Planar Property and Long-range Azimuthal Correlation in $e^+e^-$ Annihilation

TL;DR

This work investigates planar properties and Ridge-like azimuthal correlations in $e^+e^-$ annihilation by computing inclusive multi-jet production with NNLO QCD for 3-jet final states and NLO for 4-jet final states, using event-shape observables derived from the linearized sphericity tensor and a modified planarity measure, along with Ridge observables $R(\Delta\phi,\Delta\eta)$ and energy-weighted $R_{EE}$. It demonstrates strong planarity induced by perturbative jet radiation and reveals Ridge-like jet-pair correlations that are enhanced by large $|\Delta\eta|$, with fixed-order results showing convergence in key regions and highlighting the need for resummation in back-to-back configurations. The study provides high-precision QCD benchmarks for distinguishing perturbative dynamics from potential non-perturbative or collective effects, and offers a framework to compare with LEP data or future $e^+e^-$ colliders. It further validates NNLOJET-based calculations and shows Ridge signatures persist under parton showering, laying groundwork for interpreting Ridge phenomena in cleaner environments.

Abstract

The $e^+e^-$ annihilation of unpolarized beams is free from initial hadron states or initial anisotropy around the azimuthal angle, hence ideal for studying the correlations of dynamical origin via final state jets. We investigate the planar properties of the multi-jet events employing the relevant event-shape observables at next-to-next-to-leading order ($\mathcal{O}$($α_{s}^{3}$)) in perturbative QCD; particularly, the azimuthal angle correlations on the long pseudo-rapidity (polar angle) range (Ridge correlation) between the inclusive jet momenta are calculated. We illustrate the significant planar properties and the strong correlations which are natural results of the energy-momentum conservation of the perturbative QCD radiation dynamics. Our study provides benchmarks of hard strong interaction background for the investigations on the collective and/or thermal effects via the Ridge-like correlation observables for complex scattering processes.

Figures (18)

• Figure 1: An extreme planar event sample with all hadrons in the same plane. The three Sphericity axis (see Section \ref{['sec:event shape']}) are marked by the eigenvalues $\mathbf{\lambda_1}$, $\mathbf{\lambda_2}$ and $\mathbf{\lambda_3}$ (=0).
• Figure 2: The distribution of the event shape observable $\lambda_1$ in the inclusive 3-jet events at the center-of-mass energy $\sqrt{s}=m_Z$ for $y_{\text{cut}} = 10^{-2}$ (left panel) and $10^{-3}$ (right panel). The upper panels present the predictions at LO (in green), NLO (in blue), and NNLO (in red). The lower panels display the ratios of the NLO predictions and the LO predictions over the NLO predictions. The statistical errors are represented by the error bars, while the light-shaded bands indicate the systematic uncertainties arising from the scale variations according to the equation \ref{['scale']}.
• Figure 3: The distribution of the event shape observable $\lambda_2$ in the inclusive 3-jet events at the center-of-mass energy $\sqrt{s}=m_Z$ for $y_{\text{cut}} = 10^{-2}$ (left panel) and $10^{-3}$ (right panel). The upper panels present the predictions at LO (in green), NLO (in blue), and NNLO (in red). The lower panels display the ratios of the NLO predictions and the LO predictions over the NLO predictions. The statistical errors are represented by the error bars, while the light-shaded bands indicate the systematic uncertainties arising from the scale variations according to the equation \ref{['scale']}.
• Figure 4: The distribution of the event shape observable $\lambda_1$ in the inclusive 3-jet events at the center-of-mass energy $\sqrt{s}=m_Z$ for $y_{\text{cut}} = 10^{-2}$ (left panel) and $10^{-3}$ (right panel), based on PYTHIA 8.3 Bierlich:2022pfr.
• Figure 5: The distribution of the event shape observable $\lambda_2$ in the inclusive 3-jet events at the center-of-mass energy $\sqrt{s}=m_Z$ for $y_{\text{cut}} = 10^{-2}$ (left panel) and $10^{-3}$ (right panel), based on PYTHIA 8.3 Bierlich:2022pfr.