alpha_s and the tau hadronic width: fixed-order, contour-improved and   higher-order perturbation theory
  The determination of $\alpha_s$ from hadronic $\tau$ decays is revisited, with a special emphasis on the question of higher-order perturbative corrections and different possibilities of resumming the perturbative series with the renormalisation group: fixed-order (FOPT) vs. contour-improved perturbation theory (CIPT). The difference between these approaches has evolved into a systematic effect that does not go away as higher orders in the perturbative expansion are added. We attempt to clarify under which circumstances one or the other approach provides a better approximation to the true result. To this end, we propose to describe the Adler function series by a model that includes the exactly known coefficients and theoretical constraints on the large-order behaviour originating from the operator product expansion and the renormalisation group. Within this framework we find that while CIPT is unable to account for the fully resummed series, FOPT smoothly approaches the Borel sum, before the expected divergent behaviour sets in at even higher orders. Employing FOPT up to the fifth order to determine $\alpha_s$ in the $\MSb$ scheme, we obtain $\alpha_s(M_\tau)=0.320 {}^{+0.012}_{-0.007}$, corresponding to $\alpha_s(M_Z) = 0.1185 {}^{+0.0014}_{-0.0009}$. Improving this result by including yet higher orders from our model yields $\alpha_s(M_\tau)=0.316 \pm 0.006$, which after evolution leads to $\alpha_s(M_Z) = 0.1180 \pm 0.0008$. Our results are lower than previous values obtained from $\tau$ decays. 
Martin Beneke and Matthias Jamin