Analysis Of Mass Difference Of The π And ρ With Bethe-salpeter Equation

Constituent quark models have given a lot of successful descriptions of the hadron mass spectroscopy. The simple constituent quark mass plus hyperfine spin interaction model works well for the ground state mesons. The mass difference between the π and ρ mesons is huge, we have to resort to large hyperfine spin-spin interactions for explanation. We can study the fine spin-orbit and hyperfine spin-spin interaction by the single-gluon exchange potential in the QCD-inspired linear confinement potential plus one-gluon exchange potential model. However we concern why the hyperfine interaction is so large, and how to understand it in the quantum field theory. The Bethe-Salpeter equation is the traditional method to deal with the two-body relativistic bound state problems. It is widely used in the particle physics and has a lot of successful descriptions of the nature of a large number of mesons. In the framework of the Bethe-Salpeter equation, we introduce the single-gluon exchange potential and Goldstone-boson exchange potential, simplify the Bethe-Salpeter equation, and analyze the huge mass difference between π and ρ mesons. We introduce a phenomenological potential model to parameterize the coupling constants of the quark-gluon and quark-Goldstone-bosons. Firstly, we take the coupling constant as the modified Gaussian distribution, then introduce two free parameters for the coupling strength and the distribution width. Numerical results indicate that the modified Gaussian distributions lead to very well convergent integrals. Furthermore, we try to use a new potential model to parameterize the coupling constants to obtain faster convergent integrals and better numerical results. Form the simplified Bethe-Salpeter equation and the numerical results we can see that the relation between the bound state energy|Eπ|>>|Eρ|. Furthermore, we obtain similar conclusions for other pseudoscalar mesons and vector mesons, the quark-antiquark systems form more stable bound states in the pseudoscalar channels than that in the vector channels.