# On Adiabatic Renormalization with a Physically Motivated Infrared Cut-Off

Chiara Animali  Pietro Conzinu  and Giovanni Marozzi

###### Abstract

We introduce a new approach to renormalize physical quantities in curved
space-time by adiabatic subtraction. We use a comoving infrared cut-off in
defining the adiabatic counterpart of the physical quantity under
consideration, building on the fact that the adiabatic approximation is ill-
defined in the infrared tail of the spectrum. We show how this infrared cut-
off should be used to obtain a completely well-defined renormalization scheme
and how it is fundamental to avoid unphysical divergences that can be
generated by a pathological behavior of the adiabatic subtraction extended to
the infrared tail. The infrared cut-off appears as a new degree of freedom
introduced in the theory and its actual value has to be consistently fixed by
a physical prescription. As an example, we show how such degree of freedom can
be set to obtain the correct value of the conformal anomaly in the symptomatic
case of an inflationary model with gauge fields coupled to a pseudo-scalar
inflaton.

##  1 Introduction

Inflation, an early phase of accelerated expansion, was originally proposed to
solve the problems associated with the Hot Big-Bang standard theory [1, 2]. It
was then realized that inflation could also generate a nearly scale invariant
spectrum of scalar and tensor fluctuations [3, 4], already in its minimal
version with only one real scalar field.

The simplicity of the inflationary idea and its success in providing a theory
of initial conditions made inflation, from one of the possible scenarios of
the early Universe, one of the most recognized and accepted candidates,
leading to a continuous investigation of different inflationary models and to
the study of their predictions and phenomenology (see, for instance, [5], for
an exhaustive collection of inflationary models). Due to its relevance,
inflation is also a natural playground to study aspects of quantum fields in
curved space-times.

Within the framework of the inflationary paradigma, it is well-known that
correlation functions (or in general bi-linear observables) of quantum fields
on a curved background suffer from divergences. In general, the presence of
ultraviolet (UV) divergences due to fluctuations on arbitrary short scales is
a common aspect of quantum field theory [6]. In flat space and for free
theories, infinities can be removed by normal ordering, namely, by subtracting
the expectation value of the vacuum energy. This is legitimate, indeed this
contribution is not observable. Differently, in curved space-time such
divergences cannot be easily cured as in flat space [7], since the vacuum is
not unambiguously defined.

Among the several renormalization schemes proposed, the one most commonly used
in the context of inflationary cosmology is the adiabatic renormalization [8,
9, 10, 11, 12]. The adiabatic procedure to renormalize divergent quantities in
curved space-times is based on subtracting the expectation value of such
quantities associated with the adiabatic vacuum, which is the vacuum that
minimizes the creation of particles due to the presence of a time-dependent
metric. The advantage of the adiabatic renormalization relies on the fact that
on one hand its physical interpretation is clear and on the other hand its
implementation is straightforward.

From its introduction [8, 9, 10, 11] this method has been applied in various
examples with success (see [7, 13, 14] for a comprehensive review), however in
recent years its correct use and its consequences on physical observables
(like the CMB power spectrum) were subjects of several investigations and
controversy. For example, in [15, 16], it was argued that to properly evaluate
the power spectrum of inflationary fluctuations, adiabatic subtraction should
be taken into account, since such spectrum diverges at coincident points. In
particular, on one hand in [17] this idea was applied by subtracting the
adiabatic term at the Hubble exit, obtaining results that differ significantly
from the standard ones [18]. On the other hand in [19] it was suggested that
the right time to perform the subtraction is the end of inflation rather than
the Hubble exit, in which case the impact of the adiabatic subtraction on the
scalar and tensor power spectra is subleading.

As discussed in [19], the main issue in the adiabatic subtraction procedure is
that it is ill defined when the scales of interest are stretched beyond the
Hubble horizon111See also [20, 21] for further criticism on the main idea of
[15, 16, 17].. In support of this claim, in recent literature there are cases
in which the adiabatic subtraction introduces unphysical infrared (IR)
divergences (see, for example, [22, 23]) when performed over all the
k-spectrum. For example, in [22] it was shown, in the case of massless gauge
fields coupled to a pseudo-scalar inflaton, how the adiabatic renormalization
correctly removes the UV divergences but leads to IR divergences. In
particular, despite the bare values of the energy density and of the helicity
of gauge fields are not divergent in the infrared, their adiabatic
counterparts introduce logarithmic divergences associated to the behaviour of
the adiabatic approximation for massless fields in the infrared tail.

The above points suggest how the adiabatic subtraction extended over the full
IR domain leads to practical and conceptual issues. In the following we will
show how to modify the usual adiabatic renormalization procedure along this
direction, introducing a comoving IR cut-off, and we will show how to fix it
univocally, hence the renormalization scheme, by a physically motivated
prescription for the mentioned case of massless gauge fields coupled to a
pseudo-scalar inflaton.

The paper is organized as follows. In Sec. 2 we describe the adiabatic
subtraction procedure and we emphasize the proper range of application of such
adiabatic subtraction. In Sec. 3 we describe how to extend the adiabatic
renormalization scheme by inserting a comoving IR cut-off and we show how to
fix such cut-off univocally by a proper physical prescription for the
phenomenologically interesting case of a gauge field coupled with a pseudo-
scalar inflaton by a Chern-Simons-like term. In such model, in fact, the usual
adiabatic regularization method exhibits problematic aspects. We show how the
renormalization scheme can be fixed requiring to match the proper value of the
conformal anomaly of gauge fields and how this procedure leads to well-defined
finite results for their energy density and helicity integrals. Finally, in
Sec. 4 we present our final remarks and conclusions. In Appendix A we comment
on the covariant conservation of the energy momentum-tensor when adiabatic
subtraction with a comoving IR cut-off is performed, whereas Appendix B
extends the adiabatic results of Sec. 3 to the case of massive gauge fields.

##  2 Adiabatic subtraction and the need for an infrared cut-off

Let us introduce the adiabatic regularization method by studying the
pedagogical (but physically motivated) case of a test scalar field in a curved
space-time [7, 13]. We consider a Friedmann-Lemaître-Robertson-Walker (FLRW)
metric

${\rm d}s^{2}={\rm d}t^{2}-a^{2}(t)\,{\rm d}\mathbf{x}^{2}\,,$ (2.1)

and a Lagrangian density given by

$\mathcal{L}=\frac{1}{2}|g|^{1/2}\left(g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-m^{2}\phi^{2}-\xi
R\phi^{2}\right)\,,$ (2.2)

with $a(t)$ the scale factor, $\xi$ a dimensionless coupling constant and $R$
the space-time scalar curvature. It follows that the equation of motion of the
scalar field is

$\left(\Box+m^{2}+\xi R\right)\phi=0\,.$ (2.3)

In close analogy to the Minkowski case, we can proceed with the standard
quantization of the scalar field, expanding the field operator as

$\phi(x)=\sum_{\mathbf{k}}\\{A_{\mathbf{k}}f_{\mathbf{k}}(x)+A^{\dagger}_{\mathbf{k}}f^{*}_{\mathbf{k}}(x)\\}\,,$
(2.4)

where $A^{\dagger}_{\mathbf{k}}$ and $A_{\mathbf{k}}$ are the creation and
annihilation operators and the mode function $f$ is given by222V represents
the spatial volume of a box, the continuum limit is approached for
$V\rightarrow\infty$.

