Asymptotic safety in O(N) scalar models coupled to gravity

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Asymptotic

safety

in

O

(

N

)

scalar

models

coupled

to

gravity

Peter Labus

a

,

b

,

Roberto Percacci

a

,

b

,

∗

,

Gian

Paolo Vacca

c

a_{SISSA,viaBonomea265,I-34136Trieste,Italy} b_{INFN,SezionediTrieste,Italy}

c_{INFN,SezionediBologna,viaIrnerio46,I-40126Bologna,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received28May2015 Accepted8December2015 Availableonline14December2015 Editor:G.F.Giudice

We extend recent resultson scalar–tensortheories to the caseof an O(N)-invariantmultiplet.Some exact fixedpointsolutionsoftheRGflowequationsarediscussed. Wefindthatalsointhefunctional context,onemployingastandard“type-I”cutoff,toomanyscalarsdestroythegravitationalfixedpoint. Ford=3 weshowtheexistenceofthegravitationallydressedWilson–FisherfixedpointalsoforN>1. Wediscussalsotheresultsoftheanalysisforadifferent,scalar-free,coarse-grainingscheme.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

InthelastyearsmanyeffortshavebeendevotedtostudyingthepossibilityofdefiningUVcompleteQFTswhichmaydescribe gravita-tionalinteractions,possiblyinthepresenceofothermatterfields.SuchaprogramisdefinedwithintheparadigmofAsymptoticSafety[1]

andcanbe seenasa bottom-upapproach whichseeksa consistentextension to UV ofresultsoflow energyeffectivefieldtheories.It isrelatedtotheconstructionofthenon-perturbative RGflowoftheeffectiveaction,whichshouldshowaUVinteractingfixedpoint(in theoryspace)withafinitenumberofrelevantdirections.

Inarecentpaper

[2]

twoofusdiscussedfixed-functionalsolutions inascalar–tensortheorywithactioncontaining apotential V

(φ)

andagenericnon-minimalinteraction

−

F

(φ)

R.Thesefunctionsweresubjectedtoarenormalizationgroupflowdependingonacutoff k. It was foundthat in genericdimension d theflow equationsforthe dimensionlessfunctions v

(

ϕ

)

=

k−d_V

_(φ)

_{and f}

₍

_ϕ

₎

₌

_k2−d_F

_(φ)

_of thedimensionlessfield

ϕ

=

k1−d/2

_φ

_{admitsomesimpleexactsolutionswherev}

₌

_{v0isconstantand f}

₌

_f0

₊

_f2

_ϕ

2_{.Thesesolutionscan} haveinterestingapplicationstocosmologyind

=

4[3]. Unfortunatelyitseemsthatoneofthesesolutions,with f0 and f2 bothnonzero, doesnotsurvive,atleastinthesameform,when oneapplies the“renormalizationgroupimprovement”,whileanother,whichremains unchangedbythe“renormalizationgroupimprovement”,has f0

=

0 but f2

<

0,makingitsphysicalrelevancequestionable.

Inthispaperweconsiderthecasewhenthereare N scalarfields,withan O

(

N

)

-invariantactionofthegeneralform



ddx

√

g



V

(φ)

+

1 2



a

(

∇φ

a

)

2

−

F

(φ)

R



.

(1.1)

Here V and F arefunctionsoftheradialdegreeoffreedom andweusethenotation

φ

=



_a

φ

a

_φ

a _{andforthedimensionlessrescaled} field

ϕ

=



a

ϕ

a

ϕ

a. (Insection 4 itwill be proved moreconvenient tothink of v and f as functionsof thevariable

ρ

=

ϕ

2

/

2.) The remaining N

−

1 angulardegreesoffreedomwillbereferredtoastheGoldstonebosons.Oneofthemainresultsofthisnoteshows that inthecasewhenN

>

1 thereexistsalsoanothersolutionwith f0

=

0 and f2

>

0.

Theother mainresultis theconfirmation,on employingastandard cutoffof“type-I”(in theterminologyof[4]), ofthe existenceof anupperboundonthenumberofscalarsinordertoobtainan FPwithpositive f .Suchboundshavebeendiscussedearlierbutonlyfor minimallycoupledfields.Wewillseethattheexistenceofthepotentialandofthenonminimalinteractionsdoesnotsubstantiallychange thepicture.

*

Correspondingauthor.

E-mailaddresses:plabus@sissa.it(P. Labus),percacci@sissa.it(R. Percacci),vacca@bo.infn.it(G.P. Vacca).

http://dx.doi.org/10.1016/j.physletb.2015.12.022

0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

In the same scheme ford

=

3 wealso discuss the existence of a nontrivial gravitationally dressed fixed point andgive a specific solutionforN

=

2.Finallywewillemployanalternativecoarse-grainingschemeinordertotestthescheme-dependenceoftheanalytical scalingsolutions.

