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Physics Letters B 789 (2019) 480–487

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Asymptotically safe f (R)-gravity coupled to matter II: Global solutions

Natália Alkofer

Institute for Mathematics, Astrophysics, and Particle Physics (IMAPP), Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, the Netherlands

a r t i c l e i n f o a b s t r a c t

Article history:

Received

17 September 2018

Received

in revised form 3 December 2018

Accepted

24 December 2018

Available

online 2 January 2019

Editor:

A. Ringwald

Ultraviolet ﬁxed point functions of the functional renormalisation group equation for f (R)-gravity

coupled to matter ﬁelds are discussed. The metric is split via the exponential parameterisation into

abackground metric and aﬂuctuating part, the former is chosen to be the one of afour-sphere. Also

when scalar, fermion and vector ﬁelds are included global quadratic solutions exist as in the pure gravity

case for discrete sets of values for some endomorphism parameters deﬁning the coarse-graining scheme.

The asymptotic, large-curvature behaviour of the ﬁxed point functions is analysed for generic values of

these parameters. Examples for global numerical solutions are provided. A special focus is given to the

question whether matter ﬁelds might destabilise the ultraviolet ﬁxed point function. Similar to aprevious

analysis of a polynomial, small-curvature approximation to the ﬁxed point functions different classes for

such functions are found.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Searching for a viable theory of quantum gravity is one of the

most important open problems in theoretical physics. Many differ-

ent

approaches try to elucidate it from various perspectives. In this

letter, the asymptotic safety scenario for quantum gravity [1]will

be employed based on a speciﬁc ansatz for the effective action:

an investigation of f (R)-gravity minimally coupled to an arbitrary

number of scalar, Dirac, and vector ﬁelds is discussed with a spe-

cial

focus on the study of global ﬁxed functions, the generalisation

of non-Gaußian ﬁxed point (NGFP). Recently, the ﬂow equation

used herein has been derived within the functional renormalisa-

tion

group (FRG) for the effective average action, and this equation

has been solved to obtain the respective NGFP function in a poly-

nomial,

small-curvature approximation [2], see also ref. [3]. These

solutions provide the foundation of the here reported study. Re-

sults

for the NGFP function for the pure gravity case within the

employed version of the ﬂow equation have been given recently

in refs. [4,5]. Note that such solutions of NGFP functions but for

different truncations of the ﬂow equation have been obtained in

refs. [6–20]. Hereby the characteristics of the solutions for these

functions differ signiﬁcantly depending on the technical aspects of

the respective work. Given the ﬁxed functions’ importance for the

asymptotic safety scenario this requires further understanding. In

the following it will be studied whether coupling matter might

give an important hint to resolve ambiguities.

E-mail address: natalia.alkofer@gmail.com.

Coupling matter to gravity within the asymptotic safety sce-

nario

has a long history, see refs. [18,21–47], however, with mixed

results. In ref. [2] comprehensible estimates have been provided

which gravity-matter systems may give rise to NGFPs suitable for

rendering the theory asymptotically safe. In this reference the ﬂow

equation has been derived within a seven-parameter family of

non-trivial endomorphisms in the regularisation procedure. Herein

this freedom will be exploited to show the existence of global

quadratic solutions. In ref. [2]it was also shown that for vanishing

endomorphisms gravity coupled to the matter content of the stan-

dard

model of particle physics (and also many beyond the standard

model extensions) exhibit a NGFP whose properties are strikingly

similar to the case of pure gravity: there are two UV-relevant di-

rections,

and the position and critical exponents converge rapidly

when higher powers of the scalar curvature beyond the quadratic

ones are included. Building on this result numerical solutions will

be obtained for a global ﬁxed function for the pure gravity as

well as for the gravity-matter system with standard model matter

content. Hereby the discussion of the singular points of the ﬂow

equation and the asymptotic behaviour of the solution for large

scalar curvatures turns out to be the crucial element.

Based

on generic features of the employed ﬂow equation one

can show for global solutions a property already visible at the

level of the polynomial approximations, namely that addition of

fermions, stabilise an existing NGFP if the coarse graining operator

is chosen as the Laplacian. As the Standard Model is dominated

by fermions therefore a NGFP function for f (R)-gravity coupled to

the Standard Model matter content exists in this case.

https://doi.org/10.1016/j.physletb.2018.12.061

0370-2693/

SCOAP

N. Alkofer / Physics Letters B 789 (2019) 480–487 481

This letter is organised as follows: in sect. 2 the derivation of

the ﬂow equation is brieﬂy reviewed. In sect. 3 the properties of

the ﬂow equation in four dimensions are discussed. The existence

of global quadratic solutions is shown by constructing two explicit

examples. Furthermore, the asymptotic behaviour is analysed, and

numerical solutions for two selected cases are presented. In sect. 4

the

results are summarised and conclusions are provided.

