On-the-fly reduction of open loops

TL;DR

This paper introduces an on-the-fly reduction method for open-loop algorithms that simplifies one-loop amplitude calculations, significantly improving speed and numerical stability for complex particle physics computations.

Contribution

It presents a novel on-the-fly reduction technique that unifies loop construction and reduction, reducing tensor complexity and enhancing computational efficiency.

Findings

Significant speedup over previous algorithms

High numerical stability through Gram determinant expansions

Effective for multi-leg NLO and NNLO calculations

Abstract

Building on the open-loop algorithm we introduce a new method for the automated construction of one-loop amplitudes and their reduction to scalar integrals. The key idea is that the factorisation of one-loop integrands in a product of loop segments makes it possible to perform various operations on-the-fly while constructing the integrand. Reducing the integrand on-the-fly, after each segment multiplication, the construction of loop diagrams and their reduction are unified in a single numerical recursion. In this way we entirely avoid objects with high tensor rank, thereby reducing the complexity of the calculations in a drastic way. Thanks to the on-the-fly approach, which is applied also to helicity summation and for the merging of different diagrams, the speed of the original open-loop algorithm can be further augmented in a very significant way. Moreover, addressing spurious singularities of the employed reduction identities by means of simple expansions in rank-two Gram determinants, we achieve a remarkably high level of numerical stability. These features of the new algorithm, which will be made publicly available in a forthcoming release of the OpenLoops program, are particularly attractive for NLO multi-leg and NNLO real-virtual calculations.