2. Flowequations

Fortheconstruction of theflow equations we followthesame steps asin [2], whichwe briefly recall. The calculationis based on thebackgroundfieldmethod,butinsteadoftheusuallinearclassical-background-plus-quantum-fluctuation splitweusean exponential parametrizationforthemetricoftheformgμν

= ¯

gμρ

(

eh

₎

ρ

ν .Thishassomeconcretepracticaladvantagesthatwillbementionedlater,but theoriginaltheoreticalmotivationisthatitrespectsthenon-linearnatureofthemetric.Thefluctuationfieldisfurtherdecomposedinto itsirreduciblespin-two,spin-oneandspin-zerocomponentshT T_{μν ,}

ξ

μ,

σ

andh.Thenonecalculatesthesecondvariationoftheactionwith respecttothefields,whichappearsasacentralingredientintheflowequation.InordertohaveadiagonalHessianwedefinenewscalar fieldsthatinvolveamixtureof

σ

and

φ

.Wethenchooseaverysimple“unimodularphysicalgauge”whichamountssimplytosuppressing thefields

ξ

andh.WedescribebrieflythesestepsintheAppendix.Finallyweintroducea“typeI”cutoff,whichamountstoreplacingall occurrencesof

− ¯∇

2_by

_{− ¯∇}

2

₊

_R

k

(

− ¯∇

2

)

[4].Inparticularwewillusethecutoffprofilefunction Rk

(

z

)

=



k2

₋

_z

_θ



_k2

₋

_z

_[5].Notethat thiscutoffis“spectrallyadjusted”,inthesensethatitdependsonsomerunningcouplings,andalsodependsexplicitlyonthebackground scalarfield,inadditiontothebackgroundmetric.Laterweshallconsideralsoothertypesofcutoff.

Ingeneraldimensiond andforanynumberofscalars,denotingbyadotthederivativewithrespecttoRGtimet

=

log k,thefullflow equationsare

˙

v

= −dv

+

1 2

(

d

−

2

)

ϕ

v 

₊

_c d

(

d

−

1

)



d2

−

d

−

3

d

+

2

+

cd

(

d

−

2

)(

d

+

1

)

2

˙

f

− (d

−

2

)

ϕ

f



4

(

d

+

2

)

f

+

cd 2



d2

−

4

f

+ (

d

−

2

)



1

+

v

2

˙

f

−

6 f

− (

d

−

2

)

ϕ

f



+

4

(

d

−

1

)

f

2

˙

f

+ (

d

−

1

)

f

− (

d

−

2

)

ϕ

f



2

(

d

+

2

)

2

(

d

−

1

) (

f

)

2

+ (

d

−

2

)

f

(

1

+

v

)



+

cd

(

N

−

1

)

ϕ

ϕ

+

v

(

ϕ

)

(2.1)

˙

f

= (

2

−

d

)

f

+

1 2

(

d

−

2

)φ

f 

₋

_c d d6

₋

_2d5

₋

_15d4

₋

_46d3

₊

_38d2

₊

_96d

₋

₂₄ 12

(

d

+

2

) (

d

−

1

)

d (2.2)

−

cd



d5

−

17d3

−

60d2

+

4d

+

48

2

˙

f

− (d

−

2

)

ϕ

f



48

(

d

−

1

)

d

(

d

+

2

)

f

−

cd

(

d

−

2

)

f f

+

2 f2



×

×

(

d

−

2

)(

d

+

2

)

f 2

_{+ (}

_d

₋

₁

₎

_f2

₍

_d

₋

₂

_)φ

_f

₋

₂

˙

f



+

2

(

d

−

1

)

f f

(

2

−

d

)

ϕ

f

+ (

d

+

2

)

f

+

2

˙

f



(

d

+

2

)

f

2

(

d

−

1

) (

f

)

2

+ (

d

−

2

)

f

(

v

+

1

)



2

+

cd f

(

d

−

2

)

ϕ



4

(

d

−

1

)

f

+ (

d

−

2

)



v

+

1

−

8

(

d

−

1

) ˙

f



+

2

(

d

−

2

)

d f v

− ˙

f



v

+

1



24

2

(

d

−

1

) (

f

)

2

+ (d

−

2

)

f

(

v

+

1

)



−

cd d

(

N

−

1

)

ϕ

12

(

ϕ

+

v

(

ϕ

))

−

cd

(

N

−

1

)

ϕ

f

(

ϕ

)

(

ϕ

+

v

(

ϕ

))

2 (2.3)

wherec_d−1

= (

4

π

)

d/2

(

d

/

2

+

1

)

.Theonlydifference betweenthecase N

=

1 discussedin

[2]

andthecaseofgeneral N isthe addition tothebetafunctionals ofv and f of thecontributionoftheGoldstonemodes, whichcanbe easily identifiedby thefactorN

−

1.The “RG-unimproved”or“one-loop”flowequationscanbeobtainedbyreplacinginther.h.s.

˙

f

→ −(

d

−

2

)

f

+

d

−

2 2

ϕ

f 

_;

_˙

_f

_{→ −}

d

−

2 2 f 

₊

d

−

2 2

ϕ

f 

_,

_(2.4)

whichisequivalenttosettingF and

˙

F

˙

tozerointheequationforthedimensionfulfunctionsV andF .

3. Scalingsolutions

Welistheresomesimpleanalyticsolutionsthatexistinanydimension.Makingtheansatzthat v and f arebothconstantleads toa fixedpoint FP1.Usingthesimple,unimprovedequations,ithasthefollowingcoordinates:

v_∗

=

cd



(

d

−

1

)(

d

−

2

)

2d

+

N

−

1 d

,

(3.1) f_∗

= −

cd d5

₋

_4d4

₋

_7d3

₋

_50d2

₊

_60d

₊

₂₄ 24d

(

d

−

1

)(

d

−

2

)

−

cd

(

N

−

1

)

d 12

(

d

−

2

)

.

(3.2)

Wenotethatthefirstfractionin f_∗ispositiveford

<

6

.

17.ThusforN

=

1 thisisanupperboundonthedimensiondictatedbypositivity ofNewton’sconstant.Havingadditionalscalarslowersthisbound.TheboundbecomeslowerthanfourbetweenN

=

11 andN

=

12.Thus, thefixedpoint hasnegative Newton’sconstantwhentherearemorethan 11 scalars,aresultthatisinrough agreementwithprevious calculations

[6]

thatalsousedasimilarcutoff.

