T-odd generalized and quasi transverse momentum dependent parton distribution in a scalar spectator model

Generalized transverse momentum dependent parton distributions (GTMDs), as mother funtions of transverse momentum dependent parton distributions (TMDs) and generalized parton distributions (GPDs), encode the most general parton structure of hadrons. We calculate four twist-two time reversal odd GTMDs of pion in a scalar spectator model. We study the dependence of GTMDs on the longitudinal momentum fraction $x$ carried by the active quark and the transverse momentum $|\vec k_T|$ for different values of skewness $\xi$ defined as the longitudinal momentum transferred to the proton as well as the total momentum $|\vec\Delta_T|$ transferred to the proton. In addition, the quasi-TMDs and quasi-GPDs of pion have also been investigated in this paper.


I. INTRODUCTION
Exploring the partonic substructure of hadrons is still at the frontier of hadronic high energy physics research. The parton distribution functions (PDFs) make it clear how the longitudinal parton momentums in hadron are distributed. However, they only include one dimension information. Therefore probing how partons distribute in the transverse plane in both momentum and coordinate space becomes a vital topic. Typically, the transverse spatial distribution of partons inside a hadron can be quantified by Generalised parton distributions (GPDs) [1][2][3], which can be denoted as a function of longitudinal momentum fraction x carried by the parton, the longitudinal momentum ξ transferred to the hadron and the total momentum transferred t. They can be accessed through measurements in hard exclusive reactions like deep virtual Compton scattering and hard exclusive meson production [2,4]. While the transverse momentum dependent parton distributions (TMDs) [5][6][7] depending on both the longitudinal and transverse motion of partons inside a hadron can be studied by the description of various hard semi-inclusive reactions. More general distibutions than the GPDs and the TMDs, the generalized transverse momentum dependent parton distributions (GTMDs) [8][9][10] could reduce to them in specific kinematical limits, therefore serve as mother distributions. The GTMDs can directly enter the description of hard exclusive reactions. The parametrization of the generalized quarkquark correlation functions for a spin-0 and spin-1/2 hadron in terms of GTMDs are given in Refs. [8,9]. Then the authors in Ref. [10] add a complete classification of gluon GTMDs. Particularly, the correlator related to the timereversal odd (T-odd) GTMDs is contributed by the final state interactions from gauge link or Wilson line. These interactions are necessary to generate the single spin asymmetries [11].
Although PDFs are related to parton fields in QCD, it is difficult to calculate them directly in QCD since they are nonperturbative quantities. This difficulties may be overcomed by Lattice QCD method studying the PDFs from first principles. However, PDFs are usually defined on the light cone, which poses a problem for the standard Euclidean formulation, and in lattice QCD calculation only moments of distributions in x can be accessed as matrix elements of local operators [12,13]. To overcome these issues, the proposed large-momentum effective theory (LaMET) of Ji has been presented [14]. This method evaluates PDFs on the lattice through quasi-PDFs [14][15][16], whose mother correlation functions includes a spacelike operator γ z instead of the usual lightlike γ + entering the definition of the standard PDFs. These quasi-PDFs can be reached directly from the lattice QCD calculation [17] and as the quasi-PDFs depend logarithmically on P z when P z becomes large, and they need a perturbative matching in LaMET to reduce to the standard PDFs. A very recent review of LaMET is given in [18]. Many theoretical discussions and lattice simulations for quasi-PDFs and similar quantities has been performed [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. Moreover, several model calculations of quasi-PDFs have been carried out [35][36][37][38][39][40][41]. There have also been a number of works on quasi-PDFs renormalization [20,[42][43][44][45][46][47][48][49]. The approach [14] can be generalized to any light-cone correlations in hadron physics, e.g. the correlators related to GPDs and TMDs. There has been a lot of efforts on the quasi-TMDs, including their renormalization and matching to the physical TMDs [18,[50][51][52][53][54][55][56][57][58]. These quasi-TMD works have made important breakthroughs on their pinched-pole singularity issue, renormalization, evolution, soft factor subtraction, and factorization into physical TMDs. Moreover, the nonperturbative Collins-Soper evolution kernel of TMDs has been calculated in [59,60] , which is an important step in TMDs full extractions from lattice QCD. In summary, it may be useful to study the quasi-distributions such as quasi-PDFs, quasi-GPDs and quasi-TMDs.
Among the hadrons, pions are very fascinating particles and they hold a lot of information on the structure of hadrons. There has been a tremendous effort to deduce the parton distribution functions of the pion. Pions provide the force that binds the protons and neutrons inside the nuclei and they also influence the properties of the isolated nucleons. Thus understanding of matter is not complete without getting a detailed information on the role of pions. In this paper, being inspired by the previous works for quasi-distribution model calculations [61][62][63], we will probe the T-odd GTMDs, quasi-TMDs and the quasi-GPDs of the pion applying a scalar spectator model. In particular, GPDs of the pion have been obtained in various models like chiral quark model [64,65], NJL model [66,67], light-front constituent quark model [68] and lattice QCD [69][70][71][72]. We will give out the analytical results of all four twist-two T-odd GTMDs, quasi-TMDs and quasi-GPDs in the present paper, and conduct a qualitative analysis of all these distributions.
The remainder of this paper is as follows: Sec.II below describes in detail the theoretical definition of various pion parton distributions. In Sec.III we give out the analytical results of four T-odd GTMDs, quasi-TMDs and quasi-GPDs in a scalar spectator. In Sec.IV, we present our numerical studies using a group of fitted model parameters of a scalar spectator model. A brief conclusion is presented in Sect.V.

