Functional reduction of one-loop Feynman integrals with arbitrary masses

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 F-2(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 kinematic variables is considered.

Automated summary

The method of functional reduction for dimensionally regularized one-loop Feynman integrals with massive propagators is explained, based on the author's proposed functional relations. Explicit formulae are provided for simplifying one-loop scalar integrals by expressing them as ratios of polynomials in masses and kinematic invariants. It is shown that a general scalar n-point integral can be expressed as a linear combination of integrals depending only on n variables, and these integrals are explicitly given in terms of hypergeometric functions of (n - 1) dimensionless variables.