# Inflationary magnetogenesis: solving the strong coupling and its non-
Gaussian signatures

Debottam Nandi

###### Abstract

The simplest model of primordial magnetogenesis can provide scale-invariant
magnetic fields that can explain the present abundances of it in the cosmic
scales. Two kinds of solutions of the coupling function can lead to such
phenomena and both of them suffer from the problems of either strong-coupling
or large backreaction. In this work, we consider the coupling function as a
linear combination of both kinds with a model parameter. We find that the
parameter needs to be as small as $\sim 10^{-20}$ in order to evade the
backreaction problem. On the other hand, requiring that the modes above Mpc
scales do not suffer strong coupling, we also obtain a weak constraint of the
model parameter to be greater than $10^{-60}$. For the allowed range of the
model parameter, we, then, analytically evaluate the cross-correlation
functions between the magnetic fields and the curvature perturbation. We find
that such a combination preserves the consistency relation. Also, the result
leads to enhanced non-Gaussianity in equilateral as well as flattened limits
with unique signatures that characterize the novelty of this model.

##  1 Introduction

Astrophysical observations suggest the abundances of large scale coherent
magnetic fields in all cosmic scales. These fields are not only present in
bound cosmological structures e.g. stars, galaxies, and the cluster of
galaxies but they also seem to be present in the intergalactic medium and
large scale structures. In galaxies and clusters of galaxies, the strengths of
the magnetic fields have been measured to be a few micro Gauss, they are
estimated to be of the order of $10^{-17}$ Gauss in Mpc length scales in
intergalactic medium [1, 2, 3, 4, 5, 6, 7, 8, 9] and large scale structures
[10, 11, 12, 13, 14, 15, 16].

The origin of such magnetic fields is still unknown. Various explanations that
have been put forward can broadly be classified into two categories:
Astrophysical and Primordial [17, 18, 19, 20]. The primary issue with the
astrophysical sources is that, they cannot explain the coherent length of the
field. It is therefore believed that the magnetic field is generated in the
early universe and are then amplified by the galactic dynamo mechanism at late
times to give rise to such strength of the fields in those scales.

Inflationary paradigm [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] is
currently is the most successful early Universe paradigm that can explain the
Cosmic Microwave Background Radiation (CMBR) and the formation of large scale
structures. However, during this time, the standard electromagnetic (EM) field
decays very fast due to its conformal invariance structure in the four
dimensions and thus it can still not generate sufficient seed magnetic fields.
Therefore, it has been realized that to generate such strength, the necessary
condition is to break the conformal symmetry of the EM field. Several authors
have suggested many ways to break the conformal invariance of the EM field by
introducing (i) a coupling of the EM field with the Ricci/Riemann tensors,
(ii) a non-minimal coupling of the EM field with scalar/axion/fermionic field,
and (iii) compactification from higher dimensional space-time [34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 19, 47, 48, 49, 50, 51, 52, 53, 54, 55,
56, 57, 58, 59, 60, 61, 62]. Recently, a new model has also been introduced
where the conformal symmetry is broken due to the higher derivative action of
the $U(1)$ field without any coupling function that can generate sufficient
magnetic field [63].

In this work, we are interested in models where the standard EM action is
coupled with the scalar field responsible for the slow-roll inflation. The
time-dependent coupling function leads to the amplification of EM field
fluctuations and thus can generate an ample amount of magnetic field. The
coupling function is chosen in such a way that, at the end of inflation, it
becomes unity and later remains the same so that the standard EM theory is
restored. There exist two solutions of the coupling function that lead to the
scale-invariant magnetic field which is preferred. However, even with the
success, these kinds of models suffer two crucial problems: the strong
coupling problem and the backreaction problem [44, 64, 65]. The backreaction
problem arises due to the fact that the energy density of the EM field grows
faster than the background energy density, thus can break the inflationary
dynamics, which is not preferred. The second is the strong coupling problem in
which the time evolution of the coupling function during inflation leads to
the EM field being in the strong coupling regime at the beginning of
inflation. This implies that the perturbative calculation of the EM field
fluctuations can not be trusted.

In this work, we try to construct the coupling function in such a manner that
the scale-invariant spectrum of the magnetic field remains the same, and at
the same time, the above problems can also be evaded. Since the solutions are
dual, the obvious choice is to choose the coupling function as a linear
combination of both solutions. This kind of choice provides two parameters of
which, one can be fixed by using the fact that, coupling function becomes
unity at the end of inflation. This leads to one free model parameter. By
choosing such a combination, one can immediately see that the inverse of the
coupling function, i.e., the _effective coupling constant_ , while going
backward in time from the end of inflation, grows to a maximum, and at then
decays very fast, unlike the conventional case, and thus it _may_ address the
issue of the strong coupling. Also, by demanding that there is no backreaction
at the largest observable scales, we may immediately obtain constraints on
this parameter.

In order to search for its implications, it would be interesting to study the
cross-correlations between the magnetic fields and the curvature perturbation.
This has been extensively studied in the literature [66, 67, 68, 69, 70, 71,
72]. It is found that the magnitude of non-Gaussianities generated through
such correlations has been estimated to be quite large for inflation. Also, it
is found that the consistency relation: whether the non-Gaussian parameter in
the squeezed limit can be written in terms of the two-point function, is
satisfied [68, 69]. Motivated by these results, we also study the same and
found that our new choice of the coupling function preserves the consistency
relation, irrespective of the choice of the model parameter. Also, the non-
Gaussianity parameters in the other limits are obtained, which are different
than the conventional scenario studied before. The results indicate an
interesting outlook of such choice of the viable coupling function.

The article is organized as follows. In the following section, we discuss the
basic model of the primordial magnetogenesis. We obtain the preferred scale-
invariant solution of the magnetic field that can explain $\sim 10^{-12}$ G
strength of the field in Mpc scales at present time. We also discuss the two
main problems associated with this kind of solution. In Sec. 3, we introduce
the new coupling function and obtain the energy density of the EM field.
Requiring no backreaction in the furthest observable scales, we obtain the
constraint on the model parameter. We also show that the observable modes,
i.e., the modes associated with the Mpc length scales exits the Hubble horizon
and become _classical_ before the system enters into the strong coupling
regime, thus evading the problem for the modes we are interested in. In Sec.
4, we then briefly discuss the cross-correlations of the magnetic fields and
the curvature perturbation and evaluate the non-Gaussianities associated with
the new coupling function in Sec. 5. We show that, while the consistency
relation is preserved, it is highly enhanced in other limits with unique
signatures. The results also differ from previous works and we conclude our
work with a brief discussion in Sec. 6.

A few words on our conventions and notations are in order at this stage of our
discussion. In this work, we work with the natural units such that
$\hbar=c=1$, and we define the Planck mass to be $M_{{}_{\rm
Pl}}=(8\,\pi\,G)^{-1/2}$. We adopt the metric signature of $(-,+,+,+)$. Also,
we should mention that, while the Greek indices are contracted with the metric
tensor $g_{\mu\nu}$, the Latin indices contracted with the Kronecker delta
$\delta_{ij}$. Moreover, we shall denote the partial and the covariant
derivatives as $\partial$ and $\nabla$. The overdots and overprimes, as usual,
denote derivatives with respect to the cosmic time $t$ and the conformal time
$\eta$ associated with the Friedmann-Lemaître-Robertson-Walker (FLRW) line-
element, respectively.

##  2 The simple model of primordial magnetogenesis

In this section, we briefly discuss the mechanism to generate magnetic fields
in the early Universe. Let us first consider the background to be the
spatially flat, Friedmann-Lemaître-Robertson-Walker (FLRW) metric that is
described by the line-element

${\rm d}s^{2}=a^{2}(\eta)\,\left(-{\rm d}\eta^{2}+\delta_{ij}\,{\rm
d}x^{i}\,{\rm d}x^{j}\right),$ (2.1)

where $a(\eta)$ is the scale factor and $\eta$ denotes the conformal time
coordinate, and $\delta_{ij}$ is the Kronecker delta function. The standard
electromagnetic theory is given by the action:

$S_{\rm EM}=-\frac{1}{4}\int{\rm
d}^{4}x\,\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu},\quad
F_{\mu\nu}\equiv\partial_{\mu}\,A_{\nu}-\partial_{\nu}\,A_{\mu},$ (2.2)

where $A_{\mu}$ is the vector potential and the above action is gauge-
invariant. The electric and the magnetic fields are defined as

$E_{i}=-\frac{1}{a}A_{i}^{\prime},\qquad
B_{i}=\frac{1}{a}\epsilon_{ijk}\partial_{j}A_{k}$ (2.3)

The action is conformally invariant in four dimensions. As a result, one can
show that the EM field fluctuations decay as $a^{-2}$ with the expansion of
the universe, and therefore cannot explain the present abundances of the
magnetic field. Hence, one needs to break the conformal invariance for the
amplification of the vacuum fluctuations. There exist infinite possibilities,
yet the simplest and the most elegant one is to couple the action (2.2) with a
field responsible for the early Universe dynamics, that can also preserve the
gauge invariance:

$S=-\frac{1}{4}\int{\rm
d}^{4}x\,\sqrt{-g}\,f^{2}\left(\phi\right)F_{\mu\nu}F^{\mu\nu},$ (2.4)

where $\phi$ is a homogeneous scalar field which is assumed to be the inflaton
field. $f(\phi)$ is the coupling function responsible to break the conformal
symmetry. Since gauge invariance is preserved, one can choose an arbitrary
gauge. In this work, we use the Coulomb gauge where $A_{0}=0$ and
$\partial_{i}A^{i}=0$ and the action (2.4) becomes

$S=\frac{1}{2}\int{\rm d}^{3}{\rm\bf x}\,{\rm
d}\eta\,f^{2}\left(\phi\right)\left(A_{i}^{\prime}{}^{2}-\frac{1}{2}\left(\partial_{i}A_{j}-\partial_{j}A_{i}\right)^{2}\right).$
(2.5)

The above action leads to the equation of motion of the vector field $A_{i}$
as

$A_{i}^{\prime\prime}+2\,\frac{f^{\prime}}{f}A_{i}^{\prime}-\partial_{j}\partial^{j}A_{i}=0.$
(2.6)

In order to quantize the EM field, the vector potential can be Fourier
decomposed as

$\hat{A}_{i}\left(\eta,{\rm\bf x}\right)=\sum_{\sigma=1,2}\int\frac{{\rm
d}^{3}{\rm\bf k}}{\left(2\pi\right)^{3/2}}\,\epsilon_{i\,\sigma}^{{\rm\bf
k}}\,\left(\hat{b}^{\sigma}_{{\rm\bf k}}\,A_{{\rm\bf
k}}\left(\eta\right)\,e^{i\,{\rm\bf k}.{\rm\bf
x}}+\hat{b}^{\sigma\dagger}_{{\rm\bf k}}\,A^{\ast}_{{\rm\bf
k}}\left(\eta\right)\,e^{-i\,{\rm\bf k}.{\rm\bf x}}\right),$ (2.7)

where $\epsilon_{i\,\sigma}^{{\rm\bf k}}$ is the polarization vector and
$\sigma$ corresponds to the spin associated with it.
$\hat{b}^{\sigma}_{{\rm\bf k}}$ and $\hat{b}^{\sigma\dagger}_{{\rm\bf k}}$ are
the annihilation and creation operator. By definition
$\delta_{ij}{\epsilon}^{i}{\epsilon}^{j}=1$ and the following identities are
verified:

${\epsilon}_{\sigma}^{i}k_{i}=0,\quad\sum_{\sigma=1,2}{\epsilon}_{\sigma}^{i}\left(\mathbf{k}\right){\epsilon}_{j,\sigma}\left(\mathbf{k}\right)=\delta_{j}^{i}-\delta_{jl}k^{i}k^{l}/{k^{2}},$

with the usual commutation relation

$\left[\hat{b}_{\mathbf{k_{1}}}^{\sigma},\hat{b}_{\mathbf{k_{2}}}^{\sigma^{\prime}\dagger}\right]=\delta^{\left(3\right)}\left(\mathbf{k_{1}}-\mathbf{k_{2}}\right)\delta_{\sigma\sigma^{\prime}}.$
(2.8)

Let us now define the canonically normalized vector field ${\cal A}_{k}\equiv
f(\eta)\,A_{k}(\eta)$ (dropping the vector sign as the field depends only on
the amplitude of ${\rm\bf k}$). Rewriting the action (2.6) in terms of the
canonical field gives us

