KCL-PH/TH-2023-07

False vacuum decay rates, more precisely

Wen-Yuan Ai,*††*<EMAIL_ADDRESS>Jean Alexandre††††
<EMAIL_ADDRESS>and Sarben Sarkar‡††‡<EMAIL_ADDRESS>

Theoretical Particle Physics and Cosmology, King’s College London,
Strand, London WC2R 2LS, UK

###### Abstract

We develop a method for accurately calculating vacuum decay rates beyond the
thin-wall regime in a pure scalar field theory at the one-loop level of the
effective action. It accounts for radiative effects resulting from quantum
corrections to the classical bounce, including gradient effects stemming from
the inhomogeneity of the bounce background. To achieve this, it is necessary
to compute not only the functional determinant of the fluctuation operator in
the background of the classical bounce but also its functional derivative
evaluated at the classical bounce. The former is efficiently calculated using
the Gel’fand-Yaglom method. We illustrate how the latter can also be
calculated with the same method, combined with a computation of various
Green’s functions.

###### Contents

  1. 1 Introduction
  2. 2 Callan–Coleman formalism and the tree-level bounce
  3. 3 Quantum-corrected bounce and radiative corrections to the decay rate
  4. 4 Functional determinants for an $O(D)$ background and the Gel’fand-Yaglom method
  5. 5 Functional derivative of a functional determinant
  6. 6 Numerical example
  7. 7 Conclusions
  8. A Solving Green’s functions
    1. A.1 $G_{l}(r,r^{\prime};\varphi)$ and $\hat{G}_{l}(r,r^{\prime})$
    2. A.2 $\mathcal{G}(r,r^{\prime})$
  9. B Renormalisation with dimensional regularisation

##  1 Introduction

One of the most striking features of the Standard Model (SM) is that the
electroweak vacuum is metastable [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The
measured 125 GeV Higgs boson [12, 13], together with the 173 GeV top-quark
[14], suggests that the Higgs potential in the SM develops a lower minimum due
to the renormalisation running of the Higgs self-interaction coupling. The
electroweak vacuum could then decay to the true vacuum via tunneling which is
a first-order phase transition in quantum field theory. State-of-the-art
calculations suggest that the electroweak vacuum lies at the edge of the
stable region, having a lifetime longer than the present age of the Universe
[11, 15, 16, 17, 18, 19].111The scalar potential, the top Yukawa and the
electroweak gauge couplings have been extracted from data with full 2-loop
next-to-next-leading order (NNLO) precision [15, 16]. These parameters have
been extrapolated to large energies with full 3-loop NNLO renormalisation-
group-equation (RGE) precision [16].

In contrast to the running couplings, the radiative corrections to tunneling
are computed less accurately. The transition rate is sensitive to a solitonic
field configuration in Euclidean space, the so-called “bounce” [20, 21]. This
bounce configuration also encodes the information of the nucleated critical
bubble. Typically the bounce is calculated from the tree-level scalar
potential with the running coupling constants evaluated at the scale
corresponding to the bounce radius. However, the bounce is an inhomogeneous
configuration, and the beta functions for the couplings do not account for the
effects of the background inhomogeneity. The inhomogeneity of the bounce
requires us to go beyond the effective potential. Such effects from the
background inhomogeneity, as well as the corresponding higher-loop radiative
corrections, have not yet been included in the full SM computations but are
only studied in simpler scalar [22, 23, 24], Yukawa [25], and gauge [26]
theories in the planar thin-wall approximation. These studies are based on a
Green’s function method developed earlier in Ref. [27]. See also Refs. [28,
29, 30] for the case of two-dimensional spacetime.

We extend previous work by going beyond the thin-wall regime. In order to
calculate the quantum-corrected bounce, we need both the functional
determinants of the fluctuation operators and their functional derivatives
evaluated at the classical bounce (see Eq. (20) or Eq. (25) below). The
functional determinants in the $O(4)$-symmetric backgrounds can be efficiently
computed by the Gel’fand-Yaglom method [31, 32]. We show that the functional
derivative of a functional determinant can be computed with the same method,
combined with a computation of various Green’s functions. This method is
closely related to the Green’s function method used in Refs. [27, 22, 23, 25,
24, 26]. Our method is applied numerically to compute the corresponding
radiative effects in an archetypal scalar model.

The outline of the paper is as follows. In the next section, we briefly review
the standard Callan-Coleman formalism for false vacuum decay in terms of the
classical bounce. In Sec. 3, we reformulate the decay rate using the one-
particle-irreducible (1PI) effective action [33] in a way that radiative
corrections to the decay rate can be incorporated. In Sec. 4, we introduce the
powerful Gel’fand-Yaglom method of computing functional determinants. In Sec.
5, we show how the Gel’fand-Yaglom method, in combination with a Green’s
function method, is used to compute the quantum corrections to the classical
bounce and the decay rate. In Sec. 6 we give the results of numerical
calculations using our method. In Sec. 7 we give our conclusions. For
completeness, some technical details are presented in the appendices. In
Appendix A, we give a general introduction to how to solve the Green’s
functions. In Appendix B, we discuss on how to renormalise the one-loop
effective action using the dimensional renormalisation scheme. Throughout the
paper, we use $\hbar=c=1$.

##  2 Callan–Coleman formalism and the tree-level bounce

We consider the archetypal model,

$\displaystyle\mathcal{L}_{\rm
M}=\frac{1}{2}\eta^{\mu\nu}(\partial_{\mu}\Phi)(\partial_{\nu}\Phi)-U(\Phi)\,,$
(1)

where $\eta^{\mu\nu}$ is the Minkowski metric with signature $\\{+,-,-,-\\}$
and

$\displaystyle
U(\Phi)=\frac{1}{2}m^{2}\Phi^{2}-\frac{1}{3!}g\Phi^{3}+\frac{1}{4!}\lambda\Phi^{4}\,.$
(2)

The couplings $m^{2}$, $g$, $\lambda$ all take positive values. When
$g^{2}>8\lambda m^{2}/3$, the potential has two local minima. For $8\lambda
m^{2}/3<g^{2}<3\lambda m^{2}$, $\varphi=0$222We use $\varphi$ to denote the
vacuum expectation value of the quantum field $\Phi$, $\varphi=\langle
0|\Phi|0\rangle$. This should be clearer when we introduce the effective
action below. is the global minimum; for $g^{2}>3\lambda m^{2}$, the global
minimum becomes

$\displaystyle\varphi_{\rm TV}=\frac{3g+\sqrt{9g^{2}-24\lambda
m^{2}}}{2\lambda}\,.$ (3)

When $g^{2}=3\lambda m^{2}$, the two local minima become degenerate. Below we
consider the parameter region $g^{2}>3\lambda m^{2}$. The potential (see Fig.
1) has two minima at $\varphi_{\rm FV}=0$ and $\varphi_{\rm TV}$,
corresponding to the false and true vacua respectively. In the potential, we
have chosen $U(\varphi_{\rm FV})=0$ for convenience.

Figure 1: The classical potential $U(\Phi)$ for the archetypical scalar model
with false vacuum decay, given by Eqs. (1) and (2).

The decay rate is obtained from the Euclidean transition amplitude

$\displaystyle Z[0]=\langle\varphi_{\rm
FV}|\mathrm{e}^{-H\mathcal{T}}|\varphi_{\rm
FV}\rangle=\int\mathcal{D}\Phi\>\mathrm{e}^{-S[\Phi]}\,,$ (4)

where $H$ is the Hamiltonian, $\mathcal{T}$ is the amount of Euclidean time
taken by the transition and $S$ is the Euclidean action. In terms of $Z[0]$,
the decay rate per volume is given by [21]

$\displaystyle\frac{\gamma}{V}=\lim_{\mathcal{T}\rightarrow\infty}\frac{2}{V\mathcal{T}}|{\rm
Im}(\ln Z[0])|\,.$ (5)

The partition function $Z[0]$ is evaluated by using the method of steepest
descent for path integrals. The classical equation of motion in Euclidean
spacetime reads

$\displaystyle-\partial^{2}\varphi+U^{\prime}(\varphi)=0\,,$ (6)

with the boundary conditions $\varphi|_{\tau\rightarrow\pm\infty}=\varphi_{\rm
FV}$ and $\dot{\varphi}|_{\tau=0}=0$, where the dot denotes the derivative
with respect to the Euclidean time $\tau$. A trivial solution is
$\varphi(x)=\varphi_{\rm FV}$. There is a nontrivial solution—the bounce—which
is $O(4)$ symmetric in Euclidean spacetime. The bounce solution is relevant
for false vacuum decay and describes a field configuration that starts and
ends in Euclidean time in the false vacuum. Working in four-dimensional
hyperspherical coordinates, the equation of motion takes the form

$\displaystyle-\frac{\mathrm{d}^{2}\varphi(r)}{\mathrm{d}r^{2}}-\frac{3}{r}\frac{\mathrm{d}\varphi(r)}{\mathrm{d}r}+U^{\prime}(\varphi(r))=0\,,$
(7)

where $r=\sqrt{\tau^{2}+\mathbf{x}^{2}}$. The boundary conditions become
$\varphi(r\rightarrow\infty)=\varphi_{\rm FV}$ and
$\mathrm{d}\varphi(r)/\mathrm{d}r|_{r=0}=0$. We denote this classical bounce
solution as $\varphi_{b}^{(0)}$.

