Conical twist fields and null polygonal Wilson loops

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Castro Alvaredo, O., Doyon, B. & Fioravanti, D. (2018). Conical twist fields and

null polygonal Wilson loops. Nuclear Physics B, 931, pp. 146-178. doi:

10.1016/j.nuclphysb.2018.04.002

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Nuclear Physics B 931 (2018) 146–178

www.elsevier.com/locate/nuclphysb

Conical

twist

fields

and

null

polygonal

Wilson

loops

Olalla

A. Castro-Alvaredo

_,

_{Benjamin Doyon}

_,

_{Davide Fioravanti}

a_{Department of Mathematics, City, University of London, 10 Northampton Square, EC1V 0HB, UK} b_{Department of Mathematics, King’s College London, Strand, WC2R 2LS, UK}

c_{Sezione INFN di Bologna, Dipartimento di Fisica e Astronomia, Università di Bologna, Via Irnerio 46, Bologna, Italy}

Received 16October2017;receivedinrevisedform 17February2018;accepted 5April2018 Availableonline 11April2018

Editor: HubertSaleur

Abstract

UsinganextensionoftheconceptoftwistfieldinQFTtospace–time(external)symmetries,westudy conicaltwistfieldsintwo-dimensionalintegrableQFT.Thesecreateconicalsingularitiesofarbitraryexcess angle.Weshowthat,uponappropriateidentificationbetweentheexcessangleandthenumberofsheets, theyhavethesameconformaldimensionasbranch-pointtwistfieldscommonlyusedtorepresentpartition functionsonRiemannsurfaces,andthatbothfieldshavecloselyrelatedformfactors.However,weshow thatconicaltwistfieldsaretrulydifferentfrombranch-pointtwistfields.Theygeneratedifferentoperator productexpansions(shortdistanceexpansions)andformfactorexpansions(largedistanceexpansions).In fact,weverifyinfreefieldtheories,byre-summingformfactors,thattheconicaltwistfieldsoperator prod-uctexpansionsarecorrectlyreproduced.Weproposethatconicaltwistfieldsarethecorrectfieldsinorderto understandnullpolygonalWilsonloops/gluonscatteringamplitudesofplanarmaximallysupersymmetric Yang–Millstheory.

©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

In quantumfieldtheory(QFT)anysingularityinanotherwiseflatspace–timeisassociated withaquantumfieldlocalizedonthesingularity.Sincethestress-energytensorTμν _isthefirst

* _{Correspondingauthor.}

E-mail addresses:o.castro-alvaredo@city.ac.uk(O.A. Castro-Alvaredo), benjamin.doyon@kcl.ac.uk(B. Doyon),

fioravanti@bo.infn.it(D. Fioravanti).

https://doi.org/10.1016/j.nuclphysb.2018.04.002

Fig. 1.ApictorialrepresentationoftheconicaltwistfieldVα(x,τ ).Theredhorizontallinerepresentsthebranchcut

inducedbytheconicalsingularity,andthearrowsrepresentthecontinuouspathalocalfieldtakeswhencontinuedaround theposition(x,τ ).Whenreachingthecut,thelocalfieldisaffectedbyaclockwiserotationofangleα.Effectively, a wedgeofangleαisaddedtothefullcyclearoundthepoint(x,τ ).(Forinterpretationofthecoloursinthefigure(s), thereaderisreferredtothewebversionofthisarticle.)

orderresponsetoasmallmetrictransformation,onemayexpectthatmetricsingularitiescanbe representedasexponentialsofthestress-energytensor.Naturalexamplesofmetricsingularities areconicalsingularitiesinanotherwise(two-dimensional)Euclideanflatspace:pointsofinfinite curvaturewithexcessangleα=0 (thecurvatureispositive(negative)forα <0 (α >0)).The goalofthepresentpaper istopropose aquantum field,thatwe willcallaconicaltwist field, expressedintermsofthestress-energytensor,andthatrepresentsagenericconicalsingularity ofexcessangle α.InthepictureofEuclideanfieldtheory,itisafieldwhoseposition(x,τ )is theend-pointofabranchcutextendingtowardsitsright,throughwhichotherfieldsareaffected byarotationclockwiseofangleαwithrespectto(x,τ ).Thisisanextensionofthewellknown conceptoftwistfields[1–9] tospace–timesymmetries,insteadofinternalsymmetries.SeeFig.1. LetRμ(y)(withμ=1,2 representingspaceandimaginarytimerespectively)1bethe con-servedcurrentassociated torotationsymmetrywithrespecttothe point(x,τ ).Theproposed conicaltwistfield,positionedat(x,τ )andaddingawedgeofangleα >0 centred at(x,τ ),isof theform

Vα(x, τ )=

e−αx∞dyR2(y,τ )

(1)

wherethesquarebracketsindicateanappropriaterenormalizationofthefield.Wewilldefinea renormalizationprocedurethatmakestheexponentialfiniteandwelldefined.Wewillshow,by adirectcomputationof theeffectofscaletransformations,thattheresultingfieldisaspinless scalingfieldwithscalingdimension2α givenby

α=

c

24

α+2π

2π −

2π α+2π

. (2)

Thisdirectcomputationofthescalingdimensionissubtle.Asusual,oneseparatesthe renormal-izedfieldintothemultiplicativerenormalizationε−2α_{timesaregularizedexponential,andone}

showsthatthelattertransformstrivially.Buttrivialtransformationonlyoccursiftherotation cur-rentR2(y,τ )ischosentoincludeanappropriatetermproportionaltotheidentityoperator,which affectsthechoiceofthemultiplicativerenormalization.TheproofsarereportedinAppendixA

(seeinparticularAppendicesA.1andA.3).

Thisconicaltwistfieldhastheunusual-lookinghermiticityproperty

V†

α=Vα (3)

1 _{OurconventionisthatifO(x)isaspinlessfield,then[}

RdyR2(y,τ ),O(x+x,τ )]=x∂τO(x+x,τ ).Inparticular

andgeneratestheoperatorproductexpansion(OPE)

Vα(z)Vα(0)=Cα+α

α,α z−

2α−2α+2α+αV

α+α(0)+ · · · (4)

whereC_α,αα+α arethe(yetunknown)three-pointcouplings.Inparticular,

Vα(z)V−α(0)=z−2α−2−α1+. . . (5)

where we take conformal field theory (CFT) normalization to fix the structure constant to

C0

α,−α =1. Note that thanks to(1),conical twist fields can inprinciple be seenas

exponen-tialsofVirasoromodes,orVirasorovertexoperators.Suchexponentialsimplementconformal transformationsandwerestudiedinquitesomegeneralityin[10].Inthissense,theconicaltwist fieldVα(0)implementsthesingularconformaltransformationz→z1/(1+α/2π ).

In the study of QFT on branched Riemann surfaces, branchpoints of nth-root type, with integern,arenegativecurvatureconicalsingularitiesofexcessangleαthataremultiplesof2π:

α=2π(n−1). (6)

FieldsassociatedwithconicalsingularitiesonRiemannsurfaceshavebeenstudiedalongtime agointhecontextoforbifoldCFT[11–14].Morerecently,thishasbeenunderstoodingeneral 1+1-dimensionalQFT:suchsingularitiescanbestudiedusingbranch-pointtwistfieldsT (and their conjugate T˜) [15]. These are fields whichexist in an n-copy replicamodel andwhich are associatedtothe(or any)generatorofthe Zn-symmetryof thereplicamodel.Theconical

twistfieldisofadifferent(butrelated)nature,anddoesneitherrequirentobeaninteger(αto be amultiple of 2π) northe model toconsistof n replicas(it existsina singlecopyof the model).Itsscalingdimension(2) agreeswiththatofthebranch-pointtwistfieldwhen nisan integerandisrelatedtoαasin(6),althoughweemphasisethatthecomputationweprovidefor thescalingdimensionoftheconicalfieldisbasedon(1) andindependentfromtheconceptof branch-pointtwistfield.Asweshallseeinmoredetaillater,conicalfieldformfactorsagreewith aparticularsubsetofthoseofthebranch-pointtwistfield[15] whereallparticleslieonthesame copy.However,thehermiticityproperty(3) andtheOPE(4) differfromthoseofbranch-point twistfields.Branch-pointtwistfieldsaregenericallynotself-conjugate,andtheirOPEsdonot involvebranch-pointtwistfieldsatdifferentvaluesofn(asnisapropertyofthemodel,notof the field),butratherdifferentelementsof theZn-symmetry.Further,theformfactorseriesfor

the vacuumtwo-point functionT(0)T˜( )n containssumsoverreplicaindices,from 1ton,

as theseparametrizethe particlesof thereplicamodel.In contrast,thatof conicaltwistfields doesnotinvolvesuchsums.Conceptually,thedifferencebetweentheconicaltwistfieldandthe branch-point twist fieldis thatthe latterreproducesbothametricsingularityandaparticular branching(ofnth-roottype),whiletheformerreproducesonlythemetricsingularity,addingor deletingspacewithtrivialbranching.

