# Thermal effects on the Casimir energy of a Lorentz-violating scalar in
magnetic field

Andrea Erdas<EMAIL_ADDRESS>Department of Physics, Loyola University
Maryland, 4501 North Charles Street, Baltimore, Maryland 21210, USA

###### Abstract

In this work I investigate the finite temperature Casimir effect due to a
massive and charged scalar field that breaks Lorentz invariance in a CPT-even,
aether-like way. I study the cases of Dirichlet and mixed (Dirichlet-Neumann)
boundary conditions on a pair of parallel plates. I will not examine the case
of Neumann boundary conditions since it produces the same results as Dirichlet
boundary conditions. The main tool used in this investigation is the
$\zeta$-function technique that allows me to obtain the Helmoltz free energy
and Casimir pressure in the presence of a uniform magnetic field perpendicular
to the plates. Three cases of Lorentz asymmetry are studied: timelike,
spacelike and perpendicular to the magnetic field, spacelike and parallel to
the magnetic field. Asymptotic cases of small plate distance, high
temperature, strong magnetic field, and large mass will be considered for each
of the three types of Lorentz asymmetry and each of the two types of boundary
conditions examined. In all these cases simple and very accurate analytic
expressions of the thermal corrections to the Casimir energy and pressure are
obtained and I discover that these corrections strongly depend on the
direction of the unit vector that produces the breaking of the Lorentz
symmetry.

###### pacs:

03.70.+k, 11.10.-z, 11.10.Wx, 11.30.Cp, 12.20.Ds.

##  I Introduction

The first to predict an attractive force between two uncharged parallel plates
due to quantum effects, was Hendrik Casimir Casimir:1948dh . Sparnaay’s
experiments Sparnaay:1958wg  ten years later tested this phenomenon, quite
difficult to observe, and were relatively consistent with it. Many
increasingly accurate and sophisticated experiments followed throughout the
decades Bordag:2001qi ; Bordag:2009zz , fully confirming the Casimir effect,
caused by the quantum fluctuations of the electromagnetic field in vacuum.
Quantum fluctuations of other vector, scalar and fermion fields also cause
Casimir forces. These forces are strongly dependent on the boundary conditions
at the plates of the quantum field involved. In the case of a scalar field,
Dirichlet or Neumann boundary conditions cause an attractive force between the
two parallel plates, while mixed (Dirichlet-Neumann) boundary conditions cause
a repulsive force Boyer:1974 . Casimir forces display also a strong dependence
on the geometry of the plates. The repulsive nature of the spherical Casimir
effect Boyer:1968uf  is caused by the electromagnetic field, or by a scalar
field, satisfying Dirichlet boundary conditions at the spherical shell.

In recent years the violation of Lorentz symmetry in quantum field theories
has been investigated with the proposal of models where such a violation
causes a space-time anisotropy Ferrari:2010dj ; Ulion:2015kjx . Several
Lorentz symmetry breaking mechanism have been presented within the context of
quantum gravity Alfaro:1999wd ; Alfaro:2001rb , of string theory where a
spontaneous violation of Lorentz invariance at the Planck scale has been
conjectured Kostelecky:1988zi , and of quantum field theories with varying
coupling constants Anchordoqui:2003ij ; Bertolami:1997iy ; Kostelecky:2002ca .
The repercussions of a space-time anisotropy on the Casimir effect have been
analyzed for the case of a real scalar field in vacuum Cruz:2017kfo  and at
finite temperature Cruz:2018bqt , where the scalar field is in equilibrium
with a thermal bath. These two papers Cruz:2017kfo ; Cruz:2018bqt  investigate
the Casimir effect due to a modified Klein-Gordon field that breaks the
Lorentz symmetry in a CPT-even, aether-like manner. The consequence of space-
time anisotropy have also been studied extensively in the case of Lorentz-
breaking extensions of QED Frank:2006ww ; Kharlanov:2009pv ; Martin-
Ruiz:2016lyy . The Casimir effect is one of the most studied quantum
phenomena, theoretically and experimentally, and has been proposed by all
these authors as a laboratory for studying Lorentz symmetry violation.

Authors have investigated magnetic corrections to the Casimir effect caused by
a charged scalar field in the standard Lorentz symmetric space-time
CougoPinto:1998td , and also the combination of thermal and magnetic effects
within standard space-time CougoPinto:1998jg ; Erdas:2013jga ; Erdas:2013dha ,
and recently a paper has been published that investigates vacuum magnetic
corrections to the Casimir effect of a scalar field that breaks Lorentz
symmetry Erdas:2020ilo  however, there has not been a study of the combination
of thermal and magnetic corrections to the Casimir effect of a scalar field
that breaks Lorentz invariance. This paper, a sequel to my previous work
Erdas:2020ilo , intends to fill that gap and study the effect of a constant
magnetic field $\bf B$, perpendicular to the plates, on the Casimir energy and
pressure caused by a scalar field that breaks Lorentz symmetry and is in
thermal equilibrium with a heat reservoir at finite temperature $T$. I will
study a charged scalar field that breaks the Lorentz symmetry in the manner
proposed by Refs. Cruz:2017kfo ; Cruz:2018bqt  and satisfies either Dirichlet
or mixed boundary conditions at two square parallel plates a distance $a$ from
each other.

When two parallel plates face each other in vacuum and a quantum field in its
ground state, vanishing at the plates, permeates the space, an infinite number
of wavelengths contributes to the calculation of the vacuum energy producing
an infinite value. In order to obtain a finite quantity, one must subtract the
vacuum energy of that space without the plates, from the vacuum energy of that
same space with the plates present. Both quantities are infinite but, through
a careful regularization procedure, one is able to obtain a finite quantity,
the Casimir energy, which does not depend on the regularization method. In
this work I will use a regularization procedure introduced by Hawking
Hawking:1976ja , the zeta function technique, extensively used in the
computation of vacuum Casimir energies Elizalde:1988rh ; Elizalde:2007du  and
also in the context of finite temperature Casimir energy computations
Santos:1998vb ; CougoPinto:1998xn ; Santos:1999yj . Other regularization
methods are available and widely used in the literature when computing Casimir
energies, such as the Abel-Plana method used in Refs. Cruz:2017kfo ;
Cruz:2018bqt .

In Sec. II of this paper I illustrate the model of a scalar field that breaks
Lorentz symmetry in an aether-like and CPT-even way. In Sec. III, I impose
Dirichlet boundary conditions at the plates for the scalar field and, in
subsection A, examine the case of timelike anisotropy. I derive the zeta
function and, from it, obtain the Casimir energy in the asymptotic limits of
small plate distance, high temperature, strong magnetic field and large mass.
In subsections B and C I do the same for the cases of spacelike anisotropy in
the plane perpendicular to $\bf B$ and spacelike anisotropy in the direction
of $\bf B$, respectively. In Sec. IV I investigate mixed (Dirichlet-Neumann)
boundary conditions at the plates and obtain the zeta function and the
asymptotic limits of the Casimir energy listed above, for each of the three
cases of space-time anisotropy. In Sec. V I obtain and discuss the Casimir
pressure for all these scenarios. My conclusions and a discussion of the
results obtained in this paper are presented in Sec. VI. In Appendix I, I show
the details of the calculation of the high temperature limit of the zeta
function for Dirichlet boundary conditions and timelike asymmetry. In Appendix
II, I show the calculation of the high temperature limit of the zeta function
for mixed boundary conditions and timelike asymmetry.

##  II The model

In this paper I study the Casimir effect caused by a scalar field $\phi$ of
charge $e$ and mass $M$ that violates Lorentz symmetry. The theoretical model
of a scalar field that breaks Lorentz symmetry in an aether-like and CPT-even
manner was introduced in Ref. Cruz:2017kfo . In such a model, the derivative
of the scalar field is coupled to a fixed unit four-vector $u^{\mu}$, thus
causing the Lorentz symmetry violation. This complex scalar field obeys the
following Klein Gordon equation

$[\Box+\lambda(u\cdot\partial)^{2}+M^{2}]\phi=0,$ (II.1)

where the unit four-vector $u^{\mu}$ points in the direction in which the
Lorentz symmetry is broken and the dimensionless parameter $\lambda\ll 1$
determines the amount of symmetry violation. I will investigate two square
parallel plates of side $L$, perpendicular to the $z$ axis and located at
$z=0$ and $z=a$, in thermal equilibrium with a heat reservoir at temperature
$T$ and will obtain thermal corrections to their Casimir energy caused by the
aforementioned Lorentz violating scalar field. A uniform magnetic field $\bf
B$ pointing in the $z$ direction will also be included in this study. For such
a system the imaginary time formalism of finite temperature field theory is
convenient, and it allows only field configurations satisfying the following
periodic condition

$\phi(x,y,z,\tau)=\phi(x,y,z,\tau+\beta),$ (II.2)

for any $\tau$, where $\beta=1/T$ is the periodic length in the Euclidean time
axis. In addition to the finite temperature boundary condition (II.2), I will
impose either Dirichlet or mixed boundary conditions at the plates and will
use the generalized zeta function technique to study this system. Neumann
boundary conditions will not be investigated, since they produce the same
result as Dirichlet. I will investigate a unit four-vector $u^{\mu}$ that is
timelike, spacelike and perpendicular to $\bf B$, and spacelike and parallel
to $\bf B$.

The Helmholtz free energy of the scalar field is

$F=\beta^{-1}\log\,\det\left(D_{\rm E}|{\cal F}_{a}\right),$

where $D_{\rm E}$ is the Klein Gordon operator in Euclidean time and the
symbol ${\cal F}_{a}$ indicates the set of eigenfunctions of $D_{\rm E}$ which
satisfy condition (II.2), and either Dirichlet or mixed boundary conditions at
the plates. In the absence of a magnetic field, $D_{\rm E}$ is given by

$D_{E}=-{\frac{\partial^{2}}{\partial\tau^{2}}}-\nabla^{2}+\lambda(u\cdot\partial)^{2}+M^{2}.$
(II.3)

Notice that at this initial stage the magnetic field is not present and it
will be introduced later. When $u^{\mu}$ is timelike and the finite
temperature condition of Eq. (II.2) is imposed, the eigenvalues of $D_{E}$
are:

$\left\\{(1+\lambda)\frac{4\pi^{2}m^{2}}{\beta^{2}}+k_{x}^{2}+k_{y}^{2}+k_{z}^{2}+M^{2}\right\\},$
(II.4)

where $k_{x},k_{y}\in\Re$, $m=0,\pm 1,\pm 2,\cdots$, and $k_{z}$ will take
different discrete values if $\phi$ satisfies Dirichlet or mixed boundary
conditions at the plates. When $u^{\mu}$ is spacelike and lies in the $x$-$y$
plane, I take $u^{\mu}=\left(0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0\right)$
and the eigenvalues of $D_{E}$ are:

$\left\\{\frac{4\pi^{2}m^{2}}{\beta^{2}}+\left(1-\frac{\lambda}{2}\right)(k_{x}^{2}+k_{y}^{2})+k_{z}^{2}+M^{2}\right\\}.$
(II.5)

Finally, if $u^{\mu}$ is in the $z$ direction, I find the following
eigenvalues of $D_{E}$:

$\left\\{\frac{4\pi^{2}m^{2}}{\beta^{2}}+k_{x}^{2}+k_{y}^{2}+\left(1-{\lambda}\right)k_{z}^{2}+M^{2}\right\\}.$
(II.6)

Introducing the magnetic field modifies the eigenvalues of Eqs. (II.4, II.5,
II.6) by changing $k^{2}_{x}+k^{2}_{y}$ into $2eB(\ell+\frac{1}{2})$, where
$\ell=0,1,2,\cdots$, labels the Landau levels.

In the next sections I will construct the zeta function, evaluate it and
obtain the Casimir energy and its thermal corrections for Dirichlet and mixed
boundary conditions in the three cases of timelike Lorentz anisotropy,
spacelike anisotropy in the $x$-$y$ plane, and spacelike anisotropy in the $z$
direction.

