Effective approach to the Antoniadis-Mottola model:
quantum decoupling of the higher derivative terms

Wagno Cesar e Silva ***E-mail address<EMAIL_ADDRESS>and Ilya L. Shapiro
†††E-mail address<EMAIL_ADDRESS>

Departamento de Física, ICE, Universidade Federal de Juiz de Fora,

36036-900, Juiz de Fora, Minas Gerais, Brazil

###### Abstract

We explore the decoupling of massive ghost mode in the $4D$ (four-dimensional)
theory of the conformal factor of the metric. The model was introduced by
Antoniadis and Mottola in [1] and can be regarded as a close analog of the
fourth-derivative quantum gravity. The analysis of the derived one-loop
nonlocal form factors includes their asymptotic behavior in the UV and IR
limits. In the UV (high energy) domain, our results reproduce the Minimal
Subtraction scheme-based beta functions of [1]. In the IR (i.e., at low
energies), the diagrams with massive ghost internal lines collapse into
tadpole-type graphs without nonlocal contributions and become irrelevant. On
the other hand, those structures that contribute to the running of parameters
of the action and survive in the IR, are well-correlated with the divergent
part (or the leading in UV contributions to the form factors), coming from the
effective low-energy theory of the conformal factor. This effective theory
describes only the light propagating mode. Finally, we discuss whether these
results may shed light on the possible running of the cosmological constant at
low energies.

Keywords: Higher derivatives, quantum gravity, massive ghosts, cosmological
constant, decoupling, conformal anomaly

##  1 Introduction

The running of the cosmological constant at low energies represents an
interesting alternative to the numerous models of Dark Energy, as it provides
the equation of state which is close, but not identical to the
$\omega_{\Lambda}=-1$, of the cosmological constant. On the other hand, there
is no full understanding of whether such a running is possible or not, such
that this issue remains uncertain and is a subject of phenomenological
considerations, as discussed in [2, 3] and many subsequent works. The main
difficulty for the thorough theoretical investigation is that the traditional
approach to quantum decoupling [4] implies calculating the nonlocal form
factor (or its equivalent) and taking its low-energy limit. The cosmological
constant acquires physical sense only in curved spacetime and, in principle,
the corresponding form factors have to be built from covariant elements and
analysed in curved space. According to the Appelquist-Carazzone theorem [4],
heavy degrees of freedom decouple in the IR regime, and their loop corrections
are quadratically suppressed. The same effect should hold in curved spacetime,
leading to the corresponding decoupling theorems.

The described program has been fulfilled in a series of papers [2, 5, 6, 7]
where the nonlocal form factors in the vacuum (gravitational) actions were
calculated and analysed. The problem is that, these nonlocal form factors
describe the decoupling, but only for the fourth-derivative terms in the
action. Owing to covariance, the form factors depend on the d’Alembertian
operator $\operatorname{\square}$. The positive powers of this operator give
zero when acting on the cosmological constant and produce surface terms when
acting on the scalar curvature $R$. Let us note that part of the mentioned
papers, Refs. [6, 7], include the discussion of the form factors of surface
terms (see [8] for the latest discussions of the mathematical aspects of the
problem), and there may be even interesting applications of the running of
Newton constant, related to these surface terms. However, it is unclear how
one can gain information about the running of the cosmological constant in the
traditional covariant framework.

The situation changes dramatically if we perform a conformal transformation.
For instance, using the parametrization

$\displaystyle
g_{\mu\nu}\,=\,\frac{\phi^{2}}{M^{2}}\,\exp\big{\\{}\bar{h}_{\mu\nu}\big{\\}},\qquad\exp\big{\\{}\bar{h}_{\mu\nu}\big{\\}}\,=\,\eta_{\mu\nu}+\bar{h}_{\mu\nu}+\frac{1}{2}\,\bar{h}_{\mu\lambda}\bar{h}^{\lambda}_{\,\,\nu}+\frac{1}{3!}\,\bar{h}_{\mu\lambda}\bar{h}^{\lambda}_{\,\,\tau}\bar{h}^{\tau}_{\,\,\nu}+\dots\,\,,$
(1)

with the traceless $\bar{h}_{\mu\nu}$ and constant scale parameter $M$,
transforms the cosmological constant term $\sqrt{-g}$ into $\phi^{4}$ plus
$\phi^{4}\bar{h}^{n}$-vertices. It is known that there is no problem to find
nonlocal form factor and verify IR decoupling for the $\phi^{4}$-term in the
scalar theory [9] and one should expect this to be equally easy in the
gravitational version of the theory.

Unfortunately, the described approach does not constitute a comprehensive
solution of the problem of the running cosmological constant. In particular,
it is not obvious that such a non-covariant running will preserve the
structure of the $\phi^{4}\bar{h}^{n}$-vertices, such that the running can be
safely attributed to the cosmological constant and not to the artificial
scheme of reparametrization. Anyway, the running of the cosmological term in
the $4D$ (four-dimensional) theory of the conformal factor of the metric is an
attractive object of study, starting from the first proposal [10] and its
realization by Antoniadis and Mottola [1]. The model of quantum conformal
factor follows the idea to perform secondary quantization of the anomaly
induced effective action of vacuum. This action appears as a result of
integrating conformal anomaly [11, 12] coming from the quantum effects of
matter fields (see, e.g., [13] for the review or [14] for the textbook level
introduction). The simplest realization of the anomaly induced action is a
theory of a single scalar field with fourth derivatives, on a flat background.
This procedure corresponds “switching off” the $\bar{h}_{\mu\nu}$-mode in the
parametrization (1).

In the paper [1] it was shown that such a model, with additional Einstein-
Hilbert and cosmological terms, is renormalizable and, in particular,
describes the running of the cosmological term. The remaining question is
whether this running holds in the low energy domain or only in the UV. Indeed,
this is a general question that is quite relevant for all higher derivative
models of quantum gravity. These models may be renormalizable [15], or even
superrenormalizable [16] and this enables one to consistently derive the
renormalization group equations for the effective charges. In the $4D$ case,
the beta functions are partially ambiguous [17, 18, 19], while in the six- or
higher-derivative models, all beta functions do not depend on the gauge fixing
conditions [20]. However, in which physical situations the corresponding
running can be applied? The one-loop corrections behind the beta functions
come from the three different types of diagrams: (i) with internal lines of
the massless degrees of freedom (gravitons); (ii) with internal lines of
massive components, i.e., higher derivative ghosts (or ghost-like states,
ghost tachyons, etc) and normal degrees of freedom, typical for the
superrenormalizable models; (iii) with mixed (massless and massive) internal
lines. The standard approach to effective quantum gravity [21] assumes that
only the first and third types of diagrams give relevant contributions in the
IR and that these contributions are the same as in the effective model where
the propagators and vertices are constructed from the action of GR. As a
consequence, the IR limit of an arbitrary model of quantum gravity corresponds
to the quantum GR, which does not have massive degrees of freedom. As with all
reasonable assumptions, this statement has to be verified. A relevant
question, posed in [22] and, in a more explicit form, in [23], was whether it
is possible that, instead of quantum GR, the IR theory of quantum gravity may
be based on some other theory, e.g., based on a nonlocal action, such that
there are still no massive degrees of freedom propagating in the IR. Answering
this question requires making one-loop calculation in the momentum-dependent
scheme of renormalization, in a theory with higher derivatives. Such
calculation is very complicated and it looks reasonable to consider a toy
model which possesses the same main features (i.e., massive and massless
particle contents owing to higher derivatives and non-polynomial
interactions). This kind of a model would enable one to perform necessary
calculation in a more economic way.

It is easy to note that the theory of quantum conformal factor [1] represents
a nearly perfect toy model for the fourth-derivative quantum gravity. The
Lagrangian of this theory includes non-polynomial interactions in the two-
derivative and zero-derivative sectors, similar to the fourth derivative
quantum gravity. This means, the general structure of the relevant diagrams
includes all the aforementioned (i), (ii) and (iii)-types. Regardless the
calculation of the form factors in the momentum subtraction scheme in the
theory [1] are rather involved (as the reader may see in what follows), they
are still alleviated compared to the ones in a full version of quantum
gravity, where one has to face more extensive set of degrees of freedom and
complicated tensor structures, typical for diagrammatic treatment of quantum
gravity.

In the present work, we report on the derivation of nonlocal form factors in
the fourth-derivative model of quantum conformal factor and perform the
analysis of the UV and IR asymptotic behaviour of these quantum corrections.
It is important to note that the effective approach to the theory of conformal
factor induced by anomaly has an independent interest. In the recent paper
[24], it was shown that this theory provides, in the effective approach, a
propagation of a scalar mode of the gravitational field, which is not present
in GR. In our opinion, the investigation of quantum IR decoupling is useful
for a better general understanding of this model in the effective framework.

The paper is organized as follows: In sect. 2, we briefly review the four-
derivative model for the conformal factor and present the derivation of its UV
divergences using the heat-kernel method. The corresponding expression will be
used, in what follows, as a reference to verify the main result in the UV. In
sect. 3, we formulate the elements of Feynman technique, i.e., the propagator
and vertices for the model, and consider the diagrams producing ultraviolet
(UV) divergences. Furthermore, we derive the one-loop corrections, including
the nonlocal form factors in the propagator sector. Section 4 includes a
description of the asymptotic behavior of nonlocal contributions to the two-
point function in the UV and IR limits. In sect. 5 we discuss the connection
between the momentum dependence in the IR regime of the fundamental theory and
the divergences in the effective low-energy model containing only the light
(massless) mode. As usual in massless theories, the divergences define not
only UV, but also the IR behaviour of the theory and can be used for
comparison with the IR limit of the full theory. In Sec. 6 we present a
discussion of the implications of the IR decoupling for the cosmological
constant problems. Finally, in Sect. 7, we draw our conclusions and discuss
the possibilities of a subsequent work. The four Appendices complement the
main text. In the Appendix A, one can find the set of the Feynman diagrams
used in our calculations, while in Appendix B, we collect intermediate
formulas concerning the calculation of Feynman integrals in dimensional
regularization. Contributions to the two-point function from divergence-free
diagrams are shown in Appendix C, and in Appendix D we present the complete
expressions of the one-loop quantum corrections to the three- and four-point
vertices.

The notations include the Minkowski signature $(+,-,-,-)$. Also, to reduce the
size of the formulas, we avoid indicating the $+i\epsilon$ in the denominators
of the propagators. Indeed, the loop calculations were performed in Euclidean
signature.

##  2 The model

Let us start with a brief review of the model which we shall use in the
present work. The action of the model is the simplest form of the solution of
the anomaly-induced action [11, 12], with the flat fiducial metric, plus the
Einstein-Hilbert and cosmological terms,

$\displaystyle S_{\textrm{cf}}\,=\,\int
d^{4}x\left\\{2b(\square\sigma)^{2}-\left(2w+2b+3c\right)\left[\square\sigma+(\partial\sigma)^{2}\right]^{2}+\frac{3}{\kappa}e^{2\sigma}(\partial\sigma)^{2}-\frac{\Lambda}{\kappa}e^{4\sigma}\right\\}.\quad$
(2)

Here $\kappa=8\pi G$ and the coefficients $w$, $b$ and $c$ are the one-loop
semiclassical beta functions in the vacuum sector,

$\displaystyle w=\frac{1}{120(4\pi)^{2}}(N_{s}+6N_{f}+12N_{v}),$
$\displaystyle b=-\frac{1}{360(4\pi)^{2}}(N_{s}+11N_{f}+62N_{v}),$
$\displaystyle c=\frac{1}{180(4\pi)^{2}}(N_{s}+6N_{f}-18N_{v}),$ (3)

where $N_{s},\,N_{f},\,N_{v}$ are the multiplicities of the quantum conformal
matter fields of spins zero, one-half and one, respectively. The trace anomaly
which produces the induced part of the action (2) is

$\displaystyle\langle
T^{\mu}_{\,\,\mu}\rangle\,=\,-\big{(}wC^{2}+bE_{4}+c\,{\square}R\big{)}.$ (4)

The coefficient $c$ can be modified by adding a finite local term $R^{2}$ to
the action $S_{anom}$ (see [13, 25, 26, 27] for detailed discussion). This
feature will not affect our considerations, especially because we will not
need particular versions of the beta functions (3) and concentrate on the
general features of the quantum theory of conformal factor based on (2).

