DESY 16-161

CERN-TH-2016-187

Gauge-Independent Scales Related
to the Standard Model Vacuum Instability

J.R. Espinosaa,b, M. Garnyc, T. Konstandind, A. Riottoe
a Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology (BIST), Campus UAB, E-08193, Bellaterra (Barcelona),
Spain
b ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain
c CERN Theory Department, CH-1211 Geneva, Switzerland
d DESY, Notkestr. 85, 22607 Hamburg, Germany
e Department of Theoretical Physics and Center for Astroparticle Physics (CAP)
24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland

The measured (central) values of the Higgs and top quark masses indicate that
the Standard Model (SM) effective potential develops an instability at high
field values. The scale of this instability, determined as the Higgs field
value at which the potential drops below the electroweak minimum, is about
$10^{11}$ GeV. However, such a scale is unphysical as it is not gauge
invariant and suffers from a gauge-fixing uncertainty of up to two orders of
magnitude. Subjecting our system, the SM, to several probes of the instability
(adding higher order operators to the potential; letting the vacuum decay
through critical bubbles; heating up the system to very high temperature;
inflating it) and asking in each case physical questions, we are able to
provide several gauge-invariant scales related with the Higgs potential
instability.

##  1 Introduction

After the LHC runs made so far, with the discovery of a light Higgs boson [1]
and no signal of additional new physics, we face the possibility that the
Standard Model (SM) might describe physics up to very high energy scales,
possibly up to the Planck scale. The value of the Higgs mass, measured with
great precision by the LHC, which gives the combined value $M_{h}=125.09\pm
0.21$ (stat) $\pm 0.11$ (syst) GeV [2], turns out to be of particular interest
in this context. Large radiative corrections from the heavy top quark
destabilize the Higgs potential at large field values making the electroweak
(EW) vacuum metastable. For the current value (from the LHC + Tevatron
combination) $M_{t}=173.34\pm 0.27$ (stat) $\pm 0.71$ (syst) GeV [3] we most
likely live in such unstable vacuum [4]. Intriguingly, we seem to be rather
close to the boundary of stability [4, 5, 6, 7, 8, 9] and this translates into
a very long lifetime (many orders of magnitude larger than the age of the
universe) against decay by quantum tunneling. One concludes that this
metastability does not represent an inconsistency of the SM and cannot be used
to argue in favor of new physics. The potential instability has also very
interesting cosmological implications [10, 11, 12, 13] and might have a deeper
significance (for some attempts in that direction see [5, 14, 15]).

The instability scale defined as the field value at which the Higgs potential
gets lower than the EW vacuum, is quite large, of order $10^{11}$ GeV for the
central experimental values of $M_{h}$ and $M_{t}$ quoted above. However, this
instability scale turns out to be a gauge-dependent quantity (the previous
numerical value corresponds to the potential evaluated, at NNLO, in Landau
gauge [4, 5]). Such gauge dependence issues [16] are well known since the
early days of the effective potential [17] but the problem has attracted some
attention recently [18, 19, 20] in the wake of the Higgs discovery and the
realization that we might be living in a metastable vacuum. The uncertainty in
the instability scale due to this gauge dependence was estimated in [18] to be
potentially sizeable, of up to two orders of magnitude.

The goal of this paper is to address this issue by deriving physical gauge-
independent scales associated to the instability scale (with varying degrees
of how direct the connection is). Following previous discussions and to ease
the comparison with earlier literature we use Fermi (or Lorentz) gauge, using
the gauge fixing parameter $\xi$ to track the gauge dependence of our results.
The gauge dependence of the effective potential (or more fundamentally of the
effective action) is described by the so-called Nielsen identity, which we
review in Section 2. After showing explicitly the gauge dependence of the
instability scale in Fermi gauge in Section 3 we then discuss several ways of
extracting physical scales associated with the potential instability. This
instability can be cured by heavy physics that affects the potential through
non-renormalizable operators, say $\lambda_{6}|H|^{6}/\Lambda^{2}$, where
$\Lambda$ is the mass scale characterizing the heavy physics. Even though the
potential is a gauge-dependent object, the mass scale $\Lambda$ required to
stabilize the potential turns out to be gauge-independent. We discuss this
particular proposal (exploring also non-renormalizable operators of higher
orders) in Section 4. We show how in this way the instability scale can be
determined unambiguosly and lies within one order of magnitude of the naive
instability scale calculated in Landau gauge.

A different scale related to the instability can be obtained from the radius
$R_{c}$ of the critical bubble for vacuum decay. We discuss this in Section 5,
giving a gauge-independent definition of this radius, including also gravity
effects. The energy scale associated to this critical radius is the scale at
which new physics can have a direct impact on the vacuum lifetime. In the case
of the SM it is much heavier that the instability scale itself, being rather
close to the Planck scale.

In Section 6 we also study the behaviour of the unstable potential at very
high temperatures and obtain a critical temperature at which there is a
degeneracy between the EW minimum and the one at very high field values. This
temperature, which can be proven to be gauge invariant, turns out to be too
loosely related to the instability scale in the SM and not too illuminating.

Finally, in Section 7 we discuss how to probe the instability scale via
inflation, that causes fluctuations in the Higgs field proportional to the
Hubble rate $H_{I}$ and makes it probe the unstable region if $H_{I}$ is large
(comparable to the instability scale). We prove that the probability of
finding the Higgs in a certain field range, after a given number of e-folds of
inflation, is a gauge-invariant quantity. Then we discuss how to extract a
value for the Hubble rate that reflects closely the scale of the potential
instability with the advantage of being a gauge-invariant quantity.

After drawing some conclusions, we collect some technical details and results
in several Appendices. In Appendix A we calculate the renormalization group
equations for the Wilson coefficients of higher order operators added to the
effective potential. In Appendix B we derive the $\xi$-dependence of the
different functions that appear in the effective action (and the energy-
momentum tensor derived from it) when using a derivative expansion, up to
${\cal O}(\partial^{4})$. In Appendix C we discuss the validity of the Nielsen
identity at finite temperature, deriving explicit results for the SM at one-
loop.

##  2 The Nielsen Identities

The dependence of the Higgs effective potential on the gauge-fixing parameters
$\xi$ derives from the gauge dependence of the effective action $S$ itself.
Nevertheless, the potential and the effective action are very useful and it is
possible to extract from them physical quantities that are gauge-independent.

The Nielsen identity [24, 25, 26] describes the $\xi$-dependence of the
effective action and plays a central role in discussing how to obtain gauge-
independent quantities. For cases with a Higgs background only, the identity
reads

$\xi\frac{\partial}{\partial\xi}S[h(x),\xi]=-\int d^{4}y\ K[h(y)]\frac{\delta
S}{\delta h(y)}\ ,$ (1)

where $K[h(y)]$ is a known functional of $h$, given in [24]. According to this
identity, the effective action evaluated on a solution of the equation of
motion (EoM) for $h$, (that is, $\delta S/\delta h=0$) is $\xi$-independent. A
particular instance of this general result is the $\xi$-independence of the
values of the effective potential at its extrema, a well known result.

If one writes the effective action in a derivative expansion

$S[h]=\int d^{4}x\left[-V(h)+\frac{1}{2}Z(h)(\partial_{\mu}h)^{2}+{\cal
O}(\partial^{4})\right]\ ,$ (2)

a series of Nielsen identities for the coefficient-functions in this expansion
can be derived from the identity in Eq. (1). We show this explicitly, up to
fourth order in the expansion, in Appendix B. At the lowest order in this
derivative expansion, i.e. for constant field configurations, one finds a
Nielsen identity for the effective potential:

$\xi\frac{\partial V}{\partial\xi}+C(h)V^{\prime}=0\ ,$ (3)

where $C(h)$ is the value of $K[h]$ for constant $h$. As anticipated above,
this identity implies the $\xi$-independence of the values of the potential at
the extremal points. The identity (3) can be rewritten as

$\xi\frac{dV}{d\xi}=\xi\frac{\partial V}{\partial\xi}+\frac{\partial
V}{\partial h}\,\xi\frac{\partial h}{\partial\xi}=0\ ,$ (4)

showing that the explicit $\xi$-dependence of the effective potential can be
compensated by an implicit $\xi$-dependence of the field as:

$\xi\frac{dh}{d\xi}=C(h)\ .$ (5)

A change in $\xi$ is therefore equivalent to the field redefinition (5). This
way of looking a the effect of a change in $\xi$ makes it obvious that the
values of the potential at the extremal points are gauge-independent.

Next, we derive Nielsen identities for the equation of motion and for the
energy-momentum tensor $T^{\mu\nu}$ that allow us to understand the
$\xi$-dependence of the solutions of the EoM and prove the $\xi$-independence
of $T^{\mu\nu}$, both of which are needed for later discussions. We begin with
the Nielsen identity for the effective action given in Eq. (1). To derive a
Nielsen identity for the equation of motion, we take a functional derivative
with respect to $h(x)$, getting

$\xi\frac{\partial}{\partial\xi}\frac{\delta S}{\delta h(x)}+\int
d^{4}y\left(\frac{\delta^{2}S}{\delta h(y)\delta h(x)}K[h(y)]+\frac{\delta
S}{\delta h(y)}\frac{\delta K[h(y)]}{\delta h(x)}\right)=0\;.$ (6)

Evaluating the effective action for a field $\bar{h}(x,\xi)$ that solves the
EoM, $\delta S/\delta h=0$, the last term in the equation above drops, while
the first two terms can be combined into a total derivative,

$\xi\frac{d}{d\xi}\frac{\delta S[\bar{h},\xi]}{\delta h(x)}=0\;,$ (7)

provided $\bar{h}$ fulfills

$\xi\frac{d}{d\xi}\bar{h}(x,\xi)=K[\bar{h}(x)]\,.$ (8)

As the total derivative (7) is zero, this in fact shows that the solution of
the EoM for $\xi+d\xi$ is $\bar{h}+K[\bar{h}]d\log\xi$. In other words, the
solution $\bar{h}(x,\xi)$ of (8) describes how a solution of the EoM changes
when varying the gauge parameter $\xi$.111It is worth noting that the value of
the action for any field configuration, even off-shell, is invariant under the
combined $\xi$\- and field change implied by (8), as is evident from Eq. (1).

Let us next discuss the $\xi$-independence of the energy-momentum tensor, that
is defined through the variation of the action under changes of the metric
tensor, as

$T^{\mu\nu}(x)=-\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g_{\mu\nu}(x)}\ ,$
(9)

where $g$ is the determinant of the metric. We are ultimately interested in a
flat background, but keep the metric explicit in order to derive the energy-
momentum tensor. In order to derive a Nielsen identity for the energy-momentum
tensor222 This requires that the Nielsen identity not only holds in flat
Minkowski space but for an arbitrary background metric (at least in the
neighbourhood of flat space). Such generalized Nielsen identity can be
obtained by writing all terms in a general coordinate invariant form. Gauge-
fixing of gravity is not required for a non-dynamical background metric which
is all we need here to define the energy-momentum tensor. The usual proof of
the Nielsen identity is then directly carried over to the non-flat case. To
calculate explicitly $K[h,g]$ is nevertheless much more cumbersome.  we take a
derivative of (1) with respect to $g_{\mu\nu}(x)$,

$\xi\frac{\partial}{\partial\xi}T^{\mu\nu}(x)-\frac{2}{\sqrt{-g(x)}}\int
d^{4}y\left(\frac{\delta S}{\delta h(y)\delta
g_{\mu\nu}(x)}K[h(y)]+\frac{\delta S}{\delta h(y)}\frac{\delta K[h(y)]}{\delta
g_{\mu\nu}(x)}\right)=0\;.$ (10)

When evaluated for $\bar{h}(x,\xi)$, a solution of the EoM, the first two
terms can again be combined into a total derivative, while the last term
vanishes, and we get

$\xi\frac{d}{d\xi}T^{\mu\nu}(x)\Big{|}_{h=\bar{h}(x,\xi)}=0\;.$ (11)

This means that the explicit gauge parameter dependence is precisely
compensated by the change of the field value when varying $\xi$. Therefore,
the total $\xi$-dependence vanishes, such that on-shell the energy-momentum
tensor is gauge-fixing independent.

For the particular case in which we are interested, the SM in Fermi gauge, we
have in fact two $\xi$ parameters appearing in the EW gauge-fixing Lagrangian:

${\cal
L}_{gf}=-\frac{1}{2\xi_{B}}(\partial^{\mu}B_{\mu})^{2}-\frac{1}{2\xi_{W}}(\partial^{\mu}W^{a}_{\mu})^{2}\
,$ (12)

and the effective potential depends on both of them. We have a Nielsen
identity for each, with

$\xi_{i}\frac{\partial V}{\partial\xi_{i}}+C_{i}(h)V^{\prime}=0\ ,$ (13)

for $\xi_{i}=\xi_{B},\xi_{W}$, with the $C_{i}(h)$ functions given by

$\displaystyle C_{B}(h)$ $\displaystyle=$
$\displaystyle\frac{ig^{\prime}}{2}\int d^{4}y\ \langle
c(x)\chi^{0}(x)\bar{c}(y)\partial_{\mu}B^{\mu}(y)\rangle\ ,$ (14)
$\displaystyle C_{W}(h)$ $\displaystyle=$ $\displaystyle\frac{ig}{2}\int
d^{4}y\ \langle
c_{a}(x)\chi_{a}(x)\bar{c}_{b}(y)\partial_{\mu}W_{b}^{\mu}(y)\rangle\ ,$ (15)

where $c,\bar{c}$ ($c_{a},\bar{c}_{a}$) are the $U(1)_{Y}$ [$SU(2)_{L}$] ghost
fields and $\chi_{a}$ the Goldstone boson fields while $B_{\mu}$ and
$W_{\mu}^{a}$ are the $U(1)_{Y}$ and $SU(2)_{L}$ gauge bosons, respectively
and $h$ is a constant background.

