The Wigner distributions for u and d quarks in a proton are calculated using the light front wave functions (LFWFs) of the scalar quark-diquark model for nucleon constructed from the soft-wall AdS/QCD correspondence. We present a detail study of the quark orbital angular momentum(OAM) and its correlation with quark spin and proton spin. The quark density distributions, considering the different polarizations of quarks and proton, in transverse momentum plane as well as in transverse impact parameter plane are presented for both u and d quarks.
It is shown that in both the gluonic and strange sea explanations of the Ellis-Jaffe sum rule violation discovered by the European Muon Collaboration (EMC), the spin of the proton, when viewed in in its rest reference frame, could by fully provided by quarks and antiquarks within a simple quark model picture, taken into account the relativistic effect from the Wigner rotation.
We present a recent calculation of the Wigner distributions of quarks and gluons in a perturbative model. We also present the results for the orbital angular momentum and the spin-orbit correlations.
Wigner distributions contain combined position and momentum space information of the quark distributions and are related to both generalized parton distributions (GPDs) and transverse momentum dependent parton distributions (TMDs). We report on a recent model calculation of the Wigner distributions for the quark and their relation to the orbital angular momentum.
We investigate the Wigner distributions and generalized transverse momentum-dependent distributions (GTMDs) for $u$ and $d$ quarks in the proton by using light-front quark-diquark model. We consider the contribution of scalar and axial-vector diquark having spin-0 and spin-1 respectively. We take different polarization configurations of quark and proton to calculate the Wigner distributions. The Wigner distributions are studied in the impact-parameter space, momentum space and mixed space for $u$ and $d$ quarks in the proton. We also study the relation of GTMDs with longitudinal momentum fract…
We show that the diffractive forward production of two quarkonia, especially the $χ_{cJ}$ and $η_c$ states, in proton-proton or proton-ion collisions can access the Weizsäcker-Williams gluon Wigner distribution of the proton. We use the hybrid factorization approach where the collinear, double gluon PDF is applied for one of the protons and the $k_T$-dependent (Wigner or GTMD) distribution for the other. The production of quarkonia is treated in the NRQCD framework. A particularly concise formula is obtained for double $χ_{J=1}$ production.
We calculate the Wigner functions for a quark target dressed with a gluon. These give a combined position and momentum space information of the quark distributions and are related to both generalized parton distributions (GPDs) and transverse momentum dependent parton distributions (TMDs). We calculate and compare the different definitions of quark orbital angular momentum in this model. We compare our results with other model calculations.
We investigate quark Wigner distributions in a light-cone spectator model. Both the scalar and the axial-vector spectators are included. The light-cone wave functions are derived from effective quark-spectator-nucleon vertex and then generalized by adjusting the power of energy denominators. The gauge link is taken into account by introducing relative phases to the light-cone amplitudes, and the phases are estimated from one gluon exchange interactions. The mixing distributions, which describe the correlation between transverse coordinate and transverse momentum and represent quark orbital mot…
We develop the concept of quantum phase-space (Wigner) distributions for quarks and gluons in the proton. To appreciate their physical content, we analyze the contraints from special relativity on the interpretation of elastic form factors, and examine the physics of the Feynman parton distributions in the proton's rest frame. We relate the quark Wigner functions to the transverse-momentum dependent parton distributions and generalized parton distributions, emphasizing the physical role of the skewness parameter. We show that the Wigner functions allow to visualize quantum quarks and gluons us…
We investigate the Wigner distributions for $u$ and $d$ quarks in a light-front quark-diquark model of a proton to unravel the spatial and spin structure. The light-front wave functions are modeled from the soft-wall AdS/QCD prediction. We consider the contributions from both the scalar and the axial vector diquarks. The Wigner distributions for unpolarized, longitudinally polarized, and transversely polarized protons are presented in the transverse momentum plane as well as in the transverse impact parameter plane. The Wigner distributions satisfy a Soffer-bound-type inequality. We also evalu…
We investigate the quark Wigner distributions in a light-cone spectator model. The Wigner distribution, as a quasi-distribution function, provides the most general one-parton information in a hadron. Combining the polarization configurations, unpolarized, longitudinal polarized or transversal polarized, of the quark and the proton, we can define 16 independent Wigner distributions at leading twist. We calculate all these Wigner distributions for the $u$ quark and the $d$ quark respectively. In our calculation, both the scalar and the axial-vector spectators are included, and the Melosh-Wigner …
We discuss the quark phase-space or Wigner distributions of the nucleon which combine in a single picture all the information contained in the generalized parton distributions and the transverse-momentum dependent parton distributions. In particular, we present results for the distribution of unpolarized quarks in a longitudinally polarized nucleon obtained in a light-front constituent quark model. We show how the quark orbital angular momentum can be extracted from the Wigner distributions and compare it with alternative definitions.
