Explicit solutions of the multi-loop integral recurrence relations and its application

TL;DR

The paper tackles explicit solutions to IBP-like recurrence relations for multi-loop Feynman integrals by formulating a contour-integral representation for solution functions that satisfy the recurrence. This approach bypasses ad hoc reduction to master integrals and yields dimension-shifting relations, providing efficient computation, as demonstrated on 3-loop QED vacuum polarization where full D-dependence is obtainable in minutes rather than hours. It further extends the framework to non-vacuum cases with external momenta, deriving a generalized recurrence structure and indicating a determinant-based weight function that could lead to explicit formulas, albeit with further work needed. Overall, the method offers a promising route to explicit, efficient multi-loop integral solutions and potential connections to existing recurrences.

Abstract

The approach to the constructing explicit solutions of the recurrence relations for multi-loop integrals are suggested. The resulting formulas demonstrate a high efficiency, at least for 3-loop vacuum integrals case. They also produce a new type of recurrence relations over the space-time dimension.