Quark Orbital Angular Momentum Research Articles

Impressions of Parton Distribution Functions

Abstract Parton distribution functions (DFs) have long been recognised as key measures of hadron structure. Today, theoretical prediction of such DFs is becoming possible using diverse methods for continuum and lattice analyses of strong interaction (QCD) matrix elements. Recent developments include a demonstration that continuum and lattice analyses yield mutually consistent results for all pion DFs, with behaviour on the far valence domain of light-front momentum fraction that matches QCD expectations. Theory is also delivering an understanding of the distributions of proton mass and spin amongst its constituents, which varies, of course, with the resolving scale of the measuring probe. Aspects of the pion DF and proton spin developments are sketched herein, along with some novel perspectives on gluon and quark orbital angular momentum.

Probing the Quark Orbital Angular Momentum at Electron-Ion Colliders Using Exclusive π^{0} Production.

We propose to detect signals from quark orbital angular momentum (OAM) through exclusive π^{0} production in electron- (longitudinally polarized) proton collisions. Our analysis demonstrates that the sin2ϕ azimuthal angular correlation between the transverse momentum of the scattered electron and the recoil proton serves as a sensitive probe of quark OAM. Additionally, we present a numerical estimate of the asymmetry associated with this correlation for the kinematics accessible at the Electron-Ion Colliders in the U.S. and China. This study aims to pave the way for the first experimental study of quark OAM in relation to the Jaffe-Manohar spin sum rule.

Quark orbital angular momentum of ground-state octet baryons

Here we study the quark orbital angular momentum of the ground octet baryons employing an extended chiral constituent quark model, within which the baryon wave functions are taken to be superposition of the traditional $qqq$ and the $qqqq\overline{q}$ higher Fock components. Coupling between the two configurations is estimated using the $^{3}P_{0}$ quark-antiquark creation mechanism, and the corresponding coupling strength is determined by fitting the sea flavor asymmetry of the nucleon. The obtained numerical results show that the quark angular momentum of the nucleon, $\mathrm{\ensuremath{\Sigma}}, \mathrm{\ensuremath{\Lambda}}$, and $\mathrm{\ensuremath{\Xi}}$ hyperons, are in the range 0.10--0.30. In addition, the quark angular momentum of all the hyperons are a little bit smaller than that of the nucleon. And the octet baryons spin fractions taken by the intrinsic quark orbital angular momentum could be up to $60%$ in present model.

Proton momentum and angular momentum decompositions with overlap fermions

We present a calculation of the proton momentum and angular momentum decompositions using overlap fermions on a $2+1$-flavor RBC/UKQCD domain-wall lattice at 0.143 fm with a pion mass of 171 MeV which is close to the physical one. A complete determination of the momentum and angular momentum fractions carried by up, down, strange, and glue inside the proton has been done with valence pion masses varying from 171 to 391 MeV. We have utilized fast Fourier transform on the stochastic-sandwich method for connected-insertion parts and the cluster-decomposition error reduction technique technique for disconnected-insertion parts has been used to reduce statistical errors. The full nonperturbative renormalization and mixing between the quark and glue operators are carried out. The final results are normalized with the momentum and angular momentum sum rules and reported at the physical valence pion mass at $\overline{\mathrm{MS}}(\ensuremath{\mu}=2\text{ }\text{ }\mathrm{GeV})$. The renormalized momentum fractions for the quarks and glue are $⟨x{⟩}^{q}=0.491(20)(23)$ and $⟨x{⟩}^{g}=0.509(20)(23)$, respectively, and the renormalized total angular momentum fractions for quarks and glue are $2{J}^{q}=0.539(22)(44)$ and $2{J}^{g}=0.461(22)(44)$, respectively. The quark spin fraction is $\mathrm{\ensuremath{\Sigma}}=0.405(25)(37)$ from our previous work and the quark orbital angular momentum fraction is deduced from $2{L}^{q}=2{J}^{q}\ensuremath{-}\mathrm{\ensuremath{\Sigma}}$ to be 0.134(22)(44).

