aainstitutetext: Department of Physics, King’s College London, WC2RS 2LS,
United Kingdombbinstitutetext: Physik-Department, Technische Universität
München,
James-Franck-Str., 85748 Garching, Germany

# Double-well instantons in finite volume

Wen-Yuan Ai, a Jean Alexandre b,1 Matthias Carosi,111Corresponding author b
Björn Garbrecht a and Silvia Pla<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Assuming a toroidal space with finite volume, we derive analytically the full
one-loop vacuum energy for a scalar field tunnelling between two degenerate
vacua, taking into account discrete momentum. The Casimir energy is computed
for an arbitrary number of dimensions using the Abel-Plana formula, while the
one-loop instanton functional determinant is evaluated using the Green’s
functions for the fluctuation operators. The resulting energetic properties
are non-trivial: both the Casimir effect and tunnelling contribute to the Null
Energy Condition violation, arising from a non-extensive true vacuum energy.
We discuss the relevance of this mechanism to induce a cosmic bounce,
requiring no modified gravity or exotic matter.

KCL-PH-TH/2024-04
TUM-HEP-1497/24

##  1 Introduction

Quantum Field Theory (QFT) usually assumes an infinite volume, in which case
the energy levels are continuous, and tunnelling between degenerate vacua is
completely suppressed. In finite volume things are different though: energies
are quantised, and tunnelling between degenerate vacua is allowed, as in
Quantum Mechanics. The first feature leads to the known Casimir effect (see
Bordag  for a review) while the second is at the origin of convexity of the
effective potential Symanzik:1969ek ; Coleman:1974jh ; Iliopoulos:1974ur ;
Haymaker:1983xk ; Fujimoto:1982tc ; Bender:1983nc ; Hindmarsh:1985nc ;
Alexandre:2012ht ; Plascencia:2015pga ; Millington:2019nkw  and therefore
restoration of a symmetric true vacuum. In the present article, we consider
the energetic consequences of the interplay between the above finite volume
effects, that we compare in the specific situation where space has the
topology of a three-torus.

The Casimir energy of a system in a finite volume is defined as the difference
between the ground state energy calculated with discrete and continuous
momenta. It is known that the result is highly dependent on the geometry of
the confining space, as well as the boundary conditions satisfied by the field
inside this space. For a field with mass $m$, the Casimir effect is usually
suppressed as $\approx\exp(-mL)$ for $mL\gg 1$, where $L$ is a typical length
of the system. Tunnelling, on the other hand, is typically suppressed as the
exponential of the volume $\approx\exp[-(mL)^{3}]$, where $(mL)^{3}$ is the
action of a Euclidean-time-dependent and spatially homogeneous instanton
connecting the two degenerate vacua.222For this reason tunnelling between
degenerate vacua is completely suppressed in the limit of infinite volume,
where spontaneous symmetry breaking occurs instead. As a result, tunnelling is
negligible compared to the Casimir effect if we consider the situation $mL\gg
1$.333This is the case of three space dimensions, and of course things could
be different in one space dimension.

In the situation where $mL\lesssim 1$, tunnelling and Casimir effects could be
of the same order, potentially leading to new phenomena. The effective theory
taking into account tunnelling of a real scalar field between two degenerate
vacua has been studied in Alexandre:2022qxc ; Alexandre:2023iig . It was found
that the effective potential is indeed convex, with a symmetric true vacuum at
$\left<\phi\right>=0$ and a corresponding ground state energy lower than the
original degenerate vacua, similarly to what happens in Quantum Mechanics.
This result was obtained by neglecting the spatially inhomogeneous
fluctuations around the instanton configuration, as well as the Casimir
effect. In the present work, we refine this analysis by including all quantum
fluctuations and using a full analytic result for the Casimir energy.

We note here that $O(4)$-symmetric saddle points, representing bubbles of true
vacuum developing in a false vacuum, are allowed in the situation of non-
degenerate vacua only Coleman:1977py ; Callan:1977pt  (see Andreassen:2016cvx
; Ai:2019fri  for reviews). Degenerate vacua would lead to bubbles of infinite
radius, which justifies only looking at homogeneous saddle points in the
present work. We also note that the tunnelling-induced symmetry restoration
described here happens at zero temperature and is unrelated to high-
temperature symmetry restoration.

An important feature related to the above finite-volume effects is the
violation of the Null Energy Condition (NEC - see Rubakov:2014jja ;
Kontou:2020bta  for reviews), which is known in the case of the Casimir
effect, and which has been shown in Alexandre:2022qxc ; Alexandre:2023iig  for
tunnelling between degenerate vacua. NEC violation in both cases is related to
a non-trivial dependence on the typical length $L$ of the system considered,
which implies a non-extensive ground state energy and leads to remarkable
energetic properties. In this article we derive these effects precisely,
taking into account the full one-loop quantum fluctuations about the
instanton, with discrete momenta.

Section 2 sets the framework, where we define the semiclassical approximation
for the partition function of a self-interacting scalar field with a double-
well potential. We consider the same instanton dilute gas as the one used in
Quantum Mechanics, based on homogeneous and Euclidean-time-dependent
configurations Kleinert:2004ev .

In Section 3 we derive the Casimir energy for a massive scalar field in a
three-torus, using the Abel-Plana formula. This is done first in one space
dimension and then in arbitrary dimensions, based on a recursion formula which
is derived in Appendix B, together with the details of the calculation for two
and three space dimensions.

The tunnelling contribution is then described in Section 4, where the full
one-loop functional determinant for fluctuations about the instanton is
obtained based on the method of the Green’s function described therein.
Throughout the calculation, the discrete nature of the momenta is kept, and no
further approximation is made. This calculation is done for the first time,
and all the details are therefore given in this section. The true ground state
energy is then discussed, and we find that tunnelling can be of the same order
of magnitude as the Casimir effect for $mL\lesssim 1$.

In Section 5 we discuss the possibility of inducing a cosmic bounce from the
NEC violation mechanism obtained here, without modified gravity or exotic
matter, as was already described in Alexandre:2023pkk ; Alexandre:2023bih ,
but where quantum fluctuations were calculated with continuous momenta.

##  2 True ground state energy

We consider a real scalar field in a symmetric double-well potential and in
finite volume. The Quantum Mechanical analogy, corresponding to field theory
in zero spatial dimensions, is well-known from Ref. Coleman:1985rnk . Here, we
choose space to be a three-torus with size $L$ in the three directions, thus
corresponding to the volume $L^{3}$. The Euclidean action is

$S_{E}=\int_{-\beta/2}^{\beta/2}d\tau\int
d^{3}\mathbf{x}\left(\frac{1}{2}(\partial_{\tau}\phi)^{2}+\frac{1}{2}(\nabla\phi)^{2}+U(\phi)\right)~{},$
(1)

where

$\displaystyle U(\phi)=\frac{\lambda}{4!}(\phi^{2}-v^{2})^{2}+U_{0}\,,$ (2)

and $\tau=it$ is the imaginary time. We will consider the zero-temperature
limit $\beta\to\infty$. The energy density $U_{0}$, which fixes the origin of
energies, corresponds to a cosmological constant in the discussion on
Cosmology.

As discussed in the Introduction, in this context the scalar field can tunnel
between the two degenerate bare minima $\pm v$, resulting in an effective
theory with a symmetric vacuum state located at $\left<\phi\right>=0$ (see
Figure 1). The explicit effective potential has been calculated at quadratic
order in Alexandre:2022qxc ; Alexandre:2023iig , where one finds a positive
mass term, therefore showing convexity. In the present article, we focus on
the true vacuum expectation value (vev) from the beginning and thus we
consider a vanishing source. Indeed, finite volume implies a one-to-one
mapping between the source and the vev, and thus a unique Legendre transform
to obtain the one-particle-irreducible effective action. In this mapping a
vanishing source corresponds to a vanishing vev.

Figure 1: The classical potential (solid line) and the effective potential
(dashed line) in a finite volume. Tunnelling between the degenerate minima in
the classical potential restores the symmetry, giving a symmetric true vacuum
at $\phi_{c}=0$.

###  2.1 Semi-classical approximation

We focus here on the fundamental quantity, which is the true ground state
energy

$E_{0}=-\lim_{\beta\to\infty}\frac{1}{\beta}\log
Z_{E}=-\lim_{\beta\to\infty}\frac{1}{\beta}\log\int\mathcal{D}[\phi]~{}e^{-S_{E}[\phi]}~{},$
(3)

$\beta$ being the inverse temperature, $Z_{E}$ the Euclidean partition
function and $S_{E}$ the Euclidean action (1). We will evaluate $Z_{E}$ in the
context of the semi-classical approximation, where the contribution of each
dominant configuration (the saddle points) is independent from the others,
such that

$\displaystyle Z_{E}=$
$\displaystyle\int\mathcal{D}[\phi]~{}e^{-S_{E}[\phi]}\approx\sum_{\overline{\phi}\>\mathrm{saddle}}[\overline{\phi}]~{},$
(4)

where $[\overline{\phi}]$ is the one-loop approximation of the individual
path-integral corresponding to a unique saddle point $\overline{\phi}$

$\displaystyle[\overline{\phi}]$
$\displaystyle=\int\mathcal{D}[\eta]~{}e^{-S_{E}[\overline{\phi}+\eta]}$
$\displaystyle\approx
e^{-S[\overline{\phi}]}\int\mathcal{D}[\eta]~{}e^{-\frac{1}{2}\int\eta
S_{E}^{\prime\prime}[\overline{\phi}]\eta}\>=\>e^{-S_{E}[\overline{\phi}]}\left(\det
S_{E}^{\prime\prime}[\overline{\phi}]\right)^{-\frac{1}{2}}~{},$ (5)

where $S^{\prime\prime}_{E}[\bar{\phi}]$ denotes the fluctuation operator
about the saddle point $\bar{\phi}$, and we have used the simplified notation
for the integral in the exponent. Eq. (2.1) requires further treatment in case
the saddle point $\overline{\phi}$ explicitly breaks some symmetry of the
action. In this case, the operator $S_{E}^{\prime\prime}[\overline{\phi}]$ has
one or more zero modes, over which the integral is not Gaussian. The integral
over the zero modes, however, can be traded for an integral over the
collective coordinates of the saddle point.

