Distribution of Angular Momentum in the Transverse Plane
NNuclear Physics B Proceedings Supplement 00 (2018) 1–5
Nuclear Physics BProceedingsSupplement
Distribution of Angular Momentum in the Transverse Plane
L. Adhikari and M. Burkardt
Department of Physics, New Mexico State University, Las Cruces, NM 88003, USA
Abstract
Fourier transforms of GPDs describe the distribution of partons in the transverse plane. The 2nd moment of GPDs has beenidentified by X.Ji with the angular momentum (orbital plus spin) carried by the quarks - a fundamental result that is being widelyutilized in the spin decomposition of a longitudinally polarized nucleon. However, I will demonstrate that, despite the above results,the Fourier transform of the 2nd moment of GPDs does not describe the distribution of angular momentum in the transverse planefor a longitudinally polarized target.
Keywords:
GPDs, angular momentum
1. Introduction
The 2-dimensional Fourier transform of GeneralizedParton Distribution (GPD) H ( x , , t ) yields the distribu-tion of partons in the transverse plane for an unpolarizedtarget [1]. q ( x , (cid:126) b ⊥ ) = (cid:90) d (cid:126) ∆ ⊥ (2 π ) H ( x , , − (cid:126) ∆ ⊥ ) e − i (cid:126) b ⊥ · (cid:126) ∆ ⊥ (1)As a corollary, one finds that the distribution of chargein the transverse plane is given by the 2-dimensionalFourier transform of the Dirac form factor F ( t ) [2].GPDs can also be used to study the angular momen-tum carried by quarks of flavor q using the Ji-relation[3] J q = (cid:90) dx x (cid:104) H q ( x , ξ, + E q ( x , ξ, (cid:105) , (2)which requires GPDs extrapolated to momentum trans-fer t =
0. The observation that GPDs describe the dis-tribution of partons in the transverse plane led to theconjecture [5] that the Fourier transform of J q ( t ) ≡ (cid:90) dx x (cid:104) H q ( x , ξ, t ) + E q ( x , ξ, t ) (cid:105) (3) yields the distribution of angular momentum in posi-tion space. This suggested interpretation regarding thedistribution of angular momentum is frequently used inthe physics motivation for experiments as well as the 12GeV upgrade at Je ff erson Lab (see e.g. [6]).In this note, we will investigate whether such an in-terpretation is justified. For this purpose, we considerthe 2-dimensional Fourier transform of J q ( t ). AlthoughRef. [5] originally suggested taking a 3-dimensionalFourier transform, most experimental papers that quotethe idea that J q ( t ) can be used to understand the dis-tribution of angular momentum in the transverse planeconsider a 2-dimensional Fourier transform. If the 3-dimensional Fourier transform yields information aboutthe distribution in 3-dimensional space then by integrat-ing over the z coordinate one reduces the distribution tothe transverse plane. Hence, if one can demonstrate thatthe interpretation of the 2-dimensional Fourier trans-form as the distribution of angular momentum in thetransverse plane is flawed, then the interpretation of the3-dimensional Fourier transform must automatically beflawed as well.Using a scalar diquark model, we will calculatethe distribution of quark Orbital Angular Momentum(OAM) using two complementary approaches: in the1 a r X i v : . [ h e p - ph ] J u l Nuclear Physics B Proceedings Supplement 00 (2018) 1–5 first approach, we take the 2-dimensional Fourier trans-form of J q ( t ) calculated in this model. From that wesubtract the spin-distribution in the transverse planeevaluated from the same light-cone wave functions thatwere used to calculate the GPDs. In the second ap-proach we calculate the distribution of quark OAM asa function of the impact parameter also directly fromthe same light-cone wave functions used in the first ap-proach.We selected the scalar diquark model for this studynot because we think it is a good approximation forQCD, but to make a point of principle for which that factthat it is straightforward to maintain Lorentz invariancein this model is very important. Furthermore, since itis not a gauge theory, no issues arise as to whether oneshould include the vector potential in the definition ofOAM or in which gauge the calculation should be done,i.e. there is no di ff erence between Ji’s OAM (2) and thatof Ja ff e and Manohar [4].
