aainstitutetext: Tsung-Dao Lee Institute and School of Physics and Astronomy,
Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240
Chinabbinstitutetext: Shanghai Key Laboratory for Particle Physics and
Cosmology, Key Laboratory for Particle Astrophysics and Cosmology (MOE),
Shanghai Jiao Tong University, Shanghai 200240, Chinaccinstitutetext: Amherst
Center for Fundamental Interactions, Department of Physics, University of
Massachusetts, Amherst, MA 01003, USAddinstitutetext: Kellogg Radiation
Laboratory, California Institute of Technology, Pasadena, CA 91125 USA

# Tunneling Potentials for the Tunneling Action: Gauge Invariance

Suntharan Arunasalam<EMAIL_ADDRESS>a,b,c,d Michael J.
Ramsey-Musolf  mjrm<EMAIL_ADDRESS>
###### Abstract

We formulate a procedure to obtain a gauge-invariant tunneling rate at zero
temperature using the recently developed tunneling potential approach. This
procedure relies on a consistent power counting in gauge coupling and a
derivative expansion. The tunneling potential approach, while numerically more
efficient than the standard bounce solution method, inherits the gauge-
dependence of the latter when naïvely implemented. Using the Abelian Higgs
model, we show how to obtain a tunneling rate whose residual gauge-dependence
arises solely from the polynomial approximations adopted in the tunneling
potential computation.

††preprint: ACFI-T21-07

##  1 Introduction

The topic of phase transitions in the early universe has a long and rich
history at the interface of cosmology and particle physics. In recent years,
determining the thermal history of symmetry breaking in the Standard Model of
particle physics and its possible extensions has seen a resurgence of
interest. The extensive experimental program at the Relativistic Heavy Ion
Collider has confirmed the results of lattice computations that the transition
to the confined phase of quantum chromodynamics at zero baryon chemical
potential is a smooth crossover. Analogous lattice computations in the
electroweak theory imply that the electroweak symmetry-breaking (EWSB)
transition is also a smooth crossover for a Higgs boson heavier than $\sim
70-80$ GeVKajantie:1996qd ; Laine:1998jb ; Csikor:1998eu ; Farakos:1994kx ;
Kajantie:1995kf ; Kajantie:1996mn ; Farakos:1995dn ; Rummukainen:1998as ;
Gurtler:1997hr ; Aoki:1999fi – well below the measured value of $125$ GeV.
Nevertheless, there exists strong theoretical motivation to consider embedding
the Standard Model in a larger beyond the Standard Model (BSM) framework that
includes new particles with sub-TeV masses.

If such an embedding is realized in nature, then it is interesting to ask
whether the BSM particles and their interactions would qualitatively change
the nature of the EWSB transition and possibly introduce new, earlier phases
in the early universe that precede the present Higgs phase. At the same time,
the existence of a first order phase transition to the Higgs phase and/or an
earlier phase could provide the necessary preconditions for generation of the
cosmic baryon asymmetry via electroweak baryogenesis (see Ref.
Morrissey:2012db and references therein) while simultaneously giving rise to
relic gravitational radiation. The existence of such a thermal history could
in principle be inferred indirectly from searches for new particles and/or
deviations of Higgs boson properties from Standard Model expectations at the
CERN Large Hadron Collider or prospective future collidersRamsey-
Musolf:2019lsf . A direct echo of the first order phase transition could
detected in next generation gravitational wave searches (see, e.g., Refs.
Caprini:2015zlo ; Huang:2016odd ; Alves:2018jsw ; Caprini:2019egz ;
Alves:2019igs ; Zhou:2020idp ).

In this context, a definitive confrontation of theory and experiment requires
the most theoretically robust computations of the early universe
thermodynamics in a given BSM scenario and of the corresponding dynamics. The
first step is to identify the phase diagram and associated critical
temperature $T_{C}$ for each possible transition from one phase to another.
Here, one already encounters a need to go beyond perturbation theory, as the
latter cannot ascertain when a given transition is crossover (compared to a
bona fide phase transition). The second step entails analyzing the dynamics.
The mere existence of a possible transition does not in itself guarantee that
it will have occurred, even if nominally energetically favorable.

In the case of a first order transition, characterized by the presence of a
potential barrier between the two phases, one must also determine that the
transition (tunneling) rate is sufficiently large compared to the Hubble rate
at the transition temperature. Moreover, determining the frequency spectrum of
the associated gravitational radiation requires knowledge of the tunneling
rate per unit volume,

$\Gamma=Ae^{-B}$ (1)

where at $T=0$ $B$ is the four-dimensional (4d) Euclidean effective action,
$S_{E}$, while for $T>0$ one has $B=S_{3}/T$, with $S_{3}$ being the 3d
Euclidean action. The prefactor $A$ is generally estimated on dimensional
grounds and at finite-$T$ is generally taken to carry a mild $T$-dependence.
As a first order phase transition proceeds via bubble nucleation, one often
refers to $\Gamma$ as the nucleation rate, with $T_{N}$ denoting the
temperature at the onset of nucleation, with $T_{N}<T_{C}$.

The classic approach for computing $\Gamma$ at any temperature is via the
computation of the bounce action Coleman:1977py ; Callan:1977pt using the
“overshoot/undershoot” method. While this approach works particularly well for
phase transitions involving a single field, it can be intractable in the
context of multi-field problems. Over the past decade, alternate approaches
have been developed for this case, including CosmoTransitionsWainwright:2011kj
, BSMPTBasler:2018cwe ; Basler:2020nrq , and Bubble ProfilerAthron:2019nbd .
Recently, a powerful algebraic method for computing $B$ in both the single and
multi-field contexts was developed in Refs. Espinosa:2018hue ;
Espinosa:2018szu . The method relies on the introduction of an auxiliary
function known as the “tunneling potential” , $V_{t}$, and the use of
approximations that facilitate algebraic solution of the field equations to
high numerical accuracy without directly solving any differential equations.
In principle, the tunneling potential approach holds the promise of
facilitating a more efficient survey of nucleation dynamics over a wide array
of models and model parameter space as compared to pre-existing methods and
numerical packages.

In practice, a common challenge when computing the nucleation rate and $T_{N}$
is maintenance of gauge invariance. As a physical observable, $\Gamma$ is a
gauge-invariant quantity. Formally, this property is encoded in the Nielsen
identitiesNielsen:1975fs ; Fukuda:1975di , which state that the effective
action is gauge invariant when evaluated using an extremal field
configuration. Unfortunately, common computational approaches – when naïvely
implemented – do not necessarily respect this requirement. The gauge
invariance problem is especially challenging when radiative corrections
generate the potential barrier between different vacua. Well-known examples
include the Abelian Higgs model as well as the Standard Model. In both cases,
it is possible to implement gauge invariance by adopting a well-defined power-
counting in the relevant couplings and a systematic truncation
procedureMetaxas:1995ab . Alternatively, in the presence of a tree-level
barrier, it may be desirable to incorporate higher-order corrections –
including gauge boson degrees of freedom – to the manifestly gauge-invariant
tree-level computation. Indeed, recent results for the thermodynamics of the
real triplet of the SM, wherein a tree-level barrier can appear, suggests that
thermal (loop) corrections can play a more significant role than one might
otherwise expectNiemi:2020hto . Including these higher order corrections thus
introduces the gauge invariance problem. While it is possible to implement
these higher order corrections using the $\hbar$-expansion about the tree-
level Euclidean action (for relevant discussions, see e.g., Refs. Laine:1994zq
; Patel:2011th ), the practical viability of computing the relevant
fluctuation determinant remains to be fully explored.

