QCD splitting functions beyond kinematical limits

TL;DR

This work introduces a decomposition of QCD splitting functions into universal scalar dipole radiators and spin-dependent remainders, computed up to ${\alpha_s^2}$, to avoid relying on soft or collinear kinematic limits. Using axial gauge and the background field method, the authors define scalar multipoles that reproduce the leading infrared behavior while encapsulating subleading soft and azimuthal correlations in a controlled way. They provide explicit tree-level and one-loop results for both quark- and gluon-initiated splittings, and show how three-parton splittings can be assembled from dipole radiators plus remainder terms, with a careful accounting of abelian and non-abelian contributions. The framework aims to yield subtraction schemes at NNLO free of soft-collinear overlaps and to inform the development of improved parton showers with reduced hierarchical ambiguities, significantly impacting precision QCD phenomenology and jet physics.

Abstract

We present a systematic decomposition of QCD splitting functions into scalar dipole radiators and pure splitting remainders up to second order in the strong coupling. The individual components contain terms that are formally sub-leading in soft or collinear scaling parameters, but well understood and universal due to their origin in scalar QCD. The multipole radiator functions which we derive share essential features of the known double-soft and one-loop soft gluon currents, and are not based on kinematical approximations.

Figures (14)

• Figure 1: Examples of one-to-two splitting processes. Figure (a) shows the splitting of a quark into a quark and a gluon, Fig. (b) the branching of a gluon into two gluons. The right-hand side sketches the decomposition of the full vertices into scalar and spin-dependent components.
• Figure 2: Feynman diagrams leading to the $1\to 2$ parton splitting functions discussed in Sec. \ref{['sec:two-parton_tree-level']}. The shaded blob and lines to the left represent the hard process with its associated external partons. See the main text for details.
• Figure 3: Feynman diagrams leading to the $1\to 3$ quark only splitting functions discussed in Sec. \ref{['sec:one_to_three_splittings_quark']}. The shaded blob and lines to the left represent the hard process with its associated external partons. See the main text for details.
• Figure 4: Feynman diagrams leading to the $1\to 3$ quark-to-quark-gluon splitting functions discussed in Sec. \ref{['sec:one_to_three_splittings_quark']}. The shaded blob and lines to the left represent the hard process with its associated external partons. See the main text for details.
• Figure 5: Feynman diagrams leading to the $1\to 3$ gluon-to-quark-gluon splitting functions discussed in Sec. \ref{['sec:one_to_three_splittings_gluon']}. The shaded blob and lines to the left represent the hard process with its associated external partons. See the main text for details.