2. Distribution of Angular Momentum in the Trans-verse Plane
Following Ref. [5], we define ρ J ( (cid:126) b ⊥ ) = (cid:90) d (cid:126) ∆ ⊥ (2 π ) e − i (cid:126) ∆ ⊥ · (cid:126) b ⊥ J q ( − (cid:126) ∆ ⊥ ) , (4)where J q ( − (cid:126) ∆ ⊥ ) ≡ (cid:90) dxx [ H q ( x , ξ, − (cid:126) ∆ ⊥ ) + E q ( x , ξ, − (cid:126) ∆ ⊥ )] ≡
12 [ A q ( − (cid:126) ∆ ⊥ ) + B q ( − (cid:126) ∆ ⊥ )] . (5)The main goal of this work is to investigate whether it isjustified to interpret ρ J ( (cid:126) b ⊥ ) as the distribution of angularmomentum in the transverse plane.Calculating the relevant GPDs is straightforward us-ing the light-cone wave functions [7] for the scalar di-quark model ψ ↑ + (cid:16) x ,(cid:126) k ⊥ (cid:17) = (cid:18) M + mx (cid:19) φ ( x ,(cid:126) k ⊥ ) (6) ψ ↑− ( x ,(cid:126) k ⊥ ) = − k + ik x φ ( x ,(cid:126) k ⊥ ) ψ ↓ + ( x ,(cid:126) k ⊥ ) = k + ik x φ ( x ,(cid:126) k ⊥ ) ,ψ ↓− ( x ,(cid:126) k ⊥ ) = ( M + mx ) φ ( x ,(cid:126) k ⊥ )with φ ( x ,(cid:126) k ⊥ ) = g / √ − xM − (cid:126) k ⊥ + m x − (cid:126) k ⊥ + λ − x . Here g is theYukawa coupling and M / m / λ are the masses of the ‘nucleon’ / ‘quark’ / diquark respectively. Furthermore x is the momentum fraction carried by the quark and (cid:126) k ⊥ ≡ (cid:126) k ⊥ e − (cid:126) k ⊥ γ represents the relative ⊥ momentum. Theupper wave function index ↑ refers to the helicity of the‘nucleon’ and the lower index to that of the quark.For the generalized form factors needed to evaluate(5) one finds [7] A q ( − (cid:126) ∆ ⊥ ) = (cid:90) dx xH q ( x , , − (cid:126) ∆ ⊥ ) (7)where H q ( x , , − (cid:126) ∆ ⊥ ) = (cid:90) d (cid:126) k ⊥ π (cid:20) ψ ↑ ∗ + ( x ,(cid:126) k (cid:48)⊥ ) ψ ↑ + ( x ,(cid:126) k ⊥ ) + ψ ↑ ∗− ( x ,(cid:126) k (cid:48)⊥ ) ψ ↑− ( x ,(cid:126) k ⊥ ) (cid:21) (8)where (cid:126) k (cid:48)⊥ = (cid:126) k ⊥ + (1 − x ) (cid:126) ∆ ⊥ as well as B q ( − (cid:126) ∆ ⊥ ) = (cid:90) dx xE ( x , , − (cid:126) ∆ ⊥ ) (9) E q ( x , , − (cid:126) ∆ ⊥ ) = − M ∆ − i ∆ (cid:90) d (cid:126) k ⊥ π (cid:20) ψ ↑ ∗ + ( x ,(cid:126) k (cid:48)⊥ ) ψ ↓ + ( x ,(cid:126) k ⊥ ) + ψ ↑ ∗− ( x ,(cid:126) k (cid:48)⊥ ) ψ ↓− ( x ,(cid:126) k ⊥ ) (cid:21) . (10)From these GPDs one can determine the OAM as ob-tained from GPDs through the Ji relation (2) as L q = (cid:90) dx (cid:104) xH q ( x , , + xE ( x , , − ∆ q ( x ) (cid:105) , (11)where ∆ q ( x ) = (cid:90) d (cid:126) k ⊥ π (cid:34)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ↑ + ( x ,(cid:126) k ⊥ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ↑− ( x ,(cid:126) k ⊥ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:35) . (12)Since some of the above (cid:126) k ⊥ -integrals diverge, a man-ifestly Lorentz invariant Pauli-Villars regularization(subtraction with heavy scalar λ → Λ ) is always un-derstood.To evalulate relation (4), we simplify and rewrite (8)and (10) as: H ( x , , − (cid:126) ∆ ⊥ ) = g π (cid:90) d (cid:126) k ⊥ (cid:34)(cid:90) d α (1 − x )( m + xM ) [( (cid:126) k ⊥ + (1 − x ) (cid:126) ∆ ⊥ α ) + F ] + − x (cid:126) k (cid:48) ⊥ + u ) + − x (cid:126) k ⊥ + u ) − (cid:90) d α (1 − x )( u + (1 − x ) (cid:126) ∆ ⊥ )(( (cid:126) k ⊥ + (1 − x ) (cid:126) ∆ ⊥ α ) + F ) (cid:35) (13)2 Nuclear Physics B Proceedings Supplement 00 (2018) 1–5 where u = x − x + + x λ and F = (1 − x ) (cid:126) ∆ ⊥ α (1 − α ) + x − x + + x λ Similarly, E ( x , , − (cid:126) ∆ ⊥ ) = (14) Mg π (cid:90) d α − xx ( m + xM ) α (1 − α ) − xx ∆ ⊥ − M + m x + λ − x In order to describe distributions in impact parameterspace, we introduce wave functions in impact parameterspace as [9] ψ s ( x , (cid:126) b ⊥ ) ≡ π (1 − x ) (cid:90) d (cid:126) k ⊥ e i (cid:126) k ⊥· (cid:126) b ⊥ − x ψ s ( x ,(cid:126) k ⊥ ) (15)where calculating suitable prefactor π (1 − x ) is straight-forward using the following relation: (cid:90) | ψ s ( x , (cid:126) b ⊥ ) | d b ⊥ = (cid:90) | ψ s ( x ,(cid:126) k ⊥ ) | d k ⊥ . (16)Note the factor − x in the exponent which accountsfor the fact that the variable (cid:126) k ⊥ is conjugate to the dis-placement between the active quark and the spectator,while the impact parameter (cid:126) b represents the displace-ment of the active quark from the center of momentumof