Partition functionsonRiemannsurfaceswithbranchpointshavereceivedalotofattention recentlyduetotheiruse,withinthe“replicatrick”,fortheevaluationofthevonNeumannand Rényientanglemententropies[16,17,15,18,19] andthelogarithmicnegativity[20–23].Inmost of these applications,one establishes the result for integer n, and thenperforms an analytic continuationtorealvaluesofn,whichisingeneralhardtodefineuniquelyandinvolvesacertain amountofguesswork.However,inthecaseoftheentanglemententropyforasinglehalf-infinite intervalinanyQFT,oneknowstheexactreduceddensitymatrix[24–27]2:itistheexponential

2 _{Theworks[24,25],wheretheresultwasobtainedforthefirsttime,arerigorousandbasedonthealgebraicformulation}

oftheintegral oftherotationcurrent,exactlyasin(1) (asaconsequence ofglobalconformal invariance,onethenalsoknowsitfor asingleinterval ofany lengthinCFT,seefor instance [28]).Inthiscasetheanalyticcontinuationissimple,andthisexplainsvariousaspectsof the so-calledentanglementspectrum[29].Also,inthiscasethereduced densitymatrix isinfact proportionaltoaspecialcaseoftheconicaltwistfieldaswedefineitin(1) (andaverageswith respecttopowersofthereduceddensitymatrixarecorrelationfunctionswithasingleinsertion oftheconicaltwistfield).Here,westudy(1) forthefirsttimeasaquantumfield,takingproducts ofconicalfields,anddiscussingtheirOPEsandformfactors.

Ourmotivationfortheconsiderationofthisfieldinpartcomesfromrecentworksongluon scatteringamplitudesinplanar(Nc→ ∞)N =4 SuperYang–Mills(SYM)theory,whichare

equaltoexpectationvaluesof null polygonalWilsonloops(WLs) [30].In the simplest, non-trivialcase,thesixgluonamplitude,orequivalentlythehexagonalWL,wastentativelyidentified by[31],inthe strongcoupling regime,withthe formfactorseries forthe two-pointfunction

T(0)T˜( )n of a branch point twist field T and its hermitian conjugate T˜ in the massive

O(6)non-linearsigmamodel(NLSM).Strictlyspeaking,thisappliestothecontributionofthe scalarexcitationstotheWL,wherethereare differentexpressionsforall (particle)form fac-tors3[31–34],althoughtheirdetailedderivationisstillmissing.Aposteriori,thisidentification wassupportedbythefactthatashort-distancere-summationoftheseriesreproducestheright conformallimit andscalingdimension forthe twistfield[31–34] (with differentcomputation strategies,asmentionedbelow).

However,the formfactorseriesofT(0)T˜( )n,asfirstobtainedin[15],isinfact,strictly

speaking,differentfromthatfoundforsixgluonamplitudeorhexagonalWL.Indeed,as men-tioned,theformfactorexpansionofT(0)T˜( ) ncontainssumsoverreplicaindices.Incontrast,

sixgluonamplitudesorhexagonalWL,althoughformedoutofthesamebuildingblocks (mod-ulussquareof twistfieldformfactors) [31],donot containreplicasums.Furthermore, inthe amplitude/WLcasethevalueofnissetton=p₄=5₄ –throughapictorialanalogywitha pen-tagon,wheretheparticlejumpp=5 timesonthe edgestogobacktoitsstartingpoint, into whichthehexagonisdecomposed.Yetnmustbeanintegerinorderforbranch-pointtwistfields tobedefined,asitcountsthenumberofreplicas;non-integervaluesofnareobtainedbyanalytic continuationsanddon’tstrictlycorrespondtotwo-pointfunctions.

Instead,wewillarguethatsixgluonamplitudes/hexagonalWLsmaybeexpressedas two-pointfunctionsVα(x)Vα(0) oftheconicaltwistfieldsintroducedabove,withα=π/2.Indeed,

the formfactorseries for Vα(x)Vα(0) isingeneral of the sameformas the series

describ-ingthesixgluonamplitudeorhexagonalWL,inparticularwithoutsumoverreplicaindex.In addition,as perthe prescription for theform factorsof Vα,the formfactorsinthe seriesfor

Vπ/2(x)Vπ/2(0) coincidewiththoseof thetwistfieldT whereallparticlesare onthesame

copyfor the choice n= 5₄, inagreement withthe amplitude/WLresult. In [31] an OPE was identified,whichcanbewrittenas

Vπ

2(z)Vπ2(0)∼z

−4π

2+2πO_hexa+ · · ·. (7)

InthisOPE,Ohexa wasinterpretedas associatedtoan“hexagon”amplitudewithformfactors

equaltothoseofthebranchpointtwistfieldT for n=3₂ (againwithallparticlesonthesame

thennotetheknownfactthattheTTtheorypredictsthestatetosatisfytheKubo–Martin–Schwingerconditionwith respecttotheTTmodularoperator.Inphysicsparlance,thisimpliesthatthereduceddensitymatrixistheexponentialof theLorentzboostgenerator.

copy).ThisleadstotheidentificationOhexa=Vπ,whichagainagreeswiththegeneralconical

fieldOPE(4),andfurthersupportsourconjecture.Thepowerin(7) is4π

2 −2π=

c

180.In

particular, for c=5 asinthe O(6)NLSM,thisgivesexponent ₃₆1 whichhasbeenrecovered very preciselybyre-summingthe formfactorexpansionin[31–34].In [32,33] thiswasdone by applyingtotheasymptotically freeO(6)NLSMthewell-knowncumulantexpansion [35] reviewedandemployedbelow.

Inotherwords,withthedefinitiongivenhereoftheconicaltwistfieldwehavesetonsolid groundtheinterpretationoftheformfactorseriesobtainedforgluonscatteringamplitudes/null polygonalWLs.TheexactinterpretationwasmadeinthemasslesslimitoftheO(6)NLSM, re-alisedbytheinfinite’tHooftcouplinglimit.But,assupportedinthefollowingbytheformfactor propertieswederive,wewouldexpectthisidentificationtobetrueforthemassiveO(6)NLSM realisedbythestrongcouplingregime,andsimilarconicalfieldstobeinvolvedinthefullstring NLSMreplacingtheO(6)NLSM([36] andreferencestherein)atanycoupling.Further,onecan generalisetootherpolygons:intheWLpicture,k-sidedpolygonalWilsonloopscorrespondto

(k−4)-pointcorrelationfunctions(theheptagoncorrespondstothethreepointfunction,the oc-tagontothefourpointfunction,andsoon).Finally,forthefulltheoryweexpectthattheconical twistfieldformfactorsareassociatedtoanintegrabletheorywithscatteringmatrixdefinedon theGKPvacuum[37–39].

