New four-dimensional integrals by Mellin–Barnes transform

Abstract

This paper is devoted to the calculation of a special class of integrals by Mellin–Barnes transform. It contains double integrals in the position space in d=4−2ϵ dimensions, where ϵ 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 the logarithm of the ratio of space-time intervals, and the denominator is the product of powers of space-time 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 ϵ. 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 δ. 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 twofold Mellin–Barnes transform are the same as before the δ 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’s function of N=4 supersymmetric Yang–Mills theory can be expressed in terms of Usyukina–Davydychev functions.