The Bethe-Salpeter equation with fermions

Abstract

The Bethe-Salpeter (BS) equation in the ladder approximation is studied within a fermion theory: two fermion fields (constituents) with mass m interacting via an exchange of a scalar field with mass μ. The BS equation can be written in the form of an integral equation in the configuration Euclidean x-space with the symmetric kernel K for which Tr K 2 = ∞ due to the singular character of the fermion propagator. This kernel is represented in the form K = K 0 + K I . The operator K 0 with Tr K 0 2 = ∞ is of the “fall at the center” potential type and describes a continuous spectrum only. Besides the presence of this operator leads to a restriction on the value of the coupling constant. The kernel K I with Tr K I 2 < ∞ is responsible for bound fermion-fermion states.