${\cal
A}_{k}^{\prime\prime}+\left(k^{2}-\frac{f^{\prime\prime}}{f}\right){\cal
A}_{k}=0.$ (2.9)

###  2.1 Energy spectrum

In this section, we focus on the energy spectrum, which later will help up to
evaluate the amount of the magnetic field generated during the early Universe.
It will also be crucial to understand the backreaction problem of such kinds
of models. Varying the action (2.4) with respect to the metric provides us the
energy-momentum tensor of the EM field:

$T_{\mu\nu}=f^{2}(\phi)\,\left(F_{\mu}^{\
\beta}F_{\nu\beta}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right).$
(2.10)

The above energy-momentum tensor leads to the total energy density of the EM
field as

$\rho_{\rm EM}=\rho_{\rm E}+\rho_{\rm B},$ (2.11)

where $\rho_{E}$ and $\rho_{B}$ are the energy densities of the Electric and
magnetic fields, respectively:

$\displaystyle\rho_{\rm E}$ $\displaystyle\equiv$
$\displaystyle\frac{f^{2}(\phi)}{2}g^{ij}E_{i}E_{j}=\frac{f^{2}(\phi)}{2\,a^{4}(\eta)}A_{i}^{\prime}{}^{2},$
(2.12) $\displaystyle\rho_{\rm B}$ $\displaystyle\equiv$
$\displaystyle\frac{f^{2}(\phi)}{2}g^{ij}B_{i}B_{j}=\frac{f^{2}(\phi)}{4\,a^{4}(\eta)}\left(\partial_{i}A_{j}-\partial_{j}A_{i}\right)^{2}.$
(2.13)

Using (2.7) and the above expressions, one can immediately obtain the electric
and magnetic field spectra as

$\displaystyle\mathcal{P}_{\rm B}\equiv\frac{d\rho_{B}}{d\log k}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\frac{k^{5}}{a^{4}}\left|{\cal
A}\left(k,\eta\right)\right|^{2},$ (2.14) $\displaystyle\mathcal{P}_{\rm
E}\equiv\frac{d\rho_{E}}{d\log k}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{2\pi^{2}}\frac{k^{3}}{a^{4}}\left|\left(\frac{{\cal
A}\left(k,\eta\right)}{f}\right)^{\prime}\right|^{2}.$ (2.15)

Since, at the end of inflation, the coupling becomes unity, the magnetic field
at the end of inflation is simply:

$B_{\rm end}(k,a)=\sqrt{\left.\mathcal{P}_{\rm B}(k,a)\right|_{\rm end}},$
(2.16)

where, $|_{\rm end}$ implies the value at the end of inflation.

###  2.2 Inflationary solution

During slow-roll inflation, the scale factor solution can _approximately_ be
written as a de-Sitter solution: $a(\eta)\approx-\frac{1}{H\eta}$, where $H$
is the Hubble parameter which is nearly constant. There are infinite
possibilities of choosing the coupling function, however, the simplest of them
is to choose the power-law solution of the scale factor, i.e.,

$f\propto a^{\alpha}\propto(-\eta)^{-\alpha}.$

Assuming the slow-roll inflatinary Universe is governed by the canonical
scalar field minimally coupled to gravity with potential $V(\phi)$, the
coupling function in terms of the scalar field can be obtained as

$f(\phi)\propto e^{-\alpha\int\frac{V{\rm d}\phi}{V_{\phi}}},$

where $V_{\phi}\equiv\partial V/\partial\phi.$ After the end of inflation, the
Universe lies at the minima of the potential. Therefore, to restore the
standard EM theory, the function needs to be unity, which can easily be
achieve with appropriate normalization. For example, consider the potential
$V(\phi)\propto\phi^{2}$, i.e., the chaotic potential, and the coupling
function for this model becomes $f(\phi)=e^{-\alpha\phi^{2}/4}$. This
corresponds to the approximate time-dependent solution of the coupling
function as

$f(\eta)=\left\\{\begin{aligned} &(\eta/\eta_{\rm
end})^{-\alpha},\quad\eta\leq\eta_{\rm end}\\\
&0,\qquad\qquad\,\,\,\eta>\eta_{\rm end},\end{aligned}\right.$ (2.17)

where $\eta_{\rm end}$ is the time when inflation ends. With such choice, Eq.
(2.9) becomes

${\cal
A}_{k}^{\prime\prime}+\left(k^{2}-\frac{\alpha\left(\alpha+1\right)}{\eta^{2}}\right){\cal
A}_{k}=0,$ (2.18)

whose solution can be written in terms of the Bessel functions as

${\cal
A}_{k}=\sqrt{-k\eta}\,\left[C_{1}(k)J_{\alpha+1/2}(-k\eta)+C_{2}(k)J_{-\alpha-1/2}(-k\eta)\right].$
(2.19)

By imposing the Bunch-Davies initial conditions, the two integration constants
$C_{1}$ and $C_{2}$ can easily be fixed as

$C_{1}\left(k\right)=\sqrt{\frac{\pi}{4k}}\frac{e^{-i(\alpha-1)\pi/2}}{\cos\left(\alpha\pi\right)},\qquad
C_{2}\left(k\right)=\sqrt{\frac{\pi}{4k}}\frac{e^{i\alpha\pi/2}}{\cos\left(\alpha\pi\right)}.$
(2.20)

When the modes leave the Hubble horizon, i.e., when $k\eta\ll 1$, one can use
the asymptotic solution of the Bessel’s function and using that, the above
solution can be written as

$\left.{\cal A}_{k}\right|_{-k\eta\rightarrow
0}=\frac{C_{1}(k)}{2^{\alpha+\frac{1}{2}}\Gamma(\alpha+\frac{3}{2})}(-k\eta)^{1+\alpha}+\frac{C_{2}(k)}{2^{-\alpha-\frac{1}{2}}\Gamma(\frac{1}{2}-\alpha)}(-k\eta)^{-\alpha}.$
(2.21)

This implies that, as $k\eta\rightarrow 0$, for $\alpha>-1/2$, the first part
of the right-hand side of the solution dominates, whereas the second part
dominates for $\alpha<-1/2$.

###  2.3 Scale-invariant Magnetic spectrum and the present abundances

Since we obtain the solutions of the primordial magnetic field, with the help
of (2.14), one can find out the magnetic field energy density at the end of
inflation. At the super-Hubble scale, the spectrum (2.14) becomes

$\mathcal{P}_{\rm B}=\frac{\mathcal{F}(n)}{2\pi^{2}}H^{4}(-k\eta)^{4+2n}$
(2.22)

where $n=-\alpha,\quad\alpha>-1/2$ and $n=1+\alpha,\quad\alpha<-1/2$, and
$\mathcal{F}(n)=\pi/(2^{2n+1}\Gamma^{2}(n+1/2)\cos^{2}(n\pi))$ [20]. We have
also used $a(\eta)\approx-1/(H\eta)$ as the near de-Sitter solution for the
slow-roll inflationary scenario. It is then obvious that the scale-invariant
magnetic field can be obtained for $\alpha=2$ and $-3$ and in these cases, the
spectrum $\sim\delta^{2}_{\rm B}$ becomes $\mathcal{O}(1)H^{4}$. This implies
that the magnetic field generated at the end of inflation is

$\delta_{\rm B}\sim H^{2}.$

From the CMB observations [73], one can find that the constraint on the Hubble
scale at the of inflation is $H\sim 10^{-6}M_{{}_{\rm Pl}}$ which is
translated to the strength of the magnetic field to be $10^{-12}$ in Planck
scale or $10^{45}$ G at the end of inflation.

After the end of inflation, since the coupling function becomes unity and the
standard EM theory is restored, Eq. (2.9) loses the term $f^{\prime\prime}/f$.
This implies that, after the end of inflation $\delta_{\rm B}$ decays as
$a^{-2}$ and the electric field vanishes quickly [20]. Assuming instantaneous
reheating and using the entropy conservation, one can obtain how much universe
expands after the end of inflation to the present epoch and it is
approximately

$\left(\frac{a_{0}}{a_{\rm end}}\right)\sim 10^{29},$

where $a_{0}$ is the scale factor at present time. Since the magnetic field
decays as $a^{-2}$, the field value today cannot exceed $10^{-13}$ G.

###  2.4 Backreaction and strong coupling problem

Let us now focus on the backreaction problem. In order to ensure that there is
no backreaction due to the EM field, the energy density (see Eq. (2.11)) must
remain smaller than the background energy density $\rho_{\rm inf}\equiv
3H^{2}M_{{}_{\rm Pl}}^{2}$. The main contribution to the energy density comes
from the super-Hubble scales because the contribution from the subhorizon
scales is renormalized in the leading order. Then, similar to the magnetic
field spectrum, one can simply obtain the electric field spectrum for the
super-Hubble modes as

$\mathcal{P}_{\rm E}=\frac{\mathcal{G}(m)}{2\pi^{2}}H^{4}(-k\eta)^{4+2m},$
(2.23)

where, $m=1-\alpha,\quad\alpha>1/2$ and $m=\alpha,\quad\alpha<1/2$, and
$\mathcal{G}(m)=\pi/(2^{2m+3}\Gamma^{2}(m+3/2)\cos^{2}(\pi m)).$ The energy
density of the electric field can be obtained as

$\displaystyle\rho_{\rm E}$ $\displaystyle=$
$\displaystyle\int_{a_{i}H}^{a_{\rm end}H}\mathcal{P}_{\rm E}(k)\frac{{\rm
d}k}{k}$ (2.24) $\displaystyle=$ $\displaystyle\left\\{\begin{aligned}
&\frac{\mathcal{G}(m)}{2\pi^{2}(4+2m)}H^{4}\left(1-\left(\frac{a_{i}}{a_{\rm
end}}\right)^{4+2m}\right),\quad m\neq-2,\\\
&\frac{\mathcal{G}(m)}{2\pi^{2}}H^{4}\log\left(\frac{a_{i}}{a_{\rm
end}}\right),\quad m=-2,\end{aligned}\right.$

where $a_{i}$ is the scale factor at the beginning of inflation.

Now, consider the case of scale-invariant magnetic field spectrum. This can be
obtained for $\alpha=-3$ and $2$. For $\alpha=-3$, the density corresponding
to the electric field becomes

$\mathcal{O}(1)H^{4}\left(1-\left(\frac{a_{\rm
end}}{a_{i}}\right)^{2}\right).$

Assuming inflation lasts at least for 60 e-folds time, $|\rho_{E}|/\rho_{\rm
inf}\approx 10^{52}H^{2}/M_{{}_{\rm Pl}}^{2}\approx 10^{40}\gg 1$. It implies
that the electric field density dominates over the inflaton field energy
density $\rho_{\rm inf}$ and the backreaction spoils the homogeneous
background. Therefore, the solution corresponding to $\alpha=-3$ is not
allowed. However, one can evaluate the same with $\alpha=2$ and show that, in
that case, the electric field energy density remains ${\cal O}(1)H^{4}$ which
is much smaller than that of the background field. Therefore, the allowed
scale-invariant solution for magnetogenesis is $\alpha=2$.

However, unlike the curvature perturbations, the magnetic field energy density
does not need to be scale-invariant and the sole purpose of magnetogenesis is
to generate sufficient magnetic fields. Therefore, one can ask, for what value
of $\alpha$, one can get the maximum amplitude of the magnetic field without
the backreaction problem. It can easily be shown that the value is
$\alpha\simeq 2.2$ for which the magnetic field at the present epoch at Mpc
scale can be as high as $10^{-8}$ G [44].

However, these allowed range of the coupling function $f(\phi)$ possesses
another subtle yet crucial problem: strong coupling. Consider the vector field
is coupled with a charged fermion in the standard form:

${\cal
L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+i\bar{\psi}\gamma^{\mu}(\partial_{\mu}+igA_{\mu})\psi$

where $g$ is the coupling constant, then after re-scaling the vector potential
by the coupling constant $A_{\mu}\rightarrow gA_{\mu},$ we bring this
Lagrangian to the form:

${\cal
L}=-\frac{1}{4g^{2}}F_{\mu\nu}F^{\mu\nu}+i\bar{\psi}\gamma^{\mu}(\partial_{\mu}+iA_{\mu})\psi,$

which indicates that the coupling function $f$ acts like the inverse of the
coupling constant. Therefore, small values of $f$ correspond to large coupling
constant $g$, which implies that the theory lies in the uncontrollable strong
coupling region and the quantum theory cannot be trusted: known as the strong
coupling problem. Therefore, only larger values of $f$, i.e., $f\geq 1$ is
allowed in order to avoid the strong coupling problem [44].

