TheJournal’snamewillbesetbythepublisher DOI:willbesetbythepublisher (cid:13)c Ownedbytheauthors,publishedbyEDPSciences,2016 6 1 0 2 On the universality of the J = 0 fixed pole contribution in DVCS n a J 8 D.Müller1,a and K.Semenov-Tian-Shansky2,b 1TheoreticalPhysicsDivision,RudjerBoškovic´Institute,HR-10002Zagreb,Croatia ] h 2PetersburgNuclearPhysicsInstitute,Gatchina,188300,St.Petersburg,Russia p - p Abstract.S.Brodsky,F.J.Llanes-EstradaandA.SzczepaniakformulatedtheJ=0fixed e poleuniversalityhypothesisfor(deeply)virtualComptonscattering.Weshowthatinthe h Bjorkenlimitthishypothesisisequivalenttothevalidityoftheinversemomentsumrule [ fortheD-termformfactor. However,anysupplementaryD-termaddedtoageneralized 1 parton distribution (GPD) results in an additional J = 0 fixed pole contribution that v violatesuniversality.Unfortunately,onecannotprovideanyreliabletheoreticalargument 0 excludingtheexistenceofsuchsupplementaryD-term.Moreover,theviolationofJ=0 3 fixedpoleuniversalitywasrevealedinfieldtheoreticalGPDmodels. Therefore, J = 0 8 fixedpoleuniversalityhypothesisremainsjustanexternalassumptionandprobablywill 1 neverbeproventheoretically. 0 . 1 0 6 1 Introduction 1 : v StudiesofComptonscatteringoffa nucleonγ(∗)(q )+N(p ) → γ(∗)(q )+N(p )withbothphotons 1 1 2 2 i being real (q2 = q2 = 0) or with one (q2 = −Q2, q2 = 0) or two (q2 = −Q2, q2 = −Q2) virtual X 1 2 1 1 2 1 1 2 2 space-likephotonsallowtoprobenucleonstructureinlowandhighenergyregimes. Thecustomary r a theoreticalframeworkwhichhelpstoaddressComptonscatteringinhighenergyregimeistheRegge theory. AssumingtheexistenceoftheleadingReggetrajectoryα(t)(t ≡(p − p )2),thehighenergy 2 1 leading asymptotic behavior∼ sα(t) of the amplitude originatesfrom a movingpole in the plane of complexcross-channelangularmomentumJ.Besidesthesecommonmovingpoles(and/orcuts)there alsomightexistso-calledfixedpolesingularities(seee.g.ChapterIofRef.[1]),thatdonotslidewith thechangeoft andcannotberevealedbymeansof theanalyticcontinuationin J. A J = J fixed 0 polesingularitymayarisefromacrosschannelexchangewithanon-Reggeized(elementary)particle ofspinJ inthecrosschannel(orfromacontactinteractionterm).ItappearsthenastheKronecker-δ 0 singularityinthecomplexJ-plane. The discussion on the manifestation of the J = 0 fixed pole singularity in Compton scattering, whichresultsinconstantcontributioninhigh-energyasymptoticlimit,lastssincethelatesixties(see thepioneeringpaper[2]andRef.[3]aswellasreferencestherein).S.Brodsky,F.CloseandJ.Gunion identifiedthiscontributionatthediagrammaticlevel.Theyargueitoriginatesfrom“seagull”diagrams correspondingtolocaltwophotoninteraction. Morerecently,in[4]S.Brodsky,F.J.Llanes-Estrada andA.SzczepaniakemphasizedthatsuchJ =0fixedpolecontributioninvirtualComptonscattering ae-mail:[email protected] be-mail:[email protected] TheJournal’sname