Massive tree-level splitting functions beyond kinematical limits

TL;DR

The paper resolves the challenge of incorporating heavy-quark mass effects in QCD jet evolution by deriving a compact, limit-free representation of massive $1→3$ tree-level splitting functions. It introduces a framework that decomposes these functions into scalar dipole radiator functions, lower-order splitting functions, and finite pure splitting remainders, facilitated by physical polarization states and Berends-Giele currents. The authors provide explicit massive one-to-three splitting functions and scalar radiator functions, including abelian and non-abelian components, and demonstrate how to assemble them from radiators and lower-order terms. This approach yields improved numerical stability and evaluation speed, supports infrared subtraction and resummation, and offers a robust path toward higher-order calculations and more precise heavy-quark jet modeling at the LHC.

Abstract

We present a compact form of the massive 1$\to$3 tree-level QCD splitting functions, enabling fast evaluation and improved numerical stability in practical applications. We discuss a decomposition of the results in terms of lower-order splitting functions, scalar dipole antenna functions and pure splitting remainders. The two-gluon radiator functions introduced in this context generalize the expressions obtained from the double-soft approximation.

Figures (7)

• Figure 1: Feynman diagrams contributing to the one-to-two parton splittings with massive quarks computed in Sec. \ref{['sec:two-parton_tree-level_quark']} (a) and Sec. \ref{['sec:two-parton_tree-level_gluon']} (b). Double lines indicate a quark of on-shell mass $m$.
• Figure 2: Feynman diagrams contributing to the one-to-three all-quark splittings with massive quarks computed in Sec. \ref{['sec:one_to_three_splittings_quark']}. Diagram (a) shows a process with different on-shell masses for partons $1/2$ and $3$, (indicated by the double line and the thick line in the figure), while diagrams (b) and (c) show a process with identical on-shell masses for all the quarks.
• Figure 3: Feynman diagrams contributing to the one-to-three gluon emission splittings with massive quarks, computed in Sec. \ref{['sec:one_to_three_splittings_quark_rad']}. The double line indicates a quark of mass $m$.
• Figure 4: Feynman diagrams contributing to the one-to-three gluon splitting into massive quarks computed in Sec. \ref{['sec:one_to_three_splittings_gluon']}. The double line indicates a quark of mass $m$.
• Figure 5: Feynman diagrams contributing to the scalar radiator function for a single gluon emission. The double lines indicate quark masses of the scalar particles.