Therefore, for $\alpha\geq 2,$ the coupling function at the beginning of
inflation takes the value $f_{i}\simeq 10^{-52}-10^{-57},$ which indicates
that these ‘allowed’ values suffers from strong coupling problem. In fact, it
has been shown that one can generate maximum amplitude of the magnetic field
for the value of $\alpha\simeq-2.2$ _without both strong coupling and
backreaction problems_ , and in that case, the field generated today cannot
exceed $10^{-30}$ G in Mpc scales [44].

##  3 The proposed model: choosing a new coupling function

In the above section, we discussed how we can generate magnetic fields with
the coupling function that breaks the conformal symmetry of the standard EM
theory. As mentioned above, up to $10^{-8}$ G magnetic field amplitude can be
generated that can successfully answer the existence of magnetic fields in
large scales. However, again, as discussed, it severely suffers from the
strong coupling problem and the theory cannot be trusted.

Consider the scale-invariant case $\alpha=2$, i.e., $f\propto(-\eta)^{-2}$.
While this choice can generate the required primordial field, the value of it
gets extremely small toward the beginning of inflation. On the other hand, for
$\alpha=3$, while there is no strong coupling, it spoils the background energy
density as the EM field causes backreaction.

However, it should be noted that the solution with strong coupling problem
does not possess backreaction and vice-versa, and motivated by this, in this
article, we choose the coupling function as the linear combination of the
above two scale-invariant cases, i.e.,

$f(\eta)=d_{1}\left(\frac{\eta}{\eta_{\rm
end}}\right)^{3}+d_{2}\left(\frac{\eta}{\eta_{\rm end}}\right)^{-2},$ (3.1)

where $d_{1},d_{2}$ are constants. Requiring $f\rightarrow 1$ at the end of
inflation helps to re-write the above as

$f(\eta)=d\left(\frac{\eta}{\eta_{\rm
end}}\right)^{3}+(1-d)\left(\frac{\eta}{\eta_{\rm end}}\right)^{-2}.$ (3.2)

We have re-written $d_{1}$ as $d$, which is again a constant. The novelty of
such choice is mainly because the two solutions are dual, i.e., they both lead
to the similar solution of the canonical field ${\cal A}_{k}$, and therefore,
any linear combination of them can also lead to a similar solution, which
helps in evaluating everything analytically. Then, one can also ask for its
implications in the context of primordial magnetogenesis. In the next section,
we will show that in order to solve the backreaction, $d$ must be smaller than
$10^{-20}$, which, in turn, can evade the problem of strong coupling for modes
corresponding to Mpc scales.

###  3.1 Solving backreaction problem

For such choice of the coupling function, Eq. (3.2) leads to
$f^{\prime\prime}/f=6/\eta^{2}$, which corresponds solution of the canonical
vector field ${\cal A}_{k}$, i.e., Eq. (2.19) for $\alpha=2$. Note that the
solution is identical to the solution $\alpha=-3$ as $f^{\prime\prime}/f$ is
identical to that of $\alpha=2$, i.e., the two solutions are dual. In this
case, the present day magnetic field strength can be generated up to
$10^{-13}$ G, as discussed before. Now, let us focus on the electric field
spectrum. In the super Hubble scale, using the expression (2.15), it becomes

$\left.\mathcal{P}_{\rm E}\right|_{-k\eta\rightarrow
0}=\frac{H^{4}}{4\pi^{2}}\frac{1}{x^{2}}\frac{\left(-15\,d\,x^{4}+(1-d)\,x\,x_{\rm
end}\right)}{d\,x^{5}+(1-d)},$ (3.3)

where $x\equiv-k\eta,~{}x_{\rm end}\equiv-k\eta_{\rm end}$. We can now
directly evaluate the energy density of the electric field as previously done
and it becomes

$\rho_{\rm E}\simeq\frac{225\,d^{2}\,H^{4}}{8\pi^{2}}\left(\frac{a_{\rm
end}}{a_{i}}\right)^{2}.$ (3.4)

The total energy density of the EM field is dominated by the electric field
density as the magnetic field energy density becomes logarithmic for the
solution and thus can be ignored. Requiring the total EM energy density must
be smaller than the background energy density $3H^{2}M_{{}_{\rm Pl}}^{2}$, one
can find the constraint on the parameter $d$ as:

$d<\frac{2\sqrt{2}\,\pi}{5\sqrt{3}}\frac{M_{{}_{\rm
Pl}}}{H}\left(\frac{a_{i}}{a_{\rm end}}\right)$ (3.5)

Assuming inflation lasts for at least $60$ e-folds, $d$ must be smaller than
$10^{-20}.$

This is one of the main results of this work: _we have considered the linear
combination of the dual scale-invariant solution and find the constraint on
such combination._ In the next section, we will show the implications of such
choice on the strong coupling problem.

###  3.2 Solving strong coupling problem

In order to generate $10^{-13}$ G magnetic field today in Mpc scales, the
coupling function becomes smaller and smaller at earlier times, and at the
beginning of the inflation, it reaches the value $10^{-52}$, implying that the
effective coupling constant is huge. In this section, we show that the above
choice of the coupling function (3.2) can avoid such a high value of the
coupling constant and remains very briefly in the strong coupling region. This
region also lies close to the end of the inflation, where all observable modes
are already far outside the Hubble radius, thus avoid the strong coupling
problem.

In the previous section, we show that the magnitude of $d$ needs to be
extremely small, i.e., $d\ll 1$ in order to avoid the backreaction. Because of
such choice, as one can see from (3.2), as $|\eta|$ increases, initially $f$
decays as $\eta^{-2}$ and enters into the strong coupling region. Shortly
after that, it reaches a minimum and then it grows as $\eta^{3}$ and quickly
recovers from the strong coupling problem. One can easily estimate the region
for which the proposed coupling function remains in the strong coupling region
and when it gets out of that region. The minimum can easily be found by
equating the time derivative of the coupling function to zero and assuming
$d\ll 1$, it becomes

$\left(\frac{\eta_{\rm min}}{\eta_{\rm end}}\right)\sim d^{-1/5},\qquad f_{\rm
min}\sim f(\eta_{\rm min})\approx d^{2/5}.$ (3.6)

$\eta_{\rm min}$ is time when the coupling function reaches the minimum. This
implies that, the coupling function behaves as $\eta^{-2}$ in the region
$|\eta_{\rm end}|\leq|\eta|\leq d^{-1/5}|\eta_{\rm end}|$ and the minimum
value of the coupling function is $d^{2/5}$. To get the perspective, consider
$d\approx 10^{-20}$ with the assumption that the CMB pivot, i.e., $0.05$ Mpc
scale exits the Hubble radius 50 e-fold before the end of inflation, we get
$|\eta_{\rm end}|\sim 10^{-20}$ Mpc, $|\eta_{\rm min}|\sim 10^{-16}$ Mpc and
$f^{-1}_{\rm min}\sim 10^{8}$. However, after $|\eta|>|\eta_{\rm min}|,$ the
coupling function starts growing as $\eta^{3}$ and quickly reaches unity at

$\left(\frac{\eta_{\ast}}{\eta_{\rm end}}\right)\sim d^{-1/3}.$

$\eta_{\ast}$ is the time when the coupling function reaches unity, again.
This implies that the model remains in the domain of strong coupling for
$|\eta_{\rm end}|\leq|\eta|<d^{-1/3}|\eta_{\rm end}|$ with the peak of the
coupling constant $f^{-1}\sim d^{-2/5}$. For similar choice of $d$ to be
$10^{-20}$, we get the domain of strong coupling as $10^{-20}\,{\rm
Mpc}\leq|\eta|\leq 10^{-13}$ Mpc. Note that the region gets broader with the
choice of smaller values of $d$ with higher coupling constant $f^{-1}_{\rm
min}.$ Therefore, it is always preferred to choose the $d$ as large as
possible. This becomes apparent from Fig. 1.

Compared to the conventional scenario with $f\propto\eta^{-2},$ the above
example thus reduces the problem of strong coupling drastically, as the domain
of the strong coupling and the amplitude of the coupling constant $f^{-1}_{\rm
min}$ gets smaller than before. The above example also shows that, compared to
the Mpc scales, $\eta_{\ast}$ lies very close to the end of inflation. At this
time, the modes corresponding to Mpc scales become super horizon and
_classical_ , similar to the primordial perturbations. Requiring the modes
corresponding to the Mpc length scales leave the Hubble radius before the
model enters into the region of strong coupling _weakly_ provides the lower
bound of $d$ as $\sim 10^{-60}.$ Since the strong coupling problem relates to
whether the quantum origin of the EM field can be trusted, one can argue that,
for $10^{-60}<d<10^{-20},$ the problem does not arise as the observable modes
become classical, thus such choice of coupling function _solves the strong
coupling problem for modes of Mpc scale._

Figure 1: The proposed coupling function is plotted against the inflationary
e-folding number $N$. 0 denotes the end of inflation, whereas positive $N$
values denote the earlier past, and blue, red and black curve correspond to
$d=10^{-20},~{}10^{-30}$ and $10^{-40}$, respectively. From this figure, it is
evident that, while in the range $\eta_{\rm end}\leq\eta\leq\eta_{\rm min}$,
the coupling function $f$ acts as $\eta^{-2}$, it behaves as $\eta^{3}$ for
$\eta\geq\eta_{\rm min}.$ Also, among these three examples, $d=10^{-40}$ makes
the system stay in the strong coupling region (when $f<1$) for the longest
period $N<30,$ while the modes corresponding to Mpc scales leave the Hubble
horizon $N\sim 50.$

This is the main result of this work: _we introduce a new coupling function,
which corresponds to the linear combination of the dual scale-invariant
magnetic field solution and find that it can evade the strong coupling problem
for modes of Mpc scales, which we are interested in, and simultaneously can
produce $10^{-13}$ G magnetic field in Mpc scales today without any
backreaction._ In the next section, we will study the effect of such model in
higher-order cross-correlation functions.

##  4 Cross-correlations between magnetic fields and curvature perturbation

In the previous section, we showed that the proposed coupling function can
evade the problem of strong coupling for Mpc scales, and at the same time, can
provide sufficient seed magnetic field. However, it is evident that, from the
study of power spectrum, we are unable to detect new observational
information, and therefore, the obvious thing to do is to study the higher-
order cross-correlation functions between the magnetic fields and the
curvature perturbation. This has been extensively studied in the literature
[66, 67, 68, 69, 70, 71, 72]. In this section, we extend the calculation for
our model.

Let us start with the quadratic action for the curvature perturbation
$\mathcal{R}$ that provides the evolution of the mode functions. The action is

$\mathcal{S}_{\mathcal{R}}=\int{\rm d}^{3}{\rm\bf x}\,{\rm
d}\eta\,a^{2}(\eta)\,\epsilon\,\left(\mathcal{R}^{\prime}{}^{2}-\partial_{i}\mathcal{R}\partial^{i}\mathcal{R}\right),$
(4.1)

where, $\epsilon\equiv-\frac{\dot{H}}{H^{2}}$ is the slow-roll parameter. We
can write the curvature perturbation $\mathcal{R}$ in terms of the mode
functions as

$\mathcal{R}(\eta,{\rm\bf x})=\int\frac{{\rm d}^{3}{\rm\bf
x}}{(2\pi)^{3/2}}\left(\hat{a}_{{\rm\bf k}}\mathcal{R}_{\rm\bf
k}(\eta)e^{i{\rm\bf k}.{\rm\bf x}}+\hat{a}^{\dagger}_{{\rm\bf
k}}\mathcal{R}^{\ast}_{\rm\bf k}(\eta)e^{-i{\rm\bf k}.{\rm\bf x}}\right)$
(4.2)

and during slow-roll inflation, the solution of the mode function in leading
order slow-roll can be written as

$\mathcal{R}_{k}(\eta)=\frac{1}{\sqrt{2\,\epsilon}}\frac{H}{\sqrt{2\,k^{3}}}\left(1+ik\eta\right)e^{-ik\eta}.$
(4.3)

Now, in order to evaluate the cross-correlations between magnetic fields and
curvature perturbation, we first need the interaction Hamiltonian involving
the vector fields $A_{i}$ and curvature perturbation $\mathcal{R}$. This can
easily be obtained from the third order perturbed Lagrangian as the density of
the interaction Hamiltonian $\mathcal{H}_{\rm int}=-\mathcal{L}_{3}$ (see
Refs. [74, 75, 76]):

$H_{\rm int}=\frac{1}{2}\int{\rm d}^{3}{\rm\bf
x}\,f^{2}(\eta)\left[\left(\mathcal{R}-\frac{\mathcal{R}^{\prime}}{aH}\right)\left(A_{i}^{\prime}{}^{2}+F_{ij}F^{ij}\right)+\frac{2\partial_{i}\mathcal{R}}{aH}A_{j}^{\prime}F^{ij}\right].$
(4.4)

The cross-correlation between the magnetic fields and the curvature
perturbation in real space is defined as

$\displaystyle\left<\mathcal{R}(\eta,{\rm\bf x})B_{i}(\eta,{\rm\bf
x})B^{i}(\eta,{\rm\bf x})\right>=\int\frac{{\rm d}^{3}{\rm\bf k}_{1}\,{\rm
d}^{3}{\rm\bf k}_{2}\,{\rm d}^{3}{\rm\bf
k}_{3}}{\left(2\pi\right)^{9/2}}e^{i({\rm\bf k}_{1}+{\rm\bf k}_{2}+{\rm\bf
k}_{3}).{\rm\bf x}}\left<\mathcal{R}_{{\rm\bf k}_{1}}(\eta)B_{i\,{\rm\bf
k}_{2}}(\eta)B^{i}_{{\rm\bf k}_{3}}(\eta)\right>,$ (4.5)

where $B_{i}$ is related to the vector field $A_{i}$ as in (2.3).
$\mathcal{R}_{\rm\bf k}$ and $B_{i\,{\rm\bf k}}$ are the Fourier modes
associated with the curvature perturbation (see Eq. (4.3)) and i-$th$
component of the magnetic field, respectively.

