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Lower Limits on $μ\to e γ$ from new Measurements on $U_{e3}$

New data on the lepton mixing angle $θ_{13}$ imply that the $eμ$ element of the matrix $m_νm_ν^\dagger$, where $m_ν$ is the neutrino Majorana mass matrix, cannot vanish. This implies a lower limit on lepton flavor violating processes in the $eμ$ sector in a variety of frameworks, including Higgs triplet models or the concept of minimal flavor violation in the lepton sector. We illustrate this for the branching ratio of $μ\to e γ$ in the type II seesaw mechanism, in which a Higgs triplet is responsible for neutrino mass and also mediates lepton flavor violation. We also discuss processes like $μ\to e\bar{e}e$ and $μ\to e$ conversion in nuclei. Since these processes have sensitivity on the individual entries of $m_ν$, their rates can still be vanishingly small.