Leading-Color Two-Loop Amplitudes for Four Partons and a W Boson in QCD

TL;DR

This work delivers the first analytic expressions for the leading-color planar two-loop QCD amplitudes for four partons with a W boson decaying to leptons, and it extends to Z/γ* cases. The authors reconstruct the finite remainders from numerical unitarity data, expressing results in a basis of one-mass pentagon functions to enable efficient numerical evaluation. A novel reconstruction strategy combines reductions to five-point one-mass kinematics, common-denominator rationalization, univariate partial fractions, and Vandermonde sampling to manage the complexity of five-point, one-mass, two-loop kinematics. The approach leverages finite-field arithmetic and the Caravel framework, providing analytic results that can be mapped to other partonic channels and integrated into NNLO QCD predictions for W+2-jet production. The methodology and results significantly advance precision QCD for multi-jet processes at the LHC and set the stage for practical NNLO implementations.

Abstract

We present the leading-color two-loop QCD corrections for the scattering of four partons and a $W$ boson, including its leptonic decay. The amplitudes are assembled from the planar two-loop helicity amplitudes for four partons and a vector boson decaying to a lepton pair, which are also used to determine the planar two-loop amplitudes for four partons and a $Z/γ^*$ boson with a leptonic decay. The analytic expressions are obtained by setting up a dedicated Ansatz and constraining the free parameters from numerical samples obtained within the framework of numerical unitarity. The large linear systems that must be solved to determine the analytic expressions are constructed to be in Vandermonde form. Such systems can be very efficiently solved, bypassing the bottleneck of Gaussian elimination. Our results are expressed in a basis of one-mass pentagon functions, which opens the possibility of their efficient numerical evaluation.

Figures (5)

• Figure 1: Schematic representation of the two partonic processes in eq. \ref{['eq:partProcesses']}
• Figure 2: Sample diagrams for the different contributions to \ref{['eq:nfDecomp']} for $\kappa=\mathrm{g}$.
• Figure 3: Sample diagrams for the different contributions to \ref{['eq:nfDecomp']} for $\kappa=\mathrm{Q}$.
• Figure 4: A non-planar diagram that contributes in the leading-color approximation to $Z$-boson production. This type of contribution has a distinct coupling structure and is consistently dropped.
• Figure 5: The structure of diagrams contributing to axial-vector currents considered in this work. The shaded blob contains the remaining external particles, as well as loops. Each gluon that couples to the quark line adds two additional $\gamma$-matrices to the spinor chain, one from the interaction vertex and one from the additional quark propagator.