Aside from the single bounce solution, there are also multi-bounce solutions
that go back and forth from the false vacuum $N$ times, for arbitrary $N$.
These multi-bounce configurations exponentiate [21, 34, 35] in analogous to
disconnected vacuum Feynman diagrams. Integrating over the fluctuations about
the stationary points and carefully dealing with some subtleties from the
negative and zero modes for the fluctuation operator evaluated at the bounce,
one obtains the decay rate per unit volume [21]

$\displaystyle\frac{\gamma}{V}=\left(\frac{B^{(0)}}{2\pi}\right)^{2}\left|\frac{\det^{\prime}[-\partial^{2}+U^{\prime\prime}(\varphi^{(0)}_{b})]}{\det[-\partial^{2}+U^{\prime\prime}(\varphi_{\rm
FV})]}\right|^{-1/2}\,\mathrm{e}^{-B^{(0)}}\,,$ (8)

where $B^{(0)}=S[\varphi_{b}^{(0)}]-S[\varphi_{\rm FV}]$, $\det^{\prime}$
means that the zero eigenvalues are omitted from the determinant and a
prefactor $\sqrt{B^{(0)}/2\pi}$ is included for each of the four collective
coordinates that are related to the zero modes corresponding to spacetime
translations [36]. On using $S[\varphi_{\rm FV}]=0$ we have

$\displaystyle
B^{(0)}=2\pi^{2}\int_{0}^{\infty}\mathrm{d}r\,r^{3}\left[\frac{1}{2}(\partial_{r}\varphi^{(0)}_{b}(r))^{2}+U(\varphi^{(0)}_{b})\right]\,.$
(9)

##  3 Quantum-corrected bounce and radiative corrections to the decay rate

We reformulate Eq. (5) such that radiative effects can be systematically
considered, using the 1PI effective action [33]. First, we introduce the
partition function in the presence of a source $J$,

$\displaystyle Z[J]$
$\displaystyle=\int\mathcal{D}\Phi\,\mathrm{e}^{-S[\Phi]+\int\mathrm{d}^{4}x\,J(x)\Phi(x)}\,.$
(10)

The one-point function in the presence of the source $J$ is given by

$\displaystyle\varphi_{J}(x)$ $\displaystyle=\langle
0|\Phi(x)|0\rangle_{J}=\frac{\delta\ln Z[J]}{\delta J(x)}\,.$ (11)

The effective action then is defined as the Legendre transform

$\displaystyle\Gamma[\varphi_{J}]$ $\displaystyle=-\ln
Z[J]+\int\mathrm{d}^{4}x\,J(x)\varphi_{J}(x)\,.$ (12)

From the effective action, one obtains the quantum equation of motion for the
one-point function, $\delta\Gamma[\varphi_{J}]/\delta\varphi_{J}(x)=J(x)$.
Note that $\Gamma$ has no explicit dependence on $J$. Denoting
$\varphi\equiv\varphi_{J=0}$, one obtains the equation of motion for the
vacuum expectation value in the absence of sources by setting $J=0$,

$\displaystyle\frac{\delta\Gamma[\varphi]}{\delta\varphi(x)}=0\,.$ (13)

A nontrivial $O(4)$-symmetric solution to Eq. (13) would give the quantum-
corrected bounce $\varphi_{b}$. In terms of the effective action, the decay
rate per volume (5) can be written as

$\displaystyle\frac{\gamma}{V}=\frac{2\,|{\rm
Im}\,\mathrm{e}^{-\left(\Gamma[\varphi_{b}]-\Gamma[\varphi_{\rm
FV}]\right)}|}{V\mathcal{T}}\,.$ (14)

This is the formula used in Refs. [22, 37, 38, 34] where it is implicitly
assumed that $\Gamma[\varphi_{\rm FV}]=0$. Since adding constant and linear
terms in the Lagrangian does not affect the dynamics (when gravity is not
considered), in the renormalisation procedure one can always implement
$\varphi_{\rm FV}=0$ and $\Gamma[\varphi_{\rm FV}]=0$, even beyond the tree
level.

On denoting the fluctuation operator evaluated at a general configuration
$\varphi$ as

$\displaystyle\mathcal{M}[\varphi]=-\partial^{2}+U^{\prime\prime}(\varphi)\,,$
(15)

the one-loop effective action reads

$\displaystyle\Gamma^{(1)}[\varphi]=S[\varphi]+\frac{1}{2}\,\ln\frac{\det\mathcal{M}[\varphi]}{\det\widehat{\mathcal{M}}}\equiv
S[\varphi]+\frac{1}{2}D[\varphi]\,,$ (16)

where $\widehat{\mathcal{M}}\equiv\mathcal{M}[\varphi_{\rm FV}]$ and
$D[\varphi]\equiv\ln\frac{\det\mathcal{M}[\varphi]}{\det\widehat{\mathcal{M}}}$.
Above, the normalisation choice $\Gamma[\varphi_{\rm FV}]=0$ is made.

The bounce at the one loop, $\varphi_{b}^{(1)}$, satisfies

$\displaystyle\left.\frac{\delta
S[\varphi]}{\delta\varphi(x)}\right|_{\varphi_{b}^{(1)}}+\frac{1}{2}\left.\frac{\delta
D[\varphi]}{\delta\varphi(x)}\right|_{\varphi^{(1)}_{b}}=0\,.$ (17)

Expanding this equation about
$\varphi^{(0)}_{b}\equiv\varphi_{b}^{(1)}-\Delta\varphi_{b}$ gives333We assume
that the one-loop corrected bounce is close to the classical bounce. If the
false vacuum is generated by radiative corrections in the first place, it is
necessary to use the two-particle irreducible effective action [39] as
explained in Refs. [37, 38].

$\displaystyle[-\partial^{2}+U^{\prime\prime}(\varphi^{(0)}_{b})]\Delta\varphi_{b}+\frac{1}{2}\left.\frac{\delta
D[\varphi]}{\delta\varphi(x)}\right|_{\varphi^{(0)}_{b}}=0\,,$ (18)

where we ignored perturbatively small higher-order terms. Substituting
$\varphi^{(1)}_{b}=\varphi^{(0)}_{b}+\Delta\varphi_{b}$ into Eq. (16), we
obtain

$\displaystyle\Gamma^{(1)}[\varphi^{(1)}_{b}]$
$\displaystyle=S[\varphi^{(0)}_{b}]+\frac{1}{2}D[\varphi^{(0)}_{b}]+\frac{1}{2}\int\mathrm{d}^{4}x\,\Delta\varphi_{b}(x)\left[-\partial^{2}+U^{\prime\prime}(\varphi^{(0)}_{b}(x))\right]\Delta\varphi_{b}(x)$
$\displaystyle+\frac{1}{2}\int\mathrm{d}^{4}x\,\left.\frac{\delta
D[\varphi]}{\delta\varphi(x)}\right|_{\varphi_{b}^{(0)}}\Delta\varphi_{b}(x)\,,$
(19)

Using Eq. (18), we have

$\displaystyle\Gamma^{(1)}[\varphi^{(1)}_{b}]$
$\displaystyle=S[\varphi^{(0)}_{b}]+\frac{1}{2}D[\varphi^{(0)}_{b}]+\frac{1}{4}\int\mathrm{d}^{4}x\,\left.\frac{\delta
D[\varphi]}{\delta\varphi(x)}\right|_{\varphi_{b}^{(0)}}\Delta\varphi_{b}(x)\,.$
(20)

Assuming that the corrected bounce is $O(4)$-symmetric, we can consider only
$O(4)$-symmetric configurations $\varphi(r)$ in $\Gamma^{(1)}[\varphi]$. In
this case, we can treat $\varphi(r)$ as a one-dimensional function and derive
its equation of motion by directly varying the action $S[\varphi]$ and
$D[\varphi]$ with respect to the one-dimensional field $\varphi(r)$. For
example, one can derive the classical equation (7) by writing

$\displaystyle
S[\varphi]=2\pi^{2}\int\mathrm{d}r\,r^{3}\left[\frac{1}{2}(\partial_{r}\varphi(r))^{2}+U(\varphi(r))\right]$
(21)

(valid for $O(4)$-symmetric configurations) and varying it with respect to
$\varphi(r)$. Then the one-loop equation of motion reads

$\displaystyle\left.\frac{\delta
S[\varphi]}{\delta\varphi(r)}\right|_{\varphi_{b}^{(1)}}+\frac{1}{2}\left.\frac{\delta
D[\varphi]}{\delta\varphi(r)}\right|_{\varphi^{(1)}_{b}}=0$
$\displaystyle\qquad\qquad\qquad\Rightarrow
2\pi^{2}r^{3}\left[-\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}-\frac{3}{r}\frac{\mathrm{d}}{\mathrm{d}r}+U^{\prime\prime}(\varphi_{b}^{(0)})\right]\Delta\varphi_{b}(r)+\frac{1}{2}\left.\frac{\delta
D[\varphi]}{\delta\varphi(r)}\right|_{\varphi^{(1)}_{b}}=0\,.$ (22)

Note that in this procedure $\delta
D[\varphi]/\delta\varphi(r)|_{\varphi_{b}^{(0)}}$ is not equal to $\delta
D[\varphi]/\delta\varphi(x)|_{\varphi_{b}^{(0)}}$; they even have different
dimensions in mass. The reason to introduce this procedure is the following.
Below we will have explicit expressions of $D[\varphi]$ only for
$O(4)$-symmetric configurations $\varphi(r)$ instead of general four-
dimensional configurations $\varphi(x)$. As a consequence, we can take the
functional derivative of $D[\varphi]$ only with respect to $\varphi(r)$,
instead of with respect to the general four-dimensional function $\varphi(x)$.
From Eq. (22),

$\displaystyle\Delta\varphi_{b}(r)=-\frac{1}{2}\int\mathrm{d}r^{\prime}\,\mathcal{G}(r,r^{\prime})\left[\frac{1}{2\pi^{2}r^{3}}\left.\frac{\delta
D[\varphi]}{\delta\varphi(r^{\prime})}\right|_{\varphi_{b}^{(0)}}\right]\,,$
(23)

where $\mathcal{G}(r,r^{\prime})$ is the Green’s function, satisfying

$\displaystyle\left[-\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}-\frac{3}{r}\frac{\mathrm{d}}{\mathrm{d}r}+U^{\prime\prime}(\varphi_{b}^{(0)})\right]\mathcal{G}(r,r^{\prime})=\delta(r-r^{\prime})\,.$
(24)

Finally, Eq. (20) becomes

$\displaystyle\Gamma^{(1)}[\varphi^{(1)}_{b}]$
$\displaystyle=S[\varphi^{(0)}_{b}]+\frac{1}{2}D[\varphi^{(0)}_{b}]+\frac{1}{4}\int\mathrm{d}r\,\left.\frac{\delta
D[\varphi]}{\delta\varphi(r)}\right|_{\varphi_{b}^{(0)}}\Delta\varphi_{b}(r)\,,$
(25)

where the last term is a one-dimensional integral.

