Scalaron Decay in Perturbative Quantum Gravity B. Latosh12 1Bogoliubov Laboratory of Theoretical Physics JINR Dubna 141980 Russia

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Scalaron Decay in Perturbative Quantum Gravity

B. Latosh ∗1,2

1Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Russia

2Dubna State University, Universitetskaya str. 19, Dubna 141982, Russia

Abstract

A certain quadratic gravity model provides a successfully inﬂationary scenario. The inﬂation is driven by

the new scalar degree of freedom called scalaron. After the end of inﬂation the scalaron decays in matter and

dark matter degrees of freedom reheating the Universe. We study new channels by which the scalaron can

transfer energy to the matter sector. These channels are annihilation and decay via intermediate graviton

states. Results are obtained within perturbative quantum gravity. In the heavy scalaron limit only scalar

particles are produced by the annihilation channel. Scalaron decays in all types of particles are allowed. In

the light scalaron limit decay channel is strongly suppressed. Boson production via the annihilation channel

is expected to be dominant at the early stages of reheating, while fermion production will dominate later

stages.

1 Introduction

The modiﬁed gravity R+R2model has a special place in the modiﬁed gravity landscape. Firstly, the model is

healthy despite having higher derivative terms [1,2]. Secondly, the model has one additional scalar degree of

freedom which can be made explicit via a one-to-one mapping on a scalar-tensor gravity [2,3,4,5,6]. Most

importantly, the model provides an inﬂationary scenario completely consistent with the observational data

[7,8,9,10].

Opportunities to describe the universe reheating within the R+R2model were extensively studied. Perhaps

the simplest scenario is based on transmutations of the new gravitational degree of freedom – scalaron [11,12,

13,14,15]. Scalaron drives inﬂation in the slow roll regime sliding on the plane part of the potential. When

the inﬂation ends with a graceful exit, the scalaron begins to oscillate around the minimum of the potential.

These oscillations can induce an extensive particle creation which will reheat the universe. If the scalaron has

channels through which it can transmute into other degrees of freedom, then the corresponding factors, like an

annihilation cross section or a decay width, will naturally enter cosmological equations and will serve as dumping

factors suppressing the discussed oscillations. The exact mechanism responsible for the scalaron transmutation

is strongly model-dependent.

We address two aspects of such a reheating scenario. First and foremost, scalaron can annihilate to matter

(and dark matter) degrees of freedom through intermediate graviton states. Such annihilation is possible

because all types of matter, including the scalaron, are coupled to gravity. Such processes can be consistently

described within the perturbative quantum gravity and the corresponding annihilation cross sections can be

calculated. The role of this reheating channel will be discussed. Secondly, scalaron can decay directly to matter

degrees of freedom because it is universally coupled to the matter energy-momentum tensor. It receives a new

decay channel at the one-loop level, as it also can annihilate via intermediate graviton states. We will present

particular examples of such processes and discuss their contribution to reheating.

This paper is organized as follows. Firstly, we discuss the setup used in calculations. We show that after the

end of an inﬂation weak quantum gravitational eﬀects can be consistently described by perturbative quantum

gravity. We brieﬂy discuss its formalism and applicability in Section 2. In Section 3we discuss the scalaron

annihilation to matter degrees of freedom. For the sake of simplicity, we only consider decays in states with

s= 0, m6= 0 which serves as an analogy with the Higgs boson; s= 1/2, m6= 0 which serves as an analogy with

quark, lepton, and dark matter degrees of freedom; and s= 1, m= 0 which serves as an analogy with gluons

and photons. In Section 4we construct explicit examples of scalaron decays in matter degrees of freedom that

only exist at the one-loop level. We discuss the role of such processes in reheating. We bring our conclusions in

Section 5.

∗latosh@theor.jinr.ru

arXiv:2210.00781v1 [gr-qc] 3 Oct 2022

2 Perturbative quantum gravity setup

Perturbative quantum gravity provides a simple framework capable to account for quantum gravitational eﬀects

consistently [16,17,18,19,20]. The approach is based on the following premises. Firstly, the constructed theory

is eﬀective. This means that the theory applicability domain lies below the Planck scale and it should not be

extended further in the UV. Secondly, gravity is perturbative. This means that the theory only accounts for

gravitational eﬀects described by small metric perturbations.

On the practical ground these premises results in the following construction. The full spacetime metric gµν

is composite of the ﬂat background ηµν and small perturbations hµν :

gµν =ηµν +κ hµν .(1)

Here κis the gravitational coupling related with the Newton constant GNas follows:

κ2= 32 π GN.(2)

In some sense hµν plays a role of a spin-2 gauge ﬁeld propagating in a ﬂat background. Quantum dynamics of

the system is described by the corresponding generating functional

Z[Jµν ] = ZD[hαβ ] exp [iA[hαβ ] + i Jµν hµν ].(3)

Here Ais the microscopic action describing the used gravity model. The action Ashall be expanded in a

perturbative series with respect to hµν to spawn the following inﬁnite series:

A=−1

2hµν Pµναβ hαβ

+κb

Vµ1ν1µ2ν2µ3ν3

3hµ1ν1hµ2ν2hµ3ν3+κ2b

Vµ1ν1µ2ν2µ3ν3µ4ν4

4hµ1ν1hµ2ν2hµ3ν3hµ4ν4+Oκ3.

(4)

The ﬁrst term of the expansion describes the graviton propagator, and terms b

Vncorrespond to graviton interac-

tion vertices. It shall be noted that this expansion contains an inﬁnite number of interaction terms consecutively

suppressed by the Planck scale.

