aainstitutetext: CEICO, FZU — Institute of Physics of the Czech Academy of
Sciences,
Na Slovance 1999/2, 182 21 Prague 8, Czech Republicbbinstitutetext: Institute
for Theoretical Physics, Spinoza Institute & EMME$\Phi$, Utrecht University,
Buys Ballot Building, Princetonplein 5, 3584 CC Utrecht, The Netherlands

###### Abstract

We consider the classical attractor regime of the spectator Abelian Higgs
model in power-law inflation, and compute the one-loop corrections to its
evolution. For computations we utilize dimensional regularization and the
propagators in the unitary gauge. The corrections to both the scalar
condensate and the energy-momentum tensor exhibit secular ultraviolet
contributions, that tend to slow down the rolling of the scalar down its
potential, and drive it away from the classical attractor. These corrections
need not be suppressed if the $U(1)$ charge is much larger than the scalar
self-coupling, which is seen already in flat space. In addition, at late times
the secular corrections necessarily invalidate the perturbative loop
expansion. We find the late time secular corrections to be captured by the
renormalization group, which opens up the possibility to resum them past the
breakdown of perturbativity.

##  1 Introduction

The question of whether inflationary observables — expressed in terms of
scalar and tensor cosmological perturbations — can be significantly affected
by quantum loop corrections is still very much an open one. Quantum
corrections to cosmological perturbations can arise in different manners: (i)
as indirect corrections descending from the quantum corrections to the
background fields, and (ii) as direct corrections arising from interactions
between evolving perturbations. On a technical level, the indirect corrections
descend from the corrections to the one-point functions (condensates), while
the direct corrections are corrections to the connected two-point functions.
The computation of former type is simpler as typically the loop consists of a
single propagator.

While the understanding of the direct type of corrections to the cosmological
perturbations is still marred by questions of gauge dependence Miao:2017feh ,
considerable progress has been made in understanding the indirect type of
corrections, to which this work is devoted to as well. When it comes to
corrections to condensates in cosmological expanding and accelerating spaces
distinctions should be made between ultraviolet and infrared corrections. The
former are in a sense universal as they depend on the ultraviolet structure of
the theory captured by modes that are not strongly coupled to the background.
That is why ultraviolet corrections in principle can be computed in any curved
space Birrell:1982ix ; Elizalde:1993ee ; Elizalde:1993qh . These have found
applications in inflation, investigating quantum corrections to the inflaton
potential Markkanen:2012rh ; Herranen:2016xsy , particularly in Higgs
inflation Bezrukov:2007ep ; Bezrukov:2013fka ; Bezrukov:2014ipa  in connection
to the quantum corrections to the Higgs effective potential Mooij:2011fi ;
George:2012xs ; George:2013iia ; George:2015nza ; Fumagalli:2016lls ;
Fumagalli:2016sof , and in studies of how curvature corrections influence the
Higgs potential stability DiLuzio:2014bua ; Herranen:2014cua ;
Herranen:2015ima ; Figueroa:2017slm ; Markkanen:2018bfx ; Mantziris:2022fuu
in inflation and reheating (for a review see Markkanen:2018pdo ).

Infrared corrections derive from the tree-level effect of gravitational
particle production Parker:1968mv ; Parker:1969au ; Parker:1971pt  for fields
non-conformally coupled to gravity, such as scalars or gravitons. It should be
emphasized that infrared corrections can be more important that ultraviolet
corrections. These effects are innate to accelerating expanding spacetimes,
and have no analogue in flat space, as they are in effect a consequence of the
cosmological horizon. As opposed to ultraviolet corrections, infrared ones are
not universal and depend on the details of the model and the expanding
spacetime, and it is much more challenging to quantify them. That is the
reason why most works in the literature make a simplifying assumption of
approximating the expanding spacetime by exactly de Sitter space,
characterized by the constant Hubble rate. Such works date back to
Birrell:1982ix ; Shore:1979as ; Allen:1983dg ; Vilenkin:1982sg , and the
investigations of curvature corrections to symmetry breaking potentials and
the study of phase transitions. Early results on corrections to the inflaton
evolution suggested that the corrections are tiny Bilandzic:2007nb ;
Herranen:2013raa , which is due to the smallness of the couplings and the
assumed perturbative regime. However, strong infrared effects for spectator
fields can lead to symmetry restoration of the scalar models with symmetry-
breaking potentials Starobinsky:1994bd ; Serreau:2013eoa , but also to
dynamical symmetry breaking in massless scalar electrodynamics Prokopec:2002uw
; Prokopec:2002jn ; Prokopec:2003iu  and the generation of a non-
perturbatively large mass for the vector field Prokopec:2006ue ;
Prokopec:2007ak ; Prokopec:2008gw . When infrared effects are large typically
one needs to describe them using non-perturbative methods, the Starobinsky’s
stochastic formalism Starobinsky:1986fx ; Starobinsky:1994bd  being perhaps
the most prominent one.

The de Sitter space is a very often good approximation for the realistic slow-
roll inflationary background. However, primordial cosmological observables
crucially depend on the parameters measuring the deviation of primordial
inflation from the exact de Sitter space. This deviation is usually encoded in
the slow-roll parameters, where the principal one measures the rate of change
of the Hubble rate, $\epsilon\\!=\\!-\dot{H}/H^{2}$, and the higher order ones
encode the higher time derivatives. It was argued in Ref. Miao:2015oba  that
quantum corrections introduce an additional fine-tuning problem into single
scalar potential models of inflation, and presumably the same is true for
spectator fields. This is why it is paramount to understand how quantum
corrections depend on the slow-roll parameters. Efforts in that direction have
been undertaken in inflation for spectator scalars Janssen:2009pb ;
Prokopec:2015owa ; Cho:2015pwa ; Markkanen:2017edu  and for the inflaton
Liao:2018sci ; Miao:2019bnq ; Miao:2020zeh ; Kyriazis:2019xgj ;
Sivasankaran:2020dzp ; Katuwal:2021kry , and even for the period of reheating
Ema:2016dny ; Katuwal:2022szw . Particularly insightful are studies of quantum
effects in power-law inflation, characterized by constant principal slow-roll
parameter $\epsilon$, and subsequently promoted to an adiabatic function of
time. This is because analytically tractable computations are still feasible,
both for the ultraviolet and for the infrared, and one can get a clear picture
of the effects that corrections can engender. A particularly important
reference for this work is Janssen:2009pb  which considered a symmetry-
breaking scalar potential model in power-law inflation with the assumption of
the scaling solution, and it found that inclusion of quantum effects can
significantly affect the scalar potential by dynamically restoring the
symmetry of the potential in the small field regime, more rapidly than in de
Sitter.

In this work we extend the analysis of Ref. Janssen:2009pb  and compute one-
loop corrections to the condensate of the Abelian Higgs model — scalar
electrodynamics with a symmetry breaking potential. For the sake of clarity we
adopt several simplifying assumptions. We assume that the Abelian Higgs model
is a spectator, in the sense that it does not source the evolution of the
expanding background. For the background we assume it is exact power-law
inflation, characterized by the constant principal slow-roll parameter
$\epsilon$. This still allows for an analytically tractable analysis of the
problem, at least at one-loop order. We consider the scalar modulus of the
non-minimally coupled Abelian Higgs model to be in the attractor regime in
power-law inflation, where it tracks the evolution of the decreasing
background Hubble rate, $\overline{\phi}\\!\propto\\!H$. This scaling
attractor can be seen as a dynamical generalization of the symmetry breaking
minimum in equilibrium theories, as it is formed by the competition between
the non-minimal coupling and the quartic self-coupling.

The main focus of this work are quantum corrections to the evolution of the
scaling attractor. We account for loop corrections to the attractor descending
from both the charged scalar and the vector. These are quantified by computing
one-loop corrections to the scalar one-point function, and to the graviton
one-point function (that is sourced by the energy-momentum tensor. Our
analysis utilizes propagators for power-law inflation, namely the scalar one
thas was worked out in Janssen:2008px , and the vector one that was
constructed in Glavan:2020zne  in the unitary gauge. An important difference
between the analysis performed in this work and that in Janssen:2009pb  is in
that the only source of breaking of scaling we encounter descends from
contributions connected to the ultraviolet scale $\mu$ introduced by the
counterterms. The non-vanishing scalar condensate generates masses for scalar
and vector perturbations, which regulates their infrared sector and prevents
large infrared efects from developing, except in the small condensate limit
which is beyond the scope of this work. Our finding show that there is a
secular ultraviolet effect that typically slows down the rolling of the
spectator scalar down its potential, and eventually drives the condensate away
from the classical attractor. Interestingly, this effect is completely absent
in the de Sitter limit, which again points to the importance of considering
quantum effects in more realistic inflationary backgrounds.

We perform the loop computations using dimensional regularization and we adopt
a rather conservative approach to renormalization. Namely we forbid
counterterm coefficients from depending on the principal slow roll parameter
$\epsilon$, that we consider to be a geometric quantity, that describes a
particular realization of the system, and does not characterize the theory.
This severely restricts which terms can be subtracted by finite parts of the
counterterms, thus making the results more robust to renormalization
prescription dependencies. Furthermore, we shy away from using any initial
state subtractions, as these introduce renormalization scheme dependence to
the initial state, and generate corrections which anyway decay at late times
we are interested in.

This work is organized as follows. In section 2 we present the model and set
the stage for the problem. Sections 3 and 4 are devoted to the calculation of
the one-loop scalar condensate and energy-momentum tensor, where we discuss in
detail the countertems needed to renormalize the results in dimensional
regularization. Section 5 is devoted to studying various limits and comparing
them to the literature. Particular attention is paid to the analysis of
secular effects and their growth, and to the conditions under which
pertubation theory breaks. In section 6 we present a preliminary analysis of
how to emply renormalization group tools to restore perturbativity of the
results. In particular, we show there that the RG resummation allows one to
write the complete one-loop results in the tree-level form, up to an
additional term in the energy-momentum tensor, which is due to the
perturbative nonrenormalizability of quantum gravity. In section 7 we discuss
our main results.

##  2 Preliminaries

The Abelian Higgs model is a commonly utilized as a toy model for the
electroweak sector of the standard model. In this work we consider its
nonminimally coupled variant, and the quantum corrections to its dynamics in
power-law inflation. In this section we summarize the properties of the
background, and recount the main properties of the model.

###  2.1 Power-law inflation

Spatially flat expanding cosmological spaces are well described by the
Friedmann-Lemaître-Robertson-Walker (FLRW) invariant line element,

$ds^{2}=-dt^{2}+a^{2}(t)d\vec{x}^{\,2}=a^{2}(\eta)\Bigl{[}-d\eta^{2}+d\vec{x}^{\,2}\Bigr{]}\,,$
(1)

where time is conveniently given either in physical time coordinate $t$ or the
conformal time coordinate $\eta$, the two being related via the scale factor,
$dt\\!=\\!ad\eta$, and where equal time spatial hypersurfaces are Euclidean
spaces covered by Cartesian coordinates $x^{i}$. The expansion of the space is
encoded by the scale factor $a$, and its first several derivatives. The first
derivative is usually given as the physical Hubble rate $H$, or the conformal
Hubble rate $\mathcal{H}$,

$H=\frac{1}{a}\frac{da}{dt}\,,\qquad\qquad\mathcal{H}=\frac{1}{a}\frac{da}{d\eta}\,,\qquad\qquad\mathcal{H}=aH\,,$
(2)

while in inflation the second derivative is encoded by the principal slow-roll
parameter,

$\epsilon=-\frac{1}{H^{2}}\frac{dH}{dt}=1-\frac{1}{\mathcal{H}^{2}}\frac{d\mathcal{H}}{d\eta}\,.$
(3)

In this work we consider power-law inflation Lucchin:1984yf ; La:1989za  that
is defined by the constant slow-roll parameter,

$\epsilon={\tt const.}\,,\qquad\qquad 0<\epsilon<1\,.$ (4)

Even though this model is not a realistic model of inflation, as it is
excluded by observations Planck:2018vyg  that favour an adiabatically evolving
$\epsilon$, it is more realistic than the de Sitter space, $\epsilon\\!=\\!0$,
often utilized to study quantum field theoretic effects in inflation. In
addition to being almost as mathematically tractable as de Sitter, power law
inflation incorporates the effect of the non-vanishing slow-roll parameter,
and the evolving Hubble rate $H\\!=\\!H_{0}a^{-\epsilon}$. The scale factor
and the conformal Hubble rate in power-law inflation take the form,

$a(\eta)=\Bigl{[}1-(1\\!-\\!\epsilon)H_{0}(\eta\\!-\\!\eta_{0})\Bigr{]}^{-\frac{1}{1-\epsilon}}\,,\qquad\quad\mathcal{H}(\eta)=H_{0}a^{1-\epsilon}\,,$
(5)

where $\eta_{0}$ is the initial time at which $a(\eta_{0})\\!=\\!1$, and
$H_{0}\\!=\\!\mathcal{H}(\eta_{0})$. The curvature tensors in power-law
inflation are,

$\displaystyle R_{\mu\nu\rho\sigma}={}$ $\displaystyle
2H^{2}g_{\mu[\rho}g_{\sigma]\nu}+4\epsilon\bigl{(}a^{2}\delta^{0}_{[\mu}g_{\nu][\sigma}\delta^{0}_{\rho]}\bigr{)}\,,$
(6) $\displaystyle R_{\mu\nu}={}$
$\displaystyle(D\\!-\\!1\\!-\\!\epsilon)H^{2}g_{\mu\nu}+(D\\!-\\!2)\epsilon
H^{2}\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\,,$ (7)
$\displaystyle R={}$ $\displaystyle(D\\!-\\!1)(D\\!-\\!2\epsilon)H^{2}\,.$ (8)

In addition, it is useful to note that the Weyl tensor,

$C_{\mu\nu\rho\sigma}\equiv
R_{\mu\nu\rho\sigma}-\frac{4}{(D\\!-\\!2)}g_{\mu][\rho}R_{\sigma][\nu}+\frac{2R}{(D\\!-\\!1)(D\\!-\\!2)}g_{\mu[\rho}g_{\sigma]\nu}\,.$
(9)

###  2.2 Nonminimally coupled Abelian Higgs model

The Abelian Higgs model consists of an Abelian gauge field $A_{\mu}$
interacting with a charged complex scalar $\Phi$. Its action in
$D$-dimensional curved space is,

$\displaystyle S\bigl{[}A_{\mu},\Phi,\Phi^{*}\bigr{]}={}$
$\displaystyle\int\\!d^{D\\!}x\,\sqrt{-g}\,\biggl{[}-\frac{Z_{\scriptscriptstyle
A}^{0}}{4}g^{\mu\rho}g^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma}-Z_{\phi}^{0}g^{\mu\nu}\bigl{(}D_{\mu}\Phi\bigr{)}^{\\!*}\bigl{(}D_{\nu}\Phi\bigr{)}$
$\displaystyle\hskip
99.58464pt-\lambda_{0}\bigl{(}\Phi^{*}\Phi\bigr{)}^{2}-\xi_{0}R\Phi^{*}\Phi\biggr{]}\,,$
(10)

where $g_{\mu\nu}$ denotes the metric tensor
[$g^{\mu\rho}g_{\mu\nu}\\!=\\!\delta^{\rho}_{\nu}$, $g\\!=\\!{\rm
det}(g_{\mu\nu})$], $R$ is the Ricci scalar, and the bare couplings (denoted
by sub/superscripts 0) in (10) are split into their renormalized values plus
counterterms (denoted by prefix $\delta)$,

$Z_{\scriptscriptstyle A}^{0}=1+\delta Z_{\scriptscriptstyle A}\,,\quad
Z_{\phi}^{0}=1+\delta
Z_{\phi}\,,\quad\lambda_{0}=\lambda+\delta\lambda\,,\quad\xi_{0}=\xi+\delta\xi\,,\quad
q_{0}=q+\delta q\,.\quad$ (11)

The counterterms are necessary to absorb ultraviolet divergences of quantum
loops, and are organized as a power series in $\hbar$, the dependence on which
is henceforth suppressed by adopting the natural units
$\hbar\\!=\\!c\\!=\\!1$. In the unitary gauge, defined by the condition ${\rm
Im}(\Phi)\\!=\\!0$, the action (10) reads Glavan:2020zne ,

$\displaystyle S[A_{\mu},\phi]={}$
$\displaystyle\int\\!d^{D\\!}x\,\sqrt{-g}\,\biggl{[}-\frac{Z_{A}^{0}}{4}g^{\mu\rho}g^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma}-\frac{(q_{0}\phi)^{2}}{2}g^{\mu\nu}A_{\mu}A_{\nu}$
$\displaystyle\hskip
99.58464pt-\frac{Z_{\phi}^{0}}{2}g^{\mu\nu}(\partial_{\mu}\phi)(\partial_{\nu}\phi)-\frac{\lambda_{0}}{4}\phi^{4}-\frac{\xi_{0}}{2}R\phi^{2}\biggr{]}\,.$
(12)

where $\phi\\!\equiv\\!{\rm Re}(\Phi)/\sqrt{2}$ is the canonically normalized
real scalar field. Note that in the limit $q\\!\to\\!0$ the unitary gauge
action reduces to the one for the self-interacting real scalar field. This is
the sense in which all the results presented here can be seen in the limit of
vanishing charge.

Following Glavan:2020zne , we consider the Abelian Higgs model in rigid power-
law inflation. Only the scalar is supposed to have a homogeneous and isotropic
condensate, which satisfies the equation of motion,

$\Bigl{[}\mathop{\mathchoice{\displaystyle\displaystyle
3\makebox[7.7778pt]{\hbox to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox
to6.75pt{ \hrule height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern
0.0pt\vrule width=0.0pt}}}{\textstyle\displaystyle 3\makebox[7.7778pt]{\hbox
to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox to6.75pt{ \hrule
height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptstyle\displaystyle 3\makebox[5.44446pt]{\hbox
to4.72499pt{\vrule width=0.0pt\kern 0.0pt\vbox to4.72499pt{ \hrule
height=0.0pt,width=4.72499pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptscriptstyle\displaystyle 3\makebox[3.8889pt]{\hbox
to3.375pt{\vrule width=0.0pt\kern 0.0pt\vbox to3.375pt{ \hrule
height=0.0pt,width=3.375pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}}-\xi
R-\lambda\overline{\phi}^{\,2}\Bigr{]}\overline{\phi}=0\,,$ (13)

that admits an attractor solution,

$\overline{\phi}=\frac{\overline{\phi}_{0}}{H_{0}}H\,,$ (14)

with the amplitude set by the parameters of the model and the slow-roll
parameter of the background,

$\frac{\overline{\phi}_{0}}{H_{0}}=\pm\sqrt{\frac{1}{\lambda}\Bigl{[}\epsilon(D\\!-\\!1\\!-\\!2\epsilon)\\!-\\!\xi(D\\!-\\!1)(D\\!-\\!2\epsilon)\Bigr{]}}\\!\xrightarrow{D\to
4}\pm\sqrt{\frac{1}{\lambda}\Bigl{[}(1\\!-\\!6\xi)(2\\!-\\!\epsilon)\\!-\\!2(1\\!-\\!\epsilon)^{2}\Bigr{]}}\,.$
(15)

For this attractor solution to exist the amplitude above should be real, which
puts some bounds on the non-minimal coupling,

$\xi<\frac{\epsilon(3\\!-\\!2\epsilon)}{6(2\\!-\\!\epsilon)}\equiv\xi_{\rm
cr}\,,$ (16)

dependent on the slow-roll parameter. We study scalar and vector fluctuations
around this background solution. The vector field $A_{\mu}$ is assumed not to
have a classical condensate and is treated as a fluctuation, while the full
scalar field, $\phi\\!=\\!\overline{\phi}\\!+\\!\varphi$, is expanded in
fluctuations $\varphi$ around the classical attractor condensate defined in
(14) and (15). We capture the nonlinear evolution of quantized fluctuations
$\hat{\varphi}$ and $\hat{A}_{\mu}$ perturbatively, by computing corrections
to the evolution of the linearized (non-interacting) fluctuations
$\hat{\varphi}^{0}$ and $\hat{A}_{\mu}^{0}$. We organize the corrections in
powers of the linearized fields, which generates the usual loop expansion from
the interaction picture. Each power of the linearized field counts as one
power of $\sqrt{\hbar}$, which we henceforth suppress by adopting natural
units $\hbar\\!=\\!1$.