While the form factors and parton distributions provide separately the shape of the proton in coordinate and momentum spaces, a more powerful imaging of the proton structure can be obtained through phase-space distributions. Here we introduce the Wigner-type quark and gluon distributions which depict a full-3D proton at every fixed light-cone momentum, like what seen through momentum("color")-filters. After appropriate phase-space reductions, the Wigner distributions are related to the generalized parton distributions (GPD's) and transverse-momentum dependent parton distributions which are mea…
We present a calculation of Wigner distributions for gluons in light-front dressed quark model. We calculate the kinetic and canonical gluon orbital angular momentum and spin-orbit correlation of the gluons in this model.
The quark orbital angular momentum (OAM) has been recognized as an important piece of the proton spin puzzle. A lot of effort has been invested in trying to extract it quantitatively from the generalized parton distributions (GPDs) and the transverse-momentum dependent parton distributions (TMDs), which are accessed in high-energy processes and provide three-dimensional pictures of the nucleon. Recently, we have shown that it is more natural to access the quark OAM from the phase-space or Wigner distributions. We discuss the concept of Wigner distributions in the context of quantum field theor…
We calculate quark Wigner distributions using the light-front wave functions in a dressed quark model. In this model, a proton target is replaced by a simplified spin-1/2 state, namely a quark dressed with a gluon. We calculate the Wigner distributions for different polarization configuration of quark and the target state in this model.
We investigate the quark orbital angular momentum of the nucleon in the absence of gauge-field degrees of freedom, by using the concept of the Wigner distribution and the light-cone wave functions of the Fock state expansion of the nucleon. The quark orbital angular momentum is obtained from the phase-space average of the orbital angular momentum operator weighted with the Wigner distribution of unpolarized quarks in a longitudinally polarized nucleon. We also derive the light-cone wave function representation of the orbital angular momentum. In particular, we perform an expansion in the nucle…
We study the Wigner functions of the nucleon which provide multidimensional images of the quark distributions in phase space. These functions can be obtained through a Fourier transform in the transverse space of the generalized transverse-momentum dependent parton distributions. They depend on both the transverse position and the three-momentum of the quark relative to the nucleon, and therefore combine in a single picture all the information contained in the generalized parton distributions and the transverse-momentum dependent parton distributions. We focus the discussion on the distributio…
We propose a six-dimensional light-front Wigner distribution for the complete description of partonic structures of a hadron such as pion and proton, taking advantage of the recently proposed light-front variable $\tilde{z}$ by Miller and Brodsky. Quantities derived from the Wigner distribution contain the most general information of partonic structures, including also new quantities correlating longitudinal coordinate with transverse momenta or transverse coordinates, together with spins. The new Wigner distribution can be viewed as a relativistic version of the original Wigner distribution i…
We derive the consequences of the Myhrer-Thomas explanation of the proton spin problem for the distribution of orbital angular momentum on the valence and sea quarks. After QCD evolution these results are found to be in very good agreement with both recent lattice QCD calculations and the experimental constraints from Hermes and JLab.
Effect of the quark intrinsic motion on the proton spin structure functions is demonstrated. It is shown, that the covariant version of the quark-parton model taking into account the orbital motion gives the consistent picture of the proton spin structure, which is based on the valence quarks. This picture is supported by the recent data, which indicate, that the spin contributions from the sea quarks and gluons are compatible with zero.
