Scheme variations of the QCD coupling and hadronic $\tau$ decays

The Quantum Chromodynamics (QCD) coupling, $\alpha_s$, is not a physical observable of the theory since it depends on conventions related to the renormalization procedure. We introduce a definition of the QCD coupling, denoted by $\hat\alpha_s$, whose running is explicitly renormalization scheme invariant. The scheme dependence of the new coupling $\hat\alpha_s$ is parameterized by a single parameter $C$, related to transformations of the QCD scale $\Lambda$. It is demonstrated that appropriate choices of $C$ can lead to substantial improvements in the perturbative prediction of physical observables. As phenomenological applications, we study $e^+e^-$ scattering and decays of the $\tau$ lepton into hadrons, both being governed by the QCD Adler function.