sigma_{4}$, $\sigma_{5}$ are specified in table
2.
Table 2: Eigenfrequencies of the eigenmodes given in table 1.
* $i$ | $\sigma_{i}^{2}$
---|---
0 | 0
1 | $\bar{\omega}_{r0}^{2}$
2 | $\bar{\omega}_{\theta 0}^{2}$
3 | $\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta 0}^{2}$
4 | $\left\\{(2n+1)(\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta 0}^{2})-[4n(n+1)(\bar{\omega}_{\theta 0}^{2}-\bar{\omega}_{r0}^{2})^{2}+(\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta 0}^{2})^{2}]^{1/2}\right\\}/(2n)$
5 | $\left\\{(2n+1)(\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta 0}^{2})+[4n(n+1)(\bar{\omega}_{\theta 0}^{2}-\bar{\omega}_{r0}^{2})^{2}+(\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta 0}^{2})^{2}]^{1/2}\right\\}/(2n)$
Similarly, for the vertical epicyclic mode ($i=2$) we find one non-zero
coefficient corresponding to $j=3$ and receive the final expression for the
eigenfunction $W_{\theta}$ of the vertical epicyclic mode,
$W_{\theta}=a_{2}\bar{y}+\beta C_{3}\bar{x}\bar{y}+\mathcal{O}(\beta^{2})$
(42)
with $C_{3}=a_{3}b_{23}$. Again, $a_{3}$ is the normalisation constant (see
table 3) and the $b_{23}$ coefficient is given by
$\displaystyle b_{23}$ $\displaystyle=$
$\displaystyle\frac{a_{2}a_{3}\pi}{8n^{2}(n+2)(n+1)(\sigma_{2}^{2}-\sigma_{3}^{2})\bar{\omega}_{\theta
0}^{3}\bar{\omega}_{r0}^{3}}\left\\{L_{110}-2n(m-\bar{\omega}_{\theta
0})\times\right.$ (43)
$\displaystyle\left.\times\left[\mathcal{A}_{11}(\bar{\omega}_{\theta
0}-m)+2m\Omega_{11}\right]\right\\}$
with $\sigma_{2}$, $\sigma_{3}$ listed in table 2.
Table 3: Normalisation constants of the eigenmodes in table 1 (Blaes et al,
2007). They are calculated such that the eigenfunctions are normalised in the
inner product (29).
* $i$ | $a_{i}$
---|---
0 | $\left({n\bar{\omega}_{r0}\bar{\omega}_{\theta 0}/\pi}\right)^{1/2}$
1 | $a_{0}[2(n+1)\bar{\omega}_{r0}^{2}]^{1/2}$
2 | $a_{0}[2(n+1)\bar{\omega}_{\theta 0}^{2}]^{1/2}$
3 | $a_{0}[4(n+1)(n+2)\bar{\omega}_{r0}^{2}\bar{\omega}_{\theta 0}^{2}]^{1/2}$
4 | $a_{0}\left\\{{(n+2)[\sigma_{4}^{2}-(\bar{\omega}_{\theta 0}^{2}+\bar{\omega}_{r0}^{2})]/\left[2n\sigma_{4}^{2}-(2n+1)(\bar{\omega}_{\theta 0}^{2}+\bar{\omega}_{r0}^{2})\right]}\right\\}^{1/2}$
5 | $a_{0}\left\\{{(n+2)[\sigma_{5}^{2}-(\bar{\omega}_{\theta 0}^{2}+\bar{\omega}_{r0}^{2})]/\left[2n\sigma_{5}^{2}-(2n+1)(\bar{\omega}_{\theta 0}^{2}+\bar{\omega}_{r0}^{2})\right]}\right\\}^{1/2}$
Then, using (35)-(38), we find the terms
$\displaystyle\mathcal{A}^{(1)}|W_{1}^{(0)}\rangle=a_{1}\mathcal{A}_{11}\bar{x}^{2},\qquad\qquad$
$\displaystyle\mathcal{A}^{(1)}|W_{2}^{(0)}\rangle=a_{2}\mathcal{A}_{11}\bar{x}\bar{y},$
$\displaystyle\bar{\Omega}^{(1)}|W_{1}^{(0)}\rangle=a_{1}\Omega_{11}\bar{x}^{2},\qquad\qquad$
$\displaystyle\bar{\Omega}^{(1)}|W_{2}^{(0)}\rangle=a_{2}\Omega_{11}\bar{x}\bar{y},$
$\displaystyle\hat{L}^{(1)}|W_{1}^{(0)}\rangle=a_{1}(L_{107}+L_{108}\bar{x}^{2}+L_{109}\bar{y}^{2}),\qquad\qquad$
$\displaystyle\hat{L}^{(1)}|W_{2}^{(0)}\rangle=a_{2}L_{110}\bar{x}\bar{y}.$
(44)
Substituting (44) into the formula (34) for $\omega_{i}^{(1)}$, we obtain in
the inner product for both modes $i=1$ and $i=2$ odd functions of $\bar{x}$
and $\bar{y}$, such that the integration over the elliptical torus cross-
section yields zero. Therefore, to find the relevant pressure corrections to
the radial and vertical mode frequencies, one needs to extend the expansion to
second-order in torus thickness.
#### 4.2.2 Second-order corrections
The perturbation equation (20) expanded to second-order in $\beta$ reads
$\displaystyle\hat{L}^{(0)}W_{i}^{(2)}+\hat{L}^{(1)}W_{i}^{(1)}+\hat{L}^{(2)}W_{i}^{(0)}$
$\displaystyle=$
$\displaystyle-2n\left\\{\sigma_{i}^{2}W_{i}^{(2)}+(\sigma_{i}^{2}\mathcal{A}^{(1)}\right.$
(45)
$\displaystyle\left.-2m\sigma_{i}\bar{\Omega}^{(1)})W_{i}^{(1)}+\left[\sigma_{i}^{2}\mathcal{A}^{(2)}\right.\right.$
$\displaystyle\left.\left.+m^{2}\left(\bar{\Omega}^{(1)}\right)^{2}-2m\sigma_{i}\bar{\Omega}^{(2)}+2\sigma_{i}\bar{\omega}_{i}^{(2)}\right.\right.$
$\displaystyle\left.\left.-2m\sigma_{i}\mathcal{A}^{(1)}\bar{\Omega}^{(1)}\right]W_{i}^{(0)}\right\\}.$
Analogous to the first-order case one may write
$W_{i}^{(2)}=\sum_{j}{c_{ij}W_{j}^{(0)}}$ (46)
and take the inner product of (45) with a zeroth-order eigenfunction
$W_{k}^{(0)}$. For a subscript $k$ referring to the mode of interest ($k=i$)
we obtain the formula for the second-order correction,
$\displaystyle\bar{\omega}_{i}^{(2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4n\sigma_{i}}\left[\left\langle
W_{i}^{(0)}|\hat{L}^{(2)}+2nm^{2}\left(\bar{\Omega}^{(1)}\right)^{2}-4nm\sigma_{i}\bar{\Omega}^{(2)}\right.\right.$
(47)
$\displaystyle\left.\left.+2n\sigma_{i}^{2}\mathcal{A}^{(2)}-4nm\sigma_{i}\mathcal{A}^{(1)}\bar{\Omega}^{(1)}|W_{i}^{(0)}\right\rangle\right.$
$\displaystyle\left.+\sum_{j}b_{ij}\left\langle
W_{i}^{(0)}|\hat{L}^{(1)}+2n\sigma_{i}^{2}\mathcal{A}^{(1)}-4nm\sigma_{i}\bar{\Omega}^{(1)}|W_{j}^{(0)}\right\rangle\right].$
The second-order terms of $\mathcal{A}$, $\bar{\Omega}$, $f$ and the $\hat{L}$
operator take now the form
$\displaystyle\mathcal{A}^{(2)}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{21}\bar{x}^{2}+\mathcal{A}_{22}\bar{y}^{2},$ (48)
$\displaystyle\bar{\Omega}^{(2)}$ $\displaystyle=$
$\displaystyle\Omega_{21}\bar{x}^{2}+\Omega_{22}\bar{y}^{2},$ (49)
$\displaystyle f^{(2)}$ $\displaystyle=$ $\displaystyle
f_{21}\bar{x}^{4}+f_{22}\bar{x}^{2}\bar{y}^{2}+f_{23}\bar{y}^{4},$ (50)
$\displaystyle\hat{L}^{(2)}$ $\displaystyle=$
$\displaystyle\left\\{L_{201}\bar{x}^{4}+L_{202}\bar{x}^{2}+L_{203}\bar{x}^{2}\bar{y}^{2}+L_{204}\bar{y}^{2}+L_{205}\bar{y}^{4}\right\\}\partial_{\bar{x}}^{2}$
(51)
$\displaystyle+\left\\{L_{206}\bar{x}^{4}+L_{207}\bar{x}^{2}+L_{208}\bar{x}^{2}\bar{y}^{2}+L_{204}\bar{y}^{2}+L_{209}\bar{y}^{4}\right\\}\partial_{\bar{y}}^{2}$
$\displaystyle+\left\\{L_{210}\bar{x}^{3}+L_{211}\bar{x}+L_{212}\bar{x}\bar{y}^{2}\right\\}\partial_{\bar{x}}$
$\displaystyle+\left\\{L_{213}\bar{x}^{2}\bar{y}+L_{214}\bar{y}+L_{215}\bar{y}^{3}\right\\}\partial_{\bar{y}}$
$\displaystyle+\left\\{L_{216}+L_{217}\bar{x}^{2}+L_{218}\bar{y}^{2}\right\\},$
with the coefficients $\mathcal{A}_{21}$, $\mathcal{A}_{22}$, $\Omega_{21}$,
$\Omega_{22}$, $f_{21}$, $f_{22}$, $f_{23}$ and $L_{201}$ \- $L_{218}$ again
specified in the appendix.
Inserting all first and second-order expansion terms into (47) we gain a
fairly long expression for $\bar{\omega}_{1}^{(2)}$. Along with (22) it leads
to the resulting formula for the eigenfrequency $\bar{\omega}_{r}$ of the
radial mode. In order to keep the overview we write the expression in the
following form
$\displaystyle\bar{\omega}_{r}$ $\displaystyle=$
$\displaystyle\bar{\omega}_{r0}+m-\frac{\beta^{2}}{4n\sigma_{1}}\sum_{l=1}^{4}P_{l}+\mathcal{O}(\beta^{3})$
(52)
where
$\displaystyle P_{1}$ $\displaystyle\equiv$ $\displaystyle\langle
W_{1}^{(0)}|\hat{L}^{(2)}+2nm^{2}\left(\bar{\Omega}^{(1)}\right)^{2}-4nm\sigma_{1}\bar{\Omega}^{(2)}+2n\sigma_{1}^{2}\mathcal{A}^{(2)}$
$\displaystyle-4nm\sigma_{1}\mathcal{A}^{(1)}\bar{\Omega}^{(1)}|W_{1}^{(0)}\rangle$
$\displaystyle=$
$\displaystyle\frac{a_{1}^{2}\pi}{4n(n+1)(n+2)\bar{\omega}_{\theta
0}^{3}\bar{\omega}_{r0}^{5}}\left\\{\left(L_{212}+L_{218}\right)\bar{\omega}_{r0}^{2}\right.$
(53)
$\displaystyle\left.+\left[3\left(L_{210}+L_{217}+2m^{2}n\Omega_{11}^{2}\right)\right.\right.$
$\displaystyle\left.\left.+2\left(L_{211}+L_{216}\right)\left(n+2\right)\bar{\omega}_{r0}^{2}\right]\bar{\omega}_{\theta
0}^{2}\right.$
$\displaystyle\left.-4mn\left[3(\mathcal{A}_{11}\Omega_{11}+\Omega_{21})\bar{\omega}_{\theta
0}^{2}+\Omega_{22}\bar{\omega}_{r0}^{2}\right]\bar{\omega}_{r0}\right.$
$\displaystyle\left.+2n\left(3\mathcal{A}_{21}\bar{\omega}_{\theta
0}^{2}+\mathcal{A}_{22}\bar{\omega}_{r0}^{2}\right)\bar{\omega}_{r0}^{2}\right\\},$
$\displaystyle P_{2}\,$ $\displaystyle\equiv$ $\displaystyle\,b_{10}\,\langle
W_{1}^{(0)}|\hat{L}^{(1)}+2n\sigma_{1}^{2}\mathcal{A}^{(1)}-4nm\sigma_{1}^{2}\bar{\Omega}^{(1)}|W_{0}^{(0)}\rangle$
(54) $\displaystyle=$
$\displaystyle\frac{a_{0}a_{1}b_{10}\pi\bar{\omega}_{r0}\left(\mathcal{A}_{11}\bar{\omega}_{r0}-2m\Omega_{11}\right)}{(n+1)\bar{\omega}_{\theta
0}\bar{\omega}_{r0}^{3}},$
$\displaystyle P_{q-1}\,$ $\displaystyle\equiv$
$\displaystyle\,b_{1q}\,\langle
W_{1}^{(0)}|\hat{L}^{(1)}+2n\sigma_{1}^{2}\mathcal{A}^{(1)}-4nm\sigma_{1}^{2}\bar{\Omega}^{(1)}|W_{q}^{(0)}\rangle$
(55) $\displaystyle=$ $\displaystyle
b_{1q}\frac{a_{1}a_{q}\pi}{2n(n+1)(n+2)\bar{\omega}_{\theta
0}^{3}\bar{\omega}_{r0}^{5}(\bar{\omega}_{\theta
0}^{2}-\bar{\omega}_{r0}^{2})}\left\\{-2(n+1)\bar{\omega}_{r0}^{2}\bar{\omega}_{\theta
0}^{2}\times\right.$ $\displaystyle\left.\times\left[3\bar{\omega}_{\theta
0}^{2}(L_{102}-L_{105}+L_{108})+2(n+2)\bar{\omega}_{\theta
0}^{2}\bar{\omega}_{r0}^{2}(L_{101}-L_{104}\right.\right.$
$\displaystyle\left.\left.+L_{107})+\bar{\omega}_{r0}^{2}(L_{103}-L_{106}+L_{109}-L_{110})\right]+n\sigma_{q}^{2}\left\\{\bar{\omega}_{r0}^{4}\left[L_{103}\right.\right.\right.$
$\displaystyle\left.\left.\left.+L_{109}+2(2+n)\bar{\omega}_{\theta
0}^{2}(L_{101}+L_{107})\right]-\bar{\omega}_{r0}^{2}\bar{\omega}_{\theta
0}^{2}\left[L_{106}+L_{110}\right.\right.\right.$
$\displaystyle\left.\left.\left.+2(n+2)\bar{\omega}_{\theta
0}^{2}L_{104}\right]+3\bar{\omega}_{r0}^{2}\bar{\omega}_{\theta
0}^{2}(L_{102}+L_{108})-3\bar{\omega}_{\theta 0}^{4}L_{105}\right\\}+\right.$
$\displaystyle\left.n(\bar{\omega}_{\theta
0}^{2}-\bar{\omega}_{r0}^{2})\bar{\omega}_{r0}(\mathcal{A}_{11}\bar{\omega}_{r0}-2m\Omega_{11})\left[2(n+2)\bar{\omega}_{r0}^{2}\bar{\omega}_{\theta
0}^{2}\right.\right.$ $\displaystyle\left.\left.+3\bar{\omega}_{\theta
0}^{2}w_{q1}+\bar{\omega}_{r0}^{2}w_{q2}\right]\right\\}.$
Here $q=4$ or $q=5$ in case of $P_{3}$ or $P_{4}$, respectively.
For the vertical epicyclic mode we find a much shorter expression,
$\displaystyle\bar{\omega}_{\theta}$ $\displaystyle=$
$\displaystyle\bar{\omega}_{\theta
0}+m-\beta^{2}\frac{a_{2}^{2}\pi}{128n^{4}(n+1)^{2}(n+2)^{2}\left(\sigma_{2}^{3}-\sigma_{2}\sigma_{3}^{2}\right)\omega_{\theta}^{8}\omega_{r}^{6}}\times$
(56) $\displaystyle\times a_{3}^{2}\pi\left[L_{110}+2n\bar{\omega}_{\theta
0}\left(\mathcal{A}_{11}\bar{\omega}_{\theta
0}-2m\Omega_{11}\right)\right]\left\\{3L_{109}\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta
0}^{2}\left[L_{108}\right.\right.$
$\displaystyle\left.\left.+L_{110}+2L_{107}(n+2)\bar{\omega}_{r0}^{2}+2n\bar{\omega}_{\theta
0}(\mathcal{A}_{11}\bar{\omega}_{\theta 0}-2m\Omega_{11})\right]\right\\}$
$\displaystyle-8n^{2}(n+1)(n+2)\bar{\omega}_{\theta
0}^{3}\bar{\omega}_{r0}^{3}\left(\sigma_{2}^{2}-\sigma_{3}^{2}\right)\left\\{3\left(L_{215}+L_{218}\right)\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta
0}^{2}\left[L_{213}\right.\right.$
$\displaystyle\left.\left.+L_{217}+2m^{2}n\Omega_{11}^{2}+2\left(L_{214}+L_{216}\right)(n+2)\bar{\omega}_{r0}^{2}\right]-4mn\left[\left(\mathcal{A}_{11}\Omega_{11}\right.\right.\right.$
$\displaystyle\left.\left.\left.+\Omega_{21}\right)\bar{\omega}_{\theta
0}^{2}+3\Omega_{22}\bar{\omega}_{r0}^{2}\right]\bar{\omega}_{\theta
0}+2n\left(\mathcal{A}_{21}\bar{\omega}_{\theta
0}^{2}+3\mathcal{A}_{22}\bar{\omega}_{r0}^{2}\right)\bar{\omega}_{\theta
0}^{2}\right\\}$ $\displaystyle+\mathcal{O}(\beta^{3}).$
## 5 The properties of epicyclic modes for non-slender tori
The equations (39), (42), (52) and (56) represent the epicyclic mode
eigenfunctions and eigenfrequencies of non-slender tori as functions of
$r_{0}$, $a$, $m$, $n$ and $\beta$. In this section we use these formulas to
illustrate the behaviour for the axisymmetric ($m=0$) and lowest-order non-
axisymmetric ($m=\pm 1$) epicyclic modes for variable torus thickness and
varying black hole spin value. We take the polytropic index $n=3$ that refers
to a radiation-pressure dominated torus 222Note that varying $n$ makes no
relevant difference to the results..
All figures that are related to frequency behaviour (1-3, 5-7) display the
frequencies defined as $\nu=\omega/2\pi$ 333As it is commonly used, throughout
the paper we refer to quantities $\omega$ as to ’frequencies’ as well,
although in exact notation they should be called ’angular velocities’.. The
horizontal axis in all these figures starts at the radius of the marginally
stable orbit calculated for the appropriate black hole spin.
### 5.1 Axisymmetric epicyclic modes
In the axisymmetric ($m=0$) case the $\phi$-components vanish from the
equations and the problem becomes symmetric with respect to the rotational
axis ($\theta=0$). The corresponding axisymmetric radial and vertical
epicyclic frequencies are shown in figures 1, 2 and 3. The $\beta=0$ line
always refers to the epicyclic frequency of a test particle, while the
$\beta>0$ lines illustrate the behaviour of the two frequencies when the torus
becomes thicker. As a result of the topology of the equipotential surfaces
there is an upper limit on $\beta$ for a given torus pressure maximum above
which no closed equipotential surfaces and consequently no equilibrium tori
may exist. This limit is incorporated into the figures as a dash-dotted line
that defines the region of ’allowed frequencies’ above it and to its right
(inside the shaded region).
Figure 1: The axisymmetric radial epicyclic frequency as a function of the
torus pressure maximum $r_{0}$ for tori of various thickness that orbit a
black hole of mass $M=10M_{{}_{\odot}}$. The allowed frequencies corresponding
to equilibrium tori lie inside the shaded region. Left In the Schwarzschild
case $a=0$. Right For a rotating black hole of $a=0.8$.
For any black hole spin both axisymmetric frequencies decrease with increasing
torus thickness. A comparison of the left ($a=0$) and right ($a=0.8$) panels
of figures 1 and 2 and the respective panel of figure 3 ($a=0.999$)
illustrates the influence of the black hole rotation.
The radial frequency qualitatively retains for all spin values the same
profile albeit the torus thickness. The radius where it becomes zero moves,
however, away from the central object. It is interesting to note that for all
values of $a$ the frequency maximum for a fixed $\beta$ is reduced by almost
exactly the same relative amount444This does not apply to the actual thickness
of the torus, since a given $\beta$ implies for different $a$ a different
extent of the torus.. Moreover, for a non-extremely rotating black hole with
$a\lesssim 0.96$, this is more or less true for any location of the pressure
maximum $r_{0}$, while for $a\gtrsim 0.96$ the frequencies at small radii tend
to crowd together, as may be seen in figure 3.
The vertical frequency for a Schwarzschild black hole changes its profile with
increasing torus thickness from a monotonic function in the case of test
particle frequency to a function exhibiting a maximum value (see the left
panel of figure 2). For a Kerr black hole the non-monotonicity is already
present for the test particle frequency (although for low to moderate spin
values the frequency maximum is located at radii inside the marginally stable
orbit) and it remains also for a thicker torus (see the right panels of
figures 2 and 3). At very high spin values ($a\gtrsim 0.96$), the frequency
shape is modified in such a way that the frequency maxima for all tori almost
coincide with the $\beta=0$ curve (figure 3, right panel).
Figure 2: The same as figure 1, but for the axisymmetric vertical epicyclic
mode.
In order to illustrate the characteristic features of the flow for the modes
of a thicker torus we use equations (18), (39) and (42) to plot the
corresponding poloidal velocity fields. The axisymmetric radial mode shows a
relatively coherent flow with only slight deviations from the radial motion in
the outer regions of the torus (see the left panel in figure 4). The
axisymmetric vertical mode, however, shows a more complex behaviour. As seen
in the right panel of figure 4, it involves vertical motions near the inner
and outer edges of the torus, as well as smaller radial flows in regions close
to the pressure maximum. Its velocity pattern exhibits the features as one of
the lowest-order slender torus modes calculated in Blaes, Arras and Fragile
(2006), the so-called x-mode (see their figure 1). The velocity fields of both
modes keep their characteristics similar to those described above
independently of the black hole spin.
Figure 3: The same as figure 1, but for the axisymmetric radial (left) and
vertical (right) epicyclic mode in case of a near-extreme Kerr black hole of
$a=0.999$.
Figure 4: Poloidal flow velocity fields of the axisymmetric radial (left) and
vertical (right) epicyclic mode for a torus of $\beta=0.15$ with pressure
maximum at $r_{0}=8r_{\rm{g}}$ orbiting a Kerr black hole with $a=0.5$.
### 5.2 Non-axisymmetric epicyclic modes
The lowest-order non-axisymmetric ($m=\pm 1$) radial and vertical epicyclic
mode frequencies are shown in figures 5-7. Again, the $\beta=0$ curve in each
case refers to the test particle frequency and the dash-dotted line specifies
the region of allowed frequencies (inside the shaded region). Like above, the
left panels of figures 5 and 6 display the frequencies for tori that orbit a
Schwarzschild black hole ($a=0$), while the corresponding right panels show
the same for a rapidly spinning Kerr black hole ($a=0.8$). Figure 7 then
displays the frequencies for a near-extreme Kerr black hole ($a=0.999$).
Figure 5: The non-axisymmetric $m=1$ (top ) and $m=-1$ (bottom) radial
epicyclic frequency as a function of $r_{0}$ for different values of $\beta$
for a $M=10M_{{}_{\odot}}$ black hole. Left For a non-rotating black hole.
Right For a spinning black hole of $a=0.8$.
The $m=1$ radial epicyclic mode frequency of a slightly non-slender torus is
given by equation (52). For a test particle ($\beta=0$) it reduces to the sum
of $\Omega_{0}$ and the test particle axisymmetric radial epicyclic frequency
$\omega_{r0}$. For all values of $a$ this frequency decreases with increasing
torus thickness (see the top panels of figure 5 and the top left panel of
figure 7).
The $m=-1$ radial mode frequency, again given by (52), is in the test particle
case represented (in absolute value) by the difference between $\Omega_{0}$
and $\omega_{r0}$. At small radii the frequencies for all spin values increase
with growing torus thickness (see the bottom panels of figure 5 and the bottom
left panel of figure 7). With rising $r_{0}$, the non-slender torus
frequencies start oscillating about the $\beta=0$ frequency and eventually
converge to the test particle profile.
In spherically symmetric spacetimes, the orbital frequency $\Omega_{0}$ and
the axisymmetric vertical epicyclic frequency $\omega_{\theta 0}$ of a test
particle coincide. This is no longer true in case of an axially symmetric
(Kerr) spacetime or the frequencies of a non-slender torus. The non-
axisymmetric $m=1$ vertical epicyclic frequency (given by (56)) for test
particles corresponds to the sum of $\Omega_{0}$ and $\omega_{\theta 0}$. For
a non-slender torus the frequency behaves up to $a\lesssim 0.96$ similarly to
the axisymmetric $\omega_{\theta}$ (compare the top panels of figure 6 to
figure 2). Its form slightly differs only for very high spin values ($a\gtrsim
0.96$) (see the top right panel of figure 7).
The $m=-1$ vertical mode frequency (equation (56)) is for a test particle
given (in absolute value) by the difference between $\Omega_{0}$ and
$\omega_{\theta 0}$. In a Schwarzschild spacetime this difference equals zero
for test particles, but for increased torus thickness the frequency grows
(bottom left panel of figure 6). For a Kerr black hole, the frequency
converges for all spin values to the test particle frequency as the pressure
maximum $r_{0}$ moves away from the black hole. At smaller radii and for
$a\lesssim 0.96$ there is a crossing point with the $\beta=0$ frequency, such
that, in the first interval, rising $\beta$ causes the frequency to increase
in contrast to the second interval where the frequencies show opposite
behaviour (figure 6, bottom right panel). For $a\gtrsim 0.96$ the crossing
point vanishes and rising torus thickness evokes rising frequencies at all
radii (bottom right panel of figure 7).
Poloidal velocity fields of the non-axisymmetric modes are displayed in figure
8. The $m=1$ radial mode velocity field exhibits radial flows originating in
the central regions of the torus and pointing outwards in opposite directions
(top left panel). The $m=-1$ radial mode velocity field has a similar
character, except that the radial flows point inwards (bottom left panel). The
poloidal flow of the $m=1$ vertical mode has analogous features to that in the
axisymmetric case (compare the top right panel of figure 8 to the right panel
of figure 4), while for the $m=-1$ vertical mode the flow is, apart from small
variations, mainly vertical (figure 8, bottom right panel). Once again, the
shape of the velocity fields described here remains preserved for all values
of the black hole spin.
Figure 6: The same as figure 5, but for the vertical epicyclic mode.
Figure 7: The non-axisymmetric $m=1$ (top) and $m=-1$ (bottom) epicyclic mode
frequencies in case of a near-extreme Kerr black hole of $a=0.999$. Left The
radial epicyclic mode. Right The vertical epicyclic mode.
Figure 8: Poloidal velocity fields of the non-axisymmetric epicyclic modes
for the same torus as in figure 4. Top The $m=1$ radial (left) and vertical
(right) epicyclic mode. Bottom The $m=-1$ radial (left) and vertical (right)
epicyclic mode.
Blaes et al (2007) presented the properties of the axisymmetric and non-
axisymmetric radial epicyclic mode frequencies for non-slender tori in a
spherical pseudo-Newtonian potential, likewise based on calculations accurate
to second-order with respect to the torus thickness. Comparing their figures
2, 6, 7 to our figures 1, 2, 3, 5 and 7, one may distinguish the behaviour of
frequencies calculated in a pseudo-Newtonian potential from those calculated
in a fully-relativistic Schwarzschild and Kerr geometry. Our results show that
the radial and vertical mode frequencies in a rotating, axially symmetric
(Kerr) spacetime follow the same trend as those calculated in a non-rotating,
spherically symmetric potential, except that the vertical frequency has in a
rotating spacetime a non-monotonic profile already in the case of test
particles. This non-monotonicity feature is for thicker tori present in both,
axially and spherically symmetric, potentials. Generally, only extremely high
black hole spin values ($a\gtrsim 0.96$) cause slight variations in the
frequency behaviour.
In order to compare the poloidal velocity fields of the axisymmetric modes in
Kerr geometry (our figure 4) to those derived in the pseudo-Newtonian
potential see figure 5 of Blaes et al (2007).
## 6 Conclusions
We have assumed a pressure-supported torus of small radial extent in a Kerr
spacetime that performs epicyclic oscillations and studied the pressure
effects on the epicyclic modes properties, i.e., how the modes eigenfunctions
and eigenfrequencies of a torus differ from those of a free test particle. For
this purpose we calculated the relevant pressure corrections to the
axisymmetric and lowest-order non-axisymmetric epicyclic mode eigenfunctions
and eigenfrequencies within first- (eigenfunctions) and second-order
(eigenfrequencies) accuracy in torus thickness.
In the limit of an infinitely slender torus, the radial and vertical epicyclic
oscillations occur as global oscillations that correspond to purely radial and
vertical displacements of the whole torus at epicyclic frequencies of free
test particles orbiting at the position of the torus pressure maximum
(Abramowicz et al, 2006; Blaes, Arras and Fragile, 2006). Several numerical
studies (e.g. Montero et al, 2004; Rubio-Herrera and Lee, 2005a, b; Šrámková
et al, 2007) have shown that when the torus gets thicker, its (axisymmetric)
oscillations occur at lower frequencies. This has been confirmed recently by
analytic pseudo-Newtonian calculations (Blaes et al, 2007), where pressure
corrections to epicyclic modes of a small-size, constant specific angular
momentum torus in the Paczyński and Wiita (1980) potential were derived. Using
the same approach, but within the framework of general relativity, we extended
their results into the rotating Kerr geometry. For both axisymmetric and the
radial non-axisymmetric oscillations explored in Blaes et al (2007), our
calculations qualitatively confirm the trends as carried out there for a
spherically symmetric potential. As expected (and also demonstrated in Blaes
et al (2007)), the epicyclic mode eigenfunctions of thicker tori no longer
describe a purely radial or vertical displacement since there appear some
deviations of the flow in the poloidal velocity fields for both modes.
The configuration considered here is represented by the idealised model of a
non-selfgravitating, non-accreting, non-magnetised torus with constant
specific angular momentum555Uniform specific angular momentum distribution
throughout the torus is (aside from simplicity reasons) assumed because the
relativistic Papaloizou-Pringle equation for non-constant distributions does
not describe a self-adjoint eigenvalue problem (see equation (26) in Blaes,
Arras and Fragile (2006)), and we do not have the appropriate complete
orthonormal set of eigenfunctions that is necessary to apply the perturbation
method (see also discussion in Blaes et al (2007))., studied within a purely
hydrodynamical regime. These tori are widely known to be dynamically unstable
under global, non-axisymmetric perturbations (Papaloizou & Pringle, 1984).
However, this instability can be suppressed when accretion through the inner
edge of the torus takes place (Blaes, 1987; Blaes and Hawley, 1988). A
possible interpretation is that torus-like structures can be formed within the
innermost regions of a standard, nearly-Keplerian accretion disc where several
physical processes may give rise to pressure gradients that in turn form tori.
Such torus-like accretion flows seem to appear in three-dimensional global MHD
simulations (De Villiers and Hawley et al, 2003; Machida et al, 2006).
Whether purely hydrodynamic modes may exist in the presence of magnetic fields
that surely play a significant role in the physics of accretion flows is an
issue still to be investigated more deeply. A few studies in that context have
been carried out, e.g. by Montero et al (2007) who explored the oscillation
properties of relativistic tori comprising a toroidal magnetic field, and
reported similar results as obtained in previous investigations of non-
magnetised torus oscillations. Their initial set-up, however, did not allow
for development of the magneto-rotational turbulence (MRI). One of the first
attempts to investigate the effects of MRI on the properties of hydrodynamic
oscillation modes via relativistic MHD simulations was done by Fragile (2005).
In general, studying oscillations of black hole accretion discs can improve
our understanding of the origin of the observed X-ray variability. In
particular HF QPOs that are detected in the X-ray light curves of several
X-ray binaries are often attributed to disc oscillations. The results
discussed here can be applied to models that directly involve epicyclic
oscillations. Assuming a particular oscillation model, the identification of
oscillation frequencies with the frequencies of observed QPOs can provide a
precise determination of the mass or spin of the black hole (e.g. Wagoner et
al, 2001; Abramowicz and Kluźniak, 2001; Kato and Fukue, 2006). Black hole
spin estimates for several microquasars have been carried out, based on the
resonance model considering epicyclic oscillations in a thin disc that occur
at frequencies of free test particles (Török et al, 2005). Applying pressure
corrections to epicyclic frequencies, our results should be taken into account
to obtain (more) accurate estimations (Blaes et al, 2007).
This work was supported by the Polish Ministry of Science grant N203 009
31/1466 (OS) and the Czech grant LC06014 (ES). The authors would like to thank
Marek Abramowicz for initiating this work and his invaluable support, as well
as Włodek Kluźniak and Omer Blaes for their kind advice and encouragement. We
also thank Pavel Bakala and Gabriel Török for many useful discussions and
technical help. Then we would like to acknowledge the hospitality of the
Silesian University in Opava and NORDITA in Copenhagen and Stockholm where
most of the work was carried out.
## Appendix
Here we display the coefficients introduced to abbreviate the analytic terms
we derive and use in section 4.
### A.1 The first-order terms
The coefficients that appear in the first-order expansion terms of $A$,
$\bar{\Omega}$, $f$, $\hat{L}$ in subsection 4.2.1 read
$\displaystyle\mathcal{A}_{11}$ $\displaystyle=$
$\displaystyle-2L_{101}\frac{r_{0}^{1/2}(a^{2}-2ar_{0}^{1/2}+r_{0}^{2})}{(r_{0}-a^{2})(r_{0}^{3/2}+a)},$
(57) $\displaystyle\Omega_{11}$ $\displaystyle=$ $\displaystyle-
L_{101}\frac{r_{0}^{2}(2a-3r_{0}^{1/2}+r_{0}^{3/2})}{(r_{0}-a^{2})(r_{0}^{3/2}+a)},$
(58) $\displaystyle f_{11}$ $\displaystyle=$
$\displaystyle\frac{2}{r_{0}^{4}}\left[\frac{r_{0}^{2}}{a^{2}+(r_{0}-2)r_{0}}\right]^{1/2}\left\\{-a^{4}+2a^{3}r_{0}^{1/2}-4a^{2}(r_{0}-1)r_{0}\right.$
(59)
$\displaystyle\left.+2ar_{0}^{3/2}(5r_{0}-6)+r_{0}^{2}[8+(r_{0}-8)r_{0}]\right\\},$
$f_{12}=\frac{2}{r_{0}^{4}}\left[\frac{r_{0}^{2}}{a^{2}-2r_{0}+r_{0}^{2}}\right]^{1/2}\left(3a^{2}-4ar_{0}^{1/2}+r_{0}^{2}\right)\left(a^{2}-2r_{0}+r_{0}^{2}\right),$
(60) $\displaystyle L_{101}$ $\displaystyle=$
$\displaystyle\frac{2(r_{0}-a^{2})}{r_{0}^{2}}\left[\frac{r_{0}^{2}}{a^{2}+r_{0}(r_{0}-2)}\right]^{1/2},$
(61) $\displaystyle L_{102}$ $\displaystyle=$ $\displaystyle-
L_{101}\bar{\omega}_{r0}^{2}+f_{11},$ (62) $\displaystyle L_{103}$
$\displaystyle=$ $\displaystyle-L_{101}\bar{\omega}_{\theta 0}^{2}+f_{12},$
(63) $\displaystyle L_{104}$ $\displaystyle=$
$\displaystyle-2\left[\frac{r_{0}^{2}}{a^{2}+r_{0}(r_{0}-2)}\right]^{-1/2},$
(64) $\displaystyle L_{105}$ $\displaystyle=$ $\displaystyle-
L_{104}\bar{\omega}_{r0}^{2}+f_{11},$ (65) $\displaystyle L_{106}$
$\displaystyle=$ $\displaystyle-L_{104}\bar{\omega}_{\theta 0}^{2}+f_{12},$
(66) $\displaystyle L_{107}$ $\displaystyle=$
$\displaystyle\frac{r_{0}(r_{0}-1)}{(r_{0}-a^{2})}L_{101},$ (67)
$\displaystyle L_{108}$ $\displaystyle=$
$\displaystyle-(2nL_{101}+L_{107})\bar{\omega}_{r0}^{2}+3nf_{11},$ (68)
$\displaystyle L_{109}$ $\displaystyle=$ $\displaystyle-
L_{107}\bar{\omega}_{\theta 0}^{2}+nf_{12},$ (69) $\displaystyle L_{110}$
$\displaystyle=$ $\displaystyle 2n(f_{12}-L_{104}\bar{\omega}_{\theta
0}^{2}).$ (70)
### A.2 The second-order terms
The coefficients in the second-order terms of $A$, $\bar{\Omega}$, $f$,
$\hat{L}$ in subsection 4.2.2 take the following forms
$\displaystyle\mathcal{A}_{21}$ $\displaystyle=$
$\displaystyle\frac{1}{\left[2a+\left(r_{0}-3\right)r_{0}^{1/2}\right]\left[a^{2}+\left(r_{0}-2\right)r_{0}\right]\left(a+r_{0}^{3/2}\right)^{2}\,r_{0}^{3/2}}\times$
$\displaystyle\times\left[5a^{6}+2a^{5}\left(3r_{0}-2\right)r_{0}^{1/2}+a^{4}\left[\left(r_{0}+10\right)r_{0}-64\right]r_{0}\right.$
(71)
$\displaystyle\left.+4a^{3}\left[\left(6r_{0}-23\right)r_{0}+40\right]r_{0}^{3/2}+a^{2}\left\\{\left[\left(6r_{0}-67\right)r_{0}+100\right]r_{0}\right.\right.$
$\displaystyle\left.\left.-108\right\\}r_{0}^{2}+2a\left(r_{0}+16\right)r_{0}^{9/2}+\left(5r_{0}-16\right)r_{0}^{6}\right],$
$\displaystyle\mathcal{A}_{22}$ $\displaystyle=$
$\displaystyle-\frac{5a^{3}+a^{2}(r_{0}-8)r_{0}^{1/2}+3ar_{0}^{2}-r_{0}^{7/2}}{[2a+(r_{0}-3)r_{0}^{1/2}](a+r_{0}^{3/2})r_{0}^{3/2}},$
(72) $\displaystyle\Omega_{21}$ $\displaystyle=$
$\displaystyle\frac{a+(r_{0}-2)r_{0}^{1/2}}{[a^{2}+(r_{0}-2)r_{0}](a+r_{0}^{3/2})^{2}}\left[a^{3}-a^{2}(r_{0}+6)r_{0}^{1/2}\right.$
(73) $\displaystyle\left.+3a(3r_{0}+2)r_{0}+3(r_{0}-4)r_{0}^{5/2}\right],$
$\Omega_{22}=1-\frac{2a}{a+r_{0}^{3/2}},$ (74) $\displaystyle f_{21}$
$\displaystyle=$
$\displaystyle\frac{1}{2\left[2a+(r_{0}-3)r_{0}^{1/2}\right]r_{0}^{4}\left[a^{2}+(r_{0}-2)r_{0}\right]}\times\left[8a^{7}\right.$
(75)
$\displaystyle\left.+4a^{5}r_{0}(3r_{0}+8)+a^{6}r_{0}^{1/2}(4r_{0}-37)+a^{4}r_{0}^{3/2}\left[5r_{0}(2r_{0}-17)+74\right]\right.$
$\displaystyle\left.+16a^{3}r_{0}^{2}\left[r_{0}(3r_{0}-1)-5\right]+4ar_{0}^{3}\left\\{r_{0}[(124-21r_{0})r_{0}-192]+88\right\\}\right.$
$\displaystyle\left.+a^{2}r_{0}^{5/2}\left\\{r_{0}\left[r_{0}(32r_{0}-307)+500\right]-188\right\\}\right.$
$\displaystyle\left.+r_{0}^{7/2}\left(r_{0}\left\\{r_{0}\left[(77-6r_{0})r_{0}-286\right]+380\right\\}-168\right)\right],$
$\displaystyle f_{22}$ $\displaystyle=$
$\displaystyle-\frac{1}{r_{0}^{4}(2a-3r_{0}^{1/2}+r_{0}^{3/2})}\times\left(24a^{5}-85a^{4}r_{0}^{1/2}+72a^{3}r_{0}\right.$
(76)
$\displaystyle\left.+34a^{2}r_{0}^{3/2}+12a^{4}r_{0}^{3/2}-48ar_{0}^{2}+2a^{3}r_{0}^{2}-69a^{2}r_{0}^{5/2}+52ar_{0}^{3}\right.$
$\displaystyle\left.+12r_{0}^{7/2}+11a^{2}r_{0}^{7/2}-6ar_{0}^{4}-14r_{0}^{9/2}+3r_{0}^{11/2}\right),$
$\displaystyle f_{23}$ $\displaystyle=$
$\displaystyle\frac{1}{6r_{0}^{4}\left(2a-3r_{0}^{1/2}+r_{0}^{3/2}\right)}\times\left(24a^{5}-63a^{4}r_{0}^{1/2}+24a^{3}r_{0}+24a^{2}r_{0}^{3/2}\right.$
(77)
$\displaystyle\left.+12a^{4}r_{0}^{3/2}-10a^{2}r_{0}^{5/2}-24ar_{0}^{3}+8ar_{0}^{4}+9r_{0}^{9/2}-4r_{0}^{11/2}\right),$
$\displaystyle L_{201}$ $\displaystyle=$ $\displaystyle
f_{21}+L_{101}f_{11}-L_{202}\bar{\omega}_{r0}^{2},$ (78) $\displaystyle
L_{202}$ $\displaystyle=$ $\displaystyle\frac{3a^{2}-2r_{0}}{r_{0}^{2}},$ (79)
$\displaystyle L_{203}$ $\displaystyle=$ $\displaystyle
f_{22}+L_{101}f_{12}-L_{202}\bar{\omega}_{\theta
0}^{2}-L_{204}\bar{\omega}_{r0}^{2},$ (80) $\displaystyle L_{204}$
$\displaystyle=$ $\displaystyle-\frac{a^{2}}{r_{0}^{2}},$ (81) $\displaystyle
L_{205}$ $\displaystyle=$ $\displaystyle f_{23}-L_{204}\bar{\omega}_{\theta
0}^{2},$ (82) $\displaystyle\vskip 12.0pt plus 4.0pt minus 4.0ptL_{206}$
$\displaystyle=$ $\displaystyle
f_{21}+(L_{101}-L_{107})f_{11}-L_{207}\bar{\omega}_{r0}^{2},$ (83)
$\displaystyle L_{207}$ $\displaystyle=$
$\displaystyle\frac{3[a^{2}+(r_{0}-2)r_{0}]}{r_{0}^{2}},$ (84) $\displaystyle
L_{208}$ $\displaystyle=$ $\displaystyle
f_{22}+(L_{101}-L_{107})f_{12}-L_{207}\bar{\omega}_{\theta
0}^{2}-L_{204}\bar{\omega}_{r0}^{2},$ (85) $\displaystyle L_{209}$
$\displaystyle=$ $\displaystyle f_{23}-L_{204}\bar{\omega}_{\theta 0}^{2},$
(86) $\displaystyle\vskip 12.0pt plus 4.0pt minus 4.0ptL_{210}$
$\displaystyle=$ $\displaystyle
L_{101}\left(3n+\frac{r_{0}^{2}-r_{0}}{r_{0}-a^{2}}\right)f_{11}-\left[\frac{6na^{2}-2r_{0}(r_{0}+2n-2)}{r_{0}^{2}}\right]\bar{\omega}_{r0}^{2}$
(87) $\displaystyle+4nf_{21},$ $\displaystyle L_{211}$ $\displaystyle=$
$\displaystyle-2+\frac{4}{r_{0}},$ (88) $\displaystyle L_{212}$
$\displaystyle=$ $\displaystyle
2nf_{22}+L_{101}\left(n+\frac{r_{0}^{2}-r_{0}}{r_{0}-a^{2}}\right)f_{12}-L_{211}\bar{\omega}_{\theta
0}^{2}-2nL_{204}\bar{\omega}_{r0}^{2},$ (89) $\displaystyle\vskip 12.0pt plus
4.0pt minus 4.0ptL_{213}$ $\displaystyle=$ $\displaystyle
2n\left[f_{22}+(L_{101}-L_{107})f_{12}-L_{207}\bar{\omega}_{\theta
0}^{2}\right]+\bar{\omega}_{r0}^{2},$ (90) $\displaystyle L_{214}$
$\displaystyle=$ $\displaystyle-1,$ (91) $\displaystyle L_{215}$
$\displaystyle=$ $\displaystyle 4nf_{23}-(2nL_{204}-1)\bar{\omega}_{\theta
0}^{2},$ (92) $\displaystyle\vskip 12.0pt plus 4.0pt minus 4.0ptL_{216}$
$\displaystyle=$
$\displaystyle\frac{r_{0}}{a^{2}+r_{0}(r_{0}-2)}\left\\{[r_{0}^{3}+a^{2}(r_{0}+2)]\left(\omega_{i}^{(0)}\Omega_{0}\right)^{2}\right.$
(94)
$\displaystyle\left.-{m^{2}(r_{0}-2)-4ma\omega_{i}^{(0)}\Omega_{0}}\right\\},$
$\displaystyle L_{217}$ $\displaystyle=$ $\displaystyle-
L_{216}\bar{\omega}_{r0}^{2},$ (95) $\displaystyle L_{218}$ $\displaystyle=$
$\displaystyle-L_{216}\bar{\omega}_{\theta 0}^{2}.$ (96)
### A.3 Eigenfunctions-related coefficients
The coefficients $w_{41}$, $w_{42}$, $w_{51}$, $w_{52}$ introduced in table 1
have the form
$\displaystyle w_{41}=-\frac{\bar{\omega}_{r0}^{2}(2\bar{\omega}_{\theta
0}^{2}+2n\bar{\omega}_{\theta 0}^{2}-n\sigma_{4}^{2})}{\bar{\omega}_{\theta
0}^{2}-\bar{\omega}_{r0}^{2}},$ (97) $\displaystyle
w_{42}=\frac{\bar{\omega}_{\theta
0}^{2}(2\bar{\omega}_{r0}^{2}+2n\bar{\omega}_{r0}^{2}-n\sigma_{4}^{2})}{\bar{\omega}_{\theta
0}^{2}-\bar{\omega}_{r0}^{2}},$ (98) $\displaystyle
w_{51}=-\frac{\bar{\omega}_{r0}^{2}(2\bar{\omega}_{\theta
0}^{2}+2n\bar{\omega}_{\theta 0}^{2}-n\sigma_{5}^{2})}{\bar{\omega}_{\theta
0}^{2}-\bar{\omega}_{r0}^{2}},$ (99) $\displaystyle
w_{52}=\frac{\bar{\omega}_{\theta
0}^{2}(2\bar{\omega}_{r0}^{2}+2n\bar{\omega}_{r0}^{2}-n\sigma_{5}^{2})}{\bar{\omega}_{\theta
0}^{2}-\bar{\omega}_{r0}^{2}}.$ (100)
## References
## References
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* Nowak and Lehr (1998) Nowak M A and Lehr D E 1998 Theory of black hole accretion disks ed M A Abramowicz, G Björnsson and J E Pringle (Cambridge University Press) p 233
* Paczyński and Wiita (1980) Paczyński B and Wiita P J 1980 Astron. Astrophys. 88 23
* Papaloizou & Pringle (1984) Papaloizou J C B and Pringle J E 1984 Mon. Not. R. Astron. Soc. 208 721
* Proga & Begelman (2003) Proga D and Begelman 2003 Astrophys. J. 583 69
* Rezzolla et al (2003a) Rezzolla L, Yoshida S and Zanotti O 2003a Mon. Not. R. Astron. Soc. 344 978
* Rezzolla et al (2003b) Rezzolla L, Yoshida S, Maccarone T J and Zanotti O 2003b Mon. Not. R. Astron. Soc. 344 37
* Rubio-Herrera and Lee (2005a) Rubio-Herrera E and Lee W H 2005a Mon. Not. R. Astron. Soc. 357 L31
* Rubio-Herrera and Lee (2005b) Rubio-Herrera E and Lee W H 2005b Mon. Not. R. Astron. Soc. 362 789
* Šrámková et al (2007) Šrámková E, Torkelsson U and Abramowicz M A 2007 Astron. Astrophys. 467 641
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* Török et al (2005) Török G, Abramowicz M A, Kluźniak W and Stuchlík Z 2005 Astron. Astrophys. 436 1
* Török and Stuchlík (2005) Török G and Stuchlík Z 2005 Astron. Astrophys. 437 775
* Wagoner (1999) Wagoner R V 1999 Phys. Rev. 311 259
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|
arxiv-papers
| 2009-01-12T18:40:27 |
2024-09-04T02:48:59.852725
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "O. Straub, E. Sramkova",
"submitter": "Odele Straub",
"url": "https://arxiv.org/abs/0901.1635"
}
|
0901.1734
|
# Order-disorder effect of $A$-site and oxygen-vacancy on magnetic and
transport properties of Y1/4Sr3/4CoO3-δ
Shun Fukushima Tomonori Sato Daisuke Akahoshi Hideki Kuwahara
h-kuwaha@sophia.ac.jp Department of Physics, Sophia University
Chiyoda-ku, Tokyo 102-8554, JAPAN
###### Abstract
We have synthesized $A$-site ordered ($A$O)- and $A$-site disordered
($A$D)-Y1/4Sr3/4CoO3-δ (YSCO) with various oxygen deficiency $\delta$, and
have made a comparative study of the structural and physical properties. We
have found that $A$-site (Y/Sr) ordering produces the unconventional oxygen-
vacancy ordered (OO) structure, and that the magnetic and transport properties
of both $A$O- and $A$D-YSCO strongly depend on the oxygen-vacancy (or excess
oxygen) ordering pattern. $A$O-YSCO with a stoichiometric $\delta$ of 0.5 has
the unconventional OO structure reflecting Y/Sr ordering pattern. With
decreasing $\delta$ from 0.5, the overall averaged OO structure is essentially
unchanged except for an increase of occupancy ratio for the oxygen-vacant
sites. At $\delta=0.34$, excess oxygen atoms are ordered to form a novel
superstructure, which is significant for the room-temperature ferromagnetism
of $A$O-YSCO. In $A$D-YSCO, on the other hand, the quite different OO
structure, which is of a brownmillerite-type, is found only in the vicinity of
$\delta=0.5$.
###### pacs:
75.30.-m, 75.30.Kz, 75.50.Dd
## I Introduction
Transition-metal oxides with perovskite structure, $R_{1-x}Ae_{x}B$O3 ($R$:
rare-earth, $Ae$: alkaline-earth, and $B$: transition-metal), and their
derivatives exhibit rich physical properties such as high-$T_{C}$
superconductivity in Cu oxides and colossal magnetoresistance (CMR) effect in
Mn oxides.Imada_RMP_70 Perovskite-related oxides have been intensively
studied from the viewpoints of not only strongly correlated electron physics
but also potential application for correlated electron devices. In normal
perovskite structures, $R$ and $Ae$ atoms randomly occupy the $A$-sites.
Resultant $A$-site randomness often suppresses electronic phases, resulting in
a large reduction of phase transition temperatures, which makes its practical
application difficult. Therefore, it is required to reduce or remove such
$A$-site randomness for the achievement of its practical application.
Perovskite oxides with $A$-site ordered ($A$O) structures are expected as
promising candidates for correlated electron devices because they are free
from $A$-site randomness. One of the typical examples is a high-$T_{C}$
superconductor YBa2Cu3O7, which has a relatively high superconducting
transition temperature of 90 K among high-$T_{C}$ superconductors. Another
example is $A$O-$R$BaMn2O6, in which the charge- and orbital-ordering
temperature ($T_{\rm CO}=300$-480 K) is much higher than that of conventional
manganites with $A$-site disordered ($A$D) structures.Nakajima_JPCS_63 ;
Nakajima_JPSJ_71 $A$-site cation ordering not only raises the $T_{\rm CO}$
but also gives birth to an anomalous electronic phase that is not found in
$A$D perovskite manganites. In $A$O-$R$BaMn2O6, for example, the charge- and
orbital-ordering pattern and the associated magnetic structure are quite
different from those of $A$D manganites.Arima_PRB_66 ; Williams_PRB_66 ;
Uchida_JPSJ_71 ; Kageyama_JPSJ_72 Furthermore, in $A$O-$R$BaMn2O6, the
electronic phases such as the ferromagnetic metallic, charge- and orbital-
ordered, and $A$-type antiferromagnetic phases compete with each other to form
a multicritical point at $R$ = Nd. $R$/Ba disordering largely modifies the
electronic phase diagram.Akahoshi_PRL_90 ; Nakajima_JPSJ_73 In
$A$D-$R$BaMn2O6, the ferromagnetic metallic phase and charge- and orbital-
ordered one are largely suppressed, and consequently, large phase fluctuation
is enhanced near the original multicritical region ($R$ = Nd). Such large
phase fluctuation is significant for the CMR effect.Akahoshi_PRL_90 ;
Motome_PRL_91
A cobaltite with a new type of an $A$O perovskite structure, Y1-xSrxCoO3-δ has
been recently reported by Istomin et al.Istomin_Chem.Mater._15 and Withers et
al.Withers_JSSC_174 In Fig. 1(d), the crystal structure of $A$O-Y1-xSrxCoO3-δ
($x$ = 3/4, $\delta=0.50$) is displayed. Y and Sr atoms are ordered within the
$ab$-plane, and a CoO6 octahedral layer and a CoO4 tetrahedral layer
alternately stack along the $c$-axis to form four-times periodicity. Oxygen-
vacancies in the CoO4 tetrahedral layers (black spheres in Figs. 1(d) and (e))
are regularly arranged in an unconventional way: four oxygen-vacancies form a
cluster near Y atoms. The oxygen deficiency $\delta$ ($\leq 0.5$) depends on
occupancy ratio only for the oxygen-vacant sites. When the oxygen-vacant sites
are fully occupied, $\delta$ becomes zero. Kobayashi et al. reported that
$A$O-Y1-xSrxCoO3-δ with $0.75\leq x\leq 0.8$ shows a ferromagnetic behavior
above room-temperature.Kobayashi_PRB_72 Ishiwata et al. proposed that
ordering of Co $e_{g}$ orbitals causes the room-temperature
ferromagnetism.Ishiwata_PRB_75 As another characteristic of
$A$O-Y1-xSrxCoO3-δ, the physical properties are susceptible to the variation
of $\delta$. The ground state of $A$O-Y1/3Sr2/3CoO3-δ changes from an
antiferromagnetic insulator ($\delta=0.34$) to a ferromagnetic metal
($\delta=0.30$) with a slight decrease of $\delta$.Maignan_JSSC_178
In this study, we have prepared $A$O- and $A$D-Y1/4Sr3/4CoO3-δ with different
$\delta$, and have made a comparative study of their structural and physical
properties in order to reveal the effect of $A$-site and oxygen-vacancy
arrangement on the physical properties of the perovskite cobaltites.
## II Experiment
Table 1: Annealing conditions, oxygen deficiency $\delta$ determined by
iodometric titration, and corresponding Co valences of Y1/4Sr3/4CoO3-δ (YSCO).
(a) $A$O/OO-YSCO
---
annealing condition | $\delta$ | Co valence
400 atm O2 873 K (10 h) | $0.302(17)$ | 3.15(3)
1 atm O2 773 K (2 h) | $0.340(9)$ | 3.07(2)
Ar 1173 K (12 h) | $0.437(3)$ | 2.88(1)
(b) $A$D/OD-YSCO
annealing condition | $\delta$ | Co valence
400 atm O2 873 K (10 h) | $0.152(6)$ | 3.45(1)
1 atm O2 773 K (2 h) | $0.158(4)$ | 3.43(1)
4% H2 in Ar 423 K (10 min) | $0.206(7)$ | 3.34(1)
4% H2 in Ar 473 K (10 min) | $0.222(7)$ | 3.31(1)
4% H2 in Ar 523 K (10 min) | $0.272(3)$ | 3.21(1)
4% H2 in Ar 573 K (10 min) | $0.319(5)$ | 3.11(1)
(c) $A$D/OO-YSCO
annealing condition | $\delta$ | Co valence
4% H2 in Ar 573 K (6 h) | $0.470(4)$ | 2.81(1)
$A$O- and $A$D-Y1/4Sr3/4CoO3-δ (YSCO) were prepared in a polycrystalline form
by solid state reaction. Mixed powders of Y2O3, SrCO3, and CoO were heated at
1073 K in air with a few intermediate grindings, and sintered at 1423 K in
air. The sintered powder was then treated at 1173 K in Ar to enhance $A$-site
ordering. The resultant product has the $A$O-structure with $\delta=0.44$.
$A$D-YSCO was obtained by rapidly cooling the sintered powder from 1473 (air)
to 77 K (liquid nitrogen). Hereafter, we refer to the thus obtained sample as
RC-YSCO. We controlled $\delta$ of $A$O- and $A$D-YSCO by annealing the
samples under various conditions (Table I). The values of $\delta$ were
determined by iodometric titration. Each powdered sample (ca. 30 mg) was
dissolved in 1 M HCl solution (ca. 50 ml) containing excess aqueous KI. The
amount of the formed I2 was titrated with 0.01 M Na2S2O3 solution using starch
as a colorimetric indicator. The titration was performed more than three times
for each sample. Co valences were determined by assuming that the samples have
a nominal cation ratio. The average values of $\delta$ and the corresponding
Co valences are listed in Table I. Powder X-ray diffraction (XRD) measurements
were carried out at room-temperature using a RIGAKU Rint-2100 diffractometer
with Cu $K\alpha$ radiation. The crystal structures of the samples were
analyzed by Rietveld method using RIETAN-2000.rietan Magnetic and transport
properties were measured from 5 to 350 K using a Quantum Design, Physical
Property Measurement System (PPMS). Magnetization measurements above room-
temperature ($T=300$-800 K) were carried out using a Quantum Design, Magnetic
Property Measurement System (MPMS).
## III Results
### III.1 Crystal structures
Figure 1: (Color online) X-ray-diffraction patterns of (a) $A$-site and
oxygen-vacancy ordered ($A$O/OO)-Y1/4Sr3/4CoO2.66, (b) $A$-site and oxygen-
vacancy disordered ($A$D/OD)-Y1/4Sr3/4CoO2.84, and (c) $A$-site disordered and
oxygen-vacancy ordered ($A$D/OO)-Y1/4Sr3/4CoO2.53. Solid circles and solid
lines represent observed and calculated diffraction profiles, respectively.
Vertical marks under the diffraction profiles indicate calculated peak
positions. The insets show the diffraction profiles at low angles. For
comparison, diffraction profiles of $\delta=0.30$ and 0.44 ($A$O/OO) are also
depicted in the inset of (a). Crystal structure and its idealized oxygen-
vacant layer of (d),(e) $A$O/OO-Y1/4Sr3/4CoO2.5 and (f),(g)
$A$D/OO-Y1/4Sr3/4CoO2.5. Small black spheres of (d)-(g) represent oxygen-
vacancies.
Figure 1(a) shows the XRD pattern of $A$O-YSCO with $\delta=0.34$. It should
be noted that the Bragg peak around $2\theta=11.5$ deg. is a piece of evidence
that both Y/Sr and oxygen-vacancies are ordered. Except for an additional
Bragg peak indicated by an arrow in the inset of Fig. 1(a), the XRD patterns
of $A$O-YSCO with $\delta=0.30$ and 0.44 are similar to that of $\delta=0.34$
($A$O/OO), and all the XRD patterns can be fitted to the $A$O/OO structure
($I4/mmm$) shown in Fig. 1(d). Note that all $A$O/OO-YSCO prepared in this
study, whose $\delta$ is smaller than the stoichiometric composition
($\delta=0.5$), have excess oxygen atoms. The result of the XRD measurement
indicates that these excess oxygen atoms randomly occupy the oxygen-vacant
sites (or raise the occupancy ratio of the site) in the CoO4 layers with
keeping the overall averaged OO structure shown in Figs. 1(d) and (e), that
is, the OO structure is robust against the variation of $\delta$. The
additional Bragg peak observed in $\delta=0.34$ ($A$O/OO) can be indexed as
(1/4 1/4 0) in an $a_{p}\times a_{p}\times a_{p}$ setting ($a_{p}$ denotes a
pseudo-cubic perovskite cell), evidencing the existence of a four-times
superstructure along the [1 1 0] direction. Because the additional peak is
sensitive to $\delta$ and found only in the XRD pattern of $\delta=0.34$
($A$O/OO), it is reasonable to conclude that the superstructure arises from
ordering of the excess oxygen atoms occupying the oxygen-vacant sites. This
means that $A$O-YSCO probably has another oxygen stoichiometry near
$\delta=0.34$ besides the stoichiometric $\delta=0.5$. $\delta=1/3$ (or
$3/8$), which is close to 0.34, is likely to be the second stoichiometric
composition. A similar superlattice peak (1/4 1/4 0) is also observed in
$A$O-Er0.78Sr0.22CoO2.63 by use of a high-intensity synchrotron X-ray source
Ishiwata_PRB_75 , implying that $\delta=0.34$ ($A$O/OO) has the similar oxygen
superstructure to $A$O-Er0.78Sr0.22CoO2.63. The intensity of the superlattice
peak of our sample is much stronger than that of
$A$O-Er0.78Sr0.22CoO2.63fnote_ESCO , indicating that the oxygen deficiency
$\delta$ of our sample is closer to the stoichiometry than that of
$A$O-Er0.78Sr0.22CoO2.63. The detailed crystal structure of $\delta=0.34$
($A$O/OO) is now under investigation.
Then, we exhibit the XRD pattern of RC-YSCO with $\delta=0.16$ in Fig. 1(b).
What is a significant difference from the XRD patterns of $A$O-YSCO system is
that the characteristic Bragg peaks arising from $A$-site and oxygen-vacancy
ordering totally vanish. The XRD pattern can be well fitted to a simple cubic
perovskite structure ($Pm\bar{3}m$), indicating that Y and Sr atoms randomly
occupy the $A$-sites, and that oxygen-vacancies are randomly distributed in
the structure; RC-YSCO has an $A$D and oxygen-vacancy disordered (OD)
structure. All RC-YSCO with $0.15\leq\delta\leq 0.32$ have the same simple
cubic structure as RC-YSCO with $\delta=0.16$. From now on, we will refer to
these compounds as $A$D/OD-YSCO.
On the other hand, the XRD pattern of RC-YSCO with $\delta$ = 0.47 (Fig. 1(c))
is quite different from those of RC-YSCO with $0.15\leq\delta\leq 0.32$. A
characteristic Bragg peak, which implies $A$-site and/or oxygen-vacancy
ordering, is clearly seen around $2\theta=11.5$ deg. By annealing RC-YSCO with
$\delta=0.47$ in O2 at 573 K, which is low enough to prevent $A$-site cations
from moving around to rearrange, the characteristic peak totally disappears,
and the annealed compound takes a simple cubic perovskite structure.
Therefore, we conclude that only oxygen-vacancy ordering contributes to the
appearance of the characteristic peak; RC-YSCO with $\delta=0.47$ has an
$A$D/OO structure. Rietveld analysis reveals that the OO structure of
$A$D-YSCO with $\delta=0.47$ is of a brownmillerite (Ca2(Fe,Al)2O5)-type (Fig.
1(f) and (g), $Ibm2$)Colville_Acta_Cryst_B27 , which is quite different from
that of $A$O/OO-YSCO shown in Figs. 1(d) and (e).
To summarize this section, YSCO can be classified into three types of the
crystal structures: the $A$O/OO, $A$D/OD, and $A$D/OO structures. In the
following section, we demonstrate that the arrangement of Y/Sr and oxygen-
vacancies considerably affects the physical properties of YSCO.
### III.2 $A$-site and oxygen-vacancy ordered ($A$O/OO)-YSCO
Figure 2: (Color online) Temperature dependence of (a) magnetization, (b)
resistivity, and (c) inverse magnetization ($H/M$) of $A$O/OO-Y1/4Sr3/4CoO3-δ.
FC: field cooled.
Figures 2(a) and (b) show the temperature dependence of the magnetization and
resistivity of $A$O/OO-YSCO with $\delta=0.30$, 0.34, and 0.44. The
magnetization of $\delta$ = 0.34 ($A$O/OO) shows a weak ferromagnetic
transition around 330 K as previously reported by Kobayashi et
al.Kobayashi_PRB_72 Ishiwata et al. suggest that the weak ferromagnetism (the
room-temperature ferromagnetism) originates in canted antiferromagnetic or
ferrimagnetic order caused by ordering of Co $e_{g}$ orbital.Ishiwata_PRB_75
Then, the magnetization of $\delta=0.34$ ($A$O/OO) shows a sharp cusp around
300 K, implying that a magnetic or spin-state transition occurs. The
resistivity of $\delta=0.34$ ($A$O/OO) exhibits an anomaly around the weak
ferromagnetic transition temperature, below which it is insulating. Figure
2(c) shows the inverse magnetization of $A$O/OO-YSCO with $\delta=0.30$, 0.34,
and 0.44. All the inverse magnetizations approximately obey the Curie-Weiss
law above 400 K. The Weiss temperature $\theta_{W}$ and effective moment
$P_{\rm eff}$ of $\delta=0.34$ ($A$O/OO) are found to be $-60$ K and 3.3
$\mu_{\rm B}/{\rm Co}$, respectively. The magnetic and transport properties of
$\delta=0.34$ ($A$O/OO) observed in this study are consistent with those
previously reported by Kobayashi et al.Kobayashi_PRB_72 ; Kobayashi_JPSJ_75 ,
except for the magnetic cusp at 300 K. The origin of the cusp will be
discussed later.
Figure 3: (Color online) Magnetic field dependence of magnetization ($M$-$H$
curves) of $A$O/OO-Y1/4Sr3/4CoO3-δ at 5 K.
In $\delta=0.30$ ($A$O/OO), the weak ferromagnetic insulating phase above
room-temperature is considerably suppressed, while the magnetization is
continuously increasing with decreasing temperature, and the ferromagnetic
correlation is larger than that of $\delta=0.34$ ($A$O/OO) below $\sim 100$ K
(Figs. 2(a) and 3). With decreasing $\delta$ from 0.34 to 0.30, the
$\theta_{W}$ increases from $-60$ K to 60 K, that is, an antiferromagnetic
interaction between Co ions turns into a ferromagnetic one. The resistivity of
$\delta=0.30$ ($A$O/OO) largely drops in the whole temperature region compared
with that of $\delta=0.34$ ($A$O/OO). These results show that $A$O-YSCO
approaches a ferromagnetic metal with a decrease of $\delta$, i.e. with an
increase of Co valence, and that the weak ferromagnetic insulating phase
($\delta\approx 0.34$) and ferromagnetic metallic clusters (oxygen rich
region: $\delta\ll 0.34$) coexist in $\delta=0.30$ ($A$O/OO).
On the other hand, the magnetization of $\delta=0.44$ ($A$O/OO), which is
close to the stoichiometric composition of $\delta=0.5$ and has excess oxygen
atoms occupying 12 % of the oxygen-vacant sites, shows a slight increase
around 230 K, suggesting that a weak ferromagnetic transition occurs. However,
as seen from the magnetic field dependence of the magnetization ($M$-$H$
curves) (Fig. 3), the weak ferromagnetic magnetization is much smaller than
that of $\delta=0.34$ ($A$O/OO). The inverse magnetization of $\delta=0.44$
($A$O/OO) shows a clear anomaly around 350 K (Fig. 2(c)), which is attributed
to the remnant of the room-temperature ferromagnetic phase most stabilized
around $\delta=0.34$. These results indicate that the magnetic properties of
$\delta=0.44$ ($A$O/OO) can be explained by the coexistence of the matrix
phase ($\delta=0.5$) and embedded clusters ($\delta\approx 0.34$) due to the
excess oxygen atoms. The weak ferromagnetism below 230 K probably comes from
the matrix phase.
To summarize this section, the room-temperature ferromagnetism is observed
only in $\delta=0.34$ ($A$O/OO), and a slight deviation of $\delta$ from 0.34
strongly suppresses the room-temperature ferromagnetism. The sensitivity of
the room-temperature ferromagnetic phase to $\delta$ also supports our
aforementioned conclusion that $A$O/OO-YSCO has another oxygen stoichiometry
around $\delta=0.34$.
### III.3 $A$-site and oxygen-vacancy disordered ($A$D/OD)-YSCO
Figure 4: (Color online) Temperature dependence of (a) magnetization and (b)
resistivity of $A$D/OD-Y1/4Sr3/4CoO3-δ. The inset of (a) shows the inverse
magnetization of $A$D/OD-Y1/4Sr3/4CoO3-δ ($\delta=0.15$)
We display in Figs. 4(a) and (b) the temperature dependence of the
magnetization and resistivity of $A$D/OD-YSCO with $0.15\leq\delta\leq 0.32$.
In $\delta=0.15$ ($A$D/OD), the resistivity exhibits metallic transport,
except for a slight upturn at low temperatures, and the magnetization abruptly
increases below 160 K. The $\theta_{W}$ ($=190$ K) estimated from the inverse
magnetization (the inset of Fig. 4(a)) is close to the magnetic transition
temperature. The $M$-$H$ curve at 5 K exhibits a ferromagnetic behavior with a
saturation magnetization of 1.2 $\mu_{\rm B}/{\rm Co}$ (Fig. 5). These facts
indicate that $\delta=0.15$ ($A$D/OD) is a typical ferromagnetic metal. Such a
ferromagnetic metallic behavior is sometimes observed in conventional
perovskite cobaltites with $A$D structure such as La1-xSrxCoO3 ($x\geq
0.18$).Itoh_JPSJ_63 ; Senaris_JSSC_118 It is widely accepted that the origin
of the ferromagnetic metallicity of La1-xSrxCoO3 can be explained by the
double exchange interaction.Itoh_JPSJ_63 ; Kriener_PRB_69 ; Saitoh_PRB_56 ;
Yamaguchi_JPSJ_64 The saturation magnetization of $\delta=0.15$ ($A$D/OD) is
close to that of La0.8Sr0.2CoO3 ($\sim 1.3$ $\mu_{\rm B}/{\rm Co}$ at 5 K),
indicating that Co3+ and Co4+ of $\delta=0.15$ ($A$D/OD) in the ferromagnetic
metallic state have similar electronic configurations to those of
La1-xSrxCoO3.Senaris_JSSC_118 ; Kriener_PRB_69 Thus, it is reasonable to
conclude that the ferromagnetic metallic state of $A$D-YSCO with $\delta=0.15$
is stabilized via the double-exchange mechanism.
Figure 5: (Color online) Magnetic field dependence of magnetization ($M$-$H$
curves) of $A$D/OD-Y1/4Sr3/4CoO3-δ at 5 K.
With an increase of $\delta$, the ferromagnetic metallic state is steeply
suppressed (Fig. 4). In $\delta=0.27$ and 0.32 ($A$D/OD), the ferromagnetic
component is negligible as seen from the temperature dependence of the
magnetization and the $M$-$H$ curves, and the resistivity exhibits a typical
insulating behavior (Figs. 4 and 5).
### III.4 $A$-site disordered and oxygen-vacancy ordered ($A$D/OO)-YSCO
Figure 6(a) shows the temperature dependence of the magnetization and
resistivity of $A$D/OO-YSCO with $\delta=0.47$, which has the brownmillerite-
type OO structure as described in section III.1. The magnetization of $A$D/OO-
YSCO with $\delta=0.47$ abruptly increases around 130 K just like that of
$A$D/OD-YSCO with $\delta=0.15$ (Fig. 4(a)). However, the saturation
magnetization of $0.15$ $\mu_{\rm B}/{\rm Co}$ (Fig. 6(b)) is much smaller
than that of $A$D/OD-YSCO with $\delta=0.15$, and the resistivity of $A$D/OO-
YSCO with $\delta=0.47$ shows an insulating behavior in the whole temperature
region. The parent compound of $A$D/OO-YSCO, Sr2Co2O5, which also has the
brownmillerite-type OO structure, undergoes a similar weak ferromagnetic
transition at 200 K and a $G$-type antiferromagnetic transition at 537
K.Munoz_PRB_78 In $A$D/OO-YSCO with $\delta=0.47$, similarly, a $G$-type
anitiferromagnetic transition might occur far above room-
temperature.fnote_HiTemp We note that the weak ferromagnetic moment of
$A$D/OO-YSCO with $\delta=0.47$ is more than 10 times larger than that of
Sr2Co2O5. In $A$D/OO-YSCO with $\delta=0.47$, which is in a mixed valence
state of Co3+/Co2+, a ferrimagnetic transition accompanying charge-order might
occur at 130 K.
Figure 6: (a) Temperature dependence of magnetization and resistivity, and (b)
magnetic field dependence of magnetization ($M$-$H$ curve) of
$A$D/OO-Y1/4Sr3/4CoO3-δ ($\delta=0.47$).
## IV Discussion
Now we discuss the effect of the $A$-site and oxygen-vacancy arrangement on
the physical properties of YSCO. $A$O-YSCO with the stoichiometric composition
of $\delta=0.5$ has the unconventional OO structure, in which, four oxygen-
vacancies in the oxygen deficient CoO2-2δ layers form a cluster near Y atom in
the adjacent Y1/4Sr3/4O layers as shown in Figs. 1(d) and (e). Y/Sr
disordering largely modifies the arrangement of the oxygen-vacancies.
$A$D-YSCO with $\delta=0.5$ has the brownmillerite-type OO structure (Figs.
1(f) and (g)), which is often found in conventional perovskite oxides with
large oxygen deficiency such as Sr2Fe2O5Harder_BM ,
Ca2Fe2O5Colville_Acta_Cryst_B26 , and Sr2Co2O5Takeda_JPSJ_33 . From these
results, it is obvious that Y/Sr ordering gives rise to the unconventional OO
structure reflecting the Y/Sr ordering pattern. The OO structure of $A$O-YSCO
is so robust that excess oxygen atoms partially occupy the oxygen-vacant sites
with keeping the overall OO structure shown in Figs. 1(d) and (e). In
contrast, the OO structure of $A$D-YSCO (Figs. 1(f) and (g)) is so fragile
that it is easily destroyed by introduction of a small amount of excess oxygen
atoms. Consequently, the OO structure is found only in the vicinity of
$\delta=0.5$ in the case of $A$D-YSCO. The robustness of the OO structure of
$A$O-YSCO originates in periodic potential due to $A$-site ordering.
As for the magnetic properties, both $A$O/OO- and $A$D/OO-YSCO with the
stoichiometric $\delta$ (around 0.34 and 0.47 respectively) exhibit the weak
ferromagnetic behaviors as demonstrated in section III. It should be noted
that weak ferromagnetic behaviors are also observed in other oxygen
stoichiometric $A$O perovskites such as YBaCo2O5+x ($x=0.50$ and 0.44) and
YBaMn2O5+x ($x=0.50$), which have the superstructures formed by excess oxygen
atoms.Akahoshi_JSSC_156 ; Karppinen_JSSC_177 Therefore, it is probable that
oxygen-vacancy (or excess oxygen) ordering plays a significant role in the
weak ferromagnetism of both $A$O/OO- and $A$D/OO-YSCO.
Then, we propose one plausible model to explain the origin of the weak
ferromagnetism. In the following discussion, we suppose that the magnetic
interaction between Co ions is antiferromagnetic. Considering a CoO6
octahedron adjacent to a CoO4 tetrahedron (or a CoO5 square pyramid) in oxygen
deficient perovskite structures, an inversion symmetry is absent at the center
between the two Co ions. In such a case, a spin-canted moment is locally
induced through the Dzyaloshinskii-Moriya interaction.Dzyaloshinskii_JPCS_4 ;
Moriya_PhysRev_120 Another possible case is as follows: if the spin states of
CoO6 and CoO4 (or CoO5) were different from each other, the CoO6-CoO4 (or
CoO5) pair could have a local ferrimagnetic moment. In the OO structures, the
CoO6-CoO4 (or CoO5) pairs, i.e., the local net moments are regularly arranged.
As a result, the weak ferromagnetic magnetization is macroscopically observed.
On the other hand, in the OO structure with large oxygen nonstoichiometry
(excess oxygen randomly occupy the oxygen-vacant sites) or in the OD
structures ($\delta=0.27$ and 0.32 ($A$D/OD)) in which, the CoO6-CoO4 (or
CoO5) pairs are randomly distributed, the local net moments oriented randomly
are canceled out with each other. The reason why the weak ferromagnetic moment
of $\delta=0.44$ ($A$O/OO) below 230 K is very small may be attributed to
randomness due to excess oxygen atoms occupying 12 % of the oxygen-vacant
sites. Note that the ferromagnetism of $A$D/OD-YSCO with $\delta=0.15$ is
induced through the double-exchange interaction as described in section III.3.
The magnetization of $\delta=0.34$ ($A$O/OO) we prepared exhibits the sharp
cusp around 300 K (Fig. 2(a)), which is not clearly discerned in $A$O-YSCO
previously reported.Kobayashi_PRB_72 In this study, $A$O-YSCO was treated in
Ar to enhance Y/Sr ordering as mentioned in II, while $A$O-YSCO in the
previous reports was not. As a result, the degree of Y/Sr order of our sample
is higher than that of $A$O-YSCO in the previous reports, which may be the
main cause of the emergence of the cusp. Another plausible explanation for the
emergence of the cusp is that our sample is closer to the oxygen stoichiometry
than $A$O-YSCO in the previous reports.
## V Summary
We have synthesized $A$-site ordered ($A$O)- and $A$-site disordered
($A$D)-Y1/4Sr3/4CoO3-δ (YSCO) with various oxygen deficiency $\delta$, and
have made a comparative study of their structural and physical properties.
$A$O-YSCO with $\delta=0.44$, which is near the stoichiometry of $\delta=0.5$,
has the unconventional oxygen-vacancy ordered (OO) structure reflecting
$A$-site ordering pattern, and shows the weak ferromagnetic behavior below 230
K. Decreasing $\delta$ from 0.5 inducing an increase in the occupancy ratio of
the oxygen-vacant site does not essentially change the overall OO structure.
In $A$O-YSCO with $\delta=0.34$, which is close to another oxygen
stoichiometry, excess oxygen ordering in the oxygen-vacant sites causes the
room-temperature ferromagnetism. On the other hand, in $A$D-YSCO, the
different OO structure, which is of the brownmillerite-type, is found only in
the vicinity of $\delta=0.5$, and also shows the weak ferromagnetic behavior
below 130 K. $A$D/OD-YSCO with $\delta=0.15$ exhibits a ferromagnetic metallic
behavior which is attributed to the double-exchange interaction. With an
increase of $\delta$, that is, with a decrease of Co valence, the
ferromagnetic metallic phase is suppressed. Our main conclusions are as
follows: First, $A$-site ordering gives rise to the unconventional OO
structure, and makes the OO structure robust against randomness due to excess
oxygen atoms. Second, oxygen-vacancy (or excess oxygen) ordering is
indispensable for the weak ferromagnetic behaviors of both $A$O- and
$A$D-YSCO.
###### Acknowledgements.
We thank D. Vieweg and A. Loidl for discussions and their help with the
magnetization measurements using MPMS. This work was partially supported by
Iketani Science and Technology Foundation, the Mazda Foundation, the Asahi
Glass Foundation, and by Grant-in-Aid for Scientific Research (C) from Japan
Society for the Promotion of Science (JSPS).
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* (22) M. Kriener, C. Zobel, A. Reichl, J. Baier, M. Cwik, K. Berggold, H. Kierspel, O. Zabara, A. Freimuth, and T. Lorenz, Phys. Rev. B 69, 094417 (2004).
* (23) S. Yamaguchi, H. Taniguchi, H. Takagi, T. Arima, and Y. Tokura, J. Phys. Soc. Jpn. 64, 1885 (1995).
* (24) T. Saitoh, T. Mizokawa, A. Fujimori, M. Abbate, Y. Takeda, and M. Takano, Phys. Rev. B 56, 1290 (1997).
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* (26) Since $A$D/OO-YSCO is chemically unstable, we could not obtain reliable data of the physical properties at higher temperatures.
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|
arxiv-papers
| 2009-01-13T09:52:56 |
2024-09-04T02:48:59.865303
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shun Fukushima, Tomonori Sato, Daisuke Akahoshi, and Hideki Kuwahara",
"submitter": "Shun Fukushima",
"url": "https://arxiv.org/abs/0901.1734"
}
|
0901.1749
|
# Magnetic and dielectric properties of $A_{2}$CoSi2O7 ($A$=Ca, Sr, Ba)
crystals
M. Akaki J. Tozawa D. Akahoshi and H. Kuwahara Department of Physics,
Sophia University, Tokyo 102-8554, Japan m-akaki@sophia.ac.jp
###### Abstract
We have investigated the magnetic and dielectric properties of $A_{2}$CoSi2O7
($A$=Ca, Sr, and Ba) crystals with a two-dimensional network of CoO4 and SiO4
tetrahedra connected with each other through the corners. In Ca2CoSi2O7, a
weak ferromagnetic transition occurs at 5.7 K, where the dielectric constant
parallel to the $c$ axis shows a concomitant anomaly. The large
magnetocapacitance effect is observed below 5.7 K;
$\Delta\varepsilon(H)/\varepsilon(0)\equiv[\varepsilon(H)-\varepsilon(0)]/\varepsilon(0)$
reaches 13 % at 5.1 K. These results indicate a strong coupling between the
magnetism and dielectricity in Ca2CoSi2O7. Sr2CoSi2O7 shows a similar
magnetoelectric behavior to that of Ca2CoSi2O7. In contrast, in Ba2CoSi2O7,
which has the different arrangement of SiO4 and CoO4 tetrahedra from that of
Ca2CoSi2O7, the magnetocapacitance is hardly observed. The key for the
magnetocapacitance effect of $A_{2}$CoSi2O7 lies in the quasi-two-dimensional
crystal structure.
## 1 Introduction
Since the discovery of the giant magnetoelectric effect in TbMnO3 [1],
multiferroic materials that are both magnetic and dielectric have been
attracting much attention. The mechanism of the magnetic ferroelectricity is
well explained in terms of a spin-current model proposed by Katsura et al [2].
According to this model, the ferroelectricity of multiferroic materials
originates in a spiral spin structure. Therefore, materials with spin
frustration and/or nontrivial spin structures have attracted renewed interest
as promising candidates for new magnetoelectrics.
In this context, we expect $A_{2}$CoSi2O7 ($A$=Ca, Sr, and Ba) as one of such
candidates because $A_{2}$CoSi2O7 is a derivative of Ba2CuGe2O7, which has a
spiral spin structure below 3.26 K [3]. However, the magnetic transition
temperature of Ba2CuGe2O7 is rather low probably because of a spin fluctuation
of Cu2+ ($S$=1/2), which makes detailed measurements of the magnetoelectric
properties difficult. We thus select Co2+ ($S$=3/2) -based compound instead of
Cu-based one in order to raise the magnetic transition temperature. The
crystal structures of Ca2CoSi2O7 (Sr2CoSi2O7) and Ba2CoSi2O7 are shown in
Figs. 1(a) and (b), respectively [4, 5]. As seen from Figs. 1(a) and (b), both
of them have a two-dimensional structure in which SiO4 and CoO4 tetrahedra are
connected through the corners, but the arrangement of SiO4 and CoO4 tetrahedra
is different from each other. The crystal structure consists of an alternate
stack of the two-dimensional CoSi2O7 and Ca2+ (Ba2+) layers along the $c$
($b$) axis. However, little is known about the physical properties of
Ca2CoSi2O7 and Ba2CoSi2O7, and Sr2CoSi2O7 has not been investigated so far. In
this work, we have investigated the magnetic and dielectric properties of
$A_{2}$CoSi2O7.
## 2 Experiment
Single crystalline samples were grown by the floating zone method. Sample
characterization was performed by powder X-ray diffraction measurements at
room temperature. We confirmed that the obtained crystals are of single phase
and that Sr2CoSi2O7 has the same crystal structure as Ca2CoSi2O7. All the
specimens used in this study were cut along the crystallographic principal
axes into a rectangular shape by means of X-ray back-reflection Laue
technique. The magnetic properties were measured using a commercial apparatus
(Quantum Design, Physical Property Measurement System (PPMS)). The dielectric
constant was measured at 100 kHz using an LCR meter (Agilent, 4284A).
## 3 Results and Discussion
Figure 1: The crystal structures of Ca2CoSi2O7 (Sr2CoSi2O7) (tetragonal,
space-group $P\overline{4}2_{1}m$) (a) and Ba2CoSi2O7 (monoclinic, space-group
$C2/c$) (b). Temperature dependence of dielectric constant (c), (e), (g) and
magnetization (d), (f), (h) of $A_{2}$CoSi2O7 crystals. The magnetization
after zero-field-cooling and dielectric constant were measured in warming
scan. Figure 2: Magnetic field dependence of magnetocapacitance of Ca2CoSi2O7
(a), (b) and Sr2CoSi2O7 (c), (d) with different measurement configurations at
several fixed temperatures.
Figure 1 shows the temperature dependence of the dielectric constant and
magnetization of $A_{2}$CoSi2O7. The magnetization ($M_{\perp}$) of Ca2CoSi2O7
shows a jump at 5.7 K when applying magnetic fields perpendicular to the $c$
axis (Fig. 1(d)), indicating that a weak ferromagnetic (WF) transition occurs
at the temperature. On the other hand, no anomaly is found in the
magnetization ($M_{\parallel}$) measured with external magnetic fields
parallel to the $c$ axis. The dielectric constant perpendicular to the $c$
axis ($\varepsilon_{\perp}$) does not show any anomaly, while the dielectric
constant parallel to the $c$ axis ($\varepsilon_{\parallel}$) does a slight
increase below the WF transition temperature ($T_{\rm WF}$) (Fig. 1(c)). The
simultaneous change of the $M_{\perp}$ and $\varepsilon_{\parallel}$ at 5.7 K
implies that the magnetism is coupled with the dielectricity in Ca2CoSi2O7. As
seen from Figs. 1 (e) and (f), the magnetic and dielectric properties of
Sr2CoSi2O7 are similar to those of Ca2CoSi2O7. In Sr2CoSi2O7, the $T_{\rm WF}$
shifts to slightly higher temperature of 7 K, below which the
$\varepsilon_{\parallel}$ shows an abrupt increase compared with that of
Ca2CoSi2O7. On the other hand, the magnetic and dielectric properties of
Ba2CoSi2O7 are quite different from those of Ca2CoSi2O7 and Sr2CoSi2O7. The
magnetization of Ba2CoSi2O7 shows an anomaly at 5 K (Fig. 1(h)), suggesting
that some magnetic transition occurs, while the $\varepsilon_{\perp}$ shows
little change at the temperature (Fig. 1(g)). These results suggest that a
coupling between the magnetization and dielectric constant is very weak in
Ba2CoSi2O7.
In Fig. 2, we show the magnetic field dependence of the magnetocapacitance
($\Delta\varepsilon(H)/\varepsilon(0)\equiv[\varepsilon(H)-\varepsilon(0)]/\varepsilon(0)$)
at several fixed temperatures. Below the $T_{\rm WF}$, the
$\varepsilon_{\perp}$ of Ca2CoSi2O7 strongly depends on magnetic fields
parallel to the $c$ axis, and the large positive magnetocapacitance is
observed (Fig. 2(b)). With decreasing temperature, the peak position of the
magnetocapacitance curves shifts to higher magnetic fields of 5 T (5.5 K) and
8 T (5.1 K), and the magnetocapacitance effect is further enhanced;
$\Delta\varepsilon(H)/\varepsilon(0)$ reaches 13 % at 5.1 K. The
magnetocapacitance of Ca2CoSi2O7 is relatively large compared with those of
other recently discovered multiferroic materials (TbMnO3: 10 % [1], MnWO4: 4 %
[6], LiCu2O2: 0.4 % [7]). The observed large magnetocapacitance effect
provides clear evidence for a strong coupling between the magnetism and
dielectricity in Ca2CoSi2O7. Although Ca2CoSi2O7 does not show spontaneous
ferroelectric polarization in the absence of magnetic fields (not shown),
applying magnetic fields induces electric polarization below the $T_{\rm WF}$.
This is so-called ”magnetic-field-induced pyroelectricity” [8], which has not
been reported so far in other multiferroic materials to our knowledge. The
$\varepsilon_{\parallel}$ of Ca2CoSi2O7 is slightly suppressed by applying
magnetic fields perpendicular to the $c$ axis (Fig. 2(a)), i.e., the negative
magnetocapacitance is found.
In Sr2CoSi2O7, the $\varepsilon_{\parallel}$ depends on magnetic fields
perpendicular to the $c$ axis. The relatively large negative
magnetocapacitance is observed below the $T_{\rm WF}$ (Fig. 2(c));
$\mid\Delta\varepsilon(H)/\varepsilon(0)\mid$ reaches 3.5 % at 2 K. Compared
with the other multiferroic materials without magnetic-field-induced
polarization, the magnetocapacitance of Sr2CoSi2O7 is relatively large
(BiMnO3: 0.5 % [9], TeCuO3: 1.0 % [10], BaCo2Si2O7: 0.2 % [11]). The negative
magnetocapacitance is ascribed to suppression of the $\varepsilon_{\parallel}$
by applying magnetic fields perpendicular to the $c$ axis below the $T_{\rm
WF}$. In contrast to the case of Ca2CoSi2O7, the $\varepsilon_{\perp}$ of
Sr2CoSi2O7 is independent of magnetic fields parallel to the $c$ axis, and
magnetic-field-induced pyroelectricity does not show up.
In Ba2CoSi2O7, the dielectric constant is insensitive to magnetic fields (not
shown), and electric polarization does not appear. This means that the
correlation between the magnetism and dielectricity is almost negligible in
Ba2CoSi2O7. The difference among the magnetoelectric behaviors of Ca2CoSi2O7,
Sr2CoSi2O7 and Ba2CoSi2O7 is probably due to the difference in their two-
dimensional networks of CoO4 and SiO4 tetrahedra. Therefore, further
information on their crystal and magnetic structures are required for a full
understanding of the mechanism of the large magnetocapacitance effect of
$A_{2}$CoSi2O7. Synchrotron X-ray and neutron diffraction measurements are now
in progress.
## 4 Conclusion
In summary, we have investigated the magnetic and dielectric properties of
$A_{2}$CoSi2O7 ($A$=Ca, Sr, and Ba) and have observed the large
magnetocapacitance effect in Ca2CoSi2O7 and Sr2CoSi2O7 crystals. The large
magnetocapacitance effect indicates a strong coupling between the magnetism
and dielectricity in Ca2CoSi2O7 and Sr2CoSi2O7. In contrast, Ba2CoSi2O7 hardly
shows the magnetocapacitance, indicating that a coupling between the magnetism
and dielectricity is almost negligible. The arrangement of CoO4 and SiO4
tetrahedra is significant for the large magnetocapacitance of $A_{2}$CoSi2O7.
## Acknowledgment
This work was supported by Grant-in-Aid for scientific research (C) from the
Japan Society for Promotion of Science.
## References
* [1] Kimura T, Goto T, Shintani H, Ishizaka K, Arima T, and Tokura Y 2003 Nature 426 55
* [2] Katsura H, Nagaosa N, and Balatsky A V 2005 Phys. Rev. Lett. 95 057205
* [3] Zheludev A, Shirane G, Sasago Y, Kiode N, and Uchinokura K 1996 Phys. Rev. B 54 15163
* [4] Hagiya K, Ohmasa M, and Iishi K 1993 Acta Cryst. B 49 172
* [5] Adams R D, Layland R, Payen C, and Datta T 1996 Inorg. Chem. 35 3492
* [6] Taniguchi K, Abe N, Takenobu T, Iwasa Y, and Arima T 2006 Phys. Rev. Lett. 97 097203
* [7] Park S, Choi Y J, Zhang C L, and Cheong S-W 2007 Phys. Rev. Lett. 98 057601
* [8] The detailed results will be reported elsewhere.
* [9] Kimura T, Kawamoto S, Yamada I, Azuma M, Takano M, and Tokura Y 2003 Phys. Rev. B 67 180401(R)
* [10] Lawes G, Ramirez A P, Varma C M, and Subramanian M A 2003 Phys. Rev. Lett. 91 257208
* [11] Akaki M, Nakamura F, Akahoshi D, and Kuwahara H 2008 Physica B 403 1505
|
arxiv-papers
| 2009-01-13T10:25:42 |
2024-09-04T02:48:59.872141
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Akaki, J. Tozawa, D. Akahoshi, and H. Kuwahara",
"submitter": "Mitsuru Akaki",
"url": "https://arxiv.org/abs/0901.1749"
}
|
0901.1882
|
# XMM-Newton and Suzaku analysis of the Fe K complex in the Seyfert 1 galaxy
Mrk 509
G. Ponti1,2,3, M. Cappi2, C. Vignali3, G. Miniutti1,4, F. Tombesi2,3, M.
Dadina2,3, A.C. Fabian5, P. Grandi2, J. Kaastra6,7, P.O. Petrucci8, S.
Bianchi9, G. Matt9, L. Maraschi10 and G. Malaguti2
1APC Université Paris 7 Denis Diderot, 75205 Paris Cedex 13, France
2INAF–IASF Bologna, Via Gobetti 101, I–40129, Bologna, Italy
3Dipartimento di Astronomia, Università di Bologna, Via Ranzani 1, I–40127,
Bologna, Italy
4Laboratorio de Astrofísica Espacial y Física Fundamental (CAB-CSIC-INTA),
Postal Address:
LAEFF, European Space Astronomy Center, P.O. Box 78, E-28691 Villanueva de la
Cañada, Madrid
5Institute of Astronomy, Madingley Road, Cambridge CB3 0HA
6SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA
Utrecht, The Netherlands
7Astronomical Institute, University of Utrecht, Postbus 80000, 3508 TA
Utrecht, The Netherlands
8Laboratoire d’Astrophysique de Grenoble -Université Joseph–Fourier/CNRS UMR
5571 -BP 53, F–38041 Grenoble, France
9Dipartimento di Fisica, Universitá degli Studi Roma Tre, via della Vasca
Navale 84, 00146 Roma, Italy
10INAF/Osservatorio Astronomico di Brera, Via Brera 28, 20121, Milano, Italy
ponti@iasfbo.inaf.it
###### Abstract
We report on partially overlapping XMM-Newton ($\sim$260 ks) and Suzaku
($\sim$100 ks) observations of the iron K band in the nearby, bright Seyfert 1
galaxy Mrk 509. The source shows a resolved neutral Fe K line, most probably
produced in the outer part of the accretion disc. Moreover, the source shows
further emission blue–ward of the 6.4 keV line due to ionized material. This
emission is well reproduced by a broad line produced in the accretion disc,
while it cannot be easily described by scattering or emission from
photo–ionized gas at rest. The summed spectrum of all XMM-Newton observations
shows the presence of a narrow absorption line at 7.3 keV produced by highly
ionized outflowing material. A spectral variability study of the XMM-Newton
data shows an indication for an excess of variability at 6.6–6.7 keV. These
variations may be produced in the red wing of the broad ionized line or by
variation of a further absorption structure. The Suzaku data indicate that the
neutral Fe K$\alpha$ line intensity is consistent with being constant on long
timescales (of a few years) and they also confirm as most likely the
interpretation of the excess blueshifted emission in terms of a broad ionized
Fe line. The average Suzaku spectrum differs from the XMM-Newton one for the
disappearance of the 7.3 keV absorption line and around 6.7 keV, where the
XMM-Newton data alone suggested variability.
###### keywords:
galaxies: individual: Mrk 509 – galaxies: active – galaxies: Seyfert – X-rays:
galaxies
††pagerange: XMM-Newton and Suzaku analysis of the Fe K complex in the Seyfert
1 galaxy Mrk 509–LABEL:lastpage
## 1 Introduction
Deep investigations of the Fe K band in the brightest AGNs allow us to probe
the presence of highly ionized emitting/absorbing components from the
innermost regions around the central black hole. The high–sensitivity X–ray
satellites XMM-Newton and Chandra have shown that the presence of a narrow
core of the lowly ionized Fe K$\alpha$ line is nearly ubiquitous (Yaqoob &
Padhmanaban 2004; Guainazzi et al. 2006; Nandra et al. 2007) and that ionized
components of the line, generally associated with emission from photo– and/or
collisionally– ionized distant gas are also common (NGC 5506, NGC 7213, IC
4329A; Bianchi et al. 2003; Page et al. 2003; Reynolds et al. 2004; Ashton et
al. 2004; Longinotti et al. 2007; see also Bianchi et al. 2002; 2005). The
presence of broad (neutral or ionized) components of Fe K lines can only be
tested via relatively long exposures of the brightest sources (e.g., Guainazzi
et al. 2006; Nandra et al. 2007). Moreover, the observational evidence for
broad lines and their interpretation in terms of relativistic effects may be
questioned when an important absorbing ionized component is present. Spectral
variability studies help in disentangling the different, often degenerate,
spectral components (Ponti et al. 2004; Iwasawa et al. 2004; Ponti et al.
2006; Tombesi et al. 2007; Petrucci et al. 2007; DeMarco et al., in prep.).
Mrk 509 (z=0.034397) is the brightest Seyfert 1 of the hard (2–100 keV) X-ray
sky (Malizia et al. 1999; Revnivtsev et al. 2004; Sazonov et al. 2007) that is
not strongly affected by a warm absorber component (Pounds et al. 2001; Yaqoob
et al. 2003). The HETG Chandra observation confirms the presence of a narrow
component of the Fe K line with an equivalent width (EW) of 50 eV (Yaqoob et
al. 2004). The presence of a second ionized component of the Fe K line at
6.7–6.9 keV has been claimed by Pounds et al. (2001) who fitted it using a
relativistic profile, but Page et al. (2003) showed that the same spectral
feature was consistent also with a simple Compton reflection component from
distant material. The broad–band BeppoSAX spectrum and, in particular, the
soft excess, have been fitted by De Rosa et al. (2004) with a reflection
component from a ionized disc in addition to a neutral reflection component.
Finally, Dadina et al. (2005) found evidence of absorption due to transient,
relativistically red–blue– shifted ionized matter.
Here we present the spectral and variability analysis of the complex Fe K band
of Mrk 509, using the whole set of XMM-Newton and Suzaku observations. The
paper is organized as follows. Section 2 describes the observations and the
data reduction. In Sect. 3 the spectral analysis of the EPIC-pn data of the Fe
K band (using phenomenological models) is presented. In particular in Sect.
3.3, to check for the presence of an absorption line, the EPIC-MOS data have
also been considered. In Sect. 3.4 the spectral variability analysis, within
the XMM-Newton observations, is presented. Sect. 4 describes the spectral
analysis of the Fe K band of the Suzaku summed (XIS0+XIS3) data and the
detailed comparison with the spectrum accumulated during the XMM-Newton
observations. In Sect. 4.1 the HXD-pin data are introduced in order to
estimate the amount of reflection continuum present in the source spectrum.
Finally, a more physically self-consistent fit of the EPIC spectra of all the
EPIC instruments (EPIC-pn plus the two EPIC-MOS) is investigated in Sect. 5.
The results of our analysis are discussed in Sect. 6, followed by conclusions
in Sect. 7.
## 2 Observations and data reduction
Mrk 509 was observed 5 times by XMM-Newton on 2000–10–25, 2001–04–20,
2005–10–16, 2005–10–20 and 2006–04–25. All observations were performed with
the EPIC–pn CCD camera operating in small window observing mode and with the
thin filter applied. The total pn observation time is of about 260 ks. Since
the live–time of the pn CCD in small window mode is 71 per cent, the net
exposure of the summed spectrum is of about 180 ks. The analysis has been made
with the SAS software (version 7.1.0), starting from the ODF files. Single and
double events are selected for the pn data, while only single events are used
for the MOS camera because of a slight pile–up effect. For the pn data we
checked that the results obtained using only single events (that allow a
superior energy resolution) are consistent with those from the MOS, finding
good agreement. The source and background photons are extracted from a region
of 40 arcsec within the same CCD of the source both for the pn and MOS data.
Response matrices were generated using the SAS tasks RMFGEN and ARFGEN.
Suzaku observed Mrk 509 four times on 2006–04–25, 2006–10–14, 2006–11–15 and
2006–11–27. The last XMM-Newton and the first Suzaku observations overlap over
a period of $\sim$ 25 ks. Event files from version 2.0.6.13 of the Suzaku
pipeline processing were used and spectra were extracted using XSELECT.
Response matrices and ancillary response files were generated for each XIS
using XISRMFGEN and XISSIMARFGEN version 2007–05–14. The XIS1 camera data are
not considered here because of the relatively low effective area in the Fe K
energy interval, while the XIS2 is unavailable for observations performed
after November 2006. We used the data obtained during the overlapping interval
to check whether the EPIC pn and MOS data on one hand and the Suzaku XIS0 and
XIS3 data on the other hand are consistent within the inter–calibration
uncertainties. We found an overall good agreement between the data from the
two satellites, the parameters related to the main iron emission features and
the power–law continuum being the same within the errors (except for the XIS2
camera above 8 keV). The total XIS observation time is about 108 ks. The
source and background photons are extracted from a region of 4.3 arcmin within
the same CCD of the source. For the HXD/PIN, instrumental background spectra
and response matrices provided by the HXD instrument team have been used. An
additional component accounting for the CXB has been included in the spectral
fits of the PIN data.
All spectral fits were performed using the Xspec software (version 12.3.0) and
include neutral Galactic absorption (4.2$\times$1020 cm-2; Dickey & Lockman
1990), the energies are rest frame if not specified otherwise, and the errors
are reported at the 90 per cent confidence level for one interesting parameter
(Avni 1976). The sum of the spectra has been performed with the MATHPHA,
ADDRMF and ADDARF tools within the HEASOFT package (version 6.1).
## 3 Fe K band emission of Mrk 509: the XMM-Newton data
The primary goal of this investigation is the study of the Fe K line band;
therefore, in order to avoid the effects of the warm absorber (although not
strong; Yaqoob et al. 2003; Smith et al. 2007) and of the soft excess, we
concentrate on the analysis of the data in the 3.5–10 keV band only. A
detailed study of the warm absorber and its variations will be performed by
Detmers et al. (in prep), we can nevertheless anticipate that the warm
absorber has negligible effect in the Fe K energy band and thus on the results
presented here.
Figure 1: 3.5–10 keV EPIC–pn light curves of the XMM-Newton observations. The
abscissa shows the observation time in seconds. The time between the different
observations is arbitrary. The black, red, green, blue and light blue show the
light curves during the 2000–10–25, 2001–04–20, 2005–10–16, 2005–10–20 and
2006–04–25 observations, respectively.
Figure 1 shows the source light curve in the 3.5–10 keV energy band obtained
from the XMM-Newton pointings. Mrk 509 shows variations of the order of
$\sim$30 per cent over the different observations, while almost no variability
is detected within each observation. Only during the fourth observation the
source shows significant variability, with a mean fractional rms of about
0.04.
We start the analysis of the XMM-Newton data considering the spectra from the
EPIC-pn camera only (including the EPIC-MOS data only when a check of the
significance of a feature is required; see Sect. 3.3). We have fitted a simple
power law model to the 3.5–10 keV data and found that the spectral index
steepens with increasing flux. It goes from 1.54$\pm$0.03 to 1.72$\pm$0.03 for
fluxes of 2.5$\times$10-11 and 3.3$\times$10-11 erg cm-2 s-1, respectively
(3.0$\times$10-11 – 4.3$\times$10-11 erg cm-2 s-1, in the 2–10 keV band). We
firstly phenomenologically fitted the Fe K complex of each single observation
with a series of emission-absorption lines (see also $\S$3.3) and checked that
the results on the parameters of Fe K complex obtained in each observation are
consistent within the errors (not a surprising result in light of the low
statistics of the single spectra and weakness of the ionized features; see
$\S$ 3.4). Hence, we concluded that the continuum variations do not strongly
affect the observed shape of the narrow–band emission/absorption structures in
the Fe K band. Thus, in order to improve the signal–to–noise ratio and thus to
detail the fine structures of the Fe K band, the spectra of all the XMM-Newton
observations have been summed (see $\S$3.4 for the study of the source
spectral variability). The summed mean EPIC–pn spectrum has been grouped in
order to have at least 1000 counts in each data bin. Moreover, this binning
criterion ensures to have at least 30 data–points per keV in the 4–7 keV band,
where the Fe K$\alpha$ complex is expected to contribute. This guarantees a
good sampling of the energy resolution of the instrument and the possibility
of fully exploiting the spectral potentials of the EPIC instruments. Fig. 2
shows the ratio between the data and the best–fit power law. The energy band
used during the fit has been restricted to 3.5–5 and 8–10 keV, in order to
avoid the Fe K band, hence measuring the underlying continuum.
Figure 2: (Upper panel) Observed–frame 3.5–10 keV summed XMM-Newton EPIC–pn
spectrum fitted in the 3.5–5 and 8–10 keV band with a power law. (Panel a)
Data/model ratio. This ratio shows a clear evidence for a neutral Fe K
emission line and further emission from ionized Fe, as well as other
complexities around 7 keV. (Panel b) Data/model ratio when two resolved
emission lines (for the Fe K$\alpha$ and K$\beta$) are included in the
spectral fitting. Strong residuals are still present, indicative of ionized Fe
K emission, while no residual emission redward of the neutral Fe K line
appears. (Panel c) Data/model ratio when a narrow emission line is included in
the model to reproduce the ionized emission. (Panel d) Same as panel c, but
with a single broad emission line instead of a narrow line. In both cases
(panel c and d), an absorption feature is present around 7 keV. (Panel e)
Data/model ratio when an absorption component (modeled using xstar) and a
relativistic ionized line are added to the power law and the emission from
neutral Fe K.
The resulting best–fit power–law continuum has a photon index of 1.63$\pm$0.01
and very well reproduces the source emission ($\chi^{2}$=170.0 for 163 degrees
of freedom, dof) outside the Fe K band. The inclusion of the Fe K band shows
that other components are necessary to reproduce it ($\chi^{2}$=753.9 for 307
dof). The bad statistical result is explained by the presence of clear
spectral complexity in the 6–7 keV band.
### 3.1 The 6.4 keV emission line
Panel a of Fig. 2 shows the clear evidence for a prominent emission line,
consistent with a neutral Fe K$\alpha$ line at 6.4 keV. We therefore added a
Gaussian emission line to the model, obtaining a very significant improvement
of the fit ($\Delta\chi^{2}$=392.1 for the addition of 3 dof). The best–fit
energy of the line is 6.42$\pm$0.02 keV, consistent with emission from neutral
or slightly ionised material. The line has an equivalent width of 69$\pm$8 eV
and is clearly resolved ($\sigma$=0.12$\pm$0.02 keV), as shown by the contour
plot in the left panel of Fig. 3.
Figure 3: (Left panel) Contour plot of the sigma vs. intensity of the neutral
Fe K line. (Right panel) Contour plot of the energy vs. intensity of the
narrow line used to fit the ionized Fe K emission. The narrow line energy is
not consistent with emission from Fe XXV (neither with the forbidden at 6.64
keV, nor with the resonant at 6.7 keV), Fe XXVI or Fe K$\beta$, whose energy
is indicated by the vertical dotted lines (from left to right).
The residuals in panel b of Fig. 2 show no excess redward of this emission
line, which could have been indicative of emission from relativistically
redshifted neutral material.
### 3.2 The ionized Fe K emission line
An excess is, however, present in the range 6.5–7 keV (Fig. 2, panel a). If
modeled with a Fe K$\beta$ component with the expected energy (fixed at 7.06
keV) and forced to have an intensity of 0.15 of the K$\alpha$ (Palmeri et al.
2003a,b; Basko 1978; Molendi et al. 2003) and a width equal to the Fe
K$\alpha$ line (i.e. assuming that the K$\alpha$ and K$\beta$ line originate
from one and the same material), the fit improves significantly
($\Delta$$\chi^{2}$=20.3). Nonetheless, significant residuals are still
present in the 6.5–6.9 keV band (panel b of Fig.2). If this further excess is
modelled with a narrow Gaussian line ($\Delta\chi^{2}$=25 for 2 additional
dof), the feature (EW=12$\pm$4 eV) is found to peak at E=6.86$\pm$0.04 keV
(see panel c of Fig. 2 and right panel of Fig. 3). Thus, the energy centroid
is not consistent with the line being produced by either Fe XXV or Fe XXVI
(right panel of Fig. 3) in a scattering medium distant from the X-rays source
(Bianchi et al. 2002; 2004). The higher energy transition of the Fe XXV
complex is the ”resonant line” expected at 6.7 keV (see e.g. Bianchi et al.
2005). Thus, to save this interpretation, it is required that the photo-
ionized gas has a significant blueshift ($\sim$5700 km/s, if the line is
associated to Fe XXV) or redshift ($\sim$4500 km/s, for Fe XXVI). Then,
instead of fitting the ionized excess with a single line, we fitted it with
two narrow lines forcing their energies to be 6.7 and 6.96 keV. The fit
clearly worsens ($\chi^{2}$=326.7 for 302 dof, corresponding to a
$\Delta$$\chi^{2}=-10.1$ for the same dof). However, if the gas is allowed to
be outflowing, the fit improves ($\Delta$$\chi^{2}$=4.3 for the addition of 1
new parameter; $\chi^{2}$=312.3 for 301 dof; the EW are 8.9 and 12.4 eV for
the Fe XXV and Fe XXVI lines, respectively) as respect to the single narrow
emission line and it results to have a common velocity of
3500${}^{+1900}_{-1200}$ km/s.
Alternatively, the excess could be produced by a single broad line coming from
matter quite close to the source of high–energy photons (in this case the Fe K
emission is composed by Fe K$\alpha$+$\beta$ plus another Fe K line). Leaving
the width of the line free to vary, the fit improves, with $\chi^{2}$=311.1
(panel d Fig. 2) and $\Delta\chi^{2}$ of 5.5, with respect to the single
narrow ionized emission line fit, and $\Delta\chi^{2}$ of 1.3 for the same dof
with respect to the best–fit model with two narrow ionized lines. The
resulting broad ionized Fe K line has EW=23$\pm$9 eV and
$\sigma$=0.14${}^{+0.13}_{-0.08}$ keV. The best–fit energy of the line does
not change significantly (E=6.86${}^{+0.08}_{-0.16}$ keV); however, in this
case the emission is consistent (at the 99 per cent confidence level) with
either Fe XXV or Fe XXVI. Although the statistical improvement is not highly
significant, in the following we will consider that the $\sim$6.8–6.9 keV
excess is indeed associated with a resolved emission line.
### 3.3 Ionized absorption?
The XMM-Newton data also display a narrow absorption feature at E$\sim$7 keV
(observed frame; see Fig. 2, panel d). Since this feature is very close to the
broad excess we just discussed, its significance and intensity are degenerate
with the broad emission–line parameters. In order to gain some insight, we
then fixed the broad emission–line parameters at the best–fit ones obtained
before the addition of a narrow ($\sigma$ fixed at 1 eV) Gaussian absorption
line component. In this case, the line is significant at the $\sim$99 per cent
confidence level (dashed contours of Fig. 4; $\Delta\chi^{2}$=15.5 for 2
additional parameters; see also panel e of Fig. 2). Once the MOS data111 The
shapes of the emission/absorption lines in the MOS instruments appear slightly
narrower, although consistent with the values obtained with the pn instrument.
are added, the significance of this feature increases to 99.9 per cent (solid
contours of Fig. 4), in both cases, of a broad and of a narrow ionized
emission line. The best fit energy and EW of the line are
E=7.28${}^{+0.03}_{-0.02}$ keV and EW=$-$14.9${}^{+5.2}_{-5.5}$ eV,
E=7.33${}^{+0.03}_{-0.04}$ keV and EW=$-$13.1${}^{+5.9}_{-2.9}$ eV, in the pn
alone and in the pn+MOS, respectively.
Figure 4: Superposition of the pn (green), MOS1 (black) and MOS2 (red) summed
spectra of all the XMM-Newton observations. The data are fitted, in the 3.5–5
and 8–10 keV bands with a power law, absorbed by Galactic material. The same
structures are present in the three spectra. In particular, a narrow drop of
emission is present in all the instruments at the same energy (see vertical
dotted line). (Inset panel) Confidence contour plot of the intensity vs.
energy of the narrow unresolved ionized absorption when using the pn data
alone (dashed contours) and including the MOS data as well (solid contours).
The lines indicate the 68.3 (black), 90 (red), 99 (green) and 99.9 (blue) per
cent confidence levels.
### 3.4 Time resolved spectral variability and total rms spectrum
One of the goals of the present analysis is to search for time–variation of
the emission/absorption features of the Fe K complex. To measure possible
variations in the Fe K band, the mean EPIC-pn spectra of each of the 5 XMM-
Newton observations have been studied. The spectra are fitted with the same
model composed by a power law plus three emission lines for the Fe K$\alpha$,
K$\beta$ (with the width fixed at the best–fit value, $\sigma$=0.12 keV) and
the broad ionized Fe K line. The low statistics of the spectra of the single
observations prevents us from the detection of significant spectral
variability of the weak ionised emission/absorption lines. The neutral Fe K
line is better constrained and we find that its EW is anti–correlated with the
level of the continuum, as expected for a constant line.
A different, more sensitive, way to detect an excess of spectral variability
is the total rms function. The upper panel of Fig. 5 displays the shape of the
summed spectrum in the Fe K line band. The lower panel shows the total rms
spectrum (Revnivtsev et al. 1999; Papadakis et al. 2005) calculated with time
bins of $\sim$4.5 ks. The total rms is defined by the formula:
$RMS(E)=\frac{\sqrt{S^{2}(E)-<\sigma^{2}_{err}>}}{\Delta E*arf(E)}$ (1)
where S2 is the source variance in a given energy interval $\Delta$E;
$<\sigma^{2}_{err}>$ is the scatter introduced by the Poissonian noise and arf
is the telescope effective area convolved with the response matrix222The total
rms spectrum provides the intrinsic source spectrum of the variable component.
Nevertheless, we measure the variance as observed through the instrument.
Thus, the sharp features in the source spectrum, as well as the effects of the
features on the effective area, are broadened by the instrumental spectral
resolution. For this reason, to obtain the total rms spectrum, we take into
account the convolution of the effective area with the spectral response..
This function shows the spectrum of the varying component only, in which any
constant component is removed and has been computed by using the different
XMM-Newton observations as if they were contiguous. The total rms spectrum may
be reproduced by a power law with a spectral index of 2.13 ($\chi^{2}$=46.4
for 43 dof). Thus, the variable component is steeper than the observed power
law in the mean spectrum, in agreement with the mentioned observed steepening
of the photon index ($\Gamma$) with flux. The $\Gamma$–flux correlation is
commonly observed in Seyfert galaxies and has been interpreted as being due to
the flux-correlated variations of the power-law slope produced in a corona
above an accretion disc and related to the changes in the input soft seed
photons (e.g. Haardt, Maraschi & Ghisellini 1997; Maraschi & Haardt 1997;
Poutanen & Fabian 1999; Zdziarski et al. 2003). These models predict the
presence of a pivot point, that would correspond to a minimum in the total rms
spectrum. The observation of a perfect power law shape (see Fig. 5) indicates
that the pivot point (if present) has to be outside the 3–10 keV energy band.
On the other hand, the slope -flux behaviour can be explained in terms of a
two-component model (McHardy, Papadakis & Uttley 1998; Shih, Iwasawa & Fabian
2002) in which a constant-slope power law varies in normalization only, while
a harder component remains approximately constant, hardening the spectral
slope at low flux levels only, when it becomes prominent in the hard band. In
this scenario the spectral index of the variable component is equal to the one
of the total rms spectrum, that is $\Gamma$=2.13.
Moreover we note that at the energy of the neutral and ionized Fe K line
components, no excess of variability is present, in agreement with these
components being constant, while an indication for an excess of variability is
present around 6.7 keV. In order to compute the significance of this
variability feature, a narrow Gaussian line has been added to the modelling of
the total rms spectrum. The best–fit energy of the additional line is 6.69
keV, with a $\sigma$ fixed at the instrumental energy resolution, while the
resulting $\Delta\chi^{2}$ is 8.9 for the addition of 2 parameters (that
corresponds to an F–test significance of 98.8 per cent). Introducing the line,
the continuum spectral index steepens to $\Gamma\sim 2.18$. The dashed line in
Fig. 5 highlights the centroid energy of the neutral Fe K$\alpha$ line, while
the dotted line (at $\sim$6.7 keV, rest frame) is placed at the maximum of the
variability excess. This energy corresponds to a drop of emission in the real
spectrum, as we shall discuss in more detail in Section 5.
Figure 5: Lower panel: Total rms variability spectrum of the XMM-Newton
observations. The data (blue crosses) show the spectrum of the variable
component. The best–fit model is a power law with spectral index $\Gamma$=2.18
(red line) plus a Gaussian emission line (improving the fit by
$\Delta\chi^{2}$ of 8.9 for the addition of 2 parameters). The dashed line
highlights the centroid energy of the neutral Fe K$\alpha$ line, while the
dotted line is placed at the maximum of the variability excess, modeled with
the Gaussian emission line. The excess variability energy corresponds to a
drop of emission of the real spectrum.
## 4 The Suzaku view of the Fe K band emission
As mentioned in $\S$2, the source was also observed with Suzaku. The first 25
ks Suzaku observation is simultaneous with the last XMM-Newton pointing. The
source spectra of all the instruments are in very good agreement, during the
simultaneous observation. The spectrum is also consistent with the presence of
the emission and absorption lines, as observed in the mean XMM-Newton
spectrum, nevertheless, due to the low statistics of the 25 ks spectrum and
the weakness of the ionized features, it is not possible to perform a detailed
comparison. Only the presence of the strong Fe K$\alpha$ line can be
investigated, the ionized emission and absorption lines are not constrained in
the 25 ks Suzaku exposure.
Also during the 4 Suzaku pointings, Mrk 509 has shown little variability, with
flux changes lower than 10–15 per cent, hampering any spectral variability
study. Fig. 6 shows the XMM-Newton (black) and Suzaku XIS0+XIS3 (red) summed
mean spectra. The data were fitted, in the 3.5–5 and 7.5–10 keV bands, with a
simple power law and Galactic absorption: the ratio of the data to the best
fit model is shown in Fig. 6. The source emission varied between the XMM-
Newton and the Suzaku observations. The best–fit spectral index and the 3.5–10
keV band fluxes are: $\Gamma$=1.63$\pm$0.01 and $\Gamma$=1.71$\pm$0.02 and
2.63$\times$10-11 and 3.11$\times$10-11 ergs cm-2 s-1, during the XMM-Newton
and Suzaku observations, respectively. The neutral and ionized Fe K emission
lines appear constant, while some differences are present at 6.7 keV, the same
energy where the XMM-Newton data were suggesting an increase of variability.
Other more subtle differences appears at $\sim$ 7 keV, where the absorption
line imprints its presence in the XMM-Newton data only.
Figure 6: XMM-Newton (black) and Suzaku XIS0+XIS3 (red) summed mean spectra.
The data are fitted, in the 3.5–5 and 7.5–10 keV bands, with a simple power
law, absorbed by Galactic material, and the ratio of the data to the best fit
model is shown. The arrows mark absorption features in the spectrum.
The Suzaku spectrum of Mrk 509 shows, in good agreement with the XMM-Newton
one, a resolved neutral Fe K line smoothly joining with a higher energy
excess, most likely due to ionized iron emission (see Fig. 6). Given that no
absorption lines around 6.7 keV or 7.3 keV are present in the Suzaku data, the
spectrum may be useful to infer the properties of the emission lines more
clearly.
The XIS0+XIS3 Suzaku summed spectrum has been fitted in the 3.5–10 keV band
with a power law plus two resolved Gaussian emission lines to reproduce the
emission from Fe K$\alpha$+$\beta$. The parameters of the Fe K$\alpha$ line
are free to vary, while the Fe K$\beta$ ones are constrained as in $\S$3.2.
This fit leaves large residuals ($\chi^{2}$=1379.8 for 1337 dof) in the Fe K
band. In this respect, it is difficult to describe the $>$6.5 keV excess with
a single narrow ionized Fe line (either due to Fe XXV or Fe XXVI). In fact,
although the addition of a narrow line is significant ($\Delta\chi^{2}$=20.1
for 2 more parameters), it leaves residuals in the Fe K band. This remaining
excess can be reproduced ($\Delta\chi^{2}$=5.9 for 1 more parameter), in a
photoionized gas scenario, by a blend of two unresolved ionized lines,
requiring three emission lines to fit the Fe K band (FeK$\alpha$+$\beta$, Fe
XXV and Fe XXVI). In this case, such as in the analysis of the XMM-Newton mean
spectrum, a blueshift of this component is suggested
(v=2600${}^{+2800}_{-2000}$ km s-1). However, the best–fit model (this
scenario is strengthened by the lack of narrow peaks) suggests that the excess
may be in fact associated with a broad ionized Fe line (over which the $\sim$
6.7 keV and $\sim$ 7.3 keV absorption lines are most likely superimposed, but
during the XMM-Newton observation only). In fact considering a broad Fe line
instead of the two narrow lines we obtain an improvement of
$\Delta\chi^{2}$=9.7 for the same dof (see Table 1, model A).
| 3.5–10 keV | BEST–FIT | SPECTRA | | | | | |
---|---|---|---|---|---|---|---|---|---
| Suzaku | | | | | | | |
| $\Gamma$ | pl norma | ENeut. | $\sigma$Neut. | ANeut.b (EW)c | EIon. | $\sigma$Ion./rin | AIon.b (EW)c | $\chi^{2}$/dof
| | | keV | keV | | keV | keV/rg | |
A | 1.72$\pm$0.02 | 1.12$\pm$0.02 | 6.42$\pm$0.03 | $<$0.06 | 1.7$\pm$0.5 (32) | 6.54$\pm$0.09 | 0.40$\pm$0.1 | 4.6$\pm$1.2 (90) | 1344/1340
B | 1.72$\pm$0.02 | 1.12$\pm$0.02 | 6.42$\pm$0.02 | $<$0.07 | 2.1$\pm$0.5 (40) | 6.61$\pm$0.08 | 24$\pm$10 | 3.4$\pm$0.8 (79) | 1346/1340
| Self-consistent model | | | | | | | |
| XMM-Newton | | | | | | | |
| $\Gamma$ | pl norma | ENeut. | $\sigma$Neut. | ANeut.b | Incl. | $\xi$ | ARefl.Ion.b |
| | | keV | keV | | deg | erg cm s-1 | |
C | 1.70$\pm$0.01 | 0.92$\pm$0.04 | 6.41$\pm$0.01 | 0.07$\pm$0.01 | 2.2$\pm$0.3 | 47$\pm$2 | 11${}^{+200}_{-7}$ | 0.9${}^{+3.0}_{-0.5}$ |
| NHd | log($\xi$) | z | $\chi^{2}$/dof | | | | |
| 5.8${}^{+5.2}_{-4.8}$ | 5.15${}^{+1.25}_{-0.52}$ | $-$0.0484${}^{+0.012}_{-0.013}$ | 894.3/876 | | | | |
Table 1: Top panel: Best–fit values of the summed spectra (XIS0+XIS3) of all
Suzaku observations fitted in the 3.5–10 keV band. Both model A and B include
a power law and two Gaussian lines K$\alpha$+$\beta$ to fit the 6.4 keV
excess. In addition to this baseline model, either another Gaussian component
(Model A) or a DISKLINE profile (Model B) have been added to reproduce the
ionized line, respectively. In Table the best–fit power law spectral index
($\Gamma$) and normalization as well as the Fe K$\alpha$ energy, width and
normalization are reported for model A and B. The energy, width and
normalization are reported when a Gaussian profile for the ionized Fe K line
is considered (Model A), while the best fit energy, inner radius and
normalization are presented when a DISKLINE profile is fitted (Model B).
Standard disc reflectivity index, outer disc radius and disc inclination of
$\alpha=3$, r${}_{out}=400$ rg and 30∘ have been assumed for the relativistic
profile. Bottom panel: Best fit results of the summed XMM-Newton EPIC-pn and
EPIC-MOS data of Mrk 509 (fitted in the 3.5–10 keV band). The model
(wabs*zxipcf*(pow+zgaus+zgaus+pexrav+kdblur*(reflion))) consists of: i) a
power law; ii) two Gaussian emission lines for the Fe K$\alpha$ and K$\beta$
emission (this latter has energy is fixed to the expected value, 7.06 keV,
intensity and width tied to the K$\alpha$ values); iii) a neutral reflection
continuum component (pexrav in Xspec) with $R$=1 (value broadly consistent
with the pin constraints and the values previously observed; De Rosa et al.
2004), Solar abundance and high energy cut off of the illuminating power law
at 100 keV; iv) a ionized disc reflection spectrum (reflion model; Ross &
Fabian 2005) with the disc inner and outer radii and the emissivity of 6, 400
rg and $-3$, respectively. The best fit disc inclination and ionization and
the normalization of the disc reflection component are shown; v) an ionized
absorption component (zxipcf) totally covering the nuclear source. The best
fit column density, ionization parameter and outflow velocity are reported. a)
In units of 10-2 photons keV-1 cm-2 s-1 at 1 keV; b) In units of 10-5 photons
cm-2 s-1; c) In units of eV; d) In units of 1022 atoms cm-2.
Thus, the Suzaku data indicate that the broad excess at 6.5–6.6 keV is indeed
due to a broad line rather than a blend of narrow ionized Fe lines. Since
broad lines may arise because of relativistic effects in the inner regions of
the accretion flow, we tested this hypothesis by fitting the excess at 6.5–6.6
keV with a diskline profile. The statistics of the spectrum is not such to
allow us to constrain all the parameters of the ionized diskline model. Thus,
the disc reflectivity index has been fixed at the standard value ($\alpha=-3$,
where the emissivity is proportional to $r^{\alpha}$), the outer disc radius
and inclinations to 400 gravitational radii (rg) and 30∘, respectively. The
broad line is consistent with being produced in the accretion disc (Table 1,
Model B); however, the emission from the innermost part of the disc is not
required, the lower limit on the inner disc radius being 10–15 rg. As clear
from Fig. 6, the Suzaku data do not require any ionized Fe K absorption
structures.
In order to quantify the differences between the Suzaku and XMM-Newton spectra
(and, in particular, the reality of the absorption structures at 6.7 and 7.3
keV appearing in the XMM-Newton spectrum only) we fixed all the parameters of
the Suzaku model (apart from the intensity and spectral index of the direct
power law) and fit the XMM-Newton data with that model. This corresponds to
assuming that the intrinsic line shapes do not vary between the two
observations. Then, a narrow Gaussian line has been added to the XMM-Newton
model to estimate the significance of the putative absorption structures. The
improvement in the spectral fitting is evident, as indicated by the
$\Delta\chi^{2}$=28.3 and 22 in the case of a line at E=6.72$\pm$0.04 keV and
E=7.29$\pm$0.04 keV, respectively. The presence of these spectral features
only in the XMM-Newton observations is thus indicative of variability at
energies $\sim$6.6–6.7 and $\sim$7.3 keV.
### 4.1 The Suzaku pin data to constrain the reflection fraction
We add the pin data to measure the amount of reflection continuum. We note
that the pin data provide a good quality spectrum up to 50 keV. The model used
involves a direct power law plus a neutral reflection component ($pexrav$
model in $Xspec$; Magdziarz & Zdziarski 1995) plus the Fe K$\alpha$+$\beta$
resolved lines and a broad (DISKLINE) component of the line. As for model B we
fix some of the parameters of the DISKLINE profile (disc inclination=30∘,
rout=400 rg and $\alpha$=-3). Moreover we assume a high–energy cut off of 100
keV and Solar abundance. Thus, by fitting the 3–50 keV band data, we obtained
a reflection fraction $R=0.4^{+0.6}_{-0.2}$ and a spectral index
$\Gamma$=1.76${}^{+0.12}_{-0.03}$. The total EW of the emission lines above
the reflected continuum (about 1.2 keV) is broadly consistent with the
theoretical expectations (Matt et al. 1996) and with what observed in Compton
thick Seyfert 2 galaxies, where the primary continuum is absorbed and only the
reflection is observed. Nevertheless, also for this source, as already known
from previous studies (Zdziarski et al. 1999), we observe that the spectral
index and the reflection fraction are degenerate and strongly depend on the
energy band considered. In fact, if the 2–10 keV band is considered, the
reflection fraction increases, resulting to be $R$=1.1${}^{+0.2}_{-0.5}$ and
the power–law photon index of $\Gamma$=1.88${}^{+0.03}_{-0.02}$. The total EW
of the Fe emission lines above the reflected continuum are about 750 eV. Again
these values are broadly in agreement with expectations (Matt et al. 1996).
## 5 A physically self-consistent fit: Possible origin of the spectral
features
The analysis of the XMM-Newton and Suzaku data shows evidence for the presence
of: i) a resolved, although not very broad, ($\sigma\sim$0.12 keV) neutral Fe
K$\alpha$ line and associated Fe K$\beta$ emission; ii) an ionized Fe K
emission line inconsistent with emission from a distant scattering material at
rest and most likely produced in the accretion disc; iii) an absorption line
at $\sim$7.3 keV, present in the summed spectrum of all XMM-Newton
observations only; iv) an indication for an enhancement of variability - both
by considering the XMM-Newton data alone and by comparison between the two
data sets - at $\sim$6.7 keV that could be either due to the high variability
of the red wing of the broad ionized Fe K line, possibly associated with a
variation of the ionisation of the disc, or to a second ionized absorption
line.
These emission/absorption components are partially inter–connected to each
other given the limited CCD resolution onboard XMM-Newton and Suzaku. Thus we
re–fit the XMM-Newton (both the pn and MOS in the 3.5–10 keV energy band) data
with a model containing components that better describe the physical processes
occurring in the AGN. In particular, we consider two Gaussian lines for the Fe
K$\alpha$ and K$\beta$ emission plus a neutral reflection component (pexrav in
xspec) with a reflection fraction $R=1$ (consistent with the constraints given
by the Suzaku pin data). The Fe K$\alpha$ line has an equivalent width of 1
keV above the reflection continuum. Moreover, we fit the broad ionized Fe K
line with a fully self–consistent relativistic ionised disc reflection
component (reflion model in Xspec; Ross & Fabian 2005, convolved with a LAOR
kernel; KDBLUR in Xspec).
The statistics prevents us from constraining the parameters of the
relativistic profile. Standard values for the relativistic profile are
assumed, with the disc inner and outer radii and the emissivity of 6, 400 rg,
and $-$3, respectively. Finally, the $\sim$7.3 keV absorption line has been
fitted with a photoionised absorption model (zxipcf model in Xspec; Miller et
al. 2007; Reeves et al. 2008; Model C, Table 1), assuming a total covering
factor.
Table 1 shows the best–fit parameters. Once the presence of the reflection
continuum is taken into account, the power law slope becomes steeper
($\Gamma$=1.70$\pm$0.01, $\Delta\Gamma\sim$0.07) as compared to the fit with a
simple power law and emission absorption lines (see §3). The best fit energy
of the neutral Fe K$\alpha$ line is E=6.41$\pm$0.01 keV, consistent with being
produced by neutral material, and results to be narrower
($\sigma$=0.07$\pm$0.01 keV) than in the previous fits. The ionized emission
line is fitted with a ionized disc reflection model. The only free parameters
of such a component are the inclination and ionisation parameter of the disc
that result to be 47$\pm$2∘ and $\xi$=11${}^{+200}_{-7}$ erg cm s-1 (Model C,
Table 1). The material producing the 7.3 keV absorption feature in the XMM-
Newton data has to be highly ionized, as also indicated by the absence of a
strong continuum curvature. In fact, the best ionization parameter is
log($\xi$)=5.15${}^{+1.25}_{-0.52}$ and the column density
$N_{H}=5.8^{+5.2}_{-4.8}\times 10^{22}$ cm-2. Nevertheless the observed energy
of the absorption feature does not correspond to any strong absorption
features, thus there is evidence for this absorption component to be
outflowing with a shift $v=-0.0484^{+0.012}_{-0.013}$ c ($\sim
14000^{+3600}_{-4200}$ km s-1). The resulting $\chi^{2}$ is 894.3 for 876 dof.
## 6 Discussion
This study clearly shows that long exposures are need to disentangle the
different emitting/absorbing components contributing to the shape–variability
of the Fe K complex in Seyfert galaxies. Here we discuss the origin of both
neutral and ionized emission and absorption Fe lines in Mrk 509 which allow to
have insights in the innermost regions of the accretion flow.
### 6.1 Neutral/lowly ionized Fe emission line
Once the broad ionized line is fitted, the width of the Fe K$\alpha$ line
lowers to a value of 72$\pm$11 eV (see Fig. 3) that corresponds to a FWHM(Fe
K$\alpha$)=8000$\pm$1300 km s-1 (see Model C, Table 1). This value is slightly
higher than that measured by Yaqoob & Padmanabhan with a $\sim$50 ks HETG
Chandra observation (2820${}^{+2680}_{-2800}$ km s-1). The FWHM of the Fe
K$\alpha$ line is larger than the width of the H$\beta$ line
(FWHM(H$\beta$)=3430$\pm$240 km s-1; Peterson et al. 2004; Marziani et al.
2003), indicating that the Fe line is produced closer to the center than the
optical BLR and, of course, than the torus postulated in unified models; we
note that a wide range of FWHM values is observed for the BLR and the Fe K
lines in local Seyfert galaxies (Nandra 2006). However, the UV and soft X-ray
spectra of Mrk 509 show evidence for the presence of broad emission lines with
FWHM of 11000 km s-1 (Kriss et al. 2000). The origin of these UV and soft–X
lines is still highly debated, nevertheless they may indicate that the BLR
region is stratified, i.e. that these lines are not produced in the optical
BLR but in the inner part of a stratified BLR region (see also Kaastra et al.
2002; Costantini et al. 2007), possibly as close as 2000 rg from the center
(about 0.012 pc, being the mass of the black hole in Mrk 509
M${}_{BH}\simeq$1.43$\pm$0.12$\times$108 M⊙ Peterson et al. 2004; Marziani et
al. 2003). Nevertheless, if the line is produced in the innermost part of a
stratified BLR, it would require either a higher covering fraction or a higher
column density than generally derived from the optical and ultraviolet bands.
Simulations by Leahy & Creighton (1993) show that about 70 per cent of the
sky, as seen by the central source, has to be covered in order to produce the
Fe K$\alpha$ line, if the broad line clouds have column densities of about
1023 cm-2, while the typical values for the BLR clouds covering fractions are
of the order of 10–25 per cent (Davidson & Netzer 1979; Goad & Koratkar 1998).
Alternatively, the Fe K$\alpha$ line may be produced by reflection by the
outer part of the accretion disc.
### 6.2 Ionized Fe emission lines
The spectrum of Mrk 509 shows emission from ionized iron, consistent with
either Fe XXV or Fe XXVI, implying photoionized gas outflowing or inflowing
respectively. Alternatively, the ionized Fe K emission may be produced by
reflection from the inner part of the accretion disc.
In fact, both the XMM-Newton and the Suzaku data are consistent with the two
scenarios, even if a slightly better fit ($\Delta\chi^{2}$=5.5 and 9.7 for
XMM-Newton and Suzaku, respectively) is obtained in the case of broad line.
Moreover in the case of narrow emission lines the emitting gas should have a
significant outflow (for Fe XXV, v$\sim$3500 and 2600 km s-1 for XMM-Newton
and Suzaku, respectively) or inflow (for Fe XXVI, v$\sim$4500 km s-1) with
velocities higher than what generally observed (Reynolds et al. 2004;
Longinotti et al. 2007; but see also Bianchi et al. 2008 that detect an
outflow of v=900${}^{+500}_{-700}$ km s-1). On the other hand, the high
radiative efficiency of the source ($\eta$=0.12; Woo & Urry 2002) suggests
that the accretion disc is stable down to the innermost regions around the BH,
where the reflection component should be shaped by relativistic effects. For
these reasons, although an outflowing emitting gas is not excluded, the broad
line interpretation seems favoured. In fact, the profile of the line is
compatible with being shaped by relativistic effects, consistent with its
origin being in the surface of an accretion disc, in vicinity of a black hole.
Nevertheless, the width of the line is not a compelling evidence. The observed
broadening of the line can be reproduced also with the Comptonization process
occurring in the upper layer of the ionized accretion disc. Moreover, we
stress that the main evidences for the presence of a broad Fe K line comes
from the mean summed spectrum. The process of summing spectra, although is a
powerful way to extract information, might be dangerous in presence of
spectral variability and when applied to observations taken many years apart.
Thus, the final answer on the origin of these ionized lines will be obtained
with either a higher resolution observation or with significantly longer XMM-
Newton exposures.
### 6.3 Ionized Fe absorption lines
The XMM-Newton data indicate the presence of a highly ionized absorption
component, the best fit column density being
NH=5.8${}^{+5.2}_{-4.8}\times$1022 cm-2 and ionization
log($\xi$)=5.15${}^{+1.25}_{-0.52}$. Moreover, fitting the absorption with
this model, it results that the absorber has to be blueshifted by
0.0484${}^{+0.012}_{-0.013}$ c. The blueshift corresponds to an outflow
velocity of $\sim$14000 km s-1. The structure implies a significant blueshift
if the absorber is located in the core of Mrk 509 but, considering the
systemic velocity of the galaxy, its energy is also consistent with a local
absorber (McKernan et al. 2004; 2005; Risaliti et al. 2005; Young et al. 2005;
Miniutti et al. 2007; but see also Reeves et al. 2008). Nevertheless, the
observed variability between the XMM-Newton and Suzaku observations points
towards an origin within Mrk 509.
An hint of variability is observed around 6.7 keV both in the XMM-Newton data
and by comparing the XMM-Newton and Suzaku spectra. This could be due in
principle to variability in the red wing of the ionized emission line.
However, the total rms spectrum shows a peak of variability that is consistent
with being narrow, thus it may suggest an alternative explanation. Indeed, the
observed difference between the XMM-Newton and Suzaku Fe K line shapes could
be due to a further ionized absorption component, present only the XMM-Newton
observations, with a column density NH=5.4${}^{+4.8}_{-4.4}\times$1021 cm-2
and ionization parameter log($\xi$)=2.04${}^{+0.43}_{-0.60}$. When the
structure at 6.7 keV is fitted with such a component, an absorption structure
appears around 7.3 keV, nevertheless its equivalent width is not strong enough
to reproduce the total absorption feature; moreover, it appears at slightly
different energy, not completely fitting the $\sim$7.3 keV line. Thus, the
absorption structures at 6.7 and the one at 7.3 keV may be connected and they
may be indicative of another absorption screen. If this further lower
ionization absorption component is present, different absorption feature would
be expected (due to the low ionization and high column density) at lower
energies. Smith et al. (2007) analyzed the RGS data and detected two
absorption components with physical parameters similar
(log($\xi$)=2.14${}^{+0.19}_{-0.12}$ and 3.26${}^{+0.18}_{-0.27}$;
NH=0.75${}^{+0.19}_{-0.11}$ and 5.5${}^{+1.3}_{-1.4}\times 10^{21}$ cm-2) to
the ones that we infer, strengthening this interpretation. There is also
evidence for another, higher ionization, mildly relativistic, and variable
ionized component in the XMM data. The study of this more extreme component is
addressed in another paper (Cappi et al., in preparation).
The observation of highly ionized matter in the core of Mrk 509 is in line
with its high BH mass and accretion rate. In fact, we remind that at the
Eddington limit the radiation pressure equals the gravitational pull, however
the densities of the matter lowers with the BH mass (Shakura & Sunyaev 1976).
Thus the ionization of the material surrounding high accretion rate and BH
mass AGNs, such as Mrk 509, should be higher than normal. We stress, however,
that in order to detail the physical parameters of the ionised
emitter/absorber, further long observations are required.
## 7 Conclusions
The Fe K band of Mrk 509 shows a rich variety of emission/absorption
components. The XMM-Newton and Suzaku data shows evidence for the presence of:
* •
a resolved, although not very broad, ($\sigma\sim$0.07 keV) neutral Fe
K$\alpha$ line and associated Fe K$\beta$ emission. The width of the line
suggests that the 6.4 keV line is produced in the outer part of the accretion
disc (the broad line region or torus emission seem unlikely). The measured
reflection fraction is consistent in this case with the intensity of the line,
while a covering factor or column density higher than generally observed would
be required if the line were produced in the BLR or the torus;
* •
both the Suzaku and the XMM-Newton data show an excess due to ionized Fe K
emission. Both datasets show a superior fit when a broad ionized line coming
from the central parts of the accretion disc is considered. The data are
inconsistent with narrow emission from a distant scattering material at rest,
while it can not be excluded if the gas is outflowing (v$\sim$3500 km s-1)
* •
both EPIC–pn and MOS data show an absorption line at $\sim$7.3 keV, present in
the summed spectrum of all XMM-Newton observations only. This component
confirms the presence of highly ionized, outflowing (v$\sim$14000 s-1), gas
along the line of sight. The comparison between XMM-Newton and Suzaku suggests
a variability of this component;
* •
a hint of an enhancement of variability - both by considering the XMM-Newton
data alone and by comparison between the two data sets - at $\sim$6.7 keV that
could be either due to the high variability of the red wing of the broad
ionized Fe K line, possibly associated with a variation of the ionisation of
the disc, or to a second ionized absorption line.
## Acknowledgments
This paper is based on observations obtained with XMM-Newton, an ESA science
mission with instruments and contributions directly funded by ESA Member
States and NASA. This work was partly supported by the ANR under grant number
ANR-06-JCJC-0047. GP, CV and SB thank for support the Italian Space Agency
(contracts ASI–INAF I/023/05/0 and ASI I/088/06/0). GM acknowledge funding
from Ministerio de Ciencia e Innovación through a Ramón y Cajal contract. GP
thanks Regis Terrier, Andrea Goldwurm and Fabio Mattana for useful discussion.
We would like to thank the anonymous referee for the detailed reading of the
manuscript and for the comments that greatly improved the readability of the
paper.
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|
arxiv-papers
| 2009-01-13T22:28:01 |
2024-09-04T02:48:59.881091
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. Ponti (1,2,3), M. Cappi (2), C. Vignali (3), G. Miniutti (1,4), F.\n Tombesi (2,3), M. Dadina (2,3), A.C. Fabian (5), P. Grandi (2), J. Kaastra\n (6,7), P.O. Petrucci (8), S. Bianchi (9), G. Matt (9), L. Maraschi (10), G.\n Malaguti (2) ((1) Laboratoire APC, Paris, (2) INAF-IASF Bologna, (3)\n Dipartimento di Astronomia, Universita' di Bologna, (4) LAEFF, Madrid, (5)\n Institute of Astronomy, Cambridge, (6) SRON, Utrecht, (7) Astronomical\n Institute, University of Utrecht, (8) Laboratoire d'Astrophysique de\n Grenoble, (9) Dipartimento di Fisica, Universita' degli Studi Roma Tre, (10)\n INAF/Osservatorio Astronomico di Brera, Milano)",
"submitter": "Gabriele Ponti",
"url": "https://arxiv.org/abs/0901.1882"
}
|
0901.1924
|
arxiv-papers
| 2009-01-14T03:41:37 |
2024-09-04T02:48:59.892278
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhenhai Jing, Baoming Bai, Xiao Ma, Ying Li",
"submitter": "Zhen-hai Jing",
"url": "https://arxiv.org/abs/0901.1924"
}
|
|
0901.2189
|
# Raman Scattered He II $\lambda$ 6545 in the Young and Compact Planetary
Nebula NGC 6790
Eun-Ha Kang1, Byeong-Cheol Lee2 & Hee-Won Lee1 1 Department of Astronomy and
Space Science, Astrophysical Research Center for the Structure and Evolution
of the Cosmos, Sejong University, Seoul, 143-747, Korea
2 Department of Astronomy and Atmospheric Sciences, Kyungpook National
University hwlee@sejong.ac.kr
###### Abstract
We present the high resolution spectra of the young and compact planetary
nebula NGC 6790 obtained with the echelle spectrograph at Bohyunsan Optical
Astronomy Observatory and report the discovery of Raman scattered He II
$\lambda$ 6545 in this object. This line feature is formed in a thick neutral
region surrounding the hot central star, where He II$\lambda$ 1025 line
photons are scattered inelastically by hydrogen atoms. A Monte Carlo technique
is adopted to compute the line profiles with a simple geometric model, in
which the neutral region is in the form of a cylindrical shell that is
expanding from the central star. From our line profile analysis, the expansion
velocity of the H I region lies in the range $v_{exp}=15-19{\rm\ km\ s^{-1}}$.
Less stringent constraints are put on the H I column density $N_{HI}$ and
covering factor $C$, where the total flux of Raman He II$\lambda$6545 is
consistent with their product $CN_{HI}\sim 0.5\times 10^{20}{\rm\ cm^{-2}}$.
The Monte Carlo profiles from stationary emission models exhibit deficit in
the wing parts. A much better fit is obtained when the He II emission region
is assumed to take the form of a ring that slowly rotates with a rotation
speed $\sim 18{\rm\ km\ s^{-1}}$. Brief discussions are presented regarding
the mass loss processes and future observations.
planetary nebulae — planetary nebulae: individual NGC 6790 — radiative
transfer — scattering — mass loss
††slugcomment: Submitted to ApJ
## 1 Introduction
Mass loss is an important process that mainly occurs in the late stage of
stellar evolution. A star with a mass less than $8{\rm\ M_{\odot}}$ loses a
significant amount of mass in the giant stage before becoming a planetary
nebula with a hot white dwarf at its center. Considering the Chandrasekhar
limit of $1.4{\rm\ M_{\odot}}$, the mass loss process in the giant stage with
enriched heavy elements should be important in the chemical evolution of the
interstellar medium. In this regard, with a recent history of mass loss, young
planetary nebulae are interesting objects to study the mass loss process.
It is expected that around a young planetary nebula there may be a significant
amount of neutral material that was lost in the previous stage of stellar
evolution. In this case, the neutral region is exposed to the strong UV
emission line source in the vicinity of the hot central star of the planetary
nebula. Therefore, important information related with the mass loss process
can be gathered from investigations of the scattering processes of the UV
radiation originating from the center region.
Taylor, Gussie & Pottasch (1990) performed H I 21 cm radio observations for a
number of compact planetary nebulae (see also Altschuler et al. 1986, Gussie &
Taylor 1995, Schneider et al. 1987). Their target selection was made on the
basis of high radio brightness temperature, which is indicative of the nebular
compactness. They searched an absorption trough that may be formed at the
radial velocity of a compact planetary nebula when the neutral region blocks
the background H I radio emission from our Galaxy. A number of compact young
planetary nebulae including IC 5117 and NGC 6790 have been detected. Adopting
an excitation temperature $T_{HI}=100{\rm\ K}$, the typical H I column density
was determined to be of order $N_{HI}\sim 10^{20}{\rm\ cm^{-2}}$ in these
objects.
Astrophysical Raman spectroscopy involving atomic hydrogen was initiated by
Schmid (1989), who identified the mysterious broad emission bands occurring at
6825 Å and 7088 Å in many symbiotic stars (see also Nussbaumer, Schmid & Vogel
1989). He proposed that a hydrogen atom in the ground state is excited with
the absorption of an incident far UV O VI$\lambda$ 1032 photon and de-excites
into the $2s$ level with the re-emission of an optical photon at 6825 Å. An
analogous process for far UV O VI$\lambda$1038 yields optical photons at 7088
Å. The large line width and prominent linear polarization exhibited by these
scattered features strongly support his proposal (e.g. Harries & Howarth
1996). Observations made simultaneous in the UV and optical regions also
confirm the Raman scattering nature (Espey et al. 1995).
In the spectrum of the symbiotic star RR Telescopii, Van Groningen (1993)
discovered Raman scattered He II features that are formed blueward of hydrogen
Balmer emission lines. He II emission lines arising from transitions between
$n=2k$ and $n=2$ levels have wavelengths that are slightly shorter than
hydrogen Lyman lines owing to the fact that He II ions are single electron
atoms with a slightly larger two body reduced mass. The proximity to resonance
is responsible for a large scattering cross section requiring the existence of
a neutral region with $N_{HI}\sim 10^{20}{\rm\ cm^{-2}}$ around a He II
emission source. Raman scattered He II features are also reported in other
symbiotic stars including He 2-106, HM Sagittae and V1016 Cygni (Lee, Kang &
Byun 2001, Jung & Lee 2004b, Birriel 2004).
Raman scattering of He II by atomic hydrogen also operates in young planetary
nebulae. The first discovery was reported by Péquignot et al. (1997) in their
spectroscopic analysis of the young planetary nebula NGC 7027. Subsequently,
Groves et al. (2002) found the same Raman scattered He II features in the
planetary nebula NGC 6302. Recently, Lee et al. (2006) reported that the
compact planetary nebula IC 5117 also exhibits Raman scattered He II features
blueward of H$\alpha$ and H$\beta$. In these objects, it appears that the
central star is surrounded by a neutral region with a significant covering
factor. In particular, Lee et al. (2006) discussed in detail the atomic
physics of He II recombination and Raman scattering processes.
We present our high resolution spectra of the young and compact planetary
nebula NGC 6790 and report our finding of the Raman scattered He
II$\lambda$6545 feature in this object. Using the H$\alpha$ image, Tylenda et
al. (2003) measured the angular size of NGC 6790 to be $4^{\prime\prime}\times
3^{\prime\prime}$. This size estimate of NGC 6790 is consistent with the HST
image shown by Kwok, Su & Sahai (2003), who also identified two inner shells
of similar orientations in NGC 6790. The distance to NGC 6790 is poorly known.
Gathier et al. (1986) proposed that NGC 6790 is further than $\sim 0.8{\rm\
kpc}$ based on their kinematic considerations. Adopting a statistical method
Zhang (1995) suggested a distance of 5.7 kpc to NGC 6790. In their high
resolution spectroscopy of NGC 6790, Aller, Hyung & Feibelman (1996) proposed
a core mass of $0.6{\rm\ M_{\odot}}$ and an age of 6000 yr with the note that
these values are dependent on the uncertain distance to NGC 6790.
We perform Monte Carlo radiative transfer simulations in order to obtain the
geometric and kinematic information of the neutral region. In section 2, we
describe our observation and line fitting analyses and the following section
presents our results of the Monte Carlo radiative transfer. In the final
section, we discuss briefly our observation and mass loss processes of NGC
6790.
## 2 Observation and Analysis
### 2.1 Observation and Data
We observed the young planetary nebula NGC 6790 on the night of 2008 May 31
using the 1.8 m telescope at Bohyunsan Optical Astronomy Observatory (BOAO).
The spectrograph that we used is the BOES (BOAO Echelle Spectrograph), which
is a bench-mounted echelle system fed by optical fibers with various
diameters. We used the 300 micron fiber, which yields the spectral resolution
$\sim 30,000$ with the field of view of $3^{\prime\prime}$. The spectral
coverage ranges 3600 Å through 10,500 Å. We obtained two spectra with exposure
times of 600 s and 7200 s, respectively. A Th-Ar lamp was used for wavelength
calibraions. For more detailed information on BOES, one is referred to Kim et
al. (2007). Standard procedures using the IRAF packages were followed to
reduce the spectra.
In Fig. 1, we show parts of our spectra around H$\alpha$ and H$\beta$. The
vertical axis represents the relative flux density. We normalize the flux
density using [N II]$\lambda$6548, which is set to have a flux density peak of
unity. The top panel of Fig. 1 is the spectrum around H$\alpha$ with an
exposure time of 600 s. We note strong forbidden emission lines of N II at
6548 Å and 6583 Å. In this short exposure spectral image, the strongest
H$\alpha$ is unsaturated, allowing us to fit the H$\alpha$ profile. The middle
panel of Fig. 1 shows the H$\alpha$ part of the spectrum with an exposure time
of 7200 s. The strong H$\alpha$ is saturated and we can discern very faint
emission lines including He II$\lambda$6527\. He II$\lambda$6527 arises from
transitions between $n=14$ and $n=5$ levels. We also clearly notice that
around [N II]$\lambda$6548 there exists a broad bump-like feature. This
feature is not an instrumental artifact because no such feature is present
near [N II]$\lambda$ 6583, which is supposed to be 3 times stronger than [N
II]$\lambda$6548 (e.g. Osterbrock 1987). We propose that this broad feature is
Raman scattered He II$\lambda$6545.
The bottom panel of Fig. 1, we show our spectrum of NGC 6790 around H$\beta$
with the exposure time of 7200 s. If Raman scattered He II exists blueward of
H$\alpha$, we may expect a similar feature blueward of H$\beta$. Indeed, when
Péquignot et al. (1997) reported the operation of He II Raman scattering in
NGC 7027, they detected Raman scattered He II$\lambda$4850\. In this object,
Raman scattered He II$\lambda$4850 is not blended with other strong emission
lines, which is in contrast with Raman He II$\lambda$6545 that is severely
blended with [N II]$\lambda$6548\. In the bottom panel of Fig 1, no broad
feature around 4850 Å is detected with a level of any significance. The quite
strong and sharp emission feature at 4851 Å is an emission line totally
irrelevant with Raman scattering. Aller et al. (1996) identified this emission
line as a forbidden line from Fe II. Our spectrum is of insufficient quality
to confirm the existence of Raman scattered He II$\lambda$ 4850\. However,
this does not cast serious doubts of the Raman scattering nature of the 6545
feature, because Raman He II$\lambda$4850 is always weaker than Raman He
II$\lambda$6545\. More discussion on this point is presented in section 3.2.
### 2.2 Line Fitting Analysis
Single Gaussian functions in the form
$f(\lambda)=f_{0}\exp[-(\lambda-\lambda_{c})^{2}/\Delta\lambda^{2}]$ are used
to fit the permitted emission lines of H$\alpha$, He II$\lambda$6560, He
II$\lambda$6527 and the two N II forbidden lines. The least chi square method
is adopted to obtain the best fitting Gaussian functions. We use the atomic
spectral data from the website of the National Institute of Standard and
Technology(NIST), from which we note that each emission line in our spectra of
NGC 6790 appears systematically redward of atomic line center by an amount of
20.7${\rm\ km\ s^{-1}}$. In Table 1, we summarize the result of our profile
analysis. The fitting parameters are quite similar to those found for IC 5117
by Lee et al. (2006).
Fig. 2 illustrates our line fitting analysis of the emission lines in NGC
6790. The top panels show the result for H$\alpha$ and He II$\lambda$6560\.
The short exposure data are used for the H$\alpha$ emission line, which is
excellently fitted by a single Gaussian function with a width
$\Delta\lambda=0.54{\rm\ \AA}$. He II$\lambda$6560 is also well fitted by a
single Gaussian function with a considerably smaller width of
$\Delta\lambda=0.48{\rm\ \AA}$ than that for H$\alpha$.
The middle panels of Fig. 2 show our result for He II$\lambda$6527, which is
significantly weak compared with He II$\lambda$ 6560\. He II$\lambda$6527 is
strongly blended with another unidentified emission line. Because He
II$\lambda$6560 is well-fitted by a single Gaussian, He II$\lambda$6527 should
be also fitted by a single Gaussian function, which is shown by the dotted
line in panel (c). The long dashed line in panel (c) shows our Gaussian fit to
the unidentified emission line. Groves et al.(2002) noted the existence of [N
II]$\lambda$6527 redward of He II$\lambda$6527 with the wavelength difference
of 0.14 Å in their spectrum of NGC 6302. However, the unidentified emission
line in our spectrum of NGC 6790 can not be [N II]$\lambda$6527, because it
appears redward of He II$\lambda$6527 by 1 Å. Furthermore, based on the NIST
data, [N II]$\lambda$6527 has the Einstein A coefficient $A=5.45\times
10^{-7}s^{-1}$. Compared with [N II]$\lambda$6548 having $A=9.19\times
10^{-4}$, [N II]$\lambda$6527 should be weaker than [N II]$\lambda$6548 by a
factor of 1700. Based on these atomic data, we plotted [N II]$\lambda$6527
with a dot-dashed line in Fig.2. As is shown in the figure, [N
II]$\lambda$6527 is significantly weaker than He II$\lambda$6527, and hence
can not affect the over all line fitting result. The 6528Å feature is much
stronger than [N II]$\lambda$6527 and still remains to be identified. Panel
(d) shows the composite profiles of the two single Gaussian functions in panel
(c). From our profile analysis shown also in Table 1, we conclude that the
flux ratio of He II$\lambda$ 6527 and He II$\lambda$ 6560 is
$F_{6527}/F_{6560}=4.1\times 10^{-2}.$ (1)
The bottom panels of Fig. 2 show the detailed profiles of [N II] lines. It is
interesting to note that [N II]$\lambda$6548 exhibits a sharp absorption
feature centered at 6548.60 Å. The line center of [N II]$\lambda$6548 appears
at 6548.51 Å, and the sharp absorption feature is excellently fitted by a
single Gaussian function with a width of $\Delta\lambda=0.1{\rm\ \AA}$ and
center at $\lambda_{0}=6548.60{\rm\ \AA}$. We find no such absorption feature
in [N II]$\lambda$6583, which should exhibit exactly the same profile with 3
times more flux (e.g. Osterbrock 1987). To our knowledge, no plausible metal
transition is responsible for this sharp absorption. We also checked the
telluric absorption lines without finding any strong candidate. In the
spectrum of IC 5117 obtained with the 3.6 m Canada-France-Hawaii Telescope we
find no similar absorption feature, for which [N II]$\lambda$6548 exhibits
exactly the same profile as [N II]$\lambda$6583\. We tentatively propose that
this is attributed to H$\alpha$ that is redshifted by an amount of
$v_{abs}\sim 800{\rm\ km\ s^{-1}}$. However, in this work, we limit our
attention to the Raman scattered He II$\lambda$6545 with no further discussion
of this possibly interesting feature.
Lee et al.(2001) performed a line profile analysis of Raman scattered He
II$\lambda$6545 in a number of symbiotic stars. They subtracted one third of
the flux near [N II]$\lambda$6583 from the flux near [N II]$\lambda$6548 to
expose a broad Raman scattering line feature successfully. However, in view of
the existence of the unidentified absorption feature in [N II]$\lambda$6548
and more severe blending with [N II]$\lambda$6548, we took another approach,
in which the Raman scattered He II$\lambda$6545 feature is directly fitted
from our Monte Carlo data.
## 3 Monte Carlo Radiative Transfer
### 3.1 Monte Carlo Procedure
In this subsection, we describe the procedure of our Monte Carlo analysis of
the Raman scattered He II$\lambda$6545\. Many planetary nebulae exhibit
nonspherical morphology, which may have its origin in the asymmetric mass loss
processes. In the case of NGC 6790, the HST image obtained by Kwok et al.
(2003) shows elongated shells around the central star. As a first
approximation, we adopt a cylindrical shell model for neutral material, which
is schematically illustrated in Fig. 3. A similar geometry was considered in
the analysis of IC 5117 by Lee et al. (2006).
In this cylindrical shell geometry, the hot UV source is located at the center
and H I material is uniformly distributed inside the cylindrical shell with
finite height and thickness. The same geometry was adopted by Lee et al.
(2006). However, the essential difference is that we now consider the
scattering region is expanding with the constant expansion velocity $v_{exp}$.
The cylindrical region is characterized by a uniform H I density $n_{H}$, the
height $H$ and the inner and outer radii $R_{H}$ and $R_{H}+\Delta R$,
respectively. In this case, the H I column density of the cylindrical shell is
given by $N_{HI}=n_{H}\Delta R$.
Since the shell is of uniform density, instead of the physical length $l$ we
measure the distance inside the shell in terms of the scattering optical depth
$\tau$ defined by
$\tau=n_{H}\sigma_{tot}l,$ (2)
where $\sigma_{tot}$ is the sum of the cross sections for Rayleigh and Raman
scattering. Since $\sigma_{tot}$ is a sensitive function of a wavelength of
the photon being considered, a given distance may correspond to different
optical depths dependent on the wavelength. Therefore, once a photon is
generated in the Monte Carlo simulation, we assume that the wavelength does
not change as long as it is Rayleigh scattered. Considering that the
scattering region is neutral, this assumption should be reasonable.
The basic atomic physics of Raman scattering adopted in our Monte Carlo code
is explained in detail by Jung & Lee (2004a). Due to the proximity of He II
$\lambda$ 1025 to H I Ly$\beta$ resonance, the scattering cross section
increases steeply near Ly$\beta$. Yoo, Bak & Lee (2002) showed that the
branching ratio $r_{b}$ into Raman scattering increases approximately linearly
with wavelength, which is given by
$\displaystyle r_{b}$ $\displaystyle=$
$\displaystyle\sigma_{Ram}/\sigma_{tot}$ (3) $\displaystyle=$ $\displaystyle
0.1342+12.50(\lambda-\lambda_{Ly\beta})/\lambda_{Ly\beta},$
where $\sigma_{Ram}$ is the cross section for Raman scattering and
$\lambda_{Ly\beta}$ is the Ly$\beta$ center wavelength. Therefore, the Raman
conversion into the optical region is quite sensitive to the incident
wavelength, which in turn depends on the expansion velocity.
From the energy conservation, a Raman scattered He II feature is characterized
by its large width given by
${\Delta\lambda_{Ram}\over\lambda_{Ram}}=\left({\lambda_{Ram}\over\lambda_{i}}\right){\Delta\lambda_{i}\over\lambda_{i}},$
(4)
where $\lambda_{i}$ and $\lambda_{Ram}$ are wavelengths of the incident and
Raman scattered radiation (e.g. Schmid 1989, Nussbaumer et al. 1989). In the
case of Raman He II $\lambda$6545, the profile width becomes about 6 times
broader than He II$\lambda$1025, which endows a unique property that the
profile is mainly determined from the relative motion between the emitter and
the scatterer.
In our Monte Carlo calculation, we also consider the re-entry of a photon
emerging from the inner wall of the cylinder, for which we assume that this
photon travel freely until it hits the inner wall on the opposite side. We
consider a photon with a unit wavevector ${\bf\hat{k}}$ supposed to travel a
scattering optical depth $\tau$ from the position ${\bf
r}_{i}=(x_{i},y_{i},z_{i})$. If this photon emerges from the inner wall of the
cylinder, we find the two points of intersection with the inner wall of the
cylinder. This is accomplished by solving the quadratic for $\tau_{p}$
$R_{H}^{2}=|({\bf r}_{i}+\tau_{p}{\bf\hat{k}})\cdot{\hat{\rho}}|^{2},$ (5)
for which we denote the two solutions by $\tau_{p1}$ and $\tau_{p2}$ with
$\tau_{p2}>\tau_{p1}$. Here, $\hat{\rho}$ is the unit vector pointing radially
outward from the cylinder axis. The difference of the two solutions
$\Delta\tau_{p}$ is given by
$\displaystyle\Delta\tau_{p}$ $\displaystyle=$
$\displaystyle\tau_{p2}-\tau_{p1}$ (6) $\displaystyle=$
$\displaystyle{2\sqrt{R_{H}^{2}(1-k_{z}^{2})-(k_{x}y_{i}-k_{y}x_{i})^{2}}\over(1-k_{z}^{2})},$
where $k_{x},k_{y}$ and $k_{z}$ are the components of ${\bf\hat{k}}$. By
adding $\Delta\tau_{p}$ to the original photon path, we find the new
scattering site in the other side of the shell.
The incident He II$\lambda$1025 line flux and profile can be inferred from the
case B recombination theory of single electron atoms provided by Storey &
Hummer (1995). In Table 2, we show the expected He II$\lambda$1025 line flux
relative to He II$\lambda$6560 and He II$\lambda$6527 for electron number
densities $n_{e}=10^{4},10^{6}$ and $10^{8}{\rm\ cm^{-3}}$ and temperatures
$T_{e}=10^{4}$ and $2\times 10^{4}{\rm\ K}$. We note that our observed flux
ratio of He II$\lambda$ 6527 and He II$\lambda$ 6560 given in Eq. (1) is
consistent with the nebular condition of $n_{e}\sim 10^{6}{\rm\ cm^{-3}}$ and
$T_{e}=10^{4}{\rm\ K}$. However, this choice is not unique and the range of He
II$\lambda$1025 is already quite significant with the choice of parameters in
Table 2. With this caveat in mind, we fix the electron number density
$n_{e}=10^{6}{\rm\ cm^{-3}}$ and $T_{e}=10^{4}{\rm\ K}$. Adopting these values
of $n_{e}$ and $T_{e}$, the recombination theory by Storey & Hummer (1995)
gives $F_{1025}=4.2F_{6560}$, which is used for our Monte Carlo calculations.
The Monte Carlo simulation starts with a generation of He II$\lambda$1025 line
photons having the same line profile with that of observed He II$\lambda$6560,
and appropriately scaled using the recombination theory. As He II$\lambda$6560
is fitted by a single Gaussian with a width of $\Delta\lambda=0.48{\rm\ \AA}$,
we note that the line profile function $f_{UV}$ for He II$\lambda$1025 is
given by
$f_{UV}(\lambda)=f_{1025}\exp{-[(\lambda-\lambda_{1025})^{2}/\Delta\lambda_{1025}^{2}}]$
(7)
with $\Delta\lambda_{1025}=0.48\cdot 1025/6560{\rm\ \AA}=0.075{\rm\ \AA}$.
Here, the peak value $f_{1025}$ is appropriately adjusted to yield
$F_{1025}=4.2F_{6560}$.
We trace each individual He II$\lambda$1025 line photon until it escapes from
the H I region. From Eq. (4), it is noted that the profiles of the Raman
scattered features are determined from the relative kinematics between the
emission source and the H I region and almost independent of the observer’s
line of sight. Therefore, in this work, we collect all the photons
irrespective of the final direction.
### 3.2 Simulated Raman Profiles
#### 3.2.1 Spherical Emission Region
In the work of Lee et al. (2006), the analysis of Raman scattered He
II$\lambda$6545 was purely based on the atomic physics and focused on the
exact location of line center. Their computation shows that the Raman
scattered feature should be centered significantly blueward of [N
II]$\lambda$6548\. In Fig. 1, we note that the Raman He II$\lambda$6545 is
completely blended with [N II]$\lambda$6548, which implies that the neutral
scattering region should be receding from the central UV source.
In Fig. 4, we show our Monte Carlo profiles for various expansion speeds
$v_{exp}$ of the neutral scattering region with respect to the hot central
star. In this figure, the height of the cylinder is taken to be infinite so
that the covering factor of the scattering region is unity. The column density
is fixed to $N_{HI}=1\times 10^{20}{\rm\ cm^{-2}}$. The solid line shows our
observed data and the other lines show our Monte Carlo profiles corresponding
to various values of $v_{exp}$. We can clearly notice the center shift of the
Raman He II$\lambda$6545, which is highly enhanced due to the line broadening
given in Eq. (4). The top panel shows the profiles for velocities $v_{exp}\leq
40{\rm\ km\ s^{-1}}$. The bottom panel shows the profiles for velocities in
the smaller range $14{\rm\ km\ s^{-1}}\leq v_{exp}\leq 22{\rm\ km\ s^{-1}}$.
From the figure, the plausible expansion velocity is around $20{\rm\ km\
s^{-1}}$, for which the peak wavelength resides inside the [NII]$\lambda$6548
emission line.
One interesting point to note from Fig. 4 is that the strength of the Raman
feature increases sharply as $v_{exp}$ increases despite the fact that the
covering factor and $N_{HI}$ are fixed. This is explained by the fact that the
Raman scattering cross section sharply increases near H$\alpha$ due to
Ly$\beta$ resonance in the parent wavelength space. Therefore, a receding H I
region yields more Raman scattered He II$\lambda$6545 photons than when the
same region is stationary. This complicated dependence of the scattering cross
section on wavelength also results in slightly asymmetric Raman profiles,
which is barely noticeable in Fig. 4. Therefore, the Raman conversion
efficiency may be estimated accurately only after the kinematics of the
scattering region with respect to the emission source is carefully determined.
In the left panel of Fig. 5, we show the Raman profiles for various H I column
densities ranging $N_{HI}=10^{19}-1.5\times 10^{20}{\rm\ cm^{-2}}$ with the
fixed values of $H/R_{H}=2$ and $v_{exp}=20{\rm\ km\ s^{-1}}$. Within this
range of $N_{HI}$, the overall strength is nearly proportional to the H I
column density, because the H I region is mostly optically thin with respect
to Raman scattering of He II$\lambda$1025\. This expansion speed is very
similar to the value of $16{\rm\ km\ s^{-1}}$ determined from Doppler shifted
Na D absorption lines by Dinerstein, Sneden & Uglum (1995).
The right panel of Fig. 5 shows the Monte Carlo Raman profiles for various
covering factors of the cylindrical shell. As is expected, the overall
strength is also proportional to the covering factor. In both the panels of
Fig. 5, we obtain qualitatively similar profiles. This implies that the Raman
profile analysis severely suffers from the degeneracy problem involving the
covering factor and H I column density.
With this caveat in mind related with the degeneracy in $N_{HI}$ and the
covering factor, we show our best fit profile from the Monte Carlo
calculations in Fig. 6. The model parameters are $v_{exp}=19{\rm\ km\
s^{-1}}$, $N_{HI}=9\times 10^{19}{\rm\ cm^{-2}}$ and $H/R_{H}=1.7$. As in IC
5117, the H I region significantly covers the hot central star in NGC 6790.
However, in this figure we notice that the model profiles exhibit deficit both
in the blue wing and red wing parts. If this deficit is real, then it implies
that in the direction to the H I region the incident profile is broader than
in the observer’s line of sight. The next subsection discusses this point.
Jung & Lee (2004b) developed a Monte Carlo code to compute the line profile of
Raman scattered He II 4850 and analyzed their spectrum of the symbiotic star
V1016 Cyg. Using the same code, we show in Fig. 7 the Monte Carlo profile for
Raman scattered He II 4850 by a long dashed line. The same column density and
covering factor as in Fig. 6 were used in this calculation. In the figure, the
solid line shows the BOES data with the exposure time of 7200 s. Our
observational data are barely consistent with our interpretation of Raman
scattering nature. The poor quality of the current observational data hinders
a further serious quantitative analysis. A more fruitful analysis may be made
only after observational data with a better quality are secured.
#### 3.2.2 Ring-like Emission Region
In this subsection, we perform line profile analyses in the case where the
emission region takes the form of a ring that is rotating in the vicinity of
the hot central star. In the previous section, it was assumed that the He II
emission region is spherically symmetric and stationary. However, it is highly
probable that the distribution of nebular material significantly deviates from
spherical symmetry considering the non-spherical shape exhibited by most
planetary nebulae (e.g. Corradi & Schwarz 1995). In this case, the emission
region may plausibly possess an ordered motion component, which may also be
associated with the nonspherical nebular morphology. Therefore, we may expect
that ionized material is concentrated on the equatorial plane having some slow
rotation velocity component.
There exists little kinematic information available on the emission region
very near the central star. No observational data of NGC 6790 are available in
the archives of HUT and FUSE. In consideration of the absence of a unique
kinematic model accounting for all the observed emission line profiles, we
adopt a simple ring-like emission region, in which we investigate the line of
sight effect on the profiles of the He II emission and Raman scattered lines.
Depending on the line of sight of the observer, the rotation velocity
component is reduced by the factor $\sin i$, where $i$ is the inclination
angle of the ring. However, Eq. (4) dictates that the Raman profile is
determined by the velocity component of the emitter with respect to the
scatterer and fairly insensitive to the line of sight.
This proposition leads to an interesting interpretation of our profile fitting
of H$\alpha$ and He II$\lambda$6560 presented in the previous section. We may
decompose the emission profiles into a bulk component and a random component.
We further assume that the bulk component represents a slow rotation in the
equatorial plane and that the random component is attributed to a thermal
motion and a turbulent motion. For the sake of simplicity, we assume that He
II$\lambda$6560 and H$\alpha$ are formed in the same ring-like region that is
in slow rotation in the equatorial plane with the speed $v_{bulk}$.
A He ion being 4 times heavier than a hydrogen nucleus, the line width of He
II due to the thermal motion is half of that for H$\alpha$ if they are formed
in the same region. However, if the emission region possesses some turbulent
component, then overall random motion component for hydrogen is broader than
that of He II by a factor less than 2. If we denote the electron temperature
of NGC 6790 by $T_{e}=10^{4}\ T_{4}{\rm\ K}$, then the thermal velocity
associated with H$\alpha$ is given by
$v_{th,H}=\sqrt{k_{B}T_{e}\over 2m_{p}}=13\ T_{4}^{1/2}{\rm\ km\ s^{-1}},$ (8)
where $m_{p}$ is the proton mass and $k_{B}$ is the Boltzmann constant (e.g.
Rybicki & Lightman 1979). Introducing $v_{turb}$ for the turbulent velocity
scale, we denote the random velocity components of H$\alpha$ and He
II$\lambda$6560 by $v_{ran,H}$ and $v_{ran,He}$, respectively, where
$\displaystyle v_{ran,H}$ $\displaystyle=$ $\displaystyle v_{turb}+v_{th,H}$
$\displaystyle v_{ran,He}$ $\displaystyle=$ $\displaystyle
v_{turb}+(v_{th,H}/2).$ (9)
Noting that there are three model parameters, namely $i$, $v_{bulk}$ and
$v_{turb}$ for the two line widths, we also encounter a degeneracy problem.
Hoping that future observations may provide independent constraints on some of
these model parameters, we just pick out a set of values that yield a
reasonable fit to our observed data. In the top panels of Fig. 8, we show
model line profiles for He II$\lambda$6560 and H$\alpha$ from one such set
consisting of
$\displaystyle\sin i$ $\displaystyle=$ $\displaystyle 0.6,\quad
v_{bulk}=18{\rm\ km\ s^{-1}},\quad v_{turb}=14{\rm\ km\ s^{-1}},$
$\displaystyle v_{th,H}$ $\displaystyle=$ $\displaystyle 14{\rm\ km\
s^{-1}},\quad v_{th,He}=0.5v_{th,H}.$ (10)
The thermal velocity $v_{th,H}=14{\rm\ km\ s^{-1}}$ is consistent with the
electron temperature $T_{e}=10^{4}{\rm\ K}$, which is similar to that obtained
by Aller et al. (1996) from their photoionization modeling. The overall fits
to both H$\alpha$ and He II$\lambda$6560 appear quite good. The bulk velocity
component is consistent with the size of the emission ring region of order
$1{\rm\ AU}$ if we interpret the bulk motion to be Keplerian. However, the
bulk motion may not be related with the Keplerian motion but may be related
with the rotation component of the central star, for which case the physical
size of the emission region can be at best poorly constrained.
Because the H I region is also concentrated on the equatorial plane, the full
bulk velocity component should be considered without the inclination effect
for far UV He II$\lambda$1025 that is incident on the H I region. In the
bottom panel of Fig 8, the dotted line shows the He II$\lambda$6560 profile
that would be measured by a hypothetical observer in the equatorial plane. It
is excellently fitted by a single Gaussian function with a width
$\Delta\lambda=0.61{\rm\ \AA}$, which is significantly larger than the
observed value of $\Delta\lambda=0.48{\rm\ \AA}$ by a factor of 1.3. Hence,
the emission profile for He $\lambda$1025 incident on the neutral region
should also be broadened by the same factor.
In Fig. 9, we show our Monte Carlo result using the profile shown in the
bottom panel of Fig. 8 and appropriately scaled to He II$\lambda$1025\. The
other model parameters are also adjusted for better fit and they are
$v_{exp}=15{\rm\ km\ s^{-1}},N_{HI}=9\times 10^{19}{\rm\ cm^{-2}}$ and
$H/R_{H}=1.2$. A much better fit is obtained than that considered in the
previous section. However, it should also be pointed out that in constructing
the profile in Fig. 9 more model parameters have been used than in the
previous section and still the degenerate nature of the problem persists.
The expansion velocity of the H I shell in Fig. 9 is only $v_{exp}=15{\rm\ km\
s^{-1}}$, which is significantly smaller than the value $v_{exp}=19{\rm\ km\
s^{-1}}$ presented in Fig. 6. This notable discrepancy in expansion velocity
is attributed to the scattering cross section that is sharply peaked around
H$\alpha$. According to Jung & Lee (2004a), this leads to the center shift of
a Raman scattered He II feature, which is dependent on the column density. The
result shown in Fig. 9 implies that the shape or the width of the incident
profile also affects the location of the line center. A more quantitative
investigation in a significantly large parameter space is left to the future
work.
## 4 Discussion
H I Raman spectroscopy provides an accurate determination of the expansion
velocity of the H I region, for the measurement of which H I 21 cm radio
observation has been the unique tool so far. Our analysis shows that the
expansion velocity lies between $15{\rm\ km\ s^{-1}}$ and $19{\rm\ km\
s^{-1}}$, which is consistent with the value of $16{\rm\ km\ s^{-1}}$ provided
by Taylor et al. (1990). As was pointed out by Lee et al. (2006) the Raman
spectroscopy allows one to determine the H I column density whereas the
excitation temperature should be assumed before $N_{HI}$ is deduced from H I
21 cm radio observation. According to Taylor et al. (1990), $N_{HI}=2.7\times
10^{20}{\rm\ cm^{-2}}$ assuming the excitation temperature $T_{HI}=100{\rm\
K}$. Our Raman profile analysis lends support to this excitation temperature.
Our current data are of insufficient quality to lift the degeneracy of the
covering factor and H I column density, and the overall strength of the Raman
feature is determined from the product of the two quantities. However, our
Monte Carlo calculations show that Raman profiles exhibit redward asymmetry
due to enhanced scattering cross section toward H$\alpha$ resonance. With
better quality spectra that may be available from bigger telescopes, it is
hoped that tighter constraints are obtained from more refined profile
analyses. If Raman scattered He II 4850 blueward of H$\beta$ can also be used,
additional constraints can be put to break the degeneracy.
Even though the distance to NGC 6790 is highly uncertain, we may assume that
the distance is about 1 kpc for simple order of magnitude calculations.
According to Tylenda et al. (2003), the angular size of NGC 6790 is $\sim
3^{\prime\prime}$. This gives a physical size of the H I region $R\sim 5\times
10^{16}{\rm\ cm}$. If the H I region is of a thin cylindrical shell with the
height similar to its radius, the total number $N_{tot}$ of hydrogen atoms
inside the shell is approximately given by $N_{tot}=2\pi R^{2}N_{HI}\sim
6\times 10^{54}$. Here, in our order of magnitude estimate, we ignore the
inclination effect, which will overestimate the total number of hydrogen atoms
by the factor $\sin i$. The H I mass of the neutral region is inferred to be
$M_{HI}\sim 4\times 10^{-3}{\rm\ M_{\odot}}$.
Furthermore, the expansion velocity of $v_{exp}\sim 15{\rm\ km\ s^{-1}}$ and
the physical size of $R\sim 5\times 10^{16}{\rm\ cm}$ together imply the age
of order of a thousand years for NGC 6790. It should be pointed out that these
rough calculations are highly dependent on the assumed distance to NGC 6790
and still the physical size of the H I region is quite uncertain.
The origin of sharp absorption feature that appeared in [N II]$\lambda$6548 is
quite uncertain. If this absorption feature is attributed to H$\alpha$, then
it may imply the existence of clumpy components having a small covering factor
with respect to the [N II] emission region and receding with a significant
velocity of $\sim 800{\rm\ km\ s^{-1}}$. In some planetary nebulae including
M2-9 and NGC 6543, it is known that fast collimated outflows exist around the
central star with a velocity of order $1000{\rm\ km\ s^{-1}}$ (Balick 1989,
Gruendl, Chu & Guerrero 2004, Prinja et al. 2007). Ueta, Fong & Meixner (2001)
presented near IR imaging observations of AFGL 618 and reported their findings
of molecular bullet-like features moving faster than $200{\rm\ km\ s^{-1}}$.
However, it still remains a mystery whether a clumpy bullet-like object can be
ejected with so large a velocity from the center region.
It should be pointed out that a ring-like emission model may not be a unique
choice for the observed profiles of He II$\lambda$6560 and H$\alpha$. Many
kinematical models involving jet-like outflows or radial infall and/or
outflows may also yield similarly well-fitting profiles. Therefore without
convincing support from other studies such as imaging observations using
interferometry or hydrodynamical computations, it appears to be too early to
conclude about the kinematics of the He II emission region.
A ring-like emission region and H I region concentrated in the equatorial
region may provide interesting opportunities for spectropolarimetry. In
symbiotic stars, Raman scattered O VI$\lambda\lambda$6825, 7088 are known to
exhibit strong linear polarization (e.g. Harries & Howarth 1996, Schmid 1998).
The polarization structure may be closely related with the accretion and mass
loss processes that deviate from spherical symmetry (e.g. Lee & Park 1999, Lee
& Kang 2007, Ikeda et al. 2004). Because Raman scattered features consist of
purely scattered photons, they make ideal targets for linear
spectropolarimetry. Future spectropolarimetric studies may provide more
interesting information regarding the mass loss processes in AGB stars and
planetary nebulae.
We are grateful to the staffs at the Bohyunsan Optical Astronomy Observatory.
We also thank an anonymous referee for the constructive comments, which
significantly improved the presentation of our work. This research was
supported by the Astrophysical Research Center for the Structure and Evolution
of the Cosmos (ARCSEC”) funded by the Korea Science and Engineering
Foundation.
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Figure 1: High resolution spectra of NGC 6790 obtained with BOES. The top panel is a short exposure spectrum with exposure time of 600 s, and the exposure time for the middle and bottom panels is 7200 s. The relative flux density is normalized such that [N II] $\lambda$6548 has the flux density peak of unity. In the top panel, the strongest emission line H$\alpha$ is unsaturated. We also clearly see [N II] lines at 6548 Å and 6583 Å. In the middle panel, we note that around [N II] $\lambda$6548 there exists a broad wing feature. No similar feature is present around 3 times stronger [N II] $\lambda$6583, which means that the broad wing feature around [N II] $\lambda$6548 is not associated with [N II] nor is an instrumental artifacts. The bottom panel shows the H$\beta$ part of the BOES spectrum. The insufficient quality of the data hinders the clear detection of the Raman scattered He II$\lambda$4850\. Figure 2: Gaussian line fitting analysis. The solid lines show the observational data and the dotted lines show our Gaussian fits. The same flux normalization as in Fig.1 is used. The top panels show the results for H$\alpha$ and He II$\lambda$6560\. The middle panels show the line fitting result for He II$\lambda$6527 and a nearby unidentified emission line. He II$\lambda$6527 is fitted by a single Gaussian function, which is shown by the dotted line in panel (c). The long dashed line in panel (c) shows the unidentified emission line. Using the atomic data provided by NIST, we show the line contribution from [N II]$\lambda$6527 by a dot-dashed line. In panel (d) we show the composite profile from the three single Gaussians shown in panel (c). The bottom panels show the detailed views of [N II] lines. There is a sharp absorption feature in [N II]$\lambda$6548, which is also well fitted by a single Gaussian with the width $\Delta\lambda=0.1{\rm\ \AA}$ and the center at $\lambda_{c}=6548.60{\rm\ \AA}$. Figure 3: A schematic diagram of the Raman scattering geometry adopted in this work. The hot star and He II emission region are located at the center. Surrounding the UV emission region, the H I scattering region takes the form of a cylindrical shell with the inner radius $R_{H}$, the outer radius $R_{H}+\Delta R$ and the height $H$. In this work, the cylindrical shell is assumed to expand with the speed $v_{exp}$. Hydrogen atoms are distributed uniformly with a number density $n_{H}$ inside the cylindrical shell. Figure 4: Line profiles of Raman scattered He II$\lambda$6545 from our Monte Carlo simulations for various expansion speeds. The covering factor is fixed to be unity and $N_{HI}=10^{20}{\rm\ cm^{-2}}$. Due to the inelasticity of Raman scattering or Eq.(4), the location of line center is fairly sensitive to $v_{exp}$. The top panel shows the profiles for velocities in the range $v_{exp}\leq 40{\rm\ km\ s^{-1}}$ in an interval of $10{\rm\ km\ s^{-1}}$. The bottom panel shows the profiles for velocities in the range $14{\rm\ km\ s^{-1}}\leq v_{exp}\leq 22{\rm\ km\ s^{-1}}$. It is notable that the expansion velocity $v_{exp}$ affects both the location of line center and the total Raman flux. Figure 5: Monte Carlo line profiles of Raman scattered He II$\lambda$6545 for various $N_{HI}$ and covering factors. The left panel shows the Monte Carlo profiles for various $N_{HI}$ with the covering factor fixed to be $H/R_{H}=2$. The right panel shows the simulated profiles for various covering factors with fixed $N_{HI}=10^{20}{\rm\ cm^{-2}}$. Figure 6: Our best fit Monte Carlo profile of Raman scattered He II$\lambda$6545 from a stationary emission region surrounded by a cylindrical shell. The dotted line is the Monte Carlo line profile and the solid line is the observed data. The adopted parameters are $v_{exp}=19{\rm\ km\ s^{-1}}$, $H/R_{H}=1.7$, $N_{HI}=9\times 10^{19}{\rm\ cm^{-2}}$. Figure 7: BOES data around H$\beta$(solid line) and the Monte Carlo profile of Raman scattered He II$\lambda$4850 (long dashed line). The same column density and covering factor as in Fig.6 were used in the Monte Carlo calculation. The observational data are barely consistent with the Monte Carlo result. Figure 8: Line profiles of He II$\lambda$6560 and H$\alpha$ from a ring-like emission region. The axis of the ring makes an angle $i$ with the line of sight, where we take $\sin i=0.6$ as an example. The upper panels show line profiles of He II$\lambda$6560 and H$\alpha$ viewed from the observer’s line of sight. The fitting parameters are $v_{bulk}=18{\rm\ km\ s^{-1}},v_{turb}=14{\rm\ km\ s^{-1}}$ and $v_{th,H}=14{\rm\ km\ s^{-1}},v_{th,He}=0.5v_{th,H}$. See the text of the definitions of these velocities. The solid lines represent the BOES data and the dotted lines are model profiles. The lower panel shows the observed He II$\lambda$6560 profile (solid line) and the model profile that would be observed in the equatorial direction. The dotted model profile is excellently fitted by a single Gaussian with a width $\Delta\lambda=0.61{\rm\ \AA}$. Figure 9: A Monte Carlo best fit profile (dotted line) of Raman scattered He II$\lambda$6545 from a ring-like emission region considered in Fig. 7. The adopted model parameters are $\sin i=0.6$, $v_{bulk}=18{\rm\ km\ s^{-1}}$, $v_{turb}=14{\rm\ km\ s^{-1}}$, $v_{th\ H}=14{\rm\ km\ s^{-1}}$. Refer the text for the definitions of these parameters. This profile provides a much better fit than that shown in Fig. 6. It is noted that the expansion velocity of the H I shell is $v_{exp}=15{\rm\ km\ s^{-1}}$, which is significantly smaller than that considered in Fig. 6. Table 1: Single Gaussian Fit Parameters of Emission Lines Line | $\lambda_{0}$ (Å) | $f_{0}$ | $\Delta\lambda$ (Å)
---|---|---|---
H$\alpha$ 6563 | 6563.23 | 34.8 | 0.54
He II $\lambda$ 6560 | 6560.58 | 0.072 | 0.48
He II $\lambda$ 6527 | 6527.49 | 0.00295 | 0.48
${\rm[N~{}II]}\ \lambda$ 6548 | 6548.51 | 0.897 | 0.47
${\rm[N~{}II]}\ \lambda$ 6583 | 6583.90 | 2.73 | 0.48
Table 2: He II Recombination Data by Storey & Hummer (1995) Line Ratio | $T_{e}=10^{4}{\rm\ K}$ | $T_{e}=2\times 10^{4}{\rm\ K}$
---|---|---
$n_{e}=10^{4}{\rm\ cm^{-3}}$ | |
$F_{1025}/F_{6560}$ | 3.600 | 4.519
$F_{6527}/F_{6560}$ | $3.952\times 10^{-2}$ | $4.085\times 10^{-2}$
$n_{e}=10^{6}{\rm\ cm^{-3}}$ | |
$F_{1025}/F_{6560}$ | 3.804 | 4.676
$F_{6527}/F_{6560}$ | $4.098\times 10^{-2}$ | $4.152\times 10^{-2}$
$n_{e}=10^{8}{\rm\ cm^{-3}}$ | |
$F_{1025}/F_{6560}$ | 4.439 | 5.181
$F_{6527}/F_{6560}$ | $4.942\times 10^{-2}$ | $4.614\times 10^{-2}$
|
arxiv-papers
| 2009-01-15T06:15:10 |
2024-09-04T02:48:59.974235
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Eun-Ha Kang, Hee-Won Lee, Byeong-Cheol Lee",
"submitter": "Eun-Ha Kang",
"url": "https://arxiv.org/abs/0901.2189"
}
|
0901.2258
|
# Hawking Radiation due to Photon and Gravitino Tunneling
Bibhas Ranjan Majhi
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
Saurav Samanta
Narasinha Dutt College,
129, Belilious Road, Howrah-711101, India E-mail: bibhas@bose.res.inE-mail:
srvsmnt@gmail.com
> Applying the Hamilton–Jacobi method we investigate the tunneling of photon
> across the event horizon of a static spherically symmetric black hole. The
> necessity of the gauge condition on the photon field, to derive the
> semiclassical Hawking temperature, is explicitly shown. Also, the tunneling
> of photon and gravitino beyond this semiclassical approximation are
> presented separately. Quantum corrections of the action for both cases are
> found to be proportional to the semiclassical contribution. Modifications to
> the Hawking temperature and Bekenstein-Hawking area law are thereby
> obtained. Using this corrected temperature and Hawking’s periodicity
> argument, the modified metric for the Schwarzschild black hole is given.
> This corrected version of the metric, upto $\hbar$ order is equivalent to
> the metric obtained by including one loop back reaction effect. Finally, the
> coefficient of the leading order correction of entropy is shown to be
> related to the trace anomaly.
## 1 Introduction
Black holes are the solution of classical general relativity from which
nothing can escape. In 1974–75 this understanding was changed completely when
Hawking[1, 2] showed that due to quantum effects black holes can radiate
energy and the resulting spectrum is purely thermal in nature. He also showed
that the temperature of a black hole is directly proportional to its surface
gravity.
To understand Hawking radiation in a physically intuitive manner, Parikh and
Wilczek[3] described it as a quantum tunneling effect through the horizon of a
black hole. In this method one first calculates the tunneling amplitude by
exponentiating the imaginary part of the action for outgoing mode, for the
process of $s$\- wave emission and then uses the principle of detailed balance
to relate it with the Boltzmann factor. In the literature two different
approaches are available to compute the imaginary part of the action that
yields the Hawking temperature. In one method, trajectory of a radial null
geodesic is considered–this was developed in [3]. In the other method,
Hamilton-Jacobi ansatz is used–this is an extension of the complex path
analysis given in [4]. After the initial formulation of the theory it
generated a lot of interest and till now it has been applied successfully to
various types of black holes and space times [5, 6, 7, 8, 9]. It was also
noticed that in this approach there was a problem of factor $2$ in the
expression of Hawking temperature. This was later solved in [10] by taking
into account the temporal contribution to the quasi–classical amplitude.
However, most of the studies in this context have been done for spinless
scalar particles. Though there are few papers [11, 12] on spin $\frac{1}{2}$
fermion tunneling, no analysis has been done for a spin one particle like
photon. Discussion on the radiation of spin $\frac{3}{2}$ gravitino has been
done recently [13], but that study incorporates only semiclassical
approximation and does not consider quantum corrections.
In the present paper, we shall study the tunneling of photon and gravitino
from the horizon of a static spherically symmetric black hole by following the
method previously elaborated by one of us [14, 15, 12, 16] and which has been
applied later for various cases [17]. This approach is basically the
Hamilton–Jacobi method where quantum corrections to the usual semiclassical
results are taken by considering all the terms in the expansion of the action.
It was shown that all the higher order corrections are proportional to the
semiclassical contribution. Though the values of these constants depend on the
order of expansion, a general form was provided by simple dimensional
argument. By calculating the ratio between outgoing and ingoing probability of
a particle, Hawking temperature with quantum corrections was obtained. This
eventually led to the entropy of the black hole in which the first order
correction was logarithmic in nature.
Here we employ the same method to first study the tunneling of photon. First
we take the gauge fixed action in a static spherically symmetric space time
background. Variation of this action with respect to the gauge field
‘$A_{\mu}$’ gives the gauge fixed Maxwell equation. Substituting the standard
ansatz for ‘$A_{\mu}$’ and taking the semiclassical limit (i.e.
$\hbar\rightarrow 0$) we obtain the usual semiclassical Hamilton-Jacobi
equation. Solutions of this equation lead to the ingoing and outgoing
probabilities of the gauge particle. Then applying the principle of detailed
balance the usual Hawking temperature is identified.
Since later we shall extend our analysis to higher order in $\hbar$, the above
procedure is not convenient. So for simplicity we use Lorentz gauge condition
separately. Therefore we shall start with the $U(1)$ Maxwell action without
any gauge fixed term. An arbitrary variation of $A_{\mu}$ in this action gives
the standard Maxwell equation. Now substituting the previous ansatz for
‘$A_{\mu}$’ and taking $\hbar\rightarrow 0$ limit, the Hamilton-Jacobi
equation is obtained. This cannot be solved by the previous method because of
the presence of different polarization vectors. To have a relation between
these vectors we impose the Lorentz gauge condition. Substitution of the same
ansatz for ‘$A_{\mu}$’ in this gauge condition leads to another equation.
Simultaneously solving these two equations we obtain the desired Hamilton-
Jacobi equation which was derived directly from the gauge fixed action.
The photon tunneling beyond semiclassical approximation upto $\hbar$ order is
also discussed here. In this case, the action and the polarization vectors are
expanded in powers of $\hbar$. Then equating different powers of $\hbar$ on
both sides of the Maxwell equation and the gauge condition we obtain a series
of equations. These equations are simplified by using the previous equations
in a recursive manner. Here we adopt only the second formalism where Maxwell
equation and corresponding gauge condition are treated separately. Because in
the other method simplification of $\hbar$ order equation by using $\hbar^{0}$
order equation is very difficult. This analysis again convinces the usefulness
of gauge condition. After simplification we show that $\hbar$ order term of
the action is proportional to the semiclassical contribution. This study for
the photon field is completely new and has not been mentioned elsewhere.
After obtaining the explicit form of the action, we calculate the wave
function which is finally used to get the tunneling amplitude. We again apply
the detailed balance principle to get the modified Hawking temperature. The
result agrees upto $\hbar$ order with the conclusions previously obtained for
the tunneling of scalar [14, 15] and Dirac particles [12] which confirms the
robustness of the whole formalism. A point we want to mention here, is that in
our analysis, the corrected tunneling amplitude is exactly the Boltzmann
factor $e^{-\frac{\omega}{T_{h}}}$, where $T_{h}$ is the corrected Hawking
temperature. In addition, there are also approaches [18] which lead to a
different type of correction to the tunneling amplitude, that is essentially
non-thermal in nature.
We next study the gravitino tunneling beyond semiclassical approximation. For
that we consider the massless Rarita-Schwinger equation [19] in curved
geometry and follow the same formalism. Though the final results for both
photon and gravitino tunneling look similar upto $\hbar$ order, the difference
comes from the correction parameter which is later shown to be dependent on
the spin of the particle.
By using Hawking’s periodicity arguments for the temperature corrected upto
order $\hbar$, we also give the corrections of the Schwarzschild metric in our
paper. This is shown to be exactly equivalent to the result obtained in [20]
by incorporating the one loop back reaction effect in the space time. Also the
leading order correction term in the Bekenstein-Hawking area law is obtained,
which is given as the logarithmic of the usual horizon area. Finally,
application of the constant scale transformation in the metric coefficients
reveals that the coefficient of this correction is related to trace anomaly.
Before proceeding further, let us mention the organization of our paper. In
the second section we study the tunneling of photon by two different methods
in two separate subsections. In subsection 2.1 we consider the gauge fixed
action for the photon field. In the next subsection, we consider the standard
Maxwell action but impose the Lorentz gauge condition later to find the
semiclassical black hole temperature. In the third section, the first order
quantum effect to the photon tunneling is studied to find the modified Hawking
temperature. Gravitino tunneling is analyzed in the next section. The
discussion on the correction parameter is given in section 5 and the last
section is for conclusions.
## 2 Photon tunneling and Hawking temperature
In this section we study the tunneling of photon to calculate the Hawking
temperature of a black hole. This is done by following two methods in two
subsections. In the first method we start from the gauge fixed action of
Maxwell field in a curved spacetime background and then find the action by
using the Hamilton–Jacobi equation[4] to calculate the tunneling amplitude.
Finally, this is equated with the Boltzmann factor to get the black hole
temperature. In the other method we perform a similar analysis. However
instead of the gauge fixed action, we take the standard photon field action
and impose the Lorentz gauge condition later to obtain the tunneling
amplitude. In both the analysis, we use the semiclasical approximation
$\hbar\rightarrow 0$.
### 2.1 Method 1: Gauge fixed equation of motion
Throughout this paper we shall consider the background space-time to be static
and spherically symmetric in nature, i.e.
$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{g(r)}+r^{2}d\Omega^{2}$ (1)
whose horizon $r=r_{H}$ is given by $f(r_{H})=g(r_{H})=0$.
The electromagnetic field in a gravitational background is described by the
Lagrangian density
$\displaystyle\mathcal{L}=-\frac{1}{4}\sqrt{-g}F_{\mu\nu}F^{\mu\nu}$ (2)
where the field strength $F_{\mu\nu}$ is defined in terms of the gauge field
$A_{\mu}$ as,
$\displaystyle F_{\mu\nu}$ $\displaystyle=$
$\displaystyle\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$ (3) $\displaystyle=$
$\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$ (4)
Under a local gauge transformation
$\displaystyle A_{\mu}\rightarrow
A^{\prime}_{\mu}=A_{\mu}+\nabla_{\mu}\Lambda,$ (5)
the Lagrangian density (2) is invariant. In order to quantize the theory, this
symmetry is broken by adding a gauge fixing term
$\displaystyle{\mathcal{L}}_{G}=-\frac{1}{2}\xi^{-1}(\nabla_{\mu}A^{\mu})^{2}$
(6)
to (2) to get the following action
$\displaystyle
S=\int{\textrm{d}}^{4}x(\mathcal{L}+{\mathcal{L}}_{G})=-\int{\textrm{d}}^{4}x[\frac{1}{4}\sqrt{-g}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\xi^{-1}(\nabla_{\mu}A^{\mu})^{2}].$
(7)
In this subsection we shall work with this action. Variation of the above
action with respect to $A_{\mu}$ gives the equation of motion
$\displaystyle\Box
A_{\mu}+R_{\mu}^{\rho}A_{\rho}-(1-\xi^{-1})\nabla_{\mu}(\nabla_{\nu}A^{\nu})=0.$
(8)
Choosing $\xi=1$ (Feynman gauge) we get the simplified equation
$\displaystyle\Box A_{\mu}+R_{\mu}^{\rho}A_{\rho}=0.$ (9)
Above equation is written explicitly in terms of the Christoffel connection
and Ricci tensor as
$\displaystyle
g^{\rho\sigma}\Big{(}\partial_{\rho}\partial_{\sigma}A_{\mu}-2\Gamma^{\lambda}_{\sigma\mu}\partial_{\rho}A_{\lambda}-\Gamma^{\lambda}_{\rho\sigma}\partial_{\lambda}A_{\mu}-\partial_{\rho}\Gamma^{\lambda}_{\sigma\mu}A_{\lambda}+\Gamma^{\lambda}_{\rho\sigma}\Gamma^{\alpha}_{\lambda\mu}A_{\alpha}+\Gamma^{\lambda}_{\rho\mu}\Gamma^{\alpha}_{\sigma\lambda}A_{\alpha}\Big{)}+R_{\mu}^{\rho}A_{\rho}=0.$
(10)
This equation can be solved by using spherical harmonics technique as was done
in [21] for the scalar field. But here we shall follow the traditional WKB
method for tunneling.
Now in order to solve the above equation, we make the following
Hamilton–Jacobi ansatz
$\displaystyle A_{\mu}=a_{\mu}e^{-\frac{i}{\hbar}I(t,r,\theta,\phi)},$ (11)
where $a_{\mu}$ is the polarization vector and $I$ is the action. With this
ansatz, first and second order derivatives of $A_{\mu}$ in (10) can be written
as,
$\displaystyle\partial_{\sigma}A_{\mu}=\Big{(}\partial_{\sigma}a_{\mu}-\frac{i}{\hbar}a_{\mu}\partial_{\sigma}I\Big{)}e^{-\frac{i}{\hbar}I}$
(12)
$\displaystyle\partial_{\rho}\partial_{\sigma}A_{\mu}=\Big{(}\partial_{\rho}\partial_{\sigma}a_{\mu}-\frac{i}{\hbar}\partial_{\rho}a_{\mu}\partial_{\sigma}I-\frac{1}{\hbar^{2}}a_{\mu}\partial_{\rho}I\partial_{\sigma}I-\frac{i}{\hbar}a_{\mu}\partial_{\rho}\partial_{\sigma}I-\frac{i}{\hbar}(\partial_{\sigma}a_{\mu})(\partial_{\rho}I)\Big{)}e^{-\frac{i}{\hbar}I}.$
(13)
Substituting (11), (12) and (13) in (10), we get the following equation
$\displaystyle
g^{\rho\sigma}\left[(\hbar^{2}\partial_{\rho}\partial_{\sigma}a_{\mu}-i\hbar\partial_{\rho}a_{\mu}\partial_{\sigma}I-a_{\mu}\partial_{\rho}I\partial_{\sigma}I-i\hbar\partial_{\rho}\partial_{\sigma}I)\right]$
$\displaystyle-g^{\rho\sigma}\left[2\Gamma^{\lambda}_{\sigma\mu}(\hbar^{2}\partial_{\rho}a_{\lambda}-i\hbar
a_{\lambda}\partial_{\rho}I)+\Gamma^{\lambda}_{\rho\sigma}(\hbar^{2}\partial_{\lambda}a_{\mu}-i\hbar
a_{\mu}\partial_{\lambda}I)\right]$
$\displaystyle+\hbar^{2}g^{\rho\sigma}\left[-\partial_{\rho}\Gamma^{\lambda}_{\sigma\mu}a_{\lambda}+(\Gamma^{\lambda}_{\rho\sigma}\Gamma^{\alpha}_{\lambda\mu}+\Gamma^{\lambda}_{\rho\mu}\Gamma^{\alpha}_{\sigma\lambda})a_{\alpha}\right]+\hbar^{2}R_{\mu}^{\rho}a_{\rho}=0.$
(14)
Now we expand $I$ and $a_{\mu}$ in power series of $\hbar$
$\displaystyle
I(r,t,\theta,\phi)=I_{0}(r,t,\theta,\phi)+\displaystyle\sum_{i=1}^{\infty}\hbar^{i}I_{i}(r,t,\theta,\phi)$
(15) $\displaystyle a_{\mu}=a_{\mu
0}+\displaystyle\sum_{i=1}^{\infty}\hbar^{i}a_{\mu i}.$ (16)
In the above expansions the terms $I_{0}$ and $a_{\mu 0}$ are semiclassical
values whereas the remaining terms are quantum corrections involving different
powers of $\hbar$. We substitute the above equation in (14) and take the
semiclassical limit ($\hbar\rightarrow 0$) to obtain
$\displaystyle g^{\rho\sigma}a_{\mu
0}(\partial_{\rho}I_{0})(\partial_{\sigma}I_{0})=0.$ (17)
Since the tunneling occurs in the radial direction, the ($r-t$) sector of the
metric is relevant and in that case we write (17) as
$\displaystyle g^{tt}(\partial_{t}I_{0})^{2}+g^{rr}(\partial_{r}I_{0})^{2}=0.$
(18)
For our choice of metric (1), the above equation reduces to
$\displaystyle-\frac{1}{f}(\partial_{t}I_{0})^{2}+g(\partial_{r}I_{0})^{2}=0$
(19)
which is equivalently written as
$\displaystyle\partial_{t}I_{0}=\pm\sqrt{fg}\partial_{r}I_{0}.$ (20)
This is the semiclassical Hamilton-Jacobi equation. Now in order to find the
Hamilton–Jacobi solution of $I_{0}$, we note that the metric (1) that we have
taken is stationary and so it has timelike Killing vectors. Thus we take the
solution of (20) in the form
$\displaystyle I_{0}(r,t,\theta,\phi)=\Omega
t+\tilde{I}_{0}(r)+I_{0}^{\prime}(\theta,\phi)$ (21)
where $\Omega$ is the constant of motion corresponding to the timelike Killing
vectors. In a general spacetime $\Omega$ is the product of the particle’s
energy $\omega$ as measured by an arbitrary observer and the appropriate
redshift factor $V=\sqrt{-g_{tt}}$. Substituting this in (20) we get,
$\displaystyle\Omega=\pm\sqrt{fg}\frac{d\tilde{I}_{0}}{dr}.$ (22)
Integrating the above equation we find
$\displaystyle\tilde{I}_{0}(r)=\pm\Omega\int_{0}^{r}\frac{dr}{\sqrt{fg}}$ (23)
where the limits of the integration are taken such that the particle passes
through the horizon $r=r_{H}$. The $+(-)$ sign indicates that the particle is
ingoing (outgoing). Combination of (21) and (23) gives the solution for
$I_{0}(r,t)$
$\displaystyle I_{0}(r,t,\theta,\phi)=\Omega
t\pm\Omega\int_{0}^{r}\frac{dr}{\sqrt{fg}}+I_{0}^{\prime}(\theta,\phi).$ (24)
Making use of the relations (11) and (24) in the semiclassical limit we obtain
the ingoing and outgoing solutions of the Maxwell equation in curved spacetime
$\displaystyle A_{\mu{\textrm{(in)}}}\sim a_{\mu
0}{\textrm{exp}}\Big{[}-\frac{i}{\hbar}\Big{(}\Omega
t+\Omega\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}+I_{0}^{\prime}(\theta,\phi)\Big{)}\Big{]}$
(25)
and
$\displaystyle A_{\mu{\textrm{(out)}}}\sim a_{\mu
0}{\textrm{exp}}\Big{[}-\frac{i}{\hbar}\Big{(}\Omega
t-\Omega\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}+I_{0}^{\prime}(\theta,\phi)\Big{)}\Big{]}.$
(26)
When a particle tunnels through the horizon, the sign of the metric
coefficient in the $(r-t)$ sector changes. This suggests that there is an
imaginary part in the time coordinate for the crossing of the black hole
horizon and therefore a temporal contribution will appear in the expressions
of probabilities for the ingoing and outgoing particles.
Thus the ingoing and outgoing probabilities of the particle are given by,
$\displaystyle
P_{{\textrm{in}}}=|A_{\mu{\textrm{(in)}}}|^{2}\sim{\textrm{exp}}\Big{[}\frac{2}{\hbar}\Big{(}\Omega{\textrm{Im}}~{}t+\Omega{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}$
(27)
and
$\displaystyle
P_{{\textrm{out}}}=|A_{\mu{\textrm{(out)}}}|^{2}\sim{\textrm{exp}}\Big{[}\frac{2}{\hbar}\Big{(}\Omega{\textrm{Im}}~{}t-\Omega{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{)}\Big{]}.$
(28)
Note that the angular contribution $I_{0}^{\prime}(\theta,\phi)$ does not
appear in the above expressions of probabilities. In the limit
$\hbar\rightarrow 0$, everything is absorbed in the black hole and hence the
ingoing probability $P_{\textrm{in}}$ must be unity. Therefore, in this limit,
(27) yields,
$\displaystyle{\textrm{Im}}~{}t=-{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}.$
(29)
It must be noted that the above relation satisfies the classical condition
$\frac{\partial I_{0}}{\partial\Omega}=$ constant. This is understood by the
following argument. Calculating the left side of this condition from (24) we
obtain,
$\displaystyle t=\mp\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}$ (30)
where $-(+)$ sign indicates that the particle is ingoing (outgoing). So for an
ingoing particle this condition immediately yields (29). On the other hand a
naive substitution of ‘Im$~{}t$’ in (28) from (30) for the outgoing particle
gives $P_{{\textrm{out}}}=1$. But it must be noted that according to classical
general theory of relativity, a particle can be absorbed in the black hole,
while the reverse process is forbidden. In this regard, ingoing classical
trajectory exists while the outgoing classical trajectory is forbidden. Hence
use of the classical condition for outgoing particle is meaningless.
Now to find out ‘Im$~{}t$’ for the outgoing particle, we will take the help of
the Kruskal coordinates which are well behaved throughout the space-time. The
Kruskal time ($T$) and space ($X$) coordinates inside and outside the horizon
are defined as [22]
$\displaystyle T_{is}=e^{\kappa r^{*}_{is}}\cosh\\!\left(\kappa
t_{is}\right)~{}~{};\hskip 17.22217ptX_{is}=e^{\kappa
r^{*}_{is}}\sinh\\!\left(\kappa t_{is}\right)$ (31) $\displaystyle
T_{os}=e^{\kappa r^{*}_{os}}\sinh\\!\left(\kappa t_{os}\right)~{}~{};\hskip
17.22217ptX_{os}=e^{\kappa r^{*}_{os}}\cosh\\!\left(\kappa t_{os}\right)$ (32)
where $\kappa$ is the surface gravity defined by
$\displaystyle\kappa=\frac{1}{2}\sqrt{f^{\prime}(r_{H})g^{\prime}(r_{H})}~{}.$
(33)
Here ‘$is(os)$’ stands for the inside (outside) the event horizon while
$r^{*}$ is the tortoise coordinate, defined by
$\displaystyle r^{*}=\int\frac{dr}{\sqrt{f(r)g(r)}}~{}.$ (34)
These two sets of coordinates are connected through the following relations
$\displaystyle t_{is}=t_{os}-i\frac{\pi}{2\kappa}$ (35) $\displaystyle
r^{*}_{is}=r^{*}_{os}+i\frac{\pi}{2\kappa}$ (36)
so that the Kruskal coordinates get identified as $T_{is}=T_{os}$ and
$X_{is}=X_{os}$. This indicates that when a particle travels from inside to
outside the horizon, ‘$t$’ coordinate picks up an imaginary term
$-\frac{\pi}{2{\kappa}}$. This is precisely given by (29). It is noteworthy
that exactly the same imaginary temporal contribution was needed to solve the
problem of factor $2$ in the expression of black hole temperature. This was
first proposed in [10]. A more elaborate discussion on the method we follow in
the present paper may be found in [9, 23].
Therefore, using (29) in (28) we get the probability for the outgoing particle
$\displaystyle
P_{{\textrm{out}}}\sim{\textrm{exp}}\Big{[}-\frac{4}{\hbar}\Omega{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{]}.$
(37)
Now if an observer at infinity (i.e. $r\rightarrow\infty$) observes the same
tunneling process (corresponding to Hawking effect) with energy $\omega$ and
temperature $T_{H}$, then it reads the principle of “detailed balance” as
$\displaystyle\frac{P_{{\textrm{out}}}}{P_{{\textrm{in}}}}=e^{-\omega/T_{H}}.$
(38)
Since $P_{{\textrm{in}}}=1$, the above equation leads to
$\displaystyle P_{{\textrm{out}}}=e^{-\omega/T_{H}}.$ (39)
Now at $r\rightarrow\infty$, $\sqrt{-g_{tt}}=1$ and so $\Omega=\omega$.
Therefore comparing (37) and (39) we get the black hole temperature as
$\displaystyle
T_{H}=\frac{\hbar}{4}\Big{[}{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{]}^{-1}.$
(40)
This is the standard Hawking temperature obtained earlier by using tunneling
method of scalar [14] or Dirac [12] particle. This confirms that a black hole
can radiate any type of particle like a black body.
### 2.2 Method 2
In this subsection we study the same problem, namely, the tunneling of photon
using Hamilton–Jacobi method, but taking the action (7) without the gauge
fixing term. So our action reads
$\displaystyle S=-\frac{1}{4}\int F_{\mu\nu}F^{\mu\nu}\sqrt{-g}d^{4}x$ (41)
and we take care of the gauge invariance of the theory by imposing the Lorentz
gauge condition later. An arbitrary variation of $A_{\mu}$ in the action (41)
gives the equation of motion
$\displaystyle\nabla^{\mu}F_{\mu\nu}=0.$ (42)
Using the standard method of calculating the covariant derivative of a tensor
field we write the above equation as,
$\displaystyle
g^{\nu\alpha}[\partial_{\alpha}F_{\mu\nu}-\Gamma^{\lambda}_{\alpha\mu}F_{\lambda\nu}-\Gamma^{\lambda}_{\alpha\nu}F_{\mu\lambda}]=0.$
(43)
Using the definition of the field tensor $F_{\mu\nu}$ (4) in the above
equation and then substituting expressions (12) and (13) together with
expansions (15) and (16) we get order $\mathcal{O}(\hbar^{0})$ equation as
$\displaystyle g^{\nu\alpha}(-a_{\nu
0}\partial_{\alpha}I_{0}\partial_{\mu}I_{0}+a_{\mu
0}\partial_{\alpha}I_{0}\partial_{\nu}I_{0})=0.$ (44)
This is not the semiclassical Hamilton-Jacobi equation (20). Also it is not
possible to obtain solutions for $I_{0}(r,t)$ in terms of metric coefficients.
Therefore, in order to proceed further, now we impose the Lorentz gauge in the
curved spacetime
$\displaystyle\partial_{\mu}\big{(}\sqrt{-g}A^{\mu}\big{)}=0.$ (45)
This can be equivalently written as,
$\displaystyle\nabla^{\mu}A_{\mu}\equiv
g^{\mu\nu}(\partial_{\nu}A_{\mu}-\Gamma^{\sigma}_{\nu\mu}A_{\sigma})=0.$ (46)
Again using (12,13) and (15,16) in the above equation and comparing
$\hbar^{0}$ order terms on both sides we find,
$\displaystyle g^{\nu\alpha}a_{\nu 0}\partial_{\alpha}I_{0}=0.$ (47)
Due to (47), (44) simplifies to (17) which ultimately gives the desired
semiclassical Hamilton-Jacobi equation (20) obtained in the previous
subsection. Rest of the analysis to find the Hawking temperature is identical
to the previous study. Naturally, the resulting black hole temperature is
found to be (40).
## 3 Correction to the semiclassical results
So far our analysis was restricted only upto semiclassical approximation. In
the present section we shall study the effects of quantum corrections on the
black hole temperature for the tunneling of photon. To do this, we can follow
either of the methods discussed in the previous section. Since the calculation
based on first method is found to be more complicated, here we follow the
second method and improve the previous analysis by incorporating the first
order quantum effects.
Substituting (15) and (16) in (43) and then equating first order quantum
correction ($\mathcal{O}(\hbar^{1})$) on both sides, we find
$\displaystyle g^{\nu\alpha}\Big{[}-i\partial_{\alpha}a_{\nu
0}\partial_{\mu}I_{0}-a_{\nu
1}\partial_{\alpha}I_{0}\partial_{\mu}I_{0}-a_{\nu
0}\partial_{\alpha}I_{1}\partial_{\mu}I_{0}-a_{\nu
0}\partial_{\alpha}I_{0}\partial_{\mu}I_{1}$ (48) $\displaystyle+$
$\displaystyle a_{\mu 1}\partial_{\alpha}I_{0}\partial_{\nu}I_{0}+a_{\mu
0}\partial_{\alpha}I_{1}\partial_{\nu}I_{0}+a_{\mu
0}\partial_{\alpha}I_{0}\partial_{\nu}I_{1}+i\partial_{\alpha}a_{\mu
0}\partial_{\nu}I_{0}\Big{]}$ $\displaystyle-$ $\displaystyle
g^{\nu\alpha}\Gamma^{\lambda}_{\alpha\mu}[-ia_{\nu
0}\partial_{\lambda}I_{0}+ia_{\lambda
0}\partial_{\nu}I_{0}]-g^{\nu\alpha}\Gamma^{\lambda}_{\alpha\nu}[-ia_{\lambda
0}\partial_{\mu}I_{0}+ia_{\mu 0}\partial_{\lambda}I_{0}]=0.$
Using (47) and (17) we simplify (48) to get
$\displaystyle-
ig^{\nu\alpha}(\partial_{\mu}I_{0})\Big{[}\partial_{\alpha}a_{\nu 0}-ia_{\nu
1}\partial_{\alpha}I_{0}-ia_{\nu
0}\partial_{\alpha}I_{1}-\Gamma^{\lambda}_{\alpha\nu}a_{\lambda 0}\Big{]}$
(49) $\displaystyle+$ $\displaystyle
g^{\nu\alpha}\Big{[}i\partial_{\alpha}a_{\mu 0}\partial_{\nu}I_{0}+a_{\mu
0}\partial_{\alpha}I_{1}\partial_{\nu}I_{0}+a_{\mu
0}\partial_{\alpha}I_{0}\partial_{\nu}I_{1}\Big{]}$ $\displaystyle-$
$\displaystyle g^{\nu\alpha}\Gamma^{\lambda}_{\alpha\mu}[-ia_{\nu
0}\partial_{\lambda}I_{0}+ia_{\lambda
0}\partial_{\nu}I_{0}]-ig^{\nu\alpha}\Gamma^{\lambda}_{\alpha\nu}a_{\mu
0}\partial_{\lambda}I_{0}]=0.$
This equation alone is not sufficient to find the solution of $I_{1}$. As
before we need to use the gauge condition (46). Substituting (15) and (16) in
(46) and then equating the terms of the order of $\hbar^{1}$ on both sides we
get
$\displaystyle g^{\nu\alpha}\Big{(}\partial_{\alpha}a_{\nu 0}-ia_{\nu
1}\partial_{\alpha}I_{0}-ia_{\nu
0}\partial_{\alpha}I_{1}-\Gamma^{\lambda}_{\alpha\nu}a_{\lambda 0}\Big{)}=0.$
(50)
Using (50) in (49) we obtain
$\displaystyle g^{\nu\alpha}\Big{[}i\partial_{\alpha}a_{\mu
0}\partial_{\nu}I_{0}+a_{\mu
0}\partial_{\alpha}I_{1}\partial_{\nu}I_{0}+a_{\mu
0}\partial_{\alpha}I_{0}\partial_{\nu}I_{1}\Big{]}$ (51) $\displaystyle-$
$\displaystyle g^{\nu\alpha}\Gamma^{\lambda}_{\alpha\mu}[-ia_{\nu
0}\partial_{\lambda}I_{0}+ia_{\lambda
0}\partial_{\nu}I_{0}]-ig^{\nu\alpha}\Gamma^{\lambda}_{\alpha\nu}a_{\mu
0}\partial_{\lambda}I_{0}]=0.$
Since only ($r-t$) sector of the metric is relevant in our analysis, the above
expression can be expanded as
$\displaystyle-\frac{1}{f}\Big{[}i\partial_{t}a_{\mu
0}\partial_{t}I_{0}+2a_{\mu
0}\partial_{t}I_{1}\partial_{t}I_{0}\Big{]}+g\Big{[}i\partial_{r}a_{\mu
0}\partial_{r}I_{0}+2a_{\mu 0}\partial_{r}I_{1}\partial_{r}I_{0}\Big{]}$
$\displaystyle-i\frac{1}{f}\Gamma^{r}_{t\mu}[a_{t0}\partial_{r}I_{0}-a_{r0}\partial_{t}I_{0}]+ig\Gamma^{t}_{r\mu}[a_{r0}\partial_{t}I_{0}-a_{t0}\partial_{r}I_{0}]$
$\displaystyle+\frac{i}{f}\Gamma^{r}_{tt}a_{\mu
0}\partial_{r}I_{0}-ig\Gamma^{r}_{rr}a_{\mu 0}\partial_{r}I_{0}=0.$ (52)
Making use of (20) and ($r-t$) component of (47) we reduce the above equation
as
$\displaystyle-\frac{1}{f}[\pm i\sqrt{fg}\partial_{t}a_{\mu 0}\pm 2a_{\mu
0}\sqrt{fg}\partial_{t}I_{1}]+g[i\partial_{r}a_{\mu 0}+2a_{\mu
0}\partial_{r}I_{1}]$ $\displaystyle+\frac{i}{f}\Gamma^{r}_{tt}a_{\mu
0}-ig\Gamma^{r}_{rr}a_{\mu 0}=0.$ (53)
Since the terms independent of the single particle action $I$ will not
contribute to the thermodynamic quantities, we drop them from (53) to find,
$\displaystyle\partial_{t}I_{1}=\pm\sqrt{fg}\partial_{r}I_{1}.$ (54)
This equation is quite analogous to its semiclassical counterpart (20).
Comparing this with (20) we get
$\displaystyle(\partial_{t}I_{i})=\pm\sqrt{fg}(\partial_{r}I_{i})$ (55)
for $i=0$ and 1. This implies that the solution of these equations are not
independent and $I_{1}$ is proportional to $I_{0}$. Thus (15) is written as
$\displaystyle I(r,t,\theta,\phi)=I_{0}(r,t,\theta,\phi)+\hbar\gamma
I_{0}(r,t,\theta,\phi)$ (56)
where $\gamma$ is the proportionality constant. Since the action $I$ has the
dimension of $\hbar$, the proportionality constant ($\gamma$) should have the
dimension of $\hbar^{-1}$. Again in our units $G=c=k_{B}=1$ and $\hbar$ has
the dimension of mass square. So $\gamma$ is of the form
$\displaystyle\gamma=\frac{\beta_{1}}{M^{2}},$ (57)
where $M$ is the mass of the black hole, the only mass parameter that appears
in the problem. $\beta_{1}$ is some dimensionless constant having value such
that quantum correction is of the order of $\hbar$. Combining (56) and (57) we
get
$\displaystyle I=\big{(}1+\beta_{1}\frac{\hbar}{M^{2}}\big{)}I_{0}.$ (58)
Therefore to obtain a solution of $I$ upto $\hbar^{1}$ order, it is sufficient
to solve $I_{0}$. The solution for $I_{0}$ was obtained in the previous
section which is given by (24). Substituting (24) in (58) we get the action
$\displaystyle I=\big{(}1+\beta_{1}\frac{\hbar}{M^{2}}\big{)}\left[\Omega
t\pm\Omega\int_{0}^{r}\frac{dr}{\sqrt{fg}}+I_{0}^{\prime}(\theta,\phi)\right].$
(59)
Above equation contains the quantum correction together with the standard
semiclassical term. Expectedly in the limit $\hbar\rightarrow 0$, (59) reduces
to (24).
Having obtained the solution of the single particle action, we can follow the
analysis of subsection 2.1 in a straight forward manner to calculate the black
hole temperature. The modified Hawking temperature upto first order quantum
correction thus obtained is
$\displaystyle T_{h}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{4}\Big{[}\big{(}1+\beta_{1}\frac{\hbar}{M^{2}}\big{)}{\textrm{Im}}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}\Big{]}^{-1}$
(60) $\displaystyle=$ $\displaystyle
T_{H}\Big{(}1+\beta_{1}\frac{\hbar}{M^{2}}\Big{)}^{-1}$
where $T_{H}$ is the semiclassical Hawking temperature given by (40). This
expression exactly matches with earlier results upto $\hbar^{1}$ order for
scalar [14] or Dirac [12] particle tunneling.
## 4 Gravitino tunneling beyond semiclassical approximation
We follow the method discussed in the previous section to study gravitino
tunneling. As claimed in the introduction, our analysis goes beyond the
semiclassical approximation by incorporating all possible quantum corrections.
The semiclassical Hawking temperature is shown to be altered properly.
The Rarita-Schwinger equation[19] for the massless spin-$3/2$ fermion in a
curved spacetime background is given by
$\displaystyle i\gamma^{\mu}\nabla_{\mu}\psi_{\nu}=0,$ (61)
together with a constraint
$\displaystyle\gamma^{\mu}\psi_{\mu}=0$ (62)
to ensure that there is no Dirac state in $\psi$. Here
$\psi_{\nu}\equiv\psi_{\nu a}$ is a vector valued spinor and the covariant
derivative is defined in the usual way,
$\displaystyle\nabla_{\mu}=\partial_{\mu}+\frac{i}{2}\Gamma{{}^{\alpha}}{{}_{\mu}}{{}^{\beta}}\Sigma_{\alpha\beta};\,\,\
\Gamma{{}^{\alpha}}{{}_{\mu}}{{}^{\beta}}=g^{\beta\nu}\Gamma^{\alpha}_{\mu\nu};\,\,\
\Sigma_{\alpha\beta}=\frac{i}{4}\Big{[}\gamma_{\alpha},\gamma_{\beta}\Big{]}.$
(63)
We take the following representations of the $\gamma$ matrices
$\displaystyle\gamma^{t}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{f(r)}}\left(\begin{array}[]{c c}i&0\\\
0&-i\end{array}\right);\,\,\ \gamma^{r}=\sqrt{g(r)}\left(\begin{array}[]{c
c}0&\sigma^{3}\\\ \sigma^{3}&0\end{array}\right)$ (68)
$\displaystyle\gamma^{\theta}$ $\displaystyle=$
$\displaystyle\frac{1}{r}\left(\begin{array}[]{c c}0&\sigma^{1}\\\
\sigma^{1}&0\end{array}\right);\,\,\,\
\gamma^{\phi}=\frac{1}{r\textrm{sin}\theta}\left(\begin{array}[]{c
c}0&\sigma^{2}\\\ \sigma^{2}&0\end{array}\right)$ (73)
which satisfy $\\{\gamma^{\mu},\gamma^{\nu}\\}=2g^{\mu\nu}$. Since we are
working only with the radial trajectories, the $(r-t)$ sector of the metric
(1) is important. Hence (61) is expressed as
$\displaystyle
i\gamma^{\mu}\partial_{\mu}\psi_{\nu}-\frac{1}{2}\Big{(}g^{tt}\gamma^{\mu}\Gamma^{r}_{\mu
t}-g^{rr}\gamma^{\mu}\Gamma^{t}_{\mu r}\Big{)}\Sigma_{rt}\psi_{\nu}=0.$ (74)
Substituting the metric coefficients and the non-vanishing connections
$\displaystyle\Gamma^{r}_{tt}=\frac{f^{\prime}g}{2};\,\,\
\Gamma^{t}_{tr}=\frac{f^{\prime}}{2f}$ (75)
for the metric (1) in (74), we get the following equation
$\displaystyle
i\gamma^{t}\partial_{t}\psi_{\mu}+i\gamma^{r}\partial_{r}\psi_{\mu}+\frac{f^{\prime}g}{2f}\gamma^{t}\Sigma_{rt}\psi_{\mu}=0.$
(76)
We take the following ansatz for the wave function
$\displaystyle\psi_{\mu}(t,r)=\left(\begin{array}[]{c}A_{\mu}(t,r)\\\
B_{\mu}(t,r)\\\ C_{\mu}(t,r)\\\
D_{\mu}(t,r)\end{array}\right)=\left(\begin{array}[]{c}a_{\mu}\\\ b_{\mu}\\\
c_{\mu}\\\
d_{\mu}\end{array}\right){\textrm{exp}}\Big{[}-\frac{i}{\hbar}I(t,r)\Big{]}$
(85)
where $I(r,t)$ is the action. Using this ansatz and calculating the value of
$\Sigma$ from (63)
$\displaystyle\Sigma_{rt}=\frac{i}{2}\left(\begin{array}[]{c c c
c}0&0&i\sqrt{\frac{f}{g}}&0\\\ 0&0&0&-i\sqrt{\frac{f}{g}}\\\
-i\sqrt{\frac{f}{g}}&0&0&0\\\ 0&i\sqrt{\frac{f}{g}}&0&0\end{array}\right),$
(90)
we write (76) component-wise as,
$\displaystyle\frac{\hbar}{\sqrt{f}}(\partial_{t}a_{\mu})+\frac{i}{\sqrt{f}}a_{\mu}(\partial_{t}I)-i\hbar\sqrt{g}(\partial_{r}c_{\mu})+\sqrt{g}c_{\mu}(\partial_{r}I)-\frac{\hbar
f^{\prime}\sqrt{g}}{2f}c_{\mu}=0$ (91)
$\displaystyle\frac{\hbar}{\sqrt{f}}(\partial_{t}b_{\mu})+\frac{i}{\sqrt{f}}b_{\mu}(\partial_{t}I)+i\hbar\sqrt{g}(\partial_{r}d_{\mu})-\sqrt{g}d_{\mu}(\partial_{r}I)+\frac{\hbar
f^{\prime}\sqrt{g}}{2f}d_{\mu}=0$ (92)
$\displaystyle-\frac{\hbar}{\sqrt{f}}(\partial_{t}c_{\mu})-\frac{i}{\sqrt{f}}c_{\mu}(\partial_{t}I)-i\hbar\sqrt{g}(\partial_{r}a_{\mu})+\sqrt{g}a_{\mu}(\partial_{r}I)-\frac{\hbar
f^{\prime}\sqrt{g}}{2f}a_{\mu}=0$ (93)
$\displaystyle-\frac{\hbar}{\sqrt{f}}(\partial_{t}d_{\mu})-\frac{i}{\sqrt{f}}d_{\mu}(\partial_{t}I)+i\hbar\sqrt{g}(\partial_{r}b_{\mu})-\sqrt{g}b_{\mu}(\partial_{r}I)+\frac{\hbar
f^{\prime}\sqrt{g}}{2f}b_{\mu}=0.$ (94)
Here, we ignore the constraint equation (62) since they are not important for
the solution of the action. In the above equations, the terms which do not
involve the single particle action will not contribute to the thermodynamic
entities of the black hole. Therefore we drop those terms to write (91)–(94)
as
$\displaystyle-\frac{i}{\sqrt{f}}a_{\mu}(\partial_{t}I)-\sqrt{g}c_{\mu}(\partial_{r}I)=0$
(95)
$\displaystyle-\frac{i}{\sqrt{f}}b_{\mu}(\partial_{t}I)+\sqrt{g}d_{\mu}(\partial_{r}I)=0$
(96)
$\displaystyle\frac{i}{\sqrt{f}}c_{\mu}(\partial_{t}I)-\sqrt{g}a_{\mu}(\partial_{r}I)=0$
(97)
$\displaystyle\frac{i}{\sqrt{f}}d_{\mu}(\partial_{t}I)+\sqrt{g}b_{\mu}(\partial_{r}I)=0.$
(98)
From (95) and (97) we note that $a_{\mu}$ and $c_{\mu}$ will have nonvanishing
values only when
$\displaystyle{\textrm{det}}\left(\begin{array}[]{c
c}-\frac{i}{\sqrt{f}}(\partial_{t}I)&-\sqrt{g}(\partial_{r}I)\\\
-\sqrt{g}(\partial_{r}I)&\frac{i}{\sqrt{f}}(\partial_{t}I)\end{array}\right)=0.$
(101)
This condition gives the result
$\displaystyle(\partial_{t}I)^{2}=fg(\partial_{r}I)^{2}$ (102)
or equivalently,
$\displaystyle\partial_{t}I=\pm\sqrt{fg}\partial_{r}I.$ (103)
Substituting (103) in (95) we get
$\displaystyle a_{\mu}=\pm ic_{\mu}.$ (104)
The above results can also be obtained by solving (96) and (98)
simultaneously. As before, we expand $I,a_{\mu},$ and $c_{\mu}$ in power
series of $\hbar$:
$\displaystyle
I(r,t)=I_{0}(r,t)+\displaystyle\sum_{i=1}^{\infty}\hbar^{i}I_{i}(r,t)$ (105)
$\displaystyle a_{\mu}=a_{\mu
0}+\displaystyle\sum_{i=1}^{\infty}\hbar^{i}a_{\mu i};\,\,\,c_{\mu}=c_{\mu
0}+\displaystyle\sum_{i=1}^{\infty}\hbar^{i}c_{\mu i}.$ (106)
Now substituting these in (103) and (104) and then equating different powers
of $\hbar$ on both sides of equation we obtain,
$\displaystyle\partial_{t}I_{i}=\pm\sqrt{fg}\partial_{r}I_{i}$ (107)
and
$\displaystyle a_{\mu i}=\pm ic_{\mu i}$ (108)
for $i=0,1,2,\cdot\cdot\cdot$. Note that (107) is same as (55) for $i=0,1$
which was obtained order by order for the photon field.
