Towards the determination of the dimension of the critical surface in asymptotically safe gravity

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Towards

the

determination

of

the

dimension

of

the

critical

surface

in

asymptotically

safe

gravity

Kevin Falls

a

,

b

,

∗

,

Nobuyoshi Ohta

c

,

Roberto Percacci

a

,

b a_{InternationalSchoolforAdvancedStudies,viaBonomea265,34136Trieste,Italy}

b_{INFN,SezionediTrieste,Italy}

c_{DepartmentofPhysics,KindaiUniversity,Higashi-Osaka,Osaka577-8502,Japan}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received6May2020

Receivedinrevisedform7September2020 Accepted7September2020

Availableonline24September2020 Editor:B.Grinstein

Datasetlink:

https://zen-odo.org/record/4017671#.X2jWpmcza8o

WecomputethebetafunctionsofHigherDerivativeGravitywithintheFunctionalRenormalizationGroup approach, going beyondpreviouslystudiedapproximations. We findthat the presence ofanontrivial Newtonian coupling induces,in addition to the free fixed point of the one-loopapproximation, also twonontrivialfixedpoints,ofwhichonehastherightsignstobefreefromtachyons.Ourresultsare consistentwithearliersuggestionsthatthedimensionofthecriticalsurfaceforpuregravityisthree.

©2020TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Higher DerivativeGravity(HDG)isthetheory ofgravitybased onthemetricasthecarrierofdegreesoffreedom,withanaction containing termsof order zero,one andtwo inthe curvature. It contains bothdimensionfulcouplings(the cosmologicaland New-ton constant)anddimensionless ones(the coefficientsofthe HD terms).Whentreatedperturbativelyinthelatter,itis renormaliz-able [1],butnotunitary.Followingsomeearlierattempts [2,3],its one-loop betafunctions were correctly derived forthe first time in [4];for moredetails andgeneralizations,see [5,6]. Depending on the signs of the couplings, the theory can be asymptotically free,butithasghostsand/or tachyons. Therehasbeenrecentlya revival ofinterest in thistheory,andproposals to get around its problemsinvariousways [7–17].

Intheasymptoticsafetyapproachtoquantumgravity,onetries to constructa continuum limit around an interactingfixed point (FP) [18]. The main tool to investigate the gravitational renor-malizationgroup has beenthe FunctionalRenormalizationGroup Equation (FRGE), asapplied for thefirst time to gravity by Mar-tinReuter[19].Itdefinesaflowonthetheoryspaceconsistingof alldiffeomorphisminvariantfunctionalsofthemetric.Oneexpects thatataninteractinggravitationalFP,infinitelymanygravitational couplingswillbenonzero.Inspiteofthiscomplication,much ev-idence for the existence of such a FP has been collected so far [20,21].

*

Correspondingauthor.

E-mailaddresses:kfalls@sissa.it(K. Falls),

ohtan@phys.kindai.ac.jp

(N. Ohta), percacci@sissa.it(R. Percacci).

Inthe context ofasymptotic safety,when one uses the FRGE, thereis never the needto postulate the formof thebare action tobeusedinthepathintegral.Instead,onedirectlycalculatesthe flow ofthe effective action asa function of an external “coarse-graining”scale,orIR cutoff,k. Inthiscontext,the actionofHDG canbeusedasanansatzfortherunningeffectiveaction.Wewill callthisthe“HDGtruncation”.It trackstheflow ofthetheoryin afive-dimensional“theory space” parametrizedby thecouplings:

V

, ZN,

λ

,

ξ

and

ρ

, defined below. The beta functions of HDG

havebeenstudiedfromthispointofview inseveralpapers.They wereobtainedinaone-loopapproximationtotheFRGEin [22–25]. Inthese calculations, the betafunctionsof the HD couplings are asymptoticallyfree,inagreementwiththeoldperturbativeresults, butthe flow of thedimensionful couplings looks very similar to theoneoftheEinstein-Hilberttruncation,andexhibitsa nontriv-ial FP for the cosmological and Newton constant. To go beyond oneloop,onehastokeeptermsinvolvingthebetafunctionsinthe r.h.s.oftheflowequation,andthensolvethesealgebraicequations for the beta functions. We highlight this process in Section 3.1. Thisproducesnon-linearitiesthatamounttoresummations of in-finitelymanyloopdiagrams.Thishasbeencalculatedin[26,27] on ageneric Einsteinmanifold, andafullyinteractingFPwas found, butthesecalculationswerelimitedtooneortwo,outofthethree HD couplings. This may seem to be sufficient, since one of the threecouplingsisthecoefficient oftheEulerterm, thatdoesnot contribute to the local dynamics. Unfortunately, as we shall see inSect.2.1, on an Einstein manifold one computesthe beta func-tionof certain linearcombinations ofthe threecouplings, andit isactuallyimpossibletoidentifythebetafunctionofthetwo dy-namicallyinterestingones:there isan unknownmixingwiththe https://doi.org/10.1016/j.physletb.2020.135773

0370-2693/©2020TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

betafunctionoftheEulerterm. Tocompute thebetafunctionsof alltheindependentcouplingsisthemaintaskofthispaper.