the entire hadron. Using these wave functions, it isstraightforward to evaluate the quark spin distribution inthe transverse plane for a longitudinally polarized ’nu-cleon’ as ρ S ( (cid:126) b ⊥ ) = (cid:90) dx (cid:34)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ↑ + ( x , (cid:126) b ⊥ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ↑− ( x , (cid:126) b ⊥ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:35) . (17)where (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ↑ + ( x , (cid:126) b ⊥ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = g (1 + x ) π (1 − x ) (cid:34) (cid:90) ∞ dk ⊥ k ⊥ J ( | (cid:126) k ⊥ · (cid:126) b ⊥ − x | )( (cid:126) k ⊥ + u ) (cid:35) (18)and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ↑− ( x , (cid:126) b ⊥ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = g π − x (cid:34)(cid:90) ∞ dk ⊥ k ⊥ J ( | (cid:126) k ⊥ · (cid:126) b ⊥ − x | )( (cid:126) k ⊥ + u ) (cid:35) (19)If (4) can be interpreted as the angular momentumdensity then the di ff erence L q ( (cid:126) b ⊥ ) ≡ ρ J ( (cid:126) b ⊥ ) − ρ S ( (cid:126) b ⊥ ) (20)represents the orbital angular momentum density. In thefollowing section, we will investigate if that is the case.
3. Impact Parameter Space Distribution Directlyfrom Light Front Wave Functions
With the light-cone wave functions available (6), itis also straightforward to compute the orbital angularmomentum L zq of the ‘quark’ for a ’nucleon’ polarizedin the ˆ z direction directly as [8] L q = (cid:90) dx (cid:90) d (cid:126) k ⊥ π (1 − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ↑− ( x ,(cid:126) k ⊥ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (21)Evaluating the above integrals is tedious, but straight-forward, and one finds [8] L zq = L zq (22)as was expected since L zq in the scalar diquark modeldoes not contain a vector potential and therefore nogauge related issues arise (in QED for an electron L q (cid:44) L q [8]).Likewise, one can define the orbital angular momen-tum density directly using light-cone wave functions(15) as L z and b ≡ | (cid:126) b ⊥ | can be simultaneously measured.For a nucleon with spin up, only the wave function com-ponent ψ ↑− has one unit or orbital angular momentumshared between the active quark (weight factor 1 − x )and the spectator (weight factor x ) [8] and therefore L q ( b ) = (cid:90) dx (1 − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ↑− ( x , (cid:126) b ⊥ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (23)represents the orbital angular momentum density forthe active quark as a function of the distance from thecenter of momentum in a ’nucleon’ that is polarized inthe + ˆ z direction.Evaluating integrals available (20) and (23) are tid-ious but straight forward. Use of manifestly Lorentzinvariant Pauli-Villars regularization (subtraction withheavy scalar λ → Λ ) is easily understood to isolate thedivergence piece for some of k ⊥ integrals. Both L q ( b )and L q ( (cid:126) b ⊥ ) are shown in Fig. 1 and it is clear that thearea under the curve is the only feature that these twodistributions have in common. (cid:90) ∞ db b L q ( b ) = (cid:90) ∞ db b L q ( b ) (24)With the relations available (21),(11) ,and (20) ,it is alsostraight forward to show (cid:90) d (cid:126) b ⊥ L q ( (cid:126) b ⊥ ) = L q = L q = π (cid:90) ∞ db b L q ( b ) . (25)This result clearly demonstrates that L q ( (cid:126) b ⊥ ) does notrepresent the distribution of angular momentum for a3 Nuclear Physics B Proceedings Supplement 00 (2018) 1–5 L z distribution L q ( b ) (23) for a nu-cleon polarized in the + ˆ z direction, dotted line L q ( (cid:126) b ⊥ ) (20) obtainedfrom the Fourier transform of the Ji-relation at nonzero momentumtransfer ρ J ( (cid:126) b ⊥ ) after subtracting the spin distribution. longitudinally polarized target, since L q ( b ) already hasthat interpretation. As a corollary, we also concludethat the Fourier transform of J q ( t ) (3) does not repre-sent the distribution of angular momentum either - re-gardless whether the Fourier transform is two- or three-dimensional. These observations represent the main re-sult of this work.