Now following the analysis presented in the earlier sections, we can calculate
the black hole temperature due to gravitino tunneling. The result thus
obtained is
$\displaystyle
T_{h}=T_{H}\big{(}1+\sum_{i=1}^{\infty}\beta_{i}\frac{\hbar^{i}}{M^{2i}}\big{)}^{-1}$
(109)
which upon first order approximation matches with (60), though the values of
the first order correction parameter $\beta_{1}$ for photon and gravitino are
not same. This point will be examined in detail in the next section.
Some comments on the corrected form of the Hawking temperature (60) are as
follows. For the Schwarzschild black hole $f(r)=g(r)=1-\frac{2M}{r}$.
Substituting this in (60) and performing the contour integration we obtain the
first order quantum corrected Hawking temperature as
$\displaystyle T_{h}=\frac{\hbar}{8\pi
M}\Big{(}1+\beta_{1}\frac{\hbar}{M^{2}}\Big{)}^{-1}.$ (110)
Using this corrected form of temperature and exploiting the Hawking’s
periodicity arguments one can find the corrected form of the Schwarzschild
metric upto $\hbar$ order as
$\displaystyle
ds^{2}_{{\textrm{corr}}}=-\Big{[}1-\frac{2M}{r}\Big{(}1+\beta_{1}\frac{\hbar}{M^{2}}\Big{)}\Big{]}dt^{2}+\frac{dr^{2}}{\Big{[}1-\frac{2M}{r}\Big{(}1+\beta_{1}\frac{\hbar}{M^{2}}\Big{)}\Big{]}}+r^{2}d\Omega^{2}.$
(111)
For detailed discussions see [15]. Therefore the fractional change of the
metric coefficients is $-\frac{\beta_{1}\hbar}{M^{2}}$ which is precisely the
result given in [20]. The previous derivation was based on the solution of
Einstein equation including the renormalized energy-momentum tensor for the
back reaction effect in the spacetime. In that case the coefficient (which is
$\beta_{1}$ for our case) is related to the number of different types of
fields. In the next section we shall explicitly show how our result matches
with earlier work [24, 25] which incorporates the effect of all loops back
reaction in the spacetime.
Now from the first law of thermodynamics $dS_{\textrm{bh}}=\frac{dM}{T_{h}}$,
it is easy to find the corrected form of the Bekenstein-Hawking entropy which
in this case is given by,
$\displaystyle S_{\textrm{bh}}=\frac{A}{4\hbar}+4\pi\beta_{1}\ln
A+{\textrm{higher order terms in $\hbar$}}$ (112)
where $A=16\pi M^{2}$ is the area of the event horizon of the Schwarzschild
black hole. The first term is the usual semiclassical result and the second
term is the logarithmic correction [26, 27, 28, 29, 30, 14] which in this case
comes from $\hbar$ order correction to the one particle action and so on. In
the next section we will discuss a method of fixing the coefficients.
## 5 Discussions on correction parameter $\beta_{1}$
In this section we discuss about the undetermined coefficient $\beta_{1}$ for
both photon and gravitino cases. To determine this, we begin by studying the
behaviour of actions (58) and (105) for the photon tunneling first.
Apparently, one might think, for a zero rest mass field the trace of the
energy-momentum tensor ($T^{\mu}_{\mu}$) is zero. But the point is, at the
quantum level it is not possible to preserve the conformal and diffeomorphism
symmetries simultaneously. In fact, violation of the conformal invariance
leads to a nonvanishing $T^{\mu}_{\mu}$. For chiral theory, both of these
symmetries are violated and therefore both divergence and trace of energy-
momentum tensor are nonzero. This point has been rigorously studied for black
hole case in [31]. Throughout our analysis diffeomorphism symmetry is always
preserved and so we connect $\beta_{1}$ only with the trace anomaly. This is
done by simple scaling argument which was originally initiated by Hawking
[32].
Under an infinitesimal constant scale transformation, parametrized by $k$, the
metric coefficients change as,
$\displaystyle\tilde{g}{{}_{\mu\nu}}=kg_{\mu\nu}\simeq(1+\delta k)g_{\mu\nu}.$
(113)
Due to this transformation, the coefficients of $(r-t)$ sector of the metric
(1) change as $\tilde{f}=kf,\tilde{g}=k^{-1}g$. Also, to preserve the scale
invariance of the Lorentz gauge condition (45), the field $A^{\mu}$ transforms
as $\tilde{A^{\mu}}=k^{-2}A^{\mu}$. On the other hand, the action (41) for
photon field shows that $A^{\mu}$ has the dimension of mass. Since the only
mass parameter we have in this problem is the black hole mass $M$, the
infinitesimal change of it is given by,
$\displaystyle\tilde{M}=k^{-2}M\simeq(1-2\delta k)M.$ (114)
Now from (24) and (29) the imaginary part of the semiclassical contribution of
the outgoing single particle action is
$\displaystyle\textrm{Im}I{{}_{0}}{{}_{(\textrm{out})}}=-2\Omega{\textrm{Im}}~{}\int_{0}^{r}\frac{dr}{\sqrt{f(r)g(r)}}$
(115)
where for $r\rightarrow\infty$, $\Omega=\omega$ which gets identified with the
energy (i.e. mass $M$) of a stable black hole [30]. Therefore $\omega$
transforms according to $M$ under (113). Considering the imaginary part of the
term $\mathcal{O}(\hbar)$ in (59), we get, under the scale transformation,
$\displaystyle{\tilde{\cal{A}}}_{(1)}$ $\displaystyle\equiv$
$\displaystyle{\hbar}\textrm{Im}\tilde{I}{{}_{1}}{{}_{(\textrm{out})}}=\Big{(}\frac{{\hbar}\beta_{1}}{\tilde{M}^{2}}\Big{)}\textrm{Im}\tilde{I}{{}_{0}}{{}_{(\textrm{out})}}.$
Using (114) we write the above equation as
$\displaystyle{\tilde{\cal{A}}}_{(1)}$ $\displaystyle\simeq$
$\displaystyle\Big{(}\frac{\hbar\beta_{1}}{{M}^{2}}\Big{)}(1+2\delta
k)\textrm{Im}I{{}_{0}}{{}_{(\textrm{out})}}$ (116) $\displaystyle=$
$\displaystyle{\cal{A}}_{(1)}+\Big{(}\frac{\hbar\beta_{1}}{{M}^{2}}\Big{)}2\delta
k\textrm{Im}I{{}_{0}}{{}_{(\textrm{out})}}.$
Therefore the change of ${\cal{A}}_{(1)}$ is given by,
$\displaystyle\delta{\cal{A}}_{(1)}$ $\displaystyle=$
$\displaystyle{\tilde{\cal{A}}}_{(1)}-{\cal{A}}_{(1)}$ (117)
$\displaystyle\simeq$
$\displaystyle\Big{(}\frac{\hbar\beta_{1}}{{M}^{2}}\Big{)}2\delta
k\textrm{Im}I{{}_{0}}{{}_{(\textrm{out})}}$
which leads to the following equation,
$\displaystyle\frac{\delta{\cal{A}}_{(1)}}{\delta
k}=2\Big{(}\frac{\hbar\beta_{1}}{{M}^{2}}\Big{)}\textrm{Im}I{{}_{0}}{{}_{(\textrm{out})}}.$
(118)
At this point we use of the definition of energy-momentum tensor in the above
equation to get,
$\displaystyle\textrm{Im}\int
d^{4}x\sqrt{-g}T_{\mu}^{\mu}=\frac{2\delta{\cal{A}}_{(1)}}{\delta
k}=4\Big{(}\frac{\hbar\beta_{1}}{{M}^{2}}\Big{)}\textrm{Im}I{{}_{0}}{{}_{(\textrm{out})}}.$
(119)
From (119) it is clear that, in the presence of trace anomaly, the action is
not invariant under the scale transformation. Since for the Schwarzschild
black hole $f(r)=g(r)=1-\frac{2M}{r}$, from (115) we obtain
$\textrm{Im}I_{0{(\textrm{out})}}=-4\pi\omega M$. Substitution of this result
in (119) for $\omega=M$ we find
$\displaystyle\hbar\beta_{1}=-\frac{1}{16\pi}\textrm{Im}\int
d^{4}x\sqrt{-g}T_{\mu}^{\mu}.$ (120)
Since the higher loop calculations to get $T_{\mu\nu}$ (from which
$T_{\mu}^{\mu}$ is obtained) is very much complicated, usually in literature
[33] only one loop calculation for $T_{\mu\nu}$ is discussed. Thus, comparing
only the $\hbar^{1}$ order on both sides of (120), we obtain,
$\displaystyle\beta_{1}=-\frac{1}{16\pi}{\textrm{Im}}\int
d^{4}x\sqrt{-g}{T^{\mu}_{\mu}}^{(1)}.$ (121)
This relation clearly shows that $\beta_{1}$ is connected to the trace
anomaly.
The correction to the black hole entropy (which is proportional to
$\beta_{1}$) was calculated by Hawking himself and he showed it to be related
to the trace anomaly [32]. This was done by path integral approach based on
zeta function regularization where the path integral was modified by taking
into account the effect of fluctuations coming from the scalar field. The
entropy expression found was
$\displaystyle
S_{\textrm{bh}}=\frac{A}{4\hbar}-\frac{1}{2}\Big{(}{\textrm{Im}}\int
d^{4}x\sqrt{-g}T^{\mu}_{\mu}\Big{)}\ln A$ (122)
which is equivalent to our result (112). The coefficient of the logarithmic
term of the above expression matches with (121) apart from a numerical factor.
This mismatch in the numerical factor is a consequence of the fact that we
have considered the photon field instead of scalar field. Previously, it has
been established [34] that upto order $\hbar$, the result obtained from WKB
ansatz are equivalent to the path integral result. Therefore, it is not
surprising that the result obtained here from simple scaling arguments, is
consistent with the path integral approach.
Following the identical analysis for the gravitino case one can immediately
show that
$\displaystyle\beta_{1}|_{\textrm{gravitino}}=\frac{3}{8\pi}{\textrm{Im}}\int
d^{4}x\sqrt{-g}{T_{\mu}}{{}^{\mu}}^{(1)}|_{\textrm{gravitino}}$ (123)
where ${T_{\mu}}{{}^{\mu}}^{(1)}|_{\textrm{gravitino}}$ is the trace of the
renormalized energy-momentum tensor of gravitino upto one loop expansion.
Similar relations were given earlier for scalar particle [15] and spin 1/2
particle [12] where it has been shown that the coefficient $\beta_{1}$ is
related to trace anomaly. The only difference is the factor before the
integration. This agrees well with the earlier conclusion [35, 26] where using
conformal field theory technique, it was shown that $\beta_{1}$ is related to
trace anomaly and is given by,
$\displaystyle\beta_{1}=-\frac{1}{360\pi}\Big{(}-N_{0}-\frac{7}{4}N_{\frac{1}{2}}+13N_{1}+\frac{233}{4}N_{\frac{3}{2}}-212N_{2}\Big{)}.$
(124)
Here ‘$N_{s}$’ denotes the number of fields with spin ‘$s$’. For gauge field
case $N_{1}=1$ and $N_{0}=N_{\frac{1}{2}}=N_{\frac{3}{2}}=N_{2}=0$ whereas for
gravitino case $N_{\frac{3}{2}}=1$ and $N_{0}=N_{\frac{1}{2}}=N_{1}=N_{2}=0$.
## 6 Conclusions
We have shown that photon and gravitino can tunnel through the event horizon
of a black hole just like spin zero and spin half particles. Thus our present
work is a natural extension of the Hamilton–Jacobi method previously developed
in [12, 14, 15, 16]. In case of photon tunneling, presence of gauge freedom
makes the analysis more complicated than the studies for other particles.
Nevertheless we have successfully employed the Hamilton–Jacobi method to
compute the semiclassical single particle action, and from that, the tunneling
amplitude of photon. This has been done by following two different methods. In
the first method, we started from a gauge fixed action and calculated the
equations of motion for the photon field in a general curved spacetime
background. Using the Hamilton–Jacobi ansatz in this equation of motion we
obtained the single particle action and tunneling amplitude. After that,
principle of detailed balance has been used to recover the semiclassical
Hawking temperature. In the other method, starting from the standard Maxwell
action in a curved geometry, we follow the previous analysis to obtain a
differential equation of the single particle action. Only at this point we
used the gauge freedom of photon by considering the Lorentz gauge condition.
This, under semiclassical approximation, gave another differential equation.
Combination of these two equations produce the same solution of the action.
This naturally gave the same semiclasical black hole temperature.
In this paper we have also improved the semiclassical results by incorporating
first order quantum effects in the theory. For that we generalized the second
method by taking into account the $\hbar$ order equations which come from the
Lorentz gauge condition and the Maxwell equation in a gravitational
background. Interestingly, it has been found that the correction term of the
single particle action is proportional to the semiclassical contribution –
exactly as happens for the scalar and Dirac particles. By dimensional
argument, the proportionality constant was shown to be related with the mass
of black hole. The corrected action eventually led to the modified Hawking
temperature which is in complete agreement with the result obtained
earlier[12, 14]. In our knowledge, existing analysis of tunneling formalism
involved emission of spin zero, spin $\frac{1}{2}$ or spin $\frac{3}{2}$
particles from black hole, without discussing the tunneling of photon. In that
sense our work fills an important gap present in the literature.
The formalism was applied equally well to the gravitino tunneling case.
Previously this was studied [13] only upto semiclassical level. Here we have
incorporated all the quantum corrections to get the modified black hole
temperature and the Bekenstein–Hawking area law. Expectedly, the area law
involved logarithmic area correction together with the standard inverse power
of area term. Finally, the coefficients of the logarithmic term of entropy
which was related with trace anomaly were calculated for both photon and
gravitino. This completed our analysis.
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|
arxiv-papers
| 2009-01-15T13:17:54 |
2024-09-04T02:48:59.985623
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bibhas Ranjan Majhi, Saurav Samanta",
"submitter": "Bibhas Majhi Ranjan",
"url": "https://arxiv.org/abs/0901.2258"
}
|
0901.2269
|
# An Excursion-Theoretic Approach to Stability of Discrete-Time Stochastic
Hybrid Systems
Debasish Chatterjee ETL I19
Physikstrasse 3
ETH Zürich
8092 Zürich
Switzerland
Ph. +41-44-632-2326
Fax: +41-44-632-1211 chatterjee@control.ee.ethz.ch and Soumik Pal C-547
Padelford Hall
Department of Mathematics
University of Washington, Seattle
WA 98195
Ph. +1-206-543-7832 soumik@math.washington.edu
###### Abstract.
We address stability of a class of Markovian discrete-time stochastic hybrid
systems. This class of systems is characterized by the state-space of the
system being partitioned into a safe or target set and its exterior, and the
dynamics of the system being different in each domain. We give conditions for
$\boldsymbol{L}_{1}$-boundedness of Lyapunov functions based on certain
negative drift conditions outside the target set, together with some more
minor assumptions. We then apply our results to a wide class of randomly
switched systems (or iterated function systems), for which we give conditions
for global asymptotic stability almost surely and in $\boldsymbol{L}_{1}$. The
systems need not be time-homogeneous, and our results apply to certain systems
for which functional-analytic or martingale-based estimates are difficult or
impossible to get.
###### Key words and phrases:
stochastic stability, excursion theory, Markov process
###### 2000 Mathematics Subject Classification:
Primary: 93E15; Secondary: 60J05
Debasish Chatterjee’s research is partially supported by the Swiss National
Science foundation grant 200021-122072.
## 1\. Introduction
Increasing complexity of engineering systems in the modern world has led to
the hybrid systems paradigm in systems and control theory [vS00, Lib03]. A
hybrid system consists of a number of domains in the state-space and a
dynamical law corresponding to each domain; thus, at any instant of time the
dynamics of the system depends on the domain that its state is in. One would
then restrict attention to behavior of the system in individual domains, which
is typically a simpler problem. However, understanding how the dynamics in the
individual domains interact among each other is necessary in order to ensure
smooth operation of the overall system. This article is a step towards
understanding the behavior of (possibly non-Markovian) stochastic hybrid
systems which undergo excursions into different domains infinitely often. Here
we consider the simplest and perhaps the most important hybrid system,
consisting of a compact target or safe set and its exterior, with different
dynamics inside and outside the safe set. Our objective is to introduce a new
method of analysis of systems that are outside the safe set infinitely often
in course of their evolution. The analysis carried out here provides a basis
for controller synthesis of systems with control inputs—it gives clear
indications about the type of controllers to be designed in order to ensure
certain natural and basic stability properties in closed loop.
Let us look at two interesting and practically important examples of hybrid
systems with two domains—a compact safe set and its exterior, with different
dynamics in each. The first concerns optimal control of a Markov process with
state constraints. Markov control processes have been extensively studied; we
refer the reader to the excellent monographs and surveys [BS78, Bor91, HLL96,
HLL99] for further information, applications and references. For our purposes
here, consider the canonical example of a linear controlled system perturbed
by additive Gaussian noise and having probabilistic constraints on the states.
A hybrid structure of the controlled system naturally presents itself in the
following fashion. Except in the most trivial of cases, computing the
constrained optimal control over an infinite horizon is impossible, and one
resorts to a a rolling-horizon controller. (Rolling-horizon controllers are
considerably popular, for basic definitions, comparisons and references see
e.g., [Mac01] in the deterministic context, and [CHL09] and the references
therein in the stochastic context.) Computational overheads restrict the size
of the window in the rolling-horizon controller, and determine the maximal
(typically bounded) region—called the safe set—in which this controller can be
active. No matter how good the resulting controller is, the additive nature of
the Gaussian noise ensures that the states are subjected to excursions away
from the safe set infinitely often almost surely. Once outside the safe set,
the rolling-horizon controller is switched off and a recovery strategy is
activated, whose task is to bring the states back to the safe set quickly and
efficiently. This problem is of great practical interest and a subject of
current research, see e.g., [CHL09] and the references in them for possible
strategies inside the safe set, and [CCCL08] for one possible recovery
strategy. Evidently, stability of this hybrid system depends largely on the
recovery strategy, since as long as the states stay inside the safe set, they
are bounded. However, traditional methods of stability analysis do not work
well precisely because of the unlimited number of excursions. Theorem (2.2) of
this article addresses this issue, and provides a method of ensuring strong
boundedness and stability properties of the hybrid system. Intuitively it says
that under the recovery strategy there exists a well-behaved supermartingale
until the states hit the safe set, then the system state is bounded in
expectation uniformly over time. A complete picture of stability and ergodic
properties of a general controlled hybrid system is beyond the scope of the
present article, and will be reported elsewhere. We refer the reader to
[CFM05, Chapter 3] for earlier work pertaining to stability of a class of
hybrid systems, and to [MT09] for stability of general discrete-time Markov
processes.
The second example is one that we shall pursue further in this article,
namely, a class of discrete-time Markov processes called iterated function
systems [BDEG88, LM94] (ifs). They are widely applied, for instance, in the
construction of fractals [LM94], in studies on the process of generation of
red blood corpuscles [LM02, LS04], in statistical physics [Kif86], and
simulation of important stochastic processes [Wer05]. Of late they are being
employed in key problems of physical chemistry and computational biology,
namely, the behavior of the chemical master equation [Wil06, Chapter 6] (CME),
which governs the continuous-time stochastic (Markovian) reaction-kinetics at
very low concentrations (of the order of tens of molecules). Invariant
distributions, certain finite-time properties, and robustness properties with
respect to disturbances of the underlying Markov process are of interest in
modeling and analysis of unicellular organisms. It is well-known that the CME
is analytically intractable (see [JH07, ACK08] for special cases), but the
invariant distribution of the Markov process can be recovered from simulation
of the embedded Markov chain in a computationally efficient way [MA08]. This
embedded chain is an ifs taking values in a nonnegative integer lattice. From
a biological perspective, good health of a cell corresponds to the ifs
evolving in a safe region on an average, despite moderate disturbances to the
numbers of molecules involved in the key reactions. However, in most cases
compact invariant sets do not exist. It is therefore of interest to find
conditions under which, even though there are excursions of the states away
from a safe set infinitely often, the ifs is stochastically bounded, or some
strong stability properties hold. Theorem (2.2) of this article leads to
results (in §3) which address this issue.
This article unfolds as follows. §2 contains our main results—Theorem (2.2)
and (2.9), which provide conditions under which a Lyapunov function of the
states is $\boldsymbol{L}_{1}$-bounded. We establish this
$\boldsymbol{L}_{1}$-boundedness under the assumptions that a certain derived
process is a supermartingale outside a compact set, and some more minor
conditions.111It also seems conceivably possible that relaxed Foster-Lyapunov
inequalities as in [DFMS04, Condition $\mathbf{D}(\phi,V,C)$, p. 1356] arising
in the context of subgeometric convergence to a stationary distribution can be
employed in the construction of the aforementioned supermartingale; this
constitutes future work. (The supermartingale condition alone is not enough,
as pointed out in [PR99], where the authors establish variants of our results
for scalar, possibly non-Markovian processes having increments with bounded
$p$-th moments for $p>2$.) For our results to hold, the underlying process
need not be time-homogeneous or Markovian. To wit, in §2.2 we define a class
of hybrid processes that switch between two Markov processes depending on
whether they inside or outside a fixed set in the state-space, and demonstrate
that although the resulting process may be non-Markovian, our results continue
to hold. Connections to optimal stopping problems are drawn in §2.3, which
gives a systematic procedure for verifying our assumptions. In §2.4 we apply
the techniques our techniques to a class of sampled diffusion processes. In
addition to the cases considered here, the results in §2 will be of interest
in queueing theory, along the lines of the works [HLR96, BKR+01]. §3 contains
some applications of the results in §2 to stability and robustness of ifs. The
classical weak stability questions concerning the existence and uniqueness of
invariant measures of ifs, addressed in e.g., [DF99, JT01, Sza06], revolve
around average contractivity hypotheses of the constituent maps and continuity
of the probabilities. In §3.1 we look at stronger stability properties of the
ifs, namely, global asymptotic stability almost surely and in expectation, for
which we give sufficient conditions. There are no assumptions of global
contractivity or memoryless choice of the maps at each iterate; we just
require a condition resembling average contractivity in terms of Lyapunov
functions with a suitable coupling condition with the Markovian transition
probabilities. We mention that although some of the assumptions in [JT01]
resemble ours, the conditions needed to establish existence of invariant
measures in [JT01] are stronger than what we employ; see §3.1 for a detailed
comparison. We also demonstrate in §3.2 that under mild assumptions, iterated
function systems possess strong stability and robustness properties with
respect to bounded disturbances. In this subsection the exogenous bounded
disturbance is not modeled as a random process.
#### Notations
Let $\mathbb{N}\coloneqq\\{1,2,\ldots\\}$,
$\mathbb{N}_{0}\coloneqq\\{0,1,2,\ldots\\}$, and $\mathbb{R}_{\geqslant
0}\coloneqq[0,\infty[$. We let $\left\lVert\cdot\right\rVert$ denote the
standard Euclidean norm on $\mathbb{R}^{d}$. We let $\bar{B}_{r}$ denote the
closed Euclidean ball around $0$, i.e.,
$\bar{B}_{r}\coloneqq\bigl{\\{}y\in\mathbb{R}^{d}\big{|}\left\lVert
y\right\rVert\leqslant r\bigr{\\}}$. For a vector $v\in\mathbb{R}^{d}$ let
$v^{\scriptscriptstyle{\mathrm{T}}}$ denote its transpose, and $\left\lVert
v\right\rVert_{P}$ denote $\sqrt{v^{\scriptscriptstyle{\mathrm{T}}}Pv}$ for a
$d\times d$ real matrix $P$. The maximum and minimum of two real numbers $a$
and $b$ is denoted by $a\vee b$ and $a\wedge b$, respectively.
## 2\. General results
Before we get into hybrid systems, it will be simpler to follow the arguments
if we start by considering a discrete-time Markov chain.
### 2.1. Obtaining $\boldsymbol{L}_{1}$ Bound using Excursions
Let $X\coloneqq(X_{t})_{t\in\mathbb{N}_{0}}$ be a discrete time Markov chain
with a state space $\mathcal{S}$. We denote the transition kernel of this
chain by $\mathsf{P}$, i.e., for every $x\in\mathcal{S}$, the probability
measure $\mathsf{P}_{x}(\cdot)\coloneqq\mathsf{P}(x,\cdot)$ determines the law
of $X_{t+1}$, conditioned on $X_{t}=x$. At this point we only assume the state
space $\mathcal{S}$ to be any Polish space.
###### (2.1) Assumption.
There exists a nonnegative function
$\varphi:\mathbb{N}_{0}\times\mathcal{S}\longrightarrow\mathbb{R}_{\geqslant
0}$ satisfying the following.
1. (i)
There exists a subset $K\subset\mathcal{S}$ such that the process
$(Y_{t})_{t\in\mathbb{N}_{0}}$ defined by $Y_{t}=\varphi(t,X_{t})$ is a
supermartingale under $\mathsf{P}_{x_{0}}$, for every
$x_{0}\in\mathcal{S}\setminus K$ until the first time $X_{t}$ hits $K$. To
wit, if $X_{0}=x_{0}\in\mathcal{S}\setminus K$ and we define
$\tau_{K}^{\vphantom{T}}=\inf\bigl{\\{}t>0\;\big{|}\;X_{t}\in K\bigr{\\}},$
then the process
$\bigl{(}Y_{t\wedge\tau_{K}^{\vphantom{T}}}\bigr{)}_{t\in\mathbb{N}_{0}}$ is a
supermartingale under $\mathsf{P}_{x_{0}}$.
2. (ii)
There exists a nonnegative measurable real-valued function
$V:\mathcal{S}\longrightarrow\mathbb{R}$ and a positive sequence
$(\theta(t))_{t\in\mathbb{N}_{0}}$ such that
$\varphi(t,x)\geqslant V(x)/\theta(t)\quad\text{for all
}(t,x)\in\mathbb{N}_{0}\times\mathcal{S},$
and $C\coloneqq\sum_{t\in\mathbb{N}_{0}}\theta(t)<\infty$.
3. (iii)
$\delta\coloneqq\sup_{x\in K}V(x)<\infty$. $\diamondsuit$
Our objective is to prove under the above condition (and another minor
assumptions) that there exists a bound on
$\sup_{t}\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\bigr{]}$ depending on $x_{0}$.
###### (2.2) Theorem.
Consider the setup in Assumption (2.1), and assume that
((2.3)) $\beta\coloneqq\sup_{x_{0}\in
K}\mathsf{E}\bigl{[}\varphi(0,X_{1})\boldsymbol{1}_{\\{X_{1}\in\mathcal{S}\setminus
K\\}}\;\big{|}\;X_{0}=x_{0}\bigr{]}<\infty.$
Let $\gamma\coloneqq\sup_{t\in\mathbb{N}_{0}}\theta(t)$. Then we have
$\sup_{t\in\mathbb{N}_{0}}\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\bigr{]}\leqslant
C\beta+\delta+\gamma\varphi(0,x_{0}).$
In the rest of this section we prove the above theorem. Fix a time
$t\in\mathbb{N}_{0}$, and define two random times
$g_{t}\coloneqq\sup\bigl{\\{}s\in\mathbb{N}_{0}\;\big{|}\;s\leqslant
t,\;X_{s}\in K\bigr{\\}}\quad\text{and}\quad
h_{t}\coloneqq\inf\bigl{\\{}s\in\mathbb{N}_{0}\;\big{|}\;s\geqslant
t,\;X_{s}\in K\bigr{\\}}.$
We follow the standard custom of defining supremum over empty sets to be
$-\infty$, and the infimum over empty sets to be $+\infty$.
Note that $g_{t}$ is not a stopping time with respect to the natural
filtration generated by the process $X$, although $h_{t}$ is. The random
interval $[g_{t},h_{t}]$ is a singleton if and only if $X_{t}\in K$.
Otherwise, we say that $X_{t}$ is within an excursion outside $K$.
Now we have the following decomposition:
((2.4))
$\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\bigr{]}=\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\boldsymbol{1}_{\\{g_{t}=-\infty\\}}\bigr{]}+\sum_{s=0}^{t}\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\boldsymbol{1}_{\\{g_{t}=s\\}}\bigr{]}.$
Our first objective is to bound each of the expectations
$\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\boldsymbol{1}_{\\{g_{t}=s\\}}\bigr{]}$.
Before we move on, let us first prove a Lemma which follows readily from
Assumption 1.
###### (2.5) Lemma.
Let $X_{0}=x_{0}\in\mathcal{S}\setminus K$. Then
((2.6))
$\mathsf{E}_{x_{0}}\bigl{[}V(X_{s})\boldsymbol{1}_{\\{\tau^{\vphantom{T}}_{K}>s\\}}\bigr{]}\leqslant\varphi(0,x_{0})\theta(s)\qquad\text{for
}s\in\mathbb{N}_{0},$
where $(\theta(t))_{t\in\mathbb{N}_{0}}$ is defined in Assumption (2.1).
###### Proof.
This is a straightforward application of Optional Sampling Theorem (OST) for
discrete-time supermartingales. Applying OST for the bounded stopping time
$s\wedge\tau^{\vphantom{T}}_{K}$ to the supermartingale
$\bigl{(}\varphi(t,X_{t})\bigr{)}_{t\in\mathbb{N}_{0}}$, in view of
$\varphi\geqslant 0$, we have
$\begin{split}\varphi(0,x_{0})&\geqslant\mathsf{E}_{x_{0}}\\!\bigl{[}\varphi(s\wedge\tau_{K}^{\vphantom{T}},X_{s\wedge\tau_{K}^{\vphantom{T}}})\bigr{]}\geqslant\mathsf{E}_{x_{0}}\bigl{[}\varphi(s,X_{s})\boldsymbol{1}_{\\{\tau^{\vphantom{T}}_{K}>s\\}}\bigr{]}.\end{split}$
Now, by condition (i) in Assumption (2.1), we can write $\varphi(s,x)\geqslant
V(x)/\theta(s)$. Thus, substituting back, one has
$\varphi(0,x_{0})\geqslant\mathsf{E}_{x_{0}}\bigl{[}V(X_{s})\boldsymbol{1}_{\\{\tau^{\vphantom{T}}_{K}>s\\}}\bigr{]}/\theta(s).$
Since $(\theta(t))_{t\in\mathbb{N}_{0}}$ is positive, we arrive at ((2.6)). ∎
We are ready for the proof of Theorem (2.2).
###### Proof of Theorem (2.2).
Let us consider three separate cases:
Case 1. ($-\infty<g_{t}<t$). In this case $g_{t}$ can take values
$\\{0,1,2,\ldots,t-1\\}$. Now, if $s\in\\{0,1,2,\ldots,t-1\\}$, then
$\begin{split}\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})&\boldsymbol{1}_{\\{g_{t}=s\\}}\bigr{]}=\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\boldsymbol{1}_{\\{X_{s}\in
K\\}}\boldsymbol{1}_{\\{X_{i}\notin K,\;i=s+1,\ldots,t\\}}\bigr{]}\\\
&=\int_{K}\mathsf{P}^{s}(x_{0},\mathrm{d}x)\int_{\mathcal{S}\setminus
K}\mathsf{P}(x,\mathrm{d}y)\;\mathsf{E}_{y}\bigl{[}V(X_{t-s-1})\boldsymbol{1}_{\\{\tau_{K}^{\vphantom{T}}>t-s-1\\}}\bigr{]},\end{split}$
and by ((2.5)) it follows that the right-hand side is at most
$\int_{K}\mathsf{P}^{s}(x_{0},\mathrm{d}x)\int_{\mathcal{S}\setminus
K}\mathsf{P}(x,\mathrm{d}y)\;\varphi(0,y)\theta(t-s-1).$
Thus, one has
((2.7))
$\begin{split}\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\boldsymbol{1}_{\\{g_{t}=s\\}}\bigr{]}&\leqslant\theta(t-s-1)\int_{K}\mathsf{P}^{s}(x_{0},\mathrm{d}x)\int_{\mathcal{S}\setminus
K}\mathsf{P}(x,\mathrm{d}y)\;\varphi(0,y)\\\ &\leqslant\theta(t-s-1)\sup_{x\in
K}\mathsf{E}_{x}\bigl{[}\varphi(0,X_{1})\boldsymbol{1}_{\\{X_{1}\in\mathcal{S}\setminus
K\\}}\bigr{]}\\\ &=\theta(t-s-1)\beta.\end{split}$
Case 2. ($g_{t}=t$). This is easy, since $X_{t}\in K$ implies
$V(X_{t})\leqslant\delta$. Thus
$\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\boldsymbol{1}_{\\{g_{t}=t\\}}\bigr{]}\leqslant\delta\mathsf{P}_{x_{0}}(X_{t}\in
K)\leqslant\delta.$
Case 3. ($g_{t}=-\infty$). This is the case when the chain started from
outside $K$ and has not yet hit $K$, and therefore,
$\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\boldsymbol{1}_{\\{g_{t}=-\infty\\}}\bigr{]}=\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\boldsymbol{1}_{\\{\tau_{K}^{\vphantom{T}}>t\\}}\bigr{]}\leqslant\varphi(0,x_{0})\theta(t).$
Combining all three cases above, we get the bound:
((2.8))
$\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\bigr{]}\leqslant\sum_{s=0}^{t-1}\theta(t-s-1)\beta+\delta+\varphi(0,x_{0})\theta(t).$
Maximizing the right-hand side of ((2.8)) over $t$, we arrive at
$\sup_{t\in\mathbb{N}_{0}}\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\bigr{]}\leqslant\beta\sum_{s=0}^{\infty}\theta(s)+\delta+\varphi(0,x_{0})\sup_{t\in\mathbb{N}_{0}}\theta(t),$
which is the bound stated in the theorem. ∎
Often it will turn out that $\varphi(t,x)$ is a function $\psi(t,V(x))$ as in
the case of the classical Foster-Lyapunov type supermartingales [MT09]. In
that case $\varphi(t,x)=\mathrm{e}^{\alpha t}V(x)$, for some positive
$\alpha$. Thus $\varphi(t,\cdot)$ is a linear function of $V(x)$ for each
fixed $t$, with $\theta(t)=\mathrm{e}^{-\alpha t}$, which shows that the
sequence $(\theta(t))_{t\in\mathbb{N}_{0}}$ is summable. See also [FK04] and
the references therein for more general Foster-Lyapunov type conditions. For
examples which are not linear see §2.4.
### 2.2. A Class of Hybrid Processes
The preceding analysis can be extended for processes which switch their
behavior depending on whether the current value is within $K$ or not. They
constitute a particularly useful class of controlled processes in which a
controller attempts to drive the system into a _target_ or safe set
$K\subset\mathcal{S}$ whenever the system gets out of $K$ due to its inherent
randomness. Below we give a rigorous construction of such a process.
### A process $X$ that is $(Y,Z)$-hybrid with respect to $K$
Consider a pair of Markov chains $(Y,Z)$ where $Y$ is a time-homogeneous
Markov chain, and $Z$ is a (possibly) time inhomogeneous Markov chain. We
construct a _hybrid discrete-time stochastic process_ $X$ by the following
recipe:
Firstly, let the state space for the process be $\mathcal{S}^{\mathbb{N}_{0}}$
along with the natural filtration
$\mathcal{F}_{0}\subseteq\mathcal{F}_{1}\subseteq\mathcal{F}_{2}\subseteq\ldots$
generated by the coordinate maps.
Secondly, we define the sequence of stopping times
$\sigma_{0}\coloneqq\tau_{0}\coloneqq-\infty$ and
$\tau_{1}\leqslant\sigma_{1}\leqslant\tau_{2}\leqslant\sigma_{2}\leqslant\ldots$
by
$\begin{split}\tau_{i}&\coloneqq\inf\bigl{\\{}t>\sigma_{i-1}\;\big{|}\;X_{t}\in
K\bigr{\\}}\quad\text{and}\\\
\sigma_{i}&\coloneqq\inf\bigl{\\{}t>\tau_{i}\;\big{|}\;X_{t}\notin
K\bigr{\\}}\end{split}$
for $i\in\mathbb{N}_{0}$.
Finally, we define the process $X$ as follows: for a measurable
$B\subset\mathcal{S}$,
$\displaystyle\text{if }X_{t}=x,\exists\,i:\tau_{i}\leqslant t<\sigma_{i},\;$
$\displaystyle\begin{cases}X_{t}=Y_{t},\\\ \mathsf{P}\bigl{(}X_{t+1}\in
B\;\big{|}\,\mathcal{F}_{t}\bigr{)}=\mathsf{P}\bigl{(}Y_{1}\in
B\;\big{|}\;Y_{0}=x\bigr{)},\end{cases}$ $\displaystyle\text{if
}X_{t}=x,\exists\,i:\sigma_{i}\leqslant t<\tau_{i+1},\;$
$\displaystyle\begin{cases}X_{t}=Z_{t},\\\ \mathsf{P}\bigl{(}X_{t+1}\in
B\;\big{|}\,\mathcal{F}_{t}\bigr{)}=\mathsf{P}\bigl{(}Z_{t+1-\sigma_{i}}\in
B\;\big{|}\;Z_{t-\sigma_{i}}=x\bigr{)}.\end{cases}$
To wit, the process defined above behaves as the homogeneous chain $Y$
whenever it is inside $K$. Once the process $X$ exits the set $K$, a
controller alters the behavior of the chain which, until it enters $K$ again,
behaves as a copy of the inhomogeneous chain $Z$ starting from a point outside
$K$. The process $X$ is in general non-Markovian due to the possible time
inhomogeneity of $Z$. Nevertheless, it is a natural class of examples of
switching systems whose Markovian behavior switches in different regions on
the state space. We say that $X$ is _$(Y,Z)$ -hybrid with respect to $K$_.
The following generalization of Theorem (2.2) can be proved along lines of the
original proof. The only requirement is a slight modification of the condition
((2.3)) which is needed to alter the second inequality in ((2.7)).
###### (2.9) Theorem.
Consider a stochastic process $X$ that is $(Y,Z)$-hybrid with respect to a
measurable $K\subset\mathcal{S}$ for some homogeneous Markov chain $Y$ and
some possibly inhomogeneous Markov chain $Z$. Suppose Assumption (2.1) holds
for the process $Z$ and
((2.10)) $\beta:=\sup_{y_{0}\in
K}\mathsf{E}\left[\varphi(0,Y_{1})1_{\\{Y_{1}\in\mathcal{S}\setminus
K\\}}\;\big{|}\;Y_{0}=y_{0}\right]<\infty.$
If the process $X$ starts from $x_{0}\in\mathcal{S}\setminus K$, we have
((2.11))
$\sup_{t\in\mathbb{N}_{0}}\mathsf{E}_{x_{0}}\bigl{[}V(X_{t})\bigr{]}\leqslant
C\beta+\delta+\gamma\varphi(0,x_{0}).$
It is interesting to note that the right side of above bound is a total of
individual contributions by the control (for $C$), the choice of $K$ (for
$\delta$), and the initial configuration (for $x_{0}$). We stress that _the
conclusion holds even when $X$ is no longer a Markov chain due to the time
inhomogeneity of $Z$_. This is important, especially because operator-
theoretic bounds like Foster-Lyapunov, or martingale-based bounds do not work
in such a case.
### 2.3. Connection with Optimal Stopping Problems
Suppose that we are given a Markov chain $Z$ taking values in $\mathcal{S}$, a
function $V:\mathcal{S}\longrightarrow\mathbb{R}$, and a measurable target or
safe set $K\subset\mathcal{S}$. (Alternatively, we may assume that we are
given an $\mathcal{S}$-valued process $X$ that is $(Y,Z)$-hybrid with respect
to a measurable $K\subset\mathcal{S}$.) Our objective is to investigate
whether the sequence $\bigl{(}V(X_{t})\bigr{)}_{t\in\mathbb{N}_{0}}$ is
$\boldsymbol{L}_{1}$-bounded. To this end one can follow the two-step
procedure of first searching for a function $\varphi$ satisfying Assumption
(2.1), followed by an application of Theorem (2.9). A systematic procedure of
doing this is given by the following connection with Optimal Stopping
problems.
Let $(\theta(t))_{t\in\mathbb{N}_{0}}$ be some positive sequence of numbers
such that $\sum_{t\in\mathbb{N}_{0}}\theta(t)$ is finite. Define the pay-off
or the reward function as
$h(t,x)=\begin{cases}V(x)/\theta(t)&\quad\text{if}\;x\in\mathcal{S}\setminus
K,\;t\in\mathbb{N}_{0},\\\ 0&\quad\text{if}\;x\in
K,\;t\in\mathbb{N}_{0}.\end{cases}$
Recall that the Optimal Stopping problem [PS06, Chapter 1] for the process $Z$
and the reward function $h$ defined above consists of finding a stopping time
$\tau^{*}$ such that
((2.12))
$\mathsf{E}_{x}\Bigl{[}h\bigl{(}\tau^{*}\wedge\tau_{K}^{\vphantom{T}},Z_{\tau^{*}\wedge\tau_{K}^{\vphantom{T}}}\bigr{)}\Bigr{]}=\operatorname*{ess\,sup}_{\tau}\mathsf{E}_{x}\Bigl{[}h\bigl{(}\tau\wedge\tau_{K}^{\vphantom{T}},Z_{\tau\wedge\tau_{K}^{\vphantom{T}}}\bigr{)}\Bigr{]},$
where $\tau_{K}^{\vphantom{T}}$ is the hitting time to the set $K$, and
$\operatorname*{ess\,sup}$ refers to essential supremum over the set of all
possible stopping times (see [PS06, Chapter 1, Lemma 1.3]).
Define the value function as
((2.13))
$\varphi(n,x_{0})\coloneqq\operatorname*{ess\,sup}_{\tau\in\mathbb{T}_{n}}\mathsf{E}\bigl{[}h(\tau,V(Z_{\tau}))\,\big{|}\,Z_{n}=x_{0}\bigr{]},$
where $\mathbb{T}_{n}$ is the set of stopping times
$\bigl{\\{}(\tau\vee n)\wedge\tau_{K}^{\vphantom{T}}\,\big{|}\,\text{$\tau$ an
arbitrary stopping time}\bigr{\\}}.$
###### (2.14) Theorem.
Suppose that the value function $\varphi(0,x_{0})$ is finite for all
$x_{0}\in\mathcal{S}$, then
1. (i)
$\varphi(t,x_{0})$ is finite for all $t\in\mathbb{N}_{0}$ and
$\varphi(t,x_{0})\geqslant V(x_{0})/\theta(t)\quad\text{for all
}(t,x_{0})\in\mathbb{N}_{0}\times(\mathcal{S}\setminus K).$
2. (ii)
The process $(Y_{t})_{t\in\mathbb{N}_{0}}$ defined by
$Y_{t}\coloneqq\varphi\bigl{(}t\wedge\tau_{K}^{\vphantom{T}},Z_{t\wedge\tau_{K}^{\vphantom{T}}}\bigr{)}$
is a supermartingale.
###### Proof.
The proof follows from the general theory of optimal stopping. See, for
example, [CRS71, Chapter 4]. The sequence of rewards is given by the process
$V(Z_{t\wedge\tau_{K}^{\vphantom{T}}})/\theta(t)$, $t=0,1,2,\ldots$. Applying
[CRS71, Theorem 4.1, p. 66] we get
$\varphi(n,x_{0})=\bigl{(}V(x_{0})/\theta(n)\bigr{)}\vee\Bigl{(}\mathsf{E}\Bigl{[}\varphi\bigl{(}n+1,Z_{(n+1)\wedge\tau_{K}^{\vphantom{T}}}\bigr{)}\,\Big{|}\,Z_{n\wedge\tau_{K}^{\vphantom{T}}}=x_{0}\Bigr{]}\Bigr{)}.$
By considering the first of the two terms in the maximum on the right-hand
side above we obtain (i), and (ii) follows from the second. ∎
In other words, the value function $\varphi(t,x)$ defined in ((2.13))
satisfies the conditions of Theorem (2.9).
###### (2.15) Theorem.
Consider an $\mathcal{S}$-valued process $X$ that is $(Y,Z)$-hybrid with
respect to a measurable $K\subset\mathcal{S}$ as in §2.2. Suppose that for
some nonnegative integrable sequence $(\theta(t))_{t\in\mathbb{N}_{0}}$ the
optimal stopping problem ((2.12)) has a finite value function
$\varphi(t,x_{0})$. If additionally condition ((2.10)) is true, then the bound
((2.11)) holds.
Let us remark that the value function, being the envelope, is the smallest
supermartingale (hence the sharpest bound) that can satisfy Theorem (2.9).
Several methods of solving optimal stopping problems in the Markovian setting
are available and we refer the reader to [PS06] for a complete review.
###### (2.16) Remark.
There is a parallel converse result employing standard Foster-Lyapunov
techniques for the verification of $f$-ergodicity and $f$-regularity [MT09,
Chapter 14] of Markov processes. The analysis is based on the functional
inequality $\mathsf{E}[V(X_{1})\mid
X_{0}=x]-V(x)\leqslant-f(x)+b\boldsymbol{1}_{C}(x)$ for measurable functions
$V:\mathcal{S}\longrightarrow[0,\infty]$ and
$f:\mathcal{S}\longrightarrow[1,\infty[$, a scalar $b>0$, and a Borel subset
$C$ of $\mathcal{S}$; [MT09, Theorem 14.2.3] asserts that the minimal solution
to this inequality, which exists if $C$ is petite (see [MT09] for precise
details), is a “value function” given by
$G_{C}(x,f)\coloneqq\mathsf{E}\bigl{[}\sum_{t=0}^{\sigma_{C}}f(X_{t})\,\big{|}\,X_{0}=x]$,
where $\sigma_{C}$ is the first hitting-time to $C$. The proof is also based
on the existence of a certain supermartingale, and the Markov property is
employed crucially.$\vartriangleleft$
### 2.4. A Class of Sampled Diffusions
In the setting of the process $X$ being $(Y,Z)$-hybrid with respect to a given
set $K$, suppose that the state-space for the Markov chains $Y$ and $Z$ is
$\mathbb{R}^{d}$ and the safe set $K$ is compact. Observe that the only
challenge in applying Theorem (2.9) is to find a suitable function $\varphi$
given the Markov chain $Z$ and the function $V$. In applications, a natural
choice for the function $V$ is given by square of the Euclidean norm, i.e.,
$V(x)=\sum_{i=1}^{d}x_{i}^{2}$. For this choice of $V$, we describe below a
natural class of examples of Markov chains for which one can construct a
$\varphi$ that satisfies part (i) of Assumption (2.1).
Consider a diffusion with a possibly time-inhomogeneous drift function, given
by the $d$-dimensional stochastic differential equation
((2.17)) ${\mathrm{d}}X_{t}=b(t,X_{t})\mathrm{d}t+\mathrm{d}W_{t},$
where $W_{t}=(W_{t}(1),W_{t}(2),\ldots,W_{t}(d))$ is a vector of $d$
independent Brownian motions, and $b:\mathbb{R}_{\geqslant
0}\times\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d}$ is a measurable function.
We will abuse the notations somewhat and construct a function
$\varphi:\mathbb{R}_{\geqslant 0}\times\mathbb{R}_{\geqslant
0}\longrightarrow\mathbb{R}_{\geqslant 0}$ such that
$\bigl{(}\varphi(t,V(X_{t}))\bigr{)}_{t\in\mathbb{N}_{0}}$ is a
supermartingale outside a compact set $K$ and satisfies
$\varphi(t,\xi)\geqslant\xi/\theta(t)$ for some nonnegative sequence
$(\theta(t))_{t\in\mathbb{N}_{0}}$. We define
$Z_{i}=X_{i\wedge\tau_{K}^{\vphantom{T}}}$ for $i\in\mathbb{N}_{0}$; $Z$ is
the the diffusion sampled at integer time points before hitting $K$. It is
clear that $Z$ is a Markov chain such that
$\bigl{(}\varphi(i,V(Z_{i}))\bigr{)}_{i\in\mathbb{N}_{0}}$ is a
supermartingale that satisfies the Assumptions (2.1) as long as
$\sum_{t\in\mathbb{N}_{0}}\theta(t)<\infty$.
To construct such a $\varphi$, let us consider a well known family of one-
dimensional diffusion, known as the squared Bessel processes (BESQ). This
family is indexed by a single nonnegative parameter $\delta\geqslant 0$ and is
described as the unique strong solution of the SDE
((2.18))
$\mathrm{d}Y_{t}=2\sqrt{Y_{t}}\,\mathrm{d}\mathfrak{b}_{t}+\delta\,\mathrm{d}t,\qquad
Y_{0}=y_{0}\geqslant 0,$
where $\mathfrak{b}\coloneqq(\mathfrak{b}_{t})_{t\in\mathbb{N}_{0}}$ is a one-
dimensional standard Brownian motion. We have the following Lemma:
###### (2.19) Lemma.
Let $F:\mathbb{R}\longrightarrow\mathbb{R}_{\geqslant 0}$ be a nonnegative,
increasing, and convex function, and fix any terminal time $S>0$. Define the
function
((2.20))
$\varphi(t,y)\coloneqq\mathsf{E}\bigl{[}F(Y_{S})\,\big{|}\,Y_{t}=y\bigr{]},\quad
t\in[0,S],$
where $Y$ solves the SDE ((2.18)). Then $\varphi$ satisfies the following
properties:
1. (i)
$\varphi$ is increasing in $y$,
2. (ii)
$\varphi$ is convex in $y$, and
3. (iii)
$\varphi$ satisfies the partial differential equation
((2.21)) $\begin{cases}\dfrac{\partial\varphi}{\partial
t}+\delta\varphi^{\prime}+2y\varphi^{\prime\prime}=0,\quad y>0,\;t\in(0,S),\\\
\varphi(S,y)=F(y).\end{cases}$
Note that $\varphi^{\prime}$ and $\varphi^{\prime\prime}$ in the statement of
Lemma (2.18) refers to the first and second derivatives with respect to the
second argument of $\varphi$.
###### Proof.
The proof proceeds by coupling. Let us first show that $\varphi$ is increasing
as claimed in (i). Fix $S>0$. Consider any two starting points $0\leqslant
x<y$. Construct on the same sample space two copies of BESQ processes
$Y^{(1)}$ and $Y^{(2)}$ such that both of them satisfy ((2.18)) with respect
to the same Brownian motion $\mathfrak{b}$ but $Y^{(1)}_{0}=x$ and
$Y^{(2)}_{0}=y$. It is possible to do this since the SDE ((2.18)) admits a
strong solution (see [KS08, Chapter 5, Proposition 2.13]). Hence, by [KS08,
Chapter 5, Proposition 2.18], it follows that $Y^{(1)}_{t}\leqslant
Y^{(2)}_{t}$ for all $t\geqslant 0$. Since $F$ is an increasing function, we
get
$\varphi(t,x)=\mathsf{E}_{x}\Bigl{[}F\bigl{(}Y^{(1)}_{S-t}\bigr{)}\Bigr{]}\leqslant\mathsf{E}_{y}\Bigl{[}F\bigl{(}Y^{(2)}_{S-t}\bigr{)}\Bigr{]}=\varphi(t,y).$
This proves that $\varphi$ is increasing in the second argument.
For convexity of $\varphi$ claimed in (ii), we use a different coupling. We
follow arguments very similar to the one used in the proof of [Hob98, Theorem
3.1]. Consider three initial points $0<z<y<x$. And let
$\hat{X},\hat{Y},\hat{Z}$ be three independent BESQ processes that start from
$x,y$, and $z$ respectively. Define the stopping times
$\tau_{x}=\inf\Bigl{\\{}u\;\Big{|}\;\hat{Y}_{u}=\hat{X}_{u}\Bigr{\\}},\quad\tau_{z}=\inf\Bigl{\\{}u\;\Big{|}\;\hat{Y}_{u}=\hat{Z}_{u}\Bigr{\\}}.$
Fix a time $t\in[0,S]$, and let $T=S-t$. Define
$\sigma=\tau_{x}\wedge\tau_{z}\wedge T.$
Now, on the event $\sigma=\tau_{x}$, it follows from symmetry that
((2.22))
$\begin{split}\mathsf{E}\Bigl{[}\left(\hat{X}_{T}-\hat{Z}_{T}\right)F\bigl{(}\hat{Y}_{T}\bigr{)}\boldsymbol{1}_{\\{\sigma=\tau_{x}\\}}\Bigr{]}&=\mathsf{E}\Bigl{[}\left(\hat{Y}_{T}-\hat{Z}_{T}\right)F\bigl{(}\hat{X}_{T}\bigr{)}\boldsymbol{1}_{\\{\sigma=\tau_{x}\\}}\Bigr{]},\\\
\mathsf{E}\Bigl{[}\left(\hat{X}_{T}-\hat{Y}_{T}\right)F\bigl{(}\hat{Z}_{T}\bigr{)}\boldsymbol{1}_{\\{\sigma=\tau_{x}\\}}\bigr{]}&=0.\end{split}$
Similarly, on the event $\sigma=\tau_{z}$, we have
((2.23))
$\begin{split}\mathsf{E}\Bigl{[}\left(\hat{X}_{T}-\hat{Z}_{T}\right)F\bigl{(}\hat{Y}_{T}\bigr{)}\boldsymbol{1}_{\\{\sigma=\tau_{z}\\}}\Bigr{]}&=\mathsf{E}\Bigl{[}\left(\hat{X}_{T}-\hat{Y}_{T}\right)F\bigl{(}\hat{Z}_{T}\bigr{)}\boldsymbol{1}_{\\{\sigma=\tau_{z}\\}}\Bigr{]},\\\
\mathsf{E}\Bigl{[}\left(\hat{Z}_{T}-\hat{Y}_{T}\right)F\bigl{(}\hat{X}_{T}\bigr{)}\boldsymbol{1}_{\\{\sigma=\tau_{z}\\}}\bigr{]}&=0.\end{split}$
And finally, when $\sigma=T$, we must have
$\hat{Z}_{T}<\hat{Y}_{T}<\hat{X}_{T}$. We use the convexity property of $F$ to
get
((2.24))
$\displaystyle\mathsf{E}\Bigl{[}\left(\hat{X}_{T}-\hat{Z}_{T}\right)F\bigl{(}\hat{Y}_{T}\bigr{)}\boldsymbol{1}_{\\{\sigma=T\\}}\Bigr{]}$
$\displaystyle\leqslant\mathsf{E}\Bigl{[}\left(\hat{X}_{T}-\hat{Y}_{T}\right)F\bigl{(}\hat{Z}_{T}\bigr{)}\boldsymbol{1}_{\\{\sigma=T\\}}\Bigr{]}$
$\displaystyle\quad+\mathsf{E}\Bigl{[}\left(\hat{Y}_{T}-\hat{Z}_{T}\right)F\bigl{(}\hat{X}_{T}\bigr{)}\boldsymbol{1}_{\\{\sigma=T\\}}\Bigr{]}.$
Combining the three cases in ((2.22)), ((2.23)), and ((2.24)) we get
((2.25))
$\mathsf{E}\Bigl{[}\left(\hat{X}_{T}-\hat{Z}_{T}\right)F\bigl{(}\hat{Y}_{T}\bigr{)}\Bigr{]}\leqslant\mathsf{E}\Bigl{[}\left(\hat{X}_{T}-\hat{Y}_{T}\right)F\bigl{(}\hat{Z}_{T}\bigr{)}\Bigr{]}+\mathsf{E}\Bigl{[}\left(\hat{Y}_{T}-\hat{Z}_{T}\right)F\bigl{(}\hat{X}_{T}\bigr{)}\Bigr{]}.$
We now use the fact that $\hat{X},\hat{Y}$, and $\hat{Z}$ are independent.
Also, it is not difficult to see from the SDE ((2.18)) that
$\mathsf{E}_{x}\bigl{[}\hat{X}_{T}\bigr{]}-x=\mathsf{E}_{y}\bigl{[}\hat{Y}_{T}\bigr{]}-y=\mathsf{E}_{z}\bigl{[}\hat{Z}_{T}\bigr{]}-z=\delta
t$. Thus, from ((2.25)) we infer that
$(x-z)\varphi(t,y)\leqslant(x-y)\varphi(t,z)+(y-z)\varphi(t,x),\quad\text{for
all}\;0<z<y<x.$
This proves convexity of $\varphi$ in its second argument.
Finally, to see (iii), it suffices to observe that the equation ((2.21)) is
the classical generator relation for diffusions, for which we refer to [KS08,
Chapter 5.4]. The transition density of BESQ processes are smooth and have an
explicit representation that satisfy equation ((2.21)). The general case can
be obtained by differentiating under the integral with respect to $F$. ∎
Let us return to the multidimensional diffusion given by ((2.17)). We consider
the process $(\zeta_{t})_{t\in\mathbb{N}_{0}}$, where
$\zeta_{t}\coloneqq\varphi\bigl{(}t,\left\lVert
X_{t}\right\rVert^{2}\bigr{)}$, and $\varphi$ is the function in ((2.20)).
Note that, since $F$ is nonnegative, so is $\varphi$. Additionally, since
$\varphi$ is convex, we have
$\varphi(t,\xi)\geqslant\varphi(t,0)+\varphi^{\prime}(t,0+)\xi.$
Hence the sequence $(\theta(t))_{t=0}^{S}$ is given by
$\theta(t)=1/\varphi^{\prime}(t,0+),\quad t=0,1,\ldots,S.$
We have the following Theorem:
###### (2.26) Theorem.
Suppose that there exists a compact set $K\subset\mathbb{R}^{d}$ such that
that the drift function $b=(b_{1},b_{2},\ldots,b_{d})$ in the SDE ((2.17))
satisfies the sector condition
$\sum_{i=1}^{d}x_{i}b_{i}(t,x)<0\quad\text{for
}\;(t,x)\in\mathbb{R}_{\geqslant 0}\times(\mathcal{S}\setminus K).$
Fix any terminal time $T>0$. Define the process
$(\zeta_{t})_{t\in\mathbb{N}_{0}}\coloneqq\bigl{(}\varphi\bigl{(}t,\left\lVert
X_{t}\right\rVert^{2}\bigr{)}\bigr{)}_{t\in\mathbb{N}_{0}}$, , where $\varphi$
is the nonnegative, increasing, convex function defined in ((2.20)) with
$F(y)=\left\lVert y\right\rVert^{2}\quad\text{and}\quad\delta=d.$
Then, with the set-up as above, the stopped process
$\bigl{(}\zeta_{t\wedge\tau_{K}^{\vphantom{T}}\wedge T}\bigr{)}_{t\geqslant
0}$ is a (local) supermartingale.
###### Proof.
Applying Itô’s rule to $(\zeta_{t})_{t\in\mathbb{R}_{\geqslant 0}}$, we get
((2.27))
${\mathrm{d}}\zeta_{t}=\mathrm{d}M_{t}+\left[\frac{\partial\varphi}{\partial
t}+\mathcal{L}\varphi\right]\mathrm{d}t,$
where $M\coloneqq(M_{t})_{t\in\mathbb{R}_{\geqslant 0}}$ is in general a local
martingale ($M$ is a martingale under additional assumptions of boundedness on
the first derivative of $\varphi$), and $\mathcal{L}$ is the generator of $X$.
We compute
$\begin{split}\frac{\partial\varphi}{\partial
t}+\mathcal{L}\varphi&=\frac{\partial\varphi}{\partial
t}+\sum_{i=1}^{d}b_{i}\frac{\partial\varphi}{\partial
x_{i}}+\frac{1}{2}\sum_{i=1}^{d}\frac{\partial^{2}\varphi}{\partial
x^{2}_{i}}\\\ &=\frac{\partial\varphi}{\partial
t}+2\varphi^{\prime}\sum_{i=1}^{d}b_{i}x_{i}+\frac{1}{2}\left[2\mathrm{d}\varphi^{\prime}+\varphi^{\prime\prime}\sum_{i=1}^{d}4x_{i}^{2}\right]\\\
&=\frac{\partial\varphi}{\partial
t}+\mathrm{d}\varphi^{\prime}+2\left(\sum_{i}x_{i}^{2}\right)\varphi^{\prime\prime}+2\varphi^{\prime}\sum_{i=1}^{d}b_{i}x_{i}=2\varphi^{\prime}\sum_{i}b_{i}x_{i},\end{split}$
where the final equality holds since $\varphi$ satisfies ((2.21)) at
$y=\sum_{i}x_{i}^{2}$.
We know that $\varphi^{\prime}>0$ since $\varphi$ is increasing, and, by our
assumption, $\sum_{i}x_{i}b_{i}<0$ whenever $x\not\in K$. Thus,
$\frac{\partial\varphi}{\partial t}+\mathcal{L}\varphi\leqslant
0\quad\text{for }\;(t,x)\in[0,T]\times(\mathcal{S}\setminus K).$
Now the claim follows from the semimartingale decomposition given in ((2.27)).
∎
Note that the supermartingale $(\zeta_{t})_{t\in\mathbb{N}_{0}}$ has been
defined only for a bounded temporal horizon. Thus, to show that Theorem (2.9)
holds, some additional uniformity assumptions would be needed.
## 3\. Application to Discrete-Time Randomly Switched Systems
In this section we look at several cases of discrete-time randomly switched
systems (or, iterated function systems,) in which Theorem (2.2) of §2 applies
and gives useful uniform $\boldsymbol{L}_{1}$ bounds of Lyapunov functions. In
§3.1 we give sufficient conditions for global asymptotic stability almost
surely and in $\boldsymbol{L}_{1}$ of discrete-time randomly switched systems.
Assumptions of global contractivity in its standard form or memoryless choice
of the maps at each iterate are absent; we simply require a condition
resembling average contractivity in terms of Lyapunov functions with a
suitable coupling condition with the Markovian transition probabilities. In
§3.2 we demonstrate that under mild hypotheses iterated function systems
possess strong stability and robustness properties with respect to bounded
disturbances that are not modelled as random processes.222Recall the following
notation: We let $\mathcal{K}$ denote the collection of strictly increasing
continuous functions $\alpha:\mathbb{R}_{\geqslant
0}\longrightarrow\mathbb{R}_{\geqslant 0}$ such that $\alpha(0)=0$; we say
that a function $\alpha$ belongs to class-$\mathcal{K}_{\infty}$ if
$\alpha\in\mathcal{K}$ and $\lim_{r\to\infty}\alpha(r)=\infty$. A function
$\beta:\mathbb{R}_{\geqslant
0}\times\mathbb{N}_{0}\longrightarrow\mathbb{R}_{\geqslant 0}$ belongs to
class-$\mathcal{KL}$ if $\beta(\cdot,n)\in\mathcal{K}$ for a fixed
$n\in\mathbb{N}_{0}$, and if $\beta(r,n)\to 0$ as $n\to\infty$ for fixed
$r\in\mathbb{R}_{\geqslant 0}$. Recall that a function
$f:\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d}$ is locally Lipschitz
continuous if for every $x_{0}\in\mathbb{R}^{d}$ and open set $O$ containing
$x_{0}$, there exists a constant $L>0$ such that $\left\lVert
f(x)-f(x_{0})\right\rVert\leqslant L\left\lVert x-x_{0}\right\rVert$ whenever
$x\in O$.
### 3.1. Stability of Discrete-Time Randomly Switched Systems
Consider the system
((3.1)) $X_{t+1}=f_{\sigma_{t}}(X_{t}),\qquad X_{0}=x_{0},\quad
t\in\mathbb{N}_{0}.$
Here
$\sigma:\mathbb{N}_{0}\longrightarrow\mathcal{P}\coloneqq\\{1,\ldots,\mathrm{N}\\}$
is a discrete-time random process, the map
$f_{i}:\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d}$ is continuous and locally
Lipschitz, and there are points $x_{i}^{\star}\in\mathbb{R}^{d}$ such that
$f_{i}(x_{i}^{\star})=0$ for each $i\in\mathcal{P}$. The initial condition of
the system $x_{0}\in\mathbb{R}^{d}$ is assumed to be known. Our objective is
to study stability properties of this system by extracting certain nonnegative
supermartingales.
The system ((3.1)) can be viewed as an iterated function system:
$X_{t+1}=f_{\sigma_{t}}\circ\cdots\circ f_{\sigma_{1}}\circ
f_{\sigma_{0}}(x_{0})$. Varying the point $x_{0}$ but keeping the same maps
leads to a family of Markov chains initialized from different initial
conditions. The article [DF99] treats basic results on convergence and
stationarity properties of such systems with the process
$(\sigma_{t})_{t\in\mathbb{N}_{0}}$ being a sequence of independent and
identically distributed random variables taking values in $\mathcal{P}$, and
each map $f_{i}$ is a contraction. These results were generalized in [JT01]
with the aid of Foster-Lyapunov arguments.
The analysis carried out in [JT01] requires a Polish state-space, and employs
the following three principal assumptions: (a) the maps are non-separating on
an average, i.e., the average separation of the Markov chains initialized at
different points is nondecreasing over time; (b) there exists a set $C$ such
that the Markov chains started at different initial conditions contract after
the set $C$ is reached; and (c) there exists a measurable real-valued function
$V\geqslant 1$, bounded on $C$, and satisfying a Foster-Lyapunov drift
condition $QV(x)\leqslant\lambda V(x)+b\boldsymbol{1}_{C}(x)$ for some
$\lambda\in\>]0,1[$ and $b<\infty$, where $Q$ is the transition kernel. Under
these conditions the authors establish the existence and uniqueness of an
invariant measure which is also globally attractive, and the convergence to
this measure is exponential. In particular, this showed that the main results
of [DF99], which are primarily related to existence and uniqueness of
invariant probability measures, continue to hold if the contractivity
hypotheses on the family $\\{f_{i}\\}_{i\in\mathcal{P}}$ are relaxed. In this
subsection we look at stronger properties, namely, $\boldsymbol{L}_{1}$
boundedness and stability, and almost sure stability of the system ((3.1))
under Assumption (2.1). No contractivity inside a compact set is needed to
establish existence of an invariant measure under Assumption (2.1).
###### (3.2) Assumption.
The process $(\sigma_{t})_{t\in\mathbb{N}_{0}}$ is an irreducible Markov chain
with initial probability distribution $\pi^{\circ}$ and a transition matrix
$P\coloneqq[p_{ij}]_{\mathrm{N}\times\mathrm{N}}$.$\diamondsuit$
It is immediately clear that the discrete-time process
$(\sigma_{t},X_{t})_{t\in\mathbb{N}_{0}}$, taking values in the Borel space
$\mathcal{P}\times\mathbb{R}^{d}$, is Markovian under Assumption (3.2). The
corresponding transition kernel is given by
$\displaystyle Q\bigl{(}(i,x),\mathcal{P}^{\prime}\times
B\bigr{)}=\textstyle{\sum_{j\in\mathcal{P}^{\prime}}p_{ij}\boldsymbol{1}_{B}\bigl{(}f_{j}(x)\bigr{)}}\quad$
$\displaystyle\text{for }\mathcal{P}^{\prime}\subset\mathcal{P},B\text{ a
Borel subset of }\mathbb{R}^{d},$ $\displaystyle\text{and
}(i,x)\in\mathcal{P}\times\mathbb{R}^{d}.$
Our basic analysis tool is a family of Lyapunov functions, one for each
subsystem, and at different times we shall impose the following two distinct
sets of hypotheses on them.333It will be useful to recall here that the
deterministic system $x_{t+1}=f_{i}(x_{t}),\;t\in\mathbb{N}_{0},$ with initial
condition $x_{0}$ is said to be _globally asymptotically stable_ (in the sense
of Lyapunov) if (a) for every $\varepsilon>0$ there exists a $\delta>0$ such
that $\left\lVert x_{0}-x_{i}^{\star}\right\rVert<\delta$ implies $\left\lVert
x_{t}-x_{i}^{\star}\right\rVert<\varepsilon$ for all $t\in\mathbb{N}_{0}$, and
(b) for every $r,\varepsilon^{\prime}>0$ there exists a $T>0$ such that
$\left\lVert x_{0}-x_{i}^{\star}\right\rVert<r$ implies $\left\lVert
x_{t}-x_{i}^{\star}\right\rVert<\varepsilon$ for all $t>T$. The condition (a)
goes by the name of Lyapunov stability of the dynamical system (or of the
corresponding equilibrium point $x_{i}^{\star}$), and (b) is the standard
notion of global asymptotic convergence to $x_{i}^{\star}$.
###### (3.3) Assumption.
There exist a family $\\{V_{i}\\}_{i\in\mathcal{P}}$ of nonnegative measurable
functions on $\mathbb{R}^{d}$, functions
$\alpha_{1},\alpha_{2}\in\mathcal{K}$, numbers $\lambda_{\circ}\in\;]0,1[$,
$r>0$ and $\mu>1$, such that
1. (V1)
$\alpha_{1}(\left\lVert x-x_{i}^{\star}\right\rVert)\leqslant
V_{i}(x)\leqslant\alpha_{2}(\left\lVert x-x_{i}^{\star}\right\rVert)\quad$ for
all $x$ and $i$,
2. (V2)
$V_{i}(x)\leqslant\mu V_{j}(x)\quad$ whenever $\left\lVert x\right\rVert>r$,
for all $i,j$, and
3. (V3)
$V_{i}(f_{i}(x))\leqslant\lambda_{\circ}V_{i}(x)\quad$ for all $x$ and
$i$.$\diamondsuit$
###### (3.4) Assumption.
There exist a family $\\{V_{i}\\}_{i\in\mathcal{P}}$ of nonnegative measurable
functions on $\mathbb{R}^{d}$, functions
$\alpha_{1},\alpha_{2}\in\mathcal{K}$, a matrix
$[\lambda_{ij}]_{\mathrm{N}\times\mathrm{N}}$ with nonnegative entries, and
numbers $r>0$, $\mu>1$, such that (V1)-(V2) of Assumption (3.3) hold, and
1. (V3′)
$V_{i}(f_{j}(x))\leqslant\lambda_{ij}V_{i}(x)\quad$ for all $x$ and
$i,j$.$\diamondsuit$
The condition (V1) in Assumption (3.3) is standard in deterministic system
theory literature, ensuring, in particular, positive definiteness of each
$V_{i}$. (V2) stipulates that outside $\bar{B}_{r}$ the functions
$\\{V_{i}\\}_{i\in\mathcal{P}}$ are linearly comparable to each other. The
conditions (V1) and (V3) together imply that each subsystem is globally
asymptotically stable, with sufficient stability margin—the smaller the number
$\lambda_{\circ}$, the greater is the stability margin. In fact, standard
converse Lyapunov theorems show that (V1) and (V3) are necessary and
sufficient conditions for each subsystem to be globally asymptotically stable.
The only difference between Assumptions (3.3) and (3.4) is that the latter
keeps track of how each Lyapunov function evolves along trajectories of every
subsystem.
Let us define $\displaystyle{\hat{p}\coloneqq\max_{i\in\mathrm{N}}p_{ii}}$ and
$\displaystyle{\tilde{p}\coloneqq\max_{i,j\in\mathcal{P},i\neq j}p_{ij}}$.
###### (3.5) Proposition.
Consider the system ((3.1)), and suppose that either of the following two
conditions holds:
1. _(S1)_
Assumptions (3.2) and (3.3) hold, and
$\lambda_{\circ}(\hat{p}+\mu\tilde{p})<1$.
2. _(S2)_
Assumptions (3.2) and (3.4) hold, and
$\textstyle{\mu\cdot\left(\max_{i\in\mathcal{P}}\sum_{j\in\mathcal{P}}p_{ij}\lambda_{ji}\right)<1}$.
Let $\tau_{r}\coloneqq\inf\bigl{\\{}t\in\mathbb{N}_{0}\big{|}\left\lVert
X_{t}\right\rVert\leqslant r\bigr{\\}}$ and $V_{i}^{\prime}(x)\coloneqq
V_{i}(x)\boldsymbol{1}_{\mathbb{R}^{d}\setminus\bar{B}_{r}}(x)$. Suppose that
$\left\lVert x_{0}\right\rVert>r$. Then there exists $\alpha>0$ such that the
process
$\bigl{(}\mathrm{e}^{\alpha(t\wedge\tau_{r})}V^{\prime}_{\sigma_{t\wedge\tau_{r}}}(X_{t\wedge\tau_{r}})\bigr{)}_{t\in\mathbb{N}_{0}}$
is a nonnegative supermartingale.
###### (3.6) Corollary.
Consider the system ((3.1)), and assume that the hypotheses of Proposition
(3.5) hold. Then there exists a constant $c>0$ such that
$\displaystyle{\sup_{t\in\mathbb{N}_{0}}\mathsf{E}\\!\left[\vphantom{\big{|}}\alpha_{1}(\left\lVert
X_{t}\right\rVert)\vphantom{\big{|}}\right]<c}$.
It is possible to derive simple conditions for stability of the system ((3.1))
from Proposition (3.5). To this end we briefly recall two standard stability
concepts.
###### (3.7) Definition.
If $\ker(f_{i}-\mathrm{id})=\\{0\\}$ for each $i\in\mathcal{P}$, the system
((3.1)) is said to be
* $\circ$
_globally asymptotically stable almost surely_ if
1. (AS1)
$\displaystyle{\mathsf{P}\Bigl{(}\forall\,\varepsilon>0\;\;\exists\,\delta>0\text{
s.t.\ }\sup_{t\in\mathbb{N}_{0}}\left\lVert
X_{t}\right\rVert<\varepsilon\text{ whenever }\left\lVert
x_{0}\right\rVert<\delta\Bigr{)}=1}$,
2. (AS2)
$\displaystyle{\mathsf{P}\Bigl{(}\forall\,r,\varepsilon^{\prime}>0\;\;\exists\,T>0\text{
s.t.\ }\sup_{\mathbb{N}_{0}\ni t>T}\left\lVert
X_{t}\right\rVert<\varepsilon^{\prime}\text{ whenever }\left\lVert
x_{0}\right\rVert<r\Bigr{)}=1}$;
* $\circ$
_$\alpha$ -stable in $\boldsymbol{L}_{1}$_ for some $\alpha\in\mathcal{K}$ if
1. (SM1)
$\displaystyle{\forall\,\varepsilon>0\;\;\exists\,\delta>0\text{ s.t.\
}\sup_{t\in\mathbb{N}_{0}}\mathsf{E}\\!\left[\vphantom{\big{|}}\alpha(\left\lVert
X_{t}\right\rVert)\vphantom{\big{|}}\right]<\varepsilon\text{ whenever
}\left\lVert x_{0}\right\rVert<\delta}$,
2. (SM2)
$\displaystyle{\forall\,r,\varepsilon^{\prime}>0\;\;\exists\,T>0\text{ s.t.\
}\sup_{\mathbb{N}_{0}\ni
t>T}\mathsf{E}\\!\left[\vphantom{\big{|}}\alpha(\left\lVert
X_{t}\right\rVert)\vphantom{\big{|}}\right]<\varepsilon^{\prime}\text{
whenever }\left\lVert x_{0}\right\rVert<r}$.$\Diamond$
###### (3.8) Corollary.
Suppose that $\ker(f_{i}-\mathrm{id})=\\{0\\}$ for each $i\in\mathcal{P}$, and
that either of the hypotheses _(S1)_ and _(S2)_ of Proposition (3.5) holds
with $r=0$. Then
* $\circ$
there exists $\alpha>0$ such that
$\lim_{t\to\infty}\mathsf{E}\bigl{[}\mathrm{e}^{\alpha
t}V_{\sigma_{t}}(X_{t})\bigr{]}=0$, and
* $\circ$
the system ((3.1)) is globally asymptotically stable almost surely and
$\alpha_{1}$-stable in $\boldsymbol{L}_{1}$ in the sense of Definition (3.7).
The proofs of Proposition (3.5), Corollary (3.6) and Corollary (3.8) are given
after the following simple Lemma; the crude estimate asserted in it resembles
the distribution of a Binomial random variable, except that we have
$\hat{p}+\tilde{p}\geqslant 1$. For $t\in\mathbb{N}$ let the random variable
$N_{t}$ denote the number of times the state of the Markov chain changes on
the period of length $t$ starting from $0$, i.e.,
$N_{t}\coloneqq\sum_{i=1}^{t}\boldsymbol{1}_{\\{\sigma_{i-1}\neq\sigma_{i}\\}}$.
###### (3.9) Lemma.
Under Assumption (3.2) we have for $s<t$, $s,t\in\mathbb{N}_{0}$,
$\mathsf{P}\bigl{(}N_{t}-N_{s}=k\big{|}\sigma_{s}\bigr{)}\leqslant\begin{cases}\displaystyle{\left(\binom{t-s}{k}\hat{p}^{(t-s-k)}\tilde{p}^{k}\right)\wedge
1}\quad&\text{if $k=0,1,\ldots,t-s$},\\\ 0&\text{else}.\end{cases}$
###### Proof.
Fix $s<t$, $s,t\in\mathbb{N}_{0}$, and let
$\eta_{k}(s,t)\coloneqq\mathsf{P}\bigl{(}N_{t}-N_{s}=k\big{|}\sigma_{s}\bigr{)}$.
Then by the Markov property, for $k=0,1,\ldots,t-s$,
$\displaystyle\eta_{k}(s,t)$
$\displaystyle=\eta_{k}(s,t-1)\mathsf{P}\bigl{(}N_{t}-N_{s}=k\big{|}N_{t-1}-N_{s}=k,\sigma_{s}\bigr{)}$
$\displaystyle\qquad+\eta_{k-1}(s,t-1)\mathsf{P}\bigl{(}N_{t}-N_{s}=k\big{|}N_{t-1}-N_{s}=k-1,\sigma_{s}\bigr{)}$
$\displaystyle\leqslant\hat{p}\eta_{k}(s,t-1)+\tilde{p}\eta_{k-1}(s,t-1).$
The set of initial conditions $\eta_{i}(s,t)=0$ for all $i\geqslant t-s$,
follow from the trivial observation that there cannot be more than $t-s$
changes of $\sigma$ on a period of length $t-s$. This gives a well-defined set
of recursive equations, and a standard induction argument shows that
$\eta_{k}(s,t)\leqslant\binom{t-s}{k}\hat{p}^{(t-s-k)}\tilde{p}^{k}$. This
proves the assertion. ∎
###### Proof of Proposition (3.5).
First we look at the assertion under the condition (S1). Fix $s<t$,
$s,t\in\mathbb{N}_{0}$. Given
$(\sigma_{s\wedge\tau_{r}},X_{s\wedge\tau_{r}})$, from (V3) we get
$V^{\prime}_{\sigma_{s\wedge\tau_{r}}}\bigl{(}X_{(s+1)\wedge\tau_{r}}\bigr{)}\leqslant\lambda_{\circ}V^{\prime}_{\sigma_{s\wedge\tau_{r}}}\bigl{(}X_{s\wedge\tau_{r}}\bigr{)}$,
and if $\sigma_{s+1}\neq\sigma_{s}$, we employ (V2) to get
$V^{\prime}_{\sigma_{(s+1)\wedge\tau_{r}}}\bigl{(}X_{(s+1)\wedge\tau_{r}}\bigr{)}\leqslant\mu
V^{\prime}_{\sigma_{s\wedge\tau_{r}}}\bigl{(}X_{(s+1)\wedge\tau_{r}}\bigr{)}$.
Therefore,
$\displaystyle
V^{\prime}_{\sigma_{(s+1)\wedge\tau_{r}}}\bigl{(}X_{(s+1)\wedge\tau_{r}}\bigr{)}\leqslant\mu\lambda_{\circ}V^{\prime}_{\sigma_{s\wedge\tau_{r}}}\bigl{(}X_{s\wedge\tau_{r}}\bigr{)}$
$\displaystyle\text{if
}\sigma_{(s+1)\wedge\tau_{r}}\neq\sigma_{s\wedge\tau_{r}},\quad\text{and}$
$\displaystyle
V^{\prime}_{\sigma_{(s+1)\wedge\tau_{r}}}\bigl{(}X_{(s+1)\wedge\tau_{r}}\bigr{)}\leqslant\lambda_{\circ}V^{\prime}_{\sigma_{s\wedge\tau_{r}}}\bigl{(}X_{s\wedge\tau_{r}}\bigr{)}$
$\displaystyle\text{otherwise}.$
Iterating this procedure we arrive at the pathwise inequality
((3.10))
$V^{\prime}_{\sigma_{t\wedge\tau_{r}}}\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}\leqslant\mu^{N_{t\wedge\tau_{r}}-N_{s\wedge\tau_{r}}}\lambda_{\circ}^{t\wedge\tau_{r}-s\wedge\tau_{r}}V^{\prime}_{\sigma_{s\wedge\tau_{r}}}\bigl{(}X_{s\wedge\tau_{r}}\bigr{)}.$
Since $s\wedge\tau_{r}=t\wedge s\wedge\tau_{r}$, and $t\wedge\tau_{r}$ is
measurable with respect to $\mathfrak{F}_{t\wedge s\wedge\tau_{r}}$, we invoke
the Markov property of $(\sigma_{t},X_{t})_{t\in\mathbb{N}_{0}}$to arrive at
$\displaystyle\mathsf{E}\Bigl{[}V^{\prime}_{\sigma_{t\wedge\tau_{r}}}\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}$
$\displaystyle\Big{|}(\sigma_{s\wedge\tau_{r}},X_{s\wedge\tau_{r}})\Bigr{]}$
$\displaystyle\leqslant
V^{\prime}_{\sigma_{s\wedge\tau_{r}}}\bigl{(}X_{s\wedge\tau_{r}}\bigr{)}\lambda_{\circ}^{t\wedge\tau_{r}-s\wedge\tau_{r}}\mathsf{E}\Bigl{[}\mu^{N_{t\wedge\tau_{r}}-N_{s\wedge\tau_{r}}}\Big{|}(\sigma_{s\wedge\tau_{r}},X_{s\wedge\tau_{r}})\Bigr{]}.$
We now apply the estimate in Lemma (3.9) to get
$\mathsf{E}\Bigl{[}\mu^{N_{t\wedge\tau_{r}}-N_{s\wedge\tau_{r}}}\Big{|}(\sigma_{s\wedge\tau_{r}},X_{s\wedge\tau_{r}})\Bigr{]}\leqslant\sum_{k=0}^{t\wedge\tau_{r}-s\wedge\tau_{r}}\binom{t\wedge\tau_{r}-s\wedge\tau_{r}}{k}\hat{p}^{(t\wedge\tau_{r}-s\wedge\tau_{r}-k)}\tilde{p}^{k}\mu^{k}=\bigl{(}\hat{p}+\mu\tilde{p}\bigr{)}^{t\wedge\tau_{r}-s\wedge\tau_{r}}$,
and this leads to
$\mathsf{E}\Bigl{[}V^{\prime}_{\sigma_{t\wedge\tau_{r}}}\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}\Big{|}(\sigma_{s\wedge\tau_{r}},X_{s\wedge\tau_{r}})\Bigr{]}\leqslant
V^{\prime}_{\sigma_{s\wedge\tau_{r}}}\bigl{(}X_{s\wedge\tau_{r}}\bigr{)}\bigl{(}\lambda_{\circ}(\hat{p}+\mu\tilde{p})\bigr{)}^{t\wedge\tau_{r}-s\wedge\tau_{r}}.$
Since $\lambda_{\circ}(\hat{p}+\mu\tilde{p})<1$, letting
$\alpha^{\prime}\coloneqq\lambda_{\circ}(\hat{p}+\mu\tilde{p})\mathrm{e}^{\alpha}<1$,
the above inequality gives
((3.11))
$\displaystyle\mathsf{E}\Bigl{[}\mathrm{e}^{\alpha(t\wedge\tau_{r}-s\wedge\tau_{r})}V^{\prime}_{\sigma_{t\wedge\tau_{r}}}$
$\displaystyle\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}\Big{|}(\sigma_{s\wedge\tau_{r}},X_{s\wedge\tau_{r}})\Bigr{]}$
$\displaystyle\leqslant
V^{\prime}_{\sigma_{s\wedge\tau_{r}}}\bigl{(}X_{s\wedge\tau_{r}}\bigr{)}(\alpha^{\prime})^{t\wedge\tau_{r}-s\wedge\tau_{r}}\leqslant
V^{\prime}_{\sigma_{s\wedge\tau_{r}}}\bigl{(}X_{s\wedge\tau_{r}}\bigr{)}.$
This shows that
$\bigl{(}\mathrm{e}^{\alpha(t\wedge\tau_{r})}V^{\prime}_{\sigma_{t\wedge\tau_{r}}}\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}\bigr{)}_{t\in\mathbb{N}_{0}}$
is a nonnegative supermartingale.
Let us now look at the assertion of the Proposition under the condition (S2).
Fix $t\in\mathbb{N}_{0}$. Then from (V3′),
$V^{\prime}_{j}\bigl{(}f_{\sigma_{t\wedge\tau_{r}}}\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}\bigr{)}\leqslant\lambda_{j\sigma_{t\wedge\tau_{r}}}V^{\prime}_{j}\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}$
for all $j\in\mathcal{P}$, and by (V2),
$V^{\prime}_{\sigma_{(t+1)\wedge\tau_{r}}}\bigl{(}f_{\sigma_{t\wedge\tau_{r}}}\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}\bigr{)}\leqslant\lambda_{\sigma_{(t+1)\wedge\tau_{r}}\sigma_{t\wedge\tau_{r}}}V^{\prime}_{\sigma_{(t+1)\wedge\tau_{r}}}\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}\leqslant\mu\lambda_{\sigma_{(t+1)\wedge\tau_{r}}\sigma_{t\wedge\tau_{r}}}V^{\prime}_{\sigma_{t\wedge\tau_{r}}}\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}$.
This leads to
$\displaystyle\mathsf{E}\Bigl{[}V^{\prime}_{\sigma_{(t+1)\wedge\tau_{r}}}\bigl{(}X_{(t+1)\wedge\tau_{r}}\bigr{)}\Big{|}(\sigma_{t\wedge\tau_{r}},X_{t\wedge\tau_{r}})\Bigr{]}$
$\displaystyle\leqslant\mu\Biggl{(}\max_{i\in\mathcal{P}}\sum_{j\in\mathcal{P}}p_{ij}\lambda_{ji}\Biggr{)}V^{\prime}_{\sigma_{t\wedge\tau_{r}}}\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}.$
Since by hypothesis there exists $\alpha>0$ such that
$\mu\left(\max_{i\in\mathcal{P}}\sum_{j\in\mathcal{P}}p_{ij}\lambda_{ji}\right)\mathrm{e}^{\alpha}<1$,
the last inequality shows immediately that
$\bigl{(}\mathrm{e}^{\alpha(t\wedge\tau_{r})}V^{\prime}_{\sigma_{t\wedge\tau_{r}}}\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}\bigr{)}_{t\in\mathbb{N}_{0}}$
is a supermartingale. This concludes the proof. ∎
###### Proof of Corollary (3.6).
First observe that since each map $f_{i}$ is locally Lipschitz, the diameter
of the set $D_{i}\coloneqq\bigl{\\{}f_{i}(x)\big{|}x\in\bar{B}_{r}\bigr{\\}}$
is finite, and since $\mathcal{P}$ is finite, so is the diameter of
$\bigcup_{i\in\mathcal{P}}D_{i}$. Therefore, if $Q$ is the transition kernel
of the Markov process $(\sigma_{t},X_{t})_{t\in\mathbb{N}_{0}}$, then
employing (V1) and the fact that $f_{i}$ is locally Lipschitz for each $i$, we
arrive at
$\displaystyle\mathsf{E}\Bigl{[}V_{\sigma_{1}}(X_{1})$
$\displaystyle\boldsymbol{1}_{\\{X_{1}\in\mathbb{R}^{d}\setminus\bar{B}_{r}\\}}\Big{|}(\sigma_{0},X_{0})=(i,x_{0})\Bigr{]}=\sum_{j\in\mathcal{P}}p_{ij}\boldsymbol{1}_{\mathbb{R}^{d}\setminus\bar{B}_{r}}(f_{j}(x_{0}))V_{j}(f_{j}(x_{0}))$
$\displaystyle\leqslant\sum_{j\in\mathcal{P}}p_{ij}\boldsymbol{1}_{\mathbb{R}^{d}\setminus\bar{B}_{r}}(f_{j}(x_{0}))\alpha_{2}(\left\lVert
f_{j}(x_{0})\right\rVert)\leqslant\sum_{j\in\mathcal{P}}p_{ij}L\left\lVert
x_{0}\right\rVert<Lr<\infty$
for $\left\lVert x_{0}\right\rVert<r$, where $L$ is such that
$\sup_{j\in\mathcal{P},y\in\bar{B}_{r}}\left\lVert
f_{j}(y)\right\rVert\leqslant L\left\lVert y\right\rVert$. This shows that
condition (2.3) of Theorem (2.2) holds under our hypotheses, and by
Proposition (3.5) we know that there exists $\alpha>0$ such that
$\Bigl{(}\mathrm{e}^{\alpha(t\wedge\tau_{r})}V_{\sigma_{t\wedge\tau_{r}}}\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}\boldsymbol{1}_{\mathbb{R}^{d}\setminus\bar{B}_{r}}\bigl{(}X_{t\wedge\tau_{r}}\bigr{)}\Bigr{)}_{t\in\mathbb{N}_{0}}$
is a supermartingale. Theorem (2.2) now guarantees the existence of a constant
$C^{\prime}>0$ such that
$\sup_{t\in\mathbb{N}_{0}}\mathsf{E}\\!\left[\vphantom{\big{|}}V_{\sigma_{t}}(X_{t})\boldsymbol{1}_{\mathbb{R}^{d}\setminus\bar{B}_{r}}(X_{t})\vphantom{\big{|}}\right]\leqslant
C^{\prime}$, and finally, from (V1) it follows that there exists a constant
$c>0$ such that
$\sup_{t\in\mathbb{N}_{0}}\mathsf{E}\\!\left[\vphantom{\big{|}}\alpha_{1}(\left\lVert
X_{t}\right\rVert)\vphantom{\big{|}}\right]\leqslant c<\infty$, as asserted. ∎
###### Proof of Corollary (3.8).
We prove almost sure global asymptotic stability and $\alpha_{1}$ stability in
$\boldsymbol{L}_{1}$ of ((3.1)) under the condition (S1) of Proposition (3.5);
the proofs under (S2) are similar.
First observe that since $\ker(f_{i}-\mathrm{id})=\\{0\\}$ for each
$i\in\mathcal{P}$, i.e., $0$ is the equilibrium point of each individual
subsystem, $\mathsf{P}_{x_{0}}\bigl{(}\tau_{\\{0\\}}<\infty\bigr{)}=0$ for
$x_{0}\neq 0$, where $\tau_{\\{0\\}}$ is the first time that the process
$(X_{t})_{t\in\mathbb{N}_{0}}$ hits $\\{0\\}$. Indeed, since
$\ker(f_{i}-\mathrm{id})=\\{0\\}$ for each $i\in\mathcal{P}$ and $x_{0}\neq 0$
we have
$Q\bigl{(}(i,x_{0}),\mathcal{P}\times\\{0\\}\bigr{)}=\sum_{j\in\mathcal{P}}p_{ij}\boldsymbol{1}_{\\{0\\}}(f_{j}(x_{0}))=0$,
which shows that $Q^{n}\bigl{(}(i,x_{0}),\mathcal{P}\times\\{0\\}\bigr{)}=0$
whenever $x_{0}\neq 0$. The observation now follows from
$\mathsf{P}_{x_{0}}\bigl{(}\tau_{\\{0\\}}<\infty\bigr{)}=\mathsf{P}_{x_{0}}\bigl{(}\bigcup_{n\in\mathbb{N}}\bigl{\\{}\tau_{\\{0\\}}=n\bigr{\\}}\bigr{)}\leqslant\sum_{n\in\mathbb{N}}\mathsf{P}_{x_{0}}\bigl{(}\tau_{\\{0\\}}=n\bigr{)}$.
Therefore, with $\tau_{\\{0\\}}=\tau_{r}=\infty$, proceeding as in the proof
of Proposition (3.5) above, one can show that $\bigl{(}\mathrm{e}^{\alpha
t}V_{\sigma_{t}}\bigl{(}X_{t}\bigr{)}\bigr{)}_{t\in\mathbb{N}_{0}}$ is a
supermartingale for some $\alpha>0$. In particular, With $s=0$ and
$\tau_{r}=\infty$ in ((3.11)), we apply (V1) to arrive at
$\lim_{t\to\infty}\mathsf{E}\bigl{[}\mathrm{e}^{\alpha
t}V_{\sigma_{t}}\bigl{(}X_{t}\bigr{)}\bigr{]}=\lim_{t\to\infty}\mathsf{E}\Bigl{[}\mathsf{E}\bigl{[}\mathrm{e}^{\alpha
t}V_{\sigma_{t}}\bigl{(}X_{t}\bigr{)}\big{|}(\sigma_{0},x_{0})\bigr{]}\Bigr{]}\leqslant\lim_{t\to\infty}\alpha_{2}(\left\lVert
x_{0}\right\rVert)(\alpha^{\prime})^{t}=0$. Standard supermartingale
convergence results and the definition of $\tau_{\\{0\\}}$ imply that
$\mathsf{P}\Bigl{(}\lim_{t\to\infty}V_{\sigma_{t}}(X_{t})=0\Bigr{)}=1$. With
$s=0$ and $\tau_{r}=\tau_{\\{0\\}}=\infty$, the pathwise inequality ((3.10))
in conjunction with (V1) give
$V_{\sigma_{t}}(X_{t})\leqslant\alpha_{2}(\left\lVert
x_{0}\right\rVert)\mu^{N_{t}}\lambda_{\circ}^{t}$. The foregoing inequality
implies that for almost every sample path
$(\sigma_{t},X_{t}^{\prime})_{t\in\mathbb{N}_{0}}$ corresponding to initial
condition $X_{0}=x_{0}^{\prime}$ with $\left\lVert
x_{0}^{\prime}\right\rVert<\left\lVert x_{0}\right\rVert$, one has
$\lim_{t\to\infty}V_{\sigma_{t}}(X_{t}^{\prime})\leqslant\lim_{t\to\infty}\alpha_{2}(\left\lVert
x_{0}^{\prime}\right\rVert)\mu^{N_{t}}\lambda_{\circ}^{t}\leqslant\lim_{t\to\infty}\alpha_{2}(\left\lVert
x_{0}\right\rVert)\mu^{N_{t}}\lambda_{\circ}^{t}=0,$
which proves (AS2). Since the family $\\{f_{i}\\}_{i\in\mathcal{P}}$ is
finite, and each $f_{i}$ is locally Lipschitz, there exists $L>0$ such that
$\sup_{i\in\mathcal{P}}\left\lVert f_{i}(x)\right\rVert\leqslant L\left\lVert
x\right\rVert$ whenever $\left\lVert x\right\rVert\leqslant 1$. Fix
$\varepsilon>0$. By (AS2) we know that for almost all sample paths there
exists a constant $T>0$ such that $\sup_{t\geqslant T}\left\lVert
X_{t}\right\rVert<\varepsilon$ whenever $\left\lVert x_{0}\right\rVert<1$.
Then the choice of $\delta=\bigl{(}\varepsilon L^{-T}\bigr{)}\wedge 1$
immediately gives us the (AS1) property.
It remains to verify (SM1) and (SM2). Both the properties follow from ((3.11))
in the proof of Proposition (3.5), with $s=0$ and $\tau_{r}=\tau_{\\{0\\}}=0$.
Indeed, with these values of $s$ and $\tau_{r}$, ((3.11)) becomes
$\displaystyle\mathsf{E}\bigl{[}\mathrm{e}^{\alpha t}\alpha_{1}(\left\lVert
X_{t}\right\rVert)\bigr{|}(\sigma_{0},X_{0})\bigr{]}$
$\displaystyle\leqslant\mathsf{E}\bigl{[}\mathrm{e}^{\alpha
t}V_{\sigma_{t}}(X_{t})\bigr{|}(\sigma_{0},X_{0})\bigr{]}\leqslant
V_{\sigma_{0}}(X_{0})(\alpha^{\prime})^{t}\leqslant\alpha_{2}(\left\lVert
x_{0}\right\rVert)(\alpha^{\prime})^{t}$
in view of (V1), where
$\alpha^{\prime}=\lambda_{\circ}(\hat{p}+\mu\tilde{p})\mathrm{e}^{\alpha}<1$.
Therefore, given $\varepsilon>0$, we simply choose
$\delta<\alpha_{2}^{-1}(\varepsilon)$ to get (SM1). Given
$r,\varepsilon^{\prime}>0$, we simply choose
$T=0\vee\bigl{(}\ln(\alpha_{2}(r)/\varepsilon^{\prime})/\ln(\alpha^{\prime})\bigr{)}$
to get (SM2). This completes the proof. ∎
### 3.2. Robust Stability of Discrete-Time Randomly Switched Systems
Conditions for the existence of the supermartingale
$\bigl{(}\mathrm{e}^{\alpha(t\wedge\tau_{K}^{\vphantom{T}})}V\bigl{(}X_{t\wedge\tau_{K}^{\vphantom{T}}}\bigr{)}\bigr{)}_{t\in\mathbb{N}_{0}}$
in §2 can be easily expressed in terms of the transition kernel $Q$. However,
if $Q$ is not known exactly, which may happen if the model of the underlying
system generating the Markov process $(X_{t})_{t\in\mathbb{N}_{0}}$ is
uncertain, one needs different methods. We look at one such instance below.
Consider the system
((3.12)) $X_{t+1}=f_{\sigma_{t}}(X_{t},w_{t}),\qquad X_{0}=x_{0},\quad
t\in\mathbb{N}_{0},$
where we retain the definition $\sigma$ from §3.1,
$f_{i}:\mathbb{R}^{d}\times\mathbb{R}^{m}\longrightarrow\mathbb{R}^{d}$ is
locally Lipschitz continuous in both arguments with $f_{i}(0,0)=0$ for each
$i\in\mathcal{P}$, and $(w_{t})_{t\in\mathbb{N}_{0}}$ is a bounded and
measurable $\mathbb{R}^{m}$-valued disturbance sequence. We do not model
$(w_{t})_{t\in\mathbb{N}_{0}}$ as a random process; as such, the transition
kernel of ((3.12)) is not unique.
###### (3.13) Definition.
The system ((3.12)) is said to be _input-to-state stable in
$\boldsymbol{L}_{1}$_ if there exist functions
$\chi,\chi^{\prime}\in\mathcal{K}_{\infty}$ and $\psi\in\mathcal{KL}$ such
that $\mathsf{E}_{x_{0}}\bigl{[}\chi(\left\lVert
X_{t}\right\rVert)\bigr{]}\leqslant\psi(\left\lVert
x_{0}\right\rVert,t)+\sup_{s\in\mathbb{N}_{0}}\chi^{\prime}(\left\lVert
w_{s}\right\rVert)$ for all $t\in\mathbb{N}_{0}$.$\Diamond$
Our motivation for this definition comes from the concept of input-to-state
stability iss in the deterministic context [JW01]. Consider the $i$-th
subsystem of ((3.12)) $x_{t+1}=f_{i}(x_{t},w_{t})$ for $t\in\mathbb{N}_{0}$
with initial condition $x_{0}$; note that $(x_{t})_{t\in\mathbb{N}_{0}}$ is a
deterministic sequence. This nonlinear discrete-time system is said to be iss
if there exist functions $\psi\in\mathcal{KL}$ and
$\chi\in\mathcal{K}_{\infty}$ such that $\left\lVert
x_{t}\right\rVert\leqslant\psi(\left\lVert
x_{0}\right\rVert,t)+\sup_{s\in\mathbb{N}_{0}}\chi(\left\lVert
w_{s}\right\rVert)$ for $t\in\mathbb{N}_{0}$. A sufficient set of conditions
(cf. [JW01, Lemma 3.5]) for iss of this system is that there exist a
continuous function $V:\mathbb{R}^{d}\longrightarrow\mathbb{R}_{\geqslant 0}$,
$\alpha_{1},\alpha_{2}\in\mathcal{K}_{\infty}$, $\rho\in\mathcal{K}$, and a
constant $\lambda\in\;]0,1[$, such that $\alpha_{1}(\left\lVert
x\right\rVert)\leqslant V(x)\leqslant\alpha_{2}(\left\lVert x\right\rVert)$
for all $x\in\mathbb{R}^{d}$, and $V(f_{i}(x,w))\leqslant\lambda V(x)$
whenever $\left\lVert x\right\rVert>\rho(\left\lVert w\right\rVert)$.
In this framework we have the following Proposition.
###### (3.14) Proposition.
Consider the system ((3.12)), and suppose that
1. (i)
Assumption (3.2) holds,
2. (ii)
there exist continuous functions
$V_{i}:\mathbb{R}^{d}\longrightarrow\mathbb{R}_{\geqslant 0}$ for
$i\in\mathcal{P}$, $\alpha_{1},\alpha_{2},\rho\in\mathcal{K}_{\infty}$, a
constant $\mu>1$ and a matrix $[\lambda_{ij}]_{\mathrm{N}\times\mathrm{N}}$ of
nonnegative entries, such that
1. (a)
$\alpha_{1}(\left\lVert x\right\rVert)\leqslant
V_{i}(x)\leqslant\alpha_{2}(\left\lVert x\right\rVert)\qquad$ for all $x$ and
$i$,
2. (b)
$V_{i}(x)\leqslant\mu V_{j}(x)\qquad$ for all $x$ and $i,j$, and
3. (c)
$V_{i}(f_{j}(x))\leqslant\lambda_{ij}V_{i}(x)\qquad$ whenever $\left\lVert
x\right\rVert>\rho(\left\lVert w\right\rVert)$ and all $i,j$,
3. (iii)
$\mu\Bigl{(}\max_{i\in\mathcal{P}}\sum_{j\in\mathcal{P}}p_{ij}\lambda_{ji}\Bigr{)}<1$.
Then ((3.12)) is input-to-state stable in $\boldsymbol{L}_{1}$ in the sense of
Definition (3.13).
###### Proof.
We define the compact set
$K\coloneqq\bigl{\\{}(i,y)\in\mathcal{P}\times\mathbb{R}^{d}\big{|}\left\lVert
y\right\rVert\leqslant\sup_{s\in\mathbb{N}_{0}}\rho(\left\lVert
w_{s}\right\rVert)\bigr{\\}}$, and let
$\tau_{K}^{\vphantom{T}}\coloneqq\inf\bigl{\\{}t\in\mathbb{N}_{0}\big{|}X_{t}\in
K\bigr{\\}}$. In this setting we know from the preceding analysis that
$\varphi(t,\xi)=\mathrm{e}^{\alpha t}\xi$, $\theta(t)=\mathrm{e}^{-\alpha t}$,
and $C=1/(1-\mathrm{e}^{-\alpha})$. We see from the estimate ((2.8)) in the
proof of Theorem (2.2) that
$\displaystyle\mathsf{E}_{x_{0}}\bigl{[}V_{\sigma_{t}}(X_{t})\bigr{]}$
$\displaystyle\leqslant\varphi(0,V_{\sigma_{0}}(x_{0}))\theta(t)+\frac{\beta}{1-\mathrm{e}^{-\alpha}}+\delta\leqslant\alpha_{2}(\left\lVert
x_{0}\right\rVert)\mathrm{e}^{-\alpha
t}+\frac{\beta}{1-\mathrm{e}^{-\alpha}}+\delta.$
Standard arguments show that there exists some
$\chi^{\prime\prime}\in\mathcal{K}_{\infty}$ such that $\beta$ and $\delta$
are each dominated by
$\chi^{\prime\prime}\bigl{(}\sup_{s\in\mathbb{N}_{0}}\left\lVert
w_{s}\right\rVert\bigr{)}$, and therefore, there exists some
$\chi^{\prime}\in\mathcal{K}_{\infty}$ such that
$\beta/(1-\mathrm{e}^{-\alpha})+\delta$ is dominated by
$\chi^{\prime}\bigl{(}\sup_{s\in\mathbb{N}_{0}}\left\lVert
w_{s}\right\rVert\bigr{)}$. Applying (ii)(a) on the left-hand side of the last
inequality, we conclude that ((3.12)) is input-to-state stable with
$\chi=\alpha_{1}$ and $\psi(r,t)=\alpha_{2}(r)\mathrm{e}^{-\alpha t}$. ∎
## Acknowledgments
The authors thank Daniel Liberzon and John Lygeros for helpful comments,
Andreas Milias-Argeitis for useful discussions related to the chemical master
equation, and the anonymous reviewer for a thorough review of the manuscript,
several helpful comments, and drawing their attention to [MT09, Chapter 14].
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|
arxiv-papers
| 2009-01-15T14:16:30 |
2024-09-04T02:48:59.995735
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Debasish Chatterjee and Soumik Pal",
"submitter": "Debasish Chatterjee",
"url": "https://arxiv.org/abs/0901.2269"
}
|
0901.2286
|
# Comparison of filtering methods in SU(3) lattice gauge theory
F. Bruckmanna, a, C. B. Langb, M. Limmerb, T. Maurera, A. Schäfera and S.
Solbriga
a Institut für Theoretische Physik, Universität Regensburg,
D-93040 Regensburg, Germany
b Institut für Physik, FB Theoretische Physik, Karl-Franzens-Universität Graz,
A-8010 Graz, Austria
E-mail: ,
,
,
,
,
,
###### Abstract:
We systematically compare filtering methods used to extract topological
excitations from lattice gauge configurations. We show that there is a strong
correlation of the topological charge densities obtained by APE and Stout
smearing. Furthermore, a first quantitative analysis of quenched and dynamical
configurations reveals a crucial difference of their topological structure:
the topological charge density is more fragmented, when dynamical quarks are
present. This fact also implies that smearing has to be handled with great
care, not to destroy these characteristic structures.
## 1 Filtering methods
Many methods have been developed to extract the IR content from lattice data.
Unfortunately, all these methods introduce ambiguities and parameters. Thus,
to get a coherent picture of the topological structure of the QCD vacuum, it
is necessary to find ways of controlling or even removing these ambiguities.
One of the first attempts to filter out the UV “noise” has been APE smearing
[1], defined as:
$U_{\mu}^{\text{APE}}=P_{SU(N_{c})}\left\\{(1-\alpha_{APE})U_{\mu}^{\text{old}}+\frac{\alpha_{APE}}{6}(\text{staples})\right\\},$
(1)
where $\alpha_{APE}$ determines the weight of the old link and the sum of the
attached staples. The right hand side has to be projected back to the gauge
group. Unfortunately, there is no unique mapping. One approach is to take the
unitary part of the polar decomposition and normalize this matrix by its
determinant. Stout smearing [2] circumvents this projection by using the
exponential map:
$U_{\mu}^{\text{Stout}}=\exp\Big{\\{}\frac{i}{2}Q_{\mu}(U,\rho_{\mu\nu})\Big{\\}}\cdot
U_{\mu}^{\text{old}},$ (2)
where $Q_{\mu}(U,\rho_{\mu\nu})$ is a hermitian matrix constructed from all
plaquettes containing the old link $U_{\mu}$ and weighted by factors
$\rho_{\mu\nu}$. We use the common choice $\rho_{\mu\nu}=\rho_{Stout}$ for
isotropic smearing.
A relatively new method is Laplace filtering [3]. The filtered links are
obtained from a spectral sum of the lowest eigenmodes of the covariant lattice
Laplacian111The original link is reproduced for all eigenmodes, $N=N_{c}\cdot
Vol$, with no projection needed.:
$U_{\mu}^{\text{Laplace}}(x)=P_{SU(N_{c})}\left\\{-\sum_{n=1}^{N}\lambda_{n}\Phi_{n}(x)\otimes\Phi_{n}^{\dagger}(x+\hat{\mu})\right\\}.$
(3)
This procedure acts as a low-pass filter in the sense of a Fourier
decomposition. At this point it should be stressed that Laplace filtering is
completely different from smearing, because it is based on rather global
objects, namely the eigenmodes, and does not locally modify the gauge links in
contrast to smearing.
Taking the filtered links as a starting point, one can reconstruct the
topological charge density
$q(x)=\operatorname{Tr}\big{(}F_{\mu\nu}(x)\widetilde{F}_{\mu\nu}(x)\big{)}/16\pi^{2}$
from an improved field strength tensor [4].
Also the fermionic definition of the topological charge, via the eigenmodes
$\psi$ of a chiral Dirac operator, has been used to explore the IR structure
[5]. For this so called Dirac filtering one truncates the sum in
$q_{Dirac}(x)=\sum_{n=1}^{N}\bigg{(}\frac{\lambda_{n}}{2}-1\bigg{)}\psi^{\dagger}_{n}(x)\gamma_{5}\psi_{n}(x)$
(4)
and takes only the lowest $N$ modes into account. While the zero-modes
determine the total topological charge $Q=\sum q(x)$ due to the index theorem,
the non zero-modes modify the local structure of the density, leaving the
total charge unaffected.
## 2 Comparison of the different methods
In an earlier study a qualitative and quantitative similarity of the
introduced filtering methods for quenched SU(2) gauge configurations has been
observed [6]. One central element of this comparison is the correlator of two
topological charge densities $q_{A}(x)$ and $q_{B}(x)$ defined by:
$\chi_{AB}\equiv\big{(}1/V\big{)}\sum_{x}\;\big{(}q_{A}(x)-\overline{q}_{A}\big{)}\;\big{(}q_{B}(x)-\overline{q}_{B}\big{)},$
(5)
where the mean values are subtracted for convenience. From this we can
construct a quantity that reflects the “matching” of two methods:
$\Xi_{AB}\equiv\frac{\chi_{AB}^{2}}{\chi_{AA}\;\chi_{BB}}$ (6)
$\Xi_{AB}$ is obviously equal to one, if $q_{A}(x)$ is proportional to
$q_{B}(x)$ and deviates the more from one, the more the densities differ.
The main idea is now to relate different filter parameters for those
combinations where $\Xi$ is maximal. In fig. 1 the contour lines of $\Xi$ for
several methods and parameter ranges are shown. On the right hand plot two
exemplary combinations are indicated that correspond to the best matching
value for different filtering strengths.
An interesting observation is that there is an almost one-to-one
correspondence for n steps of APE and n steps of Stout smearing when
$\alpha_{APE}\approx 6\cdot\rho_{Stout}$. As seen in the plot on the lhs. of
fig. 1, $\Xi>0.95$ for a large number of smearing steps. This is consistent
with results by Capitani et al. [7], where such a relation has been derived
from perturbation theory. While they have focused on global observables with
up to 3 smearing steps, our nonperturbative result reflects the local
similarity of both methods and their strongly correlated topological charge
densities up to 50 steps.
|
---|---
Figure 1: Level curves of $\Xi=0.95,0.85,\ldots$ (starting from the diagonal)
for APE vs. Stout smearing (left) and $\Xi=0.8,0.7,\ldots$ (starting from the
inside) for APE smearing (S) vs. Laplace modes (L) (right). $\blacktriangle$
and $\bullet$ mark two examples of “matching” parameters for weak and strong
filtering respectively (from [6]).
## 3 Cluster analysis of the topological charge density
Another important challenge is to extract observables from lattice data, that
could be compared with continuum models of the vacuum. One possibility is to
analyze the cluster structure of the topological charge density. Two lattice
points belong to the same cluster, if they are nearest neighbors and have the
same sign of the topological charge density. Bruckmann et al. [6] found a
power law for the number of clusters as function of the ratio of points with
$|q(x)|$ lying above a variable cut-off $q_{cut}$ and the total number of
lattice points. The exponent $\xi$ of this power law is highly characteristic
for the topological structure of the QCD vacuum. Different models lead to
different predictions, which allows for a very sensitive test. If one has for
instance pure noise, the exponent is $1$, as every point forms its own
cluster. On the other hand one will have an exponent close to zero for very
smooth densities with large structures.
---
Figure 2: left: Exponent $\xi$ of the analysis for clusters common to APE and
Stout smearing. The solid lines show the values predicted from the dilute
instanton gas. right: Total number of distinct clusters for a constant
fraction $f=0.0755$ of points lying above the cut-off. Less than 6 steps are
not considered, as the definition of the topological charge gets ill-defined.
Errors have been calculated using an ensemble average over 10 configurations
but are partly too small to see.
To reduce ambiguities we take only those clusters into account, which are
common to different filters, whose parameters were matched according to
maximal values of $\Xi$. So, if there is an artifact coming from one method,
it is unlikely that this artifact will also be seen by the other.
The exponent for clusters common to APE and Stout can be found in fig. 2
(left). We used one quenched and one dynamical $N_{f}=2$ ensemble with equal
lattice spacing (see tab. 1). Obviously the exponents of the dynamical
configurations lie above the quenched values.
In order to interpret the cluster exponent, a model of dilute quantized
topological objects of general shape and with a size distribution
$d(\rho)\sim\rho^{\beta}$ has been considered in [6]. It leads to
$\xi=(1+4/(\beta+1))^{-1}$ (in 4 dimensions). Following this model, our
findings give a larger coefficient $\beta$ in the dynamical case. Hence,
smaller topological objects become suppressed.
Moreover, the rhs. of fig. 2 shows that for a fixed number of points, lying
above the cut-off, much more clusters are found in the dynamical case. Thus we
conclude that when fermion loops are taken into account, the topological
structure is more complex and fragmented, in the sense of larger number of
distinct objects per volume. This seems to be in accordance to the findings of
the Adelaide group, where small instantons have been seen to be suppressed in
the presence of dynamical quarks, while the total number of instantons
increased, see fig. 6 in [8].
The difference of the cluster exponents quenched vs. dynamical vanishes for
stronger smearing ($\sim$ 30 steps) and the exponents settle down to the same
plateau. So we have reasons to believe that too much smearing destroys the
impact of dynamical quarks.
On the lhs. of fig. 2 we have included for comparison the exponents
$\xi=7/11\approx 0.64$ and $\xi=23/35\approx 0.66$ for the SU(3) instanton gas
without resp. with dynamical quarks. Taking the dilute instanton gas as a
simplified model, it is obvious that the true vacuum should have a higher
exponent, as more structures are present. However, the result in fig. 2 (left)
shows that this is only the case for very few smearing steps, for slightly
stronger filtering we reach smoother configurations than predicted by the
dilute instanton gas. This is another indication of smearing artefacts.
| lat. size | lat. spacing | $\beta_{LW}$ | $m_{0}$
---|---|---|---|---
quenched | $16^{3}\cdot 32$ | 0.148 | 7.90 | –
dynamical | $16^{3}\cdot 32$ | 0.150 | 4.65 | -0.060
Table 1: Ensembles were generated with the Lüscher-Weisz gauge action and a
chirally improved Dirac operator [9]. For the dynamical simulations two
flavors of mass degenerate light quarks were used [10].
## 4 Conclusion and outlook
In conclusion, we have found a strong correlation of the topological charge
densities obtained from APE and Stout smearing. Furthermore, our first results
for dynamical quarks imply that the topological structure is more complex and
fragmented in the presence of fermion loops. But there are also indications
that smearing has to be used with great caution, especially when dealing with
dynamical configurations. The smallness of the cluster exponent is a sign that
smearing is more destructive in SU(3) than in SU(2). Preliminary results for
clusters common to APE smearing and Laplace filtering do not show such
artefacts, as Laplace filtering preserves smaller objects better. This effect
is under investigation.
## References
* [1] M. Falcioni et al., Nucl. Phys. B251 (1985) 624; M. Albanese et al., Phys. Lett. B192 (1987) 163
* [2] C. Morningstar and M. J. Peardon, Phys. Rev., D69 (2004) 054501, [hep-lat/0311018]
* [3] F. Bruckmann and E. M. Ilgenfritz. Phys. Rev. D72 (2005) 114502, [hep-lat/0509020]
* [4] S. O. Bilson-Thompson et al., Annals. Phys. 304 (2003) 1-21, [hep-lat/0203008v1]
* [5] P. Hasenfratz, V. Laliena and F. Niedermayer, Phys. Lett. B427 (1998) 125, [hep-lat/9801021]; I. Horvath et al., Phys. Rev. D67 (2003) 011501, [hep-lat/0203027]; I. Horvath et al., Phys. Rev. D68 (2003) 114505, [hep-lat/0302009]
* [6] F. Bruckmann et al., _Eur. Phys. J_., A33:333–338 (2007), [hep-lat/0612024]
* [7] S. Capitani, S. Dürr, and C. Hoelbling, JHEP, 11:028 (2006), [hep-lat/0607006]
* [8] D. Leinweber and P. J. Moran, Phys. Rev. D78 (2008) 054506, [arxiv:0801.2016]
* [9] C. Gattringer, Phys. Rev. D63 (2001) 114501, [hep-lat/0003005]; C. Gattringer, I. Hip and C. B. Lang, Nucl. Phys. D597 (2001) 451, [hep-lat/0007042]
* [10] C. B. Lang et al., PoS(LATTICE 2007)114 (2007), C. B. Lang et al., (2008), [hep-lat/0812.1681]
|
arxiv-papers
| 2009-01-15T15:43:08 |
2024-09-04T02:49:00.005885
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. Bruckmann, F. Gruber, C. B. Lang, M. Limmer, T. Maurer, A.\n Sch\\\"afer and S. Solbrig",
"submitter": "Florian Gruber",
"url": "https://arxiv.org/abs/0901.2286"
}
|
0901.2376
|
# A Limit Theorem in Singular Regression Problem
Sumio Watanabe
Precision and Intelligence Laboratory
Tokyo Institute of Technology
4259 Nagatsuta, Midoriku, Yokohama, 226-8503 AJapan
e-mail:swatanab@pi.titech.ac.jp
###### Abstract
In statistical problems, a set of parameterized probability distributions is
used to estimate the true probability distribution. If Fisher information
matrix at the true distribution is singular, then it has been left unknown
what we can estimate about the true distribution from random samples. In this
paper, we study a singular regression problem and prove a limit theorem which
shows the relation between the singular regression problem and two birational
invariants, a real log canonical threshold and a singular fluctuation. The
obtained theorem has an important application to statistics, because it
enables us to estimate the generalization error from the training error
without any knowledge of the true probability distribution.
## 1 Introduction
Let $M$ and $N$ be natural numbers, and ${\mathbb{R}}^{M}$ and
${\mathbb{R}}^{N}$ be $M$ and $N$ dimensional real Euclidean spaces
respectively. Assume that $(\Omega,{\mathcal{B}},P)$ is a probability space
and that $(X,Y)$ is an ${\mathbb{R}}^{M}\times{\mathbb{R}}^{N}$-valued random
variable which is subject to a simultaneous probability density function,
$q(x,y)=\frac{q(x)}{(2\pi\sigma^{2})^{N/2}}\exp\Bigl{(}-\frac{|y-r_{0}(x)|^{2}}{2\sigma^{2}}\Bigr{)},$
where $q(x)$ is a probability density function on ${\mathbb{R}}^{M}$,
$\sigma>0$ is a constant, $r_{0}(x)$ is a measurable function from
${\mathbb{R}}^{M}$ to ${\mathbb{R}}^{N}$, and $|\cdot|$ is the Euclidean norm
of ${\mathbb{R}}^{N}$. The function $r_{0}(x)$ is called a regression function
of $q(x,y)$. Assume that $\\{(X_{i},Y_{i});i=1,2,...,n\\}$ is a set of random
variables which are independently subject to the same probability distribution
as $(X,Y)$. Let $W$ be a subset of ${\mathbb{R}}^{d}$. Let $r(x,w)$ be a
function from ${\mathbb{R}}^{M}\times W$ to ${\mathbb{R}}^{N}$. The square
error $H(w)$ is a real function on $W$,
$H(w)=\frac{1}{2}\sum_{i=1}^{n}|Y_{i}-r(X_{i},w)|^{2}.$
An expectation operator $E_{w}[\;\;\;]$ on $W$ is defined by
$E_{w}[F(w)]=\frac{\displaystyle\int F(w)\exp(-\beta
H(w))\varphi(w)dw}{\displaystyle\int\exp(-\beta H(w))\varphi(w)dw},$ (1)
where $F(w)$ is a measurable function, $\varphi(w)$ is a probability density
function on $W$, and $\beta>0$ is a constant called an inverse temperature.
Note that $E_{w}[F(w)]$ is not a constant but a random variable because $H(w)$
depends on random variables. Two random variables $G$ and $T$ are defined by
$\displaystyle G$ $\displaystyle=$
$\displaystyle\frac{1}{2}E_{X}E_{Y}[|Y-E_{w}[r(X,w)]|^{2}],$ $\displaystyle T$
$\displaystyle=$
$\displaystyle\frac{1}{2n}\sum_{i=1}^{n}|Y_{i}-E_{w}[r(X_{i},w)]|^{2}.$
These random variables $G$ and $T$ are called the generalization and training
errors respectively. Since $E_{X,Y}[|Y-r_{0}(X)|^{2}]=N\sigma^{2}$, it is
expected on some natural conditions that both $E[G]$ and $E[T]$ converge to
$S=N\sigma^{2}/2$ when $n$ tends to infinity if there exists $w_{0}\in W$ such
that $r(x,w_{0})=r_{0}(x)$. In this paper, we ask how fast such convergences
are, in other words, our study concerns with a limit theorem which shows the
convergences $n(E[G]-S)$ and $n(E[T]-S)$, when $n\rightarrow\infty$. If Fisher
information matrix
$I_{ij}(w)=\int\partial_{i}r(x,w)\cdot\partial_{j}r(x,w)q(x)dx,$
where $\partial_{i}=(\partial/\partial w_{i})$, is positive definite for
arbitrary $w\in W$, then this problem is well known as a regular regression
problem. In fact, in a regular regression problem, convergences
$n(E[G]-S)\rightarrow d\sigma^{2}/2$ and $n(E[T]-S)\rightarrow-d\sigma^{2}/2$
hold. However, if $I(w_{0})=\\{I_{ij}(w_{0})\\}$ is singular, that is to say,
if $\det I(w_{0})=0$, then the problem is called a singular regression problem
and convergences of $n(E[G]-S)$ and $n(E[T]-S)$ have been left unknown.
In general it has been difficult to study a limit theorem for the case when
Fisher information matrix is singular. However, recently, we have shown that a
limit theorem can be established based on resolution of singularities, and
that there are mathematical relations between the limit theorem and two
birational invariants in singular density estimation [16, 17, 18]. In this
paper we prove a new limit theorem for the singular regression problem, which
enables us to estimate birational invariants from random samples. The limit
theorem proved in this paper has an important application to statistics,
because the expectation value of the generalization error $E[G]$ can be
estimated from that of the training error $E[T]$ without any knowledge of the
true probability distribution. Example Let $M=N=1$, $d=4$, $w=(a,b,c,d)$, and
$W=\\{w\in{\mathbb{R}}^{4};|w|\leq 1\\}$. If the function $r(x,w)$ is defined
by
$r(x,w)=a\sin(bx)+c\sin(dx),$
and $r_{0}(x)=0$, then the set $\\{w\in W;r(x,w)=r_{0}(x)\\}$ is not one
point, and Fisher information matrix at $(a,b,c,d)=(0,0,0,0)$ is singular. A
lot of functions used in statistics, information science, brain informatics,
and bio-informatics are singular, for example, artificial neural networks,
radial basis functions, and wavelet functions.
## 2 Main Results
We prove the main theorems based on the following assumptions. Basic
Assumptions.
(A1) The set of parameters $W$ is defined by
$W=\\{w\in{\mathbb{R}}^{d};\pi_{j}(w)\geq 0\;\;(j=1,2,...,k)\\},$
where $\pi_{j}(w)$ is a real analytic function. It is assumed that $W$ is a
compact set in ${\mathbb{R}}^{d}$ whose open kernel is not the empty set. The
probability density function $\varphi(w)$ on $W$ is given by
$\varphi(w)=\varphi_{1}(w)\varphi_{2}(w),$
where $\varphi_{1}(w)\geq 0$ is a real analytic function and
$\varphi_{2}(w)>0$ is a function of class $C^{\infty}$.
(A2) Let $s\geq 8$ be the number that is equal to 4 times of some integer.
There exists an open set $W^{*}\supset W$ such that $r(x,w)-r_{0}(x)$ is an
$L^{s}(q)$-valued analytic function on $W^{*}$, where $L^{s}(q)$ is a Banach
space defined by using its norm $|\;\;|_{s}$,
$L^{s}(q)=\\{f;|f|_{s}=\Bigl{(}\int|f(x)|^{s}q(x)dx\Bigl{)}^{1/s}<\infty\\}.$
(A3) There exists a parameter $w_{0}\in W$ such that $r(x,w_{0})=r_{0}(x)$. If
these basic assumptions are satisfied, then
$K(w)=\frac{1}{2}\int|r(x,w)-r_{0}(x)|^{2}q(x)dx$ (2)
is a real analytic function on $W^{*}$. A subset $W_{a}\subset W$ is defined
by
$W_{a}=\\{w\in W\;;\;K(w)\leq a\\}.$
Note that $W_{0}$ is the set of all points that satisfy $K(w)=0$. In general,
$W_{0}$ is not one point and it contains singularities. This paper gives a
limit theorem for such a case. Proofs of lemmas and theorems in this section
are given in section 6.
###### Lemma 1.
Assume (A1), (A2), and (A3) with $s\geq 4$. Then
$\zeta(z)=\int_{W}K(w)^{z}\varphi(w)dw$
is a holomorphic function on $Re(z)>0$ which can be analytically continued to
the unique meromorphic function on the entire complex plane whose poles are
all real, negative, and rational numbers.
###### Lemma 2.
Assume (A1), (A2), and (A3) with $s\geq 8$. Then there exists a constant
$\nu=\nu(\beta)\geq 0$ such that
$V=\sum_{i=1}^{n}\Bigl{(}E_{w}[\;|r(X_{i},w)|^{2}\;]-|\;E_{w}[r(X_{i},w)]\;|^{2}\Bigr{)}$
satisfies
$\lim_{n\rightarrow\infty}E[V]=\frac{2\nu}{\beta}.$ (3)
Based on Lemma 1 and 2, we define two important values $\lambda,\nu>0$.
###### Definition 2.1.
Let the largest pole of $\zeta(z)$ be $(-\lambda)$ and its order $m$. The
constant $\lambda>0$ is called a real log canonical threshold. The constant
$\nu=\nu(\beta)$ is referred to as a singular fluctuation.
The real log canonical threshold is an important invariant of an analytic set
$K(w)=0$. For its relation to algebraic geometry and algebraic analysis, see
[4, 5, 6, 9, 10, 11]. It is also important in statistical learning theory, and
it can be calculated by resolution of singularities [16, 3]. The singular
fluctuation is an invariant of $K(w)=0$ which is found in statistical learning
theory [15, 18], whose relation to singularity theory is still unknown. The
followings are main theorems of this paper.
###### Theorem 1.
Assume the basic assumptions (A1), (A2), and (A3) with $s\geq 8$. Let
$S=N\sigma^{2}/2$. Then
$\displaystyle\displaystyle\lim_{n\rightarrow\infty}n(E[G]-S)$
$\displaystyle=$ $\displaystyle\frac{\lambda-\nu}{\beta}+\nu\sigma^{2},$ (4)
$\displaystyle\displaystyle\lim_{n\rightarrow\infty}n(E[T]-S)$
$\displaystyle=$ $\displaystyle\frac{\lambda-\nu}{\beta}-\nu\sigma^{2}.$ (5)
This theorem shows that both the real log canonical threshold $\lambda$ and
singular fluctuation $\nu$ determine the singular regression problem.
###### Theorem 2.
Assume the basic assumptions (A1), (A2), and (A3) with $s\geq 12$. Then
$E[G]=E\Bigl{[}\Bigl{(}1+\frac{2\beta V}{nN}\Bigr{)}T\Bigr{]}+o_{n},$
where $o_{n}$ is a function of $n$ which satisfies $no_{n}\rightarrow 0$.
By this theorem, $V$ and $T$ can be calculated from random samples without any
direct knowledge of the true regression function $r_{0}(x)$. Therefore, $E[G]$
can be estimated from random samples, resulting that we can find the optimal
model or hyperparameter for the smallest generalization error. If the model is
regular, then $\lambda=\nu=d/2$ for arbitrary $0<\beta\leq\infty$, resulting
that Theorem 2 coincides with AIC [1] of a regular statistical model.
Therefore, Theorem 2 is a widely applicable information criterion, which we
can apply to both regular and singular problems. We use Theorem 2 without
checking that the true distribution is a singularity or not.
## 3 Preparation of Proof
We use notations, $S=N\sigma_{2}/2$ and
$\displaystyle S_{i}$ $\displaystyle=$ $\displaystyle Y_{i}-r_{0}(X_{i}),$
$\displaystyle f(x,w)$ $\displaystyle=$ $\displaystyle r(x,w)-r_{0}(x).$
Then $\\{S_{i}\\}$ are independent random variables which are subject to the
normal distribution with average zero and covariance matrix $\sigma^{2}I$
where $I$ is the $d\times d$ identity matrix. It is immediately derived that
$\displaystyle E[T]$ $\displaystyle=$ $\displaystyle
S-E\Bigl{[}\frac{1}{n}\sum_{i=1}^{n}S_{i}\cdot E_{w}[f(X_{i},w)]\Bigr{]}$
$\displaystyle+E\Bigl{[}\frac{1}{2n}\sum_{i=1}^{n}|E_{w}[f(X_{i},w)]|^{2}\Bigr{]},$
$\displaystyle E[G]$ $\displaystyle=$ $\displaystyle
S+\frac{1}{2}E[E_{X}[|E_{w}[f(X,w)]|^{2}]],$ $\displaystyle E[V]$
$\displaystyle=$ $\displaystyle
E\Bigl{[}\sum_{i=1}^{n}\\{E_{w}[|f(X_{i},w)|^{2}]-|E_{w}[f(X_{i},w)]|^{2}\\}\Bigr{]}.$
The function $f(x,w)$ is an $L^{s}(q)$-valued analytic function on $W^{*}$. In
eq.(1), we can define $E_{w}[\;\;]$ by replacing $H(w)$ by $H_{0}(w)$,
$H_{0}(w)=\frac{1}{2}\sum_{i=1}^{n}|f(X_{i},w)|^{2}-\sum_{i=1}^{n}S_{i}\cdot
f(X_{i},w),$
which can be rewritten as
$H_{0}(w)=nK(w)-\sqrt{n}\;\eta_{n}(w),$
where $K(w)$ is given in eq.(2), and
$\displaystyle\eta_{n}(w)$ $\displaystyle=$
$\displaystyle\eta_{n}^{(1)}(w)+\eta_{n}^{(2)}(w),$
$\displaystyle\eta_{n}^{(1)}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{i}\cdot f(X_{i},w),$
$\displaystyle\eta_{n}^{(2)}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(K(w)-\frac{1}{2}|f(X_{i},w)|^{2}).$
We define a norm $\|\;\;\|$ of a function of $f$ on $W$ by
$\|f\|=\sup_{w\in W}|f(w)|.$
Since $W$ is a compact set of ${\mathbb{R}}^{d}$, the set $B(W)$ that is a set
of all continuous and bounded function on $W$ is a Polish space, and both
$\eta_{n}^{(1)}(w)$ and $\eta_{n}^{(2)}(w)$ are $B(W)$-valued random
variables. Because $f(X,w)$ is an $L^{s}(q)$-valued analytic function,
$\\{\eta_{n}^{(1)}\\}$ and $\\{\eta_{n}^{(2)}\\}$ are tight random processes,
resulting that $\eta_{n}^{(1)}$ and $\eta_{n}^{(2)}$ weakly converge to unique
tight gaussian processes $\eta^{(1)}$ and $\eta^{(2)}$ respectively which have
the same covariance matrices as $\eta_{n}^{(1)}$ and $\eta_{n}^{(2)}$
respectively when $n\rightarrow\infty$ [13, 17, 18].
###### Lemma 3.
Assume (A1), (A2), and (A3) with $s\geq 8$. Then
$\displaystyle E[\|\eta_{n}^{(1)}\|^{s}]$ $\displaystyle<$
$\displaystyle\infty,$ $\displaystyle E[\|\eta_{n}^{(2)}\|^{s/2}]$
$\displaystyle<$ $\displaystyle\infty,$
###### Proof.
Since $f(x,w)$ is an $L^{s}(q)$-valued analytic function, it is represented by
the absolutely convergent power series $f(x,w)=\sum_{j}a_{j}(x)w^{j}$ which
satisfies $|a_{j}(x)|\leq M(x)/r^{j}$ for some function $M(x)\in L^{s}(q)$
where $r=(r_{1},..,r_{d})$ is the associative convergence radii. By using this
fact, the former inequality is proved [17, 18]. Also $K(w)-(1/2)f(x,w)^{2}$ is
an $L^{s/2}(q)$-valued analytic function, the latter inequality is proved. ∎
###### Lemma 4.
For arbitrary natural number $n$,
$\displaystyle E[E_{w}[\sqrt{n}\;\eta_{n}^{(1)}(w)]]$ $\displaystyle=$
$\displaystyle\sigma^{2}\beta E[V],$ $\displaystyle
E[E_{w}[\sqrt{n}\;\eta_{n}^{(2)}(w)]]$ $\displaystyle=$ $\displaystyle
E[E_{w}[nK(w)-\frac{1}{2}\sum_{i=1}^{n}|f(X_{i},w)|^{2}]].$
###### Proof.
The second equation is trivial. Let us prove the first equation. Let the left
hand side of the first equation be $A$. Since $\\{S_{i}\\}$ are independently
subject to the normal distribution with covariance matrix $\sigma^{2}I$,
$\displaystyle A$ $\displaystyle=$ $\displaystyle
E\Bigl{[}\sum_{i=1}^{n}S_{i}\cdot E_{w}[f(X_{i},w)]\Bigr{]}$ $\displaystyle=$
$\displaystyle\sigma^{2}E\Bigl{[}\sum_{i=1}^{n}\nabla_{S_{i}}\cdot
E_{w}[f(X_{i},w)]\Bigr{]}$ $\displaystyle=$
$\displaystyle\sigma^{2}E\Bigl{[}\sum_{i=1}^{n}\nabla_{S_{i}}\cdot\Bigl{(}\frac{\int
f(X_{i},w)\exp(-\beta H_{0}(w))\varphi(w)dw}{\int\exp(-\beta
H_{0}(w))\varphi(w)dw}\Bigr{)}\Bigr{]}$ $\displaystyle=$
$\displaystyle\beta\sigma^{2}E\Bigl{[}\sum_{i=1}^{n}E_{w}[|f(X_{i},w)|^{2}]-|E_{w}[f(X_{i},w)]|^{2}\Bigr{]},$
which is equal to the right hand side of the first equation. ∎
###### Definition 3.1.
Let us define five random variables.
$\displaystyle D_{1}$ $\displaystyle=$ $\displaystyle
nE_{w}[E_{X}[|f(X,w)|^{2}]],$ $\displaystyle D_{2}$ $\displaystyle=$
$\displaystyle nE_{X}[|E_{w}[f(X,w)]|^{2}],$ $\displaystyle D_{3}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{n}E_{w}[|f(X_{i},w)|^{2}],$
$\displaystyle D_{4}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}|E_{w}[f(X_{i},w)]|^{2},$ $\displaystyle D_{5}$
$\displaystyle=$ $\displaystyle E_{w}[\sqrt{n}\;\eta_{n}(w)].$
Then, by using Lemma 4, it follows that
$\displaystyle E[G]$ $\displaystyle=$ $\displaystyle S+\frac{1}{2n}E[D_{2}],$
(6) $\displaystyle E[T]$ $\displaystyle=$ $\displaystyle
S-\frac{\beta\sigma^{2}}{n}E[D_{3}-D_{4}]+\frac{1}{2n}E[D_{4}],$ (7)
$\displaystyle E[V]$ $\displaystyle=$ $\displaystyle E[D_{3}-D_{4}],$ (8)
$\displaystyle E[D_{5}]$ $\displaystyle=$
$\displaystyle\beta\sigma^{2}E[D_{3}-D_{4}]+(1/2)E[D_{1}-D_{3}].$ (9)
We show that five expectation values $E[D_{j}]$ $(j=1,2,3,4,5)$ converge to
constants. To show such convergences, it is sufficient to prove that each
$D_{j}$ weakly converges to some random variable and that
$E[(D_{j})^{1+\delta}]<C$ for some $\delta>0$ and constant $C>0$ [13].
###### Definition 3.2.
For a given constant $\epsilon>0$, a localized expectation operator
$E_{w}^{\epsilon}[\;\;]$ is defined by
$E_{w}^{\epsilon}[F(w)]=\frac{\displaystyle\int_{K(w)\leq\epsilon}F(w)\exp(-\beta
H_{0}(w))\varphi(w)dw}{\displaystyle\int_{K(w)\leq\epsilon}\exp(-\beta
H_{0}(w))\varphi(w)dw}.$ (10)
Let $D_{i}^{\epsilon}$ $(i=1,2,3,4,5)$ be random variables that are defined by
replacing $E_{w}[\;\;]$ by $E_{w}^{\epsilon}[\;\;]$.
###### Lemma 5.
Let $0<\delta<s/4-1$. For arbitrary $\epsilon>0$, $j=1,2,3,4,5$,
$\lim_{n\rightarrow\infty}E[|D_{j}-D_{j}^{\epsilon}|^{1+\delta}]=0.$
###### Proof.
We can prove five equations by the same way. Let us prove the case $j=3$. Let
$L(w)=\sum_{i=1}^{n}|f(X_{i},w)|^{2}$. Because $f(x,w)$ is $L^{s}(q)$-valued
analytic function, $E[(\|L\|/n)^{1+\delta}]<\infty$.
$\displaystyle|D_{3}-D_{3}^{\epsilon}|$ $\displaystyle\leq$
$\displaystyle\frac{\displaystyle\int_{K(w)\geq\epsilon}L(w)\exp(-\beta
H_{0}(w))\varphi(w)dw}{\displaystyle\int_{K(w)\leq\epsilon}\exp(-\beta
H_{0}(w))\varphi(w)dw}$ $\displaystyle\leq$
$\displaystyle\frac{\|L\|\;e^{-n\beta\epsilon+2\beta\sqrt{n}\|\eta_{n}\|}}{\int_{K(w)\leq\epsilon}\exp(-\beta
nK(w))\varphi(w)dw}$ $\displaystyle\leq$ $\displaystyle
C_{1}\;n^{d/2}\|L\|\exp(-n\beta\epsilon/2+(2\beta/\epsilon)\|\eta_{n}\|^{2})$
where we used
$2\sqrt{n}\|\eta_{n}\|\leq(n\epsilon/2+(2/\epsilon)\|\eta_{n}\|^{2})$ and
$C_{1}>0$ is a constant. From Lemma 3, $E[\|\eta_{n}\|^{s/2}]\equiv
C_{2}<\infty$, hence by using $C_{3}=(8\epsilon^{2})^{s/4}C_{2}$,
$P(\|\eta_{n}\|^{2}\geq n/(8\epsilon^{2}))\leq C_{3}/n^{s/4}.$
Let $E[F]_{A}$ be the expectation value of $F(x)I_{A}(x)$ where $I_{A}(x)$ is
the defining function of a set $A$, in other words, $I_{A}(x)=1$ if $x\in A$
or $0$ if otherwise.
$\displaystyle E[|D_{3}-D_{3}^{\epsilon}|^{1+\delta}]$ $\displaystyle=$
$\displaystyle E[|D_{3}-D_{3}^{\epsilon}|^{1+\delta}]_{\\{\|\eta_{n}\|^{2}\geq
n/(8\epsilon^{2})\\}}$
$\displaystyle+E[|D_{3}-D_{3}^{\epsilon}|^{1+\delta}]_{\\{\|\eta_{n}\|^{2}<n/(8\epsilon^{2})\\}}.$
The first term of the right hand side is not larger than
$C_{3}E[\|L\|^{1+\delta}]/n^{s/4}$ and the second term is not larger than
$E[(C_{1}\|L\|)^{1+\delta}]n^{d/2}\exp(-n\beta\epsilon/4)$. Both of them
converge to zero. ∎
## 4 Resolution of Singularities
To study the expectation on the region $W_{\epsilon}$ we need resolution of
singularities because $W_{0}$ contains singularities in general. Let
$\epsilon>0$ be a sufficiently small constant. Then by applying Hironaka’s
theorem [7] to the real analytic function
$K(w)\prod_{j=1}^{k}\pi_{j}(w)\varphi_{1}(w)$, all functions $K(w)$,
$\pi_{j}(w)$, and $\varphi_{1}(w)$ are made normal crossing. In fact, there
exist an open set $W_{\epsilon}^{*}\subset W^{*}$ which contains
$W_{\epsilon}$, a manifold $U^{*}$, and a proper analytic map
$g:U^{*}\rightarrow W_{\epsilon}^{*}$ such that in each local coordinate of
$U^{*}$,
$\displaystyle K(g(u))$ $\displaystyle=$ $\displaystyle u^{2k},$
$\displaystyle\varphi(g(u))|g(u)^{\prime}|$ $\displaystyle=$
$\displaystyle\phi(u)|u^{h}|,$
where $k=(k_{1},...,k_{d})$ and $h=(h_{1},...,h_{d})$ are multi-indices
($k_{j}$ and $h_{j}$ are nonnegative integers),
$u^{2k}=\prod_{j}u_{j}^{2k_{j}}$, $u^{h}=\prod_{j}u_{j}^{h_{j}}$,
$|g(u)^{\prime}|$ is the absolute value of Jacobian determinant of $w=g(u)$,
and $\phi(u)>0$ is a function of class $C^{\infty}$. Let
$U=g^{-1}(W_{\epsilon})$. Since $g$ is a proper map and $W_{\epsilon}$ is
compact, $U$ is also compact. Moreover, it is covered by a finite sum
$U=\cup_{\alpha}U_{\alpha},$
where each $U_{\alpha}$ can be taken to be $[0,b]^{d}$ in each local
coordinate using some $b>0$, and
$\int_{W_{\epsilon}}F(w)\varphi(w)dw=\sum_{\alpha}\int_{U_{\alpha}}F(g(u))\phi_{\alpha}(u)|u^{h}|du,$
where $\phi_{\alpha}(u)\geq 0$ is a function of class $C^{\infty}$. In this
paper, we apply these facts to analyzing the singular regression problem. For
resolution of singularities and its applications, see [7] and [4],[16]. Lemma
1 is directly proved by these facts [4, 8, 16]. Moreover, the following lemma
is simultaneously obtained.
###### Lemma 6.
The largest pole $(-\lambda)$ and its order $m$ of $\zeta(z)$ are given by
$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\min_{\alpha}\min_{j}\Bigl{(}\frac{h_{j}+1}{2k_{j}}\Bigr{)},$
(11) $\displaystyle m$ $\displaystyle=$
$\displaystyle\max_{\alpha}\\#\Bigl{\\{}j;\lambda=\frac{h_{j}+1}{2k_{j}}\Bigr{\\}},$
(12)
where, if $k_{j}=0$, $(h_{j+1}+1)/2k_{j}$ is defined to be $+\infty$ and $\\#$
shows the number of elements of the set. Let $\\{U_{\alpha^{*}}\\}$ be the set
of all local coordinates that attain both $\min_{\alpha}$ in eq.(11) and
$\max_{\alpha}$ in eq.(12). Such coordinates are referred to as the essential
coordinates.
For a given real analytic function $K(w)$, there are infinitely many different
resolutions of singularities. However, $\lambda$ and $m$ do not depend on the
pair $(U^{*},g)$. They are called birational invariants. By the definition of
$K(w)$ in eq.(2), there exists an $L^{s}(q)$-valued analytic function $a(x,u)$
on each local coordinate in $U^{*}$ such that
$f(x,u)=a(x,u)u^{k}$
and $E_{X}[|a(X,u)|^{2}]=2$. Therefore,
$H_{0}(g(u))=n\;u^{2k}-\sqrt{n}\;u^{k}\;\xi_{n}(u),$
where
$\xi_{n}(u)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}S_{i}\cdot
a(X_{i},u)+\frac{1}{\sqrt{n}}\sum_{i=1}^{n}u^{k}\Bigl{(}1-\frac{a(X_{i},u)^{2}}{2}\Bigr{)}.$
Then $E[\|\xi_{n}\|^{s/2}]<\infty$ and $E[\|\nabla\xi_{n}\|^{s/2}]<\infty$,
because both $a(x,u)$ and $\nabla a(x,u)$ are $L^{s}(q)$-valued analytic
function, where $\|\nabla\xi_{n}\|=\max_{j}\sup_{w}|\partial_{j}\xi_{n}(u)|$.
The expectation operator $E_{u}[\;\;]$ on $U$ is defined so that it satisfies
$E_{w}^{\epsilon}[F(w)]=E_{u}[F(g(u))]$. Then
$\displaystyle D_{1}^{\epsilon}$ $\displaystyle=$ $\displaystyle
nE_{u}[2u^{2k}],$ $\displaystyle D_{2}^{\epsilon}$ $\displaystyle=$
$\displaystyle nE_{X}[|E_{u}[a(X,u)u^{k}]|^{2}],$ $\displaystyle
D_{3}^{\epsilon}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}E_{u}[|a(X_{i},u)|^{2}u^{2k}],$ $\displaystyle
D_{4}^{\epsilon}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}|E_{u}[a(X_{i},u)u^{k}]|^{2},$ $\displaystyle
D_{5}^{\epsilon}$ $\displaystyle=$ $\displaystyle
E_{u}[\sqrt{n}\xi_{n}(u)u^{k}].$
###### Lemma 7.
Let $s\geq 12$ and $0<\delta<s/6-1$. For $i=1,2,3,4,5$, there exists a
constant $C>0$ such that $E[(D_{i}^{\epsilon})^{1+\delta}]<C$ holds.
###### Proof.
Since $0\leq D_{4}^{\epsilon}\leq D_{3}^{\epsilon}$, $0\leq
D_{2}^{\epsilon}\leq D_{1}^{\epsilon}$, and
$|D_{5}^{\epsilon}|\leq(\|\xi_{n}\|^{2}+2D_{1}^{\epsilon})/2$, it is
sufficient to prove $j=1,3$. The proof for $j=1,3$ can be done by the same
way. Let us prove the case $j=3$. In $l=1,2,..,d$, at least one of $k_{l}\geq
1$. By using partial integration for $du_{l}$, we can show that there exists
$c_{1}>0$ such that
$E_{u}[u^{2k}]\leq\frac{c_{1}}{n}\\{1+\|\xi_{n}\|^{2}+\|\nabla\xi_{n}\|^{2}\\}.$
(13)
Therefore by using $L=(1/n)\sum_{i=1}^{n}\|a(X_{i})\|^{2}$ and Hölder’s
inequality with $1/3+1/(3/2)=1$,
$\displaystyle E[(D_{3}^{\epsilon})^{1+\delta}]\leq
E[(c_{1}L(1+\|\xi_{n}\|^{2}+\|\nabla\xi_{n}\|^{2}))^{1+\delta}]$
$\displaystyle\leq
E[(c_{1}L)^{3+3\delta}]^{1/3}E[(1+\|\xi_{n}\|^{2}+\|\nabla\xi_{n}\|^{2})^{(3+3\delta)/2}]^{3/2}.$
Since $E[\|a(X)\|^{s}]<\infty$, $E[\|\xi_{n}\|^{s/2}]<\infty$, and
$E[\|\nabla\xi_{n}\|^{s/2}]<\infty$, this expectation is finite. ∎
## 5 Renormalized distribution
###### Definition 5.1.
For a given function $h(u)$ on $U$, the renormalized expectation operator
$E_{u,t}^{*}[\;\;|h]$ is defined by
$E_{u,t}^{*}[F(u,t)|h]=\frac{\displaystyle\sum_{\alpha^{*}}\int_{0}^{\infty}dt\int
D(du)F(u,t)t^{\lambda-1}e^{-\beta
t+\beta\sqrt{t}\;h(u)}}{\displaystyle\sum_{\alpha^{*}}\int_{0}^{\infty}dt\int
D(du)t^{\lambda-1}e^{-\beta t+\beta\sqrt{t}\;h(u)}},$
where $D(du)$ is a measure which is defined in eq.(16) and $\sum_{\alpha^{*}}$
shows the sum of all essential coordinates. Also we define
$\displaystyle D_{1}^{*}(h)$ $\displaystyle=$ $\displaystyle
E_{u,t}^{*}[2t|h],$ $\displaystyle D_{2}^{*}(h)$ $\displaystyle=$
$\displaystyle E_{X}[|E_{u,t}^{*}[a(X,u)\sqrt{t}]|^{2}|h],$ $\displaystyle
D_{5}^{*}(h)$ $\displaystyle=$ $\displaystyle E_{u,t}^{*}[h(u)\sqrt{t}|h].$
###### Lemma 8.
The following convergences in probability hold.
$\displaystyle D_{1}^{\epsilon}-D_{1}^{*}(\xi_{n})$ $\displaystyle\rightarrow$
$\displaystyle 0,$ $\displaystyle D_{2}^{\epsilon}-D_{2}^{*}(\xi_{n})$
$\displaystyle\rightarrow$ $\displaystyle 0,$ $\displaystyle
D_{3}^{\epsilon}-D_{1}^{*}(\xi_{n})$ $\displaystyle\rightarrow$ $\displaystyle
0,$ $\displaystyle D_{4}^{\epsilon}-D_{2}^{*}(\xi_{n})$
$\displaystyle\rightarrow$ $\displaystyle 0,$ $\displaystyle
D_{5}^{\epsilon}-D_{5}^{*}(\xi_{n})$ $\displaystyle\rightarrow$ $\displaystyle
0.$
###### Proof.
These five convergences can be proved by the same way. We show
$D_{3}^{\epsilon}-D_{1}^{*}(\xi_{n})\rightarrow 0$. Let
$L(u)=(1/n)\sum_{i=1}^{n}|a(X_{i},u)|^{2}$. Since $E_{X}[|a(X,u)|^{2}]=2$,
$\displaystyle|D_{3}^{\epsilon}-D_{1}^{*}(\xi_{n})|$ $\displaystyle\leq$
$\displaystyle|E_{u}[nL(u)u^{2k}]-E_{u}[E_{X}[a(X,u)^{2}]u^{2k}]|$
$\displaystyle+|E_{u}[2u^{2k}]-E_{u,t}^{*}[2t|\xi_{n}]|.$
Let the first and second terms of the left hand side of this inequality be
$D_{6}$ and $D_{7}$ respectively. Then
$D_{6}\leq\|L-a(X)\|^{2}E_{u}[nu^{2k}].$
By the convergence in probability $\|L-a(X)\|\rightarrow 0$ and eq.(13),
$D_{6}$ converges to zero in probability. From Lemma 10 and 11 in Appendix, it
is derived that
$|E_{u}[u^{2k}]-E_{u,t}^{*}[t|\xi_{n}]|\leq\frac{c_{1}}{\log
n}\frac{e^{2\beta\|\xi_{n}\|^{2}}}{\min(\phi)^{2}}\\{1+\beta\|\nabla\xi_{n}\|\\},$
(14)
which shows $D_{7}\rightarrow 0$ in probability. ∎
###### Lemma 9.
For arbitrary function $h(u)$, the following equality holds.
$D_{1}^{*}(h)=D_{5}^{*}(h)+\frac{2\lambda}{\beta}.$
###### Proof.
Let $F_{p}(u)$ be a function defined by
$F_{p}(u)=\int_{0}^{\infty}t^{p}\;t^{\lambda-1}\;e^{-\beta
t+\beta\sqrt{t}h(u)}dt.$
Then by using the partial integration of $dt$,
$F_{1}(u)=\frac{1}{2}h(u)F_{1/2}(u)+\frac{\lambda}{\beta}F_{0}(u).$
By the definition of $D_{1}^{*}(h)=E_{u,t}^{*}[2t|h]$ and
$D_{5}^{*}(h)=E_{u,t}^{*}[h(u)\sqrt{t}|h]$, we obtain the lemma. ∎
## 6 Proof of Main Theorems
### 6.1 Proof of Lemma 1
###### Proof.
Lemma 1 is already proved in section 4. ∎
### 6.2 Proof of Lemma 2
###### Proof.
By the definition, $V=D_{3}-D_{4}$. By Lemma 5 and 7,
$E[V^{1+\delta}]<\infty$. Reall that the convergence in law
$\xi_{n}\rightarrow\xi$ holds. The random variable
$D_{1}^{*}(\xi_{n})-D_{2}^{*}(\xi_{n})$ is a continuous function of $\xi_{n}$,
hence it converges to a random variable $D_{1}^{*}(\xi)-D_{2}^{*}(\xi)$ in
law. Therefore, by Lemma 5 and 8, $D_{3}-D_{4}$ converges to the same random
variable in law. Hence $E[V]$ converges to a constant when $n$ tends to
infinity. ∎
### 6.3 Proof of Theorem 1
###### Proof.
By the same way as proof of Lemma 2, both $E[D_{1}]$ and $E[D_{3}]$ converge
to $E[D_{1}^{*}(\xi)]$ whereas both $E[D_{2}]$ and $E[D_{4}]$ converge to
$E[D_{2}^{*}(\xi)]$. From eqs.(6), (7), and (8)
$\displaystyle E[n(G-S)]$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{2}E[D_{2}^{*}(\xi)],$ $\displaystyle E[n(T-S)]$
$\displaystyle\rightarrow$
$\displaystyle-2\sigma^{2}\nu+\frac{1}{2}E[D_{2}^{*}(\xi)],$ $\displaystyle
E[V]$ $\displaystyle\rightarrow$ $\displaystyle
E[D_{1}^{*}(\xi)]-E[D_{2}^{*}(\xi)],$
where we used the definition of $\nu$, that is to say,
$E[D_{1}^{*}(\xi)-D_{2}^{*}(\xi)]=2\nu/\beta$. From Lemma 9,
$E[D_{1}^{*}(\xi)]=2\sigma^{2}\nu+\frac{2\lambda}{\beta},$
resulting that
$E[D_{2}^{*}(\xi)]=2\sigma^{2}\nu+\frac{2\lambda-2\nu}{\beta},$
which completes the theorem. ∎
### 6.4 Proof of Theorem 2
###### Proof.
From Theorem 1,
$\displaystyle E[G]$ $\displaystyle=$
$\displaystyle\frac{N\sigma^{2}}{2}+\Bigl{(}\frac{\lambda-\nu}{\beta}+\nu\sigma^{2}\Bigr{)}\frac{1}{n}+o_{n},$
$\displaystyle E[T]$ $\displaystyle=$
$\displaystyle\frac{N\sigma^{2}}{2}+\Bigl{(}\frac{\lambda-\nu}{\beta}-\nu\sigma^{2}\Bigr{)}\frac{1}{n}+o_{n},$
where $no_{n}\rightarrow 0$. Therefore
$\displaystyle E[G]$ $\displaystyle=$ $\displaystyle
E[T]+\frac{2\nu\sigma^{2}}{n}+o_{n}$ $\displaystyle=$ $\displaystyle
E[T]\Bigl{(}1+\frac{2\beta E[V]}{Nn}\Bigr{)}+o_{n}.$
To prove Theorem 2, it is sufficient to show $E[VT]-E[V]E[T]\rightarrow 0$.
$E[|V(T-E[T])|]\leq E[V^{2}]^{1/2}E[(T-E[T])^{2}]^{1/2}.$
Since $s/4-1\geq 2$,
$0\leq E[V^{2}]\leq E[(D_{3})^{2}]<\infty.$
Let $S^{(n)}=\frac{1}{n}\sum_{i=1}^{n}|S_{i}|^{2}/2$, $S=\sigma^{2}N/2$. Then
$E[(T-E[T])^{2}]\leq 3E[(T-S^{(n)})^{2}+(S^{(n)}-S)^{2}+(S-E[T])^{2}].$
Firstly, from
$T-S^{(n)}=\frac{E_{w}[\eta_{n}(w)]}{\sqrt{n}}+\frac{D_{3}}{2n^{2}},$
we obtain
$E[(T-S^{(n)})^{2}]\leq\frac{2E[\|\eta\|^{2}]}{n}+\frac{E[D_{3}^{2}]}{n},$
which converges to zero. Secondly, $\\{S_{i}\\}$ are independently subject to
the normal distribution, hence $E[(S^{(n)}-S)^{2}]\rightarrow 0$. And lastly,
$T-S=\frac{D_{1}}{n},$
hence $E[(T-S)^{2}]$ also converges to zero. ∎
## 7 Conclusion
In this paper, we proved that singular regression problem is mathematically
determined by two birational invariants, the real log canonical threshold and
singular fluctuation. Moreover, there is a universal relation between the
generalization error and the training error, by which we can estimate two
birational invariants from random samples.
## Appendix
To prove eq.(14), we use the following lemmas. Let $\xi$ and $\varphi$ are
functions of $C^{1}$ class from $[0,b]^{d}$ to ${\mathbb{R}}$. Assume that
$\varphi(u)>0$, $u=(x,y)\in[0,b]^{d}$. The partition function of $\xi$,
$\varphi$, $n>1$, and $p\geq 0$ is defined by
$\displaystyle Z^{p}(n,\xi,\varphi)$ $\displaystyle=$
$\displaystyle\int_{[0,b]^{m}}dx\int_{[0,b]^{d-m}}dy\;K(x,y)^{p}\;x^{h}y^{h^{\prime}}\;\varphi(x,y)$
(15)
$\displaystyle\times\exp(-n\beta\;K(x,y)^{2}+\sqrt{n}\beta\;K(x,y)\;\xi(x,y)).$
where $K(x,y)=x^{k}y^{k^{\prime}}$. Let us use
$\displaystyle\|\xi\|$ $\displaystyle=$
$\displaystyle\max_{(x,y)\in[0,b]^{d}}|\xi(x,y)|,$
$\displaystyle\|\nabla\xi\|$ $\displaystyle=$ $\displaystyle\max_{1\leq j\leq
m}\max_{(x,y)\in[0,b]^{d}}\Bigl{|}\frac{\partial\xi}{\partial x_{j}}\Bigr{|}.$
Without loss of generality, we can assume that four multi-indices
$k,k^{\prime},h,h^{\prime}$ satisfy
$\frac{h_{1}+1}{2k_{1}}=\cdots=\frac{h_{r}+1}{2k_{m}}=\lambda<\frac{h^{\prime}_{j}+1}{2k^{\prime}_{j}}\;\;\;(j=m+1,2,...,d).$
In this appendix, we define $a(n,p)\equiv(\log n)^{m-1}/n^{\lambda+p}$.
###### Lemma 10.
There exist constants $c_{1},c_{2}>0$ such that for arbitrary $\xi$ and
$\varphi$ ($\varphi(x)>0\in[0,b]^{d}$) and an arbitrary natural number $n>1$,
$c_{1}\;a(n,p)\;e^{-\beta\|\xi\|^{2}/2}\min(\varphi)\leq
Z^{p}(n,\xi,\varphi)\leq c_{2}\;a(n,p)\;e^{\beta\|\xi\|^{2}/2}\;\|\varphi\|$
holds, where $\displaystyle\min(\varphi)=\min_{u\in[0,b]^{d}}\varphi(u)$.
Let $\xi$ and $\varphi$ be functions of class $C^{1}$. We define
$Y^{p}(n,\xi,\varphi)\equiv\gamma\;a(n,p)\int_{0}^{\infty}dt\int_{[0,b]^{s}}dy\;t^{\lambda+p-1}y^{\mu}e^{-\beta
t+\beta\sqrt{t}\xi_{0}(y)}\varphi_{0}(y),$
where we use notations,
$\gamma=b^{|h|+m-2|k|\lambda}/(2^{m}(m-1)!\prod_{j=m+1}^{d}k_{j})$,
$\xi_{0}(y)=\xi(0,y)$, $\varphi_{0}(y)=\varphi(0,y)$, $\mu=h^{\prime}-2\lambda
k^{\prime}$. A measure $D(du)$ on ${\mathbb{R}}^{d}$ is defined by
$D(du)=\gamma\delta(x)y^{\mu}.$ (16)
###### Lemma 11.
There exists a constant $c_{3}>0$ such that, for arbitrary $n>1$, $\xi$,
$\varphi$, and $p\geq 0$,
$\displaystyle|Z^{p}(n,\xi,\varphi)-Y^{p}(n,\xi,\varphi)|$
$\displaystyle\leq\frac{c_{1}\;a(n,p)}{\log
n}\;e^{\beta\|\xi\|^{2}/2}\\{\beta\|\nabla\xi\|\|\varphi\|+\|\nabla\varphi\|+\|\varphi\|\\}.$
Moreover, there exist constant $c_{4},c_{5}>0$ such that, for arbitrary $\xi$,
$\varphi$, $n>1$,
$c_{4}\;a(n,p)\;e^{-\beta\|\xi\|^{2}/2}\min(\varphi)\leq
Y^{p}(n,\xi,\varphi)\leq c_{5}\;a(n,p)\;e^{\beta\|\xi\|^{2}/2}\;\|\varphi\|.$
###### Proof.
Lemmas 10 and 11 are proved by direct but rather complicated calculations [17,
18]. Let us introduce the outline of the proof. Let $F_{p}(x,y)$ be the
integrated function in eq.(15) and $Z^{p}=Z^{p}(n,\xi,\phi)$.
$Z^{p}=\int dx\int dyF_{p}(x,y),$
which is equal to
$Z^{p}=\int_{0}^{\infty}dt\int_{[0,b]^{d}}dx\;dy\;\delta(t-K(x,y)^{2})\;F_{p}(x,y).$
(17)
Therefore, the problem results in $\delta(t-K(x,y)^{2})$. For arbitrary
function $\Psi(x,y)$ of class $C^{\infty}$, the function
$\zeta(z)=\int_{[0,b]^{d}}K(x,y)^{2z}\Psi(x,y)\;dxdy$
is the meromorphic function whose poles are $(-\lambda_{j})$ and its order
$m_{j}$, hence it has Laurent expansion,
$\zeta(z)=\zeta_{0}(z)+\sum_{j=1}^{\infty}\frac{c_{j}(\Psi)}{(z+\lambda_{j})^{m_{j}}},$
where $\zeta_{0}(z)$ is a holomorphic function and $c_{j}(\Psi)$ is a Schwartz
distribution. Since $\int\delta(t-K(x,y)^{2})\Psi(x,y)dxdy$ is the Mellin
transform of $\zeta(z)$, we have an asymptotic expansion of
$\delta(t-K(x,y)^{2})$ for $t\rightarrow+0$,
$\delta(t-K(x,y)^{2})=\sum_{j=1}^{\infty}\sum_{m=1}^{m_{j}}t^{\lambda_{j}-1}(-\log
t)^{m-1}c_{jm}(x,y),$
where $c_{jm}(x,y)$ is a Schwartz distribution. By applying this expansion to
eq.(17), we obtain two lemmas. ∎
## References
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* [4] M.F. Atiyah. Resolution of singularities and division of distributions. Communications of Pure and Applied Mathematics, Vol.13, pp.145-150. 1970.
* [5] I.N. Bernstein. The analytic continuation of generalized functions with respect to a parameter. Functional Analysis and Applications, Vol.6, pp.26-40, 1972.
* [6] I.M. Gelfand and G.E. Shilov. Generalized Functions. Academic Press, San Diego, 1964.
* [7] H. Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero. Annals of Mathematics, Vol.79, pp.109-326, 1964.
* [8] M. Kashiwara. B-functions and holonomic systems. Inventiones Mathematicae, Vol. 38, pp.33-53, 1976.
* [9] M. Mustata. Singularities of pairs via jet schemes. Journal of the American Mathematical Society, Vol.15, pp.599-615. 2002.
* [10] T. Oaku. Algorithms for b-functions, restrictions, and algebraic local cohomology groups of D-modules. Advances in Applied Mathematics, Vol.19, pp.61-105, 1997.
* [11] M. Saito. On real log canonical thresholds, arXiv:0707.2308v1, 2007\.
* [12] G. Schwarz. Estimating the dimension of a model. Annals of Statistics, Vol.6, No.2, pp.461-464. 1978.
* [13] A. W. van der Vaart, J. A. Wellner. Weak Convergence and Empirical Processes. Springer,1996.
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|
arxiv-papers
| 2009-01-16T01:00:39 |
2024-09-04T02:49:00.012566
|
{
"license": "Public Domain",
"authors": "Sumio Watanabe",
"submitter": "Sumio Watanabe",
"url": "https://arxiv.org/abs/0901.2376"
}
|
0901.2419
|
# Cr-doping effect on the orbital fluctuation of
heavily doped Nd1-xSrxMnO3 ($x$ $\approx$ 0.625)
R. Tasaki r-tasaki@sophia.ac.jp S. Fukushima M. Akaki D. Akahoshi H.
Kuwahara Department of Physics, Sophia University
Chiyoda-ku, Tokyo 102-8554, JAPAN
###### Abstract
We have investigated the Cr-doping effect of Nd0.375Sr0.625MnO3 near the phase
boundary between the $x^{2}-y^{2}$ and $3z^{2}-r^{2}$ orbital ordered states,
where a ferromagnetic correlation and concomitant large magnetoresistance are
observed owing to orbital fluctuation. Cr-doping steeply suppresses the
ferromagnetic correlation and magnetoresistance in Nd0.375Sr0.625Mn1-yCryO3
with $0\leq y\leq 0.05$, while they reappear in $0.05<y\leq 0.10$. Such a
reentrant behavior implies that a phase boundary is located at $y=0.05$, or a
phase crossover occurs across $y=0.05$.
Perovskite manganites, Colossal magnetoresistance (CMR) , Orbital Fluctuation,
Impurity Effect
###### pacs:
75.47.Gk, 75.30.Kz, 75.47.Lx
††preprint: AP-06
## .1 Introduction
Mn oxides with a perovskite structure have attracted much attention because of
the colossal magnetoresistance (CMR) effectDagotto_PR_344 ; Tokura_RPP_69 .
Since the magnetic and transport properties of the perovskite manganites are
strongly affected by ordering patterns of $x^{2}-y^{2}$ and/or $3z^{2}-r^{2}$
orbitals, a detailed investigation of the orbital-ordered (OO) states is
significant for understanding the CMR effect. In heavily doped Nd1-xSrxMnO3
(NSMO), there exist two types of OO states, which exhibit highly anisotropic
magnetic and transport propertiesKajimoto_PRB_60 . One is the $x^{2}-y^{2}$ OO
state ($0.53\leq x<0.63$), accompanying the $A$-type antiferromagnetic (AF)
order in which the $x^{2}-y^{2}$ electrons are conducting within the
ferromagnetic (F) planeKuwahara_PRB_82 . The other is the $3z^{2}-r^{2}$ OO
state ($0.63\leq x\leq 0.80$), accompanying the $C$-type AF order. These two
OO states compete with each other in a bicritical manner at
$x=0.625$Kajimoto_PRB_60 . Near the bicritical region, competition between the
two OO states causes spatial orbital fluctuation on nanometer scale, which
gives rise to the F correlation and concomitant large magnetoresistance
(MR)Akahoshi_PRB_77 ; Nagao_JPCM_19 .
It is well-known that the presence of quenched disorder in a bicritical (or
multicritical) region where a ferromagnetic metallic (FM) and AF insulating
states meet often causes phase separation phenomena, which are essential for
the CMR. In Nd0.5Ca0.5MnO3, for example, Cr-substitution on Mn-sites turns the
charge- and orbital-ordered (CO/OO) state into the FM oneKimura_PRB_62 ;
Kimura_PRL_83 .
Therefore, it can be expected that Cr-doping into NSMO near the bicritical
region ($x=0.625$) induces phase separation and/or enhances the orbital
fluctuation, which might lead to nontrivial phenomena such as the CMR. In this
study, we have investigated the Cr-doping effect of
Nd0.375Sr0.625Mn1-yCryO3(NSMCO) ($0\leq y\leq 0.10$).
## .2 Experiment
NSMCO crystals with $0\leq y\leq 0.10$ were prepared using the floating zone
method. We confirmed that all synthesized crystals are of single phase by the
powder X-ray diffraction method. Magnetic and transport properties were
measured using a Quantum Design physical property measurement system (PPMS).
We randomly cut the synthesized crystals with the size larger than twin-domain
size for measurements of magnetic and transport properties.
## .3 Results and discussion
Figure 1: (Color online) Temperature ($T$) dependence of (a) magnetization
($M$) and (b) magnetoresistance [MR(80 kOe)] of
Nd0.375Sr0.625Mn1-yCryO3(NSMCO) with $y$ = 0, 0.03, and 0.05. ZFC represents
zero field cooling process. MR(80 kOe) is defined as MR(80 kOe)
$\equiv\rho$(80 kOe) / $\rho$(0 Oe).
First, we show in Figs. 1(a) and 1(b) the temperature ($T$) dependence of the
magnetization ($M$) and MR(80 kOe) of NSMCO with $0\leq y\leq 0.05$,
respectively. Here MR(80 kOe) is defined as MR(80 kOe) $\equiv\rho$(80 kOe) /
$\rho$(0 Oe), where $\rho$(0 Oe) and $\rho$(80 kOe) are resistivities measured
in $H$ = 0 Oe and 80 kOe, respectively. In $y$ = 0, the F correlation is
observed due to the orbital fluctuation below 65 K, and the MR(80 kOe) at 5 K
is below 0.01: the resistivity drops more than two orders of magnitude by
applying a magnetic field of $H$ = 80 kOe. This result is consistent with our
previous reportAkahoshi_PRB_77 . Cr-doping steeply suppresses the F
correlation and concomitant MR, which are most suppressed at $y$ = 0.05.
Figure 2: (Color online) $T$ dependence of (a) $M$ and (b) MR(80 kOe) of
NSMCO with $y$= 0.05, 0.07, and 0.10.
Then, we exhibit the $T$ dependence of the $M$ and MR(80 kOe) of NSMCO with
$0.05\leq y\leq 0.10$ in Figs. 2(a) and 2(b). In $y>0.05$, the F correlation
and MR reappear and are evolving with an increase of $y$ from 0.05 to 0.10.
Note that the $M$ and MR of $y=0.10$ are quite similar to those of $y=0$, as
clearly seen from Figs. 1 and 2, indicating that the F correlation and MR of
NSMCO show a reentrant behavior with a change of Cr-concentration $y$.
Figure 3: Cr-doping concentration $y$ dependence of (a) $M$ under $H$=1 kOe
and (b) MR(80 kOe) at 15 K.
We plot the $M$ under $H$ = 1 kOe and MR(80 kOe) at 15K as a function of $y$
in Figs. 3(a) and 3(b), respectively. These figures, as mentioned above,
demonstrate that the F correlation and MR are most suppressed at $y$ = 0.05
and show the reentrant behavior, implying that a phase boundary is located at
$y=0.05$, or a phase crossover occurs across $y=0.05$.
Let us discuss the origin of the reentrant behavior. In $0\leq y\leq 0.05$,
the F correlation and MR are systematically reduced with increasing $y$. This
behavior is quite similar to that observed in NSMO with $x=0.625$, the F
correlation and MR of which are also suppressed with increasing hole-
concentration $x$ from 0.625Akahoshi_PRB_77 . Therefore, we interpret that Cr-
and hole-doping have almost the same effect on the magnetic and transport
properties of NSMO with $x=0.625$ in $0\leq y\leq 0.05$. This is probably
because Cr3+ and Mn4+ have the same electronic configuration of
$t_{2g}^{3}e_{g}^{0}$. In $y>0.05$, the F correlation and MR reappear; the F
correlation is developing with further increasing $y$ from 0.05. This reminds
us the fact that Cr-doping into CO/OO Nd0.5Ca0.5MnO3 produces the FM clusters
embedded in the CO/OO matrixKimura_PRB_62 ; Kimura_PRL_83 . In NSMO with
$x=0.625$ as well as Nd0.5Ca0.5MnO3, the F correlation is probably induced
around Cr3+. However, the F correlation is not so strong compared with that of
Cr-doped Nd0.5Ca0.5MnO3, the reason for which might be explained by the fact
that NSMO with $x=0.625$ is apart from the FM state, which is often found in
the low-doped ($x\leq 0.5$) perovskite manganites. The reentrant behavior
observed in Cr-doped NSMO with $x=0.625$ is perhaps due to competition between
the hole-doping effect and the FM cluster effect caused by Cr-doping. In
$0\leq y\leq 0.05$, the number (or the size) of the FM clusters is so small
that the hole-doping effect is dominant. With increasing $y$, the number (or
the size) of the FM clusters is becoming large, and the FM cluster effect
finally overcomes the hole-doping effect in $y>0.05$. As a result, the F
correlation is macroscopically observed again in the magnetic and transport
properties of NSMCO with $y>0.05$. The detailed mechanism of the reentrant
behavior is now under investigation.
## .4 Acknowledgment
We thank Y. Izuchi for her help in growing single crystals and the
measurements using PPMS. This work was partly supported by the Mazda
Foundation, the Asahi Glass Foundation, and Grant-in-Aid for scientific
research (C) from the Japan Society for Promotion of Science.
## References
* (1) E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344, 1 (2001)
* (2) Y. Tokura, Rep. Prog. Phys. 69, 797 (2006)
* (3) R. Kajimoto, H. Yoshizawa, H. Kawano, H. Kuwahara, Y. Tokura, K. Ohoyama, and M. Ohashi, Phys. Rev. B. 60, 9506 (1999)
* (4) H. Kuwahara, T. Okuda, Y. Tomioka, A. Asamitsu, and Y. Tokura, Phys. Rev. Lett. 82, 4316 (1999)
* (5) D. Akahoshi, R. Hatakeyama, M. Nagao, T. Asaka, Y. Matsui, and H. Kuwahara, Phys. Rev. B. 77, 054404 (2008)
* (6) M. Nagao, T. Asaka, D. Akahoshi, R. Hatakeyama, T. Nagai, M. Saito, K. Watanabe, M. Tanaka, A. Yamazaki, T. Hara, K. Kimoto, H. Kuwahara, and Y. Matsui, J. Phys.: Condens. Matter. 19, 492201 (2007)
* (7) T. Kimura, R. Kumai, Y. Okimoto, Y. Tomioka, and Y. Tokura, Phys. Rev. B. 62, 15021 (2000)
* (8) T. Kimura, Y. Tomioka, R. Kumai, Y. Okimoto, and Y. Tokura, Phys. Rev. Lett. 83, 3940 (1999)
|
arxiv-papers
| 2009-01-16T08:42:31 |
2024-09-04T02:49:00.020123
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. Tasaki, S. Fukushima, M. Akaki, D. Akahoshi, and H. Kuwahara",
"submitter": "Raita Tasaki",
"url": "https://arxiv.org/abs/0901.2419"
}
|
0901.2523
|
11institutetext: National Institute of Chemical Physics and Biophysics, Rävala
10, 15042 Tallinn, Estonia 22institutetext: IFISC, Instituto de Física
Interdisciplinar y Sistemas Complejos (CSIC-UIB), E-07122 Palma de Mallorca,
Spain 33institutetext: Dipartimento di Fisica, Università di Camerino, I-62032
Camerino, Italy
# Stochastic resonance in bistable confining potentials
On the role of confinement
Els Heinsalu 1122 Marco Patriarca 11 Fabio Marchesoni 33
(Received: date / Revised version: date)
###### Abstract
We study the effects of the confining conditions on the occurrence of
stochastic resonance (SR) in continuous bistable systems. We model such
systems by means of double-well potentials that diverge like $|x|^{q}$ for
$|x|\to\infty$. For super-harmonic (hard) potentials with $q>2$ the SR peak
sharpens with increasing $q$, whereas for sub-harmonic (soft) potentials,
$q<2$, it gets suppressed.
###### pacs:
05.40.-aFluctuation phenomena, random processes, noise, and Brownian motion
and 02.50.EyStochastic processes
## 1 Introduction
The simplest dynamical system displaying stochastic resonance (SR) is a
Brownian particle bound into a one-dimensional double well under the action of
a time oscillating tilt and subjected to fluctuating forces (noise)
Gammaitoni1998a ; Benzi1981a . The SR mechanism can be revealed as a maximum
in the amplitude of the periodic component of the average particle position as
a function of the noise intensity (temperature). Due to fluctuations, the
particle randomly jumps between the two potential wells with Kramers rate
borkovec that depends on the double well potential and temperature. When the
average escape time of the particle out of the potential minima (i.e., the
inverse of the Kramers rate) approximately equals the half time-period of the
applied perturbation, the noise induced interwell jumps and the periodic force
synchronize, thus leading to SR.
When studying the problem of a Brownian particle in a symmetric double well
periodically tilted in time, the corresponding potential $U(x)$ is usually
assumed to diverge like $U(x)\sim x^{4}$ at large $x$ Gammaitoni1998a ;
borkovec , so as to ensure a robust confining action. However, the divergence
of the potential for $|x|\to\infty$ strongly affects the response of the
system to an external time-periodic forcing. The goal of the present paper is
to investigate how the Brownian motion in a double well changes with the
confining strength of the one-dimensional potential $U(x)$. For simplicity we
assume that $U(x)\sim|x|^{q}$ for $x\to\pm\infty$. By studying the dependence
of a SR spectral quantifier on $q$, we conclude that bistability is a
necessary, but not sufficient condition for a one-dimensional system to
exhibit SR.
## 2 Model
The model discussed in the following represents an overdamped Brownian
particle with coordinate $x$. Its dynamics is described by the Langevin
equation,
$\eta\dot{x}=-U^{\prime}(x)+A(t)+\xi(t)\,,$ (1)
where $(\dots)^{\prime}\equiv\mathrm{d}(\dots)/\mathrm{d}x$. The confining
potential,
$U(x)=U_{0}\exp\left(-x^{2}/L_{0}^{2}\right)+k|x|^{q}/q\,,$ (2)
is obtained by superimposing a Gaussian repulsive barrier of height $U_{0}$
and width $L_{0}$, to a power-law potential well. To ensure confinement, our
analysis is restricted to $q>1$. The total potential is mirror symmetric at
$x=0$, i.e. $U(x)=U(-x)$. Depending on $q$ a potential $U(x)$ is called hard
(super-harmonic) for $q>2$, or soft (sub-harmonic) for $q<2$ zannetti . The
periodic drive $A(t)$ is chosen as
$A(t)=A_{0}\cos(\Omega t)\,,$ (3)
with amplitude, $A_{0}$, angular frequency, $\Omega\equiv 2\pi\nu$, and time
origin arbitrarily set to zero. The fluctuating force $\xi(t)$ is modeled as a
stationary zero-mean Gaussian noise with auto-correlation function
$\langle\xi(t)\xi(t^{\prime})\rangle=2\eta
k_{\mathrm{B}}T\delta(t-t^{\prime})$. Here $T$ is the temperature and $\eta$
the friction coefficient.
For numerical purposes it is convenient to choose $U_{0}$, $L_{0}$, and
$\tau\equiv\eta L_{0}^{2}/U_{0}$ as the new units respectively of energy,
space and time. Correspondingly, the variables and the parameters appearing in
Eq. (1) can be replaced by the dimensionless quantities $\tilde{x}=x/L_{0}$,
$\tilde{t}=t/\tau$, $\tilde{k}=L_{0}^{q}k/U_{0}$,
$\tilde{A}_{0}=A_{0}L_{0}/U_{0}$, $\tilde{\Omega}=\Omega\tau$, and
$\tilde{T}=k_{B}T/U_{0}$. To avoid a cumbersome notation, in the following we
omit all the tildes. In dimensionless notation the potential (2) reads,
$U(x)=\exp(-x^{2})+k|x|^{q}/q\,,$ (4)
and the Langevin equation (1) can be rewritten as
$\displaystyle\dot{x}=2{x}\exp(-x^{2})-k|{x}|^{q}/x+{A}_{0}\cos({\Omega}{t})+\sqrt{{T}}\xi(t)\,,$
(5)
after the Gaussian noise $\xi(t)$ has been further rescaled so that
$\langle\xi({t})\rangle=0$ and
$\langle\xi({t})\xi({t}^{\prime})\rangle=2\delta({t}-{t}^{\prime})$. In the
following we study how changing $q$ influences the response of the particle to
the periodic forcing signal $A(t)$. As a result of rescaling, the height,
$U_{0}$, and the width, $L_{0}$, of the potential barrier, as well as the
friction coefficient, $\eta$, have been set to one. The remaining tunable
parameter $k$ of the potential (4) will be kept fixed to $k=0.2$ throughout
the present paper. Due to the Gaussian nature of the potential barrier, the
barrier height, $\Delta U$, and the potential minima, $\pm x_{m}$, weakly
depend on $q$ (see Fig. 1); therefore, the observed residual SR dependence on
$q$ is mostly an effect of the varying confining strength of the potential.
We have simulated the behavior of the system by numerically integrating the
rescaled Langevin equation (5) through a Milshtein algorithm Kloeden1999a ;
Milstein2004a . Stochastic trajectories were simulated for different time
lengths $t_{\mathrm{max}}$ and time steps $\Delta t$, so as to ensure
appropriate numerical accuracy and transient effects subtraction. Average
quantities have been obtained as ensemble averages over at least $10^{4}$
trajectories.
Figure 1: Rescaled potential (4) for $k=0.2$ and $q$ ranging between $1.5$ and
$8$. The barrier height is approximately constant, $\Delta U\simeq 0.66$, and
the minima $\pm x_{m}$ slowly shrink with $q$ from $x_{m}\simeq 1.59$ down to
$x_{m}\simeq 1.17$.
## 3 Results
In the long time regime, after transient effects subsided, the response
$\langle x(t)\rangle$ of a particle moving in a symmetric bistable potential
$U(x)$ under the action of the signal (3) with small-amplitude,
$A_{0}x_{m}\ll\Delta U$, and low-frequency, $\Omega\ll
U^{\prime\prime}(x_{m})$, results from the interplay of inter- and the
intrawell dynamics Gammaitoni1998a . On ignoring for the time being the
intrawell dynamics, the system response at low temperatures is dominated by
its harmonic component Gammaitoni1998a ; McNamara1989a ; Presilla1989a ;
Hu1990a ; Jung1991a
$\langle x(t\\!\to\\!\infty)\rangle=\bar{x}(T)\cos[\Omega t-\bar{\phi}(T)]\,,$
(6)
with amplitude, $\bar{x}(T)$, and phase, $\bar{\phi}(T)$, approximated by
$\displaystyle\bar{x}(T)$ $\displaystyle=$ $\displaystyle\frac{A_{0}\langle
x^{2}\rangle_{0}}{T}\frac{2r}{\sqrt{4r^{2}+\Omega^{2}}},$ (7)
$\displaystyle\bar{\phi}(T)$ $\displaystyle=$
$\displaystyle\arctan(\Omega/2r)\,.$ (8)
Here $r\propto\exp(-\Delta U/T)$ is the Kramers rate and $\langle
x^{2}\rangle_{0}$ the variance of the stationary unperturbed process $x(t)$
($A_{0}=0$), both temperature dependent quantities. The amplitude $\bar{x}(T)$
can be manipulated by tuning the noise level. Note that Eqs. (6)-(8) hold in
the linear response theory limit, only, i.e., for $A_{0}x_{m}\ll T$ and
$\Omega>r$ Jung1993 ; schneidman .
According to Eq. (7), in the limit $T\to 0$ the amplitude $\bar{x}(T)$
vanishes due to the potential barrier. The rate $r$ for the particle to
overcome the potential barrier decreases to zero exponentially when lowering
the temperature, that is $r\ll\Omega$. The interwell jumps are thus inhibited
and the particle gets locked in either minima with probability $1/2$; hence
$\lim_{T\to 0}\langle{x}\rangle=0$. In contrast, for high temperatures,
$T\gg\Delta U$, $r$ may grow much larger than $\Omega$ and, consequently,
$\bar{x}(T)\simeq\langle x^{2}\rangle_{0}/T$. For a hard potential with $q>2$
we show below that $\langle x^{2}\rangle_{0}\sim T^{2/q}$, so that, again,
$\lim_{T\to\infty}\bar{x}(T)=0$. The occurrence of these limits for $T\to 0$
and $T\to\infty$ implies the existence of a maximum of $\bar{x}(T)$ for some
optimal $T\sim\Delta U$. This is the so-called spectral characterization of SR
Gammaitoni1998a .
Figure 2: Rescaled amplitude $\bar{x}(T)/A_{0}$, defined in Eq. (6), versus
$T$ for the potential (4) with $k=0.2$, $q=2$. The dashed lines represent the
intrawell oscillations, Eq. (9), with $\kappa=1/|2k\ln(k/2)|$ for $T\to 0$,
and $\kappa=k$ in the limit $T\to\infty$.
### 3.1 Harmonic confining potentials
However, even if the approximate results (6)-(8) describe correctly the
occurrence of SR in most bistable systems, Figs. 2 and 3 ($q>1$) clearly show
that for $T\to 0$ the amplitude $\bar{x}(T)$ approaches a non-zero limit
$\bar{x}(0)>0$. This is a characteristic signature of the intrawell dynamics
Jung1993 ; schneidman . Moreover, for (and only for) $q=2$ a similar behavior
occurs also in the opposite limit $T\to\infty$: the curves $\bar{x}(T)$ attain
an horizontal asymptote, see Fig. 2. The coexistence of these two asymptotes,
peculiar to $q=2$, strongly suppresses the SR peak.
The nonzero $\bar{x}(T)$ limits for $T\to 0$ and $T\to\infty$ can be explained
by noticing that an overdamped Brownian particle bound to a generic harmonic
potential well, $U(x)=\kappa(x-x_{0})^{2}/2$, responds to the signal (3) with
amplitude
$\bar{x}=A_{0}/\sqrt{\Omega^{2}+\kappa^{2}}.$ (9)
[Note also that its variance in the absence of forcing ($A_{0}=0$) is $\langle
x^{2}\rangle_{0}=T/\kappa$.]
In the low temperature limit, $T\to 0$, the particle described by the Langevin
equation (5) is locked in either the right or left potential well, where it
executes additional harmonic oscillations around the corresponding minima
$x_{0}=\pm x_{m}$ Gammaitoni1998a ; Jung1993 ; schneidman ; lowD . Such
intrawell oscillations should not be mistaken for the interwell dynamics
described by Eq. (6) Hu1990a . Their amplitude is well reproduced by Eq. (9)
with $\kappa\equiv U^{\prime\prime}(\pm x_{m})=|2k\ln(k/2)|$.
In the high temperature limit, $T\to\infty$, the fluctuations $\xi(t)$ may
grow so intense that the barrier of the bistable potential (4) becomes
ineffective; the particle is thus effectively confined into a parabolic
potential with $\kappa=k$ and centered at $x_{0}=0$. The amplitude of the
periodic component of the particle response to the external force is then
described again by Eq. (9) but with $\kappa=k$.
For small frequencies the rescaled amplitude ${\bar{x}}/A_{0}$ only depends on
the curvature of the bistable potential at $x_{0}=\pm x_{m}$ for $T\to 0$,
${\bar{x}}/A_{0}=1/|2k\ln(k/2)|$, and at $x_{0}=0$ for $T\to\infty$,
${\bar{x}}/A_{0}=1/k$.
The argument above can be easily generalized for any value of $q$ at low
temperatures, but it becomes untenable in the limit $T\to\infty$, where
nonlinearity comes into play.
Figure 3: Rescaled amplitude $\bar{x}(T)/A_{0}$, defined by Eq. (6), versus
$T$ for the potential (4) with $k=0.2$ and different $q>2$ (hard potentials).
The dashed lines are the decay power law $T^{2/q-1}$.
### 3.2 Hard confining potentials
As anticipated above, at high temperatures the presence of the central barrier
can be ignored. This implies that for $T\to\infty$ Eq. (7) simplifies to
$\frac{{\bar{x}}(T)}{A_{0}}=\frac{\langle
x^{2}\rangle_{0}}{T}=\frac{1}{T}\frac{\int_{0}^{\infty}dx~{}x^{2}~{}\exp{(-kx^{q}/qT)}}{\int_{0}^{\infty}dx~{}\exp{(-kx^{q}/qT)}}.$
(10)
In Eq. (10) we made use of the inequality $r\gg\Omega$ and of the approximate
expression $P_{0}(x)={\cal N}\exp(-kx^{q}/qT)$ for the stationary probability
density of the unperturbed process (5); ${\cal N}$ is an appropriate
normalization constant. Note that for sufficiently low $\Omega$, the condition
$r\gg\Omega$ can be consistent with the approximations in Eq. (7), whereas
suppressing the potential barrier makes the very definition of $r$
meaningless.
An explicit calculation yields
$\frac{{\bar{x}}(T)}{A_{0}}=\left(\frac{q}{k}\right)^{2/q}\frac{\Gamma(3/q)}{\Gamma(1/q)}~{}\frac{1}{T^{1-2/q}}.$
(11)
Ignoring the algebraic factors we conclude that
$\displaystyle\lim_{T\to\infty}\bar{x}(T)\sim T^{\,2/q-1}\,.$ (12)
From here one can see that $\bar{x}$ decreases with increasing $T$ only for
hard confining potentials with $q>2$. In particular, for the prototypical case
of a quartic potential, $q=4$ Gammaitoni1998a , one finds $\bar{x}(T)\sim
1/\sqrt{T}$, as confirmed by the simulation results (see Fig. 3). For $q=2$,
one recovers the harmonic limit discussed in the foregoing subsection.
The decay law of $\bar{x}(T)$, Eq. (12), is clearly a consequence of the
nonlinearity of the potential. Indeed, the same power law can be recovered by
implementing the stochastic linearization scheme of Ref. bulsara : In Gaussian
approximation, for $q$ an integer, $\lim_{|x|\to\infty}U(x)=\kappa x^{2}/2$
with $\kappa=(q-1)!!k\langle x^{2}\rangle_{0}^{q/2-1}$; from the relation
$\langle x^{2}\rangle_{0}=T/\kappa$, holding for harmonic potentials, Eq. (12)
follows.
Moreover, $\bar{x}(T)$ cannot decrease faster than $T^{-1}$, which happens for
$q\to\infty$. It should be noticed that $\bar{x}(T)\sim T^{-1}$ is the decay
law predicted in two-state model approximation McNamara1989a , where $\langle
x^{2}\rangle_{0}$ is replaced by $x_{m}^{2}$ (i.e., a constant).
Figure 4: Rescaled amplitude $\bar{x}(T)/A_{0}$ versus $T$ for the potential
(4) with $k=0.2$ and $q=1.5$ (soft potential). The dashed lines represent the
horizontal asymptotes $1/\Omega$ (see text). In place of the SR peak an
inflexion point is detectable for low $\Omega=2\pi\nu$.
### 3.3 Soft confining potentials
Equation (12) for $q<2$ suggests that $\bar{x}(T)$ may diverge at high
temperatures. However, when dealing with soft potentials, the linear theory
approximations (6)-(8) must be used with caution. In the limit $T\to 0$ the
interwell oscillation amplitude (7) is known to apply only for very small
perturbation amplitudes zannetti : This explains the residual $A_{0}$
dependence of the low $T$ plateaus reported in Fig. 4.
More importantly, in the high $T$ limit, although the barrier of a soft
potential is awash with noise, confinement gets so weak that the particle is
driven up and down the potential walls primarily by the deterministic force
$A(t)$, rather than by the noise. [For a comparison, we remind that a particle
falls from $\pm\infty$ down to $\pm x_{m}$ in a finite time for $q>2$ and in
an infinite time for $q<2$.] In conclusion, on assuming that the Brownian
particle oscillates as if it were (almost) free, its amplitude would read
$\lim_{T\to\infty}\bar{x}(T)\sim A_{0}/\Omega\,.$ (13)
$\bar{x}(T)$ is then expected to develop high $T$ plateaus also for $q<2$,
but, in contrast with the cases discussed in Sec. 3.1, such plateaus are
inverse proportional to the drive frequency (also for low frequency drives,
see Fig. 4).
In the case of sub-harmonic bistable potentials the hallmark of SR is thus the
monotonic increase of the response amplitude with T, as opposed to the
occurrence of a maximum often detected in the super-harmonic potentials. Such
a behavior resembles the phenomenon of ”SR without tuning” discussed in Ref.
Collins1995 , with the important difference that here it has been observed in
a single unit, rather than in a summing network of $N$ excitable units.
## 4 Conclusions
We conclude this note with two important remarks:
(i) The coexistence of two locally stable minima separated by a potential
barrier is commonly advocated to explain the occurrence of a SR peak in a
continuous bistable dynamics. Here we have shown that this keeps being true as
long as the confining action exerted by the potential is super-harmonic. Most
notably, for harmonic and sub-harmonic potentials the periodic component of
the system response may increase monotonically with the noise level.
(ii) In many experimental reports (see, for a review, Ref. JSP ), the authors
tried to characterize the SR peak by means of Eq. (6), without paying much
attention to the $T$ dependence of the quantity $\langle x^{2}\rangle_{0}$. In
some cases they adopted an outright two-state model with $\langle
x^{2}\rangle_{0}=x_{m}^{2}$. This led to a poor fit of the decaying tail of
$\bar{x}(T)$, whereas a more accurate fit could have given a valuable clue to
better model the system at hand EPR .
## Acknowledgments
This work has been supported by the Estonian Science Foundation through Grant
No. 7466 (M.P., E.H.), Spanish MEC and FEDER through project FISICOS
(FIS2007-60327), and ESF STOCHDYN project (E.H.).
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arxiv-papers
| 2009-01-16T16:31:52 |
2024-09-04T02:49:00.026360
|
{
"license": "Public Domain",
"authors": "Els Heinsalu, Marco Patriarca, and Fabio Marchesoni",
"submitter": "Marco Patriarca",
"url": "https://arxiv.org/abs/0901.2523"
}
|
0901.2581
|
Spontaneous Reaction Silencing in Metabolic Optimization
Takashi Nishikawa,1,4 Natali Gulbahce,2,3 Adilson E. Motter4,∗
1Division of Mathematics and Computer Science, Clarkson University, Potsdam,
NY 13699, USA.
2Department of Physics and Center for Complex Network Research, Northeastern
University, Boston, MA 02115, USA.
3Center for Cancer Systems Biology, Dana Farber Cancer Institute, Boston, MA
02115, USA.
4Department of Physics and Astronomy and Northwestern Institute on Complex
Systems, Northwestern University, Evanston, IL 60208, USA.
∗Corresponding author. Department of Physics and Astronomy, Northwestern
University, 2145 Sheridan Road, Evanston, IL 60208, USA; Tel.: +1 847 491
4611; Fax: +1 847 491 9982; E-mail: motter@northwestern.edu
Abstract
Metabolic reactions of single-cell organisms are routinely observed to become
dispensable or even incapable of carrying activity under certain
circumstances. Yet, the mechanisms as well as the range of conditions and
phenotypes associated with this behavior remain very poorly understood. Here
we predict computationally and analytically that any organism evolving to
maximize growth rate, ATP production, or any other linear function of
metabolic fluxes tends to significantly reduce the number of active metabolic
reactions compared to typical non-optimal states. The reduced number appears
to be constant across the microbial species studied and just slightly larger
than the minimum number required for the organism to grow at all. We show that
this massive spontaneous reaction silencing is triggered by the
irreversibility of a large fraction of the metabolic reactions and propagates
through the network as a cascade of inactivity. Our results help explain
existing experimental data on intracellular flux measurements and the usage of
latent pathways, shedding new light on microbial evolution, robustness, and
versatility for the execution of specific biochemical tasks. In particular,
the identification of optimal reaction activity provides rigorous ground for
an intriguing knockout-based method recently proposed for the synthetic
recovery of metabolic function.
## Author Summary
Cellular growth and other integrated metabolic functions are manifestations of
the coordinated interconversion of a large number of chemical compounds. But
what is the relation between such whole-cell behaviors and the activity
pattern of the individual biochemical reactions? In this study, we have used
flux balance-based methods and reconstructed networks of Helicobacter pylori,
Staphylococcus aureus, Escherichia coli and Saccharomyces cerevisiae to show
that a cell seeking to optimize a metabolic objective, such as growth, has a
tendency to spontaneously inactivate a significant number of its metabolic
reactions, while all such reactions are recruited for use in typical
suboptimal states. The mechanisms governing this behavior not only provide
insights into why numerous genes can often be disabled without affecting
optimal growth, but also lay a foundation for the recently proposed synthetic
rescue of metabolic function, in which the performance of suboptimally
operating cells can be enhanced by disabling specific metabolic reactions. Our
findings also offer explanation for another experimentally observed behavior,
in which some inactive reactions are temporarily activated following a genetic
or environmental perturbation. The latter is of utmost importance given that
non-optimal and transient metabolic behaviors are arguably common in natural
environments.
## Introduction
A fundamental problem in systems biology is to understand how living cells
adjust the usage pattern of their components to respond and adapt to specific
genetic, epigenetic, and environmental conditions. In complex metabolic
networks of single-cell organisms, there is mounting evidence in the
experimental [5, 6, 1, 3, 4, 2] and modeling [7, 8, 9, 10, 11, 12, 13, 14]
literature that a surprisingly small part of the network can carry all
metabolic functions required for growth in a given environment, whereas the
remaining part is potentially necessary only under alternative conditions
[15]. The mechanisms governing this behavior are clearly important for
understanding systemic properties of cellular metabolism, such as mutational
robustness, but have not received full attention. This is partly because
current modeling approaches are mainly focused on predicting whole-cell
phenotypic characteristics without resolving the underlying biochemical
activity. These approaches are typically based on optimization principles, and
hence, by their nature, do not capture processes involving non-optimal states,
such as the temporary activation of latent pathways during adaptive evolution
towards an optimal state [16, 17].
To provide mechanistic insight into such behaviors, here we study the
metabolic system of single-cell organisms under optimal and non-optimal
conditions in terms of the number of active reactions (those that are actually
used). We implement our study within a flux balance-based framework [19, 20,
23, 18, 21, 22]. Figure 1 illustrates key aspects of our analysis using the
example of Escherichia coli. For any typical non-optimal state (Fig. 1B), all
the reactions in the metabolic network are active, except for those that are
necessarily inactive due either to mass balance constraints or environmental
conditions (e.g., nutrient limitation). In contrast, a large number of
additional reactions are predicted to become inactive for any metabolic flux
distribution maximizing the growth rate (Fig. 1A). This spontaneous reaction
silencing effect, in which optimization causes massive reaction inactivation,
is observed in all four organisms analyzed in this study, H. pylori, S.
aureus, E. coli, and S. cerevisiae, which have genomes and metabolic networks
of increasing size and complexity (Materials and Methods). Our analysis
reveals two mechanisms responsible for this effect: (1) irreversibility of a
large number of reactions, which under intracellular physiological conditions
[14] is shared by more than 62% of all metabolic reactions in the organisms we
analyze (Table 1 and Note 1); and (2) cascade of inactivity triggered by the
irreversibility, which propagates through the metabolic network due to
stoichiometric and flux balance constraints. We provide experimental evidence
of this phenomenon and explore applications to data interpretation by
analyzing intracellular flux and gene activity data available in the
literature.
The drastic difference between optimal and non-optimal behavior is a general
phenomenon that we predict not only for the maximization of growth, but also
for the optimization of any typical objective function that is linear in
metabolic fluxes, such as the production rate of a metabolic compound.
Interestingly, we find that the resulting number of active reactions in
optimal states is fairly constant across the four organisms analyzed, despite
the significant differences in their biochemistry and in the number of
available reactions. In glucose media, this number is $\sim 300$ and
approaches the minimum required for growth, indicating that optimization tends
to drive the metabolism surprisingly close to the onset of cellular growth.
This reduced number of active reactions is approximately the same for any
typical objective function under the same growth conditions.
We suggest that these findings will have implications for the targeted
improvement of cellular properties [24]. Recent work predicts that the
knockout of specific enzyme-coding genes can enhance metabolic performance and
even rescue otherwise nonviable strains [25]. The possibility of such
knockouts bears on the issue of whether the inactivation of the corresponding
enzyme-catalyzed reactions would bring the whole-cell metabolic state close to
the target objective. Thus, our identification of a cascading mechanism for
inducing optimal reaction activity for arbitrary objective functions provides
a natural set of candidate genetic interventions for the knockout-based
enhancement of metabolic function [25].
## Results
### Typical Non-optimal States
We model cellular metabolism as a network of metabolites connected through
reaction and transport fluxes. The state of the system is represented by the
vector $\mathbf{v}=(v_{1},\ldots,v_{N})^{T}$ of these fluxes, including the
fluxes of $n$ internal and transport reactions, as well as $n_{\text{ex}}$
exchange fluxes for modeling media conditions. Under the constraints imposed
by stoichiometry, reaction irreversibility, substrate availability, and the
assumption of steady-state conditions, the state of the system is restricted
to a feasible solution space $M\subseteq\mathbb{R}^{N}$ (Materials and
Methods). Within this framework, we first consider the number of active
reactions in a typical non-optimal state $\mathbf{v}\in M$.
We can prove that, with the exception of the reactions that are inactive for
all $\mathbf{v}\in M$, all the metabolic reactions are active for almost all
$\mathbf{v}\in M$, i.e., for any typical state chosen randomly from $M$ (Text
S1, Section 1). Accordingly, the number $n_{+}(\mathbf{v})$ of active
reactions in a typical non-optimal state is constant, i.e.,
$n_{+}(\mathbf{v})=n_{+}^{\text{typ}},\quad\text{for almost all }\mathbf{v}\in
M.$ (1)
The reactions that are inactive for all states are so either due to mass
balance or environmental conditions, and can be identified numerically using
flux coupling [26] and flux variability analysis [9].
##### Mass balance.
Part of the metabolic reactions are forced to be inactive solely due to mass
balance, independently of the medium conditions. For example, glutathione
oxidoreductase in the E. coli reconstructed model involves oxidized
glutathione, but because there is no other metabolic reaction that can balance
the flux of this metabolite, the reaction cannot be active in any steady
state. We characterize such reactions uniquely by a linear relationship
between vectors of stoichiometric coefficients (Text S1, Section 2). Although
these reactions are inactive in any steady state, some of them may play a role
in transient dynamics (e.g., after environmental changes) [27], for which
time-dependent analysis is required [28]. Others may be part of an incomplete
pathway at an intermediate stage of the organism’s evolution or, more likely,
an artifact of the incompleteness or stoichiometric inconsistencies of the
reconstructed model. Such inconsistencies have been identified in previous
models [29], such as an earlier version of the model we use for S. cerevisiae
[30].
##### Environmental conditions.
Other reactions are constrained to be inactive due to the constraints arising
from the environmental conditions imposed by the medium. For example, all
reactions in the allantoin degradation pathway must be inactive for E. coli in
media with no allantoin available, since allantoin cannot be produced
internally. Similarly, the reactions involved in aerobic respiration are
generally inactive for any state under anaerobic growth.
#####
The results for the typical activity of each organism in glucose minimal media
(Materials and Methods) are summarized in the top bars of Fig. 2 and in Table
2. The fraction of active reactions ranges from 50%–82%, while 9%–23% are
inactive due to mass balance constraints and 9%–26% are inactive due to the
environmental conditions. Although the absolute number of active reactions
tends to increase with the size of the metabolic network, the fraction of
active reactions appears to show the opposite tendency. Figure 1B shows that
most of the subsystems of the E. coli metabolism are almost completely active,
but a few have many inactive reactions. For example, due to the incompleteness
of the network many reactions involving cofactors and prosthetic group
biosynthesis cannot be used under steady-state conditions in any environment.
In addition, many reactions in the alternate carbon metabolism, as well as
many transport and extracellular reactions, must be inactive in the absence of
the corresponding substrates in the glucose medium.
### Growth-Maximizing States
We now turn to the maximization of growth rate, which is often hypothesized in
flux balance-based approaches and found to be consistent with observation in
adaptive evolution experiments [32, 33, 34, 31]. Performing numerical
optimization in glucose minimal media (Materials and Methods), we find that
the number of active reactions in such optimal states is reduced by 21%–50%
compared to typical non-optimal states, as indicated in the middle bars of
Fig. 2. Interestingly, the absolute number of active reactions under maximum
growth is $\sim 300$ and appears to be fairly independent of the organism and
network size for the cases analyzed. We observe that the minimum number of
reactions required merely to sustain positive growth [7, 8] is only a few
reactions smaller than the number of reactions used in growth-maximizing
states (bottom bars, Fig. 2). This implies that surprisingly small metabolic
adjustment or genetic modification is sufficient for an optimally growing
organism to stop growing completely, which reveals a robust-yet-subtle
tendency in cellular metabolism: while the large number of inactive reactions
offers tremendous mutational and environmental robustness [35], the system is
very sensitive if limited only to the set of reactions optimally active. The
flip side of this prediction is that significant increase in growth can result
from very few mutations, as observed recently in adaptive evolution
experiments [36].
We now turn to mechanisms underlying the observed reaction silencing, which is
spread over a wide range of metabolic subsystems (see Fig. 1 for E. coli). The
phenomenon is caused by growth maximization via reaction irreversibility and
cascading of inactivity.
##### Irreversibility.
We identify three different scenarios in which reaction irreversibility causes
reaction inactivity under maximum growth. The simplest case is when the
reaction is part of a parallel pathway structure. While stoichiometrically
equivalent pathways lead to alternate optima [9], “non-equivalent” redundancy
can force irreversible reactions in less efficient pathways to be inactive. To
illustrate this effect, we show in Fig. 3A three alternative pathways (P1, P2,
and P3) for glucose transport and utilization in the E. coli metabolism.
Pathway P1 is active under maximum growth, while P2 and P3 are inactive
because they are stoichiometrically less efficient for cellular growth.
Indeed, we computationally predict that knocking out P1 would make P2 active,
but the growth rate would be reduced by 2.5%. Knocking out both P1 and P2
would make P3 active, but the growth rate would be reduced by more than 10%.
Here, the irreversibility of P2 and P3 is essential. For example, if P2 were
reversible, the biomass production could be increased (by about 0.05%) by
making this pathway active in the opposite direction, which creates a
metabolic cycle equivalent to a combination of the pyruvate kinase reaction
and the transport of protons out of the cell. The pyruvate kinase itself does
not contribute to the increase in biomass production (it is inactive under
maximum growth condition), but the cycle would provide a more efficient
transport of protons to balance the influx of protons accompanying the ATP
synthesis, which in turn would increase biomass production.
A different silencing scenario is identified when no clear parallel pathway
structure is recognizable. In this scenario there is a transverse pathway
that, were it reversible, could be used to increase growth by redirecting
metabolic flow from “non-limiting” pathways to those that limit the production
of biomass precursors. This includes transverse reactions that establish one-
way communication between pathways that lead to different building blocks of
the biomass (when one of them is more limiting than the others). In the E.
coli model, for example, isocitrate lyase in the glyoxylate bypass is
predicted to be inactive under maximum growth, as shown in Fig. 3B. This
prediction is consistent with the observation from growth experiments in
glucose media [17]. Again, the irreversibility of the reaction (Note 2) is
essential for this argument because, if this constraint is hypothetically
relaxed, we predict that the reaction becomes active in the opposite
direction, which leads to a slight increase in the maximum growth rate (about
0.005%).
A third scenario for the irreversibility mechanism arises when a transport
reaction is irreversible because the corresponding substrate is absent in the
medium. For example, since acetate, a possible carbon and energy source, is
absent in the given medium, the corresponding transport reaction is
irreversible; acetate can only go out of the cell (Note 3). For E. coli under
maximum growth, we computationally predict that this transport reaction is
inactive. This indicates that E. coli growing maximally in the given glucose
medium wastes no acetate by excretion, which is consistent with experimental
observation in glucose-limited culture at low dilution rate [37]. Our
predictions in the previous section, in contrast, imply that acetate transport
would be active in typical non-optimal states, suggesting that suboptimal
growth may induce behavior that mimics acetate overflow metabolism. More
generally, we predict that a suboptimal cell will activate more transport
reactions, and hence excrete larger number of metabolites than a growth-
optimized cell. This experimentally testable prediction can be further
supported by our single-reaction knockout computations considered in a
subsequent section (Experimental Evidence) to model suboptimal response to
perturbation.
We interpret these inactivation mechanisms involving reaction irreversibility
as a consequence of the linear programming property that the set of optimal
solutions $M_{\text{opt}}$ must be part of the boundary of $M$ [38]. As such,
$M_{\text{opt}}$ is characterized by a set of _binding_ constraints, defined
as inequality constraints (e.g., $v_{i}\leq\beta_{i}$) satisfying two
conditions: the equality holds ($v_{i}=\beta_{i}$) for all $\mathbf{v}\in
M_{\text{opt}}$ and $M_{\text{opt}}$ is sensitive to changes in the
constraints (changes in $\beta_{i}$). In two dimensions, for example,
$M_{\text{opt}}$ would be an edge of $M$, characterized by a single binding
constraint, or a corner of $M$, characterized by two binding constraints. In
general, at least $d-d_{\text{opt}}$ linearly independent constraints must be
binding, where $d$ and $d_{\text{opt}}$ are the dimensions of $M$ and
$M_{\text{opt}}$, respectively. Since many metabolic reactions are subject to
the irreversibility constraint ($v_{i}\geq 0$), this is expected to lead to
many inactive reactions ($v_{i}=0$). Indeed, by excluding the $k$ constraints
that are not associated with reaction irreversibility (those for the ATP
maintenance reaction and exchange fluxes), we obtain an upper bound on the
number of active reactions $n_{+}(\mathbf{v})$:
$n_{+}(\mathbf{v})\leq n_{+}^{\text{typ}}-(d-d_{\text{opt}}-k).$ (2)
##### Cascading.
The remaining set of reactions that are inactive for all $\mathbf{v}\in
M_{\text{opt}}$ is due to cascading of inactivity. On one hand, if all the
reactions that produce a metabolite are inactive, any reaction that consumes
this metabolite must be inactive. On the other hand, if all the reactions that
consume a metabolite are inactive, any reaction that produces this metabolite
must be inactive to avoid accumulation, as this would violate the steady-state
assumption. Therefore, the inactivity caused by the irreversibility mechanism
triggers a cascade of inactivity both in the forward and backward directions
along the metabolic network. In general, there are many different sets of
reactions that, when inactivated, can create the same cascading effect, thus
providing flexibility in potential applications of this effect to the design
of optimal strains [25]. The cascades in the growth-maximizing states,
however, are spontaneous, as opposed to those that would be induced in
metabolic knockout applications [25] or in reaction essentiality and
robustness studies [40, 39, 41]. Different but related to the cascades of
inactivity are the concepts of enzyme subsets [42], coupled reaction sets [26]
and correlated reaction sets [10], which describe groups of reactions that
operate together and are thus concurrently inactivated in cascades.
##### Conditional inactivity.
While the irreversibility and cascading mechanisms cause the inactivity of
many reactions for all $\mathbf{v}\in M_{\text{opt}}$, the inactivity of other
reactions can depend on the specific growth-maximizing state, whose non-
uniqueness in a given environment has been evidenced both theoretically [9,
43, 10] and experimentally [16]. To explore this dependence, we use the
duality principle of linear programming problems [38] to identify all the
binding constraints generating the set of optimal solutions $M_{\text{opt}}$
(Text S1, Section 3). This characterization is then used to count the number
$n_{+}^{\text{opt}}$ ($n_{0}^{\text{opt}}$) of reactions that are active
(inactive) for all $\mathbf{v}\in M_{\text{opt}}$, leading to rigorous bounds
for the number of active reactions $n_{+}(\mathbf{v})$:
$n_{+}^{\text{opt}}\leq n_{+}(\mathbf{v})\leq n-n_{0}^{\text{opt}}.$ (3)
Numerical values of the bounds under maximum growth are indicated by the error
bars in Fig. 2. Note that the upper bounds are consistently smaller than
$n_{+}^{\text{typ}}$ for typical non-optimal states, indicating that reaction
silencing necessarily occurs for all growth-maximizing states. For E. coli,
these results are consistent with a previous study comparing reaction
utilization under a range of different growth conditions [10]. They are also
consistent with existing results for different E. coli metabolic models [14,
12, 13] based on flux variability analysis [9]. Furthermore, we can prove
(Text S1, Section 3) that the distribution of $n_{+}(\mathbf{v})$ within the
upper and lower bounds is singular in that the upper bound is attained for
almost all optimal states:
$n_{+}(\mathbf{v})=n-n_{0}^{\text{opt}}\quad\text{for almost all
$\mathbf{v}\in M_{\text{opt}}$}.$ (4)
Numerical simulations using standard simplex methods [44] actually result in
much fewer active reactions, as shown in Fig. 2 (red middle bars), because the
algorithm finds solutions on the boundary of $M_{\text{opt}}$. This behavior
is expected, however, under the concurrent optimization of additional
metabolic objectives, which generally tend to drive the flux distribution
toward the boundary of $M_{\text{opt}}$.
#####
Figure 2 summarizes the inactivity mechanisms for the four organisms under
maximum growth in glucose media (see also Fig. 1), showing the inactive
reactions caused by the irreversibility (green) and cascading (yellow)
mechanisms, as well as those that are conditionally inactive (orange). Observe
that in sharp contrast to the number of active reactions, which depends little
on the size of the network, the number of inactive reactions (either separated
by mechanisms or lumped together) varies widely and shows non-trivial
dependence on the organisms.
### Typical Linear Objective Functions
Although we have focused so far on maximizing the biomass production rate, the
true nature of the fitness function driving evolution is far from clear [45,
47, 46, 48]. Organisms under different conditions may optimize different
objective functions, such as ATP production or nutrient uptake, or not
optimize a simple function at all. In particular, some metabolic behaviors,
such as the selection between respiration and fermentation in yeast, cannot be
explained by growth maximization [49]. Other behaviors may be systematically
different in situations mimicking natural environments [50]. Moreover, various
alternative target objectives can be conceived and used in metabolic
engineering applications. We emphasize, however, that while specific numbers
may differ in each case, all the arguments leading to Eqs. (2)–(4) are general
and apply to any objective function that is linear in metabolic fluxes. To
obtain further insights, we now study the number of active reactions in
organisms optimizing a typical linear objective function by means of random
uniform sampling.
We first note the fact (proved in Text S1, Section 4) that with probability
one under uniform sampling, the optimal solution set $M_{\text{opt}}$ consists
of a single point, which must be a “corner” of $M$, termed an extreme point in
the linear programming literature. In this case, $d_{\text{opt}}=0$, and Eq.
(2) becomes
$n_{+}(\mathbf{v})\leq n_{+}^{\text{typ}}-d+k.$ (5)
With the additional requirement that the organism show positive growth, we
uniformly sample these extreme points, which represent all distinct optimal
states for typical linear objective functions.
We find that the number of active reactions in typical optimal states is
narrowly distributed around that in growth-maximizing states, as shown in Fig.
4. This implies that various results on growth maximization extend to the
optimization of typical objective functions. In particular, we see that a
typical optimal state is surprisingly close to the onset of cellular growth
(estimated and shown as dashed vertical lines in Fig. 4). Despite the
closeness, however, the organism maintains high versatility, which we define
as the number of distinct functions that can be optimized under growth
conditions. To demonstrate this, consider the H. pylori model, which has 392
reactions that _can_ be active, among which at least 302 _must_ be active to
sustain growth (Table 3). While only a few more than 302 active reactions are
sufficient to optimize any objective function, the number of combinations for
choosing them can be quite large. For example, there are
$\frac{(392-302)!}{(392-302-5)!5!}\approx 4\times 10^{7}$ combinations for
choosing, say, 5 extra reactions to be active. Moreover, this number increases
quickly with the network size: S. cerevisiae, for example, has less than 2.5
times more reactions than H. pylori, but about 500 times more combinations
($\frac{(579-275)!}{(579-275-5)!5!}\approx 2\times 10^{10}$).
### Experimental Evidence
Our results help explain previous experimental observations. Analyzing the 22
intracellular fluxes determined by Schmidt et al. [51] for the central
metabolism of E. coli in both aerobic and anaerobic conditions, we find that
about 45% of the fluxes are smaller than 10% of the glucose uptake rate (Table
4). However, less than 19% of the reversible fluxes and more than 60% of the
irreversible fluxes are found to be in this group (Fisher exact test, one-
sided $p<0.008$). For the 44 fluxes in the S. cerevisiae metabolism
experimentally measured by Daran-Lapujade et al. [52], less than 8% of the
reversible fluxes and more than 42% of the irreversible fluxes are zero (Table
5; Fisher exact test, one-sided $p<10^{-7}$). This higher probability for
reduced fluxes in irreversible reactions is consistent with our theory and
simulation results (Table 6) combined with the assumption that the system
operates close to an optimal state. For the E. coli data, this assumption is
justified by the work of Burgard & Maranas [45], where a framework for
inferring metabolic objective functions was used to show that the organisms
are mainly (but not completely) driven by the maximization of biomass
production. The S. cerevisiae data was also found to be consistent with the
fluxes computed under the assumption of maximum growth [35].
Additional evidence for our results is derived from the inspection of 18
intracellular fluxes experimentally determined by Emmerling et al. [53] for
both wild-type E. coli and a gene-deficient strain not exposed to adaptive
evolution. It has been shown [21] that while the wild-type fluxes can be
approximately described by the optimization of biomass production, the
genetically perturbed strain operates sub-optimally. We consider the fluxes
that are more than 10% (of the glucose uptake rate) larger in the gene-
deficient mutant than in the wild-type strain. This group comprises less than
27% of the reversible fluxes but more than 45% of the irreversible fluxes
(Table 7; Fisher exact test, one-sided $p<0.12$). This correlation indicates
that irreversible fluxes tend to be larger in non-optimal metabolic states,
consistently with our predictions.
Altogether, our results offer an explanation for the temporary activation of
latent pathways observed in adaptive evolution experiments after environmental
[16] or genetic perturbations [17]. These initially inactive pathways are
transiently activated after a perturbation, but subsequently inactivated again
after adaptive evolution. We hypothesize that transient suboptimal states are
the leading cause of latent pathway activation. Since we predict that a large
number of reactions are inactive in optimal states but active in typical non-
optimal states, many reactions are expected to show temporary activation if we
assume that the state following the perturbation is suboptimal and both the
pre-perturbation and post-adaptation states are near optimality. To
demonstrate this computationally for the E. coli model, we consider the
idealized scenario where the perturbation to the growth-maximizing wild type
is caused by a reaction knockout and the initial response of the metabolic
network—before the perturbed strain evolves to a new growth-maximizing
state—is well approximated by the method of minimization of metabolic
adjustment (MOMA) [21]. MOMA assumes that the perturbed organisms minimize the
square-sum deviation of its flux distribution from the wild-type distribution
(under the constraints imposed by the perturbation).
Figure 5A shows the distribution of the number of active reactions for single-
reaction knockouts that alter the flux distribution but allow positive MOMA-
predicted growth. While the distribution is spread around 400–500 for the
suboptimal states in the initial response, it is sharply peaked around 300 for
the optimal states at the endpoint of the evolution, which is consistent with
our results on random sampling of the extreme points (Fig. 4). We thus predict
that the initial number of active reactions for the unperturbed wild-type
strain (which is 297, as shown by a dashed vertical line in Fig. 5A) typically
increases to more than 400 following the perturbation, and then decays back to
a number close to 300 after adaptive evolution in the given environment, as
illustrated schematically in Fig. 5B. A neat implication of our analysis is
that the active reactions in the early post-perturbation state includes much
more than just a superposition of the reactions that are active in the pre-
and post-perturbation optimal states, thus explaining the pronounced burst in
gene expression changes observed to accompany media changes and gene knockouts
[16, 17]. For example, for E. coli in glucose minimal medium, temporary
activation is predicted for the Entner-Doudoroff pathway after pgi knockout
and for the glyoxylate bypass after tpi knockout, in agreement with recent
flux measurements in adaptive evolution experiments [17].
Another potential application of our results is to explain previous
experimental evidence that antagonistic pleiotropy is important in adaptive
evolution [54], as they indicate that increasing fitness in a single
environment requires inactivation of many reactions through regulation and
mutation of associated genes, which is likely to cause a decrease of fitness
in some other environments [15].
## Discussion
Combining computational and analytical means, we have uncovered the
microscopic mechanisms giving rise to the phenomenon of spontaneous reaction
silencing in single-cell organisms, in which optimization of a single
metabolic objective, whether growth or any other, significantly reduces the
number of active reactions to a number that appears to be quite insensitive to
the size of the entire network. Two mechanisms have been identified for the
large-scale metabolic inactivation: reaction irreversibility and cascade of
inactivity. In particular, the reaction irreversibility inactivates a pathway
when the objective function could be enhanced by hypothetically reversing the
metabolic flow through that pathway. We have demonstrated that such pathways
can be found among non-equivalent parallel pathways, transverse pathways
connecting structures that lead to the synthesis of different biomass
components, and pathways leading to metabolite excretion. Although the
irreversibility and cascading mechanisms do not require explicit maximization
of efficiency, massive reaction silencing is also expected for organisms
optimizing biomass yield or other linear functions (of metabolic fluxes)
normalized by uptake rates [49]. Furthermore, while we have focused on minimal
media, we expect the effect to be even more pronounced in richer media. On one
hand, a richer medium has fewer absent substrates, which increases the number
of active reactions in non-optimal states. On the other hand, a richer medium
allows the organism to utilize more efficient pathways that would not be
available in a minimal medium, possibly further reducing the number of active
reactions in optimal states.
Our study carries implications for both natural and engineered processes. In
the rational design of microbial enhancement, for example, one seeks genetic
modifications that can optimize the production of specific metabolic
compounds, which is a special case of the optimization problem we consider
here and akin to the problem of identifying optimal reaction activity [24,
25]. The identification of a reduced set of active reactions also provides a
new approach for testing the existence of global metabolic objectives and
their consistency with hypothesized objective functions [47]. Such an approach
is complementary to current approaches based on coefficients of importance
[45, 46] or putative objective reactions [48] and is expected to provide novel
insights into goal-seeking dynamic concepts such as cybernetic modeling [55].
Our study may also help model compromises between competing goals, such as
growth and robustness against environmental or genetic changes [56].
In particular, our results open a new avenue for addressing the origin of
mutational robustness [58, 57, 59, 61, 60, 62]. Single-gene deletion
experiments on E. coli and S. cerevisiae have shown that only a small fraction
of their genes are essential for growth under standard laboratory conditions
[5, 6, 1]. The number of essential genes can be even smaller given that growth
defect caused by a gene deletion may be synthetically rescued by compensatory
gene deletions [25], an effect not accounted for in single-gene deletion
experiments. Under fixed environmental conditions, large part of this
mutational robustness arises from the reactions that are inactive under
maximum growth, whose deletion is predicted to have no effect on the growth
rate [35]. For example, for E. coli in glucose medium, we predict that 638 out
of the 931 reactions can be removed simultaneously while retaining the maximum
growth rate (Note 4), which is comparable to 686 computed in a minimal genome
study in rich media [11]. But what is the origin of all these non-essential
genes?
A recent study on S. cerevisiae has shown that the single deletion of almost
any non-essential gene leads to a growth defect in at least one stress
condition [15], providing substantive support for the long-standing hypothesis
that mutational robustness is a byproduct of environmental robustness [61] (at
least if we assume that the tested conditions are representative of the
natural conditions under which the organisms evolved). An alternative
explanation would be that in variable environments, which is a natural
selective pressure likely to be more important than considered in standard
laboratory experiments, the apparently dispensable pathways may play a
significant role in suboptimal states induced by environmental changes. This
alternative is based on the hypothesis that latent pathways provide
intermediate states necessary to facilitate adaptation, therefore conferring
competitive advantage even if the pathways are not active in any single fixed
environmental condition. In light of our results, this hypothesis can be
tested experimentally in medium-perturbation assays by measuring the change in
growth or other phenotype caused by deleting the predicted latent pathways in
advance to a medium change.
We conclude by calling attention to a limitation and strength of our results,
which have been obtained using steady-state analysis. Such analysis avoids
complications introduced by unknown regulatory and kinetic parameters, but
admittedly does not account for constraints that could be introduced by the
latter. Nevertheless, we have been able to draw robust conclusions about
dynamical behaviors, such as the impact of perturbation and adaptive evolution
on reaction activity. Our methodology scales well for genome-wide studies and
may prove instrumental for the identification of specific extreme pathways
[63, 64] or elementary modes [65, 66] governing the optimization of metabolic
objectives. Combined with recent studies on complex networks [67, 68, 69, 71,
70, 72, 73] and the concept of functional modularity [74], our results are
likely to lead to new understanding of the interplay between _network
activity_ and _biological function_.
### Notes
1. 1.
In addition, under steady-state conditions in the media considered in this
study, more than 77% of the reversible reactions become constrained to be
irreversible, rendering a total of more than 92% of all reactions
“effectively” irreversible.
2. 2.
This reaction is regarded in the biochemical literature as irreversible under
physiological conditions in the cell, and is constrained as such in the
modeling literature [14, 32, 75, 76].
3. 3.
Similar effective irreversibility is at work for any transport or internal
reaction that is a unique producer of one or more chemical compounds in the
cell.
4. 4.
For single-reaction knockouts, MOMA predicts that 662 out of the 931 deletion
mutants grow at more than 99% of the wild-type growth rate. Among these 662
reactions, 95% are predicted to be inactive under maximum growth.
## Materials and Methods
### Strains and media
All the stoichiometric data for the in silico metabolic systems used in our
study are available at http://gcrg.ucsd.edu/In_Silico_Organisms. For H. pylori
26695 [77], we used a medium with unlimited amount of water and protons, and
limited amount of oxygen (5 mmol/g DW-h), L-alanine, D-alanine, L-arginine,
L-histidine, L-isoleucine, L-leucine, L-methionine, L-valine, glucose, Iron
(II and III), phosphate, sulfate, pimelate, and thiamine (20 mmol/g DW-h). For
S. aureus N315 [78], we used a medium with limited amount of glucose,
L-arginine, cytosine, and nicotinate (100 mmol/g DW-h), and unlimited amount
of iron (II), protons, water, oxygen, phospate, sulfate, and thiamin. For E.
coli K12 MG1655 [75], we used a medium with limited amount of glucose (10
mmol/g DW-h) and oxygen (20 mmol/g DW-h), and unlimited amount of carbon
dioxide, iron (II), protons, water, potassium, sodium, ammonia, phospate, and
sulfate. For S. cerevisiae S288C [76], we used a medium with limited amount of
glucose (10 mmol/g DW-h), oxygen (20 mmol/g DW-h), and ammonia (100 mmol/g
DW-h), and unlimited amount of water, protons, phosphate, carbon dioxide,
potassium, and sulfate. The flux through the ATP maintenance reaction was set
to 7.6 mmol/g DW-h for E. coli and 1 mmol/g DW-h for S. aureus and S.
cerevisiae.
### Feasible solution space
Under steady-state conditions, a cellular metabolic state is represented by a
solution of a homogeneous linear equation describing the mass balance
condition,
$\mathbf{S}\mathbf{v}=\mathbf{0},$ (6)
where $\mathbf{S}$ is the $m\times N$ stoichiometric matrix and
$\mathbf{v}\in\mathbb{R}^{N}$ is the vector of metabolic fluxes. The
components of $\mathbf{v}=(v_{1},\dots,v_{N})^{T}$ include the fluxes of $n$
internal and transport reactions as well as $n_{\text{ex}}$ exchange fluxes,
which model the transport of metabolites across the system boundary.
Constraints of the form $v_{i}\leq\beta_{i}$ imposed on the exchange fluxes
are used to define the maximum uptake rates of substrates in the medium.
Additional constraints of the form $v_{i}\geq 0$ arise for the reactions that
are irreversible. Assuming that the cell’s operation is mainly limited by the
availability of substrates in the medium, we impose no other constraints on
the internal reaction fluxes, except for the ATP maintenance flux for S.
aureus, E. coli, and S. cerevisiae (see Strains and media section above). The
set of all flux vectors $\mathbf{v}$ satisfying the above constraints defines
the feasible solution space $M\subset\mathbb{R}^{N}$, representing the
capability of the metabolic network as a system.
### Maximizing growth and other linear objective functions
The flux balance analysis (FBA) [19, 20, 23, 18, 22] used in this study is
based on the maximization of a metabolic objective function
$\mathbf{c}^{T}\mathbf{v}$ within the feasible solution space $M$, which is
formulated as a linear programming problem:
maximize: $\displaystyle\mathbf{c}^{T}\mathbf{v}=\sum_{i=1}^{N}c_{i}v_{i}$ (7)
subject to:
$\displaystyle\mathbf{S}\mathbf{v}=\mathbf{0},\quad\mathbf{v}\in\mathbb{R}^{N},$
$\displaystyle\alpha_{i}\leq v_{i}\leq\beta_{i},\quad i=1,\ldots,N.$
We set $\alpha_{i}=-\infty$ if $v_{i}$ is unbounded below and
$\beta_{i}=\infty$ if $v_{i}$ is unbounded above. For a given objective
function, we numerically determine an optimal flux distribution for this
problem using an implementation of the simplex method [44]. In the particular
case of growth maximization, the objective vector $\mathbf{c}$ is taken to be
parallel to the biomass flux, which is modeled as an effective reaction that
converts metabolites into biomass.
### Finding minimum reaction set for nonzero growth
To find a set of reactions from which none can be removed without forcing zero
growth, we start with the set of all reactions and recursively reduce it until
no further reduction is possible. At each recursive step, we first compute how
much the maximum growth rate would be reduced if each reaction were removed
from the set individually. Then, we choose a reaction that causes the least
change in the maximum growth rate, and remove it from the set. We repeat this
step until the maximum growth rate becomes zero. The set of reactions we have
just before we remove the last reaction is a desired minimal reaction set.
Note that, since the algorithm is not exhaustive, the number of reactions in
this set is an upper bound and approximation for the minimum number of
reactions required to sustain positive growth.
## Acknowledgements
The authors thank Linda J. Broadbelt for valuable discussions and for
providing feedback on the manuscript. The authors also thank Jennifer L. Reed
and Adam M. Feist for providing information on their in silico models.
## Supporting Information
Text S1: Mathematical Results
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Figure 1: Optimal (A) and non-optimal (B) reaction activity in the
reconstructed metabolic network of E. coli in glucose minimal medium
(Materials and Methods). The pie charts show the fractions of active and
inactive reactions in the metabolic subsystems defined in the iJR904 database
[75]. The color code is as follows: active reactions (red), inactive reactions
due to mass balance (black) and environmental constraints (blue), inactive
reactions due to the irreversibility (green) and cascading (yellow)
mechanisms, and conditionally inactive reactions (orange), which are inactive
reactions that can be active for other growth-maximizing states under the same
medium condition. The optimal state shown in panel A was obtained by flux
balance analysis (FBA, see Materials and Methods). The network is constructed
by drawing an arrow from one subsystem to another when there are at least 4
metabolites that can be produced by reactions in the first subsystem and
consumed by reactions in the second. Larger pies represent subsystems with
more reactions. Figure 2: Number of active and inactive reactions in the
metabolic networks of H. pylori, S. aureus, E. coli, and S. cerevisiae. For
each organism, the bars correspond to a typical non-optimal state (top), a
growth-maximizing state (middle), and a state with the minimum number of
active reactions required for growth (bottom), which was estimated using the
algorithm described in Materials and Methods. The error bar represents the
upper and lower theoretical bounds, given by Eq. (3), on the number of active
reactions in any growth-maximizing state. The breakdown of inactive reactions
and their color coding are the same as in Fig. 1. All results are obtained
using glucose minimal media (Materials and Methods) and are further detailed
in Tables 2 and 3. Figure 3: Portions of E. coli metabolic network under
maximum growth condition. (A) P1, P2, and P3 are alternative pathways for
glucose transport and utilization. The most efficient pathway, P1, is active
under maximum growth in glucose minimal medium. P2 and P3 are inactive, but if
P1 is knocked out, P2 becomes active, and if both P1 and P2 are knocked out,
P3 becomes active. In both knockout scenarios, the growth is predicted to be
suboptimal. (B) Isocitrate lyase reaction in the pathway bypassing the
tricarboxylic acid (TCA) cycle is predicted to be inactive under maximum
growth due to its irreversibility. If it were to operate in the opposite
direction, it would serve as a transverse pathway which redirects metabolic
flow to growth-limiting reactions, increasing the maximum biomass production
rate slightly. In both panels, single and double arrows are used to indicate
irreversible and reversible reactions, respectively, and colors indicate the
behavior of the reactions under maximum growth: active (red), inactive due to
the irreversibility (green), and inactive due to cascading (yellow). Figure 4:
Probability distribution of the number of active reactions in nonzero-growth
states that optimize typical objective functions. The red solid lines indicate
the corresponding number in the growth-maximizing state of Fig. 2 (middle bar,
red), and the red dashed lines indicate our estimates of the minimum number of
reactions required for the organism to grow (Materials and Methods). [When the
nonzero growth requirement is relaxed, a second sharp peak (not shown) arises,
corresponding to a drop of $\sim 250$ in the number of active reactions caused
by the inactivation of the biomass reaction.] Figure 5: Distribution of the
number of active reactions in the E. coli metabolic network after a single-
reaction knockout. (A) The initial response is predicted by the minimization
of metabolic adjustment (MOMA) and the endpoint of adaptive evolution by the
maximization of the growth rate (FBA), using the medium defined in Materials
and Methods and a commercial optimization software package [79]. We consider
all 77 nonlethal single-reaction knockouts that change the flux distribution.
