Equation for the Nakanishi weight function using the inverse Stieltjes transform

TL;DR

This paper derives an integral equation for the Nakanishi weight function in bound state Bethe-Salpeter equations using the inverse Stieltjes transform, providing a new method to compute the Nakanishi function g.

Contribution

It introduces a novel integral equation for the Nakanishi weight function g using the inverse Stieltjes transform, linking the Bethe-Salpeter kernel to the Nakanishi representation.

Findings

Derived an integral equation g= Vg for the Nakanishi function.

Provided a method to obtain the kernel V from the Bethe-Salpeter kernel K.

Established a framework for calculating the Nakanishi weight function in bound states.

Abstract

The bound state Bethe-Salpeter amplitude was expressed by Nakanishi in terms of a smooth weight function g. By using the generalized Stieltjes transform, we derive an integral equation for the Nakanishi function g for a bound state case. It has the standard form g= Vg, where V is a two-dimensional integral operator. The prescription for obtaining the kernel V starting with the kernel K of the Bethe-Salpeter equation is given.