isuniversal(i.e. independentofphotonvirtualities)andcanbeexpressedbytheinversemomentof thecorrespondingt-dependentpartondistributionfunction(PDF). Ontheotherhand,theJ =0fixedpolecontributionindeeplyvirtualComptonscattering(DVCS) is closely related with the D-term form factor, which arises as the subtraction constant for deeply virtual Compton scattering amplitude within the partonic picture. The D-term [5] was originally introducedtocomplementthepolynomialityconditionforGPDswithinthedoubledistribution(DD) representation. Later it was realized that it could be implemented as an inherent part of GPD both within the modified DD representation and within the representations based on conformal partial wave(PW)expansionofGPDs. Below, followingRefs. [6, 7], we reviewthe relationbetween the J = 0 fixedpole contribution and the D-term form factor. We tie together the J-analytic propertiesof cross channel SO(3) PWs and those of GPD conformalmomentsin the conformalspin j. We show thatthe J = 0 fixed pole universalityhypothesisgenerallyremainsunprovenandonlycanbeacceptedasexternalprinciplefor buildingupphenomenologicalGPDmodels. 2 Dispersive approachfor Compton amplitude WefocusontheComptonform-factor(CFF)H(ν≡s−u,t|Q2,Q2)occurringintheparametrizationof 4 1 2 thetransversenon-flipphotonhelicityamplitudeofComptonscattering.ThisCFFhasevensignature (P= +1;C = +1)andistheanalogueoftheusualDiracelectromagneticformfactor. Intheforward kinematicstheimaginarypartofH correspondstothedeepinelasticstructurefunctionF . 1 A fixed-t dispersion relationfor the CFF H can be obtainedadoptingusualassumptionson the analytic properties of CFFs. Neglecting the Born term (which is irrelevant in the Bjorken limit) we consider the deformedintegration contour surroundingcuts on the real axis starting at the pion productionthreshold ν = Q21+Q22+t+(M+2mπ)2−M2 (see Fig. 1). Assuming that H vanishes at infinity cut 4M (lim H(ν,t|Q2,Q2)=0),weobtaintheunsubtractedDRinthestandardform, |ν|→∞ 1 2 1 ∞ 2ν′ImH(ν′,t|Q2,Q2) H(ν,t|Q2,Q2)= dν′ 1 2 . (1) 1 2 πZ ν′2−ν2−iǫ νcut OnceH doesnotvanishatinfinityonestillcanformallyconsidertheunsubtractedDRbyspecifying a regularization procedure both for the divergent integral along the real axis and (either divergent) contributionsfromlargesemicirclesofthecontouronrightpanelofFig.1. Apossiblechoiceisthe analyticregularization(seee.g. Ref.[8]). Thedispersiveintegralisreplacedbytheloopintegralin thecomplexplanethatincludesthepointν = ∞, denotedas(∞). TheunsubtractedDRforCFFH thenreads 1 (∞) 2ν′ImH(ν′,t|Q2,Q2) H(ν,t|Q2,Q2)=H (t|Q2,Q2)+ dν′ 1 2 , (2) 1 2 ∞ 1 2 πZ ν′2−ν2−iǫ νcut where the constant H , arises from the analytic regularization at ν = ∞. Within the Regge–pole ∞ expansionoftheamplitudeitisinterpretedastheJ =0fixedpolecontribution. AnalternativeformofDRemployedwithinthedeeplyvirtual(d.v.)regimeinvolvesonesubtrac- tionattheunphysicalpointν=0: 1 ∞ dν′2ν2ImH(ν′,t|Q2,Q2) H(ν,t|Q2,Q2)d=.v.H (t|Q2,Q2)+ 1 2 . (3) 1 2 0 1 2 πZ ν′ ν′2−ν2−iǫ νcut Generally, the subtraction constant H (t|Q2,Q2) represents an unknown quantity. However, in the 0 1 2 deeplyvirtualregimeonecanformulatetheexternalprincipleallowingtofixthesubtractionconstant 6thInternationalConferenceonPhysicsOpportunitiesatElectron-IonCollider ν ν −νcut νcut −νcut νcut Figure1. Leftpanel: InitialintegrationcontourinthecomplexνplanetoexpresstheCFFthroughtheCauchy integral.Rightpanel:Deformationoftheintegrationcontourineq.(1). fromtheabsorptivepartoftheamplitude. Relyingontheoperatorproductexpansiononecancheck that for equal photon virtualities Q2 = Q2 = Q2 (DIS kinematics) due to the current conservation 1 2 thesubtractionconstantvanishesto theleadingtwistaccuracy. Within thekinematicsofDVCS the subtractionconstantin(3)correspondstotheD-termformfactorH (t|Q2 =Q2,Q2 =0)d=.v.4D(t). 0 1 2 Thedispersionrelations(2) and(3)aresupposedtorepresentthesamefunction. Therefore,the J = 0 fixed pole contributionH could be related to the subtraction constantH . Pluggingin the ∞ 0 algebraicdecomposition 2ν′ = 1 2ν2 + 2 oftheCauchykernelinto(2)andcomparingitwith ν′2−ν2−iǫ ν′ν′2−ν2−iǫ ν′ (3),wereadoffthesumrule 2 (∞) dν H (t|Q2,Q2)=H (t|Q2,Q2)− ImH(ν,t|Q2,Q2), ∞ 1 2 0 1 2 πZ ν 1 2 νcut expressing the J = 0 fixed pole contribution through the subtraction constant and the analytically regularizedinversemomentoftheabsorptivepartoftheamplitude. InthedeeplyvirtualkinematicsregimetheconvenientkinematicalvariablearetheBjorken-like variableξ = Q2 ,whereQ2 =−(q1+q2)2;andthephotonasymmetryparameter1ϑ = q21−q22 + (p1+p2)·12(q1+q2) 4 q21+q22 O(t/Q2). Notethatfort = 0ϑ = 0onerecoverstheusualDISkinematics,whileϑ = 1corresponds totheDVCSsetup. WithinthesescalingvariablestheanalyticallyregularizedDR(2)andthesubtractedone(3)read asfollows: 1 1 dξ′2ξ2ImH(ξ′,t|ϑ) H(ξ,t|ϑ)= +H (t|ϑ); (4) πZ ξ′ ξ2−ξ′2−iǫ ∞ (0) 1 1 2ξ′ImH(ξ′,t|ϑ) H(ξ,t|ϑ)= dξ′ +H (t|ϑ). (5) πZ ξ2−ξ′2−iǫ 0 0 Here,thelowerintegrationlimit,ξ = 0,correspondstoν = ∞andtheupperintegrationlimitξ = cut Q2 ,hasbeensettoξ =1inthe(generalized)Bjorkenlimit. TheequivalenceofthetwoDRs(4), 2Mνcut cut (5)isensuredbythesumrule(4),whichnowreads 2 1 dξ H (t|ϑ)=H (t|ϑ)− ImH(ξ,t|ϑ). (6) ∞ 0 πZ ξ (0) Withinthesenotations,theJ =0fixedpoleuniversalityhypothesisof[4]reads: 2 1 dξ H (t|ϑ)=− ImH(ξ,t|ϑ=0). (7) ∞ πZ ξ (0) 1VariousdefinitionsofϑencounteredinliteraturedifferbyO(1/Q2)termsvanishinginthegeneralizedBjorkenlimit. TheJournal’sname 3 Fixed pole contributionin GPD framework In this section we address the issue of the subtraction constant for the dispersive representation of CFFwithin thepartonicpicture. We wouldnowliketo pointouttheoriginofadditionalfixedpole contribution∆H ,whichviolatestheuniversalityconjecturewithintheGPDframework. ∞ Withinthecollinearfactorizationapproachtotheleadingorder(LO)accuracy,theCFFH(ξ,t|ϑ) arisesfromtheelementaryamplitude 