Integrating out beyond tree level
and relativistic superfluids

Austin Joyce,a Alberto Nicolis,b Alessandro Podo,b and Luca Santonic

aKavli Institute for Cosmological Physics, Department of Astronomy and
Astrophysics,
The University of Chicago, Chicago, IL 60637, USA,

bCenter for Theoretical Physics and Department of Physics,
Columbia University, New York, NY 10027, USA

cICTP, International Centre for Theoretical Physics,
Strada Costiera 11, 34151, Trieste, Italy

###### Abstract

We revisit certain subtleties of renormalization that arise when one derives a
low-energy effective action by integrating out the heavy fields of a more
complete theory. Usually these subtleties are circumvented by matching some
physical observables, such as scattering amplitudes, but a more involved
procedure is required if one is interested in deriving the effective theory to
all orders in the light fields (but still to fixed order in the derivative
expansion). As a concrete example, we study the $U(1)$ Goldstone low-energy
effective theory that describes the spontaneously broken phase of a $\phi^{4}$
theory for a complex scalar. Working to lowest order in the derivative
expansion, but to all orders in the Goldstones, we integrate out the radial
mode at one loop and express the low-energy effective action in terms of the
renormalized couplings of the UV completion. This yields the one-loop equation
of state for the superfluid phase of (complex) $\phi^{4}$. We perform the same
analysis for a renormalizable scalar $SO(N)$ theory at finite chemical
potential, integrating out the gapped Goldstones as well, and confirm that the
effective theory for the gapless Goldstone exhibits no obvious sign of the
original $SO(N)$ symmetry.

##  1 Introduction

Within the path-integral formulation of quantum field theory, a low-energy
effective field theory can be defined through the procedure of integrating
out. Concretely, if one is given a fundamental theory involving heavy fields
$h(x)$ and light fields $\ell(x)$ which is described by an action
$S[h,\ell\,]$, the low-energy effective action $S_{\rm eff}[\ell\,]$—for the
light fields only—is formally defined by integrating out the heavy fields:

${\rm e}^{iS_{\rm eff}[\ell\,]}\equiv\int\\!Dh\,{\rm e}^{iS[h,\ell\,]}\;.$ (1)

To the extent to which path-integral manipulations make sense, this definition
is concrete and predictive. In particular, one can compute correlation
functions of the light fields by performing a path integral over them weighted
by the exponential of the effective action:

$\langle\ell(x_{1})\dots\ell(x_{n})\rangle=\frac{\displaystyle{\int}\\!D\ell
Dh\,{\rm
e}^{iS[h,\ell\,]}\,\ell(x_{1})\cdots\ell(x_{n})}{\displaystyle{\int}\\!D\ell
Dh\,{\rm e}^{iS[h,\ell\,]}}=\frac{\displaystyle{\int}\\!D\ell\,{\rm
e}^{iS_{\rm
eff}[\ell\,]}\,\ell(x_{1})\cdots\ell(x_{n})}{\displaystyle{\int}\\!D\ell\,{\rm
e}^{iS_{\rm eff}[\ell\,]}}\,.$ (2)

In fact, at this level there is no low-energy expansion whatsoever. The only
restriction is that, not surprisingly, one uses the effective action for
$\ell$ to compute correlation functions of $\ell$ only, but not of $h$. Put
another way: as long as one is only interested in correlation functions of
$\ell$, one can perform the path-integral over $h$ once and for all, which is
what eq. (1) does.

Importantly, all this continues to be true beyond tree level, where one starts
encountering ultraviolet (UV) divergences. Provided that the fundamental
theory is regulated, and the necessary counterterms to renormalize it are
included in $S[h,\ell\,]$, eq. (1) yields the action $S_{\rm eff}[\ell\,]$ for
the regulated effective theory, including the necessary counterterms to
renormalize it. This is a consequence of eq. (2); as far as correlation
functions of the light fields go, the effective theory is just an exact
rewriting of the full theory.

The power of effective field theory lies in the derivative expansion,
truncated at some finite order. From the bottom-up perspective, this is a
reflection of several dearly held principles: locality, perturbativity,
predictivity, et cetera. From the top-down perspective of eq. (1), if one
actually wants to derive the effective action from the fundamental theory, a
pragmatic motivation for the derivative expansion is that, in general, one
cannot perform the path integral in (1) for arbitrary field configurations
$\ell(x)$, even working perturbatively in the couplings of the fundamental
action $S[h,\ell\,]$.

However, at this point a problem arises: the derivative expansion completely
changes the high-energy behavior of the theory, and thus the structure of UV
divergences. As a result, the counterterms necessary to renormalize the
effective theory correctly—so that it reproduces the same results as the full
theory, when appropriate—are not those that come out of (1). In the simplest
terms, the problem has to do with expanding the propagators of the heavy
fields in momentum, which is one way to think about the derivative expansion:

$\frac{i}{q^{2}-M^{2}+i\epsilon}\quad\xrightarrow{q^{2}\ll
M^{2}}\quad-\frac{i}{M^{2}}\left[1+\frac{q^{2}}{M^{2}}+\frac{q^{4}}{M^{4}}+\dots\right]\,.$
(3)

The left-hand side decays at large $q$, while on the right-hand side the first
term is a constant, and all the others increase at large $q$, faster and
faster as we push the derivative expansion further. Clearly, loop integrals
will have completely different UV properties depending on whether or not the
propagator is expanded [1]. (For an explicit example of this at work, see
[2].)

So, if we work with the effective theory to some finite order in the
derivative expansion, it seems that eq. (1) has no predictive power beyond
tree-level. To renormalize the effective theory in a manner consistent with
the full theory—so that the two agree for low-energy observables—we have to
introduce new counterterms, because those that come out of (1) are not the
correct ones. But then, we face the following difficulty: what comes out of
the right-hand side of (1), once expanded in derivatives, is a local
functional of the light fields compatible with the symmetries of the theory,
whereas the new counterterms we have to introduce are the most general local
terms compatible with the symmetries. Functionally, this makes eq. (1)
useless.

This is one of the reasons why, beyond tree level, eq. (1) is usually bypassed
altogether. Instead of using (1), one parametrizes the most general effective
theory compatible with the symmetries to the desired order in the derivative
expansion, computes as many low-energy observables as needed to fix all the
parameters in the effective theory, and matches them to the same observables
computed in the full theory, thereby fixing the effective theory’s parameters
in terms of those of the full theory. This procedure is called “matching.”
This works well in practice for a finite number of parameters, which is
usually all we need in particle physics when we are after, say, a low-energy
amplitude with a fixed number of external legs at some finite order in the
loop expansion. In that case, we are only interested in the effective theory
to finite order in both the derivative expansion and in the number of fields.

However, there are other cases—most notably a relativistic field theory at
finite density—where we might be interested in an effective theory to fixed
order in the derivative expansion, but to all orders in the number of fields.
What are we to do then? Should we match infinitely many amplitudes? Moreover,
it is a pity that such a simple and beautiful definition as eq. (1) loses its
operational meaning beyond tree level, becoming essentially a cartoon of the
whole story. Can one salvage it, perhaps by modifying it?

Georgi [1] has proposed a formal matching procedure, whose definition does not
require expanding in the number of fields. The basic idea is to use the
quantum effective action for the light fields, $\Gamma[\ell\,]$. All
correlation functions of the light fields can be computed directly using
$\Gamma[\ell\,]$ at tree level, and so there are no more subtleties regarding
UV divergences, even upon expanding in derivatives. Thus, the all-orders
matching requirement is simply that the effective action $S_{\rm eff}[\ell\,]$
be such that the quantum effective action for $\ell$ computed in the effective
theory is the same as that computed in the full theory:

$\Gamma_{\rm EFT}[\ell\,]=\Gamma_{\rm full}[\ell\,]\equiv\Gamma[\ell\,]\;.$
(4)

At the path integral level, this corresponds to modifying (1) to read

$\bigg{(}\int D\delta\ell\,{\rm e}^{iS_{\rm
eff}[\ell+\delta\ell]}\bigg{)}_{\rm 1PI}=\bigg{(}\int Dh\,D\delta\ell\,{\rm
e}^{iS[h,\,\ell+\delta\ell]}\bigg{)}_{\rm 1PI\;for\;\delta\ell}\;.$ (5)

On the left-hand side, the path integral is restricted to one-particle-
irreducible (1PI) diagrams, and simply computes the quantum effective action
in the effective theory. On the right-hand side, the path integral is
restricted to diagrams that are 1PI for the fluctuations of $\ell$, while $h$
is integrated out completely. This computes the quantum effective action for
$\ell$ only, but in the full theory.111A somewhat different alternative to
matching has been proposed in [3].

This definition of $S_{\rm eff}[\ell\,]$ is somewhat implicit. One should
still parametrize $S_{\rm eff}[\ell\,]$ as the most general local effective
action compatible with the symmetries. Then, if one is able to compute the
quantum effective action both in the effective theory and in the full theory,
one can fix the free parameters in $S_{\rm eff}[\ell\,]$. To the best of our
knowledge, this procedure has not been attempted in practice. Our aim in this
paper is to carry it out at one loop in two simple cases, which are relevant
for relativistic field theories at finite density—in particular, for
relativistic superfluids.

Of course, in general we are not able to compute quantum effective actions for
generic field configurations, even stopping at one loop order. However, the
procedure is systematic within the derivative expansion. In particular, to
zeroth order in derivatives, it amounts to computing the Coleman–Weinberg
potential [4], which is quite easy to do with functional methods [5]. To
higher orders, the computation in general becomes substantially more involved
[6, 7], but such complications will not matter for us. In the two examples
that we will study below, by applying essentially the same technique as used
for the Coleman–Weinberg potential, we will be able to carry out the
computation to all orders in a Goldstone field, and to the order of one
derivative per field in the derivative expansion.

