Masses of the physical mesons from an effective QCD--Hamiltonian

TL;DR

This paper derives a semi-analytical model for meson masses using an effective QCD Hamiltonian, achieving good agreement with experimental data through a novel approach involving a running coupling constant.

Contribution

It introduces a new approximate solution to the effective QCD Hamiltonian in the meson sector, incorporating a running coupling and variational methods for mass calculations.

Findings

Derived semi-analytical meson mass formulas

Achieved remarkable agreement with experimental meson masses

Determined quark masses by fitting vector meson data

Abstract

The front form Hamiltonian for quantum chromodynamics, reduced to an effective Hamiltonian acting only in the $q\bar q$ space, is solved approximately. After coordinate transformation to usual momentum space and Fourier transformation to configuration space a second order differential equation is derived. This retarded Schr\"odinger equation is solved by variational methods and semi-analytical expressions for the masses of all 30 pseudoscalar and vector mesons are derived. In view of the direct relation to quantum chromdynamics without free parameter, the agreement with experiment is remarkable, but the approximation scheme is not adequate for the mesons with one up or down quark. The crucial point is the use of a running coupling constant $\alpha_s(Q^2)$, in a manner similar but not equal to the one of Richardson in the equal usual-time quantization. Its value is fixed at the Z mass and the 5 flavor quark masses are determined by a fit to the vector meson quarkonia.