Massive Feynman diagrams and inverse binomial sums

TL;DR

This paper analytically evaluates complex inverse binomial sums related to massive Feynman diagrams using hypergeometric functions and generating functions, providing new results for multi-loop diagrams and epsilon-expansions.

Contribution

It introduces a novel analytical approach to evaluate inverse binomial sums linked to hypergeometric functions, applicable to various multi-loop Feynman diagrams.

Findings

Derived new formulas for inverse binomial sums in Feynman diagram calculations.

Expressed results in terms of generalized polylogarithmic functions.

Applied methods to higher-order epsilon-expansions of physical quantities.

Abstract

When calculating higher terms of the epsilon-expansion of massive Feynman diagrams, one needs to evaluate particular cases of multiple inverse binomial sums. These sums are related to the derivatives of certain hypergeometric functions with respect to their parameters. Exploring this connection and using it together with an approach based on generating functions, we analytically calculate a number of such infinite sums, for an arbitrary value of the argument which corresponds to an arbitrary value of the off-shell external momentum. In such a way, we find a number of new results for physically important Feynman diagrams. Considered examples include two-loop two- and three-point diagrams, as well as three-loop vacuum diagrams with two different masses. The results are presented in terms of generalized polylogarithmic functions. As a physical example, higher-order terms of the epsilon-expansion of the polarization function of the neutral gauge bosons are constructed.