II. DEFINITION OF PION PARTON DISTRIBUTIONS
A. Transverse momentum dependent parton distribution h ⊥ 1π For a spinless particle, such as the pion, only two leading twist TMDs arise, in contrast to the eight found for spin-1 2 particles [73]. The TMD f 1π is simply the unpolarized quark distribution, whereas the Boer-Mulders (BM) function [74], h ⊥ 1π , describes the distribution of transversely polarized quarks in the pion. The BM function h ⊥ 1π is defined from the quark-quark distribution correlation function where P is the four-momentum of the pion moving along the z-axis with the components (P + , P − , 0 ⊥ ) in the lightcone coordinates, in which the plus and minus components of any four-vector a µ have the form a ± = (a 0 ± a 3 )/ √ 2, and transverse part a ⊥ = (a 1 , a 2 ). The quark field and momentum are denoted by k and ψ. In the correlator Eq.(1) we have Wilson lines where Here P is the path-ordering operator and A is the gluon field. The strong couping constant is denoted by g s . The BM function h ⊥ 1π can be obtained by with the pion mass denoted by M .

B. T-odd GTMDs and GPDs
To obtain GTMDs, we start from the generalized k T -dependent correlator denoted by where the initial and final state four-momentum are characterized by p and p . We use the common kinematical variables where we consider the range 0 ≤ ξ ≤ 1 of the skewness variable ξ. In general, the generalized k T -dependent correlator in Eq.(5), unlike GPDs or TMDs, are complex-valued functions. We can reach four complex-valued twist-two GTMDs where the superscripts e, o stand for T-even and T-odd part respectively. We have adopted the general definitions The T-odd twist-two GTMDs correspond to the imaginary part of complex-valued twist-two GTMDs.
The twist-two standard GPDs of quarks for a spin-0 hadron come from the integrated quark-quark correlator obtained from the correlator W in Eq.(5) by means of the projection The GPDs parameterize the Dirac traces F [Γ] of the GPD-correlator in Eq. (8) and there are only two GPDs in twist-two C. Quasi-TMD and quasi-GPD In the following we turn to the definitions of the quasi-TMD and quasi-GPD of the pion meson. Quasi-TMD h ⊥ 1π (x, k 2 T ; P z ) is defined through an equal-time spatial correlation function The Wilson lines read where The original paper on quasi-PDFs suggested to use the matrix γ 3 [14] for the unpolarized quasi-PDF f 1,Q (x; P 3 ). It was later argued that the matrix γ 0 would lead to a better suppression of higher-twist contributions [75]. Similarly, quasi-GPDs are also defined through an equal-time spatial correlation function where the Wilson line is given by Then the twist-two quasi-GPDs of the pion can be obtained through two definitions