Notation and conventions: We use natural units and the mostly minus signature
for the metric throughout. We will overuse the symbol $\mu$: as a Lorentz
index, as a chemical potential, and as a renormalization scale (though
typically we will denote the renormalization scale as $\bar{\mu}$ in
$\overline{\rm MS}$). Unfortunately, in all of these cases $\mu$ is the
conventional symbol of choice, hopefully the meaning will be clear from the
context.

##  2 Weakly coupled UV completions for relativistic superfluids

From a quantum field theory (QFT) perspective, a superfluid can be defined as
a system with a spontaneously broken internal $U(1)$ symmetry, in a state of
finite density (or finite chemical potential) for the associated charge [8,
9]. Its low-energy effective theory involves a derivatively coupled Goldstone
field $\pi(x)$, the superfluid phonon, on which the original $U(1)$ symmetry
acts as a shift symmetry,

$\pi\mapsto\pi+{\rm const}.$ (6)

The symmetry breaking pattern is conveniently parametrized in terms of a
spacetime scalar $\phi(x)$ with a time-dependent expectation value. The phonon
$\pi(x)$ describes fluctuations about this expectation value [10]:

$\phi(x)=\mu t+\pi(x)\;,$ (7)

where $\mu$ is the chemical potential. Notice that $\pi(x)$ nonlinearly
realizes not only the internal $U(1)$ symmetry, but also time translations and
Lorentz boosts.222That a single Goldstone field is responsible for nonlinearly
realizing more than one spacetime symmetry is a manifestation of the so-called
inverse Higgs effect [11]. (Though a diagonal combination of time translations
and internal shifts is linearly realized.)

To lowest order in derivatives, the low-energy effective action is simply

$S_{\rm eff}=\int{\rm d}^{4}x\,P(X)\;,\qquad X\equiv(\partial\phi)^{2}\;,$ (8)

where $P$ is a generic function associated with the equation of state. It is
the pressure as a function of the (squared) chemical potential. (We refer the
reader to refs. [8, 9, 10] for further details about this framework.)

The simplest weakly coupled UV completion of the superfluid effective theory
is the theory of a complex scalar $\Phi(x)$ with a (linearly realized) $U(1)$
symmetry (see also [12, 13]),

$S=\int{\rm
d}^{4}x\Big{(}|\partial\Phi|^{2}-m^{2}|\Phi|^{2}-\lambda|\Phi|^{4}\Big{)}\;,$
(9)

where $\phi(x)$ arises as the phase of the complex field:
$\Phi(x)={\textstyle\frac{1}{\sqrt{2}}}\,\rho(x)\,{\rm e}^{i\phi(x)}$.

If $m^{2}$ is positive, the origin is stable and the $U(1)$ symmetry is not
spontaneously broken in the absence of a chemical potential, $\mu$, for the
$U(1)$ charge. In fact, there is a minimum nonzero value of $\mu$ below which
the symmetry is unbroken and the ground state charge density is strictly zero.
At tree level, that value is simply $m$. For $\mu>m$, the origin becomes
unstable, the $U(1)$ symmetry gets broken, and at the same time the system
acquires a charge density.

On the other hand, if $m^{2}$ is negative the origin is unstable already at
$\mu=0$. The $U(1)$ symmetry is thus always spontaneously broken, and any
nonzero value of $\mu$, no matter how small, yields a nonzero charge density.

In what follows we will treat both of these cases simultaneously. In each
case, once the symmetry is broken, the only light degree of freedom is the
Goldstone, the radial mode can be integrated out, and the low-energy dynamics
are described by an action of the form (8).

A slightly more general UV completion is the theory of $N$ real scalars
$\Psi_{a}(x)$, $a=1,\cdots,N$, with an internal $SO(N)$ symmetry,

$S=\int{\rm
d}^{4}x\left(\frac{1}{2}\big{\lvert}\partial\vec{\Psi}\big{\rvert}^{2}-\frac{m^{2}}{2}\big{\lvert}\vec{\Psi}\big{\rvert}^{2}-\frac{\lambda}{4}\big{|}\vec{\Psi}\big{|}^{4}\right)\;,$
(10)

of which eq. (9) is just the $N=2$ version, with
$\Phi={\textstyle\frac{1}{\sqrt{2}}}(\Psi_{1}+i\Psi_{2})$. In order to get a
superfluid state, we should switch on a chemical potential for one of the
$SO(N)$ charges. Let’s call $J_{ab}$ the $SO(N)$ charge that generates
rotations in the $(ab)$-plane and, without loss of generality, consider
turning on a chemical potential for the charge associated to the symmetry
generator $J_{12}$.

As in the $U(1)$ case, spontaneous symmetry breaking (SSB) will occur for
$\mu>m$ if $m^{2}$ is positive, and for any value of $\mu$ if $m^{2}$ is
negative. Once the symmetry is broken, we have a gapless Goldstone associated
with $J_{12}$ and a number of gapped Goldstones associated with the generators
that do not commute with $J_{12}$ [14]. In particular, for each pair of
generators that transform as a doublet under $J_{12}$ we get one gapped
Goldstone with gap $\mu$. Such pairs are simply $(J_{1b},J_{2b})$ with $b>2$,
and there are $N-2$ of them.

Overall there is one gapless Goldstone, $N-2$ gapped ones, and one radial
mode, making up $N$ degrees of freedom—as many as there are in the original
multiplet $\vec{\Psi}$. At low energies we can integrate out all of the them
apart from the gapless Goldstone, and we end up once again with the general
framework described at the beginning of this section. It is still an open
question whether in a situation like this there is any low-energy remnant of
the full $SO(N)$ symmetry once everything but the gapless Goldstone has been
integrated out [15].

In the next section we will perform this integrating-out procedure at one-loop
order, as outlined in the Introduction. As emphasized there, the procedure is
only predictive if it can be carried out without introducing new counter-terms
beyond those needed to renormalize the UV completions. So, for future
reference, we report here the one-loop counter-terms for the theories (9) and
(10), which can be computed via standard relativistic QFT methods (see e.g.,
[16]). Working in dimensional regularization (DR) and modified minimal
subtraction ($\overline{\rm MS}$), they are

$\frac{\delta
m^{2}}{\bar{m}^{2}}=-2(N+2)\,\frac{\bar{\lambda}}{16\pi^{2}}\frac{1}{d-4}\;,\qquad\frac{\delta\lambda}{\bar{\lambda}}=-2(N+8)\,\frac{\bar{\lambda}}{16\pi^{2}}\frac{1}{d-4}\;,$
(11)

where the barred parameters are the renormalized $\overline{\rm MS}$ ones, and
the bare parameters appearing in the Lagrangian are $m^{2}=\bar{m}^{2}+\delta
m^{2}$ and $\lambda=\bar{\lambda}+\delta\lambda$. The case $N=2$ corresponds
to the $U(1)$ theory, eq. (9), and—as usual for $\phi^{4}$ theories—there is
no wave-function renormalization at one loop. Moreover, we are ignoring the
renormalization of the cosmological constant, which plays no role in the
absence of gravity.

The $\overline{\rm MS}$ scheme is convenient for computations, but the
renormalized couplings it involves are not particularly physical. In
particular, they depend on the value of the (arbitrary) renormalization scale
$\bar{\mu}$. As a nontrivial check of our final result, we will rewrite the
quantum effective action in terms of physical couplings only, in which case
the $\bar{\mu}$ dependence will drop out. To this end, we will need the
relationships between $\overline{\rm MS}$ couplings and more physical ones.
For the theories at hand they are

$\bar{m}^{2}=m^{2}_{\rm
pole}+2(N+2)\dfrac{\lambda}{16\pi^{2}}m^{2}\left(\log\dfrac{\bar{\mu}}{m}+\dfrac{1}{2}\right)\;,\qquad\bar{\lambda}=\lambda_{\rm
thr}+2(N+8)\dfrac{\lambda^{2}}{16\pi^{2}}\left(\log\dfrac{\bar{\mu}}{m}+\dfrac{1}{3}\right)\;,$
(12)

where $m^{2}_{\rm pole}$ is the pole mass in the unbroken phase at vanishing
chemical potential (in the case that this is stable), and $\lambda_{\rm thr}$
is the amplitude for two-to-two identical-particle elastic scattering at
threshold,

$i{\cal M}_{2\to 2}(s=4m^{2},t=0,u=0)\equiv-6i\,\lambda_{\rm thr}\;.$ (13)

On the other hand, keeping the $\overline{\rm MS}$ couplings around along with
the associated $\bar{\mu}$ dependence will allow us to RG-improve our result
at large chemical potential.

##  3 Integrating out at one loop

We now want to carry out the integrating out and matching procedure outlined
in the introduction. The goal is to start from a finite (renormalized)
ultraviolet theory and derive the low energy EFT by integrating out the heavy
modes at one loop.

###  3.1 The $U(1)$ theory

Let’s start with the $U(1)$ case, eq. (9). The more general $SO(N)$ case will
be an almost trivial generalization. We want to rewrite the path integral in
polar coordinates,

$\Phi(x)=\frac{1}{\sqrt{2}}\rho(x){\rm e}^{i\phi(x)}\;,$ (14)

in terms of which the action reads

$S[\rho,\phi]=\int{\rm
d}^{4}x\left(\frac{1}{2}(\partial\rho)^{2}+\frac{1}{2}\rho^{2}(\partial\phi)^{2}-\frac{m^{2}}{2}\rho^{2}-\frac{\lambda}{4}\rho^{4}\right)\;,$
(15)

and integrate out $\rho$ at one loop. We are assuming that the $U(1)$ symmetry
is spontaneously broken, so that the path integral is peaked comfortably away
from the origin. In this case, we can ignore the subtleties that $\rho$ is
constrained to be positive, and that $\phi$ is an angular variable which is
periodically identified: $\phi=\phi+2\pi$.333See, for instance, ref. [17] for
a discussion of how to deal with topological constraints like these in path
integrals.