On top of induced part, the action includes Einstein-Hilbert and cosmological
terms, which are not renormalized at the initial semiclassical theory, but
become very relevant at the second stage, when we quantize the conformal
factor.

The idea that the conformal factor can be quantum, despite it emerges as an
effective mode in the integration of matter fields, comes from Polyakov’s
approach in $2D$, related to string theory [28]. The idea of using the
equivalent metric-scalar (Liouville) model as the basis of $2D$ quantum
gravity was quite popular in 90-s. The use of the analogous theory (in curved
spacetime) as a model for $4D$ quantum gravity was proposed in [10]. In four
dimensions, the theory for the conformal factor can be regarded as a truncated
version of the four-derivative quantum gravity at large distances (i.e., for
the low energies, or IR), providing a screening mechanism for the cosmological
constant [1]. An important difference with the $2D$ induced gravity is that,
in $4D$ one can add the classical terms. Alternatively, one can make the
Einstein-Hilbert and cosmological terms to be generated in the scheme of
induced gravity [29], but this requires an independent scalar field and does
not fit our purpose to construct a simplified model to explore the decoupling
in a higher derivative quantum gravity.

As any fourth-derivative quantum gravity model, the model of our interest has
massive modes, which can be ghosts and tachyonic ghosts.111 It is worth
mentioning that in Ref. [30] it was argued that ghosts are eliminated in the
pure anomaly theory, which is equivalent to (2), by imposition of the
constraints of diffeomorphism invariance. This may be an indication that in
any theory of gravity, diffeomorphism invariance and the constraints they
impose must play a role. The question of our interest is what happens with the
contributions of these massive modes at low energies.

It proves useful to introduce notations similar to [1],

$\displaystyle\theta^{2}\equiv(2w+3c),\;\;\;\zeta\equiv\big{(}2w+2b+3c\big{)},\qquad\gamma\equiv\dfrac{3}{\kappa}\,,\qquad\textrm{and}\qquad\lambda\equiv\dfrac{\Lambda}{\kappa},$
(5)

such that the action (2) becomes

$\displaystyle S_{\textrm{cf}}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\,\Bigl{\\{}-\theta^{2}(\square\sigma)^{2}-\zeta\left[2(\partial\sigma)^{2}\square\sigma+(\partial\sigma)^{4}\right]+\gamma\,e^{2\sigma}(\partial\sigma)^{2}-\lambda\,e^{4\sigma}\Bigr{\\}}.\;\;\;\;\;$
(6)

The difference in notations with the paper by Antoniadis and Mottola is the
coefficient of the kinetic sector of higher derivative terms $\theta^{2}$,
which is denoted $Q^{2}/(4\pi)^{2}$ in [1]. On top of this, we fix the
anomalous scaling dimension $\alpha=1$ to avoid additional complications of
formulas.

The last two terms in (6) come from the Einstein-Hilbert and cosmological
constant terms. In the IR, these terms dominate over the higher derivative
terms and it proves useful to split the Lagrangian into two terms, i.e.,

$\displaystyle\mathcal{L}_{\textrm{IR}}=\gamma
e^{2\sigma}(\partial\sigma)^{2}-\lambda e^{4\sigma}$ (7)

and

$\displaystyle\mathcal{L}_{4\textrm{der}}=-\theta^{2}(\square\sigma)^{2}-\zeta\left[2(\partial\sigma)^{2}\square\sigma+(\partial\sigma)^{4}\right].$
(8)

Our plan is to evaluate the quantum corrections in full theory (6) and,
separately, for the theory based on the IR-term (7). Due to the presence of
higher derivative terms, the one-loop divergences in the full theory are
obtained using the generalized Schwinger-DeWitt technique [17, 31].

Using the background field method, the conformal factor is decomposed into
classical $\sigma$ and quantum $\rho$ counterparts,
$\sigma\rightarrow\sigma+\rho$. Then we obtain the bilinear in the quantum
field forms for the two terms,

$\displaystyle\mathcal{\mathcal{S}}_{4\textrm{der}}^{(2)}$ $\displaystyle=$
$\displaystyle-\int
d^{4}x\Big{\\{}\theta^{2}(\square\rho)^{2}+2\zeta\big{[}(\partial\rho)^{2}\square\sigma+2(\partial_{\mu}\rho)(\partial^{\mu}\sigma)\square\rho+(\partial\sigma)^{2}(\partial\rho)^{2}$
(9)
$\displaystyle+\,\,2(\partial_{\mu}\rho)(\partial_{\nu}\rho)(\partial^{\mu}\sigma)(\partial^{\nu}\sigma)\big{]}\Big{\\}}$

and

$\displaystyle\mathcal{\mathcal{S}}_{IR}^{(2)}=\int d^{4}x\Big{\\{}\gamma
e^{2\sigma}\big{[}(\partial\rho)^{2}+4\rho(\partial_{\mu}\rho)(\partial^{\mu}\sigma)+2\rho^{2}(\partial\sigma)^{2}\big{]}-8\lambda\rho^{2}e^{4\sigma}\Big{\\}}.$
(10)

The Hermitian forms for the structures (9) and (10) are obtained as

$\displaystyle\frac{\delta^{2}\mathcal{\mathcal{S}}_{4\textrm{der}}^{(2)}}{\delta\rho(y)\delta\rho(z)}$
$\displaystyle=$
$\displaystyle-\,2\theta^{2}\square^{2}+4\zeta\big{[}2(\square\sigma)\square-2\partial^{\mu}(\partial^{\nu}\sigma)\partial_{\mu}\partial_{\nu}+4(\partial_{\nu}\sigma)\partial^{\mu}(\partial^{\nu}\sigma)\partial_{\mu}$
$\displaystyle+\,\,(\partial\sigma)^{2}\square+2(\square\sigma)(\partial^{\mu}\sigma)\partial_{\mu}+2(\partial^{\mu}\sigma)(\partial^{\nu}\sigma)\partial_{\mu}\partial_{\nu}\big{]},$
$\displaystyle\frac{\delta^{2}\mathcal{\mathcal{S}}_{IR}^{(2)}}{\delta\rho(y)\delta\rho(z)}$
$\displaystyle=$ $\displaystyle-\,\,2\gamma
e^{2\sigma}\big{[}\square+2(\partial^{\mu}\sigma)\partial_{\mu}+2(\partial\sigma)^{2}+2\square\sigma\big{]}-16\lambda
e^{4\sigma}.$ (11)

So, for the complete model (6), we have

$\displaystyle\frac{\delta^{2}\mathcal{\mathcal{S}}^{(2)}}{\delta\rho(y)\delta\rho(z)}=-2\theta^{2}\hat{H},$
(12)

where the self-adjoint four-derivative minimal operator is

$\displaystyle\hat{H}=\square^{2}+V^{\mu\nu}\partial_{\mu}\partial_{\nu}+N^{\mu}\partial_{\mu}+U,$
(13)

with the elements

$\displaystyle
V^{\mu\nu}=-\frac{2\zeta}{\theta^{2}}\big{[}2\eta^{\mu\nu}\square\sigma-2\partial^{\mu}\partial^{\nu}\sigma+\eta^{\mu\nu}(\partial\sigma)^{2}+2(\partial^{\mu}\sigma)\partial^{\nu}\sigma\big{]}+\frac{\gamma}{\theta^{2}}e^{2\sigma}\eta^{\mu\nu},$
$\displaystyle
N^{\mu}=-\frac{4\zeta}{\theta^{2}}\big{[}2(\partial_{\nu}\sigma)\partial^{\mu}\partial^{\nu}\sigma+(\square\sigma)\partial^{\mu}\sigma\big{]}+\frac{2\gamma}{\theta^{2}}e^{2\sigma}(\partial^{\mu}\sigma),$
$\displaystyle
U=\frac{2\gamma}{\theta^{2}}e^{2\sigma}\big{[}(\partial\sigma)^{2}+\square\sigma\big{]}+\frac{8\lambda}{\theta^{2}}e^{4\sigma}.$
(14)

Using the standard algorithm for the fourth-order operators [17, 31], we
arrive at the expression for the divergences

$\displaystyle\bar{\Gamma}_{\textrm{div}}^{(1)}=-\frac{1}{\varepsilon}\int
d^{4}x\left\\{\frac{5\zeta^{2}}{\theta^{4}}\left[\square\sigma+(\partial\sigma)^{2}\right]^{2}+\frac{\gamma}{\theta^{2}}\Big{(}\frac{3\zeta}{\theta^{2}}+2\Big{)}(\partial\sigma)^{2}e^{2\sigma}-\Big{(}\frac{8\lambda}{\theta^{2}}-\frac{\gamma^{2}}{2\theta^{4}}\Big{)}e^{4\sigma}\right\\},$
(15)

where we introduce the useful notation $\varepsilon=(4\pi)^{2}(n-4)$ and
neglect the irrelevant surface terms. This result agrees with the previous
calculations [1, 32], except for an apparent misprint in the sign of Eq. (4)
of [32]. An additional verification can be found in the recent paper [33].

In that follows, we shall confirm the expression (15) by the calculation of
both divergent and finite nonlocal (leading logarithms) parts of the Feynman
diagrams. By considering the minimal subtraction (MS) scheme, one can easily
derive UV $\beta$-functions for the theory (6). In the next section, we will
determine the finite parts of the one-loop diagrams that produce these
divergences. In this case, the structure (15) will be useful in identifying
the diagrams that are relevant for our purposes. For example, from the
coefficients of the terms $(\partial\sigma)^{2}e^{2\sigma}$ and $e^{4\sigma}$,
we can expect that diagrams involving interaction vertices $\gamma$ and
$\gamma\zeta$ provide contributions to the Einstein-Hilbert sector, while
diagrams with vertices $\lambda$ and $\gamma^{2}$ are associated with
corrections to the cosmological constant sector. Later on, we shall see that
this identification may not hold in the effective low-energy model with two
derivatives.

For completeness, we also derived the divergences of the effective theory,
based on the IR-term, Eq. (7), separately. The result is

$\displaystyle\bar{\Gamma}_{\textrm{div,\,IR}}^{(1)}=-\frac{1}{\varepsilon}\int
d^{4}x\,\bigg{\\{}\frac{1}{2}\big{[}\square\sigma+(\partial\sigma)^{2}\big{]}^{2}-\frac{8}{3}\Lambda\,e^{2\sigma}(\partial\sigma)^{2}+\frac{32}{9}\Lambda^{2}e^{4\sigma}\bigg{\\}}.$
(16)

As it should be expected from the power counting, the fourth-derivative
counterterms are required in this theory, as it is non-renormalizable. At the
same time, neglecting the fourth-derivative terms according to the effective
approach, we arrive at the reference expression to compare with the IR limit
of the full theory.

##  3 One-loop corrections from Feynman diagrams

In a model with higher derivatives, to explore the decoupling in the loop
corrections, one has to separate massive and massless degrees of freedom. In
many cases, this can be achieved by introducing auxiliary fields (see, e.g.,
[34]). However, in the case of the theory (6), this approach is not
operational owing to our interest in the quantum corrections in the theory
that have higher derivatives in both kinetic terms and the interactions. Thus,
we shall make the separation at the level of the propagator and vertices in
the Feynman diagrams, i.e., use the method close to the one of [35].