In light of the novelty and promise of the tunneling potential approach, we
investigate here the implementation of gauge invariance in this framework.
Perhaps, unsurprisingly, implementation of the tunneling potential method
without a well-defined procedure for maintaining gauge invariance will lead to
gauge-dependent artifacts. Fortunately, one may circumvent this pitfall by
adopting a derivative expansion, as in Ref. Metaxas:1995ab ; Weinberg:1992ds .
For concreteness, we adopt the Abelian Higgs model and show how, for
$\Gamma(T=0)$, to translate the derivative expansion and power counting of
Ref. Metaxas:1995ab into the tunneling potential method. This translation
maintains the advantage of the purely algebraic procedure outlined in Refs.
Espinosa:2018hue ; Espinosa:2018szu and is otherwise relatively
straightforward. We expect that this translation will work equally well for
any scenario in which one is able to define a leading order, gauge invariant
tunneling action – whether or not the potential barrier exists at tree level
or, as in the Abelian Higgs model, occurs through gauge-boson loops.
Application of these ideas to the tunneling rate at $T>0$ will appear in a
future publication.

Our discussion is organized as follows. In Section 2 we present the Abelian
Higgs model and the relevant quantities for this approach. In Section 3, we
discuss the gauge-dependence problem in the conventional bounce solution
context as well as the tunneling potential framework. Section 4 reviews the
use of the derivative expansion to obtain a gauge-independent nucleation rate
in the bounce solution method. In Section 5, we translate the derivative
expansion into the tunneling potential approach and show how doing so yields a
gauge-independent calculation of the tunneling action. Section 6 summarizes
our results. Relevant technical details appear in two Appendices.

##  2 The model

The Lagrangian for the Abelian Higgs model is

$\displaystyle\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\left(D_{\mu}\Phi\right)^{*}D^{\mu}\Phi-
V_{0}\left(\Phi^{*}\Phi\right)$ (2)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and
$D_{\mu}=\partial_{\mu}-ieA_{\mu}$ are the standard electromagnetic tensor and
covariant derivative respectively, and where the potential is

$\displaystyle
V_{0}(\Phi^{*}\Phi)=\frac{1}{2}m^{2}\Phi^{*}\Phi+\frac{\lambda}{4}\left(\Phi^{*}\Phi\right)^{2}\
\ \ .$ (3)

We use the background field approach wherein we write the complex scalar as
$\Phi=\phi+\frac{1}{\sqrt{2}}(h+i\chi)$ where $\phi$ is the real space-time
independent background field and $h$ and $\chi$ are the Higgs and Goldstone
bosons, respectively. We now introduce the $R_{\xi}$-gauge through the gauge-
fixing and ghost Lagrangians,

$\displaystyle\mathcal{L}_{GF}$
$\displaystyle=-\frac{1}{2\xi}\left[\partial_{\mu}A^{\mu}-ie\xi\left(\phi^{\ast}\Phi-\Phi^{\ast}\phi\right)\right]^{2}$
(4) $\displaystyle\mathcal{L}_{FP}$
$\displaystyle=\partial_{\mu}\bar{c}\partial^{\mu}c-e^{2}\xi\left(\phi^{\ast}\Phi+\Phi^{\ast}\phi\right)\bar{c}c\
\ \ .$ (5)

The tree-level masses of these fields and the corresponding numbers of degrees
of freedom can be seen in Table 1.

Field | Tree level $(\mathrm{mass})^{2}$ | #dof
---|---|---
Space-like gauge polarization | $2e^{2}\phi^{2}$ | 3
Time-like gauge polarization | $2\xi e^{2}\phi^{2}$ | 1
Higgs boson | $\frac{1}{2}m^{2}+\frac{3}{2}\lambda\phi^{2}$ | 1
Goldstone boson | $\frac{1}{2}m^{2}+\frac{1}{2}\lambda\phi^{2}+2\xi e^{2}\phi^{2}$ | 1
Ghost | $2\xi e^{2}\phi^{2}$ | -2
Table 1: The tree level masses and the number of degrees of freedom of the
fields

We now impose a power counting of the form $\lambda\sim e^{4}$ and $m^{2}\sim
e^{4}\,\langle\phi\rangle^{2}$ as in Metaxas:1995ab 111We believe the
statement of power counting for $m^{2}$ in Metaxas:1995ab contains a typo and
should read as given here.. Doing so introduces a barrier in the leading order
potential, $V_{e^{4}}$, through loop effects. The effective potential can be
calculated in the usual way as

$\displaystyle V_{\text{eff}}=\sum_{\mathrm{j\in
particles}}-\frac{in_{j}}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\ln(k^{2}-m_{j}^{2})\
\ \ ,$ (6)

where $n_{j}$ are the numbers of degrees of freedom as in Table 1 and $m_{j}$
are the tree level masses. We evaluate these integrals using the
$\overline{\mathrm{MS}}$ renormalization scheme as

$\displaystyle-\frac{i}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\ln(k^{2}-m_{j}^{2})=\frac{m_{j}^{4}}{64\pi^{2}}\left[\ln\frac{m_{j}^{2}}{\mu^{2}}-C_{j}\right]$
(7)

where $C_{j}=\frac{5}{6}$ for the space-like polarizations of the gauge boson
and $\frac{3}{2}$ for all other particles and where $\mu$ is the
renormalization scale.

Consider now the gauge-coupling dependence of each term. First, the
contributions from the time-like gauge boson polarization and one ghost degree
of freedom cancel completely. The contribution due to the space-like gauge
boson is of order $e^{4}$ and is $\xi$-independent. The contributions from the
Goldstone boson and the remaining ghost degree of freedom partially cancel
against each other and produce a contribution of order $e^{6}$. Finally the
Higgs boson produces a contribution of order $e^{8}$. We may now write the
leading order potential as:

$\displaystyle
V_{e^{4}}(\phi)=\frac{m^{2}}{2}\phi^{2}+\frac{\lambda}{4}\phi^{4}+\frac{3m_{A}(\phi)^{4}}{64\pi^{2}}\left[\ln\frac{m_{A}^{2}}{\mu^{2}}-\frac{5}{6}\right]$
(8)

where $m_{A}^{2}=2e^{2}\phi^{2}$ is the mass-squared of the spacelike gauge
boson. The important point to note here is that the effective potential is
gauge-independent at this order.

