Pion $\beta$ decay and $\tau \to \pi \pi {\nu }_{\tau }$ beyond leading logarithms

Abstract

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 $\beta$ decay ${\pi }^{\pm }\to {\pi }^0{e}^{\pm }{\nu }_e$ and hadronic $\tau$ decays ${\tau }^{\pm }\to {\pi }^{\pm }{\pi }^0{\nu }_{\tau }$ -- whose decay rates depend on the so-called $\gamma W$ 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 $\beta$ decay with theory uncertainties improved by a factor of three, 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 $\tau \to \pi \pi {\nu }_{\tau }$ with negligible uncertainty from the short-distance matching, as necessary for a future $\tau$-based determination of the hadronic-vacuum-polarization contribution to the anomalous magnetic moment of the muon.

Comments: 13 pages, 2 figures; journal version