Antenna Subtraction at NNLO

TL;DR

This work develops and implements an NNLO antenna subtraction framework to compute exclusive jet observables in e+e- annihilation. It constructs a complete set of tree- and one-loop antenna functions for quark-antiquark, quark-gluon, and gluon-gluon radiators, and provides their analytic integrals over relevant antenna phase spaces. The method systematically builds subtraction terms for double real, real-virtual, and double virtual contributions, ensuring local cancellation of all infrared poles across all color structures, demonstrated explicitly for the 1/N^2 subleading color channel in e+e- → 3 jets. The results offer explicit integrated forms and a clear path to NNLO event generation for jet observables, with potential extensions to hadronic initial states and broader QCD processes.

Abstract

The computation of exclusive QCD jet observables at higher orders requires a method for the subtraction of infrared singular configurations arising from multiple radiation of real partons. We present a subtraction scheme relevant for NNLO perturbative calculations in $e^+e^- \to $ jets. The building blocks of the scheme are antenna functions derived from the matrix elements for tree-level $1\to 3$ and $1 \to 4$ and one-loop $1 \to 3$ processes. By construction, these building blocks have the correct infrared behaviour when one or two particles are unresolved. At the same time, their integral over the antenna phase space is straightforward. As an example of how to use the scheme we compute the NNLO contributions to the subleading colour QED-like contribution to $e^+e^- \to 3$ jets. To illustrate the application of NNLO antenna subtraction for different colour structures, we construct the integrated forms of the subtraction terms needed for the five-parton and four-parton contributions to $e^+e^- \to 3$ jets at NNLO in all colour factors, and show that their infrared poles cancel analytically with the infrared poles of the two-loop virtual correction to this observable.