Solution to Bethe–Salpeter equation via Mellin–Barnes transform

We consider the Mellin–Barnes (MB) transform of the triangle ladder-like scalar diagram in d=4d=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 the Belokurov–Usyukina reduction method for four-dimensional scalar integrals in position space. The result is represented in terms of the Euler ψ   function and its derivatives. We derive new formulas for the MB two-fold integration in the complex planes of two complex variables. We demonstrate that these formulas solve the Bethe–Salpeter equation. We comment on further applications of the solution to the Bethe–Salpeter equation for the vertices in N=4N=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, the theory of integral transforms, or the theory of polylogarithms in general, but has its origin in a simple recursive property of smooth functions, which may be shown by using basic methods of mathematical analysis.