# Gupta-Feynman based Quantum Field Theory of Einstein’s Gravity

A. Plastino 1,3,4, M.C.Rocca1,2,3,
1 Departamento de Física, Universidad Nacional de La Plata,
2 Departamento de Matemática, Universidad Nacional de La Plata,
3 Consejo Nacional de Investigaciones Científicas y Tecnológicas
(IFLP-CCT-CONICET)-C. C. 727, 1900 La Plata - Argentina
4 SThAR - EPFL, Lausanne, Switzerland

###### Abstract

This paper is an application to Einstein’s gravity (EG) of the mathematics
developed in A. Plastino, M. C. Rocca: J. Phys. Commun. 2, 115029 (2018). We
will quantize EG by appeal to the most general quantization approach, the
Schwinger-Feynman variational principle, which is more appropriate and
rigorous that the functional integral method, when we are in the presence of
derivative couplings We base our efforts on works by Suraj N. Gupta and
Richard P. Feynman so as to undertake the construction of a Quantum Field
Theory (QFT) of Einstein Gravity (EG). We explicitly use the Einstein
Lagrangian elaborated by Gupta [1] but choose a new constraint for the theory
that differs from Gupta’s one. In this way, we avoid the problem of lack of
unitarity for the $S$ matrix that afflicts the procedures of Gupta and
Feynman.
Simultaneously, we significantly simplify the handling of constraints. This
eliminates the need to appeal to ghosts for guarantying the unitarity of the
theory.
Our ensuing approach is obviously non-renormalizable. However, this
inconvenience can be overcome by appealing tho the mathematical theory
developed by Bollini et al. [2, 3, 4, 5, 6]
Such developments were founded in the works of Alexander Grothendieck [7] and
in the theory of Ultradistributions of Jose Sebastiao e Silva [8] (also known
as Ultrahyperfunctions).
Based on these works, we have constructed a mathematical edifice, in a lapse
of about 25 years, that is able to quantize non-renormalizable Field Theories
(FT).
Here we specialize this mathematical theory to treat the quantum field theory
of Einsteins’s gravity (EG).
Because we are using a Gupta-Feynman inspired EG Lagrangian, we are able to
evade the intricacies of Yang-Mills theories.
PACS: 11.10.-z, 03.70.+k, 03.65.Ca, 03.65.Db.
KEYWORDS: Quantum Field Theory; Einstein gravity; Non-renormalizable theories,
Unitarity.

###### Contents

  1. 1 Introduction
  2. 2 Preliminary Materials
  3. 3 The Lagrangian of Einstein’s QFT
  4. 4 The Quantization of the Theory
    1. 4.1 Undesired effects if one does not use our constraint
  5. 5 The self energy of the graviton
    1. 5.1 Self-Energy evaluation for $\nu=4$
  6. 6 Including Axions into the picture
  7. 7 The complete Self Energy of the Graviton
    1. 7.1 Self-Energy evaluation for $\nu=4$
  8. 8 Self Energy of the Axion
    1. 8.1 Self-Energy evaluation for $\nu=4$
  9. 9 Discussion

##  1 Introduction

Quantifying Einstein gravity (EG) is still an open problem, a kind of holy
grail for quantum field theory (QFT). The failure of some attempts in this
direction have failed because i) they appeal to Rigged Hilber Space (RHS) with
undefined metric, ii) problems of non-unitarity, and also iii) non-
renormalizablity issues. Here we quantize EG by appeal to the most general
quantization approach, the Schwinger-Feynman variational principle, which is
more appropriate and rigorous that the functional integral method, when we are
in the presence of derivative couplings.
Here we build up an unitary EG’s QFT in the wake of related effort by Suraj N.
Gupta [1]. We deviate from his work by using a different EG-constraint, facing
then a problem similar to that posed by Quantum Electrodynamics (QED). In
order to quantize the concomitant non-renormalizable variational problem we
appeal to mathematics developed by Bollini et al. [2, 3, 4, 5, 6], based upon
the theory of Ultradistributions de J. Sebastiao e Silva (JSS) [8], also known
as Ultrahyperfunctions. The above cited mathematics were specifically devised
to quantify non-renormalizable field theories during 25 years, culminating in
[6]. We consequently face a theory similar to QED, endowed with unitarity at
all finite orders in the power expansion in $G$ (gravitation constant) of the
EG Lagrangian. This was attempted without success first by Gupta and then by
Feynman, in his celebrated Acta Physica Polonica paper [9].

Mathematically, quantizing a non-renormalizable field theory is tantamount to
suitably defining the product of two distributions (a product in a ring with
zero-divisors in configuration space), an old problem in functional theory
tackled successfully in [2, 3, 4, 5, 6].
Remark that, in QFT, the problem of evaluating the product of distributions
with coincident point singularities is related to the asymptotic behavior of
loop integrals of propagators.

In references [2, 3, 4, 5] it was demonstrated that it is possible to define a
general convolution between the ultradistributions of JSS [8]
(Ultrahyperfunctions). This convolution yields another Ultrahyperfunction.
Therefore, we have a product in a ring with zero divisors. Such a ring is the
space of distributions of exponential type, or ultradistributions of
exponential type, obtained applying the anti-Fourier transform to the space of
tempered ultradistributions or ultradistributions of exponential type.

We must clarify at this point that the ultrahyperfunctions are the
generalization and extension to the complex plane of the Schwartz tempered
distributions and the distributions of exponential type. That is, the tempered
distributions and those of exponential type are a subset of the
ultrahyprefunctions.

In our work we do not use counter-terms to get rid of infinities, because our
convolutions are always finite. We do not want counter-terms, since a non-
renormalizable theory involves an infinite number of them .

At the same time, we conserve all extant solutions to the problem of running
coupling constants and the renormalization group. The convolution, once
obtained, converts configuration space into a ring with zero-divisors. In it,
one has now defined a product between the ring-elements. Thus, any unitary-
causal-Lorentz invariant theory quantized in such a manner becomes predictive.
The distinction between renormalizable on non-renormalizable QFT’s becomes
unnecessary now.

With our convolution, that uses Laurent’s expansions in the parameter employed
to define it, all finite constants of the convolutions become completely
determined, eliminating arbitrary choices of finite constants. This is
tantamount to eliminating all finite renormalizations of the theory. The
independent term in the Laurent expansion give the convolution value. This
translates to configuration space the product-operation in a ring with
divisors of zero.

This paper is organized as follows:

  * •

Section 2 presents preliminary materials.

  * •

Section 3 is devoted to the QFT Lagrangian for EG

  * •

In Section 4 we quantize the ensuing theory.

  * •

In Section 5 the graviton’s self-energy is evaluated up to second order.

  * •

In Section 6 we introduce axions into our picture and deal with the axions-
gravitons interaction.

  * •

In Section 7 we calculate the graviton’s self-energy in the presence of
axions.

  * •

In Section 8 we evaluate, up to second order, the axion’s self-energy.