Linearized perturbations around the attractor condensate (14) and their two-
point functions have been worked out in the unitary gauge in Glavan:2020zne .
The linearized scalar perturbations,

$S_{\scriptscriptstyle
S}^{\scriptscriptstyle(2)}[A_{\mu},\varphi]=\int\\!d^{D\\!}x\,\sqrt{-g}\,\biggl{[}-\frac{1}{2}g^{\mu\nu}(\partial_{\mu}\varphi)(\partial_{\nu}\varphi)-\frac{3\lambda}{2}\overline{\phi}^{\,2}\varphi^{2}-\frac{\xi}{2}R\varphi^{2}\biggr{]}\,.$
(17)

correspond to a spectator scalar with an effective mass
$3\lambda\overline{\phi}^{2}$ and the nonminimal coupling to gravity $\xi$.
Since the tree-level condensate (14) scales as the Hubble rate, for the
purposes of dynamics we can consider the linearized scalar perturbations to
possess just an effective mass,

$M_{\scriptscriptstyle S}^{2}\\!=\\!3\lambda\overline{\phi}^{2}\\!+\xi
R=\Bigl{[}3\epsilon(D\\!-\\!1\\!-\\!2\epsilon)\\!-\\!2\xi(D\\!-\\!1)(D\\!-\\!2\epsilon)\Bigr{]}H^{2}\\!\xrightarrow{D\to
4}\\!3\Bigl{[}-4\xi(2\\!-\\!\epsilon)+\epsilon(3\\!-\\!2\epsilon)\Bigr{]}H^{2},$
(18)

or an effective non-minimal coupling,

$\xi_{\scriptscriptstyle S}=\frac{M_{\scriptscriptstyle
S}^{2}}{R}\,\xrightarrow{D\to
4}\,-2\xi+\frac{\epsilon(3\\!-\\!2\epsilon)}{2(2\\!-\\!\epsilon)}\,.$ (19)

Nevertheless, when computing the energy-momentum tensor, it is important to
make the distinction between the two commensurate contributions in (17). The
latter of the two quantities is bound by $\xi_{\scriptscriptstyle
S}\\!>\\!(3\\!-\\!2\epsilon)\epsilon/[6(2\\!-\\!\epsilon)]$ due to (16), which
ensures there are no infrared divergences for CTBD states Ford:1977in ;
Janssen:2009nz .

Linearized vector perturbations,

$S_{\scriptscriptstyle
V}^{\scriptscriptstyle(2)}[A_{\mu}]=\int\\!d^{D\\!}x\,\sqrt{-g}\,\biggl{[}-\frac{1}{4}g^{\mu\rho}g^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma}-\frac{\bigl{(}q\overline{\phi}\bigr{)}^{\\!2}}{2}g^{\mu\nu}\\!A_{\mu}A_{\nu}\biggr{]}$
(20)

behave as though massive, with the effective mass induced by the scalar
condensate,

$M_{\scriptscriptstyle
V}^{2}=\bigl{(}q\overline{\phi}\bigr{)}^{\\!2}=\frac{q^{2}}{\lambda}\Bigl{[}\epsilon(D\\!-\\!1\\!-\\!2\epsilon)\\!-\\!\xi(D\\!-\\!1)(D\\!-\\!2\epsilon)\Bigr{]}H^{2}\\!\xrightarrow{D\to
4}\frac{q^{2}}{\lambda}\Bigl{[}-6\xi(2\\!-\\!\epsilon)\\!+\\!\epsilon(3\\!-\\!2\epsilon)\Bigr{]}H^{2}\,,$
(21)

or as effectively non-minimally coupled,

$\xi_{\scriptscriptstyle V}=\frac{M_{\scriptscriptstyle
V}^{2}}{R}\xrightarrow{D\to
4}\frac{q^{2}}{\lambda}\biggl{[}-\xi\\!+\\!\frac{\epsilon(3\\!-\\!2\epsilon)}{6(2\\!-\\!\epsilon)}\biggr{]}\,.$
(22)

Eq. (16) implies $\xi_{\scriptscriptstyle V}\\!>0$ and $M_{\scriptscriptstyle
V}^{2}>0$, which ensures stability and infrared finiteness of vector
perturbations, and distinguishes our model from vector curvaton models
Dimopoulos:2008rf .

In this work we compute the one-loop effects that interacting perturbations
impart on the evolution of the scalar condensate in the attractor (14), and on
the energy-momentum tensor; the latter accounts for the backreaction onto the
power-law inflation. This computation requires the two-point functions of
scalar and vector perturbations, and in particular their coincidence limits
collected in the following subsection.

###  2.3 Two-point functions

The two-point functions for linearized fluctuations of the model in (12) in
the unitary gauge in power-law inflation have been worked out in
Glavan:2020zne . They are expectation values of linearized fields (denoted by
superscript 0). In this work we compute the one-loop corrections to the scalar
condensate, and to the energy-momentum tensor. These are simplest one-loop
corrections, where the loop is formed by a single propagator only. That is why
we require only the dimensionally regulated coincidence limits of the
propagators from Glavan:2020zne . For the scalar this is,

$\bigl{\langle}\hat{\varphi}^{0}(x)\hat{\varphi}^{0}(x)\bigr{\rangle}=\boldsymbol{\Gamma}(\nu_{\scriptscriptstyle
S})(1\\!-\\!\epsilon)^{2}H^{2}\biggl{[}\Bigl{(}\frac{D\\!-\\!3}{2}\Bigr{)}^{\\!2}\\!-\nu_{\scriptscriptstyle
S}^{2}\biggr{]}\,,$ (23)

and for the vector,

$\displaystyle\bigl{\langle}\hat{A}_{\mu}^{0}(x)\hat{A}_{\nu}^{0}(x)\bigr{\rangle}=\boldsymbol{\Gamma}(\nu_{\scriptscriptstyle
V})(1\\!-\\!\epsilon)^{2}H^{2}\biggl{[}\Bigl{(}\frac{D\\!-\\!3}{2}\Bigr{)}^{2}\\!-\nu_{\scriptscriptstyle
V}^{2}\biggr{]}$
$\displaystyle\times\Biggl{\\{}\frac{1}{D}\biggl{[}(D\\!-\\!1)+\biggl{(}D\\!-\\!1\\!+\\!\frac{(D\\!-\\!4)\epsilon}{2(1\\!-\\!\epsilon)}\biggr{)}\biggl{(}2\\!+\\!\frac{(D\\!-\\!4)(2\\!-\\!\epsilon)}{2(1\\!-\\!\epsilon)}\biggr{)}\frac{(1\\!-\\!\epsilon)^{2}H_{0}^{2}}{(q\overline{\phi}_{0})^{2}}\biggr{]}g_{\mu\nu}$
$\displaystyle\hskip
28.45274pt+\frac{(D\\!-\\!4)\epsilon}{2(1\\!-\\!\epsilon)}\biggl{(}2\\!+\\!\frac{(D\\!-\\!4)(2\\!-\\!\epsilon)}{2(1\\!-\\!\epsilon)}\biggr{)}\frac{(1\\!-\\!\epsilon)^{2}H_{0}^{2}}{(q\overline{\phi}_{0})^{2}}\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\Biggr{\\}}\,,$
(24)

where we have defined a coefficient,

$\displaystyle\boldsymbol{\Gamma}(\nu)\equiv\frac{\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}\,\bigl{[}(1\\!-\\!\epsilon)H\bigr{]}^{D-4}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma\bigl{(}\frac{D-3}{2}\\!+\\!\nu\bigr{)}\,\Gamma\bigl{(}\frac{D-3}{2}\\!-\\!\nu\bigr{)}}{\Gamma\bigl{(}\frac{1}{2}\\!+\\!\nu\bigr{)}\,\Gamma\bigl{(}\frac{1}{2}\\!-\\!\nu\bigr{)}}$
(25) $\displaystyle\overset{D\to
4}{\scalebox{1.8}[1.0]{$\sim$}}\,\frac{\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}\,\bigl{[}(1\\!-\\!\epsilon)H\bigr{]}^{D-4}}{(4\pi)^{\frac{D}{2}}}\Biggl{\\{}1+\frac{(D\\!-\\!4)}{2}\biggl{[}\psi\Bigl{(}\frac{1}{2}\\!+\\!\nu\Bigr{)}+\psi\Bigl{(}\frac{1}{2}\\!-\\!\nu\Bigr{)}\biggr{]}+\mathcal{O}\bigl{[}(D\\!-\\!4)^{2}\bigr{]}\Biggr{\\}}\,,$

divergent in $D\\!=\\!4$ on the account of
$\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}\,\overset{D\to
4}{\scalebox{1.8}[1.0]{$\sim$}}\,2/(D\\!-\\!4)$, that depends on indices of
scalar and vector perturbations, respectively,

$\displaystyle\nu_{\scriptscriptstyle S}^{2}={}$
$\displaystyle\biggl{(}\frac{D\\!-\\!1\\!-\\!\epsilon}{2(1\\!-\\!\epsilon)}\biggr{)}^{\\!\\!2}-\frac{\xi_{\scriptscriptstyle
S}(D\\!-\\!1)(D\\!-\\!2\epsilon)}{(1\\!-\\!\epsilon)^{2}}\xrightarrow{D\to
4}\frac{25}{4}-\frac{2(1\\!-\\!6\xi)(2\\!-\\!\epsilon)}{(1\\!-\\!\epsilon)^{2}}\,,$
(26) $\displaystyle\nu_{\scriptscriptstyle V}^{2}={}$
$\displaystyle\biggl{(}\frac{D\\!-\\!3\\!-\\!\epsilon}{2(1\\!-\\!\epsilon)}\biggr{)}^{\\!\\!2}-\frac{\xi_{\scriptscriptstyle
V}(D\\!-\\!1)(D\\!-\\!2\epsilon)}{(1\\!-\\!\epsilon)^{2}}\xrightarrow{D\to
4}\frac{1}{4}+\frac{2q^{2}}{\lambda}\biggl{[}1\\!-\\!\frac{(1\\!-\\!6\xi)(2\\!-\\!\epsilon)}{2(1\\!-\\!\epsilon)^{2}}\biggr{]}\,.$
(27)

In four dimensions the range of these indices is limited by (16),

$\nu_{\scriptscriptstyle
S}^{2}\in\Bigl{(}-\infty,\frac{9}{4}\Bigr{)}\,,\qquad\qquad\nu_{\scriptscriptstyle
V}^{2}\in\Bigl{(}-\infty,\frac{1}{4}\Bigr{)}\,.$ (28)

We also need a dimensionally regulated scalar kinetic term, which is easily
computed by first taking derivatives of the scalar Wightman two-point function
and then using expressions from Sec. 7.5.3 of Glavan:2020zne , 111If the
coincidence limit is defined in terms of the Feynman propagator instead care
needs to be taken to discard possible local terms resulting from derivatives
acting on the time-ordeing. In addition, in the unitary gauge attention must
be paid to the fact that the Feynman propagator is not the Green’s function of
the theory Glavan:2020zne , and to the difference between $\mathcal{T}$ and
$\mathcal{T}^{*}$ ordered products, similarly to what was pointed out in
perturbative quantum gravity by Donoghue Donoghue:2020hoh .

$\displaystyle\bigl{\langle}\partial_{\mu}\hat{\varphi}^{0}(x)\,\partial_{\nu}\hat{\varphi}^{0}(x)\bigr{\rangle}={}$
$\displaystyle\boldsymbol{\Gamma}(\nu_{\scriptscriptstyle
S})(1\\!-\\!\epsilon)^{4}H^{4}\biggl{[}\Bigl{(}\frac{D\\!-\\!3}{2}\Bigr{)}^{\\!2}\\!-\nu_{\scriptscriptstyle
S}^{2}\biggr{]}$
$\displaystyle\times\Biggl{\\{}-\frac{1}{D}\biggl{[}\Bigl{(}\frac{D\\!-\\!1}{2}\Bigr{)}^{\\!2}\\!-\nu_{\scriptscriptstyle
S}^{2}\biggr{]}g_{\mu\nu}+\biggl{[}\frac{(D\\!-\\!2)\epsilon}{2(1\\!-\\!\epsilon)}\biggr{]}^{2}\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\Biggr{\\}}\,.$
(29)

Likewise we need the coincidence limit of the vector field strenght
correlator, obtained by taking the coincident limit of the correlator computed
in Sec. 8 of Glavan:2020zne ,

$\displaystyle\bigl{\langle}\hat{F}_{\mu\nu}^{0}(x)\hat{F}_{\rho\sigma}^{0}(x)\bigr{\rangle}=\boldsymbol{\Gamma}(\nu_{\scriptscriptstyle
V})(1\\!-\\!\epsilon)^{4}H^{4}\biggl{[}\Bigl{(}\frac{D\\!-\\!3}{2}\Bigr{)}^{\\!2}\\!-\nu_{\scriptscriptstyle
V}^{2}\biggr{]}$ (30)
$\displaystyle\times\Biggl{\\{}-\frac{4}{D}\biggl{[}\Bigl{(}\frac{D\\!-\\!1}{2}\Bigr{)}^{\\!2}\\!-\nu_{\scriptscriptstyle
V}^{2}\biggr{]}g_{\mu[\rho}g_{\sigma]\nu}-\frac{2(D\\!-\\!4)\epsilon}{(1\\!-\\!\epsilon)}\biggl{[}1-\frac{(D\\!-\\!4)\epsilon}{2(1\\!-\\!\epsilon)}\biggr{]}\bigl{(}a^{2}\delta_{[\mu}^{0}g_{\nu][\sigma}\delta_{\rho]}^{0}\bigr{)}\Biggr{\\}}\,.$

Even though for convenience we have introduced a number of parameters in this
section, it should be emphasized that there are only two independent coupling
constants in the problem, $\lambda$ and $q$, and two background parameters,
$\epsilon$ and $H_{0}$. Nevertheless, we use the parameters introduced here
interchangeably, to make the expressions more compact. Six particularly useful
combinations when presenting final results are are,

$\displaystyle(1\\!-\\!\epsilon)^{2}H^{2}\xrightarrow{D\to 4}{}$
$\displaystyle-\frac{\lambda\overline{\phi}^{\,2}}{2}+\frac{1}{2}\Bigl{(}\frac{1}{6}\\!-\\!\xi\Bigr{)}R\,,$
(31a)
$\displaystyle(1\\!-\\!\epsilon)^{2}\Bigl{(}\frac{1}{4}\\!-\\!\nu_{\scriptscriptstyle
S}^{2}\Bigr{)}H^{2}\xrightarrow{D\to 4}{}$ $\displaystyle
3\lambda\overline{\phi}^{2}-\Bigl{(}\frac{1}{6}\\!-\\!\xi\Bigr{)}R\,,$ (31b)
$\displaystyle(1\\!-\\!\epsilon)^{2}\Bigl{(}\frac{9}{4}\\!-\\!\nu_{\scriptscriptstyle
S}^{2}\Bigr{)}H^{2}\xrightarrow{D\to 4}{}$ $\displaystyle
2\lambda\overline{\phi}^{2}\,,$ (31c)
$\displaystyle(1\\!-\\!\epsilon)^{2}\Bigl{(}\frac{25}{4}\\!-\\!\nu_{\scriptscriptstyle
S}^{2}\Bigr{)}H^{2}\xrightarrow{D\to 4}{}$ $\displaystyle
2\Bigl{(}\frac{1}{6}\\!-\\!\xi\Bigr{)}R\,,$ (31d)
$\displaystyle(1\\!-\\!\epsilon)^{2}\Bigl{(}\frac{1}{4}\\!-\\!\nu_{\scriptscriptstyle
V}^{2}\Bigr{)}H^{2}\xrightarrow{D\to 4}{}$
$\displaystyle\bigl{(}q\overline{\phi}\bigr{)}^{\\!2}\,,$ (31e)
$\displaystyle(1\\!-\\!\epsilon)^{2}\Bigl{(}\frac{9}{4}\\!-\\!\nu_{\scriptscriptstyle
V}^{2}\Bigr{)}H^{2}\xrightarrow{D\to 4}{}$
$\displaystyle\bigl{(}q\overline{\phi}\bigr{)}^{\\!2}-\lambda\overline{\phi}^{2}+\Bigl{(}\frac{1}{6}\\!-\\!\xi\Bigr{)}R\,.$
(31f)

##  3 Scalar tadpole

Expanding the unitary gauge action (12) to cubic order in perturbations,

$S^{\scriptscriptstyle(3)}[A_{\mu},\varphi]=\\!\\!\int\\!\\!d^{D\\!}x\,\sqrt{-g}\,\biggl{[}-\overline{\phi}\Bigl{(}q^{2}\varphi
A^{\mu}A_{\mu}-\lambda\varphi^{3}\Bigr{)}-\varphi\Bigl{(}\delta\lambda\overline{\phi}^{\,3}+\delta\xi
R\overline{\phi}-\delta Z_{\phi}\mathop{\mathchoice{\displaystyle\displaystyle
3\makebox[7.7778pt]{\hbox to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox
to6.75pt{ \hrule height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern
0.0pt\vrule width=0.0pt}}}{\textstyle\displaystyle 3\makebox[7.7778pt]{\hbox
to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox to6.75pt{ \hrule
height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptstyle\displaystyle 3\makebox[5.44446pt]{\hbox
to4.72499pt{\vrule width=0.0pt\kern 0.0pt\vbox to4.72499pt{ \hrule
height=0.0pt,width=4.72499pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptscriptstyle\displaystyle 3\makebox[3.8889pt]{\hbox
to3.375pt{\vrule width=0.0pt\kern 0.0pt\vbox to3.375pt{ \hrule
height=0.0pt,width=3.375pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}}\overline{\phi}\Bigr{)}\biggr{]}\,,$ (32)

defines the vertices and counterterms that generate the one-loop corrections
to the tadpole, whose diagramatic representation is given in Fig. 1. Note that
the divergent counterterm coefficients count as two powers of pertrubations.

Figure 1: Diagrams depicting the one-loop corrections to the one-point
function from the scalar perturbation. Dashed line correspond to the scalar
propagator, and wavy ones to the vector propagator. The first two diagrams
descend from the vertices defined in the cubic action (32), which also defines
the counterterms represented by the last diagram.

The equation of motion for the one-loop correction to the scalar one-point
function takes the form,

$\Bigl{[}\mathop{\mathchoice{\displaystyle\displaystyle
3\makebox[7.7778pt]{\hbox to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox
to6.75pt{ \hrule height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern
0.0pt\vrule width=0.0pt}}}{\textstyle\displaystyle 3\makebox[7.7778pt]{\hbox
to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox to6.75pt{ \hrule
height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptstyle\displaystyle 3\makebox[5.44446pt]{\hbox
to4.72499pt{\vrule width=0.0pt\kern 0.0pt\vbox to4.72499pt{ \hrule
height=0.0pt,width=4.72499pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptscriptstyle\displaystyle 3\makebox[3.8889pt]{\hbox
to3.375pt{\vrule width=0.0pt\kern 0.0pt\vbox to3.375pt{ \hrule
height=0.0pt,width=3.375pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}}-\xi
R-3\lambda\overline{\phi}^{\,2}\Bigr{]}\bigl{\langle}\hat{\varphi}\bigr{\rangle}=\mathcal{S}_{\scriptscriptstyle
S}+\mathcal{S}_{\scriptscriptstyle V}+\delta\mathcal{S}\,,$ (33)

Where the scalar loop and the vector loop sources, corresponding to the first
and the second diagram in Fig. 1 respectively, are

$\mathcal{S}_{\scriptscriptstyle
S}=3\lambda\overline{\phi}\,\bigl{\langle}\hat{\varphi}^{0}\hat{\varphi}^{0}\bigr{\rangle}\,,\qquad\qquad\mathcal{S}_{\scriptscriptstyle
V}=q^{2}\overline{\phi}\,g^{\mu\nu}\bigl{\langle}\hat{A}_{\mu}^{0}\hat{A}_{\nu}^{0}\bigr{\rangle}\,,$
(34)

and the counterterms, corresponding to the last amputated diagram in Fig. 1
contribute as,

$\displaystyle\delta\mathcal{S}={}$
$\displaystyle\bigl{(}\delta\xi\\!-\\!\xi\delta
Z_{\phi}\bigr{)}R\overline{\phi}+\bigl{(}\delta\lambda\\!-\\!\lambda\delta
Z_{\phi}\bigr{)}\overline{\phi}^{\,3}$ $\displaystyle={}$
$\displaystyle\biggl{[}\bigl{(}\delta\xi\\!-\\!\xi\delta
Z_{\phi}\bigr{)}(D\\!-\\!1)(D\\!-\\!2\epsilon)H^{2}+\bigl{(}\delta\lambda\\!-\\!\lambda\delta
Z_{\phi}\bigr{)}\overline{\phi}^{\,2}\biggr{]}\overline{\phi}\,,$ (35)

where we have used the tree-level equation of motion (13) to write them in
this form. The counterterm coefficients will be chosen so as to absorb the
divergences from the scalar and the vector source (34). These divergences are
absorbed independently for each source, up to arbitrary finite parts.
Accordingly, we split the counterterm contribution to the source into three
parts,

$\delta\mathcal{S}=\bigl{[}\delta\mathcal{S}\bigr{]}_{\scriptscriptstyle
S}^{\rm div.}+\bigl{[}\delta\mathcal{S}\bigr{]}_{\scriptscriptstyle V}^{\rm
div.}+\bigl{[}\delta\mathcal{S}\bigr{]}^{\rm fin.}\,,$ (36)

and this split translates onto the analogous split of all the counterterm
coefficients. Since the renormalizability of the theory should not depend on
the background spacetime, the counterterm coefficients ahould be independent
of the packground parameters such as $\epsilon$. This breaks some redundancy
between the coefficients.

