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Pion $β$ decay and $τ\toππν_τ$ beyond leading logarithms

Vincenzo Cirigliano, Martin Hoferichter, Nicola Valori

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 $β$ decay $π^\pm\toπ^0 e^\pmν_e$ and hadronic $τ$ decays $τ^\pm\toπ^\pmπ^0ν_τ$ -- 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 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 $τ\toππν_τ$ 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.

Pion $β$ decay and $τ\toππν_τ$ 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 decay and hadronic decays -- whose decay rates depend on the so-called 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 decay with theory uncertainties improved by a factor of three, which renders theory uncertainties negligible for future determinations of even beyond the reach of the PIONEER experiment, and an evaluation of isospin-breaking corrections to with negligible uncertainty from the short-distance matching, as necessary for a future -based determination of the hadronic-vacuum-polarization contribution to the anomalous magnetic moment of the muon.
Paper Structure (12 sections, 55 equations, 2 figures)

This paper contains 12 sections, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Renormalization group and matrix elements
  3. Definitions and conventions
  4. Matching to ChPT
  5. Nonperturbative matrix element
  6. Derivation of the matching conditions and universality
  7. Applications
  8. Pion decay
  9. Conclusions
  10. Matching with the spurion method
  11. Matching at the amplitude level
  12. Renormalization group to NLL

Figures (2)

  • Figure 1: Resulting integrand $M_\pi(Q^2)$ for the lattice-QCD inputs from Ref. Feng:2020zdc (upper, $n_f=3$) and Ref. Yoo:2023gln (lower, $n_f=4$). In both cases, we show the lattice-QCD results up to $2\,\text{GeV}$, the pQCD curves above $1\,\text{GeV}$ (green dashed and solid for three- and four-loop order), and the constant one-loop result above the respective values of $Q_1$ (maroon dashed and solid). The resulting subtracted integrand as it enters in Eq. \ref{['box_EFT']} is represented by the black curves (dashed and solid for three- and four-loop pQCD between $Q_0$ and $Q_1$). The dashed vertical line indicates the value of $Q_1$ for the three-loop case.
  • Figure B.2: Diagrams contributing to the $\mathcal{O}(\alpha)$ corrections to $\beta$ decays in LEFT. Double lines represent hadronic states, single lines leptons, and wavy lines photons. The weak and electromagnetic current insertions are represented by hatched circles.