Revisiting Bremsstrahlung emission associated with Light Dark Matter annihilations

We compute the single bremsstrahlung emission associated with the pair annihilation of spin-0 particles into electrons and positrons, via the t-channel exchange of a heavy fermion. We compare our result with the work of Beacom et al. . Unlike what is stated in the literature, we show that the Bremsstrahlung cross section is not necessarily given by the tree-level annihilation cross section (for a generalized kinematics) times a factor related to the emission of a soft photon. Such a factorization appears only in the soft photon limit or in the approximation where the masses of the particles in the initial and final states are negligible with respect to the mass of the internal particle. However, in the latter case, we do not recover the same factor as for e^+ e- --> mu^+ mu^- gamma. Numerically the difference, in the hard photon limit, is as large as a factor 3.6. However the effect on the upper limit of the dark matter mass is not significant. Using gamma ray observations, we obtain an upper limit on the dark matter mass of 30 or 100 MeV (depending on the region that is considered for the 511 keV analysis) while Beacom et al. found 20 MeV.