In order to compute the higher-order correlation function during inflation, we
adopt a very useful and powerful tool of the in-in formalism [77]. In this
formalism, the expectation value of an operator $\mathcal{O}$ at time $\eta$
is given by

$\left<\Omega\left|\mathcal{O}(\eta)\right|\Omega\right>=\left<0\left|\bar{T}\left(e^{i\int_{-\infty}^{\eta}H_{\rm
int}(\eta^{\prime}){\rm
d}\eta^{\prime}}\right)\mathcal{O}(\eta)T\left(e^{-i\int_{-\infty}^{\eta}H_{\rm
int}(\eta^{\prime\prime}){\rm d}\eta^{\prime\prime}}\right)\right|0\right>,$
(4.6)

where, $\left.|\Omega\right>$ and $\left.|0\right>$ are the vacuum of the
interaction and free theories, respectively, $T$ and $\bar{T}$ are the time
and anti time ordering operators, respectively, and $H_{\rm int}$ is the
interaction Hamiltonian. Using leading order contribution of the interaction
Hamiltonian, the cross-correlation function _at the end of inflation, i.e.,
$\eta_{\rm end}$_ can simply be written as [67, 69]

$\displaystyle\left<\mathcal{R}_{{\rm\bf k}_{1}}(\eta_{\rm end})B_{i\,{\rm\bf
k}_{2}}(\eta_{\rm end})B^{i}_{{\rm\bf k}_{3}}(\eta_{\rm
end})\right>=-i\int_{\eta_{\rm i}}^{\eta_{\rm end}}{\rm
d}\eta\left<\left[\mathcal{R}_{{\rm\bf k}_{1}}(\eta_{\rm end})B_{i\,{\rm\bf
k}_{2}}(\eta_{\rm end})B^{i}_{{\rm\bf k}_{3}}(\eta_{\rm end}),H_{\rm
int}(\eta)\right]\right>.$

Now, for simplicity, we define a new quantity $G_{\mathcal{R}BB}({\rm\bf
k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})$ as

$\left<\mathcal{R}_{{\rm\bf k}_{1}}(\eta_{\rm end})B_{i\,{\rm\bf
k}_{2}}(\eta_{\rm end})B^{i}_{{\rm\bf k}_{3}}(\eta_{\rm
end})\right>\equiv\left(2\pi\right)^{-3/2}G_{\mathcal{R}BB}({\rm\bf
k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})\delta^{3}({\rm\bf k}_{1}+{\rm\bf
k}_{2}+{\rm\bf k}_{3}).$ (4.8)

Then, by using the expression (4) along with the interaction Hamiltonian
(4.4), the definition of the mode functions (4.3) and (2.7), and the Wick’s
theorem, $G_{\mathcal{R}BB}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})$ can
be expressed as

$G_{\mathcal{R}BB}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf
k}_{3})=\sum_{i=1}^{6}\mathcal{G}_{i}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf
k}_{3}),$ (4.9)

where six $\mathcal{G}$’s are defined as

$\displaystyle\mathcal{G}_{1}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})$
$\displaystyle=$ $\displaystyle
i\left(k_{1}^{2}-k_{2}^{2}-k_{3}^{2}\right)\mathcal{R}_{k_{1}}(\eta_{\rm
end})A_{k_{2}}(\eta_{\rm end})A_{k_{3}}(\eta_{\rm end})\times$ (4.10)
$\displaystyle\int^{\eta_{\rm end}}_{\eta_{i}}{\rm
d}\eta\,f^{2}(\eta)\mathcal{R}^{\ast}_{k_{1}}(\eta)A^{\ast\prime}_{k_{2}}(\eta)A^{\ast\prime}_{k_{3}}(\eta)+c.c.$
$\displaystyle\mathcal{G}_{2}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})$
$\displaystyle=$
$\displaystyle-i\left(k_{1}^{2}-k_{2}^{2}-k_{3}^{2}\right)\mathcal{R}_{k_{1}}(\eta_{\rm
end})A_{k_{2}}(\eta_{\rm end})A_{k_{3}}(\eta_{\rm end})\times$ (4.11)
$\displaystyle\int^{\eta_{\rm end}}_{\eta_{i}}{\rm
d}\eta\,\frac{f^{2}(\eta)}{aH}\mathcal{R}^{\ast\prime}_{k_{1}}(\eta)A^{\ast\prime}_{k_{2}}(\eta)A^{\ast\prime}_{k_{3}}(\eta)+c.c.$
$\displaystyle\mathcal{G}_{3}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})$
$\displaystyle=$
$\displaystyle-i\left(k_{2}^{2}k_{3}^{2}+\frac{1}{4}\left(k_{1}^{2}-k_{2}^{2}-k_{3}^{2}\right)^{2}\right)\mathcal{R}_{k_{1}}(\eta_{\rm
end})A_{k_{2}}(\eta_{\rm end})A_{k_{3}}(\eta_{\rm end})\times$ (4.12)
$\displaystyle\int^{\eta_{\rm end}}_{\eta_{i}}{\rm
d}\eta\,f^{2}(\eta)\mathcal{R}^{\ast}_{k_{1}}(\eta)A^{\ast}_{k_{2}}(\eta)A^{\ast}_{k_{3}}(\eta)+c.c.$
$\displaystyle\mathcal{G}_{4}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})$
$\displaystyle=$ $\displaystyle
i\left(k_{2}^{2}k_{3}^{2}-\frac{1}{4}\left(k_{1}^{2}-k_{2}^{2}-k_{3}^{2}\right)^{2}\right)\mathcal{R}_{k_{1}}(\eta_{\rm
end})A_{k_{2}}(\eta_{\rm end})A_{k_{3}}(\eta_{\rm end})\times$ (4.13)
$\displaystyle\int^{\eta_{\rm end}}_{\eta_{i}}{\rm
d}\eta\,\frac{f^{2}(\eta)}{aH}\mathcal{R}^{\ast\prime}_{k_{1}}(\eta)A^{\ast}_{k_{2}}(\eta)A^{\ast}_{k_{3}}(\eta)+c.c.$

$\displaystyle\mathcal{G}_{5}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})$
$\displaystyle=$ $\displaystyle
i\left(k_{2}^{2}k_{3}^{2}+\frac{1}{4}\left(k_{1}^{2}-k_{2}^{2}-k_{3}^{2}\right)\left(k_{1}^{2}+3k_{3}^{2}-k_{2}^{2}\right)\right)\mathcal{R}_{k_{1}}(\eta_{\rm
end})A_{k_{2}}(\eta_{\rm end})A_{k_{3}}(\eta_{\rm end})\times$ (4.14)
$\displaystyle\int^{\eta_{\rm end}}_{\eta_{i}}{\rm
d}\eta\,\frac{f^{2}(\eta)}{aH}\mathcal{R}^{\ast}_{k_{1}}(\eta)A^{\ast\prime}_{k_{2}}(\eta)A^{\ast}_{k_{3}}(\eta)+c.c.$
$\displaystyle\mathcal{G}_{6}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})$
$\displaystyle=$ $\displaystyle
i\left(k_{2}^{2}k_{3}^{2}+\frac{1}{4}\left(k_{1}^{2}-k_{2}^{2}-k_{3}^{2}\right)\left(k_{1}^{2}+3k_{3}^{2}-k_{2}^{2}\right)\right)\mathcal{R}_{k_{1}}(\eta_{\rm
end})A_{k_{2}}(\eta_{\rm end})A_{k_{3}}(\eta_{\rm end})\times$ (4.15)
$\displaystyle\int^{\eta_{\rm end}}_{\eta_{i}}{\rm
d}\eta\,\frac{f^{2}(\eta)}{aH}\mathcal{R}^{\ast}_{k_{1}}(\eta)A^{\ast}_{k_{2}}(\eta)A^{\ast\prime}_{k_{3}}(\eta)+c.c.$

The abbreviated form of $c.c$ is complex conjugate. Now, given the coupling
function $f(\eta)$, which fixes the solution of the vector field $A_{\rm\bf
k}(\eta)$, and the solution of the curvature perturbation $\mathcal{R}_{\rm\bf
k}$, one can easily compute the cross-correlations using the above
expressions.

In our case, with the coupling function (3.2), and the solution of the vector
field $A_{{\rm\bf k}}$, i.e.,

$A_{k}(\eta)\simeq\frac{e^{-ik\eta}}{\sqrt{2\,k}}\left(d\left(\frac{\eta}{\eta_{\rm
end}}\right)^{5}-1\right)\frac{\left(3+3ik\eta-k^{2}\eta^{2}\right)}{k^{2}\eta^{2}}$

as well as the solution of the curvature perturbations $\mathcal{R}_{\rm\bf
k}$ (4.3), we have evaluated all $\mathcal{G}$’s, and the explicit expressions
are given in Appendix A. Please note that, in doing such evaluations, we
assume that $d\ll 1$, which is evident from Sec. 3.1.

At this point, please notice the logarithmic dependence of $\mathcal{G}_{3}$
and $\mathcal{G}_{4}$. Also, in all expressions of $\mathcal{G}$’s, $\eta_{\rm
end}$ appears in the denominator. In $\mathcal{G}_{3}$ and $\mathcal{G}_{4}$,
while it is $\eta_{\rm end}^{4}$, in other four $\mathcal{G}$’s, it appears as
$\eta_{\rm end}^{6}$. Later, while evaluating the non-Gaussianity parameter in
the next section, we will show that, while $\eta_{\rm end}^{4}$ dependence
cancels out, $\eta_{\rm end}^{6}$ as well as the logarithmic dependencies, in
general, do not. And therefore, for $\eta_{\rm end}\rightarrow 0$ limit,
except for the case of squeezed limit, it contributes enormously in evaluating
the non-Gaussianity parameter.

##  5 Non-Gaussianity parameter

In the last section, we have defined the cross-correlation function between
the magnetic fields and the curvature perturbation. However, it is not
dimensionless, and therefore, to characterize the correlation function more
prominently, we introduce the non-Gaussianity parameter $b_{\rm NL}$. The
parameter is dimensionless and captures the shape and amplitude of the three-
point cross-correlation function discussed in the previous section.

