This paper describes a new algorithm that turns complicated two-loop tensor integrals into simpler scalar integrals. The work is aimed at improving automated calculations of particle collisions at the Large Hadron Collider and future colliders. The authors implemented the method inside the OpenLoops framework and report a first successful validation of the approach.

The authors split two-loop amplitude calculations into three parts: the process-dependent tensor coefficients, the loop-momentum tensor integrals, and the parts that live only in (D−4) dimensions (where D=4−2ε is the regularisation dimension used to handle infinities). They separate the genuinely four-dimensional numerator from the (D−4)-dimensional remainder and then apply a recursive tensor decomposition to the four-dimensional part. A key analytic identity for a rank-two loop-momentum tensor (q^μ q^ν) is used to reduce higher-rank tensors step by step. The remaining rank-1 pieces are handled with a Passarino–Veltman style reduction, so the whole integrand is expressed in terms of scalar integrals plus well-defined shifts of propagator powers.

The implementation mixes symbolic and numerical steps. A Mathematica generator builds the analytic reduction rules and a Fortran code evaluates them efficiently for each phase-space point. The scalar integrals produced by the reduction are then decreased to a small set of ‘‘master integrals’’ using integration-by-parts identities and the Laporta algorithm. For that step the authors use Kira with the FireFly backend and tools such as ImproveMasters.m to pick a good master basis. The master integrals in this first implementation are evaluated numerically with pySecDec; the generator prepares the required orders in ε (the small parameter in D=4−2ε) and pySecDec is called for every phase-space point.