A subtraction scheme for computing QCD jet cross sections at NNLO: integrating the doubly unresolved subtraction terms

TL;DR

The paper completes the NNLO subtraction scheme for QCD jet cross sections by computing the double soft-type contributions to the integrated doubly unresolved cross section via Mellin–Barnes representations. By deriving analytic expressions up to $O(\epsilon^{-2})$ for the insertion operator $I^{(0)}_2$ and expressing the remaining terms through a set of master integrals, the authors enable fully differential predictions for $e^+e^- \to 2$ and $3$ jets at NNLO within this framework. The approach relies on precise phase-space factorisation, double soft momentum mappings, and a careful flavour/colour decomposition of the counterterms, with numerical MB evaluations for higher-order poles. The result is a complete, process-independent subtraction scheme at NNLO, allowing the cancellation of infrared singularities and the computation of finite cross sections, at least for three jets, with extensions to higher jet multiplicities contingent on further matrix-element knowledge.

Abstract

We finish the definition of a subtraction scheme for computing NNLO corrections to QCD jet cross sections. In particular, we perform the integration of the soft-type contributions to the doubly unresolved counterterms via the method of Mellin-Barnes representations. With these final ingredients in place, the definition of the scheme is complete and the computation of fully differential rates for electron-positron annihilation into two and three jets at NNLO accuracy becomes feasible.