(B) Schematic illustration of the network reaction activity during the
adaptive evolution after a small perturbation, indicating the temporary
activation of many latent pathways.
## Tables
#### Table 1: Reversibility of metabolic reactions in the reconstructed
networks.
| H. pylori | S. aureus | E. coli | S. cerevisiae
---|---|---|---|---
Total number of reactions [$n$] : | 479 | 641 | 931 | 1149
Reversible | 165 | 220 | 245 | 430
Irreversible | 314 | 421 | 686 | 719
#### Table 2: Metabolic reactions in typical non-optimal states of the
simulated metabolisms.
| H. pylori | S. aureus | E. coli | S. cerevisiae
---|---|---|---|---
Total number of reactions [$n$] | 479 | 641 | 931 | 1149
Inactive reactions: | 87 | 222 | 322 | 570
Due to mass balance | 44 | 133 | 141 | 268
Due to environmental conditionsa | 43 | 89 | 181 | 302
Active reactions [$n_{+}^{\text{typ}}$] | 392 | 419 | 609 | 579
a These reactions are inactive due to constraints arising from the
availability of substrates in the media defined in Materials and Methods.
#### Table 3: Metabolic reactions in maximum growth states of the simulated
metabolisms.a
| H. pylori | S. aureus | E. coli | S. cerevisiae
---|---|---|---|---
Active reactions under typical non-optimal states [$n_{+}^{\text{typ}}$] | 392 | 419 | 609 | 579
Active reactions under maximum growthb: | 308 | 282 | 297 | 289
lower bound [$n_{+}^{\text{opt}}$] | 257 | 77 | 272 | 196
upper bound [$n-n_{0}^{\text{opt}}$] | 351 | 414 | 355 | 426
Minimum number of active reactions for growthc | 302 | 281 | 292e | 275
Inactive reactions under maximum growthb [$n_{0}^{\text{opt}}$]: | 171 | 359 | 634 | 860
Due to irreversibility | 29 | 3 | 147 | 72
Due to cascading | 12 | 2 | 107 | 81
Due to mass balance | 44 | 133 | 141 | 268
Due to environmental conditions | 43 | 89 | 181 | 302
Conditionally inactived | 43 | 132 | 58 | 137
a With respect to the minimal media defined in Materials and Methods.
b Based on a single optimal state found using an implementation of the simplex
method [44].
c Estimated using the algorithm described in Materials and Methods.
d Predicted to be inactive by the simplex method [44], but can be active in
some other growth-maximizing states. Likewise, some of the reactions predicted
to be active can be inactive in some other optimal states, but the number of
such reactions is expected to be small since the simplex method finds a
solution on the boundary of $M_{\text{opt}}$, which tends to have more
inactive reactions than a typical optimal solution.
e The corresponding minimum number of active reactions for maximum growth is
293.
#### Table 4: Experimentally determined fluxes of intracellular reactions
involved in the glycolysis, pentose phosphate pathway, TCA cycle, and amino
acid biosynthesis of E. coli K12 MG1655 under aerobic and anaerobic conditions
[51].
| Aerobic | Anaerobic
---|---|---
| Reversible | Irreversible | Reversible | Irreversible
Number of fluxes | 8 | 14 | 8 | 14
Number of fluxes $<$ 10% of glucose uptake rate | 1 | 7 | 2 | 10
#### Table 5: Experimentally determined fluxes of intracellular reactions
involved in the glycolysis, metabolic steps around pyruvate, TCA cycle,
glyoxylate cycle, gluconeogenesis, and pentose phosphate pathway of S.
cerevisiae strain CEN.PK113 7D grown under glucose, maltose, ethanol, and
acetate limitation [52].
| Glucose | Maltose | Ethanol | Acetate
---|---|---|---|---
| Rev. | Irr. | Rev. | Irr. | Rev. | Irr. | Rev. | Irr.
Number of fluxes | 22 | 22 | 22 | 22 | 22 | 22 | 22 | 22
Number of zero fluxes | 2 | 8 | 2 | 7 | 1 | 11 | 2 | 11
Percentage of zero fluxes | 9.1% | 36.4% | 9.1% | 31.8% | 4.5% | 50.0% | 9.1% | 50.0%
#### Table 6: Fraction of inactive reactions in the simulated metabolism of E.
coli and S. cerevisiae under maximum growth condition.a
| E. coli | S. cerevisiae
---|---|---
| Reversible | Irreversible | Reversible | Irreversible
Number of reactions | 245 | 686 | 430 | 719
Number of inactive reactions | 139 | 495 | 301 | 559
Percentage of inactive reactions | 56.7% | 72.2% | 70.0% | 77.7%
a Same states considered in Table 3.
#### Table 7: Experimentally determined fluxes of reversible and irreversible
reactions of wild-type E. coli JM101 versus its pyruvate kinase-deficient
mutant PB25 [53].
| Reversible | Irreversible
---|---|---
Number of fluxes | 30 | 24
Number of mutant fluxes that are largera by $>$ 10% of glucose uptake rate | 8 | 11
a Relative to the corresponding fluxes in the wild-type strain.
Spontaneous reaction silencing in metabolic optimization
T. Nishikawa, N. Gulbahce, A. E. Motter
Supporting Information
Text S1: Mathematical Results
## 1\. Number of active reactions in typical steady states
The mass balance constraints $\mathbf{S}\mathbf{v}=\mathbf{0}$ define the
linear subspace
$\text{Nul}\,\mathbf{S}=\\{\mathbf{v}\in\mathbb{R}^{N}\,|\,\mathbf{S}\mathbf{v}=\mathbf{0}\\}$
(the null space of $\mathbf{S}$), which contains the feasible solution space
$M$. However, the set $M$ can possibly be smaller than
$\text{Nul}\,\mathbf{S}$ because of the additional constraints arising from
the environmental conditions (the availability of substrates in the medium,
reaction irreversibility, and cell maintenance requirements). Therefore, $M$
may have smaller dimension than $\text{Nul}\,\mathbf{S}$. If we denote the
dimension of $M$ by $d$, there exists a unique $d$-dimensional linear
submanifold of $\mathbb{R}^{N}$ that contains $M$, which we denote by $L_{M}$.
We can then use the Lebesgue measure naturally defined on $L_{M}$ to make
probabilistic statements, since we can define the probability of a subset
$A\subseteq M$ as the Lebesgue measure of $A$ normalized by the Lebesgue
measure of $M$. In particular, we say that $v_{i}\neq 0$ for almost all
$\mathbf{v}\in M$ if the set $\\{\mathbf{v}\in M\,|\,v_{i}=0\\}$ has Lebesgue
measure zero on $L_{M}$. An interpretation of this is that $v_{i}\neq 0$ with
probability one for an organism in a random state under given environmental
conditions. Using this notion, we prove the following theorem on the reaction
fluxes.
###### Theorem 1.
If $v_{i}\neq 0$ for some $\mathbf{v}\in M$, then $v_{i}\neq 0$ for almost all
$\mathbf{v}\in M$.
###### Proof.
Suppose that $v_{i}\neq 0$ for some $\mathbf{v}\in M$. The set
$L_{i}:=\\{\mathbf{v}\in L_{M}\,|\,v_{i}=0\\}$ is a linear submanifold of
$L_{M}$, so we have $\dim{L_{i}}\leq\dim{L_{M}}$. If
$\dim{L_{i}}=\dim{L_{M}}$, then we have $L_{i}=L_{M}\supseteq M$, implying
that we have $v_{i}=0$ for any $\mathbf{v}\in M$, which violates the
assumption. Thus, we must have $\dim{L_{i}}<\dim{L_{M}}$, implying that
$L_{i}$ has zero Lebesgue measure on $L_{M}$. Since $M\subseteq L_{M}$, we
have $M_{i}:=\\{\mathbf{v}\in M\,|\,v_{i}=0\\}\subseteq\\{\mathbf{v}\in
L_{M}\,|\,v_{i}=0\\}=L_{i}$, and thus $M_{i}$ also has Lebesgue measure zero.
Therefore, we have $v_{i}\neq 0$ for almost all $\mathbf{v}\in M$. ∎
Theorem 1 implies that we can group the reactions and exchange fluxes into two
categories:
1. 1.
Always inactive: $v_{i}=0$ for all $\mathbf{v}\in M$, and
2. 2.
Almost always active: $v_{i}\neq 0$ for almost all $\mathbf{v}\in M$.
Consequently, the number $n_{+}(\mathbf{v})$ of active reactions satisfies
$n_{+}(\mathbf{v})=n_{+}^{\text{typ}}:=n-n_{0}^{m}-n_{0}^{e}\quad\text{for
almost all }\mathbf{v}\in M,$ (1)
where $n_{0}^{m}$ is the number of inactive reactions due to the mass balance
constraints (characterized by Theorem 2) and $n_{0}^{e}$ is the number of
additional reactions in the category 1 above, which are due to the
environmental conditions. Combining this result with the finding that optimal
states have fewer active reactions (see the main text), it follows that a
typical state $\mathbf{v}\in M$ is non-optimal.
## 2\. Inactive reactions due to mass balance constraints
Let us define the stoichiometric coefficient vector of reaction $i$ to be the
$i$th column of the stoichiometric matrix $\mathbf{S}$. We similarly define
the stoichiometric coefficient vector of an exchange flux. If the
stochiometric vector of reaction $i$ can be written as a linear combination of
the stoichiometric vector of reactions/exchange fluxes
$i_{1},i_{2},\ldots,i_{k}$, we say that $i$ is a linear combination of
$i_{1},i_{2},\ldots,i_{k}$. We use this linear relationship to completely
characterize the set of all reactions that are always inactive due to the mass
balance constraints, regardless of any additionally imposed constraints, such
as the availability of substrates in the medium, reaction irreversibility,
cell maintenance requirements, and optimum growth condition.
###### Theorem 2.
Reaction $i$ is inactive for all $\mathbf{v}$ satisfying
$\mathbf{S}\mathbf{v}=\mathbf{0}$ if and only if it is not a linear
combination of the other reactions and exchange fluxes.
###### Proof.
We denote the stoichiometric coefficient vectors of reactions and exchange
fluxes by $\mathbf{s}_{1},\ldots,\mathbf{s}_{N}$. The theorem is equivalent to
saying that there exists $\mathbf{v}$ satisfying both
$\mathbf{S}\mathbf{v}=\mathbf{0}$ and $v_{i}\neq 0$ if and only if
$\mathbf{s}_{i}$ is a linear combination of $\mathbf{s}_{k}$,
$k=1,2,\ldots,N$, $k\neq i$.
To prove the forward direction in this statement, suppose that $v_{i}\neq 0$
in a state $\mathbf{v}$ satisfying $\mathbf{S}\mathbf{v}=\mathbf{0}$. By
writing out the components of the equation $\mathbf{S}\mathbf{v}=\mathbf{0}$
and rearranging, we get
$s_{ji}v_{i}=\sum_{k\neq i}(-v_{k})s_{jk},\quad j=1,\ldots,m.$ (2)
Since $v_{i}\neq 0$, we can divide this equation by $v_{i}$ to see that
$\mathbf{s}_{i}$ is a linear combination of $\mathbf{s}_{k}$, $k\neq i$ with
coefficients $c_{k}=-v_{k}/v_{i}$.
To prove the backward direction, suppose that $\mathbf{s}_{i}=\sum_{k\neq
i}c_{k}\mathbf{s}_{k}$. If we choose $\mathbf{v}$ so that $v_{k}=c_{k}$ for
$k\neq i$ and $v_{i}=-1$, then for each $j$, we have
$(\mathbf{S}\mathbf{v})_{j}=\sum_{k}v_{k}s_{jk}=v_{i}s_{ji}+\sum_{k\neq
i}v_{k}s_{jk}=-s_{ji}+\sum_{k\neq i}c_{k}s_{jk}=0,$
so $\mathbf{v}$ satisfies $\mathbf{S}\mathbf{v}=\mathbf{0}$. ∎
## 3\. Number of active reactions in optimal states
The linear programming problem for finding the flux distribution maximizing a
linear objective function can be written in the matrix form:
maximize: $\displaystyle\mathbf{c}^{T}\mathbf{v}$ (3) subject to:
$\displaystyle\mathbf{S}\mathbf{v}=\mathbf{0},\;\mathbf{A}\mathbf{v}\leq\mathbf{b},\;\mathbf{v}\in\mathbb{R}^{N},$
where $\mathbf{A}$ and $\mathbf{b}$ are defined as follows. If the $i$th
constraint is $v_{j}\leq\beta_{j}$, the $i$th row of $\mathbf{A}$ consists of
all zeros except for the $j$th entry that is $1$, and $b_{i}=\beta_{j}$. If
the $i$th constraint is $\alpha_{j}\leq v_{j}$, the $i$th row of $\mathbf{A}$
consists of all zeros except for the $j$th entry that is $-1$, and
$b_{i}=-\alpha_{j}$. A constraint of the type $\alpha_{j}\leq
v_{j}\leq\beta_{j}$ is broken into two separate constraints and represented in
$\mathbf{A}$ and $\mathbf{b}$ as above. The inequality between vectors is
interpreted as inequalities between the corresponding components, so if the
rows of $\mathbf{A}$ are denoted by
$\mathbf{a}_{1}^{T},\mathbf{a}_{2}^{T},\ldots,\mathbf{a}_{K}^{T}$ (where
$\mathbf{a}_{i}^{T}$ denotes the transpose of $\mathbf{a}_{i}$),
$\mathbf{A}\mathbf{v}\leq\mathbf{b}$ represents the set of $K$ constraints
$\mathbf{a}_{i}^{T}\mathbf{v}\leq b_{i}$, $i=1,\ldots,K$. By defining the
feasible solution space
$M:=\\{\mathbf{v}\in\mathbb{R}^{N}\,|\,\mathbf{S}\mathbf{v}=\mathbf{0},\;\mathbf{A}\mathbf{v}\leq\mathbf{b}\\},$
(4)
the problem can be compactly expressed as maximizing
$\mathbf{c}^{T}\mathbf{v}$ in $M$.
The duality principle (Best & Ritter, 1985) expresses that any linear
programming problem (primal problem) is associated with a complementary linear
programming problem (dual problem), and the solutions of the two problems are
intimately related. The dual problem associated with problem (3) is
minimize: $\displaystyle\mathbf{b}^{T}\mathbf{u}_{1}$ (5) subject to:
$\displaystyle\mathbf{A}^{T}\mathbf{u}_{1}+\mathbf{S}^{T}\mathbf{u}_{2}=\mathbf{c},\;\mathbf{u}_{1}\geq\mathbf{0},$
$\displaystyle\mathbf{u}_{1}\in\mathbb{R}^{K},\;\mathbf{u}_{2}\in\mathbb{R}^{m},$
where $\\{\mathbf{u}_{1},\mathbf{u}_{2}\\}$ is the dual variable. A
consequence of the Strong Duality Theorem (Best & Ritter, 1985) is that the
primal and dual solutions are related via a well-known optimality condition:
$\mathbf{v}$ is optimal for problem (3) if and only if there exists
$\\{\mathbf{u}_{1},\mathbf{u}_{2}\\}$ such that
$\displaystyle\mathbf{S}\mathbf{v}=\mathbf{0},\;\mathbf{A}\mathbf{v}\leq\mathbf{b},$
(6)
$\displaystyle\mathbf{A}^{T}\mathbf{u}_{1}+\mathbf{S}^{T}\mathbf{u}_{2}=\mathbf{c},\;\mathbf{u}_{1}\geq\mathbf{0},$
(7) $\displaystyle\mathbf{u}_{1}^{T}(\mathbf{A}\mathbf{v}-\mathbf{b})=0.$ (8)
Note that each component of $\mathbf{u}_{1}$ can be positive or zero, and we
can use this information to find a set of reactions that are forced to be
inactive under optimization, as follows. For any given optimal solution
$\mathbf{v}_{0}$, Eq. (8) is equivalent to
$u_{1i}(\mathbf{a}_{i}^{T}\mathbf{v}_{0}-b_{i})=0$, $i=1,\ldots,K,$ where
$u_{1i}$ is the $i$th component of $\mathbf{u}_{1}$. Thus, if $u_{i1}>0$ for a
given $i$, we have $\mathbf{a}_{i}^{T}\mathbf{v}_{0}=b_{i}$, and we say that
the constraint $\mathbf{a}_{i}^{T}\mathbf{v}\leq b_{i}$ is binding at
$\mathbf{v}_{0}$. In particular, if an irreversible reaction ($v_{i}\geq 0$)
is associated with a positive dual variable ($u_{1i}>0$), then the
irreversibility constraint is binding, and the reaction is inactive
($v_{i}=0$) at $\mathbf{v}_{0}$. In fact, we can say much more: we prove the
following theorem stating that such a reaction is actually required to be
inactive for all possible optimal solutions for a given objective function
$\mathbf{c}^{T}\mathbf{v}$.
###### Theorem 3.
Suppose $\\{\mathbf{u}_{1},\mathbf{u}_{2}\\}$ is a dual solution corresponding
to an optimal solution of problem (3). Then, the set $M_{\text{opt}}$ of all
optimal solutions of (3) can be written as
$M_{\text{opt}}=\\{\mathbf{v}\in
M\,|\,\mathbf{a}_{i}^{T}\mathbf{v}=b_{i}\text{ for all $i$ for which
$u_{1i}>0$}\\},$ (9)
and hence every reaction associated with a positive dual component is binding
for all optimal solutions in $M_{\text{opt}}$.
###### Sketch of proof.
Let $\mathbf{v}_{0}$ be the optimal solution associated with
$\\{\mathbf{u}_{1},\mathbf{u}_{2}\\}$ and let $Q$ denote the right hand side
of (9). Any $\mathbf{v}\in Q$ is an optimal solution of (3), since
straightforward verification shows that it satisfies (6-8) with the same dual
solution $\\{\mathbf{u}_{1},\mathbf{u}_{2}\\}$. Thus, we have $Q\subseteq
M_{\text{opt}}$. Conversely, suppose that $\mathbf{v}$ is an optimal solution
of (3). Then, $\mathbf{v}$ can be shown to belong to $H$, which we define to
be the hyperplane that is orthogonal to $\mathbf{c}$ and contains
$\mathbf{v}_{0}$, i.e.,
$H:=\\{\mathbf{v}\in\mathbb{R}^{N}\,|\,\mathbf{c}^{T}(\mathbf{v}-\mathbf{v}_{0})=0\\}.$
(10)
This, together with the fact that $\mathbf{v}$ satisfies
$\mathbf{S}\mathbf{v}=\mathbf{0}$ and $\mathbf{A}\mathbf{v}\leq\mathbf{b}$,
from (6), can be used to show that $\mathbf{v}\in Q$. Therefore, any optimal
solution must belong to $Q$. Putting both directions together, we have
$Q=M_{\text{opt}}$. ∎
Thus, once we solve Eq. (3) numerically and obtain a _single_ pair of primal
and dual solutions ($\mathbf{v}_{0}$ and
$\\{\mathbf{u}_{1},\mathbf{u}_{2}\\}$), we can use the characterization of
$M_{\text{opt}}$ given in Eq. (9) to identify all reactions that are required
to be inactive (or active) for any optimal solutions. To do this we solve the
following auxiliary linear optimization problems for each $i=1,\ldots,N$:
maximize/minimize: $\displaystyle v_{i}$ (11) subject to:
$\displaystyle\mathbf{S}\mathbf{v}=\mathbf{0},\;\mathbf{A}\mathbf{v}\leq\mathbf{b},\;\mathbf{a}_{i}^{T}\mathbf{v}=b_{i}\text{
for all $i$ for which $u_{1i}>0$.}$
If the maximum and minimum of $v_{i}$ are both zero, then the corresponding
reaction is required to be inactive for all $\mathbf{v}\in M_{\text{opt}}$. If
the minimum is positive or maximum is negative, then the reaction is required
to be active. Otherwise, the reaction may be active or inactive, depending on
the choice of an optimal solution. Thus, we obtain the numbers
$n_{+}^{\text{opt}}$ and $n_{0}^{\text{opt}}$ of reactions that are required
to be active and inactive, respectively, for all $\mathbf{v}\in
M_{\text{opt}}$. The number of active reactions for any $\mathbf{v}\in
M_{\text{opt}}$ is then bounded as
$n_{+}^{\text{opt}}\leq n_{+}(\mathbf{v})\leq n-n_{0}^{\text{opt}}.$ (12)
The distribution of $n_{+}(\mathbf{v})$ within the bounds is singular: the
upper bound in Eq. (12) is attained for almost all $\mathbf{v}\in
M_{\text{opt}}$. To see this, we apply Theorem 1 with $M$ replaced by
$M_{\text{opt}}$. This is justified since we can obtain $M_{\text{opt}}$ from
$M$ by simply imposing additional equality constraints. Therefore, if we set
aside the $n_{0}^{\text{opt}}$ reactions that are required to be inactive
(including $n_{0}^{m}$ and $n_{0}^{e}$ reactions that are inactive for all
$\mathbf{v}\in M$), all the other reactions are active for almost all
$\mathbf{v}\in M_{\text{opt}}$. Consequently,
$n_{+}(\mathbf{v})=n-n_{0}^{\text{opt}}\quad\text{for almost all
}\mathbf{v}\in M_{\text{opt}}.$ (13)
We can also use Theorem 3 to further classify those inactive reactions caused
by the optimization as due to two specific mechanisms:
1. 1.
Irreversibility. The irreversibility constraint ($v_{i}\geq 0$) on a reaction
can be binding ($v_{i}=0$), which directly forces the reaction to be inactive
for all optimal solutions. Such inactive reactions are identified by checking
the positivity of dual components ($u_{1i}$).
2. 2.
Cascading. All other reactions that are required to be inactive for all
$\mathbf{v}\in M_{\text{opt}}$ are due to a cascade of inactivity triggered by
the first mechanism, which propagates over the metabolic network via the
stoichiometric and mass balance constraints.
In general, a given solution of problem (3) can be associated with multiple
dual solutions. The set and the number of positive components in
$\mathbf{u}_{1}$ can depend on the choice of a dual solution, and therefore
the categorization according to mechanism is generally not unique. As an
example, consider a metabolic network containing a chain of two simple
irreversible reactions, $A\xrightarrow{v_{1}}B\xrightarrow{v_{2}}C$. Since the
two reactions are fully coupled via the mass balance constraint ($v_{1}=v_{2}$
whenever $\mathbf{S}\mathbf{v}=\mathbf{0}$), we can show that different
combinations of dual components are possible for a given optimal solution: (i)
$u_{11}>0,u_{12}=0$; (ii) $u_{11}=0,u_{12}>0$; or (iii) $u_{11}>0,u_{12}>0$.
In each case, the set of reactions in the irreversibility category is
different, and the number of such reactions are different in case (iii). This
comes from the fact that the same result ($v_{1}=v_{2}=0$) follows from
forcing $v_{1}=0$ only, $v_{2}=0$ only, or both. Thus, we can interpret the
non-uniqueness of the categorization as the fact that different sets of
triggering inactive reactions can create the same cascading effect on the
reaction activity.
## 4\. Typical linear objective functions
Since the feasible solution space $M$ is convex, its “corner” can be
mathematically formulated as an extreme point, defined as a point
$\mathbf{v}\in M$ that cannot be written as
$\mathbf{v}=a\mathbf{x}+b\mathbf{y}$ with $a+b=1$, $0<a<1$ and ${\bf x,y}\in
M$ such that ${\bf x\neq y}$. Intuition from the two-dimensional case (Fig.
S1) suggests that for a typical choice of the objective vector $\mathbf{c}$
such that Eq. (3) has a solution, the solution is unique and located at an
extreme point of $M$.
Figure S1: Optimum is typically achieved at a single extreme point. The only
exception is when the objective vector $\mathbf{c}$ is in the direction
perpendicular to an edge, in which case all points on the edge are optimal.
We prove here that this is indeed true in general, as long as the objective
function is bounded on $M$, and hence an optimal solution exists.
###### Theorem 4.
Suppose that the set of objective vectors
$B=\\{\mathbf{c}\in\mathbb{R}^{N}\,|\,\text{$\mathbf{c}^{T}\mathbf{v}$ is
bounded on $M$}\\}$ has positive Lebesgue measure. Then, for almost all
$\mathbf{c}$ in $B$, there is a unique solution of Eq. (3), and it is located
at an extreme point of $M$.
###### Proof.
For a given $\mathbf{c}\in B$, the function $\mathbf{c}^{T}\mathbf{v}$ is
bounded on $M$, so the solution set
$M_{\text{opt}}=M_{\text{opt}}(\mathbf{c})$ of Eq. (3) consists of either a
single point or multiple points. Suppose $M_{\text{opt}}$ consists of a single
point $\mathbf{v}$ and it is not an extreme point. By definition, it can be
written as $\mathbf{v}=a{\bf x}+b{\bf y}$ with $a+b=1$, $0<a<1$ and ${\bf
x,y}\in M$ such that ${\bf x\neq y}$. Since $\mathbf{v}$ is the only solution
of Eq. (3), ${\bf x}$ and ${\bf y}$ must be suboptimal, and hence we have
$\mathbf{c}^{T}{\bf x}<\mathbf{c}^{T}\mathbf{v}$ and $\mathbf{c}^{T}{\bf
y}<\mathbf{c}^{T}\mathbf{v}$. Then,
$\displaystyle\mathbf{c}^{T}{\bf y}$ $\displaystyle=$
$\displaystyle\mathbf{c}^{T}(\mathbf{v}-a{\bf x})/b$ $\displaystyle=$
$\displaystyle(\mathbf{c}^{T}\mathbf{v}-a\mathbf{c}^{T}{\bf x})/b$
$\displaystyle>$
$\displaystyle(\mathbf{c}^{T}\mathbf{v}-a\mathbf{c}^{T}\mathbf{v})/b$
$\displaystyle=$ $\displaystyle\frac{1-a}{b}\,\mathbf{c}^{T}\mathbf{v}$
$\displaystyle=$ $\displaystyle\mathbf{c}^{T}\mathbf{v},$
and we have a contradiction with the fact that $\mathbf{v}$ is an optimum.
Therefore, if $M_{\text{opt}}$ consists of a single point, it must be an
extreme point of $M$.
We are left to show that the set of $\mathbf{c}\in B$ for which
$M_{\text{opt}}(\mathbf{c})$ consists of multiple points has Lebesgue measure
zero. By Theorem 3, for a given $\mathbf{c}$, there exists a set of indices
$I\subseteq\\{1,\ldots,K\\}$ such that
$M_{\text{opt}}(\mathbf{c})=Q_{I}:=\\{\mathbf{v}\in
M\,|\,\mathbf{a}_{i}^{T}\mathbf{v}=b_{i}\text{ for all $i\in I$}\\}$, so
$\\{\mathbf{c}\in\mathbb{R}^{N}\,|\,M_{\text{opt}}(\mathbf{c})\text{ contains
multiple
points}\\}\subseteq\bigcup_{I}\\{\mathbf{c}\in\mathbb{R}^{N}\,|\,Q_{I}=M_{\text{opt}}(\mathbf{c})\\},$
(14)
where the union is taken over all $I\subseteq\\{1,\ldots,K\\}$ for which
$Q_{I}$ contains multiple points. If $\mathbf{c}$ is in one of the sets in the
union in Eq. (14), the set $Q_{I}$, being the set of all optimal solutions, is
orthogonal to $\mathbf{c}$. Hence, $\mathbf{c}$ is in $Q_{I}^{\perp}$, the
orthogonal complement of $Q_{I}$ defined as the set of all vectors orthogonal
to $Q_{I}$. Therefore,
$\\{\mathbf{c}\in\mathbb{R}^{N}\,|\,M_{\text{opt}}(\mathbf{c})\text{ contains
multiple points}\\}\subseteq\bigcup_{I}Q_{I}^{\perp},$ (15)
Because $Q_{I}$ is convex, it contains multiple points if and only if its
dimension is at least one, implying that each $Q_{I}^{\perp}$ in the union in
Eq. (15) has dimension at most $N-1$, and hence has zero Lebesgue measure in
$\mathbb{R}^{N}$. Since there are only a finite number of possible choices for
$I\subseteq\\{1,\ldots,K\\}$, the right hand side of Eq. (15) is a finite
union of sets of Lebesgue measure zero. Therefore, the left hand side also has
Lebesgue measure zero. ∎
Reference
Best MJ, Ritter K (1985) Linear Programming: Active Set Analysis and Computer
Programs. Prentice-Hall, Engelwood Cliffs, New Jersey, USA
|
arxiv-papers
| 2009-01-16T21:35:00 |
2024-09-04T02:49:00.033625
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Takashi Nishikawa, Natali Gulbahce, Adilson E. Motter",
"submitter": "Takashi Nishikawa",
"url": "https://arxiv.org/abs/0901.2581"
}
|
0901.2613
|
# Gauge extensions of supersymmetric models and hidden valleys
Mingxing Luo and Sibo Zheng
Zhejiang Institute of Modern Physics, Department of Physics,
Zhejiang University, Hangzhou 310027, P. R. China.
E-mail luo@zimp.zju.edu.cn sibozheng.zju@gmail.com
###### Abstract:
Supersymmetric models with extended group structure beyond the standard model
are revisited in the framework of general gauge mediation. Sum rules for
sfermion masses are shown to depend genuinely on the group structure, which
can serve as important probes for specific models. The left-right model and
models with extra $U(1)$ are worked out for illustrations. If the couplings of
extra gauge groups are small, supersymmetric hidden valleys of the scale
$10-100$ GeV can be naturally constructed in companion of a TeV-scale
supersymmetric visible sector.
Supersymmetric Phenomenology, Gauge Mediation
## 1 Introduction
Recently, a general method has been proposed to calculate soft terms in gauge
mediation models (GGM) [1]. It turns out that all soft terms in a specific
model can be determined by a few parameters, which encode the information of
the hidden sector. One obtains two sum rules [2] which make a distinctive
feature in $R$-symmetry breaking gauge mediation, in comparsion with gravity
mediation [3, 4], if the breaking of supersymmetry is communicated to the
visible sector only by standard model gauge interactions. In principle, the
hidden sector can be either weakly or strongly coupled. And the formalism is
valid for both direct and non-direct gauge mediation.
In this paper, we will reconsider the gauge extended supersymmetric model
(ESM) using the GGM formalism. The ESM can easily be constructed in deformed
ISS theories [14]. In these models, the unbroken global symmetry $G_{0}$ in
the hidden sector is larger than $G_{SM}$. If there are extra gauge structures
beyond the standard model (SM) and the corresponding symmetry in $G_{0}$ is
weakly gauged at the supersymmetric breaking scale $M_{SUSY}$, extra
interactions in addition to those of $G_{SM}$ yield modifications to the soft
terms, among other things. In general, they modify the sum rules in [2]. The
extra gauge structure is assumed to be spontaneously brokon at an intermediate
scale between $M_{SUSY}$ and $M_{EW}$, with the corresponding gauge boson of
masses in the order of a few TeV. At the electro-weak scale, only the SM is
left in the visible sector. Such arrangement is similar to the
$Z^{\prime}$-mediated supersymmetric breaking [9, 10] in spirit.
To be concrete, we will consider the abelian case where the gauged part of
$G_{0}$ is $G_{SM}\times U(1)^{\prime}$, and the non-abelian supersymmertic
left-right model where the gauged part of $G_{0}$ is $SU(3)_{c}\times
SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$. We find that the sum rules in [2]
cannot be retained in either cases. The resulting modifications are easily
obtained and dependent on the couplings of extra gauge sector. If the extra
gauge couplings are comparable with the ones in the SM, the sum rules are
broken significantly. If these couplings are small enough, these sum rules can
survive as approximations. These analysis can be directly applied to theories
with more sophisticated gauge structures, with similar conclusions.
The small coupling case is then used to construct models with a particular
type supersymmetric hidden valley models [17]. In particular, we will
construct a model in which the extra $U(1)^{\prime}$ communicates between the
supersymmetry breaking sector and a hidden valley sector, which generates
supersymmetry breakings in the latter. Simultaneously, the same
$U(1)^{\prime}$ communicates the hidden valley sector to the visible sector.
If the $U(1)^{\prime}$ coupling is of the order $10^{-1}-10^{-2}$ at
$M_{SUSY}$, the soft terms in the hidden valley are two or four orders of
magnitude smaller than those in the visible sector, which implies that an
$\mathcal{O}(1-10)$ TeV-scale visible sector is accompanied by an
$\mathcal{O}(10-100)$ GeV-scale hidden valley.
The rest of the paper is organized as follows. In section II, we briefly
review and comment on the GGM formalism. In section III, we discuss
supersymmetric models with group structure beyond SM. Particular attention
will be payed to the sum rules for sfermion masses. In section IV, we propose
a class of supersymmetric hidden valleys with $U(1)^{\prime}$. Finally we
conclude in section V.
## 2 Review and comments on GGM
In this section, we briefly review the GGM formalism, with an emphasis on
calculations of soft terms and sum rules for sfermion masses in the MSSM. The
gauge current of the hidden sector is a real linear superfield,
$\displaystyle{}\bar{D}^{2}\mathcal{J}=D^{2}\mathcal{J}=0$ (2.1)
with $\partial_{\mu}j^{\mu}=0$, as required by the current conservation
condition. The two-point correlator of $\mathcal{J}$ can generally be written
as,
$\displaystyle<\mathcal{J}(p,\theta,\bar{\theta})\mathcal{J}(p^{\prime},\theta^{\prime},\bar{\theta}^{\prime})>=2\pi^{4}\delta^{4}(p+p^{\prime})I(p,p^{\prime},\theta,\theta^{\prime})$
(2.2)
(2.1) and (2.2) can be generally solved, by a set of real functions
$B_{1/2},C_{a}$ [5]. The Fourier transforms of the correlators in momentum
space are,
$\displaystyle{}\langle J(p)J(-p)\rangle$ $\displaystyle=$
$\displaystyle\tilde{C}_{0}(p^{2}/M^{2};\,M/\Lambda)$ $\displaystyle\langle
j_{\alpha}(p)\bar{j}_{\dot{\alpha}}(-p)\rangle$ $\displaystyle=$
$\displaystyle-\sigma_{\alpha\dot{\alpha}}^{\mu}p_{\mu}\tilde{C}_{1/2}(p^{2}/M^{2};\,M/\Lambda)$
$\displaystyle\langle j_{\mu}(p)j_{\nu}(-p)\rangle$ $\displaystyle=$
$\displaystyle-(p^{2}\eta_{\mu\nu}-p_{\mu}p_{\nu})\tilde{C}_{1}(p^{2}/M^{2};\,M/\Lambda)$
$\displaystyle\langle j_{\alpha}(p)j_{\beta}(-p)\rangle$ $\displaystyle=$
$\displaystyle\epsilon_{\alpha\beta}M\tilde{B}_{1/2}(p^{2}/M^{2})$ (2.3)
If SUSY is unbroken, $\tilde{C}_{0}=\tilde{C}_{1/2}=\tilde{C}_{1}$,
$\tilde{B}_{1/2}=0$. If supersymmetry is broken, these relations do not hold
in general, as now $(Q_{\alpha}+Q^{\prime}_{\dot{\alpha}})I\neq 0$ and
$(\bar{Q}_{\alpha}+\bar{Q}^{\prime}_{\dot{\alpha}})I\neq 0$.
The gauge current superfield acts as a source for visible vector superfield
via the coupling,
$\displaystyle{}\mathcal{L}_{int}=2g\int
d^{4}\theta\mathcal{J}V+\dots=g(JD-\lambda
j-\bar{\lambda}\bar{j}-j^{\mu}V_{\mu})+\cdots,$ (2.4)
Note that in writing (2.4) the Wes-Zumino gauge has been chosen for the vector
superfield. Integrating out the messenger sector, we obtain the effective
Lagrangian for the gauge supermultiplet,
$\displaystyle\delta\mathcal{L}_{eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g^{2}\tilde{C}_{0}(0)D^{2}-g^{2}\tilde{C}_{1/2}(0)i\lambda\sigma^{\mu}\partial_{\mu}\bar{\lambda}-{1\over
4}g^{2}\tilde{C}_{1}(0)F_{\mu\nu}F^{\mu\nu}$ (2.5) $\displaystyle-$
$\displaystyle{1\over 2}g^{2}(M\tilde{B}_{1/2}(0)\lambda\lambda+c.c.)+\dots$
This gives contributions to the gaugino and sfermion masses, respectively,
${}M_{r}=g_{r}^{2}M\tilde{B}_{1/2}^{(r)}(0),~{}~{}~{}~{}~{}~{}\tilde{m}_{f}^{2}=\sum_{r=1}^{r=3}g_{r}^{4}c_{2}(f;r)A_{r}$
(2.6)
where $c_{2}(f;r)$ is the Casimir of the representation $f$ under the $r$
gauge group and
$\displaystyle{}A_{r}=-\int\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{p^{2}}\left(3\tilde{C}_{1}^{(r)}(p^{2}/M^{2})-4\tilde{C}_{1/2}^{(r)}(p^{2}/M^{2})+\tilde{C}_{0}^{(r)}(p^{2}/M^{2})\right)$
(2.7)
Note that the $\mu$ and $B\mu$ terms are model dependent. They can not be
determined unless more assumptions on the Higgs sector and the hidden sector
are made.
Now a few comments are in order for (2.6). First, one can choose the
superfield formalism at the starting point (2.4). The correlator of vector
superfields takes a simple form
$<\mathcal{V}\mathcal{V}>=\delta^{4}(\theta-\theta^{\prime})/p^{2}$ with the
gauge fixing parameter ${\xi}=1$. The wave function renormalization
$\mathcal{Z}_{Q}$ in the Kahler potential $\int
d^{4}\theta\tilde{Q}e^{-2\mathcal{V}}Q$ yields exactly the soft sfermion
masses in (2.6). Furthermore, the tri-linear $A$ terms in superpotential can
be obtained by replacing $Q$ with canonically normalized $Q^{\prime}$,
$\displaystyle{}Q^{\prime}=\left(1-\frac{1}{2}\mathcal{Z}|_{\theta^{2}}\theta^{2}-\frac{1}{2}\mathcal{Z}|_{\theta^{2}}\bar{\theta}^{2}\right)Q,$
(2.8)
Second, vector superfileds in (2.4) are usually massive after the
corresponding gauge symmetries are broken. If $m_{V}>>M_{SUSY}$, they can be
integrated out and will play a minor role in the communication of
supersymmetry breaking. If $m_{V}\sim M_{SUSY}$, the effects of $m_{V}$ need
then to be taken into account. At the leading order, one would have
$\displaystyle{}A^{\prime}_{r}=-\int\frac{d^{4}p}{(2\pi)^{4}(p^{2}-m_{V}^{2})}\left(\frac{3p^{2}}{p^{2}-m_{V}^{2}}\tilde{C^{\prime}}_{1}^{(r)}(p^{2}/M^{2})-4\tilde{C^{\prime}}_{1/2}^{(r)}(p^{2}/M^{2})+\tilde{C^{\prime}}_{0}^{(r)}(p^{2}/M^{2})\right)$
In this paper, we will assume $m_{V}<<M_{SUSY}$ for simplicity and (2.7) will
be used.
The positivity of $A_{r}$ has been proved in F-term supersymmetry breaking
with $F\ll M^{2}$ [6, 5], where $A_{r}$ can be written as a derivative term.
From (2.7), one sees that there are three independent functions for all
sfermion masses in a generation. Thus, there are at least two independent
sfermion masses relations, or sum rules,
$\displaystyle{}Tr\left(Y\tilde{m}_{f}\right)=0,~{}~{}~{}~{}Tr\left((B-L)\tilde{m}_{f}\right)=0$
(2.10)
These sum rules are one of the most distinctive features in such a gauge
mediation setting, in comparsion with other mediation mechanisms.
## 3 Sum rules in ESM
In this section, we will discuss supersymmetric models with gauge groups
beyond the SM ones $G_{SM}=SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$. We will
start with the simple extension with an extra abelian $U(1)^{\prime}$, then
move on to non-abelian gauges. In particular, we will concentrate on the left-
right symmetric model, though our analysis can be easily generalized to any
theories with more elaborated groups, with similar conclusions.
### 3.1 Abelian case
Since most of phenomenological results in this section are independent of the
details in the hidden sector, we will not address the issue of realizations of
these gauge structure in this section. In literature, there have been
extensive efforts to construct viable models. For example, gauge extended
models in ISS-like theories have been discussed before [11, 12, 13]. Theories
with similar gauge structures in the hidden sector can be found in [14], where
the ISS superpotential is deformed by $W_{def}$.111 Strongly coupled ISS-like
SQCD theories can be described by weakly coupled magnetic dual theories at low
energy scale. The magnetic theories have superpotentials of the same structure
as that of generalized O’Raifeartaigh models. According to the general proof
in generalized O’Raifeartaigh models [15], the $R$-symmetry must be
spontaneously broken when $W_{def}$ comes from a set of singlet fields with
$R$-charges of neither zero or two. On the other hand, the hidden sector
discussed in section 4 will be in another paradigm [8], instead of direct
gauge mediation. Partly, it is because that there are generally unacceptable
light gauginos or LHC unaccessible heavy sfermions in direct gauge mediation,
as discussed in [21].
Here, we assume that there is an extra abelian $U(1)^{\prime}$ in both the
hidden and the visible sectors. The soft terms can be obtained by calculations
similar to the ones in the previous section. The $U(1)^{\prime}$ introduces
extra $C^{\prime}_{a}$’s (thus $A^{\prime}$) and $\tilde{B}^{\prime}_{1/2}$,
which modify the sfermion and gaugino masses
$\displaystyle{}\delta\tilde{m}^{2}_{f_{i}}=\frac{3}{5}g^{\prime
4}q^{2}_{i}A^{\prime},~{}~{}~{}~{}~{}~{}~{}\delta\tilde{M}_{\lambda_{i}}=g^{\prime
2}M\tilde{B}^{\prime}_{1/2}.$ (3.1)
where $q_{i}$ are the $U(1)^{\prime}$ charges of fermions and $g^{\prime}$ is
the gauge coupling. Putting everything together, the soft masses are,
$\displaystyle{}\left(\begin{array}[]{c}m^{2}_{Q}\\\ m^{2}_{U}\\\ m^{2}_{D}\\\
m^{2}_{L}\\\ m^{2}_{E}\\\
\end{array}\right)=\frac{1}{60}\left(\begin{array}[]{ccccc}80&45&1&36q^{2}_{Q}\\\
80&0&16&36q^{2}_{U}\\\ 80&0&4&36q^{2}_{D}\\\ 0&45&9&36q^{2}_{L}\\\
0&0&36&36q^{2}_{E}\\\
\end{array}\right)\left(\begin{array}[]{c}g_{3}^{4}A_{3}\\\ g_{2}^{4}A_{2}\\\
g_{Y}^{4}A_{Y}\\\ g^{{}^{\prime}4}A^{\prime}\\\ \end{array}\right)$ (3.16)
So the sum rules in the previous subsection is not valid in general. Instead,
one has
$\displaystyle{}Tr\left(Y\tilde{m}_{f}\right)$ $\displaystyle=$
$\displaystyle\frac{3}{5}g^{\prime
4}(q^{2}_{Q}-2q^{2}_{U}+q^{2}_{D}-q^{2}_{L}+q^{2}_{E})A^{\prime}$ (3.17)
$\displaystyle Tr\left((B-L)\tilde{m}_{f}\right)$ $\displaystyle=$
$\displaystyle\frac{3}{5}g^{\prime
4}(2q^{2}_{Q}-2q^{2}_{U}-q^{2}_{D}-2q^{2}_{L}+q^{2}_{E})A^{\prime}$ (3.18)
Without the $U(1)^{\prime}$ interaction, one gets back the original sum rules
(2.10). One the other hand, there are five soft masses and four independent
$A$’s in (3.16), from which one can deduce one sum rule for the sfermion
masses,
$\displaystyle{}0$ $\displaystyle=$
$\displaystyle\left(q^{2}_{U}-q^{2}_{D}-\frac{1}{3}q^{2}_{E}\right)m^{2}_{Q}+\left(-q^{2}_{Q}+q^{2}_{D}+q^{2}_{L}-\frac{1}{3}q^{2}_{E}\right)m^{2}_{U}$
(3.19) $\displaystyle+$
$\displaystyle\left(q^{2}_{Q}-q^{2}_{U}-q^{2}_{L}+\frac{2}{3}q^{2}_{E}\right)m^{2}_{D}-\left(q^{2}_{U}-q^{2}_{D}-\frac{1}{3}q^{2}_{E}\right)m^{2}_{L}$
$\displaystyle+$
$\displaystyle\frac{1}{3}\left(q^{2}_{Q}+q^{2}_{U}-2q^{2}_{D}-q^{2}_{L}\right)m^{2}_{D}$
Now we have a few comments on these results:
* •
The sum rule $Tr(Ym^{2})=0$ holds provided that,
$\displaystyle{}q_{E}=q_{D}=0,~{}~{}~{}q^{2}_{Q}=2q^{3}_{U}+q_{L}^{2}$ (3.20)
while the other sum rule $Tr((B-L)m^{2})=0$ holds provided that,
$\displaystyle{}q_{Q}=q_{U}=0,~{}~{}~{}q^{2}_{D}=q^{2}_{E}-2q_{L}^{2}$ (3.21)
From (3.20) and (3.21), one can see the original sum rules (2.10) cannot be
retained at the same time, except for all $U(1)^{\prime}$ charges being set to
zero.
* •
If the visible and the hidden sectors are assumed to be anomaly free
separately, neither sum rules in (2.10) can be retained.
* •
If the coupling $g^{\prime}$ is substantially smaller than the SM couplings in
magnitude, (2.10) hold approximately.
The spontaneous breaking of $U(1)^{\prime}$ can be similar to the usual
$U(1)$’s without supersymmetry. One can introduce standard model singlets $S$
to trigger the breaking and extra exotic singlets to cancel the anomalies [9].
In particular, $S$ can obtain vacuum expectation value by radiative
corrections, provided that Yukawa couplings between $S$ and exotic singlets is
large enough.
### 3.2 Non-abelian case
We now move on to the discussion of extra non-abelian gauge groups. For
concreteness, we will consider the left-right model with
$G_{0}=SU(3)_{c}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$ [16], which
breaks into $G_{SM}$ via $SU(2)_{R}\times U(1)_{B-L}\rightarrow U(1)_{Y}$ at
some scale $M_{R}$. For simplicity, we will assume that $M_{R}<<M_{SUSY}$,
though our general results do not depend on this assumption. The analysis can
be easily generalized to gauge groups of higher ranks.
The $U(1)_{B-L}$ charges can be easily read from their $U(1)_{Y}$ charges.
Explicitly, coupling $g_{Y}$ and charges $q_{Y}$ are determined by
$g_{R},g_{B-L}$ via
$\displaystyle{}g_{Y}=\frac{g_{R}g_{B-L}}{g_{R}^{2}+g_{B-L}^{2}},~{}~{}~{}~{}~{}~{}q_{Y,i}=T^{3}_{R,i}+\tilde{q}_{i}.$
(3.22)
It is straightforward to get the masses for soft sfermions,
$\displaystyle{}\left(\begin{array}[]{c}m^{2}_{Q}\\\ m^{2}_{P}\\\ m^{2}_{L}\\\
m^{2}_{E}\\\
\end{array}\right)=\frac{1}{60}\left(\begin{array}[]{ccccc}80&45&0&1\\\
80&0&45&1\\\ 0&45&0&9\\\ 0&0&0&36\\\
\end{array}\right)\left(\begin{array}[]{c}g_{3}^{4}A_{3}\\\ g_{L}^{4}A_{L}\\\
g_{R}^{4}A_{R}\\\ g_{B-L}^{4}A_{B-L}\\\ \end{array}\right)$ (3.35)
where $P=(U,D)$ carries quantum numbers of
$(\mathbf{\bar{3}},\mathbf{1},\mathbf{2},\frac{1}{6})$.
Since fermions in the visible sector fits into spinor representations of
$SO(10)\supset G_{0}$, it is anomaly free. So the hidden sector must be
anomaly free also. Generally, there can be chiral matters $S_{i}$ with quantum
numbers $(\mathbf{1},\mathbf{1},\mathbf{2},q_{S_{i}})$ $(i\geq 1)$ and $M_{j}$
with quantum numbers $(\mathbf{1},\mathbf{2},\mathbf{1},q_{M_{j}})$ $(j\geq
0)$ in the hidden sector. The quantum numbers $q$ are constrained by the
anomaly free conditions. Specifically,
$\displaystyle{}SU(2)_{R}-SU(2)_{R}-U(1)_{B-L}$ $\displaystyle:$
$\displaystyle\sum_{(doublet,S)}\tilde{q}_{i}=0$ $\displaystyle
SU(2)_{L}-SU(2)_{L}-U(1)_{B-L}$ $\displaystyle:$
$\displaystyle\sum_{(doublet,M)}\tilde{q}_{i}=0$ $\displaystyle
U(1)_{B-L}-U(1)_{B-L}-U(1)_{B-L}$ $\displaystyle:$
$\displaystyle\sum_{i=(Q,S,M)}\tilde{q}^{3}_{i}=0$ $\displaystyle Graviton-
Graviton-U(1)_{B-L}$ $\displaystyle:$
$\displaystyle\sum_{i=(Q,S,M)}\tilde{q}_{i}=0$ (3.36)
Other anomaly free conditions are automatically satisfied by the charge
assignments in (3.22). We note that
* •
The sum rules (2.10) are both broken. Actually, they are modified to be
$\displaystyle{}Tr(Ym^{2}_{\tilde{f}})=\frac{3}{4}m^{2}_{E},~{}~{}~{}~{}~{}Tr((B-L)m^{2}_{\tilde{f}})=\frac{1}{2}m^{2}_{E}$
(3.37)
These two equations are independent of specific contents of the hidden sector.
Thus, they can serve as important probes of left-right supersymmetric models.
* •
If $SU(2)_{R}\times U(1)_{B-L}\rightarrow U(1)_{Y}$, with masses of gauge
bosons $(A_{+},~{}A_{-},~{}A_{0})$ near $M_{SUSY}$, the $A_{r}$’s in (2.7)
need to be replaced by those in (2). There are then six free parameters
$(A_{3},A_{2},A_{Y},A_{+},A_{-},A_{0})$ and five sfermion masses. This implies
(3.37) is modified again in this case.
The constraints (3.2) can be satisfied by proper assignments of charges
$q_{S_{i}}$ and $q_{M_{j}}$. At least one $S$ is needed to break $G_{0}$ into
$G_{SM}$.
Other extensions of group structure beyond SM induce corresponding sum rules,
some of which can be independent of details of the hidden sector, which serve
as generic probes of such theories.
## 4 Supersymmetric hidden valleys
Usually the hidden sector is assumed to be very heavy. Actually, a light
hidden sector cannot be ruled out if its communication with the visible sector
is sufficiently suppressed. Scenarios of light hidden sectors with small
coupling with the visible sector has been recently advocated and dubbed as
hidden valleys [17].
In $U(1)$ theories, one always has $\beta_{g^{\prime}}>0$ and the
corresponding couplings decrease with the decrease of energy. It is thus
possible that the effects from $U(1)^{\prime}$s are tiny at the electro-weak
scale due to renormalization group flows. In addition, the couplings between
the visible sector and the $U(1)^{\prime}$s are suppressed further by the
massive gauge boson $m_{Z^{\prime}}$’s. So the existence of extra
$U(1)^{\prime}$s cannot be ruled out by present experiments. Naturally, extra
$U(1)^{\prime}$s has been proposed to communicate the hidden valley sector to
the visible sector [17].
Here we will construct a model in which the extra $U(1)^{\prime}$ communicates
between the supersymmetry breaking sector and a hidden valley sector, which
generates supersymmetry breakings in the latter. Simultaneously, the same
$U(1)^{\prime}$ communicates the hidden sector to the visible sector. We will
see that if the $U(1)^{\prime}$ coupling is of the order $10^{-1}-10^{-2}$ at
$M_{SUSY}$, the soft terms in the hidden valley are two or four orders of
magnitude smaller than those in the visible sector. That is to say, an
$\mathcal{O}(1-10)$ TeV-scale visible sector is accompanied by an
$\mathcal{O}(10-100)$ GeV-scale hidden valley.
To be concrete, we will consider a class of models with the following
symmetries and particle contents,
(4.6)
Specifically,
* •
The theory is composed of three parts. The hidden sector is composed of a
spurion $X$ referred to be SUSY-breaking sector and an $SU(n_{v})$ gauge
theory with $v$-quarks in its bi-fundamental representations. The $v$-sector
is referred to as hidden valley. The messenger sector contains the
$\Phi_{i}$’s, which are neutral under $SU(n_{v})$ but charged under
$G_{SM}\times U(1)^{\prime}$. The visible sector contains gauged
$U(1)^{\prime}$ extension of group structure beyond $G_{SM}$ below
$\sqrt{F_{X}}$.
* •
The gauge symmetry is $SU(n_{v})\times G_{SM}\times U(1)^{\prime}$. Shown in
Table 1 are also the quantum numbers and representations of chiral matters.
If the SUSY-breaking sector is realized in the scheme of direct gauge
mediation, there will be unacceptable light gauginos or LHC unaccessible heavy
sfermions in general [21]. Thus, we turn to the old paradigm [8] to realize
supersymmetry and R-symmetry breaking. In such a scheme, it is not necessary
to construct the hidden sector explicitly. One simply assumes that a singlet
spurion $X$ is responsible for supersymmetry breaking and $X$ almost
determines all the phenomenological features. For explicit SUSY-breaking
sectors that induce such a spurion X, see [8] and reference therein.
The Lagrangian for the model in the table reads,
$\displaystyle{}\mathcal{L}=\int d^{2}\theta W+\int d^{4}\theta K+\int
d^{2}\theta\left(\mathcal{W}_{MSSM}^{2}+\mathcal{W}^{{}^{\prime}2}+\mathcal{W}_{h}^{2}\right)$
(4.7)
where
$\displaystyle{}W$ $\displaystyle=$
$\displaystyle\lambda_{ij}X\bar{\Phi}_{i}\Phi_{j},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X=M+F_{X}\theta^{2}$
(4.8) $\displaystyle K$ $\displaystyle=$
$\displaystyle\left(\Phi^{\dagger}_{i}e^{-2V_{MSSM}-2V^{\prime}}\Phi_{i}+Q_{m}^{\dagger}e^{-2V_{MSSM}-2V^{\prime}}Q_{m}+q_{j}^{\dagger}\
e^{-2V_{h}-2V^{\prime}}q_{j}+T_{\pm}^{\dagger}\
e^{-2V^{\prime}}T_{\pm}\right)$
where $Q_{m}$ denote the chiral matter superfields in SSM sector, $V_{h}$ and
$\mathcal{W}_{h}$ denote the vector and spinor superfield of hidden valley
gauge theory respectively. $T_{\pm}$ are responsible for triggering the
spontaneously breaking of $U(1)^{\prime}$.
Figure 1: Gauge mediation with (right) and without (left) extra
$U(1)^{\prime}$. The black line indicates gauge mediation due to $G_{SM}$
while the dashed ones due to $U(1)^{\prime}$.
One can see any two of the three sectors can communicate through the gauged
$U(1)^{\prime}$ group theory, as shown in Figure.1. Both the messenger sector
and the visible sector contain $G_{SM}$ gauge interactions, which dominate the
communication between them. But the hidden valley communicates with others
only via the $U(1)^{\prime}$.
It is straightforward to work out the soft masses in both the visible sector
and the hidden valley ($\lambda_{ij}=\delta_{ij}=1$). Generically,
$\displaystyle{}\tilde{m}^{2}_{Q_{i}}$ $\displaystyle=$
$\displaystyle\sum_{a=1}^{3}\frac{g_{a}^{4}N_{mess}}{(16\pi^{2})^{2}}C_{2}(f_{i},a)\left(\frac{F_{X}}{M}\right)^{2}+\mathcal{O}(g^{\prime
4})$ $\displaystyle M_{\lambda_{a}}$ $\displaystyle=$
$\displaystyle\frac{g_{a}^{2}N_{mess}}{(16\pi^{2})}\left(\frac{F_{X}}{M}\right)+\mathcal{O}(g^{\prime
2})$ (4.10)
in the visible sector and
$\displaystyle{}\tilde{m}^{2}_{q_{i}}=\frac{3g^{\prime 4}N_{mess}q^{\prime
2}_{i}}{5(16\pi^{2})^{2}}\left(\frac{F_{X}}{M}\right)^{2},~{}~{}~{}~{}~{}\tilde{m}^{2}_{T_{\pm}}=\frac{3g^{\prime
4}N_{mess}q^{\prime
2}_{\pm}}{5(16\pi^{2})^{2}}\left(\frac{F_{X}}{M}\right)^{2}$ (4.11)
in the hidden valley and $T_{\pm}$ chiral superfields respectively, $N_{mess}$
is the number of messengers. Finally, the gaugino mass of $U(1)^{\prime}$
vector superfield reads,
$\displaystyle{}M_{\lambda_{\bar{V}}}=\frac{g^{\prime
2}N_{mess}}{(16\pi^{2})}\left(\frac{F_{X}}{M}\right)$ (4.12)
Below the scale where $U(1)^{\prime}$ is spontaneously broken at scale
$\Lambda$ with mass $M_{V^{\prime}}=4g^{\prime 2}\Lambda^{2}$, the
$V^{\prime}$ vector superfield can be integrated out, leaving following
couplings in the effective theory at leading order,
$\displaystyle{}-\frac{1}{4\Lambda^{2}}\int
d^{4}\theta\left((1+\tilde{m}^{2}_{T}\theta^{4})(\sum_{m}q^{\prime}_{m}Q^{\dagger}_{m}e^{V_{MSSM}}Q_{m}+\sum_{j}q^{\prime}_{j}q^{\dagger}_{j}e^{V_{h}}q_{j})^{2}\right)$
(4.13)
where $q^{\prime}_{m}$ are the $U(1)^{\prime}$ charges of chiral matters in
MSSM. (4.13) also induces mixing couplings between operators in MSSM and
hidden valley, which are suppressed by the $U(1)^{\prime}$ gauge bosons mass.
The tree level Higgs masses $m_{H_{u}}$ and $m_{H_{d}}$ in the visible sector
are similar to (4),
$\displaystyle\tilde{m}^{2}_{H_{u,d}}=\sum_{a=1}^{2}\frac{g_{a}^{4}N_{mess}}{(16\pi^{2})^{2}}C_{2}(H_{u,d},a)\left(\frac{F_{X}}{M}\right)^{2}\sim\tilde{m}^{2}_{Q}$
(4.14)
As is well-known, the tree-level lightest Higgs mass $m_{h}$ is always lighter
than $m_{Z}$, no matter explicit values of $m_{H_{u}}$ and $m_{H_{d}}$. This
contradicts experimental observations but $m_{h}$ can be lifted over $m_{Z}$
by taking loop corrections into account. On the other hand, $m_{h}$ may be
further lifted by including higher dimensional couplings in (4.13).
Explicitly, the correction to potential in visible sector reads,
$\displaystyle{}\delta
V=q^{\prime}_{H_{u}}\tilde{v}H^{\dagger}_{u}H_{u}+q^{\prime}_{H_{d}}\tilde{v}H^{\dagger}_{d}H_{d}+\epsilon_{1}(H_{u}^{{\dagger}}H_{u})^{2}+\epsilon_{2}(H_{d}^{{\dagger}}H_{d})^{2}+\epsilon_{3}(H^{{\dagger}}_{d}H_{u})^{2}$
(4.15)
where
$\displaystyle\tilde{v}$ $\displaystyle=$
$\displaystyle\frac{\tilde{m}^{2}_{T}}{4\Lambda^{2}}\sum_{j}q^{\prime}_{\tilde{H}_{j}}|<\tilde{H}_{j}>|^{2},$
$\displaystyle\epsilon_{1}$ $\displaystyle=$ $\displaystyle q^{\prime
2}_{H_{u}}\frac{\tilde{m}^{2}_{T}}{4\Lambda^{2}},$ $\displaystyle\epsilon_{2}$
$\displaystyle=$ $\displaystyle q^{\prime
2}_{H_{d}}\frac{\tilde{m}^{2}_{T}}{4\Lambda^{2}}$ $\displaystyle\epsilon_{3}$
$\displaystyle=$ $\displaystyle(q^{\prime
2}_{H_{u}}+q^{\prime}_{H_{u}}q^{\prime}_{H_{d}}+q^{\prime
2}_{H_{d}})\frac{\mu^{2}}{4\Lambda^{2}}+q^{\prime}_{H_{u}}q^{\prime}_{H_{d}}\frac{\tilde{m}^{2}_{T}}{4\Lambda^{2}}$
(4.16)
Here $<\tilde{H}_{j}>$ are VeVs of scalars in hidden valley. Linear
approximations $F_{H_{u}}\simeq-\mu H_{d}^{{\dagger}}$ and
$F_{H_{d}}\simeq-\mu H_{u}^{{\dagger}}$ have been used in above calculations.
For typical parameters $\Lambda\sim 10^{3}$GeV, $\tilde{m}_{T}\sim 10-100$GeV,
$<\tilde{H}_{j}>\sim 100$GeV and $\mu\sim 200$GeV, the corrections to lightest
higgs bosons are dominated by $\epsilon_{3}$,
$\displaystyle\delta_{\epsilon_{3}}m^{2}_{h}\simeq\epsilon_{3}v^{2}\sim\mathcal{O}(10~{}GeV)^{2}$
(4.17)
This correction is independent of $\tan\beta$. Thus it contributes
significantly at the large $\tan\beta$ limit, as other contributions are
usually proportional to $1/\tan\beta$ [22].
Figure 2: Spectra and decay chains of the supersymmetric hidden valley with
$F_{X}/M\sim 10^{5}$ GeV. The dashed lines refer to particles that decay into
jets/leptons. $\lambda_{\bar{V}}$ and VSSP represent the next-lightest
supersymmetric particles in the visible and hidden sectors respectively.
From (4) and (4.11), One sees that the soft masses in supersymmetric hidden
valley are two or four order of magnitude smaller than those in visible
sector. Typically, they are in the order of $10-10^{2}$ GeV, while soft masses
of the visible sector and $m_{Z^{\prime}}$ are usually around TeV. For such
mass parameters, decay chains can be expected between the visible and hidden
valley sectors. In most decaying processes, jets/lepton pairs will be
generated, as shown in Figure.2. Phenomenologically, the generations of
$v$-quarks, the decay widths and their signals at colliders follow the general
pattern discussed in [18].
Finally, we outline the phenomenological features in the visible sector:
* •
As worked out in section 3.1, the sum rules in the visible sector are expected
to hold approximately. Notice that (most of) results in section 3 are
independent of the SUSY-breaking sector.
* •
The gaugino of $U(1)^{\prime}$ vector superfield is the next-lightest
supersymmetric particles (NSSP) in the visible sector if $\mid q_{+}\mid$ is
larger than $1/\sqrt{0.6N_{mess}}$. Otherwise, $T$-scalars are NSSP. When
sfermion and SM gaugino masses taken to be LHC accessible $\mathcal{O}(1)$
TeV, NSSP is around $10-100$ GeV.
* •
At the large $\tan\beta$ limit, higher dimensional couplings arising from
(4.13) in MSSM are the main sources to correct the Higgs spectra, which can
substantially uplift the lightest Higgs masses across the lower bound at LEPII
in the typical parameter space.
It would be interesting to construct a single hidden sector, which
spontaneously breaks supersymmetry and $R$-symmetry, but has desired unbroken
gauge symmetry and a hidden valley sector. One possible realization could be
an ISS-like theory with partially unbroken gauge symmetry [12].
## 5 Conclusions
In this paper, we have analyzed supersymmetric models with extended group
structure beyond the standard model in the framework of general gauge
mediation. We have concentrated on the sum rules for sfermion masses, and they
are shown to depend genuinely on the group structure, which can serve as
important probes of the specific model. In particular, they are rather
different from those in models with SM gauge group (2.10). For definiteness,
the left-right model and models with extra $U(1)$ has been worked out in
details. When the couplings of extra gauge groups are smaller than those in
the SM, the sum rules in (2.10) hold approximately.
We have constructed a model in which the extra $U(1)^{\prime}$ communicates
between the supersymmetry breaking sector and a hidden valley sector, which
generates supersymmetry breakings in the latter. Simultaneously, the same
$U(1)^{\prime}$ communicates the hidden sector to the visible sector. If the
$U(1)^{\prime}$ coupling is of the order $10^{-1}-10^{-2}$ at $M_{SUSY}$, soft
terms in the hidden valley are a few orders smaller than those in the visible
sector, which imply an $\mathcal{O}(1-10)$ TeV-scale visible sector is
accompanied by an $\mathcal{O}(1-100)$ GeV-scale hidden valley. Also, extra
higher dimensional couplings help to uplift the mass of the lightest Higgs
particle. The model conforms to the stringent constraints from LEP and other
precision experiments, as the communication between the visible and hidden
valley sectors is suppressed by the massive gauge bosons $m_{Z^{\prime}}$, in
addition to the smallness of the gauge coupling.
## Acknowledgement
This work is supported in part by the National Science Foundation of China
(10425525) and (10875103).
## References
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, arXiv:0812.3668.
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|
arxiv-papers
| 2009-01-17T07:20:19 |
2024-09-04T02:49:00.046337
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mingxing Luo and Sibo Zheng",
"submitter": "Sibo Zheng",
"url": "https://arxiv.org/abs/0901.2613"
}
|
0901.2738
|
# Delaunay triangulations of lens spaces
François Guéritaud
(Date: January 2009.
AMS subject classification: 52B11, 57M50.
Keywords: lens space, convex hull, continued fraction, Farey, Delaunay
triangulation. )
###### Abstract.
We compute the convex hull $\Pi$ of an arbitrary finite subgroup $\Gamma$ of
${\mathbb{C}^{*}}^{2}$ — or equivalently, of a generic orbit of the action of
$\Gamma$ on $\mathbb{C}^{2}$. The basic case is
$\Gamma=\\{(e^{2ik\pi/q},e^{2ikp\pi/q})~{}|~{}0\leq k<q\\}$ where
$p\in\llbracket 2,q-2\rrbracket$ is coprime to $q$: then, $\Pi$ projects to a
canonical or “Delaunay” triangulation $\mathcal{D}$ of the lens space
$L_{p/q}=\mathbb{S}^{3}/\Gamma$ (endowed with its spherical metric), and the
combinatorics of $\mathcal{D}$ are dictated by the continued fraction
expansion of $p/q$.
## 1\. Introduction
Given a compact pointed Riemannian $3$–manifold $(M,x_{0})$, a natural object
to construct is the Voronoi domain of $x_{0}$, i.e. the set $X$ of all points
$x$ such that the shortest path from $x$ to $x_{0}$ is unique. This domain $X$
can be embedded as a contractible subset of the universal cover
$\widetilde{M}$ of $M$; if $M$ is homogeneous, then $X$ is typically (though
not always) the interior of a polyhedron whose faces are glued in pairs to
yield $M$. If so, dual to $X$ (and this gluing data) is the so-called Delaunay
decomposition $\mathcal{D}$ of $M$, which comprises one cell per vertex of
$X$, and has only one vertex, namely $x_{0}$. If $\widetilde{M}$ is
$\mathbb{S}^{3}$ or $\mathbb{R}^{3}$ or $\mathbb{H}^{3}$, it is a classical
result that $\mathcal{D}$ is itself realized by geodesic polyhedra which tile
$M$.
A strong motivation for studying the Delaunay decomposition is that it is a
combinatorial invariant of $(M,x_{0})$ that encodes all the topology of $M$;
this also suggests that computing $\mathcal{D}$ is hard in general. Jeff
Weeks’ program SnapPea [We] achieves this numerically in the cusped hyperbolic
case (taking $x_{0}$ in the cusp); for explicit theoretical predictions of
$\mathcal{D}$ in special cases, see for example [G1, ASWY, La, G2, GS].
This paper is primarily concerned (Sections 2 through 5) with the case
$M=\mathbb{S}^{3}/\varphi$, where
$\varphi(z,z^{\prime})=\left(e^{\frac{2i\pi}{q}}z,e^{\frac{2ip\pi}{q}}z^{\prime}\right)$
and $\mathbb{S}^{3}$ is seen as the unit sphere of $\mathbb{C}^{2}$. Here,
$\frac{p}{q}$ is a rational of $(0,1)$ in reduced form, and $M$ is called the
_lens space_ $L_{p/q}$. We will show that the combinatorics of $\mathcal{D}$
(and $X$) are dictated by the continued fraction expansion of $\frac{p}{q}$
(and are independent of the choice of basepoint $x_{0}$).
The lift of $\mathcal{D}$ to $\mathbb{S}^{3}$ is the Delaunay decomposition of
$\mathbb{S}^{3}$ with respect to a _finite set_
$\langle\varphi\rangle\widetilde{x}_{0}$ of vertices. Finally, in Section 6,
we extend our results to the case where $\langle\varphi\rangle$ is replaced by
an arbitrary finite subgroup of $\mathbb{S}^{1}\times\mathbb{S}^{1}$ (possibly
non-cyclic, acting possibly with fixed points on $\mathbb{S}^{3}$).
### History
After the first version of this paper was posted, Günter M. Ziegler made me
aware of Smilansky’s paper [S2] where essentially the same results were
proven. The approaches are similar, except for the key result: we prove the
convexity of a certain plane curve $\gamma$ by a big computation (Claim 12);
Smilansky in [S2] seems unaware that $\gamma$ is always convex, but has a
clever lemma (proved in [S1]) to show that $\gamma$ behaves “as though it were
convex” with respect to certain intersecting lines.
Note that Sergei Anisov has also announced similar results in [A1, A2].
### Acknowledgements
The main result (without its proof!) occurred to me during the workshop on
Heegaard splittings at AIM, Palo Alto, in December 2007. It is a pleasure to
thank the organizers of this beautiful meeting, as well as Omprakash Gnawali
for early computer experiments and Saul Schleimer for subsequent discussions
on the topic.
## 2\. Preliminaries
Let $x_{0}$ be a point of $\mathbb{S}^{3}$ and
$\mathcal{O}\subset\mathbb{S}^{3}$ its $\langle\varphi\rangle$–orbit. Suppose
that the convex hull $\Pi$ of $\mathcal{O}$ has non-empty interior. It is
well-known that the boundary of $\Pi$ then decomposes into affine cells, whose
projections to $\mathbb{S}^{3}$ (from the origin) are precisely the cells of
the Delaunay decomposition $\mathcal{D}$. Therefore, all we have to do is to
determine the faces of the convex hull $\Pi$ of $\mathcal{O}$: these are
Theorems 1 and 3 below.
### 2.1. What is the generic case?
However, if $p\equiv\pm 1~{}[\text{mod }q]$, then any orbit $\mathcal{O}$ of
$\varphi$ is a regular polygon contained in a plane of
$\mathbb{R}^{4}\simeq\mathbb{C}^{2}$, which easily implies that the Voronoi
domain $X$ of $L_{p/q}$ (for any basepoint) is bounded by only two spherical
caps (this is a special case where $X$ is _not_ a proper spherical
polyhedron). It is also easy to see that the isometry group of $L_{p/q}$ acts
transitively on $L_{p/q}$ in that case.
Therefore, we will assume $p\notin\\{1,q-1\\}$. Then, the identity component
of the isometry group of $L_{p/q}$ lifts to the group
$G=\mathbb{S}^{1}\times\mathbb{S}^{1}$ acting diagonally on $\mathbb{C}^{2}$
(of course, $\varphi\in G$). The $G$-orbits in $\mathbb{S}^{3}$ are the tori
$\\{(z,z^{\prime})~{}|~{}\frac{|z^{\prime}|}{|z|}=\kappa\\}$ for
$\kappa\in\mathbb{R}_{+}^{*}$, and the circles $C=\\{0\\}\times\mathbb{S}^{1}$
and $C^{\prime}=\mathbb{S}^{1}\times\\{0\\}$. If $x_{0}\in C\cup C^{\prime}$,
then the orbit $\mathcal{O}=\langle\varphi\rangle x_{0}$ is a plane regular
polygon, so the Voronoi domain $X$ is again bounded by two spherical caps.
Therefore, we will be concerned with the generic case
$x_{0}\in\mathbb{S}^{3}\smallsetminus(C\cup C^{\prime})$. Since changing
$x_{0}$ only modifies its orbit $\mathcal{O}$ (and therefore the polyhedron
$\Pi$) by a diagonal automorphism of $\mathbb{C}^{2}$, all basepoints
$x_{0}\notin C\cup C^{\prime}$ are equivalent as regards the combinatorics of
$\Pi$ and of the Delaunay decomposition. In fact $x_{0}$ does not even need to
belong to the _unit_ sphere: for convenience, we will take
$x_{0}=(1,0,1,0)\in\sqrt{2}\mathbb{S}^{3}$ in Theorem 1.
### 2.2. An intuitive description of the triangulation
Clearly, $L_{p/q}$ is obtained by gluing two solid tori
$\\{(z,z^{\prime})\in\mathbb{S}^{3}~{}|~{}\frac{|z|}{|z^{\prime}|}\geq
1\\}/\varphi$ and
$\\{(z,z^{\prime})\in\mathbb{S}^{3}~{}|~{}\frac{|z|}{|z^{\prime}|}\leq
1\\}/\varphi$, boundary-to-boundary. Equivalently, $L_{p/q}$ is a thickened
torus $(\mathbb{S}^{1})^{2}\times[0,1]$, attached to two thickened disks (one
for each boundary component, along possibly very different slopes
$s,s^{\prime}$) and capped off with two balls.
We now sketch a way of triangulating $L_{p/q}$ that emulates this
construction: although it will not be needed in the sequel, it might provide
some geometric intuition (the triangulation described here will turn out to be
combinatorially equivalent to the Delaunay decomposition of $L_{p/q}$).
Consider the standard unit torus $T:=\mathbb{R}^{2}/\mathbb{Z}^{2}$ decomposed
into two simplicial triangles, $(0,0)(0,1)(1,1)$ and $(0,0)(1,0)(1,1)$. We can
simplicially attach two faces of a tetrahedron $\Delta$ to $T$, so that
$\Delta$ materializes an _exchange of diagonals_ in the unit square. The union
$T\cup\Delta$ is now a (partially) thickened torus, whose top and bottom
boundaries are triangulated in two different ways. We can attach a new
tetrahedron $\Delta^{\prime}$, e.g. to the top boundary, so as to perform a
new exchange of diagonals. Iterating the process many times, we can obtain a
triangulation of (possibly a retract of) $T\times[0,1]$ with top and bottom
triangulated (into two triangles each) in two essentially arbitrary ways.
Finally, there exists a standard way of folding up the top boundary
$T\times\\{1\\}$ on itself, identifying its two triangles across an edge: this
was perhaps first formulated that way in [JR]. The result after folding-up is
a _solid torus_ , also described with many pictures in [GS]. (In that paper,
we show that such triangulated solid tori also arise naturally in the Delaunay
decompositions of many _hyperbolic_ manifolds, namely, large “generic” Dehn
fillings.) If we fold up the bottom $T\times\\{0\\}$ in a similar way, it
turns out we can get any $L_{p/q}$ with $p\equiv/\hskip 4.0pt\pm
1~{}[\text{mod }q]$.
The main theorems below (1 and 3) describe this same triangulation in a way
that is self-contained and completely explicit, although perhaps less
synthetic or helpful than the process described above. The interested reader
may infer the equivalence of the two descriptions from the proof of Theorem 3;
see also [GS].
### 2.3. Strategy
Let $\mathbb{T}:=(\mathbb{R}/2\pi\mathbb{Z})^{2}$ be the standard torus and
$\iota:\mathbb{T}\rightarrow\mathbb{C}^{2}\simeq\mathbb{R}^{4}$ denote the
standard injection, satisfying
$\iota(u,v)=(\cos u,\sin u,\cos v,\sin v).$
The subgroup
$\Gamma:=\\{\tau_{k}=(k\frac{2\pi}{q},kp\frac{2\pi}{q})\\}_{k\in\mathbb{Z}}$
of $\mathbb{T}$ is such that $\iota(\Gamma)=\mathcal{O}$, the orbit of
$(1,0,1,0)\in\mathbb{R}^{4}$ under $\varphi$. Therefore, each top-dimensional
cell (tetrahedron, as it turns out) in $\partial\Pi$ is spanned by the images
under $\iota$ of four points
$\tau,\tau^{\prime},\tau^{\prime\prime},\tau^{\prime\prime\prime}$ of
$\Gamma$.
Our main result, Theorem 1, claims that $\tau,\dots,\tau^{\prime\prime\prime}$
are the vertices of certain _parallelograms_ of $\mathbb{T}$ with the minimal
possible area, namely $\frac{(2\pi)^{2}}{q}$. To prove this, the strategy is
to consider a linear form $\rho:\mathbb{R}^{4}\rightarrow\mathbb{R}$ that
takes the same value, say $Z>0$, on
$\iota(\tau),\dots,\iota(\tau^{\prime\prime\prime})$; then look (e.g. in the
chart $[-\pi,\pi]^{2}$) at the level curve $\gamma=(\rho\circ\iota)^{-1}(Z)$.
Lemma 9 says that if $Z$ and the coefficients of $\rho$ satisfy certain
inequalities, then $\gamma$ is a _convex_ Jordan curve passing through
$\tau,\dots,\tau^{\prime\prime\prime}$. Intuitively, if the hyperplane
$\rho^{-1}(Z)$ passes _far enough_ from the origin of $\mathbb{R}^{4}$ (in a
sense depending on the direction of $\ker\rho$), it will only skim a small cap
off $\iota(\mathbb{T})$ that looks convex in the chart. Convexity is key: it
will imply that no other point of $\Gamma$ than
$\tau,\dots,\tau^{\prime\prime\prime}$ lies inside $\gamma$, i.e. in
$(\rho\circ\iota)^{-1}[Z,+\infty)$. In other words,
$\rho^{-1}(Z)\supset\iota(\\{\tau,\dots,\tau^{\prime\prime\prime}\\})$ is a
supporting plane of the convex hull of $\iota(\Gamma)=\mathcal{O}$.
Proving that $Z$ and the coefficients of $\rho$ satisfy the inequalities of
Lemma 9 will be the trickier part of the work, done in Section 5 using only
basic trigonometry.
### 2.4. Notation
Until the end of Section 5, we fix $q\geq 5$ and $p\in\llbracket
2,q-2\rrbracket$ coprime to $q$, so that $Q:=\frac{p}{q}$ is a rational of
$(0,1)$ in reduced form. We denote by $x_{0}$ the point $(1,1)$ of
$\mathbb{C}^{2}$, and by $x_{k}$ the $k$-th iterate of $x_{0}$ under the map
$\varphi:(z,z^{\prime})\mapsto(e^{\frac{2i\pi}{q}}z,e^{\frac{2ip\pi}{q}}z^{\prime})$.
Finally we let $\Pi$ be the convex hull of $x_{0},\dots,x_{q-1}$. We identify
$\mathbb{R}^{4}$ with $\mathbb{C}^{2}$ in the standard way. The transpose of a
matrix $M$ is written $M^{t}$.
By _Farey graph_ , we mean the graph obtained by connecting two rationals
$\frac{\alpha}{a},\frac{\beta}{b}$ of
$\mathbb{P}^{1}\mathbb{R}=\partial_{\infty}\mathbb{H}^{2}$ by a geodesic line
in $\mathbb{H}^{2}$ whenever $|\alpha b-\beta a|=1$ (this graph consists of
the ideal triangle $01\infty$ reflected in its sides _ad infinitum_ , and
$\text{PSL}_{2}\mathbb{\mathbb{Z}}\subset\text{PSL}_{2}\mathbb{R}\simeq\text{Isom}^{+}(\mathbb{H}^{2})$
acts faithfully transitively on oriented edges). For example, two rationals
connected by a Farey edge are called _Farey neighbors_. Refer to [Vi] for the
classical casting of continued fractions in terms of the Farey graph.
## 3\. Main result: description of the faces of $\Pi$
###### Theorem 1.
Let $A=\frac{\alpha}{a},B=\frac{\beta}{b}\in[0,1]$ be Farey neighbors such
that $Q=\frac{p}{q}$ lies strictly between $A$ and $B$, at most one of $A,B$
is a Farey neighbor of $Q$, and at most one of $A,B$ is a Farey neighbor of
$\infty$ (i.e. belongs to $\\{0,1\\}$). Then $x_{0},x_{a},x_{b},x_{a+b}$ span
a top-dimensional cell (tetrahedron) of $\Pi$.
Note that in the simplest case $\frac{p}{q}=\frac{2}{5}$, there is only one
pair $\\{\frac{\alpha}{a},\frac{\beta}{b}\\}=\\{\frac{1}{3},\frac{1}{2}\\}$.
Theorem 1 will be proved in Section 5. Meanwhile, we check (Theorem 3) that
there are no _other_ top-dimensional faces in $\partial\Pi$. Note that we make
no assumption on whether $A<B$ or $B<A$, or on whether $a<b$ or $b<a$ (all
four possibilities can arise), so we will always be able to switch $A$ and $B$
for convenience.
###### Remark 2.
It is well-known that the number of unordered pairs of rationals
$\\{\frac{\alpha}{a},\frac{\beta}{b}\\}$ satisfying the hypotheses of Theorem
1 is $n-3$, where $n$ is the sum of all coefficients of the continued fraction
expansion of $Q$. Moreover, these pairs are naturally ordered: the first pair
is $\\{\frac{0}{1},\frac{1}{2}\\}$ or $\\{\frac{1}{2},\frac{1}{1}\\}$
according to the sign of $Q-\frac{1}{2}$; the pair coming after
$\\{\frac{\alpha}{a},\frac{\beta}{b}\\}$ is either
$\\{\frac{\alpha}{a},\frac{\alpha+\beta}{a+b}\\}$ or
$\\{\frac{\alpha+\beta}{a+b},\frac{\beta}{b}\\}$. Reversing this, the pair
coming _before_ $\\{\frac{\alpha}{a},\frac{\beta}{b}\\}$ is
$\\{\frac{\min(\alpha,\beta)}{\min(a,b)},\frac{|\alpha-\beta|}{|a-b|}\\}$. The
last pair $\\{\frac{\alpha}{a},\frac{\beta}{b}\\}$ contains exactly one Farey
neighbor of $\frac{p}{q}$ and is such that $\frac{\alpha+\beta}{a+b}$ is
another Farey neighbor of $\frac{p}{q}$: therefore that last pair satisfies
either $\frac{\alpha+(\alpha+\beta)}{a+(a+b)}=\frac{p}{q}$ or
$\frac{(\alpha+\beta)+\beta}{(a+b)+b}=\frac{p}{q}$.
###### Theorem 3.
All top-dimensional faces of $\Pi$ are tetrahedra whose vertices are of the
form $x_{n}x_{n+a}x_{n+b}x_{n+a+b}$ with $a,b$ as in Theorem 1, and
$n\in\mathbb{Z}$.
###### Proof.
Assuming Theorem 1, it is enough to find a tetrahedron of the given form,
adjacent to every face of the tetrahedron $T_{a,b}:=x_{0}x_{a}x_{b}x_{a+b}$
(but possibly with a different pair $\\{a,b\\}$).
First, the faces of $T_{a,b}$ obtained by dropping $x_{0}$ or $x_{a+b}$ indeed
have neighbors:
If $T_{a,b}$ is the first tetrahedron for the ordering, Remark 2 implies
$T_{a,b}=T_{1,2}=x_{0}x_{1}x_{2}x_{3}$. The face $x_{0}x_{1}x_{2}$ of
$T_{a,b}$ (obtained by dropping $x_{3}$) is adjacent to
$\varphi^{-1}(T_{a,b})=x_{-1}x_{0}x_{1}x_{2}$, and similarly the face
$x_{1}x_{2}x_{3}$ obtained by dropping $x_{0}$ is adjacent to
$\varphi(T_{a,b})=x_{1}x_{2}x_{3}x_{4}$.
If $T_{a,b}$ is _not_ the first tetrahedron, then we can assume $a<b$ and by
Remark 2 there is a previous tetrahedron $T_{b-a,a}$. The face
$x_{0}x_{a}x_{b}$ of $T_{a,b}$ is adjacent to
$T_{b-a,a}=x_{0}x_{b-a}x_{a}x_{b}$; the face $x_{a}x_{b}x_{a+b}$ of $T_{a,b}$
is adjacent to $\varphi^{a}(T_{b-a,a})=x_{a}x_{b}x_{2a}x_{a+b}$.
Lastly, the faces of $T_{a,b}$ obtained by dropping $x_{a}$ or $x_{b}$ also
have neighbors:
If $T_{a,b}$ is the last tetrahedron, then Remark 2 implies $a+2b=q$ (up to
switching $a,b$), hence $x_{a+b}=x_{-b}$ (because $x_{q}=x_{0}$). Therefore
the face $x_{0}x_{a}x_{a+b}=x_{0}x_{a}x_{-b}$ of $T_{a,b}$ is adjacent to
$\varphi^{-b}(T_{a,b})=x_{-b}x_{a-b}x_{0}x_{a}$, and the face
$x_{0}x_{b}x_{a+b}=x_{a+2b}x_{b}x_{a+b}$ of $T_{a,b}$ is adjacent to
$\varphi^{b}(T_{a,b})=x_{b}x_{a+b}x_{2b}x_{a+2b}$.
If $T_{a,b}$ is _not_ the last tetrahedron, then up to switching $a,b$ there
is, by Remark 2, a next tetrahedron $T_{a,a+b}$. Therefore the face
$x_{0}x_{a}x_{a+b}$ of $T_{a,b}$ is adjacent to
$T_{a,a+b}=x_{0}x_{a}x_{a+b}x_{2a+b}$, and the face $x_{0}x_{b}x_{a+b}$ of
$T_{a,b}$ is adjacent to $\varphi^{-a}(T_{a,a+b})=x_{-a}x_{0}x_{b}x_{a+b}$. ∎
## 4\. Main tools
Under the assumptions of Theorem 1, and before we start its proof proper, let
us introduce some tools. These are of two types: arithmetic properties of the
integers appearing in the Farey diagram (Section 4.1), and geometric
properties of the standard embedding $\iota$ of
$\mathbb{S}^{1}\times\mathbb{S}^{1}$ into $\mathbb{R}^{2}\times\mathbb{R}^{2}$
(especially its intersections with hyperplanes), in Section 4.2.
### 4.1. Farey relationships on integers
Let $X=\frac{\xi}{x}=\frac{\alpha+\beta}{a+b}$ and
$Y=\frac{\eta}{y}=\frac{|\alpha-\beta|}{|a-b|}$ be the two common Farey
neighbors of $A$ and $B$ ($X$ is closer to $Q$ while $Y$ is closer to
$\infty=\frac{1}{0}$; we have $X,Y\in[0,1]$). We introduce the notation
$\frac{u}{v}\wedge\frac{s}{t}:=|ut-vs|$
for any two rationals $\frac{u}{v},\frac{s}{t}$ in reduced form. For example,
if $h,h^{\prime}$ are rational, then $h\wedge h^{\prime}=1$ if and only if
$h,h^{\prime}$ are Farey neighbors; moreover, the denominator of $h$ is always
equal to $h\wedge\infty$.
We thus have $\left\\{\begin{array}[]{c}a=A\wedge\infty\\\ b=B\wedge\infty\\\
x=X\wedge\infty\\\ y=Y\wedge\infty\\\ q=Q\wedge\infty\end{array}\right.$ and
we define $\left\\{\begin{array}[]{c}a^{\prime}:=A\wedge Q\\\
b^{\prime}:=B\wedge Q\\\ x^{\prime}:=X\wedge Q\\\ y^{\prime}:=Y\wedge
Q\end{array}\right.$, all positive.
###### Proposition 4.
One has $\left\\{\begin{array}[]{ccc}a+b&=&x\\\ |a-b|&=&y\end{array}\right.$,
and $\left\\{\begin{array}[]{ccc}a^{\prime}+b^{\prime}&=&y^{\prime}\\\
|a^{\prime}-b^{\prime}|&=&x^{\prime}\end{array}\right.$, and
$a^{\prime}b+b^{\prime}a=q.$
###### Proof.
The first two identities are obvious from the definitions of $X,Y$. For the
next two identities, notice that $\alpha q-ap$ and $\beta q-bp$ have opposite
signs, because $Q$ lies between $A$ and $B$ : therefore
$a^{\prime}+b^{\prime}=|\alpha q-ap|+|\beta q-bp|=|(\alpha q-ap)-(\beta
q-bp)|=|(\alpha-\beta)q-(a-b)p|=Y\wedge Q~{};$
$|a^{\prime}-b^{\prime}|=||\alpha q-ap|-|\beta q-bp||=|(\alpha q-ap)+(\beta
q-bp)|=|(\alpha+\beta)q-(a+b)p|=X\wedge Q~{}.$
For the last identity, compute
$\displaystyle a^{\prime}b+b^{\prime}a$ $\displaystyle=$
$\displaystyle(Q\wedge A)(\infty\wedge B)+(Q\wedge B)(\infty\wedge A)$
$\displaystyle=$ $\displaystyle b|q\alpha-pa|+a|q\beta-pb|$ $\displaystyle=$
$\displaystyle|b(q\alpha-pa)-a(q\beta-pb)|$ $\displaystyle=$ $\displaystyle
q|b\alpha-a\beta|=q(A\wedge B)=q~{}.$
∎
An easy consequence is that all of
$a,a^{\prime},b,b^{\prime},x,x^{\prime},y,y^{\prime}$ are integers of
$\llbracket 1,q-1\rrbracket$. Note that the properties of Proposition 4 are
invariant under the exchange of $(a,a^{\prime})$ with $(b,b^{\prime})$ and
under the exchange of $(a,b,x,y)$ with
$(a^{\prime},b^{\prime},y^{\prime},x^{\prime})$ (which actually amounts to
swapping $Q$ and $\infty$).
###### Proposition 5.
None of $a,a^{\prime},b,b^{\prime}$ is equal to $\frac{q}{2}$.
###### Proof.
Suppose $b=\frac{q}{2}$. Since $a^{\prime}b+b^{\prime}a=q$, we then have
$a^{\prime}=1$. We have $b^{\prime}a=q-a^{\prime}b=\frac{q}{2}$ so $a$ divides
$\frac{q}{2}$, but $a$ is also coprime to $b=\frac{q}{2}$ (because $A\wedge
B=1$). Therefore $a=1$ (which by the way means $A\in\\{0,1\\}$). But since
$a^{\prime}=1$, this implies that $A$ is a Farey neighbor both of $Q$ and
$\infty$, i.e. $Q$ has the form $\frac{1}{q}$ or $\frac{q-1}{q}$, which we
ruled out in the first place.
If instead of $b$ another term of $a,a^{\prime},b,b^{\prime}$ is equal to
$\frac{q}{2}$, then we can apply the same argument, up to permuting
$a,a^{\prime},b,b^{\prime}$. ∎
Notice, however, that one of $a,a^{\prime},b,b^{\prime}$ could be _larger_
than $\frac{q}{2}$.
### 4.2. Level curves on the torus
Let $\mathbb{T}:=(\mathbb{R}/2\pi\mathbb{Z})^{2}$ be the standard torus and
$\iota:\mathbb{T}\rightarrow\mathbb{C}^{2}\simeq\mathbb{R}^{4}$ denote the
standard injection, satisfying
$\iota(u,v)=(\cos u,\sin u,\cos v,\sin v).$
The subgroup
$\Gamma=\\{\tau_{k}=(k\frac{2\pi}{q},kp\frac{2\pi}{q})\\}_{k\in\mathbb{Z}}$ of
$\mathbb{T}$ lifts to an affine lattice $\Lambda$ of the universal cover
$\mathbb{R}^{2}$ of $\mathbb{T}$. The index of $2\pi\mathbb{Z}^{2}$ in
$\Lambda$ is $q$. Rationals $A,B$ are still as in Theorem 1.
###### Proposition 6.
Define the lifts $u=(a\frac{2\pi}{q},ap\frac{2\pi}{q}-2\alpha\pi)$ and
$v=(b\frac{2\pi}{q},bp\frac{2\pi}{q}-2\beta\pi)$ of $\tau_{a}$ and $\tau_{b}$
respectively. Also define the center $\overline{c}:=\frac{1}{2}(u+v)$ of the
parallelogram $D:=(0,u,u+v,v)$ of $\mathbb{R}^{2}$. Then $(u,v)$ is a basis of
the lattice $\Lambda$, and $D$ is contained in the square
$\overline{c}+(-\pi,\pi)^{2}$.
###### Proof.
Clearly, $\Lambda\subset\mathbb{R}^{2}$ has covolume $(2\pi)^{2}/q$. On the
other hand, the determinant of $(u,v)$ is $2\pi\frac{2\pi}{q}(\alpha
b-a\beta)=\pm(2\pi)^{2}/q$, so $(u,v)$ is a basis of $\Lambda$.
The abscissae of $u,v$ are clearly positive, and their sum is
$\frac{a+b}{q}2\pi=\frac{x}{q}2\pi<2\pi$.
The ordinates $2\pi a(Q-A)$ of $u$ and $2\pi b(Q-B)$ of $v$ have opposite
signs, and the sum of their absolute values is
$2\pi\left(\left|\frac{ap-\alpha q}{q}\right|+\left|\frac{bp-\beta
q}{q}\right|\right)=2\pi\frac{A\wedge Q+B\wedge
Q}{q}=2\pi\frac{y^{\prime}}{q}<2\pi~{},$
by Proposition 4. This proves the claim on $D$. ∎
###### Definition 7.
Let $c=\left(\frac{a+b}{q}\pi,[p\frac{a+b}{q}-(\alpha+\beta)]\pi\right)$
denote the projection of $\overline{c}$ to the torus
$\mathbb{T}=(\mathbb{R}/2\pi\mathbb{Z})^{2}$.
###### Proposition 8.
Let $\Lambda\subset\mathbb{R}^{2}$ be a lattice and $P$ be a strictly convex,
compact region of $\mathbb{R}^{2}$ such that $\Lambda\cap\partial P$ consists
of the four vertices of a fundamental parallelogram of $\Lambda$. Then
$\Lambda\cap P=\Lambda\cap\partial P$ (i.e. $P$ contains no other lattice
points).
###### Proof.
Without loss of generality, $\Lambda=\mathbb{Z}^{2}$ and
$\\{0,1\\}^{2}\subset\partial P$. Since $P$ is strictly convex, the horizontal
axis $\mathbb{R}\times\\{0\\}$ intersects $P$ precisely along
$[0,1]\times\\{0\\}$. A similar statement holds for each side of the unit
square. Therefore
$P\smallsetminus\\{0,1\\}^{2}\subset(0,1)\times\mathbb{R}\cup\mathbb{R}\times(0,1)$,
which contains no other vertices of $\mathbb{Z}^{2}$. ∎
The idea of the proof of Theorem 1 is to consider a linear form
$\rho:\mathbb{R}^{4}\rightarrow\mathbb{R}$ that takes the same value $Z>0$ on
$x_{0},x_{a},x_{b},x_{a+b}$ and check that $\rho<Z$ on all other $x_{i}$. This
will be achieved by looking at the level curve $\gamma$ of $\rho\circ\iota$ in
$\mathbb{T}$, of level $Z$, and checking that the lift of $\gamma$ to
$\mathbb{R}^{2}$ bounds a convex body that satisfies the hypotheses of
Proposition 8. For this, we will need the following property and its
corollary.
###### Lemma 9.
If $(U,U^{\prime}),(V,V^{\prime})\in\mathbb{R}^{2}\smallsetminus\\{(0,0)\\}$
and $Z\in\mathbb{R}_{+}^{*}$ satisfies
$\left|\sqrt{V^{2}+V^{\prime 2}}-\sqrt{U^{2}+U^{\prime
2}}\right|<Z<\sqrt{V^{2}+V^{\prime 2}}+\sqrt{U^{2}+U^{\prime 2}}~{},$
then the preimage of $Z$ under
$\begin{array}[]{rrcl}&\mathbb{R}^{2}&\rightarrow&\mathbb{R}\\\
\psi~{}:&(x,y)&\longmapsto&(U\cos x+U^{\prime}\sin x)+(V\cos y+V^{\prime}\sin
y)\end{array}$
consists of a convex curve $\gamma$ (i.e. a closed curve bounding a strictly
convex domain), together with all the translates of $\gamma$ under
$2\pi\mathbb{Z}^{2}$, which are pairwise disjoint.
###### Proof.
Up to shifting $x$ and $y$ by constants, we can assume
$U^{\prime}=V^{\prime}=0$ and $U,V>0$. Up to exchanging $x$ and $y$, we can
furthermore assume $V\geq U$, so that $0\leq V-U<Z<V+U$ and $\psi(x,y)=U\cos
x+V\cos y$. Notice that $U,V,Z$ now satisfy all three strong triangular
inequalities.
Let $C$ be the square $[-\pi,\pi]^{2}$. Let us first determine that
$\gamma:=\psi^{-1}(Z)\cap C$ is a convex curve contained in the interior of
$C$. If $(x,y)\in\gamma$ then $U\cos x\geq Z-V\in(-U,U)$ so
$|x|\leq\arccos\frac{Z-V}{U}\in(0,\pi)~{}\text{ and }~{}\pm
y=f(x):=\arccos\frac{Z-U\cos x}{V}\in[0,\pi)~{},$
since $Z-U>-V$. Clearly, $f$ vanishes at $\pm\arccos\frac{Z-V}{U}$. Moreover,
using the chain rule
$(\arccos\circ\,g)^{\prime\prime}=-\frac{g^{\prime\prime}(1-g^{2})+gg^{\prime
2}}{(1-g^{2})^{3/2}}$, computation yields
$f^{\prime\prime}(x)=\frac{-U^{2}Z}{[V^{2}-(Z-U\cos
x)^{2}]^{\frac{3}{2}}}\left[1+\frac{V^{2}-Z^{2}-U^{2}}{UZ}\cos
x+\cos^{2}x\right]$
so to show $f^{\prime\prime}<0$ it is enough to check that the discriminant of
the polynomial in $\cos x$ (in the right factor) is negative. This amounts to
$\left|\frac{V^{2}-Z^{2}-U^{2}}{UZ}\right|<2$, which in turn follows from the
triangular inequalities $(Z+U)^{2}>V^{2}$ and $(Z-U)^{2}<V^{2}$.
We have proved that $\gamma$ is a convex curve (the union of the graphs of $f$
and $-f$) contained in the interior of $C$: the rest of the lemma follows
easily. ∎
###### Corollary 10.
Under the assumptions of Lemma 9, the set $H:=\psi^{-1}[Z,+\infty)$ consists
of the disjoint union of all the convex domains bounded by $\gamma$ and its
translates.
###### Proof.
Again restricting to $U^{\prime}=V^{\prime}=0<U\leq V$, we see that $H\cap C$
contains the origin (encircled by $\gamma$, and where $\psi$ achieves its
maximum $U+V$) and does not contain $(\pi,\pi)$ (where $\psi$ achieves its
minimum $-U-V$). The theorem of intermediate values allows us to conclude. ∎
## 5\. Proof of Theorem 1
Identifying $\mathbb{C}^{2}$ with $\mathbb{R}^{4}$ in the standard way, the
matrix with column vectors $x_{0},x_{a},x_{b},x_{a+b}$ is
(1) $M:=\left(\begin{array}[]{clll}1&\cos a\frac{2\pi}{q}&\cos
b\frac{2\pi}{q}&\cos(a+b)\frac{2\pi}{q}\\\ 0&\sin a\frac{2\pi}{q}&\sin
b\frac{2\pi}{q}&\sin(a+b)\frac{2\pi}{q}\\\ 1&\cos pa\frac{2\pi}{q}&\cos
pb\frac{2\pi}{q}&\cos p(a+b)\frac{2\pi}{q}\\\ 0&\sin pa\frac{2\pi}{q}&\sin
pb\frac{2\pi}{q}&\sin p(a+b)\frac{2\pi}{q}\end{array}\right)~{}.$
We refer to $\\{x_{0},x_{a},x_{b},x_{a+b}\\}$ as our _candidate face_.
### 5.1. Candidate faces are non-degenerate
###### Proposition 11.
The determinant $D$ of the matrix $M$ is nonzero.
###### Proof.
Rotating the plane of the first two coordinates by $\frac{-a-b}{q}\pi$, and
the plane of the last two coordinates by $\frac{-a-b}{q}p\pi$, we see that
$\displaystyle D$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{llll}\cos\frac{-a-b}{q}\pi&\cos\frac{a-b}{q}\pi&\cos\frac{b-a}{q}\pi&\cos\frac{a+b}{q}\pi\\\
\sin\frac{-a-b}{q}\pi&\sin\frac{a-b}{q}\pi&\sin\frac{b-a}{q}\pi&\sin\frac{a+b}{q}\pi\\\
\cos\frac{-a-b}{q}p\pi&\cos\frac{a-b}{q}p\pi&\cos\frac{b-a}{q}p\pi&\cos\frac{a+b}{q}p\pi\\\
\sin\frac{-a-b}{q}p\pi&\sin\frac{a-b}{q}p\pi&\sin\frac{b-a}{q}p\pi&\sin\frac{a+b}{q}p\pi\end{array}\right|$
$\displaystyle=$ $\displaystyle
4\left|\begin{array}[]{llll}\cos\frac{a+b}{q}\pi&\cos\frac{a-b}{q}\pi&\cos\frac{b-a}{q}\pi&\cos\frac{a+b}{q}\pi\\\
0&0&\sin\frac{b-a}{q}\pi&\sin\frac{a+b}{q}\pi\\\
\cos\frac{a+b}{q}p\pi&\cos\frac{a-b}{q}p\pi&\cos\frac{b-a}{q}p\pi&\cos\frac{a+b}{q}p\pi\\\
0&0&\sin\frac{b-a}{q}p\pi&\sin\frac{a+b}{q}p\pi\end{array}\right|\hskip
6.0pt\text{(column operations)}$ $\displaystyle=$ $\displaystyle
4\left|\begin{array}[]{ll}\cos\frac{a+b}{q}\pi&\cos\frac{a-b}{q}\pi\\\
\cos\frac{a+b}{q}p\pi&\cos\frac{a-b}{q}p\pi\end{array}\right|\cdot\left|\begin{array}[]{ll}\sin\frac{a-b}{q}\pi&\sin\frac{a+b}{q}\pi\\\
\sin\frac{a-b}{q}p\pi&\sin\frac{a+b}{q}p\pi\end{array}\right|$
$\displaystyle=$
$\displaystyle\textstyle{4~{}(2\cos\frac{a}{q}\pi\cos\frac{b}{q}\pi\cdot\sin\frac{ap}{q}\pi\sin\frac{bp}{q}\pi-2\sin\frac{a}{q}\pi\sin\frac{b}{q}\pi\cdot\cos\frac{ap}{q}\pi\cos\frac{bp}{q}\pi)}$
$\displaystyle\textstyle{~{}(2\sin\frac{a}{q}\pi\cos\frac{b}{q}\pi\cdot\sin\frac{bp}{q}\pi\cos\frac{ap}{q}\pi-2\sin\frac{b}{q}\pi\cos\frac{a}{q}\pi\cdot\sin\frac{ap}{q}\pi\cos\frac{bp}{q}\pi)}~{},$
so we only need to prove
$\tan\frac{ap\pi}{q}\tan\frac{bp\pi}{q}\neq\tan\frac{a\pi}{q}\tan\frac{b\pi}{q}\hskip
10.0pt;\hskip
10.0pt\tan\frac{a\pi}{q}\tan\frac{bp\pi}{q}\neq\tan\frac{b\pi}{q}\tan\frac{ap\pi}{q}$
(provided all these tangents are finite). Since $ap-\alpha
q=a^{\prime}\cdot\sigma(Q-A)$ (where $\sigma$ is the sign function) and $\tan$
is $\pi$-periodic,
$\tan\frac{ap\pi}{q}=\tan\frac{ap-\alpha
q}{q}\pi=\sigma(Q-A)\tan\frac{a^{\prime}}{q}\pi$
and similarly $\tan\frac{bp\pi}{q}=\sigma(Q-B)\tan\frac{b^{\prime}}{q}\pi$.
Since $Q$ lies between $A$ and $B$, the signs of $Q-A$ and $Q-B$ are opposite,
so we only need to prove
(5)
$\tan\frac{a^{\prime}\pi}{q}\tan\frac{b^{\prime}\pi}{q}\neq-\tan\frac{a\pi}{q}\tan\frac{b\pi}{q}\hskip
10.0pt;\hskip
10.0pt\tan\frac{a\pi}{q}\tan\frac{b^{\prime}\pi}{q}\neq-\tan\frac{b\pi}{q}\tan\frac{a^{\prime}\pi}{q}~{}.$
(All these tangents _are_ finite, by Proposition 5.) If
$a,a^{\prime},b,b^{\prime}\leq\frac{q}{2}$, then all the values of “tan” in
(5) are positive, which yields the result.
If one of $a,a^{\prime},b,b^{\prime}$ is larger than $\frac{q}{2}$, say
$b>\frac{q}{2}$, then $a^{\prime}b+b^{\prime}a=q$ requires $a^{\prime}=1$,
which entails $a\geq 2$ (because $A$ is not a Farey neighbor of both $Q$ and
$\infty$), and $b^{\prime}\geq 2$ (because $A$ and $B$ are not both Farey
neighbors of $Q$). We have $ab^{\prime}=q-b<\frac{q}{2}$ and
$b=\frac{q-ab^{\prime}}{a^{\prime}}=q-ab^{\prime}$. Therefore the first
inequality of (5) can be written
$\tan\frac{\pi}{q}\tan\frac{b^{\prime}\pi}{q}\neq\tan\frac{a\pi}{q}\tan\frac{ab^{\prime}\pi}{q}~{},$
which is clearly true (both members are positive, but the right one is larger,
factor-wise, because $a\geq 2$).
Similarly, the second inequality of (5) becomes
$\tan\frac{a\pi}{q}\tan\frac{b^{\prime}\pi}{q}\neq\tan\frac{ab^{\prime}\pi}{q}\tan\frac{\pi}{q}$
(all values of “$\tan$” are still positive), i.e.
$\frac{\tan\frac{a\pi}{q}}{\tan\frac{\pi}{q}}\neq\frac{\tan\frac{ab^{\prime}\pi}{q}}{\tan\frac{b^{\prime}\pi}{q}}~{}.$
Notice that without the “$\tan$’s”, this would be an identity. To see that the
right member is larger, it is therefore enough to make sure that the function
$g:u\mapsto\frac{\tan u}{\tan(u/a)}$ is increasing on $(0,\frac{\pi}{2})$.
Computation yields
$g^{\prime}(u)=\frac{\sin(2u/a)-\sin(2u)/a}{2\sin^{2}(u/a)\cos^{2}u}~{}:$
since $a\geq 2$, the numerator is clearly positive, by strict concavity of
$\sin$ on $[0,\pi]$.
If instead of $b$ another term of $a,a^{\prime},b,b^{\prime}$ is larger than
$\frac{q}{2}$, then we can apply the same argument, up to permuting
$a,a^{\prime},b,b^{\prime}$. ∎
### 5.2. Candidate faces are faces of the convex hull
We must now show that if $\rho:\mathbb{R}^{4}\rightarrow\mathbb{R}$ is some
linear form that takes the same value $Z>0$ on each column vector
$x_{0},x_{a},x_{b},x_{a+b}$ (i.e.
$\iota(\tau_{0}),\iota(\tau_{a}),\iota(\tau_{b}),\iota(\tau_{a+b})$) of the
matrix $M$ from (1), then $\rho\circ\iota(\tau_{k})<Z$ for any $k\in\llbracket
0,q-1\rrbracket\smallsetminus\\{0,a,b,a+b\\}$. This will be done by showing
_via_ Corollary 10 that $(\rho\circ\iota)^{-1}[Z,+\infty)$ is (once lifted to
$\mathbb{R}^{2}$) a convex region of the type seen in Proposition 8.
An elementary computation shows that in coordinates,
(6)
$\left\\{\begin{array}[]{rcl}\rho&=&(-1)^{\alpha+\beta}\left(\begin{array}[]{r}-\cos\frac{a+b}{q}\pi\sin\frac{ap\pi}{q}\sin\frac{bp\pi}{q}\\\
-\sin\frac{a+b}{q}\pi\sin\frac{ap\pi}{q}\sin\frac{bp\pi}{q}\\\
\cos\frac{a+b}{q}p\pi\sin\frac{a\pi}{q}\sin\frac{b\pi}{q}\\\
\sin\frac{a+b}{q}p\pi\sin\frac{a\pi}{q}\sin\frac{b\pi}{q}\end{array}\right)^{t}=:\left(\begin{array}[]{l}U\\\
U^{\prime}\\\ V\\\ V^{\prime}\end{array}\right)^{t}\\\ &&\\\
Z&=&(-1)^{\alpha+\beta}\left(\cos\frac{a+b}{q}p\pi\sin\frac{a\pi}{q}\sin\frac{b\pi}{q}-\cos\frac{a+b}{q}\pi\sin\frac{ap\pi}{q}\sin\frac{bp\pi}{q}\right)\\\
&=&\frac{(-1)^{\alpha+\beta}}{2}\left(\cos\frac{a+b}{q}p\pi\cos\frac{a-b}{q}\pi-\cos\frac{a+b}{q}\pi\cos\frac{a-b}{q}p\pi\right)\end{array}\right.$
will do ($Z$ will turn out to be positive by Claim 12 below; so far we only
know $Z\neq 0$ by Proposition 11). The notation $U,U^{\prime},V,V^{\prime}$ is
made to fit Lemma 9. Define
$\left\\{\begin{array}[]{rclcl}U^{\prime\prime}&:=&\sqrt{U^{2}+U^{\prime
2}}&=&|\sin\frac{ap\pi}{q}\sin\frac{bp\pi}{q}|>0\\\
V^{\prime\prime}&:=&\sqrt{V^{2}+V^{\prime
2}}&=&|\sin\frac{a\pi}{q}\sin\frac{b\pi}{q}|>0~{}.\end{array}\right.$
###### Claim 12.
The point $c$ of Definition 7 is the absolute maximum of $\rho\circ\iota$ on
the torus $\mathbb{T}$. Moreover,
$Z=\cos\frac{x^{\prime}}{q}\pi\cos\frac{y}{q}\pi-\cos\frac{x}{q}\pi\cos\frac{y^{\prime}}{q}\pi~{},$
$Z$ is positive, and one has:
$|V^{\prime\prime}-U^{\prime\prime}|<Z<V^{\prime\prime}+U^{\prime\prime}$.
This claim proves Theorem 1. Indeed, assume the claim, and let $H$ denote
$[Z,+\infty)$. Let $\overline{\pi}$ denote the natural projection
$\mathbb{R}^{2}\rightarrow\mathbb{T}$. By Corollary 10, the level curve
$(\rho\circ\iota\circ\overline{\pi})^{-1}(Z)\subset\mathbb{R}^{2}$ contains a
striclty convex closed curve $\gamma$ centered around $\overline{c}$,
contained in the square $C:=\overline{c}+(-\pi,\pi)^{2}$ and passing through
the representatives of $\tau_{0},\tau_{a},\tau_{b},\tau_{a+b}$ contained in
$C$. By Proposition 6, these representatives are the vertices $0,u,v,u+v$ of
the fundamental parallelogram $D$. Corollary 10 and Proposition 8 then yield
the result: $(\rho\circ\iota)^{-1}(H)$ contains no other points $\tau_{k}$
than $\tau_{0},\tau_{a},\tau_{b},\tau_{a+b}$.
###### Proof.
(Claim 12). The maximum of $\rho\circ\iota$ on $\mathbb{T}$ is clearly
$U^{\prime\prime}+V^{\prime\prime}$. Since
$\iota(c)=\left(\begin{array}[]{c}\cos\frac{a+b}{q}\pi\\\
\sin\frac{a+b}{q}\pi\\\ \cos[p\frac{a+b}{q}-(\alpha+\beta)]\pi\\\
\sin[p\frac{a+b}{q}-(\alpha+\beta)]\pi\end{array}\right)~{},$
we can compute
$\displaystyle\rho\circ\iota(c)$ $\displaystyle=$
$\displaystyle(-1)^{\alpha+\beta}\left(-\sin\frac{ap\pi}{q}\pi\sin\frac{bp\pi}{q}\pi+(-1)^{\alpha+\beta}\sin\frac{a\pi}{q}\sin\frac{b\pi}{q}\right)$
$\displaystyle=$ $\displaystyle-\sin\frac{ap-\alpha q}{q}\pi\sin\frac{bp-\beta
q}{q}\pi+\sin\frac{a}{q}\pi\sin\frac{b}{q}\pi$ $\displaystyle=$
$\displaystyle\sin\frac{A\wedge Q}{q}\pi\sin\frac{B\wedge
Q}{q}\pi+\sin\frac{a}{q}\pi\sin\frac{b}{q}\pi$ $\displaystyle=$
$\displaystyle\sin\frac{a^{\prime}}{q}\pi\sin\frac{b^{\prime}}{q}\pi+\sin\frac{a}{q}\pi\sin\frac{b}{q}\pi$
because $ap-\alpha q$ and $bp-\beta q$ have opposite signs ($Q$ lies between
$A$ and $B$). Both terms in the last expression are positive since
$a,a^{\prime},b,b^{\prime}\in\llbracket 1,q-1\rrbracket$. In fact, since
$V^{\prime\prime}=\left|\sin\frac{a\pi}{q}\sin\frac{b\pi}{q}\right|=\sin\frac{a\pi}{q}\sin\frac{b\pi}{q}$
and
$U^{\prime\prime}=\left|\sin\frac{ap\pi}{q}\sin\frac{bp\pi}{q}\right|=\left|\sin\frac{ap-\alpha
q}{q}\pi\sin\frac{bp-\beta
q}{q}\pi\right|=\sin\frac{a^{\prime}\pi}{q}\sin\frac{b^{\prime}\pi}{q}~{},$
we have shown that $\rho\circ\iota(c)=U^{\prime\prime}+V^{\prime\prime}$, the
absolute maximum of $\rho\circ\iota$.
The computation of $Z$ follows similar lines: in the second expression for $Z$
in (6), notice that the first and last cosines can be written
$(-1)^{\alpha+\beta}\cos\frac{a+b}{q}p\pi=\cos\frac{(a+b)p-(\alpha+\beta)q}{\pi}=\cos\frac{X\wedge
Q}{q}\pi~{};$
$(-1)^{\alpha-\beta}\cos\frac{a-b}{q}p\pi=\cos\frac{(a-b)p-(\alpha-\beta)q}{\pi}=\cos\frac{Y\wedge
Q}{q}\pi$
(using Proposition 4). Together with $\frac{a+b}{q}=\frac{x}{q}$ and
$\frac{a-b}{q}=\frac{\pm y}{q}$, this yields the desired expression of
$Z=\cos\frac{x^{\prime}}{q}\pi\cos\frac{y}{q}\pi-\cos\frac{x}{q}\pi\cos\frac{y^{\prime}}{q}\pi$.
The upper bound on $Z$ is obvious from the first expression of $Z$ in (6). We
now focus on the lower bound (which will also imply $Z>0$), i.e. we aim to
show
(7)
$\cos\frac{x^{\prime}}{q}\pi\cdot\cos\frac{y}{q}\pi-\cos\frac{x}{q}\pi\cdot\cos\frac{y^{\prime}}{q}\pi>2\left|\sin\frac{a^{\prime}}{q}\pi\cdot\sin\frac{b^{\prime}}{q}\pi-\sin\frac{a}{q}\pi\cdot\sin\frac{b}{q}\pi\right|~{}.$
By Proposition 4, the right member of (7) can be written
$\left|\left(\cos\frac{x^{\prime}}{q}\pi-\cos\frac{y^{\prime}}{q}\pi\right)-\left(\cos\frac{y}{q}\pi-\cos\frac{x}{q}\pi\right)\right|~{};$
therefore we are down to proving the two identities
$\left\\{\begin{array}[]{l}\displaystyle{\left(\cos\frac{x^{\prime}}{q}\pi-1\right)\cdot\left(\cos\frac{y}{q}\pi+1\right)~{}>~{}\left(\cos\frac{x}{q}\pi+1\right)\cdot\left(\cos\frac{y^{\prime}}{q}\pi-1\right)}\\\
\\\
\displaystyle{\left(\cos\frac{x^{\prime}}{q}\pi+1\right)\cdot\left(\cos\frac{y}{q}\pi-1\right)~{}>~{}\left(\cos\frac{x}{q}\pi-1\right)\cdot\left(\cos\frac{y^{\prime}}{q}\pi+1\right)~{}.}\end{array}\right.$
Using $\cos t+1=2\cos^{2}\frac{t}{2}$ and $\cos t-1=-2\sin^{2}\frac{t}{2}$,
this in turn amounts to
$\left\\{\begin{array}[]{l}\displaystyle{\sin\left(\frac{x^{\prime}}{q}\right)\frac{\pi}{2}\cdot\cos\left(\frac{y}{q}\right)\frac{\pi}{2}~{}<~{}\cos\left(\frac{x}{q}\right)\frac{\pi}{2}\cdot\sin\left(\frac{y^{\prime}}{q}\right)\frac{\pi}{2}}\\\
\\\
\displaystyle{\cos\left(\frac{x^{\prime}}{q}\right)\frac{\pi}{2}\cdot\sin\left(\frac{y}{q}\right)\frac{\pi}{2}~{}<~{}\sin\left(\frac{x}{q}\right)\frac{\pi}{2}\cdot\cos\left(\frac{y^{\prime}}{q}\right)\frac{\pi}{2}~{},}\end{array}\right.$
or equivalently
(8)
$\left\\{\begin{array}[]{rcll}\frac{\displaystyle{\sin\left(\frac{x^{\prime}}{q}\right)\frac{\pi}{2}}}{\displaystyle{\sin\left(\frac{y^{\prime}}{q}\right)\frac{\pi}{2}}}&<&\frac{\displaystyle{\sin\left(\frac{q-x}{q}\right)\frac{\pi}{2}}}{\displaystyle{\sin\left(\frac{q-y}{q}\right)\frac{\pi}{2}}}&\hskip
20.0pt(i)\\\ \\\
\frac{\displaystyle{\sin\left(\frac{y}{q}\right)\frac{\pi}{2}}}{\displaystyle{\sin\left(\frac{x}{q}\right)\frac{\pi}{2}}}&<&\frac{\displaystyle{\sin\left(\frac{q-y^{\prime}}{q}\right)\frac{\pi}{2}}}{\displaystyle{\sin\left(\frac{q-x^{\prime}}{q}\right)\frac{\pi}{2}}}&\hskip
20.0pt(ii).\end{array}\right.$
To prove (8)-$(i)$ and (8)-$(ii)$, we will use
###### Proposition 13.
If $\displaystyle{0<s<t<\frac{\pi}{2}}$ and
$\displaystyle{0<s^{\prime}<t^{\prime}<\frac{\pi}{2}}$ satisfy $s<s^{\prime}$
and $\displaystyle{\frac{s}{t}\leq\frac{s^{\prime}}{t^{\prime}}}$, then
$\displaystyle{\frac{\sin s}{\sin t}<\frac{\sin s^{\prime}}{\sin
t^{\prime}}}$.
###### Proof.
Up to decreasing $t$, it is clearly enough to treat the case
$\frac{s}{t}=\frac{s^{\prime}}{t^{\prime}}=\frac{1-\lambda}{1+\lambda}$ (where
$0<\lambda<1$). The result then follows from the fact that
$f(u)=\frac{\sin(1-\lambda)u}{\sin(1+\lambda)u}$ is increasing on
$(0,\frac{\pi}{2(1+\lambda)}]$, which can be seen by computing
$f^{\prime}(u)=\frac{\sin(2\lambda
u)-\lambda\sin(2u)}{\sin^{2}(1+\lambda)u}~{}:$
here the numerator is positive by strong concavity of $\sin$ on
$[0,\frac{\pi}{1+\lambda}]$. ∎
We now prove (8)-$(i)$: by Proposition 13, it is enough to check
$0<x^{\prime}<y^{\prime}<q\text{ and }0<y<x<q$
(which are obvious from Proposition 4), plus
(9) $x^{\prime}<q-x~{}\text{ and
}~{}\frac{x^{\prime}}{y^{\prime}}\leq\frac{q-x}{q-y}~{}.$
The first inequality of (9) amounts, by Proposition 4, to
$|a-b|+(a+b)<a^{\prime}b+b^{\prime}a$
which can be written
$(a^{\prime}-1)(b\pm 1)+(b^{\prime}-1)(a\mp 1)>0~{}.$
If $a^{\prime}$ and $b^{\prime}$ are $>1$, then at least one of the products
in the left member is positive, and we are done. If $a^{\prime}=1$, then
$b^{\prime}>1$ (because $A,B$ are not both Farey neighbors of $Q$ in the
assumptions of Theorem 1) and $a>1$ (because $Q,\infty$ have no common Farey
neighbors, i.e. $p\notin\\{1,q-1\\}$) and we are also done. If $b^{\prime}=1$,
the argument is the same, exchanging $(A,a,a^{\prime})$ and
$(B,b,b^{\prime})$.
The second inequality of (9) amounts to
$q(y^{\prime}-x^{\prime})\geq y^{\prime}x-x^{\prime}y$
which by Proposition 4 can also be written
$y^{\prime}-x^{\prime}\geq\frac{(a^{\prime}+b^{\prime})(a+b)-|(a^{\prime}-b^{\prime})(a-b)|}{a^{\prime}b+b^{\prime}a}=:H~{}.$
Here the left member is at least 2 : indeed, by Proposition 4 it can be
written
$a^{\prime}+b^{\prime}-|a^{\prime}-b^{\prime}|=2\inf\\{a^{\prime},b^{\prime}\\}~{}.$
The right member $H$, however, is at most $2$ : indeed,
$\displaystyle 2-H$ $\displaystyle=$
$\displaystyle\frac{a^{\prime}(2b-a-b)+b^{\prime}(2a-a-b)+|(a^{\prime}-b^{\prime})(a-b)|}{a^{\prime}b+b^{\prime}a}$
$\displaystyle=$
$\displaystyle\frac{(a^{\prime}-b^{\prime})(b-a)+|(a^{\prime}-b^{\prime})(a-b)|}{a^{\prime}b+b^{\prime}a}$
and the numerator has the form $u+|u|\geq 0$. This finishes the proof of
(8)-$(i)$.
The proof of (8)-$(ii)$ is identical with that of (8)-$(i)$, exchanging
$(a,b,x,y)$ with $(a^{\prime},b^{\prime},y^{\prime},x^{\prime})$. Claim 12,
and therefore Theorem 1, are proved. ∎
## 6\. General finite subgroups of
$\iota(\mathbb{T})\subset{\mathbb{C}^{*}}^{2}$
In this last section, let $\Gamma$ be _any_ finite subgroup of
$\mathbb{T}=(\mathbb{S}^{1})^{2}=(\mathbb{R}/2\pi\mathbb{Z})^{2}$. There
exists a unique rational $Q=\frac{p}{q}\in[0,1)$ (here in reduced form) and a
unique pair $(\mu,\nu)\in\mathbb{Z}_{>0}^{2}$ such that $\Gamma$ is the
preimage of $\\{\tau_{k}=(\frac{k}{q},\frac{kp}{q})\\}_{0\leq k<q}$ under
$\begin{array}[]{rrcl}&\mathbb{T}&\longrightarrow&\mathbb{T}\\\
\psi_{\mu\nu}~{}:&(s,t)&\mapsto&(\mu s,\nu t)~{}.\end{array}$
Indeed, $\mu$ (resp. $\nu$) is just the cardinality of
$\Gamma\cap(\mathbb{S}^{1}\times\\{0\\})$ (resp.
$\Gamma\cap(\\{0\\}\times\mathbb{S}^{1})$); the order of $\Gamma$ is
$q\mu\nu$. The case $\frac{p}{q}=0$ can be put aside: it corresponds to
$\iota(\Gamma)\subset\mathbb{R}^{4}$ being (the vertices of) the Cartesian
product of a regular $\mu$-gon with a regular $\nu$-gon (the 3-dimensional
faces are then regular prisms; degeneracies occur if $\mu\leq 2$ or $\nu\leq
2$). The case $\mu=\nu=1$ was treated in the previous sections, including the
discussion of degeneracies when $p\equiv 0\text{ or }\pm 1~{}[\text{mod }q]$.
It is easy to see that if $\mu=1<\nu$ (resp. $\nu=1<\mu$) and
$\frac{p}{q}=\frac{1}{2}$, then $\iota(\Gamma)$ is contained in a
$3$-dimensional subspace of $\mathbb{R}^{4}$ — in fact, $\iota(\Gamma)$ is the
vertex set of an antiprism with $\nu$-gonal (resp. $\mu$-gonal) basis, which
in turn degenerates to a tetrahedron when $\nu=2$ (resp. $\mu=2$). Therefore,
we can make
###### Assumption 14.
Until the end of this section,
* •
at least one of the positive integers $\mu,\nu$ is larger than one;
* •
the rational $\frac{p}{q}\in(0,1)$ is not $\frac{1}{2}$ when $\mu=1$ or
$\nu=1$.
Then, we claim that faces of the convex hull of
$\iota(\gamma)\subset\mathbb{R}^{4}$ come in three types:
1. (1)
If $A,B\in[0,1]$ are rationals satisfying the hypotheses of Theorem 1, then
there is a tetrahedron spanned by the images under
$\iota:\mathbb{T}\rightarrow\mathbb{R}^{4}$ of
$\textstyle{\left(\frac{0}{q\mu},\frac{0}{q\nu}\right),\left(\frac{a}{q\mu}2\pi,\frac{ap-\alpha
q}{q\nu}2\pi\right),\left(\frac{b}{q\mu}2\pi,\frac{bp-\beta
q}{q\nu}2\pi\right),\left(\frac{a+b}{q\mu}2\pi,\frac{(a+b)p-(\alpha+\beta)q}{q\nu}2\pi\right),}$
which are clearly four points of
$\Gamma=\psi_{\mu\nu}^{-1}\\{\tau_{1},\dots,\tau_{q}\\}$. They form a
parallelogram whose center is
$c=\left(\frac{a+b}{q\mu}\pi,\frac{(a+b)p-(\alpha+\beta)q}{q\nu}\pi\right)$.
2. (2)
If $\nu>1$, add an extra tetrahedron of the type above for the pair
$\\{A,B\\}=\\{\frac{0}{1},\frac{1}{1}\\}$ (this was ruled out in Theorem 1
because $A,B$ were not allowed both to be Farey neighbors of
$\infty=\frac{1}{0}$). Similarly, if $\mu>1$, add an extra tetrahedron of the
type above for $\\{A,B\\}$ equal to the unique pair of Farey neighbors
$\frac{\alpha}{a},\frac{\beta}{b}$ such that
$\frac{\alpha+\beta}{a+b}=\frac{p}{q}$. (If $\frac{p}{q}=\frac{1}{2}$ and
$\mu,\nu\geq 2$, these two “extra” tetrahedra are in fact the same one.)
3. (3)
If $\nu>1$, add an extra cell spanned by the $2\nu$ vertices images under
$\iota$ of
$\textstyle{\left\\{\left.\left(0,\frac{k}{\nu}2\pi\right)~{}\right|~{}0\leq
k<\nu\left\\}\,\cup\left\\{\left.\left(\frac{1}{q\mu}2\pi,\frac{p+kq}{q\nu}2\pi\right)~{}\right|~{}0\leq
k<\nu\right\\}\right.\right..}$
If $\nu>2$, this cell is an antiprism with regular $\nu$-gonal basis; it
degenerates to a tetrahedron when $\nu=2$. Similarly, if $\mu>1$, add an extra
cell spanned by the $2\mu$ vertices images under $\iota$ of
$\textstyle{\left\\{\left.\left(\frac{k}{\mu}2\pi,0\right)~{}\right|~{}0\leq
k<\mu\right\\}\cup\left\\{\left.\left(\frac{p+kq}{q\mu}2\pi,\frac{1}{q\nu}2\pi\right)~{}\right|~{}0\leq
k<\mu\right\\}~{}.}$
Actually, cells of type (3) degenerate to segments when $\mu,\nu=1$.
###### Observation 15.
Let $\\{A,B\\}\subset[0,1]$ be a pair of rationals describing a face of type
(1) or (2), define $a,a^{\prime},b,b^{\prime}\in\mathbb{Z}_{>0}$ and
$x,x^{\prime},y,y^{\prime}\in\mathbb{Z}_{\geq 0}$ in the usual way, and bear
in mind Proposition 4. Then,
* •
having $a=b=1$ (i.e. $y=0$, i.e. $y^{\prime}=q$) is only allowed if $\nu>1$;
* •
having $a^{\prime}=b^{\prime}=1$ (i.e. $x^{\prime}=0$, i.e. $x=q$) is only
allowed if $\mu>1$;
* •
Proposition 5 no longer holds: some of $a,a^{\prime},b,b^{\prime}$ may be
equal to $\frac{q}{2}$.
First we prove that cells of types (1)–(2)–(3), pushed forward by $\Gamma$,
are combinatorially glued face-to-face (i.e. an analogue of Theorem 3 holds).
The proof exactly shadows that of Theorem 3 (lifting to the cover
$\psi_{\mu\nu}$), except that when $\mu>1$ (resp. $\nu>1$), we must check that
faces of type (2)–(3) also fit together correctly.
Assume $\nu>1$: the “first” tetrahedron (of type (2) in the list),
corresponding to $\\{A,B\\}=\\{\frac{0}{1},\frac{1}{1}\\}$, is spanned (up to
action of $\Gamma$) by the images under $\iota$ of
$\left(\frac{0}{q\mu},\frac{0}{q\nu}\right),\left(\frac{1}{q\mu}2\pi,\frac{p}{q\nu}2\pi\right),\left(\frac{1}{q\mu}2\pi,\frac{p-q}{q\nu}2\pi\right),\left(\frac{2}{q\mu}2\pi,\frac{2p-q}{q\nu}2\pi\right)~{}.$
The subfaces obtained by dropping the second or third of these four vertices
also belong to faces of type (1) (with
$\\{A,B\\}=\\{\frac{0}{1},\frac{1}{2}\\}$ or $\\{\frac{1}{2},\frac{1}{1}\\}$),
by the argument of the proof of Theorem 3. The face obtained by dropping the
last vertex is clearly a face of the $\nu$-antiprism of type (3). The face
obtained by dropping the first vertex is clearly a face of that same
antiprism, shifted by $(\frac{1}{q\mu},\frac{p}{q\nu})\in\Gamma$. The
antiprism and its shift, finally, are glued base-to-base along
$\iota\left\\{(\frac{1}{q\mu}2\pi,\frac{p+kq}{q\nu}2\pi)~{}|~{}0\leq
k<\nu\right\\}$.
A similar argument holds when $\mu>1$ near the “end” of the sequence of
tetrahedra: again, this just amounts to swapping $Q$ and $\infty$.
Next, we proceed to show that the candidate faces of types (1)–(2)–(3) are
indeed faces of the convex hull of $\iota(\Gamma)$.
### 6.1. Faces of type (3)
The vertices $\left(\\!\\!\begin{array}[]{c}1\\\ 0\\\ \cos\frac{2k\pi}{\nu}\\\
\sin\frac{2k\pi}{\nu}\end{array}\\!\\!\right)_{\\!0\leq k<\nu}$ and
$\left(\\!\\!\begin{array}[]{c}\cos\frac{2\pi}{q\mu}\\\
\sin\frac{2\pi}{q\mu}\\\ \cos\frac{2\pi(p+kq)}{q\nu}\\\
\sin\frac{2\pi(p+kq)}{q\nu}\end{array}\\!\\!\right)_{\\!0\leq k<\nu}$ form two
regular $\nu$-gons contained in _distinct_ planes parallel to
$\\{(0,0)\\}\times\mathbb{R}^{2}$, and are not translates of each other (they
are off by a rotation of angle
$2\pi\frac{p}{q\nu}\notin\frac{2\pi}{\nu}\mathbb{Z}$): this shows that they
are the vertices of a convex, non–degenerate antiprism. Moreover, these $2\nu$
vertices clearly maximize the linear form
$\rho=(\cos\frac{\pi}{q\mu},\sin\frac{\pi}{q\mu},0,0)$ (that is a purely
2-dimensional statement) and therefore span a face of the convex hull of
$\iota(\Gamma)$. Similarly, the vertices of the other antiprism maximize
$\rho^{\prime}=(0,0,\cos\frac{\pi}{q\nu},\sin\frac{\pi}{q\nu})$.
### 6.2. Faces of type (1) and (2)
Let $\\{A,B\\}=\\{\frac{\alpha}{a},\frac{\beta}{b}\\}$ be as in type (1) or
(2); the candidate face now is spanned by the column vectors of
$M:=\left(\begin{array}[]{cccc}1&\cos\frac{a}{\mu q}2\pi&\cos\frac{b}{\mu
q}2\pi&\cos\frac{a+b}{\mu q}2\pi\\\ 0&\sin\frac{a}{\mu q}2\pi&\sin\frac{b}{\mu
q}2\pi&\sin\frac{a+b}{\mu q}2\pi\\\ 1&\cos\frac{ap-\alpha q}{\nu
q}2\pi&\cos\frac{bp-\alpha q}{\nu q}2\pi&\cos\frac{(a+b)p-(\alpha+\beta)q}{\nu
q}2\pi\\\ 0&\sin\frac{ap-\alpha q}{\nu q}2\pi&\sin\frac{bp-\alpha q}{\nu
q}2\pi&\sin\frac{(a+b)p-(\alpha+\beta)q}{\nu q}2\pi\end{array}\right)~{}.$
We now transpose the argument of Section 5. Generally speaking, the presence
of $\mu,\nu\geq 1$ makes _even more true_ any given inequality that we have to
check, but we must check it also for the extra tetrahedra of type (2): hence
some additional care.
### Candidate faces are non-degenerate
Rotating the first two coordinates by $\frac{-a-b}{\mu q}\pi$ and the last two
by $\frac{-(a+b)p+(\alpha+\beta)q}{\nu q}\pi=\frac{-(ap-\alpha q)-(bp-\beta
q)}{\nu q}\pi$, using the method of Section 5.1, and replacing
$\frac{(ap-\alpha q)\pm(bp-\beta q)}{\nu q}$ with $\frac{a^{\prime}\mp
b^{\prime}}{\nu q}\cdot\sigma(ap-\alpha q)$, compute
$\begin{array}[]{rcl}\det M&=&\pm 4\left|\begin{array}[]{cc}\cos\frac{a+b}{\mu
q}\pi&\cos\frac{a-b}{\mu q}\pi\\\ \cos\frac{a^{\prime}-b^{\prime}}{\nu
q}\pi&\cos\frac{a^{\prime}+b^{\prime}}{\nu
q}\pi\end{array}\right|\cdot\left|\begin{array}[]{cc}\sin\frac{a-b}{\mu
q}\pi&\sin\frac{a+b}{\mu q}\pi\\\ \sin\frac{a^{\prime}+b^{\prime}}{\nu
q}\pi&\sin\frac{a^{\prime}-b^{\prime}}{\nu q}\pi\end{array}\right|\\\ &=&\pm
16(\cos\frac{a\pi}{\mu q}\cos\frac{b\pi}{\mu q}\sin\frac{a^{\prime}\pi}{\nu
q}\sin\frac{b^{\prime}\pi}{\nu q}+\sin\frac{a\pi}{\mu q}\sin\frac{b\pi}{\mu
q}\cos\frac{a^{\prime}\pi}{\nu q}\cos\frac{b^{\prime}\pi}{\nu q})\\\
&&\cdot\,(\sin\frac{a\pi}{\mu q}\cos\frac{b\pi}{\mu
q}\sin\frac{b^{\prime}\pi}{\nu q}\cos\frac{a^{\prime}\pi}{\nu
q}+\sin\frac{b\pi}{\mu q}\cos\frac{a\pi}{\mu q}\sin\frac{a^{\prime}\pi}{\nu
q}\cos\frac{b^{\prime}\pi}{\nu q})~{}.\end{array}$
To follow up the method of Section 5.1, we would divide both factors of $\det
M$ by
$\textstyle{H:=\cos\frac{a\pi}{\mu q}\cos\frac{b\pi}{\mu
q}\cos\frac{a^{\prime}\pi}{\nu q}\cos\frac{b^{\prime}\pi}{\nu q}}~{}:$
however, that number can be $0$. In that case, each factor of $\det M$ has a
vanishing summand. Let us prove that the other summand is then nonzero, so
that $\det M\neq 0$. (Note that the _sines_ in $\det M$ never vanish, only the
_cosines_ may.)
If $\cos\frac{a\pi}{\mu q}=0$, then $\mu=1$ and $a=\frac{q}{2}$. This implies
$\nu>1$ by Assumption 14, so the first factor of $\det M$ has a nonzero second
summand. Moreover, the second factor of $\det M$ has a nonzero first summand
unless $\cos\frac{b\pi}{\mu q}=0$ i.e. $b=\frac{q}{2}=a$. But $a,b$ are
coprime, so we then have $a=b=1$ and $q=2$ and $\frac{p}{q}=\frac{1}{2}$,
which is ruled out when $\mu=1$ (Assumption 14). If another factor of $H$
vanishes, the argument is similar up to switching $(a,a^{\prime})$ with
$(b,b^{\prime})$, and/or $(a,b,\mu)$ with $(a^{\prime},b^{\prime},\nu)$. In
any case, $M$ is invertible. On the other hand, if $H\neq 0$, we must make
sure that
(10) $\textstyle{\tan\frac{a^{\prime}\pi}{\nu q}\tan\frac{b^{\prime}\pi}{\nu
q}\neq-\tan\frac{a\pi}{\mu q}\tan\frac{b\pi}{\mu q}}~{}\text{ ;
}~{}\textstyle{\tan\frac{a\pi}{\mu q}\tan\frac{b^{\prime}\pi}{\nu
q}\neq-\tan\frac{b\pi}{\mu q}\tan\frac{a^{\prime}\pi}{\nu q}}~{}.$
If $\mu>1$ and $\nu>1$, all tangents in (10) are positive, so (10) holds.
Suppose $\mu=1<\nu$. Then at most one of $a^{\prime},b^{\prime}$ is equal to
$1$ (Observation 15). If $a,b<\frac{q}{2}$, the members in (10) have opposite
signs. If $a>\frac{q}{2}$, since $a^{\prime}b+b^{\prime}a=q$, we have
$b^{\prime}=1$ which implies $a^{\prime}>1$ and $a=q-a^{\prime}b$. Thus, (10)
becomes
$\textstyle{\tan\frac{a^{\prime}\pi}{\nu q}\tan\frac{\pi}{\nu
q}\neq\tan\frac{a^{\prime}b\pi}{q}\tan\frac{b\pi}{q}~{}\text{ ;
}~{}\left.\tan\frac{a^{\prime}b\pi}{q}\right/\tan\frac{b\pi}{q}\neq\left.\tan\frac{a^{\prime}\pi}{\nu
q}\right/\tan\frac{\pi}{\nu q}~{}:}$
in the first inequality, even if $b=1$, the right member is larger because
$\nu>1$. In the second inequality, even if $b=1$, the method of Section 5.1
shows that the left member is larger because $\nu>1$ and $a^{\prime}>1$.
If $b>\frac{q}{2}$, the argument is the same, exchanging $(a,a^{\prime})$ with
$(b,b^{\prime})$. Finally, if $\nu=1<\mu$, the argument is again the same,
switching $(a,b,\mu)$ with $(a^{\prime},b^{\prime},\nu)$. Therefore, the
matrix $M$ is invertible and the candidate face is non-degenerate.
### Candidate faces are faces of the convex hull
Let us now prove that if a linear form $\rho=(U,U^{\prime},V,V^{\prime})$
takes the same value $Z>0$ on each column vector of $M$, then $\rho\circ\iota$
achieves its maximum on $\mathbb{T}$ at $c$ and
$|V^{\prime\prime}-U^{\prime\prime}|<Z<V^{\prime\prime}+U^{\prime\prime}$,
where $U^{\prime\prime}=\sqrt{U^{2}+U^{\prime 2}}$ and
$V^{\prime\prime}=\sqrt{V^{2}+V^{\prime 2}}$ (by the argument after Claim 12,
this will show that the candidate face is a face of the convex hull). An
elementary computation shows that
$\left\\{\begin{array}[]{rcl}\rho&=&\left(\begin{array}[]{r}-\cos\frac{a+b}{\mu
q}\pi\sin\frac{ap-\alpha q}{\nu q}\pi\sin\frac{bp-\beta q}{\nu q}\pi\\\
-\sin\frac{a+b}{\mu q}\pi\sin\frac{ap-\alpha q}{\nu q}\pi\sin\frac{bp-\beta
q}{\nu q}\pi\\\ \cos\frac{(ap-\alpha q)+(bp-\beta q)}{\nu
q}\pi\sin\frac{a}{\mu q}\pi\sin\frac{b}{\mu q}\pi\\\ \sin\frac{(ap-\alpha
q)+(bp-\beta q)}{\nu q}\pi\sin\frac{a}{\mu q}\pi\sin\frac{b}{\mu
q}\pi\end{array}\right)^{t}=:\left(\begin{array}[]{l}U\\\ U^{\prime}\\\ V\\\
V^{\prime}\end{array}\right)^{t}\\\ &&\\\ Z&=&\cos\frac{(ap-\alpha
q)+(bp-\beta q)}{\nu q}\pi\sin\frac{a\pi}{\mu q}\sin\frac{b\pi}{\mu
q}-\cos\frac{a+b}{\mu q}\pi\sin\frac{ap-\alpha q}{\nu q}\pi\sin\frac{bp-\beta
q}{\nu q}\pi\\\ &=&\frac{1}{2}\left(\cos\frac{x^{\prime}\pi}{\nu
q}\cos\frac{y\pi}{\mu q}-\cos\frac{x\pi}{\mu q}\cos\frac{y^{\prime}\pi}{\nu
q}\right)\end{array}\right.$
will do (the second expression of $Z$ follows from the first one and from the
fact that $(ap-\alpha q)(bp-\beta q)<0$ — again, the sign of $Z$ remains to be
checked). First,
$\begin{array}[]{rcl}\rho\circ\iota(c)&=&-\sin\frac{ap-\alpha q}{\nu
q}\pi\sin\frac{bp-\beta q}{\nu q}\pi+\sin\frac{a}{\mu q}\pi\sin\frac{b}{\mu
q}\pi\\\ &=&\sin\frac{a^{\prime}}{\nu q}\pi\sin\frac{b^{\prime}}{\nu
q}\pi+\sin\frac{a}{\mu q}\pi\sin\frac{b}{\mu
q}\pi=U^{\prime\prime}+V^{\prime\prime}\end{array}$
(again because $(ap-\alpha q)(bp-\beta q)<0$), so
$\underset{\mathbb{T}}{\max}(\rho\circ\iota)=\rho\circ\iota(c)$.
The upper bound $U^{\prime\prime}+V^{\prime\prime}$ for $Z$ is clear from its
first expression; the lower bound follows lines similar to the proof of Claim
12: we just need to check
$\textstyle{2Z=\cos\frac{x^{\prime}\pi}{\nu q}\cos\frac{y\pi}{\mu
q}-\cos\frac{x\pi}{\mu q}\cos\frac{y^{\prime}\pi}{\nu
q}>2\left|\sin\frac{a\pi}{\mu q}\sin\frac{b\pi}{\mu
q}-\sin\frac{a^{\prime}\pi}{\nu q}\sin\frac{b^{\prime}\pi}{\nu q}\right|~{}.}$
The right member being $|(\cos\frac{x^{\prime}}{\nu
q}\pi-\cos\frac{y^{\prime}}{\nu q}\pi)-(\cos\frac{y}{\mu
q}\pi-\cos\frac{x}{\mu q}\pi)|$, we only need
$\textstyle{(\cos\frac{x^{\prime}}{\nu q}\pi\pm 1)\cdot(\cos\frac{y}{\mu
q}\pi\mp 1)~{}>~{}(\cos\frac{x}{\mu q}\pi\mp 1)\cdot(\cos\frac{y^{\prime}}{\nu
q}\pi\pm 1)}$
which amounts to
(11) ${\frac{\sin\frac{x^{\prime}}{\nu
q}\cdot\frac{\pi}{2}}{\sin\frac{y^{\prime}}{\nu
q}\cdot\frac{\pi}{2}}<\frac{\sin\frac{\mu q-x}{\mu
q}\cdot\frac{\pi}{2}}{\sin\frac{\mu q-y}{\mu q}\cdot\frac{\pi}{2}}\hskip
6.0pt(i)\hskip 8.0pt\text{ and }~{}\frac{\sin\frac{y}{\mu
q}\cdot\frac{\pi}{2}}{\sin\frac{x}{\mu
q}\cdot\frac{\pi}{2}}<\frac{\sin\frac{\nu q-y^{\prime}}{\nu
q}\cdot\frac{\pi}{2}}{\sin\frac{\nu q-x^{\prime}}{\nu
q}\cdot\frac{\pi}{2}}\hskip 6.0pt(ii)~{}.}$
Let us focus on (11)-$(i)$. By Proposition 13, it is enough to check
$0<x^{\prime}<y^{\prime}<\nu q$ and $0<y<x<\mu q$ (which are clear from
Proposition 4: indeed, by Observation 15, we _may_ have $y^{\prime}=q$ but
then $\nu>1$; we _may_ have $x=q$ but then $\mu>1$), plus
(12) $\textstyle{\frac{x^{\prime}}{\nu}<q-\frac{x}{\mu}~{}\text{ and
}~{}\frac{x^{\prime}}{y^{\prime}}\leq\frac{\mu q-x}{\mu q-y}~{}.}$
The first inequality of (12) can be written
$\frac{|a^{\prime}-b^{\prime}|}{\nu}+\frac{a+b}{\mu}<a^{\prime}b+b^{\prime}a$,
or equivalently,
$\textstyle{(a^{\prime}-\frac{1}{\mu})\cdot(b\pm\frac{1}{\nu})+(b^{\prime}-\frac{1}{\mu})\cdot(a\mp\frac{1}{\nu})>0~{}.}$
If $\mu,\nu>1$ this is obvious. If $\mu=1<\nu$, then at least one of
$a^{\prime},b^{\prime}$ is larger than $1$ (Observation 15), and the product
where it appears is positive: done. If $\nu=1<\mu$, then at least one of $a,b$
is larger than one and we are also done.
The second inequality of (12) can be written
$\mu(y^{\prime}-x^{\prime})\geq\frac{(a+b)(a^{\prime}+b^{\prime})-|(a-b)(a^{\prime}-b^{\prime})|}{a^{\prime}b+b^{\prime}a}$.
As in the proof of Claim 12, the left member is
$2\mu\inf\\{a^{\prime},b^{\prime}\\}\geq 2$ while the right member is at most
$2$. The proof of (11)-$(ii)$ is identical to that of (11)-$(i)$, swapping
$(a,b,x,y)$ with $(a^{\prime},b^{\prime},y^{\prime},x^{\prime})$.
## References
* [A1] Sergei Anisov, _Geometrical spines of lens manifolds_ , Journal of the Lond. Math. Soc. 74, Issue 03 (2006), 799–816.
* [A2] Sergei Anisov, _Cut loci in lens manifolds_ , C. R. Math. Acad. Sci. Paris 342, No. 8 (2006), 595–600.
* [ASWY] Hirotaka Akiyoshi, Makoto Sakuma, Masaaki Wada,Yasushi Yamashita, _Punctured torus groups and 2-bridge knot groups I_ , Lec. Notes in Math. 1909, Springer (2007).
* [G1] François Guéritaud, _Géométrie hyperbolique effective et triangulations idéales canoniques en dimension trois_ , PhD thesis, 154 pages, Orsay (2006).
* [G2] François Guéritaud, _Triangulated cores of punctured-torus groups_ , J. Diff. Geom. 81 (2009), 91–142.
* [GS] François Guéritaud and Saul Schleimer, _Canonical triangulations of Dehn fillings_ , arXiv:math.GT/0801359, 37 pages, submitted.
* [JR] Bus Jaco, Hyam Rubinstein, _Layered triangulations of 3–manifolds_ , preprint (2006), 96 pages, available at http://www.math.okstate.edu/~jaco/ .
* [La] Marc Lackenby, _The canonical decomposition of once–punctured torus bundles_ , Comment. Math. Helv. 78 (2003), 363–384.
* [S1] Zeev Smilansky, _Convex hulls of generalized moment curves_ , Israel J. Math. 52 (1985), 115–128.
* [S2] Zeev Smilansky, _Bi-cyclic 4-polytopes_ , Israel J. Math. 70 (1990), 82–92.
* [Vi] I. Vinogradov, _An Introduction to the Theory of Numbers_ , Pergamon Press, London – New York, 1955, pp. vi+155.
* [We] Jeffrey Weeks, _SnapPea_ , a software for the study of hyperbolic manifolds, http://www.geometrygames.org/SnapPea/ .
Laboratoire Paul Painlevé (UMR 8524)
CNRS – Université de Lille I
59 655 Villeneuve d’Ascq Cédex, France
Francois.Gueritaud@math.univ-lille1.fr
|
arxiv-papers
| 2009-01-18T21:10:30 |
2024-09-04T02:49:00.060813
|
{
"license": "Public Domain",
"authors": "Francois Gueritaud",
"submitter": "Fran\\c{c}ois Gu\\'eritaud",
"url": "https://arxiv.org/abs/0901.2738"
}
|
0901.2759
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# Coevolution of game and network structure: The temptation increases the
cooperator density
Shao-Meng Qin Institute of Theoretical Physics, Lanzhou University, Lanzhou
$730000$, China Guo-Yong Zhang Institute of Theoretical Physics, Lanzhou
University, Lanzhou $730000$, China Yong Chen Corresponding author. Email:
ychen@lzu.edu.cn Institute of Theoretical Physics, Lanzhou University,
Lanzhou $730000$, China
###### Abstract
Most papers about the evolutionary game on graph assume the statistic network
structure. However, social interaction could change the relationship of
people. And the changing social structure will affect the people’s strategy
too. We build a coevolutionary model of prisoner’s dilemma game and network
structure to study the dynamic interaction in the real world. Based on the
asynchronous update rule and Monte Carlo simulation, we find that, when
players prefer to rewire their links to the richer, the cooperation density
will increase. The reason of it has been analyzed.
###### pacs:
02.50.Le, 05.50.+q, 64.60.Ht, 87.23.Ge
## I introduction
Cooperation is a key aspect in the real world, ranging from biological systems
to human behavior nature1 ; nature2 . Therefore, people restore to the game
theory to study the emergency and maintenance of cooperation in biology,
psychology, computer science, and economics biology ; book1 ; book2 ; PR .
Especially, the prisoner’s dilemma game (PDG), has become a metaphor to
approach the emergency of cooperation and altruism behavior. In the tradition
PDG, each of two players chooses a strategy from cooperation ($C$) or
defection ($D$) simultaneously and gets payoff. They both receive $R$ upon
mutual $C$ and $P$ upon mutual $D$. A defector gets $T$ when it plays game
with cooperator who gets $S$. In PDG, we have $T>R>P>S$ and $2R>S+T$. Because
the mutual $C$ get the highest total income, $D$ is the better choice than $C$
no mater what the other player’s strategy. Without any mechanism for the
evolution of cooperation, natural selection favors defection. The other widely
studied games include snowdrift game SNG ; SNG2 , public good game PGG , rock-
paper-scissors game RPS , and so on.
The complex network has also attracted lots of attentions in the past few
years. The complex network is ubiquitous in nature. The human society can also
be described as the systems composed of interacting agents. The classical
social network maps the individual into the node, and the connection between
individuals into the link. The evolutionary game theory in spatial structure
has became a unifying paradigm to study how cooperation may be sustained in a
structured population Nowak . It was found that the spatial extension is one
of several natural mechanisms to enforce cooperation. Network structure will
affect the behavior of strategy density structure . In lattice network, the
cooperation is usually get together to support each other to resist the
defection lattice1 ; lattice2 ; SNG2 . Santos and Pacheco found in Scale-Free
networks the strong correlation leads to the dominating trait throughout the
entire range of parameters of both games in scale-free networks SF . And also,
there are anmount of researches on other networks, like small-world SW and
random network random .
When the player on the structure network chose the better strategy to play
game, in fact, not that the players select the proper strategy, but player’s
strategy is determined by the network structure. For example, in scale-free
networks, the large degree nodes (hubs) and the nodes which connect to hubs
tend to be occupied by $C$ SF .
The networks used in the most papers of this field are statistic. The
connection will never change once it is build. It is not realistic enough, as
the interactions themselves help shape the network SW . What is more, in the
real world, the relationship between the people is not constant. Sometimes
people cannot cut some relationship with their relatives, neighbors or
colleagues but they can end their old relationship and build a new one.
Sometimes this changing is caused by the results of the game, because people
would like to make friends in a reciprocal respect. For example, people always
like to make friends with rich one for a sake of pursuing fortune. So, when we
study the social model in network like PDG, the network structure should be
dynamical entities Arne . The nodes can remove or sustain their link in
network according to the game results.
Till now, there are few models studied the cooperative behaviors in a groups
with adaptive connections. Besides some early work eW1 ; eW , Arne build a
coevolution model of strategy and structure Arne . In this model, the
probability of forming or cutting link between node $A$ and $B$ is based on
their strategies. The changing of network structure is result from the
strategy changing in the network. Then it also affect the strategy density
back. However, the link could change even if the nodes’ strategy do not change
in their model. The rewire of link in this model is not the player’s own
decision. Li et al. also build a coevolution model that the node rewire its
link only for changing its strategy LRL . Moreover, in this model, the node
rewire its long range link based on the existed network structure, not the
playing game results.
In our opinion, a rational model for coevolution of game and network structure
should contain two features: (1) The nodes rewire their links only when agents
change their status; (2) The rewiring should be based on the playing results
of game. In this paper, we will present a coevolution model of the PDG and
network. We use PDG as a metaphor to studying cooperation between unrelated
individuals and consider a social networks with four fixed local links and one
adjustable long-range link (LRL). The agents in the network play game with
their network neighbors. They will change their strategies and adjust LRLs
according to the results of game. Then the network structure changing also
affect the cooperation density.
## II Model
We set up a system of $N$ players arranged at the nodes of a ring lattice
network. Each node is connected with four local nodes. These local
interactions will not change during the whole process of the evolution.
Besides four fixed links, every node in this lattice has an adjustable LRL
which connects to another node and self-connections and the duplicate links
are excluded. We call the LRL out-link for the node to whom it belong or in-
link for the node to whom it connect. The node can select another node to
which the out-link wires, but it cannot give up the LRL. Therefore, each node
has at least one out-link and many possible in-links. When node changes its
strategy, it will also rewire its LRL. We will discuss when and how LRLs
rewire later.
As suggested by Nowak and May Nowak , we adopt $R=1$, $T=b$ $(1<b<2)$, and
$S=P=0$. Then $b$ can be considered as the temptation to $D$ against $C$.
Every player plays the PDGs with its neighbors on network and itself and get
the total payoff $W$. After each round of the game, players are allowed to
inspect their neighbors’ total payoffs and change their strategies in the next
round. The player $i$ updates its strategy by selecting one of its neighbors
$j$ with a probability $\gamma_{ij}$,
$\gamma_{ij}=\sum_{m\in\Omega_{i}}\frac{k_{j}(t)}{k_{m}(t)},$ (1)
where $\Omega_{i}$ is the community composed of the nearest neighbors of the
player $i$, and $k_{m}(t)$ is the degree of node $m$ at time $t$. In the
spirit of preferential attachment proposed by A.-L. Barábasi and R. Albert PS
, we incorporate the preferential selection rule to model social behaviors. In
Eq. 1, player with large degree has more probability to impact his neighbors.
That is true in the society that people who have great impact often have lots
of social relations and they are also focused by their friends. Node $i$ will
follow the node $j$’s strategy by the probability,
$W=\frac{1}{1+\exp\left[(W_{i}-W_{j})/\kappa\right]},$ (2)
where $W_{i}$ and $W_{j}$ are the total payoffs of node $i$ and $j$, and
$\kappa$ indicates the noise generated by the players allowing irrational
choices ka1 ; lattice1 ; lattice3 .
If node $j$ has the same strategy with $i$ or $i$ do not mimic $j$’s strategy,
node $i$ will do nothing. Otherwise, it will rewire its LRL to a new one.
There are two rewiring rules in our model: random rewiring and preferential
rewiring. With probability $p_{c}$, the density of cooperation in the network,
node $i$ will chose a new node randomly. For the rest probability $1-p_{c}$,
node $i$ will chose a new node according to the node’s payoff. In the
preferential rewiring rule, the node rewires its link according to the payoff
of all nodes in network,
$\lambda_{ij}=\sum_{m\in G}\frac{W_{j}^{\alpha}}{W_{m}^{\alpha}},$ (3)
where $\lambda_{ij}$ is the probability of node $i$ rewiring its link to $j$
and $G$ presents all nodes in the graph. $\alpha$ is used to change the effect
of payoff. $\alpha=0$ indicates that the payoff has no effect here and the
nodes rewire their links randomly. For $\alpha>0$, the node will prefer to
connect the node with larger payoffs. So it also looks like a kind of
preferential selection rule.
## III Simulation Results
We run our simulations with varying $b$ and $\alpha$ for fixed $\kappa=0.1$
and the system size $N=1000$. All the results in this paper are obtained from
the average results with $100$ different Monte Carlo (MC) simulation trails.
We start with node linking its LRLs to other nodes randomly with equal
probability and random initial state with $p_{c}=0.5$ as the initial state.
The players update their strategies in random sequence. In every MC step, all
nodes have one chance to change their strategies and rewire their links.
### III.1 Strategy evolution
Figure 1: (Color online) Frequency of cooperators $p_{c}$ for different
$\alpha$ as functions of the advantage of defectors $b$. Figure 2: (Color
online) Frequency of cooperators $p_{c}$ evolve with $t$ for systems at
different parameters on PDGs.
Figure 1 shows the frequency of cooperators $p_{c}$ in our model as the
functions of $b$ for different $\alpha$. Similar to evolutionary game in
regular network lattice1 ; lattice2 , we also find two thresholds in our
model. Full cooperation is achieved if $b$ does not exceed the threshold
$b_{c1}$. For $b>b_{c2}$, $C$ cannot resist the temptation of $b$ and cannot
survive in the network. In the region of $b_{c1}<b<b_{c2}$, $C$ and $D$ can
coexist in the network. Compared with the case of $\alpha=0$, the position of
$b_{c1}$ does not change with $\alpha$. However, $\alpha$ affect the $b_{c2}$
conspicuously.
The probability of node using preferential selection to rewire its LRL is
$1-p_{c}$. Therefore $\alpha$ does not work at $p_{c}$ close to $1$ or $b$
close to $b_{c1}$. When $\alpha<1.6$, the qualitative results $p_{c}$ remain
unaffected by $\alpha$ that $p_{c}$ decreases monotonous with $b$. When
$\alpha>1.6$, there exists a region of $b$ promoting cooperation obviously.
This promotion starts at $b=1.64$ ($\alpha=1.6$) and this region enlarge with
increasing $\alpha$. But the effect of promotion does not increase with
$\alpha$. We observe that $p_{c}$ does not change at $1.55<b<1.65$ for
$\alpha=1.7$, $1.8$, and $1.9$. Actually, the transition is caused by the
changing of network structure. We will discuss it in the next subsection.
In order to discuss how the $\alpha$ promotes $p_{c}$ in the promotion region,
we present the time evolutions of $p_{c}$ in Fig. 2 for fixed $b=1.5$ with
different $\alpha$ values. The red, blue, and black lines are the averages of
$100$ trials for $\alpha=0$, $1.5$, and $1.9$ respectively. The green one is
the $p_{c}$ time series of one trail in the black line. For $\alpha=0$ and
$1.5$, $p_{c}$ decreases with time to its station state quickly. As shown in
Fig. 1, $p_{c}$ for $\alpha=1.5$ is a little higher than that of $\alpha=0$.
However, for $\alpha=1.9$, $p_{c}$ decreases like $\alpha=0$ firstly, and then
the evolution of network drives $p_{c}$ increasing with time to $0.76$.
Considering that the black line is the average of $100$ trails, we believe the
green line in Fig. 2 contains more details of the evolution. In the early
stage of the green line, $p_{c}$ decreases to a temporary stable state in a
manner similar to but a little larger than $\alpha=1.5$. However, at $t=2000$,
there is a sharp increasing in the green line from about $0.4$ to $0.76$ which
is also the final level of the average result (the black line). It means that
the gradually increasing of the black line is caused by the average effect of
$100$ same sharply increase at different times.
### III.2 Network structure
In this model the behavior of $p_{c}$ and the evolution of network structure
are equal important. The evolution of network structure results in the
transition of $p_{c}$.
In order to describe the network structure, we first present the degree
distribution $P(k)$ in Fig. 3. Panel (a) is $P(k)$ in the case of the stable
state of red line in Fig. 2. Here the preferential rewiring does not work and
all LRLs select the target nodes randomly. Considering the self-connection is
forbidden, we know
$P(k)=C_{N-5}^{k-5}\left(\dfrac{1}{N-4}\right)^{k-5}\left(1-\dfrac{1}{N-4}\right)^{N-k+5}.$
Here $N$ usually is large enough, so one can get
$P(k)=C_{N}^{k-5}\left(\dfrac{1}{N}\right)^{k-5}\left(1-\dfrac{1}{N}\right)^{N-k+5}.$
Figure 3(b) is $P(k)$ for the stable state of blue line in Fig. 2. $P(k)$ in
(b) is similar to that of (a) but the largest degree is $19$. Fig. 3(c) is
$P(k)$ for the stable state of gree line in Fig. 2 and (d) is for the green
line after the sharp increasing.
Both (c) and (d) in Fig. 3 are the degree distributions of one trial, but not
the cumulative stationary degree distribution of $100$ different trials. By
comparing (c) with (d), it is helpful to uncover the reason of the sharp
increasing in Fig. 2. In Fig. 3 (d), there is only one node that its degree is
larger than half of the other nodes connected to it. We name this node which
has the largest degree in the network as hub node (HN). As presented in Fig.
4, the other nodes can be divided into two types: the nodes connect their LRLs
to HN and the nodes do not. We name the first node as AN and the second one as
BN. The number of them are $N_{A}$ and $N_{B}$, respectively.
Figure 3: (Color online) The cumulative stationary degree distributions $P(k)$
in PDGs. Figure 4: (Color online) Illustration of HN, AN, and BN. Each node in
the network has four fixed links and there are five red nodes wire their LRLs
to the blue one. In order to make AN and HN prominent, we do not draw the LRLs
of other nodes. The blue node has the largest degree in this net, so blue node
is HN, and the red one is AN and the others are BN. We draw the arrows in the
figure to present these LRLs are out-links for AN and in-links for HN.
Now, we exam the detail of the network after the sharp increasing in the green
line ($\alpha=1.9$, $b=1.5$) of Fig. 2. Note that the strategy of HN is always
$C$ and the strategy of most ANs is also $C$. Before the sharp increasing or
in the case of other parameters without sharp increasing, the HNs are also
prefer to $C$. This phenomenon is also observed in some other networks with
hub nodes SF ; LRL . More detailed information of our model are listed in
Table 1.
In Table 1, $p_{Ac}$ is the cooperation density of AN and $p_{Bc}$ is for BN.
Almost all nodes of ANs chose the strategy $C$, so we do not need to present
the mean payoff of AN with $D$. What is more, it is found that $p_{Bc}=0.308$
is close to the case of $\alpha=0$ ($p_{c}=0.314$ for $b=1.5$, $p_{c}=0.235$
for $b=1.55$, and $p_{c}=0.179$ for $b=1.6$). It means that the existence of
AN does not affect the strategy density of BN. As discussed in Ref. LRL , AN
can resist the temptation of $b$ by mimicking the strategy of HN. After the
sharp increasing, the probability that AN mimics the strategy of HN is much
larger than that of other neighbors. The HN’s payoff is also larger because it
has a lots of in-link LRLs. We will discuss the details of these probabilities
in the next subsection. On the other hand, only the node with strategy $C$ can
grow into HN. If HN is occupied by $D$, HN will get higher payoff temporarily.
However, as we discussed above, AN will follow HN’s strategy and the strategy
of AN will be $D$. Then the HN cannot earn payoff from its in-link LRLs. Once
HN cannot earn enough payoff, both preferential and random rewiring will drive
ANs to rewire its LRL to other nodes. Then a new HN with strategy $C$ will
appear in the network. So it seems that strategy $C$ is a better choice for HN
because it can earn a stable higher payoff.
From Table 1, we also find that the BNs with $D$ earn the most payoff and the
payoff of BN with $C$ is close to the payoff of AN. However, although the mean
payoff of BN with $D$ is the highest, in fact, the density of cooperator
doesn’t decrease with time. It shows that the probability of $C$ mimicking $D$
strategy and $D$ mimicking $C$ strategy are the same.
Table 1: The detailed information of prisoner’s dilemma games ($\alpha=1.9$). | $b=1.5$ | $b=1.55$ | $b=1.6$
---|---|---|---
$N_{A}$ | $669$ | $621$ | $605$
$N_{B}$ | $330$ | $378$ | $394$
$p_{c}$ | $0.766$ | $0.686$ | $0.663$
$p_{Ac}$ | $0.992$ | $0.988$ | $0.987$
$p_{Bc}$ | $0.308$ | $0.191$ | $0.164$
payoff of AN | $5.213$ | $4.806$ | $4.682$
payoff of BN with $C$ | $5.222$ | $4.987$ | $5.025$
payoff of BN with $D$ | $5.713$ | $5.541$ | $5.561$
Figure 5: (Color online) Because the network structure in our model is one
dimension lattice, we can use a color line to present the snapshot of the
status of the network. The black and white dots present $C$ and $D$ in BN. The
green and red dots present $C$ and $D$ in AN. In order to know how the AN, BN,
and strategy evolve in network, we arrange these snapshots with time from top
to the botton at $1600<t<2100$ in the left panel for the green line in Fig. 2.
The right panel presents the time evolution of the number of AN at the same
time.
Each horizontal line in Fig. 5 presents a snapshot of the network. We arrange
these snapshots with time from top to the botton to show how the AN evolves
with time. So we can depict every player’s strategies in network and observe
the evolution of these strategies. The riht panel is $N_{A}$ at the same time
with the left. There is also a sharp increasing of $N_{A}$ at the same time
like the green line in Fig. 2 in looks. At $t=1750$, about $50$ MC steps
before the transition, $N_{A}$ increase gradually from about $10$ to $50$.
After the sharp increasing, $N_{A}$ still increase gradually to the final
stable state. Moreover, before the sharp increasing happened, one can observe
many black blocks (the upper part of the left panel in figure 5). It means the
model has the similar feature of PDG in regular network that the $C$ node
tends to get together for blocks to resist the $D$. These blocks start at a
few $C$s, maybe three or more, and then close to each other in the network
coincidentally. Then a block is established and it will grow to change their
neighbors’ strategies. After some MC steps, the block will shrink and then
disappears in the last. After the sharp increasing of $N_{A}$, there are too
few red dots ($D$ in AN). The green strip ($C$ in AN) indicates that the ANs
or BNs are very stable in the network. The probability of AN change to BN is
very small and vice versa.
### III.3 Discussion
Based on the results in the above context, the effect of $\alpha$ is different
from various $b$ and $\alpha$. After the sharp increasing, the nodes in
network can be divided into AN and BN. Almost all AN are $C$ and the density
of $C$ in BN is close to the case of $\alpha=0$. So we can use the mean field
theory and some basic feature of stable state to explain why the sharp
increasing happened.
After the sharp increasing in Fig. 3, the system reaches the stable state
gradually. Then we have ${dN_{A}}/{dt}=0$ or $N_{A\rightarrow
B}=N_{B\rightarrow A}$, where $N_{A\rightarrow B}$ is the average number of
nodes changed from AN to BN in one MC step and $N_{B\rightarrow A}$ is that
changed from BN to AN.
Considering that there are too few $D$s in AN, we assume that $N_{A\rightarrow
B}$ is only caused by $C\rightarrow D$ and random rewire. Here, we neglect the
preferential rewiring. Because the contribution of preferential rewiring is
only about $2\%$ of random rewiring. Then we get
$\displaystyle N_{A\rightarrow B}$ $\displaystyle=$
$\displaystyle(1-p_{c})Q_{A\rightarrow B}p_{c}N_{A}$ (4)
$\displaystyle\frac{\left(5+\frac{N-N_{A}}{N}\right)\left(4+\frac{N-N_{A}}{N}\right)}{N_{A}+\left(5+\frac{N-N_{A}}{N}\right)\left(4+\frac{N-N_{A}}{N}\right)}.$
Here, $p_{c}$ means the change happened in the random rewiring, and
$5+(N-N_{A})/N$ is the mean degree of nodes in networks. We neglect self-
connection and multi-connection forbidden and we have $N\approx N-1$ here.
Because AN has the same strategy with HN, AN only mimics the strategy from
other $4+(N-N_{A})/N$ neighbors. The big fraction is the probability of AN do
not chose HN to mimic the strategy. The last $(1-p_{c})$ is the probability of
mimicked target with strategy $D$. We assume $Q_{A\rightarrow B}$ is the
probability of success in the mimicking.
Then $N_{B\rightarrow A}$ will be more complicate. We assume that BN change to
AN because they use the preferential rewiring. The contribution of random
rewiring is about $0.2\%$ of preferential rewiring, so we neglect it and
derive the following formula,
$\displaystyle N_{B\rightarrow A}$ $\displaystyle=$
$\displaystyle\left[p_{Bc}(1-p_{c})+(1-p_{Bc})p_{c}\right]Q_{B\rightarrow
A}(1-p_{c})N_{B}$ (5)
$\displaystyle\frac{(N_{A}+5)^{\alpha}}{(N_{A}+5)^{\alpha}+N_{A}(2+4p_{c})^{\alpha}+p_{Bc}N_{B}\left(1+(5+\frac{N-N_{A}}{N})p_{c}\right)^{\alpha}+N_{B}(1-p_{Bc})\left(b(5+\frac{N-N_{A}}{N})p_{c}\right)^{\alpha}},$
where $1-p_{c}$ means the preferential rewiring, and $p_{Bc}(1-p_{c})$ is the
probability of BN with strategy $C$ to mimic its $D$ neighbor and that $D$ try
to mimic its $C$ neighbor. The fraction here is the probability of node rewire
to HN using the preferential rewiring. We assume $Q_{B\rightarrow A}$ is the
probability of success in the mimicking.
Now, one can get $p_{Bc}$ from the simulation of $\alpha=0.0$ and
$p_{c}=(p_{Bc}N_{B}+N_{A})/N$, and then we know how $N_{A}$ evolves with time
by using $N_{B\rightarrow A}-N_{A\rightarrow B}$. If $N_{B\rightarrow
A}-N_{A\rightarrow B}=0$, $N_{A}$ will not change with time. And
$N_{B\rightarrow A}-N_{A\rightarrow B}>0$ means $N_{A}$ will increase in the
next MC step. However, we do not know $Q_{A\rightarrow B}$ and
$Q_{B\rightarrow A}$ yet. According that the mean payoff of $BN$ with $D$ is
larger than mean payoff of $AN$ in Tab. 1, we conjecture $M=Q_{B\rightarrow
A}/Q_{A\rightarrow B}$ and $M>1$. Indeed, we find $M=2.0$ is fit to our model.
We will take $b=1.5$ with $\alpha=1.9$ and $\alpha=1.3$ as examples. For
$b=1.5$ and $\alpha=0$ we get $p_{c}=0.314$ from the simulation.
Fig. 6 plots ${dN_{A}}/{dt}$ as different $N_{A}$. For $\alpha=1.3$ there are
two stable points at $N_{A}=3$ and $860$, and one unstable point at
$N_{A}=23$. For $\alpha=1.9$, there is only one stable state at $N_{A}=939$.
The unstable point will decrease with the increasing of $\alpha$ and
coincident with the first stable point at $\alpha=1.69$. However, even
$\alpha=0$, the maximal degree in the network is about $12$. So the first
stable point can be discarded. When the unstable point crosses $N_{A}=12$ or
there is only one stable point, the system will reach to the second stable
point.
Figure 6: (Color online) $N_{B\rightarrow A}-N_{A\rightarrow B}$ with various
$N_{A}$. Panel (b) is an enlargement of (a).
## IV Conclusion
The coevolution of dynamics and network structure is rapidly becoming an
important field of the evolutionary game. It contains more details about the
social interaction in the real world. In this paper, we build a co-evolution
model of PDG and network structure. Each node in network has four fixed local
links and one adjustable LRL. When the node changes its strategy, it will
rewire its LRL to another node according to the node’s payoff and density of
cooperation. And we introduce a parameter $\alpha$ to denote the effect of
payoff.
Many early works eW1 ; ew; Arne ; LRL also proved that the adaptive network
can enhance the cooperation. All these enhancements are caused by the
emergency of cooperator with large degree in the network. In eW1 , the
cooperation is very sensitive to the plasticity parameter and only the
adaptive network can enhance the cooperation.
In our model, the players rewire their LRLs for any $\alpha$, but the
cooperation is enhanced only in the case of $\alpha>0$ that this enhancement
is obvious for $\alpha>1.6$ in a certain region of $b$. However, our results
show that the enhancement of cooperation only happen in the case of changing
the network structure property. In our model, for $\alpha=0$, the node will
also rewire its LRL, but the network property will not change and the
cooperation level will not be enhanced. The cooperation is enhanced only when
the node rewires its LRL according to the payoff. Similar phenomena was also
observed in our simulations with snowdrift game (SG). We found that SG is more
sensitive to $\alpha$ than PDG and the obvious enhancement is for a smaller
$\alpha$. So we conjecture that the coevolution of network structure and game
is an important mechanics to maintain the cooperation in the real society.
Different from the results in eW1 ; eW ; Arne ; LRL which the cooperation
always dominates in the adaptive network and the increasing of cooperation is
limited. That is caused by two reasons: (1) In the probability of $p_{c}$ the
player use the random rewiring. (2) The existance of four fixed links in
network can be regarded as a noise to prevent the preferential selection. In
eW , authors discussed the leaders and the global cascades. If every node
could change its strategy in a smaller probability, the global cascades of
coopertation is also observed in our model.
The analysis in this paper is based on the balance of AN and BN. However, when
the sharp increasing didn’t happen, perhaps there exist more than one HNs and
HN is changing from one node to another frequently. Because of the absense of
the information about spacial structure and Eq. 2, the presented analysis in
this model is not very precise any more. Actually, it is impossible to include
all the details for the analysis. We just hold on the main factors of the
model and it works well enough to explain the main features of our model.
###### Acknowledgements.
This work was supported by the National Natural Science Foundation of China
under Grant No. $10305005$ and by the Fundamental Research Fund for Physics
and Mathematic of Lanzhou University. This study is supported by the high-
performance computer program in Lanzhou University.
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* (11) M. A. Nowak and R. M. May, Nature (London) 359, 826 (1992); M. A. Nowak and R. M. May, Int. J. Bifurcat. Chaos 3, 35 (1993).
* (12) Z. X. Wu, X. J. Xu, Y. Chen, and Y. H. Wang, Phys. Rev. E. 71, 037103 (2005); J. Vukov, G. Szabó, and A. Szolnoki, ibid. 77, 026109 (2008).
* (13) G. Szabó and C. Töke, Phys. Rev. E 58, 69 (1998).
* (14) S. M. Qin, Y. Chen, X. Y. Zhao, and S. Jian, Phys. Rev. E 78, 041129 (2008).
* (15) F. C. Santos and J. M. Pacheco, Phys. Rev. Lett. 95, 098104 (2005); F. C. Santos, J. M. Pacheco, and T. Lenaerts, Proc. Natl. Acad. Sci. U.S.A. 103, 3490 (2006).
* (16) M. Tomassini, L. Luthi, and M. Giacobini, Phys. Rev. E 73, 016132 (2006).
* (17) J. Ren, W.-X. Wang, and F. Qi, Phys. Rev. E 75, 045101 (2007).
* (18) J. M. Pacheco, A. Traulsen, and M. A. Nowak, Phys. Rev. Lett. 97, 258103 (2006).
* (19) M. G. Zimmermann, V. M. Eguíluz, and M. S. Miguel, Phys. Rev. E 69, 065102 (2004).
* (20) M. G. Zimmermann and V. M. Eguíluz, Phys. Rev. E 72, 056118 (2005).
* (21) W. Li, X.-M Zhang, and G. Hu, Phys. Rev. E 76, 045102 (2007).
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|
arxiv-papers
| 2009-01-19T02:07:14 |
2024-09-04T02:49:00.071740
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shao-Meng Qin, Guo-Yong Zhang, and Yong Chen",
"submitter": "Yong Chen",
"url": "https://arxiv.org/abs/0901.2759"
}
|
0901.3137
|
# Microscopic theory of the Andreev gap
Tobias Micklitz1 and Alexander Altland2 1Materials Science Division, Argonne
National Laboratory, Argonne, Illinois 60439, USA
2Institut für Theoretische Physik, Universität zu Köln, Zülpicher Str. 77,
50937 Köln, Germany
###### Abstract
We present a microscopic theory of the Andreev gap, i.e. the phenomenon that
the density of states (DoS) of normal chaotic cavities attached to
superconductors displays a hard gap centered around the Fermi energy. Our
approach is based on a solution of the quantum Eilenberger equation in the
regime $t_{D}\ll t_{\rm E}$, where $t_{D}$ and $t_{\mathrm{E}}$ are the
classical dwell time and Ehrenfest-time, respectively. We show how quantum
fluctuations eradicate the DoS at low energies and compute the profile of the
gap to leading order in the parameter $t_{D}/t_{\rm E}$.
###### pacs:
03.65.Sq, 03.65.Yz, 05.45.Mt
The attachment of a superconductor to a conducting cavity leads to a
suppression of the normal density of states – the proximity effect. For
cavities with classically chaotic dynamics, a discrepancy is found between
semiclassical calculations prev and such based on random matrix theory (RMT)
rmt : Semiclassics obtains a small yet finite DoS for all excitation energies
$\epsilon$ above the Fermi level $\epsilon_{\rm F}$, while RMT predicts the
formation of a hard gap below some energy $\epsilon^{\ast}$. The origin of
this so-called ‘gap problem’ in Andreev billiards was pointed out by Lodder
and Nazarov some time ago prev : quantum corrections not captured in the
principal semiclassical approximation are expected to generate a hard spectral
gap for trajectories longer than the Ehrenfest time. Although various semi-
phenomenological realizations of this mechanism have been formulated, a fully
microscopic theory of gap formation is outstanding. The construction of such a
theory is the goal of the present paper.
Quasiclassical Eilenberger equation — Consider a two-dimensional Andreev
billiard, i.e. a chaotic normal-conducting cavity attached to a bulk
superconductor. We wish to compute the cavity DoS in a ‘semiclassical’ regime
where the quantum time scales of the problem exceed all classical scales.
Under these circumstances one expects prev the gap, $\epsilon^{\ast}$ to be
set by the inverse of the Ehrenfest time, $\epsilon^{\ast}=\pi\hbar/2t_{\rm
E}$, where $t_{\rm E}=\lambda^{-1}\ln(c^{2}/\hbar)$, $\lambda$ is the dominant
Lyapunov exponent of the system, and $c^{2}$ a classical action scale whose
detailed value is of little relevance. Heuristically, $t_{\rm E}$ is the time
a minimal wave package needs to spread over classical portions of phase space;
the dynamics at time scales beyond $t_{\rm E}$ is no longer classical.
To compute the DoS, we start out from the quantum Eilenberger equation (for
notational convenience we suppress the infinitesimal imaginary increment in
$\epsilon+i0$)
$\displaystyle\left[\epsilon\sigma_{3}-i\Delta\sigma_{2}+H\openone\stackrel{{\scriptstyle\ast}}{{,}}G\right]=0$
(1)
for the quasiclassical retarded matrix Green function, $G(\mathbb{x})$, i.e.
the Wigner transform of the Gorkov superconductor Green function. In (1),
$\sigma_{i}$ are Pauli matrices acting in particle-hole space,
$\mathbb{x}=(\mathbb{q},\mathbb{p})^{T}$ is a phase space point in the shell
of constant energy, $H(\mathbb{x})=\epsilon_{F}$, $H(\mathbb{x})$ is the
Hamilton function, and the order parameter amplitude
$\Delta=\Delta(\mathbb{q})$ is non-vanishing only at the cavity-superconductor
interface. The Green function is subject to the nonlinear constraint $G\ast
G=\openone$, and yields the DoS as
$\nu(\epsilon)=\frac{\nu_{0}}{2\Omega}\operatorname{Re}\int
d^{2}x\operatorname{tr}\left[G(\mathbb{x})\sigma_{3}\right]$, where
$\Omega=\int_{H(\mathbb{x})=\epsilon_{F}}d^{2}x\,1$ is the volume of the
energy shell and $\nu_{0}$ the normal metallic DoS. Finally, the symbol
‘$\ast$’ indicates that all products between phase space functions in Eq. (1)
are Moyal products $(A\ast B)(\mathbb{x})=\exp\big{(}{i\hbar\over
2}\partial_{\mathbb{x^{\prime}}}^{T}I\partial_{\mathbb{x}}\big{)}\big{|}_{\mathbb{x}=\mathbb{x^{\prime}}}A(\mathbb{x^{\prime}})B(\mathbb{x})$.
Figure 1: Inset: classical trajectory connecting the superconductor/normal
conductor interface of a chaotic Andreev billiard with itself. Main part:
abstract phase space representation of that trajectory and its vicinity in a
system of locally stable ($s$) and unstable ($u$) coordinates. The meaning of
the shaded areas is explained in the main text.
Classical evolution and its inconsistency— Upon Taylor expansion to lowest
orders $(A\ast B)(\mathbb{x})=A(\mathbb{x})B(\mathbb{x})+{i\hbar\over
2}\\{A,B\\}(\mathbb{x})+{\cal O}(\hbar^{2}\partial_{x}^{2})$ ($\\{\,,\\}$ is
the Poisson bracket) Eq. (1) assumes the standard form of the classical
Eilenberger equation eilenberger
$[i\epsilon\sigma_{3}+\Delta\sigma_{2},G]-\hbar{\cal L}G=0,$ (2)
where ${\cal L}=\\{H,.\\}$ generates the classical Liouville flow. However,
(finite order) Taylor expansions of the Moyal product become problematic in
cases where the function $G$ displays structure on linear scales
$\lesssim{\cal O}(\hbar)$ and higher order derivatives ${\cal
O}(\hbar^{2}\partial_{x}^{2})$ become of the same order as $\hbar{\cal L}$; as
we shall see, this is precisely what happens on the solutions $G$ supporting
the DoS in the region of the spectral gap.
The classical Eilenberger equation (2) describes the evolution of $G$ along
individual classical trajectories $\gamma$ beginning and ending at the
superconductor interface (cf. inset of Fig. 1.) Parameterizing a trajectory
$\gamma$ of length $T$ in terms of a coordinate $t\in[-T/2,T/2]$, the
Liouville operator on $\gamma$ assumes the form ${\cal L}=\partial_{t}$ and
the solution in the asymptotic limit $\epsilon/\Delta\rightarrow 0$ is prev
$\displaystyle G_{T}(t)=-i\tan\left({\epsilon
T\over\hbar}\right)\sigma_{3}+{\cos\left({2\epsilon
t\over\hbar}\right)\sigma_{2}+\sin\left({2\epsilon
t\over\hbar}\right)\sigma_{1}\over\cos\left({\epsilon T\over\hbar}\right)}.$
(3)
Denoting the $\sigma_{i}$-components of $G$ by $G_{i}$, the solution obeys the
boundary conditions prev
$\displaystyle G_{T,1}(\pm T/2)=\pm iG_{T,3}(\pm T/2),\quad G_{T,2}(\pm
T/2)=1.$ (4)
The component $G_{T,3}=\mathrm{const.}$ generates (via the identity
$\text{Im}\left(\tan\left(x+i0\right)\right)=\pi\sum_{m}\delta\left(x-(m+1/2)\pi\right)$)
a quantization condition, $\epsilon T=(m+{1\over 2})\pi\hbar$, $m=0,1,2,...$,
for the flight times of trajectories contributing to the DoS at energy
$\epsilon$. The exponential sparsity of trajectories with $T\gg t_{\text{D}}$
much larger than the average dwell time pathdist then leads to an exponential
suppression of the DoS for $\epsilon\lesssim\hbar t_{\text{D}}^{-1}$, but not
to a gap.
In view of the continuity conditions underlying the approximation (2), it is
mandatory to explore what happens as we transversally depart from an isolated
trajectory into surrounding phase space. To this end, it is useful to
interpret each trajectory as element of a corridor or band trbands1 ; effrmt ;
trbands6 which is formed by all trajectories that run through the same
sequence of scattering events. A schematic of a band is shown in the bottom
part of Fig. 1, where the straight line represents a trajectory beginning and
ending at points $\mathbb{x}$ and $\mathbb{x}^{\prime}$ in the SN interface.
We introduce Poincaré sections through the trajectory, and span them by the
locally stable and unstable coordinates, $s$ and $u$, respectively. The shaded
areas then represent the SN interface ($S_{1}$), the image of that area under
the Hamiltonian flow after time $t$ ($S_{2}$), the intersection of the image
with the interface ($S_{3}$), and the pre-image of the intersection ($S$),
respectively. Points in $S$ remain compactly confined and exit at the same
instance $T$. The image of $S$ under evolution defines a ’corridor’ of
sections across which the quasiclassical solutions $G_{T}$ is nearly constant.
While the transverse area, $us$, of the corridor is a conserved quantity, its
shape is not. At a given instance of time, $t$, its smallest linear extensions
is given by (cf. Fig. 1) $\sim{\rm const.}\times{\,\rm
min}(e^{-\lambda(T/2+t)},e^{{-\lambda(T/2-t)}})$, with a classical
proportionality constant. For trajectory times $T>t_{\mathrm{E}}$, that scale
may shrink below ${\cal O}(\hbar)$, and this is when Eq. (2) becomes
problematic: at low energies, $\epsilon\sim\hbar t_{\rm E}^{-1}$, the narrow
corridors of long trajectories $T>t_{\rm E}$ meander through the bulk of phase
space, in which trajectories are of average length $\sim t_{\rm D}\ll t_{\rm
E}$ and Green functions are ‘locked’ to the superconductor order parameter,
$G(\epsilon)\simeq\sigma_{2}$. (Here and throughout, we use the notation
$\simeq$ to indicate equality up to inconsequential corrections scaling with
some positive power of $\hbar$.) The ensuing sharp variation of the solution
$G_{T}$ over trans-corridor sections of quantum extension $\lesssim{\cal
O}(\hbar)$ conflicts with quasiclassical smoothness conditions required for
Eq. (2).
Our solution to the problem proceeds in two steps: we first transversally
extend (3) to a solution of (2) in a ‘Planck tube’ fn11
$\displaystyle Z=\bigcup_{-{T\over 2}\leq t\leq{T\over 2}}Z_{t},\quad
Z_{t}=\\{(u,s,t)|\;|us|\leq\hbar,\;|u|,|s|<c\\},$ (5)
centered around $\gamma$. This – singular – configuration will then be the
basis for the construction of a smooth configuration $G$ that solves the
quantum equation (1) up to corrections $\sim t_{D}/t_{\mathrm{E}}$. The
quantum $G$ displays a hard spectral gap.
Figure 2: Vicinity of a long trajectory in a system of locally stable (s) and
unstable (u) coordinates. Points at the boundary $|us|\sim\hbar$ belong to
trajectories $\gamma$ generically of length $T\sim t_{\mathrm{E}}$ (here
illustrated by the curved line.) The cloudy region at the ends of $\gamma$
represent generic phase-space points.
Consider, then, the corridor carried by a trajectory $\gamma$ of length $T\geq
t_{\mathrm{E}}$. (In the wide corridors of shorter trajectories the Green
function does not depend noticeably on transverse coordinates $(u,s,t)\in
Z_{t}$ and the solution (3) can be taken face value.) We assume the corridor
sections $Z_{t}$ to be small enough to afford a linearization ZurekPaz
$\displaystyle{\cal L}=\partial_{t}+\lambda u\partial_{u}-\lambda
s\partial_{s},$ (6)
where the terms $u\partial_{u}$ and $s\partial_{s}$ describe the divergence
and contraction of phase flow around $\gamma$, respectively.
Figure 3: On the length of trajectories piercing the boundary of the Planck
cell around a long reference trajectory. a) reference times corresponding to a
bulk point in the N-region, b) point close to the exit into the
superconductor, c) point close to the entrance into the N-region. Discussion,
see text.
Going forward (backward) in time, the trajectory through a point $(u,s,t)\in
Z_{t}$ will stay in the vicinity of $\gamma$ for a time $t(u)$ ($t(s)$) where
$t(x)=\lambda^{-1}\ln(c/|x|)$. (cf. Fig. 2.) Thereafter a classically short
time, typically of ${\cal O}(t_{\text{D}})$, passes before the departing
trajectory exits; up to classical corrections, the time of flight of the
trajectory through $(u,s,t)$ thus reads
$t(u)+t(s)=\lambda^{-1}\ln(c^{2}/|us|)$. Specifically, for phase-space points
on the boundary of the Planck cell $|us|\sim\hbar$ and therefore
$t(u)+t(s)\simeq t_{\mathrm{E}}$. The above consideration applies to phase
space points far away from the SN interface (cf. Fig. 3 a)). For points close
to the interface, it may happen that the trajectory through $(u,s,t)$ hits the
interface before it has diverged up to $c$, in which case the exit time is
shorter than $t(u)$ (Fig. 3 b)). Or, it has been in the system for a time
shorter than $t(s)$ before the reference point is reached (Fig. 3 c)). We
subsume these different cases, by introducing effective in- and out-times
$t_{o}(u)=\mathrm{min}(t(u),T/2-t)$ and $t_{i}(s)=\mathrm{min}(t(s),T/2+t)$,
where the function
$\mathrm{min}(t,t^{\prime})\equiv-{1\over\lambda}\ln\big{(}e^{-\lambda
t}+e^{-\lambda t^{\prime}}\big{)}$ smoothly interpolates between $t$ and
$t^{\prime}$ over a ‘microscopic’ switching interval $\sim\lambda^{-1}$. These
functions evolve uniformly, in the sense ${\cal L}(t_{i/o}(s/u))=(+/-)1$. This
means that the effective (up to corrections of ${\cal
O}(t_{D}/t_{\textrm{E}})$) duration of the trajectory through $(u,s,t)\in
Z_{t}$, is given by $T_{(u,s,t)}\equiv t_{o}(u)+t_{i}(s)$ and the trajectory
parameter by $\tau_{(u,s,t)}\equiv{1\over 2}(t_{i}(s)-t_{o}(u))$.
Substitution, $T\to T_{(u,s,t)}$ and $t\to\tau_{(u,s,t)}$ in (3) then obtains
a transverse extension
$\displaystyle G^{c}(u,s,t)\equiv G_{T_{(u,s,t)}}(\tau_{(u,s,t)})$ (7)
of (2) fn_bound . $G^{c}$ solves the Eilenberger equation in direct
consequence of the flow-uniformity of $t_{i}(s)$ and $t_{o}(u)$. By the same
token, however, the solution becomes singular at times $T\geq t_{\mathrm{E}}$
when $t_{i,o}$ begin to display structure on scales $\lesssim\hbar$. Next, we
show that this is not what happens in the full quantum dynamics.
Quantum evolution and spectral gap— Let us define a generalization,
$t^{q}_{i}(u,s,t)$, of $t_{i}(s,t)$ by requiring uniformity under the full
dynamics, $-i\hbar^{-1}[H\overset{\ast}{,}\,t_{i}^{q}]=1$, or
$\displaystyle{\cal L}\,t_{i}^{q}+[{\cal V}\overset{\ast}{,}\,t_{i}^{q}]=1,$
(8)
where $[{\cal
V}\overset{\ast}{,}\;]\equiv-i\hbar^{-1}[H\overset{\ast}{,}\;]-{\cal L}$
accounts for quantum corrections to the linearized classical dynamics. The
above equation may be solved by introducing ‘action-angle coordinates’ $I=us$,
$\phi\equiv{1\over 2}\ln(u/s)$ in terms of which ${\cal
L}=\partial_{t}+\lambda\partial_{\phi}$. The ${\cal V}$-term may now be
formally removed by ‘gauging’ Eq. (8) with
$\displaystyle U(I,\phi,t)={\cal
P}e_{\ast}^{{1\over\lambda}\int_{0}^{\phi}d\phi^{\prime}{\cal
V}(I,\phi^{\prime})},$ (9)
where $e_{\ast}^{(...)}$ is defined by a Moyal series expansion in the
exponent and ${\cal P}$ is a $\phi$-ordering prescription (see Ref. [wl1, ]
for details) accounting for the non-commutativity of ${\cal V}(I,\phi)$ at
different values of $\phi$. By construction [wl2, ], $U$ obeys ${\cal
L}U={\cal V}\ast U$, which means that Eq. (8) is solved by
$\displaystyle t^{q}_{i}(u,s,t)=(U\ast t_{i}\ast U^{-1})(u,s,t).$ (10)
In practice, both the detailed form of ${\cal V}$ and of $U$ will not be
known. This lack of knowledge, however, is not of essential concern to us; to
the logarithmic accuracy required by the present analysis, basic scaling
arguments suffice to determine the action of $U$ on $t_{i}$: describing
nonlinear corrections to the flow, the expansion of ${\cal V}$ for small $u,s$
starts as $\hbar{\cal V}=us\times{\cal O}(u^{n}s^{m}),n+m>0$. Accordingly,
$U=1+\hbar^{-1}us\times{\cal O}(u^{n}s^{m})$. This entails that for any
function $f$ that is smooth (analytic) around $u=s=0$,
$(U*f*U^{-1})(u,s)=f(u,s)+{\cal
O}(\partial_{u}fu^{n+1}s^{m},\partial_{s}fu^{n}s^{m+1})$. At the small values
of coordinates we are interested in, $|us|\sim\hbar$, the ${\cal
O}(\dots)$-terms become irrelevant, which reflects the irrelevancy of
dynamical corrections to the linearized flow close to the trajectory center.
To explore the effect of $U$ on singular functions (such as
$t_{i}(s)\sim\ln(|s|)$), we notice that for arbitrary $f(s)$,
$\displaystyle e^{-iku}\ast f(s)=f(s+\hbar k)\ast e^{-iku}.$ (11)
This identity suggests to introduce a Fourier mode decomposition
$U(u,s,t)=\int dk\,U_{(s,t)}(k)e^{-iku}$. Specifically, let us consider values
$|s|\sim\hbar$, where singularities begin to put the semiclassical theory at
risk. For these values, the support of the mode coefficients $U_{(s,t)}(k)$
extends up to ’classical’ values $k\sim\hbar^{0}$ fn_ucl . We thus obtain
$t_{i}^{q}(u,s,t)=\int dk\,t_{i}(s+\hbar k)\ast F_{(s,t)}(k)$, where the
positive indefinite but normalized ($\int dkF_{(s,t)}(k)=1$) ‘weight’ function
$F_{s}(k)=\left(e^{-iku}U_{(s,t)}(k)\right)\ast U^{-1}(u,s,t)$. A
straightforward estimate now shows that for asymptotically small $\hbar$ the
integral evaluates to
$t_{i}^{q}(u,s,t)=t_{i}(|s|+\hbar/u_{0})\simeq\mathrm{min}(t_{i}(s),t_{\mathrm{E}})$,
where $u_{0}$ is a non-universal constant. Similarly,
$t^{q}_{o}(u,s,t)\simeq\mathrm{min}(t_{o}(u),t_{\mathrm{E}})$. Summarizing, we
have found that the operators $U$ act to truncate singularities in trajectory
times in a manner independent of the detailed form of the potential ${\cal
V}$.
Building on these results it is now straightforward to construct a smooth
solution of the quantum equation (1): its general solution is given by
$C\sigma_{3}+(1-C^{2})\left(\cos\left({\epsilon\tau^{q}\over\hbar}\right)\sigma_{2}+\sin\left({\epsilon\tau^{q}\over\hbar}\right)\sigma_{1}\right)$,
where $\tau^{q}=(t^{q}_{i}-t^{q}_{o})/2$ and we used that smooth functions
$f(\tau^{q})$ (’$\sin$’, ’$\cos$’, etc.) evolve linearly,
$[H\overset{\ast}{,}\,f(\tau^{q})]\simeq
f^{\prime}(\tau^{q})[H\overset{\ast}{,}\,\tau^{q}]=i\hbar
f^{\prime}(\tau^{q})$, up to corrections of ${\cal O}(1/t_{\rm E}\lambda)$
fn_tel . The normalization function $C=C(u,s,t)$ is determined by requiring
stationarity $[H\overset{\ast}{,}\,C]=0$, and compatibility with the boundary
conditions (4). These two conditions lead to the identification
$C=i\tan(\epsilon T^{q}(u,s,t)/\hbar)$, where
$T^{q}(u,s,t)=\min(T(u,s,t),t_{\mathrm{E}})$ is the effective trajectory time.
In conclusion, we have found that the quantum equation (1) is solved by
$G_{T^{q}(u,s,t)}(\tau^{q}(u,s,t))$, which differs from the solution of the
classical equation (2) by an upper cutoff $t_{\mathrm{E}}$ limiting both the
trajectory time $T^{q}$ and the trajectory parameter $\tau^{q}$. Technically,
this is the main result of the present letter.
The above solution signals that quantum fluctuations couple narrow bands of
transverse extension $\lesssim\hbar$ to neighboring phase space. This coupling
is strongest in the terminal regions of long trajectories $|\tau|\gtrsim
t_{\mathrm{E}}$ where bands flatten in one direction. Inspection of Fig. 2
shows that the $\hbar$-neighborhood of these segments is pierced by
trajectories whose length and parameter are uniformly given by $T\simeq
t_{\mathrm{E}}$ and $|\tau|\simeq t_{\mathrm{E}}/2$, respectively. At these
values the solutions $G$ are nearly stationary (up to corrections ${\cal
O}(t_{D}/t_{\mathrm{E}})$, and this reflects in the asymptotic constancy of
the regularized solution $G_{T^{q}\gtrsim t_{\mathrm{E}}}(\tau^{q}\gtrsim\pm
t_{\mathrm{E}})\simeq G_{t_{\mathrm{E}}}(\pm t_{\mathrm{E}})$ at large
parameter values. The capping of trajectory times $T^{q}\lesssim
t_{\mathrm{E}}$ in turn implies a vanishing of the DoS for energies
$\epsilon<\epsilon^{\ast}$. (Technically,
$\mathrm{Re}(G_{3}(\mathbb{x}))=-\mathrm{Im}\tan(\epsilon T^{q}(u,s,t)/\hbar)$
is vanishing for these energies.) The fact that all trajectories of nominal
length $T>t_{\mathrm{E}}$ get reduced to the uniform effective length
$T^{q}\simeq t_{\mathrm{E}}$ implies an accumulation of spectral weight at the
gap edge $\epsilon^{\ast}=\pi\hbar/2t_{\mathrm{E}}$. A straightforward
estimate based on the classical density of long trajectories
$p(T)dT\sim\exp(-T/t_{\mathrm{D}})dT$ shows that the peak is of moderate
height $\rho(\epsilon^{\ast})/\rho_{\mathrm{cl}}(\epsilon^{\ast})={\cal
O}(1)$, where $\rho_{\mathrm{cl}}(\epsilon^{\ast})$ is the semiclassical
estimate of the DoS. Its width is of ${\cal
O}(t_{\mathrm{D}}/t_{\mathrm{E}}^{2})$ which reflects an uncertainty in the
effective trajectory times of ${\cal O}(t_{\mathrm{D}})$. In a real
environment, the position of the gap may also be affected by mesoscopic
fluctuations of system parameters silvestrov . However, such effects are
beyond the scope of the present paper.
Summary and discussion — We have solved the quantum Eilenberger equation to
leading order in the small parameter $t_{D}/t_{E}$. Our solution verifies the
existence of a gap in the DoS of clean chaotic Andreev billiards. It is
worthwhile to compare these results to two earlier approaches to dealing with
the singularities of the classical Eilenberger theory: in [effrmt, ],
Silvestrov et al. argued that on bands narrower than a Planck cell, classical
dynamics may be effectively replaced by RMT modeling. In [vavlarkin, ] Vavilov
and Larkin, coupled the system to artificial short range disorder, fine tuned
in strength to mimic quantum corrections to classical propagation. This latter
procedure renders the long time dynamics effectively stochastic, thus
preventing the build-up of sharply defined phase space structures. Our
analysis shows that phenomenological input of either type is not, in fact,
necessary. The conjunction of classical hyperbolicity and quantum uncertainty
encoded in the native Eilenberger equation automatically regularizes classical
singularities at large times. This mechanism operates under rather general
conditions and can be described at moderate theoretical efforts. We therefore
believe the concepts discussed above to be of wider applicability.
We are grateful for discussions with P. Brouwer. This work was supported by
SFB/TR 12 of the Deutsche Forschungsgemeinschaft and the U.S. Department of
Energy, Office of Science, under Contract No. DE-AC02-06CH11357.
## References
* (1) A. Lodder, Y. V. Nazarov, Phys. Rev. B 58, 5783 (1998).
* (2) J. Melsen et al., Europhys. Lett. 35, 7 (1996).
* (3) G. Eilenberger, Z. Phys. 214, 195 (1968); J. Rammer, Quantum Field Theory of Non-Equilibrium States (Cambridge, New York, 2007).
* (4) The distribution of path lengths $T$ in chaotic billiards is $p(T)\propto e^{-T/t_{\rm D}}$ dt .
* (5) W. Bauer et al., Phys. Rev. Lett. 65, 2213 (1990).
* (6) L. Wirtz et al., Phy. Rev. B, 56, 7589 (1997).
* (7) P. G. Silvestrov et al., Phys. Rev. Lett. 90, 116801 (2003).
* (8) R. S. Whitney et al. , Phys. Rev. Lett. 94, 116801 (2005).
* (9) The principal role of the Planck cell in this context has already been discussed in H. Schomerus and C. W. J. Beenakker, Phys. Rev. Lett. 82, 2951 (1999).
* (10) W. H. Zurek, J. P. Paz, Phys. Rev. Lett. 72, 2508 (1994).
* (11) D. J. Gross et al., Adv. Theor. Math. Phys. 4, 893 (2000).
* (12) At its terminal points, $G^{c}$ respects the boundary conditions (4). This is because at $\tau_{(u,s,t)}=\pm T_{(u,s,t)}/2$ the trajectory has either hit the superconductor interface (Fig. 3 b,c)), or they have left the vicinity of the reference trajectory (Fig. 3 a)), meaning that they will hit the interface after a classically short time $t_{D}$.
* (13) A detail proof of this property can be found in Appendix A of M. Chaichian et al., Phys. Lett. B 666, 199 (2008).
* (14) The (super)linear scaling of ${\cal V}$ in $s$ implies that as a function of $u$, ${\cal V}(u,s\sim\hbar)\lesssim\hbar^{0}$. By the same token, the $u$-dependence of the function $U(u,s\sim\hbar)\sim\hbar^{0}$ is classical, and the same applies to the mode spectrum.
* (15) This is seen by straightforward comparison of the action of the Moyal product’s higher order derivatives on the envelope function $f$ and its argument function $\tau^{q}$, resp.
* (16) P. G. Silvestrov, Phys. Rev. Lett. 97, 067004 (2006).
* (17) M. G. Vavilov, A. I. Larkin, Phys. Rev. B 67, 115335 (2003).
|
arxiv-papers
| 2009-01-20T20:29:09 |
2024-09-04T02:49:00.095365
|
{
"license": "Public Domain",
"authors": "T. Micklitz and Alexander Altland",
"submitter": "Tobias Micklitz",
"url": "https://arxiv.org/abs/0901.3137"
}
|
0901.3231
|
# A test of first order scaling in Nf =2 QCD:
a progress report
Scuola Normale Superiore, Dip. Fisica & INFN, Pisa, Italy
E-mail Claudio Bonati
Dip. Fisica & INFN, Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy
E-mail bonati@df.unipi.it Massimo D’Elia
Dip. Fisica & INFN, Genova, Via Dodecaneso 33, I-16146 Genova, Italy
E-mail delia@ge.infn.it Adriano Di Giacomo
Dip. Fisica & INFN, Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy
E-mail digiaco@df.unipi.it Claudio Pica
Brookhaven National Laboratory, Physics Department, Upton, NY 11973-5000, USA
E-mail pica@bnl.gov
###### Abstract:
We present the status of our analysis on the order of the finite temperature
transition in QCD with two flavors of degenerate fermions. Our new simulations
on large lattices support the hypothesis of the first order nature of the
transition, showing a preliminary two state signal. We will discuss the
implications and the next steps in our analysis.
## 1 Motivation
It’s a long standing issue whether the finite temperature transition in QCD
(at zero chemical potential) from a confined to a deconfined phase is really a
true transition or a simple crossover at finite fermion masses. This is not
just an academic question if one wants to pursue the idea of a dual symmetry
responsible for confinement, as suggested by experiments on the free quarks
abundance in nature [1]. In this respect, the most natural hypothesis is the
presence of a true phase transition even at finite masses rather than a
crossover.
For three or more degenerate fermions we know that a first order transition
occurs in a region nearby the chiral point [2, 3]. In the 2+1 case there are
some hints in favor of a crossover scenario, at physical quark masses, from
the study of the susceptibility of the chiral condensate [4, 5]. The most
debated case is QCD with two degenerate fermions: previous studies [6, 7, 8] –
performed on lattices with a modest extent – claimed that the chiral
transition belongs to the $O(4)$ universality class, thus leading to a
crossover at finite quark masses. Such a conclusion implies also the presence
of a possible second order end point in the temperature-chemical potential
plane that should have a clear experimental signature (see for example [9,
10]). Despite the efforts, no evidence for such an end point has been found so
far at heavy ions colliders (BNL-RHIC, CERN-SPS, GSI-FAIR). For all the above
reasons, to establish the order of the chiral transition in two-flavor QCD is
a particularly important problem in lattice QCD which is still open and
deserves a careful and deep analysis. Starting from our previous
investigation, we report here about the progress made during the last year,
using larger lattices.
## 2 Current status
This work is the last step of a long-term project. The first step [11] was a
direct check of a second order scaling. Pisarski and Wilczek [12] predicted,
by means of a chiral model (thus assuming that only the chiral degrees of
freedom are responsible for the order of the transition), that the QCD
transition at the chiral point should be in the $O(4)$ universality class if a
IR fixed point exists111In the case of staggered fermions this universality
class reduces to $O(2)$ at finite lattice spacing., i.e. if it is a second
order transition, not excluding the first order case.
A finite size scaling (FSS) analysis is the best way to address the problem of
determining the critical exponents of a transition in a lattice simulation.
The problem has two scales, the temperature and the bare quark mass, as shown
e.g. in the equations of the scaling of the free energy density and its
derivative, the specific heat:
$\displaystyle L/kT$ $\displaystyle\simeq$ $\displaystyle L_{s}^{d}\phi(\tau
L_{s}^{1/\nu},am_{q}L_{s}^{y_{h}})$ (1) $\displaystyle C_{v}-C_{0}$
$\displaystyle\simeq$ $\displaystyle L_{s}^{\alpha/\nu}\phi_{c}(\tau
L_{s}^{1/\nu},am_{q}L_{s}^{y_{h}})$ (2)
Our strategy is to get rid of one of the two scales, namely the second one, by
fixing its value $am_{q}L_{s}^{y_{h}}=\rm{const}$, and looking at the FSS in
the other variable. Obviously this strategy implies previous knowledge of
$y_{h}$ so one could test consistency of data with a particular scaling
hypothesis. The result of these consistency tests are shown in figure 1. If
the $O(4)$, or $O(2)$, scaling is correct then all the curves should fall on
top of each other. The conclusion based on these first observations is that a
second order universality class as predicted by chiral models is excluded.
Notice also that the universality classes considered have $\alpha<0$, i.e. the
specific heat should not grow with the volume, in contrast with data. Notice
that O(4) and O(2) critical indexes are both very close to those of the
$U(2)_{L}\times U(2)_{R}/U(2)_{V}$ universality class, predicted as a possible
alternative in case of a light $\eta^{\prime}$ meson at the transition [13].
Therefore our data exclude that universality class as well.
Figure 1: Second order consistency check. Upper row refers to $O(4)$ scaling,
lower to $O(2)$. Columns refer respectively to the specific heat and the
chiral susceptibility from left to right.
The following step is to check directly the first order scaling for which some
hints are found using approximate scaling laws in [11]. The results of such a
test [14], where $y_{h}=3$, are shown in figure 2. The specific heat scales
quite nicely with the first order hypothesis and also the chiral condensate
susceptibility is in agreement if one excludes the curve at $L_{s}=16$ arguing
that it probably lies outside the scaling region since it has the smallest
volume and the heaviest mass.
Figure 2: First order consistency check of the specific heat (left) and the
chiral susceptibility (right).
## 3 First order scaling analysis
The preceding observations were convincing enough to argue in favor of a first
order transition. However several questions still remain:
* •
where is the scaling with the volume of the peak at fixed mass, expected for
some small masses? (here is difficult to quantify the term “small”)
* •
where are the double peaks expected in the observables histograms near the
transition point?
Before looking at the simulations let’s consider again the scaling laws at
fixed bare quark mass and in particular equation 2. If a second order
transition is present at the chiral point then at finite mass everything is
analytical and no divergence can arise, i.e. in the thermodynamical limit any
dependence on $L$ should vanish. We can expand in terms of the inverse of the
second parameter and find that the leading term in the expansion of $\phi_{c}$
must be $\propto 1/(am_{q}L^{y_{h}})^{\alpha/(\nu y_{h})}$. In the case of a
first order transition, where the equations are valid if the transition is
really weak (as this is the case), a constant term (in volume) is present
because of a peculiar cancellation occurring only in this case:
$C_{V}-C_{0}\simeq am_{q}^{-1}\phi_{1}(\tau V)+V\phi_{2}(\tau V)$ (3)
the second term giving non-zero latent heat.
The relative weight of the two terms is unknown a priori. It is perfectly
possible that the singular, diverging, term is really small at the volumes
explored. Nevertheless one should observe a shrinking with the volume in the
width of the specific heat curve, in contrast with the crossover case where a
constant curve is expected.
In order to check volume effects we decided to dedicate a large part of our
computational facilities to a run with standard staggered fermions at
$am_{q}=0.01335$ on a $48^{3}\times 4$ lattice. This corresponds to a pion
mass of about twice the physical value and to a spatial size of $\sim 13-14$
fm. We present here our results at this stage.
We simulated four different temperatures around the peak of the specific heat.
In figure 3 we show the plaquette and the chiral condensate histograms.
Figure 3: Spatial plaquette (left) and chiral condensate (right) distribution
histograms.
Data seem to present a weak signal of a double peak for betas 5.2719 and
5.2720. However statistics is still too low to draw any conclusion in this
respect. A close look at the spatial plaquette susceptibility ($\sim$ specific
heat) (fig. 4) shows that the peak does not grow with the volume but the
reweighted curve shrinks with the correct factor (see right figure).
Figure 4: Spatial plaquette susceptibility (left). Rescaling of $\beta$ axis
(right) showing a good scaling of peak width. The $L_{s}=16$ lattice is out of
the scaling region.
## 4 Conclusions
The determination of the order of the $N_{f}=2$ QCD transition is an
interesting, rather tough problem, and we are still far from a conclusive
answer. We reported our progress in understanding various aspects of the
problem. First we concluded that a second order phase transition in the $O(4)$
or $O(2)$ universality classes at the chiral point has to be excluded.
Secondly, investigating the possibility of a first order nature of the
transition, we found several indications that could be the right answer. A
direct consistency check of this hypothesis gave a positive answer, especially
for the specific heat, an observable that does not imply any assumption on the
symmetry behind the transition. Looking for metastabilities proved to be a
more difficult task. Very weak signals of double peaks were found. If the
transition is really first order this implies a very small latent heat, so
that double peak structures are masked by simple thermal noise. A simple
analysis of the scaling equations shows that in the first order case we should
expect two contributions to scaling, one regular and one singular giving the
latent heat. The actual data sets suggest that the regular term is much bigger
than the singular one at the volumes and masses explored. We observe the
expected shrinking of the specific heat curve with the exponents predicted for
a first order.
We need to increase statistics to have reliable ensemble with small errors and
if our early conclusions are confirmed (eventually by simulations with an
improved action and/or finer lattice spacing) the standard crossover scenario
has to be changed.
This work was done using the apeNEXT facilities in Rome during the time of
several months.
## References
* [1] C. Amsler et al. [Particle Data Group], Phys. Lett. B 667, 1 (2008).
* [2] F. R. Brown et al., Phys. Rev. Lett. 65, 2491 (1990).
* [3] X. Liao, Nucl. Phys. Proc. Suppl. 106, 426 (2002) [arXiv:hep-lat/0111013].
* [4] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo, Nature 443, 675 (2006) [arXiv:hep-lat/0611014].
* [5] Y. Aoki, Z. Fodor, S. D. Katz and K. K. Szabo, Phys. Lett. B 643, 46 (2006) [arXiv:hep-lat/0609068].
* [6] S. Aoki et al. [JLQCD Collaboration], Phys. Rev. D 57, 3910 (1998) [arXiv:hep-lat/9710048].
* [7] F. Karsch and E. Laermann, Phys. Rev. D 50, 6954 (1994) [arXiv:hep-lat/9406008].
* [8] C. W. Bernard et al., Phys. Rev. D 49, 3574 (1994) [arXiv:hep-lat/9310023].
* [9] M. A. Stephanov, PoS LAT2006, 024 (2006) [arXiv:hep-lat/0701002].
* [10] M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev. Lett. 81, 4816 (1998) [arXiv:hep-ph/9806219].
* [11] M. D’Elia, A. Di Giacomo and C. Pica, Phys. Rev. D 72, 114510 (2005) [arXiv:hep-lat/0503030].
* [12] R. D. Pisarski and F. Wilczek, Phys. Rev. D 29, 338 (1984).
* [13] F. Basile, A. Pelissetto and E. Vicari, PoS LAT2005, 199 (2006) [arXiv:hep-lat/0509018].
* [14] G. Cossu, M. D’Elia, A. Di Giacomo and C. Pica, arXiv:0706.4470 [hep-lat].
|
arxiv-papers
| 2009-01-21T10:57:22 |
2024-09-04T02:49:00.103383
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. Bonati, G.Cossu, M. D'Elia, A. Di Giacomo, C. Pica",
"submitter": "Guido Cossu",
"url": "https://arxiv.org/abs/0901.3231"
}
|
0901.3308
|
# Description of the vector $G$-bundles over $G$-spaces with quasi-free proper
action of discrete group $G$
Mishchenko , A.S Partly supported by the grant of RFFI No.08-01-00034-a,
NSh-1562.2008.1, Program 2.1.1/5031 Morales Meléndez, Quitzeh
###### Abstract
We give a description of the vector $G$-bundles over $G$-spaces with qusi-free
proper action of discrete group $G$ in terms of the classifying space.
## 1 The setting of the problem
This problem naturally arises from the Conner-Floyd’s description ([2]) of the
bordisms with the action of a group $G$ using the so-called fix-point
construction. This construction reduces the problem of describing the bordisms
to two simpler problems: a) description of the fixed-point set (or, more
generally, the stationary point set), which happens to be a submanifold
attached with the structure of its normal bundle and the action of the same
group $G$, however, this action could have stationary points of lower rank; b)
description of the bordisms of lower rank with an action of the group $G$. We
assume that the group $G$ is discrete.
Lets $\xi$ be an $G$-equivariant vector bundle with base $M$.
$\begin{array}[]{c}\xi\\\ \Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$$}}$\hss}\\\ M\\\ \end{array}$ (1)
Lets $H<G$ be a normal finite subgroup. Assume that the action of the group
$G$ over the base $M$ reduces to the factor group $G_{0}=G/H$:
$\begin{array}[]{ccc}G\times
M&\smash{\mathop{\buildrel\over{\longrightarrow}}}&M\\\ \Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$$}}$\hss}&&\parallel\\\ G_{0}\times
M&\smash{\mathop{\buildrel\over{\longrightarrow}}}&M\\\ \end{array}$ (2)
suppose, additionally, that the action $G_{0}\times
M\smash{\mathop{\buildrel\over{\longrightarrow}}}M$ is free and there is no
more fixed points of the action of the group $H$ in the total space of the
bundle $\xi$.
So, we have the following commutative diagram
$\begin{array}[]{ccc}G\times\xi&\smash{\mathop{\buildrel\over{\longrightarrow}}}&\xi\\\
\Big{\downarrow}\hbox to0.0pt{$\vbox{\hbox{$$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$$}}$\hss}\\\ G_{0}\times
M&\smash{\mathop{\buildrel\over{\longrightarrow}}}&M\\\ \end{array}$ (3)
###### Definition 1
As in [6, p. 210], we shall say that the described action of the group $G$ is
quasi-free over the base with normal stationary subgroup $H$.
Reducing the action to the subgroup $H$, we obtain the simpler diagram:
$\begin{array}[]{ccc}H\times\xi&\smash{\mathop{\buildrel\over{\longrightarrow}}}&\xi\\\
\Big{\downarrow}\hbox to0.0pt{$\vbox{\hbox{$$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$$}}$\hss}\\\ M&=&M\\\ \end{array}$ (4)
Following [4], let
$\rho_{k}:H\smash{\mathop{\buildrel\over{\longrightarrow}}}{\bf U}(V_{k})$ be
the series of all the irreducible (unitary) representation of the finite group
$H$. Then the $H$-bundle $\xi$ can be presented as the finite direct sum:
$\xi\approx\bigoplus_{k}\left(\xi_{k}\bigotimes V_{k}\right),$ (5)
where the action of the group $H$ over the bundles $\xi_{k}$ is trivial,
$V_{k}$ denotes the trivial bundle with fiber $V_{k}$ and with fiberwise
action of the group $H$, defined using the linear representation $\rho_{k}$.
###### Lemma 1
The group $G$ acts on every term of the sum (5) separately.
Proof. Consider now the action of the group $G$ over the total space of the
bundle $\xi$. Fix a point $x\in M$. The action of the element $g\in G$ is
fiberwise, and maps the fiber $\xi_{x}$ to the fiber $\xi_{gx}$:
$\Phi(x,g):\xi_{x}\smash{\mathop{\buildrel\over{\longrightarrow}}}\xi_{gx}.$
Also, for a par of elements $g_{1},g_{2}\in G$ we have:
$\Phi(x,g_{1}g_{2})=\Phi\left(g_{2}x,g_{1}\right)\circ\Phi\left(x,g_{2}\right),$
(6)
$\begin{array}[]{ccccc}\Phi(x,g_{1}g_{2}):\xi_{x}&\smash{\mathop{\buildrel\Phi\left(x,g_{2}\right)\over{\longrightarrow}}}&\xi_{g_{2}x}&\smash{\mathop{\buildrel\Phi\left(g_{2}x,g_{1}\right)\over{\longrightarrow}}}&\xi_{g_{1}g_{2}x}\\\
\end{array}$
In particular, if $g_{2}=h\in H<G$, then $g_{2}x=hx=x$. So,
$\begin{array}[]{ccccc}\Phi(x,gh):\xi_{x}&\smash{\mathop{\buildrel\Phi\left(x,h\right)\over{\longrightarrow}}}&\xi_{x}&\smash{\mathop{\buildrel\Phi\left(x,g\right)\over{\longrightarrow}}}&\xi_{gx}\\\
\end{array}$
Analogously, if $g_{1}=h\in H<G$, then $g_{1}gx=hgx=gx$. So
$\begin{array}[]{ccccc}\Phi(x,hg):\xi_{x}&\smash{\mathop{\buildrel\Phi\left(x,g\right)\over{\longrightarrow}}}&\xi_{gx}&\smash{\mathop{\buildrel\Phi\left(gx,h\right)\over{\longrightarrow}}}\xi_{gx}\\\
\end{array}$
According to [4] the operator $\Phi\left(x,h\right)$ does not depends on the
point $x\in M$,
$\Phi(x,h)=\Psi(h):\bigoplus_{k}\left(\xi_{k,x}\bigotimes
V_{k}\right)\smash{\mathop{\buildrel\over{\longrightarrow}}}\bigoplus_{k}\left(\xi_{k,x}\bigotimes
V_{k}\right),$
here, since the action of the group $H$ is given over every space $V_{k}$
using pairwise different irreducible representations $\rho_{k}$, we have
$\Psi(h)=\bigoplus_{k}\left({\hbox{\bf Id}}\bigotimes\rho_{k}(h)\right).$
In this way, we obtain the following relation:
$\Phi(x,gh)=\Phi(x,g)\circ\Psi(h)=\Phi(x,ghg^{-1}g)=\Psi(ghg^{-1})\circ\Phi(x,g).$
(7)
Lets write the operator $\Phi(x,g)$ using matrices to decompose the space
$\xi_{x}$ as the direct sum
$\xi_{x}=\bigoplus_{k}\left(\xi_{k,x}\bigotimes V_{k}\right):$
$\Phi(x,g)=\left(\begin{array}[]{cccc}\Phi(x,g)_{1,1}&\cdots&\Phi(x,g)_{k,1}&\cdots\\\
\vdots&\ddots&\vdots&\\\ \Phi(x,g)_{1,k}&\cdots&\Phi(x,g)_{k,k}&\cdots\\\
\vdots&&\vdots&\ddots\\\ \end{array}\right)$ (8)
If $k\neq l$ then $\Phi(x,g)_{k,l}=0$, i.e. the matrix $\Phi(x,g)$ its
diagonal,
$\Phi(x,g)=\bigoplus_{k}\Phi(x,g)_{k,k}:\bigoplus_{k}\left(\xi_{k,x}\bigotimes
V_{k}\right)\smash{\mathop{\buildrel\over{\longrightarrow}}}\bigoplus_{k}\left(\xi_{k,gx}\bigotimes
V_{k}\right),$ $\Phi(x,g)_{k,k}:\left(\xi_{k,x}\bigotimes
V_{k}\right)\smash{\mathop{\buildrel\over{\longrightarrow}}}\left(\xi_{k,gx}\bigotimes
V_{k}\right),$
as it was required to prove.
## 2 Description of the particular case $\xi=\xi_{0}\bigotimes V$
Here we will consider the particular case of a $G$-vector bundle
$\xi=\xi_{0}\otimes V$ with base $M$.
$\begin{array}[]{c}\xi\\\ \Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$$}}$\hss}\\\ M\\\ \end{array}$
where the action of the group $G$ is quasi-free over the base with finite
normal stationary subgroup $H<G$.
We will assume that the group $H$ acts trivially over the bundle $\xi_{0}$. By
$V$ we denote the trivial bundle with fiber $V$ and with fiberwise action of
the group $H$ given by an irreducible linear representation $\rho$.
###### Definition 2
A canonical model for the fiber in a $G$-bundle $\xi=\xi_{0}\bigotimes V$ with
fiber $F\otimes V$ is the product $G_{0}\times\left(F\otimes V\right)$ with an
action of the group $G$
$\begin{array}[]{ccc}G\times\left(G_{0}\times\left(F\otimes
V\right)\right)&\smash{\mathop{\buildrel\phi\over{\longrightarrow}}}&G_{0}\times\left(F\otimes
V\right)\\\ \Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$$}}$\hss}\\\ G\times
G_{0}&\smash{\mathop{\buildrel\mu\over{\longrightarrow}}}&G_{0}\\\
\end{array}$
where $\mu$ denotes the natural left action of $G$ on its quotient $G_{0}$,
and
$\phi([g],g_{1}):[g]\times\left(F\otimes
V\right)\to[g_{1}g]\times\left(F\otimes V\right)$
is given by the formula
$\begin{array}[]{ll}\phi([g],g_{1})={\hbox{\bf
Id}}\otimes\rho(u(g_{1}g)u^{-1}(g)).\end{array}$ (9)
where
$u:G\smash{\mathop{\buildrel\over{\longrightarrow}}}H$
is a homomorphism of right $H$-modules by multiplication, i.e.
$u(gh)=u(g)h,\quad u(1)=1,\quad g\in G,h\in H.$
###### Lemma 2
The definition (9) of the action of $G$ is well-defined.
Proof. It is enough to prove that that a) the formula (9) defines an action,
i.e.
$\begin{array}[]{ll}\phi([g],g_{2}g_{1})=\phi([g_{1}g],g_{2})\circ\phi([g],g_{1}),\end{array}$
and b) that the formula (9) does not depends on the chosen representative
$gh\in[g]$:
${\hbox{\bf Id}}\otimes\rho(u(g_{1}g)u^{-1}(g))={\hbox{\bf
Id}}\otimes\rho(u(g_{1}gh)u^{-1}(gh))$
for every $g\in G$ and $h\in H$.
In fact,
$\begin{array}[]{ll}\phi([g],g_{2}g_{1})={\hbox{\bf
Id}}\otimes\rho(u(g_{2}g_{1}g)u^{-1}(g))=\\\ \\\ {\hbox{\bf
Id}}\otimes\rho(u(g_{2}g_{1}g)u(g_{1}g)u^{-1}(g_{1}g)u^{-1}(g))=\\\
={\hbox{\bf Id}}\otimes\rho(u(g_{2}g_{1}g)u(g_{1}g))\circ{\hbox{\bf
Id}}\otimes\rho(u^{-1}(g_{1}g)u^{-1}(g))=\\\ \\\
=\phi([g_{1}g],g_{2})\circ\phi([g],g_{1}),\end{array}$
what proves a), and, recalling the equation $u(gh)=u(g)h$ for every $g\in G$
and $h\in H$, it is clear that
$u(g_{1}gh)u^{-1}(gh)=u(g_{1}g)hh^{-1}u^{-1}(g)=u(g_{1}g)u^{-1}(g),$
which is a sufficient condition for b) to be true.
As it is well known, for the actions we are studying, we can always consider
over the base $M$ an atlas of equivariant charts $\\{O_{\alpha}\\}$,
$M=\bigcup_{\alpha}O_{\alpha},$ $[g]O_{\alpha}=O_{\alpha},\qquad\forall[g]\in
G_{0}.$
If the atlas is fine enough, then every chart can be presented as a disjoint
union of its subcharts:
$O_{\alpha}=\bigsqcup_{[g]\in G_{0}}[g]U_{\alpha}\approx U_{\alpha}\times
G_{0},$
i.e. $[g]U_{\alpha}\cap[g^{\prime}]U_{\alpha}=\emptyset$ if
$[g]\neq[g^{\prime}]$, and when $\alpha\neq\beta$, if
$U_{\alpha}\cap[g_{\alpha\beta}]U_{\beta}\neq\emptyset$, then the element
$g_{\alpha\beta}$ is the only one for which that intersection is non-empty,
i.e. if $[g]\neq[g_{\alpha\beta}]$, then
$U_{\alpha}\cap[g]U_{\beta}=\emptyset$, i.e.
$O_{\alpha}\cap
O_{\beta}\approx\left(U_{\alpha}\cap[g_{\alpha\beta}]U_{\beta}\right)\times
G_{0},$
for every $\alpha,\beta$. We use these facts and notations to formulate the
next theorem.
###### Theorem 1
The bundle $\xi=\xi_{0}\bigotimes V$ is locally homeomorphic to the cartesian
product of some chart $U_{\alpha}$ by the canonical model. More precisely, for
a fine enough atlas, there exist $G$-equivariant trivializations
$\psi_{\alpha}:O_{\alpha}\times\left(F\otimes V\right)\to\xi|_{O_{\alpha}}$
(10)
where
$O_{\alpha}\times\left(F\otimes V\right)\approx
U_{\alpha}\times\left(G_{0}\times\left(F\otimes V\right)\right)$
and the diagram
$\begin{array}[]{ccc}\xi|_{O_{\alpha}}&\smash{\mathop{\buildrel
g\over{\longrightarrow}}}&\xi|_{O_{\alpha}}\\\ \Big{\uparrow}\hbox
to0.0pt{$\vbox{\hbox{$\psi_{\alpha}$}}$\hss}&&\Big{\uparrow}\hbox
to0.0pt{$\vbox{\hbox{$\psi_{\alpha}$}}$\hss}\\\
U_{\alpha}\times\left(G_{0}\times\left(F\otimes
V\right)\right)&\smash{\mathop{\buildrel{\hbox{\bf
Id}}\times\phi(g)\over{\longrightarrow}}}&U_{\alpha}\times\left(G_{0}\times\left(F\otimes
V\right)\right)\\\ \end{array}$ (11)
is commutative where $g\in G$, ${\hbox{\bf Id}}:U_{\alpha}\to U_{\alpha},$ and
$\phi(g)$ denotes the canonical action.
Proof. Using an atlas as in the remarks at the beginning of the theorem, we
shall construct the trivialization (10) starting from an arbitrary
trivialization
$\psi_{\alpha}:U_{\alpha}\times\left(F\otimes V\right)\to\xi|_{U_{\alpha}}$
in such a way, that the diagram
$\begin{array}[]{ccc}\xi|_{U_{\alpha}}&\smash{\mathop{\buildrel
g\over{\longrightarrow}}}&\xi|_{[g]U_{\alpha}}\\\ \Big{\uparrow}\hbox
to0.0pt{$\vbox{\hbox{$\psi_{\alpha}$}}$\hss}&&\Big{\uparrow}\hbox
to0.0pt{$\vbox{\hbox{$\psi_{\alpha}$}}$\hss}\\\ U_{\alpha}\times\left(F\otimes
V\right)&\smash{\mathop{\buildrel\over{\longrightarrow}}}&[g]U_{\alpha}\times\left(F\otimes
V\right)\\\ \end{array}$
commutes for every $g\in[g]$, where the left and upper arrows are given and we
have to construct the down and right arrows.
From such a construction, the equivariance will follow automatically and the
proof of the theorem reduces to show that the constructed down arrow coincides
with that on (11).
Evidently, for a given trivialization
$\psi_{\alpha}:U_{\alpha}\times\left(F\otimes V\right)\to\xi|_{U_{\alpha}}$,
there are several ways to define a trivialization
$\psi_{\alpha}:[g]U_{\alpha}\times\left(F\otimes
V\right)\to\xi|_{[g]U_{\alpha}}$, since there are several elements $g\in G$
sending $\xi|_{U_{\alpha}}$ to $\xi|_{[g]U_{\alpha}}$.
