Functional reduction of one-loop Feynman integrals with arbitrary masses

• Published in: The journal of high energy physics Vol. 2022; no. 6; pp. 155 - 49
• Main Author: Tarasov, O. V.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 28.06.2022; Springer Nature B.V; SpringerOpen
• Subjects: Classical and Quantum Gravitation; Dimensionless analysis; Electroweak Precision Physics; Elementary Particles; High energy physics; Higher Order Electroweak Calculations; Higher-Order Perturbative Calculations; Hypergeometric functions; Integrals; Kinematics; Mathematical analysis; Physics; Physics and Astronomy; Polynomials; Polyvinylidene chlorides; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Reduction; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Higher Order Electroweak Calculations; Electroweak Precision Physics; Higher-Order Perturbative Calculations
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP06(2022)155

A bstract 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 kinematic variables is considered.