###  3.1 Scalar loop source

This is the simplest part to compute. Scalar tadpole source can be written as,
using the coincident scalar two-point function (23) and expressions from Sec.
2.2,

$\displaystyle\mathcal{S}_{\scriptscriptstyle S}={}$
$\displaystyle\overline{\phi}\,\boldsymbol{\Gamma}(\nu_{\scriptscriptstyle
S})\Biggl{\\{}\frac{3D(D\\!-\\!2)(D\\!-\\!6)\lambda}{32(D\\!-\\!1)}\biggl{[}1\\!-\\!\frac{4(D\\!-\\!1)\xi}{(D\\!-\\!2)}\biggr{]}(D\\!-\\!1)(D\\!-\\!2\epsilon)H^{2}$
$\displaystyle\hskip
113.81102pt+\frac{3(D\\!+\\!2)(8\\!-\\!D)\lambda^{2}}{8}\overline{\phi}^{\,2}+\mathcal{O}\bigl{[}(D\\!-\\!4)^{2}\bigr{]}\Biggr{\\}}\,.$
(37)

Comparing with the the counterterms (35), the simplest $\epsilon$-independent
choices are,

$\displaystyle\bigl{[}\delta\xi\\!-\\!\xi\delta
Z_{\phi}\bigr{]}_{\scriptscriptstyle S}^{\rm div.}={}$
$\displaystyle\frac{\mu^{D-4}\,\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}}{(4\pi)^{\frac{D}{2}}}\times\frac{3D(D\\!-\\!2)(6\\!-\\!D)\lambda}{32(D\\!-\\!1)}\biggl{[}1\\!-\\!\frac{4(D\\!-\\!1)\xi}{(D\\!-\\!2)}\biggr{]}\,,$
(38a) $\displaystyle\bigl{[}\delta\lambda\\!-\\!\lambda\delta
Z_{\phi}\bigr{]}_{\scriptscriptstyle S}^{\rm div.}={}$
$\displaystyle\frac{\mu^{D-4}\,\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}}{(4\pi)^{\frac{D}{2}}}\times\frac{3(D\\!+\\!2)(D\\!-\\!8)\lambda^{2}}{8}\,.$
(38b)

Taking the $D\\!\to\\!4$ limit of the scalar loop tadpole source produces a
fully renormalized result,

$\mathcal{S}_{\scriptscriptstyle
S}+\bigl{[}\delta\mathcal{S}\bigr{]}_{\scriptscriptstyle S}^{\rm
div.}\xrightarrow{D\to
4}\frac{3\lambda\overline{\phi}H^{2}}{16\pi^{2}}(1\\!-\\!\epsilon)^{2}\Bigl{(}\frac{1}{4}\\!-\\!\nu_{\scriptscriptstyle
S}^{2}\Bigr{)}\biggl{[}2\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}+\Psi(\nu_{\scriptscriptstyle
S})\biggr{]}\,,$ (39)

where we defined,

$\Psi(\nu)\equiv\psi\Bigl{(}\frac{1}{2}\\!+\\!\nu\Bigr{)}+\psi\Bigl{(}\frac{1}{2}\\!-\\!\nu\Bigr{)}\,,$
(40)

where $\psi(z)\\!=\\!\frac{{d}}{{d}z}\ln[\Gamma(z)]$ denotes the digamma
function.

###  3.2 Vector loop source

For the vector tadpole source we only need the fully contracted coincident
vector propagator from (24). Using some relations between parameters from Sec.
(2.2) we write it in the form,

$\displaystyle\mathcal{S}_{\scriptscriptstyle V}={}$
$\displaystyle\overline{\phi}\,\boldsymbol{\Gamma}(\nu_{\scriptscriptstyle
V})\Biggl{\\{}\frac{(D\\!-\\!1)q^{2}}{2}\Bigl{[}2q^{2}\\!-\\!(D\\!-\\!2)\lambda\Bigr{]}\overline{\phi}^{\,2}+3(D\\!-\\!4)q^{2}\biggl{[}\frac{3}{4}-\frac{(1\\!-\\!\epsilon)^{3}H_{0}^{2}}{(q\overline{\phi}_{0})^{2}}\biggr{]}\epsilon
H^{2}$
$\displaystyle+\frac{(D\\!-\\!2)(8\\!-\\!D)q^{2}}{16}\biggl{[}1\\!-\\!\frac{8(D\\!-\\!1)\xi}{(8\\!-\\!D)}\biggr{]}(D\\!-\\!1)(D\\!-\\!2\epsilon)H^{2}+\mathcal{O}\bigl{[}(D\\!-\\!4)^{2}\bigr{]}\Biggr{\\}}\,.$
(41)

Comparing this source with the form of the counterterms it is clear that the
two divergent terms have the same structure. Therefore, the simplest
$\epsilon$-independent choice for counterterm coefficients seems to be,

$\displaystyle\bigl{[}\delta\xi\\!-\\!\xi\delta
Z_{\phi}\bigr{]}_{\scriptscriptstyle V}^{\rm div.}={}$
$\displaystyle\frac{\mu^{D-4}\,\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}}{(4\pi)^{\frac{D}{2}}}\times\frac{(D\\!-\\!2)(D\\!-\\!8)q^{2}}{16}\biggl{[}1\\!-\\!\frac{8(D\\!-\\!1)\xi}{(8\\!-\\!D)}\biggr{]}\,,$
(42) $\displaystyle\bigl{[}\delta\lambda\\!-\\!\lambda\delta
Z_{\phi}\bigr{]}_{\scriptscriptstyle V}^{\rm div.}={}$
$\displaystyle\frac{\mu^{D-4}\,\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}}{(4\pi)^{\frac{D}{2}}}\times\frac{(D\\!-\\!1)q^{2}}{2}\Bigl{[}(D\\!-\\!2)\lambda\\!-\\!2q^{2}\Bigr{]}\,.$
(43)

This leads to the renormalized vector tadpole source,

$\displaystyle\mathcal{S}_{\scriptscriptstyle
V}+\bigl{[}\delta\mathcal{S}\bigr{]}_{\scriptscriptstyle V}^{\rm
div.}\xrightarrow{D\to 4}{}$
$\displaystyle\frac{3q^{2}\overline{\phi}H^{2}}{16\pi^{2}}\biggl{\\{}(1\\!-\\!\epsilon)^{2}\Bigl{(}\frac{9}{4}\\!-\\!\nu_{\scriptscriptstyle
V}^{2}\Bigr{)}\biggl{[}2\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}+\Psi(\nu_{\scriptscriptstyle
V})\biggr{]}$ $\displaystyle\hskip
71.13188pt+\biggl{[}\frac{3}{2}-\frac{2(1\\!-\\!\epsilon)^{3}H_{0}^{2}}{\bigl{(}q\overline{\phi}_{0}\bigr{)}^{\\!2}}\biggr{]}\epsilon\biggr{\\}}\,.$
(44)

###  3.3 Solving tadpole equation

Here we solve equation (33) for the condensate correction
$\bigl{\langle}\hat{\varphi}\bigr{\rangle}$, with sources computed in (39) and
(44). We can write the equation as,

$\displaystyle-\biggl{[}\frac{1}{a^{2}}\partial_{0}^{2}+\frac{2H}{a}\partial_{0}+6\xi_{\scriptscriptstyle
S}(2\\!-\\!\epsilon)H^{2}\biggr{]}\bigl{\langle}\hat{\varphi}\bigr{\rangle}={}$
$\displaystyle\frac{3\overline{\phi}H^{2}}{16\pi^{2}}\biggl{[}2\mathcal{A}\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}+\mathcal{B}\biggr{]}\,,$
(45)

where the coefficients are read off from the sources (39) and (44) computed in
the preceding sections. The first coefficient,
$\mathcal{A}\\!=\\!\mathcal{A}_{\scriptscriptstyle
S}\\!+\\!\mathcal{A}_{\scriptscriptstyle V}$, receives contributions from both
the scalar and vector loop source, which are, respectively, succinctly written
as,

$\mathcal{A}_{\scriptscriptstyle
S}=\lambda(1\\!-\\!\epsilon)^{2}\Bigl{(}\frac{1}{4}\\!-\\!\nu_{\scriptscriptstyle
S}^{2}\Bigr{)}\,,\qquad\qquad\mathcal{A}_{\scriptscriptstyle
V}=q^{2}(1\\!-\\!\epsilon)^{2}\Bigl{(}\frac{9}{4}\\!-\\!\nu_{\scriptscriptstyle
V}^{2}\Bigr{)}\,.$ (46)

Analogous is true for the second coefficient,
$\mathcal{B}\\!=\\!\mathcal{B}_{\scriptscriptstyle
S}\\!+\\!\mathcal{B}_{\scriptscriptstyle V}\\!+\\!\delta\mathcal{B}$, where
scalar and vector contributions are, respectively,

$\displaystyle\mathcal{B}_{\scriptscriptstyle S}={}$
$\displaystyle\lambda(1\\!-\\!\epsilon)^{2}\Bigl{(}\frac{1}{4}\\!-\\!\nu_{\scriptscriptstyle
S}^{2}\Bigr{)}\Psi(\nu_{\scriptscriptstyle S})\,,$ (47a)
$\displaystyle\mathcal{B}_{\scriptscriptstyle V}={}$ $\displaystyle
q^{2}(1\\!-\\!\epsilon)^{2}\Bigl{(}\frac{9}{4}\\!-\\!\nu_{\scriptscriptstyle
V}^{2}\Bigr{)}\Psi(\nu_{\scriptscriptstyle
V})+q^{2}\epsilon\biggl{[}\frac{3}{2}-\frac{2(1\\!-\\!\epsilon)^{3}H_{0}^{2}}{\bigl{(}q\overline{\phi}_{0}\bigr{)}^{\\!2}}\biggr{]}\,,$
(47b)

but in addition it receives a contribution from the finite parts of the
counterterms,

$\frac{3\,\delta\mathcal{B}}{16\pi^{2}}=6(2\\!-\\!\epsilon)\bigl{[}\delta\xi\\!-\\!\xi\delta
Z_{\phi}\bigr{]}^{\rm
fin.}+\frac{\overline{\phi}_{0}^{\,2}}{H_{0}^{2}}\bigl{[}\delta\lambda\\!-\\!\lambda\delta
Z_{\phi}\bigr{]}^{\rm fin.}\,.$ (48)

The solution of equation (45) is not complicated,

$\bigl{\langle}\hat{\varphi}\bigr{\rangle}=\frac{3\overline{\phi}}{16\pi^{2}}\biggl{[}2\mathscr{A}\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}+\mathscr{B}\biggr{]}+\frac{3\overline{\phi}}{16\pi^{2}}\biggl{[}\mathscr{C}_{\scriptscriptstyle+}a^{p_{+}}+\mathscr{C}_{\scriptscriptstyle-}a^{p_{-}}\biggr{]}\,,$
(49)

where the left term is a dynamical contribution, with the two coefficients
fixed by the sources of the equation of motion (45),

$\mathscr{A}=-\frac{H_{0}^{2}}{2\lambda\overline{\phi}_{0}^{\,2}}\mathcal{A}\,,\qquad\qquad\mathscr{B}=-\frac{H_{0}^{2}}{2\lambda\overline{\phi}_{0}^{\,2}}\Bigl{[}\mathcal{B}-6\epsilon(1\\!-\\!\epsilon)\mathscr{A}\Bigr{]}\,,$
(50)

while the right term contains two free constants of integration corresponding
to homogeneous solutions that redshift away, as their powers,

$p_{\scriptscriptstyle\pm}=-(1\\!-\\!\epsilon)\Bigl{(}\frac{3}{2}\pm\nu_{\scriptscriptstyle
S}\Bigr{)}\,,$ (51)

both have a negative real part (note that $\nu_{\scriptscriptstyle S}$ can be
real or imaginary). Imposing initial conditions on
$\bigl{\langle}\hat{\varphi}\bigr{\rangle}$ and its derivative at the initial
time $\eta_{0}$ fixes the free integration constants
$\mathscr{C}_{\scriptscriptstyle+}$ and $\mathscr{C}_{\scriptscriptstyle-}$ in
(49). Requiring that the condensate corrections are minimized at the initial
moment $\eta_{0}$, in the sense that both
$\bigl{\langle}\hat{\varphi}\bigr{\rangle}(\eta_{0})\\!=\\!0$ and
$\partial_{0}\bigl{\langle}\hat{\varphi}\bigr{\rangle}(\eta_{0})\\!=\\!0$,
fixes the two constants, 222The choice of
$\mathscr{C}_{\scriptscriptstyle\pm}$ in (52) is valid both when $p_{\pm}$ are
real or a pair of complex conjugate numbers. In the latter case,
$(\mathscr{C}_{\scriptscriptstyle+}a^{p_{+}})^{*}\\!=\\!\mathscr{C}_{\scriptscriptstyle-}a^{p_{-}}$
such that $\langle\hat{\varphi}\rangle$ in (49) remains real.

$\displaystyle\mathscr{C}_{\scriptscriptstyle+}={}$
$\displaystyle\frac{p_{\scriptscriptstyle-}}{(p_{\scriptscriptstyle+}\\!-p_{\scriptscriptstyle-})}\biggl{[}2\mathscr{A}\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H_{0}}{\mu}\Bigr{]}+\mathscr{B}+\frac{2\epsilon\mathscr{A}}{p_{\scriptscriptstyle-}}\biggr{]}\,,$
(52a) $\displaystyle\mathscr{C}_{\scriptscriptstyle-}={}$
$\displaystyle\frac{p_{\scriptscriptstyle+}}{(p_{\scriptscriptstyle-}\\!-p_{\scriptscriptstyle+})}\biggl{[}2\mathscr{A}\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H_{0}}{\mu}\Bigr{]}+\mathscr{B}+\frac{2\epsilon\mathscr{A}}{p_{\scriptscriptstyle+}}\biggr{]}\,.$
(52b)

The choice of such coefficients amounts to a finite renormalization of the
initial state, and is important when discussing the fine-tuning issues in
inflation Miao:2015oba . However, here we are primarily interested in
corrections descending from dynamics. Given that the state-dependent
contributions decay in time in comparison, we shall henceforth not consider
them, and set
$\mathscr{C}_{\scriptscriptstyle+}\\!=\\!\mathscr{C}_{\scriptscriptstyle-}\\!=\\!0$.
Thus, the one-loop correction to the condensate receives additive
contributions,
$\bigl{\langle}\hat{\varphi}\bigr{\rangle}\\!=\\!\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
S}\\!+\\!\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle V}$,
from the scalar and the vector loop, respectively,

$\displaystyle\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
S}={}$
$\displaystyle\\!-\frac{3\lambda\overline{\phi}}{16\pi^{2}}\biggl{[}\frac{3\lambda\overline{\phi}^{2}\\!-\\!\bigl{(}\frac{1}{6}\\!-\\!\xi\bigr{)}R}{2\lambda\overline{\phi}^{\,2}}\biggr{]}\biggl{[}2\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}+\Psi(\nu_{\scriptscriptstyle
S})+\frac{3\epsilon(1\\!-\\!\epsilon)H^{2}}{\lambda\overline{\phi}^{\,2}}\biggr{]}\,,$
(53)
$\displaystyle\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
V}={}$
$\displaystyle-\frac{3q^{2}\overline{\phi}}{16\pi^{2}}\Biggl{\\{}\biggl{[}\frac{\bigl{(}q\overline{\phi}\bigr{)}^{\\!2}\\!-\\!\lambda\overline{\phi}^{2}\\!+\\!\bigl{(}\frac{1}{6}\\!-\\!\xi\bigr{)}R}{2\lambda\overline{\phi}^{2}}\biggr{]}\biggl{[}2\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}+\Psi(\nu_{\scriptscriptstyle
V})$ $\displaystyle\hskip
56.9055pt+\frac{3\epsilon(1\\!-\\!\epsilon)H^{2}}{\lambda\overline{\phi}^{\,2}}\biggr{]}+\frac{3\epsilon
H^{2}}{4\lambda\overline{\phi}^{\,2}}\Biggr{\\}}+\frac{3\lambda\overline{\phi}}{16\pi^{2}}\\!\times\\!\frac{\epsilon(1\\!-\\!\epsilon)^{3}H^{4}}{\bigl{(}\lambda\overline{\phi}^{\,2}\bigr{)}^{\\!2}}\,,$
(54)

where the simplest choice for the finite parts of the counterterms in (48) is
$\delta\mathcal{B}\\!=\\!0$, which implies,

$\bigl{[}\delta\xi\\!-\\!\xi\delta Z_{\phi}\bigr{]}^{\rm
fin.}=0\,,\qquad\quad\bigl{[}\delta\lambda\\!-\\!\lambda\delta
Z_{\phi}\bigr{]}^{\rm fin.}=0\,,$ (55)

that we henceforth assume. Note that in expressions above the argument of the
logarithms, and the mode function indices are expressed in terms of the scalar
field and curvature scalar using (31).