The structure of the vertices and the propagator for the fundamental theory
(6) can be calculated by using the parametrization
$\sigma\rightarrow\sigma+\rho$, where $\rho$ is a small perturbation and
expanding the exponential terms in the power series in $\rho$. Collecting the
quadratic terms, we find that the propagator satisfies the equation

$\displaystyle
2\big{[}\theta^{2}\square^{2}+\gamma\square+8\lambda\big{]}G(x,y)\,=\,i\delta^{4}(x-y).$
(17)

Making the Fourier transform,

$\displaystyle
G(x,y)=\int\frac{d^{4}k}{(2\pi)^{4}}\,e^{-ik\cdot(x-y)}\widetilde{G}(k)$ (18)

and assuming $\Lambda\ll\gamma^{2}/\theta^{2}$, we get

$\displaystyle\widetilde{G}(k)\,\,=\,\,\frac{i}{2[\theta^{2}k^{4}-\gamma
k^{2}+8\lambda]}\,\,\simeq\,\,\frac{i}{2\,\theta^{2}\big{(}k^{2}-\frac{\gamma}{\theta^{2}}\big{)}\big{(}k^{2}-\frac{8}{3}\Lambda\big{)}}.$
(19)

Finally, in the same approximation, the propagator can be written as

$\displaystyle\widetilde{G}(k)\,\,=\,\,\frac{i}{2\,\theta^{2}\big{(}m^{2}-M^{2})}\bigg{[}\frac{1}{k^{2}-m^{2}}-\frac{1}{k^{2}-M^{2}}\bigg{]}.$
(20)

It is easy to identify a healthy degree of freedom with the mass
$m^{2}=8\Lambda/3$ and a ghostly mode with the Planck-scale mass,
$M^{2}=\gamma/\theta^{2}$.

We need to consider only those interaction vertices that are relevant for the
one-loop corrections to the propagator. The vertices for the $3$\- and
$4$-point functions arise from the derivative interaction terms in the part
$\mathcal{L}_{4\textrm{der}}$ and from the higher order terms in the
exponential expansion in $\mathcal{L}_{\textrm{IR}}$,

$\displaystyle\zeta\big{[}2(\partial\rho)^{2}\square\rho+(\partial\rho)^{4}\big{]},\qquad
2\gamma(\partial\rho)^{2}\big{[}\rho+\rho^{2}\big{]},,\qquad\frac{32\lambda}{3}\big{[}\rho^{3}+\rho^{4}\big{]}.$
(21)

> Figure 1: Feynman diagrams associated with the interaction vertices. The
> primes denotes derivatives acting on the propagators.

In Fig. 1, we have presented the vertices corresponding to these interaction
terms. The analytic expressions have the form

$\displaystyle V^{(3)}_{\zeta}(p,k,q)=-4i\zeta\big{[}p^{2}(k\cdot
q)+q^{2}(k\cdot p)+k^{2}(p\cdot q)\big{]},$ $\displaystyle
V^{(3)}_{\gamma}(p,k,q)=2i\gamma(k^{2}+p^{2}+q^{2}),$ $\displaystyle
V^{(3)}_{\lambda}=-64i\lambda$ (22)

and

$\displaystyle V^{(4)}_{\zeta}(p,k,q,r)=-8i\zeta\big{[}(k\cdot q)(p\cdot
r)+(k\cdot p)(q\cdot r)+(k\cdot r)(p\cdot q)\big{]},$ $\displaystyle
V^{(4)}_{\gamma}(p,k,q,r)=4i\gamma(k^{2}+p^{2}+q^{2}+r^{2}),$ $\displaystyle
V^{(4)}_{\lambda}=-256i\lambda,$ (23)

where $(k\cdot p)=k_{\mu}p^{\mu}$.

Now we are in the position to determine the one-loop contributions to the
self-energy (correction to the propagator) of the healthy (light) mode. First,
consider the diagrams producing the UV divergences (15), derived previously
using the heat-kernel method. These divergences are responsible for the MS-
scheme based beta functions and serve as the UV references for the complete
expressions.

In the theory (6) the expression for the two-point function is

$\displaystyle
G^{(2)}_{\textrm{1-loop}}(p,-p)\propto\frac{i}{(p^{2}-m^{2})}\big{(}\bar{\Sigma}_{\gamma}+\bar{\Sigma}_{\lambda}+\widetilde{\Sigma}_{\gamma\lambda}+\widetilde{\Sigma}_{\lambda^{2}}+\Sigma_{\zeta^{2}}+\Sigma_{\gamma\zeta}+\Sigma_{\gamma^{2}}+\ldots\big{)}\frac{i}{(p^{2}-m^{2})},\;$
(24)

where the omitted $(\ldots)$ terms denote contributions from the divergence-
free diagrams. On top of this, the finite contributions
$\Sigma_{\lambda\zeta},\;\Sigma_{\gamma\lambda}$ and $\Sigma_{\lambda^{2}}$
are proportional to the mass $m^{2}$ (as shown in Appendix C). As we shall see
in Sec. 4, these terms can be neglected in our analysis of the IR limit, when
we consider $m^{2}=0$.

We can write the general symbolic expression for the self-energy function, in
the second-order (in coupling constants) approximation as presented in Fig. 2,
for corrections of the type $\Sigma$, and in Fig. 3, for corrections
$\widetilde{\Sigma}$.

> Figure 2: General structure of one-loop diagrams for self-energy functions
> $\Sigma$. Solid lines indicate light degrees of freedom while dashed lines
> stand for the massive ghosts.

> Figure 3: “Tadpole” diagrams associated with the corrections
> $\widetilde{\Sigma}$.

Furthermore, in Figs. 2 and 3, the expressions for the light, ghost and mixed
sectors are, respectively,

$\displaystyle\Sigma_{\textrm{light}}\propto\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{i}{(k^{2}-m^{2})}V^{(3)}(p,-k,k-p)\frac{i}{\big{[}(k-p)^{2}-m^{2}\big{]}}V^{(3)}(-p,k,p-k),\mbox{\qquad}$
$\displaystyle\Sigma_{\textrm{ghost}}\propto\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{i}{(k^{2}-M^{2})}V^{(3)}(p,-k,k-p)\frac{i}{\big{[}(k-p)^{2}-M^{2}\big{]}}V^{(3)}(-p,k,p-k),\mbox{\qquad}$
$\displaystyle\Sigma_{\textrm{mixed}}\propto\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{i}{(k^{2}-m^{2})}V^{(3)}(p,-k,k-p)\frac{i}{\big{[}(k-p)^{2}-M^{2}\big{]}}V^{(3)}(-p,k,p-k),\mbox{\qquad}$
(25)

while the contributions of the first-order in the coupling constants have the
general structure shown in Fig. 4,

> Figure 4: “Snail” diagrams associated with the corrections $\bar{\Sigma}$.
> In this case, obviously, there are no contributions from a mixed sector.

with

$\displaystyle\bar{\Sigma}_{\textrm{light}}\propto\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{i}{(k^{2}-m^{2})}V^{(4)}(p,k,-p,-k),$
$\displaystyle\bar{\Sigma}_{\textrm{ghost}}\propto\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{i}{(k^{2}-M^{2})}V^{(4)}(p,k,-p,-k).$
(26)

The diagrammatic representation of the contributions to the two-point function
with different couplings, is presented in the Appendix A. The diagrams in Fig.
5 correspond to the fifth term in (24), and the last term is associated with
the diagrams in Fig. 7. In Fig. 6 there are shown the diagrams that correspond
to the term proportional to $\gamma\zeta$. The diagrams for the first-order
terms in the couplings $\gamma$ and $\lambda$, are depicted in Figs. 8 and 9,
respectively. In addition, the third and fourth terms in (24) are associated
with the tadpole diagrams with interaction vertices $\gamma\lambda$ shown in
Fig. 10, and $\lambda^{2}$ shown in Fig. 11.

Let us note that each diagram here represents the sum over all the
topologically equivalent diagrams with different permutations over the
external momenta and with all possible placements of derivatives on the
internal and external lines. On top of that, we omitted some tadpole-type
diagrams that do not contribute to $G^{(2)}(p,-p)$, as they include
derivatives of the propagator in a single spacetime point, and hence vanish.

To evaluate the integrals in (24) we used dimensional regularization. In the
model under consideration, this requires extending the standard list of
divergent expressions [36] for the integrals in the spacetime of $2\omega$
complex dimensions. The integrals proportional to $\zeta^{2}$, $\gamma\zeta$
and $\gamma^{2}$ read, respectively, as

$\displaystyle\Sigma^{(2\omega)}_{\zeta^{2}}(p)$ $\displaystyle=$
$\displaystyle-\,\frac{8\zeta^{2}}{\theta^{4}(m^{2}-M^{2})^{2}}\int\frac{d^{2\omega}k}{(2\pi)^{2\omega}}\,\Gamma^{(3,3)}_{\zeta^{2}}\biggl{\\{}\frac{2}{(k^{2}-m^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}$
(27)
$\displaystyle-\,\,\frac{1}{(k^{2}-m^{2})\big{[}(k-p)^{2}-m^{2}\big{]}}\,-\,\frac{1}{(k^{2}-M^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}\biggr{\\}},$
$\displaystyle\Sigma^{(2\omega)}_{\gamma\zeta}(p)$ $\displaystyle=$
$\displaystyle-\,\frac{4\gamma\zeta}{\theta^{4}(m^{2}-M^{2})^{2}}\int\frac{d^{2\omega}k}{(2\pi)^{2\omega}}\,\Gamma^{(3,3)}_{\gamma\zeta}\biggl{\\{}\frac{2}{(k^{2}-m^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}$
(28)
$\displaystyle-\,\,\frac{1}{(k^{2}-m^{2})\big{[}(k-p)^{2}-m^{2}\big{]}}\,-\,\frac{1}{(k^{2}-M^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}\biggr{\\}}$

and

$\displaystyle\Sigma^{(2\omega)}_{\gamma^{2}}(p)$ $\displaystyle=$
$\displaystyle-\,\frac{2\gamma^{2}}{\theta^{4}(m^{2}-M^{2})^{2}}\int\frac{d^{2\omega}k}{(2\pi)^{2\omega}}\,\Gamma^{(3,3)}_{\gamma^{2}}\biggl{\\{}\frac{2}{(k^{2}-m^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}$
(29)
$\displaystyle-\,\,\frac{1}{(k^{2}-m^{2})\big{[}(k-p)^{2}-m^{2}\big{]}}-\frac{1}{(k^{2}-M^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}\biggr{\\}},$

where we used the following combinations of the vertex factors:

$\displaystyle\Gamma^{(3,3)}_{\zeta^{2}}\,=\,p^{4}k^{4}-2p^{2}k^{2}(p\cdot
k)^{2}+(p\cdot k)^{4},$
$\displaystyle\Gamma^{(3,3)}_{\gamma\zeta}\,=\,p^{4}k^{2}-(p^{2}+k^{2})(p\cdot
k)^{2}+(p\cdot k)^{3}-p^{2}k^{2}(p\cdot k)+k^{4}p^{2},$
$\displaystyle\Gamma^{(3,3)}_{\gamma^{2}}\,=\,p^{4}+k^{4}+(p\cdot
k)^{2}+2p^{2}k^{2}-2(p^{2}+k^{2})(p\cdot k).$ (30)

The results of the integrations in the Euclidean space are222Some intermediate
details of the calculations can be found in Appendix B.1. The calculations
were verified using the Package-$X$ [37] in Mathematica [38].