The quasi-probabilistic Wigner distributions are the quantum mechanical analog of the classical phase-space distributions. We investigate quark Wigner distributions for a quark state dressed with a gluon, which can be thought of as a simple composite and relativistic spin-1/2 state with a gluonic degree of freedom. We calculate various polarization configurations, namely unpolarized, longitudinally polarized and transversely polarized quark and the target state using light-front wave functions in this model. At leading twist, one can define 16 quark Wigner distributions, however, we obtain onl…
Experimental processes that are sensitive to parton Wigner distributions provide a powerful tool to advance our understanding of proton structure. In this work, we compute gluon Wigner and Husimi distributions of protons within the Color Glass Condensate framework, which includes a spatially dependent McLerran-Venugopalan initial configuration and the explicit numerical solution of the JIMWLK equations. We determine the leading anisotropy of the Wigner and Husimi distributions as a function of the angle between impact parameter and transverse momentum. We study experimental signatures of these…
The orbital angular momentum ($L_q$) of the proton is studied by employing the extended constituent quark model. Contributions from different flavors, namely, up, down, strange, and charm quarks in the proton are investigated. Probabilities of the intrinsic $q\bar{q}$ pairs are calculated using a $^{3}P_{0}$ transition operator to fit the sea flavor asymmetry $I_a=\bar{d}-\bar{u}=0.118\pm0.012$ of the proton. Our numerical results lead to $L_q=0.158 \pm 0.014$, in agreement with $4/3I_a=0.157 \pm 0.016$, and consistent with findings based on various other approaches.
Covariant version of the quark-parton model is studied. Dependence of the structure functions on the 3D quark intrinsic motion is discussed. The important role of the quark orbital momentum, which is a particular case of intrinsic motion, appears as a direct consequence of the covariant description. Effect of orbital motion is substantial especially for polarized structure functions. At the same time, the procedure for obtaining the quark momentum distribution from the structure functions is suggested.
Since the announcement of the proton spin crisis by the European Muon Collaboration there has been considerable progress in unravelling the distribution of spin and orbital angular momentum within the proton. We review the current status of the problem, showing that not only have strong upper limits have been placed on the amount of polarized glue in the proton but that the experimental determination of the spin content has become much more precise. It is now clear that the origin of the discrepancy between experiment and the naive expectation of the fraction of spin carried by the quarks and …
A brief summary on the Wigner rotation effect in the understanding of the proton spin ``crisis" related with the Ellis-Jaffe sum rule violation, and on the proton-neutron isospin symmetry breaking explanation of the Gottfried sum rule violation. The proton spin could be fully provided by the spin sum of quarks if one considers the Wigner rotation effect in the measured quark helicity distributions.
We investigate the gluon Wigner distributions for unpolarized, longitudinally polarized and transversely polarized target state. Instead of a nucleon, we take the target state to be a quark dressed with a gluon at one loop and investigate the gluon Wigner distributions at leading twist. Better numerical convergence is obtained compared to an earlier study, that removes the regulator dependence of the results. We present a first calculation of the Wigner distribution for the transversely polarized target and linearly polarized gluon. We study the spin densities in momentum and impact parameter …
We study the covariant version of the quark-parton model, in which the general rules of the angular momentum composition are accurately taken into account. We demonstrate how these rules affect the relativistic interplay between the quark spins and orbital angular momenta, which collectively contribute to the proton spin. The spin structure functions $g_{1}$ and $g_{2}$ corresponding to the many-quark state $J=1/2$ are studied and it is shown they satisfy constraints and relations well compatible with the available experimental data including proton spin content $ΔΣ\lesssim1/3$. The suggested …
We present a systematic study of the proton spin structure in terms of measurable parton distributions. For a transversely-polarizedproton, we derive a polarization sum rule from the leading generalized parton distributions appearing in hard exclusive processes. For a longitudinally-polarized proton, we obtain a helicity decomposition from well-known quark and gluon helicity distributions and orbital angular-momentum contributions. The latter is shown to be related to measurable subleading generalized parton distributions and quantum-phase space Wigner distributions.
