# High-temperature expansion of the one-loop free energy of a scalar field on
a curved background

I.S. Kalinichenko<EMAIL_ADDRESS>Physics Faculty, Tomsk State University,
Tomsk, 634050 Russia  P.O. Kazinski<EMAIL_ADDRESS>Physics Faculty, Tomsk
State University, Tomsk, 634050 Russia  Department of Higher Mathematics and
Mathematical Physics, Tomsk Polytechnic University, Tomsk, 634050 Russia

###### Abstract

The complete form of the high-temperature expansion of the one-loop
contribution to the free energy of a scalar field on a stationary
gravitational background is derived. The explicit expressions for the
divergent and finite parts of the high-temperature expansion in a three-
dimensional space without boundaries are obtained. These formulas generalize
the known one for the stationary spacetime. In particular, we confirm that for
a massless conformal scalar field the leading correction to the Planck law
proportional to the temperature squared turns out to be nonzero due to non-
static nature of the metric. The explicit expression for the so-called energy-
time anomaly is found. The interrelation between this anomaly and the
conformal (trace) anomaly is established. The natural simplest Lagrangian for
the “Killing vector field” is given.

###### pacs:

04.62.+v

##  I Introduction

The high-temperature expansions of the partition functions are the classical
subject of quantum field theory on a curved background. As for the one-loop
contributions to the free energy of quantum fields, we just mention the works
DowKen ; NakFuk ; DowSch ; DowSch1 ; HuCrSt ; Kirsten ; Campor ; GusZeln ;
Fursaev1 ; Fursaev2  where the different approaches to this problem were
implemented. Surprisingly, in spite of the fact that the first attempts to
find the high-temperature expansion were undertaken over about thirty years
ago DowKen , the problem was not completely solved, to our knowledge, in its
general statement for the arbitrary stationary (non-static) gravitational
background. The present paper is aimed to fill this gap. Such a high-
temperature expansion for the free energy will be derived here. In particular,
for a massless conformal scalar field we shall obtain the leading correction
to the Planck law proportional to the temperature squared and confirm the
result of Fursaev1 ; Fursaev2 . This correction is absent for a static
gravitational background DowKen ; NakFuk ; DowSch . Also we shall derive the
explicit expression for the finite part of the high-temperature expansion,
which seems to be a new result.

Apart from the immediate implications for the Casimir effect, astrophysics,
and cosmology, the high-temperature expansion of the free energy for fermions
can be used to analyze the derivative expansion of the one-loop contribution
to the effective action at zero temperature (see, e.g., olopq ; gmse )
regularized by the energy cutoff. According to the general prescriptions of
the renormalization theory Collins , the structures appearing as the
divergencies in the effective action must be included to the initial action to
cancel these divergencies (not necessarily to zero). As a rule, these
divergencies and the finite part depend nontrivially on the Killing vector
$\xi^{\mu}$ defining the stationarity of the background and the vacuum state
of the quantum fields (psfss , see also DeWQFTcspt ; BrOtPa ; FrZel ). We
shall see in the present paper that this is indeed the case. Hence, the
analysis of the high-temperature expansion sheds a light on the low energy
quantum dynamics of the vector field $\xi^{\mu}$ as it was discussed in gmse ;
psfss . To see how this works, one may bear in mind the expansion of the
Heisenberg-Euler effective action HeisEul ; Schwing , where the first non-
trivial term describes the light by light scattering.

Another interesting point following from the results of the present paper is
the interrelation between the conformal (trace) anomaly and the energy-time
anomaly (the notion of the latter was introduced in psfss , see also below).
As we shall see, we cannot renormalize the quantum theory of a massless
conformal scalar field in such a way that both the conformal and energy-time
anomalies vanish. The elimination of the one anomaly results in the appearance
of the other and vice versa. It is noteworthy that the factor at the logarithm
of the temperature entering the high-temperature expansion and determining the
anomaly turns out to be independent of the Killing vector and coincides with
the standard expression for the conformal anomaly DowSch ; Fursaev2 . However,
the finite part of the high-temperature expansion does involve certain
contractions of the Killing vector field. Loosely speaking, the problem of
dependence of the effective action on the Killing vector comes from the
infrared modes of the gravitational field and so it is reasonable that the
renormalization group beta function defining the ultraviolet behaviour of the
theory is independent of the Killing vector. Though, of course, this fact is
not a priori obvious since both in the ultraviolet and the infrared regions we
have the mode functions defined with respect to the same Killing vector (see,
however, Fursaev2  for the possible proof).

The paper is organized as follows. Since the subject of the article is rather
technical, we partition it into the small sections distinguishing the major
successive steps of the derivation, which culminates in Sec. VII where the
results and its implications are discussed. In Sec. II, we provide general
formulas for the one-loop correction to the free energy. The main technical
tool, which we shall employ to derive the high-temperature expansion, is the
heat kernel of a Laplacian operator (see VasilHeatKer  for review). Therefore,
in Secs. III, IV, and V we reduce our problem to the evaluation of the heat
kernel expansion coefficients. At first, Sec. III, we reduce the three-
dimensional problem to the four-dimensional one to provide the explicit
general covariance to the expansion. The method, which is used here, is a
finite dimensional analog of the gauge fixing procedure in the functional
integral. Then, in Sec. IV, we resum the heat kernel expansion applying the
theorem proved in Parker . Notice that we do not use the conformal
transformation to derive the high-temperature expansion DowKen ; DowSch ;
DowSch1 ; Kirsten ; Campor ; GusZeln ; Fursaev1 ; Fursaev2 , but apply the
method directly to the Fourier transformed Klein-Gordon operator. So, our
approach is rather close to the one used in NakFuk . After that, the problem
becomes in essence the same as in a flat spacetime. In Sec. V, we evaluate the
expansions of necessary integrals employing the procedure used in olopq .
Then, in Sec. VI, a general formula for the high-temperature expansion of the
free energy is obtained. In Sec. VII, we particularize the general formulas to
the three-dimensional space and single out the divergent and finite parts of
the high-temperature expansion. At this point we essentially employ the
results of Ven , where the heat kernel coefficients were derived up to
$a_{6}$. In Appendix A, we give some useful relations for the metric
possessing the Killing vector. In Appendix B, the relevant parts of the heat
kernel coefficients borrowed from VasilHeatKer ; Ven ; Avramid ; Gilkey  are
calculated.

Despite the fact that we study the high-temperature expansion of the free
energy of quantum scalar field on a gravitational background, the results are
easily generalized to higher spins and other stationary backgrounds. The
problem lies only in the amount of calculation, which increases due to the
additional terms in the heat kernel expansion. A non-zero chemical potential
can be also included. We postpone the investigation of these problems for the
future research.

We shall use the following conventions for the curvatures and other structures
appearing in the heat kernel expansion:

$R^{\alpha}_{\
\beta\mu\nu}=\partial_{[\mu}\Gamma^{\alpha}_{\nu]\beta}+\Gamma^{\alpha}_{[\mu\gamma}\Gamma^{\gamma}_{\nu]\beta},\qquad
R_{\mu\nu}=R^{\alpha}_{\ \mu\alpha\nu},\qquad R=R^{\mu}_{\mu}.$ (1)

The square and round brackets at a pair of indices denote antisymmetrization
and symmetrization without $1/2$, respectively. The Greek indices are raised
and lowered by the metric $g_{\mu\nu}$ which has the signature $-2$. Also we
assume that the metric possesses the timelike Killing vector $\xi^{\mu}$:

$\mathcal{L}_{\xi}g_{\mu\nu}=0,\qquad\xi^{2}=g_{\mu\nu}\xi^{\mu}\xi^{\nu}>0.$
(2)

The space dimension will be denoted by $d$ and $d=3$. Nevertheless, we shall
put $d=3$ only in the final result. The system of units is chosen such that
$c=\hbar=1$.

##  II General formulas

Consider a scalar quantum field on a stationary gravitational background at a
finite reciprocal temperature $\beta$. The free energy for this system is
defined in the standard way

$e^{-\beta F}:=\operatorname{Tr}e^{-\beta\mathcal{H}},$ (3)

where $\mathcal{H}$ is the Hamiltonian of the scalar field expressed in terms
of the creation-annihilation operators. The mode functions of this field
corresponding to the energy $\omega$ span the kernel of the Klein-Gordon
operator,

$H(x,y)=(-\nabla^{2}_{x}-m^{2})\frac{\delta(x-y)}{|g|^{1/4}(x)|g|^{1/4}(y)}=\\\
=|g|^{-1/4}(x)\biggl{[}-|g|^{-1/4}(x)\partial_{\mu}\sqrt{|g|}g^{\mu\nu}\partial_{\nu}|g|^{-1/4}(x)-m^{2}\biggr{]}\frac{\delta(x-y)}{|g|^{1/4}(y)},$
(4)

where all the time derivatives should be replaced by $-i\omega$. This
operator, which we denote as $H(\omega)$, must be supplemented by the
appropriate boundary conditions. To simplify further calculations we assume
that the system considered is large enough to neglect the boundary effects or
the space represents a compact manifold without boundary. The operator
$H(\omega)$ is Hermitian with respect to the measure $\sqrt{|g|}$ on the
square-integrable functions depending on $\mathbf{x}$.

Our aim is to calculate the one-loop contribution to the free energy (3). To
this end, we use the fact that

$\partial_{\omega}\operatorname{Tr}_{d}\theta(H(\omega))=\sum_{k}\operatorname{sgn}(\varepsilon^{\prime}_{k}(\omega))\delta(\omega-\omega_{k})$
(5)

defines the signed spectral density in the $\omega$-space for the self-adjoint
operator $H(\omega)$ possessing the eigenvalues $\varepsilon_{k}(\omega)$.
Here the Heaviside step function $\theta(H(\omega)-\varepsilon)$ is the
spectral decomposition of unity associated with $H(\omega)$ and $\omega_{k}$
is the solution to the equation

$\varepsilon_{k}(\omega)=0.$ (6)

If this equation has several solutions then the sum over all such solutions
should be taken in (5). Note that $\omega$ may be any parameter of the self-
adjoint operator and not only the energy. In the case when $\omega$ is the
energy as defined above, $\varepsilon^{\prime}_{k}(\omega)$ is usually
positive for $\omega>0$ (the particle branch) and negative for $\omega<0$ (the
antiparticle branch). Differentiating the equations

$H(\omega_{k},m^{2})\psi_{k}(m^{2})=0,\qquad\langle\psi_{k}|\psi_{k}\rangle=1,$
(7)

with respect to $m^{2}$, it is easy to see that the last statement is valid
for the same cases, when the standard prescription $m^{2}\rightarrow m^{2}-i0$
defines the Feynman propagator. It follows from the quasiclassical asymptotic
of the operator $H(\omega)$, where all the derivatives entering it are
replaced by the momenta $-ip_{\mu}$, that $\varepsilon^{\prime}_{k}(\omega)$
can be negative for $\omega>0$ under the ergosphere, i.e., the antiparticle
branch appears in the positive-frequency region. This results in the particle
creation Starobin  and the reconstruction of the vacuum state. Therefore we
restrict our consideration by the region out of the ergosphere.

Assuming that the lowest particle energy is strictly positive (this is the
case when the system is placed in a finite “box”), we can write the one-loop
contribution to the free energy from the particles as

$\mp\beta
F=\int_{0}^{\infty}d\omega\partial_{\omega}\operatorname{Tr}_{d}\theta(H(\omega))\ln(1\pm
e^{-\beta\omega})=\pm\beta\int_{0}^{\infty}d\omega\frac{\operatorname{Tr}_{d}\theta(H(\omega))}{e^{\beta\omega}\pm
1},$ (8)

where plus corresponds to fermions and minus is for bosons. The contribution
from the antiparticles is given by the same integral, but with $H(-\omega)$
instead of $H(\omega)$. Hence, the one-loop contribution to the effective
action from one bosonic mode at zero temperature reads (see, e.g., olopq ;
gmse )

$\Gamma^{(1)}_{1b}/T=-\lim_{\beta\rightarrow 0}\partial_{\beta}(\beta
F)=\partial_{\beta}\Big{[}\beta\int_{0}^{\infty}d\omega\frac{\operatorname{Tr}_{d}\theta(H(-\omega))}{e^{\beta\omega}+1}\Big{]}_{\beta\rightarrow
0},$ (9)

where $\beta^{-1}$ plays the role of the energy cutoff and $T$ is the time
interval tending to infinity. The fermionic free energy for a scalar field
(i.e. (8) with the plus sign) can be also used to estimate the partition
function for fermions when the spin-gravity interaction is negligible. For
further purposes it is useful to represent the step function in the integral
form:

$\theta(H(\omega))=\int_{C}\frac{d\tau}{2\pi i}\frac{e^{-\tau
H(\omega)}}{\tau},$ (10)

where the contour $C$ runs along the imaginary axis from top to bottom and
encircles the origin from the left. The exponent on the right-hand side is the
so-called heat kernel. In order to obtain the high-temperature expansion
($\beta\rightarrow 0$), it will be sufficient to know the expansion of the
trace of the heat kernel in a series in $\tau$.