III. ANALYTICAL RESULTS IN A SCALAR SPECTATOR MODEL
In this section, being inspired by the previous works for quasi-distribution model calculations [61][62][63], we apply a scalar spectator model to reach the analytic results of the quasi-TMD h ⊥ 1π (x, k 2 T ; P z ), quasi-GPD H 1 (x, ξ, t; P z ) and T-odd GTMDs. In this model, two types of particles have to be considered: the pion target with mass M and the quark or antiquark with mass m. A pion field φ is coupled to a quark and an antiquark using a pseudo-scalar interaction. Including isospin the interaction part of the Lagrangian reads where g π is the coupling constant and τ i are the Pauli matrices. In the work [63], a point-like coupling have been adopted instead of a simple constant g π to eliminate the divergences arising after integration over large k T . Furthermore the parameters of the spectator model have been determined by the authors of Ref. [63] through fitting the model result of unpolaried parton distribution f 1π (x) to the GRV parametrization [76] for the pion. We follow the same point-like coupling form in [63] as wherex = |x|. Here g π , α, β and λ are the model parameters. By choosing the point-like coupling in Eq. (18), the applicable range of x could be −1 < x < 1. This kinematical region is of great interest for quasi-PDFs and quasi-GPDs.
A. Reuslts for TMD h ⊥ 1π , GPD H1 and T-odd GTMDs We first discuss the result for TMD h ⊥ 1π . To get nonzero results for these functions requires considering at least one-loop corrections that include effects from the Wilson line. At the leading order in g 2 s , one finds for the correlator in Eq.(1) where l + integral is realized from taking the imaginary part of the eikonal propagator: 1/(−l + +iε) → −2πiδ(l + ). The color factor satisfies C F = 4/3. Then performing the integrals for k − and l − applying contour integration together with Eq.(4), one obtains where we have used The result in Eq. (20) is the same with previous prediction in Refs. [8,77]. h ⊥ 1π is negative which agrees with previous expectations [78].
The GPD H 1 can be extracted from the integrated quark-quark correlator in Eq. (8), which reads where g ± π = g π (k T ± 1 2 ∆ T ). Performing the integrals for k − applying contour integration together with Eq.(9), we obtains where the D i in the denominator has the form Then we focus on T-odd GTMDs in the spectator model. Similar to TMD case, one needs to introduce one loop diagrams for the correlator shown as The correlator in Eq.(5) reads in the spectator model Evaluating the k − , l − -integral by contour integration, the result for the four T-odd GTMDs can be cast into where all the four T-odd GTMD only have nonvanishing analytical results in the DGLAP region. The following is a compilation of the numerators of all the leading-twist T-odd GTMDs: The denominators in Eq.(26) are given by If taking | ∆ T | = 0, we work out the l T -integral reserving the real part of the results and obtain where On the other hand, when | ∆ T | = 0, using the similar method as the derivation of Eq.(29), the the four T-odd GTMD results can be obtained as where with

B. Results for quasi-TMD and quasi-GPD
We start from the equal-time spatial correlation function in Eq. (10), which can be written in the spectator as In order to apply contour integration we can rewrite the denominator as where k 0 ± , k 0 ± are the poles for k 0 Then according to Eq.(13), the quasi-TMD reads where P + = 1/ √ 2( P 2 z + M 2 + P z ) and P 0 = P 2 z + M 2 . It can be verified that in the limit P z → ∞, the quasi-TMD in Eq.(35) reduces to the standard TMD shown as Eq. (20). At the same time, h ⊥ 1π,Q(0) (x, k 2 T ; P z ) = −h ⊥ 1π,Q(3) (x, k 2 T ; P z ). The quasi-GPD of the pion meson can be calculated in a similar way. In the spectator model, the correlator in Eq. (14) to calculate the quasi-GPD has the form: After performing the k 0 -integral using the contour integration, we write down the analytical result of the quasi-GPD as follows according to Eq.(16) where The poles coming from the denominator are given by

