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

We describe a new method 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. The full reduction to scalar integrals is unified with the amplitude construction in a single recursion within the OpenLoops framework. This approach strongly exploits the factorisation of one-loop integrands in a product of loop segments. The on-the-fly approach, which is also applied to helicity summation and the merging of different diagrams, increases the speed of the original open-loop algorithm in a very significant way. A remarkably high level of numerical stability is achieved by exploiting freedoms in reduction identities and through simple expansions in rank-two Gram determinants. These features are particularly attractive for NLO multi-leg and NNLO real-virtual calculations. The new algorithm will be made public in a forthcoming release of the OpenLoops program.