Themainmotivationforthisisthedeterminationofthe dimen-sion of the UV critical surface. There is evidence fromthe f

(

R

)

truncationsthatthescalingexponentsatthenontrivialfixedpoint arenottoodifferentfromtheclassicalones,sothatcouplingswith positivemassdimensionremainrelevantandcouplingswith neg-ativemassdimension remain irrelevantFP [28–32].The marginal coupling of the R2 term becomesrelevant, so altogether, in this truncation,thedimensionofthecriticalsurfaceseemstobethree. An attemptto include differenttensor structures hasbeen made in [31],whereactionsoftheform f1(Rμν Rμν

)

+

R f2(Rμν Rμν

)

are studied,leadingtothesameconclusion.Alimitationofthese cal-culations is that, ona spherical background,it is not possibleto properlydisentangleindependentcouplingswiththesamenumber ofcurvatures.ThecaseofRiccitensorsquaredandscalarcurvature squared actions on an Einstein manifold, has already been cited above [27]. While moregeneral than spheres, Einstein manifolds arestillnotgeneralenoughtodistinguishallinvariants. Withthis limitation, it was found again that the dimension of the critical surfaceisthree.Thissuggeststhatsomelinearcombinationofthe HDcouplingsmaybeanirrelevantoperator.Itseemspossible,and evenlikely,thatthedimensionofthecriticalsurfaceinpure grav-ity isdetermined entirely by the fateof the HD couplings, since they are not expected to remain marginal at an interacting FP.1 We find that of the three dimensionlesscouplings, one becomes relevant,oneirrelevantandone–thecoefficientoftheEulerterm –remainsmarginal.ThebetafunctionoftheEulertermisrelated tothea-function.Thea-theoremstatesthatwhentwofixedpoints arejoinedbyanRGtrajectory,thevalueofa attheIRfixedpoint islowerthantheoneattheUVfixedpoint.Wefindsomeevidence thatthismayholdalsoingravity.

Inthepresentpaperwetrytoshedsomelightontheseissues by computing thebeta functionsofall the HD couplings beyond theone-loopapproximation,takingtheanomalousdimensionsinto account. We shall do this by using the “Universal RG Machine” to compute the r.h.s. of the FRGE on an arbitrary background. Thisisatechniquebasedonnon-diagonalheatkernelcoefficients that canbe usedto evaluatefunctionaltracesinvolving covariant derivatives actingon a function of a Laplacian.The Universal RG Machine hasbeenintroduced,andappliedtothe Einstein-Hilbert truncation,in[34].Lateritwasusedtocalculatetheone-loopbeta functionsinHDG[35].Technicaldetailsaregivenin[36].Herewe bring that program one step forward by evaluating the full beta functionsofHDG,includingtheanomalous dimensions.Themain stepsofthecalculationareoutlinedinSect.2,andinSect.3we de-scribe the results.We find three fixed points, of which one has vanishing higher derivative couplings, while the others are fully interacting. Inprinciple, any ofthese could be a viable UV fixed point.Tohaveaviabletheory,onewouldalsohaveto prove uni-tarity.Forthefirstofthesefixedpoints,onecouldapplythe argu-mentsdevelopedinperturbationtheory[7–17].Fortheremaining ones, theissueismoreinvolvedandwillrequirea detailedstudy ofthetwopointfunction.

2. Beta functions

2.1. WhyEinsteinbackgroundsarenotenough

Letusmomentarily concentrateontheHD terms,thatwe can write as

L

H D

=

α

R2

+ β

R2μν

+

γ

R2μνρλ. Dueto the fact that the

1 _{Sofartheonlyindicationthatthingscouldbemorecomplicatedcomesfrom} work inprogress byKluthand Litimonactionsoftheform f1(RμνρσRμνρσ)+

R f2(RμνρσRμνρσ),whereatermcubicincurvatureseemstobecomerelevant [33].

Gauss–BonnetcombinationE

=

R2

μναβ

−

4R2μν

+

R2 istopological, oneofthesecouplingsisuninterestingasfaraslocaldynamicsis concerned.ItisthereforemoremeaningfultowritetheLagrangian as

L

H D

=

1 2

λ

C 2

₊

1

ξ

R 2

₋

1

ρ

E (2.1) where 1

ξ

=

3

α

+ β +

γ

3

,

1 2

λ

=

β

+

4

γ

2

,

−

1

ρ

= −

β

+

2

γ

2

,

(2.2) and C2

₌

_R2

μναβ

−

2R2μν

+

13R2 is the square of the Weyl

ten-sor. We are mainly interested in the beta functionsof

λ

and

ξ

. Calculations are simpler on an Einstein background. In this case E

=

Rμνρσ Rμνρσ andC2

₌

_{Rμνρσ Rμνρσ}

₋

_R2

_/

_6,so

L

H D

=



1

ξ

−

1 12

λ



R2

+



1 2

λ

−

1

ρ



E

.

