Improved analysis for $\mu^-e^-\to e^-e^-$ in muonic atoms by photonic interaction

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.