##  III Reduction to $d+1$

Of course, we can apply the formulas from VasilHeatKer ; Avramid ; Ven ;
Gilkey  directly to (10), but this leads to rather cumbersome expressions that
are not explicitly covariant because of the $(3+1)$ decomposition. Therefore,
we reduce our $3$-dimensional problem to the $4$-dimensional one preserving
explicitly the general covariance. To this end, we introduce the auxiliary
integration variable,

$e^{-\tau H(\omega)}=\int_{-\infty}^{\infty}dp_{0}e^{-\tau
H(p_{0})}\delta(p_{0}-\omega)=\int_{-\infty}^{\infty}dp_{0}e^{-\tau[H(p_{0})+\frac{1}{\xi^{2}}(\omega^{2}-p^{2}_{0})]}\delta(p_{0}-\omega),$
(11)

and employ the standard Gaussian representation of the delta function

$\delta(p_{0}-\omega)=\lim_{\lambda\rightarrow-\infty}\sqrt{\frac{\lambda}{\pi}}e^{-\tau\lambda(p_{0}-\omega)^{2}}\tau^{1/2}.$
(12)

The second term in the exponent in (11) tends to zero when $\lambda$ goes to
minus infinity. The usefulness of such an addition will become clear soon. The
expressions with a finite $\lambda$ correspond to the same system but with a
Gaussian broadening of the spectrum. Denoting

$H_{\lambda}(p_{0},\omega):=H(p_{0})+\lambda(p_{0}-\omega)^{2}+\frac{1}{\xi^{2}}(\omega^{2}-p^{2}_{0}),$
(13)

we have

$\operatorname{Tr}_{d}e^{-\tau
H(\omega)}=\lim_{\lambda\rightarrow-\infty}\sqrt{\frac{\lambda}{\pi}}\int_{-\infty}^{\infty}dp_{0}\operatorname{Tr}_{d}e^{-\tau
H_{\lambda}(p_{0},\omega)}\tau^{1/2}=\frac{1}{T}\lim_{\lambda\rightarrow-\infty}2\sqrt{\pi\lambda}\operatorname{Tr}_{D}e^{-\tau
H_{\lambda}}\tau^{1/2},$ (14)

where $D:=d+1$ is the spacetime dimension. Then we make a similarity transform
such that the diagonal of the heat operator in the $x$-representation and,
consequently, its the trace are left intact, while the Hamiltonian becomes
(see also Appendix A for the notation)

$H_{G}:=(\xi^{2})^{1/4}H_{\lambda}(\xi^{2})^{-1/4}=-G^{\mu\nu}(\tilde{\nabla}_{\mu}-i\omega
g_{\mu})(\tilde{\nabla}_{\nu}-i\omega g_{\nu})-X,$ (15)

where $g_{\mu}:=\xi_{\mu}/\xi^{2}$ and

$\begin{gathered}G_{\mu\nu}=g_{\mu\nu}-\xi^{2}g_{\mu}g_{\nu}+\lambda^{-1}g_{\mu}g_{\nu},\qquad
G^{\mu\nu}=g^{\mu\nu}-\xi^{2}(1-\lambda\xi^{2})g^{\mu}g^{\nu},\qquad\det
G_{\mu\nu}=\lambda^{-1}g^{2}\det g_{\mu\nu},\\\ X=V-E,\qquad
V=\frac{1}{4}h_{\mu}h^{\mu}-\frac{1}{2}\nabla^{\mu}h_{\mu},\qquad
E=\frac{\omega^{2}}{\xi^{2}}-m^{2},\end{gathered}$ (16)

and $h_{\mu}:=\partial_{\mu}\ln\sqrt{\xi^{2}}$. The connection
$\tilde{\nabla}_{\mu}$ is the Levi-Civita connection compatible with the
metric $G_{\mu\nu}$. The latter is negative definite so that the operator
$H_{G}$ is of the Laplacian type.

##  IV Resummation of the heat kernel expansion

Now we can use the heat kernel expansion

$\langle x|e^{-\tau
H_{G}}|y\rangle=\sum_{k=0}^{\infty}a_{k}(x,y)\frac{\tau^{k-D/2}}{(4\pi)^{D/2}}.$
(17)

It is also relevant here that the space has no boundary. Otherwise the terms
with the half-integer $k$ appear. To exploit the heat kernel expansion, we
resum it using the theorem proved in Parker

$\langle x|e^{-\tau
H_{G}}|y\rangle=e^{\tau[X(x)+\frac{1}{6}\tilde{R}(x)]}\langle
x|e^{-\tau[X(x)+\frac{1}{6}\tilde{R}(x)]}e^{-\tau
H_{G}}|y\rangle=e^{\tau[X(x)+\frac{1}{6}\tilde{R}(x)]}\sum_{k=0}^{\infty}\tilde{a}_{k}(x,y)\frac{\tau^{k-D/2}}{(4\pi)^{D/2}},$
(18)

where $\tilde{a}_{k}$ are obtained from $a_{k}$ by removing the terms
containing $X$ and $\tilde{R}$ without derivatives. As for the $X$ terms, this
theorem simply follows from the observation that

$\langle x|e^{-\tau[X(x)+\frac{1}{6}\tilde{R}(x)]}e^{-\tau H_{G}}|y\rangle$
(19)

is invariant under the transform $X\rightarrow X+const$ and so its expansion
does not contain $X$ without derivatives. On the other hand, expanding

$\langle x|e^{-\tau[X(x)+\frac{1}{6}\tilde{R}(x)]}e^{-\tau
H_{G}}|y\rangle=e^{-\tau[X(x)+\frac{1}{6}\tilde{R}(x)]}\langle x|e^{-\tau
H_{G}}|y\rangle$ (20)

in $\tau$, we see that at the same power of $\tau$:

$\tilde{a}_{k}=a_{k}+O(X,\tilde{R}),$ (21)

where $O(X,R)$ are the terms containing $X$ and $\tilde{R}$ without
derivatives and $a_{s}$ with $s<k$. If $\tilde{a}_{k}$ does not contain $X$
and $\tilde{R}$ without derivatives in virtue of summation of such terms to
the exponent then $\tilde{a}_{k}$ is obtained by a mere obliteration of the
terms in $a_{k}$ proportional to $X$ or $\tilde{R}$ without derivatives. The
fact that $\tilde{a}_{k}$ do not depend on the scalar curvature $\tilde{R}$
can be proven by a direct inspection of the heat kernel expansion coefficients
Parker  or by solving the defining equations for the heat kernel in the
Gaussian approximation (see, e.g., psfss ).

##  V Evaluation of the integrals over $\tau$ and $\omega$

For brevity, we introduce the notation

$\begin{gathered}\tilde{m}^{2}:=m^{2}+V+\frac{1}{6}\tilde{R}=m^{2}+\frac{1}{4}h^{2}+\frac{1}{6}(R-\nabla^{\mu}h_{\mu})-\frac{1}{24}(\lambda^{-1}-\xi^{2})f^{2}=\\\
=m^{2}+\frac{1}{4}h^{2}+\frac{1}{6}[R-(\lambda^{-1}-\xi^{2})R_{\mu\nu}g^{\mu}g^{\nu}-\lambda^{-1}g^{2}\nabla^{\mu}h_{\mu}],\end{gathered}$
(22)

where we have used the relations from Appendix A. The quasiclassical expansion
for the free energy (8) after the above transformations becomes

$-F=\frac{-i}{T}\lim_{\lambda\rightarrow-\infty}\sqrt{\frac{\lambda}{\pi}}\sum_{k=0}^{\infty}\int
dx\sqrt{G}\int_{0}^{\infty}\frac{d\omega\tilde{a}_{k}(\omega,x)}{e^{\beta\omega}\pm
1}\int_{C}\frac{d\tau}{(4\pi)^{D/2}}e^{-\tau(\frac{\omega^{2}}{\xi^{2}}-\tilde{m}^{2})}\tau^{k-d/2-1},$
(23)

where $\tilde{a}_{k}(x):=\tilde{a}_{k}(x,x)$. Of course, on using the heat
kernel in the form (18), we neglect all the exponentially suppressed at
$\beta\rightarrow 0$ terms in the high-temperature expansion. For the complete
analytical structure of the heat kernel in the $\tau$-plane see, e.g., psfss .
The integral over $\tau$ can be easily taken

$\int_{C}d\tau
e^{-\tau(\frac{\omega^{2}}{\xi^{2}}-\tilde{m}^{2})}\tau^{k-d/2-1}=(-1)^{k}\frac{2\pi
ie^{-i\pi
d/2}}{\Gamma(d/2-k+1)}\theta\Big{(}\frac{\omega^{2}}{\xi^{2}}-\tilde{m}^{2}\Big{)}\Big{(}\frac{\omega^{2}}{\xi^{2}}-\tilde{m}^{2}\Big{)}^{d/2-k}.$
(24)

Comparing (24), (23), and (16), we see that the imaginary units are all
canceled out and the expression (23) is real as it should be. The formula (24)
ought to be understood in a distributional sense for $k\geq d/2$. The simplest
way to take this into account is to assume that the variable $d$ is complex
and tends to its physical value $3$. This rule can be check by convolving (24)
with a test function single-valued near the real positive semiaxis $\omega\geq
0$. Only for those test functions does the prescription of the analytical
continuation work. In our case this requirement is evidently fulfilled.

Now the integral over $\omega$ has to be evaluated. The heat kernel expansion
coefficients $\tilde{a}_{k}(\omega)$ are polynomial in $\omega$:

$\tilde{a}_{k}(\omega,x)=:\sum_{j=0}^{2k}\tilde{a}_{k}^{(j)}(x)(g^{2}\omega^{2})^{j/2},$
(25)

where the upper summation limit is dictated by dimensional reasons. Observe
that the heat kernel

$\langle x|e^{-\tau H_{G}(\omega)}|y\rangle$ (26)

is Hermitian at the real $\tau$ and therefore its diagonal is real in this
case. On the other hand, from Eq. (15) we see that the complex conjugation
effectively results in a change of the sign of $\omega$. Hence, by the
uniqueness of the analytic continuation in $\tau$, the diagonal of (26) is an
even function of $\omega$ for any $\tau$. Then the polynomial (25) contains
solely the even powers of the energy $\omega$ as long as $E$ is an even
function of $\omega$.

So, we need to take the integral

$I=\int_{0}^{\infty}\frac{\beta d\omega}{e^{\beta\omega}\pm
1}\frac{\theta(\omega^{2}g^{2}-\tilde{m}^{2})}{\Gamma(d/2-k+1)}(g^{2}\omega^{2})^{d/2-k+j/2}\Big{(}1-\frac{\tilde{m}^{2}}{\omega^{2}g^{2}}\Big{)}^{d/2-k}.$
(27)

The further procedure is quite analogous to the case of a flat spacetime (see,
e.g., olopq ; DolJack ; HabWeld ; Kapusta ). We shall follow olopq .
Substituting the expansion

$\Big{(}1-\frac{m^{2}}{\omega^{2}g^{2}}\Big{)}^{d/2-k}=\sum_{n=0}^{\infty}\frac{(-1)^{n}\Gamma(d/2-k+1)}{n!\Gamma(d/2-k-n+1)}\Big{(}\frac{m^{2}}{\omega^{2}g^{2}}\Big{)}^{n}$
(28)

to the integral $I$, we reduce it to a sum of the incomplete zeta functions.
Furthermore, we expand these incomplete zeta functions making use of the
formulas GrRy :

$\begin{split}\int_{a}^{\infty}d\omega\frac{\omega^{\nu-1}}{e^{\omega}-1}&=\Gamma(\nu)\zeta(\nu)-\sum_{n=-1}^{\infty}\frac{(-1)^{n}\zeta(-n)a^{\nu+n}}{\Gamma(n+1)(\nu+n)},\\\
\int_{a}^{\infty}d\omega\frac{\omega^{\nu-1}}{e^{\omega}+1}&=(1-2^{1-\nu})\Gamma(\nu)\zeta(\nu)-\sum_{n=0}^{\infty}(1-2^{1+n})\frac{(-1)^{n}\zeta(-n)a^{\nu+n}}{\Gamma(n+1)(\nu+n)}.\end{split}$
(29)

The term with $n=-1$ is understood as the limit $n\rightarrow-1$. For the
bosonic case we obtain

$I_{b}=\sum_{n=0}^{\infty}\frac{(-1)^{n}\tilde{m}^{d-2k}\beta_{T}^{-j}}{n!\Gamma(d/2-k-n+1)}\Big{[}\Gamma(d+j-2k-2n+1)\zeta(d+j-2k-2n+1)(\beta_{T}\tilde{m})^{2k+2n-d}-\\\
-\sum_{l=-1}^{\infty}\frac{(-1)^{l}\zeta(-l)(\beta_{T}\tilde{m})^{j+l+1}}{\Gamma(l+1)(d+j-2k-2n+l+1)}\Big{]},$
(30)

where $\beta_{T}:=\sqrt{\xi^{2}}\beta$ is the Tolman reciprocal temperature
and we have assumed that $\tilde{m}^{2}\geq 0$. The sum over $n$ of the second
term in the square brackets can be taken and is expressed through the beta
function:

$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!\Gamma(d/2-k-n+1)(d+j-2k-2n+l+1)}=-\frac{\Gamma(k-(d+j+l+1)/2)}{2\Gamma\big{(}(1-j-l)/2\big{)}}.$
(31)

Whence we get

$I_{b}=\sum_{n=0}^{\infty}\frac{(-1)^{n}\Gamma(d+j-2k-2n+1)\zeta(d+j-2k-2n+1)}{n!\Gamma(d/2-k-n+1)}\tilde{m}^{2n}\beta_{T}^{2k+2n-j-d}+\\\
+\sum_{l=-1}^{\infty}\frac{(-1)^{l}\zeta(-l)\Gamma(k-(d+j+l+1)/2)}{2\Gamma(l+1)\Gamma\big{(}(1-j-l)/2\big{)}}\tilde{m}^{d-2k+j+l+1}\beta_{T}^{l+1}.$
(32)

So, we have all what we need to obtain the high-temperature expansion.

