Table Of ContentTwist 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.
References
[1] R.L.JaffeandA.Manohar,Nucl.Phys.B337(1990)509.
[2] X.D.Ji,Phys.Rev.Lett.78,610(1997)
[3] M.Burkardt,Phys.Rev.D69,057501(2004)
[4] M.Diehl,Phys.Rept.388,41(2003).
[5] C.LorceandB.Pasquini,Phys.Rev.D84(2011)014015.
[6] A.Courtoy,G.R.Goldstein,J.O.G.Hernandez,S.LiutiandA.Rajan,Phys.Lett.B731,141(2014)
[7] S.Meissner,A.MetzandM.Schlegel,JHEP0908(2009)056.
[8] D.V.KiptilyandM.V.Polyakov,Eur.Phys.J.C37(2004)105.
[9] M.Penttinen,M.V.Polyakov,A.G.ShuvaevandM.Strikman,Phys.Lett.B491,96(2000)
[10] Y.HattaandS.Yoshida,JHEP1210,080(2012)
[11] R.D.TangermanandP.J.Mulders,hep-ph/9408305
[12] A.Bacchetta,M.Diehl,K.Goeke,A.Metz,P.J.MuldersandM.Schlegel,JHEP0702,093(2007)
7
TwistThreeGeneralizedPartonDistributionsforOrbitalAngularMomentum AbhaRajan
[13] R.L.Jaffe,Lect.NotesPhys.496,178(1997)doi:10.1007/BFb0105860[hep-ph/9602236]
[14] A.Rajan,A.Courtoy,M.Engelhardt,S.Liuti,inpreparation(2016);S.Liutietal.,theseproceedings.
[15] G.R.Goldstein,J.O.HernandezandS.Liuti,Phys.Rev.D84,034007(2011)
[16] A.Bacchetta,F.ContiandM.Radici,Phys.Rev.D78,074010(2008)
[17] JaffeR.L.,Comm.Nucl.Part.Phys.,19(1990)239.
[18] M.Burkardt,Phys.Rev.D66,114005(2002)
8
