Maximum Entropy Method for Valence Quark Distributions in Exotic Hadrons: A Study of the $Z_c(3900)$ Case

TL;DR

This paper applies the Maximum Entropy Method to determine valence quark distributions in exotic hadrons, revealing shifts in distribution peaks, estimating the $Z_c(3900)$ radius, and comparing form factors with molecular state models.

Contribution

It introduces a novel application of MEM to exotic hadrons and provides initial valence quark distribution estimates for the $Z_c(3900)$ at low energy scales.

Findings

Valence quark distribution peaks shift to smaller x with more valence quarks.

Estimated radius of $Z_c(3900)$ is 1.276 fm at low Q^2.

Discrepancy found between computed charge form factor and molecular state calculations.

Abstract

In this study we demonstrate the application of the Maximum Entropy Method (MEM) to determine the valence quark distribution of exotic hadrons. Our investigation yields three key findings. Firstly, we observe a significant shift towards smaller Bjorken scale $x$ in the peak position of the valence quark distribution for hadrons with an increasing number of valence quarks, consistent with previous results by Kawamura and Kumano. Secondly, assuming that the $Z_c(3900)$ initially consists of four valence quarks, we employ MEM to determine its initial valence quark distribution, estimating a radius of $r_c=1.276$ fm at an extremely low resolution scale $Q^2$. Furthermore, we identify a notable discrepancy between our computed charge form factor $G_c(q)$ at leading order and the outcomes of hadron molecular state calculations. We propose that this form factor can be extracted from the QCD counting rule cross-section, which is grounded in Generalized Distribution Amplitudes (GDA) linked to the multi-quark states.