The homogeneous saddle points $\overline{\phi}$ are the solutions to the
classical (Euclidean) equations of motion

$\ddot{\phi}+\frac{\lambda v^{2}}{6}\phi-\frac{\lambda}{6}\phi^{3}=0\,,$ (6)

where the dot stands for the derivative with respect to the Euclidean time,
$\ddot{\phi}=\frac{\partial^{2}\phi}{\partial\tau^{2}}$. There are two trivial
solutions in this model, $\phi\equiv\pm v$. The contribution of these two
configuration reads

$[+v]=[-v]=e^{-L^{3}\beta U_{0}}\left(\det
S_{E}^{\prime\prime}[v]\right)^{-\frac{1}{2}}=e^{-L^{3}\beta
U_{0}}\left(\det(-\partial_{\tau}^{2}-\nabla^{2}+m^{2})\right)^{-\frac{1}{2}}\,,$
(7)

where $m^{2}\equiv U^{\prime\prime}(\phi)|_{\phi=\pm v}=\frac{\lambda
v^{2}}{3}$. We postpone the calculation of the functional determinant to
Section 3.

###  2.2 Instanton gas

Equation (6) also admits Euclidean-time-dependent but spatially homogeneous
solutions, the instanton and the anti-instanton, i.e., the kink ($K$) and the
anti-kink ($\overline{K}$) in Euclidean time

$\phi_{K/\overline{K}}(\tau)=\pm
v\tanh\omega(\tau-\tau_{1})\,,\qquad\mathrm{where}\qquad\omega^{2}=\frac{\lambda
v^{2}}{12}=\frac{m^{2}}{4}\,,$ (8)

and $\tau_{1}$ is the location of the (anti-)kink. The contribution of these
configurations to the partition function reads

$\left[K\right]=\left[\overline{K}\right]=\int_{-\beta/2}^{\beta/2}d\tau_{1}[I][v]=\beta[I][v]\,,$
(9)

where we isolated the contribution of the centre of the (anti-)instanton (the
“jump”)

$\displaystyle[I]=$
$\displaystyle\>\sqrt{\frac{K}{2\pi}}e^{-K}\left(\frac{\sideset{}{{}^{\prime}}{\det}S_{E}^{\prime\prime}[\phi_{K}]}{\det
S_{E}^{\prime\prime}[v]}\right)^{-\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\>\sqrt{\frac{K}{2\pi}}e^{-K}\left(\frac{\sideset{}{{}^{\prime}}{\det}\left(-\partial_{\tau}^{2}-\nabla^{2}+\frac{m^{2}}{2}\left(\frac{3\phi_{K}^{2}}{v^{2}}-1\right)\right)}{\det(-\partial_{\tau}^{2}-\nabla^{2}+m^{2})}\right)^{-\frac{1}{2}}~{},$
(10)

and $\det^{\prime}$ is the fluctuation determinant without the contribution of
the zero mode. The latter arises from the invariance of the system under
translation of the jump and leads to the integration over the variable
$\tau_{1}$ in Eq. (9). We exchange the integration of fluctuations in the
direction of the time-translational zero mode $\sim\partial_{\tau}\phi_{K}$
for an integration over the location of the jump. In fact, the jump can be
located at any point in the time interval $(-\beta/2,\beta/2)$, and we must
sum over all possible such positions. This leads to the overall factor of
$\beta$. The factor of $\sqrt{\frac{K}{2\pi}}$ is the Jacobian of the
transformation to the collective coordinate $\tau_{1}$. Finally, $K$ is the
classical (Euclidean) action of the kink configuration, without $U_{0}$,

$K=\frac{2(mL)^{3}}{\lambda}~{}.$ (11)

We must also take into account the contribution of the multi-instantons,
together with their zero modes. These correspond to back-and-forth
oscillations between the two maxima of the upside-down potential. In the
dilute instanton gas approximation, we assume that the average separation
between two successive jumps is large compared to the width of a jump

$\Delta\tau\equiv\langle\tau_{i}-\tau_{i-1}\rangle\gg\omega^{-1}~{}.$ (12)

Then, the classical field configuration corresponding to $N$ successive jumps
starting at $-v$ for $\tau\to-\beta/2$ can be approximately written as follows

$\phi_{N-\mathrm{jumps}}\approx
v(-1)^{N+1}\tanh\omega(\tau-\tau_{1})\tanh\omega(\tau-\tau_{2})\dots\tanh\omega(\tau-\tau_{N})~{},$
(13)

where $\tau_{i}$ is the location of the $i$-th jump. Eq. (13) is valid for
both even and odd $N$. The above configurations are associated with their
“anti” type ones, which differ only by a minus sign. In Appendix A, we discuss
a condition for the dilute gas approximation to be valid.

The contribution to the partition function of the $N$-jumps configuration is
given by

$\left[K_{1}\overline{K}_{2}\ldots
K_{N-1}\overline{K}_{N}\right]\approx\int_{-\frac{\beta}{2}}^{\frac{\beta}{2}}d\tau_{1}\int_{\tau_{1}}^{\frac{\beta}{2}}d\tau_{2}\dots\int_{\tau_{N-1}}^{\frac{\beta}{2}}d\tau_{N}[I]^{N}[v]=\frac{\beta^{N}}{N!}[I]^{N}[v]\,,$
(14)

where the integration region for the location of the jumps is fixed by an
instanton always being followed by an anti-instanton and vice versa, a
condition imposed by only having two degenerate minima. Note that once again,
we have isolated the contribution of fluctuations around the jumps from the
contribution of fluctuations around the constant configuration.

The full partition function of the theory at quadratic order in the
fluctuations is finally

$Z_{E}\approx
2\sum_{N=0}^{\infty}\frac{\beta^{N}}{N!}[I]^{N}[v]=2[v]e^{\beta[I]}\,,$ (15)

where the factor of 2 is introduced by the two options $\phi(-\beta/2)=\pm v$.
The ground state energy can then be expressed in terms of $[v]$ and $[I]$
using Eq. (3), we find

$E_{0}=-\lim_{\beta\to\infty}\frac{1}{\beta}\log[v]-[I]=E_{\mathrm{stat}}+E_{\mathrm{inst}}\,.$
(16)

As explained in Section 3, the contribution $E_{\text{stat}}$ of the static
saddle points to the ground state energy can be identified with the Casimir
energy once we renormalise it. The ratio of functional determinants appearing
in $[I]$ is calculated in Section 4.

##  3 Casimir energy

The energy contribution of the static saddle points $E_{\mathrm{stat}}$ can be
easily written down

$\displaystyle E_{\mathrm{stat}}=$
$\displaystyle-\lim_{\beta\to\infty}\frac{1}{\beta}\log[v]$ $\displaystyle=$
$\displaystyle
L^{3}U_{0}+\frac{1}{2}\lim_{\beta\to\infty}\frac{1}{\beta}\mathrm{tr}\log(-\partial_{\tau}^{2}-\nabla^{2}+m^{2})$
$\displaystyle=$ $\displaystyle
L^{3}U_{0}+\frac{1}{2}\sum_{\mathbf{n}\in\mathbb{Z}^{3}}\sqrt{\mathbf{k}_{n}^{2}+m^{2}}\,,$
(17)

where $\mathbf{k}_{n}=\frac{2\pi}{L}|\mathbf{n}|$. To arrive at the last
equality, we have used

$\displaystyle{\rm
tr}\log(-\partial_{\tau}^{2}-\nabla^{2}+m^{2})=\beta\int\frac{d\omega}{2\pi}\sum_{\mathbf{n}\in\mathbb{Z}^{3}}\log(\omega^{2}+\mathbf{k}_{n}^{2}+m^{2})$
(18)

and performed the integral over $\omega$. The energy $E_{\mathrm{stat}}$ is
nothing but the vacuum energy $E_{0}^{(3)}$ of a massive scalar field on a
three-dimensional torus. We can rewrite Eq. (3) as

$E_{\rm stat}=L^{3}U_{0}+E_{\rm
Casimir}^{(3)}+\frac{L^{3}}{2}\int_{\mathbb{R}^{3}}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\sqrt{\mathbf{k}^{2}+m^{2}}\,,$
(19)

where we have adopted the standard definition of the Casimir energy Bordag .
$E_{\rm Casimir}^{(3)}$ is finite. The last divergent term will be eventually
dealt with in renormalisation, see Eq. (52b) and below. To compute the Casimir
energy, we use the Abel-Plana formula

$\sum_{n=0}^{\infty}f(n)-\int_{0}^{\infty}dt\,f(t)=\frac{1}{2}f(0)+i\int_{0}^{\infty}dt\,\frac{f(it)-f(-it)}{e^{2\pi
t}-1}$ (20)

which holds for any function $f(z)$ holomorphic in the region ${\rm Re}\,z\geq
0$ that vanishes fast enough towards infinity such that both the series and
the integral are convergent. Thus, to use the formula, we must multiply the
term in the sum $\sqrt{\mathbf{k}_{n}^{2}+m^{2}}$ by a regulator $\eta(n)$,
namely a smooth function that decays fast towards infinity and such that
$\eta(0)=1$. As explained in Bordag , we never need to state our regulator
explicitly, as we will find a result that is completely independent of it. The
difference between the discrete and the continuum vacuum energy, yields the
Casimir energy due to the compactness of the space

$\displaystyle E_{\mathrm{Casimir}}^{(3)}(m;L,L,L)=\>$ $\displaystyle
E_{0}^{(3)}(m;L,L,L)-E_{\infty}^{(3)}(m;L,L,L)$ $\displaystyle=\>$
$\displaystyle\frac{1}{2}\sum_{\mathbf{n}\in\mathbb{Z}^{3}}\sqrt{\mathbf{k}_{n}^{2}+m^{2}}-\frac{L^{3}}{2}\int_{\mathbb{R}^{3}}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\sqrt{\mathbf{k}^{2}+m^{2}}\,,$
(21)

where our notation specifies that we are considering a three-torus of equal
sizes of length $L$. The details of the calculations are found in Appendix B.
In the following, we outline the main steps.

