Connection between Feynman integrals having different values of the space-time dimension

TL;DR

This paper introduces a systematic algorithm to derive recurrence relations for dimensionally regularized Feynman integrals, linking integrals in different space-time dimensions using explicit differential operators, applicable to complex multi-loop diagrams.

Contribution

It presents a novel method for obtaining recurrence relations between Feynman integrals in different dimensions, complementing existing techniques and addressing irreducible numerators.

Findings

The method works for one-, two-, and three-loop integrals.

Explicit formulas for differential operators are derived for each diagram.

The approach effectively handles irreducible numerators.

Abstract

A systematic algorithm for obtaining recurrence relations for dimensionally regularized Feynman integrals w.r.t. the space-time dimension $d$ is proposed. The relation between $d$ and $d-2$ dimensional integrals is given in terms of a differential operator for which an explicit formula can be obtained for each Feynman diagram. We show how the method works for one-, two- and three-loop integrals. The new recurrence relations w.r.t. $d$ are complementary to the recurrence relations which derive from the method of integration by parts. We find that the problem of the irreducible numerators in Feynman integrals can be naturally solved in the framework of the proposed generalized recurrence relations.