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Twist Three Generalized Parton Distributions for Orbital Angular Momentum 6 1 0 Abha Rajan∗† 2 UniversityofVirginia n E-mail: [email protected] a J Simonetta Liuti 0 3 UniversityofVirginiaandINFN,LaboratoriNazionalidiFrascati,Italy E-mail: [email protected] ] h p Westudytheorbitalangularmomentumcontributiontothespinstructureoftheproton. Itiswell - p knownthatthequarkandgluonspincontributionsdonotadduptotheprotonspin. Wemotivate e h the connection between the Generalized Transverse Momentum Distribution (GTMD) F , and 14 [ orbitalangularmomentumbyexploringtheunderlyingquarkprotonhelicityamplitudestructure. 1 ThetwistthreeGeneralizedPartonDistribution(GPD)E˜ ,wasshowntoconnecttoOAM.We v 2T 0 studythesefunctionsusingadiquarkmodelcalculation. TheGTMDF14 isuniqueinthatitcan 6 describebothJaffe-ManoharandJiOAMdependingonchoiceofgaugelink, i.e. whetherfinal 1 0 stateinteractionsareincludedornot. WeperformacalculationofF14inbothscenarios. 0 . 2 0 6 1 : v i X r a QCDEvolution2015 May26-30,2015 JeffersonLab(JLAB),NewportNewsVirginia,USA ∗Speaker. †This work is supported by U.S. Department of Energy, Office of Science, Office of Nuclear Physics contracts DE-AC05-06OR23177,DE-FG02-01ER41200 (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ TwistThreeGeneralizedPartonDistributionsforOrbitalAngularMomentum AbhaRajan 1. Introduction Orbital angular momentum is generated by the movement of quarks and and gluons inside the proton and is one of the candidates for accounting for the missing proton spin. Two famous de- compositionsofprotonspinalaJaffeManohar[1] 1 1 = ∆Σ+L +∆G+L (1.1) q g 2 2 andJi[2] 1 =J +J (1.2) q g 2 differprimarilyintheirdefinitionofOAML →i(cid:126)r×(cid:126)∂ andL →i(cid:126)r×D(cid:126). DefiningOAMusing JM Ji the covariant derivative allows the Ji decomposition to be gauge invariant. In this description, the Generalized Parton Distributions (GPDs) H and E give access to the total partonic angular momentumJ. 1(cid:90) 1 J = dxx(H (x,0,0)+E (x,0,0)) (1.3) q q q 2 −1 ToaccessJi’sOAM,thespincontribution 1∆Σneedstobesubtracted. 2 1(cid:90) 1 1(cid:90) 1 L = dxx(H (x,0,0)+E (x,0,0))− dxH˜ (x,0,0) (1.4) Ji q q q 2 2 −1 −1 1 =J − ∆Σ (1.5) q 2 GPDsdonotincludetransversemomentumandaredefinedinacollinearframework. Hence,L Ji includesastraightgaugelink. However,todescribeL weneedtoincludefinalstateinteractions JM i.e. useastaplegaugelink[3]. 2. TwistTwoandTwistThreeHelicityAmplitudes ThequarkquarkcorrelatorthatdescribestheseGPDshastheform, Wγ+ =(cid:90) dz−eixP+z−(cid:104)p(cid:48),Λ(cid:48)|ψ¯(−z/2)γ+ψ(z/2)| p,Λ(cid:105) (2.1) ΛΛ(cid:48) 2π z+=zT=0 In what follows, p = P+∆/2, p(cid:48) = P−∆/2, P = (p+p(cid:48))/2, k = k¯+∆/2, k(cid:48) = k¯−∆/2, k¯ = (k+k(cid:48))/2,andtheskewnessparameter,ξ =∆+/P+=0,hencet =∆2 (t ≡−∆2 forξ =0). This T correlator can equivalently be described using helicity amplitudes A where Λ (Λ(cid:48)) and λ Λ,λ,Λ(cid:48),λ(cid:48) (λ(cid:48)) give proton and quark initial (final) helicities respectively. The proton helicities depend on the initial and final states chosen. The quark helicities, on the other hand, are dependent on the projectionoperatorused. Hence,thegeneralformofthecorrelatorisgivenby, (cid:90) dz WΓ = −eixP+z−(cid:104)p(cid:48),Λ(cid:48)|ψ¯(−z/2)Γψ(z/2)| p,Λ(cid:105) (2.2) ΛΛ(cid:48) 2π z+=zT=0 2 TwistThreeGeneralizedPartonDistributionsforOrbitalAngularMomentum AbhaRajan Atleadingtwist,Γ: γ+,γ+γ5,iσi+ giverisetothevector,axialvector(chiraleven/quarkhelicity non flip) and the tensor GPDs (chiral odd / quark helicity flip). This can also be understood by connectingtothehelicityprojectionoperator1±γ5 [4], (cid:90) dz A = −eixP+z−(cid:104)p(cid:48),Λ(cid:48)|ψ¯(−z/2)γ+(1±γ5)ψ(z/2)| p,Λ(cid:105) (2.3) Λ,±,Λ(cid:48),± 2π z+=zT=0 Thevectoroperatorγ+ensuresnospinflip. Thisgivesusthedefinitionoftheleadingtwistnon-flip helicityamplitudes.Wγ+ andWγ+γ5 canbeexpressedintermsofhelicityamplitudesas: Λ,Λ(cid:48) Λ,Λ(cid:48) Wγ+ = 1(A +A ) (2.4) Λ,Λ(cid:48) 2 Λ,+,Λ(cid:48),+ Λ,−,Λ(cid:48),− Wγ+γ5 = 1(A −A ) (2.5) Λ,Λ(cid:48) 2 Λ,+,Λ(cid:48),+ Λ,−,Λ(cid:48),− In the case of γ+, as we are summing over the quark helicities, the quarks are unpolarized, while inthecaseofγ+γ5,thequarksarepolarized. WealsoclearlyseeherehowthePartonDistribution Function(PDF)g (x)thatparametrizesWγ+γ5 intheforwardcasegivesthequarkspincontribution 1 totheprotonspinbylookingattheunderlyinghelicityamplitudestructure: ∆Σ→A −A −A +A (2.6) ++,++ +−,+− −+,−+ −−,−− i.e. longitudinallypolarizedquarkinalongitudinallypolarizedproton. Now, if we think about the orbital angular momentum contribution, we need the quarks to be unpolarized [5, 6] (If they are polarized, the quark spin would contribute as well.) Hence, we wouldadd thequarkswithoppositehelicities. Theamplitudeswouldlooklike L →A +A −A −A (2.7) ++,++ +−,+− −+,−+ −−,−− At leading twist no PDF, GPD or TMD survives this combination of the amps. However if we includethepartonictransversemomentuminthepicture, (cid:90) dz Wγ+ = −eixP+z−−ikT.zT(cid:104)p(cid:48),Λ(cid:48)|ψ¯(−z/2)Γψ(z/2)| p,Λ(cid:105) (2.8) z+=0 2π theparametrizationofthenewcorrelatorgivesusGTMDs[7], Wγ+ = 1 u¯(p(cid:48),Λ(cid:48))[F +iσi+kTi F +iσi+∆iTF +iσijkTi ∆Tj F ]u(p,Λ) (2.9) Λ,Λ(cid:48) 2M 11 p¯ 12 p¯ 13 M2 14 + + (cid:18) k(cid:126) ×∆(cid:126) (cid:19) (cid:18)Λ∆1+i∆2 Λk1+ik2 (cid:19) T T = F +iΛ F δ + (2F −F )+ F δ (2.10) 11 M2 14 ΛΛ(cid:48) 2M 13 11 M 12 Λ−Λ(cid:48) If we look at the proton helicity non flip case (δ ) and, following (2.7), take the difference of ΛΛ(cid:48) theprotonhelicityalong+and−directions,theGTMDF survives. Itisinterestingtonotethat 14 at leading twist both the off forwardness and inclusion of transverse momentum of the partons is necessarytoaccessOAM. 3 TwistThreeGeneralizedPartonDistributionsforOrbitalAngularMomentum AbhaRajan In[8,9]PolyakovetalshowedthatthetwistthreeGPDG connectstoOAM(lateralsoshownin 2 [10]), (cid:20) ∆i ∆i γ+ γ+γ5 (cid:21) Wγ⊥i (x,ζ,t)=U¯(P(cid:48),Λ(cid:48)) (H+E)γ⊥i +2M⊥G1+γ⊥i G2+ P⊥+ G3+iε⊥ij∆⊥j P+ G4 U(P,Λ) (2.11) (cid:90) 1 1(cid:90) 1 1(cid:90) 1 − dxxG (x,0,0)= dxx(H (x,0,0)+E (x,0,0))− dxH˜ (x,0,0) (2.12) 2 q q q 2 2 −1 −1 −1 The GPD E˜ defined in [7] is equivalent to G and for uniformity, we will refer to this GPD as 2T 2 E˜ inthistext. Togetmoreinsight,wecanlookattheGTMDstructureforWγi : 2T Wγi = 1 (cid:18)ki F +∆i F +iΛεij(kTi F +∆iTF )(cid:19)δ + 1 (cid:18)M(Λδ +δ )F (2.13) ΛΛ(cid:48) P+ T 21 T 22 T M 27 M 28 ΛΛ(cid:48) P+ i1 i2 23 Λk1+ik2 Λ∆1+i∆2 (cid:19) + (ki F +∆i F )+ (2∆i F −(ki F +∆i F )) δ (2.14) M T 24 T 25 2M T 26 T 21 T 22 Λ−Λ(cid:48) The GTMDs F and F survive when we subtract the proton − and + helicity. These are also 27 28 exactlytheGTMDsthatcontributetoE˜ . 