It is easier to start by computing the Casimir energy on a one-dimensional
space with periodic boundary conditions, namely a circle. The result for it is
known (see Bordag )

$E_{\mathrm{Casimir}}^{(1)}(m;L)=\>-\frac{1}{\pi L}F_{(1,0)}(mL)\,,$ (22)

where the function $F_{(1,0)}(x)$ is defined as

$\displaystyle
F_{(1,0)}(x)=\int_{x}^{\infty}dy\,\frac{\sqrt{y^{2}-x^{2}}}{e^{y}-1}~{}.$ (23)

We highlight that $F_{(1,0)}(x)$ is a smooth function over the whole positive
real line, and has the following asymptotic behaviours

$\displaystyle F_{(1,0)}(0)=\>$ $\displaystyle\frac{\pi^{2}}{6}\,,$ (24)
$\displaystyle F_{(1,0)}(mL\gg 1)=\>$
$\displaystyle\sqrt{\frac{\pi}{2}}\sqrt{mL}e^{-mL}\left(1+\mathcal{O}((mL)^{-1})\right)\,,$
(25)

namely for very small lengths $L$ the massless behaviour dominates the Casimir
energy, while as $L$ grows an exponential suppression kicks in.

Now we can use the recursive formula proven in Appendix B, valid for the
Casimir energy of a real massive scalar field on a $D$-dimensional torus of
sizes $a_{1},\ldots,a_{D}$ where ${D>1}$

$\displaystyle E_{\mathrm{Casimir}}^{(D)}(m;a_{1},\ldots,a_{D})=\>$
$\displaystyle-\frac{1}{\pi^{D}a_{D}}\left(\prod_{i=1}^{D-1}\frac{a_{i}}{a_{D}}\right)F_{(D,0)}(ma_{D})$
$\displaystyle+\sum_{n=-\infty}^{\infty}E_{\mathrm{Casimir}}^{(D-1)}\left(\sqrt{m^{2}+\frac{4\pi^{2}n^{2}}{a_{D}^{2}}};a_{1},\ldots,a_{D-1}\right)\,,$
(26)

where

$\displaystyle
F_{(D,0)}(x)=\>\left(\prod_{i=1}^{D-1}\int_{0}^{\infty}dy_{i}\right)F_{(1,0)}\left(\scriptstyle{\sqrt{x^{2}+\sum_{i=1}^{D-1}y_{i}^{2}}}\right)=\>\int_{0}^{\infty}dyF_{(D-1,0)}\left(\sqrt{x^{2}+y^{2}}\right)\,.$
(27)

Using the formula (3), we find the Casimir energy for a real massive scalar
field on a generic three-torus in Eq. (B.2.2). Setting equal sizes
$a_{1}=a_{2}=a_{3}=L$ we finally find444From now on, we write
$E_{\mathrm{Casimir}}^{(3)}(m;L)$ instead of
$E_{\mathrm{Casimir}}^{(3)}(m;L,L,L)$ since we will only be considering a
torus with sides of equal length.

$E_{\mathrm{Casimir}}^{(3)}(m;L)=\>-\frac{1}{\pi^{3}L}\left[F_{(3,0)}(mL)+\pi
F_{(2,1)}(1,mL)+\pi^{2}F_{(1,2)}(1,1,mL)\right]\,,$ (28)

where the $F_{(i,j)}$ are defined as

$\displaystyle
F_{(i,1)}(r,x)=\>\sum_{n=-\infty}^{\infty}F_{(i,0)}(r\sqrt{x^{2}+4\pi^{2}n^{2}})\,,$
(29a) $\displaystyle
F_{(i,2)}(r_{1},r_{2},x)=\>\sum_{n=-\infty}^{\infty}F_{(i,1)}(r_{1},r_{2}\sqrt{x^{2}+4\pi^{2}n^{2}})\,.$
(29b)

##  4 The tunneling contribution

We now turn to the calculation of the tunnelling (or instanton) contribution
to the energy. In Section 2.2 we have derived

$E_{\mathrm{inst}}=\>-[I]=\>-\sqrt{\frac{K}{2\pi}}e^{-K}\left(\frac{\sideset{}{{}^{\prime}}{\det}\left(-\partial_{\tau}^{2}-\nabla^{2}+\frac{m^{2}}{2}\left(\frac{3\phi_{K}^{2}}{v^{2}}-1\right)\right)}{\det(-\partial_{\tau}^{2}-\nabla^{2}+m^{2})}\right)^{-\frac{1}{2}}\,.$
(30)

In this section, we compute the ratio of functional determinants. To do so, we
employ the resolvent method, developed in Baacke:1993jr ; Baacke:1993aj ;
Baacke:1994ix ; Baacke:2008zx  and used in, e.g., Garbrecht:2015oea ;
Ai:2018guc ; Ai:2020sru ; Ai:2019fri . The method states that the ratio of the
determinants of two operators can be written as an integral of some modified
Green’s function. Explicitly

$\log\frac{\det\hat{A}}{\det\hat{B}}=-\int_{0}^{\infty}ds\int
d^{4}x\left(G_{A}(s;x,x)-G_{B}(s;x,x)\right)\,,$ (31)

where the modified Green’s functions known as resolvents are defined through
the differential equation

$\left(\hat{A}+s\right)G_{A}(s;x,y)=\delta^{(4)}(x-y)\,,$ (32)

and similarly for the operator $\hat{B}$. To see Eq. (31), we write the
modified Green’s functions in its spectral representation, e.g.,

$\displaystyle
G_{A}(s;x,x^{\prime})=\sum_{n}\frac{\phi^{*}_{A,n}(x)\phi_{A,n}(x^{\prime})}{\lambda_{A,n}+s}\,,\qquad\text{where}\qquad\hat{A}\phi_{A,n}=\lambda_{A,n}\phi_{A,n}\,.$
(33)

Then

$\displaystyle\int_{0}^{S}ds\int
d^{4}x\,G_{A}(s;x,x)=-\sum_{n}\log\frac{\lambda_{A,n}}{\lambda_{A,n}+S}\,,$
(34)

where we used that the eigenfunctions $\phi_{A,n}$ are normalized to one.
Therefore,

$\displaystyle-\int_{0}^{S}ds\int
d^{4}x\left(G_{A}(s;x,x)-G_{B}(s;x,x)\right)=\sum_{n}\log\frac{\lambda_{A,n}}{\lambda_{B,n}}-\sum_{n}\log\frac{\lambda_{A,n}+S}{\lambda_{B,n}+S}\,.$
(35)

When taking $S\rightarrow\infty$, the second term on the RHS vanishes.

###  4.1 Finding the modified Green’s function

To compute the ratio of determinants appearing in Eq. (30) via the resolvent
method, we must find two modified Green’s functions. We will first evaluate
the more complicated one, namely for the fluctuation operator around the
instanton configuration. The equation reads

$\left(-\partial_{\tau}^{2}-\nabla^{2}+\frac{m^{2}}{2}\left(\frac{3\phi_{K}^{2}}{v^{2}}-1\right)+s\right)G_{K}(s;x,x^{\prime})=\delta^{(4)}(x-x^{\prime})\,.$
(36)

Because of invariance under spatial translations, we can make the following
ansatz

$G_{K}(s;x,x^{\prime})=\frac{1}{L^{3}}\sum_{\mathbf{n}\in\mathbb{Z}^{3}}e^{-i\mathbf{k}_{n}\cdot(\mathbf{x}-\mathbf{x}^{\prime})}F^{(K)}_{n}(s;\tau,\tau^{\prime})\,,$
(37)

where the discretised momentum is $\mathbf{k}_{n}=\frac{2\pi}{L}\mathbf{n}$.
After changing to variable $u=\tanh\omega\tau$, the equation for the
$\tau$-dependent part becomes

$\left(\frac{d}{du}(1-u^{2})\frac{d}{du}-\frac{\mu_{n}^{2}}{1-u^{2}}+6\right)\frac{m}{2}F^{(K)}_{n}(s;u,u^{\prime})=-\delta(u-u^{\prime})\,,$
(38)

where we have introduced the dimensionless quantity
$\mu_{n}=2\sqrt{1+\frac{\mathbf{k}_{n}^{2}}{m^{2}}+\frac{s}{m^{2}}}$.

The solution to the equation above can be obtained in terms of associated
Legendre functions $P_{2}^{\mu_{n}}(u)$ and $Q_{2}^{\mu_{n}}(u)$
Garbrecht:2018rqx ; Ai:2019fri . It reads Garbrecht:2015oea

$\displaystyle
F_{n}^{(K)}(s;u,u^{\prime})=\frac{2}{m}\frac{1}{2\mu_{n}}\Big{[}\theta(u-u^{\prime})\left(\frac{1-u}{1+u}\right)^{\frac{\mu_{n}}{2}}\left(\frac{1+u^{\prime}}{1-u^{\prime}}\right)^{\frac{\mu_{n}}{2}}$
$\displaystyle\qquad\quad\times\small{\left(1-3\frac{(1-u)(1+\mu_{n}+u)}{(1+\mu_{n})(2+\mu_{n})}\right)\left(1-3\frac{(1-u^{\prime})(1-\mu_{n}+u^{\prime})}{(1-\mu_{n})(2-\mu_{n})}\right)}+(u\leftrightarrow
u^{\prime})\Big{]}\,.$ (39)

Note that the function above has a pole for $\mu_{n}=2$, namely for $n=0$ and
$s=0$. This is not surprising: the instanton background breaks the time-
translational symmetry of the theory, thus inducing a zero mode that appears
as a simple pole in the physical Green’s function $G(0;x,x^{\prime})$. We need
to define the subtracted Green’s function on the space orthogonal to the zero
mode. Note that in the spectral decomposition, the modified Green’s function
is given by

$G(s;x,x^{\prime})=\sum_{\lambda}\frac{\phi_{\lambda}(x)\phi_{\lambda}^{*}(x^{\prime})}{\lambda+s}=\frac{\phi_{0}(x)\phi_{0}^{*}(x^{\prime})}{s}+\sideset{}{{}^{\prime}}{\sum}_{\lambda}\frac{\phi_{\lambda}(x)\phi_{\lambda}^{*}(x^{\prime})}{\lambda+s}\,.$
(40)

Thus, to isolate the contribution of the zero mode $\phi_{0}$ it is enough to
look at the leading $1/s$ behaviour in $F_{0}^{(K)}$

$F_{0}^{(K)}(s;u,u^{\prime})=\frac{3}{2m}\frac{(1-u^{2})(1-u^{\prime
2})}{\mu_{0}^{2}-4}+\text{finite terms for }s\to 0\,.$ (41)

By subtracting this term from expression (4.1), we find the time-dependent
part of the modified subtracted Green’s function

