  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.
