Numerical Evaluation of Feynman Loop Integrals by Reduction to Tree Graphs

TL;DR

This paper introduces a novel numerical method for evaluating Feynman loop integrals by reducing them to phase-space integrals over on-shell particles, enabling efficient NLO event generation.

Contribution

The paper presents a new approach using the Feynman Tree Theorem to evaluate loop integrals numerically, avoiding nested integrations and facilitating NLO Monte Carlo event generation.

Findings

Successfully applied to NLO Bhabha scattering in QED.

Constructed a Monte Carlo event generator at NLO.

Demonstrated efficient numerical evaluation of loop integrals.

Abstract

We present a new method for the numerical evaluation of loop integrals which is based on the Feynman Tree Theorem. The loop integrals are replaced by phase-space integration over fictitious extra on-shell particles. This integration can be performed alongside with the Monte-Carlo integration of ordinary phase space, avoiding the time-consuming nesting of loop evaluation inside the integrand, and directly leading to NLO event generation. We systematically construct subtractions, necessary to cancel both ultraviolet divergences and the extra threshold singularities in phase-space which arise in the numerical evaluation. Infrared singularities can be dealt with by standard methods. As a proof of concept, we apply the method to NLO Bhabha scattering in QED and construct the corresponding NLO Monte Carlo event generator.