##  VI High-temperature expansion

Substituting the expression (32) to (23), we eventually arrive at

$-F_{b}=\sum_{k,j=0}^{\infty}\int
d\mathbf{x}\frac{\sqrt{|g|}\tilde{a}_{k}^{(j)}}{(4\pi)^{d/2}}\Big{[}\sum_{n=0}^{\infty}\frac{(-1)^{n+k}\Gamma(d+j-2k-2n+1)\zeta(d+j-2k-2n+1)}{n!\Gamma(d/2-k-n+1)\beta_{T}^{d+j-2k-2n+1}\tilde{m}^{-2n}}+\\\
+\sum_{l=-1}^{\infty}\frac{(-1)^{l+k}\zeta(-l)\Gamma(k-(d+j+l+1)/2)}{2\Gamma(l+1)\Gamma\big{(}(1-j-l)/2\big{)}}\tilde{m}^{d+j-2k+l+1}\beta_{T}^{l}\Big{]},$
(33)

where the limit of the infinite $\lambda$ is implied. The expression standing
in the square brackets is regular, when $d$ tends to a positive integer, since
(30) is the entire function of $d$. Therefore, we can set $d$ to its physical
value in the factor at this square brackets. Besides, due to the property of
the zeta function a “half” of the terms in the sum over $l$ vanish. These are
the terms with $l=2,4,6,\ldots$ As far as the fermions are concerned, similar
calculations lead to ($\tilde{m}^{2}\geq 0$)

$-F_{f}=\sum_{k,j=0}^{\infty}\int
d\mathbf{x}\frac{\sqrt{|g|}\tilde{a}_{k}^{(j)}}{(4\pi)^{d/2}}\times\\\
\times\Big{[}\sum_{n=0}^{\infty}(1-2^{2n+2k-j-d})\frac{(-1)^{n+k}\Gamma(d+j-2k-2n+1)\zeta(d+j-2k-2n+1)}{n!\Gamma(d/2-k-n+1)\beta_{T}^{d+j-2k-2n+1}\tilde{m}^{-2n}}+\\\
+\sum_{l=0}^{\infty}(1-2^{1+l})\frac{(-1)^{l+k}\zeta(-l)\Gamma(k-(d+j+l+1)/2)}{2\Gamma(l+1)\Gamma\big{(}(1-j-l)/2\big{)}}\tilde{m}^{d+j-2k+l+1}\beta_{T}^{l}\Big{]}.$
(34)

The terms at the fixed power of $\beta$ both for bosons and fermions are the
same up to an overall numeric factor and in that sense are universal. However,
the expansion of the free energy for bosons involves one additional
contribution with $l=-1$.

Let us analyze the first and second terms in the square brackets of (33) and
(34) separately. We see that both for bosons and fermions the second terms in
the square brackets are not expanded in the increasing powers of $\beta$.
Rather, we have a derivative (or large mass) expansion of the terms at the
every fixed power of the reciprocal temperature. It is not difficult to write
out a closed form expression for a whole series at the fixed power of $\beta$.
With this end in view, we introduce the function (cf. olopq ; DowKen )

$\sigma^{l}_{\nu}(m^{2}):=\int_{C}\frac{dss^{\nu-1}}{(e^{2\pi
i\nu}-1)\Gamma(\nu)}\int_{0}^{\infty}d\omega\omega^{l}\operatorname{Tr}_{d}e^{-sH(\omega)}.$
(35)

The function under the integral over $s$ given by the integral over $\omega$
is analytic for $\operatorname{Re}s>0$. It should be analytically continued to
the imaginary axis and to the vicinity of the origin where the contour $C$
lies. Usually, this can be achieved by rotating the integration contour in the
$\omega$-plane (see the high frequency asymptotics in Eq. (23)). Notice that
the function (35) is not the generalized zeta function as long as $H(\omega)$
possesses the negative eigenvalues. Although these functions are closely
related. Repeating all the above calculations, one can convince oneself that
the expansion (33) can be rewritten as

$-F_{b}=\sum_{k,n,j=0}^{\infty}\int
d\mathbf{x}\frac{\sqrt{|g|}\tilde{a}_{k}^{(j)}}{(4\pi)^{d/2}}\frac{(-1)^{k}\Gamma(d+j-2k-2n+1)\zeta(d+j-2k-2n+1)}{n!\Gamma(d/2-k-n+1)\beta_{T}^{d+j-2k-2n+1}(-\tilde{m}^{2})^{-n}}+\\\
+\sum_{l=-1}^{\infty}\frac{(-1)^{l}\zeta(-l)}{\Gamma(l+1)}\sigma_{\epsilon}^{l}\beta^{l},$
(36)

where $\epsilon=(\bar{d}-d)/2$ is the complex number tending to zero and
$\bar{d}$ is the physical dimension ($\bar{d}=3$). The analogous formula holds
for fermions as well. The function $\sigma_{\nu}^{l}(m^{2})$ depends only on
$m^{2}$ and the background fields. Consequently, we need certain additional
assumptions apart from $\beta\rightarrow 0$ in order to obtain an explicit
expression for it. Henceforth we assume that the gravitational field varies
slowly such that the derivative expansions presented in (33) and (34) make
sense and provide reliable approximations for the functions (35). This holds
when the characteristic length of the variation of the gravitational field is
much larger than the Compton wavelength associated with the effective mass
$\tilde{m}$. In any case, these second terms give subleading contributions to
the high-temperature expansion. For fermions, they stand at the nonegative
integer powers of $\beta$, while for bosons the leading contribution from
these terms is proportional to $\beta^{-1}$. The first terms in the square
brackets in the expansions (33) and (34) are much more singular at
$\beta\rightarrow 0$.

Consider the first terms in the square brackets of Eqs. (33) and (34). There
is a finite number of such terms at any fixed power of $\beta$. Indeed, at any
fixed number $k$, the number $j$ must be less or equal to $2k$ by dimensional
reasons. However, inasmuch as we resummed the expansion and collected all the
terms without derivatives of $E$ to the exponent, the latter terms are absent
in $\tilde{a}_{k}$. Consequently, the worst terms, which contain the maximal
power of $\omega$ at the fixed dimension $2k$, are of the form

$\nabla E\nabla E\cdots\nabla E.$ (37)

If the dimension $2k$ is not a multiple of $3$ then the worst terms look like
(37), but with the additional covariant derivative, or with $\Omega^{2}$ (see,
for example, (84)). Thus, we conclude that (see also NakFuk ; BarvMukh )

$j\leq[4k/3],$ (38)

where the square brackets denote the integer part of the number. These worst
terms stand at

$\beta^{2n+2k-[4k/3]-d-1},$ (39)

and so the power of $\beta$ increases. Further, we restrict ourself by the
singular and finite parts of the high-temperature expansion as they give the
leading contribution to the free energy. From (39) we conclude that the
maximal number $k$ that we need is determined by the equality

$2k-[4k/3]=D,$ (40)

and for a three dimensional space $k\leq 6$. Fortunately, the heat kernel
coefficients $a_{k}$ with $k\leq 6$ were found in Ven  and we just can borrow
these results. The relevant parts of these coefficients are presented in
Appendix B.

Now we turn again to the second terms in the square brackets of the expansions
(33) and (34). The contributions with the positive even $l$ vanish. Moreover,
the contribution from these terms is zero when $l$ is an odd positive number
and $j$ is an even nonegative number. As we discussed above, the number $j$
must be even and nonegative. The explicit expressions for $\tilde{a}_{k}$
given in Appendix B are found to be even functions of $\omega$ for those
$\tilde{a}_{k}$ which we are interested in and confirm thereby the general
considerations.

Note that these second terms were disregarded in Fursaev1 ; Fursaev2 ; NakFuk
. It is relevant for a correct evaluation of these terms that we did not make
a conformal transformation to the optic metric and resummed the heat kernel
expansion. The resummation gives a non-perturbative expression for the heat
kernel which is more reliable in terms of the original metric rather than the
optic one (see for details psfss ). In principle, the first contributions in
the square brackets of Eqs. (33) and (34) may be evaluated in terms of the
conformally transformed metric. However, in order to provide a transparent way
to the poles cancelation in the $d$-plane they have to be expressed through
the same metric as the second terms.

##  VII Results and implications

Bearing in mind these observations, we can write for $d=3$ in the bosonic case

$-F_{b}=\int\frac{d\mathbf{x}\sqrt{|g|}}{(4\pi)^{d/2}}\sum_{k,j=0}^{\infty}(-1)^{k}\Big{\\{}\sideset{}{{}^{\prime}}{\sum}_{n=0}^{\infty}\frac{\Gamma(D+j-2k-2n)\zeta(D+j-2k-2n)\tilde{a}_{k}^{(j)}}{n!\Gamma(d/2-k-n+1)\beta_{T}^{D+j-2k-2n}(-\tilde{m}^{2})^{-n}}\\\
+\frac{(-\tilde{m}^{2})^{(D+j)/2-k}\tilde{a}_{k}^{(j)}}{4((D+j)/2-k)!\Gamma((1-j)/2)}\big{[}\ln\frac{\tilde{m}^{2}\beta_{T}^{2}e^{2\gamma}}{4\pi^{2}}-\psi((D+j)/2-k+1)+\psi((1-j)/2)\big{]}\\\
+\delta_{j,0}\Gamma(k-d/2)\frac{\tilde{a}_{k}^{(0)}}{2\beta_{T}}\tilde{m}^{d-2k}-\tilde{a}_{k}^{(j)}\tilde{m}^{D+j-2k}\frac{\Gamma(k-(D+j)/2)}{4\Gamma((1-j)/2)}\Big{\\}},$
(41)

where $\gamma$ is the Euler constant. The prime at the sum over $n$ says that
the singular terms are discarded, the second term is zero by definition when
the argument of the factorial is negative, and the last term vanishes by
definition when the gamma function entering the numerator tends to infinity.
The last term stands at the negative power of $\tilde{m}$. Later on we shall
cast out such contributions although it is these contributions which are
“protected” from the high energy physics, i.e., the particles with a large
mass (but with $m\ll\beta^{-1}$) give a negligible contribution to these
terms. The high-temperature expansion in the fermionic case looks like

$-F_{f}=\int\frac{d\mathbf{x}\sqrt{|g|}}{(4\pi)^{d/2}}\sum_{k,j=0}^{\infty}(-1)^{k}\Big{\\{}\sideset{}{{}^{\prime}}{\sum}_{n=0}^{\infty}(1-2^{2k+2n-j-d})\frac{\Gamma(D+j-2k-2n)\zeta(D+j-2k-2n)\tilde{a}_{k}^{(j)}}{n!\Gamma(d/2-k-n+1)\beta_{T}^{D+j-2k-2n}(-\tilde{m}^{2})^{-n}}\\\
-\frac{(-\tilde{m}^{2})^{(D+j)/2-k}\tilde{a}_{k}^{(j)}}{4((D+j)/2-k)!\Gamma((1-j)/2)}\big{[}\ln\frac{4\tilde{m}^{2}\beta_{T}^{2}e^{2\gamma}}{\pi^{2}}-\psi((D+j)/2-k+1)+\psi((1-j)/2)\big{]}\\\
+\tilde{a}_{k}^{(j)}\tilde{m}^{D+j-2k}\frac{\Gamma(k-(D+j)/2)}{4\Gamma((1-j)/2)}\Big{\\}}.$
(42)