First off, notice that from the path integral measure we get a nontrivial UV-
divergent determinant,

$D\Phi D\Phi^{*}=D\rho D\phi\det\rho(x)=D\rho D\phi\,{\rm
e}^{\delta^{4}(0)\int{\rm d}^{4}x\log\rho(x)}\;.$ (16)

The delta function is a pure power-law divergence,

$\delta^{4}(0)=\int\frac{{\rm d}^{4}p}{(2\pi)^{4}}=i\int\frac{{\rm
d}^{4}p_{E}}{(2\pi)^{4}}\equiv iI_{0}\;,$ (17)

which vanishes in dimensional regularization. Eventually we will work in
dimensional regularization, so we can drop this determinant. Interestingly—as
we will see—even if we were to keep it around, to the order we will be
working, all its effects would cancel against other one-loop contributions.
(Like all path-integral determinants, that in eq. (16) is formally of one-loop
order.)

Second, notice that $\phi$ only enters the action (15) in the combination
$X=(\partial\phi)^{2}$. So, when we integrate out $\rho$, the derivative
expansion will not be in terms of derivatives of $\phi$, but in terms of
derivatives of $X$. This means that configurations with slowly varying $X$,
regardless of how large $X$ itself is—that is, regardless of how large the
first derivative of $\phi$ is—can be dealt with to zeroth order in the
derivative expansion. Operationally, this means that we can compute the
effective action for the superfluid state to zeroth order in derivatives of
$X$, but to all orders in $X$, in essentially the same way that one computes a
Coleman–Weinberg potential.

Finally, as mentioned before, it turns out that integrating out a heavy field
beyond tree level is no substitute for responsible EFT matching [1]. More
precisely: if one were to keep all the non-local structure in the low-energy
effective action for the light fields, then integrating out a heavy field
would be a consistent procedure—it would correspond to doing the first step in
the path integral over all the fields that computes correlation functions
involving the light fields only. However, in general one is interested in a
truncated derivative expansion of the low-energy effective theory, and that
changes completely the UV behavior of loop integrals, and thus the structure
of UV divergences. Hence the need for matching observables directly.

Fortunately, instead of computing all correlation functions and matching them
one by one, it is possible to match to all orders in the light fields, but to
any fixed order in the derivative expansion. The idea is to use the quantum
effective action. Following [1], and specializing to our case, the rule is:
compute the one-$\phi$-irreducible quantum effective action in the full
theory, $\Gamma_{\rm full}[\phi]$, at some order in the derivative expansion,
and demand that, at the same order in the derivative expansion, the low-energy
effective action $S_{\rm EFT}[\phi]$ be such that its 1PI quantum effective
action, $\Gamma_{\rm EFT}[\phi]$, matches that computed in the full theory:

$\Gamma_{\rm EFT}[\phi]=\Gamma_{\rm full}[\phi]\equiv\Gamma[\phi]\;.$ (18)

This is a necessary and sufficient condition for all correlation functions of
$\phi$ to match between the full theory and the low-energy effective theory.

To compute the one-$\phi$-irreducible quantum effective action in the full
theory, we separate $\phi$ into a background plus fluctuations,

$\phi(x)\to\phi(x)+\pi(x)\;,$ (19)

and perform the path integral over $\pi$ and the radial mode $\rho$, keeping
only the contributions that are 1PI for $\pi$:

${\rm e}^{i\Gamma_{\rm full}[\phi]}\equiv\bigg{(}\int D\rho D\pi\,{\rm
e}^{iS[\rho,\phi+\pi]}\bigg{)}_{\rm 1PI\;for\;\pi}\,.$ (20)

At tree level, this just amounts to solving the classical equation of motion
for $\rho$ and substituting it back into the action:

$\Gamma^{\rm tree}_{\rm full}[\phi]=S[\rho_{0}[X],\phi]\;,$ (21)

where $\rho_{0}[X]$ is the saddle-point solution for $\rho$, for a given
$X=(\partial\phi)^{2}$, which obeys

$(\Box+m^{2}-X)\rho_{0}+\lambda\rho_{0}^{3}=0\;.$ (22)

When $X$ is a constant (ignoring derivatives of $X$), this has two solutions:

$\rho^{2}_{0}=0\quad\mbox{and}\quad\rho^{2}_{0}=\frac{X-m^{2}}{\lambda}\qquad(X={\rm
const}.)\;.$ (23)

The first corresponds to the unbroken phase, and is inconsistent with our
assumption of having SSB. Thus, we will not consider it further. The second
solution corresponds to SSB, and exists only when

$X>m^{2}\;,$ (24)

which, for positive $X$, is exactly the condition on $\mu$ stated in Section
2.

Substituting this solution back into the action (15), we get the tree-level
$\Gamma[\phi]$ to lowest order in derivatives:

$\Gamma^{\rm tree}_{\rm full}[\phi]=\int{\rm
d}^{4}x\left(\frac{1}{2}\rho_{0}^{2}\,X-\frac{m^{2}}{2}\rho^{2}_{0}-\frac{\lambda}{4}\rho_{0}^{4}\right)=\int{\rm
d}^{4}x\,\frac{\big{(}X-m^{2}\big{)}^{2}}{4\lambda}\;,$ (25)

in agreement with [12, 13].

At one loop, we have to perturb about the saddle point, $\rho=\rho_{0}+h$,
expand the action (15) keeping only the quadratic terms in $h$ and $\pi$,

$S_{2}[h,\pi]=\int{\rm d}^{4}x\left(\frac{1}{2}(\partial
h)^{2}+\frac{1}{2}\rho_{0}^{2}\,(\partial\pi)^{2}+2V^{\mu}\,h\partial_{\mu}\pi-\frac{1}{2}m^{2}_{\rm
eff}h^{2}\right)\;,$ (26)

where we have defined

$m^{2}_{\rm eff}\equiv m^{2}-X+3\lambda\rho_{0}^{2}\;,\qquad
V_{\mu}\equiv\rho_{0}\partial_{\mu}\phi\,\;.$ (27)

We then integrate over $h$ and $\pi$ to obtain the effective action

${\rm e}^{i\Delta\Gamma^{\rm 1\,loop}_{\rm full}[\phi]}=\int DhD\pi\,{\rm
e}^{iS_{2}[h,\pi]}\;.$ (28)

To be clear about the notation: $\rho_{0}$, $m^{2}_{\rm eff}$, and $V^{\mu}$
are all functionals of the background field $\phi(x)$. In the limit where
$\partial_{\mu}\phi$ is constant, they become local functions of
$\partial_{\mu}\phi$ itself, which then are also constant. In particular:

$m^{2}_{\rm eff}\xrightarrow{X=\,{\rm const.}}2(X-m^{2})\,.$ (29)

From now on, we will take this limit, so that $\rho_{0}$, $m^{2}_{\rm eff}$,
and $V^{\mu}$ are all constant.

To compute the path integral (28), we first change variables for $\pi$ to
canonically normalize it

$\pi=\varphi/\rho_{0}\;,\qquad
D\pi=D\varphi\big{(}\det\rho_{0}(x)\big{)}^{-1}\,.$ (30)

This exactly cancels the determinant we got from the change of variables in
(16), evaluated at the saddle point.444In fact, had we kept that determinant
around, it would have affected the position of the saddle point by an order
$\lambda$ shift. However, had we plugged the new saddle point into the action,
by stationarity of the action at the unperturbed saddle point, the value of
the action would not have changed at order $\lambda$. So, overall, at this
order, all the effects of these two determinants cancel exactly. Next, we
complete the square for $h$:

$\displaystyle\int DhD\varphi\,{\rm e}^{iS_{2}}$ $\displaystyle=\int
Dh^{\prime}D\varphi\,\exp\left[\frac{i}{2}\left(-h^{\prime}(\Box+m^{2}_{\rm
eff})h^{\prime}-\varphi\left[\Box+4\partial_{\mu}\phi\partial_{\nu}\phi\frac{\partial^{\mu}\partial^{\nu}}{\Box+m^{2}_{\rm
eff}}\right]\varphi\right)\right]\,,$ (31)

where we have defined $h^{\prime}\equiv
h-2\partial^{\mu}\phi\frac{1}{\square+m^{2}_{\rm eff}}\partial_{\mu}\varphi$.
Up to background-independent factors, we therefore have

$\displaystyle{\rm e}^{i\Delta\Gamma^{\rm 1\,loop}_{\rm full}[\phi]}$
$\displaystyle=\det\big{(}\Box+m^{2}_{\rm
eff}\big{)}^{-1/2}\times\det\left(\Box+4\partial_{\mu}\phi\partial_{\nu}\phi\,\frac{\partial^{\mu}\partial^{\nu}}{\Box+m^{2}_{\rm
eff}}\right)^{-1/2}$ (32)
$\displaystyle=\det\big{(}\Box\big{)}^{-1/2}\times\det\left(\Box+m^{2}_{\rm
eff}+4\partial_{\mu}\phi\partial_{\nu}\phi\,\frac{\partial^{\mu}\partial^{\nu}}{\Box}\right)^{-1/2}\,.$

The first determinant is background independent, so we can drop it. Using
standard functional methods [5], the second determinant yields

$\displaystyle i\Delta\Gamma^{\rm 1\,loop}_{\rm
full}[\phi]=-\frac{1}{2}\int{\rm d}^{4}x\int\frac{{\rm
d}^{4}p}{(2\pi)^{4}}\log{\left(p^{2}-m^{2}_{\rm
eff}-4\partial_{\mu}\phi\partial_{\nu}\phi\,\frac{p^{\mu}p^{\nu}}{p^{2}}\right)}\;,$
(33)

where the argument of the log is made dimensionless by an arbitrary scale,
whose value only affects the cosmological constant.