Thepaperisorganisedasfollows:insection2weprovidearigorousQFTdefinitionof coni-caltwistfieldsanddiscusstheirpropertiesatandnearconformalcriticalpoints.Inparticularwe arguethat,unlikethemorestandarddefinitionofatwistfieldinQFTasafieldassociatedwith aninternalsymmetryofthetheory,conicaltwistfieldsareassociatedwithrotationalspace–time symmetry(henceanexternalsymmetry)andinsertconicalsingularitiescorrespondingtoexcess rotationangle α.Theymaythereforebeformally expressedas (appropriatelyregularised) ex-ponentials ofintegralsoftherotationNoethercurrentoverthelinethatstartsatthetwistfield insertion pointandextendsuptoinfinity.Insection3 wecharacterisethesamefieldsthrough theirformfactorsandwritethecorrespondingformfactorequationsforintegrable1+1 dimen-sionalQFTswithdiagonalscattering.Wegivethegeneralformofthetwo-particleformfactor and,forfreetheories,givealsoclosedformulaeforallhigherparticleformfactors.Insection4

wereview thestandardcumulantexpansion of two-pointfunctionsof localfields intermsof theirformfactorsandspecializeittoconicaltwistfieldtwo-pointfunctionsinfreetheories.For freetheoriesweobtain anexactresummationoftheleadingshort-distancecontributiontothe correlators.Insection5wetestourformfactorformulaebynumericallyevaluatingouranalytic expressionsfortheleadingshort-distancebehaviourofthecorrelatorVα(0)Vα( ) andfinding

thattheyexactlymatchtheexpecteddecayfromtheconformalOPEs.Inparticular,weobserve thattheformfactorresummationbecomessubtlewhenanyof theexcessanglesisnegativein whichcasesadetailedanalysisoftheintegrands’polestructureisrequired.Thisisreminiscent ofthekindofissuesthatarisewhenanalyticallycontinuingthecorrelatorsofbranchpointtwist fieldsfromnintegerton≥1 andreal.Wepresentourconclusionsinsection6.Aderivationof fourdefiningpropertiesoftheconicaltwistfieldispresentedinappendixA.

2. Conicaltwistfieldsinquantumfieldtheory

Theconstructionoftheconicaltwistfieldisbasedonthestandardtheoryoftwistfields asso-ciatedwithinternalQFTsymmetries.Recallthatinthequantizationontheline,atwistfieldVσ,

Vσ(x, τ )O(y, τ )=

σ·O(y, τ )Vσ(x, τ ) (y>x)

O(y, τ )Vσ(x, τ ) (y<x) (8)

whereσ·OisthetransformationofOunderσ.Throughtheusualtime-orderingprescription, acorrelationfunctionwithatwistfieldinsertionmaybeevaluatedbyapathintegraloverfield configurationswithajumpconditionacrossthecut{(y,τ ):y>x}.Thejumpconditionimposes continuitybetweenanylocalfieldOjustbelowthecut,anditstransformσ·Ojustabovethecut. Sinceσ isasymmetry,theresultonlydependsonthehomotopyclassofthecutonspace–time withpuncturesatthepositionsofotherlocalfields inthecorrelationfunction.A twistfieldis localinthesensethatitcommutesatspace-likedistanceswiththestress-energytensor,as the latterisinvariantunderσ.

Letusextendthisconcepttoactionsofσ onspace–time(externalsymmetries).Specifically, letσ bearotationclockwisebyanangleα >0 withrespecttothepointz=x+iτ.Forinstance, ifO(y,τ )hasspins,anddenotingthepositionbyacomplexcoordinatew=y+iτ,wehave

σ ·O(w)=e−isαO(e−iαw).4 Letusdenote byVα(x,τ )the correspondingconical twistfield

satisfyingtheexchange relations(8).5 Theassociatedjumpconditioninthe path-integral rep-resentationimposescontinuitybetweenfieldsjustbelowthecut{(y,τ ):y>x}andclockwise

α-rotated fieldsjustabove thecut.Intuitively, aclockwiserotationbrings thefields fromjust abovethehalf-linetoarotatedpositioninan“extra”space–timewedgebelowthehalfline,thus addingawedgecenteredat(x,τ )ofangleαandcreatinganegative-curvatureconical singular-ity.SeeFig.1.

SincerotationsformaLiegroup,theyhaveanassociatedNoethercurrentRμ(y).This imme-diatelyleadstotheexponentialrepresentationofVα(x,τ )intermsofthestress-energytensor,as

in(1).Indeed,thestandardcommutationrelationsbetweentherotationNoethercurrentand lo-calfieldsthenguaranteesthatthisexponentialgeneratesthecorrectexchangerelations(8) with

σ the clockwiseα-rotation. Clearly,conical twistfields arenot local,eveninthelargesense ofcommutingwiththestress-tensoratspace-likedistances.However,sinceRμisaconserved current,thefieldVα(x,τ )doesnotdependontheshapeofthecutemanatingfromitsposition,

but,again,onlyonitshomotopyclassonspace–timewithpuncturesatthepositionsofinserted localfields.ThisisthesenseinwhichVα(x,τ )maybeunderstoodasenjoyingacertainlocality

property.Hencewehavemoregenerally

Vα(x, τ )=

e−α

dsdyμ(s)_ds μνRν(y(s)) ₍₉₎

foranycurves→yμ(s)connecting(x,τ )to∞.

Intherestofthissection,wemaketheseideasmorepreciseandstatethemainpropertiesof theresultingfield.

2.1. Rotationcurrentandsingularcurvature

Inthefollowing,itwillbeconvenienttousecomplexcoordinatesz=x+iτ,z¯=x−iτ and thestress-energytensorcomponentsT = −(π/2)Tz¯z¯ _andT¯ _{= −(π/2)T}zz_{withtheusualCFT}

normalization,T (z)T (z)(c/2)(z−z)−4,T (¯ z)¯ T (¯ z¯)(c/2)(z¯− ¯z)−4wherecisthecentral

4 _{Thisoperatorisnotnecessarilyholomorphicbutmaydependalsoonw¯whichtransformsaccordingly.} 5 _{Relation(8) definesafamilyoftwistfields.ThefieldV}

αisfurtherdefinedbyimposingappropriateminimality

chargeof theultravioletCFT.Wewillalso denoteby = −(π/2)Tμμ the“CFT-normalized”

traceofthestress-energytensor.Inparticular,wehave∂T¯ +∂=0 and∂T¯+ ¯∂=0. ForsimplicityweconcentrateontheconicaltwistfieldVα:=Vα(0,0)positionedatthe

ori-gin. Notethat inorderforthe expression(1) togeneratethe correctexchangerelations, only thecommutatorsoftherotationcurrent Rμwithotherlocalfieldsarerequired.Therefore,the choiceofthecurrentRμdefiningtheconicalfieldisambiguouswithrespecttoadditionofterms proportionaltotheidentityoperator1.ThisaffectsthenormalizationofthefieldVα.Wepropose

tolifttheambiguitybyadoptingthefollowingrotationcurrent:

iπRz= ¯zT¯−z−1

¯

zG(m|z|)1

−iπRz¯=zT− ¯z−1

zG(m|z|)1 (10)

wheremisamassscaleoftheQFTmodelandG(r)withr:=m|z|isauniversalscalingfunction with

G(0)= c

12. (11)

Attheconformalpoint,wherethereisnomassscale,thefactorG(r)istheconstantc/12 in(10). Exceptforthesingularityattheoriginin(10),givenby(11),thefunctionG(r)/risrequiredto beintegrableontheline.Itsexactformdoesnotaffecttheevaluationofcorrelationfunctions, butmightbeinvolvedintheevaluationofvacuumexpectationvaluesofconicalfieldsinmassive QFT.

