Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD

TL;DR

This work delivers analytic, leading-color two-loop five-parton QCD amplitudes in the planar limit by combining numerical unitarity with exact finite-field reconstruction. Key innovations include a dimensional-reduction treatment of $D_s$ states, a Mandelstam-variable-centric reconstruction, and a Leinartas-based multivariate partial-fraction decomposition that yields compact expressions in terms of pentagon functions. The results enable NNLO QCD predictions for three-jet production at hadron colliders and provide a framework for further analytic and numerical exploration of multi-leg, multi-loop amplitudes. Ancillary files accompany the publication, offering ready-to-use one- and two-loop amplitudes and their master-integral decompositions.

Abstract

We present the analytic form of all leading-color two-loop five-parton helicity amplitudes in QCD. The results are analytically reconstructed from exact numerical evaluations over finite fields. Combining a judicious choice of variables with a new approach to the treatment of particle states in $D$ dimensions for the numerical evaluation of amplitudes, we obtain the analytic expressions with a modest computational effort. Their systematic simplification using multivariate partial-fraction decomposition leads to a particularly compact form. Our results provide all two-loop amplitudes required for the calculation of next-to-next-to-leading order QCD corrections to the production of three jets at hadron colliders in the leading-color approximation.

Figures (9)

• Figure 1: Contraction of the open $(D_s-4)$-dimensional spinor indices for amplitudes with external quarks: each quark line closes upon itself. Indices connected by red dashed lines are traced over.
• Figure 2: Alternative contraction of $(D_s-4)$-dimensional spinor indices leading to a single trace. Indices connected by green dashed lines are traced over.
• Figure 3: 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 4: 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 5: 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 $N_f^2$.