2. RG equation for gravity and matter in f (R) truncation

To make this letter self-contained the derivation of the ﬂow

equation given in ref. [2]is brieﬂy reviewed. It is based on the

form of the FRG equations given in refs. [48,49], adapted to the

case of gravity [1],

∂



STr





(2)



−1

∂



(1)

Here, k is the RG scale and t = ln(k/k

) the RG “time” with k

be-

ing

an arbitrary reference scale. 

denotes the effective average

action and 

(2)

its second variation with respect to the ﬂuctua-

tion

ﬁelds. R

is a regulator introduced such that solving the ﬂow

equation effectively integrates out quantum ﬂuctuations with mo-

menta

above the scale k.

Throughout

this work for the gravitational part of the effective

average action a f (R) truncation on d-spheres as background will

be employed. To be explicit, the corresponding total effective ac-

tion

reads



= 

grav

+

matter

, (2)

where 

grav

is the gravitational part of the effective average action

and 

matter

contains the matter ﬁelds. The gravitational part of the

action is assumed to be given by



grav



√

(R) + 

+

, (3)

where f

(R) is an arbitrary, scale-dependent function of the Ricci

scalar R, and the action is supplemented by suitable gauge ﬁxing

and ghost terms. This sector is taken to be identical to the one

studied in ref. [5]. The matter sector is assumed to consist of N

scalar ﬁelds, N

Dirac fermions, and N

Abelian gauge ﬁelds. The

latter ones are ﬁxed to Feynman gauge, and thus on curved back-

grounds

the related ghosts are included. Matter self-interactions

as well as the RG scale dependence of the matter wave-function

renormalisations are neglected.

a next step one splits the metric into a background and a

ﬂuctuating (quantum) part. In the following the exponential split

μν

μρ

)

(4)

is used thereby avoiding any signature change even for large ﬂuc-

tuating

ﬁelds. A detailed derivation of the gravitational part of the

RG equation can be found in refs. [4,5], see also refs. [2,3].

Some

of the global solutions for ﬁxed functions presented here

can only be obtained if additional endomorphisms in the reg-

ulator

functions appearing in the RG equations are introduced.

To this end, a set of parameters α

G,M

,D,V ,T

, with the subscript la-

belling

the type of ﬁeld and the superscript the gravity or matter

sector, are introduced: the regulators are chosen to depend on

G,M

,D,V ,T

:=  − α

G,M

,D,V ,T

R where  is the Laplace operator, and

R is the positive curvature scalar of the background sphere. The

labelling of subscripts refer to: S scalar, D Dirac, V transverse vec-

tor,

and T transverse traceless symmetric tensor ﬁeld.

ref. [2]all the traces appearing in the RG equation have been

done by explicitly summing over the eigenvalues of the Laplacian

on the sphere in the different spin channels, for the respective

expressions for these eigenvalues see, e.g., ref. [50]. Using a Litim-

type

regulator [51,52]

(z) = (k

− z)θ(k

− z ), (5)

∂

(z) = 2k

θ(k

− z ),

these sums are all ﬁnite. Subsequently, an additional smoothing

operation (namely averaging over two sums performed on the up-

per

and lower limit of the resulting “staircase” function) are em-

ployed

as part of the regularisation, see ref. [2]for more details.

Two

widely used choices for the coarse graining operators

termed “Type I” and “Type II” (see [53]for detailed deﬁnitions and

a discussion of this typology) are given by the following choice of

endomorphism parameters,

Type I: α

= α

= 0 , (6a)

Type II:

=−

d(d−1)

, α

d−1

, α

=−

, α

=−

, α

= α

= 0 . (6b)

The RG equation takes the form of a partial differential equa-

tion

for the scale-dependent function f

(

R). As usual it is advan-

tageous

to formulate it in dimensionless variables r =

R/k

and

ϕ(r) = f (

R)/k

. This leads to a separation of the “classical” scale

dependence of f (

R) from the quantum one which reads in four

dimensions:

∂





√

g ∂

f (

R) (7)

= V



∂

ϕ(r) +4ϕ(r) −2rϕ



(r)



where V

= 384π

is the volume of the 4-sphere. The ﬂow

equation is then given by:

ϕ +4ϕ −2rϕ



= T

ghost

sinv

(8)

+ T

scalar

Dirac

vector

where

2(4π)

1 +







1 +



−



(9a)



1 +



−



12(4π)



+2ϕ



−2rϕ





1 +



−





1 +



−



sinv

2(4π)





−







(9b)



1 +



−





1 +





12(4π)



−2rϕ





−







1 +







1 +



−





1 +



−



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