Application and explicit solution of recurrence relations with respect to space-time dimension

TL;DR

The work develops a d-shift recurrence framework to reduce tensor Feynman integrals to a scalar basis, enabling explicit analytic solutions for one-loop n-point functions. It extends the method to two-loop propagator-type integrals and provides closed-form, hypergeometric representations (including Appell $F_1$ for the 3-point case) with attention to asymptotics at large space-time dimension. The approach leverages parametric representations and recurrence relations to express master integrals in terms of Gauss $_2F_1$, Appell $F_1$, and related functions, offering practical tools for multi-leg and multi-loop radiative-correction calculations. These results enhance analytic control over Feynman integrals and support precision computations in high-energy physics.

Abstract

A short review of the method for the tensor reduction of Feynman integrals based on recurrence relations with respect to space-time dimension d- is given. A solution of the difference equation with respect to d for the n - point one-loop integrals with arbitrary momenta and masses is presented. The result is written as multiple hypergeometric series depending on ratios of Gram determinants. For the 3-point function a new expression in terms of the Appell hypergeometric function F_1 is presented.