Near threshold heavy quarkonium photoproduction at large momentum transfer

Perturbative QCD is applied to investigate the near threshold heavy quarkonium photoproduction at large momentum transfer. We take into account the contributions from the leading three-quark Fock states of the nucleon. The dominant contribution comes from the three-quark Fock state with one unit quark orbital angular momentum (OAM) whereas that from zero quark OAM is suppressed at the threshold. From our analysis, we also show that there is no direct connection between the near threshold heavy quarkonium photoproduction and the gluonic gravitational form factors of the nucleon. Based on the comparison between our result and recent GlueX data of $J/\psi$ photoproduction, we make predictions for $\psi'$ and $\Upsilon$ (1S,2S) states which can be tested in future experiments.

Relations between generalized parton distributions and transverse momentum dependent parton distributions

We investigate the relations between transverse momentum dependent parton distributions (TMDs) and generalized parton distributions (GPDs) in a light-front quark-diquark model motivated by soft-wall AdS/QCD. Many relations are found to have similar structure in different models. It is found that a relation between the Sivers function and the GPD ${E}_{q}$ can be obtained in this model in terms of a lensing function. The quark orbital angular momentum is calculated and the results are compared with the results in other similar models. Implications of the results are discussed. Relations among different TMDs in the model are also presented.

From Ji to Jaffe-Manohar orbital angular momentum in lattice QCD using a direct derivative method

A Lattice QCD approach to quark orbital angular momentum in the proton based on generalized transverse momentum-dependent parton distributions (GTMDs) is enhanced methodologically by incorporating a direct derivative technique. This improvement removes a significant numerical bias that had been seen to afflict results of a previous study. In particular, the value obtained for Ji quark orbital angular momentum is reconciled with the one obtained independently via Ji's sum rule, validating the GMTD approach. Since GTMDs simultaneously contain information about the quark impact parameter and transverse momentum, they permit a direct evaluation of the cross product of the latter. They are defined through proton matrix elements of a quark bilocal operator containing a Wilson line; the choice in Wilson line path allows one to continuously interpolate from Ji to Jaffe-Manohar quark orbital angular momentum. The latter is seen to be significantly enhanced in magnitude compared to Ji quark orbital angular momentum, confirming previous results.

QCD evolution of the orbital angular momentum of quarks and gluons: Genuine twist-three part

We present the numerical solution of the one-loop QCD evolution equation for the genuine twist-three part of the orbital angular momentum (OAM) distributions of quarks and gluons inside a longitudinally polarized nucleon. This is based on the observation that the evolution is identical to that of the Efremov-Teryaev-Qiu-Sterman function for transverse single spin asymmetry. Together with the known evolution of the Wandzura-Wilczek part, the one-loop evolution of OAM distributions is now practically under control. We also study, for the first time, the scale dependence of the potential angular momentum defined as the difference between the Ji and Jaffe-Manohar definitions of OAM.

Lorentz invariance and QCD equation of motion relations for generalized parton distributions and the dynamical origin of proton orbital angular momentum

We derive new Lorentz invariance and equation of motion relations between twist-three generalized parton distributions (GPDs) and moments in the parton transverse momentum, ${k}_{T}$, of twist-two generalized transverse momentum-dependent distributions (GTMDs), as a function of the parton longitudinal momentum fraction $x$. Although GTMDs in principle define the observables for partonic orbital motion, experiments that can unambiguously detect them appear remote at present. The relations presented here provide a solution to this impasse in that, e.g., the orbital angular momentum density is connected to directly measurable twist-three GPDs. Out of 16 possible equation of motion relations that can be written in the $T$-even sector, we focus on three helicity configurations that can be detected analyzing specific spin asymmetries: two correspond to longitudinal proton polarization and are associated with quark orbital angular momentum and spin-orbit correlations; the third, obtained for transverse proton polarization, is a generalization of the relation obeyed by the ${g}_{2}$ structure function. We also exhibit an additional relation connecting the off-forward extension of the Sivers function to an off-forward Qiu-Sterman term.