$f_{\mathbf{k}}=(2V)^{-1/2}a(t)^{-3/2}h_{k}(t)e^{i\mathbf{k}\cdot\mathbf{x}}\,.$
(2.5)

From Eq. (2.3) it follows that the rescaled mode function $h_{k}(t)$ satisfies
the equation

$\ddot{h}_{k}+\Omega^{2}_{k}\,h_{k}=0\,,$ (2.6)

where a dot denotes a derivative w.r.t. cosmic time $t$ and the frequency is
given by 333We explicitly show the expression of the frequency but the
argument of this section is completely general.

$\Omega_{k}^{2}=\omega_{k}^{2}+\sigma\,,$ (2.7)

with

$\omega_{k}(t)=(k^{2}/a(t)^{2}+m^{2})^{1/2}\,\,,\qquad\sigma(t)=(6\xi-3/4)(\dot{a}/a)^{2}+(6\xi-3/2)\ddot{a}/a\,.$
(2.8)

The adiabatic renormalization method relies on the Wentzel-Kramer-Brillouin
(WKB) approximation of the mode function $h_{k}$

$h_{k}(t)=\frac{1}{\sqrt{2W_{k}(t)}}e^{-i\int W_{k}(t^{\prime}){\rm
d}t^{\prime}}\,,$ (2.9)

where $W_{k}(t)$ can be determined by inserting the WKB ansatz (2.9) into the
equation of motion (2.6). This leads to the non-linear differential equation

$W_{k}(t)^{2}=\Omega_{k}(t)^{2}-\left(\frac{\ddot{W}_{k}(t)}{2W_{k}(t)}-\frac{3\dot{W}_{k}(t)^{2}}{4W_{k}(t)^{2}}\right)\,,$
(2.10)

which in general is not possible to solve. However, solutions of this equation
can be obtained iteratively in the approximation of a background that slowly
changes in time (namely, under the condition of adiabatic expansion). This can
be pictured by introducing an adiabatic parameter $\epsilon$ that describes
the slowness of the time variation of the metric, and which is thus assumed to
be $\epsilon\ll 1$, and by assigning a power of $\epsilon$ to each time
derivative. In this way, the solution for $W_{k}(t)$ is obtained as a power
series in time derivatives

$W_{k}(t)=W_{k}^{(0)}(t)+\epsilon\,W_{k}^{(1)}(t)+\cdots+\epsilon^{n}\,W_{k}^{(n)}(t)\,,$
(2.11)

where $W_{k}^{(n)}$ is given by iterating the recursive equation up to order
$n$. Furthermore, let us add that time derivatives can be thought as curvature
derivatives, and thus the expansion in time also becomes an expansion in
curvature [7, 13].444This point can be better understood considering the
adiabatic approximation as the right approximation to minimize particle
creation in the limit where the single particle energy is larger with respect
to the energy scale determined by the curvature of the space-time [7, 13].
Thereby, following this procedure, the adiabatic solution for the mode
function can be obtained at each adiabatic order.

At this point, the proper renormalization based on the adiabatic method is
then realized performing the subtraction between a bare UV-divergent quantity
and its adiabatic counterpart, up to the right adiabatic order needed to
remove the UV divergences.

As a representative example, let us consider the expectation value of the
energy-momentum tensor $T_{\mu\nu}$. The finite result is given by the
subtraction

$\langle{T_{\mu\nu}}\rangle_{\text{ren}}=\langle
T_{\mu\nu}\rangle_{\text{bare}}-\langle T_{\mu\nu}\rangle_{\text{ad}}\,,$
(2.12)

where the first term on the r.h.s is the bare quantity while the second one is
its adiabatic counterpart. For this particular case of the energy-momentum
tensor one should consider the adiabatic expansion up to fourth order to be
able to cancel the UV divergences of the bare expectation value [7, 13].

From this brief introduction, it is immediate to grasp the power of the
adiabatic renormalization procedure, thanks to its intuitive physical meaning
as well as to its straightforward implementation. However, let us remark that
adiabatic renormalization (or, equivalently, regularization) concerns the
renormalization of UV divergences, essentially because the WKB ansatz for the
mode functions matches exactly the solution in the deep UV, where the space-
time is well approximated by the Minkowski one.

To better understand this point, we can consider again the WKB expression in
Eq. (2.9). As we can see it describes an oscillating solution, which is indeed
suitable only for modes that experience a negligible gravitational
interaction. In a cosmological fashion we should say that this is a good
approximation only for those modes that are sub-horizon. This last aspect is
properly the main point to keep in mind. However, in the common practice, the
adiabatic subtraction is generally extended also to the IR domain, despite
this is not properly fair, since, as said, the adiabatic approximation is not
well-defined in the IR (superhorizon) regime [19].

The argumentation is twofold, indeed let us consider the adiabatic expansion
up to a generic order $n>4$ for the energy-momentum tensor, integrating in the
momentum space between $k=0$ and a UV cut-off $\Lambda$. Since each adiabatic
order brings to a derivative term in the expansion, from a dimensional
analysis, it follows that the adiabatic expectation value $\langle
T_{\mu\nu}\rangle_{\text{ad}}$ will have the following general structure555We
explicitly checked the general structure of Eq. (2.13) for the model discussed
in Sec. 3.

$\langle
T\rangle_{\text{ad}}^{(n>4)}=H^{4}\sum_{n>4}\left(c_{n}\left(\frac{H}{m}\right)^{n-4}+c^{\prime}_{n}\left(\frac{H}{\Lambda}\right)^{n-4}\right)\,,$
(2.13)

where $H=\dot{a}/a$ is the Hubble parameter and the coefficients
$(c_{n},c^{\prime}_{n})$ are fixed by the particular model.

From the above general expression we can note two important points. In the
deep UV domain, when $\Lambda\rightarrow\infty$, the higher order terms go to
zero and we can truncate the series at the fourth adiabatic order, which is
indeed the order needed to remove the UV divergences. On the other hand, the
IR regime produces higher order terms involving $m$ which are increasingly
relevant for $m<H$ (as we generally have), and, in particular, for
$m\rightarrow 0$. Therefore, to extend the integral to such regime, to be
consistent, we should include all orders in the calculation.