IfweincludetheRGimprovementthecoordinatesofFP1changeto:

v_∗

=

cd



(

d2

−

1

)(

d

−

2

)

d

(

d

+

2

)

+

N

−

1 d

,

(3.3) f_∗

= −c

d d6

−

2d5

−

15d4

−

46d3

+

38d2

+

96d

−

24 12d

(

d

−

1

)(

d2

₋

₄

₎

−

cd

(

N

−

1

)

d 12

(

d

−

2

)

.

(3.4)

Now,for N

=

1 positivityofNewton’sconstantrequiresd

<

5

.

73,andtheboundbecomeslower thanfourbetweenN

=

14 and N

=

15. Thus,thefixedpointhasnegativeNewton’sconstantwhentherearemorethan14 scalars.

Ifwe make theansatz that v is constant and f is ofthe form f0

+

f2

ϕ

2_{, there isa solutionFP2 forthe unimproved fixed point} equations v_∗

=

cd



(

d

−

1

)(

d

−

2

)

2d

+

N

−

1 d

,

(3.5) f_∗

(

ϕ

)

=

cd



d5

−

4d4

−

7d3

−

50d2

+

84d

+

24 24d

(

d

−

1

)(

d

−

2

)

−

(

N

−

1

)(

d2

−

d

+

12

)

12

(

d

−

1

)(

d

−

2

)

+

ϕ

2 2

(

d

−

1

)

.

(3.6)

Weobservethatwhile f2 isalwayspositive, f0becomesnegativewheneitherd orN becometoolarge.ForexampleforfixedN

=

2 this happensatd

≈

5

.

8 andforfixedd

=

4 thishappensforN

≈

5

.

6.Thesolutionisprobablyunphysicalinthesecases.

The sameansatz doesnot yield a solution forthe improvedequations. Wesuspect that theremay be asolution withvery similar propertiesbutdifferentfunctionalform.Thesearchofsuchgeneralizationisbeyondthescopeofthispaper.

Ifwemaketheansatzthatv isconstantand f isproportionalto

ϕ

2 _{(i.e.that f0}

₌

_{0),thefixedpointequationisquadraticandadmits} tworealsolutionsFP3andFP4.Theyarethesamefortheimprovedandunimprovedflowequations.Theexpressionsforarbitraryd are quitelong,soweonlygiveheretheformulaeford

=

3

v_∗

=

N 18

π

2

,

f∗

= −

9N

−

80

±

√

9N2

₋

_264N

₊

₅₂₉₆ 96

(

N

−

1

)

ϕ

2 _(3.7) andford

=

4 v_∗

=

2

+

N 128

π

2

,

f∗

= −

6N

−

41

±

√

4N2

₋

_100N

₊

₁₃₂₁ 48

(

N

−

1

)

ϕ

2

_,

_(3.8)

wheretheuppersigncorrespondstoFP3andthelowersigncorresponds toFP4.WenotethatFP3hasafiniteN

→

1 limit,whileinthe caseofFP4thereisadivergence.InfactforN

=

1 thefixedpointFP3hadalreadybeenseenin

[2]

,butFP4doesnotexistinthat case. Imposingthat f

>

0 wefindthatFP3isalwaysunphysicalwhileFP4isacceptablefor0

<

N

<

15

.

33 ind

=

3 and1

<

N

<

11

.

25 ind

=

4.

4. Large-N limit

All the solutions described in the previous section have the property that the function f_∗ is not positive ifthe number of scalars exceedsacertainlimit.Inordertoconfirmthatthisbehaviorpersistsweconsiderherethelarge-N limitoftheflow.Itisveryconvenient to change variable anduse

ρ

=

ϕ

2

_/

_{2 as the argument ofthe functions v and f . It is easy to see that in the large-N limit both the} potentialsv and f aswellas

ρ

scalelinearlyin N.Inparticularinsuchalimittheflowequations,intermsofallthequantitiesalready rescaledbyN,readsimply

˙

v

= −dv

+ (d

−

2

)

ρ

v

+

cd 1

+

v

˙

f

= −(

d

−

2

)

f

+



(

d

−

2

)

ρ

−

cd

(

1

+

v

)

2



f

−

d 12 cd 1

+

v (4.1)

Thefixed pointequation forthescalarpotential v isthesameasinflatspace. Thereisasolution witha constantpotential v and one withanontrivialonewhichisknownanalytically

[7]

.InthefirstcasethesolutiontoEqs.(4.1)isgivenby

v

=

cd d

,

f

= −



d 12

(

d

−

2

)

cd

+

2a



+

2ad

−

2 cd

ρ

,

(4.2)

wherea isa constant ofintegration.Forsuch aline offixedpoints it isclearthat, ford

>

2 and foranyvalue ofa, thefunction f is unphysicalsinceeithertheconstantpartorthecoefficientof

ρ

arenegative.

