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Total decay rate of a muon bound to a light nucleus

Andrzej Czarnecki, A. O. Davydov, M. Y. Kaygorodov

TL;DR

This work resolves a discrepancy in the total decay rate of a muon bound to light nuclei by performing fully relativistic, all-order-in-$\alpha Z$ calculations using Dirac-Coulomb wavefunctions for both the bound muon and the emitted electron. The authors identify that previous disagreements with the perturbative $(\alpha Z)^2$ expansion for $Z=8$ arose from insufficient convergence of the partial-wave series, and they demonstrate convergence up to high $\kappa_{\max}$ with tail extrapolation. Their results align with the perturbative expansion corrected for finite electron mass and indicate a negative next-order $\alpha Z$ correction, providing robust benchmarks for light muonic atoms and clarifying the source of prior inconsistencies. The study also confirms that neglected effects like finite-size, screening, and QED corrections are subleading for the total rate in these light systems.

Abstract

We revisit the total decay rate of a muon being in the ground state of a Coulomb potential with atomic charge numbers $4\leq Z \leq 9$. The discrepancy between the perturbative $(αZ)^2$ result of [Phys. Rev. \textbf{119}, 365 (1960)] and the fully relativistic partial-wave calculation of [At. Data Nucl. Data Tables \textbf{54}, 165 (1993)] for oxygen ($Z=8$) is shown to originate from insufficient convergence of the partial-wave series in the latter work. Our accurate relativistic calculations restore agreement with the perturbative $αZ$~expansion and indicate a negative sign for the next-order $(αZ)^3$ correction.

Total decay rate of a muon bound to a light nucleus

TL;DR

This work resolves a discrepancy in the total decay rate of a muon bound to light nuclei by performing fully relativistic, all-order-in- calculations using Dirac-Coulomb wavefunctions for both the bound muon and the emitted electron. The authors identify that previous disagreements with the perturbative expansion for arose from insufficient convergence of the partial-wave series, and they demonstrate convergence up to high with tail extrapolation. Their results align with the perturbative expansion corrected for finite electron mass and indicate a negative next-order correction, providing robust benchmarks for light muonic atoms and clarifying the source of prior inconsistencies. The study also confirms that neglected effects like finite-size, screening, and QED corrections are subleading for the total rate in these light systems.

Abstract

We revisit the total decay rate of a muon being in the ground state of a Coulomb potential with atomic charge numbers . The discrepancy between the perturbative result of [Phys. Rev. \textbf{119}, 365 (1960)] and the fully relativistic partial-wave calculation of [At. Data Nucl. Data Tables \textbf{54}, 165 (1993)] for oxygen () is shown to originate from insufficient convergence of the partial-wave series in the latter work. Our accurate relativistic calculations restore agreement with the perturbative ~expansion and indicate a negative sign for the next-order correction.
Paper Structure (4 sections, 15 equations, 1 figure, 2 tables)

This paper contains 4 sections, 15 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical method
  3. Numerical results and discussion
  4. Conclusion

Figures (1)

  • Figure 1: Comparison of the present converged values for $1-\Gamma/\Gamma_0$ (red circles) with $1-f(x_e)\,\left[1 - (\alpha Z)^2/2\right]$ (blue solid line), where $f(x_e)$ is a correction due to the finite electron mass, and the result of Ref. Watanabe:1993emp (green square). The difference between the red circles and the solid line is interpreted as the next-order-in-$\alpha Z$ corrections.