The anisotropic $\partial\phi\partial\phi$ term inside the log makes computing
the momentum integral slightly complicated. We describe the calculation in
Appendix A. Working for simplicity in dimensional regularization, as $d\to 4$
the result is

$\displaystyle\Delta\Gamma^{\rm 1\,loop}_{\rm full}[\phi]=$
$\displaystyle\int{\rm d}^{4}x\left(-\frac{1}{32\pi^{2}(d-4)}\big{(}m_{\rm
eff}^{4}+2m_{\rm eff}^{2}X+2X^{2}\big{)}\right.$ (34) $\displaystyle\hskip
11.38092pt+\frac{1}{384\pi^{2}}\bigg{[}9m_{\rm eff}^{4}+18m_{\rm
eff}^{2}X+10X^{2}-6(m_{\rm eff}^{4}+2m_{\rm eff}^{2}X+2X^{2})\log\left(m_{\rm
eff}^{2}/\bar{\mu}^{2}\right)$ $\displaystyle\hskip
62.59596pt\left.-10\frac{X^{3}}{m_{\rm eff}^{2}}\,f\big{(}-{4X}/{m_{\rm
eff}^{2}}\big{)}\bigg{]}\right)\;,$

where $m^{2}_{\rm eff}$ depends on $X$ as in eq. (29), $\bar{\mu}$ is the
$\overline{\rm MS}$ renormalization scale, and $f$ is a generalized
hypergeometric function, which, if needed, can be rewritten in terms of
rational functions, square roots, and logarithms:

$\displaystyle f(z)$
$\displaystyle\equiv\,_{3}F_{2}\left[\left.\begin{array}[]{c}1,~{}~{}1,~{}~{}{\textstyle\frac{7}{2}}\\\
4,~{}~{}5\end{array}\right\lvert z\right]$ (35)
$\displaystyle=\frac{16}{5}\left[\frac{(3z^{2}-10z-8)(1-\sqrt{1-z}\,)}{z^{4}}+\frac{z^{2}+4}{z^{3}}-3\,\frac{z^{2}-4z+8}{z^{3}}\log{\bigg{(}\frac{1+\sqrt{1-z}}{2}}\,\bigg{)}\right]\;.$

In (34), we have collected all the divergent terms in the first line. We will
discuss renormalization of this effective action in Sec. 3.3.

###  3.2 The $SO(N)$ theory

We now consider the $SO(N)$ version of the story, described by the action eq.
(10). It is immediate to generalize the computation we just performed to this
case. In fact, if one resists the temptation to express the path integral in
spherical coordinates, and instead uses the higher dimensional version of
cylindrical coordinates, the computation is almost the same as in the $U(1)$
case, with one minor addition.

Consider rewriting the field multiplet $\vec{\Psi}$ in polar coordinates in
the (12)-plane, but in cartesian coordinates for the other components:

$\vec{\Psi}(x)=\left(\begin{array}[]{c}\rho(x)\,\cos\phi(x)\\\
\rho(x)\,\sin\phi(x)\\\ \sigma_{1}(x)\\\ \vdots\\\
\sigma_{N-2}(x)\end{array}\right)\,.$ (36)

Like in the $U(1)$ case, the path integral measure changes as

$D\vec{\Psi}\equiv D\Psi_{1}\dots D\Psi_{N}=D\rho\,D\phi\,\det\rho(x)\times
D\vec{\sigma}\,\;,$ (37)

where $\vec{\sigma}$ is the $(N-2)$-dimensional vector
$(\sigma_{1},\dots,\sigma_{N-2})$, and the action becomes

$S[\rho,\phi,\vec{\sigma}\,]=\int{\rm
d}^{4}x\left(\frac{1}{2}(\partial\rho)^{2}+\frac{1}{2}\rho^{2}(\partial\phi)^{2}+\frac{1}{2}(\partial\vec{\sigma})^{2}-\frac{m^{2}}{2}\big{(}\rho^{2}+|\vec{\sigma}|^{2}\big{)}-\frac{\lambda}{4}\big{(}\rho^{2}+|\vec{\sigma}|^{2}\big{)}^{2}\right)\;.$
(38)

Notice that $\vec{\sigma}$ does not couple directly to $\phi$. This is the
main advantage of using this parametrization of the field space.

The change in measure can be neglected for exactly the same reason as
before—it vanishes in dimensional regularization, and more generally, it
cancels against another determinant that we will encounter along the way. To
compute the quantum effective action for $\phi$, we therefore have to evaluate
the path integral

${\rm e}^{i\Gamma_{\rm full}[\phi]}\equiv\left(\int
D\rho\,D\pi\,D\vec{\sigma}\,{\rm
e}^{iS[\rho,\,\phi+\pi,\,\vec{\sigma}\,]}\right)_{\rm 1PI\;for\;\pi}\,.$ (39)

Working again to zeroth order in derivatives of $X=(\partial\phi)^{2}$, the
saddle point for $\rho$ is exactly the same as before, eq. (23), and for
$\vec{\sigma}$ it is just $\vec{\sigma}=0$. This means that the tree-level
$\Gamma[\phi]$ is identical to before, eq. (25).

We perturb the action around this saddle point as $\rho=\rho_{0}+h$ and
$\phi=\phi+\pi$ and expand up to quadratic order. After doing this, the action
becomes

$S_{2}[h,\pi,\vec{\sigma}\,]=\int{\rm d}^{4}x\left(\frac{1}{2}(\partial
h)^{2}+\frac{1}{2}\rho_{0}^{2}\,(\partial\pi)^{2}+2V^{\mu}\,h\partial_{\mu}\pi-\frac{1}{2}m^{2}_{\rm
eff}h^{2}+\frac{1}{2}(\partial\vec{\sigma})^{2}-\frac{1}{2}X|\vec{\sigma}|^{2}\right)\;.$
(40)

with $V^{\mu}$ and $m_{\rm eff}^{2}$ defined in the same way as in eq. (27).
It is now clear that, at this order, we have two decoupled sectors:

  * •

The $\boldsymbol{\pi}$-$\boldsymbol{h}$ sector: This sector is given by the
first four terms of (40) and has precisely the same action as in the $U(1)$
case, eq. (26). Correspondingly, the path integral over these fields yields
exactly the same result as in the $U(1)$ case, eq. (34).

  * •

The $\boldsymbol{\vec{\sigma}}$ sector: This sector is given by the last two
terms of (40) and is made up of $N-2$ free massive scalars with mass $X$—the
gapped Goldstones—in agreement with the general results of [9], which we
summarized briefly in Sec. 2. Up to irrelevant field-independent factors, the
path integral over this sector yields the determinant

$\det(\Box+X)^{-\frac{N-2}{2}}=\exp\left(-\frac{N-2}{2}\int{\rm
d}^{4}x\int\frac{{\rm
d}^{4}p}{(2\pi)^{4}}\log{\big{(}p^{2}-X\big{)}}\right)\;,$ (41)

which corresponds to one of the momentum integrals we have already computed
for the $U(1)$ case (see (89) in Appendix A). The contribution from the log is
then

$\displaystyle\int\frac{{\rm
d}^{4}p_{E}}{(2\pi)^{4}}\log{\big{(}p_{E}^{2}+X\big{)}}=\frac{X^{2}}{16\pi^{2}}\left(\frac{1}{d-4}-\frac{3}{4}+\frac{1}{2}\log(X/\bar{\mu}^{2})\right)\;.$
(42)

Putting everything together, the quantum effective action for $\phi$ in the
$SO(N)$ case simply reads

$\Gamma_{\rm full}^{SO(N)}[\phi]=\Gamma_{\rm full}^{U(1)}[\phi]-(N-2)\int{\rm
d}^{4}x\,\frac{X^{2}}{32\pi^{2}}\left(\frac{1}{d-4}-\frac{3}{4}+\frac{1}{2}\log(X/\bar{\mu}^{2})\right)\;,$
(43)

where $\Gamma_{\rm full}^{U(1)}[\phi]$ is the sum of the tree level effective
action, eq. (25), and of the one-loop correction in the $U(1)$ case, eq. (34).

###  3.3 Renormalization

We now come to the issue of renormalization. We will consider directly the
more general $SO(N)$ case. The $U(1)$ case can be obtained by simply setting
$N=2$.

The divergent terms in (43)—including those that come from (34)—read

$\displaystyle\Gamma_{\rm full}^{SO(N)}[\phi]$
$\displaystyle\supset-\frac{1}{32\pi^{2}(d-4)}\big{(}m_{\rm eff}^{4}+2m_{\rm
eff}^{2}X+NX^{2}\big{)}\,,$ (44) $\displaystyle\xrightarrow{m_{\rm
eff}^{2}\to\,2(X-m^{2})}\frac{1}{32\pi^{2}(d-4)}\big{(}12m^{2}X-(N+8)X^{2}\big{)}\;,$

where in the last step we dropped an $X$-independent term, since it is
associated with the cosmological constant, which we haven’t been keeping track
of.

In the $\overline{\rm MS}$ scheme, one would normally just drop these
divergent terms. However, as emphasized in the Introduction, we do not want to
introduce new counterterms besides those already needed to renormalize the UV
completion, eq. (10), which is defined in terms of two parameters only,
$m^{2}$ and $\lambda$.555In principle, one should also consider wave-function
renormalization, but in $\phi^{4}$ theories this starts mattering only at two-
loop order. So, it had better be the case that renormalizing $m^{2}$ and
$\lambda$ as in eq. (11) completely cancels the divergent terms in (44).

