The effective running quark mass from the variational solution of the 4dimensional Bethe-Salpeter equation for charmonium
V. Sauli
TL;DR
The paper addresses whether a fixed constituent charm mass can describe both charmonium spectra and leptonic decays within a Poincaré-invariant, covariant 4D Bethe–Salpeter framework. It implements a variational solution of the full 4D BSE for charmonium using a kernel that embodies an infrared running coupling and a gluon-mass scale, and extracts the effective charm mass at the scale $\xi \approx M_V^2/4$ by fitting experimental leptonic decay constants and vector masses. The results show that the running charm mass must slide with the quarkonium scale, yielding near-unique $\langle m_c(M_V/2)\rangle$ values for states like $J/\psi$, $\psi(3600)$, and $\psi(4040)$, while a single fixed mass fails to reproduce both masses and decays and ghost states are avoided through the framework. This approach demonstrates the viability of a fully covariant 4D BSE treatment for heavy quarkonia and lays groundwork for computing electromagnetic form factors and transitions within the same formalism.
Abstract
The scale dependence of the modulus of the effective charm quark mass is determined from measured values of leptonic decay constants and vector charmonium masses. Unlike nonrelativistic approximations, the Poincaré-invariant framework based on the full four-dimensional (4D) Bethe-Salpeter equation does not support a single-valued constituent quark mass approximation. To overcome the obstacles of low accuracy associated with standard methods, the variational method is used to solve the full 4D Bethe-Salpeter equations. This method achieves precise agreement with the experiment while avoiding unphysical abnormal states. The quark mass must slide to provide almost unique values for the charm running mass at the scale of individual charmonium states, achieving precise agreement with the experiment on one hand and avoiding unphysical abnormal states on the other.