Table Of ContentEur. Phys. J. C manuscript No.
(will be inserted by the editor)
Wigner distributions and orbital angular momentum of a
proton
D. Chakrabarti1, T. Maji1, C. Mondal1 · A. Mukherjee2
6
1 Received:date/Revisedversion:date
0
2
Abstract TheWignerdistributionsforuanddquarks transversity distributions, h (x). The generalized par-
l 1
u in a proton are calculated using the light front wave ton distributions(GPDs) and the transverse momen-
J
functions (LFWFs) of the scalar quark-diquark model tum dependent distributions(TMDs) encode informa-
2
for nucleon constructed from the soft-wall AdS/QCD tions about the three dimensional structure of the nu-
1
correspondence.Wepresentadetailstudyofthequark cleons.InthedeeplyvirtualComptonscattering(DVCS),
] orbital angular momentum(OAM) and its correlation deeplyvirtualmesonelectroproduction(DVMPs),amore
h
with quark spin and proton spin. The quark density general views of parton distributions, in the collinear
p
- distributions, considering the different polarizations of frame,isstudiedbyGPDs[1,2,3,4]whicharefunctions
p
quarks and proton, in transverse momentum plane as of longitudinal momentum and two transverse impact
e
well as in transverse impact parameter plane are pre- parameter coordinates. TMDs[5,6,7,8] are functions of
h
[ sented for both u and d quarks. the transverse momentum of the parton and appear
in the semi inclusive deep inelastic scattering(SIDIS)
2
v where the collinear picture is no longer enough to ex-
7 plain the single spin asymmetry(SSA).
1 1 Introduction
Wigner distributions are six dimensional distribu-
2
tions containing more general informations about the
3 A complete understanding of partonic structure of nu-
0 nucleon structure. Wigner distributions do not have
cleon is one of the challenging tasks in the particle
. probabilisticinterpretationbutincertainlimits,reduce
1 physics. Both theoretical and experimental efforts are
0 to GPDs and TMDs. The Wigner distributions are de-
going on to unravel the three dimensional distributions
6 fined as functions of three momentums and three po-
1 of the partons and their contributions to the nucleon
sitions of a parton inside a nucleon. The concept of
: spin and angular momentum. Because of the nonper-
v Wignerdistributionswasfirstintroducedin[9].In[10],
turbative nature of QCD, it is very difficult to per-
i
X fivedimensionalWignerdistributionswereproposedin
form first principle calculations of the hadron prop-
the light-front formalism with three momentum and
r erties. However a perturbative approach in light cone
a two position components of a parton. Wigner distri-
framework allows us to calculate the parton distribu-
butions integrated over transverse momentum give the
tion function(PDF), f(x), which gives the probability
GPDs at zero skewness, the TMDs are obtained by in-
ofhavingapartonwithlight-conelongitudinalmomen-
tegrating over transverse impact parameters with zero
tumfractionxinsideanucleonbutitcontainsnoinfor-
momentumtransferandtheintegrationovertransverse
mation about the transverse structure or angular mo-
momentumandtransversepositionsprovidethePDFs.
mentum distributions. The spin correlation of partons
TheWignerdistributionsafterintegratingoverthelight
are described by the helicity distribution, g (x), and
1
cone energy of the parton are interpreted as a Fourier
transform of corresponding generalized transverse mo-
Department of Physics, Indian Institute of Technology Kan-
mentum dependent distributions (GTMDs) which are
pur,Kanpur208016,India
·DepartmentofPhysics,IndianInstituteofTechnologyBom- functionsofthelightconethree-momentumofthepar-
bay,Mumbai400076,India ton as well as the momentum transfer to the nucleon.
2 D.Chakrabarti1,T.Maji1,C.Mondal1,A. Mukherjee2
AngularmomentumofaquarkisextractedfromWigner helicity λ . The xP+ and p are the longitudinal mo-
s ⊥
distributionstakingthephasespaceaverage.Thespin- mentum and transverse momentum of the active quark
spin and spin-orbital angular momentum(OAM) cor- respectively. The ψλN are the light-front wave func-
qλq
relations between a nucleon and a quark inside the tions corresponding to the nucleon helicity λ = ±
N
nucleon can also be described from phase space aver- andquarkhelicityλ =±.Weadoptthegenericansatz
q
age of Wigner distributions. Wigner distributions have for the quark-diquark model of the valence Fock state
been studied in different models e.g., in lightcone con- of the nucleon LFWFs [17], assuming vanishing quark
stituent quark model[11,12], in chiral soliton model[13, mass
14],lightfrontdressedquarkmodel[15],lightconespec- ψ+ (x,p )=φ(1)(x,p ),
q+ ⊥ q ⊥
tatormodel[16].Inthiswork,weinvestigatetheWigner
p1+ip2
distributions for unpolarized and polarized proton and ψ+ (x,p )=− φ(2)(x,p ),
q− ⊥ xM q ⊥
the orbital angular momentum(OAM) and spin-spin
p1−ip2
andspin-OAMcorrelationsinascalardiquarkmodelof ψ− (x,p )= φ(2)(x,p ), (2)
q+ ⊥ xM q ⊥
proton [17] with the light front wavefunctions modeled
from AdS/QCD prediction. ψq−−(x,p⊥)=φ(q1)(x,p⊥),
Thepaperisorganizedasfollows.Wefirstintroduce where ϕ(1)(x,p ) and ϕ(2)(x,p ) are the wave func-
q ⊥ q ⊥
the lightfront scalar diquark model in Sec.2 and the tions predicted by soft-wall AdS/QCD in [19] with the
Wigner distributions in Sec.3. The different definitions AdS/QCD scale parameter κ=0.4 GeV.
