Scheme-independent determination of the QCD running coupling at all scales from jet observables using the principle of maximum conformality and infinite-order scale setting

We present a new approach to determining the strong coupling ${\alpha }_s\left(Q\right)$, over the entire range of validity of perturbative QCD, for scales above ${\Lambda }_{\mathrm{QCD}}$ and up to the Planck scale $\sim 1.22·{10}^{19}$ GeV, with the highest precision and using the data of a single experiment. In particular, we use the results obtained for the thrust ($T$) and $C$-parameter ($C$) distributions in $e^{+}{e}^{-}$ annihilation at a single annihilation energy $\sqrt{s}=M_Z$ (i.e. at the $Z^0$ peak). This new method is based on the intrinsic conformality (iCF) and on the Infinite-Order Scale Setting, using the Principle of Maximum Conformality (i.e. the PMC${}_{\infty }$), which allows a rigorous determination of the renormalization scales for the event-shape variable distributions satisfying all of the requirements of Renormalization Group Invariance, including renormalization-scheme independence and consistency with Abelian theory in the $N_C\to 0$ limit. This new method is based on the scale-invariance of the iCF, which allows determination of ${\alpha }_s\left({\mu }_0\right)$ at any scale ${\mu }_0$, and on the Maximum Likelihood statistical approach. We propose a novel approach to determining the best-fitting range by considering all possible intervals over the entire range of bins available in the perturbative region and selecting that which returns the most-likely-lowest ${\chi }_{\min }^2$. This new method is designed to eliminate the errors that arise due to selection of the bin-interval and that have been neglected in previous analyses. In particular, using data for thrust and $C$-parameter at the $Z^0$ peak from ALEPH, OPAL, DELPHI and L3 experiments, we obtain the average value: ${\alpha }_s\left(M_Z\right)=0.{1182}_{-0.0007}^{+0.0007}$, for the strong coupling. This determination of ${\alpha }_s\left(M_Z\right)$ is consistent with the world average and has an improved precision with respect to the values obtained from the analysis of event shape observables currently used in the world average.