# Wigner distributions and GTMDs in a proton using light-front quark-diquark model

###### Abstract

We investigate the Wigner distributions and generalized transverse momentum-dependent distributions (GTMDs) for and 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 and quarks in the proton. We also study the relation of GTMDs with longitudinal momentum fraction carried by the active quark for different values of (skewness) which is defined as the longitudinal momentum transferred to the proton. Further, we study the GTMDs in the relation with for zero skewness at different values of quark transverse momentum as well as at different values of total momentum transferred to the proton .

## I Introduction

The major aim of the hadron physics is to expose the relationship between partons (basic degrees of freedom of QCD) and hadrons. The parton distribution functions (PDFs) provide the spread of the parton carrying a longitudinal momentum fraction in the hadron. On the other hand, the distribution located in the direction transverse to the motion of the hadron is explained through the generalized parton distributions (GPDs) gpd (); gpda (); gpdb () as a function of longitudinal momentum fraction carried by the parton, the longitudinal momentum transferred to the hadron and the total momentum transferred (, and respectively). Further, to describe the structure in momentum space, transverse momentum-dependent parton distributions (TMDs) were introduced tmda (); tmd (); tmd1 (); tmd2 (), which depend on the transverse momentum carried by the parton (). The GPDs and TMDs explain well the three-dimensional picture of internal structure of hadron, however, to understand the hadron structure more precisely, joint position and momentum distributions: Wigner distributions were instigated wigner (). Wigner distributions are quasi-probabilistic distributions which on application of certain limits provide the probabilistic distributions. Wigner distributions are associated with the generalized parton correlation functions (GPCFs) which when integrated over the light-cone energy of the parton reduce to the generalized transverse momentum-dependent parton distributions (GTMDs) gpcf (): mother distributions. After suitable integrations, GTMDs can further be reduced to GPDs and TMDs.

The GPDs can be experimentally obtained via hard exclusive processes namely deeply virtual Compton scattering (DVCS) dvcs (); dvcs1 (); dvcs2 (); dvcs3 (); dvcs4 (); dvcsa () where the interaction of the virtual photon with the parton of the nucleon leads to the radiation of a real photon from that parton and deeply virual meson production (DVMP) dvmpa (); dvmp (); dvmp1 () where the interaction of the virtual photon with the parton of the nucleon leads to the emission of light-vector meson from that parton. GPDs are also accessible via -meson photoproduction rho (); rhoa (), timelike Compton scattering tcs (), heavy charmonia photoproduction for the production of gluon GPDs hcp (); hcpa () and exclusive pion or photon-induced lepton pair-production epa (); epb (); ep (). The information on GPDs and nucleon structure can be extracted from the measurements of ongoing and upcoming experiments at Hall-A and Hall-B of JLab with CLAS collaboration jlaba (); jlabb (); jlabc (); jlab (); jlab1 (), J-PARC jparc (); jparca (); jparc1 () and COMPASS compass (). The TMDs are obtainable via semi-inclusive deep inelastic scattering (SIDIS) sidisa (); sidisb (); sidis () and Drell-Yan processes eetmd (); drell (). The SIDIS data is accessible from the upgraded experiments at JLab jlab2 (), electron ion collider (EIC) eic (), DESY etc. desy (). The rich data of Drell-Yan process is accessed via experiments at FNAL, BNL, J-PARC etc. etmd (); bnl ().

Even though Wigner distributions have been executed in many fields of physics like heavy ion collision, quantum information, quantum molecular dynamics, signal analysis, non-linear dynamics signal (); signal1 (); signal2 () and have been studied in some experiments exp (); exp1 () but no experiments have been done so far been done to extract Wigner distributions describing the multi-dimensional picture of the proton. Theoretical studies to understand Wigner distribution and GTMDs have however been attempted widely using light-cone spectator model spectator (), AdS/QCD quark-diquark model ads (); ads1 (); maji-AdS (), light-front dressed quark model dressed1 (); dressed (), light-cone constituent quark model, chiral soliton model constituent1 (); constituent (); soliton () etc.. Recently, Wigner distributions and GTMDs for electron have also been studied electron (). The spin-orbital angular momentum and spin-spin correlation between the polarized nucleon and quark can be determined by applying the phase space average to the Wigner distributions ads (); dressed (). Other possible versions of phase-space distributions are: Husimi distribution (smeared version of Wigner distribution) and Kirkwood distribution where the former one is real and positive definite and later is complex husimi (). It has been introduced in recent times that the gluon GTMDs can be accessible through diffractive di-jet production in deep-inelastic lepton-nucleon scattering exp_gtmd (); exp_gtmd1 (); exp_gtmd2 () and also in virtual photon-nucleus quasi-elastic scattering exp_gtmd3 (). It has also been identified that the GTMDs of gluons can be measured in proton-nucleus collisions exp_gtmd4 (). The quark GTMDs were recently measured by considering the exclusive double Drell-Yan process gtmd_DY ().

