Tests of analytical hadronisation models using event shape moments in {\epem} annihilation

TL;DR

The paper systematically tests analytical hadronisation models against moments of event-shape distributions in $\,e^+e^-$ annihilation over $14$–$209\mathrm{\,GeV}$ using JADE and OPAL data. By fitting $\alpha_{S}(M_{Z^0})$ and universal nonperturbative parameters for the dispersive model and shape function, and by applying the SDG approximation, it assesses universality, energy evolution, and the impact of higher-order corrections. The results show mean values are broadly well described across models, with the dispersive model exhibiting universality in its two main parameters, while higher moments and variances reveal model-specific limitations; the SDG approach provides the most precise $\alpha_{S}(M_{Z^0})$ measurement. Overall, moments offer a powerful, complementary test of nonperturbative QCD and highlight the need for higher-order refinements in analytical hadronisation formalisms.

Abstract

Predictions of analytical models for hadronisation, namely the dispersive model, the shape function and the single dressed gluon approximation, are compared with moments of hadronic event shape distributions measured in \epem annihilation at centre-of-mass energies between 14 and 209 GeV. In contrast to Monte Carlo models for hadronisation, analytical models require to adjust only two universal parameters, the strong coupling and a second quantity parametrising nonperturbative corrections. The extracted values of as are consistent with the world average and competitive with previous measurements. The variance of event shape distributions is compared with predictions given by some of these models. Limitations of the models, probably due to unknown higher order corrections, are demonstrated and discussed.

Figures (9)

• Figure 1: Fits of the dispersive prediction to JADE and OPAL measurements of $1-T$ moments. The solid line shows the prediction with fitted values of $\alpha_\mathrm{S}(M_{\mathrm{Z^0}})$ and $\alpha_0(\mu_I)$, the dashed line shows the pure NLO contribution. The inner error bars show the statistical uncertainties used in the fit, and the outer error bars show the combined statistical and experimental systematic errors
• Figure 2: Fits of the dispersive prediction to JADE and OPAL measurements of $B_\mathrm{W}$ moments. The solid line shows the prediction with fitted values of $\alpha_\mathrm{S}(M_{\mathrm{Z^0}})$ and $\alpha_0(\mu_I)$, the dashed line shows the pure NLO contribution. The inner error bars show the statistical uncertainties used in the fit, and the outer error bars show the combined statistical and experimental systematic errors
• Figure 3: Measurements of $\alpha_\mathrm{S}(M_{\mathrm{Z^0}})$ and $\alpha_0(\mu_I)$ from moments of six event shape variables at PETRA and LEP energies. The inner error bars---where visible---show the statistical errors, the outer bars show the total errors. The dashed lines indicate the weighted averages, the shaded bands show their errors. Only the measurements indicated by solid symbols are used for the averages
• Figure 4: Fits of the shape function prediction with $\overline\lambda_2=0$ to JADE and OPAL measurements of first and second moments of $1-T$ and $C$ and second and fourth moments of $M_\mathrm{H}$. The solid line shows the fitted prediction and the dashed line shows the pure NLO contribution. The inner error bars show the statistical uncertainties used in the fit and the outer error bars show the combined statistical and experimental systematic errors. Most of the error bars are smaller than the data points
• Figure 5: Measurements of $\alpha_\mathrm{S}(M_{\mathrm{Z^0}})$ and $\lambda_1$ from moments of three event shape variables at PETRA and LEP energies. The inner error bars---where visible---show the statistical errors, the outer bars show the total errors