The non-Gaussianity parameter $b_{\rm NL}$ in the local form is defined as

$\displaystyle\hat{B}_{i\,\bf k}(\eta)=\hat{B}_{i\,\bf{k}}^{({\rm
G})}(\eta)+b_{\rm NL}\,\int\frac{{\rm d}^{3}{\bf
k^{\prime}}}{(2\,\pi)^{3/2}}\,\hat{\mathcal{R}}^{(\rm
G)}_{\bf{k}-\bf{k^{\prime}}}(\eta)\,\hat{B}_{i\,{\bf k^{\prime}}}^{({\rm
G})}(\eta),$ (5.1)

where $\hat{B}_{i\,\bf k}(\eta)$ is the Fourier model of the magnetic field
and $(G)$ denotes the Gaussian part of the Fourier mode. Using the above
expression, one can now evaluate the cross-correlation function (4.8) that
interests us and then compare with the same that leads to

$b_{\rm NL}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf
k}_{3})=-\frac{f^{2}(\eta_{\rm end})}{8\pi^{4}a^{4}(\eta_{\rm
end})}\frac{k_{1}^{3}k_{2}^{3}k_{3}^{3}\,G_{\mathcal{R}BB}({\rm\bf
k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})}{\mathcal{P}_{\mathcal{R}}({\rm\bf
k}_{1})\left(k_{3}^{2}\mathcal{P}_{\rm B}({\rm\bf
k}_{2})+k_{2}^{2}\mathcal{P}_{\rm B}({\rm\bf k}_{3})\right)},$ (5.2)

where, $\mathcal{P}_{\rm B}$ is defined in $\eqref{eq:MagSpec}$ and
$\mathcal{P}_{\mathcal{R}({\rm\bf k})}$ is the power spectrum of the curvature
perturbation, defined as

$\mathcal{P}_{\mathcal{R}}({\rm\bf
k})\equiv\frac{k^{3}}{2\pi^{2}}\left|\mathcal{R}_{{\rm\bf k}}\right|^{2}.$

At the end of inflation, both the spectra, in our case, take the form

$\mathcal{P}_{\mathcal{R}}({\rm\bf
k})\approx\frac{H^{2}}{8\pi^{2}\,\epsilon}\qquad\mathcal{P}_{\rm B}({\rm\bf
k})\approx\frac{9H^{2}}{4\pi^{2}}.$ (5.3)

With the help of the above spectra, along with the six $\mathcal{G}$’s given
in Appendix A, one can easily evaluate $b_{\rm NL}$. Since, the expression is
enormous, as obvious from the Appendix A, one can simply obtain the same in
the three interesting limits: squeezed, equilateral and flattened and study it
extensively.

###  5.1 Squeezed limit

In the squeezed limit, one of the wavenumber is significantly smaller than the
other two. Since the wavevctors form a triangle configuration as ${\rm\bf
k}_{1}+{\rm\bf k}_{2}+{\rm\bf k}_{3}=0$, ${\rm\bf k}_{1}\rightarrow 0$ implies
that ${\rm\bf k}_{2}=-{\rm\bf k}_{3}$. In this limit, the non-Gaussianity
parameter becomes:

$\displaystyle\lim\limits_{{\rm\bf k}_{1}\rightarrow 0}b^{\rm sq}_{\rm
NL}({\rm\bf k}_{1},{\rm\bf k},-{\rm\bf k})$ $\displaystyle=$
$\displaystyle\lim\limits_{-k\eta_{\rm end}\rightarrow
0}\,\frac{3(6+k^{2}\eta_{\rm end}^{2})}{9+3k^{2}\eta_{\rm
end}^{2}+k^{4}\eta_{\rm end}^{4}}=2.$ (5.4)

Two things to note: $b_{\rm NL}$ is independent of $d$ and $k$. The result is
not surprising as it is expected because of the consistency condition which is
satisfied during inflation [68, 69]. It tells us that, in the squeezed limit,
the non-Gaussianity parameter can be written as the spectral index of the
magnetic field energy density as

$b^{\rm sq}_{\rm NL}({\rm\bf k}_{1},{\rm\bf k},-{\rm\bf k})=\frac{4-n_{\rm
B}}{2},$

where,

$n_{\rm B}\equiv\frac{{\rm d}\ln\mathcal{P}_{\rm B}}{{\rm d}\ln k}.$

Since, the choice of the coupling function (3.2) leads to scale-invariant
magnetic field spectrum, the corresponding spectral index $n_{\rm B}$ is zero,
which makes $b_{\rm NL}$ to be equal to two.

The relation has previously been obtained with the help of power-law kind of
coupling function, i.e., $f\propto(-\eta)^{\alpha}$ [64, 65]. However,
although the magnetic spectrum is scale-invariant, our proposed coupling
function is not power-law, and even after that, with this new choice (3.2), we
obtain the result: _consistency relation is satisfied_. The result is
interesting and unique as in the next section, it will be apparent that in the
other limits, the non-Gaussianity parameter depends both on $k$ and $d$. This
is another interesting result of this work.

###  5.2 Equilateral limit

Unlike the squeezed limit, in the equilateral limit, the non-Gaussianity
parameter behaves differently and possesses unique signatures. In this limit,
as it forms equilateral triangle, it implies $k_{1}=k_{2}=k_{3}$. In this
configuration, the non-Gaussianity parameter acts as

$\displaystyle b^{\rm eq}_{\rm NL}(k,k,k)=4.11-\frac{2.37\,d}{k^{2}\eta_{\rm
end}^{2}}-2.62\log|k\eta_{\rm end}|.$ (5.5)

Such form of non-Gaussianity parameter is unique: it contains two kinds of
$\eta_{\rm end}$ dependence. One kind appears with the parameter $d$ in the
denominator, while the other enters into the logarithmic. The logarithmic
dependence is natural and explored before for $f\propto(-\eta)^{-2}$, while
the former one is unexplored to date.

The above expression also provides us an interesting scenario: for a given
wavenumber, $b_{\rm NL}$ vanishes at $d_{\ast}\simeq-k^{2}\eta_{\rm
end}^{2}\log|k\eta_{\rm end}|$. When $d>d_{\ast}$, the non-Gaussian parameter
becomes negative and entirely behaves as

$b^{\rm eq}_{\rm NL}\simeq-\frac{2.37\,d}{k^{2}\eta_{\rm end}^{2}}.$

For $d>d_{\ast},$ it becomes positive and independent of $d$, and quickly
reaches to the saturated value

$b^{\rm eq}_{\rm NL}\simeq-2.62\,\log|k\eta_{\rm end}|.$

This is plotted in Fig. 2.

Figure 2: $b^{\rm eq}_{\rm NL}$ is plotted against $d$ for the pivot scale
$k=0.05$ Mpc. For $d>10^{-41},$ it behaves as $\propto d$, whereas, for
$d<10^{-41}$, it quickly saturates at $132.$

In Fig. 3, for different values of $d$, we plot the $b_{\rm NL}$ in the
equilateral limit. As obvious from (5.5), for a given value of $k$ such that
$d>-k^{2}\eta_{\rm end}^{2}\log|k\eta_{\rm end}|,$ then the non-Gaussianity
parameter behaves as $\propto k^{-2},$ whereas, for $d>-k^{2}\eta_{\rm
end}^{2}\log|k\eta_{\rm end}|,$ it becomes positive and possesses a maximum at

$k_{\rm max}=-\frac{1.34\sqrt{d}}{\eta_{\rm end}},\quad b^{\rm max}_{\rm
NL}=2.79+0.608\,d-2.62\,\log[1.34\,\sqrt{d}].$

After that, it becomes independent of $d$ and behaves logarithmic. In Fig. 4,
$k_{\rm max}$ and $b^{\rm max}_{\rm NL}$ is plotted against $d$. It becomes
apparent that, in equilateral limit, $b^{\rm max}_{\rm NL}\sim\mathcal{O}(2).$

Figure 3:  We plot $b^{\rm eq}_{NL}$ against $k$ for different values of $d$
(blue for $10^{-20},$ red for $d=10^{-30},$ green for $d=10^{-40}$, orange for
$d=10^{-49}$, purple for $d=10^{-50}$ and black for $10^{-51}$).

Figure 4: In the left, we plot $k_{\rm max}\propto\sqrt{d}$ against $d$ in the
equilateral limit. In the right, we plot $b^{\rm max}_{\rm NL}$ against $d$.
As one can see, the maximum value of the non-Gaussianity parameter decreases
with $d$.

From these figures, it is now obvious that the _in the equilateral limit, the
non-Gaussianity parameter corresponding to the pivot scale can be as large as
$10^{23}$ to zero, depending on the value of the model parameter $d$, which is
huge and never explored._

###  5.3 Flattened limit

The flattened limit implies that $k_{1}=2k_{2}=2k_{3}$, and in this limit,
similar to equilateral limit, $b_{\rm NL}$ behaves as

$b^{\rm fl}_{\rm NL}=12.7-\frac{115.6\,d}{k^{2}\eta_{\rm
end}^{2}}-12\log|k\eta_{\rm end}|$ (5.6)

Again, similar to equilateral limit, for $d=d_{\ast}\simeq-0.1\,k^{2}\eta_{\rm
end}^{2}\,\log|k\eta_{\rm end}|$, the non-Gaussianity parameter vanishes. For
$d$ greater than that value, it behaves as

$-\frac{115.6\,d}{k^{2}\eta_{\rm end}^{2}},$

while for $d<d_{\ast}$, it behaves as

$-12\log|k\eta_{\rm end}|.$

Such dependencies of $b^{\rm fl}_{\rm NL}$ have been plotted in Fig. 5 and
Fig. 6. Also, $k_{\rm max}$ and $b^{\rm max}_{\rm NL}$ for this case yield as

$k_{\rm max}\simeq-4.39\frac{\sqrt{d}}{\eta_{\rm end}},\quad b^{\rm max}_{\rm
L}\simeq 6.75-40.67\,d-12\log[4.38\sqrt{d}].$

Figure 5: $b^{\rm fl}_{\rm NL}$ is plotted against $d$ for the pivot scale
$k=0.05$ Mpc. For $d>10^{-41},$ it behaves as $\propto d$, whereas, for
$d<10^{-41}$, it quickly saturates at $\sim 600.$

Figure 6:  We plot $b^{\rm fl}_{NL}$ against $k$ for different values of $d$
(blue for $10^{-20},$ red for $d=10^{-30},$ green for $d=10^{-40}$, orange for
$d=10^{-49}$, purple for $d=10^{-50}$ and black for $10^{-51}$).

Figure 7: In the left, we plot $k_{\rm max}\propto\sqrt{d}$ against $d$ in the
equilateral limit. In the right, we plot $b^{\rm max}_{\rm NL}$ against $d$.
As one can see, the maximum value of the non-Gaussianity parameter decreases
with $d$.

These have been plotted in Fig. 7. From these figure, it becomes apparent that
the non-Gaussianity parameter in flattened limit always generate higher value
than that in the equilateral limit. For $d>d_{\ast}$, the ratio between them,
i.e., $b^{\rm fl}_{\rm NL}/b^{\rm eq}_{\rm NL}\sim 50$, whereas, for
$d<d_{\ast},$ it is $\sim 4.6$.

Similar to the equilateral limit, it is now obvious that the _in the flattened
limit, the non-Gaussianity parameter corresponding to the pivot scale can be
as large as $10^{24}$ to zero, depending on the value of the model parameter
$d$. This is another interesting result of our work._

##  6 Conclusions

Inflationary magnetogenesis has long been studied in the literature that can
explain the current magnetic field abundances in the cosmic scales. In doing
so, many mechanisms have been proposed and among them,
$-1/4f^{2}(\phi)F_{\mu\nu}F^{\mu\nu}$ model grabs the most attention. At
present time, it can safely generate $10^{-13}$ G at Mpc length scale. For two
kinds of coupling function solutions, this can be achieved. However, both of
them suffer severe issues: while one leads to high electric field energy
density that can spoil the background and the model cannot self-sustain,
another leads to what is referred to as the strong coupling problem. In this
case, the coupling function, at the beginning of inflation possesses a very
tiny value $\ll 1$, which implies that the effective coupling constant is
extremely high, i.e., $\gg 1$ which, in turn, makes the quantization process
of the vector field theory untrustworthy. Due to these issues, one can show
that such model can generate not more than $10^{-30}$ G magnetic field self-
consistently in the Mpc length scales [44].

We noticed that, although each of the scale-invariant solutions suffers from
backreaction or the strong coupling, each of them also solves the other
problem, i.e., the solution with strong coupling does not possess backreaction
and vice-versa. Motivated by this, we proposed a new coupling function which
is a linear combination of both the coupling functions. In doing so, we
introduce a model parameter that characterizes the linear combination and
showed that, in order to evade the backreaction problem, one finds the value
to be extremely small i.e., $d\lesssim 10^{-20}.$ We also found that, for such
model with a non-zero value of $d$, it contains two distinct regions:

  * 1.

for $|\eta|\gtrsim d^{-1/3}|\eta_{\rm end}|,$ there is no strong coupling,

  * 2.

for $|\eta_{\rm end}|\lesssim|\eta|\lesssim d^{-1/3}\,|\eta_{\rm end}|,$ the
model enters into the strong coupling region.