(2.3)

Thisimpliesthatifweexpandther.h.s.ofthefunctionalRG equa-tionon anEinstein background,andwe interpretthe coefficients ofR2 _andE

₌

_{Rμνρσ Rμνρσ asbetafunctions,wecanreadoffthe}

betafunctionsoftwocombinations of

λ

,

ξ

,

ρ

butweare unable tounambiguouslyidentify

β

λand

β

ξ.Todothis,weneedan addi-tionalindependentequation,thatinturnrequiresa moregeneral background.Thisiswhatwedointhispaper.

AllcalculationswillbebasedontheEuclideanaction S

=



d4x

√

−

g

[

V

−

ZNR

+

L

H D

],

(2.4)

where ZN

=

₁₆1_πG, G beingNewton’s constant,

V =

2



ZN and



isthecosmologicalconstant. Sometimeswe shalluse the combi-nations

ω

≡ −

3

λ

ξ

, θ

≡

λ

ρ

.

(2.5)

2.2.Remarkonthetopologicalterm

Before embarking in calculations, we can make a general re-markontheGauss-Bonnetterm, thatactuallyholdsindependently ofthetruncation.DuetothetopologicalcharacterofthetermE,its coefficient1

/

ρ

doesnotappearintheHessianandthereforedoes notappearinther.h.s.oftheflowequation.Thusthebetafunction of

ρ

musthavetheform

β

ρ

= −

1 16

π

2aρ

2

_,

_(2.6)

wherea isafunctionofalltheothercouplings,butnotof

ρ

itself. Inthesearchofa fixedpointone cansolvefirstthe equationsof all the other couplings, which are also independent of

ρ

. When these fixed point values are inserted in (2.6), a becomes just a number.TheUVbehavior of

ρ

isdeterminedbythevalue ofthis number.Ifa

=

0,

ρ

couldreachanyvalue intheUV.Ifa

>

0 (a

<

0),whenallothercouplingsareveryclosetoafixedpoint,itwill runlogarithmicallytozerofromabove(below).

2.3.Expansionandgaugefixing

Wesplitthemetricgμν

= ¯

gμν

+

hμν ,wheregμν is

¯

anarbitrary background.Fordetailsoftheexpansionoftheaction,wereferto [24].Thegauge-fixingandghostactioncanbewritten

L

G F+F P

/



¯

g

= −

1 2a

χ

μY μν

_χ

ν

+

i Zghc

¯

μ



(μνgh)cν

+

1 2ZYbμY μν_b ν

+

ZY

¯ζ

μYμν

ζ

μ

,

(2.7)

where cμ,

¯

cμ are complex ghosts and bμ is a real communting field,

¯ζ

μ,

ζ

μ arecomplexanticommutingfields,and

χ

μ

≡ ¯∇

λhλμ

+

b

¯∇

μh

,



(μνgh)

≡

gμν

¯∇

2

+ (

2b

+

1

) ¯

∇

μ

¯∇

ν

+ ¯

Rμν

,

Yμν

≡ ¯

gμν

¯∇

2

+

c

¯∇

μ

¯∇

ν

−

f

¯∇

ν

¯∇

μ

,

(2.8) wherea,b,c and f aregaugeparameters.Thereissomefreedom inhowwechoosethewavefunctionrenormalisations Zgh andZY

sincetheycanberescaledwhilekeeping Z2

ghZY

=

1

/

a fixed

with-outaffectingthepathintegral.Inourcalculationswefix

Zgh

=

1

,

ZY

=

1

/

a (2.9)

Wemaketheusualgaugechoice a

= λ ,

b

= −

1

+

4

ω

4

+

4

ω

,

c

=

2

3

(

1

+

ω

) ,

f

=

1

,

(2.10) leadingtoaminimalfourthorderoperatorforthefluctuations.The operatorsin(2.7) arethen



(μνgh)

≡

gμν

¯∇

2

−

σ

gh

¯∇

μ

¯∇

ν

+ ¯

Rμν

,

Yμν

≡ ¯

gμν

¯∇

2

−

σ

Y

¯∇

μ

¯∇

ν

−

Rμν

,

(2.11) with

σ

gh

= −

1

−

2b

= −

1

−

2

ω

2

(

1

+

ω

)

;

σ

Y

=

1

−

2

γ

−

α

β

+

4

γ

=

1

−

2

ω

3

.

(2.12)

Wenotethatthecancellationbetweenunphysicaldegreesof free-dom becomes exact in the “Landau gauge” limit a

→

0, which happenstobesatisfiedintheasymptoticallyfreeregime.

Then, the quadratictermsin theaction can be writtenin the form [24]

L

(2)

₌

_h

μνKμνρσ

O

ρσαβhαβ

,

(2.13)

wheretheoperator

O

is

O

= 

2

₊

_V

ρλ

¯∇

ρ

¯∇

λ

+

U

,

(2.14)

with



= − ¯∇

2,U

=

K−1_{W andwewrite}

K

=

β

+

4

γ

4



I

+

4

α

+ β

γ

−

α

P



,

K−1

=

4

β

+

4

γ



I

−

4

α

+ β

3

α

+ β +

γ

P



,

(2.15)

where

I

istheidentityinthespaceofsymmetrictensorand

P

is aprojector

I

μν,αβ

≡ δ

μν,αβ

=

1 2

(

g

¯

μαg

¯

νβ

+ ¯

gμβg

¯

να

) ,

P

μνρσ

≡

Pμνρσ

=

1 4g

¯

μν

_¯

_g ρσ

.

(2.16)

ThecoefficientsV ρλandU arefunctionsofthecurvatures,

V

and ZN,forwhoseformwereferagainto[24].

The“betafunctional”ofthetheory isthesumofthree contri-butionscomingfromgravitons,ghostsandthenewghostbμ:

˙

k

=

Tg

+

Tgh

+

TY

.

(2.17)

In orderto write thesetermsmore explicitly,we haveto choose a cutoff for each of them. Fora one-loop calculation, where the couplings inther.h.s. oftheequation are treatedasfixed, itwas mostconvenienttothinkofthecutoffasa functionofthewhole

operator

O

,



ghorY respectively(so-calledtypeIIIcutoff).Inthis

paperwewillnotignoretherunningofthecouplingsthatmaybe presentinthecutoff,soitisbesttominimizetheirpresence.This isachievedbychoosingthecutofftobeafunctionof



only (so-calledtype Icutoff). Theone-loopcalculationwiththiscutoffhas beendonebeforein[35].

2.4.Gravitoncontribution

Wechoosethegraviton cutofftohavetheform

R

=

K Rk

(

2

)

,

where Rk

(

2

)

= (

k4

− 

2

)θ (

k4

− 

2

)

and we define as usual

Pk

(

2

)

= 

2

+

Rk

(

2

)

=

k4

θ (

k4

− 

2

)

.Notethatitisconvenient

toview Rk asafunctionof



2,althoughofcourseonecouldalso

view it asa function of



. Then, writingthe kinetic operator as



2

+

V

+

U ,thegravitoncontributiontotheFRGEis Tg

=

1 2Tr

∂

t

[

K Rk

(

2

)

]

K

[

O

+

Rk

(

2

)

]

=

1 2Tr

∂

tRk

(

2

)

+

η

KRk

(

2

)

Pk

(

2

)

+

V

+

U

,

(2.18) wherewedefined

η

K

=

K−1 dK dt

.

(2.19)

Notethat

η

K isatensor.From(2.15) wefind

η

K

=

η

1

I

+

η

P

P ,

(2.20) where

η

1

= −

˙λ

λ

,

η

P

= −

ξ ˙λ

− λ˙ξ

λ(

3

λ

− ξ)

.

(2.21)

We divide V and U into various terms: V

=

V0

+

V1 and U

=

U0

+

U1

+

U2,wherethesubscriptcountsthepowerofcurvature,

andtheremainingdimensioniscarriedeitherby

V

or ZN:

V0

∼

ZN

∇∇ ;

V1

∼

R

∇∇ ;

U0

∼

V

;

U1

∼

ZNR

;

U2

∼

R2

.

Wenowhavetodecidehowtoexpandthefractionin(2.18).Since wewanttocomputethebetafunctionsofallthecouplingsin(2.4), weneedtoexpandtosecondorderincurvatures.Itwouldbe nat-ural to assume that

√

V ∼

ZN

∼

R (which implies also



∼

R),

butsuchanexpansionwouldmissimportantfeatures,asweshall discussbelow.It ispossible withouttoomuch effort tokeep the full dependenceon

V

,andwe shall doso.We will thereforenot expand in U0.It is much harder to keep all dependence on ZN,

thereforewe willexpand in V0, V1, U1 andU2,to firstorderin ZN

/

k2,independentlyofcurvatures.2Thiscorrespondsto

consider-ingatrans-Planckianregime.IfoneconsiderstheEinstein-Hilbert partoftheaction,itcorrespondtoastronggravity expansion.See [37] forarecentdiscussion.Keeping onlytermsuptolinearorder inZN wethushavetoevaluate:

Tgrav

=

1 2Tr



∂

tRk

()

+

η

KRk

()

Pk

()

+

U0

×



1

−

1 Pk

()

+

U0

(

V0

+

V1

+

U1

+

U2

)

+

1 Pk

()

+

U0 V0 1 Pk

()

+

U0 V1

+

1 Pk

()

+

U0 V1 1 Pk

()

+

U0 V0

2 _{NotethatwewroteV=2Z}

Nandtreatedasanindependentcoupling,the

+

2V0U2

(

Pk

()

+

U0

)

2

+

V2 1

(

Pk

()

+

U0

)

2

+

2V1U1

(

Pk

()

+

U0

)

2

+

3V0V12

(

Pk

()

+

U0

)

3



.

(2.22)

Inthelasttwolineswehavewrittenthetermsonlyinaschematic way, without paying attention to their order: to be precise one hasto write out severaltermswherethe projectors

P

appearin differentpositions.(Fordetails,wereferthereadertotheancillary fileonthearXivpage.)

2.5. Ghostcontribution

Tosomeextent,itispossibletotreat



gh andY together.Both

operators arenon-minimal, andoftheform

δ

_μν

+

σ

¯∇

μ

¯∇

ν

+

Bνμ (notetheoverallsignisreversed),where

σ

isaconstantdefinedin (2.12) and_Bνμ

=

s_Rνμ,

¯

wheres

= −

1 for



ghands

=

1 forY .Inthe

standardone-loopcalculations,onecanusetheknownheatkernel coefficientsfor thistype of operators.In contrast to[22–24] and coherentlywiththetreatmentofgravitons,weuseatypeIcutoff also forthe ghosts. Thistype of cutofffor ghosts hadbeen used beforein[35]. Thenoveltyofourcalculationisthat wealsotake intoaccountthecontributionsduetotheanomalousdimensions

η

gh

=

0

,

η

Y

= −β

λ

/λ .

(2.23)

ThetypeIcutoffhastheform3

R

μ_kν

=

Z

δ

ν_μRk

(),

(2.24)

where Z is given by (2.9), (2.10). Adding the cutoff, the kinetic operator(asidefromthefactor Z ) becomesPk

()δ

νμ

+

σ

¯∇

μ

¯∇

ν

+

Bνμ. In the flow equation one needs the inverse ofthis operator. We referto[35] forsome technicaldetails.The evaluationofthe tracestosecondorderincurvaturesisratherlaborious.Intheend wearriveatthefollowing

Tgh

= −

1

(

4

π

)

2



d4x



g

¯



3

−

2

σ

gh

−

2

σ

2 gh log

(

1

−

σ

gh

)



k4

−

1 12

σ

2 gh



₃

σ

gh

(

2

+

σ

gh

(

7

−

5

σ

gh

))

σ

gh

−

1

−

2

(

3

−

2

σ

gh

)

log

(

1

−

σ

gh

)



k2R

¯

−

11 90R

¯

2 μνρλ

+

43

−

2

σ

gh

(

13

+

σ

gh

)

45

(

1

−

σ

gh

)

2

¯

R2_μν

+



5 18

+

1 6

(

1

−

σ

gh

)

2



¯

R2

.

(2.25)

Note the appearance of log

(

1

−

σ

gh

)

= −

log

(

2

(

1

+

ω

)/

3

)

, which

forcesustoconsideronlythedomain

ω

>

−

1.ForY : TY

= −

1 2 1

(

4

π

)

2



d4x



g

¯



3

−

2

σ

Y

−

2

σ

2 Y log

(

1

−

σ

Y

)

+

η

Y



2

−

σ

Y

+

σ

_Y2 2

σ

_Y2

+

(

1

−

σ

Y

)

σ

_Y3 log

(

1

−

σ

Y

)



k4

+



−

2

+

σ

Y 4

σ

Y

−

3

+

2

σ

Y 6

σ

2 Y log

(

1

−

σ

Y

)

3 _{Weobservethatthecalculationoftheghostcontributionsisconsiderably} sim-plerwithaso-calledtype-IIcutoffRμ

kν=Zδ μ

νRk(+B).Theuseofthisalternative

schemefortheghostswouldleadtoonlysmallquantitativedifferencesinthefinal resultsforthefixedpointsandweshallnotdiscussthisindetail.

+

η

Y



6

−

σ

Y 12

σ

2 Y

+

3

−

2

σ

Y

−

σ

Y2 6

σ

3 Y log

(

1

−

σ

Y

)



k2R

¯

−

11 90



1

+

η

Y 2



¯

R2_μνρλ

+



43 45

+

η

Y



₂₀

₋

₂₀

σ

Y

−

39

σ

Y2

+

29

σ

Y3 120

σ

2 Y

(

σ

Y

−

1

)

−

1

−

σ

Y

−

2

σ

Y2 12

σ

3 Y log

(

1

−

σ

Y

)



¯

R2_μν

−



2 9

+

η

Y



₄

₊

σ

_Y2

+

σ

_Y3

−

3

σ

_Y4 48

(

−

1

+

σ

Y

)

σ

_Y2

−

2

−

σ

Y

−

2

σ

Y2 24

σ

3 Y log

(

1

−

σ

)



¯

R2

.

