On the Mass Difference between π and ρ Using a Relativistic Two-Body Model

Abstract

The big mass difference between the pion (π) and rho meson (ρ) possibly originates from the spin-dependent nature of the interactions in the two states since these two states are similar except for spin. Both π and ρ are quark-antiquark systems which can be treated using the two-body Dirac equations (TBDE) of constraint dynamics. This relativistic approach for two-body system has the advantage over the non-relativistic treatment in the sense that the spin-dependent nature is automatically coming out from the formalism. We employed Dirac’s relativistic constraint dynamics to describe quark-antiquark systems. Within this formalism, the 16-component Dirac equation is reduced to the 4-component 2nd-order differential equation and the radial part of this equation is simply a Schrödinger-type equation with various terms calculated from the basic radial potential. We used a modified Richardson potential for quark-antiquark systems which satisfies the conditions of confinement and asymptotic freedom. We obtained the wave functions for these two mesons which are not singular at short distances. We also found that the cancellation between the Darwin and spin-spin interaction terms occurs in the π mass but not in the ρ mass and this is the main source of the big difference in the two meson masses.