1 2x H(ξ,t|ϑ)L=O dx H(+)(x,η=ϑξ,t), (8) Z ξ2−x2−iǫ 0 where H(+)(x,η,t) ≡ H(x,η,t) − H(−x,η,t) stands for the antisymmetric charge even quark GPD combination.Notethatforallallowedvaluesof|ϑ|≤1theimaginarypartoftheCFFisgivenbythe GPDintheouterregionξ ≥η. Inserting the imaginary part of (8) into the sum rule (6) allows to expressthe J = 0 fixed pole contributionH (t|ϑ)totheLOaccuracybytheGPDintheouterregion: ∞ 1 dx H (t|ϑ)L=OH (t|ϑ)−2 H(+)(x,ϑx,t). (9) ∞ 0 Z x (0) Bypluggingtheimaginarypartof(8)intothesubtractedDR(5)andequatingitwiththeLOconvo- lutionformula(8)fortheCFF,onecanworkouttheGPDsumrule[13,14]: 1 2x H (t|ϑ)≡4D(t|ϑ)L=O dx H(+)(x,ϑx,t)−H(+)(x,ϑξ,t) (10) 0 Z x2−ξ2 0 h i forthe D-termformfactor,whichcorrespondsto thesubtractionconstantH . Itseemsthatbyfor- 0 mallytakingthehighenergylimitξ → 0 onecanprovidesa regularproofforthe J = 0fixedpole universalityconjecture(7). We arguethatonehastobeprudentwhileperformingthislimitingpro- cedureandseparatetheintegrationregionin(9)intocentralx ∈ [0;θξ]andouter x ∈ [θξ;1]regions. Omittingthecontributionfromthecentralregionmeansignoringpossibleseparate D-termaddenda toGPD.Asanillustration,weconsideredthespecificformofaDDrepresentationforGPD: 1 1−y H(+)(x,η)= dy dz (1−ax)δ(x−y−zη)−{x→−x} h(y,z), (11) Z0 Z−1+y h i whereh(y,z)isDD,symmetricinzandantisymmetriciny. Thecasea = 0correspondstotheusual DDrepresentation(withoutaD-term). Fora,0GPD(11)includesaninherentD-termpartandthe polynomialityconditionisrespectedinitscompleteform. TheGPDsumrule(10)isalsorespected andthe J =0fixedpoleuniversalityconjecture(7)isvalid. However,addingasupplementaryseparateD-termcontribution H(+)(x,η)→H(+)(x,η)+θ(|x|≤|η|)df.p.(x/|η|) (12) leadstobreakdownoftheJ =0fixedpoleuniversalityconjecturebytheJ =0fixedpolecontribution 4Df.p.(t|ϑ)= 1dx 2xϑ2 df.p.(x,t): 0 1−ϑ2x2 R 1 dx H (t|ϑ)L=O−2 q(+)(x,t)+4Df.p.(t|ϑ) . (13) ∞ Z x 0 ∆H∞(t|ϑ) | {z } 6thInternationalConferenceonPhysicsOpportunitiesatElectron-IonCollider A particular example of a field theoretical GPD model with a separate D-term contribution is providedbythecalculation[9]ofpionGPDsinthenon-localchiralquarkmodel[10]. Inthismodel theuniversalityconjecture(7)isnotvalidduetoasupplementaryD-termdf.p.(x,t)contribution,which hastobeaddedtomakeGPDsatisfythesoftpiontheorem[11]fixingpionGPDsinthelimitη→1. Anotherexampleis givenbythe photonGPD H , calculatedto one-loopaccuracyin [12], wherea 1 directcalculationoftheCFFintheξ→0limitdisprovestheconjecture(7). We concludethatthe validity of the J = 0 fixed poleuniversalityconjecturecorrespondsto the internaldualityprincipleforGPDs,relatingGPDsintheinnerandoutersupportregions(seeRef.