On-the-fly reduction of open loops

• Published in: The European physical journal. C, Particles and fields Vol. 78; no. 1; pp. 1 - 35
• Main Authors: Buccioni, Federico, Pozzorini, Stefano, Zoller, Max
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2018; Springer; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Astronomy; Astrophysics and Cosmology; Construction methods; Elementary Particles; Hadrons; Heavy Ions; Helicity; Mathematical analysis; Measurement Science and Instrumentation; Nuclear Energy; Nuclear Physics; Numerical stability; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Reduction; Regular Article - Theoretical Physics; Singularities; String Theory
• ISSN: 1434-6044; 1434-6052; 1434-6052
• DOI: 10.1140/epjc/s10052-018-5562-1

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.