  * •

Finally, in Section 9, some conclusions are drawn.

##  2 Preliminary Materials

We appeal here to the most general quantification approach, Schwinger-Feynman
variational principle [10], which is able to deal even with high order
supersymmetric theories , as exemplified by [11, 12]. Such theories can not be
quantized with the usual Dirac-brackets technique.

We introduce the action for a set of fields defined by

$\boldsymbol{\cal
S}[\sigma(x),\sigma_{0},\phi_{A}(x)]=\int\limits_{\sigma_{0}}^{\sigma(x)}{\cal
L}[\phi_{A}(\xi),\partial_{\mu}\phi_{A}(\xi),\xi]d\xi,$ (2.1)

where $\sigma(x)$ if a space-like surface passing through the point $x$.
$\sigma_{0}$ is that surface at the remote past, at which all field variations
vanish. The Schwinger-Feynman variational principle dictates that

”Any Hermitian infinitesimal variation $\delta\boldsymbol{\cal S}$ of the
action induces a canonical transformation of the vector space in which the
quantum system is defined, and the generator of this transformation is this
same operator $\delta\boldsymbol{\cal S}$”.

Accordingly, the following equality holds:

$\delta\phi_{A}=i[\delta\boldsymbol{\cal S},\phi_{A}].$ (2.2)

Thus, for a Poincare transformation we have

$\delta\boldsymbol{\cal S}=a^{\mu}\boldsymbol{\cal P}_{\mu}+\frac{1}{2}a^{\mu
v}\boldsymbol{\cal M}_{\mu v},$ (2.3)

where the field variation is given by

$\delta\phi_{a}=a^{\mu}\hat{P}_{\mu}\phi_{A}+\frac{1}{2}a^{\mu v}\hat{M}_{\mu
v}\phi_{A}.$ (2.4)

From (2.2) one gathers that

$\partial_{\mu}\phi_{A}=i[\boldsymbol{\cal P}_{\mu},\phi_{A}].$ (2.5)

Specifically,

$\partial_{0}\phi_{A}=i[\boldsymbol{\cal P}_{0},\phi_{A}].$ (2.6)

This last result will be employed in quantizing EG.

##  3 The Lagrangian of Einstein’s QFT

Our EG Lagrangian reads [1]

${\cal
L}_{G}=\frac{1}{\kappa^{2}}\boldsymbol{R}\sqrt{|g|}-\frac{1}{2}\eta_{\mu
v}\partial_{\alpha}h^{\mu\alpha}\partial_{\beta}h^{v\beta},$ (3.1)

where $\eta^{\mu\nu}=diag(1,1,1,-1)$, $h^{\mu\nu}=\sqrt{|g|}g^{\mu\nu}$ The
second term in eq. (3.1) fixes the gauge. We apply now the linear
approximation

$h^{\mu v}=\eta^{\mu v}+\kappa\phi^{\mu v},$ (3.2)

where $\kappa^{2}$ is the gravitation’s constant and $\phi^{\mu v}$ the
graviton field. We write

${\cal L}_{G}={\cal L}_{L}+{\cal L}_{I},$ (3.3)

where

${\cal L}_{L}=-\frac{1}{4}[\partial_{\lambda}\phi_{\mu
v}\partial^{\lambda}\phi^{\mu
v}-2\partial_{\alpha}\phi_{\mu\beta}\partial^{\beta}\phi^{\mu\alpha}+2\partial^{\alpha}\phi_{\mu\alpha}\partial_{\beta}\phi^{\mu\beta}],$
(3.4)

and, up to 2nd order, one has [1]:

${\cal L}_{I}=-\frac{1}{2}\kappa\phi^{\mu
v}[\frac{1}{2}\partial_{\mu}\phi^{\lambda\rho}\partial_{v}\phi_{\lambda\rho}+\partial_{\lambda}\phi_{\mu\beta}\partial^{\beta}\phi_{v}^{\lambda}-\partial_{\lambda}\phi_{\mu\rho}\partial^{\lambda}\phi_{v}^{\rho}],$
(3.5)

having made use of the constraint

$\phi_{\mu}^{\mu}=0.$ (3.6)

This constraint is required in order to satisfy gauge invariance [13] For the
graviton we have then

$\square\phi_{\mu v}=0,$ (3.7)

whose solution is

$\phi_{\mu v}=\frac{1}{(2\pi)^{\frac{3}{2}}}\int\left[\frac{a_{\mu
v}(\vec{k})}{\sqrt{2k_{0}}}e^{ik_{\mu}x^{\mu}}+\frac{a^{+}_{\mu
v}(\vec{k})}{\sqrt{2k_{0}}}e^{-ik_{\mu}x^{\mu}}\right]d^{3}k,$ (3.8)

with $k_{0}=|\vec{k}|$.

##  4 The Quantization of the Theory

We need some definitions. The energy-momentum tensor is

$T_{\rho}^{\lambda}=\frac{\partial{\cal L}}{\partial\partial^{\rho}\phi^{\mu
v}}\partial^{\lambda}\phi^{\mu v}-\delta_{\rho}^{\lambda}{\cal L},$ (4.1)

and the time-component of the four-momentum is

$\boldsymbol{\cal P}_{0}=\int T_{0}^{0}\;d^{3}x.$ (4.2)

Using (3.4) we have

$T_{0}^{0}=\frac{1}{4}[\partial_{0}\phi_{\mu v}\partial^{0}\phi^{\mu
v}+\partial_{j}\phi_{\mu v}\partial^{j}\phi^{\mu
v}-2\partial_{\alpha}\phi_{\mu
0}\partial^{0}\phi^{\mu\alpha}-2\partial_{\alpha}\phi_{\mu
j}\partial^{j}\phi^{\mu\alpha}+$
$2\partial_{\alpha}\phi^{\mu\alpha}\partial_{0}\phi_{\mu}^{0}+2\partial_{\alpha}\phi^{\mu\alpha}\partial_{j}\phi_{\mu}^{j}].$
(4.3)

Consequently,

$\boldsymbol{\cal P}_{0}=\frac{1}{4}\int|\vec{k}|\left[a_{\mu
v}(\vec{k})a^{+\mu v}(\vec{k})+a^{+\mu v}(\vec{k})a_{\mu
v}(\vec{k})\right]d^{3}k.$ (4.4)

Appeal to (2.6) leads to

$[\boldsymbol{\cal P}_{0},a_{\mu v}(\vec{k})]=-k_{0}a_{\mu v}(\vec{k})$
$[\boldsymbol{\cal P}_{0},a^{+\mu v}(\vec{k})]=k_{0}a^{+\mu v}(\vec{k}).$
(4.5)

From the last relation in (4.5) one gathers that

$|\vec{k}|a^{+\rho\lambda}(\vec{k^{{}^{\prime}}})=\frac{1}{2}\int|\vec{k}|[a_{\mu
v}(\vec{k}),a^{+\rho\lambda}(\vec{k^{{}^{\prime}}})]a^{+\mu
v}(\vec{k})\;d^{3}k.$ (4.6)

