Lower limits on the strengths of gamma ray lines from WIMP dark matter annihilation

We study the spectra of gamma ray signals that arise from dark matter annihilation in the Universe. We focus on the large class of theories where the photon spectrum includes both continuum spectrum of gamma rays that arise from annihilation into standard model states at tree level, as well as monochromatic gamma rays arising from annihilation directly into two photons at the one-loop level. In this class of theories we obtain lower bounds on the ratio of the strength of the gamma ray line relative to the gamma ray continuum as a function of the dark matter mass and spin. These limits are obtained from the unitarity relation between the tree-level amplitude of the primary annihilation channel and the imaginary part of the loop-level amplitude for annihilation directly into photons, with the primary decay products running in the loop. These results are exact in the limit that dark matter annihilation at tree level is exclusively to a single standard model species, occurs through the lowest partial wave and respects $CP$. Away from this limit the bounds are approximate. Our conclusions agree with known results in the literature for the cases of the minimal supersymmetric standard model, universal extra dimensions and the littlest Higgs with $T$ parity. We use the Fermi-LAT observations to translate these limits into upper bounds on the dark matter annihilation cross section into any specific standard model state.