The physics of hadronic tau decays

Hadronic $\tau$ decays provide a clean laboratory for the precise study of quantum chromodynamics (QCD). Observables based on the spectral functions of hadronic $\tau$ decays can be related to QCD quark-level calculations to determine fundamental quantities like the strong-coupling constant, parameters of the chiral Lagrangian $\mid V_{us}\mid$, the mass of the strange quark, and to simultaneously test the concept of quark-hadron duality. Using the best available measurements and a revisited analysis of the theoretical framework, the value ${\alpha }_s\left(m_{\tau }^2\right)=0.345\pm {0.004}_{\exp }\pm {0.009}_{\mathrm{th}}$ is obtained. Taken together with the determination of ${\alpha }_s\left(M_Z^2\right)$ from the global electroweak fit, this result leads to the most accurate test of asymptotic freedom: the value of the logarithmic slope of ${\alpha }_s^{-1}\left(s\right)$ is found to agree with QCD at a precision of 4%. The $\tau$ spectral functions can also be used to determine hadronic quantities that, due to the nonperturbative nature of long-distance QCD, cannot be computed from first principles. An example for this is the contribution from hadronic vacuum polarization to loop-dominated processes like the anomalous magnetic moment of the muon. This article reviews the measurements of nonstrange and strange $\tau$ spectral functions and their phenomenological applications.