The inverse Mellin transform via analytic continuation

Abstract

We present a method to calculate the x-space expressions of massless or massive operator matrix elements in QCD and QED containing local composite operator insertions, depending on the discrete Mellin index N, directly, without computing the Mellin-space expressions in explicit form analytically. Here N belongs either to the even or odd positive integers. The method is based on the resummation of the operators into effective propagators and relies on an analytic continuation between two continuous variables. We apply it to iterated integrals as well as to the more general case of iterated non-iterative integrals, generalizing the former ones. The x-space expressions are needed to derive the small-x behaviour of the respective quantities, which usually cannot be accessed in N-space. We illustrate the method for different (iterated) alphabets, including non-iterative 2F1 and elliptic structures, as examples. These structures occur in different massless and massive three-loop calculations. Likewise the method applies even to the analytic closed form solutions of more general cases of differential equations which do not factorize into first-order factors.

An explicit expression of generating function for one-loop tensor reduction

“…Another notable advancement is the introduction of the "generating function" [94][95][96][97][98]. In fact, the concept of a generating function is well-established in both physics and mathematics.…”

“…In [95], Kosower employed integration-by-parts (IBP) generating vectors to derive appropriate IBP reduction relations in certain special 2-loop case. More recently, these functions have also been discussed in [96], detailing how to directly proceed from these representations to the physical result.…”