The digamma functions appearing in the expressions are easily calculated using
the formulas GrRy :

$\psi\Big{(}\frac{1}{2}-n\Big{)}=\sum_{k=1}^{n}\frac{2}{2k-1}-\ln
4-\gamma,\qquad\psi(1+n)=\sum_{k=1}^{n}\frac{1}{k}-\gamma.$ (43)

More specifically, taking into account that $\tilde{a}_{0}=1$ and
$\tilde{a}_{1}=0$, we derive for bosons

$\begin{array}[]{l}-F_{b}=\int
d\mathbf{x}\sqrt{|g|}\Big{\\{}\dfrac{\pi^{2}\beta^{-4}_{T}}{90}-\dfrac{\beta^{-2}_{T}}{24}(\tilde{m}^{2}-\frac{1}{2}\tilde{a}_{2}^{(2)}-\frac{1}{4}\tilde{a}_{3}^{(4)})+\dfrac{\beta_{T}^{-1}}{12\pi}(\tilde{m}^{3}+\frac{3}{4\tilde{m}}\tilde{a}_{2}^{(0)}+\ldots)\\\
+\dfrac{1}{64\pi^{2}}\ln\dfrac{\tilde{m}^{2}\beta_{T}^{2}e^{2\gamma}}{16\pi^{2}}\big{[}\tilde{m}^{4}+\tilde{m}^{2}(\tilde{a}_{2}^{(2)}+\frac{3}{2}\tilde{a}_{3}^{(4)})+2\tilde{a}_{2}^{(0)}+\tilde{a}_{3}^{(2)}+\frac{3}{2}\tilde{a}_{4}^{(4)}+\frac{15}{4}\tilde{a}_{5}^{(6)}+\frac{105}{8}\tilde{a}_{6}^{(8)}\big{]}\\\\[10.00002pt]
-\dfrac{3}{128\pi^{2}}\big{[}\tilde{m}^{4}-\tilde{m}^{2}(\frac{2}{3}\tilde{a}_{2}^{(2)}+\frac{5}{3}\tilde{a}_{3}^{(4)})-\frac{4}{3}(\tilde{a}_{3}^{(2)}+2\tilde{a}_{4}^{(4)}+\frac{23}{4}\tilde{a}_{5}^{(6)}+22\tilde{a}_{6}^{(8)})\big{]}+\ldots\Big{\\}},\end{array}$
(44)

and for fermions

$\begin{array}[]{l}-F_{f}=\int
d\mathbf{x}\sqrt{|g|}\Big{\\{}\dfrac{7\pi^{2}\beta^{-4}_{T}}{720}-\dfrac{\beta^{-2}_{T}}{48}(\tilde{m}^{2}-\frac{1}{2}\tilde{a}_{2}^{(2)}-\frac{1}{4}\tilde{a}_{3}^{(4)})\\\
-\dfrac{1}{64\pi^{2}}\ln\dfrac{\tilde{m}^{2}\beta_{T}^{2}e^{2\gamma}}{\pi^{2}}\big{[}\tilde{m}^{4}+\tilde{m}^{2}(\tilde{a}_{2}^{(2)}+\frac{3}{2}\tilde{a}_{3}^{(4)})+2\tilde{a}_{2}^{(0)}+\tilde{a}_{3}^{(2)}+\frac{3}{2}\tilde{a}_{4}^{(4)}+\frac{15}{4}\tilde{a}_{5}^{(6)}+\frac{105}{8}\tilde{a}_{6}^{(8)}\big{]}\\\\[10.00002pt]
+\dfrac{3}{128\pi^{2}}\big{[}\tilde{m}^{4}-\tilde{m}^{2}(\frac{2}{3}\tilde{a}_{2}^{(2)}+\frac{5}{3}\tilde{a}_{3}^{(4)})-\frac{4}{3}(\tilde{a}_{3}^{(2)}+2\tilde{a}_{4}^{(4)}+\frac{23}{4}\tilde{a}_{5}^{(6)}+22\tilde{a}_{6}^{(8)})\big{]}+\ldots\Big{\\}}.\end{array}$
(45)

The dots in the expressions above denote the terms at the negative powers of
$\tilde{m}$ or the positive powers of $\beta$. The coefficients diverging at
$\beta\rightarrow 0$ provide the energy-time anomaly (psfss  and also BrOtPa ;
FrZel ) when the energy cutoff is used for the regularization of the one-loop
contribution to the effective action. Recall that the energy-time anomaly
represents the variance of the renormalized effective action under the uniform
dilatations of the Killing vector (see below). The flat spacetime limit of the
expansions (44) and (45) coincides with the known one DolJack ; Kapusta ;
HabWeld .

Now we consider the coefficients at the different powers of $\beta$ in detail.
First, we observe that

$\int
d\mathbf{x}\sqrt{|g|}\beta^{-2}_{T}(\tilde{m}^{2}-\frac{1}{2}\tilde{a}_{2}^{(2)}-\frac{1}{4}\tilde{a}_{3}^{(4)})=\beta^{-2}\int
d\mathbf{x}\sqrt{|g|}\big{[}(m^{2}+\frac{1}{6}R)\xi^{-2}+\frac{1}{12}f^{2}-\frac{1}{3}\nabla^{\mu}(g^{2}h_{\mu})\big{]}.$
(46)

If we omit the last term representing a total divergence then we confirm the
results of DowKen ; NakFuk ; DowSch  that the term at $\beta^{-2}$ is absent
for a massless conformal scalar field on a static gravitational background.
Meanwhile, we see that this is not the case for a stationary spacetime. For
such spacetimes, the leading correction to the Planck law is of the order of
$\beta^{-2}$ and proportional to the Maxwell-like term $f^{2}$. Formula (46)
coincides with Eq. (4.18) of Fursaev1  up to the integral of a total
derivative. Under certain circumstances the last term in (46) reducing to a
boundary term also considerably contributes. However, in this case other
boundary contributions that we discarded earlier must be included into the
free energy too.

In the bosonic case, the next correction is proportional to $\beta^{-1}$. For
a massless scalar field on a curved background satisfying the vacuum Einstein
equations $R_{\mu\nu}=0$, this contribution becomes purely imaginary as long
as the correction to $m^{2}$ is negative in this case psfss . There is also an
imaginary contribution to the free energy due to the logarithmic term. The
imaginary contributions to the effective action signalize about a certain
instability of the vacuum of quantum fields which was initially chosen. As a
result, we may anticipate that a non-zero vacuum expectation value of the
scalar field develops (the Bose-Einstein condensation).

Of course, if we accurately evaluate the high-temperature expansion of (8)
then there are not any imaginary contributions to the free energy as the
expression (8) is real-valued. For $\tilde{m}^{2}<0$, the additional terms
appear in (44) and (45) such that the imaginary part of the free energy
vanishes. In the fermionic case, the contributions to the real part of the
free energy standing at $\beta^{-4}$, $\beta^{-2}$, $\beta^{0}$, and
$\ln\beta$ do not change, i.e., the additional terms cancel exactly the
imaginary part coming from the logarithm and add certain contributions at the
higher powers of $\beta$. At the same time, in the bosonic case, the imaginary
part is also canceled out, but the infrared divergence arises in the term with
$\beta^{-1}$. Other contributions to the singular and finite parts of the
high-temperature expansion remain unchanged. The term at $\beta^{-1}$ entering
the free energy $F_{b}$ becomes

$\int
d\mathbf{x}\frac{\sqrt{|g|}}{(4\pi)^{d/2}}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2\beta_{T}}\bigg{\\{}\frac{|\tilde{m}|^{d-2k}\tilde{a}_{k}^{(0)}}{\Gamma(d/2-k+1)}\big{[}\ln\frac{\bar{\lambda}^{2}}{\xi^{2}|\tilde{m}|^{2}}+\gamma+\psi(d/2-k+1)\big{]}-\\\
-\sum_{j=1}^{[4k/3]}\tilde{a}_{k}^{(j)}(-1)^{j/2}\frac{|\tilde{m}|^{d+j-2k}\Gamma(j/2)}{\Gamma((d+j)/2-k+1)}\bigg{\\}}=\int\frac{d\mathbf{x}\sqrt{|g|}}{12\pi^{2}\beta_{T}}\Big{[}|\tilde{m}|^{3}\ln\frac{g^{2}\bar{\lambda}^{2}e^{8/3}}{4|\tilde{m}|^{2}}+\frac{3|\tilde{m}|}{2}(\tilde{a}^{(2)}_{2}+\tilde{a}^{(4)}_{3})+\ldots\Big{]},$
(47)

where $\bar{\lambda}$ is the infrared energy cutoff. The contributions
vanishing at $\bar{\lambda}\rightarrow 0$ were neglected and dots denote the
negative powers of $\tilde{m}$.

These corrections to (44) and (45) have their origin in the fact that at
$\tilde{m}^{2}<0$ the integral (27) can be represented as the sum of two
integrals along the contours $[\bar{\lambda},i\sqrt{\xi^{2}}|\tilde{m}|]$ and
$[i\sqrt{\xi^{2}}|\tilde{m}|,+\infty)$. The latter integral was already
evaluated and its contribution to the high-temperature expansion is given by
(41) or (42). The former integral,

$\int_{\bar{\lambda}}^{i\sqrt{\xi^{2}}|\tilde{m}|}\frac{\beta
d\omega}{e^{\beta\omega}\pm
1}\frac{(g^{2}\omega^{2})^{j/2}(g^{2}\omega^{2}-\tilde{m}^{2})^{d/2-k}}{\Gamma(d/2-k+1)},$
(48)

is readily taken if one expands $(e^{\beta\omega}\pm 1)^{-1}$ in a Laurent
series in $\beta\omega$. The infrared cutoff is only needed in the bosonic
case for the contribution at $(\beta\omega)^{-1}$ and $j=0$:

$\int_{\bar{\lambda}}^{i\sqrt{\xi^{2}}|\tilde{m}|}\frac{\beta
d\omega}{e^{\beta\omega}-1}\frac{(g^{2}\omega^{2}-\tilde{m}^{2})^{d/2-k}}{\Gamma(d/2-k+1)}\approx\int_{\bar{\lambda}}^{i\sqrt{\xi^{2}}|\tilde{m}|}\frac{d\omega}{\omega}\frac{(g^{2}\omega^{2}-\tilde{m}^{2})^{d/2-k}}{\Gamma(d/2-k+1)}.$
(49)

This integral, in turn, can be estimated at $\bar{\lambda}\rightarrow 0$
making use of the expansion of the incomplete beta function GrRy :

$\begin{split}\int_{a}^{1}dxx^{\alpha-1}(1-x)^{\beta-1}&=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}-\sum_{n=0}^{\infty}\frac{(-1)^{n}\Gamma(\beta)a^{n+\alpha}}{n!\Gamma(\beta-n)(n+\alpha)},\\\
\int_{a}^{1}\frac{dx}{x}(1-x)^{\beta-1}&=-\gamma-\psi(\beta)-\ln
a-\sum_{n=1}^{\infty}\frac{(-1)^{n}\Gamma(\beta)a^{n}}{n!\Gamma(\beta-n)n}.\end{split}$
(50)

To sum up, we see that at $\tilde{m}^{2}\leq 0$ the infrared divergencies
arise in the free energy. Therefore, a more accurate analysis of the low
energy modes is required. These divergencies may be cured by introducing the
Bose-Einstein condensate and summing the ring diagrams (see, e.g., Kapusta ;
DolJack ) or, as in the flat spacetime, by introducing the boundaries and
accounting for the finite size effects. In the case, when

$l\lesssim m^{-1}\ll L,$ (51)

where $l$ is the characteristic scale of the variations of the gravitational
field and $L$ is the size of the “box”, the boundary effects are negligible
and so the condensation is the only possible mean to remove the infrared
divergencies. Keeping in mind that the free energy obtained above is the
finite-temperature part of the effective action, the appearance of the
infrared divergencies at $m^{2}\rightarrow 0$ for a slowly varying
gravitational field is quite expectable (see, e.g., Kapusta ; DolJack ;
Weinberg ; GrPeYa ; Akhmed ; Polyak ). For the rapidly changing metric field
such divergencies may not appear GusZeln , but this fact has nothing in common
with the infrared divergencies arising in the infrared limit.