AlthoughwewillnotfullyaddresstheproblemoftheexactformofG(r)here,wenevertheless proposethatthefunctionG(r)shouldtaketheform

G(r)= c

12−2r

2_g(r)+

2

r

0

drrg(r), g(r)=m−2(x) _C,K (12)

wheretheexpectationvalue(x) _C,KisevaluatedontheplaneCwithaGaussiancurvatureK

thatissingularatthepositionoftheconicalfield,

K(x)= −2π δ(2)(x). (13)

Thesingularityin(10),withresiduespecifiedby(11),isessentialinordertoguaranteethe correctscaletransformation propertiesofthe conicalfield. Weprovidefullarguments in Ap-pendixA.A furtherjustificationforthissingularity,aswellasforthechoice(12) beyondCFT, isasfollows.Attheinsertionoftheconicalfield,aconicalsingularityemerges.Singularitiesin QFT giverisetoadditionalrenormalizations,andatthe singularpointmicroscopiceffectsare important.Inordertoaccountforthese,astandardregularizationisperformedbymakingahole inspace–timearoundthesingularity(seebelow).The limitisthentakeninwhichtheholeis madeinfinitesimallysmall.Attheboundaryofthehole,thetraceofthestress-energytensor ac-quiresanonzeroexpectationvalueproportionaltothe(negative)linearcurvature.Theboundary integralofthislinearcurvatureis−2π.Inthelimitwheretheholebecomesapoint,thismaybe replaced(inaccordancewiththeGauss–Bonnettheorem)byapunctualsingularGaussian curva-tureK(x)= −2π δ(2)(x).Weaccountforthisbyshiftingthestress-tensortracebyitsexpectation valueonthespaceCwiththesingularcurvatureK,

Theshifthasadelta-functioncontributionattheorigin,whichisinferredfromthe CFTtrace anomalyformula

(x) _C,K CF T

= −cK(x)

24 . (15)

The otherelements of the stress-energytensor, Tand T¯,are shifts of T and T¯ respectively, determinedbytheconservationequations ∂¯T+∂Θ=0 and∂T¯ + ¯∂Θ=0.Accountingfor the delta-functionsingularityusing∂(¯ 1/z)=∂(1/z)¯ =π δ(2)(x),wethenobtain

T=T− c

12z2f (r)1, T¯ = ¯T − c

12z¯2f (r)1, Θ=+ cm2

12 h(r)1

where(cm2/12)h(r)= −(x) _C,K andf (0)=1 (and limr→0h(r)<∞).The conservation

equationsimplythat,awayfromtheorigin,f andhsatisfyr2h(r)=f(r).Therotationcurrent takesthestandardformusingtheseshiftedstress-energytensorcomponents,iπRz= ¯zT¯ −zΘ

and−iπRz¯=zT− ¯zΘ,whichgivestheuniversalscalingfunctionG(r)=(c/12)(f (r)+r2h(r))

inagreementwith(12).

Notethatthe integrationmeasureinvolvedin(9) isdxμμνRν(x)=(dzRz−dz¯Rz¯)/(2i),

whereμνistheanti-symmetricsymbol.InCFT,thisspecializesto

dxμμνRν(x)= −

1 2π

zTdz+ ¯zT¯dz¯− c

12

dz

z +

dz¯

¯

z

1

(CFT). (16)

In the nextsubsection we provide furthercalculations showing that the choice of rotation current(16) is correctinCFT.Away formtheconformal point, theexpression(10) with(12) constitutesaconjecture.

2.2. Theconicalfield

Besidesthechoiceofthecurrent,theright-handsideof(1) requiresaregularization/ renor-malizationprocedure.Thetheorydevelopedin[10] allowsinprincipletoregularizesuch Vira-sorovertexoperatorsbyseparatingtheexponentialintoaproductofexponentialswithpositive andnegativeVirasorogenerators.However,makingthefullconnectionwith[10] isbeyondthe scopeofthispaper,andinsteadweconcentrateonamoredirectrenormalization,ifmoredifficult tocontrol.Weusean“angularrenormalization”procedurebasedonanangle-splitting prescrip-tion.Thistakesinspirationfrom[40,41] whereangularquantizationwasusedinordertostudy

U (1)twistfieldsintheDiractheory.Herenoexplicitquantizationschemeisusedandwe con-centrateoncorrelationfunctions.

Wefirstdefinearegularizedexponentialofalineintegralstartingat0inthecomplexplane. WeregularizetheTaylorexpansionasfollows:(a) a circularholeofradiusε >0 ismadearound theorigin,ontheboundaryofwhichweimposeconformalboundaryconditions,modifyingthe lowerlimit ofthe integral toaposition onthisboundary;(b) powersof theintegral are split intoproductsofintegralsalongraysatdifferentangles.Wethentakethelimitwhereallangles areequaltoeach other.Thatis,anε-regularizedexponentialisdefined,insideanycorrelation functionsonanopensetD {0},as

e

L

0 dxμfμ(x)

ε· · ·

D=

∞

j=0

1

j!{φlimk→0}

_j

k=1

Leiφk

εeiφk

dxμfμ(x)· · ·

D\εD

whereL> εandweusecomplexnumbersinordertorepresentcoordinatesintheintegration limits.Heretheellipsis· · · representpotentialinsertionsoflocalfieldsatfixedpositions.

In ordertodefinetheconical twistfields, we thentakearenormalizedlimit whereε→0 oftheregularizedexponential.Ontheplane,thefieldcorrespondingtotheinsertionofasingle conicalsingularityofexcessangleαattheoriginis:

Vα:=lim ε→0ε

−2α

e−α

_∞

0 dxμμνRν(x)

ε (18)

whereαisgivenby(2).InCFTthissuffersfrominfrareddivergences.Onemayregulatethese

infrared divergencies byafinite-volume cutoff. Forinstance, onthe disk Dof radius ,we definetheconicalfieldatposition0as

Vα_D:=lim ε_→0ε

−2α

e−α

0dxμμνRν(x)

ε. (19)

ThelatterisinfactvalidbothinmassiveQFTandinCFT,andthelimit → ∞canbetakenin massivemodelsandreproduces(18).

RecallthatRμisaconservedNoethercurrentassociatedtotherotationsymmetry.Therefore, theintegral_εe eiφiφdxμμνRν(x),insidecorrelationfunctionsontheannulusofinnerradiusεand

outerradius withconformalboundaryconditions,isinvariantunderachangeoftheangleof the path φ,as longas the pathdoes not crossthe positions of otherlocalfields inserted. As a consequence,the limits onthe angles φk in(17),on the right-hand sidesof (18) and (19),

exist:thecorrelationfunctionsareindependentoftheanglesφkiftheyarenearenoughtoeach

other.

Notethat,againusingcurrentconservation,itisnotnecessarytotakepathsthatarestraight rays: for thepurpose of theconical twistfield, the regularizedexponentialcanbe definedby taking,for thekth termof the Taylorexpansion, anycollection ofk non-crossing pathsfrom oneboundary totheotherofthe annulus,andbytaking thelimit,orderbyorder,wherethey accumulatetoasinglerayfromεto .

In AppendixA,weprovide fieldtheoryargumentsfor thefollowingstatements.PointIIis expressedhere,andshowninAppendixA,inthecaseα∈ [0,2π )forsimplicity.

I. Thelimitε→0 in(18) and(19) exitsandisfiniteinsidecorrelationfunctions.

II. ThefieldVα (onthediskortheplane)implementsaconicalsingularityattheoriginofangle

2π+α:it makesacut onL=(0, )andinsertsthereawedge inadisjoint copyof D boundedbythesegmentsL₊=(0, )andL₋=e−iα(0, ),byestablishingatLcontinuity frombelowtoL₋,andfromabovetoL₊(take → ∞fortheplane).

III. ThefieldVα (onthediskortheplane)behavesasaspinlessscalingfieldatposition0with

scalingdimension2α givenby(2).

IV. Thefieldisconformallynormalized,thatis

Vα _D= −2α (CFT). (20)

NotethatPropertiesIIandIIIimplytheOPEs(4) (and(5)):astwoconicalfieldsapproacheach other,theirassociatedexcessanglesmustaddup,andtheensuingsingularityisdeterminedby thescalingdimensionsofthefieldsinvolved.

