Numerical evaluation of phase space integrals by sector decomposition

TL;DR

The paper tackles the challenge of NNLO double real emission contributions to $e^+e^-\to$ jets by applying iterated sector decomposition to phase-space integrals, enabling numerical extraction of infrared poles without explicit subtraction schemes. It shows that IR divergences cancel when summing over all cuts of a renormalized topology, with explicit UV renormalization constants used to control subdivergences. Analytic results are given for the lower-multiplicity cuts, while the $1\to4$ phase-space integral is evaluated numerically, including a detailed treatment of singularities and a demonstration of pole cancellation to high precision. The work extends the applicability of sector decomposition to full matrix elements and complex topologies, providing a flexible pathway toward fully differential NNLO predictions for jet observables.

Abstract

In a series of papers we have developed the method of iterated sector decomposition for the calculation of infrared divergent multi-loop integrals. Here we apply it to phase space integrals to calculate a contribution to the double real emission part of the e+e- -> 2 jets cross section at NNLO. The explicit cancellation of infrared poles upon summation over all possible cuts of a given topology is worked out in detail for a specific example.

Figures (2)

• Figure 1: Sample three loop topology related to $e^+e^-\to 2$ jets at NNLO.
• Figure 2: Cut topology with $1/\epsilon^4$ poles.