Consistent Measurements of alpha_s from Precise Oriented Event Shape Distributions

TL;DR

The paper addresses a high-precision determination of the strong coupling $\alpha_s$ from oriented event-shape distributions in $e^+e^-$ annihilation at the Z pole. It analyzes DELPHI 1994 data with 18 infrared- and collinear-safe observables, employing ${\cal O}(\alpha_s^2)$ QCD including event orientation and experimentally optimized renormalization scales, and cross-validates with NLLA, Padé approximants, and heavy-quark mass effects. The study finds a stable $\alpha_s(M_Z^2)$ around 0.1168–0.1180, with the Jet Cone Energy Fraction (JCEF) providing the smallest total uncertainty; results are consistent across methods, reinforcing confidence in scale-setting procedures. Overall, the work demonstrates robust, multi-observable consistency for $\alpha_s$ determinations at the Z mass and highlights the value of scale-optimization and oriented observables for precision QCD tests.

Abstract

An updated analysis using about 1.5 million events recorded at $\sqrt{s} = M_Z$ with the DELPHI detector in 1994 is presented. Eighteen infrared and collinear safe event shape observables are measured as a function of the polar angle of the thrust axis. The data are compared to theoretical calculations in ${\cal O} (α_s^2)$ including the event orientation. A combined fit of $α_s$ and of the renormalization scale $x_μ$ in $\cal O(α_s^2$) yields an excellent description of the high statistics data. The weighted average from 18 observables including quark mass effects and correlations is $α_s(M_Z^2) = 0.1174 \pm 0.0026$. The final result, derived from the jet cone energy fraction, the observable with the smallest theoretical and experimental uncertainty, is $α_s(M_Z^2) = 0.1180 \pm 0.0006 (exp.) \pm 0.0013 (hadr.) \pm 0.0008 (scale) \pm 0.0007 (mass)$. Further studies include an $α_s$ determination using theoretical predictions in the next-to-leading log approximation (NLLA), matched NLLA and $\cal O(α_s^2$) predictions as well as theoretically motivated optimized scale setting methods. The influence of higher order contributions was also investigated by using the method of Padé approximants. Average $α_s$ values derived from the different approaches are in good agreement.

Figures (3)

• Figure 1: Results of the QCD fits applying experimentally optimized scales for 18 event shape distributions. The error bars indicated by the solid lines are the quadratic sum of the experimental and the hadronization uncertainty. The error bars indicated by the dotted lines include also the additional uncertainty due to the variation of the renormalization scale due to scale variation around the central value $x_{\mu}^{exp}$ in the range $0.5 \cdot x_{\mu}^{exp} \le x_{\mu} \le 2 \cdot x_{\mu}^{exp}$. Also shown is the correlated weighted average (see text). The $\chi^{2}$-value is given before readjusting according to Eq. \ref{['chisquare']}.
• Figure 2: Results of the QCD fits applying a fixed renormalization scale $x_\mu = 1$ . The error bars indicated by the solid lines are the quadratic sum of the experimental and the hadronization uncertainty. The error bars indicated by the dotted lines include also the additional uncertainty due to the variation of the renormalization scale around the central value $x_{\mu}^{exp}$ from $0.5 \cdot x_{\mu}^{exp} \le x_{\mu} \le 2 \cdot x_{\mu}^{exp}$. Also shown is the correlated weighted average. It has been calculated assuming the same effective correlation $\rho_{{\rm eff}} = 0.635$ as for the fit results applying experimentally optimized scales. The $\chi^2 / n_{df}$ for the weighted average is 71/17, where the $\chi^{2}$ given corresponds to the value before adjusting $\rho_{{\rm eff}}$. In order to yield $\chi^2 / n_{df} = 1$, the errors have to be scaled by a factor $\rm f_{err} =3.38$.
• Figure 3: $\alpha_s(M_{\rm Z}^2)$ and $\rm \Delta \chi^2 = \chi^2 - \chi^2_{min}$ for the distribution of the Jet Cone Energy Fraction as a function of $x_{\mu}$ from QCD fits applying $\cal O$($\alpha_s^2$) prediction, $\cal O$($\alpha_s^3$) in Padé Approximation and the Padé Sum Approximation. Additionally, the $\chi^2$ minimum for the $\cal O$($\alpha_s^2$) fit and the renormalization scale value $x_{\mu} = 1$ have been indicated in the $\alpha_s(M_{\rm Z}^2)$ curves.