Let us write the tree-level potential as

$V_{0}(h)=-\frac{1}{2}m^{2}h^{2}+\frac{1}{4}\lambda h^{4}\ .$ (16)

The renormalized one-loop potential (in $\overline{\rm MS}$ scheme) is
obtained as [27]

$V_{1}(\phi)=\sum_{\alpha}N_{\alpha}J^{0}_{\alpha}(M_{\alpha}^{2})=\frac{\kappa}{4}\sum_{\alpha}N_{\alpha}M_{\alpha}^{4}\left[\log\frac{M_{\alpha}^{2}}{\mu^{2}}-C_{\alpha}\right]\
,$ (17)

where $\kappa=1/(16\pi^{2})$, $\mu$ is the renormalization scale and the index
$\alpha$ runs over different particle species, with $N_{\alpha}$ degrees of
freedom (taken negative for fermions). The squared masses $M_{\alpha}^{2}$ are
the corresponding masses in an $h$ background. The main contributions to the
potential come from:

$\begin{array}[]{rll}{\rm Top\ quarks:}&N_{t}=-12\
,&M_{t}^{2}=\frac{1}{2}y_{t}^{2}h^{2}\ ,\\\ W^{\pm}\ {\rm bosons}:&N_{W}=6\
,&M_{W}^{2}=\frac{1}{4}g^{2}h^{2}\ ,\\\ Z^{0}\ {\rm bosons}:&N_{Z}=3\
,&M_{Z}^{2}=\frac{1}{4}(g^{2}+{g^{\prime}}^{2})h^{2}\ ,\\\ {\rm Higgs\
bosons}:&N_{h}=1\ ,&M_{h}^{2}=-m^{2}+3\lambda h^{2}\ ,\\\ {\rm Neutral\
Goldstones}:&N_{B_{\pm}}=1\
,&M_{B_{\pm}}^{2}=\frac{1}{2}[M_{G}^{2}\pm\sqrt{M_{G}^{4}-4(\xi_{W}M_{W}^{2}+\xi_{B}M_{B}^{2})M_{G}^{2}}]\
,\\\ {\rm Charged\ Goldstones}:&N_{A_{\pm}}=2\
,&M_{A_{\pm}}^{2}=\frac{1}{2}(M_{G}^{2}\pm\sqrt{M_{G}^{4}-4\xi_{W}M_{G}^{2}M_{W}^{2}})\
,\end{array}$ (18)

where we use the auxiliary squared masses

$M_{B}^{2}\equiv\frac{1}{4}{g^{\prime}}^{2}h^{2}\ ,\quad
M_{G}^{2}\equiv\frac{1}{h}\frac{\partial V_{0}}{\partial h}=-m^{2}+\lambda
h^{2}\ .$ (19)

Finally, the $C_{\alpha}$s are constants: $C_{\alpha}=3/2$ for scalars and
fermions and $C_{\alpha}=5/6$ for gauge bosons.

Fermi gauge is afflicted by infrared divergences beyond those generic in all
gauges (with massless Goldstone bosons) that remain even after the resummation
cure that fixes the latter [21, 22]. For a discussion of this issue (more
precisely an IR divergence in the first derivative of the effective potential)
and its solution(s), see [23]. In order to avoid this complication we use a
Fukuda-Kugo IR regulator $\mu_{IR}$ as described in Subsection 4.2 of [23]. In
this case one needs to add to the potential also ghost contributions with
$N_{c_{Z}}=-2,N_{c_{W}}=-4,M_{c_{Z}}^{2}=\mu_{IR}M_{Z},M_{c_{W}}^{2}=\mu_{IR}M_{W},C_{c_{i}}=3/2$.
We checked that none of our results depends on the choice of $\mu_{IR}$.

The functions $C_{B,W}(h)$ that enter in the Nielsen identity in Eq. (3),
calculated at one loop, are

$\displaystyle C^{(1)}_{B}(h)$ $\displaystyle=$
$\displaystyle\frac{\xi_{B}}{8}{g^{\prime}}^{2}h\left[\frac{I^{0}_{B_{+}}-I^{0}_{B_{-}}}{M^{2}_{B_{+}}-M^{2}_{B_{-}}}\right]\
,$ (20) $\displaystyle C^{(1)}_{W}(h)$ $\displaystyle=$
$\displaystyle\frac{\xi_{W}}{8}{g}^{2}h\left[\frac{I^{0}_{B_{+}}-I^{0}_{B_{-}}}{M^{2}_{B_{+}}-M^{2}_{B_{-}}}+2\frac{I^{0}_{A_{+}}-I^{0}_{A_{-}}}{M^{2}_{A_{+}}-M^{2}_{A_{-}}}\right]\
,$ (21)

with

$I^{0}_{\alpha}(M_{\alpha}^{2})\equiv 2\frac{\partial}{\partial
M_{\alpha}^{2}}J^{0}_{\alpha}(M_{\alpha}^{2})=\kappa
M_{\alpha}^{2}\left(\log\frac{M_{\alpha}^{2}}{\mu^{2}}-C_{\alpha}+\frac{1}{2}\right)\
.$ (22)

With the previous expressions it is straightforward to check that the one-loop
Nielsen identities

$\xi_{i}\frac{\partial V_{1}(h)}{\partial\xi_{i}}+C^{(1)}_{i}\frac{\partial
V_{0}(h)}{\partial h}=0\ ,$ (23)

are indeed fulfilled. For the numerical analyses in the following sections we
set $\xi_{B}=\xi_{W}=\xi$ at the EW scale (more precisely at $\mu=M_{t}$) as
was done in [18], and discuss the dependence of different quantities with
$\xi$.

##  3 Gauge Dependence of the Instability Scale

As discussed in the Introduction, for the measured values of the Higgs and top
quark masses, the Standard Model develops an instability at high field values.
One can take as the scale of the instability the field value $h_{I}$ at which
the effective potential drops below the value of the electroweak minimum.
Given the order of magnitude of the scales involved this corresponds in
practice to $V(h_{I})=0$, with $h_{I}\simeq 10^{11}$ GeV for the central
values of $M_{h}$ and $M_{t}$. It is well-known that $h_{I}$ is not a gauge-
independent quantity and it was calculated recently, using Fermi and $R_{\xi}$
gauges [18], that the gauge dependence leads to an uncertainty of up to two
orders of magnitude in $h_{I}$ (if one takes extreme values $\xi\sim 300$, at
the limit of validity of the perturbative regime). For illustration, the gauge
dependence of $h_{I}$ in Fermi gauge is shown in Fig. 1. The black solid line
corresponds to the result extracted from the one-loop RG improved effective
potential, taking into account also the running of the gauge fixing
parameters. It turns out that even when fixing $\xi_{B}=\xi_{W}=\xi$ to large
values at the EW scale, the running significantly reduces their value in the
UV [18] reducing the impact of gauge dependence. This is illustrated by the
black dashed line, for which the running of the gauge-fixing parameters has
been neglected. These two lines reproduce the results of [18].

Figure 1: Gauge dependence of several quantities normalized to their value for
$\xi=0$. Black lines give the scale $h_{I}(\xi)$ at which the effective
potential drops below the value of the electroweak minimum taking into account
the running of the gauge fixing parameters (solid) or keeping them fixed
(dashed). For comparison, we also show the value of the effective potential at
the maximum, $V_{max}^{1/4}$, with or without resummation of IR divergences,
as indicated.

One might ask whether the gauge dependence, especially for very large values
of $\xi$, signals a poor perturbative description rather than the expected
gauge dependence of $h_{I}$. To address this question we also show the value
of the potential evaluated at the maximum in Fig. 1, which is a gauge-
independent quantity. Its residual gauge dependence due to the perturbative
computation of the potential is indeed small (blue dotted line). This
dependence is further reduced to the few $\%$ level for the effective
potential with IR resummed Goldstone mass parameter [21, 22] (orange dot-
dashed line) in agreement with expectations [23]. In addition, we checked that
the gauge dependence of $h_{I}$ agrees, at the same level of accuracy, with
the one expected from the Nielsen identity, obtained by solving the
differential equation Eq. (8). Note that the gauge dependence of $h_{I}$ in
$R_{\xi}$ gauges is of similar magnitude, but goes in the opposite direction
[18], such that the value of $h_{I}$ varies over a large range.

One pragmatic attitude concerning the gauge dependence of $h_{I}$ is simply to
ignore the problem altogether: an order of magnitude estimate of the
instability scale, especially when it is so high, might be good enough and the
Landau gauge calculation should give a quite reasonable estimate of that scale
(why would one take such large values of the gauge fixing parameter as
$\xi\sim 300$?). Moreover, the same problem affects the vacuum expectation
value of the Higgs in the electroweak vacuum. That is, the usual $v_{EW}=246$
GeV is also a gauge dependent quantity. However, as long as one calculates
physical quantities the gauge dependence should drop out. Residual gauge
dependence might still be left over from truncating perturbative expansions,
in which case taking large values of the gauge fixing parameters would be ill
advised. Nevertheless, as a point of principle, it is interesting to think of
what physical scales are there that can be associated to the presence of the
instability of the potential. We discuss several possibilities in the
following Sections.

##  4 Stabilization by Irrelevant Operators

In this section we explore one physical scale, independent of the gauge-fixing
parameter, that reflects in a very clean way the underlying instability scale
in the SM effective potential. The basic idea is to probe the behaviour of the
theory at high field values by adding a fictitious higher-dimensional operator
to the Higgs potential, characterized by a suppression scale $\Lambda$.333
Generically, if there is really new physics at the scale $\Lambda$ the use of
an effective theory to study the impact on the heavy physics on the
instability scale, requires a wide separation between the two scales, see [28]
for a discussion of this point. See also [29] for a recent analysis (based on
functional RG methods) of the potential instability in the presence of higher-
dimensional operators. At tree-level the potential is

$V_{0}(h)=-\frac{1}{2}m^{2}h^{2}+\frac{\lambda}{4}h^{4}+c_{n}\frac{h^{n}}{2^{n/2}\Lambda^{n-4}}\,,$
(24)

and we consider only even powers of $n$, with $n\geq 6$, as the non-
renormalizable operator should arise from powers of $(H^{\dagger}H)$.

After including radiative corrections the SM part of the potential develops
the instability at some scale [at LO this is captured by $\lambda(h)$ turning
negative] but, for positive $c_{n}$, the new term stabilizes the effective
potential at even larger field values. By varying $\Lambda$ (keeping $c_{n}$
fixed) it is possible to find a critical value of $\Lambda_{c}$ for which the
high-scale minimum at $h_{c}$ [$V^{\prime}(h_{c},\xi;\Lambda_{c})=0$] and the
EW minimum at $v_{EW}$ are degenerate,

$V(h_{c},\xi;\Lambda_{c})=V(v_{EW},\xi;\Lambda_{c})\approx 0\,.$ (25)

Since the shape of the effective potential depends on the particular value of
the gauge-fixing parameter $\xi$ (as indicated) one might expect that the
critical value $\Lambda_{c}$ required to fullfill the conditions above would
also be $\xi$-dependent. Interestigly this is not the case and $\Lambda_{c}$
is a gauge-independent scale. This follows from the fact that the value of the
potential at an extremal point is independent of $\xi$. The argument is
illustrated by Fig. 2: a change in $\xi$ affects the unstable potential
$V(h,\xi)$ shown in the upper left panel as a field redefinition and
$V(h,\xi+\Delta\xi)$ is shown in green in the upper right corner. The lower
left panel shows instead the effect of adding to the original potential a non-
renormalizable term adjusted to the critical value $V(h,\xi;\Lambda_{c})$,
such that the potential has two degenerate minima. It is then obvious that the
$\xi$ transformation of this potential, $V(h,\xi+\Delta\xi;\Lambda_{c})$,
still has two degenerate minima and therefore $\Lambda_{c}$ is also the
critical scale for the potential $V(h,\xi+\Delta\xi)$ and one concludes that
$\Lambda_{c}$ is $\xi$-independent. On the other hand, the field value $h_{c}$
at which the high scale minima appears does depend on the gauge fixing. This
suggests the use of $\Lambda_{c}$ instead of $h_{I}$ as a physical measure of
the instability scale.

Figure 2: Illustration of the $\xi$-independence of the scale $\Lambda_{c}$
defined in the text. Upper left panel: $V(h,\xi)$. Upper right panel:
$V(h,\xi+\Delta\xi)$ in green, shows the rescaling effect of changing
$\xi\to\xi+\Delta\xi$. Lower left panel: $V(h,\xi;\Lambda_{c})$ shows the
effect of the non-renormalizable term suppressed by $\Lambda_{c}$ that gives
two degenerate minima. Lower right panel: shows in green the effect of $\xi$
change on previous potential. The degenerate minima stay degenerate.

We describe below how to implement this idea in practice. The effect of the
higher-dimensional operator in the one-loop effective potential is
straightforward to take into account, simply using the modified Higgs and
Goldstone field-dependent squared masses

$\displaystyle M_{H}^{2}(h)$ $\displaystyle=$ $\displaystyle
V_{0}^{\prime\prime}(h)=-m^{2}+3\lambda
h^{2}+\frac{n(n-1)c_{n}h^{n-2}}{2^{n/2}\Lambda^{n-4}}\ ,$ $\displaystyle
M_{G}^{2}(h)$ $\displaystyle=$
$\displaystyle\frac{1}{h}V_{0}^{\prime}(h)=-m^{2}+\lambda
h^{2}+\frac{nc_{n}h^{n-2}}{2^{n/2}\Lambda^{n-4}}\ ,$ (26)

where primes denote $h$-derivatives. In addition, we take into account IR
resummation of the Goldstone mass, as discussed in [23], to minimize residual
gauge dependences. At the order we work, this amounts to adding to $M_{G}^{2}$
the one-loop Goldstone self-energy $\Pi_{g}$ given by

$\frac{1}{\kappa}\Pi_{g}=\frac{dM_{H}^{2}}{dh^{2}}M_{H}^{2}(L_{H}-1)-6y_{t}^{2}M_{t}^{2}(L_{t}-1)+\frac{3}{2}g^{2}M_{W}^{2}\left(L_{W}-\frac{1}{3}\right)+\frac{3}{4}(g^{2}+{g^{\prime}}^{2})M_{Z}^{2}\left(L_{Z}-\frac{1}{3}\right)\
,$ (27)

where $L_{X}=\log(M_{X}^{2}/\mu^{2})$.

The renormalization group (RG) improved effective potential resums large
logarithms between the electroweak and high-scale minima in the usual way (see
e.g. [4]). For our purpose it is enough to use the one-loop potential (with IR
resummed Goldstone mass) with SM parameters running at two loops (the
renormalization group equations in Fermi gauge can be found in [18]) while we
treat the effect of the non-renormalizable operator at leading order, with
$c_{n}$ running at one loop. The one-loop beta function for $c_{n}$,
$\beta_{c_{n}}\equiv dc_{n}/d\log{\mu}$, neglecting corrections of order
$m^{2}/\Lambda^{2}\ll 1$, is given by (see Appendix A)

$\beta_{c_{n}}=3\kappa\left(n\lambda+y_{t}^{2}-\frac{{g^{\prime}}^{2}}{4}-\frac{3g^{2}}{4}\right)nc_{n}+\frac{\kappa}{8}\sum_{m=6}^{n-2}m\
p(n,m)\left[m\ p(n,m)+n\right]c_{m}c_{p(n,m)}\ ,$ (28)

where $p(n,m)\equiv n+4-m$. Once we introduce the higher order operator in the
SM potential the theory is not renormalizable and further operators of even
higher order are generated radiatively. This is reflected in the RG equation
above, through the $c_{m}c_{p(n,m)}$ terms. We impose the boundary condition
$c_{n}(\Lambda)=1$, $c_{m\not=n}(\Lambda)=0$. Close to the scale $\Lambda$ the
operator $h^{n}$ dominates and we can safely neglect the rest for our
analysis.

The non-renormalizable couplings are completely irrelevant at the electroweak
scale. For the renormalizable couplings we impose boundary conditions at the
electroweak scale, more precisely at $\mu_{0}\equiv M_{t}=173.34\,$GeV, using
the best-fit values given in [5]. We first consider the lowest choice $n=6$,
and then discuss what happens for higher $n$.

Figure 3: Renormalization group improved (and IR resummed) effective potential
$V(h,\xi)$ normalized to $h^{4}$ for the SM in Fermi gauge. The different
curves in each panel show the dependence on the gauge fixing parameter
$\xi=0,2,4,8,\dots,256$. The four panels correspond to different choices of
the scale $\Lambda$ of the dimension-six operator $\propto h^{6}/\Lambda^{2}$,
as indicated.