$\displaystyle\Sigma_{\zeta^{2}}(p)$ $\displaystyle=$
$\displaystyle\frac{i\zeta^{2}p^{4}}{(4\pi)^{2}\theta^{4}}\,\bigg{\\{}5\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{m^{2}}\Big{)}\bigg{]}-\frac{1}{4}\big{[}9A^{2}-5(ab)^{2}-37\big{]}-\frac{1}{2(ab)^{2}c^{5}}\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}$
(31)
$\displaystyle-\,\frac{1}{2(ab)^{2}d^{5}}\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}-\bigg{[}\frac{1}{2}(ab)^{3}+\frac{5}{2}ab\Big{(}ab+\frac{a}{2}+2\Big{)}+\frac{15a}{4}\Big{(}1+\frac{1}{4b}\Big{)}$
$\displaystyle+\,5\Big{(}2+\frac{3}{4b}+\frac{1}{2ab}\Big{)}\bigg{]}\,\mbox{ln}\,(1+4b)+\frac{A^{5}}{2(ab)^{2}}\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}\bigg{\\}},$
$\displaystyle\Sigma_{\gamma\zeta}(p)$ $\displaystyle=$
$\displaystyle-\,\frac{i\gamma\zeta
p^{2}}{(4\pi)^{2}\theta^{4}}\,\bigg{\\{}3\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{m^{2}}\Big{)}\bigg{]}+\frac{1}{2c(ab)^{2}}\bigg{[}\frac{a}{2}\Big{(}2-\frac{1}{c^{2}}\Big{)}+1\bigg{]}\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}$
(32)
$\displaystyle+\,\frac{1}{2d(ab)^{2}}\bigg{[}\frac{a}{2}(4b+1)\Big{(}2-\frac{1}{d^{2}}\Big{)}+1\bigg{]}\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}-\Big{(}ab+\frac{a}{2}-6\Big{)}$
$\displaystyle-\,\frac{A}{2}\bigg{[}\Big{(}\frac{1}{ab}-ab\Big{)}\Big{(}1+\frac{1}{2b}+\frac{1}{ab}\Big{)}-\frac{2}{b}\Big{(}1+\frac{1}{4b}\Big{)}\bigg{]}\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}$
$\displaystyle-\,\bigg{[}\frac{ab}{2}\Big{(}ab+\frac{a}{2}+2\Big{)}+3\Big{(}1+\frac{a}{2}\Big{)}\Big{(}1+\frac{1}{4b}\Big{)}-\frac{1}{ab}\bigg{]}\,\mbox{ln}\,(1+4b)\bigg{\\}},$
$\displaystyle\Sigma_{\gamma^{2}}(p)$ $\displaystyle=$
$\displaystyle\frac{i\gamma^{2}}{(4\pi)^{2}\theta^{4}}\,\bigg{\\{}2\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{m^{2}}\Big{)}\bigg{]}+3-\frac{1}{2c(ab)^{2}}\Big{(}\frac{a^{2}}{4}-\frac{1}{c^{2}}+2\Big{)}\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}$
(33)
$\displaystyle+\,\frac{A}{2(ab)^{2}}\Big{(}ab+\frac{a}{2}-1\Big{)}^{2}\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}-\frac{1}{2}\bigg{[}ab+a\Big{(}1+\frac{1}{4b}\Big{)}+3$
$\displaystyle+\,\frac{1}{2b}\Big{(}1-\frac{2}{a}\Big{)}\bigg{]}\,\mbox{ln}\,(1+4b)-\frac{1}{2d(ab)^{2}}\bigg{[}\frac{a^{2}}{4}(4b+1)^{2}-\frac{1}{d^{2}}+2\bigg{]}\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}\bigg{\\}},\hskip
22.76228pt$

where

$\displaystyle\dfrac{1}{\epsilon}\equiv\dfrac{1}{2-\omega}-\gamma_{\textrm{E}}+\,\mbox{ln}\,(4\pi),\qquad
a=\dfrac{4m^{2}}{p^{2}},\qquad b=\dfrac{M^{2}-m^{2}}{4m^{2}}$ (34)

and $\gamma_{\textrm{E}}\approx 0.577$ is the Euler-Mascheroni constant. In
the limit $\omega\rightarrow 2$, the results (31), (32) and (33) represent
divergent and finite parts. Note that the finite part of these expressions has
a very complicated dependence on the external momentum. For these nonlocal
structures, in the mixed sector, we used the notation

$\displaystyle A\,=\,\sqrt{(1+ab)^{2}+a}.$ (35)

Furthermore, the notations used in the light and ghost sectors, include,
respectively,

$\displaystyle c^{2}=\dfrac{p^{2}}{p^{2}+4m^{2}},\qquad
d^{2}=\dfrac{p^{2}}{p^{2}+4M^{2}}.$ (36)

It is worth explaining how the definitions (35) and (36) can be used in
identifying contributions coming from the different sectors. E.g., the
logarithmic form factors involving $c$, in the results presented above, come
from the “pure” loops with only the propagators of the light degrees of
freedom (equivalently, for the other momentum-dependent logarithmic
structures). Of course, there are terms that result from the combination of
contributions from the three sectors (i.e., light, massive ghost and mixed),
such as $\,\mbox{ln}\,(1+4b)$ and those with only polynomial dependencies on
the external momentum.

In top of this, the relevant corrections involving the quartic vertices, are
given by

$\displaystyle\bar{\Sigma}_{\gamma}(p)\,=\,\frac{2i\gamma
p^{2}}{\theta^{2}}\,I_{\textrm{quad}}\qquad\mbox{and}\qquad\bar{\Sigma}_{\lambda}(p)=\frac{64i\lambda}{\theta^{2}}\,I_{\textrm{quad}},$
(37)

where the integral is

$\displaystyle I_{\textrm{quad}}$ $\displaystyle=$
$\displaystyle\frac{1}{(m^{2}-M^{2})}\int\frac{d^{2\omega}k}{(2\pi)^{2\omega}}\,\,\biggl{\\{}\frac{1}{(k^{2}+m^{2})}-\frac{1}{(k^{2}+M^{2})}\biggr{\\}}$
(38) $\displaystyle=$
$\displaystyle-\,\frac{1}{(4\pi)^{2}}\,\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{m^{2}}\Big{)}\,+\,1\,-\,\frac{M^{2}}{(m^{2}-M^{2})}\,\mbox{ln}\,\Big{(}\frac{m^{2}}{M^{2}}\Big{)}\bigg{]}.$

The results (37) are tadpole-type contributions, which do not produce a
nonlocal form factor. Therefore, these corrections are not relevant to our
analysis at low energies and were included just of completeness. Of course,
the same consideration applies to second-order tadpole-type corrections, which
in principle contribute to divergences in the Einstein-Hilbert and
cosmological constant sectors,

$\displaystyle\widetilde{\Sigma}_{\gamma\lambda}(p)\,=\,-\frac{16i\gamma\lambda
p^{2}}{\theta^{4}m^{2}M^{2}}\,I_{\textrm{quad}}\qquad\mbox{and}\qquad\widetilde{\Sigma}_{\lambda^{2}}(p)=-\frac{512i\lambda^{2}}{\theta^{4}m^{2}M^{2}}\,I_{\textrm{quad}},$
(39)

with $m^{2}M^{2}=8\lambda/\theta^{2}$. Actually, the corrections in (39) can
be disregarded because tadpole diagrams, such as those presented in Figs. 10
and 11, normally are eliminated using renormalization conditions (see, e.g.,
[39] or the Chapter 11 of [40] for more details).

Let us note that here we presented only the results for the self-energy. The
lower-order vertices were also derived and produce qualitatively the same
picture. The nonlocal parts of these contributions follow a standard
logarithmic structure, as those for propagator corrections. Since the
corresponding formulas are relatively bulky, they are separated in Appendix D.

##  4 Asymptotic behavior

In this section, we explore the asymptotic behavior of the one-loop
contributions (31), (32), and (33). Our main interest is to verify how these
expressions interpolate between the UV and IR regions of the fundamental
theory. In this way, we have a chance to understand what happens to the
nonlocal form factors of the contribution of loops with the massive degrees of
freedom (massive ghosts) in the IR.

We start with the limit $p^{2}\rightarrow\infty$ that corresponds to the UV
regime $p^{2}\gg M^{2}\gg m^{2}$. In this case, Eqs. (32), (31), and (33)
simplify and we arrive at the expressions

$\displaystyle\Sigma^{\textrm{UV}}_{\zeta^{2}}(p^{2}\rightarrow\infty)$
$\displaystyle=$
$\displaystyle\frac{i\zeta^{2}p^{4}}{(4\pi)^{2}\theta^{4}}\,\Bigg{\\{}5\bigg{[}\frac{1}{\epsilon}-\,\mbox{ln}\,\bigg{(}\frac{p^{2}}{\mu^{2}}\bigg{)}\bigg{]}+3-\frac{15(M^{2}+m^{2})}{p^{2}}$
(40)
$\displaystyle+\,\,\frac{10(m^{4}+m^{2}M^{2}+M^{4})}{p^{4}}\,\mbox{ln}\,\Big{(}\frac{p^{2}}{M^{2}}\Big{)}+\frac{35(M^{2}+m^{2})}{6p^{4}}$
$\displaystyle+\,\,\frac{40M^{2}m^{2}}{3p^{4}}+\frac{10m^{6}}{p^{4}M^{2}}\,\mbox{ln}\,\Big{(}\frac{m^{2}}{M^{2}}\Big{)}+\mathcal{O}\Big{(}\frac{M^{6}}{p^{6}}\Big{)}\bigg{\\}},$
$\displaystyle\Sigma^{\textrm{UV}}_{\gamma\zeta}(p^{2}\rightarrow\infty)$
$\displaystyle=$ $\displaystyle-\,\frac{i\gamma\zeta
p^{2}}{(4\pi)^{2}\theta^{4}}\,\Bigg{\\{}3\bigg{[}\frac{1}{\epsilon}-\,\mbox{ln}\,\Big{(}\frac{p^{2}}{\mu^{2}}\Big{)}\bigg{]}+7-\frac{9(M^{2}+m^{2})}{p^{2}}$
(41)
$\displaystyle-\,\,\frac{6(M^{2}+m^{2})}{p^{2}}\,\mbox{ln}\,\Big{(}\frac{p^{2}}{M^{2}}\Big{)}-\frac{6m^{4}}{p^{2}M^{2}}\,\mbox{ln}\,\Big{(}\frac{m^{2}}{M^{2}}\Big{)}+\mathcal{O}\Big{(}\frac{M^{4}}{p^{4}}\Big{)}\bigg{\\}},$
$\displaystyle\Sigma^{\textrm{UV}}_{\gamma^{2}}(p^{2}\rightarrow\infty)$
$\displaystyle=$
$\displaystyle\frac{i\gamma^{2}}{(4\pi)^{2}\theta^{4}}\,\Bigg{\\{}2\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\Big{(}\frac{p^{2}}{\mu^{2}}\Big{)}+2\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{M^{2}}\Big{)}\bigg{]}+5$
(42)
$\displaystyle+\,\,\frac{4m^{2}}{M^{2}}\,\mbox{ln}\,\Big{(}\frac{m^{2}}{M^{2}}\Big{)}+\mathcal{O}\Big{(}\frac{M^{2}}{p^{2}}\Big{)}\bigg{\\}}.$

As expected in the UV regime, the leading logarithmic terms in the form
factor, i.e., the terms with $\,\mbox{ln}\,\big{(}p^{2}/\mu^{2}\big{)}$, are
proportional to the corresponding divergences. It is easy to verify that, when
returning to the coordinate representation, the divergent part of the
expressions above, together with (37) in the UV, correspond to the result
(15), obtained from the heat-kernel technique.

On the other hand, assuming $m^{2}=0$ in the formulas of
$\Sigma_{\zeta^{2}},\;\Sigma_{\gamma\zeta}$ and $\Sigma_{\gamma^{2}}$ (see
part B.2 of Appendix B), the analysis of the IR regime $M^{2}\gg p^{2}$ of
these corrections provides

$\displaystyle\Sigma^{\textrm{IR}}_{\zeta^{2}}(M^{2}\gg
p^{2})\bigg{|}_{m^{2}=0}$ $\displaystyle=$
$\displaystyle\frac{i\zeta^{2}p^{4}}{(4\pi)^{2}\theta^{4}}\,\Bigg{\\{}5\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{M^{2}}\Big{)}\bigg{]}-\frac{1}{6}\bigg{(}7+\frac{35p^{2}}{2M^{2}}-\frac{9p^{4}}{2M^{4}}\bigg{)}$
(43)
$\displaystyle+\,\,\frac{p^{4}}{2M^{4}}\,\mbox{ln}\,\Big{(}\frac{M^{2}}{p^{2}}\Big{)}+\mathcal{O}\Big{(}\frac{p^{6}}{M^{6}}\Big{)}\bigg{\\}},$
$\displaystyle\Sigma^{\textrm{IR}}_{\gamma\zeta}(M^{2}\gg
p^{2})\bigg{|}_{m^{2}=0}$ $\displaystyle=$ $\displaystyle-\frac{i\gamma\zeta
p^{2}}{(4\pi)^{2}\theta^{4}}\,\Bigg{\\{}3\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{M^{2}}\Big{)}\bigg{]}-\frac{1}{2}+\frac{2p^{2}}{3M^{2}}$
(44)
$\displaystyle-\,\,\frac{p^{4}}{M^{4}}\bigg{[}\frac{7}{20}-\frac{1}{2}\,\mbox{ln}\,\Big{(}\frac{p^{2}}{M^{2}}\Big{)}\bigg{]}+\mathcal{O}\Big{(}\frac{p^{6}}{M^{6}}\Big{)}\bigg{\\}},$
$\displaystyle\Sigma^{\textrm{IR}}_{\gamma^{2}}(M^{2}\gg
p^{2})\bigg{|}_{m^{2}=0}$ $\displaystyle=$
$\displaystyle\frac{i\gamma^{2}}{(4\pi)^{2}\theta^{4}}\,\Bigg{\\{}2\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\bigg{(}\frac{\mu^{2}}{M^{2}}\bigg{)}\bigg{]}+\frac{13}{6}\frac{p^{2}}{M^{2}}$
(45)
$\displaystyle-\,\,\frac{p^{4}}{2M^{4}}\bigg{[}\frac{8}{5}+\,\mbox{ln}\,\Big{(}\frac{p^{2}}{M^{2}}\Big{)}\bigg{]}+\mathcal{O}\bigg{(}\frac{p^{6}}{M^{6}}\bigg{)}\bigg{\\}}.$