The solution of this integral equation is

$[a_{\mu
v}(\vec{k}),a^{+\rho\lambda}(\vec{k^{{}^{\prime}}})]=\left[\delta_{\mu}^{\rho}\delta_{v}^{\lambda}+\delta_{v}^{\rho}\delta_{\mu}^{\lambda}\right]\delta(\vec{k}-\vec{k^{{}^{\prime}}}).$
(4.7)

As customary, the physical state $|\psi>$ of the theory is defined via the
equation

$\phi_{\mu}^{\mu}|\psi>=0.$ (4.8)

We use now the the usual definition

$\Delta_{\mu\nu}^{\rho\lambda}(x-y)=<0|T[\phi_{\mu\nu}(x)\phi^{\rho\lambda}(y)]|0>.$
(4.9)

The graviton’s propagator then turns out to be

$\Delta_{\mu\nu}^{\rho\lambda}(x-y)=\frac{i}{(2\pi)^{4}}(\delta_{\mu}^{\rho}\delta_{v}^{\lambda}+\delta_{v}^{\rho}\delta_{\mu}^{\lambda})\int\frac{e^{ik_{\mu}(x^{\mu}-y^{\mu})}}{k^{2}-i0}d^{4}k.$
(4.10)

As a consequence, we can write

$\boldsymbol{\cal P}_{0}=\frac{1}{4}\int|\vec{k}|\left[a_{\mu
v}(\vec{k})a^{+\mu v}(\vec{k^{{}^{\prime}}})+a^{+\mu
v}(\vec{k^{{}^{\prime}}})a_{\mu
v}(\vec{k})\right]\delta(\vec{k}-\vec{k^{{}^{\prime}}})d^{3}kd^{3}k^{{}^{\prime}},$
(4.11)

or

$\boldsymbol{\cal P}_{0}=\frac{1}{4}\int|\vec{k}|\left[2a^{+\mu
v}(\vec{k^{{}^{\prime}}})a_{\mu
v}(\vec{k})+\delta(\vec{k}-\vec{k^{{}^{\prime}}})\right]\delta(\vec{k}-\vec{k^{{}^{\prime}}})d^{3}kd^{3}k^{{}^{\prime}}.$
(4.12)

Thus, we obtain

$\boldsymbol{\cal P}_{0}=\frac{1}{2}\int|\vec{k}|a^{+\mu v}(\vec{k})a_{\mu
v}(\vec{k})d^{3}k,$ (4.13)

where we have used the fact that the product of two deltas with the same
argument vanishes [2], i.e.,
$\delta(\vec{k}-\vec{k^{{}^{\prime}}})\delta(\vec{k}-\vec{k^{{}^{\prime}}})=0$.
This illustrates the fact that using Ultrahyperfunctions is here equivalent to
adopting the normal order in the definition of the time-component of the four-
momentum

$\boldsymbol{\cal P}_{0}=\frac{1}{4}\int|\vec{k}|:\left[a_{\mu
v}(\vec{k})a^{+\mu v}(\vec{k})+a^{+\mu v}(\vec{k})a_{\mu
v}(\vec{k})\right]:d^{3}k.$ (4.14)

Now, we must insist on the fact that the physical state should satisfy not
only Eq. (4.8) but also the relation (see [1])

$\partial_{\mu}\phi^{\mu v}|\psi>=0.$ (4.15)

The ensuing theory is similar to the QED-one obtained via the quantization
approach of Gupta-Bleuler. This implies that the theory is unitary for any
finite perturbative order. In this theory only one type of graviton emerges,
$\phi^{12}$, while in Gupta’s approach two kinds of graviton emerge.
Obviously, this happens for a non-interacting theory, as remarked by Gupta.

###  4.1 Undesired effects if one does not use our constraint

If we do NOT use the constraint (4.8), we have

$\boldsymbol{\cal P}_{0}=\frac{1}{2}\int|\vec{k}|\left[a^{+\mu
v}(\vec{k})a_{\mu
v}(\vec{k})-\frac{1}{2}a_{\mu}^{+\mu}(\vec{k})a_{v}^{v}(\vec{k})\right]d^{3}k,$
(4.16)

and, appealing to the Schwinger-Feynman variational principle we find

$|\vec{k}|a_{\rho\lambda}^{+}(\vec{k^{{}^{\prime}}})=$
$\frac{1}{2}\int|\vec{k}|\left\\{a^{+\mu v}(\vec{k})[a_{\mu
v}(\vec{k}),a_{\rho\lambda}^{+}(\vec{k^{{}^{\prime}}})]-\frac{1}{2}a_{\mu}^{+\mu}(\vec{k})[a_{v}^{v}(\vec{k}),a_{\rho\lambda}^{+}(\vec{k^{{}^{\prime}}})]\right\\}\;d^{3}k,$
(4.17)

whose solution is

$[a_{\mu
v}(\vec{k}),a_{\rho\lambda}^{+}(\vec{k^{{}^{\prime}}})]=\left[\eta_{\mu\rho}\eta_{v\lambda}+\eta_{v\rho}\eta_{\mu\lambda}-\eta_{\mu
v}\eta_{\rho\lambda}\right]\delta(\vec{k}-\vec{k^{{}^{\prime}}}).$ (4.18)

The above is the customary graviton’s quantification, that leads to a theory
whose $S$ matrix in not unitary [1, 9].

##  5 The self energy of the graviton

To evaluate the graviton’s self-energy (SF) we start with the interaction
Hamiltonian ${\cal H}_{I}$. Note that the Lagrangian contains derivative
interaction terms.

${\cal H}_{I}=\frac{\partial{\cal
L}_{I}}{\partial\partial^{0}\phi^{\mu\nu}}\partial^{0}\phi^{\mu\nu}-{\cal
L}_{I}.$ (5.1)

