Feynman integrals in two dimensions and single-valued hypergeometric functions

TL;DR

This paper demonstrates that all massless two-dimensional Feynman integrals can be expressed using single-valued hypergeometric functions, linking them to intersection theory and providing explicit formulas for various loop integrals.

Contribution

It introduces a unified framework expressing two-dimensional Feynman integrals in terms of single-valued hypergeometric functions and their bilinear representations, connecting physics and advanced mathematics.

Findings

All one-loop integrals are expressed via Lauricella functions.

L-loop ladder integrals relate to generalized hypergeometric functions.

The approach uses intersection numbers in twisted homology.

Abstract

We show that all Feynman integrals in two Euclidean dimensions with massless propagators and arbitrary non-integer propagator powers can be expressed in terms of single-valued analogues of Aomoto-Gelfand hypergeometric functions. The latter can themselves be written as bilinears of hypergeometric functions, with coefficients that are intersection numbers in a twisted homology group. As an application, we show that all one-loop integrals in two dimensions with massless propagators can be written in terms of Lauricella $F_D^{(r)}$ functions, while the $L$-loop ladder integrals are related to the generalised hypergeometric ${}_{L+1}F_L$ functions.