Pion β Decay and τ → π π ν τ beyond Leading Logarithms

The consistent matching of short-distance contributions and hadronic matrix elements is crucial for precise predictions of weak processes involving hadrons. In this Letter, we address this point for charged-current processes involving two pions-pion β decay π^{±}→π^{0}e^{±}ν_{e} and hadronic τ decays τ^{±}→π^{±}π^{0}ν_{τ}-whose decay rates depend on the so-called γW box correction. Using recent results from lattice QCD, we show how to formulate the matching beyond leading-logarithmic accuracy, in particular, how to cancel the dependence on the scheme choice for evanescent operators. As main results, we obtain a prediction for the decay rate of pion β decay with theory uncertainties improved by a factor of 3, which renders theory uncertainties negligible for future determinations of V_{ud} even beyond the reach of the PIONEER experiment, and an evaluation of isospin-breaking corrections to τ→ππν_{τ} with negligible uncertainty from the short-distance matching, as necessary for a future τ-based determination of the hadronic-vacuum-polarization contribution to the anomalous magnetic moment of the muon.