  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.