Inordertodealwiththesecondcasewiththenontrivial solutionforv,itsequationisbestwritten,ford notaneveninteger,interms ofw

(

ρ

)

=

v

(

ρ

)

intheimplicitform

ρ

=

cd d 42F1



2

,

1

−

d 2

;

2

−

d 2

; −

w



(4.3)

ThereforethescalarsectorpresentstheWilson–Fisher typeglobalscalingsolutionfor2

<

d

<

4.Alsothefixedpointequationfor f can befurthersimplifiedintermsofw anddefininganewfunction g

(

w

)

=

f

(

ρ

)

onehas

0

= −(

d

−

2

)

g

+

2w g

−

d 12

cd

1

+

w (4.4)

whichcanbesolvedanalyticallyandgivestheimplicitsolution:

f

(

ρ

)

=

g

(

w

)

= −

d cd 12

(

d

−

2

)

2F1



1

,

1

−

d 2

;

2

−

d 2

; −

w



(4.5) which is always negative for d

>

2 and therefore such a fixed point for 2

<

d

<

4 appears in the large N limit to lead to negative gravitationalinteractionsinthecurrentformulation.Wealsonotethatintheasymptoticregion

ρ

→ ∞

onehas f

(

ρ

)

= −

_3d(d2₋₂₎

ρ

.

5. Stabilityanalysis

Wediscussherethelinearizationoftheflowind

=

4 aroundthefixed pointFP1.Webeginwiththesimpler“RG-unimproved”case, whichyieldsthefollowinglinearizedequations:

0

= −(λ +

4

)δ

v

+



ϕ

−

N

−

1 32

π

2

_ϕ



δ

v

−

1 32

π

2

δ

v (5.1) 0

= −(λ +

2

)δ

f

+



ϕ

−

N

−

1 32

π

2

_ϕ



δ

f

+

N

−

1 96

π

2

_ϕ

δ

v

+

1 96

π

2

δ

v

−

1 32

π

2

δ

f (5.2)

Inthiscasethecriticalexponentsareequaltotheirclassicalvalues:

θ

1

=

4

,

wt1

= (δ

u

, δ

f

)

1

= (

1

,

0

)

θ

3

=

2

,

wt3

= (δ

u

, δ

f

)

3

= (

0

,

1

)

θ

5

=

0

,

wt5

= (δ

u

, δ

f

)

5

=



0

,

−

N 32

π

2

+

ϕ

2



(5.3)

The solution FP1 for the full fixed point equations in d

=

4 has coordinates

v∗

=

N+4 128π2

,

f∗

=

169−12N 2304π2



. Linearizing the full flow equationsaroundthissolutionwegettheeigenvalueequationsfortheeigenperturbations:

0

= −(λ +

4

)δ

v

+

72

λ

169

−

12N

δ

f

+



ϕ

−

N

−

1 32

π

2

_ϕ



δ

v

+

72

ϕ

12N

−

169

δ

f 

₋

1 32

π

2

δ

v 0

=



3

λ(

45

−

4N

)

12N

−

169

−

2



δ

f

+



3

(

4N

−

45

)

ϕ

12N

−

169

−

N

−

1 32

π

2

_ϕ



δ

f

+

N

−

1 96

π

2

_ϕ

δ

v

+

1 96

π

2

δ

v

−

1 32

π

2

δ

f (5.4)

Wecanstudythespectrumoftheleadingeigenvaluesanalytically.WefindfourrelevantandonemarginaldirectionforN

<

45₄,while for 45₄

<

N

<

169₁₂ thereareonlytworelevantandonemarginaldirections,since

θ

2and

θ

4 becomenegative.Inparticular

θ

1

=

4

,

(δ

v

, δ

f

)

1

= (

1

,

0

)

θ

2

=

2

+

68 3

(

45

−

4N

)

,

(δ

v

, δ

f

)

2

=



72 12N

−

101

,

1



θ

3

=

2

,

(δ

v

, δ

f

)

3

=



−

29N 544

π

2

+

ϕ

2

_,

N

(

169

−

12N

)

3264

π

2



θ

4

=

68 3

(

45

−

4N

)

,

(δ

v

, δ

f

)

4

=



−

3N

(

48N

(

3N

−

76

)

+

22475

)

16

π

2

₍

_4N

₋

₄₅

₎₍

_6N

₋

₅₉

₎₍

_12N

₋

₁₀₁

₎

+

72 12N

−

101

ϕ

2

_,

−

N

(

12N

−

169

)(

12N

−

125

)

96

π

2

₍

_4N

₋

₄₅

₎₍

_12N

₋

₁₀₁

₎

+

ϕ

2



θ

5

=

0

,

(δ

v

, δ

f

)

5

=



29N

(

N

+

2

)

17408

π

4

−

29

(

N

+

2

)

272

π

2

ϕ

2

₊

_ϕ

4

_,

N

(

N

+

2

)(

12N

−

227

)

52224

π

4

−

(

N

+

2

)(

12N

−

169

)

1632

π

2

ϕ

2



(5.5) Wenotethattheeigenvaluescomeingroupswithinwhichtheyareshiftedbytwo.Thisbehaviorhadalreadybeenobservedandexplained in

[8]

.

Fig. 1. Themaximalvalueofthefieldreachedinthenumericalevolutionbeforeencounteringasingularity,asafunctionoftheinitialconditionsv(ϕ=0)and f(ϕ=0), forthecased=3 andN=1 (leftpanel)andN=2 (rightpanel).Aclearspikecanbeseeninthecenterofeachfigure.

6. ThegravitationallydressedWilson–Fisherfixedpoint

In

[2]

welookedforscalingsolutionsind

=

3 withapotentialresemblingthat oftheWilson–Fisherfixedpoint.Weconcentratedon theunimproved equations, andfound a solutionforsufficiently small

ϕ

whose potential is almost indistinguishablefromthe Wilson– Fisher potential, and witha function f that starts out positive at

φ

=

0 buthas negative second derivative, such that it crosseszero atsomecriticalvalue

ϕ

≈

0

.