Let us be more precise concerning Property II. Recall that we may construct a manifold

Mα, withaconicalsingularityasfollows;forsimplicitywerestrictourattentiontoα∈(0,π ).

boundaryofthe disk.Theresultholdsas wellontheplanebytaking → ∞,with( D)_→∞

identifiedwiththe plane C.Let D(j ),j =0,1,2 be threecopies of the disk. Considertwo copiesof the diskminussegments,Dˇ(1)= D(1)\ [0, )and Dˇ(2)= D(2)\e−iα[0, ),anda copy of the punctured disk Dˇ(0) = D(0)\ {0}. Forany 0≤a < b≤2π, denotethe subsets

{z∈ ˇD(j ):arg(z)∈(a,b)}byDˇ(j )_(a,b).Themanifoldcanbecoveredbythreepatches:thetwocut disksDˇ(1)andDˇ(2),andawedgeDˇ(₍0₋)_α,α).Thetransitionfunctionsbetweenthesepatchesare

U(21): ˇD(₍₀2)_,2π−α)→ ˇD(₍1₀)_,2π−α), U(10): ˇD₍(1₋)_α,0)→ ˇD₍(₋0)_α,0), U(02): ˇD(₍0₀)_,α)→ ˇD(₍2₋)_α,0)

(21)

withU(21)(z)=z,U(10)(z)=zandU(02)(z)=e−iαz.ItisclearthatMα, hasexcessangleα

attheorigin:itisamanifoldwithaconical singularityattheorigin. PointIIabove maythen beexpressedasfollows.Consideraproductoflocalfields_jOj(xj)fordisjointcoordinates

xj∈ D\ {0}lyinginthediskofradius minustheorigin.Then,

Vα(0)jOj(xj) _D

Vα(0) D =

j

Oj(U (xj))

Mα,

(22)

forany Oj(xj),whereU: D\ {0}→Mα, maps D\ [0, )into Dˇ(1) andmaps(0, )into

ˇ

D(0)_{,inbothcasesasU (z)=z.InsertionsoflocalfieldsinotherregionsofM}

α, areobtained

bycontinuationinthepositionsxj.

Remark.Whentheangleαisanintegermultipleof2π,itispossibletoconnectconicalfieldsVα

withordinarytwistfields,inareplicamodel,associatedwithspecialelementsofthepermutation group.Considerforinstancethetwo-pointfunctionVα(x)Vα(0) forα=2π(n−1)andα=

2π(n−1)withn,n∈N.Considerareplicamodelcomposedofm=n+n−1 copiesofthe originalmodel, andthe twistfieldsT(1···n) andT(n···m) associatedtothepermutation elements

(1· · ·n)and(n· · ·m) that,respectively,cyclicallypermutethecopies1,. . . ,n,andthe copies

n,. . . ,m.ThenVα(x)Vα(0) = T(1···n)(x)T(n···m)(0).Ineffect,thecopynumbernisidentified

withtheoriginalplane,andtheextracopiesbelowandabovenwiththeextraspaceintroducedby theconicalsingularities.Suchconstructionshoweveronlyworkforexcessanglesthatareinteger multiplesof2π,althoughanaturalanalyticcontinuationinnandmofT(1···n)(x)T(n···m)(0) will

givethecorrectcontinuationtoothervaluesofαandα.

3. Formfactorapproachtoconicaltwistfields

InmassiveQFTanalternativewaytodefinetheconicaltwistfieldsistofullycharacterisetheir matrixelements.Wethereforeturntothedescriptionof1+1-dimensionalQFTonMinkowski space–timeintermsofitsHilbertspaceofasymptoticrelativisticparticles.Wewillverifythat (intheUVorshort-distancelimit)thecorrectOPEs(4),andinparticular(5),arerecovered,thus lendingsupporttothefactthatthefieldwhoseformfactorswedescribehereisindeedtheconical twistfieldintroducedabove.

Inthecontextof1+1-dimensionalQFT,formfactorsaredefinedastensorvaluedfunctions representingmatrixelementsofsome(local)operatorO(x)locatedattheoriginx=0 between amulti-particlein-stateandthevacuum:

FO|μ1...μk

Fig. 2.Apictorialrepresentationoftheconicaltwistfieldk-particleformfactor.Theredhorizontallinerepresentsthe branchcutinducedbytheconicalsingularity.

Here|θ1,. . . ,θk inμ1,...,μkrepresentthephysical“in”asymptoticstatesofmassiveQFTand0|is

thevacuumstate.Multi-particlestatescarryindicesμi,whicharequantumnumbers

character-izingthevariousparticlespecies,anddependontherealparametersθi,whicharetheassociated

rapidities.Therapiditiescharacterizetheenergyandmomentaofparticlesin1+1 dimensions throughthewell-knownrelationsEμ=mμcoshθ andpμ=mμsinhθ,wheremμ isthemass

of particle μ. The form factorsare defined for all rapiditiesby analytically continuing from someorderingoftherapidities;afixedorderingprovidesacompletebasisofstates,forinstance

θ1> θ2>· · ·> θk isthestandardchoicetodescribein-states.

In1+1-dimensionalintegrableQFTthereisawell-knownapproachknownastheformfactor programmewhichprovidesasystematicmeansto(apriori)computeallmatrixelements(23) for anylocalfieldO [4,7].The “standard”formfactorprogrammegenerallyrelies onlocalityof operators. Nonetheless, we willnow see that we canalso developsuch aprogrammefor the conicaltwistfieldsbyslightlyalteringandadaptingtheformfactoraxiomsfoundin[4,7].

Letusdenoteby

FVα|μ1...μk

k (θ1, . . . , θk):= 0|Vα(0)|θ1, . . . , θk inμ1,...,μk (24)

the formfactorsof theconical twistfieldwithexcessangle α.Wemayrepresentthese func-tionspictorially asshowninFig.2.Becauseof relativisticinvarianceandspinlessness ofthe conicaltwistfields,theseformfactorsmustonlydependontherapiditydifferences;inthe two-particlecase,theythenbecomefunctionsofonlyonevariable,θ=θ1−θ2andsowewillwrite FVα|μ1μ2

2 (θ )forthetwo-particleformfactor.

TheformfactorprogrammeisaRiemann–Hilbertproblemfortheformfactors(2),thatisaset ofconsistencyequationswhichestablishtheirmonodromypropertiesandsingularitystructure. Itwasshownin[15] thatthestandardformfactorequations[4,7] canbemodifiedtoencompass (local)branchpointtwistfieldsdefinedonreplicatheories.Notethattheformfactorequations derived in[15] are infact very similartoequations foundpreviouslyin[42],althoughinthe latteradifferentmotivation(i.e. describingtheUnruheffect)wastaken,withentirelydifferent interpretationofthesolutions.

Fig. 3. A pictorial representation of Watson’s equation (25).

Fig. 4. A pictorial representation of Watson’s equation (26).

Fig. 5. A pictorial representation of the kinematic residue equation (27).

Fig. 6. A pictorial representation of the kinematic residue equation (28).

3.1. Theconicalfieldformfactorequations

Itisintuitivelynottoohardtoseehowtheequationsfrombranchpointtwistfieldsmaybe adaptedtothenon-localconicaltwistfielddefinedearlier.Letthetwo-particlescatteringmatrix betweenparticlesμ1,μ2beSμ1μ2(θ )(weassumeforsimplicitythatthereisnobackscattering)

andletμ¯representtheanti-particleofμ.Weproposethefollowingconicaltwistfieldformfactor equationsandrepresentthempictoriallyinFigs.3, 4, 5and 6:

FVα|...μiμi+1...

k (. . . , θi, θi+1, . . .)=Sμiμi+1(θi i+1)F

Vα|...μi+1μi...

k (. . . , θi+1, θi, . . .), (25)

FVα|μ1μ2...μk

Res

¯

θ0=θ0

FVα| ¯μμμ1...μk

k+2 (θ¯0+iπ, θ0, θ1. . . , θk)=iFV

α|μ1...μk

k (θ1, . . . , θk), (27)

Res

¯

θ0=θ0

FVα| ¯μμμ1...μk

k+2 (θ¯0+iπ, θ0−iα, θ1. . . , θk)

= −i

k

i=1

Sμμi(θ0i−iα)F

Vα|μ1...μk

k (θ1, . . . , θk). (28)

Itisworthpointingoutthatequation(28) isactuallynotindependentfrom(27) (thisisalso trueforthetwokinematicresidueequationsforthebranchpointtwistfieldgivenin[15]).Itis easytoshowthat(28) becomes(27) afterrepeateduseof(25),(26) andcertainbasicproperties ofthescatteringmatrixsuchasunitaritySμ1μ2(θ )Sμ2μ1(−θ )=1 andcrossingSμ1μ2(θ+iπ )= Sμ2μ¯1(−θ ).However, bothequations (28) and(27) are physicallyimportant sincetheyimply

thepresenceoftwo(ratherthanone)kinematicpoles,andthishasabearingindeterminingthe analyticstructureoftheformfactors.Furtheruseof(25),(26),(27) and(28) doesnotgenerate morekinematicpoles.Asusual,therewillanadditionalresidueequationinthecasewhenbound state polesare present.Thiswill be identicalto thebranchpoint twistfieldbound state pole equationthatcanbefoundin[15] (equation (6.8))whenrestrictedtoparticlesonthesamecopy. LetusnowconsiderWatson’sequations(25)–(26) forthetwo-particleformfactors.Asnoted earlier,thesearefunctionsoftherapiditydifferencesonlysowecanwrite