A close-up of the instability region of the RG-improved effective potential
for $n=6$ is shown in Fig. 3. The upper left plot corresponds to the Standard
Model (without higher-dimensional operators, i.e., $\Lambda\to\infty$) for
several values of $\xi$ from $0$ to 256. We see that the scale $h_{I}$ [at
which $V(h_{I})=0$] is indeed $\xi$-dependent. In the other plots we added to
the potential the $c_{6}h^{6}/(8\Lambda^{2})$ operator, with
$c_{6}(\Lambda)=1$. For $\Lambda=5\times 10^{12}$ GeV (upper right plot), the
value of the potential at the high-scale minimum is negative, _i.e._ deeper
than the electroweak minimum. For $\Lambda=10^{12}$ GeV the potential is
positive at the high-scale minimum, such that the electroweak vacuum becomes
the absolute minimum. For $\Lambda=\Lambda_{c}=2.4\times 10^{12}$ GeV both
minima are exactly degenerate (lower left plot). It is apparent that this
degeneracy occurs for all values of $\xi$ at the same value of $\Lambda_{c}$,
as discussed above.444The value of the potential at the minimum is gauge
invariant for any value of $\Lambda$. The apparent gauge dependence in the
upper and lower right plots is due to the fact that the minimum of the ratio
$V/h^{4}$ shown in the plots is offset from the true minimum of $V$ for
$V\not=0$. On the other hand, the field value $h_{c}$ at which the minimum
occurs does depend on $\xi$, as expected.

In Fig. 4 we compare the critical scale $\Lambda_{c}$ obtained for $n=6$ with
the instability scale $h_{I}$ at which $V=0$, comparing directly with the
results in [18], reproduced by the black lines. Even when varying the gauge
fixing parameter over a large range, $\xi=(0,250)$, we find that the
$\Lambda_{c}$ scale determined from the RG improved one-loop potential is
stable at the level of $6\%$, with the residual gauge dependence due to the
perturbative truncation of the potential. This is to be contrasted with the
gauge dependence of the instability scale defined via the field value, that
varies by half an order of magnitude over the same range of $\xi$. Let us also
mention that we checked that (i) the effect of the running of the $c_{m>6}$
higher-dimensional operators generated by the running even when only $c_{6}$
is present at the scale $\Lambda$, is minor, affecting $\Lambda_{c}$ by less
than $2\%$. (ii) The value of $\Lambda_{c}$ is also stable at the percent
level when varying (by a factor $2$ up or down) the renormalization scale used
to resum large logarithms.

Figure 4: Gauge (in)dependence of the instability scale of the Standard Model,
as a function of the gauge fixing parameter $\xi$. The black lines reproduce
the results from [18], and correspond to the field value $h_{I}$ for which
$V(h_{I})=0$ (the dashed line is obtained if the running of $\xi$ is
neglected). The red line shows the critical value $\Lambda_{c}$ of a
dimension-six operator that stabilizes the SM potential giving a large scale
minimum degenerate with the EW one. The residual $\xi$-dependence of
$\Lambda_{c}$ is $\sim 6\%$ within the range shown.  Figure 5: Critical scale
$\Lambda_{c}$ of a $h^{n}/\Lambda^{n-4}$ operator that stabilizes the SM
potential giving a large scale minimum degenerate with the EW one, for various
values of $n$. The residual $\xi$-dependence for $\xi\in(0,250)$ is below the
$10\%$ level for all $n$. For large $n$ the critical scale approaches
$\Lambda_{I}\simeq 10^{11}$ GeV.

There is freedom to choose the dimension $n$ of the non-renormalizable
operator we use to extract the scale $\Lambda_{c}$. In Fig. 5 we show the
critical value $\Lambda_{c}$ obtained for various choices of $n=6,8,\dots$.
For large $n$ the critical value asymptotes to $\Lambda_{I}\simeq 10^{11}$
GeV. The reason is that in this limit the effect of the higher-dimensional
operator can be considered almost as a step function. Therefore, one may
identify the asymptotic large-$n$ value, that we call $\Lambda_{I}$ as the
physical scale beyond which the electroweak vacuum becomes unstable.
Irrespective of the precise physical interpretation of this scale, it is
important that it can be determined in an unambiguous way, independent of the
gauge fixing. It is interesting to note that $\Lambda_{I}$ is actually rather
close to the naive instability scale $h_{I}$ obtained in Landau gauge by
demanding $V(h_{I})=0$. As we show below this is not a coincidence but depends
on some of the assumptions we have made, in particular for the scaling of
$c_{n}$ with $n$.

Let us approximate the potential in the instability region as

$V(h,\Lambda)\simeq\frac{1}{4}\lambda(h)h^{4}+c_{n}\frac{h^{n}}{2^{n/2}\Lambda^{n-4}}\
,$ (29)

where the bulk of the radiative corrections to the potential are captured in
the running quartic $\lambda(h)$, (that includes also finite one-loop
contributions), evaluated at the renormalization scale $\mu=h$. This
approximation is sufficient to determine the leading scaling of $\Lambda_{c}$
and its limiting value $\Lambda_{I}$ with different parameters. In particular
this approximation ignores the $\xi$-dependence of the radiatively corrected
potential, but this is not an issue once we have a general argument for the
$\xi$-independence of $\Lambda_{c}$. From the conditions
$V(h_{c},\Lambda_{c})=0$ and $V^{\prime}(h_{c},\Lambda_{c})=0$ we get

$\lambda(h_{c})\simeq\frac{\beta_{\lambda}(h_{c})}{n-4}\,,\quad\quad\Lambda_{c}\simeq
h_{c}\left[\frac{-4nc_{n}}{2^{n/2}\beta_{\lambda}(h_{c})}\right]^{1/(n-4)}\ .$
(30)

Taking the large $n$ limit we see that $\lambda(h_{c})\rightarrow 0$, so that
$h_{c}\to h_{I}$, the point at which the quartic coupling crosses zero:
$\lambda(h_{I})=0$. To take the large $n$ limit of $\Lambda_{c}$, which
determines $\Lambda_{I}$, we need to specify how $c_{n}$ scales with $n$. The
generic expectation, based on simple $\hbar$ power counting, is that at tree-
level $c_{n}\sim g^{n-2}$ where $g$ represents a generic coupling between the
Higgs field and the physics at the scale $\Lambda$. With this $n$ dependence
included we get the limit

$\Lambda_{I}\simeq\frac{g}{\sqrt{2}}h_{I}\ .$ (31)

In the numerical analysis of Fig. 5 we simply took $g=1$ but in general one
does expect some dependence on the coupling strength $g$. After all, in order
for the new physics represented by the non-renormalizable operator to
stabilize the potential, it should matter how strong is the coupling of that
new physics to the Higgs field.555In fact, how the scale of new physics able
to stabilize the potential could vary depending on the coupling strength has
been analyzed before, using particular models, see [30]. Up to that
unavoidable model-dependence we are nevertheless able to extract a gauge-
independent scale to be very closely associated with the scale of instability
of the SM potential.

In this context, it is interesting to ask what is the highest scale where some
(perturbative) new physics has to appear if one demands absolute vacuum
stability. The highest scale is obtained for the lowest order $n=6$, and we
take $c_{6}\sim g^{4}$ in accordance with the expected scaling with the
coupling strength to the new physics [the validity of the effective
description requires $g\gtrsim{\cal O}(1)$, so as to keep sufficiently
separate $\Lambda$ and the potential minimum]. For the central value of the SM
parameters one obtains (see Fig. 3) $\Lambda_{c}\lesssim 2.4\times 10^{12}$
GeV $\times\,\,g^{2}\lesssim 3.8\times 10^{14}$ GeV, where we assumed $g<4\pi$
to get this rough estimate of the largest possible scale where new physics has
to appear to ensure absolute vacuum stability.666The presence of the
dimension-$5$ (Weinberg) operator with a similar suppression scale would not
spoil this upper bound. As is well-known [41], heavy Majorana neutrinos tend
to make the potential more unstable and therefore would call for even lower
values of $\Lambda_{c}$.

##  5 Gauge Invariance and the Tunneling Critical Bubble

In quantum field theory, false vacuum decay proceeds by the nucleation of
bubbles that probe the lower-energy phase, are large enough to grow (with the
bulk energy gain overcoming the surface tension), and eventually engulf the
whole of space [31]. The tunneling probability density (which determines the
false vacuum lifetime) actually measures how likely it is to nucleate such
bubbles per unit time and unit volume.

This decay rate $\Gamma$ is calculated by first finding the so-called bounce
solution $h_{b}$ to the Euclidean EoM [32]. This solution is $O(4)$
spherically symmetric, $h_{b}(\rho)$, with $\rho^{2}=t_{E}^{2}+r^{2}$, where
$t_{E}$ is Euclidean time and $r^{2}=\vec{x}^{2}$. It has boundary conditions
$h_{b}=v_{f}$ at $t_{E}\rightarrow\pm\infty$ (where $v_{f}$ is the expectation
value of the field in the false vacuum) and a turning point (assigned without
loss of generality to $t_{E}=0$) with $\partial h_{b}/\partial t_{E}=0$ for
all $\vec{x}$. The decay rate scales as $\Gamma\sim e^{-S_{4}}$, where $S_{4}$
is the Euclidean action for this bounce solution, $S_{4}=S_{E}[h_{b}]$ (see
e.g. [33] for a nice introduction). The gauge-independence of $S_{4}$ and
$\Gamma$ has been discussed in the literature before [34, 35] and follows the
usual pattern: the explicit $\xi$-dependence that appears in the effective
action, described by the Nielsen identity (now for the Euclidean action):

$\xi\frac{\partial S_{E}}{\partial\xi}+\int d^{4}xK_{E}[h(x)]\frac{\delta
S_{E}}{\delta h}=0\ ,$ (32)

is compensated by an implicit dependence on $\xi$ of the solutions of the
EoM/bounce equation, given by

$\xi\frac{dh_{b}}{d\xi}=K_{E}[h_{b}]\ ,$ (33)

and leading to

$\frac{dS_{4}}{d\xi}=0\ .$ (34)

Here, rather than on $S_{4}$, we are interested in the radius of the critical
bubble, which provides another physical scale associated to the potential
instability. The critical bubble, the most likely bubble profile for vacuum
decay, corresponds to the bounce solution evaluated at $t_{E}=0$,
$h_{B}(r)\equiv h_{b}(\sqrt{0^{2}+r^{2}})$. It is an $O(3)$ spherically
symmetric bubble configuration with zero total energy (as the tunneling
process must conserve energy). In fact, the analytic continuation of the
Euclidean bounce to Minkowski spacetime (with
$\rho^{2}=t_{E}^{2}+r^{2}\rightarrow-t^{2}+r^{2}$, where $r^{2}=\vec{x}^{2}$
is the square of the distance to the bubble center) gives the real time
evolution of the nucleated bubble (see [43] for details). The field profile,
$h_{B}(r)$, could be used to define the radius $R_{c}$ of the critical bubble,
e.g. as the radius at which the field is half-way between the false vacuum
value at infinity and its value at the center of the bubble,
$h_{B}(R_{c})=[h_{B}(0)-v_{f}]/2$. However, it is clear that such definition
leads to a $\xi$-dependent $R_{c}$ as the bounce solution depends on $\xi$ in
a non-trivial way, according to Eq. (33).777 In practice, the $\xi$-dependence
of the critical radius defined in this way might be small in the SM case,
given that $C(h)\propto h$ up to small logarithmic effects [see Eqs. (20,21)].
However, we would like to give a definition of the critical radius that is
$\xi$-independent from first principles.

A $\xi$-independent critical radius can be defined if, instead of using the
field bubble profile, one uses the energy density profile. Let us show how
this works using a derivative expansion approximation for the effective
action, although it is clear that the general derivation does not rely on such
expansion.

Let us assume then that the derivative expansion of the effective action
converges and gives a sufficiently good approximation at second order in
derivatives. The Euclidean effective action for the $O(4)$ symmetric bounce
$h_{b}(\rho)$ reads

$S_{E}=2\pi^{2}\int_{0}^{\infty}\,d\rho\,\rho^{3}\left[\frac{1}{2}Z(h_{b})(\partial_{\rho}h_{b})^{2}+V(h_{b})-V_{f}\right]\
,$ (35)

where $V_{f}=V(v_{f})$ is the potential value in the false vacuum. The bounce
solution $h_{b}(\rho)$ satisfies the Euclidean EoM

$Z(h_{b})\mathop{}\\!\mathbin{\Box}_{E}h_{b}+\frac{1}{2}Z^{\prime}(h_{b})\
(\partial_{\rho}h_{b})^{2}=V^{\prime}(h_{b})\ ,$ (36)

with
$\mathop{}\\!\mathbin{\Box}_{E}h_{b}\equiv\partial_{\rho}(\rho^{3}\partial_{\rho}h_{b})/\rho^{3}$
and depends on $\xi$ because both $Z$ and $V$ are gauge-dependent objects.
Their gauge dependence is described by the corresponding Nielsen identities
that are obtained by expanding the functional $K_{E}[h(x)]$ of Eq. (32) in
gradients888 To avoid confusion, note that our definition of $D(h)$ and
$\tilde{D}(h)$ is different form that used in [36].

$K_{E}[h(x)]=C(h)+D(h)(\partial_{\mu}h\
\partial^{\mu}h)+\tilde{D}(h)(\partial_{\mu}\partial^{\mu}h)+O(\partial^{4})\,,$
(37)

with spacetime indices contracted using the Euclidean metric. As the Nielsen
identity of Eq. (32) holds for generic fields $h(x)$, one can extract the
$\xi$-dependence of $V$ and $Z$ by cancelling the terms of order
$\partial^{0}$, $(\partial h)^{2}$ and $(\partial^{2}h)$ separately. One gets
the usual relation for the effective potential

$\xi\frac{\partial V}{\partial\xi}+C(h,\xi)V^{\prime}=0\,,$ (38)

where primes denote field derivatives, while for $Z$ one gets

$\xi\frac{\partial
Z}{\partial\xi}=-CZ^{\prime}-2ZC^{\prime}-2DV^{\prime}+2(\tilde{D}V^{\prime})^{\prime}\,.$
(39)

That the functions $C$, $D$ and $\tilde{D}$ indeed fulfill these relations as
a consequence of the Nielsen identity can be checked on a case by case basis
(see e.g. [34, 36] and Appendix C of [23] where this is shown explicitly at
one-loop for the Abelian Higgs model in Fermi gauge). In addition, the bounce
solution depends on $\xi$ according to Eq. (33), which in this spherically
symmetric case takes the form

$\xi\frac{dh_{b}(\rho)}{d\xi}=C(h_{b})+D(h_{b})(\partial_{\rho}h_{b})^{2}+\tilde{D}(h_{b})\mathop{}\\!\mathbin{\Box}_{E}h_{b}+O(\partial^{4})\
.$ (40)

Making use of the relations above it is then straightforward to show that

$\xi\frac{d}{d\xi}S_{E}[h_{b}]=0\ ,$ (41)

to order ${\cal O}(\partial^{4})$, as expected. The proof is extended to
fourth order in the derivative expansion in Appendix B and of course holds in
full generality even when the derivative expansion is not applicable.