##  4 Energy-momentum tensor

In order to have a well defined one-loop energy-momentum tensor, in addition
to the action for the model in (12), we need to consider purely geometrical
higher-derivative counterterms tHooft:1974toh ,

$S_{\rm
ctm.}[g_{\mu\nu}]=\int\\!d^{D\\!}x\,\sqrt{-g}\,\Bigl{[}\alpha_{\scriptscriptstyle
R}R^{2}+\alpha_{\scriptscriptstyle
C}\mathcal{C}^{2}+\alpha_{\scriptscriptstyle G}\mathcal{G}\Bigr{]}\,,$ (56)

where the first counterterm is the square of the Ricci tensor, the second is
the square of the Weyl tensor (9),
$C^{2}\\!=\\!C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}$, and the last one is
the Gauss-Bonnet invariant
$\mathcal{G}\\!=\\!R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}\\!-\\!4R^{\mu\nu}R_{\mu\nu}\\!+\\!R^{2}$.
Only the first of these counterterms contributes to renormalization on FLRW
background. The variation of the $\mathcal{C}^{2}$ term is proportional to the
Weyl tensor which vanishes for conformally flat spacetimes (9), and the Gauss-
Bonnet part is only a surface term that does not contribute in the bulk. Thus,
the full energy-momentum tensor operator is defined as

$\displaystyle\hat{T}_{\mu\nu}={}$
$\displaystyle\frac{-2}{\sqrt{-g}}\frac{\delta S_{*}}{\delta
g_{\mu\nu}}\biggr{|}_{\begin{subarray}{c}\phi\to\hat{\phi}\\\
A\to\hat{A}\end{subarray}}=Z_{A}\biggl{[}\delta_{(\mu}^{\rho}\delta_{\nu)}^{\sigma}\\!-\\!\frac{1}{4}g_{\mu\nu}g^{\rho\sigma}\biggr{]}g^{\alpha\beta}\hat{F}_{\rho\alpha}\hat{F}_{\sigma\beta}$
(57)
$\displaystyle+\bigl{(}q_{0}\hat{\phi}\bigr{)}^{2}\biggl{[}\delta_{(\mu}^{\rho}\delta_{\nu)}^{\sigma}\\!-\\!\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\biggr{]}\hat{A}_{\rho}\hat{A}_{\sigma}+Z_{\phi}\biggl{[}\delta_{(\mu}^{\rho}\delta_{\nu)}^{\sigma}\\!-\\!\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\biggr{]}\bigl{(}\partial_{\rho}\hat{\phi}\bigr{)}\bigl{(}\partial_{\sigma}\hat{\phi}\bigr{)}$
$\displaystyle-\frac{\lambda_{0}}{4}\hat{\phi}^{\,4}g_{\mu\nu}+\xi_{0}\Bigl{[}G_{\mu\nu}+g_{\mu\nu}\mathop{\mathchoice{\displaystyle\displaystyle
3\makebox[7.7778pt]{\hbox to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox
to6.75pt{ \hrule height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern
0.0pt\vrule width=0.0pt}}}{\textstyle\displaystyle 3\makebox[7.7778pt]{\hbox
to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox to6.75pt{ \hrule
height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptstyle\displaystyle 3\makebox[5.44446pt]{\hbox
to4.72499pt{\vrule width=0.0pt\kern 0.0pt\vbox to4.72499pt{ \hrule
height=0.0pt,width=4.72499pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptscriptstyle\displaystyle 3\makebox[3.8889pt]{\hbox
to3.375pt{\vrule width=0.0pt\kern 0.0pt\vbox to3.375pt{ \hrule
height=0.0pt,width=3.375pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}}-\nabla_{\\!\mu}\nabla_{\\!\nu}\Bigr{]}\hat{\phi}^{\,2}+\alpha_{\scriptscriptstyle
R}H_{\mu\nu}^{\scriptscriptstyle R}\,,$

where the last term is,

$\displaystyle H_{\mu\nu}^{\scriptscriptstyle R}={}$ $\displaystyle
4\nabla_{\mu}\nabla_{\nu}R-4g_{\mu\nu}\mathop{\mathchoice{\displaystyle\displaystyle
3\makebox[7.7778pt]{\hbox to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox
to6.75pt{ \hrule height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern
0.0pt\vrule width=0.0pt}}}{\textstyle\displaystyle 3\makebox[7.7778pt]{\hbox
to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox to6.75pt{ \hrule
height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptstyle\displaystyle 3\makebox[5.44446pt]{\hbox
to4.72499pt{\vrule width=0.0pt\kern 0.0pt\vbox to4.72499pt{ \hrule
height=0.0pt,width=4.72499pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptscriptstyle\displaystyle 3\makebox[3.8889pt]{\hbox
to3.375pt{\vrule width=0.0pt\kern 0.0pt\vbox to3.375pt{ \hrule
height=0.0pt,width=3.375pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}}R-4RR_{\mu\nu}+g_{\mu\nu}R^{2}$
$\displaystyle\xrightarrow{\epsilon={\rm
const.}}(D\\!-\\!1)(D\\!-\\!2\epsilon)(D\\!-\\!4\\!-\\!6\epsilon)H^{4}\Bigl{[}(D\\!-\\!1\\!-\\!4\epsilon)g_{\mu\nu}-4\epsilon\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\Bigr{]}\,.$
(58)

We expand the energy-momentum tensor operator (57) up to quadratic order in
fluctuating fields,

$\hat{T}_{\mu\nu}=\overline{T}_{\mu\nu}+\hat{t}_{\mu\nu}^{\scriptscriptstyle\,(1)}+\hat{t}_{\mu\nu}^{\scriptscriptstyle\,(2)}+\dots$
(59)

where higher orders do not contribute at one-loop level. The three parts are
defined and their expectation values computed in the following three
subsections.

###  4.1 Tree-level part

The classical contribution to the energy-momentum tensor derives from the
scalar condensate (14) only,

$\displaystyle\overline{T}_{\mu\nu}={}$
$\displaystyle\biggl{[}\delta_{\mu}^{\rho}\delta_{\nu}^{\sigma}\\!-\\!\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\biggr{]}\bigl{(}\partial_{\rho}\overline{\phi}\bigr{)}\bigl{(}\partial_{\sigma}\overline{\phi}\bigr{)}-\frac{\lambda}{4}\overline{\phi}^{\,4}g_{\mu\nu}+\xi\Bigl{[}G_{\mu\nu}+g_{\mu\nu}\mathop{\mathchoice{\displaystyle\displaystyle
3\makebox[7.7778pt]{\hbox to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox
to6.75pt{ \hrule height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern
0.0pt\vrule width=0.0pt}}}{\textstyle\displaystyle 3\makebox[7.7778pt]{\hbox
to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox to6.75pt{ \hrule
height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptstyle\displaystyle 3\makebox[5.44446pt]{\hbox
to4.72499pt{\vrule width=0.0pt\kern 0.0pt\vbox to4.72499pt{ \hrule
height=0.0pt,width=4.72499pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptscriptstyle\displaystyle 3\makebox[3.8889pt]{\hbox
to3.375pt{\vrule width=0.0pt\kern 0.0pt\vbox to3.375pt{ \hrule
height=0.0pt,width=3.375pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}}-\nabla_{\\!\mu}\nabla_{\\!\nu}\Bigr{]}\overline{\phi}^{\,2}\,.$
(60)

Evaluated on the attractor solution (14) it reads,

$\displaystyle\overline{T}_{\mu\nu}={}$
$\displaystyle\frac{(1\\!-\\!6\xi)\epsilon+(D\\!-\\!4)\xi}{4}\Bigl{[}-(D\\!-\\!1\\!-\\!4\epsilon)g_{\mu\nu}+4\epsilon\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\Bigr{]}\bigl{(}\overline{\phi}H\bigr{)}^{\\!2}$
$\displaystyle\xrightarrow{D\to
4}\frac{(1\\!-\\!6\xi)\epsilon}{4}\biggl{[}-(3\\!-\\!4\epsilon)g_{\mu\nu}+4\epsilon\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\biggr{]}\bigl{(}\overline{\phi}H\bigr{)}^{\\!2}\,,$
(61)

and it is indeed covariantly conserved,
$\nabla^{\mu}\overline{T}_{\mu\nu}\\!=\\!0$. It describes an effective ideal
fluid with an equation of state dependent on the background,

$\overline{w}=-1+\frac{4\epsilon}{3}\,.$ (62)

The energy-momentum tensor of the condensate, which is a spectator from the
perspective of the expansion, redshifts away faster
($\rho\\!\propto\\!H^{4}\\!\propto\\!a^{-4\epsilon}$) than some fluid driving
the expansion of the power-law inflation ($\rho\propto H^{2}\propto
a^{-2\epsilon}$)

###  4.2 Part linear in fluctuations

Graviton couples to linearized fluctuations too. This would be the
contribution to the energy-momentum tensor coming from the correction to the
scalar tadpole. However, this split should only go so far on the account of it
not being conserved by itself! Therefore, contributions cannot be split in a
physical way.

Figure 2: Diagrams depicting the one-loop corrections to the graviton one-
point function descending from the one-loop corrections to the scalar one-
point function in Fig. 1. Curly lines correspond to the graviton propagator,
dashed line to the scalar propagator, and wavy lines to the vector propagator,
while encircled crosses stand for the classical condensate insertions. The
last diagram stands for counterterms. Amputating the graviton propagator
leaves the contribution to the one-loop energy-momentum tensor.

$\bigl{\langle}\hat{t}_{\mu\nu}^{\,\scriptscriptstyle(1)}\bigr{\rangle}=\Bigl{[}2\delta_{(\mu}^{\rho}\delta_{\nu)}^{\sigma}\\!-\\!g_{\mu\nu}g^{\rho\sigma}\Bigr{]}\bigl{(}\partial_{\rho}\overline{\phi}\bigr{)}\partial_{\sigma}\bigl{\langle}\hat{\varphi}\bigr{\rangle}-\lambda\overline{\phi}^{\,3}\bigl{\langle}\hat{\varphi}\bigr{\rangle}g_{\mu\nu}+2\xi\Bigl{[}G_{\mu\nu}\\!+\\!g_{\mu\nu}\\!\mathop{\mathchoice{\displaystyle\displaystyle
3\makebox[7.7778pt]{\hbox to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox
to6.75pt{ \hrule height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern
0.0pt\vrule width=0.0pt}}}{\textstyle\displaystyle 3\makebox[7.7778pt]{\hbox
to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox to6.75pt{ \hrule
height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptstyle\displaystyle 3\makebox[5.44446pt]{\hbox
to4.72499pt{\vrule width=0.0pt\kern 0.0pt\vbox to4.72499pt{ \hrule
height=0.0pt,width=4.72499pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptscriptstyle\displaystyle 3\makebox[3.8889pt]{\hbox
to3.375pt{\vrule width=0.0pt\kern 0.0pt\vbox to3.375pt{ \hrule
height=0.0pt,width=3.375pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}}-\nabla_{\mu}\nabla_{\nu}\Bigr{]}\overline{\phi}\bigl{\langle}\hat{\varphi}\bigr{\rangle}\,.$
(63)

This contribution is not covariantly conserved by itself. In fact, from the
tree-level equation of motion (13), and the one-loop tadpole equation (33) it
follows,

$\nabla^{\mu}\bigl{\langle}\hat{t}_{\mu\nu}^{\,\scriptscriptstyle(1)}\bigr{\rangle}=\bigl{(}\partial_{\nu}\overline{\phi}\bigr{)}\Bigl{[}\mathcal{S}_{\scriptscriptstyle
S}+\mathcal{S}_{\scriptscriptstyle V}+\delta\mathcal{S}\Bigr{]}\,.$ (64)

This non-conservation has nothing to do with quantum anomalies, or quantum
physics for that matter. Rather, it is a consequence of the part quadratic in
fluctuations, discussed in the following section, contributing to the energy-
momentum tensor at the same order as the part linear in fluctuations discussed
in this section.

Plugging in the tree-level attractor solution for the condensate (14), and the
one-loop condensate correction (49) into (63) gives the tadpole contribution
to the one-loop energy-momentum tensor. Just like the condensate correction it
receives dynamically generated contributions, and the initial state dependent
contributions,

$\bigl{\langle}\hat{t}_{\mu\nu}^{\,\scriptscriptstyle(1)}\bigr{\rangle}=\bigl{\langle}\hat{t}_{\mu\nu}^{\,\scriptscriptstyle(1)}\bigr{\rangle}_{\rm
dyn.}+\mathcal{T}_{\mu\nu}^{\scriptscriptstyle+}+\mathcal{T}_{\mu\nu}^{\scriptscriptstyle-}\,.$
(65)

where the former contribution is,

$\displaystyle\bigl{\langle}\hat{t}_{\mu\nu}^{\,\scriptscriptstyle(1)}\bigr{\rangle}_{\rm
dyn.}={}$
$\displaystyle\frac{3}{8\pi^{2}}\biggl{[}2\mathscr{A}\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}+\mathscr{B}\biggr{]}\biggl{[}\overline{T}_{\mu\nu}-\frac{\lambda}{4}\overline{\phi}^{\,4}g_{\mu\nu}\biggr{]}$
(66)
$\displaystyle-\frac{\bigl{(}\overline{\phi}H\bigr{)}^{\\!2}\epsilon\mathscr{A}}{8\pi^{2}}\biggl{\\{}\Bigl{[}(1\\!-\\!6\xi)(2\\!-\\!5\epsilon)\\!-\\!2(1\\!-\\!\epsilon)\Bigr{]}g_{\mu\nu}-\Bigl{[}(1\\!-\\!6\xi)(1\\!+\\!5\epsilon)\\!-\\!(1\\!-\\!\epsilon)\Bigr{]}\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\biggr{\\}},$

while the latter contributions are,

$\displaystyle\mathcal{T}_{\mu\nu}^{\scriptscriptstyle\pm}={}$
$\displaystyle\frac{3\mathscr{C}_{\scriptscriptstyle\pm}a^{p_{\pm}}\bigl{(}\overline{\phi}H\bigr{)}^{\\!2}}{16\pi^{2}}\biggl{\\{}g_{\mu\nu}\\!\times\\!\bigl{(}3\\!-\\!3\epsilon\\!+\\!p_{\scriptscriptstyle\pm}\bigr{)}\Bigl{[}2\xi\bigl{(}1\\!+\\!2\epsilon\\!-\\!p_{\scriptscriptstyle\pm}\bigr{)}\\!-\\!\epsilon\,\Bigr{]}$
$\displaystyle\hskip
28.45274pt+\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\\!\times\\!2\Bigl{[}\epsilon\bigl{(}\epsilon\\!-\\!p_{\scriptscriptstyle\pm}\bigr{)}+\xi
p_{\scriptscriptstyle\pm}-\xi\bigl{(}3\epsilon\\!-\\!p_{\scriptscriptstyle\pm}\bigr{)}\bigl{(}2\epsilon\\!-\\!p_{\scriptscriptstyle\pm}\bigr{)}\Bigr{]}\biggr{\\}}\,.$
(67)

These initial state dependent contributions are conserved by themselves,
$\nabla^{\mu}\mathcal{T}_{\mu\nu}^{\scriptscriptstyle\pm}\\!=\\!0$. They
redshift away compared to the dynamical contribution and we do not consider
them further.

###  4.3 Part quadratic in fluctuations

Figure 3: Diagrams depicting the one-loop corrections to the graviton one-
point function descending from vacuum fluctuations. Curly lines correspond to
the graviton propagator, dashed line to the scalar propagator, and wavy lines
to the vector propagator. The last diagram stands for counterterms. Amputating
the graviton propagator leaves the contribution to the one-loop energy-
momentum tensor.

The renormalized expectation value of the quadratic part of the energy-
momentum tensor is a sum of all three contributions from Fig. 3,

$\bigl{\langle}\hat{t}_{\mu\nu}^{\,\scriptscriptstyle(2)}\bigr{\rangle}=\bigl{\langle}\hat{t}_{\mu\nu}^{\,\scriptscriptstyle(2)}\bigr{\rangle}_{\scriptscriptstyle
S}+\bigl{\langle}\hat{t}_{\mu\nu}^{\,\scriptscriptstyle(2)}\bigr{\rangle}_{\scriptscriptstyle
S}+\delta T_{\mu\nu}\,,$ (68)

where the first contribution comes from the leftmost digram with a scalar
loop,

$\displaystyle\bigl{\langle}\hat{t}_{\mu\nu}^{\,\scriptscriptstyle(2)}\bigr{\rangle}_{\scriptscriptstyle
S}={}$
$\displaystyle\biggl{[}\delta_{(\mu}^{\rho}\delta_{\nu)}^{\sigma}\\!-\\!\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\biggr{]}\bigl{\langle}\partial_{\rho}\hat{\varphi}^{0}\,\partial_{\sigma}\hat{\varphi}^{0}\bigr{\rangle}-\frac{3\lambda}{2}\overline{\phi}^{\,2}\bigl{\langle}\hat{\varphi}^{0}\hat{\varphi}^{0}\bigr{\rangle}g_{\mu\nu}$
$\displaystyle+\xi\Bigl{[}G_{\mu\nu}\\!+\\!g_{\mu\nu}\mathop{\mathchoice{\displaystyle\displaystyle
3\makebox[7.7778pt]{\hbox to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox
to6.75pt{ \hrule height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern
0.0pt\vrule width=0.0pt}}}{\textstyle\displaystyle 3\makebox[7.7778pt]{\hbox
to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox to6.75pt{ \hrule
height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptstyle\displaystyle 3\makebox[5.44446pt]{\hbox
to4.72499pt{\vrule width=0.0pt\kern 0.0pt\vbox to4.72499pt{ \hrule
height=0.0pt,width=4.72499pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptscriptstyle\displaystyle 3\makebox[3.8889pt]{\hbox
to3.375pt{\vrule width=0.0pt\kern 0.0pt\vbox to3.375pt{ \hrule
height=0.0pt,width=3.375pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}}-\nabla_{\\!\mu}\nabla_{\\!\nu}\Bigr{]}\bigl{\langle}\hat{\varphi}^{0}\hat{\varphi}^{0}\bigr{\rangle}\,,$
(69)

the second contribution comes from the middle diagram with the vector loop,

$\bigl{\langle}\hat{t}_{\mu\nu}^{\,\scriptscriptstyle(2)}\bigr{\rangle}_{\scriptscriptstyle
V}=\biggl{[}\delta_{(\mu}^{\rho}\delta_{\nu)}^{\sigma}g^{\alpha\beta}\\!-\\!\frac{1}{4}g_{\mu\nu}g^{\rho\sigma}g^{\alpha\beta}\biggr{]}\bigl{\langle}\hat{F}_{\rho\alpha}^{0}\hat{F}_{\sigma\beta}^{0}\bigr{\rangle}+\bigl{(}q\overline{\phi}\bigr{)}^{2}\biggl{[}\delta_{(\mu}^{\rho}\delta_{\nu)}^{\sigma}\\!-\\!\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\biggr{]}\bigl{\langle}\hat{A}_{\rho}^{0}\hat{A}_{\sigma}^{0}\bigr{\rangle}\,,$
(70)

and the remaining contribution comes from counterterms, denoted by the
rightmost diagram,

$\displaystyle\delta T_{\mu\nu}={}$ $\displaystyle\delta
Z_{\phi}\biggl{[}\delta_{(\mu}^{\rho}\delta_{\nu)}^{\sigma}\\!-\\!\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\biggr{]}\bigl{(}\partial_{\rho}\overline{\phi}\bigr{)}\bigl{(}\partial_{\sigma}\overline{\phi}\bigr{)}-\frac{\delta\lambda}{4}\overline{\phi}^{\,4}g_{\mu\nu}$
$\displaystyle\hskip
42.67912pt+\delta\xi\Bigl{[}G_{\mu\nu}+g_{\mu\nu}\mathop{\mathchoice{\displaystyle\displaystyle
3\makebox[7.7778pt]{\hbox to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox
to6.75pt{ \hrule height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern
0.0pt\vrule width=0.0pt}}}{\textstyle\displaystyle 3\makebox[7.7778pt]{\hbox
to6.75pt{\vrule width=0.0pt\kern 0.0pt\vbox to6.75pt{ \hrule
height=0.0pt,width=6.75pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptstyle\displaystyle 3\makebox[5.44446pt]{\hbox
to4.72499pt{\vrule width=0.0pt\kern 0.0pt\vbox to4.72499pt{ \hrule
height=0.0pt,width=4.72499pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}{\scriptscriptstyle\displaystyle 3\makebox[3.8889pt]{\hbox
to3.375pt{\vrule width=0.0pt\kern 0.0pt\vbox to3.375pt{ \hrule
height=0.0pt,width=3.375pt\vss\hrule height=0.0pt}\kern 0.0pt\vrule
width=0.0pt}}}}-\nabla_{\\!\mu}\nabla_{\\!\nu}\Bigr{]}\overline{\phi}^{\,2}+\delta\alpha_{\scriptscriptstyle
R}H_{\mu\nu}^{\scriptscriptstyle R}\,.$ (71)

This contribution is also not conserved on its own,

$\nabla^{\mu}\bigl{\langle}\hat{t}_{\mu\nu}^{\,\scriptscriptstyle(2)}\bigr{\rangle}=-\bigl{(}\partial_{\nu}\overline{\phi}\bigr{)}\Bigl{[}\mathcal{S}_{\scriptscriptstyle
S}+\mathcal{S}_{\scriptscriptstyle V}+\delta\mathcal{S}\Bigr{]}\,,$ (72)

but rather its non-conservation precisely cancels the one in (64), so that the
entire one-loop energy momentum tensor is covariantly conserved, as it should
be,

$\nabla^{\mu}\Bigl{(}\bigl{\langle}\hat{t}_{\mu\nu}^{\,\scriptscriptstyle(1)}\bigr{\rangle}+\bigl{\langle}\hat{t}_{\mu\nu}^{\,\scriptscriptstyle(2)}\bigr{\rangle}\Bigr{)}=0\,.$
(73)

The counterterm part, evaluated for power-law inflation reads,

$\displaystyle\delta T_{\mu\nu}={}$ $\displaystyle
g_{\mu\nu}\\!\times\\!\biggl{\\{}-\frac{\delta\lambda}{4}\overline{\phi}^{4}+\frac{\epsilon^{2}}{2}\delta
Z_{\phi}\bigl{(}\overline{\phi}H\bigr{)}^{\\!2}+\delta\alpha_{\scriptscriptstyle
R}(D\\!-\\!1)(D\\!-\\!2\epsilon)(D\\!-\\!1\\!-\\!4\epsilon)(D\\!-\\!4\\!-\\!6\epsilon)H^{4}$
$\displaystyle\hskip
28.45274pt-\frac{\delta\xi}{2}\Bigl{[}(D\\!-\\!1)(D\\!-\\!2)-6(D\\!-\\!2)\epsilon+12\epsilon^{2}\Bigr{]}H^{2}\overline{\phi}^{\,2}\biggr{\\}}$
$\displaystyle+\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\\!\times\\!\biggl{\\{}\bigl{(}\delta
Z_{\phi}\\!-\\!6\delta\xi\bigr{)}\epsilon^{2}\bigl{(}\overline{\phi}H\bigr{)}^{\\!2}+24\delta\alpha_{\scriptscriptstyle
R}(D\\!-\\!1)(D\\!-\\!2\epsilon)\epsilon^{2}H^{4}$ $\displaystyle\hskip
56.9055pt+(D\\!-\\!4)\delta\xi\epsilon\bigl{(}\overline{\phi}H\bigr{)}^{\\!2}-4(D\\!-\\!4)\delta\alpha_{\scriptscriptstyle
R}(D\\!-\\!1)(D\\!-\\!2\epsilon)\epsilon H^{4}\biggr{\\}}\,.$ (74)

Just as for the tadpole, the scalar and the vector loops are renormalized
independently in the energy-monetum tensor, and we split the contributions,

$\delta T_{\mu\nu}=\bigl{[}\delta T_{\mu\nu}\bigr{]}_{\scriptscriptstyle
S}^{\rm div.}+\bigl{[}\delta T_{\mu\nu}\bigr{]}_{\scriptscriptstyle V}^{\rm
div.}+\bigl{[}\delta T_{\mu\nu}\bigr{]}^{\rm fin.}\,.$ (75)

####  4.3.1 Scalar loop contribution

For the scalar part the scalar parts of counterterms from before are enough.
This is because the vector parts of counterterms vanish for $q\\!\to\\!0$.