FVα|μ1μ2

2 (θ12)=Sμ1μ2(θ12)FV α|μ2μ1

2 (−θ12) (29)

withθ12:=θ1−θ2and

FVα|μ1μ2

2 (θ12+2πi)=FV

α|μ2μ1

2 (−θ12+iα), (30)

or,combiningbothequations:

FVα|μ1μ2

2 (θ12)=Sμ1μ2(θ12)FV α|μ2μ1

2 (−θ12)=FV

α|μ2μ1

2 (−θ12+i(2π+α)). (31)

Thekinematicresidueequationsontheotherhandtellusthat

Res

¯

θ0=θ0 FVα| ¯μμ

2 (θ¯0−θ0+iπ )=iFV

α

0 , (32)

and

Res

¯

θ0=θ0 FVα| ¯μμ

2 (θ¯0−θ0+i(α+π ))= −iFV

α

0 , (33)

whereFVα

0 = Vα isthevacuumexpectationvalueoftheconicaltwistfield.Itturnsoutthat,

undertheidentification(6),theseareexactlythesameequationsassatisfiedbythetwo-particle formfactorofabranchpointtwistfieldonann-copyreplicamodel,iftheparticlesμ1,μ2,μ,μ¯

areallsettolieinthesamecopyofthereplicamodel.Thismeansthatthesamegeneralsolution foundin[15] willalsosolvetheequationsabovegiving:

FVα|μ1μ2

2 (θ )=

Vα sinπ_n

2nsinh

iπ−θ

2n

sinh

iπ+θ

2n

FV

α|μ1μ2

min (θ )

FVα|μ1μ2

min (iπ )

, (34)

whereαisrelatedtonasper(6),andwhereFVα|μ1μ2

min (θ )isaminimalformfactor,thatis,a

Herewehaveexpressed(34) intermsofthevariablenasformulaearesimpler.Wemaynow testthenewformfactorequationsandtheirsolutionsbyinvestigatingthe(massive)correlators ofthefieldswhoseconformalOPEsweregivenin(4)–(5).Hereforsimplicityandinorderto illustratetheconceptsintheclearestfashionpossible,wewilldothisforfreetheories,forwhich alsohigherparticleformfactorscanbeobtained.WewillshowthattheCFTscalingbehaviour isexactlyrecoveredatshort-distancesfromaformfactorexpansion.

3.2. Freetheories

Thesimplestmodelstoconsiderareofcoursefreetheories,thatisthefreeMajoranaFermion andfree Klein–Gordon Boson theories withrelativisticscattering matrix S(θ )= ∓1, respec-tively. Inthiscase thereisonlyone particletype(so we maydrop theparticle indices μi in

theformfactors).BoththesetheorieshaveaninternalZ2symmetrieswhichimpliesonly even-particleformfactorsarenon-vanishing.Thetwo-particleformfactorisgivenby(34) with

FVα

min(θ )=

−isinh₂θ_n for the free Fermion

1 for the free Boson. (35)

Clearly,wehave

FVα

2 (θ )=

−FVα

2 (−θ ) for the free Fermion FVα

2 (−θ ) for the free Boson.

(36)

Duetothefreenatureofthesetheories,andinparticularthefactthattheconicaltwistfieldisan exponentialofquadraticexpressionsinfreefields,the2k-particleformfactorsadmitremarkably simpleexpressions.Theycanbeexpressedas

FVα

2k (θ1, . . . , θ2k)=

Vα Pf(K) for the free Fermion

Vα Hf(K) for the free Boson, (37)

wherePf isthePfaffianandHf istheHaffnianofthematrixKdefinedas6

Kij=

FVα

2 (θi−θj)

Vα

. (38)

RecallthatthePfaffianofanantisymmetricmatrixKisgivenby

Pf(K)=det(K) (39)

whereastheHaffnianisgivenby

Hf(K)= 1 k!2k

σ∈S2k

k

i=1

Kσ (2i−1) σ (2i), (40)

whereS2kisthesetofallpermutationsofZ2k= {1,2,. . . ,2k}.Thusthetwo-particleformfactor

(38) isthebuildingblockofallhigherparticleformfactorsoftheconicaltwistfieldVα.Since

6 _{Notethatthesameobjectwas(mistakenly)namedthePermanentin[51] eventhoughthesamedefinition(40) was}

wewillbeusingtheexpression(38) repeatedlyinthefollowingcomputations,wewillfromnow onusetheshorternotation:

Kij:=f (θij;α), (41)

withαgivenin(6) andθij=θj−θj.

4. Cumulantexpansion

In1+1-dimensionalQFTwehavethatthe(normalized)logarithmofthetwo-pointfunction oflocalfieldsO1,O2admitsanexpansionoftheform[35,49,50]

log

O1(0)O2( )

O1 O2

=∞

j=1

c_j12( ), (42)

with

c12_j ( )= 1 j!(2π )j

N

μ1,...,μj=1

∞

−∞ dθ1· · ·

∞

−∞ dθj h

12|μ1...μj

j (θ1,· · ·, θj)e−

m j_i=1coshθi_,

(43)

wherethe functionsh12_j |μ1...μj(θ1,· · ·,θj)aregivenintermsof theformfactorsof thefields

involved,N isthenumberofparticlesinthespectrumandμi representtheparticle’squantum

numbers.Forexample:

h12₁|μ(θ )= FO1|μ

1 (θ )

FO

† 2|μ

1 (θ )

_∗

O1 O2

h12|μ1μ2

2 (θ1, θ2)=

FO1|μ1μ2

2 (θ1, θ2)

FO

† 2|μ1μ2

2 (θ1, θ2)

_∗

O1 O2 − h12|μ1

1 (θ1)h

12|μ2

1 (θ2), (44)

andsoon.Herewehaveusedthegenericproperty:

μj···μ1θj. . . θ1|O2(0)|0 = 0|O

†

2(0)|θ1. . . θj ∗μ1···μj=:F

O† 2|μ1...μj

j (θ1, . . . , θj)∗. (45)

The expansion (43) with(44) isusually referredtoas thecumulantexpansion of the two-pointfunction(seee.g. [35,49,50])anditisparticularlywellsuitedfor extractingthe leading log behaviourofthetwo-pointfunctionsform 1 (wheremisamassscale)providedthat thecumulantsh12_j |μ1...μj(θ1,· · ·,θj)satisfycertainasymptoticpropertiesintherapiditiessothat

theintegrals(43) arefinite.Forsimplicityletusconsidertemporarilythecaseofasingleparticle specie,andoperatorsO1,O2thatarespinless.Ifallformfactorsareknown,onemayextractthe

leadingUVbehaviourbyemployingthefactthatrelativisticinvarianceimpliesthatallform fac-torsdependonlyonrapiditydifferences.Asaconsequence,oneoftherapiditiesintheintegrals (43) maybeintegratedover,leadingto

c12_j ( )= 2 j!(2π )j

∞

−∞ dθ2· · ·

∞

−∞

whereK0(x)isaBesselfunctionand

d_j2= ⎛ ⎝j

p=2

coshθp+1

⎞ ⎠

2

−

⎛

⎝j

p=2

sinhθp

⎞ ⎠

2

. (47)

Providedthefunctionsh12_j (0,θ2,· · ·,θj)vanishforlargeθks,wemay,form 1,expandthe

BesselfunctionasK0(m dj)= −log −γ +log 2−log(mdj)+ · · · whereγ=0.5772157...

istheEuler–Mascheroniconstant.Form 1 weexpectthebehaviour

log

O1(0)O2( )

O1 O2

m 1

= −x12log −K12+ · · · (48)

wherex12isaconstantthatcapturestheshort-distancepowerlawofthecorrelator(describedby

CFT)andK12 isaconstantgenerallyrelatedtotheexpectationvaluesandconformalstructure

constantsofO1andO2.