The coefficient of the logarithmic term is closely related with the conformal
(trace) anomaly DowKen ; DowSch . Substituting the expressions for
$\tilde{a}_{k}^{(j)}$ from Appendix B, we come to DowKen ; DowSch ; NakFuk ;
GusZeln ; Fursaev2

$\begin{array}[]{l}\tilde{m}^{4}+\tilde{m}^{2}(\tilde{a}_{2}^{(2)}+\frac{3}{2}\tilde{a}_{3}^{(4)})+2\tilde{a}_{2}^{(0)}+\tilde{a}_{3}^{(2)}+\frac{3}{2}\tilde{a}_{4}^{(4)}+\frac{15}{4}\tilde{a}_{5}^{(6)}+\frac{105}{8}\tilde{a}_{6}^{(8)}=\\\
=m^{4}+\frac{1}{3}m^{2}R+\frac{1}{90}(R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-R^{\mu\nu}R_{\mu\nu}+6\nabla^{2}R)+\frac{1}{36}R^{2}.\end{array}$
(52)

A great number of cancelations happens in calculating this expression. This
fact can be considered as the indirect verification of the correctness of the
coefficients entering the expansion above. Remarkably, this expression does
not depend on the Killing vector field. The nonminimal coupling adds to the
anomaly (52) the term,

$-2m^{2}\kappa
R+(\kappa^{2}-\frac{\kappa}{3})R^{2}-\frac{\kappa}{6}\nabla^{2}R,$ (53)

as one can easily deduce from the substitution rule (76). In the conformal
case $m^{2}=0$, $\kappa=1/6$, the coefficient of the logarithm coincides with
the standard expression for the conformal anomaly.

Some comments on the conformal transformations are in order to caution the
interested reader. In the paper DowSch , the high-temperature expansion of the
free energy of a scalar field on a static gravitational background was
obtained by the use of the conformal transform from the ultrastatic spacetime.
Despite the fact that the final result (Eq. (18), DowSch ) for the terms at
$\beta^{-4}$, $\beta^{-2}$, $\beta^{-1}$, and $\ln\beta$ is correct for the
static gravitational field, the method for its derivation seems to be invalid
at the point where the scaling property of the generalized zeta function (the
equation before Eq. (10) of DowSch ) was employed. This scaling property holds
only for the conformally invariant operator ($m^{2}=0$ and
$\kappa_{D}=(D-2)/4(D-1)$) and does not take place in a general case (see,
e.g., DowHKEscal ). Of course, one can always make a transformation

$\begin{array}[]{l}g_{\mu\nu}\rightarrow\bar{g}_{\mu\nu}=e^{2\rho}g_{\mu\nu},\\\
|\bar{g}|^{1/4}\bar{\nabla}^{2}|\bar{g}|^{-1/4}-\kappa\bar{R}+m^{2}=\\\
=e^{-\rho}\Big{[}|g|^{1/4}\nabla^{2}|g|^{-1/4}-\kappa
R+(D-1)(\kappa-\kappa_{D})\big{(}2\nabla^{2}\rho+(D-2)\nabla_{\mu}\rho\nabla^{\mu}\rho\big{)}+m^{2}e^{2\rho}\Big{]}e^{-\rho},\end{array}$
(54)

with $\rho$ independent of time. Nevertheless, the variation of this
expression with respect to $\rho$ gives not only the term arising in the
scaling property of the zeta function DowSch , but also the contribution from
the variation of the expression standing in the square brackets (54). This
contribution is not trivial in the case $\kappa\neq\kappa_{D}$ and $m^{2}\neq
0$.

On the other hand DowSch1 , one can indeed make the substitution (54) from the
beginning (4) and apply the whole machinery of the heat kernel expansion (17)
to the operator standing in the square brackets. The spectrum associated with
this operator is obviously the same as the spectrum associated with the
initial operator, and so the high-temperature expansions must coincide. If the
function $\rho$ is of a general type, such that, in particular, the
coefficients at $\omega^{2}$ and $m^{2}$ in $H(\omega)$ are not constants,
then this function disappears from the high-temperature expansion upon
restoration of the dependence $g_{\mu\nu}$ on $\rho$. However, for a certain
particular choice of $\rho$ yielding a vanishing of some terms in the
coefficients $a_{k}(\omega,x)$ of the heat kernel expansion, for example, for
$\rho=\ln\sqrt{\xi^{2}}$ leading to the constant coefficient at $\omega^{2}$
in $H(\omega)$, to the optical metric, and to the needlessness of the
resummation of the expansion for a static metric DowSch ; DowSch1 , the
dependence of the free energy on this $\rho$ does not disappear, in general.
In making the conformal transformations, one should take into account that the
coefficients $a^{(j)}_{k}$, $j\geq 2$, become nonvanishing and they also
contribute to the free energy or, in other words, the additional terms arise
in the scaling relation for the generalized zeta function DowHKEscal . This,
of course, is just a manifestation of the fact that the Klein-Gordon equation
is not conformally invariant for the arbitrary $\kappa$ and $m^{2}$.

Now we return to the discussion of the properties of the expansions (44) and
(45). The contribution to the finite part of the free energy at
$\beta\rightarrow 0$, which determines, in particular, the Casimir force at
zero temperature, does depend on the Killing vector. The whole expression is
rather huge, and we do not write it here (see Eq. (88)), but the terms at
$m^{2}$ take a compact form:

$\tilde{m}^{4}-\tilde{m}^{2}(\frac{2}{3}\tilde{a}_{2}^{(2)}+\frac{5}{3}\tilde{a}_{3}^{(4)})+\ldots=m^{4}+\frac{m^{2}}{3}(R+\frac{5}{3}\xi^{2}R_{\mu\nu}g^{\mu}g^{\nu}+\frac{11}{6}h^{2})+\ldots$
(55)

These terms give the leading contribution to the finite part when $m^{-1}$ is
much smaller than the characteristic scale of variations of the gravitational
field. The finite part does not coincide with the answers presented in DowSch
and DowSch1  (compare the coefficient at $h^{2}$). Recalling that the masses
of all the massive particles are generated by the Higgs mechanism, we see that
the term at $m^{2}$ gives a contribution to the effective potential of the
Higgs field. This results in the additional correction to the gravitational
mass-shift effect discussed in gmse .

If one regards the expression (45) as the regularized version of the one-loop
contribution to the effective action at zero temperature with $\beta$ playing
the role of the regularization parameter (see Eq. (9)), then an interesting
relation between the conformal and energy-time anomalies can be observed.
Recall that the corresponding global symmetries lead to the (formal) Noether
theorem

$g_{\mu\nu}(x)\frac{\delta\Gamma}{\delta
g_{\mu\nu}(x)}=2\sqrt{|g|}\nabla_{\mu}D^{\mu},\qquad\xi^{\mu}(x)\frac{\delta\Gamma}{\delta\xi^{\mu}(x)}=\sqrt{|g|}\nabla_{\mu}T^{\mu},$
(56)

where $\Gamma[g_{\mu\nu},\xi^{\mu},\Phi]$ is the renormalized effective
action, and $D^{\mu}$ and $T^{\mu}$ are the corresponding currents. Usually,
these relations are broken by the anomaly terms resulting from the quantum
corrections. In our particular case (the massless conformal scalar field), we
can renormalize the regularized contribution (45) to the effective action in
several ways.

The first one: we add the following counterterms to the initial Lagrangian

$\beta^{-4}(g_{\mu}g^{\mu})^{2},\qquad\beta^{-2}f^{2},\qquad[\tilde{m}^{4}+\tilde{m}^{2}(\tilde{a}_{2}^{(2)}+\frac{3}{2}\tilde{a}_{3}^{(4)})+\cdots]\ln(\beta^{-2}g^{2}).$
(57)

As for the last counterterm, such a choice is unnatural and would be
prohibited if the quantum theory of the vector field $g_{\mu}$ will be
renormalizable (cf. the Coleman-Weinberg potential ColWein  containing the
term $\phi^{4}\ln\phi^{2}$). Nevertheless, if we add such counterterms, the
global conformal invariance of the renormalized effective action is broken as
long as this logarithmic counterterm is not conformally invariant. The
conformal anomaly is given by the factor at the logarithm. On the other hand,
the second relation in (56) holds on a quantum level provided the power-like
divergencies are completely canceled out. The finite part and the coefficient
of the logarithm are invariant under the stretching of the Killing vector.

The second way: we add to the initial Lagrangian the counterterms

$\beta^{-4}(g_{\mu}g^{\mu})^{2},\qquad\beta^{-2}f^{2},\qquad[\tilde{m}^{4}+\tilde{m}^{2}(\tilde{a}_{2}^{(2)}+\frac{3}{2}\tilde{a}_{3}^{(4)})+\cdots]\ln(\beta^{-2}),$
(58)

the coefficients at the power-like divergencies being renormalized to finite
constants (not necessarily to zero). Then the second relation in (56) is
broken, while the first one is left untouched DeWQFTcspt . All the
counterterms are invariant even under the local conformal transformations up
to total derivatives.

So, we face with a choice what the symmetry we want to preserve on a quantum
level. Whatever the case, the dependence of the effective action on the
Killing vector cannot be completely removed. The imaginary part of the
effective action, which is responsible for the Hawking particle production
Hawk , depends on the Killing vector.

The field $g_{\mu}$ plays a distinguished role in defining the state of
quantum fields. The structure of the counterterms suggests the simplest form
of the classical action for this field. Its Lagrangian density reads gmse ;
KostelBB ; Will ; DeWQFTcspt

$\mathcal{L}=-\frac{1}{4}f^{2}-\frac{1}{2}(\bar{\alpha}|\phi|^{2}+\bar{\beta}+\bar{\kappa}R)g^{2}-\frac{\bar{\gamma}}{4}g^{4},\qquad\bar{\gamma}>0,$
(59)

where $\phi$ is the Higgs field ($SU(2)$ doublet), $\bar{\alpha}$,
$\bar{\gamma}$, and $\bar{\kappa}$ are the dimensionless constants, and
$\bar{\beta}$ has the mass dimension $2$. This action resembles the simplest
model of a ferromagnetic LandLifshECM  near the Curie point in the Landau
theory of phase transitions. The constants $\bar{\alpha}$ and $\bar{\beta}$
are such that the potential possesses the minimum at $g^{2}>0$. It is natural
to suppose that the characteristic magnitude of the vacuum expectation value
of the field $g_{\mu}$ is of the order of the Planck scale. From the
renormalization theory viewpoint, it is also appealing to suppose that
$\bar{\alpha}g^{2}=\mu^{2}$, where $\mu^{2}$ is the parameter of the Higgs
potential (see gmse  for the notation). In this case, the constant
$\bar{\beta}$ can be put to zero and all the coupling constants of the model
become dimensionless, apart from the Newton and the cosmological constants. In
particular, we have the estimations for the constants determining the
potential of the field $g_{\mu}$: $\bar{\alpha}\approx\eta^{2}/m_{Pl}^{2}$ and
$\bar{\gamma}\approx\bar{\alpha}^{2}$, where $\eta$ is the vacuum expectation
value of the Higgs field. It should be noted that the dynamics of the field
$g_{\mu}$ are determined not from the least action principle, but from the
self-consistency requirement gmse  imposed on the effective action. The field
$g_{\mu}$ is at the minimum of the effective potential only for the Minkowski
spacetime (see gmse ). The exploration of the viability of the above model and
its phenomenological consequences will be given elsewhere. Also note that the
fields with higher spins can induce additional terms in (59). The spinor
fields, for example, add the contribution proportional to $h^{2}$ to the terms
standing in round brackets in (59).

##  VIII Conclusion

In this paper, we derived the high-temperature expansion of the free energy in
a complete form (36). The main results of the paper are collected and
discussed in Sec. VII. In particular, the explicit expressions for the
divergent and finite parts of the high-temperature expansions are given in
Eqs. (44), (45) in the case of the four-dimensional spacetime. We generalized
the results of the papers Fursaev1 ; Fursaev2  concerning the high-temperature
expansion of the free energy on stationary spacetimes and found the additional
contributions to the high-temperature expansion that were disregarded in those
papers. In that part of the expansion which was presented in Fursaev1 ;
Fursaev2  our expression for the free energy coincides with given in Fursaev1
; Fursaev2  up to integrals of the total derivatives. The method used in the
present paper is straightforwardly generalized to the case of the higher spin
fields, to the fields with the dispersion law differing from the relativistic
one, and to the grand canonical ensembles.