This seems to indicate that in order to extend the adiabatic subtraction also
to the IR regime, the series cannot in general be truncated at the given order
needed to remove the infinities in the UV, but it should be considered up to
all orders.

Therefore, in light of the above arguments, we suggest that the procedure of
adiabatic regularization should be always performed on a proper domain which
excludes the IR tail of the spectrum. Namely, the adiabatic subtraction should
be considered only up to a comoving IR cut-off $c=\beta a(t)H(t)$. This IR
cut-off is associated to the scale at which the adiabatic solution is not
anymore a good approximation for the mode functions, condition that happens
when the modes start to feel the curvature of space-time. In other words, it
is related to the horizon "exit" of modes, and the coefficient $\beta$, which
is a new free parameter introduced by the renormalization method, should be
determined by a proper physical prescription, fully in line with the spirit of
each renormalization scheme [6]. It should be noted that this IR cut-off is
related only to the adiabatic counterpart, while the bare quantity is free to
run in the total momentum space.

##  3 Infrared cut-off and conformal anomaly matching

The extension of the adiabatic renormalization method here proposed has an
interesting and remarkable application in the study of the energy density and
helicity of gauge fields coupled with an axion-like inflaton ( see e.g. [22,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] for the rich
phenomenology associated with axion-like inflationary models and the
production of gauge fields in this context). As shown in [22], the standard
adiabatic renormalization of these two quantities, despite correctly removes
the divergences in the UV, also introduces unphysical IR divergences, leading
to not well-defined final results.

Following for example [22], the Lagrangian of the model is given by

$\mathcal{L}=-\frac{1}{2}(\nabla\phi)^{2}-V(\phi)-\frac{1}{4}(F^{\mu\nu})^{2}-\frac{g\phi}{4}F^{\mu\nu}\tilde{F}_{\mu\nu}\,,$
(3.1)

where
$\tilde{F}^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}/2=\epsilon^{\mu\nu\alpha\beta}(\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha})/2$,
$\nabla$ is the covariant derivative, and the coupling constant $g$ can be
expressed in terms of the axion decay constant $f$ by the relation
$g=\alpha/f$, with $\alpha$ a dimensionless parameter. Finally, the background
is assumed to be described by a FLRW metric as given in Eq. (2.1).

Due to the coupling with the inflaton field $\phi$ in Eq. (3.1), quantum
fluctuations of the gauge field $A_{\mu}$ are amplified. In this context, two
interesting quantities that can be considered are the following. The first is
the vacuum expectation value of the energy density of the produced gauge
fields

$\frac{\langle\mathbf{E}^{2}+\mathbf{B}^{2}\rangle}{2}=\int\frac{{\rm
d}k}{(2\pi)^{2}a(\tau)^{4}}k^{2}\left[|A^{\prime}_{+}|^{2}+|A^{\prime}_{-}|^{2}+k^{2}\left(|A_{+}|^{2}+|A_{-}|^{2}\right)\right]\,.$
(3.2)

This is given by the $(0,0)$ component of the associated energy-momentum
tensor

$T_{\mu\nu}^{(F)}=F_{\rho\mu}F^{\rho}_{\nu}+g_{\mu\nu}\frac{\mathbf{E}^{2}-\mathbf{B}^{2}}{2}\,,$
(3.3)

and it enters in the Friedmann equations in the following way

$H^{2}={1\over 3M_{p}^{2}}\left[{\dot{\phi}^{2}\over
2}+V(\phi)+{\langle\mathbf{E}^{2}+\mathbf{B}^{2}\rangle\over
2}\right]\,,\qquad\quad\dot{H}=-{1\over
2M_{p}^{2}}\left[\dot{\phi}^{2}+{2\over
3}\langle\mathbf{E}^{2}+\mathbf{B}^{2}\rangle\right]\,.$ (3.4)

The second is the so-called helicity integral (following the notation of
[22]), given by

$\left\langle{\mathbf{E}\cdot\mathbf{B}}\right\rangle=-\int\frac{{\rm
d}k}{(2\pi)^{2}a(\tau)^{4}}k^{3}\frac{\partial}{\partial\tau}\left(|A_{+}|^{2}-|A_{-}|^{2}\right)\,,$
(3.5)

which affects the equation of motion of the pseudo-scalar field as

$\ddot{\phi}+3H\dot{\phi}+V_{\phi}=g\left\langle{\mathbf{E}\cdot\mathbf{B}}\right\rangle\,.$
(3.6)

In all the above expressions $\mathbf{E}$ and $\mathbf{B}$ are the electric
and magnetic fields associated to the gauge field $A_{\mu}$ and the
expectation values in Eqs. (3.2) and (3.5) are explicitly expressed in terms
of the mode functions $A_{\pm}$ (where the basis of circular polarization has
been chosen). Moreover, a prime stands for derivative w.r.t. the conformal
time $\tau$ (${\rm d}\tau={\rm d}t/a$).

The Fourier mode functions $A_{\pm}$ of the gauge fields satisfy the equation
of motion

$\frac{{\rm d}^{2}}{{\rm d}\tau^{2}}A_{\pm}(\tau,k)+\left(k^{2}\mp
kg\phi^{\prime}\right)A_{\pm}(\tau,k)=0\,.$ (3.7)

Under the assumption of de Sitter expansion for the background (i.e.
$a(\tau)=-1/(H\tau)$ with $\tau<0$, $H=\text{const.}$ and
$\dot{\phi}=\text{const.}$), we can rewrite the above equation in terms of the
constant parameter $\xi\equiv g\phi^{\prime}/(2a(\tau)H)=g\dot{\phi}/(2H)$. In
this approximation the analytical solution of Eq.(3.7) is given in terms of
Whittaker $W-$ functions

$A_{\pm}(\tau,k)=\frac{1}{\sqrt{2k}}e^{\pm\pi\xi/2}W_{\pm
i\xi,\frac{1}{2}}(-2ik\tau)\,.$ (3.8)

The bare integrals in Eqs. (3.2) and (3.5) can then be computed analytically
by using the mode functions (3.8), after imposing a comoving UV cut-off
$\Lambda\,a(\tau)$ in order to identify the UV divergences. We report in the
following the final results for the bare energy density and helicity integrals
evaluated in [22]

