Analysis of mass difference of the $\pi$ and $\rho$ with Bethe-Salpeter equation

TL;DR

This paper investigates the mass difference between the π and ρ mesons using the Bethe-Salpeter equation, incorporating both one-pion and one-gluon exchange forces to clarify their bound energy disparity.

Contribution

It introduces a combined force model in the Bethe-Salpeter framework to analyze meson mass differences, highlighting the dominant energy of the π meson.

Findings

The π meson has significantly larger bound energy than the ρ meson.

Inclusion of both exchange forces clarifies mass difference origins.

Explicit projection simplifies the Bethe-Salpeter equation analysis.

Abstract

In this article, we take into account the one-pion exchange force besides the one-gluon exchange force to study the mass difference of the $\pi$ and $\rho$ mesons with the Bethe-Salpeter equation. After projecting the Bethe-Salpeter equation into an simple form, we can see explicitly that the bound energy $|E_\pi|\gg |E_\rho|$.