Remarks on the convergence properties of the conformal block expansion

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Remarks on the convergence properties of the conformal block expansion

Slava Rychkov, Pierre Yvernay

To cite this version:

Slava Rychkov, Pierre Yvernay. Remarks on the convergence properties of the conformal block ex-

pansion. Physics Letters B, Elsevier, 2016, 753, pp.682-686. �10.1016/j.physletb.2016.01.004�. �hal-

01253827v2�

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Remarks on the convergence properties of the conformal block expansion

Slava Rychkov^a,b,c,∗,Pierre Yvernay^d,a

aCERN,TheoreticalPhysicsDepartment,Geneva,Switzerland

bLaboratoiredePhysiqueThéoriquedel’ÉcoleNormaleSupérieure(LPTENS),Paris,France cSorbonneUniversités,UPMCUniv. Paris06,FacultédePhysique,Paris,France dENSTAParisTech,Palaiseau,France

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

Articlehistory:

Received17November2015 Accepted5January2016 Availableonline8January2016 Editor:N.Lambert

WeshowhowtorefinetheconformalblockexpansionconvergenceestimatesfromarXiv:1208.6449.In doingsowefindanovelexplicitformulaforthe3dconformalblocksontherealaxis.

©^{2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introductionandformulationoftheproblem

Many interesting new results about conformal field theories (CFTs)ind>2 dimensionshavebeenrecentlyobtainedusingthe bootstrap approach. In thismethod one uses the operator prod- uct expansion(OPE) associativityto constrain theCFT data(local operatordimensionsandtheirOPEcoefficients).Operationally,one expandsthe four point functionsinconformal partial waves and imposes that expansionsin different channels give the same re- sult.

Theresulting“conformalbootstrap equations”haveanicefea- turethatthey are totallymathematicallywell defined,expressing the agreement betweenconvergent power serieswith nonempty andoverlappingregionsofconvergence[1].Inpracticeonewould liketoknowhowfasttheseseriesconverge.Thisquestionwas al- readystudiedin[1],andherewewillgiveitafreshlook,butfirst let usexplain whythis isimportant. Forusthe main interestin thisquestioncomesfromtheneedtoputonsolidgroundGliozzi’s approachtothebootstrap.

Recallthattherearecurrentlytwocompetingapproachestothe numericalbootstrap.Inthefirstone,knownasthelinear/semidef- inite (LSD) programming[2–10],¹ one actually does not truncate theexpansion, orrathertruncatesit atsuch highdimension and spinoftheexchangedoperatorsthatthetruncationerrorisabso- lutelynegligible.Theobtainedresultstakeformofrigorousbounds

* Correspondingauthorat:LaboratoiredePhysiqueThéoriquedel’ÉcoleNormale Supérieure(LPTENS),Paris,France.

1 Wejustciteafewpaperswhereimportantdevelopmentstepsofthemethod weremade.

on thespaceofCFTdata.It isusingthisapproachthat mostnu- merical results were obtained. Here we will only highlight the study of the 3d Ising model which yielded world’s most precise estimatesonitsleadingcriticalexponents[6,8–10].

In the second approach,by Gliozzi andcollaborators [11–13], one does truncate the expansion pretty severely, keeping just a handfuloflow-dimensionoperators.Onethencompletelyneglects thetruncationerrorand,expandingaroundtheusualmiddlepoint to a finite order, gets a system of nonlinear equations to solve for the dimensions of the retained operators and their OPE co- efficients. This methodhas severaladvantages over the LSD pro- gramming: it is more intuitive, not so heavy on the numerical side, andis applicablefor bothunitary andnon-unitary theories.

On the other hand it is not as systematic, lacking as of now a built-inmechanismtoestimatethetruncation-inducederror.There aree.g. smallbutnoticeabledifferencesbetweenthe3dIsingcriti- calexponentsdeterminedbyusingGliozzi’sapproach[12]andthe LSD programming [8,10]. The LSD results are rigorous; they also agreewithMonteCarlo(whilebeingmoreprecise),whileGliozzi’s approachdoesnot.

We consideritanimportantopenproblemtofindamodifica- tion ofGliozzi’s approachwhichwouldkeepitsabove-mentioned positive features, while allowing toestimate an error induced by thetruncation.Havingagoodcontrolovertherateofconvergence oftheconformalblockexpansionisaprerequisiteforthistask.

