Double parton distribution of valence quarks in the pion in chiral quark models

Abstract

The valence double parton distribution of the pion is analyzed in the framework of chiral quark models, where in the chiral limit factorization between the longitudinal and transverse degrees of freedom occurs. This feature leads, at the quark-model scale, to a particularly simple distribution of the form D(x1, x2, y) = δ(1−x1 −x2)F (y), where x1,2 are the longitudinal momentum fractions carried the valence quark and antiquark, and y is the separation of their transverse positions. For y = 0 this result complies immediately to the Gaunt-Sterling sum rules. The DGLAP evolution to higher scales is carried out in terms of the Mellin moments. We then explore its role on the longitudinal correlation quantified with the ratio of the double distribution to the product of single distributions, D(x1, x2, y)/D(x1)D(x2). We point out that the ratios of moments x n 1 x m 2 / x n 1 x m 2 are independent of the evolution, providing particularly suitable measures to be tested in the upcoming lattice simulations. The transverse form factor F (y) is obtained in variants of the Nambu-Jona-Lasinio model with the spectral and Pauli-Villars regularizations. Interestingly, with the spectral regularization of the model, the effective cross section for the double parton scattering of pions is exactly equal to the geometric cross section, σ eff = π y 2 and yields about 20 mb.

Two-current correlations in the pion in the Nambu and Jona-Lasinio model

“…This exercise has been effective to examine the validity of commonly used approximations for which dPDFs could be expressed in terms of GPDs -and were found to be violated, especially at low momentum scale and in the valence region. These results have been recently confirmed in a similar NJL framework by an independent calculation [25].…”

Sum rule improved double parton distributions in position space

“…It is hence essential to develop realistic models for DPDs. Considerable efforts have been made to compute them in quark models [51][52][53][54][55][56][57][58][59][60][61][62], and lattice calculations of Mellin moments of DPDs are underway [63,64]. In addition, there are important theoretical constraints on DPDs.…”

Applications of the Nambu–Jona-Lasinio model to the partonic structure of the pion

“…and link the marginal projections/moments of dPDFs to sPDFs in a way typical of probability distributions. They follow from a decomposition of the parton operators in a basis of the light-front wave functions [17] 7(taken from [6]).…”

“…An efficient method is based on the Mellin moments, similarly to the case of sPDFs. Details are presented in [6,21].…”

“…These results, amended with the necessary QCD evolution which generates radiatively the gluon degrees of freedom, compares very favorably to the available experimental and lattice data. We briefly review some of these results and then pass to a recent topic of double parton distribution functions (dPDF) in the pion [4][5][6] evaluated in chiral quark models followed with the DGLAP evolution. We discuss the issue of partonic correlations and the proposed measures based on the Mellin moments of dPDFs [6].…”