Transition magnetic moments of decuplet to octet baryons in the chiral constituent quark model

In light of the developments of the chiral constituent quark model (χCQM) in studying low energy hadronic matrix elements of the ground-state baryons, we extend this model to investigate their transition properties. The magnetic moments of transitions from the decuplet to octet baryons are calculated with explicit valence quark spin, sea quark spin and sea quark orbital angular momentum contributions. Since the experimental data is available for only a few transitions, we compare our results with the results of other available models. The implications of other complicated effects such as chiral symmetry breaking and SU(3) symmetry breaking arising due to confinement of quarks are also discussed.

Twist-3 generalized parton distributions in deeply-virtual Compton scattering

The sub-leading power of the scattering amplitude for deeply-virtual Compton scattering (DVCS) off the nucleon contains leading-twist and twist-3 generalized parton distributions (GPDs). We point out that in DVCS, at twist-3 accuracy, one cannot address any individual twist-3 GPD. This complication appears on top of the deconvolution issues familiar from the twist-2 DVCS amplitude. Accessible are exclusively linear combinations involving both vector and axial-vector twist-3 GPDs. This implies, in particular, that the (kinetic) orbital angular momentum of quarks can hardly be constrained by twist-3 DVCS observables. Moreover, using the quark-target model, we find that twist-3 GPDs can be discontinuous. The discontinuities however cancel in the DVCS amplitude, which further supports the hypothesis of factorization at twist-3 accuracy.

Proton Spin Puzzle and Confinement of Quarks

It is shown that the confinement effect that is significant at great distances from the nucleon center is capable of inducing the appreciable quark helicity flip. A simultaneous change in projections of the orbital angular momentum of quarks on the direction of their motion, which follows from the Dirac equation, ensures the conservation of the total angular momentum of the system. Thus, a part of the proton spin, measured in experiments with deep inelastic scattering of leptons, may be hidden in the orbital angular momentum of quarks.

Valence-Quark Structure of N* Resonances from DSEs

We present a unified Quantum Chromodynamics (QCD)-based description of elastic and transition electromagnetic form factors involving the nucleon and its resonances. We compare predictions made using a framework built upon a Faddeev equation kernel and interaction vertices that possess QCD-like momentum dependence with results obtained using a confining, symmetry-preserving treatment of a vector$\,\otimes\,$vector contact-interaction in a widely-used leading-order (rainbow-ladder) truncation of QCD's Dyson-Schwinger equations. This comparison explains that the contact-interaction framework produces hard form factors, curtails some quark orbital angular momentum correlations within a baryon, and suppresses two-loop diagrams in the elastic and transition electromagnetic currents. Such defects are rectified in our QCD-based approach and, by contrasting the results obtained for the same observables in both theoretical schemes, shows those objects which are most sensitive to the momentum dependence of elementary quantities in QCD.

Feasibility Studies for Single Transverse-Spin Asymmetry Measurements at a Fixed-Target Experiment Using the LHC Proton and Lead Beams (AFTER@LHC)

The measurement of Single Transverse-Spin Asymmetries, A_N, for various quarkonium states and Drell–Yan lepton pairs can shed light on the orbital angular momentum of quarks and gluons, a fundamental ingredient of the proton-spin puzzle. The AFTER@LHC proposal combines a unique kinematic coverage and large luminosities thanks to the Large Hadron Collider beams to deliver precise measurements, complementary to the knowledge provided by collider experiments such as at RHIC. In this paper, we report on sensitivity studies for J/psi , varUpsilon and Drell–Yan A_N done using the performance of LHCb-like or ALICE-like detectors, combined with polarised gaseous hydrogen and helium-3 targets. In particular, such analyses will provide us with new insights and knowledge about transverse-momentum-dependent parton distribution functions for quarks and gluons and on twist-3 collinear matrix elements in the proton and the neutron.

Quark orbital dynamics in the proton from lattice QCD: From Ji to Jaffe-Manohar orbital angular momentum

Given a Wigner distribution simultaneously characterizing quark transverse positions and momenta in a proton, one can directly evaluate their cross-product, i.e., quark orbital angular momentum. The aforementioned distribution can be obtained by generalizing the proton matrix elements of quark bilocal operators which define transverse momentum-dependent parton distributions (TMDs); the transverse momentum information is supplemented with transverse position information by introducing an additional nonzero momentum transfer. A gauge connection between the quarks must be specified in the quark bilocal operators; the staple-shaped gauge link path used in TMD calculations yields the Jaffe-Manohar definition of orbital angular momentum, whereas a straight path yields the Ji definition. An exploratory lattice calculation, performed at the pion mass m_pi = 518 MeV, is presented which quasi-continuously interpolates between the two definitions and demonstrates that their difference can be clearly resolved. The resulting Ji orbital angular momentum is confronted with traditional evaluations based on Ji's sum rule. Jaffe-Manohar orbital angular momentum is enhanced in magnitude compared to its Ji counterpart.

