# Bound state equation for the Nakanishi weight function

## TL;DR

This paper derives a new integral equation for the Nakanishi weight function in bound states, linking Bethe-Salpeter and Light-Front formalisms through complex analysis and integral transforms.

## Contribution

It introduces a novel integral equation for the Nakanishi weight function applicable to any kernel from an irreducible Feynman amplitude.

## Key findings

- Derived a new integral equation for the Nakanishi weight function.
- Expressed the weight function in terms of Light-Front wave functions.
- Provided a method to obtain the kernel from Bethe-Salpeter kernels.

## 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 $g= N g$, 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.