Comingback tothe generalcasewithpotentially manyparticlespecies, usingthe leading termintheBesselfunctionexpansionintheseries(42) oneobtains

x12= ∞

j=1

2

j!(2π )j N

μ1,...,μj=1

∞

−∞ dθ2· · ·

∞

−∞ dθjh

12|μ1...μj

j (0, θ2,· · ·, θj). (49)

AsimilarexpressioncanbefoundfortheconstantK12asshownin [50].

Wenotethatthecumulantexpansionmethodwasappliedinthecontextofgluonscattering amplitudes/WL in[32,33].In this contextthe O(6) NLSMis considered,and yields double-logarithmicterms(log log)in(48).Similardouble-logarithmtermsarealsofoundinthe branch-pointtwistfieldtwo-pointfunctionof thefreebosonmodel[51,53].Nevertheless,theleading logarithmisstillpresent,andtheconstantx12isstillrelatedtotheshort-distancedominantpower

law.

4.1. Cumulantexpansionforconicaltwistfieldsinfreetheories

Weusethecumulantexpansionabove,andobtain

log

Vα(0)Vα( )

Vα Vα

=∞

j=1

cαα_2j( ) (50)

with

c₂αα_j( )= 1 (2j )!(2π )2j

∞

−∞ dθ1· · ·

∞

−∞

dθ2j hαα

2j (θ1,· · ·, θ2j)e

−m

2j

i=1

coshθi

. (51)

Comparedto(46) wehavedroppedtheμ1,. . . ,μj super-indicesinhj asfromnowonwewill

workonly withone particletype. Wealso sum over evenparticle numbersonly, as for free theoriesallothertwistfieldformfactorsarezero.

ustowritesimplecloseformulaeforthecumulantexpansion(51).Inordertobetterunderstand thisstructureletusconsideraparticular example,namelythefour-particlecontributiontothe expansion(51).Weneedtocompute

h₄αα(θ1,· · ·, θ4)=F4Vα(θ1, θ2, θ3, θ4)F4Vα(θ1, θ2, θ3, θ4)∗−F2Vα(θ12) 2

FVα

2 (θ34) 2

−FVα

2 (θ13)

2FVα

2 (θ24)

2−FVα

2 (θ14)

2FVα

2 (θ23)

2. (52)

Astandardpropertyoffreetheoriesisthatwhenthefour-particlecontributionaboveiswritten intermsoftwo-particleformfactors(using(37))thethreetwo-particlecontributionsaboveare identicallycancelledandonlysixtermsremainthatcanbewrittenas

hαα₄ (θ1, θ2, θ3, θ4)int=6f (θ12;α)f (θ23;α)∗f (θ34;α)f (θ14;α)∗, (53)

wherethe superscript“int”means equality underintegrationinallrapidities(inotherwords, thesixtermsareallequalunderrelabellingoftherapidities).Thispropertyholdsbothforfree FermionsandfreeBosonsandgeneralisestoallfunctionshαα(θ1,· · ·,θ2j)withtheprefactor6

abovegeneralisingto(2j−1)!sothatwemaywrite

hαα_2j(θ1,· · ·, θ2j)

int

=(2j−1)!f (θ12;α)f (θ1 2j;α)∗ j−1

k=1

f (θ2k+1 2k+2;α)f (θ2k2k+1;α)∗

int

=(2j−1)!f (θ12;α)f (θ2j1;α)

j−1

k=1

f (θ2k+1 2k+2;α)f (θ2k+1 2k;α),

(54)

up tothe identificationθ2j+1≡θ1. Inthe secondline we haveused thefact that f (θ;α)∗=

−f (θ;α)=f (−θ;α)forfreeFermionsandf (θ;α)∗=f (θ;α)=f (−θ;α)forfreeBosons. Wecanthereforewrite

cαα_2j( )= 1 (2j )(2π )2j

∞

−∞ dθ1· · ·

∞

−∞

dθ2je−m

2j

i=1coshθi_{f (θ}

12;α)f (θ2j1;α)

×

j−1

k=1

f (θ2k+1 2k+2;α)f (θ2k+1 2k;α), (55)

whichisvalidbothforfreeFermionsandBosons.

4.2. Exactshort-distanceasymptoticsofcorrelationfunctions

Asshownbyequation(49) andbycomparingtotheshort-distanceCFTpredictions(4)–(5) wemayextracttheshortdistanceasymptotics

log

Vα(0)Vα( )

Vα Vα

m 1

= −xααlog +log

Vα+α Cα+α

αα

Vα Vα

. (56)

Comparingto(4)–(5) weexpectthat

Ontheotherhand,combiningtheformfactorexpansions(55) withthegeneralresults(46) and (49) weobtain

xαα=

∞

j=1

1

j (2π )2j

∞

−∞ dx2· · ·

∞

−∞

dx2jf (−

2j

k=2

xk;α)f (x2j;α)

×

j−1

k=1

f (x2k+1;α)f (−x2k;α) (58)

wherewechangedvariablesto

xk=θk,k+1 for k=2, . . . ,2j−1, (59)

andx2j=θ2j wehave,inparticularthat

θ2= 2j

k=2

xk. (60)

Theexpressionsabovecanbesimplifiedfurtherbyfactoringintegralsdependingonodd-labelled andeven-labelledvariablesafterintroducinganewvariabley=j_k=1x2k sothat

xαα=

∞

j=1

1

j (2π )2j

∞ −∞ dy ∞ −∞ dx2· · ·

∞

−∞

dx2j−1f (−y−

j−1

k=1

x2k+1;α)f (y−

j−1

k=1 x2k;α)

×

j

k=2

f (x2k₋1;α)f (−x2k;α)= ±

∞

j=1

1

j (2π )2j

∞

−∞

dy Gj(y;α)Gj(y;α), (61)

wheretheplussignisforthefreeBoson,theminussigncorrespondstothefreeFermion.Here

Gj(y;α)=

∞

−∞

ds1. . . dsj−1f (y+

j−1

k=1 sk;α)

j−1

k=1

f (sk;α)

=

∞

−∞

dx Gj−1(y+x;α)f (x;α) (62)

for j >1 andG1(y;α)=f (y;α). Theintroduction of the functionsGj(y;α) togetherwith

therecursiverelationaboveprovidesapowerfulnumericalrecipefortheevaluationofthesum (58).Indeedasimilarrecursivestructurewasexploitedin[54,51] inordertofindanevenmore efficientwaytoevaluatetheintegrals(61).Thisisbasedonusingtherelation

∞

−∞

dy Gj(y;α)eiys= ˆf (s;α)j, (63)

wheref (sˆ ;α)istheFourier-transformedtwo-particleformfactor

ˆ

f (s;α)= ∞

−∞

Thisinturnmeansthatwecanwrite

Gj(y;α)=

1 2π

∞

−∞

dsf (sˆ ;α)je−isy, (65)

sothattheevaluationofeveryfunctionGj(y;α)isreducedtothecomputationofasingle

inte-gral.Furthermore,oncewehavewrittenGj(y;α)intheform(65) theinfinitesum(61) canbe

computedexactlyto:

xαα= ∓

1

(2π )2 ∞ −∞ dy ∞ −∞ ds1 ∞ −∞ ds2log

1−f (sˆ 1;α)f (sˆ 2;α

₎

(2π )2

e−iy(s1+s2)

= ∓ 1

2π ∞ ∞ dslog

1−f (sˆ ;α)f (ˆ−s;α

₎

(2π )2

, (66)

whereweusedthedefinition δ(x−a)=₂1_π−∞∞ dye−iy(x−a) oftheDiracdeltafunction.The minussigncorrespondstothefreeBosontheoryandtheplussigntothefreeFermiontheory.All thatisleftnowistocomputethefunctionf (sˆ ;α).Thisisfeasibleforfreetheories.Thefunction isnotparticularlysimplebutitisveryeasytoevaluatenumericallyinthecontextoftheintegral above.Wehave

