The unpolarized TMD valence quark distributions of light mesons in the statistical covariant parton model approach

We investigate the unpolarized transverse momentum dependent (TMD) structure of light mesons at low Q2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q^2$$\end{document} scales in nonperturbative region of quantum chromodynamics (QCD). To this end, we calculate the unpolarized TMD valence quark distribution functions of pion and kaon based on the covariant parton model in which the flavour dependent covariant momentum distribution of quarks is needed. We apply the statistical model for this momentum distribution, for the first time, and also consider the quarks to be massive. The results of TMD valence distributions of light mesons show convenient general properties. We also obtain the integrated valence quark distributions of pion and kaon and compare our results with those of other models.


Introduction
Consistency of high-energy scattering processes data, for instance, Deep Inelastic Scattering (DIS) [1,2] and Drell-Yan processes (DY) [3,4] with theoretical researches, allows to explain hadron in terms of parton distribution functions (PDFs). In comparison with other known experiments, the best information on light quarks with no constraints on sea distributions, is obtained from charged lepton scattering off hadrons [5]. The PDFs of hadrons can be explored in two regions of quantum chromodynamics, perturbative and nonperturbative QCD [6][7][8][9][10]. Formulation of the covariant parton model (CPM), based on naive parton model, have provided a convenient framework for hadron PDFs [11][12][13][14][15][16]. There are different QCD inspired models in which PDFs are approximated using numerous parameters in polynomials [17,18] while some of them are not able to determine a e-mail: p_abolhadi@phy.uk.ac.ir b e-mail: myazdan@uk.ac.ir c e-mail: hnematollahi@uk.ac.ir (corresponding author) flavour separation [19]. Furthermore, the efforts for offering a complete description of the polarized and unpolarized hadron structure led to introduction of a statistical way [20] which generated compatible PDFs. The other achievements obtained applying this way, established it as a successful approach [19,[21][22][23][24][25][26]. The statistical approach can predict the spin and flavour features of hadronic structure which usually in global fitting are determined by adding some theoretical constraints [26].
In this work, we calculate the unpolarized TMD valence quark distributions of light mesons based on the covariant parton model in nonperturbative region of QCD. In this model the quark covariant momentum distribution of these mesons is needed. We use the result of statistical model for this covariant momentum distribution. The organization of this paper is as follows: in Sect. 2 we explain the covariant parton model and obtain the unpolarized TMD quark distribution. The statistical model is reviewed in Sect. 3. The integrated and TMD valence quark distributions of light mesons are calculated based on the statistical covariant parton model in Sect. 4 and the obtained results for pion and kaon are given in Sects. 4.1 and 4.2, respectively. We give our conclusion in Sect. 5.