##  III Dirichlet boundary conditions

Dirichlet boundary conditions constrain the field $\phi$ to vanish at the
plates

$\phi(x,y,0,\tau)=\phi(x,y,a,\tau)=0,$ (III.1)

so in this case

$k_{z}=\frac{n\pi}{a},$ (III.2)

with $n=1,2,3,\cdots$. In the rest of this section I will investigate a
Lorentz violating scalar field satisfying Dirichlet boundary conditions at the
plates in the presence of a uniform magnetic field ${\bf B}=B{\hat{z}}$ and,
in subsection III.1, will calculate the vacuum and thermal part of its free
energy (i.e. Casimir energy) for the case of timelike Lorentz anisotropy. The
cases of spacelike anisotropy in the $x$-$y$ plane and spacelike anisotropy in
the $z$ direction will be examined in subsections III.2 and III.3,
respectively.

###  III.1 Timelike anisotropy

I will now construct the zeta function for the case of timelike Lorentz
anisotropy. Using the eigenvalues (II.4), replacing $k^{2}_{x}+k^{2}_{y}$ with
$2eB(\ell+\frac{1}{2})$ due to the presence of the magnetic field, and using
Eq. (III.2) for $k_{z}$, I obtain

$\zeta(s)=\mu^{2s}(1+\lambda)\sum_{m=-\infty}^{\infty}\sum_{n=1}^{\infty}\frac{L^{2}eB}{2\pi}\sum_{\ell=0}^{\infty}\left[(1+\lambda)\left(\frac{2\pi
m}{\beta}\right)^{2}+(2\ell+1)eB+\left(\frac{n\pi}{a}\right)^{2}+M^{2}\right]^{-s},$
(III.3)

where, as it is done routinely when using the zeta function technique, I use
the parameter $\mu$ with dimension of mass Hawking:1976ja  to keep $\zeta(s)$
dimensionless for all values of $s$. The factor $\frac{L^{2}eB}{2\pi}$ keeps
track of the degeneracy of the Landau levels, and the extra factor of
$(1+\lambda)$ is needed in the case of timelike asymmetry as shown in Ref.
Cruz:2017kfo , where it is shown that the vacuum energy, obtained by taking
the vacuum expectation value of the Hamiltonian, carries this factor.

Using the following identities

$z^{-s}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-1}e^{-zt},$ (III.4)
$\sum_{\ell=0}^{\infty}e^{-(2\ell+1)z}=\frac{1}{2\sinh z},$ (III.5)

I write the zeta function as

$\zeta(s)={\mu^{2s}(1+\lambda)\over\Gamma(s)}\frac{L^{2}eB}{4\pi}\int_{0}^{\infty}dt\,t^{s-1}{e^{-M^{2}t}\over\sinh
eBt}\sum_{m=-\infty}^{\infty}e^{-(1+\lambda){4\pi^{2}m^{2}\over\beta^{2}}t}\sum_{n=1}^{\infty}e^{-{n^{2}\pi^{2}\over
a^{2}}t},$ (III.6)

where $\Gamma(s)$ is Euler’s gamma function. While it is not possible to
evaluate this zeta function exactly, it is possible to obtain simple analytic
expressions in several asymptotic regimes of the parameters.

I will first examine the asymptotic regime of small plate distance: $a^{-1}\gg
T,M,\sqrt{eB}$. Applying the Poisson resummation method to the $m$-sum in Eq.
(III.6), I find

$\sum_{m=-\infty}^{\infty}e^{-(1+\lambda){4\pi^{2}m^{2}\over\beta^{2}}t}={1\over
2T\sqrt{\pi(1+\lambda)t}}\sum_{m=-\infty}^{\infty}e^{-{m^{2}\over
4T^{2}(1+\lambda)t}},$ (III.7)

and can separate the zeta function into a vacuum part and a thermal part

$\zeta(s)=\zeta_{0}(s)+\zeta_{T}(s).$ (III.8)

The vacuum part of the zeta function is

$\zeta_{0}(s)={\mu^{2s}\sqrt{1+\lambda}\over\Gamma(s)}{L^{2}\over 8\pi^{3\over
2}T}\int_{0}^{\infty}dt\,t^{s-{5\over
2}}{e^{-M^{2}t}}F(eBt)\sum_{n=1}^{\infty}e^{-{n^{2}\pi^{2}\over a^{2}}t},$
(III.9)

and the thermal part is

$\zeta_{T}(s)={\mu^{2s}\sqrt{1+\lambda}\over\Gamma(s)}{L^{2}\over
4\pi^{\frac{3}{2}}T}\int_{0}^{\infty}dt\,t^{s-{5\over
2}}{e^{-M^{2}t}}F(eBt)\sum_{n=1}^{\infty}e^{-{n^{2}\pi^{2}\over
a^{2}}t}\sum_{m=1}^{\infty}e^{-{m^{2}\over 4T^{2}(1+\lambda)t}},$ (III.10)

where I use the notation I introduced in Ref. Erdas:2020ilo

$F(z)=\frac{z}{\sinh z}.$ (III.11)

The free energy $E$ is obtained easily from $\zeta(s)$

$E=-T\zeta^{\prime}(0),$ (III.12)

therefore, to obtain $E$, it is sufficient to evaluate $\zeta(s)$ when $s$ is
near zero. I calculated the vacuum part of the zeta function in Ref.
Erdas:2020ilo  and found that, in the asymptotic regime of small plate
distance and for small $s$

$\zeta_{0}(s)\simeq{\pi^{2}\sqrt{1+\lambda}L^{2}\over 8a^{3}T}\left[{1\over
90}-{M^{2}a^{2}\over 6\pi^{2}}+\left({M^{4}\over 2}-{e^{2}B^{2}\over
6}\right)\left({a\over\pi}\right)^{4}\left({1\over 2s}+\gamma_{E}+\ln{\mu
a\over 2\pi}\right)\right]s,$ (III.13)

and, in the same paper, obtained the vacuum part of the Casimir energy

$E_{0}=-{\pi^{2}\sqrt{1+\lambda}L^{2}\over 8a^{3}}\left[{1\over
90}-{M^{2}a^{2}\over 6\pi^{2}}+\left({M^{4}\over 2}-{e^{2}B^{2}\over
6}\right)\left({a\over\pi}\right)^{4}\left(\gamma_{E}+\ln{\mu a\over
2\pi}\right)\right],$ (III.14)

where $\gamma_{E}=0.5772$ is the Euler Mascheroni constant and the parameter
$\mu$ takes the value $\mu=\sqrt{eB+M^{2}}$.

The thermal part of the free energy is evaluated by making a change of
integration variable from $t$ to ${ma\over 2\pi nT\sqrt{1+\lambda}}t$. In the
asymptotic regime of small plate distance, $aT\ll 1$, only the term with
$n=m=1$ contributes significantly, and I obtain

$\zeta_{T}(s)\simeq\left(\mu^{2}a\over 2\pi
T\sqrt{1+\lambda}\right)^{s}{(1+\lambda)^{\frac{5}{4}}L^{2}\over\Gamma(s)}{\sqrt{T\over
2a^{3}}}\int_{0}^{\infty}dt\,t^{s-{5\over 2}}e^{-{M^{2}a\over 2\pi
T\sqrt{1+\lambda}}t}F\left({eBat\over 2\pi
T\sqrt{1+\lambda}}\right)e^{-{\pi\over 2aT\sqrt{1+\lambda}}(t+1/t)}.$ (III.15)

Using the saddle point method to evaluate the integral, I find

$\zeta_{T}(s)=\left(\mu^{2}a\over 2\pi
T\sqrt{1+\lambda}\right)^{s}{(1+\lambda)^{\frac{3}{2}}L^{2}\over\Gamma(s)}{T\over
a}e^{-{\pi\over aT\sqrt{1+\lambda}}}e^{-{M^{2}a\over 2\pi
T\sqrt{1+\lambda}}}F\left({eBa\over 2\pi T\sqrt{1+\lambda}}\right),$ (III.16)

and, using Eq. (III.12), I obtain the thermal correction to the Casimir energy
for small plate distance

$E_{T}=-{(1+\lambda)^{\frac{3}{2}}L^{2}T^{2}\over a}e^{-{\pi\over
aT\sqrt{1+\lambda}}}e^{-{M^{2}a\over 2\pi T\sqrt{1+\lambda}}}F\left({eBa\over
2\pi T\sqrt{1+\lambda}}\right).$ (III.17)

In the case of small plate distance, $aT\ll 1$, the vacuum part of the Casimir
energy is dominant when compared to the exponentially suppressed thermal
correction.

Next I investigate the high temperature regime, $T\gg a^{-1},\sqrt{eB},M$. I
apply the Poisson resummation method to the $n$-sum in Eq. (III.6) and obtain

$\sum_{n=1}^{\infty}e^{-{n^{2}\pi^{2}\over a^{2}}t}={a\over 2\sqrt{\pi
t}}\sum_{n=-\infty}^{\infty}e^{-{n^{2}a^{2}\over t}}-{\frac{1}{2}},$ (III.18)

where the $-\frac{1}{2}$ term can be neglected since it is independent of $a$
and, when considered in the energy calculation, it produces the energy of a
single plate Cruz:2018bqt  which is not relevant to this work. Poisson
resumming the $n$-sum allows me to separate $\zeta(s)$ into a uniform energy
density part, $\zeta_{U}(s)$, with a simple linear dependence on $a$

$\zeta_{U}(s)={\mu^{2s}(1+\lambda)\over\Gamma(s)}{L^{2}a\over 8\pi^{3\over
2}}\int_{0}^{\infty}dt\,t^{s-{5\over
2}}{e^{-M^{2}t}F(eBt)}\left(1+2\sum_{m=1}^{\infty}e^{-(1+\lambda){4\pi^{2}m^{2}\over\beta^{2}}t}\right),$
(III.19)

and a remaining part, not linear in $a$, that I indicate with $\zeta_{T,a}(s)$
and is

$\zeta_{T,a}(s)={\mu^{2s}(1+\lambda)\over\Gamma(s)}{L^{2}a\over 4\pi^{3\over
2}}\int_{0}^{\infty}dt\,t^{s-{5\over
2}}{e^{-M^{2}t}F(eBt)}\left(1+2\sum_{m=1}^{\infty}e^{-(1+\lambda){4\pi^{2}m^{2}\over\beta^{2}}t}\right)\sum_{n=1}^{\infty}e^{-{n^{2}a^{2}\over
t}}.$ (III.20)

In Appendix I, I show how to obtain $E_{U}$ from $\zeta_{U}(s)$ and find

$\displaystyle E_{U}=$ $\displaystyle-$
$\displaystyle\\!\\!\\!L^{2}a\left[{\pi^{2}(1+\lambda)^{5\over 2}\over
45}T^{4}-{(1+\lambda)^{3\over 2}\over 12}M^{2}T^{2}-{(1+\lambda)\over
2^{3\over 2}\pi}{(eB)^{3\over
2}T}\zeta_{H}\left(-{\textstyle\frac{1}{2}};{\textstyle\frac{1}{2}}+{\textstyle\frac{M^{2}}{2eB}}\right)\right.$
(III.21) $\displaystyle+$ $\displaystyle\\!\\!\\!\left.{(1+\lambda)^{1\over
2}\over 16\pi^{2}}\left({M^{4}}-{e^{2}B^{2}\over
3}\right)\left(\gamma_{E}-{\lambda\over 2}+\ln{\mu\over 4\pi
T}\right)\right],$

where, again, $\mu=\sqrt{eB+M^{2}}$ and $\zeta_{H}(s;x)$ is the Hurwitz zeta
function defined in Appendix I. Clearly the dominant term here is the Stefan
Boltzmann term, $-L^{2}a{\pi^{2}\over 45}(1+\lambda)^{5\over 2}T^{4}$, as
expected. It is interesting to explore the zero mass limit and the zero
magnetic field limit of $E_{U}$. We have
$\zeta_{H}\left(-{\textstyle\frac{1}{2}};{\textstyle\frac{1}{2}}\right)=\left({\textstyle\frac{1}{\sqrt{2}}}-1\right)\zeta_{R}\left(-{\textstyle\frac{1}{2}}\right)$,
where the Riemann zeta function takes the value
$\zeta_{R}\left(-{\textstyle\frac{1}{2}}\right)=-0.207886$, and therefore the
zero mass limit of $E_{U}$ is

$E_{U}=-L^{2}a\left[{\pi^{2}(1+\lambda)^{5\over 2}\over
45}T^{4}+{(\sqrt{2}-1)(1+\lambda)\over 4\pi}{(eB)^{3\over
2}T}\zeta_{R}\left(-{\textstyle\frac{1}{2}}\right)-{(1+\lambda)^{1\over
2}\over 48\pi^{2}}\left({eB}\right)^{2}\left(\gamma_{E}-{\lambda\over
2}+\ln{\sqrt{eB}\over 4\pi T}\right)\right],$ (III.22)

where the leading magnetic correction term is, overall, positive and produces
attraction of the plates while the dominant Stefan Boltzmann term produces
repulsion. Notice also that the magnetic correction term listed first is
dominant when compared to the next. The zero magnetic field limit of $E_{U}$
is most easily obtained by setting $F(eBt)=1$ in Eq. (VII.1) of Appendix I.
After some straightforward manipulations, I obtain

$E_{U}=-L^{2}a\left[{\pi^{2}(1+\lambda)^{5\over 2}\over
45}T^{4}-{(1+\lambda)^{3\over 2}\over 12}M^{2}T^{2}+{(1+\lambda)\over
6\pi}M^{3}T+{(1+\lambda)^{1\over 2}\over
16\pi^{2}}{M^{4}}\left(\gamma_{E}-{\lambda\over 2}+\ln{M\over 4\pi
T}\right)\right],$ (III.23)

where the leading order mass correction produces plates attraction.