$\displaystyle F_{n}^{(K)\,\perp}(s;u,u^{\prime})=$
$\displaystyle\frac{2}{m}\frac{1}{2\mu_{n}}\left[\theta(u-u^{\prime})\left(\frac{1-u}{1+u}\right)^{\frac{\mu_{n}}{2}}\left(\frac{1+u^{\prime}}{1-u^{\prime}}\right)^{\frac{\mu_{n}}{2}}\right.$
$\displaystyle\times\left.\left(1-3\frac{(1-u)(1+\mu_{n}+u)}{(1+\mu_{n})(2+\mu_{n})}\right)\left(1-3\frac{(1-u^{\prime})(1-\mu_{n}+u^{\prime})}{(1-\mu_{n})(2-\mu_{n})}\right)+(u\leftrightarrow
u^{\prime})\right]$
$\displaystyle-\delta_{n,0}\frac{3}{2m}\frac{(1-u^{2})(1-u^{\prime
2})}{\mu_{n}^{2}-4}\,.$ (42)

The coincident limit of the modified Green’s function, needed for the
resolvent method, is

$\displaystyle G_{K}^{\perp}(s;x,x)=\>$
$\displaystyle\frac{2}{mL^{3}}\Bigg{\\{}\frac{1}{2\mu_{0}}+\frac{3(1-u^{2})}{2\mu_{0}}\left(\frac{u^{2}}{\mu_{0}^{2}-1}-\frac{1}{2}\frac{1-u^{2}}{\mu_{0}+2}\right)+$
$\displaystyle\qquad+\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}\in\mathbb{Z}^{3}}\frac{1}{2\mu_{n}}\left(1+3(1-u^{2})\sum_{\kappa=1}^{2}(-1)^{\kappa}\frac{\kappa-1-u^{2}}{\mu_{n}^{2}-\kappa^{2}}\right)\Bigg{\\}}\,,$
(43)

where the primed sum runs over $\mathbb{Z}^{3}\backslash\\{0,0,0\\}$.

We must now compute the modified Green’s function for the fluctuation operator
around the static saddle points. The differential equation reads

$(-\partial_{\tau}^{2}-\nabla^{2}+m^{2}+s)G_{v}(s;x,x^{\prime})=\delta^{(4)}(x-x^{\prime})\,.$
(44)

Using the same ansatz and the same change of variable we performed above, we
get to the equation for the time-dependent part

$\left(\frac{d}{du}(1-u^{2})\frac{d}{du}-\frac{\mu_{n}^{2}}{1-u^{2}}\right)\frac{m}{2}F^{(v)}_{n}(s;u,u^{\prime})=-\delta(u-u^{\prime})\,,$
(45)

which can be solved in terms of associated Legendre functions of the first and
second kind $P_{0}^{\mu_{n}}(u)$ and $Q_{0}^{\mu_{n}}(u)$. See for example
Garbrecht:2018rqx . We have Ai:2019fri

$\displaystyle F^{(v)}_{n}(s;u,u^{\prime})=$
$\displaystyle\frac{2}{m}\frac{1}{2\mu_{n}}\left[\theta(u-u^{\prime})\left(\frac{1-u}{1+u}\right)^{\frac{\mu_{n}}{2}}\left(\frac{1+u^{\prime}}{1-u^{\prime}}\right)^{\frac{\mu_{n}}{2}}+(u\leftrightarrow
u^{\prime})\right]\,,$ (46)

and in the coincident limit

$G_{v}(s;x,x)=\>\frac{2}{mL^{3}}\sum_{\mathbf{n}\in\mathbb{Z}^{3}}\frac{1}{2\mu_{n}}\,.$
(47)

###  4.2 Computing the renormalised ratio of functional determinants

We have all the ingredients we need to apply the resolvent method in Eq.(31).
We have

$\displaystyle\log\frac{\sideset{}{{}^{\prime}}{\det}S_{E}^{\prime\prime}[\phi_{K}]}{\det
S_{E}^{\prime\prime}[v]}=$ $\displaystyle-\int_{0}^{\infty}ds\int
d^{3}\mathbf{x}\int_{-\infty}^{\infty}d\tau\left(G_{K}^{\perp}(s;x,x)-G_{v}(s;x,x)\right)$
$\displaystyle=$
$\displaystyle-\int_{0}^{\infty}ds\>\int_{-1}^{1}\frac{2du}{m(1-u^{2})}\frac{2}{m}\Bigg{\\{}\frac{3(1-u^{2})}{2\mu_{0}}\left(\frac{u^{2}}{\mu_{0}^{2}-1}-\frac{1}{2}\frac{1-u^{2}}{\mu_{0}+2}\right)+$
$\displaystyle\qquad+\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}\in\mathbb{Z}^{3}}\frac{1}{2\mu_{n}}3(1-u^{2})\sum_{\kappa=1}^{2}(-1)^{\kappa}\frac{\kappa-1-u^{2}}{\mu_{n}^{2}-\kappa^{2}}\Bigg{\\}}$
$\displaystyle=$ $\displaystyle-\log
3+\frac{4}{m^{2}}\int_{0}^{\infty}ds\>\frac{1}{\mu_{0}(\mu_{0}+2)}$
$\displaystyle-2\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}\in\mathbb{Z}^{3}}\log\frac{3m^{2}+2\mathbf{k}_{n}^{2}+3m\sqrt{\mathbf{k}_{n}^{2}+m^{2}}}{|\mathbf{k}_{n}|\sqrt{4\mathbf{k}_{n}^{2}+3m^{2}}}\,.$
(48)

The expression above still contains a divergent integral

$\frac{4}{m^{2}}\int_{0}^{S}ds\>\frac{1}{\mu_{0}(\mu_{0}+2)}=-\log 4m^{2}+\log
S+\mathcal{O}(S^{-1})\to\infty\,.$

This arises because of the mismatch in the number of eigenmodes between the
two operators in the ratio, a consequence of having subtracted the zero mode
from the fluctuation operator around the kink. Specifically, this can be seen
from Eq. (35). Removing the zero mode from $G_{A}$ would lead to a divergent
contribution $\log(\lambda_{B,0}+S)\sim\log S$ for $S\rightarrow\infty$ from
the second term on the RHS. Therefore, this divergence is a mere artefact of
removing a zero mode in the resolvent-formula for the functional determinants
Eq. (31) and should be dropped out. We find an expression for the un-
renormalised logarithm of the ratio of determinants

$\displaystyle\frac{1}{2}\log\frac{\sideset{}{{}^{\prime}}{\det}S_{E}^{\prime\prime}[\phi_{K}]}{\det
S^{\prime\prime}[v]}=$ $\displaystyle-\frac{1}{2}\log
12m^{2}-\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}\in\mathbb{Z}^{3}}\log\frac{3m^{2}+2\mathbf{k}_{n}^{2}+3m\sqrt{\mathbf{k}_{n}^{2}+m^{2}}}{|\mathbf{k}_{n}|\sqrt{4\mathbf{k}_{n}^{2}+3m^{2}}}\,.$
(49)

We can identify this quantity as the one-loop correction to the 1PI effective
action Jackiw:1974cv  around the kink configuration, having subtracted the
correction around the static one.

The divergences from the last term in Eq. (19) and in (49) can only be removed
via renormalisation of the one-loop effective action $\Gamma^{(1)}[\phi]$.
With the counter-terms, we have explicitly

$\displaystyle\Gamma^{(1)}[\phi_{K}]$
$\displaystyle-\Gamma^{(1)}[v]=\>S[\phi_{K}]-S[v]+\frac{1}{2}\log\frac{\det
S^{\prime\prime}_{E}[\phi_{K}]}{\det
S^{\prime\prime}_{E}[v]}+S_{c.t.}[\phi_{K}]-S_{c.t.}[v]$ $\displaystyle=\>$
$\displaystyle K+\frac{1}{2}\log\frac{\det S^{\prime\prime}_{E}[\phi]}{\det
S^{\prime\prime}_{E}[v]}+\int_{0}^{\infty}d\tau\int
d^{3}\mathbf{x}\left(\frac{\delta
m^{2}}{2}(\phi_{K}^{2}-v^{2})+\frac{\delta\lambda}{4!}(\phi_{K}^{4}-v^{4})\right)\,.$
(50)

Note that once the counter-terms have been added, all the coupling constants,
including the mass and $U_{0}$, are the renormalised ones and are to be
intended as such in what follows. For clarity, we still use the same notations
as for the bare quantities. Then the instanton contribution to the energy is

$E_{\mathrm{inst}}=\>-e^{-\left(\Gamma^{(1)}[\phi_{K}]-\Gamma^{(1)}[v]\right)}\,.$
(51)

Since the zero-point energy $U_{0}$ is field independent, it is cancelled out
in the difference. The UV divergences are sensitive to neither the background
nor the topology of space. Therefore, we can find $\delta m^{2}$ and
$\delta\lambda$ by simply renormalising the one-loop effective potential in
the infinite volume limit that is obtained from the effective action for
constant field backgrounds,

$\displaystyle\qquad\qquad\qquad\left.\Gamma^{(1)}[\phi]\right|_{\phi\text{ is
constant}}=\int d^{4}x\,U_{\rm eff}(\phi)\,,$ (52a) $\displaystyle\Rightarrow
U_{\rm
eff}(\phi)=U(\phi)+\frac{1}{2}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\sqrt{\mathbf{k}^{2}+U^{\prime\prime}(\phi)}+\frac{\delta
m^{2}}{2}\phi^{2}+\frac{\delta\lambda}{4!}\phi^{4}+\delta U_{0}\,.$ (52b)

The integral above is divergent, and we need to choose a regularisation
procedure that we will use to regulate both the integral in the continuum
limit and its corresponding series. We use a smooth-regulator

$\eta\left(\frac{|\mathbf{k}|}{\Lambda}\right)=\>e^{-\frac{|\mathbf{k}|^{2}}{\Lambda^{2}}}\,,$
(53)

where $\Lambda$ is some large cut-off scale. With this, the integral reads

$\displaystyle\frac{1}{2}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\sqrt{\mathbf{k}^{2}+U^{\prime\prime}(\phi)}\,e^{-\frac{\mathbf{k}^{2}}{\Lambda^{2}}}$
$\displaystyle\qquad=\frac{[U^{\prime\prime}(\phi)]^{2}}{128\pi^{2}}\left[\frac{16\Lambda^{4}}{(U^{\prime\prime}[\phi])^{2}}+\frac{8\Lambda^{2}}{U^{\prime\prime}(\phi)}+1+2\gamma_{E}+2\log\frac{U^{\prime\prime}(\phi)}{4\Lambda^{2}}\right]+\mathcal{O}\left(\frac{\sqrt{U^{\prime\prime}(\phi)}}{\Lambda}\right)\,.$
(54)