As explained above, the critical bubble is given by
$h_{B}(r)=h_{b}(t_{E}=0,r)$, where $r$ is the distance in 3D space to the
center of the bubble. The real time evolution of the bubble, after nucleation,
is given by the analytic continuation of the $t_{E}=0$ bounce,
$h_{B}(t,r)=h_{b}(\sqrt{-t^{2}+r^{2}})$, which automatically satisfies the
Minkowski EoM

$-Z(h_{B})\mathop{}\\!\mathbin{\Box}h_{B}-\frac{1}{2}Z^{\prime}(h_{B})\
[(\partial_{t}h_{B})^{2}-(\partial_{r}h_{B})^{2}]=V^{\prime}(h_{B})\ ,$ (42)

with
$\mathop{}\\!\mathbin{\Box}h_{B}=\partial_{t}^{2}h_{B}-\partial_{r}(r^{2}\partial_{r}h_{B})/r^{2}$.
At $t=0$, $\partial_{t}h_{B}=0$ but
$\partial_{t}^{2}h_{B}=-\partial_{r}h_{B}/r$ so that Eq. (42) does agree with
the bounce equation (36). Notice also that the $\xi$-dependence of $h_{B}(r)$
follows directly from (40) and is given by

$\xi\frac{dh_{B}(r)}{d\xi}=C(h_{B})+D(h_{B})(\partial_{r}h_{B})^{2}-\tilde{D}(h_{B})\mathop{}\\!\mathbin{\Box}h_{B}+O(\partial^{4})\
.$ (43)

The energy density for the critical bubble is

$E_{B}(r)=\frac{1}{2}Z(h_{B})(\partial_{r}h_{B})^{2}+V(h_{B})-V_{f}\ .$ (44)

Using Eqs. (38), (39) and (43) we get

$\xi\frac{d}{d\xi}E_{B}(r)=(\tilde{D}V^{\prime})^{\prime}(\partial_{r}h_{B})^{2}-\tilde{D}V^{\prime}\mathop{}\\!\mathbin{\Box}h_{B}={\cal
O}(\partial^{4})\ ,$ (45)

where, for the last step, we use the EoM for $h_{B}$, Eq. (42), and its
$r$-derivative to show that $V^{\prime}$ and $V^{\prime\prime}$ are quantities
of order ${\cal O}(\partial^{2})$. This completes the check that the energy
density profile is $\xi$-independent to the order we work. The previous
discussion, based on the derivative expansion at second order in derivatives,
can be extended to higher orders as shown in Appendix B.

Even when the derivative expansion is not convergent, the energy density
profile must be $\xi$-independent in any self-consistent calculation: as shown
in Section 2, the energy-momentum tensor is $\xi$-independent, and therefore,
the energy density of the critical bubble must be $\xi$-independent too. As
the total energy density of the critical bubble integrates to zero,999To prove
this explicitly, integrate by parts to show
$\int_{0}^{\infty}r^{2}(V[h_{B}]-V_{f})dr=-\int_{0}^{\infty}r^{3}V^{\prime}\partial_{r}h_{B}dr/3$
and use the EoM to substitute $V^{\prime}$ and obtain in this way a relation
between potential and surface energies.

$E_{B,\text{tot}}=4\pi^{2}\int_{0}^{\infty}r^{2}E_{B}(r)dr=0\ ,$ (46)

with $E_{B}(r)$ starting negative at $r=0$, turning positive at some $r$ and
going to zero at $r\rightarrow\infty$, there must exist a distance $r=R_{c}$
at which $E_{B}(r)$ reaches a maximum. That is,

$\left.\frac{dE_{B}(r)}{dr}\right|_{R_{c}}=0\
,\quad\quad\left.\frac{d^{2}E_{B}(r)}{dr^{2}}\right|_{R_{c}}<0\ .$ (47)

This value $R_{c}$ is our definition of radius of the critical bubble and is
$\xi$-independent by construction. More explicitly, if we use the energy
density expression in Eq. (44), taking its $r$-derivative and simplifying it
using the field equation of motion, we arrive at the simple (implicit) result
for the critical radius

$\left.R_{c}=\frac{3Z\partial_{r}h_{B}}{2V^{\prime}}\right|_{R_{c}}\ .$ (48)

It can be checked, using the $\xi$-dependence of $Z,V$ and $h_{B}$ that indeed
one has $dR_{c}/d\xi=0$, to ${\cal O}(\partial^{2})$. After nucleation at
$t=0$, the bubble radius can still be defined via the maximum of the energy
density, which at $t>0$ includes also a nonzero contribution from the kinetic
term $Z(h_{B})(\partial_{t}h_{B})^{2}/2$, and is $\xi$-invariant at all times.
Although this definition is frame-dependent, for macroscopic values $t,r\gg
R_{c}$, it can be shown that $R(t)\simeq\sqrt{R^{2}_{c}+t^{2}}\simeq t$ (in
agreement with the fact that $h_{B}(r,t)$ is a function of $r^{2}-t^{2}$
[43]).

###  5.1 The Thin-Wall Critical Bubble

We discuss now the case of a thin-wall critical bubble which, as usual, allows
a good analytic understanding of the parametric dependence of several
quantities of interest. The critical bubble wall is thin when the energy
difference between false and true vacua is small compared to the potential
barrier that separates them. In this case the field profile of the bubble
interpolates between the two vacua (true vacuum inside the bubble, false
outside) with a very rapid transition, such that the wall thickness is much
smaller than the bubble size.

Without making assumptions about the derivative expansion we write the general
expression for the Euclidean action for the bounce solution $h_{b}(\rho)$ as

$S_{E}=2\pi^{2}\int_{0}^{\infty}\,d\rho\,\rho^{3}\left[{\cal
L}_{K}(h_{b})+V(h_{b})-V_{f}\right]\ ,$ (49)

where ${\cal L}_{K}$ contains all the derivative terms beyond the simple
$Z(h)(\partial h)^{2}/2$. We have to assume that the action $S_{E}$ is to a
certain extent ’local’. For example the two-point function
$\delta^{2}S_{E}/\delta h^{2}$ falls off exponentially on a length scale of
the order of the inverse Higgs mass. This does not imply that the derivative
expansion converges since the bubble thickness is typically of the same scale.

In the thin-wall case it is easy to see heuristically that the critical radius
should be $\xi$-independent: the effective action for the inner and outer
parts of the bubble depend only on the constant pieces of the bounce profile.
The $\xi$-dependence of these parts is given by the function $C$ as in Eq.
(40) evaluated for a constant field value, with negligible contributions from
the derivative terms (important only in the bubble wall). Since $\xi$-changes
map the inner and outer parts of the critical bubble onto themselves, the
bubble size is then $\xi$-independent up to the negligible bubble wall
thickness.

This is of course consistent with the analytic estimate of the critical radius
in the thin-wall case. The total Euclidean action receives a contribution from
the inside region of the bubble which is approximately given by

$S_{E,{\rm in}}\simeq
2\pi^{2}\int_{0}^{R_{c}}d\rho\,\rho^{3}\left[V(h_{b})-V_{f}\right]\
\simeq-\frac{1}{2}\pi^{2}R_{c}^{4}\,\epsilon_{V}\,,$ (50)

where $\epsilon_{V}\equiv V_{f}-V_{t}$, is the potential difference between
the true and false minima. On the other hand, the wall contributes an amount
proportional to the surface of the critical bubble. We can write it in terms
of the wall tension as

$S_{E,{\rm
wall}}=2\pi^{2}\int_{R_{c}-\Delta}^{R_{c}+\Delta}d\rho\,\rho^{3}\left[{\cal
L}_{K}(h_{b})+V(h_{b})-V_{f}\right]\equiv 2\pi^{2}R_{c}^{3}\,\sigma\,,$ (51)

which can be considered as the definition of $\sigma$, and gives the right
scaling with $R_{c}$. All the (possibly complicated) dependence of ${\cal
L}_{K}(h_{b})$ on field gradients is modelled by the wall tension. Finally,
the exterior of the bubble, for which $V(h_{b})-V_{f}\simeq 0$ gives a zero
contribution to the total action, $S_{E,{\rm out}}\simeq 0$.

Extremizing the Euclidean action $S_{4}=S_{E,{\rm in}}+S_{E,{\rm
wall}}+S_{E,{\rm out}}$ with respect to the size of the critical bubble then
yields the standard results

$R_{c}=\frac{3\sigma}{\epsilon_{V}}\ ,\quad
S_{4}=\frac{27\pi^{2}\sigma^{4}}{2\epsilon_{V}^{3}}\ .$ (52)

Obviously $\epsilon_{V}$ is gauge-independent, as the values of the potential
at its minima are $\xi$-independent. Since the action $S_{4}$ is also
$\xi$-independent, this is then also true for the wall tension $\sigma$ and
the size of the critical bubble $R_{c}$. The same result for $R_{c}$ can be
obtained by examining the total energy of the critical bubble

$E_{B,\text{tot}}=4\pi\int_{0}^{\infty}r^{2}E_{B}(r)dr\simeq 4\pi\left(\sigma
R_{c}^{2}-\frac{1}{3}\epsilon_{V}R_{c}^{3}\right)=0\ ,$ (53)

which can be solved for $R_{c}$ with the same result as in (52). This critical
radius coincides with the location of the maximum in the energy density
profile (that we used to define $R_{c}$ in the general case) as

$E_{B}(r)\simeq\left\\{\begin{array}[]{ccl}-\epsilon_{V}&&(r<R_{c})\ ,\\\
\sigma\delta(r-R_{c})&&(r=R_{c})\ ,\\\ 0&&(r>R_{c})\ ,\end{array}\right.$ (54)

peaks at $r=R_{c}$.

###  5.2 The SM Case

The energy scale $\mu_{d}\sim 1/R_{c}$ associated to vacuum decay and
determined by the critical bubble radius is of relevance for discussing the
impact of heavy physics beyond the SM on the lifetime of the EW vacuum. In
particular, if new physics starts to be relevant at such scale, then it can
impact the lifetime of the false vacuum even if $\mu_{d}$ were much higher
than the instability scale $\Lambda_{I}$. In fact this is precisely what
happens in the SM.

As we have seen, at field values much higher than the EW scale the SM
effective potential can be well approximated as

$V(h)\simeq\frac{1}{4}\lambda_{\rm eff}h^{4}\ ,$ (55)

with $\lambda_{\rm eff}<0$. As is well known [37], the bounce solution for
such a potential can be calculated analytically (assuming a constant
$\lambda_{\rm eff}$) and is

$h_{b}(\rho)=\frac{2\sqrt{2}}{R\sqrt{-\lambda_{\rm
eff}}}\left(\frac{1}{1+\rho^{2}/R^{2}}\right)\ ,$ (56)

where $R$, that determines the critical bubble size, is arbitrary. For any
value of $R$ one gets the same Euclidean action for the bounces, with
$S_{E}[h_{b}]=8\pi^{2}/(-3\lambda_{\rm eff})$. The energy profile for the
critical bubble is

$E_{B}(r)=\frac{16(r^{2}/R^{2}-1)}{(-\lambda_{\rm
eff})R^{4}(1+r^{2}/R^{2})^{4}}\ ,$ (57)

and has a maximum at $r=\sqrt{5/3}R\simeq 1.3R$ so that $R$ basically
coincides with the critical radius defined through the maximum of $E_{B}(r)$.

As appreciated long ago [38], the scale invariance of the bounce solution is
broken by radiative effects: $\lambda_{\rm eff}$ is not really constant but
receives potentially large radiative corrections enhanced by powers of large
logarithms $\propto\log(h/\mu)$, where $\mu$ is the renormalization scale. In
order to resum such logarithms one chooses $\mu\simeq h$, in practice
evaluating a running $\lambda_{\rm eff}(\mu)$ at $\mu=h$. The Euclidean action
that suppresses the EW vacuum decay is minimal at the field value (or
renormalization scale) at which $\lambda_{\rm eff}$ reaches its most negative
value, that is, $d\lambda_{\rm eff}/d\log\mu\equiv\beta_{\lambda_{\rm
eff}}=0$, with $\lambda_{\rm eff}<0$. The scale at which that happens singles
out one special size for the critical bubble, with $1/R_{c}\sim\mu_{d}$ and
$\beta_{\lambda_{\rm eff}}(\mu_{d})=0$.

Figure 6: Spatial profile of the energy density (44) for the critical bubble
for two different values $\xi=0,200$ in Fermi gauge. These results are
obtained solving the bounce equation (42) numerically (using the RG-improved
and IR resummed 1-loop effective potential and the corresponding correction to
the kinetic term) and choosing the field value in the bubble center
$h_{B}(r=0)$ such that the bounce action is minimized.  Figure 7: Dependence
of the inverse critical radius $1/R_{c}$ and the field value at the center of
the critical bubble $h_{B}(r=0)$ on the gauge-fixing parameter. The bubble
radius varies at the $10\%$ level for $\xi=0,200$, consistent with the
expected gauge independence and residual perturbative uncertainties. The value
$h_{B}(r=0)$, on the other hand, is gauge dependent. For comparison we also
show the radius $1/R_{1/2}$ at which the field value has dropped to
$h_{B}(r=0)/2$, which in this particular case is nearly $\xi$-independent
also, see footnote 7.

For the measured $M_{h}$ and $M_{t}$ (at the central values of their
experimental intervals) one gets [4] $\mu_{d}\simeq 10^{17}$ GeV, to be
compared with the instability scale $\Lambda_{I}\simeq 10^{11}$ GeV. Figure 6
shows the energy-density profile of the critical bubble, for two values of
$\xi$ illustrating also the $\xi$-independence of the profile. The critical
radius derived from it, is therefore also $\xi$-independent, as show in Figure
7, as a function of $\xi$. For comparison, notice that the value of the field
at the center of the bubble is $\xi$-dependent, as expected. The hierarchical
separation between $\mu_{d}$ and $\Lambda_{I}$ is a consequence of the nearly
scale independence of the effective potential and the Euclidean action for the
bounce (see [39, 40] for some recent discussions on this). From this fact it
follows that BSM physics modifying the effective potential at the scale
$\mu_{BSM}$ with $\mu_{BSM}\leq\mu_{d}$ can have a large impact on the EW
vacuum lifetime even if $\mu_{BSM}\gg\Lambda_{I}$.

The simplest (and best motivated) example of such scenario has been discussed
long ago: a see-saw scenario with heavy right-handed neutrinos of mass
$M_{\nu_{R}}$ [41]. This model requires large neutrino Yukawa couplings (to
fit light neutrino masses $m_{\nu}$) that cause the running $\lambda_{\rm
eff}$ to become even more negative above the $M_{\nu_{R}}$ threshold, further
destabilizing the EW vacuum. In fact, the prediction of a too-short lived EW
vacuum can be used [7] to rule out $M_{\nu_{R}}\geq 10^{13-14}$ GeV for
$m_{\nu}\simeq 0.1-1$ eV.