##  Appendix A Useful relations with the Killing vector

In this appendix, we collect some formulas regarding the differential calculus
on the Riemannian manifold with the Killing vector. Let us introduce the
notation:

$g_{\mu}=\frac{\xi_{\mu}}{\xi^{2}},\qquad
h_{\mu}=\partial_{\mu}\ln\sqrt{\xi^{2}},\qquad
f_{\mu\nu}=\partial_{[\mu}g_{\nu]},$ (60)

where $\xi^{\mu}$ is the time-like Killing vector of the metric $g_{\mu\nu}$.
Then the following useful relations hold

$\begin{gathered}f_{\mu\lambda}g^{\lambda}=0,\qquad
g_{\lambda}h^{\lambda}=0,\qquad\nabla_{\mu}g_{\nu}=\frac{1}{2}f_{\mu\nu}-h_{(\mu}g_{\nu)},\\\
\nabla_{\mu}h_{\nu}g^{\nu}=-\frac{1}{2}f_{\mu\nu}h^{\nu}+g_{\mu}h^{2},\qquad
g^{\mu}g^{\nu}\nabla_{\mu}h_{\nu}=g^{2}h^{2},\\\
g^{\lambda}\nabla_{\lambda}f_{\mu\nu}=-g_{[\mu}f_{\nu]\lambda}h^{\lambda},\qquad\nabla_{\mu}f_{\nu\lambda}g^{\lambda}=\frac{1}{2}f^{2}_{\mu\nu}+g_{\mu}f_{\nu\lambda}h^{\lambda},\qquad\nabla^{\lambda}f_{\lambda\mu}g^{\mu}=-\frac{1}{2}f^{2},\end{gathered}$
(61)

and for the curvature

$\begin{gathered}g^{\lambda}R_{\lambda\nu\sigma\mu}=\frac{1}{2}\nabla_{\nu}f_{\sigma\mu}-\frac{1}{2}h_{[\sigma}f_{\mu]\nu}+h_{[\sigma}g_{\mu]}h_{\nu}+f_{\sigma\mu}h_{\nu}-g_{[\sigma}\nabla_{\mu]}h_{\nu},\\\
g^{\lambda}R_{\lambda\mu}=f_{\mu\lambda}h^{\lambda}-\frac{1}{2}\nabla^{\lambda}f_{\lambda\mu}-g_{\mu}\nabla_{\lambda}h^{\lambda},\qquad
R_{\mu\nu}g^{\mu}g^{\nu}=\frac{1}{4}f^{2}-g^{2}\nabla_{\lambda}h^{\lambda},\end{gathered}$
(62)

where $f^{2}:=f_{\mu\nu}f^{\mu\nu}=-\operatorname{Tr}f^{2}$.

In the course of the derivation of the high-temperature expansion, it was
useful to introduce the negative definite metric $G_{\mu\nu}$ (see Eq. (16)).
The corresponding Christoffel symbols are

$\tilde{\Gamma}^{\rho}_{\mu\nu}=\Gamma^{\rho}_{\mu\nu}+\frac{1}{2}(\lambda^{-1}-\xi^{2})g_{(\mu}f_{\nu)}^{\
\rho}-\xi^{2}(h_{(\mu}g_{\nu)}g^{\rho}-g_{\mu}g_{\nu}h^{\rho})=:\Gamma^{\rho}_{\mu\nu}+\gamma^{\rho}_{\mu\nu},$
(63)

where $\Gamma^{\rho}_{\mu\nu}$ are the Christoffel symbols for the metric
$g_{\mu\nu}$. Hereinafter all the quantities associated to the metric
$G_{\mu\nu}$ are designated by tildas. From (63) we see that the connection
$\tilde{\Gamma}^{\rho}_{\mu\nu}$ tends to the projected connection (see, e.g.,
psfss ; LandLifshCTF ; Fursaev1 ; Fursaev2 ) when $\lambda$ goes to infinity.
The curvature tensor for the connection $\tilde{\Gamma}^{\rho}_{\mu\nu}$ is
defined in the standard way

$\tilde{R}^{\rho}_{\ \nu\sigma\mu}=R^{\rho}_{\
\nu\sigma\mu}+\nabla_{[\sigma}\gamma_{\mu]\nu}^{\rho}+\gamma_{[\sigma\lambda}^{\rho}\gamma_{\mu]\nu}^{\lambda}.$
(64)

After tedious calculations, we arrive at

$\tilde{R}^{\rho}_{\ \nu\sigma\mu}=R^{\rho}_{\
\nu\sigma\mu}+\frac{1}{2}(\lambda^{-1}-\xi^{2})\Bigl{(}f_{\sigma\mu}f_{\nu}^{\
\rho}+g_{[\mu}\nabla_{\sigma]}f_{\nu}^{\
\rho}-g_{\nu}\nabla^{\rho}f_{\sigma\mu}-\frac{1}{2}f_{\nu[\sigma}f_{\mu]}^{\
\rho}\Bigr{)}\\\
-\xi^{2}\Bigl{(}g_{[\mu}h_{\sigma]}h_{\nu}g^{\rho}-g_{[\mu}h_{\sigma]}h^{\rho}g_{\nu}+\frac{1}{2}h_{[\mu}f_{\sigma]\nu}g^{\rho}-\frac{1}{2}g_{[\mu}f_{\sigma]\nu}h^{\rho}+g_{[\mu}\nabla_{\sigma]}h_{\nu}g^{\rho}-g_{[\mu}\nabla_{\sigma]}h^{\rho}g_{\nu}+f_{\sigma\mu}(h_{\nu}g^{\rho}-h^{\rho}g_{\nu})+\\\
+f_{\nu}^{\ \rho}h_{[\sigma}g_{\mu]}-\frac{1}{2}h_{[\mu}f_{\sigma]}^{\
\rho}g_{\nu}+\frac{1}{2}g_{[\mu}f_{\sigma]}^{\
\rho}h_{\nu}\Bigr{)}+\frac{1}{4}(\lambda^{-1}-\xi^{2})^{2}g_{\nu}g_{[\sigma}f_{\mu]}^{\
\lambda}f_{\lambda}^{\
\rho}-\frac{1}{2}(\lambda^{-1}-\xi^{2})\xi^{2}g_{[\sigma}f_{\mu]}^{\
\lambda}h_{\lambda}g^{\rho}g_{\nu}.$ (65)

The Ricci curvature becomes

$\tilde{R}_{\mu\nu}=R_{\mu\nu}+\frac{1}{2}(\lambda^{-1}-\xi^{2})[f^{2}_{\mu\nu}-g_{(\mu}\nabla^{\rho}f_{\rho\nu)}-g_{(\mu}f_{\nu)\rho}h^{\rho}]-\xi^{2}[g_{\mu}g_{\nu}(h^{2}-\nabla_{\rho}h^{\rho})+g_{(\mu}f_{\nu)\rho}h^{\rho}]+\\\
+\frac{1}{4}(\lambda^{-1}-\xi^{2})^{2}g_{\mu}g_{\nu}f^{2}+h_{\mu}h_{\nu}+\nabla_{\mu}h_{\nu}.$
(66)

Consequently, the scalar curvature of the metric $G_{\mu\nu}$ is written as

$\tilde{R}=G^{\mu\nu}\tilde{R}_{\mu\nu}=R+2\nabla_{\rho}h^{\rho}-\frac{1}{4}(\lambda^{-1}-\xi^{2})f^{2}.$
(67)

##  Appendix B Relevant parts of the heat kernel expansion coefficients

Here we provide the parts of the heat kernel expansion coefficients relevant
to the high-temperature expansion of the one-loop free energy in a three
dimensional space. It is these coefficients that determine the divergent and
finite parts of the high-temperature expansion when $\beta\rightarrow 0$. We
just employ the known expressions VasilHeatKer ; Avramid ; Ven ; Gilkey  for
the heat kernel coefficients adapted to the operator $H_{G}$ (15). Notice that
we did not check the general formulas for the coefficients $a_{k}$ presented
in VasilHeatKer ; Avramid ; Ven ; Gilkey  save the terms at small $k$.

As follows from Eqs. (33) and (34), the relevant parts of the heat kernel
coefficients $a_{k}$ satisfy the inequality $j\geq k-4$ (for $d=3$), where $j$
is the power of $\omega$ entering the expression. In our case, using the
notation of VasilHeatKer , we should put

$\begin{gathered}E\rightarrow V-E,\qquad\Omega_{\mu\nu}\rightarrow-i\omega
f_{\mu\nu},\\\ V=\frac{1}{4}h^{2}-\frac{1}{2}\nabla^{\mu}h_{\mu},\qquad
E=\frac{\omega^{2}}{\xi^{2}}-m^{2}.\end{gathered}$ (68)

So, $E$ has the second power in $\omega$, the curvature of the gauge fields
$\Omega$ is of the first power, and the other structures are of the zeroth
order in $\omega$. Besides, as we resummed the heat kernel expansion (see Eq.
(18)) and collected the terms $(V-E)$ and $\tilde{R}/6$ without derivatives to
the exponent, such terms in $a_{k}$ must be obliterated. We should to warn the
reader that the Riemann tensor used in VasilHeatKer  differs from our one by
the sign, while the Ricci tensors coincide.

Denoting the relevant part of the coefficient by tilde, we have

$\tilde{a}_{0}=1,\qquad\tilde{a}_{1}=0,$ (69)
$\begin{array}[]{l}\tilde{a}_{2}=\frac{1}{2!}\Big{[}\frac{1}{15}\tilde{\nabla}^{2}\tilde{R}-\frac{1}{90}\tilde{R}_{\mu\nu}\tilde{R}^{\mu\nu}+\frac{1}{90}\tilde{R}_{\alpha\beta\mu\nu}\tilde{R}^{\alpha\beta\mu\nu}+\frac{1}{3}\tilde{\nabla}^{2}V-\frac{1}{3}\tilde{\nabla}^{2}E+\frac{1}{6}\Omega^{2}\Big{]},\end{array}$
(70)

where

$\begin{array}[]{l}\tilde{\nabla}^{2}\tilde{R}=(\nabla^{2}-h^{\rho}\nabla_{\rho})[R+2\nabla^{\mu}h_{\mu}-\frac{1}{4}(\lambda^{-1}-\xi^{2})f^{2}],\\\
\tilde{R}^{\mu\nu}\tilde{R}_{\mu\nu}=R^{\mu\nu}R_{\mu\nu}+2R^{\mu\nu}[h_{\mu}h_{\nu}+\nabla_{\mu}h_{\nu}+\frac{1}{2}(\lambda^{-1}-\xi^{2})f^{2}_{\mu\nu}]+\frac{1}{4}(\lambda^{-1}-\xi^{2})^{2}[\operatorname{Tr}f^{4}+\frac{1}{4}(f^{2})^{2}]+\\\
+\frac{1}{2}(\lambda^{-1}-\xi^{2})[(\nabla^{\mu}f_{\mu\nu})^{2}+hf^{2}h+2\nabla^{\mu}h^{\nu}f^{2}_{\mu\nu}]+(\lambda^{-1}+\xi^{2})\nabla^{\rho}f_{\rho\mu}f^{\mu\nu}h_{\nu}+\\\
+\frac{\xi^{2}}{2}f^{2}(\nabla^{\mu}h_{\mu}-h^{2})+h^{\mu}\nabla_{\mu}h^{2}+2h^{2}\nabla^{\mu}h_{\mu}+\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}-(\nabla^{\mu}h_{\mu})^{2},\\\
\tilde{R}^{\mu\nu\rho\sigma}\tilde{R}_{\mu\nu\rho\sigma}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-\frac{3}{2}(\lambda^{-1}-\xi^{2})R^{\rho\nu\sigma\mu}f_{\rho\nu}f_{\sigma\mu}+(\lambda^{-1}-\xi^{2})(\nabla_{\rho}f_{\mu\sigma}\nabla^{\rho}f^{\mu\sigma}+2hf^{2}h)-\\\
-2\xi^{2}(\nabla^{\mu}h^{\nu}f^{2}_{\mu\nu}-hf^{2}h+\frac{3}{2}h^{\mu}\nabla_{\mu}f^{2}+3h^{2}f^{2}+2g^{2}\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}+2g^{2}h^{\mu}\nabla_{\mu}h^{2})+\\\
+\frac{1}{8}(\lambda^{-1}-\xi^{2})^{2}[3(f^{2})^{2}+5\operatorname{Tr}f^{4}],\\\
\tilde{\nabla}^{2}V=(\nabla^{2}-h^{\rho}\nabla_{\rho})(\frac{1}{4}h^{2}-\frac{1}{2}\nabla^{\mu}h_{\mu}),\\\
\tilde{\nabla}^{2}E=\omega^{2}g^{2}(6h^{2}-2\nabla^{\mu}h_{\mu}),\\\
\Omega^{2}\equiv\Omega^{\mu\nu}\Omega_{\mu\nu}=-\omega^{2}f^{2}.\end{array}$
(71)

In the expression (70) for $\tilde{a}_{2}$ and further for $\tilde{a}_{k}$
(Eqs. (73), (79), (84), and (87)), the indices are raised and lowered by the
use of the metric $G_{\mu\nu}$. Collecting all the above terms, we obtain