We choose the following renormalisation conditions

$\displaystyle U_{\mathrm{eff}}(\phi)\rvert_{\phi=\pm v}=\>$ $\displaystyle
U_{0}\,,$ (55a)
$\displaystyle\frac{d^{2}U_{\mathrm{eff}}(\phi)}{d\phi^{2}}\rvert_{\phi=\pm
v}=\>$ $\displaystyle m^{2}\,,$ (55b)
$\displaystyle\frac{d^{4}U_{\mathrm{eff}}(\phi)}{d\phi^{4}}\rvert_{\phi=\pm
v}=\>$ $\displaystyle\lambda\,.$ (55c)

Imposing the renormalisation conditions fixes the counter-terms

$\displaystyle\delta U_{0}=$
$\displaystyle-\frac{m^{4}}{256\pi^{2}}\left(143+\gamma_{E}-\frac{8\Lambda^{2}}{m^{2}}+\frac{32\Lambda^{4}}{m^{4}}-\log\frac{4\Lambda^{2}}{m^{2}}\right)\,,$
(56a) $\displaystyle\delta m^{2}=$ $\displaystyle-\frac{\lambda
m^{2}}{64\pi^{2}}\left(-31-\gamma_{E}+\frac{4\Lambda^{2}}{m^{2}}+\log\frac{4\Lambda^{2}}{m^{2}}\right)\,,$
(56b) $\displaystyle\delta\lambda=$
$\displaystyle-\frac{3\lambda^{2}}{32\pi^{2}}\left(5+\gamma_{E}-\log\frac{4\Lambda^{2}}{m^{2}}\right)\,.$
(56c)

With the above renormalisation procedure, the divergent term in Eq. (19) has
essentially been regularised by the counter-term $\delta U_{0}$, leaving the
renormalised, physical $U_{0}$ through the condition (55a).

The contribution of the counter-terms to the effective action reads

$\displaystyle S_{c.t.}[\phi_{K}]-S_{c.t.}[v]=\>$
$\displaystyle\int_{-\infty}^{\infty}d\tau\int
d^{3}\mathbf{x}\left(\frac{\delta
m^{2}}{2}(\phi_{K}^{2}-v^{2})+\frac{\delta\lambda}{4!}(\phi_{K}^{4}-v^{4})\right)$
$\displaystyle=\>$ $\displaystyle-2m^{3}L^{3}\left(\frac{3\delta
m^{2}}{m^{2}\lambda}+\frac{\delta\lambda}{\lambda^{2}}\right)$
$\displaystyle=\>$
$\displaystyle+\frac{3m^{3}L^{3}}{32\pi^{2}}\left(-21+\gamma_{E}+\frac{4\Lambda^{2}}{m^{2}}-\log\frac{4\Lambda^{2}}{m^{2}}\right)\,.$
(57)

The resulting expression for the instanton contribution to the ground state
energy is

$\displaystyle E_{\mathrm{inst}}=$
$\displaystyle-\sqrt{12m^{2}}\sqrt{\frac{m^{3}L^{3}}{\pi\lambda}}\exp\bigg{\\{}-\frac{2m^{3}L^{3}}{\lambda}$
$\displaystyle\qquad+\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}\in\mathbb{Z}^{3}}e^{-\frac{k_{n}^{2}}{\Lambda^{2}}}\log\frac{3m^{2}+2k_{n}^{2}+3m\sqrt{k_{n}^{2}+m^{2}}}{k_{n}\sqrt{4k_{n}^{2}+3m^{2}}}$
$\displaystyle\qquad-\frac{3m^{3}L^{3}}{32\pi^{2}}\left(-21+\gamma_{E}+\frac{4\Lambda^{2}}{m^{2}}-\log\frac{4\Lambda^{2}}{m^{2}}\right)\bigg{\\}}\,.$
(58)

We can check that these counter-terms correctly renormalise the series by
considering the continuum limit of the latter, and extracting its UV
divergences. We find

$\displaystyle\sideset{}{{}^{\prime}}{\sum}_{\mathbf{n}\in\mathbb{Z}^{3}}e^{-\frac{\mathbf{k}_{n}^{2}}{\Lambda^{2}}}$
$\displaystyle\log\frac{3m^{2}+2\mathbf{k}_{n}^{2}+3m\sqrt{\mathbf{k}_{n}^{2}+m^{2}}}{|\mathbf{k}_{n}|\sqrt{4\mathbf{k}_{n}^{2}+3m^{2}}}$
$\displaystyle\longrightarrow\>$ $\displaystyle
L^{3}\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}e^{-\frac{\mathbf{k}^{2}}{\Lambda^{2}}}\log\frac{3m^{2}+2\mathbf{k}^{2}+3m\sqrt{\mathbf{k}^{2}+m^{2}}}{|\mathbf{k}|\sqrt{4\mathbf{k}^{2}+3m^{2}}}$
$\displaystyle=\>$
$\displaystyle\frac{L^{3}}{2\pi^{2}}\int_{\epsilon}^{\infty}d|\mathbf{k}|\>\mathbf{k}^{2}e^{-\frac{\mathbf{k}^{2}}{\Lambda^{2}}}\left(\frac{3m}{2|\mathbf{k}|}-\frac{3m^{3}}{8|\mathbf{k}|^{3}}+\mathcal{O}\left(\frac{m^{5}}{|\mathbf{k}|^{5}}\right)\right)$
$\displaystyle=\>$
$\displaystyle\frac{3m^{3}L^{3}}{32\pi^{2}}\left(\frac{4\Lambda^{2}}{m^{2}}-\log\frac{4\Lambda^{2}}{m^{2}}\right)+\text{finite
terms}\,,$ (59)

which are indeed cancelled by the counter-term contribution in Eq. (4.2).

The result in Eq. (4.2) includes the full one-loop contribution of the
fluctuations around the jump. In Refs. Alexandre:2023iig ; Alexandre:2023pkk ,
the instanton contribution to the ground state energy is calculated by only
keeping homogeneous fluctuations. This can be obtained from the result in Eq.
(4.2) by neglecting the contributions from $\mathbf{n}\neq\mathbf{0}$
fluctuations, namely

$E^{(\mathrm{hom})}_{\mathrm{inst}}=-\sqrt{12m^{2}}\sqrt{\frac{m^{3}L^{3}}{\pi\lambda}}\exp\left\\{-\frac{2m^{3}L^{3}}{\lambda}\right\\}\,.$
(60)

The calculation of this result resembles one of a tunnelling problem in
Quantum Mechanics (QM). Therefore, we will refer to Eqs. (60) and (4.2) as QM
and QFT results, respectively. The inhomogeneous fluctuations yield additional
one-loop corrections, which we expect to be perturbatively small compared to
the QM result, at least for small coupling $\lambda$. In Figure 2, we plot the
QM and the QFT results. As expected, the smaller $\lambda$ the smaller the
change in the instanton energy. In particular, for $\lambda\ll 1$, the QM
result approximates the full QFT result really well. This justifies the
approximations made in Refs. Alexandre:2023iig ; Alexandre:2023pkk , where the
spatial dependence of the quantum fluctuations was ignored. Interestingly, for
larger $\lambda$ the peak of the energy in the QFT result shrinks and moves to
smaller values of $mL$ when compared to the QM one.

(a) $\lambda=0.001$

(b) $\lambda=0.01$

(c) $\lambda=0.1$

(d) $\lambda=1$

Figure 2: Comparison between the quantum mechanical approximation and the full
QFT result of $E_{\rm inst}$, as defined in Eq. (60) and Eq. (4.2)
respectively.

###  4.3 Comparing with the Casimir energy

We can now compare the relative contribution of the Casimir effect and the
tunnelling to the ground state energy. We show the comparison in Figure 3. At
first glance, we see that the Casimir energy is generally larger than the
instanton one. However, for $mL\sim 1$ the latter can become of the same order
of magnitude as the former, as can be seen in the bottom panel of Figure 3.

(a)

(b)

Figure 3: Comparison of the contributions to the ground state energy as
defined in Eq. (28) and (4.2) for various values of $\lambda$.

Because of its relevance for violating the Null Energy Condition in Eq. (63),
we are primarily interested in the sum of energy density and pressure, which
in the thermodynamic limit can be written as

$\rho+p=\>\frac{E_{0}}{V}-\frac{\mathrm{d}E_{0}}{\mathrm{d}V}\,,$ (61)

where $E_{0}$ is the renormalised ground state energy and $V=L^{3}$ is the
spatial volume. Specifically, the energy $E_{0}$ reads

$E_{0}=L^{3}U_{0}+E_{\textrm{Casimir}}^{(3)}+E_{\textrm{inst}}\,,$ (62)

where $E_{\textrm{Casimir}}^{(3)}$ is given in Eq. (28) and
$E_{\textrm{inst}}$ is given in Eq. (4.2). In Figure 4, we plot the relative
contribution of the Casimir effect and the instantons to $\rho+p$. The top
panel clearly shows that for large enough coupling $\lambda$ the effect of
instantons becomes of the same order as the Casimir effect. In the bottom
panel $\rho+p$ is plotted, both ignoring and including instanton effects for
$\lambda=1$. The instanton effects result in a steeper curve, thus making the
NEC violation stronger.

(a)

(b)

Figure 4: Comparison of the contributions to $\rho+p$ as defined in Eq. (61)
from the tunnelling and Casimir effects, respectively. Panel (a): the ratio of
the contribution of the instantons over the contribution of the Casimir effect
is shown for various values of $\lambda$. Panel (b): the sum of energy density
and pressure is shown with and without instanton effects for $\lambda=1$.

