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 ψ-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.

Zastavnyi operators and positive definite radial functions

“…We now invoke the well known relationship [2], and, by abuse of notation, we now write z := z . We have (2.9)…”

Multi-fold contour integrals of certain ratios of Euler gamma functions from Feynman diagrams: orthogonality of triangles

“…We have used here the definition of the Mellin-Barnes transform D (z2,z3) [ν 1 , ν 2 , ν 3 ] from our Ref. [1],…”

“…The formulas for the Fourier transforms of the massless propagators may be found, for example, in Ref. [1]. The formula we need is…”

“…Mellin-Barnes (MB) transforms of the momentum integrals corresponding to ladder diagrams at any loop order have been studied in Refs. [7,1,8,9] in four-dimensional case. In Ref.…”

“…and is given for the first time in Ref. [1], where D (u,v) [ν 1 , ν 2 , ν 3 ] is MB transform of the one-loop massless triangle scalar momentum integral, ε 1 , ε 2 , ε 3 are three complex numbers which satisfy the condition ε 1 + ε 2 + ε 3 = 0, and J is a factor that depends on ε 1 , ε 2 , ε 3 only. Eq.…”