$\displaystyle\bigl{\langle}\hat{t}^{\scriptscriptstyle\,(2)}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
S}=g_{\mu\nu}\times\boldsymbol{\Gamma}(\nu_{\scriptscriptstyle
S})(1\\!-\\!\epsilon)^{2}H^{2}\biggl{[}\Bigl{(}\frac{D\\!-\\!3}{2}\Bigr{)}^{\\!2}\\!-\nu_{\scriptscriptstyle
S}^{2}\biggr{]}$ (76) $\displaystyle\hskip
28.45274pt\times\frac{1}{D}\Biggl{\\{}-\frac{(D\\!-\\!2)^{2}}{4}\biggl{[}1-\frac{4(D\\!-\\!1)\xi}{(D\\!-\\!2)}\biggr{]}(D\\!-\\!1\\!-\\!D\epsilon)\epsilon
H^{2}-3\lambda\overline{\phi}^{\,2}\Biggr{\\}}$
$\displaystyle+\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\times\boldsymbol{\Gamma}(\nu_{\scriptscriptstyle
S})\biggl{[}1\\!-\\!\frac{4(D\\!-\\!1)\xi}{(D\\!-\\!2)}\biggr{]}\frac{(D\\!-\\!2)^{2}\epsilon^{2}H^{2}}{32}\times\Biggl{\\{}(D\\!+\\!2)(8\\!-\\!D)\lambda\overline{\phi}^{\,2}$
$\displaystyle\hskip
42.67912pt-\frac{D(D\\!-\\!2)(6\\!-\\!D)}{4}\biggl{[}1\\!-\\!\frac{4(D\\!-\\!1)\xi}{(D\\!-\\!2)}\biggr{]}(D\\!-\\!2\epsilon)H^{2}+\mathcal{O}\bigl{[}(D\\!-\\!4)^{2}\bigr{]}\Biggr{\\}}\,.$

Since the factor $\boldsymbol{\Gamma}(\nu_{\scriptscriptstyle S})$ defined in
(25) is divergent in $D\\!=\\!4$, so are most of the contributions to the
naive expectation value above. These need to be absorbed by the counterterms
in (74), by judiciously choosing four counterterm coefficients. Note that we
already have two conditions on these coefficients (38) from renormalizing the
scalar contribution to the tadpole. Therefore, there are only two independent
conditions that we can require here. They are most conveniently found by
requiring that the divergences of the
$\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}$ part are absorbed,
since that is accomplished by a simple comparison with (74), and the simplest
choice is

$\displaystyle\bigl{[}\delta
Z_{\phi}\\!-\\!6\delta\xi\bigr{]}_{\scriptscriptstyle S}^{\rm div.}={}$
$\displaystyle\frac{\mu^{D-4}\,\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}}{(4\pi)^{\frac{D}{2}}}\\!\times\\!\frac{(D\\!-\\!2)^{2}(D\\!+\\!2)(D\\!-\\!8)\lambda}{32}\biggl{[}1\\!-\\!\frac{4(D\\!-\\!1)\xi}{(D\\!-\\!2)}\biggr{]}\,,$
(77) $\displaystyle\bigl{[}\delta\alpha_{\scriptscriptstyle
R}\bigr{]}_{\scriptscriptstyle S}^{\rm div.}={}$
$\displaystyle\frac{\mu^{D-4}\,\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}}{(4\pi)^{\frac{D}{2}}}\times\frac{D(D\\!-\\!2)^{2}}{768(D\\!-\\!1)}\biggl{[}1\\!-\\!\frac{4(D\\!-\\!1)\xi}{(D\\!-\\!2)}\biggr{]}^{2}\,.$
(78)

In addition to fixing $\delta\alpha_{\scriptscriptstyle R}$, this fixes the
three counterterms,

$\displaystyle\bigl{[}\delta\xi\bigr{]}_{\scriptscriptstyle S}^{\rm div.}={}$
$\displaystyle\frac{\mu^{D-4}\,\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}}{(4\pi)^{\frac{D}{2}}}\times\frac{3D\lambda\bigl{[}1\\!-\\!(D\\!-\\!1)(D\\!-\\!2)\xi\bigr{]}}{8(D\\!-\\!1)}\,,$
(79a) $\displaystyle\bigl{[}\delta\lambda\bigr{]}_{\scriptscriptstyle S}^{\rm
div.}={}$
$\displaystyle\frac{\mu^{D-4}\,\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}}{(4\pi)^{\frac{D}{2}}}\times\biggl{[}-\frac{9D\lambda^{2}}{4}\biggr{]}\,,$
(79b) $\displaystyle\bigl{[}\delta Z_{\phi}\bigr{]}_{\scriptscriptstyle
S}^{\rm div.}={}$
$\displaystyle\frac{\mu^{D-4}\,\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}}{(4\pi)^{\frac{D}{2}}}\times\Bigl{[}-3(D\\!-\\!4)\lambda\Bigr{]}\,,$
(79c)

which agrees (up to finite contributions) with the counterterms found in
Janssen:2009pb . Note that the wavefunction renormalization is finite, which
agrees with what was also found in Ref. Canevarolo:2022dvh  in a different
setting. Now we have for the renormalized scalar part of the energy-momentum
tensor,

$\displaystyle\bigl{\langle}\hat{t}^{\scriptscriptstyle\,(2)}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
S}\\!+\\!\bigl{[}\delta T_{\mu\nu}\bigr{]}_{\scriptscriptstyle S}^{\rm
div.}\\!\xrightarrow{D\to
4}-g_{\mu\nu}\\!\times\\!\frac{3\lambda\overline{\phi}^{\,2}}{64\pi^{2}}(1\\!-\\!\epsilon)^{2}H^{2}\Bigl{(}\frac{1}{4}\\!-\\!\nu_{\scriptscriptstyle
S}^{2}\Bigr{)}\biggl{[}2\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}\\!+\Psi(\nu_{\scriptscriptstyle
S})\\!-\frac{1}{2}\biggr{]}$
$\displaystyle-\frac{(1\\!-\\!6\xi)H^{4}}{64\pi^{2}}(1\\!-\\!\epsilon)^{2}\Bigl{(}\frac{1}{4}\\!-\\!\nu_{\scriptscriptstyle
S}^{2}\Bigr{)}\Biggl{\\{}g_{\mu\nu}\\!\times\\!\biggl{[}(3\\!-\\!4\epsilon)\epsilon\biggl{(}2\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}+\Psi(\nu_{\scriptscriptstyle
S})\biggr{)}\\!+\\!1\\!-\\!\frac{17\epsilon}{6}\biggr{]}$ $\displaystyle\hskip
85.35826pt-\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\\!\times\\!4\epsilon\biggl{[}\epsilon\biggl{(}2\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}+\Psi(\nu_{\scriptscriptstyle
S})\biggr{)}+\frac{1}{3}\biggr{]}\Biggr{\\}}\,.$ (80)

The covariant contribution in the first line accounts for the non-
conservation, while the remainder is conserved by itself.

####  4.3.2 Vector loop contribution

$\displaystyle\bigl{\langle}\hat{t}^{\scriptscriptstyle\,(2)}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
V}={}$ $\displaystyle
g_{\mu\nu}\times\boldsymbol{\Gamma}(\nu_{\scriptscriptstyle
V})(1\\!-\\!\epsilon)^{4}H^{4}\biggl{[}\Bigl{(}\frac{D\\!-\\!3}{2}\Bigr{)}^{\\!2}\\!-\\!\nu_{\scriptscriptstyle
V}^{2}\biggr{]}$ $\displaystyle\hskip
14.22636pt\times\Biggl{\\{}-\frac{(D\\!-\\!1)}{D}\Bigl{(}\frac{9}{4}\\!-\nu_{\scriptscriptstyle
V}^{2}\Bigr{)}-\frac{3(D\\!-\\!4)(3\\!-\\!\epsilon)}{8(1\\!-\\!\epsilon)}+\mathcal{O}\bigl{[}(D\\!-\\!4)^{2}\bigr{]}\Biggr{\\}}\,.$
(81)

As before, the two remaining conditions for the counterterm coefficients are
most easily read off from the
$\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}$ part, and the most
convenient choice seems to be,

$\bigl{[}\delta Z_{\phi}\\!-\\!6\delta\xi\bigr{]}_{\scriptscriptstyle V}^{\rm
div.}=\mathcal{O}\bigl{[}(D\\!-\\!4)^{0}\bigr{]}\,,\qquad\qquad\bigl{[}\delta\alpha_{\scriptscriptstyle
R}\bigr{]}_{\scriptscriptstyle V}^{\rm div.}=0\,,$ (82)

which then uniquely fixes all the divergent parts of the counterterm
coefficients tied to vector loops,

$\displaystyle\bigl{[}\delta\xi\bigr{]}_{\scriptscriptstyle V}^{\rm div.}={}$
$\displaystyle\frac{\mu^{D-4}\,\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}}{(4\pi)^{\frac{D}{2}}}\times\frac{(D\\!-\\!2)(D\\!-\\!8)q^{2}}{16}\,,$
(83a) $\displaystyle\bigl{[}\delta\lambda\bigr{]}_{\scriptscriptstyle V}^{\rm
div.}={}$
$\displaystyle\frac{\mu^{D-4}\,\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}}{(4\pi)^{\frac{D}{2}}}\times\Bigl{[}-(D\\!-\\!1)q^{4}\Bigr{]}\,,$
(83b) $\displaystyle\bigl{[}\delta Z_{\phi}\bigr{]}_{\scriptscriptstyle
V}^{\rm div.}={}$
$\displaystyle\frac{\mu^{D-4}\,\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}}{(4\pi)^{\frac{D}{2}}}\times\biggl{[}-\frac{(D\\!-\\!1)(D\\!-\\!2)q^{2}}{2}\biggr{]}\,,$
(83c) $\displaystyle\bigl{[}\delta\alpha_{\scriptscriptstyle
R}\bigr{]}_{\scriptscriptstyle V}^{\rm div.}={}$
$\displaystyle\frac{\mu^{D-4}\,\Gamma\bigl{(}\frac{2-D}{2}\bigr{)}}{(4\pi)^{\frac{D}{2}}}\times
0\,,$ (83d)

such that $\bigl{[}\delta Z_{\phi}\\!-\\!6\delta\xi\bigr{]}^{\rm
div.}\\!\\!=\\!-7/(32\pi^{2})$, in agreement with (82). Inserting the values
for these coefficients in (74) and adding it to (81) produces a finite
renormalized result upon taking the four-dimensional limit,

$\displaystyle\bigl{\langle}\hat{t}^{\scriptscriptstyle\,(2)}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
V}\\!+\bigl{[}\delta T_{\mu\nu}\bigr{]}_{\scriptscriptstyle V}^{\rm
div.}\\!\xrightarrow{D\to 4}\\!{}$
$\displaystyle-\\!\frac{3\bigl{(}q\overline{\phi}\bigr{)}^{\\!2}}{64\pi^{2}}H^{2}\Biggl{\\{}\Biggl{[}(1\\!-\\!\epsilon)^{2}\Bigl{(}\frac{9}{4}\\!-\\!\nu_{\scriptscriptstyle
V}^{2}\Bigr{)}\biggl{(}\\!2\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}\\!\\!+\\!\Psi(\nu_{\scriptscriptstyle
V})\\!-\\!\frac{1}{2}\biggr{)}$
$\displaystyle-2\epsilon\biggl{(}\frac{(1\\!-\\!\epsilon)^{3}H^{2}}{\bigl{(}q\overline{\phi}\bigr{)}^{\\!2}}\\!-\\!\frac{3}{4}\biggr{)}-\frac{(3\\!-\\!4\epsilon)(2\\!+\\!7\epsilon)}{6}\Biggr{]}\\!\times\\!g_{\mu\nu}+\frac{2(2\\!+\\!7\epsilon)\epsilon}{3}\\!\times\\!\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\Biggr{\\}}\,.$
(84)

####  4.3.3 Full one-loop results

Adding up the scalar loop contribution to the linear (66) and quadratic (80)
parts of the energy-momentum tensor, and writing them in terms of the scalar
perturbation mass (18) gives,

$\displaystyle\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
S}={}$
$\displaystyle-\frac{\lambda}{4\pi^{2}}\biggl{[}\frac{3\lambda\overline{\phi}^{2}\\!-\\!\bigl{(}\frac{1}{6}\\!-\\!\xi\bigr{)}R}{2\lambda\overline{\phi}^{\,2}}\biggr{]}\Biggl{\\{}\biggl{[}2\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}+\Psi(\nu_{\scriptscriptstyle
S})+\frac{2\\!+\\!15\epsilon}{6\epsilon}$ $\displaystyle\hskip
56.9055pt-\frac{(1\\!-\\!\epsilon)}{2\epsilon(1\\!-\\!6\xi)}+\frac{9\epsilon(1\\!-\\!\epsilon)H^{2}}{2\lambda\overline{\phi}^{\,2}}\biggr{]}\overline{T}_{\mu\nu}-\frac{g^{\rho\sigma}\overline{T}_{\rho\sigma}}{8(1\\!-\\!\epsilon)}g_{\mu\nu}\Biggr{\\}}\,.$
(85)

and adding up the vector loop contributions from (66) and (84), and writing
them in terms of scalar and vector masses (18) and (21) gives,

$\displaystyle\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
V}={}$
$\displaystyle-\frac{3q^{2}}{8\pi^{2}}\Biggl{\\{}\biggl{[}\frac{\bigl{(}q\overline{\phi}\bigr{)}^{\\!2}\\!-\\!\lambda\overline{\phi}^{\,2}\\!+\\!\bigl{(}\frac{1}{6}\\!-\\!\xi\bigr{)}R}{2\lambda\overline{\phi}^{\,2}}\biggr{]}\biggl{[}2\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}+\Psi(\nu_{\scriptscriptstyle
V})+\frac{1\\!+\\!5\epsilon}{3\epsilon}$
$\displaystyle-\frac{(1\\!-\\!\epsilon)}{3\epsilon(1\\!-\\!6\xi)}+\frac{3\epsilon(1\\!-\\!\epsilon)H^{2}}{\lambda\overline{\phi}^{\,2}}\biggr{]}\overline{T}_{\mu\nu}+\biggl{[}\frac{(2\\!+\\!7\epsilon)}{12\epsilon(1\\!-\\!6\xi)}+\frac{3\epsilon
H^{2}}{4\lambda\overline{\phi}^{\,2}}\biggr{]}\overline{T}_{\mu\nu}$
$\displaystyle-\biggl{[}\frac{\bigl{(}q\overline{\phi}\bigr{)}^{\\!2}\\!-\\!\lambda\overline{\phi}^{\,2}\\!+\\!\bigl{(}\frac{1}{6}\\!-\\!\xi\bigr{)}R}{2\lambda\overline{\phi}^{\,2}}\biggr{]}\frac{g^{\rho\sigma}\overline{T}_{\rho\sigma}}{8(1\\!-\\!\epsilon)}g_{\mu\nu}\Biggr{\\}}+\frac{3\lambda}{8\pi^{2}}\\!\times\\!\frac{\epsilon(1\\!-\\!\epsilon)^{3}H^{4}}{\bigl{(}\lambda\overline{\phi}^{\,2}\bigr{)}^{\\!2}}\overline{T}_{\mu\nu}\,,$
(86)

where $\overline{T}_{\mu\nu}$ is the tree level result given in (61).
Recalling that $\nabla^{\mu}\overline{T}_{\mu\nu}\\!=\\!0$, one can easily
check that both (85) and (86) are separately covariantly conserved, 333Useful
relations are
$\nabla_{\nu}\bigl{(}g^{\rho\sigma}\overline{T}_{\rho\sigma}\bigr{)}\\!=\\!-4\epsilon
aH\delta_{\nu}^{0}\bigl{(}g^{\rho\sigma}\overline{T}_{\rho\sigma}\bigr{)}$,
and
$\nabla^{\mu}\bigl{[}\ln(a)\overline{T}_{\mu\nu}\bigr{]}\\!=\\!-(H/a)\overline{T}_{0\nu}$.
as they should be. We emphasize this is true only after linear and quadratic
contributions are added up. The final expressions (85) and (86) are rather
complicated, and one may be tempted to subtract some of the terms by a
judicious choice of finite counterterms. A closer look at the finite part of
the energy-momentum tensor counterterms in (74),

$\bigl{[}\delta T_{\mu\nu}\bigr{]}^{\rm fin.}=\biggl{\\{}\bigl{[}\delta
Z_{\phi}\bigr{]}^{\rm
fin.}+\frac{144\lambda(2\\!-\\!\epsilon)}{(1\\!-\\!6\xi)}\frac{H^{2}}{\lambda\overline{\phi}^{\,2}}\bigl{[}\delta\alpha_{\scriptscriptstyle
R}\bigr{]}^{\rm fin.}\biggr{\\}}\\!\times\\!\overline{T}_{\mu\nu}\,.$ (87)

where we took account of the simple choice for the condensate correction (55),
reveals that further simplifications are rather limited if we restrict
ourselves to counterterm coefficients being independent of $\epsilon$.
Therefore, we shall not pursue this, and instead we shall focus our analysis
on the role of secular effects, and perturbativity of the one-loop results
(85)–(86).

##  5 Various limits

In this section we discuss various limits of the one-loop corrections to the
condensate (53) and (54), and to the energy momentum tensor (85), and (86),
and compare them with the results from the literature when possible.