Thus, consider a set-theoretic cross-section
$p^{\prime}:G_{0}\smash{\mathop{\buildrel\over{\longrightarrow}}}G,$
to the projection $p$ in the exact sequence of groups
${\bf
1}\smash{\mathop{\buildrel\over{\longrightarrow}}}H\smash{\mathop{\buildrel\over{\longrightarrow}}}G\smash{\mathop{\buildrel
p\over{\longrightarrow}}}G_{0},$ $p\circ p^{\prime}={\hbox{\bf
Id}}:G_{0}\smash{\mathop{\buildrel
p^{\prime}\over{\longrightarrow}}}G\smash{\mathop{\buildrel
p\over{\longrightarrow}}}G_{0}.$
Put
$g^{\prime}=p^{\prime}\circ
p:G\smash{\mathop{\buildrel\over{\longrightarrow}}}G.$
Without loss of generality, we can take $g^{\prime}(1)=1$.
In this case
$g^{\prime}(g)=gu^{-1}(g),$
where
$u:G\smash{\mathop{\buildrel\over{\longrightarrow}}}H$
is a homomorphism of right $H$-modules by multiplication, i.e.
$u(gh)=u(g)h,\quad g\in G,h\in H.$
In particular, this means that
$g^{\prime}(gh)=g^{\prime}(g),\quad h\in H.$
Lets
$\tilde{\psi}_{\alpha}:U_{\alpha}\times
F\smash{\mathop{\buildrel\over{\longrightarrow}}}\xi_{0}|_{U_{\alpha}}$
be some trivialization. We define the trivialization $\psi_{\alpha}$ in (10)
by the rule: if $[g]x_{\alpha}\in[g]U_{\alpha}$, i.e. $x_{\alpha}\in
U_{\alpha}$, then, the map
$\psi_{\alpha}([g]x_{\alpha}):[g]x_{\alpha}\times\left(F\otimes
V\right)\smash{\mathop{\buildrel\over{\longrightarrow}}}\xi_{[g]x_{\alpha}}\otimes
V$
is given by the formula
$\begin{array}[]{ll}\psi_{\alpha}([g]x_{\alpha})&=\Phi(x_{\alpha},g^{\prime}(g))\circ\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf
Id}}\right)=\\\
&=\Phi(x_{\alpha},gu^{-1}(g))\circ\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf
Id}}\right).\\\ \end{array}$ (12)
where, from the first equality, it is clear that the definition does not
depend on the representative $g\in[g]$.
In particular, for $[g]=1$, we recover the initial trivialization
$\psi_{\alpha}(x_{\alpha})=\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf
Id}}$
since $\Phi(x,g^{\prime}(1))=\Phi(x,1)=1$.
Using this trivialization the action of the group $G$ can be carried to the
cartesian product $O_{\alpha}\times\left(F\otimes V\right)$:
$\Phi_{\alpha}(g):O_{\alpha}\times\left(F\otimes
V\right)\smash{\mathop{\buildrel\over{\longrightarrow}}}O_{\alpha}\times\left(F\otimes
V\right).$
Lets $x_{\alpha}\in U_{\alpha}$, $g\in G$, then
$\Phi_{\alpha}([g]x_{\alpha},g_{1}):[g]x_{\alpha}\times\left(F\otimes
V\right)\smash{\mathop{\buildrel\over{\longrightarrow}}}[g_{1}g]x_{\alpha}\times\left(F\otimes
V\right)$
is given by the formula
$\Phi_{\alpha}([g]x_{\alpha},g_{1})=\left(\psi_{\alpha}([g_{1}g]x_{\alpha})\right)^{-1}\Phi([g]x_{\alpha},g_{1})\psi_{\alpha}([g]x_{\alpha}).$
Applying (12), we obtain
$\begin{array}[]{ll}\Phi_{\alpha}([g]x_{\alpha},g_{1})=&\left(\Phi(x_{\alpha},g_{1}gu^{-1}(g_{1}g))\circ\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf
Id}}\right)\right)^{-1}\circ\\\
&\circ\Phi([g]x_{\alpha},g_{1})\circ\Phi(x_{\alpha},gu^{-1}(g))\circ\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf
Id}}\right)=\\\ \\\ &=\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf
Id}}\right)^{-1}\circ\\\
&\circ\Phi(x_{\alpha},g_{1}gu^{-1}(g_{1}g))^{-1}\circ\Phi([g]x_{\alpha},g_{1})\circ\Phi(x_{\alpha},gu^{-1}(g))\circ\\\
&\circ\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf Id}}\right)=\\\
\\\ &=\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf
Id}}\right)^{-1}\circ\\\
&\circ\Phi(x_{\alpha},u^{-1}(g_{1}g))^{-1}\circ\Phi(x_{\alpha},g_{1}g)^{-1}\circ\Phi([g]x_{\alpha},g_{1})\circ\\\
&\circ\Phi(x_{\alpha},g)\circ\Phi(x_{\alpha},u^{-1}(g))\circ\\\
&\circ\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf Id}}\right)=\\\
\\\ &=\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf
Id}}\right)^{-1}\circ\\\
&\circ\Phi(x_{\alpha},u^{-1}(g_{1}g))^{-1}\circ\Phi(x_{\alpha},u^{-1}(g))\circ\\\
&\circ\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf Id}}\right);\\\
\\\ \end{array}$
$\begin{array}[]{ll}\Phi_{\alpha}([g]x_{\alpha},g_{1})=&\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf
Id}}\right)^{-1}\circ\\\ &\circ\left({\hbox{\bf
Id}}\otimes\rho(u(g_{1}g))\right)\circ\left({\hbox{\bf
Id}}\otimes\rho(u^{-1}(g))\right)\circ\phantom{aaaaaaaaaaaaaaaaa}\\\
&\circ\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf Id}}\right)=\\\
\\\ &=\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf
Id}}\right)^{-1}\circ\\\ &\circ\left({\hbox{\bf
Id}}\otimes\left(\rho(u(g_{1}g)u^{-1}(g))\right)\right)\circ\\\
&\circ\left(\tilde{\psi}_{\alpha}(x_{\alpha})\otimes{\hbox{\bf Id}}\right)=\\\
\\\ &={\hbox{\bf Id}}\otimes\rho(u(g_{1}g)u^{-1}(g)).\end{array}$
The operator
$\begin{array}[]{ll}\Phi_{\alpha}([g]x_{\alpha},g_{1})={\hbox{\bf
Id}}\otimes\rho(u(g_{1}g)u^{-1}(g))=\phi(g_{1},[g]).\end{array}$
does not depend on the point $x_{\alpha}\in U_{\alpha}$. So, the theorem is
proved.
By $\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes V\right)\right)$ we denote
the group of equivariant automorphisms of the space $G_{0}\times\left(F\otimes
V\right)$ as a vector $G$-bundle with base $G_{0}$, fiber $F\otimes V$ and
canonical action of the group $G$.
###### Corollary 1
The transition functions on the intersection
$O_{\alpha}\cap
O_{\beta}\approx\left(U_{\alpha}\cap[g_{\alpha\beta}]U_{\beta}\right)\times
G_{0},$
i.e. the homomorphisms $\Psi_{\alpha\beta}$ on the diagram
$\begin{array}[]{ccc}\left(U_{\alpha}\cap[g_{\alpha\beta}]U_{\beta}\right)\times\left(G_{0}\times\left(F\otimes
V\right)\right)&\smash{\mathop{\buildrel\Psi_{\alpha\beta}\over{\longrightarrow}}}&\left(U_{\alpha}\cap[g_{\alpha\beta}]U_{\beta}\right)\times\left(G_{0}\times\left(F\otimes
V\right)\right)\\\ \Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$$}}$\hss}\\\
\left(U_{\alpha}\cap[g_{\alpha\beta}]U_{\beta}\right)\times
G_{0}&\smash{\mathop{\buildrel{\hbox{\bf
Id}}\over{\longrightarrow}}}&\left(U_{\alpha}\cap[g_{\alpha\beta}]U_{\beta}\right)\times
G_{0}\\\ \end{array}$ (13)
are equivariant with respect to the canonical action of the group $G$ over the
product of the base by the canonical model, i.e.
$\Psi_{\alpha\beta}(x)\circ\phi(g_{1},[g])=\phi(g_{1},[g])\circ\Psi_{\alpha\beta}(x)$
for every $x\in U_{\alpha}\cap[g_{\alpha\beta}]U_{\beta},\;g_{1}\in G,\;[g]\in
G_{0}$, In other words,
$\Psi_{\alpha\beta}(x)\in\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes
V\right)\right).$
Now we give a more accurate description of the group
$\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes V\right)\right)$. By
definition, an element of the group
$\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes V\right)\right)$ is an
equivariant mapping $\mathbf{A}^{a}$, such that the pair $(\mathbf{A}^{a},a)$
defines a commutative diagram
$\begin{array}[]{ccc}\left(G_{0}\times\left(F\otimes
V\right)\right)&\smash{\mathop{\buildrel\mathbf{A}^{a}\over{\longrightarrow}}}&G_{0}\times\left(F\otimes
V\right)\\\ \Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$$}}$\hss}\\\ G_{0}&\smash{\mathop{\buildrel
a\over{\longrightarrow}}}&G_{0},\\\ \end{array}$
which commutes with the canonical action, i.e. the map
$a\in\mathrm{Aut}_{G}(G_{0})$ satisfies the condition
$a\in\mathrm{Aut}_{G}(G_{0})\approx G_{0},\quad a[g]=[ga],\;[g]\in G_{0},$
and the mapping $\mathbf{A}^{a}=(A^{a}[g])_{[g]\in G_{0}}$,
$A^{a}[g]:[g]\times(F\otimes V)\to[ga]\times(F\otimes V)$
satisfies a commutation condition with respect to the action of the group $G$:
$\begin{array}[]{ccc}[g]\times(F\otimes V)&\smash{\mathop{\buildrel
A^{a}[g]\over{\longrightarrow}}}&[ga]\times(F\otimes V)\\\
\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\phi(g_{1},[g])$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\phi(g_{1},[ga])$}}$\hss}\\\ {[g_{1}g]\times(F\otimes
V)}&\smash{\mathop{\buildrel
A^{a}[g_{1}g]\over{\longrightarrow}}}&[g_{1}ga]\times(F\otimes
V)\end{array}\quad,$
$\phi(g_{1},[ga])\circ A^{a}[g]=A^{a}[g_{1}g]\circ\phi(g_{1},[g])$ (14)
i.e.
$({\hbox{\bf
Id}}\otimes\rho(u(g_{1}ga)u^{-1}(ga)))A^{a}[g]=A^{a}[g_{1}g]({\hbox{\bf
Id}}\otimes\rho(u(g_{1}g)u^{-1}(g)))$ (15)
where $[g]\in G_{0},\quad g_{1}\in G$.
###### Lemma 3
One has an exact sequence of groups
${\bf 1}\to
GL(F)\smash{\mathop{\buildrel\over{\longrightarrow}}}\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes
V\right)\right)\smash{\mathop{\buildrel\over{\longrightarrow}}}G_{0}\to{\bf
1}.$ (16)
Proof. To define a projection
$pr:\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes
V\right)\right)\smash{\mathop{\buildrel\over{\longrightarrow}}}G_{0}$
we send the fiberwise map
$\mathbf{A}^{a}:G_{0}\times\left(F\otimes
V\right)\smash{\mathop{\buildrel\over{\longrightarrow}}}G_{0}\times\left(F\otimes
V\right)$
to its restriction over the base $a:G_{0}\to G_{0}$, i.e.
$a\in\mathrm{Aut}_{G}(G_{0})\approx G_{0}$. So, this is a well-defined
homomorphism.
We need to show that $pr$ is an epimorphism and that its kernel is isomorphic
to $GL(F)$. Lets calculate the kernel.
For $[a]=[1]$ we have
$({\hbox{\bf
Id}}\otimes\rho(u(g_{1}g)u^{-1}(g)))A^{1}[g]=A^{1}[g_{1}g]({\hbox{\bf
Id}}\otimes\rho(u(g_{1}g)u^{-1}(g)))$ (17)
In the case $g_{1}=h\in H$, we obtain
$({\hbox{\bf Id}}\otimes\rho(u(hg)u^{-1}(g)))A^{1}[g]=A^{1}[g]({\hbox{\bf
Id}}\otimes\rho(u(hg)u^{-1}(g)))$
Since the representation $\rho$ is irreducible, by Schur’s lemma, we have
$A^{1}[g]=B^{1}[g]\otimes{\hbox{\bf Id}}.$
On the other side, assuming in (17) that $g=1$, we have
$({\hbox{\bf Id}}\otimes\rho(u(g)))A^{1}[1]=A^{1}[g]({\hbox{\bf
Id}}\otimes\rho(u(g))),$
i.e.
$({\hbox{\bf Id}}\otimes\rho(u(g)))(B^{1}[1]\otimes{\hbox{\bf
Id}})=(B^{1}[g]\otimes{\hbox{\bf Id}})({\hbox{\bf Id}}\otimes\rho(u(g))),$
or
$(B^{1}[g]\otimes{\hbox{\bf Id}})=(B^{1}[1]\otimes{\hbox{\bf Id}}).$
So, the kernel $\ker pr$ is isomorphic to the group $GL(F)$.
In the generic case, i.e. $[a]\neq 1$, we can compute the operator $A^{a}[g]$
in terms of its value at the identity $A^{a}[1]$ from the formula (15):
assuming $g=1$, we obtain (changing $g_{1}$ by $g$):
$({\hbox{\bf Id}}\otimes\rho(u(ga)u^{-1}(a)))A^{a}[1]=A^{a}[g]({\hbox{\bf
Id}}\otimes\rho(u(g))),$ (18)
i.e.
$A^{a}[g]=({\hbox{\bf Id}}\otimes\rho(u(ga)u^{-1}(a)))A^{a}[1]({\hbox{\bf
Id}}\otimes\rho(u^{-1}(g))),$ (19)
Therefore, the operator is completely defined by its value
$A^{a}[1]:[1]\times(F\otimes V)\to[a]\times(F\otimes V)$
at the identity $g=1$.
Now we describe the operator $A^{a}[1]$ in terms of the representation $\rho$
and its properties.
We have a commutation rule with respect to the action of the subgroup $H$:
$\begin{array}[]{ccc}[1]\times(F\otimes V)&\smash{\mathop{\buildrel
A^{a}[1]\over{\longrightarrow}}}&[a]\times(F\otimes V)\\\
\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\phi(h,[1])$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\phi(h,[a])$}}$\hss}\\\ {[1]\times(F\otimes
V)}&\smash{\mathop{\buildrel
A^{a}[1]\over{\longrightarrow}}}&[a]\times(F\otimes V)\end{array}\quad,$
Equivalently
$A^{a}[1]\circ\phi(h,[1])=\phi(h,[a])\circ A^{a}[1],$
i.e.
$A^{a}[1]\circ({\hbox{\bf Id}}\otimes\rho(h))=({\hbox{\bf
Id}}\otimes\rho(g^{\prime-1}(a)hg^{\prime}(a)))\circ A^{a}[1],$
i.e.
$A^{a}[1]\circ({\hbox{\bf Id}}\otimes\rho(h))=({\hbox{\bf
Id}}\otimes\rho_{g^{\prime}(a)}(h))\circ A^{a}[1].$
The last equation means that the operator should $A^{a}[1]$ permute these
representations, or equivalently, such an operator exists only when the
representations $\rho$ and $\rho_{g^{\prime}(a)}$ are equivalent. Recalling
the commutation rule (7), we see that this is the case we are been
considering.
Thus, if the representations $\rho$ and $\rho_{g}$ are equivalent, we have an
(inverse) splitting operator $C(g)$, satisfying the equation
$\rho_{g}(h)=\rho\left(g^{-1}hg\right)=C(g)\rho(h)C^{-1}(g).$ (20)
for every $g\in G$. The operator $C(g)$ is defined up to multiplication by a
scalar operator $\mu_{g}\in\SS^{1}\subset{\bf C}^{1}$.
So
$A^{a}[1]\circ({\hbox{\bf Id}}\otimes\rho(h))=({\hbox{\bf Id}}\otimes
C(g^{\prime}(a))\circ\rho(h)\circ C^{-1}(g^{\prime}(a)))\circ A^{a}[1],$
or
$({\hbox{\bf Id}}\otimes C^{-1}(g^{\prime}(a)))\circ A^{a}[1]\circ({\hbox{\bf
Id}}\otimes\rho(h))=({\hbox{\bf Id}}\otimes\rho(h))\circ({\hbox{\bf
Id}}\otimes C^{-1}(g^{\prime}(a)))\circ A^{a}[1],$
Then, by the Schur’s lemma,
$({\hbox{\bf Id}}\otimes C^{-1}(g^{\prime}(a)))\circ
A^{a}[1]=B^{a}[1]\otimes{\hbox{\bf Id}},$
i.e.
$A^{a}[1]=B^{a}[1]\otimes C(g^{\prime}(a)),$
Using the formula (19), we obtain
$A^{a}[g]=({\hbox{\bf Id}}\otimes\rho(u(ga)u^{-1}(a)))(B^{a}[1]\otimes
C(g^{\prime}(a)))({\hbox{\bf Id}}\otimes\rho(u^{-1}(g))),$
i.e.
$A^{a}[g]=B^{a}[1]\otimes(\rho(u(ga)u^{-1}(a))\circ
C(g^{\prime}(a))\circ\rho(u^{-1}(g))).$ (21)
This means, that by defining the matrix $B^{a}[1]$, it is possible to obtain
all the operators $A^{a}[g]$ satisfying the equation (19).
It remains to verify the commutation rule (15), i.e. in the formula
$({\hbox{\bf
Id}}\otimes\rho(u(g_{1}ga)u^{-1}(ga)))A^{a}[g]=A^{a}[g_{1}g]({\hbox{\bf
Id}}\otimes\rho(u(g_{1}g)u^{-1}(g)))$
we substitute the expression (21):
$\begin{array}[]{c}({\hbox{\bf
Id}}\otimes\rho(u(g_{1}ga)u^{-1}(ga)))\circ(B^{a}[1]\otimes(\rho(u(ga)u^{-1}(a))\circ
C(g^{\prime}(a))\circ\rho(u^{-1}(g))))=\\\ \\\
=(B^{a}[1]\otimes(\rho(u(g_{1}ga)u^{-1}(a))\circ
C(g^{\prime}(a))\circ\rho(u^{-1}(g_{1}g))))\circ({\hbox{\bf
Id}}\otimes\rho(u(g_{1}g)u^{-1}(g)))\end{array}$
that is
$\begin{array}[]{c}B^{a}[1]\otimes\rho(u(g_{1}ga)u^{-1}(ga)))\circ(\rho(u(ga)u^{-1}(a))\circ
C(g^{\prime}(a))\circ\rho(u^{-1}(g))))=\\\ \\\
=B^{a}[1]\otimes(\rho(u(g_{1}ga)u^{-1}(a))\circ
C(g^{\prime}(a))\circ\rho(u^{-1}(g_{1}g))))\circ(\rho(u(g_{1}g)u^{-1}(g)))\end{array}$
Note that this identity does not depend on the particular matrix $B^{a}[1]$,
thus, this means that we only need to verify the identity for arbitrary $a,g$
and $g_{1}$:
$\begin{array}[]{c}\rho(u(g_{1}ga)u^{-1}(ga)))\circ(\rho(u(ga)u^{-1}(a))\circ
C(g^{\prime}(a))\circ\rho(u^{-1}(g))))=\\\ \\\
=(\rho(u(g_{1}ga)u^{-1}(a))\circ
C(g^{\prime}(a))\circ\rho(u^{-1}(g_{1}g))))\circ(\rho(u(g_{1}g)u^{-1}(g))),\end{array}$
which is obvious, after the natural simplifications
$\begin{array}[]{c}\rho(u(g_{1}ga)u^{-1}(a))\circ
C(g^{\prime}(a))\circ\rho(u^{-1}(g))))=\\\ \\\
=(\rho(u(g_{1}ga)u^{-1}(a))\circ
C(g^{\prime}(a))\circ\rho(u^{-1}(g))),\end{array}$
So, it follows, that for every element $[a]\in G_{0}$ there exist an element
$(A^{a},a)\in\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes V\right)\right)$.
This means that the homomorphism
$\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes
V\right)\right)\smash{\mathop{\buildrel pr\over{\longrightarrow}}}G_{0}$
is in fact an epimorphism, and the lemma is proved.
It is clear that there is an equivalence between $G$-vector bundles with fiber
$G_{0}\times\left(F\otimes V\right)$ over a (compact) base $X$, where $G$ acts
trivially over the base and canonically over the fiber, and homotopy classes
of mappings from $X$ to the space
$B\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes V\right)\right)$.
Lets denote by $\mathrm{Vect}_{G}(M,\rho)$ the category of $G$-equivariant
vector bundles $\xi=\xi_{0}\otimes V$ with base $M$, where the action of the
group $G$ is quasi-free over the base with finite normal stationary subgroup
$H<G$, the group $H$ acts trivially over the bundle $\xi_{0}$ and $V$ denotes
the trivial bundle with fiber $V$ and with fiberwise action of the group $H$
given by an irreducible linear representation $\rho$. Here we need to require
for the representations $\rho_{g}(h)=\rho(g^{-1}hg)$ to be equivalent for
every $g\in G$, in the other case, in view of the commutation rule, this
category may be void.
This is a category because, in fact, we are just taking vector bundles over
the space $M$, then applying tensor product by the fixed bundle $V$ and
defining some action of the group $G$ over the resulting spaces. The inclusion
$GL(F)\hookrightarrow\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes
V\right)\right)$ from lemma 2 ensures that the identities are included.
Denote by $\mathrm{Bundle}(X,L)$ the category of principal $L$-bundles over
the base $X$.
###### Theorem 2
There is a monomorphism
$\mathrm{Vect}_{G}(M,\rho)\longrightarrow\mathrm{Bundle}(M/G_{0},\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes
V\right)\right)).$ (22)
Proof. By corollary 3, every element $\xi\in\mathrm{Vect}_{G}(M,\rho)$ is
defined by transition functions
$\Psi_{\alpha\beta}:\;\left(U_{\alpha}\cap[g_{\alpha\beta}]U_{\beta}\right)\to\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes
V\right)\right)$
where by construction, when $[g]\neq[g_{\alpha\beta}]$, we have
$U_{\alpha}\cap[g]U_{\beta}=\emptyset$ and if $[g]\neq 1$, then
$U_{\alpha}\cap[g]U_{\alpha}=\emptyset$ and
$U_{\beta}\cap[g]U_{\beta}=\emptyset$. This means that the sets $U_{\alpha}$
and $U_{\beta}$ project homeomorphically to open sets under the natural
projection $M\to M/G_{0}$. So, these transition functions are well-defined
over an atlas of the quotient space $M/G_{0}$ and they form a $G$-bundle with
fiber $G_{0}\times\left(F\otimes V\right)$ over this quotient space.
By the same arguments, it is obvious that every $G$-equivariant map
$h_{\alpha}:O_{\alpha}\times\left(F\otimes V\right)\to
O_{\alpha}\times\left(F\otimes V\right)$ (23)
can be interpreted as a map
$h_{\alpha}:U_{\alpha}\times\left(G_{0}\times\left(F\otimes V\right)\right)\to
U_{\alpha}\times\left(G_{0}\times\left(F\otimes V\right)\right)$ (24)
by means of the homeomorphism $O_{\alpha}\approx U_{\alpha}\times G_{0}$,
where the set $U_{\alpha}$ can be thought as an open set of the space
$M/G_{0}$. Equivalently,
$h_{\alpha}:U_{\alpha}\to\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes
V\right)\right)$ (25)
where $U_{\alpha}$ is homeomorphic to an open set of the space $M/G_{0}$.
Therefore, the map (22) is well defined.
Conversely, if we start from mappings of the form (25) where the sets
$U_{\alpha}$ are open in $M/G_{0}$, by refining the atlas, if it is necessary,
we can always think that the inverse image of the open sets $U_{\alpha}$ under
the quotient map $M\to M/G_{0}$ are homeomorphic to the product
$U_{\alpha}\times G_{0}$ and then obtain mappings of the form (23). Therefore,
the map (22) is a monomorphism.
Of course, the map (22) its not in general an epimorphism, since, when we
define the category $\mathrm{Vect}_{G}(M,\rho)$, we are automatically fixing a
bundle $M\to M/G_{0}$, or equivalently, a homotopy class in
$[M/G_{0},BG_{0}]$.
###### Theorem 3
If the space $X$ is compact, then
$\mathrm{Bundle}(X,\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes
V\right)\right))\approx\bigsqcup_{M\in\mathrm{Bundle}(X,G_{0})}\mathrm{Vect}_{G}(M,\rho).$
(26)
Proof. By theorem 5, there is an inclusion
$\bigcup_{M\in\mathrm{Bundle}(X,G_{0})}\mathrm{Vect}_{G}(M,\rho)\hookrightarrow\mathrm{Bundle}(X,\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes
V\right)\right)).$ (27)
Now we will construct an inverse to the map (27), so the fact that the last
union is disjoint will follow. Let
$\Psi_{\alpha\beta}:\;\left(U_{\alpha}\cap
U_{\beta}\right)\to\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes
V\right)\right)$
be the transition functions of a bundle
$\xi\in\mathrm{Bundle}(X,\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes
V\right)\right))$. By lemma 2, there is a continuous projection of groups
$pr:\mathrm{Aut}_{G}\left(G_{0}\times\left(F\otimes V\right)\right)\to G_{0}$.
So, by composition with $pr$ we obtain a bundle with the discrete fiber
$G_{0}$, and it is well known that $G_{0}$ acts fiberwise and freely over the
total space $M$ of this bundle and that $M/G_{0}=X$.
Also, we can assume that we have chosen an atlas such that there is a
homeomorphism
$M\approx\underset{\alpha}{\bigcup}\left(U_{\alpha}\times
G_{0}\right)\approx\underset{\alpha}{\bigcup}\left(\bigsqcup_{[g]\in
G_{0}}[g]U_{\alpha}\right)$
where the intersections are defined by the rule
$[1]U_{\alpha}\cap[g_{\alpha\beta}]U_{\beta}\approx U_{\alpha}\cap U_{\beta}$
where $[g_{\alpha\beta}]=pr\circ\Psi_{\alpha\beta}$.
On the other hand, we have
$\xi\approx\underset{\alpha}{\bigcup}\left(U_{\alpha}\times\left(G_{0}\times\left(F\otimes
V\right)\right)\right)$
where $U_{\alpha}\times\left(G_{0}\times\left(F\otimes V\right)\right)$
intersects $U_{\beta}\times\left(G_{0}\times\left(F\otimes V\right)\right)$ on
the points $(x,g,f\otimes v)=(x,\Psi_{\alpha\beta}([g],f\otimes
v))=(x,[g_{\alpha\beta}g],A_{\alpha\beta}[g](f\otimes v))$ where $x\in
U_{\alpha}\cap U_{\beta}$ and, once again, we are using lemma 2 for the
description of the operators $\Psi_{\alpha\beta}$.
Taking into account the homeomorphism
$U_{\alpha}\times G_{0}\approx\bigsqcup_{[g]\in G_{0}}[g]U_{\alpha}$
we can rewrite
$([g]x,f\otimes v)=([gg_{\alpha\beta}]x,A_{\alpha\beta}[g](f\otimes v))$
.
Therefore, the projection
$\left(U_{\alpha}\times G_{0}\right)\times\left(F\otimes V\right)\to
U_{\alpha}\times G_{0}$
extends to a well-defined and continuous projection
$\xi\to M.$
It is clear by the preceding formulas, that this projection will be
$G$-equivariant, if $G$ acts canonically over the fibers and in by left
translations on $G_{0}$ under the quotient map $G\to G/H=G_{0}$. So, we have
$\xi\in\mathrm{Vect}_{G}(M,\rho)$.
To end the proof, we make the remark that, by the theory of principal
$G_{0}$-bundles, the construction of the space $M$ is up to equivariant
homeomorphism. This means that the inverse to (27) is well defined.
## References
* [1] Luke G., Mishchenko A. S., Vector Bundles And Their Applications. Kluwer Academic Publishers Group (Netherlands), 1998. ISBN: 9780792351542
* [2] P. Conner, E. Floyd. Differentiable periodic maps. Berlin, Springer-Verlag 1964.
* [3] Palais R.S. On the Existence of Slices for Actions of Non-Compact Lie Groups Ann. Math., 2nd Ser., Vol. 73, No. 2. (1961), pp. 295-323.
* [4] Atiyah M.F., K-theory. Benjamin, New York, (1967).
* [5] Serre J.P., Representations lineáires des groupes finis. Hermann, Paris. 1967.
* [6] Levine M., Serpé C.,On a spectral sequence for equivariant K-theory K-Theory (2008) 38 pp. 177222
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arxiv-papers
| 2009-01-21T16:37:27 |
2024-09-04T02:49:00.109837
|
{
"license": "Public Domain",
"authors": "Alexander S. Mishchenko, Quitzeh Morales Mel\\'endez",
"submitter": "Alexander Mishchenko",
"url": "https://arxiv.org/abs/0901.3308"
}
|
0901.3350
|
# Non-WKB Models of the FIP Effect: Implications for Solar Coronal Heating and
the Coronal Helium and Neon Abundances
J. Martin Laming Space Science Division, Naval Research Laboratory Code
7674L, Washington, D.C. 20375
###### Abstract
We revisit in more detail a model for element abundance fractionation in the
solar chromosphere, that gives rise to the “FIP Effect” in the solar corona
and wind. Elements with first ionization potential below about 10 eV, i.e.
those that are predominantly ionized in the chromosphere, are enriched in the
corona by a factor 3-4. We model the propagation of Alfvén waves through the
chromosphere using a non-WKB treatment, and evaluate the ponderomotive force
associated with these waves. Under solar conditions, this is generally pointed
upwards in the chromosphere, and enhances the abundance of chromospheric ions
in the corona. Our new approach captures the essentials of the solar coronal
abundance anomalies, including the depletion of He relative to H, and also the
putative depletion of Ne, recently discussed in the literature. We also argue
that the FIP effect provides the strongest evidence to date for energy fluxes
of Alfvén waves sufficient to heat the corona. However it appears that these
waves must also be generated in the corona, in order to preserve the rather
regular fractionation pattern without strong variations from loop to loop
observed in the solar corona and slow speed solar wind.
Sun:abundances – Sun:chromosphere – turbulence – waves
## 1 Introduction
Since about 1985 it has been recognized that the composition of the solar
corona and the photosphere are not the same. In the corona, the ratios of
abundances of elements with first ionization potentials (FIP) less than about
10 eV relative to abundances of elements with a FIP greater than about 10 eV
are about a factor of 3-4 times higher than in the photosphere. Those elements
with a FIP greater than 10 eV appear to have a photospheric composition in
general (with respect to hydrogen) in the corona, and the low FIP elements are
enhanced in abundance there. This fractionation has recently been explained
(Laming, 2004a) as being due to the ponderomotive force in the chromosphere
from Alfvén waves. This is usually directed upwards, and acts on chromospheric
ions, but not neutrals. Elements that are predominantly ionized in the
chromosphere (low FIP elements like Al, Mg, Si, Ca, and Fe) can be enhanced in
abundance as they flow up to the corona, whereas high FIP elements such as C,
N, O, Ne, and Ar that are largely neutral appear essentially unaffected. The
abundance of sulfur (FIP = 10.36 eV) is between photospheric and coronal.
It was recognized very early that the most plausible site for FIP
fractionation was the chromosphere, where low FIP elements are generally
ionized and where high FIP elements are at least partially neutral. Henoux
(1995, 1998) reviewed the early models. Those which rely on ion-neutral
separation by diffusion in magnetic fields, in temperature gradients, or in
upward plasma flows suffer from problems of the speed of the process
(diffusion is inherently slow) or the choice of boundary conditions. Realistic
FIP effect models must include some form of external force that acts upon the
plasma ions but not upon the neutrals. The first effort along these lines
(Antiochos 1994) considered the cross-B thermoelectric electric field
associated with the downward heat flux carried by electrons which gives rise
to chromospheric evaporation. This draws ions into the flux tube, enhancing
their abundance over neutrals. The absence of a FIP effect in coronal holes
arises naturally, but in coronal regions where FIP fractionation occurs, a
mass dependence is predicted, which is not observed.
Henoux & Somov (1992) proposed that cross-B pressure gradients arising in
current carrying loops could enhance the ion abundance by a “pinch” force.
Azimuthal motions of the partially ionized photosphere at flux tube boundaries
generate a system of currents flowing in opposite directions, such that the
azimuthal component of the field vanishes at infinity. Details of the
fractionation (mass dependence, degree, etc) remain to be worked out, but it
is thought to begin just above the temperature minimum region at about 4000 K,
and continue until temperatures where all elements are ionized. More recent
suggestions have been that chromospheric ions, but not neutrals, are heated by
either reconnection events (Arge & Mullan 1998), or by waves that can
penetrate down to loop footpoints from the corona (Schwadron, Fisk & Zurbuchen
1999). Strengths and weaknesses of these two models are discussed in some
detail by Laming (2004). For the time being we comment that although both
models seem capable of producing mass independent fractionation of about the
right degree, they, in common with the others mentioned above, only predict
positive FIP effects. In these models there is no possibility of an “inverse
FIP” effect such as seen in the coronae of active stars (see e.g. Feldman &
Laming 2000, Laming 2004 and references therein).
This consideration led Laming (2004) to consider the action of ponderomotive
forces due to Alfvén waves propagating up through the chromosphere and either
transmitting into or being reflected from coronal loops. This force can in
principle be either upward or downward and is given approximately by (see
derivation in Appendix A)
$F={q^{2}\over 4m\left(\Omega^{2}-\omega^{2}\right)}{\partial\delta
E_{\perp}^{2}\over\partial z}$ (1)
where $\Omega$ and $\omega$ are the particle cyclotron frequency and the wave
frequency respectively, $q$ and $m$ the charge and mass, and $E_{\perp}$ is
the wave peak transverse electric field. The dependence on the Alfvén speed,
$V_{A}$, means that the ponderomotive force is usually strongest at the top of
the chromosphere. The ponderomotive acceleration, $F/m$, is independent of the
ion mass, leading to the essentially mass independent fractionation that is
observed. Using an analytic model solar loop from Hollweg (1984), upward
ponderomotive forces on ions are much more common in solar conditions, and for
typical density gradients in the chromosphere can be larger than the
gravitational force downward. The magnitude of the FIP fractionation is
dictated by the resonant properties of the coronal loop and the corresponding
wave energy density. Large loops with resonant frequencies similar to the
chromospheric period of 200-300 seconds admit strong Alfvén wave fluxes with
correspondingly large FIP fractionation. Open field lines, or coronal holes,
with formally infinite wave periods or unresolved fine structures (i.e. the
chromosphere and lower transition region away from coronal loop footpoints,
Feldman 1983, 1987) with much shorter wave periods do not have much Alfvén
wave transmission into them and so would have very low FIP fractionation, as
observed. Using model chromospheres from Vernazza, Avrett, & Loeser (1981),
most of the FIP fractionation is found to occur at the top of the chromosphere
at altitudes with strong density gradients near the plateau regions, where low
FIP elements are essentially completely ionized and high FIP elements are
typically at least 50% ionized. The Laming (2004) model thus comes about as a
natural extension of existing work on Alfvén wave propagation in the solar
atmosphere with essentially no extra physics required.
In this paper we revisit the Laming (2004a) model using a numerical treatment
of Alfvén wave propagation in a coronal loop rooted in the chromosphere at
each footpoint. Section 2 describes this model and the improvements over
Laming (2004a) with a series of illustrative examples. Section 3 gives a more
complete tabulation of the FIP fractionation expected in a variety of
elements, and section 4 gives some discussion of the implications of the new
models, both for the solar coronal abundances of helium and neon, and for MHD
wave origin and propagation in the solar atmosphere, before section 5
concludes.
## 2 Ponderomotive Driving of the FIP Effect
### 2.1 Introduction and Formalism
Laming (2004) used a WKB approximation to treat the case of strong
transmission of Alfvén waves into a coronal loop, and hence evaluate the FIP
fractionation. Here we have extend this work to a non-WKB treatment. The
procedure follows that described in detail by Cranmer & van Ballegooijen
(2005), but applied to closed rather than open magnetic field structures. The
transport equations are (see Appendix B for a derivation)
${\partial I_{\pm}\over\partial t}+\left(u\pm V_{A}\right){\partial
I_{\pm}\over\partial z}=\left(u\pm V_{A}\right)\left({I_{\pm}\over
4H_{D}}+{I_{\mp}\over 2H_{A}}\right),$ (2)
where $I_{\pm}=\delta v\pm\delta B/\sqrt{4\pi\rho}$ are the Elsässer variables
for inward and outward propagating Alfvén waves respectively. The Alfvén speed
is $V_{A}$, the upward flow speed in $u$ and the density is $\rho$. The signed
scale heights are $H_{D}=\rho/\left(\partial\rho/\partial z\right)$ and
$H_{A}=V_{A}/\left(\partial V_{A}/\partial z\right)$. In the solar
chromosphere and in closed loops we may take $u<<V_{A}$.
We use the same chromospheric model as before in Laming (2004), but this time
also embedding it in a 2D force-free magnetic field computed from formulae
given in Athay (1981), and shown in Figure 1. We take a scale here of 1 unit
to 1,000 km, and place the bottom of the plot at 500km altitude in the
chromosphere. The region where the plasma beta (ratio of gas pressure to
magnetic pressure) equals unity is taken at 650 km altitude, or at $y=0.15$ in
the figure. This region is where upcoming acoustic waves from the convection
zone can convert to Alfvén waves by mode or parametric conversion, or where
downgoing Alfvén waves can convert back to acoustic waves. We adopt an
altitude of 650 km as the boundary of our simulations where Alfvén waves are
launched upwards. We consider a loop model similar to that of Hollweg (1984),
the coronal portion of which is illustrated in Figure 2. Waves from beneath
impinge on the right hand side chromosphere-corona boundary and are either
reflected back down, or transmitted into the loop. Waves in the loop section
bounce back and forth, with a small probability of leaking back into the
chromosphere at either end. The magnetic field in the coronal loop section is
uniform, and is simply the extension of chromospheric force-free field into
the corona. In the current work, the non-uniformity of the magnetic field in
the low chromosphere has no effect on the FIP effect, since the fractionation
in this work appears towards the top of the chromosphere where the magnetic
field is almost parallel. The loop density is similarly extrapolated, but with
a density scale height taken here to be equal to the loop length. All models
presented here assume waves propagating up the $y$-axis at $x=0$.
Equations (2) are integrated from a starting point in the left hand side
chromosphere (hereafter chromosphere “A”) where Alfv’en waves leak down into
the chromosphere, back through the corona to the right hand side (chromosphere
“B”) where waves are fed up from below. In this way the reflection and
transmission of Alfvén waves at the loop footpoints and elsewhere is naturally
reconstructed. The velocity and magnetic field perturbations are calculated
from
$\displaystyle\delta v$ $\displaystyle={I_{+}+I_{-}\over 2}$ (3)
$\displaystyle{\delta B\over\sqrt{4\pi\rho}}$ $\displaystyle={I_{+}-I_{-}\over
2}.$ (4)
The wave energy density and positive and negative going energy fluxes are
$\displaystyle U$ $\displaystyle={\rho\delta v^{2}\over 2}+{\delta B^{2}\over
8\pi}={\rho\over 4}\left(I_{+}^{2}+I_{-}^{2}\right)$ (5) $\displaystyle F_{+}$
$\displaystyle={\rho\over 4}I_{+}^{2}V_{A}$ (6) $\displaystyle F_{-}$
$\displaystyle={\rho\over 4}I_{-}^{2}V_{A},$ (7)
and the wave peak electric field appearing in equation (1) is
$\delta E_{\perp}^{2}={B^{2}\over 2c^{2}}\left(I_{+}^{2}+I_{-}^{2}\right).$
(8)
Given the ponderomotive acceleration from equation (1), the FIP fractionations
are calculated in a similar manner to Laming (2004a), but with one important
modification. In the momentum equations (10) and (11) of Laming (2004a), we
add the motion in the wave to the ion and neutral partial pressures, so that
$P_{s,i,n}=\left(k_{\rm
B}T/m_{s}+v_{turb}^{2}+v_{wave}^{2}\right)\rho_{s,i,n}/2$ where the first two
terms in parentheses represent the ion thermal velocity and the microturbulent
velocity in the model chromosphere, and the third term represents the motion
of the ion in the Alfvén wave. In true collisionless plasma, neutrals would
not respond to the wave. However the solar chromosphere is sufficiently
collisional that neutrals move with the ions in the wave motion (e.g. Vranjes
et al., 2008) for wave frequencies well below the charge exchange rate that
couples neutrals and ions, and so neutrals require the same form for their
partial pressure as the ions. Following the derivation through, writing
$\nu_{s,n}\partial P_{s,i}/\partial z+\nu_{s,i}\partial P_{s,n}/\partial z$
from equations (10) and (11) in Laming (2004a) (with the ponderomotive term in
$\partial\rho_{s,i}/\partial z\rightarrow 0$, i.e. $b=0$) we find
${\partial\over\partial z}\left[{\rho_{s}\over 2}\left({k_{\rm B}T\over
m_{s}}+v_{turb}^{2}+v_{wave}^{2}\right)\right]+\rho_{s}\left[g+\nu_{eff}\left(u_{s}-u\right)\right]+\rho_{s}a\xi_{s}\nu_{eff}/\nu_{si}=0$
(9)
where
$\nu_{eff}=\nu_{s,i}\nu_{s,n}/\left[\xi_{s}\nu_{s,n}+\left(1-\xi_{s}\right)\nu_{s,i}\right]$,
with $\xi_{s}$ being the ionization fraction of element $s$, and $\nu_{s,i}$
and $\nu_{s,n}$ the collision rates of ions and neutrals respectively of
element $s$ with the ambient gas, and $a$ the ponderomotive acceleration. This
leads to
${\rho_{s}\left(z_{u}\right)\over\rho_{s}\left(z_{l}\right)}={v_{s}\left(z_{l}\right)^{2}\over
v_{s}\left(z_{u}\right)^{2}}\exp\left(2\int_{z_{l}}^{z_{u}}{-g-\nu_{eff}\left(u_{s}-u\right)+\xi_{s}a\nu_{eff}/\nu_{s,i}\over
v_{s}^{2}}dz\right)$ (10)
where $v_{s}^{2}=kT/m_{s}+v_{turb}^{2}+v_{wave}^{2}$. We argue that in the
absence of the ponderomotive acceleration $a$, the effect of turbulence should
be to fully mix the element composition. Thus we choose $u_{s}$ in equation
(7) such that
${v_{s}\left(z_{l}\right)^{2}\over
v_{s}\left(z_{u}\right)^{2}}\exp\left(2\int_{z_{l}}^{z_{u}}{-g-\nu_{eff}\left(u_{s}-u\right)\over
v_{s}^{2}}dz\right)=1$ (11)
to yield the fractionation by the ponderomotive force as
${\rho_{s}\left(z_{u}\right)\over\rho_{s}\left(z_{l}\right)}=\exp\left(2\int_{z_{l}}^{z_{u}}{\xi_{s}a\nu_{eff}/\nu_{s,i}\over
v_{s}^{2}}dz\right).$ (12)
As mentioned previously (Laming, 2004a), the solar chromosphere is undoubtedly
more active and dynamic than represented by equations (6-9). However this
choice allows us to model a chromosphere which in the absence of Alfvén waves
is completely mixed, presumably by hydrodynamic turbulence, and upon which the
ponderomotive force acts to selectively accelerate chromospheric ions.
Chromospheric simulations excluding turbulence find huge (and unobserved)
variations in various coronal element abundances due to ambipolar and thermal
diffusion (Killie & Lie-Svendson, 2007). It would appear that such abundance
variations are unavoidable in the situation that the chromosphere remains
undisturbed for a sufficient length of time (days to weeks in the case of
Killie & Lie-Svendson, 2007). We argue that hydrodynamic turbulence acts on
timescales much shorter than this (but still longer than that required to
establish the FIP effect) to leave a fully mixed chromosphere in the absence
of ponderomotive forces. We model the ponderomotive acceleration as acting
only on the chromospheric ions, since the ponderomotive acceleration divided
by the flow velocity $a/u\sim 10-1000$ s-1 is much greater than the charge
exchange rate (of order 1 s-1). This rate should also be sufficiently greater
than the turbulent mixing rate discussed above.
We also use a more recent chromospheric model (model C7 in Avrett & Loeser,
2008), introduced as an update to the older VALC model (Vernazza, Avrett, &
Loeser, 1981) used previously in Laming (2004a). We continue to evaluate
photoionization rates using incident spectra based on Vernazza & Reeves (1978)
with the extensions and modifications outlined in Laming (2004a). In most
cases, the “very active region” spectrum is used, being the most consistent
with the underlying chromospheric model.
### 2.2 A Loop On Resonance
In the following, Figures 3, 4, and 5 illustrate the solutions for a loop
100,000 km long, with coronal magnetic field $B=9.9$ G. This yields a
wavelength for 3 minute period waves approximately the same as twice the loop
length, and therefore such waves can be transmitted into the coronal loop
section from the chromosphere. We concentrate on 3 minute waves since unlike 5
minutes waves, these require no special conditions to propagate up into the
corona (de Pontieu et al., 2005). Figure 3 shows the coronal section of the
loop. From top to bottom the three panels give the amplitudes of $\delta v$
and $\delta B/\sqrt{4\pi\rho}$ in units of km s-1. Real and imaginary parts
are given as black and gray lines respectively, with $\delta
B/\sqrt{4\pi\rho}$ and $\delta v$ given by solid and dashed line respectively.
The wave amplitude has been chosen to be $\sim 30$ km s-1 in the corona,
giving a typical spectral line FWHM consistent with observations (e.g.
McIntosh et al., 2008). The second panel gives the wave actions (energy
fluxes) for the left going (solid) and right going (dashed) lines, and their
difference divided by the magnetic field as a dotted line, in arbitrary units.
This last quantity should be a straight horizontal line if energy is properly
conserved in the calculation. The third panel gives the ponderomotive
acceleration, in cm s-2. Throughout the coronal section of the loop, it is
significantly lower than the gravitational acceleration. Solid lines indicate
positive, i.e. right going, and dashed lines indicate left going
accelerations. The oscillation amplitude has been chosen to give mass motions
within observational constraints (Chae et al., 1998; McIntosh et al., 2008),
as measured from line profiles.
Figure 4 shows the same three plots for the left hand side chromosphere “A”,
where waves leak down from the corona, together with a fourth panel showing
the degree of fractionation for the abundance ratios Fe/H, O/H, and He/H. The
ponderomotive acceleration in the chromosphere is much larger than in the
corona, especially towards the top. The most significant fractionation occurs
here, increasing Fe/H in this case by a factor of 1.4 over photospheric
values, O/H by a factor of around 1.25, with He/H remaining nearly unchanged.
Finally Figure 5 shows the same four panels for the chromosphere “B” on the
right hand side, where the upgoing Alfvén waves originate. The ponderomotive
acceleration is still pointed upwards (though is negative in the coordinate
system used here), giving the same FIP fractionations as before. In exact
resonance, the chromospheric ponderomotive force behaves the same as at the
opposite footpoint already shown in Figure 4.
### 2.3 A Loop Off Resonance
Figures 6, 7 and 8 show the same variables as before, but for a loop 100,000
km long and with magnetic field $B=19.8$ G. Now the loop is a quarter
wavelength long, and almost complete reflection of the incident Alfvén waves
on the right hand side takes place. The simulation has been normalized so that
the incident Alfvén wave flux coming up from the chromosphere is about the
same as for the on resonance case. Figure 6 shows that the coronal loop
oscillation is now much weaker than before, by about a factor 20. In the left
hand chromosphere “A” (Figure 7) negligible Alfvén wave flux leaks through and
no FIP fractionation occurs. In the right hand chromosphere “B” (Figure 8),
the behavior is quite different to the previous case. The ponderomotive force
is now downwards pointing for most heights in the chromosphere, and it is
still very small, also giving essentially no FIP fractionation. The downward
directed ponderomotive force might be of interest in cases where the
turbulence is stronger. As reviewed in Laming (2004a), the coronae of various
active stars exhibit an inverse FIP effect, where the low FIPs are depleted in
the corona instead of being enhanced. The reversal of the ponderomotive force
under these conditions is a plausible mechanism for such abundance anomalies.
### 2.4 Loops with Stronger Turbulence
The first case above was designed to give a coronal nonthermal mass motion
within observational limits, i.e. a root mean square $\delta v\simeq 30$ km
s-1. The upgoing energy flux of Alfvén waves at the loop footpoint is $\sim
10^{5}$ ergs cm-2 s-1, and is insufficient to power the coronal radiation
power loss by one to two orders of magnitude. In this subsection we consider
the same loop as in the first case, but with an Alfvén wave upward energy flux
of about $2\times 10^{6}$ ergs cm-2 s-1; sufficient to power radiation from a
100,000 km loop with a density of $10^{8}-10^{9}$ cm-3. The predicted
nonthermal mass motions in the corona are now unphysically high, in excess of
100 km s-1, unless we are able to argue that only a small region of the corona
oscillates with this speed. We discuss this further in subsection 4.1. This is
less of a problem in the transition region where the “classical” transition
region that connects a coronal loop with the chromosphere has only recently
been identified in observations (Peter, 2001), being otherwise masked by
“unresolved fine structures” (Feldman, 1983, 1987). Peter (2001) in fact
observed nonthermal line broadening in what he interprets as the “classical”
transition region approaching the values modeled in this section, and suggests
that they arise from the passage of an Alfvén wave with sufficient energy flux
to heat the corona. In this strong wave field, the behavior of the FIP
fractionation is now subtly different, as shown in Figures 9 and 10 for the
left and right hand side chromospheres (“A” and “B”) respectively. On each
side, Fe is somewhat more enhanced than in the previous case, at 3 - 3.3. O
has a FIP fractionation of about 1.7-1.8, similar to before, but He/H is now
at about 0.8 of its photospheric value.
This new behavior can be understood with reference to equation 5, and the
denominator in the integral, $v_{s}^{2}=kT/m_{s}+v_{turb}^{2}+v_{wave}^{2}$.
In the case that the first two terms in $v_{s}^{2}$ dominate, i.e. weak Alfvén
turbulence, all elements (high FIPs as well as low FIPs) are fractionated
positive to H because H has the largest thermal velocity in the denominator.
When the Alfvénic velocity dominates, the fractionation changes and is
determined solely by the numerator in the integral. In this case the element
that stays neutral the longest, He, as expected, has the lowest abundance in
the corona, being depleted with respect to H. This occurs because H
experiences a stronger ponderomotive enhancement. O/H is unchanged, again as
expected because O and H have very similar ionization potentials and their
ionization structures are locked by charge exchange reactions between them. Fe
remains fractionated with respect to H, by a similar amount as before. The
inclusion of the Alfvén turbulence in $v_{s}$ leads to a natural saturation of
the FIP effect, at about the level observed. Thus for a wide range of
turbulence levels, and FIP effect of around 3 should be expected.
The decrease in He/H is especially interesting. It might be relevant to the He
abundance in the solar wind, of around 4-5% (e.g. Aellig et al., 2001; Kasper
et al., 2007) compared with a photospheric abundance of 8%, also seen in
coronal holes and quiet solar corona (Laming & Feldman, 2001, 2003), and is
discussed further below.
## 3 More Realistic Examples
We put three Alfvén waves with angular frequencies 0.025, 0.022, and 0.016 rad
s-1, with relative intensities 1:0.5:0.25 in the left hand chromosphere
designed to match the network power spectrum displayed in Figure 1 of Muglach
(2003). The loop is 100,000 km long as before, with a magnetic field of 7.1 G,
which puts the 0.025 rad $s^{-1}$ on resonance. In the first case, FIP
fractionations are computed for the left hand chromosphere “A”, using the very
active region spectrum of Vernazza & Reeves (1978), and are given in Table 1.
This is the region where waves leak into the chromosphere from the corona
before being reflected back up again, and should give FIP fractionation. The
corresponding model is shown in Figure 11. There are three important
differences from the tabulation given previously (Table 2 in Laming, 2004a).
The first is that with increasing wave energy flux, the FIP fractionations now
appear to saturate at levels corresponding to a fractionation of low FIP
elements overabundant with respect to high FIP elements by a factor of about
3, and does not increase without limit. This arises from the inclusion of the
term in $v_{wave}^{2}$ in the ion and neutral partial pressures discussed
above, and means that for a wide range of turbulent energy densities, similar
fractionated abundances should result. The other new features, already
mentioned briefly above, are the depletion in the He abundance, and at higher
energy fluxes also the Ne abundance relative to H. These also stem from the
modification to the partial pressures.
These new calculations are compared in Table 1 with observations from
Zurbuchen et al. (2002), Bryans et al. (2008) and Giammanco et al. (2008).
Zurbuchen et al. (2002) give abundances measured in the slow speed solar wind
during 1997/8 relative to O, relative to photospheric abundances given by
Grevesse & Sauval (1998). Bryans et al. (2008) give abundances observed
spectroscopically in a region of quiet solar corona, again tabulated relative
to the photospheric composition of Grevesse & Sauval (1998), with small
modifications by Feldman & Laming (2000). With the exceptions of Mg and K, the
calculated abundances agree well with those observed for a wave energy flux
between one and four times that shown in Figure 11, both for the elements that
are depleted, like He and Ne, and for those enriched. We predict stronger
fractionation in Mg than is in fact observed, and stronger than in Laming
(2004a). The reason for this has been tracked to the use of the newer
chromospheric model from Avrett & Loeser (2008), where H retains a higher
degree of ionization lower in the chromosphere than in the previous VAL
models. This then in turn renders the ionization of Mg probably spuriously
high because of charge transfer ionization with the ambient protons. The
difference in ionization fraction between 0.99 and 0.95 makes a considerable
impact on the fractionations that result. Other low FIP elements, Si, and Fe,
do not have charge transfer ionization rates tabulated by Kingdon & Ferland
(1996), and so are unaffected by this change. The cause of the discrepancy for
K is less clear. Like Na, K is very highly ionized throughout the chromosphere
due to its very low FIP, and should be expected to fractionate strongly,
though Bryans et al. (2008) do comment that their analysis only includes one
line of K IX.
The results of the calculation for the right hand side chromosphere “B” are
given in Figure 12. The loop model chosen is resonant with the 0.025 angular
frequency wave, and this is the component transmitted into the corona. However
the FIP fractionation is significantly reduced by the presence of the other
wave frequencies which are reflected from the corona. In the right hand
chromosphere “B” the weaker components on the left are now the strongest. This
does not produce much change in the ponderomotive acceleration, but increases
the term $v_{wave}^{2}$ in the denominator of the integrand in equation 6,
thereby reducing the fractionation. A wave source in the chromosphere is
unlikely to be monochromatic, and so this situation of partial transmission
and partial reflection with the reduced FIP fractionation will be ubiquitous
in the solar atmosphere. This does not agree with observations, for which
chromosphere “A” is a much better match. We therefore argue that if the FIP
effect is due to the ponderomotive force of Alfvén waves in the chromosphere,
these must have a source in the corona. We return to this thought in
subsection 4.1.
Although this calculation has been done for a closed loop, we expect that this
chromospheric wave pattern will also arise at the footpoint of an open field
line in a coronal hole. In fact the character of our chromospheric solution
matches well with that found in the open field case by Cranmer & van
Ballegooijen (2005), subsequently shown in Cranmer et al. (2007) to exhibit
FIP fractionation similar to that observed in the fast solar wind. We
emphasize this point by showing in Figures 14 and 15 the coronal and
chromospheric portions of an open field flux tube. In this case we start the
integration at an altitude of $5\times 10^{5}$ km with purely outgoing waves,
and work back to the solar surface. This restricts us to the region where the
solar wind outflow speed is still much lower than the Alfvén speed, in keeping
with our assumption of $u<<V_{A}$ above. We take magnetic field from
Banaskiewicz et al. (1998), modified by Cranmer & van Ballegooijen (2005), and
choose a density scale height to match the observed and modeled density
profiles in Laming (2004b). Figure 14 shows $\delta v$ and $\delta
B/\sqrt{4\pi\rho}$, chosen to match the observational and modeling constraints
in Cranmer & van Ballegooijen (2005). Figure 15 shows the extension of these
variables into chromosphere “B”, in a similar manner to the previous figures.
While there is much more to be said about the wave properties in open field
lines, the important point to be made here is that the ponderomotive force
naturally produces a very small fractionation in this geometry. This is
consistent with observed abundances in the fast solar wind (Zurbuchen et al.,
2002) and in coronal holes (Feldman et al., 1998). A tabulation of coronal
hole fractionations is given in Table 2, in a similar format to that in Table
1, using the coronal hole incident spectrum of Vernazza & Reeves (1978).
## 4 Discussion
### 4.1 Alfvén Wave Energy Fluxes
It appears from the forgoing that the abundance anomalies observed in various
regions of the solar corona may yield inferences on the energy fluxes of
Alfvén waves in the chromosphere. Our initial considerations then imply that
wave energy fluxes sufficient to heat the solar corona or accelerate the solar
wind are necessary to produce the correct fractionation. Energy fluxes
observed in slow mode and fast mode waves are not sufficient to heat the solar
corona (Erdélyi & Fedun, 2007). The detection of Alfvén waves, the favored
mode for transporting energy to the solar corona, is much harder, since they
are incompressible and can only be revealed through Doppler shifts or motions,
which become hard to see in inhomogeneous conditions where Alfvén waves on
neighboring flux surfaces can propagate at different speeds and lose phase
coherence. However Tomczyk et al. (2007) claim the detection of Alfvén waves
in the solar corona, albeit with insufficient energy flux to heat the corona.
van Doorsselaere et al. (2008) argue that the detected waves are in fact kink
mode waves, for the reasons suggested above. de Pontieu et al. (2007) observe
transverse waves in the chromosphere, which they argue should be interpreted
as Alfvén waves in the absence of a chromospheric waveguide. However these
waves are inferred from the observed oscillations of spicules, which clearly
have radial structure. The energy flux detected by these authors $\sim 10^{5}$
ergs cm-2 s-1 is close to being sufficient to heat the solar corona or
accelerate the solar wind.
In this paper we argue that the FIP effect is due to the ponderomotive force
associated with transverse waves in the chromosphere. Longitudinal MHD waves
do not generate electric field. In order to generate the observed FIP
fractionation, the energy fluxes associated with these waves need to be of
order $10^{6}-10^{7}$ ergs cm-2 s-1, much closer to those required for coronal
heating. It is clear that FIP fractionation is associated with the
transmission of waves between the chromosphere and the corona, and
correspondingly we argue that the required transverse waves should be
identified as Alfvén waves to meet this condition. The fast mode totally
internally reflects somewhere in the transition region or low corona (Schwartz
& Leroy 1982, Leroy & Schwartz 1982).
The nonthermal mass motions predicted in the coronal section of this loop are
higher than observed. In the transition region, this is not necessarily a
problem since ample evidence exists to show that the “classical” transition
regions of coronal loops are rarely observed, being masked as they are by a
population of smaller “unresolved fine structures” (Feldman, 1983, 1987;
Peter, 2001). In the corona, this may also be true if the heating occurs in
thin filament or shells as in Alfvén resonance models (e.g. Terradas et al.,
2008) while the rest of the emitting loop undergoes much slower oscillations.
Another possibility might be the generation of turbulence following coronal
reconnection events associated with nanoflares (Dahlburg et al., 2005). In
each case it is likely that the turbulence would actually be produced in the
coronal section of the loop, not in the chromosphere, and will also be in
resonance with the loop. This also appears to be the conclusion to be drawn
from section 3. Chromosphere “A” where waves leak down from the corona before
being reflected back again gives stronger and more consistent FIP
fractionations for a wide variety of wave spectra than chromosphere “B”, where
waves are incident upwards on the loop from the chromosphere below. We
therefore argue that the FIP effect is more likely to arise with a coronal
source of Alfvén waves, rather than a chromospheric source as originally
conceived in Laming (2004a), and that this inference will constrain the means
by which the corona may be heated.
### 4.2 Helium and Neon in the Solar Corona
The fractionations computed in this paper differ from those in Laming (2004a)
in three notable ways. First, as the turbulent energy density increases, the
fractionation does not increase without limit but saturates at values broadly
consistent with those observed. This is due to a refinement in our formalism
discussed above, where the wave oscillation velocity is included in the ion
and neutral partial pressures in equation 3.
The second is that at high turbulence levels, He becomes significantly
depleted relative to H. Comparing Tables 1 and 2, we find a stronger depletion
in the coronal loop, representative of the slow speed solar wind, than we
would in a coronal hole, the source of the fast wind. The abundance ratio He/H
in the fast solar wind is fairly constant at about 5% (Aellig et al., 2001;
Zurbuchen et al., 2002), or a depletion of 0.59 from the photospheric value of
8.5%. He/H in the slow speed solar is lower, and generally more variable.
Aellig et al. (2001) and Kasper et al. (2007) find He/H varying with wind
speed, with these variations being more pronounced at solar minimum, where
He/H $\sim 1\%$ for speeds below 300 km s-1, approaching 4.5% for speeds above
500 km s-1. At solar maximum, He/H is always in the range 3.5 - 5%. Kasper et
al. (2007) also find a dependence on heliographic latitude during periods of
solar minimum, with lower He/H being found closer to the heliographic equator.
Table 1 gives values of He/H down to about 3.5%. Overall though, our modeled
values for the abundance ratio He/H are very encouragingly consistent with
observations, lending confidence to our approach.
The third, and most controversial new feature is the similar depletion
predicted for Ne. This was originally suggested by Drake & Testa (2005) from a
survey of the Ne/O abundance ratio in a sample of late-type stellar coronae,
as a solution to the problem in helioseismology presented by the reduction in
the solar photospheric abundance of O (see Caffau et al., 2008, and references
therein). Specifically, (Basu & Antia, 2004) the depth of the solar convection
zone demands a metallicity higher than that coming from the standard solar
composition, with the O abundance revised downwards by nearly a factor of 1.5
(Asplund et al., 2004) from Grevesse & Sauval (1998). Ne, having no
photospheric lines on which to base an abundance measurement, was suggested as
the element most likely to resolve this by having a higher postulated
abundance (e.g. Bahcall et al., 2005; Basu & Antia, 2008). Drake & Testa
(2005) find coronal Ne/O typically $\sim 0.4$ in stars which exhibit either no
FIP effect or an inverse FIP effect, and argued that the general consistency
of Ne/O among their sample of 21 stars suggests no significant fractionation
between Ne and O here between photosphere and corona. The solar coronal
abundance ratio Ne/O, measured at $\simeq 0.15-0.18$ (Schmelz et al., 2005;
Young, 2005) would imply therefore that Ne is depleted in the solar corona
relative to the photosphere, similarly to He. Our calculations in Table 1
provide some support to this view, especially at higher turbulence levels,
where Ne/O is about 0.5 of its photospheric value.
### 4.3 Fractionation in the Low Chromosphere
One main feature of Laming (2004) model and the calculations presented above
is that the fractionation is predicted to occur relatively high up in the
chromosphere, at altitudes greater than 2000 km. However in the literature
there are already indications that, at least in active regions and flares,
that fractionation should set in lower down.
In an analysis of HRTS II (the Naval Research Laboratory’s High Resolution
Telescope and Spectrograph) data, Athay (1994) observed variations in the C I
1561Å /Fe II 1563 Å line intensity ratio. Compared to plage regions around a
sunspot, the sunspot itself has a higher ratio C I/Fe II, while surrounding C
I dark flocculi have a lower ratio. Similar results are found by Doschek, Dere
& Lund (1991) and Feldman, Widing & Lund (1990). This absence of fractionation
in the sunspot presumably relates to the absence of acoustic waves in sunspots
(e.g. Muglach, Hofmann, & Staude, 2005) because convection is inhibited by the
strong magnetic field (Parchevsky & Kosovichev, 2007). The fact that this is
observed in lines of C I and Fe II suggests that fractionation must set in at
lower altitudes than originally modeled by Laming (2004; see Figure 1 (left
and right panels), where O and C are becoming ionized in the region of
fractionation, and one would expect neutral O I and C I to be emitted from
lower, unfractionated layers).
The existence of fractionation at these low altitudes also offers a possible
explanation of the observation by Phillips et al. (1994) who found rather
small difference in the abundances of Fe determined from soft X-ray flare
plasma, compared with that lower down in the atmosphere, determined from the
Fe K$\beta$ fluorescent line, rather than the strong FIP effect expected. More
recently Murphy & Share (2005) studied $\gamma$-ray emission from flares.
Protons accelerated into the chromosphere by the flare excite $\gamma$-ray
emission from the ambient plasma when its density reaches about
$10^{14}-10^{15}$ cm-3. Element abundances determined from the resulting
$\gamma$-ray spectrum show the presence of a FIP fractionation. The densities
at which this occurs correspond to the low chromosphere where sound and Alfvén
speeds are approximately equal, and certainly not the Lyman $\alpha$ plateau
region where fractionation is expected in the Laming (2004) model. A search
for FIP fractionation in photospheric lines i.e. below the chromosphere
(Sheminova & Solanki 1999) reveals very little, if any fractionation. Thus all
available observational evidence suggests that the low chromosphere as another
plausible place for FIP fractionation to occur.
We speculate that the growth of Alfvén waves from sound waves near the
$\beta=1$ layer will give an extra ponderomotive force in this region that can
account for this. Zaqarashvili & Roberts (2006) give a treatment of the
parametric conversion of sound waves into Alfvén waves which requires
$\beta=1$ when both are traveling in the same direction along the magnetic
field. This distinguishes it from the phenomenon of mode conversion, which
requires nonzero wavevector perpendicular to the magnetic field to proceed
(e.g. McDougall & Hood 2007), and parametric conversion lower down in high
$\beta$ plasma where the sound waves must be oblique (Zaqarashvili & Roberts,
2002). Waves impinging on the chromosphere from below are fast magnetoacoustic
waves from the high $\beta$ (gas pressure/magnetic pressure) solar interior.
In the absence of mode conversion, these retain their acoustic character
propagating as a slow mode wave when $\beta<<1$ further up in the chromosphere
(McDougall & Hood 2007). At the altitude where $\beta\simeq 1$ (i.e. where the
phase speeds of magnetic and acoustic waves are similar) these waves can mode
convert into other MHD wave modes (Bogdan et al. 2003). This would be
consistent with the findings of Sheminova & Solanki (1999), who find
essentially no FIP effect at photospheric altitudes. Acoustic waves will
produce no ponderomotive force, and only once mode conversion to the other MHD
modes has occurred can fractionation proceed.
## 5 Conclusions
In conclusion then we have refined the model of Laming (2004a) for FIP
fractionations arising from the ponderomotive force as Alfvén waves propagate
through the chromosphere. We have implemented a non-WKB treatment of the wave
transport, which can be further modified to include the effects of wave growth
and damping, and made a correction to the previous formalism to include the
Alfvén wave transverse velocity in the chromospheric ion and neutral partial
pressures. The new effects are a saturation of the FIP effect at the correct
level, and predicted depletions in the coronal abundances of He and Ne, again
consistent with observations. We find the best match to the observed coronal
or solar wind element abundances arises for models with an Alfvén wave energy
fluxes sufficient to heat the corona or accelerate the solar wind. The
inference that a coronal source of Alfvén waves provides a FIP effect better
matching the observations suggests that coronal abundance anomalies may
provide novel insights into the coronal heating mechanism(s).
This work was supported by NASA Contract NNG05HL39I, and by basic research
funds of the Office of Naval Research. I thank Daniel Savin, Cara Rakowski and
an anonymous referee for comments on the manuscript.
## Appendix A The Ponderomotive Force
The ponderomotive force arises from the effects of wave refraction in an
inhomogeneous plasma. In a nonmagnetic plasma, the refractive index,
$\sqrt{\epsilon}$, is given by $\epsilon=1-\omega_{p}^{2}/\omega^{2}$ where
$\omega_{p}$ is the plasma frequency. Waves are refracted to high refractive
index, which means low plasma density. The increased wave pressure can then
expel even more plasma from the low density region, leading to ducting
instabilities. In magnetic plasma,
$\epsilon=1-\omega_{p}^{2}/\left(\omega^{2}-\Omega^{2}\right)$, where $\Omega$
is the ion cyclotron frequency. Thus waves refract to high density regions,
and plasma is attracted to regions of high wave energy density. A simple
expression for the ponderomotive force on an ion may be derived as follows.
The Lagrangian density for a system of thermal plasma of density $n$ with
particle mass $m$ and waves is
$L=\sum_{i}{1\over
2}m_{i}\left(v_{thi,i}^{2}+v_{osc,i}^{2}\right)+\sum_{i}{q_{i}\over
c}\left({\bf v}_{th,i}+{\bf v}_{osc,i}\right)\cdot\delta{\bf
A}+{\epsilon\delta E^{2}-\delta B^{2}\over 8\pi}$ (A1)
where $v_{th,i}$ is the thermal speed and $v_{osc,i}$ is the oscillatory speed
induced by the wave of particle $i$, with mass $m_{i}$, and charge $q_{i}$.
Wave electric and magnetic fields are given by $\delta{\bf E}$ and $\delta{\bf
B}$ respectively, and $\delta{\bf A}$ is the vector potential. We have omitted
the interaction term involving the electrostatic potential, since this is
constant in a neutral plasma. Putting $\delta
B^{2}/8\pi=\sum_{i}mv_{osc,i}^{2}/2+\delta E^{2}/8\pi$ and ${\bf
v}_{osc,i}\cdot\delta{\bf A}=0$ for MHD waves, then
$L=\sum_{i}{1\over 2}mv_{thi,i}^{2}+\sum_{i}{q_{i}\over c}{\bf
v}_{th,i}\cdot{\bf A}+{\left(\epsilon-1\right)\delta E^{2}\over
8\pi}=\sum_{i}{1\over 2}mv_{thi,i}^{2}+\sum_{i}{q_{i}\over c}{\bf
v}_{th,i}\cdot{\bf A}+\sum_{i}{q_{i}^{2}\over
2m_{i}\left(\Omega_{i}^{2}-\omega^{2}\right)}{\delta E^{2}}.$ (A2)
The “$z$” Euler-Lagrange equation gives
${d\over dt}\left(mv_{th,iz}\right)={q_{i}^{2}\over
2m_{i}\left(\Omega_{i}^{2}-\omega^{2}\right)}{d\delta E^{2}\over dz},$ (A3)
neglecting the spatial variation of $B$ and hence $\Omega_{i}$, and evaluating
for the component of $v_{th,i}$ orthogonal to ${\bf A}$ and ${\bf B}$. This is
the same as the expression derived by Landau, Lifshitz & Pitaevskii (1984),
and agrees with earlier work (e.g. Lee & Parks 1983) if $\delta E^{2}=\delta
E_{p}^{2}/2$, where $\delta E_{p}$ is the peak electric field in the wave,
giving a ponderomotive force
$F_{i}={q_{i}^{2}\over 4m_{i}\left(\Omega_{i}^{2}-\omega^{2}\right)}{d\delta
E_{p}^{2}\over dz}.$ (A4)
When $\omega<<\Omega_{i}$, the ponderomotive acceleration is thus independent
of ion mass, which is one crucial property relevant to obtaining an almost
mass independent fractionation as observed. It is also independent of ion
change, so long as the ion is charged (and not neutral). Litwin & Rosner
(1998) give a similar expression derived from the ${\bf j}\times{\bf B}$ term
in the MHD momentum equation.
## Appendix B The Non-WKB Transport Equations
We start from the linearized MHD force and induction equations,
$\rho{\partial\delta{\bf v}\over\partial t}+\nabla\left(\rho{\bf
u}\cdot\delta{\bf v}\right)={\left(\nabla\times\delta{\bf B}\right)\times{\bf
B}\over 4\pi}={\left({\bf B}\cdot\nabla\right)\delta{\bf
B}-\left(\nabla\delta{\bf B}\right)\cdot{\bf B}\over 4\pi},$ (B1)
and
${\partial\delta{\bf B}\over\partial t}=\nabla\times\left(\delta{\bf
v}\times{\bf B}\right)+\nabla\times\left({\bf u}\times\delta{\bf
B}\right)=\left({\bf B}\cdot\nabla\right)\delta{\bf v}-\delta{\bf
B}\nabla\cdot{\bf u}-\left({\bf u}\cdot\nabla\right)\delta{\bf B},$ (B2)
where ${\bf u}$ and ${\bf B}$ are the unperturbed velocity and magnetic field,
$\delta{\bf v}$ and $\delta{\bf B}$ are the perturbations, and $\rho$ is the
density. Equation (B1) is rewritten using $\nabla\left(\rho{\bf
u}\cdot\delta{\bf v}\right)=\rho{\bf u}\times\nabla\times\delta{\bf
v}+\delta{\bf v}\times\nabla\times\left(\rho{\bf u}\right)+\left(\rho{\bf
u}\cdot\nabla\right)\delta{\bf v}+\left(\delta{\bf
v}\cdot\nabla\right)\rho{\bf u}$ to yield
${\partial\delta{\bf v}\over\partial t}+\left({\bf
u}\cdot\nabla\right)\delta{\bf v}={\bf V}_{A}\cdot\nabla\left(\delta{\bf
B}\over\sqrt{4\pi\rho}\right)+{\delta{\bf B}\over\sqrt{4\pi\rho}}{{\bf
V}_{A}\cdot\nabla\rho\over 2\rho}+{\left(\nabla{\bf B}\right)\cdot\delta{\bf
B}\over 4\pi\rho}-{\delta{\bf v}\cdot\nabla\left(\rho{\bf u}\right)\over\rho}$
(B3)
where ${\bf V}_{A}={\bf B}/\sqrt{4\pi\rho}$ is the Alfvén velocity. Writing
$\left(\nabla{\bf B}\right)\cdot\delta{\bf B}=\left(\partial B_{x}/\partial
x\right)\delta{\bf B}=-\left(\partial B_{z}/\partial z\right)\delta{\bf B}/2$
since $\nabla\cdot{\bf B}=0$ (assuming $\partial B_{x}/\partial x=\partial
B_{y}/\partial y$), and similarly for $\left(\nabla\rho{\bf
u}\right)\cdot\delta{\bf v}$, and using $\partial\left(\rho
u_{z}/B_{z}\right)/\partial z=0$ gives
${\partial\delta{\bf v}\over\partial t}+\left({\bf
u}\cdot\nabla\right)\delta{\bf v}={\bf V}_{A}\cdot\nabla\left(\delta{\bf
B}\over\sqrt{4\pi\rho}\right)+{\delta{\bf B}\over\sqrt{4\pi\rho}}{V_{A}\over
2H_{D}}-{\delta{\bf B}\over\sqrt{4\pi\rho}}{V_{A}\over 2H_{B}}+\delta{\bf
v}{u\over 2H_{B}}.$ (B4)
Here $1/H_{B}=\partial\ln B_{z}/\partial z$, $1/H_{D}=\partial\ln\rho/\partial
z$, and below $1/H_{A}=\partial\ln V_{A}/\partial z$. Similar manipulations
give the induction equation in the form
${\partial\over\partial t}\left(\delta{\bf
B}\over\sqrt{4\pi\rho}\right)+\left({\bf u}\cdot\nabla\right){\delta{\bf
B}\over\sqrt{4\pi\rho}}=\left({\bf V}_{A}\cdot\nabla\right)\delta{\bf
v}+{\delta{\bf B}\over\sqrt{4\pi\rho}}{u\over 2H_{D}}+\delta{\bf v}{V_{A}\over
2H_{B}}-{\delta{\bf B}\over\sqrt{4\pi\rho}}{u\over 2H_{B}}.$ (B5)
Taking equation (B4) plus or minus equation (B5) and rearranging gives the
final result,
${\partial I_{\pm}\over\partial t}+\left(u\pm V_{A}\right){\partial
I_{\pm}\over\partial z}=\left(u\pm V_{A}\right)\left({I_{\pm}\over
4H_{D}}+{I_{\mp}\over 2H_{A}}\right),$ (B6)
where $I_{\pm}=\delta{\bf v}\pm\delta{\bf B}/\sqrt{4\pi\rho}$, representing
waves propagating in the $\mp$ z-directions.
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Table 1: Coronal FIP Fractionations ratio | relative wave energy flux | obs. |
---|---|---|---
| 1/64 | 1/16 | 1/4 | 1 | 4 | 16 | 64 | a | b | c
He/H | 1.0 | 1.1 | 1.1 | 0.79 | 0.55 | 0.46 | 0.44 | 0.68 | |
C/H | 1.1 | 1.3 | 1.5 | 1.2 | 0.84 | 0.66 | 0.60 | 1.36 | |
N/H | 1.1 | 1.3 | 1.5 | 1.2 | 0.86 | 0.72 | 0.69 | 0.72 | |
O/H | 1.1 | 1.4 | 1.9 | 1.8 | 1.3 | 1.1 | 1.1 | 1.00 | |
Ne/H | 1.1 | 1.2 | 1.4 | 1.1 | 0.76 | 0.63 | 0.60 | 0.58 | |
Na/H | 1.2 | 1.9 | 4.6 | 9.8 | 13. | 13. | 13. | | 7.8${+13\atop-5}$ | 2.1${+2\atop-1}$
Mg/H | 1.2 | 1.7 | 3.5 | 6.1 | 6.9 | 6.6 | 6.3 | 2.58 | 2.8${+2.3\atop-1.3}$ | 3.0${+1.7\atop-1.1}$
Al/H | 1.2 | 1.6 | 2.8 | 3.6 | 3.1 | 2.7 | 2.5 | | 3.6${+1.7\atop-1.2}$ | 6.8${+4.0\atop-2.5}$
Si/H | 1.2 | 1.7 | 3.0 | 4.2 | 3.8 | 3.2 | 3.1 | 2.49 | 5.1${+3\atop-1.9}$ |
S/H | 1.1 | 1.4 | 2.0 | 1.9 | 1.4 | 1.1 | 1.0 | 1.62 | 2.2$\pm 0.2$ | 2.3${+1.3\atop-0.8}$
Ar/H | 1.1 | 1.4 | 1.8 | 1.5 | 1.1 | 0.91 | 0.87 | | |
K/H | 1.3 | 2.3 | 8.0 | 25. | 38. | 41. | 42. | | 1.8${+0.4\atop-0.6}$ | 4.2${+6.3\atop-2.5}$
Ca/H | 1.2 | 1.6 | 2.6 | 3.0 | 2.3 | 1.9 | 1.8 | | 3.5${+4.3\atop-1.9}$ | 3.1${+1.8\atop-1.1}$
Fe/H | 1.2 | 1.6 | 2.8 | 3.3 | 2.6 | 2.2 | 2.1 | 2.28 | 4.4$\pm 0.5$ |
Ni/H | 1.2 | 1.6 | 2.4 | 2.5 | 1.9 | 1.5 | 1.5 | | |
Kr/H | 1.1 | 1.4 | 1.9 | 1.7 | 1.3 | 1.1 | 1.0 | | |
Rb/H | 1.3 | 2.3 | 7.4 | 19. | 25. | 25. | 25. | | |
W/H | 1.3 | 2.3 | 6.9 | 16. | 18. | 17. | 17. | | |
Table 2: Coronal Hole FIP Fractionations ratio | relative wave energy flux | obs.
---|---|---
| 1/64 | 1/16 | 1/4 | 1 | 4 | 16 | 64 |
He/H | 1.0 | 1.0 | 1.0 | 1.0 | 0.95 | 0.92 | 0.91 | 0.58
C/H | 1.0 | 1.0 | 1.1 | 1.1 | 0.99 | 0.96 | 0.95 | 1.41
N/H | 1.0 | 1.0 | 1.1 | 1.1 | 1.0 | 0.97 | 0.96 | 0.93
O/H | 1.0 | 1.1 | 1.1 | 1.2 | 1.1 | 1.0 | 1.0 | 1.00
Ne/H | 1.0 | 1.0 | 1.1 | 1.0 | 0.98 | 0.94 | 0.93 | 0.47
Na/H | 1.2 | 1.3 | 1.4 | 1.4 | 1.3 | 1.2 | 1.2 |
Mg/H | 1.1 | 1.2 | 1.3 | 1.2 | 1.2 | 1.1 | 1.1 | 1.92
Al/H | 1.1 | 1.1 | 1.1 | 1.1 | 1.1 | 1.0 | 1.0 |
Si/H | 1.1 | 1.1 | 1.1 | 1.1 | 1.1 | 1.0 | 1.0 | 1.86
S/H | 1.0 | 1.0 | 1.1 | 1.1 | 1.0 | 0.98 | 0.97 | 1.56
Ar/H | 1.0 | 1.1 | 1.1 | 1.1 | 1.0 | 1.0 | 0.99 |
K/H | 1.2 | 1.2 | 1.3 | 1.3 | 1.2 | 1.2 | 1.2 |
Ca/H | 1.0 | 1.1 | 1.1 | 1.1 | 1.1 | 1.0 | 1.0 |
Fe/H | 1.0 | 1.1 | 1.1 | 1.1 | 1.1 | 1.0 | 1.0 | 1.67
Ni/H | 1.0 | 1.1 | 1.1 | 1.1 | 1.1 | 1.0 | 1.0 |
Kr/H | 1.0 | 1.1 | 1.1 | 1.1 | 1.1 | 1.0 | 0.99 |
Rb/H | 1.2 | 1.2 | 1.3 | 1.3 | 1.3 | 1.2 | 1.2 |
W/H | 1.1 | 1.1 | 1.2 | 1.2 | 1.1 | 1.1 | 1.1 |
Figure 1: Force free magnetic field, computed from Athay (1981), from the
center of a network segment ($x=0$) to the center of a supergranule cell
($x=1$). We take $x=1$ to represent 1000 km, and y=0 to represent an altitude
of 500 km above the photosphere. The solid lines represent magnetic lines of
force, and dashed lines are logarithmically spaced contours of the Alfvén
speed, assuming the density falls off exponentially with height. FIP
fractionation in this work occurs towards the top of the chromosphere, where
the magnetic field is nearly parallel. Figure 2: Cartoon illustrating the
model. Alfvén waves are incident on the coronal loop from below on the right
hand side. Waves are either transmitted into the loop or reflected back down
again. Waves in the coronal loop bounce back and forth, with some leakage at
each footpoint. The magnetic field is taken to be uniform in the coronal
section of the loop (illustrated), while it varies according to Figure 1
within the chromosphere (not shown on this figure). Figure 3: Coronal section
of loop, length 100,000 km, magnetic field 9.9 G, (half wavelength long)
showing from top: Elsässer variables in km s-1 ($\delta B/\sqrt{4\pi\rho}$
solid lines, $\delta v$ dashed lines), with black lines for real parts and
gray lines for imaginary parts. The loop is approximately half a wavelength
long. Middle; wave energy fluxes in ergs cm-2 s-1, the thin solid line shows
the difference in energy fluxes divided by the magnetic field strength and
should be a horizontal line if energy is properly conserved. Bottom, the
ponderomotive acceleration in cm s-2. Positive acceleration means positive
along the $z$ axis, which is upwards pointing near $z=0$ and downwards near
$z=100,000$. Figure 4: Same as figure 3 giving the first three panels for the
left hand chromosphere “A”, where waves leak down from the corona. The extra
bottom right panel shows the FIP fractionation for Fe, O, and He, relative to
H. Figure 5: Same as figure 4 for the right hand side chromosphere “B”, where
Alfvén waves are launched up from the convection zone. Figure 6: Coronal
section of loop, length 100,000 km, magnetic field 19.8 G, showing from top:
Elsässer variables in km s-1 ($\delta B/\sqrt{4\pi\rho}$ solid lines, $\delta
v$ dashed lines), with black lines for real parts and gray lines for imaginary
parts. The loop is now a quarter wavelength long and reflects most Alfvén
waves incident from below, and consequently has much smaller nonthermal
motions than the previous resonant case. Middle; wave energy fluxes in ergs
cm-2 s-1. Bottom, the ponderomotive acceleration in cm s-2. Figure 7: Same as
figure 6 giving the first three panels for the left hand chromosphere “A”,
where waves leak down from the corona. The extra bottom right panel shows the
FIP fractionation for Fe, O, and He, relative to H. In the case of a loop off
resonance, no waves are transmitted through to chromosphere “A”, and no
fractionation occurs. Figure 8: Same as figure 7 for the right hand side
chromosphere “B”, where Alfvén waves are launched up from the convection zone.
Almost complete reflection of Alfvén waves occurs from the loop footpoint,
leading to no fractionation. Figure 9: Same as figure 4 (on resonance case)
giving the first three panels for the left hand chromosphere “A”, where waves
leak down from the corona. The extra bottom right panel shows the FIP
fractionation for Fe, O, and He, relative to H. The wave energy flux has been
increased by a factor 20, leading to stronger coronal nonthermal motions, and
stronger fractionation. Helium is now depleted in the corona relative to the
chromosphere. Figure 10: Same as figure 5 (on resonance case) for the right
hand side chromosphere “B”, where Alfvén waves are launched up from the
convection zone. The wave energy flux has been increased by a factor 20,
leading to stronger coronal nonthermal motions, and stronger fractionation.
Figure 11: Same as figure 8 (off resonance case) for the right hand side
chromosphere “B”, where Alfvén waves are launched up from the convection zone.
The wave energy flux has been increased by a factor 20, leading to stronger
coronal nonthermal motions, and stronger fractionation. Waves are now
reflected, and a small inverse FIP effect results. Figure 12: Same as figure 9
giving the first three panels for the left hand chromosphere “A”, where waves
leak down from the corona. The extra bottom right panel shows the FIP
fractionation for Fe, O, and He, relative to H. Three wave frequencies are now
introduced to simulate more nearly a realistic chromospheric power spectrum.
Figure 13: Same as figure 11 for the right hand side chromosphere “B”, where
three Alfvén waves are launched up from the convection zone. The FIP fractions
are reduced from those with a single incident wave, even though one of the
waves here is on resonance. The contributions to the partial pressure of the
other waves “dilute” the fractionation, by increasing the value of $v_{s}^{2}$
in the denominator of the integral in equation 6, without increasing the
ponderomotive acceleration. Figure 14: Coronal section of open field region up
to 500,000 km altitude, showing from top: Elsässer variables in km s-1
($\delta B/\sqrt{4\pi\rho}$ solid lines, $\delta v$ dashed lines), with black
lines for real parts and gray lines for imaginary parts. Middle; wave energy
fluxes in ergs cm-2 s-1. Bottom, the ponderomotive acceleration in cm s-2.
Figure 15: Same as figure 11 for the right hand side chromosphere “B”, where
Alfvén waves are launched up from the convection zone into an open field
region. The FIP fractionations are evaluated with an incident coronal hole
spectrum, as opposed to that for an active region, and show the absence of
strong fractionation consistent with observations of the fast solar wind and
coronal holes.
|
arxiv-papers
| 2009-01-21T20:46:26 |
2024-09-04T02:49:00.118017
|
{
"license": "Public Domain",
"authors": "J. Martin Laming",
"submitter": "Martin Laming",
"url": "https://arxiv.org/abs/0901.3350"
}
|
0901.3384
|
# A Boundary Approximation Algorithm for Distributed Sensor Networks
Michael I. Ham Marko A. Rodriguez Theoretical Division - Center for
Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico
87545
###### Abstract
We present an algorithm for boundary approximation in locally-linked sensor
networks that communicate with a remote monitoring station. Delaunay
triangulations and Voronoi diagrams are used to generate a sensor
communication network and define boundary segments between sensors,
respectively. The proposed algorithm reduces remote station communication by
approximating boundaries via a decentralized computation executed within the
sensor network. Moreover, the algorithm identifies boundaries based on
differences between neighboring sensor readings, and not absolute sensor
values. An analysis of the bandwidth consumption of the algorithm is presented
and compared to two naive approaches. The proposed algorithm reduces the
amount of remote communication (compared to the naive approaches) and becomes
increasingly useful in networks with more nodes.
††preprint: LA-UR-09-00111
## I Introduction
Sensor networks consist of a set of sensor devices that communicate with each
other through a wired or wireless communication network. Such systems often
communicate with a remote station and are a promising technology for remotely
detecting physical phenomena such as forest fires, chemical leaks, or
radioactive clouds. For many applications, it is necessary that the network
not only identify a phenomenon, but also determine the boundary of the
detected phenomenon. For example, by establishing the boundary of a forest
fire, a sensor network can help fire fighters determine where to concentrate
their efforts.
Adding more sensors to a network increases the accuracy of any boundary
approximation algorithm, but consequently, increases the amount of data
generated. Therefore, if all data is processed at the remote station, the
required bandwidth is proportional to the size of the network. On the other
hand, if only the nodes that sense the phenomenon report back, the required
bandwidth is proportional to the size of the phenomenon. As an alternative to
both of these naive approaches, we present a decentralized algorithm for
boundary identification that limits remote station communication by
determining the boundary segments of a phenomenon via a distributed
computation that is carried out within the sensor network. Moreover, only
sensors that identify a boundary ultimately communicate with the remote
station. Therefore, the amount of remote communication is proportional to the
size of the phenomenon’s boundary.
## II Two Naive Boundary Approximation Algorithms
A naive solution to boundary approximation would be for each sensor to report
its internal state to the remote station. Once the state of each sensor
reaches the remote station, the station calculates the boundary using either a
centralized version of the proposed method (to follow) or any other known
centralized algorithm. Assuming $n$ nodes in the sensor network and each
sensor has a cost of $\beta$ for a long-range transmission, the cost of this
approach is $n\beta$. However, a complication with this approach is that the
remote station must have the capacity to receive information from each node
simultaneously in order to ensure an accurate snapshot of the phenomenon’s
location. Regardless, as the size of network increases, this approach becomes
prohibitive since the cost scales with the number of sensors.
A second naive solution would be for only those sensors that detect the
phenomenon to report back to the remote station. If $m$ is the number of nodes
sensing a phenomenon, where $m\leq n$, then $m\beta$ would be cost of this
algorithm. Using this method, remote communication scales with the size of the
phenomenon, not the size of the network. A simple real-world example
demonstrates a potential fault with this approach for certain phenomena.
Imagine a sensor network whose function is to sense a gray-scale environment.
Given a constant light source, a sensation threshold can be determined. In
such cases, only those sensors that sense a high enough gray-scale value would
report to the remote station. However, once the light source is reduced beyond
the preset sensation threshold, the phenomenon is no longer detected even
though there exists a relative difference in the readings of the sensors at
the boundary. Using the algorithm presented next, a relative analysis
determines the boundary regardless of the strength of the light source
(assuming there exists more light than absolute dark). Sensors use local
communication to detect a boundary by comparing neighboring measurements and
only sensors that identify a boundary communicate with the remote station.
Finally, the number of sensors reporting scales with the size of the
phenomenon’s boundary. In many cases, the area of a phenomenon is likely to be
significantly larger than the boundary.
## III The Proposed Boundary Approximation Algorithm
Given a sensor network with $n$ nodes, a Delaunay triangulation is used to
determine the neighbors of each node in the network delaunay:lee1980 . Next, a
Voronoi diagram is generated to determine boundary segments between
neighboring sensors. Such diagrams create cells with boundaries (segments),
where all points on the cell boundary are equidistant between the two
neighboring sensor nodes voronoi:aurenhammer1991 . Figure 1 presents a
visualization of a Delaunay triangulation (Figure 1a) and Voronoi diagram
(Figure 1b) for $100$ randomly distributed nodes within a 2D space. Because
there exists no distributed algorithm for calculating a Delaunay
triangularization and a Voronoi diagram cover:li2003 , the sensor network’s
remote station can be used to calculate these. This one time calculation
occurs only after sensors have been distributed and assumes that the remote
station knows the exact location of each sensor.
Figure 1: a. A Delaunay triangulation and b. a Voronoi diagram for 100
randomly distributed sensors. Sensors $i$ and $j$ are identified as well as
the $(i,j)$ Voronoi cell boundary that is equidistant between $i$ and $j$.
Assume that a given sensor $i$ takes a measurement $\psi_{i}\in[0,1]$. In
order to accomplish a local, distributed calculation of a phenomenon’s
boundary, $i$ must communicate with each of its neighbors and compare its
measurement with the measurements taken by those neighbors. If a particular
neighbor $j$ of $i$ has a $\psi_{j}$ that is significantly different than
$\psi_{i}$, then $i$ can assume that the phenomenon’s boundary exists
somewhere between itself and $j$. This threshold of difference is defined by
$\theta\in[0,1]$ and a boundary exists when $\psi_{i}-\psi_{j}>\theta$. Since
sensors have spatial gaps between them, the location of the boundary cannot be
known exactly. The best approximation of the phenomenon’s boundary is
determined to be the line directly equidistant from $i$ and $j$. Conveniently,
this line is the segment $(i,j)$ as defined by the Voronoi diagram. Therefore,
once $(i,j)$ is determined to be a boundary segment, only this information
needs to be transmitted to the remote station. Thus, only those sensors at the
boundary of the phenomenon are communicating with the remote station.
Moreover, the aggregate of all their reports is the approximated boundary.
Figure 2 presents two simulated phenomenon: one with a linear boundary and the
other with a circular boundary. Each phenomena exist within a $100$ node
sensor network. Table 1 presents the cost of each boundary detection approach
for both phenomena, where $\epsilon$ denotes the relatively low cost of all
inter-node communication comm:wierelthier2000 . It should be noted that for
certain types of networks, especially radio wireless networks, the cost of
local communication can be orders of magnitude less than long-range, remote
communication. It is in these situations where the proposed algorithms is most
efficient.
Figure 2: A representation of the boundary approximated by the proposed algorithm for phenomena with a. linear and b. circular boundaries. Gray-scale shading denotes the phenomena. The boundary of the phenomena is the black line and the approximated boundary is the dashed line. The approximated boundary is always a collection of Voronoi cell segments. boundary | 1${}^{\text{st}}$ naive | 2${}^{\text{nd}}$ naive | proposed
---|---|---|---
linear | $100\beta$ | $80\beta$ | $11\beta+\epsilon$
circular | $100\beta$ | $38\beta$ | $24\beta+\epsilon$
Table 1: The cost of each approach for simulated phenomena with linear and
circular boundaries (see Figure 2). $\beta$ denotes the cost for remote
communication and $\epsilon$ is the total cost for inter-network
communication.
## IV Monte Carlo Simulation
Monte Carlo simulations provide a means to test a system with many degrees of
freedom, where an exhaustive parameter sweep is considered intractable
metropolis_1949 . We utilize a Monte Carlo simulation of sensor networks
containing $3$, $4$, $5$, $10$, $25$, $100$, $200$, $500$, and $1000$ nodes.
For each population of nodes, one hundred different 2D space configurations
are created within a fixed area. In each of the one hundred configurations, we
activate a random set of nodes. The number of activated nodes is sequentially
increased from $1$ to $n$. This random selection of nodes is done 100 times.
For each resulting pattern, we calculate the number of sensors that would
report back to the central station using the various boundary detection
algorithms previously described. Figure 3 presents the results for networks of
$10$, $100$, and $1000$ nodes using the first naive approach (top horizontal
line), the second naive approach (black diagonal line), and our proposed
approach (gray cloud). Finally, Table 2 demonstrates for all networks tested,
the maximum number of nodes reporting to the remote station. The results of
Table 2 demonstrate that the proposed algorithm becomes more efficient as more
nodes are added to the network. It should be noted, that for both naive
approaches, the maximum number of reporting nodes is 100%.
Figure 3: A Monte Carlo simulation of the number of nodes (as a percent of the whole population) observing a phenomenon vs. the number (as a percent of the whole population) reporting back to the remote station in networks with $10$, $100$, and $1000$ nodes. For the proposed algorithm, as the number of nodes increases, the maximum number of nodes reporting to the remote station decreases. number of nodes | max reporting (%)
---|---
3 | 100
4 | 100
10 | 90
25 | 84
100 | 72
200 | 68
500 | 63.6
1000 | 61.5
Table 2: A Monte Carlo simulation identifies the maximum number of reporting
nodes for the proposed algorithm.
## V Conclusions
Related work on boundary approximation in sensor networks relies mainly on
local communication and distributed computation edge:liao2003 ; local:chin2003
; image:deva2003 ; bound:nowak2003 . However, most boundary approximation
algorithms do not determine the boundary of the phenomenon, only the sensors
that lie at the boundary. By knowing which sensor’s lie at the boundary, the
remote station can then estimate the actual line defining the boundary of the
phenomenon. In contrast, the proposed algorithm computes the phenomenon’s
boundary internal to the network without reliance on the remote station. Local
communication is used to identify pairs of nodes with readings whose
difference is greater than $\theta$. One of the two nodes transmits the pair’s
Voronoi segment to the remote station. The aggregation of all these segments
is the approximated boundary of the phenomenon.
It should be noted that a boundary can never be determined exactly since
spatial gaps exist between sensors. Therefore, any calculation of the
phenomenon’s boundary is only an approximation. To reduce boundary location
uncertainty, more sensors can be added to the network. As sensor networks
increase in size, it is important to keep costs to a minimum. Costs can be
reduced by utilizing low bandwidth communication and energy efficient
processors with moderate clock speeds and small amounts of on-board memory.
The proposed algorithm helps achieve one of these objectives by reducing
remote station communication. The algorithm may prove useful in wireless
sensor networks where radio communication over long distances requires
significantly more energy than local communication comm:wierelthier2000 .
## Acknowledgements
We would like to thank Levi Larkey and Vadas Gintautas for their contributions
to this article. This research was funded by a U.S. Department of Education
GAANN Fellowship and an IC Postdoctoral Fellowship. Further support was
provided by the Los Alamos National Laboratory.
## References
* (1) Franz Aurenhammer. Voronoi diagrams - a survey of fundamental geometric data structure. ACM Computing Surveys (CSUR), 23(3):345–405, 1991.
* (2) Krishna Kant Chintalapudi and Ramesh Govindan. Localized edge detection in sensor fields. Ad-hoc Networks Journal, 2003.
* (3) Divya Devaguptapu and Bhaskar Krishnamachari. Applications of localized image processing techniques in wireless sensor networks. In Edward M. Carapezza, editor, Unattended Ground Sensor Technologies and Applications V, volume 5090, pages 247–256. SPIE, 2003\.
* (4) D.T. Lee and B.J. Schachter. Two algorithms for constructing a delaunay triangulation. International Journal of Computer and Information Sciences, 9(3):219–242, 1980.
* (5) Xiang-Yang Li, Peng-Jun Wan, and Ophir Frieder. Coverage in wireless ad hoc sensor networks. IEEE Transactions on Computers, 52(6):753–763, 2003.
* (6) Pei-Kai Liao, Min-Kuan Chang, and C.C. Jay Kuo. Distributed edge detection with composite hypothesis test in wireless sensor networks. In Proceedings of the Global Telecommunications Conference, pages 129–133. IEEE, 2004.
* (7) N. Metropolis and S. Ulam. The Monte Carlo method. Journal of the American Statistical Association, 44:335–341, 1949\.
* (8) Robert Nowak and Urbashi Mitra. Boundary estimation in sensor networks: Theory and methods. In Information Processing in Sensor Networks: Second Internationl Workshop IPSN, pages 80–95. Springer-Verlag, 2003.
* (9) Jeffrey .E Wieselthier, Gam D. Nguyen, and Anthony Ephremides. On the construction of energy-efficient broadcast and multicasttrees in wireless networks. In INFOCOM ’2000: Proceedings of the 19th Annual Joint Conference of the IEEE Computer and Communications Societies, pages 585–594, Washington, DC, USA, 2000. IEEE Computer Society.
|
arxiv-papers
| 2009-01-22T00:59:01 |
2024-09-04T02:49:00.129095
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael I. Ham and Marko A. Rodriguez",
"submitter": "Marko A. Rodriguez",
"url": "https://arxiv.org/abs/0901.3384"
}
|
0901.3443
|
arxiv-papers
| 2009-01-22T09:57:51 |
2024-09-04T02:49:00.135038
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lino Miramonti",
"submitter": "Lino Miramonti",
"url": "https://arxiv.org/abs/0901.3443"
}
|
|
0901.3480
|
# Extreme events in discrete nonlinear lattices
A. Maluckov$\ {}^{1}$, Lj. Hadžievski$\ {}^{2}$, N. Lazarides$\ {}^{3,4}$, and
G. P. Tsironis$\ {}^{3}$ $\ {}^{1}$Faculty of Sciences and Mathematics,
Department of Physics, P. O. Box 224, 18001 Niš, Serbia
$\ {}^{2}$Vinča Institute of Nuclear Sciences, P. O. Box 522, 11001 Belgrade,
Serbia
$\ {}^{3}$Department of Physics, University of Crete, and Institute of
Electronic Structure and Laser, Foundation for Research and Technology –
Hellas, P. O. Box 2208, 71003 Heraklion, Greece
$\ {}^{4}$Department of Electrical Engineering, Technological Educational
Institute of Crete, P. O. Box 140, Stavromenos, 71500, Heraklion, Crete,
Greece
###### Abstract
We perform statistical analysis on discrete nonlinear waves generated though
modulational instability in the context of the Salerno model that interpolates
between the intergable Ablowitz-Ladik (AL) equation and the nonintegrable
discrete nonlinear Schrödinger (DNLS) equation. We focus on extreme events in
the form of discrete rogue or freak waves that may arise as a result of rapid
coalescence of discrete breathers or other nonlinear interaction processes. We
find power law dependence in the wave amplitude distribution accompanied by an
enhanced probability for freak events close to the integrable limit of the
equation. A characteristic peak in the extreme event probability appears that
is attributed to the onset of interaction of the discrete solitons of the AL
equation and the accompanied transition from the local to the global
stochasticity monitored through the positive Lyapunov exponent of a nonlinear
map.
###### pacs:
63.20.Ry; 47.20.Ky; 05.45+a
Introduction.- The motivation of the present work stems from observations of
the sudden appearance of extremely large amplitude sea waves referred to as
rogue or freak waves Kharif . These waves appear very suddenly in relatively
calm seas, reach amplitudes of over $20m$ and may destroy or sink small as
well as large vessels Muller . Theoretical analysis of ocean freak waves has
been linked to nonlinearities in the waver wave equations, studied though the
nonlinear Schrödinger (NLS) equation and shown that the probability of their
appearance is not insignificant Onorato . A scenario for freak wave generation
in NLS is through a Benjamin-Feir (modulational) instability, resulting in
self-focusing effects and subsequent formation of freak waves Zakharov .
Modulational instability (MI) induces local exponential growth in the wave
train amplitude Onorato1 ; Shukla that has been confirmed experimentally and
numerically Ruban .
Intriguingly, there are completely different physical systems that possess the
required nonlinear characteristics which favour the appearance of rogue waves.
Recent observation of optical rogue waves in a microstructured optical fiber
was reported Solli in a regime near the threshold of soliton-fission
supercontinuum generation, i.e., in a region where MI plays a key role in the
dynamics. A generalized NLS equation was used successfully to model the
generation of optical rogue waves while, additionally, control and
manipulation of rogue soliton formation was also discussed Dudley . The
mechanism of the rogue waves creation, or, more generally of extreme events,
has become an issue of principal interest in various other contexts as well,
since rogue waves can signal catastrophic phenomena such as an earthquake, a
thunderstorm, or a severe financial crisis. Knowledge of the probability of
occurrence of extreme events and the capability to predict the time at which
such an event may take place is of a great value. Such events are usually
rare, and they exhibit ”extreme-value” statistics, typically characterized by
heavy-tailed probability distributions. Experimental observation of optical
rogue-wave-like fluctuations in fiber Raman amplifiers show that the
probability distribution of their peak power follows a power law Hammani .
In this work we focus on the discrete counterparts of rogue waves that may
appear in nonlinear lattices as a result of discrete soliton or breather
induction and their mutual interactions. Specifically we investigate the role
of integrability in the formation of discrete rogue waves (DRW) and the
resulting extreme event statistics. Their appearance may affect dramatically
the physical systems. We use the Salerno model Salerno that through a unique
parameter interpolates between a fully integrable discrete lattice, viz. the
Ablowitz-Ladik (AL) lattice Ablowitz , and the nonintegrable DNLS equation
ELS ; MT . One of the basic questions to be addressed below is the probability
of occurrence of a DRW as a function of the degree of integrability of the
lattice and thus study the role of the latter in the production of extreme
lattice events Nicolis .
The Salerno model.- The Salerno model (SM) is given through the following set
of equations
$\displaystyle
i\frac{d\psi_{n}}{dt}=-(1+\mu|\psi_{n}|^{2})(\psi_{n+1}+\psi_{n-1})-\gamma|\psi_{n}|^{2}\psi_{n}$
(1)
where $\mu$ and $\gamma$ are two nonlinearity parameters. When $\mu=0$ the
model becomes the DNLS equation while for $\gamma=0$ it reduces to AL. Several
properties of the model such as integrability Rumpf and stability of
localized travelling waves Cai ; Hennig have been analyzed. Both the norm $N$
and the Hamiltonian $H$ of the model are conserved quantities. They are given
by
$\displaystyle N$ $\displaystyle=$
$\displaystyle\frac{1}{\mu}\sum_{n}\ln{|1+\mu|\psi_{n}|^{2}|},$ (2)
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{n}\left[\frac{\gamma}{\mu^{2}}\ln|1+\mu|\psi_{n}|^{2}|-\frac{\gamma}{\mu}|\psi_{n}|^{2}-2Re[\psi_{n}\psi_{n+1}^{*}]\right].$
(3)
It is also known that Eq. (1) exhibits MI, which may give rise to stationary,
spatially localized solutions in the form of discrete breathers (DBs), i.e.,
periodic and spatially localized nonlinear excitations DBs . The MI induced
DBs appear in random lattice locations and may be mobile. High-amplitude DBs
tend to absorb low-amplitude ones, resulting after some time in a small number
of very high amplitude excitations, which may get pinned at a specific lattice
site due to the Peierls-Nabarro potential barrier in nonintegrable lattices
Kivshar1 . In general, high-amplitude DBs are virtual bottlenecks which slow
down the relaxation processes in nonlinear lattices Tsironis ; Rasm , and it
has been proposed that they may serve as models for freak waves Dysthe . The
development of MI in Eq. (1) can be analyzed with the linear stability
analysis of its the plane wave solutions perturbed by small phase and
amplitude perturbations Maluckov . The interplay between the on-site and
intersite nonlinear terms (i.e., according to the variation of their relative
strength through $\mu$ and $\gamma$), may change MI properties and,
consequently, the conditions for the DBs to exist in the lattice Kivshar . The
SM has recently found applications in modelling Bose-Einstein condensates of
dipolar atoms in a strong periodic potential Gomez , dilute Bose-Einstein
condensates trapped in a periodic potential Trombettoni , and even biological
systems Salerno1 .
For later convenience in the numerical simulation, the variables $\psi_{n}$ in
Eq. (1) are rescaled as $\psi_{n}=\xi_{n}/\sqrt{\mu}$, so that in terms of
$\xi_{n}$ the dynamic equations read
$\displaystyle
i\frac{d\xi_{n}(t)}{dt}=-(1+|\xi_{n}(t)|^{2})(\xi_{n+1}+\xi_{n-1})-\Gamma|\xi_{n}(t)|^{2}\xi_{n},$
(4)
where $\Gamma=\gamma/\mu$. Therefore, the whole two-dimensional parameter
space $(\gamma,\mu)$ can be scaled by $\mu=1$, leaving $\gamma$ as a free
parameter. With that scaling we may go as close to the DNLS limit as we want
to, by simply let $\Gamma$ to attain very large values. However, the exact
DNLS limit $\mu=0$ has to be calculated separately.
Figure 1: Evolution of the scaled amplitudes $|\xi_{n}|$ for a lattice of
size $N=101$, with $\Gamma$ ($\mu$ and $\gamma$ in the DNLS case), is shown on
the figure. The initial conditions for all cases are $\xi_{n}=1$ for any $n$
(uniform) plus a small amount of white noise.
Statistics of extreme events.- We integrate numerically the system of Eqs. (4)
with periodic boundary conditions using a sixth order Runge-Kutta algorithm
with fixed time-stepping $\Delta t=10^{-4}$. We started simulations with
different initial conditions (the plane wave, uniform background with white
noise and Gaussian noise) which gave similar results. Here we present
calculations in which the initial condition is uniform, $\xi_{n}=1$ for any
$n$, with the addition of a small amount of white noise to accelerate the
development of the MI. The uniform solution is chosen in the interval where it
is known from linear stability analysis that it is unstable. By varying the
nonlinearity parameters we identify broadly three regimes of DRWs that are
shown as spatiotemporal evolutions in Fig. 1. For the purely integrable AL
lattice ($\Gamma=0$) DBs are mobile and essentially noninteracting; as a
result we do not observe significant formation of high DRWs (Fig. 1a). In the
vicinity of the AL limit, i.e. for small $\Gamma$ ($\Gamma\sim 0.1$), there is
an onset of weak interaction of the localized modes of the AL lattice leading
to a significant increase in DRW formation that are mobile (Fig. 1b,c). In
this regime the DBs are highly mobile indicating that DB merging could be
responsible for creation of high-amplitude localized waves. For $\Gamma>>0.1$
on the other hand, DNLS-type behavior dominates the SM and localized
structures that are initially created through the MI become easily trapped in
the lattice (Fig. 1d).
The three regimes mentioned previously are probed by calculating the time-
averaged height distributions $P_{h}$. We first define the forward (backward)
height at the $n-$th site as the difference between two successive minimum
(maximum) and maximum (minimum) values of $|\xi_{n}(t)|$. We use then both the
forward and the backward heights for the calculation of the local height
distribution; after spatial averaging the latter results in the height
probability densities (HPDs) shown in Fig. 2. We note that the tails of the
HPDs are related to extreme events and the appearance of DRWs. For $\Gamma$
finite the HPDs are sharply peaked but have extended tails indicating that
extreme events are more than several times as large as the mean distribution
height. In the DNLS limit ($\Gamma\gg 1$) the obtained HPD is very close to
the Rayleigh distribution whose tails decay very fast VanKampen , indicating
negligible probability for the occurrence of extreme events (dotted curve in
Fig. 2). In all the other cases the decay of the tails of the HPDs is much
slower.
In order to probe further the onset of extreme discrete events we employ the
practice used in water waves and define a DRW as one that has a height greater
than $h_{th}=2.2h_{s}$, with $h_{s}$ being the significant wave height. The
latter is defined as the average height of the one-third higher waves in the
height distribution. As a result, the probability of occurrence of extreme DRW
events $P_{ee}=P_{h}(h>h_{th})$ is obtained by integration of the (normalized)
HPD from $h=h_{th}$ up to infinity. By evaluating several HPDs as those in
Fig. 2 we may estimate the probability of occurrence of DRWs $P_{ee}$ as a
function of the parameter $\Gamma$ (the results are shown in Fig. 3). We note
that the probability for the occurrence of a DRW has a certain value in the AL
case, subsequently peaks for small values of $\Gamma$ and decays precipitously
when $\Gamma>>1$. This behavior of the probability $P_{ee}$ is compatible with
the DB picture outlined earlier, viz. in the very weakly nonintegrable regime
the AL modes may interact leading to DB fusion and DRW generation. On the
other hand, as nonintegrability becomes stronger, the scattering of the AL
modes is more chaotic leading to a suppression of DRW formation.
Figure 2: The normalized height probability density $P_{h}(h)$ for several
values of $\Gamma$ and for the DNLS limit (with $\gamma=6$). The line with
slope $-1$ is added to assist comparisons and corresponds to $P_{h}\sim 1/h$.
Approximately vertical drop corresponds to the DNLS limit with an exponential
tail. The increase of $\Gamma(\mu=1)$ leads to the decrease of the slope and
appearance of plateau on the $P_{h}$ curve; the latter increases the extreme
event probability leading a maximum at $\Gamma=0.07$ (Fig. 3.) Figure 3: The
normalized probability $P_{ee}=P_{h}(h\geq h_{th})$ for the occurrence of
extreme events as a function of the integrability parameter $\Gamma$. All data
present averaged results of five numerical measurements differing in the
initial conditions.
Map approach.- In order to probe deeper on the formation of DRWs we substitute
$\psi_{n}=\phi_{n}\exp(-i\omega t)$ into Eq. (1), with $\phi_{n}$ a real-
valued function of the lattice site $n$, and obtain the stationary equation
$\displaystyle\omega\phi_{n}+(1+\mu|\phi_{n}|^{2})(\phi_{n+1}+\phi_{n-1})+\gamma|\phi_{n}|^{2}\phi_{n}=0,$
(5)
which can be transformed in the two-dimensional map
$\displaystyle x_{n+1}=-\frac{\omega+\gamma x_{n}^{2}}{1+\mu
x_{n}^{2}}x_{n}-y_{n},\qquad y_{n+1}=x_{n},$ (6)
where we have defined $x_{n}=\phi_{n}$ and $y_{n}=\phi_{n-1}$. Eqs. (6)
represent a real analytic area-preserving map Hennig ; Hennig1 with the
lattice index $n$ playing the role of discrete ’time’.
The phase portraits of the map Eq. (6) for several $\Gamma$-values are shown
in Fig. 4. In the AL limit, the phase space consists of perfectly disconnected
separatrices while for non-zero $\Gamma$, the stable and unstable manifolds
intersect transversely, resulting in the generation of a homoclinic tangle.
With increasing $\Gamma$ the motion near separatrices becomes exceedingly
complicated and the trajectories wander irregularly before approaching an
attracting set (Figs. 4b and 4c). Moreover, for any $\Gamma\neq 0$, the
position of separatrices in phase space changes in time, resulting in
overlapping of neighboring separatrices and diffusion in those regions which
have been traversed by a separatrix. The sharp peak of the probability of
occurrence of extreme events $P_{ee}(h>h_{th})$ in the SM (Fig. 3) can be
associated with the opening of a stochasticity web, when orbits fast explore
all extended narrow stochasticity regions leading to an anomalous relaxation
phase Rumpf ; VanKampen . This event signs the transition from the local to
global stochasticity Lichtenberg in SM. On the other hand, the decrease of
$P_{ee}(h>h_{th})$ for larger $\Gamma$’s is related to the increasingly longer
trapping time in more developed stochasticity region.
The Melnikov analysis in the SM Hennig shows that the magnitude of the
separatrix splitting and the consequent development of stochasticity depends
on the $\Gamma/|\omega|$ ratio. The conjecture that $P_{ee}$ is associated
with the complexity of the phase portraits of the corresponding maps implies
that $P_{ee}$ should also depend on the $\Gamma/|\omega|$ ratio. In our case
$|\omega|$ is related to the modulation frequency of the initially uniform
solution $U$ with the relation $|\omega|=(\gamma+2\mu)U^{2}+2$, which, through
the MI process it transformed into a train of localized DB-like
configurations. We have checked numerically that for fixed ratio
$\Gamma/|\omega|$ and different values of $U$ and $\Gamma$ we obtain the same
HPD. As a consequence, the probability of extreme events $P_{ee}$ as a
function of the $\Gamma/|\omega|$ is qualitatively the same with that of
$P_{ee}$ as a function of $\Gamma$ shown in Fig. 3.
The degree of nonintegrability in the SM model can be quantified by
calculating the Lyapunov exponents of the corresponding maps Maluckov1 . We
have thus calculated the maximum Lyapunov exponent $L$ Lichtenberg for the
map Eq. (6), for the parameters used in the calculation of the phase portrait
shown in the left panels of Fig. 4. It is observed that homoclinic orbits
which correspond to perfect separatrices are characterized by vanishing
Lyapunov exponent (Fig. 4a). With increasing stochasticity, $L$ tends to a
finite positive value which generally depends on the values of the parameters
and the initial conditions (Figs. 4a and 4b).
Figure 4: Orbits started at different initial positions in the neighborhood
of map origin and corresponding the one-dimensional Liapunov exponents.
Conclusions.- The probability of occurrence of extreme events $P_{ee}$ in the
SM results from the competition between the self-focusing and the energy
transport mechanisms which are implicitly correlated with the degree of
integrability of the model Rumpf . Through modulational instability and
starting from a slightly perturbed uniform background we can generate high-
amplitude localized moving structures of the DB type that lead to the
formation of extreme events of DRW type. Depending on their number, amplitude
and life-time, they may prevent of facilitate the energy flow in the lattice,
affecting thus the probability of extreme event formation $P_{ee}$. We find
that the latter probability depends strongly on $\Gamma$ that affects the
degree of integrability of the lattice: DRW are much more probable very close
to the integrable SM limit rather than in the nonintegrable one. We find a
resonance-like maximum in $P_{ee}(\Gamma)$ that, through a nonlinear map
approach, is linked to separatrix breaking and the onset of global
stochasticity. This regime corresponds physically to weak interaction between
the quasi-integrable modes of the system.
A. M. and Lj.H. acknowledge support from the Ministry of Science of Serbia
(Project 141034). One of us (GPT) acknowledges discussions with Oriol Bohigas.
## References
* (1) C. Kharif and E. Pelinovsky, Eur. J. Mech. B Fluids 22, 603 (2003).
* (2) P. Müller, C. Garrett, and A. Osborne, Oceanography 18, 66 (2005).
* (3) M. Onorato, A. R. Osborne, M. Serio, and S. Bertone, Phys. Rev. Lett. 86, 5831 (2001).
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* (5) M. Onorato, A. R. Osborne, and M. Serio, Phys. Rev. Lett. 96, 014503 (2006).
* (6) P. K. Shukla, I. Kourakis, B. Eliasson, M. Marklund, and L. Stenflo, Phys. Rev. Lett. 97, 094501 (2006).
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* (16) B. Rumpf and A. C. Newell, Physica D 184, 162 (2003).
* (17) D. Cai, A. R. Bishop, and N. Grønbech-Jensen, Phys. Rev. Lett. 72, 591 (1994).
* (18) D. Hennig , K. Ø. Rasmussen, H. Gabriel and A. Bülow, Phys. Rev. E 54, 5788 (1996); D. Hennig, N. G. Sun, H. Gabriel and G. P. Tsironis, Phys. Rev. E 52, 255 (1995).
* (19) S. Flach and C. R. Willis, Phys. Rep. 295, 181 (1998); S. Flach and A. V. Gorbach, Phys. Rep. 467, 1 (2008); and references therein.
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* (24) A. Maluckov, Lj. Hadžievski, and B. Malomed, Phys. Rev. E 76, 046605 (2007).
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* (26) J. Gomez-Gardees, B. A. Malomed, L. M. Floria, and A. R. Bishop, Phys. Rev. E 73, 036608 (2006).
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|
arxiv-papers
| 2009-01-22T14:10:54 |
2024-09-04T02:49:00.139638
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Maluckov, Lj. Hadzievski, N. Lazarides, G. P. Tsironis",
"submitter": "Aleksandra Maluckov",
"url": "https://arxiv.org/abs/0901.3480"
}
|
0901.3491
|
# Laboratory tests for the cosmic neutrino background using beta-decaying
nuclei
Bob McElrath CERN, Geneva 23, CH-1211 Switzerland
###### Abstract
We point out that the Pauli blocking of neutrinos by cosmological relic
neutrinos can be a significant effect. For zero-energy neutrinos, the standard
parameters for the neutrino background temperature and density give a
suppression of approximately $1/2$. We show the effect this has on three-body
beta decays. The size of the effect is of the same order as the recently
suggested neutrino capture on beta-decaying nuclei.
###### keywords:
## 1 Introduction
The last remaining remnant of the big bang, which is composed of a known
particle is the Cosmic Neutrino Background (CNB). It decouples from thermal
equilibrium at $T\sim 2$ MeV. It gives information about the universe at a
time significantly before the decoupling of photons at $T\sim 1$ eV. The
processes responsible for their creation and decoupling are well understood
nuclear physics.
Today these neutrinos are expected to be extremely cold ($1.952{\rm
K}=1.68\times 10^{-4}{\rm eV}$). As such, they are extremely difficult to
detect due to the fact that weak interaction cross sections scale as
$(G_{F}E)^{2}$. They have a density of $\rho_{\nu}=3/22n_{\gamma}=56/{\rm
cm}^{3}$ per species of neutrino and anti-neutrino, corresponding to a
luminosity $\mathcal{L}=1.7\times 10^{13}/{\rm cm^{2}s}$ if the neutrinos were
massless. [1, 2] These numbers rely on a specific cosmological model which
could be substantially modified, if neutrinos cluster gravitationally, or if
they have nontrivial dynamics after freeze-out. [3] It has also recently been
shown that these neutrinos are a quantum liquid, and their fluctuations have
the quantum numbers of a graviton, opening the prospect that measurements of
relic neutrinos could then be compared with gravitational constants. [4]
Nuclei that undergo $\beta$-decay are a precise and specific laboratory to
look for the CNB. There exists a vast array of nuclei that can emit or absorb
neutrinos at a wide range of energies. A signal seen using $\beta$-decaying
nuclei constitutes a specific test because they are already known to emit or
absorb lepton number in specific ways. All other proposals could not in
principle tell if the effect was due to an object with lepton number. [1, 2]
High-energy neutrinos (e.g. Z-burst) can be absorbed by things which do not
carry lepton number, and anomalous forces could have a variety of sources that
have nothing to do with lepton number (for instance, a Dark Matter wind).
Finally the effects of neutrinos on the CMB cannot be disentangled from other
relativistic species that are not be fermionic or do not carry lepton number.
Rates using decaying nuclei are in principle much higher than other proposals
such as coherent scattering, because the energy for the observation is coming
from nuclear mass differences, and not from the neutrinos themselves. The
energy $Q$ in nuclear transitions is $\mathcal{O}({\rm MeV})$. Giving the CNB
neutrinos this energy in a coherent experiment would require moving with a
velocity corresponding to a boost factor $\gamma=Q/T_{\nu}\simeq 10^{10}$. For
comparison, $\gamma$ at the LHC with protons is about 15000, or 5500 with
Lead.
There are two ways to see an effect of the neutrino background using a beta-
decaying nucleus: add a neutrino to it or remove a neutrino from it. Both were
suggested by Weinberg in 1962. [5]
Adding a neutrino to the background is suppressed for momenta which are
already occupied by the CNB thermal distribution, due to the fact that
neutrinos are fermions and their chemical potential and average energy are
similar. This is an $\mathcal{O}(1)$ effect, if one can create neutrinos
having the correct energy.
Removing a neutrino from the CNB using nuclei is known as neutrino capture
($\nu$C). Capture of reactor neutrinos is the original mode used to discover
the neutrino. Recently there has been a surge of interest in this mode for
detecting the CNB using $\beta$-decaying nuclei, which can have zero
threshold.[6, 7, 8]
## 2 Pauli Blocking by The Cosmic Neutrino Background
The CNB is a thermal distribution in a particular frame $u^{\alpha}$ which we
assume to be coincident with the dipole from the Cosmic Microwave Background,
which points in the direction $(264.85\pm 0.10)^{\circ},(48.25\pm
0.04)^{\circ}$ in galactic coordinates, with velocity $368\pm 2$ km/s. Its
thermal distribution is then
$\displaystyle
F_{i}(\vec{p})=\left[e^{(p^{\alpha}u_{\alpha}-\mu_{i})/kT}+1\right]^{-1}$ (1)
for each species of neutrino and anti-neutrino $i$, having mass $m_{i}$ and
chemical potential $\mu_{i}$ and four-momentum $p^{\alpha}$ in the cosmic rest
frame $u^{\alpha}=(1,\vec{0})$ and this reduces to the usual non-relativistic
Fermi-Dirac distribution. The relativistic chemical potential is the Fermi
energy at zero temperature, $\mu_{i}=E_{F}=\sqrt{m^{2}+p_{F}^{2}}$, and the
nominal Fermi momentum predicted by the standard cosmological model is
$p_{F}=\sqrt[3]{3\pi^{2}\rho_{\nu}}=\sqrt[3]{3\pi^{2}\rho_{\overline{\nu}}}=3.6\times
10^{-5}$ eV, where $\rho_{\nu}$ is the number density per flavor. We will
refer to this as the “standard” chemical potential.
Figure 1: Sum of suppression factor $1-F_{i}(p)$ for three mass eigenstates
vs. neutrino momentum for several values of the neutrino mass, assuming a
standard chemical potential.
Figure 2: The differential event rate as a function of neutrino momentum for
several choices of neutrino mass and normal/inverted hierarchy, assuming a
standard chemical potential and the kinematics of Tritium.
A process which emits neutrinos has a suppression $[1-F_{i}(\vec{p})]$ due to
Pauli blocking from this thermal distribution. This is independent of whether
the neutrinos are described as a localized classical gas having small
uncertainty $\Delta x\ll n^{-1/3}$ or a quantum liquid $\Delta x\gg n^{-1/3}$
for number density $n$. For a beta decay this is
$d\Gamma=2\pi\sum_{i}\int|\mathcal{M}_{i}|^{2}\xi_{i}^{2}[1-F_{i}(\vec{p})]dPS$
(2)
where $dPS$ is the differential phase space, $\xi_{i}$ is the eigenvector
component of neutrino mass eigenstate $i$ in the electron neutrino direction,
$\mathcal{M}$ is the matrix element of the emission of an electron-type
neutrino with mass $m_{i}$, and one sums over the mass eigenstates since final
state emitted particles must be in a mass eigenstate. This suppression factor
is experimentally indistinguishable from $1$ except in a region in which the
emitted neutrino has the same energy as the CNB. We plot the suppression
factor, summed over flavors in Fig.2
The Matrix Element for the beta decay is
$\displaystyle|\mathcal{M}|^{2}=\frac{G_{F}^{2}}{128\pi^{3}M_{I}^{2}}\big{[}$
$\displaystyle(g_{V}+g_{A})^{2}(p_{J}\cdot p_{e})(p_{I}\cdot p_{\nu})$
$\displaystyle+$ $\displaystyle(g_{V}-g_{A})^{2}(p_{I}\cdot p_{e})(p_{J}\cdot
p_{\nu})$ $\displaystyle+$ $\displaystyle(g_{V}^{2}-g_{A}^{2})(p_{I}\cdot
p_{J})(p_{e}\cdot p_{\nu})\big{]}$
where $g_{V}$ and $g_{A}$ are the vector and axial vector weak charges of the
atom. For $I$=neutron, $g_{V}=g_{A}=-1/2$. The matrix element also reaches a
minimum $|\mathcal{M}|^{2}=0$ at $p_{\nu}$=0, so the event rate at $p_{\nu}=0$
is zero. Because of this, the minimum in Fig.2 is deceptive. The differential
event rate including the matrix element is plotted in Fig.2 for tritium,
assuming a standard chemical potential.
To place a neutrino into the background with significant suppression, we need
that the invariant $p^{\alpha}u_{\alpha}/kT_{\nu}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}E_{F}/kT_{\nu}$.
Since our velocity with respect to the cosmic rest frame is small
($\beta=\frac{v}{c}=1.23\times 10^{-3}$), one can ignore our velocity and
pretend we can do the experiment in the cosmic rest frame. 111One must use
Special relativity and not Galilean relativity here. The atoms and electron
may be non-relativistic, but the neutrino is relativistic.
In a normal $\beta^{\pm}$ decay an atom $I$ decays to atom $J$ by emitting an
electron or positron and an anti-neutrino or neutrino. If one can precisely
measure the momenta of $I$, $J$, and $e^{\pm}$, one can solve for the neutrino
momenta. This requires momentum resolution on each of order $\delta
p\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}\sqrt{2mkT_{\nu}}$. If the initial state $I$ is at
rest, this corresponds to a temperature
$T=\frac{2}{3}\frac{m_{\nu}}{M_{I}}T_{\nu}\simeq 1.40\times
10^{-9}\left(\frac{m_{\nu}}{\rm eV}\right)\left(\frac{\rm
amu}{M_{I}}\right)\rm K.$
Stated another way, the de Broglie wavelength of the neutrinos is 1.2 mm.
Since the uncertainty on momentum scales with momentum, the initial state must
have a similar de Broglie wavelength. Modern atomic Bose-Einstein Condensate
and Degenerate Fermi Gas experiments using laser and evaporative cooling
routinely reach $10^{-9}$ K today. Another promising technology to get to
these precisions in the initial state is “Crystallized Beams”. [9]
The final state of the decay is the atom $J$ almost exactly back-to-back with
the electron or positron. Similarly these final state particles must be
measured with a precision
$\frac{\delta p}{p}\simeq\sqrt{\frac{2mkT_{\nu}}{Q(Q+2m_{e})}}\simeq
1.33\times 10^{-7}\sqrt{\frac{m_{\nu}}{eV}}\sqrt{\frac{18{\rm keV}}{Q}}.$
where in the last term we assume $Q<2m_{e}$.
One might wonder if the effect shown here can impact experiments such as
KATRIN which attempt to measure the neutrino mass using the highest energy
electrons in a beta decay. KATRIN attempts to measure mass due to the change
in slope and rate suppression near the endpoint, and they do not have the
resolution to see the actual endpoint itself. Their resolution is
approximately $\Delta E_{e}\sim 1$ eV. The effects here only effect the
highest (electron) energy bin, and reduce the number of events there by
$\mathcal{O}(10^{-18}N)$ where $N$ is the total decays they see. Existing
experiments simply do not have the rate for this effect to be a concern.
## 3 Acknowledgements
We thank Patrick Huber and Mats Landroos for fruitful discussions.
## References
* [1] A. Ringwald, arXiv:hep-ph/0505024.
* [2] G. B. Gelmini, Phys. Scripta T121 (2005) 131 [arXiv:hep-ph/0412305].
* [3] A. Ringwald and Y. Y. Y. Wong, JCAP 0412 (2004) 005 [arXiv:hep-ph/0408241].
* [4] B. McElrath, arXiv:0812.2696 [gr-qc].
* [5] S. Weinberg, Phys. Rev. 128 (1962) 1457.
* [6] A. G. Cocco, G. Mangano and M. Messina, JCAP 0706 (2007) 015 [arXiv:hep-ph/0703075].
* [7] R. Lazauskas, P. Vogel and C. Volpe, J. Phys. G 35 (2008) 025001 [arXiv:0710.5312 [astro-ph]].
* [8] M. Blennow, arXiv:0803.3762 [astro-ph].
* [9] H. Danared, A. Källberg, K.-G. Rensfelt, and A. Simonsson, Phys. Rev. Lett. 88, 174801 (2002).
|
arxiv-papers
| 2009-01-22T15:07:11 |
2024-09-04T02:49:00.146292
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bob McElrath",
"submitter": "Bob McElrath",
"url": "https://arxiv.org/abs/0901.3491"
}
|
0901.3549
|
# Effective Optical Response of Metamaterials
Guillermo P. Ortiz gortiz@exa.unne.edu.ar Departamento de Física, Facultad de
Ciencias Exactas, Universidad Nacional del Nordeste,
Av. Libertad 5400 Campus-UNNE, W3404AAS Corrientes, Argentina. Brenda E.
Martínez-Zérega Centro Universitario de los Lagos, Universidad de
Guadalajara, Enrique Díaz de León SN, Paseos de la Montaña, Lagos de Moreno,
Jalisco, C.P. 47460, México. Division of Photonics, Centro de Investigaciones
en Optica,
León, Guanajuato, México Bernardo S. Mendoza bms@cio.mx Division of
Photonics, Centro de Investigaciones en Optica,
León, Guanajuato, México W. Luis Mochán mochan@fis.unam.mx Instituto de
Ciencias Físicas, Universidad Nacional Autónoma de México,
Apdo. Postal 48-3, 62251 Cuernavaca, Morelos, México
###### Abstract
We use a homogenization procedure for Maxwell’s equations in order to obtain
in the local limit the frequency dependent macroscopic dielectric response
tensor $\epsilon^{M}_{ij}(\omega)$ of metamaterials made of a matrix with
inclusions of any geometrical shape repeated periodically with any lattice
structure. We illustrate the formalism calculating $\epsilon^{M}_{ij}(\omega)$
for several structures. For dielectric rectangular inclusions within a
conducting material we obtain an anisotropic response which may change from
conductor-like at low $\omega$ to dielectric-like with resonances at large
$\omega$, attaining a very small reflectance at intermediate frequencies which
can be tuned through geometrical tailoring. A simple explanation allowed us to
predict and confirm similar behavior for other shapes, even isotropic, close
to the percolation threshold.
###### pacs:
78.67.Bf, 77.22.Ch, 78.20.Ci, 78.20.Bh
## I Introduction
Metamaterials are typically binary composites of conventional materials: a
matrix with inclusions of a given shape, arranged in a periodic structure. A
theoretical model to predict their macroscopic optical properties is very
desirable. Since the times of Maxwell, Lord Rayleigh and Maxwell-Garnet up to
today, many authors have contributed to the calculation of the bulk
macroscopic response in terms of the dielectric properties of its constituents
(for example, see Refs.[J.C.Garland and D.B.Tanner, 1978; Mochán and
R.G.Barrera, 1994; Milton et al., 2003]) employing various approaches such as
variational theories or completely general theories.Hashin and Shtrikman
(1962) The macroscopic effective response can be obtained by defining the
microscopic response of a composite, averaging the microscopic fields and
eliminating the contribution of the fluctuating fields to the average of the
the microscopic response.W.L.Mochán and R.G.Barrera (1985a) Furthermore, the
accuracy of the computational method may be confirmed by using general
theorems such as Keller’s reciprocal theorem.J.B.Keller (1963, 1964); Nevard
and J.B.Keller (1985)
Recent technologies allow the manufacture of ordered composite materials with
periodic structures. For instance, high resolution electron beam lithography
and its interferometric version have been used in order to make particular
designs of nano-structured composites, producing various shapes with
nanometric sizes.Akahane et al. (2003); Grigorenko et al. (2005) Moreover, ion
milling techniques are capable of producing high quality air hole periodic and
non-periodic two-dimensional (2D) arrays, where the holes can have different
geometrical shapes.Koerkamp et al. (2004); Gordon et al. (2004) Therefore, it
is possible to build devices with novel macroscopic optical properties.Pendry
(2000) For example, a negative refractive index has been predicted and
observedV.M.Shalaev et al. (2005) for a periodic composite structure of a
dielectric matrix with noble metal inclusions of trapezoidal shape.Kildishev
et al. (2006)
These advances in metamaterial design have motivated a renewed interest in the
study of their optical properties, although the study of the optical
properties of composites is not new, and several important schemes have been
developed in the past. For example, the macroscopic responses of a
bidimensional periodic array of infinite cylinders was calculated in 1959 in
terms of the Hertz’s potential for a two-dimensional scattering
problem.Khizhnyak (1959) Rayleigh’s extended method was applied in order to
predict the optical properties of a disordered array of spheres.R.C.McPhedran
and D.R.McKenzie (1977) The variation of the conductivity with the filling
fraction of an ordered array of conducting spheres on an insulating matrix has
been studied too,W.T.Doyle (1977) and the multipolar effects due to the
inhomogeneities of the local field have been analyzedClaro (1984) for
dielectric spheres at high filling fraction, yielding criteria for their
importance as a function of interparticle separation.Rojas and Claro (1986)
Furthermore, a general theory was developed to describe the electromagnetic
response without any reference to a specific representation, resulting on a
powerful tool to calculate the macroscopic dielectric response.W.L.Mochán and
R.G.Barrera (1985b) For periodic composites, a Fourier representation is most
fitting and expressions for the bulk macroscopic response may be written in
terms of the Fourier coefficients of the microscopic response.Li (1997); R.Tao
et al. (1990); Shen et al. (1990); A.A.Krokhin et al. (2002); Halevi et al.
(1999); Datta et al. (1993) On the other hand, a spectral representation
theory has allowed the separation of geometric from material properties,Fuchs
(1977); Milton (1981); Bergman and Dunn (1992) and it has been employed to
study the transport properties of several systems.Mochán and R.G.Barrera
(1994); Milton et al. (2003)
In connection with nano-structured metallic films there has been some
important development as well. An exact eigenfunction formulation,Sheng et al.
(1982) and an approximate modal formalism,Lochbihler and Depine (1993) were
used to explain resonances in the zeroth diffraction order of silver square-
wave gratings,Sheng et al. (1982) and gold-wire gratings.Lochbihler (1994) In
these works, it was found that resonances might appear due to the excitation
of surface modes. Such modes can be excited if their momentum matches that of
the incident light after being diffracted by some reciprocal lattice vector of
the periodically structured metal surface. Thus, surface plasmon-polariton
(SPP) modes are excited on the metal-air interface yielding several related
phenomena such as an enhancement of optical transmission through sub-
wavelength holes.Ghaemi et al. (1998) Beside the single coupling to SPP modes,
double resonant conditionsDarmanyan and Zayats (2003) and waveguide modesPorto
et al. (1999) seem to play an important role in the enhancement for metallic
gratings with very narrow slits and for compound gratings.Skigin and Depine
(2005)
A very strong polarization dependence in the optical response of periodic
arrays of oriented sub-wavelength holes on metal hosts has been recently
reported,Koerkamp et al. (2004); Gordon et al. (2004) as well as for a single
rectangular inclusion within a perfect conductor.García-Vidal et al. (2005)
The studies above do not rely on SPP excitation as a mechanism to explain the
optical results.
In this work we obtain the macroscopic dielectric response of a periodic
composite, using a homogenization procedure first proposed by Mochán and
Barrera W.L.Mochán and R.G.Barrera (1985a) within the context of the local
field effect at crystals, liquids and disordered composites. In this procedure
the macroscopic response of the system is obtained from its microscopic
constitutive equations by eliminating the spatial fluctuations of the field
with the use of Maxwell’s equations and solving for the macroscopic
displacement in terms of the macroscopic electric field. Besides the average
dielectric function, the formalism above incorporates the effects that the
rapidly varying Fourier components of the microscopic response has on the
macroscopic response. An equivalent procedure suitable for periodic systems
was recently proposed by P. Halevi and F. Pérez-RodríguezHalevi and Pérez-
Rodríguez (2006) and applied to photonic crystals and metamaterials. Although
developed independently, it may be considered an extension of the generalized
local field effect theory developed previously by Mochán and BarreraW.L.Mochán
and R.G.Barrera (1985a) and it has been applied to the dielectric, magnetic
and in general, the bi-anisotropic response of photonic crystals. Similar
homogenization procedures are also found in Refs. A.A.Krokhin et al., 2002;
Halevi et al., 1999; Datta et al., 1993. We further restrict ourselves to the
local limit, in which we neglect the dependence of the response on the
wavevector, or more precisely, on the Bloch’s vector. The macroscopic optical
response is obtained in terms of the geometrical shape of the inclusions,
their periodic arrangement, and the dielectric function of the host and the
inclusions. The proposed scheme is straightforward, requiring standard
numerical computations. It has the advantage of fully accounting for the
detailed geometry of the system. For systems with periods much smaller than
the wavelength of the incoming light, the local limit becomes the exact
response while it accounts for the local field effect, i.e., the interaction
among parts of the system through the spatially fluctuating electromagnetic
field. We reproduce, previously reported results,Milton et al. (1981); R.Tao
et al. (1990); Bergman and Dunn (1992) and novel effects resulting solely from
the geometrical shape of the inclusions, namely, the existence of transparency
windows within metal-dielectric metamaterials slightly above the percolation
threshold of the metallic phase.
The article is organized as follows. In Sec. II we present the theoretical
approach used for the calculation of the macroscopic dielectric response of
the composite. In Sec. III we validate our formalism comparing it with
previous schemes, yielding very good agreement. Then, we present results for
two-dimensional periodic structure consisting of a gold host with dielectric
rectangular prism or circular inclusions. Finally, in Sec. IV we present our
conclusions.
## II Theoretical Approach
In order to calculate the macroscopic dielectric response of a metamaterial we
follow the steps of Ref. W.L.Mochán and R.G.Barrera, 1985a. We start by
defining appropriate average and fluctuation idempotent projectors
$\hat{P}_{a}$ and $\hat{P}_{f}=\hat{1}-\hat{P}_{a}$ such that $\hat{P}_{a}$
acting on any microscopic field $\mathbf{F}$ produces its macroscopic
projection $\mathbf{F}^{M}\equiv\mathbf{F}_{a}\equiv\hat{P}_{a}\mathbf{F}$,
while $\hat{P}_{f}$ acting on the same field yields the spatially fluctuating
part $\mathbf{F}_{f}\equiv\hat{P}_{f}\mathbf{F}=\mathbf{F}-\mathbf{F}^{M}$
which we wish to eliminate. The constitutive equation
$\mathbf{D}=\hat{\epsilon}\mathbf{E}$, where $\hat{\epsilon}$ is the
dielectric operator (in the general case, a complex tensorial integral
operator for each frequency), may be split into macroscopic and spatially
fluctuating parts. Thus we write
$\mathbf{D}^{M}=\hat{\epsilon}_{aa}\mathbf{E}^{M}+\hat{\epsilon}_{af}\mathbf{E}_{f},$
(1)
where $\hat{O}_{\alpha\beta}=\hat{P}_{\alpha}\hat{O}\hat{P}_{\beta}$
($\alpha,\beta=a,f$) for any operator $\hat{O}$ and we used the idempotency of
the projectors. Furthermore, the fluctuating part of the wave equation for a
non-magnetic material is given by
$\nabla\times(\nabla\times\mathbf{E}_{f})=k_{0}^{2}\mathbf{D}_{f}=k_{0}^{2}(\hat{\epsilon}_{fa}\mathbf{E}^{M}+\hat{\epsilon}_{ff}\mathbf{E}_{f}),$
(2)
where $k_{0}=\omega/c=2\pi/\lambda_{0}$ and $\lambda_{0}$ are the free space
wavenumber and wavelength corresponding to frequency $\omega$ and we assumed
that the external sources have no spatial fluctuations (otherwise, a
homogenization procedure would prove useless). We solve Eq. (2) for the
fluctuating electric field
$\mathbf{E}_{f}=-\left(\hat{\epsilon}_{ff}-\frac{1}{k_{0}^{2}}(\nabla\times\nabla\times)_{ff}\right)^{-1}\hat{\epsilon}_{fa}\mathbf{E}^{M}$
(3)
where $\nabla\times\nabla\times$ denotes the operator
($\mathrm{grad}\,\mathrm{div}-\nabla^{2})$ and the inverse on the RHS may be
interpreted in real space as a Green’s function, i.e., an integral operator
whose kernel obeys a differential equation with a singular source. The inverse
in the second term in the RHS of Eq. (3) is performed after the projections
onto the space of fluctuating fields, denoted by the two subscripts $ff$.
Finally, we substitute Eq. (3) into Eq. (1) to obtain the macroscopic relation
$\mathbf{D}^{M}=\hat{\epsilon}^{M}\mathbf{E}^{M}$ where we identify the
macroscopic dielectric operator
$\hat{\epsilon}^{M}=\hat{\epsilon}_{aa}-\hat{\epsilon}_{af}\left(\hat{\epsilon}_{ff}-\frac{1}{k_{0}^{2}}(\nabla\times\nabla\times)_{ff}\right)^{-1}\hat{\epsilon}_{fa}.$
(4)
The first term in the RHS of Eq. (4) represents the average dielectric
response, while the second term incorporates the effect of the interactions
through the small-lengthscale spatial fluctuations of the field on the
macroscopic response.
We rewrite Eq. (4), which corresponds to Eq. (21) of Ref. W.L.Mochán and
R.G.Barrera, 1985a, as
$\hat{\epsilon}^{M}=\hat{\epsilon}_{aa}-\hat{\epsilon}_{af}\hat{\Phi}_{fa},$
(5)
where $\hat{\Phi}_{fa}$ is defined through
$\hat{\mathcal{W}}_{ff}\hat{\Phi}_{fa}=\hat{\epsilon}_{fa}$ (6)
and we introduced the wave operator
$\hat{\mathcal{W}}=\hat{\epsilon}-\frac{1}{k_{0}^{2}}\nabla\times\nabla\times.$
(7)
For a periodic system, we can use Bloch’s theorem to represent the fields and
operators through their Fourier components
$\mathbf{F}_{\mathbf{q}}(\mathbf{r})=\sum_{\mathbf{G}}\mathbf{F}_{\mathbf{q}}(\mathbf{G})e^{i(\mathbf{q}+\mathbf{G})\cdot\mathbf{r}},$
(8) ${\cal
O}_{\mathbf{q}}(\mathbf{r},\mathbf{r}^{\prime})=\sum_{\mathbf{G}\mathbf{G}^{\prime}}{\cal
O}_{\mathbf{q}}(\mathbf{G},\mathbf{G}^{\prime})e^{i[(\mathbf{q}+\mathbf{G})\cdot\mathbf{r}-(\mathbf{q}+\mathbf{G}^{\prime})\cdot\mathbf{r}^{\prime}]},$
(9)
where $\mathbf{F}_{\mathbf{q}}(\mathbf{r})$ denotes an arbitrary position
dependent field with a given Bloch vector $\mathbf{q}$, ${\cal
O}_{\mathbf{q}}(\mathbf{r},\mathbf{r}^{\prime})$ is the kernel corresponding
to an arbitrary operator $\hat{{\cal O}}$ for the same Bloch vector, and
$\mathbf{G}$, $\mathbf{G}^{\prime}$ are reciprocal vectors. In this case we
can chose $\hat{P}_{a}$ as a truncation operator in reciprocal space that
eliminates the Fourier components outside of the first Brillouin zone, which
can be represented by a Kronecker’s delta
$\hat{P}_{a}\to\delta_{\mathbf{G}0}$. Also, we can identify $\nabla$ with a
diagonal block matrix
$i(\mathbf{q}+\mathbf{G})\delta_{\mathbf{G}\mathbf{G}^{\prime}}$. Thus, we
rewrite Eqs. (5)-(7) as
$\left[\epsilon^{M}_{\mathbf{q}}\right]_{ik}=\left[\epsilon_{\mathbf{q}}(\mathbf{0},\mathbf{0})\right]_{ik}-\sum_{j}\sum_{\mathbf{G}\neq
0}\left[\epsilon_{\mathbf{q}}(\mathbf{0},\mathbf{G})\right]_{ij}\left[\Phi_{\mathbf{q}}(\mathbf{G},\mathbf{0})\right]_{jk},$
(10) $\sum_{j}\sum_{\mathbf{G}^{\prime}\neq
0}\left[{\mathcal{W}}_{\mathbf{q}}(\mathbf{G},\mathbf{G}^{\prime})\right]_{ij}\left[\Phi_{\mathbf{q}}(\mathbf{G}^{\prime},\mathbf{0})\right]_{jk}=\left[\epsilon_{\mathbf{q}}(\mathbf{G},\mathbf{0})\right]_{ik},$
(11)
and
$\left[{\mathcal{W}}_{\mathbf{q}}(\mathbf{G},\mathbf{G}^{\prime})\right]_{ij}=\left[\epsilon_{\mathbf{q}}(\mathbf{G},\mathbf{G}^{\prime})\right]_{ij}+\frac{1}{k_{0}^{2}}\delta_{\mathbf{G}\mathbf{G}^{\prime}}\sum_{kl}\delta_{il}^{kj}(q_{k}+G_{k})(q_{l}+G_{l}).$
(12)
As the fields are vector valued for each reciprocal vector, our operators are
matrix valued for each pair of reciprocal vectors. Thus, in the equations
above we introduced explicitly the Cartesian indices $ijkl$. We also
introduced the usual four-index delta function
$\delta_{il}^{kj}=\delta_{ik}\delta_{lj}-\delta_{ij}\delta_{lk}$. Notice that
$\mathbf{G}$ and $\mathbf{G}^{\prime}$ are different from zero in Eqs.
(10)-(12), as they involve the fluctuating fields. Our Eqs. (10)-(12) are
closely related to Eq. (35) of Ref. Halevi and Pérez-Rodríguez, 2006. We
remark that in the long wavelength limit $G/k_{0}\gg 1$, so that the
transverse part of the RHS of Eq. (12) is dominated by its large second term.
Thus, from Eq. (11), the transverse part of
$\Phi_{\mathbf{q}}(\mathbf{G},\mathbf{0})$ becomes small, of the order of
$k_{0}^{2}/G^{2}$. Nevertheless, the second term on the RHS of Eq. (12) does
not affect the longitudinal part of
${\mathcal{W}}_{\mathbf{q}}(\mathbf{G},\mathbf{G}^{\prime})$, so that the
longitudinal part of $\Phi_{\mathbf{q}}(\mathbf{G},\mathbf{0})$ becomes
dominant in this limit. This means that in the long wavelength limit, the
fluctuations are mostly longitudinalW.L.Mochán and R.G.Barrera (1985a) and we
may neglect retardation in their calculation.
We consider now a two-component system made up of a homogeneous host with a
local isotropic dielectric function $\epsilon_{h}$, in which arbitrarily
shaped particles with a local isotropic dielectric function $\epsilon_{p}$ are
periodically included. Then,
$\left[\epsilon_{\mathbf{q}}(\mathbf{G},\mathbf{G}^{\prime})\right]_{ij}=\left[\epsilon_{h}\delta_{\mathbf{G},\mathbf{G}^{\prime}}+\epsilon_{ph}S(\mathbf{G}-\mathbf{G}^{\prime})\right]\delta_{ij},$
(13)
where $\epsilon_{ph}\equiv\epsilon_{p}-\epsilon_{h}$. The Fourier coefficients
$S(\mathbf{G})=\frac{1}{\Omega}\int
S(\mathbf{r})e^{i\mathbf{r}\cdot\mathbf{G}}d\mathbf{r}=\frac{1}{\Omega}\int_{v}e^{i\mathbf{r}\cdot\mathbf{G}}d\mathbf{r},$
(14)
characterize completely the shape of the particle, as the integrals are over
the volume $v$ occupied by the inclusions within a single unit cell whose
total volume is $\Omega$. Here, we introduced the characteristic function
$S(\mathbf{r})$ whose value is $S(\mathbf{r})=1$ within $v$ and
$S(\mathbf{r})=0$ outside $v$. In particular,
$S(\mathbf{G}=\mathbf{0})=v/\Omega\equiv f,$ (15)
with $f$ the filling fraction of the inclusions, and
$\left[\epsilon_{q}(\mathbf{0},\mathbf{0})\right]_{ij}=(\epsilon_{h}+\epsilon_{ph}f)\delta_{ij}.$
(16)
Notice that for local media
$[\epsilon_{\mathbf{q}}(\mathbf{G},\mathbf{G}^{\prime})]_{ij}$ depends only on
the difference $\mathbf{G}-\mathbf{G}^{\prime}$ and it does not depend on
$\mathbf{q}$.
Finally, substituting Eq. (13) in Eq. (10) and taking the local
$\mathbf{q}\to\mathbf{0}$ limit, we obtain
$\epsilon^{M}_{ij}\equiv[\epsilon^{M}_{\mathbf{0}}]_{ij}=(\epsilon_{h}+\epsilon_{ph}f)\delta_{ij}-\epsilon_{ph}\sum_{\mathbf{G}\neq
0}S(-\mathbf{G})[\Phi_{\mathbf{0}}(\mathbf{G},\mathbf{0})]_{ij},$ (17)
where $[\Phi_{\mathbf{0}}(\mathbf{G},\mathbf{0})]_{ij}$ is obtained by solving
Eq. (11) after substituting Eq. (13) and
$\left[\mathcal{W}_{\mathbf{0}}(\mathbf{G},\mathbf{G}^{\prime})\right]_{ij}=\left[\epsilon_{h}\delta_{\mathbf{G},\mathbf{G}^{\prime}}+\epsilon_{ph}S(\mathbf{G}-\mathbf{G}^{\prime})\right]\delta_{ij}-\frac{1}{k_{0}^{2}}(G^{2}\delta_{ij}-G_{i}G_{j})\delta_{\mathbf{G}\mathbf{G}^{\prime}}$
(18)
from Eq. (12). Notice that in principle we could take the local limit
$\mathbf{q}\to\mathbf{0}$ without also taking the long wavelength limit
$k_{0}\to 0$, although it is advisable to verify that
$\epsilon^{M}_{\mathbf{q}}$ is close to
$\epsilon^{M}_{\mathbf{0}}=\epsilon^{M}$ for the relevant wavevectors
$\mathbf{q}$ that appear in each particular application. We remark that the
first term on the RHS of Eq. (17) is isotropic as it is simply the average of
the response of the constituents, which we took to be local, piecewise
homogeneous and isotropic. Nevertheless, the second term includes information
on the geometry of the system, including both the shapes of the particles and
their periodic arrangement. Thus, in general it yields a non-isotropic
contribution to the macroscopic dielectric tensor.
In the following section we show several examples of this procedure to
calculate the macroscopic dielectric tensor $\epsilon^{M}_{ij}$.
## III Results
### III.1 Comparison to Previous Work
In this section we apply our results to light moving across a 2D square array
of infinite square dielectric prisms with diagonals aligned with the sides of
the square primitive cell, a system previously proposed by Milton et al.Milton
et al. (1981) We chose the parameters $\epsilon_{p}=5.0$, $\epsilon_{h}=1.0$,
and $f=0.3$. We take a finite free-space wavelength $\lambda_{0}=10L$, with
$L$ the lattice parameter, so that, according to Eq. (18) we expect only small
retardation effects of the order of $(L/\lambda_{0})^{2}=1/100$. We choose the
polarization normal to the prisms axis so that in our local limit the system
is effectively isotropic in 2D. We truncated our matrices in reciprocal space
by setting a maximum value $2\pi n_{\mathrm{max}}/L$ for the magnitude
$|G_{x}|$ and $|G_{y}|$ of the components of the reciprocal vectors, so for a
field polarization within the plane the number of rows and columns for the
matrix
$\left[\mathcal{W}_{\mathbf{0}}(\mathbf{G},\mathbf{G}^{\prime})\right]_{ij}$
in Eq. (18) is given by $8n_{\mathrm{max}}(n_{\mathrm{max}}+1)$. To test the
convergence of our computational procedure, in Fig. 1 we show our results for
$\epsilon^{M}_{ij}\equiv\epsilon^{M}\delta_{ij}$ as a function of the maximum
index $n_{\mathrm{max}}$.
Figure 1: (color online) We show the normal-to-the-axis macroscopic dielectric
function $\epsilon^{M}$ obtained through Eq. (17) for a 2D square array with
lattice parameter $L$ of square dielectric prisms with response
$\epsilon_{p}=5.0$ placed in vacuum with filling fraction $f=0.3$ for a free-
space wavelength $\lambda_{0}=10L$ as a function of the largest reciprocal
vector index vs. $1/n_{\mathrm{max}}$. We show a linear extrapolation of our
results toward $1/n_{\mathrm{max}}\to 0$ and we indicate the values predicted
by Maxwell-Garnet formula and by some other authors mentioned in the text.
From Fig. 1 we see that $\epsilon^{M}$ converges approximately as
$1/n_{\mathrm{max}}$, and values of the order around $n_{\mathrm{max}}=40$ are
needed to obtain an accuracy better than 0.5% without extrapolating, yielding
large matrices of more than $13000\times 13000$ elements. In the same figure
we have indicated the response obtained by Milton et al.,Milton (1981), Tao et
al.,R.Tao et al. (1990) and Bergman et al.Bergman and Dunn (1992) which
studied the same composite. As we see, linear extrapolation of our results
towards $1/n_{\mathrm{max}}\to 0$ converge to those Bergman et al. and of
Milton et al., whereas the result of Tao et al. differs slightly. Finally, we
also compare our results with those of mean-field theory, as embodied in
Maxwell-Garnet’s (MG) formulae
$\epsilon^{M}=\epsilon_{h}+f\epsilon_{ph}-\frac{\epsilon_{ph}^{2}f(1-f)}{\gamma\epsilon_{h}+\epsilon_{ph}(1-f)},$
(19)
with $\gamma=2$ for our 2D system.Datta et al. (1993) As we see in the figure,
the MG results differ from ours and the other authors’ results, mainly due to
its intrinsic limitations.Datta et al. (1993) We have checked our results with
other set of parameters also reported by the same cited authors and we have
obtained similar agreement as mentioned above. The rate of convergence of our
method is similar to that reported in Ref. Sozuer et al., 1992 when written in
terms of $n_{\mathrm{max}}$.
We can also test the convergence of our results above using Keller’s
theorem,J.B.Keller (1963, 1964); Nevard and J.B.Keller (1985) which we may
write as $K=K_{x}K_{y}=1$, where we define Keller’s coefficients along
principal axes $x,y$ as
$K_{x}=(\epsilon^{M}_{x}\tilde{\epsilon}^{M}_{x})/(\epsilon_{h}\epsilon_{p})$
and
$K_{y}=(\epsilon^{M}_{y}\tilde{\epsilon}^{M}_{y})/(\epsilon_{h}\epsilon_{p})$.
Here, we introduced the macroscopic response $\tilde{\epsilon}^{M}_{x}$ and
$\tilde{\epsilon}^{M}_{y}$ of the reciprocal system that is obtained from the
original system by interchanging $\epsilon_{h}\leftrightarrow\epsilon_{p}$.
Indeed, for our isotropic system we expect $K_{x}=1$ as $K_{x}=K_{y}$. In Fig.
2 we show $K_{x}-1$ vs. $1/n_{\mathrm{max}}$ for the system corresponding to
Fig. 1.
Figure 2: (color online) Keller’s coefficient $K_{x}-1$ as a function of
$1/n_{\mathrm{max}}$ and its linear extrapolation towards
$n_{\mathrm{max}}\to\infty$ for the same system as in Fig. 1 in the cases of
$\lambda_{0}/L=10$ and 100.
We see clearly that $K_{x}-1$ decreases linearly in $1/n_{\mathrm{max}}$.
However, its extrapolation towards $n_{\mathrm{max}}\to\infty$ does not attain
the value $K_{x}-1=0$ as expected from Keller’s theorem. The reason for the
small discrepancy is that our calculation includes retardation effects which
we expect to be of order $(L/\lambda_{0})^{2}$, while Keller’s theorem is
strictly valid only in systems with no retardation. To confirm this statement,
we also display in Fig. 2 the results of a calculation for $\lambda_{0}=100L$,
showing that in this case, the discrepancy between the extrapolated and the
expected value is negligible. Thus, we have verified that our calculation is
consistent with Keller’s theorem in the absence of retardation and has an
error that goes to zero as $1/n_{\mathrm{max}}$ when $\lambda_{0}/L\to\infty$.
Figure 3: (color online) $\epsilon^{M}$ versus the filling fraction for the
same system as shown on Fig. 1. Our result was obtained from Eqs. (17) and
(18) employing a $13120\times 13120$ matrix
$\left[\mathcal{W}_{\mathbf{0}}(\mathbf{G},\mathbf{G}^{\prime})\right]_{ij}$.
In Fig. 3 we show nearly converged (error $<0.5\%$) results for $\epsilon^{M}$
as a function of the filling fraction $f$ for the same system as in Fig. 1. We
can see again an excellent agreement of our results with those obtained by
Milton et al., and Bergman et al., and, to a lesser extent, with those of Tao
et al. The MG results are noticeably lower, with a discrepancy that increases
with filling fraction. We have obtained results identical to ours in Figs. 1
and 3 using Eq. (35) of Ref. Halevi and Pérez-Rodríguez, 2006, confirming that
our formalism is equivalent to that of Halevi and Pérez-Rodríguez. In
conclusion, our approach does indeed reproduce the results of other works, and
thus we have validated our numerical scheme and can be confident on the
accuracy of our results.
### III.2 2D array
Having confirmed our calculation procedure through comparison to earlier works
and convergence tests, we proceed to evaluate the optical properties of a
metamaterial.
Figure 4: Unit cell of a 2D rectangular array of rectangular prisms with
response $\epsilon_{p}$ within a host with response $\epsilon_{h}$. The aspect
ratio of the rectangles is determined by the point
$A=(L_{x}\xi_{x},L_{y}f/\xi_{x})$ that lies on the dashed line from
$(L_{x}f,L_{y})$ to $(L_{x},L_{y}f)$.
We choose a 2D rectangular lattice of rectangular prisms, assuming
translational symmetry along $z$ (see Fig. 4). The unit cell has lengths
$L_{x}$ and $L_{y}$ along the $x$ and $y$ directions, and the inclusions have
corresponding sizes $a_{x}$ and $a_{y}$. The shape of the lattice is
controlled by a parameter $\eta$ defined through
$L_{x}=\eta L_{y},$ (20)
and we define
$\xi_{i}\equiv\frac{a_{i}}{L_{i}}\quad i=x,y.$ (21)
Then,
$\mathbf{G}=n_{x}\frac{2\pi}{L_{x}}\hat{x}+n_{y}\frac{2\pi}{L_{y}}\hat{y}=\frac{2\pi}{L_{x}}(n_{x}\hat{x}+\eta
n_{y}\hat{y}),$ (22)
for integer $n_{x}$ and $n_{y}$, and
$f=\xi_{x}\xi_{y}.$ (23)
From Eq. (14) we obtain
$S(\mathbf{G})=\mbox{sinc}(\frac{G_{x}a_{x}}{2})\mbox{sinc}(\frac{G_{y}a_{y}}{2})=\mbox{sinc}(\pi\xi_{x}n_{x})\mbox{sinc}(\pi\frac{f}{\xi_{x}}n_{y}),$
(24)
where $\mbox{sinc}(x)=\sin(x)/x$. We can vary the shape of the inclusion
keeping the filling fraction fixed by simply changing $\xi_{x}$ within the
bounds
$f\leq\xi_{x}\leq 1.$ (25)
The array is square if $L_{x}=L_{y}$. Furthermore, if
$\xi_{x}=\xi_{y}=\sqrt{f}$ the inclusions have a square cross section, while
for $\xi_{x}>\sqrt{f}$ ($\xi_{x}<\sqrt{f}$) they become elongated along $x$
($y$). For $\xi_{x}=1$, $\xi_{y}=f$ ($\xi_{x}=f$, $\xi_{y}=1$) the inclusions
fully occupy the unit cell along $x$ ($y$), contacting neighbor inclusions, so
that the systems becomes an effectively one dimensional system of slabs with
surfaces normal to $y$ ($x$).
To interpret the results easily we consider a semi-infinite slab $z>0$ cut out
of our metamaterial and we calculate its normal incidence reflectance
$R_{\zeta}=\left|\frac{\sqrt{\epsilon^{M}_{\zeta\zeta}}-1}{\sqrt{\epsilon^{M}_{\zeta\zeta}}+1}\right|^{2},\quad(\zeta=x,y)$
(26)
corresponding to a $\zeta=x,y$ linearly polarized incoming beam propagating
through empty space along $z$ and impinging upon the interface which we locate
at $z=0$. In this equation we have neglected the possibility of a magnetic
permeability $\mu\neq 1$, which may be expected even when the constituents of
the system are non-magnetic due to the possible non-locality of the
macroscopic dielectric response $\epsilon^{M}_{\mathbf{q}}$, as may be
obtained from Eq. (10). The non-locality may be partially accounted for by a
local dielectric function $\epsilon^{M}=\epsilon^{M}_{\mathbf{0}}$ and a local
magnetic permeability $\mu$ which is of the order of
$\mu-1\sim(\epsilon^{M}_{\mathbf{q}}-\epsilon^{M}_{\mathbf{0}})/q^{2}$.Halevi
and Pérez-Rodríguez (2006) From Eq. (12), we expect $\mu-1\sim k_{0}^{2}L^{2}$
where $L$ is of the order of the periodicity of the system. Another criteria
that has been developed for conducting structures states that the magnetic
response may be neglected as long as the cross section of the particles is
much smaller than the penetration depth.A.A.Krokhin et al. (2007) Thus, in the
examples that follow we may safely neglect the magnetic permeability.
In the following figures we choose a square unit cell with $L_{x}=L_{y}=40$ nm
with gold in the interstitial region, for which we use the experimentally
determined response $\epsilon_{h}=\epsilon\mbox{(Au)}$,E.D.Palik (1985) and
with dielectric inclusions for which we chose $\epsilon_{p}=4$. For different
values for the filling fraction $f$ we control the rectangular geometry of the
inclusion with the parameter $\xi_{x}$. The value of $n_{\mathrm{max}}$ is set
to 50 which gives good converged results.111The numerical burden of such large
matrices has to be surmounted with the use of ScaLAPACK subroutines
(http://www.netlib.org/scalapack/) to efficiently solve Eq. (11) on a high-end
computer cluster. Typical time on 40-processor grid is 1.3 hours per energy
point.
We start with the extreme case $\xi_{x}=1$, for which we have a system of
alternating conductor and dielectric flat slabs piled up along the $y$
direction. In this case, the non-retarded macroscopic dielectric response is
given exactly by
$\epsilon^{M}_{xx}=f\epsilon_{p}+(1-f)\epsilon_{h},$ (27)
and
$\frac{1}{\epsilon^{M}_{yy}}=\frac{f}{\epsilon_{p}}+\frac{1-f}{\epsilon_{h}}.$
(28)
The latter expression can be written as
$\epsilon^{M}_{yy}=\epsilon_{h}+f\epsilon_{ph}-\frac{\epsilon_{ph}^{2}f(1-f)}{\epsilon_{h}+\epsilon_{ph}(1-f)},$
(29)
which is the MG result for one dimension, i.e., Eq. (19) with $\gamma=1$.
Figure 5: (color online) Reflectance $R_{\zeta}$ ($\zeta=x,y$) vs. photon
energy for $\xi_{x}=1$, i.e., for a 1D multi-layer of alternating slabs of
gold ($\epsilon_{h}=\epsilon(Au)$) and a dielectric ($\epsilon_{p}=4$), with
$L=40$ nm, and $f=0.5$. The slabs are normal to the $y$ direction and the
incoming light propagates along the $z$ direction. We compare numerical
results obtained from Eq. (17) with the exact non-retarded results obtained
through Eq. (27) for $R_{x}$ and Eq. (28) or Eq. (29) (Maxwell-Garnet in 1D)
for $R_{y}$.
In Fig. 5 we show $R_{\zeta}$ vs. the photon energy $\hbar\omega$ as obtained
through our numerical scheme (Eq. (17)), and compare them to the exact non-
retarded results (Eq. (27) and Eq. (29)). We remark that both numerical and
exact results agree closely. Actually, in the appendix we show analytically
that in this case our formalism coincides exactly with Eq. (27) and Eqs.
(28-29) and that Eq. (27) holds also along the translational invariant
direction of 2D systems.A.A.Krokhin et al. (2002)
The system is highly anisotropic ($R_{x}\neq R_{y}$) so that the 1D MG results
are only applicable along the $y$ direction ($R_{y}$). Notice that the
behavior of the system at low frequencies is metallic for $\zeta=x$, with a
very high reflectance, while it is dielectric-like for $\zeta=y$, as electric
the current may flow unimpeded through the Au layers in the $x$ direction, but
it would be interrupted along the $y$ direction by the dielectric layers.
Figure 6: (color online) Reflectance $R_{x}$ (thin lines) and $R_{y}$ (thick
lines) vs the photon energy, for a gold host with inclusions of
$\epsilon_{p}=4$ and a fixed $\xi_{y}=0.5$, for three different values of
$\xi_{x}$=0.5, 0.7, and 0.9 (see text for details). We also show the
reflectivity of gold.
Having checked that our approach coincides with two well-known analytic
limits, we proceed to show results for other choices of $\xi_{x}$ and $f$. In
Fig. 6 we show the reflectance $R_{\zeta}$ for inclusion with three
rectangular cross sections with a fixed $\xi_{y}=0.5$ for several choices of
$\xi_{x}$=0.5, 0.7, and 0.9, with corresponding values of $f$=0.25, 0.35, and
0.45. Thus, we include square and rectangular prisms. As could be expected,
$R_{x}=R_{y}$ for the square isotropic case $\xi_{x}=\xi_{y}$, while for
rectangular sections the reflectance becomes strongly dependent on the
polarization $\zeta=x$ or $y$; the anisotropy increases as $\xi_{x}$ moves
away from $\xi_{y}$. As $\epsilon_{p}$ and $\epsilon_{h}$ are isotropic, the
anisotropy $\epsilon^{M}_{x}\neq\epsilon^{M}_{y}$ of the macroscopic response
arises from the last term of Eq. (17). Thus, the source of the anisotropy is
the local-field interaction among the inclusions, linked to the geometry of
the system.
We notice that $R_{\zeta}$ for $\zeta=x$ polarization, along the elongated
side of the rectangles, is qualitatively similar to the isotropic case, as
well as to that of gold (shown in the same Fig. 6). To wit, for low energies
the reflectance is very large, as gold behaves like a Drude metal and most of
the light is reflected. For higher energies and especially above the
interband-transitions threshold of Gold ($\sim 2.44$ eV), the reflectance
diminishes as gold deviates from the pure Drude-like behavior and dissipation
mechanisms beyond ohmic heating appear. It is important to note that the
surface and bulk plasma frequencies ($\sim$ 5, 6 eV) are still higher up in
energy than such threshold.
However, we notice an interesting effect for $\zeta=y$ polarization, along the
short side of the rectangles. At some energies $R_{y}$ deviates strongly from
the isotropic case as $\xi_{x}$ increases, and shows a counterintuitive
behavior, developing a deep minimum which approaches zero reflectance for some
values of the photon energy. This may appear surprising, as gold is very
reflective in the infrared region. Nevertheless the geometry of the inclusions
changes this behavior dramatically. It is also interesting to note that above
the interband threshold the anisotropy is drastically reduced as $R_{x}\sim
R_{y}$.
Figure 7: (color online) Real (top panel) and imaginary (bottom panel) part of
$\epsilon^{M}_{yy}$ vs. photon energy for the same system as that presented on
Fig. 6.
To explain the surprising behavior of the reflectance, in Fig. 7 we show the
real and imaginary part of the macroscopic response $\epsilon^{M}_{yy}$ for
the same system as the one presented in Fig. 6. For $\xi_{x}>\xi_{y}$ the
response along $y$ is dielectric like, with a positive Re($\epsilon^{M}_{yy}$)
larger than unity, not unlike the 1D layered system presented in Fig. 5.
Nevertheless, as the dielectric prisms are completely isolated from each other
by the metallic interstices, the Au region percolates and the behavior at low
enough frequencies is metallic, with a negative $\epsilon^{M}_{yy}$. Thus,
there is a photon energy where Re($\epsilon^{M}_{yy}$) crosses through unity.
This energy is red-shifted as $\xi_{x}$ grows and the metallic behavior
disappears completely at the limit $\xi_{x}=1$. Thus, for appropriate values
of $\xi_{x}$, the crossing may be situated at frequencies too low to excite
interband transitions in Au, but large enough so that ohmic losses become
unimportant. For that frequency at which $\mbox{Re}(\epsilon_{yy})\approx 1$,
and $\mbox{Im}(\epsilon_{yy})\ll 1$ there is a good impedance matching between
vacuum and the material, and thus, there is a small reflectance which
approaches zero. When this conditions holds, the transmittance of a finite
slab approaches unity. Our results show that this is the case at
$\hbar\omega\approx 1.25$ (1.7) eV for $\xi_{x}=0.9$ ($\xi_{x}=0.7$).
Figure 8: (color online) Reflectance $R_{x}=R_{y}=R$ vs. photon energy for a
system made up of circular cylinders with $\epsilon_{p}=4$ within an Au matrix
$\epsilon_{h}=\epsilon(\mbox{Au})$ for four values of the filling fraction
$f=0.25,0.5,0.6,$ and 0.7. We also show the reflectance of gold for
comparison.
We explained the different behaviors between the $x$ and $y$ response of a
system of rectangular prisms for different values of $\xi_{x}$ and different
frequencies in terms of a low-frequency metallic and high-frequency insulating
behavior of the composite, which in turn is related to the percolation of the
metallic host. We may confirm these ideas by studying other systems with
dielectric inclusions within a percolating conducting host, such as a square
array of dielectric cylinders within an Au host. For a filling fraction
$f>\pi/4$ the dielectric cylinders would touch each other impeding the flow of
current between the conducting regions that would become isolated from each
other, and the system would behave as an insulator. Thus, we study the case
$f<\pi/4$ for which we expect the low frequency behavior to be metallic like
with a transition into a dielectric like behavior at higher frequencies as in
the rectangular case. This system is however isotropic, $R_{x}=R_{y}=R$. To
perform the calculation we only require the Fourier coefficients
$S(\mathbf{G})=2fJ_{1}(Gr_{c})/Gr_{c}$, with $r_{c}$ the radius of the
inclusions, and $J_{1}$ the $J$-Bessel function of order one. Fig. 8 shows
that $R$ indeed attains large values at low frequencies, corresponding to a
metallic behavior, and then attains a minimum corresponding to the expected
transition into a dielectric behavior. The transition frequency is red-shifted
and becomes broader and deeper as $f$ increases.
Our examples show that the reflection goes rapidly from almost one to almost
zero at a frequency in the near infrared which may be tuned by choosing the
filling fraction and, in the case of rectangular inclusions, by changing the
aspect ratio. A square array of cylinders is isotropic within the $x-y$ plane,
so, in a sense, the array of rectangular prisms is richer, as it allows us to
change the behavior from conducting-like to insulating-like by simply rotating
the polarization.
We remark that the behavior of the reflection discussed above is induced
solely by the geometry of the metamaterial and is not simply connected to the
structure of the response functions of the constituent materials, that is, to
resonances in $\epsilon_{p}$ and/or in $\epsilon_{h}$. Similar resonances,
mainly related to the geometry of the metamaterial, were already predicted by
Khizhnyak back in 1958.Khizhnyak (1959) Our results, clearly show the huge
difference that the shape of the inclusions makes on the optical properties of
the system.Koerkamp et al. (2004); Gordon et al. (2004); García-Vidal et al.
(2005)
## IV Conclusions
We have developed a systematic scheme to calculate the complex frequency
dependent macroscopic dielectric function for metamaterials. Starting from
Maxwell’s equations and employing a long wavelength approximation we have
derived an expression for the macroscopic dielectric function
$\epsilon^{M}_{ij}$ that depends on the dielectric functions of the host
$\epsilon_{h}$ and particles $\epsilon_{p}$, and on the geometry of both the
unit cell and the inclusions. The calculation is setup through expansions of
the microscopic fields in plane wave components, and in general a large number
of reciprocal vectors $\mathbf{G}$ are required to achieve convergence of the
results. We validated our formalism through convergence tests and through
comparison of our results to those from previous calculations, founding an
excellent agreement. Then, we calculated macroscopic response and the normal-
incidence reflectivity for systems made up of dielectric rectangular prisms
and cylinders arranged in a 2D square lattice within a gold host. Although the
host and the inclusions are intrinsically isotropic, we found that, if the
inclusion is geometrically anisotropic, so is the macroscopic optical
response. For rectangular prisms of high aspect ratio we found a very
anisotropic optical response, where the infrared reflectance is almost unity
when the field is polarized along the long axis, while it can attain values
very close to zero when the field is polarized along the short axis. We
explained this behavior in terms of a transition from a low-frequency
conducting behavior to a high-frequency dielectric behavior for systems not
too far from percolation into the non-conducting phase. The transition may
occur at frequencies in the infrared frequencies for which one would naively
expect very low values for the transmittance. We verified this explanation
through the calculation of the reflectance of a square array of cylindrical
prisms, which shows an isotropic but otherwise similar behavior as we approach
the percolation threshold $f=\pi/4$. Our formalism may be employed to explore
and design of very diverse systems with a tailored optical response. We hope
this work would motivate the construction of such systems and their optical
characterization for the experimental verification of our results.
###### Acknowledgements.
We acknowledge inspiring discussions with Peter Halevi and Felipe Pérez-
Rodríguez. This work was partially supported by DGAPA-UNAM grant IN120909
(WLM), by CONACyT grants 48915-F (BMS) and J49731-F (BEMZ) and by ANPCyT grant
190-PICTO-UNNE (GPO).
## Appendix
We show that our formalism, as embodied in Eq. (11), Eq. (17) and Eq. (18) are
equivalent in the non-retarded limit to the analytical results (27) and Eq.
(28) for the case of periodically alternating isotropic thin flat slabs. We
chose the $y$ axis normal to the slabs, so that the reciprocal vectors
$\mathbf{G}=G\hat{y}$ lie all along $y$. In this case, both
$[\mathcal{W}_{0}(G,G^{\prime})]_{ij}$ and $[\epsilon_{0}(G,0)]_{ij}$ are
diagonal, so we can consider separately the cases of polarization along the
$x$ and the $y$ axes.
For $x$ polarization we rewrite Eq. (11) as
$\sum_{G^{\prime}\neq
0}\left(\epsilon_{h}\delta_{GG^{\prime}}+\epsilon_{ph}S(G-G^{\prime})-\frac{G^{2}}{k_{0}^{2}}\delta_{GG^{\prime}}\right)[\Phi_{0}(G^{\prime},0)]_{xx}=\epsilon_{ph}S(G),$
(30)
whose solution is
$[\Phi_{0}(G^{\prime},0)]_{xx}=0+\mathcal{O}(k_{0}^{2}/G^{2}).$ (31)
Substitution in Eq. (17) yields immediately Eq. (27) to order 0 in the small
quantities $k_{0}/G$ in the non-retarded limit. Notice that the argument above
is valid for any system which has translational invariance along one or more
directions whenever the polarization direction points along those directions,
since Eq. (30) holds when all the reciprocal vectors $\mathbf{G}$ are
perpendicular to the polarization direction. In particular, for systems which
have texture only along two dimensions, the macroscopic dielectric function
along the third dimension is simply the volume average of the microscopic
dielectric functions.A.A.Krokhin et al. (2002)
For $y$ polarization we rewrite Eq. (11) as
$\sum_{G^{\prime}\neq
0}\left(\epsilon_{h}\delta_{GG^{\prime}}+\epsilon_{ph}S(G-G^{\prime})\right)[\Phi_{0}(G^{\prime},0)]_{yy}=\epsilon_{ph}S(G),$
(32)
as $G^{2}-G_{y}G_{y}=0$. Although $[\Phi_{0}(G,0)]_{yy}$ is only defined for
$G\neq 0$, we can extend its definition to $G=0$ by choosing
$[\Phi_{0}(0,0)]_{zz}\equiv 0$ and extending Eq. (32) to include the $G=0$
term. For consistency, we have to add an unknown term to its RHS which only
applies to the $G=0$ term, i.e.,
$\sum_{G^{\prime}}\left(\epsilon_{h}\delta_{GG^{\prime}}+\epsilon_{ph}S(G-G^{\prime})\right)[\Phi_{0}(G^{\prime},0)]_{yy}=\epsilon_{ph}S(G)+C\delta_{G,0},$
(33)
where the sum includes now all values of $G^{\prime}$. Taking the Fourier
transform of Eq. (33) we obtain
$\epsilon_{h}[\Phi_{0}(y)]_{yy}+\epsilon_{ph}S(y)[\Phi_{0}(y)]_{yy}=\epsilon_{ph}S(y)+C,$
(34)
which yields
$[\Phi_{0}(y)]_{yy}=\frac{\epsilon_{ph}S(y)+C}{\epsilon(y)}.$ (35)
The constant $C$ must be chosen so that the spatial average of
$[\Phi_{0}(y)]_{yy}$ vanishes,
$0=[\Phi_{0}(G=0,0)]_{yy}=\frac{1-f}{\epsilon_{h}}C+\frac{f}{\epsilon_{p}}(\epsilon_{ph}+C),$
(36)
vanishes. Substituting the result in (35) we obtain
$[\Phi_{0}(y)]_{yy}=\frac{\epsilon_{ph}}{\epsilon(y)}\left(S(y)-\frac{f\epsilon_{h}}{\epsilon_{h}+\epsilon_{ph}(1-f)}\right).$
(37)
Now we extend the sum in Eq. (17) to include the $G=0$ contribution, allowing
us to employ the convolution theorem to obtain
$\sum_{G}S(-G)[\Phi_{0}(G,0)]_{yy}=\frac{1}{L_{y}}\int
dy\,S(y)[\Phi_{0}(y)]_{yy}=f\frac{\epsilon_{ph}}{\epsilon_{p}}\left(1-\frac{f\epsilon_{h}}{\epsilon_{h}+\epsilon_{ph}(1-f)}\right),$
(38)
which we substitute into Eq. (17) to finally obtain Eq. (29).
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|
arxiv-papers
| 2009-01-22T20:38:28 |
2024-09-04T02:49:00.152731
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guillermo P. Ortiz, Brenda E. Mart\\'inez-Z\\'erega, Bernardo S.\n Mendoza, W. Luis Moch\\'an",
"submitter": "Luis Moch\\'an",
"url": "https://arxiv.org/abs/0901.3549"
}
|
0901.3589
|
# The Search for Heavy Majorana Neutrinos
Anupama Atre1,2, Tao Han2,3,4, Silvia Pascoli5, Bin Zhang4
1Fermi National Accelerator Laboratory, MS106, P.O.Box 500, IL 60510, U.S.A.
2Kavli Institute of Theoretical Physics, University of California, Santa
Barbara, CA 93107, U.S.A.
3Department of Physics, University of Wisconsin, 1150 University Ave, Madison,
WI 53706, U.S.A.
4Center for High Energy Physics, Department of Physics, Tsinghua University,
Beijing 100084, P.R. China
5Institute for Particle Physics Phenomenology, Department of Physics, Durham
University, Durham DH1 3LE, United Kingdom avatre@fnal.gov,
than@hep.wisc.edu, silvia.pascoli@durham.ac.uk, zb@mail.tsinghua.edu.cn
(Communication author)
###### Abstract:
The Majorana nature of neutrinos can be experimentally verified only via
lepton-number violating processes involving charged leptons. We study $36$
lepton-number violating ($LV$) processes from the decays of tau leptons and
pseudoscalar mesons. These decays are absent in the Standard Model but, in
presence of Majorana neutrinos in the mass range $\sim 100\mbox{ }\rm MeV$ to
$5\mbox{ }\rm GeV$, the rates for these processes would be enhanced due to
their resonant contribution. We calculate the transition rates and branching
fractions and compare them to the current bounds from direct experimental
searches for $\Delta L=2$ tau and rare meson decays. The experimental non-
observation of such $LV$ processes places stringent bounds on the Majorana
neutrino mass and mixing and we summarize the existing limits. We also extend
the search to hadron collider experiments. We find that, at the Tevatron with
$8\ \mbox{fb}^{-1}$ integrated luminosity, there could be $2\sigma$
($5\sigma$) sensitivity for resonant production of a Majorana neutrino in the
$\mu^{\pm}\mu^{\pm}$ modes in the mass range of $\sim 10-180\ \mbox{\rm GeV}\
(10-120\ \mbox{\rm GeV})$. This reach can be extended to $\sim 10-375\
\mbox{\rm GeV}\ (10-250\ \mbox{\rm GeV})$ at the LHC of 14 TeV with $100\
\mbox{fb}^{-1}$. The production cross section at the LHC of 10 TeV is also
presented for comparison. We study the $\mu^{\pm}e^{\pm}$ modes as well and
find that the signal could be large enough even taking into account the
current bound from neutrinoless double-beta decay. The signal from the gauge
boson fusion channel $W^{+}W^{+}\rightarrow\ell^{+}_{1}\ell^{+}_{2}$ at the
LHC is found to be very weak given the rather small mixing parameters. We
comment on the search strategy when a $\tau$ lepton is involved in the final
state.
††preprint: FERMILAB-PUB-08-086-T, NSF-KITP-08-54, MADPH–06–1466, DCPT/07/198,
IPPP/07/99
## 1 Introduction
In the Standard Model (SM) of strong and electroweak interactions, neutrinos
are strictly massless due to the absence of right-handed chiral states
($N_{R}$) and the requirement of $SU(2)_{L}$ gauge invariance and
renormalizability. Recent neutrino oscillation experiments have conclusively
shown that neutrinos are massive [1]. This discovery presents a pressing need
to consider physics beyond the SM. It is straightforward to obtain a Dirac
mass term $m_{D}(\overline{\nu_{L}}N_{R}+\mathrm{h.c.})$ for a neutrino by
including the right-handed state, just like the treatment for all other
fermions via Yukawa couplings to the Higgs doublet in the SM. However, a
profound question arises: Since $N_{R}$ is a SM gauge singlet, why should a
gauge-invariant Majorana mass term ${1\over 2}MN_{R}N_{R}$ not exist in the
theory? In fact, there is strong theoretical motivation for the Majorana mass
term to exist since it could naturally explain the smallness of the observed
neutrino masses via the so-called “see-saw” mechanism [2]
$m_{\nu}\approx{m^{2}_{D}\over M}.$ (1)
From a model-building point of view, there are many scenarios that could
incorporate the Majorana mass. Examples include Left-Right symmetric gauge
theories [3]; $SO(10)$ Supersymmetric (SUSY) grand unification [4] and other
grand unified theories [5]; models with exotic Higgs representations [6, 7];
R-parity violating interactions ($\Delta L=1$) in Supersymmetry (SUSY) [8] and
theories with extra dimensions [9]. There are other proposals to generate
Majorana masses for neutrinos at a higher scale $M$ without relying on the
right-handed state $N_{R}$ [10, 11]. According to the scheme in generating the
mass scale $M$ in Eq. (1), it has been customary to call them Type I [2], Type
II [10] or Type III [11].
Within the context of the SM, there is only one gauge-invariant operator [12]
that is relevant to the neutrino mass,
${\kappa\over\Lambda}l_{L}H\ l_{L}H,$ (2)
where $l_{L}$ and $H$ are the SM lepton and Higgs doublets, respectively. The
constant $\kappa$ is a model-dependent effective coupling and $\Lambda$ is the
new physics cut-off scale. It is a dimension-5 non-renormalizable operator,
and leads to Majorana neutrino masses of the order $\kappa v^{2}/\Lambda$,
after the Higgs field acquires a vacuum expectation value $v$, in accordance
with the see-saw scheme. Higher dimensional operators that give rise to
Majorana neutrino masses have also been constructed in a model-independent
manner [13]. The challenging task is to look for experimental evidence to
probe the new physics scale $\Lambda$ and to distinguish the underlying
theoretical models mentioned above.
In the neutrino sector, besides the rich phenomena of neutrino flavor
oscillations and the possible existence of new sources of CP-violation, lepton
number violation by two units ($\Delta L=2$), as implied by a Majorana mass
term, plays a crucial role. Not only may it result in important consequences
in particle physics, nuclear physics and cosmology but it would also guide us
in understanding the fundamental symmetries of physics beyond the SM. Although
the prevailing theoretical prejudice prefers Majorana neutrinos,
experimentally testing the nature of neutrinos and lepton-number violation
($LV$) in general, is of fundamental importance. In accelerator-based
experiments, neutrinos in the final state are undetectable by the detectors,
leading to the so-called “missing energy” and therefore missing lepton numbers
as well. One is thus forced to look for charged leptons in the final state.
The basic process with $\Delta L=2$ can be generically expressed by
$\displaystyle W^{-}W^{-}\rightarrow\ell^{-}_{1}\ell^{-}_{2},$ (3)
where $W^{-}$ is a virtual SM weak boson and $\ell_{1,2}=e,\mu,\tau$. By
coupling fermion currents to the $W$ bosons as depicted in Fig. 1, and
arranging the initial and final states properly, one finds various physical
processes that can be experimentally searched for. The best known example is
neutrinoless double-beta decay ($0\nu\beta\beta$) [14, 15, 16], which proceeds
via the parton-level subprocess $dd\to uu\ W^{-*}W^{-*}\to uu\ e^{-}e^{-}$.
Other interesting classes of $LV$ processes involve tau decays such as
$\tau^{-}\to\ell^{+}M_{1}^{-}M_{2}^{-}$ [17, 18] where the light mesons
$M_{1},M_{2}$ are $\pi,K$, rare meson decays such as
$M^{+}_{1}\to\ell^{+}_{1}\ell^{+}_{2}M^{-}_{2}$ [19, 20, 18] and hyperon
decays such as $\Sigma^{-}\to\Sigma^{+}e^{-}e^{-}$, $\Xi^{-}\to
p\mu^{-}\mu^{-}$ etc. [21]. One could also explore additional processes like
$e^{-}\to\mu^{+}$ [22], $\mu^{-}\to e^{+}$ [18, 23] and $\mu^{-}\to\mu^{+}$
conversion [18, 24]. One may also consider searching for signals at
accelerator and collider experiments via $e^{-}e^{-}\to W^{-}W^{-}$ [25],
$e^{+}e^{-}\to Z^{0}\to N+X$ [26],
$e^{\pm}p\to\nu_{e}(\overline{\nu_{e}})\ell_{1}^{\pm}\ell_{2}^{\pm}X$ [27],
neutrino nucleon scattering $\nu_{\ell}(\overline{\nu_{\ell}}){\cal
N}\to\ell^{\mp}\ell_{1}^{\pm}\ell_{2}^{\pm}X$ [28],
$pp\to\ell^{+}_{1}\ell^{+}_{2}X$ [29, 30, 31, 32, 33, 34], top-quark decays
$t\to b\ell^{+}_{1}\ell^{+}_{2}W^{-}$ [35], charged-Higgs production
$e^{\pm}e^{\pm}\rightarrow H^{\pm}H^{\pm}$ [36], and in the decay
$N\to\ell^{\pm}H^{\mp}$ [37].
Figure 1: A generic diagram for $\Delta L=2$ processes via Majorana neutrino
exchange.
The dynamics for $\Delta L=2$ processes as in Eq. (3) is dictated by the
properties of the exchanged neutrinos. For a Majorana neutrino that is light
compared to the energy scale in the process, the transition rates for $LV$
processes are proportional to the product of two flavor mixing matrix elements
among the light neutrinos and a $LV$ mass insertion
$\displaystyle\left<m\right>_{\ell_{1}\ell_{2}}^{2}=\biggl{|}\sum_{m=1}^{3}U_{\ell_{1}m}U_{\ell_{2}m}m_{\nu_{m}}\biggr{|}^{2},$
(4)
where $\left<m\right>_{\ell_{1}\ell_{2}}$ are the “effective neutrino masses”.
If the neutrinos are heavy compared to the energy scale involved, then the
contribution scales as
$\left|\sum_{m^{\prime}=4}^{3+n}\frac{V_{\ell_{1}m^{\prime}}V_{\ell_{2}m^{\prime}}}{m_{N_{m^{\prime}}}}\right|^{2},$
(5)
where $V$ is the mixing matrix between the light flavor and heavy neutrinos.
Unfortunately, both situations encounter a severe suppression either due to
the small neutrino mass like $m_{\nu_{m}}^{2}/M_{W}^{2}$, or due to the small
mixing $\left|V_{\ell_{1}m^{\prime}}V_{\ell_{2}m^{\prime}}\right|^{2}$. An
important observation is that when the heavy neutrino mass is kinematically
accessible, a process may undergo resonant production of the heavy neutrino.
The transition rate can be substantially enhanced and goes like
${\Gamma(N_{m^{\prime}}\to i)\ \Gamma(N_{m^{\prime}}\to f)\over
m_{N_{m^{\prime}}}\Gamma_{N_{m^{\prime}}}},$ (6)
where $i,f$ refer to the initial and final state during the transition.
The possible existence of sterile neutrinos in the mass range relevant for
resonant enhancement of $\Delta L=2$ processes studied in this paper is
motivated in several scenarios. Models which implement the see-saw mechanism
at low energies have been recently considered [38, 39]: the neutrino masses
generated are accidentally small and active-sterile mixing can be as large as
few percent. See-saw models at the electroweak scale can explain neutrino
masses if appropriate symmetries are imposed and at the same time provide an
appealing mechanism for baryon asymmetry generation via resonant leptogenesis
[40]. In theories with dynamical electroweak symmetry breaking, sterile
neutrinos with masses in the $100$s of MeV to GeV range are invoked to explain
light neutrino masses [41]. Sterile neutrinos can also play a role in
understanding the flavour problem in the leptonic sector. It has been shown
that mixing with sterile neutrinos can be at the origin of the large angles in
the neutrino sector [42].
Heavy, mostly-sterile neutrinos have been investigated for their role in
cosmology and astrophysics, in particular in Big Bang Nucleosynthesis, Large
Scale Structure formation [43], cosmic microwave background, diffuse
extragalactic background radiation, supernovae [44] and as dark matter
candidates [45, 46, 47] (for a review on MeV sterile neutrinos, see Ref.
[48]). A keV sterile neutrino is a viable dark matter candidate [45, 47],
which can also explain the origin of pulsar kicks [49]. Decays of heavy,
mostly-sterile neutrinos have been proposed to explain the early ionization of
the Universe [50]. Due to mixing, dark matter sterile neutrinos would decay
radiatively contributing to the Diffuse Extragalactic Background Radiation and
inducing x-ray emission from galaxy clusters [51, 52]. A large coupling
between sterile neutrinos and light dark matter scalars can be at the origin
of neutrino masses and of the observed dark matter abundance [53]. A model
with sterile neutrinos in the keV-GeV mass range has been proposed to explain
the dark matter of the Universe as well as baryogenesis [54, 55]. Its
phenomenological and astrophysical signatures have been considered in detail
in Refs. [56, 52]. This model assumes the existence of one sterile neutrino
with keV mass for dark matter and two heavier neutrinos with quasi-degenerate
GeV masses for successful baryogenesis. The required mixing $|V_{\ell
m^{\prime}}|^{2}$ of the latter neutrinos with the active ones is mass
dependent and lies in the range $10^{-11}-10^{-8}$, for a mass of $1$ GeV.
Additional constraints on the heavy neutrino mass and mixing angles can be
derived from astrophysical observations. Sterile neutrinos mixed with active
ones would be efficiently produced in supernovae cores, escaping from it and
depleting substantially the supernova core energy, and, therefore, might
modify the supernova evolution. Recently, it was shown that sterile neutrinos
in the mass range $\sim 0.2\ \mathrm{GeV}$ and small mixing angle with $\mu$
and $\tau$ neutrinos could enhance the energy transport from the core to the
stalled shock and favor the supernova explosion [44]. They could also explain
the high velocity of pulsars if the momentum carried away by heavy sterile
neutrinos is emitted asymmetrically [49]. Detailed reviews and discussions of
heavy neutrinos in the Early Universe and their present bounds can be found,
e.g., in Refs. [48, 57, 58].
Cosmological and astrophysical constraints on sterile neutrinos are typically
very strong but are not as robust as the ones from laboratory searches as they
typically depend on the production mechanism of sterile neutrinos in the Early
Universe and on the cosmological evolution. For example, they can be
significantly weakened or evaded if the reheating temperature is low [47, 59],
if their density in the Early Universe is diluted by entropy injected at late
times [55] or if they have non-standard interactions. In these cases, much
larger mixing angles with active neutrinos are allowed by cosmological
observations and can be tested in terrestrial experiments. Therefore, it is
important to perform experimental searches of heavy sterile neutrinos with
increased sensitivity and, specifically for Majorana neutrinos, to consider
$\Delta L=2$ processes. If a positive signal is found and is incompatible with
the cosmological and/or astrophysical observations, one would need to consider
modifications to the standard cosmological scenario and/or would gain new
insight on the evolution of astrophysical objects.
In this paper, we study resonant contributions of heavy Majorana neutrinos to
$\Delta L=2$ processes involving two charged leptons in accelerator-based
experiments. We establish our conventions and discuss the current constraints
on the mass and mixing of heavy neutrinos in Sec. 2. In Section 3 we lay out
the general expressions for the heavy neutrino contributions to low energy
$LV$ decays and study two classes of $\Delta L=2$ processes,
* (a)
tau decays $\tau^{-}\rightarrow\ell^{+}M_{1}^{-}M_{2}^{-}$,
* (b)
rare meson decays $K^{+},\ D^{+},\ D^{+}_{s},\ B^{+}\rightarrow\ell_{1}^{+}\
\ell_{2}^{+}\ M_{2}^{-}$.
We calculate the enhanced transition rates and branching fractions and compare
them to the bounds set by direct experimental searches. A non-observation of
such $\Delta L=2$ processes places stringent constraints on the mass and
mixing of Majorana neutrinos which are also presented in this section. The
resonant production of Majorana neutrinos at hadron colliders, namely the
Tevatron and LHC are studied and updated in Sec. 4. We draw our conclusions in
Sec. 5. We discuss in detail the formalism, the decay modes and the total
decay width of heavy Majorana neutrinos and the transition rates of $LV$
processes in the Appendices.
## 2 Majorana neutrinos in extension of the standard model
To set up our notation and convention, we first discuss the formalism for the
simplest extension of the SM which includes right handed singlets. Also in
this section, we present the current constraints on the mass and mixing of a
heavy neutrino from various direct detection experiments, accelerator searches
and electroweak precision constraints.
### 2.1 Formalism for Heavy Neutrino Mixing
The leptonic content in our simplest extension of the SM includes three
generations of left-handed SM $SU(2)_{L}$ doublets and $n$ right-handed SM
singlets
$L_{aL}=\left(\begin{array}[]{c}\nu_{a}\\\ l_{a}\end{array}\right)_{L},\quad
N_{bR},$ (7)
where $a=1,2,3$ and $b=1,2,\cdots,n$. The gauge-invariant Yukawa interactions
lead to Dirac masses for the charged leptons and neutrinos after the Higgs
field develops a vacuum expectation value $v$. It is also possible for the
singlet neutrinos to have a heavy Majorana mass term. The full neutrino mass
terms as well as the diagonalized eigenvalues can be expressed as
$\displaystyle-{\cal L}_{m}^{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\ \sum_{a=1}^{3}\sum_{b=1}^{n}\
(\overline{\nu_{aL}}\ m^{\nu}_{ab}\ N_{bR}+\overline{N^{c}_{bL}}\
m^{\nu*}_{ba}\ \nu^{c}_{aR})+\sum_{b,b^{\prime}=1}^{n}\ \overline{N^{c}_{bL}}\
B_{bb^{\prime}}\ N_{b^{\prime}R}\right)+\mathrm{h.c.}$ (8) $\displaystyle=$
$\displaystyle{1\over 2}\left(\sum_{m=1}^{3}m_{\nu_{m}}\ \overline{\nu_{mL}}\
\nu^{c}_{mR}+\sum_{m^{\prime}=4}^{3+n}M_{N_{m^{\prime}}}\
\overline{N^{c}_{m^{\prime}L}}\ N_{m^{\prime}R}\right)+\mathrm{h.c.}$
with the mixing relations between the gauge and mass eigenstates
$\displaystyle\nu_{aL}=\sum_{m=1}^{3}U_{am}\nu_{mL}+\sum_{m^{\prime}=4}^{3+n}V_{am^{\prime}}N^{c}_{m^{\prime}L},$
(9) $\displaystyle UU^{\dagger}+VV^{\dagger}=I.$ (10)
In terms of the mass eigenstates, the gauge interaction Lagrangian can be
written as
$\displaystyle-{\cal L}$ $\displaystyle=$
$\displaystyle\frac{g}{\sqrt{2}}W^{+}_{\mu}\left(\sum_{\ell=e}^{\tau}\sum_{m=1}^{3}U^{*}_{\ell
m}\
\overline{\nu_{m}}\gamma^{\mu}P_{L}\ell+\sum_{\ell=e}^{\tau}\sum_{m^{\prime}=4}^{3+n}V^{*}_{\ell
m^{\prime}}\
\overline{N^{c}_{m^{\prime}}}\gamma^{\mu}P_{L}\ell\right)+\mathrm{h.c.}$ (11)
$\displaystyle+$
$\displaystyle\frac{g}{2\cos\theta_{W}}Z_{\mu}\left(\sum_{\ell=e}^{\tau}\sum_{m=1}^{3}U^{*}_{\ell
m}\ \overline{\nu_{m}}\gamma^{\mu}P_{L}\
\nu_{\ell}+\sum_{\ell=e}^{\tau}\sum_{m^{\prime}=4}^{3+n}V^{*}_{\ell
m^{\prime}}\ \overline{N^{c}_{m^{\prime}}}\gamma^{\mu}P_{L}\
\nu_{\ell}\right)+\mathrm{h.c.}$
Further details about the mixing formalism are given in Appendix A.
A few important remarks are in order before the detailed considerations. First
of all, parameterically, the light neutrino masses $m_{\nu,\ diag}$ are of the
order of magnitude $(m^{\nu}_{D})^{2}/B$, while the heavy neutrino masses are
$M_{N,\ diag}\simeq B$. Secondly, the mixing parameters would typically scale
as $U^{\dagger}U\approx I$ and $V^{\dagger}V\approx m_{\nu}/M_{N}$. Thirdly,
the Majorana mass term for the flavor states $\nu_{aL}$, absent in Eq. (8) and
corresponding to the null entry $0_{3\times 3}$ in Eq. (108), may receive non-
zero contributions as Majorana masses for the light active neutrinos, for
instance from higher dimensional $\Delta L=2$ operators or in theories with a
triplet Higgs field. The general formalism presented here remains the same. In
this paper, we will take a phenomenological approach toward these parameters.
We will simply take the masses and mixing elements of the heavy neutrino as
free parameters, only subject to some constraints from experimental
observations. The assumption that the masses and mixing elements are not
rigorously related by the see-saw relations is feasible from a model-building
point of view, since some fine-tuning or some ansatz of the neutrino mass
matrix can always alter the general relations. Several scenarios where it is
possible to have rather low mass of the heavy neutrino were mentioned in the
previous section. Here and henceforth, we consider the case when only one
heavy Majorana neutrino is kinematically accessible and denote it by $N_{4}$,
with the corresponding mass $m_{4}$ and mixing with charged lepton flavors
$V_{\ell 4}$. If we stick with this simple parameterization, the SM Higgs
boson will couple to the heavy neutrinos as well. We present the couplings in
Appendix A. When appropriate, we will include this effect. As noted above,
some fine-tuning [60] would be needed to avoid excessive contributions to the
light neutrino mass.
### 2.2 Current Constraints on $N_{4}$ Masses And Mixing
In laboratory searches, no positive evidence of sterile neutrinos has been
found so far,111Indications of the existence of a neutrino with 17 keV mass
were subsequently shown to be non valid. For a review, see Ref. [61]. Studying
interactions of neutrinos from $\pi$ and $\mu$ decays, an anomaly in time
distribution was found [62]. It could be interpreted as the existence of a
neutrino emitted in pion decays with mass of 33.9 MeV. Searches for this
neutral fermion have not given any positive signature and have allowed to
constrain the mixing to be $|V_{\mu 4}|^{2}<9.2\times 10^{-8}$ at 95% C.L.
[63]. in the mass range of interest, 100 eV–100 GeV.222For sterile neutrinos
with smaller masses a rather complete analysis of the bounds can be found in
Ref. [64]. See also the implications of the recent results from the MiniBooNE
collaboration [65, 66].
A very powerful probe of the mixing of heavy neutrinos with both $\nu_{e}$ and
$\nu_{\mu}$ are peak searches in leptonic decays of pions and kaons [67]. If a
heavy neutrino is produced in such decays, the lepton spectrum would show a
monochromatic line at
$E_{\ell}=\frac{m_{M}^{2}+m_{\ell}^{2}-m_{4}^{2}}{2m_{M}},$ (12)
where $E_{\ell}$ and $m_{\ell}$ are respectively the lepton energy and mass,
$m_{M}$ is the meson mass. The mixing angle controls the branching ratio of
this process as:
$\frac{\Gamma\big{(}M^{+}\rightarrow\ell^{+}N_{4}\big{)}}{\Gamma\big{(}M^{+}\rightarrow\ell^{+}\nu_{\ell}\big{)}}=\frac{|V_{\ell
4}|^{2}}{\sum^{3}_{m=1}|U_{\ell m}|^{2}}\rho\approx|V_{\ell 4}|^{2}\rho~{},$
(13)
where $\rho$ is a kinematical factor [67]:
$\rho=\frac{\sqrt{1+\mu^{2}_{\ell}+\mu^{2}_{4}-2\big{(}\mu_{\ell}+\mu_{4}+\mu_{\ell}\mu_{4}\big{)}}\Big{(}\mu_{\ell}+\mu_{4}-\big{(}\mu_{\ell}-\mu_{4}\big{)}^{2}\Big{)}}{\mu_{\ell}\big{(}1-\mu_{\ell}\big{)}^{2}},$
(14)
with $\mu_{i}=m_{i}^{2}/m_{M}^{2}$. For large $m_{4}$, the helicity
suppression of the $\pi,K\rightarrow\ell\nu_{\ell}$ decays weakens and there
is an enhancement for $M^{+}\rightarrow\ell^{+}N_{4}$ by a relative factor
$m_{4}^{2}/m_{\ell}^{2}$, reaching up to $10^{4}-10^{5}$ compared to that of
$\pi\rightarrow e\nu_{e}$ and $K\rightarrow e\nu_{e}$ in the SM, respectively.
These bounds are very robust because they rely only on the assumption that a
heavy neutrino exists and mixes with $\nu_{e}$ and/or $\nu_{\mu}$.
Another strategy to constrain heavy neutrinos mixed with $\nu_{e}$,
$\nu_{\mu}$ and $\nu_{\tau}$, is via searches of the products of their decays.
If kinematically allowed, $N_{4}$ would be produced in every process in which
active neutrinos are emitted with a branching fraction proportional to the
mixing parameter $|V_{\ell 4}|^{2}$. They would subsequently decay via Charged
Current (CC) and Neutral Current (NC) interactions into neutrinos and other
“visible” particles, such as electrons, muons and pions. Searches for these
“visible” decay-products were performed and were used to constrain the mixing
parameters. In beam dump experiments, $N_{4}$ would be produced in meson
decays, with the detector located far away from the production site. The
suppression of the flux of $N_{4}$ needs to be taken into account if the decay
length is very short and, therefore, typically both an upper and a lower bound
on the mixing angle can be set. Otherwise, the production can happen in the
detector itself, as for the limits obtained from a reanalysis of LEP data,
using the possible decays of the $Z^{0}$ [26] into heavy neutrinos. In this
case, large values of the mixing angle can be excluded. These bounds are less
robust than the ones previously discussed. In fact, if the heavy neutrinos
have other dominant decay channels into invisible particles, these bounds
would be weakened, if not completely evaded. For example, a coupling of the
type $gN\nu\phi$ (see Ref. [68, 53]), with $\phi$ a scalar, can induce very
fast decays, which might dominate over the ones induced by CC and NC
interactions. In this case, if the decay length is very short due to these
strong interactions, the flux of $N_{4}$ might be suppressed at the far
detector and the bound would not apply. If the production happens in the
detector itself, the bounds would need to be recomputed considering the
branching fraction into “visible” channels. Notice that we do not report here
the bounds from Ref. [69] as they do not apply to the heavy neutrinos under
consideration. In these analyses it was assumed that heavy neutrinos were
produced via $Z^{0}\rightarrow\mbox{$N_{4}$}\bar{\mbox{$N_{4}$}}$ with the
same strength as an active neutrino. In our scheme, this would correspond to
having mixing angle equal to 1. Then, the search for $N_{4}$ decays in the
detector was used to constrain the heavy neutrino parameters. These data
should be reanalyzed considering that the production of $N_{4}$ is suppressed
by $|V_{\ell 4}|^{2}$. Comparing the expected number of events with the
backgrounds we estimate that typically bounds of order $|V_{\ell 4}|^{2}<{\rm
few}\ \times 10^{-3}$–$10^{-2}$ could be deduced. However, a detailed analysis
should be performed and we do not report these limits in our figures.
For masses above the production threshold, additional constraints can be
obtained from lepton universality as the decay rates for muons, pions, taus as
well as the invisible decay width for the $Z^{0}$ boson are modified with
respect to the SM predictions [70, 71, 72]. Flavour changing neutral current
processes such as $\mu\rightarrow e\gamma$, $\mu\rightarrow ee^{+}e^{-}$ and
$\mu$–$e$ conversion in nuclei are affected by the existence of heavy sterile
neutrinos and strong limits can be obtained on the mixing with active
neutrinos [73, 74, 75]. These bounds are reported in Section 2.2.4.
Finally, in Section 2.2.5 we discuss the very strong constraints on
$|V_{e4}|^{2}$ which can be obtained from the non-observation of neutrinoless
double beta decay. It should be noticed that in the presence of more than one
sterile neutrino, possible cancellations between the contributions to the
decay rate can be achieved and the bounds would be consequently much weaker.
Next, we review the laboratory constraints on the mixing between heavy and
active neutrinos, depending on flavour and the mass of sterile neutrinos.
#### 2.2.1 Mixing with $\nu_{e}$
The mixing parameter $V_{e4}$ can be tested in searches of kinks in the
$\beta$-decay spectrum, of peaks in the spectrum of electrons in meson decays
and, finally, of $N_{4}$ decays in reactor and accelerator neutrino
experiments.
For masses $30\ {\rm eV}\simeq m_{4}\simeq 1$ MeV, the most sensitive probe is
the search for kinks in the $\beta$-decay spectra [67]. In the presence of
heavy neutrinos mixed with $\nu_{e}$, the Kurie plot would be given by the
contributions of the decays into light neutrinos as well as into heavy ones.
This induces a kink in the Kurie plot at the end point electron energy $E_{e}$
$E_{e}=\frac{M_{i}^{2}+m_{e}^{2}-(M_{f}+m_{4})^{2}}{2M_{i}},$ (15)
where $M_{i,f}$ are the mass of the initial and final nuclei, respectively,
and $m_{e}$ is the electron mass. In Fig. 2 we report the most stringent
limits, obtained by using different nuclei [76, 77, 78, 79, 80]. In reactors
and in the Sun only low mass, $m_{4}<$ few MeV, heavy neutrinos can be
produced. The constraints obtained by looking for their decays into electron-
positron pairs [81, 82] are reported in Fig. 2 with solid (cyan) contour
labeled Bugey and short dashed (blue) contour labeled Borexino. The region
with long dash dotted (grey) contour, labelled $\pi\rightarrow e\nu$, is
excluded by peak searches [83].
Figure 2: Bounds on $|V_{e4}|^{2}$ versus $m_{4}$ in the mass range 10 eV–10
MeV. The excluded regions with contours labeled 187Re [76], 3H [77] , 63Ni
[78] , 35S [79] , 20F and Fermi2 [80] refer to the bounds from kink searches.
All the limits are given at 95% C.L. except for the ones from Ref. [80] which
are at 90% C.L.. The areas delimited by short dashed (blue) contour labeled
Borexino and solid (cyan) contour labeled Bugey are excluded at 90% C.L. by
searches of $N_{4}$ decays from the Borexino Counting Test facility [81] and
Ref. [82] respectively. The region with long-dash-dotted (grey) contour,
labelled $\pi\rightarrow e\nu$, is excluded by peak searches [83]. The dotted
(maroon) line labeled $0\nu\beta\beta$ indicates the bound from searches of
neutrinoless double beta-decay [84].
For heavier masses peak searches give the most stringent bounds, shown in Fig.
3. Notice that, due to the weakened helicity suppression of the $\pi$ decay,
the sensitivity on $V_{e4}$ increases with $m_{4}$ till phase space becomes
relevant at $m_{4}>80$ MeV, for $\pi\rightarrow e\nu_{e}$. The excluded
region, at 90% C.L., from Ref. [83], is indicated with the solid (black) line
labeled $\pi\rightarrow e\nu$. For heavier masses, stringent bounds are
obtained by looking at the electron spectrum in $K$ decays [85] and are
indicated by the double dash dotted (purple) line labeled $K\rightarrow e\nu$
in Fig. 3. Assuming that only CC and NC interactions are at play, stringent
bounds have been obtained on $|V_{e4}|^{2}$ and are reported in Fig. 3 by the
rest of the contours (except dotted (maroon) line labeled $0\nu\beta\beta$).
In particular, the limits at 90% C.L. from Refs. [86, 87, 88], assume the
production of $N_{4}$ in meson decays and look for visible channels in a
detector located some distance from the source. The limits at 95% C.L. in
Refs. [89, 90] analyse the data from DELPHI and L3 detectors, looking for
$N_{4}$ from $Z^{0}$-decays. In Fig. 3 we also report the excluded region from
neutrinoless double beta-decay experiments [91, 84], bounded by dotted
(maroon) line, valid if the heavy neutrinos are Majorana particles (see
further).
Figure 3: Bounds on $|V_{e4}|^{2}$ versus $m_{4}$ in the mass range 10
MeV–100 GeV. The areas with solid (black) contour labeled $\pi\rightarrow
e\nu$ and double dash dotted (purple) contour labeled $K\rightarrow e\nu$ are
excluded by peak searches [83, 85]. Limits at 90% C.L. from beam-dump
experiments are taken from Ref. [86] (PS191), Ref. [87] (NA3) and Ref. [88]
(CHARM). The limits from contours labeled DELPHI and L3 are at 95% C.L. and
are taken from Refs. [89] and [90] respectively. The excluded region with
dotted (maroon) contour is derived from a reanalysis of neutrinoless double
beta decay experimental data [84].
#### 2.2.2 Mixing with $\nu_{\mu}$
The bounds on $|V_{\mu 4}|^{2}$ come from searches of peaks in the spectrum of
muons in pion and kaon decays and of the decays of $N_{4}$ produced in
neutrino beams and $e^{+}e^{-}$ collisions.
As already discussed in the case of mixing with $\nu_{e}$, peak searches
provide very robust and stringent bounds, by looking at pion decays for masses
up to 34 MeV, and at kaon decays for higher masses. A detailed review is given
in Figs. 1 and 2 in Ref. [92] and for masses larger than 100 MeV the limits
are reported in Fig. 4.
Figure 4: Limits on $|V_{\mu 4}|^{2}$ versus $m_{4}$ in the mass range 100
MeV–100 GeV come from peak searches and from $N_{4}$ decays. The area with
solid (black) contour labeled $K\rightarrow\mu\nu$ [92] is excluded by peak
searches. The bounds indicated by contours labeled by PS191 [86], NA3 [87],
BEBC [93], FMMF [94], NuTeV [95] and CHARMII [96] are at 90% C.L., while
DELPHI [89] and L3 [90] are at 95% C.L. and are deduced from searches of
visible products in $N_{4}$ decays. For the beam dump experiments, NA3, PS191,
BEBC, FMMF and NuTeV we give an estimate of the upper limit for the excluded
values of the mixing angle.
The other limits on $|V_{\mu 4}|^{2}$ are found in decay searches and are also
shown in Fig. 4. They come from beam dump experiments [87, 86, 93, 94, 95] and
from direct $N_{4}$ production in the detectors DELPHI [89], L3 [90] and CHARM
[96].
#### 2.2.3 Mixing with $\nu_{\tau}$
Heavy neutrinos mixed with $\tau$ neutrinos can be produced either via CC
interactions if a $\tau$ is produced or in NC interactions. The only limits
come from searches of $N_{4}$ decays and are reported in Fig. 5. The bounds at
90% C.L. from CHARM [97] and NOMAD [98] assume production via $D$ and $\tau$
decays. The DELPHI bound at 95% C.L. [89] assumes $N_{4}$ production in
$Z^{0}$ decays and with respect to the bound on $|V_{e4}|^{2}$ and $|V_{\mu
4}|^{2}$ there is $\tau$-production kinematical suppression for low masses
which weakens the constraint for masses in the range $m_{4}\sim 2$–3 GeV.
Figure 5: Bounds on $|V_{\tau 4}|^{2}$ versus $m_{4}$ from searches of decays
of heavy neutrinos, given in Ref. [97] (CHARM) and in Ref. [98] (NOMAD) at 90%
C.L., and in Ref. [89] (DELPHI) at 95% C.L.
#### 2.2.4 Electroweak Precision Tests
The presence of heavy neutral fermions affects processes below their mass
threshold due to their mixing with standard neutrinos [70] and significant
bounds can be set by precision electroweak data. The effective $\mu$-decay
constant $G_{\mu}$, measured in muon decays, is modified with respect to the
SM value and can be related to the fundamental coupling $G_{F}$ as:
$G_{\mu}=G_{F}\sqrt{(1-|V_{e4}|^{2})(1-|V_{\mu 4}|^{2})}~{}.$ (16)
The $\mu-e$ universality test, done by comparing the decay rate of pions into
$e\bar{\nu}$ and $\mu\bar{\nu}$, can be used to constrain the ratio
$\frac{1-|V_{e4}|^{2}}{1-|V_{\mu 4}|^{2}},$ (17)
for $m_{4}>m_{\pi}$ [70, 71]. The analysis of experimental data leads to
$\frac{1-|V_{\mu 4}|^{2}}{1-|V_{e4}|^{2}}=1.0012\pm 0.0016$ [71], which
implies $|V_{e4}|^{2}<0.004$ at $2\sigma$ for the least conservative case of
$|V_{\mu 4}|^{2}=0$. For $m_{4}>m_{\tau}$, the $\mu-\tau$ universality sets
limits on:
$\frac{1-|V_{\tau 4}|^{2}}{1-|V_{\mu 4}|^{2}},$ (18)
and can be tested by looking at the $\tau$ leptonic and hadronic decays which
give $|V_{\tau 4}|^{2}-|V_{\mu 4}|^{2}=0.0057\pm 0.0065$ [71] and $|V_{\tau
4}|^{2}-|V_{e4}|^{2}=0.0054\pm 0.0064$ [71]. The most constraining bound on
$|V_{\tau 4}|^{2}$ is obtained for $|V_{e4}|^{2},|V_{\mu 4}|^{2}=0$ and reads
$|V_{\tau 4}|^{2}<0.018$ at $2\sigma$. The unitarity constraint on the first
row of the CKM matrix [99] reads
$\sum_{i=1,2,3}|V^{\rm CKM}_{ui}|^{2}=\frac{1}{1-|V_{\mu 4}|^{2}}=0.9992\pm
0.0011,$ (19)
and translates into a very strong bound on $|V_{\mu 4}|^{2}$, $|V_{\mu
4}|^{2}<0.0003\ (0.0014)$, at $1\ (2)\sigma$, which holds for sterile
neutrinos heavier than the $\Lambda$ baryon.
In the presence of heavy singlet neutrinos heavier than half the $Z^{0}$ mass,
the invisible decay rate of $Z^{0}$ would be reduced with respect to the SM
one, $\Gamma_{Z\rightarrow\mathrm{inv}}^{\rm SM}$, as:
$\frac{\Gamma_{Z\rightarrow\mathrm{inv}}}{\Gamma_{Z\rightarrow\mathrm{inv}}^{\rm
SM}}\simeq(1-\frac{1}{6}|V_{e4}|^{2}-\frac{1}{6}|V_{\mu
4}|^{2}-\frac{2}{3}|V_{\tau 4}|^{2}).$ (20)
By a standard model fit to LEP data, the effective number of neutrinos is now
determined to be $N_{\nu}=2.984\pm 0.008$ [99] and provides a bound on
$|V_{\ell 4}|^{2}$ similar to but somewhat weaker than the ones obtained by
lepton-universality.
A combined analysis of an old set of unitarity bounds [71], which does not
include the one from the CKM matrix determination, leads to the following
limits at 90% C.L. $|V_{e4}|^{2}<0.012,|V_{\mu 4}|^{2}<0.0096\ \mbox{and}\
|V_{\tau 4}|^{2}<0.016.$ If the CKM matrix constraint is included and partial
cancellations between the contributions of different flavors are taken into
account, a previous combined study [70] then gives the more robust limits at
$90\%$ C.L., $|V_{e4}|^{2}<0.0066,|V_{\mu 4}|^{2}<0.0060$ and $|V_{\tau
4}|^{2}<0.018$. A very recent analysis [72] has updated these results using
the latest electroweak precision data, except for the CKM observables. They
find at 90% C.L.
$|V_{e4}|^{2}<0.003,\qquad|V_{\mu 4}|^{2}<0.003,\qquad|V_{\tau
4}|^{2}<0.006~{}.$ (21)
If the constraints from CKM observables are included, we expect the bounds to
become somewhat stronger, given by $|V_{e4}|^{2}<0.002,|V_{\mu 4}|^{2}<4\times
10^{-5},|V_{\tau 4}|^{2}<0.006$ [100]. In the following, we take the bound
$|V_{\mu 4}|^{2}<0.0060$ as a conservative reference limit on the mixing for
comparison with the results of our study.
Indirect limits on the parameters characterizing heavy sterile neutrinos can
be obtained from searches for flavour changing neutral current processes such
as $\mu\rightarrow e\gamma$, $\mu\rightarrow ee^{+}e^{-}$ and $\mu-e$
conversion in nuclei [75]. The branching fraction for $\mu\rightarrow e\gamma$
induced by the mixing with heavy singlet neutrinos is given by [73, 74, 75]:
${\rm Br}(\mu\rightarrow
e\gamma)=\frac{3\alpha}{8\pi}\left|\sum_{m^{\prime}}V_{em^{\prime}}V_{\mu
m^{\prime}}^{\ast}\
g\left(\frac{m_{N_{m^{\prime}}}^{2}}{m^{2}_{W}}\right)\right|^{2}~{},$ (22)
where $m^{\prime}$ indicates the heavy sterile neutrinos with mass
$m_{N_{m^{\prime}}}$, and $m_{W}$ is the mass of the $W$ boson. The function
$g(x)$ is given by
$g(x)=\frac{x(1-6x+3x^{2}+2x^{3}-6x^{2}\ln(x))}{2(1-x)^{4}},$ (23)
where $g(x)$ goes from 0 to 1 as $x$ varies from 0 to infinity. At present,
the branching fraction is constrained to be ${\rm Br}(\mu\rightarrow
e\gamma)<1.2\times 10^{-11}$ [101] at 90% C.L. implying that, for one extra
sterile neutrino, $|V_{e4}V_{\mu 4}^{\ast}|<0.015\ (3.5\times 10^{-4})\
[1.2\times 10^{-4}]$ for $m_{4}=10\ {\rm GeV}\ (100\ {\rm GeV})\ [1000\ {\rm
GeV}]$. Similar constraints are imposed by searches for the processes $\mu-e$
conversion in nuclei and $\mu\rightarrow ee^{+}e^{-}$ [75]. The current
strongest bound comes from the search for $\mu-e$ conversion in $\mathrm{Ti}$
for which the branching ratio with respect to the total nuclear muon capture
rate is constrained to be ${\rm Br}(\mu\,{\rm Ti}\rightarrow e\,{\rm
Ti})<4.3\times 10^{-12}$ at 90% C.L. [102]. For one sterile neutrino, this
translates into a bound on the following quartic combination of mixing angles
$\left|V_{e4}V_{\mu 4}^{\ast}\sum_{\ell}|V_{\ell 4}|^{2}\right|<1.3\times
10^{-3}\left(100\,{\rm GeV}/m_{4}\right)^{2}$, which is weaker than the bounds
from $\mu\rightarrow e\gamma$ searches but becomes important at very high
values of the masses, $m_{4}\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}10\ {\rm TeV}$. In
the presence of more than one sterile neutrino, partial cancellations between
their contributions are possible, with a consequent weakening of the bounds.
Future more sensitive searches will further improve these limits.
The most stringent EW precision constraints are compiled in Table 1 and
include the bounds reported above from universality tests and lepton flavor
changing processes. These bounds are obtained barring cancellations between
mixing angles and therefore could be weakened if some parameters are of the
same order.
Table 1: Most stringent model-independent constraints on the mixing elements of the heavy neutrino from precision electro-weak measurements. The bounds on $|V_{\ell 4}|^{2}$, $\ell=e,\mu,\tau$ and on $|V_{e4}V_{\mu 4}|$ at 90% C.L.. See text for details. Mixing element | Range of $m_{4}$ | EW Measurement
---|---|---
$|V_{e4}|^{2}$ | $m_{4}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}{\cal O}(m_{\pi})$ | $<0.003~{}\cite[cite]{[\@@bibref{}{delAguila:2008pw}{}{}]}$
$|V_{\mu 4}|^{2}$ | $m_{4}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}{\cal O}(m_{\Lambda})$ | $<0.003~{}\cite[cite]{[\@@bibref{}{delAguila:2008pw}{}{}]}$
$|V_{\tau 4}|^{2}$ | $m_{4}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}{\cal O}(m_{\tau})$ | $<0.006~{}\cite[cite]{[\@@bibref{}{delAguila:2008pw}{}{}]}$
$|V_{e4}V_{\mu 4}|$ | 10 GeV (100 GeV) [1000 GeV] | $<0.015\ (3.5\times 10^{-4})\ [1.2\times 10^{-4}]\ $
#### 2.2.5 Neutrinoless Double Beta Decay ($0\nu\beta\beta$)
The most well studied among $\Delta L=2$ processes is neutrinoless double beta
decay ($0\nu\beta\beta$) and the constraints from it deserve special
attention. The constraints on $|V_{e4}|^{2}$ for a wide range of heavy
neutrino masses ($10\ \rm MeV\leq m_{4}\leq 100\ \rm GeV$) are shown in Figs.
2 and 3. For heavy neutrinos with mass, $m_{N_{m^{\prime}}}\gg 1~{}{\rm GeV}$,
the bound is [91, 84]
$\displaystyle\sum_{m^{\prime}}\frac{\left|V_{em^{\prime}}\right|^{2}}{m_{N_{m^{\prime}}}}<5\times
10^{-5}~{}{\rm TeV}^{-1}.$ (24)
The constraint above is very strong and makes it impossible to observe at
colliders the like-sign dilepton signature with electrons (see Sec. 4).
## 3 Lepton-Number Violating Decays
The key point for the search of lepton-number violating processes in this
paper is to consider the substantial enhancement via resonant neutrino
production. One thus needs to evaluate the decay widths of $N_{4}$ to various
channels. We consider the decay width of the heavy Majorana neutrino in two
regimes: when the mass is much smaller than that of the $W$ boson and when the
mass is larger than the mass of the $W$ boson. Based on this, we then compute
the $\Delta L=2$ decay branching fractions for $\tau$ lepton and $K,D,D_{s}$
and $B$ mesons.
### 3.1 Decay Modes of Heavy Majorana Neutrino
#### 3.1.1 Decay Modes of Heavy Majorana Neutrino with mass $m_{4}\ll m_{W}$
For the $LV$ low energy tau decays and rare meson decays the resonant
contribution is from a heavy Majorana neutrino with mass of order $\rm MeV$ to
$\rm GeV$. In this section we discuss the decay modes of a Majorana neutrino
which is lighter than the $W$ boson, so that $m_{4}\ll m_{W}$. The heavy
neutrino decays via charged and neutral current interactions to the modes
listed below. The partial decay widths of the heavy Majorana neutrino with the
leading terms in mixing and in the massless limit of the final state particles
are given below. The full detailed expressions for the same are given in
Appendix C.
$\displaystyle\Gamma^{\ell P}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\ell^{-}P^{+})=\frac{G^{2}_{F}}{16\pi}f^{2}_{P}\
|V_{q\bar{q}^{\prime}}|^{2}\ |V_{\ell 4}|^{2}\ m^{3}_{4},$ (25)
$\displaystyle\Gamma^{\nu_{\ell}P}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\nu_{\ell}P^{0})=\frac{G^{2}_{F}}{64\pi}f^{2}_{P}\
|V_{\ell 4}|^{2}\ m^{3}_{4},$ (26) $\displaystyle\Gamma^{\ell V}$
$\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\ell^{-}V^{+})=\frac{G^{2}_{F}}{16\pi}f^{2}_{V}\
|V_{q\bar{q}^{\prime}}|^{2}\ |V_{\ell 4}|^{2}\ m^{3}_{4},$ (27)
$\displaystyle\Gamma^{\nu_{\ell}V}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\nu_{\ell}V^{0})={\frac{G^{2}_{F}}{2\pi}}{\kappa^{2}_{V}}\
{f^{2}_{V}}\ {{|V_{\ell 4}|}^{2}}\ {m^{3}_{4}},$ (28)
$\displaystyle\Gamma^{\ell_{1}\ell_{2}\nu_{\ell_{2}}}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\ell^{-}_{1}\ell^{+}_{2}\nu_{\ell_{2}})=\frac{G^{2}_{F}}{192\pi^{3}}\
{|V_{\ell_{1}4}|}^{2}\ m^{5}_{4},$ (29)
$\displaystyle\Gamma^{\nu_{\ell_{1}}\ell_{2}\ell_{2}}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\nu_{\ell_{1}}\ell^{-}_{2}\ell^{+}_{2})=\frac{G^{2}_{F}}{96\pi^{3}}\
{|V_{\ell_{1}4}|}^{2}\ m^{5}_{4}\
[\alpha_{1}+\delta_{\ell_{1}\ell_{2}}\alpha_{2}],$ (30)
$\displaystyle\Gamma^{\nu_{\ell_{1}}\nu\nu}$ $\displaystyle\equiv$
$\displaystyle\sum_{\ell_{2}=e}^{\tau}\Gamma(N_{4}\rightarrow\nu_{\ell_{1}}\nu_{\ell_{2}}\overline{\nu_{\ell_{2}}})=\frac{G^{2}_{F}}{96\pi^{3}}\
|V_{\ell_{1}4}|^{2}\ m^{5}_{4},$ (31)
where $P^{+(0)}$ and $V^{+(0)}$ are charged (neutral) pseudoscalar and vector
mesons, $f_{M}$ are the meson decay constants and $V_{q\bar{q}^{\prime}}$ are
the CKM matrix elements.
All the decay modes listed above contribute to the total decay width of the
heavy Majorana neutrino which is given by:
$\displaystyle\Gamma_{N_{4}}$ $\displaystyle=$
$\displaystyle\sum_{\ell,P}{\Gamma^{\nu_{\ell}P}}+\sum_{\ell,V}{\Gamma^{\nu_{\ell}V}}+\sum_{\ell,P}{2\Gamma^{\ell
P}}+\sum_{\ell,V}{2\Gamma^{\ell V}}$ (32) $\displaystyle+$
$\displaystyle\sum_{\ell_{1},\ell_{2}(\ell_{1}\neq\ell_{2})}{2\Gamma^{\ell_{1}\ell_{2}\nu_{\ell_{2}}}}+\sum_{\ell_{1},\ell_{2}}{\Gamma^{\nu_{\ell_{1}}\ell_{2}\ell_{2}}}+\sum_{\ell_{1}}{\Gamma^{\nu_{\ell_{1}}\nu\nu}},$
where $\ell,\ell_{1},\ell_{2}=e,\mu,\tau$. For a Majorana neutrino, the
$\Delta L=0$ process $N_{4}\rightarrow\ell^{-}P^{+}$ as well as its charge
conjugate $\Delta L=2$ process $N_{4}\rightarrow\ell^{+}P^{-}$ are possible
and have the same width $\Gamma^{\ell P}$. Hence the factor of 2 associated
with the decay width of this mode in Eq. (32). Similarly, the $\Delta L=0$ and
its charge conjugate $\Delta L=2$ process are possible for the decay modes
$N_{4}\rightarrow\ell^{-}V^{+}$ and
$N_{4}\rightarrow\ell^{-}_{1}\ell^{+}_{2}\nu_{\ell_{2}}$ and hence have a
factor of 2 associated with their width in Eq. (32).
For the low energy $LV$ tau decays and rare meson decays we consider, the mass
of the heavy neutrino is in the range $140\ {\rm MeV}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}m_{4}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}5278\ \rm MeV$.
For this mass range we list all the possible decay channels for $N_{4}$ in
Table 6 in Appendix C. The mass and decay constants of pseudoscalar and vector
mesons used in the calculation of partial widths given in Eqs. (25)$-$(31) are
listed in Table 7 in Appendix E.