From Eqs. (23) and (25), we see that, at the one-loop level of the effective
action, a self-consistent bounce solution and the corrected decay rate are
derived from $D[\varphi^{(0)}_{b}]$ and the functional derivative $\delta
D[\varphi]/\delta\varphi(r)|_{\varphi_{b}^{(0)}}$ (or $\delta
D[\varphi]/\delta\varphi(x)|_{\varphi_{b}^{(0)}}$ if one uses Eqs. (18) and
(20))).

Directly using the definition for $D[\varphi]$, one has

$\displaystyle\left.\frac{\delta
D[\varphi]}{\delta\varphi(x)}\right|_{\varphi^{(0)}_{b}}=\mathcal{M}^{\prime}[\varphi_{b}^{(0)}(x)]G(x,x;\varphi^{(0)}_{b})\,,$
(26)

where $G(x,y;\varphi)$ is the Green’s function to the operator
$\mathcal{M}[\varphi]$ and a prime on $\mathcal{M}$ denotes the derivative
with respect to the field. In this case, one needs to calculate the Green’s
function $G(x,y;\varphi^{(0)}_{b})$ and take the coincident limit $x=y$.

Below, we use the Gel’fand-Yaglom method to compute the functional determinant
$D[\varphi]$ as well as its functional derivative.

##  4 Functional determinants for an $O(D)$ background and the Gel’fand-Yaglom
method

The powerful Gel’fand-Yaglom method [31] is widely employed in calculations
for tunneling in theoretical as well as phenomenological studies, see e.g.
Refs. [7, 40, 19, 18, 32, 41, 42]. It is used in BubbleDet [43], a Python
package that aims to compute functional determinants released recently.

Although we are primarily interested in false vacuum decay in four-dimensional
spacetime, we consider $D$-dimensional spacetime in this section to keep the
discussion more general. For a general $O(D)$-symmetric configuration
$\varphi$ that approaches zero (the false vacuum $\varphi_{\rm FV}$) for
$r\rightarrow\infty$, we have the decomposition

$\displaystyle D[\varphi]=\sum_{l=0}^{\infty}{\rm
deg}(l;D)\ln\left(\frac{{\det}\mathcal{M}_{l}[\varphi]}{{\det}\widehat{\mathcal{M}}_{l}}\right)\,,$
(27)

where444The operators $\mathcal{M}_{l}[\varphi]$ are obtained when
substituting $\Psi(r,\mathbf{\theta})=\psi_{l}(r)Y_{l}(\mathbf{\theta})$,
where $Y_{l}(\mathbf{\theta})$ are the hyperspherical Harmonics, into the
eigenequations for $\mathcal{M}[\varphi]$. It is the same for
$\widehat{\mathcal{M}}_{l}$.

$\displaystyle\mathcal{M}_{l}[\varphi]$
$\displaystyle=-\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}-\frac{D-1}{r}\frac{\mathrm{d}}{\mathrm{d}r}+\frac{l\left(l+D-2\right)}{r^{2}}+U^{\prime\prime}(\varphi(r))\,,$
(28a) $\displaystyle\widehat{\mathcal{M}}_{l}$
$\displaystyle=-\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}-\frac{D-1}{r}\frac{\mathrm{d}}{\mathrm{d}r}+\frac{l\left(l+D-2\right)}{r^{2}}+m^{2}\,,$
(28b)

and

$\displaystyle{\rm deg}(l;D)=\frac{(l+D-3)!(2l+D-2)}{l!(D-2)!}\,.$ (29)

For each $l$, one can calculate the functional determinant ratio using the
Gel’fand-Yaglom theorem [31]. Define the functions $\psi_{l}(r)$ and
$\hat{\psi}_{l}(r)$ as the solutions to the following equations

$\displaystyle\mathcal{M}_{l}[\varphi]\psi_{l}(r;\varphi)=0\,,\qquad\widehat{\mathcal{M}}_{l}\hat{\psi}_{l}(r)=0\,,$
(30)

with the same leading regular behavior at $r=0$. The Gel’fand-Yaglom theorem
states that

$\displaystyle\frac{{\det}\mathcal{M}_{l}[\varphi]}{{\det}\widehat{\mathcal{M}}_{l}}=\frac{\psi_{l}(\infty;\varphi)}{\hat{\psi}_{l}(\infty)}\equiv
T_{l}(\infty;\varphi)\,.$ (31)

We can first have an asymptotic analysis for $\hat{\psi}_{l}(r)$ and
$\psi_{l}(r)$. For $r\rightarrow 0$, one can ignore the potential term in the
operators $\widehat{\mathcal{M}}_{l}$ and $\mathcal{M}_{l}[\varphi]$.
Therefore, the solutions regular at $r=0$ are proportional to $r^{l}$. Without
loss of generality, one can choose the factors such that both
$\hat{\psi}_{l}(r)$ and $\psi_{l}(r)$ are normalised to $r^{l}$ as
$r\rightarrow 0$. Then we obtain

$\displaystyle\hat{\psi}_{l}(r)=\left(\frac{2}{m}\right)^{l+D/2-1}\Gamma\left(l+\frac{D}{2}\right)\,r^{1-D/2}I_{l+D/2-1}(mr)\,,$
(32)

where $\Gamma(x)$ is the gamma function. $I_{l+D/2-1}(x)$ are the modified
Bessel functions of the first kind, satisfying

$\displaystyle\left[-\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}-\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}-\left(m^{2}+\frac{(l+D/2-1)^{2}}{r^{2}}\right)\right]I_{l+D/2-1}(mr)=0\,.$
(33)

$I_{l+D/2-1}(mr)\sim r^{l+D/2-1}(m/2)^{l+D/2-1}/\Gamma(l+D/2)$ for
$r\rightarrow 0$. Substituting $\psi_{l}=T_{l}\hat{\psi}_{l}$ into Eq. (30),
one obtains

$\displaystyle\left[-\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}-\left(2m\frac{I_{l+D/2-2}(mr)}{I_{l+D/2-1}(mr)}-\frac{2l+D-3}{r}\right)\frac{\mathrm{d}}{\mathrm{d}r}+W(\varphi(r))\right]T_{l}(r;\varphi)=0\,,$
(34)

where

$\displaystyle W(\varphi)=U^{\prime\prime}(\varphi)-m^{2}\,.$ (35)

The boundary conditions are

$\displaystyle
T_{l}(0;\varphi)=1\,,\quad\left.\frac{\mathrm{d}T_{l}(r;\varphi)}{\mathrm{d}r}\right|_{r=0}=0\,.$
(36)

We note that Eq. (34) can be written in a slightly different form due to the
recursion relations of the modified Bessel functions.

Below, we take $D=4$.

### Divergences and renormalisation

The sum

$\displaystyle\sum_{l=0}^{\infty}(l+1)^{2}\ln T_{l}(\infty;\varphi)$ (37)

is divergent. This is not surprising as we are calculating the one-loop
effective action whose divergence can only be removed by counterterms.
Formally, we can write the renormalised one-loop effective action as

$\displaystyle S^{\rm ren}_{1}[\varphi]=S_{0}[\varphi]+\Delta
S[\varphi]+S_{\rm ct}[\varphi]\,,$ (38)

where $S_{0}[\varphi]$ is the classical action, $\Delta
S[\varphi]=D[\varphi]/2$, and $S_{\rm ct}[\varphi]$ is the contribution from
the counterterms. In the renormalised effective action, all the couplings
(including mass parameters) are now renormalised ones. Different
renormalisation schemes define different counterterms and renormalised
couplings. In the $\overline{\rm MS}$ scheme, we have

$\displaystyle S^{\rm ren}_{1}[\varphi]=S_{0}^{\overline{\rm
MS}}[\varphi]+\left(\Delta S[\varphi]-(\Delta S[\varphi])^{\rm
pole}\right)\equiv S_{0}^{\overline{\rm MS}}[\varphi]+(\Delta S[\varphi])^{\rm
reg}\,,$ (39)

where $S_{0}^{\overline{\rm MS}}$ is the lowest-order action expressed in
terms of the renormalised $\overline{\rm MS}$ couplings, and $(\Delta
S[\varphi])^{\rm pole}$ is pole of the divergent part of $\Delta S$ defined
according to the $\overline{\rm MS}$ renormalisation prescription.

In the literature, two slightly different methods of obtaining a $(\Delta
S[\varphi])^{\rm reg}$ are used. The first one was used by Dunne and Min [40]
(see also [44, 45]), and followed by Refs. [41, 42, 43]. The second one was
used even earlier by Baacke and Kiselev [46], and followed in Refs. [7, 47,
48, 19]. Below, we shall show that when the functional derivative with respect
to $\varphi$ is taken, the first method gives a UV divergent result. This
means that the first method is at most valid only “on shell”, i.e., only for
the classical bounce solution but not for variations about it. The second
method still gives a UV convergent functional derivative. However, we observe
that the regularized one-loop functional determinant contains a UV-finite
tadpole term (linear term in $\varphi$) for our model which would shift the
position of the false vacuum from zero. For self-consistency, we therefore
propose removing such a nonvanishing tadpole by a linear counterterm. This
would modify one of the finite terms ($A_{\rm fin}^{(1)}$) to be discussed
below.

The first method gives [40, 44]

$\displaystyle(\Delta S[\varphi])_{\rm DM}^{\rm
reg}=\frac{1}{2}\sum_{l=0}^{\infty}(l+1)^{2}\left\\{\ln
T_{l}(\infty;\varphi)-\frac{\int_{0}^{\infty}\mathrm{d}r\,rW(\varphi)}{2(l+1)}+\frac{\int_{0}^{\infty}\mathrm{d}r\,r^{3}W(\varphi)(W(\varphi)+2m^{2})}{8(l+1)^{3}}\right\\}$
$\displaystyle-\frac{1}{16}\int_{0}^{\infty}\mathrm{d}r\,r^{3}W(\varphi)\left(W(\varphi)+2m^{2}\right)\left[\log\left(\frac{\mu
r}{2}\right)+\gamma_{\rm E}+1\right]\,,$ (40)

where $\gamma_{\rm E}$ is the Euler’s constant, and $\mu$ is a renormalisation
scale.

