An upper limit on the total annihilation cross section of dark matter (DM) has recently been derived from the observed atmospheric neutrino background. We show that comparable bounds are obtained for DM masses around the TeV scale by observations of the diffuse gamma-ray flux by EGRET, because electroweak bremsstrahlung leads to non-negligible electromagnetic branching ratios, even if DM particles only couple to neutrinos at tree level. A better mapping and the partial resolution of the diffuse gamma-ray background into astrophysical sources by the GLAST satellite will improve this bound in the near future.
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