GPDs and Orbital Angular Momentum

Generalized Parton Distributions provide information on the distribution of quarks in impact parameter space. For transversely polarized nucleons, these impact parameter distributions are transversely distorted and this deviation from axial symmetry leads on average to a net transverse force from the spectators on the active quark in a DIS experiment. This force when acting along the whole trajectory of the active quark leads to transverse single-spin asymmetries. For a longitudinally polarized nucleon target, the transverse force implies a torque acting on the quark Orbital Angular Momentum (OAM). The resulting change in OAM as the quark leaves the target equals the difference between the Jaffe–Manohar and Ji OAMs. In the context of the scalar diquark model we demonstrate that the effect from that torque contributes at the same order as single-spin asymmetries.

Contribution of orbital angular momentum to the nucleon spin

We have calculated the orbital angular momentum of quarks and gluons in the nucleon. The calculations are carried out in the next to leading order utilizing the so-called valon model. It is found that the average quark orbital angular momentum is positive, but small, and the average gluon orbital angular momentum is negative and large. We also report on some regularities about the total angular momentum of the quarks and the gluon, as well as on the orbital angular momentum of the separate partons. We have also provided partonic angular momentum, [Formula: see text] as a function of [Formula: see text].

Quark orbital angular momentum in the MIT bag model

We present the results for the Generalized Transverse Momentum Distribution related to quark Orbital Angular Momentum, {\it i.e.} $F_{14}$, in the MIT bag model. This model has been modified to include the Peierls--Yoccoz projection to restore translational invariance. Such a modification allows to fulfill more satisfactorily basic sum rules, that would otherwise be less elegantly carried out with the original version. Using the same model, we have calculated the twist-$3$ GPD that corresponds to Orbital Angular Momentum \`a la Ji, through the Penttinen--Polyakov--Shuvaev--Strikman sum rule. Recently, a new relation between the two definitions of the quark Orbital Angular Momentum at the density level has been proposed, which we illustrate here within the model. The sum rule is fulfilled. Still within the framework of the MIT bag model, we analyze the Wandzura--Wilczek expression for the GPD of interest. The genuine quark-gluon contribution is evaluated directly thanks to the equation of motion of the bag, which allows for a direct control of the kinematical contributions to the twist-3 GPD.

Baryons and chiral symmetry

The relevance of chiral symmetry in baryons is highlighted in three examples in the nucleon spectroscopy and structure. The first one is the importance of chiral dynamics in understanding the Roper resonance. The second one is the role of chiral symmetry in the lattice calculation of [Formula: see text] term and strangeness. The third one is the role of chiral [Formula: see text] anomaly in the anomalous Ward identity in evaluating the quark spin and the quark orbital angular momentum. Finally, the chiral effective theory for baryons is discussed.

Exclusive ω meson muoproduction on transversely polarised protons

Exclusive production of ω mesons was studied at the COMPASS experiment by scattering 160 GeV/c muons off transversely polarised protons. Five single-spin and three double-spin azimuthal asymmetries were measured in the range of photon virtuality 1 (GeV/c)2<Q2<10 (GeV/c)2, Bjorken scaling variable 0.003<xBj<0.3 and transverse momentum squared of the ω meson 0.05 (GeV/c)2<pT2<0.5 (GeV/c)2. The measured asymmetries are sensitive to the nucleon helicity-flip Generalised Parton Distributions (GPD) E that are related to the orbital angular momentum of quarks, the chiral-odd GPDs HT that are related to the transversity Parton Distribution Functions, and the sign of the πω transition form factor. The results are compared to recent calculations of a GPD-based model.