Distribution amplitudes and functions of ground-state scalar and pseudoscalar charmonia

TL;DR

This paper uses continuum Schwinger function methods to analyze ground-state scalar and pseudoscalar charmonia, revealing complex internal structures and nontrivial distribution amplitudes and functions that challenge simple hydrogen-like models.

Contribution

It demonstrates that charmonia have intricate wave functions with negative and positive domains, and provides detailed distribution amplitudes and functions as benchmarks for future theoretical studies.

Findings

The $ ext{chi}_{c0}$ distribution amplitude is not positive definite.

The $ ext{chi}_{c0}$ and $ ext{eta}_c$ distribution functions become similar under scale evolution.

The glue momentum fraction in these states is 10 ext% less than in the pion.

Abstract

Charmonia are often supposed to provide simple hydrogen-like ``atomic'' systems that can be used to obtain insights into heavier-quark QCD. We use continuum Schwinger function methods to analyse this hypothesis in connection with ground-state scalar and pseudoscalar charmonia and find that a more complex picture of these states may be necessary. For instance, considering orbital angular momentum, the $\chi_{c0}$ is not a simple $P$-wave system; similarly, the $\eta_c$ wave function contains more than merely $S$-wave contributions. The distribution amplitudes (DAs) and distribution functions (DFs) of these mesons are also nontrivial. For instance, the $\chi_{c0}$ DA is not positive definite: owing to QCD symmetries, it possesses domains of balanced negative and positive support. This feature is also expressed in the $\chi_{c0}$ DF, but differences between $\chi_{c0}$ and $\eta_c$ DFs diminish under scale evolution. Notably, the light-front momentum fraction carried by glue is the same in both states: it is 10\% less than the in-pion glue momentum fraction. Whilst experimental confirmation of the predictions herein is unlikely, our results should serve as benchmarks for complementary theory attempts to understand local and global structural features of heavier-quark hadrons.