Next, we get a rough numerical estimate of the total width of the heavy
neutrino with the decay modes discussed in Eqs. (25)$-$(31). To do this, we
consider the massless limit of the decay products of the heavy neutrino,
include only leading terms in mixing of ${\cal O}({|V_{\ell 4}|^{2}})$ and
ignore small factors like $\pi$ and $|V^{CKM}|^{2}$ in calculating the partial
decay widths. We can only get a rough estimate of the width in this
approximation, but it is sufficient to see that it warrants the use of narrow
width approximation. The two body decays of the heavy neutrino have a general
form
$\Gamma^{2body}\sim\frac{G^{2}_{F}f^{2}_{M}m^{3}_{4}}{10\pi}{|V_{\ell
4}|^{2}}\sim\frac{G^{2}_{F}f^{2}_{M}m^{3}_{4}}{10}{|V_{\ell
4}|^{2}}\sim(10^{-13}\mbox{ }{|V_{\ell 4}|^{2}})\mbox{ }\rm GeV,$ (33)
where typical values of $m_{4}\sim 1\ \rm GeV$, $f_{M}\sim 0.1\ \rm GeV$ and
$G_{F}\sim 10^{-5}\ \rm GeV^{-2}$ have been used. The three body decays of the
heavy neutrino have a general form
$\Gamma^{3body}\sim\frac{G^{2}_{F}m^{5}_{4}}{100\pi^{3}}{|V_{\ell
4}|^{2}}\sim\frac{G^{2}_{F}m^{5}_{4}}{1000}{|V_{\ell
4}|^{2}}\sim(10^{-13}\mbox{ }{|V_{\ell 4}|^{2}})\mbox{ }\rm GeV,$ (34)
where typical values of $m_{4}\sim 1\ \rm GeV$ and $G_{F}\sim 10^{-5}\ \rm
GeV^{-2}$ have been used. The total width of the heavy neutrino is then given
by
$\displaystyle\Gamma_{N_{4}}$ $\displaystyle=$ $\displaystyle\mbox{(number of
decay modes)}\times(\Gamma^{2body}+\Gamma^{3body})$ (35) $\displaystyle\sim$
$\displaystyle 50\times(10^{-13}\mbox{ }\rm GeV+10^{-13}\mbox{ }\rm GeV)\mbox{
}{|V_{\ell 4}|^{2}}\sim(10^{-11}\mbox{ }{|V_{\ell 4}|^{2}})\mbox{ }\rm GeV.$
As shown above, the width of the heavy neutrino $\sim{\cal O}(10^{-11}\mbox{
}{|V_{\ell 4}|^{2}})\mbox{ }\rm GeV$ is much smaller than the mass of the
heavy neutrino $\sim{\cal O}(1\ \rm GeV)$ and we can use the narrow width
approximation to an excellent approximation.
Now we look at the lifetime of the heavy Majorana neutrino to determine the
decay length. The lifetime is given by
$\displaystyle\tau_{N_{4}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma_{N_{4}}}\sim\frac{1}{10^{-11}\ {|V_{\ell
4}|^{2}}\ \rm GeV}\ ,$ (36) $\displaystyle\sim$ $\displaystyle 10^{11}\
|V_{\ell 4}|^{-2}\ \rm GeV^{-1}\sim 6.58\times 10^{-14}\ |V_{\ell 4}|^{-2}\
s,$
which gives a typical decay length $c\tau_{N_{4}}\sim 1\times 10^{-5}\
|V_{\ell 4}|^{-2}\ \mathrm{m}$. Note that for a very small mixing, $|V_{\ell
4}|^{2}<{\cal O}(10^{-5}),$ the $N_{4}$ may escape from the detector if it is
not much heavier than a GeV. We will take this effect into account in the
following studies.
#### 3.1.2 Decay Modes of Heavy Majorana Neutrino with mass $m_{4}>m_{W}$
In this section we discuss the decay modes of the Majorana neutrino which is
heavier than the $W$ gauge boson, so that $m_{4}>m_{W}$. The decay modes of
the heavy Majorana neutrino are to a $W$ or a $Z$ gauge boson plus the
corresponding SM lepton. The partial decay widths for longitudinal and
transverse gauge bosons $W^{\pm},Z^{0}$ in static heavy neutrino frame are
$\displaystyle\Gamma^{\ell W_{L}}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\ell^{-}W^{+}_{L})=\Gamma(N_{4}\rightarrow\ell^{+}W^{-}_{L})=\frac{g^{2}}{64\pi
M^{2}_{W}}\left|V_{\ell 4}\right|^{2}\ m^{3}_{4}\ (1-\mu_{W})^{2},$ (37)
$\displaystyle\Gamma^{\ell W_{T}}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\ell^{-}W^{+}_{T})=\Gamma(N_{4}\rightarrow\ell^{+}W^{-}_{T})=\frac{g^{2}}{32\pi}\left|V_{\ell
4}\right|^{2}\ m_{4}\ (1-\mu_{W})^{2},$ (38)
$\displaystyle\Gamma^{\nu_{\ell}Z_{L}}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\nu_{\ell}Z_{L})=\frac{g^{2}}{64\pi
M^{2}_{W}}{{|V_{\ell 4}|}^{2}}\ m^{3}_{4}\ (1-\mu_{Z})^{2},$ (39)
$\displaystyle\Gamma^{\nu_{\ell}Z_{T}}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\nu_{\ell}Z_{T})=\frac{g^{2}}{32\pi\cos^{2}_{W}}{{|V_{\ell
4}|}^{2}}\ m_{4}\ (1-\mu_{Z})^{2},$ (40)
where $\mu_{i}$ are the masses of the gauge bosons scaled by the mass of the
heavy neutrino and are given by $\mu_{i}=m^{2}_{i}/m^{2}_{4}$. To obtain the
total decay width for $N_{4}$, we sum over the charged leptons $\ell$ and as
discussed earlier include the $\Delta L=0$ process
$N_{4}\rightarrow\ell^{-}W_{L,T}^{+}$ as well as the charge conjugate $\Delta
L=2$ process $N_{4}\rightarrow\ell^{+}W_{L,T}^{-}$. Hence the factor of $2$
associated with the decay width of these modes in the expression for the total
width below.
$\Gamma_{N_{4}}=\sum_{\ell}{\Bigl{(}2\Gamma^{\ell W_{L}}}+{2\Gamma^{\ell
W_{T}}}+{\Gamma^{\nu_{\ell}Z_{L}}}+{\Gamma^{\nu_{\ell}Z_{T}}}\Bigr{)}.$
In Eqs. (39)$-$(3.1.2), we have used the relation (see Appendix for details)
$\sum_{m=1}^{3}\left|U^{\nu N}_{m4}\right|^{2}\
=\Bigl{[}\sum_{\ell=e}^{\tau}{{|V_{\ell
4}|}^{2}}\Bigl{(}1-\sum_{\ell_{1}=e}^{\tau}{|V_{\ell_{1}4}|}^{2}\Bigr{)}\Bigr{]},\
\ {\rm since}\ \ UU^{\dagger}+VV^{\dagger}=I.$ (41)
Ignoring terms of order $\left|V_{\ell 4}\right|^{4}$ we have
$\displaystyle\sum_{m}\left|U^{\nu
N}_{m4}\right|^{2}\approx\sum_{\ell}\left|V_{\ell 4}\right|^{2}.$ (42)
In this approximation, the total width of a heavy Majorana neutrino can be
written as
$\Gamma_{N_{4}}\left\\{\begin{array}[]{ll}\displaystyle\approx\sum_{\ell}\left|V_{\ell
4}\right|^{2}\frac{3G_{F}m^{3}_{4}}{8\pi\sqrt{2}}&{\rm for}\ \
m_{4}>m_{W},\\\\[17.07164pt] \displaystyle\propto\sum_{\ell}\left|V_{\ell
4}\right|^{2}G_{F}^{2}m^{3}_{4}(f^{2}_{M}+m^{2}_{4})&{\rm for}\ \ m_{4}\ll
m_{W},\end{array}\right.$ (43)
where the expression when $m_{4}\ll m_{W}$ is obtained from Eq. (32) and
$f_{M}$ are the meson decay constants. We note that the approximate form of
the total width as given in Eq. (43) is only for intuitive purposes to infer
the general behaviour of the total width as a function of mass. The precise
expressions for the total width of the heavy Majorana neutrino as given in
Eqs. (32), (3.1.2) and (141) have been used in the numerical analysis.
Figure 6: (a) Top: decay width and (b) bottom: decay length (normalized by
$\sum_{\ell}\left|V_{\ell 4}\right|^{2}$) versus mass of heavy Majorana
neutrino for real and virtual weak bosons with the inclusion of Higgs decay
channel for $m_{H}=120$ GeV .
Figure 7: (a) Left: branching fractions for decay of heavy Majorana neutrino
into $W^{*}$ and $Z^{*}$ bosons with varying heavy neutrino mass; (b) right:
branching fractions for decay of heavy Majorana neutrino into longitudinal and
transverse gauge bosons in static heavy neutrino frame with the inclusion of
Higgs decay channel for $m_{H}=120$ GeV .
It should be noted that in the SM, if $N_{4}$ is heavier than the Higgs boson,
then the decay to a Higgs will be present and the partial width is given by
$\displaystyle\Gamma^{\nu
H}\equiv\Gamma(N_{4}\rightarrow\nu_{\ell}H)={g^{2}\over 64\pi m_{W}^{2}}\
\left|V_{\ell 4}\right|^{2}\ m^{3}_{4}\ (1-\mu_{H})^{2}.$ (44)
In Fig. 6 we plot the decay width of the heavy Majorana neutrino versus its
mass normalized by the common factor $\sum_{\ell}\left|V_{\ell 4}\right|^{2}$.
We can see in Fig. 6(a) that for a heavy neutrino with mass $m_{4}>m_{W}$, the
decay width increases as $G_{F}m^{3}_{4}$ as given in Eq. (43). Given the
rather small mixing parameter, the width remains narrow even for
$m_{4}\sim{\cal O}$(1 TeV). For a lighter neutrino with $m_{4}\ll m_{W}$, the
width can be very small. The proper decay length is presented in Fig. 6(b). We
see from this that for $m_{4}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}20$ GeV and
$|V_{\mu 4}|^{2}\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}10^{-4}$ from Fig. 4, we have $c\tau\sim 1\
\mu\mathrm{m}$.
In Fig. 7(a) we plot the branching fractions of the heavy Majorana neutrino
decay to $W\ell$ and $Z\nu$ versus varying heavy neutrino mass $m_{4}$. In
Fig. 7(b) we plot the branching fractions for the decays into longitudinal and
transverse gauge bosons in static heavy neutrino frame. When the neutrino mass
is large, it mainly decays to longitudinal gauge bosons and
$\mathrm{Br}(N_{4}\rightarrow W^{+}\ell^{-})\simeq\mathrm{Br}(N_{4}\rightarrow
Z\nu)=\mathrm{Br}(N_{4}\rightarrow H\nu)=25\%.$ In terms of the search at
hadron colliders, we prefer to adopt the $W\ell$ mode since we wish to
reconstruct the full event including the lepton number.
### 3.2 Lepton-Number Violating Tau Decays
In this section we examine tau decays into an anti-lepton and two mesons
$\tau^{-}(p_{1})\rightarrow\ell^{+}(p_{2})\ M_{1}^{-}(q_{1})\
M_{2}^{-}(q_{2})$ (45)
which is a process with $\Delta L=-2$. The decay amplitude for the above
process is given by
${i\cal M}=2G_{F}^{2}V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}f_{M_{1}}f_{M_{2}}{V_{\tau
4}^{*}}{V^{*}_{\ell 4}}\
m_{4}\Biggl{[}\frac{\overline{v_{\tau}}\not{\hbox{\kern-4.0pt$q$}}_{1}\not{\hbox{\kern-4.0pt$q$}}_{2}P_{R}v_{\ell}}{(p_{1}-q_{1})^{2}-m_{4}^{2}+i\Gamma_{N_{4}}m_{4}}\Biggr{]}+(q_{1}\leftrightarrow
q_{2}),$ (46)
where $V^{CKM}_{M_{i}}$ and $f_{M_{i}}$ are the quark flavor mixing element
and the decay constant for the meson $M_{i}$ respectively. From this decay
amplitude, we can calculate the transition rate $\Gamma^{\tau}_{LV}$ and the
branching fraction normalised by the tau decay width. In Appendix D, we give
the calculations and the full expressions for the decay branching fraction of
the process (45) in terms of the mass of heavy neutrino, $m_{4}$, and the
mixing $|V_{\tau 4}V_{\ell 4}|^{2}$. To understand the physical picture, we
can express the branching fraction in an intuitive form, in the massless limit
of the final state particles, as
$\displaystyle\mathrm{Br}$ $\displaystyle=$
$\displaystyle\frac{\Gamma^{\tau}_{\\!\\!\\!\mbox{}_{LV}}}{\Gamma_{\tau}}=\Gamma^{\tau}_{\\!\\!\\!\mbox{}_{LV}}\Bigl{(}\frac{192\pi^{3}}{G^{2}_{F}m^{5}_{\tau}}\Bigr{)},$
(47) $\displaystyle\sim$
$\displaystyle\frac{3}{2}\pi(1-\frac{1}{2}\delta_{M_{1}M_{2}})G^{2}_{F}f^{2}_{M_{1}}f^{2}_{M_{2}}|V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}|^{2}\mbox{
}|V_{\tau 4}V_{\ell
4}|^{2}\Bigl{(}1-\frac{m^{2}_{4}}{m^{2}_{\tau}}\Bigr{)}\Bigl{(}\frac{m_{4}}{\Gamma_{N_{4}}}\Bigr{)},$
$\displaystyle\sim$ $\displaystyle 10^{-3}\mbox{
}|V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}|^{2}\mbox{ }|V_{\tau 4}V_{\ell 4}|,$
where we have used typical values of $m_{4}\sim 1\ \rm GeV$, $f_{M_{i}}\sim
0.1\ \rm GeV$, $G_{F}\sim 1\times 10^{-5}\ \rm GeV^{-2}$ and
$\Gamma_{N_{4}}\sim 10^{-11}\ |V_{\ell 4}|^{2}\ \rm GeV$. From the simple
expression given above one can easily make a rough estimate of the required
sensitivity and hence the feasibility of observation in terms of the mixing
parameters for a given model.
A direct search for $LV$ tau decays has been made at the BaBar detector and
the limits on the branching fractions were reported in Ref. [103]. The
experimental limits for various decay modes are typically of the order of
$10^{-7}$, as given in Table 2. From the non-observation of the $LV$ tau decay
modes one can determine bounds on the mixing parameters ${|V_{\ell 4}V_{\tau
4}|}^{2}$ as a function of the heavy neutrino mass $m_{4}$. To do this in a
comprehensive manner, we carry out a Monte Carlo sampling of the mixing
parameters and the mass of the heavy neutrino. For simplicity, the mixing
elements $V_{e4},V_{\mu 4}$ and $V_{\tau 4}$ are allowed to vary in the range
from 0 to 1. The ranges of mass sampled for the heavy neutrino are listed in
Table 2 for the various tau decay modes. We only sample the range of masses
that lead to a resonant enhancement of the width as the other mass regions
have very small transition rates as discussed earlier. We then calculate the
transition rates and branching fractions over the entire range of mixing and
mass of the heavy neutrino and the results of the Monte Carlo sampling are
discussed next.
Table 2: Mass and mixing elements of heavy neutrino and the decay mode constraining them with the corresponding experimental bounds on branching fractions. Bounds for $\Delta L=2$ tau decays are from Ref. [103] Mixing element | Range of $m_{4}\ (\rm MeV)$ | Decay mode | $B_{exp}$
---|---|---|---
| 140 - 1637 | $\tau^{-}\rightarrow e^{+}\pi^{-}\pi^{-}$ | $2.7\times 10^{-7}$
$|V_{e4}V_{\tau 4}|$ | 140 - 1637 | $\tau^{-}\rightarrow e^{+}\pi^{-}K^{-}$ | $1.8\times 10^{-7}$
| 494 - 1283 | $\tau^{-}\rightarrow e^{+}K^{-}K^{-}$ | $1.5\times 10^{-7}$
| 245 - 1637 | $\tau^{-}\rightarrow\mu^{+}\pi^{-}\pi^{-}$ | $0.7\times 10^{-7}$
$|V_{\mu 4}V_{\tau 4}|$ | 245 - 1637 | $\tau^{-}\rightarrow\mu^{+}\pi^{-}K^{-}$ | $2.2\times 10^{-7}$
| 599 - 1283 | $\tau^{-}\rightarrow\mu^{+}K^{-}K^{-}$ | $4.8\times 10^{-7}$
The relevant mixing parameters ${|V_{e4}V_{\tau 4}|}$ and ${|V_{\mu 4}V_{\tau
4}|}$ are probed as a function of the heavy neutrino mass $m_{4}$ and are
shown in Fig. 8(a) and Fig. 8(b), respectively. Under the assumption that the
detector was able to reconstruct all the signal events, the region above the
curves is excluded by the current direct experimental search for $LV$ tau
decays. The most stringent bound on ${|V_{e4}V_{\tau 4}|}$ is of ${\cal
O}(10^{-6})$ and comes from $\tau^{-}\rightarrow e^{+}\pi^{-}\pi^{-}$. The
most stringent bound on ${|V_{\mu 4}V_{\tau 4}|}$ is also of ${\cal
O}(10^{-6})$ and comes from $\tau^{-}\rightarrow\mu^{+}\pi^{-}\pi^{-}$. This
is three orders of magnitude more sensitive than the limits from precision
electroweak data which constrain the square of the mixing ${|V_{\ell 4}|}^{2}$
to be less than few times $10^{-3}$. In the absence of detection of $LV$
processes the constraints on mixing from peak searches, accelerator
experiments, reactor experiments and others (collectively called laboratory
constraints here and henceforth) described in Fig. 2 $-$ Fig. 5 are also
applicable here. In the mass region probed by $LV$ tau decays the most
stringent current constraints are $|V_{e4}|^{2}<10^{-7}-10^{-8}$, $|V_{\mu
4}|^{2}<10^{-6}-10^{-8}$ and $|V_{\tau 4}|^{2}<10^{-1}-10^{-4}$. This would
roughly translate into constraints on $|V_{e4}V_{\tau 4}|<10^{-4}-10^{-6}$ and
$|V_{\mu 4}V_{\tau 4}|<10^{-4}-10^{-6}$ which are comparable to the limits
from $LV$ tau decay modes. We explore more combinations of mixing elements and
also provide better constraints on mixing in some mass regions. To summarize,
the constraints on mixing from $LV$ tau decays are always competitive with or
better than precision EW constraints and laboratory constraints in the
corresponding mass region. The experimental bounds can improve in future and
an order of magnitude improvement in the experimental branching fraction will
give approximately an order of magnitude improvement in the constraints for
the mixing parameters ${|V_{\ell 4}V_{\tau 4}|}$. More importantly, a
detection in one of the laboratory experiments implies the existence of a
sterile neutrino while a detection in one of the modes studied in our analysis
would imply $LV$ and hence the existence of a Majorana neutrino.
It should be noted that the intermediate heavy Majorana neutrino is treated as
a real particle which propagates before decaying. If it exits the experimental
apparatus prior to decaying, then the signal corresponding to the $\Delta L=2$
process cannot be reconstructed and no bound could be deduced from the non-
observation of such events. In Figs. 8(a) and 8(b), we provide an estimate of
the bound on the mixing parameters which takes into account the probability of
the heavy Majorana neutrino to decay within the detector of size
$L_{\mathrm{exp}}$. This probability is given by
$P=1-\exp(-L_{\mathrm{exp}}\Gamma_{N})$ (48)
and for small masses and/or small mixing parameters and consequently long
decay lengths, it can be approximated with $P\simeq
L_{\mathrm{exp}}\Gamma_{N}$. We take $L_{\mathrm{exp}}=10$ m, the typical size
of the detectors used in the experiments under consideration. For simplicity,
we take $N_{4}$ to be relativistic but we keep its gamma factor $\gamma=1$, as
a more precise value requires a full understanding of the experimental setup.
We assume $|V_{e4}|=|V_{\mu 4}|=|V_{\tau 4}|$. An estimate of the realistic
bound on the mixing parameter $|V_{e4}V_{\tau 4}|$ is then given by
$|V_{e4}V_{\tau
4}|(=|V_{e4}|^{2})=\sqrt{|V_{e4}|^{2}_{\infty}/(L_{\mathrm{exp}}\Gamma_{N0})},$
(49)
where $|V_{e4}|^{2}_{\infty}$ is the bound obtained assuming that all the
$N_{4}$ decay in the detector and discussed above, and $\Gamma_{N0}$ is the
decay rate for a fully active heavy Majorana neutrino, i.e. when the mixing
parameter $|V_{\ell 4}|=1$ for $\ell=e,\mu,\tau$. The bounds remain unchanged
for large values of the mixing angle and /or large values of $m_{4}$, as the
decay length in these cases is very short. However, the most sensitive limit
on $|V_{e4}V_{\tau 4}|(=|V_{e4}|^{2})$ coming from $\tau\rightarrow e\pi\pi$
searches gets weakened to $\sim 4\times 10^{-4}\ (4\times 10^{-5})\ (1\times
10^{-5})$ for $m_{4}=0.2\ (0.5)\ (1.0)$ GeV. Similarly, the searches for
$\tau\rightarrow\mu\pi\pi$ allows to set a bound on $|V_{\mu 4}V_{\tau
4}|(=|V_{\mu 4}|^{2})$ which weakens to $|V_{\mu 4}V_{\tau 4}|<1\times
10^{-4}\ (2\times 10^{-5})\ (1\times 10^{-5})$ for $m_{4}=300\ (600)\ (900)$
MeV. A detailed analysis taking into account the experimental setup should be
performed in order to obtain more precise bounds.
|
---|---
Figure 8: (a) Left: excluded regions above the curves for $|V_{e4}V_{\tau 4}|$
versus $m_{4}$; (b) right: same as (a) but for $|V_{\mu 4}V_{\tau 4}|$. The
thin black lines correspond to the estimate of the bound (for $\tau\rightarrow
e\pi\pi$ and $\tau\rightarrow\mu\pi\pi$) once the probability of $N_{4}$ decay
in the detector is taken into account.
### 3.3 Lepton-Number Violating Rare Meson Decays
Table 3: Same as Table 2 but for $\Delta L=2$ rare meson decays. The experimental bounds are from Ref. [99], the bounds for $D^{+}\rightarrow e^{+}e^{+}\pi^{-}(K^{-})$ are from Ref. [104]. Mixing element | Range of $m_{4}\ (\rm MeV)$ | Decay mode | $B_{exp}$
---|---|---|---
| 140 - 493 | $K^{+}\rightarrow e^{+}e^{+}\pi^{-}$ | $6.4\times 10^{-10}$
| 140 - 1868 | $D^{+}\rightarrow e+e^{+}\pi^{-}$ | $3.6\times 10^{-6}$
| 494 - 1868 | $D^{+}\rightarrow e^{+}e^{+}K^{-}$ | $4.5\times 10^{-6}$
| 140 - 1967 | $D^{+}_{s}\rightarrow e^{+}e^{+}\pi^{-}$ | $6.9\times 10^{-4}$
$|V_{e4}|^{2}$ | 494 - 1967 | $D^{+}_{s}\rightarrow e^{+}e^{+}K^{-}$ | $6.3\times 10^{-4}$
| 140 - 5278 | $B^{+}\rightarrow e^{+}e^{+}\pi^{-}$ | $1.6\times 10^{-6}$
| 494 - 5278 | $B^{+}\rightarrow e^{+}e^{+}K^{-}$ | $1.0\times 10^{-6}$
| 776 - 5278 | $B^{+}\rightarrow e^{+}e^{+}\rho^{-}$ | $2.6\times 10^{-6}$
| 892 - 5278 | $B^{+}\rightarrow e^{+}e^{+}K^{*-}$ | $2.8\times 10^{-6}$
| 245 - 388 | $K^{+}\rightarrow\mu^{+}\mu^{+}\pi^{-}$ | $3.0\times 10^{-9}$
| 245 - 1763 | $D^{+}\rightarrow\mu^{+}\mu^{+}\pi^{-}$ | $4.8\times 10^{-6}$
| 599 - 1763 | $D^{+}\rightarrow\mu^{+}\mu^{+}K^{-}$ | $1.3\times 10^{-5}$
| 881 - 1763 | $D^{+}\rightarrow\mu^{+}\mu^{+}\rho^{-}$ | $5.6\times 10^{-4}$
| 997 - 1763 | $D^{+}\rightarrow\mu^{+}\mu^{+}K^{*-}$ | $8.5\times 10^{-4}$
$|V_{\mu 4}|^{2}$ | 245 - 1862 | $D^{+}_{s}\rightarrow\mu^{+}\mu^{+}\pi^{-}$ | $2.9\times 10^{-5}$
| 599 - 1862 | $D^{+}_{s}\rightarrow\mu^{+}\mu^{+}K^{-}$ | $1.3\times 10^{-5}$
| 997 - 1862 | $D^{+}_{s}\rightarrow\mu^{+}\mu^{+}K^{*-}$ | $1.4\times 10^{-3}$
| 245 - 5173 | $B^{+}\rightarrow\mu^{+}\mu^{+}\pi^{-}$ | $1.4\times 10^{-6}$
| 599 - 5173 | $B^{+}\rightarrow\mu^{+}\mu^{+}K^{-}$ | $1.8\times 10^{-6}$
| 881 - 5173 | $B^{+}\rightarrow\mu^{+}\mu^{+}\rho^{-}$ | $5.0\times 10^{-6}$
| 997 - 5173 | $B^{+}\rightarrow\mu^{+}\mu^{+}K^{*-}$ | $8.3\times 10^{-6}$
| 140 - 493 | $K^{+}\rightarrow e^{+}\mu^{+}\pi^{-}$ | $5.5\times 10^{-10}$
| 140 - 1868 | $D^{+}\rightarrow e^{+}\mu^{+}\pi^{-}$ | $5.0\times 10^{-5}$
| 494 - 1868 | $D^{+}\rightarrow e^{+}\mu^{+}K^{-}$ | $1.3\times 10^{-4}$
| 140 - 1862 | $D^{+}_{s}\rightarrow e^{+}\mu^{+}\pi^{-}$ | $7.3\times 10^{-4}$
$|V_{e4}V_{\mu 4}|$ | 494 - 1967 | $D^{+}_{s}\rightarrow e^{+}\mu^{+}K^{-}$ | $6.8\times 10^{-4}$
| 140 - 5278 | $B^{+}\rightarrow e^{+}\mu^{+}\pi^{-}$ | $1.3\times 10^{-6}$
| 494 - 5278 | $B^{+}\rightarrow e^{+}\mu^{+}K^{-}$ | $2.0\times 10^{-6}$
| 776 - 5278 | $B^{+}\rightarrow e^{+}\mu^{+}\rho^{-}$ | $3.3\times 10^{-6}$
| 892 - 5278 | $B^{+}\rightarrow e^{+}\mu^{+}K^{*-}$ | $4.4\times 10^{-6}$
We now investigate the $LV$ processes in which a meson decays into two like-
sign leptons and another meson
$M_{1}^{+}(q_{1})\rightarrow\ell^{+}(p_{1})\ \ell^{+}(p_{2})\
M_{2}^{-}(q_{2}).$ (50)
These decays are similar to the tau decay modes described in the previous
section. The decay amplitude for the above process is given by
$\displaystyle i{\cal M}^{P}$ $\displaystyle=$ $\displaystyle
2G_{F}^{2}V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}f_{M_{1}}f_{M_{2}}{V_{\ell_{1}4}}{V_{\ell_{2}4}}\
m_{4}$ (51) $\displaystyle\times$
$\displaystyle\Biggl{[}\frac{\overline{u_{\ell_{1}}}\not{\hbox{\kern-4.0pt$q$}}_{1}\not{\hbox{\kern-4.0pt$q$}}_{2}P_{R}v_{\ell_{2}}}{(q_{1}-p_{1})^{2}-{m_{4}}^{2}+i\Gamma_{N_{4}}m_{4}}\Biggr{]}+(p_{1}\leftrightarrow
p_{2}),$ $\displaystyle i{\cal M}^{V}$ $\displaystyle=$ $\displaystyle
2G_{F}^{2}V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}f_{M_{1}}f_{M_{2}}{V_{\ell_{1}4}}{V_{\ell_{2}4}}\
m_{4}\ m_{M_{2}}$ (52) $\displaystyle\times$
$\displaystyle\Biggl{[}\frac{\overline{u_{\ell_{1}}}\not{\hbox{\kern-4.0pt$q$}}_{1}\not\epsilon^{\lambda}(q_{2})P_{R}v_{\ell_{2}}}{(q_{1}-p_{1})^{2}-{m_{4}}^{2}+i\Gamma_{N_{4}}m_{4}}\Biggr{]}+(p_{1}\leftrightarrow
p_{2}),$
where $i{\cal M}^{P}$ and $i{\cal M}^{V}$ are the decay amplitudes when the
meson $M_{2}$ is a pseudoscalar or vector meson respectively and
$V^{CKM}_{M_{i}}$ and $f_{M_{i}}$ are the quark flavor mixing element and the
decay constant for the meson $M_{i}$ respectively. From this decay amplitude,
we can calculate the transition rate $\Gamma^{M_{1}}_{\\!\\!\\!\mbox{}_{LV}}$
and the branching fraction normalised by the decay width of the meson $M_{1}$.
In Appendix E, we give the calculations for the decay branching fraction of
the process (50) in terms of the mass of heavy neutrino, $m_{4}$, and the
mixing $|V_{\ell_{1}4}V_{\ell_{2}4}|$. We can express the branching fraction
in an intuitive form, in the massless limit of the final state particles, as
$\displaystyle\mathrm{Br}$ $\displaystyle=$
$\displaystyle\frac{\Gamma^{M_{1}}_{\\!\\!\\!\mbox{}_{LV}}}{\Gamma_{M_{1}}}=\Gamma^{M_{1}}_{\\!\\!\\!\mbox{}_{LV}}\mbox{
}\tau_{M_{1}},$ (53) $\displaystyle\sim$
$\displaystyle\frac{1}{64\pi^{2}}(1-\frac{1}{2}\delta_{\ell_{1}\ell_{2}})G^{4}_{F}f^{2}_{M_{1}}f^{2}_{M_{2}}|V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}|^{2}\mbox{
}|V_{\ell_{1}4}V_{\ell_{2}4}|^{2}\Bigl{(}1-\frac{m^{2}_{4}}{m^{2}_{\tau}}\Bigr{)}m^{5}_{M_{1}}\tau_{M_{1}}\Bigl{(}\frac{m_{4}}{\Gamma_{N_{4}}}\Bigr{)},$
$\displaystyle\sim$ $\displaystyle(10^{-16}\ \rm GeV)\ \tau_{M_{1}}\mbox{
}|V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}|^{2}\mbox{ }|V_{\ell_{1}4}V_{\ell_{2}4}|,$
where we have used typical values of $m_{4}\sim 1\ \rm GeV$, $f_{M_{i}}\sim
0.1\ \rm GeV$, $G_{F}\sim 1\times 10^{-5}\ \rm GeV^{-2}$, $\Gamma_{N_{4}}\sim
10^{-11}\ |V_{\ell 4}|^{2}\ \rm GeV$ and $\tau_{M_{1}}$ is in seconds. Using
the values for the lifetimes of the mesons in Appendix E, the branching
fractions for the various mesons are given by
$\displaystyle\mathrm{Br}(K)$ $\displaystyle\sim$
$\displaystyle|V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}|^{2}\mbox{
}|V_{\ell_{1}4}V_{\ell_{2}4}|,$ (54) $\displaystyle\mathrm{Br}(D,\ B)$
$\displaystyle\sim$ $\displaystyle 10^{-4}\ \mbox{
}|V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}|^{2}\mbox{ }|V_{\ell_{1}4}V_{\ell_{2}4}|,$
(55) $\displaystyle\mathrm{Br}(D_{s})$ $\displaystyle\sim$ $\displaystyle
10^{-5}\mbox{ }|V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}|^{2}\mbox{
}|V_{\ell_{1}4}V_{\ell_{2}4}|.$ (56)
As mentioned earlier, with the simple expressions above one can easily make a
rough estimate of the required sensitivity and hence the feasibility of
observation in terms of the mixing parameters for a given model.
Searches for rare meson decay modes have been made in numerous experiments.
Table 3 summarizes the current experimental limits on branching fractions
given by Refs. [99, 104]. From the non-observation of these $LV$ rare meson
decay modes one can determine constraints on mixing parameters
${|V_{\ell_{1}4}V_{\ell_{2}4}|}$ as a function of the heavy neutrino mass
$m_{4}$. To do this in a comprehensive manner, we carry out a Monte Carlo
sampling of the mixing parameters and the mass of the heavy neutrino similar
to tau decay. The mixing elements $V_{e4},V_{\mu 4}$ and $V_{\tau 4}$ are
allowed to range from 0 to 1 for simplicity. Only the range of mass that leads
to a resonant enhancement of the width is sampled for the heavy neutrino and
listed in Table 3 for the various meson decay modes. The transition rates and
branching fractions are then calculated over the entire range of mixing and
mass of the heavy neutrino and the results of the Monte Carlo sampling are
discussed next.
For the various decay modes, the mixing parameters probed are
${|V_{e4}|}^{2}$, ${|V_{e4}V_{\mu 4}|}$ and ${|V_{\mu 4}|}^{2}$ depending on
the final state leptons. Again, we plot the excluded region of the mixing
parameters as a function of neutrino mass, as shown in Fig. 9 for
${|V_{e4}|}^{2}$, Fig. 10 for ${|V_{e4}V_{\mu 4}|}$ and in Fig. 11 for
${|V_{\mu 4}|}^{2}$. The regions above the curves are excluded by the current
direct experimental searches for $LV$ meson decays. First we plot the limits
which can be derived assuming that all $N_{4}$ decay in the detector and give
a positive signature. The most stringent constraints are from the
$K^{+}\rightarrow\ell^{+}_{1}\ell^{+}_{2}\pi^{-}$ mode with mixings of ${\cal
O}(10^{-9})$ excluded for ${|V_{e4}|}^{2}$, ${|V_{e4}V_{\mu 4}|}$ and
${|V_{\mu 4}|}^{2}$. This is six orders of magnitude more sensitive than the
limits from precision electroweak data which constrains the square of the
mixing ${|V_{\ell 4}|}^{2}$ to be less than few times $10^{-3}$. Next in
sensitivity are the $D$ and $D_{s}$ decay modes with constraints of order few
times $10^{-3}$ which are similar to the constraints from precision
electroweak data. The bounds for the same mixing elements are much weaker in
the mass range above $2\ \rm GeV$. Even though the limits are weak in this
region, it is important not to neglect the experimental study of these
processes. It only implies that there is a large parameter space available for
the mass and mixing of heavy neutrinos.
As discussed for the $\Delta L=2$ tau decays, in the absence of detection of
$LV$ processes the laboratory constraints on mixing described in Fig. 2 $-$
Fig. 5 are also applicable here. In the mass region probed by $LV$ meson
decays the most stringent laboratory bounds are
$|V_{e4}|^{2}<10^{-7}-10^{-8}$, $|V_{\mu 4}|^{2}<10^{-6}-10^{-8}$ for
$m_{4}<2$ GeV and $|V_{\mu 4}|^{2}<10^{-4}$ for $m_{4}>2$ GeV. This would
roughly translate into constraints on $|V_{e4}V_{\mu 4}|<10^{-6}-10^{-8}$ for
$m_{4}<2$ GeV and $|V_{e4}V_{\mu 4}|<10^{-5}-10^{-6}$ for $m_{4}>2$ GeV. It
should be noted that, if these experiments were able to fully reconstruct the
signal, the limits from $K$ meson decays would be better than the laboratory
constraints by at least an order of magnitude in the corresponding mass
region. In fact, the constraints on ${|V_{e4}|}^{2}$ from the kaon decay mode
$K^{+}\rightarrow e^{+}e^{+}\pi^{-}$ would be more stringent than even the
constraints from $0\nu\beta\beta$ shown in Fig. 3. Usually $0\nu\beta\beta$
experiments have the best sensitivity as they have an advantage of a large
“effective luminosity” resulting from the large number of nuclei available for
decay. The meson (and tau) experiments on the other hand have a small
luminosity coming from a limited number of mesons (taus) produced in
accelerators compared to the number of nuclei in $0\nu\beta\beta$ experiments.
It is interesting to note that the resonant enhancement in the case of the $K$
meson decay is able to match or improve over the large “effective luminosity”
of $0\nu\beta\beta$ experiments. In conclusion, the constraints on mixing from
$LV$ meson decays are competitive with the precision EW constraints and all
the laboratory constraints, potentially even $0\nu\beta\beta$, in some mass
regions. But again, we emphasize that the aim of our analysis is to study $LV$
processes and hence Majorana neutrinos.
We have also taken into account the fact that, for small mixing, only part of
the heavy sterile neutrinos produced will decay in the detector. We have
considered $L_{\mathrm{exp}}=10$ m, $|V_{e4}|=|V_{\mu 4}|=|V_{\tau 4}|$ and
the gamma factor of $N_{4}$, $\gamma=1$, for simplicity. In this case, as
discussed for the $\Delta L=2$ tau decays, the bounds get sensibly weakened.
An estimate of these bounds is reported in Figs. 9, 10 and 11 by thin black
lines. We see that the bounds get significantly weakened by few orders of
magnitude for $K\rightarrow ee\pi$, $K\rightarrow e\mu\pi$ and
$K\rightarrow\mu\mu\pi$ and a careful analysis of these searches should be
performed to find the detailed bounds on the mixing angles.
The sensitivity of current direct experimental searches are not adequate to
constrain mixings for some decay modes. The theoretically allowed branching
fraction versus mass $m_{4}$ for such modes is given in Fig. 12. As we can
deduce from Table 3 and Fig. 12 all the modes are very close to start being
probed by direct experimental searches. The experimental bounds on branching
fractions can improve in future and similar to tau decay modes, an order of
magnitude improvement in the experimental branching fraction will give
approximately an order of magnitude improvement in the constraints for the
mixing parameters ${|V_{\ell_{1}4}V_{\ell_{2}4}|}$. Currently we do not have
any constraints on the mixing parameter ${|V_{\tau 4}|}^{2}$ from $LV$ rare
meson decay modes. Only very weak constraints for $\mathrm{BR}(B\to
X\tau^{+}\tau^{-})<\cal{O}(\mbox{5}\%)$ exist in a theoretical analysis [105].
The similar signature $B^{+}\rightarrow\tau^{+}\tau^{+}M^{-}$ is a possible
decay mode that would bound ${|V_{\tau 4}|}^{2}$ and should be pursued.
Figure 9: Excluded regions above the curves for ${|V_{e4}|}^{2}$ versus
$m_{4}$ from $M_{1}^{+}\rightarrow e^{+}e^{+}M_{2}^{-}$ searches. The thin
black line corresponds to an estimate of the bound from $K^{+}\rightarrow
e^{+}e^{+}\pi^{-}$ once the probability of decay of $N_{4}$ in the detector is
taken into account. Figure 10: Same as Fig. 9 but for $|V_{e4}V_{\mu 4}|$
from $M_{1}^{+}\rightarrow e^{+}\mu^{+}M_{2}^{-}$ searches. Figure 11: Same as
Fig. 9 but for ${|V_{\mu 4}|}^{2}$ from
$M_{1}^{+}\rightarrow\mu^{+}\mu^{+}M_{2}^{-}$ searches. Figure 12: Branching
fraction versus heavy neutrino mass $m_{4}$ for decay modes
$M_{1}^{+}\rightarrow\ell_{1}^{+}\ell_{2}^{+}M_{2}^{-}$ not yet constrained by
direct experimental searches. The regions below the curve are theoretically
allowed.
## 4 Collider Signatures
Figure 13: (a) Left: Feynman diagram for like-sign dilepton signature via $WW$
fusion in hadronic collisions; (b) right: the exchanged coherent diagram which
is same as heavy neutrino production and decay.
In this section we study heavy Majorana neutrinos at hadron colliders. The
most distinctive channels of the signal involve like-sign di-leptons. It was
first proposed in Ref. [29] in the context of the left-right symmetric model,
and subsequently studied in Ref. [30, 31, 32, 33] We discuss the signatures
for a heavy Majorana neutrino and the sensitivity to probe the parameters
$m_{4}$ and $V_{\ell 4}$ at the Tevatron and the LHC.
As for the production of a heavy Majorana neutrino at hadron colliders, the
representative diagrams at the parton level are depicted in Fig. 13, with the
exchange of final state leptons implied. The first diagram is via $WW$ fusion
with a $t$-channel heavy neutrino $N_{4}$ exchange, directly analogous to the
process of $0\nu\beta\beta$. The second diagram is via $s$-channel $N_{4}$
production and subsequent decay. Although in our full calculations, we have
coherently counted for all the contributing diagrams of like-sign dilepton
production including possible identical particle crossing, it is informative
to separately discuss these two classes of diagrams due to their
characteristically different kinematics.
The scattering amplitude for the process in Fig. 13(a) is proportional to
$V_{\ell_{1}4}V_{\ell_{2}4}$ and the cross section can be expressed as
$\sigma\left(pp\rightarrow
W^{\pm}W^{\pm}\to\ell_{1}^{\pm}\ell_{2}^{\pm}X\right)=\left(2-\delta_{\ell_{1}\ell_{2}}\right)\left|V_{\ell_{1}4}V_{\ell_{2}4}\right|^{2}\sigma_{0}(WW),$
(57)
where $\sigma_{0}(WW)$ is the “bare cross section”, independent of the mixing
parameters. We show the bare cross section at the LHC energy of 14 TeV versus
the heavy neutrino mass in Fig. 14. This cross section can be at the order of
tens of femtobarns. However, due to the large suppression of the small flavor
mixing to the fourth power, the cross section is rather small. This process
was calculated in Ref. [20] under the effective vector boson approximation.
The authors of Ref. [20] obtained significantly more optimistic results than
ours. Further scrutiny indicated that they missed a factor of
${G^{2}_{F}m^{4}_{W}}/{8}$ and their result should be scaled down by this
factor. The corresponding curve for the Tevatron is not shown in Fig. 14 as
the bare cross section is smaller by nearly two orders of magnitude. Including
the small mixing element (to the fourth power) further reduces the cross
section drastically with no hope of detection at the Tevatron via this mode.
Figure 14: The bare cross section $\sigma_{0}(WW)$ versus mass of the heavy
neutrino $m_{4}$.
By far, the dominant production process of heavy Majorana neutrino in hadronic
collisions is the diagram shown in Fig. 13(b). We calculate the exact process,
but it turns out to be an excellent approximation to parameterize the cross
section as
$\sigma(pp\rightarrow\ell_{1}^{\pm}\ \ell_{2}^{\pm}\
W^{\mp})\approx\left(2-\delta_{\ell_{1}\ell_{2}}\right)\sigma(pp\rightarrow\ell_{1}^{\pm}N_{4})Br(N_{4}\rightarrow\ell_{2}^{\pm}W^{\mp})\propto\frac{|V_{\ell_{1}4}V_{\ell_{2}4}|^{2}}{\sum_{\ell=e}^{\tau}\left|V_{\ell
4}\right|^{2}}.$ (58)
This observation allows us to study the process in a model-independent way. We
can rewrite the cross section in a factorized form
$\sigma(pp\rightarrow\ell_{1}^{\pm}\ \ell_{2}^{\pm}\
W^{\mp}\rightarrow\ell_{1}^{\pm}\ \ell_{2}^{\pm}\ j\
j^{\prime})=\left(2-\delta_{\ell_{1}\ell_{2}}\right)\ S_{\ell_{1}\ell_{2}}\
\sigma_{0}(N_{4}),$ (59)
where $\sigma_{0}(N_{4})$, called the “bare cross section”, is only dependent
on the mass of heavy neutrino and is independent of all the mixing parameters
when the heavy neutrino decay width is narrow. As seen in Fig. 6, this is
indeed the case for $m_{4}\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}1$ TeV once we fold in the constraints $|V_{\ell
4}|^{2}<{\cal O}(10^{-3})$ from precision EW measurements. We calculate the
exact cross section for the dilepton production and use the definition Eq.
(59) to find the bare cross sections $\sigma_{0}(N_{4})$, which are shown in
Fig. 15 at the Tevatron and the LHC energies versus the mass of the heavy
Majorana neutrino. Due to the fact that the LHC will start its operation at 10
TeV, we have calculated the cross sections at both 10 and 14 TeV c.m. energy.
The production rate is increased at the higher energy by a factor of 1.5, 2.0,
2.5 for $m_{4}=100,\ 550$ and 1000 GeV, respectively. We will mainly present
our results at 14 TeV for the rest of the paper. An obvious feature of the
cross sections is the transition near the $W$ mass. For $m_{4}<M_{W}$ the
cross section is nearly a constant due to an on-shell $W$ production via the
Drell-Yan mechanism with its subsequent leptonic decay to $\ell^{\pm}N_{4}$.
For $m_{4}>M_{W}$ the cross section falls off sharply versus $m_{4}$ and the
on-shell decay goes like $N_{4}\to\ell^{\pm}W^{\mp}\to\ell^{\pm}\ j_{1}j_{2}$.
Figure 15: The bare cross section $\sigma_{0}(N_{4})$ versus mass of heavy
Majorana neutrino $m_{4}$ for the Tevatron ($p\bar{p}$ at 1.96 TeV, solid
curve) and the LHC ($pp$ at 10 and 14 TeV, dotted and dashed curves,
respectively).
The flavor information of the final state leptons is parameterized by
$S_{\ell_{1}\ell_{2}}=\frac{\left|V_{\ell_{1}4}V_{\ell_{2}4}\right|^{2}}{\sum_{\ell=e}^{\tau}\left|V_{\ell
4}\right|^{2}},$ (60)
In general the two final state charged leptons can be of any flavor
combination, namely,
$e^{\pm}e^{\pm},\ \ e^{\pm}\mu^{\pm},\ \ e^{\pm}\tau^{\pm},\ \
\mu^{\pm}\mu^{\pm},\ \ \mu^{\pm}\tau^{\pm}\quad{\rm
and}\quad\tau^{\pm}\tau^{\pm}.$ (61)
The constraint from $0\nu\beta\beta$ as given in Eq. (24) is very strong and
makes it difficult to observe like-sign di-electrons $e^{\pm}e^{\pm}$. The
events with $\tau$ leptons will be challenging to reconstruct experimentally.
We will thus concentrate on clean dilepton channels of $\mu^{\pm}\mu^{\pm}$
and $\mu^{\pm}e^{\pm}$, although we will comment on our proposal to include
the $\tau$ modes. The corresponding mixing parameters in our notation will be
$S_{\mu\mu}=\frac{\left|V_{\mu
4}\right|^{4}}{\sum_{\ell=e}^{\tau}\left|V_{\ell 4}\right|^{2}},\quad
S_{e\mu}=\frac{\left|V_{e4}V_{\mu
4}\right|^{2}}{\sum_{\ell=e}^{\tau}\left|V_{\ell 4}\right|^{2}},$ (62)
respectively. Given the smallness of $|V_{e4}|^{2}$, we can further simplify
our study by exploring only two cases: an optimistic case $|V_{\mu
4}|^{2}\gg|V_{\tau 4}|^{2},\ |V_{e4}|^{2}$ and a generic case $|V_{\mu
4}|^{2}\approx|V_{\tau 4}|^{2}\gg|V_{e4}|^{2}$, which lead to
$S_{\mu\mu}=\left\\{\begin{array}[]{c}|V_{\mu 4}|^{2}\quad{\rm(optimistic)}\\\
{1\over 2}|V_{\mu 4}|^{2}\quad{\rm(generic)}\end{array}\right.,\qquad S_{\mu
e}=\left\\{\begin{array}[]{c}|V_{e4}|^{2}\quad{\rm(optimistic)}\\\ {1\over
2}|V_{e4}|^{2}\quad{\rm(generic)}\end{array}\right..$ (63)
### 4.1 Search for Like-sign Dilepton Signals at the Tevatron
We now consider the search for $N_{4}$ at the Fermilab Tevatron, which is
currently running at a c.m. energy of 1.96 TeV in $p\bar{p}$ collisions. We
concentrate on the clean like-sign $\mu^{\pm}\mu^{\pm}$ mode
$p\bar{p}\to\mu^{\pm}\mu^{\pm}\ j_{1}j_{2}\ X,$ (64)
where $X$ is some inclusive hadronic activities common in hadronic collisions.
To quantify the signal observability, we first impose the basic acceptance
cuts on leptons and jets to simulate the CDF/D0 detector coverage
$\displaystyle p_{T}^{\mu}>5{\,\rm GeV},\quad|\eta^{\mu}|<2.0,\quad
p_{T}^{j}>10{\,\rm GeV},\quad|\eta^{j}|<3.0.$ (65)
We also smear the lepton momentum by a tracking resolution and the jet energy
by hadronic calorimeter resolution as
$\displaystyle{\Delta p_{T}^{\mu}\over p_{T}^{\mu}}=1.5\times 10^{-3}\
p_{T}^{\mu},\quad{\Delta E_{j}\over E_{j}}={75\%\over\sqrt{E_{j}}}\oplus 3\%,$
(66)
where $p_{T}^{\mu}$ and $E_{j}$ are in units of GeV.
Figure 16: Normalized distributions $\sigma^{-1}d\sigma/dX$ for $m_{4}=20,\
50$ and 100 GeV at the Tevatron for (a) upper left: the minimal isolation
$\Delta R_{\ell j}^{min}$; (b) upper right: the missing transverse momentum
$p\\!\\!\\!/_{T}$; (c) bottom left: the $2\ell 2j$ system invariant mass
$m(\ell\ell jj)$; (d) bottom right: the di-jet invariant mass $m(jj)$.
The signal events we are searching for have very unique kinematical features.
For the purpose of illustration, we choose $m_{4}=20,\ 50\ \rm GeV$ (below
$m_{W}$ threshold) and 100 GeV (above $m_{W}$). First of all, there are two
well-isolated like-sign charged leptons. This is shown in Fig. 16(a) by a
normalized distribution of the minimal isolation $\Delta R_{\ell
j}=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}$. Second, there is essentially no
missing transverse energy. However, realistically, the detectors have finite
resolutions as simulated by the Gaussian smearing given in Eq. (66).
Consequently, there is always some misbalance in the energy-momentum
measurements, which is attributed to the missing transverse energies and is
plotted in Fig. 16(b). Thirdly, due to the existence of an on-shell $W^{\pm}$
in the signal process, one would expect to reconstruct it by an invariant mass
either from the $2\ell 2j$ system $m(\ell\ell jj)$ (in the case of DY
production) or from the di-jets $m_{(}jj)$ (in the case of $N_{4}$ decay).
This is demonstrated in Figs. 16(c) and (d), respectively. The above
kinematical features motivate us to impose the following event selection cuts
$\displaystyle\Delta R^{min}_{\ell j}>0.5,$ (67) $\displaystyle 60\ {\rm
GeV}<\ {\rm either}\ m(\ell\ell jj)\ \ {\rm or}\ m(jj)<100\ \rm GeV,$ (68)
$\displaystyle p\\!\\!\\!/_{T}<20\ \rm GeV.$ (69)
These cuts are highly efficient in selecting the signal events. We illustrate
this in Table 4, in which we calculate the signal rates with the consecutive
cuts for $m_{4}=60$ GeV and $\left|V_{\mu 4}\right|^{2}=\left|V_{\tau
4}\right|^{2}=5\times 10^{-3}\gg\left|V_{e4}\right|^{2}$. Note that the choice
of mixing elements used in the illustration is motivated by constraints from
precision EW measurements. However this is for illustration purposes only and
in our full analysis we have kept $S_{\mu\mu}$ as a free parameter.
At the Tevatron energies, the SM contribution to the like-sign dilepton events
is rather small. The leading background of this type comes from the top-quark
production and its cascade decay via the chain
$\displaystyle t\to W^{+}b\to\ell^{+}\nu_{\ell}\ b,$ (70)
$\displaystyle\bar{t}\rightarrow W^{-}\bar{b}\to W^{-}\ \bar{c}\ \nu_{\ell}\
\ell^{+}.$ (71)
The background rates and survival probabilities with the consecutive cuts are
also given in Table 4. We see that the $t\bar{t}$ background is essentially
eliminated by the selective cuts. We have also considered other SM backgrounds
coming from the production of $W^{\pm}W^{\pm}jj,\ W^{\pm}Zjj$. After the
selective cuts, all these backgrounds are negligibly small.
Table 4: The representative signal and background cross sections at the Tevatron, for $\mu^{\pm}\mu^{\pm}jj$ and the efficiencies with the consecutive cuts. For illustration, we have used $m_{4}=60$ GeV, $\left|V_{\mu 4}\right|^{2}=\left|V_{\tau 4}\right|^{2}=5\times 10^{-3}\gg\left|V_{e4}\right|^{2}$. | No cut | Basic cut (65) | $+\Delta R$ (67) | $+m(jj),m(\ell\ell jj)$ (68) | +$p\\!\\!\\!/_{T}$ (69)
---|---|---|---|---|---
Signal | | | | |
$\sigma$ (fb) | 319 | 108 | 99 | 96 | 96
eff. | - | 33% | 92% | 97% | 100%
$t\bar{t}$ Bkg | | | | |
$\sigma$ (fb) | 78.4 | 58.2 | 1.85 | 0.04 | 0.005
eff. | - | 74% | 3.2% | 2.2% | 12.5%
Figure 17: $\sigma_{0}(N_{4})$ with varying heavy neutrino mass $m_{4}$ after
all the cuts. The two cases correspond to muon rapidity acceptance at D0 and
CDF.
In Fig. 17, we plot the bare cross section $\sigma_{0}(N_{4})$ with the basic
cuts of Eq. (65) as well as the selection cuts Eqs. (67)$-$(69). The reduction
in rate is mainly due to the basic acceptance cuts. For comparison, we have
also included two choices of pseudo-rapidity cut $|\eta(\mu)|<2$ and
$|\eta(\mu)|<1.5$. We now consider the statistical significance of the signal
observation. In the absence of background events, we use Poisson statistics to
determine the search sensitivity. We take a signal with ${95\%}$ Confidence
Level (as this is very close to $2\sigma$ we call it a 2$\sigma$ effect
henceforth) to be 3 events. We can thus translate this to the sensitivity to
the mixing parameter
$(2-\delta_{\ell_{1}\ell_{2}})\sigma_{0}(N_{4})\ S_{\ell_{1}\ell_{2}}\ {\it
L}\geq 3,$ (72)
where $\it L$ is the integrated luminosity.
The CDF collaboration at the Tevatron has successfully studied the events with
like-sign dileptons in a different context [106]. Given our event selection,
in Fig. 18 we estimate the sensitivity reach for the mixing parameters versus
$m_{4}$ at the $2\sigma$ (solid curves) and $5\sigma$ (dashed curves) level at
the Tevatron. In Figs. 18(a$-$b) (upper-left and upper-right), the sensitivity
is shown for $S_{\mu\mu}$ with 2 and 8 fb-1 integrated luminosity. The
horizontal dotted lines are the constraint on $S_{\mu\mu}\simeq|V_{\mu
4}|^{2}<6\times 10^{-3}$ from an analysis of precision EW measurements [70].
The DELPHI [89] and L3 [90] bounds are also given for comparison. We find that
the Tevatron has the potential to reach the following sensitivity for the mass
of the heavy neutrino
$m_{4}\sim\left\\{\begin{array}[]{c}40-130\ {\rm GeV}\ \quad{\rm for}\
2\sigma\ \ {\rm with\ 2\ fb}^{-1};\\\ 10-180\ {\rm GeV}\ (50-120\ {\rm
GeV})\quad{\rm for}\ 2\sigma\ (5\sigma)\ {\rm with\ 8\ fb}^{-1}.\\\
\end{array}\right.$ (73)
Alternatively, the sensitivity for the mixing parameter can be
$S_{\mu\mu}\sim\left\\{\begin{array}[]{c}2\times 10^{-5}\ \qquad{\rm for}\
2\sigma\ \ {\rm with\ 2\ fb}^{-1};\\\ 5\times 10^{-6}\ (2\times
10^{-5})\quad{\rm for}\ 2\sigma\ (5\sigma)\ {\rm with\ 8\ fb}^{-1}.\\\
\end{array}\right.$ (74)
Similar to Figs. 18(a$-$b), Figs. 18(c$-$d) (lower-left and lower-right) show
the results for $S_{e\mu}$ instead. The lower dotted curve in Fig. 18(d) is
the bound on $S_{e\mu}\simeq|V_{e4}|^{2}$ from $0\nu\beta\beta$. We have
assumed the same detection efficiencies for $\mu$ and $e$. With this
assumption, the slightly better reach for $S_{e\mu}$ compared to $S_{\mu\mu}$
is due the factor of two difference in total rate with identical and
nonidentical particles as evident from Eq. (59). With 2 fb-1 luminosity, the
sensitivity to $|V_{e4}|^{2}$ is not close to the stringent bound from the
$0\nu\beta\beta$ decay as seen in Fig. 3. We see from Fig. 18(d) that with 8
fb-1 luminosity, the Tevatron sensitivity for $S_{e\mu}$ may reach the level
of the current bound from $0\nu\beta\beta$. From Eq. (72), it is
straightforward to obtain future sensitivity to mixing parameters
($S_{\mu\mu},S_{e\mu}$) by a simple scaling of the luminosity.
Figure 18: The Tevatron sensitivity to the mixing parameters versus $m_{4}$
(a) upper-left: $2\sigma$ and $5\sigma$ sensitivity of $S_{\mu\mu}$ with 2
fb-1 integrated luminosity; (b) upper-right: same as (a) but with 8 fb-1
integrated luminosity; (c) lower-left: $2\sigma$ and $5\sigma$ sensitivity of
$S_{e\mu}$ with 2 fb-1 integrated luminosity; (d) lower-right: same as (c) but
with 8 fb-1 integrated luminosity. The horizontal dotted lines in (a) and (b)
are the constraint on $S_{\mu\mu}\simeq|V_{\mu 4}|^{2}<6\times 10^{-3}$ from
an analysis of precision EW measurements [70]. The DELPHI [89] and L3 [90]
bounds are also given here for comparison. The lower dotted curve in (d) is
the bound on $S_{e\mu}\simeq|V_{e4}|^{2}$ from $0\nu\beta\beta$.
### 4.2 Search for Like-sign Dilepton Signals at the LHC
At the LHC with a c.m. energy of 14 TeV in $pp$ collisions, we adopt the basic
acceptance cuts on leptons and jets as
$\displaystyle p_{T}^{\ell}>10{\,\rm GeV},\quad|\eta^{\ell}|<2.5,\quad
p_{T}^{j}>15{\,\rm GeV},\quad|\eta^{j}|<2.5.$ (75)
The efficiency of these cuts increases with heavy neutrino mass and is $50\%$
for $m_{4}=200\ \rm GeV$ and $80\%$ for $m_{4}=800\ \rm GeV$. The smearing
parameters to simulate the ATLAS/CMS detectors are [107]
$\displaystyle{\Delta p_{T}^{\mu}\over p_{T}^{\mu}}=36\times 10^{-5}\
p_{T}^{\mu},\quad{\Delta E_{j}\over E_{j}}={1\over\sqrt{E_{j}}}\oplus 5\%,$
(76)
where $p_{T}^{\mu}$ and $E_{j}$ are in units of GeV.
Figure 19: Normalized distributions $\sigma^{-1}d\sigma/dX$ for $m_{4}=60,\
100,\ 200$ and 500 GeV at the LHC for (a) upper left: the minimal isolation
$\Delta R_{\ell j}^{min}$; (b) upper right: the missing transverse momentum
$p\\!\\!\\!/_{T}$; (c) bottom left: the $2\ell 2j$ system invariant mass
$m(\ell\ell jj)$; (d) bottom right: the di-jet invariant mass $m(jj)$.
We again present the characteristic kinematical distributions for the signal.
Fig. 19(a) shows the normalized distribution of the minimal isolation $\Delta
R_{\ell j}$. The simulated missing transverse momentum after the energy-
momentum smearing is plotted in Fig. 19(b). The invariant masses of the $2\ell
2j$ system $m(\ell\ell jj)$ and the di-jets $m_{(}jj)$ are demonstrated in
Figs. 19(c) and (d), respectively. We thus design the selection cuts at the
LHC as
$\displaystyle\Delta R^{min}_{\ell j}>0.5,$ (77) $\displaystyle 60\ {\rm
GeV}<\ {\rm either}\ m(\ell\ell jj)\ \ {\rm or}\ m(jj)<100\ \rm GeV,$ (78)
$\displaystyle p\\!\\!\\!/_{T}<25\ \rm GeV.$ (79)
These cuts are highly efficient in selecting the signal events. We illustrate
this in Table 5, in which we calculate the signal rates with the consecutive
cuts for $m_{4}=200$ GeV and $\left|V_{\mu 4}\right|^{2}=\left|V_{\tau
4}\right|^{2}=5\times 10^{-3}\gg\left|V_{e4}\right|^{2}$. Again the choice of
mixing elements is motivated by constraints from precision EW measurements.
However as discussed earlier this is for illustration purposes only and in our
full analysis we have kept $S_{\mu\mu}$ and $S_{\mu e}$ as free parameters.
Table 5: The representative signal and background cross sections at the LHC, for $\mu^{\pm}\mu^{\pm}jj$ and the efficiencies with the consecutive cuts. For illustration, we have used $m_{4}=200$ GeV, $\left|V_{\mu 4}\right|^{2}=\left|V_{\tau 4}\right|^{2}=5\times 10^{-3}\gg\left|V_{e4}\right|^{2}$, and $m_{H}=120,\ 300$ GeV. | No cut | Basic cut | $+p\\!\\!\\!/_{T}$ cut | $+\Delta R$ cut | $+m(jj),m(\ell\ell jj)$ cut
---|---|---|---|---|---
| | (75) | (79) | (77) | (78)
Signal | | | | |
$\sigma$ (fb) | 0.86 | 0.42 | 0.37 | 0.35 | 0.33
eff. | - | 48% | 88% | 96% | 94%
$t\bar{t}$ Bkg | | | | |
$\sigma$ (fb) | 29.6 | 16.9 | 2.7 | 0.075 | 0.002
eff. | - | 57% | 16% | 2.8% | 2.7%
$W^{\pm}W^{\pm}W^{\mp}$ | $m_{H}=$120 GeV | | | |
$\sigma$ | 1.01 | 0.42 | 0.057 | 0.052 | 0.050
eff. | - | 42% | 14% | 91% | 96%
| $m_{H}=$300 GeV | | | |
$\sigma$ (fb) | 1.28 | 0.58 | 0.066 | 0.061 | 0.058
eff. | - | 45% | 11% | 92% | 95%
$W^{\pm}W^{\pm}jj$ | $m_{H}=$120 GeV | | | |
$\sigma$ (fb) | 4.2 | 1.3 | 0.29 | 0.17 | 0.019
eff. | - | 31% | 22% | 59% | 11%
| $m_{H}=$300 GeV | | | |
$\sigma$ (fb) | 4.4 | 1.4 | 0.34 | 0.19 | 0.025
eff. | - | 32% | 24% | 56% | 13%
Figure 20: The bare cross section $\sigma_{0}(N_{4})$ versus heavy neutrino
mass $m_{4}$ after all the cuts at the LHC (14 TeV). The solid (dotted) line
correspond to the exclusion (inclusion) of the Higgs decay channel for
$m_{H}=120$ GeV.
In Fig. 20, we plot the bare cross section $\sigma_{0}(N_{4})$ with the basic
cuts of Eq. (75) as well as the selection cuts Eqs. (77)$-$(79) at 14 TeV. The
solid (dotted) curves correspond to the bare cross section without (with) the
Higgs decay channel for $m_{H}=120$ GeV. The reduction in rate is mainly due
to the basic acceptance cuts. We note that the cross section with the cuts at
14 TeV is higher than that at 10 TeV by a factor of 1.4$-$1.6 for
$m_{4}=100-500$ GeV. The sensitivity reach for the mixing parameters to be
presented later will be scaled down roughly according to this factor for LHC
with c.m. energy of 10 TeV.
As discussed in the previous section, a large SM background comes mainly from
top quark production and decay via the chain decay $t\rightarrow b\rightarrow
c\ \ell^{+}\ \nu_{\ell}$. Fortunately, after all the selective cuts in Eqs.
(75)$-$(79), the top-quark decay background is essentially eliminated and has
no remaining events for the expected luminosity of 100 fb-1 at LHC.
There are several other SM backgrounds coming from like-sign $W$ boson
production at the LHC energies. First of all, the triple gauge-boson
production process
$pp\rightarrow W^{\pm}W^{\pm}W^{\mp}\rightarrow\ell^{\pm}\ell^{\pm}\nu\nu\
jj,$ (80)
leads to the irreducible background with two like-sign leptons plus jets.
Next, the same final state can be produced via the process
$pp\rightarrow W^{\pm}W^{\pm}\ jj\rightarrow\ell^{\pm}\ell^{\pm}\nu\nu\ jj,$
(81)
where the two jets may come from either QCD scattering or from the gauge-boson
fusion process. However these backgrounds have two missing neutrinos and can
be suppressed by a combination of cuts on the missing transverse energy and
invariant mass. We also analysed the backgrounds coming from $Z$ boson
production
$pp\rightarrow jjZZ,\quad pp\rightarrow jjZW.$ (82)
in which some charged leptons are missing in the detection so that they lead
to like-sign dilepton events. The backgrounds are very small after the cuts.
We list the number of background events and efficiency of cuts in Table 5 for
a luminosity of 100 fb-1 at LHC. The total background is about $7-8$ events
for 100 fb-1 at the LHC. The main background is from the
$W^{\pm}W^{\pm}W^{\mp}$ channel and can be further suppressed if a tighter
missing energy cut could be exploited. For instance, the background events may
be reduced by half, leaving about $3-4$ events with $p\\!\\!\\!/_{T}<15$ GeV.
The last but not least important feature of the signal is the direct
reconstruction of the resonant mass of $N_{4}$ in the final state
$\ell^{\pm}jj$. This is shown in Fig. 21 for the SM background and the signal
with $m_{4}=200,\ 400$ GeV. We see the effective reconstruction of the
resonant mass. For a given mass $m_{4}$ in the search, one can further make
the event selection on $m(\ell jj)$
$0.8\ m_{4}<m(\ell jj)<1.2\ m_{4},$ (83)
to estimate the significance of the signal observation. This loose cut has
little effect on the signal, but reduces the total background to $0-4$ events
for 100 fb-1 in the range of $m_{4}$ as shown in Fig. 22. We once again adopt
Poisson statistics to determine the search sensitivity. The number of signal
events needed for $2\sigma$ significance would be $3-11$; and $15-44$ for
$5\sigma$ significance. In Fig. 23(a) and Fig. 23(b), we summarize the
sensitivity for $S_{\mu\mu}$ and $S_{e\mu}$ versus $m_{4}$, respectively. The
solid (dashed) curves correspond to $2\sigma$ ($5\sigma$) limits on
$S_{\ell\ell^{\prime}}$ with the exclusion of the Higgs decay channel. The
dotted (dash dotted) curves are similar but with the inclusion of the Higgs
decay channel for $m_{H}=120$ GeV. The horizontal dotted line corresponds to
constraints on $|V_{\mu 4}|^{2}<6\times 10^{-3}$ from precision EW
measurements [70]. In Fig. 23(b) the dashed line at the bottom corresponds to
the limit from $0\nu\beta\beta$.
Figure 21: Invariant mass distributions of $m(\ell jj)$ for the signal with
$m_{4}=200,\ 400$ GeV and background processes. Figure 22: Number of
background events vs mass of the heavy neutrino, $m_{4}$.
Figure 23: (a) Left: $2\sigma$ and $5\sigma$ sensitivity for $S_{\mu\mu}$
versus $m_{4}$ at the LHC with 100 fb-1 integrated luminosity; (b) right: same
as (a) but for $S_{e\mu}$ . The solid and dashed (dotted and dash dotted)
curves correspond to limits with the exclusion (inclusion) of the Higgs decay
channel for $m_{H}=120$ GeV. The horizontal dotted line corresponds to the
constraint on $S_{\mu\mu}\simeq|V_{\mu 4}|^{2}<6\times 10^{-3}$ from precision
EW measurements [70].
In the optimistic case, we assume that $\left|V_{\tau
4}\right|^{2}\ll\left|V_{\mu 4}\right|^{2}$ and $S_{\mu\mu}\simeq\left|V_{\mu
4}\right|^{2}\leq 6\times 10^{-3}$. The detection sensitivity on heavy
neutrino mass can be
$\displaystyle m_{4}\sim\left\\{\begin{array}[]{c}375\
\rm{GeV}\,\,\,\,\,\,\rm{for}\,\,\,2\sigma;\\\ 250\
\rm{GeV}\,\,\,\,\,\,\rm{for}\,\,\,5\sigma.\end{array}\right.$ (86)
Or alternatively, the mixing parameter can be probed to
$\displaystyle S_{\mu\mu}\sim\left\\{\begin{array}[]{c}7\times
10^{-7}\,\,\,\,\,\,\rm{for}\,\,\,2\sigma;\\\ 3\times
10^{-6}\,\,\,\,\,\,\rm{for}\,\,\,5\sigma.\end{array}\right.$ (89)
In particular, even with the very stringent bound on $|V_{e4}|^{2}$ from
$0\nu\beta\beta$ as indicated by the dashed curve in Fig. 23(b), one may still
have $2\sigma$ sensitivity if $m_{4}\approx m_{W}$.
Figure 24: (a) Left: same as Fig. 20 but with the tighter cuts of Eqs. (90)
and (91); (b) right: same as Fig. 23(a) but with the tighter cuts of Eqs. (90)
and (91).
Our calculations for hadron colliders have been based on parton-level
simulations. A recent study [33] pointed out that there may be other
backgrounds to be considered when detector effects are included. One of them
is the faked like-sign dileptons from the $b\bar{b}$ cascade decay. The other
is due to the QCD multi-jet radiation to degrade the reconstruction of $W\to
jj$. Those backgrounds can not be easily simulated in particular at the
parton-level. A preliminary analysis including full CMS detector simulations
cannot support their claim [34]. Nevertheless, we may consider to design more
stringent acceptance cuts to further discriminate against the backgrounds.
First, common wisdom suggests to tighten up the charged lepton isolation
requirement
$\Delta R^{min}_{\ell j}>0.8,$ (90)
which would remove the backgrounds from $b,c$ decays substantially, but a full
assessment can be made only when real data become available and after the
detectors are fully understood. Next, we may increase the jet threshold to
suppress the initial state QCD jet radiation to purify the $W\to jj$ sample.
Our estimate based on a PYTHIA simulation shows that the kinematics of a DY-
type electroweak process can be largely preserved with the appropriate jet
threshold. We thus examine the cut
$p_{T}^{j}>25\ {\rm GeV},$ (91)
which results in only about $17\%$ of the events with potential jet
contamination. The results with the tightened cuts are given in Fig. 24. We
see that the stringent cuts severely hurt the low mass region, but the effect
on the high mass region is modest.
### 4.3 Like-sign Dilepton Signals with $\tau$ in the Final States
So far we have only presented the results with electron and muon final states
and ignored the taus. This is due to the experimental challenge of $\tau$
reconstruction. Given the importance to cover all the lepton flavors, one must
strive to include taus on the search list. Besides the known experimental
practice for $\tau$ identification at the Tevatron [108], there are proposals
to identify $\tau$ events in connection with the neutrino sector [109]. The
central issue is to reconstruct the missing momenta from $\tau$ decays. We can
generalize our requirement for the charged leptons to the isolated charged
tracks presumably from the $\tau$ decays ($e,\mu$, or one-prong and three-
prong charged hadrons)
$p_{T}({\rm track})>10{\,\rm GeV},\quad|\eta({\rm track})|<2.5.$ (92)
This assures that the parent taus are very energetic. For events with one
$\tau$ and no other sources of missing particles, the missing momentum will be
along the direction of the charged track. We thus have
${\vec{p}}\ ({\rm invisible})=\kappa\vec{p}\ ({\rm track}),$ (93)
where the proportionality constant $\kappa$ is determined from the
$E\\!\\!\\!/_{T}$ measurement by assigning $E\\!\\!\\!/_{T}=\kappa p_{T}({\rm
track})$. For events with two taus, we generalize it to
${\vec{p}}\ ({\rm invisible})=\kappa_{1}\vec{p}\ ({\rm
track}_{1})+\kappa_{2}\vec{p}\ ({\rm track}_{2}).$ (94)
As long as the two $\tau$ tracks are not linearly dependent, $\kappa_{1}$ and
$\kappa_{2}$ can be determined again from the $E\\!\\!\\!/_{T}$ measurement.
The missing momenta, as well as the $\tau$ kinematics, are thus fully
reconstructed.
Although we believe that the $N_{4}$ signals in the modes of
$e^{\pm}\tau^{\pm},\ \mu^{\pm}\tau^{\pm}$ and $\tau^{\pm}\tau^{\pm}$ would be
very promising for observation, the background analyses will be considerably
more involved due to the complication of $\tau$ reconstruction. Since our
simulations are performed at the parton level, we are unable to adequately
address the background suppression and to quantify the signal observability.
We thus leave this for future studies.
## 5 Summary and Conclusions
The observation of a $LV$ process would show that neutrino is a Majorana
particle unambiguously. Apart from light neutrinos, $LV$ processes involving
SM particles can receive a contribution from heavy Majorana neutrinos due to
mixing. In fact, this contribution can be resonantly enhanced for appropriate
masses of the heavy neutrino. In the absence of observation of $LV$
interactions, the rates for these processes can constrain the mixing elements
${|V_{\ell_{1}4}V_{\ell_{2}4}|}$ as a function of the mass $m_{4}$ of the
heavy Majorana neutrino. We considered two classes of $LV$ violating
processes: (a) low energy $\Delta L=2$ tau decays and rare meson decays and
(b) collider signals for like-sign dilepton production with no missing energy
implying the existence of Majorana neutrinos. We emphasize the necessity of
involving two charged leptons and no neutrinos in the initial and final
states, to be conclusive about lepton-number violation.
For the low energy interactions we evaluated the transition rates and
branching fractions as a function of the mass and mixing of the heavy
neutrinos. We then translated the current experimental bounds from direct
searches into limits on ${|V_{\ell_{1}4}V_{\ell_{2}4}|}$ as a function of the
mass $m_{4}$ of the heavy neutrino. Amongst the rare meson decays, the
$K^{+}\rightarrow\ell_{1}^{+}\ell_{2}^{+}\pi^{-}$ decay mode currently gives
the most sensitive experimental limits on ${|V_{\ell_{1}4}V_{\ell_{2}4}|}$.
Potentially, these constraints are six orders of magnitude more stringent than
the constraints from precision electroweak data which limit $|V_{\ell 4}|^{2}$
to few times $10^{-3}$. As the intermediate heavy sterile neutrino is a real
particle which might exit the detector if the decay length is longer than the
detector size, for very small mixing angles the bounds get weakened but are
still much more stringent than the electroweak precision constraints. This
effect should be taken into account and a detailed analysis of past
experimental data is required in order to find the precise limits on the
mixing angles. Next in sensitivity are the $D$ and $D_{s}$ meson decay modes
with constraints of the order of $10^{-3}$. Again, these are competitive with
if not better than constraints from EW precision data. The other processes (in
other mass ranges) have very weak experimental limits, weaker than EW
precision data and essentially do not impose any meaningful bounds on
${|V_{\ell_{1}4}V_{\ell_{2}4}|}$. This implies that more accurate experimental
studies on those rare decays should be strongly encouraged. In particular,
many interesting processes of $D,\ B$ decays have not even been experimentally
probed as well as those with a $\tau$ lepton in the final state. Among the
$\tau$-decay modes the best limits come from
$\tau^{-}\rightarrow\ell^{+}\pi^{-}\pi^{-}$. The other $\tau$-decay modes have
sensitivity of order $10^{-3}$ to $10^{-5}$. Again, the constraints from
$\tau$ decay modes are competitive with or better than constraints from
precision EW data by 2 to 3 orders of magnitude. The experimental bound on
$LV$ processes is expected to improve in the future. The future sensitivity of
the square of the mixing parameter will increase approximately by an order of
magnitude for every order of magnitude improvement in experimental bounds on
branching fractions. We have shown that the low energy $\Delta L=2$ $\tau$
decays and rare meson decays can be very strong probes to discover or
constrain the mass and mixing of heavy Majorana neutrinos. Even those decay
modes which do not impose strong constraints should not be neglected. It only
implies that a large range of the parameter space is available for
exploration.
In addition to analyzing the $LV$ tau and meson decay modes and precision EW
measurements we also compiled the constraints on the mixing elements
($|V_{e4}|^{2},|V_{\mu 4}|^{2}$ and $|V_{\tau 4}|^{2}$) from peak searches,
accelerator experiments, reactor experiments and others - collectively called
laboratory constraints. In the absence of detection of $LV$, the laboratory
constraints and the ones from $\Delta L=2$ processes can be compared. The
constraints on mixing from $LV$ tau decays are always competitive with or
better than laboratory constraints in the corresponding mass region while the
constraints from $LV$ meson decays are competitive with laboratory constraints
only in some mass regions. We note that we explore more combinations of mixing
elements and also provide better constraints on mixing in some mass regions.
More importantly, a detection in one of the experiments analyzed to obtain
laboratory constraints implies the existence of a sterile neutrino while a
detection in one of the $\Delta L=2$ tau or meson decay modes studied in our
analysis would imply $LV$ and hence the existence of a Majorana neutrino. We
pointed out the fact, often ignored in the literature when analyzing low-
energy processes, that a heavy neutrino might decay outside the detector if it
becomes long-lived for low mass (less than a GeV) and/or very small mixing.
For the collider signals of heavy Majorana neutrinos we looked for the
definitive lepton-number violating like-sign dilepton production and no
missing energy. Such signals have low backgrounds and have the potential for
discovery of heavy Majorana neutrinos. At the Tevatron, with the current and
future integrated luminosities, we find that the mass of the heavy Majorana
neutrinos can be probed up to
$m_{4}\sim\left\\{\begin{array}[]{c}10-130\ {\rm GeV}\ (10-75\ {\rm
GeV})\quad{\rm for}\ 2\sigma\ (5\sigma)\ {\rm with\ 2\ fb}^{-1};\\\ 10-180\
{\rm GeV}\ (10-120\ {\rm GeV})\quad{\rm for}\ 2\sigma\ (5\sigma)\ {\rm with\
8\ fb}^{-1}.\\\ \end{array}\right.$
Alternatively, the sensitivity for the mixing parameter can be
$S_{\mu\mu}\sim|V_{\mu 4}|^{2}\sim\left\\{\begin{array}[]{c}2\times 10^{-5}\
(10^{-4})\qquad{\rm for}\ 2\sigma\ (5\sigma)\ {\rm with\ 2\ fb}^{-1};\\\
5\times 10^{-6}\ (2\times 10^{-5})\quad{\rm for}\ 2\sigma\ (5\sigma)\ {\rm
with\ 8\ fb}^{-1}.\\\ \end{array}\right.$
This will surpass the DELPHI [89] and L3 [90, 110] $95\%$ C.L. bounds.
The sensitivity for heavy Majorana neutrinos can be extended significantly at
the LHC. With 100 fb-1 of integrated luminosity,
$\displaystyle m_{4}\sim 375\ (250)\
\rm{GeV}\,\,\,\,\,\,\rm{for}\,\,\,2\sigma\,\,(5\sigma);$ (95)
or alternatively, the mixing parameter can be probed to
$\displaystyle S_{\mu\mu}\sim|V_{\mu 4}|^{2}\sim 7\times 10^{-7}\ \ (3\times
10^{-6})\,\,\,\,\,\rm{for}\,\,\,2\sigma\ (5\sigma).$ (96)
The sensitivity at LHC will go well beyond the DELPHI and L3 $95\%$ C.L.
bounds in both mass reach and mixing, and beyond the current bound on
$|V_{e4}|^{2}$ from $0\nu\beta\beta$. In summary, there is a rich avenue of
possibilities for discovering or constraining the elusive Majorana neutrinos.
###### Acknowledgments.
We would like to thank Pavel Fileviez Perez for comments on the draft. This
research was supported in part by the U.S. DOE under Grants No.DE-
FG02-95ER40896, W-31-109-Eng-38, and in part by the Wisconsin Alumni Research
Foundation. Fermilab is operated by Fermi Research Alliance, LLC under
Contract No. DE-AC02-07CH11359 with the United States Department of Energy.
The work at KITP was supported in part by the National Science Foundation
under Grant No. PHY05-51164. The work of B. Z. is supported by the National
Science Foundation of China under Grant No. 10705017. SP would like to thank
the Theoretical Physics Department at Fermilab and the PH-TH Unit at CERN for
hospitality.
## Appendix A Lepton mixing formalism
In this appendix, we illustrate the parameterization for the lepton sector,
although we have not followed the relations literally, assuming that new
physics beyond this minimal formalism exists. The leptonic content in the
theory includes three generations of left-handed SM $SU(2)_{L}$ doublets and
$n$ right-handed SM singlets:
$L_{aL}=\left(\begin{array}[]{c}\nu_{a}\\\ l_{a}\end{array}\right)_{L},\quad
N_{bR},$ (97)
where $a=1,2,3$ and $b=1,2,3,\cdots,n$ ($n\geq 2$ for at least two massive
neutrinos).
The leptonically universal gauge interactions involving neutrinos are of the
form
$\displaystyle-{\cal
L}=\left(\frac{g}{\sqrt{2}}W^{+}_{\mu}\sum_{a=1}^{3}\overline{{\nu_{a}}_{L}}\
\gamma^{\mu}l_{aL}+\mathrm{h.c.}\right)+\frac{g}{2\cos_{W}}Z_{\mu}\sum_{a=1}^{3}\overline{{\nu_{a}}_{L}}\
\gamma^{\mu}{\nu_{a}}_{L}.$ (98)
The gauge-invariant Yukawa interactions are
$\displaystyle-{\cal L}_{Y}=\left(\sum_{a,b=1}^{3}f^{l}_{ab}\
\overline{L_{aL}}\ H{l_{bR}}+\sum_{a=1}^{3}\sum_{b=1}^{n}\ f^{\nu}_{ab}\
\overline{L_{aL}}\ \hat{H}N_{bR}\right)+\mathrm{h.c.}$ (99)
where $H$ is the SM Higgs doublet and $\hat{H}=i\tau_{2}H^{*}$. After the
Higgs field develops a vev $\langle H\rangle\to v/\sqrt{2}$, the Yukawa
interactions lead to Dirac masses for the leptons
$-{\cal L}_{m}^{D}=\left(\sum_{a,b=1}^{3}\overline{l_{aL}}\ m^{l}_{ab}\
{l_{bR}}+\sum_{a=1}^{3}\sum_{b=1}^{n}\ \overline{\nu_{aL}}\ m^{\nu}_{ab}\
N_{bR}\right)+\mathrm{h.c.}$ (100)
where the mass matrices are given by the vev times the corresponding Yukawa
couplings $m^{l,\nu}_{ab}=f^{l,\nu}_{ab}v/\sqrt{2}$.
The $3\times 3$ mass matrix $m^{l}$ can be diagonalized by two unitary
rotations among the gauge interaction eigenstates $l_{L},\ l_{R}$
$\displaystyle O_{L}^{\dagger}\ m^{l}\ O_{R}={\rm
diag}(m_{e},m_{\mu},m_{\tau}),\quad l_{a}=O_{a\ell}\ \ell,$ (101)
where $\ell=e,\mu,\tau$ are the mass eigenstates, which define the charged
lepton flavors. The Dirac masses as well as interactions with the Higgs boson
for the charged leptons now have the standard form
$-{\cal L}_{Y}^{\ell}=\sum_{\ell=e}^{\tau}m_{\ell}\ (1+{H\over v})\
\overline{\ell}\ {\ell}.$ (102)
If the Yukawa interactions of Eq. (99) are the whole source for neutrino mass,
then we would have min($n,3$) massive Dirac neutrinos.
To complete the neutrino mass sector, there is also a possible heavy Majorana
mass term
$-{\cal L}_{m}^{M}=\frac{1}{2}\sum_{b,b^{\prime}=1}^{n}\overline{N^{c}_{bL}}\
B_{bb^{\prime}}\ N_{b^{\prime}R}+\mathrm{h.c.}$ (103)
where a charge conjugate state is defined as $\psi^{c}=C\bar{\psi}^{T}$
($\overline{\psi^{c}}={\psi}^{T}C$), and a chiral state satisfies
$(\psi^{c})_{\tau}=(\psi_{-\tau})^{c},$ with $\tau=-,+$ for $L,R$. The full
neutrino mass terms thus read
$\displaystyle-{\cal L}_{m}^{\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\ \sum_{a=1}^{3}\sum_{b=1}^{n}\
(\overline{\nu_{aL}}\ m^{\nu}_{ab}\ N_{bR}+\overline{N^{c}_{bL}}\
m^{\nu}_{ba}\ \nu^{c}_{aR})+\sum_{b,b^{\prime}=1}^{n}\ \overline{N^{c}_{bL}}\
B_{bb^{\prime}}\ N_{b^{\prime}R}\right)+\mathrm{h.c.}$ (108) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\overline{\nu_{L}}\ \
\overline{N^{c}_{L}}\right)\left(\begin{array}[]{cc}0_{3\times
3}&m^{\nu}_{3\times n}\\\ m^{\nu T}_{n\times 3}&B_{n\times
n}\end{array}\right)\left(\begin{array}[]{c}\nu^{c}_{R}\\\
N_{R}\end{array}\right)+\mathrm{h.c.}$
where we have used the identity $\overline{\nu_{aL}}\ m_{ab}\
N_{bR}=\overline{N^{c}_{bL}}\ m_{ba}\ \nu^{c}_{aR}$,
The mass matrix can be diagonalized by one unitary transformation
${\mathbb{L}}^{\dagger}\left(\begin{array}[]{cc}0&m^{\nu}\\\ m^{\nu
T}&B\end{array}\right){\mathbb{L}}^{*}=\left(\begin{array}[]{cc}m^{\nu}_{diag}&0\\\
0&M^{N}_{diag}\end{array}\right)$ (109)
where the mass eigenvalues are of the order
$\displaystyle m^{\nu}_{diag}\approx{m_{\nu}^{2}\over B},\quad
M^{N}_{diag}\approx B.$ (110)
${\mathbb{L}}$ is a $(3+n)\times(3+n)$ unitary matrix and can be parameterized
as
${\mathbb{L}}=\left(\begin{array}[]{cc}U_{3\times 3}&V_{3\times n}\\\
X_{n\times 3}&Y_{n\times n}\end{array}\right).$ (111)
The relation between the gauge interaction eigenstates and the mass
eigenstates are given by
$\left(\begin{array}[]{c}\nu_{L}\\\
N^{c}_{L}\end{array}\right)={\mathbb{L}}\left(\begin{array}[]{c}\nu_{L}\\\
N^{c}_{L}\end{array}\right)_{m},$ (112)
with the mass eigenstates $\nu_{m}\ (m=1,2,3),\ N_{m^{\prime}}\
(m^{\prime}=4,\cdots,3+n).$ The diagonalized (Majorana) mass terms of Eq.
(108) thus read
$\displaystyle-{\cal L}_{m}^{\nu}={1\over 2}\left(\sum_{m=1}^{3}m^{\nu}_{m}\
\overline{\nu_{mL}}\ \nu^{c}_{mR}+\sum_{m^{\prime}=4}^{3+n}M^{N}_{m^{\prime}}\
\overline{N^{c}_{m^{\prime}L}}\ N_{m^{\prime}R}\right)+\mathrm{h.c.}\ ,$ (113)
with the mixing relations between the gauge and mass eigenstates
$\displaystyle\nu_{aL}$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{3}U_{am}\nu_{mL}+\sum_{m^{\prime}=4}^{3+n}V_{am^{\prime}}N^{c}_{m^{\prime}L},\
\
N^{c}_{bL}=\sum_{m=1}^{3}X_{bm}\nu_{mL}+\sum_{m^{\prime}=4}^{3+n}Y_{bm^{\prime}}N^{c}_{m^{\prime}L},$
(114) $\displaystyle\nu^{c}_{aR}$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{3}U^{*}_{am}\nu^{c}_{mR}+\sum_{m^{\prime}=4}^{3+n}V^{*}_{am^{\prime}}N_{m^{\prime}R},\
\
N_{bR}=\sum_{m=1}^{3}X^{*}_{bm}\nu_{mR}^{c}+\sum_{m^{\prime}=4}^{3+n}Y^{*}_{bm^{\prime}}N_{m^{\prime}R}.$
(115)
Note that the unitarity condition for $\mathbb{L}$ leads to the relations
$\displaystyle UU^{\dagger}+VV^{\dagger}=U^{\dagger}U+X^{\dagger}X=I_{3\times
3},$ (116) $\displaystyle
XX^{\dagger}+YY^{\dagger}=V^{\dagger}V+Y^{\dagger}Y=I_{n\times n}.$ (117)
Parametrically, $UU^{\dagger}$ and $\ Y^{\dagger}Y\sim{\cal{O}}(1),$
$VV^{\dagger}$ and $X^{\dagger}X\sim{\cal{O}}(m_{\nu}/M_{N}).$
In terms of the mass eigenstates, the gauge interaction lagrangian Eq. (98)
can be written as
$\displaystyle-{\cal L}$ $\displaystyle=$
$\displaystyle\frac{g}{\sqrt{2}}W^{+}_{\mu}\left(\sum_{\ell=e}^{\tau}\sum_{m=1}^{3}(U^{\dagger}O_{L})_{m\ell}\
\overline{\nu_{m}}\gamma^{\mu}P_{L}\ell+\sum_{\ell=e}^{\tau}\sum_{m^{\prime}=4}^{3+n}(V^{\dagger}O_{L})_{m^{\prime}\ell}\
\overline{N^{c}_{m^{\prime}}}\gamma^{\mu}P_{L}\ell\right)+\mathrm{h.c.}$ (118)
$\displaystyle+$
$\displaystyle\frac{g}{2\cos_{W}}Z_{\mu}\left(\sum_{m_{1},m_{2}=1}^{3}(U^{\dagger}U)_{m_{1}m_{2}}\
\overline{\nu_{m_{1}}}\gamma^{\mu}P_{L}\nu_{m_{2}}+\sum_{m_{1}^{\prime},m_{2}^{\prime}=4}^{3+n}(V^{\dagger}V)_{m_{1}^{\prime}m_{2}^{\prime}}\overline{N^{c}_{m_{1}^{\prime}}}\gamma^{\mu}P_{L}N^{c}_{m_{2}^{\prime}}\right)$
$\displaystyle+$
$\displaystyle\frac{g}{2\cos_{W}}Z_{\mu}\left(\sum_{m_{1}=1}^{3}\sum_{m_{2}^{\prime}=4}^{3+n}(U^{\dagger}V)_{m_{1},m_{2}^{\prime}}\overline{\nu_{m_{1}}}\gamma^{\mu}P_{L}N^{c}_{m_{2}^{\prime}}+\mathrm{h.c.}\right).$
To make the couplings more intuitive, we define the combination matrices by
$\displaystyle U^{l\nu}=O_{L}^{\dagger}U,\quad V^{lN}=O_{L}^{\dagger}V,\quad
U^{\nu N}=U^{\dagger}V,\quad U^{\nu\nu}=U^{\dagger}U,\quad
V^{NN}=V^{\dagger}V.$ (119)
We thus rewrite the gauge interaction lagrangian by one mixing matrix for each
term
$\displaystyle-{\cal L}$ $\displaystyle=$
$\displaystyle\frac{g}{\sqrt{2}}W^{+}_{\mu}\left(\sum_{\ell=e}^{\tau}\sum_{m=1}^{3}U^{l\nu*}_{\ell
m}\
\overline{\nu_{m}}\gamma^{\mu}P_{L}\ell+\sum_{\ell=e}^{\tau}\sum_{m^{\prime}=4}^{3+n}V^{lN*}_{\ell
m^{\prime}}\
\overline{N^{c}_{m^{\prime}}}\gamma^{\mu}P_{L}\ell\right)+\mathrm{h.c.}$ (120)
$\displaystyle+$
$\displaystyle\frac{g}{2\cos_{W}}Z_{\mu}\left(\sum_{m_{1},m_{2}=1}^{3}U^{\nu\nu}_{m_{1}m_{2}}\
\overline{\nu_{m_{1}}}\gamma^{\mu}P_{L}\nu_{m_{2}}+\sum_{m_{1}^{\prime},m_{2}^{\prime}=4}^{3+n}V^{NN}_{m_{1}^{\prime}m_{2}^{\prime}}\
\overline{N_{m_{1}^{\prime}}}\gamma^{\mu}P_{L}N_{m_{2}^{\prime}}\right)$
$\displaystyle+$
$\displaystyle\frac{g}{2\cos_{W}}Z_{\mu}\left(\sum_{m_{1}=1}^{3}\sum_{m_{2}^{\prime}=4}^{3+n}U^{\nu
N}_{m_{1}m_{2}^{\prime}}\
\overline{\nu_{m_{1}}}\gamma^{\mu}P_{L}N^{c}_{m_{2}^{\prime}}+\mathrm{h.c.}\right).$
These couplings along with the mixing matrices Eq. (119) give the most general
leptonic interactions of the charged and neutral currents in terms of the mass
eigenstates. Alternatively, the neutral current interactions can be aligned
along with that of the charged currents when rotating left-handed neutrinos in
the same way as the charged leptons,
$\displaystyle\nu_{aL}=(O_{L})_{a\ell}\ \nu_{\ell L},\ {\rm or}\ \nu_{\ell L}$
$\displaystyle=$ $\displaystyle\sum_{m=1}^{3}(O_{L}^{\dagger}U)_{\ell
m}\nu_{mL}+\sum_{m^{\prime}=4}^{3+n}(O_{L}^{\dagger}V)_{\ell
m^{\prime}}N^{c}_{m^{\prime}L}.$ (121)
It may be convenient in certain practical calculations to rewrite the neutral
current interactions in terms of their flavor eigenstates
$\displaystyle-{\cal L}$ $\displaystyle=$
$\displaystyle\frac{g}{\sqrt{2}}W^{+}_{\mu}\left(\sum_{\ell=e}^{\tau}\sum_{m=1}^{3}U^{*}_{\ell
m}\
\overline{\nu_{m}}\gamma^{\mu}P_{L}\ell+\sum_{\ell=e}^{\tau}\sum_{m^{\prime}=4}^{3+n}V^{*}_{\ell
m^{\prime}}\
\overline{N^{c}_{m^{\prime}}}\gamma^{\mu}P_{L}\ell\right)+\mathrm{h.c.}$ (122)
$\displaystyle+$
$\displaystyle\frac{g}{2\cos\theta_{W}}Z_{\mu}\left(\sum_{\ell=e}^{\tau}\sum_{m=1}^{3}U^{*}_{\ell
m}\ \overline{\nu_{m}}\gamma^{\mu}P_{L}\
\nu_{\ell}+\sum_{\ell=e}^{\tau}\sum_{m^{\prime}=4}^{3+n}V^{*}_{\ell
m^{\prime}}\ \overline{N^{c}_{m^{\prime}}}\gamma^{\mu}P_{L}\
\nu_{\ell}\right)+\mathrm{h.c.}+...\qquad$
where we have dropped the superscripts for $U,\ V$ defined in Eq. (119), for
simplicity as adopted throughout the text.
Figure 25: Feynman rules for the charged current vertices in terms of the
neutrino mass eigenstates, as given in Eq. (A.22).
For the reader’s convenience, we give most of the corresponding Feynman rules
for the interaction vertices, listed in Fig. 25 for the charged currents, and
in Fig. 26 for the neutral currents. The Feynman rules for the other diagrams
can be easily deduced from the ones that are explicitly written down in Fig.
25 and Fig. 26.
Figure 26: Feynman rules for the neutral current vertices in terms of the
neutrino mass eigenstates, as given in Eqs. (A.20).
Finally, the heavy neutrino interactions with the Higgs boson read
$\displaystyle-{\cal
L_{H}}=\frac{H}{v}\sum_{\ell=e}^{\tau}\sum_{m^{\prime}=4}^{3+n}V^{*}_{\ell
m^{\prime}}\ M^{N}_{m^{\prime}}\ \overline{N^{c}_{m^{\prime}}}P_{L}\
\nu_{\ell}+\mathrm{h.c.}+...$ (123)
The corresponding Feynman rule for the interaction vertex is given in Fig. 27.
Figure 27: Feynman rule for the Higgs vertex in terms of the heavy neutrino
mass eigenstates, as given in Eq. (A.23).
## Appendix B General amplitude of $\Delta L=2$ processes
The charged current interaction lagrangian in terms of neutrino mass
eigenstates is
${\cal
L}_{cc}=-\frac{g}{\sqrt{2}}W^{+}_{\mu}\Bigl{(}\sum_{\ell=e}^{\tau}\sum_{m=1}^{3}U^{l\nu*}_{\ell
m}\
\overline{\nu_{m}}\gamma^{\mu}P_{L}\ell+\sum_{\ell=e}^{\tau}\sum_{m^{\prime}=4}^{3+n}V^{lN*}_{\ell
m^{\prime}}\
\overline{N^{c}_{m^{\prime}}}\gamma^{\mu}P_{L}\ell\Bigr{)}+\mathrm{h.c.}$
(124)
where $P_{L}=\frac{1}{2}(1-\gamma_{5})$. The leptonic $\Delta L=2$ subprocess
$W^{-}W^{-}\rightarrow\ell_{1}^{-}\ell_{2}^{-}$ is induced by the product of
two charged currents
${{\cal
M}_{lep}^{\mu\nu}}\propto\sum_{m=1}^{3}U^{l\nu}_{\ell_{1}m}U^{l\nu}_{\ell_{2}m}\
(\overline{\ell_{1}}\gamma^{\mu}P_{L}\nu_{m})(\overline{\ell_{2}}\gamma^{\nu}P_{L}\nu_{m})+\sum_{m^{\prime}=4}^{3+n}V^{lN}_{\ell_{1}m^{\prime}}V^{lN}_{\ell_{2}m^{\prime}}\
(\overline{\ell_{1}}\gamma^{\mu}P_{L}N_{m^{\prime}})(\overline{\ell_{2}}\gamma^{\nu}P_{L}N_{m^{\prime}}),$
(125)
which can be rewritten using charge conjugation as
${{\cal
M}_{lep}^{\mu\nu}}\propto{\sum_{m=1}^{3}}U^{l\nu}_{\ell_{1}m}U^{l\nu}_{\ell_{2}m}\
(\overline{\ell_{1}}\gamma^{\mu}P_{L}\nu_{m})(\overline{\nu_{m}}\gamma^{\nu}P_{R}\ell^{c}_{2})+\sum_{m^{\prime}=4}^{3+n}V^{lN}_{\ell_{1}m^{\prime}}V^{lN}_{\ell_{2}m^{\prime}}\
(\overline{\ell_{1}}\gamma^{\mu}P_{L}N_{m^{\prime}})(\overline{N_{m^{\prime}}}\gamma^{\nu}P_{R}\ell^{c}_{2}).$
(126)
The Majorana neutrino fields can be contracted to form a neutrino propagator,
and the transition matrix element is thus given by
$\displaystyle{{\cal M}_{lep}^{\mu\nu}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2}{\sum_{m=1}^{3}}U^{l\nu}_{\ell_{1}m}U^{l\nu}_{\ell_{2}m}\
({\overline{\ell_{1}}}\gamma^{\mu}P_{L})\frac{\not{\hbox{\kern-4.0pt$q$}}+m_{\nu_{m}}}{q^{2}-m_{\nu_{m}}^{2}+i\Gamma_{\nu_{m}}m_{\nu_{m}}}(\gamma^{\nu}P_{R}{\ell^{c}_{2}})$
(127) $\displaystyle+$
$\displaystyle\frac{g^{2}}{2}{\sum_{m^{\prime}=4}^{3+n}}V^{lN}_{\ell_{1}m^{\prime}}V^{lN}_{\ell_{2}m^{\prime}}\
({\overline{\ell_{1}}}\gamma^{\mu}P_{L})\frac{\not{\hbox{\kern-4.0pt$q$}}+m_{N_{m^{\prime}}}}{q^{2}-m_{N_{m^{\prime}}}^{2}+i\Gamma_{N_{m^{\prime}}}m_{N_{m^{\prime}}}}(\gamma^{\nu}P_{R}{\ell^{c}_{2}}),$
where $q$ is the momentum exchange carried by the neutrino. The
$\not{\hbox{\kern-4.0pt$q$}}$ term vanishes due to the chirality flip.
Including the crossed diagram ($\ell_{1}\leftrightarrow\ell_{2}$) the leptonic
amplitude then becomes
$\displaystyle{{\cal M}_{lep}^{\mu\nu}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2}{\sum_{m=1}^{3}}U^{l\nu}_{\ell_{1}m}U^{l\nu}_{\ell_{2}m}\
{m_{\nu_{m}}}{\overline{u_{1}}}\Biggl{(}\frac{\gamma^{\mu}\gamma^{\nu}}{q^{2}-m^{2}_{\nu_{m}}+i\Gamma_{\nu_{m}}m_{\nu_{m}}}+\frac{\gamma^{\nu}\gamma^{\mu}}{q^{\prime
2}-m^{2}_{\nu_{m}}+i\Gamma_{\nu_{m}}m_{\nu_{m}}}\Biggr{)}P_{R}v_{2}$ (128)
$\displaystyle+$
$\displaystyle\frac{g^{2}}{2}{\sum_{m^{\prime}=4}^{3+n}}V^{lN}_{\ell_{1}m^{\prime}}V^{lN}_{\ell_{2}m^{\prime}}\
{m_{N_{m^{\prime}}}}\times$
$\displaystyle{\overline{u_{1}}}\Biggl{(}\frac{\gamma^{\mu}\gamma^{\nu}}{q^{2}-m^{2}_{N_{m^{\prime}}}+i\Gamma_{N_{m^{\prime}}}m_{N_{m^{\prime}}}}+\frac{\gamma^{\nu}\gamma^{\mu}}{q^{\prime
2}-m^{2}_{N_{m^{\prime}}}+i\Gamma_{N_{m^{\prime}}}m_{N_{m^{\prime}}}}\Biggr{)}P_{R}v_{2}.$
For the light Majorana neutrinos, namely, $m=1,2,3$ the masses
$m_{\nu_{m}}\sim\cal{O}(\mbox{eV})$ [111] and for the heavy Majorana neutrinos
, the masses $m_{N_{m^{\prime}}}\sim\cal{O}(\rm MeV-\rm GeV)$ for the low
energy processes we consider. The heavy Majorana neutrino contribution has a
resonant enhancement when $q^{2},q^{\prime 2}\approx m^{2}_{N_{m^{\prime}}}$
and is the dominant one. The light Majorana neutrino contribution however
encounters a severe suppression due to the small neutrino mass like
${m^{2}_{\nu_{m}}}/{M^{2}_{W}}$. Hence we can neglect the contributions of the
light Majorana neutrinos and the $\sum_{m=1}^{3}$ part of the amplitude drops
out.
In principle all the heavy Majorana neutrinos will contribute to the amplitude
but in our analysis we only consider the contribution of one of the heavy
neutrinos, in particular the lightest one for simplicity. So the amplitude can
now be written as
${{\cal M}_{lep}^{\mu\nu}}=\frac{g^{2}}{2}V_{\ell_{1}4}V_{\ell_{2}4}\
{m_{4}}{\overline{u_{1}}}\Biggl{(}\frac{\gamma^{\mu}\gamma^{\nu}}{q^{2}-m_{4}^{2}+i\Gamma_{N_{4}}m_{4}}+\frac{\gamma^{\nu}\gamma^{\mu}}{q^{\prime
2}-m_{4}^{2}+i\Gamma_{N_{4}}m_{4}}\Biggr{)}P_{R}v_{2}.$ (129)
We can rewrite the above amplitude as
$\displaystyle{{\cal M}_{lep}^{\mu\nu}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2}V_{\ell_{1}4}V_{\ell_{2}4}\
{m_{4}}\frac{\overline{u_{1}}\gamma^{\mu}\gamma^{\nu}P_{R}v_{2}}{q^{2}-m_{4}^{2}+i\Gamma_{N_{4}}m_{4}}+\frac{g^{2}}{2}V_{\ell_{1}4}V_{\ell_{2}4}\
{m_{4}}\frac{\overline{u_{1}}\gamma^{\nu}\gamma^{\mu}P_{R}v_{2}}{q^{\prime
2}-m_{4}^{2}+i\Gamma_{N_{4}}m_{4}}$ (130) $\displaystyle=$ $\displaystyle{\cal
M}_{1}+{\cal M}_{2}.$
When $q^{2}\approx m^{2}_{4}$, ${\cal M}_{1}$ has a resonant contribution and
when $q^{\prime 2}\approx m^{2}_{4}$, ${\cal M}_{2}$ has a resonant
contribution. In general, $q\neq q^{\prime}$, and it is convenient to split up
the individual resonant contributions by the Single-Diagram-Enhanced multi-
channel integration method [112]. To do this, define the functions
$f_{i}=\frac{|{\cal M}_{i}|^{2}}{\sum_{i}|{\cal
M}_{i}|^{2}}\Bigg{|}\sum_{i}{\cal M}_{i}\Bigg{|}^{2}$ (131)
Then the amplitude squared is given by
$\Bigg{|}\sum_{i}{\cal M}_{i}\Bigg{|}^{2}=\sum_{i}f_{i}$ (132)
The amplitude squared splits up into the functions $f_{i}$ defined above and
the phase space integration can be done for each $f_{i}$ separately. This
helps to make convenient simplifications for the phase space integration and
the computation can be carried out in parallel. The contributions from each
$f_{i}$ can be added up after phase space integration.
## Appendix C Decay modes of heavy Majorana neutrino
In this section we will discuss in detail the decay modes of the heavy
Majorana neutrino $N_{4}$, with mass $m_{4}$ much smaller than the mass of the
W boson, $m_{W}$. From EW precision measurements the mixing elements $|V_{\ell
4}|^{2}\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}{\cal O}(10^{-3})$ and the higher order terms in
mixing would be very small and can be ignored. Hence the widths are presented
only up to leading terms in mixing. The charged current and neutral current
vertices of $N_{4}$ with the mixing elements are given in Fig. 25 and Fig. 26.
With increasing mass of the heavy neutrino new decay channels open up and can
be classified into two body and three body decays. The decay width scales as
the third and the fifth power of the mass($m_{4}$) for two and three body
decays respectively.
1) $N_{4}\rightarrow\ell^{-}P^{+}$ where $\ell=e,\mu,\tau$ and $P^{+}$ is a
charged pseudoscalar meson. This decay mode has charged current interactions
only as shown in Fig. 25 and the decay width is given by
$\displaystyle\Gamma^{\ell P}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\ell^{-}P^{+})=\frac{G^{2}_{F}}{16\pi}f^{2}_{P}\
|V_{q\bar{q}^{\prime}}|^{2}\ |V_{\ell 4}|^{2}\ m^{3}_{4}\mbox{
}I_{1}(\mu_{\ell},\mu_{P}),$ $\displaystyle I_{1}(x,y)$ $\displaystyle=$
$\displaystyle[(1+x-y)(1+x)-4x]\lambda^{\frac{1}{2}}(1,x,y),$
$\displaystyle\lambda(a,b,c)$ $\displaystyle=$ $\displaystyle
a^{2}+b^{2}+c^{2}-2ab-2bc-2ca,$ (133)
where $f_{P}$ is the meson decay constant and $V_{q\bar{q}^{\prime}}$ are the
CKM matrix elements. $\mu_{\ell}$ and $\mu_{P}$ are the masses scaled by the
mass of the heavy neutrino and are given by $\mu_{i}=m^{2}_{i}/m^{2}_{4}$.
2) $N_{4}\rightarrow\nu_{\ell}P^{0}$ where
$\nu_{\ell}=\nu_{e},\nu_{\mu},\nu_{\tau}$ and $P^{0}$ is a neutral
pseudoscalar meson. This decay mode has neutral current interactions only as
shown in Fig. 26 and the decay width is given by
$\Gamma^{\nu_{\ell}P}\equiv\Gamma(N_{4}\rightarrow\nu_{\ell}P^{0})=\frac{G^{2}_{F}}{64\pi}f^{2}_{P}\
|V_{\ell 4}|^{2}\ m^{3}_{4}\ (1-\mu_{P})^{2},$ (134)
where $f_{P}$ is the meson decay constant, $\mu_{P}$ is the mass of the
neutral meson scaled by the mass of the heavy neutrino and is given by
$\mu_{P}=m^{2}_{P}/m^{2}_{4}$. The mass of the light neutrino $\sim{\cal
O}(\rm eV)$ [18] is much smaller than the mass of $N_{4}\sim{\cal O}(\rm
MeV-\rm GeV)$ and can be neglected to a very good approximation. We have set
the mass of the light neutrino to zero in the expression for the width above
and henceforth.
3) $N_{4}\rightarrow\ell^{-}V^{+}$ where $\ell=e,\mu,\tau$ and $V^{+}$ is a
charged vector meson. This decay mode has charged current interactions only as
shown in Fig. 25 and the decay width is given by
$\displaystyle\Gamma^{\ell V}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\ell^{-}V^{+})=\frac{G^{2}_{F}}{16\pi}f^{2}_{V}\
|V_{q\bar{q}^{\prime}}|^{2}\ |V_{\ell 4}|^{2}\ m^{3}_{4}\mbox{
}I_{2}(\mu_{\ell},\mu_{V}),$ $\displaystyle I_{2}(x,y)$ $\displaystyle=$
$\displaystyle[(1+x-y)(1+x+2y)-4x]\lambda^{\frac{1}{2}}(1,x,y),$ (135)
where $f_{V}$ is the vector meson decay constant and $V_{q\bar{q}^{\prime}}$
are the CKM matrix elements. $\mu_{\ell}$ and $\mu_{V}$ are the masses of the
lepton and the vector meson scaled by the mass of the heavy neutrino and are
given by $\mu_{i}=m^{2}_{i}/m^{2}_{4}$.
4) $N_{4}\rightarrow\nu_{\ell}V^{0}$ where
$\nu_{\ell}=\nu_{e},\nu_{\mu},\nu_{\tau}$ and $V^{0}$ is a neutral vector
meson. This decay mode has neutral current interactions only as shown in Fig.
26 and the decay width is given by
$\displaystyle\Gamma^{\nu_{\ell}V}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\nu_{\ell}V^{0})={\frac{G^{2}_{F}}{2\pi}}{\kappa^{2}_{V}}\
{f^{2}_{V}}\ {{|V_{\ell 4}|}^{2}}\ {m^{3}_{4}}\
I_{3}(\mu_{\nu_{\ell}},\mu_{V}),$ $\displaystyle I_{3}(x,y)$ $\displaystyle=$
$\displaystyle(1+2y)(1-y)\lambda^{\frac{1}{2}}(1,x,y),$ (136)
where $f_{V}$ is the meson decay constant, $\mu_{V}$ is the mass of the
neutral meson scaled by the mass of the heavy neutrino and is given by
$\mu_{V}=m^{2}_{V}/m^{2}_{4}$. $\kappa_{V}$ is the vector coupling associated
with the meson and is expressed in terms of $x_{w}=\sin^{2}\theta_{w}$, where
$\theta_{w}$ is the Weinberg angle. The values of $\kappa$ for the various
vector mesons are: $\kappa=\frac{1}{3}x_{w}$ for $\rho^{0}$ and $\omega$;
$\kappa=(-\frac{1}{4}+\frac{1}{3}x_{w})$ for $K^{*0},\overline{K}^{*0}$ and
$\phi$; and $\kappa=(\frac{1}{4}-\frac{2}{3}x_{w})$ for
$D^{*0},\overline{D}^{*0}$ and $J/\psi$.
5) $N_{4}\rightarrow\ell^{-}_{1}\ell^{+}_{2}\nu_{\ell_{2}}$ where
$\ell_{1},\ell_{2}=e,\mu,\tau$ with $\ell_{1}\neq\ell_{2}$. This decay mode
has charged current interactions only as shown in Fig. 25 and the decay width
is given by
$\displaystyle\Gamma^{\ell_{1}\ell_{2}\nu_{\ell_{2}}}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\ell^{-}_{1}\ell^{+}_{2}\nu_{\ell_{2}})=\frac{G^{2}_{F}}{192\pi^{3}}m^{5}_{4}\
{|V_{\ell_{1}4}|}^{2}\ I_{1}(x_{\ell_{1}},x_{\nu_{\ell_{2}}},x_{\ell_{2}}),$
$\displaystyle I_{1}(x,y,z)$ $\displaystyle=$ $\displaystyle
12\int\limits_{(x+y)^{2}}^{(1-z)^{2}}\frac{ds}{s}(s-x^{2}-y^{2})(1+z^{2}-s)\lambda^{\frac{1}{2}}(s,x^{2},y^{2})\lambda^{\frac{1}{2}}(1,s,z^{2}),$
(137)
where $I_{1}(0,0,0)=1$, $x_{i}$ are the masses scaled by the mass of the heavy
neutrino and are given by $x_{i}=m_{i}/m_{4}$. The mass of the light neutrino
$\sim{\cal O}(\rm eV)$ is much smaller than the mass of $N_{4}\sim{\cal O}(\rm
MeV-\rm GeV)$ and hence can be neglected compared to the mass of $N_{4}$. We
have set the mass of the light neutrino to zero with very good approximation
in the expression for the width above and henceforth.
6) $N_{4}\rightarrow\nu_{\ell_{1}}\ell^{-}_{2}\ell^{+}_{2}$ where
$\ell_{1},\ell_{2}=e,\mu,\tau$. Both charged current and neutral current
interactions as shown in Fig. 25 and Fig. 26 are relevant for this mode and
the decay width is given by
$\displaystyle\Gamma^{\nu_{\ell_{1}}\ell_{2}\ell_{2}}$ $\displaystyle\equiv$
$\displaystyle\Gamma(N_{4}\rightarrow\nu_{\ell_{1}}\ell^{-}_{2}\ell^{+}_{2})=\frac{G^{2}_{F}}{96\pi^{3}}{|V_{\ell_{1}4}|}^{2}\
m^{5}_{4}\times\biggl{[}\Bigl{(}g^{\ell}_{L}g^{\ell}_{R}+\delta_{\ell_{1}\ell_{2}}g^{\ell}_{R}\Bigr{)}I_{2}(x_{\nu_{\ell_{1}}},x_{\ell_{2}},x_{\ell_{2}})$
(138) $\displaystyle+$
$\displaystyle\Bigl{(}{(g^{\ell}_{L})}^{2}+{(g^{\ell}_{R})}^{2}+\delta_{\ell_{1}\ell_{2}}(1+2g^{\ell}_{L})\Bigr{)}I_{1}(x_{\nu_{\ell_{1}}},x_{\ell_{2}},x_{\ell_{2}})\biggr{]},$
$\displaystyle I_{2}(x,y,z)$ $\displaystyle=$ $\displaystyle
24yz\int\limits_{(y+z)^{2}}^{(1-x)^{2}}\frac{ds}{s}(1+x^{2}-s)\lambda^{\frac{1}{2}}(s,y^{2},z^{2})\lambda^{\frac{1}{2}}(1,s,x^{2}),$
(139)
where $I_{2}(0,0,0)=1$, $I_{1}(x,y,z)$ has been defined in Eq. (C), $x_{i}$
are the masses scaled by the mass of the heavy neutrino and are given by
$x_{i}=m_{i}/m_{4}$, $g^{\ell}_{L}=-\frac{1}{2}+x_{w}$, $g^{\ell}_{R}=x_{w}$
and $x_{w}=\sin^{2}\theta_{w}=0.231$, where $\theta_{w}$ is the Weinberg
angle.
7) $N_{4}\rightarrow\nu_{\ell_{1}}\nu\overline{\nu}$ where
$\nu_{\ell_{1}}=\nu_{e},\nu_{\mu},\nu_{\tau}$. This decay mode has neutral
current interactions only as shown in Fig. 26. Using the massless
approximation for the neutrinos as described above the decay width has a
simple form given by
$\Gamma^{\nu_{\ell_{1}}\nu\nu}\equiv\sum_{\ell_{2}=e}^{\tau}\Gamma(N_{4}\rightarrow\nu_{\ell_{1}}\nu_{\ell_{2}}\overline{\nu_{\ell_{2}}})=\frac{G^{2}_{F}}{96\pi^{3}}|V_{\ell_{1}4}|^{2}\
m^{5}_{4}.$ (140)
All the decay modes listed above contribute to the total decay width of the
heavy Majorana neutrino which is given by:
$\displaystyle\Gamma_{N_{4}}$ $\displaystyle=$
$\displaystyle\sum_{\ell,P}{\Gamma^{\nu_{\ell}P}}+\sum_{\ell,V}{\Gamma^{\nu_{\ell}V}}+\sum_{\ell,P}{2\Gamma^{\ell
P}}+\sum_{\ell,V}{2\Gamma^{\ell V}}$ (141) $\displaystyle+$
$\displaystyle\sum_{\ell_{1},\ell_{2}(\ell_{1}\neq\ell_{2})}{2\Gamma^{\ell_{1}\ell_{2}\nu_{\ell_{2}}}}+\sum_{\ell_{1},\ell_{2}}{\Gamma^{\nu_{\ell_{1}}\ell_{2}\ell_{2}}}+\sum_{\nu_{\ell_{1}}}{\Gamma^{\nu_{\ell_{1}}\nu\nu}},$
where $\ell,\ell_{1},\ell_{2}=e,\mu,\tau$. For a Majorana neutrino, the
$\Delta L=0$ process $N_{4}\rightarrow\ell^{-}P^{+}$ as well as its charge
conjugate $|\Delta L|=2$ process $N_{4}\rightarrow\ell^{+}P^{-}$ are possible
and have the same width, $\Gamma^{\ell P}$. Hence the factor of 2 associated
with the decay width of this mode in Eq. (141). Similarly, the $\Delta L=0$
and its charge conjugate $|\Delta L|=2$ process are possible for the decay
modes $N_{4}\rightarrow\ell^{-}V^{+}$ and
$N_{4}\rightarrow\ell^{-}_{1}\ell^{+}_{2}\nu_{\ell_{2}}$ and hence have a
factor of 2 associated with their width in Eq. (141).
As mentioned earlier, new channels open with increasing mass of the heavy
neutrino. For the low energy $LV$ tau decays and rare meson decays we
consider, the mass of the heavy neutrino is in the range $140\ \rm
MeV\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}m_{4}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}5278\ \rm MeV$.
For this mass range we list all the possible decay channels for $N_{4}$ in
Table 6. The mass and decay constants of pseudoscalar and vector mesons used
in the calculation of partial widths given in Eqs. (C -140) are listed in
Table 7 in Appendix E.
Table 6: Decay modes of heavy Majorana neutrino based on its mass $m_{4}$. Mass of heavy | Decay mode of | Mass of heavy | Decay mode of
---|---|---|---
neutrino ($\rm MeV$) | heavy neutrino | neutrino ($\rm MeV$) | heavy neutrino
$\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}\sum_{m}\nu_{m}=10^{-6}$ | $N_{4}\rightarrow\nu_{\ell_{1}}\nu_{\ell_{2}}\overline{\nu_{\ell_{2}}}$ | $>m_{\mu}+m_{\tau}=1880$ | $N_{4}\rightarrow\mu^{-}\tau^{+}\nu_{\tau}+c.c$
| | | $N_{4}\rightarrow\tau^{-}\mu^{+}\nu_{\mu}+c.c$
$>2m_{e}=1.02$ | $N_{4}\rightarrow\nu_{\ell}e^{-}e^{+}$ | $>m_{\tau}+m_{\pi}=1920$ | $N_{4}\rightarrow\tau^{-}\pi^{+}+c.c$
$>m_{e}+m_{\mu}=106$ | $N_{4}\rightarrow e^{-}\mu^{+}\nu_{m}+c.c$ | $>m_{e}+m_{D_{s}}=1970$ | $N_{4}\rightarrow e^{-}D^{+}_{s}+c.c$
| $N_{4}\rightarrow\mu^{-}e^{+}\nu_{e}+c.c$ | |
$>m_{\pi^{0}}=135$ | $N_{4}\rightarrow\nu_{\ell}\pi^{0}$ | $>m_{\mu}+m_{D}=1980$ | $N_{4}\rightarrow\mu^{-}D^{+}+c.c$
$>m_{e}+m_{\pi}=140$ | $N_{4}\rightarrow e^{-}\pi^{+}+c.c$ | $>m_{D^{*0}}=2010$ | $N_{4}\rightarrow\nu_{\ell}D^{*0}$
$>2m_{\mu}=211$ | $N_{4}\rightarrow\nu_{\ell}\mu^{-}\mu^{+}$ | $>m_{\overline{D}^{*0}}=2010$ | $N_{4}\rightarrow\nu_{\ell}\overline{D}^{*0}$
$>m_{\mu}+m_{\pi}=245$ | $N_{4}\rightarrow\mu^{-}\pi^{+}+c.c$ | $>m_{e}+m_{D^{*}}=2010$ | $N_{4}\rightarrow e^{-}D^{*^{+}}+c.c$
$>m_{e}+m_{K}=494$ | $N_{4}\rightarrow e^{-}K^{+}+c.c$ | $>m_{\mu}+m_{D_{s}}=2070$ | $N_{4}\rightarrow\mu^{-}D^{+}_{s}+c.c$
$>m_{\eta}=548$ | $N_{4}\rightarrow\nu_{\ell}\eta$ | $>m_{e}+m_{D^{*}_{s}}=2110$ | $N_{4}\rightarrow e^{-}D^{*+}_{s}+c.c$
$>m_{\mu}+m_{K}=599$ | $N_{4}\rightarrow\mu^{-}K^{+}+c.c$ | $>m_{\mu}+m_{D^{*}}=2120$ | $N_{4}\rightarrow\mu^{-}D^{*+}+c.c$
$>m_{\rho^{0}}=776$ | $N_{4}\rightarrow\nu_{\ell}\rho^{0}$ | $>m_{\mu}+m_{D^{*}_{s}}=2220$ | $N_{4}\rightarrow\mu^{-}D^{*+}_{s}+c.c$
$>m_{e}+m_{\rho}=776$ | $N_{4}\rightarrow e^{-}\rho^{+}+c.c$ | $>m_{\tau}+m_{K}=2270$ | $N_{4}\rightarrow\tau^{-}K^{+}+c.c$
$>m_{\omega}=783$ | $N_{4}\rightarrow\nu_{\ell}\omega$ | $>m_{\tau}+m_{\rho}=2550$ | $N_{4}\rightarrow\tau^{-}\rho^{+}+c.c$
$>m_{\mu}+m_{\rho}=882$ | $N_{4}\rightarrow\mu^{-}\rho^{+}+c.c$ | $>m_{\tau}+m_{K}^{*}=2670$ | $N_{4}\rightarrow\tau^{-}K^{*+}+c.c$
$>m_{e}+m_{K^{*}}=892$ | $N_{4}\rightarrow e^{-}K^{*+}+c.c$ | $>m_{\eta_{c}}=2980$ | $N_{4}\rightarrow\nu_{\ell}\eta_{c}$
$>m_{K^{*0}}=896$ | $N_{4}\rightarrow\nu_{\ell}K^{*0}$ | $>m_{J/\psi}=3100$ | $N_{4}\rightarrow\nu_{\ell}J/\psi$
$>m_{\overline{K}^{*0}}=896$ | $N_{4}\rightarrow\nu_{\ell}\overline{K}^{*0}$ | $>2m_{\tau}=3550$ | $N_{4}\rightarrow\nu_{\ell}\tau^{-}\tau^{+}$
$>m_{\eta^{\prime}}=958$ | $N_{4}\rightarrow\nu_{\ell}\eta^{\prime}$ | $>m_{\tau}+m_{D}=3650$ | $N_{4}\rightarrow\tau^{-}D^{+}+c.c$
$>m_{\mu}+m_{K^{*}}=997$ | $N_{4}\rightarrow\mu^{-}K^{*+}+c.c$ | $>m_{\tau}+m_{D_{s}}=3750$ | $N_{4}\rightarrow\tau^{-}D^{+}_{s}+c.c$
$>m_{\phi}=1019$ | $N_{4}\rightarrow\nu_{\ell}\phi$ | $>m_{\tau}+m_{D^{*}}=3790$ | $N_{4}\rightarrow\tau^{-}D^{*+}+c.c$
$>m_{e}+m_{\tau}=1780$ | $N_{4}\rightarrow e^{-}\tau^{+}\nu_{\tau}+c.c$ | $>m_{\tau}+m_{D^{*}_{s}}=3890$ | $N_{4}\rightarrow\tau^{-}D^{*+}_{s}+c.c$
| $N_{4}\rightarrow\tau^{-}e^{+}\nu_{e}+c.c$ | |
$>m_{e}+m_{D}=1870$ | $N_{4}\rightarrow e^{-}D^{+}+c.c$ | |
## Appendix D Lepton-number violating tau decay
The decay amplitude for lepton number violating tau decays can be separated
into leptonic and hadronic parts,
${i\cal M}={({\cal M}_{lep})_{\mu\nu}}{({\cal M}_{had})^{\mu\nu}}.$ (142)
For the tree level amplitude, the hadronic part can be expressed in terms of
the decay constants of the mesons in a model independent way. The box diagram
includes hadronic matrix elements which cannot be simplified in terms of decay
constants and needs to be evaluated in a model dependent way. We expect the
tree level amplitude to dominate and do not include the box diagram. It has
been argued that in certain cases for rare meson decays sub-leading
contributions may be appreciable [113, 20]. Even in such a scenario the
difference will not be important at the current level of sensitivities and we
include the more conservative limit from tree level diagrams only. The tau
decays and the rare meson decays are crossed versions of each other and the
above arguments are true for both.
The leptonic part of the subprocess $\tau^{-}\rightarrow\ell^{+}W^{-*}W^{-*}$
is obtained by crossing the amplitude in (129)
${{\cal M}_{lep}^{\mu\nu}}=\frac{g^{2}}{2}V^{*}_{\tau 4}V^{*}_{\ell 4}\
{\overline{v_{\tau}}}\frac{m_{4}}{q^{2}-m_{4}^{2}+i\Gamma_{N_{4}}m_{4}}\gamma^{\mu}\gamma^{\nu}P_{R}v_{\ell}.$
(143)
Combining the hadronic and leptonic parts, the decay amplitude for
$\tau^{-}(p_{1})\rightarrow\ell^{+}(p_{2})\ M_{1}^{-}(q_{1})\
M_{2}^{-}(q_{2})$ (144)
is given by
$\displaystyle{i\cal M}$ $\displaystyle=$ $\displaystyle{({\cal
M}_{lep})_{\mu\nu}}{{\cal M}_{M_{1}}^{\mu}}{{\cal
M}_{M_{2}}^{\nu}}+(M_{1}\leftrightarrow M_{2})$ $\displaystyle=$
$\displaystyle
2G_{F}^{2}V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}f_{M_{1}}f_{M_{2}}{V_{\tau
4}^{*}}{V^{*}_{\ell 4}}\
m_{4}\Biggl{[}\frac{\overline{v_{\tau}}\not{\hbox{\kern-4.0pt$q$}}_{1}\not{\hbox{\kern-4.0pt$q$}}_{2}P_{R}v_{\ell}}{(p_{1}-q_{1})^{2}-m_{4}^{2}+i\Gamma_{N_{4}}m_{4}}\Biggr{]}$
$\displaystyle+$ $\displaystyle
2G_{F}^{2}V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}f_{M_{1}}f_{M_{2}}{V_{\tau
4}^{*}}{V^{*}_{\ell 4}}\ m_{4}\Biggl{[}\frac{\overline{v_{\tau}}\not q_{2}\not
q_{1}P_{R}v_{\ell}}{(p_{1}-q_{2})^{2}-m_{4}^{2}+i\Gamma_{N_{4}}m_{4}}\Biggr{]},$
(146) $\displaystyle=$ $\displaystyle{\cal M}_{1}+{\cal M}_{2},$ (147)
where $V^{CKM}_{M_{i}}$ are the quark flavor-mixing matrix elements for the
mesons and $f_{M_{i}}$ are meson decay constants. Then the functions, $f_{1}$
and $f_{2}$ defined in Eq. (131) are given by
$\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle\Biggl{(}\frac{F_{\tau}A}{a^{2}_{1}+b^{2}}\Biggr{)}\Biggl{[}\frac{(a^{2}_{2}+b^{2})A+(a_{1}a_{2}+b^{2})C}{(a^{2}_{2}+b^{2})A+(a^{2}_{1}+b^{2})B}+(q_{1}\leftrightarrow
q_{2})\Biggr{]},$ (148) $\displaystyle f_{2}$ $\displaystyle=$ $\displaystyle
f_{1}(q_{1}\leftrightarrow q_{2}),$ (149) $\displaystyle A(p_{i},q_{j})$
$\displaystyle=$ $\displaystyle 8(p_{1}\cdot q_{1})(p_{2}\cdot
q_{2})(q_{1}\cdot q_{2})-4m^{2}_{M_{1}}(p_{1}\cdot q_{2})(p_{2}\cdot q_{2})$
(150) $\displaystyle-$ $\displaystyle 4m^{2}_{M_{2}}(p_{1}\cdot
q_{1})(p_{2}\cdot q_{1})+2m^{2}_{M_{1}}m^{2}_{M_{2}}(p_{1}\cdot p_{2}),$
$\displaystyle B(p_{i},q_{j})$ $\displaystyle=$ $\displaystyle
A(q_{1}\leftrightarrow q_{2}),$ (151) $\displaystyle C(p_{i},q_{j})$
$\displaystyle=$ $\displaystyle 4(p_{1}\cdot p_{2})(q_{1}\cdot
q_{2})^{2}-A(p_{i},q_{j}),$ (152) $\displaystyle D(p_{i},q_{j})$
$\displaystyle=$ $\displaystyle C(q_{1}\leftrightarrow q_{2}),$ (153)
$\displaystyle F_{\tau}$ $\displaystyle=$ $\displaystyle
4G^{4}_{F}f_{M_{1}}^{2}f_{M_{2}}^{2}|V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}|^{2}{|V_{\tau
4}V_{\ell 4}|}^{2}m^{2}_{4},$ (154) $\displaystyle a_{1,2}(p_{i},q_{j})$
$\displaystyle=$ $\displaystyle(p_{1}-q_{1,2})^{2}-m^{2}_{4};\ \ \
b=\Gamma_{N_{4}}m_{4}.$ (155)
The decay width for the $LV$ tau decay is then given by
$\displaystyle\Gamma^{\tau}_{LV}$ $\displaystyle=$ $\displaystyle(1-{1\over
2}\delta_{M_{1}M_{2}})\frac{1}{128\pi^{5}m_{\tau}}\Biggl{[}\int
f_{1}dPS_{31}+\int f_{2}dPS_{32}\Biggr{]},$ (156) $\displaystyle dPS_{31}$
$\displaystyle=$
$\displaystyle\frac{\pi^{2}}{4m^{2}_{\tau}}\lambda^{\frac{1}{2}}(m^{2}_{\tau},m^{2}_{M_{1}},m^{2}_{c1})\lambda^{\frac{1}{2}}(m^{2}_{c1},m^{2}_{\ell},m^{2}_{M_{2}})\frac{dm^{2}_{c1}}{m^{2}_{c1}}dy_{1}dy_{2}dy_{3}dy_{4},$
(157) $\displaystyle dPS_{32}$ $\displaystyle=$ $\displaystyle
dPS_{31}(q_{1}\leftrightarrow q_{2}),$ (158)
where $dPS_{31}$ and $dPS_{32}$ are the phase space factors obtained by
conveniently clustering two different sets of particles to enable applying the
narrow-width approximation easily. $y_{1}$ to $y_{4}$ are rescaled angular
variables with integration limits $0\leq y_{i}\leq 1$. As seen in Sec. 3.1.1,
the width of the heavy neutrino is very small compared to the mass and hence
we can apply the narrow-width approximation.
$\int\frac{dm^{2}_{c_{i}}}{(m^{2}_{c_{i}}-m^{2}_{4})^{2}+\Gamma^{2}_{N_{4}}m^{2}_{4}}\Bigg{|}_{{\Gamma_{N_{4}}\rightarrow\
0}}=\int\delta(m^{2}_{c_{i}}-m^{2}_{4})dm^{2}_{c_{i}}\frac{\pi}{\Gamma_{N_{4}}m_{4}}$
(159)
Applying the narrow-width approximation as described above and integrating
over the $\delta-$function we get
$\displaystyle\int f_{1}dPS_{31}$ $\displaystyle=$
$\displaystyle\int\Biggl{(}\frac{F_{\tau}A\pi^{3}}{4m^{2}_{\tau}m^{3}_{4}\Gamma_{N_{4}}}\Biggl{[}\frac{(a^{2}_{2}+b^{2})A+b^{2}(B+C+D)}{(a^{2}_{2}+b^{2})A+b^{2}B}\Biggr{]}$
(160)
$\displaystyle\lambda^{\frac{1}{2}}(m^{2}_{\tau},m^{2}_{M_{1}},m^{2}_{4})\lambda^{\frac{1}{2}}(m^{2}_{4},m^{2}_{\ell},m^{2}_{M_{2}})\Biggr{)}dy_{1}dy_{2}dy_{3}dy_{4},$
$\displaystyle\int f_{2}dPS_{32}$ $\displaystyle=$ $\displaystyle\int
f_{1}dPS_{31}(q_{1}\leftrightarrow q_{2})$ (161)
Now, we can find the decay rate from Eq. (156). Normalized to the $\tau$ decay
width $\Gamma_{\tau}=G_{F}^{2}m_{\tau}^{5}/192\pi^{3}$, the corresponding
branching fraction is
$\mathrm{Br}=\Gamma^{\tau}_{\\!\\!\\!\mbox{}_{LV}}/\Gamma_{\tau}$. The masses
and decay constants of mesons are listed in Table 7. The CKM matrix elements
and $\tau$ mass are taken from the Particle Data Group (PDG) [99]:
$m_{\tau}=1777\ {\rm MeV},\ |V_{ud}|=0.9738,\ |V_{us}|=0.2200.$
## Appendix E Rare meson decay
The rare meson decays
$M_{1}^{+}(q_{1})\rightarrow\ell^{+}(p_{1})\ \ell^{+}(p_{2})\
M_{2}^{-}(q_{2})$
have the same Feynman diagrams as tau decay. The meson $M_{2}$ can be a
pseudoscalar or vector meson. The decay amplitude when $M_{2}$ is a
pseudoscalar meson is given by
$\displaystyle i{\cal M}^{P}$ $\displaystyle=$ $\displaystyle
2G_{F}^{2}V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}f_{M_{1}}f_{M_{2}}{V_{\ell_{1}4}}{V_{\ell_{2}4}}\
m_{4}\Biggl{[}\frac{\overline{u_{\ell_{1}}}\not{\hbox{\kern-4.0pt$q$}}_{1}\not{\hbox{\kern-4.0pt$q$}}_{2}P_{R}v_{\ell_{2}}}{(q_{1}-p_{1})^{2}-{m_{4}}^{2}+i\Gamma_{N_{4}}m_{4}}\Biggr{]}$
(162) $\displaystyle+$ $\displaystyle
2G_{F}^{2}V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}f_{M_{1}}f_{M_{2}}{V_{\ell_{1}4}}{V_{\ell_{2}4}}\
m_{4}\Biggl{[}\frac{\overline{u_{\ell_{1}}}\not q_{2}\not
q_{1}P_{R}v_{\ell_{2}}}{(q_{1}-p_{2})^{2}-{m_{4}}^{2}+i\Gamma_{N_{4}}m_{4}}\Biggr{]}$
$\displaystyle=$ $\displaystyle{\cal M}^{P}_{1}+{\cal M}^{P}_{2}.$
Next we consider the case where $M_{2}$ is a vector meson. The decay amplitude
is given by
$\displaystyle i{\cal M}^{V}$ $\displaystyle=$ $\displaystyle
2G_{F}^{2}V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}f_{M_{1}}f_{M_{2}}{V_{\ell_{1}4}}{V_{\ell_{2}4}}\
m_{4}\
m_{M_{2}}\Biggl{[}\frac{\overline{u_{\ell_{1}}}\not{\hbox{\kern-4.0pt$q$}}_{1}\not\epsilon^{\lambda}(q_{2})P_{R}v_{\ell_{2}}}{(q_{1}-p_{1})^{2}-{m_{4}}^{2}+i\Gamma_{N_{4}}m_{4}}\Biggr{]}$
(163) $\displaystyle+$ $\displaystyle
2G_{F}^{2}V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}f_{M_{1}}f_{M_{2}}{V_{\ell_{1}4}}{V_{\ell_{2}4}}\
m_{4}\
m_{M_{2}}\Biggl{[}\frac{\overline{u_{\ell_{1}}}\not\epsilon^{\lambda}(q_{2})\not
q_{1}P_{R}v_{\ell_{2}}}{(q_{1}-p_{2})^{2}-{m_{4}}^{2}+i\Gamma_{N_{4}}m_{4}}\Biggr{]}$
$\displaystyle=$ $\displaystyle{\cal M}^{V}_{1}+{\cal M}^{V}_{2}.$
Similar to tau decay, we define the functions $f^{P}_{1},f^{P}_{2}$ and
$f^{V}_{1},f^{V}_{2}$ for the pseudoscalar and vector mesons respectively as
given in Eq. (131). $f^{P}_{i}$ and $f^{V}_{i}$ turn out to have the same form
and are given below.
$\displaystyle f^{P(V)}_{1}$ $\displaystyle=$
$\displaystyle\Biggl{(}\frac{F_{M}A^{P(V)}}{a^{2}_{1}+b^{2}}\Biggr{)}\Biggl{[}\frac{(a^{2}_{2}+b^{2})A^{P(V)}+(a_{1}a_{2}+b^{2})C^{P(V)}}{(a^{2}_{2}+b^{2})A^{P(V)}+(a^{2}_{1}+b^{2})B^{P(V)}}+(p_{1}\leftrightarrow
p_{2})\Biggr{]},$ (164) $\displaystyle f^{P(V)}_{2}$ $\displaystyle=$
$\displaystyle f^{P(V)}_{1}(p_{1}\leftrightarrow p_{2}),$ (165) $\displaystyle
A^{P}(p_{i},q_{j})$ $\displaystyle=$ $\displaystyle 8(p_{1}\cdot
q_{1})(p_{2}\cdot q_{2})(q_{1}\cdot q_{2})-4m^{2}_{M_{1}}(p_{1}\cdot
q_{2})(p_{2}\cdot q_{2})$ (166) $\displaystyle-$ $\displaystyle
4m^{2}_{M_{2}}(p_{1}\cdot q_{1})(p_{2}\cdot
q_{1})+2m^{2}_{M_{1}}m^{2}_{M_{2}}(p_{1}\cdot p_{2}),$ $\displaystyle
A^{V}(p_{i},q_{j})$ $\displaystyle=$ $\displaystyle 8(p_{1}\cdot
q_{1})(p_{2}\cdot q_{2})(q_{1}\cdot q_{2})-4m^{2}_{M_{1}}(p_{1}\cdot
q_{2})(p_{2}\cdot q_{2})$ (167) $\displaystyle+$ $\displaystyle
4m^{2}_{M_{2}}(p_{1}\cdot q_{1})(p_{2}\cdot
q_{1})-2m^{2}_{M_{1}}m^{2}_{M_{2}}(p_{1}\cdot p_{2}),$ $\displaystyle
B^{P(V)}(p_{i},q_{j})$ $\displaystyle=$ $\displaystyle
A^{P(V)}(p_{1}\leftrightarrow p_{2}),$ (168) $\displaystyle
C^{P}(p_{i},q_{j})$ $\displaystyle=$ $\displaystyle 4(p_{1}\cdot
p_{2})(q_{1}\cdot q_{2})^{2}-A^{P}(p_{i},q_{j}),$ (169) $\displaystyle
C^{V}(p_{i},q_{j})$ $\displaystyle=$ $\displaystyle 4(p_{1}\cdot
p_{2})(q_{1}\cdot q_{2})^{2}-4m^{2}_{M_{1}}m^{2}_{M_{2}}(p_{1}\cdot
p_{2})-A^{V}(p_{i},q_{j}),$ (170) $\displaystyle D^{P(V)}(p_{i},q_{j})$
$\displaystyle=$ $\displaystyle C^{P(V)}(p_{1}\leftrightarrow p_{2}),$ (171)
$\displaystyle F_{M}$ $\displaystyle=$ $\displaystyle
4G^{4}_{F}f_{M_{1}}^{2}f_{M_{2}}^{2}|V^{CKM}_{M_{1}}V^{CKM}_{M_{2}}|^{2}{|V_{\ell_{1}4}V_{\ell_{2}4}|}^{2}m^{2}_{4},$
(172) $\displaystyle a_{1,2}(p_{i},q_{j})$ $\displaystyle=$
$\displaystyle(q_{1}-p_{1,2})^{2}-m^{2}_{4};\ \ \ b=\Gamma_{N_{4}}m_{4}.$
(173)
The decay rate for $LV$ rare meson decay is then given by
$\displaystyle\Gamma^{M_{1}}_{LV}$ $\displaystyle=$ $\displaystyle(1-{1\over
2}\delta_{\ell_{1}\ell_{2}})\frac{1}{64\pi^{5}m_{M_{1}}}\Biggl{[}\int
f^{P(V)}_{1}dPS_{31}+\int f^{P(V)}_{2}dPS_{32}\Biggr{]},$ (174) $\displaystyle
dPS_{31}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}}{4m^{2}_{M_{1}}}\lambda^{\frac{1}{2}}(m^{2}_{M_{1}},m^{2}_{\ell_{1}},m^{2}_{c1})\lambda^{\frac{1}{2}}(m^{2}_{c1},m^{2}_{\ell_{2}},m^{2}_{M_{2}})\frac{dm^{2}_{c1}}{m^{2}_{c1}}dy_{1}dy_{2}dy_{3}dy_{4},$
(175) $\displaystyle dPS_{32}$ $\displaystyle=$ $\displaystyle
dPS_{31}(p_{1}\leftrightarrow p_{2}),$ (176)
where $dPS_{31}$ and $dPS_{32}$ are the phase space factors obtained by
conveniently clustering two different sets of particles to enable applying the
narrow-width approximation easily. $y_{1}$ to $y_{4}$ are rescaled angular
variables with integration limits $0\leq y_{i}\leq 1$. The width of the heavy
neutrino is very small compared to the mass and hence we can apply the narrow-
width approximation.
$\int\frac{dm^{2}_{c_{i}}}{(m^{2}_{c_{i}}-m^{2}_{4})^{2}+\Gamma^{2}_{N_{4}}m^{2}_{4}}\Bigg{|}_{{\Gamma_{N_{4}}\rightarrow\
0}}=\int\delta(m^{2}_{c_{i}}-m^{2}_{4})dm^{2}_{c_{i}}\frac{\pi}{\Gamma_{N_{4}}m_{4}}$
(177)
Applying the narrow-width approximation as described above and integrating
over the $\delta-$function we get
$\displaystyle\int f^{P(V)}_{1}dPS_{31}$ $\displaystyle=$
$\displaystyle\int\Biggl{(}\frac{F_{M}A^{P(V)}\pi^{3}}{4m^{2}_{M1}m^{3}_{4}\Gamma_{N_{4}}}\Biggl{[}\frac{(a^{2}_{2}+b^{2})A^{P(V)}+b^{2}(B^{P(V)}+C^{P(V)}+D^{P(V)})}{(a^{2}_{2}+b^{2})A^{P(V)}+b^{2}B^{P(V)}}\Biggr{]}$
(178)
$\displaystyle\lambda^{\frac{1}{2}}(m^{2}_{M_{1}},m^{2}_{\ell_{1}},m^{2}_{4})\lambda^{\frac{1}{2}}(m^{2}_{4},m^{2}_{\ell_{2}},m^{2}_{M_{2}})\Biggr{)}dy_{1}dy_{2}dy_{3}dy_{4},$
$\displaystyle\int f^{P(V)}_{2}dPS_{32}$ $\displaystyle=$ $\displaystyle\int
f^{P(V)}_{1}dPS_{31}(p_{1}\leftrightarrow p_{2}),$ (179)
Now, we can find the decay rate from Eq. (174). The branching fraction is then
given by $\mathrm{Br}=\tau_{M_{1}}\Gamma^{M_{1}}_{\\!\\!\\!\mbox{}_{LV}}$.
The CKM matrix elements and the lifetimes of mesons used in our calculations
are taken from PDG [99] and are listed below.
$\displaystyle|V_{ub}|=0.00367,\ |V_{cd}|=0.224,\ |V_{cs}|=0.996;$
$\displaystyle\tau_{K}=1.2384\times 10^{-8}\ {\rm s},\ \tau_{D}=1.040\times
10^{-12}\ {\rm s},\ \tau_{D_{s}}=4.9\times 10^{-13}\ {\rm s},\
\tau_{B}=1.671\times 10^{-12}\ {\rm s}.$
The mass and decay constants of pseudoscalar and vector mesons used in our
calculations are listed in Table 7.
Table 7: Mass and decay constants of pseudoscalar and vector mesons used. Pseudoscalar | Mass | Decay Constant | Vector | Mass | Decay Constant
---|---|---|---|---|---
Meson | $(\rm MeV)$[99] | $(\rm MeV)$ [99] | Meson | $(\rm MeV)$[99] | $(\rm MeV)$[114]
$\pi^{\pm}$ | 139.6 | 130.7 | $\rho^{\pm}$ | 775.8 | 220
$K^{\pm}$ | 493.7 | 159.8 | $K^{*\pm}$ | 891.66 | 217
$D^{\pm}$ | 1869.4 | 222.6 [115] | $D^{*}$ | 2010 | 310
$D^{\pm}_{s}$ | 1968.3 | 266 | $D^{*\pm}_{s}$ | 2112.1 | 315
$B^{\pm}$ | 5279 | 190 [116] | $\omega$ | 782.59 | 195
$\pi^{0}$ | 135 | 130 | $K^{*0},\overline{K}^{*0}$ | 896.10 | 217
$\eta$ | 547.8 | 164.7 [117] | $\phi$ | 1019.456 | 229
$\eta^{\prime}$ | 957.8 | 152.9 [117] | $D^{*0},\overline{D}^{*0}$ | 2006.7 | 310
$\eta_{c}$ | 2979.6 | 335.0 [118] | $J/\psi$ | 3096.916 | 459 [119]
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|
arxiv-papers
| 2009-01-23T02:30:42 |
2024-09-04T02:49:00.165964
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anupama Atre, Tao Han, Silvia Pascoli, Bin Zhang",
"submitter": "Bin Zhang",
"url": "https://arxiv.org/abs/0901.3589"
}
|
0901.3803
|
# Search for the Decays ${B^{0}_{(s)}\rightarrow e^{+}\mu^{-}}$ and
${B^{0}_{(s)}\rightarrow e^{+}e^{-}}$ in CDF Run II
T. Aaltonen Division of High Energy Physics, Department of Physics,
University of Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki,
Finland J. Adelman Enrico Fermi Institute, University of Chicago, Chicago,
Illinois 60637 T. Akimoto University of Tsukuba, Tsukuba, Ibaraki 305, Japan
B. Álvarez Gonzálezs Instituto de Fisica de Cantabria, CSIC-University of
Cantabria, 39005 Santander, Spain S. Amerioy Istituto Nazionale di Fisica
Nucleare, Sezione di Padova-Trento, yUniversity of Padova, I-35131 Padova,
Italy D. Amidei University of Michigan, Ann Arbor, Michigan 48109 A.
Anastassov Northwestern University, Evanston, Illinois 60208 A. Annovi
Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare,
I-00044 Frascati, Italy J. Antos Comenius University, 842 48 Bratislava,
Slovakia; Institute of Experimental Physics, 040 01 Kosice, Slovakia G.
Apollinari Fermi National Accelerator Laboratory, Batavia, Illinois 60510 A.
Apresyan Purdue University, West Lafayette, Indiana 47907 T. Arisawa Waseda
University, Tokyo 169, Japan A. Artikov Joint Institute for Nuclear
Research, RU-141980 Dubna, Russia W. Ashmanskas Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 A. Attal Institut de Fisica d’Altes
Energies, Universitat Autonoma de Barcelona, E-08193, Bellaterra (Barcelona),
Spain A. Aurisano Texas A&M University, College Station, Texas 77843 F.
Azfar University of Oxford, Oxford OX1 3RH, United Kingdom P. Azzurriz
Istituto Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity
of Siena and bbScuola Normale Superiore, I-56127 Pisa, Italy W. Badgett
Fermi National Accelerator Laboratory, Batavia, Illinois 60510 A. Barbaro-
Galtieri Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley,
California 94720 V.E. Barnes Purdue University, West Lafayette, Indiana
47907 B.A. Barnett The Johns Hopkins University, Baltimore, Maryland 21218
V. Bartsch University College London, London WC1E 6BT, United Kingdom G.
Bauer Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
P.-H. Beauchemin Institute of Particle Physics: McGill University, Montréal,
Québec, Canada H3A 2T8; Simon Fraser University, Burnaby, British Columbia,
Canada V5A 1S6; University of Toronto, Toronto, Ontario, Canada M5S 1A7; and
TRIUMF, Vancouver, British Columbia, Canada V6T 2A3 F. Bedeschi Istituto
Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena
and bbScuola Normale Superiore, I-56127 Pisa, Italy D. Beecher University
College London, London WC1E 6BT, United Kingdom S. Behari The Johns Hopkins
University, Baltimore, Maryland 21218 G. Bellettiniz Istituto Nazionale di
Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena and bbScuola
Normale Superiore, I-56127 Pisa, Italy J. Bellinger University of Wisconsin,
Madison, Wisconsin 53706 D. Benjamin Duke University, Durham, North Carolina
27708 A. Beretvas Fermi National Accelerator Laboratory, Batavia, Illinois
60510 J. Beringer Ernest Orlando Lawrence Berkeley National Laboratory,
Berkeley, California 94720 A. Bhatti The Rockefeller University, New York,
New York 10021 M. Binkley Fermi National Accelerator Laboratory, Batavia,
Illinois 60510 D. Biselloy Istituto Nazionale di Fisica Nucleare, Sezione di
Padova-Trento, yUniversity of Padova, I-35131 Padova, Italy I. Bizjakee
University College London, London WC1E 6BT, United Kingdom R.E. Blair
Argonne National Laboratory, Argonne, Illinois 60439 C. Blocker Brandeis
University, Waltham, Massachusetts 02254 B. Blumenfeld The Johns Hopkins
University, Baltimore, Maryland 21218 A. Bocci Duke University, Durham,
North Carolina 27708 A. Bodek University of Rochester, Rochester, New York
14627 V. Boisvert University of Rochester, Rochester, New York 14627 G.
Bolla Purdue University, West Lafayette, Indiana 47907 D. Bortoletto Purdue
University, West Lafayette, Indiana 47907 J. Boudreau University of
Pittsburgh, Pittsburgh, Pennsylvania 15260 A. Boveia University of
California, Santa Barbara, Santa Barbara, California 93106 B. Braua
University of California, Santa Barbara, Santa Barbara, California 93106 A.
Bridgeman University of Illinois, Urbana, Illinois 61801 L. Brigliadori
Istituto Nazionale di Fisica Nucleare, Sezione di Padova-Trento, yUniversity
of Padova, I-35131 Padova, Italy C. Bromberg Michigan State University, East
Lansing, Michigan 48824 E. Brubaker Enrico Fermi Institute, University of
Chicago, Chicago, Illinois 60637 J. Budagov Joint Institute for Nuclear
Research, RU-141980 Dubna, Russia H.S. Budd University of Rochester,
Rochester, New York 14627 S. Budd University of Illinois, Urbana, Illinois
61801 S. Burke Fermi National Accelerator Laboratory, Batavia, Illinois
60510 K. Burkett Fermi National Accelerator Laboratory, Batavia, Illinois
60510 G. Busettoy Istituto Nazionale di Fisica Nucleare, Sezione di Padova-
Trento, yUniversity of Padova, I-35131 Padova, Italy P. Bussey Glasgow
University, Glasgow G12 8QQ, United Kingdom A. Buzatu Institute of Particle
Physics: McGill University, Montréal, Québec, Canada H3A 2T8; Simon Fraser
University, Burnaby, British Columbia, Canada V5A 1S6; University of Toronto,
Toronto, Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British Columbia,
Canada V6T 2A3 K. L. Byrum Argonne National Laboratory, Argonne, Illinois
60439 S. Cabrerau Duke University, Durham, North Carolina 27708 C. Calancha
Centro de Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040
Madrid, Spain M. Campanelli Michigan State University, East Lansing,
Michigan 48824 M. Campbell University of Michigan, Ann Arbor, Michigan 48109
F. Canelli14 Fermi National Accelerator Laboratory, Batavia, Illinois 60510
A. Canepa University of Pennsylvania, Philadelphia, Pennsylvania 19104 B.
Carls University of Illinois, Urbana, Illinois 61801 D. Carlsmith
University of Wisconsin, Madison, Wisconsin 53706 R. Carosi Istituto
Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena
and bbScuola Normale Superiore, I-56127 Pisa, Italy S. Carrillon University
of Florida, Gainesville, Florida 32611 S. Carron Institute of Particle
Physics: McGill University, Montréal, Québec, Canada H3A 2T8; Simon Fraser
University, Burnaby, British Columbia, Canada V5A 1S6; University of Toronto,
Toronto, Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British Columbia,
Canada V6T 2A3 B. Casal Instituto de Fisica de Cantabria, CSIC-University of
Cantabria, 39005 Santander, Spain M. Casarsa Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 A. Castrox Istituto Nazionale di Fisica
Nucleare Bologna, xUniversity of Bologna, I-40127 Bologna, Italy P.
Catastiniaa Istituto Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa,
aaUniversity of Siena and bbScuola Normale Superiore, I-56127 Pisa, Italy D.
Cauzdd Istituto Nazionale di Fisica Nucleare Trieste/Udine, I-34100 Trieste,
ddUniversity of Trieste/Udine, I-33100 Udine, Italy V. Cavaliereaa Istituto
Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena
and bbScuola Normale Superiore, I-56127 Pisa, Italy M. Cavalli-Sforza
Institut de Fisica d’Altes Energies, Universitat Autonoma de Barcelona,
E-08193, Bellaterra (Barcelona), Spain A. Cerri Ernest Orlando Lawrence
Berkeley National Laboratory, Berkeley, California 94720 L. Cerritoo
University College London, London WC1E 6BT, United Kingdom S.H. Chang Center
for High Energy Physics: Kyungpook National University, Daegu 702-701, Korea;
Seoul National University, Seoul 151-742, Korea; Sungkyunkwan University,
Suwon 440-746, Korea; Korea Institute of Science and Technology Information,
Daejeon, 305-806, Korea; Chonnam National University, Gwangju, 500-757, Korea
Y.C. Chen Institute of Physics, Academia Sinica, Taipei, Taiwan 11529,
Republic of China M. Chertok University of California, Davis, Davis,
California 95616 G. Chiarelli Istituto Nazionale di Fisica Nucleare Pisa,
zUniversity of Pisa, aaUniversity of Siena and bbScuola Normale Superiore,
I-56127 Pisa, Italy G. Chlachidze Fermi National Accelerator Laboratory,
Batavia, Illinois 60510 F. Chlebana Fermi National Accelerator Laboratory,
Batavia, Illinois 60510 K. Cho Center for High Energy Physics: Kyungpook
National University, Daegu 702-701, Korea; Seoul National University, Seoul
151-742, Korea; Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute
of Science and Technology Information, Daejeon, 305-806, Korea; Chonnam
National University, Gwangju, 500-757, Korea D. Chokheli Joint Institute for
Nuclear Research, RU-141980 Dubna, Russia J.P. Chou Harvard University,
Cambridge, Massachusetts 02138 G. Choudalakis Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139 S.H. Chuang Rutgers University,
Piscataway, New Jersey 08855 K. Chung Carnegie Mellon University,
Pittsburgh, PA 15213 W.H. Chung University of Wisconsin, Madison, Wisconsin
53706 Y.S. Chung University of Rochester, Rochester, New York 14627 T.
Chwalek Institut für Experimentelle Kernphysik, Universität Karlsruhe, 76128
Karlsruhe, Germany C.I. Ciobanu LPNHE, Universite Pierre et Marie
Curie/IN2P3-CNRS, UMR7585, Paris, F-75252 France M.A. Ciocciaa Istituto
Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena
and bbScuola Normale Superiore, I-56127 Pisa, Italy A. Clark University of
Geneva, CH-1211 Geneva 4, Switzerland D. Clark Brandeis University, Waltham,
Massachusetts 02254 G. Compostella Istituto Nazionale di Fisica Nucleare,
Sezione di Padova-Trento, yUniversity of Padova, I-35131 Padova, Italy M.E.
Convery Fermi National Accelerator Laboratory, Batavia, Illinois 60510 J.
Conway University of California, Davis, Davis, California 95616 M. Cordelli
Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare,
I-00044 Frascati, Italy G. Cortianay Istituto Nazionale di Fisica Nucleare,
Sezione di Padova-Trento, yUniversity of Padova, I-35131 Padova, Italy C.A.
Cox University of California, Davis, Davis, California 95616 D.J. Cox
University of California, Davis, Davis, California 95616 F. Crescioliz
Istituto Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity
of Siena and bbScuola Normale Superiore, I-56127 Pisa, Italy C. Cuenca
Almenaru University of California, Davis, Davis, California 95616 J. Cuevass
Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005
Santander, Spain R. Culbertson Fermi National Accelerator Laboratory,
Batavia, Illinois 60510 J.C. Cully University of Michigan, Ann Arbor,
Michigan 48109 D. Dagenhart Fermi National Accelerator Laboratory, Batavia,
Illinois 60510 M. Datta Fermi National Accelerator Laboratory, Batavia,
Illinois 60510 T. Davies Glasgow University, Glasgow G12 8QQ, United Kingdom
P. de Barbaro University of Rochester, Rochester, New York 14627 S. De Cecco
Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1, ccSapienza
Università di Roma, I-00185 Roma, Italy A. Deisher Ernest Orlando Lawrence
Berkeley National Laboratory, Berkeley, California 94720 G. De Lorenzo
Institut de Fisica d’Altes Energies, Universitat Autonoma de Barcelona,
E-08193, Bellaterra (Barcelona), Spain M. Dell’Orsoz Istituto Nazionale di
Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena and bbScuola
Normale Superiore, I-56127 Pisa, Italy C. Deluca Institut de Fisica d’Altes
Energies, Universitat Autonoma de Barcelona, E-08193, Bellaterra (Barcelona),
Spain L. Demortier The Rockefeller University, New York, New York 10021 J.
Deng Duke University, Durham, North Carolina 27708 M. Deninno Istituto
Nazionale di Fisica Nucleare Bologna, xUniversity of Bologna, I-40127 Bologna,
Italy P.F. Derwent Fermi National Accelerator Laboratory, Batavia, Illinois
60510 G.P. di Giovanni LPNHE, Universite Pierre et Marie Curie/IN2P3-CNRS,
UMR7585, Paris, F-75252 France C. Dionisicc Istituto Nazionale di Fisica
Nucleare, Sezione di Roma 1, ccSapienza Università di Roma, I-00185 Roma,
Italy B. Di Ruzzadd Istituto Nazionale di Fisica Nucleare Trieste/Udine,
I-34100 Trieste, ddUniversity of Trieste/Udine, I-33100 Udine, Italy J.R.
Dittmann Baylor University, Waco, Texas 76798 M. D’Onofrio Institut de
Fisica d’Altes Energies, Universitat Autonoma de Barcelona, E-08193,
Bellaterra (Barcelona), Spain S. Donatiz Istituto Nazionale di Fisica
Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena and bbScuola Normale
Superiore, I-56127 Pisa, Italy P. Dong University of California, Los
Angeles, Los Angeles, California 90024 J. Donini Istituto Nazionale di
Fisica Nucleare, Sezione di Padova-Trento, yUniversity of Padova, I-35131
Padova, Italy T. Dorigo Istituto Nazionale di Fisica Nucleare, Sezione di
Padova-Trento, yUniversity of Padova, I-35131 Padova, Italy S. Dube Rutgers
University, Piscataway, New Jersey 08855 J. Efron The Ohio State University,
Columbus, Ohio 43210 A. Elagin Texas A&M University, College Station, Texas
77843 R. Erbacher University of California, Davis, Davis, California 95616
D. Errede University of Illinois, Urbana, Illinois 61801 S. Errede
University of Illinois, Urbana, Illinois 61801 R. Eusebi Fermi National
Accelerator Laboratory, Batavia, Illinois 60510 H.C. Fang Ernest Orlando
Lawrence Berkeley National Laboratory, Berkeley, California 94720 S.
Farrington University of Oxford, Oxford OX1 3RH, United Kingdom W.T. Fedorko
Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637 R.G.
Feild Yale University, New Haven, Connecticut 06520 M. Feindt Institut für
Experimentelle Kernphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany
J.P. Fernandez Centro de Investigaciones Energeticas Medioambientales y
Tecnologicas, E-28040 Madrid, Spain C. Ferrazzabb Istituto Nazionale di
Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena and bbScuola
Normale Superiore, I-56127 Pisa, Italy R. Field University of Florida,
Gainesville, Florida 32611 G. Flanagan Purdue University, West Lafayette,
Indiana 47907 R. Forrest University of California, Davis, Davis, California
95616 M.J. Frank Baylor University, Waco, Texas 76798 M. Franklin Harvard
University, Cambridge, Massachusetts 02138 J.C. Freeman Fermi National
Accelerator Laboratory, Batavia, Illinois 60510 I. Furic University of
Florida, Gainesville, Florida 32611 M. Gallinaro Istituto Nazionale di
Fisica Nucleare, Sezione di Roma 1, ccSapienza Università di Roma, I-00185
Roma, Italy J. Galyardt Carnegie Mellon University, Pittsburgh, PA 15213 F.
Garberson University of California, Santa Barbara, Santa Barbara, California
93106 J.E. Garcia University of Geneva, CH-1211 Geneva 4, Switzerland A.F.
Garfinkel Purdue University, West Lafayette, Indiana 47907 K. Genser Fermi
National Accelerator Laboratory, Batavia, Illinois 60510 H. Gerberich
University of Illinois, Urbana, Illinois 61801 D. Gerdes University of
Michigan, Ann Arbor, Michigan 48109 A. Gessler Institut für Experimentelle
Kernphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany S. Giagucc
Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1, ccSapienza
Università di Roma, I-00185 Roma, Italy V. Giakoumopoulou University of
Athens, 157 71 Athens, Greece P. Giannetti Istituto Nazionale di Fisica
Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena and bbScuola Normale
Superiore, I-56127 Pisa, Italy K. Gibson University of Pittsburgh,
Pittsburgh, Pennsylvania 15260 J.L. Gimmell University of Rochester,
Rochester, New York 14627 C.M. Ginsburg Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 N. Giokaris University of Athens, 157 71
Athens, Greece M. Giordanidd Istituto Nazionale di Fisica Nucleare
Trieste/Udine, I-34100 Trieste, ddUniversity of Trieste/Udine, I-33100 Udine,
Italy P. Giromini Laboratori Nazionali di Frascati, Istituto Nazionale di
Fisica Nucleare, I-00044 Frascati, Italy M. Giuntaz Istituto Nazionale di
Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena and bbScuola
Normale Superiore, I-56127 Pisa, Italy G. Giurgiu The Johns Hopkins
University, Baltimore, Maryland 21218 V. Glagolev Joint Institute for
Nuclear Research, RU-141980 Dubna, Russia D. Glenzinski Fermi National
Accelerator Laboratory, Batavia, Illinois 60510 M. Gold University of New
Mexico, Albuquerque, New Mexico 87131 N. Goldschmidt University of Florida,
Gainesville, Florida 32611 A. Golossanov Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 G. Gomez Instituto de Fisica de
Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain G. Gomez-
Ceballos Massachusetts Institute of Technology, Cambridge, Massachusetts
02139 M. Goncharov Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139 O. González Centro de Investigaciones Energeticas
Medioambientales y Tecnologicas, E-28040 Madrid, Spain I. Gorelov University
of New Mexico, Albuquerque, New Mexico 87131 A.T. Goshaw Duke University,
Durham, North Carolina 27708 K. Goulianos The Rockefeller University, New
York, New York 10021 A. Greseley Istituto Nazionale di Fisica Nucleare,
Sezione di Padova-Trento, yUniversity of Padova, I-35131 Padova, Italy S.
Grinstein Harvard University, Cambridge, Massachusetts 02138 C. Grosso-
Pilcher Enrico Fermi Institute, University of Chicago, Chicago, Illinois
60637 R.C. Group Fermi National Accelerator Laboratory, Batavia, Illinois
60510 U. Grundler University of Illinois, Urbana, Illinois 61801 J.
Guimaraes da Costa Harvard University, Cambridge, Massachusetts 02138 Z.
Gunay-Unalan Michigan State University, East Lansing, Michigan 48824 C.
Haber Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley,
California 94720 K. Hahn Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139 S.R. Hahn Fermi National Accelerator Laboratory,
Batavia, Illinois 60510 E. Halkiadakis Rutgers University, Piscataway, New
Jersey 08855 B.-Y. Han University of Rochester, Rochester, New York 14627
J.Y. Han University of Rochester, Rochester, New York 14627 F. Happacher
Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare,
I-00044 Frascati, Italy K. Hara University of Tsukuba, Tsukuba, Ibaraki 305,
Japan D. Hare Rutgers University, Piscataway, New Jersey 08855 M. Hare
Tufts University, Medford, Massachusetts 02155 S. Harper University of
Oxford, Oxford OX1 3RH, United Kingdom R.F. Harr Wayne State University,
Detroit, Michigan 48201 R.M. Harris Fermi National Accelerator Laboratory,
Batavia, Illinois 60510 M. Hartz University of Pittsburgh, Pittsburgh,
Pennsylvania 15260 K. Hatakeyama The Rockefeller University, New York, New
York 10021 C. Hays University of Oxford, Oxford OX1 3RH, United Kingdom M.
Heck Institut für Experimentelle Kernphysik, Universität Karlsruhe, 76128
Karlsruhe, Germany A. Heijboer University of Pennsylvania, Philadelphia,
Pennsylvania 19104 J. Heinrich University of Pennsylvania, Philadelphia,
Pennsylvania 19104 C. Henderson Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139 M. Herndon University of Wisconsin, Madison,
Wisconsin 53706 J. Heuser Institut für Experimentelle Kernphysik,
Universität Karlsruhe, 76128 Karlsruhe, Germany S. Hewamanage Baylor
University, Waco, Texas 76798 D. Hidas Duke University, Durham, North
Carolina 27708 C.S. Hillc University of California, Santa Barbara, Santa
Barbara, California 93106 D. Hirschbuehl Institut für Experimentelle
Kernphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany A. Hocker Fermi
National Accelerator Laboratory, Batavia, Illinois 60510 S. Hou Institute of
Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China M. Houlden
University of Liverpool, Liverpool L69 7ZE, United Kingdom S.-C. Hsu Ernest
Orlando Lawrence Berkeley National Laboratory, Berkeley, California 94720
B.T. Huffman University of Oxford, Oxford OX1 3RH, United Kingdom R.E.
Hughes The Ohio State University, Columbus, Ohio 43210 U. Husemann Yale
University, New Haven, Connecticut 06520 M. Hussein Michigan State
University, East Lansing, Michigan 48824 J. Huston Michigan State
University, East Lansing, Michigan 48824 J. Incandela University of
California, Santa Barbara, Santa Barbara, California 93106 G. Introzzi
Istituto Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity
of Siena and bbScuola Normale Superiore, I-56127 Pisa, Italy M. Ioricc
Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1, ccSapienza
Università di Roma, I-00185 Roma, Italy A. Ivanov University of California,
Davis, Davis, California 95616 E. James Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 D. Jang Carnegie Mellon University,
Pittsburgh, PA 15213 B. Jayatilaka Duke University, Durham, North Carolina
27708 E.J. Jeon Center for High Energy Physics: Kyungpook National
University, Daegu 702-701, Korea; Seoul National University, Seoul 151-742,
Korea; Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute of
Science and Technology Information, Daejeon, 305-806, Korea; Chonnam National
University, Gwangju, 500-757, Korea M.K. Jha Istituto Nazionale di Fisica
Nucleare Bologna, xUniversity of Bologna, I-40127 Bologna, Italy S.
Jindariani Fermi National Accelerator Laboratory, Batavia, Illinois 60510 W.
Johnson University of California, Davis, Davis, California 95616 M. Jones
Purdue University, West Lafayette, Indiana 47907 K.K. Joo Center for High
Energy Physics: Kyungpook National University, Daegu 702-701, Korea; Seoul
National University, Seoul 151-742, Korea; Sungkyunkwan University, Suwon
440-746, Korea; Korea Institute of Science and Technology Information,
Daejeon, 305-806, Korea; Chonnam National University, Gwangju, 500-757, Korea
S.Y. Jun Carnegie Mellon University, Pittsburgh, PA 15213 J.E. Jung Center
for High Energy Physics: Kyungpook National University, Daegu 702-701, Korea;
Seoul National University, Seoul 151-742, Korea; Sungkyunkwan University,
Suwon 440-746, Korea; Korea Institute of Science and Technology Information,
Daejeon, 305-806, Korea; Chonnam National University, Gwangju, 500-757, Korea
T.R. Junk Fermi National Accelerator Laboratory, Batavia, Illinois 60510 T.
Kamon Texas A&M University, College Station, Texas 77843 D. Kar University
of Florida, Gainesville, Florida 32611 P.E. Karchin Wayne State University,
Detroit, Michigan 48201 Y. Katol Osaka City University, Osaka 588, Japan R.
Kephart Fermi National Accelerator Laboratory, Batavia, Illinois 60510 J.
Keung University of Pennsylvania, Philadelphia, Pennsylvania 19104 V.
Khotilovich Texas A&M University, College Station, Texas 77843 B. Kilminster
Fermi National Accelerator Laboratory, Batavia, Illinois 60510 D.H. Kim
Center for High Energy Physics: Kyungpook National University, Daegu 702-701,
Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon, 305-806, Korea; Chonnam National University, Gwangju,
500-757, Korea H.S. Kim Center for High Energy Physics: Kyungpook National
University, Daegu 702-701, Korea; Seoul National University, Seoul 151-742,
Korea; Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute of
Science and Technology Information, Daejeon, 305-806, Korea; Chonnam National
University, Gwangju, 500-757, Korea H.W. Kim Center for High Energy Physics:
Kyungpook National University, Daegu 702-701, Korea; Seoul National
University, Seoul 151-742, Korea; Sungkyunkwan University, Suwon 440-746,
Korea; Korea Institute of Science and Technology Information, Daejeon,
305-806, Korea; Chonnam National University, Gwangju, 500-757, Korea J.E. Kim
Center for High Energy Physics: Kyungpook National University, Daegu 702-701,
Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon, 305-806, Korea; Chonnam National University, Gwangju,
500-757, Korea M.J. Kim Laboratori Nazionali di Frascati, Istituto Nazionale
di Fisica Nucleare, I-00044 Frascati, Italy S.B. Kim Center for High Energy
Physics: Kyungpook National University, Daegu 702-701, Korea; Seoul National
University, Seoul 151-742, Korea; Sungkyunkwan University, Suwon 440-746,
Korea; Korea Institute of Science and Technology Information, Daejeon,
305-806, Korea; Chonnam National University, Gwangju, 500-757, Korea S.H. Kim
University of Tsukuba, Tsukuba, Ibaraki 305, Japan Y.K. Kim Enrico Fermi
Institute, University of Chicago, Chicago, Illinois 60637 N. Kimura
University of Tsukuba, Tsukuba, Ibaraki 305, Japan L. Kirsch Brandeis
University, Waltham, Massachusetts 02254 S. Klimenko University of Florida,
Gainesville, Florida 32611 B. Knuteson Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139 B.R. Ko Duke University, Durham,
North Carolina 27708 K. Kondo Waseda University, Tokyo 169, Japan D.J. Kong
Center for High Energy Physics: Kyungpook National University, Daegu 702-701,
Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon, 305-806, Korea; Chonnam National University, Gwangju,
500-757, Korea J. Konigsberg University of Florida, Gainesville, Florida
32611 A. Korytov University of Florida, Gainesville, Florida 32611 A.V.
Kotwal Duke University, Durham, North Carolina 27708 M. Kreps Institut für
Experimentelle Kernphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany J.
Kroll University of Pennsylvania, Philadelphia, Pennsylvania 19104 D. Krop
Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637 N.
Krumnack Baylor University, Waco, Texas 76798 M. Kruse Duke University,
Durham, North Carolina 27708 V. Krutelyov University of California, Santa
Barbara, Santa Barbara, California 93106 T. Kubo University of Tsukuba,
Tsukuba, Ibaraki 305, Japan T. Kuhr Institut für Experimentelle Kernphysik,
Universität Karlsruhe, 76128 Karlsruhe, Germany N.P. Kulkarni Wayne State
University, Detroit, Michigan 48201 M. Kurata University of Tsukuba,
Tsukuba, Ibaraki 305, Japan S. Kwang Enrico Fermi Institute, University of
Chicago, Chicago, Illinois 60637 A.T. Laasanen Purdue University, West
Lafayette, Indiana 47907 S. Lami Istituto Nazionale di Fisica Nucleare Pisa,
zUniversity of Pisa, aaUniversity of Siena and bbScuola Normale Superiore,
I-56127 Pisa, Italy S. Lammel Fermi National Accelerator Laboratory,
Batavia, Illinois 60510 M. Lancaster University College London, London WC1E
6BT, United Kingdom R.L. Lander University of California, Davis, Davis,
California 95616 K. Lannonr The Ohio State University, Columbus, Ohio 43210
A. Lath Rutgers University, Piscataway, New Jersey 08855 G. Latinoaa
Istituto Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity
of Siena and bbScuola Normale Superiore, I-56127 Pisa, Italy I. Lazzizzeray
Istituto Nazionale di Fisica Nucleare, Sezione di Padova-Trento, yUniversity
of Padova, I-35131 Padova, Italy T. LeCompte Argonne National Laboratory,
Argonne, Illinois 60439 E. Lee Texas A&M University, College Station, Texas
77843 H.S. Lee Enrico Fermi Institute, University of Chicago, Chicago,
Illinois 60637 S.W. Leet Texas A&M University, College Station, Texas 77843
S. Leone Istituto Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa,
aaUniversity of Siena and bbScuola Normale Superiore, I-56127 Pisa, Italy
J.D. Lewis Fermi National Accelerator Laboratory, Batavia, Illinois 60510
C.-S. Lin Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley,
California 94720 J. Linacre University of Oxford, Oxford OX1 3RH, United
Kingdom M. Lindgren Fermi National Accelerator Laboratory, Batavia, Illinois
60510 E. Lipeles University of Pennsylvania, Philadelphia, Pennsylvania
19104 A. Lister University of California, Davis, Davis, California 95616
D.O. Litvintsev Fermi National Accelerator Laboratory, Batavia, Illinois
60510 C. Liu University of Pittsburgh, Pittsburgh, Pennsylvania 15260 T.
Liu Fermi National Accelerator Laboratory, Batavia, Illinois 60510 N.S.
Lockyer University of Pennsylvania, Philadelphia, Pennsylvania 19104 A.
Loginov Yale University, New Haven, Connecticut 06520 M. Loretiy Istituto
Nazionale di Fisica Nucleare, Sezione di Padova-Trento, yUniversity of Padova,
I-35131 Padova, Italy L. Lovas Comenius University, 842 48 Bratislava,
Slovakia; Institute of Experimental Physics, 040 01 Kosice, Slovakia D.
Lucchesiy Istituto Nazionale di Fisica Nucleare, Sezione di Padova-Trento,
yUniversity of Padova, I-35131 Padova, Italy C. Lucicc Istituto Nazionale di
Fisica Nucleare, Sezione di Roma 1, ccSapienza Università di Roma, I-00185
Roma, Italy J. Lueck Institut für Experimentelle Kernphysik, Universität
Karlsruhe, 76128 Karlsruhe, Germany P. Lujan Ernest Orlando Lawrence
Berkeley National Laboratory, Berkeley, California 94720 P. Lukens Fermi
National Accelerator Laboratory, Batavia, Illinois 60510 G. Lungu The
Rockefeller University, New York, New York 10021 L. Lyons University of
Oxford, Oxford OX1 3RH, United Kingdom J. Lys Ernest Orlando Lawrence
Berkeley National Laboratory, Berkeley, California 94720 R. Lysak Comenius
University, 842 48 Bratislava, Slovakia; Institute of Experimental Physics,
040 01 Kosice, Slovakia D. MacQueen Institute of Particle Physics: McGill
University, Montréal, Québec, Canada H3A 2T8; Simon Fraser University,
Burnaby, British Columbia, Canada V5A 1S6; University of Toronto, Toronto,
Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British Columbia, Canada V6T
2A3 R. Madrak Fermi National Accelerator Laboratory, Batavia, Illinois 60510
K. Maeshima Fermi National Accelerator Laboratory, Batavia, Illinois 60510
K. Makhoul Massachusetts Institute of Technology, Cambridge, Massachusetts
02139 T. Maki Division of High Energy Physics, Department of Physics,
University of Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki,
Finland P. Maksimovic The Johns Hopkins University, Baltimore, Maryland
21218 S. Malde University of Oxford, Oxford OX1 3RH, United Kingdom S.
Malik University College London, London WC1E 6BT, United Kingdom G. Mancae
University of Liverpool, Liverpool L69 7ZE, United Kingdom A. Manousakis-
Katsikakis University of Athens, 157 71 Athens, Greece F. Margaroli Purdue
University, West Lafayette, Indiana 47907 C. Marino Institut für
Experimentelle Kernphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany
C.P. Marino University of Illinois, Urbana, Illinois 61801 A. Martin Yale
University, New Haven, Connecticut 06520 V. Martink Glasgow University,
Glasgow G12 8QQ, United Kingdom M. Martínez Institut de Fisica d’Altes
Energies, Universitat Autonoma de Barcelona, E-08193, Bellaterra (Barcelona),
Spain R. Martínez-Ballarín Centro de Investigaciones Energeticas
Medioambientales y Tecnologicas, E-28040 Madrid, Spain T. Maruyama
University of Tsukuba, Tsukuba, Ibaraki 305, Japan P. Mastrandrea Istituto
Nazionale di Fisica Nucleare, Sezione di Roma 1, ccSapienza Università di
Roma, I-00185 Roma, Italy T. Masubuchi University of Tsukuba, Tsukuba,
Ibaraki 305, Japan M. Mathis The Johns Hopkins University, Baltimore,
Maryland 21218 M.E. Mattson Wayne State University, Detroit, Michigan 48201
P. Mazzanti Istituto Nazionale di Fisica Nucleare Bologna, xUniversity of
Bologna, I-40127 Bologna, Italy K.S. McFarland University of Rochester,
Rochester, New York 14627 P. McIntyre Texas A&M University, College Station,
Texas 77843 R. McNultyj University of Liverpool, Liverpool L69 7ZE, United
Kingdom A. Mehta University of Liverpool, Liverpool L69 7ZE, United Kingdom
P. Mehtala Division of High Energy Physics, Department of Physics, University
of Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki, Finland
A. Menzione Istituto Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa,
aaUniversity of Siena and bbScuola Normale Superiore, I-56127 Pisa, Italy P.
Merkel Purdue University, West Lafayette, Indiana 47907 C. Mesropian The
Rockefeller University, New York, New York 10021 T. Miao Fermi National
Accelerator Laboratory, Batavia, Illinois 60510 N. Miladinovic Brandeis
University, Waltham, Massachusetts 02254 R. Miller Michigan State
University, East Lansing, Michigan 48824 C. Mills Harvard University,
Cambridge, Massachusetts 02138 M. Milnik Institut für Experimentelle
Kernphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany A. Mitra
Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China
G. Mitselmakher University of Florida, Gainesville, Florida 32611 H. Miyake
University of Tsukuba, Tsukuba, Ibaraki 305, Japan N. Moggi Istituto
Nazionale di Fisica Nucleare Bologna, xUniversity of Bologna, I-40127 Bologna,
Italy C.S. Moon Center for High Energy Physics: Kyungpook National
University, Daegu 702-701, Korea; Seoul National University, Seoul 151-742,
Korea; Sungkyunkwan University, Suwon 440-746, Korea; Korea Institute of
Science and Technology Information, Daejeon, 305-806, Korea; Chonnam National
University, Gwangju, 500-757, Korea R. Moore Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 M.J. Morelloz Istituto Nazionale di
Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena and bbScuola
Normale Superiore, I-56127 Pisa, Italy J. Morlock Institut für
Experimentelle Kernphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany P.
Movilla Fernandez Fermi National Accelerator Laboratory, Batavia, Illinois
60510 J. Mülmenstädt Ernest Orlando Lawrence Berkeley National Laboratory,
Berkeley, California 94720 A. Mukherjee Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 Th. Muller Institut für Experimentelle
Kernphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany R. Mumford The
Johns Hopkins University, Baltimore, Maryland 21218 P. Murat Fermi National
Accelerator Laboratory, Batavia, Illinois 60510 M. Mussinix Istituto
Nazionale di Fisica Nucleare Bologna, xUniversity of Bologna, I-40127 Bologna,
Italy J. Nachtman Fermi National Accelerator Laboratory, Batavia, Illinois
60510 Y. Nagai University of Tsukuba, Tsukuba, Ibaraki 305, Japan A. Nagano
University of Tsukuba, Tsukuba, Ibaraki 305, Japan J. Naganoma University of
Tsukuba, Tsukuba, Ibaraki 305, Japan K. Nakamura University of Tsukuba,
Tsukuba, Ibaraki 305, Japan I. Nakano Okayama University, Okayama 700-8530,
Japan A. Napier Tufts University, Medford, Massachusetts 02155 V. Necula
Duke University, Durham, North Carolina 27708 J. Nett University of
Wisconsin, Madison, Wisconsin 53706 C. Neuv University of Pennsylvania,
Philadelphia, Pennsylvania 19104 M.S. Neubauer University of Illinois,
Urbana, Illinois 61801 S. Neubauer Institut für Experimentelle Kernphysik,
Universität Karlsruhe, 76128 Karlsruhe, Germany J. Nielseng Ernest Orlando
Lawrence Berkeley National Laboratory, Berkeley, California 94720 L. Nodulman
Argonne National Laboratory, Argonne, Illinois 60439 M. Norman University of
California, San Diego, La Jolla, California 92093 O. Norniella University of
Illinois, Urbana, Illinois 61801 E. Nurse University College London, London
WC1E 6BT, United Kingdom L. Oakes University of Oxford, Oxford OX1 3RH,
United Kingdom S.H. Oh Duke University, Durham, North Carolina 27708 Y.D.
Oh Center for High Energy Physics: Kyungpook National University, Daegu
702-701, Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon, 305-806, Korea; Chonnam National University, Gwangju,
500-757, Korea I. Oksuzian University of Florida, Gainesville, Florida 32611
T. Okusawa Osaka City University, Osaka 588, Japan R. Orava Division of
High Energy Physics, Department of Physics, University of Helsinki and
Helsinki Institute of Physics, FIN-00014, Helsinki, Finland K. Osterberg
Division of High Energy Physics, Department of Physics, University of Helsinki
and Helsinki Institute of Physics, FIN-00014, Helsinki, Finland S. Pagan
Grisoy Istituto Nazionale di Fisica Nucleare, Sezione di Padova-Trento,
yUniversity of Padova, I-35131 Padova, Italy E. Palencia Fermi National
Accelerator Laboratory, Batavia, Illinois 60510 V. Papadimitriou Fermi
National Accelerator Laboratory, Batavia, Illinois 60510 A. Papaikonomou
Institut für Experimentelle Kernphysik, Universität Karlsruhe, 76128
Karlsruhe, Germany A.A. Paramonov Enrico Fermi Institute, University of
Chicago, Chicago, Illinois 60637 B. Parks The Ohio State University,
Columbus, Ohio 43210 S. Pashapour Institute of Particle Physics: McGill
University, Montréal, Québec, Canada H3A 2T8; Simon Fraser University,
Burnaby, British Columbia, Canada V5A 1S6; University of Toronto, Toronto,
Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British Columbia, Canada V6T
2A3 J. Patrick Fermi National Accelerator Laboratory, Batavia, Illinois
60510 G. Paulettadd Istituto Nazionale di Fisica Nucleare Trieste/Udine,
I-34100 Trieste, ddUniversity of Trieste/Udine, I-33100 Udine, Italy M.
Paulini Carnegie Mellon University, Pittsburgh, PA 15213 C. Paus
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 T.
Peiffer Institut für Experimentelle Kernphysik, Universität Karlsruhe, 76128
Karlsruhe, Germany D.E. Pellett University of California, Davis, Davis,
California 95616 A. Penzo Istituto Nazionale di Fisica Nucleare
Trieste/Udine, I-34100 Trieste, ddUniversity of Trieste/Udine, I-33100 Udine,
Italy T.J. Phillips Duke University, Durham, North Carolina 27708 G.
Piacentino Istituto Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa,
aaUniversity of Siena and bbScuola Normale Superiore, I-56127 Pisa, Italy E.
Pianori University of Pennsylvania, Philadelphia, Pennsylvania 19104 L.
Pinera University of Florida, Gainesville, Florida 32611 K. Pitts
University of Illinois, Urbana, Illinois 61801 C. Plager University of
California, Los Angeles, Los Angeles, California 90024 L. Pondrom University
of Wisconsin, Madison, Wisconsin 53706 O. Poukhov111Deceased Joint Institute
for Nuclear Research, RU-141980 Dubna, Russia N. Pounder University of
Oxford, Oxford OX1 3RH, United Kingdom F. Prakoshyn Joint Institute for
Nuclear Research, RU-141980 Dubna, Russia A. Pronko Fermi National
Accelerator Laboratory, Batavia, Illinois 60510 J. Proudfoot Argonne
National Laboratory, Argonne, Illinois 60439 F. Ptohosi Fermi National
Accelerator Laboratory, Batavia, Illinois 60510 E. Pueschel Carnegie Mellon
University, Pittsburgh, PA 15213 G. Punziz Istituto Nazionale di Fisica
Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena and bbScuola Normale
Superiore, I-56127 Pisa, Italy J. Pursley University of Wisconsin, Madison,
Wisconsin 53706 J. Rademackerc University of Oxford, Oxford OX1 3RH, United
Kingdom A. Rahaman University of Pittsburgh, Pittsburgh, Pennsylvania 15260
V. Ramakrishnan University of Wisconsin, Madison, Wisconsin 53706 N. Ranjan
Purdue University, West Lafayette, Indiana 47907 I. Redondo Centro de
Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040 Madrid,
Spain P. Renton University of Oxford, Oxford OX1 3RH, United Kingdom M.
Renz Institut für Experimentelle Kernphysik, Universität Karlsruhe, 76128
Karlsruhe, Germany M. Rescigno Istituto Nazionale di Fisica Nucleare,
Sezione di Roma 1, ccSapienza Università di Roma, I-00185 Roma, Italy S.
Richter Institut für Experimentelle Kernphysik, Universität Karlsruhe, 76128
Karlsruhe, Germany F. Rimondix Istituto Nazionale di Fisica Nucleare Bologna,
xUniversity of Bologna, I-40127 Bologna, Italy L. Ristori Istituto Nazionale
di Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena and
bbScuola Normale Superiore, I-56127 Pisa, Italy A. Robson Glasgow
University, Glasgow G12 8QQ, United Kingdom T. Rodrigo Instituto de Fisica
de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain T.
Rodriguez University of Pennsylvania, Philadelphia, Pennsylvania 19104 E.
Rogers University of Illinois, Urbana, Illinois 61801 S. Rolli Tufts
University, Medford, Massachusetts 02155 R. Roser Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 M. Rossi Istituto Nazionale di Fisica
Nucleare Trieste/Udine, I-34100 Trieste, ddUniversity of Trieste/Udine,
I-33100 Udine, Italy R. Rossin University of California, Santa Barbara,
Santa Barbara, California 93106 P. Roy Institute of Particle Physics: McGill
University, Montréal, Québec, Canada H3A 2T8; Simon Fraser University,
Burnaby, British Columbia, Canada V5A 1S6; University of Toronto, Toronto,
Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British Columbia, Canada V6T
2A3 A. Ruiz Instituto de Fisica de Cantabria, CSIC-University of Cantabria,
39005 Santander, Spain J. Russ Carnegie Mellon University, Pittsburgh, PA
15213 V. Rusu Fermi National Accelerator Laboratory, Batavia, Illinois 60510
B. Rutherford Fermi National Accelerator Laboratory, Batavia, Illinois 60510
H. Saarikko Division of High Energy Physics, Department of Physics,
University of Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki,
Finland A. Safonov Texas A&M University, College Station, Texas 77843 W.K.
Sakumoto University of Rochester, Rochester, New York 14627 O. Saltó
Institut de Fisica d’Altes Energies, Universitat Autonoma de Barcelona,
E-08193, Bellaterra (Barcelona), Spain L. Santidd Istituto Nazionale di
Fisica Nucleare Trieste/Udine, I-34100 Trieste, ddUniversity of Trieste/Udine,
I-33100 Udine, Italy S. Sarkarcc Istituto Nazionale di Fisica Nucleare,
Sezione di Roma 1, ccSapienza Università di Roma, I-00185 Roma, Italy L.
Sartori Istituto Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa,
aaUniversity of Siena and bbScuola Normale Superiore, I-56127 Pisa, Italy K.
Sato Fermi National Accelerator Laboratory, Batavia, Illinois 60510 A.
Savoy-Navarro LPNHE, Universite Pierre et Marie Curie/IN2P3-CNRS, UMR7585,
Paris, F-75252 France P. Schlabach Fermi National Accelerator Laboratory,
Batavia, Illinois 60510 A. Schmidt Institut für Experimentelle Kernphysik,
Universität Karlsruhe, 76128 Karlsruhe, Germany E.E. Schmidt Fermi National
Accelerator Laboratory, Batavia, Illinois 60510 M.A. Schmidt Enrico Fermi
Institute, University of Chicago, Chicago, Illinois 60637 M.P.
Schmidt††footnotemark: Yale University, New Haven, Connecticut 06520 M.
Schmitt Northwestern University, Evanston, Illinois 60208 T. Schwarz
University of California, Davis, Davis, California 95616 L. Scodellaro
Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005
Santander, Spain A. Scribanoaa Istituto Nazionale di Fisica Nucleare Pisa,
zUniversity of Pisa, aaUniversity of Siena and bbScuola Normale Superiore,
I-56127 Pisa, Italy F. Scuri Istituto Nazionale di Fisica Nucleare Pisa,
zUniversity of Pisa, aaUniversity of Siena and bbScuola Normale Superiore,
I-56127 Pisa, Italy A. Sedov Purdue University, West Lafayette, Indiana
47907 S. Seidel University of New Mexico, Albuquerque, New Mexico 87131 Y.
Seiya Osaka City University, Osaka 588, Japan A. Semenov Joint Institute
for Nuclear Research, RU-141980 Dubna, Russia L. Sexton-Kennedy Fermi
National Accelerator Laboratory, Batavia, Illinois 60510 F. Sforza Istituto
Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena
and bbScuola Normale Superiore, I-56127 Pisa, Italy A. Sfyrla University of
Illinois, Urbana, Illinois 61801 S.Z. Shalhout Wayne State University,
Detroit, Michigan 48201 T. Shears University of Liverpool, Liverpool L69
7ZE, United Kingdom P.F. Shepard University of Pittsburgh, Pittsburgh,
Pennsylvania 15260 M. Shimojimaq University of Tsukuba, Tsukuba, Ibaraki 305,
Japan S. Shiraishi Enrico Fermi Institute, University of Chicago, Chicago,
Illinois 60637 M. Shochet Enrico Fermi Institute, University of Chicago,
Chicago, Illinois 60637 Y. Shon University of Wisconsin, Madison, Wisconsin
53706 I. Shreyber Institution for Theoretical and Experimental Physics,
ITEP, Moscow 117259, Russia A. Sidoti Istituto Nazionale di Fisica Nucleare
Pisa, zUniversity of Pisa, aaUniversity of Siena and bbScuola Normale
Superiore, I-56127 Pisa, Italy P. Sinervo Institute of Particle Physics:
McGill University, Montréal, Québec, Canada H3A 2T8; Simon Fraser University,
Burnaby, British Columbia, Canada V5A 1S6; University of Toronto, Toronto,
Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British Columbia, Canada V6T
2A3 A. Sisakyan Joint Institute for Nuclear Research, RU-141980 Dubna,
Russia A.J. Slaughter Fermi National Accelerator Laboratory, Batavia,
Illinois 60510 J. Slaunwhite The Ohio State University, Columbus, Ohio 43210
K. Sliwa Tufts University, Medford, Massachusetts 02155 J.R. Smith
University of California, Davis, Davis, California 95616 F.D. Snider Fermi
National Accelerator Laboratory, Batavia, Illinois 60510 R. Snihur Institute
of Particle Physics: McGill University, Montréal, Québec, Canada H3A 2T8;
Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6; University
of Toronto, Toronto, Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British
Columbia, Canada V6T 2A3 A. Soha University of California, Davis, Davis,
California 95616 S. Somalwar Rutgers University, Piscataway, New Jersey
08855 V. Sorin Michigan State University, East Lansing, Michigan 48824 J.
Spalding Fermi National Accelerator Laboratory, Batavia, Illinois 60510 T.
Spreitzer Institute of Particle Physics: McGill University, Montréal, Québec,
Canada H3A 2T8; Simon Fraser University, Burnaby, British Columbia, Canada V5A
1S6; University of Toronto, Toronto, Ontario, Canada M5S 1A7; and TRIUMF,
Vancouver, British Columbia, Canada V6T 2A3 P. Squillaciotiaa Istituto
Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena
and bbScuola Normale Superiore, I-56127 Pisa, Italy M. Stanitzki Yale
University, New Haven, Connecticut 06520 R. St. Denis Glasgow University,
Glasgow G12 8QQ, United Kingdom B. Stelzer Institute of Particle Physics:
McGill University, Montréal, Québec, Canada H3A 2T8; Simon Fraser University,
Burnaby, British Columbia, Canada V5A 1S6; University of Toronto, Toronto,
Ontario, Canada M5S 1A7; and TRIUMF, Vancouver, British Columbia, Canada V6T
2A3 O. Stelzer-Chilton Institute of Particle Physics: McGill University,
Montréal, Québec, Canada H3A 2T8; Simon Fraser University, Burnaby, British
Columbia, Canada V5A 1S6; University of Toronto, Toronto, Ontario, Canada M5S
1A7; and TRIUMF, Vancouver, British Columbia, Canada V6T 2A3 D. Stentz
Northwestern University, Evanston, Illinois 60208 J. Strologas University of
New Mexico, Albuquerque, New Mexico 87131 G.L. Strycker University of
Michigan, Ann Arbor, Michigan 48109 D. Stuart University of California,
Santa Barbara, Santa Barbara, California 93106 J.S. Suh Center for High
Energy Physics: Kyungpook National University, Daegu 702-701, Korea; Seoul
National University, Seoul 151-742, Korea; Sungkyunkwan University, Suwon
440-746, Korea; Korea Institute of Science and Technology Information,
Daejeon, 305-806, Korea; Chonnam National University, Gwangju, 500-757, Korea
A. Sukhanov University of Florida, Gainesville, Florida 32611 I. Suslov
Joint Institute for Nuclear Research, RU-141980 Dubna, Russia T. Suzuki
University of Tsukuba, Tsukuba, Ibaraki 305, Japan A. Taffardf University of
Illinois, Urbana, Illinois 61801 R. Takashima Okayama University, Okayama
700-8530, Japan Y. Takeuchi University of Tsukuba, Tsukuba, Ibaraki 305,
Japan R. Tanaka Okayama University, Okayama 700-8530, Japan M. Tecchio
University of Michigan, Ann Arbor, Michigan 48109 P.K. Teng Institute of
Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China K. Terashi
The Rockefeller University, New York, New York 10021 J. Thomh Fermi National
Accelerator Laboratory, Batavia, Illinois 60510 A.S. Thompson Glasgow
University, Glasgow G12 8QQ, United Kingdom G.A. Thompson University of
Illinois, Urbana, Illinois 61801 E. Thomson University of Pennsylvania,
Philadelphia, Pennsylvania 19104 P. Tipton Yale University, New Haven,
Connecticut 06520 P. Ttito-Guzmán Centro de Investigaciones Energeticas
Medioambientales y Tecnologicas, E-28040 Madrid, Spain S. Tkaczyk Fermi
National Accelerator Laboratory, Batavia, Illinois 60510 D. Toback Texas A&M
University, College Station, Texas 77843 S. Tokar Comenius University, 842
48 Bratislava, Slovakia; Institute of Experimental Physics, 040 01 Kosice,
Slovakia K. Tollefson Michigan State University, East Lansing, Michigan
48824 T. Tomura University of Tsukuba, Tsukuba, Ibaraki 305, Japan D.
Tonelli Fermi National Accelerator Laboratory, Batavia, Illinois 60510 S.
Torre Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica
Nucleare, I-00044 Frascati, Italy D. Torretta Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 P. Totarodd Istituto Nazionale di Fisica
Nucleare Trieste/Udine, I-34100 Trieste, ddUniversity of Trieste/Udine,
I-33100 Udine, Italy S. Tourneur LPNHE, Universite Pierre et Marie
Curie/IN2P3-CNRS, UMR7585, Paris, F-75252 France M. Trovato Istituto
Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena
and bbScuola Normale Superiore, I-56127 Pisa, Italy S.-Y. Tsai Institute of
Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China Y. Tu
University of Pennsylvania, Philadelphia, Pennsylvania 19104 N. Turiniaa
Istituto Nazionale di Fisica Nucleare Pisa, zUniversity of Pisa, aaUniversity
of Siena and bbScuola Normale Superiore, I-56127 Pisa, Italy F. Ukegawa
University of Tsukuba, Tsukuba, Ibaraki 305, Japan S. Vallecorsa University
of Geneva, CH-1211 Geneva 4, Switzerland N. van Remortelb Division of High
Energy Physics, Department of Physics, University of Helsinki and Helsinki
Institute of Physics, FIN-00014, Helsinki, Finland A. Varganov University of
Michigan, Ann Arbor, Michigan 48109 E. Vatagabb Istituto Nazionale di Fisica
Nucleare Pisa, zUniversity of Pisa, aaUniversity of Siena and bbScuola Normale
Superiore, I-56127 Pisa, Italy F. Vázquezn University of Florida,
Gainesville, Florida 32611 G. Velev Fermi National Accelerator Laboratory,
Batavia, Illinois 60510 C. Vellidis University of Athens, 157 71 Athens,
Greece M. Vidal Centro de Investigaciones Energeticas Medioambientales y
Tecnologicas, E-28040 Madrid, Spain R. Vidal Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 I. Vila Instituto de Fisica de
Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain R. Vilar
Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005
Santander, Spain T. Vine University College London, London WC1E 6BT, United
Kingdom M. Vogel University of New Mexico, Albuquerque, New Mexico 87131 I.
Volobouevt Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley,
California 94720 G. Volpiz Istituto Nazionale di Fisica Nucleare Pisa,
zUniversity of Pisa, aaUniversity of Siena and bbScuola Normale Superiore,
I-56127 Pisa, Italy P. Wagner University of Pennsylvania, Philadelphia,
Pennsylvania 19104 R.G. Wagner Argonne National Laboratory, Argonne,
Illinois 60439 R.L. Wagner Fermi National Accelerator Laboratory, Batavia,
Illinois 60510 W. Wagnerw Institut für Experimentelle Kernphysik, Universität
Karlsruhe, 76128 Karlsruhe, Germany J. Wagner-Kuhr Institut für
Experimentelle Kernphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany T.
Wakisaka Osaka City University, Osaka 588, Japan R. Wallny University of
California, Los Angeles, Los Angeles, California 90024 S.M. Wang Institute
of Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China A.
Warburton Institute of Particle Physics: McGill University, Montréal, Québec,
Canada H3A 2T8; Simon Fraser University, Burnaby, British Columbia, Canada V5A
1S6; University of Toronto, Toronto, Ontario, Canada M5S 1A7; and TRIUMF,
Vancouver, British Columbia, Canada V6T 2A3 D. Waters University College
London, London WC1E 6BT, United Kingdom M. Weinberger Texas A&M University,
College Station, Texas 77843 J. Weinelt Institut für Experimentelle
Kernphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany H. Wenzel Fermi
National Accelerator Laboratory, Batavia, Illinois 60510 W.C. Wester III
Fermi National Accelerator Laboratory, Batavia, Illinois 60510 B. Whitehouse
Tufts University, Medford, Massachusetts 02155 D. Whitesonf University of
Pennsylvania, Philadelphia, Pennsylvania 19104 A.B. Wicklund Argonne
National Laboratory, Argonne, Illinois 60439 E. Wicklund Fermi National
Accelerator Laboratory, Batavia, Illinois 60510 S. Wilbur Enrico Fermi
Institute, University of Chicago, Chicago, Illinois 60637 G. Williams
Institute of Particle Physics: McGill University, Montréal, Québec, Canada H3A
2T8; Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6;
University of Toronto, Toronto, Ontario, Canada M5S 1A7; and TRIUMF,
Vancouver, British Columbia, Canada V6T 2A3 H.H. Williams University of
Pennsylvania, Philadelphia, Pennsylvania 19104 P. Wilson Fermi National
Accelerator Laboratory, Batavia, Illinois 60510 B.L. Winer The Ohio State
University, Columbus, Ohio 43210 P. Wittichh Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 S. Wolbers Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 C. Wolfe Enrico Fermi Institute,
University of Chicago, Chicago, Illinois 60637 T. Wright University of
Michigan, Ann Arbor, Michigan 48109 X. Wu University of Geneva, CH-1211
Geneva 4, Switzerland F. Würthwein University of California, San Diego, La
Jolla, California 92093 S. Xie Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139 A. Yagil University of California, San Diego,
La Jolla, California 92093 K. Yamamoto Osaka City University, Osaka 588,
Japan J. Yamaoka Duke University, Durham, North Carolina 27708 U.K. Yangp
Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637 Y.C.
Yang Center for High Energy Physics: Kyungpook National University, Daegu
702-701, Korea; Seoul National University, Seoul 151-742, Korea; Sungkyunkwan
University, Suwon 440-746, Korea; Korea Institute of Science and Technology
Information, Daejeon, 305-806, Korea; Chonnam National University, Gwangju,
500-757, Korea W.M. Yao Ernest Orlando Lawrence Berkeley National
Laboratory, Berkeley, California 94720 G.P. Yeh Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 J. Yoh Fermi National Accelerator
Laboratory, Batavia, Illinois 60510 K. Yorita Waseda University, Tokyo 169,
Japan T. Yoshidam Osaka City University, Osaka 588, Japan G.B. Yu
University of Rochester, Rochester, New York 14627 I. Yu Center for High
Energy Physics: Kyungpook National University, Daegu 702-701, Korea; Seoul
National University, Seoul 151-742, Korea; Sungkyunkwan University, Suwon
440-746, Korea; Korea Institute of Science and Technology Information,
Daejeon, 305-806, Korea; Chonnam National University, Gwangju, 500-757, Korea
S.S. Yu Fermi National Accelerator Laboratory, Batavia, Illinois 60510 J.C.
Yun Fermi National Accelerator Laboratory, Batavia, Illinois 60510 L.
Zanellocc Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1, ccSapienza
Università di Roma, I-00185 Roma, Italy A. Zanetti Istituto Nazionale di
Fisica Nucleare Trieste/Udine, I-34100 Trieste, ddUniversity of Trieste/Udine,
I-33100 Udine, Italy X. Zhang University of Illinois, Urbana, Illinois 61801
Y. Zhengd University of California, Los Angeles, Los Angeles, California 90024
S. Zucchellix Istituto Nazionale di Fisica Nucleare Bologna, xUniversity of
Bologna, I-40127 Bologna, Italy
###### Abstract
We report results from a search for the lepton flavor violating decays
$B^{0}_{(s)}\rightarrow e^{+}\mu^{-}$, and the flavor-changing neutral-current
decays $B^{0}_{(s)}\rightarrow e^{+}e^{-}$. The analysis uses data
corresponding to ${\rm 2\;fb^{-1}}$ of integrated luminosity of $p\bar{p}$
collisions at $\sqrt{s}=1.96~{}{\rm TeV}$ collected with the upgraded Collider
Detector (CDF II) at the Fermilab Tevatron. The observed number of
$B^{0}_{(s)}$ candidates is consistent with background expectations. The
resulting Bayesian upper limits on the branching ratios at 90% credibility
level are $\mathcal{B}(B^{0}_{s}\rightarrow e^{+}\mu^{-})<2.0\times 10^{-7}$,
$\mathcal{B}(B^{0}\rightarrow e^{+}\mu^{-})<6.4\times 10^{-8}$,
$\mathcal{B}(B^{0}_{s}\rightarrow e^{+}e^{-})<2.8\times 10^{-7}$ and
$\mathcal{B}(B^{0}\rightarrow e^{+}e^{-})<8.3\times 10^{-8}$. From the limits
on $\mathcal{B}(B^{0}_{(s)}\rightarrow e^{+}\mu^{-})$, the following lower
bounds on the Pati-Salam leptoquark masses are also derived:
${M_{LQ}}(B^{0}_{s}\rightarrow e^{+}\mu^{-})>47.8\;{\rm TeV/c^{2}}$, and
${M_{LQ}}(B^{0}\rightarrow e^{+}\mu^{-})>59.3\;{\rm TeV/c^{2}}$, at 90%
credibility level.
###### pacs:
13.20.He 13.30.Ce 12.15.Mm 12.60.-i
CDF Collaboration222With visitors from aUniversity of Massachusetts Amherst,
Amherst, Massachusetts 01003, bUniversiteit Antwerpen, B-2610 Antwerp,
Belgium, cUniversity of Bristol, Bristol BS8 1TL, United Kingdom, dChinese
Academy of Sciences, Beijing 100864, China, eIstituto Nazionale di Fisica
Nucleare, Sezione di Cagliari, 09042 Monserrato (Cagliari), Italy, fUniversity
of California Irvine, Irvine, CA 92697, gUniversity of California Santa Cruz,
Santa Cruz, CA 95064, hCornell University, Ithaca, NY 14853, iUniversity of
Cyprus, Nicosia CY-1678, Cyprus, jUniversity College Dublin, Dublin 4,
Ireland, kUniversity of Edinburgh, Edinburgh EH9 3JZ, United Kingdom,
lUniversity of Fukui, Fukui City, Fukui Prefecture, Japan 910-0017 mKinki
University, Higashi-Osaka City, Japan 577-8502 nUniversidad Iberoamericana,
Mexico D.F., Mexico, oQueen Mary, University of London, London, E1 4NS,
England, pUniversity of Manchester, Manchester M13 9PL, England, qNagasaki
Institute of Applied Science, Nagasaki, Japan, rUniversity of Notre Dame,
Notre Dame, IN 46556, sUniversity de Oviedo, E-33007 Oviedo, Spain, tTexas
Tech University, Lubbock, TX 79609, uIFIC(CSIC-Universitat de Valencia), 46071
Valencia, Spain, vUniversity of Virginia, Charlottesville, VA 22904,
wBergische Universität Wuppertal, 42097 Wuppertal, Germany, eeOn leave from J.
Stefan Institute, Ljubljana, Slovenia,
Rare particle decays that are either forbidden within the standard model of
particle physics (SM), or are expected to have very small branching ratios
provide excellent signatures with which to look for new physics and allow to
probe subatomic processes that are beyond the reach of direct searches. The
decays $B^{0}_{(s)}\rightarrow e^{+}\mu^{-}$ qconj are forbidden within the
SM, in which lepton number and lepton flavor are conserved. However the
observation of neutrino oscillations indicates that lepton flavor is not
conserved. To date, no lepton flavor violating (LFV) decays in the charged
sector such as $B^{0}_{(s)}\rightarrow e^{+}\mu^{-}$ have been observed. These
decays are allowed in models where the SM has been extended by heavy singlet
Dirac neutrinos DIRAC . The LFV decays are also allowed in some physics
scenarios beyond the SM, such as the Pati-Salam model pati-salam and
supersymmetry (SUSY) models SUSY . The grand-unification theory by J. Pati and
A. Salam predicts a new interaction to mediate transitions between leptons and
quarks via exchange of spin-1 gauge bosons, which are called Pati-Salam
leptoquarks (LQ), that carry both color and lepton quantum numbers pati-salam
. The lepton and quark components of the leptoquarks are not necessarily from
the same generation scott ; Blanke , and the decays $B^{0}_{s}\rightarrow
e^{+}\mu^{-}$ and $B^{0}\rightarrow e^{+}\mu^{-}$ can be mediated by different
types of leptoquarks. Processes involving flavor-changing neutral currents
(FCNCs) can occur in the SM only through higher-order Feynman diagrams where
new physics contributions can provide a significant enhancement. Compared to
$B^{0}_{(s)}\rightarrow\mu^{+}\mu^{-}$ btomumu , the FCNC decays of
$B^{0}_{(s)}\rightarrow e^{+}e^{-}$ are further suppressed by the square of
the ratio of the electron and muon masses $(m_{e}/m_{\mu})^{2}$. The SM
expectations for branching ratios of $B^{0}_{(s)}\rightarrow e^{+}e^{-}$ are
of the order of $10^{-15}$ Misiak .
In this Letter we report on a search for the LFV decays
$B^{0}_{(s)}\rightarrow e^{+}\mu^{-}$ and the FCNC decays
$B^{0}_{(s)}\rightarrow e^{+}e^{-}$, using a data sample corresponding to 2
$\mathrm{fb}^{-1}$of integrated luminosity collected in $p\bar{p}$ collisions
at $\sqrt{s}=1.96$ TeV. With no evidence for either the LFV or FCNC decays, we
set upper limits on their branching ratios using the common reference decay
$B^{0}\rightarrow K^{+}\pi^{-}$, which has a precisely-known branching ratio.
This is the first time a search for $B^{0}_{s}\rightarrow e^{+}e^{-}$ has been
performed.
A detailed description of the CDF II detector can be found in Ref. JpsiPRD .
Here we give a brief description of the detector elements most relevant to
this analysis. Charged particle tracking is provided by a silicon microstrip
detector together with the surrounding open-cell wire drift chamber (COT),
both immersed in a 1.4 T axial magnetic field. The tracking system provides
precise vertex and momentum measurement for charged particles in the
pseudorapidity range $\left|\eta\right|<1.0$ csys . Surrounding the tracking
system are electromagnetic (CEM) and hadronic sampling calorimeters, arranged
in a projective geometry. Drift chambers and scintillation counters are
located behind the calorimeters to detect muons within $\left|\eta\right|<0.6$
(CMU) and $0.6<\left|\eta\right|<1.0$ (CMX).
We use a data sample enriched in two-body $B$–decays selected by a three-level
trigger system using the extremely fast tracker XFT at level-1, and the
silicon vertex trigger SVT at level-2. The trigger requires two oppositely-
charged tracks, each with a transverse momentum $p_{T}>2~{}{\rm GeV/c}$, and
an impact parameter IP $0.1<d_{0}<1$ mm. It also requires the scalar sum of
the transverse momenta of the two tracks to be greater than 5.5 GeV/c, the
difference in the azimuthal angles of the tracks
$20^{\circ}<\Delta\varphi<135^{\circ}$, and a transverse decay length lxy
$L_{xy}>200$ $\mu$m. At the level-3 trigger stage, and in the offline
analysis, the trigger selections are enforced with a more accurate
determination of the same quantities. In the off-line analysis, additionally
we require: the $B$–meson isolation ${\rm I}>0.675$ isodef , the pointing
angle $\Delta\phi<6.3^{\circ}$ poidef , and a tighter selection of
$L_{xy}>375$ $\mu$m. These three thresholds were optimized in an unbiased way
to obtain the best sensitivity for the searches using the procedure described
in Ref. punzi .
Electron and muon identification is applied in the selection of
${B^{0}_{(s)}\rightarrow e^{+}\mu^{-}}$ and ${B^{0}_{(s)}\rightarrow
e^{+}e^{-}}$ decay modes. The electron identification eID requires that both
the specific ionization ($\mathit{dE/dx}$) measured in the COT, and the
transverse and longitudinal shower shape as measured in the CEM, be consistent
with the hypothesis that the particle is an electron. The performance of
electron identification is optimized using pure electron samples reconstructed
from $\gamma\rightarrow e^{+}e^{-}$ conversions and hadron and muon samples
from $D^{0}\rightarrow K^{-}\pi^{+}$, $\Lambda\rightarrow p\pi^{-}$, and
$J/\psi\rightarrow\mu^{+}\mu^{-}$ decays. We find the identification
efficiency to be around 70% for electrons. The muon identification starts from
tracks in the COT that are extrapolated into the muon detectors and are
required to match hits in the muon systems. The muon selection is fully
efficient for muons with $p_{T}>$ 2 GeV/c in CMU or CMX.
The mass resolution $\sigma_{m}$ of fully-reconstructed $B$–meson decays to
two charged particles is about 28 MeV/c2. Energy loss due to bremsstrahlung by
electrons generates a tail on the low side of the mass distribution. This tail
is more prominent for the $B^{0}_{(s)}\rightarrow e^{+}e^{-}$ channels, where
two electrons are involved. We define search windows of (5.262–5.477) GeV/c2
for $B^{0}_{s}\rightarrow e^{+}\mu^{-}$ and (5.171–5.387) GeV/c2 for
$B^{0}\rightarrow e^{+}\mu^{-}$. These correspond to a window around the
nominal values of the $B^{0}_{s}$ and $B^{0}$ masses PDG of approximately
$\pm 3\sigma_{m}$. To recover some of the acceptance loss due to electron
bremsstrahlung for the $B^{0}_{(s)}\rightarrow e^{+}e^{-}$ channels, we choose
wider and asymmetric search windows ranging from 6 $\sigma_{m}$ below to 3
$\sigma_{m}$ above the nominal values of the $B^{0}_{s}$ and $B^{0}$ masses.
The search windows are (5.154–5.477) GeV/c2 for the $B^{0}_{s}$ and
(5.064–5.387) GeV/c2 for the $B^{0}$. The sideband regions (4.800–5.028)
GeV/c2 and (5.549–5.800) GeV/c2 are used to estimate the combinatorial
backgrounds.
The background contributions considered include combinations of random track
pairs and partial $B$ decays that accidentally meet the selection requirement
(combinatorial), and hadronic two-body $B$ decays in which both final
particles are misidentified as leptons. The combinatorial background is
evaluated by extrapolating the normalized number of events found in the
sidebands to the signal region. The double-lepton misidentification rate is
determined by applying electron and muon misidentification probabilities to
the number of two-body decays found in the search window.
Figure 1 shows the invariant mass distribution for $e^{+}\mu^{-}$ candidates.
We observe one event in the $B^{0}_{s}$ mass window, and two events in the
$B^{0}$ mass window, consistent with the estimated total background of $0.8\pm
0.6$ events in the $B^{0}_{s}$ search window, and $0.9\pm 0.6$ in the $B^{0}$
window. The combinatorial background in both channels is estimated to be
$0.7\pm 0.6$ events. The number of events where two tracks are misidentified
as electron and muon is estimated to be $0.09\pm 0.02$ for the $B^{0}_{s}$
case and $0.22\pm 0.04$ for the $B^{0}$ case.
Figure 2 shows the invariant mass distributions for $e^{+}e^{-}$ candidate
pairs where both tracks were identified as electrons. We observe one event in
the $B^{0}_{s}$ mass window, and two events in the $B^{0}$ mass window. We
estimate the total background contributions to be $2.7\pm 1.8$ events in both
the $B^{0}_{s}$ and $B^{0}$ mass windows. The dominant contribution comes from
combinatorial background: $2.7\pm 1.8$ compared to the contribution where both
tracks are misidentified as electrons: $0.038\pm 0.008$ for both $B^{0}_{s}$
or $B^{0}$.
We use the reference decay $B^{0}\rightarrow K^{+}\pi^{-}$ to set a limit on
$\mathcal{B}(B^{0}_{s}\rightarrow e^{+}\ell^{-})$ (where $\ell$ is either $e$
or $\mu$), using the following expression:
$\displaystyle\mathcal{B}(B^{0}_{s}\rightarrow e^{+}\ell^{-})=$
$\displaystyle\frac{N(B^{0}_{s}\rightarrow
e^{+}\ell^{-})\cdot\mathcal{B}(B^{0}\rightarrow K^{+}\pi^{-})\cdot
f_{d}/f_{s}}{\epsilon^{rel}_{B^{0}_{s}\rightarrow e^{+}\ell^{-}}\cdot
N(B^{0}\rightarrow K^{+}\pi^{-})}.$
The expression for the $B^{0}$ channels is identical, except that the ratio of
$b$–quark fragmentation probabilities: $f_{d}/f_{s}$ is not present. In the
expression, $N(B^{0}_{s}\rightarrow e^{+}\ell^{-})$ is the calculated upper
limit on the number of $B^{0}_{s}\rightarrow e^{+}\ell^{-}$ events,
$N(B^{0}\rightarrow K^{+}\pi^{-})$ is the observed number of events from the
reference channel $B^{0}\rightarrow K^{+}\pi^{-}$,
$\mathcal{B}(B^{0}\rightarrow K^{+}\pi^{-})=(19.4\pm 0.6)\times 10^{-6}$ PDG
is the branching ratio for the $B^{0}\rightarrow K^{+}\pi^{-}$ decay, and
$\epsilon^{rel}_{B^{0}_{s}\rightarrow e^{+}\ell^{-}}$ is the detector
acceptance and event selection efficiency for reconstructing
$B^{0}_{s}\rightarrow e^{+}\ell^{-}$ decays relative to that for
$B^{0}\rightarrow K^{+}\pi^{-}$. The value of $f_{d}/f_{s}$ is $3.86\pm 0.59$,
where the (anti-)correlation between the uncertainties has been accounted for
HFAG2006 . To calculate the detector acceptance, we use simulated events with
a detailed simulation of the CDF II detector and event selection. We obtain
$\epsilon^{rel}_{B^{0}_{s}\rightarrow e^{+}\mu^{-}}=0.207\pm 0.016$,
$\epsilon^{rel}_{B^{0}\rightarrow e^{+}\mu^{-}}=0.210\pm 0.012$,
$\epsilon^{rel}_{B^{0}_{s}\rightarrow e^{+}e^{-}}=0.129\pm 0.011$, and
$\epsilon^{rel}_{B^{0}\rightarrow e^{+}e^{-}}=0.128\pm 0.011$. The
uncertainties listed above are the combined statistical and systematic
uncertainties. The later include uncertainties from detector fiducial
coverage, electron and muon identification efficiencies, detector material
determination, $B^{0}_{(s)}$ $p_{T}$ spectrum, and $B^{0}_{(s)}$ lifetimes.
The reference channel $B^{0}\rightarrow K^{+}\pi^{-}$ has been reconstructed
using the same selection criteria except lepton identification. We find
$6387\pm 214$ $B^{0}\rightarrow K^{+}\pi^{-}$ events, using a fitting
procedure similar to that described in Ref. b2hh-180pb .
The upper limit on the branching ratio in each search window is obtained using
the Bayesian approach PDG , assuming a flat prior, and incorporating Gaussian
uncertainties into the limit. The total systematic uncertainties, listed in
Table 1, are used as input for the limit calculation. Table 2 lists the upper
limits we obtain on the branching ratios at 90% (95%) credibility level
(C.L.).
Table 1: Values used to calculate the limits on $\mathcal{B}(B^{0}_{(s)}\rightarrow e^{+}\mu^{-})$ and $\mathcal{B}(B^{0}_{(s)}\rightarrow e^{+}e^{-})$ and their uncertainties. Source | Values | $\mathcal{B}(B^{0}_{s}\rightarrow e^{+}\mu^{-})$ | $\mathcal{B}(B^{0}\rightarrow e^{+}\mu^{-})$ | $\mathcal{B}(B^{0}_{s}\rightarrow e^{+}e^{-})$ | $\mathcal{B}(B^{0}\rightarrow e^{+}e^{-})$
---|---|---|---|---|---
$N(B^{0}\rightarrow K^{+}\pi^{-})$ | $6387\pm 214$ | 3.4% | 3.4% | 3.4% | 3.4%
$\mathcal{B}(B^{0}\rightarrow K^{+}\pi^{-})$ | $(19.4\pm 0.6)\times 10^{-6}$ | 3.1% | 3.1% | 3.1% | 3.1%
$f_{B^{0}}/f_{B^{0}_{s}}$ | $3.86\pm 0.59$ | 15.3% | - | 15.3% | -
$\epsilon^{rel}_{B^{0}_{s}\rightarrow e^{+}\mu^{-}}$ | $0.207\pm 0.016$ | 7.6% | - | - | -
$\epsilon^{rel}_{B^{0}\rightarrow e^{+}\mu^{-}}$ | $0.210\pm 0.012$ | - | 5.9% | - | -
$\epsilon^{rel}_{B^{0}_{s}\rightarrow e^{+}e^{-}}$ | $0.129\pm 0.011$ | - | - | 8.9% | -
$\epsilon^{rel}_{B^{0}\rightarrow e^{+}e^{-}}$ | $0.128\pm 0.011$ | - | - | - | 8.9%
Total | | 17.7% | 7.5% | 18.3% | 10.0%
Table 2: Branching ratio limits at 90(95) % C.L.
$\mathcal{B}(B^{0}_{s}\rightarrow e^{+}\mu^{-})<2.0~{}(2.6)\times 10^{-7}$
---
$\mathcal{B}(B^{0}\rightarrow e^{+}\mu^{-})<6.4~{}(7.9)\times 10^{-8}$
$\mathcal{B}(B^{0}_{s}\rightarrow e^{+}e^{-})<2.8\times 10^{-7}$
$\mathcal{B}(B^{0}\rightarrow e^{+}e^{-})<8.3\times 10^{-8}$
Within the Pati-Salam leptoquark model, the following relationship between the
$\mathcal{B}(B^{0}_{(s)}\rightarrow e^{+}\mu^{-})$ and the leptoquark mass
($M_{LQ}$) can be derived scott :
$\displaystyle\mathcal{B}(B^{0}_{(s)}\rightarrow e^{+}\mu^{-})$
$\displaystyle=$
$\displaystyle\pi\alpha^{2}_{s}(M_{LQ})\frac{1}{M^{4}_{LQ}}F^{2}_{B^{0}_{(s)}}m^{3}_{B^{0}_{(s)}}R^{2}\cdot\frac{\tau_{B^{0}_{(s)}}}{\hbar},$
where
$R=\frac{m_{B^{0}_{(s)}}}{m_{b}}\left(\frac{\alpha_{s}(M_{LQ})}{\alpha_{s}(m_{t})}\right)^{-\frac{4}{7}}\left(\frac{\alpha_{s}(m_{t})}{\alpha_{s}(m_{b})}\right)^{-\frac{12}{23}}$.
The values and uncertainties of the quantities used in the calculation of
$M_{LQ}$ are the following PDG : the top-quark mass $m_{t}$ (171.2 $\pm$ 2.1
GeV/c2), the bottom quark mass $m_{b}$ (4.20 $\pm$ 0.17 GeV/c2), the charm
quark mass $m_{c}$ (1.27 $\pm$ 0.11 GeV/c2), the $B^{0}$-meson mass
$m_{B^{0}}$ (5.27953 $\pm$ 0.00033 GeV/c2), the $B^{0}_{s}$-meson mass
$m_{B^{0}_{s}}$ (5.3663 $\pm$ 0.0006 GeV/c2), the $B^{0}$-meson lifetime
$\tau_{B^{0}}$ (1.530 $\pm$ 0.009 ps), the $B^{0}_{s}$-meson lifetime
$\tau_{B^{0}_{s}}$ (1.470 $\pm$ 0.027 ps), the coupling strength $F_{B^{0}}$
(0.178 $\pm$ 0.014 GeV), and $F_{B^{0}_{s}}$ (0.200 $\pm$ 0.014 GeV)constant .
For the strong coupling constant we use $\alpha_{s}(M_{Z^{0}})$ = 0.115, which
is evolved to $M_{LQ}$ using the Marciano approximation marciano assuming no
colored particles exist with masses between $m_{t}$ and $M_{LQ}$. Using the
limits on the branching ratios listed in Table 2, we calculate limits on the
masses of the corresponding Pati-Salam leptoquarks of
${M_{LQ}}(B^{0}_{s}\rightarrow e^{+}\mu^{-})>47.8~{}(44.9)\;{\rm TeV/c^{2}}$
and ${M_{LQ}}(B^{0}\rightarrow e^{+}\mu^{-})>59.3~{}(56.3)\;{\rm TeV/c^{2}}$
at 90 (95)% C.L. Figure 3 shows the limit and the relation between the
leptoquark mass and the branching ratio for the $B^{0}_{s}$ meson.
Figure 1: Invariant mass distribution of $e^{+}\mu^{-}$ pairs for events where
one track passed the electron identification and the other track the muon
identification. The $B^{0}_{s}$ ($B^{0}$) search window is indicated by the
solid (dotted) line. The sideband regions are indicated by the dashed lines.
Figure 2: Invariant mass distributions of $e^{+}e^{-}$ pairs for events where
both tracks passed the electron identification. The $B^{0}_{s}$ ($B^{0}$)
search window is indicated by the solid (dotted) line. The sideband regions
are indicated by dashed lines. Figure 3: Leptoquark mass limit corresponding
to the 90 (95) % C.L. on $\mathcal{B}(B^{0}_{s}\rightarrow e^{+}\mu^{-})$. The
error band is obtained by varying the values entering the theoretical
calculation within their uncertainties. The uncertainties stemming from
approximating $\alpha_{s}$ are not included.
In summary, we report on a search for the lepton flavor violating decays
$B^{0}_{(s)}\rightarrow e^{+}\mu^{-}$ and the flavor changing neutral current
decays $B^{0}_{(s)}\rightarrow e^{+}e^{-}$ using data corresponding to 2
$\mathrm{fb}^{-1}$ of integrated luminosity collected in $p\overline{p}$
collisions at $\sqrt{s}=1.96$ TeV. This is the first search for
$B^{0}_{s}\rightarrow e^{+}e^{-}$ decays. We observe no evidence for these
decays and set limits that are the most stringent to date. These results
represent a significant improvement compared to the previous measurement
CDF_RUNI by CDF and the best results from $B$-Factories babar07 ; belle03 ;
CLEO2 .
We thank the Fermilab staff and the technical staffs of the participating
institutions for their vital contributions. This work was supported by the
U.S. Department of Energy and National Science Foundation; the Italian
Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Culture,
Sports, Science and Technology of Japan; the Natural Sciences and Engineering
Research Council of Canada; the National Science Council of the Republic of
China; the Swiss National Science Foundation; the A.P. Sloan Foundation; the
Bundesministerium für Bildung und Forschung, Germany; the Korean Science and
Engineering Foundation and the Korean Research Foundation; the Science and
Technology Facilities Council and the Royal Society, UK; the Institut National
de Physique Nucleaire et Physique des Particules/CNRS; the Russian Foundation
for Basic Research; the Ministerio de Ciencia e Innovación, and Programa
Consolider-Ingenio 2010, Spain; the Slovak R&D Agency; and the Academy of
Finland.
## References
* (1) Throughout this Letter inclusion of charge conjugate reactions is implied.
* (2) A. Ilakovac, Phys. Rev. D 62, 036010 (2000).
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* (6) M. Blanke et al. J. High Energy Phys. 05 (2007) 103.
* (7) T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. 100, 101802 (2008).
* (8) M. Misiak and J. Urban, Phys. Lett. B 451, 161 (1999); G. Buchalla and A. J. Burn, Nucl. Phys. B548, 309 (1999).
* (9) D. Acosta et al. (CDF Collaboration), Phys. Rev. D 71, 032001 (2005); and references therein.
* (10) The polar angle $(\theta)$ in cylindrical coordinates is measured with respect to the proton beam direction, which defines the $z$-axis. Pseudorapidity $(\eta)$ is defined as $\eta=-\ln(\tan\frac{\theta}{2})$.
* (11) E.J. Thomson et al., IEEE Trans. Nucl. Sci. 49, 1063 (2002).
* (12) W. Ashmanskas et al., Nucl. Instrum. Methods A518, 532 (2004).
* (13) The impact parameter $d_{0}$ is the distance of closest approach of the track to the beam line.
* (14) F. Abe et al. (CDF Collaboration), Phys. Rev. D 57, 5382 (1998).
* (15) Due to the hard $b$-quark fragmentation, $B$-mesons carry most of the momentum of the $b$-quark. The isolation is defined as $\mathit{I}=p_{T}(B)/(\sum_{i}p^{i}_{T}+p_{T}(B))$, where $p_{T}(B)$ is the transverse momentum of the $B$ candidate, and the sum runs over all other tracks within a cone of radius 1 in $\eta-\phi$ space around the $B$ flight direction.
* (16) For track pairs coming from the two-body decay of a $B$, the vector pointing from the primary vertex to the $B$ decay vertex in the transverse plane $\vec{l}_{xy}$ should point in the same direction as the transverse momentum vector $\vec{p}_{\rm{T}}$(B) of the $B$ candidate. $\Delta\phi$ is defined as the angle between $\vec{l}_{xy}$ and $\vec{p}_{\rm{T}}(B)$.
* (17) G. Punzi, arXiv:physics/0308063v2.
* (18) A. Abulencia et al. (CDF Collaboration), Phys. Rev. Lett. 97, 012002 (2006).
* (19) C. Amsler et al., Physics Letters B667, 1 (2008)
* (20) Heavy Flavor Averaging Group, arXiv:hep-ex/0704.3575v1.
* (21) A. Abulencia et al. (CDF Collaboration), Phys. Rev. Lett. 97, 211802 (2006).
* (22) J. Bordes et al. J. High Energy Phys. 12 (2004) 064.
* (23) W. J. Marciano, Phys. Rev. D 29, 580 (1984).
* (24) F. Abe et al. (CDF Collaboration), Phys. Rev. Lett. 81, 5742 (1998).
* (25) B. Aubert et al. (B AB AR Collaboration), Phys. Rev. Lett. 99, 251803 (2007).
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|
arxiv-papers
| 2009-01-24T00:39:41 |
2024-09-04T02:49:00.190329
|
{
"license": "Public Domain",
"authors": "The CDF Collaboration: T. Aaltonen, et al",
"submitter": "Hans Wenzel",
"url": "https://arxiv.org/abs/0901.3803"
}
|
0901.3837
|
# Electroweak Chiral Lagrangian for a Hypercharge-universal Topcolor Model
Jun-Yi Lang1, Shao-Zhou Jiang1, Qing Wang1,2111Corresponding author at:
Department of Physics, Tsinghua University, Beijing 100084, P.R.China
Email address: wangq@mail.tsinghua.edu.cn(Q.Wang). 1Department of
Physics,Tsinghua University,Beijing 100084,P.R.China
2Center for High Energy Physics, Tsinghua University, Beijing 100084,
P.R.china
(Jan 24, 2009)
###### Abstract
Electroweak chiral Lagrangian for a hypercharge-universal topcolor model is
investigated. We find that the assignments of universal hypercharge improve
the results obtained previously from K.Lane’s prototype natural TC2 model by
allowing a larger $Z^{\prime}$ mass resulting in a very small T parameter and
the S parameter is still around the order of $+1$.