In principle, even new physics at $\mu_{BSM}>\mu_{d}$ could impact the EW
vacuum lifetime, as recently emphasized in [42]. All that is required is that
the BSM physics makes the Higgs effective potential even more unstable at
$h>\mu_{d}$ (and this implies already Planckian physics) so that bounces with
even smaller Euclidean action are possible. However, it is difficult to find
plausible reasons why physics at the Planck scale (presumably more
fundamental) should worsen the potential instability (unlike what happened in
the well-motivated see-saw scenario where one understands the origin of the
instability as due to new Yukawa couplings). In connection to this, it has
also been noticed [40] that, while BSM physics with $\mu_{BSM}\gg\Lambda_{I}$
can easily make the EW vacuum more unstable, it is much harder to make the
potential more stable. The reason is that increasing the Euclidean action at
high scales suppresses bounces that cause tunneling to such large scales but
does not modify the bounce at the scale $\mu_{d}$, which still dominates
vacuum decay and therefore leads to the standard EW vacuum lifetime.

###  5.3 Including Gravity

We now extend our considerations by including gravity [43, 44]. This is
particularly relevant since in the SM the tunneling bubbles do not experience
a flat geometry, but an anti-de Sitter one. For discussions on the effect of
gravitational corrections on the EW vacuum decay see [55, 15, 46, 56, 57, 54].
We start by writing the most general rotationally invariant Euclidean metric

$ds^{2}=d\zeta^{2}+\rho^{2}(\zeta)d\Omega_{3}^{2},$ (58)

where $\zeta$ is a radial coordinate measuring distances along radial curves
normal to three-spheres, $d\Omega_{3}^{2}$ is the line element angular
distance on a unit $3$-sphere and $\rho$ is the radius of curvature of each
three-sphere. Of this metric we need the corresponding Ricci scalar

$R=\frac{6}{\rho^{2}}\left(-\rho\rho^{\prime\prime}-\rho^{\prime 2}+1\right),$
(59)

where primes stand for differentiation with respect to $\zeta$. The EoM/bounce
equation for $h$ is $\xi$-independent also in curved space-time. The reason is
that the background metric is obtained through Einstein’s equations and is
sourced by the energy-momemtum tensor. Being the latter $\xi$-independent, so
is the metric. This implies that the EoM in curved space-time differs from the
flat space-time one only by an object, the metric, that is invariant under
$\xi$ changes.

Vacuum decay is controlled by the difference $S_{E}[h_{b}]-S_{E,f}$ between
two Euclidean actions, the one for the bounce and the background one in the
false vacuum. The bounce action can be written as

$S_{E}[h_{b}]=2\pi^{2}\int d\zeta\,\rho^{3}\left({\cal
L}_{K}+V-\frac{1}{2}m_{P}^{2}R\right),$ (60)

where $m_{P}^{2}=1/(8\pi G)$ is the reduced Planck mass (squared) and $G$ is
Newton’s constant. ${\cal L}_{K}$ and $V$ are the Lagrangian containing the
derivative terms and the effective potential term, respectively. Although our
results below hold beyond the derivative expansion, for simplicity we will
take ${\cal L}_{K}=Z(h){h^{\prime}}^{2}/2$, with $h^{\prime}\equiv dh/d\zeta$
and give below many expressions simplified to this order of the expansion. The
background action is $S_{E,f}=-24\pi^{2}m_{P}^{4}/V_{f}$ for a de Sitter false
vacuum ($V_{f}>0$, where $V_{f}$ is the false vacuum potential) as in the SM
case and $S_{E,f}=0$ for a Minkowski one.

By inserting Eq. (59) into Eq. (60), we obtain

$\displaystyle S_{E}[h_{b}]$ $\displaystyle=$ $\displaystyle 2\pi^{2}\int
d\zeta\,\left[\rho^{3}\left({\cal
L}_{K}+V\right)+3m_{P}^{2}\left(\rho^{2}\rho^{\prime\prime}+\rho\rho^{\prime
2}-\rho\right)\right]$ (61) $\displaystyle=$ $\displaystyle 2\pi^{2}\int
d\zeta\,\left[\rho^{3}\left({\cal
L}_{K}+V\right)-3m_{P}^{2}\rho\left(1+\rho^{\prime 2}\right)\right],$

where in the last step we have simplified the curvature term by integrating by
parts.101010Possible boundary terms play no role in this calculation and we
simply ignore them throughout. From this action one gets the Euclidean EoM for
the field, which gives the bounce equation

$Zh^{\prime\prime}+3\frac{\rho^{\prime}}{\rho}Zh^{\prime}+\frac{1}{2}Z^{\prime}{h^{\prime}}^{2}=V^{\prime}\
,$ (62)

where, with an abuse of notation we use $Z^{\prime}\equiv\partial Z/\partial
h$ and $V^{\prime}\equiv\partial V/\partial h$, while $\rho^{\prime}\equiv
d\rho/d\zeta$ and $h^{\prime}\equiv dh/d\zeta$.

The next step is to employ the $\zeta\zeta$-component of the Euclidean
Einstein’s equations

$G_{\zeta\zeta}=\frac{3}{\rho^{2}}\left(1-\rho^{\prime
2}\right)=-\frac{1}{m_{P}^{2}}\,T_{\zeta\zeta}\ ,$ (63)

where $G_{\zeta\zeta}$ is the $\zeta\zeta$-component of Einstein’s tensor and
$T_{\zeta\zeta}$ is the $\zeta\zeta$-component of the energy-momentum tensor,
with $T_{\zeta\zeta}=Z{h^{\prime}}^{2}/2-V$. Using now Eq. (63) in Eq. (60),
we can further simplify the action as

$\displaystyle S_{E}[h_{b}]$ $\displaystyle=$ $\displaystyle 2\pi^{2}\int
d\zeta\,\left[\rho^{3}\left({\cal
L}_{K}+V-\,T_{\zeta\zeta}\right)-6\rho\,m_{P}^{2}\right]$ (64)
$\displaystyle=$ $\displaystyle 4\pi^{2}\int
d\zeta\,\left(\rho^{3}V-3\rho\,m_{P}^{2}\right)\ .$

The important remark at this point is that, since the energy-momentum tensor
is $\xi$-independent, so must be Einstein’s equations and the various
components of the metric. Having previously shown that the effective action
calculated on the bounce is $\xi$-independent provided that it is evaluated on
configurations such that $\xi\,dh_{b}/d\xi=K[h_{b}]$, by reasoning as in the
flat case, we conclude that the action is $\xi$-independent. The background
action, $S_{E,f}=-24\pi^{2}m_{P}^{4}/V_{f}$, is trivially $\xi$-independent as
the potential value in the false vacuum, $V_{f}$, is $\xi$-independent.

Concerning the $\xi$-independence of the critical bubble radius in the
presence of gravity, one can still use a bubble profile that is
$\xi$-independent if, instead of using the field itself, one resorts to the
profile of $T_{\zeta\zeta}$. Although defining a local energy density is in
general not possible in the presence of gravity, spherical symmetry allows one
to give a well-behaved definition [45]. The total energy/mass of the critical
bubble is given by the integral (see e.g. [46] for a recent discussion)

$E=4\pi\int_{0}^{\infty}d\rho\,\rho^{2}\,T_{\zeta\zeta}=4\pi\int_{0}^{\infty}d\rho\,\rho^{2}\,\left(\frac{1}{2}Z{h^{\prime}}^{2}+V-V_{f}\right)\
,$ (65)

where $T_{\zeta\zeta}$ is now the Minkowski one. Although $T_{\zeta\zeta}$
comes from matter alone, $E$ includes gravitational self-interaction
contributions (a similar formula is used in defining the total mass of a
spherically symmetric star). In this respect, it is instructive to apply (65)
to the thin-wall case taking the radius $R$ of the bubble as arbitrary (and
setting $Z=1$ and $V_{f}=0$ here for simplicity). Defining the wall tension as

$\sigma\equiv\int_{0}^{\infty}d\xi\left(\frac{1}{2}{h^{\prime}}^{2}+V\right)\
,$ (66)

we can use ${h^{\prime}}^{2}/2+V=\sigma\delta(\xi-\bar{\xi})$, where
$\bar{\xi}$ corresponds to the position of the bubble wall,
$R=\rho(\bar{\xi})$. The total mass/energy of such bubble is then

$\displaystyle E(R)$ $\displaystyle=$ $\displaystyle
4\pi\int_{0}^{R}d\rho\,\rho^{2}(-\epsilon_{V})+4\pi\int_{\bar{\xi}-\delta}^{\bar{\xi}+\delta}d\xi\,\rho^{2}\rho^{\prime}\,\sigma\delta(\xi-\bar{\xi})$
(67) $\displaystyle=$
$\displaystyle-\frac{4\pi}{3}\epsilon_{V}R^{3}+4\pi\sigma
R^{2}\frac{1}{2}(\rho^{\prime}_{+}+\rho^{\prime}_{-})\ ,$

where $-\epsilon_{V}$ is the value of the potential inside the bubble, and, in
evaluating the second (wall) integral we have taken care of the fact that
$\rho^{\prime}$, given by Eq. (63), jumps at the wall with

$\displaystyle\rho^{\prime}_{-}$ $\displaystyle\equiv$
$\displaystyle\rho^{\prime}(\bar{\xi}-\delta)=\sqrt{1+8\pi
G\epsilon_{V}R^{2}/3}$ $\displaystyle\rho^{\prime}_{+}$ $\displaystyle\equiv$
$\displaystyle\rho^{\prime}(\bar{\xi}+\delta)=\sqrt{1-2GE(R)/R}\ ,$ (68)

with $\rho^{\prime}_{+}$ corresponding to the Schwarzschild solution with
total mass $E(R)$. Solving for $E(R)$ one gets

$E(R)=-\frac{4\pi}{3}\epsilon_{V}R^{3}+4\pi\sigma R^{2}\sqrt{1+8\pi
G\epsilon_{V}R^{2}/3}-8\pi^{2}G\sigma^{2}R^{3}\,$ (69)

which shows explicitly the gravitational contributions to the total energy of
the bubble (and reproduces the result given in [47]).

The critical radius in this thin-wall case can then be obtained by solving
$E(R_{c})=0$, with the result

$R_{c}=\frac{R_{c,0}}{1-2\pi G\epsilon_{V}R_{c,0}^{2}/3}\ ,$ (70)

where $R_{c,0}=3\sigma/\epsilon_{V}$ is the critical bubble radius without
gravitational effects, as in Eq. (52). This $R_{c}$ agrees with the original
result of Coleman-De Luccia [43] and shows how gravitational effects force
$R_{c}>R_{c,0}$. This result highlights one important difference with respect
to the flat case: gravity contributes a negative energy density inside the
bubble, where the geometry of space is distorted and for large bubble radius
the enclosed volume grows only like $R^{2}$ rather than $R^{3}$. While energy
conservation requires critical bubbles to always have zero energy, it is not
guaranteed that a critical bubble exists (just making its radius large enough)
so that it becomes possible that the gravitational contribution prevents the
tunneling to happen through critical bubbles [43].

It is straightforward to show that the total energy of the critical bubble
[46] in the general case, without the thin-wall assumption, as given in Eq.
(65), is zero. Simply integrate by parts as explained in footnote 9, paying
attention now to the $\zeta$-dependence of $\rho$ and use the bounce equation
(62) to arrive at a relation between the kinetic and potential contributions,
that ensures $E=0$. The radius defined as the maximum of $T_{\zeta\zeta}$,
$\partial_{\rho}T_{\zeta\zeta}=(1/\rho^{\prime})\partial_{\xi}T_{\zeta\zeta}=0$
is then obtained, after using the bounce equation (62), as

$R_{c}=\left.\rho^{\prime}\frac{3Zh^{\prime}_{B}}{2V^{\prime}}\right|_{R_{c}}\
,$ (71)

which generalizes the result of Eq. (48) including gravity effects. Here

$\left.\rho^{\prime}\right|_{R_{c}}=\sqrt{\left.1+\frac{8\pi
G}{3}R_{c}^{2}\left(\frac{1}{2}Z{h_{B}^{\prime}}^{2}-V\right)\right|_{R_{c}}}\
.$ (72)

We see once again that $R_{c}$ thus defined is indeed $\xi$-independent, as
can be explicitly checked using the $\xi$-derivatives of $Z$, $V$ and $h_{B}$
extracted from the Nielsen identity.111111In the derivation above we have
implicitly assumed that $\rho^{\prime}\neq 0$ , which is correct when the
false vacuum is Minkowski. In the case of a dS false vacuum (like the SM EW
vacuum) one can have $\rho^{\prime}=0$ at some $\zeta$ with important
implications for the bounce properties and the vacuum decay rate. For a
thorough discussion of this point see [15, 46].

Concerning the impact of gravity on $R_{c}$ it would be wrong to identify the
$\rho^{\prime}$ factor as the only effect, as the value of the ratio
$3Zh^{\prime}_{B}/(2V^{\prime})$ has to be evaluated at $R_{c}$, not
$R_{c,0}$, and also because gravity affects the bounce equation (62), thus
modifying $3Zh^{\prime}_{B}/(2V^{\prime})$ as a function of radial distance.

Finally, when gravitational effects make a false dS vacuum stable and there is
no possibility of vacuum decay via nucleation of bubbles of the true phase
there is still the possibility of decay via Hawking-Moss instantons [48]. In
that case the vacuum decay is controlled by the value of the potential at the
maximum separating the two vacua, and this being a $\xi$-independent quantity
so is the Hawking-Moss decay rate. There is no radius associated to this type
of instanton, or more precisely, the transition occurs in a whole Hubble
patch, of size determined by the false vacuum value, $V_{f}>0$, again a
$\xi$-independent quantity.

##  6 Stabilization by Thermal Effects

As is well known, thermal corrections tend to restore broken gauge symmetries
[49, 50]. It is therefore natural to ask at what temperature the deep minimum
of the potential at high field values would be made degenerate with the low-
scale minimum (that at such high $T$ would already be at the origin $h=0$ in
field space) by thermal corrections.121212Usually the thermal change of the
potential, e.g. across a phase transition, is pictured keeping the origin
fixed while the broken minimum is raised as $T$ increases. A more correct
depiction should keep the potential at $h\gg T$ fixed (as thermal corrections
in that field range are Boltzmann suppressed) while the symmetric minimum gets
deeper as $T$ increases. Of course both pictures are directly related by a
field-independent term in the potential, but that term is $T$-dependent and
physical. That critical temperature $T_{c}$ provides yet another physical
scale associated with the potential instability.

The proof that $T_{c}$ is $\xi$-independent requires the generalization of the
Nielsen identity for the thermally corrected effective potential. The fact
that Nielsen identities also hold at finite temperature (suitably modified to
take thermal effects into account) ultimately follows from the fact that the
partition function respects the BRST symmetry so that Ward identities still
hold at finite $T$. The same applies to the Nielsen identity, that can be
regarded as a Ward identity for the effective potential. It is in fact
straightforward to generalize the Nielsen identity (3) to include thermal
corrections (see e.g. [51, 36]). One has

$\xi\frac{\partial V_{T}}{\partial\xi}+C_{T}(h,T,\xi)V_{T}^{\prime}=0\,,$ (73)

where $V_{T}=V(h,T,\xi)$ is the thermally corrected potential and
$C_{T}(h,T,\xi)$ generalizes to finite temperature the function $C(h,\xi)$ of
Eq. (3). Appendix C gives explicit details about the calculation of $V_{T}$
and $C_{T}$ in the SM.

