Bound state equation for the Nakanishi weight function

Abstract

The bound state Bethe–Salpeter amplitude was expressed by Nakanishi using a two-dimensional integral representation, in terms of a smooth weight function g, which carries the detailed dynamical information. A similar, but one-dimensional, integral representation can be obtained for the Light-Front wave function in terms of the same weight function g. By using the generalized Stieltjes transform, we first obtain g in terms of the Light-Front wave function in the complex plane of its arguments. Next, a new integral equation for the Nakanishi weight function g is derived for a bound state case. It has the standard form g=Ng, where N is a two-dimensional integral operator. We give the prescription for obtaining the kernel N starting with the kernel K of the Bethe–Salpeter equation. The derivation is valid for any kernel given by an irreducible Feynman amplitude.

Keywords

• Bethe–Salpeter equation
• Nakanishi representation
• Light-Front