On-the-fly reduction of open loops

Abstract

Building on the open-loop algorithm we introduce a new method for the automated construction of oneloop 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.

A new method for one-loop amplitude generation and reduction in Openloops

“…. ,D 3 from the 4 This is also valid for triangles in renormalisable theories [1,6] if we set terms involvingD 3 , p 3 to zero. 5 The terms ∝q 2 , whereq =q − q is (D − 4)-dimensional, lead to rational terms of type R 1 [6].…”

“…2) The reduction step is then followed by the next dressing steps and further reduction steps for each resulting topology until the loop is fully dressed and final reduction steps are performed at integral level (for details see [1]). The freedom to chooseD 0 , .…”

“…Here we identified a simple kinematical configuration responsible for numerical instabilities due to a vanishing rank-two Gram determinant ∆ and perform an analytical expansion in ∆ [1]. In this way we completely avoid Gram determinant instabilities in our algorithm, which leads to excellent numerical stability (see section 3).…”

“…This new helicity treatment leads to an additional gain of a factor two-three in speed, depending on the process (see [1]). …”

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We describe a new method [1] for the automated construction of one-loop amplitudes based on the open-loop algorithm, where various operations are performed on-the-fly while constructing the integrand. In particular, an on-the-fly reduction interleaved with the construction steps of the amplitude keeps the maximum tensor rank in the loop momentum at two throughout the algorithm, thus drastically reducing the complexity of the calculation.

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