PACS numbers: 12.60.Nz; 11.10.Lm, 11.30.Rd, 12.10.Dm
††preprint: TUHEP-TH-09167
Topcolor-assisted technicolor (TC2) is a class of new physics models which
combines technicolor and topcolor together to realize the electroweak symmetry
breaking dynamically. In these theories, a technicolor condensate provides the
masses to the weak vector bosons and an extended technicolor (ETC) sector
gives masses to the light quarks and leptons, and a bottom-quark-sized mass to
the top. The majority of the top-quark mass is due to the formation of a top-
quark condensate through the dynamics of an extended color gauge sector. The
typical gauge group of the TC2 models is
$\displaystyle SU(N)_{TC}\otimes SU(3)_{1}\otimes SU(3)_{2}\otimes
SU(2)_{L}\otimes U(1)_{Y_{1}}\otimes U(1)_{Y_{2}}$ (1)
, in which the extended color and hypercharge groups $SU(3)_{1}\otimes
SU(3)_{2}\otimes U(1)_{Y_{1}}\otimes U(1)_{Y_{2}}$ spontaneously break to
their diagonal subgroup $SU(3)_{C}\otimes U(1)_{Y}$ at a few TeVs and the
remaining electroweak groups $SU(2)_{L}\otimes U(1)_{Y}$ spontaneously break
to their electromagnetic subgroup $U(1)_{\mathrm{em}}$ at the electroweak
scale due to a combination of a top-quark condensate and a technifermion
condensate. In the original TC2 model Hill95 ; Lane95 , the extended hyper-
charge sector $U(1)_{Y_{1}}\otimes U(1)_{Y_{2}}$ is usually arranged
nonuniversal in flavor to ensure that the bottom-quark and other light quarks
and leptons do not condensate. Recently a new type of TC2 model with a flavor-
universal extended hyper-charge sector is proposed in Ref.Sekhar , the authors
there have examined various experimental and theoretical constraints, finding
that precision electroweak measurements yield the strongest bounds on the
model and the goodness of fit to all available Z-pole and LEP2 data for
hypercharge-universal topcolor is comparable to that of the standard model
(SM). In contrast, TC2 models with a flavor nonuniversal hypercharge sector
are markedly disfavored by the data. The similar result on the nonuniversal
hypercharge TC2 models is also obtained from our works HongHao08 ; JunYi09 ,
where we have computed the coefficients of the bosonic part of electroweak
chiral Lagrangian (EWCL) up to the order $p^{4}$ and found an upper bound for
the mass of flavor nonuniversal $Z^{\prime}$ boson. For Hill’s schematic TC2
model Hill95 , $Z^{\prime}$ mass $M_{Z^{\prime}}$ is a few TeVs and the S
parameter can be either positive or negative depending on whether the
$M_{Z^{\prime}}$ is large or small HongHao08 . While for K.Lane’s prototype
natural TC2 model Lane95 , $M_{Z^{\prime}}$ must be smaller than 400GeV and
the S parameter is around order of $+1$ JunYi09 . Since Ref.Sekhar already
shows explicitly the experiment fit of the TC2 model due to the changes from
the nonuniversal to the universal assignments for hypercharge sector, it is
worthwhile to apply our formulation developed in Ref.HongHao08 to the flavor-
universal hypercharge topcolor model proposed in Ref.Sekhar to examine the
improvements from an alternative point of view. Our formulation offers an
upper bound on nonuniversal $Z^{\prime}$ mass previously, while Ref.Sekhar
gives a lower bound of universal $Z^{\prime}$ mass of roughly 2TeV. We expect
that applying our formulation to flavor-universal hypercharge topcolor model
produces an upper bound on universal $Z^{\prime}$ mass which will compensate
the lower bound for the mass of universal $Z^{\prime}$ boson obtained from
Ref.Sekhar . In fact, from EWCL point of view, except technicolor and
$Z^{\prime}$ contributions, there are many other different sources to
influence EWCL coefficients. In Ref.JunYi09 , we have made efforts to
investigate the effective four-fermion interactions induced by extended
technicolor (ETC). We find that their effects are small and we further point
out that the walking technicolor (WTC) effects are worth future investigation.
Considering that the authors in Ref.Sekhar assume that WTC effects do not
generate large precision electroweak corrections, up to present stage, we
ignore WTC effects in this work.
In this paper, we are mainly interested in the effects from flavor-universal
hypercharge sector, to reduce the computations and to be convenient for
comparison with flavor-nonuniversal hypercharge model, we base our
calculations on the K.Lane’s prototype natural TC2 model Lane95 discussed in
Ref.JunYi09 , but change its hypercharge assignments to that given in
Ref.Sekhar . The gauge charges are shown as Table I.
TABLE I. Gauge charge assignments of techniquarks for hypercharge universal
TC2 model discussed in present paper. These techniquarks are $SU(3)_{1}\otimes
SU(3)_{2}$ singlets.
field | $T_{L}^{l}$ | $U_{R}^{l}$ | $D_{R}^{l}$ | $T_{L}^{t}$ | $U_{R}^{t}$ | $D_{R}^{t}$ | $T_{L}^{b}$ | $U_{R}^{b}$ | $D_{R}^{b}$
---|---|---|---|---|---|---|---|---|---
$SU(N)$ | N | N | N | N | N | N | N | N | N
$SU(2)_{L}$ | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1
$U(1)_{\mbox{\tiny$Y_{1}$}}$ | 0 | $\frac{1}{2}$ | -$\frac{1}{2}$ | 0 | $\frac{1}{2}$ | -$\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $-\frac{1}{2}$
$U(1)_{\mbox{\tiny$Y_{2}$}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
In later numerical computations, technicolor group representation will be
taken to be $N=3$.
The action of the symmetry breaking sector is
$\displaystyle S_{\rm
SBS}[G_{\mu}^{\alpha},A_{1\mu}^{A},A_{2\mu}^{A},W_{\mu}^{a},B_{1\mu},B_{2\mu},\bar{T}^{l},T^{l},\bar{T}^{t},T^{t},\bar{T}^{b},T^{b}]$
$\displaystyle=\int d^{4}x({\cal L}_{\mathrm{gauge}}+{\cal
L}_{\mathrm{techniquark}}+\mathcal{L}_{\mathrm{breaking}}+\mathcal{L}_{\mathrm{4T}})\;,~{}~{}$
(2)
with ${\cal L}_{\mathrm{techniquark}}$, $\mathcal{L}_{\mathrm{breaking}}$ and
$\mathcal{L}_{\mathrm{4T}}$ being the same as those in Ref.JunYi09 and the
modified techinquark Lagrangian with flavor-universal hypercharge is
$\displaystyle\mathcal{L}_{\mathrm{techniquark}}$ $\displaystyle=$
$\displaystyle\bar{T}^{l}(i\not{\partial}-g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}-g_{2}\frac{\tau^{a}}{2}\not{W}^{a}P_{L}-\frac{1}{2}q_{1}\not{B}_{1}\tau^{3}P_{R})T^{l}+\bar{T}^{t}(i\not{\partial}-g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}-g_{2}\frac{\tau^{a}}{2}\not{W}^{a}P_{L}$ (3)
$\displaystyle-\frac{1}{2}q_{1}\not{B}_{1}\tau^{3}P_{R})T^{t}+\bar{T}^{b}(i\not{\partial}-g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}-g_{2}\frac{\tau^{a}}{2}\not{W}^{a}P_{L}-\frac{1}{2}q_{1}\not{B}_{1}\tau^{3}P_{R})T^{b}\;.$
Rotating hypercharge gauge fields $B_{1\mu}$ and $B_{2\mu}$ as
$\displaystyle\begin{pmatrix}B_{1\mu}&B_{2\mu}\end{pmatrix}=\begin{pmatrix}Z_{\mu}^{\prime}&B_{\mu}\end{pmatrix}\begin{pmatrix}\cos\theta^{\prime}&-\sin\theta^{\prime}\\\
\sin\theta^{\prime}&\cos\theta^{\prime}\end{pmatrix}\;,\hskip
28.45274ptg_{1}=q_{1}\sin\theta^{\prime}=q_{2}\cos\theta^{\prime}\;.~{}~{}~{}~{}$
(4)
The techinquark Lagrangian (3) is then reduced to
$\displaystyle{\cal
L}_{\mathrm{techniquark}}=\bar{\psi}(i\not{\partial}-g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}+\not{V}+\not{A}\gamma^{5})\psi\;,~{}~{}$ (5)
where all three doublets techniquarks are arranged in one by six matrix
$\psi=(U^{l},D^{l},U^{t},D^{t},U^{b},D^{b})^{T}$ and
$\displaystyle
V_{\mu}=(-\frac{1}{2}g_{2}\frac{\tau^{a}}{2}W_{\mu}^{a}-\frac{1}{2}g_{1}\frac{\tau^{3}}{2}B_{\mu})\otimes\mathbf{I}+Z_{V\mu}\hskip
28.45274ptA_{\mu}=(\frac{1}{2}g_{2}\frac{\tau^{a}}{2}W_{\mu}^{a}-\frac{1}{2}g_{1}\frac{\tau^{3}}{2}B_{\mu})\otimes\mathbf{I}+Z_{A\mu}\;,$
(6)
with $\mathbf{I}=\mathrm{diag}(1,1,1)$,
$Z_{V\mu}=\mathrm{diag}(Z_{V\mu}^{l},Z_{V\mu}^{t},Z_{V\mu}^{b})$,
$Z_{A\mu}=\mathrm{diag}(Z_{A\mu}^{l},Z_{A\mu}^{t},Z_{A\mu}^{b})$ and
$\displaystyle
Z_{V\mu}^{l}=Z_{V\mu}^{t}=Z_{V\mu}^{b}=Z_{A\mu}^{l}=Z_{A\mu}^{t}=Z_{A\mu}^{b}=-\frac{1}{4}g_{1}\cot\theta^{\prime}Z^{\prime}_{\mu}\tau^{3}$
(7)
As done in Ref.JunYi09 , the EWCL for present model is
$\displaystyle\exp\bigg{(}iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]\bigg{)}$
$\displaystyle=$
$\displaystyle\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}G_{\mu}^{\alpha}\mathcal{D}Z^{\prime}_{\mu}e^{iS_{\mathrm{SBS}}[G_{\mu}^{\alpha}\\!,0,0,W_{\mu}^{a}\\!,B_{1\mu}\\!,B_{2\mu}\\!,\bar{T}^{l}\\!,T^{l}\\!,\bar{T}^{t}\\!,T^{t}\\!,\bar{T}^{b}\\!,T^{b}]}~{}~{}$
(8) $\displaystyle=$
$\displaystyle\mathcal{N}[W_{\mu}^{a},B_{\mu}]\int\mathcal{D}\mu(U)\exp\bigg{(}iS_{\mathrm{eff}}[U,W_{\mu}^{a},B_{\mu}]\bigg{)}\;,$
where $U(x)$ is a dimensionless unitary unimodular matrix field in EWCL, and
${\cal D}\mu(U)$ denotes the normalized functional integration measure on $U$.
The normalization factor $\mathcal{N}[W_{\mu}^{a},B_{\mu}]$ is determined
through the requirement that when the technicolor interactions are switched
off, $S_{\mathrm{eff}}[U,W_{\mu}^{a},B_{\mu}]$ must vanish.
The following computation procedure is exactly the same as those given in
Ref.JunYi09 , in which we integrated out the technigluons, the techniquarks
and the colorons. We abbreviate the detailed process and only write down the
resulted action,
$\displaystyle\int{\cal D}G_{\mu}^{\alpha}{\cal D}\bar{\psi}{\cal D}\psi{\cal
D}Z^{\prime}_{\mu}e^{iS_{\mathrm{SBS}}\big{|}_{A^{A}_{1\mu}=A^{A}_{2\mu}=0}}=\int\mathcal{D}\mu(U){\cal
D}Z^{\prime}_{\mu}e^{iS_{\mathrm{Z^{\prime}}}[U,W_{\mu}^{a},B_{\mu},Z^{\prime}_{\mu}]}\;,$
(9)
with
$\displaystyle
S_{\mathrm{Z^{\prime}}}[U,W_{\mu}^{a},B_{\mu},Z^{\prime}_{\mu}]$
$\displaystyle=$
$\displaystyle-i\mathrm{Tr}\log(i\not{\partial}+\not{V}+\not{A}\gamma^{5})+\int
d^{4}x\bigg{[}-\frac{1}{4}W_{\mu\nu}^{a}W^{a,\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}Z^{\prime}_{\mu\nu}Z^{\prime\mu\nu}$
(10)
$\displaystyle+\frac{1}{2}M_{0}^{2}Z^{\prime}_{\mu}Z^{\prime\mu}+3\mathrm{tr}_{f}\bigg{(}(F_{0}^{1D})^{2}a^{2}-\mathcal{K}_{1}^{1D,\Sigma\neq
0}(d_{\mu}a^{\mu})^{2}-\mathcal{K}_{2}^{1D,\Sigma\neq
0}(d_{\mu}a_{\nu}-d_{\nu}a_{\mu})^{2}$
$\displaystyle+\mathcal{K}_{3}^{1D,\Sigma\neq
0}(a^{2})^{2}+\mathcal{K}_{4}^{1D,\Sigma\neq
0}(a_{\mu}a_{\nu})^{2}-\mathcal{K}_{13}^{1D,\Sigma\neq
0}V_{\mu\nu}V^{\mu\nu}+i\mathcal{K}_{14}^{1D,\Sigma\neq
0}a_{\mu}a_{\nu}V^{\mu\nu}\bigg{)}\bigg{]}$
$\displaystyle+\mathcal{O}(p^{6})\;,$
where $M_{0}$ is the bare mass of $Z^{\prime}$ boson from spontaneously
breaking of $SU(3)_{1}\otimes SU(3)_{2}\otimes U(1)_{Y_{1}}\otimes
U(1)_{Y_{2}}\Rightarrow SU(3)_{C}\otimes U(1)_{Y}$, as in Ref.JunYi09 its
relation with vacuum expectation value $\tilde{v}$ causing breaking is
$M_{0}^{2}=\frac{25}{36}\frac{g_{1}^{2}\tilde{v^{2}}}{\sin^{2}\\!\theta^{\prime}\cos^{2}\\!\theta^{\prime}}$.
The coefficients $F_{0}^{1D}$, $\mathcal{K}_{i}^{1D,\Sigma\neq 0}$ for
$i=1,2,3,4,13,14$ are strong interaction coefficients for one doublet
technicolor model which depend on techniquark self energy and are already
computed numerically in Ref.HongHao08 ; JunYi09 . Further
$\displaystyle v_{\mu}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{2}(g_{2}\frac{\tau^{a}}{2}W_{\xi\mu}^{a}+g_{1}\frac{\tau^{3}}{2}B_{\xi\mu})-\frac{1}{4}g_{1}\cot\theta^{\prime}Z^{\prime}_{\mu}\tau^{3}\;,$
(11) $\displaystyle a_{\mu}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}(g_{2}\frac{\tau^{a}}{2}W_{\xi\mu}^{a}-g_{1}\frac{\tau^{3}}{2}B_{\xi\mu})-\frac{1}{4}g_{1}\cot\theta^{\prime}Z^{\prime}_{\mu}\tau^{3}\;,$
(12)
in which $W_{\xi\mu}^{a}$ and $B_{\xi\mu}$ are rotated electroweak gauge
fields given in Eq.(26) and (27) in Ref.JunYi09 which absorb Goldstone field
$U$ into the definition of gauge fields.
We can further decompose (10) into
$\displaystyle
S_{\mathrm{Z^{\prime}}}[U,W_{\mu}^{a},B_{\mu},Z^{\prime}_{\mu}]=\tilde{S}_{\mathrm{Z^{\prime}}}[U,W_{\mu}^{a},B_{\mu},Z^{\prime}_{\mu}]+S_{\mathrm{Z^{\prime}}}[U,W_{\mu}^{a},B_{\mu},0]\;,$
(13)
where
$\tilde{S}_{\mathrm{Z^{\prime}}}[U,W_{\mu}^{a},B_{\mu},Z^{\prime}_{\mu}]$ is
the $Z^{\prime}$ dependent part of
$S_{\mathrm{eff}}[U,W_{\mu}^{a},B_{\mu},Z^{\prime}_{\mu}]$. We find that the
$Z^{\prime}$ independent part
$S_{\mathrm{Z^{\prime}}}[U,W_{\mu}^{a},B_{\mu},0]$ is just the same as that
given in Ref.JunYi09 which is three times of the one-doublet technicolor
model result given in Ref.HongHao08 . Similar as Ref.JunYi09
$\tilde{S}_{\mathrm{Z^{\prime}}}[U,W_{\mu}^{a},B_{\mu},Z^{\prime}_{\mu}]$ has
the structure
$\displaystyle\tilde{S}_{\mathrm{Z^{\prime}}}[U,W_{\mu}^{a},B_{\mu},Z^{\prime}_{\mu}]=\int
d^{4}x~{}[\frac{1}{2}Z^{\prime}_{R,\mu}D_{Z}^{-1,\mu\nu}Z^{\prime}_{R,\nu}+Z_{R}^{\prime,\mu}J_{Z,\mu}+Z_{R}^{2}Z_{R,\mu}^{\prime}J^{\mu}_{3Z}+g_{4Z}\frac{g_{1}^{4}}{c_{Z^{\prime}}^{4}}Z_{R}^{\prime,4}]\;,~{}~{}~{}~{}$
(14)
where
$D_{Z}^{-1,\mu\nu}=g^{\mu\nu}(\partial^{2}+M^{2}_{Z^{\prime}})-(1+\lambda_{Z})\partial^{\mu}\partial^{\nu}+\Delta^{\mu\nu}_{Z}(X)$
and to normalize $Z^{\prime}$ field correctly, we introduce normalized field
$Z^{\prime}_{R,\mu}$ as
$Z^{\prime}_{\mu}=\frac{1}{c_{Z^{\prime}}}Z^{\prime}_{R,\mu}$. Due to the
present universal assignment of hypercharge, parameters appeared in
$\tilde{S}_{\mathrm{Z^{\prime}}}[U,W_{\mu}^{a},B_{\mu},Z^{\prime}_{\mu}]$ are
different from those in Ref.JunYi09 ,
$\displaystyle c_{Z^{\prime}}^{2}$ $\displaystyle=$ $\displaystyle
1+3\mathcal{K}g_{1}^{2}\cot^{2}\theta^{\prime}+\frac{3}{2}\mathcal{K}_{2}^{1D,\Sigma\neq
0}g_{1}^{2}\cot^{2}\theta^{\prime}+\frac{3}{2}\mathcal{K}_{13}^{1D,\Sigma\neq
0}g_{1}^{2}\cot^{2}\theta^{\prime}\;,$ (15) $\displaystyle M_{Z^{\prime}}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{c_{Z^{\prime}}^{2}}\\{M_{0}^{2}+\frac{3g_{1}^{2}\cot^{2}\theta^{\prime}}{4}(F_{0}^{1D})^{2}\\}\;,$
(16) $\displaystyle\lambda_{Z}$ $\displaystyle=$
$\displaystyle-\frac{3g_{1}^{2}\cot^{2}\theta^{\prime}}{4c_{Z^{\prime}}^{2}}\mathcal{K}_{1}^{1D,\Sigma\neq
0}\;,$ (17) $\displaystyle\Delta^{\mu\nu}_{Z}(X)$ $\displaystyle=$
$\displaystyle\frac{g_{1}^{2}\cot^{2}\theta^{\prime}}{16c_{Z^{\prime}}^{2}}\bigg{[}(-12\mathcal{K}_{1}^{1D,\Sigma\neq
0}-3\mathcal{K}_{3}^{1D,\Sigma\neq 0}+6\mathcal{K}_{13}^{1D,\Sigma\neq
0}-3\mathcal{K}_{14}^{1D,\Sigma\neq
0})\mathrm{tr}[X_{\mu}\tau^{3}]\mathrm{tr}[X^{\nu}\tau^{3}]$ (18)
$\displaystyle+(24\mathcal{K}_{1}^{1D,\Sigma\neq
0}-6\mathcal{K}_{4}^{1D,\Sigma\neq 0}-12\mathcal{K}_{13}^{1D,\Sigma\neq
0}+6\mathcal{K}_{14}^{1D,\Sigma\neq 0})\mathrm{tr}[X_{\mu}X^{\nu}]$
$\displaystyle+g^{\mu\nu}(-3\mathcal{K}_{3}^{1D,\Sigma\neq
0}+3\mathcal{K}_{4}^{1D,\Sigma\neq 0}+12\mathcal{K}_{13}^{1D,\Sigma\neq
0}-6\mathcal{K}_{14}^{1D,\Sigma\neq 0})\mathrm{tr}[X_{k}X^{k}]$
$\displaystyle+g^{\mu\nu}(-3\mathcal{K}_{4}^{1D,\Sigma\neq
0}-6\mathcal{K}_{13}^{1D,\Sigma\neq 0}+3\mathcal{K}_{14}^{1D,\Sigma\neq
0})\mathrm{tr}[X_{k}\tau^{3}]\mathrm{tr}[X^{k}\tau^{3}]\bigg{]}\;,$
$\displaystyle J_{Z}^{\mu}$ $\displaystyle=$ $\displaystyle
J_{Z0}^{\mu}+\frac{g_{1}^{2}\gamma}{c_{Z^{\prime}}}\partial^{\nu}B_{\mu\nu}+\tilde{J}_{Z}^{\mu}\;,$
(19) $\displaystyle J_{Z0\mu}$ $\displaystyle=$
$\displaystyle\frac{3g_{1}\cot\theta^{\prime}}{4c_{Z^{\prime}}}i(F_{0}^{1D})^{2}\mathrm{tr}[X_{\mu}\tau^{3}]\;,$
(20) $\displaystyle\gamma$ $\displaystyle=$
$\displaystyle-3\mathcal{K}\cot\theta^{\prime}-\frac{3}{2}(\mathcal{K}_{2}^{1D,\Sigma\neq
0}+\mathcal{K}_{13}^{1D,\Sigma\neq 0})\cot\theta^{\prime}\;,$ (21)
$\displaystyle\tilde{J}_{Z}^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{g_{1}\cot\theta^{\prime}}{4c_{Z^{\prime}}}\bigg{[}\mathcal{K}_{1}^{1D,\Sigma\neq
0}\\{3i\mathrm{tr}[U^{{\dagger}}(D^{\nu}D_{\nu}U)U^{{\dagger}}D^{\mu}U\tau^{3}]-3i\mathrm{tr}[U^{{\dagger}}(D^{\nu}D_{\nu}U)\tau^{3}U^{{\dagger}}D^{\mu}U]$
(22)
$\displaystyle-3i\partial^{\mu}\mathrm{tr}[U^{{\dagger}}(D^{\nu}D_{\nu}U)\tau^{3}]\\}+(-6\mathcal{K}_{2}^{1D,\Sigma\neq
0}+6\mathcal{K}_{13}^{1D,\Sigma\neq
0})\partial_{\nu}\mathrm{tr}[\overline{W}^{\mu\nu}\tau^{3}]$
$\displaystyle+(\frac{3i}{4}\mathcal{K}_{3}^{1D,\Sigma\neq
0}-\frac{3i}{4}\mathcal{K}_{4}^{1D,\Sigma\neq
0}-3\mathcal{K}_{13}^{1D,\Sigma\neq
0}+\frac{3i}{2}\mathcal{K}_{14}^{1D,\Sigma\neq
0})\mathrm{tr}[X^{\nu}X_{\nu}]\mathrm{tr}[X^{\mu}\tau^{3}]$
$\displaystyle+(\frac{3i}{2}\mathcal{K}_{4}^{1D,\Sigma\neq
0}+3\mathcal{K}_{13}^{1D,\Sigma\neq
0}-\frac{3i}{2}\mathcal{K}_{14}^{1D,\Sigma\neq
0})\mathrm{tr}[X^{\mu}X_{\nu}]\mathrm{tr}[X^{\nu}\tau^{3}]$
$\displaystyle+(-3\mathcal{K}_{13}^{1D,\Sigma\neq
0}+\frac{3}{4}\mathcal{K}_{14}^{1D,\Sigma\neq
0})\mathrm{tr}[\overline{W}^{\mu\nu}(X_{\nu}\tau^{3}-\tau^{3}X_{\nu})]$
$\displaystyle+(6i\mathcal{K}_{13}^{1D,\Sigma\neq
0}-\frac{3}{2}i\mathcal{K}_{14}^{1D,\Sigma\neq
0})\partial_{\nu}\mathrm{tr}[X^{\mu}X^{\nu}\tau^{3}]\bigg{]}\;,$
$\displaystyle g_{4Z}$ $\displaystyle=$
$\displaystyle(\mathcal{K}_{3}^{1D,\Sigma\neq
0}+\mathcal{K}_{4}^{1D,\Sigma\neq 0})\frac{3\cot^{4}\theta^{\prime}}{128}\;,$
(23) $\displaystyle J_{3Z}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{3ig_{1}^{3}\cot^{3}\theta^{\prime}}{32c_{Z^{\prime}}^{3}}(\mathcal{K}_{3}^{1D,\Sigma\neq
0}+\mathcal{K}_{4}^{1D,\Sigma\neq 0})\mathrm{tr}[X^{\mu}\tau_{3}]\;,$ (24)
where
$\displaystyle\mathcal{K}=-\frac{1}{48\pi^{2}}\left(\log\frac{\kappa^{2}}{\Lambda^{2}}+\gamma\right)\hskip
28.45274pt\Lambda,\kappa\mbox{: ultraviolet and infrared cutoffs}\;.$ (25)
With similar procedure of Ref.JunYi09 to integrate out the $Z^{\prime}$
field, we find that $S_{\mathrm{eff}}[U,W_{\mu}^{a},B_{\mu}]$ defined in (8)
has exactly the standard structure of EWCL given by Ref.EWCL , from which we
can read out coefficients up to order of $p^{4}$ as follows,
$\displaystyle f^{2}$ $\displaystyle=$ $\displaystyle
3(F_{0}^{1D})^{2}\;,\hskip
56.9055pt\beta_{1}=\frac{3(F_{0}^{1D})^{2}g_{1}^{2}\cot^{2}\theta^{\prime}}{8M^{2}_{0}+6(F_{0}^{1D})^{2}g_{1}^{2}\cot^{2}\theta^{\prime}}\;,$
(26) $\displaystyle\alpha_{1}$ $\displaystyle=$ $\displaystyle
3L_{10}^{1D}+\frac{3(F_{0}^{1D})^{2}}{2M_{Z^{\prime}}^{2}}\beta_{1}+2\beta_{1}\tan\theta^{\prime}\gamma-6\beta_{1}L_{10}^{1D}\;,$
$\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}L_{9}^{1D}+\frac{3(F_{0}^{1D})^{2}}{2M_{Z^{\prime}}^{2}}\beta_{1}+2\beta_{1}\tan\theta^{\prime}\gamma+3\beta_{1}L_{9}^{1D}\;,$
$\displaystyle\alpha_{3}$ $\displaystyle=$
$\displaystyle(-\frac{3}{2}+3\beta_{1})L_{9}^{1D}\;,$
$\displaystyle\alpha_{4}$ $\displaystyle=$ $\displaystyle
3L_{2}^{1D}+6\beta_{1}L_{9}^{1D}+\frac{3(F_{0}^{1D})^{2}}{2M_{Z^{\prime}}^{2}}\beta_{1}\;,$
$\displaystyle\alpha_{5}$ $\displaystyle=$
$\displaystyle\frac{3}{2}L_{3}^{1D}+3L_{1}^{1D}-\frac{3(F_{0}^{1D})^{2}}{2M_{Z^{\prime}}^{2}}\beta_{1}-6\beta_{1}L_{9}^{1D}\;,$
$\displaystyle\alpha_{6}$ $\displaystyle=$
$\displaystyle-\frac{3(F_{0}^{1D})^{2}}{2M_{Z^{\prime}}^{2}}\beta_{1}+24\beta_{1}^{2}L_{1}^{1D}-6\beta_{1}(4L_{1}^{1D}+L_{9}^{1D})\;,$
(27) $\displaystyle\alpha_{7}$ $\displaystyle=$
$\displaystyle\frac{3(F_{0}^{1D})^{2}}{2M_{Z^{\prime}}^{2}}\beta_{1}+6\beta_{1}^{2}(L_{3}^{1D}+2L_{1}^{1D})-2\beta_{1}(3L_{3}^{1D}+6L_{1}^{1D}-3L_{9}^{1D})\;,$
$\displaystyle\alpha_{8}$ $\displaystyle=$
$\displaystyle-\frac{3(F_{0}^{1D})^{2}}{2M_{Z^{\prime}}^{2}}\beta_{1}+12\beta_{1}L_{10}^{1D}\;,$
$\displaystyle\alpha_{9}$ $\displaystyle=$
$\displaystyle-\frac{3(F_{0}^{1D})^{2}}{2M_{Z^{\prime}}^{2}}\beta_{1}+6\beta_{1}(L_{10}^{1D}-L_{9}^{1D})\;,$
$\displaystyle\alpha_{10}$ $\displaystyle=$
$\displaystyle-4\beta_{1}^{2}(-18L_{1}^{1D}-3L_{3}^{1D})+32\beta_{1}^{4}\cot^{4}\theta^{\prime}g_{4Z}-\frac{3}{2}\beta_{1}^{3}(96L_{1}^{1D}+16L_{3}^{1D})\;,$
$\displaystyle\alpha_{11}$ $\displaystyle=$
$\displaystyle\alpha_{12}=\alpha_{13}=\alpha_{14}=0\;,$
where $L_{i}^{1D}$ for $i=1,3,9,10$ are EWCL coefficients for one doublet
technicolor model discussed in Ref.HongHao08 .
The features of these results which are the same as those in K.Lane’s model
are:
1. 1.
The contributions to the $p^{4}$ order coefficients are divided into two
parts: the three doublets technicolor model contribution and the $Z^{\prime}$
contribution.
2. 2.
All corrections from the $Z^{\prime}$ particle are at least proportional to
$\beta_{1}$ which vanish if the mixing disappear by $\theta^{\prime}=0$.
3. 3.
Since $L_{10}^{\mathrm{1D}}<0$, combining with positive $\beta_{1}$, (27) then
tells us $\alpha_{8}$ is negative. Then $U=-16\pi\alpha_{8}$ coefficient given
in Ref.EWCL is always positive in present model.
Since $\alpha_{1}$ and $\alpha_{2}$ depend on $\gamma$ which from (21) further
rely on an extra parameter $\mathcal{K}$. We can combine (26),(15) and (16)
together to fix $\mathcal{K}$,
$\displaystyle\frac{3(F_{0}^{1D})^{2}g_{1}^{2}\cot^{2}\theta^{\prime}}{8\beta_{1}M_{Z^{\prime}}^{2}}$
$\displaystyle=$ $\displaystyle
1+3\mathcal{K}g_{1}^{2}\cot^{2}\theta^{\prime}+\frac{3}{2}\mathcal{K}_{2}^{1D,\Sigma\neq
0}g_{1}^{2}\cot^{2}\theta^{\prime}+\frac{3}{2}\mathcal{K}_{13}^{1D,\Sigma\neq
0}g_{1}^{2}\cot^{2}\theta^{\prime}\;.$ (28)
Once $\mathcal{K}$ is fixed, with the help of (25), we can determine the ratio
of infrared cutoff $\kappa$ and ultraviolet cutoff $\Lambda$, in Fig.2, we
draw the $\kappa/\Lambda$ as functions of $T$ and $M_{Z^{\prime}}$, we find
natural criteria $\Lambda>\kappa$ offers stringent constraints on the allowed
region for $T$ and $M_{Z^{\prime}}$ that present theory prefers small $T$
parameter. The upper bound for $Z^{\prime}$ mass increases as the value of $T$
decrease, for example, upper bound is below 1TeV for $T$ being order of
$10^{-3}$ and 2-3TeV for $T$ being order of $10^{-5}$. In Fig.2, we draw
$Z^{\prime}$ mass as a function of $T$ parameter and the gray region is the
forbidden zone where $\kappa\geq\Lambda$.
Figure 1: The ratio of infrared cutoff and ultraviolet cutoff $\kappa/\Lambda$
as functions of the $T$ parameter and $Z^{\prime}$ mass in unit of TeV.
Figure 2: Upper bound of $Z^{\prime}$ mass in unit of TeV as a function of
the $T$ parameter and $\kappa/\Lambda$.
Not like K.Lane’s model discussed in Ref.JunYi09 where we have the upper
bound of $Z^{\prime}$ mass 400GeV, now this upper bound is pushed higher as
long as we have a very small $T$ parameter. Considering that Ref.Sekhar
already gives lower bound of $M_{Z^{\prime}}=2.08$TeV, from Fig.2 we find it
corresponds to $T<7.09\times 10^{-5}$. With this constraints on
$M_{Z^{\prime}}$, in Fig.3 we further draw the S parameter in terms of $T$ and
$M_{Z^{\prime}}$. From this graph, we find that the S parameter in the region
of $T<7.09\times 10^{-5}$ and $M_{Z^{\prime}}>2$TeV is still at order of $+1$
which implies present model is still not fully matching with the experiment
data. Compared to previous result for K.Lane’s natural TC2 model with
nonuniversal hypercharge assignments, we find that the value of the S
parameter does decrease due to the universal hypercharge. For example,
$S\approx 1.1$ at $T=10^{-2}$ for K.Lane’s model, while $S\approx 0$ at
$T=10^{-2}$ for present model, this is compatible with result obtained in
Ref.Sekhar , but for more smaller T parameter, $S$ increases and finally for
$M_{Z^{\prime}}$ at 2-3TeV, $S$ is still at order of $+1$. Finally for
completion of our discussion, we depict all nonzero coefficients $\alpha_{i}$.
Fig.4 is the graph for $\alpha_{1}$ and $\alpha_{2}$, Fig.5 is for
$\alpha_{3}$, $\alpha_{4}$ and $\alpha_{7}$, Fig.6 is for $\alpha_{5}$,
$\alpha_{6}$, $\alpha_{9}$ and $\alpha_{8}$, Fig.7 is for $\alpha_{10}$. In
all these diagrams, we find that the curves are not sensitive to
$M_{Z^{\prime}}$ when $M_{Z^{\prime}}>1-2$TeV, therefore we do not label the
$M_{Z^{\prime}}$ on the graph. For Fig.5, Fig.6 and Fig.7, the T axis starts
from $10^{-3}$ instead of $10^{-6}$, since below $T=10^{-3}$, all curves
approach to zero.
Figure 3: The S parameter as functions of $T$ and $M_{Z^{\prime}}$
Figure 4: $\alpha_{1}$ and $\alpha_{2}$ as functions of $T$
Figure 5: $\alpha_{3}$, $\alpha_{4}$ and $\alpha_{7}$ as functions of $T$
Figure 6: $\alpha_{5}$,$\alpha_{6}$,$\alpha_{8}$ and $\alpha_{9}$ as functions
of $T$
Figure 7: $\alpha_{10}$ as a function of $T$
To summarize, we apply the formulation developed in Ref.HongHao08 to a
hypercharge-universal topcolor model, compute all the coefficients of the
bosonic part of EWCL up to the order of $p^{4}$. We find that the universal
hypercharge does improve the model from the original nonuniversal hypercharge
assignments by allowing a larger $Z^{\prime}$ mass resulting in a very small T
parameter, but the S parameter is still kept at order of $+1$.
## Acknowledgments
This work was supported by National Science Foundation of China (NSFC) under
Grant No. 10875065.
## References
* (1) C.T.Hill, Phys.Lett.B 345, 483(1995)
* (2) K.Lane and E.Eichten, Phys.Lett. B 352, 382(1995)
* (3) F.Braam, M.Flossdorf, R.S.Chivukula, S.D.Chiara and E.H.Simmons, Phys. Rev. D 77, 055005(2008)
* (4) H.H.Zhang, S.Z.Jiang, J.Y.Lang and Q.Wang, Phys. Rev. D. 77, 055003(2008)
* (5) J.Y.Lang, S.Z.Jiang and Q.Wang, Phys. Rev. D. 79, 015002(2009)
* (6) T.Appelquist and G-H. Wu, Phys. Rev. D48, 3235(1993); D51, 240(1995)
|
arxiv-papers
| 2009-01-24T14:26:01 |
2024-09-04T02:49:00.199833
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun-Yi Lang, Shao-Zhou Jiang, Qing Wang",
"submitter": "Wang Qing",
"url": "https://arxiv.org/abs/0901.3837"
}
|
0901.3929
|
# Revisiting the Age of Enlightenment from a
Collective Decision Making Systems Perspective
Marko A. Rodriguez and Jennifer H. Watkins
Los Alamos National Laboratory
Los Alamos, New Mexico 87545 M.A. Rodriguez is with T-5/Center for Nonlinear
Studies, Los Alamos National Laboratory, Los Alamos, NM 87545 USA e-mail:
marko@lanl.gov.Jennifer H. Watkins is with the International and Applied
Technology Group, Los Alamos National Laboratory, Los Alamos, NM 87545 USA
e-mail: jhw@lanl.gov.This research was conducted by the Collective Decision
Making Systems (CDMS) project at the Los Alamos National Laboratory
(http://cdms.lanl.gov).Rodriguez, M.A., Watkins, J.H., “Revisiting the Age of
Enlightenment from a Collective Decision Making Systems Perspective,” First
Monday, volume 14, number 8, ISSN:1396-0466, LA-UR-09-00324, University of
Illinois at Chicago Library, August 2009.
###### Abstract
The ideals of the eighteenth century’s Age of Enlightenment are the foundation
of modern democracies. The era was characterized by thinkers who promoted
progressive social reforms that opposed the long-established aristocracies and
monarchies of the time. Prominent examples of such reforms include the
establishment of inalienable human rights, self-governing republics, and
market capitalism. Twenty-first century democratic nations can benefit from
revisiting the systems developed during the Enlightenment and reframing them
within the techno-social context of the Information Age. This article explores
the application of social algorithms that make use of Thomas Paine’s (English:
1737–1809) representatives, Adam Smith’s (Scottish: 1723–1790) self-interested
actors, and Marquis de Condorcet’s (French: 1743–1794) optimal decision making
groups. It is posited that technology-enabled social algorithms can better
realize the ideals articulated during the Enlightenment.
###### Index Terms:
collective decision making, computational governance, e-participation,
e-democracy, computational social choice theory.
## I Introduction
Eighteenth century Europe is referred to as The Age of Enlightenment, a period
when prominent thinkers began to question traditional forms of authority and
power and the moral standards that supported these forms. One of the most
significant and enduring contributions of the time was the notion that a
government’s existence should be predicated on protecting and supporting the
natural, immutable rights of its citizens. Among these rights are the right to
self-governance, autonomy of thought, and equality. The inherent virtue of
these ideas forced many European nations to relinquish time-honored
aristocratic and monarchic systems. Moreover, it was the philosophy of the
Enlightenment that inspired the formalization of a governing structure that
would define a new nation: the United States of America.
Natural rights exposed during the Enlightenment are immutable. That is, they
are rights not granted by the government, but instead are rights inherent to
man. However, the systems that maintain and support these rights merit no such
permanence. While modern democratic governments strive to achieve the ideals
of the Enlightenment, it is put forth that governments can better serve them
by making greater use of the technological advances of the present day
Information Age. The technological infrastructure that now supports modern
nations removes the physical restrictions that dictated many of the design
choices of these early government architects. As such, many of today’s
government structures are remnants of the technological constraints of the
eighteenth century. Modern nations have an obligation to improve their systems
so as to better ensure the fulfillment of the rights of man. Inscribed at the
Jefferson Memorial is this statement by Thomas Jefferson (American:
1743–1826), another thinker of the Enlightenment: “[…] institutions must go
hand in hand with the progress of the human mind. As that becomes more
developed, more enlightened […] institutions must advance also to keep pace
with the times.” To move in this direction, the principle of citizen
representation as articulated by Thomas Paine (English: 1737–1809) and the
principle of competitive actors for the common good as articulated by Adam
Smith (Scottish: 1723–1790) are considered from a techno-social, collective
decision making systems perspective. Moreover, the rationale for these
principles can be understood within the mathematical formulations of Marquis
de Condorcet’s (French: 1743–1794) requirements for optimal decision making.
## II The Condorcet Jury Theorem: Ensuring Optimal Decision Making
Marquis de Condorcet (portrayed in Figure 1) ardently supported equal rights
and free and universal public education. These ideals were driven as much by
his ethics as they were by his mathematical investigations into the
requirements for optimal decision making.
Figure 1: A portrait of Marquis de Condorcet. This is a public domain
photograph courtesy of Wikimedia Commons.
One of his most famous results is the Condorcet statement and its associated
theorem. In his 1785 Essai sur l’Application de l’Analyse aux Probabilités des
Decisions prises à la Pluralité des Voix (english translation: Essay on the
Application of Analysis to the Probability of Majority Decisions), Condorcet
states that when a group of “enlightened” decision makers chooses between two
options under a majority rule, then as the size of the decision making
population tends toward infinity, it becomes a certainty that the best choice
is rendered [1]. The first statistical proof of this statement is the
Condorcet jury theorem. The model is expressed as follows. Imagine there
exists $n$ independent decision makers and each decision maker has a
probability $p\in[0,1]$ of choosing the best of two options in a decision. If
$p>0.5$, meaning that each individual decision maker is enlightened, and as
$n\rightarrow\infty$, the probability of a majority vote outcome rendering the
best decision approaches certainty at $1.0$. This is known as the “light side”
of the Condorcet jury theorem. The “dark side” of the theorem states that if
$p<0.5$ and as $n\rightarrow\infty$, the probability of a majority vote
outcome rendering the best decision approaches $0.0$. Figure 2 plots the
relationship between $p$ and $n$, where the gray scale values denote a range
from 100% probability of the group rendering the best decision (white) to a 0%
probability (black).
Figure 2: The relationship between $p\in[0,1]$ and $n\in(1,2,\ldots,100)$
according to the Condorcet jury theorem model. Darker values represent a lower
probability of a majority vote rendering the best decision and the lighter
values represent a higher probability of a majority vote rendering the best
decision.
The Condorcet jury theorem is one of the original formal justifications for
the application of democratic principles to government. While the theorem does
not reveal any startling conditions for a successful democracy, it does
distill the necessary conditions to two variables (under simple assumptions).
If a decision making group has a large $n$ and a $p>0.5$, then the group is
increasing its chances of optimal decision making. Unfortunately, the theorem
does not suggest a means to achieve these conditions, though in practice many
mechanisms do exist that strive to meet them. For instance, democracies do not
rely on a single decision maker, but instead use senates, parliaments, and
referendums to increase the size of their voting population. Moreover, for
general elections, equal voting rights facilitate large citizen participation.
Furthermore, democratic nations tend to promote universal public education so
as to ensure that competent leaders are chosen from and by an enlightened
populace. It is noted that the practices employed by democratic nations to
ensure competent decision making are implementation choices, and a society
must not value the implementation of its government. Tradition must be forgone
if another implementation would serve better. Implementations of government
should be altered and amended so as to better realize the ideals of the
nation.
Technology-enabled social algorithms may provide a means by which to reliably
achieve the conditions of the “light” side of the Condorcet jury theorem, thus
ensuring optimal decision making. Furthermore, modern algorithms have the
potential to do so in a manner that better honors the right of each citizen to
participate in government decision making as such algorithms are not
constrained by eighteenth century technology. Present day social algorithms,
in the form of information retrieval and recommendation services, already
contribute significantly to the augmentation of human and social intelligence
[2]. In line with these developments, this article presents two social
algorithms that show promise as mechanisms for governance-based collective
decision making. One algorithm exaggerates Thomas Paine’s citizen
representation in order to accurately simulate the behavior of a large
decision making population ($n\rightarrow\infty$), and the other employs Adam
Smith’s market philosophy to induce participation by the enlightened within
that population ($p\rightarrow 1$). Both algorithms utilize the Condorcet jury
theorem to the society’s advantage.
## III Dynamically Distributed Democracy: Simulating a Large Decision Making
Population
Figure 3: An oil portrait of Thomas Paine painted by Auguste Millière in
1880. This is a public domain photograph courtesy of Wikimedia Commons.
Thomas Paine (portrayed in Figure 3) was born in England, but in his middle
years, he relocated to America on the recommendation of Benjamin Franklin. It
was in America, in the time leading up to the American Revolution, that his
enlightened ideals were well received. In 1776, the year in which the
Declaration of Independence was written, Thomas Paine wrote a widely
distributed pamphlet entitled Common Sense which outlined the values of a
democratic society [3]. This pamphlet discussed the equality of man and the
necessity for all those at stake to partake in the decision making processes
of the group. As a formal justification of this value, the Condorcet jury
theorem would hold that a direct democracy would be the most likely democracy
to yield optimal decisions as the voting population is the largest it can
possibly be for a nation. In practice, the desire for a direct democracy is
tempered by the tremendous burden that constant voting would impinge on
citizens (not to mention the logistical problems such a model would incur
within present day voting infrastructures). For this reason, representation is
required. Thomas Paine states that when populations are small “some convenient
tree will afford them a State house”, but as the population increases it
becomes a necessity for representatives to act on behalf of their
constituents. Moreover, the central tenet of political representation is that
representatives “act in the same manner as the whole body would act were they
present.” The remainder of this section presents a social algorithm that
simulates the manner in which the whole population would act without requiring
pre-elected, long-standing representatives.
Assuming a two-option majority rule, an individual citizen’s judgement can be
placed along a continuum between the two options such that the “political
tendency” of citizen $i$ is denoted $\mathbf{x}_{i}\in[0,1]$. For example,
given United States politics, a political tendency of $0$ represents a fully
Republican perspective, a tendency of $1$ represents a fully Democratic
perspective, and a tendency of $0.5$ denotes a moderate. Given this
definition, there are two ways to quantify the population as a whole. One way
is to calculate the average tendency of all citizens. That is
$d^{\text{tend}}=\frac{1}{n}\sum_{i=1}^{i\leq n}\mathbf{x}_{i}$, where
$d^{\text{tend}}\in[0,1]$ is the collective tendency of the population. Given
a uniform distribution of political tendency within $\mathbf{x}$, the
collective tendency approaches $0.5$ as the size of the population increases
toward infinity. The other way to quantify the group is to require that the
citizen’s tendency be reduced to a binary option (i.e. a two option vote). If
a citizen has a political tendency that is less than $0.5$, then they will
vote $0$. For a tendency greater than $0.5$, they will vote $1$. If they have
a tendency equal to $0.5$ then a fair coin toss will determine their vote.
This majority wins vote is denoted $d^{\text{vote}}\in\\{0,1\\}$.
Imagine a direct democracy in the purest sense, where a raise of hands or a
shout of voices is replaced by an Internet architecture and a sophisticated
error- and fraud-proof ballot system. All citizens have the potential to vote
on any decisions they wish; if they cannot vote on a particular decision for
whatever reason, they abstain from participating. In practice, not every
decision will be voted on by all $n$ citizens. Citizens will be constrained by
time pressures to only participate in those votes in which they are most
informed or most passionate. If we assume that all citizens have a tendency,
whether they vote or not, how would the collective tendency and collective
vote change as citizen participation waned? Let
$d_{100}^{\text{tend}}\in[0,1]$ and $d_{100}^{\text{vote}}\in\\{0,1\\}$ denote
the collective tendency and vote given by 100% participation. Let
$d_{k}^{\text{tend}}\in[0,1]$ and $d_{k}^{\text{vote}}\in\\{0,1\\}$ denote the
collective tendency and vote if only $k$-percent of the population
participates. The error in the collective tendency for $k$-percent
participation is calculated as
$e_{k}^{\text{tend}}=|d_{100}^{\text{tend}}-d_{k}^{\text{tend}}|.$
The further away the active voters’ collective tendency is from the full
population’s collective tendency, the higher the error. The gray line in
Figure 4 plots the relationship between $k$ and $e_{k}^{\text{tend}}$. As
citizen participation wanes, the ability for the remaining, active
participants to reflect the tendency of the whole becomes more difficult.
Next, the error in the collective vote is calculated as the proportion of
voting outcomes that are different than what a fully participating population
would have voted and is denoted $e_{k}^{\text{vote}}$. The gray line in Figure
5 plots the relationship between $k$ and $e_{k}^{\text{vote}}$. As
participation wanes, the proportion of decisions that differ from what would
have occurred given full participation decreases. As with collective tendency,
a small active voter population is unable to replicate the voting behavior of
the whole.
Figure 4: The relationship between $k$ and $e_{k}^{\text{tend}}$ for direct
democracy (gray line) and dynamically distributed democracy (black line). The
plot provides the average error over a simulation that was run with 1000
artificially generated networks composed of 100 citizens each. Figure 5: The
relationship between $k$ and $e_{k}^{\text{vote}}$ for direct democracy (gray
line) and dynamically distributed democracy (black line). The plot provides
the proportion of identical, correct decisions over a simulation that was run
with 1000 artificially generated networks composed of 100 citizens each.
Dynamically distributed democracy is a social representation algorithm that
provides a means by which any subset of the population can accurately simulate
the decision making results of the whole population [4]. As such, the
algorithm reflects the primary tenet of representation as originally outlined
by Thomas Paine. The argument for the use of the algorithm as a mechanism for
representation goes as follows. Not everyone in a population needs to vote as
others in that same population more than likely have a nearly identical
political tendency and thus, identical vote. What does need to be recorded is
the frequency of that sentiment in the population. If an active, voting
citizen is similar in tendency to $10$ non-active citizens, then the active
citizen’s ballot can be weighted by $10$ to reflect the tendencies of the non-
participating citizens. Dynamically distributed democracy accomplishes this
weighting through a similarity- or trust-based social network that is used to
propagate voting “power” to active voters so as to mitigate the error incurred
by waning citizen participation.
As previously stated, let $\mathbf{x}\in[0,1]^{n}$ denote the political
tendency of each citizen in this population, where $\mathbf{x}_{i}$ is the
tendency of citizen $i$ and, for the purpose of simulation, is determined from
a uniform distribution. Assume that every citizen in a population of $n$
citizens uses some social network-based system to create links to those
individuals that they believe reflect their tendency the best. In practice,
these links may point to a close friend, a relative, or some public figure
whose political tendencies resonate with the individual. In other words,
representatives are any citizens, not political candidates that serve in
public office. Let $\mathbf{A}\in[0,1]^{n\times n}$ denote the link matrix
representing the network, where the weight of an edge, for the purpose of
simulation, is denoted
$\mathbf{A}_{i,j}=\begin{cases}1-\left|\mathbf{x}_{i}-\mathbf{x}_{j}\right|&\text{if
link exists}\\\ 0&\text{otherwise}.\end{cases}$
In words, if two linked citizens are identical in their political tendency,
then the strength of the link is $1.0$. If their tendencies are completely
opposing, then their trust (and the strength of the link) is $0.0$. Note that
a preferential attachment network growth algorithm is used to generate a
degree distribution that is reflective of typical social networks “in the
wild” (i.e. scale-free properties). Moreover, an assortativity parameter is
used to bias the connections in the network towards citizens with similar
tendencies. The assumption here is that given a system of this nature, it is
more likely for citizens to create links to similar-minded individuals than to
those whose opinions are quite different. The resultant link matrix
$\mathbf{A}$ is then normalized to be row stochastic in order to generate a
probability distribution over the weights of the outgoing edges of a citizen.
Figure 6 presents an example of an $n=100$ artificially generated trust-based
social network, where red denotes a tendency of $0.0$, purple a tendency of
$0.5$, and blue a tendency of $1.0$.
Figure 6: A visualization of a network of trust links between citizens. Each
citizen’s color denotes their “political tendency”, where full red is $0$,
full blue is $1$, and purple is $0.5$. The layout algorithm chosen is the
Fruchterman-Reingold layout.
Given this social network infrastructure, it is possible to better ensure that
the collective tendency and vote is appropriately represented through a
weighting of the active, participating population. Every citizen, active or
not, is initially provide with $\frac{1}{n}$ “vote power” and this is
represented in the vector $\mathbf{\pi}\in\mathbb{R}_{+}^{n}$, such that the
total amount of vote power in the population is $1$. Let
$\mathbf{y}\in\mathbb{R}_{+}^{n}$ denote the total amount of vote power that
has flowed to each citizen over the course of the algorithm. Finally,
$\mathbf{a}\in\\{0,1\\}^{n}$ denotes whether citizen $i$ is participating
($\mathbf{a}_{i}=1$) in the current decision making process or not
($\mathbf{a}_{i}=0$). The values of $\mathbf{a}$ are biased by an unfair coin
that has probability $k$ of making the citizen an active participant and $1-k$
of making the citizen inactive. The iterative algorithm is presented below,
where $\circ$ denotes entry-wise multiplication and $\epsilon\approx 1$.
$\mathbf{\pi}\leftarrow 0$
while _$\sum_{i=1}^{i\leq n}\mathbf{y}_{i} <\epsilon$_ do
$\mathbf{y}\leftarrow\mathbf{y}+(\mathbf{\pi}\circ\mathbf{a})$
$\mathbf{\pi}\leftarrow\mathbf{\pi}\circ(1-\mathbf{a})$
$\mathbf{\pi}\leftarrow\mathbf{A}\mathbf{\pi}$
end
In words, active citizens serve as vote power “sinks” in that once they
receive vote power, from themselves or from a neighbor in the network, they do
not pass it on. Inactive citizens serve as vote power “sources” in that they
propagate their vote power over the network links to their neighbors
iteratively until all (or $\epsilon$) vote power has reached active citizens.
At this point, the tendency in the active population is defined as
$\delta^{\text{tend}}=\mathbf{x}\cdot\mathbf{y}$. Figure 4 plots the error
incurred using dynamically distributed democracy (black line), where the error
is defined as
$e_{k}^{\text{tend}}=|d_{100}^{\text{tend}}-\delta^{\text{tend}}_{k}|.$
Next, the collective vote $\delta_{k}^{\text{vote}}$ is determined by a
weighted majority as dictated by the vote power accumulated by active
participants. Figure 5 plots the proportion of votes that are different from
what a fully participating population would have rendered (black line). In
essence, if a citizen, for any reason, is unable to participate in a decision
making process, then they may abstain from participating knowing that the
underlying social network will accurately distribute their vote power to their
neighbor or neighbor’s neighbor. In this way, representation is dynamic,
distributed, and democratic.
Thomas Paine outlines that representatives should maintain “fidelity to the
public” and believes this is accomplished through frequent elections [3]. The
utilization of an Internet-based social network system affords repeated
“elections” in the form of citizens creating outgoing links to other citizens
as they please, when they please, and to whom them please. That is, citizens
can dynamically choose representatives who need not be picked from only a
handful of candidates. Moreover, if a selected representative falters in their
ability to represent a citizen, incoming links can be immediately retracted
from them. Such an architecture turns the representative’s status from that of
elected public official to that of a self-intentioned citizen.
While many countries have political institutions that are set up according to
a left, right, and moderate agenda, the individual perspectives of a citizen
may be more complex. In many cases, a citizen’s political tendency may only be
amenable to a multi-dimensional representation. In a multi-relational social
network, the links are augmented with labels in order to denote the type of
trust one citizen has for another. In this way, voting power propagates over
the links in a manner that is biased to the domain of the decision. For
example, citizen $i$ may trust citizen $j$ in the domain of “education” but
not in the domain of “health care”. This design has been articulated in [5].
Supporting systems, including the means by which ballots are proposed and
issues are discussed, is presented in [6].
With the Internet, supporting Web technologies, and dynamically distributed
democracy, it is possible to dynamically determine a representative-layer of
government that accurately reflects a full direct democracy. In this respect,
the larger population helps to ensure, according to the Condorcet jury
theorem, that the decisions are either definitely right or definitely wrong.
Other technologies can be utilized to induce participation by only those that
are more likely than not to choose the optimal decision.
## IV Decision Markets: Incentivizing an Enlightened Majority
Figure 7: An etching of Adam Smith originally created by Cadell and Davies
(1811), John Horsburgh (1828), or R.C. Bell (1872). This is a public domain
photograph courtesy of Wikimedia Commons.
Adam Smith (portrayed in Figure 7) was a Scottish moral and economic
philosopher who is best known for his two most famous works entitled The
Theory of Moral Sentiments (1759) and An Inquiry into the Nature and Causes of
the Wealth of Nations (1776). In the latter work, Adam Smith outlines the
economic benefits of a division of labor within a society. Each citizen in the
population serves a particular specialized function, and only in the
dependency relationships amongst these specialists does an efficient,
decentralized economy emerge. With many suppliers and consumers, the
production requirements of a society as a whole is difficult to know. Adam
Smith appreciated markets for their ability to expose these requirements
through the “natural price” of goods. Moreover, he understood that competition
within the market was a necessary driving force guaranteeing an accurate
representation of commodity prices. Adam Smith states that when a citizen
pursues “his own interest he frequently promotes that of the society more
effectually than when he really intends to promote it” [7].
Market mechanisms are not only useful for determining commodity prices as they
can be generally applied to information aggregation and ultimately, to
collective decision making. Such markets are called decision markets [8].
Similar to a division of labor, the knowledge required to make optimal
decisions for a society is dispersed throughout the population. For difficult
problems, it is naïve to think that a single individual has the requisite
knowledge to yield an optimal decision, much like it is naïve to think a
single merchant will offer the optimal price. A decision market functions
because it guarantees a return on investment for quality information. In this
respect, a decision market is a tool for attracting a population of
knowledgeable citizens much like a commodity market is a tool for attracting
knowledgeable speculators. In short, a decision market is a self-selection
mechanism that incentivizes participation from those who have knowledge
regarding the problem and are confident in their knowledge and discourages
participation from others without forbidding it.
Decision markets reward individuals for buying low and selling high, thus
encouraging those who believe they know which way the market will move to
contribute their information in the form of the price at which they purchase
and sell shares. A decision market differs from commodity markets (such as the
New York Stock Exchange) in that stocks represent objective states about the
world that can ultimately be determined, but are presently unknown. For
example, given the market question “Will decisions markets be used in U.S.
government by the year 2013?”, shares of stocks in a “yes” outcome and in a
“no” outcome are purchased and sold on the market. A high market price for a
stock indicates that the collective believes this outcome to be true with a
high likelihood. The purpose of the market is to incentivize knowledgeable
citizens to contribute to the decision by rewarding them for useful
contributions and conversely to inflict a penalty for contributing poor
information.
In order to demonstrate the benefits of incentives in decision making, a
simulation is provided. Suppose there exists $n$ citizens and a
$d$-dimensional “knowledge space”. Each citizen is represented as a point in
this space. That is, citizens have different degrees of knowledge in the
various dimensions (i.e. domains) of the space. A citizen’s point in this
space is generated by a normal distribution with a mean of $p\in[0,1]$ and a
variance of $(p(1-p))^{2}$. Next, there exists an objective truth in this
spaced called the environment. For the purpose of simulation, the environment
$\mathbf{e}$ is the largest valued point in the knowledge space (i.e.
$\mathbf{e}_{i}=1:1\leq i\leq d$). There also exists a market $\mathbf{m}$
which denotes the collective’s subjective understanding of the objective
environment. For the purpose of simulation, the market starts as the smallest
valued point in the knowledge space (i.e. $\mathbf{m}_{i}=0:1\leq i\leq d$).
Each citizen participates in the market, moving the market closer or further
away from the environment. The closer the market is to the environment, the
more accurate the collective decision. There are two markets in the
simulation: an incentive-free market and an incentive market. The results of
these two markets are compared in order to demonstrate the benefits of using
incentives.
Figure 8: The states of the incentive-free and incentive markets (purple) and
the objective state of the environment (green) are diagrammed in a
$3$-dimensional knowledge space. There exists two paths: the incentive-free
market path (red) and the incentive market path (blue). The dotted cubes
denote the range of an incentive-free market (red - $0.5$) and incentive
market (blue - $0.75$) for a $p=0.5$. Refer to the text for a description of
the diagrammed market paths.
Before presenting the results of a larger simulation, a small diagrammed
example is provided to better elucidate the simulation rules. Figure 8
diagrams a $3$-dimensional knowledge space with both markets (bottom left
purple point) and an environment (top right green point). The behavior of the
citizens denotes the market paths (red and blue arrows). Also, there exists a
$p=0.5$ population of $3$ citizens, where citizen
$\mathbf{c}^{1}=[0.7,0.5,0.4]$, citizen $\mathbf{c}^{2}=[0.5,0.6,0.3]$, and
citizen $\mathbf{c}^{3}=[0.3,0.5,0.7]$. At time step $t=0$, both the
incentive-free and incentive markets are at $[0,0,0]$. At $t=1$, citizen
$\mathbf{c}^{1}$ participates in both markets. In the incentive-free market,
citizen $\mathbf{c}^{1}$ has no incentive to contribute his best knowledge and
thus, randomly chooses a dimension in which to move the market. According to
the diagram, the citizen’s random choice moves the market in the
$3^{\text{rd}}$ dimension by $0.4$. In the incentivized market, citizen
$\mathbf{c}^{1}$ chooses the dimension in which he has the most knowledge
(i.e. the dimension with the maximum value). Moreover, a biased coin toss
determines whether he participates or not, where $\mathbf{c}^{1}$ has a $70$%
chance of participating in the incentive market. Assuming the coin toss
permits it, $\mathbf{c}^{1}$ moves the incentivized market in the
$1^{\text{st}}$ dimension to $0.7$. This process continues in sequence for
citizens $\mathbf{c}^{2}$ and $\mathbf{c}^{3}$. Assuming that all citizens
participate in both markets, at the end, the incentive-free market is located
at point $[0.5,0.5,0.4]$, while the incentive market is located at point
$[0.7,0.6,0.7]$. The market error is calculated as the normalized Euclidean
distance between the final market position and the environment for a given
$p$,
$e_{p}^{\text{dist}}=\frac{1}{\sqrt{d}}\sqrt{\sum_{i=1}^{i\leq
d}(\mathbf{e}_{i}-\mathbf{m}_{i})^{2}}.$
The incentive-free market has an error of $0.287$ and the incentive market has
an error of $0.113$. Thus, the incentive market is closest to the environment.
There are two distinctions between the markets. In the incentive-free market,
there is no benefit to producing an enlightened solution, so the citizen makes
a contribution without comparing his knowledge against the environment. In the
incentive market, there are two incentive structures. The first incentive is
to participate along the dimension in which the citizen is most knowledgeable.
The second incentive is to participate only if the citizen has a satisfactory
degree of knowledge. This means that poor information is excluded from the
market and that the most valuable knowledge of the citizen is included.
To demonstrate the effects of an incentive-free and incentive market on a
larger population, over various values of $p$, and in a $50$-dimensional
knowledge space simulation results are provided. Figure 9 depicts the
normalized Euclidean distance error of the incentive-free market (gray line)
and the incentive market (black line) for varying $p$. Next, Figure 10
provides the proportion of correct collective decisions. A decision is either
correct or incorrect. While the market yields a point in $[0,1]^{d}$, rounding
the dimension values of the point to either $1$ or $0$ provides the final
decision made by the citizens. For a given $p$, the proportion of times that
the market rounds to the environment is the proportion of correct decisions
and is denoted $e_{p}^{\text{deci}}\in[0,1]$.
Figure 9: The relationship between $p$ and $e_{p}^{\text{dist}}$ for an
incentive-free market (gray line) and an incentive market (black line). The
plot provides the average error over $1000$ simulations with $d=50$ and
$n=1000$. Figure 10: The relationship between $p$ and $e_{p}^{\text{deci}}$
for an incentive-free market (gray line) and an incentive market (black line).
The plot provides the proportion of correct decisions over $1000$ simulations
with $d=50$ and $n=1000$.
It is the principle of self-selection, and therefore citizen choice, that
provides the mechanism by which knowledge is aggregated. Choice is manifested
in a number of ways. First, citizens choose whether or not to participate in
the market at all. This reduces the amount of poor information that enters the
market. Second, citizens choose how often to participate. The market,
therefore, induces citizens to become more knowledgeable so as to gain from
the market. Finally, citizens choose the extent of their participation. If a
citizen has knowledge that is not well reflected in the market, suggesting
that their knowledge is unique and therefore valuable, the citizen is
incentivized to participate more so than if the market closely mimics their
knowledge. In decision markets, it is the pricing mechanism of the market that
serves the incentivizing role. However, the asset traded in the market need
not be money. To maintain the egalitarian nature of self-selection, the market
can be based in virtual money with rewards, reputation, or other social
inducements as the backing. It has been demonstrated that virtual money is
able to preserve the accuracy of decision markets [9].
As presented in the simulation the decisions of a society are multi-
dimensional. It is likely that no single citizen has the requisite knowledge
in all dimensions to make informed decisions. The ability to reach an optimal
decision is dependent on the many dimensions such that ignorance of one
dimension may lead to a suboptimal conclusion. The probability parameter of
the Condorcet jury theorem model is misleading. It is not through probability
that one achieves an optimal decision, but through the careful application of
knowledge to the decision. The use of a market is not a guarantee that
decision makers have $p>0.5$. The market is a guarantee that citizen knowledge
has been thoughtfully applied to the decision.
## V Conclusion
The purpose of a democratic government is to preserve and support the ideals
of its population. The ideals established during the Enlightenment are general
in nature: life, liberty, and the pursuit of happiness. In articulating these
values, the founders of modern democracies provided a moral heritage that
remains highly regarded in societies today. However, it should be remembered
that it is the ideals that are valuable, not the specific implementation of
the systems that protect and support them. If there is another implementation
of government that better realizes these ideals, then, by the rights of man,
it must be enacted. It was the great thinkers of the eighteenth century
Enlightenment who provided the initial governance systems. It is the challenge
and the mandate of the Information Age to redesign these governance systems in
light of present day technologies.
## References
* [1] M. de Condorcet, “Essai sur l’application de l’analyse á la probabilité des décisions rendues á la pluralité des voix,” Paris, France, 1785.
* [2] J. H. Watkins and M. A. Rodriguez, _Evolution of the Web in Artificial Intelligence Environments_ , ser. Studies in Computational Intelligence. Berlin, DE: Springer-Verlag, 2008, ch. A Survey of Web-Based Collective Decision Making Systems, pp. 245–279. [Online]. Available: http://repositories.cdlib.org/hcs/WorkingPapers2/JHW2007-1/
* [3] T. Paine, “Common sense,” American colonies, 1776.
* [4] M. A. Rodriguez and D. J. Steinbock, “A social network for societal-scale decision-making systems,” in _Proceedingss of the North American Association for Computational Social and Organizational Science Conference_ , Pittsburgh, PA, 2004. [Online]. Available: http://arxiv.org/abs/cs.CY/0412047
* [5] M. A. Rodriguez, “Social decision making with multi-relational networks and grammar-based particle swarms,” in _Proceedings of the Hawaii International Conference on Systems Science_. Waikoloa, Hawaii: IEEE Computer Society, January 2007, pp. 39–49. [Online]. Available: http://arxiv.org/abs/cs.CY/0609034
* [6] M. Turrof, S. R. Hiltz, H.-K. Cho, Z. Li, and Y. Wang, “Social decision support system (SDSS),” in _Proceedings of the Hawaii International Conference on Systems Science Hawaii International Conference on Systems Science_. Waikoloa, Hawaii: IEEE Computer Society, January 2002, pp. 81–90.
* [7] A. Smith, _An Inquiry into the Nature and Causes of the Wealth of Nations_. London, England: W. Strahan and T. Cadell, Londres, 1776.
* [8] R. Hanson, “Decision markets,” _IEEE Intelligent Systems_ , vol. 14, no. 3, pp. 16–19, May 1999.
* [9] E. Servan-Schrieber, J. Wolfers, D. M. Pennock, and B. Galebach, “Prediction markets: Does money matter?” _Electronic Markets_ , vol. 14, no. 3, pp. 243–251, September 2004.
|
arxiv-papers
| 2009-01-25T22:50:41 |
2024-09-04T02:49:00.209242
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marko A. Rodriguez and Jennifer H. Watkins",
"submitter": "Marko A. Rodriguez",
"url": "https://arxiv.org/abs/0901.3929"
}
|
0901.3939
|
Related Worksec2_rel In this section, we will discuss related work.
Existing Related Systems. Much work has been done in the field of map data
processing. A system used for acquisition, storage, indexing, and retrieval of
map images is described in samet98ijdar. The inputs to their system are raster
images of maps. These images are then stored as a record containing map
related information in a relational database. Advanced database techniques
from the field of spatial databases are adopted to build indices. Queries are
posed to the system using an SQL-like language. In lim02jcdl, a G-Portal
system is set up to identify, classify, and organize geo-spatial and geo-
referenced content on the World Wide Web. A digital library can access these
data by using a map-based interface. A legend-driven map interpretation
system, MARCO (MAp Retrieval by COntent), is proposed in samet96itpami to
convert map images from their physical representation to their logical
representation. The logical representation is then stored and used to build
index. R. P. Futrelle has set up a diagram understanding system to understand
diagrams in technical documents futrelle92computer,futrelle07tr. This is the
first such system to fully parse a variety of actual diagrams drawn from the
research literature. Digmap
systemhttp://code.google.com/p/digmap/http://code.google.com/p/digmap/gir07martins
is a geographic IR system based on the historical digitized maps.
Our work is different from those we described above. In their approaches, the
information is dynamically extracted from either paper-based maps or digital
maps on the Web. They consider maps in isolation, i.e., excluding other
content in digital documents. In contrast, we extract maps from digital
documents and utilize the “context” of the maps like captions and references
in the text, as well as other context-based boosting factors depending on the
digital documents, to get hints on understanding their contents. Our
experiment results show that inclusion of these factors can significantly
improve the map retrieval performance. The DIGMAP system takes into account
some textual description of the maps. However, these descriptions are provided
along with the maps. DIGMAP does not consider the problem of extracting maps
and the related map metadata from documents, as well as the problem of
distinguishing maps from other images in a document. Both of these problems
are critical because a large number of maps are embedded in documents.
There is another online map search online
system††http://scilsresx.rutgers.edu/ gelern/maps/ jcdl08gelernter, which was
motivated in part by discussions with our team, in which maps are also
extracted from some Web documents in PDF format The maps are indexed based on
maps’ content in three dimensions, region, time period, and theme. However,
this project is just starting and no details of the design or implementation
are discussed in jcdl08gelernter.
Multimodal/Structured Document Retrieval. Our work is also related to indexing
and retrieving images for multimodal digital documents. Multimodal documents
convey information using both text and images. Different Information Retrieval
(IR) techniques have been used to build up, index and retrieval functions.
Some of these existing works deal with only the text in the document and
ignore all the image-related information witten99mg. Some of them deal with
only images, as in content-based image retrieval (CBIR) smeulders00tpami.
Others use both text and image objects mitra00ir. Current OCR tools were not
very effective in extracting text from maps in our preliminary experiments. In
the future, we will extract and index text lying inside maps. Once such
information is extracted, our current framework can index and utilize them
seamlessly.
As discussed in Section sec6_index, we consider the map metadata as various
information fields for a map, and propose a structured document retrieval
technique to build a map index. Therefore, we now discuss related work on
structured document retrieval briefly. A document is said to be structured
when it contains multiple fields. A document’s field structure is commonly
used to improve retrieval performance in practice. The most commonly used
approach for structured document retrieval is a score/rank linear combination
wilkinson94sigir,myaeng98sigir,lalmas00tr,wu07ipm, which treats each field as
a separate document and computes a combined scores/ranks. In computing scores
for each field, any ranking function for unstructured document retrieval can
be adopted. Another approach is to essentially combine term frequencies
instead of scores. In robertson04cikm, an extension of the BM25 formula is
introduced and is a simple and efficient method which combines term
frequencies instead of field scores. In ogilvie03sigir, within the language
model framework, this approach proposes a separate language model for each
field and then combines them linearly. Our work adopts and extends these two
pieces of work and proposes novel retrieval functions for the map search
system.
|
arxiv-papers
| 2009-01-26T02:14:29 |
2024-09-04T02:49:00.217391
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qingzhao Tan, Prasenjit Mitra, C. Lee Giles",
"submitter": "Qingzhao Tan",
"url": "https://arxiv.org/abs/0901.3939"
}
|
0901.3964
|
# Robust photon-spin entangling gate using a quantum-dot spin in a microcavity
C.Y. Hu1 chengyong.hu@bristol.ac.uk W.J. Munro2,3 J.L. O’Brien1 J.G. Rarity1
1Department of Electrical and Electronic Engineering, University of Bristol,
University Walk, Bristol BS8 1TR, United Kingdom 2Hewlett-Packard
Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdom
3National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo
101-8430, Japan
###### Abstract
Semiconductor quantum dots (known as artificial atoms) hold great promise for
solid-state quantum networks and quantum computers. To realize a quantum
network, it is crucial to achieve light-matter entanglement and coherent
quantum-state transfer between light and matter. Here we present a robust
photon-spin entangling gate with high fidelity and high efficiency (up to 50
percent) using a charged quantum dot in a double-sided microcavity. This gate
is based on giant circular birefringence induced by a single electron spin,
and functions as an optical circular polariser which allows only one
circularly-polarized component of light to be transmitted depending on the
electron spin states. We show this gate can be used for single-shot quantum
non-demolition measurement of a single electron spin, and can work as an
entanglement filter to make a photon-spin entangler, spin entangler and photon
entangler as well as a photon-spin quantum interface. This work allows us to
make all building blocks for solid-state quantum networks with single photons
and quantum-dot spins.
###### pacs:
78.67.Hc, 03.67.Mn, 42.50.Pq, 78.20.Ek
## I Introduction
A quantum network cirac97 utilizes matter quantum bits (qubits) to store and
process quantum information at local nodes, and light qubits (photons) for
long-distance quantum state transmission between different nodes. Quantum
networks can be used for distributed quantum computing or for large scale and
long distance quantum communications between spatially remote parties. There
are several physical systems based on cavity quantum electrodynamics (cavity-
QED), which could be used for quantum networks with high success probability
for quantum-state transfer or processing. One is the atom-cavity system in
which single photon sources kuhn02 ; mckeever04 ; wilk07a , light-atom
entanglement sherson06 ; boozer07 and a single-photon single-atom quantum
interface wilk07 have been recently demonstrated. But it is far from a
trivial task to scale up and to trap the atoms. Another one is the
superconducting qubit-cavity system which attracts great interest in recent
years. Single photon generation in the microwave frequency region houck07 and
a quantum bus allowing distant qubits to interact at will sillanpaa07 ;
majer07 have been implemented recently in this system.
The third one is the semiconductor quantum dot (QD)-cavity system loss98 ;
imamoglu99 ; calarco03 ; yao05 ; clark07 . Firstly, triggered single-photon
sources or polarization-entangled photon pair sources based on semiconductor
QDs have been demonstrated with high quantum efficiency, high photon
indistinguishability, and low multi-photon emission probability moreau01 ;
yuan02 ; santori02 ; stevenson06 ; akopian06 . These deterministic photon
sources are key ingredients for secure quantum networks. Secondly,
semiconductor QD spins are promising candidates to construct qubits for
storing and processing quantum states awschalom02 ; atature06 due to the long
electron spin coherence time ($T_{2}\sim\mu$s) petta05 ; greilich06 and spin
relaxation time ($T_{1}\sim$ms) kroutvar04 . Moreover, self-assembled QDs can
be embedded in various high-finesse optical microcavities or nanocavities, so
cavity-QED can be exploited to engineer QD emissions or related optical
transitions as demanded reithmaier04 ; yoshie04 ; peter05 ; hennessy07 ;
press07 ; reitzenstein07 . The most attractive feature is its compatibility
with standard semiconductor processing techniques. Therefore, the QD-cavity
system holds great promise for compact and scalable solid-state quantum
networks and quantum computers. However, the photon-spin entanglement and
quantum state transfer between photon and QD spin have not yet been
demonstrated yao05 ; flindt07 .
Here we propose a robust photon-spin entangling gate using a charged QD in a
double-sided microcavity, and show this gate can be used as photon-spin
entangler, spin entangler, photon entangler as well as reversible and coherent
quantum-state transfer between single photons and QD spins. This gate is based
on giant circular birefringence induced by a single electron spin, and is
ideal for an optical quantum non-demolition (QND) measurement of a single
electron spin in a double-sided microcavity. This gate is robust and flexible
compared to our previous gate using a charged QD in a single-sided microcavity
hu08a ; hu08b .
The paper is organized as follows: In Sec.II, the photon-spin entangling gate
is introduced. In Sec. III we show this gate can be used for single-shot QND
measurements of a single QD spin. After that, we show a spin entangler in Sec.
IV, a photon entangler in Sec. V and a photon-spin quantum interface in Sec.
VI by applying this photon-spin entangling gate. Finally, we present our
conclusions and outlook in Sec. VII.
## II Photon-spin entangling gate
We consider a singly charged QD, e.g., a self-assembled In(Ga)As QD or a GaAs
interfacial QD, or even a semiconductor nanocrystal inside an optical cavity,
such as a micropillar or microdisk microcavity, or a photonic crystal
nanocavity. Fig. 1a shows a micropillar microcavity where the two
GaAs/Al(Ga)As distributed Bragg reflectors (DBR) and the transverse index
guiding provide the three-dimensional confinement of light. The two DBRs are
made symmetric in order to achieve high resonant transmission of light. Both
DBRs are partially reflective allowing light into and out of the cavity (i.e.,
a double-sided cavity). The circular cross section of the micropillar supports
the circularly polarized light. The QD is located at the antinodes of the
cavity field to achieve optimized light-matter coupling.
Figure 1: (a) A charged QD inside a micropillar microcavity with circular
cross section. (b) Spin selection rule for optical transitions of negatively-
charged exciton $X^{-}$ (see text).
The optical properties of singly charged QDs are dominated by the optical
resonances of the negatively-charged exciton $X^{-}$ (also called trion) which
consists of two electrons bound to one hole warburton97 . Due to the Pauli’s
exclusion principle, $X^{-}$ shows spin-dependent optical transitions (see
Fig. 1b)hu98 : the left circularly polarized photon (marked by $|L\rangle$ or
L-photon) only couples the electron in the spin state $|\uparrow\rangle$ to
$X^{-}$ in the spin state $|\uparrow\downarrow\Uparrow\rangle$ with the two
electron spins antiparallel; the right circularly polarized photon (marked by
$|R\rangle$ or R-photon) only couples the electron in the spin state
$|\downarrow\rangle$ to $X^{-}$ in the spin state
$|\downarrow\uparrow\Downarrow\rangle$. Here $|\uparrow\rangle$ and
$|\downarrow\rangle$ represent electron spin states $|\pm\frac{1}{2}\rangle$,
$|\Uparrow\rangle$ and $|\Downarrow\rangle$ represent heavy-hole spin states
$|\pm\frac{3}{2}\rangle$. The light-hole sub-band and the split-off sub-band
are energetically far apart from the heavy-hole sub-band and can be neglected.
The spin is quantized along the normal direction of the cavity, i.e., the
propagation direction of the input (or output) light. This spin selection rule
for $X^{-}$ is also called the Pauli blocking effect warburton97 ; calarco03 .
The reflection and transmission coefficients of this $X^{-}$-cavity structure
can be investigated by solving the Heisenberg equations of motion for the
cavity field operator $\hat{a}$ and $X^{-}$ dipole operator $\sigma_{-}$, and
the input-output equationswalls94 :
$\begin{cases}&\frac{d\hat{a}}{dt}=-\left[i(\omega_{c}-\omega)+\kappa+\frac{\kappa_{s}}{2}\right]\hat{a}-\text{g}\sigma_{-}\\\
&~{}~{}~{}~{}~{}~{}-\sqrt{\kappa}\hat{a}_{in}-\sqrt{\kappa}\hat{a}^{\prime}_{in}+\hat{H}\\\
&\frac{d\sigma_{-}}{dt}=-\left[i(\omega_{X^{-}}-\omega)+\frac{\gamma}{2}\right]\sigma_{-}-\text{g}\sigma_{z}\hat{a}+\hat{G}\\\
&\hat{a}_{r}=\hat{a}_{in}+\sqrt{\kappa}\hat{a}\\\
&\hat{a}_{t}=\hat{a}^{\prime}_{in}+\sqrt{\kappa}\hat{a}\\\ \end{cases}$ (1)
where $\omega$, $\omega_{c}$, and $\omega_{X^{-}}$ are the frequencies of the
input photon, cavity mode, and $X^{-}$ transition, respectively. g is the
$X^{-}$-cavity coupling strength given by
$\text{g}=(e^{2}f/4\epsilon_{r}\epsilon_{0}m_{0}V_{eff})^{1/2}$ where $f$ is
the $X^{-}$ oscillator strength and $V_{eff}$ is the effective modal volume,
$\gamma/2$ is the $X^{-}$ dipole decay rate, and $\kappa$, $\kappa_{s}/2$ are
the cavity field decay rate into the input/output modes, and the leaky modes,
respectively. The background absorption can also be included in
$\kappa_{s}/2$. $\hat{H}$, $\hat{G}$ are the noise operators related to
reservoirs. $\hat{a}_{in}$, $\hat{a}^{\prime}_{in}$ and $\hat{a}_{r}$,
$\hat{a}_{t}$ are the input and output field operators.
In the approximation of weak excitation, i.e., less than one photon inside the
cavity per cavity lifetime so that QD is in the ground state at most time, we
take $\langle\sigma_{z}\rangle\approx-1$. The reflection and transmission
coefficients in the steady state can be obtained
$\begin{split}&r(\omega)=1+t(\omega)\\\
&t(\omega)=\frac{-\kappa[i(\omega_{X^{-}}-\omega)+\frac{\gamma}{2}]}{[i(\omega_{X^{-}}-\omega)+\frac{\gamma}{2}][i(\omega_{c}-\omega)+\kappa+\frac{\kappa_{s}}{2}]+\text{g}^{2}}.\end{split}$
(2)
By taking $\text{g}=0$, we get the reflection and transmission coefficients
for an empty cavity where the QD does not couple to the cavity
$\begin{split}&r_{0}(\omega)=\frac{i(\omega_{c}-\omega)+\frac{\kappa_{s}}{2}}{i(\omega_{c}-\omega)+\kappa+\frac{\kappa_{s}}{2}}\\\
&t_{0}(\omega)=\frac{-\kappa}{i(\omega_{c}-\omega)+\kappa+\frac{\kappa_{s}}{2}}\end{split}$
(3)
The reflection and transmission spectra versus the frequency detuning
$\omega-\omega_{c}$ are presented in Fig. 2a for different coupling strength
g. With increasing g (e.g. by reducing the effective modal volume
$V_{\text{eff}}$), the cavity mode splits into two peaks due to the quantum
interference in the “one dimensional atom”regime with
$\kappa<4\text{g}^{2}/\kappa<\gamma$ waks06 ; garnier07 (which has been
experimentally demonstrated recently englund07 ), and the vacuum Rabi
splitting in the strong coupling regime with $\text{g}>(\kappa,\gamma)$
reithmaier04 ; yoshie04 ; peter05 ; hennessy07 ; press07 ; reitzenstein07 . We
notice that the transmittance or reflectance are different between the empty
cavity ($\text{g}=0$) and the coupled cavity ($\text{g}\neq 0$) (the coupled
$X^{-}$-cavity system is called coupled cavity hereafter). This enables us to
make a photon-spin entangling gate as discussed below.
Figure 2: Calculated transmission and reflection spectra of the $X^{-}$-cavity
system. (a) Transmission (solid curves) and reflection (dotted curves) spectra
vs the frequency detuning $(\omega-\omega_{c})/\kappa$ for different coupling
strength. (b) The gate fidelity vs the frequency detuning in the strong
coupling regime ($\text{g}=2.4\kappa$ is taken). High fidelity can be achieved
if $|\omega-\omega_{c}|<\kappa<\text{g}$. (c) Transmittance $|t(\omega_{0})|$
(solid curve) and reflectance $|r(\omega_{0})|$ (dotted curve) vs the
normalized coupling strength. (d) The gate fidelity vs the normalized coupling
strength. (e) Transmittance $|t(\omega_{0})|$ (solid curve) and reflectance
$|r(\omega_{0})|$ (dotted curve) vs the normalized side leakage rate. (f) The
gate fidelity vs the normalized side leakage rate.
$\omega_{c}=\omega_{X^{-}}=\omega_{0}$ is assumed for (a)-(f). $\kappa_{s}=0$
and $\gamma=0.1\kappa$ are taken for (a)-(d).
If the single excess electron in the QD lies in the spin state
$|\uparrow\rangle$, the L-photon feels a coupled cavity with reflectance
$|r(\omega)|$ and the transmittance $|t(\omega)|$, whereas the R-photon feels
the empty cavity with the reflectance $|r_{0}(\omega)|$ and transmittance
$|t_{0}(\omega)|$; Conversely, if the electron lies in the spin state
$|\downarrow\rangle$, the R-photon feels a coupled cavity, whereas the
L-photon feels the empty cavity. The difference in transmission and reflection
between right and left circularly polarized light, which can be called giant
circular birefringence, means we have created a circular polariser controlled
by the electron spin. For any quantum input we can define a transmission
operator
$\begin{split}\hat{t}(\omega)=&t_{0}(\omega)(|R\rangle\langle
R|\otimes|\uparrow\rangle\langle\uparrow|+|L\rangle\langle
L|\otimes|\downarrow\rangle\langle\downarrow|)\\\ &+t(\omega)(|R\rangle\langle
R|\otimes|\downarrow\rangle\langle\downarrow|+|L\rangle\langle
L|\otimes|\uparrow\rangle\langle\uparrow|),\end{split}$ (4)
where $t_{0}(\omega)$, $t(\omega)$ are the transmission coefficients of the
empty cavity and coupled cavity, respectively.
In the strong coupling regime, i.e., $\text{g}>(\kappa,\gamma)$ and in the
central frequency regime $|\omega-\omega_{c}|\ll\text{g}$, we have
$|t(\omega)|\rightarrow 0$ (see Fig. 2a), thus the transmission operator can
be simplified as
$\hat{t}(\omega)\simeq t_{0}(\omega)(|R\rangle\langle
R|\otimes|\uparrow\rangle\langle\uparrow|+|L\rangle\langle
L|\otimes|\downarrow\rangle\langle\downarrow|).$ (5)
Obviously, this transmission operator is now constructed from the empty cavity
only. We show later how this operator can be used as a photon-spin entangling
gate. Here however we can define an operator fidelity (based on amplitude)
from equations (4) and (5) as
$F=\frac{|t_{0}(\omega)|}{\sqrt{|t_{0}(\omega)|^{2}+|t(\omega)|^{2}}}.$ (6)
Near-unity fidelity is reached when $|t(\omega)|\rightarrow 0$ which is only
achieved within a small frequency window $|\omega-\omega_{c}|<\kappa$ (see
Fig. 2b) and in the strong coupling regime with $\text{g}>(\kappa,\gamma)$
(see Fig. 2c and Fig. 2d). The strongly coupled QD-cavity has been
demonstrated in various microcavities and nanocavities recently reithmaier04 ;
yoshie04 ; peter05 ; hennessy07 ; press07 ; reitzenstein07 . For micropillars
with diameter around $1.5~{}\mu$m, the coupling strength $\text{g}=80~{}\mu$eV
and the quality factor more than $4\times 10^{4}$ (corresponding to
$\kappa=33~{}\mu$eV) have been reported reithmaier04 ; reitzenstein07 ,
indicating $\text{g}/\kappa=2.4$ is achievable for the In(Ga)As QD-cavity
system. $\gamma$ is about several $\mu$eV. Our calculations in Fig. 2 are
based on these experimental parameters.
A practical optical cavity can have some side leakage, which induces a
decrease in the transmittance of the empty cavity and the gate fidelity (see
Fig. 3e and Fig. 3f). However, the improvement of fabrication techniques can
suppress the side leakage reitzenstein07 . When the side leakage is made
negligible compared with the main cavity decay into the input/output modes, we
get $|t_{0}(\omega_{0})|=1$ and unity gate fidelity.
For a realistic QD, the spin selection rule discussed earlier is not perfect
if we take the heavy-light hole mixing into account. This can reduce the gate
fidelity by a few percent as the hole mixing in the valence band is in the
order of a few percent bester03 ; calarco03 [e.g., for self-assembled
In(Ga)As QDs]. The hole mixing could be reduced by engineering the shape and
size of QDs or using different types of QDs.
As discussed above, the photon-spin entangling gate requires the weak
excitation condition, i.e., the input light intensity has to be less than one
photon per cavity lifetime. This condition can be satisfied by single photons,
e.g. QD single photon sources which can be triggered electrically or optically
moreau01 ; yuan02 ; santori02 . Recently there are lots of experimental
efforts to develop high-quality QD single-photon sources with high
efficiencies, small multi-photon events and time-bandwidth limited photon
pulses shields07 .
This photon-spin entangling gate can also work in the reflection geometry, but
its application is more complicated and we leave the discussions elsewhere. As
a result, the photon-spin entangling gate in the transmission geometry is only
$50\%$ efficient.
In the following, we show that this photon-spin entangling gate can be used
for QND measurement of a single electron spin, and also can work as photon-
spin entangler, spin entangler, or photon entangler. With this gate,
reversible quantum state transfer between photon and spin can be implemented.
Compared with our previous gate hu08a ; hu08b and Turchette et al’s
conditional phase shift gate using a single-sided cavity turchette95 , this
photon-spin entangling gate using a double-sided cavity is more robust and
flexible. We notice that other photon-spin entangling gates was also reported
recently flindt07 ; hu08a ; hu08b ; lindner08 .
## III Single-shot optical QND measurement of a single spin
If we prepare the input photon in a linear polarization state
$|H\rangle=(|R\rangle+|L\rangle)/\sqrt{2}$ and the electron spin in the state
$|\psi^{s}\rangle=|\uparrow\rangle$, according to equation (5) the state
transformation is
$|H\rangle\otimes|\uparrow\rangle\xrightarrow{\hat{t}(\omega)}\frac{t_{0}(\omega)}{\sqrt{2}}|R\rangle|\uparrow\rangle.$
(7)
So only the right-handed circularly polarized component is transmitted (see
Fig. 3a). Similarly, if the electron spin is in the state
$|\downarrow\rangle$, only the left-handed circularly polarized component is
transmitted (see Fig. 3b). Obviously, this is a circular polariser which
allows only one circular polarized light to be transmitted depending on the
spin state. This feature enables us to detect the electron spin by measuring
the helicity of the transmitted light using a $\lambda/4$ wave plate and a
polarizing beam splitter (see Fig. 3c).
Figure 3: QND measurement of a single QD spin. (a) The right-circularly
polarized component of a linearly polarized light is transmitted if the
electron spin in the $|\uparrow\rangle$ state. (b) The left-circularly
polarized component of a linearly polarized light is transmitted if the
electron spin in the $|\downarrow\rangle$ state. (c) Both the right- and left-
circularly polarized component of a linearly polarized light are transmitted
if the electron spin in a superposition state. PBS (polarizing beam splitter),
D1 and D2 (photon detectors), and $\lambda/4$ (quarter-wave plate).
If the electron spin is in an arbitrary superposition state
$|\psi^{s}\rangle=\alpha|\uparrow\rangle+\beta|\downarrow\rangle$ (see Fig.
3c), the state transformation is
$|H\rangle\otimes(\alpha|\uparrow\rangle+\beta|\downarrow\rangle)\xrightarrow{\hat{t}(\omega)}\frac{t_{0}(\omega)}{\sqrt{2}}(\alpha|R\rangle|\uparrow\rangle+\beta|L\rangle|\downarrow\rangle).$
(8)
Thus after transmission, the light polarization state becomes entangled with
the spin state. This is why we call this gate a photon-spin entangling gate.
If we measure the light in $|R\rangle$ (or $|L\rangle$) polarization, the
electron spin collapses to $|\uparrow\rangle$ (or $|\downarrow\rangle$) state.
Although this gate work in the near resonance region, the weak excitation
condition means nearly no real excitation occurs in the $X^{-}$-cavity system.
As a result, the disturbance to the electron spin system due to the light
input is quite small. Within the spin relaxation time ($\sim$ms) kroutvar04 ,
repeated measurements will yield the same results, so this single-shot spin
detection method is a QND measurement grangier98 , in contrast to other
single-spin detection methods by the time-averaged Faraday rotation or Kerr
rotation measurement reported recently berezovsky06 ; atature07 . In parallel,
a QND measurement of single photon polarization state could also be
implemented using the above spin QND measurement. QND measurement is critical
for scalable quantum information processing liu05 ; nemoto05 .
The QD spin eigen-state can be prepared, for example, by optical spin pumping
atature06 ; xu07 . From the above discussions, we see the single-shot QND
measurement of single spin can be also used to prepare the spin eigen state
and cool the spin via photon detection liu05 . From the spin basis state,
there are two ways to create the spin superposition state: either via spin-
flip Raman transitions atature06 , or by performing single spin rotations
using nanosecond ESR microwave pulses petta05 . Recently, ultrafast optical
coherent control of electron spins has been reported in quantum wells on
femtosecond time scales gupta01 and in QDs on picosecond time scales
berezovsky08 ; press08 , which is much shorter than the QD spin coherence time
($T_{2}\sim\mu$s). This allows ultrafast $\pi/2$ spin rotation which is
required in our schemes for spin state preparation or spin Hadamard operation.
## IV Entangle remote spins via a single photon
We show here that the photon-spin entangling gate can be used to generate
entanglement between remote spins in different cavities via a single photon
(see Fig. 4a). In the first $X^{-}$-cavity system, the spin is prepared in the
state
$|\psi^{s}\rangle_{1}=\alpha_{1}|\uparrow\rangle_{1}+\beta_{1}|\downarrow\rangle_{1}$
and transmission operator is $\hat{t}_{1}(\omega)$; In the second
$X^{-}$-cavity system, the spin is prepared in the state
$|\psi^{s}\rangle_{2}=\alpha_{2}|\uparrow\rangle_{2}+\beta_{2}|\downarrow\rangle_{2}$
and transmission operator is $\hat{t}_{2}(\omega)$. Both $X^{-}$-cavity
systems work in the strong coupling regime to get high gate fidelity, but the
parameters g, $\kappa$, $\kappa_{s}$, $\omega_{c}$ and $\omega_{X^{-}}$ for
this two systems are not necessary to be the same.
A single photon in $|H\rangle$ polarization passes through the first cavity,
then through the second cavity, after which its polarization is checked (see
Fig. 4a). The corresponding state transformation is
$\begin{split}&|H\rangle\otimes(\alpha_{1}|\uparrow\rangle_{1}+\beta_{1}|\downarrow\rangle_{1})\otimes(\alpha_{2}|\uparrow\rangle_{2}+\beta_{2}|\downarrow\rangle_{2})\xrightarrow{\hat{t}_{1,2}(\omega)}\\\
&\frac{t_{10}(\omega)t_{20}(\omega)}{\sqrt{2}}(\alpha_{1}\alpha_{2}|R\rangle|\uparrow\rangle_{1}|\uparrow\rangle_{1}+\beta_{1}\beta_{2}|L\rangle|\downarrow\rangle_{1}|\downarrow\rangle_{2})\end{split}$
(9)
By applying the Hadamard gate on the photon state using a polarizing beam
splitter, we obtain entangled spin states
$|\Phi^{s}_{12}\rangle=\alpha_{1}\alpha_{2}|\uparrow\rangle_{1}|\uparrow\rangle_{2}\pm\beta_{1}\beta_{2}|\downarrow\rangle_{1}|\downarrow\rangle_{2}$
(10)
on detecting the photon in the $|H\rangle$ state (for “+”), or in
$|V\rangle=(|R\rangle-|L\rangle)/\sqrt{2}$ state (for “-”). On setting the
coefficients $\alpha_{1,2}$ and $\beta_{1,2}$ to $1/\sqrt{2}$, we get
maximally entangled spin states.
We see the single photon works as a quantum bus to couple or entangle remote
spins on demand, but the two spins in two cavities can be slightly different
in their transition frequencies. However, if the cavity mode frequency
$\omega_{c}$ and the $X^{-}$ transition frequency $\omega_{X^{-}}$ match with
the photon frequency $\omega$ for the two $X^{-}$-cavity systems, the success
probability $|t_{10}(\omega)t_{20}(\omega)|^{2}/2$ to achieve the spin
entanglement can be increased. As discussed earlier, if the side leakage can
be made significantly small, $|t_{10}(\omega)|$ and $|t_{20}(\omega)|$ can
both reach unity and we get the maximal success probability of $50\%$. But we
know for certain we have succeeded in entangling the spins when a photon is
detected. The schemes based on quantum interference of emitted photons can
generate remote atomic entanglement chou05 ; moehring07 , and could be
extended to entangle distant spins childress06 ; simon07a . However these
schemes suffer from low success probability, and require identical atoms or
spins moehring07 . There are also some other schemes based on Faraday rotation
leuenberger05 ; hu08a and the probabilistic schemes based on the dispersive
spin-photon interactions grond08 using bright coherent light as proposed by
van Loock et al and Ladd et al loock06 .
The above scheme can be easily extended to generate multi-spin entangled
states, such as Greenberger-Horne-Zeilinger (GHZ) states greenberger90 by
passing the single photon through all cavities and finally checking the photon
polarization. On setting all $\alpha^{\prime}s$ and $\beta^{\prime}s$ to
$1/\sqrt{2}$, we get maximally entangled spin GHZ states:
$|\text{GHZ}^{s}\rangle_{N}=\frac{1}{\sqrt{N}}(|\uparrow\rangle_{1}|\uparrow\rangle_{2}\cdot\cdot\cdot|\uparrow\rangle_{N}\pm|\downarrow\rangle_{1}|\downarrow\rangle_{2}\cdot\cdot\cdot|\downarrow\rangle_{N})$
(11)
Alternatively, starting from entangled spin pairs, we could build higher-order
entangled spin states such as GHZ states or cluster states briegel01 with N
unlimited. The success probability is $1/2^{k}$ depending on the number k of
single photons used. Again the detection of the photons heralds a successful
entanglement operation.
We point out here that the influence of photon reflection between cavities can
be removed by utilizing suitable timing system. Once we have created entangled
spin states, either optical or electrical pumping can be used to excite
$X^{-}$ in QDs. Spin entanglement is then transferred to photon polarization
entanglement via $X^{-}$ emissions due to the same optical spin selection rule
of $X^{-}$ as discussed earlier. However, we show another scheme below - a
photon entangler which can entangle independent photons with different
frequencies or different pulse length.
Figure 4: Schematic diagram of a spin / photon entangler. (a) A proposed
scheme to entangle remote spins in different microcavities via a single
photon. PBS (polarizing beam splitter) and D1 and D2 (photon detectors). (b) A
proposed scheme to entangle independent photons via a single spin in a
microcavity.
## V Entangle independent photons via a single spin
As shown in Fig. 4b, photon 1 in the state
$|\psi^{ph}\rangle_{1}=\alpha_{1}|R\rangle_{1}+\beta_{1}|L\rangle_{1}$ and
photon 2 in the state
$|\psi^{ph}\rangle_{2}=\alpha_{2}|R\rangle_{2}+\beta_{2}|L\rangle_{2}$ are
input into the cavity in sequence. The two independent photons can have
different frequencies, but both are in the frequency window
$|\omega-\omega_{c}|<\kappa$. The electron spin is prepared in a superposition
state
$|\psi^{s}\rangle=\frac{1}{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)$.
The transmission operator $\hat{t}(\omega)$ for the $X^{-}$-cavity system is
again described by equation (5).
After transmission, the state transformation is
$\begin{split}(\alpha_{1}&|R\rangle_{1}+\beta_{1}|L\rangle_{1})\otimes(\alpha_{2}|R\rangle_{2}+\beta_{2}|L\rangle_{2})\otimes|\psi^{s}\rangle\\\
~{}\xrightarrow{\hat{t}(\omega)}&\frac{t_{0}(\omega_{1})t_{0}(\omega_{2})}{\sqrt{2}}\left(\alpha_{1}\alpha_{2}|R\rangle_{1}|R\rangle_{2}|\uparrow\rangle+\beta_{1}\beta_{2}|L\rangle_{1}|L\rangle_{2}|\downarrow\rangle\right).\end{split}$
(12)
By applying a Hadamard gate on the electron spin (e.g., using a $\pi/2$
microwave or optical pulse), the right side of equation (12) becomes
$\begin{split}\frac{t_{0}(\omega_{1})t_{0}(\omega_{2})}{2}&\\{(\alpha_{1}\alpha_{2}|R\rangle_{1}|R\rangle_{2}+\beta_{1}\beta_{2}|L\rangle_{1}|L\rangle_{2})|\uparrow\rangle\\\
&+(\alpha_{1}\alpha_{2}|R\rangle_{1}|R\rangle_{2}-\beta_{1}\beta_{2}|L\rangle_{1}|L\rangle_{2})|\downarrow\rangle\\}\end{split}$
(13)
Next, the electron spin eigen-state can be detected by the QND measurement as
discussed earlier using a weak coherent light (or single photons) in H
polarization. Depending on the detected spin state in $|\uparrow\rangle$ or
$|\downarrow\rangle$, we get the entangled photon states
$\Phi^{ph}_{12}=(\alpha_{1}\alpha_{2}|R\rangle_{1}|R\rangle_{2}\pm\beta_{1}\beta_{2}|L\rangle_{1}|L\rangle_{2})$
(14)
On setting the coefficients $\alpha_{1,2}$ and $\beta_{1,2}$ to $1/\sqrt{2}$,
maximally entangled photon states can be generated.
Although photon 1 and photon 2 never meet before, each of them gets entangled
with the electron spin after sequentially interacting with the spin. The spin
measurement then projects the two photons into entangled states. This
entanglement-by-projection scheme does not require photon indistinguishability
or photon interference as demanded by other schemes using photon mixing on a
beam splitter fattal04 . This kind of single-photon pulses can come from QD
single photon sources moreau01 ; yuan02 ; santori02 .
Recent experiments have shown GaAs or In(Ga)As single QDs have long electron
spin coherence time ($T_{2}\sim\mu$s)petta05 ; greilich06 and spin relaxation
time ($T_{1}\sim$ms) kroutvar04 . Due to the spin decoherence, the density
matrix of the electron spin in the initial state
$|\psi^{s}\rangle=\frac{1}{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)$
evolves at time t ($t\ll T_{1}$)
$\rho(t)=\begin{pmatrix}1/2&e^{-t/T_{2}}/2\\\
e^{-t/T_{2}}/2&1/2\end{pmatrix},$ (15)
which represents a spin mixed state. As a result, the entanglement fidelity
with respect to equation (14) becomes
$F=\frac{1}{2}(1+e^{-t/T_{2}}),$ (16)
which decreases with t. Therefore high fidelity photon entanglement can only
be achieved when the time interval between two photons is much shorter than
the spin coherence time ($T_{2}\sim\mu$s) in the QD. This entanglement between
photons with different arrival time is ideal for quantum relay type
applications.
If increasing $|t_{0}(\omega)|$ to one by optimizing the cavity, the success
probability for the photon entanglement generation can reach $25\%$, so
coincidence measurement of photons is required to post-select the entangled
state.
We could also extend this scheme to generate multi-photon GHZ states by
passing all photons through the cavity in sequence and finally checking the
spin state after applying a Hadamard gate on the spin. An alternative way to
generate GHZ greenberger90 or cluster states briegel01 is to start from the
generation of entangled photon pairs and then repeat this procedure to
increase the size such that the photon number N can be unlimited. On setting
all $\alpha^{\prime}s$ and $\beta^{\prime}s$ to $1/\sqrt{2}$, we get maximally
entangled photon GHZ states:
$|\text{GHZ}^{ph}\rangle_{N}=\frac{1}{\sqrt{N}}(|R\rangle_{1}|R\rangle_{2}\cdot\cdot\cdot|R\rangle_{N}\pm|L\rangle_{1}|L\rangle_{2}\cdot\cdot\cdot|L\rangle_{N}).$
(17)
The maximal success probability is then $1/2^{N}$.
Figure 5: Schematic diagram of a photon-spin quantum interface. (a) State
transfer from a photon to a spin. (b) State transfer from a spin to a photon.
PBS (polarizing beam splitter), D1 and D2 (photon detectors), and $\lambda/4$
(quarter-wave plate).
## VI Photon-spin quantum interface
Quantum interface is a critical component for quantum networks. Here we show
reversible and coherent quantum-state transfer between photon and spin using
the photon-spin entangling gate. In Fig. 5a, a photon in an arbitrary state
$|\psi^{ph}\rangle=\alpha|R\rangle+\beta|L\rangle$ is input to the cavity with
the electron spin prepared in the state
$|\psi^{s}\rangle=\frac{1}{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)$.
After transmission, the photon and the spin become entangled, i.e,
$(\alpha|R\rangle+\beta|L\rangle)\otimes|\psi^{s}\rangle\xrightarrow{\hat{t}(\omega)}\frac{t_{0}(\omega)}{\sqrt{2}}\left(\alpha|R\rangle|\uparrow\rangle+\beta|L\rangle|\downarrow\rangle\right).$
(18)
By applying a Hadamard gate on the photon state using a polarizing beam
splitter, we obtain a spin state
$|\Phi^{s}_{1}\rangle=\alpha|\uparrow\rangle\pm\beta|\downarrow\rangle$ on
detecting a photon in the $|H\rangle$ or $|V\rangle$ state. Therefore, the
photon state is transferred to the electron spin state.
In Fig. 5b, a photon in the polarization state
$|\psi^{ph}\rangle=(|R\rangle+|L\rangle)/\sqrt{2}$ is input to the cavity with
the electron spin in an arbitrary state
$|\psi^{s}\rangle=\alpha|\uparrow\rangle+\beta|\downarrow\rangle$. After
transmission, the photon and the spin become entangled, i.e,
$|\psi^{ph}\rangle\otimes(\alpha|\uparrow\rangle+\beta|\downarrow\rangle)\xrightarrow{\hat{t}(\omega)}\frac{t_{0}(\omega)}{\sqrt{2}}\left(\alpha|R\rangle|\uparrow\rangle+\beta|L\rangle|\downarrow\rangle\right).$
(19)
After applying a Hadamard gate on the electron spin (e.g., using a $\pi/2$
microwave or optical pulse), the spin eigen-state is detected by the QND
measurement as discussed earlier. On detecting the electron spin in the
$|\uparrow\rangle$ or $|\downarrow\rangle$ state, the photon is then projected
in the state $|\Phi^{ph}_{1}\rangle=\alpha|R\rangle\pm\beta|L\rangle$. So the
spin state is transferred to the photon state.
In contrast to the original teleportation protocol which involves three qubits
bennett93 , our state transfer scheme requires only two qubits thanks to the
tunable amount of entanglement. The success probability is
$|t_{0}(\omega)|^{2}/2$, which can be increased to $50\%$ by optimizing the
cavity. The state transfer fidelity is determined by the gate fidelity as
described by equation (6).
## VII Conclusions
Entanglement is a fundamental resource in quantum information science. With
the proposed photon-spin entangling gate, it is possible to generate almost
all kinds of local or remote entanglement among photons and QD spins with high
fidelity. This entanglement would find wide applications in quantum
communications such as quantum cryptography and quantum teleportation.
Moreover, this entanglement is essential to implement a quantum bus, quantum
interface, quantum memories and quantum repeaters, all of which are critical
building blocks for quantum networks. The high-order multiparticle
entanglement could be used for entanglement-enhanced quantum measurement
giovannetti04 , or cluster-state based quantum computing raussendorf01 ;
nielsen06 .
This gate can also work as an active device such as a polarization-controlled
single photon source moreau01 ; yuan02 ; santori02 , which could be driven by
the electron spin dynamics. These single photons on demand can be sent back to
the gate to get entangled photons based on our schemes. Techniques for
manipulating single photons have been well developed, and significant progress
on fast QD-spin cooling and manipulating has been made recently atature06 ;
xu07 ; petta05 ; berezovsky08 . Together with this work, we believe a charged
QD in an optical cavity is promising for solid-state quantum networks and
quantum computing.
## Acknowledgements
C.Y.H. thanks M. Atatüre, S. Bose, and S. Popescu for helpful discussions.
J.G.R. acknowledges support from the Royal Society. This work is partly funded
by EPSRC-GB IRC in Quantum Information Processing, QAP (Contract No. EU
IST015848), and MEXT from Japan.
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|
arxiv-papers
| 2009-01-26T10:15:19 |
2024-09-04T02:49:00.223072
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C.Y.Hu, W.J.Munro, J.L.O'Brien, J.G. Rarity",
"submitter": "Chengyong Hu",
"url": "https://arxiv.org/abs/0901.3964"
}
|
0901.4058
|
# Radiation from relativistic jets in turbulent magnetic fields
K.-I. Nishikawa M. Medvedev B. Zhang P. Hardee J. Niemiec Å. Nordlund J.
Frederiksen Y. Mizuno H. Sol G. J. Fishman
###### Abstract
Using our new 3-D relativistic electromagnetic particle (REMP) code
parallelized with MPI, we have investigated long-term particle acceleration
associated with an relativistic electron-positron jet propagating in an
unmagnetized ambient electron-positron plasma. The simulations have been
performed using a much longer simulation system than our previous simulations
in order to investigate the full nonlinear stage of the Weibel instability and
its particle acceleration mechanism. Cold jet electrons are thermalized and
ambient electrons are accelerated in the resulting shocks. The acceleration of
ambient electrons leads to a maximum ambient electron density three times
larger than the original value. Behind the bow shock in the jet shock strong
electromagnetic fields are generated. These fields may lead to the afterglow
emission. We have calculated the time evolution of the spectrum from two
electrons propagating in a uniform parallel magnetic field to verify the
technique.
###### Keywords:
Weibel instability, magnetic field generation, synchrotron radiation
###### :
98.70.Rz gamma-ray sources; gamma-ray bursts
## 1 RPIC simulations
Particle-in-cell (PIC) simulations can shed light on the physical mechanism of
particle acceleration that occurs in the complicated dynamics within
relativistic shocks. Recent PIC simulations of relativistic electron-ion and
electron-positron jets injected into an ambient plasma show that acceleration
occurs within the downstream jet nishi03 ; nishi05 ; Hededal & Nishikawa
(2005); nishi06 ; ram07 . In general, these simulations have confirmed that
relativistic jets excite the Weibel instability. The Weibel instability
generates current filaments and associated magnetic fields medv99 , and
accelerates electrons nishi03 ; nishi05 ; Hededal & Nishikawa (2005); nishi06
; ram07 .
Pair Jets Injected into Unmagnetized Pair Plasmas using a Large System
We have performed simulations using a system with ($L_{\rm x},L_{\rm y},L_{\rm
z})=(4005\Delta,105\Delta,105\Delta)$ and a total of $\sim 1$ billion
particles (12 particles$/$cell$/$species for the ambient plasma) in the active
grid zones nishi08a . In the simulations the electron skin depth,
$\lambda_{\rm ce}=c/\omega_{\rm pe}=10.0\Delta$, where $\omega_{\rm pe}=(4\pi
e^{2}n_{\rm e}/m_{\rm e})^{1/2}$ is the electron plasma frequency and the
electron Debye length $\lambda_{\rm e}$ is half of the grid size. Here the
computational domain is six times longer than in our previous simulations
nishi06 ; ram07 . The electron number density of the jet is $0.676n_{\rm e}$,
where $n_{\rm e}$ is the ambient electron density and $\gamma=15$. The
electron/positron thermal velocity of the jet is $v^{\rm e}_{\rm
j,th}=0.014c$, where $c=1$ is the speed of light. Jets are injected in a plane
across the computational grid at $x=25\Delta$ in the positive $x$ direction in
order to eliminate effects associated with the boundary conditions at
$x=x_{\rm\min}$. Radiating boundary conditions were used on the planes at
$x=x_{\min}~{}{\&}~{}x_{\max}$. Periodic boundary conditions were used on all
transverse boundaries. The ambient and jet electron-positron plasma has mass
ratio $m_{\rm e^{-}}/m_{\rm e^{+}}=1$. The electron/positron thermal velocity
in the ambient plasma is $v^{\rm e}_{\rm a,th}=0.05c$.
Figure 1 shows the averaged (in the $y-z$ plane) electron density and
electromagnetic field energy along the jet at $t=2000\omega_{\rm pe}^{-1}$ and
$3750\omega_{\rm pe}^{-1}$. The resulting profiles of jet (red), ambient
(blue), and total (black) electron density are shown in Fig. 1a. The ambient
electrons are accelerated by the jet electrons and pile up towards the front
part of jet. At the earlier time the ambient plasma density increases linearly
behind the jet front as shown by the dashed blue line in Fig. 1a. At the later
time the ambient plasma shows a rapid increase to a plateau behind the jet
front, with additional increase to a higher plateau farther behind the jet
front. The jet density remains approximately constant except near the jet
front.
Figure 1: The averaged values of electron density (a) and field energy (b)
along the $x$ at $t=3750\omega_{\rm pe}^{-1}$ (solid lines) and
$2000\omega_{\rm pe}^{-1}$ (dashed lines). Fig. 1a shows jet electrons (red),
ambient electrons (blue), and the total electron density (black). Fig. 1b
shows electric field energy (blue), magnetic field energy (red), and the total
field energy (black) divided by the total kinetic energy.
The Weibel instability remains excited by continuously injected jet particles
and the electromagnetic fields are kept at a high level, about four times that
seen in a previous much shorter grid simulation system ($L_{\rm
x}=640\Delta$). At the earlier simulation time ($t=2000\omega_{\rm pe}^{-1}$)
a large electromagnetic oscillating structure is generated and accelerates the
ambient plasma. As shown in Fig. 1b, at the later simulation time the
oscillating structure extends up to $x/\Delta=1100$, then becomes more uniform
and the magnetic field energy becomes larger than the electric field energy.
These strong electromagnetic fields become very small beyond $x/\Delta=2000$
in the shocked ambient region nishi06 ; ram07 .
The acceleration of ambient electrons becomes visible when jet electrons pass
about $x/\Delta=500$. The maximum density of accelerated ambient electrons is
attained at $t=1750\omega_{\rm pe}^{-1}$. The maximum density gradually
reaches a plateau as seen in Fig. 1a. The maximum electromagnetic field energy
is located at $x/\Delta=700$ as shown in Fig. 1b. The location of this maximum
remains in this region at large simulation times.
### 1.1 New Numerical Method of Calculating Synchrotron and Jitter Emission
from Electron Trajectories in Self-consistently Generated Magnetic Fields
Let a particle be at position ${\bf{r}_{0}}(t)$ at time $t$ nishi08 ; Hededal
(2005). At the same time, we observe the electric field from the particle from
position $\bf{r}$. However, because of the finite propagation velocity of
light, we observe the particle at an earlier position
$\bf{r}_{0}(\rm{t}^{{}^{\prime}})$ where it was at the retarded time
$t^{{}^{\prime}}=t-\delta t^{{}^{\prime}}=t-\bf{R}(\rm{t}^{{}^{\prime}})/c$.
Here $\bf{R}(\rm{t}^{{}^{\prime}})=|\bf{r}-\bf{r}_{0}(\rm{t}^{{}^{\prime}})|$
is the distance from the charge (at the retarded time $t^{{}^{\prime}}$) to
the observer.
After some calculation and simplifying assumptions the total energy $W$
radiated per unit solid angle per unit frequency from a charged particle
moving with instantaneous velocity $\bm{\beta}$ under acceleration
$\bm{\dot{\beta}}$ can be expressed as
$\displaystyle\frac{d^{2}W}{d\Omega d\omega}$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}cq^{2}}{16\pi^{3}}\left|\int^{\infty}_{\infty}\frac{\bf{n}\times[(\bf{n}-\bm{\beta})\times\bm{\dot{\beta}}]}{(1-\bm{\beta}\cdot\bf{n})^{2}}e^{i\omega(t^{{}^{\prime}}-\bf{n}\cdot\bf{r}_{0}({\rm
t}^{{}^{\prime}})/{\rm c})}dt^{{}^{\prime}}\right|^{2}$ (1)
Here,
$\bf{n}\equiv\bf{R}(\rm{t}^{{}^{\prime}})/|\bf{R}(\rm{t}^{{}^{\prime}})|$ is a
unit vector that points from the particle’s retarded position towards the
observer. The choice of unit vector $\bf{n}$ along the direction of
propagation of the jet (hereafter taken to be the $x$-axis) corresponds to
head-on emission. For any other choice of $\bf{n}$ (e.g., $\theta=1/\gamma$),
off-axis emission is seen by the observer. The observer’s viewing angle is set
by the choice of $\bf{n}$ ($n_{\rm x}^{2}+n_{\rm y}^{2}+n_{\rm z}^{2}=1$).
In order to calculate radiation from relativistic jets propagating along the
$x$ direction nishi08 we consider a test case which includes a parallel
magnetic field ($B_{\rm x}$), and jet velocity of $v_{\rm j1,2}=0.99c$. Two
electrons are injected with different perpendicular velocities ($v_{\perp
1}=0.1c,v_{\perp 2}=0.12c$). A maximum Lorenz factor of
$\gamma_{\max}=\\{(1-(v_{\rm j2}^{2}+v_{\perp 2}^{2})/c^{2}\\}^{-1/2}=13.48$
accompanies the larger perpendicular velocity.
Figure 2: The paths of two electrons moving helically along the $x-$direction
in a homogenous magnetic ($B_{\rm x}$) field shown in the $x-y$-plane (a). The
two electrons radiate a time dependent electric field. An observer situated at
great distance along the n-vector sees the retarded electric field from the
moving electrons (b). The observed power spectrum at different viewing angles
from the two electrons (c). Frequency is in units of $\omega_{\rm pe}^{-1}$.
Figure 2 shows electron trajectories in the $x-y$ plane (a: left panel), the
radiation (retarded) electric field (red: $v_{\perp 1}=0.12c$, blue: $v_{\perp
1}=0.1c$) (b: middle panel), and spectra (right panel) for the case $B_{\rm
x}=3.70$. The two electrons are propagating left to right with gyration in the
$y-z$ plane (not shown). The gyroradius is about $0.44\Delta$ for the electron
with the larger perpendicular velocity. The power spectra were calculated at
the point $(x,y,z)=(64,000,000\Delta,43.0\Delta,43.0\Delta)$. The seven curves
show the power spectrum at viewing angles of 0∘ (red), 10∘ (orange), 20∘
(yellow), 30∘ (moss green), 45∘ (green), 70∘ (light blue), and 90∘ (blue). The
higher frequencies become stronger at the $10^{\circ}$ viewing angle . The
critical angle for off-axis radiation $\theta_{\gamma}=\gamma_{\max}^{-1}$ for
this case is 13.48∘. As shown in this panel, the spectrum at a larger viewing
angle ($>20^{\circ}$) has smaller amplitude.
Since the jet plasma has a large velocity $x$-component in the simulation
frame, the radiation from the particles (electrons and positrons) is heavily
beamed along the $x$-axis (jitter radiation) Medvedev (2006).
In order to obtain the spectrum of synchrotron (jitter) emission, we consider
an ensemble of electrons selected in the region where the Weibel instability
has grown fully and electrons are accelerated in the generated magnetic
fields. We will calculate emission from about 20,000 electrons during the
sampling time $t_{\rm s}=t_{\rm 2}-t_{\rm 1}$ with Nyquist frequency
$\omega_{\rm N}=1/2\Delta t$ where $\Delta t$ is the simulation time step and
the frequency resolution $\Delta\omega=1/t_{\rm s}$.
Emission obtained with the method described above is self-consistent, and
automatically accounts for magnetic field structures on the small scales
responsible for jitter emission. By performing such calculations for
simulations with different parameters, we can investigate and compare the
quite different regimes of jitter- and synchrotron-type emission Medvedev
(2006). The feasibility of this approach has already been demonstrated Hededal
(2005); Hededal & Nordlund (2005), and its implementation is straightforward.
Thus, we should be able to address the low frequency GRB spectral index
violation of the synchrotron spectrum line of death Medvedev (2006).
This work is supported by AST-0506719, AST-0506666, NASA-NNG05GK73G,
NNX07AJ88G, NNX08AG83G, NNX08AL39G, and NNX09AD16G. JN was supported by MNiSW
research projects 1 P03D 003 29 and N N203 393034, and The Foundation for
Polish Science through the HOMING program, which is supported through the EEA
Financial Mechanism.Simulations were performed at the Columbia facility at the
NASA Advanced Supercomputing (NAS). and IBM p690 (Copper) at the National
Center for Supercomputing Applications (NCSA) which is supported by the NSF.
Part of this work was done while K.-I. N. was visiting the Niels Bohr
Institute. Support from the Danish Natural Science Research Council is
gratefully acknowledged.
## References
* (1)
* (2) K.-I. Nishikawa, P. Hardee, G. Richardson, R. Preece, H. Sol, H., and G. J. Fishman, _ApJ_ , 595, 555–563 (2003)
* (3) K.-I. Nishikawa, P. Hardee, G. Richardson, R. Preece, H. Sol, H., and G. J. Fishman, _ApJ_ , 623, 927–937 (2005)
* Hededal & Nishikawa (2005) C. B. Hededal, and K.-I. Nishikawa, 2005, _ApJ_ , 623, L89–L92, (2005)
* (5) K.-I. Nishikawa, P. Hardee, C. B. Hededal, and G. J. Fishman, _ApJ_ , 642, 1267–1274 (2006)
* (6) E. Ramirez-Ruiz, K.-I. Nishikawa, and C. B. Hededal, _ApJ_ , 671, 1877–1885 (2007)
* (7) K.-I. Nishikawa, J. Niemiec, H. Sol, M. Medvedev, et al. in Proceedings of The 4th Heidelberg International Symposium on High Energy Gamma-Ray Astronomy, July 7-11, 2008, in Heidelberg, Germany (2008) (arXiv:0809.5067)
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* (9) K.-I. Nishikawa, J. Niemiec, M. Medvedev, H. Sol, P. E. Hardee, Y. Mizuno, B. Zhang, M. Pohl, and M. Oka, _ApJ_ , in preparation (2008)
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* Hededal & Nordlund (2005) C.B. Hededal, and Å. Nordlund, _ApJL_ , submitted (2005) (arXiv:astro-ph/0511662)
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|
arxiv-papers
| 2009-01-26T17:48:35 |
2024-09-04T02:49:00.232944
|
{
"license": "Public Domain",
"authors": "K.-I. Nishikawa, M. Medvedev, B. Zhang, P. Hardee, J. Niemiec, A.\n Nordlund, J. Frederiksen, Y. Mizuno, H. Sol, G. J. Fishman",
"submitter": "Ken-Ichi Nishikawa",
"url": "https://arxiv.org/abs/0901.4058"
}
|
0901.4113
|
# Beneficial effects of intercellular interactions between pancreatic islet
cells in blood glucose regulation
Junghyo Jo
Laboratory of Biological Modeling,
National Institute of Diabetes and Digestive and Kidney Diseases,
National Institutes of Health, Bethesda, MD 20892, U.S.A.
Moo Young Choi
Department of Physics and Astronomy and Center for Theoretical Physics,
Seoul National University, Seoul 151-747, Korea
Corresponding author. Address: Department of Physics and Astronomy, Seoul
National University, Seoul 151-747, Korea. E-mail: mychoi@snu.ac.kr Duk-Su
Koh
Department of Physiology and Biophysics,
University of Washington, Seattle, WA 98195, U.S.A
###### Abstract
Glucose homeostasis is controlled by the islets of Langerhans which are
equipped with $\alpha$-cells increasing the blood glucose level, $\beta$-cells
decreasing it, and $\delta$-cells the precise role of which still needs
identifying. Although intercellular communications between these endocrine
cells have recently been observed, their roles in glucose homeostasis have not
been clearly understood. In this study, we construct a mathematical model for
an islet consisting of two-state $\alpha$-, $\beta$-, and $\delta$-cells, and
analyze effects of known chemical interactions between them with emphasis on
the combined effects of those interactions. In particular, such features as
paracrine signals of neighboring cells and cell-to-cell variations in response
to external glucose concentrations as well as glucose dynamics, depending on
insulin and glucagon hormone, are considered explicitly. Our model predicts
three possible benefits of the cell-to-cell interactions: First, the
asymmetric interaction between $\alpha$\- and $\beta$-cells contributes to the
dynamic stability while the perturbed glucose level recovers to the normal
level. Second, the inhibitory interactions of $\delta$-cells for glucagon and
insulin secretion prevent the wasteful co-secretion of them at the normal
glucose level. Finally, the glucose dose-responses of insulin secretion is
modified to become more pronounced at high glucose levels due to the
inhibition by $\delta$-cells. It is thus concluded that the intercellular
communications in islets of Langerhans should contribute to the effective
control of glucose homeostasis.
Key words: glucose homeostasis, islets of Langerhans, feedback, diabetes
## 1 Introduction
Homeostasis, maintenance of the constant physiological state, is one of the
main characteristics of life. In particular, glucose homeostasis is critical
because glucose is the energy source for our bodies; the malfunctioning of
this process causes several disease states including diabetes mellitus and
brain coma.
In order to understand glucose homeostasis, we first need to examine the
tissue controlling the blood glucose level, the islet of Langerhans in the
pancreas. It consists mainly of three types of endocrine cells: $\alpha$-cells
which secrete glucagon hormone increasing the glucose level, $\beta$-cells
which secrete insulin decreasing the glucose level, and $\delta$-cells which
secrete somatostatin, known to inhibit activities of $\alpha$\- and
$\beta$-cells. The hormone secretion of a cell influences the behavior of
neighboring cells, and is thus tightly correlated with the islet structure
[Hopcroft et al., 1985, Pipeleers et al., 1982]. In rodents, an islet contains
about 1,000 endocrine cells on average: $\beta$-cells, occupying the most
volume (70 to 80%) of an islet, populate largely in its core, whereas
non-$\beta$-cells are located on the mantle [Brissova et al., 2005].
To the first approximation, $\alpha$\- and $\beta$-cells should be sufficient
for glucose control because $\alpha$-cells can increase the glucose level
whereas $\beta$-cells can decrease the level. The importance of this bi-
hormonal mechanism for glucose homeostasis has been well recognized
[Cherrington et al., 1976]. However, it should be noted that endocrine cells
in the islet interact with each other rather than act independently. For
example, the electrical coupling between $\beta$-cells through gap-junctions
is known to enhance insulin secretion of coupled $\beta$-cells [Jo et al.,
2005, Pipeleers et al., 1982, Sherman et al., 1988]. In addition, it has been
recently reported that chemical interactions between neighboring cells through
hormones [Cherrington et al., 1976, Franklin et al., 2005, Orci & Unger, 1975,
Ravier & Rutter, 2005, Samols et al., 1965, Samols & Harrison, 1976, Soria et
al., 2000] and neurotransmitters [Brice et al., 2002, Franklin & Wollheim,
2004, Gilon et al., 1991, Moriyama & Hayashi, 2003, Rorsman et al., 1989,
Wendt et al., 2004], termed “paracrine interaction,” affect glucose
regulation.
Among these intercellular communications, enhancement of insulin secretion by
glucagon [Brereton et al., 2007, Samols et al., 1965, Soria et al., 2000]
seems to be paradoxical because $\alpha$-cells, playing the reciprocal role to
$\beta$-cells in glucose regulation, promote the activity of $\beta$-cells. In
contrast, insulin, secreted by $\beta$-cells, inhibits glucagon secretion of
$\alpha$-cells [Cherrington et al., 1976, Franklin et al., 2005, Ravier &
Rutter, 2005, Samols & Harrison, 1976, Soria et al., 2000], which appears
natural. Furthermore, the role of the third cell-type, $\delta$-cells, is
still not completely known although there have been reports that somatostatin
hormone, secreted by $\delta$-cells, suppresses the hormone secretion of both
$\alpha$\- and $\beta$-cells [Cherrington et al., 1976, Daunt et al., 2006,
Orci & Unger, 1975, Soria et al., 2000].
What is then the raison d’etre of the paradoxical interactions between
$\alpha$\- and $\beta$-cells and the inhibitory action by $\delta$-cells?
Despite previous studies as to these questions over the last thirty years
[Orci & Unger, 1975, Pipeleers, 1987, Soria et al., 2000, Unger & Orci, 1977],
there still lacks concrete understanding of the role of these interactions in
terms of glucose homeostasis. The primary difficulty in understanding these
interactions lies in the complexity of the islet system which includes many
interactions between different coexisting cell-types working in different
conditions.
In this paper, we analyze the interactions between $\alpha$-, $\beta$-, and
$\delta$-cells, which contribute to the precise control of the glucose level,
by means of a mathematical model incorporating experimentally known
interactions between islet cells (see above). As a result, our model predicts
that the intracellular interactions modify insulin and glucagon secretion in a
way to control the blood glucose level more efficiently.
## 2 Islet model
### 2.1 Activity of islet cells
We begin with a simplified model in which cells of each type can take one of
two states (active and silent). The state of a cell, represented by “Ising
spin” $\sigma$, is defined to be active ($\sigma=+1$) when the cell secretes
islet hormone; otherwise the state is defined as silent ($\sigma=-1$).
Accordingly, the state of an islet consisting of $\alpha$-, $\beta$-, and
$\delta$-cells can be represented by
($\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta}$). There is a total of
$2^{3}$ possible states of the islet, among which ($+1,-1,-1$) and
($-1,+1,+1$) describe the islet state at low and high glucose levels,
respectively. The main source of changing cell states is the blood glucose
level $\tilde{G}$, which globally influences all cells. In addition, the
paracrine interaction $\tilde{J}$ from neighboring cells locally affects cell
states. In this manner we obtain a simple Ising-type model, generally
characterizing two-state dynamics in statistical physics: The glucose level
$\tilde{G}$ corresponds to the external magnetic field and the paracrine
interaction $\tilde{J}$ to the local interaction between spins.
¿From the known cellular interactions illustrated in Fig. 1, one may determine
local stimuli $G_{\alpha}$, $G_{\beta}$, and $G_{\delta}$, which change the
states of $\alpha$-, $\beta$-, and $\delta$-cells, respectively, in the forms:
$\displaystyle G_{\alpha}$ $\displaystyle=$
$\displaystyle-G-\frac{1+\sigma_{\beta}}{2}J_{\alpha\beta}-\frac{1+\sigma_{\delta}}{2}J_{\alpha\delta}$
$\displaystyle G_{\beta}$ $\displaystyle=$ $\displaystyle
G+\frac{1+\sigma_{\alpha}}{2}J_{\beta\alpha}-\frac{1+\sigma_{\delta}}{2}J_{\beta\delta}$
$\displaystyle G_{\delta}$ $\displaystyle=$ $\displaystyle mG,$ (1)
where $G\equiv\tilde{G}-\tilde{G}_{0}$ measures the excess glucose level from
the basal glucose level $\tilde{G}_{0}$ during the fasting period. The
reciprocal nature of $\alpha$\- and $\beta$-cells in the responses to glucose
is manifested by the opposite signs in front of $G$ in the first equation (for
$G_{\alpha}$) and the second one (for $G_{\beta}$) of Eq. 2.1. In addition,
the asymmetric interaction between these two cell types is also reflected in
the second terms involving $J_{\alpha\beta}$ and $J_{\beta\alpha}$ of the
equations. For simplicity, we assume that the interaction strength
$J_{\beta\alpha}$ from $\alpha$\- to $\beta$-cells is the same as
$J_{\alpha\beta}$ from $\beta$\- to $\alpha$-cells and given by $J_{1}$, i.e.,
$J_{\alpha\beta}=J_{\beta\alpha}=J_{1}$. The last terms involving
$J_{\alpha\delta}$ and $J_{\beta\delta}$ describe the inhibition effects of
$\delta$-cells on $\alpha$\- and $\beta$-cells, both with negative signs.
Although the endogenous strengths of the interactions from $\delta$-cells to
$\alpha$\- and $\beta$-cells are not known, the exogenous stimulus of
somatostatin has been reported to inhibit both insulin and glucagon secretion
to a similar degree [Cherrington et al., 1976]. As a first approximation, it
is thus assumed that both interactions have the same strength:
$J_{\alpha\delta}=J_{\beta\delta}=J_{2}$. Here the interaction strengths
$J_{1}$ and $J_{2}$ are expressed in terms of the relative effects to glucose
stimulation, and therefore have the unit of mM corresponding to the hormonal
stimulus $\tilde{J}$, namely, a given amount of stimulus $\tilde{J}$ by
hormone is considered to produce the same effects on a cell as a certain
amount $J$ of glucose stimulation. In our simplified model, $\delta$-cells are
not influenced by neighboring $\alpha$\- and $\beta$-cells but stimulated
solely by glucose; therefore, $G_{\delta}$ depends only on $G$ in Eq. 2.1.
Like $\beta$-cells, $\delta$-cells become active, and secrete somatostatin
above a threshold level of glucose. The glucose sensitivity of $\delta$-cells
is expected to have a value between zero and unity, i.e., $0<m<1$ because the
threshold level for the activation of $\delta$-cells is lower than that of
$\beta$-cells [Efendić et al., 1979, Nadal et al., 1999]. We thus choose the
value $m=0.5$ in this study; the overall behavior does not depend
qualitatively on the value of $m$.
For given local stimulus $G_{\alpha}$ considering glucose stimulus $G$ and
effects of insulin and somatostatin, the transition rate of an $\alpha$-cell
from state $\sigma_{\alpha}$ to state $-\sigma_{\alpha}$ depends on the states
of other cell types as well as its own state, and is denoted as
$w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$. This transition
rate should satisfy the detailed balance condition between two $\alpha$-cell
states $\sigma_{\alpha}$ and $-\sigma_{\alpha}$ at equilibrium:
$w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})P(\sigma_{\alpha})=w_{\alpha}(-\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})P(-\sigma_{\alpha}),$
(2)
where the probability $P(\sigma_{\alpha})$ for state $\sigma_{\alpha}$ follows
the Boltzmann distribution $\exp[-G_{\alpha}(1+\sigma_{\alpha})/2\Theta]$ with
respect to the quantity $G_{\alpha}(1+\sigma_{\alpha})/2$ for the local
stimulus $G_{\alpha}$. Namely, $\alpha$-cells favor the state minimizing the
quantity $G_{\alpha}(1+\sigma_{\alpha})/2$. Here $\Theta$ measures the amount
of uncertainty, which is inevitable in biological systems. The origin may be
the heterogeneous glucose sensitivity of cells and/or the environmental noise
including thermal fluctuations. It is obvious that
$G_{\alpha}(1+\sigma_{\alpha})/2$ and $\Theta$ correspond to the energy and
the temperature, respectively, in statistical physics. In this study, the
“temperature” is taken to be unity ($\Theta=1$) in units of the “energy”,
which is biologically tantamount to the fluctuations caused by 1 mM change of
glucose stimulation.
The ratio between the reciprocal transition rates thus reads
$\displaystyle\frac{w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})}{w_{\alpha}(-\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})}$
$\displaystyle=$
$\displaystyle\exp\Bigg{[}-\frac{1}{\Theta}G_{\alpha}\sigma_{\alpha}\Bigg{]}$
(3) $\displaystyle=$
$\displaystyle\exp\Bigg{[}\frac{1}{\Theta}\Bigg{(}G_{\alpha}^{\text{eff}}\sigma_{\alpha}+\frac{J_{1}}{2}\sigma_{\alpha}\sigma_{\beta}+\frac{J_{2}}{2}\sigma_{\delta}\sigma_{\alpha}\Bigg{)}\Bigg{]}$
with $G_{\alpha}^{\text{eff}}\equiv G+J_{1}/2+J_{2}/2$, where Eq. 2.1 has been
used to obtain the second line. There the three stimulation terms represent
effective glucose stimulation, paracrine interaction from $\beta$-cells, and
another from $\delta$-cells, respectively. Assuming that these stimuli affect
independently the $\alpha$-cell state, we write the transition rate in the
form
$\displaystyle w_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{2\tau}\left[1+\tanh\left(\frac{G_{\alpha}^{\text{eff}}}{2\Theta}\right)\sigma_{\alpha}\right]\left[1+\tanh\left(\frac{J_{1}}{4\Theta}\right)\sigma_{\alpha}\sigma_{\beta}\right]$
(4)
$\displaystyle\times\left[1+\tanh\left(\frac{J_{2}}{4\Theta}\right)\sigma_{\delta}\sigma_{\alpha}\right],$
where $\tau$ measures the characteristic time of the transition and it has
been noted that $\tanh(y\sigma)=\sigma\tanh y$ for $\sigma=\pm 1$. Note that
among possible transition rates satisfying Eq. 2, we adopt the Glauber
dynamics [Glauber, 1963] to choose the specific form of Eq. 4, which exhibits
the sigmoidal form ubiquitously describing response functions in biological
systems. However, the behavior of the system in general does not depend
qualitatively on the specific form of the transition rate satisfying Eq. 2.
Similarly, we obtain the transition rates $w_{\beta}$ and $w_{\delta}$ of
$\beta$\- and $\delta$-cells.
The transition rates of three cell types can be summarized as
$\displaystyle w_{x}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$
$\displaystyle=$
$\displaystyle\frac{1}{2\tau}\left[w^{x}+w^{x}_{\alpha}\sigma_{\alpha}+w^{x}_{\beta}\sigma_{\beta}+w^{x}_{\delta}\sigma_{\delta}+w^{x}_{\alpha\beta}\sigma_{\alpha}\sigma_{\beta}\right.$
(5)
$\displaystyle\left.+w^{x}_{\beta\delta}\sigma_{\beta}\sigma_{\delta}+w^{x}_{\delta\alpha}\sigma_{\delta}\sigma_{\alpha}+w^{x}_{\alpha\beta\delta}\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\right],$
with $x=\alpha,\beta,$ and $\delta$, where the coefficients are given in
Tables 1 and 2. The master equation, describing the evolution of the
probability $P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$ for the islet
in state $(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$, reads
$\displaystyle\frac{d}{dt}P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$
$\displaystyle~{}~{}=w_{\alpha}(-\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})P(-\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})+w_{\beta}(\sigma_{\alpha},-\sigma_{\beta},\sigma_{\delta})P(\sigma_{\alpha},-\sigma_{\beta},\sigma_{\delta})$
$\displaystyle~{}~{}~{}~{}~{}+w_{\delta}(\sigma_{\alpha},\sigma_{\beta},-\sigma_{\delta})P(\sigma_{\alpha},\sigma_{\beta},-\sigma_{\delta})-w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$
$\displaystyle~{}~{}~{}~{}~{}-w_{\beta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})-w_{\delta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$
(6)
with the transition rates $w_{\alpha}$, $w_{\beta}$, and $w_{\delta}$ in Eq.
5. Note that Eq. 2.1 describes the net flux to state
$(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$ simply given by the
difference between the in-flux to state
$(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$ from other states
$(-\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$,
$(\sigma_{\alpha},-\sigma_{\beta},\sigma_{\delta})$, and
$(\sigma_{\alpha},\sigma_{\beta},-\sigma_{\delta})$ and the out-flux from
state $(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$ to others.
¿From this master equation, it is straightforward to obtain the time evolution
of the ensemble averages of the cell states and their correlations. For
example, multiplying both sides of Eq. 2.1 by $\sigma_{\alpha}$ and summing
over all configurations, we obtain the evolution equation for the average
$\langle\sigma_{\alpha}\rangle\equiv\sum_{\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta}}\sigma_{\alpha}P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$
of the state of $\alpha$-cells:
$\frac{d}{dt}\langle\sigma_{\alpha}\rangle=-2\langle\sigma_{\alpha}w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle$
(7)
and similarly,
$\displaystyle\frac{d}{dt}\langle\sigma_{\beta}\rangle$ $\displaystyle=$
$\displaystyle-2\langle\sigma_{\beta}w_{\beta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle$
(8) $\displaystyle\frac{d}{dt}\langle\sigma_{\delta}\rangle$ $\displaystyle=$
$\displaystyle-2\langle\sigma_{\delta}w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle.$
(9)
Note that $1+\langle\sigma_{\alpha}\rangle$ gives twice the average activity
of $\alpha$-cells, etc.
Among the eight equations for the probability
$P(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})$ corresponding to the eight
possible states of the islet, only seven are independent, due to the
normalization condition
$\sum_{\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta}}p(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})=1$.
Therefore, there exist four more equations in addition to the above three
describing the average of cell states. Those are evolution equations for
correlations of two cell states and of three cell states. The equation for the
correlation function $\langle\sigma_{\alpha}\sigma_{\beta}\rangle$ of the
$\alpha$-cell and $\beta$-cell states can again be derived from Eq. 2.1,
multiplied by $\sigma_{\alpha}\sigma_{\beta}$ and summed over all
configurations:
$\frac{d}{dt}\langle\sigma_{\alpha}\sigma_{\beta}\rangle=-2\langle\sigma_{\alpha}\sigma_{\beta}w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle-2\langle\sigma_{\alpha}\sigma_{\beta}w_{\beta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle.$
(10)
The equations for $\langle\sigma_{\beta}\sigma_{\delta}\rangle$ and
$\langle\sigma_{\delta}\sigma_{\alpha}\rangle$ are also obtained in the same
way:
$\displaystyle\frac{d}{dt}\langle\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle=$
$\displaystyle-2\langle\sigma_{\beta}\sigma_{\delta}w_{\beta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle-2\langle\sigma_{\beta}\sigma_{\delta}w_{\delta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle$
$\displaystyle\frac{d}{dt}\langle\sigma_{\delta}\sigma_{\alpha}\rangle$
$\displaystyle=$
$\displaystyle-2\langle\sigma_{\delta}\sigma_{\alpha}w_{\delta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle-2\langle\sigma_{\delta}\sigma_{\alpha}w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle.$
(11)
Note that correlations of two cell states represent the relative activity of
the two cells. Accordingly, it makes a good measure of the different responses
between two cells. Similarly, the equation for correlations of three cell
states is given by
$\displaystyle\frac{d}{dt}\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle=$
$\displaystyle-2\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}w_{\alpha}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle-2\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}w_{\beta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle$
(12)
$\displaystyle-2\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}w_{\delta}(\sigma_{\alpha},\sigma_{\beta},\sigma_{\delta})\rangle.$
Substituting the transition rates in Eq. 5 into Eqs. 7 to 12, we finally
obtain equations for the states of the three cell types and their
correlations:
$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\alpha}\rangle$ $\displaystyle=$
$\displaystyle-w^{\alpha}_{\alpha}-w^{\alpha}\langle\sigma_{\alpha}\rangle-w^{\alpha}_{\alpha\beta}\langle\sigma_{\beta}\rangle-w^{\alpha}_{\delta\alpha}\langle\sigma_{\delta}\rangle-w^{\alpha}_{\beta}\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle-w^{\alpha}_{\alpha\beta\delta}\langle\sigma_{\beta}\sigma_{\delta}\rangle-w^{\alpha}_{\delta}\langle\sigma_{\delta}\sigma_{\alpha}\rangle+w^{\alpha}_{\beta\delta}\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\beta}\rangle$ $\displaystyle=$
$\displaystyle-w^{\beta}_{\beta}-w^{\beta}_{\alpha\beta}\langle\sigma_{\alpha}\rangle-w^{\beta}\langle\sigma_{\beta}\rangle-w^{\beta}_{\beta\delta}\langle\sigma_{\delta}\rangle-w^{\beta}_{\alpha}\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle-w^{\beta}_{\delta}\langle\sigma_{\beta}\sigma_{\delta}\rangle-w^{\beta}_{\alpha\beta\delta}\langle\sigma_{\delta}\sigma_{\alpha}\rangle-w^{\beta}_{\delta\alpha}\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\delta}\rangle$ $\displaystyle=$
$\displaystyle-w^{\delta}_{\delta}-w^{\delta}_{\delta\alpha}\langle\sigma_{\alpha}\rangle-w^{\delta}_{\beta\delta}\langle\sigma_{\beta}\rangle-w^{\delta}\langle\sigma_{\delta}\rangle-w^{\delta}_{\alpha\beta\delta}\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle-w^{\delta}_{\beta}\langle\sigma_{\beta}\sigma_{\delta}\rangle-w^{\delta}_{\alpha}\langle\sigma_{\delta}\sigma_{\alpha}\rangle-w^{\delta}_{\alpha\beta}\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle=$
$\displaystyle-(w^{\alpha}_{\alpha\beta}+w^{\beta}_{\alpha\beta})-(w^{\alpha}_{\beta}+w^{\beta}_{\beta})\langle\sigma_{\alpha}\rangle-(w^{\alpha}_{\alpha}+w^{\beta}_{\alpha})\langle\sigma_{\beta}\rangle$
$\displaystyle-(w^{\alpha}_{\alpha\beta\delta}+w^{\beta}_{\alpha\beta\delta})\langle\sigma_{\delta}\rangle-(w^{\alpha}+w^{\beta})\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle-(w^{\alpha}_{\delta\alpha}+w^{\beta}_{\delta\alpha})\langle\sigma_{\beta}\sigma_{\delta}\rangle-(w^{\alpha}_{\beta\delta}+w^{\beta}_{\beta\delta})\langle\sigma_{\delta}\sigma_{\alpha}\rangle$
$\displaystyle-(w^{\alpha}_{\delta}+w^{\beta}_{\delta})\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle=$
$\displaystyle-(w^{\beta}_{\beta\delta}+w^{\delta}_{\beta\delta})-(w^{\beta}_{\alpha\beta\delta}+w^{\delta}_{\alpha\beta\delta})\langle\sigma_{\alpha}\rangle-(w^{\beta}_{\delta}+w^{\delta}_{\delta})\langle\sigma_{\beta}\rangle$
$\displaystyle-(w^{\beta}_{\beta}+w^{\delta}_{\beta})\langle\sigma_{\delta}\rangle-(w^{\beta}_{\delta\alpha}+w^{\delta}_{\delta\alpha})\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle-(w^{\beta}+w^{\delta})\langle\sigma_{\beta}\sigma_{\delta}\rangle-(w^{\beta}_{\alpha\beta}+w^{\delta}_{\alpha\beta})\langle\sigma_{\delta}\sigma_{\alpha}\rangle$
$\displaystyle-(w^{\beta}_{\alpha}+w^{\delta}_{\alpha})\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\delta}\sigma_{\alpha}\rangle$
$\displaystyle=$
$\displaystyle-(w^{\delta}_{\delta\alpha}+w^{\alpha}_{\delta\alpha})-(w^{\delta}_{\delta}+w^{\alpha}_{\delta})\langle\sigma_{\alpha}\rangle-(w^{\delta}_{\alpha\beta\delta}+w^{\alpha}_{\alpha\beta\delta})\langle\sigma_{\beta}\rangle$
$\displaystyle-(w^{\delta}_{\alpha}+w^{\alpha}_{\alpha})\langle\sigma_{\delta}\rangle-(w^{\delta}_{\beta\delta}+w^{\alpha}_{\beta\delta})\langle\sigma_{\alpha}\sigma_{\beta}\rangle$
$\displaystyle-(w^{\delta}_{\alpha\beta}+w^{\alpha}_{\alpha\beta})\langle\sigma_{\beta}\sigma_{\delta}\rangle-(w^{\delta}+w^{\alpha})\langle\sigma_{\delta}\sigma_{\alpha}\rangle$
$\displaystyle-(w^{\delta}_{\beta}+w^{\alpha}_{\beta})\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle\tau\frac{d}{dt}\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle=$
$\displaystyle-(w^{\alpha}_{\alpha\beta\delta}+w^{\beta}_{\alpha\beta\delta}+w^{\delta}_{\alpha\beta\delta})-(w^{\alpha}_{\beta\delta}+w^{\beta}_{\beta\delta}+w^{\delta}_{\beta\delta})\langle\sigma_{\alpha}\rangle$
(13)
$\displaystyle-(w^{\alpha}_{\delta\alpha}+w^{\beta}_{\delta\alpha}+w^{\delta}_{\delta\alpha})\langle\sigma_{\beta}\rangle-(w^{\alpha}_{\alpha\beta}+w^{\beta}_{\alpha\beta}+w^{\delta}_{\alpha\beta})\langle\sigma_{\delta}\rangle$
$\displaystyle-(w^{\alpha}_{\delta}+w^{\beta}_{\delta}+w^{\delta}_{\delta})\langle\sigma_{\alpha}\sigma_{\beta}\rangle-(w^{\alpha}_{\alpha}+w^{\beta}_{\alpha}+w^{\delta}_{\alpha})\langle\sigma_{\beta}\sigma_{\delta}\rangle$
$\displaystyle-(w^{\alpha}_{\beta}+w^{\beta}_{\beta}+w^{\delta}_{\beta})\langle\sigma_{\delta}\sigma_{\alpha}\rangle-(w^{\alpha}+w^{\beta}+w^{\delta})\langle\sigma_{\alpha}\sigma_{\beta}\sigma_{\delta}\rangle.$
### 2.2 Glucose homeostasis
Heretofore we have focused on the cellular interactions at a given glucose
level. To study dynamics of glucose homeostasis, however, we should also take
into account the change of the glucose level and incorporate another equation
for glucose regulation into the model. Based on the fact that $\alpha$\- and
$\beta$-cells secrete glucagon and insulin, respectively, raising and reducing
the glucose level, the equation for the glucose level $G$ is taken to be
$\tau_{G}\frac{dG}{dt}=\frac{1+\langle\sigma_{\alpha}\rangle}{2}-\frac{1+\langle\sigma_{\beta}\rangle}{2},$
(14)
where $\tau_{G}$ is the characteristic time for the hormones to regulate the
glucose level. It is expected that $\tau_{G}$ is larger than the
characteristic time $\tau$ of the change in cell states. Equation 14 describes
the decrease or increase of the glucose level when $\alpha$-cells or
$\beta$-cells are active ($\sigma_{\alpha}=1$ or $\sigma_{\beta}=1$). Here,
for simplicity, we have used the same characteristic time $\tau_{G}$ for
glucagon and insulin to regulate glucose levels. Having different time
constants turns out merely to shift the stationary level of blood glucose. To
sum, we have a total of eight differential equations given by Eqs. 13 and 14,
which describe the process of glucose homeostasis.
## 3 Results
### 3.1 Asymmetric interactions between $\alpha$\- and $\beta$-cells
In our model, activities of $\alpha$-, $\beta$-, and $\delta$-cells are
determined by the external glucose level together with feedback loops of
intercellular interactions. A given cell, subject to a glucose stimulus,
secretes hormone which influences the behavior of neighboring cells. In
response, the neighboring cells reversely influence the given cell. These
mutual interactions through hormones constitute the feedback loop which is
widely employed for advanced system control in engineering [Bechhoefer, 2005].
The interactions between $\alpha$\- and $\beta$-cells are asymmetric: While
glucagon secreted from $\alpha$-cells enhances insulin secretion of
$\beta$-cells [Brereton et al., 2007, Samols et al., 1965, Soria et al.,
2000], insulin inhibits glucagon secretion [Cherrington et al., 1976, Franklin
et al., 2005, Ravier & Rutter, 2005, Samols & Harrison, 1976, Soria et al.,
2000]. The former positive interaction to the counterpart cells may seem
strange, but it eventually contributes to the construction of a negative
feedback loop for both cells. At low glucose levels, $\alpha$-cells secrete
glucagon, which enhances insulin secretion. In turn, insulin inhibits the
glucagon secretion of $\alpha$-cells. Therefore, their interactions as a whole
tend to suppress the glucagon secretion from $\alpha$-cells. Similar negative
feedback operates when $\beta$-cells are activated by high glucose
concentration. It is noteworthy that this feedback works more efficiently in
case that the glucose level varies. At a static glucose level, it should be
difficult for the mutual interactions between $\alpha$\- and $\beta$-cells to
arise simultaneously because insulin and glucagon are secreted at different
glucose levels.
In general, a negative feedback favors stability of a system because it
attenuates overaction of the system such as overshoot or undershoot. The
negative feedbacks in an islet system contribute to the stable recovery to the
normal glucose level $G_{\infty}$, when the system is externally perturbed by
stimuli such as a glucose dose. The normal glucose level $G_{\infty}$, reached
by $G\,(\equiv\tilde{G}-\tilde{G}_{0})$ at stationarity, depends on the
cellular interactions shown in Fig. 2. The asymmetric interaction $J_{1}$
lowers the basal glucose level because $\alpha$-cells activate $\beta$-cells
which secret insulin and thus reduces the glucose level. In addition, the
inhibitory interaction $J_{2}$ of $\delta$-cells, albeit the same for
$\alpha$\- and $\beta$-cells, suppresses the activity of $\beta$-cells more
than that of $\alpha$-cells at the normal glucose level, because the activity
of $\beta$-cells is higher than that of $\alpha$-cells resulting from the
asymmetric interaction between $\alpha$\- and $\beta$-cells. Accordingly, the
basal glucose level tends to increase as the strength $J_{2}$ of the
inhibitory interaction is increased.
Figure 3 demonstrates the smooth recovery of the glucose level in the presence
of cellular interactions (solid line), compared with the somewhat erratic
recovery, once reaching low glucose levels, in the absence of the interactions
(dashed line). For comparison, we also consider the behavior in the case of
symmetric interactions between $\alpha$\- and $\beta$-cells, i.e., where
glucagon inhibits insulin secretion and vice versa, only to find even more
erratic recovery (see the dotted line). Shown here is the recovery from the
high glucose state [$G=1$ mM (or $\tilde{G}=\tilde{G}_{0}$ \+ 1 mM),
$\langle\sigma_{\alpha}\rangle=-1$, and $\langle\sigma_{\beta}\rangle=1$]. The
recovery from a low glucose state gives the same results (data not shown)
although such erratic recovery is more pronounced for the glucose level
starting from a higher value.
To examine the stability in approaching the normal glucose level, we define
the balance function
$b(G)\equiv\tau_{G}\frac{dG}{dt}=\frac{1+\langle\sigma_{\alpha}\rangle}{2}-\frac{1+\langle\sigma_{\beta}\rangle}{2},$
(15)
which describes the glucose level change during the characteristic time. Since
the activity of cells represents their hormone secretion, $b(G)$ appropriately
describes the effectiveness of the glucose regulation by $\alpha$\- and
$\beta$-cells. If the characteristic time $\tau_{G}$ of glucose regulation is
much larger than the characteristic time $\tau$ of cell responses in Eq. 13,
i.e., $\tau\ll\tau_{G}$, the glucose level should be in a quasi-stationary
state at time $t$ shorter than $\tau_{G}$. Then the fast dynamics of cell
states in Eq. 13 saturates rapidly at a given glucose level and the seven
variables, activities and correlations, reach their fixed points depending on
the glucose level $G$. In particular $\langle\sigma_{\alpha}\rangle$ and
$\langle\sigma_{\beta}\rangle$ depend on $G$, giving the balance function in
Eq. 15 as a function of $G$, with a fixed point at $G=G_{\infty}$ (see Fig.
4). At low glucose levels ($G<G_{\infty}$), we have the balance function
greater than zero ($b>0$), or $dG/dt>0$, thus the glucose level grows with
time. At high glucose levels ($G>G_{\infty}$), the opposite behavior arises.
The resulting flow of the balance function is illustrated by the arrows in
Fig. 4 and it is concluded that the balance function correctly describes
glucose homeostasis. Further, the slope of $b(G)$ near the fixed point
$G=G_{\infty}$ represents how smoothly the glucose level approaches the normal
level: The slope of the balance function for the asymmetric interaction is
small at $G=G_{\infty}$, which results in the smooth recovery of the normal
glucose level shown in Fig. 3. This result is more evident with the
interaction strength $J_{1}$ larger and the characteristic time $\tau_{G}$
shorter.
If $\alpha$\- and $\beta$-cells would inhibit each other, how should the
result change? As suggested already [Saunders et al., 1998], the bidirectional
inhibitory interactions seem to be optimal in view of that $\alpha$\- and
$\beta$-cells play opposite roles in glucose regulation. Remarkably, however,
such symmetric interactions turn out to result in dynamically unstable
responses, as shown by the dotted line in Fig. 3. If this were the case,
glucagon secreted by $\alpha$-cells at low glucose levels would suppress
$\beta$-cells from secreting insulin. As the secretion of insulin decreases,
so would the inhibitory effects of insulin on the glucagon secretion diminish.
It should thus follow that glucagon secretion is not negatively controlled,
implying more glucagon secretion. Such an apparent positive feedback loop,
enhancing hormone secretion, gives rise to an instability in the islet system
(see Fig. 3).
### 3.2 Inhibitory interactions of $\delta$-cells
#### 3.2.1 Suppression of co-secretion from $\alpha$\- and $\beta$-cells
There is basal hormone secretion from $\alpha$\- and $\beta$-cells even at the
normal glucose level [Cherrington et al., 1976], where it is not necessary to
change the blood glucose concentration with the help of glucagon or insulin.
Obviously, the simultaneous secretion of glucagon and insulin at the normal
level should be minimized because the opposite effects of the two would cancel
out, nullifying the net effects on the glucose level. Such wasteful co-
secretion of counteracting hormones can be prevented by $\delta$-cells
secreting somatostatin, which inhibits secretion of both glucagon and insulin.
In our model, the average activity of cells is given by
$(1+\langle\sigma\rangle)/2$. Accordingly, the average cell state
$\langle\sigma\rangle=\pm 1$ means that all cells are active/silent; in
particular $\langle\sigma\rangle=0$ corresponds to half of the cells being
active. In the absence of the inhibitory interaction of $\delta$-cells, Fig.
5(a) shows that both $\langle\sigma_{\alpha}\rangle$ and
$\langle\sigma_{\beta}\rangle$ take values greater than $-1$ even at the
normal glucose level. Namely, fluctuations associated with the biological
uncertainty $\Theta$ have some fraction of cells still active, leading to
basal hormone secretion. Here the presence of inhibitory interactions of
$\delta$-cells lowers the basal activity of $\alpha$\- and $\beta$-cells, as
shown in Fig. 5(b), which reduces co-secretion of the counteracting hormones,
glucagon and insulin.
Figure 6 displays the relation between $\langle\sigma_{\alpha}\rangle$ and
$\langle\sigma_{\beta}\rangle$, in the absence ($J_{2}=0$ mM) and presence
($J_{2}=2$ mM) of the inhibitory interaction of $\delta$-cells. The system at
low or high glucose levels is described by the upper left or lower right parts
of the curves on the
$(\langle\sigma_{\beta}\rangle,\langle\sigma_{\alpha}\rangle)$ plane,
respectively. Namely, when the glucose concentration is low, $\alpha$\- and
$\beta$-cells are in high and in low activity, respectively
($\langle\sigma_{\alpha}\rangle>0$ and $\langle\sigma_{\beta}\rangle<0$); this
is reversed at high glucose concentrations. It is manifested that the
inhibitory interaction of $\delta$-cells reduces simultaneous activation of
$\alpha$\- and $\beta$-cells. Compared with the result for $J_{2}=0$ mM
(dashed line), the result for $J_{2}=2$ mM (solid line) shows that the
activity of $\beta$\- or $\alpha$-cells is reduced substantially at high or
low glucose levels. In particular $\alpha$-cells remain almost silent
($\langle\sigma_{\alpha}\rangle\approx-1$) at high glucose levels. Note,
however, that those endocrine cells are not totally silent at given glucose
levels and still exhibit residual activity, which results from fluctuations in
the glucose responses of the cells. Interestingly, it was suggested that such
basal hormone secretion also plays an effective role: The minimal basal
secretion of glucagon compensates the glucose uptake in the liver while basal
secretion of insulin inhibits over-secretion of the basal glucagon
[Cherrington et al., 1976].
#### 3.2.2 Enhancement of glucose dose-responses of $\beta$-cells
Another consequence of the inhibitory interaction of $\delta$-cells is the
shift of glucose dose-responses for insulin secretion to the right direction.
This is associated with the increased control of $\beta$-cells by
$\delta$-cells at high glucose levels. Figure 7 indeed shows that the shift
leads to more conspicuous glucose responses of $\beta$-cells at high glucose
levels. In general $\beta$-cells are coupled with each other through gap-
junction channels, which help the cells synchronize their behaviors [Sherman &
Rinzel, 1991]. A $\beta$-cell cluster thus tends to produce all-or-none
glucose responses [Soria et al., 2000]. In the real islet, on the other hand,
$\delta$-cells, with their inhibitory interactions depending on the glucose
level, can modify the glucose dose-response of $\beta$-cells. Accordingly,
insulin response can be more pronounced at high glucose levels ($G>0$).
It is observed that some primitive animals have only $\beta$\- and
$\delta$-cells in their islets, unlike the mammals whose islets contain
$\alpha$-cells as well as $\beta$\- and $\delta$-cells [Falkmer, 1985]. This
difference could perhaps be attributed to an evolutionary adaptation. At early
evolutionary stages, the islet might be a passive system: Without
$\alpha$-cells directly increasing the glucose level, the glucose level should
increase passively as a result of the decrease in insulin secretion. Still,
the precise glucose dose-responses at high glucose levels could be possible
with $\delta$-cells. At later stages, equipped with $\alpha$-cells, the islet
became an active system with regard to glucose regulation. It is of interest
that this evolutionary change is correlated with the fact that $\beta$\- and
$\delta$-cells are closer to each other than $\alpha$-cells in the development
of a stem cell [Kemp et al., 2003]. In addition, $\beta$\- and $\delta$-cells
have functional similarities of using ATP-dependent K+ channels in glucose
responses [Quesada et al., 2006, Quesada et al., 1999].
## 4 Discussions
The islet of Langerhans is a precise system that controls the glucose level
through the use of three main types of endocrine cells. Here it is of interest
to investigate whether the existing interactions between those cells are
beneficial for glucose homeostasis. There are some evidence for the critical
role of the interactions, which may not obviously be addressed by probing
$\alpha$\- and $\beta$-cells separately. The molecular mechanism of how
$\alpha$-cells regulate glucagon secretion at variable glucose levels is still
not clearly understood [Gromada et al., 2007]. Several works attempted to
explain this by means of the interactions between $\alpha$\- and
$\beta$-cells: At high glucose levels, glucagon secretion is inhibited by
insulin, GABA, or Zn2+ secreted from $\beta$-cells [Gromada et al., 2007,
Ishihara et al., 2003]. There is also a hypothesis that glucose has direct
effects on $\alpha$-cells through endoplasmic reticulum Ca2+ storage [Vieira
et al., 2006]. Another evidence for the role of cellular interactions in
glucose homeostasis comes from hyperglucagonomia, which occurs in diabetics at
abnormally high glucose levels. It appears paradoxical that the glucagon
levels of such patients are high even though the blood glucose levels are high
enough to make $\alpha$-cells silent [Gromada et al., 2007]. This puzzling
result can be explained on the basis of cellular interactions in an islet
[Franklin et al., 2005, Rorsman et al., 1989, Takahashi et al., 2006]. Note
that there is also another explanation of this phenomenon in terms of the
peculiar glucose dose-responses of (rat) $\alpha$-cells [Kemp et al., 2003].
In contrast, there also exist a few reports that some cellular interactions
may not exist and are not necessary for glucose homeostasis: It has been
proposed that the microcirculation from $\beta$\- to peripheral $\alpha$\- and
$\delta$-cells prohibits the paracrine action from non-$\beta$ to
$\beta$-cells [Wayland, 1997]. In addition, it has recently been reported that
islet transplantation is successful in recovering from hyperglycaemia with
only $\beta$-cell clusters [King et al., 2007].
Nevertheless, the existence of the receptors of signalling molecules such as
insulin, glucagon, somatostatin, glutamate, and GABA, which are expressed in
pancreatic endocrine cells, apparently implicates their physiological roles in
the fine control of glucose levels [Gromada et al., 2007, Strowski & Blake,
2008]. A better understanding of this tissue, therefore, will contribute to
more advanced medical treatment of diabetes than the current one relying
mostly on insulin. For example, it is conceivable to use other hormones such
as glucagon and somatostatin for more active and precise glucose control.
A variety of complicated interactions in an islet makes it difficult to
recognize their roles, and existing experiments as to those interactions have
focused mostly on static responses of the endocrine cells. However, it is
likely that the cellular interactions actually contribute to dynamical
responses to glucose. In this study, therefore, to understand the role of
intercellular interactions between $\alpha$-, $\beta$-, and $\delta$-cells, we
have proposed an islet model and investigated the effects of integrated
intercellular communications between those cells in glucose homeostasis. Our
mathematical model can systematically include all the cellular interactions
and identify their effects on static and/or dynamic responses to external
glucose changes. It also takes individual heterogeneities into consideration,
e.g., in glucose sensitivity; the basal hormone secretion at the normal
glucose level reflects that some cells can be active to secrete hormones even
though most of the cells are silent at that glucose level. The small
variations in glucose responses among homologous cells may contribute
crucially to the cellular interactions between heterologous cells, which are
actually activated in quite different glucose concentrations, because the
heterogeneous responses of homologous cells can lead to an overlap in the
activation between the heterologous cells. ¿From this model, it has been
revealed that the interactions give more stable, efficient, and accurate
control of glucose: First, asymmetric interactions between $\alpha$\- and
$\beta$-cells contribute to the dynamic stability when the glucose level,
perturbed from the normal level, recovers to the latter. Second, the
interactions of somatostatin for glucagon and insulin secretion prevent their
wasteful co-secretion at the normal glucose level. In addition, at high
glucose levels, the inhibition by $\delta$-cells modifies glucose dose-
responses of insulin secretion. For a more realistic and accurate
understanding, it would be necessary to know the physiological values of the
model parameters. In particular, the relative effects of the direct glucose
stimulus $G$ and paracrine interactions $J$ on the states of endocrine cells
should be identified.
Here it is proposed that these predictions can be verified in experiment. As
for the role of $\delta$-cells, our results may be confirmed through the use
of cell clusters of different compositions of cell-types, for which the
culture method was used in the existing study [Pipeleers et al., 1982].
Another prediction related with the asymmetric interactions between $\alpha$\-
and $\beta$-cells needs to be verified in vivo experiment on transgenic mice
because the effects should arise in the dynamics of whole-body glucose
regulation. Note that the specific cellular interaction may be blocked
selectively in knockout mice lacking specific hormone receptors in an
endocrine cell [Diao et al., 2005, Sorensen et al., 2006].
Beyond the interactions between endocrine cells analyzed in this study, there
exist reports that $\delta$-cells are also influenced by $\alpha$\- and
$\beta$-cells [Unger & Orci, 1977] and these paracrine interactions should be
considered with the microcirculation of hormones in an islet as described
above [Wayland, 1997]. It has also been reported that there exist autocrine
interactions via which a cell is affected by its own hormone secretion
[Aspinwall et al., 1999, Cabrera et al., 2008]. Furthermore, input from
exocrine cells [Bertelli et al., 2001, Bishop & Polak, 1997, Wayland, 1997]
and glucose-sensing neurons [Schuit et al., 2001] have been suggested. There
may thus be more complex communications in the pancreas for glucose
homeostasis, which are left for further study. Finally, we also point out that
the mathematical model proposed can be generalized to describe cellular
interactions in other systems, e.g., neural networks consisting of excitatory
and inhibitory couplings.
Acknowledgments
We thank D. Gardner-Hofatt and W. Heuett for useful comments on the
manuscript. M.Y.C. thanks Asia Pacific Center for Theoretical Physics, where
part of this work was performed, for hospitality. This work was supported in
part by the KOSEF/MOST grant through National Core Research Center for Systems
Bio-Dynamics and by the KOSEF-CNRS Cooperative Program.
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## Tables
Table 1: Coefficients in the transition rate. Here $k^{x}\equiv(g^{x}+j_{1}^{x}+j_{2}^{x}+g^{x}j_{1}^{x}j_{2}^{x})(1+g^{x}j_{1}^{x}+j_{1}^{x}j_{2}^{x}+j_{2}^{x}g^{x})^{-1}$ with $x$ denoting $\alpha$, $\beta$, or $\delta$. Parameters $g^{x}$, $j_{1}^{x}$, and $j_{2}^{x}$ are given in Table 2. Coefficient | Value
---|---
$w^{x}$ | $1$
$w^{x}_{\alpha}$ | $k^{x}$
$w^{x}_{\beta}$ | $k^{x}j^{x}_{1}$
$w^{x}_{\delta}$ | $k^{x}j^{x}_{2}$
$w^{x}_{\alpha\beta}$ | $j^{x}_{1}$
$w^{x}_{\beta\delta}$ | $j^{x}_{1}j^{x}_{2}$
$w^{x}_{\delta\alpha}$ | $j^{x}_{2}$
$w^{x}_{\alpha\beta\delta}$ | $k^{x}j^{x}_{1}j^{x}_{2}$
Table 2: Parameters in the coefficients of the transition rate. $x$ | $\alpha$ | $\beta$ | $\delta$
---|---|---|---
$g^{x}$ | ${\rm tanh}(G/2\Theta)$ | ${\rm tanh}(-G/2\Theta)$ | ${\rm tanh}(-mG/2\Theta)$
$j_{1}^{x}$ | ${\rm tanh}(J_{1}/4\Theta)$ | ${\rm tanh}(-J_{1}/4\Theta)$ | $0$
$j_{2}^{x}$ | ${\rm tanh}(J_{2}/4\Theta)$ | ${\rm tanh}(J_{2}/4\Theta)$ | $0$
## Figure Legends
#### Figure 1.
Schematic diagram of cellular interactions between $\alpha$-, $\beta$-, and
$\delta$-cells. The arrow represents enhancement while bars represent
inhibition. Here the intercellular interactions between two cells are present
only when both are in active states.
#### Figure 2.
Stationary glucose level $G_{\infty}$, depending on the cellular interaction
strengths $J_{1}$ and $J_{2}$. Note that $G_{\infty}$ measures the stationary
level relative to the fasting glucose level $\tilde{G}_{0}$ in the absence of
cellular interactions.
#### Figure 3.
Time evolution of the glucose level $G$, depending on the interactions between
$\alpha$\- and $\beta$-cells. Starting from the high-glucose state
[$\langle\sigma_{\alpha}\rangle=-1$ and $\langle\sigma_{\beta}\rangle=1$ at
$G=1$ mM (or $\tilde{G}=\tilde{G}_{0}$ \+ 1 mM)], the system recovers
eventually the normal glucose level $G_{\infty}$. The time constants are taken
to be $\tau=\tau_{G}=1$ for simplicity and the cellular interactions have
strengths $J_{1}=2$ mM and $J_{2}=0$ mM.
#### Figure 4.
Balance function for glucose regulation, depending on the interactions between
$\alpha$\- and $\beta$-cells. The strengths of cellular interactions are
$J_{1}=2$ mM and $J_{2}=0$ mM.
#### Figure 5.
Time evolution of the average states $\langle\sigma_{\alpha}\rangle$ and
$\langle\sigma_{\beta}\rangle$ of $\alpha$\- and $\beta$-cells, starting
initially from the high-glucose state $\langle\sigma_{\alpha}\rangle=-1$ and
$\langle\sigma_{\beta}\rangle=1$ at $G=1$ mM, during the recovery to the
normal glucose level. The asymmetric interactions between $\alpha$\- and
$\beta$-cells have the strength $J_{1}=2$ mM whereas the inhibitory
interactions of $\delta$-cells are absent in (a) $J_{2}=0$ mM but present in
(b) $J_{2}=2$ mM.
#### Figure 6.
The average state $\langle\sigma_{\alpha}\rangle$ of $\alpha$-cells versus
$\langle\sigma_{\beta}\rangle$ of $\beta$-cells for the asymmetric interaction
$J_{1}=2$ mM, in the absence ($J_{2}=0$ mM) and presence ($J_{2}=2$ mM) of the
inhibitory interaction of $\delta$-cells. The dotted line along the diagonal
represents the stationary condition
$\langle\sigma_{\alpha}\rangle=\langle\sigma_{\beta}\rangle$.
#### Figure 7.
Glucose dose-responses in the activity of $\beta$-cells for the inhibitory
interaction $J_{2}=0$ and $2$ mM. The asymmetric interactions are taken to
have the strength $J_{1}=2$ mM.
Figure 1:
Figure 2:
Figure 3:
Figure 4:
(a) (b)
Figure 5:
Figure 6:
Figure 7:
|
arxiv-papers
| 2009-01-26T21:24:00 |
2024-09-04T02:49:00.239667
|
{
"license": "Public Domain",
"authors": "Junghyo Jo, Moo Young Choi, and Duk-Su Koh",
"submitter": "Junghyo Jo",
"url": "https://arxiv.org/abs/0901.4113"
}
|
0901.4145
|
# Approximate, analytic solutions of the Bethe equation for charged particle
range
Damian C. Swift dswift@llnl.gov PLS-CMMD, Lawrence Livermore National
Laboratory, 7000 East Avenue, Livermore, California 94550, USA James M.
McNaney PLS-CMMD, Lawrence Livermore National Laboratory, 7000 East Avenue,
Livermore, California 94550, USA
(December 16, 2008, revised February 10, 2009 – LLNL-JRNL-410093)
###### Abstract
By either performing a Taylor expansion or making a polynomial approximation,
the Bethe equation for charged particle stopping power in matter can be
integrated analytically to obtain the range of charged particles in the
continuous deceleration approximation. Ranges match reference data to the
expected accuracy of the Bethe model. In the non-relativistic limit, the
energy deposition rate was also found analytically. The analytic relations can
be used to complement and validate numerical solutions including more detailed
physics.
charged particle, energy loss, stopping power, ion implantation
###### pacs:
34.50.Bw, 52.77.Dq, 85.40.Ry
## I Introduction
The deceleration of ions as they pass through matter is important in a wide
range of fields: medical ion radiation therapy, such as the treatment of
tumors Brahme2004 ; radiography with ions Li2006 ; radiolysis of chemical
compounds Chitose1999 ; ion implantation in material processing and
semiconductor doping Shockley1954 ; gas discharge plasmas Tsendin1995 ;
locality of energy deposition in nuclear fusion plasmas, including ion beam
heating for controlled thermonuclear fusion Keefe1982 ; the design of
radiation shielding for nuclear reactors Normand1989 and spacecraft
Wilson1997 ; and particle physics experiments Mulhearn2004 . Energy loss for
different combinations of ion specie, ion energy, and decelerating material
has been measured since the early 20th century Crowther1906 , with a
corresponding development in theoretical work. Calculations of ion energy
loss, and hence range, are very frequently performed using variants of the
Bethe Bethe1930 and Bethe-Bloch Bloch1933 relations. Experimental and
theoretical developments have focused on making corrections to the original
Bethe relation to account for details of interactions with bound electrons and
crystal structures at low energies Barkas1956 ; Ziegler1985 , and quantum-
mechanical limits to the transfer of energy under extreme relativistic
conditions Jackson1999 . The original Bethe relation is still used widely,
particularly when a reliable, approximate result is needed rapidly, or in the
fairly wide range of energies where the Bethe relation is adequately accurate
Bichsel2004 .
The Bethe relation describes the stopping power: the rate at which a moving
ion loses energy to the surrounding material. It is not trivial to use this
relation to obtain the range of an ion, i.e. the distance for it to lose all
of its kinetic energy. In practice, ion ranges are calculated using numerical
integration in multi-physics computer programs, or from scaling laws
normalized to the range for other energies or masses. However, reliance on
sophisticated computer programs for infrequent calculations, without the
ability to make a compact analytic estimate, can lead to errors. Analytic
solutions are also valuable for validating computer programs reproducing the
same physics. Here we point out analytic solutions to accurate approximations
of the Bethe relation, which can give good estimates of ion ranges in matter.
## II Range from stopping power
In the continuous deceleration approximation, charged particles traversing
matter lose kinetic energy $E$ at a rate depending on their instantaneous
energy and the local material. Expressed as the energy loss rate per distance
traveled, the stopping power $dE/dx$ can be used to determine the range $l$ of
the particle, by integrating the deposition until the particle is stationary:
$\int_{0}^{l}\frac{dE(x)}{dx}\,dx=-E_{0}.$ (1)
However, $dE/dx$ is expressed naturally in terms of $E$ rather than $x$.
Rearranging,
$l=\int_{E_{0}}^{0}\frac{dx}{dE}\,dE.$ (2)
The integral can be found numerically for arbitrary stopping powers, or
analytically for stopping powers of sufficiently simple form.
The Bethe equation Bethe1930 describes the deceleration of charged particles
by interaction with the electrons in matter:
$\frac{dE}{dx}=-\frac{4\pi}{m_{e}c^{2}}\frac{NZz^{2}}{\beta^{2}}\left(\frac{q^{2}}{4\pi\epsilon_{0}}\right)^{2}\left[\ln\frac{2m_{e}c^{2}\beta^{2}}{\bar{I}\left(1-\beta^{2}\right)}-\beta^{2}\right]$
(3)
where $\beta=v/c$, $v$ is the ion speed, $\bar{I}$ is the effective ionization
of the target material, $Z$ and $z$ are the atomic numbers of the target and
ion species respectively, $N$ is the number density of target nuclei, $m_{e}$
is the mass of an electron, $\epsilon_{0}$ is the permittivity of free space,
and $c$ is the speed of light.
To find the range of charged particles from Eq. 3, Eq. 2 can be integrated
numerically, though this is not straightforward because of a singularity at
low energies. We have not found an analytic solution for the integral.
## III Taylor expansion
$-dx/dE$ can be expanded as a Taylor series to make it more tractable for
integration. This can be done for Eq. 3 with a relativistic expression for
$\beta(E)$; we do it also for a non-relativistic $\beta(E)$ because the
resulting integral is more amenable to subsequent manipulation.
### III.1 Relativistic
The relativistic relation between $\beta$ and kinetic energy $E$ is
$\beta(E)=\frac{\sqrt{E(E+2m_{i}c^{2})}}{E+m_{i}c^{2}}$ (4)
where $m_{i}$ is the rest-mass of the moving ion. For later convenience, we
scale key quantities to be dimensionless. Substituting into Eq. 3 and
expanding about zero,
$-\frac{dx}{dE}=\frac{4\pi\epsilon_{0}^{2}m_{e}c^{2}}{q^{4}z^{2}NZ}\left[\frac{2\hat{E}}{L}-\frac{3\hat{E}^{2}\left(L-1\right)}{L^{2}}\right]+O(\hat{E}^{3})$
(5)
where
$\hat{E}\equiv\frac{E}{m_{i}c^{2}},\quad\hat{I}\equiv\frac{\bar{I}}{2m_{e}c^{2}}$
(6)
are the scaled kinetic energy and mean ionization, and
$L\equiv\ln\frac{2\hat{E}}{\hat{I}}.$ (7)
Integrating Eq. 2, the range is
$l=\frac{4\pi\epsilon_{0}^{2}m_{e}c^{2}m_{i}c^{2}}{q^{4}z^{2}NZ}\left[\frac{1}{2}\hat{I}^{2}\mbox{Ei}\left(2L\right)+\frac{3}{4}\hat{I}^{3}\mbox{Ei}\left(3L\right)-\frac{3\hat{E}^{3}}{L}\right]$
(8)
where $\mbox{Ei}(z)$ is the exponential integral function,
$\mbox{Ei}(z)\equiv-\int_{-z}^{\infty}\frac{e^{-t}}{t}\,dt.$ (9)
### III.2 Non-relativistic
In the non-relativistic limit, $\beta(E)=\sqrt{2E/m_{i}}$. Following the same
procedure as above, we find that the non-relativistic form of each of $-dx/dE$
and $l$ is simply the first term of the corresponding relativistic relation.
## IV Mixed-species targets
For a target comprising multiple elements, the stopping power can be estimated
from the combination of stopping powers from each element $s$
$\frac{dE}{dx}\simeq\sum_{s}\frac{dE}{dx}(Z_{s},N_{s})$ (10)
– the Bragg addition rule. This relation is approximate because of chemical
bond formation, which alters the effective ionization. For ion energies much
greater than the bond energies, the approximation should be accurate.
Expanding as before and gathering terms, the range can be expressed very
similarly to that for single-element targets. In the relativistic case,
$l=\frac{4\pi\epsilon_{0}^{2}m_{e}c^{2}m_{i}c^{2}}{q^{4}z^{2}\tilde{Z}}\left[\frac{1}{2}\tilde{I}^{2}\mbox{Ei}(2\tilde{L})+\frac{3}{4}\tilde{I}^{3}\mbox{Ei}(3\tilde{L})-\frac{3\hat{E}^{3}}{\tilde{L}}\right]$
(11)
where
$\tilde{I}\equiv\exp\frac{\sum_{s}N_{s}Z_{s}\ln\hat{I}_{s}}{\tilde{Z}},\quad\tilde{L}\equiv\ln\frac{2\hat{E}}{\tilde{I}}\\\
$ (12)
and
$\tilde{Z}\equiv\sum_{s}N_{s}Z_{s}$ (13)
is the total electron density in the target.
In the non-relativistic case,
$l=\frac{2\pi\epsilon_{0}^{2}m_{e}c^{2}m_{i}c^{2}}{q^{4}z^{2}\tilde{Z}}\tilde{I}^{2}\mbox{Ei}\left(2\tilde{L}\right),$
(14)
which is again simply the first term of the relativistic relation.
## V Polynomial fit to the stopping distance scale
Although well-characterized numerical approximations to the exponential
integral exist Pecina1986 , they are not available as standard functions in
mainstream computer languages, and require significant effort to implement
from scratch. However, the logarithmic terms in the stopping power and range
vary slowly compared with the powers of $E$; over wide ranges of energy, the
stopping power can be approximated accurately by low-order polynomials. The
use of polynomial approximations avoids the need to evaluate the exponential
integral function. We define a stopping distance scale
$D\equiv-Edx/dE,$ (15)
which is particularly well-behaved above the low-energy singularity as it
tends to zero with $E$, and increases monotonically (Fig. 1). This quantity
can be used as a crude, $O(1)$, (over)estimate of particle range, without
requiring any series expansion or integration. Approximating $D$ by a
polynomial
$D_{p}(E)=\sum_{j}a_{j}E^{j},$ (16)
the range (Eq. 2) is simply
$l_{p}\simeq a_{0}\ln E+\sum_{j>0}\frac{a_{j}E^{j}}{j}.$ (17)
$a_{0}$ must be zero, since $E^{n}/\ln(\alpha E)\rightarrow 0$ as
$E\rightarrow 0$.
Figure 1: Example calculation of stopping distance scale: protons in water.
The curve can be fitted well by a quadratic.
To find the charged particle range in a specific substance, it is
straightforward to tabulate $D(E)$ using Eq. 3 (and Eq. 10 for a multi-species
target), fit a polynomial $D_{p}(E)$, and evaluate Eq. 17. However, it is also
possible to find a universal polynomial fit. Defining for convenience a
different scaled energy
$F\equiv\frac{4m_{e}E}{\bar{I}m_{i}}=\frac{2\hat{E}}{\hat{I}}$ (18)
the stopping distance scale is (using Eq. 5)
$D(F)=\frac{\pi\epsilon_{0}^{2}}{2m_{e}q^{4}}\frac{m_{i}\bar{I}^{2}}{z^{2}NZ}\left[\frac{F^{2}}{\ln
F}+\frac{3}{8}\frac{\bar{I}}{m_{e}c^{2}}\frac{1-\ln F}{(\ln
F)^{2}}F^{3}\right],$ (19)
where the first term is the non-relativistic approximation. The prefactor
comprises universal constants and a simple problem-specific factor
$m_{i}\bar{I}^{2}/z^{2}NZ$. The relative magnitude of the relativistic term to
the non-relativistic term depends only on $\bar{I}$. Thus, by finding
polynomial approximations
$\displaystyle-\frac{F^{2}}{\ln F}\simeq P_{NR}(F)=\sum_{j}n_{j}F^{j}$ (20)
$\displaystyle\frac{\ln F-1}{(\ln F)^{2}}F^{3}\simeq
P_{R}(F)=\sum_{j}r_{j}F^{j}$ (21)
it is straightforward to find the polynomial coefficients for any problem:
$a_{j}=\frac{\pi\epsilon_{0}^{2}}{2m_{e}q^{4}}\frac{m_{i}\bar{I}^{2}}{z^{2}NZ}\left(\frac{4m_{e}}{\bar{I}m_{i}}\right)^{j}\left(n_{j}+\frac{3}{8}\frac{\bar{I}}{m_{e}c^{2}}r_{j}\right)$
(22)
and hence the range through Eq. 17.
The derivation presented above is valid for any choice of units. The Bethe
relation breaks down when the logarithm changes sign, i.e. when $E$ approaches
$\bar{I}m_{i}/4m_{e}$. Physically meaningful distances are obtained for
greater energies. Using the Bloch estimate Bloch1933 for the effective
ionization of the material,
$\bar{I}\simeq 10Zq,$ (23)
the relations are valid for $E>4600Zq$. Polynomial fits were calculated over a
range suitable for hadrons of energy $\sim$MeV to GeV (Table 1). The
relativistic term diverges rapidly outside the fitting region.
Table 1: Polynomial fits to stopping distance scale functions. parameter | $5\leq F\leq 100$ | $100\leq F\leq 10000$
---|---|---
$n_{1}$ | $2.17423$ | $1.64377\times 10$
$n_{2}$ | $2.29035\times 10^{-1}$ | $1.31696\times 10^{-1}$
$n_{3}$ | $-3.36317\times 10^{-4}$ | $-4.55336\times 10^{-6}$
$n_{4}$ | | $2.07676\times 10^{-10}$
| $10\leq F\leq 200$ | $500\leq F\leq 10000$
$r_{2}$ | $-1.33946\times 10$ | $-9.92931\times 10^{3}$
$r_{4}$ | $2.04195$ | $3.49521\times 10$
$r_{6}$ | $1.58588\times 10^{-1}$ | $1.00674\times 10^{-1}$
$r_{8}$ | $-7.67337\times 10^{-5}$ | $-7.28447\times 10^{-7}$
## VI Energy deposition profile
Given $E(x)$, the profile of energy deposition $-dE(x)/dx$ (Bragg curve) can
be calculated. $E(x)$ is the inverse of the range, $l(E)$.
For the non-relativistic range relation, $E(x)$ can be expressed in terms of
the inverse of the exponential integral function,
$-\frac{dE(x)}{dx}\simeq\frac{q^{4}z^{2}NZ}{4\pi\epsilon_{0}^{2}\bar{I}}\exp\left[\frac{1}{2}\mbox{Ei}^{-1}(\alpha
x)\right]\mbox{Ei}^{-1}(\alpha x)$ (24)
where
$\alpha\equiv\frac{2m_{e}q^{4}z^{2}NZ}{\pi\epsilon_{0}^{2}\bar{I}^{2}m_{i}}.$
(25)
If the stopping distance scale is represented locally in energy by a
sufficiently simple polynomial, then it too may be used to calculate $-dE/dx$.
For example, taking a local quadratic fit
$D_{p}=a_{1}E+a_{2}E^{2}\quad\Rightarrow\quad
l_{p}=a_{1}E+\frac{a_{2}}{2}E^{2},$ (26)
one obtains
$-\frac{dE}{dx}\simeq\frac{1}{\sqrt{a_{1}^{2}+2a_{2}x}}$ (27)
(Fig. 2).
Figure 2: Example calculation of Bragg curve via a polynomial (quadratic) fit
to $D(E)$: protons in water. The curve is presented backward from its usual
form: as if the particles are accelerating from rest. Conceptually, the curve
can be continued to arbitrarily high energies, i.e. long ranges.
## VII Example calculations
Reference calculations are used to validate radiation protection simulations
using different computer programs. Here we compare the analytic solutions of
the Bethe relation with results from widely-used programs SRIM SRIM , which
uses numerical solutions of more detailed stopping powers developed from the
Bethe relation, and MCNP MCNP , which collects Monte-Carlo statistics for the
simulated interaction of individual particles. Trial calculations were made
for protons and $\alpha$-particles stopping in Al and water.
We use the Bloch estimate, Eq. 23 for the effective ionization of the target
material. More accurate calculations have been developed more recently, but
the original Bloch estimate serves to demonstrate the correctness of our
analysis.
The results are consistent with the accuracy of the Bethe relation itself
(Table 2), and are consistent with direct numerical integration of the Bethe
equation without being affected by the low energy singularity. The greatest
difference was for relativistic protons in Al. In this regime, radiative
losses and nuclear reactions become significant Bichsel2004 and the Bethe
relation requires additional corrections.
Table 2: Comparison between analytic calculation and computer simulations of ion ranges (in millimeters). system | analytic | MCNP 5 | SRIM
---|---|---|---
p $\rightarrow$ Al | | |
10 MeV | 0.59 | 0.62 | 0.63
100 MeV | 36 | 37 | 37
1 GeV | 180 | 1510 | 1530
$\alpha$ $\rightarrow$ Al | | |
10 MeV | 0.054 | 0.062 | 0.061
100 MeV | 3.0 | 3.2 | 3.1
1 GeV | 170 | 180 | 180
p $\rightarrow$ water | | |
10 MeV | 1.1 | 1.2 | 1.2
100 MeV | 73 | 77 | 76
1 GeV | 300 | 320 | 320
$\alpha$ $\rightarrow$ water | | |
10 MeV | 0.097 | 0.110 | 0.110
100 MeV | 6.0 | 6.3 | 6.2
1 GeV | 350 | 380 | 375
## VIII Conclusions
Analytic solutions were found to power series expansions and polynomial fits
to the Bethe relation. These solutions provide a convenient way to calculate
ion ranges and energy deposition in regimes where the Bethe relation is valid,
i.e. kinetic energies of roughly 1-100 MeV/u, without depending on numerical
integration. The use of a Taylor series restricts the accuracy at high energy;
the relativistic expansion thus incorporates relativistic contributions to the
range but is not valid to arbitrarily high energies. However, the analytic
solutions can readily be used with more accurate formulations of the effective
ionization. The accuracy was demonstrated by comparison with simulations from
widely-used computer programs of ion ranges in Al and water.
## Acknowledgments
The authors would like to acknowledge the contribution of Tim Goorley (Los
Alamos National Laboratory) for providing reference calculations of stopping
distance using SRIM and MCNP, and Sergei Kucheyev (Lawrence Livermore National
Laboratory) for helpful discussions. This work was performed in support of
Laboratory-Directed Research and Development project 09-ERD-037 under the
auspices of the U.S. Department of Energy under contract DE-AC52-07NA27344.
## References
* (1) For instance, A. Brahme, Int. J. Radiat. Oncol. Biol. Phys. 58, 2, pp 603-616 (2004).
* (2) For instance, T. Li, Z. Liang, J.V. Singanallur, T.J. Satogata, D.C. Williams, and R.W. Schulte, Med. Phys. 33, 3, pp 699-706 (2006).
* (3) For instance, N. Chitose, Y. Katsumura, M. Domae, Z. Zuo, T. Murakami, and J.A. LaVerne, J. Phys. Chem. A 103, 24, pp 4769-4774 (1999).
* (4) For instance, W. Shockley, U.S. patents 2,666,814 (1949) and 2,787,564 (1954) and many more recent developments.
* (5) L.D. Tsendin, Plasma Sources Sci. Technol. 4, pp 200-211 (1995).
* (6) For instance, D. Keefe, Ann. Rev. Nuc. Part. Sci. 32, pp 391-441 (1982).
* (7) For instance, E. Normand and W.R. Doherty, IEEE Trans. Nuc. Sci. 36, 6, pp 2349-2355 (1989).
* (8) For instance, J.W. Wilson, J. Miller, A. Konradi, and F.A. Cucinotta, NASA Conference Publication 3360 (1997).
* (9) For instance, M.J. Mulhearn, ‘A direct search for Dirac magnetic monopoles,’ D.Phil. thesis, Massachusetts Inst. of Technol. (2004).
* (10) J.A. Crowther, Phil. Mag. 12, 379 (1906).
* (11) H. Bethe, Ann. Phys. 397, 3, pp 325-400 (1930).
* (12) F. Bloch, Ann. Phys. 16, 287 (1933).
* (13) W.H. Barkas, W. Birnbaum, and F.M. Smith, Phys. Rev. 101, 778 (1956).
* (14) J.F. Ziegler, J.P. Biersack, and U. Littmark, “The Stopping and Range of Ions in Matter” vol. 1 (Pergamon, New York, 1985).
* (15) J.D. Jackson, Phys. Rev. D 59, 017301 (1999).
* (16) H. Bichsel, D.E. Groom, and S.R. Klein, Rev. part. phys. sec. 27, Phys. Lett. B 592 (1-4) pp 1-1109 (2004).
* (17) For example, P. Pecina, Bull. Astron. Inst. Czechoslovakia 37, pp 8-12 (1986).
* (18) J.F. Ziegler, ‘SRIM’ computer program, http://www.srim.org (2008).
* (19) ‘MCNP’ computer program, http://mcnp-green.lanl.gov (2008).
|
arxiv-papers
| 2009-01-26T23:33:59 |
2024-09-04T02:49:00.249722
|
{
"license": "Public Domain",
"authors": "Damian C. Swift, James M. McNaney",
"submitter": "Damian Swift",
"url": "https://arxiv.org/abs/0901.4145"
}
|
0901.4741
|
# Development of Vertically Integrated Circuits for ILC Vertex Detectors
Ronald Lipton
for the Fermilab Pixel R&D Group
Fermilab
P.O. Box 500 Batavia Illinois USA
###### Abstract
We report on studies of vertically interconnected electronics (3D) performed
by the Fermilab pixel group over the past two years. These studies include
exploration of interconnect technology, backside thinning and laser annealing,
the production of the first 3D chip for particle physics, the VIP, and plans
for a commercial two-tier 3D fabrication run. Studies of Direct bond
Interconnect (DBI) oxide bonding and Silicon-on-Insulator based technologies
are presented in other talks in this conference.
## 1 3D electronics
3D electronics is generally defined as consisting of multiple layers of
electronics, thinned, bonded and interconnected to form a monolithic circuit.
This technology has become an area of intense focus in the electronics
industry as a way to improve circuit performance without the expense and
complexity of smaller feature size [2] [3]. 3D electronics provides the
ability to integrate heterogeneous technologies, reduce interconnect lengths,
and expand bus width. These technologies are particularly interesting for
pixel detectors, where they offer new techniques for integrating sensors and
electronics, and provide substantially more processing power per pixel than
conventional technologies.
Fabrication of a 3D stack depends on the development of several techniques
including:
* •
Bonding between layers including oxide to oxide fusion, copper/tin bonding,
copper/copper bonding, polymer/adhesive bonding
* •
Wafer thinning using grinding, lapping, etching, and Chemical Mechanical
Polishing (CMP)
* •
Through wafer via formation and metalization either with isolation using
Through Silicon Vias (TSVs) or without isolation as in Silicon-on-Insulator
(SOI) devices.
* •
High precision alignment
The technology can also be separated into techniques which form the via before
3D integration is performed (via first) and those which form vias afterwards
(via last).
## 2 VIP Chip
The VIP chip [5] was intended as a demonstration of 3D technology as applied
to an ILC vertex detector. It was fabricated in 0.18 micron SOI CMOS
technology by MIT-Lincoln Labs as part of a DARPA-sponsored multiproject run.
The chip consists of three tiers of electronics interconnected by 3D vias. The
MIT-LL technology utilizes oxide bonding to bond tiers together. This wafer
bonding technology mates activated, planarized silicon oxide surfaces to form
a robust inter-wafer bond [4]. The top ”handle” silicon in the bonded wafer
stack is thinned to $70\mu m$ and the remainder of the silicon is etched away
to expose the oxide surface. Vias are then formed by etching thorough the
insulating oxide to contact internal metalization layers. The process is
repeated to form the required number of tiers.
Figure 1: Schematic of the VIP chip showing the three tiers of electronics.
The drawing on the right shows the metalization for the three layers.
The VIP incorporates an amplifier/disriminator with double correlated sample
and hold in tier three, both a 5-bit digital and an analog time stamp in tier
2, and sparsification and digital logic in tier one, all within a $20\mu m$
square pixel. In ILC operation time stamped hits are stored within the pixel
during the bunch train. The chip utilizes part of the 199 ms period between
trains for readout, while the front end current is reduced to save power. The
front end is designed to consume less than 4 mW/mm${}^{2}\times f$ where $f$
is the front end duty factor. A token passing readout scheme stores addresses
on the periphery, minimizing the logic on the pixel. Figure 1 shows a
schematic of the chip as well as a visualization of the metal layers.
The chip was submitted in October 2006 and received in October 2007. Initial
tests showed that the overall yield was low, with only a few chips showing the
ability to propagate the readout token through the full 64 x 64 matrix. The
single best-performing chip was selected for full testing. Figure 2 shows the
results of the most complete system test, where a pattern of test pulses are
injected into the front-end amplifiers, and a sparse scan is performed,
reading out those channels where the discriminator fired and latched the
readout flag.
Figure 2: Pattern of 119 pixels injected with a test charge (left) and read
out in the subsequent sparse scan.
Our testing has demonstrated the basic functionality of the chip including
propagation of the readout token, threshold scans, input test charge scans,
verification of digital and analog time stamping, full sparsified data
readout, and fixed pattern and temporal noise measurements. No problems could
be found associated with the 3D vias between tiers. Although the chip was
fully functional, we were not fully satisfied with yield and performance.
Performance problems stemmed from large leakage currents in the protection
diodes and transistors, and poor matching of current mirrors. Many of these
issues can be traced to the sensitivity of mixed mode designs in fully
depleted SOI to the transistor environment and process variations [6] [7].
Intrinsic SOI process problems were exacerbated by our aggressive design,
which made extensive use of minimum feature size transistors and dynamic
logic, which is sensitive to transistor leakage current.
An new version of the chip, the VIP2, was submitted to the third DARPA
sponsored 3D multiproject run in October 2008. As a result of useful
interaction with MIT-LL on SOI analog design the overall quality should be
considerably improved. Changes to the chip include:
* •
Different power and grounding layout
* •
Larger transistor sizes ($0.18\to 0.45\mu m$) equivalent feature size
* •
Larger pixels (30 x 30 microns)
* •
Redundant vias and larger traces in critical paths
* •
Redesign of current mirrors to reduce thermal effects
* •
Removal of dynamic logic due to leakage current problems
Changes were also made to improve the overall functionality including
increasing the digital time stamp from 5 to 7 bits. We expect that the changes
will lead to a much more reliable chip which can be bonded to sensors for test
beam studies.
## 3 Commercial 3D Technologies
An R&D process, such as the one provided by MIT-LL has disadvantages of long
turn-around time and process uncertainties. We are now exploring alternative
3D processes implemented as part of a high volume commercial process. Tezzaron
(Naperville Ill) has developed a 3D technology implemented in the high volume
0.13 micron Chartered (Singapore) process [3]. This is a ”via first” process
where through-silicon ”supercontacts” are formed after transistor fabrication
but before any metalization processing. The 6 micron deep by 1 micron diameter
supercontacts are filled with tungsten at the same time as the transistor
contacts are formed. Wafers are finished normally, however there is a top
layer of thin patterned copper which forms both the bond between wafers and
the wafer-to-wafer electrical interconnection. Wafers are then bonded face-to-
face with moderate pressure and temperature. Silicon on the top wafer is
ground down to the supercontacts and the contacts are metalized to form either
external bond pads or to provide connections to the next tier.
Fermilab is organizing a multiproject run in the Tezzaron/Chartered process
with submission expected in Spring of 2009\. The run includes designs from 13
institutes. The Fermilab designs will include a two tier version of the VIP
chip, the VIP2b, as well as test devices for the CMS upgrade and X-ray
imaging. Standard commercial CMOS should provide a reliable process with low
noise, multiple transistor options, better rad hardness, less wasted via area,
faster turn-around, as well as the availability of full wafers for sensor
integration.
## 4 Sensor Integration
The bonding, thinning and lithography process used to build 3D tiers can also
be used for sensor integration with readout. Both Tezzaron and Ziptronix
provide 3D processes, based on copper-copper and oxide bonding respectively,
which can be used to include sensors as a base tier in a 3D stack. 3D
processes offer finer pitch and more robust mechanical interconnection than is
available in solder-based bump bonding. The high planarity and strong
interlayer bonds allow bonded readout ICs to be thinned to 25 $\mu m$ or less,
which can provide access to integrated through-silicon vias.
We have explored two sensor bonding techniques: Cu-Sn and DBI oxide bonding.
The DBI technique is described in another contribution to this conference [8].
We have contracted with RTI to explore Cu-Sn bonding with sparse contacts to
minimize contact mass. These technologies do not have the contact bridging
problems that limit pitch for solder bumps. However they are not self-aligning
which requires special care in aligning the bonded surfaces. In that study
successful Cu and Cu-Sn bump structures were fabricated that were compatible
with $20\mu m$ I/O pitch. Electrical tests indicated that the bonding yield
was $>99$ % for both metallurgies. Samples were also destructively tested to
evaluate bond strength. In both cases the bond strength is considerably higher
than comparable solder bump arrays [9].
## 5 Thinning and Laser Annealing
Wafer thinning is an important part of 3D technology, and the ability to
process and handle thinned silicon is crucial to the goal of constructing a
very low mass vertex detector. In some 3D and SOI technologies it will be
important to thin the devices after topside processing. After thinning a
backside contact ohmic contact must also be formed. The contact is usually
fabricated by a high temperature anneal of an ion implantation. This high
temperature step is unacceptable for fully processed electronics, where the
temperature must be kept below 450 degrees C. to protect the topside
metalization.
We have developed a thinning/implantation/annealing process which limits the
maximum temperature of the topside to below 100 deg C. The wafer is first
bonded to a pyrex carrier using a 3M UV release adhesive designed for wafer
thinning applications. The wafer is then thinned and polished using standard
techniques. The bonded wafer is ion implanted, taking care to ground the
silicon edges and controlling the implantation rate to limit the temperature.
The implantation is then annealed using a eximer laser system which melts the
silicon locally on the backside to a depth of $300$ nm, keeping the topside
close to room temperature.
Figure 3: Secondary Ion Mass Spectrometry (SIMS) phosphorus dose profile of
strip detector before (red) and after (blue) laser annealing.
Initial studies were performed using individual strip sensors, which were
thinned to remove the backside ohmic implant, re-implanted and laser annealed.
All of these devices showed acceptable performance at depletion, with some
variation of leakage current depending on laser dose and annealing
environment. A scan of the dopant concentration before and after the annealing
process is shown in Figure 3. This work was followed by studies using 6” test
wafers donated by Micron Semiconductor. These wafers were thinned to 50
microns on the pyrex handle, implanted and laser annealed at MIT-Lincoln Labs.
Depletion voltage was reduced from 80 to 2.5 volts with acceptable leakage
current. Work is ongoing at Cornell to determine the optimal implantation and
annealing parameters.
## 6 Ladder design
3D technologies provide the ability to construct a low mass, dense, tiled
array of chips, which can be used to fabricate ladder and disk planes for the
ILC. Figure 4 shows an example of such a structure. Multi-tier readout ICs are
fabricated utilizing through-silicon vias. These ICs are bonded to an
independently fabricated sensor wafer with a fine pitch technology such as DBI
or cu-cu. Once bonded, the Readout ICs are thinned to reveal the topside TSVs.
Topside interconnections are patterned using standard lithography and wirebond
or other contacts are made. Connections to external power and signal cables
could be made at the ends of the sensor, which could have the appropriate
interconnection patterns. This technique has a number of advantages:
* •
The sensor can be a fully depleted detector with charge collection by drift
rather than diffusion.
* •
The sensor wafer serves as a base for the ROICs, obviating the need for
reticle stitching.
* •
The 3D ROICs can include power control tiers.
* •
Known good ROIC die can be used.
Final sensor thinning would have to occur after the topside processing. This
could be done by backgrinding a sensor with an imbedded ohmic contact, either
as part of an epitaxial stack or using the SOI technique demonstrated by the
Max Plank Institute [10]. This would avoid the additional implant and laser
annealing steps needed if the backside were not pre-processed.
Figure 4: Conceptual drawing of a thinned ladder(left) based on 3D
interconnects (right). Readout chips could be bonded to the sensor wafer using
an oxide bonding process (DBI) then thinned to $\approx 25\mu m$ to expose
through-silicon vias to provide interconnections. The sensor wafer could be
pre-thinned and mounted on a handle wafer during the DBI processing.
## 7 Conclusions
Fermilab has produced and tested the VIP, the first 3D chip designed for
particle physics applications. The chip demonstrated the required
functionality but suffered from low yield and compromised performance. An
improved version of the chip has been submitted to MIT-LL. The VIP2b, a two-
tier 0.13 micron CMOS chip implemented in the Tezzaron 3D process, will be
submitted this spring. This submission will extend the development of this
technology to applications at super-LHC and in x-ray imaging. We are
continuing to develop wafer thinning, interconnection, and post-processing
technologies aimed at demonstrating the ability to build precise, low mass,
low power vertex detector systems.
## References
* [1] Presentation:
`http://ilcagenda.linearcollider.org/contributionDisplay.py?contribId=42&sessionId=8&confId=2628`
* [2] IBM Journal of Research and Development, Volume 52, No. 6, 2008. Issue devoted to 3D technology.
* [3] Philip Garrou, Christopher Bower, Peter Ramm, Handbook of 3D Integration Technology and Applications of 3D Integrated Circuits, Wiley-VCH, 2008.
* [4] IEEE Transactions on Electron Devices, Vol. 53, No. 10, October 2006.
* [5] A Vertically Integrated Pixel Readout Device for the Vertex Detector at the International Linear Collider, FERMILAB-PUB-08-564
* [6] M. Connell et al., 2007 IEEE/SEMI Advanced Semiconductor Manufacturing Conference.
* [7] Tenbroek et al., Solid-State Circuits Conference, 1997, pp. 276-279, 16-18 Sept. 1997.
* [8] `http://ilcagenda.linearcollider.org/contributionDisplay.py?contribId=207&sessionId=21&confId=2628`
* [9] Allan Huffman, Fabrication, Assembly, and Evaluation of Cu-Cu Bump bonding Arrays for Ultra-fine Pitch Hybridization and 3D Integration, Pixel 2008, Fermilab, Batavia, Illinois, September 22-26, 2008.
* [10] P. Fischer et al., Nucl. Instrum. Meth. A 582, 843 (2007).
|
arxiv-papers
| 2009-01-29T18:27:28 |
2024-09-04T02:49:00.271242
|
{
"license": "Public Domain",
"authors": "Ronald Lipton (for the Fermilab Pixel R&D Group)",
"submitter": "Ronald Lipton",
"url": "https://arxiv.org/abs/0901.4741"
}
|
0901.4790
|
# PAIRING SYMMETRY AND PAIRING STATE IN FERROPNICTIDES: THEORETICAL OVERVIEW
I.I. Mazina and J. Schmalianb Code 6391, Naval Research Laboratory,
Washington, DC 20375 Iowa State University and Ames Laboratory, Ames, IA,
50011
(February 17, 2009)
###### Abstract
We review the main ingredients for an unconventional pairing state in the
ferropnictides, with particular emphasis on interband pairing due to magnetic
fluctuations. Summarizing the key experimental prerequisites for such pairing,
the electronic structure and nature of magnetic excitations, we discuss the
properties of the $s^{\pm}$ state that emerges as a likely candidate pairing
state for these materials and survey experimental evidence in favor of and
against this novel state of matter.
One fist of iron, the other of steel
If the right one don’t get you, then the left one will
Merle Travis, 16 tons
## 1 Introduction
The discovery of cuprate superconductors has changed our mentality in many
ways. In particular, the question that would have sounded moot to most before
1988, what is the symmetry of the superconducting state, is now the first
question to be asked when a new superconductor has been discovered. The pool
of potential candidates, before considered at best a mental Tetris for
theorists, had acquired a practical meaning. It has been demonstrated that
superconductivity in cuprates is $d$-wave, while in MgB2 it is multi-gap
$s$-wave with a large gap disparity. There is considerable evidence that
Sr2RuO4 is a $p$-wave material. Other complex order parameters are routinely
discussed for heavy fermion systems or organic charge transfer salts. It is
likely that the newly discovered ferropnictides represent another
superconducting state, not encountered in experiment before.
Besides the general appreciation that pairing states may be rather nontrivial,
it has also been recognized that unconventional pairing is likely due, at
least to some extent, to electronic (Coulomb or magnetic) mechanisms and,
conversely, electronic mechanisms are much more likely to produce
unconventional pairing symmetries than the standard uniform-gap $s$-wave. It
has been appreciated that the actual symmetry is very sensitive to the
momentum dependence of the pairing interaction, as well as to the underlying
electronic structure (mostly, fermiology).
Therefore we have structured this overview so that it starts with a layout of
prerequisites for a meaningful discussion of the pairing symmetry. First of
all, we shall describe the gross features of the fermiology according to
density-functional (DFT) calculations, as well as briefly assess verification
of such calculations via ARPES and quantum oscillations experiments. Again,
detailed discussion of these can be found elsewhere in this volume. We will
also point out where one may expect caveats in using the DFT band structure:
it is in our view misleading to assume that these compounds are uncorrelated.
While not necessarily of the same nature as in cuprates, considerable
electron-electron interaction effects cannot be excluded and are even
expected.
We will then proceed to discuss the role of magnetic fluctuations as well as
other excitations due to electron-electron interactions. We discuss the
special role the antiferromagnetic (AFM) ordering vector plays for the pairing
symmetry and address the on-site Coulomb (Hubbard correlations), to the extent
of their possible effect on the pairing symmetry, and possible overscreeining
(Ginzburg-Little) interactions. We also discuss puzzling issues that are
related to the magnetoelastic interaction in these systems. As for a
discussion of the electron-phonon interaction we refer to the article by Boeri
et al in this volume. The final part of this review consists of a summary of
theoretical aspects of the pairing state, along with a discussion of its
experimental manifestations.
## 2 Prerequisites for addressing the Cooper pairing
### 2.1 Electronic structure and fermiology
#### 2.1.1 Density functional calculations
The two families of the Fe-based superconductors are $1111$ systems ROFeAs
with rare earth ions R[1, 2] and the $122$ systems AFe2As2 with alkaline earth
element A[3]. Both families have been studied in much detail by first
principles DFT calculations. Here and below, unless specifically indicated, we
use a 2D unit cell with two Fe per cell, and the corresponding reciprocal
lattice cell; the $x$ and $y$ directions are along the next-nearest-neighbor
Fe-Fe bond. It appears that all materials share the same common motif: two or
more hole-like Fermi surfaces near the $\Gamma$ point [$\mathbf{k=}(0,0)$],
and two electron-like surfaces near the M point [$\mathbf{k=}(\pi,\pi)$] (Fig.
1-5). This is true, however, in strictly non-magnetic calculations only, when
the magnetic moment on each Fe is restricted to zero. As discussed below, this
is not necessarily a correct picture.
Figure 1: (color online) The Fermi surface of the non-magnetic LaAsFeO for 10%
e-doping [4] Figure 2: (color online) The Fermi surface of the non-magnetic
BaFe2As2 for 10% e-doping (Co doping, virtual crysatl approximation)[4]
If, however, we neglect this potential caveat, and concentrate on the two best
studied systems, 1111 and 122, the following relevant characteristics can be
pointed out: First, the density of states (DOS) for holes and electrons is
comparable for undoped materials; with doping, respectively one or the other
becomes dominant. For instance, for Ba0.6K0.4Fe2As2 the calculated DOS (in the
experimental structure) for the three hole bands varies between $1.1$
st/eV/f.u. and $1.3$ st/eV/f.u., the inner cylinder having, naturally, the
smallest DOS and the outer the largest. For the electron bands the total DOS
is $1.2$ st/eV/f.u., that is, two to three times smaller than the total for
the hole bands[4]. We shall see later that this is important. Another
interesting effect is that in the 122 family doping in either direction
strongly reduces the dimensionality compared to undoped compounds (in the 1111
family this effect exists, but is much less pronounced), see Fig. 4. This
suggests that the reason that doping destroys the long-range magnetic order
(it is believed by many that such a destruction is prerequisite for
superconductivity in ferropnictides) is not primarily due to the change in the
2D electronic structure, as it was initially anticipated[5], but rather due to
the destruction of magnetic coupling between the layers. Indeed the most
striking difference between the undoped 1111 and undoped 122 electronic
structure is quasi two-dimensionality of the former and a more 3D character of
the latter (the difference is clear already in the paramagnetic calculations,
but is particularly drastic in the antiferromagnetic state), while at the same
time the observed magnetism in the 122 family is at least three times stronger
than in LaFeAsO (in the mean-field DFT calculation the difference is quite
small).
Figure 3: (color online) The Fermi surface of the non-magnetic BaFe2As2 for
10% h-doping (20% Cs doping, virtual crysatl approximation.[4]
The fact that the nesting is very imperfect is crucial from the point of view
of an SDW instability, making the material stable against infinitesimally
small magnetic perturbation. For superconductivity, however, it is less
important, as discussed later in the paper.
Figure 4: (color online) The Fermi surface of BaFe2As2 for 20% h-doping
(corresponding to Ba1.6K0.4Fe2As2, calculated as 40% Cs doping in the virtual
crystal approximation) [4]. Figure 5: (color online) The Fermi surface of
undoped nonmagnetic FeTe. [4]
#### 2.1.2 Experimental evidence
Experimental evidence regarding the band structure and fermiology of these
materials comes, basically, from two sources: Angular resolved photoemission
spectroscopy (ARPES) and quantum oscillations measurements. The former has an
additional advantage of being capable of probing the electronic structure in
the superconducting state, assessing the amplitude and angular variation of
the superconducting gap. A potential disadvantage is that it is a surface
probe, and pnictides, especially the 122 family, are much more three-
dimensional than cuprates. This means that, first, the in-plane bands as
measured by ARPES, strongly depend on the normal momentum, $k_{\perp},$ and,
second, there is a bigger danger of surface effects in the electronic
structure than in the cuprates. There are indications that the at least in
1111 compounds the surface is charged, that is to say, the doping level in the
bulk is different from that on the surface. Additionally, LDA calculations
suggest that in the magnetic prototypes, the band structure depends
substantially on interlayer magnetic ordering, again, not surprisingly, mostly
in the 122 compounds, as Fig.6 illustrates. Of course, there is no guarantee
that the last two layers order in the same way as the bulk (or even with the
same moment).
Figure 6: (color online) Band structure of the orthorhombic antiferromagnetic
BaFe2As2 calculated for two different interlayer ordering pattern: the
experimental antiferromagnetic one (space group #66, broken green) and the
hypothetical ferromagnetic (still antiferromagnetic in plane, space group #67,
solid red). In both cases the magnetic moment on Fe was artifically suppressed
to 1 $\mu_{B}$ by aplying a fictitious negative Hubbard U [4]. The point N is
above the point Y.
These caveats notwithstanding, ARPES has already provided invaluable
information. ARPES measurements have been performed for both 1111[6, 7] and
122 materials[8, 9, 10, 11]. These measurements demonstrated the existence of
a well-defined Fermi surface that consists of hole and electron pockets, in
qualitative agreement with the predictions of electronic structure
calculations. Thus, one can say that the topology of the Fermi surface,
including the location and the relative size of the individual Fermi surface
sheets agrees with the LDA expectation — which is most important for the
pairing models. Similarly, it is rather clear that the ARPES bandwidth is
reduced from the LDA one by a factor of 2–2.5, similar to materials with
strong itinerant magnetic fluctuations (cf., for instance, Sr2RuO4 near a
magnetic quantum critical point[12]). These findings are also consistent with
the deduced normal state linear specific heat coefficient in 1111 materials
(e.g., $4-6$ mJ/mol K2 in Ref. [13]) corresponding to a factor 1–2 compared to
the bare LDA value[14]. However, in the 122 compound a specific heat
coefficient 63 mJ/mol K2 was reported[13], to be compared with roughly
11.5mJ/mol K2 from the LDA calculations[4]. While a renormalization of 5.5 is
not consistent with either ARPES or quantum oscillations, consistency among
different experimental publications for the 122 systems is lacking as well
[15, 13].
Another experimental probe of the electronic structure is based on quantum
oscillations that measure extremal cross-section areas of the FS (ideally, for
different directions of the applied field) and the effective masses. Such
measurements are very sensitive to the sample quality, therefore so far only a
handful of results are available. However, data on the P-based 1111 compound
agree reasonably well with band structure calculations[16], and indicate the
same mass renormalization as ARPES[17]
Importantly, quantum oscillations measurements on AFM 122 compounds[18, 19]
indicate that even the undoped pnictides are well defined Fermi liquids, even
though a significant portion of the Fermi surface disappears due to the
opening of a magnetic gap. The frequencies of the magneto-oscillations then
suggest that the ordered magnetic state has small Fermi surface pockets
consistent with the formation of a spin-density wave. Thus, the electronic
structure of the pnictides is consistent with a metallic state with well
defined Fermi surfaces.
Besides determining the overall shape of the Fermi surface sheets, ARPES is
able to yield crucial information about the momentum dependence of the
superconducting gap. Several groups performed high quality ARPES measurements
of this effect[7, 8, 9, 10]. In some cases significant differences in the size
of the gap amplitude for different Fermi surface sheets have been observed.
However, there seems to be a consensus between all ARPES groups that the gap
amplitude on an individual Fermi surface sheet depends weakly on the
direction. While this seems to favor a pairing state without nodes, one has to
keep in mind that all measurements so far have been done for fixed values of
the momentum $k_{\perp}$, perpendicular to the planes. While it might be
premature to place too much emphesis on the relative magnitude of the gaps
observed in different bands in ARPES experiments, it is worth noting that most
experimentalists agree that in the hole-doped 122 material the inner hole
barrel and the electron barrel have comparable (and large) superconducting
gaps, while the outer hole barrel has about twice smaller gap. On the other
hand, there are first data[20] indicating that in the electron doped
BaFe1.85Co0.15As2 the hole and the electron bands have about the same gap
despite the hole pockets shrinking, and electron pocket extending. Even more
interesting, the most natural interpretation of the measured fermiology is
that the hole FS in BaFe1.85Co0.15As2 actually corresponds to the outer
($xz/yz)$ barrel in Ba0.6K0.4Fe2As2 that has a small gap in that compound.
#### 2.1.3 Role of spin fluctuations in electronic structure
As is clear from the above discussion, strong spin fluctuations have a
substantial effect upon the band structure. First of all, they dress one-
electron excitations providing mass renormalization, offering an explanation
for the factor 2–2.5. This is in fact a relatively modest renormalization: it
is believed that, for instance, in He3 or in Sr2RuO4 itinerant spin
fluctuations provide renormalization of a factor of 4 or larger. However, it
is likely that the effect goes beyond simple mass renormalziation. As will be
discussed in detail below, there is overwhelming evidence of large local
moments on Fe, mostly from the fact that the Fe-As bond length corresponds to
a fully magnetic (large) Fe ion. There is also evidence that the in-plane
moments are rather well correlated in the planes, and the apparent loss of the
long-range ordering above $T_{N}$ is mainly due to a loss of 3D coherency
between the planes[21]. It is only natural to expect a similar situation to be
true when magnetism is suppressed by doping.
If that is the case, the electronic structure in the paramagnetic parts of the
phase diagram, at least in the vicinity of the transition, should not be
viewed as dressed nonmagnetic band, but rather as an average between the bands
corresponding to various magnetic 3D stackings (cf. Fig. 6). Fig. 6,
corresponding to the $T=0$ magnetic moment of 1 $\mu_{B},$ is probably
exaggerating this effect, but it is still likely that in a considerable range
of temperatures and doping near the observed magnetic phase boundary a
nonmagnetic band structure is not a good starting point, and a theory based on
magnetic precursors is needed. More experiments, particularly using diffuse
scattering, and more theoretical work are needed to clarify the issue. A
discussion to this effect may be found in Ref. [22]. See also Section 2.3
below.
### 2.2 Magnetic excitations
#### 2.2.1 Experimental evidence
Compared to cuprates and other similar compounds, two peculiarities strike the
eye. First, the parent compounds of the pnictide superconductors assume an
antiferromagnetic structure, where neighboring Fe moments are parallel along
one direction withinin the FeAs plane and antiparallel along the other.
Neutron scattering data yield ordered moments per Fe of $0.35\mu_{B}$ for
LaFeAsO[23], $0.25\mu_{B}$ for NdFeAsO[24], $0.8\mu_{B}$ for CeFeAsO[25], and
$0.9$ $\mu_{B}$ for BaFe2As2[26]. Intriguingly, in NdFeAsO the ordered moment
at very low temperatures increases by a factor of 3 to 4 at the temperature
corresponding to the ordering of Nd-spins[27]. Note that the correct magnetic
structure has been theoretically predicted by DFT calculations[5, 28], which,
moreover, consistently overestimated the tendency to magnetism (as opposed to
the cuprates). Second, the magnetically ordered state remains metallic. As
opposed to cuprates or other transition metal oxides, the undoped systems
exhibit a small but well established Drude conductivity[29], display magneto-
oscillations[18] and have Fermi surface sheets of a partially gapped metallic
antiferromagnetic state[30]. Above the magnetic ordering temperature a sizable
Drude weight, not untypical for an almost semimetal has been observed.
Further, the ordered Fe magnetic moment in the 1111 systems depends
sensitively on the rare earth ion, very different from YBa2Cu3O6 where yttrium
can be substituted by various rare earth elements with hardly any effect on
the Cu moment. Note that the rare earth sites project onto the centers of the
Fe plaquettes and thus do not exchange-couple with the latter by symmetry.
Finally, the magnetic susceptibility of BaFe2As2 single crystals[31] above the
magnetic transition shows no sign for an uncoupled local moment behavior.
#### 2.2.2 Itinerant versus local magnetism
The vicinity of superconductivity to a magnetically ordered state is the key
motivation to consider pairing mechanisms in the doped systems that are linked
to magnetic degrees of freedom. Similar to cuprate superconductors, proposals
for magnetic pairing range from quantum spin fluctuations of localized
magnetic moments to fluctuations of paramagnons as expected in itinerant
electron systems. To judge whether the magnetism of the parent compounds is
localized or itinerant (or located in the crossover regime between these two
extremes) is therefore crucial for the development of the correct description
of magnetic excitations and possibly the pairing interactions in the doped
systems.
In our view the case at hand is different from such extreme cases as undoped
cuprate on one end and weak itinerant magnets like ZrZn2 on the other. While
being metals with partially gapped Fermi surface, there is evidence that Fe
ions are in a strongly magnetic states with strong Hund rule coupling for Fe.
This results in a large magnetic moment — but only for some particular
ordering patterns (for comparison, in FeO and similar materials LDA produce
large magnetic moment regardless of the imposed long range order). While it is
obvious that ferropnictides are not Mott insulators with localized spins,
interacting solely with near neighbors, a noninteracting electron system may
be not a perfect starting approximation either. To make progress we have to
decide what is the lesser of two evils and use it, even realizing the problems
with the selected approach. Given the above mentioned experimental facts, our
preference is that these systems are still on the itinerant side.
A feature that has attracted much interest is the quasi-nesting between the
electron and the hole pockets. The word “quasi” is instrumental here: even the
arguably most nested undoped LaFeAsO is very far from the ideal nesting and
even worse in the (more magnetic) BaFe2As${}_{2}.$ Indeed, it has been
observed that in the LDA calculations the nonmagnetic structure in either
compound is stable with respect to an infinitesimally small AFM perturbations,
but strongly unstable with respect to finite amplitude perturbations. This can
be understood from the point of view of the Stoner theory, applied to a finite
wave vector Q: the renormalized static spin susceptibility (in the DFT the RPA
approximation is formally exact) can be written as
$\chi_{LDA}(\mathbf{Q)}=\frac{\chi_{0}(\mathbf{Q})}{1-I\chi_{0}(\mathbf{Q})},$
(1)
where $I$ is the Stoner factor of iron, measuring the intra-atomic Hund
interaction (in the DFT, it is defined by the second variation of the
exchange-correlation functional with respect to the spin density). While the
denominator in Eq. 1 provides a strong enhancement of $\chi$, albeit not
exactly at $\mathbf{Q=}(\pi,\pi)$, but at a range of the wave vectors near
$\mathbf{Q}$), it does not by itself generate an instability. One can say that
an infinitesimally weak magnetization can only open a gap over a very small
fraction of the Fermi surface. However, a large-amplitude spin density wave
opens a gap of the order of the exchange splitting, $IM$, where $M$ is the
magnetic moment on iron, and, obviously, affects most of the conducting
electrons. In other words, the magnetism itself is generated by the strong
Hund rule coupling on Fe (just as in the metal iron), but the topology of the
Fermi surface helps select the right ordering pattern. Formation of the
magnetic moments is local; arranging them into a particular pattern is
itinerant.
There are several corollaries of this fact that are important for pairing and
superconductivity. First, despite the fact that the overall physics of these
materials is more on the itinerant side than on the localized side (see a
discussion to this effect later in the paper), it is more appropriate to
consider magnetic moments on Fe as local rather than itinerant (as for
instance in the classical spin-Peierls theory). Note that the same is true for
the metal iron as well. Second, the interaction among these moments is not
local, as for instance in superexchange systems (it appears impossible to map
the energetics of the DFT calculations onto a two nearest neighbor Heisenberg
model[32]). The AFM vector is not determined by local interactions in real
space (as for instance in the $J_{1}+J_{2}$ models, see below), but by the
underlying electronic structure in reciprocal space. Third, since the energy
gain due to formation of the SDW mainly occurs at finite (and large, $IM$ is
on the order of eV) energies, looking solely at the FS may be misleading.
Indeed, FeTe is one compound where the Fe moments apparently do not order into
a $\mathbf{Q=}(\pi,\pi)$ SDW, but in a more complex structure corresponding to
a different ordering vector[33], despite the fact that the FS shows about the
same degree of nesting (Fig.5) as LaFeAsO and a noticeably better nesting than
BaFe2As${}_{2}.$ DFT calculations correctly identify the ground state in all
these cases, and the origin can be traced down again to the opening of a
partial gap: in both 1111 and 122 compounds the $\mathbf{Q=}(\pi,\pi)$ is
about the only pattern that opens such a gap around the Fermi level, while in
FeTe comparable pseudogaps open in both magnetic structures (and the
calculated energies are very close, the actual experimental structure being
slightly lower[34]).
#### 2.2.3 Perturbative itinerant approach
Even if one accepts the point of view that the magnetism in the Fe-pnictides
is predominantly itinerant, the development of an adequate theory for the
magnetic fluctuation spectrum is still highly nontrivial. As pointed out
above, there are strong arguments that the driving force for magnetism is not
Fermi surface nesting but rather a significant local Hund’s and exchange
coupling. This can be quantitatively described in terms of a multiband Hubbard
type interaction of the Fe-$3d$ states
$\displaystyle H_{int}$
$\displaystyle=U\sum_{i,a}n_{ia\uparrow}n_{ia\downarrow}+U^{\prime}\sum_{i,a>b}n_{ia}n_{ib}$
$\displaystyle-
J_{H}\sum_{i,a>b}\left(2\mathbf{s}_{ia}\cdot\mathbf{s}_{ib}+\frac{1}{2}n_{ia}n_{ib}\right)$
$\displaystyle+J\sum_{i,a>b,\sigma}d_{ia\sigma}^{\dagger}d_{ia\overline{\sigma}}^{\dagger}d_{ib\overline{\sigma}}d_{ib\sigma},$
(2)
with intra- and inter-orbital Coulomb interaction $U$ and $U^{\prime}$, Hund’s
coupling $J_{H}$ and exchange coupling $J$, respectively. Here $a$, $b$ refer
to the orbitals in a Wannier type orbital at site $i$. $X$-ray absorption
spectroscopy measurements support large values for the Hund’s couplings that
lead to a preferred high spin configuration,[35] leading to larger values of
$J_{H}$. The importance of the Hund coupling for the normal state behavior of
the pnictides was recently stressed in Ref.[36].
Weak coupling expansions in these interaction parameters may not capture
quantitative aspects of the magnetism in the pnictides. Nevertheless, it is
instructive to summarize the main finding of the result of weak coupling
expansions, in particular as they demonstrate the very interesting and
nontrivial aspects that results from interband interactions with almost nested
hole and electron Fermi-surfaces[37, 38, 39]. For an ideal semimetal (two
identical hole and electron bands with the Fermi energies $E_{h}$ and $E_{e})$
all susceptibilities at the nesting vector Q diverge as $\log|E_{h}/E_{e}-1|$.
Depending on the details of electron-electron interaction this signals an
instability, at $E_{h}=E_{e},$ to a spin density wave state or to a
superconducting state for infinitesimal interaction. The corresponding
interference between particle-hole and particle-particle scattering events can
be analyzed by using a renormalization group approach. For $J_{H}=J=0$, the
authors of Ref.[38] find that at low energies the interactions are dominated
by Cooper pair-hopping between the two bands, favoring an
$s^{\pm}$-superconducting state that is fully gapped on each Fermi surface
sheet, but with opposite sign on the two sheets. It is worth pointing out that
this pairing mechanism is due to very generic interband scattering, not
necessarily due to _spin-fluctuations_ , as all particle-hole and particle-
particle scattering events enter in essentially the same matter. An
$s^{\pm}$-state was also obtained using a functional renormalization group
approach[37], where the authors argue that the pairing mechanism is due to
collective spin fluctuations that generate a pairing interaction at low
energies. The appeal of these calculations is clearly that controlled and thus
robust conclusions can be drawn. On the other hand, as discussed below, the
Fermi surface nesting is less crucial as is implied by these calculations.
Attempts to include sizable electron-electron interactions within an itinerant
electron theory are based on the partial summation of ladder and bubble
diagrams, in the spirit of Eq.1. This leads to the RPA type theory of Ref.[40,
41, 42, 43] and the fluctuation exchange approximation of multiband
systems[44, 45]. RPA calculations yield a magnetic susceptibility that is
peaked at or near $\mathbf{Q=}\left(\pi,\pi\right)$. For parameters where the
Fermi surface around $\Gamma$ is present, the dominant pairing channel is
again the $s^{\pm}$-state, while $d$-wave pairing occurs as one artificially
eliminates this sheet of the Fermi surface. The exchange of paramagnons
between Fermi surface sheets is shown to be an efficient mechanism for spin
fluctuation induced pairing. The fluctuation exchange (FLEX) approach is to
some extent a self consistent version of the RPA theory[46]. While the method
is not very reliable to address high energy features, the description of the
low energy dynamics spin response, the low energy electronic band
renormalization and, the nature of the pairing instabilityare rather reliable.
The fact that several orbitals matter in the FeAs systems is also of help as
FLEX type approaches can be formulated as theories that become exact in the
limit of large fermion flavor[47]. Refs.[44, 45] performed FLEX calculations
for the FeAs systems and find once again that the dominant pairing state is an
$s^{\pm}$-state, even though Ref.[44] also find a $d$-wave state in a regime
where the magnetic fluctuation spectrum is peaked at vectors away from
$\mathbf{Q=}\left(\pi,\pi\right)$. These authors find a solution that is
numerically close to a compact form
$\Delta\left(\mathbf{k}\right)=\Delta_{0}\cos(ak_{x})\cos(ak_{y}),$ (3)
but this form is neither required by symmetry nor can be consistently deduced
from any low-energy theory (where pairing occurs at or near the Fermi
surface). We will come back to this issue later in this review.
To summarize, numerous calculations that start from an itinerant description
of the magnetic interactions yield an $s^{\pm}$ pairing state caused by the
exchange of collective interband scattering or paramagnons.
#### 2.2.4 J1-J2 model
The initially assumed (although later refuted by the experiment[49]) absence
of the Drude weight in undoped ferropnictides has been taken as evidence for
the fact that they are in the vicinity of a Mott transition and should be
considered as bad metals with significant incoherent excitations[48]. If
correct, it is clearly appropriate to start from a theory of localized spins,
analogous to what is believed to be correct in the cuprate superconductors[50,
51] (it is worth noting that proximity to a Mott transition is a sufficient,
but not necessary condition for existence of local moments). If the dominant
magnetic interactions are between nearest and next nearest neighbor Fe-spins,
the following model describes the localized spins:
$H=J_{1}\sum_{\left\langle
i,j\right\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+J_{2}\sum_{\left\langle\left\langle
i,j\right\rangle\right\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}$ (4)
Here, $J_{1}$ and $J_{2}$ are the superexchange interactions between two
nearest-neighbor and next-nearest-neighbor Fe sites, respectively. A
geometrical argument can be made[52, 48] that indeed the two superexchange
paths $via$ As have comparable strength (however, this argument fails to
recognize that the direct overlap between Fe orbitals in pnictides is very
large[53], thus leading to a strong enhancement of the nearest neighbor
antiferromagnetic exchange in the localized picture[54], and that in metals
superexchange is not the only and usually not the most important magnetic
interaction). When $J_{1}>2J_{2}$ the conventional Neel state has the lowest
energy, when $J_{1}<2J_{2}$ the stripe order emerging in the experiment is the
lowest magnetic state. The system is frustrated if $J_{1}=2J_{2}.$
Upon doping the poor metal (strictly the insulator) described by Eq. 4 with
charge carriers can be investigated for superconductivity, with pairing
stabilized by strong quantum spin fluctuations. In Ref.[55] a single band of
carriers was investigated leading to either $d_{x^{2}-y^{2}}+id_{xy}$ or
$d_{xy}$-pairing, depending on the carrier concentration and the precise ratio
of $J_{1}$ and $J_{2}$. A more realistic theory for the pairing in the
$J_{1}$-$J_{2}$ model in the pnictides must of course include at least two
bands and was developed in Ref.[56]. For sufficiently large $J_{2}$, the
$s^{\pm}$-state is once again the dominating pairing state. It may seem
strange that this strong coupling theory based upon the (unlikely, from the
experimental point of view) proximity to a Mott transition has essentially the
same pairing solutions ($d$-wave for one Fermi surface sheet and
$s^{\pm}$-wave for two Fermi surface sheets separated by $\mathbf{Q}$), as the
RPA calculation of [40]. In Section 3 we will explain that this is not
surprising at all and that even a totally unphysical theory may lead to
perfectly sensible results for superconductivity, as long as it has the same
structure of magnetic excitations in the reciprocal space.
