The functional fR approximation Tim R. Morris and Dalius Stulga T.R.Morrissoton.ac.uk D.Stulgasoton.ac.uk

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The functional f(R) approximation

Tim R. Morris and Dalius Stulga

T.R.Morris@soton.ac.uk, D.Stulga@soton.ac.uk

STAG Research Centre, Department of Physics and Astronomy,

University of Southampton, Highﬁeld, Southampton, SO17 1BJ, U.K.

October 21, 2022

Abstract

This article is a review of functional f(R) approximations in the asymptotic safety approach

to quantum gravity. It mostly focusses on a formulation that uses a non-adaptive cutoﬀ, resulting

in a second order diﬀerential equation. This formulation is used as an example to give a detailed

explanation for how asymptotic analysis and Sturm-Liouville analysis can be used to uncover

some of its most important properties. In particular, if deﬁned appropriately for all values

−∞ < R < ∞, one can use these methods to establish that there are at most a discrete number

of ﬁxed points, that these support a ﬁnite number of relevant operators, and that the scaling

dimension of high dimension operators is universal up to parametric dependence inherited from

the single-metric approximation. Formulations using adaptive cutoﬀs, are also reviewed, and

the main diﬀerences are highlighted.

Keywords— Quantum gravity, Renormalization group, Asymptotic safety, f(R) approximation, Sturm-

Liouville, Asymptotic analysis

arXiv:2210.11356v1 [hep-th] 20 Oct 2022

Contents

1 Introduction 3

2 Flow equations 5

3 Cutoﬀ functions 9

4 Flow equations with adaptive cutoﬀ 11

5 Evaluating traces 16

5.1 Sphere................................................. 16

5.2 Hyperboloid.............................................. 16

5.3 Flatspace............................................... 17

6 Fixed point solutions 17

6.1 Asymptoticanalysis ......................................... 18

6.2 Numericalsolution .......................................... 21

7 Eigenoperators 22

7.1 Asymptoticanalysis ......................................... 22

7.2 Sturm-Liouvilletheory........................................ 24

References 28

1 Introduction

One attempted route to a quantum theory of gravity is through the asymptotic safety programme [1–4].

Although quantum gravity based on the Einstein-Hilbert action is plagued by ultraviolet inﬁnities that are

perturbatively non-renormalizable (implying the need for an inﬁnite number of coupling constants), a sensible

theory of quantum gravity might be recovered if there exists a suitable ultraviolet ﬁxed point [1].

The task is not just that of searching for an ultraviolet ﬁxed point. They must also have the correct

properties. Perturbatively renormalizable ones exist for example “Conformal gravity”, based on the square

of the Weyl tensor, which thus corresponds to a Gaussian ultraviolet ﬁxed point [5]. It is apparently not

suitable however, because the theory is not unitary. Suitable unitary ﬁxed points, if they exist, have to be

non-perturbative. They must also satisfy phenomenological constraints, for example they have to allow a

renormalized trajectory with classical-like behaviour in the infrared, since General Relativity is conﬁrmed

by observation across many phenomena and to impressive precision. Of particular relevance for this chapter

is that there should be a ﬁxed point with a ﬁnite number of relevant directions (otherwise it would be no

more predictive than the perturbatively deﬁned theory). Preferably the theory should have only one ﬁxed

point, or at least only a ﬁnite number (otherwise again we lose predictivity).

Functional RG (renormalization group) equation [6–11] studies, ﬁrst introduced by Wilson and Wegner

many years ago [6,7] (and called by them the “exact RG”), have ﬂourished into a powerful approach for

investigating this possibility. These equations describe the ﬂow of the Wilsonian eﬀective action for some

quantum ﬁeld theory, under changes in an eﬀective cutoﬀ scale k. The asymptotic safety literature uses

almost exclusively the ﬂow equation for Γkwhich is, modulo minor details, the Legendre eﬀective action (the

generator of one-particle irreducible diagrams) cut oﬀ in the infrared by k. It was also formulated long ago [9]

(in the sharp cutoﬀ limit) and then rediscovered for smooth cutoﬀs much later in refs. [10,11]. Following

ref. [10], Γkis sometimes called the “eﬀective average action”, however in this chapter it will simply be called

an eﬀective action.

It is not practical to solve the full functional RG equations exactly. In a situation such as this, where

there are no useful small parameters, one can only proceed by considering model approximations. These

always proceed from the following observation: Wilsonian eﬀective actions can be written as a sum over

operators, where the coeﬃcients are the couplings for these operators and they evolve with the scale k.

In fact this sum should be restricted to local operators. This is the requirement of quasi-locality, which

comes from the short range nature of the Kadanoﬀ blocking step in Wilsonian RG [6], when implemented

in the continuum [12,13]. A related point is that the Wilsonian RG is performed in euclidean signature, so

that “short range” has a sensible meaning.

The problem is that for any general solution, this sum is inﬁnite, over all possible local operators allowed

by the symmetries (the space of all such couplings being known as “theory space”). However, this motivates

the simplest model approximation which is to truncate drastically the inﬁnite dimensional theory space to a

handful of operators. An example is the original truncation studied by Reuter [2]:

Γk[gµν ] = Zd4x√gu0(k) + u1(k)R,(1)

which retains only the cosmological constant term and the scalar curvature Rterm. For obvious reasons this

is called the “Einstein-Hilbert truncation”. Classically u0=−λcc/(8πG) and u1=−1/(16πG), where λcc

is the cosmological constant and Gis Newton’s constant, but after quantum corrections these couplings run

with kin the functional RG. The minus sign in u1comes from working in euclidean signature.

