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Valence Quark Distributions in Pions: Insights from Tsallis Entropy

Jingxuan Chen, Xiaopeng Wang, Yanbing Cai, Xurong Chen, Qian Wang

TL;DR

The paper addresses how to determine pion valence quark PDFs at a low scale by incorporating valence-quark correlations through Tsallis entropy within a maximum entropy framework. It parameterizes the initial valence input under sum rules, fixes remaining degrees of freedom by entropy maximization, and evolves the distributions with a GLR-MQ-ZRS corrected DGLAP equation using a saturating infrared-safe coupling. Key findings show a preferred non-unity Tsallis index $q$ (e.g., $q\approx0.915$) indicating correlations, with evolved distributions describing E615 data and moments at $Q^2=4\,\mathrm{GeV}^2$ in agreement with other models, thereby validating the approach. The results underscore the significance of including valence-quark correlations in non-perturbative inputs for hadron PDFs and demonstrate a consistent bridge between MEM-based inference and QCD evolution for pion structure.

Abstract

We investigate the valence quark distributions of pions at a low initial scale ($Q^2_0$) by employing Tsallis entropy, a non-extensive measure that effectively captures long-range correlations among internal constituents. Utilizing the maximum entropy approach, we adopt two distinct functional forms and fit experimental data through the elegant GLR-MQ-ZRS evolution equation to derive the model parameters. Our findings indicate that the resulting valence quark distributions provide an optimal fit to experimental data, with the values of the $q$ parameter deviating from unity. This deviation indicates the significant role that correlations among valence quarks play in shaping our understanding of pion internal structure. Additionally, our computations of the first three moments of pion quark distributions at $ Q^2 = 4 \, \mathrm{GeV}^2$ display consistency with other theoretical models, thereby reinforcing the importance of incorporating valence quark correlations within this analytical framework.

Valence Quark Distributions in Pions: Insights from Tsallis Entropy

TL;DR

The paper addresses how to determine pion valence quark PDFs at a low scale by incorporating valence-quark correlations through Tsallis entropy within a maximum entropy framework. It parameterizes the initial valence input under sum rules, fixes remaining degrees of freedom by entropy maximization, and evolves the distributions with a GLR-MQ-ZRS corrected DGLAP equation using a saturating infrared-safe coupling. Key findings show a preferred non-unity Tsallis index (e.g., ) indicating correlations, with evolved distributions describing E615 data and moments at in agreement with other models, thereby validating the approach. The results underscore the significance of including valence-quark correlations in non-perturbative inputs for hadron PDFs and demonstrate a consistent bridge between MEM-based inference and QCD evolution for pion structure.

Abstract

We investigate the valence quark distributions of pions at a low initial scale () by employing Tsallis entropy, a non-extensive measure that effectively captures long-range correlations among internal constituents. Utilizing the maximum entropy approach, we adopt two distinct functional forms and fit experimental data through the elegant GLR-MQ-ZRS evolution equation to derive the model parameters. Our findings indicate that the resulting valence quark distributions provide an optimal fit to experimental data, with the values of the parameter deviating from unity. This deviation indicates the significant role that correlations among valence quarks play in shaping our understanding of pion internal structure. Additionally, our computations of the first three moments of pion quark distributions at display consistency with other theoretical models, thereby reinforcing the importance of incorporating valence quark correlations within this analytical framework.
Paper Structure (4 sections, 15 equations, 3 figures)

This paper contains 4 sections, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Non-perturbative input for valence quarks of pion and maximum entropy method with Tsallis entropy
  3. Results and discussions
  4. Summary

Figures (3)

  • Figure 1: The Tsallis entropy of valence quark nonperturbative input as functions of $B_\pi$ and $q$.
  • Figure 2: The dependence of Tsallis entropy of valence quarks on $B_\pi$ at $q=0.8$ (blue solid curve), $q=0.9$ (black dashed curve), $q=1.1$ (red dot-dashed curve), $q=1.2$ (pink dotted curve) at initial scale.
  • Figure 3: The predicted up valence quark distribution function is presented in comparison with the E615 data Conway:1989fs and E615R data Aicher:2010cb at a scale of $Q^2=20\,\mathrm{GeV^2}$.