[13] foradetaileddiscussion). It is extremely convenient to address this property within the approach [15] based on operator product expansion over the conformal basis. In this case there turns to be two kinds of analytical propertiesrelevantforGPDsandassociatedCFFs: • analyticityofCFFsinthecrosschannelangularmomentumJ; • analyticityofGPDGegenbauer/Mellinmomentsinthevariable j,labelingtheconformalspin j+2 ↔ ↔ oftwist-2quarkconformalbasisoperators2Oaj = Γ2(3jΓ/2(3)Γ/2(1++j)j) (i∂+)j ψ¯λaγ+C3j/2 D↔+ ψ. (cid:18)∂+(cid:19) In order to show explicitly the J-analytic propertieswe employ the Froissart-Gribovprojection [16] ofthecrosschannelSO(3)PWs oftheCFF: a (t|ϑ) ≡ 1 1d(cosθ)P (cosθ)H(+)(cosθ,t|ϑ), J 2 −1 t J t t where(neglectingthethresholdcorrections)thecosineofthet-cRhannelscatteringanglereads:cosθ = t −1/(ϑξ)+O(1/Q2). ForJ >0PWstheFroissart-GribovprojectionprovidestotheLOaccuracy 1 Q (1/x) a (t|ϑ)L=O2 dx J H(+)(x,ϑx,t), (14) J>0 Z x2 0 whereQ (1/x)denotetheLegendrefunctionsofthesecondkind.ForJ =0oneobtains J 1 Q (1/x) 1 a (t|ϑ)L=O2 dx 0 − H(+)(x,ϑx,t)+4D(t|ϑ). (15) J=0 Z " x2 x# 0 Indeed,asclearlyseenfromEqs.(14)and(15),onecannotrecovera (t|ϑ)bymeansoftheana- J=0 lyticcontinuationofa (t|ϑ)toJ =0.Therefore,analyticityinthecrosschannelangularmomentum J>0 Jisaffectedbythepresenceofa J =0fixedpolecontribution 1 dx af.p.(t|ϑ)≡H (t|ϑ)L=O4D(t|ϑ)−2 H(+)(x,ϑx,t). (16) J=0 ∞ Z x (0) Within the conformaloperatorproductexpansionapproachthe analytic propertiesin the conformal spin j of the Gegenbauer moments of GPDs play the crucial role. In particular, the presence of a J = 0fixedpolecontribution(16)totheCFFH isadirectconsequenceofabsenceofnon-singular conformal operators with the Lorentz spin J ≡ j+1 = 0. Such a j = −1 contribution has to be subtracted from the J = 0 PW (see the 1/x momentin the integrand of Eq. (15)). The analogous cancelationappearsalsointheframeworkofdualparametrizationofGPDs[6]. Moreover,assuming maximalanalyticityintheconformalspin j(whichcorrespondstoabsenceofthe j = −1fixedpole singularitiesmanifestastheKroneckerδ termsorsingularoperatorcontributionsinthemomentum j,−1 ↔ ↔ 3 2Herethecovariantderivative Dandthetotalderivative ∂ arecontractedwiththelight-conevectorn,C2 standforthe j usualGegenbauerpolynomialswiththeindex3/2. TheJournal’sname fractionrepresentation)leadstothevalidityoftheinternaldualityprincipleforGPDs(see[13]).This correspondstheimplementationofJ =0fixedpoleuniversalityhypothesiswithadefiniteJ =0fixed polecontributiongivenbytheinversemomentsumrule 1 dx H (t|ϑ)L=O−2 H(+)(x,0,t). (17) ∞ Z x (0) ItisworthemphasizingthattheinversemomentofPDF(17)cannotbeextractedformthesumrule (10) for D-term form factor as it simply cancels out. This sum rule is only potentially sensitive to possiblenon-universalcontributionintoH (t|ϑ). ∞ Conclusions TheJ =0fixedpoleuniversalityhypothesisremainsanattractivetheoreticalassumptionforbuilding upconsistentGPD modelsintendedfordescriptionofComptonscatteringindeeplyvirtualregime. However, at the moment this hypothesis lacks theoretical proof. 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