To check this, rewrite the bare couplings in the tree-level effective action
(25) as

$m^{2}=\bar{m}^{2}+\delta
m^{2}\;,\qquad\qquad\lambda=\bar{\lambda}+\delta\lambda\;,$ (45)

where $\bar{m}^{2}$ and $\bar{\lambda}$ are the $\overline{\rm MS}$
renormalized couplings (the relevant counterterms are given by (11)). Then,
expand eq. (25) to first order in $\delta m^{2}$ and $\delta\lambda$. It is a
matter of simple algebra to check that the only effect of this procedure is to
cancel the divergent terms (44), apart again from a cosmological constant-
related, $X$-independent term.

In conclusion, the quantum effective action for $\phi$ to one-loop order and
to zeroth order in derivatives of $X$ is simply eq. (43), with the divergent
terms removed and with all the couplings (included the $m^{2}$ inside
$m^{2}_{\rm eff}$) now interpreted as renormalized $\overline{\rm MS}$
couplings. Admittedly, its explicit form—which we display in the next
subsection—is not particularly illuminating. Before discussing different
limits and special cases to extract some physical insights, we briefly address
the issue of matching to the low-energy effective theory—an issue which, at
this order, happens to be trivial in dimensional regularization.

###  3.4 Matching

The matching prescription suggested by [1], which we briefly reviewed in the
Introduction, instructs us to engineer a low-energy effective action for
$\phi$ that—at one loop and to zeroth-order in derivatives of $X$—yields the
same quantum effective action as the one we derived above starting from the UV
completion.

Let’s start then with the most general lowest-order effective action for
$\phi$, eq. (8), and define the one-loop quantum effective action associated
with it:

${\rm e}^{i\Gamma[\phi]}=\bigg{(}\int D\pi\,{\rm e}^{iS_{\rm
eff}[\phi+\pi]}\bigg{)}_{\rm 1PI}={\rm e}^{iS_{\rm eff}[\phi]}\int D\pi\,{\rm
e}^{iS_{2}[\pi]}\;,$ (46)

where $S_{2}$ is the quadratic action for the fluctuations of $\phi$:

$S_{2}[\pi]=\int{\rm
d}^{4}x\,\frac{1}{2}Z^{\mu\nu}\,\partial_{\mu}\pi\partial_{\nu}\pi\;\;,\quad{\rm
with}\quad Z^{\mu\nu}\equiv
2P^{\prime}(X)\eta^{\mu\nu}+4P^{\prime\prime}(X)\partial^{\mu}\phi\partial^{\nu}\phi\;.$
(47)

We thus get (up to field-independent factors)

$\displaystyle\int D\pi\,{\rm e}^{iS_{2}[\pi]}$
$\displaystyle=\det(Z^{\mu\nu}\partial_{\mu}\partial_{\nu})^{-1/2}$ (48)
$\displaystyle=\exp\left(-\frac{1}{2}\int{\rm d}^{4}x\int{\rm
d}^{4}p\,\log\big{(}Z^{\mu\nu}p_{\mu}p_{\nu}\big{)}\right)\;,$ (49)

where in the last step we have used that, for constant $\partial_{\mu}\phi$,
the matrix $Z^{\mu\nu}$ is constant.

The momentum integral above vanishes in dimensional regularization, simply
because it’s a scale-less integral. Explicitly, denoting the integrand by
$I(p)$ and using standard properties of dimensional regularization [18], we
have that a rescaling of the integration variable yields

$\int{\rm d}^{d}p\,I(p)=\alpha^{d}\int{\rm d}^{d}p\,I(\alpha
p)=\alpha^{d}\int{\rm
d}^{d}p\,\big{(}I(p)+2\log\alpha\big{)}=\alpha^{d}\int{\rm d}^{d}p\,I(p)\;,$
(50)

which implies that the integral of $I(p)$ must vanish.

This implies that, if one uses dimensional regularization, at one-loop order
and to zeroth order in derivatives of $X$, the quantum effective action is the
same as the tree-level low-energy action:666This is also consistent with the
analyses of [19, 20, 21] which show that, in dimensional regularization, the
first loop corrections scale schematically as ${\cal L}\sim\partial^{4}X^{n}$.

$\Gamma[\phi]=S_{\rm eff}[\phi]\;\qquad\qquad\mbox{(in dim. reg.)}\;.$ (51)

Reading this equation backwards, we see that, to this order, the low-energy
effective action we should use for $\phi$ is simply the $\Gamma[\phi]$ we have
computed in the previous subsections.

In conclusion, integrating out at one loop all the heavy fields in the $SO(N)$
theory (10) yields the Goldstone effective theory

$\displaystyle P(X)$
$\displaystyle=\frac{\big{(}X-m^{2}\big{)}^{2}}{4\lambda}$
$\displaystyle\hskip 14.22636pt+\,\frac{1}{384\pi^{2}}\bigg{[}-8X^{2}+(m_{\rm
eff}^{4}+2m_{\rm eff}^{2}X+2X^{2})\big{(}9-6\log(m_{\rm
eff}^{2}/\bar{\mu}^{2})\big{)}-10\frac{X^{3}}{m_{\rm
eff}^{2}}\,f\left(-\frac{4X}{m_{\rm eff}^{2}}\right)$ $\displaystyle\hskip
71.13188pt+(N-2)X^{2}\big{(}9-6\log(X/\bar{\mu}^{2})\big{)}\bigg{]}\;,$ (52)

where $m^{2}_{\rm eff}=2(X-m^{2})$, $f$ is given in eq. (35), and $m^{2}$ and
$\lambda$ are to be interpreted as the renormalized $\overline{\rm MS}$
couplings of theory (10), evaluated at $\bar{\mu}$. The first line is the
tree-level contribution. The second line is the one-loop correction in the
$U(1)$ case. The third line is the only term that, at one-loop, differentiates
the $SO(N)$ case from the $U(1)$ one.

##  4 Some physics

The final result for the Goldstone effective action, eq. (52), is not
particularly illuminating in and of itself. We can however explore some
interesting physical questions and different regimes using it. These
explorations will also serve as consistency checks.

###  4.1 $P(X)$ in terms of physical parameters

For constant $X=\mu^{2}$, the function $P(X)$ is associated with the zero-
temperature equation of state of the relativistic superfluid, and gives the
pressure as a function of the chemical potential. Since this is an observable
quantity, the apparent dependence on the renormalization scale $\bar{\mu}$
must disappear once we express the final result (52) in terms of physical
quantities.

We present the result using physical parameters in the unbroken phase at zero
chemical potential, which are meaningful for theories with positive $m^{2}$.
We do so by trading the (unphysical) $\overline{\rm MS}$ mass and coupling
parameters for the pole mass and the two-to-two scattering amplitude at
threshold, as in equation (12).

After straightforward manipulations, we obtain:

$\displaystyle P(X)=$ $\displaystyle\frac{\big{(}X-m^{2}_{\rm
pole}\big{)}^{2}}{4\lambda_{\rm
thr}}+\,\frac{1}{384\pi^{2}}\Bigg{[}-8(6X^{2}-4m^{2}_{\rm pole}X-m^{4}_{\rm
pole})$ (53) $\displaystyle\hskip 35.56593pt+\,2(5X^{2}-6m^{2}_{\rm
pole}X+2m^{4}_{\rm pole})\big{(}9-6\log(m_{\rm eff}^{2}/m^{2}_{\rm
pole})\big{)}-10\frac{X^{3}}{m_{\rm eff}^{2}}\,f\left(-\frac{4X}{m_{\rm
eff}^{2}}\right)$ $\displaystyle\hskip 35.56593pt-(N-2)\Big{(}\,4(X-m^{2}_{\rm
pole})(X+2m^{2}_{\rm pole})-X^{2}\left[9-6\log(X/m^{2}_{\rm
pole})\right]\Big{)}\Bigg{]}\;,$

where we have discarded a constant term that can be reabsorbed into the
cosmological constant.777From now on, all the expressions for $P(X)$ are
understood to be given up to a cosmological constant term. As expected, all
the $\bar{\mu}$ dependent terms of equation (52) cancel in a nontrivial way,
confirming the consistency of our result.

###  4.2 SSB for positive $m^{2}$

As reviewed in Sec. 2, our theories exhibit SSB at the classical level either
for negative $m^{2}$, or, if $m^{2}$ is positive, when the chemical potential
exceeds $m$. This matches the fact that for relativistic free bosons at zero
temperature, Bose–Einstein condensation (BEC) happens precisely at $\mu=m$.

How do these statements get modified at one-loop? In a forthcoming paper, we
will study in detail the relationship between finite charge density and SSB
for scalar theories at the quantum level. For the time being, we content
ourselves with showing that, in cases with positive $m^{2}$, the one-loop
Goldstone effective theory is well-behaved if and only if $X>m^{2}_{\rm
pole}$, where $m_{\rm pole}$ is the one-loop pole mass. Since $X$ can be
thought of as the local value of the (squared) chemical potential—see Sec.
2—and since a healthy Goldstone effective theory can be taken as a sign of
having SSB, this suggests that at the quantum level the criterion for having
SSB in scalar theories is that the chemical potential exceeds the physical
mass of the bosons at hand. Which, although not obvious, is at least
reassuring from an intuitive BEC perspective.888We refer the reader to our
forthcoming paper for further discussions about this point.