The resulting one-loop contributions to the $\mathcal{O}(e^{6})$ effective
potential yield a gauge-dependent terms

$\displaystyle V_{e^{6}}^{\xi}(\phi)$
$\displaystyle=\frac{m_{\chi}^{4}(\phi)}{64\pi^{2}}\left[\ln\frac{m_{\chi}^{2}(\phi)}{\mu^{2}}-\frac{3}{2}\right]$
(9)
$\displaystyle-\frac{m_{FP}^{4}(\phi)}{64\pi^{2}}\left[\ln\frac{m_{FP}^{2}(\phi)}{\mu^{2}}-\frac{3}{2}\right]+\cdots\
\ \ ,$

where $m_{\chi}$ is the Goldstone boson mass, $m_{FP}$ is the ghost mass, and
the $+\cdots$ denote additional $\mathcal{O}(e^{6})$ contributions arising
from two-loop diagrams. As shown in Metaxas:1995ab , it is possible to take
into account a subset of these diagrams by replacing the tree level Goldstone
boson mass by

$\displaystyle m^{2}_{\chi}(\phi)$
$\displaystyle\to\frac{1}{2\phi}\frac{\partial V_{e^{4}}}{\partial\phi}+2\xi
e^{2}\phi^{2}$ (10) $\displaystyle=\frac{1}{2}m^{2}$
$\displaystyle+\frac{3}{2}\lambda\phi^{2}+2\xi
e^{2}\phi^{2}+\frac{e^{4}\phi^{4}}{8\pi^{2}}\left[3\ln\frac{2e^{2}\phi^{2}}{\mu^{2}}-1\right]$

We refer to this redefined $m_{\chi}$ as the dressed Goldstone boson mass. We
discuss below that its inclusion is essential to obtain a gauge independent
action at $\mathcal{O}(e^{6})$. The remaining 2-loop diagrams sum to a gauge-
independent contributionMetaxas:1995ab (see also Andreassen:2016cvx ). For
simplicity, we do not include these additional gauge-independent contributions
further.

To obtain a gauge-invariant effective action beyond $\mathcal{O}(e^{4})$ one
must also include higher order contributions to the wavefunction
renormalization constant, $Z(\phi)$. To do so, we proceed as in Garny:2012cg
and as discussed in Appendix A. Using the results listed there and using
$m^{2}_{\chi}=m^{2}_{FP}+\mathcal{O}(e^{4})$ and $m^{2}_{FP}=\xi
m^{2}_{A}=2\xi e^{2}\phi^{2}$, we obtain

$\displaystyle Z(\phi)$
$\displaystyle=1+\frac{e^{2}}{16\pi^{2}}\left[\xi\log\left(\frac{2e^{2}\phi^{2}\xi}{\mu^{2}}\right)+3\log\left(\frac{2e^{2}\phi^{2}}{\mu^{2}}\right)+\xi\right]$
$\displaystyle+\mathcal{O}(e^{4})$ (11)

In the following sections, we will use these quantities calculated above to
construct a gauge independent effective action.

##  3 The gauge-dependence problem

Before discussing the procedure needed to obtain a gauge invariant tunneling
rate, we first demonstrate the problem using the standard bounce solution and
the new tunneling potential approaches when naïvely implemented. For purposes
of illustration, we study a concrete numerical example in parallel with the
general, analytic discussion.

###  3.1 Standard bounce solution approach

To be concrete, we take the following numerical values for the independent
parameters: $e=0.07,\ \lambda=3.8\times 10^{-6},\ m^{2}=3.2\times 10^{-5}$ and
$\mu=2$, where the units of the dimensionful parameters are arbitrary. Note
that our choice introduces large logarithms in $V$, as needed to generate a
perceptible barrier in the presence of the concavity of the tree-level
potential222Presumably one could re-sum these large logarithms using the
renormalization group. We defer such a study to future work.. We obtain the
bounce solution to the full equations of motion, using
$V_{e^{4}}(\phi)+V_{e^{6}}^{\xi}(\phi)$, the tree-level mass for $m_{\chi}$,
and $Z(\phi)=1$. Using the Mathematica package, FindBounce Guada:2020xnz , we
calculate the action over a range of values of $\xi$ in $[0,100]$ and find
$\sim 13$% variation, as indicated by the red dashed curve in Figure 1.

Figure 1: The action evaluated as a function of the gauge dependence
parameter, $\xi$. The red dashed curve gives the action computed using the
standard bounce solution approach with the potential
$V_{e^{4}}(\phi)+V_{e^{6}}^{\xi}(\phi)$, the tree-level Goldstone mass, and
$Z(\phi)=1$. The blue line gives the action calculated following the naïve
tunneling potential method as outlined in Section 3.2.

The presence of $\xi$-dependence is not surprising, given the gauge dependence
in $V_{e^{6}}^{\xi}(\phi)$. As we now show, the naïve application of the
tunneling potential method inherits this $\xi$-dependence.

###  3.2 Tunneling potential approach

To review the logic of the tunneling potential introduced in Ref.
Espinosa:2018hue , we start with the standard approach Coleman:1977py , which
entails first finding the radially symmetric bounce solution in 4 dimensions
that satisfies the equation:

$\displaystyle\ddot{\phi}_{b}+\frac{3}{r}\dot{\phi}_{b}=\frac{\partial
V}{\partial\phi}$ (12)

with the boundary conditions:

$\displaystyle\dot{\phi}_{b}(0)=0,\ \ \phi_{b}(\infty)=\phi_{+}$ (13)

where $\phi_{+}$ is the symmetric vacuum. One usually uses a shooting method
to find the $\phi_{0}\equiv{\phi_{b}(0)}$ that satisfies the boundary
conditions. In the method proposed in Espinosa:2018hue , however, the
essential quantity is the tunneling potential, $V_{t}(\phi)$ given by

$\displaystyle V_{t}(\phi)\equiv V(\phi)-\frac{1}{2}\dot{\phi}_{b}^{2}$ (14)

The idea behind introducing this quantity is to remove all explicit reference
to the spacetime co-ordinates in the bounce equation and the effective action.
Immediately, one may use Eq. (14) to remove references to the derivative,
$\dot{\phi}_{b}$, through

$\displaystyle\dot{\phi}_{b}=-\sqrt{2\left[V(\phi)-V_{t}(\phi)\right]}$ (15)

Furthermore, one may rearrange the bounce equation to solve for the radial co-
ordinate, $r$:

$r=3\sqrt{2\left(V-V_{t}\right)/\left(V_{t}^{\prime}\right)^{2}}$ (16)

Differentiating this expression with respect to $r$, one obtains:

$\displaystyle\left(4V_{t}^{\prime}-3V^{\prime}\right)V_{t}^{\prime}=6\left(V_{t}-V\right)V_{t}^{\prime\prime}$
(17)

where primes denote the derivatives with respect to $\phi$.

