A formalism for constructing the QCD power spectrum with finite sampling

TL;DR

The paper addresses how finite particle sampling and detector effects distort the QCD angular power spectrum obtained from energy correlations. It introduces shape functions to replace delta-function energy deposits, effectively low-pass filtering the event’s energy flow so that the angular power spectrum $H_\ell$ retains meaningful information up to an event-dependent angular resolution $\xi_{\min}$ while remaining infrared- and collinear-safe. By deriving $H_\ell$ for extensive objects and incorporating measurement uncertainties, the authors provide a robust, event-by-event, fully-correlated description of QCD radiation that accounts for detector geometry and object extents. The approach promises improved insights into jet substructure and hadronization, with potential extensions to hadron colliders and integration with pileup mitigation strategies, offering a principled alternative to conventional Fox–Wolfram moment analyses.

Abstract

Recent progress in the study of QCD phenomena with energy correlators motivates novel approaches to explore the information contained in the QCD radiation spectrum. Fox-Wolfram moments are a set of observables that characterize the angular distribution of energy flow in high-energy collisions. Contrary to their conventional application, they are a class of correlated moments that cannot be reduced to one or several characteristic ones. We present a formalism to extract their fully correlated information content, while systematically discarding small-angle sampling noise, on an event-by-event basis. We show that our approach circumvents a common misspecification of the data in terms of $δ$-distributions, and is essential in keeping the power spectrum infrared and collinear safe. Our formalism introduces a means of accounting for the varying spatial extent of objects that enter the calculation of the power spectrum, with which experimental artifacts such as detector element geometries and measurement uncertainties can readily be incorporated.

Figures (4)

• Figure 1: The power spectra for two $e^{+}e^{-}\to q\bar{q}g$ events at $\sqrt{S}=\text{250} \, \text{GeV}$ which are (a) 2-jet-like and (b) 3-jet-like. Lines connect even and odd moments at integer $\ell$. Each figure shows $H_\ell$ for (upper) $n=3$ initiating partons and (lower) $N=\mathcal{O}(60)$ measurable particles (after showering and hadronization). A dashed line shows the value of $\left\langle f|f\right\rangle$ for each series, and the inset depicts the energy and orientation of the original partons.
• Figure 2: The power spectrum of a random sampling of isotropically distributed particles. Each series is labeled by the number of objects $N^*$ detected, with (a) the entire sample detected as charged tracks, and (b) the entire sample detected as neutral towers. A horizontal tick on the right edge depicts each sample's $\left\langle f|f\right\rangle$.
• Figure 3: The squared coefficient $\bar{h}^2_\ell$ for the pseudo-normal shape function, with several smearing angles $\lambda$ on a (a) linear and (b) log-linear scale.
• Figure 4: The angular correlation function for two $e^+e^-\to q\bar{q}g$ events at $\sqrt{S}=\text{250}\,\text{GeV}$ which are (a) 2-jet-like and (b) 3-jet-like.