TMD distribution function and quark correlator
Three dimensional (3D) investigations have accoutered us to present a realistic expression for experimental observations. In CPM framework, in which there is no restriction on frames for the on-shell quarks, instead of the infinite momentum frame (IMF) that is a common reference framework, we choose the rest frame of hadron applying the light front language. As the first step for theoretical calculation, the exploration of correlation function in terms of light coordinates allows one to calculate the PDFs and TMDs in different twists. The quark correlator which is expressed by [36,46], includes ψ q (ξ ) that demonstrates the quark and antiquark fields as the functions of spacetime coordinates. The color gauge invariance of the correlation function is ensured by the Wilson line, W(0, ξ; path), whose path in the spacetime depends on the considered experimental process [33,34,36]. The unpolarized T-even TMD distribution function at LO accuracy is determined by the first term of the following relation [23,43,46,58], in which q denotes different quark flavours, while p, x and p T are four-momentum, longitudinal fraction and transverse component of quark momentum, respectively. γ and φ q (x, p T , S), are the gamma vector matrix [46] and the quark-quark transverse momentum dependent correlation function. We need the given quark correlator in the CPM as [46], where i and j refer to the Dirac indices. Moreover,ū j ( p) and u i ( p) are the Dirac spinors and m is the quark mass. The quark correlation function includes the theta and delta functions (P 0 ) δ(p 2 − m 2 ), which ensure on-shell condition of quarks, and G q ( p P) as a scalar function is the covariant momentum distribution of the unpolarized quarks in different flavours. The TMD correlation function is defined by, in terms of light cone coordinates (M, S and P are the mass, spin and momentum of hadron) [46]. In the rest frame, p = ( p 0 , p 1 , p T ) and P = (M, 0), using the light cone coordinates definitions, the delta function becomes δ(x − p 0 + p 1 M ) and replacing Eq. (3) in Eq. (4) one can rewrite Eq. (2) as, By applying the properties of tracing gamma matrix, one can obtain, as the unpolarized TMD distribution function in the leading twist [23,43,44]. From p 2 = m 2 , for the massive quarks, one can find [23], that are the roots of the function x − p 0 + p 1 M . Equation (7) can be used for writing another form of δ function as [23,44], Inserting Eq. (8) in Eq. (6) gives [23,44], More than different Q 2 dependent parameters in the unpolarized TMD function, mass of quark (m) is appeared inp 0 . We have inquired the dependence of the unpolarized TMD distribution on the mass of valence quarks.

Statistical model
From Eq. (9) one can find the importance of covariant quark distribution in momentum space. This function in different models explains how the longitudinal and transversal degrees of freedom are distributed. In this paper we apply the model which is parametrized based on the idea that a hadron can be regarded as a partonic gas in a finite volume and this gaseous system is in a thermal equilibrium [5,22,25]. Due to the Pauli exclusion principle, individual treatment of valence and sea fermions can be expressed in terms of Fermi-Dirac distribution. In this regard we use the following unpolarized covariant distribution, that is introduced in the statistical model [22]. The physical parameters include: color-spin degeneracy g q = 6 for three light quarks (u, d, s), volume (V ) and temperature (T ) which have the same value for all flavours. μ denotes chemical potential which satisfies the relation μq = −μ q for quark and antiquark based on the chiral essence of QCD. For each flavour of quarks and antiquarks in hadrons, inserting the fermion distribution of Eq. (10) in Eq. (9), we will get the unpolarized TMD quark distribution: It should be pointed that the parameters in Eq. (11) such as volume and chemical potential are Q 2 dependent, also using Eq. (7), the explicit dependences of TMD quark distribution function on x and p T are given. Now we are able to calculate the unpolarized TMD quark distribution functions of hadrons using Eq. (11).

Unpolarized integrated and TMD valence distributions of light mesons
In this section we focus on the light pseudoscalar mesons which play an essential role in the description of the structure of nucleons and nuclei and also in the realization of confinement within the standard model due to their responsibility for the dynamical chiral symmetry breaking. We study the unpolarized TMD structure of these light mesons, pion and kaon, at low Q 2 scales in nonperturbative region of QCD. In this region the contributions of sea quarks and gluon to the structure of mesons can be neglected.