Eq. (17) replaces the role of the bounce equation. One must now find a
$\phi_{0}$ and $V_{t}(\phi)$ that satsify 17 as well as the boundary
conditions

$\displaystyle V_{t}(\phi_{+})=V(\phi_{+}),\ \ V_{t}(\phi_{0})=V(\phi_{0}).$
(18)

The Euclidean effective action can be written in terms of $V_{t}$ as:

$\displaystyle S_{E}$ $\displaystyle=S_{K}+S_{V}$ (19) $\displaystyle S_{K}$
$\displaystyle=2\pi^{2}\int_{0}^{\infty}\frac{1}{2}\dot{\phi}_{b}^{2}r^{3}dr=108\pi^{2}\int_{\phi_{0}}^{\phi_{+}}\frac{\left(V-V_{t}\right)^{2}}{\left(V_{t}^{\prime}\right)^{3}}d\phi$
(20) $\displaystyle S_{V}$
$\displaystyle=2\pi^{2}\int_{0}^{\infty}V(\phi_{b})r^{3}dr=108\pi^{2}\int_{\phi_{0}}^{\phi_{+}}\frac{\left(V-V_{t}\right)}{\left(V_{t}^{\prime}\right)^{3}}Vd\phi$
(21)

One may now use the scaling relations, $S_{K}=2S_{E}$ and $S_{V}=-S_{E}$ to
write

$\displaystyle
S_{E}=\frac{1}{2}S_{K}=54\pi^{2}\int_{\phi_{0}}^{\phi_{+}}\frac{\left(V-V_{t}\right)^{2}}{\left(V_{t}^{\prime}\right)^{3}}d\phi$
(22)

As discussed in Appendix B, these relations apply only the presence of the
tree-level form of the effective action and break down in the presence of
higher order derivative terms.

At face value, solving Eq. (17) is more challenging than solving the bounce
equation. However, the power of this method lies in the approximations one can
make for $V_{t}$. As shown in Espinosa:2018hue , one may choose for $V_{t}$,
without much loss of accuracy, the quartic polynomial, $V_{t4}$ that satisfies
Eq. (18) and solves Eq. (17) only at the points, $\phi_{0},\ \phi_{+}$, and
$\phi_{T}$, where the latter corresponds to the top of the barrier. This
approximation is given by:

$\displaystyle V_{t4}(\phi)$ $\displaystyle=$ $\displaystyle
V_{t3}(\phi)+a_{4}\phi^{2}\left(\phi-\phi_{0}\right)^{2}$ (23) $\displaystyle
V_{t3}(\phi)$ $\displaystyle=$
$\displaystyle\frac{V_{0}}{\phi_{0}}\phi+\frac{1}{4\phi_{0}^{2}}\left(3\phi_{0}V_{0}^{\prime}-4V_{0}\right)(\phi-\phi_{0})\phi$
$\displaystyle+\frac{1}{4\phi_{0}^{3}}\left(3\phi_{0}V_{0}^{\prime}-8V_{0}\right)\phi(\phi-\phi_{0})^{2}\
\ \ ,$

where

$\displaystyle a_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{c}\left(a_{0T}-\sqrt{a_{0T}^{2}-cU_{t3T}}\right),$
$\displaystyle c$ $\displaystyle\equiv$ $\displaystyle
4\phi_{T}^{2}\phi_{0T}^{2}\left(\phi_{0}^{2}+2\phi_{0T}\phi_{T}\right)$
$\displaystyle U_{t3T}$ $\displaystyle\equiv$ $\displaystyle
4\left(V_{t3T}^{\prime}\right)^{2}+6\left(V_{T}-V_{t3T}\right)V_{t3T}^{\prime\prime}$
(25) $\displaystyle a_{0T}$ $\displaystyle=$
$\displaystyle-6\left(V_{T}-V_{t3T}\right)\left(\phi_{0}^{2}-6\phi_{0T}\phi_{T}\right)$
$\displaystyle-8\phi_{T}\left(\phi_{0T}-\phi_{T}\right)\phi_{0T}V_{t3T}^{\prime}$
$\displaystyle+3\phi_{T}^{2}\phi_{0T}^{2}V_{t3T}^{\prime\prime}$

with $V_{0}\equiv V(\phi_{0})$, $\phi_{0T}\equiv\phi_{0}-\phi_{T}$ and
$V_{t3T}\equiv V_{t3}\left(\phi_{T}\right)$. We then obtain the best
approximation of $S_{E}$ by finding the $\phi_{0}$ that minimises the integral
in Eq. (22). Hence, the difficult requirement of solving the bounce equation
is replaced by solving a set of algebraic equations and minimising an
integral.

We now show the gauge dependence of the above discussed tunneling potential
approach and provide a prescription to eliminating this gauge dependence.
Suppose, firstly, that we define $V=V_{e^{4}}+V_{e^{6}}^{\xi}$ without using
the dressed Goldstone mass. The naïve tunneling potential approach entails the
following steps:

  1. 1.

Calculate the quartic approximation to the tunneling potential, $V_{t4}$,
using Eq. (23)

  2. 2.

Find the $\phi_{0}$ that minimizes the action as defined in Eq. (22)

  3. 3.

Evaluate this action for this value of $\phi_{0}$

Following this approach, however, leads to a gauge-dependent result that
accurately reflects the $\xi$-dependence of the naïve bounce solution method,
as one may observe by comparing the blue and red-dashed curves in Fig. 1. It
is clearly advantageous to translate the remedies for the gauge-dependence of
the bounce action into the framework of the tunneling potential approach.

##  4 Gauge-independent nucleation rate using the derivative expansion

###  4.1 Overview

To set the stage, we review the approach of Metaxas and Weinberg
Metaxas:1995ab in using the derivative expansion of the Nielsen identity to
obtain a gauge independent nucleation rate. The gauge dependence of the
effective action is controlled by the Nielsen identity,

$\displaystyle\xi\frac{\partial S_{\text{eff}}}{\partial\xi}$ $\displaystyle=$
$\displaystyle-\int d^{4}x\Bigl{\\{}\frac{\delta S_{\text{eff}}}{\delta
A_{\mu}(x)}C_{A^{\mu}}(x)$ $\displaystyle+\frac{\delta
S_{\text{eff}}}{\delta\phi(x)}C_{\phi}(x)+\frac{\delta
S_{\text{eff}}}{\delta\phi^{*}(x)}C_{\phi^{*}}(x)\Bigr{\\}}.$

This identity follows directly from the BRST invariance of the effective
action. In computing $S_{\text{eff}}$ , one assumes the vector potential,
$A_{\mu}$, is set to zero. Furthermore, performing this computation along the
real axis, $\Phi=\Phi^{*}$, one obtains the identity,

