Dirac–Schrödinger equation for quark–antiquark bound states and derivation of its interaction kernel

Abstract

The four-dimensional Dirac–Schrödinger equation satisfied by quark–antiquark bound states is derived from quantum chromodynamics. Different from the Bethe–Salpeter equation, the equation derived is a kind of first-order differential equation of Schrödinger-type in the position space. Especially, the interaction kernel in the equation is given by two different closed expressions. One expression which contains only a few types of Green's functions is derived with the aid of the equations of motion satisfied by some kinds of Green's functions. Another expression, which is represented in terms of the quark, antiquark and gluon propagators and some kinds of proper vertices, is derived by means of the technique of irreducible decomposition of Green's functions. The kernel derived not only can easily be calculated by the perturbation method, but also provides a suitable basis for nonperturbative investigations. Furthermore, it is shown that the four-dimensional Dirac–Schrödinger equation and its kernel can be directly reduced to rigorous three-dimensional forms in the equal-time Lorentz frame and the Dirac–Schrödinger equation can be reduced to an equivalent Pauli–Schrödinger equation which is represented in the Pauli spinor space. To show the applicability of the closed expressions derived and to demonstrate the equivalence between the two different expressions of the kernel, the t-channel and s-channel one gluon exchange kernels are chosen as an example to show how they are derived from the closed expressions. In addition, the connection of the Dirac–Schrödinger equation with the Bethe–Salpeter equation is discussed.