of orbital angular momentum are discussed in Sec.4. (cid:114)
4π log(1/x)
Then in Sec.5, both analytical and numerical results ϕ(qi)(x,p⊥)=Nq(i) κ 1−x xa(qi)(1−x)b(qi)
in our model are discussed in detail. The correlation
(cid:20) p2 log(1/x)(cid:21)
betweenthequarkandprotonspinsandquarkspinand exp − ⊥ , (3)
OAM correlations are discussed in Sec.6. The results 2κ2 (1−x)2
are also compared with other models. The GTMDs in Thevaluesoftheparametersa(i),b(i)andN(i)arefixed
q q q
this model are briefly discussed in Sec.7 and finally we in [20,21] by fitting the nucleon form-factor data. For
conclude in Sec.8. completeness, we list the parameters in Table 1. This
parameters a(1) a(2) b(1) b(2) N(1) N(2)
2 Light-front diquark model uquark 0.02 1.05 0.022 -0.15 2.055 1.322
dquark 0.1 1.07 0.38 -0.2 1.7618 -2.4827
Inthediquarkspectatormodel,oneofthethreevalence Table1 Theparametersinthelightfrontdiquarkmodelfor
quarksinteractswithexternalphotonandothertwova- κ=0.4GeV.
lence quarks are considered as a diquark state of spin-
0(scalar diquark) or spin-1 (vector diquark). Therefore
is a very simplistic model of the proton. It describes
the proton state |P ;S(cid:105) can be treated as a two par-
the proton by a scalar diquark and a quark and does
ticle state in the Fock-state expansion. In this paper
not assume the SU(4) symmetry of the usual diquark
we consider the scalar diquark model developed in [17,
modelswherebothscalarandaxialvectordiquarksare
18].Theaveragelight-frontmomentumofthescalardi-
quark is P = (cid:0)(1−x)P+,P−,−p (cid:1), where x is the considered.
X X ⊥
longitudinal momentum fraction carried by the struck
quark. 3 Wigner distribution
The two-particle fock-state expansion for Jz = ±1
2
are given by Inthelight-frontframework,the5-dimensionalWigner
distribution is defined as [22] :
|P;±(cid:105)=(cid:88)q (cid:90) 2(2πd)3x(cid:112)d2xp(1⊥−x) ρq[Γ](b⊥,p⊥,x;S)=(cid:90) d(22∆π)⊥2 e−i∆⊥.b⊥Wq[Γ](∆⊥,p⊥,x;S()4)
(cid:20)
1
ψ± (x,p )|+ ,0;xP+,p (cid:105) Where the correlator W[Γ] at at ∆+ = 0 and fixed
q+ ⊥ 2 ⊥
light-cone time z+ =0, is given by[10]:
(cid:21)
1
+ψ± (x,p )|− ,0;xP+,p (cid:105) , (1) 1(cid:90) dz− d2z
q− ⊥ 2 ⊥ Wq[Γ](∆ ,p ,x;S)= T eip.z
⊥ ⊥ 2 (2π)(2π)2
Wherethe|λ ,λ ;xP+,p (cid:105)representsa2-particlestate (cid:12)
withaquarkqofhselicityλq⊥,andadiquark(spectator)of (cid:104)P(cid:48)(cid:48);S|ψ¯q(−z/2)ΓW[−z/2,z/2]ψq(z/2)|P(cid:48);S(cid:105)(cid:12)(cid:12)(cid:12) (5)
z+=0
Wignerdistributionsandorbital angularmomentumofa proton 3
with the Dirac structure Γ e.g, γ+,γ+γ5. The P(cid:48) = These four distributions are related with the Fourier
(P+,P(cid:48)−,∆⊥) and the P(cid:48)(cid:48) =(P+,P(cid:48)(cid:48)−− ∆⊥) are the transforms of the GTMDs as:
2 2
initial and final momentum of proton. The W[Γ] de-
ρq (b ,p ,x)=Fq (x,0,p2,p .b ,b2), (15)
pends on the average momentum P = 1(P(cid:48)(cid:48) +P(cid:48)) of UU ⊥ ⊥ 1,1 ⊥ ⊥ ⊥ ⊥
2 1
proton, average quark momentum p⊥ = 21(p(cid:48)⊥(cid:48) +p(cid:48)⊥), ρqLU(b⊥,p⊥,x)=−M2(cid:15)i⊥jpi⊥
the proton helicity S and the transverse momentum
∂
tlirnaensWfe[r−zt/o2,tzh/2e] epnrosutornes∆t⊥he=gau(Pge⊥(cid:48)(cid:48)i−nvaPr⊥(cid:48)ia)n.cTehoef tWheilsoopn- ∂bj⊥F1q,4(x,0,p2⊥,p⊥.b⊥,b2⊥), (16)
erator.Wechoosethesymmetricframewherethecom- ρq (b ,p ,x)= 1 (cid:15)ijpi
ponents of 4-momentums, with skewness ξ =0, are UL ⊥ ⊥ M2 ⊥ ⊥
∂
P =[P+,P−,0 ], (6) Gq (x,0,p2,p .b ,b2), (17)
⊥ ∂bj 1,1 ⊥ ⊥ ⊥ ⊥
p=[xP+,p−,p ], (7) ⊥
⊥
ρq (b ,p ,x)=Gq (x,0,p2,p .b ,b2). (18)
∆=[0,0,∆ ], (8) LL ⊥ ⊥ 1,4 ⊥ ⊥ ⊥ ⊥
⊥
withP− = 1(P(cid:48)(cid:48)−+P(cid:48)−)= 4M2+∆2⊥ (weusethenota- Where the χq = F1q,1,F1q,4,G1q,1,G1q,4 can be expressed
2 4P+ as Fourier transform of corresponding GTMDs Xq =
tion v± =v0±v3). We calculate the matrix element of
Fq ,Fq ,Gq ,Gq .