Armed with the Nielsen identity, we can again interpret it as telling how the
explicit $\xi$-dependence of the potential $V_{T}$ is compensated by an
implicit $\xi$-dependence of the field given by

$\xi\frac{dh}{d\xi}=C_{T}(h,T,\xi)\ ,$ (74)

so that changing $\xi$ is equivalent to a field redefinition that does not
change the physics. It is then clear that the value of the critical
temperature, determined by the degeneracy of the potential at its two minima,
is $\xi$-independent: a change of $\xi$ modifies the location of the minima
but not the value of the potential there.

In the case of the critical $T$ for the SM instability, its numerical value
turns out to be many orders of magnitude larger than the instability scale for
the central value of $M_{h}$ and $M_{t}$, that we take to be a typical case.
We find $T_{c}\simeq 10^{29}$ GeV, which is even higher than the Planck scale
and therefore of little interest: at $m_{P}$ one certainly expects
gravitational physics to change the potential in any case. The fact that
$T_{c}$ is so large is related to the fact that we are probing the potential
at the scale associated to the non-standard minimum, and this scale is much
larger than the instability scale (in fact the scale of the minimum is of
order $10^{30}$ GeV (at $T=T_{c}$).

Of course, for values of $M_{h}$ and $M_{t}$ that bring the SM potential
closer to being stable (although such values are experimentally disfavored)
the critical $T_{c}$ could be much smaller. In principle it is even possible
to make $T_{c}$ lower that the instability scale $\Lambda_{I}$ if the non-
standard minimum at high field values is sufficiently shallow. (In fact, when
the value of the potential at the non-EW minimum goes to zero, also
$T_{c}\rightarrow 0$).

An alternative possibility for using a thermal probe of the potential
instability arises from the realization that thermal fluctuations in the early
universe can also trigger the decay of the EW vacuum [52, 10, 53, 54]. This
process is the thermal analogue of the vacuum decay by quantum fluctuations
and also proceeds via nucleation of a critical bubble with the decay rate
$\Gamma$ controlled by an $O(3)$ symmetric bounce solution of energy
$E_{3}(T)$, with $\Gamma\sim e^{-E_{3}(T)/T}$. With the use of the Nielsen
identity at finite $T$ one can then prove that $E_{3}(T)$ is $\xi$-invariant
in a way that parallels the proof for the $T=0$ decay. One can then ask what
is the maximal temperature that our EW vacuum can survive without decaying
(and this can be used to set an upper bound to the reheating temperature, see
[10]). However, for the central values of $M_{h}$ and $M_{t}$ thermal effects
stabilize the potential without destabilizing the EW vacuum and no such limit
can be set [7, 53].

##  7 Probing the Instability Scale with Inflation

Another way to characterize the instability scale of the SM Higgs potential in
a $\xi$-independent way is to consider the SM in a de Sitter background with a
constant Hubble rate $H_{I}$. This de Sitter stage can be the one taking place
at primordial epochs, namely during the inflationary stage (believed to solve
the main problems of standard cosmology), or it can be regarded as a purely
fictitious one with the goal of defining a $\xi$-independent instability
scale.

We assume that the Higgs is minimally coupled to gravity and hence is
effectively massless during inflation. (In a realistic inflationary stage, we
assume no direct coupling of the Higgs to the inflaton field). Under these
circumstances, the Higgs field develops fluctuations with amplitude
proportional to $H_{I}$ [10, 11, 12]. Large amplitude fluctuations of the
Higgs field during the de Sitter phase are dangerous as the SM Higgs can
fluctuate beyond the instability region of the effective potential. The higher
is the Hubble rate, the larger is the probability for this phenomenon to
happen, thus allowing us to define a value of the Hubble rate as a
$\xi$-independent measure of the SM instability scale.

The fluctuations of the Higgs are governed by a Langevin-like equation
obtained by the following procedure. One takes the equation of motion (in an
inflationary background) for the Higgs field and splits the latter in short
and long wavelengths, where the separation length scale is roughly the Hubble
radius. Integrating out the short modes, one obtains a Langevin equation [58,
59] for the long mode field (that we keep calling $h$) in which the effect of
the short modes is to generate a white noise $\eta$ sourcing the long mode
fluctuations. More precisely, the time derivative of the original field is
split into $dh/dt-\eta$. Since one is interested in wavelengths larger than
the Hubble radius, one can neglect the gradients of the long modes (and
therefore truncate the starting equation at the lowest order in derivatives)
obtaining the Langevin equation:

$\hbox{\sc
Langevin}[h]\equiv\sqrt{Z}\left(\frac{dh}{dt}-\eta\right)+\frac{1}{3H_{I}\sqrt{Z}}V^{\prime}=0\
,$ (75)

where $Z=Z(h)$ is the function that multiplies the kinetic term of $h$ in the
effective action expanded in derivatives, as in Eq (2). To arrive at Eq. (75),
terms which are second order in derivatives, like $(dh/dt)^{2}$ or
$d^{2}h/dt^{2}$, are neglected. The two-point correlation function for the
noise term satisfies

$\langle\eta(t)\eta(t^{\prime})\rangle=\frac{H_{I}^{3}}{4\pi^{2}Z}\delta(t-t^{\prime})\
.$ (76)

Since we previously showed that the equation of motion for the Higgs field is
$\xi$-independent and the Langevin equation stems from the equation of motion
by a simple splitting of modes, the Langevin equation turns out to be
$\xi$-independent:

$\xi\frac{d}{d\xi}\hbox{\sc Langevin}[h]=0\ .$ (77)

This can be checked explicitly using the derivative expansion of the Nielsen
identity (see Appendix B), paying attention to the fact that we also have to
split in long and short modes the equation, $\xi dh/d\xi=C(h)$, that controls
the implicit $\xi$-dependence of the solutions of the equations of motion.
This splitting leads to131313Note also that the result for $\xi d\eta/d\xi$ in
(78) is consistent with the correlator in (76) and the $\xi$-dependence of $Z$
to the same order of approximation (see Appendix B).

$\xi\frac{dh_{L}}{d\xi}=C(h_{L})\
,\quad\quad\xi\frac{d\eta}{d\xi}=C^{\prime}(h_{L})\eta\ .$ (78)

In addition, the $\xi$-derivative of the Langevin equation leads to terms
(proportional to the $D$ and $\tilde{D}$ functions of the derivative expansion
in Appendix B) that contain two powers of potential field derivatives. Such
terms are of order higher than the linear order kept for the Langevin
equation, and can be neglected.

The first equation in (78) implies [13] that, if we have a solution
$h_{L}(\xi)$ of the Langevin equation for a given value of $\xi$, we
automatically obtain a solution for $\xi+d\xi$ by the shift
$h_{L}(\xi)+C[h_{L}(\xi)]d\xi/\xi$. This is true also because the other
parameter entering the Langevin equation, namely the Hubble rate $H_{I}$, is
$\xi$-independent. Indeed, $H_{I}$ is determined by Einstein’s equations and
the $00$-component of the energy momentum tensor, both of which are
$\xi$-independent objects.

Figure 8: Illustration of the $\xi$-dependence of the fluctuating Higgs field
during an inflationary period with $N=60$ efolds and $H_{I}=5\times 10^{10}$
GeV and $\xi=0$ and $200$. The histograms correspond to $10^{4}$ runs of the
Langevin equation and the probability distribution function shown (dashed
lines) is a Gaussian approximation of width $\sqrt{N}H_{I}/(2\pi\sqrt{Z})$,
which describes well the numerical result of the Langevin runs. For
comparison, the corresponding Higgs potentials are shown by the solid lines.
The integrated probability between the potential maxima, $\sim 0.6$ in this
example, is $\xi$-independent.

Alternatively, instead of using many times the stochastic Langevin equation to
sample the behaviour of the Higgs field, one can define a probability density
function ${\cal P}(h,t)$, so that the probability of finding at a given time
$t$ the Higgs field in the interval $(h,h+dh)$ is ${\cal P}(h,t)dh$. The
function ${\cal P}(h,t)$ is obtained as a solution to the Fokker-Planck
equation

$\hbox{\sc FokkerPlanck}[{\cal
P}(h,t)]\equiv\frac{1}{\sqrt{Z}}\frac{\partial}{\partial
h}\left\\{\frac{1}{\sqrt{Z}}\left[\frac{\partial}{\partial
h}\left(\frac{H_{I}^{3}}{8\pi^{2}}\frac{\cal
P}{\sqrt{Z}}\right)+\frac{1}{3H_{I}}\frac{{\cal
P}V^{\prime}}{\sqrt{Z}}\right]\right\\}-\frac{1}{\sqrt{Z}}\frac{\partial{\cal
P}}{\partial t}=0\ ,$ (79)

which follows directly141414To obtain (79) from (75), one must use
Stratonovich’s prescription rather than Itô’s, see [60] for details. from the
Langevin equation (75). As this equation describes the same physics as the
Langevin equation, it is also $\xi$-invariant:

$\frac{\partial}{\partial\xi}\hbox{\sc FokkerPlanck}[{\cal P}(h,t)]=0\ ,$ (80)

where we use only a partial derivative because in the Fokker-Planck equation
$h$ is a dummy variable, without any implicit dependence on $\xi$. To show
this $\xi$ independence explicitly (and the previous remarks for the Langevin
equation also apply here) one needs to know how ${\cal P}(h,t)$ changes when
$\xi\rightarrow\xi+d\xi$. This can be calculated as follows: for an arbitrary
function $F(h)$ we can define its average as

$\langle F(h)\rangle=\int F(h^{\prime}){\cal P}(h^{\prime},t,\xi)dh^{\prime}\
.$ (81)

To find how this average depends on $\xi$ remember that the Langevin solutions
change as $h_{L}\rightarrow h_{L}+C(h_{L})d\log\xi$ and therefore, for
$\xi+d\xi$ the equation above reads

$\langle F(h)\rangle+\langle F^{\prime}(h)C(h)\rangle d\log\xi=\int
F(h^{\prime})\left[{\cal P}(h^{\prime},t,\xi)+\xi\frac{\partial{\cal
P}(h^{\prime},t,\xi)}{\partial\xi}d\log\xi\right]dh^{\prime}\ ,$ (82)

from which, after integration by parts in the LHS integral we get151515The
same result can be obtained noting that the probability ${\cal P}(h,\xi)dh$
must equal the probability ${\cal
P}(h+C(h)d\log\xi,\xi+d\xi)d(h+C(h)d\log\xi)$.

$\xi\frac{\partial{\cal P}}{\partial\xi}=-\frac{\partial}{\partial h}[{\cal
P}C]\ .$ (83)

We see that, contrary to what happens with the effective potential, ${\cal P}$
is not $\xi$-independent. However, the integrated probability, defined as

$P(h,t)=\int^{h}_{-\infty}{\cal P}(h^{\prime},t)dh^{\prime}\ ,$ (84)

is $\xi$-independent, provided the field interval of integration is changed
according to the usual rule $\xi dh/d\xi=C(h)$. This implies, in particular,
that the probability of finding the Higgs field after a given time within the
interval $(-h_{\rm max},h_{\rm max})$, where $h_{\rm max}$ is the field value
corresponding to the potential maximum of the barrier separating the EW vacuum
from the instability region) is $\xi$-independent [13], even though the value
of $h_{\rm max}$ itself depends on $\xi$. Figure 8 shows the Higgs potential
(solid lines) for two values of $\xi=0,200$ and the Higgs probability
distribution (dashed lines), after $N=60$ efolds with $H_{I}=5\times 10^{10}$
GeV with the Higgs field starting at the origin. Both the potential and the
Higgs probability distribution change with $\xi$ but we have checked that the
integrated probability in $(-h_{\rm max},h_{\rm max})$ is $\xi$-independent
with $\sim 9\%$ precision. Figure 9 illustrates this $\xi$-independence of the
integrated probability.

Figure 9: Integrated probability for the Higgs to remain within the potential
maxima, $\int_{-h_{max(\xi)}}^{h_{max}(\xi)}P(h,\xi)dh$ for a period of $N=60$
e-folds of inflation, as a function of the Hubble rate $H_{I}$ (left plot) and
as a function of $\xi$ for several values of $H_{I}$ (right plot). In the left
plot, the error bars show the residual gauge-fixing uncertainty when varying
$\xi=(0,200)$.

This result allows to define a $\xi$-independent measure of the instability
scale based on the Hubble rate of inflation, $H_{I}$. One can, for instance,
determine the value of $H_{I}$ for which the integrated probability between
two Higgs values is larger (or smaller) than some critical number within a
given amount of time. One can choose for instance the critical probability to
be larger than $e^{-3N}$, ($N$ being the number of $e$-folds), so that it is
likely to find the Higgs outside the safe interval $(-h_{\rm max},h_{\rm
max})$ in any of the $\sim e^{3N}$ causally independent regions that are
formed during the de Sitter stage. One gets [13] that, for $N=60$ and $h_{\rm
max}\simeq 5\times 10^{10}$ GeV (the value calculated in Landau gauge for the
central values of $M_{h}$ and $M_{t}$), this happens if $H_{I}\geq 0.2\times
10^{10}$ GeV, a scale that is quite close to the naive instability scale
calculated in Landau gauge161616 Note that the bound is quite insensitive to
the choice $e^{-3N}$ of the critical probability. If one uses instead an
${\cal O}(1)$ number the bound changes by a factor $\sim\sqrt{3N}\sim{\cal
O}(10)$ to $H_{I}\geq 2\times 10^{10}$ GeV. Using $e^{-3N}$ corresponds to the
most conservative bound, and ${\cal O}(1)$ to the less conservative one. A
more precise evaluation would require a detailed knowledge about the spatial
distribution of the Higgs fluctuations with $|h|>h_{\rm max}$ at the end of
inflation. We stress that this uncertainty is unrelated to the gauge
dependence. The bound is gauge fixing independent for any choice of the
critical probability..

We close this section with a final remark on the use of the Fokker-Planck
equation in this context. In order to use this equation to describe the
fluctuations of the Higgs field during inflation in the SM unstable potential,
it is necessary to use suitable boundary conditions to describe the
probability leakage towards the instability region. In practice this is done
by identifying the point $h_{*}$ beyond which the fluctuating field rolls down
to the instability and never moves back towards the stable region and setting
${\cal P}(h_{*})=0$ there. The original discussion of this problem [10] used
the barrier maximum, $h_{*}=h_{\rm max}$, for this purpose while a more
correct analysis [12] identified $h_{*}$ with the field value beyond $h_{\rm
max}$ at which quantum fluctuations uphill [controlled by the term $\propto
H_{I}^{3}$ in (79)] can no longer beat the classical rolling downhill
[controlled by the term $\propto V^{\prime}$ in (79)]. This analysis was
further validated by using the Langevin approach in [13]. The field value
$h_{*}$ corresponds to the point at which the two terms in the Fokker-Planck
equation compensate each other giving $\partial{\cal P}/\partial t=0$ and can
be roughly estimated as the value for which the quantum jumps $(\Delta
h)_{q}\sim H_{I}/(2\pi\sqrt{Z})$ in a time interval $\Delta t\sim 1/H_{I}$
equal the classical displacement $(\Delta h)_{cl}\sim
V^{\prime}/(3H_{I}^{2}Z)$, leading to the condition

$\frac{V^{\prime}}{Z}\simeq\frac{3H_{I}^{3}}{2\pi}\ ,$ (85)

to determine $h_{*}$. As one would expect, $h_{*}$ is a gauge-dependent
quantity but it can be shown, using the leading $\xi$-dependence of $V$ and
$Z$ that the condition (85) is $\xi$-independent if $h_{*}$ is transformed in
the usual way, with $\xi dh_{*}/d\xi=C(h_{*})$. Formally one can also check
that the Fokker-Planck equation at $h_{*}$, using further that ${\cal
P}(h_{*})=0$, and $\partial{\cal P}/\partial t=0$, leads to the condition

$\frac{H_{I}^{3}}{8\pi^{2}}\left[{\cal
P}^{\prime\prime}-\frac{3Z^{\prime}{\cal
P}^{\prime}}{2Z}\right]=-\frac{V^{\prime}{\cal P}^{\prime}}{3H_{I}}\ .$ (86)

which can also be shown to be $\xi$-invariant, if one further uses the
$\xi$-dependence of ${\cal P}$ obtained in (83).