$\begin{array}[]{l}\tilde{a}_{2}=\frac{1}{180}\Big{[}R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}+\frac{3\xi^{2}}{2}R^{\rho\nu\sigma\mu}f_{\rho\nu}f_{\sigma\mu}-R^{\mu\nu}(R_{\mu\nu}+2h_{\mu}h_{\nu}+2\nabla_{\mu}h_{\nu}-\xi^{2}f^{2}_{\mu\nu})+6(\nabla^{2}-h^{\mu}\nabla_{\mu})R\\\
+\frac{\xi^{4}}{16}(5(f^{2})^{2}+6\operatorname{Tr}f^{4})+\frac{\xi^{2}}{2}(\nabla^{\mu}f_{\mu\nu})^{2}-\xi^{2}\nabla_{\mu}f_{\nu\rho}\nabla^{\mu}f^{\nu\rho}-\xi^{2}\nabla^{\rho}f_{\rho\mu}f^{\mu\nu}h_{\nu}+\frac{\xi^{2}}{2}hf^{2}h-\xi^{2}\nabla^{\mu}h^{\nu}f^{2}_{\mu\nu}\\\
+\frac{5\xi^{2}}{2}f^{2}(\nabla^{\mu}h_{\mu}-h^{2})-2h^{2}\nabla^{\mu}h_{\mu}+(\nabla^{\mu}h_{\mu})^{2}+10\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}-\frac{25}{2}h^{\mu}\nabla_{\mu}h^{2}+15h^{\mu}\nabla^{2}h_{\mu}-3\nabla^{2}\nabla^{\mu}h_{\mu}\\\
+3h^{\mu}\nabla_{\mu}\nabla_{\nu}h^{\nu}+\frac{3\xi^{2}}{2}(h^{\mu}\nabla_{\mu}f^{2}+\nabla^{2}f^{2})\Big{]}+\omega^{2}g^{2}\Big{[}\frac{1}{3}\nabla^{\mu}h_{\mu}-h^{2}-\frac{\xi^{2}}{12}f^{2}\Big{]}.\end{array}$
(72)

The relevant part of the next heat kernel expansion coefficient is (see also
Avramid ; Gilkey )

$\begin{array}[]{l}\tilde{a}_{3}=\frac{1}{3!}\Big{[}\frac{2}{15}\tilde{\nabla}_{\mu}\Omega_{\nu\rho}\tilde{\nabla}^{\mu}\Omega^{\nu\rho}+\frac{1}{30}(\tilde{\nabla}^{\nu}\Omega_{\nu\mu})^{2}+\frac{1}{5}\Omega^{\mu\nu}\tilde{\nabla}^{2}\Omega_{\mu\nu}+\frac{1}{10}\tilde{R}_{\alpha\beta\mu\nu}\Omega^{\alpha\beta}\Omega^{\mu\nu}+\frac{1}{15}\tilde{R}^{\mu\nu}\Omega^{2}_{\mu\nu}-\\\
-\frac{1}{10}(\tilde{\nabla}^{2})^{2}E-\tilde{\nabla}_{\mu}V\tilde{\nabla}^{\mu}E-\frac{1}{15}\tilde{R}_{\mu\nu}\tilde{\nabla}^{\mu}\tilde{\nabla}^{\nu}E-\frac{1}{5}\tilde{\nabla}^{\mu}\tilde{R}\tilde{\nabla}_{\mu}E+\frac{1}{2}\tilde{\nabla}_{\mu}E\tilde{\nabla}^{\mu}E\Big{]},\end{array}$
(73)

where

$\begin{array}[]{l}\tilde{\nabla}_{\mu}\Omega_{\nu\rho}\tilde{\nabla}^{\mu}\Omega^{\nu\rho}=-\omega^{2}[\nabla_{\mu}f_{\nu\rho}\nabla^{\mu}f^{\nu\rho}+2hf^{2}h+\frac{1}{2}(\lambda^{-1}-\xi^{2})\operatorname{Tr}f^{4}],\\\
(\tilde{\nabla}^{\nu}\Omega_{\nu\mu})^{2}=-\omega^{2}[(\nabla^{\nu}f_{\nu\mu})^{2}+\frac{1}{4}(\lambda^{-1}-\xi^{2})(f^{2})^{2}+2\nabla^{\nu}f_{\nu\mu}f^{\mu\rho}h_{\rho}-hf^{2}h],\\\
\Omega^{\mu\nu}\tilde{\nabla}^{2}\Omega_{\mu\nu}=-\omega^{2}[f^{\mu\nu}\nabla^{2}f_{\mu\nu}-\frac{1}{2}(\lambda^{-1}-\xi^{2})\operatorname{Tr}f^{4}-2hf^{2}h-\frac{1}{2}h^{\rho}\nabla_{\rho}f^{2}],\\\
\tilde{R}_{\alpha\beta\mu\nu}\Omega^{\alpha\beta}\Omega^{\mu\nu}=-\omega^{2}[R_{\alpha\beta\mu\nu}f^{\alpha\beta}f^{\mu\nu}-\frac{1}{2}(\lambda^{-1}-\xi^{2})((f^{2})^{2}+\operatorname{Tr}f^{4})],\\\
(\tilde{\nabla}^{2})^{2}E=-2\omega^{2}g^{2}[\nabla^{2}\nabla^{\mu}h_{\mu}-6\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}-6h^{\mu}\nabla^{2}h_{\mu}+15h^{\mu}\nabla_{\mu}h^{2}-5h^{\mu}\nabla_{\mu}\nabla_{\nu}h^{\nu}-2(\nabla^{\mu}h_{\mu}-3h^{2})^{2}],\\\
\tilde{R}^{\mu\nu}\Omega^{2}_{\mu\nu}=-\omega^{2}[R^{\mu\nu}f^{2}_{\mu\nu}+\frac{1}{2}(\lambda^{-1}-\xi^{2})\operatorname{Tr}f^{4}+hf^{2}h+f^{2}_{\mu\nu}\nabla^{\mu}h^{\nu}],\\\
\tilde{\nabla}_{\mu}V\tilde{\nabla}^{\mu}E=\omega^{2}g^{2}[h^{\mu}\nabla_{\mu}\nabla_{\nu}h^{\nu}-\frac{1}{2}h^{\mu}\nabla_{\mu}h^{2}],\\\
\tilde{R}_{\mu\nu}\tilde{\nabla}^{\mu}\tilde{\nabla}^{\nu}E=\omega^{2}g^{2}[R_{\mu\nu}(4h^{\mu}h^{\nu}-2\nabla^{\mu}h^{\nu})+(3\lambda^{-1}-\xi^{2})hf^{2}h+6h^{4}+h^{\mu}\nabla_{\mu}h^{2}-2\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}-\\\
-2h^{2}\nabla^{\mu}h_{\mu}-(\lambda^{-1}-\xi^{2})(f^{2}_{\mu\nu}\nabla^{\mu}h^{\nu}+\nabla^{\mu}f_{\mu\nu}f^{\nu\rho}h_{\rho})+\frac{1}{2}h^{2}\xi^{2}f^{2}],\\\
\tilde{\nabla}^{\mu}\tilde{R}\tilde{\nabla}_{\mu}E=-2\omega^{2}g^{2}h^{\nu}\nabla_{\nu}[R+2\nabla^{\mu}h_{\mu}-\frac{1}{4}(\lambda^{-1}-\xi^{2})f^{2}],\\\
\tilde{\nabla}_{\mu}E\tilde{\nabla}^{\mu}E=4\omega^{4}g^{4}h^{2}.\end{array}$
(74)

And so

$\begin{array}[]{l}\tilde{a}_{3}=-\frac{\omega^{2}g^{2}}{90}\Big{[}\frac{3\xi^{2}}{2}R^{\alpha\beta\mu\nu}f_{\alpha\beta}f_{\mu\nu}+R^{\mu\nu}(\xi^{2}f^{2}_{\mu\nu}+4h_{\mu}h_{\nu}-2\nabla_{\mu}h_{\nu})+\frac{3\xi^{4}}{4}(\operatorname{Tr}f^{4}+\frac{5}{6}(f^{2})^{2})\\\
+2\xi^{2}\nabla_{\mu}f_{\nu\rho}\nabla^{\mu}f^{\nu\rho}+\frac{\xi^{2}}{2}(\nabla^{\nu}f_{\nu\mu})^{2}+3\xi^{2}f^{\mu\nu}\nabla^{2}f_{\mu\nu}-\frac{5\xi^{2}}{2}hf^{2}h+2\xi^{2}\nabla^{\nu}f_{\nu\mu}f^{\mu\rho}h_{\rho}+3\xi^{2}h^{\mu}\nabla_{\mu}f^{2}\\\
+2\xi^{2}\nabla^{\mu}h^{\nu}f^{2}_{\mu\nu}-3\nabla^{2}\nabla^{\mu}h_{\mu}+16\nabla^{\mu}h^{\nu}\nabla_{\mu}h_{\nu}+18h^{\mu}\nabla^{2}h_{\mu}-\frac{103}{2}h^{\mu}\nabla_{\mu}h^{2}+18h^{\mu}\nabla_{\mu}\nabla_{\nu}h^{\nu}+6(\nabla^{\mu}h_{\mu})^{2}\\\
-38h^{2}\nabla^{\mu}h_{\mu}+60h^{4}-\frac{5\xi^{2}}{2}h^{2}f^{2}-6h^{\mu}\nabla_{\mu}R\Big{]}+\frac{1}{3}\omega^{4}g^{4}h^{2}.\end{array}$
(75)

If we had included the nonminimal coupling $\kappa R$ of the scalar field with
gravity to the Klein-Gordon equation (4), this just would have resulted in a
change of the coefficients of the scalar curvature entering the expressions
for $\tilde{a}_{2}$ and $\tilde{a}_{3}$ and a redefinition of the effective
mass $\tilde{m}$:

$\tilde{m}^{2}\rightarrow\tilde{m}^{2}-\kappa
R,\qquad\tilde{a}_{2}\rightarrow\tilde{a}_{2}-\frac{\kappa}{6}(\nabla^{2}-h^{\mu}\nabla_{\mu})R,\qquad\tilde{a}_{3}\rightarrow\tilde{a}_{3}-\frac{\omega^{2}g^{2}\kappa}{3}h^{\mu}\nabla_{\mu}R.$
(76)

The higher coefficients are encrypted in Ven . Notice that formula (6.6) for
$\textsf{Z}_{(2)}$ in Ven  contains a spurious term $R^{2}/12$. Making use of
the notation from that paper, we write out the relevant parts of the
coefficients:

$\begin{array}[]{l}\tilde{a}_{4}=\frac{1}{4!}\Big{[}\frac{2}{5}\textsf{Z}_{(1}^{\dagger}\hat{\textsf{Z}}\textsf{Z}_{1)}+\frac{2}{5}\\{\textsf{Z}^{\dagger}_{(1}\textsf{Z}_{3)}\\}+\frac{2}{5}\textsf{Z}_{(2}^{\dagger}\textsf{Z}_{2)}+\frac{1}{5}\textsf{Z}^{2}_{(2)}\Big{]},\end{array}$
(77)

where

$\begin{array}[]{l}\textsf{Z}_{(1}^{\dagger}\hat{\textsf{Z}}\textsf{Z}_{1)}\approx-\frac{1}{3}\tilde{R}^{\mu\nu}X_{\mu}X_{\nu}+\frac{2}{3}X^{\mu}\Omega_{\mu\nu}\Omega^{\nu\lambda}_{\
\ ;\lambda},\\\
\\{\textsf{Z}^{\dagger}_{(1}\textsf{Z}_{3)}\\}\approx\frac{4}{3}X^{\mu}X^{\
\nu}_{\mu\ \nu}+\frac{2}{3}X^{\mu}X^{\nu}_{\
\nu\mu}+X^{\mu}(\Omega^{2})_{;\mu}-\frac{4}{3}X^{\mu}\Omega_{\mu\nu}\Omega^{\nu\lambda}_{\
\ ;\lambda},\\\
\textsf{Z}_{(2}^{\dagger}\textsf{Z}_{2)}\approx\frac{1}{3}[(X^{\mu}_{\
\mu})^{2}+2X^{\mu\nu}X_{\mu\nu}-2X^{\mu\nu}\Omega^{2}_{\mu\nu}+X^{\mu}_{\
\mu}\Omega^{2}+\frac{1}{4}(\Omega^{2})^{2}+\frac{1}{2}\operatorname{Tr}\Omega^{4}],\\\
\textsf{Z}^{2}_{(2)}\approx(X^{\mu}_{\ \mu})^{2}+X^{\mu}_{\
\mu}\Omega^{2}+\frac{1}{4}(\Omega^{2})^{2}.\end{array}$ (78)

Here $X=V-E$ as above, the indices of $X$ and semicolon denote the covariant
differentiation with respect to the Levi-Civita connection
$\tilde{\Gamma}^{\rho}_{\mu\nu}$, and the approximate equality means that the
relevant part of the expression is retained. Therefore we have (see also
Avramid )

