Functional reduction of one-loop Feynman integrals with arbitrary masses

TL;DR

This paper introduces a functional reduction method for one-loop Feynman integrals with arbitrary masses, simplifying complex integrals into forms involving hypergeometric functions, thus aiding calculations in quantum field theory.

Contribution

The paper presents a novel functional reduction technique that expresses complex one-loop integrals with many variables as simpler integrals with fewer variables involving hypergeometric functions.

Findings

Explicit formulas for reducing scalar integrals to hypergeometric functions.

Reduction of n-point integrals to integrals depending on only n variables.

Analytic expressions for 2-, 3-, and 4-point integrals using hypergeometric functions.

Abstract

A method of functional reduction for the dimensionally regularized one-loop Feynman integrals with massive propagators is described in detail. The method is based on a repeated application of the functional relations proposed by the author. Explicit formulae are given for reducing one-loop scalar integrals to a simpler ones, the arguments of which are the ratios of polynomials in the masses and kinematic invariants. We show that a general scalar $n$-point integral, depending on $n(n+1)/2$ generic masses and kinematic variables, can be expressed as a linear combination of integrals depending only on $n$ variables. The latter integrals are given explicitly in terms of hypergeometric functions of $(n-1)$ dimensionless variables. Analytic expressions for the 2-, 3- and 4-point integrals, that depend on the minimal number of variables, were also obtained by solving the dimensional recurrence relations. The resulting expressions for these integrals are given in terms of Gauss' hypergeometric function $_2F_1$, the Appell function $F_1$ and the hypergeometric Lauricella - Saran function $F_S$. A modification of the functional reduction procedure for some special values of kinematical variables is considered.