Functional reduction of one-loop Feynman integrals with arbitrary masses

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 2F1, the Appell function F1 and the hypergeometric Lauricella — Saran function FS. A modification of the functional reduction procedure for some special values of kinematic variables is considered.

The massless non-adjacent double off-shell scalar box integral -- branch cut structure and all-order epsilon expansion

“…[5] furthermore suggests that the adjacent double off-shell box integral is of the same complexity as the general case. The method of functional equations has recently been generalized to arbitrary internal masses [10]. These results might be suitable starting points for generalizing the program of the present paper towards an all-order ε-expansion with explicit real and imaginary parts for kinematics beyond the scope of the present work.…”