A typical term reads

$\Sigma_{G\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}(k)=k_{\alpha_{1}}k_{\alpha_{2}}(\rho-i0)^{-1}\ast
k_{\alpha_{3}}k_{\alpha_{4}}(\rho-i0)^{-1}.$ (5.2)

where $\rho=k_{1}^{2}+k_{2}^{2}+k_{3}^{2}-k_{0}^{2}$

In $\nu$ dimensions, the Fourier transform of (5.2) is

${\cal F}\\{[k_{\alpha_{1}}k_{\alpha_{2}}(\rho-i0)^{-1}\ast
k_{\alpha_{3}}k_{\alpha_{4}}(\rho-i0)^{-1}]_{\nu}\\}=$
$-\frac{2^{2\nu-2}}{(2\pi)^{\nu}}\pi^{\nu}\left[\Gamma\left(\frac{\nu}{2}\right)\right]^{2}\eta_{{\alpha_{1}}{\alpha_{2}}}\eta_{{\alpha_{3}}{\alpha_{4}}}(x+i0)^{-\nu}+$
$\frac{2^{2\nu-1}}{(2\pi)^{\nu}}\pi^{\nu}\Gamma\left(\frac{\nu}{2}\right)\Gamma\left(\frac{\nu}{2}+1\right)(\eta_{\alpha_{1}\alpha_{2}}x_{\alpha_{3}}x_{\alpha_{4}}+\eta_{\alpha_{3}\alpha_{4}}x_{\alpha_{1}}x_{\alpha_{2}})(x+i0)^{-\nu-1}-$
$\left.\frac{2^{2\nu}}{(2\pi)^{\nu}}\pi^{\nu}\left[\Gamma\left(\frac{\nu}{2}+1\right)\right]^{2}x_{\alpha_{1}}x_{\alpha_{2}}x_{\alpha_{3}}x_{\alpha_{4}}(x+i0)^{-\nu-2}\right\\}$
(5.3)

where $x=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{0}^{2}$

Anti-transforming the above equation we have

$[k_{\alpha_{1}}k_{\alpha_{2}}(\rho-i0)^{-1}\ast
k_{\alpha_{3}}k_{\alpha_{4}}(\rho-i0)^{-1}]_{\nu}=$
$i\frac{\pi^{\frac{\nu}{2}}}{4}\frac{\left[\Gamma\left(\frac{\nu}{2}+1\right)\right]^{2}}{\Gamma(\nu+2)}(\eta_{\alpha_{1}\alpha_{2}}\eta_{\alpha_{3}\alpha_{4}}+\eta_{\alpha_{2}\alpha_{3}}\eta_{\alpha_{1}\alpha_{4}}+\eta_{\alpha_{2}\alpha_{4}}\eta_{\alpha_{1}\alpha_{3}})\Gamma\left(-\frac{\nu}{2}\right)(\rho-i0)^{\frac{\nu}{2}}+$
$\left\\{i\frac{\pi^{\frac{\nu}{2}}}{2}\frac{\Gamma\left(\frac{\nu}{2}\right)\Gamma\left(\frac{\nu}{2}+1\right)}{\Gamma(\nu+1)}(\eta_{\alpha_{1}\alpha_{2}}k_{\alpha_{3}}k_{\alpha_{4}}+\eta_{\alpha_{3}\alpha_{4}}k_{\alpha_{1}}k_{\alpha_{2}})-\right.$
$i\frac{\pi^{\frac{\nu}{2}}}{2}\frac{\left[\Gamma\left(\frac{\nu}{2}+1\right)\right]^{2}}{\Gamma(\nu+2)}(\eta_{\alpha_{1}\alpha_{2}}k_{\alpha_{3}}k_{\alpha_{4}}+\eta_{\alpha_{1}\alpha_{3}}k_{\alpha_{2}}k_{\alpha_{4}}+\eta_{\alpha_{1}\alpha_{4}}k_{\alpha_{2}}k_{\alpha_{3}}+\eta_{\alpha_{3}\alpha_{4}}k_{\alpha_{1}}k_{\alpha_{2}}+$
$\left.\eta_{\alpha_{2}\alpha_{3}}k_{\alpha_{1}}k_{\alpha_{4}}+\eta_{\alpha_{2}\alpha_{4}}k_{\alpha_{1}}k_{\alpha_{3}})\right\\}\Gamma\left(1-\frac{\nu}{2}\right)(\rho-i0)^{\frac{\nu}{2}-1}+$
$i\pi^{\frac{\nu}{2}}\frac{\left[\Gamma\left(\frac{\nu}{2}+1\right)\right]^{2}}{\Gamma(\nu+2)}k_{\alpha_{1}}k_{\alpha_{2}}k_{\alpha_{3}}k_{\alpha_{4}}\Gamma\left(2-\frac{\nu}{2}\right)(\rho-i0)^{\frac{\nu}{2}-2}$
(5.4)

###  5.1 Self-Energy evaluation for $\nu=4$

We appeal now to a $\nu$-Laurent expansion and retain there the $\nu-4$
independent term [6]. Thus, we Laurent-expand (5.4) around $\nu=4$ and find

$[k_{\alpha_{1}}k_{\alpha_{2}}(\rho-i0)^{-1}\ast
k_{\alpha_{3}}k_{\alpha_{4}}(\rho-i0)^{-1}]_{\nu}=$
$i\frac{\pi^{2}}{\nu-4}\left\\{\frac{1}{5!}(\eta_{\alpha_{1}\alpha_{2}}\eta_{\alpha_{3}\alpha_{4}}+\eta_{\alpha_{2}\alpha_{3}}\eta_{\alpha_{1}\alpha_{4}}+\eta_{\alpha_{2}\alpha_{4}}\eta_{\alpha_{1}\alpha_{3}})\rho^{2}\right.-$
$\left[\frac{2}{4!}(\eta_{\alpha_{1}\alpha_{2}}k_{\alpha_{3}}k_{\alpha_{4}}+\eta_{\alpha_{3}\alpha_{4}}k_{\alpha_{1}}k_{\alpha_{2}})\right.-$
$\frac{1}{6!}(\eta_{\alpha_{1}\alpha_{2}}k_{\alpha_{3}}k_{\alpha_{4}}+\eta_{\alpha_{3}\alpha_{4}}k_{\alpha_{1}}k_{\alpha_{2}}+\eta_{\alpha_{1}\alpha_{3}}k_{\alpha_{2}}k_{\alpha_{4}}+\eta_{\alpha_{1}\alpha_{4}}k_{\alpha_{2}}k_{\alpha_{3}}+$
$\eta_{\alpha_{2}\alpha_{3}}k_{\alpha_{1}}k_{\alpha_{4}}+\eta_{\alpha_{2}\alpha_{4}}k_{\alpha_{1}}k_{\alpha_{3}})]\left.\rho+\frac{8}{5!}k_{\alpha_{1}}k_{\alpha_{2}}k_{\alpha_{3}}k_{\alpha_{4}}\right\\}-$
$\frac{i\pi^{2}}{5!2}(\eta_{\alpha_{1}\alpha_{2}}\eta_{\alpha_{3}\alpha_{4}}+\eta_{\alpha_{2}\alpha_{3}}\eta_{\alpha_{1}\alpha_{4}}+\eta_{\alpha_{2}\alpha_{4}}\eta_{\alpha_{1}\alpha_{3}})\left[\ln(\rho-i0)+\ln\pi+C-\frac{46}{15}\right]\rho^{2}+$
$i\frac{\pi^{2}}{4!}\left\\{(\eta_{\alpha_{1}\alpha_{2}}k_{\alpha_{3}}k_{\alpha_{4}}+\eta_{\alpha_{3}\alpha_{4}}k_{\alpha_{1}}k_{\alpha_{2}})\left[\ln(\rho-i0)+\ln\pi+C-\frac{8}{3}\right]\right.-$
$\frac{1}{24}(\eta_{\alpha_{1}\alpha_{2}}k_{\alpha_{3}}k_{\alpha_{4}}+\eta_{\alpha_{3}\alpha_{4}}k_{\alpha_{1}}k_{\alpha_{2}}+\eta_{\alpha_{1}\alpha_{3}}k_{\alpha_{2}}k_{\alpha_{4}}+\eta_{\alpha_{1}\alpha_{4}}k_{\alpha_{2}}k_{\alpha_{3}}+\eta_{\alpha_{2}\alpha_{3}}k_{\alpha_{1}}k_{\alpha_{4}}+$
$\eta_{\alpha_{2}\alpha_{4}}k_{\alpha_{1}}k_{\alpha_{3}})\left.\left[\ln(\rho-i0)+\ln\pi+2C-\frac{101}{15}\right]\right\\}\rho-$
$\left.i\frac{\pi^{2}}{30}k_{\alpha_{1}}k_{\alpha_{2}}k_{\alpha_{3}}k_{\alpha_{4}}\left[\ln(\rho-i0)+\ln\pi+C-\frac{47}{30}\right]+\sum\limits_{n=1}^{\infty}a_{n}(\nu-4)^{n}\right\\}.$
(5.5)