Eq.(5)inscalardiquarkmodelusingthewavefunctions 1,1 1,4 1,1 1,4
predictedbysoft-wallAdS/QCD.TheWignerdistribu- (cid:90) d2∆
tions, with the proton helicity Λ and the quark helicity χq(x,0,p2⊥,p⊥.b⊥,b2⊥)= (2π)⊥2 e−i∆⊥.b⊥
λ, for unpolarized and longitudinally polarized proton Xq(x,0,p2,p .∆ ,∆2),(19)
⊥ ⊥ ⊥ ⊥
is defined as:
1 Integratingoverallthevariables,theWignerdistri-
ρq (b ,p ,x)= [ρq[γ+](b ,p ,x;ΛSˆ )+
Λλ ⊥ ⊥ 2 ⊥ ⊥ z butions give
λρq[γ+γ5](b⊥,p⊥,x;ΛSˆz)], (9) (cid:90) dxd2p d2b ρq (b ,p ,x)=n , (20)
⊥ ⊥ UU ⊥ ⊥ q
which can be decomposed as:
(cid:90)
ρq (b ,p ,x)= 1[ρq (b ,p ,x)+ dxd2p⊥d2b⊥ρqLU(b⊥,p⊥,x)=0, (21)
Λλ ⊥ ⊥ 2 UU ⊥ ⊥
(cid:90)
ΛρqLU(b⊥,p⊥,x)+λρqUL(b⊥,p⊥,x) dxd2p⊥d2b⊥ρqUL(b⊥,p⊥,x)=0, (22)
+Λλρq (b ,p ,x)]. (10)
LL ⊥ ⊥ (cid:90)
dxd2p d2b ρq (b ,p ,x)=∆q. (23)
corresponding to the proton spin Λ =↑,↓ and quark ⊥ ⊥ LL ⊥ ⊥
spin λ =↑,↓ (where ↑ and ↓ are corresponding to +1
Where the n is the flavor factors, n =2, n =1 and
and −1 respectively). Where the Wigner distribution q u d
the ∆q is the axial charge.
ρq (b ,p ,x) of unpolarized quarks in an unpolar-
UU ⊥ ⊥ Wigner distributions cannot have a direct proba-
ized proton, and the distortions ρq (b ,p ,x) due to
LU ⊥ ⊥ bilisticinterpretation,howeverintegratingovermomen-
unpolarizedquarksinalongitudinallypolarizedproton,
tum and position, Wigner distributions can be reduced
ρq (b ,p ,x) due to longitudinally polarized quarks
UL ⊥ ⊥ to probability distributions. Integrating over b with
in an unpolarized proton and ρq (b ,p ,x) due to ⊥
LL ⊥ ⊥ ∆ =0, the Wigner distributions reduce to the trans-
longitudinally polarized quark in a longitudinally po- ⊥
versemomentumdependentpartondistributions(TMDs).
larized proton, are defined as
At z = 0, the p integration of Wigner distributions
1 ⊥ ⊥
ρq (b ,p ,x)= [ρq[γ+](b ,p ,x;+Sˆ ) give generalized parton distributions(GPDs). The un-
UU ⊥ ⊥ 2 ⊥ ⊥ z polarized TMD fq(x,p2) and GPD Hq(x,0,∆2) can
+ρq[γ+](b⊥,p⊥,x;−Sˆz)], (11) be extracted as 1 ⊥ ⊥
1
ρqLU(b⊥,p⊥,x)= 2[ρq[γ+](b⊥,p⊥,x;+Sˆz) f1q(x,p2⊥)=F1q,1(x,0,p2⊥,0,0), (24)
(cid:90)
−ρq[γ+](b⊥,p⊥,x;−Sˆz)], (12) Hq(x,0,∆2⊥)= d2p⊥F1q,1(x,0,p2⊥,p⊥.∆⊥,∆2⊥),(25)
1
ρq (b ,p ,x)= [ρq[γ+γ5](b ,p ,x;+Sˆ )
UL ⊥ ⊥ 2 ⊥ ⊥ z and the TMD gq (x,p2) and GPD H˜q(x,0,∆2) can
1L ⊥ ⊥
+ρq[γ+γ5](b ,p ,x;−Sˆ )], (13) be expressed as:
⊥ ⊥ z
ρqLL(b⊥,p⊥,x)= 21[ρq[γ+γ5](b⊥,p⊥,x;+Sˆz) g1qL(x,p2⊥)=G(cid:90)q1,4(x,0,p2⊥,0,0), (26)
−ρq[γ+γ5](b⊥,p⊥,x;−Sˆz)]. (14) H˜q(x,0,∆2⊥)= d2p⊥Gq1,4(x,0,p2⊥,p⊥.∆⊥,∆2⊥)(.27)
4 D.Chakrabarti1,T.Maji1,C.Mondal1,A. Mukherjee2
The p and b integration of the ρq and ρq give From Eq.(15) we see that
⊥ ⊥ LU UL
zero.So,therearenoTMDandGPDcorrespondingto (cid:90)
F1,4 and G1,1 GTMDs. dxd2p⊥d2b⊥(b⊥×p⊥)zρqUU(b⊥,p⊥,x)=0, (35)
The Wigner distributions can also be reduced to
whichsatisfiestheangularmomentumsumruleforun-
three dimensional quark densities by integrating over
polarized proton, the total angular momentum of con-
two mutually orthogonal components of transverse po-
stituentssumuptozero.UsingEq.(16)andEq.(19),the
sitionandmomentum,e,g.b andp (b andp ),which
y x x y
twist-2 canonical quark OAM in the light-front gauge
are not constraint by Heisenberg uncertainty principle
is
as:
(cid:90) (cid:90) p2
dbydpxρq[Γ](b⊥,p⊥,x;S)=ρ˜q[Γ](bx,py,x;S), (28) (cid:96)qz =− dxd2p⊥M⊥2F1q,4(x,0,p2⊥,0,0). (36)
with∆y =zx =0.Notethattheintegrationoverother The Jaffe-Manohar decomposition ( Eq.(29)) is not
mixed transverse components bx and py gives the same gauge invariant. Ji proposed a gauge invariant decom-
quark density as Eq.(28), with a opposite momentum position of nucleon spin as[24]
i.e, ρ˜q[Γ](b ,p ,x;S) = ρ˜q[Γ](b ,−p ,x;S). These rela-
y x x y 1
tions are true only when there is axial symmtery i.e., Sq+Lq+Jg = , (37)
2
for unpolarized or longitudinally polarized proton.
where Lq is the kinetic OAM for the quark q. How-
ever, Chen et al.[25] proposed an idea to decompose
4 Orbital angular momentum thegaugefieldA intoapuregaugepart,Apure,anda
µ µ
physical part, Aphy to give a gauge invariant definition
µ
JaffeandManoharshowedinthelight-conegaugethat of the Jaffe-Manohar decomposition.