$\displaystyle\xi\frac{\partial S_{\text{eff}}}{\partial\xi}=-\int d^{4}x\
C(x)\frac{\delta S_{\text{eff}}}{\delta\phi(x)}$ (27)

where

$\displaystyle C(x)$ $\displaystyle=\frac{ie}{2\sqrt{2}}\int
d^{4}y\Bigl{\langle}\bar{c}(x)\chi(x)c(y)\times$ (28)
$\displaystyle\Bigl{(}\partial_{\mu}A^{\mu}(y)-\sqrt{2}e\xi\bar{\phi}\chi(y)\Bigr{)}\Bigr{\rangle}.$

Now, one may perform a gradient expansion of both $S_{\text{eff}}$ and $C(x)$
as:

$\displaystyle S_{\text{eff}}$ $\displaystyle=\int
d^{4}x\left[V_{\text{eff}}(\phi)+\frac{1}{2}Z(\phi)\left(\partial_{\mu}\phi\right)^{2}+\mathcal{O}\left(\partial^{4}\right)\right]$
(29) $\displaystyle C(x)$
$\displaystyle=C_{0}(\phi)+D(\phi)\left(\partial_{\mu}\phi\right)^{2}-\partial^{\mu}\left[\tilde{D}(\phi)\partial_{\mu}\phi\right]+\mathcal{O}\left(\partial^{4}\right)$
(30) $\displaystyle\frac{\delta S_{\text{eff}}}{\delta\phi(x)}$
$\displaystyle=\frac{\partial
V_{\mathrm{eff}}(\phi)}{\partial\phi}+\frac{1}{2}\frac{\partial
Z}{\partial\phi}\left(\partial_{\mu}\phi\right)^{2}-\partial_{\mu}\left[Z(\phi)\partial_{\mu}\phi\right]+\mathcal{O}\left(\partial^{4}\right)$
(31)

Note that the total derivative term involving $\tilde{D}(\phi)$ was not
included in the derivative expansion of $C(x)$ in Metaxas:1995ab but could
contribute to this identity as pointed out in Garny:2012cg . Expanding Eq.
(27) using Eqs. (29-31), we obtain the Nielsen identity at orders
$\partial^{0}$ and $\partial^{2}$, respectively, as

$\displaystyle\xi\frac{\partial V_{\text{eff}}}{\partial\xi}$
$\displaystyle=-C_{0}\frac{\partial V_{\text{eff}}}{\partial\phi}$ (32)
$\displaystyle\xi\frac{\partial Z}{\partial\xi}$
$\displaystyle=-C_{0}\frac{\partial Z}{\partial\phi}-2D\frac{\partial
V_{\mathrm{eff}}}{\partial\phi}-2\tilde{D}\frac{\partial^{2}V_{\mathrm{eff}}}{\partial\phi^{2}}-2Z\frac{\partial
C_{0}}{\partial\phi}.$ (33)

For a detailed derivation of the Nielsen identity, one may refer to
Nielsen:1975fs ; Fukuda:1975di ; Aitchison:1983ns ; Metaxas:1995ab ;
Garny:2012cg .

The approach of Ref. Metaxas:1995ab for obtaining a gauge-independent
nucleation rate using the aforementioned derivative expansion proceeds as
follows. Using the power counting introduced in Section 2 leads to the gauge
coupling expansion

$\displaystyle V_{\text{eff}}$
$\displaystyle=V_{e^{4}}+V_{e^{6}}+\mathcal{O}\left(e^{8}\right)$ (34)
$\displaystyle Z$ $\displaystyle=1+Z_{e^{2}}+\mathcal{O}\left(e^{4}\right)$
(35)

As we show explicitly below, the leading order contribution to $C_{0}$ is of
order $e^{2}$. As pointed out in Metaxas:1995ab , the leading order
contributions to $D$ and $\tilde{D}$ are of order $e^{0}$. Thus, the leading
and sub-leading order contributions to Eqs. (32) and (33) are given by:

$\displaystyle\xi\frac{\partial V_{e^{4}}}{\partial\xi}$ $\displaystyle=0$
(36) $\displaystyle\xi\frac{\partial V_{e^{6}}}{\partial\xi}$
$\displaystyle=-C_{e^{2}}\frac{\partial V_{e^{4}}}{\partial\phi}$ (37)
$\displaystyle\xi\frac{\partial Z_{e^{2}}}{\partial\xi}$
$\displaystyle=-2\frac{\partial C_{e^{2}}}{\partial\phi}$ (38)

One may further write the leading and subleading order terms of $B$ in Eq. (1)
as

$\displaystyle B_{\text{eff}}$ $\displaystyle=B_{0}+B_{1}$ (39) $\displaystyle
B_{0}$ $\displaystyle=\int
d^{4}x\left(\frac{1}{2}\left(\partial_{\mu}\phi_{b}\right)^{2}+V_{e^{4}}(\phi_{b})\right)$
(40) $\displaystyle B_{1}$ $\displaystyle=\int
d^{4}x\left(\frac{1}{2}Z_{e^{2}}\left(\partial_{\mu}\phi_{b}\right)^{2}+V_{e^{6}}(\phi_{b})\right)$
(41)

As shown in Metaxas:1995ab , Eqs. (36-38) ensure that $S_{0}$ and $S_{1}$ are
gauge independent when evaluated at the leading order bounce solution,
satisfying the equation:

$\displaystyle\Box\phi_{b}=\frac{\partial V_{e^{4}}}{\partial\phi}.$ (42)

The gauge independence of $B_{0}$ can be seen immediately as Eq. (36) directly
implies the gauge independence of $V_{e^{4}}$ and consequently, the bounce
solution. Using Eqs. (37) and (38), we see that

$\displaystyle\xi\frac{\partial B_{1}}{\partial\xi}=-\int
d^{4}x\left[\frac{\partial
C_{e^{2}}}{\partial\phi}\left(\partial_{\mu}\phi_{b}\right)^{2}+C_{e^{2}}(\phi_{b})\frac{\partial
V_{e^{4}}(\phi_{b})}{\partial\phi}\right]$ (43)

Noting that $\frac{\partial
C_{e^{2}}}{\partial\phi}\partial_{\mu}\phi=\partial_{\mu}C_{e^{2}}$ and
integrating by parts, one obtains

$\displaystyle\xi\frac{\partial B_{1}}{\partial\xi}=\int d^{4}x\
C_{e^{2}}(\phi_{b})\left[\Box\phi_{b}-\frac{\partial
V_{e^{4}}(\phi_{b})}{\partial\phi}\right]$ (44)

Hence, the bounce equation, Eq. (42), guarantees the gauge independence of
$B_{1}$.

