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

TL;DR

Tarasov introduces a systematic set of recurrences that connect Feynman integrals across space-time dimensions by treating $d$ as an additional recurrence parameter, producing relations between $d$ and $d-2$ that complement traditional IBP methods. The framework uses the parametric representation with the polynomial $D(oldsymbol{eta})$ and a differential operator to shift dimensions, extending to tensor integrals and solving irreducible-numerator issues within the same generalized system. The paper provides explicit $d$-shift relations for one-, two-, and three-loop diagrams, derives new one-loop $n$-point reductions, and demonstrates practical evaluation and $oldsymbol{ ext{ε}}$-expansion techniques by transferring known results between dimensions. A key application shows how $d$-recurrences yield the $oldsymbol{ ext{ε}}$-expansion for a two-loop self-energy in $d=6-2oldsymbol{ ext{ε}}$ from the $d=4-2oldsymbol{ ext{ε}}$ basis, indicating broad potential for efficient, dimension-aware diagram computations.

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.