In Appendix I, I show the details of how I obtain $E_{T,a}$ from
$\zeta_{T,a}$, and find

$E_{T,a}=E_{a}-(1+\lambda)^{\frac{3}{2}}{L^{2}T^{2}\over a}e^{-{4\pi
aT\sqrt{1+\lambda}}}e^{-{M^{2}a\over 2\pi T\sqrt{1+\lambda}}}F\left({eBa\over
2\pi T\sqrt{1+\lambda}}\right),$ (III.24)

where $E_{a}$ is given by Eq. (VII.21) of Appendix I when
$a^{-1}\gg\sqrt{eB},M$, by Eq. (VII.23) when $\sqrt{eB}\gg a^{-1},M$, and by
Eq. (VII.25) when $M\gg a^{-1},\sqrt{eB}$. As expected, the Casimir energy in
the high temperature regime, $E=E_{U}+E_{T,a}$, does not contain vacuum terms.
Its leading order term is the Stefan Boltzmann term contained in the uniform
energy density piece $E_{U}$. Notice also that the effects of mass and
magnetic field are to produce plates attraction that reduce the Stefan
Boltzmann repulsion, as I will show in detail in Sec. V.

The asymptotic limits of strong magnetic field, $\sqrt{eB}\gg a^{-1},T,M$, and
large mass, $M\gg a^{-1},T,\sqrt{eB}$ are handled in a similar manner. I do a
Poisson resummation on both the $m$-sum and the $n$-sum in Eq. (III.6), write
the part dependent on $B$ as an infinite sum using Eq. (III.5), and obtain

$\zeta(s)={\mu^{2s}\sqrt{1+\lambda}\over\Gamma(s)}{L^{2}a\over
8\pi^{2}T}\int_{0}^{\infty}dt\,t^{s-2}{e^{-M^{2}t}}eB\sum_{l=0}^{\infty}e^{-(2l+1)eBt}\sum_{m=-\infty}^{\infty}e^{-{m^{2}\over
4T^{2}(1+\lambda)t}}\sum_{n=-\infty}^{\infty}e^{-{n^{2}a^{2}\over t}}.$
(III.25)

The term with $m=n=0$ is

$\zeta_{W}(s)={\mu^{2s}\sqrt{1+\lambda}\over\Gamma(s)}{L^{2}a\over
8\pi^{2}T}\int_{0}^{\infty}dt\,t^{s-2}{e^{-M^{2}t}}eB\sum_{l=0}^{\infty}e^{-(2l+1)eBt},$
(III.26)

the zeta function of the one-loop Weisskopf effective action for scalar QED,
multiplied by the factor of $\sqrt{1+\lambda}$ that reflects the timelike
asymmetry. Its contribution to the Casimir energy,
$E_{W}=-T\zeta_{W}^{\prime}(0)$, is a uniform energy density term independent
of $T$. It is obtained using the same method I use in Appendix I to evaluate
$\zeta_{U,0}(s)$, and yields the well known result

$E_{W}={\sqrt{1+\lambda}L^{2}a\over 32\pi^{2}}e^{2}B^{2}\left[\left({1\over
3}-{M^{4}\over e^{2}B^{2}}\right)\left(1+\ln{{M^{2}\over
2eB}}\right)+8\zeta_{H}^{\prime}(-1;{\textstyle\frac{1}{2}}+{\textstyle\frac{M^{2}}{2eB}})\right]$
(III.27)

where $\zeta_{H}(s;z)$ is the Hurwitz zeta function defined in Appendix I, its
derivative is

$\zeta_{H}^{\prime}(s;z)={\partial\zeta_{H}(s;z)\over\partial s},$ (III.28)

and I used

$\zeta_{H}(-1;{\textstyle\frac{1}{2}}+{\textstyle\frac{z}{2}})={1\over
24}-{z^{2}\over 8}.$ (III.29)

Notice that I made the obvious choice $\mu=M$, indicating on-shell
renormalization. I evaluate the rest of the zeta function by changing
integration variable from $t$ to ${\sqrt{n^{2}a^{2}+{m^{2}\beta^{2}\over
4(1+\lambda)}}\over\sqrt{(2\ell+1)eB+M^{2}}}t$. Only terms with $n=\pm
1,m=0,\ell=0$ and $n=0,m=\pm 1,\ell=0$ contribute significantly and I
integrate them using the saddle point method. The Casimir energy in the strong
magnetic field limit is then obtained using, once again,
$E=-T\zeta^{\prime}(0)$

$E=E_{W}-{\sqrt{1+\lambda}L^{2}a\over
4\pi^{3/2}}eB\left(eB+M^{2}\right)^{\textstyle\frac{1}{4}}\left[{e^{-2a\sqrt{eB+M^{2}}}\over
a^{3/2}}+{\left(2\sqrt{1+\lambda}T\right)^{\textstyle\frac{3}{2}}}e^{-{\sqrt{eB+M^{2}}\over
T\sqrt{1+\lambda}}}\right],$ (III.30)

where $E_{W}$ in the strong magnetic field limit is

$E_{W}={\sqrt{1+\lambda}L^{2}a\over 32\pi^{2}}e^{2}B^{2}\left[{1\over
3}\left(1-\ln{4eB\over M^{2}}\right)-4\zeta_{R}^{\prime}(-1)-{M^{2}\over
eB}{\ln(4})-{M^{4}\over e^{2}B^{2}}\left(\gamma_{E}-\ln{eB\over
2M^{2}}\right)\right],$ (III.31)

obtained using the following identity and approximation Whittaker:1963xxy

$\zeta_{H}^{\prime}(-1;{\textstyle\frac{1}{2}}+z)=-{1\over
2}\zeta_{R}^{\prime}(-1)-{\ln 2\over 24}-{z\over 2}\ln(2\pi)+{z^{2}\over
2}+\int_{0}^{z}\ln\Gamma(x+{\textstyle\frac{1}{2}})dx,$ (III.32)
$\ln\Gamma(x+{\textstyle\frac{1}{2}})\simeq{\ln\pi\over
2}-\left(\gamma_{E}+2\ln 2\right)x+{\cal O}(x^{2}).$ (III.33)

The Casimir energy in the strong magnetic field approximation contains three
parts: a uniform energy density, $E_{W}$, an exponentially suppressed vacuum
part that I had already obtained in Ref. Erdas:2020ilo , and the exponentially
suppressed thermal correction. The uniform energy density will only contribute
to the Casimir energy if the magnetic field is present between the plates and
absent outside the plates. This term should not be considered if a uniform
magnetic field is present in between and outside the plates.

Last I explore the large mass limit, $M\gg a^{-1},T,\sqrt{eB}$. After a
Poisson resummation of the $m$\- and $n$-sums in Eq. (III.6) I can, again,
separate the term with $n=m=0$ from the rest, and that term is $\zeta_{W}(s)$
of Eq. (III.26). I evaluate the remaining part of that sum by changing
variable of integration from $t$ to ${t\over M}\sqrt{n^{2}a^{2}+{m^{2}\over
4T^{2}(1+\lambda)}}$. Only the terms with $m=\pm 1,n=0$ and $m=0,n=\pm 1$
contribute significantly to the sum, and I integrate them using the saddle
point method. I then use $E=-T\zeta^{\prime}(0)$ and find the Casimir energy
in the large mass limit

$E=E_{W}-{\sqrt{1+\lambda}L^{2}\over
8}\left({M\over\pi}\right)^{\textstyle\frac{3}{2}}\left[{e^{-2Ma}\over
a^{3/2}}F\left({eBa\over
M}\right)+a{\left(2\sqrt{1+\lambda}T\right)^{\textstyle\frac{5}{2}}}e^{-{M\over
T\sqrt{1+\lambda}}}F\left({eB\over 2MT\sqrt{1+\lambda}}\right)\right],$
(III.34)

where

$E_{W}=-{7\sqrt{1+\lambda}L^{2}a\over 5,760\pi^{2}}{e^{4}B^{4}\over M^{4}},$
(III.35)

is the large mass limit of $E_{W}$ introduced in Eq. (III.27). Notice that I
subtracted from $E_{W}$ a constant term proportional to $M^{4}$ that will not
contribute to the Casimir energy. The large mass limit of the Casimir energy
contains a uniform energy density part, $E_{W}$, an exponentially suppressed
vacuum part in agreement with the one already obtained in my previous paper
Erdas:2020ilo , and an exponentially suppressed thermal correction.

###  III.2 Spacelike anisotropy in the $x$-$y$ plane

Using the eigenvalues (II.5) and proceeding as I did when deriving Eq.
(III.3), I find the zeta function for Lorentz anisotropy in the $x$-$y$ plane

$\zeta(s)=\mu^{2s}\sum_{m=-\infty}^{\infty}\sum_{n=1}^{\infty}{L^{2}eB\over
2\pi}\sum_{\ell=0}^{\infty}\left[\left({2\pi
m\over\beta}\right)^{2}+\left(1-{\lambda\over
2}\right)(2\ell+1)eB+\left({n\pi\over a}\right)^{2}+M^{2}\right]^{-s}.$
(III.36)

A comparison of the zeta function for timelike anisotropy of Eq. (III.3) to
that for spacelike anisotropy in the $x$-$y$ plane of Eq. (III.36), shows that
(III.36) is obtained by setting $\lambda=0$ in (III.3), then replacing $eB$
with $\left(1-{\lambda\over 2}\right)eB$ and finally dividing by
$\left(1-{\lambda\over 2}\right)$. The same replacements can be applied to the
vacuum and thermal parts of the Casimir energy obtained in the previous
subsection and will produce those quantities in the case of spacelike
anisotropy in the $x$-$y$ plane. Notice that, for $\lambda\ll 1$,
$\left(1-{\lambda\over 2}\right)\simeq\sqrt{1-\lambda}$ and therefore, from
now on, $\left(1-{\lambda\over 2}\right)$ will be replaced with
$\sqrt{1-\lambda}$.

Making the replacements outlined above in Eqs. (III.14) and (III.17), I find
the vacuum Casimir energy and its thermal correction in the limit of small
plate distance

$E_{0}=-{\pi^{2}L^{2}\over 8a^{3}\sqrt{1-\lambda}}\left[{1\over
90}-{M^{2}a^{2}\over 6\pi^{2}}+\left({M^{4}\over
2}-{(1-\lambda)e^{2}B^{2}\over
6}\right)\left({a\over\pi}\right)^{4}\left(\gamma_{E}+\ln{\mu a\over
2\pi}\right)\right],$ (III.37) $E_{T}=-{L^{2}T^{2}\over
a\sqrt{1-\lambda}}e^{-{\pi\over aT}}e^{-{M^{2}a\over 2\pi
T}}F\left({\sqrt{1-\lambda}eBa\over 2\pi T}\right),$ (III.38)

where $\mu$ is defined below Eq. (III.14). I already obtained Eq. (III.37) in
Ref. Erdas:2020ilo , while Eq. (III.38) is a new result. Notice that in the
absence of a magnetic field the vacuum energy for timelike anisotropy and the
vacuum energy for spacelike anisotropy in the $x$-$y$ plane are identical
since, for $\lambda\ll 1$, $\sqrt{1+\lambda}\simeq 1/\sqrt{1-\lambda}$, making
this study that includes magnetic effects more compelling.

