A massive Feynman integral and some reduction relations for Appell functions

TL;DR

This paper derives explicit, symmetric expressions for a one-loop two-point Feynman integral in terms of Appell functions and establishes new reduction relations to hypergeometric functions, enhancing mathematical understanding of these integrals.

Contribution

It provides new explicit formulas for the Feynman integral using Appell functions and introduces novel reduction relations to hypergeometric functions.

Findings

Explicit symmetric expressions for the Feynman integral in terms of Appell functions.

New reduction relations between Appell functions and Gauss hypergeometric functions.

Mathematical results that facilitate the evaluation of Feynman integrals.

Abstract

New explicit expressions are derived for the one-loop two-point Feynman integral with arbitrary external momentum and masses $m_1^2$ and $m_2^2$ in D dimensions. The results are given in terms of Appell functions, manifestly symmetric with respect to the masses $m_i^2$. Equating our expressions with previously known results in terms of Gauss hypergeometric functions yields reduction relations for the involved Appell functions that are apparently new mathematical results.