###  4.2 Explicit verification of the Nielsen identities

For the purposes of this paper, it is useful to explicitly verify Eqs. (37)
and (38) – the key ingredients in maintaining gauge invariance of $B_{1}$. The
Nielsen functional $C_{e2}$ receives a sole contribution from the diagram in
Fig. 2:

$\displaystyle
C_{e^{2}}=\frac{-ie^{2}\xi\phi}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{\left(k^{2}-m_{\chi}^{2}\right)\left(k^{2}-m^{2}_{FP}\right)}.$
(45)

Recalling that

$\frac{\partial m_{\chi}^{2}}{\partial\xi}=\frac{\partial
m_{FP}^{2}}{\partial\xi}=2e^{2}\phi^{2}\ \ \ ,$ (46)

and

$V_{e^{6}}^{\xi}=\frac{i}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\left[\ln\left(k^{2}-m_{FP}^{2}\right)-\ln\left(k^{2}-m_{\chi}^{2}\right)\right]$
(47)

we see that

$\displaystyle\xi\frac{\partial V_{e^{6}}}{\partial\xi}$ $\displaystyle=$
$\displaystyle\xi\frac{\partial}{\partial\xi}\int\frac{d^{4}k}{(2\pi)^{4}}\Bigl{[}-\frac{i}{2}\ln\left(k^{2}-m_{\chi}^{2}\right)+\frac{i}{2}\ln\left(k^{2}-m_{FP}^{2}\right)\Bigr{]}$
$\displaystyle=$ $\displaystyle i\xi
e^{2}\phi^{2}\left(m^{2}_{\chi}-m^{2}_{FP}\right)\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{\left(k^{2}-m_{\chi}^{2}\right)\left(k^{2}-m^{2}_{FP}\right)}=-C_{e^{2}}\frac{\partial
V_{e^{4}}}{\partial\phi}\ \ \ ,$

as required by Eq. (37). Importantly, to obtain the last equality, we have
used the dressed Goldstone mass squared $m_{\chi}^{2}$ given in Eq. (10). Use
of the tree-level mass would not allow the the Nielsen identiy at this order
to be satisfied.

[scale=2,transform shape][layered layout, horizontal=b to c] b [crossed
dot,black] – [scalar, half left,edge label = $\chi$] c [crossed dot] – [ghost,
half left,edge label = $c$] b ;

Figure 2: The diagram contributing to $C_{e^{2}}$

To verify Eq. (38), the most convenient approach is to evaluate the correlator
integral. Using the $\overline{MS}$ renormalization scheme, one obtains

$\displaystyle
C_{e^{2}}=e^{2}\xi\phi\frac{m_{FP}^{2}\left[\log\left(\frac{m^{2}_{FP}}{\mu^{2}}\right)-1\right]-m^{2}_{\chi}\left[\log\left(\frac{m^{2}_{\chi}}{\mu^{2}}\right)-1\right]}{32\pi^{2}(m^{2}_{\chi}-m^{2}_{FP})}$
(48)

Keeping only terms of order $e^{2}$, one obtains

$\displaystyle
C_{e^{2}}=-\frac{e^{2}\xi\phi}{32\pi^{2}}\log\left(\frac{2e^{2}\xi\phi^{2}}{\mu^{2}}\right)+\mathcal{O}\left(e^{4}\right)$
(49)

Using Eqs. (49) and (11) leads immediately to verification of Eq. (38).

###  4.3 Gauge invariance with the derivative expansion

We now implement carefully the gradient expansion of Ref. Metaxas:1995ab
discussed in the previous sections. The three key modifications of the naïve
computation are:

  1. 1.

Using the bounce solution corresponding to $V_{e^{4}}$ as seen in Eq. (42)

  2. 2.

Using a dressed mass for $m_{\chi}$ when evaluating the action. This is
essential for the Nielsen identity at $\mathcal{O}(\partial^{0})$, Eq. (37),
to be satisfied.

  3. 3.

Including the wavefunction renormalization, $Z_{e^{2}}$. As shown in Eq. (44),
this is required to cancel the gauge dependence in the effective potential.

Fig. 3 illustrates the impact of each modification, using the same numerical
inputs for the parameters as employed in obtaining Fig. 1. Implementing the
first modification slightly reduces the gauge dependence as the bounce
solution itself is no longer gauge-dependent. Numerically calculating the
action with this modification in the interval, $\xi\in[0,100]$, we found a
variation of 11%. Using the dressed mass for $m_{\chi}$ significantly
suppresses the gauge-dependence of the effective potential as this causes the
potential evaluated at the radiatively generated minimum of $V_{e^{4}}$ to be
gauge independent. We found a significantly lower variation of $0.9\%$.
Finally, incorporating the wavefunction renormalization as well, one obtains a
variation of $\lesssim 0.001\%$. This residual gauge dependence arises because
the full analytic expression for $V_{e^{6}}^{\xi}$ – which we use for
computational convenience – contains gauge-dependent higher order terms in
$e$. In principle, had we truncated $V_{e^{6}}^{\xi}$ at $\mathcal{O}(e^{6})$
then $B_{1}$ would be exactly $\xi$-independent.

Figure 3: The gauge dependence of the effective action using the modifications
proposed by the derivative expansion. The blue curve corresponds to the first
modification. The orange curve corresponds to both the first two
modifications. The green curve depicts the action incorporating all three
modifications proposed in Metaxas:1995ab

##  5 Gauge-independent nucleation rate using tunneling potentials

###  5.1 Gauge independence and the tunneling potential approach

In this section, we show how to translate the derivative expansion approach
into the tunneling potential framework in order to eliminate the gauge-
dependence exhibited by the blue curve in Fig. 1.

We start by adopting the power-counting in $e$ given in Section 2 and perform
the same three modifications as listed in Section 4.3. The key steps are as
follows.

Step 1: Calculate the leading order action, $B_{0}$, using the naïve tunneling
potential method. Specifically, we compute the quartic approximation of the
tunneling potential using Eq. (23) with $V_{0}=V_{e^{4}}(\phi_{0})$ and
$\phi_{T}$ being the maximum of $V_{e^{4}}$. We then find the $\phi_{0}$ that
minimizes the integral,

$\displaystyle
B_{0,\text{approx}}=54\pi^{2}\int_{\phi_{0}}^{\phi_{+}}\frac{\left(V_{e^{4}}-V_{t4}\right)^{2}}{\left(V_{t4}^{\prime}\right)^{3}}d\phi$
(50)

and compute the integral using this value of $\phi_{0}$. Note that so far, no
gauge dependence has been introduced as $V_{e^{4}}$ is gauge-independent.

