A comparison of efficient methods for the computation of Born gluon amplitudes

TL;DR

This paper evaluates four recursive algorithms for computing Born-level gluon amplitudes in QCD as the number of external partons $n$ grows, focusing on efficiency and numerical stability in critical phase-space regions. It compares Berends-Giele off-shell-current recursions, scalar-diagram recursions, MHV-vertex CSW recursions, and BCF on-shell momentum-shift recursions, using the leading-color quantity ${\cal M}_n$ as a performance metric. The study finds that Berends-Giele is fastest for large $n$ (e.g., $n\ge 9$), while the on-shell BCF method is most efficient for smaller $n$; MHV and scalar-diagram methods are generally slower but stable. All methods agree to about $10^{-12}$ on non-exceptional points up to $n=12$, and they remain numerically stable in soft/collinear regions, with Berends-Giele offering slightly better resilience to spurious singularities. The results guide algorithm choice for LO/NLO Monte Carlo programs at the LHC, balancing $n$ and helicity configurations to optimize computation time.

Abstract

We compare four different methods for the numerical computation of the pure gluonic amplitudes in the Born approximation. We are in particular interested in the efficiency of the various methods as the number n of the external particles increases. In addition we investigate the numerical accuracy in critical phase space regions. The methods considered are based on (i) Berends-Giele recurrence relations, (ii) scalar diagrams, (iii) MHV vertices and (iv) BCF recursion relations.

Figures (4)

• Figure 1: Ratio of the sum of squared helicity amplitudes to its factorised form for a set of 7-gluon configurations where one gluon becomes soft. $x$ is the energy fraction of the soft gluon. Key: $\diamond$ Berends-Giele $\star$ scalar diagrams $\triangle$ MHV rules $\Box$ on-shell.
• Figure 2: Ratio of the sum of squared helicity amplitudes to its factorised form for a set of 7-gluon configurations where two gluons becomes collinear. $p_T/E$ is the transverse momentum of the pair of gluons, normalised to the total energy. Key: $\diamond$ Berends-Giele $\star$ scalar diagrams $\triangle$ MHV rules $\Box$ on-shell.
• Figure 3: Fractional error in the sum of squared helicity amplitudes computed with the on-shell recursion relations for a set of 6-gluon configurations where $k_1 + k_2$ becomes collinear to $k_3 + k_4$. $p_T/E$ is the transverse momentum between the 2 pairs of gluons, normalised to the total energy.
• Figure 4: Fractional error in the sum of squared helicity amplitudes as the reference vector ($q$) used in the definition of each recursion relation becomes collinear with an external momentum ($k$). Key: $\diamond$ Berends-Giele $\star$ scalar diagrams $\triangle$ MHV rules.