Resummation of $(\beta_0 \alpha_s)^n$ Corrections in QCD: Techniques and Applications to the $\tau$ Hadronic Width and the Heavy Quark Pole Mass

We propose to resum exactly any number of one-loop vacuum polarization insertions into the scale of the coupling of lowest order radiative corrections. This makes maximal use of the information contained in one-loop perturbative corrections combined with the one-loop running of the effective coupling and provides a natural extension of the familiar BLM scale-fixing prescription to all orders in the perturbation theory. It is suggested that the remaining radiative corrections should be reduced after resummation. In this paper we implement this resummation by a dispersion technique and indicate a possible generalization to incorporate two-loop evolution. We investigate in some detail higher order perturbative corrections to the $\tau$ decay width and the pole mass of a heavy quark. We find that these corrections tend to reduce $\alpha_s(m_\tau)$ determined from $\tau$ decays by approximately 10\% and increase the difference between the bottom pole and $\MS$-renormalized mass by 30\%.