Local Unitarity: cutting raised propagators and localising renormalisation

TL;DR

This paper advances the Local Unitarity formalism to handle higher-loop calculations by generalizing to raised propagators and implementing a local renormalisation scheme, enabling fully numerical NNLO cross-section computations.

Contribution

It introduces a generalisation of the LU representation for graphs with raised propagators and a local renormalisation procedure, facilitating practical higher-loop calculations.

Findings

First fully numerical NNLO cross-sections for $oldsymbol{ ext{γ}^* ightarrow jj}$ and $oldsymbol{ ext{γ}^* ightarrow tar{t}}$.

Demonstrates all-order construction for hybrid $ ext{MS}$ and On-Shell schemes.

Provides semi-inclusive results up to N3LO for supergraphs.

Abstract

The Local Unitarity (LU) representation of differential cross-sections locally realises the cancellations of infrared singularities predicted by the Kinoshita-Lee-Nauenberg theorem. In this work we solve the two remaining challenges to enable practical higher-loop computations within the LU formalism. The first concerns the generalisation of the LU representation to graphs with raised propagators. The solution to this problem results in a generalisation of distributional Cutkosky rules. The second concerns the regularisation of ultraviolet and spurious soft singularities, solved using a fully automated and local renormalisation procedure based on Bogoliubov's R-operation. We detail an all-order construction for the hybrid $\overline{\text{MS}}$ and On-Shell scheme whose only analytic input is single-scale vacuum diagrams. Using this novel technology, we provide (semi-)inclusive results for two multi-leg processes at NLO, study limits of individual supergraphs up to N3LO and present the first physical NNLO cross-sections computed fully numerically in momentum-space, namely for the processes $\gamma^* \rightarrow j j$ and $\gamma^* \rightarrow t \bar{t}$.