###  5.1 De Sitter limit

The de Sitter limit is defined by the constant physical Hubble rate,
$H\\!=\\!H_{0}$, obtained in the limit of vanishing principal slow-roll
parameter, $\epsilon\\!\to\\!0$, where the Ricci scalar is
$R_{0}\\!=\\!12H_{0}^{2}$. At tree level the condensate (15) is constant,
while the energy-momentum tensor (61) vanishes,

$\overline{\phi}\xrightarrow{\epsilon\to 0}\overline{\phi}_{0}=\pm
H_{0}\sqrt{\frac{-12\xi}{\lambda}}\,,\qquad\qquad\overline{T}_{\mu\nu}\xrightarrow{\epsilon\to
0}0\,.$ (88)

The effective masses of fluctuations are also constant in de Sitter,

$M_{\scriptscriptstyle S}^{2}\xrightarrow{\epsilon\to 0}m_{\scriptscriptstyle
S}^{2}=-24\xi H_{0}^{2}\,,\qquad\quad M_{\scriptscriptstyle
V}^{2}\xrightarrow{\epsilon\to 0}m_{\scriptscriptstyle V}^{2}=-\frac{12\xi
q^{2}}{\lambda}H_{0}^{2}=\frac{q^{2}m_{\scriptscriptstyle
S}^{2}}{2\lambda}\,.$ (89)

Consequently, the condensate one-loop corrections (53) and (54) are also
constant, and reduce to,

$\displaystyle\frac{\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
S}}{\overline{\phi}_{0}}={}$
$\displaystyle\\!-\frac{3\lambda}{16\pi^{2}}\biggl{[}\frac{3\lambda\overline{\phi}_{0}^{2}-\bigl{(}\frac{1}{6}\\!-\\!\xi\bigr{)}R_{0}}{2\lambda\overline{\phi}_{0}^{\,2}}\biggr{]}\biggl{[}2\ln\Bigl{(}\frac{H_{0}}{\mu}\Bigr{)}+\Psi\bigl{(}\nu_{\scriptscriptstyle
S}^{0}\bigr{)}\biggr{]}\,,$ (90)
$\displaystyle\frac{\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
V}}{\overline{\phi}_{0}}={}$
$\displaystyle-\frac{3q^{2}}{16\pi^{2}}\biggl{[}\frac{\bigl{(}q\overline{\phi}_{0}\bigr{)}^{\\!2}-\lambda\overline{\phi}_{0}^{2}+\bigl{(}\frac{1}{6}\\!-\\!\xi\bigr{)}R_{0}}{2\lambda\overline{\phi}_{0}^{2}}\biggr{]}\biggl{[}2\ln\Bigl{(}\frac{H_{0}}{\mu}\Bigr{)}+\Psi\bigl{(}\nu_{\scriptscriptstyle
V}^{0}\bigr{)}\biggr{]}\,,$ (91)

where the mode function indices are,

$\nu_{\scriptscriptstyle S}^{2}\xrightarrow{\epsilon\to
0}\bigl{(}\nu_{\scriptscriptstyle
S}^{0}\bigr{)}^{2}=\frac{9}{4}-24\xi=\frac{9}{4}-\frac{m_{\scriptscriptstyle
S}^{2}}{H_{0}^{2}}\,,\qquad\quad\nu_{\scriptscriptstyle
V}^{2}\xrightarrow{\epsilon\to 0}\bigl{(}\nu_{\scriptscriptstyle
V}^{0}\bigr{)}^{2}=\frac{1}{4}-\frac{12q^{2}\xi}{\lambda}=\frac{1}{4}-\frac{m_{\scriptscriptstyle
V}^{2}}{H_{0}^{2}}\,.$ (92)

For some purposes it is more convenient to express the de Sitter limit in
terms of the effective masses of fluctuations,

$\displaystyle\frac{\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
S}}{\overline{\phi}_{0}}\xrightarrow{\epsilon\to 0}{}$
$\displaystyle-\frac{3\lambda}{16\pi^{2}}\biggl{[}\frac{m_{\scriptscriptstyle
S}^{2}\\!-\\!2H_{0}^{2}}{m_{\scriptscriptstyle
S}^{2}}\biggr{]}\biggl{[}2\ln\Bigl{(}\frac{H_{0}}{\mu}\Bigr{)}+\Psi\bigl{(}\nu_{\scriptscriptstyle
S}^{0}\bigr{)}\biggr{]}\,,$ (93)
$\displaystyle\frac{\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
V}}{\overline{\phi}_{0}}\xrightarrow{\epsilon\to 0}{}$
$\displaystyle-\frac{3q^{2}}{16\pi^{2}}\biggl{[}\frac{m_{\scriptscriptstyle
V}^{2}+2H_{0}^{2}}{m_{\scriptscriptstyle
S}^{2}}\biggr{]}\biggl{[}2\ln\Bigl{(}\frac{H_{0}}{\mu}\Bigr{)}+\Psi\bigl{(}\nu_{\scriptscriptstyle
V}^{0}\bigr{)}\biggr{]}\,.$ (94)

Note that no secular corrections remain, as they are proportional to
$\epsilon$ and vanish in the exact de Sitter limit. For the energy-momentum
tensor we first observe that, even though the classical tree-level
contribution vanishes in the de Sitter limit, the the singular limit,

$\frac{1}{\epsilon}\overline{T}_{\mu\nu}\xrightarrow{\epsilon\to
0}-\frac{3(1\\!-\\!6\xi)}{4}\bigl{(}\overline{\phi}_{0}H_{0}\bigr{)}^{\\!2}g_{\mu\nu}\,,$
(95)

is finite, so that de Sitter limits of one-loop corrections (85) and (86) are,

$\displaystyle\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
S}\xrightarrow{\epsilon\to 0}{}$
$\displaystyle\frac{1}{128\pi^{2}}\Bigl{(}m_{\scriptscriptstyle
S}^{2}-2H_{0}^{2}\Bigr{)}^{\\!2}g_{\mu\nu}\,,$ (96)
$\displaystyle\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
V}\xrightarrow{\epsilon\to 0}{}$
$\displaystyle\frac{3}{128\pi^{2}}m_{\scriptscriptstyle
V}^{2}\Bigl{(}m_{\scriptscriptstyle V}^{2}+4H_{0}^{2}\Bigr{)}g_{\mu\nu}\,.$
(97)

Curiously, the de Sitter limit of the one-loop energy-momentum tensor exhibits
no dependence on the renormalization scale $\mu$, which implies the absence of
UV divergences. This is a peculiarity of the nonminimally coupled Abelian
Higgs model (2.1) that we consider here, where it is the typically negative
non-minimal coupling, satisfying Eq. (16), that allows for the existence of
the symmetry-breaking condensate in the attractor regime.

One-loop energy momentum tensors of spectator scalars and vectors have been
computed in the literature, but only in the limit of vanishing condensate.
Therefore they cannot be compared directly to our result (96) and (96), that
comprise of the tadpole contribution from Sec. 4.2 and the quadratic
perturbation contribution of Sec. 4.3. Each of the contributions are
separately conserved in the de Sitter limit, as opposed to the general case
when only their sum is conserved. We can only compare the quadratic parts to
the existing results in the literature.

Expression (17), from which the one-loop energy-momentum tensor descends from,
suggests that the proper comparison for the scalar loop contribution is
comparing the de Sitter limit of (80),

$\bigl{\langle}\hat{t}^{\scriptscriptstyle\,(2)}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
S}+\bigl{[}\delta T_{\mu\nu}\bigr{]}_{\scriptscriptstyle S}^{\rm
div.}\\!\xrightarrow{D\to
4}\frac{(1\\!+\\!12\xi)H_{0}^{4}}{32\pi^{2}}\Biggl{\\{}-36\xi\biggl{[}2\ln\Bigl{(}\frac{H_{0}}{\mu}\Bigr{)}+\Psi\bigl{(}\nu_{\scriptscriptstyle
S}^{0}\bigr{)}\biggr{]}\\!+(1\\!+\\!12\xi)\Biggr{\\}}g_{\mu\nu}\,.$ (98)

to the result for the spectator scalar field with non-minimal coupling $\xi$
and mass $m^{2}\\!=\\!3\lambda\overline{\phi}_{0}^{2}\\!=\\!-36\xi H_{0}^{2}$.
This was first computed in Dowker:1975tf , and the reported results agree with
our, up to renormalization scheme ambiguities, and the conformal anomaly. The
de Sitter limit of the vector loop contribution to the quadratic part in (70)
suggests we should compare the de Sitter limit of the expectation value (84),

$\bigl{\langle}\hat{t}^{\scriptscriptstyle\,(2)}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
V}+\bigl{[}\delta T_{\mu\nu}\bigr{]}_{\scriptscriptstyle V}^{\rm
div.}\\!\xrightarrow{D\to 4}-\frac{3m_{\scriptscriptstyle
V}^{2}}{64\pi^{2}}\Biggl{\\{}\Bigl{(}m_{\scriptscriptstyle
V}^{2}\\!+\\!2H_{0}^{2}\Bigr{)}\\!\biggl{[}2\ln\Bigl{(}\frac{H_{0}}{\mu}\Bigr{)}\\!+\\!\Psi\bigl{(}\nu_{\scriptscriptstyle
V}^{0}\bigr{)}\biggr{]}\\!-\\!\frac{m_{\scriptscriptstyle
V}^{2}}{2}\\!-\\!2H_{0}^{2}\Biggr{\\}}g_{\mu\nu}\,.$ (99)

to the Stueckelberg model with mass $m_{\scriptscriptstyle
V}^{2}\\!=\\!\bigl{(}q\overline{\phi}_{0}\bigr{)}^{\\!2}$ computed in
Belokogne:2015etf ; Belokogne:2016dvd , where the same result was reported, up
to renormalization scheme ambiguities, and the conformal anomaly.

###  5.2 Flat space limit

The Minkowski limit is strictly speaking not accessible naively. However, we
can take it by expressing the results in terms of effective scalar masses
first, and then take $H_{0}\\!\to\\!0$ with keeping $m_{\scriptscriptstyle S}$
constant. It is most convenient to do this from the de Sitter limit of the
preceding subsection. Taking into account that digamma functions in reduce to
the following forms in the flat space limit, 444This follows from
$\nu_{\scriptscriptstyle S}^{0}\,\overset{H_{0}\to
0}{\scalebox{1.8}[1.0]{$\sim$}}\,im_{\scriptscriptstyle S}/H_{0}$ and
$\nu_{\scriptscriptstyle V}^{0}\,\overset{H_{0}\to
0}{\scalebox{1.8}[1.0]{$\sim$}}\,im_{\scriptscriptstyle V}/H_{0}$, and the
limit,
$\Psi(i|z|)=\psi\Bigl{(}\frac{1}{2}\\!+\\!i|z|\Bigr{)}+\psi\Bigl{(}\frac{1}{2}\\!-\\!i|z|\Bigr{)}\
\overset{z\to\infty}{\scalebox{1.8}[1.0]{$\sim$}}\ 2\ln(|z|)\,.$

$\displaystyle\Psi\bigl{(}\nu_{\scriptscriptstyle S}^{0}\bigr{)}\
\overset{H_{0}\to 0}{\underset{m_{\scriptscriptstyle S}={\tt
const.}}{\scalebox{1.8}[1.0]{$\sim$}}}\
2\ln\Bigl{(}\frac{m_{\scriptscriptstyle
S}}{H_{0}}\Bigr{)}\,,\qquad\qquad\Psi\bigl{(}\nu_{\scriptscriptstyle
V}^{0}\bigr{)}\ \overset{H_{0}\to 0}{\underset{m_{\scriptscriptstyle S}={\tt
const.}}{\scalebox{1.8}[1.0]{$\sim$}}}\
2\ln\Bigl{(}\frac{m_{\scriptscriptstyle V}}{H_{0}}\Bigr{)}\,,$ (100)

the one-loop tadpole corrections are,

$\frac{\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
S}}{\overline{\phi}_{0}}\xrightarrow[m_{\scriptscriptstyle S}={\tt
const.}]{H_{0}\to
0}-\frac{3\lambda}{8\pi^{2}}\ln\Bigl{(}\frac{m_{\scriptscriptstyle
S}}{\mu}\Bigr{)}\,,\qquad\frac{\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
V}}{\overline{\phi}_{0}}\xrightarrow[m_{\scriptscriptstyle S}={\tt
const.}]{H_{0}\to 0}-\frac{3q^{2}}{8\pi^{2}}\frac{m_{\scriptscriptstyle
V}^{2}}{m_{\scriptscriptstyle S}^{2}}\ln\Bigl{(}\frac{m_{\scriptscriptstyle
V}}{\mu}\Bigr{)}\,.$ (101)

These correspond to the results of Minkowski space computations Irges:2017ztc
, up to renormalization scheme dependent terms.

While we are able to infer the flat space limit from the condensate correction
in de Sitter space, we are not able to do so for the energy-momentum tensor.
The reason is that the mass term and the non-minimal coupling term are not
distinguishable in de Sitter for the evolution of the scalar. However, they
are distinguished by the energy-momentum tensor which involves the variational
derivative with respect to the metric, which treats the mass term and the non-
minimal coupling terms differently.

###  5.3 Large field limit/large non-minimal coupling/large masses

The large field limit $\overline{\phi}/H\\!\gg\\!1$ in our model corresponds
to the limit of large negative nonminimal coupling $\xi\\!\ll\\!-1$.555We
still assume the non-minimal coupling is small enough not to induce
significant backreaction on the expansion rate. Here we have for the tree-
level condensate (15) and the energy-momentum tensor (61),

$\frac{\overline{\phi}}{H}\ \overset{\xi\ll-1}{\scalebox{1.8}[1.0]{$\sim$}}\
\pm\sqrt{\frac{-6\xi(2\\!-\\!\epsilon)}{\lambda}}\,,\qquad\overline{T}_{\mu\nu}\
\overset{\xi\ll-1}{\scalebox{1.8}[1.0]{$\sim$}}\
\frac{-3\xi\epsilon}{2}\biggl{[}-(3\\!-\\!4\epsilon)g_{\mu\nu}+4\epsilon\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\biggr{]}\bigl{(}\overline{\phi}H\bigr{)}^{\\!2}\,.$
(102)

For fluctuations this is essentially the limit of large effective masses,

$\frac{M_{\scriptscriptstyle S}^{2}}{H^{2}}\
\overset{\xi\ll-1}{\scalebox{1.8}[1.0]{$\sim$}}-\\!12\xi(2\\!-\\!\epsilon)\gg
1\,,\qquad\qquad\frac{M_{\scriptscriptstyle V}^{2}}{H^{2}}\
\overset{\xi\ll-1}{\scalebox{1.8}[1.0]{$\sim$}}-\frac{6\xi
q^{2}(2\\!-\\!\epsilon)}{\lambda}\gg 1\,,$ (103)

provided the ratio of couplings $q^{2}/\lambda$ is not hierarchical. Given
that the mode function indices in this limit reduce to,

$\displaystyle\nu_{\scriptscriptstyle S}^{2}\
\overset{\xi\ll-1}{\scalebox{1.8}[1.0]{$\sim$}}\
\frac{12\xi(2\\!-\\!\epsilon)}{(1\\!-\\!\epsilon)^{2}}\ll-1\,,\qquad\quad\nu_{\scriptscriptstyle
V}^{2}\ \overset{\xi\ll-1}{\scalebox{1.8}[1.0]{$\sim$}}\
\frac{q^{2}}{\lambda}\times\frac{6\xi(2\\!-\\!\epsilon)}{(1\\!-\\!\epsilon)^{2}}\ll-1\,,$
(104)

which implies that,

$\Psi(\nu_{\scriptscriptstyle S})\
\overset{\xi\ll-1}{\scalebox{1.8}[1.0]{$\sim$}}\
2\ln\Bigl{[}\frac{M_{\scriptscriptstyle
S}}{(1\\!-\\!\epsilon)H}\Bigr{]}\,,\qquad\qquad\Psi(\nu_{\scriptscriptstyle
V})\ \overset{\xi\ll-1}{\scalebox{1.8}[1.0]{$\sim$}}\
2\ln\Bigl{[}\frac{M_{\scriptscriptstyle V}}{(1\\!-\\!\epsilon)H}\Bigr{]}\,,$
(105)

and produces the result for the one-loop condensate corrections,

$\displaystyle\frac{\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
S}}{\overline{\phi}}\
\overset{\xi\ll-1}{\scalebox{1.8}[1.0]{$\sim$}}-\\!\frac{3\lambda}{8\pi^{2}}\ln\Bigl{(}\frac{M_{\scriptscriptstyle
S}}{\mu}\Bigr{)}\,,\qquad\qquad\frac{\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
V}}{\overline{\phi}}\
\overset{\xi\ll-1}{\scalebox{1.8}[1.0]{$\sim$}}-\\!\frac{3q^{2}}{8\pi^{2}}\frac{M_{\scriptscriptstyle
V}^{2}}{M_{\scriptscriptstyle S}^{2}}\ln\Bigl{(}\frac{M_{\scriptscriptstyle
V}}{\mu}\Bigr{)}\,.$ (106)

that are a direct non-equilibrium generalization of the Coleman-Weinberg
result Coleman:1973jx  with time-dependent effective masses.

For the one-loop energy-momentum tensor the large field limit is,

$\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle S}\
\overset{\xi\ll-1}{\scalebox{1.8}[1.0]{$\sim$}}{}-\frac{\lambda}{2\pi^{2}}\ln\Bigl{(}\frac{M_{\scriptscriptstyle
S}}{\mu}\Bigr{)}\\!\times\\!\overline{T}_{\mu\nu}\,,\qquad\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
V}\ \overset{\xi\ll-1}{\scalebox{1.8}[1.0]{$\sim$}}\
-\frac{3q^{2}}{4\pi^{2}}\frac{M_{\scriptscriptstyle
V}^{2}}{M_{\scriptscriptstyle S}^{2}}\ln\Bigl{(}\frac{M_{\scriptscriptstyle
V}}{\mu}\Bigr{)}\\!\times\\!\overline{T}_{\mu\nu}\,,$ (107)

where we neglected constant corrections to the logs (that can nonetheless be
enhanced by a relative factor $1/\epsilon$ in comparison when
$\epsilon\\!\ll\\!1$.) The expressions above are not conserved on the account
of time dependence of masses of fluctuations. However, the non-conservation is
only at the subleading limit that is countered by the subleading terms we
neglected.

