[Paper Review] Numerical evaluation of Feynman loop integrals by reduction to tree graphs

Kleinschmidt, Tobias|arXiv (Cornell University)|Dec 1, 2007

Particle physics theoretical and experimental studies82 references13 citations

TL;DR

This paper introduces a novel numerical method for evaluating Feynman loop integrals by leveraging the Feynman Tree Theorem to map them onto phase-space integrals over fictitious on-shell particles. By embedding loop evaluations directly into Monte Carlo phase-space integration and systematically applying subtractions to cancel ultraviolet and threshold singularities, the approach enables efficient NLO event generation, demonstrated via a full NLO Monte Carlo generator for Bhabha scattering in QED.

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.

Motivation & Objective

• To develop a numerically efficient method for evaluating loop integrals in quantum field theory without nested integration.
• To eliminate the computational bottleneck of nesting loop evaluations within Monte Carlo phase-space integrands.
• To systematically handle ultraviolet and threshold singularities arising from fictitious on-shell particles in the reduction process.
• To enable practical NLO event generation by integrating loop corrections directly into standard Monte Carlo frameworks.
• To validate the method through a full implementation for NLO Bhabha scattering in QED.

Proposed method

• The Feynman Tree Theorem is applied to replace loop integrals with phase-space integrals over fictitious on-shell particles.
• The loop evaluation is embedded directly into the Monte Carlo phase-space integration, avoiding nested numerical loops.
• Subtraction terms are systematically constructed to cancel ultraviolet divergences and artificial threshold singularities introduced by the on-shell mapping.
• Infrared singularities are handled using standard factorization techniques, preserving the numerical stability of the method.
• The method is implemented in a full NLO Monte Carlo event generator for Bhabha scattering in QED as a proof of concept.

Experimental results

Research questions

• RQ1 Can loop integrals be efficiently evaluated by transforming them into phase-space integrals via the Feynman Tree Theorem?
• RQ2 How can ultraviolet and threshold singularities introduced by the on-shell mapping be systematically canceled in numerical integration?
• RQ3 Can this approach be integrated into standard Monte Carlo event generators without requiring nested loop evaluations?
• RQ4 Is the method numerically stable and accurate enough to produce reliable NLO results for physical processes?
• RQ5 Can the method be successfully applied to a realistic QED process like Bhabha scattering at NLO?

Key findings

• The method successfully replaces loop integrals with phase-space integrals over fictitious on-shell particles, enabling direct integration within Monte Carlo frameworks.
• The systematic subtraction of ultraviolet and threshold singularities ensures numerical stability and convergence of the integration.
• The approach avoids the computationally expensive nesting of loop evaluations inside phase-space integrands, significantly improving performance.
• Infrared singularities are effectively managed using established techniques, maintaining the method's reliability.
• A complete NLO Monte Carlo event generator for Bhabha scattering in QED was constructed, demonstrating the method's feasibility and practicality.