Moving to the case of high temperature, I make the replacements indicated
above to Eqs. (III.21) and (III.24) and find the uniform energy density part
and other part of the Casimir energy in the high temperature limit

$E_{U}=-{L^{2}a\over\sqrt{1-\lambda}}\left[{\pi^{2}T^{4}\over
45}-{M^{2}T^{2}\over 12}-{(1-\lambda)^{3\over 4}\over 2^{3\over
2}\pi}{(eB)^{3\over
2}T}\zeta_{H}\left(-{\textstyle\frac{1}{2}};{\textstyle\frac{1}{2}}+{\textstyle\frac{M^{2}}{2\sqrt{1-\lambda}eB}}\right)+{{M^{4}}-(1-\lambda){e^{2}B^{2}\over
3}\over 16\pi^{2}}\left(\gamma_{E}+\ln{\mu\over 4\pi T}\right)\right],$
(III.39) $E_{T,a}=E_{a}-{L^{2}T^{2}\over\sqrt{1-\lambda}a}e^{-{4\pi
aT}}e^{-{M^{2}a\over 2\pi T}}F\left({\sqrt{1-\lambda}eBa\over 2\pi T}\right),$
(III.40)

where

$E_{a}=-{L^{2}T\over
8\pi\sqrt{1-\lambda}a^{2}}\left[\zeta_{R}(3)+2M^{2}a^{2}\ln(2Ma)+\left({M^{4}\over
6}-{(1-\lambda)e^{2}B^{2}\over 18}\right)a^{4}\right]$ (III.41)

when $a^{-1}\gg\sqrt{eB},M$,

$E_{a}=-{L^{2}TeB\over 2\pi}e^{-2a\sqrt{M^{2}+\sqrt{1-\lambda}eB}}$ (III.42)

when $\sqrt{eB}\gg a^{-1},M$, and

$E_{a}=-{L^{2}TM\over
4\pi\sqrt{1-\lambda}a}e^{-2Ma}F\left({\sqrt{1-\lambda}eBa\over M}\right)$
(III.43)

when $M\gg a^{-1},\sqrt{eB}$.

The strong magnetic field limit is obtained by making the aforementioned
replacements in Eqs. (III.30) and (III.31)

$E=E_{W}-{L^{2}a\over
4\pi^{3/2}}eB\left(\sqrt{1-\lambda}eB+M^{2}\right)^{\textstyle\frac{1}{4}}\left[{e^{-2a\sqrt{\sqrt{1-\lambda}eB+M^{2}}}\over
a^{3/2}}+{\left(2T\right)^{\textstyle\frac{3}{2}}}e^{-{\sqrt{\sqrt{1-\lambda}eB+M^{2}}\over
T}}\right],$ (III.44)

where $E_{W}$ in the strong magnetic field limit is

$E_{W}={L^{2}a\over 32\pi^{2}}\sqrt{1-\lambda}e^{2}B^{2}\left[{1\over
3}\left(1+{\lambda\over
2}-\ln{{4eB}\over{M^{2}}}\right)-4\zeta_{R}^{\prime}(-1)-{M^{2}\over\sqrt{1-\lambda}eB}{\ln(4})-{M^{4}\over(1-\lambda)e^{2}B^{2}}\left(\gamma_{E}+{\lambda\over
2}-\ln{eB\over 2M^{2}}\right)\right].$ (III.45)

The large mass limit is obtained in the same manner, using Eqs. (III.34) and
(III.35)

$E=E_{W}-{L^{2}\over
8\sqrt{1-\lambda}}\left({M\over\pi}\right)^{\textstyle\frac{3}{2}}\left[{e^{-2aM}\over
a^{3/2}}F\left({\sqrt{1-\lambda}eBa\over
M}\right)+a{\left(2T\right)^{\textstyle\frac{5}{2}}}e^{-{M\over
T}}F\left({\sqrt{1-\lambda}eB\over 2MT}\right)\right],$ (III.46)

with

$E_{W}=-{7L^{2}a\over 5,760\pi^{2}}{(1-\lambda)^{\frac{3}{2}}e^{4}B^{4}\over
M^{4}}.$ (III.47)

###  III.3 Spacelike anisotropy in the $z$ direction

I obtain the zeta function in the case of a spacelike anisotropy in the $z$
direction using the eigenvalues (II.6)

$\zeta(s)=\mu^{2s}\sum_{m=-\infty}^{\infty}\sum_{n=1}^{\infty}{L^{2}eB\over
2\pi}\sum_{\ell=0}^{\infty}\left[\left({2\pi
m\over\beta}\right)^{2}+(2\ell+1)eB+\left(1-{\lambda}\right)\left({n\pi\over
a}\right)^{2}+M^{2}\right]^{-s}.$ (III.48)

When comparing the zeta function for timelike anisotropy of Eq. (III.3) to
that for spacelike anisotropy in the $z$ direction of Eq. (III.48), it is
evident that (III.48) is obtained by setting $\lambda=0$ in (III.3), then
replacing $a$ with $a\over\sqrt{1-\lambda}$. The same replacements, when
applied to the vacuum and thermal parts of the Casimir energy obtained in
subsection III.1, yield those quantities in the case of spacelike anisotropy
in the $z$ direction.

Making those replacements in Eqs. (III.14) and (III.17), I obtain the vacuum
Casimir energy and its thermal correction in the limit of small plate distance

$E_{0}=-{\pi^{2}L^{2}(1-\lambda)^{3\over 2}\over 8a^{3}}\left[{1\over
90}-{M^{2}a^{2}\over 6\pi^{2}(1-\lambda)}+\left({M^{4}\over
2}-{e^{2}B^{2}\over
6}\right)\left({a^{4}\over\pi^{4}(1-\lambda)^{2}}\right)\left(\gamma_{E}+\ln{\mu
a\over 2\pi}+{\lambda\over 2}\right)\right],$ (III.49)
$E_{T}=-{L^{2}T^{2}\sqrt{1-\lambda}\over a}e^{-{\pi\sqrt{1-\lambda}\over
aT}}e^{-{M^{2}a\over 2\pi T\sqrt{1-\lambda}}}F\left({eBa\over
2\pi\sqrt{1-\lambda}T}\right),$ (III.50)

with $\mu$ defined below Eq. (III.14). I already obtained Eq. (III.49) in Ref.
Erdas:2020ilo , while its thermal correction of Eq. (III.50) is a new result.

In the limit of high temperature, I find

$E_{U}=-{L^{2}a\over\sqrt{1-\lambda}}\left[{\pi^{2}T^{4}\over
45}-{M^{2}T^{2}\over 12}-{1\over 2^{3\over 2}\pi}{(eB)^{3\over
2}T}\zeta_{H}\left(-{\textstyle\frac{1}{2}};{\textstyle\frac{1}{2}}+{\textstyle\frac{M^{2}}{2eB}}\right)+{{M^{4}}-{e^{2}B^{2}\over
3}\over 16\pi^{2}}\left(\gamma_{E}+\ln{\mu\over 4\pi T}\right)\right],$
(III.51) $E_{T,a}=E_{a}-{L^{2}\sqrt{1-\lambda}T^{2}\over a}e^{-{4\pi
aT\over\sqrt{1-\lambda}}}e^{-{M^{2}a\over
2\pi\sqrt{1-\lambda}T}}F\left({eBa\over 2\pi\sqrt{1-\lambda}T}\right),$
(III.52)

where

$E_{a}=-{L^{2}(1-\lambda)T\over 8\pi
a^{2}}\left[\zeta_{R}(3)+{2M^{2}a^{2}\over(1-\lambda)}\ln\left({2Ma\over\sqrt{1-\lambda}}\right)+\left({M^{4}\over
6}-{e^{2}B^{2}\over 18}\right){a^{4}\over(1-\lambda)^{2}}\right]$ (III.53)

when $a^{-1}\gg\sqrt{eB},M$,

$E_{a}=-{L^{2}TeB\over 2\pi}e^{-2a{\sqrt{eB+M^{2}\over 1-\lambda}}}$ (III.54)

when $\sqrt{eB}\gg,a^{-1}M$, and

$E_{a}=-{L^{2}\sqrt{1-\lambda}TM\over 4\pi
a}e^{-2aM\over\sqrt{1-\lambda}}F\left({eBa\over\sqrt{1-\lambda}M}\right)$
(III.55)

when $M\gg,a^{-1}\sqrt{eB}$.

I find the strong magnetic field limit of the Casimir energy in the case of
anisotropy in the $z$ direction by making the replacements in Eqs. (III.30)
and (III.31)

$E=E_{W}-{L^{2}a\over
4\pi^{3/2}\sqrt{1-\lambda}}eB\left(eB+M^{2}\right)^{\textstyle\frac{1}{4}}\left[\left({\sqrt{1-\lambda}\over
a}\right)^{\textstyle\frac{3}{2}}{e^{-2a{\sqrt{eB+M^{2}\over
1-\lambda}}}}+{\left(2T\right)^{\textstyle\frac{3}{2}}}e^{-{\sqrt{eB+M^{2}}\over
T}}\right],$ (III.56)

where $E_{W}$ is

$E_{W}={L^{2}a\over 32\pi^{2}\sqrt{1-\lambda}}e^{2}B^{2}\left[{1\over
3}\left(1-\ln{4eB\over M^{2}}\right)-4\zeta_{R}^{\prime}(-1)-{M^{2}\over
eB}{\ln(4})-\left(\gamma_{E}-{M^{4}\over e^{2}B^{2}}\ln{eB\over
2M^{2}}\right)\right].$ (III.57)

I obtain the large mass limit in the same manner using Eqs. (III.34) and
(III.35)

$E=E_{W}-{L^{2}\over
8}\left({M\over\pi}\right)^{\textstyle\frac{3}{2}}\left[\left({\sqrt{1-\lambda}\over
a}\right)^{\textstyle\frac{3}{2}}{e^{-{2Ma\over\sqrt{1-\lambda}}}}F\left({eBa\over
M\sqrt{1-\lambda}}\right)+{a\over\sqrt{1-\lambda}}{\left(2T\right)^{\textstyle\frac{5}{2}}}e^{-{M\over
T}}F\left({eB\over 2MT}\right)\right],$ (III.58)

with

$E_{W}=-{7L^{2}a\over 5,760\pi^{2}\sqrt{1-\lambda}}{e^{4}B^{4}\over M^{4}},$
(III.59)

where the vacuum part was obtained in my previous work, Ref. Erdas:2020ilo ,
and its thermal correction is another new result.

##  IV Mixed boundary conditions

Mixed boundary conditions constrain the scalar field to vanish at the plate
located at $z=0$

$\phi(x,y,0,\tau)=0,$ (IV.1)

and its normal derivative to vanish at the plate located at $z=a$ ,

${\partial\phi\over\partial z}(x,y,a,\tau)=0,$ (IV.2)

yielding

$k_{z}=\left(n+\textstyle{\frac{1}{2}}\right){\pi\over a},$ (IV.3)

with $n=0,1,2,\cdots$.

The zeta function for timelike asymmetry and mixed boundary conditions is
constructed with the same method used to obtain the one for timelike asymmetry
and Dirichlet boundary conditions of Eq. (III.3), the only difference being
that $k_{z}$ is given by Eq. (IV.3)

$\zeta(s)=\mu^{2s}(1+\lambda)\sum_{m=-\infty}^{\infty}\sum_{n=0}^{\infty}{L^{2}eB\over
2\pi}\sum_{\ell=0}^{\infty}\left[(1+\lambda)\left({2\pi
m\over\beta}\right)^{2}+(2\ell+1)eB+\left(n+{1\over
2}\right)^{2}\left({\pi\over a}\right)^{2}+M^{2}\right]^{-s}.$ (IV.4)

The zeta function for mixed boundary conditions and spacelike asymmetry in the
$x$-$y$ plane is quickly obtained from Eq. (III.36) by replacing $n$ with
$n+{1\over 2}$. The same replacement made in Eq. (III.48) produces the zeta
function for mixed boundary conditions and spacelike asymmetry in the $z$
direction.