(2.26)

Bothagreewith[35] ifweput

η

Y

=

0. 3. Results

3.1. Betafunctions

Forthestudyoftheflow,thedimensionfulcouplings

V

andZN

havetobereplacedbytheirdimensionlesscounterparts

˜V = V/

k4

andZ

˜

N

=

ZN

/

k2,ortherelatedquantitiesG

˜

=

Gk2,

˜ = /

k2.The

beta functions are too complicated to be written here (they are givenina Mathematicanotebook [38]),butthey simplifyin two cases.Expandingforsmall

λ

weobtaintheuniversalone-loopbeta functions

β

λ

= −

133

λ

2 160

π

2

+

O

λ

3



(3.1)

β

ω

= −

λ(

200

ω

2

₊

₁₀₉₈

_ω

₊

₂₅

₎

960

π

2

+

O

λ

2



(3.2)

β

θ

=

7

(

56

−

171

θ )

1440

π

2

λ

+

O

λ

2



(3.3)

while the non-universal beta functions for G and

˜

˜

agree with those found in the one-loop calculation [35] at

λ

=

0. Explicitly theyaregivenby

β

_G˜

=

2G

˜

+ ˜

G2



−

c1 72

π

(

1

−

2

ω

)

+

c2log

(

2(1+₃ω)

)

12

π

(

1

−

2

ω

)

2



+

O

(λ)

(3.4)

β

_˜

= −

2

˜+

G

˜

72

π



c3

+ ˜

c4 1

−

2

ω

+

6

(

c5

+ ˜

c6

)

log

(

2(13+ω)

)

(

1

−

2

ω

)

2



+

O

(λ)

(3.5) withthecoefficientsc1

=

35

−

2

ω

(

109

+

176

ω

)

,c2

=

65

+

4

ω

(

7

+

2

ω

)

,c3

=

162

−

540

ω

,c4

= −

35

+

218

ω

+

352

ω

2,c5

=

6

−

96

ω

−

48

ω

2_,c

6

=

65

+

28

ω

+

8

ω

2.

Our calculation differs from one-loop calculations in that we takeintoaccounttheanomalousdimensions.Forexample,wesee

η

Y appearingexplicitlyin(2.26), whichgivescontributionstothe

betafunctionsofallthecouplings.Equation(2.23) tellsusthat

η

Y

isproportional to

β

λ. Thus, comparing theterms proportional to C2 _{on both sidesof the FRGE, we obtain a relation ofthe form}

β

i

=

Bi

+

Ci j

β

j. At one loop one just keeps

β

i

=

Bi. Solving the

algebraicequationsgivesbetafunctionsthatcontaincontributions witharbitrarilyhighlooporder.

However,fromthedefinitions,the anomalousdimensionsata fixedpointareknownaprioritobe

Table 1

Fixedpointsintheapproximation ˜V =0.

λ∗ ξ∗ ρ∗ ω∗ Z˜N∗ G˜∗

FP1 0 0 0 −0.02286 0.00833 2.388 FP2 29.26 −220.2 0 0.4040 0.01318 1.509 FP3 52.61 1672 0 −0.0944 0.00761 2.614

So,inthesearchoffixedpoints,onecanusesimplifiedbeta func-tions where these values are used: the full expressions for the anomalous dimensions are only neededwhen one calculates the scalingexponents.Itiseasytoseethatifwehadassumedthatall terms in V and U areofthe sameorder, namely

√

V ∼

ZN

∼

R,

then allthetermscontaining V0 andU1 wouldnotcontribute to

the betafunctionsof

λ

,

ξ

and

ρ

.Therefore, thesebetafunctions wouldnot contain ZN andwouldbe exactlythesame asin[35].