For concreteness and simplicity, in this paper we will study a conformal four point function of identical Hermitean primary scalaroperators:

φ (^x1)φ (x₂)φ (x₃)φ (x₄) =^g(u,v)/(x²₁₂x²₃₄)^φ. (1.1)

http://dx.doi.org/10.1016/j.physletb.2016.01.004

0370-2693/©^{2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

The function g(u,v) depends onthe usual conformally invariant crossratios andcan be expandedin conformal blocksof theex- changedprimaries:

g(u,v)=

O

λ²_Og_O(u,v) . (1.2)

We are interested in theconvergence rateof thisexpansion. We willwork intheEuclideanspacealthough theMinkowskicaseis also interesting [14–17]. We will consider here the unitary case whenallOPEcoefficientsarerealandhenceλ²_O>0.

Theconvergence question was alreadystudied in[1].To state their result we need some CFT kinematics. It is standard to parametrizeanyfourpointfunctionconfigurationintheEuclidean spaceintermsofacomplexnumberz∈C\(1,+∞).Thisisdone by mapping the four points to a plane, and then moving them withinthisplanetopositions 0,z,1,∞^{.Therelationbetweenu,} v andzis

u= |^z|², v= |¹−^z|². (1.3)

Itisveryconvenienttofurthermaptherangeofztotheunitdisk byintroducingthevariable

ρ=^z/(1+√

1−^z)², |ρ|<1. (1.4) Thishasthemeaningofplacingthepointsatthepositions ρ,−ρ, 1,−^1.Theinversetransformationisz=⁴ρ/(1+ρ)².Fromnowon weswitch from u, v touse ρ asour mainconformally invariant parameter.Ifneeded,onecangobacktou,v via(1.3),(1.4).

Theconvergenceboundprovedin[1]states²:

O:_O>_∗

λ²_Og_O(ρ)

(2_∗)^4φ

(4φ+1)|ρ|^∗, _∗→ ∞. (1.5) Thisshouldbe readasfollows.Inthelhswe havethetailofthe conformal block expansion corresponding to the primary opera- torsofdimensionabovesomecutoffdimension_∗(andallspins).

This is the error we will make in the four point function if we dropall suchoperators. We nowhold ρ fixed andtakethe limit _∗ 1.³ Sinceany configurationcorresponds to |ρ_|<1,we see thatforlarge_∗ thetailbecomesexponentiallysmall, becauseof thefactor |ρ_|^∗.One configurationparticularly importantforthe bootstrapanalysis isz= ¹₂^{,which gives}ρ₌3−²√

2≈⁰.17.We seethattheconvergenceatthispointisquitefast.

SimplenumericalexperimentsingaussianCFTswherethecon- formalblockexpansionisexactlyknowncanbeusedtocheckthat theexponentiallydecreasingfactor |ρ|^{∗ in(1.5) isbestpossible.}

Ontheotherhand,thesameexperimentsindicatethatthepower of_∗ intheprefactorisnotoptimal.Improvingtheprefactorwill bethemaingoalofourpaper.

2. Reviewof[1]

Thattheprefactorin(1.5)isnotoptimalhasasimpleoriginin thewaythatestimate wasderived in[1].Let’sreviewthederiva- tion,andthenseehowitcanbeimproved.

Step1.Oneobservesthattheconformalblockshavean expan- sionoftheform[18]

g_O(ρ)=

δ,j

f_δ,jC_j(cosφ)r^δ, (2.1)

2 a≈b andab (x→x0)meanlimx→x₀(a/b)=1 andlim supx→x0(a/b)1, respectively.

3 Moreprecisely,thebound(1.5)setsinfor∗ φ/(1− |ρ|)^[1].

whereinthelhs ρ=^{reiφ,intherhs}δ, j arethedimensionsand spinsoftheprimaryoperatorO=O,l andofitsdescendants.In particularδ=+^n,n∈N0,while j rangesbetweenl±^{n.The C}j areGegenbauerpolynomialsincosφ.Theonlyimportantthingfor usisthattheytaketheirmaximalvalue,normalizedto1,atφ=^0.