$\begin{array}[]{l}\tilde{a}_{4}=\frac{1}{4!}\Big{[}\frac{2}{5}\tilde{R}_{\mu\nu}\tilde{\nabla}^{\mu}E\tilde{\nabla}^{\nu}E+\frac{4}{15}\tilde{\nabla}^{\mu}\Omega_{\mu\nu}\Omega^{\nu\rho}\tilde{\nabla}_{\rho}E+\frac{4}{5}\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}\tilde{\nabla}^{2}E-\frac{2}{5}\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}\Omega^{2}+\frac{1}{3}(\tilde{\nabla}^{2}E)^{2}-\\\
-\frac{1}{3}\tilde{\nabla}^{2}E\Omega^{2}+\frac{1}{12}(\Omega^{2})^{2}+\frac{4}{15}\tilde{\nabla}^{\mu}\tilde{\nabla}^{\nu}E\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}E+\frac{4}{15}\tilde{\nabla}^{\mu}\tilde{\nabla}^{\nu}E\Omega^{2}_{\mu\nu}+\frac{1}{15}\operatorname{Tr}\Omega^{4}\Big{]},\end{array}$
(79)

where

$\begin{array}[]{l}\tilde{R}_{\mu\nu}\tilde{\nabla}^{\mu}E\tilde{\nabla}^{\nu}E=2\omega^{4}g^{4}[2R_{\mu\nu}h^{\mu}h^{\nu}+(\lambda^{-1}-\xi^{2})hf^{2}h+2h^{4}+h^{\mu}\nabla_{\mu}h^{2}],\\\
\tilde{\nabla}^{\mu}\Omega_{\mu\nu}\Omega^{\nu\rho}\tilde{\nabla}_{\rho}E=2\omega^{4}g^{2}(\nabla^{\mu}f_{\mu\nu}f^{\nu\rho}h_{\rho}-hf^{2}h),\\\
\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}\tilde{\nabla}^{2}E=4\omega^{4}g^{4}(h^{\mu}\nabla_{\mu}\nabla_{\nu}h^{\nu}-2h^{2}\nabla^{\mu}h_{\mu}+6h^{4}-3h^{\mu}\nabla_{\mu}h^{2}),\\\
\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}\Omega^{2}=2\omega^{4}g^{2}h^{\mu}\nabla_{\mu}f^{2},\\\
(\tilde{\nabla}^{2}E)^{2}=4\omega^{4}g^{4}(3h^{2}-\nabla_{\mu}h^{\mu})^{2},\\\
\tilde{\nabla}^{2}E\Omega^{2}=2\omega^{4}g^{2}f^{2}(\nabla_{\mu}h^{\mu}-3h^{2}),\\\
(\Omega^{2})^{2}=\omega^{4}(f^{2})^{2},\\\
\tilde{\nabla}^{\mu}\tilde{\nabla}^{\nu}E\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}E=4\omega^{4}g^{4}[\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}-2h^{\mu}\nabla_{\mu}h^{2}+3h^{4}-\frac{1}{2}(\lambda^{-1}-\xi^{2})hf^{2}h],\\\
\tilde{\nabla}^{\mu}\tilde{\nabla}^{\nu}E\Omega^{2}_{\mu\nu}=2\omega^{4}g^{2}(\nabla^{\mu}h^{\nu}f^{2}_{\mu\nu}-2hf^{2}h),\\\
\operatorname{Tr}\Omega^{4}=\omega^{4}\operatorname{Tr}f^{4}.\end{array}$ (80)

Summing up, we obtain

$\begin{array}[]{l}\tilde{a}_{4}=\frac{\omega^{4}g^{4}}{15}\Big{[}R_{\mu\nu}h^{\mu}h^{\nu}+\frac{\xi^{4}}{96}(5(f^{2})^{2}+4\operatorname{Tr}f^{4})-\frac{7\xi^{2}}{6}hf^{2}h+\frac{\xi^{2}}{3}\nabla^{\mu}h^{\nu}f^{2}_{\mu\nu}+\frac{\xi^{2}}{3}\nabla^{\mu}f_{\mu\nu}f^{\nu\rho}h_{\rho}-\frac{\xi^{2}}{2}h^{\mu}\nabla_{\mu}f^{2}\\\
-\frac{5\xi^{2}}{12}f^{2}(\nabla^{\mu}h_{\mu}-3h^{2})+\frac{45}{2}h^{4}-\frac{41}{6}h^{\mu}\nabla_{\mu}h^{2}+2h^{\mu}\nabla_{\mu}\nabla_{\nu}h^{\nu}-9h^{2}\nabla^{\mu}h_{\mu}+\frac{5}{6}(\nabla^{\mu}h_{\mu})^{2}+\frac{2}{3}\nabla_{\mu}h_{\nu}\nabla^{\mu}h^{\nu}\Big{]}.\end{array}$
(81)

Further,

$\begin{array}[]{l}\tilde{a}_{5}=\frac{1}{5!}\Big{[}\frac{1}{3}\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}{}^{2}\textsf{Z}_{1)}+\frac{9}{7}\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}_{2}\textsf{Z}_{1)}+\frac{6}{7}\\{\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}_{1}\textsf{Z}_{2)}\\}+\frac{1}{3}\\{\textsf{Z}^{\dagger}_{(1}\textsf{Z}_{1)}\textsf{Z}_{(2)}\\}\Big{]},\end{array}$
(82)

where

$\begin{array}[]{l}\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}{}^{2}\textsf{Z}_{1)}\approx
X^{\mu}\Omega^{2}_{\mu\nu}X^{\nu},\\\
\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}_{2}\textsf{Z}_{1)}\approx\frac{2}{3}X^{\mu}X_{\mu\nu}X^{\nu}+\frac{1}{3}X^{\mu}X_{\mu}X^{\nu}_{\
\nu}+\frac{1}{6}X^{\mu}X_{\mu}\Omega^{2}-\frac{1}{3}X^{\mu}\Omega^{2}_{\mu\nu}X^{\nu},\\\
\\{\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}_{1}\textsf{Z}_{2)}\\}\approx
2\textsf{Z}^{\dagger}_{(1}\hat{\textsf{Z}}_{2}\textsf{Z}_{1)},\\\
\\{\textsf{Z}^{\dagger}_{(1}\textsf{Z}_{1)}\textsf{Z}_{(2)}\\}\approx
2[X^{\mu}X_{\mu}X^{\nu}_{\
\nu}+\frac{1}{2}X^{\mu}X_{\mu}\Omega^{2}].\end{array}$ (83)

Hence, substituting (83) to (82), we come to

$\begin{array}[]{l}\tilde{a}_{5}=\frac{1}{5!}\Big{[}-2\tilde{\nabla}^{\mu}E\tilde{\nabla}^{\nu}E\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}E-\frac{5}{3}\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}E\tilde{\nabla}^{2}E+\frac{5}{6}\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}E\Omega^{2}-\frac{2}{3}\tilde{\nabla}^{\mu}E\tilde{\nabla}^{\nu}E\Omega^{2}_{\mu\nu}\Big{]},\end{array}$
(84)

where

$\begin{array}[]{l}\tilde{\nabla}^{\mu}E\tilde{\nabla}^{\nu}E\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}E=4\omega^{6}g^{6}(4h^{4}-h^{\mu}\nabla_{\mu}h^{2}),\\\
\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}E\tilde{\nabla}^{2}E=8\omega^{6}g^{6}h^{2}(3h^{2}-\nabla^{\mu}h_{\mu}),\\\
\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}E\Omega^{2}=-4\omega^{6}g^{4}h^{2}f^{2},\\\
\tilde{\nabla}^{\mu}E\tilde{\nabla}^{\nu}E\Omega^{2}_{\mu\nu}=-4\omega^{6}g^{4}hf^{2}h.\end{array}$
(85)

Collecting all the terms together, we get

$\begin{array}[]{l}\tilde{a}_{5}=-\frac{4\omega^{6}g^{6}}{5!}\Big{[}18h^{4}-2h^{\mu}\nabla_{\mu}h^{2}-\frac{10}{3}h^{2}\nabla^{\mu}h_{\mu}+\frac{5\xi^{2}}{6}h^{2}f^{2}-\frac{2\xi^{2}}{3}hf^{2}h\Big{]}.\end{array}$
(86)

The last coefficient that we need is

$\begin{array}[]{l}\tilde{a}_{6}=\frac{1}{6!}\Big{[}\frac{4}{7}X_{(1}X_{1)}X_{(1}X_{1)}+\frac{27}{14}X_{(1}X_{1}X_{1}X_{1)}\Big{]}\approx\frac{1}{6!}\frac{5}{2}(\tilde{\nabla}^{\mu}E\tilde{\nabla}_{\mu}E)^{2}=\frac{40}{6!}\omega^{8}g^{8}h^{4}.\end{array}$
(87)

In conclusion we give the explicit expression for the finite part of the high-
temperature expansion of the one-loop contribution to the free energy (see
Eqs. (44), (45)) retaining only the terms at the nonegative powers of $m^{2}$:

$\begin{array}[]{l}\tilde{m}^{4}-\tilde{m}^{2}(\frac{2}{3}\tilde{a}_{2}^{(2)}+\frac{5}{3}\tilde{a}_{3}^{(4)})-\frac{4}{3}(\tilde{a}_{3}^{(2)}+2\tilde{a}_{4}^{(4)}+\frac{23}{4}\tilde{a}_{5}^{(6)}+22\tilde{a}_{6}^{(8)})=m^{4}+m^{2}(\frac{1}{3}R+\frac{5\xi^{2}}{36}f^{2}-\frac{5}{9}\nabla^{\mu}h_{\mu}+\frac{11}{18}h^{2})\\\
+\frac{1}{45}\Big{[}\xi^{2}R^{\alpha\beta\mu\nu}f_{\alpha\beta}f_{\mu\nu}+\frac{2}{3}R^{\mu\nu}(\xi^{2}f^{2}_{\mu\nu}-2\nabla_{\mu}h_{\nu}-2h_{\mu}h_{\nu})+\frac{\xi^{4}}{6}(\operatorname{Tr}f^{4}+\frac{35}{32}(f^{2})^{2})+\frac{\xi^{2}}{3}(\nabla^{\mu}f_{\mu\nu})^{2}\\\
+\frac{4\xi^{2}}{3}(\nabla_{\mu}f_{\nu\rho}\nabla^{\mu}f^{\nu\rho}-\nabla^{\nu}f_{\nu\mu}f^{\mu\rho}h_{\rho}-f^{2}_{\mu\nu}\nabla^{\mu}h^{\nu})+2\xi^{2}(f^{\mu\nu}\nabla^{2}f_{\mu\nu}+h^{\mu}\nabla_{\mu}f^{2})-\frac{5\xi^{2}}{16}h^{2}f^{2}+\frac{15\xi^{2}}{8}f^{2}\nabla^{\mu}h_{\mu}\\\
-2\nabla^{2}\nabla^{\mu}h_{\mu}+\frac{16}{3}\nabla^{\mu}h^{\nu}\nabla_{\mu}h_{\nu}+8h^{\mu}\nabla^{2}h_{\mu}+\frac{1}{4}(\nabla^{\mu}h_{\mu})^{2}+\frac{5}{4}h^{2}\nabla^{\mu}h_{\mu}-\frac{8}{3}h^{\mu}\nabla_{\mu}h^{2}-\frac{109}{48}h^{4}\Big{]}\\\
-\frac{4}{45}h^{\mu}\nabla_{\mu}R+\frac{1}{6}R\Big{(}\frac{11}{18}h^{2}-\frac{5}{9}\nabla^{\mu}h_{\mu}+\frac{5\xi^{2}}{36}f^{2}\Big{)}+\frac{1}{36}R^{2}.\end{array}$
(88)

The nonminimal coupling adds to this part the following terms:

$-\kappa
R(\frac{11}{18}h^{2}-\frac{5}{9}\nabla^{\mu}h_{\mu}+\frac{5\xi^{2}}{36}f^{2}+2m^{2})+(\kappa^{2}-\frac{\kappa}{3})R^{2}+\frac{4\kappa}{9}h^{\mu}\nabla_{\mu}R.$
(89)

In the conformal case, $m^{2}=0$, $\kappa=1/6$, this contribution cancels
completely the terms proportional to the scalar curvature $R$ (without
derivatives) entering the finite part.

###### Acknowledgements.

We are grateful to D. V. Vassilevich for the fruitful correspondence
elucidating certain aspects of the heat kernel expansion technique. ISK
appreciates the Dynasty Foundation for financial support. The work is done
partially under the project 2.3684.2011 of Tomsk State University. It is also
supported by the RFBR grants No. 12-02-31071-mol-a and No. 13-02-00551, and by
the Russian Ministry of Education and Science, contracts No. 14.B37.21.0911
and No. 14.B37.21.1298.

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