New four-dimensional integrals by Mellin-Barnes transform

TL;DR

This paper develops a Mellin-Barnes transform method to evaluate complex four-dimensional integrals relevant to N=4 supersymmetric Yang-Mills theory, confirming their expression in terms of UD functions.

Contribution

It introduces a novel technique for calculating four-dimensional integrals with logarithmic numerators using Mellin-Barnes transform and contour modifications, extending previous methods.

Findings

Reproduces known results with Gegenbauer polynomial technique

Shows the Mellin-Barnes contour method applies with logarithmic numerators

Confirms Green functions can be expressed in terms of UD functions

Abstract

This paper is devoted to the calculation by Mellin-Barnes transform of a especial class of integrals. It contains double integrals in the position space in d = 4-2e dimensions, where e is parameter of dimensional regularization. These integrals contribute to the effective action of the N = 4 supersymmetric Yang-Mills theory. The integrand is a fraction in which the numerator is a logarithm of ratio of spacetime intervals, and the denominator is the product of powers of spacetime intervals. According to the method developed in the previous papers, in order to make use of the uniqueness technique for one of two integrations, we shift exponents in powers in the denominator of integrands by some multiples of e. As the next step, the second integration in the position space is done by Mellin-Barnes transform. For normalizing procedure, we reproduce first the known result obtained earlier by Gegenbauer polynomial technique. Then, we make another shift of exponents in powers in the denominator to create the logarithm in the numerator as the derivative with respect to the shift parameter delta. We show that the technique of work with the contour of the integral modified in this way by using Mellin-Barnes transform repeats the technique of work with the contour of the integral without such a modification. In particular, all the operations with a shift of contour of integration over complex variables of two-fold Mellin-Barnes transform are the same as before the delta modification of indices, and even the poles of residues coincide. This confirms the observation made in the previous papers that in the position space all the Green function of N = 4 supersymmetric Yang-Mills theory can be expressed in terms of UD functions.