the spin of the nucleon can be decomposed into the The kinetic OAM of quark appearing in the Ji sum
quarkspin,quarkOAM,gluonspinandgluonOAM[23]. rule is defined in terms of GPDs as[24]:
Sq+(cid:96)q+Sg+(cid:96)g = 12. (29) Lqz = 12(cid:90) dx(cid:20)x(cid:18)Hq(x,0,0)+Eq(x,0,0)(cid:19)−H˜q(x,0,0)((cid:21)38,)
Forthediquarkmodel,theabovesumrulecanbewrit-
ten as where Hq(x,ξ,t) and Eq(x,ξ,t) are unpolarized GPDs
1 and H˜q(x,ξ,t) is the helicity dependent GPD. In our
Sq+(cid:96)q+SD+(cid:96)D = , (30)
2 model calculation, the explicit expressions are given in
where the super-script D is for diquark, and for scalar Sec.5. A comparative study between longitudinal com-
diquark SD = 0. The canonical OAM operator for ponent of canonical OAM and kinetic OAM are shown
quark is defined as in the Fig. 1 and the values are given in Table 2. Note
1(cid:90) dz−d2z that the above relation (Eq.38) does not hold for den-
(cid:96)ˆqz(b−,b⊥,p+,p⊥)= 2 (2π)3⊥eip.z sity level interpretation in the transverse plane [26].
The spin-orbit correlation is given by the operator
ψ¯q(b−− z2−,b⊥)γ+ (b⊥×(−i ↔∂⊥))ψq(b−+ z2−,b⊥). Cq(b−,b ,p+,p )= 1(cid:90) dz−d2z⊥eip.z
(31) z ⊥ ⊥ 2 (2π)3
FOrAomM tdheensditeyfinopiteiorantoorfcWanigbneerexoppreersasteodra(sEq.(5)), the ψ¯q(b−− z2−,b⊥)γ+γ5(b⊥×(−i ↔∂⊥))ψq(b−+ z2−,b⊥).
(39)
(cid:96)ˆq =2(b ×p )Wˆq[γ+]. (32)
z ⊥ ⊥
The correlation between quark spin and quark OAM
Thus in light-front gauge the average canonical OAM
can be expressed with Wigner distributions ρq and
for quark is written in terms of Wigner distribution as. UL
equivalently in terms of GTMD as:
(cid:90) d∆+d2∆
(cid:96)q = ⊥(cid:104)P(cid:48)(cid:48);S|(cid:96)ˆq|P(cid:48);S(cid:105) (cid:90)
z 2P+(2π)3 z Cq = dxd2p d2b (b ×p ) ρq (b ,p ,x)
z ⊥ ⊥ ⊥ ⊥ z UL ⊥ ⊥
(cid:90)
= dxd2p⊥d2b⊥(b⊥×p⊥)zρq[γ+](b⊥,p⊥,x,Sˆz)(.33) =(cid:90) dxd2p p2⊥Gq (x,0,p2,0,0). (40)
⊥M2 1,1 ⊥
Where,thedistributionρq[γ+](b ,p ,x,Sˆ )canbe
⊥ ⊥ z
Where Cq > 0 implies the quark spin and OAM tend
written from Eqs.(11,12) as: z
to be aligned and Cq <0 implies they are anti-aligned.
ρq[γ+](b⊥,p⊥,x,+Sˆz)=ρqUU(b⊥,p⊥,x) Inourmodel,thequzarkspinandOAMtendtobeanti-
+ρq (b ,p ,x) (34) aligned for both u and d quarks.
LU ⊥ ⊥
Wignerdistributionsandorbital angularmomentumofa proton 5
One can see from Eq.(18), a similar correlator with Where
ρqLL vanishes (cid:115)
(cid:90) dxd2p⊥d2b⊥(b⊥×p⊥)zρqLL(b⊥,p⊥,x)=0. (41) Aq(i)(x)=Nq(i)4κπ lo(1g(−1/xx))xa(qi)(1−x)b(qi), (50)
log(1/x)
a˜(x)= , (51)
2κ2(1−x)2
5 Results
∆2
p˜2 =p2 + ⊥(1−x)2. (52)
WecalculatetheWignerdistributionsofprotoninlight- ⊥ ⊥ 4
front AdS/QCD quark-diquark model. Using Eq.(1) in At the limit ξ =0, the GTMDs are
Eq.(5)thequark-quarkcorrelator,Wq[Γ](∆ ,p ,x;S),
⊥ ⊥
can be expressed in terms of LFWFs as. Fq (x,∆2,p2)= 1 (cid:20)|A(1)(x)|2+(cid:18)p2 − ∆2⊥(1−x)2(cid:19)
(cid:20) 1,1 ⊥ ⊥ 16π3 q ⊥ 4
1
Wq[γ+](∆ ,p ,x;±Sˆ )= ψ±†(x,p(cid:48)(cid:48))ψ± (x,p(cid:48) ) (cid:21)
⊥ ⊥ z 16π3 q+ ⊥ q+ ⊥ 1 |A(2)(x)|2 exp(cid:2)−2a˜(x)p˜2(cid:3), (53)
(cid:21) M2x2 q ⊥
+ψ±†(x,p(cid:48)(cid:48))ψ± (x,p(cid:48) ) ,(42) (cid:20) (cid:21)
q− ⊥ q− ⊥ 1 (1−x)
Fq (x,∆ ,p2)=− |A(2)(x)|2
(cid:20) 1,4 ⊥ ⊥ 16π3 x2 q
1
Wq[γ+γ5](∆⊥,p⊥,x;±Sˆz)= 16π3 ψq±+†(x,p(cid:48)⊥(cid:48))ψq±+(x,p(cid:48)⊥) exp(cid:2)−2a˜(x)p˜2⊥(cid:3), (54)
(cid:21) 1 (cid:20)(1−x) (cid:21)
−ψ±†(x,p(cid:48)(cid:48))ψ± (x,p(cid:48) ) .(43) Gq (x,∆2,p2)=− |A(2)(x)|2
q− ⊥ q− ⊥ 1,1 ⊥ ⊥ 16π3 x2 q
for the Dirac structures Γ =γ+,γ+γ5. In the symmet- exp(cid:2)−2a˜(x)p˜2⊥(cid:3), (55)
ricframetheinitialandfinalmomentumsofthestruck 1 (cid:20) (cid:18) ∆2 (cid:19)
Gq (x,∆ ,p2)= |A(1)(x)|2− p2 − ⊥(1−x)2
quark are 1,4 ⊥ ⊥ 16π3 q ⊥ 4
p(cid:48)⊥ =p⊥−(1−x)∆2⊥, (44) M12x2|A(q2)(x)|2(cid:21)exp(cid:2)−2a˜(x)p˜2⊥(cid:3). (56)
∆
p(cid:48)⊥(cid:48) =p⊥+(1−x) 2⊥ (45) We find Fq = Gq to the leading order as found in
1,4 1,1
respectively. Using the wave functions from Eq.(2,3) in [27] for scalar diquark model. Thus the distributions
Eqs.(42,43), the explicit expressions for Wigner distri- ρq =−ρq .