92.The analysisofthesolutionbeyondthiscriticalvalueproved hardandwe wereunable toestablishits globalexistence.Perhapsmoreimportant,theHessianbecomesill-definedatthecriticalpoint,sothattheequationsthemselvesbecome unreliable. Alittlelater, usingmorepowerfulnumericaltechniques,a globalsolution wasfound fortheimprovedRGequation

[9]

. The fixed pointpotential forthissolution isagainvery similar tothe Wilson–Fisher potential,butthe function f nowhas positivesecond derivativeandispositiveeverywhere,avoidingtheissuesmentionedabove.Withhindsightthissolutioncanbeseenalsowiththesimpler techniquesusedin

[2]

,suchasTaylorexpansionandtheshootingmethod.Neartheoriginithastheexpansion

v

=

0

.

00935540

−

0

.

0292660

ϕ

2

+

0

.

00359119

ϕ

4

+

1

.

14530

ϕ

6

+ · · ·

f

=

0

.

0686040

+

0

.

172245

ϕ

2

−

0

.

132631

ϕ

4

+

0

.

390317

ϕ

6

+ · · ·

We havelooked forgeneralizations ofthis solutionfor N

>

1. Treating N as a continuous variable,candidate scaling solutions canbe foundwiththeshootingmethod.In

Fig. 1

weshow,asanexample, thecasesN

=

1 andN

=

2.Aspiralingstructureappearsclosetothe fixedpointwhichischaracterizedbyaverysharprelativepeak.

ForN

=

2 thesolutioncanbeapproximatedneartheoriginby

v

=

0

.

0146691

−

0

.

00148208

ϕ

2

−

0

.

438516

ϕ

4

+

4

.

35302

ϕ

6

+ · · ·

f

=

0

.

053294

+

0

.

26337

ϕ

2

−

0

.

570919

ϕ

4

+

1

.

43572

ϕ

6

+ · · ·

Theshapeofthepotential v showsthatthescalingsolutionischaracterized byabrokenphase. Theasymptoticbehaviorofthesolutionforlarge

ϕ

is

uas

(

ϕ

)

=

A

ϕ

6

+

23 1440

π

4_B

_ϕ

2

+

240

π

4_B2

₍

_16B

₊

_5N

₋

₄

₎

₋

_{1357 A} 216000

π

6_{A B}2

_ϕ

4

+ · · ·

fas

(

ϕ

)

=

B

ϕ

2

+

23 36

π

2

+

1219 25920

π

4_B

_ϕ

2

−

71921 A

+

2160

π

4_B2

₍

_16B

₊

_5N

₋

₄

₎

4665600

π

6_{A B}2

_ϕ

4

+ · · ·

(6.1)

ForthecaseN

=

2 there isavery goodmatchbetweenthe numericalsolutionfoundwiththeshootingmethodaroundthe originand theasymptoticbehaviorgivenabovefor A

5

.

149 and B

0

.

273 around

ϕ

0

.

65.Thereforeweconsiderthisasagoodcandidatefora globalscalingsolutionforthegravitationallydressedO

(

2

)

scalarmodel.

The critical exponents associated to this fixed point (N

=

2) have not been determined with sufficient precision using polynomial methods.WecanonlyreportthattheyarenottoofarfromthevaluesobtainedfortheN

=

1 case.Moresophisticatednumericalmethods mustbeused.

7. Scalar-freecutoff

Sofar wehaveconsidered atype-I cutoffwhich inthegravitationalsector hasthegeneralform F

( ¯

φ)

Rk

(

− ¯∇

2

)

.Thepresence ofthe prefactorF isusefulbecausetheeffectofaddingthecutoffresultssimplyinthereplacementof

− ¯∇

2_{by P}

Hessians.However,thisadvantagecomesataprice:thecutoffdependsonrunningcouplingsandthereisanexplicitbreakingofthescalar splitsymmetry

¯φ

a

→ ¯φ

a

+

a_,

_δφ

a

_{→ δφ}

a

₋

a_{.Asdiscussedin}

_[10]

_{,alreadyinpurescalartheory thepresenceofthescalarbackground} field inthecutoff actioncan lead tounphysical results.Thisargumentcasts doubtsonthe useofthesecutoffs, andit isimportantto explore alsocutoffs that do not have a prefactor F . Inthe Einstein–Hilbert truncation,where F

=

1

/(

16

π

G

)

, such cutoffs were called “pure”

[11]

.Hereweshallcallthem“scalar-free”.Itisimportanttokeepinmindthatanycutoffstilldependsonthebackgroundmetric, andthatthisdependencewillhavetobedealtwithbyothermeans,seee.g.

[12,13]

.Hereweonlygetridofthe

¯φ

-dependenceandhave thescalar splitWard identity automaticallyfulfilled. Inparticular we shalladd in thediagonal entriesfor hT T

μν and

σ

 ofthe Hessian giveninEq.(A.8)acutoff

γ

kd−2

₍

_k2

₋

_z

_{)θ (}

_k2

₋

_z

₎

_{,withthesamesignoftheLaplacianterm,toimplementthecoarse-grainingprocedure.} Foralltheothertermsweshallusethemorecommon

(

k2

₋

_z

_{)θ (}

_k2

₋

_z

₎

_.

