Power Counting to Better Jet Observables

TL;DR

This work demonstrates that parametric power counting of soft and collinear QCD modes can guide the design and interpretation of jet substructure observables. By focusing on energy correlation functions $e_{2}^{(\beta)}$ and $e_{3}^{(\beta)}$, the authors identify a clear 1‑prong vs 2‑prong boundary and establish $D_{2}^{(\beta)} = \frac{e_{3}^{(\beta)}}{(e_{2}^{(\beta)})^{3}}$ as the natural discriminant for boosted Z discrimination, robust to boosts and mass cuts. Through Monte Carlo studies with PythIA8 and Herwig++, the power-counting predictions are validated, including the behavior under pile-up where $D_{2}^{(\beta)}$ shows improved resilience relative to $C_{2}^{(\beta)}$. The paper also discusses the limitations of power counting in quark vs gluon discrimination, where no parametric separation exists, highlighting the role of order‑1 effects and MC tuning. Overall, the approach provides analytic insight and a practical framework for designing robust jet observables with potential extensions to grooming and multi-prong tagging.

Abstract

Optimized jet substructure observables for identifying boosted topologies will play an essential role in maximizing the physics reach of the Large Hadron Collider. Ideally, the design of discriminating variables would be informed by analytic calculations in perturbative QCD. Unfortunately, explicit calculations are often not feasible due to the complexity of the observables used for discrimination, and so many validation studies rely heavily, and solely, on Monte Carlo. In this paper we show how methods based on the parametric power counting of the dynamics of QCD, familiar from effective theory analyses, can be used to design, understand, and make robust predictions for the behavior of jet substructure variables. As a concrete example, we apply power counting for discriminating boosted Z bosons from massive QCD jets using observables formed from the n-point energy correlation functions. We show that power counting alone gives a definite prediction for the observable that optimally separates the background-rich from the signal-rich regions of phase space. Power counting can also be used to understand effects of phase space cuts and the effect of contamination from pile-up, which we discuss. As these arguments rely only on the parametric scaling of QCD, the predictions from power counting must be reproduced by any Monte Carlo, which we verify using Pythia8 and Herwig++. We also use the example of quark versus gluon discrimination to demonstrate the limits of the power counting technique.

Figures (15)

• Figure 1: a) 1-prong jet, dominated by collinear (blue) and soft (green) radiation. The angular size of the collinear radiation is $R_{cc}$ and the $p_T$ fraction of the soft radiation is $z_s$. b) 2-prong jet resolved into two subjets, dominated by collinear (blue), soft (green), and collinear-soft (orange) radiation emitted from the dipole formed by the two subjets. The subjets are separated by an angle $R_{12}$ and the $p_T$ fraction of the collinear-soft radiation is $z_{cs}$.
• Figure 2: Phase space defined by the measurement of the energy correlation functions $e_{2}$ and $e_{3}$. The phase space is divided into 1- and 2-prong regions with a boundary corresponding to the curve $e_{3}\sim (e_{2})^3$.
• Figure 3: Contours of constant $C_{2}^{(\beta)}$ (left) and $D_{2}^{(\beta)}$ (right) in the phase space defined by $e_{2}^{(\beta)},e_{3}^{(\beta)}$. The 1- and 2-prong regions of phase space are labeled, with their boundary corresponding to the curve $e_{3}\sim (e_{2})^3$.
• Figure 4: Phase space defined by the energy correlation functions $e_{2}^{(2)},e_{3}^{(2)}$ in the presence of a mass cut. Contours of constant $C_{2}^{(2)}$ (left) and $D_{2}^{(2)}$ (right) are shown for reference.
• Figure 5: Phase space defined by the energy correlation functions $e_{2}^{(\beta)},e_{3}^{(\beta)}$, for $\beta<2$, in the presence of a mass cut. Contours of constant $C_{2}^{(2)}$ (left) and $D_{2}^{(2)}$ (right) are shown for reference.