LU UL
butions are From Eq.(36), the canonical OAM can be written
1 (cid:90) d∆ as
ρq (b ,p ,x)= ⊥∆ J (|∆ ||b |)
UU ⊥ ⊥ 16π3 2π ⊥ 0 ⊥ ⊥ (cid:90)
(cid:20) (cid:96)q = dx(cid:96)q(x). (57)
exp(cid:0)−2a˜(x)p˜2(cid:1) |A(1)(x)|2+ z z
⊥ q
Using Eq.(54), (cid:96)q(x) can be written as
(cid:18) ∆2 (cid:19) 1 (cid:21) z
p2 − ⊥(1−x)2 |A(2)(x)|2 (,46)
⊥ 4 M2x2 q κ2
(cid:96)q(x)= Fq(x)(1−x)5. (58)
1 ∂ (cid:20) 1 (cid:90) d∆ z M2log(1/x) 2
ρq (b ,p ,x)=− (cid:15)ijpi − ⊥
LU ⊥ ⊥ M2 ⊥ ⊥∂bj 16π3 2π
⊥ In this model, (cid:96)q can also be related with pretzelosity
(cid:21) z
∆ J (|∆ ||b |)exp(cid:0)−2a˜(x)p˜2(cid:1)(1−x)|A(2)(x)|2(4,7) h⊥1T as
⊥ 0 ⊥ ⊥ ⊥ x2 q
ρqUL(b⊥,p⊥,x)= M12(cid:15)i⊥jpi⊥∂∂bj (cid:20)− 161π3 (cid:90) d2∆π⊥ (cid:96)qz =−(cid:90) dxd2p⊥2pM2⊥2(1−x)hq1⊥T(x,p2⊥). (59)
⊥
∆ J (|∆ ||b |)exp(cid:0)−2a˜(x)p˜2(cid:1)(1−x)|A(2)(x)|2(cid:21)(4,8) hq1⊥T(x,p2⊥) is one of the eight leading twist TMDs. In
⊥ 0 ⊥ ⊥ ⊥ x2 q thislight-frontscalardiquarkmodelhq⊥(x,p2)iswrit-
1T ⊥
1 (cid:90) d∆ ten as[21]
ρq (b ,p ,x)= ⊥∆ J (|∆ ||b |)
LL ⊥ ⊥ 16π3 2π ⊥ 0 ⊥ ⊥
2log(1/x) (cid:20) p2 log(1/x)(cid:21)
exp(cid:0)−2a˜(x)p˜2(cid:1)(cid:20)|A(1)(x)|2− hq1⊥T(x,p2⊥)=− πκ2 exp − κ⊥2(1−x)2 F2q(x()6.0)
⊥ q
(cid:18) ∆2 (cid:19) 1 (cid:21) whereFq(x)isgiveninEq(67).UsingEq.(53)andEq.(56)
p2⊥− 4⊥(1−x)2 M2x2|A(q2)(x)|2 (.49) in Eq.(225) and (27), the GPDs H and H˜ can be ex-
6 D.Chakrabarti1,T.Maji1,C.Mondal1,A. Mukherjee2
0.5 0.6
up down !q(x)
z
0.4
Lq(x)
z
0
0.2
M M
A A 0
O−0.5 O
rk rk−0.2
a a
u u
Q −1 Q−0.4
!q(x) −0.6
z
−1.5 Lqz(x) −0.8
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
(a) x (b) x
Fig. 1 The variation of canonical OAM (cid:96)q(x) and kinetic OAM Lq(x) with longitudinal momentum fraction x, for (a) u
z z
quarkand(b) dquark.
pressed as 5.1 Unpolarized proton
(cid:20) κ2 (cid:21)
Hq(x,0,t)= Fq(x)(1−x)2+Fq(x)(1−x)4 In our numerical study, we have considered the active
1 2 M2log(1/x)
quark to be either a u or d quark, the spectator al-
(cid:20) (cid:21)
|t| |t|
(1− log(1/x))exp − log(1/x) , (61)ways being a diquark. In other words, when we calcu-
4κ2 4κ2
late the Wigner distribution for the u quark, we have
H˜q(x,0,t)=(cid:20)Fq(x)(1−x)2−Fq(x)(1−x)4 κ2 (cid:21)not incorporated any contribution from the u quark
1 2 M2log(1/x) that is part of the diquark. The first Mellin moment of
|t| (cid:20) |t| (cid:21) ρq (b ,p ,x)isshowninFig.2.Fig.2(a)andFig.2(b)
(1+ log(1/x))exp − log(1/x) . (62) UU ⊥ ⊥
4κ2 4κ2 representthedistributionsintransversemomentumplane
for u quark and d quark respectively. The fixed impact
In the AdS/QCD light-front scalar diquark model the
parameter b is taken along yˆ and b = 0.4 fm. The
helicity flip GPD E is given[18] as ⊥ y
variation of ρq (b ,p ) in the transverse impact pa-
(cid:20) |t| (cid:21) UU ⊥ ⊥
Eq(x,0,t)=2Fq(x)(1−x)3exp − log(1/x) .(63) rameter plane are shown in Fig.2(c) and Fig.2(d) for
3 4κ2
u and d quark respectively, with fixed transverse mo-
Where Q2 = −q2 = −t, the square of the momentum mentum p along yˆ for p = 0.3 GeV. The distribu-
⊥ y
transferred in the process and is taken to be zero for tions ρu and ρd are circularly symmetric, in trans-
UU UU
OAM calculation. versemomentumplaneaswellastransverseimpactpa-
ThekineticOAMofquarks(Eq.(38))canbewritten rameter plane, with a positive maxima at the centre
as (p = p = 0), (b = b = 0) and gradually decrease
x y x y
(cid:90)
towards periphery, for both u and d quarks. The peak
Lq = dxLq(x). (64)
z z of the distribution for u quark is large compare to d
Where in this model, using Eqs.(61), (62) and Eq.(63) quark in both the planes.
