On the Numerical Evaluation of One-Loop Amplitudes: the Gluonic Case

TL;DR

This paper presents a polynomial-time algorithm for numerically evaluating one-loop gluonic amplitudes with many external particles, implemented in the Rocket program, and validated against known analytical results.

Contribution

It introduces a new polynomial complexity algorithm for one-loop amplitude evaluation using recursive tree amplitudes and an on-shell cut method, applicable to arbitrary external gluons.

Findings

Successfully computed amplitudes for up to twenty gluons

Achieved good agreement with analytical results

Analyzed algorithm's time efficiency and accuracy

Abstract

We develop an algorithm of polynomial complexity for evaluating one-loop amplitudes with an arbitrary number of external particles. The algorithm is implemented in the Rocket program. Starting from particle vertices given by Feynman rules, tree amplitudes are constructed using recursive relations. The tree amplitudes are then used to build one-loop amplitudes using an integer dimension on-shell cut method. As a first application we considered only three and four gluon vertices calculating the pure gluonic one-loop amplitudes for arbitrary external helicity or polarization states. We compare our numerical results to analytical results in the literature, analyze the time behavior of the algorithm and the accuracy of the results, and give explicit results for fixed phase space points for up to twenty external gluons.