Comparing efficient computation methods for massless QCD tree amplitudes: Closed Analytic Formulae versus Berends-Giele Recursion

TL;DR

This paper compares the computational efficiency and accuracy of closed analytic formulae and Berends-Giele recursion for evaluating massless QCD tree amplitudes, finding that analytic methods are faster for simpler cases while recursion excels for more complex amplitudes.

Contribution

It provides a detailed comparison of two methods for calculating QCD tree amplitudes, highlighting their relative efficiencies and accuracies across different amplitude complexities.

Findings

Analytic formulae are faster for MHV and NMHV amplitudes.

Berends-Giele recursion becomes more efficient for NNMHV amplitudes.

Both methods maintain good accuracy, with analytic formulas generally more precise.

Abstract

Recent advances in our understanding of tree-level QCD amplitudes in the massless limit exploiting an effective (maximal) supersymmetry have led to the complete analytic construction of tree-amplitudes with up to four external quark-anti-quark pairs. In this work we compare the numerical efficiency of evaluating these closed analytic formulae to a numerically efficient implementation of the Berends-Giele recursion. We compare calculation times for tree-amplitudes with parton numbers ranging from 4 to 25 with no, one, two and three external quark lines. We find that the exact results are generally faster in the case of MHV and NMHV amplitudes. Starting with the NNMHV amplitudes the Berends-Giele recursion becomes more efficient. In addition to the runtime we also compared the numerical accuracy. The analytic formulae are on average more accurate than the off-shell recursion relations though both are well suited for complicated phenomenological applications. In both cases we observe a reduction in the average accuracy when phase space configurations close to singular regions are evaluated. We believe that the above findings provide valuable information to select the right method for phenomenological applications.