at t=0 limit, the Lqz(x) reads The average quadrupole distortions Qibj(p⊥) and
1(cid:20) Qipj(b⊥) are defined as[10]
Lq(x)= −Fq(x)(1−x)3+Fq(x)(1−x)4(1+x)
z 2 1 2 (cid:82) d2b (2bi bj −δijb2)ρ (b ,p )
κ2 (cid:21) Qibj(p⊥)= ⊥(cid:82) d2⊥b⊥b2ρ (⊥b U,pU )⊥ ⊥ (69)
+2Fq(x)x(1−x)3 . (65) ⊥ ⊥ UU ⊥ ⊥
M2log(1/x) 3 (cid:82) d2p (2pi pj −δijp2)ρ (b ,p )
Where Qipj(b⊥)= ⊥(cid:82) d2⊥p⊥p2ρ (⊥b ,UpU )⊥ ⊥ . (70)
⊥ ⊥ UU ⊥ ⊥
F1q(x)=|Nq(1)|2x2a(q1)(1−x)2b(q1)−1, (66) Inthismodel,theaveragequadrupoledistortionisfound
F2q(x)=|Nq(2)|2x2a(q2)−2(1−x)2b(q2)−1, (67) tobezero.Sincethewavefunctionsinsoft-wallAdS/QCD
modelareofgaussiantype,theρ andρ areevenin
F3q(x)=Nq(1)Nq(2)xa(1)+a(q2)−1(1−x)b(q1)+b(q2)−1. (68) p⊥ and b⊥ resulting tothe zeroUqUuadrupoLlLe distortion.
ThevariationofthequarkOAMs(cid:96)q(x)andLq(x)with Aswediscussedbefore,thethreedimensionalquark
z z
longitudinal momentum fraction x is sown in Fig.1 for densities can be extracted from the Wigner distribu-
u and d quark. tions by integrating over one transverse momentum p
x
Wignerdistributionsandorbital angularmomentumofa proton 7
ρu [GeV−2.fm−2] ρd [GeV−2.fm−2]
UU UU
b =0.4fm 5.5 b =0.4fm 2.2
⊥ ⊥
0.4 5 0.4 2
4.5 1.8
0.2 4 0.2 1.6
] 3.5 ] 1.4
V V
e 3 e 1.2
G 0 G 0
[ 2.5 [ 1
y y
p 2 p 0.8
−0.2 −0.2
1.5 0.6
1 0.4
−0.4 0.5 −0.4 0.2
−0.4 −0.2 0 0.2 0.4 −0.4 −0.2 0 0.2 0.4
p [GeV] p [GeV]
(a) x (b) x
ρu [GeV−2.fm−2] ρd [GeV−2.fm−2]
UU UU
1 p =0.3GeV 2.5 1 p =0.3GeV 1.2
⊥ ⊥
1
0.5 2 0.5
0.8
m] 1.5 m]
[f 0 [f 0 0.6
y y
b 1 b
0.4
−0.5 −0.5
0.5
0.2
−1 −1
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
b [fm] b [fm]
(c) x (d) x
Fig. 2 TheWignerdistributionsofunpolarizedquarksinanunpolarizedprotoninthetransversemomentumplane(a,b)with
b = 0.4yˆ fm and in the transverse impact parameter plane(c,d) with p = 0.3yˆ GeV for u quarks(left column) and d
⊥ ⊥
quarks(rightcolumn).
ρ˜u [GeV−1.fm−1] ρ˜d [GeV−1.fm−1]
UU UU
5.5
0.4 5 0.4 1.8
4.5 1.6
0.2 4 0.2 1.4
V] 3.5 V] 1.2
e 3 e
G 0 G 0 1
[ 2.5 [
y y 0.8
p 2 p
−0.2 −0.2 0.6
1.5
1 0.4
−0.4 0.5 −0.4 0.2
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
b [fm] b [fm]
(a) x (b) x
Fig. 3 ρ˜q (b ,p )inmixed transverseplanefor uquark(a)anddquark(b).
UU x y
and one transverse position b variables(see Eq.(28)). positiveconstantsandweobserveα>β forbothuand
y
ρ˜ (b ,p )inmixedtransverseplaneareshowninFig.3 d quarks .
UU x y
for u and d quarks. We find that the distributions are
axiallysymmetric.Therefore,thereisnofavoredconfig- TheWignerdistributionsρqUL(b⊥,p⊥),inthetrans-
urationbetweenb ⊥p andb (cid:107)p unlikethelight- verse momentum plane, are shown in Fig.4(a) and (b)
⊥ ⊥ ⊥ ⊥
cone constituent quark model(LCCQM) [11] or chiral for u and d quarks respectively. The fixed transverse
quark soliton model(χQSM) [13]. At bx = py = 0, the impactparameterb⊥ isalongyˆwithby =0.4fm.The
probability density for u and d quark is maximum and Fig.4(c)and(d)representthedistributionρqUL(b⊥,p⊥)
decreases as e−αp2y and e−βb2x. Where the α and β are intransverseimpactparameterplane,foruanddquark
for p = p yˆ = 0.3 GeV. We observe a dipolar dis-
⊥ y
8 D.Chakrabarti1,T.Maji1,C.Mondal1,A. Mukherjee2
ρu [GeV−2.fm−2] ρd [GeV−2.fm−2]
UL UL
b =0.4fm 0.1 b =0.4fm
⊥ ⊥
0.4 0.4 0.3
0.05 0.2
0.2 0.2
0.1
] ]
V V
e 0 e 0
G 0 G 0
[ [ −0.1
y y
p p
−0.2 −0.05 −0.2 −0.2
−0.3
−0.1
−0.4 −0.4 −0.4
−0.4 −0.2 0 0.2 0.4 −0.4 −0.2 0 0.2 0.4
(a) px [GeV] (b) px [GeV]
ρu [GeV−2.fm−2] ρd [GeV−2.fm−2]
UL UL
1.2 1.2
p =0.3GeV 0.06 p =0.3GeV
⊥ ⊥ 0.2
0.8 0.04 0.8
0.1
0.4 0.02 0.4
m] m]
0 0
f 0 f 0
[ [
y −0.02 y
b b −0.1
−0.4 −0.4
−0.04
−0.2
−0.8 −0.06 −0.8
−1.2 −0.08 −1.2 −0.3
−1.2 −0.8 −0.4 0 0.4 0.8 1.2 −1.2 −0.8 −0.4 0 0.4 0.8 1.2
(c) bx [fm] (d) bx [fm]
Fig. 4 Thedipolarbehaviorofρq inthetransversemomentumplane(a,b)withb =0.4yˆfmandinthetransverseimpact
UL ⊥
parameterplane(c,d)with p =0.3yˆGeV foruquarks(leftcolumn)and dquarks(right column).