The exact value of the convolution we are interested in, i.e., the left hand
side of (5.5), is given by the independent term in the above expansion, as it
is well-known. If the reader is not familiar with this situation, see for
instance [6]. We reach

$\Sigma_{G\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}(k)=k_{\alpha_{1}}k_{\alpha_{2}}(\rho-i0)^{-1}\ast
k_{\alpha_{3}}k_{\alpha_{4}}(\rho-i0)^{-1}=-$
$\frac{i\pi^{2}}{5!2}(\eta_{\alpha_{1}\alpha_{2}}\eta_{\alpha_{3}\alpha_{4}}+\eta_{\alpha_{2}\alpha_{3}}\eta_{\alpha_{1}\alpha_{4}}+\eta_{\alpha_{2}\alpha_{4}}\eta_{\alpha_{1}\alpha_{3}})\left[\ln(\rho-i0)+\ln\pi+C-\frac{46}{15}\right]\rho^{2}-$
$i\frac{\pi^{2}}{4!}\left\\{(\eta_{\alpha_{1}\alpha_{2}}k_{\alpha_{3}}k_{\alpha_{4}}+\eta_{\alpha_{3}\alpha_{4}}k_{\alpha_{1}}k_{\alpha_{2}})\left[\ln(\rho-i0)+\ln\pi+C-\frac{8}{3}\right]\right.-$
$\frac{1}{24}(\eta_{\alpha_{1}\alpha_{2}}k_{\alpha_{3}}k_{\alpha_{4}}+\eta_{\alpha_{3}\alpha_{4}}k_{\alpha_{1}}k_{\alpha_{2}}+\eta_{\alpha_{1}\alpha_{3}}k_{\alpha_{2}}k_{\alpha_{4}}+\eta_{\alpha_{1}\alpha_{4}}k_{\alpha_{2}}k_{\alpha_{3}}+\eta_{\alpha_{2}\alpha_{3}}k_{\alpha_{1}}k_{\alpha_{4}}+$
$\eta_{\alpha_{2}\alpha_{4}}k_{\alpha_{1}}k_{\alpha_{3}})\left.\left[\ln(\rho-i0)+\ln\pi+2C-\frac{101}{15}\right]\right\\}\rho-$
$\left.i\frac{\pi^{2}}{30}k_{\alpha_{1}}k_{\alpha_{2}}k_{\alpha_{3}}k_{\alpha_{4}}\left[\ln(\rho-i0)+\ln\pi+C-\frac{47}{30}\right]\right\\}.$
(5.6)

We have to deal with 1296 diagrams of this kind.

##  6 Including Axions into the picture

Axions are hypothetical elementary particles postulated by the Peccei–Quinn
theory in 1977 to tackle the strong CP problem in quantum chromodynamics. If
they exist and have low enough mass (within a certain range), they could be of
interest as possible components of cold dark matter [14].We include now a
massive scalar field (axions) interacting with the graviton. The Lagrangian
becomes

${\cal
L}_{GM}=\frac{1}{\kappa^{2}}\boldsymbol{R}\sqrt{|g|}-\frac{1}{2}\eta_{\mu
v}\partial_{\alpha}h^{\mu\alpha}\partial_{\beta}h^{v\beta}-\frac{1}{2}[h^{\mu
v}\partial_{\mu}\phi\partial_{v}\phi+m^{2}\phi^{2}].$ (6.1)

We can now recast the Lagrangian in the fashion

${\cal L}_{GM}={\cal L}_{L}+{\cal L}_{I}+{\cal L}_{LM}+{\cal L}_{IM},$ (6.2)

where

${\cal
L}_{LM}=-\frac{1}{2}[\partial_{\mu}\phi\partial^{\mu}\phi+m^{2}\phi^{2}],$
(6.3)

so that ${\cal L}_{IM}$ becomes the Lagrangian for the axion-graviton action

${\cal
L}_{IM}=-\frac{1}{2}\kappa\phi^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi.$
(6.4)

The new term in the interaction Hamiltonian is

${\cal H}_{IM}=\frac{\partial{\cal
L}_{IM}}{\partial\partial^{0}\phi}\partial^{0}\phi-{\cal L}_{IM}.$ (6.5)

##  7 The complete Self Energy of the Graviton

The presence of axions generates a new contribution to the graviton’s self
energy

$\Sigma_{GM\mu rvs}(k)=k_{\mu}k_{r}(\rho+m^{2}-i0)^{-1}\ast
k_{v}k_{s}(\rho+m^{2}-i0)^{-1}.$ (7.1)

So as to compute it we appeal to the usual $\nu$ dimensional integral together
with the Feynman-parameters denoted by the letter x. After a Wick rotation we
obtain

$[k_{\mu}k_{r}(\rho+m^{2}-i0)^{-1}\ast
k_{v}k_{s}(\rho+m^{2}-i0)^{-1}]_{\nu}=i\int\limits_{0}^{1}\int\frac{k_{\mu}k_{r}(p_{v}-k_{v})(p_{s}-k_{s})}{[(k-px)^{2}+a]^{2}}d^{\nu}kdx,$
(7.2)

where

$a=p^{2}x-p^{2}x^{2}+m^{2}.$ (7.3)