The last formulas show that, in the IR limit, the divergences and momentum
dependence do not correlate with each other, exactly as it is expected [4]
(see also [2, 5, 7] for the semiclassical theory). We have found that this
basic feature holds also for the “mixed” diagrams, such that the Appelquist-
Carazzone theorem is valid for the fourth-derivative model with non-polynomial
interactions. In the expressions (43) and (44), the nonlocal part with
logarithmic form factor $\,\mbox{ln}\,\big{(}p^{2}/M^{2}\big{)}$ is suppressed
by powers of $M^{2}$, whereas in (45) this is not the case, as the factor
$\gamma^{2}$ (remember that $\gamma=\theta^{2}M^{2}$) cancels this suppression
in the terms proportional to $p^{4}$. Although it may not be obvious, one can
check that these nonlocal structures represent the contributions from the
light sector alone. The one-loop diagrams with mixed (light and massive ghost)
internal lines and (of course) the pure ghost contributions collapse and
produce only trivial dependencies on the external momentum. All in all, we
verified the quadratic decoupling of the heavy mode in the Feynman diagrams
with the mixed contents.

##  5 One-loop corrections in the effective theory

The last element of our investigation is the comparison between what remains
from the logarithmic form factors of the full theory in the IR and the leading
logarithms in the effective (initially local) theory without heavy degrees of
freedom. According to the existing expectations [21], the two expression
should demonstrate a perfect correlation. This result would mean, in
particular, that the quantum general relativity can serve as a universal low-
energy model in any renormalizable or superrenormalizable approach to quantum
gravity.

So, let us evaluate the quantum corrections to the propagator in the effective
low-energy model of (6), containing only the light mode. We consider a
scenario in which the energy scale is much smaller than the Planck mass.
Therefore, we can assume that the EH and cosmological constant terms dominate
over the higher derivative terms, leaving only the part
$\mathcal{L}_{\textrm{IR}}$. Under these considerations, the tree-level
propagator of the conformal factor boils down to

$\displaystyle\widetilde{G}_{\textrm{eff}}(k)\,=\,-\,\frac{i}{2\gamma\big{(}k^{2}-m^{2}\big{)}},$
(46)

where $m^{2}$ is defined in (20). The vertices are the same as those in (22)
and (23). Since we are dealing with an effective model, we are not concerned
that $\mathcal{L}_{\textrm{IR}}$ is non-renormalizable, as we may ignore the
higher-derivative divergences333According to the logic of the pioneer work
[21] (see also [41]) the divergences in quantum gravity are local expressions
and, therefore, have no direct relation to the long-distance regime
corresponding to the IR limit.. Thus, our interest is to explore the
contributions to the cosmological constant and the Einstein-Hilbert terms.
These formulas can be compared with the structures found in the IR limit of
the full theory (6).

In the low-energy effective theory, the relevant contribution is given by

$\displaystyle\Sigma^{\textrm{eff}}_{\gamma^{2}}(p)$ $\displaystyle=$
$\displaystyle\frac{ip^{4}}{(4\pi)^{2}}\biggl{\\{}\Big{(}\frac{1}{2}-\frac{5}{4}\,a+\frac{3}{8}\,a^{2}\Big{)}\,\bigg{[}\frac{1}{\epsilon}-\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{m^{2}}\Big{)}\bigg{]}+\Big{(}1-\frac{7}{4}\,a+\frac{1}{2}\,a^{2}\Big{)}$
(47)
$\displaystyle-\,\,\frac{1}{2c}\Big{(}\frac{1}{4}\,a^{2}-\frac{1}{c^{2}}+2\Big{)}\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}\biggr{\\}}.$

To make the comparison more explicit, consider the particular case $\Lambda=0$
(or, equivalently, $p^{2}\rightarrow\infty$). Then the last expression reduces
to a simpler form,

$\displaystyle\Sigma^{\textrm{eff}}_{\gamma^{2}}(p)\Big{|}_{\Lambda=0}$
$\displaystyle=$
$\displaystyle\frac{ip^{4}}{2(4\pi)^{2}}\bigg{[}\frac{1}{\epsilon}-\,\mbox{ln}\,\Big{(}\frac{p^{2}}{\mu^{2}}\Big{)}+2\bigg{]}.$
(48)

Note that the nonlocal contribution $\,\mbox{ln}\,\big{(}p^{2}/M^{2}\big{)}$
involving the interaction vertex $\gamma^{2}$ in the IR regime of the
“fundamental” theory (6), Eq. (45), correlates with the logarithmic term of
the result (48), regardless in the IR of the fundamental theory there is no UV
divergence. From these results, it is possible to establish the one-loop match
between these two scenarios, i.e., fundamental and effective. Let us note that
the emergence of $p^{4}$ factor in the effective approach and its
identification with the cosmological constant term has been discussed in the
recent literature. In particular, this issue was described in detail, exactly
as a reaction to the attribution of the part of the gravitational form factor
to the cosmological constant in [43]. It reality, the corresponding terms
appear as part of the expansion of the nonlocal form factors of the
$R_{\mu\nu}R^{\mu\nu}$ and $R^{2}$ terms and have no direct relation to the
cosmological constant term [42].

In order to establish the match between the fundamental and effective
scenarios, we introduce the relation

$\displaystyle\Sigma^{\textrm{IR}}_{\gamma^{2}}$ $\displaystyle=$
$\displaystyle\Sigma^{\textrm{eff}}_{\gamma^{2}}+\delta^{\textrm{eff}}_{\gamma^{2}},$
(49)

where $\delta^{\textrm{eff}}_{\gamma^{2}}$ is an additional term which
represents, at the one-loop level, the difference between the correction of
the fundamental theory in the low energies and the correction of the effective
theory. Considering the collapse of the diagrams with massive ghost internal
lines in the IR regime of the fundamental theory, as we saw in the previous
section, we can identify that the additional term in (49) is composed of
contributions arising from the collapse of loops in the mixed sector. These
collapsed diagrams reduce to the tadpole-type graphs, and the remaining part,
related to pure ghost loops, i.e.,

$\displaystyle\delta^{\textrm{eff}}_{\gamma^{2}}$ $\displaystyle=$
$\displaystyle
c^{(1)}_{\gamma^{2},\,\textrm{mixed}}+c^{(1)}_{\gamma^{2},\,\textrm{ghost}}.$
(50)

The contribution of the tadpole-type in (50) is proportional to the mass
$m^{2}$, and hence vanishes in the simplification adopted for the IR, namely
assuming $c^{(1)}_{\gamma^{2},\,\textrm{mixed}}\big{|}_{m^{2}=0}=0$. Using the
results (45) and (48) in relation (49), we find the leading logarithmic terms
in the form

$\displaystyle\delta^{\textrm{eff}}_{\gamma^{2}}\Big{|}_{m^{2}=0}$
$\displaystyle=$
$\displaystyle\frac{i}{(4\pi)^{2}}\,\Bigg{\\{}2M^{4}\,\mbox{ln}\,\bigg{(}\frac{\mu^{2}}{M^{2}}\bigg{)}+\frac{13}{6}p^{2}M^{2}-\frac{p^{4}}{2}\bigg{[}\frac{18}{5}+\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{M^{2}}\Big{)}\bigg{]}\Bigg{\\}}.$
(51)

In the above expression we present only the finite part, since the divergences
can be removed by a suitable renormalization procedure. As it should be
expected, the nonlocal part with momentum-dependent logarithmic form factor is
canceled and the IR matching condition, which ensure the equivalence with the
result (45), is satisfied with $\delta^{\textrm{eff}}_{\gamma^{2}}$ contains
only terms with trivial dependencies on the external momentum.

##  6 Implications for the cosmological constant problem

The cosmological constant problem is one of the main unsolved issues in the
present-day theoretical physics. The problem was formulated by Weinberg in
[44] as the need to explain the extremely precise fine tuning between the
original cosmological constant density in the vacuum action and the huge
induced contributions. One can reformulate the problem in terms of the
renormalization of the vacuum term [45] but this does not help too much in its
resolution. There are also many other interesting aspects of the problem,
related to cosmology (see, e.g., [46, 47]). Along with the main problem, in
the quantum field theory framework we need to understand whether the
cosmological constant density and the Newton constant are really constants or
these parameters can be slowly varying with the energy scale, as predicted,
e.g., by the four-derivative model of [1] (see also the examples of
discussions based on the extended models [48, 49, 32] and supersymmetric
generalization [50]) and, of course, in the full fourth-derivative [17, 18,
22] or even higher-derivative models [20] of quantum gravity.

As we saw in the previous sections, the naive Minimal Subtraction \- based
approach to the renormalization group for the cosmological constant term is
not operational, as it ignores the decoupling of the massive (ghost or
healthy, in some models) degrees of freedom. Assuming that all massive degrees
of freedom have typical masses of the Planck order of magnitude, all the
cosmological applications occur at the deep IR, where the Appelquist-Carazzone
- type decoupling changes the beta functions. The question is what remains
from these beta functions in the theory with both massive and massless degrees
of freedom [22, 23]?

The result which we got for the quantum theory of conformal factor is that, in
the deep IR, there remain the contributions (43), (44) and (45), which fit the
ones of effective low-energy quantum theory based on the local model. This
provides a positive answer to one of the main questions posed in [23] and
confirms the hypothesis of [21]. We got a strong confirmation that the IR
limit of a higher derivative model of quantum gravity, which has massive
degrees of freedom (those may be healthy modes or ghosts) corresponds to the
loop corrections in the effective quantum gravity based on general relativity.
The result comes from the model of [1] which has non-polynomial interactions,
very close to the case of real models of quantum gravity. In this sense, the
new confirmation is more explicit than the results obtained previously in the
framework of simplified models, such as [51]). Looking at the remnant
expressions of the form factors in the IR, we note that the terms without
$p^{4}$ or $p^{2}$ have only $\log(\mu^{2}/M^{2})$ coefficients and no
momentum-dependent logarithmic terms. This is consistent with the previous
analysis of the possible quantum corrections, indicating that the logarithmic
(in Euclidean momentum) form factors cannot be inserted in the cosmological
constant term [2]. Recently, similar observations were done, e.g., in [24, 52,
53]). This situation confirms the general expectation that there cannot be
physical running of the cosmological constant term, detectable by means of
flat-space calculations, as discussed in [2] and more recently in [42]. Let us
stress that this does not mean that the cosmological constant running is
impossible in general, it just cannot be detected in the flat-space
calculations [3].

It is worth noting that one can observe the running of the cosmological
constant term using the non-covariant parametrization such as (1), just as a
decoupling in the beta function of the $\phi^{4}$-interaction. Up to a certain
extent, the corresponding calculations were already developed in [9] and can
be generalized to other theories, including quantum gravity. This would be
certainly an interesting way to extend the present work. However, it is
important to be careful with the expectations to the results of such an
extension, as there will always remain a question about the physical
interpretation of the result obtained by means of non-covariant methods.

##  7 Conclusions and discussions

We have considered the detailed renormalization in the theory of the conformal
factor. Assuming a small cosmological constant, such a theory possesses two
mass scales with a strong hierarchy between them. On top of that, the theory
has non-polynomial interactions and is renormalizable [1]. These features make
the model qualitatively similar to the higher derivative quantum gravity.
Previously, the one-loop calculations in this model were performed in the
Minimal Subtraction scheme and we performed a more detailed analysis in the
momentum subtraction scheme of renormalization.