Finally f_δ,j arerelativecoefficientsofdescendantswithrespectto thatoftheprimary.Let’snormalizetheconformalblockbysetting theprimarycoefficienttoone: f,=^{1.Therestofthecoefficients} arethenfixedbyconformalsymmetry.It’simportantthattheyare allnonnegative:

f_δ,j0. (2.2)

Thisconditionissatisfiedaslongastheprimaryfieldisabovethe unitaritybound(whichistruesinceweassumewehaveaunitary theory).

Step2.Twosimpleconsequencesof(2.1)are:

|^gO(re^iφ)||^gO(r)|, g_O(r) >0. (2.3) Thismeansthat it’senoughtostudyconvergenceatreal ρ,since fornonzeroφitwillbeonlyfaster.

Step3.Sofromnowonwespecializetotherealaxis.Consider two seriesrepresentationsof thesamefunction g(r): theconfor- malblockexpansion

g(r)=

O

λ²_Og_O(r) (2.4)

and g(r)=

δ

p_δr^δ. (2.5)

To obtain the second series we simply plug in the series repre- sentation (2.1)of each conformal block into (2.4) and collect all powers of r withtheir respective coefficients. So p_δ=

λ²_Of_δ,j wherethesumisover alldescendants havingthescalingdimen- sion δ (no matter what j and the primary are). Clearly p_δ ^0.

We will call a power series whose all coefficients are positive a coefficient-positiveseries.

Nowthetailofthefirstseriesisstrictlysmallerthanthetailof thesecondseries:

O:_O>_∗

λ²_Og_O(r) <

δ>_∗

p_δr^δ. (2.6)

This is because the tail on the rhs contains all terms which are present on the lhs, and in addition it contains contributions of descendants of dimension δ > _∗ coming from primaries of di- mension_O∗.

As a matter of fact, Ref. [1]established theconvergence esti- mate onthe tailof thesecond series,andused (2.6) to saythat thefirsttailsatisfiesthesameestimate.Itisherethattheprefac- toroptimalityislost.Herewewilltrytodobetter,butfirstletus completethereview.

Step4.Noticethatthefullfourpointfunctionwithpointsposi- tionedatr,−^r,1,−^{1 hasinthelimitr}→^{1 the}asymptotics

φ (^r)φ (−^r)φ (1)φ (−¹) ≈¹/(1−^r)^4φ, (2.7) which follows fromapplying the OPE for the two pairs of oper- ators which becomeclose toeach other.Notice that asusual we normalize theoperators withthe unit two point function coeffi- cient.Given(1.1),thisgivestheasymptoticsforg(r):

g(r)≈^24φ/(1−^r)^4φ (r 1) . (2.8)

Putting together this fact with the power series representa- tion(2.5),wefindourselveswithintheassumptionsoftheHardy–

Littlewood(HL)tauberian theorem,whichestablishes theasymp- toticsfortheintegratedcoefficientsp_δ:

P(E):=

δ^E

p_δ≈ (2E)^4φ

(4_φ+¹), E→ ∞. (2.9) Aphysicistwouldnormallytrytofitapowerlawassumptionabout p_δ totheasymptotics(2.8),obtaining

p_E≈^24φE4φ−1

(4φ) ^(naive), (2.10)

andthenby integration(2.9). Noticehoweverthat inaCFT p_δ is notasmoothfunctionbutadiscretesequence.It’snotatotallyob- vious matterto showthat the result(2.9)still holds underthese circumstances. The assumption pδ^{0 is crucial.The naive argu-} mentationgives aquickwaytorecover theanswerbutit’s nota substitutetotheproof.Infactit’sknowntofailcompletelyforthe subleadingtermsintheasymptotics.Seethebook[19]forathor- oughreviewoftheHLtheoremanditsramifications.

From(2.9),thetailcanbeestimatedasfollows(denoter=^e−t, t>0):

δ>_∗

p_δe^−tδ=^t ∞

∗

dE[^P(E)−^P(_∗)]^{e−t E}

t ∞

_∗

dE P(E)e^{−t E}

≈^t ∞ _∗

dE (2E)^4φ (4_φ+¹)^e

−t E

(2_∗)^4φ (4_φ+¹)^e

−t_∗. (2.11)

Here we integrated by parts in the first line, then dropped the negative term −^P(_∗), used the HL estimate on P(E), and the asymptotic behavior of the incomplete gamma function to con- clude.Thelaststeprequires_∗ φ/t.

