Energy Correlators Beyond Angles

TL;DR

This paper introduces Energy Weighted Observable Correlations (EWOCs), a subjet-level generalization of the Energy-Energy Correlator (EEC) that enables direct probing of non-angular correlations within jets. By computing pairwise observables on subjets with tunable radius $r_ ext{sub}$ and energy weighting, EWOCs preserve collinear safety and suppress non-perturbative effects, while expanding the scope beyond angular correlations to quantities such as pairwise invariant mass. The mass EWOC is demonstrated as a practical tool for extracting mass scales from hadronic decays, showing a robust peak near the $W$ boson mass in $pp o W^+W^-$ events and favorable resilience to hadronization, underlying event, and detector effects relative to angular EEC and groomed jet masses. LO and LL perturbative analyses in $e^+e^- o$ hadrons corroborate the analytic structure, and comparisons with Pythia validate qualitative behavior. The framework generalizes to other pairwise observables and higher-point EWOCs, offering a versatile pathway for precision measurements and potential BSM applications in jet substructure.

Abstract

Energy correlators are theoretically simple and physically intuitive observables that bridge experimental and theoretical particle physics. They have for example enabled the most precise jet substructure determination of the strong coupling constant to date, and recent proposals suggest that they may be used to precisely determine of the top quark mass with calculable, small theoretical uncertainties. However, existing energy correlators all measure correlations in angles between particles, from which other observables such as mass must be inferred through potentially complicated procedures. In this work, we generalize energy correlators to enable straightforward measurements of non-angular correlations, which we call Energy Weighted Observable Correlations (EWOCs). To enforce collinear safety, EWOCs quantify correlations between subjets rather than particles. The subjet radius can be tuned to control both the physical scales probed by EWOCs and their sensitivity to non-perturbative physics. We focus on the phenomenologically relevant example of the mass EWOC, which measures mass correlations between pairs of subjets, in the task of extracting mass scales from jets. In jet substructure determinations of the mass of a hadronically-decaying W boson, we show that the mass EWOC outperforms the angle-based energy correlator, and performs comparably to the soft-drop groomed jet mass. As a first exploration of the theoretical properties of EWOCs, we also calculate the mass EWOC on light-quark jets and compare to results obtained with Pythia.

Figures (9)

• Figure 1: Visualizations of the hadronic decay of a $W$ boson in 14 TeV LHC events simulated with Pythia 8.309. \ref{['fig:m_ewoc:pp_to_ww:with_cartoon']} The mass EWOC, introduced in this work, which captures the mass of the of $W$ boson directly from the pairwise mass of subjets (the red cones) of radius $r_\text{sub}=0.3$. \ref{['fig:eec:pp_to_ww:with_cartoon']} The EEC evaluated on pairs of particles; we tune the constant $c \sim 2.5$ to extract $m_W$ from the peak of the EEC distribution. The full width at half maximum (FWHM) of the mass EWOC (about 10 GeV) is significantly smaller than the mass difference associated with the FWHM of the EEC (corresponding to about 50 GeV), suggesting that the peak of the mass EWOC will provide a more precise estimation of $m_W$.
• Figure 2: Cartoons of collinear splittings (colored lines inside the black hemispheres, for which the brown quark line splits into the pink gluon line and the green quark line) producing the same subjets (colored bars outside of the hemispheres) and the collinear safety properties of \ref{['fig:EWOCs:cartoon:angle_irc']} the EEC (see eq. (\ref{['eq:eec_safe']})), \ref{['fig:EWOCs:cartoon:mass_irc']} the mass EWOC (see eq. (\ref{['eq:m_ewoc_unsafe']})), and \ref{['fig:EWOCs:cartoon:energyweight_irc']} the EWOCs with non-unity energy weights of eq. (\ref{['eq:weights_defn']}). \ref{['fig:EWOCs:cartoon:angle_irc']} The EEC is collinear safe at the level of both particles and subjets: the collinear splitting does not change the set of angles between particles in a jet (see eq. (\ref{['eq:eec_safe']})). \ref{['fig:EWOCs:cartoon:mass_irc']} The particle-level mass EWOC is collinear unsafe (as are non-angular EWOCs in general) because the set of masses of particle pairs changes after a collinear splitting: $m \neq m' \neq m"$ (see eq. (\ref{['eq:m_ewoc_unsafe']})). The subjet-level mass EWOC, however, is collinear safe as long as subjets are unchanged by collinear splittings. For the same reason, \ref{['fig:EWOCs:cartoon:energyweight_irc']} generic subjet-level EWOCs remain collinear-safe even in the presence of non-unity energy weights.
• Figure 3: Mass EWOCs for $W$-boson pair production at the LHC at $\sqrt{s} = 14$ TeV for anti-$k_T$ jets with $k_T$ subjets. \ref{['fig:money:rsubs']} Mass EWOCs for several subjet radii $r_\text{sub}$; the peak at the $W$ mass is most pronounced if $r_\text{sub}$ is near the mean angular separation between the decay products of the $W$ boson, $\Delta \theta \sim 0.3$. \ref{['fig:money:mpi']} The mass EWOC for $r_\text{sub} = 0.3$ compared to the distribution of the Soft-Drop-groomed $W$-jet mass. The mass EWOC near $m_W$ is more robust to the presence of the underlying event (multiple parton interactions) than the groomed jet mass, though it experiences large corrections due to UE in the small-mass region.
• Figure 4: Variation in \ref{['fig:m_ewoc:pp_to_ww:compare-pT']} the mass EWOC with $r_\text{sub}=0.3$ and \ref{['fig:eec:pp_to_ww:compare-pT']} the EEC, for simulated LHC $W$-boson pair production events as the cut on the minimum jet $p_T$ is varied. The peak position of the mass EWOC is invariant to the choice of minimum $p_T$ of the $W$ jets, while the peak position of the EEC changes as the minimum $p_T$ is varied. Nonetheless, the peak of the EEC is well approximated by $R_\text{peak} \approx \, c \,\, m_W / \left\langle p_{T,\,\text{jet}}\right\rangle$, where $\left\langle p_{T,\,\text{jet}}\right\rangle$ is a function of the minimum $p_T$ cut and $c \sim 2.5$ is the same for each value of the minimum $p_T$.
• Figure 5: Non-perturbative hadronization and UE (MPI) effects on \ref{['fig:m_ewoc:p_v_h_v_mpi:rsub_3']} the mass EWOC with $r_\text{sub}=0.3$, and \ref{['fig:eec:p_v_h_v_mpi']} the EEC. Both have peaks which are resilient to each source of non-perturbative corrections.