Massive One-loop Conformal Feynman Integrals and Quadratic Transformations of Multiple Hypergeometric Series

TL;DR

This paper introduces a Mellin-Barnes integral technique to derive series representations of massive one-loop conformal Feynman integrals, confirming and extending previous results, and proving conjectures related to hypergeometric series transformations.

Contribution

The authors develop a novel Mellin-Barnes integral method for massive one-loop conformal integrals, proving conjectures and revealing new quadratic transformations in hypergeometric functions.

Findings

Derived series representations for conformal integrals in various configurations.

Proved two conjectures relating integrals to hypergeometric series.

Identified spurious contributions in Yangian bootstrap approaches.

Abstract

The computational technique of $N$-fold Mellin-Barnes (MB) integrals, presented in a companion paper by the same authors, is used to derive sets of series representations of the massive one-loop conformal 3-point Feynman integral in various configurations. This shows the great simplicity and efficiency of the method in nonresonant cases (generic propagator powers) as well as some of its subtleties in the resonant ones (for unit propagator powers). We confirm certain results in the physics and mathematics literature and provide many new results, some of them dealing with the more general massive one-loop conformal $n$-point case. In particular, we prove two recent conjectures that give the massive one-loop conformal $n$-point integral (for generic propagator powers) in terms of multiple hypergeometric series. We show how these conjectures, that were deduced from a Yangian bootstrap analysis, are related by a tower of new quadratic transformations in Hypergeometric Functions Theory. Finally, we also use our MB method to identify spurious contributions that can arise in the Yangian approach.