###  5.4 Small field limit

The small field limit $\overline{\phi}/H\\!\ll\\!1$ is implemented by taking
the critical value limit for the nonminimal coupling,

$\xi=\xi_{\rm
cr}-\Delta\xi=\frac{\epsilon(3\\!-\\!2\epsilon)}{6(2\\!-\\!\epsilon)}-\Delta\xi\,,\qquad\qquad
0<\Delta\xi\ll 1\,,$ (108)

such that the condition (16) is close to saturation, where the tree-level
condensate field value is vanishing small,

$\frac{\lambda\overline{\phi}^{\,2}}{H^{2}}\ \overset{\Delta\xi\to
0}{\scalebox{1.8}[1.0]{$\sim$}}\ 6(2\\!-\\!\epsilon)\Delta\xi\,.$ (109)

The mode function indices in the small field limit are,

$\nu_{\scriptscriptstyle
S}^{2}=\frac{9}{4}-\frac{12(2\\!-\\!\epsilon)\Delta\xi}{(1\\!-\\!\epsilon)^{2}}\,,\qquad\qquad\nu_{\scriptscriptstyle
V}^{2}=\frac{1}{4}-\frac{q^{2}}{\lambda}\frac{6(2\\!-\\!\epsilon)\Delta\xi}{(1\\!-\\!\epsilon)^{2}}\,,$
(110)

so that the digamma functions contribute as,

$\Psi(\nu_{\scriptscriptstyle S})\ \overset{\Delta\xi\to
0}{\scalebox{1.8}[1.0]{$\sim$}}\
-\frac{(1\\!-\\!\epsilon)^{2}}{4(2\\!-\\!\epsilon)\Delta\xi}+\frac{7}{3}-2\gamma_{\scriptscriptstyle
E}\,,\qquad\Psi(\nu_{\scriptscriptstyle V})\ \overset{\Delta\xi\to
0}{\scalebox{1.8}[1.0]{$\sim$}}\
-\frac{\lambda}{q^{2}}\frac{(1\\!-\\!\epsilon)^{2}}{6(2\\!-\\!\epsilon)\Delta\xi}+1-2\gamma_{\scriptscriptstyle
E}\,.$ (111)

This sufficies to evaluate the limiting behaviour of our results. The
condensate corrections (53) and (53) diverge in this limit,

$\displaystyle\frac{\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
S}}{\overline{\phi}}\ \overset{\Delta\xi\to 0}{\scalebox{1.8}[1.0]{$\sim$}}{}$
$\displaystyle-\frac{\lambda(1\\!-\\!3\epsilon)(1\\!-\\!\epsilon)^{3}}{128\pi^{2}(2\\!-\\!\epsilon)^{2}\Delta\xi^{2}}+\mathcal{O}\bigl{(}1/\Delta\xi\bigr{)}\,,$
(112)
$\displaystyle\frac{\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
V}}{\overline{\phi}}\ \overset{\Delta\xi\to 0}{\scalebox{1.8}[1.0]{$\sim$}}{}$
$\displaystyle\ \frac{(\lambda\\!-\\!3\epsilon
q^{2})(1\\!-\\!\epsilon)^{3}}{192\pi^{2}(2\\!-\\!\epsilon)^{2}\Delta\xi^{2}}+\mathcal{O}\bigl{(}1/\Delta\xi\bigr{)}\,,$
(113)

which is actually a consequence of the singular definition of the variable. In
the unitary gauge we emply here the scalar field is actually the modulus of
the complex scalar field, which is not defined at the origin. In fact, a
closer look at the leading relative condensate correction above reveals it to
be negative, meaning that the meaningful description breaks down before the
singular point. Similar behaviour is observed for the energy-momentum tensor
corrections,

$\displaystyle\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
S}\overset{\Delta\xi\to 0}{\scalebox{1.8}[1.0]{$\sim$}}\ {}$
$\displaystyle-\frac{\epsilon(1\\!-\\!\epsilon)^{5}(1\\!-\\!4\epsilon)}{32\pi^{2}(2\\!-\\!\epsilon)^{2}\Delta\xi}H^{4}\Bigl{[}-(3\\!-\\!4\epsilon)g_{\mu\nu}\\!+\\!4\epsilon\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\Bigr{]}\\!+\mathcal{O}\bigl{(}\Delta\xi^{0}\bigr{)}\,,$
(114)
$\displaystyle\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
V}\overset{\Delta\xi\to 0}{\scalebox{1.8}[1.0]{$\sim$}}\ {}$
$\displaystyle\frac{\epsilon(1\\!-\\!\epsilon)^{5}}{32\pi^{2}(2\\!-\\!\epsilon)^{2}\Delta\xi}\Bigl{[}1\\!-\\!\frac{3\epsilon
q^{2}}{\lambda}\Bigr{]}H^{4}\Bigl{[}-(3\\!-\\!4\epsilon)g_{\mu\nu}\\!+\\!4\epsilon\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\Bigr{]}\\!+\mathcal{O}\bigl{(}\Delta\xi^{0}\bigr{)}\,,$
(115)

that are finite only in the de Sitter limit,

$\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
S}\xrightarrow[\epsilon\to 0]{\Delta\xi\to
0}\frac{H^{4}}{32\pi^{2}}g_{\mu\nu}\,,\qquad\qquad\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
V}\xrightarrow[\epsilon\to 0]{\Delta\xi\to 0}0\,,$ (116)

where the scalar loop contribution reduces to the contribution of the
massless, minimally coupled scalar, supplemented by the finite contribution
obtained from taking the singular limit $\xi\\!\to\\!0$ of the nonminimally
coupled case (consistent with Glavan:2013mra ; Glavan:2014uga ; Glavan:2015cut
), while the vector loop contribution reduces to the vanishing one of a
massless photon Glavan:2022pmk ; Glavan:2022dwb ; Glavan:2022nrd . However,
behaviour of the results for nonvanishing $\epsilon$ points to the need of
examining this regime in a different gauge, that is adapted to the vanishing
condensate limit, unlike the unitary gauge we use here.

###  5.5 Hierarchal couplings

Interesting limit is provided by a hierarchy between the scalar self-coupling
constant and the $U(1)$ charge,

$q^{2}/\lambda\gg 1\,,$ (117)

when the vector loop provides a parametrically larger contribution than the
scalar one. Furthermore, given that in this limit,

$\Psi(\nu_{\scriptscriptstyle V})\
\overset{q^{2}\gg\lambda}{\scalebox{1.8}[1.0]{$\sim$}}\
2\ln\Bigl{[}\frac{q\overline{\phi}}{(1\\!-\\!\epsilon)H}\Bigr{]}\,,$ (118)

the dominant contribution to the condensate correction,

$\frac{\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
V}}{\overline{\phi}}\
\overset{q^{2}\gg\lambda}{\scalebox{1.8}[1.0]{$\sim$}}-\frac{3q^{4}}{16\pi^{2}\lambda}\ln\Bigl{(}\frac{q\overline{\phi}}{\mu}\Bigr{)}\,,$
(119)

and to the energy-momentum tensor correction,

$\displaystyle\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
V}=-\frac{3q^{4}}{8\pi^{2}\lambda}\ln\Bigl{(}\frac{q\overline{\phi}}{\mu}\Bigr{)}\\!\times\\!\overline{T}_{\mu\nu}\,.$
(120)

derive from the vector loop. We see that the hierarchy (117) of coupling
constants can lead to a big enhancement of the loop corrections. This is a
feature of the model that is present already in flat space. However, the
secular logarithm multiplying this correction is innate to curved spacetimes,
which is discussed in detail in Sec. 6.

###  5.6 Vanishing $U(1)$ charge

Even though the limit $q\\!\to\\!0$ is a singular limit of the unitary gauge
Glavan:2020zne , the scalar field $\phi$ in the unitary gauge is a gauge-
independent observable, and it is not sensitive to the singularity of the this
limit. Therefore, we may infer the result in the limit of vanishing $U(1)$
charge. Interestingly, the vector loop contributions do not vanish, neither
for the condensate,

$\displaystyle\frac{\bigl{\langle}\hat{\varphi}\bigr{\rangle}_{\scriptscriptstyle
V}}{\overline{\phi}}\xrightarrow{q\to 0}{}$
$\displaystyle\frac{3\lambda}{16\pi^{2}}\\!\times\\!\frac{\epsilon(1\\!-\\!\epsilon)^{3}H^{4}}{\bigl{(}\lambda\overline{\phi}^{\,2}\bigr{)}^{\\!2}}\,,$
(121)

nor for the energy-momentum tensor,

$\displaystyle\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}_{\scriptscriptstyle
V}\xrightarrow{q\to 0}{}$
$\displaystyle\frac{3\lambda}{8\pi^{2}}\\!\times\\!\frac{\epsilon(1\\!-\\!\epsilon)^{3}H^{4}}{\bigl{(}\lambda\overline{\phi}^{\,2}\bigr{)}^{\\!2}}\\!\times\\!T_{\mu\nu}^{\scriptscriptstyle(0)}\,.$
(122)

Nevertheless, inorder to be sure of this limit the computation should be
redone in a gauge regular in the vanishing $U(1)$ charge limit.

##  6 Late time limit and an RG explanation

The time dependence of the Hubble rate in the logarithms of the one-loop
corrections implies the secular breakdown of perturbation theory after long
enough time, when,

$\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)H}{\mu}\Bigr{]}\
\overset{a\to\infty}{\scalebox{1.8}[1.0]{$\sim$}}-\epsilon\ln(a)\gg 1\,,$
(123)

becomes large enough. This term dominates when $a\\!\gg\\!e^{1/\epsilon}$. In
inflation this number is very big, but our results are valid for finite
$\epsilon$. Therefore, the late time limit of the one-loop corrections are,

$\bigl{\langle}\hat{\phi}\bigr{\rangle}=\overline{\phi}\times\Bigl{[}1+\frac{3\lambda}{16\pi^{2}}C_{\phi}\times\epsilon\ln(a)\Bigr{]}\,,\qquad\bigl{\langle}\hat{T}_{\mu\nu}\bigr{\rangle}=\overline{T}_{\mu\nu}\times\Bigl{[}1+\frac{3\lambda}{8\pi^{2}}C_{\scriptscriptstyle
T}\times\epsilon\ln(a)\Bigr{]}\,,$ (124)

where the the scalar and vector loops contribute differently to each constant
factor,

$\displaystyle C_{\phi}={}$
$\displaystyle\frac{3\lambda\overline{\phi}^{2}\\!-\\!\bigl{(}\frac{1}{6}\\!-\\!\xi\bigr{)}R}{\lambda\overline{\phi}^{\,2}}+\frac{q^{2}}{\lambda}\biggl{[}\frac{\bigl{(}q\overline{\phi}\bigr{)}^{\\!2}\\!-\\!\lambda\overline{\phi}^{2}\\!+\\!\bigl{(}\frac{1}{6}\\!-\\!\xi\bigr{)}R}{\lambda\overline{\phi}^{2}}\biggr{]}=\frac{q^{4}}{\lambda^{2}}+\frac{1}{\Xi\\!-\\!1}\Bigl{(}\frac{q^{2}}{\lambda}-1\Bigr{)}+2\,,$
(125) $\displaystyle C_{\scriptscriptstyle T}={}$
$\displaystyle\frac{2}{3}\biggl{[}\frac{3\lambda\overline{\phi}^{2}\\!-\\!\bigl{(}\frac{1}{6}\\!-\\!\xi\bigr{)}R}{\lambda\overline{\phi}^{\,2}}\biggr{]}\\!+\frac{q^{2}}{\lambda}\biggl{[}\frac{\bigl{(}q\overline{\phi}\bigr{)}^{\\!2}\\!-\\!\lambda\overline{\phi}^{\,2}\\!+\\!\bigl{(}\frac{1}{6}\\!-\\!\xi\bigr{)}R}{\lambda\overline{\phi}^{\,2}}\biggr{]}=\frac{q^{4}}{\lambda^{2}}+\frac{1}{\Xi\\!-\\!1}\Bigl{(}\frac{q^{2}}{\lambda}-\frac{2}{3}\Bigr{)}+\frac{4}{3}\,.$
(126)

where $\Xi\\!=\\!(1\\!-\\!6\xi)(2\\!-\\!\epsilon)/[2(1\\!-\\!\epsilon)^{2}]$.
It is worth noting that had we applied the effective potential approximation
for our computation, and was utilized in Katuwal:2021kry , we would have
missed the terms of order $\mathcal{O}(\epsilon)$ in the coefficients above
that appear in the dominant late-time contribution. This can be tracked down
to the classical equation (13) that the tree-level condensate is assumed to
satisfy. The effective potential approximation in our case would correspond to
neglecting the d’Alembertian term in that equation. Even though this drops the
time derivatives, the condensate is still time-dependent because of the non-
minimal coupling, and retains the same scaling, but its amplitude does not
account correctly for the $\mathcal{O}(\epsilon)$ terms.

The factors (125) and (126) that multiply the secular corrections in (124)
depend only on the two independent combinations of model parameters,
$q^{2}/\lambda$ and $\Xi$, and this dependence is depicted in Fig. 4.

Figure 4:  Density plots of the coefficient (125) multiplying the late time
correction of the condensate (left), and of the coefficient (126) multiplying
the late time correction of the energy-momentum tensor (right). Thin black
curves denote points of equal value of the coefficient; curves are equidistant
in values of the coefficients. The value increases from the dark shading to
the light shading. The thick, black, nearly straight lines denote where the
coefficient vanishes, and the values the coefficients to the left of these
lines are negative, while values to the right are positive.

The case of hierarchal couplings $q^{2}\\!\gg\\!\lambda$ from Sec. (5.5) is
particularly interesting. It is captured by a simple expression independent of
the non-minimal coupling,

$\bigl{\langle}\hat{\phi}\bigr{\rangle}\ \overset{a\gg
1}{\scalebox{1.8}[1.0]{$\sim$}}\
\overline{\phi}\biggl{[}1+\frac{q^{4}}{\lambda}\frac{3\epsilon}{16\pi^{2}}\ln(a)\biggr{]}\,,\qquad\qquad\bigl{\langle}\hat{T}_{\mu\nu}\bigr{\rangle}\
\overset{a\gg 1}{\scalebox{1.8}[1.0]{$\sim$}}\
\overline{T}_{\mu\nu}\biggl{[}1+\frac{q^{4}}{\lambda}\frac{3\epsilon}{8\pi^{2}}\ln(a)\biggr{]}\,,$
(127)

where the hierarchy of coupling constants can make up for the
$\epsilon$-suppression in inflation.

Looking at the late time-expressions (125), in particular the condensate
correction on the left, it might be tempting to resum the secular correction,
$\bigl{\langle}\hat{\varphi}\bigr{\rangle}\\!\to\\!\overline{\phi}_{0}a^{-\epsilon(1-\delta)}$
into a renormalized slow-roll parameter
$\epsilon\\!\to\\!\epsilon(1\\!-\\!\delta)$, where $\delta\\!=\\!3\lambda
C_{\phi}/(16\pi^{2})$. This resummation would be correct provided that the
equation of motion for $\bigl{\langle}\hat{\varphi}\bigr{\rangle}$ is linear
and autonomous, and that the secular logarithm is an artifact of the time-
dependent perturbation theory. This is very often what is implicitly or
explicitly assumed when the so-called Dynamical renormalization group is
invoked to resum the secular corrections (see e.g. Burgess:2009bs ;
Kamenshchik:2020yyn ). However, the original construction of the Dynamical
renormalization group Chen:1995ena  is essentially a reformulation of multiple
scale analysis, and consequently necessitates knowing the dynamical equation
before anything can be resummed, and not only the first few perturbative
corrections. The fact that this resummation does not work here is obviated by
the energy-momentum correction that cannot be resummed by the same slow-roll
parameter correction on the account of the coefficients in (125) and (126)
being different. In fact, the secular correction in (124) is an effect of
ultraviolet physics in a nonequilibrium setting. The appropriate framework to
describe (and eventually resum) them is the standard renormalization group
formalism, implemented in a nonequilibrium setting. The following section is
devoted to discussing some aspects of it.

###  6.1 Renormalization group explanation

The secular corrections in (124) at late times dominate over the tree-level
result, and thus invalidate the perturbative approach. The first step towards
extending the results past this point is to understand which equation governs
these secular corrections. It is that equation that ultimately provides the
resummation scheme. Here we demonstrate that, at the perturbative level, the
secular corrections are governed by the standard renormalization group
equations applied to a nonequilibrium setting.

The utility of the renormalization group hinges on our ability to recognize
the relevant reference scale $\mu_{0}$ that appears together with the
arbitrary renormalization scale $\mu$, as a logarithm of a dimensionless
ratio, $\ln(\mu_{0}/\mu)$. The appropriate choice of the reference scale
ultimately allows the renormalization group formalism to effectively resum all
such logarithms, thus extending the validity of perturbative results. For
systems with multiple physical scales, such as the one we consider here,
different regimes will have different reference scales Elizalde:1993qh .
Moreover, for dynamical systems the reference scale will in general be time-
dependent. Looking at the final one-loop corrections (53)–(54) for the
condensate, and (85)–(86) for the energy-momentum tensor, the natural choice
for the reference scale seems to be $\mu_{0}\\!=\\!(1\\!-\\!\epsilon)H$.
However, as various limits considered in Sec. 5 demonstrate, the relevant
reference scale depends on the parameter regime. It is the digamma functions,
that are inextricably tied to the logarithms they appear with, that set the
correct reference scale, without having to choose it by hand. This is perhaps
most evident in the hierarchal limit of Sec. 5.5, where the reference scale is
recognized as $\mu_{0}\\!=\\!q\overline{\phi}$.

The system at hand is a multiscale problem, for which in general it is not
possible to choose a single reference scale which captures the large
logarithms in all regimes. Here this is essentially due to corrections
descending from two independent loops — the scalar and the vector one.
Potentially large logarithms are captured by the reference scale if it
accounts for the digamma function accompanying the logarithm. In our case we
can make two choices which capture either the full large logarithms descending
from the scalar loop, or ones descending from the vector loop, respectively,

$\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)^{2}H^{2}}{\mu^{2}}\Bigr{]}+\Psi(\nu_{\scriptscriptstyle
S})\longrightarrow\ln\Bigl{(}\frac{\mu_{0}^{2}}{\mu^{2}}\Bigr{)}\,,\quad{\tt
or}\quad\ln\Bigl{[}\frac{(1\\!-\\!\epsilon)^{2}H^{2}}{\mu}\Bigr{]}+\Psi(\nu_{\scriptscriptstyle
V})\longrightarrow\ln\Bigl{(}\frac{\mu_{0}^{2}}{\mu^{2}}\Bigr{)}\,,$ (128)

where
$(1\\!-\\!\epsilon)^{2}H^{2}\\!=\\!\frac{1}{2}\big{[}\big{(}\frac{1}{6}\\!-\\!\xi\big{)}R-\lambda\overline{\phi}^{\,2}\big{]}\\!>\\!0$.
However, we cannot do both simultaneously. That would require utilizing the
full machinery of multiscale renormalization group Einhorn:1983fc ;
Ford:1994dt ; Ford:1996yc ; Ford:1996hd . Nevertheless, this is not necessary
for our purposes, as the late time behaviour is not marred by such subtleties.
In fact, there is universality in the late-time secular corrections (124).
This is because the tree-level quantities that make up different reference
scales all scale like the Hubble rate. Therefore, whatever $\mu_{0}$ precisley
is, at late times we always have,

$\ln\Bigl{(}\frac{\mu_{0}}{\mu}\Bigr{)}\
\overset{a\to\infty}{\scalebox{1.8}[1.0]{$\sim$}}-\epsilon\ln(a)\ll-1\,,$
(129)

even if $m_{\scriptscriptstyle S}\\!\gg\\!H$ or $m_{\scriptscriptstyle
V}\\!\gg\\!H$.