4. Discussion
We have demonstrated within the context of a scalarYukawa diquark model that although J q ( t ) yields, in thelimit t →
0, the ˆ z component of the quark angular mo-mentum for a target polarized in the + ˆ z direction, the 2-dimensional Fourier transform of its t -dependence doesnot yield the distribution of angular momentum in im-pact parameter space.This result is best understood by recalling thatLorentz / rotational invariance is heavily used when a re-lation between the angular momentum operator, whichis not only leading twist, and twist-2 GPDs. The useof Lorentz invariance appears implicitly in the originalpaper [3], where it imposes constraints on the allowedtensor structure. In Ref. [10], Eq. (2) was rederived byconsidering the transverse deformation of parton distri-butions in a transversely polarized target. In this ap-proach, the momentum density in the ˆ z direction wascorrelated with the distribution in the transverse direc-tion for a transversely polarized target (see also Ref. [11]). While T z x comprises only half the angular mo-mentum tensor T z x − T x z , the two terms in the lat-ter turn out to yield identical contributions - providedthe target is invariant under rotations about the ˆ y axis.Therefore, as long as one considers a target with rota-tional symmetry about the ˆ y axis, one can identify theangular momentum in the ˆ y direction with the expecta-tion value of 2 T z x , which in turn can be identified witho ff forward matrix elements of the twist two operator T ++ (2). Finally, as long as considering the However, asrotational invariance has been heavily used in this pro-cess, the resulting relation (2) should hold for any com-ponent of the quark angular momentum for a nucleonpolarized in the corresponding direction. Hence one canrelate the quark angular momentum in the ˆ z direction,although it is not a priory twist-2, to matrix elements oftwist-2 operators.Our explicit calculation has shown that the Fouriertransform of J q ( t ) does not yield the distribution of an-gular momentum in the transverse plane for a longitu-dinally polarized target. However, from the discussionabove it should also be clear that it cannot be interpretedas the distribution of transverse angular momentum in atransversely polarized target: the Fourier transform of J q ( t ) yields the distribution of xT ++ . Using rotationalsymmetry arguments, that are applicable only after in-tegration over the position, that can be related to the ma-trix element of xT z and hence also of − zT x . However,this is not possible for the local (unintegrated) densities. Acknowledgements:
This work was supported bythe DOE under grant number DE-FG03-95ER40965.LA is very grateful for the generous support from theGary McCartor memorial fund, which enabled him toparticipate LC2012 in Delhi, where this work was pre-sented.
Appendix A. Di ff erent types of integrals used (cid:90) d (cid:126) k ⊥ (cid:126) k ⊥ + u ( λ ) = −−−−−−−−−−−−−−−−→ using subtraction = π log (cid:34) u ( ∧ = u ( λ = (cid:35) (A.1) (cid:90) d (cid:126) k ⊥ (cid:126) k ⊥ + F ( λ ) ) = −−−−−−−−−−−−→ using subtcts . = π (cid:34) F ( λ = − F ( ∧ = (cid:35) (A.2) J ( | (cid:126) k ⊥ · (cid:126) b ⊥ | ) = π (cid:90) d φ e i (cid:126) k ⊥ · (cid:126) b ⊥ (A.3) (cid:90) d (cid:126) b ⊥ e i ( (cid:126) k ⊥− (cid:126) k (cid:48)⊥ ) · (cid:126) b ⊥ − x = (2 π ) (1 − x ) δ ( (cid:126) k ⊥ − (cid:126) k (cid:48)⊥ )(A.4)4 Nuclear Physics B Proceedings Supplement 00 (2018) 1–5 Appendix B. Part of calculation for relation (25) (cid:90) d (cid:126) b ⊥ L q ( (cid:126) b ⊥ ) = (cid:90) d b (cid:126)ρ J ( b ⊥ ) − (cid:90) d b (cid:126)ρ S ( (cid:126) b ⊥ )(B.1) (cid:90) d (cid:126) b ⊥ (cid:126)ρ J ( b ⊥ ) ≡ J q (0) =
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