Perturbative method to estimate meson masses in the framework of Bethe-Salpeter equation beyond its dominant interaction

Abstract

We propose a novel method to calculate the meson mass in the framework of Dyson-Schwinger equation and Bethe-Salpeter equation, once their dominant interactions are identified. The method is based on the perturbation theory of matrix, which is widely used in quantum mechanics. Taking interactions other than the dominant ones as perturbations, we derive the first order correction of quark propagator. Implementing the perturbation on BSE, the mass correction at first order is then given. We exemplify this method with the well known Rainbow-Ladder (RL) truncation, and go beyond the RL using a simple model, i.e., the Munczek-Nemirovsky~(MN) model, by studying the pion and $\rho$ mesons mass shift. The results are all in good agreement with those obtained by fully solving the BSE beyond RL. Our perturbative method therefore can be used to give a semi-quantitative estimate of meson mass correction in cases when the BSEs are complicated by interactions that go beyond the dominant one.