##  5 Relevance to Cosmology

Singularity theorems in general relativity show that singularities are
inevitable under very general circumstances Hawking:1970zqf . However, they
have in their assumptions some restrictions on the energy-momentum tensor for
matter fields $T_{\mu\nu}$. In particular, it is assumed to satisfy the Null
Energy Condition $T_{\mu\nu}\ell^{\mu}\ell^{\nu}\geq 0$ where $\ell^{\mu}$ is
a null vector. For a perfect fluid, the Null Energy Condition reduces to

$\rho+p\geq 0\,,$ (63)

where $\rho$ and $p$ are the energy density and the pressure of the fluid
respectively. Therefore, identifying new NEC-violating mechanisms within
quantum field theory holds particular significance in cosmology since the
classical singularity theorems do not apply, and the cosmological singularity
can be potentially avoided. In particular, the mechanism described here can
induce a semi-classical bounce when the effective theory is coupled to gravity
in a closed universe.555Some recent works in cosmology have considered the
evidence for a closed universe, see for example Refs. DiValentino:2019qzk ;
2021PhRvD.103d1301H .

The analysis of the Casimir effect in a cosmological context was introduced
for the first time in Ref. Zeldovich:1984vk . Since then, several authors
studied its possible effects in cosmology (see Ref. Bordag  for a review). In
particular, in Ref. Herdeiro:2005zj  the backreaction problem is discussed
when considering the Casimir energy of a massless field, and static solutions
to the Friedmann equations are derived for a closed universe with positive
curvature. Other works based on the Casimir effect, have considered the
possibility to induce an accelerating universe Szydlowski:2007bg ;
Godlowski:2007gx ; Herdeiro:2005zj , but did not consider tunnelling. We also
mention here works involving the scalar Casimir effect and related to non-
trivial universe topology. The vacuum energy for a scalar field is evaluated
in either a de Sitter universe Saharian:2007ts  or an FLRW universe
Saharian:2010nep ; Pavlov:2020utn , with one or more compact extra dimensions.
Also, the stabilisation of the Einstein universe via the Casimir force is
discussed in Ref. Herdeiro:2007eb .

The effect of quantum tunnelling in cosmology was recently analysed in Ref.
Alexandre:2023pkk . Ignoring the Casimir energy and the space-dependence of
the quantum fluctuations for the “jump” part of the instanton, it is found
that tunnelling can indeed dynamically generate an effective equation of state
that supports a cosmological bounce when coupled to gravity. The energy
density and the pressure of the effective ground state were obtained, upon
assuming that the tunnelling rate is large compared to the expansion rate of
the universe,

$|H|\equiv\Big{|}\frac{\dot{a}}{a}\Big{|}\ll[I^{(0)}]\,,\quad\text{where}\quad[I^{(0)}]=\sqrt{12m^{2}}\sqrt{\frac{m^{3}L^{3}}{\pi\lambda}}\exp\bigg{\\{}-\frac{2m^{3}L^{3}}{\lambda}\bigg{\\}}~{},$
(64)

which allows the calculation of quantum fluctuations for a constant scale
factor. The time-dependence of the latter is then restored when coupling the
effective theory to gravity. In this context, the energy density and the
pressure of the ground state fluid read

$\displaystyle\rho^{(0)}(t)$
$\displaystyle=\frac{E_{0}^{(0)}(t)}{L(t)^{3}}=U_{0}-\sqrt{12m^{2}}\sqrt{\frac{m^{3}}{\pi\lambda
L(t)^{3}}}e^{-\frac{2m^{3}}{\lambda}L(t)^{3}}\,,$ (65a) $\displaystyle
p^{(0)}(t)$
$\displaystyle=-\frac{1}{3L(t)^{2}}\frac{\textrm{d}E_{0}^{(0)}(t)}{\textrm{d}L(t)}$
$\displaystyle=-U_{0}+\sqrt{12m^{2}}\sqrt{\frac{m^{3}}{\pi\lambda
L(t)^{3}}}\left(\frac{1}{2}-\frac{2m^{3}L(t)^{3}}{\lambda}\right)e^{-\frac{2m^{3}}{\lambda}L(t)^{3}}\,,$
(65b)

where $L(t)=a(t)L_{0}$, with $L_{0}$ being an arbitrary “length-cell”. It can
be easily seen that the NEC is violated for any value of the comoving volume
$a^{3}L_{0}^{3}$, and therefore a bouncing solution is obtained in FLRW
independently of the initial conditions as long as $H(t_{0})<0$.

An advantage of these finite-volume mechanisms is that they intrinsically
favour expansion over contraction since they are exponentially suppressed
after a period of expansion (i.e., for large volumes). Also, they do not rely
on modified gravity or exotic matter and are dynamically generated by a
shrinking co-volume.

Including the Casimir energy and one-loop corrections to the instanton jump
should not modify the qualitative results in Ref. Alexandre:2023pkk  since the
NEC is still violated for any value of the comoving volume, which means that
we expect to find a bouncing solution to the Friedmann equations if we start
from a contracting phase. However, quantitative results can be substantially
different since the Casimir contribution is very large. In Ref.
Alexandre:2023bih , an estimation for the size of the universe at the bounce
$L_{\textrm{bounce}}$ was given. Given that the NEC is always violated, the
cosmic bounce will happen when

$H^{2}=\frac{\dot{a}^{2}}{a^{2}}=0\,,\qquad\to\qquad
E_{0}=L^{3}U_{0}+E_{\textrm{Casimir}}^{(3)}+E_{\textrm{inst}}=0\,.$ (66)

From this equation, we can estimate $L_{\textrm{bounce}}$. For example, in the
simplified model analysed in Refs. Alexandre:2023pkk ; Alexandre:2023bih ,
where $E_{\textrm{cas}}^{(3)}=0$ and $E_{\rm inst}=E_{\rm
inst}^{(\mathrm{hom})}$ with $E_{\rm inst}^{(\mathrm{hom})}$ given in Eq.
(60), it was found $L_{\textrm{bounce}}\simeq 10^{-5}$ metres for
$U_{0}=\kappa^{-1}\Lambda=10^{-122}\,\ell_{p}^{-4}$, $m=10^{-3}$ eV and
$\lambda\sim 1$. Repeating the computation with the one-loop ground state
energy $E_{0}$ as obtained in this work, we obtain $L_{\textrm{bounce}}\simeq
10^{-2}$ metres instead.

##  6 Conclusion and outlook

A quantum field system in a finite volume and an infinite volume can have very
different behaviours. It is well known that a finite volume can generally
induce a negative contribution to the energy of the system, known as the
Casimir energy. When considering a field potential with degenerate minima, the
finite volume allows tunnelling between those degenerate minima, which would
otherwise be prohibited in the infinite-volume limit. The mentioned tunnelling
also induces a negative ground state energy when the potential of the
degenerate minima is normalised to zero and meanwhile restores the symmetry
that is classically broken by the degenerate minima. In this paper, we have
extensively studied both the Casimir and tunnelling effects for a scalar field
theory with a double-well potential in a three-torus.

We have computed the contributions to energy from the Casimir and the
tunnelling effects at the one-loop level. We derived analytic expressions for
the Casimir energy for an arbitrary $D$-dimensional torus using the Abel-Plana
formula. The tunnelling between the degenerate minima is described by a
homogeneous but time-dependent Euclidean bounce solution. At the one-loop
level, one has to integrate the fluctuations about the bounce solution as well
as the classical degenerate vacua, which then need to be properly
renormalised. This is achieved through the resolvent method Baacke:1993jr ;
Baacke:1993aj ; Baacke:1994ix ; Baacke:2008zx  and renormalisation via the
one-loop effective potential.

The finite volume constraint has remarkable energetic effects on the confined
system, allowing the violation of the Null Energy Condition. In addition to
the known Casimir effect, we show that the tunnelling effect also contributes
to NEC violation, extending the results in Ref. Alexandre:2023pkk . Depending
on the coupling constant, the tunnelling effect can be of a similar order of
magnitude as the Casimir effect. The violation of the NEC could induce a
cosmic bounce Alexandre:2023bih .

The present work may have several generalisations. The presence of fermions
(satisfying anti-periodic boundary conditions) could decrease or cancel the
Casimir contribution from the scalar field, provided the appropriate number of
degrees of freedom is considered.666The extension of the Casimir effect to
Supergravity in relation to Cosmology has been studied in Ref.
Goncharov:1987tz . In this situation, tunnelling would become the main origin
of NEC violation. Other geometries or topologies could be considered, for
example, a three-sphere instead of a three-torus, which may be relevant to
Cosmology.

###### Acknowledgements.

The work of W.Y.A. and J.A. is supported by the Engineering and Physical
Sciences Research Council (grant No. EP/V002821/1). The work of J.A. and S.P.
is supported by the Leverhulme Trust (grant No. RPG-2021-299). J.A. is also
supported by the Science and Technology Facilities Council (grant No. STFC-
ST/X000753/1). For the purpose of Open Access, the authors have applied a CC
BY public copyright licence to any Author Accepted Manuscript version arising
from this submission.

##  Appendix A Validity of the dilute instanton gas approximation

The dilute instanton gas approximation is valid if consecutive jumps are
widely separated, namely if the average separation is much larger than the
jump width

$\Delta\tau\gg\omega^{-1}\,.$ (67)

The average separation is given by the total length of the Euclidean time
interval divided by the average number of jumps in said interval

$\Delta\tau=\frac{\beta}{\langle N\rangle}\,.$ (68)

The average number of jumps is

$\langle
N\rangle=\left(\sum_{N=0}^{\infty}\frac{([I]\beta)^{N}}{N!}\right)^{-1}\sum_{N=0}^{\infty}N\frac{([I]\beta)^{N}}{N!}=\beta[I]~{},$
(69)

where the probability density $[I]=-E_{\mathrm{inst}}$ of one instanton to
form is given by Eq. (4.2) and has to be assumed renormalised. As a
consequence, the dilute gas approximation is valid if

$[I]\ll\omega~{}.$ (70)

The quantity $[I]/\omega$ is plotted in Figure 5. We must compare this with
the plot in the top panel of Figure 4. The effect of the instantons becomes
relevant for $\lambda=1$ and $mL\sim 1$. For these values of the parameters,
the instanton rate $[I]$ takes values $\sim 0.3\omega$, meaning that the
dilute instanton gas approximation is decently good.

Figure 5: The instanton rate $[I]$ in units of $\omega$ for various values of
the self-coupling $\lambda$.

##  Appendix B Details on the calculation of the Casimir energy

In this section, we go through the details of the calculation of the Casimir
energy, which has only been outlined in Section 3.

