Functional reduction of Feynman integrals

• Published in: The journal of high energy physics Vol. 2019; no. 2; pp. 1 - 34
• Main Author: Tarasov, O. V.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2019; Springer Nature B.V; SpringerOpen
• Subjects: Classical and Quantum Gravitation; Elementary Particles; Functional equations; High energy physics; Hypergeometric functions; Integrals; Kinematics; Mathematical analysis; NLO Computations; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Reduction; Regular Article - Theoretical Physics; Relativity Theory; String Theory; NLO Computations
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP02(2019)173

A bstract A method for reducing Feynman integrals, depending on several kinematic variables and masses, to a combination of integrals with fewer variables is proposed. The method is based on iterative application of functional equations proposed by the author. The reduction of the one-loop scalar triangle and box integrals with massless internal propagators to simpler integrals is described in detail. The triangle integral depending on three variables is represented as a sum over three integrals depending on two variables. By solving the dimensional recurrence relations for these integrals, an analytic expression in terms of the 2 F 1 Gauss hypergeometric function and the logarithmic function was derived. By using the functional equations, the one-loop box integral with massless internal propagators, which depends on six kinematic variables, was expressed as a sum of 12 terms. These terms are proportional to the same integral depending only on three variables different for each term. For this integral with three variables, an analytic result in terms of the F 1 Appell and 2 F 1 Gauss hypergeometric functions was derived by solving the recurrence relation with respect to the spacetime dimension d . The reduction equations for the box integral with some kinematic variables equal to zero are considered.