A Novel Method for the Precise Determination of the QCD Running Coupling from Event Shapes Distributions in Electron-Positron Annihilation

We present a novel method for precisely determining the running QCD coupling constant $\alpha_s(Q^2)$ over a wide range of $Q^2$ from event shapes for electron-positron annihilation measured at a single annihilation energy $\sqrt{s}$. The renormalization scale $Q^2$ of the running coupling depends dynamically on the virtuality of the underlying quark and gluon subprocess and thus the specific kinematics of each event. The determination of the renormalization scale for event shape distributions is obtained by using the Principle of Maximum Conformality (PMC), a rigorous scale-setting method for gauge theories which satisfies all the requirements of Renormalization Group Invariance, including renormalization-scheme independence and consistency with Abelian theory in the $N_C \to 0$ limit. In this paper we apply the PMC to two classic event shapes measured in $e^+ e^-$ annihilation: the thrust ($T$) and C-parameter ($C$). The PMC renormalization scale depends differentially on the values of $T$ and $C$. The application of PMC scale-setting determines the running coupling $\alpha_s(Q^2)$ to high precision over a wide range of $Q^2$ from $10~\rm{GeV}^2 $ to $250~\rm{GeV}^2$ from measurements of the event shape distributions at the $Z^0$ peak. The extrapolation of the running coupling using pQCD evolution gives the value $\alpha_s(M^2_Z)=0.1185\pm0.0012$ from the thrust, and $\alpha_s(M^2_Z)=0.1193^{+0.0021}_{-0.0019}$ from the C-parameter in the $\overline {\rm{MS}}$ scheme. These determinations of $\alpha_s(M^2_Z)$ are consistent with the world average and are more precise than the values obtained from analyses of event shapes currently used in the world average. The highly-consistent results for the $T$ and $C$ event-shape distributions provides an additional verification of the applicability of the PMC to pQCD.