ˆ

f (s;α)= 1 ns+i

e−iπn _2F1(1,1−_ins,2−_ins;_e−inπ₎−_einπ _2F1(1,1−_ins,2−_isn;_eiπn₎

− 1

ns−i

e−iπn _2F1(1,1+_ins,2+_ins;_e− iπ

n₎−_e iπ

n _2F1(1,1+_ins,2+_ins;_e iπ

n₎

,

(67)

forthefreeBosonand

ˆ

f (s;α)= csc

π

2n

2ins−3

e−iπn _2F1(1,3

2−ins, 5

2−ins;e

−iπ n₎

−eiπn _2F1(1,3

2−ins, 5

2−isn;e

iπ n₎

− csc

π

2n

1+2nis

e−inπ _2F1(1,1

2+ins, 3

2+ins;e

−iπ n₎

−eiπn _2F1(1,1

2+ins, 3

2+isn;e

iπ n₎

+ csc

π

2n

1−2nis

e−inπ _2F1(1,1

2−ins, 3

2−ins;e

−iπ n₎

−eiπn _2F1(1,1

2−ins, 3

2−isn;e

iπ n₎

+ csc

π

2n

2ins+3

e−iπn _2F1(1,3

2+ins, 5

2+ins;e

−iπ n₎

−eiπn _2F1(1,3

2+ins, 5

2+isn;e

iπ

n_{) , (68)}

Althoughtheformulaearemoreinvolved[50,51] itisalsopossibletowritedownaformfactor expansionoftheconstantterm ontherighthandsideof(56).Thiswouldprovideinformation aboutparticularratiosof expectationvaluesandstructureconstants.Wewillnotconsiderthis quantityherebutitwillbeinterestingtostudyitinthefutureandtounderstanditssignificance withintheapplicationofconicalfieldstothecomputationofscatteringamplitudesofgluonsin planarN =4 SYMtheory.

5. Numericalresults

Inthissection we presentacomparisonbetweenthe numericalevaluationof theformulae givenintheprevioussectionandtheanalyticalformulaepredictedfromCFT.

5.1. Thecaseα,α≥0

Wemaynowstudythesum(58) inmoredetailforfreeFermionsandBosonsandfor partic-ularchoicesofαandα.Wewillfocusfirstonthecaseα,α≥0 whichcorrespondstotaking

n, n≥1. Thecaseα=α isofparticularinterestasthiswasthecaseconsideredin[31]. Al-thoughthe fullre-summation (66) canbe done infree models,it is instructive,from amore generalperspective,towritexαα=

_∞

j=1x

(j )

αα andconsiderthefirst fewcontributionstoxαα

accordingto(61).Forj=1 wehave

x_αα(1)= 1 (2π )2

∞

−∞

dx f (x;α)f (−x;α)= ⎧ ⎨ ⎩

1 2π n

n−1

nsinπ n −

1

π

for the free Fermion

1 2π n

n−1

ntanπ_n + 1

π

for the free Boson.

(69)

Forj=2

x_αα(2)= ± 1

2(2π )4 ∞

−∞

dy G2(y;α)2 where

G2(y;α)= ∞

−∞

dx f (y+x;α)f (x;α), (70)

whereagaintheplussigncorrespondstothefreeBosonandtheminussigncorrespondstothe freeFermion.ItispossibletocomputethefunctionG2(y;α)exactly:

G2(y;α)=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

2"4π(n−1)tan"₂π_n#cosh"₂y_n#−ycsch"₂y_n#"cosh"y_n#−2 cos"π_n#+1##

n2_sec2"π 2n # cos 2π n

−cosh"y_n# for the free Fermion

4π (1−n)cot"π_n#−2ycoth"₂y_n#

n2_csc2"π n

#

cos2_nπ−cosh"y_n# for the free Boson.

(71)

Theseanalyticalexpressionscanthebeusedtoevaluatetheintegralforxαα(2)numerically.

Fig. 7.Thesolidcurvesrepresentthefunctionxαα=4α−22αforthefreeBosonandfreeFermion,respectively.

Theyareevaluatedexactlyfromtheformula(2) fordifferentvaluesofn=₂α_π+1.Thedashedcurvesrepresentdifferent approximationskmax

k=1x

(k)

αα ofthesevaluesthroughtheformfactorexpansion(61).Thereddashedcurvecorresponds

tokmax=1,thegreendashedcurveisthevaluekmax=2,thecyandashedcurveisthevaluekmax=3 andthedashed bluecurveisthevaluekmax=4.InthefreeBosoncaseweobserveveryclearconvergencetothepredictedvalue.In thefreeFermionconvergenceisevenbetter,buttheserieskmax

k=1x

(k)

αα isalternatingsothataddingtermstotheseries

alternativelyovershootsandundershootstheexactvalue.

Fig. 8.Thesolidcurvesrepresentthefunctionxαα=2α+2α−2α+αforα=2π(n−1)andα=2π(2n−1)for

thefreeBosonandfreeFermion,respectively.Theyareevaluatedexactlyfromtheformula(2) fordifferentvaluesofn. Thedashedcurvesrepresentdifferentapproximationskmax

k=1x

(k)

αα ofthesevaluesthroughtheformfactorexpansion

(61).Thereddashedcurvecorrespondstokmax=1,thegreendashedcurveisthevaluekmax=2,thebluedashedcurve isthevaluekmax=3.TheconvergencepatternissimilartothatinFig.7.

Fig. 9.Thesolidcurvesrepresentthefunctionx_αα=2α+2α−2α+αforα=2π(n−1)andα=2π(5n−1)for

Fig. 10.Thesolidcurvesrepresentthefunctionxαα=2α+2α−2α+αfordifferentvaluesofαandαforthe

freeBosonandfreeFermion,respectively.Thesolidcirclesarethenumericaloutputsfromintegrating(66) forthefree Bosonandthesolidtrianglesarethenumericalvaluesof(66) forthefreeFermion.AgreementwiththeCFTprediction isperfect.

Astrongertestcanbecarriedoutbyemployingdirectlythere-summedexpression(66).In thiscase,theagreementisperfectoverthewholerangeofvaluesofn≥1.Someexamplesare givenbelow,seeFig.10.

5.2. Thecaseα,α<0

Intheexamplesabovewe havealwaysconsideredpositiveexcessangles.Howeverwecan alsoconsidersituationswheretheexcessanglesarenegative.Negativeexcessanglescorrespond tovaluesofn<1 andthisgivesrisestosomedifficultieswhentryingtoevaluatetheformfactor expansion(61).Wecaneasilyappreciatethisifwetrytoevaluatethefirstcontributiontoxα,−α.

In thiscase,if α=2π(n−1)withn≥1 thenα=2π(1−n)<0. Letus computejustthe leadingcontributiontoxα,−α,givenby

x_α,(1₋)_α= 1 (2π )2

∞

−∞

dx f (x;α)f (−x; −α). (72)

ThisfunctioncanbeevaluatednumericallyandtheresultcanbecomparedtotheCFTprediction

x_α,(1)_−α =2α+2−α= c(n−1)

2

6(n−2)n. Note that for n=1 we have α=0 sothere is no conical

singularityandtheconicaltwistfieldsarejusttheidentity(sothepowerlawisexactly0).For

n=2 ontheotherhand−α= −2π whichmakesnogeometricsense,hencethesingularityin thescalingdimensions.Letusjustconsidervalues1≤n<2.TheresultsareshowninFig.11. TheoscillationsobservedinFig.11revealsomethingabout thestructure ofthe integral(72). The picksoccur for n=2−₂1_m with m=1,2,· · · Theseare the values of n for whichthe functionf (−x;−α)hasapoleatx=0.Asweknow,thefunctionsf (x;α)haveakinematic polestructurewithpolesontheextendedphysicalsheetatx=iπandx=iπ(2n−1)=i(π+α)

and,moregenerallyatallpointsoftheform

xk=iπ(2nk±1) for k=0,±1, . . . (73)

Forn>1 noneofthepolesabovecrossestherealline:xk=0 forn= ±₂1_k whichisalwaysless

than1.However,thesituationisdifferentforthefunctionf (−x;−α).Inthiscasethepolesare