Four jet event shapes in electron-positron annihilation

TL;DR

This work delivers a comprehensive NLO QCD analysis of four-jet event shapes in e+e- annihilation using the EERAD2 Monte Carlo, combining virtual γ*→4 partons and real γ*→5 partons via a hybrid antenna-based subtraction-slicing scheme. It defines and computes new NLO coefficients for the light hemisphere mass, narrow hemisphere broadening, Aplanarity, and jet-transition variables across JADE and Geneva algorithms, and benchmarks these against existing Monte Carlo results. The study finds large NLO corrections (K ~ 1.5–2) and sizable renormalization-scale uncertainties, with data typically exceeding fixed-order predictions; this underscores the importance of resummation of infrared logs and non-perturbative power corrections for a faithful description of four-jet observables. Overall, the results validate EERAD2, reveal the necessity of higher-order and non-perturbative effects for four-jet physics, and mirror patterns already seen in three-jet observables.

Abstract

We report next-to-leading order perturbative QCD predictions of 4 jet event shape variables for the process e+e- \to 4 jets obtained using the general purpose Monte Carlo EERAD2. This program is based on the known `squared' one loop matrix elements for the virtual γ^* \to 4 parton contribution and squared matrix elements for 5 parton production. To combine the two distinct final states numerically we present a hybrid of the commonly used subtraction and slicing schemes based on the colour antenna structure of the final state which can be readily applied to other processes. We have checked that the numerical results obtained with EERAD2 are consistent with next-to-leading order estimates of the distributions of previously determined four jet-like event variables. We also report the next-to-leading order scale independent coefficients for some previously uncalculated observables; the light hemisphere mass, narrow jet broadening, Aplanarity and the 4 jet transition variables with respect to the JADE and Geneva jet finding algorithms. For each of these observables, the next-to-leading order corrections calculated at the physical scale significantly increase the rate compared to leading order (the K factor is approximately 1.5 -- 2). With the exception of the 4 jet transition variables, the published DELPHI data lies well above the O(α_s^3) predictions. The renormalisation scale uncertainty is still large and in most cases the data prefers a scale significantly smaller than the physical scale. This situation is reminiscent of that for three jet shape variables, and should be improved by the inclusion of power corrections and resummation of large infrared logarithms.

Figures (9)

• Figure 1: The K-factors defined according to eq. (\ref{['eq:Kfac']}) for four jet event shapes, the light hemisphere mass (solid), narrow jet broadening (long-dashed), Aplanarity (dot-dashed) and jet transition variables in the JADE (short-dashed) and Geneva (dotted) schemes. Each variable has a different kinematic range.
• Figure 2: The leading order (dashed) and next-to-leading order (solid) predictions evaluated at the physical scale $\mu = \sqrt{s} = M_Z$ for (a) $1/\sigma_{\rm had} \cdot d\sigma/dB_{\rm min}$ compared to the published DELPHI data 4jetdata and (b) the difference between data and NLO theory (normalised to NLO). The short-dashed line shows the next-to-leading order prediction using the FAC scale (see eq. (\ref{['eq:facscale']})).
• Figure 3: The leading order (dashed) and next-to-leading order (solid) predictions evaluated at the physical scale $\mu = \sqrt{s} = M_Z$ for (a) $1/\sigma_{\rm had} \cdot d\sigma/d(M_L^2/s)$ compared to the published DELPHI data 4jetdata and (b) the difference between data and NLO theory (normalised to NLO). The short-dashed line shows the next-to-leading order prediction using the FAC scale (see eq. (\ref{['eq:facscale']})).
• Figure 4: The leading order (dashed) and next-to-leading order (solid) predictions evaluated at the physical scale $\mu = \sqrt{s} = M_Z$ for (a) $1/\sigma_{\rm had} \cdot d\sigma/dA$ compared to the published DELPHI data 4jetdata and (b) the difference between data and NLO theory (normalised to NLO). The short-dashed line shows the next-to-leading order prediction using the FAC scale (see eq. (\ref{['eq:facscale']})).
• Figure 5: The leading order (dashed) and next-to-leading order (solid) predictions evaluated at the physical scale $\mu = \sqrt{s} = M_Z$ for (a) $1/\sigma_{\rm had} \cdot d\sigma/dy_4^J$ compared to the published DELPHI data 4jetdata and (b) the difference between data and NLO theory (normalised to NLO). The short-dashed line shows the next-to-leading order prediction using the FAC scale (see eq. (\ref{['eq:facscale']})).