We do not report herethe form of the flow equations in any dimension, which contains hypergeometric functions. Ford

=

4 the equationsread:

˙

v

= −

4v

+

ϕ

v

−

1 8

π

2

−

f

f

(

v

+

1

)

log

_f(3 f_v₊2₁₎

+

1



−

3 f2



144

π

2_f4

+

3

γ

−

γ

2

₊

₄

_γ

_f

₋

₂

_γ

₍

_γ

₋

_{2 f}

₎

_log

f γ



−

3 f2



16

π

2

₍

_γ

₋

_f

₎

3

+

(

N

−

1

)

ϕ

32

π

2

₍

_v

₊

_ϕ

₎

(7.1)

˙

f

= −

2 f

+

ϕ

f

+

19 384

π

2

+

f



6 f2_{f f}_+f2 3 f2_+f(v₊1)

−



2 f f

+

3 f2

log

3 f2 f(v+1)

+

1



288

π

2_f4

+

10

γ

(

γ

−

2 f

)(

γ

−

f

)

−

5

γ



γ

2

₋

₃

_γ

_f

₊

_{4 f}2

_log

f γ



96

π

2

₍

_γ

₋

_f

₎

3

+

γ

−

γ

+ (

γ

−

2 f

)

log

f γ



+

f



96

π

2

₍

_γ

₋

_f

₎

2

−

(

N

−

1

)

ϕ



3 f

+

v

(

ϕ

)

+

ϕ

96

π

2

₍

_v

₊

_ϕ

₎

2 (7.2)

Unlike(2.1), (2.2), theseare transcendental equations andcannot be solved analytically. For a comparisonwe discuss onlythe scaling solutionFP1whichisdefinedby v

(

ϕ

)

=

v0 and f

(

ϕ

)

=

f0.Withthisansatz thefixed pointconditionsreducetothefollowingalgebraic equations v0

=

1 128

π

2

₍

_γ

₋

_f 0

)

2



(

N

−

10

)

γ

2

−

2

(

N

−

13

)

γ

f0

+ (

N

−

4

)

f02

−

12

γ

2

γ

−

2 f0

γ

−

f0 log



f0

γ



0

= −

2 f0

−

4N

−

55 384

π

2

−

f0

(

γ

+

9 f0

)

96

π

2

₍

_γ

₋

_f 0

)

2

−

γ



2

γ

2

₋

₆

_γ

_f 0

+

9 f₀2

log

f0 γ



48

π

2

₍

_γ

₋

_f 0

)

3 (7.3) Thesecanbesolvednumerically.ThelinearizationoftheflowequationaroundFP1yieldsthefollowingequations:

0

= −(λ +

4

)δ

v

+



ϕ

−

N

−

1 32

π

2

_ϕ



δ

v

−



3

γ



3 f₀2

+

5

γ

f0

−

2

γ

2

16

π

2_f 0

(

f0

−

γ

)

3

+

3

γ

2

₍

_γ

₋

_{4 f} 0

)

8

π

2

₍

_f 0

−

γ

)

4 log



f0

γ



δ

f

−

1 32

π

2

δ

v (7.4) 0

= −(λ +

2

)δ

f

+



γ



37 f₀2

−

11

γ

f0

+

4

γ

2

96

π

2_f 0

(

f0

−

γ

)

3

−

γ

f0

(

2

γ

+

3 f0

)

16

π

2

₍

_f 0

−

γ

)

4 log



f0

γ



δ

f

+



ϕ

−

N

−

1 32

π

2

_ϕ



δ

f

+

N

−

1 96

π

2

_ϕ

δ

v

+

1 96

π

2

δ

v

−

1 32

π

2

δ

f (7.5)

For N

=

1 and

γ

=

1 we have v0_∗

=

0

.

03314, f0_∗

=

0

.

01552. The relevant eigenperturbations around this solution can be found numerically and the corresponding eigenvalues are

−

4,

−

2

.

272,

−

2,

−

0

.

272, 0. There are therefore four relevant and one marginal couplings.For

γ

=

0

.

006 wehavev0∗

=

0

.

00353, f0∗

=

0

.

00670,whichareveryclosetothevaluesfoundwiththeothercutoffin

[2]

.The relevanteigenperturbationsaroundthissolutionhaveeigenvalues

−

4,

−

2

.

542,

−

2,

−

0

.

542,0,whicharealsocloserto theothercutoff.

ForN

=

4 and

γ

=

1 thefixedpointisat v0_∗

=

0

.

03635, f0_∗

=

0

.

01413.Thelinearperturbationanalysisaroundthefixedpointgives thefollowingeigenvalues:

−

4,

−

2

.

299,

−

2,

−

0

.

299,0.Ifinsteadwechoose

γ

=

0

.

006,wehave v0∗

=

0

.

00669, f0∗

=

0

.

00549, andthe eigenvaluesare

−

4,

−

2

.

682,

−

2,

−

0

.

682, 0.

Inthisscalar-freecutoffschemeitisinterestingtoinvestigatethesolutionsofEqs.(7.3)forFP1inthelarge N limit.Itiseasytosee thatinthislimittheybecome

v0

≈

16N

−

205

512

π

2

,

f0

≈

γ

exp

55

−

4N

16

.

(7.6)

We seethat this behavior is very differentfromthe one obtained previously: in particular f0 becomes exponentially smallbut never changessign. Thusthereis apparentlynoupperbound inN from therequirementofhavingapositive f0.Thisbehavior isinducedby

theinterplaybetweenthe log singularityin f0 andthelinear dependencein N intheequations.Thisresultisprobablynot physically correctforthefollowingreason:intheERGEoneshouldusetheHessiandefinedasthesecondderivativeoftheEffectiveAverageAction (EAA)withrespecttothequantumfield.Insteadinordertoclosetheflowequation,weareusingthesecondderivativeoftheEAAwith respecttothebackgroundfield.Eventhoughthefunction F doesnotappearinthecutoff,itdoesappearinthedenominator,wherethe coefficientof

−∇

2is F

−

γ

kd−2.Thistermisabsentwiththetype-Icutoffanditisitspresencewiththescalar-freecutoffthatgivesrise tothelogarithmic termsintheflow equation.Ina properbi-metric calculationthecoefficient of

−∇

2 inthedenominator oftheERGE wouldbe Zh

−

γ

kd−2,where Zh is thegraviton wave function renormalizationconstant. The argumentofthe logarithmic termwould thenbe Zh

/

γ

andthebetafunctionof f wouldprobablyberegularfor f

=

0.