### 2.3 Magneto-elastic coupling
The parent compounds exhibit a structural and a magnetic transition, strongly
suggesting that magnetoelastic coupling plays a role in the physics of
pnictides in general and in superconductivity in particular. Electronic
structure calculations for a non-magnetic state indicate that the electron-
phonon interaction in the pnictides is rather modest and definitely not
sufficient to explain superconducting transition temperatures of $50$ K[57,
5]. However, as these calculations were based on the nonmagnetic electronic
structure, effects of local magnetism on iron were entirely neglected. Indeed,
the equilibrium position of As calculated under this assumption are quite
incorrect and the force constant for the Fe-As bond is $30\%$ higher than it
should be. On the other hand, fully magnetic AFM calculations, while
overestimating the ordered moment, produce highly accurate equilibrium
structures and the force constant in agreement with experiment[22]. It was
pointed out that including soft magnetism in the calculation, i.e. magnetism
with directional and amplitude fluctuations, may substantially enhance the
electron-phonon coupling[58]. The emphasis is on “soft” : additional reduction
of the force constants of the Fe-As bonds does not come from the fact that the
moment exists, but from the fact that the amplitude of the moment depends on
the bond length. Intriguingly, in the 1111 systems the AFM transition occurs
somewhat below a structural phase transition. Both transitions seem to be of
the second order, or of very weakly first order[59]. In 122 compounds the
structural and magnetic orders emerge simultaneously through a strong first
order transition[60, 61].
In the ordered state, Fe spins are parallel along one direction and
antiparallel along the other. Since we expect the bond length for parallel and
antiparallel Fe-spin polarization to be distinct, magnetism couples strongly
to the shear strain
$\varepsilon_{\mathrm{shear}}=\varepsilon_{xy}-\varepsilon_{yx}$. Thus,
$\varepsilon_{\mathrm{shear}}\neq 0$ should invariably occur below the Neel
temperature. Experiment finds that the ferromagnetic bonds are shorter than
antiferromagnetic bonds. From the point of view of superexchange interaction
it seems somewhat surprising that ferromagnetic bonds shorten and the
superexchange-satisfied bonds expand. Yet this behavior is exactly the same as
the DFT calculations had predicted[52], and it can be traced down to one-
electron energy (the observed sign of the orthorhombic distortion simply
lowers the one-electron DOS at the Fermi level)[101].
What remains puzzling is however why in the 1111 family the structural
transition occurs above $T_{N}.$ Naively, this fact could be taken as evidence
for a hypothesis that elastic degrees of freedom are the driving force and
that magnetism is secondary. There are strong quantitative and qualitative
arguments against this view. First, numerous DFT calculations[62, 63, 22]
converge to the correct orthorhombic structure (with correct sign and
magnitude of the distortion), if performed with AFM magnetic ordering, and to
a tetragonal solution if done without magnetism. On the other hand, the
antiferromagnetism is obtained even without allowing for a structural
distortion. In other words, magnetism is essential for the distortion, but the
distortion is not needed for the magnetism.
There exists also a very general argument that demonstrates that the magnetism
is indeed primary and the structural distortion secondary. Historically the
relevant physics was first encountered in the 2D $J_{1}$-$J_{2}$ model[64],
and applied to ferropnictides in Refs.[50, 51]. Below we will reformulate this
argument form a general point of view. We begin with a unit cell that contains
two Fe sites (just as the actual cristallographic unit cell for the FeAs
trilayer). The most natural choice of the origin is in the middle between
these two Fe cites (Fig. 7a ). The coordinates of the atoms are
$\mathbf{r}_{ij}^{+}=\mathbf{R}_{ij}+\mathbf{d}$,
$\mathbf{r}_{ij}^{-}=\mathbf{R}_{ij}-\mathbf{d,}$
$\mathbf{d}=(\frac{1}{4},\frac{1}{4}),$ where $\mathbf{R}_{ij}$ ($i,j$
integer) are the coordinates of the centers of the unit cells. This naturally
implies partitioning the entire lattice into two sublattives, shown as open
and solid dots in Fig. 7a.
Both ferro- and antiferromagnetic checkerboard orderings correspond to a
$\mathbf{Q}=(0,0)$ perturbation of the uniform state, since in both cases all
unit cells remain identical. The Fourier transform of either patter contains
only momenta corresponding to the reciprocal lattice vectors. Conversely, a
spin density wave with the quasi-momentum $\mathbf{Q}=(\pi,\pi)$ corresponds
to flipping all spins in every other unit cell, as illustrated in Fig. 7b,c by
shading colors (blue cells have the magnetization density opposite to that of
the pink cells). It is evident from Fig. 7b and c that this imposes no
requirement upon the mutual orientation of the two sublattices. Again, one can
say that the susceptibility as a function of quasimomentum $\mathbf{q}$ inside
the first Brillouin zone does not describe fluctuations of the magnetic moment
of two ions in the same unit cell with respect to each other, for that purpose
one needs to know the linear response at all momenta $\mathbf{q+G}$, where
$\mathbf{G}$ is an arbitrary reciprocal lattice vector.
Figure 7: (color online) (a)Fe2 lattice with the fully symmetric unit cells
shown. The full circles denote one sublattice, the hollow ones the other.
Shading shows ordering corersponding to the vector
$\mathbf{Q=}\left(\pi,\pi\right)$ in the Fe2 lattice; for each ssublattice,
spins in the pink unit cells are opposite to the spins in the blue cells, but
relative orientation of the two sublattices is arbitrary. (b) Ordered state
with $\mathbf{Q=}\left(\pi,\pi\right)$ and with parallel orientation of the
spins in the unit cell ($\sigma=1$). (c) Same ordering vector
$\mathbf{Q=}\left(\pi,\pi\right)$, but with antiparallel orientation of the
spins in the unit cell ($\sigma=-1$).
Let us assume that the most stable mean field phase corresponds to Néel order
in each of the two sublattices. In the $J_{1}$-$J_{2}$ language that
corresponds to $J_{2}>J_{1}/2,$ in the itinerant language to an instability in
$\chi$ at $\mathbf{Q}=(\pi,\pi)$. Moreover, it is obvious from Fig. 7b,c that
in the classical ground state one sublattice does not exchange-couple at all
to the other, so the classical ground state is infinitely degenerate. this is
however not important for the following discussion, what matters is that the
two extreme cases are always degenerate, the one where two spin in the same
cell are parallel (Fig. 7b) or antiparallel (Fig. 7c). In the $J_{1}+J_{2}$
model the infinite degeneracy is reduced by quantum fluctuations, but the
double degeneracy remains, while in the LDA it is only double degenerate
already on the mean-field level[65].
It is instructive [64] to introduce two order parameters corresponding to the
Neel (checkerboard) ordering for each sublattice,
$\mathbf{m}_{\pm}=\sum_{ij}(-1)^{i+j}\mathbf{M}_{ij}^{\pm},$where
$\mathbf{M}_{ij}^{\pm}$ are the magnetic moments of the two Fe’s in the unit
cell $ij.$ Following Ref. [64] one can introduce the third (scalar) order
parameter,
$\sigma=\sum_{ij}\sigma_{ij}=\sum_{ij}\mathbf{M}_{ij}^{+}\cdot\mathbf{M}_{ij}^{-}$.
Now $\sigma>0$ corresponds to parallel orientation of the magnetization inside
the unit cell (Fig. 7b) while $\sigma<0$ refers to antiparallel orientation
(Fig. 7c). In the former case $\sigma>0$, neighboring Fe spins are parallel
along the diagonal and antiparallel along the counter-diagonal. The situation
is reversed for $\sigma<0$. These two configurations are degenerate and
correspond to the frequently discussed ’stripe’ magnetic order. In two
dimensions, according to the Mermin-Wagner theorem, $\sigma$ is the only order
parameter that can be finite at finite temperature. Therefore the presumably
largest energy scale of the system, the mean field transition temperature of
each sublattice, $T^{\ast}$ ($\sim J_{2}$ in the local model, and the energy
difference $E_{FM}-E_{AFM}$ in the itinerant picture), does not generate any
phase transition, but rather starts a crossover regime where the correlation
length $\xi_{m}$ for the $\mathbf{m}_{\pm}$ order parameter becomes much
longer that the lattice parameter.
In this regime, one can investigate a possibility of a phase transition
corresponding to the $\sigma$ order parameter. It is important to realize that
$\sigma$ does not have to change sign along a domain wall of the
magnetization. This ensures that $\sigma$ can order even though the sublattice
magnetization vanishes. $\sigma$ does couple to the (long-range) fluctuations
of $\mathbf{m;}$ integrating these fluctuations out one will obtain an
effective Hamiltonian coupling $\sigma_{ij}$ and
$\sigma_{i^{\prime}j^{\prime}}$ as far as $\xi_{m},$ meaning that even very
small coupling between $\mathbf{m}_{+}$ and $\mathbf{m}_{-}$ will produce a
phase transition to a finite $\sigma$ at a temperature $T_{s}\sim
J_{1}\xi_{m}^{2}(T_{s})\sim J_{1}\exp(J_{2}/T_{s})$. Solving this for $T_{s}$,
one gets $T_{S}\sim J_{2}/\log(J_{2}/J_{1})$. Note that here again $J_{1}$ and
$J_{2}\sim T^{\ast}$ just characterize the relevant energy scales and by no
means require the validity of the $J_{1}+J_{2}$ model.
As mentioned above $\sigma$ is positive (negative) for ferromagnetic
(antiferromagnetic) bonds, see Fig.8. Thus $\sigma$ couples bilinearly to the
order parameter of the orthorhombic structural transition
$F_{c}=\gamma\varepsilon_{\mathrm{shear}}\sigma.$ (5)
When the expectation value of $\sigma$ is nonzero below a transition
temperature $T_{s}$, the tetragonal symmetry is spontaneously broken leading
to $\varepsilon_{\mathrm{shear}}\neq 0$. We see that $T_{s}$ is suppressed
from $T^{\ast}$ rather weakly (logarithmically) and that even a weak coupling
between the two sublattices would produce a structural phase transition.
Figure 8: (color online) Magnetoelastic coupling: The two atoms per unit cell
are denoted by filled and open circles. A ferromagnetic bond leads to a
shortening of the nearest neighbor lattice constant (bold dashed lines), while
an antiferromagnetic bond leads to a longer lattice constanti (thin dashed
lines). Depending on the relative orientation of the two sublattices (i.e. the
sign of $\sigma$), two distortions with opposite sign of
$\varepsilon_{\mathrm{shear}}$ are possible.
The third energy scale existing in the problem is set by the interlayer
magnetic coupling, $J_{\perp}.$ In the DFT we found $J_{\perp}\lesssim 1$ meV
in LaFeAsO and $J_{\perp}\sim 16$ meV in BaFe2As2[4]. This huge difference
defines the different behavior of these two compounds. In the former the Neel
transition temperature for a sublattice ordering is on the order of
$T^{\ast}/\log(T^{\ast}/J_{\perp}),$ logarithmically smaller than $T_{s},$
while in the latter one expects a much larger $T_{N}$, and likely larger than
the $T_{s}$ for an individual FeAs plane.
The phase between $T_{N}$ and $T_{s},$ if $T_{s}>T_{N},$ was dubbed “nematic”
in Refs. [50, 51], as the order parameter $\left\langle\sigma\right\rangle\neq
0$ even though $\left\langle\mathbf{M}_{ij}\right\rangle=0$, as expected for
an axial, as opposed to vectorial order parameter. The first order nature of
the transition in the 122 systems is then likely a consequence of the coupling
to soft elastic degrees of freedom, and/or of nonlinear interactions. A more
rigorous treatment of the described physics will be published elsewhere[66].
There is another interesting experimental evidence for the unconventional
nature of the magneto-elastic coupling in these systems. In the 122 systems
the structural distortion $\propto\varepsilon_{\mathrm{shear}}$ and the
sublattice magnetization seem to be proportional to each other.[67] At a
second order transition, symmetry arguments imply however that the former
should be proportional to the square of the sublattice magnetization. At a
first order transition, no such strict connection can be established, however
one expects that the generic behavior is recovered as the strength of the
first order transition gets smaller, realizable via alcaline earth
substitution. Experiments show that the mentioned linear behavior is similar
for Ca, Ba or Sr[68]. In our view this behavior is evidence for the fact that
the first order transition in the 122 systems is never close to being weak.
Arguments that the first order character of the magneto-elastic phase
transition originates from the lattice instabilities near the onset of spin-
density wave order were recently given in Ref.[69]. However, further
discussion clearly goes beyond the limit of this review.
The fact that at the structural transition (and even above), magnetic
correlations in plane are already well established, with large correlation
lengths, explains many otherwise mysterious observations. A more detailed
discussion can be found in Ref. [22].
This picture is not without ramifications for superconductivity. First and
foremost, it implies that at superconducting composition ferropnictides,
especially the 1111 family, are not really paramagnetic, bat rather systems
with a large in-plane magnetic correlation length, much larger than the
lattice parameter and likely much larger than the superconducting correlation
length. Second, the excitation structure in such a system is unusual and
cannot be entirely described in terms of $\chi(\mathbf{Q),}$ where
$\mathbf{Q}=(\pi,\pi),$ since such a description loses the physics associated
with the parameter $\sigma.$ Finally, it implies that the lattice and spin
degrees of freedom do not fluctuate independently and are naturally connected
to each other. Therefore a detailed quantitative theory for the pairing state
will have to include lattice vibrations. Conversely, experiments that find
evidence for a lattice contribution to the pairing mechanism should not be
considered as evidence against magnetic pairing.
### 2.4 Other excitations
While everybody’s attention is attracted to magnetic pairing mechanisms and
spin fluctuations, it would be premature and preposterous to exclude any other
excitations from consideration. First of all, it might be still too early to
discard the venerable phonons. While there is no question that the
calculations performed so far [57, 5] were accurate and the linear response
technique used had proved very reliable before (MgB${}_{2},$ CaC6 $etc.),$
these calculation by definition do not take into account any effects of the
magnetism. As discussed above, it is very likely that the ground state even in
the so-called nonmagnetic region of the phase diagram is characterized by an
AFM correlation length long enough compared to the inverse Fermi vector. In
this case, the amplitude of the magnetic moment of Fe (even though its
direction fluctuates in time) is nonzero and the electronic structure is
sensitive to it. Calculations suggest that a phonon stretching the Fe-As bond
will strongly modulate this magnetic moment and thus affect the electronic
structure at the Fermi level more than for a nonmagnetic compound (or, for
that matter, a magnetic compound with a hard magnetic moment). Softness of the
Fe moments, variationally, provides an additional route for electron-phonon
coupling and should therefore always enhance the overall coupling constant.
Whether this is a weak or a strong effect, and whether the resulting coupling
is stronger in the intraband channel (enhancing the $s_{\pm}$
superconductivity) or in the interband channel (with the opposite effect), is
an open question. Only preliminary results are available[58].
Besides the phonons and the spin fluctuation, charge (polarization)
fluctuations can also, in principle, be pairing agents. To the great surprise
of the current authors, nobody has yet suggested an acoustic plasmon mechanism
for ferropnictides, a mechanism that was unsuccessfully proposed for cuprates,
for MgB2 and for CaC${}_{6}.$ Presumably the apparent lack of strong transport
anisotropy in 122 and the absence of carriers with largely disparate mass
prevented these usual suspects from being discussed.
It is not only the harsh condition on the very existence of acoustic plasmons,
but a very general malady (better known in the case of acoustic plasmons, but
generally existing for any sort of exciton pairing) that prevents plasmonic
superconductivity in most realistic cases: lattice stability. Basically,
efficient pairing of electrons via charge excitations of electronic origin
requires overscreening of electrostatic repulsion — which by itself does not
constitute a problem. But since the ion-ion interaction is screened by the
same polarization operator as electron-electron interaction, there is an
imminent danger that the former is overscreened as well. This is an
oversimplified picture (electron-electron susceptibility differs from the
response to an external field on the level of vertex corrections), but it
captures the essential physics.
This danger was appreciated by the early proponents of the excitonic
superconductivity, W. Little[70] and V. Ginzburg[71], therefore they proposed
space separation between a highly polarizable insulating media, providing
excitons, and a metallic layer or string where the superconducting electrons
live. The sandwich structure of the As-Fe-As trilayer reminds us of the
Ginzburg’s “sandwich” (“Ginzburger” ) and tempts to revisit his old proposal.
This was done recently by Sawatzky and collaborators[72] who pointed out that
As is a large ion (Pauling radius for As4- is 2.2 Å) and ionic polarizability
grows with the radius cube. Since the conducting electrons are predominantly
of Fe origin, they suggested pairing of Fe d electrons $via$ polarization of
As ions. So far, this proposal was received with a skepticism that can be
summarized as follows. (1) Analyzing the muffin-tin projected character of the
valence bands, as it was done in Ref. [72] is generally considered to be an
unreliable way to estimate the hybridization between different ions; indeed
the largest part of the electronic wave function refers to the interstitial
space, which is naturally identified as mostly As-like. (2) Removal of the As
orbitals from the basis leads to a strong reduction of the valence band width,
indicating that hybridization between Fe and As is about as strong as direct
Fe-Fe hopping. (3) When Bloch functions are projected upon the Fe-only Wannier
functions, the latter come out very diffuse and extend way beyond the Fe ionic
radius. That is to say, negligible hybridization between Fe and As, that is
prerequisite for the scenario promoted in Ref. [72], appears to be a rather
questionable proposition. Besides, above-mentioned calculations of the phonon
spectra and electron-phonon coupling implicitly account for the large
susceptibility of the As-4 ions (which comes mostly from the outer, valence
shell) yet they find no manifestation of strong As polarization: neither
particular phonon softening nor strong coupling with any phonon.
## 3 Pairing symmetry: general considerations
### 3.1 Geometrical consideration: excitation vectors and Fermi surface
Given such disparate views that different researchers hold about the origin of
magnetism in ferropnictides and of the character of spin fluctuations there,
it may seem strange that a great majority of model calculations predict the
same pairing symmetry, $s_{\pm},$ with full gaps in both electron and hole
bands, but with the opposite signs of the order parameters between the two. In
fact, this is not surprising at all. To begin with, let us point out that the
sign of the interaction mediated by boson exchange is always positive
(attraction) for charge excitations (phonons, plasmons, polarization
excitons), since the components of a Cooper pair have the same charge, but can
be either positive (for triplet pairing, where the electrons in the pair have
the same spin) or negative (repulsion) for singlet pairing, for spin
excitations. That is to say, exchange of spin fluctuations mediates repulsion.
A quick glance at the anisotropic BCS equation reveals that repulsive
interactions can be pairing when, and only when the wave vector of such a
fluctuation spans parts of the Fermi surface(s) with opposite signs of the
order parameter (equivalently, one can say that an interaction that is
repulsive everywhere in the momentul space, can be partially attractive in the
real space, for instance, for electrons located an nearest lattice sites).
This can be illustrated on a popular model of high-$T_{c}$ cuprates, which
considers a simplified cylindrical Fermi surface nearly touching the edge of
the Brillouin zone and superexchange-driven spin fluctuations with the wave
vector $(\pi,0)$. As Fig. 9a illustrates, such an interaction is pairing in
the $d_{x^{2}-y^{2}}$ symmetry, because it spans nearly perfectly the lobes of
the order parameter with the opposite signs.
Figure 9: (color online) (a) A cartoon illustrating how a repulsive
interaction corresponding to superexchange spin fluctuations $Q=(\pi,\pi$) may
generate $d$-wave pairing in cuprates. (b) The same, for an $s_{\pm}$ state
and spin fluctuations with $Q=(\pi,0)$ (in a Brillouin zone corresponding to
one Fe per cell). (c) If the central hole pocket is absent, the superexchange
interaction favors a nodeiless $d$ state.
Most models used for ferropnictides assume a simplified fermiology with one or
more hole FSs and one or more electron FSs displaced by the SDW vector
($\pi,0$) (in this Section, we use the notations corresponding to the
Brillouin zone with one Fe per cell). Any spin-fluctuation induced interaction
with this wave vector, no matter what the origin of these fluctuations (FS
nesting, frustrated superexchange, or anything else) unavoidably leads to a
superconducting state with the opposite signs of the order parameter for the
electrons and for the holes. Depending on the details of the model the ground
state maybe isotropic or anisotropic and the gap magnitudes on the different
sheets may be the same or may be different, but the general extended $s$
symmetry with the sign-reversal of the order parameter (an $s_{\pm}$ state) is
predetermined by the fermiology and the spin fluctuation wave vector (Fig.
9b).
It is worth noting that while most (but not all) models consider spin
fluctuations corresponding to the observed instability to be the leading
pairing agent, some include spin fluctuations of different nature [for
instance, nearest neighbor superexchange or nesting between the “X” and “Y”
electron pockets, both corresponding to the same wave vector, ($\pi,\pi)$ in
the unfolded zone and $(0,0)$ in the conventional zone], or phonons, or direct
Coulomb repulsion; these additional interactions may modify the gap ratios and
anisotropies (in extreme cases, creating nodes on some surfaces), but, for a
realistic choice of parameters, unlikely to change the symmetry.
Moreover, if the radius of the largest FS pocket is larger than the magnetic
vector, spin fluctuations start to generate an intraband pair-breaking
interaction, which by itself will lead to an angular anisotropy and possible
gap nodes.
The above reasoning, however, is heavily relying upon an assumption that the
topology predicted by the DFT is correct. So far, as discussed above, the
evidence from ARPES and from quantum oscillations has been favorable. It is
still of interest to imagine, for instance, electron-doped compounds not
having hole pockets at all or having them so small that the pairing energy for
them is negligible. It was pointed out[40, 73] that in this case spin
fluctuations with different momentum vectors dominate and create a nodeless
$d$-wave state in the electron pockets, as Fig. 9c illustrates.
### 3.2 General properties of the $s_{\pm}$ state
Since the $s_{\pm}$ states constitute the most popular candidate for the
superconducting symmetry of pnictides, it is worth recapitulating the physics
of this state. Let us start with the simplest possible case: two bands (two
Fermi surfaces) and interband repulsive interaction between the two. Let the
interaction strength be $-V,$ and the DOSs $N_{1}\neq N_{2}.$ To be specific,
let $N_{2}=\alpha N_{1},$ $\alpha\geq 1.$ Then in the weak coupling limit the
BCS equations read
$\displaystyle\Delta_{1}$ $\displaystyle=-\int
d\epsilon\frac{N_{2}V\Delta_{2}\tanh(E_{2}/2k_{B}T)}{2E_{2}}$
$\displaystyle\Delta_{2}$ $\displaystyle=-\int
d\epsilon\frac{N_{1}V\Delta_{1}\tanh(E_{1}/2k_{B}T)}{2E_{1}}$ (6)
where $E_{i}$ is the usual quasiparticle energy in band $i$ given by
$\sqrt{(\epsilon-\mu)^{2}+\Delta_{i}^{2}}.$ Near $T_{c}$ linearization gives
$\displaystyle\Delta_{1}$
$\displaystyle=\Delta_{2}\lambda_{12}\log(1.136\omega_{c}/T_{c})$
$\displaystyle\Delta_{2}$
$\displaystyle=\Delta_{1}\lambda_{21}\log(1.136\omega_{c}/T_{c}),$ (7)
where $\lambda_{12}=N_{2}V$, the dimensionless coupling constant, with a
similar expression for $\lambda_{21}.$ These equations readily yield
$\lambda_{eff}=\sqrt{\lambda_{12}\lambda_{21}}$ and
$-\Delta_{1}/\Delta_{2}=\sqrt{N_{2}/N_{1}}\equiv\sqrt{\alpha}.$ Note that the
Fermi surface with the larger DOS has a smaller gap. It can also be shown that
the gap ratio at zero temperature in the weak coupling limit is also given by
$\sqrt{N_{2}/N_{1}},$ and strong coupling effects tend to reduce the disparity
between the gaps.
The situation becomes more interesting for more than two orbitals with
distinct gaps. Let us consider a model for the hole-doped 122 compound. The
calculated FS (Fig.4) shows three sets of sheets: Two e-pockets at the corner
of the zone, two outer h-pockets, formed by the $xz$ and $yz$ orbitals
(degenerate at $\Gamma$ without the spin-orbit), and the inner pocket formed
by $x^{2}-y^{2}.$ In the DFT calculations all three hole cylinders are
accidentally close to each other, however, ARPES shows two distinct sets, the
inner barrel, one of which presumably corresponding to $x^{2}-y^{2}$ band, and
the outer one, presumably $xz/yz.$ The pairing interaction between the
e-pockets and the two different types of the h-pockets need not be the same
(by virtue of the the matrix elements). Using the same partial DOS as listed
above for Ba1.6K0.6Fe2As2 (both total and individual DOS depend weakly on the
position of the Fermi level, reflecting the 2D character of the band structure
at this doping), roughly 1.2 st/eV for each hole band and the same for the two
e-band together, we get the coupling matrix
$\left(\begin{array}[]{ccc}0&0&-\lambda_{1}\nu_{1}\\\
0&0&-\lambda_{2}\nu_{2}\\\ -\lambda_{1}&-\lambda_{2}&0\end{array}\right),$ (8)
where $\nu_{1,2}$ is the ratio of DOS of the first ($xz/yz)$ and the second
($x^{2}-y^{2})$ hole bands to that of the electron bands. Note that
$\nu_{1}\sim 2$ and $\nu_{2}\sim 1.$ Diagonalizing this matrix we find the gap
ratios to be
$\Delta_{1}:\Delta_{2}:\Delta_{e}=\lambda_{1}:\lambda_{2}:\sqrt{\lambda_{1}^{2}\nu_{1}+\lambda_{2}^{2}\nu_{2}.}$
The latest ARPES measurements[11] imply that $\Delta_{i}:\Delta_{o}\approx
2:1,$ where $i$ and $o$ stand for the inner and outer sets of hole Fermi
surfaces. This would mean that the two coupling constants are twice larger
that the other (although we do not know which), which is fairly possible.
However, that implies that the electron FS has a gap that is larger than that
of the largest hole band by at least a factor of $\sqrt{1.5}=1.22$ (assuming
that the outer FSs in the calculations, are formed by the $xz/yz$ bands; the
opposite assumptions leads to an even larger electron-band gap). This is in
some disagreement with the ARPES data that suggest that $\Delta_{e}$ is on the
order of $\Delta_{i}$ or slightly smaller. However, this is a small
discrepancy, which can be easily corrected by introducing small intraband
electron-phonon coupling for the hole bands, and/or taking into account
possible gap suppression by impurities in the electron band. It is also worth
noting that the spread of the measured values, depending on the sample and on
the location on the FS, is on the order of 10%.
### 3.3 Coulomb avoidance
It was realized quite some time ago that a $d$-wave pairing has an additional
advantage compared to an $s$-wave, namely that the electrons in a Cooper pair
avoid each other (the pair wave function has zero amplitude at
$\mathbf{r-r}^{\prime}=0$), strongly reducing their local Coulomb repulsion.
The leading contribution to the pairing interaction in the single band Hubbard
model $U\sum_{\mathbf{k}}\left\langle
c_{\mathbf{k\uparrow}}c_{-\mathbf{k\downarrow}}\right\rangle$ is repulsive,
but vanishes as $\sum_{\mathbf{k}}\Delta_{\mathbf{k}}=0$ due to the symmetry
of the $d$-wave state. Thus, a contact Coulomb repulsion does not affect
$d$-wave superconductivity.
The simplest possible $s^{\pm}$-wave function is given by Eq.3. In this case,
the sum over the Brillouin zone vanishes again due to nodes at $\pm ak_{x}\pm
ak_{y}=\pi/2$. This description is however somewhat misleading because it may
produce a false impression that there is a symmetry reason for the vanishing
of the Coulomb repulsion in the $s^{\pm}$state, or that this particular
functional form is essential for avoiding the Coulomb repulsion. To illustrate
that this is not the case, it is instructive to consider a toy problem in
reciprocal space. In the weak coupling regime, the effective coupling matrix
$\Lambda_{\mathbf{kk}^{\prime}}$ (note that the band index is uniquely defined
by the wave vector) is
$\Lambda_{\mathbf{kk}^{\prime}}=\lambda_{\mathbf{kk}^{\prime}}-\mu_{\mathbf{kk}^{\prime}}^{\ast},$
(9)
where $\lambda$ is the original coupling matrix in orbital space and
$\mu_{\mathbf{kk}^{\prime}}^{\ast}$ is the renormalized Coulomb
pseudopotential. The critical temperature is determined by the largest
eigenvalue of the matrix $\Lambda,$ and the $\mathbf{k}$ dependence of the
order parameter $\Delta_{\mathbf{k}}$ is given by the corresponding
eigenvector. If $\mu^{\ast}$ is a constant and
$\sum_{\mathbf{k}}\Delta_{\mathbf{k}}=0$ (as in the $d$-wave case), any
eigenvector of the matrix $\lambda$ is also an eigenvector of $\Lambda,$ with
the same eigenvalue. This proves that Coulomb avoidance takes place for any
superconductor where the order parameter averages to zero over the entire FS,
and not only for the $d$-wave symmetry.
Let us now consider a specific $s^{\pm}$ superconductor. For simplicity, let
us take two bands with the same DOS, $N_{1}=N_{2}=N$ and with an interband
coupling only:
$\lambda_{ij}=\left(\begin{array}[]{cc}0&-VN\\\ -VN&0\end{array}\right).$ (10)
We shall also assume that the Coulomb repulsion $U$ is a contact interaction,
so that $\mu_{ij}^{\ast}=UN$ is the same for all matrix elements. The maximal
eigenvalue of $\Lambda$, which corresponds to the effective coupling constant
$\lambda_{\mathrm{eff}}$, is indeed simply $VN$ and _independent_ of $U$. The
corresponding eigenvector is $\Delta_{1}=-\Delta_{2}$, i.e. the $s^{\pm}$
state. The Coulomb interaction is irrelevant, just like in case of $d$-wave
pairing. The effect is however a consequence of the assumed symmetry of the
two bands. In general, unlike the d-wave, no symmetry requires that
$\sum_{\mathbf{k}}\Delta_{\mathbf{k}}=0$. This can already be seen if one
considers a model with distinct densities of states: $N_{2}=\alpha
N_{1}=\alpha N$. We have
$\lambda_{ij}=\left(\begin{array}[]{cc}0&-\alpha VN\\\
-VN&0\end{array}\right).$ (11)
and the weak-coupling gap ratio near $T_{c}$ is $\sqrt{\alpha}$. Now the
effect of the Coulomb repulsion is not nullified, but is still strongly
suppressed. The eigenvalues are easily determined. The key result is that the
maximal eigenvalue remains positive for all finite $\alpha$. Even the extreme
limit $\lambda_{\mathrm{eff}}^{\pm}(U\rightarrow\infty)=2VN\alpha/(1+\alpha)$
is for realistic $\alpha$ only somewhat reduced compared to
$\lambda_{\mathrm{eff}}^{\pm}(U=0)=\sqrt{\alpha}VN$. This is qualitatively
different from the regular ($s_{++})$ interband-only pairing with an
attractive interband interaction of the same strength. In this case,
$\lambda_{\mathrm{eff}}^{++}(U>V/2)<0$, and the Coulomb interaction dominates
over the attractive interband pairing interaction. In the linear in $UN$
regime, the suppression rate of $\lambda_{eff}(U)$ is $(\sqrt{\alpha}-1)/2$
for $s^{\pm}$ and $(\sqrt{\alpha}+1)/2$ for $s^{++}$ pairing. For example, for
the DOSs ratio of $4$ (the gap ratio is then $2$) $\mu^{\ast}\approx
0.25\lambda_{eff}\left(U=0\right)$ will suppress an $s^{++}$ superconductivity
entirely, while in the $s^{\pm}$ case the effective coupling will be reduced
only by 8%.
The efficiency of the Coulomb avoidance is neither limited to the assumption
of a uniform Coulomb interaction among and within the bands, nor is a result
of the weak coupling approach. Strong coupling FLEX type calculations also
find pairing states with very small repulsive contribution due to Coulomb
interaction[44, 45].
## 4 Pairing symmetry: experimental manifestations
### 4.1 Parity
Since we want to review the experimental situation regarding the pairing
symmetry, the first question to ask is, whether superconductivity is singlet
or triplet? Fortunately, this question can be answered relatively confidently.
Measurements of the Knight shift on single crystals of the Co-doped BaFe2As2
superconductor[74] clearly indicate full suppression of spin susceptibility in
the superconducting state in all directions, incompatible with a triplet
pairing in a tetragonal crystal. For other compounds only polycrystalline,
direction-averaged data exist, but they fully agree with the above result,
virtually excluding triplet superconductivity. This leaves, of all possible
scenarios, essentially three: conventional $s$ (presumably multigap),
$s_{\pm}$ and $d$.
### 4.2 Gap amplitude
All experiments that distinguish between different pairing states can be,
roughly speaking, grouped into two classes: those probing the gap amplitude
and those probing the gap symmetry. The advantage of the former is that they
are comparatively easier to perform. The temperature dependence of any
observable sensitive to the excitation gap is sensitive to the presence of
nodes or multiple gaps. The disadvantage is that only a measurement of the
relative phase of the wave function will unambiguously determine the pairing
state, including its symmetry.
Important and very transparent probes of the gap amplitude are thermodynamic
measurements. The early reports of the specific heat leaned towards power-law
behavior characteristic of nodal superconductivity. The latest data [13, 15]
suggest a fully gapped superconductivity, or a dominant fully gapped component
with possible small admixture of a nodal state. While the experimental
situation is still far from consensus, especially regarding the 1111 family, a
few observations may be in place: (i) The specific heat jump in the h-doped
BaFe2As2 is strong and sharp, and in 1111 compounds is weak and poorly
expressed. This cannot be ascribed to a difference in calculated band
structures. This is either due to sample quality issues or possibly to the
more isotropic character of superconducting and magnetic properties in 122
systems. (ii) In no case can specific heat temperature dependence be fitted
with one gap. Multiple gap fits, having more parameters, are of course less
reliable. (iii) Another, usually more reliable signature of nodal
superconductivity is a square-root dependence of the specific heat coefficient
on the magnetic field. Existing reports[13] however show a clear linear
dependence, characteristic of a fully gapped superconductor.
Another popular probe is temperature dependence of the NMR relaxation rate.
Extensive studies have been done in this aspect (see other articles in this
volume). In all studied systems, the relaxation rate is non-exponential. The
initial impression was that the relaxation rate is cubic in temperature,
$1/T_{1}\propto T^{3},$ consistent with nodal lines[75, 76]. Later it was
argued that the data cannot be described by a single power law as in the
cuprates[77, 78]. These results were obtained for the 1111 systems. The
situation with the 122 family is even less clear. Published data[79, 74] do
not show exponential decay either, but the results are equally far from any
single power law behavior. Even more puzzling, the only paper reporting on the
low-$T_{c}$ LaFePO superconductor claims that the relaxation rate does not
decrease below $T_{c}$ at all[80].
The third relevant experiment is measuring the London penetration depth.
Reports are again contradictory. For instance, in Pr-based 1111 compound the
penetration depth was found[81] to barely change between $\approx 0.05T_{c}$
and $T^{\ast}\approx 0.35T_{c},$ and than increase roughly as
$(T-T^{\ast})^{2}$ between $T^{\ast}$ and $\approx 0.65T_{c},$ a picture
roughly consistent with a multi-gap nodeless superconductor. Malone $et$
$al$[82] measured Sm-based 1111 and were able to fit their data very well in
the entire interval from $T_{c}/30$ and $T_{c}$ using two full gaps. In Nd-
based 1111 the penetration depth was measured at $T>0.1T_{c}$ and fitted with
a single anisotropic gap for $0.1T_{c}<T<T_{c}/3$,[83] however, the latest
result from the same authors, taken at lower temperature, can be better fitted
with a quadratic law[84]. Similar quadratic behavior has been clearly seen in
the 122 compounds[85]. At the same time, the low-$T_{c}$ LaFePO is again odd:
it shows a linear behavior[86].
To summarize, the thermodynamic data on average lean towards a nodal
superconductivity. However, some data are not consistent with the gap nodes,
and there is no clear correlation with the sample quality either way.
Moreover, while some data suggest line nodes, others are consistent only with
point nodes, in the clean limit. One can say with a reasonable degree of
confidence that the entire corpus of the data cannot be described by any one
scenario in the clean limit. On the other hand, essentially any temperature
dependence of thermodynamic characteristics can be fitted if a particular
distribution of impurity scattering is assumed in an intermediate regime
between the Born and the unitary scattering, and a particular relation between
the intra- and interband scattering (there have been a number of paper doing
exactly that for the NMR relaxation rate, for instance, Ref. [87], or for the
penetration depth, for instance, Ref. [88]). However, the fact that all these
papers rely upon specific combinations of parameters, while the phenomena they
seek to describe are rather universal, calls for caution. Besides, except in
the pure unitary regime, scattering is accompanied by a $T_{c}$ suppression
and most papers do not find any correlation between thermodynamic probes and
$T_{c}$ among different samples. Another possibility is that required
scattering is provided not by impurities, but by intrinsic defects that are
thermodynamically or kinetically necessarily present in all samples (for
example, dynamic domain walls introduced in Ref. [22]). More measurements at
the lower temperature and on clean samples will probably clarify the matter.
At the moment one cannot consider this problem solved.
Close to the thermodynamic measurements are tunneling type experiments. As of
now, these have been nearly exclusively point-contact Andreev reflection
probes. Here, again, the experimental reports are quite inconsistent,
moreover, the situation is in some sense worse than in thermodynamic probes,
since uncontrollable surface properties enter the picture. Interpretation
generally includes fitting one curve with a large number of parameters, and
the procedure is not always well defined. Generally speaking, three types of
results have been reported: $d$-wave like, single full gap-like, and multigap.
Interpretation is particularly difficult because within the $s^{\pm}$ picture
formation of subgap Andreev bound states was predicted (e.g., Refs. [89, 90])
that can be easily mistaken for multiple gaps.
### 4.3 Phase-sensitive probes
In view of all that, experiments directly probing the gap symmetry are highly
desirable. The paramagnetic Meissner effect, also known as Wohlleben effect,
occurs in a polycrystalline sample when inter-grain weak links have random
order parameter phase shifts, $0$ or $\pi.$ It has been routinely observed in
cuprates and is considered a key signature of $d$-wave superconductivity. The
effect does not exist in conventional, even anisotropic and multi-gap
superconductors, even though sometimes it can be emulated by impurity effects
in the junctions. For $d-$wave superconductors without pronounced
crystallographic texture the Wohlleben effect is expected, and its absence can
be taken as evidence against $d$-wave. Finally, in the $s^{\pm}$ scenario the
phase is the same by symmetry for $(100)$ and $(010)$ grain boundaries, and
there are good reasons to expect the same phase for $(110)$ boundaries as
well. There may or may not be a $\pi$ phase shift for phase boundaries at some
specific orientation, likely for a narrow range of angles[91], but probably
not enough to produce a measurable Wohlleben effect. The absence of the effect
in experiment[92] is a significant argument against $d$-wave, but hardly helps
to distinguish $s$ from $s^{\pm}.$
Similarly, the $c$-axis tunneling provides evidence against the $d$-wave,
where the Josephson current strictly parallel to the crystallographic $c$
direction vanishes by symmetry. Experimentally a sizable current was
found[93].
Recalling the cuprates again, the ultimate argument in favor of the $d$-wave
was provided by the corner Josephson junction experiments that probe directly
the phase shift between two separate junctions; in cuprates, with their
$d_{x^{2}-y^{2}}$ symmetry, these junction were to be along the $(100)$ and
$(010)$ directions. Similarly, a potential $d_{xy}$ state could be detected by
the combination of $(110)$ and $(\bar{1}10)$ directions. On the other hand, a
conventional $s$ state would not produce a phase shift for any combination of
contacts. Again, the case of $s_{\pm}$ superconductivity is nontrivial. While
symmetry does not mandate a $\pi$ shift for any direction, it can be shown
that, depending on the electronic structure parameters and properties of the
interface, there may exist intermediate angles (between $0$ and $45^{o})$
where a $\pi$ shift is possible[91]. It also may be possible if the two
junctions have different tunneling properties, so that one of them filters
through only hole-pocket electrons, and the other only electron-pockets. It is
not as bizarre as it may seem, and some possibilities were discussed in Ref.
[91]. Probably the most promising design involves “sandwiches” of various
geometries. The first proposal of that kind was by Tsoi et al[90], who
suggested an $s/s^{\pm}/s^{\prime}$ trilayer, where $s$ is a conventional
quai-2D superconductor with a large Fermi surface that has no overlap with the
hole FS of the $s^{\pm}$ layer (equivalently, a superconductor with small
Fermi surfaces centered around the M points), and $s^{\prime}$ is a
conventional superconductor with a small FS centered around $\Gamma.$ This was
followed by another proposal of a bilayer of hole-doped and electron-doped 122
materials[91]. In both cases the idea is that the current through the top of
the sandwich will be dominated by the electron FS, and through the bottom by
the hole one. Both proposals require momentum conservation in the interfacial
plane, that is, basically, epitaxial or very high quality interface. The
former proposal has an additional disadvantage of requiring two high-quality
interfaces with very special conventional superconductors, particularly the
one that should filter through the electron FS is rather difficult to find. As
of now, no experiments have been reported pursuing any of the above
suggestions, but with better single crystals and thin films it should become
increasingly doable. It should be stressed, however, that in this case, unlike
the cuprates, an absence of the $\pi$ shifts in any of the proposed geometries
does not disprove the $s^{\pm}$ scenario, since the effect here is
quantitative rather than qualitative, but the presence of the sought effect
would be a very strong argument in favor of it. On the other hand, standard
90o corner junction experiments similar to cuprates are also important, as
they could prove unambiguously that the symmetry is not $d$-wave (even though
they cannot distinguish between $s$ and $s^{\pm}).$
Further properties of interfaces between an $s^{\pm}$ superconductor and
normal metal or conventional superconductor are now actively being studied
theoretically, encouraging further experimental research. Probably we will see
first results within the next year.
### 4.4 Coherence factor effects
Other signatures of the $s^{\pm}$ state are based on the fact, previously
pointed out by many in connection with the cuprates, that the coherence
factors are “reversed” for electronic transitions involving order parameters
of the opposite sign. In the conventional BCS theory, as is well known,
coherence factors of two kinds appear. The first kind, sometimes called “Type
I” or “minus” coherence factor, is given by the expression
$(1-\Delta_{\mathbf{k}}\Delta_{\mathbf{k}^{\prime}}/E_{\mathbf{k}}E_{\mathbf{k}^{\prime}}),$
where
$E_{\mathbf{k}}=\sqrt{\Delta_{\mathbf{k}}^{2}+\varepsilon_{\mathbf{k}}^{2}},$
and $\varepsilon_{\mathbf{k}}$ in the normal state excitation. The other kind,
Type II or the “plus” coherence factor has the opposite sign in front of the
fraction. If both order parameters entering this formula have the same sign,
the Type I factor is destructive, in the sense that it goes to zero when
$\varepsilon\rightarrow 0,$ and cancels out the peak in the superconducting
DOS. Type I factors appear, for instance, in the polarization operator, and as
a result there are no coherence peaks in phonon renormalization (as measured
by ultrasound attenuation, for instance) and in spin susceptibility (including
the Knight shift). Type II factors appear, for instance, in the NMR relaxation
rate, and they are constructive, resulting in the famous Hebel-Slichter peak
below $T_{c}.$
Obviously, if $\Delta_{\mathbf{k}}$ and $\Delta_{\mathbf{k}^{\prime}}$ have
opposite signs, the meaning of the coherence factors is reversed; the Type I
factors are now constructive and the Type II destructive. There are several
straightforward ramifications of that. For instance, as it was pointed out
already in the first paper proposing the $s^{\pm}$ scenario[5], the spin
susceptibility at the SDW wave vector should show resonance enhancement just
below $T_{c}$. For explicit calculations of this effect see for example
Refs.[94, 95]. There are indeed some reports of this effect, as measured by
neutron scattering[96]. In principle, one can expect a similar effect in the
phonon line-width, for the phonons with the same wave vector, just below
$T_{c},$ but this is really hard to observe.
Less straightforward are cases of the quantities that involve averaging over
the entire Brillouin zone, in which case the answer, essentially, depends on
which processes play a more dominant role in the measured quantity, those
involving intra-, or interband scattering. The answer usually depends on
additional assumptions about the matrix elements involved, which can rarely be
calculated easily from first principles. An example is electronic Raman
scattering; a possibility of a resonant enhancement in some symmetries has
been discussed recently[97].
## 5 Role of impurities
Impurity and defect scattering is believed to play an important role in
pnictide superconductors. Proximity to a magnetic instability implies that
ordinary defects may induce static magnetic moments on the neighboring Fe
sites and thus trigger magnetic scattering. If, as is nearly universally
believed, an order parameter with both signs is present, nonmagnetic
impurities are also pair-breaking. Thus the anticipation is that in regular
samples, and maybe in samples of much higher quality, impurity-induced pair
breaking will play a role.
Our intuition regarding the impurity effects in superconductors is largely
based upon the Abrikosov-Gorkov theory of Born-scattering impurities in BCS
superconductors. There was an observation at that time that folklore ascribes
to Mark Azbel: Soviet theorists do what can be done as good as it should be
done, and American ones do what shall be done as good as it could be done. For
many years the approach to the impurity effects in superconductors was largely
Soviet: most researchers refine the Abrikosov-Gorkov theory, applying it to
anisotropic gaps and to unconventional superconductors, and relatively little
has been done beyond the Born limit — despite multiple indications that most
interesting superconductors, from cuprates to MgB2 to pnictides are in the
unitary limit or in an intermediate regime.
The physics of the nonmagnetic scattering in the two different limits is quite
different. In the Born limit, averaging over all scattering events yields a
spatially uniform superconducting state and tries to reduce the variation of
the order parameter over the FS. Ultimately, for sufficiently strong
scattering, the order parameter becomes a constant, corresponding to the DOS-
weighted average over the FS. Note that unless this average is zero by
symmetry (like in d-wave) the suppression of $T_{c},$ while linear at small
concentrations, is never complete. As pointed out by Mishra $et$ $al.$ [98],
this effect should manifest itself most clearly in an extended s-wave pairing
with accidental nodes in the order parameter. Indeed, while in $d$-wave
superconductors impurities broadens nodes into finite gapless spots, in an
extended $s$ case it is likely that the order parameter of one particular sign
dominates a given FS pocket, in which case Born impurities will first make the
parts of the FS with the “wrong” order parameter gapless, and then lead to a
fully gapped superconductivity. Of course, this only holds for nonmagnetic
impurities. Isotropic magnetic impurities will be just pair-breaking as they
are in conventional superconductors, with the only interesting new physics
being that magnetic impurities cease being pair-breakers if they scatter a
pair such that the sign of the order parameter is flipping. The rule of thumb
is that a scattering path for which magnetic scattering is pair-breaking (no
change of sign of the order parameter), nonmagnetic scattering will not be
pair-breaking, and $vice$ $versa.$
The physics of the unitary limit is quite different. In that limit, the
concentration of impurities is relatively low, but the scattering potential of
an individual impurity is strong, $N(0)v_{imp}\gg 1.$ In that case rather than
suppressing superconductivity uniformly each impurity creates a bound state at
the chemical potential, thus creating a zero energy peak in the density of
states, without substantial suppression of the bulk superconductivity.
Increasing the impurity concentration broadens the peak, while increasing its
strength barely has any effect at all [99]. In an intermediate case between
the Born limit and the unitary limit, the bound state is formed inside the gap
at a finite energy and is the broader the closer it is to the gap (that is,
closer to the Born limit).
The principal difference from the point of view of the experiment is that the
unitary or intermediate scattering can create subgap density of states at
arbitrary low energy at any temperature, without a drastic suppression of
$T_{c}.$ It was shown in Ref. [87] that any standard code for solving the
Eliashberg equations in the Born limit can be easily modified, with minor
changes, to treat the unitary limit, as well as any intermediate regime.
Therefore we anticipate an imminent shift in the community from the “Soviet”
approach to the “Western” approach, with more quantitative understanding of
the effect beyond the Born approximation.
## 6 Conclusions
In this article we presented a brief overview of some proposals that have been
made for the pairing state in the Fe-pnictide superconductors. In particular,
we summarized arguments that support the view that the vicinity of
superconductivity and magnetism in these systems is not accidental. The
obvious appeal of this, and essentially any other electronic pairing mechanism
is, of course, that the involved energy scales, and thus $T_{c}$, can in
principle be larger if compared to pairing due to electron-phonon interaction.
Electronic mechanisms also promise a new level of versatility in the design of
new superconductors.
At this early stage in the research on the iron pnictide family, experiments
have not conclusively determined the pairing symmetry, the detailed pairing
state or the microscopic pairing mechanism. Still, in our view a plausible
picture emerges where superconductivity is caused by magnetic fluctuations.
Only two ingredients are vital to arrive at a rather robust conclusion for the
pairing state. First, pnictides need to have Fermi surface sheets of two
kinds, one near the center of the Brillouin zone, and the other near the
corner. Second, the typical momentum for the magnetic fluctuations should be
close to the ordering vectors $\mathbf{Q=}\left(\pi,\pi\right)$ of the parent
compounds. Then, magnetic interactions lead quite naturally to an efficient
inter-band coupling that yields an $s^{\pm}$ pairing state. This result is
general in the sense that it is obtained regardless of whether one develops a
theory based on localized quantum magnetism or itinerant paramagnons. There is
evidence that the two needed ingredients are present in the pnictides. Fermi
surface sheets at the appropriate locations have been predicted in non-
magnetic LDA calculations and seen in ARPES experiments. The magnetic ordering
vector has been determined via neutron scattering, even though we have to
stress that a clear identification of magnetic fluctuations for
superconducting systems without long range magnetic order is still lacking.
The resulting $s^{\pm}$ pairing state has a number of interesting properties.
As far as the a group theoretic classification is concerned, its symmetry is
the same as that for a conventional $s$-wave pairing, where the gap-function
has same sign on all sheets of the Fermi surface. However, there are
significant differences between the two states. The sign change in the gap
affects the coherence factors, leading to the resonance peak in the dynamic
spin susceptibility and the absence of a Hebel-Slichter peak in NMR.
Nonmagnetic impurities affect the $s^{\pm}$-state just like magnetic
impurities do in an ordinary $s$-wave state, i.e. here a behavior more akin to
$d$-wave superconductors. Another implication of the sign change in the
$s^{\pm}$-state leads to rather efficient Coulomb avoidance.
The presence of nodes in the superconducting gap in still an open issue. In
$d$-wave or $p$-wave pairing states, nodal lines or points are fixed by
symmetry. This is different for the $s^{\pm}$-state. In its most elementary
version, the sign change of the gap corresponds to a node located between two
Fermi surface sheets. This is the case for the $\Delta\left(\mathbf{k}\right)$
given in Eq.3. Energetic arguments favor such a gapless state as long as the
momentum transfer $\mathbf{Q}$ couples efficiently to large parts of distinct
Fermi surface sheets and Coulomb avoidance is efficient. However, as there is
no symmetry constraint for the location of the nodes, it is in principle
possible that there are nodes on some Fermi surface sheets.
Next to the nature of the pairing state, the microscopic understanding of the
magnetism of the Fe-pnictides is one of the most interesting aspects of these
materials. Are these systems made up of localized spins that interact via
short ranged, nearest neighbor exchange interactions or, are they better
described in terms of itinerant magnetism? While we emphasized that many
aspects of the pairing state emerge regardless of which of these points of
view is correct, this is really only true for the most elementary aspects of
the theory. As our understanding of these materials deepens, dynamical aspects
of the pairing state will become more and more important, and the details of
the magnetic degrees of freedom will matter. In our view, the most sensible
description starts from itinerant electrons, however with significant
electron-electron interaction. In detail, we find numerous arguments that
emphasize the role of magneto-elastic couplings and that favor a sizable Hund
coupling, i.e. the multi orbital character and the corresponding local multi-
orbital interactions are important to understand the magnetism and
superconductivity alike. Regardless of whether this specific point of view is
correct or not, it is already evident that the ferropnictides make up a whole
new class of materials that stubbornly refuse to behave according to one of
the simple minded categories of condensed matter theory.
## 7 Acknowledgements
This research was supported by the Ames Laboratory, operated for the U.S.
Department of Energy by Iowa State University under Contract No. DE-
AC02-07CH11358 (J.S.), and by the Office of Naval Research (I.I.M.). The
authors wish to thank all their friends and collaborators, without whom this
works could not be accomplished, and their numerous colleagues who read the
manuscript and sent us many useful and insightful comments.
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|
arxiv-papers
| 2009-01-29T21:57:07 |
2024-09-04T02:49:00.278411
|
{
"license": "Public Domain",
"authors": "I.I. Mazin and J. Schmalian",
"submitter": "Igor Mazin",
"url": "https://arxiv.org/abs/0901.4790"
}
|
0901.4854
|
# The $\rho\to\gamma\pi$ and $\omega\to\gamma\pi$ decays in quark-model
approach and estimation of coupling for pion emission by quark
A V Anisovich, V V Anisovich, L G Dakhno, M A Matveev,
V A Nikonov and A V Sarantsev Petersburg Nuclear Physics Institute, 188300,
Gatchina, Russia
###### Abstract
In the framework of the relativistic and gauge invariant spectral integral
technique, we calculate radiative decays $\rho(770)\to\gamma\,\pi(140)$ and
$\omega(780)\to\gamma\,\pi(140)$ supposing all mesons ($\pi$, $\rho$ and
$\omega$) to be quark–antiquark states. The $q\bar{q}$ wave functions found
for mesons and photon lead to a reasonably good description of data
($\Gamma^{(exp)}_{\rho^{\pm}\to\gamma\pi^{\pm}}=68\pm 30$ keV,
$\Gamma^{(exp)}_{\rho^{0}\to\gamma\pi^{0}}=77\pm 28$ keV,
$\Gamma^{(exp)}_{\omega\to\gamma\pi^{0}}=776\pm 45$ keV) that makes it
possible to estimate the coupling for the bremsstrahlung emission of pion by
quarks $g_{\pi}\equiv g_{\pi}(u\to d\pi)$. We have found two values for the
pion bremsstrahlung coupling: $|g_{\pi}|=16.7\pm 0.3\ ^{+0.1}_{-2.3}$
(Solution I) and $|g_{\pi}|=3.0\pm 0.3\ ^{+0.1}_{-2.1}$ (Solution II). Within
SU(6)-symmetry for nucleons, Solution I gives us for $\pi NN$ coupling the
value $16.4\leq g_{\pi NN}^{2}/(4\pi)\leq 23.2$ that is in qualitative
agreement with the $\pi N$ scattering data, $g_{\pi NN}^{2}/(4\pi)\simeq 14$.
For excited states, we have estimated the partial widths in Solution I as
follows: $\Gamma(\rho_{2S}^{\pm}\to\gamma\pi)\simeq 10-130$ keV,
$\Gamma(\rho_{2S}^{0}\to\gamma\pi)\simeq 10-130$ keV,
$\Gamma(\omega_{2S}\to\gamma\pi)\simeq 60-1080$ keV. The large uncertainties
emphasise the necessity to carry out measurements of the meson radiative
processes in the region of large masses.
###### pacs:
12.39.Mk, 12.38.-t, 14.40.-n
††: J. Phys. G: Nucl. Phys.
## 1 Introduction
The radiative decay amplitude is a necessary element for the study of the
quark–gluon structure of hadrons. In this paper, we present the calculation of
the radiative decays of quark–antiquark states $(q\bar{q})_{in}=\rho,\omega$
into $\gamma\pi$. In this way, we continue the calculations initiated in [1]
where radiative transitions of quarkonium states
$(Q\bar{Q})_{in}\to\gamma(Q\bar{Q})_{out}$ were studied, with the production
of massive outgoing states $(Q\bar{Q})_{out}$. Considering the production of
the $\gamma\pi$ system, a particular necessity is to take into account,
together with the annihilation $q\bar{q}\to\pi$, an additional process of the
bremsstrahlung type, namely, $q\to q\pi$.
We treat the meson decay amplitude as triangle diagram of constituent quarks
(additive quark model) calculated in terms of the spectral integration
technique, see [2] and references therein. The spectral integral technique is
rather profitable for the description of composite particles, for the content
of a composite system is thus strictly controlled. Besides, this technique is
rather convenient for the description of high spin states.
The equation for the composite $q\bar{q}$ systems in the spectral integration
technique was suggested in [3], it is a direct generalisation of the
dispersion $N/D$ equation [4] when the $N$-function was represented as an
infinite sum of separable vertices, see [2] for detail. In terms of this
equation, the $b\bar{b}$ and $c\bar{c}$ quarkonia were considered in [5],
while the light-quark $q\bar{q}$ mesons were studied in [6].
In [6], the levels of the one-component $q\bar{q}$ systems (with $I=1$ or
$I=0$ which are almost pure $s\bar{s}$ or
$n\bar{n}=(u\bar{u}+d\bar{d})/\sqrt{2}$ states) were reconstructed as well as
their wave functions. The $q\bar{q}$ systems are formed at distances, where
perturbative QCD does not work ($r\sim 0.5-1.0$ fm). In this region (the
region of soft interactions), we deal with constituent quarks and effective
massive gluons (with mass of the order of 700–1000 MeV [7, 8, 9, 10, 11]). It
means that quark–antiquark interactions undergone a significant changes as
compared to small distances; besides, at large distances the confinement
forces work. Therefore, interactions in the soft region should be
reconstructed on the basis of experimental studies – in [6], the $q\bar{q}$
interaction was reconstructed on the basis of available data for
$q\bar{q}$-levels and the $q\bar{q}$-meson radiative decays.
The standard way to investigate quark–antiquark systems is to apply the Bethe-
Salpeter equation [12] written in terms of Feynman integrals. One may find the
examples of such a study of light quark–antiquark systems in [13, 14, 15, 16,
17] and for heavy quarkonia ($c\bar{c}$ and $b\bar{b}$) in [17, 18, 19, 20,
21, 22], see also references therein.
However, one should keep in mind an important difference between the standard
Bethe–Salpeter equation and that written in terms of the spectral integral
[3]. In the dispersion relation technique, the constituents in the
intermediate state are mass-on-shell, $k^{2}_{i}=m^{2}$, while in the Feynman
technique, which is used in the Bethe–Salpeter equation, $k^{2}_{i}\neq
m^{2}$. So, in the spectral integral equation, when the high spin state
structures are calculated, we have a numerical factor $k^{2}_{i}=m^{2}$, while
in the Feynman technique one should write $k^{2}_{i}=m^{2}+(k_{i}^{2}-m^{2})$.
Here, the first term in the right-hand side provides us the contribution
similar to that used in the spectral integration technique, while the second
term cancels one of denominators of the kernel of the Bethe–Salpeter equation,
that results in the penguin or tadpole type diagrams – let us call them zoo-
diagrams. A particular property of the spectral integral technique is the
exclusion of zoo-diagrams from the equation for composite systems.
The spectral integral equation [3] gives us a unique solution for the
quark–antiquark levels and their wave functions, provided the interquark
interaction is known. Let us emphasize that the equation works for both
instantaneous interactions and the $t$-channel exchanges with retardation, and
even for the energy-dependent interactions: this follows from the fact that
the equation itself is the modified dispersion relation for the amplitude. For
solving the inverse problem, that is, for reconstructing the interaction, it
is not enough to know the meson masses — one should know wave functions of
quark–antiquark systems. Such an information is contained in the hadronic form
factors and radiative decay amplitudes. Therefore, in the approach of refs.
[3, 5, 6], we consider simultaneously the meson levels in terms of the
spectral integral equations and the meson radiative transitions in terms of
the double dispersion relations over $q\bar{q}$ states (or over corresponding
meson masses) — in this way all calculations are carried out within compatible
methods.
The calculation of radiative transition amplitudes in terms of the double
dispersive integrals was performed for some selected reactions in [23, 24, 25,
26, 27] — the basic points of the method of operator expansion used in the
calculation of double dispersive integrals can be found in [1, 2, 28].
The analyses of the light $q\bar{q}$ systems [6] and heavy $Q\bar{Q}$
quarkonia [5] in terms of the spectral integral equation differ from one
another in certain respect, because the available experimental data are of
different sort: for the $Q\bar{Q}$ systems the only known are low-lying states
(with an exception for the $1^{--}$ quarkonia $\Upsilon$ and $\psi$ where a
long series of vector states was discovered in the $e^{+}e^{-}$ annihilation).
At the same time, for the low-lying states there exists a rich set of data on
radiative decays: $(Q\bar{Q})_{\rm in}\to\gamma(Q\bar{Q})_{\rm out}$ and
$(Q\bar{Q})_{\rm in}\to\gamma\gamma$. For the light quark sector ($q\bar{q}$
systems), there exists an abundant information on masses of highly excited
states with different $J^{PC}$ (see [29, 30, 31, 32, 33] and surveys [2, 34,
35]), but the knowledge of radiative decays is rather poor.
Despite the scarcity of data on radiative decays, the light $q\bar{q}$ states
have been studied in [6], relying upon our knowledge of linear trajectories in
the $(n,M^{2})$-plane, where $n$ is the radial quantum number of the
$q\bar{q}$-meson with mass $M$ (see [2, 36]). We hope that it may somehow
compensate the lack of information on the wave functions. In the fitting
procedure [6], the main attention was paid to the states with large masses,
expecting to extract the confinement interaction. We obtained that the strong
$t$-channel interaction (which, as we think, determines the confinement)
should exist in both scalar $I\otimes I$ and vector
$\gamma_{\mu}\otimes\gamma_{\mu}$ channels. The fitting results point rather
reliably to the equality of these $t$-channel interactions [6].
Obviously, the fitting results presented in [6] should be checked (and, if
necessary, improved) by investigating the other radiative decays – following
to this program we consider here the decays $\rho\to\gamma\pi$ and
$\omega\to\gamma\pi$. Small mass of the pion requires to take into account not
only the process of photon emission with a subsequent quark–antiquark
annihilation $q\bar{q}\to\pi$ (triangle diagram of the additive quark model,
Fig. 1) but also the bremsstrahlung-type emission of pion $q\to\pi q$, with
subsequent quark–antiquark annihilation into photon $q\bar{q}\to\gamma$, see
Fig. 2. Therefore, the key points in the calculation of the
$\rho,\omega\to\gamma\pi$ decays is to know $q\bar{q}$ wave functions of pion
and vector mesons ($\rho$ and $\omega$) as well as $q\bar{q}$ wave function of
the photon $\gamma\to q\bar{q}$. Also the fitting procedure calls us to
determine the pion bremsstrahlung constant for the process $q\to\pi q$.
Figure 1: a), c) Triangle diagrams for radiative transition
$(q\bar{q})_{in}\to\gamma\pi$ with the emission of photon by quark; here
$p^{2}=M^{2}_{in}$, $p^{\prime 2}=M^{2}_{\pi}$ and $(p-p^{\prime})^{2}=q^{2}$.
b), d) Cuttings of the triangle diagrams 1a and 1c signify the double
discontinuity of the spectral integral with intermediate-state momentum
squared $P^{2}=s$, $P^{\prime 2}=s^{\prime}$ and $(P-P^{\prime})^{2}=q^{2}$.
Figure 2: a), c) Triangle diagrams for radiative transition
$(q\bar{q})_{in}\to\gamma\pi$ with the emission of pion by quark; here
$p^{2}=M^{2}_{in}$, $p^{2}_{\pi}=M^{2}_{\pi}$. b), d) Cuttings of the triangle
diagrams 2a and 2c for getting double discontinuity of the spectral integral
with $P^{2}=s$, $P^{\prime 2}=s^{\prime}$ and
$(P-P^{\prime})^{2}=p^{2}_{\pi}$.
### 1.1 Photon wave function
For the region $0\mathrel{\mathchoice{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptscriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}}Q^{2}\mathrel{\mathchoice{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptscriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}}1$
(GeV/c)2 (here $Q^{2}=-q^{2}$), the light-quark components of the photon wave
function $\gamma^{*}(Q^{2})\to q\bar{q}$ ($q=u,d,s$) are determined in [37]
(see also [2]) on the basis of data for the transitions
$\pi^{0},\eta,\eta^{\prime}\to\gamma\gamma^{*}(Q^{2})$ and reactions of
$e^{+}e^{-}$-annihilation: $e^{+}e^{-}\to\rho^{0},\omega,\phi$ and
$e^{+}e^{-}\to hadrons$ at $1<E_{e^{+}e^{-}}<3.7$ GeV (in a more rough
approximation the wave function $\gamma(Q^{2})\to q\bar{q}$ was found in
[38]).
Conventionally, one may consider two pieces of the photon wave function: soft
and hard ones. Hard component relates to the point-like vertex $\gamma\to
q\bar{q}$, it is responsible for the production of quark–antiquark pair at
high virtuality. At high energies of the $e^{+}e^{-}$ system, the ratio of
cross sections $R=\sigma(e^{+}e^{-}\to
hadrons)/\sigma(e^{+}e^{-}\to\mu^{+}\mu^{-})$ is determined by the hard
component of photon wave function, while soft component is responsible for the
production of low-energy quark–antiquark vector states such as $\rho^{0}$,
$\omega$, $\phi(1020)$, and their excitations.
In the spectral integral technique, the quark wave function of the photon,
$\gamma^{*}(Q^{2})\to q\bar{q}$, is defined as follows:
$\psi_{\gamma^{*}(Q^{2})\to q\bar{q}}(s)=\frac{G_{\gamma\to
q\bar{q}}(s)}{s+Q^{2}}\ ,$ (1)
where $G_{\gamma\to q\bar{q}}(s)$ is the vertex for the transition of photon
into $q\bar{q}$ state, depending on the invariant energy squared, $s$, of
$q\bar{q}$ system. In terms of the light-cone variables
$s=(m^{2}+k_{\perp}^{2})/[x(1-x)]$, where $m$ is the quark mass, ${k}_{\perp}$
and $x$ are the light-cone characteristics of quarks: transverse momentum and
a part of longitudinal momentum.
Rather schematically, the vertex function $G_{\gamma\to q\bar{q}}(s)$ may be
divided into two terms. The first term is responsible for the soft component
which is due to the transition of photon to vector $q\bar{q}$ meson $\gamma\to
V\to q\bar{q}$, while the second one describes the point-like interaction in
the hard domain. The principal characteristics of the soft component is the
threshold value of the vertex and the rate of its decrease with energy. The
hard component of the vertex is characterized by the energy where the point-
like interaction becomes dominant.
In [38], the photon wave function has been found assuming the quark relative
momentum dependence to be the same for all quark vertices: $g_{\gamma\to
u\bar{u}}(k^{2})=g_{\gamma\to d\bar{d}}(k^{2})$ $=$ $g_{\gamma\to
s\bar{s}}(k^{2})$, where we redenoted $G_{\gamma\to
q\bar{q}}(s)\longrightarrow g_{\gamma\to q\bar{q}}(k^{2})$ with
$k^{2}=s/4-m^{2}$. The hypothesis of the vertex universality for $u$ and $d$
quarks used in [37],
$G_{\gamma\to u\bar{u}}(s)=G_{\gamma\to d\bar{d}}(s)\equiv G_{\gamma}(s)\ ,$
(2)
looks rather trustworthy because of the degeneracy of $\rho$ and $\omega$
states, though the similarity in the $k$-dependence for non-strange and
strange quarks may be violated. Using experimental data on the transitions
$\gamma\gamma^{*}(Q^{2})\to\pi^{0},\eta,\eta^{\prime}$ only, one cannot
determine the parameters ($C,b,s_{0}$ – see below Eqs. (3) and (1.1)) for both
$G_{\gamma\to s\bar{s}}(s)$ and $G_{\gamma}(s)$. We also add the $e^{+}e^{-}$
annihilation data for the determination of wave functions, that is
$e^{+}e^{-}\to\gamma^{*}\to\rho^{0},\omega,\phi(1020)$, together with the
ratio $R(E_{e^{+}e^{-}})=\sigma(e^{+}e^{-}\to
hadrons)/\sigma(e^{+}e^{-}\to\mu^{+}\mu^{-})$ at $E_{e^{+}e^{-}}>1$ GeV. The
reactions $e^{+}e^{-}\to\gamma^{*}\to\rho^{0},\omega,\phi(1020)$ are rather
sensitive to the parameters $C_{a},b_{a}$, while the data on
$R(E_{e^{+}e^{-}})$ allow us to fix the parameter $s_{0}$.
The transition vertices for $u\bar{u},d\bar{d}\to\gamma$ have been chosen in
the form:
$\displaystyle u\bar{u},\,d\bar{d}:\qquad G_{\gamma}(s)$ $\displaystyle=$
$\displaystyle
c_{\gamma}\bigg{(}\exp({-b^{\gamma}_{1}s})+c^{\gamma}_{2}\exp({-b^{\gamma}_{2}s})\bigg{)}+\frac{1}{1+e^{-b^{\gamma}_{0}(s-s_{0}^{\gamma})}}\
,$ (3)
and the following parameter values have been found [2, 37]:
$\displaystyle u\bar{u},\,d\bar{d}:\,\,$ $\displaystyle
c^{\gamma}=32.506,\;c^{\gamma}_{2}=-0.0187,\;b^{\gamma}_{1}=4\,{\rm
GeV}^{-2},\;b^{\gamma}_{2}=0.8\,{\rm GeV}^{-2},\;$ $\displaystyle
b^{\gamma}_{0}=15\,{\rm GeV}^{-2},\;s_{0}^{\gamma}=1.614\,{\rm GeV}^{2}\ .$
With these parameters, we have a good description of the available
experimental data for $V\to e^{+}e^{-}$ and two-photon decays, see [2, 6].
### 1.2 The $\rho$, $\omega$ and $\pi$ wave functions
We characterise $q\bar{q}$-states by the following momentum-dependent wave
functions:
$\psi^{(S,L,J)}_{n}(k^{2})=\frac{G^{(S,L,J)}_{n}(k^{2})}{s-\left(M_{n}^{(S,L,J)}\right)^{2}}\
,$ (5)
where $S$, $L$, $J$ are the spin, orbital momentum and total momentum of the
$q\bar{q}$ system with mass $M^{(S,L,J)}_{n}$.
#### 1.2.1 $\rho(nL)$ and $\omega(nL)$ states
We introduce spin-orbital operators and wave functions for the states with
dominant $L=0,2$ as follows:
$\displaystyle\begin{array}[]{l|l|r}\qquad
L=0&0^{-+}&i\gamma_{5}\psi^{(0,0,0)}_{n}(k^{2})\\\ {\rm
dominant}\,L=0&1^{--}&\gamma_{\mu}^{\perp}\psi^{(1,0,1)}_{n}(k^{2})\\\
\hline\cr{\rm
dominant}\,L=2&1^{--}&3/\sqrt{2}\cdot\left(k^{\perp}_{\mu}\hat{k}^{\perp}-\frac{1}{3}k_{\perp}^{2}\gamma^{\perp}_{\mu}\right)\psi^{(1,2,1)}_{n}(k^{2}).\end{array}$
(9)
Here $k^{\perp}$ is the relative quark–antiquark momentum,
$k^{\perp}_{\mu}=(g_{\mu\mu^{\prime}}-P_{\mu}P_{\mu^{\prime}}/P^{2})k_{1\mu^{\prime}}\equiv
g_{\mu\mu^{\prime}}^{\perp}k_{1\mu^{\prime}}=-g_{\mu\mu^{\prime}}^{\perp}k_{2\mu^{\prime}}$,
so $k^{\perp}\perp P=k_{1}+k_{2}$; likewise,
$\gamma_{\mu}^{\perp}=g_{\mu\mu^{\prime}}^{\perp}\gamma_{\mu^{\prime}}$.
Definition of spin–momentum operators for other states can be found in [2,
28].
Generally, the states with different $L$ mix with each other:
$\displaystyle\hat{\psi}^{V(n,1)}_{\mu}(s)=C_{10}^{(n)}\gamma_{\mu}^{\perp}\psi^{(1,0,1)}_{n}(k^{2})+C_{12}^{(n)}\frac{3}{\sqrt{2}}\left(k^{\perp}_{\mu}\hat{k}^{\perp}-\frac{1}{3}k_{\perp}^{2}\gamma^{\perp}_{\mu}\right)\psi^{(1,2,1)}_{n}(k^{2}),$
(10)
$\displaystyle\hat{\psi}^{V(n,2)}_{\mu}(s)=C_{20}^{(n)}\gamma_{\mu}^{\perp}\psi^{(1,0,1)}_{n}(k^{2})+C_{22}^{(n)}\frac{3}{\sqrt{2}}\left(k^{\perp}_{\mu}\hat{k}^{\perp}-\frac{1}{3}k_{\perp}^{2}\gamma^{\perp}_{\mu}\right)\psi^{(1,2,1)}_{n}(k^{2}).$
But, according to [6], we have with a good accuracy
$C_{12}^{(n)}=C_{20}^{(n)}=0$, so below we put $C_{10}^{(n)}=C_{22}^{(n)}=1$.
We parameterise the $q\bar{q}$ wave functions of $\rho$, $\omega$ states,
$\psi^{(S,L,J)}_{(n)}(k^{2})$, with the following formula:
$\displaystyle\psi^{(S,L,J)}_{(n)}(k^{2})=e^{-\beta|k|^{2}}\sum\limits_{i=1}^{11}c_{i}(S,L,J;n)|k|^{i-1}\,,$
(11)
with cutting parameter $\beta=1.2$ GeV-2. In Eq. (11), we use the notation
$|k|=\sqrt{s/4-m^{2}}$ ($m$ is the mass of the light constituent quark,
$m\simeq 350$ MeV).
The constants $c_{i}(S,L,J;n)$, in GeV units, for mesons with $L=0$,
$\psi^{(S,L=0,J)}_{n}(k^{2})$, and $L=2$, $\psi^{(S,L=2,J)}_{n}(k^{2})$, are
presented in Eq. (9) and (10).
Figure 3: Trajectory for $\rho_{nS}$ and $\omega_{nS}$ states found in [6]
($M_{\rho(nS)}=M_{\omega(nS)}$). Experimental values of the masses on $\rho$\-
and $\omega$-trajectories are equal to: [$\rho_{1S}(775\pm 10)$,
$\rho_{2S}(1460\pm 20)$, $\rho_{3S}(1870\pm 70)$, $\rho_{4S}(2110\pm 35)$] and
[$\omega_{1S}(782)$, $\omega_{2S}(1430\pm 50)$, $\omega_{3S}(\sim 1830)$,
$\omega_{4S}(2205\pm 40)$].
In the solution found in [6], the $\rho_{nS}$ and $\omega_{nS}$ mesons are
degenerated: $M_{\rho(nS)}=M_{\omega(nS)}$, see Fig. 3. Coefficients
$c_{i}(S=1,L=0,J=1;n)$ for $n\leq 4$ (recall that $n$ is radial excitation
number) read:
| $\rho(1S),\omega(1S)$ | $\rho(2S),\omega(2S)$ | $\rho(3S),\omega(3S)$ | $\rho(4S),\omega(4S)$
---|---|---|---|---
$i$ | $\psi_{1}^{(1,0,1)}$ | $\psi_{2}^{(1,0,1)}$ | $\psi_{3}^{(1,0,1)}$ | $\psi_{4}^{(1,0,1)}$
1 | 44.2 | -47.0 | 34.4 | 256.1
2 | 147.9 | 96.4 | 367.3 | -3816.4
3 | -2576.7 | 1694.4 | -6627.1 | 21285.8
4 | 10145.9 | -8835.1 | 31300.6 | -61891.6
5 | -20331.5 | 18954.3 | -72495.7 | 106967.9
6 | 23805.7 | -21715.0 | 95497.7 | -115547.6
7 | -16569.8 | 13585.9 | -73882.6 | 77608.2
8 | 6338.4 | -3952.2 | 31633.5 | -29980.2
9 | -941.1 | 119.3 | -5588.5 | 4927.5
10 | -59.0 | 26.4 | -333.1 | 258.1
11 | -16.0 | 88.7 | 43.2 | -25.9
(12)
For $\rho$ and $\omega$ mesons with dominant $L=2$ we have the following
$c_{i}(S=1,L=2,J=1;n)$:
| $\rho(1D),\omega(1D)$ | $\rho(2D),\omega(2D)$ | $\rho(3D),\omega(3D)$ | $\rho(4D),\omega(4D)$
---|---|---|---|---
$i$ | $\psi_{1}^{(1,2,1)}$ | $\psi_{2}^{(1,2,1)}$ | $\psi_{3}^{(1,2,1)}$ | $\psi_{4}^{(1,2,1)}$
1 | 32.6 | 1.9 | 295.8 | 1109.3
2 | -297.9 | -20.8 | -2587.2 | -9686.9
3 | 1030.3 | 85.0 | 8635.8 | 32404.0
4 | -1720.3 | -207.3 | -13721.7 | -52043.5
5 | 1257.2 | 242.8 | 9530.7 | 36934.5
6 | 68.1 | 4.0 | 206.3 | 1219.6
7 | -702.1 | -203.4 | -4305.9 | -18749.1
8 | 419.2 | 125.4 | 2314.3 | 10789.0
9 | -113.3 | -25.0 | -521.0 | -2650.0
10 | 68.2 | 16.0 | 378.0 | 1715.0
11 | -58.4 | -16.6 | -340.7 | -1533.5
(13)
#### 1.2.2 Pion wave function
For the $\pi(140)$-meson wave function $\psi_{1}^{(0,0,0)}$, the solution
obtained by spectral integral equation is rather satisfactory, it is given by
the coefficients $c_{i}(S=0,L=0,J=0;n)$ which can be found in [6].
Still, the pion can be more precisely described by the wave function found
phenomenologically, using the pion form factor data [37]. The phenomenological
wave function and its parameters are as follows:
$\displaystyle\psi_{\pi}(s)=c_{\pi}\bigg{(}\exp({-b_{1\pi}s})+\beta\exp({-b_{2\pi}s})\bigg{)},$
$\displaystyle c_{\pi}=209.36{\rm GeV}^{-2},\;b_{1\pi}=3.57{\rm
GeV}^{-2},\;b_{2\pi}=0.4{\rm GeV}^{-2},\;\beta=0.01381.$ (14)
It should be noted that the difference between the wave function of Eq.
(1.2.2) and that found in [6] is observed either at rather small relative
momenta ($k^{2}=(s/4-m^{2})<0.1$ GeV2) and or at very large ones.
#### 1.2.3 Pion emission constant
The pion–quark coupling $g_{\pi}$ for the pion emission $q\to\pi+q$ (diagrams
of Fig. 2 type) is given by the quark form factor $g_{\pi(140)\to
q\bar{q}}(s)$ at $s=M^{2}_{\pi}$, namely, $g_{\pi}=g_{\pi(140)\to
q\bar{q}}(s=M^{2}_{\pi})$. However, the spectral integral equation does not
determine the vertices at $s\leq 4m^{2}$, so in our present fit $g_{\pi}$ is a
free parameter.
Describing the widths of $\rho(770)\to\gamma\pi(140)$ and
$\omega(780)\to\gamma\pi(140)$ with the use of vector meson (13) and pion wave
functions (1.2.2), we have found two values for the pion bremsstrahlung
coupling:
$\displaystyle{\rm Solution\,I}:$ $\displaystyle|g_{\pi}|=16.7\pm 0.3\
^{+0.1}_{-2.3}\ ,$ (15) $\displaystyle{\rm Solution\,II}:$
$\displaystyle|g_{\pi}|=3.0\pm 0.3\ ^{+0.1}_{-2.1}\ .$
The pion emission coupling, as is well known, was a subject of investigation
in physics of low-energy pion–nucleon interactions and as well as in nuclear
physics. For the pion–nucleon coupling, which is determined as $g_{\pi
NN}\bigg{(}\bar{\psi}\,^{\prime}_{N}(\vec{\tau}\vec{\varphi}_{\pi})i\gamma_{5}\psi_{N}\bigg{)}$,
the estimations give $g_{\pi NN}^{2}/(4\pi)\simeq 14$ (see, for example, [39,
40, 41] and references therein).
We can turn the description of pion–nucleon vertex into the quark language
using quark model for nucleons:
$g_{\pi NN}\bigg{(}\bar{\psi}\,^{\prime}_{N}(\vec{\tau}\vec{\varphi}_{\pi})\
i\gamma_{5}\psi_{N}\bigg{)}\longrightarrow g_{\pi
qq}\bigg{(}\bar{\psi}\,^{\prime}_{q}(\vec{\tau}\vec{\varphi}_{\pi})\
i\gamma_{5}\psi_{q}\bigg{)}\ ,$ (16)
see Appendix A for more detail. In Eq. (15), we determine the vertex
$u\to\gamma\pi$ which is a part of the quark-language Lagrangian:
$\displaystyle g_{\pi
qq}\bigg{(}\bar{\psi}\,^{\prime}_{q}(\vec{\tau}\vec{\varphi}_{\pi})\
i\gamma_{5}\psi_{q}\bigg{)}$ $\displaystyle\to$ $\displaystyle\sqrt{2}g_{\pi
qq}\,\varphi^{+}_{\pi^{+}}\bigg{(}\bar{\psi}\,^{\prime}_{d}\
i\gamma_{5}\psi_{u}\bigg{)}=g_{\pi}\,\varphi^{+}_{\pi^{+}}\bigg{(}\bar{\psi}\,^{\prime}_{d}\
i\gamma_{5}\psi_{u}\bigg{)},$ $\displaystyle\sqrt{2}g_{\pi qq}$
$\displaystyle=$ $\displaystyle g_{\pi}\,.$ (17)
In Appendix A, we show that, making use of the SU(6)-symmetry for nucleons,
one has $g_{\pi NN}=(5/3)g_{\pi qq}$. So, the SU(6)-symmetry provides us with
$g_{\pi NN}=(5/3\sqrt{2})g_{\pi}$. It means that Solution I does not
contradict the value $g_{\pi NN}^{2}/(4\pi)\simeq 14$ [39, 40, 41], thus
giving us
$16.4\leq g_{\pi NN}^{2}/(4\pi)\leq 23.2\;.$
Note that in (15) we have included systematical errors which are due to
uncertainties in the reconstruction of wave functions in the fit [6].
## 2 Gamma–pion decays of vector states $V\to\gamma\pi$
Here we present formulae which are used below for $\rho\to\gamma\pi$ and
$\omega\to\gamma\pi$ decays.
### 2.1 Polarisation vectors, amplitude and partial width for decays
$V\to\gamma\pi$
Let us introduce notations for the momenta and polarisation vectors and define
the amplitudes and decay partial widths.
#### 2.1.1 Polarisation vectors of the massive vector particle $V$ and photon
Polarisations of the vector meson, $\epsilon^{(V)}_{\mu}$, and of virtual
photon, $\epsilon^{(\gamma^{*})}_{\alpha}$, are the transverse vectors:
$\displaystyle\epsilon^{(V)}_{\beta}p_{\beta}\ =\ 0\
,\qquad\epsilon^{(\gamma^{*})}_{\alpha}q_{\alpha}\ =\ 0\ ,$ (18)
where $q$ is the virtual photon four-momentum ($q^{2}\neq 0$) and $p$ is that
of the vector meson ($p^{2}=M^{2}_{V}$). Polarisation of the vector meson
obeys the completeness condition as follows:
$\displaystyle-\sum_{a=1,2,3}\epsilon^{(V)}_{\mu}(a)\epsilon^{(V)+}_{\mu^{\prime}}(a)\
=\ g^{\perp V}_{\mu\mu^{\prime}}\equiv g^{\perp p}_{\mu\mu^{\prime}}\ ,\qquad
g^{\perp p}_{\mu\mu^{\prime}}\ =\
g_{\mu\mu^{\prime}}-\frac{p_{\mu}p_{\mu^{\prime}}}{p^{2}}\ ,$ (19)
where $g^{\perp p}_{\mu\mu^{\prime}}$ is the metric tensor operating in the
space orthogonal to the momentum $p$.
For virtual photon, $(q^{2}\neq 0)$, the completeness condition for
polarisation vectors is written in three-dimensional space:
$\displaystyle-\sum_{a=1,2,3}\epsilon^{(\gamma^{*})}_{\alpha}(a)\,\epsilon^{(\gamma^{*})+}_{\alpha^{\prime}}(a)=g^{\perp\gamma^{*}}_{\alpha\alpha^{\prime}}\
,\qquad g^{\perp\gamma^{*}}_{\alpha\alpha^{\prime}}\equiv g^{\perp
q}_{\alpha\alpha^{\prime}}=g_{\alpha\alpha^{\prime}}-\frac{q_{\alpha}q_{\alpha^{\prime}}}{q^{2}}\,.$
(20)
The polarisation vector of the real photon $(q^{2}=0)$ denoted as
$\epsilon^{\gamma}_{\alpha}$ has two independent components only, they are
orthogonal to the reaction plane:
$\epsilon^{(\gamma)}_{\alpha}q_{\alpha}=0\
,\qquad\epsilon^{(\gamma)}_{\alpha}p_{\alpha}=0\ .$ (21)
Likewise, the completeness condition for the real photon reads:
$\displaystyle-\sum_{a=1,2}\epsilon^{(\gamma)}_{\alpha}(a)\epsilon^{(\gamma){\bf+}}_{\alpha^{\prime}}(a)\
=\ g^{\perp\perp}_{\alpha\alpha^{\prime}}\ ,$ (22) $\displaystyle
g^{\perp\perp}_{\alpha\alpha^{\prime}}\ =\
g_{\alpha\alpha^{\prime}}-\frac{p_{\alpha}p_{\alpha^{\prime}}}{p^{2}}-\frac{q^{\perp}_{\alpha}q^{\perp}_{\alpha^{\prime}}}{q^{2}_{\perp}},\qquad
q^{\perp}_{\alpha}\equiv g^{\perp
V}_{\alpha\alpha^{\prime}}q_{\alpha^{\prime}}=q_{\alpha}-\frac{(pq)}{p^{2}}\,p_{\alpha}\
.$
#### 2.1.2 Amplitude for the decay $V\to\gamma\pi$
The decay amplitude $V\to\gamma\pi$ is written as a product of the spin
structure and form factor:
$\displaystyle A_{V\to\gamma\pi}$ $\displaystyle=$
$\displaystyle\epsilon_{\alpha}^{(\gamma)}\epsilon_{\mu}^{(V)}A^{(V\to\gamma\pi)}_{\alpha\mu}\
,$ $\displaystyle A^{(V\to\gamma\pi)}_{\alpha\mu}$ $\displaystyle=$
$\displaystyle
e\,S^{(V\to\gamma\pi)}_{\alpha\mu}(p,q)F^{V\to\gamma\pi}(0,M^{2}_{\pi})\ ,$
(23)
with
$\displaystyle S^{(V\to\gamma\pi)}_{\alpha\mu}(p,q)\ =\ \varepsilon_{\alpha\mu
pq}\equiv\varepsilon_{\alpha\mu\nu_{1}\nu_{2}}p_{\nu_{1}}q_{\nu_{2}}\ .$ (24)
In (23), the electron charge $e$ is singled out, and in (24) the tensor
$\varepsilon_{\alpha\mu\nu_{1}\nu_{2}}$ is the wholly antisymmetrical. Let us
emphasise the specific role of the spin operator $\varepsilon_{\alpha\mu pq}$.
Since $\varepsilon_{\alpha\mu pp}=0$, this spin operator is valid for the
reaction with both real ($\gamma$) and virtual ($\gamma^{*}$) photons, so Eq.
(23) can be used for the transition with virtual photon, with corresponding
substitution: $F^{V\to\gamma\pi}(0)\to F^{V\to\gamma\pi}(q^{2})$.
#### 2.1.3 Partial width for $V\to\gamma\pi$
The partial width for the decay $V\to\gamma\pi$ is determined as follows:
$\displaystyle M_{V}\Gamma_{V\to\gamma\pi}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\int
d\Phi_{2}(p;q,p_{\pi})\left|\sum_{\alpha\mu}A^{(V\to\gamma\pi)}_{\alpha\mu}\right|^{2}\
=$ $\displaystyle=$
$\displaystyle\frac{\alpha}{24}\frac{(M_{V}^{2}-M_{\pi}^{2})^{3}}{M_{V}^{2}}\
|F^{V\to\gamma\pi}(0,M^{2}_{\pi})|^{2}\ ,$ $\displaystyle
d\Phi_{2}(p;q,p_{\pi})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{d^{3}q}{(2\pi)^{3}\,2q_{0}}\frac{d^{3}p_{\pi}}{(2\pi)^{3}\,2p_{\pi
0}}(2\pi)^{4}\delta^{(4)}(p-q-p_{\pi})\ .$ (25)
The summation is carried out over the photon and vector meson polarisation s,
and $(\varepsilon_{\alpha\mu pq})^{2}=(M_{V}^{2}-M_{\pi}^{2})^{2}/2$. In the
final expression $\alpha=e^{2}/4\pi=1/137$.
### 2.2 Double spectral integral representation of the triangle diagrams with
photon emission
To derive double spectral integral for the form factors with photon emission
by quark and antiquark,
$F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(q^{2})$ and
$F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(q^{2})$, see Fig. 1, one
needs to calculate the double discontinuities of the triangle diagrams.
#### 2.2.1 Double discontinuities of the triangle diagrams
First, consider the photon emission by quark, see Fig. 1a. Corresponding
cuttings for the calculation of double discontinuity are shown in Fig. 1b.
In the dispersion representation, the invariant energy in the intermediate
state differs from that in the initial and final states. Because of that, at
the double discontinuity $P\neq p$ and $P^{\prime}\neq p_{\pi}$. The following
requirements are imposed on the momenta shown in the diagram of Fig. 1b [23,
38]:
$\displaystyle(k_{1}+k_{2})^{2}\ =\ P^{2}\equiv s>4m^{2}\
,\qquad(k^{\prime}_{1}+k_{2})^{2}\ =\ P^{\prime 2}\equiv s^{\prime}>4m^{2}\ .$
(26)
The momentum squared of the photon, $q^{2}$, is fixed:
$\displaystyle(p-p_{\pi})^{2}=(P-P^{\prime})^{2}\ =\
(k_{1}-k_{1}^{\prime})^{2}\ =\ q^{2}\ .$ (27)
When cutting Feynman diagram, the propagators should be substituted by the
residues in the poles. This is equivalent to the replacement as follows:
$(m^{2}-k^{2}_{1})^{-1}\to\delta(m^{2}-k^{2}_{1})$,
$(m^{2}-k^{2}_{2})^{-1}\to\delta(m^{2}-k^{2}_{2})$ and $(m^{2}-k^{\prime
2}_{1})^{-1}\to\delta(m^{2}-k^{\prime 2}_{1})$, so the intermediate-state
quarks are mass-on-shell:
$\displaystyle k^{2}_{1}=k^{2}_{2}=k^{\prime 2}_{1}=m^{2}.$ (28)
Then, for the diagram with photon emitted by quark (Fig. 1a), the double
discontinuity of the amplitude (Fig. 1b) becomes proportional to the three
factors:
$\displaystyle{\rm disc}_{s}{\rm
disc}_{s^{\prime}}A^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangleup^{\gamma})\sim
Z^{V\to\gamma\pi}_{\bigtriangleup^{\gamma}}G_{V(L)}(s)G_{\pi}(s^{\prime})$
$\displaystyle\times
d\Phi_{2}(P;k_{1},k_{2})d\Phi_{2}(P^{\prime};k^{\prime}_{1},k^{\prime}_{2})(2\pi)^{3}2k_{20}\delta^{3}(\vec{k}^{\prime}_{2}-\vec{k}_{2})$
$\displaystyle\times{\rm
Sp}\left[Q^{V(L)}_{\mu}(k)(\hat{k}_{1}+m)Q^{(\gamma)}_{\alpha}(\hat{k}^{\prime}_{1}+m)Q^{(\pi)}(-\hat{k}_{2}+m)\right]\
.$ (29)
The first factor in the right-hand side of (2.2.1) consists of the following
vertices: the quark charge factor
$Z^{V\to\gamma\pi}_{\bigtriangleup^{\gamma}}$ as well as transition vertices
$V(L)\to q\bar{q}$ and $\pi\to q\bar{q}$ which are denoted as $G_{V(L)}(s)$
and $G_{\pi}(s^{\prime})$.
The second factor contains space volumes of the two-particle states,
$d\Phi_{2}(P;k_{1},k_{2})$ and
$d\Phi_{2}(P^{\prime};k^{\prime}_{1},k^{\prime}_{2})$, which correspond to two
cuts shown in the diagram of Fig. 1b (the space volume is determined in
(2.1.3)). The factor
$(2\pi)^{3}2k_{20}\delta^{3}(\vec{k}^{\prime}_{2}-\vec{k}_{2})$ takes into
account the fact that one quark line is cut twice.
The third factor in (2.2.1) is the trace coming from the summation over the
quark spin states. Since the spin factor in the transition $V\to q\bar{q}$ may
be of two types (with dominant $S$\- or dominant $D$-wave), we have the
following operators for virtual photon, $Q^{V(L)}_{\mu}$, see Eq. (9):
$\displaystyle Q^{V(L=0)}_{\mu}(k)=\gamma^{\perp V}_{\mu}=\gamma^{\perp
P}_{\mu}\equiv g^{\perp P}_{\mu\mu^{\prime}}\gamma_{\mu^{\prime}},$
$\displaystyle Q^{V(L=2)}_{\mu}(k)\ =\
\sqrt{2}\gamma_{\mu^{\prime}}X^{(2)}_{\mu^{\prime}\mu}(k)=\frac{3}{\sqrt{2}}\left[k_{\mu}\hat{k}-\frac{1}{3}k^{2}\gamma^{\perp
P}_{\mu}\right]\,\ ,$ (30)
and for the pion:
$Q^{(\pi)}=i\gamma_{5}\ .$ (31)
Here, $k=(k_{1}-k_{2})/2$ is the relative momentum of the incoming quarks,
$k\perp P=k_{1}+k_{2}$, i.e. $k=k_{1}^{\perp P}=-k_{2}^{\perp P}$.
For real photon, we replace:
$\displaystyle Q^{(\gamma)}_{\alpha}\to
Q^{\perp\perp}_{\alpha}\equiv\gamma^{\perp\perp}_{\alpha}(P,P^{\prime})=g^{\perp\perp}_{\alpha\alpha^{\prime}}(P,P^{\prime})\gamma_{\alpha^{\prime}}\
,$ $\displaystyle
g^{\perp\perp}_{\alpha\alpha^{\prime}}(P,P^{\prime})P_{\alpha^{\prime}}=0,\quad
g^{\perp\perp}_{\alpha\alpha^{\prime}}(P,P^{\prime})P^{\prime}_{\alpha^{\prime}}=0\
,$ (32)
where $(P-P^{\prime})^{2}=0$. The metric tensor
$g^{\perp\perp}_{\alpha\alpha^{\prime}}(P,P^{\prime})$ works in the space
orthogonal to the intermediate state momenta:
$g^{\perp\perp}_{\alpha\alpha^{\prime}}(P,P^{\prime})=g_{\alpha\alpha^{\prime}}-P_{\alpha}P_{\alpha^{\prime}}/P^{2}-P^{\prime\perp
P}_{\alpha}P^{\prime\perp P}_{\alpha^{\prime}}/P^{\prime\perp
P}_{\alpha^{\prime\prime}}P^{\prime\perp P}_{\alpha^{\prime\prime}}$.
Actually, for the real photon we can use simpler oprator, say,
$Q^{(\gamma)}_{\alpha}=\gamma_{\alpha}^{\perp}$, because in the considered
decay we should have the same result for both choices, $Q^{(\gamma)}_{\alpha}$
or $Q^{\perp\perp}_{\alpha}$, due to the spin operator structure (24).
However, here we use (2.2.1) to emphasise an important point for this type of
reactions: the amplitude for transversely polarized photons is determined by
the spectral integral with transversely polarized photons in the intermediate
states as well.
For the photon emission, there are two diagrams: the second one is similar to
that of Fig. 1a but with the emission of photon by antiquark, it is shown in
Fig. 1c. The double discontinuity of the corresponding amplitude is determined
by cuttings shown in Fig. 1d:
$\displaystyle{\rm disc}_{s}{\rm
disc}_{s^{\prime}}A^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangledown_{\gamma})\sim
Z_{V\to\gamma\pi}(\bigtriangledown_{\gamma})G_{V(L)}(s)G_{\pi}(s^{\prime})$
$\displaystyle\times
d\Phi_{2}(P;k_{1},k_{2})d\Phi_{2}(P^{\prime};k^{\prime}_{1},k^{\prime}_{2})(2\pi)^{3}2k_{10}\delta^{3}(\vec{k}^{\prime}_{1}-\vec{k}_{1})$
$\displaystyle\times{\rm
Sp}\left[Q^{V(L)}_{\mu}(k)(\hat{k}_{1}+m)Q^{(\pi)}(-\hat{k}^{\prime}_{2}+m)Q^{(\gamma)}_{\alpha}(-\hat{k}_{2}+m)\right]\
.$ (33)
Likewise, there are two traces for two transitions with photon emission by
quark and antiquark:
$\displaystyle
Sp^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangleup^{\gamma})=-{\rm
Sp}\left[Q^{V(L)}_{\mu}(k)(\hat{k}_{1}+m)Q^{(\gamma)}_{\alpha}(\hat{k}^{\prime}_{1}+m)Q^{(\pi)}(-\hat{k}_{2}+m)\right]\
,$ $\displaystyle
Sp^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangledown_{\gamma})=-{\rm
Sp}\left[Q^{V(L)}_{\mu}(k)(\hat{k}_{1}+m)Q^{(\pi)}(-\hat{k}^{\prime}_{2}+m)Q^{(\gamma)}_{\alpha}(-\hat{k}_{2}+m)\right].$
To calculate the invariant form factors
$F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(0)$ and
$F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(0)$, we should extract from
(2.2.1) the intermediate-state spin operator:
$\displaystyle S^{(V\to\gamma\pi)}_{\alpha\mu}(P,\widetilde{q})\ =\
\varepsilon_{\alpha\mu P\widetilde{q}}\ ,\qquad\widetilde{q}=P-P^{\prime}\ .$
(35)
Therefore, we have:
$\displaystyle Sp^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangleup^{\gamma})$
$\displaystyle=$ $\displaystyle
S^{(V\to\gamma\pi)}_{\alpha\mu}(P,\widetilde{q})S^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(s,s^{\prime},q^{2})\
,$ $\displaystyle
Sp^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangledown_{\gamma})$ $\displaystyle=$
$\displaystyle
S^{(V\to\gamma\pi)}_{\alpha\mu}(P,\widetilde{q})S^{V(L)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(s,s^{\prime},q^{2})\
,$ (36)
where
$\displaystyle\frac{\left(Sp^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangleup^{\gamma})S^{(V\to\gamma\pi)}_{\alpha\mu}(P,\widetilde{q})\right)}{\left(S^{(V\to\gamma\pi)}_{\alpha\mu}(P,\widetilde{q})\right)^{2}}$
$\displaystyle=$ $\displaystyle
S^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(s,s^{\prime},q^{2})\ ,$
$\displaystyle\frac{\left(Sp^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangledown_{\gamma})S^{(V\to\gamma\pi)}_{\alpha\mu}(P,\widetilde{q})\right)}{\left(S^{(V\to\gamma\pi)}_{\alpha\mu}(P,\widetilde{q})\right)^{2}}$
$\displaystyle=$ $\displaystyle
S^{V(L)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(s,s^{\prime},q^{2})\ .$ (37)
Taking into account the expression ${\rm
Sp}[\gamma_{5}\gamma_{\alpha_{1}}\gamma_{\alpha_{2}}\gamma_{\alpha_{3}}\gamma_{\alpha_{4}}]=4i\varepsilon_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}$
we obtain:
$\displaystyle
S^{V(L=0)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(s,s^{\prime},q^{2})=S^{V(0)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(s,s^{\prime},q^{2})=-4m\
,$ (38) $\displaystyle
S^{V(L=2)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(s,s^{\prime},q^{2})=S^{V(2)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(s,s^{\prime},q^{2})=-\frac{m}{\sqrt{2}}\left[2m^{2}+\\!s+\\!\frac{6ss^{\prime}q^{2}}{\lambda(s,s^{\prime},q^{2})}\right],$
with
$\displaystyle\lambda(s,s^{\prime},q^{2})=(s-s^{\prime})^{2}-2q^{2}(s+s^{\prime})+q^{4}.$
(39)
The photon emission amplitude, being determined by two diagrams of Fig. 1a and
Fig. 1c, reads
$\displaystyle
A^{V(L)\to\gamma\pi}_{(\bigtriangleup^{\gamma}+\bigtriangledown_{\gamma})\alpha\mu}\\!=\\!e\,\varepsilon_{\alpha\mu
pq}\\!\left[Z^{V\to\gamma\pi}_{\bigtriangleup^{\gamma}}\\!F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(q^{2},M^{2}_{\pi})\\!+\\!Z^{V\to\gamma\pi}_{\bigtriangledown_{\gamma}}F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(q^{2},M^{2}_{\pi})\right]\\!,$
(40)
while the double discontinuities of the form factors in (40) are equal to:
$\displaystyle{\rm disc}_{s}{\rm
disc}_{s^{\prime}}F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(q^{2},M^{2}_{\pi})=G_{V(L)}(s)G_{\pi}(s^{\prime})$
$\displaystyle\times
d\Phi_{2}(P;k_{1},k_{2})d\Phi_{2}(P^{\prime};k^{\prime}_{1},k^{\prime}_{2})(2\pi)^{3}2k_{20}\delta^{3}(\vec{k}^{\prime}_{2}-\vec{k}_{2})S^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(s,s^{\prime},q^{2}),$
$\displaystyle{\rm disc}_{s}{\rm
disc}_{s^{\prime}}F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(q^{2},M^{2}_{\pi})=G_{V(L)}(s)G_{\pi}(s^{\prime})$
$\displaystyle\times
d\Phi_{2}(P;k_{1},k_{2})d\Phi_{2}(P^{\prime};k^{\prime}_{1},k^{\prime}_{2})(2\pi)^{3}2k_{10}\delta^{3}(\vec{k}^{\prime}_{1}-\vec{k}_{1})S^{V(L)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(s,s^{\prime},q^{2}).$
(41)
#### 2.2.2 The double spectral integral for the form factors with photon
emission by quark and antiquark
The equation (2.2.1) defines the form factor through the dispersion integral
as follows:
$\displaystyle
F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(q^{2},M^{2}_{\pi})=\int\limits^{\infty}_{4m^{2}}\frac{ds}{\pi}\int\limits^{\infty}_{4m^{2}}\frac{ds^{\prime}}{\pi}\frac{{\rm
disc}_{s}{\rm
disc}_{s^{\prime}}F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(q^{2},M^{2}_{\pi})}{(s-M^{2}_{V(L)})(s^{\prime}-M^{2}_{\pi})}\
,$ $\displaystyle
F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(q^{2},M^{2}_{\pi})=\int\limits^{\infty}_{4m^{2}}\frac{ds}{\pi}\int\limits^{\infty}_{4m^{2}}\frac{ds^{\prime}}{\pi}\frac{{\rm
disc}_{s}{\rm
disc}_{s^{\prime}}F^{V(L)\to\gamma\pi}_{\bigtriangledown^{\gamma}}(q^{2},M^{2}_{\pi})}{(s-M^{2}_{V(L)})(s^{\prime}-M^{2}_{\pi})}\
.$ (42)
We have
$F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(q^{2},M^{2}_{\pi})=F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(q^{2},M^{2}_{\pi})$
(43)
at equal masses of the quark and antiquark – just this case is considered
here. In (42), we omit subtraction terms, assuming that the convergence of
(42) is guaranteed by the vertices $G_{V(L)}(s)$ and $G_{\pi}(s^{\prime})$.
Furthermore, we define the wave functions of the $q\bar{q}$ systems:
$\psi_{V(L)}(s)=G_{V(L)}(s)/(s-M_{V(L)}^{2})$ and
$\psi_{\pi}(s^{\prime})=G_{\pi}(s^{\prime})/(s^{\prime}-M_{\pi}^{2})$.
After integrating over the momenta in accordance with (2.2.1), one can
represent (42) in the following form:
$\displaystyle
F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(q^{2},M^{2}_{\pi})$
$\displaystyle=$ $\displaystyle
F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(q^{2},M^{2}_{\pi})=\int\limits_{4m^{2}}^{\infty}\frac{dsds^{\prime}}{16\pi^{2}}\psi_{V(L)}(s)\psi_{\pi}(s^{\prime})$
(44) $\displaystyle\times$
$\displaystyle\frac{\Theta\left(-ss^{\prime}q^{2}-m^{2}\lambda(s,s^{\prime},q^{2})\right)}{\sqrt{\lambda(s,s^{\prime},q^{2})}}S^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(s,s^{\prime},q^{2})\
,$
where $\Theta(X)$ is the step-function: $\Theta(X)=1$ at $X\geq 0$ and
$\Theta(X)=0$ at $X<0$.
#### 2.2.3 Z-factors for photon emission
For the $\rho^{+}$ meson, the photon emission is determined by two diagrams,
see Figs. 4a and 4b, which give us the following charge factors:
$Z^{\rho^{+}\to\gamma\pi^{+}}_{\bigtriangleup^{\gamma}}=e_{u}=\frac{2}{3},\qquad
Z^{\rho^{+}\to\gamma\pi^{+}}_{\bigtriangledown_{\gamma}}=e_{d}=-\frac{1}{3}\
.$ (45)
Figure 4: Diagrams for the determination of Z-factors in the reaction
$\rho^{+}\to\gamma\pi^{+}$ with photon emission .
For neutral vector mesons ($\rho^{0}$, $\omega$), we have four diagrams, see
Fig. 5, which result in the charge factors as follows:
$\displaystyle
Z^{\rho^{0}\to\gamma\pi^{0}}_{\bigtriangleup^{\gamma}}=Z^{\rho^{0}\to\gamma\pi^{0}}_{\bigtriangledown_{\gamma}}=\frac{1}{2}(e_{u}+e_{d})=\frac{1}{6}\
,$ $\displaystyle
Z^{\omega\to\gamma\pi^{0}}_{\bigtriangleup^{\gamma}}=Z^{\omega\to\gamma\pi^{0}}_{\bigtriangledown_{\gamma}}=\frac{1}{2}(e_{u}-e_{d})=\frac{1}{2}\
.$ (46)
In (2.2.3), we use the standard flavour wave functions for $(I=1,I_{3}=0)$ and
$(I=0,I_{3}=0)$ states: $\rho^{0}=\pi^{0}=(u\bar{u}-d\bar{d})/\sqrt{2}$ and
$\omega=(u\bar{u}+d\bar{d})/\sqrt{2}$.
Figure 5: Diagrams for the determination of Z-factors in the reactions
$\rho^{0}\to\gamma\pi^{0}$ and $\omega^{0}\to\gamma\pi^{0}$ with photon
emission.
#### 2.2.4 Decay form factors at $Q^{2}=-q^{2}\to 0$
To calculate the integral at small $Q^{2}$, we substitute:
$s=\Sigma+\frac{1}{2}zQ,\quad s^{\prime}=\Sigma-\frac{1}{2}zQ,\quad
q^{2}=-Q^{2}\,.$ (47)
In the region $Q^{2}\ll 4m^{2}$, the form factors (44) can be written as
$\displaystyle
F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(-Q^{2},M^{2}_{\pi})$
$\displaystyle=$ $\displaystyle
F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(-Q^{2},M^{2}_{\pi})=\int\limits_{4m^{2}}^{\infty}\frac{d\Sigma}{\pi}\psi_{V(L)}(\Sigma)\psi_{\pi}(\Sigma)$
$\displaystyle\times$
$\displaystyle\int\limits_{-b}^{+b}\frac{dz}{\pi}\;\frac{S^{V(L)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(\Sigma+\frac{1}{2}zQ,\Sigma-\frac{1}{2}zQ,-Q^{2})}{16\sqrt{\Lambda(\Sigma,z,Q^{2})}}\
,$
$b=\sqrt{\Sigma(\frac{\Sigma}{m^{2}}-4)},\qquad\Lambda(\Sigma,z,Q^{2})=(z^{2}+4\Sigma)Q^{2}\
.$ (48)
After integrating over $z$ and substituting $\Sigma\to s$, the form factors
for $L=0,2$ read:
$\displaystyle F^{V(0)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(0,M^{2}_{\pi})$
$\displaystyle=$ $\displaystyle
F^{V(0)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(0,M^{2}_{\pi})=-4m\int\limits_{4m^{2}}^{\infty}\frac{ds}{16\pi^{2}}\psi_{\pi}(s)\psi_{V(0)}(s)$
$\displaystyle\times$
$\displaystyle\ln{\frac{s+\sqrt{s(s-4m^{2})}}{s-\sqrt{s(s-4m^{2})}}}\ ,$
$\displaystyle F^{V(2)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(0,M^{2}_{\pi})$
$\displaystyle=$ $\displaystyle
F^{V(2)\to\gamma\pi}_{\bigtriangledown_{\gamma}}(0,M^{2}_{\pi})=-m/\sqrt{2}\int\limits_{4m^{2}}^{\infty}\frac{ds}{4\pi^{2}}\psi_{\pi}(s)\psi_{V(2)}(s)$
(49) $\displaystyle\times$
$\displaystyle\left[(2m^{2}+s)\ln\frac{\sqrt{s}+\sqrt{s-4m^{2}}}{\sqrt{s}-\sqrt{s-4m^{2}}}+3\sqrt{s(s-4m^{2})}\right]\\!.$
Remind that wave functions $\psi_{V(0)}(s)=\psi^{(1,0,1)}_{n}(k^{2})$,
$\psi_{V(2)}(s)=\psi^{(1,2,1)}_{n}(k^{2})$ and $\psi_{\pi}(s)$ are presented
in Section I.
#### 2.2.5 Normalisation conditions for the wave functions $\psi_{\pi}(s)$
and $\psi_{V(L=0,2)}(s)$
It is convenient to write the normalisation conditions for $\psi_{\pi}(s)$ and
$\psi_{V(L)}(s)$ using the charge form factor of a meson:
$\displaystyle F_{charge}(0)\ =\ 1\ .$ (50)
The amplitude of the charge factor is defined by the photon-emission triangle
diagram with $(q\bar{q})_{in}=(q\bar{q})_{out}$. For the pion, the amplitude
takes the form:
$\displaystyle A_{\alpha}(q)\ =\ e(p+p_{\pi})_{\alpha}F_{charge}(q^{2})\ ,$
(51)
while $F_{charge}(q^{2})$ can be calculated in the same way as the transition
form factors considered above. The normalisation condition for pion reads:
$\displaystyle 1$ $\displaystyle=$
$\displaystyle\int\limits_{4m^{2}}^{\infty}\frac{ds}{16\pi^{2}}\
\psi_{\pi}^{2}(s)\ 2s\ \sqrt{\frac{s-4m^{2}}{s}}\ .$ (52)
For vector meson $V(L)$, the normalisation condition may be determined by
averaging over spins of the massive vector particle, see [2, 3, 42, 43] for
detail. Then, the normalisation condition reads:
$\displaystyle 1$ $\displaystyle=$
$\displaystyle\frac{1}{3}\int\limits_{4m^{2}}^{\infty}\frac{ds}{16\pi^{2}}\
\psi_{V(0)}^{2}(s)\ 4\left(s+2m^{2}\right)\sqrt{\frac{s-4m^{2}}{s}}\ ,$
$\displaystyle 1$ $\displaystyle=$
$\displaystyle\frac{1}{3}\int\limits_{4m^{2}}^{\infty}\frac{ds}{16\pi^{2}}\
\psi_{V(2)}^{2}(s)\
\frac{(8m^{2}+s)(s-4m^{2})^{2}}{8}\sqrt{\frac{s-4m^{2}}{s}}\ .$ (53)
Recall that here $\psi_{V(0)}(s)=\psi^{(1,0,1)}_{n}(k^{2})$ and
$\psi_{V(2)}(s)=\psi^{(1,2,1)}_{n}(k^{2})$ with $k^{2}=s/4\,-m^{2}$ .
#### 2.2.6 Vector mesons: normalisation condition in case of two-component
wave functions
In the solution found in [6], the wave functions $\psi_{V(0)}$ and
$\psi_{V(2)}$ are orthogonal to each other with a good accuracy. Generally,
vector states may mix. Then the vector mesons have two-component wave
functions, see (10), and normalisation condition reads:
$\displaystyle\delta_{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\int\limits_{4m^{2}}^{\infty}\frac{ds}{16\pi^{2}}\
C_{a0}^{(n)}C_{b0}^{(n)}\bigg{(}\psi^{(1,0,1)}_{n}(k^{2})\bigg{)}^{2}4\left(s+2m^{2}\right)\sqrt{\frac{s-4m^{2}}{s}}\
$ (54) $\displaystyle+$
$\displaystyle\frac{1}{3}\int\limits_{4m^{2}}^{\infty}\frac{ds}{16\pi^{2}}\
\bigg{(}C_{a0}^{(n)}C_{b2}^{(n)}+C_{b0}^{(n)}C_{a2}^{(n)}\bigg{)}\psi^{(1,0,1)}_{n}(k^{2})\psi^{(1,2,1)}_{n}(k^{2})$
$\displaystyle\times\sqrt{2}\
\frac{(s-4m^{2})^{2}}{6}\sqrt{\frac{s-4m^{2}}{s}}\ $ $\displaystyle+$
$\displaystyle\frac{1}{3}\int\limits_{4m^{2}}^{\infty}\frac{ds}{16\pi^{2}}\
C_{a2}^{(n)}C_{b2}^{(n)}\bigg{(}\psi^{(1,2,1)}_{n}(k^{2})\bigg{)}^{2}\ $
$\displaystyle\times\frac{(8m^{2}+s)(s-4m^{2})^{2}}{8}\sqrt{\frac{s-4m^{2}}{s}}\
.$
## 3 Double spectral integral representation of the triangle diagrams with
pion emission
Here, we calculate the double spectral integral for the transition form
factors with the emission of pion by quark,
$F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,M^{2}_{\pi})$ (diagram of Fig.
2a) and antiquark,
$F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\pi}}(0,M^{2}_{\pi})$ (diagram of
Fig. 2c).
#### 3.0.1 Double discontinuities of the triangle diagrams
For the diagram of Fig. 2a, the cuttings are shown in Fig. 2b, with the
following notations:
$\displaystyle k^{2}_{1}=k^{2}_{2}=k^{\prime 2}_{1}=m^{2},$
$\displaystyle(k_{1}+k_{2})^{2}\ =\ P^{2}\equiv s>4m^{2}\
,\qquad(k^{\prime}_{1}+k_{2})^{2}\ =\ P^{\prime 2}\equiv s^{\prime}>4m^{2},$
$\displaystyle(P-P^{\prime})^{2}\ =\ (k_{1}-k^{\prime}_{1})^{2}\ =\
p_{\pi}^{2}=M^{2}_{\pi}\ .$ (55)
For the diagram of Fig. 2a, the double discontinuity, determined by Fig. 2b,
contains three factors:
$\displaystyle{\rm disc}_{s}{\rm
disc}_{s^{\prime}}A^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangleup^{\pi})\sim
Z_{V\to\gamma\pi}(\bigtriangleup^{\pi})g_{\pi}G_{V(L)}(s)G_{\gamma}(s^{\prime})$
$\displaystyle\times
d\Phi_{2}(P;k_{1},k_{2})d\Phi_{2}(P^{\prime};k^{\prime}_{1},k^{\prime}_{2})(2\pi)^{3}2k_{20}\delta^{3}(\vec{k}^{\prime}_{2}-\vec{k}_{2})$
$\displaystyle\times{\rm
Sp}\left[Q^{V(L)}_{\mu}(k)(\hat{k}_{1}+m)Q^{(\pi)}(\hat{k}^{\prime}_{1}+m)Q^{(\gamma_{\perp})}_{\alpha}(-\hat{k}_{2}+m)\right]\
.$ (56)
The right-hand side of (3.0.1) is determined by the the quark charge factor
$Z_{V\to\gamma\pi}(\bigtriangleup^{\pi})$, the transition vertices $V(L)\to
q\bar{q}$ and $\gamma\to q\bar{q}$ and pion–quark coupling $g_{\pi}$. The
trace in (3.0.1) contains the operators $Q^{(\pi})$ and
$Q^{(\gamma_{\perp})}_{\alpha}$ which are determined in (2.2.1):
$Q^{(\gamma_{\perp})}_{\alpha}=\gamma^{\perp\perp}_{\alpha}(P,P^{\prime})$ and
$Q^{(\pi)}=i\gamma_{5}$.
The diagram with the emission of pion by antiquark is shown in Fig. 2c. The
double discontinuity of the corresponding amplitude, Fig. 2d, is written
similarly to (3.0.1). We have:
$\displaystyle{\rm disc}_{s}{\rm
disc}_{s^{\prime}}A^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangledown_{\pi})\sim
Z_{V\to\gamma\pi}(\bigtriangledown_{\pi})g_{\pi}G_{V(L)}(s)G_{\gamma}(s^{\prime})$
$\displaystyle\times
d\Phi_{2}(P;k_{1},k_{2})d\Phi_{2}(P^{\prime};k^{\prime}_{1},k^{\prime}_{2})(2\pi)^{3}2k_{10}\delta^{3}(\vec{k}^{\prime}_{1}-\vec{k}_{1})$
$\displaystyle\times{\rm
Sp}\left[Q^{V(L)}_{\mu}(k)(\hat{k}_{1}+m)Q^{(\gamma_{\perp})}_{\alpha}(-\hat{k}^{\prime}_{2}+m)Q^{(\pi)}(-\hat{k}_{2}+m)\right]\
.$ (57)
Correspondingly, we have two traces for two transitions with pion emission by
the quark and antiquark:
$\displaystyle Sp^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangleup^{\pi})$
$\displaystyle=$ $\displaystyle\\!-\\!{\rm
Sp}\left[Q^{V(L)}_{\mu}(k)(\hat{k}_{1}+m)Q^{(\pi)}(\hat{k}^{\prime}_{1}+m)Q^{(\gamma_{\perp})}_{\alpha}(-\hat{k}_{2}+m)\right]$
$\displaystyle=$ $\displaystyle
S^{(V\to\gamma\pi)}_{\alpha\mu}(P,P-P^{\prime})S^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(s,s^{\prime},(P-P^{\prime})^{2})\
,$ $\displaystyle Sp^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangledown_{\pi})$
$\displaystyle=$ $\displaystyle\\!-\\!{\rm
Sp}\left[Q^{V(L)}_{\mu}(k)(\hat{k}_{1}\\!+\\!m)Q^{(\gamma_{\perp})}_{\alpha}(-\hat{k}^{\prime}_{2}+m)Q^{(\pi)}(-\hat{k}_{2}+m)\right]$
(58) $\displaystyle=$ $\displaystyle
S^{(V\to\gamma\pi)}_{\alpha\mu}(P,P-P^{\prime})S^{V(L)\to\gamma\pi}_{\bigtriangledown_{\pi}}(s,s^{\prime},(P-P^{\prime})^{2})\,,$
$\displaystyle
S^{(V\to\gamma\pi)}_{\alpha\mu}(P,P-P^{\prime})=\varepsilon_{\alpha\mu
P(P-P^{\prime})}\ =\ -\varepsilon_{\alpha\mu PP^{\prime}}\ .$
Here,
$\displaystyle\frac{\left(Sp^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangleup^{\pi})S^{(V\to\gamma\pi)}_{\alpha\mu}(P,P-P^{\prime})\right)}{\left(S^{(V\to\gamma\pi)}_{\alpha\mu}(P,P-P^{\prime})\right)^{2}}$
$\displaystyle=$ $\displaystyle
S^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(s,s^{\prime},(P-P^{\prime})^{2})\
,$
$\displaystyle\frac{\left(Sp^{V(L)\to\gamma\pi}_{\alpha\mu}(\bigtriangledown_{\pi})S^{(V\to\gamma\pi)}_{\alpha\mu}(P,P-P^{\prime})\right)}{\left(S^{(V\to\gamma\pi)}_{\alpha\mu}(P,P-P^{\prime})\right)^{2}}$
$\displaystyle=$ $\displaystyle
S^{V(L)\to\gamma\pi}_{\bigtriangledown_{\pi}}(s,s^{\prime},(P-P^{\prime})^{2})\
.$ (59)
As a result, we obtain:
$\displaystyle
S^{V(0)\to\gamma\pi}_{\bigtriangleup^{\pi}}(s,s^{\prime},(P-P^{\prime})^{2})$
$\displaystyle=$ $\displaystyle
S^{V(0)\to\gamma\pi}_{\bigtriangledown_{\pi}}(s,s^{\prime},(P-P^{\prime})^{2})=4m\
,$ $\displaystyle
S^{V(2)\to\gamma\pi}_{\bigtriangleup^{\pi}}(s,s^{\prime},(P-P^{\prime})^{2})$
$\displaystyle=$ $\displaystyle
S^{V(2)\to\gamma\pi}_{\bigtriangledown_{\pi}}(s,s^{\prime},(P-P^{\prime})^{2})$
(60) $\displaystyle=$
$\displaystyle\frac{m}{\sqrt{2}}\left[2m^{2}+s+\frac{6ss^{\prime}(P-P^{\prime})^{2}}{\lambda(s,s^{\prime},(P-P^{\prime})^{2})}\right]\
.$
Let us note that spin factors
$S^{V(0)\to\gamma\pi}_{\bigtriangleup^{\pi}}(s,s^{\prime},(P-P^{\prime})^{2})$
and
$S^{V(2)\to\gamma\pi}_{\bigtriangleup^{\pi}}(s,s^{\prime},(P-P^{\prime})^{2})$
differ by the sign only from those for photon emission
$S^{V(0)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(s,s^{\prime},q^{2})$ and
$S^{V(2)\to\gamma\pi}_{\bigtriangleup^{\gamma}}(s,s^{\prime},q^{2})$, given by
(38). The pion emission amplitude, considered as a function of $q^{2}$ and
$p_{\pi}^{2}$, is determined by two processes (Figs. 2a, 2c):
$\displaystyle
A^{(V(L)\to\gamma\pi)}_{\alpha\mu}(\bigtriangleup^{\pi}+\bigtriangledown_{\pi})=e\,\varepsilon_{\alpha\mu
pq}$
$\displaystyle\times\left[Z^{V\to\gamma\pi}_{\bigtriangleup^{\pi}}F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(q^{2},p_{\pi}^{2})+Z^{V\to\gamma\pi}_{\bigtriangledown_{\pi}}F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\pi}}(q^{2},p_{\pi}^{2})\right],$
(61)
with
$F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(q^{2},p_{\pi}^{2})=F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\pi}}(q^{2},p_{\pi}^{2})$
(62)
due to the equality (3.0.1)
${\rm disc}_{s}{\rm
disc}_{s^{\prime}}F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(s^{\prime},p_{\pi}^{2})={\rm
disc}_{s}{\rm
disc}_{s^{\prime}}F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\pi}}(s^{\prime},p_{\pi}^{2}).$
(63)
#### 3.0.2 The double spectral integral for the form factors with pion
emission
The form factors read:
$\displaystyle F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(q^{2},p_{\pi}^{2})$
$\displaystyle=$ $\displaystyle
F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\pi}}(q^{2},p_{\pi}^{2})$ (64)
$\displaystyle=$
$\displaystyle\int\limits^{\infty}_{4m^{2}}\frac{ds}{\pi}\int\limits^{\infty}_{4m^{2}}\frac{ds^{\prime}}{\pi}\frac{{\rm
disc}_{s}{\rm
disc}_{s^{\prime}}F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(s^{\prime},p_{\pi}^{2})}{(s-M^{2}_{V(L)})(s^{\prime}-q^{2})}\
.$
As in (42), we assume that the convergence of (64) is guaranteed by the
vertices $G_{V(L)}(s)$ and $G_{\gamma}(s^{\prime})$.
Futhermore, we consider the production of photon, $q^{2}=0$, and use the
photon wave function
$\psi_{\gamma}(s^{\prime})=G_{\gamma}(s^{\prime})/s^{\prime}$. After
integrating over intermediate-state quark momenta, one can represent (64) for
$p_{\pi}^{2}\leq 0$ in the following form:
$\displaystyle F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,p_{\pi}^{2})$
$\displaystyle=$ $\displaystyle
F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\pi}}(0,p_{\pi}^{2})$ (65)
$\displaystyle=$ $\displaystyle
g_{\pi}\int\limits_{4m^{2}}^{\infty}\frac{dsds^{\prime}}{16\pi^{2}}\psi_{V(L)}(s)\psi_{\gamma}(s^{\prime})$
$\displaystyle\times$
$\displaystyle\frac{\Theta\left(-ss^{\prime}p_{\pi}^{2}-m^{2}\lambda(s,s^{\prime},p_{\pi}^{2})\right)}{\sqrt{\lambda(s,s^{\prime},p_{\pi}^{2})}}S^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(s,s^{\prime},p_{\pi}^{2}).$
The step-function $\Theta(X)$ was defined in (44).
Let us emphasise once again that Eq. (65) is valid in the region
$p_{\pi}^{2}\leq 0$ only. To obtain form factors at $p_{\pi}^{2}=M^{2}_{\pi}$,
one needs to continue Eq. (65) to the region $p_{\pi}^{2}>0$. Since the form
factors are analytical functions in the vicinity of $p_{\pi}^{2}=0$, the
straightforward way is to expand them in a series over $p_{\pi}^{2}$ keeping
constant and linear terms only:
$F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,p_{\pi}^{2})=F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,0)+p_{\pi}^{2}\frac{d}{dp_{\pi}^{2}}F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,0)\
.$ (66)
One can approximate $p_{\pi}^{2}\cdot
F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,0)/dp_{\pi}^{2}=F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,0)-F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,-M^{2}_{\pi})$
(here $p_{\pi}^{2}=-M^{2}_{\pi}$). Then
$\displaystyle F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,M^{2}_{\pi})$
$\displaystyle=$ $\displaystyle
F^{V(L)\to\gamma\pi}_{\bigtriangledown_{\pi}}(0,M^{2}_{\pi})$ (67)
$\displaystyle=$ $\displaystyle
2F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,0)-F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,-M^{2}_{\pi})\
.$
Both form factors, $F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,0)$ and
$F^{V(L)\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,-M^{2}_{\pi})$, are calculated
according to Eq. (65).
#### 3.0.3 Z-factors for pion emission
The charge factors for the pion emission in the decays
$\rho^{+}\to\gamma\pi^{+}$, $\rho^{0}\to\gamma\pi^{0}$,
$\omega\to\gamma\pi^{0}$ (see Figs. 6, 7) are equal to those for photon
emission as follows:
$\displaystyle
Z^{\rho^{+}\to\gamma\pi^{+}}_{\bigtriangleup^{\pi}}=Z^{\rho^{+}\to\gamma\pi^{+}}_{\bigtriangledown_{\gamma}}=e_{d},\qquad
Z^{\rho^{+}\to\gamma\pi^{+}}_{\bigtriangledown_{\pi}}=Z^{\rho^{+}\to\gamma\pi^{+}}_{\bigtriangleup^{\gamma}}=e_{u}\
,$ $\displaystyle
Z^{\rho^{0}\to\gamma\pi^{0}}_{\bigtriangleup^{\pi}}=Z^{\rho^{0}\to\gamma\pi^{0}}_{\bigtriangledown_{\pi}}=Z^{\rho^{0}\to\gamma\pi^{0}}_{\bigtriangleup^{\gamma}}=Z^{\rho^{0}\to\gamma\pi^{0}}_{\bigtriangledown_{\gamma}}=\frac{1}{2}(e_{u}+e_{d}),$
$\displaystyle
Z^{\omega\to\gamma\pi^{0}}_{\bigtriangleup^{\pi}}=Z^{\omega\to\gamma\pi^{0}}_{\bigtriangledown_{\pi}}=Z^{\omega\to\gamma\pi^{0}}_{\bigtriangleup^{\gamma}}=Z^{\omega\to\gamma\pi^{0}}_{\bigtriangledown_{\gamma}}=\frac{1}{2}(e_{u}-e_{d}).$
(68)
In the calculation of $Z$-factors (3.0.3), we take into account that pion
emission by quark is a two-step process (see Fig. 6c ): the initial quark (for
example, in Fig. 6a) emits gluons (they have isospin $I_{gluons}=0$) which
produce quark–antiquark pairs, $u\bar{u}$ or $d\bar{d}$, with equal
amplitudes, and then we face the transition $u\bar{d}\to\pi^{+}$. The block of
Fig. 6c is denoted as a coupling $g_{\pi}$.
In the process of Fig. 7a, the gluons produce $u\bar{u}$ pair with the same
amplitude as in the previous case but then we face the transiton
$u\bar{u}\to\pi^{0}$ resulting in the factor $1/\sqrt{2}$ (recall that
$\pi^{0}=(u\bar{u}-d\bar{d})/\sqrt{2}$). In the process of Fig. 7c, the
$d\bar{d}$ pair is produced, and the transiton $d\bar{d}\to\pi^{0}$ gives the
factor $-1/\sqrt{2}$ (for more detailed presentation of the quark
combinatorial rules see [2] and references therein).
Figure 6: Diagrams for $Z$-factors in the reaction
$\rho^{+}\to\gamma\pi^{+}$: a) $Z=e_{d}=-\frac{1}{3}$ and b)
$Z=e_{u}=\frac{2}{3}$.
Figure 7: Diagrams for $Z$-factors in the reactions
$\rho^{0}\to\gamma\pi^{0}$ and $\omega^{0}\to\gamma\pi^{0}$: a)
$Z(\rho^{0}\to\gamma\pi^{0})=\frac{1}{3}$,
$Z(\omega\to\gamma\pi^{0})=\frac{1}{3}$;
b)$Z(\rho^{0}\to\gamma\pi^{0})=\frac{1}{3}$,
$Z(\omega^{0}\to\gamma\pi^{0})=\frac{1}{3}$;
c)$Z(\rho^{0}\to\gamma\pi^{0})=-\frac{1}{6}$,
$Z(\omega^{0}\to\gamma\pi^{0})=\frac{1}{6}$;
d)$Z(\rho^{0}\to\gamma\pi^{0})=-\frac{1}{6}$,
$Z(\omega^{0}\to\gamma\pi^{0})=\frac{1}{6}$. Recall that
$\rho^{0}=\pi^{0}=\frac{u\bar{u}-d\bar{d}}{\sqrt{2}},\quad$ and
$\omega^{0}=\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}$.
#### 3.0.4 Partial width
In terms of the calculated form factors, the partial width reads:
$\displaystyle M_{V}\Gamma_{V\to\gamma\pi}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\cdot\frac{\alpha}{4}\frac{M^{2}_{V}-M^{2}_{\pi}}{M^{2}_{V}}\frac{\lambda(M^{2}_{V},M^{2}_{\pi},0)}{2}$
(69) $\displaystyle\times$
$\displaystyle\left[Z^{V\to\gamma\pi}_{\bigtriangleup^{\gamma}}F^{V\to\gamma\pi}_{\bigtriangleup^{\gamma}}(0,M^{2}_{\pi})+Z^{V\to\gamma\pi}_{\bigtriangleup^{\pi}}F^{V\to\gamma\pi}_{\bigtriangleup^{\pi}}(0,M^{2}_{\pi})\right.$
$\displaystyle\left.+Z^{V\to\gamma\pi}_{\bigtriangledown_{\gamma}}F^{V\to\gamma\pi}_{\bigtriangledown_{\gamma}}(0,M^{2}_{\pi})+Z^{V\to\gamma\pi}_{\bigtriangledown_{\pi}}F^{V\to\gamma\pi}_{\bigtriangledown_{\pi}}(0,M^{2}_{\pi})\right]^{2}.$
Here, the factor $1/3$ is due to the averaging over initial vector meson spin
states, the term
$\alpha/4\ \cdot(M^{2}_{V}-M^{2}_{\pi})/M^{2}_{V}$ is given by the phase space
integration, and
$\lambda(M^{2}_{V},M^{2}_{\pi},0)/2=(M^{2}_{V}-M^{2}_{\pi})^{2}/2$ is due to
the spin factor (24). The $Z$-factors are as follows:
$Z^{\rho^{0}\to\gamma\pi^{0}}_{\bigtriangleup^{\gamma}}=1/6$,
$Z^{\rho^{0}\to\gamma\pi^{0}}_{\bigtriangledown_{\gamma}}=1/6$,
$Z^{\rho^{0}\to\gamma\pi^{0}}_{\bigtriangleup^{\pi^{0}}}=1/6$,
$Z^{\rho^{0}\to\gamma\pi^{0}}_{\bigtriangledown_{\pi^{0}}}=1/6$,
$Z^{\omega\to\gamma\pi^{0}}_{\bigtriangleup^{\gamma}}=1/2$,
$Z^{\omega\to\gamma\pi^{0}}_{\bigtriangledown_{\gamma}}=1/2$,
$Z^{\omega\to\gamma\pi^{0}}_{\bigtriangleup^{\pi^{0}}}=1/2$,
$Z^{\omega\to\gamma\pi^{0}}_{\bigtriangledown_{\pi^{0}}}=1/2$.
## 4 Results and discussion
The fitting to the partial widths
$\Gamma^{(exp)}_{\rho^{\pm}\to\gamma\pi^{\pm}}=68\pm 30$ keV,
$\Gamma^{(exp)}_{\rho^{0}\to\gamma\pi^{0}}=77\pm 28$ keV,
$\Gamma^{(exp)}_{\omega\to\gamma\pi^{0}}=776\pm 45$ keV leads to the following
values of the pion emission coupling:
$\displaystyle{\rm Solution\,I}$ $\displaystyle:$
$\displaystyle\qquad\;\;\;16.7\pm 0.3\ ^{+0.1}_{-2.3}\ ,$ $\displaystyle{\rm
Solution\,II}$ $\displaystyle:$ $\displaystyle\qquad-3.0\pm 0.3\
^{+2.1}_{-0.1}\ .$ (70)
In Eq. (4), we have included systematical errors ($(+0.1/-2.3)$ for Solution I
and $(+2.1/-0.1)$ for Solution II) which are caused by the uncertainties of
the fit of $q\bar{q}$ wave functions in the spectral integral equation (see
Section 1.2).
So, we have regions of positive and negative $g_{\pi}$. However, one should
take into account that the sign of $g_{\pi}$ in (4) is rather conventional: it
depends on signs of wave functions of photon and mesons involved into
calculation. Because of that, being precise, we should state that for
$g_{\pi}$ we determine absolute values only, see (15).
Solution I gives us the value of the of pion–nucleon coupling; recall that it
is determined as a factor in the phenomenological Lagrangian: $g_{\pi
NN}\bigg{(}\bar{\psi}\,^{\prime}_{N}(\vec{\tau}\vec{\varphi}_{\pi})i\gamma_{5}\psi_{N}\bigg{)}$).
It is in agreement with the results for pion–nucleon scattering $g_{\pi
NN}^{2}/4\pi\simeq 14$ [39, 40, 41]. Namely, dealing with pion–nucleon
interaction in terms of the quark model, we use the Lagrangian:
$\displaystyle g_{\pi
qq}\bigg{(}\bar{\psi}\,^{\prime}_{q}(\vec{\tau}\vec{\varphi}_{\pi})\
i\gamma_{5}\psi_{q}\bigg{)}$ $\displaystyle=$ $\displaystyle\sqrt{2}g_{\pi
qq}\,\varphi^{+}_{\pi^{+}}\bigg{(}\bar{\psi}\,^{\prime}_{d}\
i\gamma_{5}\psi_{u}\bigg{)}+{\rm other\,terms}$ (71) $\displaystyle=$
$\displaystyle
g_{\pi}\,\varphi^{+}_{\pi^{+}}\bigg{(}\bar{\psi}\,^{\prime}_{d}\
i\gamma_{5}\psi_{u}\bigg{)}+{\rm other\,terms},$
that gives us $\sqrt{2}g_{\pi qq}=g_{\pi}$.
In Appendix A, using SU(6)-symmetry for nucleons, we demonstrate that $g_{\pi
NN}=(5/3)g_{\pi qq}$. So, in terms of SU(6)-symmetry, we have:
$g_{\pi NN}=\frac{5}{3\sqrt{2}}g_{\pi}.$ (72)
We see that Solution I, being in agreement with data [39, 40, 41], gives us
$g_{\pi NN}^{2}/(4\pi)=22.2\pm 0.8^{+0.2}_{-5.0}.$ (73)
For Solution II, we have found $0.03\leq g_{\pi NN}^{2}/(4\pi)\leq 1$, that is
far from the experimental value.
### 4.1 Predictions for excited vector states
For $\rho^{\pm}(2S)$ , $\rho^{0}(2S)$ and $\omega(2S)$ mesons, we have found
the following partial widths (in keV units):
$\displaystyle\Gamma(\rho_{2S}^{\pm}\to\gamma\pi)\simeq 10-130\,,$
$\displaystyle\Gamma(\rho_{2S}^{0}\to\gamma\pi)\simeq 10-130\,,$
$\displaystyle\Gamma(\omega_{2S}\to\gamma\pi)\simeq 60-1080\,.$ (74)
The other wave functions of highly exited states have too large uncertainties
to provide us with reliable widths. This points to the necessity to carry out
mesurements of radiative processes with mesons in the region of large masses.
#### Acknowledgement
We thank B.L. Birbrair for helpful remarks. This paper was supported by the
RFFI grant 07-02-01196-a.
## Appendix A: Nucleon pion emission vertex
in the SU(6) quark model
Here we derive the relations between couplings in phenomenological Lagrangian
for pions and nucleons, $g_{\pi
NN}\bigg{(}\bar{\psi}\,^{\prime}_{N}(\vec{\tau}\vec{\varphi}_{\pi})i\gamma_{5}\psi_{N}\bigg{)}$,
and those for quarks, $g_{\pi
qq}\bigg{(}\bar{\psi}\,^{\prime}_{q}(\vec{\tau}\vec{\varphi}_{\pi})i\gamma_{5}\psi_{q}\bigg{)}$.
To be definite, we consider transitions
$p^{\uparrow}\to\pi^{+}+n^{\downarrow}$ and
$u^{\uparrow}\to\pi^{+}+d^{\downarrow}$. We use the following SU(6) wave
functions (see, for example, Appendix D in Ref. [44]):
$\displaystyle\psi_{p}\equiv
p^{\uparrow}\bigg{(}q(1)q(2)q(3)\bigg{)}=\frac{\sqrt{2}}{3}(u^{\uparrow}u^{\uparrow}d^{\downarrow}+d^{\downarrow}u^{\uparrow}u^{\uparrow}+u^{\uparrow}d^{\downarrow}u^{\uparrow})$
$\displaystyle-\frac{1}{3\sqrt{2}}(u^{\uparrow}u^{\downarrow}d^{\uparrow}+d^{\uparrow}u^{\uparrow}u^{\downarrow}+u^{\downarrow}d^{\uparrow}u^{\uparrow}+u^{\downarrow}u^{\uparrow}d^{\uparrow}+d^{\uparrow}u^{\downarrow}u^{\uparrow}+u^{\uparrow}d^{\uparrow}u^{\downarrow}),$
$\displaystyle\bar{\psi}_{n}\equiv
n^{\downarrow}\bigg{(}q(1)q(2)q(3)\bigg{)}=\frac{\sqrt{2}}{3}(d^{\downarrow}d^{\downarrow}u^{\uparrow}+u^{\uparrow}d^{\downarrow}d^{\downarrow}+d^{\downarrow}u^{\uparrow}d^{\downarrow})$
(75)
$\displaystyle-\frac{1}{3\sqrt{2}}(d^{\downarrow}d^{\uparrow}u^{\downarrow}+u^{\downarrow}d^{\downarrow}d^{\uparrow}+d^{\uparrow}u^{\downarrow}d^{\downarrow}+d^{\uparrow}d^{\downarrow}u^{\downarrow}+u^{\downarrow}d^{\uparrow}d^{\downarrow}+d^{\downarrow}u^{\downarrow}d^{\uparrow}).$
Recall that for baryon quarks we use notation of the type
$d^{\downarrow}u^{\downarrow}d^{\uparrow}\equiv
d^{\downarrow}(1)u^{\downarrow}(2)d^{\uparrow}(3)$.
The isospin block reads:
$(\vec{\tau}\vec{\varphi}_{\pi})=\sqrt{2}\ \frac{\tau_{1}+i\tau_{2}}{2}\
\frac{\varphi^{(1)}_{\pi}-i\varphi^{(2)}_{\pi}}{\sqrt{2}}+\sqrt{2}\
\frac{\tau_{1}-i\tau_{2}}{2}\
\frac{\varphi^{(1)}_{\pi}+i\varphi^{(2)}_{\pi}}{\sqrt{2}}+\tau_{3}\varphi^{(3)}_{\pi}.$
(76)
Transition $p^{\uparrow}\to\pi^{+}+n^{\downarrow}$ is given by the following
terms in nucleon and quark spaces:
$\displaystyle g_{\pi NN}\langle\pi^{+}n^{\downarrow}|\,\sqrt{2}\
\frac{\tau_{1}+i\tau_{2}}{2}\
\frac{\varphi^{(1)}_{\pi}-i\varphi^{(2)}_{\pi}}{\sqrt{2}}\,i\gamma_{5}|p^{\uparrow}\rangle$
$\displaystyle=g_{\pi
qq}\langle\pi^{+}n^{\downarrow}\bigg{(}q(1)q(2)q(3)\bigg{)}|\,\sqrt{2}$
$\displaystyle\times\sum\limits_{j=1,2,3}\frac{\tau_{1}(j)+i\tau_{2}(j)}{2}\
\frac{\varphi^{(1)}_{\pi}-i\varphi^{(2)}_{\pi}}{\sqrt{2}}\,i\gamma_{5}(j)\;|p^{\uparrow}\bigg{(}q(1)q(2)q(3)\bigg{)}\rangle\
,$ (77)
where $\tau_{1}(j)$, $\tau_{2}(j)$ and $\gamma_{5}(j)$ act on $q(j)$.
In the non-relativistic limit, which we use for nucleons and constituent
quarks,
$i\gamma_{5}\to(-i)(\vec{\sigma}\vec{q})$
and direct calculations give:
$g_{\pi NN}=g_{\pi qq}\cdot 3\cdot\frac{5}{9}\ .$ (78)
To simplify the calculations which lead to (78), one can fix the direction of
photon momentum, for example, $\vec{q}=(q_{x},0,0)$ and then use
$\bigg{(}\vec{q}\vec{\sigma}(j)\bigg{)}\to q_{x}\sigma_{1}(j)$.
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|
arxiv-papers
| 2009-01-30T10:05:05 |
2024-09-04T02:49:00.295055
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.V. Anisovich, V.V. Anisovich, L.G. Dakhno, M.A. Matveev, V.A.\n Nikonov and A.V. Sarantsev",
"submitter": "Vladimir V. Anisovich",
"url": "https://arxiv.org/abs/0901.4854"
}
|
0901.4878
|
# Ferromagnetism in Co7(TeO3)4Br6: A byproduct of complex antiferromagnetic
order and single-ion anisotropy
M. Prester Institute of Physics, P.O.B.304, HR-10 000, Zagreb, Croatia I.
Živković Institute of Physics, P.O.B.304, HR-10 000, Zagreb, Croatia O.
Zaharko Laboratory for Neutron Scattering, ETHZ & PSI, CH-5232, Villigen,
Switzerland D. Pajić Department of Physics, Faculty of Science, Bijenička
c.32 , HR-10 000 Zagreb, Croatia P. Tregenna-Piggott Laboratory for Neutron
Scattering, ETHZ & PSI, CH-5232, Villigen, Switzerland H. Berger Institute
of Physics of Complex Matter, EPFL, 1015 Lausanne, Switzerland
###### Abstract
Pronounced anisotropy of magnetic properties and complex magnetic order of a
new oxi-halide compound Co7(TeO3)4Br6 has been investigated by powder and
single crystal neutron diffraction, magnetization and ac susceptibility
techniques. Anisotropy of susceptibility extends far into the paramagnetic
temperature range. A principal source of anisotropy are anisotropic properties
of the involved octahedrally coordinated single Co2+ ions, as confirmed by
angular-overlap-model calculations presented in this work. Incommensurate
antiferromagnetic order sets in at $T_{N}$=34 K. Propagation vector is
strongly temperature dependent reaching ${\bf{k}_{1}}$=(0.9458(6), 0,
0.6026(5)) at 30 K. A transition to a ferrimagnetic structure with
${\bf{k}_{2}}$=0 takes place at $T_{C}$=27 K. Magnetically ordered phase is
characterized by very unusual anisotropy as well: while $M-H$ scans along
$b$-axis reveals spectacularly rectangular but otherwise standard
ferromagnetic hysteresis loops, $M-H$ studies along other two principal axes
are perfectly reversible, revealing very sharp spin flop (or spin flip)
transitions, like in a standard antiferromagnet (or metamagnet).
Altogether, the observed magnetic phenomenology is interpreted as an evidence
of competing magnetic interactions permeating the system, first of all of the
single ion anisotropy energy and the exchange interactions. Different
coordinations of the Co2+-ions involved in the low-symmetry C2/c structure of
Co7(TeO3)4Br6 render the exchange-interaction network very complex by itself.
Temperature dependent changes in the magnetic structure, together with an
abrupt emergence of a ferromagnetic component, are ascribed to continual spin
reorientations described by a multi-component, but yet unknown, spin
Hamiltonian.
###### pacs:
12.34, 56.78
## I Introduction
Enormous diversity of magnetic phenomena relies on equal diversity of
interactions permeating real magnetic materials. In cases of one interaction
dominating by far over the others (typically valid for exchange interaction)
an elementary insight into magnetism of such a system can indeed be acquired
on basis of a simple, one-interaction Hamiltonian. However, any profound
knowledge of magnetism of even such a simple system implies taking into
account other interactions present in the system, whatever weak they could be.
For example, basic static properties (like magnetization, susceptibility,
magnetic structure…) of long-range ordered ferromagnets and antiferromagnets,
especially if they rely on strong exchange interaction, can be well-
interpreted within one-interaction models. Understanding dynamic features of
the same systems, like their spin excitation spectra for example, requires
however at least magnetocrystalline anisotropy (thus, in turn, several
inevitably involved specific interactions) to be included as well. The most
interesting situation, for fundamental research as well for applications,
arises in cases involving presence of magnetic interactions competing each
other in size and/or in sign while residing on energy scale of thermal
excitations within the usual experimental window $2-300$ K. Antiferromagnetic
exchange competing with single ion anisotropy can, for example, render the
certainty of long-range ordering questionable, as shown decades ago for
archetypal antiferromagnet Moriya1960PRB NiF2. In a more recently developed
general framework of Quantum Magnetism Lhuillier2002 the subject of competing
interactions is recognized as a key ingredient introducing the quantum phase
transition point into the respective phase diagrams. In another aspect,
competing interactions are responsible for complex/incommensurate magnetic
structures and emergence of ferromagnetism in the ground states of many
nominally antiferromagnetic systems Bogdanov2002 . In this category there is a
particularly interesting group of systems revealing spin reorientations taking
place within the system-specific temperature intervals. The prominent examples
are the rare-earth elements Dysprosium (Dy) and Terbium (Tb) Nagamiya1967 ,
transition-metals sesquioxides Artman65 (notably, hematite Besser1967 ,
$\alpha$-Fe2O3) and orthoferrites Shapiro1974 . From applicative side there is
renewed interest for these systems, particularly for orthoferrites Kimel2004 :
spins in these systems are subject to combined effect of antiferromagnetic
exchange and magnetocrystalline anisotropy which enables ultrafast spin
manipulation Kimel2004 . Ultrafast dynamics is a key issue for exchange-bias
devices.
In this article we present magnetic structure and properties of a recently
discovered Becker2006 magnetic system, Co7(TeO3)4Br6, which, as shown
herewith, reveals remarkable manifestations of competing magnetic
interactions. In its ground state there is a complex noncolinear long-range
magnetic order while pronounced magnetic anisotropy characterizes both
paramagnetic and magnetically ordered phase. In the temperature range
$27K-34K$ the incipient incommensurate magnetic order continuously changes by
cooling, ending up at $27K$ by an abruptly intruding ferromagnetic component
along one crystallographic axis. In the spin reorientation, underlying
emergence of ferromagnetism, antiferromagnetic backbone remains conserved: we
show that the key hallmark of antiferromagnetic order, spin flop/flip
transition, keeps characterizing magnetism of Co7(TeO3)4Br6 in measurements
along other two principal axes. The results are interpreted by invoking
competition of exchange interactions primarily with single ion anisotropy
energy but also with several other possible sources of magnetocrystalline
anisotropy.
The article is organized as follows. The involved experimental techniques are
introduced in Section II. In Section III results of neutron diffraction, dc/ac
susceptibility as well as of magnetic hysteresis/M-H studies are presented.
Results of theoretical modeling of paramagnetic susceptibility and of the
ground state properties, based on angular-overlap-model calculations, are also
presented in Section III. Both experimental and theoretical results provide
firm evidence of exchange interactions and single ion anisotropy energy ruling
magnetic properties of Co7(TeO3)4Br6. In Section IV these results are
discussed in a more general framework of competing interactions of various
sorts, knowledge of which was accumulating during the decades. In Section V
appropriate conclusions have been presented.
## II Experimental details
The single crystals of Co7(TeO3)4Br6 were synthesized via chemical vapor
transport reactions. The details of the synthesis can be found elsewhere
Becker2006 . The single crystals grow in platelet geometry and the typical
samples used in the magnetization/susceptibility studies had approximate
dimensions $4.0\times 2.0\times 0.1~{}mm^{3}$. The plane of the platelet-like
samples corresponds to the crystallographic $bc$-plane.
Neutron powder diffraction data have been collected in the temperature range 3
K - 45 K on the DMC instrument at SINQ, Paul Scherrer Institute, Villigen,
Switzerland, with neutron wavelength of 2.453 Å and 4.2 Å. Incommensurate
(ICM) wave vector has been determined on the single crystal diffractometer
TriCS at SINQ with neutron wavelength of 2.32 Å using the area detector.
Magnetization/susceptibility measurements were performed on oriented single
crystals in applied magnetic fields directed parallel to the crystallographic
$a^{\ast}$-, $b$-, and $c$-axis. Samples were oriented by the use of X-ray
diffractometer. The choice of $a^{\ast}$-axis sample orientation, instead of
the preferable $a$-axis one, is imposed by sample morphology. For DC
magnetization studies up to 5.5 T a Quantum Design superconducting quantum
interference device (SQUID) magnetometer was used, covering the temperature
range 2 – 300 K. AC susceptibility studies were performed using a CryoBIND ac
susceptibility system employing measuring ac field of 3 Oe and frequency of
430 Hz.
## III Results
### III.1 Neutron diffraction
Refinement of the neutron powder pattern collected at 45 K confirmed that the
sample is a single phase with a monoclinic _C2/c_ space group and unit cell
parameters $a=20.590(2)$ Å, $b=8.4998(7)$ Å, $c=14.631(1)$ Å,
$\beta=125.202(6)^{\circ}$, in close agreement with the original structural
work Becker2006 . As pointed out therein the crystal structure can be
described as layered in the $bc$-plane. The layers are built of networks
comprising three types of Co2+-ion distorted octahedra ([Co(1)O4Br2],
[Co(2)O4Br2], [Co(3)O4Br2]) and [TeO3E] tetrahedra, while the layers are
interconnected along the $a$-axis by the fourth Co2+-ion octahedron type,
[Co(4)O2Br4].
The collected neutron powder diffraction patterns confirmed the existence of
two sequential magnetic orderings, as claimed by the original work on basis of
magnetic susceptibility studies Becker2006 . Below $T_{N}=34$ K weak magnetic
peaks occur. They cannot be indexed as simple multiples of the crystal unit
cell revealing an incommensurate wave vector. Moreover, the incommensurability
appears in two directions, $a^{\ast}$ and $c^{\ast}$, which has been clarified
by a single crystal study using the 2D detector of TriCS. The position of the
ICM peaks is strongly temperature dependent and in Fig. 1 we show the
difference neutron powder diffraction patterns T-45 K in the range T= 25 – 32
K. Each pattern presented in Fig. 1 has been analyzed using the profile match
option of the Fullprof programRodriguez1993 and the two ICM components
$(k_{x},0,k_{z})$ of the wave vector has been refined. The temperature
dependence of $k_{x}$ and $k_{z}$ is shown in Fig. 2.
Figure 1: (Color online) Difference neutron powder diffraction $T-45$ K of
Co7(TeO3)4Br6 with $T$ in the range 25 – 32 K (DMC instrument, $\lambda=4.2$
Å). Figure 2: (Color online) Temperature dependence of the ICM magnetic vector
components refined from the DMC data, $\lambda=2.453$ Å.
At 27.5 K a second set of additional strong magnetic peaks appears, coexisting
in a short temperature interval with the ICM peaks. Further temperature
lowering leads to weakening of the ICM peaks and their transformation into
diffuse scattering, as shown in the bottom pattern of Fig. 1. Below 26 K only
the second set of magnetic peaks remains.
Figure 3: (Color online) Observed 5 K - 45 K magnetic difference pattern,
calculated and difference patterns of Co7(TeO3)4Br6 denoted by crosses, red
solid, and green dotted lines, respectively. Inset: The choice of the
$XYZ=a^{*}bc$ orthogonal system and morphology of the single crystal used.
This set can be indexed with the wave vector k2 = 0. Some of the new magnetic
peaks (i.e. (200), (110)) overlap with the nuclear reflections implying
ferromagnetic contribution, others (i.e. (001), (-201)) appear at the
positions extinct in the paramagnetic pattern, as expected from an
antiferromagnetic component. The systematic extinctions observed in the 5 K -
45 K magnetic difference pattern reveal that the C-translation and the glide
plane $c$ not combined with the time reversal are retained in magnetic
symmetry ($hkl:h+k=2n$ and $h0l:l\neq 2n$).
Representation analysis implemented in the Fullprof program Rodriguez1993 has
been used to determine the k2 = 0 magnetic structure. The Fourier coefficients
describing possible spin configurations can be written as linear combinations
of irreducible representations (IR) of the wave vector group (little group).
The magnetic representations for the 4a and 8f sites, occupied by Co(4) and
Co(1-3), can be decomposed in IR’s:
$\Gamma$(4a)= 3 $\Gamma_{1}$ \+ 3 $\Gamma_{3}$
$\Gamma$(8f)= 3 $\Gamma_{1}$ \+ 3 $\Gamma_{2}$ \+ 3 $\Gamma_{3}$ \+ 3
$\Gamma_{4}$
Note that there are four ions of the Co(4) set and eight ions in each of the
Co(1), Co(2) and Co(3) sets. In the $\Gamma_{1}$ and $\Gamma_{3}$
representations magnetic moments of the Co(1)-Co(4) sets may attain
independent values and directions, while the moments of the ions within the
same site are constrained by the symmetry relations presented in Table 1. So
all together there are twelve independent parameters ($m_{x}$, $m_{y}$ and
$m_{z}$ of four sets of Co2+ ions) in $\Gamma_{1}$ and $\Gamma_{3}$. In
$\Gamma_{2}$, $\Gamma_{4}$ IR’s the magnetic moments of the Co(4) set must be
zero.
Table 1: Irreducible representations of the wave vector group for ${\bf{k}_{2}}$= 0 in the space group $C2/c$. The notation Co($ij$) is used with the index $i$=1,…,4 labeling the set, and the index $j$=1,…,4 labeling the ions within the site. The ions of the same site have coordinates: $j$=1 (x y z), $j$=2 (1-x y -z+1/2), $j$=3 (-x+1/2 -y+1/2 1-z), $j$=4 (x 1-y z+1/2). The ions generated by the C-lattice translation have the same magnetic moment as the generating atom and are therefore omitted. The u,v,w coefficients are fixed by the symmetry for one set, but are independent for different sets. Ion | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{4}$
---|---|---|---|---
4 a
Co(41) | u v w | - | u v w | -
Co(43) | -u v -w | - | u -v w | -
$i$=1,3 | 8 f
Co(i1) | u v w | u v w | u v w | u v w
Co(i2) | -u v -w | -u v -w | u -v w | u -v w
Co(i3) | u v w | -u -v -w | u v w | -u -v -w
Co(i4) | -u v -w | u -v w | u -v w | -u v -w
Refinement of the models has been performed with Fullprof Rodriguez1993 . The
best agreement with experimental data ($R_{M}=6.6\%$, Fig. 3) is obtained for
a 3-dimensional canted $\Gamma_{1}$ model presented in Table 2 and Fig. 4. The
$a^{*}bc$ orthogonal coordinate system defined in Inset to Fig. 3 has been
used. The Co moments reach the values of 4.21(7) $\mu_{B}$/Co(1), 4.4(1)
$\mu_{B}$/Co(2), 3.8(1) $\mu_{B}$/Co(3) and 4.5(1) $\mu_{B}$/Co(4) at 5 K.
These values are larger than the spin only component (3 $\mu_{B}$) of the Co2+
ion confirming incomplete quenching of the orbital moment for all cobalt ions
in this compound Becker2006 .
Table 2: Refined parameters for the ${\bf{k}_{2}}$= 0 magnetic structure. M[$\mu_{B}$] is the ordered magnetic moment, with the Mxyz components defined in the $a^{*}bc$ orthogonal coordinate system. The Co(1-4) sets are represented by the ions $j$=1 (x y z) and $j$=2 (1-x y -z+1/2). $\alpha$ [deg] is the canting angle between the magnetic moments of the 1st and the following ions. Ions | Mx | My | Mz | $\mu_{B}$/Co | $\alpha_{1i}$
---|---|---|---|---|---
Co(11) | 2.64(9) | 1.86(6) | 2.77( 9) | 4.25(7) | 0
Co(12) | -2.64(9) | 1.86(6) | -2.77( 9) | 4.25(7) | 125(2)
Co(21) | -3.08(7) | -1.1( 1) | 3.08(7) | 4.4(1) | 95(2)
Co(22) | 3.08(7) | -1.1( 1) | -3.08(7) | 4.4(1) | 97(2)
Co(31) | -0.43(8) | -3.1( 1) | 2.22(8) | 3.8(1) | 93(2)
Co(32) | 0.43(8) | -3.1( 1) | -2.22(8) | 3.8(1) | 131(2)
Co(41) | -3.51(8) | 1.33( 9) | 2.48(8) | 4.5(1) | 90(2)
Co(42) | 3.51(8) | 1.33( 9) | -2.48(8) | 4.5(1) | 75(2)
Figure 4: (Color online) The $ac$-projection of the low-temperature (5 K)
magnetic structure of Co7(TeO3)4Br6. The four crystallographic sets of Co2+
ions are shown by violet - Co(1), red - Co(2), green - Co(3) and orange -
Co(4). The Co(21) and Co(32) ions, as well as Co(22) and Co(32), superimpose
on this projection. The pairs Co($ij$)/Co($ij$+1) have opposite Mx and Mz
magnetic components, but the same My component. The sign of My is shown near
the symbol of each ion. Spatial orientations of local magnetic moments are
illustrated by three-dimensional vectors.
The Co7(TeO3)4Br6 system could be identified as a canted antiferromagnet (weak
ferromagnet) – the angle between moments of the similar magnitude is smaller
than 180∘, giving rise to a ferromagnetic component. Each Co set ($i=1,...,4$)
has a different magnitude of the $M_{y}$ component. Ferromagnetic components
associated with the Co(1) and Co(4) sets point in one direction, while those
associated with Co(2) and Co(3) point in the opposite direction. The net
moment along the $b$ axis given by the sum of the $M_{y}$ values in Table 2
amounts to $\approx 0.5\mu_{B}/Co$, which is in close agreement with the
remanent moment obtained from the magnetization measurements (see, section
$D$). As elaborated in the discussion section, the complex non-collinear
canted magnetic structure of Co7(TeO3)4Br6 (Fig. 4) stems from different
involved interactions, first of all from the competing single ion anisotropy
and exchange interactions.
### III.2 Magnetic susceptibility
#### III.2.1 High-temperature dependence
Figure 5: (Color online) Main Panel: Temperature dependence of real component
of ac susceptibility (3 Oe, 430 Hz) for measuring field oriented along three
principal axes. DC susceptibility in small applied field (100 Oe) provides
almost identical result. Black curve represents the Curie plot for free
$S=3/2$, g=2.5, ions (Curie constant C=20.5 emuK/mol) Inset Top: Product
$\chi_{DC}\cdot T$ vs. temperature. Inset Bottom: Susceptibility inverse
$\chi_{DC}^{-1}$ vs. temperature.
Results of ac and dc susceptibility measurements in three crystallographic
directions $a^{\ast}$, $b$ and $c$ are shown in Fig. 5. Extending the original
susceptibility report Becker2006 these results document pronounced
susceptibility anisotropy characterizing Co7(TeO3)4Br6 in a broad temperature
range. For all three sample orientations the results reveal the presence of
magnetic transition at $T_{N}=34$ K, while the measurement along one axis only
($b$ axis) documents the presence of an additional, very pronounced transition
at $T_{C}=27$ K, in agreement with the original study. The present study shows
that in the paramagnetic range the particular directional susceptibilities are
remarkably different, the difference extending far above the transition
temperature range. Unlike very anisotropic susceptibility the effective
$g$-factor value, as determined from the high-temperature limit of the Curie-
Weiss (CW) plot, was found to be pretty isotropic in the room temperature
range. The values for $g$ were found to be very close to $g=2.5$ (with S=3/2)
for all three sample-to-field orientations, in full accordance with the
original powder-sample data Becker2006 . The value of the Weiss parameter
$\theta$ of the CW plot varies depending on the chosen axis. We elaborate
below that the behavior of the susceptibility is dominantly influenced by the
single-ion anisotropy so the values of $\theta$ cannot be used to estimate the
type and the strength of the involved exchange interactions. From the
experimental side we note that there were only marginal sample-to-sample and
batch-to-batch variations: The reported results thus rely only on the
intrinsic crystal structure of the compound.
Temperature dependence of the imaginary component of susceptibility is shown
in Fig. 6. These data are striking in two aspects. Firstly, imaginary
susceptibility signal is present in the direction of the $b$-axis only.
Secondly, the size and the sharpness of the imaginary susceptibility peak is,
to the best of our knowledge, not observed in other magnetically ordered
systems: under particular conditions (see Fig. 6) imaginary peak is more than
two times bigger than the peak in real susceptibility. Traditionally, peak in
imaginary susceptibility is ascribed to dissipative ferromagnetic-domain
dynamics, setting in simultaneously with the formation of the domains
immediately below $T_{c}$. Due to its extraordinary properties imaginary
susceptibility of Co7(TeO3)4Br6 will be subject of a separate publication.
Figure 6: (Color online) Temperature dependence of out-of-phase (imaginary)
component of ac susceptibility for three principal axes. Above the transition
range there is overlap of the data points for all three measurement
directions. The data for $\chi^{\prime}$ (Fig. 5) and $\chi^{\prime\prime}$
were taken within the same experimental runs. Inset: Zoomed transition
temperature range. Residual signal in measurement along c-axis (green symbols)
is ascribed to imperfect sample alignment and/or non-vanishing cross-talk
between the in- and out-of-phase components in the phase sensitive detection.
#### III.2.2 Susceptibility in magnetically ordered phase
At $T_{N}=34$ K a long-range magnetic order sets in. Both magnetic
susceptibility and neutron diffraction data are consistent with
antiferromagnetic ordering. Relative to the susceptibility maxima at $T_{N}$
there is a rapid drop of both $c-$ and $\mbox{$a^{\ast}$}-$ axis
susceptibility by cooling below $T_{N}$ (Fig. 5), while $b-$ axis
susceptibility simultaneously builds up. These findings are incompatible with
simple uniaxial magnetic AF, which obviously does not take place in
Co7(TeO3)4Br6. Three-dimensionally-canted and incommensurate order, documented
by neutron diffraction results presented above, is fully compatible with
susceptibility data. One notes that at $T_{C}$, marked by a very sharp peak of
$b-$ axis susceptibility, $c-$ and $\mbox{$a^{\ast}$}-$ axis susceptibility
exhibit a sudden drop (Fig. 5). The most natural explanation is that at
$T_{C}$ an abrupt magnetic moment reorientation takes place such that a
growing ferromagnetic component builds up at the expense of magnetic moments
participating in the ICM ordering at higher temperatures. In accordance with
the low-temperature magnetic structure (Fig. 4) the latter observation shows
that in magnetic ordering all magnetic degrees of freedom participate
cooperatively, i.e., even if there exist separate magnetic fractions they
contribute in mutual accord into the total susceptibility. One has to point
out that low-field susceptibility studies reveal a remarkable feature of the
system that one could not figure out from the crystallographic or low-
temperature magnetic structures alone: in spite of complexity of magnetic
structure susceptibility data shows that there is a distinct axis, aligned
approximately along $c-$ axis, which defines the direction of preferable spin
orientation fus0 . (A more precise determination of the preferred axis
orientation is presented in Subsection III.4.) A fact that in the low-
temperature magnetic structure the spins are aligned along very different
directions, thus not along $c$\- nor any other axis, is a consequence of
different competing interactions ruling the spin geometry in the ground state.
### III.3 Anisotropic magnetic properties in Angular Overlap Model
calculations
Angular Overlap Model (AOM) calculations were performed to estimate the single
ion magnetic anisotropy arising from the crystallographically inequivalent
cobalt(II) centres. AOM parameters for the Co-O and Co-Br bonding interactions
were estimated from values of the ligand field splitting parameters tabulated
for octahedral Co(H2O)${}_{6}^{2+}$ and tetrahedral CoBr${}_{4}^{2-}$
complexes Srivatsa1991 ; Cotton1961 assuming $e_{\pi}=0.2e_{\sigma}$. The
parameter $e_{\sigma}$ was assumed to vary with distance Tregenna03 as a
function of $1/r^{5}$ and $e_{\pi}$ as a function of $1/r^{6}$. The Racah and
spin-orbit coupling parameters were fixed at 80 % of their free-ion values,
and the orbital Zeeman interaction reduced accordingly. The AOM matrices were
constructed using LIGFIELD ligfield with the AOM parameters and angular
coordinates calculated from structural data as input, employing all 120
functions of the $3d^{7}$ electronic configuration. For the calculation of
magnetic susceptibility curves, the AOM matrices were imported into the
program MagPropTregennaNist .
The magnetic moment per ion was calculated from the expression
$M_{ion}=\frac{\sum_{n}\Bigl{(}-\frac{dE_{n}}{dB}\Bigr{)}exp\Bigl{(}-\frac{E_{n}}{k_{B}T}\Bigr{)}}{\sum_{n}exp\Bigl{(}-\frac{E_{n}}{k_{B}T}\Bigr{)}},$
(1)
and the paramagnetic molar susceptibility from
$\chi_{p}=N_{A}\frac{M_{ion}}{B}.$ (2)
In these equations $B$ designates the external magnetic field, $k_{B}$ the
Bolzmann constant and $N_{A}$ Avagadro s number. The sum is over the n
eigenstates of the Hamiltonian whose energies are designated by the symbol
$E_{n}$. The derivative in equation (1) was found according to the Hellman-
Feynman theorem,
$\frac{dE_{n}}{dB}=\langle\psi_{n}\mid\frac{d\hat{H}}{dB_{0}}\mid\psi_{n}\rangle.$
(3)
The onset of short-range ferromagnetic order was incorporated by expressing
the susceptibility as
$\frac{1}{\chi_{M}}=\frac{1}{\chi_{p}}-\lambda,$ (4)
where $\lambda$ is the molecular field parameter.
It is seen from Fig.7 that both experiment and AOM-based susceptibility
calculations identify the component of the susceptibility tensor along the
c-axis to be larger than the components along the a*- and b-axes. This would
suggest that single-ion anisotropy indeed plays a major role in determining
the observed magnetic anisotropy. Note, however, that the calculated relative
magnitudes of the susceptibility tensor along the a*- and b-axes are not in
accordance with experiment. This could arise from our rather crude estimate of
the AOM bonding parameters or may reflect the fact that no magnetic
interaction paths have been introduced into the model. In upgrading the model
with appropriate interaction paths it would be natural to assume that the
intralayer bc-plane exchange coupling dominates over the coupling along the
out-of-plane a*-axis direction. Such corrections would certainly make the
results of the model calculations in closer agreement with experiment.
Figure 7: Calculated products $\chi\cdot T$ for each of the four Co(i)
octahedra employing full single-ion Hamiltonian and AOM. Calculated results
for real single crystal as a whole is shown at bottom (plot at left). Results
corrected for weak ferromagnetic interactions ($\lambda=2$, see text) is also
shown, (plot at bottom right), to be compared with the experimental results
for $\chi\cdot T$, Fig.5.
As a result of the low-symmetry ligand field and spin-orbit coupling the
4T1g(Oh) ground term is split into 6 Kramers doublets that are well-separated
in energy. In particular the first-excited Kramers doublet lies between 100
and 300 cm-1 above the ground state for the four crystallographically
inequivalent cobalt centres. At sufficiently low temperatures, therefore, the
electronic structure may be approximated as a pseudo- S=1/2 system, this being
an approximation that is commonly employed for octahedrally co-ordinated
cobalt(II) complexesMabbs92 .
The $g^{2}$ tensor was calculated according to a method described in detail
previously Scheifele08 in which the energies of the states of the lowest
lying Kramers doublet are modeled by the eigenvalues of the S=1/2 spin-
Hamiltonian,
$\hat{H}_{s}=\beta\bf{B}\cdot g\cdot S.$ (5)
The orientation of the $g^{2}$ -tensor in the a*bc reference frame and the
ligand arrangement of four Co2+ sets are shown in Fig.8.
Table 3: Calculated components of the $\bf g$-matrix in the $a^{*}bc$ reference frame and $\bf g^{\prime}$ in the eigen coordinate frame. $g_{ij}$ | Co(1) | $j$=1,..,3 | Co(2) | $j$=1,..,3
---|---|---|---|---
$i$=1 | 5.2251 | 1.0467 | 2.4724 | 2.8165 | 0.1922 | -0.1903
2 | 1.0468 | 2.1070 | 0.5217 | 0.1922 | 3.4392 | 1.8252
3 | 2.4724 | 0.5217 | 4.1107 | -0.1903 | 1.8252 | 5.6682
$g^{\prime}_{jj}$ | 2.2740 | 1.7214 | 7.4476 | 2.9437 | 2.2866 | 6.6936
$g_{ij}$ | Co(3) | $j$=1,..,3 | Co(4) | $j$=1,..,3
$i$=1 | 1.9333 | -1.0734 | 0.8493 | 3.7126 | 0.8268 | -7.2514
2 | -1.0734 | 4.8725 | -2.2980 | 0.1097 | 0.0012 | -1.2788
3 | 0.8493 | -2.2980 | 4.2329 | -1.8364 | -0.9816 | 3.9631
$g^{\prime}_{jj}$ | 2.2383 | 1.5742 | 7.2262 | 0.4946 | -0.4940 | 9.3942
Figure 8: The alignment of $g^{2}$-tensor (violet ellipsoid) and the ligands
(oxygen in red, bromine in green) of four different Co set in the
XYZ=$a^{*}bc$ orthogonal coordinate system.
A sizable c-axis susceptibility growth over the Curie margin (Fig. 5) is
interpreted therefore as a combined effect of single-ion anisotropy and short
range ferromagnetic correlations. On one side, it would be reasonable to
assume that these correlations rely on weak ferromagnetism that, in principle,
could accompany growth of the short-range antiferromagnetic order as the
temperature approaches $T_{N}$ from above. This possibility has to be
abandoned, however, as there is no trace of any ferromagnetic order (in the
form of magnetic hysteresis, peak in imaginary susceptibility, see. Fig. 6)
setting in at, or immediately below, $T_{N}$. Ferromagnetic order, setting in
below $T_{C}$, can only partially be related to high-temperature ferromagnetic
correlations. As shown below, low-temperature ferromagnetism is manifested
along b-axis only (thus not along the preferred c-axis) and its evolution is
primarily correlated with the transformation/reorientation of the preformed
ICM matrix.
### III.4 M-H studies
The most common experimental hallmarks of magnetically ordered ferromagnets
and antiferromagnets are hysteris and spin-flop transition, respectively,
known to characterize the respective magnetization vs. field (M-H)
characteristics. Here we present the results of comprehensive M-H studies on
Co7(TeO3)4Br6 demonstrating presence of _both_ of the mentioned hallmarks:
which one of the two gets activated depends on the chosen orientation of
principal axes with respect to the applied dc field. In brief, the field
component along preferred axis, as imposed by single ion anistropy energy
(approximately $c$-axis), activates antiferromagnetic response while the field
component along $b$-axis activates ferromagnetic response.
#### III.4.1 Antiferromagnetic response
In Fig. 9 we first show the $M-H$ scans for the dc field applied parallel to
the effective preferred axis, $c$-axis. For $T>T_{N}$ the $M-H$ curves are
closed (i.e., do not show up hysteretic loops) and reveal no saturation for
high fields indicating a paramagnetic state of the system. For $T_{C}<T<T_{N}$
a sharp magnetization jump shows up around 20kOe. Magnitude of the
magnetization jumps increases as $T$ approaches $T_{C}$. The field value at
which a jump occurs, $H_{SF}^{c}$, does not change within this temperature
interval. For $T<T_{C}$, the magnetization transition additionally sharpen and
$H_{SF}^{c}$ starts to grow by cooling and reaching $H_{SF}^{c}$=40 kOe at 5
K. Magnetization jumps are naturally ascribed to the spin-flop type of spin
reorientation. The transition region is itself very narrow - at 20 K the
transition is about 200 Oe wide.
Figure 9: (Color online) Magnetization (per one Co ion) vs. field
characteristics for $c-$ axis direction in a broad temperature range. M(H)
curves reveal almost switching but perfectly reversible behavior of
magnetization in magnetically ordered phase.
With the field applied along $a^{\ast}$-axis (Fig. 10) a very similar behavior
to the case of $H\parallel c$ has been obtained. In Fig. 10 we show just
several characteristic $M-H$ scans (out of many measured) taken at
temperatures below and immediately above $T_{c}$.
Qualitatively, the $M-H$ characteristics for $H\parallel c,\mbox{$a^{\ast}$}$
can be analyzed as a switching/spin reorientation (spin-flop) phenomenon
superimposed over some field-dependent background. The background is
attributed to coherent rotation of sublatices’ magnetization and, below
$T_{c}$, to contributions from the $b$-axis ferromagnetism (s., next section).
The background is more pronounced in the case $H\parallel\mbox{$a^{\ast}$}$
and one notes a drastic change of the background slope by cooling below
$T_{c}$, Fig. 10. A rather pronounced background present in
$H\parallel\mbox{$a^{\ast}$}$ orientation is a consequence of imperfect
crystal alignment allowing a mixing-in of ferromagnetic $H\parallel b$
component. Otherwise, the major difference between the $M-H$ characteristics
measured along $c$\- and $a^{\ast}$\- axis are about factor of 3-4 bigger
spin-flop fields $H_{SF}^{\mbox{$a^{\ast}$}}$, in comparison with $H_{SF}^{c}$
at the same temperatures. The difference is attributed to geometrical reasons:
in spin-flop transition the effective field component is only the one which is
aligned along the preferred axis fus1 . From the spin-flop field values
measured at 20 K ($H_{SF}^{\mbox{$a^{\ast}$}}=4.7$ T and $H_{SF}^{c}=1.3$ T)
and crystal axes geometry (Inset to Fig. 3) one easily determines that the
axis compatible with the minimum spin-flop field closes in the
($\mbox{$a^{\ast}$},c$)-plane the angle $\phi$
($\tan\phi=\frac{H_{SF}^{c}}{H_{SF}^{\mbox{$a^{\ast}$}}}$) with the $c$-axis.
From the latter observation one concludes that the effective preffered axis is
actually declined from the $c$-axis for the angle $\phi(\approx 15^{0})$.
The observed sharpness of the spin-flop transition (Fig. 9) represents a
direct consequence of a sizable magnetic anisotropy characterizing the system.
Generally, in a spin flop transition the spins of the sublatices rotate at
$H_{SF}$ to the direction perpendicular to the field (and the preferred axis)
direction. By subsequent field increase spins are coherently rotated to become
aligned with the field only at the field value $H_{c}$. At $T=0K$, the ratio
of the two critical fields is expected to obey the relation Carlin1977
$\frac{H_{c}(0)}{H_{SF}(0)}=(2\frac{H_{E}}{H_{A}}-1)^{1/2}$. Here, $H_{E}$ and
$H_{A}$ represents the mean-field exchange field and the anisotropy field,
respectively. Hence, in cases with inherently big anisotropy, such that
$H_{A}$ approaches $H_{E}$, $H_{c}$ is not much bigger than $H_{SF}$ implying
almost direct reorientation of a sublatice spin into the direction of
preferred (i.e., magnetic field) axis. Such a transition is usually referred
to as _spin flip_ and represents a generic feature of metamagnets Carlin1977 .
Co7(TeO3)4Br6 may therefore be considered as a typical metamagnet system.
Temperature dependence of the field $H_{SF}$ is shown in Fig. 11. Although
there is no known generic analytic form of $H_{SF}(T)$, for metamagnets and
antiferromagnets in general, the observed approximately linear temperature
dependence is not very common Carlin1977 . We note however that in the case of
Co7(TeO3)4Br6 spin flip transition takes place within the magnetic structure
which involves a ferromagnetically ordered component (along b-axis, next
section) thus deviation from behavior known for other
antiferromagnets/metamagnets should not be any surprising. Also, we point out
that, in view of complex three-dimensional magnetic structure (Fig. 4) the
term ‘spin flip’ cannot be applied literary in its text-book meaning, i.e., to
mimic complete reversal of pairs originally anti-parallel oriented spins into
the direction of preferred axis. From the magnitude of magnetization jump one
cannot associate spin-flipping to any particular ion pair belonging to the two
sublatices, which could perform, hypothetically, spin reversal isolated from
the rest of structure. Instead, it’s more probable that in the energy
landscape of magnetic structure as a whole there are two neighboring minima
becoming equal in energy by the application of magnetic field $H_{SF}$ along
the effective preferred axis.
Figure 10: (Color online) Magnetization (per one Co ion) vs. field curves for
$a^{\ast}$ direction at three characteristic temperatures. Spin-flop
transitions are marked with arrows. Figure 11: (Color online) Temperature
dependence of spin-flip field $H_{SF}^{c}$, high field saturation
magnetization $M_{sat}$ and the slope in $MH$ curve at high fields
$\chi_{HF}$, all for magnetic field applied along $c$-axis.
#### III.4.2 Ferromagnetic response
When the magnetic field is applied along the $b$ axis, along which the
ferromagnetic component exists, remarkably different behavior is observed,
Fig. 12. Below $T_{N}$ a small kink develops in the $M-H$ curves but contrary
to the $H\parallel\mbox{$a^{\ast}$},c$ cases it shifts towards lower values as
temperature is decreased. On further cooling below $T_{C}$ a narrow, almost
rectangular hysteresis opens up around zero, indicating a formation of
ferromagnetic domains. Initial magnetization also shows ferromagnetic
character, achieving the saturation value in the virgin curve by applying only
100 Oe at 10 K.
Two characteristic features of a hysteresis loop are remanence and coercivity.
Below $T_{C}$ the remanent moment is practically constant with a value
$\mu_{rem}\approx 0.625\mu_{B}$ (per one Co ion). On the other hand, the
coercive field shows a linear dependence on temperature in the log-lin plot,
as indicated in the inset of Fig. 13. The relation
Figure 12: (Color online) Main Panel: Magnetic hysteresis loops for the
$b$-axis direction at different temperatures. Magnetization is scaled to one
Co ion. Inset: Virgin hysteresis curve at 2 K. At higher temperatures there is
a more abrupt magnetization transition to quasi-saturation by the application
of magnetic field of the order of 100 Oe or less.
$H_{C}(T)=H_{C}^{0}\cdot e^{-\alpha T}$ (6)
has been found to describe well the behavior of many nanostructured magnetic
systems, like thin magnetic films Vertesy1998 and amorphous systems Ribas1995
; Read1984 ; Cresswell1990 ; Pajic2007 . The common feature in those systems
was the presence of magnetic clusters with the well defined anisotropy
barrier, where jumps of magnetic moments of the clusters over the barriers are
temperature assisted.
The prominent feature of ferromagnetic order in Co7(TeO3)4Br6 is a very sharp
transitions between the two saturation states, giving rise to almost
rectangular hysteresis. Very often, rectangular hysteresis is observed in
multilayers Nakajima1993 ; Weller2001 where magnetic and nonmagnetic layers
are stacked on top of each other (Co and Pt for example Weller2001 ). Besides
the choice of the constituent materials, magnetic properties of multilayers
depend strongly on thickness of the individual layers, as well as on the
growth process. A sudden reversal of magnetization occurs when for a critical
field one nucleation site for a reversed domain is generated and the avalanche
effect is propagated throughout the material via strong exchange coupling. To
the best of our knowledge, Co7(TeO3)4Br6 is the first ‘non-multilayer’
material exhibiting the effect of rectangular hysteresis. In view of its
layered structure (stack of bc-plane layers), embedding the ferromagnetic
component within the layers, the latter interpretation seems at least as a
consistent possibility.
Alternatively, the rectangular hysteresis might actually be underlined by
Stoner-Wohlfarth single-domain model Morrish2001 ; Blundell2001 . Our single-
crystalline Co7(TeO3)4Br6 samples are certainly to big to represent a
monodomain below $T_{C}$, as one easily verifies from the fact that below
$T_{C}$ the net magnetization along $b$-axis (as well as along any other axis)
in $H_{dc}=0$ is $M=0$. From the initial (virgin) curve (Fig. 12) one notes
however that a quasi-saturation is achieved already in a very small applied
field rendering the sample practically monodomain in any bigger fields. For
this reason consideration of $M-H$ hysteresis in terms of Stoner-Wohlfarth
model makes sense. In the latter model the total magnetic energy consists of
the two terms, the energy of uniform magnetization in the field sweeping up
and down and the uniaxial anisotropy energy of magnetization $M$ as it gets
deflected from the preferred axis. Total energy minimization results with the
hysteretic $M-H$ curves, the shapes of which are precisely determined by the
direction of applied field with respect to magnetization/preferred axis
Morrish2001 . In the geometry of magnetic field aligned with the
magnetization, being realized in this study (Fig. 12), Stoner-Wohlfarth model
generates strictly rectangular hysteresis loops characterized by coercive
field equal to anisotropy field $H_{C}=2K/M$ ($K$ is a constant of uniaxial
anisotropy). At this field value the energy of magnetization in ramping
applied field just overcomes the anisotropy energy enabling spin reversal to
take place.
Figure 13: Temperature dependence of the coercive field for b-axis field
direction. Solid line is guide for the eye only.
In the particular case of Co7(TeO3)4Br6 it is interesting to note that the
effective preferred axis is oriented approximately along $c$-axis, not along
the magnetization axis ($b$-axis in our case) as is the case in the original
formulation of Stoner-Wohlfarth model. In the case of Co7(TeO3)4Br6 one has to
bear in mind that ferromagnetic moment is just a component of canted three-
dimensional order thus representing, as pointed out in this article’s title, a
byproduct of global antiferromagnetism. It is also interesting to compare
values of the two critical fields $H_{SF}$ and $H_{C}$ at the same
temperatures. $H_{SF}$ is systematically bigger than $H_{C}$ for,
approximately, an order of magnitude. This finding is consistent with
elementary understanding of dynamics of magnetically ordered systems: To
manipulate ferromagnetically ordered spins one has to apply field at the order
of anisotropy field $H_{A}$ while to do the same with antiferromagnetically
ordered spins one has to apply much larger, exchange-enhanced field
$(H_{A}H_{E})^{1/2}$ (one bears in mind that $H_{E}\gg H_{A}$). The latter
aspect has recently been pointed out Kimel2004 in the context of requirements
for ultrafast spin reorientation technologies.
## IV Discussion
In order to discus the results let us first recapitulate the main observations
presented herewith.
At $T_{N}$, a long-range magnetic ordering of Co7(TeO3)4Br6 takes place, with
a temperature dependent and incommensurate propagation wave vectorfus2
$k_{ICM}$. In direct space the latter temperature dependence might be
associated with global spin reorientations giving rise to ferromagnetic
component along the $b$-axis, setting in at $T_{C}$. Antiferromagnetic
response along the two other perpendicular axes ($a^{\ast}$\- and $c$-axis) is
kept unchanged, however. Upon lowering the temperature below $T_{C}$, the
commensurate (CM) structure is stabilized and coexists with the ICM structure
in the short temperature interval. The antiferromagnetic backbone of the CM
structure provides a natural explanation for the spin flop (or spin flip)
transitions and rectangular-shaped hysteresis loops observed in sweeping the
dc field along $a^{\ast}$,$c$-axis, and $b$ axis, respectively. When the first
nucleation center for the reversed domain is created, the backbone structure
becomes unstable relative to the 1800 rotation of all the moments, reversing
the total magnetization in an extremely narrow field interval ($\sim 100$ Oe
at 20 K and $\sim 400$ Oe at 5 K).
In an attempt to interpret these observation one has to point out that low
C2/c symmetry and 4 different (distorted) environments for magnetic ions make
the modeling for Co7(TeO3)4Br6 extremely difficult. Instead of attempting the
latter here we just elaborate magnetic interactions and mechanism found
responsible for complex magnetic ordering of Co7(TeO3)4Br6. Most obviously,
there are at least two equally important interactions ruling magnetism of this
compound: exchange interaction and single ion anisotropy energy.
Exchange interactions provide a necessary framework for magnetic ordering to
set in. As the order established at $T_{N}$ is incommensurate the exchange
interactions $J_{ij}$ between magnetic moments of Co2+ ions necessarily
extends beyond nearest neighbors, differing in size as well in sign. Single
ion anisotropy plays pronounced role in ruling anisotropic paramagnetic
susceptibility. In the ordered phase it provides dominant, or at least
sizable, contribution to macroscopic magnetocrystalline anisotropy, $K$.
(Other possible contributions to $K$ rely on magnetic-dipolar anisotropy and
exchange-interaction anisotropy, see, e.g. Ref. Besser1967, ). In the ordered
phase $K$ underlays the phenomenon of spin fl(o)ip, introducing the
temperature through temperature dependence of magnetization Zener1954 .
Now, just on ground of specific exchange interaction network $J_{ij}$
compatible with the incommensurate order and the explicit temperature
dependence of magnetocrystalline energy one is able to interpret, at least
qualitatively, several important features of Co7(TeO3)4Br6 in the ordered
state, like the temperature dependence of $k_{ICM}$ and a sudden appearance of
ferromagnetism at $T_{C}$.
Depending on whether the exchange integrals $J_{ij}$ are considered as
isotropic or anisotropic there are two possible scenarios. In the first, one
notes a striking similarity of magnetic ordering patterns of Co7(TeO3)4Br6 and
rare-earth metals Dysprosium (Dy) and Terbium (Tb). The latter elements
acquire incommensurate-helical order at their particular $T_{N}$ featuring
temperature dependent $k_{ICM}$ and ferromagnetic transition, taking place a
few degrees below $T_{N}$. By keeping aside itinerant-magnetism peculiarities,
generally important for Dy and Tb, the ordering phenomenology itself can
convincingly be interpreted Nagamiya1967 on basis of an interplay between the
isotropic exchange integrals and temperature dependent magnetocrystalline
energy. Closer examination of the driving force responsible for ferromagnetism
of Dy and Tb showed however that magnetostriction plays perhaps a more direct
role than the temperature dependence of single ion anisotropy Cooper1967 .
Whether magnetostriction sets in at $T_{c}$ in Co7(TeO3)4Br6 as well is not
fully resolved as yet. A high-resolution diffraction study is needed to detect
magnetostriction-related structural changes in vicinity of $T_{c}$.
In the second scenario the exchange interaction is considered as being
anisotropic, either due to directional dependence of exchange integrals or due
to antisymmetric form of the related spin-spin operator in the interaction
Hamiltonian. In the former case the scalars $J_{ij}$ are replaced by the
appropriate tensor. Competition of anisotropic exchange with uniaxial single
ion anisotropy, as analyzed in mean-field approximation deNeef1974 , gives
rise to different types of possible spin orders minimizing the Gibbs free
energy, allowing for spin reorientations below corresponding
temperaturesLevison1969 . In the latter case anisotropic exchange interaction
is manifested as an antisymmetric (Dzyaloshinsky-Morya, DM) spin-spin
interaction. The DM interactionMoriya1960PRL is directly responsible for
numerous cases of systems revealing weak ferromagnetism in canted
antiferromagnetsCarlin1977 ; Blundell2001 . As ferromagnetism in Co7(TeO3)4Br6
is certainly of canted type (Fig. 4), emerging abruptly from global
antiferromagnetism, it is plausible to assume relevance of DM interaction, for
Co7(TeO3)4Br6 as well. With this respect the mechanism ruling the spin
reorientation in Co7(TeO3)4Br6 could be closely related to spontaneous spin-
flip (Morin transition), taking place deep in antiferromagnetic phase of
hematite, $\alpha-Fe_{2}O_{3}$, being accompanied by a DM-based ferromagnetic
component. A driving force for the Morin transition is identified in
competition of single-ion anisotropy with long-range dipolar anisotropy term
Artman65 .
As pointed out by Becker and coworkers Becker2006 , Co7(TeO3)4Br6 is composed
of chains running along the $b$ axis. The chains contain Co(2) and Co(3) ions
and are interconnected with Co(1) ions to form the $bc$ layers. The layers are
linked through the Co(1) - Co(4) - Co(1) connection. As indicated by AOM
calculations, Co(1) and Co(4) ions are well described in the framework of
single-ion anisotropy even in the low temperature limit. On the other hand,
Co(2) and Co(3) ions seem to be substantially influenced by the exchange
interactions through the ligands, indicating a good connection along the chain
direction and rather weak perpendicular to them. The preliminary inelastic
neutron scattering experiments along the $c$ direction indicate the presence
of the dispersionless mode around 4 meV, pointing to a weak coupling along the
$c$ direction, corroborating the AOM results.
## V Conclusions
In its ground state Co7(TeO3)4Br6 is a three-dimensional canted magnetically
ordered system revealing antiferromagnetically compensated sublattices in all,
but the $b$-axis direction. The low temperature magnetic structure is
stabilized through temperature dependent incommensurate wave vector
accompanied by an abrupt emergence of ferromagnetic component along $b$-axis.
Magnetic susceptibility studies demonstrate extreme anisotropy characterizing
Co7(TeO3)4Br6. Although the ferromagnetic component establishes strictly along
the $b$-axis, the effective preferred axis, imposed by a sizable single ion
anisotropy term of octahedrally coordinated Co2+ ions, is directed
approximately along $c$-axis. Hidden in a three-dimensionally canted spin
arrangement the latter axis represents, as clearly shown in susceptibility and
$M-H$ studies, a real ‘backbone’ of antiferromagnetic order. Accordingly,
magnetic field (or its component) has to be applied along this axis to induce
entirely reversible spin flop transition. Closer examination of $M-H$ scans
taken along $c$\- and $a^{\ast}$\- axis shows that the effective preferred
axis is actually declined from the $c$-axis for an angle of approximately
$15^{0}$. From the extreme sharpness of the transition it is concluded that a
phenomenon of spin flip (instead of a spin flop) better describes the
observations, classifying Co7(TeO3)4Br6 into the category of metamagnets.
Ferromagnetic response, restricted to the direction of $b$-axis only, has been
related to the phenomenology of multilayers and/or of the Stoner-Wohlfarth
model due to strikingly rectangular hysteresis loops. Co7(TeO3)4Br6 obviously
represents a remarkable magnetic system manifesting competition of various
magnetic interactions. In the competition there is primarily a complex network
of exchange interactions and a single ion anisotropy energy. Most probably the
single ion anisotropy represents just one possible component of a more complex
magnetocrystalline anisotropy, relevant for energy balance in the ordered
phase, and in the article other alternatives are discussed along the lines of
knowledge accumulated during the decades.
## VI Acknowledgments
M.P., I. Ž., and D.P. acknowledge financing from the projects 035-0352843-2845
and 119-1191458-1017 of the Croatian Ministry of Science, Education and Sport.
M.P. thanks Djuro Drobac, Institute of Physics, Zagreb, for help in ac
susceptibility measurements. D.P. is grateful to Krešo Zadro, Dept. of
Physics, Faculty of Science, University of Zagreb, for advices related to
magnetization measurements. H.B. thanks the NCCR research pool MaNEP of the
Swiss National Science Foundation for support in sample preparation. Neutron
facilities of SINQ, Paul Scherrer Institute, Villigen, Switzerland are also
gratefully acknowledged.
## References
* (1) T. Moriya, Phys. Rev. 117 635 (1960).
* (2) See, e.g., C.Lhuillier and G.Misguich in _Frustrated Quantum Magnets_ , Lecture Notes in Physics, Springer Berlin/Heilderberg, Volume 595, (2002), and references therein.
* (3) See, e.g., A.N. Bogdanov, U.K.Rőssler, M.Wolf, and K.-H. Műller, Phys.Rev. B 66, 214410 (2002), and references therein.
* (4) T. Nagamiya, in Solid State Physics, edited by F.Seitz and D.Turnball, Academic Press, New York, Vol. 20 (1967).
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* (6) P.J. Besser, A.H. Morrish and C.W. Searle, Phys. Rev. 153, 632 (1967)
* (7) S.M. Shapiro, J.D. Axe and J.P. Remeika, Phys.Rev.B 10, 2014 (1974), and references therein.
* (8) R. Becker, M. Johnsson, H. Berger, M. Prester, I. Zivkovic, D. Drobac., M. Miljak, M. Herak, Solid State Sci. 8, 836 (2006).
* (9) J. Rodriguez-Carvajal, Physica B 192, 55 (1993).
* (10) O. Kahn, Molecular Magnetism, Wiley-VCH Line, New York (1993).
* (11) R.L. Carlin and A.J. van Duyneveldt, Magnetic properties of Transition Metal Compounds, Springer-Verlag, New York (1977).
* (12) A.V.Kimel, A.Kirilyuk, A. Tsvetkov, R.V. Pisarev, and Th. Rasing, Nature 429, 850 (2004).
* (13) R. Becker, M. Prester, H. Berger, P.H. Lin, M. Johnsson, Dj. Drobac, I. Zivkovic, J. Solid State Chem. 180, 1051 (2007).
* (14) O. Waldmann, M. Ruben, U. Ziener, P. Müller, J.M. Lehn, Inorg. Chem. 45, 6535 (2006).
* (15) The attempts to study anisotropy of g-factor directly by ESR failed because of intrinsically fast relaxation of Co ion (private communication with Andrej Zorko).
* (16) To avoid confusion, we intentionally avoid to term this _preferred_ axis _easy_ axis. The reason is the presence of another axis, $b$-axis, defining the directions of ferromagnetic component, which might be colloquially termed, in jargon of practical ferromagnetism, the _easy_ axis.
* (17) G. Vértesy and I. Tomáš, Acta Phys. Slovaca 48, 663 (1998).
* (18) The spin flop phase is naturally absent if the field is perpendicular to the preferred axis, s., e.g., Ref.Carlin1977,
* (19) R. Ribas, B. Dieny, B. Barbara and A. Labarta, J. Phys.: Condens. Matter 7, 3301 (1995).
* (20) D.A. Read, T. Moyo and G.C. Hallam, J. Magn. Magn. Mater. 44, 279 (1984).
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* (23) S. Blundell, Magnetism in Condensed Matter, Oxford University Press, New York (2001).
* (24) Due to the strong neutron absorption cross section of Co ions and the specific, plate-like shape of the crystals, the determination of the low-symmetry ICM structure for $T_{C}<T<T_{N}$ was not successful.
* (25) J. Nakajima, A. Takahashi, K,. Ohta and T. Ishikawa, J. Appl. Phys. 73, 7612 (1993).
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* (27) See, e.g., A.H. Morrish, in The Physical Principles of Magnetism, IEEE Press, (An IEEE Press Classical Reissue), New York, p.344 (2001).
* (28) C. Zener, Phys. Rev. 96, 1335 (1954).
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* (32) P. L. W. Tregenna-Piggott, H. Weihe, and A.-L. Barra, Inorg. Chem., 42, 8504-8508 (2003).
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* (34) Q.Scheifele, C.Riplinger, F Neese,; H.;Weihe, A.-L.Barra, F.Juranyi, A. Podlesnyak,; P. L. W. Tregenna-Piggott,. Inorg. Chem., 47, 439-447 (2008).
* (35) F. E. Mabbs, D. Collison, Electron Paramagnetic Resonance of Transition Metal Compounds, Elsevier, New York (1992).
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|
arxiv-papers
| 2009-01-30T13:25:41 |
2024-09-04T02:49:00.305550
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.Prester, I.Zivkovic, O.Zaharko, D.Pajic, P.Tregenna-Piggott, and\n H.Berger",
"submitter": "Mladen Prester",
"url": "https://arxiv.org/abs/0901.4878"
}
|
0902.0032
|
# The emergence of gravity as a retro-causal post-inflation macro-quantum-
coherent holographic vacuum Higgs-Goldstone field
Jack Sarfatti1 and Creon Levit2 1 Internet Science Education Project 2 NASA
Ames Research Center adastra1@mac.com creon.levit@nasa.gov
###### Abstract
We present a model for the origin of gravity, dark energy and dark matter:
Dark energy and dark matter are residual pre-inflation false vacuum random
zero point energy ($w\\!=\\!-1$) of large-scale negative, and short-scale
positive pressure, respectively, corresponding to the “zero point”
(incoherent) component of a superfluid (supersolid) ground state. Gravity, in
contrast, arises from the 2nd order topological defects in the post-inflation
virtual “condensate” (coherent) component. We predict, as a consequence, that
the LHC will never detect exotic real on-mass-shell particles that can explain
dark matter $\Omega_{\mathrm{DM}}\approx 0.23$. We also point out that the
future holographic dark energy de Sitter horizon is a total absorber (in the
sense of retro-causal Wheeler-Feynman action-at-a-distance electrodynamics)
because it is an infinite redshift surface for static detectors. Therefore,
the advanced Hawking-Unruh thermal radiation from the future de Sitter horizon
is a candidate for the negative pressure dark vacuum energy.
## 1 Gravity from topological singularities in the quantum vacuum
We consider the possibility that the Einstein-Cartan 1-forms consistent with
1915 General Relativity (GR) are local macro-quantum emergent supersolid [1]
c-number fields. We mean this in the same sense that $v$ (the locally
irrotational superflow 3D Galilean relativity group velocity 1-form in
superfluid 4He) is emergent, with quantized circulation. The single-valuedness
of the associated $S^{1}$ macroquantum coherent Higgs-Goldstone order
parameter $\Psi=|\Psi|e^{i\Theta}$ emerges from an effective spontaneous
broken [2] non-electromagnetic $U(1)=O(2)$ ground state gauge symmetry111This
corresponds to Hagen Kleinert’s multi-valued singular phase transformations
(discussed elsewhere in these proceedings). .
$v=\frac{1}{2\pi}\frac{h}{m}\mathrm{d\Theta}$ (1) $\oint
v=\frac{1}{2\pi}\frac{h}{m}\oint d\Theta=n\frac{h}{m}\\\ $ (2) $n=\pm 1,\pm
2,...$ (3)
In analogy to the above, we use a phenomenological model for the moment of
inflation with eight macroquantum coherent relative phase 0-forms $\Theta^{I}$
and $\Phi^{I}$ that form two Lorentz group 4-vectors with magnitudes $\Theta$
and $\Phi$, respectively. These magnitudes, in turn, are the phases of a
$S^{2}$ vacuum order parameter manifold. This $S^{2}$ fiber bundle over real
spacetime supports stable point monopole topological defects (simultaneous
nodes of the three corresponding Higgs fields) [3].
These GeoMetroDynamic (GMD) point monopoles are the lattice points in
spacelike slices of Hagen Kleinert’s “world crystal lattice” [4]. They
correspond to a non-trivial 2nd homotopy group of emergent effective post-
inflation field $O(3)$ mappings of surrounding non-bounding 2-cycles S23D in
3D physical space to the vacuum manifold S2.
The inhomogeneities in all eight phases $\Theta^{I}$ and $\Phi^{J}$ form the
emergent GMD tetrad field $A^{I}$. For details, see equations 14-20, below.
In the world hologram conjecture [5][6], with total hologram screen area A,
the mean separation $\Delta$L of the lattice points is for our pocket universe
c/H${}_{0}=\sqrt{N}L_{P}=10^{29}$ cm on the cosmic landscape of the megaverse,
given by[7]:
$\Delta
L=\left({\mathrm{L{}}}_{P}{}^{2}\sqrt{A}\right){}^{1/3}=\left({\mathrm{L{}}}_{P}{}^{2}\frac{c}{{\mathrm{H{}}}_{0}}\right){}^{1/3}\
\approx 10{}^{-13}\mathrm{cm}\\\ $ (4)
$N=\frac{A}{{4\mathrm{L{}}}_{P}{}^{2}}=\frac{1}{4\Lambda{{\mathrm{L{}}}_{P}{}^{2}}}=\frac{A^{\left(3/2\right)}}{\left(\Delta
L\right){}^{3}}=\frac{V}{\left(\Delta L\right){}^{3}}\\\ $ (5) $A=\partial V$
(6)
where $\partial$ is the quasi-boundary operator because the surrounding future
light cone surface is a non-bounding de Rham 2-cycle. This 2-cycle encloses
$N$ GMD point monopole defects in the three effective real $O(3)$ Higgs field
macroquantum coherent Penrose-Onsager off-diagonal-long-range-order post-
inflation vacuum parameters[8].
Wrapping once around S23D through solid angle 4$\pi$ wraps an integer $N$
times round the vacuum manifold S2. This is analogous to the global quantized
circulation vortices in superfluid 4He that are the stable topological defects
in the first homotopy group for an $O(2)$ mapping with only a single relative
Goldstone phase $\Theta$ and two real Higgs scalars $\Psi$1, $\Psi$2 (instead
of the three $\Psi$1, $\Psi$2,$\Psi$3 with two relative Goldstone phases
$\Theta$, $\Phi$ over 3D spacelike slices of physical spacetime in our toy
model).
This wrapping integer $N$ is the explanation for the Bekenstein bit quantized
areas of null black hole event horizons and observer-dependent cosmological
(e.g. dark energy future de Sitter) horizons222In this paper we use the term
“de Sitter horizon” informally. A more precise description would be “future
cosmological event horizon”. of area A that obey Hawking’s entropy and
temperature formulas
$S={\mathrm{k{}}}_{B}A/4{\mathrm{L{}}}_{P}{}^{2}={\mathrm{Nk{}}}_{B}\\\ $ (7)
$T=\frac{\partial E}{\partial
S}=\frac{\mathrm{hc}}{{\mathrm{k{}}}_{B}\sqrt{A}}=\frac{\mathrm{hc}\sqrt{\Lambda}}{{\mathrm{k{}}}_{B}}=\frac{\mathrm{hc}}{{\mathrm{k{}}}_{B}{\mathrm{L{}}}_{P}\sqrt{N}}$
(8)
$\displaystyle\int\mkern-7.2mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}
\end{picture}\mkern-7.0mu\int 2d\Theta\wedge d\Phi=4\pi
N=A/4{\mathrm{L{}}}_{P}{}^{2}=10{}^{124}\ \mathrm{Bekenstein}\ \mathrm{BITs,}$
(9)
where the double integral around the vacuum manifold is induced by a single
wrap around the future asymptotic de Sitter horizon. The de Sitter horizon is
a surrounding (but non-bounding [9]) closed 2-cycle at lightlike conformal
infinity. It is also a stretched thermal horizon for comoving observers in the
accelerating Hubble flow [5].
The remaining six Goldstone phase angles form the Calabi-Yau space of string
theory - the same field Gennady Shipov calls the “oriented point” [10].
Einstein’s 1915 curvature field is simply the local gauge field from the
4-parameter translation universal spacetime symmetry group $T(4)$ for all
matter fields (i.e., strong equivalence principle) with the constraint of zero
torsion.
Locally gauging the 10-parameter Poincare group $P(10)$ of Einstein’s 1905
special relativity gives the Einstein-Cartan theory of (dislocation defect)
torsion[4] in addition to (disclination defect) curvature333There are two
classes of defects: The monopole defects which form the “atoms” of the
supersolid world crystal lattice, and the disclinations and dislocations in
this lattice, which account for, respectively, the curvature and torsion of
spacetime. of the symmetric Levi-Civita connection. Indeed, this local gauge
field model can be reinterpreted in terms of the eight multi-valued Goldstone
phases of the coherent post-inflation vacuum field. The Calabi-Yau space seems
to be simply the torsion field in disguise.
## 2 Dark energy from the future
The future de Sitter event horizon world hologram is “our past light cone at
the end of time”[11]. It can be pictured as a pixelated spherical shell of
area NLP2 infinitely far from our detectors (in proper time) on their future
light cone, with thickness LP and duration LP/c. This shell, or “screen”, has
4D volume NLP4 with dark energy density hc/(4DVolume Hologram Screen). This
screen projects the voxels of our accelerating expanding 3D space hologram
image back from the future - indeed, back to the moment of inflation 13.7
billion years ago in what Igor Novikov calls a “globally self-consistent”
strange loop in time.
To summarize: The area of an observer’s future de Sitter horizon
holographically determines the dark energy density seen by that observer.
For a static local-non-inertial-frame (LNIF) observer (with covariant
acceleration $g=c^{2}/\sqrt{N}L_{P}=cH_{0}\approx 10^{-9}$ m/sec2 relative to
the $\Lambda\\!=\\!0$, $k\\!=\\!0$ spatially flat post-inflation background
Friedman metric) the Hawking-Unruh temperature of the future de Sitter horizon
is proportional to his acceleration. This is in similar to a static outside
observer adiabatically approaching the horizon of a black hole – being “slowly
lowered down on a cable” – who measures a temperature which approaches the
Planck temperature hc/LPkB as he approaches the horizon.
Of course, the locally coincident geodesic observer relative to the Friedman
metric sees no heat radiation - only $w=-1$ positive dark zero point energy
density vibrations of equal but opposite negative pressure per large space
dimension, as required by the Einstein Equivalence Principle (EEP).
In contrast the NASA WMAP isotropic black body radiation in the comoving
Friedman frame is coming along our past light cone from the surface of last
contact 380,000 years after the post-inflation reheating of the Big Bang, and
its energy density weakens $a(t)^{-4}$ as space expands because it has
$w=+1/3$ ratio of pressure to energy density444There is a lack of consensus
and clarity in the literature on who sees what. The Unruh effect in globally
flat Minkowski spacetime is: A covariantly proper accelerating local detector
(not on a timelike geodesic, which by definition has zero covariant proper
4-acceleration) sees thermal equilibrium blackbody radiation whose temperature
is proportional to its covariant 4-acceleration magnitude. In contrast, a
momentarily coincident non-accelerating detector sees only zero point vacuum
fluctuations instead of the thermal radiation. That is, some of the vacuum
fluctuation energy is converted into thermal radiation in the rest frame of
the intrinsically accelerating detector. Static detectors outside the event
horizon of an ideal Schwarzschild black hole are covariantly properly
accelerating in order to “stand still” at a fixed Schwarzschild radial
coordinate $r$. This is in accord with the actual Pound-Rebka Harvard Tower
experiment showing the gravity redshift using the nuclear Mossbauer effect.
Furthermore, the comoving detectors in the Robertson-Walker representation at
constant $\chi$ are analogous to the previous case at constant $r$ [11]:
$ds^{2}=-c^{2}dt^{2}+R(t)^{2}[d\chi^{2}+S_{k}^{2}(\chi)d\psi^{2}],$ However,
e.g., Davies & Davis [12] write “the response of a particle detector
travelling along a geodesic in a de Sitter invariant vacuum state; the
detector behaves as if immersed in a bath of thermal radiation” Of
course,“geodesic” depends on choice of the local GCT frame invariant gravity
tetrad field. Thus, the detector on a geodesic in the de Sitter gravity tetrad
field is actually properly accelerating with respect to the geodesic in the
zero cosmological constant Robertson-Walker gravity tetrad field. This is the
point of view we take in this paper and it agrees operationally with how the
dark energy data is actually interpreted. .
We propose that the dark energy zero point vacuum fluctuations measured by
non-rotating covariantly unaccelerated Local Inertial Frame (LIF) detectors
(on timelike geodesics relative to the physical spacetime multi-valued
connection local gauge field $\Gamma$$\mu$νλ that forms curved and torsioned
spacetime) appear as advanced Wheeler-Feynman quasi-thermal blackbody “Unruh
radiation”. It comes back from the future de Sitter horizon “perfect absorber”
with temperature that has order of magnitude
$T=\frac{\mathrm{hg}}{{\mathrm{ck{}}}_{B}}\rightarrow\frac{\mathrm{hc}}{{\mathrm{k{}}}_{B}{\mathrm{L{}}}_{P}\sqrt{N}}\approx\frac{\mathrm{hc}}{{\mathrm{k{}}}_{B}{\mathrm{L{}}}_{P}10{}^{62}}$
(10)
for covariantly accelerated Local Non-Inertial Frame (LNIF) detectors off
timelike geodesics.
For example, a static observer outside the event horizon of a non-rotating
black hole must covariantly accelerate away from the black hole radially with
$g=-\frac{GM}{r^{2}}\frac{1}{\sqrt{1-\frac{2GM}{c^{2}r}}}\sim T$ (11)
in order to stay at fixed r in the curved spacetime outside the black hole.
They need to fire their rocket engines in order to remain static. These static
LNIF observers see the event horizon as a “stretched membrane” with Unruh
temperature $T$. Coincident LIF observers do not see this at all. This is an
example of what Leonard Susskind calls “horizon complementarity” [5], in
analogy with Niels Bohr’s quantum complementarity of wave-particle duality
from the non-commutativity of the Lie algebra of observable operators on qubit
Hilbert space fibers over classical field configuration space.
The comoving observers that see an approximately isotropic WMAP Cosmic
Microwave Background (CMB) in our accelerating expanding universe are
analogous to the static LNIF observers in the Schwarzschild model where now
there is a universal acceleration $g\approx 10^{-9}$ m/sec2, which is the same
order of magnitude of the anomalous Sun-centered radial accelerations of the
two NASA Pioneer space probes beyond the orbit of Jupiter. This is a curious
coincidence that possibly has deeper significance, although it is surprising
to find Hubble’s parameter $H_{0}$ appearing on such a short scale local
metric field. In contrast LIF detectors see this advanced quasi-thermal Unruh
Wheeler-Feynman radiation as zero point vacuum fluctuation energy density
$\rho{}_{\mathrm{DE}}={\mathrm{string\ tension}}\times{\mathrm{vacuum\
curvature}}=\frac{\mathrm{string\ tension}}{\mathrm{area\ of\ future\ cosmic\
horizon}}$ (12)
$=\frac{\mathrm{hc}}{{\mathrm{NL{}}}_{P}{}^{4}}=\frac{\mathrm{hc}}{\left(10{}^{-2}\mathrm{cm}\right){}^{4}}=0.73\times{10}^{-29}\mathrm{grams}/\mathrm{cc}.$
It is as if there is an effective high frequency cutoff at
$c/10{}^{-2}\mathrm{cm}=3\times 10^{12}$ Hz for the $w=-1$ zero point dark
energy virtual photon vibrations with critical wavelength equal to the
geometric mean of the future de Sitter horizon scale and the Planck scale. The
world hologram model posits that the number N of interior 3D voxels N of size
$\Delta$L equals the number of 2D pixels of size LP on the world hologram
future de Sitter horizon.
## 3 Calabi-Yau from torsion. Brane theory from the 1970s
The four tetrad 1-form fields eI are the General Coordinate Invariant (GCI)
gravitational fields in Einstein’s 1915 GR. They form a single 4-vector under
the 6-parameter homogeneous Lorentz group $SO(1,3)$. The non-trivial
curvilinear 4D General Coordinate Transformations (GCT) connect covariantly
accelerating coincident LNIFs with g-forces on its rest detectors. A non-
gravity force is required to create a translational covariant acceleration.
Conservation of angular momentum maintains a rotating LNIF in the absence of
friction in deep space once the external torque is removed. The Lorentz group
transformations connect coincident covariantly non-accelerating LIFs with
vanishing g-forces. The tetrad field components e$\mu$I and their inverses eIμ
connect locally coincident LIFs with LNIFs. The LNIF curvilinear metric field
is gμν. The coincident LIF Center Of Mass (COM) metric
$\eta$${}_{{}_{\mathrm{IJ}}}$ is that of Minkowski space-time of Einstein’s
1905 SR. The Strong Equivalence Principle (SEP) implies for the absolute
differential local frame invariant ds
${\mathrm{ds{}}}^{2}=g{}_{\mu\nu}(\mathrm{LNIF}){\mathrm{dx{}}}^{\mu}{\mathrm{dx{}}}^{\nu}=\eta{}_{{}_{\mathrm{IJ}}}(\mathrm{LIF}){\mathrm{e{}}}^{I}{\mathrm{e{}}}^{J}$
(13)
The multi-valued Goldstone phase transformations in our toy model form a 4x4
M-Matrix of non-closed 1-forms where the non-trivial parts of the four
curvature-only tetrad 1-forms $A^{I}$ and the six non-trivial torsion field
spin connection 1-forms $\varpi$IJ = -$\varpi$JI are the diagonals and
antisymmetrized off-diagonal M-Matrix elements.
${\mathrm{M{}}}^{\mathrm{IJ}}={\mathrm{d\Theta{}}}^{I}\wedge{\Phi{}}^{J}-{\Theta{}}^{I}\wedge{\mathrm{d\Phi{}}}^{J}$
(14)
${\mathrm{dM{}}}^{\mathrm{IJ}}=-2{\mathrm{d\Theta{}}}^{I}\wedge{\mathrm{d\Phi{}}}^{J}$
(15) ${\mathrm{d{}}}^{2}=0$ (16)
${\mathrm{A{}}}^{I}=diag({\mathrm{M{}}}^{\mathrm{IJ}})$ (17)
${\mathrm{e{}}}^{I}={\mathrm{I{}}}^{I}+{\mathrm{A{}}}^{I}={\mathrm{e{}}}^{I}{}_{\mu}{\mathrm{e{}}}^{\mu}{}_{\mathrm{LNIF}}={\mathrm{e{}}}^{\mu}{}_{I}{\mathrm{e{}}}^{I}{}_{\mathrm{LIF}}$
(18) ${\mathrm{I{}}}^{\mu}{}_{I}={\delta{}}^{\mu}{}_{I}$ (19)
${\varpi{}}^{\mathrm{IJ}}={\mathrm{M{}}}^{\left[I,J\right]}$ (20)
The Einstein 1915 zero torsion field i.e. ${\varpi{}}^{\mathrm{IJ}}$= 0,
curvature field 2-form is
${\mathrm{R{}}}^{\mathrm{IJ}}={\mathrm{D\omega{}}}^{\mathrm{IJ}}={\mathrm{d\omega{}}}^{\mathrm{IJ}}+{\omega{}}^{I}{}_{K}\wedge{\omega{}}^{K}{}^{J}\\\
$ (21)
Where the torsion field 2-form in Einstein-Cartan theory beyond 1915 GR would
be
${\varpi{}}^{I}={\mathrm{De{}}}^{I}={\varpi{}}^{I}{}_{K}\wedge{\mathrm{e{}}}^{K}\\\
$ (22)
The 1915 GR Einstein-Hilbert pure gravity field action density is the 0-form
${\mathrm{L{}}}_{G}={\epsilon{}}_{\mathrm{IJKL}}{\mathrm{R{}}}^{\mathrm{IJ}}\wedge{\mathrm{e{}}}^{K}\wedge{\mathrm{e{}}}^{L}\\\
$ (23) $S_{G}=\int L_{G}d{}^{4}x.$ (24) $\frac{{\delta S{}}_{G}}{{\delta
e{}}^{I}}=0\\\ $ (25)
is the pre-Feynman action principle. It is the critical point pure gravity
vacuum classical field equation in the absence of matter field sources, that
in the usual tensor notation is
${\mathrm{R{}}}_{\mu\nu}=0.$ (26)
Adding the on-mass-shell matter fields of real quanta plus their off-mass-
shell virtual quanta, which contribute to the cosmological scalar
$\Lambda$${}_{\mathrm{zpf\ }}$field, replaces the above classical vacuum
curvature field equation. The vanishing functional derivative of the total
action $S$ with respect to the 4 GCT invariant tetrad 1-forms555The tetrad
1-forms are the compensating gauge fields from localizing the global space-
time universal translation symmetry group acting equally on all matter field
actions. i.e:
$\frac{\delta S}{{\delta e{}}^{I}}=0$ (27)
$S={\mathrm{S{}}}_{G}+{\mathrm{S{}}}_{M}+{\mathrm{S{}}}_{\mathrm{zpf}}$ (28)
${\mathrm{R{}}}_{\mu\nu}-\left(\frac{1}{2}R+\Lambda{}_{\mathrm{zpf}}\right){\mathrm{g{}}}_{\mu\nu}=-\frac{8\pi
G}{{\mathrm{c{}}}^{4}}{\mathrm{T{}}}_{\mu\nu}$ (29)
We can no longer assume the zero torsion field limit
($\varpi^{\mathrm{IJ}}=0$) of the Bianchi identity of 1915 GR. And so we dream
that the final theory will use a more general connection than that of Levi-
Civita. This more general connection is induced by locally gauging a more
general spacetime symmetry group (e.g. the Poincare[13], de Sitter or perhaps
the conformal group) instead of gauging $T(4)$ as is usually done in GR.
We use “;” and “$|$” to denote covariant differentiation with respect to the
LC connection and the more general connection, respectively, and so rewrite
the divergence of the Einstein equation as:
${\mathrm{G{}}}_{\mu\nu}{}^{|\nu}+\frac{\partial\Lambda{}_{\mathrm{zpf}\ \
}}{\partial{\mathrm{x{}}}^{\nu}}g^{\nu}_{\mu}+\frac{8\pi
G}{{\mathrm{c{}}}^{4}}{\mathrm{T{}}}_{\mu\nu}{}^{|\nu}=0$ (30)
This more general divergence (flow) equation with its additional stress-energy
currents suggests new channels inter-connecting the incoherent random vacuum
zero point fluctuations to the smooth coherent (generalized) curvature field.
However, it must be supplemented by a similar torsion field equation that
comes from the vanishing functional derivative of the total action S with
respect to the 6 dynamically independent spin connection $\varpi^{IJ}$
1-forms. The $\varpi^{IJ}$ form the compensating gauge field induced from
localizing the 6-parameter homogeneous Lorentz group $SO(1,3)$ that also acts
equally on all matter field actions, i.e.,
$\frac{\delta S}{{\delta\varpi{}}^{\mathrm{IJ}}}=0$ (31)
The extra-dimensional brane M-Theory, when mature, should predict a
renormalization group flow to larger gravity coupling strength in the short-
distance limit. This, when combined with the world hologram conjecture,
suggests Abdus Salam’s 1973 bi-metric f-gravity [14] formula
$G={\mathrm{G{}}}_{\mathrm{Newton}}(1+{\alpha e{}}^{-r/\Delta L})$ (32)
$\alpha\gg 1$
has been rediscovered by brane theorists [15] looking at the Calabi-Yau
(torsion) field.
## 4 Conclusions
In this model there is no quantum gravity in the usual sense of starting with
a classical field and quantizing it. Rather, we go the opposite way in the
spirit, though not the letter, of Sakharov’s 1967 proposal [16]. What hitherto
was called the classical gravity field is seen to be really an emergent
effective macro-quantum coherent c-number post-inflation vacuum field.
We claim the residual random negative-zero-point-pressure advanced virtual
bosons back-from-the-future manifest as the anti-gravitating universally
repulsive dark energy. This is because the future de Sitter horizon for a co-
moving observer in our universe is a Wheeler-Feynman perfect absorber – an
infinite red shift surface – just like a black hole event horizon is for a
static LNIF observer.
In contrast, we claim the universally attracting dark matter comes from
residual positive-zero-point-pressure virtual fermion-antifermion pairs. In
this picture, looking for real on-mass-shell dark matter particles in the LHC
or in underground WIMP detectors is like looking for the motion of the Earth
through the mechanical aether of Galilean relativity using Michelson and
Morley’s Victorian interferometer.
This model pocket universe, relative to our Earth-bound detectors, is created
nonlocally and self-consistently by what Igor Novikov and Kip Thorne call “a
loop in time” [17]. Advanced Hawking radiation is emitted from the future
cosmic (“de Sitter”) horizon at all times and is blue-shifted as it travels
into the past. However, only the advanced radiation emitted at time
$t_{trigger}\approx 4$ Gyr arrives back at $t=0$, where it ignites the big
bang (see figure 1.1 in [11]).
Advanced information (Hawking radiation) flows from the infinite future proper
time “Omega” $S^{2}$ horizon of the observer’s world line (shown in the lower
“conformal time” ($\tau$) diagram in [11] Fig 1.1) down the null cosmic event
horizon to its intersection with the null particle horizon (at
$t_{trigger}\approx 4$Gyr, $\tau\approx 32$Gyr). It flows back along the
particle horizon, winding up at the initial “Alpha” moment of inflation
completing what John Cramer calls a “transaction”[18].
Furthermore, looking at Fig 5.1 of [11] we see that the future cosmic de
Sitter horizon area $A_{c}\sim N(t)$, where $t$ is the proper cosmic
Robertson-Walker time (corresponding to spacelike hypersurfaces of maximal CMB
isotropy), rises from “zero” (1 Bit) to de Sitter asymptote
$N(t\to\inf)\approx 10^{124}$ Bits rather quickly. The advanced dark energy
thermal Hawking radiation reaching us now backward-through-time along our
future light cone is very close to the asymptotic value.
The final entropy of our (retro)-causal[18][19] universe is $Nk_{B}\approx
10^{124}$ Bekenstein bits, as mandated by the area of the future de Sitter
horizon, whereas the initial entropy of the universe is exactly $k_{B}=1$ bit,
as mandated by the (Planck) area of the initial singularity.
This shows very clearly why the cosmological arrow of time is aligned with the
thermodynamic arrow of time, solving Roger Penrose’s main objection [20] to
inflationary cosmology: why the early universe has relatively low entropy. We
need both retrocausality and the world hologram principle to properly
understand the Arrow of Time of the Second Law of Thermodynamics.
## 5 Background related reading
The works of John Wheeler and Richard Feynman [21], Fred Hoyle and Jayant
Narlikar [22], James Woodward [23], Michael Ibison [24], Robert Becker [25]
and John Cramer [18] on advanced electromagnetic waves, radiation reaction and
vacuum fluctuations, Leonard Susskind [5] and Jack Ng’s work on the world
hologram [7], Gennady Shipov on torsion fields [10], and finally Hagen
Kleinert’s work on singular multi-valued phase transformations as local gauge
transformations [4] are essential background reading. We have also found
Chapter 2 of Rovelli’s lectures on quantum gravity [26] very clear regarding
the use of Cartan forms in gravity theory.
During the preparation of this paper we be became aware of a publication
entitled: “Is dark energy from cosmic Hawking radiation?”[27] that relates
$w\\!=-1$ dark energy to observer-dependent Hawking radiation. However, those
authors do not clearly specify which horizon they are referring to. They omit
the (key) notion of retro-causality. We claim retrocausality is necessarily
implied when the correct cosmic horizon - the future lightlike de Sitter
horizon – is specified. Tamara Davis’s thesis [11] (especially her figures 1.1
and 5.1) clarifies much of the rampant confusion over this subtle issue.
## References
## References
* [1] Sarfatti J (1969) “Destruction of superflow in unsaturated 4He films and the prediction of a new crystalline phase of 4He with Bose-Einstein condensation” Phys. Lett. A 30 300
* [2] Sarfatti J and Stoneham M (1967) “The Goldstone theorem in the Jahn-Teller effect” Proc. Phys. Soc. 91 214\.
* [3] Thouless D (1998) Topological Quantum Numbers in Nonrelativistic Physics (River Edge, NJ: World Scientific)
* [4] Kleinert H (2008) Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation (River Edge, NJ: World Scientific)
* [5] Susskind L and Lindesay J (2005) An Introduction to Black Holes, Information and the String Theory Revolution (River Edge, NJ: World Scientific)
* [6] Hogan C (2008) “Measurement of quantum fluctuations in geometry” Phys. Rev. D 77 104031
* [7] Ng Y (2008) “Spacetime foam and dark energy” preprint arXiv:0808.1261v1 [gr-qc]
* [8] Yang C (1962) “Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors” Rev. Mod. Phys. 34 694
* [9] Kiehn R (2008) Non-equilibrium Thermodynamics from the Perspective of Continuous Topological Evolution: Vol 1 Non-Equilibrium Systems and Irreversible Processes (http:www.lulu.com/kiehn)
* [10] Shipov G (1998) A Theory of Physical Vacuum (Moscow: Russian Academy of Natural Sciences)
* [11] Davis T (2004) Fundamental Aspects of the Expansion of the Universe and Cosmic Horizons Ph.D dissertation (University of New South Wales: arXiv:astro-ph/0402278v1)
* [12] Davies P and Davis T (2003) “How far can the generalized second law be generalized?” preprint arXiv:astro-ph/0310522v1
* [13] Kibble T (1961) “Lorentz invariance and the gravitational field” J. Math. Phys. 2 212
* [14] Salam A and Strathdee J (1977) “Class of solutions for the strong-gravity equations” Phys. Rev. D 16 2668
* [15] Esposito-Farese G (2005) “Tests of Alternative Theories of Gravity” in proc. 33rd SLAC Summer Inst. on Particle Physics: Gravity In The Quantum World And The Cosmos ed Hewett J at al (Menlo Park, CA: SLAC) preprint http://www.slac.stanford.edu/econf/C0507252/proceedings.htm
* [16] Sakharov A (1968) “Vacuum quantum fluctuations in curved space and the theory of gravitation” Sov. Phys. Doklady 12 1040
* [17] Hawking S, Thorne K, Novikov I, Ferris T, Lightman A and Price R 2002 The Future of Spacetime (New York: W. W. Norton)
* [18] Cramer J (1986) “The transactional interpretation of quantum mechanics” Rev. Mod. Phys. 58 647
* [19] Price H (2008) “Toy Models for Retrocausality”preprint arXiv:0802.3230v1 [quant-ph]
* [20] Penrose R (2005) The Road to Reality: A Complete Guide to the Laws of the Universe (New York: Alfred A. Knopf)
* [21] Wheeler J and Feynman R (1945) “Interaction with the absorber as the mechanism of radiation” Rev. Mod. Phys. 17 157 and (1949) “Classical electrodynamics in terms of direct interparticle action” Rev. Mod. Phys. 21 425
* [22] Hoyle F and Narlikar J (1995) “Cosmology and action-at-a-distance electrodynamics” Rev. Mod. Phys. 67 113
* [23] Woodward J (1996) “Killing time” Found. Phys. Lett. 9 1
* [24] Ibison M (2006) “Are Advanced Potentials Anomalous?” in AAAS Conf. Proc. on Reverse Causation (preprint arXiv:0705.0083v1 [physics.gen-ph])
* [25] Becker, R. (2003) “A Gravitational Archipelago” in Proc. Int. High Frequency Gravitational Wave Working Group, P. Murad, R. M. L. Baker Jr. eds. (McLean, VA: Mitre Corp.)
* [26] Rovelli C. (2007) Quantum Gravity (Cambridge UK: Cambridge University Press)
* [27] Lee L et al (2008) “Is dark energy from cosmic Hawking radiation?” preprint arXiv:0803.1987v4 [hep-th]
|
arxiv-papers
| 2009-01-31T00:37:28 |
2024-09-04T02:49:00.321367
|
{
"license": "Public Domain",
"authors": "Jack Sarfatti and Creon Levit",
"submitter": "Creon Levit",
"url": "https://arxiv.org/abs/0902.0032"
}
|
0902.0095
|
# Hyperdeformation in the Cd isotopes: a microscopic analysis
H. Abusara and A. V. Afanasjev Department of Physics and Astronomy,
Mississippi State University, MS 39762, USA
###### Abstract
A systematics search for the nuclei in which the observation of discrete
hyperdeformed (HD) bands may be feasible with existing detector facilities has
been performed in the Cd isotopes within the framework of cranked relativistic
mean field theory. It was found that the 96Cd nucleus is a doubly magic HD
nucleus due to large proton $Z=48$ and neutron $N=48$ HD shell gaps. The best
candidate for experimental search of discrete HD bands is 107Cd nucleus
characterized by the large energy gap between the yrast and excited HD bands,
the size of which is only 15% smaller than the one in doubly magic HD 96Cd
nucleus.
###### pacs:
21.60.Jz,27.60.+j,21.10.Ma
## I Introduction
Hyperdeformation (HD) is one of critical phenomena in nuclear structure, the
study of which will considerably advance our knowledge of nuclei at extreme
conditions of very large deformation and fast rotation CRMF-HD ; Dudek . The
studies of HD will also contribute into understanding of the crust of neutron
stars, where extremely deformed nuclear structures are expected (see Ref.
stars and references therein). Although some experimental evidences of the
existence of HD at low K.98 ; C12 and high spin 152Dy-exp-1 ; 152Dy-exp-2 ;
Hetal.06 ; 60Zn-1 exist, the current experimental knowledge of HD is very
limited. New generation of detectors such as GRETA Greta and AGATA Agata
will definitely allow to study this phenomenon in more details. However, these
detectors will become functional only in the middle of next decade. Thus, it
is very important to understand whether new experimental information on HD can
be obtained with existing detectors such as GAMMASPHERE Gamma .
Theoretical efforts to study HD at high spin both in macroscopic+microscopic
(MM) method and in self-consistent approaches were reviewed in Ref. CRMF-HD .
Our recent study of HD within the framework of the cranked relativistic mean
field (CRMF) theory in the Z=40-58 part of nuclear chart CRMF-HD represents
the first ever systematic investigation of HD within the self-consistent
theory. The general features of the HD bands at high spin have been analysed
in Ref. CRMF-HD . In particular, it was concluded that the density of the HD
states in the vicinity of the yrast line is the major factor which decides
whether or not discrete HD bands can be observed. The high density of near-
yrast HD states will lead to a situation in which the feeding intensity will
be redistributed among many low-lying bands, thus drastically reducing the
intensity with which each individual band is populated. For such densities,
the feeding intensity of an individual band will most likely drop below the
observational limit of the modern experimental facilities. On the contrary,
the large energy gap between the yrast and excited HD configurations will lead
to an increased population of the yrast HD band, thus increasing the chances
of its observation.
The analysis of Ref. CRMF-HD , based on the energy gap between the last
occupied and first unoccupied routhians in the yrast HD configurations,
suggests that the density of the HD bands in the spin range where they are
yrast is high in the majority of the cases. It also indicates the Cd isotopes
as the best candidates for a search of discrete HD bands. However, one has to
remember that this type of analysis may be too simplistic because the
polarization effects induced by particle-hole excitations are neglected. In
particular, it can overestimate the size of the energy gap between the yrast
and excited HD configurations. Realistic analysis of the density of the HD
bands should include significant number of the HD configurations calculated in
a fully self-consistent manner with all polarization effects included. Such
analysis is time-consuming in computational sense and has been performed only
for 124Xe in Ref. CRMF-HD , but its extension to other nuclei is needed. Thus,
the goals of the current manuscript are (i) to perform a fully self-consistent
analysis of the density of the HD bands in the Cd isotopes and (ii) to find
the best nuclei in which experimental study of discrete HD bands can be
feasible with existing experimental facilities.