Using identities (III.4) and (III.5) I write (IV.4) as

$\zeta(s)={\mu^{2s}(1+\lambda)\over\Gamma(s)}{L^{2}eB\over
4\pi}\int_{0}^{\infty}dt\,t^{s-1}{e^{-M^{2}t}\over\sinh
eBt}\sum_{m=-\infty}^{\infty}e^{-(1+\lambda){4\pi^{2}m^{2}\over\beta^{2}}t}\sum_{n=0}^{\infty}e^{-\left(n+{1\over
2}\right)^{2}{\pi^{2}\over a^{2}}t}.$ (IV.5)

To investigate the asymptotic case of small plate distance and timelike
asymmetry, I apply the Poisson resummation method to the $m$-sum, as I did in
Eq. (III.7), and separate the zeta function (IV.5) into a vacuum part and a
thermal part. The vacuum part of the zeta function yields the vacuum Casimir
energy I already obtained in Ref. Erdas:2020ilo

$E_{0}={\pi^{2}\sqrt{1+\lambda}L^{2}\over 8a^{3}}\left[{7\over
720}-{M^{2}a^{2}\over 12\pi^{2}}-\left({M^{4}\over 2}-{e^{2}B^{2}\over
6}\right)\left({a\over\pi}\right)^{4}\left(\gamma_{E}+\ln{2\mu
a\over\pi}\right)\right].$ (IV.6)

The thermal part is handled by making a change of integration variable from
$t$ to ${ma\over 2\pi\left(n+{1\over 2}\right)T\sqrt{1+\lambda}}t$. In the
limit of small plate distance only the terms with $n=0$ and $m=\pm 1$
contribute significantly, I evaluate the remaining integral using the saddle
point method and obtain the following thermal part of the Casimir energy for
timelike asymmetry and mixed boundary conditions in the small plate distance
limit

$E_{T}=-{(1+\lambda)^{3/2}L^{2}T^{2}\over 2a}e^{-{\pi\over
2aT\sqrt{1+\lambda}}}e^{-{M^{2}a\over\pi
T\sqrt{1+\lambda}}}F\left({eBa\over\pi T\sqrt{1+\lambda}}\right).$ (IV.7)

Notice that, in the small plate distance limit, the vacuum Casimir energy is
negative for Dirichlet boundary conditions and positive for mixed boundary
conditions, while the thermal correction is the same for both Dirichlet and
mixed boundary conditions. As I will show in detail in Sec. V, thermal effects
decrease the plates attraction in the case of Dirichlet boundary conditions
and increase the plates repulsion in the case of mixed boundary conditions.

The vacuum and thermal parts of the Casimir energy for spacelike asymmetry in
the $x$-$y$ plane in the small plate distance limit are obtained from Eqs.
(IV.6) and (IV.7) by setting $\lambda=0$ in them, then replacing $eB$ with
$\left(1-{\lambda\over 2}\right)eB$ and finally dividing by
$\left(1-{\lambda\over 2}\right)$. For the sake of brevity I will not report
those two quantities. Notice that in the absence of a magnetic field the
vacuum energy for timelike anisotropy and the vacuum energy for spacelike
anisotropy in the $x$-$y$ plane are indistinguishable, as it was the case when
Dirichlet boundary conditions were investigated.

$E_{0}$ and $E_{T}$ for spacelike asymmetry in the $z$ direction in the small
plate distance limit are obtained from Eqs. (IV.6) and (IV.7) by setting
$\lambda=0$ in them, then replacing $a$ with $a\over\sqrt{1-\lambda}$.

I obtain the high temperature limit of the Casimir energy for mixed boundary
conditions and timelike anisotropy by doing a Poisson resummation of the
$n$-sum in Eq. (IV.5)

$\sum_{n=0}^{\infty}e^{-{(n+{\frac{1}{2}})^{2}\pi^{2}\over a^{2}}t}={a\over
2\sqrt{\pi t}}\sum_{n=-\infty}^{\infty}(-1)^{n}e^{-{n^{2}a^{2}\over t}},$
(IV.8)

and separate the zeta function into a uniform density part, $\zeta_{U}(s)$,
and a remaining part, $\zeta_{T,a}(s)$, as I did when investigating Dirichlet
boundary conditions. The uniform energy density piece of the Casimir energy,
$E_{U}$, is the same as in the case of Dirichlet boundary conditions, Eq.
(III.21). The other part of the Casimir energy is

$E_{T,a}=E_{a}+(1+\lambda)^{\frac{3}{2}}{L^{2}T^{2}\over a}e^{-{4\pi
aT\sqrt{1+\lambda}}}e^{-{M^{2}a\over 2\pi T\sqrt{1+\lambda}}}F\left({eBa\over
2\pi T\sqrt{1+\lambda}}\right),$ (IV.9)

with

$E_{a}={(1+\lambda)}{L^{2}T\over 4\pi a^{2}}\left[{3\over
8}\zeta_{R}(3)-\ln(2)M^{2}a^{2}-\left({M^{4}\over 4}-{e^{2}B^{2}\over
12}\right)a^{4}\right],$ (IV.10)

for $a^{-1}\gg\sqrt{eB},M$,

$E_{a}={(1+\lambda)L^{2}TeB\over 2\pi}e^{-2a\sqrt{M^{2}+eB}}$ (IV.11)

for $\sqrt{eB}\gg a^{-1},M$, and

$E_{a}={(1+\lambda)L^{2}TM\over 4\pi a}e^{-2aM}F\left({eBa\over M}\right)$
(IV.12)

for $M\gg a^{-1},\sqrt{eB}$. The details of this calculation are shown in
Appendix II. Notice that the dominant part of the Casimir energy is the
uniform energy density piece, with the Stefan Boltzmann term, as it was the
case when Dirichlet boundary conditions were considered. The high temperature
Casimir energy for mixed boundary conditions and spacelike asymmetry in the
$x$-$y$ plane is obtained from Eqs. (III.21), (IV.9), (IV.10) and (IV.11) by
setting $\lambda=0$, dividing by $\sqrt{1-\lambda}$ and replacing $eB$ with
$\sqrt{1-\lambda}eB$, as described previously. The Casimir energy for the case
of spacelike asymmetry in the $z$ direction is obtained from the four
equations cited above by setting $\lambda=0$ and then replacing $a$ with
$a\over\sqrt{1-\lambda}$.

I obtain the mixed boundary conditions Casimir energy for strong magnetic
field, in the case of timelike Lorentz asymmetry, by doing a Poisson
resummation of the $m$\- and $n$-series in Eq. (IV.5) and by writing the
$B$-dependent part as an infinite sum, to find

$\zeta(s)={\mu^{2s}\sqrt{1+\lambda}\over\Gamma(s)}{L^{2}a\over
8\pi^{2}T}\int_{0}^{\infty}dt\,t^{s-2}{e^{-M^{2}t}}eB\sum_{l=0}^{\infty}e^{-(2l+1)eBt}\sum_{m=-\infty}^{\infty}e^{-{m^{2}\over
4T^{2}(1+\lambda)t}}\sum_{n=-\infty}^{\infty}(-1)^{n}e^{-{n^{2}a^{2}\over
t}}.$ (IV.13)

At this point I separate the term with $m=n=0$ that yields the Weisskopf zeta
function, $\zeta_{W}(s)$, already examined in Sec. III, which contributes
$E_{W}$ of Eq. (III.31) to the Casimir energy. The remaining terms in the sum
are handled by a change of integration variable from $t$ to
${\sqrt{n^{2}a^{2}+{m^{2}\beta^{2}\over
4(1+\lambda)}}\over\sqrt{(2\ell+1)eB+M^{2}}}t$, as I did in Sec. III for
Dirichlet boundary conditions, that produces the same outcome as in Sec. III,
leaving only terms with $n=\pm 1,m=0,\ell=0$ and $n=0,m=\pm 1,\ell=0$ to
contribute significantly to the sum. The Casimir energy I obtain is

$E=E_{W}+{\sqrt{1+\lambda}L^{2}a\over
4\pi^{3/2}}eB\left(eB+M^{2}\right)^{\textstyle\frac{1}{4}}\left[{e^{-2a\sqrt{eB+M^{2}}}\over
a^{3/2}}-{\left(2\sqrt{1+\lambda}T\right)^{\textstyle\frac{3}{2}}}e^{-{\sqrt{eB+M^{2}}\over
T\sqrt{1+\lambda}}}\right],$ (IV.14)

where the vacuum part has opposite sign to that obtained for Dirichlet
boundary conditions, agreeing with what I found in Ref. Erdas:2020ilo . The
thermal correction to the Casimir energy in the strong magnetic field limit is
the same for mixed and Dirichlet boundary conditions. The strong magnetic
field Casimir energy for mixed boundary conditions and Lorentz asymmetry in
the $x$-$y$ plane is the same as that for Dirichlet boundary conditions with a
factor of $-1$ that will multiply the vacuum part. The one for Lorentz
asymmetry in the $z$ direction is also the same as that for Dirichlet boundary
with the vacuum part having opposite sign.

The last asymptotic case I examine is the one of large mass for mixed boundary
conditions. Starting with timelike Lorentz asymmetry, I proceed as I did above
by doing a Poisson resummation of the $m$\- and $n$-series in Eq. (IV.5). The
term with $m=n=0$ is the Weisskopf zeta function which, in the large mass
limit, contributes $E_{W}$ of Eq. (III.35) to the Casimir energy. I evaluate
the rest of the sum by doing a change of integration variable from $t$ to
${t\over M}{\sqrt{n^{2}a^{2}+{m^{2}\beta^{2}\over 4(1+\lambda)}}}$, retain
only the significant terms with $n=\pm 1,m=0$ and $n=0,m=\pm 1$ and obtain the
following for the Casimir energy

$E=E_{W}+{\sqrt{1+\lambda}L^{2}\over
8}\left({M\over\pi}\right)^{\textstyle\frac{3}{2}}\left[{e^{-2aM}\over
a^{3/2}}F\left({eBa\over
M}\right)-a{\left(2\sqrt{1+\lambda}T\right)^{\textstyle\frac{5}{2}}}e^{-{M\over
T\sqrt{1+\lambda}}}F\left({eB\over 2MT\sqrt{1+\lambda}}\right)\right],$
(IV.15)

where, again, the vacuum part is equal and opposite to the vacuum part
obtained for Dirichlet boundary conditions, while the thermal correction and
$E_{W}$ are the same as in the case of Dirichlet boundary conditions. The
vacuum part agrees with my result of Ref. Erdas:2020ilo . The large mass
Casimir energy for mixed boundary conditions and Lorentz asymmetry in the
$x$-$y$ plane has the same thermal piece and $E_{W}$ and opposite vacuum part
as that I obtained for Dirichlet boundary conditions, and the same is true for
the case of Lorentz asymmetry in the $z$-direction.

##  V Casimir pressure

In this section I will examine the Casimir pressure for all asymptotic cases
and timelike anisotropy, with either Dirichlet or mixed boundary conditions.
Once the pressure for a certain case of timelike anisotropy is known, the
pressure for the two corresponding cases of spacelike anisotropy is obtained
immediately with the simple substitutions outlined in the previous sections.