Assuming the pivot scale $k=0.05$ Mpc-1 exiting the Hubble horizon 50 e-folds
before the end of inflation, we find that, for $10^{-20}\lesssim d\lesssim
10^{-60}$, the modes corresponding to the Mpc length scales become super-
Horizon and _classical_ , similar to inflationary perturbations before the
system enters into the strong coupling region. Therefore, _one can argue that,
while very short wavelength modes suffer the problem of strong coupling, the
high wavelength modes that we are interested in do not suffer the same, thus
solving the problem of strong coupling for primordial magnetogenesis._ In this
way, we showed that, today, up to $10^{-13}$ G strength of the magnetic field
can be generated self-consistently in the Mpc scales, which earlier was only
$\sim 10^{-30}$ G [78, 64, 65].

We, further extended our work to find the non-Gaussian signature of this
model. We found that the model satisfies the consistency relation, i.e., in
the squeezed limit, the non-Gaussianity parameter can be written in terms of
the magnetic field spectra index as $b_{\rm NL}=(4-n)/2.$ We also found the
unique signatures of $b_{\rm NL}$ in the equilateral and flattened limit as:

$b_{\rm NL}(k)\sim-C_{1}\frac{d}{k^{2}\eta_{\rm end}^{2}}-C_{2}\log|k\eta_{\rm
end}|.$

For a certain range of $k$, while it is negative and $\propto k^{-2}$, for the
rest, it becomes positive and $\propto\log|k\eta_{\rm end}|$ with the
flattened limit producing greater amplitude than the same in equilateral
limit. This implies that _the non-Gaussianity parameter can be as high as
$10^{20}$ and as small as $\mathcal{O}(1)$ depending on the model parameter
$d$._ These unique signatures can easily be verified from the future
experiments which can detect these kinds of non-linearity.

The scope of such model is huge. One immediate extension is to study the
effect of similar coupling functions in the context of helical magnetic field
generation, which has been studied with the help of the auxiliary scalar field
in Ref. [72]. Inflationary Universe, too, suffers from many issues, for which,
there is a growing interest in finding alternatives to inflation and among
them, the classical bouncing scenario is the most popular one. It would also
thus be interesting to investigate the effect of such coupling function in the
bouncing scenario. Recently, in Ref. [71], it has been extensively studied
with the help of simple auxiliary scalar field coupling of $f\propto
a(\eta)^{\alpha}.$ The auxiliary field has been used due to the fact that the
curvature perturbations in the bouncing model do not satisfy the observational
constraint. However, in Refs. [79, 80], with the help of non-minimal coupling,
the first viable bouncing model has been constructed (also see Refs. [81, 82,
83] in these regard). These works are now in progress.

## Acknowledgements

The author wishes to thank L. Sriramkumar and Kinjalk Lochan for useful
discussion. The author also thanks Sagarika Tripathy for the help in
evaluating the non-Gaussian parameter.