The second method gives [47]

$\displaystyle(\Delta S[\varphi])_{\rm BL}^{\rm reg}=$
$\displaystyle\frac{1}{2}\sum_{l=0}^{\infty}(l+1)^{2}\left\\{\ln
T_{l}(\infty;\varphi)-h_{l}^{(1)}(\infty;\varphi)-\left[h_{l}^{(2)}(\infty;\varphi)-\frac{1}{2}\left(h_{l}^{(1)}(\infty;\varphi)\right)^{2}\right]\right\\}$
$\displaystyle+\frac{1}{2}\left(A_{\rm fin}^{(1)}-\frac{1}{2}A_{\rm
fin}^{(2)}\right)\,,$ (41)

where $h_{l}^{(1)}$, $h_{l}^{(2)}$ satisfy

$\displaystyle\left[-\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}-\left(2m\frac{I_{l}(mr)}{I_{l+1}(mr)}-\frac{2l+1}{r}\right)\frac{\mathrm{d}}{\mathrm{d}r}\right]h_{l}^{(1)}(r;\varphi)+W(\varphi)=0\,,$
(42a)
$\displaystyle\left[-\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}-\left(2m\frac{I_{l}(mr)}{I_{l+1}(mr)}-\frac{2l+1}{r}\right)\frac{\mathrm{d}}{\mathrm{d}r}\right]h_{l}^{(2)}(r;\varphi)+W(\varphi)h_{l}^{(1)}(r;\varphi)=0\,,$
(42b)

with boundary conditions
$h_{l}^{(1,2)}(0;\varphi)=0,\mathrm{d}h_{l}^{(1,2)}(r;\varphi)/\mathrm{d}r|_{r=0}=0$
. Above,

$\displaystyle A^{(1)}_{\rm fin}$
$\displaystyle=-\frac{m^{2}}{8}\left[1+\ln\left(\frac{\mu^{2}}{m^{2}}\right)\right]\int_{0}^{\infty}\mathrm{d}r\,r^{3}W(\varphi(r))\,,$
(43a) $\displaystyle A^{(2)}_{\rm fin}$
$\displaystyle=\frac{1}{8}\ln\left(\frac{\mu^{2}}{m^{2}}\right)\int\mathrm{d}r\,r^{3}[W(\varphi(r))]^{2}$
$\displaystyle+\frac{1}{16\pi^{2}}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}|\widetilde{W}(k)|^{2}\left[2-\frac{\sqrt{k^{2}+4m^{2}}}{|k|}\ln\left(\frac{\sqrt{k^{2}+4m^{2}}+|k|}{\sqrt{k^{2}+4m^{2}}-|k|}\right)\right]\,,$
(43b)

where

$\displaystyle\widetilde{W}(k)=\int\mathrm{d}^{4}x\,\mathrm{e}^{\mathrm{i}kx}W(\varphi(x))\,.$
(44)

In Appendix B, we derive the above equations. Since $\varphi(x)$ depends on
$r=|x|$ only, we can write

$\displaystyle\widetilde{W}(k)=\frac{4\pi^{2}}{|k|}\int_{0}^{\infty}\mathrm{d}r\,r^{2}W(\varphi(r))J_{1}(|k|r)\equiv\widetilde{W}(|k|)\,,$
(45)

where $J_{1}$ is the Bessel function of the first kind, defined as

$\displaystyle
J_{1}(|k|r)=\frac{|k|r}{\pi}\int_{0}^{\pi}\mathrm{d}\theta\,\sin^{2}\theta\,\mathrm{e}^{\mathrm{i}|k|r\cos\theta}\,.$
(46)

We observe that $A_{\rm fin}^{(1)}$ contains a linear term in $\varphi$ for
our model since $W(\varphi)=-g\varphi+\lambda\varphi^{2}/2$. As a consequence,

$\displaystyle\frac{\delta A_{\rm
fin}^{(1)}(\varphi)}{\delta\varphi(r)}=-\frac{m^{2}}{8}r^{3}U^{\prime\prime\prime}(\varphi(r))=\frac{m^{2}g}{8}r^{3}-\frac{m^{2}\lambda}{8}r^{3}\varphi(r)\,,$
(47)

is not zero even for $\varphi=0$. This means that the position of the false
vacuum would be shifted if there is no other linear terms in Eq. (4). We show
below that such additional linear terms are absent. To be self-consistent, we
have to add a linear counterterm to remove the observed tadpole term by
requiring

$\displaystyle\left.\frac{\delta S_{1}^{\rm
ren}[\varphi]}{\delta\varphi(x)}\right|_{\varphi=0}=0\,.$ (48)

This is achieved by simply doing the following replacement,

$\displaystyle A_{\rm fin}^{(1)}\rightarrow\overline{A_{\rm
fin}^{(1)}}=-\frac{m^{2}}{8}\left[1+\ln\left(\frac{\mu^{2}}{m^{2}}\right)\right]\int_{0}^{\infty}\mathrm{d}r\,r^{3}\,\overline{W(\varphi(r))}\,,$
(49)

where an overline means the subtraction of tadpole terms. Note that the term
$m^{2}gr^{3}/8$ in $\delta A_{\rm fin}^{(1)}(\varphi)/\delta\varphi(r)$ itself
is not a problem for the numerical calculations discussed below as the factor
$r^{3}$ will be canceled out in Eq. (23).

For false vacuum decay, it is natural to take $\mu=m$ (as we will do in the
following) so that $\overline{A^{(1)}_{\rm fin}}$ and $A^{(2)}_{\rm fin}$
simplify further.

One can numerically solve Eq. (42a) to obtain $h_{l}^{(1)}(r;\varphi)$ (and
hence $h_{l}^{(1)}(\infty;\varphi)$), while $h_{l}^{(2)}(\infty;\varphi)$ as
well as its functional derivative with respect to $\varphi$ can be computed
via a Green’s function method with known $h_{l}^{(1)}(r;\varphi)$, see below.

#### Zero and negative modes

When taking $\varphi=\varphi^{(0)}_{b}$, there are four zero modes in the
sector $l=1$. The integral over the fluctuations in the zero-mode directions
can be traded for an integral over the collective coordinates of the bounce,
giving [44, 43]

$\displaystyle\mathrm{e}^{-\frac{1}{2}(1+1)^{2}\ln
T_{1}(\infty;\varphi^{(0)}_{b})}\rightarrow\underbrace{(V\mathcal{T})\left(\frac{S_{\rm
E}[\varphi_{b}^{(0)}]}{2\pi}\right)^{2}}_{\rm zero-mode\
contribution}\left(\frac{\det^{\prime}\mathcal{M}_{1}[\varphi^{(0)}_{b}]}{\det\widehat{\mathcal{M}}_{1}}\right)^{-2}=(V\mathcal{T})\left[2\pi
A|\partial_{r}^{2}\varphi^{(0)}_{b}(0)|\right]^{2}\,,$ (50)

where $A$ is defined by the normalisation of the asymptotic large $r$ behavior
of the classical bounce: $\varphi_{b}^{(0)}(r)\sim Ar^{-1}K_{1}(r)$. Here
$K_{1}(r)$ is the modified Bessel function of the second kind. Using the
equation of motion for $\varphi_{b}^{(0)}$, Eq. (7), one has

$\displaystyle\left.\frac{\mathrm{d}^{2}\varphi_{b}^{(0)}(r)}{\mathrm{d}r^{2}}\right|_{r=0}=\frac{U^{\prime}(\varphi_{b}^{(0)}(0))}{4}\,,$
(51)

where we have used
$\partial_{r}\varphi_{b}^{(0)}(r)/r\rightarrow\partial^{2}_{r}\varphi_{b}^{(0)}(r)$
as $r\rightarrow 0$. So, formally, we have the replacement

$\displaystyle\ln
T_{1}(\infty;\varphi_{b}^{(0)})\rightarrow-\frac{1}{2}\ln(V\mathcal{T})-\ln\left[(\pi
AU^{\prime}(\varphi_{b}^{(0)}(0))/2)^{2}\right]\,.$ (52)

Note that the factor of $V\mathcal{T}$ in Eq. (50) will be cancelled by the
same factor in $\gamma/V$ (cf. Eq. (14)), leading to a finite decay rate per
volume.

The lowest eigenvalue mode in the $l=0$ sector is a negative mode and is
responsible for the instability of the false vacuum. Naively, the negative
mode leads to divergence in the Gaussian functional integral about the bounce.
However, this divergence is simply due to an improper application of the
method of steepest descent [21, 49, 32]. With a suitable deformation of the
contour into the complex plane whose real axis is generated by the negative
mode, one actually obtains the following replacement

$\displaystyle
T_{0}(\infty;\varphi_{b}^{(0)})\rightarrow\quad\left(\pm\frac{\mathrm{i}}{2}\right)^{-2}\left|T_{0}(\infty;\varphi_{b}^{(0)})\right|\,,$
(53)

where the additional factor is due to the contour deformation and gives rise
to an imaginary part of the effective action with a particular factor of
$1/2$. The sign depends on the direction in which one deforms the contour into
the complex plane. Usually, one chooses the way that leads to a positive decay
rate. But in our formula (14), we have taken the absolute value, and the sign
in Eq. (53) does not matter.

Below, we show how to take the functional derivative in Eqs. (4) and (4).