The analysis of the nonlocal form factors shows that in the UV, we meet a
correspondence with the Minimal Subtraction scheme results. On the other hand,
in the IR we met a strong deviation from this simplified scheme and, as it was
anticipated, a good agreement with the calculation in the effective model that
ignores the massive degree of freedom. One of the new details is that the
“mixed” diagrams, with the internal lines of both small-mass and large-mass
fields, transform into tadpoles. These diagrams contribute to the UV
divergences, but not to the nonlocal form factors. This means, the diagrams
with the large-mass internal lines collapse and become irrelevant in the IR.
In particular, nonlocal structures that survive in this regime and contribute
to the cosmological constant sector, are correlated with the UV divergent part
in an effective version of the model containing only the light mode.

It would be certainly interesting to extend the analysis which was presented
above, for the models of “real” quantum gravity, i.e., the theory of quantum
metric. As we mentioned in the Introduction, this is a technically more
challenging problem because such a theory has gauge invariance and complicated
tensor structures in the sectors of quantum metric and ghost. However, the
results presented above show that there are very good chances to meet the
expectation of universality of quantum general relativity as an effective
theory of quantum gravity, at least in the fourth derivative [15] and,
probably, all polynomial models introduced in [16], where all extra degrees of
freedom have the masses of the Planck order of magnitude [54].

At the same time, the situation may be more complicated in the non-local
theories of quantum gravity [55, 56, 57, 58] (many further references can be
found in the last review). The most popular version of nonlocal models are
free from massive degrees of freedom at the tree-level. On the other hand,
starting from the one-loop level, the structure of the propagator changes and
there are infinitely many complex-energy and complex-mass ghost-like states
with the quasi-continuous mass spectrum [59]. In this case, the universality
of general relativity as the IR quantum gravity theory is rather uncertain.
This means, there are still many interesting issues to explore in the area of
the present work.

The last point is that we have found a good correspondence between the IR
limit of the theory with massless and large-mass degrees of freedom and the UV
limit of the effective theory without the massive particles. This
correspondence extends to the contributions in the cosmological constant
sector of the gravitational action. However, these contributions are not
momentum-dependent, confirming the general no-go statement [2] concerning the
detection of the cosmological constant running by means of the flat-space
calculations. On the other hand, this output does not means that such a
running is impossible by itself. Instead, it should be interpreted as a
challenge to develop new methods of effective field theory calculations which
would be appropriate for clarifying this issue.

## Acknowledgments

W.C.S. is grateful to CAPES for supporting his PhD project. The work of I.Sh.
is partially supported by Conselho Nacional de Desenvolvimento Científico e
Tecnológico - CNPq under the grant 303635/2018-5.

## Appendices

###  A Feynman diagrams

In this appendix, we present the set of one-loop Feynman diagrams that
correspond to the corrections to the two-point function.

Figure 5: Diagrams for the two-point function that provide one-loop
contributions to the renormalization of the coupling $\zeta$. Solid lines
indicate light degrees of freedom, dashed lines stand for the massive ghosts,
and the primes denotes derivatives acting on the propagators. Figure 6:
Diagrams for the two-point function with the interaction vertex $\gamma\zeta$.
Figure 7: Diagrams for the two-point function with the interaction vertex
$\gamma^{2}$. Figure 8: Diagrams for the two-point function with the
interaction vertex $\gamma$. Figure 9: Diagrams for the two-point function
with the interaction vertex $\lambda$. Figure 10: Diagrams (tadpole) for the
two-point function with the interaction vertex $\gamma\lambda$. Figure 11:
Diagrams (tadpole) for the two-point function with the interaction vertex
$\lambda^{2}$.

Diagrams that contribute only finite corrections to the two-point function are
shown in Figs. 12, 13 and 14.

Figure 12: Divergence-free diagrams for the two-point function with the vertex
$\lambda\zeta$. Figure 13: Divergence-free diagrams for the two-point function
with the vertex $\lambda\gamma$. Figure 14: Divergence-free diagrams for the
two-point function with the vertex $\lambda^{2}$.

###  B Intermediate results

In this appendix, we present some intermediate results related to the
calculation of Feynman integrals in sect. 3.

####  B.1 Feynman integrals

By using the Feynman parametrization

$\displaystyle\frac{1}{ab}\,=\,\int_{0}^{1}\frac{dx}{\big{[}(a-b)x+b\big{]}^{2}},$
(52)

and performing the following shift of integration variable $k=q+px$, we can
rewrite the integrals related to the mixed sector in the expressions (27),
(28) and (29), respectively, as

$\displaystyle\Sigma^{(2\omega)}_{\textrm{mixed},\,\zeta^{2}}(p)$
$\displaystyle=$
$\displaystyle-\,\frac{2\zeta^{2}\,p^{4}}{\theta^{4}(m^{2}-M^{2})^{2}}\frac{(4\omega^{2}-1)}{\omega(1+\omega)}\int^{1}_{0}dx\,I_{4},$
(53) $\displaystyle\Sigma^{(2\omega)}_{\textrm{mixed},\,\gamma\zeta}(p)$
$\displaystyle=$
$\displaystyle-\,\frac{2\gamma\zeta\,p^{2}}{\theta^{4}(m^{2}-M^{2})^{2}}\frac{(2\omega-1)}{\omega}\int^{1}_{0}dx\,\Big{[}\big{(}x^{2}-x+1\big{)}p^{2}I_{2}+I_{4}\Big{]}$
(54)

and

$\displaystyle\Sigma^{(2\omega)}_{\textrm{mixed},\,\gamma^{2}}(p)$
$\displaystyle=$
$\displaystyle-\,\frac{2\gamma^{2}}{\theta^{4}(m^{2}-M^{2})^{2}}\int^{1}_{0}dx\,\bigg{[}I_{4}+\frac{1+4\omega+4\big{(}x^{2}-x\big{)}(\omega+1)}{2\omega}p^{2}I_{2}$
(55) $\displaystyle+\,\big{(}x^{2}-x+1\big{)}^{2}p^{4}I_{1}\bigg{]},$

where, in Minkowski space,

$\displaystyle
I_{1}\,=\,\int\frac{d^{2\omega}q}{(2\pi)^{2\omega}}\frac{1}{(q^{2}-\Delta)^{2}}\,=\,\frac{i}{(4\pi)^{\omega}}\Gamma(2-\omega)\Delta^{\omega-2},$
(56) $\displaystyle
I_{2}\,=\,\int\frac{d^{2\omega}q}{(2\pi)^{2\omega}}\frac{q^{2}}{(q^{2}-\Delta)^{2}}\,=\,-\frac{i}{(4\pi)^{\omega}}\omega\Gamma(1-\omega)\Delta^{\omega-1},$
(57) $\displaystyle
I_{4}\,=\,\int\frac{d^{2\omega}q}{(2\pi)^{2\omega}}\frac{q^{4}}{(q^{2}-\Delta)^{2}}\,=\,\frac{i}{(4\pi)^{\omega}}\omega(1+\omega)\Gamma(-\omega)\Delta^{\omega}.$
(58)

Here we define $\Delta\equiv p^{2}x(x-1)+(M^{2}-m^{2})x+m^{2}$. For the
integrals in the other sectors, we have the same results as above with
$\Delta=\Delta_{\textrm{ghost}}=p^{2}x(x-1)+M^{2}$ in the case of the ghost
sector, and $\Delta=\Delta_{\textrm{light}}=p^{2}x(x-1)+m^{2}$ for the light
sector.

####  B.2  Corrections $\Sigma_{\zeta^{2}},\;\Sigma_{\gamma\zeta}$ and
$\Sigma_{\gamma^{2}}$ for the case of $m^{2}=0$

The contributions from the diagrams shown in Figs. 5, 6 and 7, assuming
$m^{2}=0$, can be written, respectively, as

$\displaystyle\Sigma_{\zeta^{2}}(p)\bigg{|}_{m^{2}=0}$ $\displaystyle=$
$\displaystyle\frac{i\zeta^{2}}{(4\pi)^{2}\theta^{4}}\,\Bigg{\\{}\frac{5p^{4}}{\epsilon}+\alpha^{(2)}_{\zeta}(p)\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{M^{2}}\Big{)}+\xi^{(2)}_{\zeta}(p)+\beta^{(2)}_{\zeta,\,\textrm{light}}(p)\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{p^{2}}\Big{)}$
(59)
$\displaystyle+\,\,\beta^{(2)}_{\zeta,\,\textrm{ghost}}(p)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}+\beta^{(2)}_{\zeta,\,\textrm{mixed}}(p)\,\mbox{ln}\,\Big{(}\frac{M^{2}}{M^{2}+p^{2}}\Big{)}\bigg{\\}},$
$\displaystyle\Sigma_{\gamma\zeta}(p)\bigg{|}_{m^{2}=0}$ $\displaystyle=$
$\displaystyle-\,\frac{i\gamma\zeta}{(4\pi)^{2}\theta^{4}}\,\Bigg{\\{}\frac{3p^{2}}{\epsilon}+\alpha^{(2)}_{\gamma\zeta}(p)\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{M^{2}}\Big{)}+\xi^{(2)}_{\gamma\zeta}(p)+\beta^{(2)}_{\gamma\zeta,\,\textrm{light}}(p)\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{p^{2}}\Big{)}$
(60)
$\displaystyle+\,\,\beta^{(2)}_{\gamma\zeta,\,\textrm{ghost}}(p)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}+\beta^{(2)}_{\gamma\zeta,\,\textrm{mixed}}(p)\,\mbox{ln}\,\Big{(}\frac{M^{2}}{M^{2}+p^{2}}\Big{)}\bigg{\\}}$

and

$\displaystyle\Sigma_{\gamma^{2}}(p)\bigg{|}_{m^{2}=0}$ $\displaystyle=$
$\displaystyle\frac{i\gamma^{2}}{(4\pi)^{2}\theta^{4}}\,\Bigg{\\{}\frac{2}{\epsilon}+\alpha^{(2)}_{\gamma}(p)\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{M^{2}}\Big{)}+\xi^{(2)}_{\gamma}+\beta^{(2)}_{\gamma,\,\textrm{light}}(p)\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{p^{2}}\Big{)}$
(61)
$\displaystyle+\,\,\beta^{(2)}_{\gamma,\,\textrm{ghost}}(p)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}+\beta^{(2)}_{\gamma,\,\textrm{mixed}}(p)\,\mbox{ln}\,\Big{(}\frac{M^{2}}{M^{2}+p^{2}}\Big{)}\bigg{\\}},$

where $\beta$’s are coefficients of the nonlocal part with momentum-dependent
logarithmic form factor, decomposed according to the light, ghost and mixed
sectors, while $\alpha$’s and $\xi$’s are coefficients of contributions
involving the combination of different sectors:

$\displaystyle\beta^{(2)}_{\zeta,\,\textrm{light}}(p)\,=\,\frac{p^{8}}{2M^{4}},\hskip
28.45274pt\beta^{(2)}_{\gamma\zeta,\,\textrm{light}}(p)\,=\,-\,\frac{p^{6}}{2M^{4}},\hskip
28.45274pt\beta^{(2)}_{\gamma,\,\textrm{light}}(p)\,=\,\frac{p^{4}}{2M^{4}},$
$\displaystyle\beta^{(2)}_{\zeta,\,\textrm{ghost}}(p)\,=\,-\,\frac{p^{8}}{2M^{4}d^{5}},\hskip
104.4216pt\beta^{(2)}_{\gamma\zeta,\,\textrm{ghost}}(p)\,=\,-\,\frac{p^{2}(8M^{4}-2M^{2}p^{2}-p^{4})}{2M^{4}d},$
$\displaystyle\beta^{(2)}_{\gamma,\,\textrm{ghost}}(p)\,=\,-\,\frac{(2M^{2}-p^{2})^{2}}{2M^{4}d},\hskip
75.3998pt\beta^{(2)}_{\zeta,\,\textrm{mixed}}(p)\,=\,-\,\frac{(M^{2}+p^{2})^{5}}{M^{4}p^{2}},$
$\displaystyle\beta^{(2)}_{\gamma\zeta,\,\textrm{mixed}}(p)\,=\,-\,\frac{(M^{2}-p^{2})(M^{2}+p^{2})^{3}}{M^{4}p^{2}},\hskip
22.76219pt\beta^{(2)}_{\gamma,\,\textrm{mixed}}(p)\,=\,-\,\frac{(M^{2}+p^{2})(M^{2}-p^{2})^{2}}{M^{4}p^{2}},$
$\displaystyle\alpha^{(2)}_{\zeta}(p)\,=\,p^{4}\bigg{(}5-\frac{p^{4}}{2M^{4}}\bigg{)},\hskip
45.80894pt\alpha^{(2)}_{\gamma\zeta}(p)\,=\,p^{2}\bigg{(}3+\frac{p^{4}}{2M^{4}}\bigg{)},\hskip
22.76219pt\alpha^{(2)}_{\gamma}(p)\,=\,2-\frac{p^{4}}{2M^{4}},$
$\displaystyle\xi^{(2)}_{\zeta}(p)\,=\,7p^{4}-\frac{9M^{2}p^{2}}{2}-M^{4},\hskip
22.76219pt\xi^{(2)}_{\gamma\zeta}(p)\,=\,6p^{2}-M^{2},\hskip
50.6459pt\xi^{(2)}_{\gamma}\,=\,3.$ (62)