We examine the single-spin asymmetry (SSA) caused by the five-quark components of the proton for semi-inclusive electroproduction of charged pions in deep-inelastic scattering on a transversely polarized hydrogen target. The large SSA is considered to have close relation with quark orbital motion in the proton and suggests that the quark orbital angular momentum is nonzero. For the five-quark $qqqq\bar{q}$ components of the proton, the lowest configurations with $qqqq$ system orbitally excited and the $\bar{q}$ in the ground state would give spin-orbit correlations naturally for the quarks in …
We propose to constrain the gluon Wigner distribution in the nucleon by studying the exclusive diffractive dijet production process in ultraperipheral proton-nucleus collisions (UPCs) at RHIC and the LHC. Compared to the previous proposal in Ref. [Y. Hatta, B. W. Xiao, and F. Yuan, Phys. Rev. Lett. 116, 202301 (2016).] to study the same observable in lepton-nucleon scattering, the use of UPCs has a few advantages: not only is the cross section larger, but the extraction of the Wigner distribution from the data also becomes simpler, including its elliptic angular dependence. We compute the corr…
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 connec…
We study the three-quark light-cone amplitudes of the proton including quarks' transverse momenta. We classify these amplitudes using a newly-developed method in which light-cone wave functions are constructed from a class of light-cone matrix elements. We derive the constraints on the amplitudes from parity and time-reversal symmetries. We use the amplitudes to calculate the physical observables which vanish when the quark orbital angular momentum is absent. These include transverse-momentum dependent parton distributions $Δq_T(x, k_\perp)$, $q_T(x, k_\perp)$, $δq(x, k_\perp)$, and $δq_L(x,k_…
We study the Wigner functions of the nucleon which provide multidimensional images of the quark distributions in phase space and combine in a single picture all the information contained in the generalized parton distributions (GPDs) and the transverse-momentum dependent parton distributions (TMDs). In particular, we present results for the distribution of unpolarized quark in a longitudinally polarized nucleon. Treating this distribution as it was a classical distribution, we also obtain the expectation value of the orbital angular momentum operator and compare the corresponding results from …
Definitions of orbital angular momentum based on Wigner distributions are used as a framework to discuss the connection between the Ji definition of the quark orbital angular momentum and that of Jaffe and Manohar.We find that the difference between these two definitions can be interpreted as the change in the quark orbital angular momentum as it leaves the target in a DIS experiment. The mechanism responsible for that change is similar to the mechanism that causes transverse single-spin asymmetries in semi-inclusive deep-inelastic scattering.
We discuss two different definitions of partonic orbital angular momentum given in the literature in terms of the Fourier transform of a Wigner distribution, $F_{14}$, and a twist three generalized parton distribution, $\tilde{E}_{2T}$, respectively. We derive a relation between the two definitions which reflects their specific spin and intrinsic transverse momentum/transverse space correlations as well as their gauge link structure.
A striking prediction of several extensions of the constituent quark model, including the unquenched quark model, the pion cloud model and the chiral quark model, is a proportionality relationship between the quark sea asymmetry and the orbital angular momentum of the proton. We investigate to which extent a relationship of this kind is corroborated by the experiment, through a systematic comparison between expectations based on models and predictions obtained from a global analysis of hard-scattering data in perturbative Quantum Chromodynamics. We find that the data allows the angular momentu…
We study the orbital angular momentum contribution to the spin structure of the proton. It is well known that the quark and gluon spin contributions do not add up to the proton spin. We motivate the connection between the Generalized Transverse Momentum Distribution (GTMD) $F_{14}$, and orbital angular momentum by exploring the underlying quark proton helicity amplitude structure. The twist three Generalized Parton Distribution (GPD) $\tilde{E}_{2T}$, was shown to connect to OAM. We study these functions using a diquark model calculation. The GTMD $F_{14}$ is unique in that it can describe bot…
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 b…
Definitions of orbital angular momentum based on Wigner distributions are used as a framework to discuss the connection between the Ji definition of the quark orbital angular momentum and that of Jaffe and Manohar. We find that the difference between these two definitions can be interpreted as the change in the quark orbital angular momentum due to final state interactions as it leaves the target in a DIS experiment.
