Solution to Bethe-Salpeter equation via Mellin-Barnes transform

TL;DR

This paper develops a method using Mellin-Barnes transforms to solve the Bethe-Salpeter equation for scalar diagrams, providing new formulas and demonstrating applications in supersymmetric Yang-Mills theory.

Contribution

It introduces a reduction technique for multi-fold Mellin-Barnes transforms to two-fold transforms, solving the Bethe-Salpeter equation with new integral formulas.

Findings

Reduction of multi-fold MB transforms to two-fold transforms

Derivation of new formulas for complex plane integration

Solution of Bethe-Salpeter equation using these formulas

Abstract

We consider Mellin-Barnes transform of triangle ladder-like scalar diagram in d=4 dimensions. It is shown how the multi-fold MB transform of the momentum integral corresponding to an arbitrary number of rungs is reduced to the two-fold MB transform. For this purpose we use Belokurov-Usyukina reduction method for four-dimensional scalar integrals in the position space. The result is represented in terms of Euler psi-function and its derivatives. We derive new formulas for the MB two-fold integration in complex planes of two complex variables. We demonstrate that these formulas solve Bethe-Salpeter equation. We comment on further applications of the solution to the Bethe-Salpeter equation for the vertices in N=4 supersymmetric Yang-Mills theory. We show that the recursive property of the MB transforms observed in the present work for that kind of diagrams has nothing to do with quantum field theory, theory of integral transforms, or with theory of polylogarithms in general, but has an origin in a simple recursive property for smooth functions which can be shown by using basic methods of mathematical analysis.