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.

13 pages, 2 figures; journal version

References in corpus (16)

• Measurement of the fine-structure constant as a test of the Standard Model
• The $g-2$ of charged leptons, $\alpha (M_Z^2)$ and the hyperfine splitting of muonium
• Adler Function, Bjorken Sum Rule, and the Crewther Relation to Order alpha_s^4 in a General Gauge Theory
• High-precision quark masses and QCD coupling from $n_f=4$ lattice QCD
• The anomalous magnetic moment of the muon in the Standard Model: an update
• Electroweak Corrections to the Charm Quark Contribution to K+ -> pi+ nu nu-bar
• The strong coupling from a nonperturbative determination of the $\Lambda$ parameter in three-flavor QCD
• First-principles calculation of electroweak box diagrams from lattice QCD
• Long-distance radiative corrections to the di-pion tau lepton decay
• The Realistic Lattice Determination of alpha_s(M_Z) Revisited
• Isospin-breaking effects in the two-pion contribution to hadronic vacuum polarization
• First-Generation New Physics in Simplified Models: From Low-Energy Parity Violation to the LHC
• A Minimal Explanation of Flavour Anomalies: B-Meson Decays, Muon Magnetic Moment, and the Cabibbo Angle
• Electroweak Radiative Corrections to Semileptonic Tau Decays
• Radiative corrections to $\tau \to \pi \pi {\nu }_{\tau }$
• Improved calculation of radiative corrections to $\tau \to \pi \pi {\nu }_{\tau }$ decays