Pion
In comparison with other mesons, pion (ud) has the smallest mass and plays a dominant role in mediating nuclear force [59]. Because of the small difference between u andd masses, we assume that each of these two valence quarks has the same contribution in the structure of pion. We should note that we use the notation f q (x, p T ) ≡ q(x, p T ) in this section.
In the first step of our calculations, we obtain the up valence quark distribution of pion (u π v (x)) by integrating Eq. (11) over transverse momentum. Then by considering this fact that this valence distribution should satisfy the following sum rules, we determine the numerical values of some parameters within Eq. (11). We take the pion mass M π = 140 MeV and the effective mass of u quark in pion m π u = 30 MeV. In Fig. 1 we have presented the result of our calculations for valence quark density of pion, xu π v (x). We have compared our result with those of Refs. [50,60] at Q 0 2 = 0.15 GeV 2 and Q 0 2 = 0.11 GeV 2 , respectively. It is found that the result of our work is in good agreement with those of these references.
We have also calculated the Mellin moments of u π v based on the following relation, for n = 2, n = 3 and n = 4. The results have been listed in Table 1.  In further step of our calculations, we have obtained the TMD valence quark distribution of pion based on the statistical covariant parton model (Eq. (11)). In Fig. 2, we have displayed the three dimensional representation of u π v with respect to x and p T . Such 3D representation can provide us an appropriate insight into the general properties of TMD distribution function.
The unpolarized TMD valence distribution of pion has been plotted in Fig. 3 with respect to transverse momentum ( p T ) at three fixed values of x; x = 0.12, 0.15 and 0.18. As can be found from this figure the width of p T -distribution is dependent on x-value. This width increases by increasing the value of x.
Finally, we have depicted the x-distribution of u π v at four values of transverse momentum, p T = 0.04 GeV,  Fig. 4. It is found that the probability of finding the valence quarks in pion decreases by increasing p T value as expected.

Kaon
In this subsection we probe the TMD structure of kaon as a simple bound state of a valence up quark (u) and a valence strange antiquark (s). Since kaon has a heavier strange quark inside, studying its structure is interesting for indicating the importance of mass especially in nonperturbative region of QCD [59].
In similar to what we have done for pion, we have first computed the integrated valence distributions of kaon, u K v (x) ands K v (x), by integrating Eq. (11) over p T at low Q 2 0 scale. Then by using the sum rules of this meson as, we have determined the values of some needed parameters of Eq. (11). It should be noted that the valence quarks have the dominant role in the structure of kaon at low energy scales and we take the kaon mass M K = 494 MeV and the effective masses of u quark and s antiquark in kaon m K u = 151 MeV and m K s = 188 MeV. We have plotted the results of valence densities of kaon in Fig. 5. In this figure we have compared our results by those of Refs. [50,52]. As can be seen in Fig. 5 the results of our work have the shifts to smaller values of x in comparison with those of Refs. [50,52]. We have also calculated the Mellin moments of u K v ands K v and presented the results in Tables 2   [61,62].
In further step, we have calculated the TMD valence distributions of kaon using Eq. (11) and given the 3D representations of these TMD distributions in Fig. 6.
The TMD distributions of u K v ands K v have been plotted in Fig. 7 with respect to p T at three values of x; x = 0.12, 0.15  Fig. 8. By comparing Figs. 4 and 8, one can see that the probability of finding valence quark and antiquark approach to zero within p T > 0.07GeV and p T > 0.2 GeV for pion and kaon, respectively. This difference is due to the mass difference between these mesons.

Conclusion
Studying the structure of light mesons, pion and kaon, is of great interest particularly in nonperturbative QCD because of their role as Goldstone bosons on chiral symmetry breaking in this region. Probing the integrated PDFs of pion and kaon have been done in several works but the investigation of the transverse momentum dependence of their structure is poor yet. We have studied the unpolarized TMD valence quark distributions of pion and kaon at low Q 2 scales in noneperturbative QCD. We have linked the covariant parton model and the statistical model to calculate these TMD distributions, for the first time. In covariant parton model the quarks are often considered to be massless but we have regarded them massive in this article.
The unpolarized integrated and TMD valence distributions of pion and kaon have been obtained and the results have been given in previous section. The results of integrated PDFs are in good agreement with the results of Refs. [50,52,60] and those of TMD ones show appropriate general properties. The differences between some properties of TMD results for pion and kaon are due to the mass difference between these two mesons and it shows the important effect of meson mass (M) on the unpolarized TMD distribution function (please see Eq. (11)).
We hope to study the polarized TMDs of light mesons and report the results in future. Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors' comment: The needed data can be found in the text and references of this paper.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 . SCOAP 3 supports the goals of the International Year of Basic Sciences for Sustainable Development.