⊥
ρ˜u [GeV−1.fm−1] ρ˜d [GeV−1.fm−1]
UL UL
0.04
0.4 0.03 0.4
0.1
0.02
0.2 0.2 0.05
0.01
] ]
V V
e 0 e 0
G 0 G 0
[ −0.01 [
y y −0.05
p p
−0.02
−0.2 −0.2
−0.1
−0.03
−0.4 −0.04 −0.4 −0.15
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
b [fm] b [fm]
(a) x (b) x
Fig. 5 ρ˜q (b ,p )inmixed transverseplanecorrespondingtouquark(a)anddquark(b).
UL x y
tributions having same polarity for u and d quarks. 5.2 Longitudinally Polarized Proton
ρ˜q (b ,p ) in the transverse mixed plane are shown
UL x y
in Fig.5. We find a quadrupole distribution for u and d
quarks. Using Eq.(55) in Eq.(40) we calculate the Cq, The Wigner distributions ρqLU(b⊥,p⊥) are shown in
z
Fig.6foruanddquarks.Fig.6(a)and(b)showthevari-
the correlation between quark spin and quark OAM.
The values are: Cu = −0.0348 for u quark and Cd = ation of ρqLU(b⊥,p⊥) in transverse momentum plane
z z
for u and d quarks respectively with b is along yˆand
−0.1201fordquarks.Thereforeinthismodel,thequark ⊥
OAM tends to be anti-aligned(Cu < 0,Cd < 0) to by = 0.4 fm. The variation of ρqLU(b⊥,p⊥) in trans-
z z
verse impact parameter plane is shown in Fig.6(c) and
quark spin for both u and d quarks.
(d)withfixedp alongyˆ,p =0.3GeV.Wefinddipo-
⊥ y
lardistributionsforuanddquarks.Thepolarityofthe
Wignerdistributionsandorbital angularmomentumofa proton 9
ρu [GeV−2.fm−2] ρd [GeV−2.fm−2]
LU LU
b =0.4fm 0.1 b =0.4fm
0.4 ⊥ 0.4 ⊥ 0.3
0.05 0.2
0.2 0.2
0.1
] ]
V V
e 0 e 0
G 0 G 0
[ [ −0.1
y y
p p
−0.2 −0.05 −0.2 −0.2
−0.3
−0.1
−0.4 −0.4 −0.4
−0.4 −0.2 0 0.2 0.4 −0.4 −0.2 0 0.2 0.4
p [GeV] p [GeV]
(a) x (b) x
ρu [GeV−2.fm−2] ρd [GeV−2.fm−2]
LU LU
p =0.3GeV p =0.3GeV
1 ⊥ 0.06 1 ⊥ 0.2
0.04
0.5 0.5 0.1
0.02
m] m]
0 0
f 0 f 0
[ [
y −0.02 y
b b −0.1
−0.5 −0.04 −0.5
−0.2
−0.06
−1 −1
−0.08 −0.3
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
b [fm] b [fm]
(c) x (d) x
Fig. 6 Thedipolarbehaviorofρq inthetransversemomentumplane(a,b)withb =0.4yˆfmandinthetransverseimpact
LU ⊥
parameterplane(c,d)with p =0.3yˆGeV foruquarks(leftcolumn)and dquarks(right column).
⊥
ρ˜u [GeV−1.fm−1] ρ˜d [GeV−1.fm−1]
LU LU
0.04
0.4 0.03 0.4
0.1
0.02
0.2 0.2 0.05
0.01
] ]
V V
e 0 e 0
G 0 G 0
[ −0.01 [
y y −0.05
p p
−0.02
−0.2 −0.2
−0.1
−0.03
−0.4 −0.04 −0.4 −0.15
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
b [fm] b [fm]
(a) x (b) x
Fig. 7 Theρ˜q (b ,p )inmixed transverseplanecorrespondingtouquark(a)anddquark(b).
LU x y
dipolar distribution ρq is opposite to the polarity of q-OAM u d
ρq .ThemaximumvaLlUueofρq (b ,p )foruquarkis Canonical (cid:96)qz 0.0348 0.1201
UL LU ⊥ ⊥ Kinetic Lq -0.3812 -0.4258
z
less than that for d quarks in both the planes. Fig.7(a)
and (b) represent the distribution ρ˜q (b ,p ) in the Table 2 In the light-front AdS/QCD scalar diquark model,
LU x y thevaluesofcanonicalOAM(cid:96)qz andthekineticOAMLqz for
mixedtransverseplaneforuanddquarksrespectively. uanddquark.
We observe quadrupole distributions for both u and
d quarks. The quadrupole structures in ρq (b ,p )
LU ⊥ ⊥
and ρq (b ,p ) are found due to the presence of the
UL ⊥ ⊥
From Eqs.(57) and (64), we calculate the canonical
derivative terms in Eq.(16) and Eq.(17).