To begin with, let’s see how we can recover the fact that we have SSB only for
$X>m^{2}$ from the tree-level result,

$P_{\rm tree}(X)=\frac{(X-m^{2})^{2}}{4\lambda}\;.$ (54)

The idea is to check the stability of a configuration with constant
$X=\mu^{2}$. We separate the Goldstone field into background plus fluctuations
as

$\phi(x)=\mu t+\pi(x)\;,$ (55)

and then expand the effective action to quadratic order in the perturbations
$\pi(x)$. For a generic $P(X)$, we get

$P(X)\supset\left[P^{\prime}(\mu^{2})+2P^{\prime\prime}(\mu^{2})\mu^{2}\right]\dot{\pi}^{2}-P^{\prime}(\mu^{2})(\vec{\nabla}\pi)^{2}\equiv\alpha\,\dot{\pi}^{2}-\beta\,(\vec{\nabla}\pi)^{2}\;.$
(56)

Stability corresponds to having both $\alpha$ and $\beta$ be positive. The
sound speed is $c_{s}^{2}=\beta/\alpha$, and so subluminality corresponds to
$\alpha\geq\beta$. Overall, the combined conditions to have both stability and
subluminality thus read

$P^{\prime}(\mu^{2})>0\quad\mbox{and}\quad P^{\prime\prime}(\mu^{2})\geq 0\;.$
(57)

For the tree level effective action (54), we have

$P_{\rm tree}^{\prime}(\mu^{2})=\frac{(\mu^{2}-m^{2})}{2\lambda}\ ;\qquad
P_{\rm tree}^{\prime\prime}(\mu^{2})=\frac{1}{2\lambda}\;.$ (58)

From this, we see that second condition in (57) ($P^{\prime\prime}>0$) is
always obeyed, but the first condition (that $P^{\prime}>0$) is obeyed only
for $\mu^{2}>m^{2}$, as expected.

Let’s now check what happens at one-loop. Since in perturbation theory the
one-loop effective action is close to the tree-level one, we can restrict to
the neighborhood of $X=m^{2}$, and see what the stability and subluminality
conditions look like in that neighborhood. Expanding eq. (52) to quadratic
order in $(X-m^{2})$, choosing for simplicity the renormalization scale as
$\bar{\mu}=m$, we get

$P_{\rm
1-loop}(X)\simeq\frac{1}{\lambda}\left[\frac{\lambda}{32\pi^{2}}(N+2)m^{2}(X-m^{2})+\frac{1}{4}(X-m^{2})^{2}\left(1+\frac{\lambda}{4\pi^{2}}\right)\right]\;,\qquad(\bar{\mu}^{2}=m^{2})\;.$
(59)

From this we can compute the first and second derivatives:

$\displaystyle P^{\prime}_{\rm 1-loop}(X)$
$\displaystyle\simeq\frac{1}{\lambda}\left[\frac{\lambda}{32\pi^{2}}(N+2)m^{2}+\frac{1}{2}(X-m^{2})\left(1+\frac{\lambda}{4\pi^{2}}\right)\right]\,,$
(60) $\displaystyle P^{\prime\prime}_{\rm 1-loop}(X)$
$\displaystyle\simeq\frac{1}{2\lambda}\left(1+\frac{\lambda}{4\pi^{2}}\right)\;.$
(61)

From this we see that, once again, the second condition in (57) is always
obeyed, but now $P^{\prime}>0$ is obeyed only for (to first order in
$\lambda$)

$X>m^{2}\left(1-\frac{\lambda}{16\pi^{2}}(N+2)\right)\;,$ (62)

which is precisely the pole mass, eq. (12), for $\bar{\mu}=m$.999Recall that,
compared to eq. (12), we are now using a lighter notation whereby $m$ and
$\lambda$ stand for the $\overline{\rm MS}$ couplings $\bar{m}$ and
$\bar{\lambda}$ evaluated at scale $\bar{\mu}$.

The same result can be recovered working directly in terms of the physical
parameters. Using the $P(X)$ of eq. (53), and expanding around $X=m^{2}_{\rm
pole}$ we obtain equation (64) below. Performing a stability analysis along
the same lines, we find that $P^{\prime}(\mu^{2})>0$ if and only if
$X>m^{2}_{\rm pole}$.

###  4.3 Low densities

An interesting limit to consider is that of low charge densities. In this
limit, the two cases $m^{2}<0$ and $m^{2}>0$ behave in strikingly different
ways:

  * •

Negative $\boldsymbol{m^{2}}$: In this case we have SSB already for $X=0$,
which corresponds to a Poincaré invariant ground state with vanishing charge
density. For small $X\ll-m^{2}$, one gets a small charge density of order $X$.
Notice that this case just corresponds to a Lorentz-invariant theory of
Goldstones, expanded about a small Lorentz-violating background. The radial
mode that has been integrated out had a mass $m^{2}_{\rm eff}\simeq-2m^{2}$,
and so this is the scale that controls the derivative expansion in the $U(1)$
case. For the more general $SO(N)$ case, if one insists on integrating out the
gapped Goldstones as well—like we have done—the regime of validity of the
effective theory is much narrower, since the derivative expansion is
controlled by the gap of the gapped Goldstones, which is $X\ll-m^{2}$. Related
to this, the effective theory (52) for the gapless Goldstone $\phi$ is not
analytic at $X=0$,

$\displaystyle P(X)\simeq\frac{1}{\lambda}\bigg{[}$
$\displaystyle-\frac{1}{2}m^{2}X\left(1+7\frac{\lambda}{16\pi^{2}}\right)+\frac{1}{4}X^{2}\left(1+\frac{11}{4}\frac{\lambda}{16\pi^{2}}\right)$
(63) $\displaystyle\hskip
14.22636pt~{}~{}+\left(\dfrac{N-2}{4}\right)\dfrac{\lambda}{16\pi^{2}}X^{2}\left(\dfrac{3}{2}-\log\left(-\dfrac{X}{2m^{2}}\right)\right)\bigg{]}\,,\qquad\quad(\bar{\mu}^{2}=-2m^{2})\;,$

where for simplicity we set $\bar{\mu}^{2}=-2m^{2}$. The non-analyticity $\sim
X^{2}\log X$ is a sign that we have integrated out degrees of freedom that
become gapless as $X\to 0$.

  * •

Positive $\boldsymbol{m^{2}}$: As we saw in Sec. 4.2, in this case we have SSB
and an associated charge density only for $X>m^{2}_{\rm pole}$. It is
convenient in this case to express the $P(X)$ in terms of the on-shell
parameters. Close to that lower limit we have

$P(X)=\left(\dfrac{1}{4\lambda_{\rm
thr}}-\dfrac{N+2}{96\pi^{2}}\right)(X-m_{\rm
pole}^{2})^{2}-\dfrac{\sqrt{2}}{15\pi^{2}}\dfrac{(X-m^{2}_{\rm
pole})^{5/2}}{m_{\rm pole}}+\mathcal{O}\left((X-m^{2}_{\rm
pole})^{3}\right)\;.$ (64)

Now the charge density is of order $(X-m^{2}_{\rm pole})$. However, in
contrast to the negative $m^{2}$ case, here the Goldstone effective theory is
highly non-relativistic at low densities. In particular, the speed of sound,

$c_{s}^{2}=\frac{P^{\prime}(X)}{P^{\prime}(X)+2P^{\prime\prime}(X)X}\;,$ (65)

is found to be

$c_{s}^{2}\simeq\dfrac{X-m_{\rm pole}^{2}}{3X-m_{\rm
pole}^{2}}\simeq\dfrac{X-m_{\rm pole}^{2}}{2m_{\rm pole}^{2}},$ (66)

which goes to zero when the density goes to zero, while in the negative
$m^{2}$ case it goes to unity in that limit, since there one recovers Lorentz
invariance.

The structure of the derivative expansion is also different: here the gap of
the radial mode is $m^{2}_{\rm eff}\simeq 2(X-m^{2})$, which is small, while
the gap of the gapped Goldstones is $X\simeq m^{2}$, which is not small. The
leading non-analytic effect is thus associated with the radial mode, which has
been integrated out, and is the branch cut of $(X-m^{2}_{\rm pole})^{5/2}$.
Notice that the structure of non-analyticities is different from the negative
$m^{2}$ case: the $\log(m^{2}_{\rm eff})$ in eq. (53) is cancelled exactly by
another one that emerges in the small-$m^{2}_{\rm eff}$ expansion of
$f(-4X/m^{2}_{\rm eff})$—see eq. (35). Such a perfect cancellation is quite
mysterious, and we are not sure what its physical interpretation is.

###  4.4 High densities: RG improvement

An advantage of having the effective action to all orders in $X$ is that we
can push it to very large values of $X$—or equivalently very high
densities—where things drastically simplify. In fact, the derivative expansion
becomes better and better in that limit, since the radial mode (with gap
$m^{2}_{\rm eff}\simeq 2X$) and the gapped Goldstones (with gap $X$) become
very heavy. From (52), we find that the leading terms in this limit are

$P(X)\simeq\frac{1}{4\lambda}X^{2}\left[1+\frac{\lambda}{96\pi^{2}}\Big{(}82-5\,\kappa-60\log(2X/\bar{\mu}^{2})+(N-2)\left[9-6\log(X/\bar{\mu}^{2})\right]\Big{)}\right]\;,$
(67)

where
$\kappa={}_{3}F_{2}\left[\left.\begin{array}[]{c}1,~{}~{}1,~{}~{}{\textstyle\frac{7}{2}}\\\
4,~{}~{}5\end{array}\right\lvert-2\right]=0.771885\dots$ is a constant.

Notice that, given the one-loop renormalization group (RG) equation for
$\lambda$,

$\bar{\mu}\frac{{\rm d}\lambda}{{\rm
d}\bar{\mu}}=\Big{(}20+2(N-2)\Big{)}\frac{\lambda^{2}}{16\pi^{2}}\,,$ (68)

combined with the fact that there is no wave-function renormalization for
$\phi$, we have

$\bar{\mu}\frac{{\rm d}}{{\rm d}\bar{\mu}}P(X)=0\;,$ (69)

as expected. Then, when the log in (67) becomes large, we can use the RG and
resum the leading logs to obtain

$P(X)\simeq\frac{X^{2}}{4\,\lambda(2X)}\;,$ (70)

where $\lambda(2X)$ is the running coupling evaluated at scale
$\bar{\mu}^{2}=2X$. This is the same as the leading high-$X$ behavior of the
tree-level effective action, (54), but with the coupling constant replaced by
its running value at the appropriate scale.