$\begin{split}\frac{1}{2}\,\langle\mathbf{E}^{2}+\mathbf{B}^{2}\rangle_{\text{bare}}\,=&\,\frac{\Lambda^{4}}{8\pi^{2}}+\frac{H^{2}\Lambda^{2}\xi^{2}}{8\pi^{2}}+\frac{3H^{4}\xi^{2}(5\xi^{2}-1)\log{(2\Lambda/H)}}{16\pi^{2}}\\\
&+\frac{H^{4}\xi^{2}(-79\xi^{4}+22\xi^{2}+29)}{64\pi^{2}(1+\xi^{2})}+\frac{H^{4}\xi(30\xi^{2}-11)\sinh{(2\pi\xi)}}{64\pi^{3}}\\\
&+\frac{3iH^{4}\xi^{2}(5\xi^{2}-1)(\psi^{(1)}(1-i\xi)-\psi^{(1)}(1+i\xi))\sinh{(2\pi\xi)}}{64\pi^{3}}\\\
&-\frac{3H^{4}\xi^{2}(5\xi^{2}-1)(\psi(-1-i\xi)+\psi(-1+i\xi))}{32\pi^{2}}\,,\end{split}$
(3.9)

$\begin{split}\qquad\langle\mathbf{E}\cdot\mathbf{B}\rangle_{\text{bare}}=&\,-\frac{H^{2}\Lambda^{2}\xi}{8\pi^{2}}-\frac{3H^{4}\xi\left(5\xi^{2}-1\right)\log\left(2\Lambda/H\right)}{8\pi^{2}}\\\
&+\frac{H^{4}\xi(47\xi^{2}-22)}{16\pi^{2}}-\frac{H^{4}(30\xi^{2}-11)\sinh{(2\pi\xi)}}{32\pi^{3}}\\\
&-\frac{3iH^{4}\xi\left(5\xi^{2}-1\right)\left(\psi^{(1)}(1-i\xi)-\psi^{(1)}(1+i\xi)\right)\sinh(2\pi\xi)}{32\pi^{3}}\\\
&+\frac{3H^{4}\xi\left(5\xi^{2}-1\right)\left(\psi(1-i\xi)+\psi(1+i\xi)\right)}{16\pi^{2}}\,,\end{split}$
(3.10)

where $\psi(x)$ is the Digamma function and $\psi^{(1)}(x)\equiv{\rm
d}\psi(x)/{\rm d}x$.

These integrals, as expected for averaged quantities involving quadratic
combinations of fields in curved space-times, show UV divergences. In
particular we have quartic, quadratic and logarithmic UV divergences for the
energy density of Eq. (3.9), and only quadratic and logarithmic UV divergences
for the helicity integral of Eq. (3.10). On the other hand, they are well-
behaved in the infrared, not exhibiting IR divergences.

###  3.1 Adiabatic regularization

To remove the UV divergences that affect the averaged energy density and
helicity integrals of gauge fields, we subtract from the bare divergent
quantities their respective adiabatic counterparts, following the procedure of
adiabatic regularization highlighted above.

According to the standard convention, a mass regulator $m$ is added to the
equation of motion (3.7) 666The mass term regulator $m$ is added to be in line
with what is commonly done in literature performing adiabatic regularization
for massless fields. However, it is worth noting that, according to our
adiabatic procedure with IR cut-off, the addition of this mass regulator can
be avoided ab initio, being a redundancy in the regularization of the IR
domain. Let us stress anyway that this easily leads us to extend our adiabatic
results also to cases in which the physical mass of gauge fields is instead
different from zero (see Appendix B).

$\frac{{\rm d}^{2}}{{\rm
d}\tau^{2}}A_{\pm}^{\text{WKB}}(\tau,k)+\left(k^{2}\mp
gk\phi^{\prime}+\frac{m^{2}}{H^{2}\tau^{2}}\right)A_{\pm}^{\text{WKB}}(\tau,k)=0\,,$
(3.11)

where the adiabatic mode function of gauge fields, for each polarization
$\lambda=\pm$ is given by

$A_{\lambda}^{\text{WKB}}(k,\tau)=\frac{1}{\sqrt{2\Omega_{\lambda}(k,\tau)}}e^{-i\int\Omega_{\lambda}(k,\tau^{\prime}){\rm
d}\tau^{\prime}}\,.$ (3.12)

Inserting this solution into the equation of motion (3.11) we obtain the exact
equation for the WKB frequency

$\Omega_{\lambda}^{2}(k,\tau)=\bar{\Omega}^{2}_{\lambda}(k,\tau)+\frac{3}{4}\left(\frac{\Omega^{\prime}_{\lambda}(k,\tau)}{\Omega_{\lambda}(k,\tau)}\right)^{2}-\frac{1}{2}\frac{\Omega^{\prime\prime}_{\lambda}(k,\tau)}{\Omega_{\lambda}(k,\tau)}\,,$
(3.13)

where

$\bar{\Omega}_{\lambda}^{2}=\omega^{2}(k,\tau)-\lambda
kg\phi^{\prime}(\tau)\,,\qquad\omega^{2}(k,\tau)=k^{2}+m^{2}a^{2}(\tau)\,.$
(3.14)

As described in the previous section, by solving (3.13) iteratively, we can
obtain the $n$-order adiabatic WKB frequencies. In the cases under
consideration, the adiabatic expansion up to the fourth order is needed to
remove the UV divergences.

Thus, we obtain for the frequency up to the fourth adiabatic order

$\Omega_{\lambda}(k,\tau)=\bar{\Omega}_{\lambda}(k,\tau)+\epsilon^{2}\,\Omega^{(2)}_{\lambda}(k,\tau)+\epsilon^{4}\,\Omega^{(4)}_{\lambda}(k,\tau)\,,$
(3.15)

with, omitting for convenience the time and momentum dependence,

$\Omega^{(2)}_{\lambda}=\frac{3}{8}\frac{(\bar{\Omega}^{\prime}_{\lambda})^{2}}{\bar{\Omega}_{\lambda}^{3}}-\frac{1}{4}\frac{\bar{\Omega}_{\lambda}^{\prime\prime}}{\bar{\Omega}_{\lambda}^{2}}\,,$
(3.16)

and

$\Omega^{(4)}_{\lambda}=-\frac{1}{2}\frac{(\Omega_{\lambda}^{(2)})^{2}}{\bar{\Omega}_{\lambda}}-\frac{3}{4}\frac{\Omega_{\lambda}^{(2)}(\bar{\Omega}_{\lambda}^{\prime})^{2}}{\bar{\Omega}_{\lambda}^{4}}+\frac{3}{4}\frac{\bar{\Omega}_{\lambda}^{\prime}\Omega_{\lambda}^{(2)^{\prime}}}{\bar{\Omega}_{\lambda}^{3}}+\frac{1}{4}\frac{\Omega_{\lambda}^{(2)}\bar{\Omega}_{\lambda}^{\prime\prime}}{\bar{\Omega}_{\lambda}^{3}}-\frac{1}{4}\frac{\Omega_{\lambda}^{(2)^{\prime\prime}}}{\bar{\Omega}_{\lambda}^{2}}\,,$
(3.17)

where we should further Taylor-expand $\bar{\Omega}_{\lambda}(k,\tau)$ in
power of $\epsilon$ around ${\omega}(k,\tau)$ considering the term $(-\lambda
kg\phi^{\prime}(\tau))$ of adiabatic order one and discarding all the
resulting terms of adiabatic order larger than four in the final result.