##  8 Conclusions

It has been known since a long time that the SM might develop an instability
at large values of the Higgs field, possibly signaling the appearance of new
physics in that range of energies, if the Higgs mass turned out to be low
enough. With the discovery of the Higgs at the LHC, and the theoretical
refinements needed for the stability bound calculations, this possibility
turned out to be realized, with the Higgs mass very close (but below) the
value needed for stability.

The naive definition of the instability scale as the value of the Higgs field
at which the Higgs effective potential drops below the value of the
electroweak minimum ($\sim 10^{10}$ GeV in Landau gauge) is not physical as it
depends on the choice of the gauge-fixing and results in an uncertainty of two
orders of magnitude. While such gauge dependence has also been known for a
long time, it has now become more relevant and has attracted some attention in
recent literature.

Current direct and indirect experimental probes do not show any evidence of
new physics beyond the SM and it becomes more and more likely that the latter
might be valid up to energy scales much higher than the TeV, maybe up to the
Planck scale. It is therefore timely to provide gauge-invariant and physical
descriptions of the scales associated to the SM vacuum instability.

In this paper we have proposed several ways to characterize in a physical way
such scales. In particular, we have shown that the mass scale $\Lambda$
required to stabilize the effective potential through non-renormalizable
operators is a gauge-independent quantity. We have shown that for non-
renormalizable operators of high order, the scale $\Lambda$ is close to the
naive instability scale calculated in Landau gauge. We have also demonstrated
the gauge-invariance of three other scales: the inverse of the critical radius
of the critical bubbles that mediate vacuum decay, the critical temperature at
which our electroweak minimum and the one at very large Higgs field values
become degenerate and finally the scale associated to the Higgs instability
during inflation. Being all these scales gauge-invariant, they allow to draw
physical conclusions on the associated physics, for instance on how new
physics influences the EW vacuum lifetime.

Although we have focused on the SM potential instability, which offers the
main motivation for this work, our results are of wider relevance and can be
applied to other models (with gauge degrees of freedom) that feature similar
instabilities and/or several potential vacua.

## Acknowledgments

We thank Kaladi S. Babu, Pepe Barbón, Enrico Bertuzzo, Joan Elias-Miró, Dani
Figueroa, Jeff Fortin, Luca di Luzio, Werner Porod, Alessandro Strumia, Nikos
Tetradis, James Unwin and especially Gian Giudice for very useful discussions.
J.R.E. thanks the CERN TH-Division for hospitality and partial financial
support during several stages of this project. This work has been partly
supported by the ERC grant 669668 – NEO-NAT – ERC-AdG-2014, the Spanish
Ministry MINECO under grants FPA2013-44773-P and FPA2014-55613-P, the Severo
Ochoa excellence program of MINECO (grant SEV-2012-0234) and by the
Generalitat grant 2014-SGR-1450. MG and TK acknowledge partial support by the
Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of
excellence “Origin and Structure of the Universe”. We also acknowledge support
by the German Science Foundation (DFG) within the Collaborative Research
Center (SFB) 676 ‘Particles, Strings and the Early Universe. A.R. is supported
by the Swiss National Science Foundation (SNSF), project Investigating the
Nature of Dark Matter, project number: 200020-159223.

##  Appendix A RGES for Higher-Dimensional Operators

Consider a tree-level Higgs potential of the form

$V_{0}(h)=-\frac{1}{2}m^{2}h^{2}+\frac{\lambda}{4}h^{4}+\sum_{n}\frac{c_{n}h^{n}}{2^{n/2}\Lambda^{n-4}}\,,$
(A.1)

where summation runs over $n=6,8,\dots$. The radiatively corrected potential
is independent of the renormalization scale, as described by the Callan-
Symanzik equation

$\frac{dV}{d\mu}=\left(\mu\frac{\partial}{\partial\mu}+\sum_{i}\beta_{\lambda_{i}}\frac{\partial}{\partial\lambda_{i}}+\gamma
h\frac{\partial}{\partial h}\right)V=0\ ,$ (A.2)

where the $\lambda_{i}$’s represent all couplings and mass parameters entering
the potential (including the gauge-fixing $\xi$ parameters) and $\gamma$ is
the Higgs anomalous dimension ($\gamma\equiv d\log h/d\log\mu$). The one-loop
RG equation for $c_{n}$ can be obtained from this equation via the explicit
$\mu$-dependence of the one-loop effective potential and knowledge of the
anomalous dimension $\gamma$, which at one loop is not affected by the
irrelevant operators. In Fermi gauge we obtain

$\sum_{n}\frac{\partial V_{0}}{\partial
c_{n}}\beta_{c_{n}}=\frac{\kappa}{2}\left[M_{H}^{4}(h)+3M_{G}^{4}(h)\right]+\gamma_{r}hV_{0}^{\prime}(h)\,,$
(A.3)

where the field-dependent Higgs and Golstone masses are given in Eq. (26) and
$\gamma_{r}\equiv\gamma-\kappa(\xi_{B}{g^{\prime}}^{2}+3\xi_{W}g^{2})/4$ in
the Standard Model. Using the one-loop anomalous dimension in Fermi gauge [18]
one finds for the Standard Model

$\gamma_{r}=\kappa\left(3y_{t}^{2}-\frac{3}{4}{g^{\prime}}^{2}-\frac{9}{4}g^{2}\right)\,.$
(A.4)

Note that the $\xi$-dependence cancels in $\gamma_{r}$, such that the beta
functions are gauge-independent, as expected. Inserting (A.4) and the
potential $V_{0}(h)$ in Eq. (A.3) and Taylor expanding in $h$, one gets the
one-loop beta function given in (28) by matching same powers of $h$ in both
sides of the equation (and taking the limit $m^{2}\to 0$).

##  Appendix B Gauge Dependence and Derivative Expansion

In this Appendix we assume that a derivative expansion of the action is
applicable and discuss how the gauge parameter (generically denoted as $\xi$)
enters the action, the equation of motion, and the energy-momentum tensor up
to higher order corrections in the expansion. We give explicit results up to
fourth order in derivatives for the case of a single scalar field $\phi$,
therefore extending previous work done to ${\cal O}(\partial^{2})$ [36, 34].

We start from the expanded effective action

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\left[\frac{1}{2}Z(h)(\partial\phi)^{2}-V(\phi)\right.$ (B.1)
$\displaystyle+\left.\frac{1}{4}Z_{2}(\phi)(\partial\phi)^{4}+\frac{1}{2}Z_{3}(\phi)(\partial\phi)^{2}(\partial^{2}\phi)+\frac{1}{2}Z_{4}(\phi)(\partial^{2}\phi)^{2}+{\cal
O}(\partial^{6})\right]\ .$

There are only three $Z_{i}(\phi)$ functions at fourth order in derivatives as
other possible terms can be eliminated integrating by parts. The
$\xi$-dependence of this action is dictated by the Nielsen identity that we
write as

$\xi\frac{\partial S}{\partial\xi}+\int d^{4}xK[\phi(x)]\frac{\delta
S}{\delta\phi}=0\ .$ (B.2)

The functional $K[\phi(x)]$ can also be expanded as

$K[\phi]=C(\phi)-D(\phi)(\partial\phi)^{2}-\tilde{D}(\phi)(\partial^{2}\phi)+{\cal
O}(\partial^{4})\ ,$ (B.3)

and, for our purposes below, it is enough to keep up to second order in
derivatives only. The explicit form of the equation of motion for $\phi$, is

$\frac{\delta S}{\delta\phi}\equiv{\rm EoM}[\phi]=0\ ,$

with

$\displaystyle{\rm EoM}[\phi]=$
$\displaystyle-V^{\prime}+\frac{1}{2}Z^{\prime}(\partial\phi)^{2}-\partial_{\mu}[Z\partial^{\mu}\phi]+\frac{1}{4}Z_{2}^{\prime}(\partial\phi)^{4}-\partial_{\mu}[Z_{2}\partial^{\mu}\phi(\partial\phi)^{2}]+\frac{1}{2}Z_{3}^{\prime}(\partial^{2}\phi)(\partial\phi)^{2}$
(B.4)
$\displaystyle-\partial_{\mu}[Z_{3}\partial^{\mu}\phi(\partial^{2}\phi)]+\frac{1}{2}\partial^{2}[Z_{3}(\partial\phi)^{2}]+\frac{1}{2}Z_{4}^{\prime}(\partial^{2}\phi)^{2}+\partial^{2}[Z_{4}(\partial^{2}\phi)]+{\cal
O}(\partial^{6})\ ,$

where primes denote field derivatives, $V^{\prime}=\partial V/\partial\phi$,
etc. This shows explicitly that, evaluated on solutions of the EoM,
$V^{\prime}$ counts as being ${\cal O}(\partial^{2})$.

Plugging all the previous expansions in the Nielsen identity, which is
satisfied for generic $\phi(x)$, one can derive the following $\xi$-dependence
of the functions appearing in the action (B.1):

$\displaystyle\xi\frac{\partial V}{\partial\xi}$ $\displaystyle=$
$\displaystyle-CV^{\prime}\ ,$ $\displaystyle\xi\frac{\partial
Z}{\partial\xi}$ $\displaystyle=$ $\displaystyle-
CZ^{\prime}-2C^{\prime}Z-2DV^{\prime}+2(\tilde{D}V^{\prime})^{\prime}\ ,$
$\displaystyle\xi\frac{\partial Z_{2}}{\partial\xi}$ $\displaystyle=$
$\displaystyle-
CZ_{2}^{\prime}-4C^{\prime}Z_{2}-2C^{\prime\prime}Z_{3}-2DZ^{\prime}+...\ ,$
$\displaystyle\xi\frac{\partial Z_{3}}{\partial\xi}$ $\displaystyle=$
$\displaystyle-
CZ_{3}^{\prime}-3C^{\prime}Z_{3}-2C^{\prime\prime}Z_{4}-2DZ^{\prime}-\tilde{D}Z^{\prime}+...\
,$ $\displaystyle\xi\frac{\partial Z_{4}}{\partial\xi}$ $\displaystyle=$
$\displaystyle-CZ_{4}^{\prime}-2C^{\prime}Z_{4}-2\tilde{D}Z^{\prime}+...$
(B.5)

The terms neglected in the last three equations involve $V^{\prime}$ and
fourth-derivative terms in the expansion (B.3) of $K[\phi]$: they contribute
only at ${\cal O}(\partial^{6})$ in our discussions below.

Under a change $\xi\rightarrow\xi+d\xi$, all the functions in the action
change according to the equations (B.5) and therefore the EoM in (B.4) is also
modified. If $\bar{\phi}(\xi)$ is a solution of the original EoM, then a
solution of the EoM for $\xi\rightarrow\xi+d\xi$ is
$\bar{\phi}(\xi+d\xi)=\bar{\phi}(\xi)+d\bar{\phi}$ with

$\xi\frac{d\bar{\phi}}{d\xi}=K[\bar{\phi}(x)]\ .$ (B.6)

This fact is obvious from the invariance of the action itself, as the
dependence above gives $\xi dS/d\xi=0$, and is one clear example of the power
of using an action to discuss the symmetries in a physical problem. The
explicit check using the EoM expanded to fourth order in the derivative
expansion, as given in Eq. (B.4), is more involved, but straightforward using
(B.5) and (B.6). In performing this check, one needs to evaluate the
$\xi$-dependence of field-derivatives of the functions $V,Z$ and $Z_{i}$. This
can be done simply by taking field-derivatives of Eqs. (B.5), which are
identities that hold for generic $\phi(x)$. For instance, one gets

$\xi\frac{\partial
V^{\prime}}{\partial\xi}=-C^{\prime}V^{\prime}-CV^{\prime\prime}\ ,$ (B.7)

and so on. In this way one is able to eliminate all $\xi$ derivatives. In
order to complete the check one must also get rid of field derivatives of the
potential. The first derivative, $V^{\prime}$, is eliminated by using the EoM
(B.4). Higher order derivatives like $V^{\prime\prime}$ and
$V^{\prime\prime\prime}$ can then be eliminated by taking spacetime
derivatives of the EoM (which holds at every spacetime point). In this way one
can get rid of the combinations
$\partial_{\mu}V^{\prime}=V^{\prime\prime}\partial_{\mu}\bar{\phi}$ and
$\partial^{2}V^{\prime}=V^{\prime\prime}\partial^{2}\bar{\phi}+V^{\prime\prime\prime}(\partial\bar{\phi})^{2}$,
etc. After carrying through this program one gets

$\frac{d}{d\xi}{\rm EoM}[\bar{\phi}]=0\ ,$ (B.8)

to order ${\cal O}(\partial^{6})$. This completes the proof presented in [13]
to ${\cal O}(\partial^{4})$ with one important improvement: we have shown now
that no assumption about the subleading role of the functions $D(\phi)$ and
$\tilde{D}(\phi)$ is needed. Note also that, to order ${\cal
O}(\partial^{4})$, it is enough to use $\xi d\bar{\phi}/d\xi=C(\bar{\phi})$.

