Observational limits on the high-energy neutrino background have been used to place general constraints on dark matter that annihilates only into standard model particles. Dark matter particles that annihilate into neutrinos will also inevitably branch into electromagnetic final states through higher-order tree and loop diagrams that give rise to charged leptons, and these charged particles can transfer their energy into photons via synchrotron radiation or inverse Compton scattering. In the context of effective field theory, we calculate the loop-induced branching ratio to charged leptons and show that it is generally quite large, typically >1%, when the scale of the dark matter mass exceeds the electroweak scale, M_W. For a branching fraction >3%, the synchrotron radiation bounds on dark matter annihilation are currently stronger than the corresponding neutrino bounds in the interesting mass range from 100 GeV to 1 TeV. For dark matter masses below M_W, our work provides a plausible framework for the construction of a model for neutrinos only dark matter annihilations.
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