Closed Expression of the Interaction Kernel in the Bethe-Salpeter Equation for Quark-Antiquark Bound States

Abstract

The interaction kernel in the Bethe-Salpeter equation for quark-antiquark bound states is derived from QCD in the general case where the quark and the antiquark can be of different flavors or the same flavor. The kernel is derived with the aid of the Bethe-Salpeter equations satisfied by the quark-antiquark four-point Green function. The latter equations are established based on the equations of motion obeyed by the quark and antiquark propagators, the quark-antiquark four-point Green function and some other kinds of Green functions in which the gluon field is involved. The interaction kernel derived is given a closed and explicit expression. Since the kernel is represented in terms of only a few types of Green functions, it is not only convenient for perturbative calculations, but also suitable for nonperturbative investigations.