  If light new physics with lepton-flavor-violating couplings exists, the prime
discovery channel might not be $\ell\to\ell'\gamma$ but rather $\ell\to\ell'
X$, where the new boson $X$ could be an axion, majoron, familon or Z' gauge
boson. The most conservative bound then comes from $\ell\to\ell'+\mathrm{inv}$,
but if the on-shell $X$ can decay back into leptons or photons,
displaced-vertex searches could give much better limits. We show that only a
narrow region in parameter space allows for displaced vertices in muon decays,
$\mu\to e X, X\to \gamma\gamma, ee$, whereas tauon decays can have much more
interesting signatures.
