Subtraction terms at NNLO

TL;DR

The paper presents a comprehensive subtraction framework for NNLO calculations to cancel infrared divergences in fully differential observables. Using the leading-colour $e^+ e^-\to 2$ jets process as a concrete example, it derives explicit subtraction terms for both double and single unresolved limits, including the tree, one-loop, and two-loop amplitudes and the required ultraviolet renormalization. It introduces a hierarchical set of counterterms $d\alpha^{(l,k)}$ built from dipoles and higher-order splitting functions, organized into topologies that reproduce the singular structure of the real emission pieces and their mix with virtual corrections. The work demonstrates detailed cancellations across double and single unresolved regions and with one-virtual-one-real contributions, paving the way for fully differential NNLO Monte Carlo predictions and offering a general framework applicable to other parton splittings and initial-state configurations.

Abstract

Perturbative calculations at next-to-next-to-leading order for multi-particle final states require a method to cancel infrared singularities. I discuss the subtraction method at NNLO. As a concrete example I consider the leading-colour contributions to e+ e- --> 2 jets. This is the simplest example which exhibits all essential features. For this example, explicit subtraction terms are given, which approximate the four-parton and three-parton final states in all double and single unresolved limits, such that the subtracted matrix elements can be integrated numerically.

Figures (8)

• Figure 1: Modelling of jets at next-to-next-to-leading order. The cone labeled with $y_{cut}$ represents the experimental cuts. At next-to-next-to-leading order a jet is modelled either by one, two or three partons.
• Figure 2: Diagrams contributing to the colour-ordered partial amplitude $A^{(0)}_4(q_1,g_2,g_3,\bar{q}_4)$.
• Figure 3: Examples of one-loop diagrams, contributing to different primitive amplitudes: Diagram 2a contributes to $A_3^{(1),L,[1]}$, diagram 2b contributes to $A_3^{(1),R,[1]}$ and diagram 2c contributes to $A_3^{(1),L,[1/2]}$.
• Figure 4: Diagrams contributing to the triple collinear limit $q \rightarrow q g g$.
• Figure 5: Splitting topologies for NLO subtraction. At each splitting the emitted particle is directed towards the spectator.