##  5 Functional derivative of a functional determinant

To compute the quantum corrections to the bounce as well as the corresponding
corrections to the decay rate, we still need to compute $\delta
D[\varphi]/\delta\varphi(r)|_{\varphi^{(0)}_{b}}$ (cf. Eqs. (23) and (25)). We
will use Eq. (4), which leads

$\displaystyle\frac{\delta(\Delta S[\varphi])_{\rm BL}^{\rm
reg}}{\delta\varphi(r)}=$
$\displaystyle\frac{1}{2}\sum_{l=0}^{\infty}(l+1)^{2}\left\\{\frac{1}{T_{l}(\infty;\varphi)}\frac{\delta
T_{l}(\infty;\varphi)}{\delta\varphi(r)}-\frac{\delta
h_{l}^{(1)}(\infty;\varphi)}{\delta\varphi(r)}-\frac{\delta
h_{l}^{(2)}(\infty;\varphi)}{\delta\varphi(r)}\right.$
$\displaystyle\qquad\qquad\qquad\left.+h_{l}^{(1)}(\infty;\varphi)\frac{\delta
h_{l}^{(1)}(\infty;\varphi)}{\delta\varphi(r)}\right\\}+\frac{1}{2}\frac{\delta\overline{A_{\rm
fin}^{(1)}}(\varphi)}{\delta\varphi(r)}-\frac{1}{4}\frac{\delta A_{\rm
fin}^{(1)}(\varphi)}{\delta\varphi(r)}\,.$ (54)

At the end of this section, we shall comment on the problem if one uses Eq.
(4).

We need to compute $\delta T_{l}(\infty;\varphi)/\delta\varphi$, $\delta
h_{l}^{(1,2)}(\infty;\varphi)/\delta\varphi$. To simplify the notation, we
define

$\displaystyle\mathcal{H}_{l}[\varphi]\equiv\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}+\left(2m\frac{I_{l}(mr)}{I_{l+1}(mr)}-\frac{2l+1}{r}\right)\frac{\mathrm{d}}{\mathrm{d}r}-W(\varphi(r))\,,$
(55a)
$\displaystyle\widehat{\mathcal{H}}_{l}\equiv\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}+\left(2m\frac{I_{l}(mr)}{I_{l+1}(mr)}-\frac{2l+1}{r}\right)\frac{\mathrm{d}}{\mathrm{d}r}\,.$
(55b)

Then

$\displaystyle\mathcal{H}_{l}[\varphi]T_{l}(r;\varphi)=0\,,\qquad\widehat{\mathcal{H}}_{l}h_{l}^{(1)}(r;\varphi)=W(\varphi(r))\,,\qquad\widehat{\mathcal{H}}_{l}h^{(2)}_{l}(r;\varphi)=W(\varphi(r))h_{l}^{(1)}(r;\varphi)\,.$
(56)

Taking the functional derivative of the first equation with respect to
$\varphi(r)$ gives

$\displaystyle\mathcal{H}_{l}[r^{\prime};\varphi]\frac{\delta
T_{l}(r^{\prime};\varphi)}{\delta\varphi(r)}=-\frac{\delta\mathcal{H}_{l}[r^{\prime};\varphi]}{\delta\varphi(r)}T_{l}(r^{\prime};\varphi)=\delta(r^{\prime}-r)U^{\prime\prime\prime}(\varphi(r^{\prime}))T_{l}(r^{\prime};\varphi)\,.$
(57)

Viewing $r$ as a fixed number, the above equation in the coordinate
$r^{\prime}$ can then be solved by the Green’s function method and we obtain

$\displaystyle\frac{\delta T_{l}(r^{\prime};\varphi)}{\delta\varphi(r)}$
$\displaystyle=\int\mathrm{d}r^{\prime\prime}\,G_{l}(r^{\prime},r^{\prime\prime};\varphi)\delta(r^{\prime\prime}-r)U^{\prime\prime\prime}(\varphi(r^{\prime\prime}))T_{l}(r^{\prime\prime};\varphi)=G_{l}(r^{\prime},r;\varphi)U^{\prime\prime\prime}(\varphi(r))T_{l}(r;\varphi)\,,$
(58)

where $G_{l}(r,r^{\prime};\varphi)$ satisfies

$\displaystyle\mathcal{H}_{l}[\varphi(r)]G_{l}(r,r^{\prime};\varphi)=\delta(r-r^{\prime})\,,$
(59)

with the boundary conditions555To see these are the correct boundary
conditions, one can analyse the equation
$\mathcal{H}_{l}[\varphi+\delta\varphi]T_{l}(r;\varphi+\delta\varphi)=0$ by
writing $T_{l}(r;\varphi+\delta\varphi)=T_{l}(r;\varphi)+\delta
T_{l}(r;\varphi)$. Since $T_{l}(r;\varphi)$ and
$T_{l}(r;\varphi+\delta\varphi)$ satisfy the same boundary conditions imposed
in the Gel’fand-Yaglom theorem, $\delta T_{l}(r;\varphi)$ has the boundary
conditions $\delta T_{l}(0;\varphi)=0\,,\mathrm{d}\delta
T_{l}(r;\varphi)/\mathrm{d}r|_{r=0}=0$ which are shared by $\delta
T_{l}(r;\varphi)/\delta\varphi$ and thus the Green’s function $G_{l}$.

$\displaystyle
G_{l}(0,r^{\prime};\varphi)=0\,,\quad\partial_{r}G_{l}(r,r^{\prime};\varphi)|_{r=0}=0\,.$
(60)

Taking $r^{\prime}=\infty$ one obtains

$\displaystyle\frac{\delta
T_{l}(\infty;\varphi)}{\delta\varphi(r)}=G_{l}(\infty,r;\varphi)U^{\prime\prime\prime}(\varphi(r))T_{l}(r;\varphi)\,.$
(61)

One can apply a similar analysis for the second and third equations in (56).
But it is more straightforward to take the functional derivative with explicit
expressions of $h_{l}^{(1,2)}(\infty;\varphi)$. In this case, we need to solve
the Green’s function

$\displaystyle\widehat{\mathcal{H}}_{l}\hat{G}_{l}(r,r^{\prime})=\delta(r-r^{\prime})\,,$
(62)

with the same boundary conditions as in Eq. (60). Then

$\displaystyle
h_{l}^{(l)}(\infty;\varphi)=\int\mathrm{d}r^{\prime}\,\hat{G}_{l}(\infty,r^{\prime})W(\varphi(r^{\prime}))\,,$
(63a) $\displaystyle
h_{l}^{(2)}(\infty;\varphi)=\int\mathrm{d}r^{\prime}\hat{G}_{l}(\infty,r^{\prime})W(\varphi(r^{\prime}))h_{l}^{(1)}(r^{\prime};\varphi)\,.$
(63b)

The same expressions are valid if one replaces $\infty$ with $r$ above. Taking
the functional derivative of these expressions, one obtains

$\displaystyle\frac{\delta
h_{l}^{(1)}(\infty;\varphi)}{\delta\varphi(r)}=\hat{G}_{l}(\infty,r)U^{\prime\prime\prime}(\varphi(r))\,,$
(64a) $\displaystyle\frac{\delta
h_{l}^{(2)}(\infty;\varphi)}{\delta\varphi(r)}=\hat{G}_{l}(\infty,r)U^{\prime\prime\prime}(\varphi(r))h_{l}^{(1)}(r;\varphi)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\int\mathrm{d}r^{\prime}\,\hat{G}_{l}(\infty,r^{\prime})W(\varphi(r^{\prime}))\hat{G}_{l}(r^{\prime},r)U^{\prime\prime\prime}(\varphi(r))\,.$
(64b)

$\hat{G}_{l}(r,r^{\prime})$ can be obtained analytically (see Appendix A) and
reads

$\displaystyle\hat{G}_{l}(r,r^{\prime})=\theta(r-r^{\prime})\frac{r^{\prime}I_{l+1}(mr^{\prime})}{I_{l+1}(mr)}\left[I_{l+1}(mr)K_{l+1}(mr^{\prime})-I_{l+1}(mr^{\prime})K_{l+1}(mr)\right]\,,$
(65)

where $K_{l+1}(x)$ is the modified Bessel function of the second kind. The
Green’s functions $G_{l}(r,r^{\prime};\varphi)$ depend on $\varphi$ and can
only be obtained numerically.

From Appendix A, we can see

$\displaystyle G_{l}(r,\infty;\varphi)=\hat{G}_{l}(r,\infty)=0\,.$ (66)

Therefore, one immediately has

$\displaystyle\left.\frac{\delta
T_{l}(\infty;\varphi)}{\delta\varphi(r)}\right|_{r=\infty}=0\,.$ (67)

If one uses Eq. (4), one would obtain a divergent and cutoff-dependent result
from the second and third terms in the sum over $l$ at $r=\infty$. Therefore,
Eq. (4) cannot give a meaningful result once the functional derivative with
respect to $\varphi(r)$ is taken. On the contrary, Eq. (4) gives a finite
result because we also have

$\displaystyle\left.\frac{\delta
h_{l}^{(1)}(\infty;\varphi)}{\delta\varphi(r)}\right|_{r=\infty}=\left.\frac{\delta
h_{l}^{(2)}(\infty;\varphi)}{\delta\varphi(r)}\right|_{r=\infty}=0\,,$ (68)

due to the properties in Eq. (66). This also shows that there is no additional
tadpole term in the sum over $l$ in Eq. (4), otherwise one would obtain a term
proportional to $r^{3}$ in the functional derivative, which is not vanishing
for $r\rightarrow\infty$.

At last, the functional derivatives of $\overline{A^{(1)}_{\rm fin}}$ and
$A^{(2)}_{\rm fin}$ with respect to $\varphi(r)$ read

$\displaystyle\frac{\delta\overline{A_{\rm
fin}^{(1)}}(\varphi)}{\delta\varphi(r)}=$
$\displaystyle-\frac{m^{2}\lambda}{8}r^{3}\varphi(r)\,,$ (69a)
$\displaystyle\frac{\delta A_{\rm fin}^{(2)}(\varphi)}{\delta\varphi(r)}=$
$\displaystyle\,\frac{r^{2}}{16\pi^{2}}U^{\prime\prime\prime}(\varphi(r))\int\mathrm{d}|k|\,|k|^{2}J_{1}(|k|r)\widetilde{W}(|k|)$
$\displaystyle\times\left[2-\frac{\sqrt{|k|^{2}+4m^{2}}}{|k|}\ln\left(\frac{\sqrt{|k|^{2}+4m^{2}}+|k|}{\sqrt{|k|^{2}+4m^{2}}-|k|}\right)\right]\,,$
(69b)

where $\widetilde{W}(|k|)$ is defined in Eq. (45).

