$\mu^- \to e^-\gamma$ in a muonic atom as a probe for effective lepton flavor violating operators involving photon fields

We propose the $\mu^-\to e^-\gamma$ process in a muonic atom as a novel means to investigate charged lepton flavor violation (CLFV). We demonstrate its sensitivity in probing effective CLFV operators associated with both single and double photon fields. In comparison to $\mu^+\to e^+\gamma$ using free positive muon decays at rest, the emitted electron and photon in $\mu^-\to e^-\gamma$ exhibit non-monochromatic spectra, presenting non-trivial case. We derive the decay rate formula and demonstrate that the potential rate of the process is significant enough to motivate exploration in future muon CLFV experiments.