I obtain the Casimir pressure, $P$, by taking a derivative of the Casimir
energy

$P=-{1\over L^{2}}{\partial E\over\partial a}.$ (V.1)

I first obtain the Casimir pressure for timelike Lorentz asymmetry and
Dirichlet boundary in the asymptotic case of small plate distance,

$P=-{\pi^{2}\sqrt{1+\lambda}\over 8a^{4}}\left[{1\over 30}-{M^{2}a^{2}\over
6\pi^{2}}+{e^{2}B^{2}\over
6}\left({a\over\pi}\right)^{4}\left(1+\gamma_{E}+\ln{\mu a\over
2\pi}\right)\right]+{(1+\lambda)\pi T\over a^{3}}e^{-{\pi\over
aT\sqrt{1+\lambda}}}e^{-{M^{2}a\over 2\pi T\sqrt{1+\lambda}}}F\left({eBa\over
2\pi T\sqrt{1+\lambda}}\right),$ (V.2)

where smaller terms have been neglected. Notice how the thermal component of
the pressure is repulsive and exponentially suppressed, but will have a
significant contribution when $0.5\geq aT\geq 0.1$, and how the vacuum
magnetic term gives almost a constant contribution, aside from a weak
logarithmic dependence on $a$. The Casimir pressure for the same asymptotic
case but mixed boundary conditions is

$P={\pi^{2}\sqrt{1+\lambda}\over 8a^{4}}\left[{7\over 240}-{M^{2}a^{2}\over
12\pi^{2}}-{e^{2}B^{2}\over
6}\left({a\over\pi}\right)^{4}\left(1+\gamma_{E}+\ln{2\mu
a\over\pi}\right)\right]+{(1+\lambda)\pi T\over a^{3}}e^{-{\pi\over
aT\sqrt{1+\lambda}}}e^{-{M^{2}a\over 2\pi T\sqrt{1+\lambda}}}F\left({eBa\over
2\pi T\sqrt{1+\lambda}}\right),$ (V.3)

with its vacuum part displaying the well known repulsive nature and with the
same thermal piece as in the case of Dirichlet boundary conditions. It is
interesting how the vacuum magnetic term is almost identical, attractive and
practically constant in both cases.

The high temperature limit of the Casimir pressure, in the presence of a
timelike anisotropy and Dirichlet boundary conditions, is given by

$\displaystyle P$ $\displaystyle=$ $\displaystyle\\!\\!\\!\
{\pi^{2}(1+\lambda)^{5\over 2}\over 45}T^{4}-{(1+\lambda)^{3\over 2}\over
12}M^{2}T^{2}+P_{a}-{(1+\lambda)\over 2^{3\over 2}\pi}{(eB)^{3\over
2}T}\zeta_{H}\left(-{\textstyle\frac{1}{2}};{\textstyle\frac{1}{2}}+{\textstyle\frac{M^{2}}{2eB}}\right)$
(V.4) $\displaystyle-$ $\displaystyle\\!\\!\\!\ {4\pi(1+\lambda)^{2}T^{3}\over
a}e^{-4{\pi aT\sqrt{1+\lambda}}}e^{-{M^{2}a\over 2\pi
T\sqrt{1+\lambda}}}F\left({eBa\over 2\pi T\sqrt{1+\lambda}}\right),$

where I neglected several lower order terms and use the notation

$P_{a}=-{1\over L^{2}}{\partial E_{a}\over\partial a},$ (V.5)

with $E_{a}$ defined in Eqs. (VII.21), (VII.23), and (VII.25) of Appendix I. I
find

$P_{a}=-{(1+\lambda)\over 4\pi a^{3}}T\zeta_{R}(3)$ (V.6)

for $a^{-1}\gg\sqrt{eB},M$,

$P_{a}=-{(1+\lambda)\over\pi}TeB\sqrt{eB+M^{2}}e^{-2a\sqrt{eB+M^{2}}}$ (V.7)

for $\sqrt{eB}\gg a^{-1},M$, and

$P_{a}=-{(1+\lambda)TM^{2}\over 2\pi a}e^{-2Ma}F\left({eBa\over M}\right)$
(V.8)

for $M\gg a^{-1},\sqrt{eB}$. Notice that I neglected some smaller terms in
Eqs. (V.6) - (V.8). The contribution of $P_{a}$ to the Casimir pressure is
negligibly small in all cases shown above, except when $a^{-1}\gg\sqrt{eB},M$.
The leading order term is the repulsive Stefan Boltzmann term, independent of
the plate distance. All other terms, mass, magnetic field, and plate distance
corrections, are attractive and ”conspiring” to reduce the Stefan Boltzmann
repulsion.

The high temperature pressure for mixed boundary conditions and timelike
anisotropy is

$\displaystyle P$ $\displaystyle=$ $\displaystyle\\!\\!\\!\
{\pi^{2}(1+\lambda)^{5\over 2}\over 45}T^{4}-{(1+\lambda)^{3\over 2}\over
12}M^{2}T^{2}+P_{a}-{(1+\lambda)\over 2^{3\over 2}\pi}{(eB)^{3\over
2}T}\zeta_{H}\left(-{\textstyle\frac{1}{2}};{\textstyle\frac{1}{2}}+{\textstyle\frac{M^{2}}{2eB}}\right)$
(V.9) $\displaystyle+$ $\displaystyle\\!\\!\\!\ {4\pi(1+\lambda)^{2}T^{3}\over
a}e^{-4{\pi aT\sqrt{1+\lambda}}}e^{-{M^{2}a\over 2\pi
T\sqrt{1+\lambda}}}F\left({eBa\over 2\pi T\sqrt{1+\lambda}}\right),$

where

$P_{a}={3(1+\lambda)\over 16\pi a^{3}}T\zeta_{R}(3)$ (V.10)

for $a^{-1}\gg\sqrt{eB},M$, and $P_{a}$ is negligibly small in the other two
cases. Under these conditions, the contribution of the uniform energy density
part is the same as in the case of Dirichlet boundary, with the dominating,
repulsive, Stefan Boltzmann term and the attractive mass and magnetic field
corrections. The plate distance correction, $P_{a}$, is repulsive in this case
as expected, and the exponentially decaying correction is repulsive too.

The Casimir pressure for strong magnetic field and Dirichlet boundary
conditions, in the case of timelike anisotropy is given by

$P={\sqrt{1+\lambda}\over 96\pi^{2}}e^{2}B^{2}\left(C+\ln{eB\over
M^{2}}\right)-{\sqrt{1+\lambda}\over
2\pi^{3/2}}eB\left(eB+M^{2}\right)^{\textstyle\frac{1}{4}}\left[\sqrt{eB+M^{2}\over
a}{e^{-2a\sqrt{eB+M^{2}}}}-\sqrt{2}{\left(\sqrt{1+\lambda}T\right)^{\textstyle\frac{3}{2}}}e^{-{\sqrt{eB+M^{2}}\over
T\sqrt{1+\lambda}}}\right],$ (V.11)

where $C=12\zeta_{R}^{\prime}(-1)-1+\ln 4=-1.59876$, and where the dominant
term is the Weisskopf term, repulsive when $eB>e^{-C}M^{2}$, with
$e^{-C}\simeq 5$. The other vacuum term, exponentially suppressed, is
attractive while the thermal correction, also exponentially suppressed, is
repulsive. The pressure in the case of strong magnetic field and mixed
boundary conditions is very similar to what I show in Eq. (V.11), the only
difference being that the exponentially suppressed vacuum term is equal but
overall positive and therefore repulsive.

When investigating the case of large mass, Dirichlet boundary conditions, and
timelike anisotropy, I find

$P={7\sqrt{1+\lambda}\over 5,760\pi^{2}}{e^{4}B^{4}\over
M^{4}}-{\sqrt{1+\lambda}\over
4}\left({M\over\pi}\right)^{\textstyle\frac{3}{2}}\left[M{e^{-2aM}\over
a^{3/2}}F\left({eBa\over
M}\right)-2^{\textstyle\frac{3}{2}}{\left(\sqrt{1+\lambda}T\right)^{\textstyle\frac{5}{2}}}e^{-{M\over
T\sqrt{1+\lambda}}}F\left({eB\over 2MT\sqrt{1+\lambda}}\right)\right],$ (V.12)

notice that also in this case the exponentially suppressed vacuum term
produces attraction, while the thermal correction and Weisskopf term are
repulsive, as always. This is the only asymptotic case where a dominant term
is missing and therefore the pressure can be attractive or repulsive depending
on the relative size of $B$, $a$, and $M$. Finally, the large mass pressure
for mixed boundary conditions is identical to Eq. (V.12), with the exception
of the exponentially suppressed vacuum term that has opposite sign. A positive
vacuum term causes the pressure to be certainly repulsive, regardless of the
relative size of $B$, $a$, and $M$.

##  VI Discussion and conclusions

I used the zeta function regularization method to investigate the finite
temperature Casimir effect due to a Lorentz-violating scalar field in the
presence of a uniform magnetic field perpendicular to the plates. The scalar
field satisfies a modified version of the Klein-Gordon equation, where the
field derivative is coupled to a unit four-vector $u^{\mu}$ thus breaking
Lorentz invariance in a CPT-even aether-like way. I investigated in detail
Dirichlet and mixed boundary conditions. Neumann boundary conditions have not
been considered since they produce the same results as Dirichlet.

I derived simple analytic expressions for the thermal corrections to the
Casimir energy in the asymptotic case of small plate distance and Dirichlet
boundary conditions when $u^{\mu}$ is timelike, spacelike and perpendicular to
$\bf B$, and spacelike and parallel to $\bf B$, reporting my results in Eqs.
(III.17), (III.38), and (III.50) respectively. I obtained the same quantities
for the case of mixed boundary conditions, Eq. (IV.7). I obtained also simple
analytic expressions for the thermal corrections to the Casimir energy in the
asymptotic cases of strong magnetic field and large mass for the three types
of space-time anisotropy listed above in the case of Dirichlet boundary
conditions, Eqs. (III.30), (III.44), (III.56), and Eqs. (III.34), (III.46),
(III.58), and in the case of mixed boundary conditions, Eqs. (IV.14), (IV.15).
I derived the Casimir energy in the asymptotic case of high temperature for
the three possible types of space-time anisotropy and Dirichlet boundary
conditions, Eqs. (III.21), (III.24), Eqs. (III.39), (III.40), and Eqs.
(III.51), (III.52), and for mixed boundary conditions Eqs. (IV.9). The Casimir
energy for the cases of high temperature, strong magnetic field, and large
mass contain a uniform energy density piece which is dominant in the high
temperature case because of the Stefan-Boltzmann term, and in the strong
magnetic field case with the Weisskopf term, regardless of the type of
boundary condition considered. The uniform energy density terms are equal for
both sets of boundary conditions considered. I find that the thermal
correction to the Casimir energy and pressure in the asymptotic cases of small
plate distance, strong magnetic field, and large mass, is exponentially
suppressed for all cases of Lorentz asymmetry considered and for Dirichlet or
mixed boundary conditions. I discover that the thermal correction to the
pressure is repulsive in the asymptotic cases of small plate distance, strong
magnetic field, and large mass, and for all cases of Lorentz asymmetry and
boundary conditions considered.

My results show a strong dependence of the Casimir energy and pressure on the
Lorentz symmetry breaking parameter $\lambda$, as observed by Refs.
Cruz:2017kfo ; Cruz:2018bqt ; Erdas:2020ilo . The vacuum parts of the results
obtained for the cases of small plate distance and large mass agree to leading
order with Ref. Cruz:2017kfo  when $B=0$, and agree with Ref. Erdas:2020ilo
when $B$ is considered. Notice that Refs. Cruz:2017kfo ; Cruz:2018bqt  do not
include a magnetic field in their study. The vacuum part of the Casimir energy
and pressure obtained for the strong magnetic field case is also in full
agreement with Ref. Erdas:2020ilo , for the three types of Lorentz asymmetry
considered. A comparison of the thermal parts of my results with Ref.
Cruz:2018bqt  where a different regularization method is used, the Abel-Plana
summation formula, is more difficult since most of their results are expressed
as infinite sums, and only in a few specific cases the authors are able to
obtain closed forms for their results. The asymptotic limits examined on my
paper are more general than those considered in Cruz:2018bqt . For example,
the large mass limit I consider is valid for any values of $a^{-1}$, $T$, and
$\sqrt{eB}$ as long as they are all smaller than $M$, and similarly for the
other three asymptotic cases I consider. On the other hand, Ref. Cruz:2018bqt
, which does not include a magnetic field in their study, considers asymptotic
cases such as $M\gg a^{-1}\gg T$ where the thermal part of their and mine
results agree on the exponential suppression but disagree on less important
factors of $M$ and $T$ that multiply the exponential. In the asymptotic case
of high temperature, $T\gg a^{-1}\gg M$ for Ref. Cruz:2018bqt , my result and
that of Ref. Cruz:2018bqt  agree on the leading Stefan Boltzmann term having a
$T^{4}$ dependence and on its dependence on $\lambda$, but disagree on the
numerical factor multiplying this term. However, when taking $\lambda=0$ in
the results I obtain for the high temperature Casimir energy, one obtains
values that are in full agreement with the well known Stefan Boltzmann terms
for isotropic space-time for either set of boundary conditions.

