Generalized Recurrence Relations for Two-loop Propagator Integrals with Arbitrary Masses

TL;DR

Tarasov introduces a comprehensive method to reduce arbitrary-mass two-loop propagator integrals to a finite master basis by combining tensor decomposition with generalized integration-by-parts recurrence relations and dimension-shifting operators. The approach yields a minimal set of master integrals, notably $F^{(d)}_{11111}$, $V^{(d)}_{1111}$, and $J^{(d)}_{111}(q^2)$, from which all scalar and tensor integrals can be derived, including cases with irreducible numerators. The framework handles all mass/momentum configurations, accounts for special kinematic determinants, and provides explicit recurrences to lower indices and dimensions, with an implementation in FORM demonstrating practical runtimes. This work significantly improves the analytic and semi-analytic tractability of mass-dependent two-loop calculations, and sets the stage for extending the method to two-loop vertex functions. The results offer a clear pathway to organized, scalable evaluations of two-loop SM corrections across diverse mass spectra.

Abstract

An algorithm for calculating two-loop propagator type Feynman diagrams with arbitrary masses and external momentum is proposed. Recurrence relations allowing to express any scalar integral in terms of basic integrals are given. A minimal set consisting of 15 essentially two-loop and 15 products of one-loop basic integrals is found. Tensor integrals and integrals with irreducible numerators are represented as a combination of scalar ones with a higher space-time dimension which are reduced to the basic set by using the generalized recurrence relations proposed in Ref.[1] (Phys.Rev.D54 (1996) 6479).