Abstract:
We propose a method for using functional equations to calculate Feynman integrals analytically. We describe the algorithm for solving the functional equations and show that a solution of a functional equation for the Feynman integral is a combination of several integrals with fewer kinematic variables. In some cases, using the functional equations, we can also reduce these integrals to integrals with even fewer variables. Such a stepwise application of the functional equations leads to integrals that can be calculated more simply than the original integral. We apply the proposed method to several one-loop integrals. For the three-point and four-point integrals with massless propagators and an arbitrary space dimension d, we obtain analytic expressions in terms of hypergeometric functions.
This research was supported in part by the German
Research Foundation DFG (2012–2016) in the framework of the Research
Collaboration SFB 676 Particles, Strings, and the Early Universe: the Structure of Matter and Space–Time.
This publication is cited in the following 9 articles:
S. I. Bezrodnykh, O. V. Dunin-Barkovskaya, “Estimation of the Remainder Terms of Certain Horn Hypergeometric Series”, Comput. Math. and Math. Phys., 64:12 (2024), 2737
B. Ananthanarayan, S. Bera, T. Pathak, “AlgRel.wl: Algebraic relations for the product of propagators in Feynman integrals”, Nuclear Physics B, 995 (2023), 116345
S. I. Bezrodnykh, “Constructing basises in solution space of the system of equations for the Lauricella Function FD(N)”, Integral Transforms and Special Functions, 34:11 (2023), 813–834
S. I. Bezrodnykh, “Analytic continuation of Lauricella's function F(N)D for large in modulo variables near hyperplanes {zJ=zl}”, Integral Transform. Spec. Funct., 33:4 (2022), 276–291
I S. Bezrodnykh, “Analytic continuation of Lauricella's function F(N)D for variables close to unit near hyperplanes {zj=zl}”, Integral Transform. Spec. Funct., 33:5 (2022), 419–433
S. I. Bezrodnykh, “Formulas for analytic continuation of Horn functions of two variables”, Comput. Math. Math. Phys., 62:6 (2022), 884–903
S. I. Bezrodnykh, “Formulas for computing the Lauricella function in the case of crowding of variables”, Comput. Math. Math. Phys., 62:12 (2022), 2069–2090
S. I. Bezrodnykh, “Analytic continuation of the Kampé de Fériet function and the general double Horn series”, Integral Transforms and Special Functions, 33:11 (2022), 908
Bezrodnykh S.I., “Horn'S Hypergeometric Functions With Three Variables”, Integral Transform. Spec. Funct., 32:3 (2021), 207–223
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