We thus define the adiabatic mode functions in Eq. (3.12) up to fourth order
by using the adiabatic frequency in Eq. (3.15), and use them to compute the
adiabatic counterparts of the energy density and helicity integrals by Eqs.
(3.2) and (3.5).

We proceed now performing the adiabatic integrals, introducing the same
comoving UV cut-off to regularize the UV-divergent terms. Moreover, according
to the proposed renormalization approach, a comoving IR cut-off $\beta
a(\tau)H$ is considered, in order to take into account the fact that the
adiabatic approximation breaks down at small wavenumbers. In such a way we
obtain the following results for the adiabatic counterparts of the energy
density and helicity integrals (which match the ones in [22] if also there
$c=\beta a(\tau)H$ is considered)

$\begin{split}\frac{1}{2}\,\langle\mathbf{E}^{2}+\mathbf{B}^{2}\rangle_{\text{ad}}^{c=\beta
Ha(\tau)}=&\,\,\frac{\Lambda^{4}}{8\pi^{2}}+\frac{H^{2}\Lambda^{2}\xi^{2}}{8\pi^{2}}+\frac{3H^{4}\xi^{2}(5\xi^{2}-1)\log{(2\Lambda/H)}}{16\pi^{2}}\\\
&-\frac{\beta^{4}H^{4}}{8\pi^{2}}-\frac{\beta^{2}H^{4}\xi^{2}}{8\pi^{2}}-\frac{3H^{4}\xi^{2}(5\xi^{2}-1)\log{(2\beta)}}{16\pi^{2}}\,,\end{split}$
(3.18)
$\begin{split}\langle\mathbf{E}\cdot\mathbf{B}\rangle_{\text{ad}}^{c=\beta
Ha(\tau)}=&\,-\frac{H^{2}\Lambda^{2}\xi}{8\pi^{2}}-\frac{3H^{4}\xi(5\xi^{2}-1)\log{(2\Lambda/H)}}{8\pi^{2}}\quad\;\\\
&+\frac{\beta^{2}H^{4}\xi}{8\pi^{2}}+\frac{3H^{4}\xi(5\xi^{2}-1)\log{(2\beta)}}{8\pi^{2}}\,.\end{split}$
(3.19)

We can immediately see that the terms proportional to the UV cut-off $\Lambda$
correctly reproduce the UV divergences of the bare quantities, so that, after
subtraction, these infinities are removed.

The above expressions are obtained considering the $m\rightarrow 0$ limit,
which is indeed well-defined thanks to the introduction of the IR cut-off.
Namely, no pathological dependencies on the mass term regulator $m$ manifest
on the adiabatic results with a IR cut-off, as we can also see from the
general case of $m\neq 0$ reported in Appendix B.

###  3.2 Matching the conformal anomaly

In order to fix univocally the free parameter $\beta$, we can observe that in
the conformal limit obtained for $m\rightarrow 0$ and $\xi\rightarrow 0$, the
adiabatic expectation value of the energy density should provide the term
connected to the conformal anomaly of gauge fields [38, 39, 40]. Indeed, in
the conformal limit, a proper renormalization scheme should provide the
conformal anomaly induced by quantum effects [40, 41, 42, 43]

$\langle T^{\mu}_{\phantom{m}\mu}\rangle_{\text{phys}}=-\langle
T^{\mu}_{\phantom{\mu}\mu}\rangle_{\text{reg}}\,,$ (3.20)

where $\langle T^{\mu}_{\phantom{\mu}\mu}\rangle_{\text{reg}}$ is the trace
contribution to the energy-momentum tensor given by the particular
renormalization method applied.

In our case, since we are renormalizing physical quantities following the
adiabatic subtraction, it follows that $\langle
T^{\mu}_{\phantom{\mu}\mu}\rangle_{\text{reg}}=\langle
T^{\mu}_{\phantom{\mu}\mu}\rangle_{\text{ad}}$. Therefore, by requiring the
right value of the conformal anomaly in the proper limit, the parameter
$\beta$, which appears as a free parameter in the adiabatic expression of the
energy density, can be fixed without ambiguities. Moreover, this physically
motivated prescription for the value of the new degree of freedom immediately
allows to obtain univocally defined finite results for the quantities of
interest, after the subtraction has been performed.

In the particular case of conformally coupled massless gauge fields the
expected value of the conformal anomaly should be twice the result of the
conformal anomaly for a massless conformally coupled scalar field, namely
$2\times H^{4}/(960\pi^{2})=H^{4}/(480\pi^{2})$ [44, 7, 22, 13]. This because
the two helicities of the gauge fields are equivalent to two conformally
coupled massless scalar fields for $\xi=0$.

By performing the $m\rightarrow 0$ and $\xi\rightarrow 0$ limits for the case
of gauge fields we obtain

$\lim_{\xi\rightarrow 0,\,m\rightarrow 0}\langle
T^{\mu}_{\phantom{\mu}\mu}\rangle_{\text{ad}}=\lim_{\xi\rightarrow 0,\
m\rightarrow
0}\frac{\langle\mathbf{E}^{2}+\mathbf{B}^{2}\rangle_{\text{ad}}^{c=\beta
Ha(\tau)}}{2}=-\frac{\beta^{4}H^{4}}{8\pi^{2}}\,,$ (3.21)

and this term should reproduce the expected value of the trace anomaly.
Accordingly to (3.20), the matching procedure gives

$\frac{\beta^{4}H^{4}}{8\pi^{2}}=\frac{H^{4}}{480\pi^{2}}\implies\beta=\frac{1}{\sqrt{2}\times
15^{1/4}}\approx 0.359\,.$ (3.22)

Therefore, we have a physically motivated prescription that is able to fix
univocally the renormalization scheme. As a consequence, after the adiabatic
subtraction is performed, we are able to obtain univocal finite results for
the averaged energy density and helicity of gauge fields. Most importantly,
our adiabatic renormalization method succeeds in providing the conformal
anomaly in the proper limit, where instead the standard adiabatic procedure
fails, leading to pathological infrared divergences.