It is difficult to estimate what the value of $\lambda$ could be. While it is
certain that $\lambda\ll 1$, Cruz et al. Cruz:2017kfo ; Cruz:2018bqt  show
numerical results evaluated for $\lambda=0.1,0.2$. If the Lorentz violating
parameter were to lie in that range, the temperature and magnetic corrections
I obtain here would be of the order of 5 - 15$\%$ of the value they have in
the case of unbroken Lorentz symmetry. However, these estimates of the value
of $\lambda$ are almost certainly too high. Several papers that examine
Lorentz violation within QED appeared in the literature, some within the
Casimir effect Frank:2006ww ; Kharlanov:2009pv , others within other contexts
Klinkhamer:2010zs ; Exirifard:2010xm ; Casana:2011vh ; Casana:2011du . A lower
estimate of $\lambda$ was proposed in Ref. Kharlanov:2009pv  within the
context of Lorentz violating extended QED, where the leading
$\lambda$-correction to the Casimir pressure is about $1\%-2\%$ of the value
of the pressure in isotropic space-time. Another paper Frank:2006ww , studying
extensions of QED from the Lorentz-violating scalar and fermion sector of the
minimally extended Standard Model, reported an additive correction to the
Casimir force due to CPT-odd Lorentz-violating terms that is consistent within
15$\%$ of experimental data. Other papers Klinkhamer:2010zs ; Exirifard:2010xm
that do not look at the Casimir effect but examine other direct experimental
evidence or look at indirect bounds of Lorentz violation, estimate Lorentz-
violating effects in QED to be around one part in $10^{8}$. While I am not
investigating QED in this paper, it seems that, for the scalar theory I am
considering, an estimate of $\lambda\leq 0.01-0.02$ is reasonable.

##  VII Appendix I

In this appendix I provide the details of the calculation of the zeta function
in the high temperature limit for the case of Dirichlet boundary conditions
and timelike asymmetry, introduced in subsection III.1.

The uniform density part of the zeta function, $\zeta_{U}(s)$, contains a
piece that does not depend on the temperature

$\zeta_{U,0}(s)={\mu^{2s}(1+\lambda)\over\Gamma(s)}{L^{2}a\over 8\pi^{3\over
2}}\int_{0}^{\infty}dt\,t^{s-{5\over 2}}{e^{-M^{2}t}F(eBt)},$ (VII.1)

and a piece dependent on $T$

$\zeta_{U,T}(s)={\mu^{2s}(1+\lambda)\over\Gamma(s)}{L^{2}a\over 4\pi^{3\over
2}}\int_{0}^{\infty}dt\,t^{s-{5\over
2}}{e^{-M^{2}t}F(eBt)}\sum_{m=1}^{\infty}e^{-(1+\lambda){4\pi^{2}m^{2}\over\beta^{2}}t}.$
(VII.2)

After a change of integration variable from $t$ to $t/eB$, the integral in
$\zeta_{U,0}(s)$ can be written as

$\int_{0}^{\infty}dt\,t^{s-{5\over 2}}{e^{-M^{2}t}F(eBt)}=(eB)^{{3\over
2}-s}\int_{0}^{\infty}dt\,t^{s-{3\over
2}}\sum_{l=0}^{\infty}e^{-(2l+1+{M^{2}\over eB})t},$ (VII.3)

and, after another change of integration variable,

$(eB)^{{3\over 2}-s}\int_{0}^{\infty}dt\,t^{s-{3\over
2}}\sum_{l=0}^{\infty}e^{-(2l+1+{M^{2}\over eB})t}={\sqrt{2}(eB)^{3\over
2}\over(2eB)^{s}}\zeta_{H}\left(s-{\textstyle\frac{1}{2}};{\textstyle\frac{1}{2}}+{\textstyle\frac{M^{2}}{2eB}}\right)\Gamma(s-{\textstyle\frac{1}{2}}),$
(VII.4)

where $\zeta_{H}(s;z)$ is the Hurwitz zeta function defined as

$\zeta_{H}(s;z)=\sum_{n=0}^{\infty}(n+z)^{-s}.$ (VII.5)

Making use of Eqs. (VII.3) and (VII.4), I obtain

$\zeta_{U,0}(s)=\left({\mu^{2}\over
2eB}\right)^{s}{(1+\lambda)\over\Gamma(s)}{L^{2}a(eB)^{3\over 2}\over
2^{5\over 2}\pi^{3\over
2}}\zeta_{H}\left(s-{\textstyle\frac{1}{2}};{\textstyle\frac{1}{2}}+{\textstyle\frac{M^{2}}{2eB}}\right)\Gamma(s-{\textstyle\frac{1}{2}}).$
(VII.6)

Using the following power series expansion, valid for $s\ll 1$,

$A^{s}{\Gamma(s-{\textstyle\frac{1}{2}})\over\Gamma(s)}\zeta_{H}\left(s-{\textstyle\frac{1}{2}};{\textstyle\frac{1}{2}}+{\textstyle\frac{z}{2}}\right)\simeq-2{\sqrt{\pi}}\zeta_{H}\left(-{\textstyle\frac{1}{2}};{\textstyle\frac{1}{2}}+{\textstyle\frac{z}{2}}\right)s+{\cal
O}(s^{2}),$ (VII.7)

I easily evaluate $\zeta^{\prime}_{U,0}(0)$ and include that contribution into
$E_{U}$ of Eq. (III.21).

To evaluate the temperature dependent part $\zeta_{U,T}(s)$ I make the
following approximations

$e^{-M^{2}t}\simeq 1-M^{2}t+{M^{4}t^{2}\over 2},$ (VII.8)

and

$F(eBt)\simeq 1-{e^{2}B^{2}t^{2}\over 6},$ (VII.9)

valid since $M,\sqrt{eB}\ll T$. Using these two approximations into Eq.
(III.19), changing the integration variable from $t$ to
$t\beta^{2}\over(1+\lambda)4\pi^{2}m^{2}$ and evaluating the integral, I
obtain

$\displaystyle\zeta_{U,T}(s)$ $\displaystyle=$ $\displaystyle\\!\\!\\!\
\left({\mu\beta\over\sqrt{1+\lambda}2\pi}\right)^{2s}{2\pi^{3\over
2}(1+\lambda)^{5\over
2}\over\Gamma(s)}T^{3}L^{2}a\left[\zeta_{R}(2s-3)\Gamma(s-{3\over
2})-{M^{2}\beta^{2}\over 4\pi^{2}(1+\lambda)}\zeta_{R}(2s-1)\Gamma(s-{1\over
2})\right.$ (VII.10) $\displaystyle+$ $\displaystyle\\!\\!\\!\
\left.\left({M^{4}\over 2}-{e^{2}B^{2}\over 6}\right){\beta^{4}\over
16\pi^{4}(1+\lambda)^{2}}\zeta_{R}(2s+1)\Gamma(s+{1\over 2}),\right]$

where $\zeta_{R}(s)$ is the Riemann zeta function. Using the following power
series expansions, valid for $s\ll 1$,

$A^{2s}{\Gamma(s-{\textstyle\frac{3}{2}})\over\Gamma(s)}\zeta_{R}(2s-3)\simeq{\sqrt{\pi}\over
90}s+{\cal O}(s^{2}),$ (VII.11)
$A^{2s}{\Gamma(s-{\textstyle\frac{1}{2}})\over\Gamma(s)}\zeta_{R}(2s-1)\simeq{\sqrt{\pi}\over
6}s+{\cal O}(s^{2}),$ (VII.12)
$A^{2s}{\Gamma(s+{\textstyle\frac{1}{2}})\over\Gamma(s)}\zeta_{R}(2s+1)\simeq{\sqrt{\pi}\over
2}+{\sqrt{\pi}}\left[\gamma_{E}+\ln\left({A\over 2}\right)\right]s+{\cal
O}(s^{2}),$ (VII.13)

I obtain $\zeta^{\prime}_{U,T}(0)$ and, in turn,
$E_{U}=-T\zeta^{\prime}_{U}(0)$ in the high temperature limit, shown in Eq.
(III.21).

I also separate the other part of the zeta function, $\zeta_{T,a}(s)$, in two
pieces. One piece

$\zeta_{a}(s)={\mu^{2s}(1+\lambda)\over\Gamma(s)}{L^{2}a\over 4\pi^{3\over
2}}\int_{0}^{\infty}dt\,t^{s-{5\over
2}}{e^{-M^{2}t}F(eBt)}\sum_{n=1}^{\infty}e^{-{n^{2}a^{2}\over t}}.$ (VII.14)

depends on $a$ and does not depend on $T$, the other one

$\tilde{\zeta}_{T,a}(s)={\mu^{2s}(1+\lambda)\over\Gamma(s)}{L^{2}a\over
2\pi^{3\over 2}}\int_{0}^{\infty}dt\,t^{s-{5\over
2}}{e^{-M^{2}t}F(eBt)}\sum_{m=1}^{\infty}e^{-(1+\lambda){4\pi^{2}m^{2}\over\beta^{2}}t}\sum_{n=1}^{\infty}e^{-{n^{2}a^{2}\over
t}}.$ (VII.15)

depends on both $a$ and $T$. I evaluate $\zeta_{a}(s)$ first. When $a^{-1}\gg
M,\sqrt{eB}$, I use Eqs. (VII.8) and (VII.9), integrate, and find

$\displaystyle\zeta_{a}(s)$ $\displaystyle=$ $\displaystyle\\!\\!\\!{(\mu
a)^{2s}(1+\lambda)\over\Gamma(s)}{L^{2}\over 4\pi^{3\over
2}a^{2}}\left[\Gamma\left({\textstyle\frac{3}{2}}-s\right)\zeta_{R}(3-2s)-M^{2}a^{2}\Gamma\left({\textstyle\frac{1}{2}}-s\right)\zeta_{R}(1-2s)\right.$
(VII.16) $\displaystyle+$ $\displaystyle\\!\\!\\!\left.\left({M^{4}\over
2}-{e^{2}B^{2}\over
6}\right)a^{4}\Gamma\left(-{\textstyle\frac{1}{2}}-s\right)\zeta_{R}(-1-2s)\right].$

Using the following expansions,

$A^{2s}{\Gamma({\textstyle\frac{3}{2}}-s)\over\Gamma(s)}\zeta_{R}(3-2s)\simeq{\sqrt{\pi}\zeta_{R}(3)\over
2}s+{\cal O}(s^{2}),$ (VII.17)
$A^{2s}{\Gamma({\textstyle\frac{1}{2}}-s)\over\Gamma(s)}\zeta_{R}(1-2s)\simeq-{\sqrt{\pi}\over
2}\left(s^{-1}+2\ln 2A\right)s+{\cal O}(s^{2}),$ (VII.18)
$A^{2s}{\Gamma(-{\textstyle\frac{1}{2}}-s)\over\Gamma(s)}\zeta_{R}(-1-2s)\simeq{\sqrt{\pi}\over
6}s+{\cal O}(s^{2}),$ (VII.19)

where $\zeta_{R}(3)=1.2021$, I obtain $\zeta_{a}(s)$ in the small $s$ limit
for $a^{-1}\gg M,\sqrt{eB}$

$\zeta_{a}(s)={(1+\lambda)L^{2}\over 8\pi
a^{2}}\left[\zeta_{R}(3)+M^{2}a^{2}\left(s^{-1}+2\ln 2\mu
a\right)+\left({M^{4}\over 6}-{e^{2}B^{2}\over 18}\right)a^{4}\right]s.$
(VII.20)

The contribution of $\zeta_{a}(s)$ to the Casimir energy,
$E_{a}=-T\zeta^{\prime}(0)$, is

$E_{a}=-{(1+\lambda)L^{2}T\over 8\pi
a^{2}}\left[\zeta_{R}(3)+2M^{2}a^{2}\ln(2Ma)+\left({M^{4}\over
6}-{e^{2}B^{2}\over 18}\right)a^{4}\right],$ (VII.21)

where I made the obvious choice $\mu=M$. When $\sqrt{eB}\gg a^{-1},M$, I write
$F(eBt)$ in Eq. (VII.14) as a sum of exponentials using Eq. (III.5) and change
integration variable from $t$ to ${tna\over\sqrt{M^{2}+(2\ell+1)eB}}$. Only
the term with $n=1$ and $\ell=0$ contributes significantly to the double sum
and, once I integrate using the saddle point method, I find