One can also check explicitly, in the derivative expansion approximation, that
the energy-momentum tensor for solutions of the EoM is $\xi$-independent.
Consider an action of the form

$S=\int d^{4}x\sqrt{-g}{\cal
L}(\phi,g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi,\Box\phi)\,,$ (B.9)

where

$\Box\phi=D_{\mu}D^{\mu}\phi=\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}\partial^{\mu}\phi)=\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi)\,,$
(B.10)

is the covariant d’Alembert operator applied to a scalar field. We are
ultimately interested in a flat background, but keep the metric in order to
derive the energy momentum tensor. The equation of motion will only be needed
on a Minkowski background, where it reads

${\cal L}_{1}-\partial^{\mu}\left(2{\cal
L}_{2}\partial_{\mu}\phi\right)+\Box{\cal L}_{3}=0\,,$ (B.11)

where we used the short-hand notation ${\cal L}_{i}=\partial{\cal
L}(X_{1},X_{2},X_{3})/\partial X_{i}$. To derive the energy-momentum tensor we
use the standard relations

$\displaystyle\frac{\partial}{\partial g_{\rho\sigma}(y)}g^{\mu\nu}(x)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}(g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho})\delta(x-y)\,,$
(B.12) $\displaystyle\frac{\partial}{\partial
g_{\rho\sigma}(y)}(\sqrt{-g})^{n}$ $\displaystyle=$
$\displaystyle\frac{n}{2}(\sqrt{-g})^{n}g^{\rho\sigma}\delta(x-y)\,.$ (B.13)

This gives, on a Minkowski background,

$\frac{\partial}{\partial
g_{\rho\sigma}(y)}\Box\phi(x)=-2\delta(x-y)\partial^{\rho}\partial^{\sigma}\phi-\partial^{\rho}\delta(x-y)\partial^{\sigma}\phi-\partial^{\sigma}\delta(x-y)\partial^{\rho}\phi+g^{\rho\sigma}\partial_{\eta}\delta(x-y)\partial^{\eta}\phi\;.$
(B.14)

Using these relations one obtains

$T_{\mu\nu}[\phi]=-\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta
g_{\mu\nu}}=2\partial_{\mu}\phi\partial_{\nu}\phi\,{\cal
L}_{2}-\partial_{\mu}{\cal L}_{3}\partial_{\nu}\phi-\partial_{\nu}{\cal
L}_{3}\partial_{\mu}\phi-g_{\mu\nu}\left[{\cal L}-\partial_{\eta}({\cal
L}_{3}\partial^{\eta}\phi)\right]\;.$ (B.15)

Using the equation of motion one can check that $\partial^{\mu}T_{\mu\nu}=0$.
For the action (B.1), one gets

$\displaystyle T^{\mu\nu}$
$\displaystyle=g^{\mu\nu}\left[V-\frac{Z}{2}(\partial\phi)^{2}-\frac{Z_{2}}{4}(\partial\phi)^{4}+Z_{3}\partial_{\rho}\phi\
\partial_{\sigma}\phi\ \partial^{\rho\sigma}\varphi+Z_{4}\partial_{\rho}\phi\
\partial^{\rho}\partial^{2}\phi+Z^{\prime}_{4}(\partial^{2}\phi)(\partial\phi)^{2}\right.$
$\displaystyle\left.+\frac{Z_{4}}{2}(\partial^{2}\phi)^{2}+\frac{Z^{\prime}_{3}}{2}(\partial\phi)^{4}\right]-Z_{3}\left(\partial^{\mu}\phi\
\partial^{\nu\rho}\phi+\partial^{\nu}\phi\
\partial^{\mu\rho}\phi\right)\partial_{\rho}\phi-Z_{4}(\partial^{\mu}\phi\
\partial^{\nu}+\partial^{\nu}\phi\ \partial^{\mu})\partial^{2}\phi$
$\displaystyle-Z_{4}(\partial^{\mu}\phi\ \partial^{\nu}+\partial^{\nu}\phi\
\partial^{\mu})\partial^{2}\phi+\partial^{\mu}\phi\
\partial^{\nu}\phi\left[Z+(Z_{2}-Z_{3}^{\prime})(\partial\phi)^{2}+(Z_{3}-2Z^{\prime}_{4})(\partial^{2}\phi)\right]+{\cal
O}(\partial^{6}).$ (B.16)

Using the same relations discussed above regarding the EoM, and in particular,
the $\xi$-dependence of the solutions of the EoM given by (B.6), one is able
to prove that

$\frac{d}{d\xi}T^{\mu\nu}[\bar{\phi}]=0\ ,$ (B.17)

to order ${\cal O}(\partial^{6})$, so that the energy and momentum densities
are $\xi$-independent quantities.

The discussion above has been performed for the Minkowski action and EoM but,
for applications to tunneling rates and bounce solutions one uses instead the
Euclidean action and EoM. However, the good $\xi$-independence properties
carry over to the Euclidean case. To see this explicitly, note first that in
the Minkowskian proofs of $\xi$ independence presented above we never had to
deal with the exact form of the metric. However, the Euclidean action is
obtained by the replacement $t\rightarrow-it_{E}$, under which
$(\partial\phi)^{2}\rightarrow-(\partial_{E}\phi)^{2}$,
$(\partial^{2}\phi)\rightarrow-\partial_{E}^{2}\phi$. These sign flips could
be assigned to a parity transformation under which

$Z\rightarrow-Z,\quad Z_{i}\rightarrow Z_{i},\quad V\rightarrow V,\quad
C\rightarrow C,\quad D\rightarrow-D,\quad\tilde{D}\rightarrow-\tilde{D}\ ,$
(B.18)

with $i=2,3,4$. It is immediate to see that Eqs. (B.5), (B.8) and (B.17) are
invariant under this parity transformation so that the $\xi$-invariance they
express also holds in the Euclidean case.

##  Appendix C Nielsen Identity at Finite Temperature

The Nielsen identity describing the gauge dependence of the effective
potential extended to finite temperature takes the form given in Eq. (73).
Here we show the explicit forms of the effective potential $V_{T}$ and the
$C_{T}$ function at finite $T$ for the SM in Fermi gauge.

The calculation of $V_{T}$ is by now standard. At one-loop order one gets

$V_{T}(h)=V_{0}(h)+V_{1}(h,T)\ .$ (C.1)

Here $V_{0}(h)$ is the tree-level potential of Eq. (16). The one-loop term
$V_{1}(h,T)$ includes both the $T=0$ radiative corrections, as explicitly
given in Eq. (17), and the free-gas approximation for the thermal effects and
depends only on the $h$-dependent particle masses $M_{\alpha}$ and their
number of degrees of freedom $N_{\alpha}$ (taken negative for fermions), with
$\alpha$ labelling different particle species, see (18).

Both $T=0$ and finite $T$ contributions to the one-loop potential
$V_{1}(h,T)=V_{1}(h,0)+\Delta_{T}V_{1}(h,T)$, arise in a compact way in the
imaginary-time formalism expression

$V_{1}(h,T)=\sum_{\alpha}N_{\alpha}J_{\alpha}(M^{2}_{\alpha},T)\ .$ (C.2)

where the functions $J_{\alpha}$, defined as

$J_{\alpha}(M^{2}_{\alpha},T)\equiv\frac{1}{2}\sum_{K}\mspace{-25.0mu}\int\;\log(K^{2}+M_{\alpha}^{2})\
,$ (C.3)

are the finite-$T$ generalization of the $J^{0}_{\alpha}(M^{2}_{\alpha})$
functions introduced in Eq. (17). Inside the logarithm $K^{2}$ stands for the
Euclidean momentum, after $k_{0}\rightarrow ik_{0}$, while the integral-sum
stands for

$\sum_{K}\mspace{-25.0mu}\int\;\equiv
T\sum_{n}\int\frac{d^{3-2\epsilon}k}{(2\pi)^{3-2\epsilon}}\ ,$ (C.4)

with $\epsilon$ used for dimensional regularization. For nonzero $T$, $k_{0}$
takes discrete values: $k_{0}=2\pi nT$ for bosons and $k_{0}=(2n+1)\pi T$ for
fermions, so that the $k_{0}$ integral is replaced by a sum with $n$ running
over all integers. Note that for $T\rightarrow 0$ we have
$T\sum_{n}\rightarrow\int dk_{0}/(2\pi)$ and one recovers the $T=0$ result
(after Wick rotation), so that in fact the function (C.3) contains the finite
temperature piece on top of the $T=0$ one in a single formula. The thermal
correction is

$\Delta_{T}V_{1}(h,T)=\sum_{\alpha}N_{\alpha}T\int
d^{3}k\log\left[1+s_{\alpha}e^{-\sqrt{k^{2}+M_{\alpha}^{2}}}\right]\ ,$ (C.5)

with $s_{\alpha}=-$ ($+$) for bosons (fermions). When $T\gg M_{\alpha}$, a
high-$T$ expansion for the potential gives

$\displaystyle V_{1}(h,T)$ $\displaystyle=$ $\displaystyle\sum_{\
\alpha\,{\rm(B)}\
}N_{\alpha}\left[-\frac{\pi^{2}T^{4}}{90}+\frac{1}{24}M_{\alpha}^{2}T^{2}-\frac{1}{12\pi}M_{\alpha}^{3}T+\frac{M_{\alpha}^{4}}{64\pi^{2}}\log\frac{c_{B}T^{2}}{\mu^{2}}+{\cal
O}\left(\frac{M_{\alpha}^{6}}{T^{2}}\right)\right]$ (C.6) $\displaystyle+$
$\displaystyle\sum_{\ \alpha\,{\rm(F)}\
}N_{\alpha}\left[\frac{7\pi^{2}T^{4}}{720}-\frac{1}{48}M_{\alpha}^{2}T^{2}+\frac{M_{\alpha}^{4}}{64\pi^{2}}\log\frac{c_{F}T^{2}}{\mu^{2}}+{\cal
O}\left(\frac{M_{\alpha}^{6}}{T^{2}}\right)\right]\ ,$

where bosonic and fermionic contributions are indicated separately by (B) and
(F) respectively and $c_{B}=3/2+2\log(4\pi)-2\gamma_{E}$,
$c_{F}=3/2+2\log\pi-2\gamma_{E}$ with $\gamma_{E}\simeq 0.577$ the Euler-
Mascheroni constant. In many cases of interest (like in the study of the
electroweak phase transition) a resummation of finite $T$ IR divergent terms
(from $n=0$ bosonic modes) is necessary. The main effect, after resumming the
so-called ring diagrams is to replace the mass in the cubic term of the
bosonic expansion above by a thermally screened one,
$M_{\alpha}^{2}\rightarrow M_{\alpha}^{2}+{\cal O}(T^{2})$, see e.g. [61] for
details.

The expressions for $C_{T}$ in the SM (for Fermi gauge), generalize the $T=0$
ones given in Eqs. (14,15). Going to momentum space and using the imaginary
time formalism as before we arrive at the thermally corrected expressions (at
one loop),

$\displaystyle C^{(1)}_{B,T}(h)$ $\displaystyle=$
$\displaystyle-\frac{g^{\prime}}{2}\sum_{K}\mspace{-25.0mu}\int\;P_{c}(-K)P_{\chi^{0},\mu}(K)K^{\mu}=-\frac{\xi_{B}}{8}{g^{\prime}}^{2}h\sum_{K}\mspace{-25.0mu}\int\;\frac{1}{(K^{2}+M^{2}_{B_{+}})(K^{2}+M^{2}_{B_{-}})}\
,$ $\displaystyle C^{(1)}_{W,T}(h)$ $\displaystyle=$
$\displaystyle-\frac{g}{2}\sum_{K}\mspace{-25.0mu}\int\;P_{c_{a}}(-K)P_{\chi^{a},\mu}(K)K^{\mu}$
$\displaystyle=$
$\displaystyle-\frac{\xi_{W}}{8}{g}^{2}h\sum_{K}\mspace{-25.0mu}\int\;\left[\frac{1}{(K^{2}+M^{2}_{B_{+}})(K^{2}+M^{2}_{B_{-}})}+\frac{2}{(K^{2}+M^{2}_{A_{+}})(K^{2}+M^{2}_{A_{-}})}\right]\
,$

where the intermediate expressions are written in terms of the ghost
propagators and the Goldstone-gauge mixed ones and $M^{2}_{A_{\pm}}$,
$M^{2}_{B_{\pm}}$ are given in Eq. (18).

It is convenient to rewrite the previous expressions in terms of the functions
$I_{\alpha}(M_{\alpha}^{2},T)$ defined by

$I_{\alpha}(M_{\alpha}^{2},T)\equiv 2\frac{\partial}{\partial
M_{\alpha}^{2}}J_{\alpha}(M_{\alpha}^{2},T)=\sum_{K}\mspace{-25.0mu}\int\;\frac{1}{K^{2}+M_{\alpha}^{2}}\
.$ (C.8)

Then we get

$\displaystyle C^{(1)}_{B,T}(h)$ $\displaystyle=$
$\displaystyle\frac{1}{8}\xi_{B}{g^{\prime}}^{2}h\left[\frac{I_{B_{+}}-I_{B_{-}}}{M^{2}_{B_{+}}-M^{2}_{B_{-}}}\right]\
,$ $\displaystyle C^{(1)}_{W,T}(h)$ $\displaystyle=$
$\displaystyle\frac{1}{8}\xi_{W}{g}^{2}h\left[\frac{I_{B_{+}}-I_{B_{-}}}{M^{2}_{B_{+}}-M^{2}_{B_{-}}}+2\frac{I_{A_{+}}-I_{A_{-}}}{M^{2}_{A_{+}}-M^{2}_{A_{-}}}\right]\
,$ (C.9)

a straightforward generalization of the $T=0$ results of Eqs. (20,21). The
high-$T$ expansions of these expressions are

$\displaystyle C^{(1)}_{B,T}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{8}\xi_{B}{g^{\prime}}^{2}h\left[\log\frac{\mu^{2}}{c_{B}T^{2}}+\frac{3}{2}+\frac{4\pi
T}{M_{B+}+M_{B_{-}}}+{\cal
O}\left(\frac{M^{2}_{B_{\pm}}}{T^{2}}\right)\right]\ ,$ (C.10) $\displaystyle
C^{(1)}_{W,T}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{8}\xi_{W}g^{2}h\left[2\log\frac{\mu^{2}}{c_{B}T^{2}}+3+\frac{4\pi
T}{M_{B+}+M_{B_{-}}}+\frac{8\pi T}{M_{A+}+M_{A_{-}}}+{\cal
O}\left(\frac{M^{2}_{B_{\pm}}}{T^{2}},\frac{M^{2}_{A_{\pm}}}{T^{2}}\right)\right]\
,$

which agree with the leading term result presented in [36] (translated to
Fermi gauge).

Finally, noting that

$\displaystyle\xi_{B}\frac{\partial V_{1}(h,T)}{\partial\xi_{B}}$
$\displaystyle=$ $\displaystyle\frac{\partial J_{B_{+}}}{\partial
M^{2}_{B_{+}}}\ \xi_{B}\frac{\partial
M^{2}_{B_{+}}}{\partial\xi_{B}}+(B_{+}\rightarrow B_{-})\ ,$
$\displaystyle\xi_{W}\frac{\partial V_{1}(h,T)}{\partial\xi_{W}}$
$\displaystyle=$ $\displaystyle\frac{\partial J_{B_{+}}}{\partial
M^{2}_{B_{+}}}\ \xi_{W}\frac{\partial
M^{2}_{B_{+}}}{\partial\xi_{W}}+2\frac{\partial J_{A_{+}}}{\partial
M^{2}_{A_{+}}}\ \xi_{W}\frac{\partial
M^{2}_{A_{+}}}{\partial\xi_{W}}+(B_{+}\rightarrow B_{-},A_{+}\rightarrow
A_{-})\ ,$

and using the expressions for $M^{2}_{B_{\pm}}$ and $M^{2}_{A_{\pm}}$ given in
(18) plus the relation (C.8) and $\partial V_{0}/\partial h=M_{G}^{2}h$, it is
straightforward to check the one-loop Nielsen identities

$\displaystyle\xi_{B}\frac{\partial
V_{1}(h,T)}{\partial\xi_{B}}+C^{(1)}_{B,T}\frac{\partial V_{0}(h)}{\partial
h}=0\ ,$ (C.11) $\displaystyle\xi_{W}\frac{\partial
V_{1}(h,T)}{\partial\xi_{W}}+C^{(1)}_{W,T}\frac{\partial V_{0}(h)}{\partial
h}=0\ .$ (C.12)

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