Let us finally add a further remark. According to our proposal, it is clear
that the IR cut-off should be introduced in any case when performing adiabatic
renormalization, for the reasons explained in Sec. 3 . Here we show how the
use of this comoving cut-off does not spoil required results already obtained
within the standard approach, as one would expect. To this purpose, let us
consider the standard case of a conformally coupled massless scalar field,
where no IR divergences appear in the usual adiabatic renormalization
procedure of the energy-momentum tensor. The standard adiabatic subtraction
seems to be not problematic, indeed one is able to obtain the conformal
anomaly, one of the main results required for a renormalization procedure.
However, by explicit calculation we obtain no dependence on the new IR cut-off
within our approach, so we are able to obtain the well-known result for the
conformal anomaly

$\begin{split}\langle
T^{\mu}_{\phantom{\mu}\mu}\rangle_{\text{phys}}=&-\langle
T^{\mu}_{\phantom{\mu}\mu}\rangle_{\text{ad}}^{c=\beta Ha(t)}=\\\
&-\frac{\dot{a}(t)^{2}\ddot{a}(t)}{160\pi^{2}a(t)^{3}}+\frac{\ddot{a}(t)^{2}}{480\pi^{2}a(t)^{2}}+\frac{\dot{a}(t)\dddot{a}(t)}{160\pi^{2}a(t)^{2}}+\frac{\ddddot{a}(t)}{480\pi^{2}a(t)}\,.\end{split}$
(3.23)

This simple example shows how the adiabatic procedure here introduced produces
the correct result also in this case. The investigation of the impact of this
renormalization scheme on observables evaluated in other inflationary models
will be the subject of future works.

###  3.3 Renormalized results and comparison with minimal subtraction scheme

We can now perform the proper renormalization procedure, by subtracting to the
bare results of the energy density in Eq. (3.9) and of the helicity integral
in Eq. (3.10) their adiabatic counterparts in Eqs. (3.18) and (3.19)
respectively, where $\beta$ has to be fixed according to Eq. (3.22). The final
renormalized result for the energy density is thus

$\begin{split}\frac{1}{2}\,\langle\mathbf{E}^{2}+\mathbf{B}^{2}\rangle_{\beta}=&\,\frac{2H^{4}}{960\pi^{2}}+\frac{H^{4}\xi^{2}\left(-1185\xi^{4}+(330+4\sqrt{15})\xi^{2}+435+4\sqrt{15}\right)}{960\pi^{2}\left(1+\xi^{2}\right)}\\\
&-\frac{3H^{4}\xi^{2}\left(5\xi^{2}-1\right)\log\left(15/4\right)}{64\pi^{2}}+\frac{H^{4}\xi\left(30\xi^{2}-11\right)\sinh(2\pi\xi)}{64\pi^{3}}\\\
&-\frac{3H^{4}\xi^{2}\left(5\xi^{2}-1\right)(\psi^{(0)}(-1-i\xi)+\psi^{(0)}(-1+i\xi))}{32\pi^{2}}\\\
&+\frac{3iH^{4}\xi^{2}\left(5\xi^{2}-1\right)(\psi^{(1)}(1-i\xi)-\psi^{(1)}(1+i\xi))\sinh(2\pi\xi)}{64\pi^{3}}\,,\end{split}$
(3.24)

and the one for the helicity integral is

$\begin{split}\langle\mathbf{E}\cdot\mathbf{B}\rangle_{\beta}=&\,\frac{H^{4}\xi\left(705\xi^{2}-330-\sqrt{15}\right)}{240\pi^{2}}+\frac{3H^{4}\xi\left(5\xi^{2}-1\right)\log\left(15/4\right)}{32\pi^{2}}\\\
&+\frac{3H^{4}\xi\left(5\xi^{2}-1\right)(\psi^{(0)}(1-i\xi)+\psi^{(0)}(1+i\xi))}{16\pi^{2}}\\\
&+\frac{3iH^{4}\xi\left(5\xi^{2}-1\right)(-\psi^{(1)}(1-i\xi)+\psi^{(1)}(1+i\xi))\sinh(2\pi\xi)}{32\pi^{3}}\\\
&+\frac{H^{4}\left(11-30\xi^{2}\right)\sinh(2\pi\xi)}{32\pi^{3}}\,.\end{split}$
(3.25)

It is instructive to compare the above results with the ones obtained by a
minimal subtraction (MS) scheme, where only the UV divergences are removed (as
in [22])

$\begin{split}\frac{1}{2}\,\langle\mathbf{E}^{2}+\mathbf{B}^{2}\rangle_{\text{MS}}=&\,\frac{H^{4}\xi^{2}(-79\xi^{4}+22\xi^{2}+29)}{64\pi^{2}(1+\xi^{2})}+\frac{H^{4}\xi(30\xi^{2}-11)\sinh{(2\pi\xi)}}{64\pi^{3}}\\\
&+\frac{3iH^{4}\xi^{2}(5\xi^{2}-1)(\psi^{(1)}(1-i\xi)-\psi^{(1)}(1+i\xi))\sinh{(2\pi\xi)}}{64\pi^{3}}\\\
&-\frac{3H^{4}\xi^{2}(5\xi^{2}-1)(\psi(-1-i\xi)+\psi(-1+i\xi))}{32\pi^{2}}\,,\end{split}$
(3.26)

$\begin{split}\langle\mathbf{E}\cdot\mathbf{B}\rangle_{\text{MS}}=&+\frac{H^{4}\xi(47\xi^{2}-22)}{16\pi^{2}}-\frac{H^{4}(30\xi^{2}-11)\sinh{(2\pi\xi)}}{32\pi^{3}}\\\
&-\frac{3iH^{4}\xi\left(5\xi^{2}-1\right)\left(\psi^{(1)}(1-i\xi)-\psi^{(1)}(1+i\xi)\right)\sinh(2\pi\xi)}{32\pi^{3}}\\\
&+\frac{3H^{4}\xi\left(5\xi^{2}-1\right)\left(\psi(1-i\xi)+\psi(1+i\xi)\right)}{16\pi^{2}}\,.\end{split}$
(3.27)

Figure 1: We compare the adiabatic and MS renormalization schemes for the
energy density (left panel) and for the helicity integral (right panel). The
inset on the left panel shows the behavior of the energy density for $\xi$
close to zero. Notice that only the adiabatic scheme reproduces the correct
value of the conformal anomaly (horizontal dashed line).

In Fig. 1 we plot the renormalized energy density in the adiabatic (3.24) and
MS (3.26) scheme (left panel) and, similarly, the renormalized helicity
integral in the adiabatic (3.25) and MS (3.27) scheme (right panel) in units
of $(2\pi)^{2}/H^{4}$. As one can see, the differences between the two schemes
are of order $\mathcal{O}(1)$ at small couplings. The leading asymptotic
behaviors for $\xi\gg 1$ in the adiabatic scheme are given by

$\frac{1}{2}\,\langle\mathbf{E}^{2}+\mathbf{B}^{2}\rangle_{\beta}\sim\frac{9H^{4}\sinh(2\pi\xi)}{1120\pi^{3}\xi^{3}}\,,\qquad-\langle\mathbf{E}\cdot\mathbf{B}\rangle_{\beta}\sim\frac{9H^{4}\sinh(2\pi\xi)}{560\pi^{3}\xi^{4}}\,,$
(3.28)

and reproduce, as expected, the asymptotic behaviors in the MS scheme [22].
The adiabatic subtraction introduces further power law corrections that are
however subleading. As expected, the inset on the left panel shows that only
the adiabatic scheme provides the correct value of the conformal anomaly for
$\xi\to 0$.

