Direct numerical integration of one-loop Feynman diagrams for N-photon amplitudes

TL;DR

This paper presents a novel numerical method for directly integrating one-loop Feynman diagrams in N-photon scattering amplitudes without Feynman parameters, using contour deformation, demonstrated for N=6 and 8.

Contribution

It introduces a direct contour deformation technique for numerical integration of one-loop diagrams, bypassing Feynman parameterization, improving computational efficiency.

Findings

Successfully computed N=6 and N=8 photon amplitudes

Demonstrated feasibility of direct contour deformation method

Enhanced numerical integration approach for loop calculations

Abstract

One approach to the calculation of cross sections for infrared-safe observables in high energy collisions at next-to-leading order is to perform all of the integrations, including the virtual loop integration, by Monte Carlo numerical integration. In a previous paper, two of us have shown how one can perform such a virtual loop integration numerically after first introducing a Feynman parameter representation. In this paper, we perform the integration directly, without introducing Feynman parameters, after suitably deforming the integration contour. Our example is the N-photon scattering amplitude with a massless electron loop. We report results for N = 6 and N = 8.