Valence quark distribution of rho meson using light-front quark model

TL;DR

This work provides a comprehensive LFQM-based analysis of the rho meson's partonic structure, computing both leading- and subleading-twist PDFs and the full set of leading-twist TMDs, including tensor polarization effects unique to spin-1. By using two spin-wave constructions (S-1 and S-2), the study reveals how tensor structures enter the distributions, with $f_{1LL}(x)$ vanishing for S-1 but nonzero for S-2, and twist-3 PDFs emerging only in the S-2 case. The leading-twist PDFs and TMDs are evolved to higher scales at NLO with DGLAP, Mellin moments are obtained via a neural-network fit and show agreement with other theoretical predictions, and all positivity constraints are satisfied. Overall, the tensor polarization plays a crucial role in shaping the rho's three-dimensional partonic structure, with results that align with alternative approaches and highlight distinctive spin-1 effects relevant for future experiments at facilities like JLab, NICA, and the EIC.

Abstract

We investigate the partonic structure of the $ρ$ meson, the lightest spin-$1$ vector meson, within the light-front quark model (LFQM). To explore the sensitivity to model assumptions, we employ two distinct types of spin wave functions in the LFQM. Using light-front helicity wave functions, we derive explicit expressions for the leading-twist and subleading-twist quark parton distribution functions (PDFs), and evolve the leading-twist PDFs to higher scales with next-to-leading order (NLO) Dokshitzer--Gribov--Lipatov--Altarelli--Parisi (DGLAP) evolution. We have also calculated the Mellin moment from the evolved PDFs using a simple neural network frame and compared with available theoretical predictions. Furthermore, we compute the full set of nine leading-twist transverse-momentum-dependent distributions (TMDs) for the valence quark in the $ρ$ meson, including three tensor TMDs that arise from spin-$1$ tensor polarization of the hadron. Positivity constraints for the PDFs and TMDs are examined within this framework. Our findings highlight the crucial role of tensor polarization in shaping the three-dimensional partonic structure of vector mesons.

Figures (9)

• Figure 1: Quark PDFs for S-1 and S-2 type spin wave functions at the model scale $Q^2 = 0.20~\mathrm{GeV}^2$ as functions of the longitudinal momentum fraction $x$: (a)-(b) unpolarized $f_1(x)$, (c)-(d) helicity $g(x)$, (e)-(f) transversity $h(x)$, and (g)-(h) tensor-polarized $f_{1LL}(x)$.
• Figure 2: Evolved quark PDFs of S-1 type spin wave function at $Q^2 = 2.4$, $5$, and $20~\mathrm{GeV}^2$ obtained through NLO DGLAP evolution: (a) $x f(x)$, (b) $x g(x)$, and (c) $x h(x)$ as functions of $x$.
• Figure 3: Evolved quark PDFs of the S-2 type spin wave function at $Q^2 = 2.4$, $5$, and $20~\mathrm{GeV}^2$ obtained through NLO DGLAP evolution: (a) $x f(x)$, (b) $x g(x)$, (c) $x h(x)$, and (d) $x f_{1LL}(x)$ as functions of $x$.
• Figure 4: Positivity constraints (PS) of the PDFs for the S-1 and S-2 type spin wave function.
• Figure 5: Sub-leading twist quark PDFs for S-1 and S-2 spin wave functions at the model scale $Q^2 = 0.20~\mathrm{GeV}^2$ as functions of $x$: (a) $e(x)$, (b) $x e(x)$, (c) $g_T(x)$, (d) $x g_T(x)$, (e) $h_L(x)$, (f) $x h_L(x)$, (g) $e_{LL}(x)$, and (h) $f_{LT}(x)$. The $e_{LL}(x)$, and $f_{LT}(x)$ PDFs are zero for the case of S-1 spin wave functions.