Two-loop tensor integral coefficients in OpenLoops

TL;DR

This paper introduces a fully automated, efficient algorithm for constructing two-loop scattering amplitude integrands, significantly advancing computational methods in quantum field theory for Standard Model processes.

Contribution

It generalizes the open-loops method to two loops, enabling automated, process-independent construction of two-loop integrands with optimized CPU performance.

Findings

CPU cost scales linearly with the number of two-loop diagrams

Algorithm outperforms naive solutions by two orders of magnitude

Implemented in OpenLoops for Standard Model processes

Abstract

We present a new and fully general algorithm for the automated construction of the integrands of two-loop scattering amplitudes. This is achieved through a generalisation of the open-loops method to two loops. The core of the algorithm consists of a numerical recursion, where the various building blocks of two-loop diagrams are connected to each other through process-independent operations that depend only on the Feynman rules of the model at hand. This recursion is implemented in terms of tensor coefficients that encode the polynomial dependence of loop numerators on the two independent loop momenta. The resulting coefficients are ready to be combined with corresponding tensor integrals to form scattering probability densities at two loops. To optimise CPU efficiency we have compared several algorithmic options identifying one that outperforms naive solutions by two orders of magnitude. This new algorithm is implemented in the OpenLoops framework in a fully automated way for two-loop QED and QCD corrections to any Standard Model process. The technical performance is discussed in detail for several $2\to2$ and $2\to 3$ processes with up to order $10^5$ two-loop diagrams. We find that the CPU cost scales linearly with the number of two-loop diagrams and is comparable to the cost of corresponding real-virtual ingredients in a NNLO calculation. This new algorithm constitutes a key building block for the construction of an automated generator of scattering amplitudes at two loops.