###  B.1 $D=1$

Let us begin by computing the Casimir energy on a circle. This has already
been calculated in the past, however we perform the necessary steps here to
make our calculation in higher dimensions more transparent. We start by
writing the vacuum energy on the circle

$E_{0}^{(1)}(m;L)=\sum_{n=-\infty}^{\infty}\frac{1}{2}\omega^{(1)}(m;n)\,,$
(71)

where $\omega^{(1)}(m;n)$ is the dispersion relation

$\omega^{(1)}(m;n)=\sqrt{\left(\frac{2\pi n}{L}\right)^{2}+m^{2}}$ (72)

and $m$ is the mass. To use the Abel-Plana formula (20), we must have a sum
that goes from 0 to $\infty$. The dispersion relation is symmetric, namely
$\omega^{(1)}(m;n)=\omega^{(1)}(m;-n)$, thus we write

$\displaystyle\frac{1}{2}\sum_{n=-\infty}^{\infty}\omega^{(1)}(m;n)=\frac{1}{2}\sum_{n=0}^{\infty}(2-\delta_{n,0})\omega^{(1)}(m;n)$
$\displaystyle=\frac{1}{2}\left(2\int_{0}^{\infty}dt\,\omega^{(1)}(m;t)+\omega^{(1)}(m;0)+2i\int_{0}^{\infty}dt\,\frac{\omega^{(1)}(m;it)-\omega^{(1)}(m;-it)}{e^{2\pi
t}-1}-\omega^{(1)}(m;0)\right)$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}dt\,\omega^{(1)}(m;t)+i\int_{0}^{\infty}dt\,\mathrm{Disc}_{+}\left[\frac{\omega^{(1)}(m;it)}{e^{2\pi
t}-1}\right]\,,$ (73)

where in the last step we used that the function $\omega^{(1)}(m;it)$ has a
branch cut on $[t_{+},+\infty)$, where $t_{+}=\frac{mL}{2\pi}$.777To see this,
first recall that when $f(t)$ has branch cuts on the imaginary axis, one can
replace $it$ and $-it$ in the last term of the Abel-Plana formula with
$it+\epsilon$, $-it+\epsilon$ (with $\epsilon$ being a positive infinitesimal)
respectively. Second, it is useful to write
$\omega^{(1)}(m;t)=e^{1/2*\ln(m^{2}+t^{2})}$. Then one can show that
${\omega^{(1)}(m;it+\epsilon)-\omega^{(1)}(m;-it+\epsilon)=2i\sqrt{t^{2}-m^{2}}\theta(t-t_{+})\equiv{\rm
Disc}_{+}[\omega^{(1)}(m;it)]}$ using the analytic behavior of complex
logarithm.

A few comments are in place. First, we observe that there is no boundary term
left, namely a term independent of $L$. This is because we are computing the
energy on a circle, which has no boundary at all. This is a feature that will
hold true in higher-dimensional cases as well. Second, note how the divergent
integral is exactly the continuous limit of the original sum. In particular

$\frac{1}{2}\int_{-\infty}^{\infty}dt\,\omega^{(1)}(m;t)=\>\frac{L}{2}\int_{-\infty}^{\infty}\frac{dk}{2\pi}\sqrt{k^{2}+m^{2}}=\>E_{\infty}^{(1)}(m;L)\,,$
(74)

which is the vacuum energy on a circle of size $L$ when computed in the
continuum limit.

We are left with computing the integral over the branch cut. We have

$\displaystyle
i\int_{0}^{\infty}dt\,\mathrm{Disc}_{+}\left[\frac{\omega^{(1)}(m;it)}{e^{2\pi
t}-1}\right]=$ $\displaystyle
i\int_{0}^{\infty}dt\,2i\frac{\sqrt{\left(\frac{2\pi
t}{L}\right)^{2}-m^{2}}}{e^{2\pi t}-1}\Theta(t-t_{+})$ $\displaystyle=$
$\displaystyle-\frac{1}{\pi
L}\int_{mL}^{\infty}dy\,\frac{\sqrt{y^{2}-(mL)^{2}}}{e^{y}-1}$
$\displaystyle=$ $\displaystyle-\frac{1}{\pi L}F_{(1,0)}(mL)\,,$ (75)

where we have defined the function

$F_{(1,0)}(x)=\int_{x}^{\infty}dy\,\frac{\sqrt{y^{2}-x^{2}}}{e^{y}-1}\,.$ (76)

The function above can also be written in terms of modified Bessel functions
of the second type Bordag

$F_{(1,0)}(x)=x\sum_{n=1}^{\infty}\frac{K_{1}(nx)}{n}\,.$ (77)

We can write the vacuum energy for the massive scalar field on a circle

$E_{0}^{(1)}(m;L)=E_{\infty}^{(1)}(m;L)-\frac{1}{\pi L}F_{(1,0)}(mL)\,,$ (78)

and for the Casimir energy we obtain

$E_{\mathrm{Casimir}}^{(1)}(m;L)=-\frac{1}{\pi L}F_{(1,0)}(mL)\,.$ (79)

###  B.2 Arbitrary number of dimensions: a recursive formula

When we go higher in the number of dimensions we have more and more sums, and
the procedure which we just employed above becomes increasingly cumbersome.
Fortunately, we are able to give a recursive formula that will greatly
simplify our calculations.

_Ansatz:_ Let $D>1$ be the number of spatial dimensions. Then, the Casimir
energy for the real massive scalar field on the $D$-torus of sides
$\\{a_{1},\ldots,a_{D}\\}$ is given by

$\displaystyle E_{\mathrm{Casimir}}^{(D)}(m;a_{1},\ldots,a_{D})=\>$
$\displaystyle-\frac{1}{\pi^{D}a_{D}}\left(\prod_{i=1}^{D-1}\frac{a_{i}}{a_{D}}\right)F_{(D,0)}(ma_{D})$
$\displaystyle+\sum_{n=-\infty}^{\infty}E_{\mathrm{Casimir}}^{(D-1)}\left(\sqrt{m^{2}+\frac{4\pi^{2}n^{2}}{a_{D}^{2}}};a_{1},\ldots,a_{D-1}\right)\,,$
(80)

where

$F_{(D,0)}(x)=\>\left(\prod_{i=1}^{D-1}\int_{0}^{\infty}dy_{i}\right)F_{(1,0)}\left(\scriptstyle{\sqrt{x^{2}+\sum_{i=1}^{D-1}y_{i}^{2}}}\right)=\>\int_{0}^{\infty}dyF_{(D-1,0)}\left(\sqrt{x^{2}+y^{2}}\right)\,.$
(81)

We prove the formula (3) by construction. Let us assume we know the Casimir
energy on the $D-1$-dimensional torus of sides $\\{a_{1},\ldots,a_{D-1}\\}$.
The dispersion relation is

$\omega^{(D-1)}(m;n_{1},\ldots,n_{D-1})=\>\sqrt{\sum_{i=1}^{D-1}\left(\frac{2\pi
n_{i}}{a_{i}}\right)^{2}+m^{2}}\,.$ (82)

We then compute the Casimir energy on the $D$-torus of sides
$\\{a_{1},\ldots,a_{D}\\}$. The dispersion relation is

$\displaystyle\omega^{(D)}(m;n_{1},\ldots,n_{D})=\>$
$\displaystyle\sqrt{\sum_{i=1}^{D}\left(\frac{2\pi
n_{i}}{a_{i}}\right)^{2}+m^{2}}$ $\displaystyle=\>$
$\displaystyle\omega^{(D-1)}\left(\sqrt{m^{2}+\frac{4\pi^{2}n_{D}^{2}}{a_{D}^{2}}};n_{1},\ldots,n_{D-1}\right)\,.$
(83)

Using this, we can write the vacuum energy

$\displaystyle E_{0}^{(D)}(m;a_{1},\ldots,a_{D})=\>$
$\displaystyle\sum_{n_{D}=-\infty}^{\infty}\left(\sum_{n_{1},\ldots,n_{D-1}=-\infty}^{\infty}\frac{1}{2}\omega^{(D)}(m;n_{1},\ldots,n_{D-1},n_{D})\right)$
$\displaystyle=\>$
$\displaystyle\sum_{n_{D}=-\infty}^{\infty}E_{0}^{(D-1)}\left(\sqrt{m^{2}+\frac{4\pi^{2}n_{D}^{2}}{a_{D}^{2}}};a_{1},\ldots,a_{D-1}\right)$
$\displaystyle=\>$
$\displaystyle\sum_{n_{D}=-\infty}^{\infty}\Bigg{[}E_{\infty}^{(D-1)}\left(\sqrt{m^{2}+\frac{4\pi^{2}n_{D}^{2}}{a_{D}^{2}}};a_{1},\ldots,a_{D-1}\right)$
$\displaystyle+E_{\mathrm{Casimir}}^{(D-1)}\left(\sqrt{m^{2}+\frac{4\pi^{2}n_{D}^{2}}{a_{D}^{2}}};a_{1},\ldots,a_{D-1}\right)\Bigg{]}\,.$
(84)

The last line above already reproduces correctly the second term in our
ansatz. We are left with evaluating the first term.

$\displaystyle\sum_{n_{D}=-\infty}^{\infty}E_{\infty}^{(D-1)}\Bigg{(}\sqrt{m^{2}+\frac{4\pi^{2}n_{D}^{2}}{a_{D}^{2}}};a_{1},\ldots,a_{D-1}\Bigg{)}$
$\displaystyle=\>$
$\displaystyle\left(\prod_{i=1}^{D-1}a_{i}\int_{-\infty}^{\infty}\frac{dk_{i}}{2\pi}\right)\sum_{n_{D}=-\infty}^{\infty}\frac{1}{2}\sqrt{\frac{4\pi^{2}n_{D}^{2}}{a_{D}^{2}}+m^{2}+\sum_{i=1}^{D-1}k_{i}^{2}}$
$\displaystyle=\>$
$\displaystyle\left(\prod_{i=1}^{D-1}a_{i}\int_{-\infty}^{\infty}\frac{dk_{i}}{2\pi}\right)E_{0}^{(1)}\left(\sqrt{m^{2}+\sum_{i=1}^{D-1}k_{i}^{2}};a_{D}\right)$
$\displaystyle=\>$
$\displaystyle\left(\prod_{i=1}^{D-1}a_{i}\int_{-\infty}^{\infty}\frac{dk_{i}}{2\pi}\right)\Bigg{[}E_{\infty}^{(1)}\left(\sqrt{m^{2}+\sum_{i=1}^{D-1}k_{i}^{2}};a_{D}\right)+E_{\mathrm{Casimir}}^{(1)}\left(\sqrt{m^{2}+\sum_{i=1}^{D-1}k_{i}^{2}};a_{D}\right)\Bigg{]}$
$\displaystyle=\>$ $\displaystyle
E_{\infty}^{(D)}(m;a_{1},\ldots,a_{D})+\left(\prod_{i=1}^{D-1}a_{i}\int_{-\infty}^{\infty}\frac{dk_{i}}{2\pi}\right)\left(-\frac{1}{\pi
a_{D}}\right)F_{(1,0)}\left(a_{D}\sqrt{m^{2}+\sum_{i=1}^{D}k_{i}^{2}}\right)$
$\displaystyle=\>$ $\displaystyle
E_{\infty}^{(D)}(m;a_{1},\ldots,a_{D})-\frac{1}{\pi^{D}a_{D}}\left(\prod_{i=1}^{D-1}\frac{a_{i}}{a_{D}}\right)F_{(D,0)}(ma_{D})\,,$
(85)

which concludes the proof. Note that as
$E_{\mathrm{Casimir}}^{(1)}(m;L)\propto e^{-mL}$ as $mL\gg 1$, it can be
easily argued that the integral and the series appearing in formula (3) are
convergent for any $D$.

