Asymptotic structure of perturbative series for $\tau$ lepton observables

Abstract.

We analyze $\tau$ lepton decay observables, namely the moments of the hadronic spectral density in the finite energy interval $(0,M_\tau)$, within finite order perturbation theory including $\alpha_{\mathrm{s}}^4$ corrections. The start of the asymptotic growth of the perturbation theory series is found at this order in a scheme-invariant manner. We establish the ultimate accuracy of finite order perturbation theory predictions and discuss the construction of optimal observables.