We would like to conclude this section bringing the attention to the fact that
the adiabatic procedure is not just a tool to identify the UV-divergent terms
of quantities involving expectation values of quantum fields in curved space-
times, but it is a proper renormalization prescription. This means, in
particular, that it should provide univocally defined finite results.
Moreover, it is properly the subtraction of the contribution of the adiabatic
vacuum that introduces surprising effects in the renormalization of physical
quantities. A remarkable example is the generation of conformal anomalies
which break the classical conformal symmetry at the quantum level and that
accompany the renormalization of the energy-momentum tensor of conformally
coupled fields in cosmological space-times. This last aspect notably
underlines how the totality of the adiabatic counterpart has to be subtracted
to the bare divergent quantities. As pointed out in the introduction, there
are cases in which the standard adiabatic regularization leads to not well-
defined adiabatic terms in the infrared regime, forcing to fix a different
prescription, as the minimal scheme, for the subtraction, loosing the
uniqueness of the finite term. In particular, as shown in the analyzed case of
gauge fields, the conformal anomaly was not recovered within the minimal
scheme. This further reinforces the idea that such minimal subtraction is not
the proper prescription for adiabatic regularization.

##  4 Discussion and conclusions

In this manuscript we rediscussed the adiabatic renormalization in curved
space-time. In line with previous literature, we questioned about the correct
use of the adiabatic subtraction and, by examining its foundations and its
essential properties, we showed how its range of validity has to be restricted
to the UV regime. Consequently we evinced how the adiabatic subtraction should
be considered only for modes up to a scale comparable with the Hubble horizon.
This not only fully meets the essence of the adiabatic approximation, but is
also needed, otherwise problematic features can manifest in the deep IR, which
could be plagued by unphysical divergences. Furthermore, we pointed out that,
otherwise, in order to extend the adiabatic renormalization up to the IR, one
should take into account all the adiabatic orders for the subtraction and not
only those that ensures the removal of the UV infinities, at the price of
loosing the predictivity of the adiabatic approach.

Accordingly, we suggested to supplement the adiabatic renormalization frame
with the introduction of a comoving IR cut-off of the form $c=\beta aH$, which
stops the adiabatic subtraction approximatively around the horizon crossing of
modes. The new parameter $\beta$ has then to be fixed by a suitable physical
prescription, making the renormalization scheme univocally defined. In
particular we emphasized that, as part of a well-defined renormalization
scheme, the conformal anomaly for massless conformally coupled (quantum)
fields in curved space-times should be always reproduced. Let us further
underline how this request lets no ambiguities in the choice of the infrared
cut-off for the adiabatic counterpart and returns us unique renormalized
physical quantities.

In light of this perspective, we applied this new approach to the particular
and considerable case of a $U(1)$ gauge field coupled to an axion-like
inflaton, which is specially suitable to show how our procedure is able to
provide a well-defined renormalized energy-momentum tensor also in the IR,
where the standard adiabatic procedure failed. In this case, using the fact
that the conformal anomaly is $2\times H^{4}/(960\pi^{2})=H^{4}/(480\pi^{2})$,
since the two physical states $A_{\pm}$ of the gauge field are equivalent to
two conformally coupled massless scalar fields for $\xi=0$, we obtained the
value for the new degree of freedom introduced by the IR cut-off
$\beta=1/(\sqrt{2}\times 15^{1/4})\approx 0.359$, fixing univocally the finite
part and thus the renormalization scheme.

Moreover, this new procedure of adiabatic renormalization is also needed for
the study of the backreaction of gauge fields on the axion-like (inflaton)
evolution, due also to the contribution of the helicity integral in the
equation of motion of the inflaton field. This backreaction impacts on the
dynamics of the inflaton, possibly changing the duration of the inflationary
phase. We hope to investigate such aspects in a future work.

Let us finally recall that the need of a physical prescription to fix
univocally the renormalization scheme is a fundamental ingredient of all
renormalization schemes also in flat space-time quantum field theories. In
particular, since IR divergences are not universal as the UV ones, but depend
on the choice of the state of reference, this seems to indicate an interesting
link between our adiabatic renormalization prescription and algebraic
renormalization techniques, where the choice among the possible vacuum
Hadamard states is fixed at the level of physical observables (see [45, 46,
47] for a review on the subject).

Within this work we tried to underline the necessity of a new prescription for
the adiabatic renormalization scheme. As shown, this has strong effects in
particular for such models where otherwise the standard adiabatic approach
fails, producing unphysical IR divergences. The latter is the case for the
model presented here of an axion-like inflaton coupled to U(1)-gauge fields,
but is also the case for all other models with the same properties in the IR
(see e.g. [37, 36, 23, 35] ). We plan to apply our renormalization procedure
for those particular models in future works.

## Acknowledgments

The authors are thankful to Matteo Braglia and Nicola Pinamonti for useful
discussions and correspondence. The authors are supported in part by INFN
under the program TAsP (Theoretical Astroparticle Physics).

##  Appendix A Conservation of $T_{\mu\nu}$

According to [48], performing the adiabatic subtraction only over high
momentum modes by the introduction of a time-dependent cut-off, leads to the
violation of the energy-momentum covariant conservation by changing the time
derivative of the energy density (or, more in general, of the $T_{\mu\nu}$
components). Here we show how this is not true within our approach, which
therefore leads to a renormalized energy-momentum tensor that does not spoil
any conservation.

As an illustrative example, let us consider a generic expression for the
adiabatic counterpart of the energy density $\rho$ calculated using a comoving
IR cut-off. In a complete general way this quantity can be expressed as

$\rho=\int_{\beta a(t)H(t)}{\rm
d}k\,\frac{k^{n}}{(a(t)H(t))^{n+1}}\,g\left(\frac{k}{a(t)H(t)},t\right)\,,$
(A.1)

with $g$ an adimensional function of $k/(a(t)H(t))$ and of the proper time
$t$. It is easy to see that the introduction of a comoving IR cut-off
proportional to $a(t)H(t)$ does not change the time dependence of the
integral. Indeed, setting $x=k/(a(t)H(t))$ the integral becomes

$\rho=\int_{\beta}{\rm d}x\,x^{n}g(x,t)\,,$ (A.2)

and performing a time derivative one obtains

$\dot{\rho}=\int_{\beta}{\rm d}x\,x^{n}\frac{{\rm d}}{{\rm d}t}g(x,t)\,.$
(A.3)

We can therefore conclude that the criticism ilustrated in [48] does not apply
in our case, since the apparent time dependence of the introduced cut-off is
an artefact of the comoving coordinate choice, which can be removed by
rewriting all in terms of the physical wavenumber scaled by the Hubble factor
$k/(a(t)H(t))$. To conclude, if for $T_{\mu\nu}$ there is conservation mode-
by-mode, namely for each $k$ and $t$, than there will be also for each $x$ and
$t$ and the presence of the comoving IR cut-off will not affect such
conservation.