Step 2: Calculate $B_{1}$ starting with the expression:

$\displaystyle B_{1,\text{approx}}$ $\displaystyle=S_{K,1}+S_{V,1}$ (51)
$\displaystyle=108\pi^{2}\left(\int_{\phi_{0}}^{\phi_{+}}\left[\frac{\left(V_{e^{4}}-V_{t4}\right)^{2}}{\left(V_{t4}^{\prime}\right)^{3}}\right]_{e^{4}}Z_{e^{2}}(\phi)\
d\phi+\int_{\phi_{0}}^{\phi_{+}}\left[\frac{\left(V_{e^{4}}-V_{t4}\right)}{\left(V_{t4}^{\prime}\right)^{3}}\right]_{e^{4}}V_{e^{6}}(\phi)\
d\phi\right)$ (52)

where the quantities in the square brackets and $\phi_{0}$ are the ones
obtained from minimizing $B_{0,\text{approx}}$. This expression satisfies the
three modifications listed in Section 4.3 in the following way:

  * •

The leading order bounce solution: The role of the bounce solution is replaced
by $V_{t}$ and $\phi_{0}$ in the tunneling potential approach as indicated in
Step 1. Hence, evaluating the quantities in the square brackets using the
$\phi_{0}$ found by minimising $B_{0,\text{approx}}$ is the equivalent to
evaluating the action using the leading order bounce solution.

  * •

Using the dressed Goldstone propagator: This is achieved by introducing the
$S_{V,1}$ term with the dressed propagator included in $V_{e^{6}}$. Note that
with our choice of $V_{t}$, the scaling relation $S_{K}=2S_{E}$ holds only at
$\mathcal{O}(e^{4})$. Hence, we must explicitly compute the kinetic and
potential contributions to $B_{1}$. Satisfying Eq. (37) then requires the use
of the dressed Goldstone propagator, as discussed above.

  * •

Including the wavefunction renormalization: This is achieved by the inclusion
of $Z_{e^{2}}(\phi)$ in $S_{K,1}$.

We note that should one have in hand an exact $V_{t}$ that satisfies Eq. (17),
the above steps would result in a gauge independent calculation of $B_{1}$.
However, there is an additional numerical concern that arises due to the use
of the quartic approximation for the tunneling potential. This will be
discussed in the next section.

###  5.2 Gauge dependence of the tunneling potential approach: a numerical
study

In this section, we numerically investigate the use of the procedure outlined
in the previous section. Using the values for the parameters as in Section
3.1, we begin by investigating the accuracy of the calculation of $B_{0}$.
Computing the effective action using Eq. (22) and minimizing with respect to
$\phi_{0}$, one obtains an effective action that agrees with the results of
the FindBounce package with an error of approximately 0.3%. This can be seen
in Fig. 4.

Figure 4: The estimate of the tunneling action by calculating the integral in
Eq. (22 )using the quartic approximation in Eq. (23). We normalize $\phi_{0}$
by the broken minimum of $V_{e^{4}}$ and the action to the action computed
using Findbounce. The black dot represents the minimum of the integral in Eq.
(50).

Similarly, calculating $B_{1}$ using Eq. (52), one obtains Fig. 5. We find
that there is an error of $0.4\%-0.5\%$ when compared to the tunneling rate
calculated in Section 4.3 and a gauge dependence of order 0.1%.

Figure 5: The action calculated using the procedure outlined in Section 5.1.
These are normalized using the corresponding values found using the bounce
solution in Section 4.3.

.

One can further reduce the gauge-dependence of this approximation by looking
more deeply into the scaling relation, $S_{K}=-\frac{1}{2}S_{V}$. Suppose one
uses the value of $\phi_{0}$ that minimizes the integral Eq. (50) and
calculates the action using $S_{E}=-S_{V}$, we find a relatively large error
of approximately 15$\%$. The source of this error is the sensitivity of
$S_{V}$ to small variations in $\phi_{0}$. In contrast, $S_{K}$ is quite
robust to these variations, as this $\phi_{0}$ is a stationary point. This is
evident in Fig. 6 ,where we see that $\frac{1}{2}S_{K}$ is almost constant
while $-S_{V}$ varies dramatically with $\phi_{0}$. We note that the large
error in $S_{V}$ is artifact of using the quartic approximation. In the case
of exactly solvable tunneling potentials, the minimum of $\frac{1}{2}S_{K}$
exactly corresponds the point where the scaling relation is satisfied.

To further reduce the $\xi$-dependence for a general potential, we propose,
instead, that one should use the $\phi_{0}$ that satisfies
$\frac{1}{2}S_{K}=-S_{V}$ (the point of intersection in Fig. 6) to compute
$B_{0}$ and $B_{1}$. Calculating $B_{0}$ in this way, we find an error of
approximately $0.32\%$ with respect to the action calculated using FindBounce.
This is only a slight increase in the error for $B_{0}$ when compared to
minimizing the integral in Eq. (50).

Figure 6: The actions computed using $S_{E}=\frac{1}{2}S_{K}$(blue) and
$S_{E}=-S_{V}$ (red). The point of intersection is the point where the scaling
relations are satisfied while the black dot represents the minimum of $S_{K}$.

Following the foregoing steps and evaluating
$B_{0,\text{approx}}+B_{1,\text{approx}}$ for $\xi\in[1,100]$, we obtain Fig.
7. Comparing with Fig. 5, we see a slightly larger error for small $\xi$ by
using this value of $\phi_{0}$ rather than the one which minimizes the
integral Eq. (50). However, the gauge dependence when using this approach is
of order $0.05\%$ in this range. This makes sense as the tunneling potential
now satisfies the scaling relation. The residual gauge dependence occurs
because the $\xi$-dependence of $B_{1}$ is only zero if the bounce equation is
satisfied exactly, as seen in Eq. (44), whereas the quartic approximation
being used for $V_{t}$ only satisfies the bounce equation at the three points,
$\phi_{+},\phi_{0}$ and $\phi_{T}$. Nevertheless, the efficiency and
convenience of this approach justifies its use when compared to solving the
full bounce equation.

Figure 7: This shows the approximate effective action as a function of $\xi$
using the tunneling potential approach as defined in Eqs. (50) and (52). These
are normalized using the corresponding values found using the bounce solution
in Section 4.3.

##  6 Summary

Computing the nucleation rate is a key step in exploring the possibility of
first order phase transitions in the early universe. The problem can be
particularly challenging for scenarios involving multiple scalar fields, such
as Higgs portal models that may lead to a first order electroweak phase
transition. The recent development of the tunneling potential method provides
a computationally efficient and numerically accurate approach for addressing
this challenge. It is well-known that in conventional approaches, care must be
taken to ensure that the resulting nucleation rate is gauge-invariant as
required for a physical observable. We are, thus, motivated to investigate
gauge-invariance in the tunneling potential framework.

In this paper, we have shown a way to calculate a gauge independent nucleation
rate at $T=0$ using tunneling potentials in the Abelian Higgs theory with a
radiatively induced barrier. We first reviewed how using a derivative
expansion and a power counting of $\lambda\sim e^{4}$ allows one to obtain a
gauge-independent nucleation rate. We then showed how the tunneling potential
approach can be modified to include these expansions consistently. In
particular, one must compute the tunneling potential, $V_{t}$, and the initial
field value, $\phi_{0}$, at leading order before using these to calculate
higher order corrections. Inclusion of the dressed Goldstone mass and the
$\mathcal{O}(e^{2})$ contribution to the wavefunction renormalization is
essential. We also showed that when using the quartic approximation to the
tunneling potential, the scaling relation between the kinetic and potential
parts of the action must be directly imposed due to the numerical sensitivity
of $S_{V}$ to $\phi_{0}$. Application of these ideas to the nucleation rate at
$T>0$ will appear in a forthcoming publication.