##  6 Numerical example

With the analysis given in the previous sections, here we discuss the
procedure for numerical calculations and provide a numerical example. We leave
a more delicate phenomenological study for future work.

Denote

$\displaystyle\frac{1}{2}D[\varphi_{b}^{(0)}]=B^{(1)}-\ln\left(\pm\frac{\mathrm{i}}{2}\right)-\ln(V\mathcal{T})\,,\quad
B^{(2)}=\frac{1}{4}\int\mathrm{d}r\,\left.\frac{\delta
D[\varphi]}{\delta\varphi(r)}\right|_{\varphi_{b}^{(0)}}\Delta\varphi_{b}(r)\,,$
(70)

so that

$\displaystyle\frac{\gamma}{V}=\frac{2\,{\rm
Im}\,\mathrm{e}^{-\Gamma[\varphi_{b}^{(1)}]}}{V\mathcal{T}}=\mathrm{e}^{-B^{(0)}-B^{(1)}-B^{(2)}}\,.$
(71)

We summarise the procedure of computing the radiative corrections to the
bounce and decay rate as follows.

  * •

Step 1: Solve the classical EoM (7) and obtain the classical bounce
$\varphi_{b}^{(0)}$. Substitute it into Eq. (9) to obtain the classical bounce
action $B^{(0)}$.

  * •

Step 2: Substitute $\varphi_{b}^{(0)}$ into Eq. (34) to obtain
$T_{l}(r;\varphi_{b}^{(0)})$, in particular its limiting value at
$r\rightarrow\infty$, $T_{l}(\infty;\varphi_{b}^{(0)})$.

  * •

Step 3: Substituting $\hat{G}_{l}(\infty,r^{\prime})$ (cf. Eq. (65)) and
$\varphi_{b}^{(0)}$ into Eq. (63a) to obtain
$h_{l}^{(1)}(r;\varphi_{b}^{(0)})$ and
$h_{l}^{(1)}(\infty;\varphi_{b}^{(0)})$. Substituting
$h_{l}^{(1)}(r;\varphi_{b}^{(0)})$ into Eq. (63b), one obtains
$h_{l}^{(2)}(\infty;\varphi_{b}^{(0)})$.

The zero and negative modes in computing $T_{l}(\infty;\varphi_{b}^{(0)})$
need to be properly dealt with as discussed above. With these three steps, one
can obtain $B^{(0)}$ and $B^{(1)}$ and thus the Callan-Coleman decay rate. To
obtain the decay rate at the full one-loop level of the effective action, we
need three further steps.

  * •

Step 4: Substituting $\varphi_{b}^{(0)}$ into Eq. (59), one can obtain
$G_{l}(\infty,r^{\prime};\varphi_{b}^{(0)})$. With
$G_{l}(\infty,r^{\prime};\varphi_{b}^{(0)})$ and $T_{l}(r;\varphi_{b}^{(0)})$,
one obtains $\delta
T_{l}(\infty;\varphi)/\delta\varphi(r)|_{\varphi_{b}^{(0)}}$ through Eq. (61).

Further, with $\hat{G}_{l}(r,r^{\prime})$ and
$h_{l}^{(1)}(r;\varphi_{b}^{(0)})$, one can obtain $\delta
h_{l}^{(1)}(\infty;\varphi)/\delta\varphi(r)|_{\varphi_{b}^{(0)}}$ and
$\delta h_{l}^{(2)}(\infty;\varphi)/\delta\varphi(r)|_{\varphi_{b}^{(0)}}$
through Eqs. (64).

Substituting all of them and Eqs. (69) into Eq. (5), one can finally obtain
$\delta D[\varphi]/\delta\varphi(r)|_{\varphi_{b}^{(0)}}$.

  * •

Step 5: Compute the Green’s function in Eq. (24) and substitute the obtained
Green’s function $\mathcal{G}(r,r^{\prime})$ and $\delta
D[\varphi]/\delta\varphi(r)|_{\varphi_{b}^{(0)}}$ into Eq. (23) to obtain
$\Delta\varphi_{b}(r)$.

  * •

Step 6: Finally, substitute the obtained $\Delta\varphi_{b}(r)$ and $\delta
D[\varphi]/\delta\varphi(r)|_{\varphi_{b}^{(0)}}$ into the second equation in
(70) to obtain $B^{(2)}$.

Again, in calculating the functional derivatives with respect to $\varphi$,
the zero and negative modes need special treatment. We assume that Eq. (53) is
still valid for $\varphi$ close to $\varphi_{b}^{(0)}$, so that one can obtain
$\delta T_{0}(\infty;\varphi)/\delta\varphi(r)|_{\varphi_{b}^{(0)}}$ from that
replacement. It is not fully clear to us how to deal with the zero mode. We
thus will neglect the zero-mode contribution to the functional derivative.
However, we have checked that $\delta
h_{1}^{(1,2)}/\delta\varphi(r)|_{\varphi_{b}^{(0)}}$ are indeed negligible in
the final result of $\delta D[\varphi]/\delta\varphi(r)|_{\varphi_{b}^{(0)}}$.

In the numerical calculations, all the parameters that have mass dimension one
take values in the unit of $m$ so that $m=1$. As a benchmark point, we choose

$\displaystyle g=0.66\,,\qquad\lambda=0.1\,.$

The profile of the potential is shown in Fig. 1. The classical bounce is
obtained via a shooting algorithm and is shown in Fig. 2. The classical bounce
action is $B^{(0)}=12942.7$.

Figure 2: The profile of the classical bounce $\varphi_{b}^{(0)}(r)$.

For convenience of later discussion, we denote

$\displaystyle f_{l}[\varphi]\equiv\ln
T_{l}(\infty;\varphi)-h_{l}^{(1)}(\infty;\varphi)-\left[h_{l}^{(2)}(\infty;\varphi)-\frac{1}{2}\left(h_{l}^{(1)}(\infty;\varphi)\right)^{2}\right]\,.$
(72)

In calculating $B^{(1)}$, we first compute $f_{l}[\varphi_{b}^{(0)}]$ for
$l\geq 2$. We find that $l_{\rm max}=50$ is already a good cutoff. We plot
$f_{l}[\varphi_{b}^{(0)}]$ in Fig. 3 where we compare the numerical result
with $-50/l^{4}$. Clearly $|f_{l}[\varphi_{b}^{(0)}]|$ drops as fast as or
faster than $1/l^{4}$, ensuring a convergent result for the sum. We obtain

$\displaystyle\frac{1}{2}\sum_{l\geq
2}(l+1)^{2}f_{l}[\varphi_{b}^{(0)}]=-10.981\,.$ (73)

For the contribution from $l=0,1$, we find

$\displaystyle
B^{(1)}\supset\frac{1}{2}\ln\left|T_{0}(\infty;\varphi_{b}^{(0)})\right|=-1.449\,,$
(74a) $\displaystyle B^{(1)}\supset-\ln\left[(\pi
AU^{\prime}(\varphi_{b}^{(0)}(0))/2)^{2}\right]=-7.388\,,$ (74b)
$\displaystyle
B^{(1)}\supset-\frac{1}{2}\sum_{l=0,1}\left[h_{l}^{(1)}(\infty;\varphi)+h_{l}^{(2)}(\infty;\varphi)-\frac{1}{2}\left(h_{l}^{(1)}(\infty;\varphi)\right)^{2}\right]=-4.671\,.$
(74c)

Finally, $\overline{A_{\rm fin}^{(1)}}$ and $A_{\rm fin}^{(2)}$ are obtained
as

$\displaystyle\overline{A_{\rm fin}^{(1)}}=-15.177\,,\ A_{\rm
fin}^{(2)}=-5.322\quad\Rightarrow\quad\frac{1}{2}\left(\overline{A_{\rm
fin}^{(1)}}-\frac{1}{2}A_{\rm fin}^{(2)}\right)=-6.258\,.$ (75)

In total, we have $B^{(1)}=-30.747$.

Figure 3: The behavior of $f_{l}$ as a function of $l$.

The correction to the classical bounce $\Delta\varphi_{b}(r)$ is shown Fig. 4.
The profile of $\Delta\varphi_{b}(r)$ is negligible compared with
$\varphi_{b}^{(0)}(r)$. However, the additional correction to the decay rate
is obtained as $B^{(2)}=-16.42$, which is $53.4\%$ of $B^{(1)}$. The
correction $\Delta\varphi(r)$ to the classical bounce is maximal around
$rm\simeq 3.5$, which also corresponds to roughly the steepest part of the
bubble wall profile. This indicates that these higher-order corrections
receive large contributions from the gradients of the bubble wall. This
observation is consistent with that made in Ref. [25] for thin-wall bubbles.

Figure 4: The correction to the classical bounce, $\Delta\varphi(r)$.

##  7 Conclusions

FVD is a very important phenomenon in quantum field theory and there continue
to be many developments both conceptual and methodological [32, 50, 51, 52,
53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. In this paper, we propose a
systematic procedure to compute the self-consistent bounce solution and its
resulting radiative correction to the decay rate at the one-loop level of the
effective action. This is achieved by evaluating the one-loop effective action
at the one-loop saddle point configuration for the bounce (the self-consistent
bounce), instead of the classical one. The method requires the computation of
both conventional functional determinants and their functional derivative,
both evaluated at the classical bounce. Our procedure provides an improved
one-loop determination of the tunnelling rate, via a partial resummation of
two-loop contributions. Earlier studies on this are restricted to the thin-
wall regime [22, 25, 26] and are usually based on the “resolvent method” of
calculating functional determinants by treating Green’s function as a spectral
sum [65, 66, 67, 68, 32]. In this work, we have demonstrated that the powerful
Gel’fand-Yaglom method can be used to compute both the functional determinants
and their functional derivatives, making the computation extremely efficient.
In particular, for $O(4)$-symmetric configurations that asymptotically
approach the false vacuum, an explicit expression for the renormalised one-
loop effective action in pure scalar field theory is known. We have shown how
it is directly used in the computation.