##  Appendix B Adiabatic results for U(1)-gauge massive fields

In Sec. 3 we focused on the case of massless gauge fields, both because it has
a pathological behavior in the IR within the standard adiabatic procedure,
making particularly evident the usefulness of our method, and because it is of
phenomenological interest. In any case, our procedure should be applied in
general, hence also to the massive gauge field case (see e.g. [49] for some
applications).

We report here the full results for the adiabatic integrals of the energy
density and helicity extended to the case of massive gauge fields.777Notice
that in the massive case the integral in Eq. (3.2) becomes
$\frac{\langle\mathbf{E}^{2}+\mathbf{B}^{2}\rangle}{2}=\int\frac{{\rm
d}k}{(2\pi)^{2}a(\tau)^{4}}k^{2}\left[|A^{\prime}_{+}|^{2}+|A^{\prime}_{-}|^{2}+\left(k^{2}+m^{2}a(\tau)^{2}\right)\left(|A_{+}|^{2}+|A_{-}|^{2}\right)\right]\,.$
(B.1) .

$\begin{split}\frac{1}{2}\,\langle\mathbf{E^{2}+B^{2}}\rangle_{\text{ad},m}^{c=\beta
Ha(\tau)}=\,&\frac{\Lambda^{4}}{8\pi^{2}}+\frac{\Lambda^{2}\left(m^{2}+H^{2}\xi^{2}\right)}{8\pi^{2}}\\\
&-\frac{\left(m^{4}+6H^{2}m^{2}\xi^{2}+3H^{4}\xi^{2}\left(1-5\xi^{2}\right)\right)\log\left(2\Lambda/H\right)}{16\pi^{2}}\\\
&+\frac{15m^{4}+20H^{2}m^{2}\left(1+15\xi^{2}\right)-2H^{4}\left(1-270\xi^{2}+690\xi^{4}\right)}{960\pi^{2}}\\\
&-\frac{H^{13}\beta^{9}\left(-1+60\beta^{4}+270\xi^{2}+60\beta^{2}\xi^{2}-690\xi^{4}\right)}{480\pi^{2}\left(m^{2}+H^{2}\beta^{2}\right)^{9/2}}\\\
&-\frac{H^{11}m^{2}\beta^{7}\left(660\beta^{4}+20\beta^{2}\left(1+42\xi^{2}\right)-9\left(1-150\xi^{2}+540\xi^{4}\right)\right)}{960\pi^{2}\left(m^{2}+H^{2}\beta^{2}\right)^{9/2}}\\\
&-\frac{H^{9}m^{4}\beta^{5}\left(27+1000\beta^{4}+940\xi^{2}-4320\xi^{4}+40\beta^{2}\left(1+36\xi^{2}\right)\right)}{640\pi^{2}\left(m^{2}+H^{2}\beta^{2}\right)^{9/2}}\\\
&-\frac{H^{7}m^{6}\beta^{3}\left(1+480\beta^{4}+208\xi^{2}-1040\xi^{4}+16\beta^{2}\left(1+44\xi^{2}\right)\right)}{256\pi^{2}\left(m^{2}+H^{2}\beta^{2}\right)^{9/2}}\\\
&-\frac{H^{5}m^{8}\beta\left(60\beta^{4}+9\xi^{2}-45\xi^{4}+\beta^{2}\left(1+78\xi^{2}\right)\right)}{48\pi^{2}\left(m^{2}+H^{2}\beta^{2}\right)^{9/2}}\\\
&-\frac{H^{3}m^{10}\left(7\beta^{3}+6\beta\xi^{2}\right)}{16\pi^{2}\left(m^{2}+H^{2}\beta^{2}\right)^{9/2}}-\frac{Hm^{12}\beta}{16\pi^{2}\left(m^{2}+H^{2}\beta^{2}\right)^{9/2}}\\\
&+\frac{\left(m^{4}+6H^{2}m^{2}\xi^{2}+3H^{4}\xi^{2}\left(1-5\xi^{2}\right)\right)\log\left(\beta+(\sqrt{m^{2}+H^{2}\beta^{2}})/H\right)}{16\pi^{2}}\,,\end{split}$
(B.2) $\begin{split}\qquad\qquad\langle\mathbf{E\cdot
B}\rangle_{\text{ad},m}^{c=\beta
Ha(\tau)}=&-\frac{H^{2}\Lambda^{2}\xi}{8\pi^{2}}+\frac{3H^{2}\xi\left(3m^{2}+H^{2}\left(1-5\xi^{2}\right)\right)\log\left(2\Lambda/H\right)}{8\pi^{2}}\\\
&+\frac{3H^{2}\xi\left(-3m^{2}+H^{2}\left(-1+5\xi^{2}\right)\right)\log\left(\beta+(\sqrt{m^{2}+H^{2}\beta^{2}})/H\right)}{8\pi^{2}}\\\
&+\frac{H^{2}\xi\left(-21m^{2}+H^{2}\left(-19+56\xi^{2}\right)\right)}{16\pi^{2}}+\frac{H^{13}\beta^{9}\xi\left(19+2\beta^{2}-56\xi^{2}\right)}{16\pi^{2}\left(m^{2}+H^{2}\beta^{2}\right)^{9/2}}\\\
&+\frac{H^{11}m^{2}\beta^{7}\xi\left(49+60\beta^{2}-344\xi^{2}\right)}{32\pi^{2}\left(m^{2}+H^{2}\beta^{2}\right)^{9/2}}+\frac{H^{9}m^{4}\beta^{5}\xi\left(43+96\beta^{2}-216\xi^{2}\right)}{16\pi^{2}\left(m^{2}+H^{2}\beta^{2}\right)^{9/2}}\\\
&+\frac{H^{7}m^{6}\beta^{3}\xi\left(13+64\beta^{2}-65\xi^{2}\right)}{8\pi^{2}\left(m^{2}+H^{2}\beta^{2}\right)^{9/2}}+\frac{9H^{3}m^{10}\beta\xi}{8\pi^{2}\left(m^{2}+H^{2}\beta^{2}\right)^{9/2}}\\\
&+\frac{3H^{5}m^{8}\beta\xi\left(1+13\beta^{2}-5\xi^{2}\right)}{8\pi^{2}\left(m^{2}+H^{2}\beta^{2}\right)^{9/2}}\,.\end{split}$
(B.3)

Naturally, the parameter $\beta$ is still fixed by Eq. (3.22) , due to the
physical property that for $\xi$ and $m\rightarrow 0$ one should match the
conformal anomaly.

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