The dynamical front-form framework was introduced lc () to describe the constituent picture of hadron. A remarkable advantage of light-front dynamics is the simple light-front vacuum in QCD where the massive fluctuations are completely absent in the ground state. The absence of square root in the Hamiltonian simplifies the dynamical structure. The boost invariant light-front wavefunctions provide the inherent information about the structure of hadron lc1 (); lc2 (); lc3 (). One of the important model which finds application in non-perturbative regime of QCD is the light-front quark-diquark model tmd1 (). This model is a phenomenological approach to the work done in Ref. model (). In light-front quark-diquark model, the proton can be considered as a bound state of a quark and a diquark with a diquark spin to be 0 (scalar diquark) or 1 (axial-vector diquark). Using this model, all the T-even and T-odd TMDs of the proton are calculated using scalar and axial-vector diquarks and at nucleon-quark-diquark vertex where different choices of the form factors are considered tmd1 (). The standard parton distribution functions and quasi-parton distribution functions are successfully explained in Ref. quasi-pdf (). In this model, the proton wavefunction does not exhibit spin-isospin symmetry because of the non-vanishing relative orbital angular momentum of the quark-diquark system in its ground state. Using light-front quark-diquark model, GPDs, TMDs and spin transverse asymmetries for electron and hadron have already been evaluated tmd1 (); di-quark (); di-quark-nkumar (); bacchetta (). However, the Wigner distributions and GTMDs which provide the maximum information of internal structure of proton have not been evaluated so far.

Considering the above developments of the light-front quark-diquark model in studying internal structure of the hadrons through GPDs and TMDs, it becomes desirable to extend this model to investigate the Wigner distributions and GTMDs of the hadrons. In the present work, we have investigated the Wigner distributions in light-front quark-diquark model by considering different polarization configurations of quark and proton. We have also studied the GTMDs of quark contained by the proton for the case with different values of the longitudinal momentum transferred to the proton (non-zero skewness) as well as for the case where the longitudinal momentum transferred to the proton is zero (zero skewness). For the case with , we have further studied the variation of GTMDs with longitudinal momentum fraction . The implications of different values of quark transverse momentum and different values of the momentum transferred have also been discussed.

The paper is organized as follows. In Section II, we have given the essential details of the light-front framework and light-front quark-diquark model. The basic introduction to the Wigner distributions with the different polarization considerations and the explicit expressions of the quark Wigner distributions have been given in Section III. We have then presented the relation between the quark-quark correlators and the GTMDs and presented the analytical results for the 16 quark GTMDs for non-zero skewness in Section IV. In Section V, we have given the graphical interpretations to the Wigner distributions corresponding to the analytical results discussed in Section IV for the unpolarized proton, longitudinal-polarized proton and transversely-polarized proton. The results of the 16 quark GTMDs have also been presented in this section. Finally, the results have been summarized in Section VI.

## Ii Light-front quark-diquark model

### ii.1 General framework

In the light-cone frame lc2 (), we consider two light-like four-vectors which disintegrate a general four-vector to

(1) |

satisfying , . The transverse tensor can be expressed as

(2) |

with . The co-ordinates of a general four-vector are defined as

(3) |

For convenience, we take a frame where the four-momenta are defined as

(4) |

where , , and are the average momentum of hadron, momenta of active quark, momenta of diquark and the four-vector momentum transferred to the proton respectively. Here and are the initial and final momenta of the proton and is the mass of proton. The momentum transferred to the proton and momentum fraction carried by the active quark in longitudinal direction are denoted as (skewness) and respectively. The quark transverse momentum and diquark transverse momentum are expressed in terms of the relative transverse momentum of quark and proton transverse momentum as follows

(5) |

The total momentum transferred to the proton when the proton transverse momentum is not zero () can be expressed as

(6) |

### ii.2 Light-cone wave functions

The proton state can be defined as a superposition of the quark-diquark states and we have

(7) |

where , and are defined as scalar isoscalar diquark, vector isoscalar diquark and vector isovector states respectively.