###  C Results of the finite contributions $\Sigma_{\lambda\zeta}$,
$\Sigma_{\lambda\gamma}$ and $\Sigma_{\lambda^{2}}$

We collect here the results of the self-energy corrections that are free of
divergences. The contributions from the diagrams shown in Figs. 12, 13 and 14
are, respectively,

$\displaystyle\Sigma_{\lambda\zeta}(p)$ $\displaystyle=$
$\displaystyle-\frac{16i\lambda\zeta}{(4\pi)^{2}\theta^{4}}\,\bigg{\\{}2+\frac{1}{(ab)^{2}c^{3}}\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}+\frac{1}{(ab)^{2}d^{3}}\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}$
(63)
$\displaystyle+\,\dfrac{1}{ab}\bigg{(}A^{2}+ab+\dfrac{a}{2}+2\bigg{)}\,\mbox{ln}\,(1+4b)-\frac{A^{3}}{(ab)^{2}}\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}\bigg{\\}},$
$\displaystyle\Sigma_{\lambda\gamma}(p)$ $\displaystyle=$
$\displaystyle\frac{16i\lambda\gamma}{(4\pi)^{2}\theta^{4}p^{2}}\,\bigg{\\{}\frac{1}{(ab)^{2}c}\bigg{(}\dfrac{a}{2}-1\bigg{)}\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}+\frac{1}{(ab)^{2}d}\bigg{(}2ab+\dfrac{a}{2}-1\bigg{)}\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}$
(64)
$\displaystyle+\,\bigg{(}1+\dfrac{1}{2b}\bigg{)}\,\mbox{ln}\,(1+4b)-\frac{A}{(ab)^{2}}\bigg{(}ab+\dfrac{a}{2}-1\bigg{)}\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}\bigg{\\}}$

and

$\displaystyle\Sigma_{\lambda^{2}}(p)$ $\displaystyle=$
$\displaystyle-\frac{512i\lambda^{2}}{(4\pi)^{2}\theta^{4}p^{4}}\,\bigg{\\{}\dfrac{1}{ab}\,\mbox{ln}\,(1+4b)+\frac{1}{(ab)^{2}c}\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}+\frac{1}{(ab)^{2}d}\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}$
(65)
$\displaystyle-\,\frac{A}{(ab)^{2}}\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}\bigg{\\}}.$

###  D One-loop corrections to the three- and four-point vertices

This appendix is devoted to the one-loop corrections to the vertices. In case
of the three-point function, the relevant corrections are associated with the
diagrams in Figures 15, 16 and 17.

Figure 15: Diagrams for the three-point function with the interaction vertex
$\zeta^{2}$. Figure 16: Diagrams for the three-point function with the
interaction vertex $\gamma\zeta$. Figure 17: Diagrams for the three-point
function with the interaction vertex $\gamma^{2}$.

In the dimensional regularization scenario, these corrections are given by
following integrals,

$\displaystyle\Gamma^{(3)}_{\zeta^{2}}(p,r)\big{|}^{(2\omega)}$
$\displaystyle=$
$\displaystyle-\,\frac{4\zeta^{2}}{\theta^{4}(m^{2}-M^{2})^{2}}\int\frac{d^{2\omega}k}{(2\pi)^{2\omega}}\,\Gamma^{(3,4)}_{\zeta^{2}}\biggl{\\{}\frac{2}{(k^{2}-m^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}$
(66)
$\displaystyle-\,\,\frac{1}{(k^{2}-m^{2})\big{[}(k-p)^{2}-m^{2}\big{]}}\,-\,\frac{1}{(k^{2}-M^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}\biggr{\\}}$
$\displaystyle+\,\,t\textrm{- and }u\textrm{-channel contributions},$
$\displaystyle\Gamma^{(3)}_{\gamma\zeta}(p,r)\big{|}^{(2\omega)}$
$\displaystyle=$
$\displaystyle\frac{2\gamma\zeta}{\theta^{4}(m^{2}-M^{2})^{2}}\int\frac{d^{2\omega}k}{(2\pi)^{2\omega}}\,\Big{[}\Gamma^{(3,4)}_{\gamma\zeta}+\Gamma^{(3,4)}_{\zeta\gamma}\Big{]}\biggl{\\{}\frac{2}{(k^{2}-m^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}$
(67)
$\displaystyle-\,\,\frac{1}{(k^{2}-m^{2})\big{[}(k-p)^{2}-m^{2}\big{]}}\,-\,\frac{1}{(k^{2}-M^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}\biggr{\\}}$
$\displaystyle+\,\,t\textrm{- and }u\textrm{-channel contributions}$

and

$\displaystyle\Gamma^{(3)}_{\gamma^{2}}(p,r)\big{|}^{(2\omega)}$
$\displaystyle=$
$\displaystyle-\,\frac{\gamma^{2}}{\theta^{4}(m^{2}-M^{2})^{2}}\int\frac{d^{2\omega}k}{(2\pi)^{2\omega}}\,\Gamma^{(3,4)}_{\gamma^{2}}\biggl{\\{}\frac{2}{(k^{2}-m^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}$
(68)
$\displaystyle-\,\,\frac{1}{(k^{2}-m^{2})\big{[}(k-p)^{2}-m^{2}\big{]}}\,-\,\frac{1}{(k^{2}-M^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}\biggr{\\}}$
$\displaystyle+\,\,t\textrm{- and }u\textrm{-channel contributions},$

where the combinations of the vertex factors (for the $s$-channel diagrams)
are

$\displaystyle\Gamma^{(3,4)}_{\zeta^{2}}\,=\,-2\big{[}k^{2}(p\cdot
r)-k^{2}r^{2}-2(k\cdot p-k\cdot r)(p\cdot r-k\cdot r)-p^{2}(k\cdot
r)+r^{2}(k\cdot p)\big{]}$ $\displaystyle\hskip
41.25641pt\,\,\times\big{[}(p\cdot k)^{2}-p^{2}k^{2}\big{]},$
$\displaystyle\Gamma^{(3,4)}_{\gamma\zeta}+\Gamma^{(3,4)}_{\zeta\gamma}\,=\,4\big{[}(k\cdot
p)^{2}-k^{2}p^{2}\big{]}\big{(}k^{2}-k\cdot p+p^{2}-p\cdot
r+r^{2}\big{)}-2\big{[}k^{2}(p\cdot r-r^{2})$ $\displaystyle\hskip
41.25641pt\,\,-p^{2}(k\cdot r)-2(k\cdot r)(k\cdot r-p\cdot r)+r^{2}(k\cdot
p)+2(k\cdot p)(k\cdot r-p\cdot r)\big{]}$ $\displaystyle\hskip
41.25641pt\,\,\times\big{(}k^{2}-k\cdot p+p^{2}\big{)},$
$\displaystyle\Gamma^{(3,4)}_{\gamma^{2}}\,=\,4\big{[}k^{2}-(k\cdot
p)+p^{2}\big{]}\big{(}k^{2}-k\cdot p+p^{2}-p\cdot r+r^{2}\big{)}.$ (69)

Taking these integrals, we write the contributions to the three-point function
as

$\displaystyle\Gamma^{(3)}_{\zeta^{2}}(p,r)$ $\displaystyle=$
$\displaystyle\frac{i\zeta^{2}}{(4\pi)^{2}\theta^{4}}\,\bigg{\\{}20\big{[}(p\cdot
r)^{2}-p^{2}r^{2}\big{]}\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{m^{2}}\Big{)}\bigg{]}+\alpha^{(3)}_{\zeta^{2}}(p,r)\,\mbox{ln}\,(1+4b)$
(70)
$\displaystyle+\,\xi^{(3)}_{\zeta^{2}}(p,r)+\bigg{[}\beta^{(3)}_{\zeta^{2},\,\textrm{light}}(p,r)\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}+\beta^{(3)}_{\zeta^{2},\,\textrm{ghost}}(p,r)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}$
$\displaystyle+\,\beta^{(3)}_{\zeta^{2},\,\textrm{mixed}}(p,r)\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}+(p\leftrightarrow-r)+(p\leftrightarrow
r-p)\bigg{]}\bigg{\\}},\hskip 22.76228pt$
$\displaystyle\Gamma^{(3)}_{\gamma\zeta}(p,r)$ $\displaystyle=$
$\displaystyle-\frac{i\gamma\zeta}{(4\pi)^{2}\theta^{4}}\,\bigg{\\{}6\big{[}p^{2}+r^{2}-(p\cdot
r)\big{]}\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{m^{2}}\Big{)}\bigg{]}+\alpha^{(3)}_{\gamma\zeta}(p,r)\,\mbox{ln}\,(1+4b)$
(71)
$\displaystyle+\,\xi^{(3)}_{\gamma\zeta}(p,r)+\bigg{[}\beta^{(3)}_{\gamma\zeta,\,\textrm{light}}(p,r)\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}+\beta^{(3)}_{\gamma\zeta,\,\textrm{ghost}}(p,r)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}$
$\displaystyle+\,\beta^{(3)}_{\gamma\zeta,\,\textrm{mixed}}(p,r)\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}+(p\leftrightarrow-r)+(p\leftrightarrow
r-p)\bigg{]}\bigg{\\}},\hskip 22.76228pt$
$\displaystyle\Gamma^{(3)}_{\gamma^{2}}(p,r)$ $\displaystyle=$
$\displaystyle\frac{i\gamma^{2}}{(4\pi)^{2}\theta^{4}}\,\bigg{\\{}12\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{m^{2}}\Big{)}\bigg{]}+\alpha^{(3)}_{\gamma^{2}}(p,r)\,\mbox{ln}\,(1+4b)+18$
(72)
$\displaystyle+\,\bigg{[}\beta^{(3)}_{\gamma^{2},\,\textrm{mixed}}(p,r)\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}+\beta^{(3)}_{\gamma^{2},\,\textrm{light}}(p,r)\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}$
$\displaystyle+\,\beta^{(3)}_{\gamma^{2},\,\textrm{ghost}}(p,r)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}+(p\leftrightarrow-r)+(p\leftrightarrow
r-p)\bigg{]}\bigg{\\}},\hskip 22.76228pt$

where $\alpha$’s, $\beta$’s and $\xi$’s are coefficients with polynomial
dependencies on the external momenta. The full explicit form of these
expressions are very bulky and we do not present them here. On the other hand,
since they are polynomials the corresponding contributions are local and these
explicit expression is not really important for our analysis. Remember that
the notations $a$, $b$, $A$, $c$, $d$ are defined in (34), (35) and (36),
respectively.

It is important that the most essential, non-local parts of the expressions
have standard logarithmic structures, similar to those already evaluated in
section 4. Therefore, it should be expected that the corrections above
represent asymptotic behavior in the IR, similar to the case of the propagator
corrections considered in the main part of the paper. Let us note that we
verified and confirmed the quadratic decoupling of the heavy mode in the
vertex terms. Furthermore, the same behavior is observed for the four-point
vertex corrections. In this case, the diagrams of interest are depicted in
Figures 18, 19 and 20.