##  Appendix A Evaluating the $\mathcal{G}$’s

$\displaystyle\mathcal{G}_{1}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})$
$\displaystyle=$
$\displaystyle-\frac{H^{2}}{8\,\epsilon\,k_{1}^{3}\,k_{2}^{5}\,k_{3}^{5}\,k_{T}^{6}\,\eta_{\rm
end}^{6}}(k_{1}^{2}-k_{2}^{2}-k_{3}^{2})(k_{2}^{2}k_{3}^{2}k_{T}^{4}\eta_{\rm
end}^{2}(-9(k_{1}^{3}+2k_{1}^{2}(k_{2}+k_{3})+$ (A.1) $\displaystyle
2k_{1}(k_{2}^{2}+k_{2}k_{3}+k_{3}^{2})+(k_{2}+k_{3})(k_{2}^{2}+k_{2}k_{3}+k_{3}^{2}))-3(2k_{1}k_{2}^{2}k_{3}^{2}+$
$\displaystyle
k_{2}^{2}k_{3}^{2}(k_{2}+k_{3})+2k_{1}^{3}(k_{2}^{2}+k_{3}^{2})+2k_{1}^{2}(k_{2}+k_{3})(k_{2}^{2}+k_{2}k_{3}+k_{3}^{2}))\eta_{\rm
end}^{2}+$ $\displaystyle
k_{2}^{2}k_{3}^{2}(-4k_{1}^{3}+2k_{1}k_{2}k_{3}-k_{1}^{2}(k_{2}+k_{3})+k_{2}k_{3}(k_{2}+k_{3}))\eta_{\rm
end}^{4}+$ $\displaystyle
k_{1}^{2}k_{2}^{3}k_{3}^{3}(k_{1}+k_{2}+k_{3})\eta_{\rm
end}^{6})+5dk_{1}^{2}(-3(k_{2}+k_{3})(k_{2}^{4}-k_{2}^{3}k_{3}+k_{2}^{2}k_{3}^{2}-k_{2}k_{3}^{3}+$
$\displaystyle
k_{3}^{4})(3k_{2}^{4}+18k_{2}^{3}k_{3}+38k_{2}^{2}k_{3}^{2}+18k_{2}k_{3}^{3}+3k_{3}^{4})\eta_{\rm
end}^{2}-k_{2}^{2}k_{3}^{2}(k_{2}+k_{3})^{3}(3k_{2}^{4}+$ $\displaystyle
9k_{2}^{3}k_{3}-8k_{2}^{2}k_{3}^{2}+9k_{2}k_{3}^{3}+3k_{3}^{4})\eta_{\rm
end}^{4}-k_{2}^{4}k_{3}^{4}(k_{2}+k_{3})^{5}\eta_{\rm
end}^{6}+3k_{1}(k_{2}+k_{3})^{2}$
$\displaystyle(9(k_{2}+k_{3})^{4}-6(k_{2}^{6}+k_{2}^{5}k_{3}-4k_{2}^{4}k_{3}^{2}-2k_{2}^{3}k_{3}^{3}-4k_{2}^{2}k_{3}^{4}+k_{2}k_{3}^{5}+k_{3}^{6})\eta_{\rm
end}^{2}-$ $\displaystyle
k_{2}^{2}k_{3}^{2}(k_{2}+k_{3})^{2}(k_{2}^{2}-4k_{2}k_{3}+k_{3}^{2})\eta_{\rm
end}^{4})+k_{1}^{5}(27(k_{2}^{2}+k_{3}^{2})+18(k_{2}^{4}+k_{2}^{2}k_{3}^{2}+$
$\displaystyle k_{3}^{4})\eta_{\rm
end}^{2}+9k_{2}^{2}k_{3}^{2}(k_{2}^{2}+k_{3}^{2})\eta_{\rm
end}^{4}+4k_{2}^{4}k_{3}^{4}\eta_{\rm
end}^{6})+3k_{1}^{4}(k_{2}+k_{3})(54(k_{2}^{2}+k_{3}^{2})+$ $\displaystyle
3(7k_{2}^{4}+5k_{2}^{3}k_{3}+7k_{2}^{2}k_{3}^{2}+5k_{2}k_{3}^{3}+7k_{3}^{4})\eta_{\rm
end}^{2}+k_{2}^{2}k_{3}^{2}(11k_{2}^{2}+7k_{2}k_{3}+$ $\displaystyle
11k_{3}^{2})\eta_{\rm end}^{4}+5k_{2}^{4}k_{3}^{4}\eta_{\rm
end}^{6})+2k_{1}^{2}(k_{2}+k_{3})(81(k_{2}+k_{3})^{4}+9(k_{2}^{6}+8k_{2}^{5}k_{3}$
$\displaystyle+17k_{2}^{4}k_{3}^{2}+19k_{2}^{3}k_{3}^{3}+17k_{2}^{2}k_{3}^{4}+8k_{2}k_{3}^{5}+k_{3}^{6})\eta_{\rm
end}^{2}+3k_{2}^{2}k_{3}^{2}(3k_{2}^{4}+15k_{2}^{3}k_{3}$
$\displaystyle+14k_{2}^{2}k_{3}^{2}+15k_{2}k_{3}^{3}+3k_{3}^{4})\eta_{\rm
end}^{4}+5k_{2}^{4}k_{3}^{4}(k_{2}+k_{3})^{2}\eta_{\rm
end}^{6})+2k_{1}^{3}(45(3k_{2}^{4}+$ $\displaystyle
9k_{2}^{3}k_{3}+8k_{2}^{2}k_{3}^{2}+9k_{2}k_{3}^{3}+3k_{3}^{4})+3(k_{2}^{2}+k_{3}^{2})(12k_{2}^{4}+45k_{2}^{3}k_{3}+56k_{2}^{2}k_{3}^{2}+$
$\displaystyle 45k_{2}k_{3}^{3}+12k_{3}^{4})\eta_{\rm
end}^{2}+k_{2}^{2}k_{3}^{2}(21k_{2}^{4}+72k_{2}^{3}k_{3}+82k_{2}^{2}k_{3}^{2}+72k_{2}k_{3}^{3}+$
$\displaystyle 21k_{3}^{4})\eta_{\rm
end}^{4}+10k_{2}^{4}k_{3}^{4}(k_{2}+k_{3})^{2}\eta_{\rm end}^{6}))),$
$\displaystyle\mathcal{G}_{2}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})$
$\displaystyle=$
$\displaystyle-\frac{H^{2}}{8\,\epsilon\,k_{1}\,k_{2}^{5}\,k_{3}^{5}\,k_{T}^{7}\,\eta_{\rm
end}^{6}}(k_{1}^{2}-k_{2}^{2}-k_{3}^{2})(k_{2}^{2}k_{3}^{2}k_{T}^{4}\eta_{\rm
end}^{2}(-9(k_{1}^{2}+3k_{1}(k_{2}+k_{3})+$ (A.2) $\displaystyle
2(k_{2}^{2}+3k_{2}k_{3}+k_{3}^{2}))+3(3k_{1}k_{2}k_{3}(k_{2}+k_{3})+k_{1}^{2}(k_{2}^{2}+k_{3}^{2})-(k_{2}^{2}+k_{3}^{2})(k_{2}^{2}+$
$\displaystyle 3k_{2}k_{3}+k_{3}^{2}))\eta_{\rm
end}^{2}+(3k_{1}k_{2}^{3}(k_{1}+k_{2})^{2}+9k_{1}k_{2}^{3}(k_{1}+k_{2})k_{3}+k_{2}^{2}(k_{1}+k_{2})$
$\displaystyle(8k_{1}+k_{2})k_{3}^{2}+3k_{1}(k_{1}^{2}+3k_{1}k_{2}+3k_{2}^{2})k_{3}^{3}+(6k_{1}^{2}+9k_{1}k_{2}+k_{2}^{2})k_{3}^{4}+3k_{1}k_{3}^{5})$
$\displaystyle\eta_{\rm
end}^{4}+k_{2}^{2}k_{3}^{2}(k_{1}+k_{2}+k_{3})(2k_{1}k_{2}(k_{1}+k_{2})+(k_{1}+k_{2})(2k_{1}+k_{2})k_{3}+(2k_{1}+$
$\displaystyle k_{2})k_{3}^{2})\eta_{\rm
end}^{6})-5d(18k_{1}^{3}(3k_{2}^{2}(k_{1}+k_{2})^{2}(k_{1}+5k_{2})+21k_{2}^{2}(k_{1}+k_{2})(k_{1}+5k_{2})k_{3}+$
$\displaystyle(k_{1}+4k_{2})(3k_{1}^{2}+9k_{1}k_{2}+70k_{2}^{2})k_{3}^{2}+7(3k_{1}^{2}+18k_{1}k_{2}+40k_{2}^{2})k_{3}^{3}+3(11k_{1}+$
$\displaystyle
35k_{2})k_{3}^{4}+15k_{3}^{5})+3(21k_{1}^{2}(k_{2}+k_{3})^{2}(k_{2}^{2}+k_{3}^{2})(k_{2}^{4}+2k_{2}^{3}k_{3}+4k_{2}^{2}k_{3}^{2}+2k_{2}k_{3}^{3}+$
$\displaystyle
k_{3}^{4})+2(k_{2}+k_{3})^{2}(k_{2}^{4}-k_{2}^{3}k_{3}+k_{2}^{2}k_{3}^{2}-k_{2}k_{3}^{3}+k_{3}^{4})(3k_{2}^{4}+18k_{2}^{3}k_{3}+38k_{2}^{2}k_{3}^{2}+$
$\displaystyle
18k_{2}k_{3}^{3}+3k_{3}^{4})+3k_{1}^{6}(7k_{2}^{4}+6k_{2}^{2}k_{3}^{2}+7k_{3}^{4})+3k_{1}^{5}(k_{2}+k_{3})(25k_{2}^{4}+24k_{2}^{3}k_{3}+$
$\displaystyle
18k_{2}^{2}k_{3}^{2}+24k_{2}k_{3}^{3}+25k_{3}^{4})+2k_{1}^{3}(k_{2}+k_{3})(k_{2}^{2}+k_{3}^{2})(27k_{2}^{4}+120k_{2}^{3}k_{3}+$
$\displaystyle
209k_{2}^{2}k_{3}^{2}+120k_{2}k_{3}^{3}+27k_{3}^{4})+2k_{1}^{4}(k_{2}^{2}+k_{3}^{2})(48k_{2}^{4}+189k_{2}^{3}k_{3}+242k_{2}^{2}k_{3}^{2}+$
$\displaystyle
189k_{2}k_{3}^{3}+48k_{3}^{4})+k_{1}(k_{2}+k_{3})(15k_{2}^{8}+48k_{2}^{7}k_{3}+34k_{2}^{6}k_{3}^{2}+92k_{2}^{5}k_{3}^{3}+22k_{2}^{4}k_{3}^{4}+$
$\displaystyle
92k_{2}^{3}k_{3}^{5}+34k_{2}^{2}k_{3}^{6}+48k_{2}k_{3}^{7}+15k_{3}^{8}))\eta_{\rm
end}^{2}+k_{T}^{2}(9k_{1}k_{2}^{5}(k_{1}+k_{2})^{4}+45k_{1}k_{2}^{5}$
$\displaystyle(k_{1}+k_{2})^{3}k_{3}+3k_{2}^{4}(k_{1}+k_{2})^{2}(13k_{1}^{2}+24k_{1}k_{2}+2k_{2}^{2})k_{3}^{2}+15k_{2}^{4}(k_{1}+k_{2})$
$\displaystyle(13k_{1}^{2}+2k_{2}^{2})k_{3}^{3}+k_{2}^{2}(39k_{1}^{4}+195k_{1}^{3}k_{2}+50k_{1}^{2}k_{2}^{2}+64k_{1}k_{2}^{3}+26k_{2}^{4})k_{3}^{4}+$
$\displaystyle(9k_{1}^{5}+45k_{1}^{4}k_{2}+150k_{1}^{3}k_{2}^{2}+195k_{1}^{2}k_{2}^{3}+64k_{1}k_{2}^{4}+4k_{2}^{5})k_{3}^{5}+(36k_{1}^{4}+135k_{1}^{3}k_{2}+$
$\displaystyle
189k_{1}^{2}k_{2}^{2}+30k_{1}k_{2}^{3}+26k_{2}^{4})k_{3}^{6}+3(18k_{1}^{3}+45k_{1}^{2}k_{2}+28k_{1}k_{2}^{2}+10k_{2}^{3})k_{3}^{7}+$
$\displaystyle
3(12k_{1}^{2}+15k_{1}k_{2}+2k_{2}^{2})k_{3}^{8}+9k_{1}k_{3}^{9})\eta_{\rm
end}^{4}+k_{2}^{2}k_{3}^{2}k_{T}^{4}(2k_{2}^{2}k_{3}^{2}(k_{2}+k_{3})^{2}+3k_{1}^{3}$
$\displaystyle(k_{2}^{3}+k_{3}^{3})+k_{1}^{2}(6k_{2}^{4}+9k_{2}^{3}k_{3}+22k_{2}^{2}k_{3}^{2}+9k_{2}k_{3}^{3}+6k_{3}^{4})+3k_{1}(k_{2}^{5}+3k_{2}^{4}k_{3}+$
$\displaystyle 3k_{2}k_{3}^{4}+k_{3}^{5}))\eta_{\rm
end}^{6}+k_{1}k_{2}^{4}k_{3}^{4}(k_{2}+k_{3})k_{T}^{6}\eta_{\rm end}^{8})),$
$\displaystyle\mathcal{G}_{3}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})$
$\displaystyle=$
$\displaystyle\frac{9\,H^{2}}{32\,\epsilon\,k_{1}^{3}\,k_{2}^{5}\,k_{T}^{2}k_{3}^{5}\,\eta_{\rm
end}^{4}}(k_{1}^{4}+k_{2}^{4}+6k_{2}^{2}k_{3}^{2}+k_{3}^{4}-2k_{1}^{2}(k_{2}^{2}+k_{3}^{2}))((k_{1}+k_{2})^{2}$
(A.3) $\displaystyle(3(\gamma_{\rm
E}-1)k_{1}^{3}-3k_{1}^{2}k_{2}-k_{2}^{3})+(k_{1}+k_{2})((6\gamma_{\rm
E}-9)k_{1}^{3}-6k_{1}^{2}k_{2}-2k_{2}^{3})k_{3}+$ $\displaystyle(3(\gamma_{\rm
E}-3)k_{1}^{3}-6k_{1}^{2}k_{2}-2k_{1}k_{2}^{2}-2k_{2}^{3})k_{3}^{2}-2(2k_{1}^{2}+k_{1}k_{2}+k_{2}^{2})k_{3}^{3}-$
$\displaystyle
2(k_{1}+k_{2})k_{3}^{4}-k_{3}^{5}+3k_{1}^{3}k_{T}^{2}\,\log[-k_{T}\eta_{\rm
end}]),$ $\displaystyle\mathcal{G}_{4}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf
k}_{3})$ $\displaystyle=$
$\displaystyle\frac{-9\,H^{2}}{32\,\epsilon\,k1k_{2}^{5}k_{T}^{2}k_{3}^{5}\,\eta_{\rm
end}^{4}}(k_{1}-k_{2}-k_{3})(k_{1}+k_{2}-k_{3})(k_{1}-k_{2}+k_{3})$ (A.4)
$\displaystyle(3(k_{1}+k_{2})^{2}(3(\gamma_{\rm E}-1)k_{1}^{2}+3(\gamma_{\rm
E}-2)k_{1}k_{2}-2k_{2}^{2})+9(k_{1}+k_{2})$ $\displaystyle((3\gamma_{\rm
E}-4)k_{1}^{2}+(3\gamma_{\rm E}-7)k_{1}k_{2}-2k_{2}^{2})k_{3}+(3(9\gamma_{\rm
E}-17)k_{1}^{2}+$ $\displaystyle 27(\gamma_{\rm
E}-3)k_{1}k_{2}-22k_{2}^{2})k_{3}^{2}+3((3\gamma_{\rm
E}-10)k_{1}-6k_{2})k_{3}^{3}-6k_{3}^{4}+$ $\displaystyle
9k_{1}k_{T}^{3}\,\log[-k_{T}\eta_{\rm end}]),$
$\displaystyle\mathcal{G}_{5}({\rm\bf k}_{1},{\rm\bf k}_{2},{\rm\bf k}_{3})$
$\displaystyle=$
$\displaystyle-\frac{H^{2}}{32\,\epsilon\,k_{1}^{3}\,k_{2}^{5}\,k_{3}^{5}\,k_{T}^{7}\,\eta_{\rm
end}^{6}}((k_{1}^{2}-k_{2}^{2})^{2}+2(k_{1}^{2}+k_{2}^{2})k_{3}^{2}-3k_{3}^{4})(k_{2}^{2}k_{T}^{4}\eta_{\rm
end}^{2}$
