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

Abstract

A systematic algorithm for obtaining recurrence relations for dimensionally regularized Feynman integrals with respect to the space-time dimension $d$ is proposed. The relation between $d$- and ($d\ensuremath{-}2$)-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 with respect to $d$ 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.