Although the theoretical developments here are general, we provide an explicit
numerical study of our archetypal model. Our chosen benchmark point for the
mass and coupling parameters is such that the bounce is far away from the
thin-wall regime. In this example, the correction $B^{(1)}$ (the decay rate
due to the functional determinant evaluated at the classical bounce) is small
compared with the classical bounce action $B^{(0)}$. However, the additional
correction due to the consideration of the self-consistent bounce, $B^{(2)}$,
is of the same order of magnitude (about 50%) as the purely one-loop
correction $B^{(1)}$. This indicates that in some precision phenomenological
studies, when the conventional functional determinants are important, one may
be required to consider the self-consistent bounce as well.

Another motivation for considering the self-consistent bounce is to obtain a
gauge-independent tunneling rate when gauge fields are involved. It is known
that the effective action is gauge-dependent. By the Nielsen identity [69, 70,
34],

$\displaystyle\left(\xi\frac{\partial\Gamma[\varphi;\xi]}{\partial\xi}+K(\varphi;\xi)\frac{\delta\Gamma[\varphi;\xi]}{\delta\varphi}\right)=0\,,$
(76)

where $\xi$ is the gauge parameter, the effective action is only gauge
independent at its extrema. Then Eq. (14) gives a gauge-independent decay rate
in principle only at the self-consistent bounce [34]. Since the effective
action must be truncated, this issue is usually more complicated. For more
discussions related to this topic, see Refs. [71, 72, 73, 74, 75, 76, 77].

Our method may find applications in physics of the early Universe (see, e.g.,
Refs. [78, 79, 80]), where bubble nucleation can occur at finite temperature
[81, 82, 83, 84, 85].666Gravitational effects could also be very important in
some situations [86, 87, 88, 89, 90]. The calculation presented in this paper
can be easily generalized to lower-dimensional bounce configurations, e.g.,
$O(3)$-symmetric static bounces that are relevant for thermal first-order
phase transitions. With proper generalizations, the method developed in this
article can thus be applied to study radiative corrections in thermal first-
order phase transitions [91]. However, at finite temperatures, there are many
more theoretical uncertainties [92, 93]. The quantum corrections due to the
self-consistent bounce might be detectable in experiments involving analogue
systems in condensed matter [94, 95, 96, 97].

## Acknowledgments

The authors of this work are supported by the Engineering and Physical
Sciences Research Council (grant No. EP/V002821/1). JA is also supported by
the Leverhulme Trust (grant No. RPG-2021-299) and by the Science and
Technology Facilities Council (grant No. STFC-ST/X000753/1).

##  Appendix A Solving Green’s functions

In this appendix, we outline the procedure on solving Green’s functions
numerically. Suppose we want to solve the following Green’s function

$\displaystyle L[r]G(r,r^{\prime})=\delta(r-r^{\prime})\,,$ (A1)

where $L[r]$ is a second-order differential operator and
$\\{r,r^{\prime}\\}\in[0,\infty)$.

###  A.1 $G_{l}(r,r^{\prime};\varphi)$ and $\hat{G}_{l}(r,r^{\prime})$

For $G_{l}(r,r^{\prime};\varphi)$ and $\hat{G}_{l}(r,r^{\prime})$, the
boundary conditions are of the following type,

$\displaystyle G(0,r^{\prime})=0\,,\quad\left.\frac{\partial
G(r,r^{\prime})}{\partial r}\right|_{r=0}=0\,.$ (A2)

Assume that $\\{f_{1}(r),f_{2}(r)\\}$ are two independent (numerical)
solutions to the equation $L[r]f(r)=0$. Then the Green’s function can be
written as

$\displaystyle
G(r,r^{\prime})=\theta(r-r^{\prime})[A_{1}^{>}(r^{\prime})f_{1}(r)+A_{2}^{>}(r^{\prime})f_{2}(r)]+\theta(r^{\prime}-r)[A_{1}^{<}(r^{\prime})f_{1}(r)+A_{2}^{<}(r^{\prime})f_{2}(r)]\,,$
(A3)

where $A_{1}^{\gtrless}(r^{\prime})$ and $A_{2}^{\gtrless}(r^{\prime})$ shall
be determined by the boundary conditions and matching conditions.

The matching conditions are from the following two requirements on
$G(r,r^{\prime})$: $(i)$ $G(r,r^{\prime})$ is continuous at $r=r^{\prime}$
when viewed as a function of $r$; $(ii)$ $\partial_{r}G(r,r^{\prime})$ has a
finite jump discontinuity at $r=r^{\prime}$. The sign and magnitude of this
jump depends on the coefficient of the second-derivative term in the operator
$L[r]$. We will assume that this coefficient is $1$ to be consistent with the
operators $\mathcal{H}_{l}[\varphi]$ and $\widehat{\mathcal{H}}_{l}$. Thus we
have

$\displaystyle[A_{1}^{>}(r^{\prime})f_{1}(r^{\prime})+A_{2}^{>}(r^{\prime})f_{2}(r^{\prime})]-[A_{1}^{<}(r^{\prime})f_{1}(r^{\prime})+A_{2}^{<}(r^{\prime})f_{2}(r^{\prime})]=0\,,$
(A4a)
$\displaystyle[A_{1}^{>}(r^{\prime})f^{\prime}_{1}(r^{\prime})+A_{2}^{>}(r^{\prime})f^{\prime}_{2}(r^{\prime})]-[A_{1}^{<}(r^{\prime})f^{\prime}_{1}(r^{\prime})+A_{2}^{<}(r^{\prime})f^{\prime}_{2}(r^{\prime})]=1\,.$
(A4b)

After some algebra, one obtains

$\displaystyle
A_{1}^{>}(r^{\prime})-A_{1}^{<}(r^{\prime})=-\frac{f_{2}(r^{\prime})}{\mathcal{W}[f_{1}(r^{\prime}),f_{2}(r^{\prime})]}\,,$
(A5a) $\displaystyle
A_{2}^{>}(r^{\prime})-A_{2}^{<}(r^{\prime})=\frac{f_{1}(r^{\prime})}{\mathcal{W}[f_{1}(r^{\prime}),f_{2}(r^{\prime})]}\,,$
(A5b)

where

$\displaystyle\mathcal{W}[f_{1}(r),f_{2}(r)]=f_{1}(r)f^{\prime}_{2}(r)-f^{\prime}_{1}(r)f_{2}(r)$
(A6)

is the Wronskian.

The functions $f_{1}$ and $f_{2}$ can be determined by the boundary
conditions. We can choose, for example,777As a standard procedure for
numerical calculations, one cannot take the boundary conditions strictly at
$r=0$, but instead at a value $r_{\rm min}$ very close to zero. The boundary
conditions at $r_{\rm min}$ can be obtained via the Taylor expansion $f(r_{\rm
min})\approx f(0)+f^{\prime}(0)r_{\rm min}$ and $f^{\prime}(r_{\rm
min})=f^{\prime}(0)+f^{\prime\prime}(0)r_{\rm rmin}$, where
$f^{\prime\prime}(0)$ can be obtained by taking the limit $r\rightarrow 0$ for
the equation to be solved. The infinity would be replaced by a large value
$r_{\rm max}$ depending on the numerical bounce solution.

$\displaystyle f_{1}(0)=1\,,f_{1}^{\prime}(r)|_{r=0}=0\,;\quad
f_{2}(r)|_{r\rightarrow\infty}\rightarrow
0\,,f^{\prime}_{2}(r)|_{r\rightarrow\infty}\rightarrow 0\,.$ (A7)

In general, the boundary conditions for $f_{2}(r)$ lead to $f_{2}(0)\neq 0$,
$f_{2}^{\prime}(r)|_{r=0}\neq 0$. Then the boundary conditions for
$G(r,r^{\prime})$ require

$\displaystyle A_{2}^{<}(r^{\prime})=0\,,\quad A^{<}_{1}(r^{\prime})=0\,.$
(A8)

Substituting the above into Eqs. (A5), one obtains $A_{1}^{>}(r^{\prime})$ and
$A_{2}^{>}(r^{\prime})$ and thus also the Green’s function

$\displaystyle
G(r,r^{\prime})=\theta(r-r^{\prime})\left[-\frac{f_{1}(r)f_{2}(r^{\prime})}{\mathcal{W}[f_{1}(r^{\prime}),f_{2}(r^{\prime})]}+\frac{f_{1}(r^{\prime})f_{2}(r)}{\mathcal{W}[f_{1}(r^{\prime}),f_{2}(r^{\prime})]}\right]\,.$
(A9)

Apparently, $G(r,r)=0$.

For $\hat{G}_{l}(r,r^{\prime})$, we have $L[r]=\widehat{\mathcal{H}}_{l}$. One
can take

$\displaystyle f_{1}(r)=1\,,\qquad
f_{2}(r)=\,\frac{K_{l+1}(mr)}{I_{l+1}(mr)}\,.$ (A10)

To see that $\widehat{\mathcal{H}}_{l}f_{2}(r)=0$, we recall that the operator
$\widehat{\mathcal{H}}_{l}$ is obtained via the following procedure

$\displaystyle\widehat{\mathcal{M}}_{l}\psi_{l}=0\quad\Rightarrow\quad\widehat{\mathcal{H}}_{l}\left(\frac{\psi_{l}}{\hat{\psi}_{l}}\right)=0\,,$
(A11)

where $\hat{\psi}_{l}\sim r^{-1}I_{l+1}(mr)$ (cf. Eq. (32)) is a solution to
$\widehat{\mathcal{M}}_{l}\psi_{l}=0$. On the other hand, it is known that
$\widehat{\mathcal{M}}_{l}\psi_{l}=0$ has another solution regular at
$r\rightarrow\infty$, $r^{-1}K_{l+1}(mr)$. One immediately sees that
$\widehat{\mathcal{H}}_{l}f_{2}(r)=0$. With the basis solutions in Eq. (A10),
we get Eq. (65).