Thisiswhyitis importanttokeeptheexpansion in ZN separate

fromtheexpansionin R.4

Even the simplified beta functions with(3.6) are too compli-catedto bereportedin detail.However, weshallsee aposteriori that

˜V

isverysmallatfixedpoints.Ifweput

˜V =

0,theequations fortheremainingvariablesbecomesimpleenough:

β

λ

= −

133 160

π

2

λ

2

_{+ ˜}

_Z N

λ

3 251

ξ

−

58

λ

120

π

2

_ξ

(3.7)

β

ξ

= −

5

(

72

λ

2

−

36

λξ

+ ξ

2

)

576

π

2

+ ˜

ZN 9720

λ

3

−

1980

λ

2

ξ

+

489

λξ

2

−

14

ξ

3 6480

π

2 (3.8)

β

ρ

= −

49 180

π

2

ρ

2

_{+ ˜}

_Z N

λ

ρ

2 233

ξ

−

58

λ

240

π

2

_ξ

(3.9)

β

_Z˜ N

=



−

2

+

(

30

λ

− ξ)(

4

λ

+ ξ)

192

π

2

_ξ



˜

ZN

+

−

3168

λ

2

+

654

λξ

+

35

ξ

2 1152

π

2

_ξ(

₆

_λ

_{+ ξ)}

−

72

λ

2

−

84

λξ

+

65

ξ

2 192

π

2

₍

₆

_λ

_{+ ξ)}

2 log



2 3

−

2

λ

ξ



.

(3.10) 3.2. Fixedpoints

Nowwerecallthatalreadyintheone-loopcalculation,thebeta functions of Z

˜

N (and also

˜V

) have a nontrivial fixed point. This

nonzero valueof ZN entersinthe betafunctionsof(3.7)-(3.9) in

such awaythatbesidestheasymptotically freefixedpoint, there arenowtwo(andonlytwo)newones.Theircoordinatesaregiven inTable1.

The firstfixed point isfound alsointhe one-loop approxima-tion, and it isa non-trivialfact that it persists also when Z

˜

N is

presentinthebetafunctionsof

λ

and

ξ

.5Notethatintheone-loop approximation there is also another fixed point with

λ

= ξ =

0,

ω

= −

5

.

467,whichhoweverisexcludedbyourcondition

ω

>

−

1 (otherwiseitgivesacomplex Z

˜

N).Theremainingtwofixedpoints

are“fullyinteracting”.Itisworthnotingthatifwetreat Z

˜

N asan

externalparameter inthebetafunctionsof

λ

and

ξ

,we findthat

λ

∗ and

ξ

∗ gotoinfinityfor Z

˜

N

→

0.6

4 _{ItwouldobviouslybeevenbetternottoexpandinZ}

Natall,butthiswouldbe

technicallymuchmorechallenging. 5

Actually,thisfixedpointisbeststudiedusingthevariableωinsteadofξ.It correspondstolettingλandξgotozerowithaparticularratio,andisdifferent fromsettinge.g.firstλ=0 andthenξ=0.

6 _{andtozeroforZ}˜

N→ ∞,butthisisoutsidethedomainofourapproximation.

Wethencometothesolutionofthefullflowequations,where wetake intoaccount alsotherunningof

˜V

.There arenowmore fixedpoints, andwereport inTable2the propertiesofthemost interestingones.

Weseethatinallcasesthefixedpointvalueof

˜V

isverysmall, justifyingtheearlierapproximation

V =

0.Infact,by considering onlythebetafunctionsof

λ

,

ξ

andZ

˜

N,andtreating

˜V

asa

param-eter,andlettingthisparametervarybetweenzeroand0

.

004575, or 0

.

006928, we can see that FP2 and FP3 change continuously

fromthevaluesofTable1tothoseofTable2.Wemaythus iden-tifythefirstthreefixedpointsofTable2withthoseofTable1.

Thereareseveralotherfixedpointswith

λ

=

0,ofwhichFP4is

arepresentativeexample.Welistithereforreasonsthatwill be-comeclearlater.Theremayalsoexistothernon-trivialfixedpoints with

λ

=

0, but this would require a more extensive numerical search that we have not undertaken. Besides, these fixed points areprobablyartifactsofthetruncation,asareknown tooccurin othersimilarcases.

We note that also Z

˜

N∗ is small, and thisjustifies aposteriori theexpansionin Z

˜

N thatwe usedthroughoutourcalculations.If

we change variable from Z

˜

N to G

˜

N and set

λ

=

0,then asseen

from(3.4) thereisafixedpointatG

˜

=

0.Ontheotherhand,ifwe firstset G

˜

=

0, thereis no acceptablefixed point forthe dimen-sionless couplings. In any case, since we have expanded in Z

˜

N,

anyresultnearG

˜

=

0 isunreliable. Thisisunfortunate,becauseit meansthatwe cannotcheck whetherthereexistsa RGtrajectory joiningoneofthefixed points listedabove tothestandard weak gravityregimeintheIR.

3.3.Scalingexponents

Ifwerescalethefluctutionfieldhμν byafactor

√

λ

,sothatthe prefactorofitskinetictermiscanonical,thefixedpointFP1isseen

tobe aGaussianfixed point,andindeedwefindthat thescaling exponentsaregivenbythecanonicaldimensions:4,2,0,0,0.The scalingexponentsofFP2,listedfrommoretolessrelevant,are

θ

1,2

=

2

.

35191

±

1

.

67715i

,

θ

3

=

1

.