2T (cid:90) k .∆ 2 d2k T TF +F =E˜ (2.15) T ∆2 27 28 2T T TwoGPDsHandEentertheparametrizationattwistthreeinananalogouswayasthePDFg (x) 1 enters the definition of higher twist PDF g (x)=g (x)+g (x) [11, 12]. Hence, the GPD E˜ is T 1 2 2T the vector counterpart of g (x) for the off forward case. The combination that contributes to E˜ T 2T orF andF is 27 28 Wγ1 −iWγ2 −Wγ1 +iWγ2 (2.16) ++ ++ −− −− Aswearelookingnowat thetwistthreescenario, thequarkoperators(γi inthiscase)projectout the “bad” quark field components which is a quark gluon combination [13]. Since the gluon has spinone, thehelicityofthequarkisoppositetothatofthebadcomponent. Wefindthatthetwist threehelicityamplitudesthatcorrespondtoG aregivenby[14], 2 A −A −A +A (2.17) +−∗,++ ++,+−∗ −−∗,−+ −+,−−∗ Here, the starred helicities represent the bad component of the quark field. Due to the mixing of goodandbadcomponents,theinterpretationofthehelicityamplitudesasaprobabilitydistribution ismorecomplicatedattwistthree. 3. DiquarkModelCalculations The helicity amplitudes give us enormous physical insight and are very useful to perform model calculations. We use the parametrization of the spectator diquark model from [15]. The initial proton dissociates into a quark and a recoiling fixed mass system with the quantum numbers of a diquark. Thecouplingatthetheproton-quark-diquarkvertexisgivenby, 4 TwistThreeGeneralizedPartonDistributionsforOrbitalAngularMomentum AbhaRajan Figure1: TreeLevelSpectatorDiquarkModel[15](left)Interferencebetweengluonexchangediagramand treeleveldiagram[16](right) k2−m2 Γ=g (3.1) s(k2−m2)2 Λ g beingaconstant. Althoughthisisascalarcoupling,bothscalarandvectordiquarkconfigurations s can be achieved by varying the mass parameters. For the GTMDs, the quark - proton helicity amplitudesaredefinedas: A =φ∗ (k(cid:48),P(cid:48))φ (k,P) (3.2) Λλ,Λ(cid:48)λ(cid:48) Λ(cid:48)λ(cid:48) Λλ whereφ (k,P)andφ∗ (k(cid:48),P(cid:48))aregivenby Λλ Λ(cid:48)λ(cid:48) u¯(k,λ)U(P,Λ) φ (k,P)=Γ(k) (3.3) Λλ k2−m2 U¯(P(cid:48),Λ(cid:48))u(k(cid:48),λ(cid:48)) φ∗ (k(cid:48),P(cid:48))=Γ(k(cid:48)) (3.4) Λ(cid:48)λ(cid:48) k(cid:48)2−m2 Forthecase∆+=0, Γ(k)(cid:20) 1 (cid:18)−(m+xM)2 M2+k2 (cid:19)(cid:21) φ (k,P)= √ (m+xM)+ +M2(1+x)+2mM− x ⊥ (3.5) ΛΛ D x 2P+ x 1−x (cid:18) (cid:19) Γ(k) m+xM φ (k,P)=− √ (Λk1+ik2) 1+ (3.6) Λ−Λ D x 2xP+ Γ(k(cid:48))(cid:20) 1 (cid:18) M2+k2 (m+xM)2 φ∗ (k(cid:48),P(cid:48))= √ (m+xM)+ 2mM+M2(1+x)− x T − (3.7) Λ(cid:48)Λ(cid:48) D(cid:48) x 2P+ 1−x x (cid:19)(cid:21) +2k .∆ −∆2(1−x)+2iΛ(cid:48)∆ ×k (3.8) T T T T T Γ(k(cid:48)) m+xM φ∗ (k(cid:48),P(cid:48))= √ (−Λk˜1+ik˜2)(1− ) (3.9) Λ(cid:48)−Λ(cid:48) D(cid:48) x 2xP+ where, k˜ = k−(1−x)∆, D = k2−m2 and D(cid:48) = k(cid:48)2−m2. In general, we can represent φ → φTw2+φTw3 i.e. as a sum of a twist two and a suppressed twist three term. At twist two we onlyusetheleadingorderterms. ThetermssuppressedbyP+ representthebadcomponentofthe quark field and enter at twist three. To calculate F (x) we look at the amplitude combination in 14 5 TwistThreeGeneralizedPartonDistributionsforOrbitalAngularMomentum AbhaRajan Figure2: e(x)andxe(x)foruanddquarksindiquarkspectatormodel (2.7)andusing(3.5),arriveatthefollowing, M2 F =N . (3.10) 14 x(k2−m2)2((k−∆)2−m2)2 Λ Λ Next, we include final state interactions between the active quark and the spectator diquark [3]. Similarly to the calculation of the Sivers function in forward kinematics [16], this is achieved by by considering the interference between the tree-level scattering amplitude and the single-gluon- exchangescatteringamplitudeintheeikonalapproximationwherethegluonmomentumisalmost allalongthelightcone(Fig.1,right), F =N (cid:48) M2(1−x) (cid:90) d2lT (1+lTl.T2kT) . (3.11) 14 x(k2−m2)2 (2π)2((l −k )2−L2(m2)−2xl2)2 Λ T T x T Here, N and N (cid:48) are normalization constants and l represents the gluon momentum. L2(m2)= x xM2+(1−x)m2−x(1−x)M2 where,M isthediquarkmass. x x The calculation of GPDs and PDFs involves helicity amplitudes where the partonic transverse momentumisintegratedover, (cid:90) A = d2k A . (3.12) Λλ,Λ(cid:48)λ(cid:48) T Λλ,Λ(cid:48)λ(cid:48) FortwistthreecalculationsweneedtoconsiderinterferenceofφTw2 andφTw3 terms, A =φ∗Tw2φTw3 (3.13) Λλ∗,Λ(cid:48)λ(cid:48) Λ(cid:48)λ(cid:48) Λλ A =φTw3φ∗Tw2, (3.14) Λλ,Λ(cid:48)λ∗(cid:48) Λ(cid:48)λ(cid:48) Λλ wherethestarrednotationonthelhsrepresentsthebadcomponentsofthequarkfield. InFigure2weshow,asanexamplethetwistthreePDF,e(x),thatparametrizesthescalarcompo- nentofthecorrelationfunction. e(x)correspondstothefollowinghelicityamplitudecombination, A +A +A +A +A +A +A +A . (3.15) ++∗,++ ++,++∗ +−∗,+− +−,+−∗ −+∗,−+ −+,−+∗ −−∗,−− −−,−−∗ 6 TwistThreeGeneralizedPartonDistributionsforOrbitalAngularMomentum AbhaRajan Results for the GPD E˜ calculated using (2.7) will be presented in a forthcoming publication 2T [14]. In this case it is important to notice that the function’s normalization can be fixed using the connectiontothetwistthreereducedmatrixelementd [8,17,18], 2 (cid:90) 2 dxx2E˜Tw3=− d , (3.16) 2T 3 2 where E˜Tw3 is the genuine twist three contribution to E˜ . Lorentz invariant relations can be de- 2T 2T rived between the k2 moment of the twist two GTMDs and twist three GPDs[12]. A key step is T to look at the substructure of these functions with respect to functions that parametrize the non integratedcorrelator(k−)akaGeneralizedPartonCorrelationFunctions(GPCFs). Ourapproachis verysimilartothatin[11,12]. Therelationwefindisthefollowing, (cid:90) k2 (cid:90) 1 d2k T F (y,k2,0,0,0)=− dyE˜ (y,0,0)+H(y,0,0)+E(y,0,0) (3.17) TM2 14 T 2T x The moment in transverse momentum k is important. In some sense, it mimics the quark gluon T interactionsthatwouldbenecessarytogenerateorbitalangularmomentuminacollinearpicture. 4. Conclusions The diquark model calculations of F and E˜ allow us to study both the Jaffe Manohar and 14 2T Ji definitions of orbital angular momentum. Inclusion of final state interactions in the picture is relevant for OAM and for other non-perturbative effects such as single spin asymmetries in the proton. We have derived a relation between the GTMD (via F ) and GPD (E˜ ) definitions of 14 2T OAM. It is essential that these objects be measured experimentally and this is something we are currentlyworkingon. Muchinterestingphysicsremainstobeexploredinthefutureincludingthe evolutionofthesefunctions,andtheroleofthegenuinetwistthreecontribution. 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