The constructed theory is non-renormalizable [16,21,22]. Within the discussed approach the absence of

renormalizability is explained by the ﬁnite applicability domain. An application of the standard renormalization

procedure for perturbative gravity would require an inﬁnite number of renormalized constants. This only shows

that the data on the UV behavior of gravity is missing from the theory, which is already established by the

constraint on its applicability domain.

Despite these disadvantages, the theory provides a consistent way to calculate amplitudes in a controllable

way. At any order of the perturbation theory a set of relevant interactions can be identiﬁed uniquely together

with Feynman graphs contributing to a given matrix element. Feynman rules for graviton and matter interac-

tions, which are the core of the theory, can be evaluated analytically [23,24,25]. In paper [26] an algorithm

evaluating Feynman rules for gravity was proposed. It was implemented in FeynGrav package which extends

FeynCalc [26,27,28]. In this paper all calculations are performed with FeynGrav.

The perturbative approach can be used to account for quantum gravitational eﬀects after the end of an

inﬂationary phase. First and foremost, it is safe to assume that all processes involving scalaron, matter, and

dark matter degrees of freedom in a post-inﬂationary universe occur at a time scale much smaller than the

scale of the cosmological expansion. To put it otherwise, such processes are well localized both in space and

in time, therefore, one can decouple them from the cosmological background and account only for small local

metric perturbations describe by the perturbative theory. Secondly, R+R2gravity admits a unique mapping

of a scalar-tensor gravity [1,2]. This mapping diagonalizes the model Lagrangian and allows one to use the

standard formalism free from higher derivative terms.

Consequently, the perturbative approach to quantum gravity provides a consistent controllable way to ac-

count for quantum gravitational eﬀects. Its implementations for a post-inﬂationary universe are not obstructed

and can be used to study processes involving gravitational, scalaron, matter, and dark matter degrees of freedom.

3 Scalaron annihilation

Let us turn to a discussion of scalaron annihilation to matter degrees of freedom. We consider annihilation of

a pair of scalarons (which are scalars with mass M) to light scalars s= 0, 0 < m M; light Dirac fermions

s= 1/2, 0 < mfM; massless vectors s= 1, mv= 0. These processes are chosen because such degrees of

freedom can be associated both with the standard model degrees of freedom and with dark matter degrees of

freedom. Namely, the light scalars can be associated with the Higgs boson, fermionic degrees of freedom can

be associated with quarks and leptons, and massless vector degrees of freedom can be associated either with

photons or with gluons.

In this section, ﬁrstly, we calculate all the discussed amplitudes and the corresponding cross sections. After

this, we analyze the physical implications of the obtained results. All calculations are done with packages “Feyn-

Calc” [27,28], “FeynGrav” [26], “Package-X” [29,30], and “FeynHelpers” [31]. The corresponding publications

contain detailed descriptions of their usage, so we will not discuss them further.

Kinematics of such annihilation processes is given by the following:











pµ

1=pM2+p20 0 p,

pµ

2=pM2+p20 0 −p,

qµ

1=pm2+q2qsin θ0qcos θ,

qµ

2=pm2+q2−qsin θ0−qcos θ,

q=pM2+p2−m2











s= (p1+p2)2= 4 M2+p2,

t= (p1−q1)2=−(p+q)2+ 4 p q cos2θ

u= (p1−q2)2=−(p+q)2+ 4 p q sin2θ

(5)

Here p1and p2are in-going on-shell momenta of scalarons; q1and q2are out-going momenta of produced

particles; pis the center-of-mass spacial momentum; mis the mass of produced degrees of freedom; s,t, and u

are the standard Mandelstam variables [32,33].

A given annihilation matrix element Mis related with the diﬀerential cross section by the standard formula:

dσ =1

j|M|2(2π)4δ(4)(p1+p2−q1−q2)1

2E(p1)

2E(p2)

d3q1

(2π)32E(q1)

d3q2

(2π)32E(q2).(6)

Here jis the factor normalizing the cross section for a unit ﬂow:

j= pp1·p2−M2M2

1p0

2!−1

=M2+p2

q(M2+ 2 p2)2−M4

.(7)

The other factors normalize free initial and ﬁnial states, and perform an integration over the two-body phase

space. This formula is discussed in classical textbooks [34,35,36] in great details so we will not discuss it further.

For the given case the formula provides the following relation between a matrix element and the diﬀerential

cross section:

dσ

dΩ=1

256 π2

pq1 + p

M2−m

M2

1 + p

M2|M|2.(8)

Here dΩ = sin θ dθ dϕ is the solid angle.

The annihilation of two scalarons in two light scalars with mass 0 < m Mis given by the following matrix

element: q1q2

p1p2

=MSS→ss =iκ2

st u −(M2+m2)(t+u) + m4+M4+ 4 m2M2.(9)

Here and below in this section notations (5) are used. Therefore, p1and p2are in-going on-shell scalaron mo-

menta, q1and q2are out-going on-shell momenta of scalars, and these momenta are subjected to the conservation

law p1+p2=q1+q2. The amplitude produces the following diﬀerential cross section:

dσSS→ss

dΩ=1

16 (GM)2M

pq1 + p

M2−m

M2

h1 + p

M2i3"1 + p

M22 + p

M2+m

M2

−p

M21 + p

M2−m

M2cos 2θ#2

(10)

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ScalaronDecayinPerturbativeQuantumGravityB.Latosh*1,21BogoliubovLaboratoryofTheoreticalPhysics,JINR,Dubna141980,Russia2DubnaStateUniversity,Universitetskayastr.19,Dubna141982,RussiaAbstractAcertainquadraticgravitymodelprovidesasuccessfullyinationaryscenario.Theinationisdrivenbythenewscalardegreeoffr...

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Scalaron Decay in Perturbative Quantum Gravity B. Latosh12 1Bogoliubov Laboratory of Theoretical Physics JINR Dubna 141980 Russia

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