For $G_{l}(r,r^{\prime};\varphi)$, we have $L[r]=\mathcal{H}[\varphi]$. In
this case, to make the equation regular at $r\rightarrow\infty$ we can take

$\displaystyle f_{2}(r)=g_{2}(r)\frac{K_{l+1}(mr)}{I_{l+1}(mr)}\,,$ (A12)

which gives

$\displaystyle\frac{\mathrm{d}^{2}g_{2}(r)}{\mathrm{d}r^{2}}-\left[2m\frac{K_{l}(mr)}{K_{l+1}(mr)}+\frac{2l+1}{r}\right]\frac{\mathrm{d}g_{2}(r)}{\mathrm{d}r}-W(\varphi(r))g_{2}(r)=0\,.$
(A13)

The boundary conditions are $g_{2}(r)|_{r\rightarrow\infty}\rightarrow 1$,
$\mathrm{d}g_{2}(r)/\mathrm{d}r|_{r\rightarrow\infty}\rightarrow 0$.

###  A.2 $\mathcal{G}(r,r^{\prime})$

According to the boundary condition for $\Delta\varphi_{b}$, one has the
boundary condition for the Green’s function

$\displaystyle\mathcal{G}(r=\infty,r^{\prime})=0\,,\quad\left.\frac{\partial\mathcal{G}(r,r^{\prime})}{\partial
r}\right|_{r=0}=0\,.$ (A14)

In this case, we can choose two independent solutions
$\\{f_{1}(r),f_{2}(r)\\}$ satisfying, e.g.,

$\displaystyle f_{1}(0)=1\,,f_{1}^{\prime}(r)|_{r=0}=0\,;\quad
f_{2}(r)|_{r\rightarrow\infty}\rightarrow
0\,,f^{\prime}_{2}(r)|_{r\rightarrow\infty}\rightarrow 1\,.$ (A15)

A similar analysis gives

$\displaystyle\mathcal{G}(r,r^{\prime})=-\theta(r-r^{\prime})\frac{f_{1}(r^{\prime})f_{2}(r)}{\mathcal{W}[f_{1}(r^{\prime}),f_{2}(r^{\prime})]}-\theta(r^{\prime}-r)\frac{f_{2}(r^{\prime})f_{1}(r)}{\mathcal{W}[f_{1}(r^{\prime}),f_{2}(r^{\prime})]}\,,$
(A16)

Letting

$\displaystyle f_{1}(r)=g_{1}(r)\frac{I_{1}(mr)}{r}\,,\quad
f_{2}(r)=g_{2}(r)\frac{K_{1}(mr)}{r}\,,$ (A17)

we have

$\displaystyle-
g_{1}^{\prime\prime}(r)+\left[\frac{1}{r}-2m\frac{I_{0}(mr)}{I_{1}(mr)}\right]g^{\prime}_{1}(r)+W(\varphi_{b}^{(0)}(r))g_{1}(r)=0\,,$
(A18a) $\displaystyle-
g_{2}^{\prime\prime}(r)+\left[\frac{1}{r}+2m\frac{K_{0}(mr)}{K_{1}(mr)}\right]g^{\prime}_{2}(r)+W(\varphi^{(0)}_{b}(r))g_{2}(r)=0\,,$
(A18b)

for which the boundary conditions can be chosen as, e.g., $g_{1}(0)=2/m$,
$g^{\prime}_{1}(r)|_{r=0}=0$ and $g_{2}(r)|_{r\rightarrow\infty}=0$,
$g^{\prime}_{2}(r)|_{r\rightarrow\infty}=1$. Note that $I_{1}(mr)/r\rightarrow
m/2$ for $r\rightarrow 0$.

##  Appendix B Renormalisation with dimensional regularisation

In this appendix, we rederive the expression (4) of the regularized one-loop
functional determinant originally presented in Ref. [47].

We start with Eq. (39). The divergent parts in $\Delta S$ are contained in
$\Delta S^{(2)}$,

$\displaystyle\Delta S^{(2)}$
$\displaystyle=\frac{1}{2}\left[\ln\det(-\partial^{2}+m^{2}+W)-\ln\det(-\partial^{2}+m^{2})\right]_{\mathcal{O}(W^{2})}\,,$
(B19)

so that we can write

$\displaystyle S_{1}=S_{0}^{\overline{\rm MS}}+\underbrace{\left[\Delta
S-\Delta S^{(2)}\right]}_{\rm numerical\ part}+\underbrace{\left[\Delta
S^{(2)}-(\Delta S)_{\rm pole}\right]}_{\rm analytical\ part}\,.$ (B20)

Our goal is to find two different expressions of $\Delta S^{(2)}$ such that
the finite quantity $[\Delta S-\Delta S^{(2)}]$ can be computed numerically
while the other finite quantity $[\Delta S^{(2)}-(\Delta S)_{\rm pole}]$ can
be computed analytically. For this purpose, we assign the perturbation $W$
with a bookkeeping factor $\varepsilon$ and expand
$\ln\det(-\partial^{2}+m^{2}+\varepsilon W)$ in $\varepsilon$.

Numerically, we can write

$\displaystyle\Delta S^{(2)}=\frac{1}{2}\sum_{l=0}^{\infty}(l+1)^{2}\ln
T_{l}(\infty;\varepsilon)|_{\mathcal{O}(\varepsilon^{2})}\,.$ (B21)

where $T_{l}(\infty)$ is viewed as a function of $\varepsilon$ and expanded in
powers of $\varepsilon$,

$\displaystyle T_{l}(\infty;\varepsilon)=1+\varepsilon
h_{l}^{(1)}(\infty)+\varepsilon^{2}h_{l}^{(2)}(\infty)+\mathcal{O}(\varepsilon^{3})\,.$
(B22)

This gives

$\displaystyle\ln
T_{l}(\infty;\varepsilon)|_{\mathcal{O}(\varepsilon^{2})}=h_{l}^{(1)}(\infty)+h_{l}^{(2)}(\infty)-\frac{1}{2}(h_{l}^{(1)}(\infty))^{2}\,.$
(B23)

To obtain $h_{l}^{(1)}(\infty)$ and $h_{l}^{(2)}(\infty)$, we substitute
$T_{l}(r;\varepsilon)=1+\varepsilon
h_{l}^{(1)}(r)+\varepsilon^{2}h_{l}^{(2)}(r)+\mathcal{O}(\varepsilon^{3})$
into Eq. (34)

$\displaystyle\left[-\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}-\left(2m\frac{I_{l+D/2-2}(mr)}{I_{l+D/2-1}(mr)}-\frac{2l+D-3}{r}\right)\frac{\mathrm{d}}{\mathrm{d}r}+\varepsilon
W(\varphi(r))\right]T_{l}(r;\varphi)=0\,,$ (B24)

where the perturbation term is associated with a factor of $\varepsilon$ now.
Tracing the equations at the order of $\mathcal{O}(\varepsilon)$ and
$\mathcal{O}(\varepsilon^{2})$, we then obtain Eqs. (42) for $h_{l}^{(1)}(r)$
and $h_{l}^{(2)}(r)$.

To obtain the analytical part $[\Delta S^{(2)}-(\Delta S)_{\rm pole}]$, we
note that Eq. (B19) can be written as

$\displaystyle\Delta S^{(2)}=\frac{1}{2}{\rm
Tr}\left[(-\partial^{2}+m^{2})^{-1}W\right]-\frac{1}{4}{\rm
Tr}\left[(-\partial^{2}+m^{2})^{-1}W(-\partial^{2}+m^{2})W\right]\,.$ (B25)

The first term is

$\displaystyle\frac{1}{2}\int\mathrm{d}^{4}x\,\langle
x|(-\partial^{2}+m^{2})^{-1}W(\hat{x})|x\rangle=\frac{1}{2}\left(\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{1}{k^{2}+m^{2}}\right)\int\mathrm{d}^{4}x\,W(x)\,.$
(B26)

To do the $\overline{\rm MS}$ regularization, we perform the momentum integral
in $d$-dimensional spacetime, using the useful formula

$\displaystyle\mu^{4-d}\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}+m^{2})^{\alpha}}=\mu^{4-d}\frac{1}{(4\pi)^{d/2}}\frac{\Gamma(\alpha-\frac{d}{2})}{\Gamma(\alpha)}\frac{1}{m^{2\alpha-d}}\,.$
(B27)

Expanding the integral result around $d=4$ (with $d=4-2\epsilon$), one then
obtains the pole. Substracting the pole term according to the standard
$\overline{\rm MS}$ rule, one then obtains $A_{\rm fin}^{(1)}$.

The second term on the RHS of Eq. (B25) can be written as

$\displaystyle-\frac{1}{4}\int\mathrm{d}^{4}x\mathrm{d}^{4}y\,W(x)W(y)\int\frac{\mathrm{d}^{4}q}{(2\pi)^{4}}\mathrm{e}^{\mathrm{i}q(x-y)}\left[\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{1}{(k+q)^{2}+m^{2}}\frac{1}{k^{2}+m^{2}}\right]$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\int\frac{\mathrm{d}^{4}q}{(2\pi)^{4}}\widetilde{W}(q)\widetilde{W}(-q)\times\left[\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{1}{(k+q)^{2}+m^{2}}\frac{1}{k^{2}+m^{2}}\right]\,.$
(B28)

Again, the momentum integral inside the square brackets can be performed in
$d$-dimensional spacetime by using the Feynman-parameter method, which in our
case is

$\displaystyle\frac{1}{AB}=\int_{0}^{1}\mathrm{d}s\,\frac{1}{sA+(1-s)B}\,.$
(B29)

The final result of the momentum integral inside the square brackets is

$\displaystyle\frac{1}{16\pi^{2}}\int_{0}^{1}\mathrm{d}s\,\left[\frac{1}{\epsilon}-\gamma_{\rm
E}+\ln 4\pi+\ln\frac{\mu^{2}}{s(1-s)q^{2}+m^{2}}\right]\,.$ (B30)

Finally, subtracting the pole term ($1/\epsilon$ together with $-\gamma_{\rm
E}+\ln 4\pi$) and doing the integral over $s$ gives $A_{\rm fin}^{(2)}$.

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