8. Discussion

Thispaperextendsearlierinvestigations ofscalar–tensorgravity, wherewe usedtheexponential parametrizationofthemetric.The apriori motivation of thisparametrization is that it respects thenonlinear structure of the spaceof Riemannian metrics. It appears a posteriorithatitleadstobetterbehavedflows:therearenoinfraredsingularities

[2,14]

andtheequationscanbemadegauge-independent evenoff-shell

[14]

.Quitegenerally,theuseoftheexponentialparametrizationinconjunctionwiththephysicalgauge

ξ

_μ

= ∇

μh

=

0 leads to simplerequations.In thetheory considered here, they are sufficiently simplethat analytic scaling solutions canbe found. We have givenheretheformofsuchsolutionsinanydimension andforanynumberofscalarfields.Relativetoourearlierwork,theadvantageof havingmorethanonescalarfieldisthatthereexistsaglobalsolution(FP4)with f

≥

0 everywhere.

One interesting by-product ofthis analysisisthe confirmation of upperboundson the numberof scalars, forcompatibilitywitha fixedpointwithpositive f .Thisresultisnotconfirmedwhenoneusesascalar-free cutoffbutwehavearguedthatthisisprobablynot physicallycorrect.

Asisoftenthecasewhentakingthelarge N limitwehavebeenabletoconstructtheexactanalytical solutionsandshow thatthey areunphysicalinthetype-Icutoffscheme.

Wehavealsoinvestigatedind

=

3 the existenceofthegravitationallydressedWilson–Fisherscaling solutionforvarious finiteN. In particularwehaveshownthatphysicallyacceptablesolutiondoexistforsmallN andgivendetailsforthecaseN

=

2.

We hope that thesetools will prove useful also in the search of globalsolutions for f

(

R

)

truncations, where severalresults have been obtainedrecently [15], but a clear picture is still missing. Another important directionof investigation is the inclusion of other matter fields, in particular fermions [16–18]. If the picture we have found up to now will be confirmed also in more sophisticated truncations,the possibilitytodefine a consistentQFT ofmatter–gravity interactions willbe aguidancein theunderstanding ofseveral aspectsoffundamental/effectivephysics.Wefindalreadyinterestingtostarttoinvestigatepossibleimplicationsatphenomenologicallevel forexampleincosmology.

Acknowledgements

WewouldliketothankAstridEichhornandJanPawlowskifordiscussions.

Appendix A. Hessian

TheHessianforgravity coupledto asingle scalarhasbeengivenin[2].Here wediscussthechanges introduced byhaving ascalar multiplet forthemodel ofEq.(1.1).The scalarfield is decomposedintoa backgroundand fluctuationpart

φ

a

= ¯φ

a

+ δφ

a.The partof theHessianwhichcontains adependenceonthe scalarfluctuation

δφ

a isasumoftwo pieces.The firstisquadraticinthescalarfield fluctuationsandreads:



ddx



g

¯

1 2

δφ

a



_{− ¯∇}

2

_{+ ¯}

_V

_{− ¯}

_F_R

¯



_PR ab

+



− ¯∇

2

₊

1

¯φ

( ¯

V

− ¯

FR

¯

)



P⊥_ab

δφ

b (A.1)

wherewehaveintroducedtheprojectorsPab_R

= ¯φ

a

¯φ

b

/ ¯

φ

2andPab_⊥

= δ

ab

−

Pab_R.Thesecondmixesscalarfieldfluctuationswithmetricscalar fluctuations:



ddx



g

¯

δφ

aP_abR

¯φ

b1

¯φ



1 2h

( ¯

V 

_{− ¯}

_F_R

_¯

₎

_{− ¯}

_F

_{( ¯}

_∇

μ

¯∇

μhμν

− ¯∇

2h

− ¯

Rμνhμν

)

(A.2) Withoutlossofgeneralitywecanseparate theradialfieldcomponentfromtheGoldstonebosonsbyfixingthebackgroundtobe

¯φ

a

=

0, a

=

1

,

. . . ,

N

−

1,

¯φ

N

_{= ¯}

_ϕ

_anddefine

_δφ

N

_{= δ}

_ϕ

_{.AftertheYorkdecomposition}

hμν

=

hT Tμν

+ ¯∇

μ

ξ

ν

+ ¯∇

ν

ξ

μ

+ ¯∇

μ

¯∇

ν

σ

−

1

d

¯

gμν

¯∇

2

_σ

₊

1

dg

¯

μνh (A.3)

andtheredefinition

ξ

_μ

=



− ¯∇

2

₋

R

¯

d

ξ

μ

;

σ



₌



_{− ¯∇}

2



− ¯∇

2

₋

R

¯

d

−

1

σ

,

(A.4)

thesecondpiececanberewritten



ddx



g

¯

(

d

−

1

)

d F

¯



_δ

_ϕ



₋



− ¯∇

2



− ¯∇

2

₋

R

¯

d

−

1



σ



+



− ¯∇

2

₋

(

d

−

2

) ¯

R 2

(

d

−

1

)

+

dV

¯

 2

(

d

−

1

) ¯

F



h

(A.5)