It is interesting to compute the speed of sound (65). To this order, we get

$c_{s}^{2}\simeq\frac{1}{3}+\frac{1}{9}\beta_{1}\,\lambda(2X)\;,$ (71)

where $\beta_{1}$ is the one-loop $\beta$-function coefficient,
$\beta_{1}=\frac{20+2(N-2)}{16\pi^{2}}$. Notice that we are slightly above the
famous value of $1/3$ for a conformally invariant fluid, and more and more so
as we run towards the UV, since $\lambda$ grows. This behavior can be
understood by noticing that, for constant $\lambda$, the effective lagrangian
(70) yields a scale invariant theory if one assigns $\phi(x)$ scaling
dimension zero. Indeed, it corresponds to the effective action of a conformal
superfluid in $3+1$ dimensions [22, 23]. The system is thus conformally
invariant, up to scaling-violating effects induced by the running of the
couplings. This allows us to relate the corrections to the sound speed to the
trace anomaly.

The energy momentum tensor of a general superfluid is given by

$T_{\mu\nu}=2P^{\prime}(X)\partial_{\mu}\phi\,\partial_{\nu}\phi-\eta_{\mu\nu}P(X)\;,$
(72)

from which we can read off its energy density and pressure as

$\varrho=2P^{\prime}(X)X-P(X),\qquad\qquad p=P(X).$ (73)

For the effective action of eq. (70), the trace of $T_{\mu\nu}$ reads

$T^{\mu}{}_{\mu}=-\beta_{1}\lambda(2X)P(X)\;,$ (74)

so that the system has a trace anomaly proportional to the $\beta$-function of
$\lambda$. On the other hand, the trace of the energy momentum tensor can be
expressed in terms of the energy density and pressure as
$T^{\mu}{}_{\mu}=\varrho-3p$, so that its $X$ derivative is directly related
to $c_{s}^{2}$’s departure from $1/3$:101010For fluids and superfluids alike,
the sound speed is related to the energy density and the pressure by
$c_{s}^{2}=\frac{{\rm d}p}{{\rm d}\varrho}$.

$\frac{{\rm d}T^{\mu}{}_{\mu}}{{\rm d}X}=-3\frac{{\rm d}\varrho}{{\rm
d}X}\,\left(c_{s}^{2}-\frac{1}{3}\right)=3\frac{{\rm d}p}{{\rm
d}X}\left(\dfrac{1}{3c_{s}^{2}}-1\right).$ (75)

Equating this to the $X$ derivative of eq. (74) and working to lowest order in
$\lambda$—and so, in particular, neglecting the $X$ dependence of $\lambda$—we
reproduce exactly eq. (71).

###  4.5 A cross-check

We can also use some of the results of [24] as a nontrivial check of ours. In
particular, one of those results is that for a superfluid with
$P(X)=\frac{1}{2}X$ coupled to a heavy field $h(x)$ as

$S=\int{\rm d}^{4}x\left(\frac{1}{2}X+Xh+\frac{1}{2}(\partial
h)^{2}-\frac{M^{2}}{2}h^{2}\right)\;,$ (76)

the one loop correction to the expectation value of the stress-energy tensor
to first order in the background $X$ obeys111111We are writing eqs. (76) and
(77) not in the original notation of [24], but rather in variables that will
make the comparison with our results more straightforward.

$\Delta(\varrho+p)\simeq-\frac{XM^{2}}{4\pi^{2}\,d}\bigg{(}\frac{M^{2}}{{\rm
e}^{\gamma_{E}}\bar{\mu}^{2}}\bigg{)}^{\frac{d-4}{2}}\Gamma\bigg{(}\frac{2-d}{2}\bigg{)}\;,$
(77)

where $\gamma_{E}$ is the Euler–Mascheroni constant. Now, the action in the
$U(1)$ case, before we integrate out the radial mode, looks very much like
(76). In particular, if we consider the quadratic expansion (26) about a non-
trivial background $\phi(x)$ and the associated saddle point for $\rho(x)$,
and if we canonically normalize $\pi$, we get

$S_{2}[h,\pi]=\int{\rm
d}^{4}x\left(\frac{1}{2}(\partial\pi)^{2}+\frac{1}{2}(\partial
h)^{2}+2\partial^{\mu}\phi\,h\partial_{\mu}\pi-\frac{1}{2}m^{2}_{\rm
eff}h^{2}\right)\;.$ (78)

This is exactly the same as the quadratic expansion of (76), if we formally
treat $m^{2}_{\rm eff}$—which secretly depends on $X$—as a constant:
$m^{2}_{\rm eff}\to M^{2}$. In this case, the one-loop correction to the
quantum effective action (34), to first order in the background $X$ and
neglecting a cosmological constant, reads

$\Delta\Gamma(X)\simeq\alpha\,M^{2}X\;,\quad{\rm
with}\quad\alpha\equiv-\frac{1}{16\pi^{2}(d-4)}+\frac{3-2\log\big{(}M^{2}/\bar{\mu}^{2}\big{)}}{64\pi^{2}}\;.$
(79)

The stress-energy tensor that we get by formally treating this as a classical
Lagrangian is nothing but the one-loop correction to the expectation value of
the stress energy tensor, simply because for a general field theory one has

$\langle T_{\mu\nu}(x)\rangle=\frac{2}{\sqrt{-g}}\frac{\delta\Gamma}{\delta
g^{\mu\nu}(x)}\;,$ (80)

as can be checked straightforwardly using standard functional methods.

From (79) we thus get

$\Delta(\varrho+p)=2\,\Delta\Gamma^{\prime}(X)\,X\simeq 2\alpha\,M^{2}X\;,$
(81)

which is exactly the $d\to 4$ limit of eq. (77). Notice that the fact that we
consistently discarded contributions to the cosmological constant does not
affect our ability to compute the value of $\varrho+p$ precisely because such
a combination is insensitive to the cosmological constant, which always has
$\varrho+p=0$ as a consequence of Lorentz invariance

The result (77) was derived in [24] with methods that are so different from
ours, that we find our being able to reproduce it to be a very non-trivial
check of the validity of our analysis (and of that of [24] as well).

##  5 What happened to $SO(N)$?

Consider once again the generic $SO(N)$ case. After renormalization, the low-
energy effective theory is given by eq. (52). Recall that the only difference
with the $U(1)$ case is the relatively simple term in the last line,
explicitly proportional to $N-2$. This creates a puzzle, or at least an
uncomfortable situation, already pointed out in [15]: what happened to the
full $SO(N)$ symmetry? It seems extremely unlikely that adding a simple term
like the $(N-2)$ one in the last line of (52) can promote a quite complicated
structure like that in the first two lines from being only symmetric under
$U(1)$ to being symmetric under the full $SO(N)$—a much bigger, more
complicated (i.e., non-abelian) group.

The only sensible answer at the moment seems to be that, in the low-energy
effective theory without the gapped Goldstones, there is simply no trace of
the original $SO(N)$ symmetry. To be clear, as we already emphasized in Sec.
(4.3), the fact that the last line of (52) is not analytic at $X=0$ is a sign
that some degrees of freedom that have been integrated out become gapless at
$X=0$, and maybe it can be argued that the $N-2$ in front is a sign that there
are $N-2$ of them. But it is certainly not clear how one could conclude from
these observations that these degrees of freedom must be the gapped Goldstones
of a spontaneously broken $SO(N)$ theory, simply because it is not clear in
what sense the effective action (52) is invariant under $SO(N)$.

Notice that, with our conventions (see Sec. 2), a state with a nontrivial $X$
corresponds to having a finite chemical potential for the $J_{12}$ generator
of $SO(N)$. Of the other $SO(N)$ generators, $2(N-2)$ of them don’t commute
with $J_{12}$, but the remaining ones do. In fact, the ones that do are not
spontaneously broken, and so all the Goldstones can be taken to be in linear
representations of them. In particular, the gapless Goldstone that we retain
in the low-energy theory is neutral under them, since it has the same quantum
numbers as $J_{12}$, which commutes with those generators. So, for the
generators that commute with $J_{12}$ there is no mystery; there is no trace
of these symmetries in the effective theory for the gapless Goldstone, because
it so happens that they only act nontrivially on gapped fields. Nothing in the
low-energy theory transforms under them. This is the same, for instance, as
noticing that there is no trace of baryon number in the pions’ chiral
Lagrangian, simply because all baryons are heavy and do not appear in the low-
energy theory.

On the other hand, the $2(N-2)$ generators of $SO(N)$ that do not commute with
$J_{12}$ do act non-trivially on the gapless Goldstone, but they do so in a
way that depends on the gapped Goldstones. When these are integrated out, it’s
not clear how to make those generators act on the gapless Goldstone. In fact,
there is no known way to realize non-linearly $SO(N)$ with just a single
field, even accounting for the subtleties associated with the spontaneous
breaking of spacetime symmetries. That is, sometimes the number of Goldstones
needed to realize a given symmetry breaking pattern can be reduced by so-
called inverse Higgs constraints, which can be physically interpreted as the
result of integrating out certain gapped Goldstones. But the rules of when
that is possible are well understood, and this is not one of those cases (for
a review, see for instance [25]).

