Status of two-loop automation in OpenLoops

TL;DR

This paper reviews the current progress of OpenLoops in automating two-loop amplitude calculations, crucial for high-precision predictions at NNLO in particle physics, highlighting recent algorithmic and implementation advances.

Contribution

It presents the development status of OpenLoops for two-loop amplitude calculations, including tensor coefficient construction and the integration of rational counterterms for NNLO automation.

Findings

Automated tensor coefficient construction for QED and QCD corrections.

Implementation of two-loop rational counterterms for UV divergence handling.

Validation of the renormalisation procedure within OpenLoops.

Abstract

The calculation of hard scattering amplitudes up to NLO is automated in numerical tools, such as OpenLoops. The LHC and future experiments, however, demand high-precision predictions at NNLO and beyond for a wide range of particle processes. Hence, the development of a fully automated tool for numerical NNLO calculations is an important goal. In order to perform a numerical calculation, we decompose $D$-dimensional two-loop amplitudes into Feynman integrals with four-dimensional numerators and $(D-4)$-dimensional remainders, which contribute to the finite result through the interaction with the poles of Feynman integrals and are reconstructed during the subtraction procedure for these poles from universal rational terms. The integrals with four-dimensional numerators are further decomposed into loop momentum tensor integrals and tensor coefficients. We present the status of OpenLoops with respect to these building blocks. The algorithm for the construction of the tensor coefficients is implemented for QED and QCD corrections to the SM in a fully automated way. Recently, the renormalisation procedure and the reconstruction of the interplay of $(D-4)$-dimensional numerator parts with UV poles through two-loop rational counterterms has been implemented and validated using an in-house library for the reduction of simple tensor integrals.