The Spectrum and Confinement for the Bethe-Salpeter Equation

Abstract

The problem of calculating of the mass spectrum of the two-body Bethe-Salpeter equation is studied with no reduction to the three-dimensional ("quasipotential") equation. The method to find the ground state and excited states for a channel with any quantum numbers is presented. The problem of the confining interaction for the Bethe-Salpeter equation is discussed from the point of view of formal properties of the bound state spectrum, but with only an inspiration from the QCD. We study the kernel that is non-vanishing at large Euclidian intervals, $R_E\to \infty$, which is constructed as a special limiting case of a sum of the covariant one-boson-exchange kernels. When the usual attractive interaction is added, it is found that this kernel is similar in its effect to the non-relativistic potential in coordinate space, $V(r)$, with $V(r\to \infty) \to V_{\infty}$. The positive real constant $V_{\infty}$ gives the scale that define the limit of bound state spectrum compared to the sum of the constituent masses, $M < 2m + V_{\infty}$. At the same time the self-energy corrections remove the singularities from the propagators of the constituents, i.e. constituents do not propagate as free particles. Combination of these features of the solutions allows an interpretation of this type of interaction as a confining one. The illustrative calculations are presented for a model of massive scalar particles with scalar interaction, i.e. the "massive Wick model".