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Towards e+e- --> 3 jets at NNLO by sector decomposition

G. Heinrich

TL;DR

This work addresses the challenge of NNLO predictions for $e^+e^-\to 3$ jets by applying sector decomposition to isolate and numerically integrate infrared poles arising from double real radiation. The method yields a finite, differential cross section that can be combined with an infrared-safe measurement function in a Monte Carlo framework, demonstrated through 3-, 4-, and 5-jet rates for a sample topology. A detailed UV renormalisation analysis up to ${\cal O}(\alpha_s^3)$ shows pole cancellations in accordance with the KLN theorem, with the key pole structure verified for the 1→5 contribution. The results establish a scalable, universal approach for NNLO real radiation, enabling differential observables and paving the way for the full matrix-element treatment and extensions to massive final states.

Abstract

A method based on sector decomposition has been developed to calculate the double real radiation part of the process e+e- to 3 jets at next-to-next-to-leading order. It is shown in an example that the numerical cancellation of soft and collinear poles works well. The method is flexible to include an arbitrary measurement function in the final Monte Carlo program, such that it allows to obtain differential distributions for different kinds of observables. This is demonstrated by showing 3-, 4- and 5-jet rates at order alpha_s^3 for a subpart of the process.

Towards e+e- --> 3 jets at NNLO by sector decomposition

TL;DR

This work addresses the challenge of NNLO predictions for jets by applying sector decomposition to isolate and numerically integrate infrared poles arising from double real radiation. The method yields a finite, differential cross section that can be combined with an infrared-safe measurement function in a Monte Carlo framework, demonstrated through 3-, 4-, and 5-jet rates for a sample topology. A detailed UV renormalisation analysis up to shows pole cancellations in accordance with the KLN theorem, with the key pole structure verified for the 1→5 contribution. The results establish a scalable, universal approach for NNLO real radiation, enabling differential observables and paving the way for the full matrix-element treatment and extensions to massive final states.

Abstract

A method based on sector decomposition has been developed to calculate the double real radiation part of the process e+e- to 3 jets at next-to-next-to-leading order. It is shown in an example that the numerical cancellation of soft and collinear poles works well. The method is flexible to include an arbitrary measurement function in the final Monte Carlo program, such that it allows to obtain differential distributions for different kinds of observables. This is demonstrated by showing 3-, 4- and 5-jet rates at order alpha_s^3 for a subpart of the process.

Paper Structure

This paper contains 10 sections, 31 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Cancellation of Divergences
  3. UV renormalisation
  4. Combining the renormalised diagrams
  5. Calculation of the 5-particle contribution
  6. Inclusion of a measurement function
  7. Conclusions and Outlook
  8. Massless 5-parton Phase Space
  9. Phase space $1\to 5$ in $D\not=4$ dimensions
  10. Phase space $1\to 5$ in $D=4$ dimensions

Figures (4)

  • Figure 1: The ladder diagram
  • Figure 2: Cancellation of IR divergences in the sum over all cuts of the renormalised graph
  • Figure 3: UV renormalisation of the quark propagator at ${\cal O}(\alpha_s^3)$
  • Figure 4: 3--, 4-- and 5--jet rates at order $\alpha_s^3$ for the sample matrix element