  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.
