Three-Prong Distribution of Massive Narrow QCD Jets

TL;DR

This work derives semi-analytic, leading-order jet functions for massive, highly boosted QCD jets, focusing on planar flow (Pf) as a three-prong substructure observable. Using collinear factorization and 1-to-3 splitting functions, the authors obtain analytic leading-log results for jet mass and Pf, and perform a semi-analytic, numerical evaluation of J_f(m^2, Pf; p_T; R) across Pf values. They compare these jet functions to parton-shower predictions, study scale uncertainties, and assess hadronization and non-collinear corrections, finding qualitative agreement in a region Pf ~ 0.4–0.95 and highlighting the importance of higher-order effects and resummation for precise modeling. The results inform boosted-top discrimination strategies and motivate improved matching and filtering/template methods to isolate the hard jet core from soft contamination.

Abstract

We study the planar-flow distributions of narrow, highly boosted, massive QCD jets. Using the factorization properties of QCD in the collinear limit, we compute the planar-flow jet function from the one-to-three splitting function at tree-level. We derive the leading-log behavior of the jet function analytically. We also compare our semi-analytic jet function with parton-shower predictions using various generators.

Figures (9)

• Figure 1: Comparison of semi-analytic jet functions (dots) with the purely analytic leading-log form of Eq. \ref{['JsLL']} (solid lines) and with a fit to the leading-log form (dashed lines). The quark functions are the lower (black) curves and the gluon ones are the upper (green) curves. Two ranges of the planar-flow observable are shown: \ref{['fig:fit-small-pf']}$10^{-3} < \text{Pf} < 10^{-2}$ and \ref{['fig:fit-large-pf']}$0.1 < \text{Pf} < 0.8$. The curve parameters are listed in table \ref{['fit-parameters']}; the purely analytic parameters are given in column 1, the fit parameters for \ref{['fig:fit-small-pf']} in column 2, and the fit parameters for \ref{['fig:fit-large-pf']} in column 3. In the purely analytic form, $A_f$ is given by Eq. \ref{['AfAnalytic']} , and $B_f=1$. In the fit to the leading-log form, $A_f$ is determined by fitting in both \ref{['fig:fit-small-pf']} and \ref{['fig:fit-large-pf']}, while $B_f=1$ is set to 1 in (a) and is determined by fitting in (b). The jet momentum is $p=1~\mathrm{TeV}$, with mass $m=180~\mathrm{GeV}$ and size $R=0.4$.
• Figure 2: The ratio of semi-analytic jet functions evaluated with a splitting kernel approximated by an iterated $1\to 2$ splittings, to those with the original $1\rightarrow3$ kernel. The parameters are $p=1~\mathrm{TeV}$, $m=180~\mathrm{GeV}$ and $R=0.4$, and the couplings are evaluated at the jet-mass scale.
• Figure 3: Semi-analytic jet function near $\text{Pf}=1$. The dashed line shows the fit to the leading-log form of Eq. \ref{['JsLL']} with the coefficients given in Sect. \ref{['numeval']}.
• Figure 4: Ratios of gluon jet functions with different choices for $\mu$. The jet functions are evaluated at $p=1~\mathrm{TeV}$, $m=180~\mathrm{GeV}$ and $R=0.4$, and they are divided by the jet function that uses the hybrid (dipole) scale.
• Figure 5: Jet functions from parton-shower simulations, and our predictions of subsection \ref{['numeval']} using $1\to3$ splitting functions for the quark and gluon planar-flow jet functions. The upper (green) solid curve shows the gluon jet function, and the lower (black) solid curve the quark one. The dashed (green) curve is the average jet function with gluon fraction $x=0.24$. The region of expected validity of the semi-analytic form is highlighted. The points with error bars show the parton-shower results, in the highlighted region from top to bottom: matched SHERPA (red), unmatched SHERPA (yellow), matched MadGraph (dark blue), and unmatched MadGraph (light blue).