Apart from RG symmetry, these truncations destroy pretty well all the properties that ought to hold.

For example scheme independence (i.e. independence on choice of cutoﬀ, or more generally universality),

and modiﬁed BRST invariance [2,14] (which encodes diﬀeomorphism invariance for the quantum ﬁeld under

inﬂuence of the cutoﬀ) cannot then be recovered. Furthermore, only by keeping an inﬁnite number of local

operators can the non-local long-range nature of the (one-particle irreducible) Green’s functions be recovered

(see e.g. ref. [15]). One has to trust that by considering ever less restrictive truncations the description gets

closer to the truth. There are some examples that go well beyond the Einstein-Hilbert truncation by keeping

a large number of operators [16–19]. These are based around polynomial truncations, i.e. where everything

is discarded except powers of some suitable local operators, typically the scalar curvature Ragain, up to

some maximum degree. They appear to show convergence, in particular the number of relevant operators is

found to be three.

Another approximation in the asymptotic safety literature that is necessary in order to formulate diﬀeo-

morphism invariant truncations, such as eqn. (1), conﬂates the true (quantum) metric with the background

metric. It is called the “single metric” or “background ﬁeld” approximation, and will be described in the

next section. It is harder to relax this approximation in any substantive way, although see refs. [20–28] for

some approaches.

Whilst very encouraging results are found from multiple studies of such ﬁnite order truncations (see e.g.

the review [29]), successful implementations of more powerful approximations would build conﬁdence in the

scenario. The next step is to keep an inﬁnite number of operators. Arguably the simplest such truncation is

to keep a full function f(R), making the ansatz [22–24,30–43]

Γk[g] = Zd4x√g fk(R).(2)

This is the functional f(R)approximation which is the subject of this chapter. It is achieved by specialising

to a maximally symmetric background manifold, either a four-sphere or four-hyperboloid.

Closely related approximations have been studied in scalar-tensor [44,45] and unimodular [46] gravity,

and in three space-time dimensions [37]. In fact, the high order ﬁnite dimensional truncations [16–19]

were developed by taking examples of these f(R) equations and then further approximating to polynomial

truncations.

Note that the functional f(R) approximation actually goes beyond keeping a countably inﬁnite number

of couplings, the Taylor expansion coeﬃcients gn=f(n)(0), because a priori the large ﬁeld parts of f(R)

contain degrees of freedom that are unrelated to all these gn. For example suppose that at large Rone

ﬁnds that f(R)≈exp−a/R2, where a > 0 is some parameter. Such an f(R) is in the form of a standard

counter-example in mathematical analysis. It has the property that gn= 0 for all n.

As ref. [33] emphasised, the truncation (2) is as close as one can get to the Local Potential Approximation

(LPA) [47,48], a successful approximation for scalar ﬁeld theory in which only a general potential V(ϕ) is

kept for a scalar ﬁeld ϕ(see e.g. [47–52]). The LPA can be viewed as the start of a systematic derivative

expansion [49], in which case this lowest order corresponds to regarding the ﬁeld ϕas constant. In rough

analogy, an approximation of form (2) may be derived by working on a euclidean signature space of maximal

symmetry, where the scalar curvature Ris constant. (Typically a four-sphere is chosen.) In particular,

techniques that have proved successful in scalar ﬁeld theory [48–53] have been adapted to this very diﬀerent

context, and used to gain substantial insight [43,54–57].

The functional truncation (2) still has the problems that were highlighted earlier for its ﬁnite dimensional

counterparts. However, again one can hope that it is closer to the truth. One hint that this is in fact the

case is covered at the end of this chapter. Assuming that the most recent version [43] does have a ﬁxed

point solution, then it turns out that operators with high scaling dimension do begin to display universality

– unfortunately up to an annoying parameter that remains which is clearly caused by the single-metric

approximation.

In this chapter, it will be explained how to construct functional f(R) approximations and how to interpret

them. Important properties of formulations that use an adaptive cutoﬀ [22–24,30–41] will be reviewed. These

result in third order diﬀerential equations, with ﬁxed singularities and problematic asymptotic behaviour.

Mostly the chapter will focus on a non-adaptive cutoﬀ formulation [42,43] that results in a second order

diﬀerential equation, using it as an example to give a detailed exposition of the techniques, especially

asymptotic analysis and Sturm-Liouville analysis, that can be used to prove properties of functional f(R)

approximations. In particular, if the second order formulation is taken to apply to only one of the two spaces

(sphere or hyperboloid), the ﬁxed point solutions form a continuous set and the eigenoperator spectrum is

not quantised. However, if these spaces are joined together smoothly (through ﬂat space at their boundary),

these methods establish that there are at most a discrete number of ﬁxed points, that the ﬁxed points support

a ﬁnite number of relevant operators, and yield the result above for operators of high scaling dimension.

They do not establish that such ﬁxed points actually exist however. Such a demonstration requires more

powerful numerical analysis and/or simpler ﬁxed point formulations [43].

2 Flow equations

The starting point is the functional RG ﬂow equation [10,11]:

∂tΓk=1

2STrh(Γ(2)

k+Rk)−1∂tRki,(3)

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Thefunctionalf(R)approximationTimR.MorrisandDaliusStulgaT.R.Morris@soton.ac.uk,D.Stulga@soton.ac.ukSTAGResearchCentre,DepartmentofPhysicsandAstronomy,UniversityofSouthampton,Higheld,Southampton,SO171BJ,U.K.October21,2022AbstractThisarticleisareviewoffunctionalf(R)approximationsintheasymptoticsafety...

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