76672

,

θ

4

=

0

,

θ

5

= −

3

.

20030

,

whilethoseofFP3 are

θ

1,2

=

2

.

03270

±

1

.

52155i

,

θ

3

=

1

.

23742

,

θ

4

=

0

,

θ

5

= −

5

.

27685

.

The marginal coupling is

ρ

, the (inverse of the) coefficient of the topological term. At the non-Gaussian fixed points, we find

β

ρ

=

A

ρ

2 withA

=

0

.

01736 atFP2and A

=

0

.

02258 atFP3.Thus,

atbothfixed points,

ρ

is marginallyrelevantwhen itis negative andmarginallyirrelevantwhenitispositive.Wethusarriveatthe conclusionthat alsointhepresentapproximation, thedimension ofthecriticalsurface ofpuregravity isthree,upto themarginal topologicalterm.

3.4.Thea-function

Thebetafunctionof

ρ

isgivenby(2.6).InanordinaryCFT,the coefficienta appearsinthetraceanomalyas



Tμμ

=

1 16

π

2

cCμνρσCμνρσ

−

aE



.

(3.11)

Forexample,forafreetheorywithNS scalars,Nf Diracfieldsand

Table 2

Selectedfixedpointsincluding ˜V.

λ∗ ξ∗ ρ∗ ω∗ Z˜N∗ ˜V∗ G˜∗ ˜∗ a FP1 0 0 0 −0.02286 0.00833 0.006487 2.388 0.3894 4.356 FP2 24.91 −287.1 0 0.2603 0.01635 0.004575 1.217 0.1399 −2.741 FP3 28.24 175.6 0 −0.4825 0.01499 0.006928 1.327 0.2310 −3.566 FP4 0 −312.2 0 0 0.009222 0.006092 2.157 0.3303 4.357 a

=

1 360

(

NS

+

11Nf

+

62NV

) ,

c

=

1 120

(

NS

+

6Nf

+

12NV

) .

(3.12) Accordingtothea-theorem,ifthereisaRGtrajectoryjoiningtwo fixedpoints,a ishigherattheUVfixedpoint[39–41].Thisaccords to theintuition that a isa measure ofthenumber ofdegreesof freedom of thetheory.There is no knowna-theorem forgravity. However, we can view ourcalculation asa quantumfield theory inacurvedbackground,andfromthispoint ofview thetheorem should be applicable.7 At FP1 we have a

=

196₄₅. The values of a

at the other fixed points can be calculated numerically and are reportedinthelastcolumnofTable2.

Since FP2 andFP3 havea unique irrelevant direction, there is

onlyoneRGtrajectoryleavingthesefixedpoints,thatcanbe inte-gratednumericallyinthedirectionofincreasingt

=

log k andends up (intheUV)atanotherfixedpoint. Inthiswaywehavefound anRGtrajectorythatgoesfromFP1 toFP3 andonethatgoesfrom

FP4 toFP2.Thevalueofa decreasesalongthesetrajectories,in

ac-cordancewiththetheorem.Ontheotherhand,allthefixedpoints with

λ

=

0 haveverysimilar valuesofa and thereisa trajectory thatgoesfromFP4 toanotherfixedpointwith

λ

=

0 andaslightly

largervalueofa,incontradictiontothetheorem.Sinceitis doubt-fulthattheseadditionalfixedpointsdoexist,themeaningofthis resultisnotveryclear,andwillhavetobeinvestigatedmore care-fullyinthefuture.

3.5. Spectrum

Theappearanceofseveralnon-trivialfixedpointsisnota nov-elty inthis kindof calculations. Severalofthese are likelyto be spurious, but we donot see anyreasons whyFP1 or FP2 should

be rejected a priori, or to prefer one over the other. Regarding the spectrum, we recall that in order to avoid tachyons in the expansion around flat space, the action forgravity in Lorentzian signature8_{musthaveanegativeWeylsquaredtermandapositive} R2term.AnaiveWickrotationofthelinearizedactionaroundflat space leads to aLorentzian actionthat only differs fromthe Eu-clideanonebyanoverallsign.Therefore,FP2hasthecorrectsigns

to avoidtachyons. Although this is not sufficient to guarantee a healthytheory,itgivesussomemoreroominthesearchofone.

Noteadded:After thispaperwas submittedtothejournalthe workreferredtoinfootnote1hasappearedonthearXiv[44]. Declaration of competing interest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Acknowledgement

We would liketo thank Dario Benedetti,Taichiro Kugo, Frank Saueressigand OmarZanussoforvaluable discussions. Thiswork

7 _{Similarcalculationsinvolvinggravityhavebeenreportedin[42,43].} 8 _{weusetheLorentziansignature−+ ++.}

was supportedin partby theGrant-in-Aid forScientificResearch FundoftheJSPS(C)No.16K05331.

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