The Energy Distribution of Subjets and the Jet Shape

TL;DR

This work develops a comprehensive SCET-based framework to describe the energy distribution of subjets inside jets, capturing both inclusive and axis-centered subjets. It achieves systematic resummation of logarithms in the jet radii, including $\ln R$ and $\ln(r/R)$, via a series of factorization theorems and matching relations, with distinct treatments for the winner-take-all and standard jet axes. Key contributions include corrected one-loop cone-jet results, a detailed NLO treatment of the inclusive subjet function, and structured resummation strategies for central subjets, complemented by phenomenological predictions for proton-proton collisions. The work also clarifies recoil effects tied to the standard jet axis and connects jet-shape observables to TMD fragmentation, offering a path toward precision jet substructure studies and applications in boosted-object tagging and heavy-ion physics.

Abstract

We present a framework that describes the energy distribution of subjets of radius $r$ within a jet of radius $R$. We consider both an inclusive sample of subjets as well as subjets centered around a predetermined axis, from which the jet shape can be obtained. For $r \ll R$ we factorize the physics at angular scales $r$ and $R$ to resum the logarithms of $r/R$. For central subjets, we consider both the standard jet axis and the winner-take-all axis, which involve double and single logarithms of $r/R$, respectively. All relevant one-loop matching coefficients are given, and an inconsistency in some previous results for cone jets is resolved. Our results for the standard jet shape differ from previous calculations at next-to-leading logarithmic order, because we account for the recoil of the standard jet axis due to soft radiation. Numerical results are presented for an inclusive subjet sample for $pp \to {\rm jet}+X$ at next-to-leading order plus leading logarithmic order.

Figures (7)

• Figure 1: Illustration of a subjet with radius $r$ (red) inside a jet with a radius $R$ (green). We focus on describing the energy fraction $z_r$ of the jet that is carried by the subjet.
• Figure 2: The three configurations that enter for the quark semi-inclusive jet function at ${\cal O}(\alpha_s)$: (A) the quark and gluon are inside the jet, (B) only the quark is inside the jet, (C) only the gluon is inside the jet.
• Figure 3: The five configurations that enter for the quark subjet function at ${\cal O}(\alpha_s)$: (A) the quark and gluon are inside the jet and subjet, (B) only the quark is inside the subjet but both partons are in the jet, (C) only the gluon is inside the subjet but both partons are in the jet, (D) only the quark is inside the jet and subjet, (E) only the gluon is inside the jet and subjet.
• Figure 4: The nonperturbative corrections $1-J_q^{\rm NP}$ to the semi-inclusive quark jet function for Mellin moments $N=2$ (black), 3 (blue), 4 (green), 5 (red) as function of $1/r$. The points were extracted from parton-level and hadron-level Pythia data, using $e^+e^-$ collisions at a center-of-mass energy of $\omega= 500$ GeV, with the $e^+e^-$ anti-k$_T$ algorithm. In the left panel we restrict to large values of $r$ and show a fit to $c/r^2$ (solid lines). In the right panel, the asymptotic approach to the fragmentation limit is shown. The nonperturbative corrections for $N=2$ vanish due to eq. \ref{['eq:sumrule']}.
• Figure 5: The subjet distribution measured on an inclusive jet sample $pp\to (\mathrm{jet}\, j_r)+X$, using anti-k$_T$ with jet radius $R=0.6$ and subjet radius $r=0.2$, for representative LHC kinematics $\sqrt{s}=13$ TeV, $|\eta|<1.2$. Shown are the NLO+LL$_R$+LL$_{r/R}$ results for four different intervals of the jet transverse momentum $[25,50],\; [50,100],\; [100,200],\; [200,500]$ GeV.