Abstract
Studies of the charged lepton flavor violating process of $\mu^-e^-\to e^-e^-$ in muonic atoms by the four Fermi interaction [Y. Uesaka \textit{et al}., Phys. Rev. D {\bf 93}, 076006 (2016)] are extended to include the photonic interaction. The wave functions of a muon and electrons are obtained by solving the Dirac equation with the Coulomb interaction of a finite nuclear charge distribution. We find suppression of the $\mu^-e^-\to e^-e^-$ rate over the initial estimation for the photonic interaction, in contrast to enhancement for the four Fermi interaction. It is due to the Coulomb interaction of scattering states and relativistic lepton wave functions. This finding suggests that the atomic number dependence of the $\mu^-e^-\to e^-e^-$ rate could be used to distinguish between the photonic and the four Fermi interactions.
Highlights
We find suppression of the μ−e− → e−e− rate over the initial estimation for the photonic interaction, in contrast to enhancement for the four Fermi interaction
It has been well recognized that charged lepton flavor violation (CLFV) is important to search for new physics beyond the standard model
Together with our previous analysis [20] for the contact interaction and the present work for the photonic interaction, we find that the relativistic treatment of the emitted electrons and bound leptons is essentially important for their qualitative understanding the rate, in particular the atomic number Z dependence of the rate and the angular and energy distribution of electrons
Summary
It has been well recognized that charged lepton flavor violation (CLFV) is important to search for new physics beyond the standard model. For heavy atoms with large atomic numbers (Z), large enhancement of the μ−e− → e−e− rate due to the Coulomb attraction of the lepton wave functions to a nucleus is expected Another advantage for μ−e− → e−e− is that it can probe both the four Fermi contact and the photonic interactions, as in the μþ → eþe−eþ decay and μ− → e− conversion. The initial work [8] showed that the atomic number (Z) dependence of the μ−e− → e−e− transition rate is expected to be of Z3, owing to the probability density of the wave functions of the Coulomb-bound electrons at origin.
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