  The charged lepton flavor violating (CLFV) processes of $\mu^-e^-\rightarrow
e^-e^-$ decay by four Fermi contact interactions in a muonic atom for various
atoms are investigated. The wave functions of bound and scattering state
leptons are properly treated by solving Dirac equations with Coulomb
interaction of the finite nuclear charge distributions. This new effect
contributes significantly in particular for heavier atoms, where the obtained
decay rate is about one order of magnitude larger than the previous estimation
for $^{208}$Pb. We find that, as the atomic number $Z$ increases, the
$\mu^-e^-\rightarrow e^-e^-$ decay rates increase more rapidly than the result
of the previous work of $Z^3$, suggesting this decay as one of the promising
processes to search for CLFV interaction.
