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.

52 pages, 1 figure

References in corpus (5)

• Feynman Diagrams and Differential Equations
• Gauss hypergeometric function: reduction, epsilon-expansion for integer/half-integer parameters and Feynman diagrams
• New relationships between Feynman integrals
• Scalar 1-loop Feynman integrals as meromorphic functions in space-time dimension d
• q-derivatives of multivariable q-hypergeometric function with respect to their parameters

Cited by in corpus (7)

• $\epsilon$-Expansion of Multivariable Hypergeometric Functions Appearing in Feynman Integral Calculus
• Analytic continuations and numerical evaluation of the Appell $F_1$, $F_3$, Lauricella $F_D^{(3)}$ and Lauricella-Saran $F_S^{(3)}$ and their Application to Feynman Integrals
• Analytic continuations of the Horn $H_1$ and $H_5$ functions
• Expansion of hypergeometric functions in terms of polylogarithms with nontrivial variable change
• $\text{AlgRel.wl}$: Algebraic Relations for the Product of Propagators in Feynman integrals
• Calculation of one-loop integrals for four-photon amplitudes by functional reduction method
• The massless non-adjacent double off-shell scalar box integral -- branch cut structure and all-order epsilon expansion