Energy Correlation Functions for Jet Substructure

TL;DR

This work introduces generalized energy correlation functions that encode N-prong jet substructure without explicit subjet finding by combining energies and pairwise angles into multi-point correlators, single-parameterized by an IRC-safe exponent β. The authors define energy correlation double ratios C_N^(β) and demonstrate, through LL/NLL perturbative analysis and Monte Carlo studies, that C_1 provides strong quark/gluon discrimination (optimal at β ≈ 0.2), while C_2 enhances identification of boosted two-prong resonances and C_3 offers modest top-tagging capability. Case studies with QCD jets, boosted W/Z/H decays, and boosted tops show distinct β-dependence and mass-ratio effects, highlighting the observables’ complementarity to existing subjet-based techniques. The framework is implemented as a FastJet add-on, enabling practical, recoil-insensitive jet substructure analyses with potential for Monte Carlo tuning and data-driven studies.

Abstract

We show how generalized energy correlation functions can be used as a powerful probe of jet substructure. These correlation functions are based on the energies and pair-wise angles of particles within a jet, with (N+1)-point correlators sensitive to N-prong substructure. Unlike many previous jet substructure methods, these correlation functions do not require the explicit identification of subjet regions. In addition, the correlation functions are better probes of certain soft and collinear features that are masked by other methods. We present three Monte Carlo case studies to illustrate the utility of these observables: 2-point correlators for quark/gluon discrimination, 3-point correlators for boosted W/Z/Higgs boson identification, and 4-point correlators for boosted top quark identification. For quark/gluon discrimination, the 2-point correlator is particularly powerful, as can be understood via a next-to-leading logarithmic calculation. For boosted 2-prong resonances the benefit depends on the mass of the resonance.

Figures (15)

• Figure 1: Example kinematics with soft wide-angle radiation. Left: recoil of the jet axis (dashed) away from the hard jet core ($E_1$) due to soft wide-angle radiation ($E_2$), which is relevant for small values of $\beta$. Right: a three-particle configuration that highlights the difference between $C_{2}$ and $\tau_{2,1}$.
• Figure 2: Left: Quark/gluon discrimination curves using $C^{(\beta)}_{1}$, calculated at NLL order matched to fixed order for various values of $\beta$. The $\beta$-independent LL prediction is shown for comparison. Right: Gluon rejection rates at 50% quark efficiency, as a function of $\beta$, demonstrating that $\beta \simeq 0.2$ is optimal at NLL order (for smaller values of $\beta$, non-perturbative effects become important). Also shown is an analytic approximation from Eq. (\ref{['eq:summaryequation']}) ($C_{1}$ Approx.) that includes the most important physics that enters at NLL.
• Figure 3: Left: Distribution of $C^{(0.2)}_{1}$ for quark jets (purple) and gluon jets (orange) using Pythia dijet samples. The sample consists of anti-$k_T$ jets with radius $R=0.6$ and transverse momentum in the range $[400,500]~\textrm{GeV}$. Right: Quark versus gluon discrimination curves using $C^{(\beta)}_{1}$ for several values of $\beta$ in Pythia. Also plotted is the leading log approximation for the discrimination curve, Eq. (\ref{['eq:rocqvg']}).
• Figure 4: Gluon rejection rates at 50% quark efficiency in Pythia, as a function of $\beta$. Left: fixing the $p_T$ range to be $[400,500]~\textrm{GeV}$ and sweeping the value of $R_0$. Right: fixing $R_0 = 0.6$ and sweeping the $p_T$ range. For all of these cases, small values of $\beta$ yield the best discrimination.
• Figure 5: Left: Quark/gluon discrimination curves using jet angularities $\tau_1^{(\beta)}$ (i.e. 1-subjettiness measured with respect to the jet axis), for several values of $\beta$ in Pythia. Also plotted is the leading log approximation for the discrimination curve from Eq. (\ref{['eq:rocqvg']}) and the discrimination curve for $C^{(0.2)}_{1}$. The jet sample is the same as used in Fig. \ref{['fig:qvg_roc']}. Right: Gluon rejection rate for 50% quark efficiency as a function of $\beta$, for angularities, 1-subjettiness measured with respect to the broadening axis, and $C^{(\beta)}_{1}$. The broadening axis is defined as the axis which minimizes the $\beta = 1$ measure in $N$-subjettiness. The latter two observables are recoil-free, and therefore give better discrimination power for small values of $\beta$.