Eur. Phys. J. C (2018) 78: 70
https://doi.org/10.1140/epjc/s10052-018-5562-1

Regular Article - Theoretical Physics

On-the-fly reduction of open loops

Physik-Institut, Universität Zürich, 8057, Zurich, Switzerland

^* e-mail: zoller@physik.uzh.ch

Received: 12 December 2017
Accepted: 13 January 2018
Published online: 25 January 2018

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.