## II Theoretical framework and the details of the calculations
The calculations in the present manuscript are performed in the framework of
the CRMF theory without pairing KR.89 ; A150 using numerical scheme of Ref.
CRMF-HD . The CRMF equations for the HD states are solved in the basis of an
anisotropic three-dimensional harmonic oscillator in Cartesian coordinates
characterized by the deformation parameters $\beta_{0}=1.0$ and
$\gamma=0^{\circ}$ and oscillator frequency $\hbar\omega_{0}=41A^{-1/3}$ MeV
(see Ref. CRMF-HD for details). The truncation of basis is performed in such
a way that all states belonging to the shells up to fermionic $N_{F}$=14 and
bosonic $N_{B}$=20 are taken into account; this truncation scheme provides
sufficient numerical accuracy CRMF-HD . The NL1 parametrization of the RMF
Lagrangian RRM.86 is used in most of our calculations since it provides a
good description of the moments of inertia of the rotational bands in unpaired
regime in the SD and ND minima A150 ; ALR.98 ; A60 ; VRAL.05 , the single-
particle energies for the nuclei around the valley of $\beta$ stability ALR.98
; A250 and the excitation energies of the SD minima LR.98 . Other
parametrizations such as NL3 NL3 , NLSH NLSH , NLZ NLZ and NL3* NL3* are
used only to check the size of the HD gaps in the nuclei of interest.
Figure 1: Proton (top panel) and neutron (bottom panel) single-particle
energies (routhians) in the self-consistent rotating potential as a function
of the rotational frequency $\Omega_{x}$. They are given along the deformation
path of the yrast HD configuration in 107Cd and obtained in the calculations
with the NL1 parametrization of the RMF Lagrangian. Long-dashed, solid, dot-
dashed and dotted lines indicate $(\pi=+,r=+i)$, $(\pi=+,r=-i)$,
$(\pi=-,r=+i)$ and $(\pi=-,r=-i)$ orbitals, respectively. Solid (open) circles
indicate the orbitals occupied (emptied). The dashed box indicates the
frequency range corresponding to the spin-range $I=55-80\hbar$ in this
configuration. The arrows indicate the particle-hole excitations leading to
excited HD configurations.
Single-particle orbitals are labeled by $[Nn_{z}\Lambda]\Omega^{sign}$.
$[Nn_{z}\Lambda]\Omega$ are the asymptotic quantum numbers (Nilsson quantum
numbers) of the dominant component of the wave function at $\Omega_{x}=0.0$
MeV. The superscripts sign to the orbital labels are used sometimes to
indicate the sign of the signature $r$ for that orbital $(r=\pm i)$.
The excited HD configurations were built from the yrast HD configurations
obtained in the previous study CRMF-HD by exciting either one proton or one
neutron or both together. Proton and neutron configurations generated in this
way are labeled by $\pi$i and $\nu$j, where $i=0,1,2,...$ and $j=0,1,2,...$
are integers indicating the corresponding configurations. $\pi$0 $\otimes$
$\nu$0 represents the yrast HD configuration. Total excited configurations
$\pi$i $\otimes$ $\nu$j are constructed from all possible combinations of
proton $\pi$i and neutron $\nu$j configurations excluding the one with $i=0$
and $j=0$. The selection of excited configurations is also constrained by the
condition that the energy gap between the orbital from which the particle is
excited and the orbital into which it is excited do not exceed 2.5 MeV in the
routhian diagram for the yrast HD configuration. All configurations are
calculated in a fully self-consistent manner so that their total energies are
defined as a function of spin.
Fig. 1 illustrates the selection of excited configurations. It shows the
occupation of the proton and neutron orbitals in the yrast HD configuration in
107Cd. According to our criteria only three proton excitations across the
$Z=48$ HD gap are considered. On the contrary, more neutron $ph$-excitations
are allowed across the $N=59$ HD shell gap. Table 1 shows their detailed
structure. For example, the $\nu$1 configuration is created by exciting one
neutron from the [770]1/2+ into [413]7/2+ orbitals. One can notice that we
only consider the $ph$-excitations between the states which do not have the
same combination ($\pi$,$r$) of parity $\pi$ and signature $r$. The computer
code in general can handle the excitations between the states with the same
($\pi$, $r$), but the configurations based on such excitations are less
numerically stable and require more computational time. Because of this reason
and the fact that they do not alter significantly the results for the density
of the HD states, it was decided to neglect them in the calculations. However,
in the cases of large energy gaps between the yrast and excited HD
configurations, they are taken into account.
Table 1: Neutron particle-hole excitations in 107Cd shown in Fig. 1. label | Excitation
---|---
$\nu$1 | [770]1/2+ $\rightarrow$ [413]7/2+
$\nu$2 | [770]1/2+ $\rightarrow$ [413]7/2-
$\nu$3 | [532]3/2- $\rightarrow$ [413]7/2+
$\nu$4 | [532]3/2- $\rightarrow$ [413]7/2-
$\nu$5 | [651]3/2- $\rightarrow$ [413]7/2+
$\nu$6 | [651]3/2+ $\rightarrow$ [413]7/2-
Figure 2: (Color online) Energies of the calculated HD configurations in even-
even 96-106Cd nuclei relative to a smooth liquid drop reference $AI(I+1)$,
with the inertia parameter A=0.01. In each nucleus, the yrast and lowest
excited proton configurations are shown by solid lines. Dot-dashed and dotted
lines represent the yrast lines at low spin built from normal-deformed (ND)
and superdeformed (SD) states, respectively.
## III Discussion
Figs. 2 and 3 show the density of the HD states in even-even 96-108Cd and odd
mass 107,109Cd nuclei studied using above outlined procedure. The energy gap
between the yrast HD configuration and lowest excited HD configurations is
around 1.5 MeV in 96Cd (Fig. 2a). It is comparable with the energy gap between
the yrast and excited SD configurations in doubly magic SD nucleus 152Dy (Fig.
7 in Ref. A150 ). This energy gap in 96Cd is due to large energy cost of
particle-hole excitations across the $Z=48$ and $N=48$ HD shell gaps which
have similar size (see Fig. 1 and Table 2). All that together indicates that
the 96Cd is a doubly magic HD nucleus. Only proton excitations to the
$[420]1/2^{-}$ orbital above the $Z=48$ HD shell gap result in bound excited
proton configurations, the excitations to other orbitals located above the
$Z=48$ HD shell gap produce the proton-emitting states. The doubly magic
nature of 96Cd nucleus is confirmed also in the calculations with other RMF
parametrizations (Table 2). It is interesting to mention that the RMF
parametrizations aimed at the description of the nuclei far from stability
such as NL3, NL3*, NLSH show larger $Z=48$ and $N=48$ HD shells gaps in
96,107-109Cd than the parametrizations NL1 and NLZ fitted predominantly to
$\beta$-stability nuclei (Table 2).
Figure 3: (Color online) The same as in Fig. 2 but for 107,108,109Cd. The
yrast HD line in 108Cd is built from two signature-degenerated configurations.
With increasing neutron number the energy gap between the yrast and excited HD
configurations disappears (Fig. 2). This is due to relatively high density of
the neutron states above the $N=48$ HD shell gap (Fig. 1). Indeed, many
excited neutron configurations are located below the lowest excited proton
configurations (Fig. 2). One can also see that even-even 100-104Cd nuclei are
characterized by appreciable density of the HD states in the vicinity of the
yrast HD line (Fig. 2). The analysis of the single-particle structure in these
nuclei indicates that similar density of the HD bands is expected also in odd
mass nuclei 99-105Cd. In no way these nuclei have to be considered as good
candidates for a search of discrete HD bands since the feeding intensity will
be redistributed among many low-lying HD bands. As a result, the feeding
intensity of an individual HD band will most likely drop below the
observational limit of modern experimental facilities. Although there is some
energy gap between the lowest four HD configurations and other excited
configurations in 106Cd, this nucleus does not appear to be a good candidate
for a search of discrete HD bands because the presence of four low-lying HD
configurations will lead to a fragmentation of feeding intensity. This is one
of possible reasons why the HD bands have not been observed in this nucleus
F.05 .
On the other hand, the high density of the HD bands in above discussed nuclei
will most likely favor the observation of the rotational patterns in the form
of ridge structures in three-dimensional rotational mapped spectra Hetal.06 .
The study of these patterns as a function of neutron number can provide a
valuable information about HD at high spin.
Figure 4: (Color online) Dymanic moments of inertia $J^{(2)}$ (panel (a)),
transition quadrupole moments (panel (b)), and mass hexadecapole moments
$Q_{40}$ (panel (c)) of the yrast HD bands in the nuclei under study. All
these bands have the proton $\pi 6^{2}$ configuration, their neutron
configurations are shown in the right panel. The configuration of the yrast HD
band in 106Cd is shown in panel (a).
Further increase of the neutron number brings the neutron Fermi level to the
region of low density of the neutron states characterized by the large $N=59$
and $N=61$ HD shell gaps (Fig. 1) with the combined size of these two gaps
being around 2.5 MeV (Table 2). As a result, the 107-109Cd nuclei show
appreciable energy gap between the yrast and lowest excited HD configurations
(Fig. 3). This gap is especially pronounced in the case of 107Cd for which it
is around 1.3 MeV. Note that the size of this gap is defined by the size of
the $Z=48$ HD shell gap, since the lowest excited configuration is based on
proton excitation (Fig. 3a). Similar or even larger energy gap between the
yrast and excited HD configurations is expected in the NLZ, NL3, NL3* and NLSH
parametrizations for which the size of the $Z=48$ and $N=59$ HD shell gaps is
at least 1.7 MeV in 107Cd (Table 2). The energy gaps between the yrast and
excited HD configurations at the spins where the HD configurations become
yrast are somewhat lower in 108,109Cd being around 0.9 and 1.1 MeV. This
energy gap in 108Cd is dictated by the size of the $N=61$ HD shell gap since
lowest excited HD configurations are based on neutron excitations. Thus, in
108Cd it will be smaller (similar) in the case of the NLZ (NL3, NL3*)
parametrizations and larger in the NLSH parametrization as compared with the
one obtained in the NL1 parametrization (Table 2). In the case of 109Cd, the
energy gap between the yrast and excited configurations will be larger
(smaller) in the NL3, NL3* and NLSH (NLZ) parametrizations (Table 2).
Table 2: The size of the Z=48, N=59, and N=61 HD shell gaps (in MeV) obtained
with different parametrizations of the RMF Lagrangian for the yrast HD
configurations in 96,107,108,109Cd. They are given at rotational frequency
$\Omega_{x}=1.00$ MeV approximately corresponding to the spin at which the HD
bands become yrast. The lowest (among the different parametrizations) value of
the shell gap is shown by bold style. ’59+61’ line shows the combined size of
the $N=59$ and $N=61$ HD shell gaps. RMF Parametrizations
---
Nucleus | Gap | NL1 | NLZ | NL3 | NL3* | NLSH
96Cd | Z=48 | 1.75 | 1.93 | 2.43 | 2.27 | 2.71
| N=48 | 2.00 | 2.07 | 2.59 | 2.44 | 3.03
108Cd | Z=48 | 1.62 | 1.66 | 2.23 | 1.99 | 2.06
| N=59 | 1.30 | 1.70 | 1.50 | 1.46 | 1.20
| N=61 | 1.20 | 0.74 | 1.20 | 1.19 | 1.50
| 59+61 | 2.50 | 2.44 | 2.70 | 2.65 | 2.70
107Cd | Z=48 | 1.70 | 1.73 | 2.22 | 2.18 | 2.27
| N=59 | 1.89 | 2.16 | 2.08 | 2.04 | 1.74
109Cd | Z=48 | 1.52 | 1.61 | 1.89 | 1.84 | 1.54
| N=61 | 1.37 | 1.16 | 1.83 | 1.74 | 2.16
Two factors make the observation of discrete HD bands in 108Cd 111Two bands
with very extended shapes observed in 108Cd in Refs. Cd108-1 ; Cd108-2 were
assigned as superdeformed in Ref. Cd108 . with existing facilities less
probable than in odd-mass 107,109Cd nuclei. First, the yrast HD line in this
nucleus is built from two signature degenerate configurations (Fig. 3b) in
which the last neutron is placed into one of the signatures of the $[413]7/2$
orbital (see Fig. 1 and Ref. Cd108 ). This reduces the feeding intensity of
each of these bands by factor of 2 as compared with the case when the yrast HD
line is built from single configuration. Second, the energy gap between the
yrast and excited HD configurations decreases with increasing spin (Fig. 2b).
As a result, further reduction of feeding intensity of the yrast HD bands is
expected if the bands are populated at spins higher than the spin at which
they become yrast. On the contrary, the energy gap between the yrast and
excited HD configurations is more constant as a function of spin in 109Cd and
especially in 107Cd. All these results strongly suggest that the 107Cd nucleus
is the best candidate for the experimental search of the discrete HD bands.
This conclusion is also supported by detailed analysis of the single-particle
routhians in the yrast HD configurations of even-even nuclei studied in Ref.
CRMF-HD ; this analysis does not suggest any alternative case which would
provide similar or larger gap between the yrast and excited HD configurations
in even-even, odd and odd-odd nuclei of the $Z=40-58$ part of the nuclear
chart.
The calculated properties of the yrast HD bands in studied nuclei are shown in
Fig. 4. The HD shapes undergo a centrifugal stretching that result in an
increase of the transition quadrupole moments $Q_{t}$ with increasing
rotational frequency. This process also reveals itself in the dynamic moments
of inertia: they increase with increasing rotational frequency in the
frequency range of interest. On the other hand, the mass hexadecapole moments
$Q_{40}$ do not show a clear trend as a function of rotational frequency and
stay nearly constant in the majority of the HD bands. Unpaired band crossings
due to interaction of different single-particle orbitals are seen in the
configurations of the yrast HD bands in 100,102,106Cd nuclei. For example, the
interaction between the $(r=+i)$ signatures of the $\nu[770]1/2$ and
$\nu[532]5/2$ orbitals is responsible for the crossing seen at $\Omega_{x}\sim
1.05$ MeV in the yrast HD band in 106Cd. This crossing may be an extra factor
(in addition to the density of the near-yrast HD bands) which complicates the
observation of the HD bands in 106Cd: such bands have not been observed in
experiment of Ref. F.05 .
The current study clearly shows that the polarization effects in time-even and
time-odd mean fields have an important impact on the density of the HD states
and especially on the energy gap between the yrast and excited HD states. The
latter quantity is appreciably smaller (by up to $\sim 0.5$ MeV; compare Figs.
2 and 3 with Table 2) than the respective HD shell gap in the routhian
diagram.
The role of time-odd mean fields in the definition of the energy gap between
the yrast and excited HD configurations is quite complicated. This is
illustrated by the fact that the energy gap between the yrast HD and the
lowest excited proton and neutron HD configurations is larger by $\approx 0.2$
MeV in the calculations without NM than in the ones with NM at spins where the
HD configurations become yrast $(I\approx 67\hbar)$. This fact reflects two
different mechanisms by which the time-odd mean fields affect the relative
energies of different rotational bands. In the first mechanism, the angular
momentum content of the single-particle orbitals is modified in the presence
of time-odd mean fields, see Ref. AR.00 for details. There are two important
consequences of this mechanism. First, the same total angular momentum of the
system is built at rotational frequency which is by $\sim 25\%$ lower in the
calculations with NM than in the calculations without NM. Second, the changes
of the single-particle angular momenta of the single-particle orbitals
surrounding the HD gaps of interest (the $\pi[420]1/2$ and $\pi[541]1/2$
orbitals for proton subsystem and $\nu[413]7/2$ and $\nu[651]3/2$ for neutron
subsystem (Fig. 1)) induced by NM modify the single-particle energies of these
orbitals. As a result, these gaps are smaller by $\sim 0.12$ MeV in the
calculations with NM at $I=67\hbar$. The second mechanism is related to
additional binding due to time-odd mean fields. The time-odd mean fields are
stronger in the excited HD configuration than in the yrast HD configuration.
Thus, additional binding due to NM is stronger in excited HD configuration
than in the yrast HD configuration. This also leads to the decrease of the
energy gap between the yrast and excited HD configurations in the calculations
with NM as compared with the ones without NM.
The presence of time-odd mean fields reveals itself also in the energy
splitting of the opposite signatures of the $\nu[770]1/2$ orbital visible at
$\Omega_{x}=0.0$ MeV (Fig. 1); the occupied orbital is more bound than
unoccupied one in the RMF theory (Ref. A150 ).
When considering theoretical predictions one has to keep in mind that they are
subject of the errors in the description of the energies of the single-
particle states, which exist in the RMF theory at spherical shape RBRMG.98 ,
normal deformation A250 and quite likely at superdeformation Cd108 . The
extrapolation from spherical and normal deformation towards HD is itself a
potential source of errors since it is not know how well the response of the
mean field (or the single-particle potential and liquid drop in the MM method)
to the extreme elongation of the nucleus is reproduced in model calculations.
Such errors are not restricted to the self-consistent models; they are also
expected in the phenomenogical potentials (used in the MM method) which
describe single-particle energies at normal deformation better than self-
consistent models. However, several facts support the results and
interpretations given above. First, all RMF parametrizations used in this
study lead to the same HD configurations in 96,107-109Cd nuclei which become
yrast at similar spins (see Ref. CRMF-HD for comparison of the results
obtained with NL1 and NL3) and to similar sizes of the proton and neutron HD
shell gaps (Table 2). Second, the large size of the $Z=48$ and $N=59$ (and
especially of combined neutron $59+61$ gap) HD shell gaps reduces the
importance of the errors in the description of the energies of specific
single-particle states. Third, the MM results of Ref. A100 suggest similar
conclusions for the nuclei around 108Cd. Indeed, large $Z=48$ shell gap and
low density of the single-particle states in the vicinity of the $N=59$ and
$N=61$ HD shell gaps is clearly visible in Figs. 4 and 5 of Ref. A100 . The
$N=59$ and $N=61$ shell gaps are separated by the signature-degenerated
$7/2^{+}$ state (Fig. 5 in Ref. A100 ). Thus, similar to our case, the yrast
HD line in 108Cd will be formed from two signature degenerated configurations
in the MM calculations.
## IV Conclusions
In summary, a systematic analysis of hyperdeformation in the Cd isotopes has
been performed in the cranked relativistic mean field theory. The density of
the HD states has been analysed with the goal to find the best cases for
experimental search of the discrete HD bands. Our analysis indicates 96Cd as a
doubly magic HD nucleus in this part of nuclear chart; its magicity is due to
large $Z=48$ and $N=48$ HD shell gaps. However, experimental study of HD in
this nucleus is problematic with existing facilities due to its $N=Z$ status.
The low density of the neutron single-particle states in the vicinity of the
$N=59$ and 61 HD shell gaps and sizable $Z=48$ HD shell gap lead to
appreciable gaps between the yrast and excited HD bands in 107-109Cd nuclei,
thus offering better opportunities to observe discrete HD bands. Among these
three nuclei, the best candidate for observing the discrete HD bands with
existing facilities is 107Cd nucleus. The MM calculations of Refs. A100 ;
SDH.07 indicate that the fission barriers are sufficiently large in the
nuclei around 108Cd so that the HD minimum could survive fission for a
significant range of angular momentum. The stability of the HD minimum is
defined by its depth, the fission barrier height and the height of the barrier
between the HD and normal-deformed/superdeformed minima Dudek ; SDH.07 . Our
study clearly indicates that the HD minimum is localized in the potential
energy surface. However, future studies of the HD in this mass region have to
provide more quantitative answers on these properties of the HD minima in a
fully self-consistent framework.
The work was supported by the U.S. Department of Energy under grant DE-
FG02-07ER41459.
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|
arxiv-papers
| 2009-02-01T00:19:54 |
2024-09-04T02:49:00.328569
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.V. Afanasjev and H. Abusara",
"submitter": "Anatoli Afanasjev",
"url": "https://arxiv.org/abs/0902.0095"
}
|
0902.0098
|
# Band terminations in density functional theory.
A. V. Afanasjev Department of Physics and Astronomy, Mississippi State
University, MS 39762, USA
###### Abstract
The analysis of the terminating bands has been performed in the relativistic
mean field framework. It was shown that nuclear magnetism provides an
additional binding to the energies of the specific configuration and this
additional binding increases with spin and has its maximum exactly at the
terminating state. This suggests that the terminating states can be an
interesting probe of the time-odd mean fields provided that other effects can
be reliably isolated. Unfortunately, a reliable isolation of these effects is
not that simple: many terms of the density functional theories contribute into
the energies of the terminating states and the deficiencies in the description
of those terms affect the result. The recent suggestion ZSW.05 that the
relative energies of the terminating states in the $N\neq Z,\ A\sim 44$ mass
region given by $\Delta E$ provide unique and reliable constraints on time-odd
mean fields and the strength of spin-orbit interaction in density functional
theories has been reanalyzed. The current investigation shows that the $\Delta
E$ value is affected also by the relative placement of the states with
different orbital angular momentum ${\it l}$, namely, the placement of the $d$
(${\it l}=2$) and $f$ (${\it l}=3$) states. This indicates the dependence of
the $\Delta E$ value on the properties of the central potential.
###### pacs:
PACS:
## I Introduction
The density functional theory (DFT) in its non-relativistic BHR.03 and
relativistic VRAL realizations is a standard tool of modern nuclear structure
studies. However, providing global description of atomic nuclei, it still
suffers from the fact that many channels of effective interaction are not
uniquely defined: this is a reason for a large variety of the DFT
parametrizations, the quality of many of which is poorly known. The spin-orbit
interaction and the time-odd mean fields are of particular interest in this
context, since there are considerable variations for these quantities (see,
for example, Refs. DD.95 ; BRRMG.99 ; AR.00 ). The spin-orbit interaction
plays a crucial role in the definition of the shell structure of nuclei, and,
thus its accurate description is required so that theoretical tools have
predictive power for nuclei beyond known regions. The time-odd mean fields (or
nuclear magnetism (NM) in the language of the relativistic mean field (RMF)
theory KR.90 ) contribute to the single-particle Hamiltonian only in
situations where the intrinsic time-reversal symmetry is broken and Kramers
degeneracy of time-reversal counterparts of the single-particle levels is
removed. The rotating nuclei and odd and odd-odd mass nuclei are typical
examples of such situations, see Refs. BHR.03 ; VRAL and references quoted
therein.
It was recently suggested in Ref. ZSW.05 that the set of terminating states
in the $N\neq Z,\ A\sim 44$ mass region provides unique and reliable
constraints on time-odd mean fields and the strength of spin-orbit interaction
in Skyrme density functional theory (SDFT), see also Refs. S.07 ; ZS.07 .
Later this procedure (called as ’TS-method’ in this manuscript, where ’TS’
refers to ’terminating states’) has been used in the analysis of terminating
states in this mass region within the RMF theory BWSMG.06 which is one of the
versions of covariant density functional (CDFT) theory VRAL .
The authors of Refs. ZSW.05 ; S.07 ; ZS.07 ; BWSMG.06 claim that the TS-
method is free from the drawbacks of standard methods of defining spin-orbit
interaction based on measuring the single-particle energies of the spin-orbit
partner orbitals in spherical nuclei. As a consequence, it is stated that it
allows to define very accurately both isoscalar and isovector channels of
spin-orbit interaction ZSW.05 ; BWSMG.06 ; the feature which was impossible in
the previously existing methods.
The conclusions obtained within the TS-method are drastically different from
the ones previously obtained in the SDFT and RMF frameworks. For example,
based on the comparison of the calculated and experimental energies of spin-
orbit partner orbitals, it was shown in Ref. BRRMG.99 that the experimental
spin-orbit splittings are better reproduced in the RMF approach than in the
SDFT (see Fig. 2 in Ref. BRRMG.99 ). On the contrary, the results obtained in
Refs. ZSW.05 ; BWSMG.06 within the TS-method show that the SDFT provides
better description of spin-orbit splittings than the RMF: it was suggested in
Ref. ZSW.05 that only 5% reduction in isoscalar ls strength is needed in the
SDFT approach in order to reproduce experimental data. Considering the
conflict of these results and the importance of the spin-orbit interaction in
nuclei it is necessary to understand to which extent the basis of the
suggested TS-method is sound and justified. The goal of the present manuscript
is the study of the properties of terminating bands and their terminating
states in the RMF framework. In particular, the question of whether all DFT
contributions have been correctly accounted in the realization of the TS-
method in Refs. ZSW.05 ; BWSMG.06 is addressed is the current manuscript.
The manuscript is organized as follows. Time-odd mean fields in terminating
bands are studied in Sect. II. The basis of the TS-method, its realization in
self-consistent DFT and in the Nilsson potential are discussed in Sect. III.
Sect. IV analyses the contributions of different DFT terms into the relative
energies of terminating states in the $A\sim 44$ mass region. The discussion
of the energy scale, its connection to the effective mass of the nucleon and
their impact on the relative energies of terminating states is presented in
Sect. V. Finally, Sect. VI summarizes main conclusions.
## II Time-odd mean fields in terminating bands: test case of 20Ne
Previous DFT investigations of the modifications of the moments of inertia
KR.93 ; CRMF ; DD.95 ; YM.00 and single-particle properties AR.00 in
rotating nuclei caused by the time-odd mean fields (nuclear magnetism) were
restricted to the superdeformed (SD) bands. However, these bands are far away
from the termination and are characterized by a relatively stable deformation.
In order to understand how NM affects the properties of the terminating bands,
the ground state configuration in 20Ne has been studied. This band is a
classical example of band termination CNS . It has the
$\pi(d_{5/2})^{2}_{4}\nu(d_{5/2})^{2}_{4}$ configuration relative to the 16O
core with maximum spin $I_{max}=8^{+}$. The selection of this configuration
has been guided by its simplicity, which allows us to understand the role of
time-odd mean fields in terminating bands in greater details. Although the
terminating bands were observed also in heavier nuclei, it is difficult to
trace them from low spin up to band termination in the self-consistent
approaches VRAL without special techniques such as used in the cranked
Nilsson-Strutinsky (CNS) approach CNS .
Figure 1: (Color online) Calculated total angular momentum (panel (a)), total
binding energy (panel (b)), kinematic and dynamic moments of inertia (panels
(c) and (d)) as a function of rotational frequency $\Omega_{x}$ in the ground
state band of 20Ne. The results obtained with and without nuclear magnetism
are presented. Note that the band termination takes place at rotational
frequency above which the subsequent increase of rotational frequency does not
modify neither total angular momentum nor total binding energy. This point
also corresponds to $\gamma=60^{\circ}$ (Fig. 3c).
The investigation of NM in 20Ne has been carried out within the framework of
the cranked relativistic mean field (CRMF) theory KR.89 ; CRMF following the
formalism of Ref. AR.00 , where similar study has been performed for the yrast
SD band in 152Dy. In the CRMF calculations of this manuscript, all fermionic
and bosonic states belonging to the shells up to $N_{F}=12$ and $N_{B}=16$ are
taken into account in the diagonalization of the Dirac equation and the matrix
inversion of the Klein-Gordon equation, respectively. The detailed
investigation indicates that this truncation scheme provides very good
numerical accuracy. The NL1 NL1 parametrization of the RMF Lagrangian is
employed in the study of 20Ne, while the studies of terminating states in the
$A\sim 44$ mass region (Sects. III and IV) are performed mostly with the NL1
and NL3 NL3 parametrizations. The pairing is neglected in calculations.
Fig. 1 shows the total angular momentum, total binding energy $E$ and
kinematic (J(1)) and dynamic (J(2)) moments of inertia of the ground state
configuration in 20Ne as a function of rotational frequency obtained in the
calculations with and without NM (the later will be further denoted as WNM).
The band crossing caused by the interaction of the $r=+i$ signatures of the
[220]1/2 and [211]3/2 orbitals both in the proton and neutron subsystems takes
place at lower frequency in the calculations with NM; this is in line with
previous finding that the NM shifts the band crossing frequencies AFR.02 . The
NM also increases the moments of inertia before band crossing (Fig. 1c,d);
similar effect has been seen before in the SD bands (see Ref. AR.00 and
references therein). It also leads to a faster alignment of angular momentum
with rotational frequency (Fig. 1a); full alignment at $I_{max}=8^{+}$
corresponding to band termination takes place at lower frequency in the
calculations with NM.
In the context of study of terminating states two results are important.
First, at the band termination the NM does not modify neither total angular
momentum (Fig. 1a) nor the expectation values of the single-particle angular
momenta $<j_{x}>_{i}$ (Fig. 2). At lower frequency, the impact of NM on
$<j_{x}>_{i}$ is similar to the one previously studied in the SD band of 152Dy
AR.00 , and, thus, it will not be discussed in detail. However, one should
mention that when analyzing the impact of NM on $<j_{x}>_{i}$, the region of
band crossing and the region close to the band termination have to be excluded
from consideration because considerable differences in the deformations of the
NM and WNM solutions at given frequency distort their comparison. Second, the
NM provides an additional binding to the energies of the specific
configuration and this additional binding increases with spin and has its
maximum exactly at the terminating state (Fig. 1b and 3d)). This suggests that
the terminating states can be an interesting probe of the time-odd mean fields
provided that other effects can be reliably isolated.
When the results of the NM and WNM calculations are compared as a function of
total angular momentum, one can see that the quadrupole deformation
$\beta_{2}$ (Fig. 3a), mass hexadecapole moment $Q_{40}$ (Fig. 3b), and
$\gamma$-deformation (Fig. 3c) are almost the same in both calculations. The
only difference is seen in the total binding energies (Fig. 3d), where the NM
solution is more bound than the WNM solution. These results give a hint why
the cranked models based on the phenomenological potentials like Woods-Saxon
or Nilsson, which do not include time-odd mean fields DD.95 , are so
successful in the description of experimental data. When considered as a
function of spin the deformation properties of the rotating system are only
weakly affected by the time-odd mean fields, and the proper renormalization of
the moments of inertia CNS takes care of the $E$ versus angular momentum
curve.
Figure 2: (Color online) The expectation values of single-particle angular
momenta $<j_{x}>_{i}$ of the neutron orbitals occupied at low frequency in the
ground state configuration of 20Ne given along the deformation path of this
configuration. The single-particle orbitals are labeled by means of the
asymptotic quantum numbers $[Nn_{z}\Lambda]\Omega$ (Nilsson quantum numbers)
of their dominant component of the wave function at $\Omega_{x}=0.0$ MeV. The
orbitals with signature $r=+i$ and $r=-i$ are shown in right and left panels,
respectively. The results of the calculations with and without NM are shown by
solid and dashed lines, respectively. The region of the band crossing is
located between the dashed lines. Figure 3: Calculated quadrupole deformation
$\beta_{2}$ (panel (a)), mass hexadecapole moment $Q_{40}$ (panel (b)),
$\gamma$-deformation (panel (c)), and total binding energy $E$ (panel (d)) as
a function of angular momentum. The results obtained with (NM) and without
(WNM) nuclear magnetism are presented.
## III The TS-method
The TS-method suggested in Ref. ZSW.05 employs the terminating states in the
$A\sim 44$ mass region with the proton $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ and
$f_{7/2}^{n}$ structure, but, in general, according to Ref. ZS.07 can be
employed to terminating states in any mass region. The difference $\Delta
E^{exp}=E(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E(f_{7/2}^{n})$ between the excitation
energies $E(d_{3/2}^{-1}\,f_{7/2}^{n+1})$ and $E(f_{7/2}^{n})$ of the
terminating states with the structure $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ and
$f_{7/2}^{n}$ is dominated by the size of the magic gap 20 which is surrounded
by the $d_{3/2}$ and $f_{7/2}$ spherical subshells (see Fig. 4).
### III.1 The spin-orbit splittings in the TS-method
The principal difference between the standard and the TS-methods of defining
the strength of spin-orbit interaction is schematically shown in Fig. 4. The
standard method requires that both partners of spin-orbit ${\it ls}$ doublet
with $\it j=l\pm\frac{1}{2}$ are observed in experiment since the spin-orbit
splitting $\Delta E^{SO}$ is related to the strength of spin-orbit
interaction. This severely restricts the possibilities to study spin-orbit
interaction since both partners should be located in the vicinity of the Fermi
level to be observed: this condition is very difficult to satisfy for high-$j$
orbitals since they are characterized by large spin-orbit splittings, see, for
example, Refs. BRRMG.99 ; IEMSF.02 . On the contrary, the TS-method employs
the terminating states based on the particle-hole excitations involving the
single-particle states with $\it j=l-\frac{1}{2}$ and $\it
j^{\prime}=l^{\prime}+\frac{1}{2}$ which emerge from different $N$-shells and
are characterized by the energy splitting $\Delta E^{TS}$ (Fig. 4). Since
these states are located in the vicinity of the Fermi level, the TS-method
provides also information on the spin-orbit interaction of high-$j$ orbitals
according to Ref. ZSW.05 .
It is necessary to recognize that both methods of defining the spin-orbit
interaction are not free from important drawbacks. The experimental single-
particle states in spherical nuclei used in the standard method are strongly
affected by the couplings with vibrations in many cases MBBD.85 . On the other
hand, the $\Delta E^{TS}$ value used in the TS-method depends not only on the
spin-orbit splitting but also on how well the positions of the single-particle
states with different orbital momenta $l$ and $l^{\prime}$ (Fig. 4) are
described in the DFT calculations. The later fact has been neglected in Ref.
ZSW.05 using an analogy with the Nilsson potential, the validity of which is
questioned below.
Figure 4: Schematic comparison of the standard and TS-methods of defining the
strength of spin-orbit interaction. In the left and right panels, the single-
particle spectra without (on left) and with (on right) spin-orbit interaction
are compared.
### III.2 The TS-method in self-consistent approaches
In the self-consistent calculations, the $\Delta
E^{SC}=E(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E(f_{7/2}^{n})$ quantity is defined as
the difference of the binding energies of the corresponding terminating
states. Without going into the details of specific DFT (nonrelativistic or
relativistic), one can conclude that $\Delta E^{SC}$ depends on
* •
the energy scale of the single-particle spectra which is related to the
effective mass $m^{*}(k_{F})/m$ of the nucleon at the Fermi surface,
* •
the spin-orbit interaction,
* •
the relative placement of states with different angular momentum $l$,
* •
the time-odd mean fields (see Sects. II and IV.2),
* •
the polarization effects (both in time-even and time-odd channels) on going
from the $f_{7/2}^{n}$ to the $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ terminating state.
For simplicity of discussion, they will occasionally be called as ’ingredients
of $\Delta E^{SC}$’.
$\Delta E^{SC}$ can also be split into the terms which depend on time-even
(TE) and time-odd (TO) mean fields
$\displaystyle\Delta E^{SC}=\Delta E^{SC}_{TE}-\Delta E^{SC}_{TO}$ (1)
The minus sign in front of $\Delta E^{SC}_{TO}$ reflects the fact that NM
always decreases the size of $\Delta E$.
#### III.2.1 Coupling constant dependence of $\Delta E^{SC}$
The ingredients of $\Delta E^{SC}$ depend in a complicated way on different
terms of the DFT with at least one term contributing into each of four first
ingredients of $\Delta E^{SC}$ within the nonrelativistic SDFT. In the RMF
theory, the spin-orbit interaction is defined in a natural way without
additional coupling constant VRAL . The time-odd mean fields related to NM are
defined through the Lorentz covariance VRAL and also do not require an
additional coupling constant. However, both these terms depend in an indirect
way on the coupling constants of other terms of the RMF Lagrangian.
Considering the uncertainties of the description of the ’ingredients of
$\Delta E^{SC}$’ in the DFT, it is not obvious that simple fit (to
experimental $\Delta E^{exp}$ values) of the coupling constants of the DFT
terms related to a pair of ingredients of $\Delta E^{SC}$ (such as time-odd
mean fields and spin-orbit interaction as in Ref. ZSW.05 ; the later treated
perturbatively) will allow to define these constants in a unique way. This is
especially true considering that physical observables depend on many (or
sometimes all) coupling constants simultaneously within the DFT, and the
effect of varying one or two coupling constants may be either enhanced or
cancelled by a variation of others. Strictly speaking, the quality of such
perturbative fits involving only one or two terms of the DFT is not known
until the results of global fit including all the DFT terms are available. For
example, it was shown in Ref. LBBDM.07 that perturbative studies of tensor
terms allow only very limited conclusions.
#### III.2.2 Polarization effects
Figure 5: (Color online) (right part) Single-particle energies at spherical
shape in 46Ti obtained in the RMF calculations with the NL1 parametrization
and the Nilsson potential with standard set of parameters BR.85 (columns
indicated as “RMF-spher” and “Nilsson”). In order to facilitate the
comparison, the RMF states are given at the calculated energies, while the
energies of the states obtained in the Nilsson potential are shifted in such a
way that the average energy of the $1f_{7/2}$ and $1d_{3/2}$ states is the
same in both calculations. (left part) The magnitudes $\Delta E$ of the
$f_{7/2}-d_{3/2}$ splittings as extracted from the energies of terminating
states ($\Delta E=E(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E(f_{7/2}^{n})$) are shown by
arrows in columns “exp” (experiment), “CRMF-NM” and “CRMF-WNM” (CRMF
calculations with and without NM). The middle points of arrows are located at
the average energy of the $1f_{7/2}$ and $1d_{3/2}$ RMF states. The
terminating states are deformed in the CRMF calculations. Note that the value
of $\Delta E=5.1$ MeV (not shown in figure) obtained in the cranked Nilsson-
Strutinsky calculations compares favorably with experiment ($\Delta
E^{exp}=5.51$ MeV).
Fig. 5 illustrates the polarization effects present in the CRMF calculations
of terminating states. Since fully stretched states with spin $I_{max}$ are
reasonably well described by a single Slater determinant ZSNSZ.07 , the
comparison with experiment is performed without angular momentum restoration
as in all DFT studies of the terminating $I_{max}$ states in this mass region,
see, for example, Refs. ZSW.05 ; BWSMG.06 . At spherical shape, the
$f_{7/2}-d_{3/2}$ energy gap ($\Delta E^{sph}$) in the single-particle spectra
considerably exceeds $\Delta E^{exp}$. The $\Delta E^{sph}$ value is a very
good approximation to the results of the spherical RMF calculations without NM
in which the gap between these states is defined as the difference of the
binding energies of the $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ and $f_{7/2}^{n}$
states: the difference between two results for all nuclei under study does not
exceed 40 keV. When deformation polarization effects denoted as $\Delta
E^{def-pol}$ are taken into account (the “CRMF-WNM” column in Fig. 5), this
gap becomes even larger and reaches $\Delta E=7.57$ MeV exceeding by 37% the
experimental value. The inclusion of NM decreases the difference between
experiment and calculations considerably (by 1.14 MeV) (column “CRMF-NM” in
Fig. 5). Note that mass and charge quadrupole and hexadecapole moments change
only by $\sim 10^{-4}\%$ on going from the CRMF-WNM to CRMF-NM solutions.
Thus, the deformation differences between these two solutions are almost non-
existant and can be neglected. Based on the consideration of polarization
effects, the $\Delta E^{SC}$ can be approximated as
$\displaystyle\Delta E^{SC}\approx\Delta E^{sph}+\Delta E^{def-pol}-\Delta
E^{SC}_{TO}$ (2)
Considering that the terminating states of interest are close to spherical
(Fig. 6), this approximation which corresponds to a perturbative treatment of
the deformation polarization effects should be quite reasonable. This
approximation also allows to use the results of spherical RMF calculations in
subsequent analysis of $\Delta E$ (Sect. IV).
Figure 6: (Color online) Quadrupole deformations of the terminating states
obtained in the CRMF calculations with the NL1 parametrization. For each
nucleus, the deformation polarization energies $\Delta E^{def-pol}$ are shown
between the calculated deformation points. Three nuclei with the largest
$\Delta E^{def-pol}$ values are indicated by arrows.
### III.3 The Nilsson potential analogy of Ref. ZSW.05
In order to overcome the problems discussed in Sect. III.2.1, the authors of
Ref. ZSW.05 use the analogy with the simple form of the Nilsson potential
N.55 for the $\Delta E^{SC}_{TE}$ part. This form of the Nilsson potential is
similar to the one given below (Eq. (5)), but with the parameters $\kappa$ and
$\mu$ independent on principal quantum number $N$. However, no proof (analogy
is not a proof) is provided whether such a transition from self-consistent DFT
to the Nilsson potential is valid and whether the dependence of the energies
of the single-particle states on quantum numbers $N,j,l,$ and $s$ is the same
in the self-consistent DFT and in the Nilsson potential. Fig. 5 clearly shows
that the later is not a case.
In the simple form of the Nilsson potential, the magnitude of the
$f_{7/2}-d_{3/2}$ splitting related to the magic gap 20 is given by
$\displaystyle\Delta E^{Nil}=\hbar\omega_{0}(1-6\kappa-2\kappa\mu).$ (3)
Thus, it depends on three major factors: (i) the energy scale of the single-
particle potential characterized by $\hbar\omega_{0}$, (ii) the flat-bottom
and surface properties entering through the orbit-orbit term $\sim\mu$, and
(iii) the strength of the spin-orbit term $\kappa$. Then, the authors of Ref.
ZSW.05 using the fact that in light nuclei the Nilsson potential resembles
the pure harmonic oscillator potential, which leads to $\mu\sim 0$, conclude
that the magnitude of the $f_{7/2}-d_{3/2}$ splitting is given by
$\displaystyle\Delta E^{Nil}=\hbar\omega_{0}(1-6\kappa).$ (4)
Thus, in this approximation, the $\Delta E^{Nil}$ splitting is defined only by
the energy scale $\hbar\omega_{0}$ and the strength of the
$\it{ls}$-potential. Arguing that the energy scale is rather well constrained
by the data not only for the Nilsson but also for the self-consistent
approaches, the authors of Ref. ZSW.05 conclude that the $f_{7/2}-d_{3/2}$
splitting is directly related to the strength of the $\it{ls}$-potential. Or
alternatively, this approximation corresponds to the situation when the
$\Delta E^{Nils}$ does not depend on orbital motion of the nucleons.
### III.4 An alternative form of the Nilsson potential
One question of paramount importance we have to ask is whether the simple form
of the Nilsson potential given in Ref. N.55 and used in Ref. ZSW.05 is
unique and how well it describes the experimental data. It turns out that the
modern versions of the Nilsson potential employ the parameters $\kappa$ and
$\mu$ which are dependent on the main oscillator quantum number $N$ and on
nucleon type (proton or neutron), thus facilitating the study of wide range of
nuclei with the same set of single-particle parameters and with comparable
accuracy BR.85 ; Ragbook ; Rag-priv . Since the Nilsson potential is
phenomenological in nature, this procedure is well justified. The accuracy of
the description of different physical quantities such as, for example,
rotational properties and relative energies of different single-particle
configurations BR.85 ; Ragbook ; CNS ; GBI.86 is considerably improved when
different values of $\kappa$ and $\mu$ are used for different $N$-shells.
Although some variations between the parametrizations exist, this approach is
used in almost all parametrizations of the Nilsson potential developed from
the middle of the 80ties of the last century GBI.86 ; BR.85 ; A150 . The
studies employing this description of the Nilsson potential are abundant and
provide systematic information on the accuracy of the description of physical
observables in different mass regions. Even more sophisticated dependence of
the $\kappa$ and $\mu$ parameters on the principal quantum number $N$ and
orbital angular momentum $l$ is introduced in Ref. S.86 and employed in a
number of studies, see, for example, Refs. PRet.97 ; BZ.95 .
#### III.4.1 Terminating states in the $A\sim 44$ mass region
In the Nilsson potential with $\kappa_{N}$ and $\mu_{N}$ parameters dependent
on the principal oscillator quantum number $N$ BR.85 ; Ragbook
$\displaystyle\hat{H}^{Nil-N-dep}-\frac{3}{2}\hbar\omega_{0}=$ (5)
$\displaystyle=$
$\displaystyle\hbar\omega_{0}\\{N-\kappa_{N}[2{\bm{l}}{\bm{s}}+\mu_{N}(\bm{l}^{2}-<\bm{l}^{2}>_{N})]\\},$
the magnitude of the $f_{7/2}-d_{3/2}$ splitting related to the energy
difference of the $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ and $f_{7/2}^{n}$ terminating
states is given by
$\displaystyle\Delta E^{Nil-N-dep}=$ $\displaystyle=$
$\displaystyle\hbar\omega_{0}(1-3[\kappa_{2}+\kappa_{3}]-3\kappa_{3}\mu_{3}-\kappa_{2}\mu_{2}).$
The superscript ${}^{\prime}Nil-N-dep^{\prime}$ is used to indicate the
dependence of these expressions on the main oscillator quantum number $N$.
Table 1: The standard parametrization of the Nilsson potential from Ref. BR.85 ; Ragbook . Only the parameters for the $N=2$ and 3 shells are shown. | Protons | Neutrons
---|---|---
N | $\kappa$ | $\mu$ | $\kappa$ | $\mu$
2 | 0.105 | 0.00 | 0.105 | 0.00
3 | 0.090 | 0.30 | 0.090 | 0.25
The $\kappa_{N}$ and $\mu_{N}$ parameters of the so-called standard
parametrization of the Nilsson potential are shown for the shells of interest
in Table 1. For the protons, the value $\Delta E^{Nil-N-
dep}=0.415\hbar\omega_{0}$ is obtained in the calculations employing the
$\kappa_{N}$ parameters from the Table 1 but assuming the $\mu_{N}=0$ as it
was done in the derivation of Eq. (4). This value can be compared with the
$\Delta E^{Nil-N-dep}=0.334\hbar\omega_{0}$ value obtained with the
$\kappa_{N},\mu_{N}$ parameters from the Table 1. One can see that these two
values differ by approximately 25% and this difference is solely attributed to
the non-zero value of the $\mu_{3}$ parameter. Considering that $\Delta
E^{exp}\sim 5.5$ MeV (see Fig. 7), 25% difference correspond to 1.4 MeV; this
difference definetely cannot be ignored when experimental data is compared
with experiment.
#### III.4.2 Concluding remarks
Even for terminating states in the $A\sim 44$ mass region one cannot ignore
the dependence of the energies of the (N,l) and (N’,l’) states, from which the
${\it j=l-\frac{1}{2}}$ and ${\it j^{\prime}=l^{\prime}+\frac{1}{2}}$ states
used in the TS-method emerge (see Fig. 4), on the orbital angular momentum.
This dependence enters through the $\mu_{N}(\bm{l}^{2}-<\bm{l}^{2}>_{N})]$
term of the Nilsson potential (Eq. (5)). This is contrary to the approximation
made in the derivation of Eq. (4) which has a consequence that the energy
difference $\Delta E^{Nils}$ depends only on the energy scale
$\hbar\omega_{0}$ and the strength of the spin-orbit term $\kappa$. The
dependence of $\Delta E^{Nils}$ on the orbital angular momentum in the case of
terminating states involving single-particle states from higher $N$-shells has
been recognized in Ref. ZS.07 .
## IV Terminating states in the $A\sim 44$ mass region: what we can learn
from the comparison with experiment
Figure 7: (Color online) The experimental and calculated magnitudes $\Delta
E$ of the $f_{7/2}-d_{3/2}$ splittings as extracted from the energies of
terminating states ($\Delta E=E(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E(f_{7/2}^{n})$).
The results of the CRMF calculations are shown for the NL1 and NL3
parametrizations of the RMF Lagrangian. The experimental data corrected for
the deformation polarization effects (as obtained in the NL1 parametrization)
is shown by open circles. Note that the experimental and deformation
polarization corrected values of $\Delta E$ coincide in the case of 47V.
Fig. 7 compares the results of calculations with experiment. The same data set
as in Refs. ZSW.05 ; BWSMG.06 is used in this comparison, but it is shown as
a function of $n$, where $n$ stands for the number of the $f_{7/2}$ protons in
the $f_{7/2}^{n}$ terminating state. In addition, this figure compares the
absolute values and not the differences between experimental results and
calculations as it was done in Refs. ZSW.05 ; BWSMG.06 : the differences
normalized to 44Ca are compared in Fig. 8a. Since the $n$ value is the same
for each isotope chain ($n=0$ for the Ca isotopes, $n=1$ for the Cs isotopes,
$n=2$ for the Ti isotopes, and $n=3$ for the V isotopes), the isospin
dependence of $\Delta E$ is clearly visible. Different isotope chains show
different isospin dependencies and they are well reproduced in the
calculations (Fig. 7 and Fig. 8a). On the other hand, the calculations
overestimate the absolute value of $\Delta E$ (Fig. 7), and the difference
between the calculated and experimental $\Delta E$ values show pronounced
$n$-dependence (Fig. 8a).
One source of the discrepancy between theory and experiment is related to the
impact of effective mass of the nucleon on the single-particle spectra (see
Sect. V): in general, it should lead to an overestimate of experimental
$\Delta E$ in the calculations. The other sources of these discrepancies are
analyzed in detail in this Section.
Figure 8: (Color online) (a) The difference $(\Delta E^{th}-\Delta
E^{exp})_{norm}$ between the calculated and experimental values of $\Delta E$
(based on the results of Fig. 7). This difference is normalized to zero for
44Ca. (b) The calculated difference
$E_{TO}(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E_{TO}(f_{7/2}^{n})$ shown as a function
of $n$ for the indicated RMF parametrizations (based on the results of Fig.
9).
### IV.1 Deformation polarization effects
The deformation polarization effects discussed in Sect. III.2.2 are
characterized by the $\Delta E^{def-pol}$ energies. The sign of $\Delta
E^{def-pol}$ depends on relative deformations of the $f_{7/2}^{n}$ and
$d_{3/2}^{-1}f_{7/2}^{n+1}$ terminating states (Fig. 6). It is positive
(negative) when the $\beta_{2}$-deformation of the $d_{3/2}^{-1}f_{7/2}^{n+1}$
state is larger (smaller) than the one of the $f_{7/2}^{n}$ state. The $\Delta
E^{def-pol}$ values almost do not depend on the parametrization of the RMF
Lagrangian: the difference in their values is below 10 keV if the results of
the NL1 and NL3 parametrizations are compared. If the experimental data are
corrected for these deformation polarization effects, then smooth trend (the
curve ’exp. (def-cor)’ in Fig. 7) as a function of $n$ emerges. The $\Delta E$
value along this curve decreases by $\sim 0.25$ MeV on going from $n=0$ to
$n=3$. Assuming that these effects are reasonably well described in the
calculations, one can conclude that the nucleus-dependent fluctuations in
experimental value of $\Delta E$ (the curve ’exp.’ in Fig. 7) are due to
deformation polarization effects.
### IV.2 Nuclear magnetism (time-odd mean fields) in the terminating states
of the $A\sim 44$ mass region
Fig. 9 shows the additional bindings $E_{TO}(state)$ to the energies of
terminating states due to NM. This quantity increases with the increase of the
$n$-value for the $f_{7/2}^{n}$ and $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ terminating
states. The increase of $E_{TO}$ with isospin within specific isotope chain is
associated with the increase of the number of the occupied neutron $f_{7/2}$
states and corresponding increase in spin. The increase of the values of
$E_{TO}$ correlates with the increase of the spin of the terminating states:
for example, the $f_{7/2}^{n}$ and $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ terminating
states have $I_{max}=6^{+}$ and $I_{max}=11^{-}$ in 42Ca and
$I_{max}=\frac{31}{2}^{-}$ and $I_{max}=\frac{35}{2}^{+}$ in 47V,
respectively.
The results of the calculations confirm previous conclusion obtained in 20Ne
(Sect. II) that the additional binding due to NM is considerably enhanced in
the terminating states. At no rotation, the additional binding due to NM to
the energies of the single-particle configurations in odd-mass nuclei is in
average around $\sim 100$ keV and seldom reaches 200 keV in the mass region of
interest AA.08 . This is much smaller than the additional binding observed in
the terminating states in which it reaches 4 MeV for $n=3$ (Fig. 9).
Because of their magnitude, the $E_{TO}$ values in terminating states are also
a good measure of how well the time-odd mean fields are defined in the
specific version of DFT. The $E_{TO}$ values obtained with different
frequently used non-linear parametrizations of the RMF Lagrangian such as NL1
NL1 , NL3 NL3 , NLSH NLSH , NLRA1 NLRA1 and NLZ NLZ are shown in Fig. 9.
With increasing $E_{TO}$ and $n$, the absolute variations in the $E_{TO}$
values calculated with different RMF parametrizations increase. However, they
are still within 15% of the absolute value of $E_{TO}$. This result suggests
that within the non-linear versions of the RMF Lagrangian NM is defined with
similar accuracy.
Figure 9: (Color online) Additional bindings $E_{TO}(state)$ (in absolute
value) to the energies of terminating states due to NM shown for the
terminating states of interest. The results are shown for the indicated
parametrizations of the RMF Lagrangian as a function of $n$.
This value can be used to estimate the uncertainty in the definition of the
moments of inertia in the CRMF calculations due to the uncertainty in NM.
Dependent on the nuclear system, the NM contribution to the total kinematic
moment of inertia is approximately 10-25% CRMF ; AA.08 . Thus, the uncertainty
of the definition of the absolute value of the total kinematic moments of
inertia due to the uncertainty in the definition of NM is modest being in
range of 1.5-3.75%.
It follows from Fig. 8 that the portion of the $n$-independent part of the
discrepancy between experimental and calculated $\Delta E$ values may be
related to the uncertainties in NM since the difference
$E_{TO}(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E_{TO}(f_{7/2}^{n})$ somewhat (within
$\approx 200$ keV) depends on the RMF parametrization. The contribution of NM
into the $n$-dependent part of this discrepancy is discussed in Sect. IV.3.3.
### IV.3 The dependence of $\Delta E$ on orbital angular momentum and spin-
orbit interaction
#### IV.3.1 The impact of density modifications on the single-particle
properties
It is well known fact that the modifications of the central nucleonic
potential and its surface properties affect the single-particle states with
different angular momentum l in a different way (see Refs. MBBD.85 ; RBRMG.98
and references therein). They also alter the spin-orbit potential and lead to
the changes in the spin-orbit splittings. In order to check how big this
effect is in the nuclei under study, the proton density distributions and
single-particle spectra at spherical shape are compared in Fig. 10 for the
42Ca and 47V nuclei. These nuclei represent the lower and upper mass ends of
the data set under investigation. The configuration of 47V has 2 additional
$f_{7/2}$ neutrons and 3 additional $f_{7/2}$ protons as compared with the
configuration of 42Ca. The filling of these high-$j$ orbitals increases the
density near the surface (Fig. 10a). These modifications of the density change
the central and spin-orbit nucleonic potentials (in a similar fashion as it
was discussed in Refs. TPC.04 ; AF.05 ) leading to the modifications of the
single-particle spectra (Fig. 10b).
Figure 10: (Color online) (a) Proton density distribution in the 42Ca and 47V
nuclei as obtained for ground state in the spherical RMF calculations with the
NL1 parametrization of the Lagrangian. (b) Corresponding energies of the
single-particle states. The states in 42Ca are shown at the calculated
energies, while all states in 47V are shifted by constant value in such a way
that the average energy of the $1d_{3/2}$ and $1f_{7/2}$ states is the same as
in 42Ca.
On going from 42Ca to 47V, the spin-orbit splitting in the $d_{5/2}-d_{3/2}$
doublet decreases by 0.12 MeV (from 6.72 MeV to 6.60 MeV), while the one in
the $f_{7/2}-f_{5/2}$ doublet increases by 0.57 MeV (from 7.0 MeV to 7.57
MeV). If the modifications in the single-particle spectra would be restricted
only to the spin-orbit splittings and their modifications would evenly be
redistributed between the $j=l+1/2$ and $j=l-1/2$ members of the spin-orbit
doublet, this would decrease the $1f_{7/2}-1d_{3/2}$ splitting by 0.22 MeV.
However, the calculations show that the $f_{7/2}-d_{3/2}$ splitting is
increased by 0.34 MeV (Figs. 10). Assuming that the energy scale does not
change on going from 42Ca to 47V, this can only be explained by the change of
the relative positions of the $d$ (${\it l}=2$) and $f$ (${\it l}=3$) states
from which the $d_{3/2}$ and $f_{7/2}$ states emerge. Unfortunately, there is
no straightforward way in the RMF calculations to get an access to the $(N,l)$
states (in sense of Fig. 4). Thus, in order to illustrate the dependence on l,
the centroid energy (denoted as “centr(state)” in Fig. 10b) and defined as an
average energy of the members of spin-orbit doublet is used. Fig. 10b shows
that the energy gap between the centroids of the $d$ and $f$ spin-orbit
doublets increases by 0.55 MeV on going from 42Ca to 47V. As a consequence of
this increase and the above discussed changes in the spin-orbit splittings,
the $1f_{7/2}-1d_{3/2}$ splitting increases by 0.34 MeV. This value represents
more than half of the increase of $(\Delta E^{th}-\Delta E^{exp})_{norm}$ on
going from 42Ca to 47V (Fig. 8a).
#### IV.3.2 Relative placements of the states with different angular momentum
$l$
The fact that the relative placement of states with different orbital angular
momentum $l$ (especially, of high-$l$ states) is not well reproduced in non-
relativistic and relativistic mean field models is well known, see Refs.
MBBD.85 ; RBRMG.98 ; LBBDM.07 . The origin of this problem is connected with
the surface profile of the mean field and kinetic terms. Microscopic
considerations indicate that the effective mass of the nucleon has a
pronounced surface profile which is insufficiently parametrized in the present
mean field models MBBD.85 . In Refs. ZSW.05 ; BWSMG.06 , this fact has been
ignored and no proof has been provided that the placement of the $d$ and $f$
states, from which the $d_{3/2}$ and $f_{7/2}$ states, emerge is correct.
It turns out that the difference in absolute value of $\Delta E$ obtained in
the NL1 and NL3 parametrizations (Fig. 8) is well explained by the differences
in the relative energies of the $d$ and $f$ states in these parametrizations.
The energy gap between the centroids of the $d$ and $f$ spin-orbit doublets is
larger in the NL3 parametrization as compared with the NL1 one by
approximately 400 keV. If one corrects the NL1 results by this energy gap, one
gets the results indicated as NL1cor in Fig. 7. The NL1cor results are very
close to the NL3 ones, which strongly suggests that the difference between the
NL1 and NL3 results is predominantly due to different relative energies of the
$l=3$ and $l=2$ states in these parametrizations of the RMF Lagrangian.
Ref. BWSMG.06 has attributed the fact that the NL1 and NL3 parametrizations
differ in the description of the absolute $\Delta E$ value (Fig. 7) to the
magnitude of the iso-scalar spin-orbit potential. The current investigation
does not support this interpretation.
These results suggest that instead of readjusting the isoscalar strength of
the spin-orbit interaction as it was done in Ref. ZSW.05 , one can attempt to
readjust the coupling constants of the DFT terms influencing the relative
energies of the $l=2$ and $l=3$ states with the same effect on $\Delta E$.
Indeed, 5% reduction of the isoscalar strength of the spin-orbit interaction
introduced in Ref. ZSW.05 reduces $\Delta E$ by $\sim 350$ keV and this
change in $\Delta E$ almost does not depend on nucleus (Fig. in Ref. SW.08 ).
On the other hand, the CRMF results suggests that the same effect can be
achieved if the relative distance of the $d$ and $f$ states is modified.
Indeed, the $\Delta E$ values obtained in the CRMF calculations decrease by
approximately the same amount on going from the NL3 to NL1 parametrization of
the RMF Lagrangian and this decrease only weakly depends on the nucleus (Fig.
7).
#### IV.3.3 The $n$-dependence of $(\Delta E^{th}-\Delta E^{exp})_{norm}$
The $(\Delta E^{th}-\Delta E^{exp})_{norm}$ quantity shows pronounced
dependence on $n$ (Fig. 8a) and its trend (if normalized to a single nucleus)
almost does not depend on the RMF parametrization. In order to understand
which ingredients of $\Delta E^{SC}$ contribute into this $n$-dependence, the
variations $\delta E_{i}=\Delta E_{i}(nucleus)-\Delta E_{i}(^{47}V)$ of
different ($i$-th) terms contributing to $\Delta E^{SC}$ are studied below.
Contrary to Fig. 8a, 47V is selected as a reference in order to get a picture
less disturbed by large fluctuations of some variations in the vicinity of
42Ca (Fig. 11). The $\delta
E_{TO}=\delta(E_{TO}(d_{3/2}^{-1}\,f_{7/2}^{n+1})-E_{TO}(f_{7/2}^{n}))$
variation is obtained in the deformed CRMF calculations (from Fig. 8b), while
other variations shown in Fig. 11 are calculated in spherical RMF
calculations. Thus, I effectively employ the approximation given in Eq. (2)
assuming that the deformation polarization effects are the same both in theory
and experiment. All the results presented here are based on the calculations
with the NL1 parametrization, but it was checked that the NL3 results are
similar.
The $\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$ variation indicates that
the difference between the calculated and experimental $\Delta E$ values
decreases with decreasing $n$. Note that for a given $n$ it is almost constant
indicating only weak isospin dependence of this variation. The largest changes
as a function of nucleus amongst the calculated variations are seen in the
energy gap between the centroids of the $d$ and $f$ spin-orbit doublets (the
curve denoted as “$\delta E({\it l}-{\rm centroids})$” in Fig. 11). It has the
same trend as $\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$ as a function of
$n$. For a given $n$, it shows very large dependence on isospin. The second
largest variation is seen in the spin-orbit splitting of the $f_{7/2}-f_{5/2}$
spin-orbit doublet (the curve denoted as “$\delta E_{\it
ls}(f_{7/2}-f_{5/2})/2$” in Fig. 11). The factor 1/2 is used in this variation
since only one half of the total variation of spin-orbit splitting contributes
into the $f_{7/2}-d_{3/2}$ splitting (see Sect. IV.3.1). The $\delta E_{\it
ls}(f_{7/2}-f_{5/2})/2$ variation has the wrong trend as compared with
$\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$. The $\delta E_{\it
ls}(d_{5/2}-d_{3/2})/2$ variation of the spin-orbit splitting in the
$d_{5/2}-d_{3/2}$ doublet is quite small. It has the correct trend as compared
with $\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$.
The $\delta E(f_{7/2}-d_{3/2})$ variation of the $f_{7/2}-d_{3/2}$ splitting
approximately satisfies the relation
$\displaystyle\delta E(f_{7/2}-d_{3/2})=\delta E({\it l}-{\rm centroids})+$
$\displaystyle\delta E_{\it ls}(f_{7/2}-f_{5/2})/2+\delta E_{\it
ls}(d_{5/2}-d_{3/2})/2$ (7)
The isospin dependencies seen in $\delta E({\it l}-{\rm centroids})$ and
$\delta E_{\it ls}(f_{7/2}-f_{5/2})/2$ act is opposite directions, thus,
reducing the isospin dependence of $\delta E(f_{7/2}-d_{3/2})$ as compared
with the one of $\delta E({\it l}-{\rm centroids})$. However, the $\delta
E(f_{7/2}-d_{3/2})$ variation (Sect. IV.3.1) cannot completely account neither
for absolute value nor for isospin dependence (for a given $n$) of the
$\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$ variation.
Only when the $\delta E(f_{7/2}-d_{3/2})$ variation is combined with the
$\delta E_{TO}$ variation due to NM by
$\displaystyle\delta E^{sum}=\delta E(f_{7/2}-d_{3/2})+\delta E_{TO}$ (8)
a better description of the $\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$
variation emerges. For a given $n$, the isospin dependence of the
$\delta((\Delta E^{th}-\Delta E^{exp})_{norm})$ is well described by $\delta
E^{sum}$. The absolute value of $\delta((\Delta E^{th}-\Delta
E^{exp})_{norm})$ for the Ti nuclei is well described by $\delta E^{sum}$.
However, for the Ca and Sc nuclei, the difference between these two quantities
reaches $30\%$ of the absolute value of $\delta((\Delta E^{th}-\Delta
E^{exp})_{norm})$. The part of this discrepancy is definitely related to the
limitations of the approximation given by Eq. (2).
Thus, the current study clearly shows that the modifications of the relative
placement of the states with different angular momentum $l$, the spin-orbit
splittings and time-odd mean fields on going from 47V to 42Ca contribute into
the $n$-dependence of the difference between the calculated and experimental
$\Delta E$ values (the $\Delta E^{th}-\Delta E^{exp})_{norm}$ quantity).
Previously, this $n$-dependence of $(\Delta E^{th}-\Delta E^{exp})_{norm}$,
expressed in a different form (Fig. 1 in Ref. BWSMG.06 , has been solely
attributed to the deficiency of the iso-vector term of the spin-orbit
interaction BWSMG.06 , but the current investigation does not support such an
interpretation.
Figure 11: (Color online) The variations $\delta E_{i}$ of different terms
contributing to the $\delta(\Delta E^{th}-\Delta E^{exp})_{norm})$ variation,
see text for detail.
## V The energy scale and the effective mass of the nucleon
The terminating states are expected to be of predominantly single-particle
nature CNS ; ZSNSZ.07 : the $d_{3/2}^{-1}\,f_{7/2}^{n+1}$ terminating states
are obtained from the $f_{7/2}^{n}$ terminating state by particle-hole (p-h)
excitation from the $d_{3/2}$ state into the $f_{7/2}$ state. The energy of
this p-h excitation depends on the energies of above mentioned states, and,
thus, it is affected by the energy scale of the single-particle spectra which
is related to the effective mass $m^{*}(k_{F})/m$ of the nucleon at the Fermi
surface.
In the RMF theory, the spin-orbit interaction is effectively scaled by the
effective mass of the nucleon (Ref. R.89 ), and that is a reason why
experimental data on spin-orbit splittings are well described in the
calculations RBRMG.98 ; BRRMG.99 . This scaling also explains why the spin-
orbit splittings of the $1p_{3/2}-1p_{1/2}$, $1d_{5/2}-1d_{3/2}$ and
$1f_{7/2}-1f_{5/2}$ spin-orbit partner orbitals are almost the same in the RMF
and the Nilsson potential calculations (Fig. 5). Note, that the Nilsson
potential is characterized by the effective mass $m^{*}(k_{F})/m\sim 1$ which
is typical for experimental density of the quasiparticle states. Only in the
case of the $2p_{3/2}-2p_{1/2}$ doublet, the spin-orbit splitting is smaller
in the RMF calculations.
On the other hand, the energies of the centroids of the spin-orbit doublets
are stretched out in the RMF calculations as compared with the Nilsson
potential: the difference between the RMF and Nilsson centroid energies
increases on going away from the Fermi level (Fig. 5). Thus, the stretching
out of the single-particle spectra due to low effective mass of the nucleon
shows up mostly for orbital motion of particles and affects the relative
placement of the levels with different angular momentum l. The origin of this
problem has been discussed in Sect. IV.3.2.
The effective mass of nucleon at the Fermi surface (Lorentz mass in the
notation of Ref. JM.89 for the case of the RMF theory) is $m^{*}(k_{F})/m\sim
0.65$ in the RMF theory BRRMG.99 , $\sim 0.7$ in the case of the Hartree-Fock
(HF) approach based on the Gogny forces BHR.03 , and varies in the range
$0.6-1.0$ in the HF approach based on the Skyrme forces BHR.03 revealing much
larger flexibility of this type of the DFT with respect of effective mass. As
a consequence of low effective mass, the calculated spectra are less dense
than the experimental ones: the well known fact in non-relativistic and
relativistic models both for spherical MBBD.85 ; LR.06 ; RBRMG.98 and
deformed systems A250 ; BBDH.03 . This study shows that the $\Delta E^{SC}$
quantity differs from the $f_{7/2}-d_{3/2}$ energy gap in the spherical
single-particle spectra only by the effects of time-odd mean fields and
deformation polarization effects (Sect. III.2.2). These facts pose an open
problem on how to compare the experimental data on $\Delta E$ with the results
of the DFT calculations (especially, those with low effective mass) since the
experimental data on $\Delta E^{exp}$ is expected to be characterized by
$m^{*}(k_{F})/m\sim 1$. The implicit assumption used in Refs. ZSW.05 ;
BWSMG.06 that the DFT reproduces the empirical $\Delta E$ values relatively
well, say within $\sim 10\%$ SW.08 , may be too optimistic especially for the
DFT with low effective mass.
## VI Conclusions
In conclusion, the following results were obtained in the study of band
termination within the DFT framework:
* •
At band termination, the NM does not modify neither total angular momentum nor
the expectation values of the single-particle angular momenta $<j_{x}>_{i}$ of
the single-particle orbitals. NM provides an additional binding to the
energies of the specific configuration and this additional binding increases
with spin and has its maximum exactly at the terminating state. This suggests
that the terminating states can be an interesting probe of the time-odd mean
fields related to NM provided that other effects can be reliably isolated.
* •
The realization of the TS-method in Refs. ZSW.05 ; BWSMG.06 is based on the
analogy with simple form of the Nilsson potential which allows to neglect the
deficiences in the relative placement of the states with different angular
momentum $l$. This approximation is not valid for terminating states in the
$A\sim 44$ mass region in modern and most frequently used versions of the
Nilsson potential
* •
The impact of the relative placement of the states with different angular
momentum $l$ on $\Delta E^{SC}$ is also clearly visible in the RMF
calculations. The difference in absolute $\Delta E^{SC}$ values obtained in
the CRMF calculations with the NL1 and NL3 parametrizations of the Lagrangian
is defined by the different relative energies of the $l=3$ and $l=2$ states in
these parametrizations. The modifications of the relative distance of the
states with different angular momentum $l$ on going from 47V to 42Ca
contribute into the $n$-dependence of the difference between the calculated
and experimental $\Delta E$ values (the $\Delta E^{th}-\Delta E^{exp})_{norm}$
quantity) in addition to the ones due to the spin-orbit interaction and time-
odd mean fields.
The detailed analysis of the TS-method in the RMF framework reveals the
picture which is more complicated than the one suggested in Refs. ZSW.05 ;
BWSMG.06 . The relative placement of the states with different angular
momentum ${\it l}$, defined by the properties of central potential, has to be
taken into account in addition to the DFT terms discussed in these references
when the $\Delta E$ quantity is analyzed. Considering the similarities of the
RMF theory and SDFT, it is very likely that these conclusions are also valid
in the SDFT framework. The current investigation calls for a detailed study of
the impact of the relative placement of the states with different orbital
angular momentum ${\it l}$ on the $\Delta E^{SC}$ quantity in the SDFT
framework.
Existing results for superdeformed bands in 32S MDD.00 ; Pingst-A30-60 and
low-spin states in odd mass nuclei AA.08 point to the time-odd mean fields as
a major point of the difference between SDFT and RMF. For example, the
additional binding due to time-odd mean fields and the energy separation
between different signatures of the SD bands are considerable stronger in SDFT
as compared with RMF MDD.00 . The current study clearly shows that the
correlations induced by time-odd mean fields are large: additional binding due
to NM reaches 4 MeV for $n=3$ (Fig. 9), which is by order of magnitude larger
than those seen before in the RMF calculations at low spin. It also has a
considerable impact on $\Delta E^{SC}$: $\Delta E_{TO}\sim 1.2$ MeV (Fig. 8b).
These results call for a comparative study of time-odd mean fields in the
Skyrme DFT and RMF frameworks. Such study is necessary in order to make a
significant progress towards a better understanding of the role of time-odd
mean fields. The work in this direction is in progress and the results will be
presented in a forthcoming manuscript AA.08 .
## VII Acknowledgements
The material is based upon work supported by the Department of Energy under
Award Number DE-FG02-07ER41459.
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|
arxiv-papers
| 2009-02-01T00:48:56 |
2024-09-04T02:49:00.335622
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. V. Afanasjev",
"submitter": "Anatoli Afanasjev",
"url": "https://arxiv.org/abs/0902.0098"
}
|
0902.0099
|
# Hyperdeformation in the cranked relativistic mean field theory:
the $Z=40-58$ part of nuclear chart
A. V. Afanasjev and H. Abusara Department of Physics and Astronomy,
Mississippi State University, MS 39762, USA
###### Abstract
The systematic investigation of hyperdeformation (HD) at high spin in the
$Z=40-58$ part of the nuclear chart has been performed in the framework of the
cranked relativistic mean field theory. The properties of the moments of
inertia of the HD bands, the role of the single-particle and necking degrees
of freedom at HD, the spins at which the HD bands become yrast, the
possibility to observe discrete HD bands etc. are discussed in detail.
###### pacs:
21.60.Jz, 27.50.+e, 27.60.+j, 21.10.Ft, 21.10.Ma
## I Introduction
Since the discovery of superdeformation (SD) in 152Dy two decades ago Dy152 ,
nuclear SD has been in the focus of attention of the nuclear structure
community; it has been discovered in different mass regions and extensively
studied experimentally SD-sys and theoretically (see, for example, Refs.
BHN.95 ; A150 ; Dudek and references therein). New phenomena such as
identical bands BHN.95 were discovered, and rich variety of experimental data
allowed to test modern theoretical tools under extreme conditions of large
deformation and fast rotation.
It was known for a long time from harmonic oscillator studies BM.75 that even
more elongated shapes, called as hyperdeformed (HD) and characterized by the
semi-axis ratio of around 3:1, are possible. The existence of such stable
shapes was later confirmed in the macroscopic+microscopic (MM) method
BRAGSP.87 ; A180-HD-Chassman ; Yb168 ; A.93 ; CNSPJ.94 ; WD.95 ; CR.95 ; JA.97
; C.01 . Theoretical results on the states located in third (HD) minima are
also available in self-consistent Hartree-Fock+Bogoliubov (HFB) approaches
based on the Skyrme and Gogny forces (see Refs. ERC.97 ; SDN.06 ; HG.07 and
references quoted therein), and relativistic mean field approach RMRG.95 .
However, these results are restricted to spin zero states, which are difficult
to measure in experiment. To our knowledge, the description of the HD states
at high spin within the self-consistent approach has been attempted only in
108Cd AF.05-108Cd [within the cranked relativistic mean field (CRMF) method]
and in four $A\sim 40$ mass nuclei IMYM.02 [within the cranked Skyrme-
Hartree-Fock approach]. The general feature of all these calculations is the
fact that the semi-axis ratio of the HD shapes is less than 3:1 Dudek .
Figure 1: (Color online) The chart of nuclei in the $Z=40-58$ region. Only
experimentally known nuclei are shown. Experimental data on superdeformed
nuclei are taken from Ref. SD-sys . The nuclei in which the search for HD
structures has been performed in the HLHD experiment are taken from Ref.
Hetal.06 .
Let us mention two examples of such studies: one at spin zero, another at high
spin. In actinide nuclei, the HD states are so-called third minima states
around 232Th Th232 ; Th232-1 ; CNSPJ.94 . In these nuclei, the second saddle
point is split, leading to the excited reflection-symmetric and reflection
asymmetric configurations with large quadrupole and octupole deformations,
$\beta_{2}\sim 0.9$ and $\beta_{3}\sim 0.35$. The density distribution at the
HD minimum resembles a di-nucleus consisting of a nearly-spherical nucleus
around the doubly-magic nucleus 132Sn and a well-deformed fragment from the
neutron-rich $A\sim 100$ region CNSPJ.94 . Unfortunately, it is very difficult
to study the HD states at low spin in experiment. In order to overcome this
problem, one should use the fact that the larger moment of inertia connected
with the larger deformation drives the nucleus towards larger deformations
with increasing angular momentum; the HD minimum is thus favored by rotation
and becomes ultimately yrast at high spin. For example, cranked Nilsson-
Strutinsky calculations suggested the existence of very elongated high-spin
minima in nuclei around 168Yb Yb168 . These HD bands are expected to become
yrast at spin around 80$\hbar$.
On the experimental side, very little was known about hyperdeformation apart
from some indications of this phenomenon at low spin in the uranium nuclei
K.98 and light nuclei like 12C C12 and the observation of the HD ridge
structures at high spin in the $A\sim 150$ mass region 152Dy-exp-1 ; 152Dy-
exp-2 . Recent observation of the very extended shapes in 108Cd Cd108-1 ;
Cd108-2 , strongly motivated by earlier calculations of Ref. WD.95 and more
recent studies of Ref. C.01 , has renewed interest in the study of
hyperdeformation at high spin. Although the hyperdeformed nature of the bands
in this nucleus has not been confirmed in the subsequent cranked relativistic
mean field analysis of Ref. AF.05-108Cd (see also Sect. VB in Ref. SDH.07 ),
this experiment provided a strong motivation for subsequent experimental
searches in the $A\sim 125$ mass region (see Refs. HD-exp-1 ; HD-exp-2 ;
Hetal.06 ) and theoretical studies of Refs. Dudek ; SDH.07 within the
framework of the MM method. These experiments revealed rotational patterns in
the form of ridge-structures in three-dimensional (3D) rotational mapped
spectra with dynamic moments of inertia $J^{(2)}$ ranging from 63 to 111 MeV-1
in 12 different nuclei Hetal.06 ; the values around 110 MeV-1 observed in
118Te, 124Xe and 124,125Cs suggest that the HD structures were populated in
these experiments. However, no discrete rotational HD bands have been
identified. It is also necessary to mention that several previous attempts to
search for high spin HD structures in 147Gd 147Gd-exp-1 ; 147Gd-exp-2 , 152Dy
152Dy-exp-1 ; 152Dy-exp-2 , and 168Yb 168Yb-exp-1 did not lead to convincing
evidences for discrete HD bands.
So far, theoretical investigations of HD at high spin were carried out mainly
in the framework of the MM method. One of the main goals of the current
manuscript is to perform for the first time a systematic study of HD within
the framework of fully self-consistent theory, the CRMF theory. Fig. 1 shows
the part of the nuclear chart where our studies are performed. We restrict our
investigation to even-even nuclei; the only exceptions are odd-mass nuclei
111I [in which extremely SD doubly magic band has been found] and 123,124Xe,
123I and 125Cs [which are used in the study of the relative properties of the
HD bands]. In each isotope chain we consider nuclei ranging from the most
proton-rich ones up to the ones located at the neutron-rich side of the
$\beta$-stability valley. Neutron-rich nuclei beyond the valley of the
$\beta$-stability are excluded from consideration because of the experimental
difficulties of studying them at high spins relevant for HD. With the goal to
guide future experimental explorations and to find the nuclei in which the HD
may be studied with current and future experimental facilities, we define the
spins at which the HD bands become yrast in these nuclei. In addition,
available experimental data on the HD ridge-structures in the Te, Xe, and Cs
nuclei are analyzed. The general features of the HD bands are outlined.
The role of the single-particle degrees of freedom at hyperdeformation has not
been studied in detail till now. One of the major goals of the current
manuscript is the study of their role, and it is motivated by the desire to
understand to what extent theoretical methods developed in the study of the SD
bands are also applicable to the HD bands. It is very unlikely that the spins,
parities and excitations energies of the HD bands will be known in the initial
stage of their experimental study. The direct test of the structure of the
wave functions of the single-nucleonic orbitals (e.g. via magnetic moments)
will also not be possible at that stage. Thus, similar to the case of
superdeformation BRA.88 ; Rag.93 ; BHN.95 ; ALR.98 , the relative properties
of different HD bands may play an important role in the interpretation of
their structure. In this context, it is important to understand which changes
of the single-particle orbitals are involved in going from one HD band to
another, and how they affect physical observables like dynamic moments of
inertia $J^{(2)}$, transition quadrupole moments $Q_{t}$, total spin $I$, etc.
In particular, we will study whether the theoretical methods which were
systematically used in the configuration assignment of the SD bands are also
applicable to the HD bands. These include the methods based on the relative
properties of the dynamic moments of inertia $J^{(2)}$ BRA.88 ; BHN.95 , on
the effective alignments $i_{eff}$ Rag.93 ; BHN.95 ; ALR.98 and on the
relative transition quadrupole moments $\Delta Q_{t}$ SDDN.96 ; MADLN.07 .
The manuscript is organized as follows. The definition of physical observables
and the details of numerical calculations are discussed in Sect. II. The spins
at which the HD bands become yrast, the regions of nuclear chart where the
experimental search for the HD structures may be successful and the general
properties of the HD bands are outlined in Sect. III. The data obtained in the
search of the HD structures in the $A\sim 120$ mass region and the single-
particle degrees of freedom are also analysed in this section. Sec. IV is
devoted to the analysis of extremely superdeformed (ESD) structure in 111I.
The calculations predict the existence of doubly magic ESD structure in this
nucleus with the deformations being close to HD, which may be observed with
the current generation of $\gamma$-ray detectors. Finally, Sect. V contains
the main conclusions of our work.
## II The details of the calculations
In the relativistic mean field (RMF) theory the nucleus is described as a
system of point-like nucleons, Dirac spinors, coupled to mesons and to the
photons SW.86 ; Reinh ; VRAL.05 . The nucleons interact by the exchange of
several mesons, namely a scalar meson $\sigma$ and three vector particles
$\omega$, $\rho$ and the photon. The cranked relativistic mean field (CRMF)
theory KR.89 ; KR.90 ; KR.93 ; A150 represents the extension of RMF theory to
the rotating frame. It has successfully been tested in a systematic way on the
properties of different types of rotational bands in the regime of weak
pairing such as normal-deformed AF.05 , superdeformed A60 ; A150 as well as
smooth terminating bands VRAL.05 .
In the current study, we restrict ourselves to reflection symmetric shapes
since previous calculations in the MM method show no indications that odd-
multipole (octupole, …) deformations play a role in the SD and HD bands of the
nuclei covered by our study C.01 and in the HD bands of the $A\sim 110-125$
S.08 mass region.
### II.1 Physical observables
Similar to the case of the SD bands, it is reasonable to expect that the HD
bands will not be linked to the low-spin level scheme for a long period of
time. Thus, the spins and parities of the HD bands will not be known and it
will not be possible to define the kinematic moment of inertia $J^{(1)}$ since
it depends on the absolute values of the spin. In such a situation, the
dynamic moment of inertia $J^{(2)}$ will play an important role in our
understanding of the structure of the HD bands. This is similar to the case of
the SD bands (see Refs. BRA.88 ; BHN.95 ). Other observables, such as
transition quadrupole moments $Q_{t}$ and effective (relative) alignments
$i_{eff}$, will also be important.
In the CRMF calculations, the rotational frequency ${\mathit{\Omega}}_{x}$,
the kinematic moment of inertia $J^{(1)}$ and the dynamic moment of inertia
$J^{(2)}$ are defined by
$\displaystyle{\mathit{\Omega}}_{x}=\frac{dE}{dJ},$ (1) $\displaystyle
J^{(1)}({\mathit{\Omega}}_{x})$ $\displaystyle=$ $\displaystyle
J\left\\{\frac{dE}{dJ}\right\\}^{-1},$ (2) $\displaystyle
J^{(2)}({\mathit{\Omega}}_{x})$ $\displaystyle=$
$\displaystyle\left\\{\frac{d^{2}E}{dJ^{2}}\right\\}^{-1}.$ (3)
The charge quadrupole $Q_{0}$ and mass hexadecupole $Q_{40}$ moments are
calculated by using the expressions
$\displaystyle Q_{0}$ $\displaystyle=$ $\displaystyle
e\sqrt{\frac{16\pi}{5}}\sqrt{\left\langle
r^{2}Y_{20}\right\rangle_{p}^{2}+2\left\langle
r^{2}Y_{22}\right\rangle_{p}^{2}},$ (4) $\displaystyle Q_{40}$
$\displaystyle=$ $\displaystyle\left\langle
r^{4}Y_{40}\right\rangle_{p}+\left\langle r^{4}Y_{40}\right\rangle_{n},$ (5)
where the labels $p$ and $n$ are used for protons and neutrons, respectively,
and $e$ is the electrical charge. At axially symmetric shapes, typical for the
hyperdeformed states, the transition quadrupole moment $Q_{t}$ is equal to
$Q_{0}$.
The quadrupole deformation $\beta_{2}$ for axially-symmetric shapes is
frequently defined in self-consistent calculations from calculated and/or
experimental quadrupole moments using simple relation HG.07 ; A250 ; SGP.05
$\displaystyle\beta_{2}=\frac{1}{XR^{2}}\sqrt{\frac{5\pi}{9}}Q_{0}^{X},$ (6)
where $R=1.2A^{1/3}$ fm is the radius of the nucleus, and $Q_{0}^{X}$ is a
quadrupole moment of the $X$-th (sub)system expressed in fm2. Here $X$ refers
either to proton ($X=Z$) or neutron ($X=N$) subsystem or represents total
nuclear system ($X=A$). This expression, however, neglects the higher powers
of $\beta_{2}$ and higher multipolarity deformations $\beta_{4},\beta_{6},...$
NR.96 , which play an important role at hyperdeformation.
Considering that the definition of the deformation is model dependent NR.96 ,
and that this quantity is not experimentally measurable, we prefer to use
transition quadrupole moment $Q_{t}$ for the description of deformation
properties of hyperdeformed states. This is experimentally measurable
quantity, so in the future our predictions can be directly compared with
experiment. The deformation properties of the yrast SD band in 152Dy (which is
one of the most deformed SD bands Cd108-1 ) are used as a reference. This is
done by introducing normalized transition quadrupole moment
$Q_{t}^{norm}(Z,A)$ in the $(Z,A)$ system
$\displaystyle Q_{t}^{norm}(Z,A)=\frac{ZA^{2/3}}{100.36}\,\,\,e{\rm b}$ (7)
This equation is based on the ratio $Q_{t}^{norm}(Z,A)/Q_{t}(^{\rm 152}{\rm
Dy})$ calculated using Eq. (6) under the assumption that the
$\beta_{2}$-values in the $(Z,A)$ system and in 152Dy are the same. We use the
value $Q_{t}(^{152}{\rm Dy})=18.73$ $e$b obtained in the CRMF calculations
with the NL1 parametrization of the RMF Lagrangian for the yrast SD band in
152Dy at $I=60\hbar$ in Ref. A150 . Thus, in first approximation (neglecting
the higher powers of $\beta_{2}$ and higher multipolarity deformations
$\beta_{4},\beta_{6},...)$) the equilibrium deformation of the band in the
$(Z,A)$ system having the $Q_{t}^{norm}(Z,A)$ value is the same as in the
yrast SD band of 152Dy. We describe the band as hyperdeformed if its $Q_{t}$
value exceeds $Q_{t}^{norm}(Z,A)$ by at least 40%. This criteria is somewhat
relaxed in the $Z=40,\,\,42,\,\,44$ nuclei for which the band is defined as HD
if its $Q_{t}$ value exceeds $Q_{t}^{norm}(Z,A)$ by at least 30%.
The effective (relative) alignment $i_{eff}$ between two bands is defined as
the difference between the spins of two levels in bands A and B at the same
rotational frequency $\Omega_{x}$ Rag.93 :
$\displaystyle i_{eff}^{B,A}(\Omega_{x})=I_{B}(\Omega_{x})-I_{A}(\Omega_{x})$
(8)
This quantity has been used frequently in the analysis of the single-particle
structure of the SD bands and the configuration assignment (see Refs. Rag.93 ;
ALR.98 and references quoted therein). It depends on both the alignment
properties of the single-particle orbitals(s) by which the two bands differ
and the polarization effects induced by the particles in these orbitals AR.00
. The latter are in part related to nuclear magnetism.
Because the pairing correlations are relatively weak in the HD bands of
interest (see Sect. III.3), their intrinsic structure can be described by
means of the dominant single-particle components of the hyperintruder states
occupied. The calculated configurations will be labeled by $[p,n_{1}n_{2}]$,
where $p$, $n_{1}$ and $n_{2}$ are the number of proton $N=7$ and neutron
$N=7$ and $N=8$ hyperintruder orbitals occupied, respectively. For most of the
HD configurations, neutron $N=8$ orbitals are not occupied, so the label
${n_{2}}$ will be omitted in the labeling of such configurations.
Single-particle orbitals are labeled by $[Nn_{z}\Lambda]\Omega^{sign}$.
$[Nn_{z}\Lambda]\Omega$ are the asymptotic quantum numbers (Nilsson quantum
numbers) of the dominant component of the wave function at $\Omega_{x}=0.0$
MeV. The superscripts sign to the orbital labels are used to indicate the sign
of the signature $r$ for that orbital $(r=\pm i)$.
The spins at which the SD and HD configurations become yrast in the
calculations are defined as crossing spins $I_{cr}^{SD}$ and $I_{cr}^{HD}$,
respectively.
Figure 2: (Color online) Potential energy surfaces (PES) obtained in the
axially symmetric RMF calculations without pairing in the 142Ce nucleus. The
results of calculations with $N_{F}=12$, 14 and 26 are shown. In all these
calculations, $N_{B}$ is fixed at 26. The results with $N_{F}=26$ correspond
to a fully converged solution: the binding energies do not change with further
increase of $N_{F}$. The gaps in the PES lines are due to jumps of the
solution from one single-particle configuration to another. The same single-
particle configurations are compared at the same value of charge quadrupole
moment. The normalized value of transition quadrupole moment $Q^{norm}_{t}$
corresponding to the deformation of the yrast SD band in 152Dy is indicated by
arrow. The range of hyperdeformation is also indicated.
### II.2 Numerical scheme of the CRMF calculations
The CRMF equations are solved in the basis of an anisotropic three-dimensional
harmonic oscillator in Cartesian coordinates characterized by the deformation
parameters $\beta_{0}$ and $\gamma$ and oscillator frequency
$\hbar\omega_{0}=41A^{-1/3}$ MeV, for details see Refs. KR.89 ; A150 . The
truncation of basis is performed in such a way that all states belonging to
the shells up to fermionic $N_{F}$ and bosonic $N_{B}$ are taken into account.
The impact of the truncation of basis on the numerical accuracy of the
calculations has first been studied in the axially symmetric RMF code, see
Fig. 2. In the mass region of interest, the calculations with $N_{F}=12$
provide a reasonable approximation to the fully convergent $N_{F}=26$ solution
up to a deformation typical for the SD shapes. However, this truncation scheme
becomes a poor approximation when the quadrupole moment appreciably exceeds
the one corresponding to the lower limit of HD; the difference between the
$N_{F}=12$ and $N_{F}=26$ solutions increases rapidly with the increase of
quadrupole moment (see Fig. 2). On the other hand, in this quadrupole moment
range the results of the calculations with $N_{F}=14$ are closer to exact
solution, although still exceeding it by $\sim 1-2$ MeV at the upper end of
the calculated quadrupole moment range. It was tested that with the decrease
of the mass, the difference between the $N_{F}=14$ and $N_{F}=26$ solutions
will also decrease as well, so that the difference falls within the range of 1
MeV for the majority of the nuclei under study.
Figure 3: (Color online) The same as in Fig. 2, but for the results obtained
in the axially symmetric RMF calculations with pairing using different
parametrizations of the RMF Lagrangian and $N_{F}=26$. Figure shows the
binding energies normalized with respect to the lowest energy of the lowest
potential energy curve. Figure 4: (Color online) The crossing spins (left
panels) at which the SD (solid circles) and HD (open squares) configurations
become yrast and their transition quadrupole moments $Q_{t}$ (right panels)
for the Ce, Ba and Xe isotopes. The values for the SD configurations are shown
only in the cases when they become yrast at lower spins than the HD
configurations. The normalized transition quadrupole moments $Q^{norm}_{t}$
corresponding to the deformation of the yrast SD band in 152Dy are also shown.
These conclusions have also been tested in triaxial CRMF calculations. It was
concluded that physical observables of interest are described with sufficient
numerical accuracy when $N_{F}=12$ is used for the SD and ND states and
$N_{F}=14$ for the HD states. Thus, we employ a hybrid calculational scheme in
which the CRMF solutions in the ND- and SD minima are sought using $N_{F}=12$,
while the ones in the HD minima using $N_{F}=14$. In all CRMF calculations, we
use $N_{B}=20$. In order to eliminate the numerical inaccuracies in the
definition of the crossing spin $I_{cr}^{HD}$, the yrast ND/SD configurations,
which are crossed by the yrast HD configuration, were recalculated in the
crossing region using $N_{F}=14$, and only then the crossing spin was defined.
One should keep in mind that even with $N_{F}=14$ the spins at which the HD
configurations become yrast in the calculations may be overestimated by
$1-2\hbar$ when the deformation of the HD configurations exceeds appreciably
the one corresponding to the lower limit of HD.
When searching for different types of rotational structures it is important to
find the solutions in all local minima which are close to the yrast line in
order to properly define the crossing spins between the rotational structures
of different nature. This is easily achievable in the macroscopic+microscopic
approach by creating potential energy surfaces (PES) in the deformation space
covering quadrupole and triaxial deformations WD.95 ; PhysRep . However, the
computational cost to create similar PES in the self-consistent models is
enormous, thus, it has never been attempted in rotating nuclei. In order to
overcome this problem, we use the fact that in self-consistent approaches
without pairing the deformation of the basis defines to a large extent the
local minima where the solutions will be obtained. Thus, the solutions in the
ND minima, including triaxial ones, are searched using three combinations of
the deformation of basis: $(\beta_{0}=0.30,\gamma=-30^{\circ})$,
$(\beta_{0}=0.30,\gamma=0^{\circ})$, and
$(\beta_{0}=0.30,\gamma=+30^{\circ})$. In a similar way, the solutions in the
SD minima are searched using the following combinations of the deformations of
basis $(\beta_{0}=0.65,\gamma=-30^{\circ})$,
$(\beta_{0}=0.65,\gamma=0^{\circ})$, $(\beta_{0}=0.65,\gamma=+30^{\circ})$,
and $(\beta_{0}=0.8,\gamma=0^{\circ})$. The latter deformation of basis also
leads frequently to the HD solutions. The deformation of basis
$(\beta_{0}=1.0,\gamma=0^{\circ})$ has been used for the search of the
solutions in the HD minima. Non-zero $\gamma$-deformations of basis at large
$\beta_{0}$ lead either to the same solution as $\gamma=0^{\circ}$ or to the
highly excited configurations. For each of the above mentioned values of the
deformation of basis, the lowest in energy solutions are calculated as a
function of spin, and the yrast line is formed from these solutions.
Figure 5: (Color online) The same as in Fig. 4, but for the Te, Sn, and Pd
isotopes.
### II.3 The selection of the RMF parametrization.
The NL1 parametrization of the RMF Lagrangian NL1 is used in the majority of
the calculations in the current manuscript. As follows from previous studies,
this parametrization provides a good description of the moments of inertia of
the rotational bands in unpaired regime in the SD and ND minima A150 ; ALR.98
; A60 ; VRAL.05 , the single-particle energies for the nuclei around the
valley of $\beta$ stability ALR.98 ; A250 and the excitation energies of the
SD minima LR.98 . NL3 NL3 is an alternative parametrization, the quality of
which has been tested in rotating nuclei (but less extensively than in the
case of NL1) ALR.98 ; Zn68 ; A60 ; AF.05 . Some results with this
parametrization will be presented. Few results obtained with the NLSH NLSH
and NLZ NLZ parametrizations will be shown in Sect. III.3 in order to
illustrate the possible spread of calculated quantities. It is necessary to
keep in mind that the quality of the NLSH parametrization in respect of the
description of rotational properties of the nuclei as well as their single-
particle energies is not as good as that of the NL1 and NL3 A60 ; ALR.98 ;
A250 , and the force NLZ has not been tested in that respect.
The spins at which the rotational structures belonging to different minima in
potential energy surfaces become yrast depend in general on the relative
energies of these minima and on the moments of inertia of rotational
structures in these minima. Previous experience shows that different
parametrizations of the RMF Lagrangian give similar moments of inertia for the
same configuration ALR.98 ; A60 ; Zn68 ; VRAL.05 (see also Fig. 13 below).
Fig. 3 also illustrates that the potential energy surfaces at spin zero as a
function of charge quadrupole moment obtained with the NL1 and NL3
parametrizations are similar in shape. These two facts suggest that the HD
configurations should become yrast at approximately the same spins in both
parametrizations: this conclusion is confirmed in Sect. III.1. It is
interesting to note that the NL3 curve in Fig. 3 is similar to the one
obtained with recently developed density-dependent meson-exchange effective
interaction DD-ME2 LNVR.05 , which represents a new class of the RMF
parametrizations as compared with NL1 and NL3. However, so far this
interaction has not been used in the studies of rotating nuclei, thus, it is
not employed in the current study since its reliability in the description of
rotational properties is not known.
Figure 6: (Color online) The same as in Fig. 4, but for the Cd isotopes. The
results of the calculations with the NL1 (HD - open squares, SD - solid
squares) and NL3 (HD - solid triangles up, SD - open triangles down)
parametrizations of the RMF Lagrangian are presented. Note that the
calculations with NL3 were performed only for selected nuclei.
## III Hyperdeformation at high spin: where to expect and its general
features
### III.1 The systematics of crossing spins and transition quadrupole moments
of the HD bands
Figs. 4, 5, 6 and 7 display the spins at which the SD and HD configurations
become yrast (crossing spins) in the CRMF calculations. In addition, the
calculated transition quadrupole moments of these configurations at spin
values close to the crossing spins are shown. The calculated HD configurations
are near-prolate. One can see that the crossing spins $I_{cr}^{HD}$ are
typically lower for proton-rich nuclei. Such a feature is seen in most of the
isotope chains; by going from the $\beta$-stability valley toward the proton-
drip line, one can lower $I_{cr}^{HD}$ by approximately $10\hbar$. The minimum
of crossing spins $I^{HD}_{cr}$ is reached at $N\approx Z+10$ in the Pd, Te
and Ru isotope chains (see Figs. 5e, 5a and 7a), and the Mo isotope chain
(Fig. 7c) shows almost no dependence of $I^{HD}_{cr}$ on mass number. In other
isotope chains, the minima in crossing spins $I^{HD}_{cr}$ appear in most
proton-rich nuclei. Considering that the sensitivity of modern $\gamma$-ray
detectors allows to study discrete rotational bands only up to $\approx
65\hbar$ in medium mass nuclei Dy156 ; 152Dy-link ; Ce132-131 , and that the
observation of higher spin states will most likely require a new generation of
$\gamma$-ray tracking detectors such as GRETA or AGATA, these features of
crossing spins $I^{HD}_{cr}$ represent an important constraint.
As suggested by the studies of the Jacobi shape transition in Ref. SDH.07 ,
the coexistence of the SD and HD minima at the feeding spins may have an
impact on the survival of the HD minima because of the decay from the HD to SD
configurations. If this mechanism is active, then only the nuclei in which the
HD minimum is lower in energy than the SD one at the feeding spin and/or the
nuclei characterized by the large barrier between the HD and SD minima will be
the reasonable candidates for a search of the HD bands. Figs. 4, 5, 6 and 7
show that the HD configurations become yrast at lower spin than the SD ones
only in a specific mass range which depends on the isotope chain. This range
can be narrow as in the case of Te isotopes (Fig. 5a) or wide as in the case
of Ce isotopes (Fig. 4a). The question of the population of the HD bands
within the RMF framework definitely deserves an additional study, but such a
study is beyond the scope of the present manuscript.
Fig. 6 compares the results of the calculations for Cd isotopes obtained with
the NL1 and NL3 parametrizations of the RMF Lagrangian. One can see that both
parametrizations predict similar crossing spins $I^{SD}_{cr}$ and
$I^{HD}_{cr}$ and similar transition quadrupole moments. However, in average,
the crossing spins $I^{HD}_{cr}$ calculated with NL3 are somewhat lower (by
$1-2\hbar$) than the ones obtained in the calculations with NL1.
### III.2 The $A\sim 120$ region: the analysis of experimental data
Table 1: The values of the dynamic moment of inertia $J^{(2)}_{exp}$ of ridge structures measured in the HLHD experiment Hetal.06 . Theoretical results obtained in the MM calculations SDH.07 are shown in the last column. Nucleus | $J^{(2)}_{exp}$ | $J^{(2)}_{MM}$
---|---|---
126Ba | 77 | 118
123Xe | 71 |
122Xe | 77 | 108
121Xe | 63 |
120Te | 71 |
118Te | 111 | 97
125Cs | 100 | 106
124Cs | 111 |
124Xe | 111 | 111
122I | 71 |
121I | 77 | 102
126Xe | 83 | 110
Figure 7: (Color online) The same as in Fig. 4, but for the Ru, Mo and Zr
isotopes.
Recent Hyper-Long-HyperDeformed (HLHD) experiment at the EUROBALL-IV
$\gamma$-detector array revealed some features expected for HD nuclei HD-exp-1
; HD-exp-2 ; Hetal.06 . Although no discrete HD rotational bands have been
identified, rotational patterns in the form of ridge-structures in three-
dimensional (3D) rotational mapped spectra are identified with dynamic moments
of inertia $J^{(2)}$ ranging from 71 to 111 MeV-1 in 12 different nuclei
selected by charged particle- and/or $\gamma$-gating (see Table 1). The four
nuclei, 118Te, 124Cs, 125Cs and 124Xe, found with moment of inertia
$J^{(2)}\sim 110$ MeV-1 are most likely hyperdeformed 111The HD ridges in
152Dy are characterized by $J^{(2)}\sim 130$ MeV-1 152Dy-exp-2 . while the
remaining nuclei with smaller values of $J^{(2)}$ are expected to be
superdeformed. The width in energy of the observed ridges indicates that there
are $\approx 6-10$ transitions in the HD cascades, and a fluctuation analysis
shows that the number of bands in the ridges exceeds 10. The HD ridges are
observed in the frequency range of about 650 to 800 keV, and their dynamic
moments of inertia have typical uncertainty of 10% (e.g. $111\pm 11$ MeV-1 in
124Xe) Hub-private .
Figure 8: (Color online) Calculated kinematic and dynamic moments of inertia
(top panels) and transition quadrupole moments (bottom panels) as a function
of rotational frequency for the lowest HD solutions in 118Te, 124,125Cs and
124Xe. The structure of calculated configurations is indicated at bottom
panels. Experimental data for dynamic moments of inertia of ridge structures
are shown in top panels. Figure 9: (Color online) Energies of the calculated
configurations relative to a smooth liquid drop reference $AI(I+1)$, with the
inertia parameter $A=0.01$. The ND and SD yrast lines are shown by dotted and
dot-dot-dashed lines, respectively. Solid and dot-dashed lines are used for
the [1,2] and [1,21] HD configurations, respectively. Dashed lines represent
excited HD configurations.
The experimental data show unusual features never before seen in the studies
of the SD bands. For example, the addition of one neutron on going from 124Cs
to 125Cs decreases the experimental $J^{(2)}$ value by $\sim 10\%$ (from 111
MeV-1 down to 100 MeV-1, see Table 1). A similar situation is also seen in the
SD minimum: the addition of one neutron on going from 121Xe to 122Xe increases
the experimental $J^{(2)}$ value by $\sim 22\%$ (from 63 MeV-1 to 77 MeV-1,
see Table 1). It is impossible to find an explanation for such a big impact of
the single particle on the properties of nuclei: previous studies in the SD
minima in different parts of the nuclear chart never showed such features. The
case of the pair of 123Xe and 124Xe is even more intriguing: a single particle
triggers the transition from the SD to HD minima (see Table 1). Considering
the fact that the ridges corresponding to the SD and HD minima are observed in
neighboring nuclei, it is difficult to understand why the ridges corresponding
to both minima have not been seen in the same nucleus.
The calculated kinematic and dynamic moments of inertia as well as transition
quadrupole moments of the lowest HD solutions in the candidate HD nuclei are
shown in Fig. 8. The calculated $J^{(2)}$ moments of inertia somewhat
underestimate experimental data. The results of the MM calculations for 118Te,
124Xe and 125Cs (see Table 1) are closer to experimental data, but they are
obtained at fixed quadrupole deformation $\beta_{2}$ while other deformation
parameters $\beta_{4}$, $\beta_{6}$ and $\beta_{8}$ are automatically
readjusted so as to minimize the total free Routhian for the vacuum
configuration.
Figure 10: Proton (top panel) and neutron (bottom panel) single-particle
energies (routhians) in the self-consistent rotating potential as a function
of the rotational frequency $\Omega_{x}$. They are given along the deformation
path of the yrast HD configuration (the [1,2] conf. in Fig. 9) in 124Xe and
obtained in the calculations with the NL1 parametrization of the RMF
Lagrangian. Long-dashed, solid, dot-dashed and dotted lines indicate
$(\pi=+,r=+i)$, $(\pi=+,r=-i)$, $(\pi=-,r=+i)$ and $(\pi=-,r=-i)$ orbitals,
respectively. At $\Omega_{x}=0.0$ MeV, the single-particle orbitals are
labeled by the asymptotic quantum numbers $[Nn_{z}\Lambda]\Omega$ (Nilsson
quantum numbers) of the dominant component of the wave function. Solid (open)
circles indicate the orbitals occupied (emptied) in the [1,2] configuration.
The dashed box indicates the frequency range corresponding to the spin-range
$I=60-85\hbar$ in this configuration.
In the MM calculations, the kinematic moments of inertia of the configurations
in the HD minimum decrease smoothly with the spin, while their dynamic moments
of inertia are nearly constant (see Figs. 10 and 11 in Ref. SDH.07 ). The
behaviour of these observables as a function of rotational frequency (or spin)
is completely different in the self-consistent CRMF calculations (see Figs. 8,
11 and Fig. 15 below). The kinematic moment of inertia is either nearly
constant or very gradually increases with rotational frequency. The dynamic
moment of inertia gradually increases over the calculated frequency range
showing the features typical to the SD bands in the $A\sim 190$ mass region
which are affected by pairing WS.95 ; BHN.95 : this is despite the fact that
pairing is neglected in the CRMF calculations. The transition quadrupole
moment $Q_{t}$ is also increasing with rotational frequency; such a feature
has not been seen before in the calculations without pairing for the SD bands.
The microscopic origin of these unusual features will be discussed in more
details in Sect. III.3.
### III.3 124Xe nucleus
The results of the CRMF calculations for some HD configurations in 124Xe are
displayed in Fig. 9. The HD minimum becomes lowest in energy at spin
$82\hbar$, and the [1,2] configuration is the yrast HD configuration in the
spin range of interest. The occupation of the single-particle orbitals in this
configuration is presented in Fig. 10. The excited HD configurations displayed
in Fig. 9 are built from this configuration by exciting either one proton or
one neutron or simultaneously one proton and one neutron. The total number of
excited HD configurations shown is 35. It interesting to mention that the
configuration involving the lowest $N=8$ neutron orbital (the [1,21] conf. in
Fig. 9) is calculated at low excitation energy.
The calculations reveal a high density of the HD configurations which will be
even higher if the additional calculations for the excited configurations
would be performed starting from the low-lying excited HD configurations, such
as the [1,21] configuration. This high density is due to two facts: relatively
small $Z=54$ and $N=70$ HD shell gaps in the frequency range of interest (see
Fig. 10) and the softness of the potential energy surfaces in the HD minimum.
Fig. 11b illustrates the latter feature: the particle-hole excitations
discussed above, characterised by low excitation energy, lead to appreciable
changes in the transition quadrupole moments $Q_{t}$. It is interesting to
mention that there are large similarities between the single-particle
routhians in the vicinity of the $Z=54$ and $N=70$ HD shell gaps obtained in
the CRMF calculations for yrast HD configuration in 124Xe (Fig. 10) and the
ones obtained in the Woods-Saxon calculations for the HD minimum in 122Xe
employing the so-called universal parametrization of the Woods-Saxon potential
(see Figs. 8 and 9 in Ref. SDH.07 ). As a consequence, the high density of the
excited HD states in 124Xe is also expected in the MM calculations based on
the formalism of Ref. SDH.07 .
Figure 11: (Color online) Dynamic moments of inertia $J^{(2)}$ (panel (a)) and
transition quadrupole moments $Q_{t}$ (panel (b)) of the HD configurations in
124Xe shown in Fig. 9. They are displayed as a function of rotational
frequency $\Omega_{x}$. The regions of band crossings are excluded in these
plots.
The high density of the HD configurations may question our neglect of pairing.
This is because there are numerous possibilities to scatter proton and neutron
pairs and this process is energetically inexpensive due to the high density of
the calculated configurations. In order to test the impact of pairing on the
moments of inertia and binding energies, the comparative studies of the vacuum
HD configuration and its unpaired analog in 124Xe and of the vacuum SD
configuration and its unpaired analog in 152Dy have been performed within the
cranked relativistic Hartree+Bogoliubov (CRHB) CRHB and CRMF approaches. An
approximate particle number projection by means of the Lipkin-Nogami method is
employed in the CRHB approach. Note that unpaired analog of the vacuum HD
configuration in 124Xe (built from the [1,2] configuration by the excitation
of the proton from the $\pi[770]1/2(r=+i)$ orbital into the
$\pi[420]1/2(r=+i)$ orbital, see Fig. 10) is non-yrast in the spin range of
interest. As follows from this study, in both nuclei the pairing has a similar
impact on the moments of inertia of the configurations under consideration.
Taking into account that the SD bands in the $A\sim 150$ mass region are well
described in the calculations without pairing A150 ; ALR.98 , it is reasonable
to expect that the neglect of pairing is a valid approximation for the moments
of inertia of the HD bands in 124Xe. Pairing leads to an additional binding of
$\sim 500$ keV in the case of yrast SD band in 152Dy; this additional binding
slightly exceeds 1 MeV in the case of the vacuum HD configuration in 124Xe.
The dominant effects in the quenching of pairing correlations are the Coriolis
antipairing effect and the quenching due to shell gaps: the latter effect
being more pronounced in the SD bands of the $A\sim 150$ mass region because
of the larger size of the SD shell gaps (see Fig. 4 in Ref. A150 ). The third
mechanism of the decrease of pairing is the blocking effect Ring-book . Due to
this effect the impact of pairing on physical observables will be even lower
in the HD bands of 124Xe based on the excitation(s) of one (two) particles
considered in Fig. 9. Thus, although weak pairing will somewhat modify the
relative energies of different configurations, in no way will it create an
energy gap between the vacuum and excited configurations.
Figure 12: The self-consistent neutron density $\rho_{n}(y,z)$ as a function
of $y$\- and $z$\- coordinates for the [1,2] configuration in 124Xe at
rotational frequency $\Omega_{x}=0.75$ MeV. Top and bottom panels show 2- and
3-dimensional plots of the density distribution, respectively. In the top
panel, the densities are shown in steps of 0.01 fm-3 starting from
$\rho_{n}(y,z)=0.01$ fm-3. Figure 13: (Color online) Kinematic ($J^{(1)}$) and
dynamic ($J^{(2)}$) moments of inertia as well as transition quadrupole
$Q_{t}$ and mass hexadecupole $Q_{40}$ moments of the [1,2] configuration in
124Xe calculated with different parametrizations of the RMF Lagrangian.
The calculations suggest that it will be difficult to observe discrete HD
bands in 124Xe since their high density will lead to a situation in which the
feeding intensity will be redistributed among many low-lying bands, thus
drastically reducing the intensity with which each individual band is
populated. On the other hand, the high density of the HD bands may favor the
observation of the rotational patterns in the form of ridge-structures in
three-dimensional rotational mapped spectra as it has been seen in the HLHD
experiment Hetal.06 .
Fig. 8 shows that the HD shapes undergo a centrifugal stretching that result
in an increase of the transition quadrupole moments $Q_{t}$ with increasing
rotational frequency. This process also reveals itself in the moments of
inertia: the kinematic moments of inertia are either nearly constant or
slightly increase with increasing rotational frequency, while the dynamic
moments of inertia increase continuously and substantially over the frequency
region of interest. On the contrary, the dynamic moments of inertia of the HD
bands are almost constant as a function of rotational frequency in the MM
calculations (see Figs. 10 and 20 in Ref. SDH.07 ), which is most likely a
consequence of fixed quadrupole deformation. The above mentioned features are
general ones for the HD bands in the $A\sim 120$ mass region, see Figs. 8, 11
and 15. They are in complete contract to the features of the SD bands in
unpaired regime, in which the $Q_{t}$, $J^{(1)}$ and $J^{(2)}$ values (apart
from the unpaired band crossing regions) decrease with increasing rotational
frequency (see Refs. BRA.88 ; A150 ; A60 ; VRAL.05 and references therein).
Systematic analysis of the yrast/near-yrast HD configurations in the part of
the nuclear chart under investigation shows that the centrifugal stretching is
a general feature. At the spins, where the HD minimum is lowest in energy, it
reveals itself (with very few exceptions) by the increase of transition
quadrupole $Q_{t}$ and mass hexadecapole $Q_{40}$ moments. Only in a few HD
bands, characterized by the modest transition quadrupole moment, at low
rotational frequencies these quantities decrease with increasing $\Omega_{x}$.
However, even in these bands the $Q_{t}$ and $Q_{40}$ values start to increase
above specific value of rotational frequency. Similar features are also seen
in the dynamic moments of inertia; with a few exceptions the $J^{(2)}$ values
increase in the spin range of interest. The variations (both the increases and
decreases) in the kinematic moments of inertia are rather small ($\sim 2\%$ of
absolute value) in the frequency range of interest.
Figure 14: The weights $a_{N}^{2}$ of different $N$-components in the
structure of the wave functions of the indicated orbitals. They are shown as a
function of rotational frequency. For simplicity, the region of the crossing
between the $\nu[880]1/2^{-}$ and $\nu[411]3/2^{-}$ orbitals at
$\Omega_{x}\sim 0.55$ MeV is removed; dotted lines are used in panel (b) to
connect the weights corresponding to the $\nu[880]1/2^{-}$ orbital before and
after crossing. Figure 15: (Color online) Dynamic moments of inertia $J^{(2)}$
of selected configurations in 124Xe and neighbouring nuclei. Dynamic moment of
inertia of the [1,2] configuration A in 124Xe is shown by a thick solid line
in each panel. The $J^{(2)}$ values of the configurations in the nucleus
indicated on the panel are displayed by the lines of other types. These
configurations differ from the [1,2] configuration A in 124Xe in the
occupation of the orbitals shown in the panels. Vertical dashed lines indicate
the frequency range corresponding to the spin range $I=60-85\hbar$ in the
[1,2] configuration of 124Xe.
The basis of the CRMF model is sufficiently large to see if there is a
tendency for the development of necking. Fig. 12 shows some indications of the
necking and the clusterization of the density into two fragments in the [1,2]
configuration of 124Xe, but this effect is not very pronounced in this
nucleus.
The kinematic and dynamic moments of inertia as well as the transition
quadrupole and mass hexadecapole moments of the [1,2] configuration in 124Xe
are shown for different parametrizations of the RMF Lagrangian in Fig. 13. The
gradual increase of all physical observables is due to centrifugal stretching.
The NLZ (NLSH) parametrizations provide the largest (smallest) values of the
above mentioned physical observables, while the results obtained with NL1 and
NL3 are in between those results. Similar relations between the results
obtained with these parametrizations also exist in other regions of nuclear
chart studied so far in the CRMF or CRHB frameworks, namely, in the $A\sim 60$
ARR.99 , $A\sim 150$ ALR.98 and $A\sim 190$ CRHB regions of superdeformation
and in the $A\sim 250$ A250 region of normal deformation. The NL1 and NL3
parametrizations, which have been extensively used in the previous studies of
rotating systems and superdeformation VRAL.05 , give the values of physical
observables of interest which differ only by few %. It is known that the NLSH
parametrization somewhat underestimates the experimental moments of inertia
ARR.99 ; ALR.98 . The NLZ parametrization has not been used in the previous
studies of rotating systems, so it is unknown how well it describes such
systems.
### III.4 Single-particle properties at hyperdeformation: an example of
neighbourhood of 124Xe.
The role of the single-particle degrees of freedom at hyperdeformation was
mainly overlooked in the previous studies. It has been studied to some extent
only within the MM method in Refs. A.93 ; SDH.07 . However, the studies of
Ref. SDH.07 suggest that the 124Xe nucleus is very rigid in the HD minimum:
the dynamic moments of inertia of different HD bands differ by no more than
2%, and their changes as a function of spin are very small (see Fig. 10 in
Ref. SDH.07 ). Similar results were obtained for HD bands in 146Gd and 152Dy
in Ref. A.93 .
Figure 16: (Color online) Effective alignments $i_{eff}$ extracted from the
calculated configurations for the orbitals active in the vicinity of the
$Z=54/55$ and $N=70$ HD shell gaps (see Fig. 10). The calculated
configurations are the [1,2] conf. in 124Xe and the configurations in
neighboring nuclei (shown in Fig. 15) obtained by adding or removing a single
particle (proton or neutron). The effective alignment between configurations X
and Y is indicated as “X/Y”. The configuration X in the lighter nucleus is
taken as a reference, so the effective alignment measures the effect of the
additional particle. The compared configurations differ in the occupation of
the orbitals shown in the panels. Note that the vertical scale of different
panels is different. Vertical dashed lines indicate the frequency range
corresponding to the spin range $I=60-85\hbar$ in the conf. A of 124Xe.
On the contrary, the CRMF calculations for the dynamic moment of inertia of
the yrast and excited HD configurations in 124Xe show much larger spread and
much larger variations as a function of rotational frequency, see Fig. 11a. In
addition, large variations in the calculated transition quadrupole moments
$Q_{t}$ of these configurations are clearly seen in Fig. 11b. This suggests
that the HD minimum is relatively soft and that the individual properties of
the single-particle orbitals play an important role in the definition of the
properties of the HD bands. One of our goals is to investigate the impact of
the particle in a specific single-particle orbital on the properties of the HD
bands and to study whether the methods of configuration assignment based on
the relative properties of different bands are also applicable at HD.
#### III.4.1 The structure of the wave function
The structure of the wave function at HD is analysed on the example of a few
single-particle orbitals of the [1,2] configuration in 124Xe (Fig. 14). The
evolution of these orbitals in energy with rotational frequency is displayed
in Fig. 10. The wave function $\Psi$ is expanded into the basis states by
$\displaystyle\Psi=\sum_{N,\alpha}c_{N,\alpha}|N\alpha>$ (9)
where $N$ and $\alpha$ represent the principal quantum number and the set of
additional quantum numbers specifying the basis state, respectively. We
specify the weight $a_{N}^{2}$ of the basis states belonging to the specific
value of $N$ in the structure of the wave function as
$\displaystyle a_{N}^{2}=\sum_{N-{\rm fixed},\alpha}c^{2}_{N,\alpha}$ (10)
with the condition $\sum_{N}a_{N}^{2}=1$ following from the orthonormalization
of the wave function of the single-particle orbital.
Hyperdeformation leads to a considerable fragmentation of the wave function
over $N$, which is much larger than in the case of SD. In the regions away
from the band crossing the weight $a_{N}^{2}$ of the dominant $N$-component of
the wave function does not exceed 0.8 while the weight of second largest
component is typically around 0.2 (Fig. 14). Very strong fragmentation of the
wave function is seen in the case of the $\nu[761]3/2^{+}$ orbital: before the
band crossing the weights of the $N=7$ and $N=5$ components of the wave
function are approximately 0.6 and 0.3, respectively. Even stronger
fragmentation is seen in the region of the band crossing of the
$\nu[761]3/2^{+}$ and $\nu[301]3/2^{+}$ orbitals at $\Omega_{x}\sim 0.7$ MeV
(Figs. 10) where they strongly interact and gradually exchange their character
(Figs. 14a and c). Similar fragmentation is also seen for the
$\pi[770]1/2^{+}$ orbital (Fig. 14) which interacts strongly with the
$\pi[532]5/2^{+}$ orbital in the band crossing region at $\Omega_{x}\sim 0.8$
MeV (Fig. 10).
Figure 17: (Color online) Relative transition quadrupole moments $\Delta
Q_{t}=Q_{t}(A+1)-Q_{t}(A)$ [$A$ is the mass of the nucleus] extracted from the
calculated configurations in indicated nuclei. The compared configurations are
shown as “X/Y”: the configuration X in the lighter nucleus is taken as a
reference, so the $\Delta Q_{t}$ measures the effect of the additional
particle placed in the orbitals shown in the panels. Vertical dashed lines
indicate the frequency range corresponding to the spin range $I=60-85\hbar$ in
the [1,2] configuration of 124Xe.
#### III.4.2 The methods of configuration assignment
The HD bands in nuclei neighboring to 124Xe, which differ by either one proton
or one neutron from the [1,2] configuration in 124Xe, and their relative
properties with respect of the [1,2] configuration in 124Xe are studied in
order to investigate the applicability of different methods of configuration
assignment at HD.
The dynamic moments of inertia for the four HD bands in each of these nuclei
are compared with the one of the [1,2] configuration in 124Xe in Fig. 15. The
difference between the dynamic moments of inertia of the configurations in
nuclei with masses $A$ and $A\pm 1$ is due to the impact of the particle in
the specific single-particle orbital by which two compared configurations
differ. The results of the calculations question conventional wisdom BRA.88
that the largest impact on the dynamic moment of inertia is coming from the
particles in the intruder orbitals. Indeed, the impact of the neutron in the
hyperintruder $\nu[880]1/2^{-}$ orbital on the dynamic moments of inertia
(Fig. 15d) is comparable to the one of non-intruder $\nu[642]5/2^{+}$ orbital
or even smaller by a factor of $\sim 2$ than the impact due to the neutron in
non-intruder $\nu[532]3/2^{+}$ orbital (Fig. 15b). A similar situation is also
seen for protons, where, for example, the impact of the proton in the
hyperintruder $\pi[770]1/2^{+}$ orbital is smaller than its impact in the non-
intruder $\pi[420]1/2^{-}$ orbital. This suggests that not only angular
momentum, carried by the particle in specific single-particle orbital, but
also polarization effects it induces into time-even and time-odd mean fields
AR.00 are important when considering relative properties of two
configurations. Based on this example, one can conclude that the configuration
assignment of the HD bands, based only on the relative properties of the
dynamic moments of inertia of two compared bands, is unreliable.
Figure 18: (Color online) Proton and neutron single-particle energies in
108Cd, as a function of charge quadrupole moment $Q$, obtained in the axially
symmetric RMF calculations. Solid and dashed lines denote positive and
negative parity orbitals, respectively. The Fermi energy ${\rm E_{F}}$ is
shown by dotted line. The single-particle orbitals are labeled by the Nilsson
quantum numbers. Large shell gaps are indicated.
The configuration assignments at SD have been mostly based on the effective
alignment approach (see Refs. Rag.93 ; ALR.98 ; ARR.99 and references
therein). The success of this method is due to the fact that it was possible
to separate intruder and non-intruder orbitals since the former show
pronounced dependence of the effective alignments $i_{eff}$ on the rotational
frequency (see, for example, Figs. 2, 3, 5, 6 and 8 in Ref. ALR.98 ). On the
contrary, the effective alignments of non-intruder orbitals are typically
constant as a function of rotational frequency. It also follows from the
studies in the $A\sim 140-150$ region of superdeformation that the change of
effective alignment by $\approx 1\hbar$ within the observed frequency range
allows to identify aligning intruder orbitals with a high level of confidence.
A configuration assignment based on the effective alignments depends on how
accurately these alignments can be predicted. For example, the application of
the effective alignment approach in the $A\sim 140-150$ region of
superdeformation requires an accuracy in the prediction of $i_{eff}$ on the
level of $\sim 0.3\hbar$ and $\sim 0.5\hbar$ for nonintruder and intruder
orbitals, respectively ALR.98 ; Rag.93 ; BHN.95 . In the highly deformed and
SD bands from the $A\sim 60-80$ mass region, these requirements for accuracy
are somewhat relaxed ARR.99 ; AF.05 . We expect that in the $A\sim 125$ mass
region of HD, the effective alignments should be predicted with a precision
similar to that in the $A\sim 140-150$ region for a reliable configuration
assignment.
Figure 19: The energy gaps between the last occupied and first unoccupied
single-particle orbitals shown as a function of neutron number for different
isotope chains. They are extracted from the routhian diagrams of the lowest HD
configurations at the spin values where these configurations become yrast. The
bars are used for the energy gaps in the Cd isotopes.
Our analysis shows that a reliable configuration assignment for the HD bands
based solely on the effective alignment approach will be problematic (at least
in the $A\sim 125$ mass region) because of several reasons. First, the
hyperintruder orbitals do not show appreciable variations of $i_{eff}$ with
rotational frequency. Fig. 16 shows that the effective alignments of the
hyperintruder orbitals such as $\pi[770]1/2^{+}$ and $\nu[880]1/2^{-}$ show
little variations with rotational frequency (see Fig. 16a,d). On the contrary,
the effective alignments of the $\nu[532]3/2^{+}$ and $\nu[530]1/2^{-}$
orbitals show much larger variations reaching $1.5\hbar$ in the spin range
$I=60-85\hbar$ in the case of the latter orbital (see Fig. 16b). However, the
variations of $i_{eff}$ as a function of rotational frequency are small for
the majority of the orbitals in the spin range of interest. Thus, contrary to
the case of SD, it will be more difficult to distinguish between
hyperintruder, intruder and non-intruder orbitals based on the variations of
$i_{eff}$ with rotational frequency. This situation will become even more
complicated if the suggestion of Ref. SDH.07 that the spin range over which
the HD bands are expected to be observed ($24\hbar$ at the most; this is
shorter than in the case of SD) is true. These two features (small variations
of $i_{eff}$ and expected spin (frequency) range of the HD bands) will lead to
a situation where the $i_{eff}$ values for many orbitals will look alike
within the typical ’error bars’ of the description of $i_{eff}$ by theoretical
models, so that it will be difficult to distinguish between them within the
framework of the effective alignment approach.
Similar to the case of SD SDDN.96 ; MADLN.07 , additional information on how
the single particle affects the properties of the HD bands can be extracted
from the relative transition quadrupole moments $\Delta Q_{t}$. Fig. 17 shows
that the hyperintruder $\pi[770]1/2^{+}$ and $\nu[880]1/2^{-}$ orbitals with
$\Delta Q_{t}\approx 2$ $e$b and $\Delta Q_{t}\approx 1.25$ $e$b have the
largest impact on the transition quadrupole moments among the studied proton
and neutron orbitals. One has to keep in mind that the addition of a proton
changes the proton number by one. This change contributes approximately 0.5
$e$b in relative transition quadrupole moment $\Delta Q_{t}$ of the proton
orbitals. This effect is not present in the $\Delta Q_{t}$ values of the
neutron orbitals.
The $\Delta Q_{t}$ values were used only as a complimentary tool of the
configuration assignment at SD. This is because of the difficulty to measure
them in experiment 146Gd-exp ; A130-exp1 and the fact that they show little
variation as a function of rotational frequency, thus providing less
information than $i_{eff}$. The same features are also valid at HD; see Fig.
17 for the variations of the $\Delta Q_{t}$ values. In addition, some single-
particle orbitals such as $\pi[422]3/2^{-}$ and $\pi[303]7/2^{-}$ (Fig. 17c)
show very similar $\Delta Q_{t}$ values. This will not allow to make a unique
configuration assignment even if the experimental $\Delta Q_{t}$ values for
these orbitals are available. On the other hand, their $i_{eff}$ values differ
by $\sim 1\hbar$ (Fig. 16c), and this fact can be used in the configuration
assignment.
However, the fact that in general the effective alignment approach fails to
provide a unique configuration assignment at HD increases the role of the
method of configuration assignment based on relative transition quadrupole
moments. Our analysis shows that only simultaneous application of these two
methods by comparing experimental and theoretical $(i_{eff},\Delta Q_{t})$
values will lead to a reliable configuration assignment at HD.
Let us illustrate this on the hypothetical example of two “experimental”
bands; one in 123I and another in 124Xe. In this example, the [1,2]
configuration is assigned to the band in 124Xe. Let us assume that the
effective alignments in the 123I/124Xe pair of the bands increase from
$4.0\hbar$ to $4.25\hbar$ in the frequency range 0.62-0.87 MeV under selected
spins of these bands. Under these conditions, the “experimental” bands differ
in the occupation of the $\pi[770]1/2^{+}$ orbital (Fig. 16a). However, it is
reasonable to expect that the spins of “experimental” bands will not be fixed,
so these changes in effective alignment should be from $(4.0+n)\hbar$ to
$(4.25+n)\hbar$, where $n=0,\pm 1,\pm 2,...$. Assuming that the accuracy of
the description of effective alignments in theoretical calculations is around
$0.4\hbar$, one can conclude that for $n=-3$ the “experimental” bands can also
differ in the occupation of either the $\pi[532]5/2^{-}$ or $\pi[651]3/2^{+}$
orbitals (Fig. 16a). In a similar way to the $A\sim 150$ region of SD Rag.93 ;
ALR.98 , the systematic studies of the pairs of the bands which differ by one
proton may narrow the choice of the orbitals involved. On the other hand, the
$\Delta Q_{t}$ values for these orbitals are drastically different; $\Delta
Q_{t}\approx 2.0$ $e$b for the $\pi[770]1/2^{+}$ orbital, $\Delta Q_{t}\approx
1.4$ $e$b for $\pi[651]3/2^{+}$, and $\Delta Q_{t}\approx 0.7$ $e$b for
$\pi[532]5/2^{-}$ (see Fig. 17). So, if both quantities, $i_{eff}$ and $\Delta
Q_{t}$, are measured simultaneously, a unique configuration assignment for
“experimental” band in 123I will be possible.
The band crossing features of the HD bands provide an additional tool of
configuration assignment which can be used more frequently than in the case of
the SD bands because of strong mixing between the different $N$-shells at HD.
The large peaks in $J^{(2)}$ of the $\nu A$ and $\nu B$ configurations in
125Xe (Fig. 15d) are due to the band crossings with a strong interaction.
These crossings are also visible in the effective alignments $i_{eff}$ (Fig.
16d) and relative transition quadrupole moments $\Delta Q_{t}$ (Fig. 17d).
They originate from the crossing of the same signatures of the $\nu[301]3/2$
and $\nu[761]3/2$ orbitals, where $\nu A$ and $\nu B$ have signatures $r=+i$
and $r=-i$, respectively. The former orbital is occupied before band crossing,
the latter after band crossing. An unusual feature of these band crossings is
the fact that they originate from the interaction of the orbitals, the
dominant $N$-components of which differ by $\Delta N=4$. At SD, the crossings
between the orbitals dominated by different $N$-shells have been characterized
by a weak interaction leading to a sharp jump in $J^{(2)}$ A150 ; Sm142b ;
Paris98 . The observed unpaired SD band crossings with strong interaction are
between the orbitals with the same dominant $N$-shells and they were observed
in the nuclei around 147Gd R.91 ; A150 .
### III.5 General observations: the density of the HD bands and the necking
degree of freedom
As discussed in Sect. III.3 on the example of 124Xe, the high density of the
HD bands is one of the major obstacles for the observation of discrete HD
bands. It will lead to a situation where the feeding intensity will be
redistributed among many low-lying HD bands, thus, drastically reducing the
intensity with which each individual band is populated. As a consequence, the
feeding intensity of an individual HD band will drop below the observational
limit of experimental facility; this fact has to be taken into account when
planning future experiments for a search of discrete HD bands.
Figure 20: The self-consistent proton density $\rho_{p}(y,z)$ as a function of
$y$\- and $z$\- coordinates for the HD configurations. They are displayed at
spin values at which these configurations become yrast. For each isotope
chain, the densities in two nuclei (typically, most proton- and neutron rich
ones included in calculations) are shown. The densities are displayed in steps
of 0.01 fm-3 starting from $\rho_{p}(y,z)=0.01$ fm-3.
Two factors contribute to the high density of the HD bands, namely, relatively
small proton and neutron HD shell gaps in the frequency range of interest and
the softness of the potential energy surfaces in the HD minimum (see Sect.
III.3). Systematic mapping of the density of the HD states as a function of
the proton and neutron numbers is too costly in the computational sense
because it involves the calculation of the lowest in energy particle-hole
excitations. Thus, we decided to look at the problem of the density of the HD
states in a somewhat simplistic way by considering the proton and neutron
energy gaps between the last occupied and the first unoccupied states in the
yrast HD configurations; the small size of these gaps will most likely point
to the high density of the HD bands.
Figure 21: The same as in Fig. 12, but for yrast megadeformed state in 102Pd
at rotational frequency $\Omega_{x}=0.95$ MeV. Two top panels show
2-dimensional plots of the proton and neutron density distribution. Figure 22:
(Color online) Energies of the calculated configurations relative to a smooth
liquid drop reference $AI(I+1)$, with the inertia parameter $A=0.01$. Normal
deformed (ND), SD and HD configurations are shown by dotted, dot-dashed and
solid lines, respectively. Configuration A is shown by long-dashed line.
The analysis of the Nilsson diagrams in Fig. 18 already reveals some HD gaps
in the single-particle spectra. At the values of $Q_{0}\sim 17-20$ $e$b
typical for the HD configurations in Cd isotopes (Fig. 6b), there are very
large proton $Z=48$ and neutron $N=48$ HD shell gaps and smaller neutron gaps
at $N=58$ and 60. In general, this figure suggests that the hyperdeformation
will be more favoured in the nuclei with a similar number of protons and
neutrons because the proton and neutron shell effects for the HD shapes will
act coherently; this trend has already been seen in the crossings spins
$I_{cr}^{HD}$ for different isotope chains in Sect. III.1.
The size of these gaps and their presence will be altered (especially, for
medium and small size energy gaps) when the rotation and the self-consistent
readjustment of the neutron and proton densities with the change of particle
number are taken into account. Indeed, this is seen in Fig. 19 which shows the
energy gaps between the last occupied and first unoccupied single-particle
orbitals as a function of the neutron number for different isotope chains. The
largest proton gap at $Z=48$ is seen in Cd isotopes; its size is around 1.5
MeV in proton-rich nuclei and it increases up to 3 MeV with the increase of
neutron number. In other isotope chains, the size of the proton energy gap is
smaller than in Cd isotopes and it fluctuates around 1 MeV. For the majority
of the nuclei, the size of the neutron energy gap fluctuates around 1 MeV.
However, its size increases up to 1.5 MeV in some nuclei and in 96Cd it
reaches 2 MeV (see Fig. 19 for details).
Taking into account that the proton and neutron HD shell gaps in 124Xe are
around 1 MeV (Fig. 10) and considering the results for the density of the HD
states in this nucleus as a reference (Sect. III.3), one can conclude that the
analysis of the energy gaps suggests that in most of the nuclei the density of
the HD bands will be high. For these nuclei, the observation of discrete HD
bands using existing facilities is most likely not possible. The only
exceptions are Cd nuclei and a few nuclei in which the size of at least one
gap reaches 1.5 MeV (see Fig. 19 for details). For example, in Cd nuclei the
large size of the $Z=48$ HD shell gap (especially, for nuclei in the valley of
the $\beta$-stability) will make proton particle-hole excitations
energetically expensive. As a consequence, the density of the HD bands has to
be lower in Cd isotopes as compared with the one in other isotopes.
Figure 23: The same as in Fig. 10, but for the configuration A in 111I. Solid
(open) circles indicate the orbitals occupied (emptied). The dashed box
indicates the frequency range corresponding to the spin range $I=50-75\hbar$
in this configuration.
One has to remember that the high density of the HD bands is not necessarily a
negative factor. It favors the observation of the rotational patterns in the
form of ridge-structures in three-dimensional rotational mapped spectra as it
has been seen in the HLHD experiment for a few nuclei Hetal.06 . The
observation of ridge-structures as a function of proton and neutron number,
which seems to be feasible with existing experimental facilities such as
GAMMASPHERE, will provide invaluable information about HD at high spin.
The importance of the necking degree of freedom for the high-spin HD states
has been studied in the MM approach in Refs. A180-HD-Chassman ; C.01 .
However, this degree of freedom has not been investigated in detail at high
spin in self-consistent approaches so far. In order to fill this gap in our
knowledge, the systematics of the self-consistent proton density distributions
in the HD states obtained in the CRMF calculations are shown in Fig. 20. One
can see that in some nuclei such as 124Te, 130Xe, 132Ba the necking degree of
freedom plays an important role, while others (for example, 100Mo and 136Ce)
show no necking. The neck is typically less pronounced in the HD states of the
lighter nuclei because of their smaller deformation (see also Fig. 5 in Ref.
AF.05-108Cd ). It becomes even more important in extremely deformed structures
which according to the language of Ref. Dudek can be described as
megadeformed. Fig. 21 shows an example of density distribution for the
megadeformed state in 102Pd, which becomes yrast at $I\sim 85\hbar$ in the
CRMF calculations. The neck is more pronounced in the proton subsystem than in
the neutron one both in the HD and megadeformed structures due to the Coulomb
repulsion of the segments. This is illustrated in Fig. 21. Our self-consistent
calculations indicate that the shell structure is also playing a role in a
formation of neck. For example, the neck is visible in 132Ba but is not seen
in 116Ba (Fig. 20). This is contrary to the fact that the calculated
transition quadrupole moments of the HD states in these nuclei (Fig. 4d) and
their density elongations (Fig. 20) are comparable. These results indicate
that, in general, the necking degree of freedom is important in the HD states
and that it should be treated within the self-consistent approach which, in
particular, allows different necking for the proton and neutron subsystems.
## IV 111I nucleus: a candidate for a doubly magic extremely SD band.
Figure 24: (Color online) The same as in Fig. 15, but for dynamic moments of
inertia $J^{(2)}$ of the configurations used in Fig. 25 below. Dynamic moments
of inertia of the configuration A in 111I are shown by solid line in each
panel. Vertical dashed lines indicate the frequency range corresponding to the
spin range $I=50-75\hbar$ in the configuration A of 111I. The $\pi[4+6]1/2$
label in panel (a) indicates the orbital with strong mixing of the $N=4$ and
$N=6$ shells: this mixing predominantly emerges from the interaction of the
$\pi[420]1/2$ and $\pi[660]1/2$ states.
The results of the CRMF calculations for the configurations forming the yrast
line or located close to it in energy are shown in Fig. 22. According to the
calculations, normal- and highly-deformed bands, many of which show the high
triaxiality that is indicative of approaching band termination PhysRep ,
dominate the yrast line up to $I\approx 64\hbar$. At higher spin, more
deformed structures become yrast. The configuration A has the structure $\pi
6^{1}\nu 6^{2}$ and is yrast in the spin range $I=64-73\hbar$: no
hyperintruder $N=7$ orbitals are involved in its structure. In this spin range
it is characterized by the transition quadrupole moment $Q_{t}\sim 15.7$ $e$b
and by the $\gamma$-deformation of $\sim 1^{\circ}$. The normalized transition
quadrupole moment in this system is $Q^{norm}_{t}=11.7$ $e$b, thus, this band
is approximately 35% more deformed than the SD band in 152Dy. As a
consequence, in terms of deformation, this band can be characterized as an
extremely superdeformed (ESD) band which is only slightly less deformed than
the HD bands.
Table 2: The size of the Z=53 and N=58 ESD shell gaps [in MeV] obtained with different parametrizations of the RMF Lagrangian for the configuration A in ${}^{111}I$ at spin $I=60\hbar$ (rotational frequency $\Omega_{x}\approx 0.96$ MeV). | NL1 | NL3 | NLZ | NLSH
---|---|---|---|---
Z=53 | 1.45 | 1.25 | 1.65 | 0.70
N=58 | 1.75 | 1.85 | 1.60 | 2.00
In addition, the configuration A is well separated from the excited SD/HD
configurations below $I\sim 73\hbar$ (see Fig. 22). This is due to the
presence of the large $Z=53$ and $N=58$ ESD shell gaps in the single-particle
spectra (see Fig. 23). In this configuration, all single-particle states below
the $Z=53$ and $N=58$ ESD shell gaps are occupied by protons and neutrons,
respectively. Thus, this ESD band is a doubly-magic one. This band appears as
doubly-magic also in the calculations with widely used NL3 NL3 and NLZ NLZ
parametrizations of the RMF Lagrangian, see Table 2. Extensive calculations
with the NL3 parametrization (similar to the ones presented in Fig. 22) show
that this band become yrast at $I\sim 62\hbar$. The $Z=53$ ESD shell gap is
smaller than 1 MeV only in the NLSH NLSH parametrization of the RMF
Lagrangian (see Table 2). However, it is known that the single-particle
energies are not well described in this parametrization A250 . One should
note, however, that the size of the ESD gaps in the configuration A of 111I is
somewhat smaller than the one for the yrast SD band in 152Dy (compare Fig. 22
in the present manuscript with Fig. 3 in Ref. A150 ; see also Figs. 4, 11, 12
in Ref. ALR.98 obtained with different parametrizations of the RMF Lagrangian
and relevant for 151Tb).
The dynamic moments of inertia of the configuration A in 111I and the
configurations in neighboring nuclei are shown in Fig. 24. The increase of
$J^{(2)}$ at $\Omega_{x}\sim 1.2$ MeV is in part due to unpaired band crossing
caused by the interaction of the occupied $\nu[413]7/2^{-}$ and unoccupied
$\nu[651]3/2^{-}$ orbitals (Fig. 23). A centrifugal stretching may also
contribute to this increase of $J^{(2)}$. The effect of the occupation of a
single proton (neutron) intruder orbital on the properties of the ESD bands is
much more pronounced than that in the HD bands of the nuclei around 124Xe (see
Sect. III.4); the changes induced into dynamic moment of inertia reach at
least 10% of its absolute value for the $\pi[660]1/2^{+}$ (Fig. 24c),
$\pi[4+6]1/2^{+}$ (Fig. 24a), $\nu[651]3/2^{+}$ (Fig. 24d) and
$\nu[651]3/2^{-}$ (Fig. 24d) orbitals. In a similar way, the effective
alignments of these orbitals as well as of the $\pi[541]1/2^{+}$ orbital show
appreciable variations as a function of rotational frequency (see Fig. 25),
reaching at least $1\hbar$ in the spin range of interest. This suggests that
the configuration assignment based on the effective alignment method will be
more reliable in the case of ESD bands as compared with the HD bands in the
nuclei around 124Xe (see Sect. III.4 for a discussion of these methods).
Relative properties of the dynamic moments of inertia of two compared bands
will also play a complimentary role in the configuration assignment.
Figure 25: (Color online) The same as in Fig. 16, but for effective alignments
of the single-particle orbitals in the vicinity of the $Z=53$ and $N=58$ SD
shell gaps (see Fig. 23). The effective alignments are defined with respect to
the configuration A in 111I. Vertical dashed lines indicate the frequency
range corresponding to the spin range $I=50-75\hbar$ in the configuration A of
111I.
## V Conclusions
For the first time, the hyperdeformation at high spin has been studied in a
systematic way within the framework of a fully self-consistent theory: the
cranking relativistic mean field theory. The study covers even-even nuclei in
the $Z=40-58$ part of nuclear chart. The main results can be summarized as
follows:
* •
The crossing spins $I_{cr}^{HD}$, at which the HD configurations become yrast,
are lower for proton-rich nuclei. This is a feature seen in the most of
studied isotope chains; by going from the $\beta$-stability valley towards the
proton-drip line one can lower $I_{cr}^{HD}$ by approximately $10\hbar$.
* •
The density of the HD bands in the spin range where they are yrast or close to
yrast is high in the majority of the cases. For such densities, the feeding
intensity of an individual HD band will most likely drop below the
observational limit of modern experimental facilities. This fact has to be
taken into account when planning the experiments for a search of discrete HD
bands. Our calculations indicate Cd isotopes and few other nuclei with large
shell gaps (see Sect. III.5 for details) as the best candidates for a search
of discrete HD bands. An alternative candidate is the doubly magic extremely
superdeformed band in 111I, the deformation of which is only slightly lower
than that of the HD bands, and which may be observed with existing
experimental facilities.
* •
The high density of the HD bands will most likely favor the observation of the
rotational patterns in the form of ridge-structures in three-dimensional
rotational mapped spectra. The study of these patterns as a function of proton
and neutron numbers, which seems to be possible with existing facilities, will
provide a valuable information about hyperdeformation at high spin.
* •
With a very few exceptions, the HD shapes undergo a centrifugal stretching
that results in an increase of the values of the transition quadrupole $Q_{t}$
and mass hexadecapole $Q_{40}$ moments as well as the dynamic moments of
inertia $J^{(2)}$ with increasing rotational frequency. The kinematic moments
of inertia $J^{(1)}$ show very small variations in the frequency range of
interest. These are general features of the HD bands which distinguish them
from the normal- and superdeformed bands. Such features have not been seen
before in the calculations without pairing. In unpaired regime, the $Q_{t}$,
$J^{(2)}$ and $J^{(1)}$ values decrease with rotational frequency in the SD
configurations; the only exceptions are the regions of unpaired bands
crossings.
* •
The individual properties of the single-particle orbitals are not lost at HD.
In the future, they will allow the assignment of the configurations to the HD
bands using the relative properties of different bands. Such methods of
configuration assignment were originally developed for superdeformation. In
contrast to the case of SD, our analysis in the $A\sim 125$ mass region shows
that only simultaneous application of the methods based on effective
alignments and relative transition quadrupole moments by comparing
experimental and theoretical $(i_{eff},\Delta Q_{t})$ values will lead to a
reliable configuration assignment for the HD bands. Moreover, additional
information on the structure of the HD bands will be obtained from the band
crossing features; the cases of strong interaction of the bands in unpaired
regime at HD will be more common as compared with the situation at SD.
The physics of hyperdeformation at high spin is also defined by the fission
barriers; the competition with fission certainly makes the population of the
HD states difficult. It is an important issue, which, however, goes beyond the
scope of the current manuscript. It is likely that the fission barriers are
small or non-existent at the spins around $80-90\hbar$ in some of the studied
nuclei; the observation of the HD bands then will not be possible in these
systems. This problem definitely deserves a deeper attention; the study of the
fission barriers at high spin typical for HD within the framework of the
cranked relativistic Hartree-Bogoliubov theory is in its initial stage and the
results will be presented in a forthcoming manuscript.
## VI Acknowledgements
The help of C. W. Jang and J. Begnaud in performing numerical calculations is
highly appreciated. The work was supported by the U.S. Department of Energy
under grant DE-FG02-07ER41459. Stimulating discussions with Robert Janssens
are gratefully acknowledged.
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|
arxiv-papers
| 2009-02-01T01:13:40 |
2024-09-04T02:49:00.346864
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.V.Afanasjev and H.Abusara",
"submitter": "Anatoli Afanasjev",
"url": "https://arxiv.org/abs/0902.0099"
}
|
0902.0196
|
# Completeness of the bispectrum on compact groups††thanks: This work was
supported by NTU Research Support Grant M58020000.
Ramakrishna Kakarala School of Computer Engineering, Nanyang Technological
University, Singapore.
###### Abstract
This paper derives completeness properties of the bispectrum for compact
groups and their homogeneous spaces. The bispectrum is the Fourier transform
of the triple correlation, just as the magnitude-squared spectrum is the
Fourier transform of the autocorrelation. The bispectrum has been applied in
time series analysis to measure non-Gaussianity and non-linearity. It has also
been applied to provide orientation and position independent character
recognition, as well as to analyze statistical properties of the cosmic
microwave background radiation; in both cases, the data may be defined on a
sphere. On the real line, it is known that the bispectrum is not only
invariant under translation of the underlying function, but in many cases of
interest, it is also complete, in that the function may be recovered uniquely
up to a translation from its bispectrum. This paper extends the completeness
theory of the bispectrum to compact groups and their homogeneous spaces,
including the sphere. The main result, which depends on Tannaka-Krein duality
theory, shows that every function whose Fourier coefficient matrices are
always nonsingular is completely determined by its bispectrum, up to a single
group action. Furthermore, algorithms are described for reconstructing
functions defined on $SU(2)$ and $SO(3)$ from their bispectra.
###### keywords:
bispectrum, triple correlation, pattern recognition, invariants, Tannaka-Krein
duality.
###### AMS:
68T10,43A77,14L24
## 1 Introduction
The triple correlation of a complex-valued function on the real line is the
integral of that function multiplied by two independently-shifted copies of
itself:
$a_{3,f}(s_{1},s_{2})=\int_{-\infty}^{\infty}f^{*}(x)f(x+s_{1})f(x+s_{2})dx.$
(1)
It is easily seen that the triple correlation does not change if the function
is translated. The Fourier transform of triple correlation is the bispectrum.
If the Fourier transform of $f$ is denoted $F={\cal F}\\{f\\}$, then the
bispectrum is
$A_{3,f}(u,v)={\cal F}\\{a_{3,f}(s_{1},s_{2})\\}=F(u)F(v)F^{*}(u+v)$ (2)
The triple correlation extends the concept of autocorrelation (denoted here
$a_{2,f}$) which correlates a function with a single shifted copy of itself,
thereby enhancing the function’s latent periodicities:
$a_{2,f}(s)=\int_{-\infty}^{\infty}f^{*}(x)f(x+s)dx.$ (3)
The Fourier transform of the autocorrelation is $A_{2,f}={\cal
F}\\{a_{2,f}\\}=|F|^{2}$, which obviously lacks phase information and
therefore provides a limited analysis of a function’s structure. In contrast,
eq. (2) shows that the bispectrum contains both magnitude and phase
information, while still being invariant to translation. These properties
suggest applications in invariant matching for pattern recognition. More
importantly for matching, the bispectrum in many cases of interest is not only
invariant but also provides a complete description of the function: the
function may be reconstructed from it, up to a single unknown translation
[11].
The bispectrum was perhaps first investigated by statisticians examining the
cumulant structure of non-Gaussian random processes [3]. It is well known that
the third order cumulant, of which the triple correlation is a sample, has
zero expected value for a Gaussian process. Hence, the bispectrum is a tool
for measuring non-Gaussianity. The bispectrum was also independently studied
by physicists as a tool for spectroscopy. H. Gamo [6] in 1963 described an
apparatus for measuring the triple correlation of a laser beam, and also
showed how phase information can be completely recovered from the real part of
the bispectrum—up to sign reversal and linear offset. Gamo’s is perhaps the
first completeness result along the lines of what is explored in detail in
this paper. However, Gamo’s method implicitly requires the Fourier transform
to never be zero at any frequency. This requirement was relaxed, and the class
of functions which are known to be completely identified by their bispectra
was considerably expanded, by the study of Yellott and Iverson [11]: for
example, every integrable function with compact support is completely
determined, up to a translation, by its bispectrum.
The statistical and physical applications described above are for data defined
on Euclidean domains $\mathbb{R}^{n}$. The bispectrum also finds applications
on non-Euclidean domains such as the sphere $S^{2}$. For example, in
astrophysics, the Cosmic Background Radiation (CBR) may be modelled as a
function defined on a sphere. X. Luo [16] calculates the bispectrum of
spherical CBR functions and examines the properties the bispectral
coefficients have under cosmological scenarios such as inflation and late-time
phase transitions.
The bispectrum’s invariance and completeness have motivated researchers in
pattern recognition to apply it for template matching and shape recognition.
For example, R. Kondor [14, Ch. 8] shows how position and orientation
independent optical character recognition can be accomplished by projecting
the character from the plane on to the sphere, and subsequently using the
bispectrum on the sphere for invariant matching. Kondor’s results are
discussed further in Section 5.
Despite the interest in applying the bispectrum for non-Euclidean domains,
little has been published about important properties such as completeness. The
contribution of this paper is to derive the completeness theory of the
bispectrum for (noncommutative) compact groups and their homogeneous spaces.
In order to construct the bispectrum on groups, we require concepts from
harmonic analysis using group representation theory. Those concepts are
presented in the next section. Then, a matrix form of the bispectrum which
proves convenient for analysis is demonstrated. The matrix formulation allows
a relatively simple criterion for completeness: it is shown that functions
defined on compact groups that have nonsingular Fourier transform coefficients
are completely determined by their bispectra. This result depends on the well-
known Tannaka-Krein duality theory of compact group representations. The
completeness result is extended to homogeneous spaces using the Iwahori-
Sugiura duality theorem [9]. Reconstruction algorithms for functions defined
on $SU(2)$ and $SO(3)$ in particular are described, expanding on material in a
previous paper [13].
## 2 Preliminaries
Let us review the basic concepts of representation theory for compact groups.
For more details, the reader may consult Chevalley ([4], Chapter VI). Let $G$
be a compact group. A $n$-dimensional unitary representation of $G$ is a
continuous homomorphism $D$ of $G$ into the group $U(n,\mathbb{C})$ of
$n\times n$ unitary matrices. The following operations are all defined on
representations: complex-conjugation, direct sum, and tensor product. The
complex conjugate of any representation is simply that representation $D^{*}$
whose matrices are complex-conjugates of those of $D$. Let $D_{1}$, $D_{2}$ be
respectively a $n$-dimensional and a $m$-dimensional representation; their
direct-sum $D_{1}\oplus D_{2}$ is the $(n+m)$-dimensional representation which
maps each element $g$ to the block-diagonal matrix $D_{1}(g)\oplus D_{2}(g)$
with $D_{1}(g)$ in the upper left corner and $D_{2}(g)$ in the lower right
corner. Similarly, the tensor product $D_{1}\otimes D_{2}$ of $D_{1}$ and
$D_{2}$ is $nm$-dimensional representation whose matrices are $n\times n$
blocks, each block of size $m\times m$, where for each $g$ the $(i,j)$-th
block is the matrix $D_{2}(g)$ multiplied by the $(i,j)$-th coefficient of
$D_{1}(g)$. We reserve the symbol ${\bf 1}$ for the trivial representation
that maps all of $G$ into the number $1$.
We now define equivalence and reducibility of representations. Two
representations $D_{1}$ and $D_{2}$ are equivalent if there exists a unitary
matrix $C$ such that $D_{1}(g)=CD_{2}(g)C^{\dagger}$ for all $g\in G$, with
$\dagger$ denoting the matrix conjugate transpose. A representation $D$ is
reducible if there exists a matrix $C$ and two representations $D_{1}$,
$D_{2}$ such that $D(g)=C\left[D_{1}(g)\oplus D_{2}(g)\right]C^{\dagger}$ for
all $g\in G$; otherwise $D$ is irreducible. Every representation $D$ is the
direct sum of irreducible unitary representations.
The set ${\cal G}$ of all equivalence classes of irreducible unitary
representations of $G$ is called the dual object of $G$. Intuitively, ${\cal
G}$ is the “frequency” domain of $G$. In general, ${\cal G}$ is not a group,
unless $G$ is abelian. In what follows, we denote elements of ${\cal G}$,
which are equivalence classes, by Greek letters such as $\alpha$, $\beta$,
etc. For each $\alpha\in{\cal G}$, let ${\rm dim}(\alpha)$ denote the common
dimension of the representations in $\alpha$. Let
$\\{D_{\alpha}\\}_{\alpha\in{\cal G}}$ denote any set of representations that
contain exactly one member in each equivalence class in ${\cal G}$. We call
any such set a selection. A given selection has two properties, both
consequences of the classical result known as the Peter-Weyl theorem: (1)
there are at most countably many representations in any selection, and in
particular there are finitely many if, and only if, $G$ is finite, i.e.,
${\cal G}$ is finite if and only if $G$ is finite; (2) the set of all matrix
coefficients $d_{\alpha}^{pq}(\cdot)$, where $\alpha\in{\cal G}$ and $1\leq
p,q\leq{\rm dim}(\alpha)$, from any selection forms an orthogonal basis for
the Hilbert space $L_{2}(G)$.
### 2.1 Duality theory
Our main tool for proving completeness results is the duality theorem due to
T. Tannaka and M. Krein. There are several formulations of that result, of
which Chevalley’s [4, pp 188-203] is the most convenient for our purposes. The
formulation is as follows. Let $\Theta(G)$ be the representative algebra of
$G$, which is the algebra of complex-valued functions on $G$ that is generated
by the set of matrix coefficients of any selection
$\\{D_{\alpha}\\}_{\alpha\in{\cal G}}$. In fact, $\Theta(G)$ is independent of
the selection that is used. It is well-known that every function
$f\in\Theta(G)$ may be expressed in exactly one way as a finite linear
combination of the set of matrix coefficients from any given selection ([4, pg
189]). The structure of $\Theta(G)$ may be understood by considering algebra
homomorphisms, which are maps $\omega:\Theta(G)\rightarrow\mathbb{C}$ that are
both linear and multiplicative, i.e.,
$\omega(c_{1}f_{1}+c_{2}f_{2})=c_{1}\omega(f_{1})+c_{2}\omega(f_{2})$ and
$\omega(f_{1}f_{2})=\omega(f_{1})\omega(f_{2})$ for all scalars $c_{1}$,
$c_{2}$, and functions $f_{1}$, $f_{2}$ in $\Theta(G)$. The set of all algebra
homomorphisms is denoted $\Omega(G)$. Clearly, for every $g\in G$, the map
$\omega_{g}(f)=f(g)$ is an algebra homomorphism. Note that
$\omega_{g}(f^{*})=\omega_{g}(f)^{*}$, i.e., $\omega_{g}$ preserves complex-
conjugation. In fact, the converse is also true, an identification that is
essential to the duality between groups and their representations; see [4, pg
211] for details and proofs.
###### Theorem 1 (Tannaka-Krein).
To every algebra homomorphism $\omega\in\Omega(G)$ that preserves complex-
conjugation, i.e., $\omega(f^{*})=\omega(f)^{*}$, there corresponds a unique
element $g\in G$ such that $\omega(f)=f(g)$ for all $f\in\Theta(G)$.
We recast this duality theorem in a slightly different form (see also [20, pp
303-306]). Let $\\{D_{\alpha}\\}_{\alpha\in{\cal G}}$ be any selection of
irreducible representations, and let $\\{U(\alpha)\\}_{\alpha\in{\cal G}}$ be
a corresponding sequence of unitary matrices, such that for each $\alpha$ the
matrix $U(\alpha)$ has the same dimension as the representation $D_{\alpha}$.
We determine the necessary and sufficient conditions under which the latter
sequence arises from the former by an element of $\Theta(G)$, i.e., when
$U(\alpha)=\omega(D_{\alpha})$ for some fixed homomorphism
$\omega\in\Omega(G)$. Consider the tensor product $D_{\sigma}\otimes
D_{\delta}$ of any two representations in our selection; that representation
is, in general, reducible, and we write its decomposition into irreducibles
(taken from our selection) as follows:
$D_{\sigma}\otimes
D_{\delta}=C_{\sigma\delta}\left[D_{\alpha_{1}}\oplus\cdots\oplus
D_{\alpha_{k}}\right]C_{\sigma\delta}^{\dagger}.$ (4)
The indices $\alpha_{1}$, $\ldots$, $\alpha_{k}$ appearing on the right are
unique up to permutation ([4, pg 175]). Suppose now that there exists $\omega$
such that $U(\alpha)=\omega(D_{\alpha})$ for all $\alpha$. By applying
$\omega$ to both sides of eq. (4), and using the fact that $\omega$ is both
linear and multiplicative, we obtain that (writing $U(\alpha)$ for
$\omega(D_{\alpha})$)
$U(\sigma)\otimes
U(\delta)=C_{\sigma\delta}\left[U(\alpha_{1})\oplus\cdots\oplus
U(\alpha_{k})\right]C_{\sigma\delta}^{\dagger}$ (5)
Equation (5) is not only necessary, but also sufficient, as the following
result shows.
###### Theorem 2.
Let $\\{D_{\alpha}\\}_{\alpha\in{\cal G}}$ and
$\\{U(\alpha)\\}_{\alpha\in{\cal G}}$ be as above. If, whenever eq. (4) is
true, we have that eq. (5) is also true for same $\sigma$, $\delta$, and
matrix $C_{\sigma\delta}$, then there exists a fixed $g\in G$ such that
$U(\alpha)=D_{\alpha}(g)$ for all $\alpha$.
###### Proof.
To any sequence $\\{U(\alpha)\\}_{\alpha{\cal G}}$, there exists a unique
linear map $\omega:\Theta(G)\rightarrow\mathbb{C}$ such that
$U(\alpha)=\omega(D_{\alpha})$. To see this, note that the set of matrix
coefficients in any selection is linearly independent, and furthermore, any
function $f\in\Theta(G)$ may be written uniquely as a finite linear
combination of the matrix coefficients. Thus it is always possible to
construct a linear map $\omega:\Theta(G)\rightarrow\mathbb{C}$ that gives any
desired set of values to the corresponding coefficient functions; in
particular, there exists an $\omega$ such that $\omega(D_{\alpha})=U(\alpha)$.
We now show that $\omega$ is multiplicative and conjugate preserving. Applying
the linear map $\omega$ to both sides of eq. (4) results in the following
identity:
$\omega(D_{\sigma}\otimes
D_{\delta})=C_{\sigma\delta}\left[\omega(D_{\alpha_{1}})\oplus\cdots\oplus\omega(D_{\alpha_{k}})\right]C_{\sigma\delta}^{\dagger}.$
(6)
Substituting the matrices $U$ on the right side and using (5) reveals that
$\omega(D_{\sigma}\otimes
D_{\delta})=\omega(D_{\sigma})\otimes\omega(D_{\delta}).$ (7)
Consequently, $\omega$ is multiplicative. To prove that $\omega$ preserves
conjugation, note that the equation $D_{\alpha}D_{\alpha}^{\dagger}=I$ implies
that $\omega(D_{\alpha})\omega(D_{\alpha}^{\dagger})=I$ for all $\alpha$.
Since the matrices $U(\alpha)=\omega(D_{\alpha})$ are unitary, we also have
$\omega(D_{\alpha})\omega(D_{\alpha})^{\dagger}=I$. Matrix inverses are
unique, and thus $\omega(D_{\alpha}^{\dagger})=\omega(D_{\alpha}^{\dagger})$,
showing that $\omega$ preserves conjugation. By the Tannaka-Krein theorem,
there exists a unique $g\in G$ such that
$U(\alpha)=\omega(D_{\alpha})=D_{\alpha}(g)$ for all $\alpha$. ∎
## 3 Bispectrum
We use this result to establish sufficient conditions for a function to be
described uniquely by its bispectrum. It is convenient to establish the
Fourier transform domain for compact groups. Let
$\\{D_{\alpha}\\}_{\alpha\in{\cal G}}$ be any selection of irreducible
representations. The Fourier transform of any $f$ in $L_{1}(G)$ is the matrix-
valued function $F$, such that for each $\alpha\in{\cal G}$, we have
$F(\alpha)=\int_{G}f(g)D_{\alpha}(g)^{\dagger}dg.$ (8)
Here the integral uses the Haar measure $dg$ on $G$; because $G$ is compact,
$dg$ is both left and right invariant.
We use two important properties of the Fourier transform in what follows ([5,
pp 73-78]): (i) the Fourier transform of any $f\in L_{1}(G)$ determines $f$
uniquely up to a set of Haar measure zero; (ii) $s(g)=r(xg)$ for all $g$ if,
and only if, $S(\alpha)=R(\alpha)D_{\alpha}(x)$ for all $\alpha$.
For $f\in L_{1}(G)$, the triple correlation $a_{3,f}$ is defined as follows:
$a_{3,f}(g_{1},g_{2})=\int_{G}f(g)^{*}f(gg_{1})f(gg_{2})dg.$ (9)
(Compare with eq. (1)). Note that the triple-correlation is invariant under
left-translation, i.e., if there exists $x$ such that $r(g)=s(xg)$ for all
$g$, then $a_{3,r}=a_{3,g}$. This follows directly from the left-invariance of
the Haar measure $dg$. Similarly, we may define a right-translation invariant
version of eq. (9) by integrating $f(g)^{*}f(g_{1}g)f(g_{2}g)$, but we will
not pursue this minor variation in what follows.
Because $f\in L_{1}(G)$, we have that $a_{3,f}$ is a function in
$L_{1}(G\times G)$. It is known that any irreducible representation of
$G\times G$ is equivalent to a tensor product $D_{\sigma}\otimes D_{\delta}$,
where $D_{\sigma}\otimes D_{\delta}$ are irreducible representations of $G$
([17, pg 45]). Thus the Fourier transform of $a_{3,f}$ with respect to the
selection $\\{D_{\alpha}\\}_{\alpha\in{\cal G}}$ is the function on ${\cal
G}\times{\cal G}$ that is defined as follows:
$A_{3,f}(\sigma,\delta)=\int_{G}\int_{G}a_{3,f}(g_{1},g_{2})\left[D_{\sigma}(g_{1})^{\dagger}\otimes
D_{\delta}(g_{2})^{\dagger}\right]dg_{1}dg_{2}.$ (10)
There exists a convenient formula for computing $A_{3,f}$.
###### Lemma 3.
For any pair $\sigma$, $\delta$, let $C_{\sigma\delta}$ be the matrix and let
$\alpha_{1}$, $\ldots$, $\alpha_{k}$ be the indices appearing in eq. (4). Then
$A_{3,f}(\sigma,\delta)=\left[F(\sigma)\otimes
F(\delta)\right]C_{\sigma\delta}\left[F(\alpha_{1})^{\dagger}\oplus\cdots\oplus
F(\alpha_{k})^{\dagger}\right]\,C_{\sigma\delta}^{\dagger}.$ (11)
###### Proof.
Since $a_{3,f}$ is integrable, we use the Fubini theorem to interchange the
order of integration in the following derivation:
$\displaystyle A_{3,f}(\sigma,\delta)$ $\displaystyle=$
$\displaystyle\int_{G}\int_{G}a_{3,f}(g_{1},g_{2})\left[D_{\sigma}(g_{1})^{\dagger}\otimes
D_{\sigma}(g_{2})^{\dagger}\right]dg_{1}\,dg_{2},$ $\displaystyle=$
$\displaystyle\int_{G}\int_{G}\int_{G}f(g)^{*}f(gg_{1})f(gg_{2})\left[D_{\sigma}(g_{1})^{\dagger}\otimes
D_{\delta}(g_{2})^{\dagger}\right]dg\,dg_{1}\,dg_{2},$ $\displaystyle=$
$\displaystyle\int_{G}f(g)^{*}\int_{G}\int_{G}f(gg_{1})f(gg_{2})\left[D_{\sigma}(g_{1})^{\dagger}\otimes
D_{\delta}(g_{2})^{\dagger}\right]dg_{1}\,dg_{2}\,d_{g}.$
By making a change of variables, we find that the double integral inside
simplifies as follows:
$\int_{G}\int_{G}f(gg_{1})f(gg_{2})\left[D_{\sigma}(g_{1})^{\dagger}\otimes
D_{\delta}(g_{2})^{\dagger}\right]dg_{1}\,dg_{2}=\left[F(\sigma)\otimes
F(\delta)\right]\left[D_{\sigma}(g)\otimes D_{\delta}(g)\right].$
Upon substituting into the expression for $A_{3,f}$, we find that
$A_{3,f}=\left[F(\sigma)\otimes
F(\delta)\right]\int_{G}f(g)^{*}\left[D_{\sigma}(g)\otimes
D_{\delta}(g)\right]dg.$ (12)
Upon substituting the tensor product decomposition (4) into the above, we
obtain
$A_{3,f}(\sigma,\delta)=\left[F(\sigma)\otimes
F(\delta)\right]C_{\sigma\delta}\left[\int_{G}f(g)^{*}\left(D_{\alpha_{1}}(g)\oplus\cdots\oplus
D_{\alpha_{k}}(g)\right)dg\right]C_{\sigma\delta}^{\dagger}.$ (13)
After evaluating the integral, the result (11) follows. ∎
The lemma helps to quickly establish the basic completeness result for the
bispectrum on compact groups.
###### Theorem 4.
Let $G$ be any compact group, and let $r$ in $L_{1}(G)$ be such that its
Fourier coefficients $R(\alpha)$ are nonsingular for all $\alpha\in{\cal G}$.
Then $a_{3,s}=a_{3,r}$ for some $s\in L_{1}(G)$ if and only if there exists
$x\in G$ such that $s(g)=r(xg)$ for all $g$.
###### Proof.
If $s(g)=r(xg)$, then the translation-invariance of the triple correlation
implies that $a_{3,r}=a_{3,s}$. We now prove the converse. Let $s$ be such
that $a_{3,s}=a_{3,r}$; then $A_{3,r}=A_{3,s}$, and by Lemma 3, we obtain that
for $\sigma$, $\delta$ that
$\displaystyle\left[R(\sigma)\otimes
R(\delta)\right]C_{\sigma\delta}\left[R(\alpha_{1})^{\dagger}\oplus\cdots\oplus
R(\alpha_{k})^{\dagger}\right]$ $\displaystyle=$
$\displaystyle\left[S(\sigma)\otimes
S(\delta)\right]C_{\sigma\delta}\left[S(\alpha_{1})^{\dagger}\oplus\cdots\oplus
S(\alpha_{k})^{\dagger}\right]$ (14)
Set $\sigma=\delta={\bf 1}$, where ${\bf 1}$ is the trivial representation
$g\mapsto 1$ of $G$. Both $R({\bf 1})$ and $S({\bf 1})$ are complex numbers,
and the equality above becomes
$R({\bf 1})R({\bf 1})R({\bf 1})^{*}=S({\bf 1})S({\bf 1})S({\bf 1})^{*}.$ (15)
Thus $R({\bf 1})=S({\bf 1})$. Now set $\delta={\bf 1}$; for any $\sigma$, we
have $D_{\sigma}\otimes D_{\bf 1}=D_{\sigma}$, and thus eq. (14) becomes
$\left[R(\sigma)\otimes R({\bf
1})\right]R(\sigma)^{\dagger}=\left[S(\sigma)\otimes S({\bf
1})\right]S(\sigma)^{\dagger}.$ (16)
By assumption $R({\bf 1})=S({\bf 1})$ is a non-zero scalar, and we cancel it
from both sides to obtain that
$R(\sigma)R(\sigma)^{\dagger}=S(\sigma)S(\sigma)^{\dagger}$ for all $\sigma$.
Such an equality between matrices holds if and only if there exists a unitary
matrix $U(\sigma)$ such that $S(\sigma)=R(\sigma)U(\sigma)$. Substituting for
$S$ in (14) yields, upon rearranging terms,
$\displaystyle\left[R(\sigma)\otimes
R(\delta)\right]C_{\sigma\delta}\left[R(\alpha_{1})^{\dagger}\oplus\cdots\oplus
R(\alpha_{k})^{\dagger}\right]C_{\sigma\delta}^{\dagger}=$
$\displaystyle\left[R(\sigma)\otimes R(\delta)\right]\left[U(\sigma)\otimes
U(\delta)\right]C_{\sigma\delta}\left[U(\alpha_{1})^{\dagger}\oplus\cdots\oplus
U(\alpha_{k})^{\dagger}\right]\left[R(\alpha_{1})^{\dagger}\oplus\cdots\oplus
R(\alpha_{k})^{\dagger}\right]C_{\sigma\delta}^{\dagger}$
We cancel the nonsingular matrices $R(\sigma)$ from both sides and rearrange
the remaining terms to obtain the identity
$U(\sigma)\otimes
U(\delta)=C_{\sigma\delta}\left[U(\alpha_{1})\oplus\cdots\oplus
U(\alpha_{k})\right]C_{\sigma\delta}^{\dagger}$ (17)
Since the identity above holds for all $\sigma$, $\delta$, Theorem 2 implies
that there exists $x\in G$ such that $U(\sigma)=D_{\sigma}(x)$ for all
$\sigma$. Thus $S(\sigma)=R(\sigma)D_{\sigma}(x)$ for all $\sigma$, and the
translation property of the Fourier transform now implies that $s(g)=r(xg)$
for all $g$. ∎
The hypothesis that all coefficients $R(\sigma)$ are nonsingular is satisfied
generically, in the sense that almost every $n\times n$ matrix is nonsingular
with respect to the Lebesgue measure on the set of $n\times n$ matrices.
Nevertheless, it is desireable to weaken the hypothesis, to include for
example functions on $G$ that are invariant under the translations of a normal
subgroup $N$ of $G$. We prove a result for this case.
We review some facts concerning group representations and normal subgroups
([8, pg 64]). Let $N$ be a closed normal subgroup of $G$. Any irreducible
representation $\widetilde{D}$ of the quotient group $G/N$ extends to an
irreducible representation $D$ of $G$ by composition:
$D=\widetilde{D}\circ\pi$, where $\pi$ is the canonical coset map
$\pi:G\rightarrow G/N$. The converse is also true: any representation $D$ of
$G$ such that $D(n)=I$ for all $n\in N$ is of the form
$D=\widetilde{D}\circ\pi$ for some representation $\widetilde{D}$ of $G/N$.
Moreover, letting $\widehat{(G/N)}$ represent the dual object of the group
$G/N$, the set
${\cal
G}[N]=\\{D=\widetilde{D}\circ\pi,\widetilde{D}\in\left(\widehat{G/N}\right)\\},$
(18)
is closed under both conjugation and tensor-product decomposition, i.e., the
tensor product of any two representations from the set decomposes into
irreducible representations that are also contained in the set. Conversely, to
each subset $\hat{A}$ of ${\cal G}$ that contains ${\bf 1}$ and that is closed
under both conjugation and tensor-product decomposition, there corresponds a
unique closed and normal subgroup $N$ of $G$ such that $\hat{A}={\cal G}[N]$.
Now let $f$ be a function in $L_{2}(G)$ that is invariant under $N$, i.e.,
$f(ng)=f(gn)=f(g)$ for all $n\in N$. If $\alpha\not\in{\cal G}[N]$, then the
Fourier coefficient matrix $F(\alpha)$ is a zero matrix. To prove this, note
that the Peter-Weyl theorem implies that the matrix coefficients
$\widetilde{d}_{\alpha}^{pq}(\cdot)$ from any selection
$\\{\widetilde{D}_{\alpha}\\}_{\alpha\in(\widehat{G/N})}$ form an orthogonal
basis for $L_{2}(G/N)$; thus the corresponding functions
$d_{i}^{pq}=\widetilde{d}_{i}^{pq}\circ\pi$ on $G$ form an orthogonal basis
for the closed subspace in $L_{2}(G)$ of functions invariant under $N$.
Consequenctly, any $N$-invariant function in $L_{2}(G)$ has zero inner product
with the coefficients of $D_{\alpha}$ when $\alpha\not\in{\cal G}[N]$. We use
those facts to produce a stronger version of Theorem 4.
###### Theorem 5.
Let $r\in L_{2}(G)$ be such that its Fourier coefficients $R$ satisfy the
following conditions:
1. 1.
Each $R(\alpha)$ is either zero or nonsingular;
2. 2.
The set of $\alpha$ such that $R(\alpha)$ is non-singular includes ${\bf 1}$,
and is closed under conjugation and tensor product decomposition.
Then there exists a normal subgroup $N$ of $G$ such that $r$ is $N$-invariant,
and furthermore $r$ is uniquely determined up to left translation by its
bispectrum $A_{3,f}$.
###### Proof.
As discussed above, the set of $\alpha$ such that $R(\alpha)$ is nonsingular
corresponds to ${\cal G}[N]$ for some normal subgroup $N$ of $G$. Furthermore,
$r=\widetilde{r}\circ\pi$ for a unique function $\widetilde{r}$ on $G/N$. We
obtain $A_{3,\widetilde{r}}$ from $A_{3,r}$ by restricting the latter to the
arguments $(\sigma,\delta)$ for which $R(\sigma)$ and $R(\delta)$ are
nonsingular. Theorem 4 now shows that $\widetilde{r}$ is uniquely determined
up to a left translation by $A_{3,\widetilde{r}}$, and thus
$r=\widetilde{r}\circ\pi$ is uniquely determined up to a left translation by
$A_{3,r}$. ∎
Remark. The hypotheses of Thms 4 and 5 have an interesting interpretation in
the context of the Tauberian theorems for compact groups. The latter theorems
determine what functions lie in the span of translates of a single function
$f$ in $L_{1}(G)$. Edwards ([5, pp 121-125]) describes one such result: If
$f_{1}$, $f_{2}$ in $L_{1}(G)$ have Fourier transforms $F_{1}$, $F_{2}$, such
that $F_{2}(\alpha)=F_{1}(\alpha)M(\alpha)$ for each $\alpha$, where
$M(\alpha)$ is an arbitrary matrix whose dimensions match that of
$F_{1}(\alpha)$, then $f_{2}$ lies in the span of left translates of $f_{1}$,
i.e., $f_{2}$ may be approximated arbitrarily closely in $L_{1}$ by linear
combinations of left translates of $f_{1}$. Suppose now that $f_{1}$ satisfies
the hypothesis of Theorem 4, i.e., $F_{1}(\alpha)$ is nonsingular for all
$\alpha$. Then the aforementioned Tauberian theorem implies that any function
$f_{2}\in L_{1}(G)$ lies in the span of translates of $f_{1}$, i.e., the
translates of $f_{1}$ span $L_{1}(G)$. Similarly, if $f_{1}$ satisfies the
hypothesis of Theorem 5, then the translates of $f_{1}$ span the closed
subspace of $L_{1}(G)$ that consists of functions invariant under some fixed
normal subgroup $N$. As our theorems show, the bispectrum of $f_{1}$
identifies exactly which functions are its translates.
## 4 Homogeneous spaces
The definition of a homogeneous space is as follows. Let $G$ be any
topological group and $X$ any topological space. We say that $G$ acts (on the
right) on $X$ if for each $g\in G$ there exists a homeomorphism
$\tau_{g}:X\rightarrow X$, such that $\tau_{e}(x)=x$ for the identity $e$ in
$G$, and furthermore, for $g_{1}$, $g_{2}$ in $G$, we have
$\tau_{g_{1}g_{2}}(x)=\tau_{g_{2}}\left(\tau_{g_{1}}(x)\right)$. The group $G$
acts transitively on $X$ if for each $x_{1}$, $x_{2}$ in $X$, there exists
$g\in G$ such that $\tau_{g}(x_{1})=x_{2}$. The space $X$ is a homogeneous
space for $G$ if $G$ acts on $X$ transitively and continuously. An important
example of a homogeneous space is the quotient space of right cosets
$G\backslash H=\left\\{Hg:g\in G\right\\}$ of a closed subgroup $H$ in $G$. In
fact, it is a theorem that any locally compact homogeneous space $X$ of a
separable and locally compact group $G$ can be represented as a quotient space
$G\backslash H$ for some closed subgroup $H$ of $G$ ([2, pg 124]).
Our goal in this section is to investigate the bispectrum’s completeness for
functions on arbitrary homogeneous spaces of compact groups. By the result
cited above, we lose no generality by focusing on spaces of the form
$G\backslash H$, where $G$ is some compact group and $H$ some closed subgroup
of $G$. To any function $\widetilde{f}$ on $G\backslash H$ there corresponds a
unique function $f$ on $G$ such that $f=\widetilde{f}\circ\pi$, where
$\pi:G\rightarrow G\backslash H$ is the canonical coset map; conversely, to
any function $f$ on $G$ that is invariant under left $H$-translations, i.e.,
$f(hg)=f(g)$ for all $g\in G$ and $h\in H$, there corresponds a unique
function $\widetilde{f}$ on $G\backslash H$ such that
$f=\widetilde{f}\circ\pi$. Thus we lose no generality by further restricting
our study of functions on homogeneous spaces to functions on $G$ that are left
$H$-invariant for some closed subgroup $H$.
Our main tool for proving completeness results is the Iwahori-Sugiura duality
theorem for homogeneous spaces of compact groups [9]. Let $G$ be any compact
group, $\\{D_{\alpha}\\}_{\alpha\in{\cal G}}$ be any selection of irreducible
representations, and $\Theta(G)$ the representative algebra of $G$. For any
closed subgroup $H$ of $G$, let $\Theta_{H}(G)$ denote the subalgebra of
$\Theta(G)$ consisting of functions that are invariant under left
$H$-translations. For each $f\in\Theta_{H}(G)$, let $f(Hg)$ denote the common
value given to elements of the coset $Hg$ by $f$. The algebraic structure of
$\Theta_{H}(G)$ is revealed to a large extent by the multiplicative linear
functionals $\omega:\Theta_{H}(G)\rightarrow\mathbb{C}$, i.e., algebra
homomorphisms of $\Theta_{H}(G)$. The Iwahori-Sugiura theorem characterizes
those algebra homomorphisms that preserve conjugation.
###### Theorem 6 (Iwahori-Sugiura).
To each algebra homomorphism $\omega:\Theta_{H}(G)\rightarrow\mathbb{C}$ that
preserves conjugation, there corresponds a unique coset $Hg$ in the quotient
space $G\backslash H$ such that for all $f\in\Theta_{H}(G)$,
$\omega(f)=f(Hg).$ (19)
We describe an equivalent formulation of the Iwahori-Sugiura theorem that is
analogous to Theorem 2. Several preliminary results are required for the new
formulation, with some of the longer proofs being relegated to the appendices.
###### Lemma 7.
Any function $f\in\Theta_{H}(G)$ can be expressed as a unique finite linear
combination of the left $H$-invariant matrix coefficients of a given
selection.
The proof is given in Appendix A.
Let $G$, $H$, and $\\{D_{\alpha}\\}_{\alpha\in{\cal G}}$ be as before. Let us
define a corresponding sequence of matrices $\\{P_{\alpha}\\}_{\alpha\in{\cal
G}}$ as follows:
$P_{\alpha}=\int_{H}D_{\alpha}(h)dh,$ (20)
where $dh$ denotes the normalized Haar measure on $H$. It is easy to show that
each $P_{\alpha}$ is a projection, i.e., a self-adjoint matrix such that
$P_{\alpha}P_{\alpha}=P_{\alpha}$ ([8, pg 190]). Moreover, the projection
matrices as defined above inherit some of the tensor product properties of the
corresponding representations ([8, pg 190]).
###### Lemma 8.
Let $\\{P_{\alpha}\\}_{\alpha\in{\cal G}}$ be as above. For each $\sigma$,
$\delta$, let $C_{\sigma\delta}$ be the Clebsch-Gordan matrix and
$\alpha_{1}$, $\ldots$, $\alpha_{k}$ be the indices in the tensor product
decomposition in eq. (4). Then
$\displaystyle P_{\sigma}\otimes P_{\delta}$ $\displaystyle=$ $\displaystyle
C_{\sigma\delta}\left[P_{\alpha_{1}}\oplus\cdots\oplus
P_{\alpha_{k}}\right]C_{\sigma\delta}^{\dagger}\left[P_{\sigma}\otimes
P_{\delta}\right],$ $\displaystyle=$ $\displaystyle\left[P_{\sigma}\otimes
P_{\delta}\right]C_{\sigma\delta}\left[P_{\alpha_{1}}\oplus\cdots\oplus
P_{\alpha_{k}}\right]C_{\sigma\delta}^{\dagger}.$
It proves convenient to apply the following similarity transformations to the
$P$ matrices. For each $\alpha$, let ${\rm rank}(\alpha)$ denote the rank of
$P_{\alpha}$, and let $I({\rm rank}(\alpha))$ be the diagonal matrix whose
first ${\rm rank}(\alpha)$ diagonal entries (from the upper left) are $1$, and
the rest are $0$. Then there exists a unitary matrix $U(\alpha)$ such that
([15, pg 195]):
$U(\alpha)P_{\alpha}U(\alpha)^{\dagger}=I({\rm rank}(\alpha)),$ (21)
If we apply the same similarity transformation to the representation
$D_{\alpha}$, then it is easily seen that
$U(\alpha)D_{\alpha}(h)U(\alpha)^{\dagger}=\left[\oplus_{q=1}^{{\rm
rank}(\alpha)}{\bf 1}(h)\right]\oplus D_{\alpha}^{H}(h),\quad h\in H.$ (22)
In the decomposition above, ${\bf 1}$ is the trivial representation of $H$,
and the last term $D_{\alpha}^{H}$ is some unitary representation of $H$ that
does not contain ${\bf 1}$.
Rather than starting with an arbitrary selection of
$\\{D_{\alpha}\\}_{\alpha\in{\cal G}}$, suppose now that we choose one in
which each matrix $D_{\alpha}(h)$ is exactly equal to a direct sum where the
first ${\rm rank}(\alpha)$ representations that appear in the sum are ${\bf
1}$, i.e.,
$D_{\alpha}(h)=\left[\oplus_{q=1}^{{\rm rank}(\alpha)}{\bf 1}(h)\right]\oplus
D_{\alpha}^{H}(h),\quad h\in H.$ (23)
We always obtain such a convenient selection (that is what we shall call it
henceforth) from a given one by applying similarity transformations as
described above. For a convenient selection, the projection matrices in eq.
(20) are simply $P_{\alpha}=I({\rm rank}(\alpha))$ for all $\alpha$.
###### Lemma 9.
Let $\\{D_{\alpha}\\}_{\alpha\in{\cal G}}$ be a convenient selection and
$\\{P_{\alpha}\\}_{\alpha\in{\cal G}}$ be its projections. The nonzero
coefficients in the matrices $\\{P_{\alpha}D_{\alpha}\\}$ are precisely those
coefficients of the selection that are left $H$-invariant.
###### Proof.
Each matrix $P_{\alpha}D_{\alpha}=I({\rm rank}(\alpha))D_{\alpha}$ has its
first ${\rm rank}(\alpha)$ rows equal to those of $D_{\alpha}$, while the
remaining rows are identically zero. Moreover,
$P_{\alpha}D_{\alpha}(hg)=P_{\alpha}D_{\alpha}(g)$ for all $h$ and $g$ (simply
substitute $\int_{H}D_{\alpha}dh$ for $P_{\alpha}$ and use the translation
invariance of the Haar measure $dh$), and thus the nonzero coefficient
functions in each $P_{\alpha}D_{\alpha}$ are left $H$-invariant. We now show
the converse: any left $H$-invariant coefficient $d_{\alpha}^{pq}$ of
$D_{\alpha}$ is one of the nonzero coefficients in $P_{\alpha}D_{\alpha}$.
Left $H$-invariance requires that
$d_{\alpha}^{pq}(g)=d_{\alpha}^{pq}(hg)=\sum_{\ell=1}^{{\rm
dim}(\alpha)}d_{\alpha}^{p\ell}(h)d_{\alpha}^{\ell q}(g).$ (24)
The linear independence of the coefficients implies that
$d_{\alpha}^{p\ell}(h)=1$ for all $h$ if $\ell=p$, and
$d_{\alpha}^{p\ell}(h)=0$ for all $h$ if $\ell\neq p$. But the assumption on
$D_{\alpha}$ requires that $d_{\alpha}^{pp}(h)=1$ on $H$ only if $p\leq{\rm
rank}(\alpha)$, and thus any left $H$-invariant coefficient $d_{\alpha}^{pq}$
must appear in one of the first ${\rm rank}(\alpha)$ rows of
$P_{\alpha}D_{\alpha}$. ∎
Since the left $H$-invariant coefficients are a basis for $\Theta_{H}(G)$, any
linear map $\omega:\Theta_{H}(G)\rightarrow\mathbb{C}$ is uniquely determined
by the values that it gives to those coefficients. For each matrix
$P_{\alpha}D_{\alpha}$, the map $\omega$ produces a corresponding matrix
$\omega(P_{\alpha}D_{\alpha})$. We now determine conditions in terms of the
matrices $\omega(PD)$ under which $\omega$ is not only linear but also
multiplicative and conjugate-preserving. In the following, we use the standard
inner product $<\zeta_{1},\zeta_{2}>=\zeta_{1}\zeta_{2}^{\dagger}$ for
complex-valued row vectors $\zeta_{1}$, $\zeta_{2}$, and the standard norm
$\|\zeta\|=(<\zeta,\zeta>)^{\frac{1}{2}}$.
###### Theorem 10.
Let $\\{D_{\alpha}\\}_{\alpha\in{\cal G}}$ be a convenient selection and
$\\{P_{\alpha}\\}_{\alpha\in{\cal G}}$ be its projections. Any linear map
$\omega:\Theta_{H}(G)\rightarrow\mathbb{C}$ is both multiplicative and
conjugate-preserving if and only if the following two conditions hold for all
$\sigma$, $\delta$, $\alpha$ in ${\cal G}$:
$\displaystyle\quad\quad\quad\omega(P_{\sigma}D_{\sigma})\otimes\omega(P_{\sigma}D_{\sigma})$
$\displaystyle=$ $\displaystyle\left[P_{\sigma}\otimes
P_{\delta}\right]C_{\sigma\delta}\left[\omega(P_{\alpha_{1}}D_{\alpha_{1}})\oplus\cdots\oplus\omega(P_{\alpha_{k}}D_{\alpha_{k}})\right]C_{\sigma\delta}^{\dagger};$
(25)
$\displaystyle\omega(P_{\alpha}D_{\alpha})\omega(P_{\alpha}D_{\alpha})^{\dagger}$
$\displaystyle=$ $\displaystyle P_{\alpha}.$ (26)
In eq. (25), the matrix $C_{\sigma\delta}$ and the indices
$\alpha_{1}$,…,$\alpha_{k}$ are as in eq. (4).
The proof is given in Appendix B.
Let $f$ be a function in $L_{1}(G)$ such that $f(hg)=f(g)$ for all $h$ in a
given closed subgroup $H$ of $G$. The translation property of the Fourier
transform ensures that each Fourier coefficient $F(\alpha)$ satisfies the
identity $F(\alpha)=F(\alpha)D_{\alpha}(h)$ for all $h$ in $H$. Integrating
over $h$, we find that $F(\alpha)=F(\alpha)P_{\alpha}$ for all $\alpha$. We
say that each Fourier coefficient $F(\alpha)$ is of maximal $H$-rank if the
rank of $F(\alpha)$ equals the rank of $P_{\alpha}$. We now show that if $f$
is any left $H$-invariant function whose Fourier coefficients $F$ all have
maximal rank, then $f$ is uniquely determined by its bispectrum $A_{3,f}$ up
to a left translation. The proof of our assertion uses the standard notation
from linear algebra [15]. For each matrix $A$, let ${\rm image}(A)$ and ${\rm
ker}(A)$ denote respectively the image and kernel of $A$. For each
$\alpha\in{\cal G}$, let ${\cal H}_{\alpha}$ denote the Hilbert space on which
the corresponding representations $D_{\alpha}$ act.
###### Theorem 11.
Let $G$ be any compact group, and let $H$ be any closed subgroup of $G$. Let
$r\in L_{1}(G)$ be invariant under left $H$-translations. If the Fourier
coefficients $\\{R(\alpha)\\}_{\alpha\in{\cal G}}$ all have maximal $H$-rank,
then $a_{3,r}=a_{3,s}$ for some $s\in L_{1}(G)$ if and only if there exists
$x\in G$ such that $s(g)=r(xg)$ for all $g$.
The proof is given in Appendix C.
In the theorem above, we did not require that the function $s$ also be left
$H$-invariant. (Equality of bispectra may hold regardless of whether both
functions are $H$-invariant.) Suppose now that two left $H$-invariant
functions $r$, $s$ are such that both have maximal $H$-rank coefficients and
both have exactly the same bispectrum. The theorem just proved demonstrates
that under those conditions, there exits $x\in G$ such that $s(g)=r(xg)$ for
all $g$. Yet the element $x$ cannot be arbitrary, for $s$ is left
$H$-invariant, and thus $s(hg)=s(g)$, implying that $r(xhg)=r(xg)$ for all
$h\in H$ and $g\in G$. But since $r$ is also left $H$-invariant, we must have
$r(xg)=r(hxg)$, and thus $r(xhg)=r(hxg)$ for all $g$ and $h$. The last
identity is always satisfied if $x$ lies in the normalizer of $H$ in $G$,
which is the subgroup $N_{H}$ of $G$ defined as follows:
$N_{H}=\\{x\in G:xH=Hx\\}.$ (27)
(The normalizer of $H$ is the largest subgroup $N_{H}$ of $G$ such that $G$
itself is a normal subgroup of $N_{H}$.) In fact, we show that $x$ must lie in
$N_{H}$ in the following theorem.
###### Theorem 12.
Let $r$, $s$ in $L_{1}(G)$ be two left $H$-invariant functions whose Fourier
coefficients $R(\alpha)$ and $S(\alpha)$ both have maximal $H$-rank for all
$\alpha$. Then $a_{3,r}=a_{3,s}$ if and only if $s(g)=r(xg)$ for some $x\in
N_{H}$.
###### Proof.
The “if” assertion is shown above, so we prove the “only if” part. Suppose
that $a_{3,r}=a_{3,s}$, and that $r$, $s$, both have maximal $H$-rank
coefficients. Under those conditions, Theorem 11 shows that there exits $x\in
G$ such that $r(g)=s(xg)$ for all $g$. Then $R(\alpha)=S(\alpha)D_{\alpha}(x)$
for all $\alpha\in{\cal G}$. Furthermore, the left invariance of $r$ implies
that $R(\alpha)=R(\alpha)P_{\alpha}$ for each $\alpha$, Thus
$S(\alpha)D_{\alpha}(x)=S(\alpha)D_{\alpha}(x)P_{\alpha}$ for each $\alpha$,
and combining that with the identity $S(\alpha)=S(\alpha)P_{\alpha}$ yields
$S(\alpha)P_{\alpha}D_{\alpha}(x)=S(\alpha)P_{\alpha}D_{\alpha}(x)P_{\alpha}$,
and thus
$S(\alpha)\left[P_{\alpha}D_{\alpha}(x)-P_{\alpha}D_{\alpha}(x)P_{\alpha}\right]=0$.
By the maximal $H$-rank hypothesis, we obtain that
$P_{\alpha}D_{\alpha}(x)=P_{\alpha}D_{\alpha}(x)P_{\alpha}.$ (28)
Since $P_{\alpha}=I({\rm rank}(\alpha))$ for a convenient selection, the
element $x$ satisfies the above equality if and only if the unitary matrix
$D_{\alpha}(x)$ is the direct sum of two smaller unitary matrices, the first
with dimensions ${\rm rank}(\alpha)\times{\rm rank}(\alpha)$ and the second
with dimensions $(n-{\rm rank}(\alpha))\times(n-{\rm rank}(\alpha))$. For such
an $x$, it follows for any $h\in H$ that
$P_{\alpha}D_{\alpha}(x)^{\dagger}D_{\alpha}(h)D_{\alpha}(x)=P_{\alpha}D_{\alpha}(x^{-1}hx)=P_{\alpha}.$
(29)
But we now see by Lemma 9 that $P_{\alpha}D_{\alpha}(x^{-1}hx)=P_{\alpha}$ if
and only if $x^{-1}hx\in H$. The last inclusion holds for all $h\in H$, and
thus $x^{-1}Hx=H$, or equivalently, $x\in N_{H}$. ∎
In the interesting special case when $G$ is the group $SO(3)$ and $H$ is the
subgroup of rotations that fix the $z$-axis, we have that $N_{H}=H$. In that
case, if $r$ is any left $H$-invariant function with maximal $H$-rank
coefficients, then there are no other left $H$-invariant functions with the
same bispectrum besides $r$ itself. However, that does not mean that the
bispectrum uniquely determines $r$: any function $s$ such that $s(g)=r(xg)$ on
$G$ has the same bispectrum, although $s$ is not necessarily $H$-invariant.
If $G=SO(3)$ and $H$ as above, then the maximal $H$-rank condition is easy to
satisfy. Here it is well-known that ${\mathop{\operator@font
rank}\nolimits}(P_{\alpha})=1$ for all $\alpha\in{\cal G}$ ([19]). Thus an
arbitrary left $H$-invariant function $r$ has maximal $H$-rank coefficients if
for all $\alpha$, the matrix $R(\alpha)$ contains at least one nonzero
coefficient. That is evidently true if any noise is present in measuring $r$.
## 5 Reconstruction algorithms
The completeness theory for arbitrary compact groups in the preceeding
sections can be refined further for the special case when the group is
$SU(2)$, which is the group of all $2\times 2$ unitary matrices with
determinant $+1$. The group $SU(2)$ arises frequently in applications because
it is a double-covering of the rotation group $SO(3)$, and in many problems it
is more convenient to model three-dimensional rotations by elements of $SU(2)$
rather than the corresponding elements of $SO(3)$–one reason is that the
addition of rotations is much simpler for $SU(2)$ (Cayley-Klein parameters)
than for $SO(3)$ (Euler parameterization). The representation theory of
$SU(2)$ is known in extensive detail, and we take advantage of the special
properties of $SU(2)$’s irreducible representations to analyze the bispectrum
of bandlimited functions. The latter are functions whose Fourier coefficients
are identically zero except for a finite set of indices. One reason why
bandlimited functions are important is that any $L_{2}$-function can be
approximated as closely as desired in the $L_{2}$-metric by a bandlimited
function.
The irreducible representations of $SU(2)$ have several properties that
simplify our analysis of the bandlimited case ([18, Chapter 2]). First, there
exists one and only one irreducible representation (up to equivalence) in each
dimension. It is thus possible to index the set of all irreducible
representations (modulo equivalence) by the nonnegative integers, in such a
way that for each $\ell\geq 0$, the representation $D_{\ell}$ has dimension
$\ell+1$. With that indexing, $D_{0}$ is the trivial representation $g\mapsto
1$, and $D_{1}$ is the self represntation $g\mapsto g$. Furthermore, for any
nonnegative integers $p$, $q$, the tensor product $D_{p}\otimes D_{q}$ reduces
explicitly as follows:
$D_{p}\otimes D_{q}=C_{pq}\left[D_{p+q}\oplus D_{p+q-2}\oplus
D_{p+q-4}\oplus\cdots\oplus D_{|p-q|}\right]C_{pq}^{\dagger}.$ (30)
The unitary matrix $C_{pq}$ above is the Clebsch-Gordan matrix for $p$ and
$q$.
We now review briefly the methods of Fourier analysis on $SU(2)$. Let $f$ be
any $L_{2}$ function on $SU(2)$. Its Fourier coefficients are the matrices
$F({\ell})=\int_{G}f(g)D_{\ell}(g)^{\dagger}dg;\quad\ell\geq 0.$ (31)
The function $f$ is the limit in $L_{2}$ of the series
$\sum_{\ell=0}^{\infty}(\ell+1){\rm Tr}\left[F(\ell)D_{\ell}(g)\right].$ (32)
By Lemma 3 we know that the bispectrum on $SU(2)$ has the form
$A_{3,f}(p,q)=\left[F(p)\otimes F(q)\right]C_{pq}\left[F(p+q)^{\dagger}\oplus
F(p+q-2)^{\dagger}\oplus\cdots\oplus
F(|p-q|)^{\dagger}\right]C_{pq}^{\dagger}.$
We now devise an algorithm for recovering any real-valued bandlimited function
with nonsingular coefficients from its bispectrum. Our algorithm makes use of
the following facts from matrix theory. First, any positive definite matrix
$H$ has a unique “positive square root”, i.e., a positive-definite matrix
$H_{+}^{\frac{1}{2}}$ such that $H_{+}^{\frac{1}{2}}H_{+}^{\frac{1}{2}}=H$
([15, pg 181]). The square root is constructed explicitly by diagonalizing
$H$, i.e., finding a unitary matrix $U$ such that $H=UDU^{\dagger}$, where $D$
is the diagonal matrix of eigenvalues (here, all nonnegative), and setting
$H_{+}^{\frac{1}{2}}=UD^{\frac{1}{2}}U^{\dagger}$, where $D^{\frac{1}{2}}$ is
the diagonal matrix containing the positive square roots of the eigenvalues.
Although the positive square root $H_{+}^{\frac{1}{2}}$ is unique, there are
in fact several possible matrix square roots, each formed by setting
$H_{Q}^{\frac{1}{2}}=UQU^{\dagger}$, where $U$ is the unitary matrix reducing
$H$ to diagonal form, and $Q$ is a diagonal matrix whose entries are either
the positive or negative square roots of the eigenvalues $\lambda_{1}$,
$\ldots$, $\lambda_{n}$ of $H$:
${\rm diag}\left[\pm\sqrt{\lambda_{1}},\ldots,\pm\sqrt{\lambda_{n}}\right].$
(33)
It is easy to see that $H_{Q}^{\frac{1}{2}}H_{Q}^{\frac{1}{2}}=H$ for any such
$Q$.
The second fact from matrix theory that we use is that any nonsingular matrix
$A$ has a unique polar decomposition $A=H_{+}U$, where
$H_{+}=(AA^{\dagger})_{+}^{\frac{1}{2}}$, and $U$ is a unitary matrix. The
polar decomposition is unique in the sense that if
$A=H_{+}U=H_{+}^{\prime}U^{\prime}$ for positive definite matrices $H_{+}$,
$H_{+}^{\prime}$, and unitary matrices $U$, $U^{\prime}$, then
$H_{+}=H_{+}^{\prime}$ and $U=U^{\prime}$. It is easy to see that $H_{+}$ as
chosen above is such that ${\rm det}(H_{+})=|{\rm det}A|$. If the determinant
of $A$ is real, then we may choose a square root $H$ of $(AA^{\dagger})$ such
that $A=HU$, where $U$ is unitary and ${\rm det}(H)={\rm det}(A)$. The last
observation becomes important in our analysis of $SO(3)$ below.
We require one last fact: the coefficient matrix $F(1)$ of any real-valued
function $f$ on $SU(2)$ has nonnegative determinant. To see that, recall from
above that $D_{1}$ is the self-representation of $SU(2)$, and thus
$F(1)=\int_{G}f(g)D_{1}{g}^{\dagger}dg=\int_{G}f(g)\left[\begin{array}[]{cc}d_{1}^{11}(g)^{*}&-d_{1}^{21}(g)^{*}\\\
d_{1}^{21}(g)&d_{1}^{11}(g)\end{array}\right]dg.$ (34)
By evaluating the matrix coefficients above, and using the assumption of real
$f$, we find that ${\rm det}[F(1)]\geq 0$.
Putting all the facts above together, we obtain the following result.
###### Proposition 13.
Let $L>0$, and let $f$ be any real-valued function on $SU(2)$ whose Fourier
coefficients are such that $F(\ell)$ is a nonsingular matrix for each
$\ell\leq L$, and furthermore, $F(\ell)=0$ if $\ell>L$. Then $f$ can be
uniquely recovered up to a left translation from its bispectrum $A_{3,f}$.
###### Proof.
Since $f$ is real-valued, it follows that $F(0)$ is a real number. Equation
(5) shows that $A_{3,f}(0,0)=F(0)^{3}$, and thus we obtain $F(0)$ by taking
cube roots. By assumption, $F(0)$ is nonzero, and thus we obtain from (5) that
$\frac{A_{3,f}(1,0)}{F(0)}=F(1)F(1)^{\dagger}.$ (35)
The matrix on the right hand side above is positive definite. Let
$\hat{F}(1)=\left(\frac{A_{3,f}(1,0)}{F(0)}\right)_{+}^{\frac{1}{2}}$ be the
positive square-root as constructed above. In polar form $F(1)=\hat{F}(1)U$,
and thus $\hat{F}(1)=F(1)U^{\dagger}$. The determinant of $F(1)$ is positive,
as is the determinant of $\hat{F}(1)$, and thus ${\rm
det}\left[U^{\dagger}\right]=+1$. Consequently, $U^{\dagger}\in SU(2)$, and we
may write $\hat{F}(1)=F(1)D_{1}(x)$ for $x=U^{\dagger}$. If $L=1$, then we are
done. Otherwise, the following algorithm produces matrices
$\hat{F}(2),\ldots,\hat{F}(L)$, such that $\hat{F}(\ell)=F(\ell)D_{\ell}(x)$
for the same $x$ and for all $2\leq\ell\leq L$. Since we know $\hat{F}(1)$ and
$A_{3,f}(1,1)$, we obtain $\hat{F}(2)$ from the upper-left $3\times 3$
submatrix of the following $4\times 4$ matrix:
$C_{11}^{\dagger}\left[\hat{F}(1)^{-1}\otimes\hat{F}(1)^{-1}\right]A_{3,f}(1,1)C_{11}.$
(36)
The reason we use the matrix above is as follows. All terms above are known,
and if we substitute for $\hat{F}(1)$ and $A_{3,f}(1,1)$, then we find that
$\displaystyle
C_{11}^{\dagger}\left[\hat{F}(1)^{-1}\otimes\hat{F}(1)^{-1}\right]A_{3,f}C_{11}=$
$\displaystyle C_{11}^{\dagger}\left[D_{1}(x)^{\dagger}\otimes
D_{1}(x)^{\dagger}\right]\left[F(1)^{-1}\otimes
F(1)^{-1}\right]\left[F(1)\otimes F(1)\right]C_{11}\left[F(2)^{\dagger}\oplus
F(0)\right]C_{11}^{\dagger}C_{11}.$
Cancelling terms, and using the reduction formula (30), shows that
$C_{11}^{\dagger}\left[\hat{F}(1)^{-1}\otimes\hat{F}(1)^{-1}\right]A_{3,f}(1,1)C_{11}=\left[D_{2}(x)^{\dagger}F(2)^{\dagger}\right]\oplus
F(0).$ (37)
The upper left $3\times 3$ submatrix of the right hand side is exactly the
matrix $[F(2)D_{2}(x)]^{\dagger}$, and we set its adjoint equal to
$\hat{F}(2)$. Having obtained $\hat{F}(2)$ in that way, we obtain
$\hat{F}(\ell)$ for any $\ell>2$ from the upper $(\ell+1)\times(\ell+1)$
submatrix of the following matrix
$C_{(\ell-1)1}^{\dagger}\left[\hat{F}(\ell-1)^{-1}\otimes\hat{F}(1)^{-1}\right]A_{3,f}(\ell-1,1)C_{(\ell-1)1}$
(38)
The same argument as above shows that
$C_{(\ell-1)1}^{\dagger}\left[\hat{F}(\ell-1)^{-1}\otimes\hat{F}(1)^{-1}\right]A_{3,f}(\ell-1,1)C_{(\ell-1)1}=\left[D_{\ell}(x)^{\dagger}F(\ell)^{\dagger}\right]\oplus\hat{F}(\ell-2)^{\dagger}.$
We set $\hat{F}(\ell)$ equal to the adjoint of the upper
$(\ell+1)\times(\ell+1)$ submatrix of the right hand side, and thus obtain
that $\hat{F}(\ell)=F(\ell)D_{\ell}(x)$. On doing so for all $\ell\leq L$, the
function $\hat{f}$ on $SU(2)$ obtained by Fourier series expansion with the
coefficients $F(0)$, $\hat{F}(1)$, $\ldots$, $\hat{F}(L)$ is such that
$\hat{f}(g)=f(xg)$ for all $g$. ∎
It is easy to prove the same completeness result for bandlimited functions on
$SO(3)$. We describe the few differences that exist, drawing on standard facts
about representations of $SO(3)$ ([18, Chap II]). First, the irreducible
representations of $SO(3)$ occur only in odd dimensions, and there is exactly
one representation (modulo equivalence) in each odd dimension. Thus we may
list any selection of irreducible representations as
$\\{D_{\ell}\\}_{\ell=0}^{\infty}$, where for each $\ell$, the representation
$D_{\ell}$ has dimension $(2\ell+1)$. In that indexing, $D_{0}$ is the trivial
representation, and $D_{1}$ is equivalent to the self-representation $g\mapsto
g$ of $SO(3)$, i.e., $D_{1}(g)=UgU^{\dagger}$ for some unitary matrix $U$.
(Recall that $SO(3)$ is the set of all real-valued $3\times 3$ orthogonal
matrices with determinant $+1$.) For each $n$, $m$, the tensor-product
$D_{n}\otimes D_{m}$ reduces explicitly as follows:
$D_{n}\otimes D_{m}=C_{nm}\left[D_{n+m}\oplus D_{n+m-1}\oplus\cdots\oplus
D_{|n-m|}\right]C_{nm}^{\dagger}.$ (39)
With the formula above, it is easy to see that the recursive algorithm given
in the proof of Proposition 13 generalizes to recover all real-valued
bandlimited functions on $SO(3)$. To initialize the algorithm, we require an
estimate $\hat{F}(1)$ of the first coefficient $F(1)$ from the data
$F(1)F(1)^{\dagger}$, such that $\hat{F}(1)=F(1)D_{1}(g)$ for some element $g$
of $SO(3)$. Assuming that $F(1)$ is nonsingular, we obtain the estimate as
follows. The representation $D_{1}$ is such that $D_{1}(g)=UgU^{\dagger}$,
where $U$ is fixed as $g$ varies in $SO(3)$. Thus
$\displaystyle F(1)$ $\displaystyle=$
$\displaystyle\int_{G}f(g)D_{1}(g)^{\dagger}dg,$ $\displaystyle=$
$\displaystyle U\left[\int_{G}f(g)g^{\dagger}\right]U^{\dagger}.$
Let $F_{s}(1)$ denote the matrix that results by evaluating the integral in
brackets. Since $f$ is real-valued, and every matrix $g$ has real
coefficients, the matrix $F_{s}(1)$ has only real coefficients. Thus the
determinant of $F(1)=UF_{s}(1)U^{\dagger}$ is a real number. Assume for the
moment that ${\rm det}\left[F(1)\right]={\rm det}\left[F_{s}(1)\right]>0$. Let
$\hat{F}(1)$ and $\hat{F}_{s}(1)$ denote respectively the (unique) positive
square roots of $F(1)F(1)^{\dagger}$ and $F_{s}(1)F_{s}(1)^{\dagger}$. Since
$F(1)F(1)^{\dagger}=UF_{s}(1)F_{s}(1)^{\dagger}U^{\dagger}$, it is easily seen
that $\hat{F}(1)=U\hat{F}_{s}(1)U^{\dagger}$. Now consider the polar
decomposition $F_{s}(1)=HV$, where $H$ is positive definite and $V$ is
unitary. Recall from the earlier discussion for $SU(2)$ that
$H=\left(F_{s}(1)F_{s}(1)^{\dagger}\right)_{+}^{\frac{1}{2}}$, and thus
$H=\hat{F}_{s}(1)$. Since $F_{s}(1)$ is real-valued, $V$ must be real-valued
orthogonal matrix. Matching determinants on both sides of the equation
$F_{s}(1)=\hat{F}_{s}(1)V$ reveals that ${\rm det}[V]=+1$, and thus $V=g$, for
some $g\in SO(3)$. Substitution reveals that
$\hat{F}(1)=U\hat{F}_{s}(1)U^{\dagger}=UF_{s}(1)gU^{\dagger}=UF_{s}(1)U^{\dagger}UgU^{\dagger}=F(1)D_{1}(g).$
(40)
The assumption that ${\rm det}[F(1)]>0$ is not critical. We use it only to
obtain that ${\rm det}[V]=+1$, where $V=\hat{F}_{s}(1)^{-1}F_{s}(1)$. Instead
of selecting $\hat{F}(1)$ to be the positive definite square root of
$F(1)F(1)^{\dagger}$, we may choose $\hat{F}(1)$ to be any square root such
that ${\rm det}[\hat{F}(1)]={\rm det}[F(1)]$, e.g., by multiplying the top row
of the positive definite square root matrix by $-1$ if necessary. We do not
know ${\rm det}[F(1)]$ a priori, but if we store it as “side information”
along with the bispectrum, then we obtain a complete rotation-invariant
description for any real-valued bandlimited function on $SO(3)$. Note that
${\rm det}[F(1)]$ remains invariant under translation on $SO(3)$, i.e., if
$f(g)=s(hg)$, then $F(1)=S(1)D_{1}(h)$, but since ${\rm det}[D_{1}(h)]=+1$, we
obtain that ${\rm det}[F(1)]={\rm det}[S(1)]$. To sum up, any real-valued
bandlimited function $f$ on $SO(3)$, whose coefficient matrices are all
nonsingular up to the bandlimit, can be recovered completely—up to a single
translation on $SO(3)$—if both its bispectrum and the value of ${\rm
det}[F(1)]$ is known, and the algorithm described above is used.
## 6 Applications
As mentioned in the introduction, the invariance and completeness properties
of the bispectrum lend themselves to applications in pattern matching
problems. One particular application is described here. R. Kondor [14]
demonstrates how translation- and rotation- invariant matching of hand-written
characters is accomplished with bispectral invariants. To do so, Kondor notes
that, for practical purposes, the characters themselves may defined as
intensity-valued functions on a compact patch on $\mathbb{R}^{2}$ of radius
$1$. A transformation may be constructed that maps the planar patch to the
upper hemisphere of the sphere $S^{2}$ as follows:
$\omega:(r,\phi_{\mathbb{R}^{2}})\mapsto(\theta,\phi_{S^{2}})\quad\quad{\rm
with}\quad r=\sin(\theta);\quad\phi_{\mathbb{R}^{2}}=\phi_{S^{2}}.$ (41)
The subscripts denote the domain of the angle involved, whether it be the
plane $\mathbb{R}^{2}$ or the sphere $S^{2}$. Kondor shows that rigid body
motions on the patch, each of which consists of a rotation by an angle
$\alpha$ and a translation by a vector $T=(t_{x},t_{y})$ with $\|T\|\leq 1$,
may be mapped to 3-D rotations through the use of Euler angles
$(\theta,\phi,\psi)$ as follows
$\alpha=\psi;\quad t_{x}=\sin\theta\cos\phi;\quad t_{y}=\sin\theta\sin\phi.$
(42)
This mapping produces a local isomorphism between planar rigid motions and
spherical rotations.
By using the transformation (41), every intensity function defined on the
planar patch may be converted to a function on $S^{2}$. The problem of finding
rigid-motion invariants on $\mathbb{R}^{2}$ now becomes one of finding
rotation invariants on the sphere $S^{2}$. Since the sphere is a homogeneous
space for $SO(3)$, every function $\widetilde{f}$ on $S^{2}$ may in turn be
lifted to a function $f$ on $SO(3)$ using the “north-pole” mapping: if
$z=[0,0,1]$, then $f(R)=\widetilde{f}(Rz)$ for every $R\in SO(3)$. We may now
construct the bispectrum of $f$ from eq. (11) using the Fourier transform on
$SO(3)$, which may be calculated using spherical harmonic basis functions.
Kondor calculates bispectral invariants in this way, and shows, in an
experiment using $1000$ hand-written characters from a standard dataset, that
the invariants perform well in matching over arbitrary orientations and
starting positions of the characters.
A second application of the bispectrum occurs in astrophysical models of
primordial fluctuations, as mentioned in the introduction. Cosmic inflation
[7] predicts a Gaussian pattern of temperature anisotropies in the cosmic
microwave background radiation (CBR). The CBR anisotropy is a function defined
on $S^{2}$, and therefore we may calculate its bispectrum using eq. (11). If
the anisotropy is Gaussian, then the expected value of the angular bispectrum
is zero. However, as X. Luo [16] shows, the stochastic nature of anisotropies
means that cosmic variance makes it difficult to extract non-Gaussian
structure from CBR data. In that paper, as in much of the physics literature,
expressions of the bispectrum follow the “summation notation”, which
implicitly focuses attention at the level of individual elements of bispectral
matrices. It is hoped that the approach of this paper, including the matrix
form derived in (11), proves useful in allowing insight into higher level
properties, such as matrix rank, decomposition, and completeness.
Both of the applications mentioned apply the bispectrum to functions on the
sphere. Healy et al [10] describe a fast “divide-and-conquer” discrete
Legendre transform, which leads to an “FFT” on $S^{2}$. They show how this
transform leads to efficient computation of bispectrum on the sphere.
## 7 Summary and future directions
This paper derives completeness properties of the bispectrum for functions
defined on compact groups and their homogeneous spaces. A matrix form of the
bispectrum is derived, and it is shown that every function with nonsingular
coefficients is completely determined, up to a group translation, by its
bispectrum. A reconstruction algorithm for functions defined on the groups
$SU(2)$ and $SO(3)$ is described.
Results similar to those in this paper may be established for non-compact,
non-commutative groups [12]. Those results rely on the duality theorem of
Tatsuuma. The Tannaka-Krein duality theorem, which is central to this paper,
has been extended to compact groupoids [1]. It would be interesting to see if
a corresponding bispectral theory may be constructed there.
## Appendix A Proof of Lemma 7
Since $\Theta_{H}\subset\Theta(G)$, each $f\in\Theta_{H}(G)$ is a unique
finite linear combination of matrix coefficients $d_{\alpha}^{pq}$ (not
necessarily $H$-invariant):
$f(g)=\sum_{\alpha,p,q}c_{\alpha}^{pq}d_{\alpha}^{pq}(g).$ (43)
We set $f(hg)=f(g)$ to obtain
$\sum_{\alpha,p,q}c_{\alpha}^{pq}d_{\alpha}^{pq}(g)=\sum_{\alpha,p,q}c_{\alpha}^{pq}d_{\alpha}^{pq}(hg).$
(44)
The multiplication rule $D_{\alpha}(hg)=D_{\alpha}(h)D_{\alpha}(g)$ for
representation matrices reveals that
$d_{\alpha}^{pq}(hg)=\sum_{\ell=1}^{{\rm
dim}(\alpha)}d_{\alpha}^{p\ell}(h)d_{\alpha}^{\ell q}(g),$ (45)
and thus
$\sum_{\alpha,p,q}c_{\alpha}^{pq}d_{\alpha}^{pq}(g)=\sum_{\alpha,p,q}c_{\alpha}^{pq}d_{\alpha}^{pq}(h)d_{\alpha}^{pq}(g)+\sum_{\alpha,p,q}c_{\alpha}^{pq}\sum_{\ell\neq
p}d_{\alpha}^{p\ell}(h)d_{\alpha}^{\ell q}(g).$ (46)
The linear independence of the matrix coefficients implies that in the
equation above, $d_{\alpha}^{pp}(h)=1$ and $d_{\alpha}^{p\ell}(h)=0$ if
$\ell\neq p$. Thus $d_{\alpha}^{pq}(hg)=d_{\alpha}^{pq}(g)$ for every
coefficient function in eq. (43).
## Appendix B Proof of Theorem 10
If $\omega$ preserves multiplication and complex-conjugation, then the
Iwahori-Sugiura theorem shows that there exists a unique coset $Hg$ such that
$\omega(P_{\alpha}D_{\alpha})=P_{\alpha}D_{\alpha}(Hg)$ for all $\alpha$. From
this, equations (25) and (26) follow immediately. Suppose now that $\omega$ is
some linear map that also satisfies eq. (25). Applying Lemma 8 to both sides
of the tensor product decomposition in eq. (4) yields
$(P_{\sigma}D_{\sigma})\otimes(P_{\delta}D_{\delta})=\left[P_{\sigma}\otimes
P_{\delta}\right]C_{\sigma\delta}\left[(P_{\alpha_{1}}D_{\alpha_{1}})\oplus\cdots\oplus(P_{\alpha_{k}}D_{\alpha_{k}})\right]C_{\sigma\delta}^{\dagger}.$
(47)
Now apply $\omega$ to both sides to obtain
$\omega\left((P_{\sigma}D_{\sigma})\otimes(P_{\delta}D_{\delta})\right)=\left[P_{\sigma}\otimes
P_{\delta}\right]C_{\sigma\delta}\left[\omega(P_{\sigma_{1}}D_{\alpha_{1}})\oplus\cdots\oplus\omega(P_{\alpha_{k}}D_{\alpha_{k}})\right]C_{\sigma\delta}^{\dagger}.$
Because eq. (25) holds, we obtain that for all $\sigma$, $\delta$,
$\omega\left((P_{\sigma}D_{\sigma})\otimes(P_{\delta}D_{\delta})\right)=\omega(P_{\sigma}D_{\sigma})\otimes\omega(P_{\delta}D_{\delta}).$
(48)
Thus $\omega$ is multiplicative. Suppose now that the linear and
multiplicative map $\omega$ also satisfies eq. (26). Applying $\omega$ to both
sides of the identity
$(P_{\sigma}D_{\sigma})(P_{\alpha}D_{\sigma})^{\dagger}=P_{\alpha}$ yields
$\omega(P_{\alpha}D_{\alpha})\omega\left((P_{\alpha}D_{\alpha})^{\dagger}\right)=P_{\alpha}.$
(49)
We show that
$\omega\left((P_{\alpha}D_{\alpha})^{\dagger}\right)=\omega(P_{\alpha}D_{\alpha})^{\dagger}$,
proving that $\omega$ preserves conjugation. Let $\zeta_{\alpha}$ be any
nonzero row of the matrix $P_{\alpha}D_{\alpha}$. We establish the following
three equalities:
$\displaystyle<\omega(\zeta_{\alpha}),\omega(\zeta_{\alpha})>$
$\displaystyle=$ $\displaystyle 1,$ (50)
$\displaystyle<\omega(\zeta_{\alpha}),\omega(\zeta_{\alpha}^{*})^{*}>$
$\displaystyle=$ $\displaystyle 1,$ (51)
$\displaystyle<\omega(\zeta_{\alpha}^{*}),\omega(\zeta_{\alpha}^{*})>$
$\displaystyle=$ $\displaystyle 1.$ (52)
The first equality (50) follows from (26) (recall that we are working with a
convenient selection, for which $P_{\alpha}=I({\rm rank}(\alpha))$). The
second is derived from eq. (49). The final equality requires more work, but is
a straightforward consequence of (26) and the linearity of $\omega$. We give
its proof later, but for now assume that it is true. The three inequalities
above imply that
$\displaystyle<\omega(\zeta_{\alpha}),\omega(\zeta_{\alpha}^{*})^{*}>=\|\omega(\zeta_{\alpha})\|\|\omega(\zeta_{\alpha}^{*})^{*}\|.$
(53)
The Cauchy-Schwartz inequality shows that the identity above holds if and only
if $\omega(\zeta_{\alpha})=c\omega(\zeta_{\alpha}^{*})^{*}$, and from eq.(51)
we see that $c=1$. Thus
$\omega(\zeta_{\alpha})=\omega(\zeta_{\alpha}^{*})^{*}$. Since the preceeding
argument applies to any nonzero-row $\zeta_{\alpha}$ of any matrix
$P_{\alpha}D_{\alpha}$, it follows that $\omega$ preserves conjugation.
Now to prove (52). For any representation $D_{\alpha}$ in our selection, the
conjugate representation $D_{\alpha}^{*}$ is also irreducible, and there are
two cases: (i) $D_{\alpha}^{*}=A_{\alpha}D_{\alpha}A_{\alpha}^{\dagger}$ for
some unitary matrix $A_{\alpha}$; (ii)
$D_{\alpha}^{*}=A_{\beta}D_{\beta}A_{\beta}^{\dagger}$ where
$\beta\neq\alpha$. Assume that the first case is true. It is easy to show that
any matrix $A_{\alpha}$ expressing the equivalence of conjugate
representations is symmetric, and thus $A_{\alpha}^{\dagger}=A_{\alpha}^{*}$
([5, pg 15]). Furthermore, only the first ${\rm rank}(\alpha)$ rows of
$D_{\alpha}$ are $H$-invariant, and that must also be true of the matrix
$D_{\alpha}^{*}=A_{\alpha}D_{\alpha}A_{\alpha}^{*}$. Thus $A_{\alpha}$
transforms the first ${\rm rank}(\alpha)$ rows among themselves, which means
that $A_{\alpha}$ must have the block-diagonal form
$A_{\alpha}=A_{\alpha,1}\oplus A_{\alpha,2}$, where $A_{\alpha,1}$ is a
symmetric unitary matrix with dimensions ${\rm rank}(\alpha)\times{\rm
rank}(\alpha)$. Thus $P_{\alpha}A_{\alpha}=A_{\alpha}P_{\alpha}$. Putting
those facts together, we obtain the following identity by virtue of $\omega$’s
linearity:
$\omega(P_{\alpha}D_{\alpha}^{*})=\omega(P_{\alpha}A_{\alpha}D_{\alpha}A_{\alpha}^{*})=A_{\alpha}\omega(P_{\alpha}D_{\alpha})A_{\alpha}^{*}.$
(54)
By using the identity above and eq. (26), we find that
$\omega(P_{\alpha}D_{\alpha}^{*})\omega(P_{\alpha}D_{\alpha}^{*})^{\dagger}=P_{\alpha}$.
Noting that $P_{\alpha}=I({\rm rank}(\alpha))$, the previous equality for
matrices implies eq. (52) for their nonzero rows. Case (ii) is similar.
## Appendix C Proof of Theorem 11
If $r$ and $s$ are left translates of each other, then $a_{3,r}=a_{3,s}$, as
follows from the definition of triple correlation and the left invariance of
Haar measure. Now suppose that $a_{3,r}=a_{3,s}$. Lemma 3 shows that for all
$\sigma$, $\delta$,
$\displaystyle\left[R(\sigma)\otimes
R(\delta)\right]C_{\sigma\delta}\left[R(\alpha_{1})^{\dagger}\oplus\cdots\oplus
R(\alpha_{k})^{\dagger}\right]C_{\sigma\delta}^{\dagger}$ $\displaystyle=$
$\displaystyle\left[S(\sigma)\otimes
S(\delta)\right]C_{\sigma\delta}\left[S(\alpha_{1})^{\dagger}\oplus\cdots\oplus
S(\alpha_{k})^{\dagger}\right]C_{\sigma\delta}^{\dagger}$ .
If we set $\sigma=\delta={\bf 1}$ and apply the same argument used in the
proof of Theorem 4, we obtain $R({\bf 1})=S({\bf 1})$. The maximal $H$-rank
assumption implies that the scalar $R({\bf 1})=S({\bf 1})$ is nonzero. Now set
$\delta={\bf 1}$ in (C) above. Cancelling $R({\bf 1})=S({\bf 1})$ from both
sides shows that
$R(\sigma)R(\sigma)^{\dagger}=S(\sigma)S(\sigma)^{\dagger}.$ (56)
Hence, we have for each $\sigma$ that $S(\sigma)=R(\sigma)U(\sigma)$ for some
unitary matrix $U(\sigma)$. Substituting into eq. (C) reveals that
$\displaystyle\left[R(\sigma)\otimes
R(\delta)\right]C_{\sigma\delta}\left[R(\alpha_{1})^{\dagger}\oplus\cdots\oplus
R(\alpha_{k})^{\dagger}\right]C_{\sigma\delta}^{\dagger}=$
$\displaystyle\left[R(\sigma)\otimes
R(\delta)\right]C_{\sigma\delta}\left[U(\alpha)^{\dagger}\oplus\cdots\oplus
U(\alpha_{k})^{\dagger}\right]\left[R(\alpha_{1})^{\dagger}\oplus\cdots\oplus
R(\alpha_{k})^{\dagger}\right]C_{\sigma\delta}^{\dagger}$ .
We cancel $C_{\sigma\delta}^{\dagger}$ from both sides. The identity
$R(\alpha)=R(\alpha)P_{\alpha}$ implies that
$R(\alpha)^{\dagger}=P_{\alpha}R(\alpha)^{\dagger}$, and thus ${\rm
image}(R(\alpha)^{\dagger})\subset P_{\alpha}({\cal H}_{\alpha})$. But the
rank of $R(\alpha)^{\dagger}$ equals that of $P_{\alpha}$, and thus ${\rm
image}(R(\alpha)^{\dagger})=P_{\alpha}({\cal H}_{\alpha})$. The last identity
implies that
$\displaystyle\left[R(\sigma)\otimes
R(\delta)\right]C_{\sigma\delta}\left[P_{\alpha_{1}}\oplus\cdots\oplus
P_{\alpha_{k}}\right]=$ $\displaystyle\left[R(\sigma)\otimes
R(\delta)\right]\left[U(\sigma)\otimes
U(\delta)\right]C_{\sigma\delta}\left[U(\alpha_{1})^{\dagger}\oplus\cdots\oplus
U(\alpha_{k})^{\dagger}\right]\left[P_{\alpha_{1}}\oplus\cdots\oplus
P_{\alpha_{k}}\right].$
Multiplying both sides from the right by
$C_{\sigma\delta}^{\dagger}[P_{\sigma}\otimes P_{\delta}]$ and using Lemma 8
reveals that
$\displaystyle\left[R(\sigma)\otimes R(\delta)\right]\left[P_{\sigma}\otimes
P_{\delta}\right]=$ $\displaystyle\left[R(\sigma)\otimes
R(\delta)\right]\left[U(\sigma)\otimes
U(\delta)\right]C_{\sigma\delta}\left[U(\alpha_{1})^{\dagger}\oplus\cdots\oplus
U(\alpha_{k})^{\dagger}\right]C_{\sigma\delta}^{\dagger}\left[P_{\sigma}\otimes
P_{\delta}\right].$
We substitute $R(\sigma)=R(\sigma)P_{\sigma}$ and
$R(\delta)=R(\delta)P_{\sigma}$ into the leftmost tensor product term on the
right hand side of the equation above and simplify, to obtain
$\displaystyle\left[R(\sigma)\otimes R(\delta)\right]\left[P_{\sigma}\otimes
P_{\delta}\right]=$ $\displaystyle\left[R(\sigma)\otimes
R(\delta)\right]\left[P_{\sigma}\otimes
P_{\delta}\right]\left[U(\sigma)\otimes
U(\delta)\right]C_{\sigma\delta}\left[U(\alpha_{1})^{\dagger}\oplus\cdots\oplus
U(\alpha_{k})^{\dagger}\right]C_{\sigma\delta}^{\dagger}\left[P_{\sigma}\otimes
P_{\delta}\right].$
For each $\alpha$, the identity $R(\alpha)P_{\alpha}=R(\alpha)$ together with
the assumption that $R(\alpha)$ has maximal $H$-rank imply that $R(\alpha)$ is
one-to-one on $P_{\alpha}({\cal H}_{\alpha})$. Thus the equation above implies
that
$P_{\sigma}\otimes P_{\delta}=\left[P_{\sigma}\otimes
P_{\delta}\right]\left[U(\sigma)\otimes
U(\delta)\right]C_{\sigma\delta}\left[U(\alpha_{1})^{\dagger}\oplus\cdots\oplus
U(\alpha_{k})^{\dagger}\right]C_{\sigma\delta}^{\dagger}\left[P_{\sigma}\otimes
P_{\delta}\right]$
The matrix in between the two orthogonal projections on the right hand side is
unitary; it is easily seen that for any unitary matrix $U$ and any orthogonal
projection $P$, the matrix equation $P=PUP$ holds only if $UP=P$. Thus we
obtain
$P_{\sigma}\otimes P_{\delta}=\left[U(\sigma)\otimes
U(\delta)\right]C_{\sigma\delta}\left[U(\alpha_{1})^{\dagger}\oplus\cdots\oplus
U(\alpha_{k})^{\dagger}\right]C_{\sigma\delta}^{\dagger}\left[P_{\sigma}\otimes
P_{\delta}\right]$ (57)
Rearranging terms, we obtain the following identity:
$\left[U(\sigma)^{\dagger}\otimes U(\delta)^{\dagger}\right][P_{\sigma}\otimes
P_{\delta}]=C_{\sigma\delta}\left[U(\alpha_{1})^{\dagger}\oplus\cdots\oplus
U(\alpha_{k})^{\dagger}\right]C_{\sigma\delta}^{\dagger}[P_{\sigma}\otimes
P_{\delta}].$
Substituting from Lemma 8 in the right hand side, and subsequently taking the
matrix adjoint of both sides, reveals that
$\left[(P_{\sigma}U(\sigma))\otimes(P_{\delta}U(\delta))\right]=\left[P_{\sigma}\otimes
P_{\delta}\right]C_{\sigma\delta}\left[(P_{\alpha_{1}}U(\alpha_{1}))\oplus\cdots\oplus(P_{\alpha_{k}}U(\alpha_{k}))\right]C_{\sigma\delta}^{\dagger}.$
Theorem 10, together with the Iwahori-Sugiura Theorem, shows that the identity
above holds if and only if there exists a coset $Hx$ such that
$P_{\alpha}U(\alpha)=P_{\alpha}D_{\alpha}(Hx)$ for all $\alpha\in{\cal G}$.
Thus for each $\alpha$ we have the string of identities
$S(\alpha)=R(\alpha)U(\alpha)=R(\alpha)P_{\alpha}U(\alpha)=R(\alpha)P_{\alpha}D_{\alpha}(Hx)=R(\alpha)D_{\alpha}(x).$
(58)
The translation property of the Fourier transform now shows that $s(g)=r(xg)$
for all $g$.
Acknowledgments. I thank the numerous people who wrote for a copy of my Ph.D.
dissertation [12], in which this work was first presented. This work was
influenced in many ways by the suggestions and insights of my late supervisor,
Professor Bruce M. Bennett.
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|
arxiv-papers
| 2009-02-02T04:35:09 |
2024-09-04T02:49:00.362045
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ramakrishna Kakarala",
"submitter": "Ramakrishna Kakarala",
"url": "https://arxiv.org/abs/0902.0196"
}
|
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