Perturbative relations between $e^+e^-$ annihilation and $τ$ decay observables including resummation effects

By exploiting the analyticity properties of the two-point current-current correlator we obtain numerical predictions for the $e^+e^-$ moments in terms of the $\tau$ decay rate. We perform a partial resummation of the pertinent perturbative series expansion by solving the renormalization group equation for Adler's function. Our predictions are renormalization scheme independent but depend on the order of the perturbative $\beta$-function expansion. The analysis involves the unknown five-loop coefficient $k_3$ for which we give some new estimates.