Koike et al. proposed the charged lepton flavor violation (CLFV) decay of the muonic atom $mu^-e^-rightarrow e^-e^-$ as one of the promising processes to search for new physics beyond the standard model. It was found that the attractive interaction of leptons with the nucleus enhances the transition rate of the $mu^-e^-rightarrow e^-e^-$ process. We report on our improved analysis of this process by taking into account the distortion of the out-going electrons in the nuclear Coulomb potential and the relativistic treatment of the muon and the electrons. As results, we found significant enhancement of the transition rate. The transition rate for $^{208}$Pb becomes about 7 times larger than the previous estimation, which enhances the sensitivity of this process to discover the CLFV interaction. We also report on the energy spectrum of the out-going electron.
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