High precision heavy-boson-jet substructure with energy correlators

TL;DR

This work develops a high-precision framework for energy-energy correlators (EECs) on boosted heavy-boson jets, focusing on hadronically decaying $Z$ bosons. The key insight is that the peak in the boosted EEC arises from Sudakov resummation around the boosted Born kinematics, not a resonance; this allows the observable to be computed by boosting the well-measured $Z$-pole EEC, yielding predictions at N$^3$LL$'$ that agree with both lepton- and hadron-collider simulations. A Lorentz-invariant EEC shape function $\mathcal{F}_{\mathcal{E}\mathcal{E}}$ is derived from energy-flow symmetry, enabling direct translation from $e^+e^-$ data (e.g., OPAL) to boosted $Z$-jet observables in $pp$ and $e^+e^-$ environments. The authors also extend the formalism to higher-point correlators and to three-body decays, providing a clean, frame-independent route to precision jet substructure studies and potential top-quark mass applications. Overall, the results establish robust, lepton-collider-like precision for heavy-jet substructure at the LHC and future colliders, with a clear path to further refinements and broader applications.

Abstract

Energy-correlator-based jet substructure has gained significant attention in recent years. One of the notable applications has been the study of multi-scale jets, where distinct physical scales manifest as features localised in different angular regions of the correlator. In this article, we present the first high-precision study of energy correlators on the simplest multi-scale jets: heavy boson jets. In such systems, the boson mass $M$ introduces an additional scale, generating a sharp peak at angles $\sim M/p_T^{\rm jet}$. We show that this feature can be computed directly by boosting the EEC spectrum measured in $e^+e^- \rightarrow {\rm hadrons}$ at the $Z$ pole. We identify that the peak arises from boosting the well-studied Sudakov factorisation governing the back-to-back limit of the two-point correlator. As a result, the feature is controlled by Sudakov resummation, not a Breit-Wigner-like structure in the $Z$ decay, and is therefore calculable with exceptional precision. We provide predictions at N$^3$LL$'$ accuracy for both $pp$ $Z$-tagged jets and $e^+e^-$ di-$Z$ production, and compare them to Herwig and Pythia simulations, finding close agreement. We also demonstrate that the boosted-$Z$ spectrum can be constructed directly by boosting OPAL measurements at the $Z$ pole. In this light, energy-correlator jet substructure on the hadronic decays of heavy bosons at the LHC provide access to clean, lepton-collider-like measurements across a wide range of effective centre-of-mass energies set by the boson jet transverse momentum.

Figures (9)

• Figure 1: The EEC spectrum computed in $e^+e^-$ at the $Z$ pole as a function of $z$, compared with OPAL data OPAL:1990reb. The left panel uses logarithmic axes to emphasise the behaviour near the back-to-back region, while the right panel shows the same distributions on linear axes. Results are shown both with (blue) and without (green) non-perturbative contributions as described in the text. For the perturbative curves, the bands correspond to scale variations by a factor of 2. The bands on the non-perturbative curves are found by adding in quadrature the bands from scale variation with the bands from the uncertainties on the non-perturbative ingredients.
• Figure 2: The EEC shape function extracted from the $e^+ e^-$ EEC spectrum at the $Z$ pole. As per \ref{['fig:rest_frame']}, left panel displays the distribution on logarithmic axes to highlight the back-to-back behaviour, while the right panel shows the same result using linear axes. Curves for both the purely perturbative (green) and perturbative plus non-perturbative corrections (blue) are presented. For the perturbative curves, the bands correspond to scale variations by a factor of 2. The bands on the non-perturbative curves quadrature the bands from scale variation with the bands from the uncertainties on the non-perturbative ingredients.
• Figure 3: The spectrum $\mathrm{d} \Sigma_{\rm ee} / \mathrm{d} \theta$ for $e^+e^- \rightarrow ZZ \rightarrow {\rm hadrons} + \ell^+\ell^-$ at several centre-of-mass energies. The curves shown correspond to the evaluation of eq. \ref{['eq:boostonee']} with a shape function computed with NLO+N$^3$LL+NP accuracy (the blue curve in \ref{['fig:CSF']}). The boost of the $Z$ bosons shifts the Sudakov peak to smaller angles, following the relation $z_{\rm peak} \approx 1/(\beta\gamma)^2$ for large boosts.
• Figure 4: Comparison of the prediction for $\mathrm{d} \Sigma_{\rm ee} / \mathrm{d} \theta$ using the factorisation in eq. \ref{['eq:boostonee']} at $\sqrt{s}=930~\text{GeV}$ against Herwig and Pythia simulations. Left: perturbative comparison with hadronisation turned off. Right: full comparison including non-perturbative effects. The Herwig and Pythia curves are produced from events with a $e^+ e^- \rightarrow ZZ \rightarrow q \bar{q} q'\bar{q}'$ hard process. To increase computational speed, jets are reconstructed using anti-$k_T$ clustering with $R=0.6$, and the EEC is computed on the constituents of the two leading jets within the fiducial selection. The jet selection has no effect on the measurement in the Sudakov region.
• Figure 5: As in the right panel of \ref{['fig:eeZZ']}, but using the EEC shape function extracted directly from OPAL data taken at the $Z$ pole. The magenta band represents the propagated experimental uncertainties from the extraction.