Bound state equation for 4 or more relativistic particles

TL;DR

This paper extends a 3D reduction method for the N-particle Bethe-Salpeter equation to four particles, addressing overcounting issues with counterterms and proposing a modified reduction approach applicable to any N.

Contribution

It introduces a novel approach to handle the 4-particle Bethe-Salpeter equation with counterterms and proposes a modified reduction method for arbitrary N.

Findings

Successfully applied the method to 4-particle case with counterterms.

Developed a modified reduction method that directly yields the 3D potential.

Addresses overcounting issues in multi-particle Bethe-Salpeter equations.

Abstract

We apply the 3D reduction method we recently proposed for the N-particle Bethe-Salpeter equation to the 4-particle case. We find that the writing of the Bethe-Salpeter equation is not a straightforward task when N is larger or equal to 4, owing to the presence of mutually unconnected interactions, which could lead to an overcounting of some diagrams in the resulting full propagator. We overcome this difficulty in the N=4 case by including three counterterms in the Bethe-Salpeter kernel. The application of our 3D reduction method to the resulting Bethe-Salpeter equation suggests us a modified 3D reduction method, which gives directly the 3D potential, without the need of writing the Bethe-Salpeter kernel explicitly. The modified reduction method is usable for all N.