OAM and kinetic OAM of quarks in this model. The
values of quark OAM are given in Table.2. Note that
10 D.Chakrabarti1,T.Maji1,C.Mondal1,A. Mukherjee2
inquark-diquarkmodel,thetotalprotonOAMisgiven The Wigner distributions for longitudinally polar-
by the sum of quark and diquark angular momenta, so izedquarkinalongitudinallypolarizedproton,ρq (b ,p ),
LL ⊥ ⊥
unlikethequarkmodelsuanddquarkcontributionsdo areshowninFig.8.TheFig.8(a)and(b)representρq (b ,p )
LL ⊥ ⊥
notadduptothetotalprotonOAMandhencethesum intransversemomentumplanewithfixedb =0.4fmyˆ
⊥
of kinetic OAM of u and d in Table.2 is not the same and Fig.8(c) and (d) show the plots in the transverse
astotalcanonicalOAMoftheuandd.Thecorrelation impactparameterplanewithp =0.3GeV yˆ.Thedis-
⊥
between the canonical OAM of quark and proton spin tributions are circularly symmetric for u and d quarks
canbeunderstoodfromthesignofthe(cid:96)q.Inourmodel inboththeplanes.Thecircularsymmetryimpliesthat
z
calculation, the positive values of (cid:96)q for both u and d theρ cannotcontributetothequarkOAMasshown
z LL
implythattheprotonspintendstobealignedtoquark inEq.(41).Thepicksofthedistributionsareatthecen-
OAM for both u and d quarks. The spin contribution tre(0,0)inboththeplanes.Thereforethequarkpolar-
of the quark to the proton spin is given by[10] ization and the proton polarization tend to be parallel
for u and d quarks. Fig.9 represents the distribution
sq = 1∆q = 1(cid:90) dxH˜q(x,0,0) ρ˜qLL(bx,py) in a mixed transverse plane. The distribu-
2 2
tions are axially symmetric for both u and d quarks.
= 12(cid:90) dx d2p⊥Gq1,4(x,0,p2⊥,0,0) (71) andTFhieg.d1i1stwribituhtitohnespρoqΛlλa(rbiz⊥a,tipo⊥n)oafreprsohtoownnΛin=F↑iga.n1d0
quark polarization λ =↑,↓( Eq.(10)). Figs.10(a-d) rep-
where ∆q is the axial charge. In our model, we get
resent the variation of ρq (b ,p ) in the transverse
su = 0.946 and sd = 0.396. It is well known that Λλ ⊥ ⊥
momentum plane for u and d quarks. We observe a
thespectatordiquarkmodelhasitsownlimitations[28].
circular symmetry for Λ = λ but for Λ (cid:54)= λ the distri-
ThoughthefunctionalbehaviorsoftheGPDsandGT-
butionsgetdistortedalongp forbothuanddquarks.
MDsarewellreproducedinourmodel,theaxialcharges x
This is because, in Eq(9), the contributions from ρ
for both u and d quarks are over estimated. The model LU
andρ (ρ =−ρ )getcancelledforΛ=λ,whereas
is defined at a very low scale Q2 ≈0.09 GeV2. The ax- UL LU UL
0 for Λ (cid:54)= λ, the contributions add up and causes the
ial charge is scale dependent and known to be negative
distortion. We have shown the distributions for Λ =↑,
at larger scales. In [17], the authors have extended the
the other possible spin combinations in transverse mo-
result to an arbitrary scale Q2 and studied the evolu-
mentum plane can be found from ρq (b ,p ,p ) =
tionofunpolarizedpdfsinthismodel.Ourresultagrees ↓λ(cid:48) ⊥ x y
ρq (b ,−p ,p ), where λ(cid:48) (cid:54)= λ. Figs.10(e-h) show the
closely with theirs, in spite of the fact that the fit pa- ↑λ ⊥ x y
variation of ρq (b ,p ) in transverse impact param-
rameters are slightly different. In their model [29], the Λλ ⊥ ⊥
eter plane for u and d quarks. The distributions are
pdfs are slightly smaller in magnitude. When polarized
circularly symmetric in transverse impact parameter
pdfsorhelicitydistributionsarecomputed,thedquark
space for Λ = λ but the distributions get distorted
helicity distribution comes out to be positive, although
for Λ (cid:54)= λ, due to the same reason as described in
it is expected to be negative from the recent fit of the
caseoftransversemomentumplane.Similartothemo-
data[30].In[31],ithasbeenshownthatNNPDFallows
mentum space, the other possible spin combinations
forapositivetotal∆d(x)/d(x)(where∆d(x)standsfor
in the transverse impact parameter plane are found
helicity distribution)for larger values of x, this is also
as ρq (b ,b ,p ) = ρq (−b ,b ,p ), where λ(cid:48) (cid:54)= λ.
obtained in some other models, for example in [32], ↓λ(cid:48) x y ⊥ ↑λ x y ⊥
Themixedtransversedensitiesρ˜q (b ,p )areshownin
the above ratio was calculated in perturbative QCD Λλ x y
Fig.11 for u and d quarks. Again, for Λ=λ the contri-
taking into account the valence Fock components with
butionfromquadrupoledistortions(Fig.(5,7))ρ˜ and
non-vanishing orbital angular momentum and it was UL
ρ˜ get cancelled resulting the axial symmetry but for
found that ∆d(x)/d(x) is positive as x ≈ 0.75 and ap- LU
Λ (cid:54)= λ the contributions add up. The maxima of ρ˜
proaches 1 as x → 1. Positive values of this ratio was UU
and ρ˜ are nearly equal ( Figs. 3 and 9). As a result,
also found in an SU(6) breaking quark model calcu- LL
for Λ(cid:54)=λ, the destructive interference of these two dis-
lation in [33]. Another way to parametrize the model
tributionsgivealmostzeroatthecentre(b =0,p =0)
would be to fit the data of the helicity distributions x y
in Fig.11(c,d).
with the model parameters, instead of the form fac-
tors and the GPDs. Since in the scalar diquark model,
(cid:96)q +sq +(cid:96)D = 1/2 (as sD = 0), the diquark contribu- 6 Spin-Spin and Spin-OAM Correlation
z z
tiontothecanonicalOAMis(cid:96)D =−0.484forustruck-
z
quark and (cid:96) =−0.016 for d struck-quark. The contri- In Fig. 4 and Fig. 6, we observe that the quark OAM
z
butions of different partial waves to the quark OAM in tendstobeanti-alignedwithquarkspinandalignedto
LCCQM have been studied in [12]. theprotonspinforbothuanddquarks.Thecorrelation