After the variables-change $u=k-px$ we find

$[k_{\mu}k_{r}(\rho+m^{2}-i0)^{-1}\ast
k_{v}k_{s}(\rho+m^{2}-i0)^{-1}]_{\nu}=i\int\limits_{0}^{1}\int\frac{f(u,x,\mu,r,v,s)}{(u^{2}+a)^{2}}d^{\nu}udx,$
(7.4)

where

${f(u,x,\mu,r,v,s)}=u_{\mu}u_{r}p_{v}p_{s}(1-x)^{2}+u_{\mu}u_{r}u_{v}u_{s}-u_{\mu}u_{s}p_{r}p_{v}x(1-x)-$
$u_{\mu}u_{v}p_{r}p_{s}x(1-x)-u_{r}u_{s}p_{\mu}p_{v}x(1-x)-u_{r}u_{v}p_{\mu}p_{s}x(1-x)+$
$p_{\mu}p_{r}p_{v}p_{s}x^{2}(1-x)^{2}+u_{v}u_{s}p_{\mu}p_{r}x^{2}.$ (7.5)

After evaluation of the pertinent integrals we arrive at

$[k_{\mu}k_{r}(\rho+m^{2}-i0)^{-1}\ast k_{v}k_{s}(\rho+m^{2}-i0)^{-1}]_{\nu}=$
$i\frac{(\eta_{\mu
r}k_{v}k_{s}+\eta_{vs}k_{\mu}k_{r})m^{\nu-2}\pi^{\frac{\nu}{2}}}{8}\Gamma\left(1-\frac{\nu}{2}\right)\times$
$\left[F\left(1,1-\frac{\nu}{2},\frac{3}{2};-\frac{\rho}{4m^{2}}\right)+\frac{1}{3}F\left(1,1-\frac{\nu}{2},\frac{5}{2};-\frac{\rho}{4m^{2}}\right)\right]+$
$i(\eta_{\mu r}\eta_{vs}+\eta_{\mu v}\eta_{rs}+\eta_{\mu
s}\eta_{vr})\frac{\pi^{\frac{\nu}{2}}m^{\nu}}{4}\Gamma\left(\frac{\nu}{2}\right)\Gamma\left(-\frac{\nu}{2}\right)F\left(1,-\frac{\nu}{2},\frac{3}{2};-\frac{\rho}{4m^{2}}\right)-$
$i(\eta_{\mu s}k_{r}k_{v}+\eta_{\mu
v}k_{r}k_{s}+\eta_{rs}k_{\mu}k_{v}+\eta_{rv}k_{\mu}k_{s})\frac{m^{\nu-2}\pi^{\frac{\nu}{2}}}{48}\times$
$\Gamma\left(1-\frac{\nu}{2}\right)F\left(2,1-\frac{\nu}{2},\frac{5}{2};-\frac{\rho}{4m^{2}}\right)+$
$ik_{\mu}k_{r}k_{v}k_{s}\frac{m^{\nu-4}\pi^{\frac{\nu}{2}}}{12}\Gamma\left(2-\frac{\nu}{2}\right)F\left(2,2-\frac{\nu}{2},\frac{5}{2};-\frac{\rho}{4m^{2}}\right).$
(7.6)

###  7.1 Self-Energy evaluation for $\nu=4$

We need again a Laurent’s expansion and face

$[k_{\mu}k_{r}(\rho+m^{2}-i0)^{-1}\ast k_{v}k_{s}(\rho+m^{2}-i0)^{-1}]_{\nu}=$
$-i\frac{\pi^{2}}{\nu-4}\left\\{m^{2}(\eta_{\mu
r}k_{v}k_{s}+\eta_{vs}k_{\mu}k_{r})\left[\frac{1}{3}+\frac{1}{5}\frac{\rho}{4m^{2}}\right]-\right.$
$2{m^{4}}(\eta_{\mu r}\eta_{vs}+\eta_{\mu v}\eta_{rs}+\eta_{\mu
s}\eta_{rv})\times$
$\left[\frac{1}{8}+\frac{1}{6}\frac{\rho}{4m^{2}}+\frac{1}{15}\left(\frac{\rho}{4m^{2}}\right)^{2}\right]-$
$\frac{m^{2}}{4m^{2}+k^{2}-i0}(\eta_{\mu s}k_{r}k_{v}+\eta_{\mu
v}k_{r}k_{s}+\eta_{rs}k_{\mu}k_{v}+\eta_{rv}k_{\mu}k_{s})\times$
$\frac{k^{2}-m^{2}}{12}+\frac{m^{2}}{4}+\frac{k^{2}-m^{2}}{30}\frac{\rho}{4m^{2}}-\left.\frac{1}{6}k_{\mu}k_{r}k_{v}k_{s}\right\\}+$
$i\frac{m^{2}\pi^{2}}{2}(\eta_{\mu r}k_{v}k_{s}+\eta_{vs}k_{\mu}k_{r})\times$
$\left[\frac{1}{3}(\ln
m^{2}+\ln\pi+C-1)+\frac{1}{5}\frac{\rho}{4m^{2}}\left(\ln
m^{2}+\ln\pi+C\right)\right]+$ $i\frac{m^{2}\pi^{2}}{30}(\eta_{\mu
r}k_{v}k_{s}+\eta_{vs}k_{\mu}k_{r})\frac{\rho}{4m^{2}}\times$
$\left[F\left(1,1,\frac{7}{2};-\frac{\rho}{4m^{2}}\right)+\frac{1}{7}F\left(1,1,\frac{9}{2};-\frac{\rho}{4m^{2}}\right)\right]+$
$-i2\pi^{2}m^{4}(\eta_{\mu r}\eta_{vs}+\eta_{\mu v}\eta_{rs}+\eta_{\mu
s}\eta_{vr})\times$
$\left\\{\left[\frac{1}{8}-\frac{1}{6}\frac{\rho}{4m^{2}}-\frac{1}{15}\left(\frac{\rho}{4m^{2}}\right)^{2}+\right]\right.\times$
$\left(\ln
m^{2}+\ln\pi+1\right)-\left.\frac{1}{2}\left[\frac{3}{32}-\frac{1}{3}\left(\frac{\rho}{4m^{2}}\right)\right]\right\\}-$
$i\frac{2\pi^{2}m^{4}}{105}(\eta_{\mu r}\eta_{vs}+\eta_{\mu
v}\eta_{rs}+\eta_{\mu
s}\eta_{vr})\left(\frac{\rho}{4m^{2}}\right)^{3}F\left(1,1,\frac{9}{2};-\frac{\rho}{4m^{2}}\right)-$
$i\frac{\pi^{2}m^{2}(k^{2}-m^{2})}{12(4m^{2}+k^{2}-i0}(\eta_{\mu
s}k_{r}k_{v}+\eta_{\mu
v}k_{r}k_{s}+\eta_{rs}k_{\mu}k_{v}+\eta_{rv}k_{\mu}k_{s})\times$
$\left[\frac{1}{2}\left(\ln
m^{2}+\ln\pi+C-\frac{1}{4}\right)+\frac{1}{5}\left(\ln
m^{2}+\ln\pi+C\right)\frac{k^{2}}{4m^{2}}\right]-$
$i\frac{\pi^{2}m^{2}}{8(4m^{2}+k^{2}-i0}(\eta_{\mu s}k_{r}k_{v}+\eta_{\mu
v}k_{r}k_{s}+\eta_{rs}k_{\mu}k_{v}+\eta_{rv}k_{\mu}k_{s})\times$
$m^{2}\left[\left(\ln
m^{2}+\ln\pi+C-\frac{1}{4}\right)+\frac{k^{2}}{6}+\frac{k^{2}}{15}\frac{k^{2}}{4m^{2}}\right]-$
$i\frac{\pi^{2}m^{2}}{10}(\eta_{\mu s}k_{r}k_{v}+\eta_{\mu
v}k_{r}k_{s}+\eta_{rs}k_{\mu}k_{v}+\eta_{rv}k_{\mu}k_{s})\times$
$\frac{k^{2}-m^{2}}{21(4m^{2}+k^{2}-i0)}F\left(1,1,\frac{9}{2};-\frac{\rho}{4m^{2}}\right)\left(\frac{k^{2}}{4m^{2}}\right)^{2}-$
$i\frac{\pi^{2}}{12}k_{\mu}k_{r}k_{v}k_{s}\left[\left(\ln
m^{2}+\ln\pi\right)+\frac{k^{2}}{4m^{2}+k^{2}-i0}\right]-$
$i\frac{\pi^{2}m^{2}}{30}k_{\mu}k_{r}k_{v}k_{s}\frac{k^{2}-m^{2}}{4m^{2}+k^{2}-i0}\frac{k^{2}}{4m^{2}}F\left(1,1,\frac{7}{2};-\frac{k^{2}}{4m^{2}}\right)+$
$\sum\limits_{n=0}^{\infty}a_{n}(\nu-4)^{n}.$ (7.7)