For proton spin component, the two-particle Fock state expansion with spin-0 diquark can be expressed in terms of light-cone wave functions (LCWFs) with and denoting the helicities of proton and quark respectively. We have

(8) | |||||

The LCWFs emerging in above equation are defined as

(9) |

with

(10) |

where , and are masses of quark, proton and spin-0 diquark respectively and is the coupling constant.

The expansion of two-particle Fock state where corresponds to the flavor index , and describes the axial-vector diquark with isospin-0 or 1 in the frame defined in Eq. (4) for proton spin component with spin-1 diquark, can be expressed in terms of LCWFs . Here , and denote the helicities of proton, quark and axial-vector diquark respectively. We have

(11) | |||||

The LCWFs appearing in the above equation are further expressed as tmd1 ()

(12) |

with

(13) |

where , and are masses of quark, proton and spin-1 diquark respectively and is the coupling constant.

Similarly, for , the two particle Fock state expansion with spin-1 diquark can be written as

(14) | |||||

The LCWFs in this case are given as

(15) |

The nucleon tree-level cut amplitude is used for the calculation of quark-quark correlation function, where is considered as either scalar or axial-vector diquark. In the case of the other spectator diquark models used in literature, the proton wavefunction assumes an spin-isospin symmetry leading to the probabilistic weights of the scalar isoscalar (quark with scalar-diquark), vector isoscalar (quark with axial-vector diquark) and vector isovector (quark with axial-vector diquark) to be 3:1:2. The overall size of the couplings are balanced such that the total number of quarks become three. In the present work, because of the non-vanishing relative orbital angular momentum of the quark-diquark system in its ground state, the proton wave-function does not exhibit spin-isospin symmetry and the coefficients become 3 times smaller. Here the total number of quarks “seen” explicitly is only one, whereas the other two are hidden inside the diquark. It would be important to mention here that the total number of quarks in this case is also three. Also, the diquark which is not an elementary particle and is composed of two quarks can be probed by a photon.

## Iii Quark Wigner distribution

Wigner distributions of the quark relate to GTMDs through the quark-quark correlator or the Wigner operator as dressed (); soliton (); ads ()

(16) |

where the Wigner operator at fixed light-cone time is defined as

(17) |

Here and are initial and final momenta of proton state, is the spin of proton and refers to the specific Dirac -matrices , , , where = 1 or 2.

On integrating over the impact-parameter space , the Wigner distributions reduce to TMD correlators in the absence of total momentum transfer to the hadron ads (); soliton (). We have

(18) | |||||

On the other hand, the Wigner distributions reduce to two-dimensional Fourier transformations of GPD correlators , when integrated over the transverse momentum component . In the absence of light-front transverse co-ordinates i.e. at , we have

(19) |

with

(20) |

Further, the Wigner distribution reduce to three-dimensional quark densities by integrating over the two orthogonal directions of transverse plane, i.e. over and or and . At , we have

(21) |

and at , we have

(22) |

The combinations of different polarization configurations of the proton and the quark describe 16 independent twist-2 quark Wigner distributions ads1 (); electron (). We have the Wigner distribution for an unpolarized quark in the unpolarized proton as

(23) |

for a longitudinally-polarized quark in the unpolarized proton as

(24) |

for a transversely-polarized quark in the unpolarized proton as

(25) |

for an unpolarized quark in the longitudinally-polarized proton as

(26) |

for a longitudinally-polarized quark in the longitudinally-polarized proton as

(27) |

for a transversely-polarized quark in the longitudinally-polarized proton as

(28) |

for an unpolarized quark in the transversely-polarized proton as

(29) |

for a longitudinally-polarized quark in the transversely-polarized proton as

(30) |

for a transversely-polarized quark in the transversely-polarized proton as

(31) |

and finally the pretzelous Wigner distribution as

(32) |

The Wigner distributions for the scalar diquarks using Eqs. (16) and (17) are

(33) | |||||

(34) | |||||

(35) | |||||

(36) | |||||

(37) | |||||

(38) | |||||

(39) | |||||

(40) | |||||

(41) | |||||

(42) | |||||

For the axial-vector diquarks we have

(43) | |||||

(44) | |||||

(45) | |||||

(46) | |||||

(47) | |||||

(48) | |||||

(49) | |||||