Solution to Bethe-Salpeter equation via Mellin-Barnes transform

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.