Distribution functions of a radially excited pion

Z.-N. Xu; Z.-Q. Yao; D. Binosi; M. Ding; C. D. Roberts; J. Rodríguez-Quintero

doi:10.1140/epjc/s10052-025-14053-9

2024 Impact factor 4.8

Particles and Fields

Open Access

Eur. Phys. J. C (2025) 85: 330
https://doi.org/10.1140/epjc/s10052-025-14053-9

Regular Article - Theoretical Physics

Distribution functions of a radially excited pion

Z.-N. Xu¹, Z.-Q. Yao¹^,2, D. Binosi³, M. Ding⁴^,5, C. D. Roberts⁴^,5^a and J. Rodríguez-Quintero¹

¹ Dpto. Ciencias Integradas, Centro de Estudios Avanzados en Fis., Mat. y Comp., Fac. Ciencias Experimentales, Universidad de Huelva, 21071, Huelva, Spain
² Dpto. Sistemas Físicos, Químicos y Naturales, Univ. Pablo de Olavide, 41013, Sevilla, Spain
³ European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), Villa Tambosi, Strada delle Tabarelle 286, 38123, Villazzano, TN, Italy
⁴ School of Physics, Nanjing University, 210093, Nanjing, Jiangsu, China
⁵ Institute for Nonperturbative Physics, Nanjing University, 210093, Nanjing, Jiangsu, China

^a cdroberts@nju.edu.cn

Received: 4 February 2025
Accepted: 10 March 2025
Published online: 22 March 2025

Abstract

A nonperturbatively-improved, symmetry-preserving approximation to the quantum field equations relevant in calculations of meson masses and interactions is used to deliver predictions for all distribution functions (DFs) of the ground state pion, $π_{0}$ $\pi _0$ , and its first radial excitation, $π_{1}$ $\pi _1$ , viz. valence, glue, and sea. Regarding Mellin moments of the valence DFs, the $m = 0, 1$ $$m=0,1$$ moments in both states are identical; but for each $m \geq 2$ $m\ge 2$ , that in the $π_{0}$ $\pi _0$ is greater than its partner in the $π_{1}$ $\pi _1$ . Working with such information, pointwise reconstructions of the hadron-scale $π_{0, 1}$ $\pi _{0,1}$ valence DFs are developed. The predicted $π_{0}$ $\pi _0$ valence DF is consistent with extant results. The $π_{1}$ $\pi _1$ valence DF is novel: it possesses three-peaks, with the central maximum partnered by secondary peaks on either side, each separated from the centre by a zero: the zeroes lie at $x \approx 0.2, 0.8$ $x\approx 0.2,0.8$ and the secondary peaks at $x \approx 0.1, 0.9$ $x\approx 0.1,0.9$ . Evolution to $ζ = 3.2$ $\zeta =3.2\,$ GeV, a typical scale for nonperturbative calculations, is accomplished using an evolution scheme for parton DFs that is all-orders exact. At this higher scale, differences between the $π_{0, 1}$ $\pi _{0,1}$ valence DFs remain significant, but analogous differences between glue and sea DFs are far smaller. This analysis confirms that, owing to constraints imposed by chiral symmetry and the pattern by which it is broken in Nature, there are noticeable differences between the structural properties of the pion ground state and its radial excitations.

© The Author(s) 2025

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