We calculate the leading-twist helicity-dependent generalized parton distributions (GPDs) of the proton at finite skewness in the Nambu--Jona-Lasinio (NJL) model of quantum chromodynamics (QCD). From these (and previously calculated helicity-independent GPDs) we obtain the spin decomposition of the proton, including predictions for quark intrinsic spin and orbital angular momentum. The inclusion of multiple species of diquarks is found to have a significant effect on the flavor decomposition, and resolving the internal structure of these dynamical diquark correlations proves essential for the …
We have recently examined the static properties of the baryon octet (magnetic moments and axial vector coupling constants) in a generalized quark model in which the angular momentum of a polarized nucleon is partly spin $\langle S_z \rangle$ and partly orbital $\langle L_z \rangle$. The orbital momentum was represented by the rotation of a flux-tube connecting the three constituent quarks. The best fit is obtained with $\langle S_z \rangle = 0.08\pm 0.15$, $\langle L_z \rangle = 0.42\pm 0.14$. We now consider the consequences of this idea for the $q^2$-dependence of the magnetic and axial vect…
We investigate the Generalized Transverse Momentum-dependent Distributions (GTMDs) describing the parton structure of proton using the light-front scalar-diquark model. In particular, we study the Wigner distributions for unpolarized quark in unpolarized, longitudinally-polarized and transversely-polarized proton. We also investigate the twist-2 GTMDs for unpolarized quark $(Γ=γ^+)$ in scalar-diquark model.
A new picture is given of generalized parton distributions probed in experiments in which the probe scale $Q^{2}$ and the momentum transfer $\DD^{2}$ are well separated. Application of this picture to the $Q^{2}$ dependence of the form factors $F_{1}, F_{2}$ shows that gauge invariant quark orbital angular momentum can be measured and indeed {\it localized} in the transverse profile of the proton. A previous prediction that $\sqrt{Q^{2}}F_{2}(Q^{2})/F_{1}(Q^{2}) \sim const. $ is generalized to GPD language. This prediction appears to have been confirmed by recent CEBAF data.
The extension of the statistical parton distributions to include their transverse momentum dependence (TMD) is revisited by considering that the proton target has a finite longitudinal momentum. The TMD will be generated by means of a transverse energy sum rule. The new results are mainly relevant for electron-proton inelastic collisions in the low $Q^2$ region. We take into account the effects of the Melosh-Wigner rotation for the helicity distributions.
We study the Wigner functions of the nucleon which provide multidimensional images of the quark distributions in phase space and combine in a single picture all the information contained in the generalized parton distributions (GPDs) and the transverse-momentum dependent parton distributions (TMDs). In particular, we present results for the distribution of unpolarized quarks in a longitudinally polarized nucleon obtained in a light-cone constituent quark model. Treating the Wigner distribution as it was a classical distribution, we obtain the quark orbital angular momentum and compare it with …
The quark orbital angular momentum component of proton spin, $L_q$, can be defined in QCD as the integral of a Wigner phase space distribution weighting the cross product of the quark's transverse position and momentum. It can also be independently defined from the operator product expansion for the off-forward Compton amplitude in terms of a twist-three generalized parton distribution. We provide an explicit link between the two definitions, connecting them through their dependence on partonic intrinsic transverse momentum. Connecting the definitions provides the key for correlating direct ex…
The orbital angular momenta $L^u$ and $L^d$ of up and down quarks in the proton are estimated as functions of the energy scale as model-independently as possible, on the basis of Ji's angular momentum sum rule. This analysis indicates that $L^u - L^d$ is large and negative even at low energy scale of nonperturbative QCD, in contrast to Thomas' similar analysis based on the refined cloudy bag model. We pursuit the origin of this apparent discrepancy and suggest that it may have a connection with the fundamental question of how to define quark orbital angular momenta in QCD.
Definitions of orbital angular momentum based on Wigner distributions are used to discuss the connection between the Ji definition of the quark orbital angular momentum and that of Jaffe and Manohar. The difference between these two definitions can be interpreted as the change in the quark orbital angular momentum as it leaves the target in a DIS experiment. The mechanism responsible for that change is similar to the mechanism that causes transverse single-spin asymmetries in semi-inclusive deep-inelastic scattering.
