Explicit solutions of the 3--loop vacuum integral recurrence relations

TL;DR

The paper addresses the computational bottleneck of 3-loop vacuum integrals by deriving explicit solutions to the IBP recurrence relations through a diagonalization approach and a six-variable generating function. This yields D-shifted recurrences and compact coefficient expressions, enabling direct and efficient calculations, as demonstrated in a four-mass/two-massless case and applied to QED vacuum polarization to obtain new and known constants. The method dramatically reduces intermediate expression growth compared to traditional recursive strategies and suggests potential generalization to more complex multi-loop or non-vacuum integrals. Overall, it provides a practical framework for high-order loop integrals with substantial computational advantages and broader applicability.

Abstract

Explicit formulas for the solutions of the recurrence relations for 3--loop vacuum integrals are suggested. This formulas can be used for direct calculations and demonstrate a high efficiency. They also produce a new type of recurrence relations over the space--time dimension.