Fragmentation of a Jet with Small Radius

TL;DR

The paper develops the fragmentation function to a jet (FFJ) within SCET to address large logarithms arising from small jet radius $R$ by showing FFJs obey DGLAP evolution with an $R$-dependent scale, enabling $ ln R$ resummation. It presents explicit NLO results for quark and gluon FFJs, confirming their UV structure matches standard jet-function behavior and that momentum sum rules are satisfied. A factorization theorem is formulated for fragmentation inside a jet, linking FFJs with hadron-in-jet fragmentation and jet splitting kernels to resum $ ln R$, and this formalism is extended to subjet fragmentation inside a fat jet, yielding kernels that resum $ ln(R'/r')$. The framework unifies hadron fragmentation inside jets with jet substructure observables and provides a path toward phenomenology in jet physics and radius-ratio observables, with avenues for future higher-order studies.

Abstract

In this paper we consider the fragmentation of a parton into a jet with small jet radius $R$. Perturbatively, logarithms of $R$ can appear, which for narrow jets can lead to large corrections. Using soft-collinear effective theory, we introduce the fragmentation function to a jet (FFJ), which describes the fragmentation of a parton into a jet. We discuss how these objects are related to the standard jet functions. Calculating the FFJ to next-to-leading order, we show that these objects satisfy the standard Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations, with a natural scale that depends upon $R$. By using the standard renormalization group evolution, we can therefore resum logarithms of $R$. We further use the soft-collinear effective theory to prove a factorization theorem where the FFJs naturally appear, for the fragmentation of a hadron within a jet with small $R$. Finally, we also show how this formalism can be used to resum the ratio of jet radii for a subjet to be emitted from within a fat jet.

Figures (7)

• Figure 1: Jet fragmentation at NLO in $\alpha_s$. Diagram (a) shows the jet merging, so the contribution to FFJ should be proportional to $\delta(1-z)$. Diagram (b) shows the jet splitting, which has a contribution with a fraction $z<1$.
• Figure 2: Feynman diagrams for quark jet splitting contribution at NLO in $\alpha_s$. Here the dashed lines represent the unitary cuts. The gluon in the final state is outside the jet. Diagram (a) has its Hermitian conjugate contribution.
• Figure 3: Feynman diagrams for gluon jet splitting contribution at NLO in $\alpha_s$. Here the dashed lines represent the unitary cuts. The quark in the final state is outside the jet. Diagram (a) has its Hermitian conjugate contribution.
• Figure 4: Feynman diagrams for jet splitting contributions to jet fragmentation initiated by gluon parton. Diagram (a) has its Hermitian conjugate contribution. Diagram (a) and (b) represents $g\to J_g$ splitting, and Diagram (c) represents $g\to J_q$ splitting.
• Figure 5: Fragmentation process from the parton $(p)$ to the hadron $(p_H)$ through the jet $(p_J)$.