Planar Two-Loop Five-Parton Amplitudes from Numerical Unitarity

TL;DR

The paper develops a numerical unitarity framework to compute planar two-loop, five-parton amplitudes in QCD including massless fermion loops, by embedding external fermions in dimensional regularization and decomposing the integrand into master and surface terms. A tensor-basis approach in Ds is used to extract the relevant coefficients, with exact finite-field and floating-point arithmetic to determine master-integral coefficients and combine them with known master integrals. The method is validated by recomputing four-parton amplitudes analytically and delivering new five-parton results, including N_f corrections, while ensuring correct infrared pole structure. These results constitute a crucial step toward automated NNLO predictions for three-jet production at hadron colliders in the leading-color approximation and pave the way for broader applications to multi-loop QCD amplitudes.

Abstract

We compute a complete set of independent leading-color two-loop five-parton amplitudes in QCD. These constitute a fundamental ingredient for the next-to-next-to-leading order QCD corrections to three-jet production at hadron colliders. We show how to consistently consider helicity amplitudes with external fermions in dimensional regularization, allowing the application of a numerical variant of the unitarity approach. Amplitudes are computed by exploiting a decomposition of the integrand into master and surface terms that is independent of the parton type. Master integral coefficients are numerically computed in either finite-field or floating-point arithmetic and combined with known analytic master integrals. We recompute two-loop leading-color four-parton amplitudes as a check of our implementation. Results are presented for all independent four- and five-parton processes including contributions with massless closed fermion loops.

Figures (3)

• Figure 1: Representative Feynman diagrams for leading-color $\mathcal{A}^{(2)}(g,g,g,g,g)$ amplitudes, contributing at order $N_f^0$, $N_f^1$ and $N_f^2$.
• Figure 2: Representative Feynman diagrams for leading-color $\mathcal{A}^{(2)}(q,\bar{q},g,g,g)$ amplitudes, contributing at order $N_f^0$, $N_f^1$ and $N_f^2$.
• Figure 3: Representative Feynman diagrams for leading-color $\mathcal{A}^{(2)}(q,\bar{q},Q,\bar{Q},g)$ amplitudes, contributing at order $N_f^0$, $N_f^1$ and $N_f^2$.