####  B.2.1 $D=2$

We can use the formula (3) to obtain the Casimir energy for a massive scalar
field on a two-torus. For the sake of generality, let us take a torus of
linear sizes $a_{1}=a$ and $a_{2}=b$. The formula then reads

$\displaystyle E_{\mathrm{Casimir}}^{(2)}(m;a,b)=\>$
$\displaystyle-\frac{1}{\pi^{2}b}\frac{a}{b}F_{(2,0)}(mb)+\sum_{n=-\infty}^{\infty}E_{\mathrm{Casimir}}^{(1)}\left(\sqrt{m^{2}+\frac{4\pi^{2}n^{2}}{b^{2}}};a\right)$
$\displaystyle=$
$\displaystyle\>-\frac{1}{\pi^{2}b}\frac{a}{b}F_{(2,0)}(mb)-\frac{1}{\pi
a}\sum_{n=-\infty}^{\infty}F_{(1,0)}(\frac{a}{b}\sqrt{m^{2}b^{2}+4\pi^{2}n^{2}})$
$\displaystyle=$
$\displaystyle\>-\frac{1}{\pi^{2}a}\left[\frac{a^{2}}{b^{2}}F_{(2,0)}(mb)+\pi
F_{(1,1)}\left(\frac{a}{b},mb\right)\right]\,.$ (86)

Although not obvious, our result is indeed symmetric under exchanging $a$ and
$b$. We have defined the function

$F_{(1,1)}(r,x)=\>\sum_{n=-\infty}^{\infty}F_{(1,0)}(r\sqrt{x^{2}+4\pi^{2}n^{2}})\,.$
(87)

####  B.2.2 $D=3$

Having derived the result for $D=2$, we can finally give a result for the
Casimir energy for $D=3$. Again, we compute the Casimir energy of a massive
scalar field on a three-torus with sizes $a_{1}=a$, $a_{2}=b$ and $a_{3}=c$.
Applying the formula (3), we get

$\displaystyle E_{\mathrm{Casimir}}^{(3)}(m;a,b,c)=\>$
$\displaystyle-\frac{1}{\pi^{3}c}\frac{a}{c}\frac{b}{c}F_{(3,0)}(mc)+\sum_{n=-\infty}^{\infty}E_{\mathrm{Casimir}}^{(2)}\left(\sqrt{m^{2}+\frac{4\pi^{2}n^{2}}{c^{2}}};a,b\right)$
$\displaystyle=\>$
$\displaystyle-\frac{1}{\pi^{3}c}\frac{a}{c}\frac{b}{c}F_{(3,0)}(mc)+\sum_{n=-\infty}^{\infty}\left(-\frac{1}{\pi^{2}a}\right)\Bigg{[}\frac{a^{2}}{b^{2}}F_{(2,0)}(\frac{b}{c}\sqrt{m^{2}c^{2}+4\pi^{2}n^{2}})$
$\displaystyle+\pi
F_{(1,1)}\left(\frac{a}{b},\frac{b}{c}\sqrt{m^{2}c^{2}+4\pi^{2}n^{2}}\right)\Bigg{]}$
$\displaystyle=\>$
$\displaystyle-\frac{1}{\pi^{3}a}\Bigg{[}\frac{a^{2}}{c^{2}}\frac{b}{c}F_{(3,0)}(mc)+\pi\frac{a^{2}}{b^{2}}F_{(2,1)}\left(\frac{b}{c},mc\right)+\pi^{2}F_{(1,2)}\left(\frac{a}{b},\frac{b}{c},mc\right)\Bigg{]}\,.$
(88)

We have introduced two new functions

$\displaystyle F_{(2,1)}(r,x)=\>$
$\displaystyle\sum_{n=-\infty}^{\infty}F_{(2,0)}(r\sqrt{x^{2}+4\pi^{2}n^{2}})\,,$
(89) $\displaystyle F_{(1,2)}(r_{1},r_{2},x)=\>$
$\displaystyle\sum_{n=-\infty}^{\infty}F_{(1,1)}(r_{1},r_{2}\sqrt{x^{2}+4\pi^{2}n^{2}})\,.$
(90)

###  B.3 Comparing with the numerical renormalisation

When computing the renormalised ratio of functional determinants appearing in
the instanton contributions, we cannot use the Abel-Plana formula to obtain an
analytic result. Instead, we will subtract the ultraviolet (UV) divergences
via counter-terms, which amounts to subtracting the continuum limit from the
series. Therefore, it makes sense to compare this numerical procedure to the
result we obtained for the Casimir energy, given that we only have analytical
control for the latter. In order to do so, define the numerically renormalised
Casimir energy as follows

$E_{\mathrm{Casimir}}^{(3)}(m;L)=\>\frac{1}{2}\sum_{n,\ell,\kappa=-\infty}^{\infty}\sqrt{\mathbf{k}_{\\{n,\ell,\kappa\\}}^{2}+m^{2}}e^{-\frac{\mathbf{k}_{\\{n,\ell,\kappa\\}}^{2}}{\Lambda^{2}}}-\frac{L^{3}}{2}\int_{\mathbb{R}^{3}}\frac{d\mathbf{k}^{3}}{(2\pi)^{3}}\sqrt{\mathbf{k}^{2}+m^{2}}e^{-\frac{\mathbf{k}^{2}}{\Lambda^{2}}}\,.$
(91)

We have introduced the regularising function

$\eta(|\mathbf{k}|)=e^{-\frac{\mathbf{k}^{2}}{\Lambda^{2}}}\,,$ (92)

where $\Lambda$ is some large cut-off. Also, we have denoted the discretised
momentum by
$\mathbf{k}_{\\{n,\ell,\kappa\\}}^{2}=\frac{4\pi^{2}}{L^{2}}(n^{2}+\ell^{2}+\kappa^{2})$.
The continuum integral can be computed analytically and expanded for large
$\Lambda/m$

$\displaystyle\frac{L^{3}}{2}\int_{\mathbb{R}^{3}}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\sqrt{\mathbf{k}^{2}+m^{2}}e^{-\frac{\mathbf{k}^{2}}{\Lambda^{2}}}=\>m\frac{(mL)^{3}}{16\pi^{2}}e^{\frac{m^{2}}{2\Lambda^{2}}}\frac{\Lambda^{2}}{m^{2}}K_{1}\left(\frac{m^{2}}{2\Lambda^{2}}\right)$
$\displaystyle\qquad\qquad\qquad\qquad=\>m\frac{(mL)^{3}}{128\pi^{2}}\left[\frac{16\Lambda^{4}}{m^{4}}+\frac{8\Lambda^{2}}{m^{2}}+1+2\gamma_{E}+2\log\frac{m^{2}}{4\Lambda^{2}}\right]+\mathcal{O}\left(\frac{m}{\Lambda}\right)\,.$
(93)

As for the series, it is useful to introduce the degeneracy function
$r_{3}(j)$, which returns the number of ways in which $j$ can be written as a
sum of three squared integers. For example $r_{3}(0)=1$ and $r_{3}(1)=6$. We
can then write

$E_{\mathrm{Casimir}}^{(3)}(m;L)=\>\frac{1}{2}\sum_{j=0}^{j_{\mathrm{max}}}r_{3}(j)\sqrt{\frac{4\pi^{2}}{L^{2}}j+m^{2}}e^{-\frac{4\pi^{2}j}{L^{2}\Lambda^{2}}}+\mathrm{c.t.}\
$ (94)

We introduced a cut-off $j_{\mathrm{max}}$ to the series. This has nothing to
do with the regularisation cut-off $\Lambda$, and it is introduced to cut the
numerical sum at a certain value. To make sure that it does not affect the
renormalisation procedure, we must choose $j_{\mathrm{max}}$ such that

$\frac{4\pi^{2}}{L^{2}}j_{\mathrm{max}}\gg\Lambda^{2}\gg\frac{4\pi^{2}}{L^{2}}\,,$
(95)

where the last inequality means that $\Lambda$ must be large enough so that
only the UV modes are affected by the regulator. Then, we can compare the
Casimir energy when obtained analytically versus numerically. This is shown in
Figure 6. The matching is pretty good but for the asymptotic regions $mL\ll 1$
and $mL\gg 1$. This is a feature of the numerical procedure. To understand
this, let us rewrite our conditions on the numerical cut-off
$j_{\mathrm{max}}$ and renormalisation cut-off $\Lambda$ in a different way

$j_{\mathrm{max}}\gg\frac{m^{2}L^{2}}{4\pi^{2}}\frac{\Lambda^{2}}{m^{2}}\gg
1\,.$ (96)

From the first of these inequalities, we see that as $mL$ grows, we must
increase $j_{\mathrm{max}}$ more and more, making our numerical procedure more
expensive. On the other hand, the second inequality tells us that what is a
good renormalisation cut-off $\Lambda$ for large $mL$ becomes a bad cut-off
for small $mL$. Better agreement in the asymptotic regions can be achieved by
splitting the numerical function into two regions and defining appropriate
$j_{\mathrm{max}}$ and $\Lambda$ independently in either region. This is
allowed since the result is cut-off independent.

Figure 6: Comparison between the analytical and numerical renormalisation
procedure for the Casimir energy.

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