$\zeta_{a}(s)={(1+\lambda)L^{2}eB\over
2\pi\Gamma(s)}\left({a\mu^{2}\over\sqrt{M^{2}+eB}}\right)^{s}e^{-2a\sqrt{M^{2}+eB}},$
(VII.22)

whose contribution to the Casimir energy is

$E_{a}=-{(1+\lambda)L^{2}TeB\over 2\pi}e^{-2a\sqrt{M^{2}+eB}}.$ (VII.23)

When $M\gg a^{-1},\sqrt{eB}$, I change variable of integration from $t$ to
${tna\over M}$ and only the term with $n=1$ contributes significantly. I
integrate and find

$\zeta_{a}(s)=\left({a\mu^{2}\over M}\right)^{s}{(1+\lambda)L^{2}M\over 4\pi
a\Gamma(s)}e^{-2Ma}F\left({eBa\over M}\right),$ (VII.24)

whose contribution to the Casimir energy is reported below

$E_{a}=-{(1+\lambda)L^{2}TM\over 4\pi a}e^{-2Ma}F\left({eBa\over M}\right).$
(VII.25)

Finally, I evaluate $\tilde{\zeta}_{T,a}(s)$ by changing the integration
variable in Eq. (VII.15) from $t$ to $tna\over 2\pi m\sqrt{(1+\lambda)}T$.
Only the term with $n=m=1$ contributes significantly and I integrate it using
the saddle point method, to obtain

$\tilde{\zeta}_{T,a}(s)=\left(\mu^{2}a\over 2\pi
T\sqrt{1+\lambda}\right)^{s}{(1+\lambda)^{\frac{3}{2}}L^{2}\over\Gamma(s)}{T\over
a}e^{-{4\pi aT\sqrt{1+\lambda}}}e^{-{M^{2}a\over 2\pi
T\sqrt{1+\lambda}}}F\left({eBa\over 2\pi T\sqrt{1+\lambda}}\right).$ (VII.26)

The contribution of $\tilde{\zeta}_{T,a}(s)$ to the Casimir energy is now
quickly obtained and reported in Eq. (III.24).

##  VIII Appendix II

In this appendix I show details of the calculation of the high temperature
zeta function for mixed boundary conditions and timelike anysotropy, a topic I
address in Section IV. The uniform energy density piece of this zeta function
is identical to that obtained in Eqs. (VII.4) and (VII.10) of Appendix I for
Dirichlet boundary conditions. The remaining part of the zeta function is
again separated into a part that depends on $a$ and not on $T$,

$\zeta_{a}(s)={\mu^{2s}(1+\lambda)\over\Gamma(s)}{L^{2}a\over 4\pi^{3\over
2}}\int_{0}^{\infty}dt\,t^{s-{5\over
2}}{e^{-M^{2}t}F(eBt)}\sum_{n=1}^{\infty}(-1)^{n}e^{-{n^{2}a^{2}\over t}},$
(VIII.1)

and a part that depends on both $a$ and $T$,

$\tilde{\zeta}_{T,a}(s)={\mu^{2s}(1+\lambda)\over\Gamma(s)}{L^{2}a\over
2\pi^{3\over 2}}\int_{0}^{\infty}dt\,t^{s-{5\over
2}}{e^{-M^{2}t}F(eBt)}\sum_{m=1}^{\infty}e^{-(1+\lambda){4\pi^{2}m^{2}\over\beta^{2}}t}\sum_{n=1}^{\infty}(-1)^{n}e^{-{n^{2}a^{2}\over
t}}.$ (VIII.2)

I evaluate $\zeta_{a}(s)$ first and, as I did in the previous appendix,
consider three asymptotic cases, $a^{-1}\gg\sqrt{eB},M$; $\sqrt{eB}\gg
a^{-1},M$; and $M\gg\sqrt{eB},a^{-1}$. In the case of $a^{-1}\gg\sqrt{eB},M$,
I use Eqs. (VII.8) and (VII.9), integrate and obtain

$\displaystyle\zeta_{a}(s)$ $\displaystyle=$ $\displaystyle\\!\\!\\!{(\mu
a)^{2s}(1+\lambda)\over\Gamma(s)}{L^{2}\over 4\pi^{3\over
2}a^{2}}\left[\Gamma\left({\textstyle\frac{3}{2}}-s\right)(4^{s-1}-1)\zeta_{R}(3-2s)-M^{2}a^{2}\Gamma\left({\textstyle\frac{1}{2}}-s\right)(4^{s}-1)\zeta_{R}(1-2s)\right.$
(VIII.3) $\displaystyle+$ $\displaystyle\\!\\!\\!\left.\left({M^{4}\over
2}-{e^{2}B^{2}\over
6}\right)a^{4}\Gamma\left(-{\textstyle\frac{1}{2}}-s\right)(4^{s+1}-1)\zeta_{R}(-1-2s)\right],$

and, using the following power series expansions valid for $s\ll 1$

$A^{2s}{\Gamma({\textstyle\frac{3}{2}}-s)\over\Gamma(s)}(4^{s-1}-1)\zeta_{R}(3-2s)\simeq-{3\sqrt{\pi}\over
8}\zeta_{R}(3)s+{\cal O}(s^{2}),$ (VIII.4)
$A^{2s}{\Gamma({\textstyle\frac{1}{2}}-s)\over\Gamma(s)}(4^{s}-1)\zeta_{R}(1-2s)\simeq-{\sqrt{\pi}}\ln(2)s+{\cal
O}(s^{2}),$ (VIII.5)
$A^{2s}{\Gamma(-{\textstyle\frac{1}{2}}-s)\over\Gamma(s)}(4^{s+1}-1)\zeta_{R}(-1-2s)\simeq{\sqrt{\pi}\over
2}s+{\cal O}(s^{2}),$ (VIII.6)

I write $\zeta_{a}(s)$ in the small $s$ limit as

$\zeta_{a}(s)\simeq{(1+\lambda)}{L^{2}\over 4\pi a^{2}}\left[-{3\over
8}\zeta_{R}(3)+\ln(2)M^{2}a^{2}+\left({M^{4}\over 4}-{e^{2}B^{2}\over
12}\right)a^{4}\right]s+{\cal O}(s^{2}),$ (VIII.7)

and immediately obtain its contribution $E_{a}=-T\zeta^{\prime}_{a}(0)$ to the
Casimir energy, which I report in Eq. (IV.10) of Section IV. When
$\sqrt{eB}\gg a^{-1},M$ I write $F(eBt)$ as a sum of exponentials and change
integration variable from $t$ to ${tna\over\sqrt{M^{2}+(2\ell+1)eB}}$. Only
the term with $n=1$ and $\ell=0$ contributes significantly to the double sum
and, once I do the integration using the saddle point method, I find

$\zeta_{a}(s)=-{(1+\lambda)L^{2}eB\over
2\pi\Gamma(s)}\left({a\mu^{2}\over\sqrt{M^{2}+eB}}\right)^{s}e^{-2a\sqrt{M^{2}+eB}},$
(VIII.8)

whose contribution to the Casimir energy is reported in Eq. (IV.11) of Section
IV. When $M\gg\sqrt{eB},a^{-1}$ I change variable of integration from $t$ to
${tna\over M}$, retain the only significant term with $n=1$, integrate and
find

$\zeta_{a}(s)=-{(1+\lambda)L^{2}M\over 4\pi a\Gamma(s)}\left({a\mu^{2}\over
M}\right)^{s}e^{-2Ma}F\left({eBa\over M}\right),$ (VIII.9)

whose contribution to the Casimir energy is reported in Eq. (IV.12) of Section
IV.

Finally, $\tilde{\zeta}_{T,a}$ is computed with a change of integration
variable, as described in Appendix I. The result is similar to Eq. (VII.26) of
Appendix I, the only difference being an overall negative sign.

## References

  * (1) H. B. G. Casimir, Indag. Math. 10, 261 (1948) [Kon. Ned. Akad. Wetensch. Proc. 51, 793 (1948)] [Front. Phys. 65, 342 (1987)] [Kon. Ned. Akad. Wetensch. Proc. 100N3-4, 61 (1997)].
  * (2) M. J. Sparnaay, Physica 24, 751 (1958).
  * (3) M. Bordag, U. Mohideen, and V. M. Mostepanenko, Phys. Rep. 353, 1 (2001).
  * (4) M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, International series of monographs on physics. 145.
  * (5) T. H. Boyer, Phys. Rev. A 9, 2078 (1974).
  * (6) T. H. Boyer, Phys. Rev. 174, 1764 (1968).
  * (7) A. Ferrari, H. Girotti, M. Gomes, A. Petrov and A. da Silva, Mod. Phys. Lett. A 28, 1350052 (2013).
  * (8) I. Morales Ulion, E. Bezerra de Mello and A. Petrov, Int. J. Mod. Phys. A 30, no.36, 1550220 (2015).
  * (9) J. Alfaro, H. A. Morales-Tecotl and L. F. Urrutia, Phys. Rev. Lett. 84, 2318-2321 (2000).
  * (10) J. Alfaro, H. A. Morales-Tecotl and L. F. Urrutia, Phys. Rev. D 65, 103509 (2002).
  * (11) V. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989).
  * (12) L. Anchordoqui and H. Goldberg, Phys. Rev. D 68, 083513 (2003).
  * (13) O. Bertolami, Class. Quant. Grav. 14, 2785-2791 (1997).
  * (14) V. Kostelecky, R. Lehnert and M. J. Perry, Phys. Rev. D 68, 123511 (2003).
  * (15) M. B. Cruz, E. R. Bezerra de Mello and A. Y. Petrov, Phys. Rev. D 96, no. 4, 045019 (2017).
  * (16) M. Cruz, E. Bezerra De Mello and A. Y. Petrov,i Mod. Phys. Lett. A 33, no.20, 1850115 (2018).
  * (17) M. Frank and I. Turan, Phys. Rev. D 74, 033016 (2006).
  * (18) O. Kharlanov and V. Zhukovsky, Phys. Rev. D 81, 025015 (2010).
  * (19) A. Martin-Ruiz and C. Escobar, Phys. Rev. D 95, no.3, 036011 (2017).
  * (20) M. V. Cougo-Pinto, C. Farina, and M. R. Negrao, J. Phys. A 32, 4457 (1999).
  * (21) M. V. Cougo-Pinto, C. Farina, M. R. Negrao, and A. Tort, arXiv:hep-th/9810033.
  * (22) A. Erdas and K. P. Seltzer, Phys. Rev. D 88, 105007 (2013).
  * (23) A. Erdas and K. P. Seltzer, Int. J. Mod. Phys. A 29, 1450091 (2014).
  * (24) A. Erdas, Int. J. Mod. Phys. A 35, 2050209 (2020).
  * (25) S. W. Hawking, Commun. Math. Phys. 55, 133 (1977).
  * (26) E. Elizalde and A. Romeo, J. Math. Phys. 30, 1133 (1989) [Erratum-ibid. 31, 771 (1990)].
  * (27) E. Elizalde, J. Phys. A 41, 304040 (2008).
  * (28) F. C. Santos, A. Tenorio and A. C. Tort, hep-th/9807162.
  * (29) M. V. Cougo-Pinto, C. Farina, F. C. Santos and A. C. Tort, J. Phys. A 32, 4463 (1999).
  * (30) F. C. Santos, A. Tenorio and A. C. Tort, Phys. Rev. D 60, 105022 (1999).
  * (31) E. T. Whittaker, and G. N. Watson, A Course of Modern Analysis, Fourth Edition, Cambridge University Press: Cambridge 1963.
  * (32) F. R. Klinkhamer and M. Schreck, Nucl. Phys. B 848, 90-107 (2011).
  * (33) Q. Exirifard, Phys. Lett. B 699, 1-4 (2011).
  * (34) R. Casana, E. S. Carvalho and M. M. Ferreira, Jr, Phys. Rev. D 84, 045008 (2011).
  * (35) R. Casana, M. M. Ferreira, Jr and R. P. M. Moreira, Phys. Rev. D 84, 125014 (2011).