IV. NUMERICAL CALCULATION
In order to fix the parameters of the spectator model, the authors of Ref. [63] fit the model result of unpolaried parton distribution f 1π (x) to the GRV parametrization [76] for the pion. We adopt the fitted values for the parameters g π = 6.316, λ = 0.855, α = 0 and β = 1. We make a preliminary estimate for choosing the strong coupling α s ≈ 0.3 and adopting the quark mass m = 0.3GeV.   T-odd GTMD has nonvanishing value. In the upper, left panel of Fig.1, we plot the T-odd GTMD F o 1 (x, ξ, k 2 T , ∆ 2 T ) at ξ = 0.1 and | ∆ T | = 0. Note that in this case the result of F o 1 (x, ξ, k 2 T , ∆ 2 T ) is negative. One can also find that as the x value becomes larger, the maximum value of k T resulting in nonvanishing F o 1 (x, ξ, k 2 T , ∆ 2 T ) becomes smaller and at x = 0.1, the maximum value of k T is about 0.8GeV. For a fixed x value, as the value of k T increases, the F o 1 (x, ξ, k 2 T , ∆ 2 T ) result becomes smaller first and then larger. At a fixed value x = 0.1, the F o 1 (x, ξ, k 2 T , ∆ 2 T ) reachs the minimum when k T is about 0.3. We depict the F o 1 (x, ξ, k 2 T , ∆ 2 T ) at ξ = 0.1 and | ∆ T | = 0.5 in the upper, right panel of Fig.1. Roughly speaking, it can be seen that the x-k T regions below straight line y = −0.25x + 0.2 are related to positive F o 1 (x, ξ, k 2 T , ∆ 2 T ). Then we compare the two upper panels in Fig.1 and find that the F o 1 (x, ξ, k 2 T , ∆ 2 T ) result in the right panel is larger than that in the left panel at the same x-k T point. Comparing the upper panels with the lower panels, we emphasize that the two corresponding contours at x > 0.5 have nearly the same shape.  In the following we turn to the results of quasi-TMD h ⊥ 1π,Q(3) and quasi-GPD H 1,Q(3) . In Fig.5, we plot the quasi-TMD h ⊥ 1π,Q(3) (x, k 2 T ; P z ) as functions of x and k T at different values of P z and TMD h ⊥ 1π (x, k 2 T ) as functions of x and k T . We can see from the upper left panel that the absolute value of quasi-TMD h ⊥ 1π,Q(3) (x, k 2 T ; P z ) becomes larger as k T close to zero. In some region with negative x, the corresponding h ⊥ 1π (x, k 2 T ) has nonvanishing values. As x increases from −0.3 to 1, the resulting h ⊥ 1π (x, k 2 T ) absolute value gets larger first and then becomes smaller. For a nonvanishing h ⊥ 1π (x, k 2 T ) result, the allowable maximum value of k T is around 0.38. Comparing the three panels in Fig.5 depicting the quasi-TMD h ⊥ 1π,Q(3) (x, k 2 T ; P z ) with P z = 1, 3, 5 GeV, we can find that as the P z value increases, the minimum of x inside the contours becomes larger. In the lower left panel with P z = 5 GeV, the quasi-TMD h ⊥ 1π,Q(3) (x, k 2 T ; P z ) result stay the same at a fixed k T value except the case x close to 0 or 1. The TMD h ⊥ 1π (x, k 2 T ) as a negative function of x and k T is shown in the lower right panel of Fig.5. The absolute value of h ⊥ 1π (x, k 2 T ) can reach 0.28 in almost all the x range at a very small value of k T . Finally, we display the quasi-GPD H 1,Q(3) (x, ξ, ∆ 2 T ; P z ) as functions of x and P z for different ξ and | ∆ T | values in Fig.6. This quasi-GPD is negative. In the upper left panel, it is desired to mention that when P z > 3 GeV, the quasi-GPD H 1,Q(3) (x, ξ, ∆ 2 T ; P z ) result hardly depends on the value of P z . After comparing the two upper panels, we can acquire that the allowable range of x for the nonvanishing quasi-GPD H 1,Q(3) (x, ξ, ∆ 2 T ; P z ) becomes larger when | ∆ T | slightly increases from zero. Two panels have the very similar contour shape. When ξ = 0.5, the quasi-GPD H 1,Q(3) (x, ξ, ∆ 2 T ; P z ) result is shown in two lower panels of Fig.6. The absolute values of H 1,Q(3) (x, ξ, ∆ 2 T ; P z ) in two lower panels are smaller than those in two upper panels. We also find that when P z > 3 GeV, the quasi-GPD H 1,Q(3) (x, ξ, ∆ 2 T ; P z ) result hardly depends on the value of P z .

V. CONCLUSION
In this paper we have computed four T-odd GTMDs, quasi-TMD and quasi-GPD in a scalar spectator model. We have present the results for four T-odd GTMDs. To get nonzero results for these functions requires considering at least one-loop corrections that include effects from the Wilson line. We have studied the relation of GTMDs for different values of skewness ξ defined as the longitudinal momentum transferred to the proton and the total momentum transferred to the proton | ∆ T |. We found only in x > ξ region, the T-odd GTMDs has nonvanishing value. Generally, the four T-odd GTMDs are negative in x-k T space. However, theG o 1 (x, ξ, k 2 T , ∆ 2 T ) results in certain parameter space are positive. Note that the T-odd GTMD H ∆,o 1 (x, ξ, k 2 T , ∆ 2 T ) becomes zero when | ∆ T | = 0. We have also considered the distributions of quasi-TMD and quasi-GPD. For the contours of quasi-TMD, we can find that as the P z value increases, the minimum of x inside the contours becomes larger. We also find that when P z > 3 GeV, the quasi-GPD H 1,Q(3) (x, ξ, ∆ 2 T ; P z ) result hardly depends on the value of P z .