Again, the exact result for our four-dimensional convolution becomes

$\Sigma_{GM\mu vrs}(k)=k_{\mu}k_{r}(\rho+m^{2}-i0)^{-1}\ast
k_{v}k_{s}(\rho+m^{2}-i0)^{-1}=$ $i\frac{m^{2}\pi^{2}}{2}(\eta_{\mu
r}k_{v}k_{s}+\eta_{vs}k_{\mu}k_{r})\times$ $\left[\frac{1}{3}(\ln
m^{2}+\ln\pi+C-1)+\frac{1}{5}\frac{\rho}{4m^{2}}\left(\ln
m^{2}+\ln\pi+C\right)\right]+$ $i\frac{m^{2}\pi^{2}}{30}(\eta_{\mu
r}k_{v}k_{s}+\eta_{vs}k_{\mu}k_{r})\frac{\rho}{4m^{2}}\times$
$\left[F\left(1,1,\frac{7}{2};-\frac{\rho}{4m^{2}}\right)+\frac{1}{7}F\left(1,1,\frac{9}{2};-\frac{\rho}{4m^{2}}\right)\right]+$
$-i2\pi^{2}m^{4}(\eta_{\mu r}\eta_{vs}+\eta_{\mu v}\eta_{rs}+\eta_{\mu
s}\eta_{vr})\times$
$\left\\{\left[\frac{1}{8}-\frac{1}{6}\frac{\rho}{4m^{2}}-\frac{1}{15}\left(\frac{\rho}{4m^{2}}\right)^{2}+\right]\right.\times$
$\left(\ln
m^{2}+\ln\pi+1\right)-\left.\frac{1}{2}\left[\frac{3}{32}-\frac{1}{3}\left(\frac{\rho}{4m^{2}}\right)\right]\right\\}+$
$i\frac{2\pi^{2}m^{4}}{105}(\eta_{\mu r}\eta_{vs}+\eta_{\mu
v}\eta_{rs}+\eta_{\mu
s}\eta_{vr})\left(\frac{\rho}{4m^{2}}\right)^{3}F\left(1,1,\frac{9}{2};-\frac{\rho}{4m^{2}}\right)-$
$i\frac{\pi^{2}m^{2}(k^{2}-m^{2})}{12(4m^{2}+k^{2}-i0}(\eta_{\mu
s}k_{r}k_{v}+\eta_{\mu
v}k_{r}k_{s}+\eta_{rs}k_{\mu}k_{v}+\eta_{rv}k_{\mu}k_{s})\times$
$\left[\frac{1}{2}\left(\ln
m^{2}+\ln\pi+C-\frac{1}{4}\right)+\frac{1}{5}\left(\ln
m^{2}+\ln\pi+C\right)\frac{k^{2}}{4m^{2}}\right]-$
$i\frac{\pi^{2}m^{2}}{8(4m^{2}+k^{2}-i0}(\eta_{\mu s}k_{r}k_{v}+\eta_{\mu
v}k_{r}k_{s}+\eta_{rs}k_{\mu}k_{v}+\eta_{rv}k_{\mu}k_{s})\times$
$m^{2}\left[\left(\ln
m^{2}+\ln\pi+C-\frac{1}{4}\right)+\frac{k^{2}}{6}+\frac{k^{2}}{15}\frac{k^{2}}{4m^{2}}\right]-$
$i\frac{\pi^{2}m^{2}}{10}(\eta_{\mu s}k_{r}k_{v}+\eta_{\mu
v}k_{r}k_{s}+\eta_{rs}k_{\mu}k_{v}+\eta_{rv}k_{\mu}k_{s})\times$
$\frac{k^{2}-m^{2}}{21(4m^{2}+k^{2}-i0)}F\left(1,1,\frac{9}{2};-\frac{\rho}{4m^{2}}\right)\left(\frac{k^{2}}{4m^{2}}\right)^{2}-$
$i\frac{\pi^{2}}{12}k_{\mu}k_{r}k_{v}k_{s}\left[\left(\ln
m^{2}+\ln\pi\right)-\frac{k^{2}}{4m^{2}+k^{2}-i0}\right]-$
$i\frac{\pi^{2}m^{2}}{30}k_{\mu}k_{r}k_{v}k_{s}\frac{k^{2}-m^{2}}{4m^{2}+k^{2}-i0}\frac{k^{2}}{4m^{2}}F\left(1,1,\frac{7}{2};-\frac{k^{2}}{4m^{2}}\right)$
(7.8)

We have to deal with 9 diagrams of this kind.
Accordingly, our desired self-energy total is a combination of
$\Sigma_{G\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}(k)$ and
$\Sigma_{GM\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}(k)$.

