Axion-like Particles and Lepton Flavor Violation in Muonic Atoms

We explore the potential of the Mu2e experiment to probe the lepton-flavor-violating process $\mu^- e^- \to e^- e^-$ in a muonic atom within a simplified axion-like particle (ALP) framework featuring flavor-violating $e$-$\mu$ couplings and a flavor-diagonal pseudoscalar coupling to electrons, which also allows for possible invisible ALP decays into a dark sector. We compute the ALP-mediated contribution to the transition rate and show that, at fixed couplings, the branching ratio increases for lighter mediators and scales as $(Z-1)^3$, favoring heavier nuclei. We compare the model against constraints from $\mu\to e\gamma$, $\mu\to 3e$, $\mu\to e\gamma\gamma$, $\mu\to e+\mathrm{inv}$, and muonium-antimuonium conversion, as well as from the anomalous magnetic moments of the electron and muon. Additional astrophysical and beam-dump limits on the electron coupling are also discussed. A key result is that $\Delta a_e$ provides one of the most stringent probes of the parameter space and, in the global scan, excludes the largest fraction of sampled points. After applying the laboratory constraints used in the scan, the viable branching ratio for $\mu^- e^- \to e^- e^-$ in aluminum drops to at most $\mathcal{O}(10^{-20})$, while the resonant region $2m_e<m_a<m_\mu-m_e$ is much more heavily suppressed. The highest achievable values are closely tied to $\mathcal{B}(\mu\to 3e)$ near its current limit, indicating that the upcoming Mu3e experiment will explore the most promising region relevant for this muonic-atom signal. Our analysis shows that, although a light ALP can parametrically enhance $\mu^- e^- \to e^- e^-$ at fixed couplings, existing bounds -- especially $\Delta a_e$, $\mu\to 3e$, $\mu\to e\gamma$, and muonium oscillations -- severely limit the observable rate.