Fixingthegaugeh

=

const,

ξ

=

0 andalsoneglectingtheresidualconstantmodeofh,thefullHessianbecomes

_k(2)

=



ddx



¯

g



1 4F h

¯

T T μν



− ¯∇

2

₊

2R

¯

d

(

d

−

1

)



hT Tμν

−

(

d

−

2

)(

d

−

1

)

4 d2 F

¯

σ



₍

_−∇

2

₎

_σ



+

1 2

δ

ϕ

(

− ¯∇

2

_{+ ¯}

_V

_{− ¯}

_F_R

¯

_)δ

_ϕ

₊

1 2 N

_

−1 a=1

δφ

a



− ¯∇

2

₊

1

¯φ

( ¯

V

− ¯

FR

¯

)



δφ

a

−

(

d

−

1

)

d F

¯



_δ

_ϕ



− ¯∇

2



− ¯∇

2

₋

R

¯

d

−

1



σ





.

(A.6)

TheHessiancanthenbediagonalized bythetransformation

σ



=

σ



−

2d d

−

2

¯

F

¯

F



− ¯∇

2

₋

R¯ d−1

− ¯∇

2

δ

ϕ

,

(A.7)

andthediagonalHessianreads

_k(2)

=



ddx



¯

g



1 4F h

¯

T T μν



− ¯∇

2

₊

2R

¯

d

(

d

−

1

)



hT Tμν

−

(

d

−

2

)(

d

−

1

)

4 d2 F

¯

σ



₍

_−∇

2

₎

_σ



+

1 2

δ

ϕ



− ¯∇

2

_{+ ¯}

_V

_{− ¯}

_F_R

¯

₊

₂d

−

1 d

−

2

¯

F2

¯

F



− ¯∇

2

₋

R

¯

d

−

1



δ

ϕ

+

1 2 N

_

−1 a=1

δφ

a



− ¯∇

2

₊

1

¯φ

( ¯

V

− ¯

FR

¯

)



δφ

a



.

(A.8)

Theghosttermsinducedbythegaugefixingaregivenby:

Sgh:h

=



ddx



g c

¯

(

− ¯∇

2

)

c

,

Sgh:ξ

=



ddx



¯

g cμ

¯

gμν



− ¯∇

2

₋

R

¯

d



cν

.

(A.9) References

[1]S. Weinberg, in: S.W. Hawking, W. Israel (Eds.), General Relativity, Cambridge University Press, 1979, pp. 790–831. [2]R. Percacci, G.P. Vacca, Eur. Phys. J. C 75 (2015) 188, arXiv:1501.00888 [hep-th].

[3]T. Henz, J. Pawlowski, A. Rodigast, C. Wetterich, Phys. Lett. B 727 (2013) 298–302, arXiv:1304.7743 [hep-th]. [4]A. Codello, R. Percacci, C. Rahmede, Ann. Phys. 324 (2009) 414, arXiv:0805.2909 [hep-th].

[5]D.F. Litim, Phys. Rev. D 64 (2001) 105007, arXiv:hep-th/0103195.

[6]P. Donà, A. Eichhorn, R. Percacci, Phys. Rev. D 89 (2014) 084035, arXiv:1311.2898 [hep-th]. [7]E. Marchais, Ph.D. thesis, University of Sussex, 2012.

[8]G. Narain, R. Percacci, Class. Quantum Gravity 27 (2010) 075001, arXiv:0911.0386 [hep-th]. [9]J. Borchardt, B. Knorr, arXiv:1502.07511 [hep-th].

[10]I. Hamzaan Bridle, J.A. Dietz, Tim R. Morris, J. High Energy Phys. 1403 (2014) 093, arXiv:1312.2846 [hep-th]. [11]G. Narain, R. Percacci, Acta Phys. Pol. B 40 (2009) 3439–3457, arXiv:0910.5390 [hep-th].

[12]J.A. Dietz, T.R. Morris, arXiv:1502.07396 [hep-th].

[13]D. Becker, M. Reuter, Ann. Phys. 350 (2014) 225–301, arXiv:1404.4537 [hep-th]. [14]K. Falls, arXiv:1501.05331 [hep-th].

[15]D. Benedetti, F. Caravelli, J. High Energy Phys. 1206 (2012) 017, arXiv:1204.3541 [hep-th];

D. Benedetti, F. Caravelli, J. High Energy Phys. 1210 (2012) 157 (Erratum); D. Benedetti, Europhys. Lett. 102 (2013) 20007, arXiv:1301.4422 [hep-th];

M. Demmel, F. Saueressig, O. Zanusso, J. High Energy Phys. 1211 (2012) 131, arXiv:1208.2038 [hep-th]; J.A. Dietz, T.R. Morris, J. High Energy Phys. 1301 (2013) 108, arXiv:1211.0955 [hep-th];

J.A. Dietz, T.R. Morris, J. High Energy Phys. 1307 (2013) 064, arXiv:1306.1223 [hep-th];

M. Demmel, F. Saueressig, O. Zanusso, J. High Energy Phys. 1211 (2012) 131, arXiv:1504.07656 [hep-th].

[16]O. Zanusso, L. Zambelli, G.P. Vacca, R. Percacci, Phys. Lett. B 689 (2010) 90, arXiv:0904.0938 [hep-th];

G.P. Vacca, O. Zanusso, Phys. Rev. Lett. 105 (2010) 231601, arXiv:1009.1735 [hep-th].

[17]A. Eichhorn, H. Gies, New J. Phys. 13 (2011) 125012, arXiv:1104.5366 [hep-th].