##  8 Self Energy of the Axion

Here the self-energy is

$\Sigma^{\mu s}(k)=(\eta^{\mu r}\eta^{vs}+\eta^{\mu
s}\eta^{vr})k_{v}k_{r}(\rho+m^{2}-i0)^{-1}\ast(\rho-i0)^{-1}.$ (8.1)

In $\nu$ dimensions one has

$[k_{v}k_{r}(\rho+m^{2}-i0)^{-1}\ast(\rho-i0)^{-1}]_{\nu}=\int\frac{k_{v}k_{r}}{(k^{2}+m^{2}-i0)[(p-k)^{2}-i0]}d^{\nu}k.$
(8.2)

With the Feynman parameters used above we obtain

$[k_{v}k_{r}(\rho+m^{2}-i0)^{-1}\ast(\rho-i0)^{-1}]_{\nu}=i\int\limits_{0}^{1}\int\frac{k_{v}k_{r}}{[(k-px)^{2}+a]^{2}}d^{\nu}kdx,$
(8.3)

where

$a=(p^{2}+m^{2})x-p^{2}x^{2}.$ (8.4)

We evaluate the integral (8.3) and find

$[k_{v}k_{r}(\rho+m^{2}-i0)^{-1}\ast(\rho-i0)^{-1}]_{\nu}=$
$i\frac{\eta_{vr}m^{\nu-2}\pi^{\frac{\nu}{2}}}{\nu}\Gamma\left(1-\frac{\nu}{2}\right)F\left(1,1-\frac{\nu}{2},\frac{\nu}{2}+1;-\frac{\rho}{m^{2}}\right)+$
$\frac{2ik_{v}k_{r}m^{\nu-4}\pi^{\frac{\nu}{2}}}{\nu+2}\Gamma\left(2-\frac{\nu}{2}\right)F\left(1,2-\frac{\nu}{2},\frac{\nu}{2}+2;-\frac{\rho}{m^{2}}\right).$
(8.5)

###  8.1 Self-Energy evaluation for $\nu=4$

Once again, we Laurent-expand, this time (8.5) around $\nu=4$, encountering

$[k_{v}k_{r}(\rho+m^{2}-i0)^{-1}\ast(\rho-i0)^{-1}]_{\nu}=$
$i\pi^{2}\left\\{\frac{1}{\nu-4}\left(\frac{\eta_{vr}m^{2}}{2}-2k_{v}k_{r}\right)\right.+$
$\frac{\eta_{vr}m^{2}}{4}\left[\left(1+\frac{1}{3}\frac{\rho}{m^{2}}\right)\left(\ln
m^{2}+\ln\pi+C-\frac{1}{2}\right)-\right.$
$\left.\left(1+\frac{1}{9}\frac{\rho}{m^{2}}\right)\right]-\frac{k_{v}k_{r}}{3}\left(\ln
m^{2}+\ln\pi+C-\frac{1}{2}\right)+$
$\frac{1}{4}\left(\frac{\rho}{m^{2}}\right)\left[\frac{\eta_{vr}m^{2}}{12}\frac{\rho}{m^{2}}-\frac{k_{v}k_{r}}{3}\right]F\left(1,1,5;-\frac{\rho}{m^{2}}\right)+$
$\left.\sum\limits_{n=1}^{\infty}a_{n}(\nu-4)^{n}\right\\}$ (8.6)

The $\nu$-independent term gives the exact convolution result we are looking
for:

$\Sigma_{vr}(k)=k_{v}k_{r}(\rho+m^{2}-i0)^{-1}\ast(\rho-i0)^{-1}=$
$i\pi^{2}\left\\{\frac{\eta_{vr}m^{2}}{4}\left[\left(1+\frac{1}{3}\frac{\rho}{m^{2}}\right)\left(\ln
m^{2}+\ln\pi+C-\frac{1}{2}\right)-\right.\right.$
$\left.\left(1+\frac{1}{9}\frac{\rho}{m^{2}}\right)\right]-\frac{k_{v}k_{r}}{3}\left(\ln
m^{2}+\ln\pi+C-\frac{1}{2}\right)+$
$\left.\frac{1}{4}\left(\frac{\rho}{m^{2}}\right)\left[\frac{\eta_{vr}m^{2}}{12}\frac{\rho}{m^{2}}-\frac{k_{v}k_{r}}{3}\right]F\left(1,1,5;-\frac{\rho}{m^{2}}\right)\right\\}$
(8.7)

##  9 Discussion

We have developed above the quantum field theory (QFT) of Eintein’s gravity
(EG), that is both unitary and finite. Our results critically depend on the
use of a rather novel constraint the we introduced in defining the EG-
Lagrangian. Laurent expansions were an indispensable tool for us. In order to
quantize the theory we appealed to the variational principle of Schwinger-
Feynman’s. This process leads to just one graviton type $\phi^{12}$. The
underlying mathematics used in this effort has been developed by Bollini et
al. [2, 3, 4, 5, 6]. This mathematics is powerful enough so as to be able to
quantize non-renormalizable field theories [2, 3, 4, 5, 6]. We have evaluated
here in finite and exact fashion, for the first time as far as we know,
several quantities:

  * •

the graviton’s self-energy in the EG-field. This requires full use of the
theory of distributions, appealing to the possibility of creating with them a
ring with divisors of zero.

  * •

the above self-energy in the added presence of a massive scalar field (axions,
for instance). Two types of diagram ensue: the original ones of the pure EG
field plus the ones originated by the addition of a scalar field.

  * •

The axion’s self-energy.

  * •

Our central results revolve around Eqs. (5.6) and (7.8), corresponding to the
graviton’s self-energy, without and with the added presence of axions. Also,
we give the axion’s self-energy.

## References

  * [1] S. N. Gupta: Proc. Pys. Soc. A 65, 161 (1956); ibid 65 608 (1956). Phys. Rev. 172, 1303 (1968).
  * [2] C. G. Bollini, T. Escobar and M. C. Rocca : Int. J. of Theor. Phys. 38, 2315 (1999).
  * [3] C. G. Bollini and M. C. Rocca : Int. J. of Theor. Phys. 43, 1019 (2004).
  * [4] C. G. Bollini and M. C. Rocca : Int. J. of Theor. Phys. 43, 59 (2004).
  * [5] C. G. Bollini, P. Marchiano and M. C. Rocca : Int. J. of Theor. Phys. 46, 3030 (2007).
  * [6] A. Plastino, M. C. Rocca: J. Phys. Commun. 2, 115029 (2018).
  * [7] A. Grothendieck: Produits Tensoriels Topologiques et Espaces Nucleaires Mem. Amer. Math Soc. 16 1955\.
  * [8] J. Sebastiao e Silva : Math. Ann. 136, 38 (1958).
  * [9] R. P. Feynman: Acta Phys. Pol. 24, 697 (1963).
  * [10] A.Visconti:Quantum Field Theory. Pergamon Press (1969).
  * [11] R. Delbourgo and V. B. Prasad: J. Phys. G: Nuclear Physics 1, 377 (1975).
  * [12] D. G. Barci, C.G. Bollini, M. C. Rocca: Il Nuovo Cimento 108, 797 (1995).
  * [13] H. Kleinert Particles and Quantum Fields Free web version (2016).
  * [14] R. D. Peccei, The Strong CP Problem and Axions, in Kuster, Markus; Raffelt, Georg; Beltrán, Berta (eds.) Axions: Theory, Cosmology, and Experimental Searches, Lecture Notes in Physics, 741 (2008) 3–17.