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 2 F 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.13.2.1 Series representation 13.2.2 Integral representations 13.3 Differential relations for the Lauricella-Saran function F S

On the evaluation of the Appell F2 double hypergeometric function

“…Let us illustrate this situation with a particular case, by considering the relation [38] F 1 1, 1, 1 2 ; 2;…”

“…which appears in the course of the derivation of the mathematical expression of the one-loop 3-point scalar Feynman integral with arbitrary masses and external momenta, using the functional reduction method of [38] (see Eq. (5.21) of this reference).…”

$ε$-Expansion of Multivariable Hypergeometric Functions Appearing in Feynman Integral Calculus

“…To give a few examples, two loop massive sunset diagram [12,13], three loop vacuum diagram [14], one loop two and three point scalar functions [15,16] are evaluated in terms of multivariable hypergeometric functions. Functional relations are used to find one loop Feynman integrals (see [17] and references within) in terms of multivariable hypergeometric functions. Feynman integrals can also be realized as GKZ hypergeometric system [18][19][20].…”