At least in perturbation theory, the technical reason we lose the non-abelian
$SO(N)$ symmetry after integrating out the gapped Goldstones seems to be the
following. The gapped Goldstones transform non-linearly under $SO(N)$. So,
whatever boundary conditions we choose for them cannot be invariant under
$SO(N)$. Since integrating out at tree level is the same as solving the
classical equations of motion and plugging back into the action, and since we
need to specify boundary conditions to solve the equations of motion, the fact
that these are not invariant implies that the effective action is not
invariant either.121212This is to be contrasted with what happens when we
integrate out a full multiplet in a linear representation of a symmetry: zero
is always an invariant boundary condition. Beyond tree-level the problem
remains; within perturbation theory, higher orders cannot cure this zeroth-
order problem.

##  6 Concluding remarks and outlook

Our underlying motivation to explore the questions addressed here is related
to a more general and more fundamental question, which we discuss at length in
a separate paper: for relativistic theories at zero temperature, under what
conditions does having a finite charge density imply that the associated
symmetry is spontaneously broken?

For free bosons, the answer is provided by the phenomenon of Bose–Einstein
condensation: the ground state at finite charge density exhibits SSB. Notably
this is not the case for free fermions: the ground state at finite density
breaks Lorentz boosts, but does not break translations, rotations, or the
particle number $U(1)$ symmetry.

Beyond free theories, it becomes harder to answer the question in generality.
For fermions, if Fermi liquid theory applies, then the situation is
qualitatively the same as for free fermions [26]. For bosons—in particular for
scalars—at tree level one finds that a finite density implies SSB, simply
because the charge density operator is bilinear in a complex field $\Phi$, so
that for example
$J_{\mu}=i\,\Phi^{*}\overset{\leftrightarrow}{\partial}_{\mu}\Phi$, and so it
is impossible to have nonzero $J^{0}$ with vanishing $\Phi$.

However, beyond tree-level, in principle it appears to be possible to have a
nonzero expectation value for $J^{0}$ while having a vanishing one for $\Phi$:

$\langle\Phi^{*}\overset{\leftrightarrow}{\partial}_{0}\Phi\rangle\neq 0\
,\qquad\langle\Phi\rangle=0\;.$ (82)

The approach that we have taken has SSB built-in from the start—we are
deriving the effective action for the gapless Goldstone, which is a consistent
procedure only in the case of SSB. Still, the results of Sec. 4.2 and 4.3
provide circumstantial evidence that, at least at one loop, the possibility
outlined by eq. (82) is not realized. Indeed, consider the positive $m^{2}$
case (when $m^{2}<0$, the symmetry is always spontaneously broken, even at
vanishing charge density). On the one hand, Sec. 4.2 shows that our SSB
assumption is consistent only for large enough chemical potentials,
$\mu^{2}\geq m^{2}_{\rm pole}$. On the other hand, Sec. 4.3 shows that for
$\mu$ approaching that lower bound from above, the charge density goes to
zero. This strongly suggests that the charge density remains zero when $\mu$
is below that bound, that is, when we have no SSB.

We tackle the question head-on, without assuming SSB, in forthcoming work.

#### Acknowledgements

We are grateful to Paolo Creminelli, Gabriel Cuomo, Luca Delacrétaz, Walter
Goldberger and Riccardo Rattazzi for useful discussions. AN would also like to
thank Paolo Creminelli for earlier collaboration on related topics
(unpublished work). Our work is partially supported by the US DOE (award
number DE-SC011941) and by the Simons Foundation (award number 658906).

##  Appendix A Dimensional regularization integrals

Here we describe the evaluation of the integral (33) in dimensional
regularization. The starting point is the integral

$\Delta\Gamma^{\rm 1\,loop}_{\rm full}[\phi]=\frac{i}{2}\int{\rm
d}^{4}x\int\frac{{\rm d}^{4}p}{(2\pi)^{4}}\log{\left(p^{2}-m^{2}_{\rm
eff}-4\partial_{\mu}\phi\partial_{\nu}\phi\,\frac{p^{\mu}p^{\nu}}{p^{2}}\right)}\;,$
(83)

for generic (but constant) $\partial_{\mu}\phi$ and $m^{2}_{\rm eff}$. The
$\partial\phi\partial\phi$ structure inside the log prevents us from using
spherical coordinates, so instead we will expand in powers of
$\partial_{\mu}\phi$, compute the individual integrals, and then try to resum
the series. Notice that, as long as $m^{2}_{\rm eff}\neq 0$, the integral (83)
is very convergent in the IR, and so the IR-ambiguous $p^{\mu}p^{\nu}/p^{2}$
structure is harmless.

From expanding the log in powers of $\partial\phi$, we have

$\log\left(p^{2}-m^{2}_{\rm
eff}-\partial_{\mu}\phi\partial_{\nu}\phi\frac{p^{\mu}p^{\nu}}{p^{2}}\right)=\log{\big{(}p^{2}-m^{2}_{\rm
eff}\big{)}-\sum_{n=1}^{\infty}\frac{4^{n}}{n}\partial_{\mu_{1}}\phi\partial_{\nu_{1}}\phi\cdots\partial_{\mu_{n}}\phi\partial_{\nu_{n}}\phi\frac{p^{\mu_{1}}p^{\nu_{1}}\cdots
p^{\mu_{n}}p^{\nu_{n}}}{p^{2n}(p^{2}-m^{2}_{\rm eff})^{n}}}\;.$ (84)

Upon integrating in $p$, one can replace the product of momenta $p^{\mu}$ in
the numerator with a totally symmetrized combination of $n$ $\eta^{\mu\nu}$
tensors. The correct normalization factor in $d$ dimensions is [18]

$\frac{p^{\mu_{1}}p^{\nu_{1}}\cdots
p^{\mu_{n}}p^{\nu_{n}}}{p^{2n}}\longrightarrow\frac{\Gamma(d/2)\Gamma(n+1/2)}{\Gamma(1/2)\Gamma(n+d/2)}\,\eta^{(\mu_{1}\nu_{1}}\dots\eta^{\mu_{n}\nu_{n})}\,.$
(85)

Thus, the right-hand side of (84) can be replaced by

$\log\big{(}p^{2}-m^{2}_{\rm
eff}\big{)}-\frac{\Gamma(d/2)}{\Gamma(1/2)}\sum_{n=1}^{\infty}\frac{1}{n}(4X)^{n}\,\frac{\Gamma(n+1/2)}{\Gamma(n+d/2)}\frac{1}{(p^{2}-m^{2}_{\rm
eff})^{n}}\,.$ (86)

Or, upon Wick rotating, we can write

$\log{\big{(}p_{E}^{2}+m^{2}_{\rm
eff}\big{)}}-\frac{\Gamma(d/2)}{\Gamma(1/2)}\sum_{n=1}^{\infty}\frac{(-4X)^{n}}{n}\,\frac{\Gamma(n+1/2)}{\Gamma(n+d/2)}\frac{1}{(p_{E}^{2}+m^{2}_{\rm
eff})^{n}}\;.$ (87)

In order to evaluate the integral of this expression, we need the standard
dim-reg integrals

$I_{n}\equiv\mu_{\rm MS}^{4-d}\int\frac{{\rm
d}^{d}p_{E}}{(2\pi)^{d}}\frac{1}{(p_{E}^{2}+m^{2}_{\rm
eff})^{n}}=\frac{(m^{2}_{\rm eff})^{2-n}}{16\pi^{2}}\left(\frac{m^{2}_{\rm
eff}}{4\pi\mu_{\rm MS}^{2}}\right)^{d/2-2}\frac{\Gamma(n-d/2)}{\Gamma(n)}\,,$
(88)

for positive integer $n$, as well as the integral

$I_{\rm log}\equiv\mu_{\rm MS}^{4-d}\int\frac{{\rm
d}^{d}p_{E}}{(2\pi)^{d}}\log\big{(}p_{E}^{2}+m^{2}_{\rm
eff}\big{)}=-\frac{\partial}{\partial n}I_{n}\bigg{|}_{n\to
0}=-\frac{m^{4}_{\rm eff}}{16\pi^{2}}\left(\frac{m^{2}_{\rm eff}}{4\pi\mu_{\rm
MS}^{2}}\right)^{d/2-2}\Gamma(-d/2)\;,$ (89)

where $\mu_{\rm MS}$ denotes the minimal subtraction renormalization scale. Of
all these, only $I_{\rm log}$, $I_{1}$, and $I_{2}$ diverge for $d\to 4$.

Then, to compute (83) we expand out the log using (84), perform the integrals
at each order using (88) and (89) and then resum the series to obtain

$\displaystyle\Delta\Gamma^{\rm 1\,loop}_{\rm full}[\phi]$
$\displaystyle=\int{\rm d}^{4}x\bigg{(}-\frac{1}{32\pi^{2}(d-4)}\big{(}m_{\rm
eff}^{4}+2m_{\rm eff}^{2}X+2X^{2}\big{)}$ (90) $\displaystyle\hskip
11.38092pt+\frac{1}{384\pi^{2}}\bigg{[}9m_{\rm eff}^{4}+18m_{\rm
eff}^{2}X+10X^{2}-6(m_{\rm eff}^{4}+2m_{\rm eff}^{2}X+2X^{2})\log(m_{\rm
eff}^{2}/\bar{\mu}^{2})$ $\displaystyle\hskip 62.59596pt-10\frac{X^{3}}{m_{\rm
eff}^{2}}\,{}_{3}F_{2}\left[\begin{array}[]{c}1,~{}~{}1,~{}~{}{\textstyle\frac{7}{2}}\\\
4,~{}~{}5\end{array}\left\lvert-\frac{4X}{m_{\rm
eff}^{2}}\right.\right]\bigg{]}\bigg{)}\;,$

where as usual $\bar{\mu}^{2}=4\pi{\rm e}^{-\gamma_{E}}\mu_{\rm MS}^{2}$,
where $\gamma_{E}$ is the Euler–Mascheroni constant and ${}_{3}F_{2}$ is a
generalized hypergeometric function. This is precisely the expression (34)
used in the main text.

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