Puttingtogetherthisestimateand(2.6)weobtain(1.5)forreal r<1.ByStep2,thesameboundisvalidforallcomplex ρ=^reiφ. It’s instructive to compare (2.11) with an estimate we would have obtained if we just approximated the sequence p_δ by the naivepowerlaw(2.10):

δ>_∗

p_δe^−tδ≈ ∞

_∗

dE p_Ee^{−t E}

≈^24φ4_∗^φ−1 t(4φ) ^e

−^t_∗ (naive). (2.12) Thereasonwhythisnaive estimatehasa betterprefactor canbe tracedbackto havingdropped −^P(_∗) inthechain ofestimates leadingto(2.11).IfP(E)isanicepowerlaw,thereisanearcance- lationbetween P(E)and−^P(_∗)forE closeto_∗,whichsharp- ens the bound.⁴ Tojustify thiscancelation in general, we would need some sort of subleading asymptotics for P(E), and unfor- tunately this is not available; see the discussion after Eq. (4.20) in[1].So(2.11)isthebestwecurrentlyhaveinfullgenerality.

4 WethankPetrKravchukandHirosiOogurifordiscussionsconcerningthispoint.

3. 3dconformalblocksontherealaxis

As indicated in the previous section, the lossin the prefactor of the convergence rate estimate of [1] comes fromtreating the primaries and thedescendants in the conformal blocks on equal footing. There are manyterms in the rhs of (2.6) whichare not present in the lhs. To improve the estimate we should think of a conformal block as a whole, instead of separating it into con- stituents. However, conformal blocks in general are complicated specialfunctions,whiletheHLtheoremisforsumsofpowersofr.

While therearegeneralizationsoftheHLtheoremvalidformore general functionsofr,herewe will demonstratea more simple- mindedapproach.

Westumbledonthepossibilityofthisapproachwhilestudying the conformalblocksin3d.We willthereforestart bypresenting theseresults,whichareofindependentinterest.

As shown in [20], conformal blocks on the real axis satisfy an ordinary differential equation. This is not obvious since as a function ofcomplex ρ they satisfy a partialdifferential equation.

HoweverthereareinfacttwoPDEs:thewell-knownsecondorder one coming fromthe quadraticCasimir, andin additiona fourth order one comingfromthe quarticCasimir.Taken together these twoPDEsimplyanODEontherealaxis.

The relevant ODE is obtained by specializing to the case of equalexternaloperatordimensionsbysettingP=^S=^{0 in(4.10a)} of[20].Ittakestheform

D4g_,l(r)=^{0 (3.1)}

with

D4=(r−¹)³(r+¹)⁴r⁴d⁴ dr⁴

+²(r−¹)²(r+¹)³r³{(2 +⁵)r²+² −¹}^d3 dr³

−²(r−¹)(r+¹)²r²

(c2−( +⁴)(2 +³))r⁴

−2(2 ²+c2+3 −5)r²−2 ²+c2+ _d² dr²

−2(r+1)r

(2 +3)(c2−2( +1))r⁶ +(4(−^{2 2}+ +³)−^c2(2 +⁵))r⁴

+(c₂+²( −¹)) (1−² )r²+(1+² )c₂

_d

dr

+(1−^r){(2(2 +¹)c2−^c4)r⁶+²(−^c4+²(2 +¹)c2)r⁵ +(c₄+²(6 −¹)c₂)r⁴+^4r3(c₄+^2c2(2 −¹))

+(c4+^2c2(6 −¹))r²+²(2c2(2 +¹)−^c4)r +^2c2(2 +¹)−^c4}.

Here =^d/2−^{1 while c}2 and c4 are the quadratic andquartic Casimireigenvalues expressedintermsoftheprimary dimension andspin:

c₂=¹₂[^l(l+² )+(−²−² )],

c4=l(l+2 )(−1)(−1−2 ). (3.2) Itturnsoutthattheaboveequationcanbesolvedind=^{3 bya} judicioussubstitution. Toguessthesubstitution,considerfirstthe conformalblocksofthe“leadingtwist”operatorsforwhich

=^l+² , l=⁰,1,2. . . (3.3)