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.
Highlights
An important example is given by the real–virtual contributions to next-to-next-to leading order (NNLO) calculations, which require very fast and highly stable one-loop amplitudes in deeply infrared regions of phase space
The essence of the open-loop method [9] consists of a numerical recursion that generates cut-open loop diagrams, called open loops, by multiplying, one after the other, the various building blocks that are connected through loop propagators
We confirm that runtimes tend to grow linearly with the number of one-loop diagrams up to 2 → 4 processes [9], and we find that this scaling behaviour persists up to 2 → 5 processes
Summary
Based on the integrand reduction method by del Aguila and Pittau [2], we will introduce an on-the-fly technique for the reduction of open loops In this way, we will promote OpenLoops to an algorithm that combines the construction and the reduction of loop amplitudes in a unified numerical recursion. For what concerns numerical stability, in order to avoid severe instabilities that result from squared inverse Gram determinants in the reduction identities of [2], we present a method that isolates such instabilities in certain triangle topologies and circumvents them via analytic expansions in the limit of small Gram determinants In this way we obtain the first integrand-reduction algorithm that is essentially free from Gram-determinant instabilities.
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