The observations above can be formalized using the renormalization group
machinery. Our results depend on the arbitrary renormalization scale $\mu$,

$\bigl{\langle}\hat{\phi}\bigr{\rangle}=\bigl{\langle}\hat{\phi}\bigr{\rangle}\Bigl{(}H,\bigl{\\{}\lambda_{n}\bigr{\\}},\ln\bigl{[}\tfrac{\mu_{0}}{\mu}\bigr{]}\Bigr{)}\,,\qquad\quad\bigl{\langle}\hat{T}_{\mu\nu}\bigr{\rangle}=\bigl{\langle}\hat{T}_{\mu\nu}\bigr{\rangle}\Bigl{(}H,\bigl{\\{}\lambda_{n}\bigr{\\}},\ln\bigl{[}\tfrac{\mu_{0}}{\mu}\bigr{]}\Bigr{)}\,,$
(130)

where
$\bigl{\\{}\lambda_{n}\bigr{\\}}\\!=\\!\\{\lambda,\xi,Z_{\phi},\alpha_{\scriptscriptstyle
R}\\}$ stands for all the coupling constants. Note that we have chosen to
treat the scalar wavefunction renormalization as a coupling constant with its
associated $\beta$-function for convenience, instead of treating it as the
anomalous dimension $\gamma$ of the field; this distinction is essentially
immaterial Coleman:1973jx . The renormalization scale $\mu$ does not have a
physical meaning by itself. What does have physical sense is performing
several measurements at some scale $\mu_{*}$. This process determines the
couplings as functions of the ratio $\mu/\mu_{*}$

$\lambda_{n}\rightarrow\lambda_{n}\bigl{(}\ln\bigl{[}\tfrac{\mu_{*}}{\mu}\bigr{]}\bigr{)}\,,$
(131)

such that the dependence on $\mu$ completely disappears from the result,

$\displaystyle\mu\frac{d}{d\mu}\bigl{\langle}\hat{\phi}\bigr{\rangle}\Bigl{(}H,\bigl{\\{}\lambda_{n}\bigl{(}\ln\bigl{[}\tfrac{\mu_{*}}{\mu}\bigr{]}\bigr{)}\bigr{\\}},\ln\bigl{[}\tfrac{\mu_{0}}{\mu}\bigr{]}\Bigr{)}=0\,,$
(132)
$\displaystyle\mu\frac{d}{d\mu}\bigl{\langle}\hat{T}_{\mu\nu}\bigr{\rangle}\Bigl{(}H,\bigl{\\{}\lambda_{n}\bigl{(}\ln\bigl{[}\tfrac{\mu_{*}}{\mu}\bigr{]}\bigr{)}\bigr{\\}},\ln\bigl{[}\tfrac{\mu_{0}}{\mu}\bigr{]}\Bigr{)}=0\,,$
(133)

which is guaranteed by the running couplings satisfying the running equations,

$\mu\frac{d}{d\mu}\lambda_{n}=\beta_{n}\bigl{(}\\{\lambda_{m}\\}\bigr{)}\,,$
(134)

where $\beta_{n}$ are the $\beta$-functions associated to couplings
$\lambda_{n}$, that are determined by the structure of divergences. The one-
loop $\beta$-functions are obtained by acting
$-\mu\frac{\partial}{\partial\mu}$ on the counterterm coefficients given in
sections 4.3.1 and 4.3.2, and taking the limit $D\\!\rightarrow\\!4$,

$\beta_{\xi}=-\frac{\lambda(1\\!-\\!6\xi)}{16\pi^{2}}+\frac{q^{2}}{16\pi^{2}}\,,\quad\
\beta_{\lambda}=\frac{9\lambda^{2}}{8\pi^{2}}+\frac{3q^{4}}{8\pi^{2}}\,,\quad\
\beta_{Z_{\phi}}=\frac{3q^{2}}{8\pi^{2}}\,,\quad\
\beta_{{\alpha}_{\scriptscriptstyle
R}}=-\frac{(1\\!-\\!6\xi)^{2}}{1152\pi^{2}}\,.$ (135)

It is straightforward to check that our one-loop corrections satisfy the
Callan-Symanzik equations to the given order,

$\mu\frac{\partial}{\partial\mu}\bigl{\langle}\hat{\varphi}\bigr{\rangle}=-\beta_{n}\frac{\partial}{\partial\lambda_{n}}\overline{\phi}\,,\qquad\qquad\mu\frac{\partial}{\partial\mu}\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}=-\beta_{n}\frac{\partial}{\partial\lambda_{n}}\overline{T}_{\mu\nu}\,,$
(136)

where we have reintroduced an arbitrary $Z_{\phi}$ and
$\alpha_{\scriptscriptstyle R}$ dependence in the tree-level expressions,

$\displaystyle\overline{\phi}\bigl{(}H,\xi,\lambda,Z_{\phi}\bigr{)}={}$
$\displaystyle\pm
H\sqrt{\frac{1}{\lambda}\Bigl{[}Z_{\phi}\epsilon(3\\!-\\!2\epsilon)-6\xi(2\\!-\\!\epsilon)\Bigr{]}}\,,$
(137)
$\displaystyle\overline{T}_{\mu\nu}\bigl{(}H,\xi,\lambda,Z_{\phi},\alpha_{\scriptscriptstyle
R}\bigr{)}={}$ $\displaystyle\biggl{[}-\frac{1}{4}(3\\!-\\!4\epsilon)\epsilon
g_{\mu\nu}+\epsilon^{2}\bigl{(}a^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\bigr{)}\biggr{]}$
$\displaystyle\hskip
56.9055pt\times\biggl{[}(Z_{\phi}\\!-\\!6\xi)\bigl{(}\overline{\phi}H\bigr{)}^{\\!2}+144\alpha_{R}(2\\!-\\!\epsilon)H^{4}\biggr{]}\,.$
(138)

Then, having correctly identified the relevant late-time reference scale, as
described at the beginning of this section, we immediately infer the late-time
limit,

$\bigl{\langle}\hat{\varphi}\bigr{\rangle}\
\overset{a\to\infty}{\scalebox{1.8}[1.0]{$\sim$}}\
\epsilon\ln(a)\times\beta_{n}\frac{\partial}{\partial\lambda_{n}}\overline{\phi}\,,\qquad\qquad\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}\
\overset{a\to\infty}{\scalebox{1.8}[1.0]{$\sim$}}\
\epsilon\ln(a)\times\beta_{n}\frac{\partial}{\partial\lambda_{n}}\overline{T}_{\mu\nu}\,.$
(139)

Apart from efficiently reproducing the late-time correction obtained from the
loop computation, the real power of the renormalization group is in its
ability to resum the large logarithms. This is accomplished by first making a
variable substitution, and adopting the dimensionless ratio that includes the
reference scale $\mu_{0}$ as the new variable,

$s=\ln\Bigl{(}\frac{\mu_{0}}{\mu}\Bigr{)}\,,$ (140)

such that the Callan-Symanzik equations read,

$\frac{d}{ds}\bigl{\langle}\hat{\phi}\bigr{\rangle}\Bigl{(}H,\lambda_{n}\bigl{(}\ln\bigl{[}\tfrac{\mu_{*}}{\mu_{0}}\bigr{]}\\!+\\!s\bigr{)},s\Bigr{)}=0\,,\qquad\frac{d}{ds}\bigl{\langle}\hat{T}_{\mu\nu}\bigr{\rangle}\Bigl{(}H,\lambda_{n}\bigl{(}\ln\bigl{[}\tfrac{\mu_{*}}{\mu_{0}}\bigr{]}\\!+\\!s\bigr{)},s\Bigr{)}=0\,,$
(141)

and the equations for the running couplings,

$\frac{d}{ds}\lambda_{n}=\beta_{n}\bigl{(}\\{\lambda_{m}\\}\bigr{)}\,.$ (142)

Once we have solved for the running couplings as functions of $s$, the Callan-
Symanzik equations allow us to express the results choosing any $s$.
Particularly useful is the choice that removes as much of the explicit
$s$-dependence from the results, and translates it to the $s$-dependence of
the couplings. This choice corresponds to the following,

$\bigl{\langle}\hat{\phi}\bigr{\rangle}=\bigl{\langle}\hat{\phi}\bigr{\rangle}\Bigl{(}H,\lambda_{n}\bigl{(}\ln\bigl{[}\tfrac{\mu_{*}}{\mu_{0}}\bigr{]}\bigr{)},0\Bigr{)}\,,\qquad\quad\bigl{\langle}\hat{T}_{\mu\nu}\bigr{\rangle}=\bigl{\langle}\hat{T}_{\mu\nu}\bigr{\rangle}\Bigl{(}H,\lambda_{n}\bigl{(}\ln\bigl{[}\tfrac{\mu_{*}}{\mu_{0}}\bigr{]}\bigr{)},0\Bigr{)}\,.$
(143)

provided we had identified the relevant reference scale $\mu_{0}$. What (143)
accomplishes when compared with (141) is that the explicit dependence on the
Hubble rate via $s$ is moved to the coupling constants. Working at the one-
loop level corresponds to solving Eqs. (142) to linear order in $s$,

$\displaystyle\xi(s)={}$
$\displaystyle\xi(0)-\biggl{[}\frac{\lambda(0)\big{[}1\\!-\\!6\xi(0)\big{]}}{16\pi^{2}}-\frac{q^{2}(0)}{16\pi^{2}}\biggr{]}s\,,$
(144a) $\displaystyle\lambda(s)={}$
$\displaystyle\lambda(0)+\biggl{[}\frac{9\lambda^{2}(0)}{8\pi^{2}}+\frac{3\,q^{4}(0)}{8\pi^{2}}\biggr{]}s\,,$
(144b) $\displaystyle Z_{\phi}(s)={}$ $\displaystyle
Z_{\phi}(0)+\frac{3\,q^{2}(0)}{8\pi^{2}}s\,,$ (144c)
$\displaystyle\alpha_{\scriptscriptstyle R}(s)={}$
$\displaystyle\alpha_{\scriptscriptstyle
R}(0)-\frac{\big{[}1\\!-\\!6\xi(0)\big{]}^{2}}{1152\pi^{2}}s\,.$ (144d)

It is straightforward to see that when these solutions are plugged into the
tree-level result they reproduce exactly the late-time behaviour,

$\displaystyle\bigl{\langle}\hat{\phi}\bigr{\rangle}={}$
$\displaystyle\overline{\phi}+\bigl{\langle}\hat{\varphi}\bigr{\rangle}\equiv\overline{\phi}\bigl{(}H,\xi(s),\lambda(s),Z_{\phi}(s)\bigr{)}\,,$
(145) $\displaystyle\bigl{\langle}\hat{T}_{\mu\nu}\bigr{\rangle}={}$
$\displaystyle\overline{T}_{\mu\nu}+\bigl{\langle}\hat{t}_{\mu\nu}\bigr{\rangle}\equiv\overline{T}_{\mu\nu}\bigl{(}H,\xi(s),\lambda(s),Z_{\phi}(s),\alpha_{\scriptscriptstyle
R}(s)\bigr{)}\,.$ (146)

Note that, even though all of the time-dependence has been absorbed into
effective running of the coupling constants, this does not imply that the
effect is not observable. On the contrary, quantum corrections will cause the
ratio $\bigl{\langle}\hat{\phi}\bigr{\rangle}/H$ to acquire time dependence,
which is a physical effect. We emphasize that the classical form in
(145)–(146) is in general achieved only at asymptotically late times. At
intermediate times, other logarithmically enhanced contributions can in
general occur. However, since we are here primarily interested in the question
of restoring perturbativity at late times, this analysis suffices.

The analysis of this section opens up an exciting possibility of resumming the
corrections past the breakdown of the perturbative expansion, and obtaining
reliable late-time behaviour utilizing the machinery of the renormalization
group. Accomplishing this requires supplementing the equations for the running
couplings (142) with an additional two for the $U(1)$ charge $q$, and for the
vector wavefunction renormalization $Z_{\scriptscriptstyle A}$, and then
solving the entire system exactly. For this we first need the one-loop
$\beta$-functions for $q$ and $Z_{\scriptscriptstyle A}$ in the unitary gauge,
which do not appear at the level of one-loop analysis we performed here, but
are important for resumming the late-time behaviour correctly. This is left
for future work. Finally, we emphasize that the renormalization group can
explain the large ultraviolet logarithms, but not the infrared logarithms
Woodard:2008yt , which in inflation are captured by a variant of Starobinsky’s
stochastic formalism Starobinsky:1986fx ; Starobinsky:1994bd , as was
accomplished in Miao:2021gic ; Glavan:2021adm .

##  7 Discussion

Quantum effects in inflation are widely studied utilizing de Sitter space as
an idealized model space for inflation, and quantum loop corrections have been
calculated in many models of interest. On the other hand, comparatively little
is known about quantum effects in more realistic models of inflation, where
the expansion rate is an adiabatic function of time. Of particular interest
are models which exhibit novel secular effects that do not occur in de Sitter
limit, and this work is devoted to one such system. We consider power-law
inflation, characterized by a constant principal slow-roll parameter
$\epsilon\\!=\\!-\dot{H}/H^{2}$, as a mathematically tractable model of
inflating background, on which we studied the behaviour of a spectator non-
minimally coupled Abelian Higgs model. The Abelian Higgs model is considered
to be in a classical attractor regime, characterized by the scaling behaviour,
$\overline{\phi}/H\\!=\\!{\rm const.}$, in which the condensate tracks the
evolution of the Hubble rate. This attractor can be seen as a dynamical
generalization of a symmetry-breaking minimum. We computed dimensionally
regulated one-loop corrections to the condensate depicted in Fig. 1, and to
the energy-momentum tensor depicted in Figs. 2 and 3. The tadpole
contributions to the energy-momentum tensor in Fig. 2 are often not
considered, but are of essential importance because, only when they are added
to the contributions from vacuum fluctuations in Fig. 3, is the one-loop
energy-momentum tensor conserved. The computations are performed in the
unitary gauge,666 We find no obstacle in computing dimensionally regulated
loop correction in the unitary gauge, conforming with the experience from flat
space equilibrium computations Irges:2017ztc . However, Ref. Mooij:2011fi
reports problems with the unitary gauge for dynamical scalar condensates, when
compared to the covariant gauge computations. This issue should be resolved by
repeating the explicit computations of this paper in some analogue of the
$R_{\xi}$ gauge.  in which both quantities of our interest are manifestly
gauge-independent observables. Our results (53)–(54) and (85)–(86) capture
both the infrared and ultraviolet effects, and they go beyond the effective
potential approximation, in that they descend from the full one-loop effective
action corrections, which include kinetic term corrections that need not be
small for finite $\epsilon$ backgrounds.

Particularly interesting are the secular corrections to both the condensate
and the energy-momentum tensor given in (124)–(126), which dominate at late
times. The corrected late time result takes the form of the tree-level result
multiplied by a time-dependent amplitude. Since this amplitude grows in time,
the corrections can be seen as driving the evolution away from the classical
attractor given in (14) and (15), pointing to the quantum instability of the
attractor. Classically the condensate rolls down the potential tracking the
evolution of the Hubble rate. The late time one-loop correction is enhanced by
a logarithm of the scale factor (the number of e-foldings) multiplied by the
coefficient that takes positive values for the most range of the model
parameters, as seen from Fig. 4. Thus the one-loop correction tends to slow
down the rolling of the spectator scalar down its potential.

It is interesting that the secular corrections we find are multiplied by the
principal slow-roll parameter $\epsilon$, and thus are suppressed in
inflation, and vanish in the exact de Sitter limit $\epsilon\\!=\\!0$, that is
often employed as a tractable model of inflation. This points to the fact that
power-law inflation, and other classes of inflating spacetimes closer to more
realistic slow-roll inflation can harbour qualitatively different, and
potentially important effects. While in inflation the slow-roll parameter acts
as a suppression factor, our results are valid for finite constant (or
adiabatically evolving) $\epsilon$, including instances where it does not act
as a suppression. Nonetheless, given enough time the secular correction will
necessarily overcome the possible suppressions coming from the slow-roll
parameter or the coupling constants, leading to the breakdown of perturbation
theory. Similar types of corections have also been noticed 777 It is
interesting that for quantum-gravitational corrections found in Frob:2018tdd ;
Lima:2020prb  the one-loop correction is actually decaying with respect to the
tree-level result. Even though there is a $\ln(a)$ enhancement factor
descending from the UV corrections, just like the one we find here, the
correction decays as a power law as it is additionally multiplied by a
decaying $H^{2}$ that has to be there to form a dimensionless product with the
dimensionful loop-counting parameter $\kappa^{2}\\!=\\!16\pi
G_{\scriptscriptstyle N}$, where $G_{N}$ is the Newton’s constant. in
Janssen:2009pb ; Frob:2018tdd ; Lima:2020prb ; Katuwal:2021kry . This serves
as a very good motivation for further studies of quantum corrections to
observables in power-law inflation, using both perturbative Janssen:2009pb ;
Glavan:2019uni , and non-perturbative Prokopec:2015owa ; Cho:2015pwa ;
Markkanen:2017edu  methods.

Another observation to be made is that the possible hierarchy between the
coupling constants, $q^{2}\\!\gg\\!\lambda$, discussed in Sec. 5.5, leads to
an unsuppressed loop correction. This correction is furthermore made bigger at
late times by the secular logarithm, exacerbating the problem of applying
perturbation theory. Unlike the secular correction, the coupling hierarchy
enhancement survives in the de Sitter limit. In fact it is already seen in
flat space Coleman:1973jx , where the vector loop moves the minimum of the
effective potential by an amount larger than tree-level. Despite that one
expects perturbativity to be respected at two- and higher loops Ford:1992pn ;
Nielsen:2022oec . In cosmological expanding spaces, on the other hand, this
perturbativity will necessarily be spoiled at late times due to secular
corrections we report here.

The secular corrections we find are an ultraviolet effect, and can ultimately
be accounted for by the renormalization group, as we show in Sec. 6.1. This is
because the secular logarithm always appears together with the logarithm of
the renormalization scale $\mu$ in the form
$\ln\bigl{[}(1\\!-\\!\epsilon)H/\mu\bigr{]}$ given in (123). Consequently, the
secular correction can be captured by an effective time-dependent running of
the coupling constants given by the renormalization group equations. Note that
this does not imply that all the physical scales are simply rescaled
differently at different points in time. It is the fact that the quantum-
corrected condensate no longer scales as the Hubble rate, and it is their
ratio that has a physical meaning. It would be interesting to investigate the
full power of the renormalization group to resum the late time behaviour, and
provide reliable results beyond the breakdown of perturbation theory. For the
case at hand, that would first require computing the beta functions for the
$U(1)$ charge and the vector field wavefunction renormalization, which do not
appear in the strictly perturbative computation at hand. Early efforts in this
direction are undertaken in Refs. Miao:2021gic ; Glavan:2021adm , but a lot
more is needed to fully develop the formalism, especially for its higher loop
implementation.

It would, furthermore, be interesting to examine quantum corrections to the
condensate when the scalar field is the inflaton Janssen:2008dw  as well as
the quantum backreaction from cosmological perturbations Mukhanov:1996ak ;
Abramo:1997hu ; Abramo:1998hi ; Abramo:1998hj ; Abramo:2001dc . A notable
model is Higgs inflation Bezrukov:2007ep ; Bezrukov:2013fka , and stability of
the quantum model is of particular interest, which can be addressed without
Bezrukov:2014ipa , or with taking into account the background curvature
corrections Herranen:2014cua ; Herranen:2015ima ; Figueroa:2017slm  and
additional fine tuning problems Miao:2015oba ; Liao:2018sci ; Miao:2019bnq ;
Miao:2020zeh ; Kyriazis:2019xgj ; Sivasankaran:2020dzp ; Katuwal:2021kry .
Important questions to be addressed are how the secular corrections discussed
in this work affect stability of Higgs inflationary models, as well as the
amplitude of the scalar and tensor spectra. Secular corrections to the
reheating period following Higgs inflation is also of great interest
Herranen:2015ima ; Figueroa:2017slm ; Ema:2016dny ; Katuwal:2022szw .

## Acknowledgements

We thank Arttu Rajantie and Markus Fröb for discussions on the topic. DG was
supported by the European Union and the Czech Ministry of Education, Youth and
Sports (Project: MSCA Fellowship CZ FZU I — CZ.02.01.01/00/22_010/0002906).
This work is part of the Delta ITP consortium, a program of the Netherlands
Organisation for Scientific Research (NWO) that is funded by the Dutch
Ministry of Education, Culture and Science (OCW) — NWO project number
24.001.027.

## References

  * (1) S. P. Miao, T. Prokopec and R. P. Woodard, “Deducing Cosmological Observables from the S-matrix,” Phys. Rev. D 96 (2017) no.10, 104029 [arXiv:1708.06239 [gr-qc]].
  * (2) N. D. Birrell and P. C. W. Davies, “Quantum Fields in Curved Space,” Cambridge Univ. Press, 1984, ISBN 978-0-521-27858-4, 978-0-521-27858-4 doi:10.1017/CBO9780511622632
  * (3) E. Elizalde and S. D. Odintsov, “Renormalization group improved effective potential for gauge theories in curved space-time,” Phys. Lett. B 303 (1993), 240-248 [arXiv:hep-th/9302074 [hep-th]].
  * (4) E. Elizalde and S. D. Odintsov, “Renormalization group improved effective potential for interacting theories with several mass scales in curved space-time,” Z. Phys. C 64 (1994), 699-708 [arXiv:hep-th/9401057 [hep-th]].
  * (5) T. Markkanen and A. Tranberg, “Quantum Corrections to Inflaton and Curvaton Dynamics,” JCAP 11 (2012), 027 [arXiv:1207.2179 [gr-qc]].
  * (6) M. Herranen, A. Hohenegger, A. Osland and A. Tranberg, “Quantum corrections to inflation: the importance of RG-running and choosing the optimal RG-scale,” Phys. Rev. D 95 (2017) no.2, 023525 [arXiv:1608.08906 [hep-ph]].
  * (7) F. L. Bezrukov and M. Shaposhnikov, “The Standard Model Higgs boson as the inflaton,” Phys. Lett. B 659 (2008), 703-706 [arXiv:0710.3755 [hep-th]].
  * (8) F. Bezrukov, “The Higgs field as an inflaton,” Class. Quant. Grav. 30 (2013), 214001 [arXiv:1307.0708 [hep-ph]].
  * (9) F. Bezrukov, J. Rubio and M. Shaposhnikov, “Living beyond the edge: Higgs inflation and vacuum metastability,” Phys. Rev. D 92 (2015) no.8, 083512 [arXiv:1412.3811 [hep-ph]].
  * (10) S. Mooij and M. Postma, “Goldstone bosons and a dynamical Higgs field,” JCAP 09 (2011), 006 [arXiv:1104.4897 [hep-ph]].
  * (11) D. P. George, S. Mooij and M. Postma, “Effective action for the Abelian Higgs model in FLRW,” JCAP 11 (2012), 043 [arXiv:1207.6963 [hep-th]].