A General Algorithm for Calculating Jet Cross Sections in NLO QCD

TL;DR

This work introduces a general, subtraction-based algorithm for calculating jet cross sections at next-to-leading order in QCD for arbitrary processes. Central to the method are universal dipole factorization formulae that reproduce soft and collinear limits, enabling exact analytic integration of the subtraction terms and a four-dimensional numerical implementation. The framework handles jets with no identified partons, those with identified hadrons, and cases with initial-state hadrons, by systematically organizing infrared and collinear singularities into I, P, K, and H insertion operators and related factorization kernels. The approach yields finite, process-independent expressions that can be implemented in general-purpose Monte Carlo programs, with broad applicability and clear pathways for extensions to fragmentation, heavy flavors, polarization, and eventually NNLO. The paper also provides explicit analytic results for the required dipole integrals and final formulae for various process classes, forming a practical blueprint for NLO jet physics computations.

Abstract

We present a new general algorithm for calculating arbitrary jet cross sections in arbitrary scattering processes to next-to-leading accuracy in perturbative QCD. The algorithm is based on the subtraction method. The key ingredients are new factorization formulae, called dipole formulae, which implement in a Lorentz covariant way both the usual soft and collinear approximations, smoothly interpolating the two. The corresponding dipole phase space obeys exact factorization, so that the dipole contributions to the cross section can be exactly integrated analytically over the whole of phase space. We obtain explicit analytic results for any jet observable in any scattering or fragmentation process in lepton, lepton-hadron or hadron-hadron collisions. All the analytical formulae necessary to construct a numerical program for next-to-leading order QCD calculations are provided. The algorithm is straightforwardly implementable in general purpose Monte Carlo programs.

Figures (4)

• Figure 1: Diagrammatic representation of the external-leg insertion rule. The blobs denote the tree-level matrix elements and their complex conjugate. The dots on the right-hand side stand for non-singular terms both in the soft and collinear limits.
• Figure 2: Pictorial representation of the dipole factorization procedure. When the partons $i$ and $j$ become soft and/or collinear, the singularities are factorized into the term ${\bf V}_{ij,k}$ (the dashed box on the right-hand side) which embodies correlations with a single additional parton $k$.
• Figure 3: Effective diagrams for the different dipole formulae introduced in Sect. \ref{['dff']}. The blobs denote the $m$-parton matrix element. Incoming and outgoing lines respectively stand for initial-state and final-state partons.
• Figure 4: Altarelli-Parisi insertion operators for the time-like (Eq. (\ref{['pfindef']})) and the space-like (Eq. (\ref{['pdef']})) cases.