Comparing efficient computation methods for massless QCD tree amplitudes: Closed analytic formulas versus Berends-Giele recursion

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-antiquark pairs. In this work we compare the numerical efficiency of evaluating these closed analytic formulas to a numerically efficient implementation of the Berends-Giele recursion. We compare calculation times for color-ordered tree amplitudes with parton numbers ranging from 4 to 25 with no, one, two, and three external quark lines. We find that the analytic results are generally faster in the case of maximally helicity-violating and next-to-maximally helicity-violating amplitudes. Starting with the next-to-next-to-maximally helicity-violating amplitudes the Berends-Giele recursion becomes more efficient. In addition to the runtime we also compare the numerical accuracy. The analytic formulas 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. In summary, our findings show that for up to nine gluons the closed analytic formulas perform best.

Figure 1

Average time required per phase-space point for the evaluation of pure gluon amplitudes as a function of the parton multiplicity (GGT analytic formulas, BG Berends-Giele).

Figure 2

Evaluation time per phase-space point for amplitudes with a quark-antiquark pair and $n-2$ gluons (GGT analytic formulas, BG Berends-Giele).

Figure 3

Evaluation time per phase-space point for amplitudes with two quark-antiquark pairs (different flavors) and $n-4$ gluons (GGT analytic formulas, BG Berends-Giele).

Figure 4

Evaluation time per phase-space point for amplitudes with three quark-antiquark pairs (different flavors) and $n-6$ gluons (GGT analytic formulas, BG Berends-Giele).

Figure 5

Accuracy $\delta$ for the 25-gluon amplitude for the purely numerical evaluation based on the Berends-Giele recursion (BG) and for analytic formulas (GGT) as described in Sec. 2a. The phase-space generation is via sequential splitting.

Figure 6

Average accuracy $|\delta |$ for the pure-gluon amplitudes as a function of the gluon multiplicity (GGT analytic formulas, BG Berends-Giele). The phase-space generation is via sequential splitting.

Figure 7

The average accuracy of different MHV amplitudes is plotted against $⟨{log}_{10}(|⟨ij⟩{|}^4/s^2)⟩$, where $i$ and $j$ are the positions of the negative-helicity gluons. The data points have been determined using RAMBO and sequential splitting for multiplicities from $n=5$ to $n=25$. The thick grey line is the theoretical prediction of Eq. (25).

Figure 8

Average accuracy for separate helicity amplitudes involving quarks. Phase-space generation is via sequential splitting (GGT analytic formulas, BG Berends-Giele).

Figure 9

Probability density of the collinearity of phase-space points generated by RAMBO.

Figure 10

Probability density of the collinearity of phase-space points generated by sequential splitting.