$\displaystyle(9(3(k_{1}+k_{2})^{2}(k_{1}^{2}+k_{1}k_{2}+k_{2}^{2})+9(k_{1}+k_{2})(k_{1}^{2}+k_{1}k_{2}+k_{2}^{2})k_{3}+$
$\displaystyle
2(5k_{1}^{2}+6k_{1}k_{2}+5k_{2}^{2})k_{3}^{2}+6(k_{1}+k_{2})k_{3}^{3}+2k_{3}^{4})+3(6k_{1}^{2}k_{2}^{2}(k_{1}+k_{2})^{2}+$
$\displaystyle
18k_{1}^{2}k_{2}^{2}(k_{1}+k_{2})k_{3}+(3k_{1}^{4}+9k_{1}^{3}k_{2}+26k_{1}^{2}k_{2}^{2}+6k_{1}k_{2}^{3}+2k_{2}^{4})k_{3}^{2}+$
$\displaystyle
3(3k_{1}^{3}+6k_{1}^{2}k_{2}+4k_{1}k_{2}^{2}+2k_{2}^{3})k_{3}^{3}+2(4k_{1}^{2}+3k_{1}k_{2}+3k_{2}^{2})k_{3}^{4}+3(k_{1}+k_{2})k_{3}^{5}+$
$\displaystyle k_{3}^{6})\eta_{\rm
end}^{2}+k_{3}^{2}(3k_{1}^{2}k_{2}^{2}(k_{1}+k_{2})(2k_{1}+k_{2})+9k_{1}^{2}k_{2}^{2}(2k_{1}+k_{2})k_{3}+(3k_{1}^{4}+$
$\displaystyle
9k_{1}^{3}k_{2}+16k_{1}^{2}k_{2}^{2}+3k_{1}k_{2}^{3}+k_{2}^{4})k_{3}^{2}+3(k_{1}+k_{2})(2k_{1}^{2}+k_{1}k_{2}+k_{2}^{2})k_{3}^{3}+$
$\displaystyle(3k_{1}^{2}+2k_{2}^{2})k_{3}^{4})\eta_{\rm
end}^{4}+2k_{1}^{2}k_{2}^{2}k_{3}^{4}(k_{1}+k_{3})(k_{1}+k_{2}+k_{3})\eta_{\rm
end}^{6})-$ $\displaystyle
5d(9k_{1}^{3}(9k_{1}^{5}+63k_{1}^{4}(k_{2}+k_{3})+210k_{1}^{2}(k_{2}+k_{3})(k_{2}^{2}+3k_{2}k_{3}+k_{3}^{2})+$
$\displaystyle
6k_{1}^{3}(29k_{2}^{2}+63k_{2}k_{3}+29k_{3}^{2})+35k_{1}(3k_{2}^{4}+18k_{2}^{3}k_{3}+38k_{2}^{2}k_{3}^{2}+18k_{2}k_{3}^{3}+3k_{3}^{4})+$
$\displaystyle
5(k_{2}+k_{3})(3k_{2}^{4}+18k_{2}^{3}k_{3}+38k_{2}^{2}k_{3}^{2}+18k_{2}k_{3}^{3}+3k_{3}^{4}))+3(3k_{2}^{2}(k_{1}+k_{2})^{4}$
$\displaystyle(3k_{1}^{4}+9k_{1}^{3}k_{2}-4k_{1}^{2}k_{2}^{2}+3k_{1}k_{2}^{3}+k_{2}^{4})+21k_{2}^{2}(k_{1}+k_{2})^{3}(3k_{1}^{4}+9k_{1}^{3}k_{2}-$
$\displaystyle
4k_{1}^{2}k_{2}^{2}+3k_{1}k_{2}^{3}+k_{2}^{4})k_{3}+(k_{1}+k_{2})^{2}(9k_{1}^{6}+45k_{1}^{5}k_{2}+243k_{1}^{4}k_{2}^{2}+477k_{1}^{3}k_{2}^{3}-$
$\displaystyle
196k_{1}^{2}k_{2}^{4}+154k_{1}k_{2}^{5}+56k_{2}^{6})k_{3}^{2}+7(k_{1}+k_{2})(9k_{1}^{6}+45k_{1}^{5}k_{2}+99k_{1}^{4}k_{2}^{2}+$
$\displaystyle
45k_{1}^{3}k_{2}^{3}-4k_{1}^{2}k_{2}^{4}+10k_{1}k_{2}^{5}+8k_{2}^{6})k_{3}^{3}+(150k_{1}^{6}+672k_{1}^{5}k_{2}+1001k_{1}^{4}k_{2}^{2}+$
$\displaystyle
287k_{1}^{3}k_{2}^{3}+21k_{1}k_{2}^{5}+21k_{2}^{6})k_{3}^{4}+(159k_{1}^{5}+441k_{1}^{4}k_{2}+239k_{1}^{3}k_{2}^{2}+42k_{1}^{2}k_{2}^{3}+$
$\displaystyle
21k_{1}k_{2}^{4}+6k_{2}^{5})k_{3}^{5}+21(k_{1}+k_{2})(4k_{1}^{3}+3k_{1}^{2}k_{2}+5k_{1}k_{2}^{2}+k_{2}^{3})k_{3}^{6}+(45k_{1}^{3}+$
$\displaystyle
168k_{1}^{2}k_{2}+266k_{1}k_{2}^{2}+56k_{2}^{3})k_{3}^{7}+14(3k_{1}^{2}+9k_{1}k_{2}+4k_{2}^{2})k_{3}^{8}+21(k_{1}+k_{2})k_{3}^{9}+$
$\displaystyle 3k_{3}^{10})\eta_{\rm
end}^{2}+(k_{1}+k_{2}+k_{3})^{2}(9k_{1}^{2}k_{2}^{4}(k_{1}+k_{2})^{4}+45k_{1}^{2}k_{2}^{4}(k_{1}+k_{2})^{3}k_{3}+$
$\displaystyle
3k_{2}^{2}(k_{1}+k_{2})^{2}(3k_{1}^{4}+9k_{1}^{3}k_{2}+17k_{1}^{2}k_{2}^{2}+3k_{1}k_{2}^{3}+k_{2}^{4})k_{3}^{2}+15k_{2}^{2}(k_{1}+k_{2})$
$\displaystyle(3k_{1}^{4}+9k_{1}^{3}k_{2}-7k_{1}^{2}k_{2}^{2}+3k_{1}k_{2}^{3}+k_{2}^{4})k_{3}^{3}+(9k_{1}^{6}+45k_{1}^{5}k_{2}+114k_{1}^{4}k_{2}^{2}+30k_{1}^{3}k_{2}^{3}+$
(A.5) $\displaystyle
88k_{1}^{2}k_{2}^{4}+5k_{1}k_{2}^{5}+13k_{2}^{6})k_{3}^{4}+(36k_{1}^{5}+135k_{1}^{4}k_{2}+138k_{1}^{3}k_{2}^{2}-60k_{1}^{2}k_{2}^{3}+5k_{1}k_{2}^{4}+$
$\displaystyle
2k_{2}^{5})k_{3}^{5}+(54k_{1}^{4}+135k_{1}^{3}k_{2}+72k_{1}^{2}k_{2}^{2}+60k_{1}k_{2}^{3}+13k_{2}^{4})k_{3}^{6}+3(12k_{1}^{3}+15k_{1}^{2}k_{2}+$
$\displaystyle
5k_{1}k_{2}^{2}+5k_{2}^{3})k_{3}^{7}+3(3k_{1}^{2}+k_{2}^{2})k_{3}^{8})\eta_{\rm
end}^{4}+k_{2}^{2}k_{3}^{2}(k_{1}+k_{2}+k_{3})^{4}(3k_{1}k_{2}^{2}k_{3}^{2}(k_{2}+k_{3})+$
$\displaystyle
k_{2}^{2}k_{3}^{2}(k_{2}+k_{3})^{2}+3k_{1}^{4}(k_{2}^{2}+k_{3}^{2})+3k_{1}^{3}(k_{2}+k_{3})(2k_{2}^{2}+k_{2}k_{3}+2k_{3}^{2})+k_{1}^{2}(3k_{2}^{4}+$
$\displaystyle
9k_{2}^{3}k_{3}-16k_{2}^{2}k_{3}^{2}+9k_{2}k_{3}^{3}+3k_{3}^{4}))\eta_{\rm
end}^{6}+k_{1}^{2}k_{2}^{4}k_{3}^{4}(k_{1}+k_{2}+k_{3})^{6}\eta_{\rm
end}^{8})),$ $\displaystyle\mathcal{G}_{6}({\rm\bf k}_{1},{\rm\bf
k}_{2},{\rm\bf k}_{3})$ $\displaystyle=$
$\displaystyle-\frac{H^{2}}{32\,\epsilon\,k_{1}^{3}\,k_{2}^{5}\,k_{3}^{5}\,k_{T}^{7}\,\eta_{\rm
end}^{6}}((k_{1}^{2}-k_{2}^{2})^{2}+2(k_{1}^{2}+k_{2}^{2})k_{3}^{2}-3k_{3}^{4})(k_{3}^{2}(k_{1}+k_{2}+k_{3})^{4}\eta_{\rm
end}^{2}$ (A.6)
$\displaystyle(9(3k_{1}^{4}+9k_{1}^{3}(k_{2}+k_{3})+2k_{1}^{2}(5k_{2}^{2}+9k_{2}k_{3}+6k_{3}^{2})+3k_{1}(2k_{2}^{3}+4k_{2}^{2}k_{3}+6k_{2}k_{3}^{2}+$
$\displaystyle
3k_{3}^{3})+(k_{2}+k_{3})(2k_{2}^{3}+4k_{2}^{2}k_{3}+6k_{2}k_{3}^{2}+3k_{3}^{3}))+3(k_{2}^{2}(k_{1}+k_{2})(3k_{1}^{3}+6k_{1}^{2}k_{2}+$
$\displaystyle
2k_{1}k_{2}^{2}+k_{2}^{3})+3k_{2}^{2}(3k_{1}^{3}+6k_{1}^{2}k_{2}+2k_{1}k_{2}^{2}+k_{2}^{3})k_{3}+2(3k_{1}^{4}+9k_{1}^{3}k_{2}+13k_{1}^{2}k_{2}^{2}+$
$\displaystyle
6k_{1}k_{2}^{3}+3k_{2}^{4})k_{3}^{2}+6(2k_{1}^{3}+3k_{1}^{2}k_{2}+k_{1}k_{2}^{2}+k_{2}^{3})k_{3}^{3}+2(3k_{1}^{2}+k_{2}^{2})k_{3}^{4})\eta_{\rm
end}^{2}+$ $\displaystyle
k_{2}^{2}(3k_{1}^{2}k_{2}^{2}(k_{1}+k_{2})^{2}+9k_{1}^{2}k_{2}^{2}(k_{1}+k_{2})k_{3}+2(k_{1}+k_{2})(3k_{1}^{3}+6k_{1}^{2}k_{2}+2k_{1}k_{2}^{2}+$
$\displaystyle
k_{2}^{3})k_{3}^{2}+3(k_{1}+k_{2})(3k_{1}^{2}+k_{2}^{2})k_{3}^{3}+(3k_{1}^{2}+k_{2}^{2})k_{3}^{4})\eta_{\rm
end}^{4}+2k_{1}^{2}k_{2}^{4}(k_{1}+k_{2})k_{3}^{2}$
$\displaystyle(k_{1}+k_{2}+k_{3})\eta_{\rm
end}^{6})-5d(9k_{1}^{3}(9k_{1}^{5}+63k_{1}^{4}(k_{2}+k_{3})+210k_{1}^{2}(k_{2}+k_{3})(k_{2}^{2}+$
$\displaystyle
3k_{2}k_{3}+k_{3}^{2})+6k_{1}^{3}(29k_{2}^{2}+63k_{2}k_{3}+29k_{3}^{2})+35k_{1}(3k_{2}^{4}+18k_{2}^{3}k_{3}+38k_{2}^{2}k_{3}^{2}+$
$\displaystyle
18k_{2}k_{3}^{3}+3k_{3}^{4})+5(k_{2}+k_{3})(3k_{2}^{4}+18k_{2}^{3}k_{3}+38k_{2}^{2}k_{3}^{2}+18k_{2}k_{3}^{3}+3k_{3}^{4}))+$
$\displaystyle
3(3k_{2}^{2}(k_{1}+k_{2})^{4}(3k_{1}^{4}+9k_{1}^{3}k_{2}-4k_{1}^{2}k_{2}^{2}+3k_{1}k_{2}^{3}+k_{2}^{4})+21k_{2}^{2}(k_{1}+k_{2})^{3}$
$\displaystyle(3k_{1}^{4}+9k_{1}^{3}k_{2}-4k_{1}^{2}k_{2}^{2}+3k_{1}k_{2}^{3}+k_{2}^{4})k_{3}+(k_{1}+k_{2})^{2}(9k_{1}^{6}+45k_{1}^{5}k_{2}+243k_{1}^{4}k_{2}^{2}+$
$\displaystyle
477k_{1}^{3}k_{2}^{3}-196k_{1}^{2}k_{2}^{4}+154k_{1}k_{2}^{5}+56k_{2}^{6})k_{3}^{2}+7(k_{1}+k_{2})(9k_{1}^{6}+45k_{1}^{5}k_{2}+$
$\displaystyle
99k_{1}^{4}k_{2}^{2}+45k_{1}^{3}k_{2}^{3}-4k_{1}^{2}k_{2}^{4}+10k_{1}k_{2}^{5}+8k_{2}^{6})k_{3}^{3}+(150k_{1}^{6}+672k_{1}^{5}k_{2}+$
$\displaystyle
1001k_{1}^{4}k_{2}^{2}+287k_{1}^{3}k_{2}^{3}+21k_{1}k_{2}^{5}+21k_{2}^{6})k_{3}^{4}+(159k_{1}^{5}+441k_{1}^{4}k_{2}+239k_{1}^{3}k_{2}^{2}+$
$\displaystyle
42k_{1}^{2}k_{2}^{3}+21k_{1}k_{2}^{4}+6k_{2}^{5})k_{3}^{5}+21(k_{1}+k_{2})(4k_{1}^{3}+3k_{1}^{2}k_{2}+5k_{1}k_{2}^{2}+k_{2}^{3})k_{3}^{6}+$
$\displaystyle(45k_{1}^{3}+168k_{1}^{2}k_{2}+266k_{1}k_{2}^{2}+56k_{2}^{3})k_{3}^{7}+14(3k_{1}^{2}+9k_{1}k_{2}+4k_{2}^{2})k_{3}^{8}+$
$\displaystyle 21(k_{1}+k_{2})k_{3}^{9}+3k_{3}^{10})\eta_{\rm
end}^{2}+(k_{1}+k_{2}+k_{3})^{2}(9k_{1}^{2}k_{2}^{4}(k_{1}+k_{2})^{4}+$
$\displaystyle
45k_{1}^{2}k_{2}^{4}(k_{1}+k_{2})^{3}k_{3}+3k_{2}^{2}(k_{1}+k_{2})^{2}(3k_{1}^{4}+9k_{1}^{3}k_{2}+17k_{1}^{2}k_{2}^{2}+3k_{1}k_{2}^{3}+$
$\displaystyle
k_{2}^{4})k_{3}^{2}+15k_{2}^{2}(k_{1}+k_{2})(3k_{1}^{4}+9k_{1}^{3}k_{2}-7k_{1}^{2}k_{2}^{2}+3k_{1}k_{2}^{3}+k_{2}^{4})k_{3}^{3}+(9k_{1}^{6}+$
$\displaystyle
45k_{1}^{5}k_{2}+114k_{1}^{4}k_{2}^{2}+30k_{1}^{3}k_{2}^{3}+88k_{1}^{2}k_{2}^{4}+5k_{1}k_{2}^{5}+13k_{2}^{6})k_{3}^{4}+(36k_{1}^{5}+135k_{1}^{4}k_{2}+$
$\displaystyle
138k_{1}^{3}k_{2}^{2}-60k_{1}^{2}k_{2}^{3}+5k_{1}k_{2}^{4}+2k_{2}^{5})k_{3}^{5}+(54k_{1}^{4}+135k_{1}^{3}k_{2}+72k_{1}^{2}k_{2}^{2}+60k_{1}k_{2}^{3}+$
$\displaystyle
13k_{2}^{4})k_{3}^{6}+3(12k_{1}^{3}+15k_{1}^{2}k_{2}+5k_{1}k_{2}^{2}+5k_{2}^{3})k_{3}^{7}+3(3k_{1}^{2}+k_{2}^{2})k_{3}^{8})\eta_{\rm
end}^{4}+$ $\displaystyle
k_{2}^{2}k_{3}^{2}(k_{1}+k_{2}+k_{3})^{4}(3k_{1}k_{2}^{2}k_{3}^{2}(k_{2}+k_{3})+k_{2}^{2}k_{3}^{2}(k_{2}+k_{3})^{2}+3k_{1}^{4}(k_{2}^{2}+k_{3}^{2})+$
$\displaystyle
3k_{1}^{3}(k_{2}+k_{3})(2k_{2}^{2}+k_{2}k_{3}+2k_{3}^{2})+k_{1}^{2}(3k_{2}^{4}+9k_{2}^{3}k_{3}-16k_{2}^{2}k_{3}^{2}+9k_{2}k_{3}^{3}+$
$\displaystyle 3k_{3}^{4}))\eta_{\rm
end}^{6}+k_{1}^{2}k_{2}^{4}k_{3}^{4}(k_{1}+k_{2}+k_{3})^{6}\eta_{\rm
end}^{8})).$

$\gamma_{\rm E}$ is the Euler-Mascheroni constant and $k_{T}\equiv
k_{1}+k_{2}+k_{3}.$

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