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.