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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.
To validate the full chain, the authors tested a two-to-two scattering example with a pentagon–triangle two-loop topology. They set the overall energy variable s=1, used t and an external mass m^2 as kinematic variables, and chose massless internal lines while keeping the external legs massive to regulate infrared divergences. As a nontrivial sample they reduced a rank‑4 two-loop integral and rewrote a contracted version of it in terms of scalar integrals. According to the authors, this produced a first successful validation computation. They note, however, that the final result depends on the numerical precision of the externally computed master integrals.
Important caveats are mentioned. Rational terms that originate from infrared singularities are still under investigation and are not yet fully settled in this framework. The current master-integral pipeline relies on pySecDec calls per phase-space point, which affects performance and ties the final accuracy to that tool. The authors plan further optimisations: shifting parts of the integration-by-parts reduction into the numerical code, exploring alternative master-integral tools, and adjusting requested precision depending on prefactors. The work is a step toward fully automated next-to-next-to-leading-order tools, but some pieces remain to be improved and tested more widely.