Optimizing the Reduction of One-Loop Amplitudes

TL;DR

The paper addresses the computational bottleneck of reducing one-loop amplitudes to master integrals by exploiting the polynomial structure of the cut-integrand and introducing projection-based coefficient extraction inspired by the Discrete Fourier Transform. By selecting cut-solution variables as roots of unity and applying one-dimensional projections, the authors recover all polynomial coefficients with a number of evaluations equal to the number of coefficients, significantly speeding up the extraction of 3- and 2-point function coefficients within the OPP framework. The approach is demonstrated on NLO QCD corrections to $u\bar{d}\to W^+W^-W^+$, where it yields substantial speedups (roughly 40% overall), without compromising numerical stability, and is framed as generalizable to multi-variable polynomials arising in loop-reduction problems. The results suggest that complex-phase parametrizations and projection techniques can enhance efficiency in unitarity- and factorization-based loop computations for collider phenomenology.

Abstract

We present an optimization of the reduction algorithm of one-loop amplitudes in terms of master integrals. It is based on the exploitation of the polynomial structure of the integrand when evaluated at values of the loop-momentum fulfilling multiple cut-conditions, as emerged in the OPP-method. The reconstruction of the polynomials, needed for the complete reduction, is rended very versatile by using a projection-technique based on the Discrete Fourier Transform. The novel implementation is applied in the context of the NLO QCD corrections to u d-bar --> W+ W- W+.

Figures (1)

• Figure 1: Box and pentagon diagrams contributing to virtual QCD corrections to $u \bar{d} \to W^+ W^- W^+$. Wavy lines can be either Z or photons, dashed lines represent $W^+$ and $W^-$. Diagrams involving an exchange in the two final $W^+$ should also be considered.