###### Acknowledgements.

We thank J. Löfgren and T.V.I. Tenkanen for helpful discussions and J. Löfgren
for a careful reading of this manuscript. This work was supported in part
under National Natural Science Foundation of China grant No. 19Z103010239.

##  Appendix A Wavefunction Renormalization

This may be calculated in an analogous way to what is done in Garny:2012cg by
finding the leading order $\mathcal{O}(p^{2})$ contributions to the diagrams
in Figure 8. Writing the kinetic part of the effective action as:

$\displaystyle S_{K}=\int
d^{4}x\frac{1}{2}Z(\phi)\left(\partial_{\mu}\phi\right)^{2}$ (53)

we find that $Z(\phi)$ is given by:

$\displaystyle Z(\phi)$
$\displaystyle=1+Z_{a}(\phi)+Z_{b}(\phi)+Z_{c}(\phi)+Z_{d}(\phi)+\mathcal{O}(e^{4})$
$\displaystyle Z_{a}(\phi)$
$\displaystyle=\frac{5e^{2}}{32\pi^{2}}+\frac{11\xi
e^{2}}{96\pi^{2}}-\frac{3e^{2}}{16\pi^{2}}\frac{m_{FP}^{2}\ln\left(\frac{m^{2}_{FP}}{m_{A}^{2}}\right)}{m_{FP}^{2}-m_{A}^{2}}$
$\displaystyle Z_{b}(\phi)$
$\displaystyle=\frac{e^{2}\xi}{48\pi^{2}}\left(\frac{m_{FP}^{2}}{m_{\chi}^{2}}\right)$
$\displaystyle Z_{c}(\phi)$ $\displaystyle=-\frac{e^{2}\xi}{24\pi^{2}}$
$\displaystyle Z_{d}(\phi)$ $\displaystyle=$
$\displaystyle\frac{e^{2}\xi}{16\pi^{2}}$
$\displaystyle\frac{m^{2}_{FP}\left(\log\left(\frac{m^{2}_{FP}}{\mu^{2}}\right)-\frac{3}{2}\right)-m^{2}_{\chi}\left(\log\left(\frac{m^{2}_{\chi}}{\mu^{2}}\right)-\frac{3}{2}\right)}{(m^{2}_{FP}-m^{2}_{\chi})}$
$\displaystyle+\frac{3e^{2}}{16\pi^{2}}$
$\displaystyle\frac{m^{2}_{A}\left(\log\left(\frac{m^{2}_{A}}{\mu^{2}}\right)-\frac{5}{6}\right)-m^{2}_{\chi}\left(\log\left(\frac{m^{2}_{\chi}}{\mu^{2}}\right)-\frac{5}{6}\right)}{(m^{2}_{A}-m^{2}_{\chi})}$

[scale=1.3,transform shape][layered layout, horizontal=b to c] a [] –
[scalar,edge label=$h$] b [] – [boson, half left,edge label = $A_{\mu}$] c []
– [boson, half left,edge label = $A_{\mu}$] b, c – [scalar,edge label=$h$] d ;

(a)

[scale=1.3,transform shape][layered layout, horizontal=b to c] a [] –
[scalar,edge label=$h$] b [] – [scalar, half left,edge label = $\chi$] c [] –
[scalar, half left,edge label = $\chi$] b, c – [scalar,edge label=$h$] d ;

(b)

[scale=1.3,transform shape][layered layout, horizontal=b to c] a [] –
[scalar,edge label=$h$] b [] – [ghost, half left,edge label = $c$] c [] –
[ghost, half left,edge label = $\overline{c}$] b, c – [scalar,edge label=$h$]
d ;

(c)

[scale=1.3,transform shape][layered layout, horizontal=b to c] a [] –
[scalar,edge label=$h$] b [] – [boson, half left,edge label = $A_{\mu}$] c []
– [scalar, half left,edge label = $\chi$] b, c – [scalar,edge label=$h$] d ;

(d)

Figure 8: The $\mathcal{O}(p^{2})$ terms in these diagrams form the leading
order contributions to $Z(\phi)$

##  Appendix B Scaling relations

The scaling relations $S_{E}=S_{K}/2=-S_{V}$ follow from the logic given in
Ref. Derrick:1964ww . Recalling that the leading order Euclidean action
$S_{E}^{0}$ is given by

$\displaystyle S_{K}^{0}$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}\
r^{3}dr\,\frac{1}{2}\dot{\phi}_{b}^{2}$ (54) $\displaystyle S_{V}^{0}$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\
r^{3}dr\,V_{e^{4}}(\phi_{b})$ (55)

where $\phi_{b}$ minimizes $S_{E}^{0}$. Now rescale $x^{\mu}=\lambda r^{\mu}$,
so that

$\displaystyle S_{K}^{0}$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda^{2}}\,I_{K}$ (56) $\displaystyle S_{V}^{0}$
$\displaystyle=$ $\displaystyle\frac{1}{\lambda^{4}}\,I_{V}\ \ \ ,$ (57)

where

$\displaystyle I_{K}$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}\
x^{3}dx\,\frac{1}{2}\dot{\phi}_{b}^{2}$ (58) $\displaystyle I_{V}$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\
x^{3}dx\,V_{e^{4}}(\phi_{b})\ \ \ .$ (59)

Thus,

$S_{E}^{0}=\frac{1}{\lambda^{2}}\,I_{K}+\frac{1}{\lambda^{4}}\,I_{V}\ \ \ .$
(60)

Now,

$-\frac{dS_{E}^{0}}{d\lambda}=\frac{2}{\lambda^{3}}\,I_{K}+\frac{4}{\lambda^{5}}\,I_{V}\
\ \ .$ (61)

Since $\phi_{b}$ extremizes $S_{E}^{0}$ we must have that

$\left(\frac{dS_{E}^{0}}{d\lambda}\right)_{\lambda=1}=0\ \ \ .$ (62)

Thus, we obtain $I_{K}=-2I_{V}$, so that $S_{K}^{0}=2S_{E}^{0}=-2S_{V}^{0}$.

It is now straightforward to see how the scaling relations can be broken by
the inclusion of higher order contributions to the effective action. First,
recall that Eq. (62) only holds for an extremal field configuration. In the
present context, the bounce solution $\phi_{b}$ extremizes $S_{E}$ in the
presence of $V_{e^{4}}$ but not when next-to-next-to leading order
contributions are included via $Z_{e^{2}}$ and $V_{e^{6}}$. Second – and more
generally – should the effective action receive contributions beyond second
order in derivatives, $S_{K}$ will no longer scale as $1/\lambda^{2}$ upon the
change of variables $x^{\mu}=\lambda r^{\mu}$.

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