Figure 18: Diagrams for the four-point function with the interaction vertex
$\zeta^{2}$. Figure 19: Diagrams for the four-point function with the
interaction vertex $\gamma\zeta$. Figure 20: Diagrams for the four-point
function with the interaction vertex $\gamma^{2}$.

The analytic expressions corresponding to these diagrams are

$\displaystyle\Gamma^{(4)}_{\zeta^{2}}(p,r,q)\big{|}^{(2\omega)}$
$\displaystyle=$
$\displaystyle-\,\frac{8\zeta^{2}}{\theta^{4}(m^{2}-M^{2})^{2}}\int\frac{d^{2\omega}k}{(2\pi)^{2\omega}}\,\Gamma^{(4,4)}_{\zeta^{2}}\biggl{\\{}\frac{2}{(k^{2}-m^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}$
(73)
$\displaystyle-\,\,\frac{1}{(k^{2}-m^{2})\big{[}(k-p)^{2}-m^{2}\big{]}}\,-\,\frac{1}{(k^{2}-M^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}\biggr{\\}}$
$\displaystyle+\,\,t\textrm{- and }u\textrm{-channel contributions},$
$\displaystyle\Gamma^{(4)}_{\gamma\zeta}(p,r,q)\big{|}^{(2\omega)}$
$\displaystyle=$
$\displaystyle\frac{4\gamma\zeta}{\theta^{4}(m^{2}-M^{2})^{2}}\int\frac{d^{2\omega}k}{(2\pi)^{2\omega}}\,\Gamma^{(4,4)}_{\gamma\zeta}\biggl{\\{}\frac{2}{(k^{2}-m^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}$
(74)
$\displaystyle-\,\,\frac{1}{(k^{2}-m^{2})\big{[}(k-p)^{2}-m^{2}\big{]}}\,-\,\frac{1}{(k^{2}-M^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}\biggr{\\}}$
$\displaystyle+\,\,t\textrm{- and }u\textrm{-channel contributions},$
$\displaystyle\Gamma^{(4)}_{\gamma^{2}}(p,r,q)\big{|}^{(2\omega)}$
$\displaystyle=$
$\displaystyle-\,\frac{2\gamma^{2}}{\theta^{4}(m^{2}-M^{2})^{2}}\int\frac{d^{2\omega}k}{(2\pi)^{2\omega}}\,\Gamma^{(4,4)}_{\gamma^{2}}\biggl{\\{}\frac{2}{(k^{2}-m^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}$
(75)
$\displaystyle-\,\,\frac{1}{(k^{2}-m^{2})\big{[}(k-p)^{2}-m^{2}\big{]}}\,-\,\frac{1}{(k^{2}-M^{2})\big{[}(k-p)^{2}-M^{2}\big{]}}\biggr{\\}}$
$\displaystyle+\,\,t\textrm{- and }u\textrm{-channel contributions},$

where, for the $s$-channel diagrams,

$\displaystyle\Gamma^{(4,4)}_{\zeta^{2}}\,=\,\big{[}r^{2}(k\cdot
q)-k^{2}(r\cdot q)+2(k\cdot q)\big{(}r\cdot q\big{)}+q^{2}(k\cdot r)-2(k\cdot
r)\big{(}k\cdot q-r\cdot q\big{)}\big{]}$ $\displaystyle\hskip
41.25641pt\,\,\times\big{[}k^{2}\big{(}p^{2}-p\cdot r-p\cdot
q\big{)}-p^{2}(k\cdot r+k\cdot q)+2(k\cdot r+k\cdot q)(p\cdot r+p\cdot q)$
$\displaystyle\hskip 41.25641pt\,\,+(r^{2}+q^{2})(k\cdot p)-2(k\cdot
p)\big{(}k\cdot r+k\cdot q+p\cdot r+p\cdot q-r\cdot q\big{)}+2(k\cdot
p)^{2}\big{]},$
$\displaystyle\Gamma^{(4,4)}_{\gamma\zeta}\,=\,-2\big{[}k^{2}(r\cdot
q)-r^{2}(k\cdot q)-2(k\cdot q)\big{(}r\cdot q\big{)}+2(k\cdot r)\big{(}k\cdot
q-r\cdot q\big{)}-q^{2}(k\cdot r)\big{]}$ $\displaystyle\hskip
41.25641pt\,\,\times\big{[}k^{2}-k\cdot r-k\cdot q+p^{2}-p\cdot r-p\cdot
q+r^{2}+2(r\cdot q)+q^{2}\big{]},$
$\displaystyle\Gamma^{(4,4)}_{\gamma^{2}}\,=\,4\big{[}k^{2}-k\cdot r-k\cdot
q+p^{2}-p\cdot r-p\cdot q+r^{2}+2(r\cdot q)+q^{2}\big{]}$ $\displaystyle\hskip
41.25641pt\,\,\times\big{(}k^{2}-k\cdot r-k\cdot q+r^{2}+r\cdot
q+q^{2}\big{)}.$ (76)

The analytic expressions of the four-point vertex corrections involving the
couplings $\zeta^{2}$, $\gamma\zeta$ and $\gamma^{2}$ are, respectively,

$\displaystyle\Gamma^{(4)}_{\zeta^{2}}(p,r,q)$ $\displaystyle=$
$\displaystyle-\frac{i\zeta^{2}}{(4\pi)^{2}\theta^{4}}\,\bigg{\\{}20\big{[}p^{2}(r\cdot
q)-r^{2}(p\cdot q)-2(p\cdot q)(r\cdot q)-q^{2}(p\cdot r)$ (77)
$\displaystyle+\,2(p\cdot r)\big{(}p\cdot q-r\cdot
q\big{)}\big{]}\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{m^{2}}\Big{)}\bigg{]}+\alpha^{(4)}_{\zeta^{2}}(p,r,q)\,\mbox{ln}\,(1+4b)$
$\displaystyle+\,\beta^{(4,\,s)}_{\zeta^{2},\,\textrm{mixed}}(p,r,q)\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}\bigg{|}_{p\leftrightarrow
r+q}+\xi^{(4)}_{\zeta^{2}}(p,r,q)$
$\displaystyle+\,\beta^{(4,\,s)}_{\zeta^{2},\,\textrm{light}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}\bigg{|}_{p\leftrightarrow
r+q}+\beta^{(4,\,s)}_{\zeta^{2},\,\textrm{ghost}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}\bigg{|}_{p\leftrightarrow
r+q}$
$\displaystyle+\,\beta^{(4,\,t)}_{\zeta^{2},\,\textrm{light}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}\bigg{|}_{p\leftrightarrow
q-p}+\beta^{(4,\,t)}_{\zeta^{2},\,\textrm{ghost}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}\bigg{|}_{p\leftrightarrow
q-p}$
$\displaystyle+\,\beta^{(4,\,t)}_{\zeta^{2},\,\textrm{mixed}}(p,r,q)\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}\bigg{|}_{p\leftrightarrow
q-p}$
$\displaystyle+\,\beta^{(4,\,u)}_{\zeta^{2},\,\textrm{light}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}\bigg{|}_{p\leftrightarrow
r-p}+\beta^{(4,\,u)}_{\zeta^{2},\,\textrm{ghost}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}\bigg{|}_{p\leftrightarrow
r-p}$
$\displaystyle+\,\beta^{(4,\,u)}_{\zeta^{2},\,\textrm{mixed}}(p,r,q)\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}\bigg{|}_{p\leftrightarrow
r-p}\bigg{\\}},$ $\displaystyle\Gamma^{(4)}_{\gamma\zeta}(p,r,q)$
$\displaystyle=$
$\displaystyle\frac{i\gamma\zeta}{(4\pi)^{2}\theta^{4}}\,\bigg{\\{}12(p\cdot
r+p\cdot q-r\cdot
q)\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{m^{2}}\Big{)}\bigg{]}+\xi^{(4)}_{\gamma\zeta}(p,r,q)$
(78)
$\displaystyle+\,\alpha^{(4)}_{\gamma\zeta}(p,r,q)\,\mbox{ln}\,(1+4b)+\beta^{(4,\,s)}_{\gamma\zeta,\,\textrm{light}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}\bigg{|}_{p\leftrightarrow
r+q}$
$\displaystyle+\,\beta^{(4,\,s)}_{\gamma\zeta,\,\textrm{mixed}}(p,r,q)\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}\bigg{|}_{p\leftrightarrow
r+q}$
$\displaystyle+\,\beta^{(4,\,s)}_{\gamma\zeta,\,\textrm{ghost}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}\bigg{|}_{p\leftrightarrow
r+q}+\beta^{(4,\,t)}_{\gamma\zeta,\,\textrm{light}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}\bigg{|}_{p\leftrightarrow
q-p}$
$\displaystyle+\,\beta^{(4,\,t)}_{\gamma\zeta,\,\textrm{mixed}}(p,r,q)\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}\bigg{|}_{p\leftrightarrow
q-p}$
$\displaystyle+\,\beta^{(4,\,t)}_{\gamma\zeta,\,\textrm{ghost}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}\bigg{|}_{p\leftrightarrow
s-p}+\beta^{(4,\,u)}_{\gamma\zeta,\,\textrm{light}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}\bigg{|}_{p\leftrightarrow
r-p}$
$\displaystyle+\,\beta^{(4,\,u)}_{\gamma\zeta,\,\textrm{mixed}}(p,r,q)\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}\bigg{|}_{p\leftrightarrow
r-p}$
$\displaystyle+\,\beta^{(4,\,u)}_{\gamma\zeta,\,\textrm{ghost}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}\bigg{|}_{p\leftrightarrow
r-p}\bigg{\\}}$

and

$\displaystyle\Gamma^{(4)}_{\gamma^{2}}(p,r,q)$ $\displaystyle=$
$\displaystyle\frac{i\gamma^{2}}{(4\pi)^{2}\theta^{4}}\,\bigg{\\{}24\bigg{[}\frac{1}{\epsilon}+\,\mbox{ln}\,\Big{(}\frac{\mu^{2}}{m^{2}}\Big{)}\bigg{]}+\alpha^{(4)}_{\gamma^{2}}(p,r,q)\,\mbox{ln}\,(1+4b)+36$
(79)
$\displaystyle+\,\beta^{(4,\,s)}_{\gamma^{2},\,\textrm{light}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}\bigg{|}_{p\leftrightarrow
r+q}+\beta^{(4,\,s)}_{\gamma^{2},\,\textrm{ghost}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}\bigg{|}_{p\leftrightarrow
r+q}$
$\displaystyle+\,\beta^{(4,\,s)}_{\gamma^{2},\,\textrm{mixed}}(p,r,q)\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}\bigg{|}_{p\leftrightarrow
r+q}$
$\displaystyle+\,\beta^{(4,\,t)}_{\gamma^{2},\,\textrm{light}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}\bigg{|}_{p\leftrightarrow
q-p}+\beta^{(4,\,t)}_{\gamma^{2},\,\textrm{ghost}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}\bigg{|}_{p\leftrightarrow
q-p}$
$\displaystyle+\,\beta^{(4,\,t)}_{\gamma^{2},\,\textrm{mixed}}(p,r,q)\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}\bigg{|}_{p\leftrightarrow
q-p}$
$\displaystyle+\,\beta^{(4,\,u)}_{\gamma^{2},\,\textrm{light}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+c}{1-c}\Big{)}\bigg{|}_{p\leftrightarrow
r-p}+\beta^{(4,\,u)}_{\gamma^{2},\,\textrm{ghost}}(p,r,q)\,\mbox{ln}\,\Big{(}\frac{1+d}{1-d}\Big{)}\bigg{|}_{p\leftrightarrow
r-p}$
$\displaystyle+\,\beta^{(4,\,u)}_{\gamma^{2},\,\textrm{mixed}}(p,r,q)\,\mbox{ln}\,\Big{[}\frac{(A+1)^{2}-(ab)^{2}}{(A-1)^{2}-(ab)^{2}}\Big{]}\bigg{|}_{p\leftrightarrow
r-p}\bigg{\\}}.$

The indices in the coefficients $\beta^{(4)}(p,r,q)$ denote $s-,\,t-$ and
$u-$channel contributions.

It is easy to note that these expressions are in a good qualitative agreement
with the self-energy corrections and ones for the three-point vertices.

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