On the universality of the $J=0$ fixed pole contribution in DVCS

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by  Dieter Müller

TheJournal’snamewillbesetbythepublisher
DOI:willbesetbythepublisher
(cid:13)c Ownedbytheauthors,publishedbyEDPSciences,2016
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On the universality of the J = 0 fixed pole contribution in DVCS
n
a
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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
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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]
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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
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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.
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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 }
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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.
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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. Phenomenologicalverification of
this conjecture from the GPD sum rule (9) is in principle possible but is biased by the necessity of
extrapolationofexperimentalresultsintohighenergyasymptoticregimeandthereforewillhardlybe
consideredasdecisive.
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