Constraining the dark matter interpretation of the positron excess with $\gamma$-ray data

The particle origin of dark matter (DM) is still one of the main puzzles in modern physics. One of the most promising search strategy to detect DM at laboratories is through the indirect search of cosmic particles that are produced from DM annihilation in space. In particular, the flux of cosmic positrons has been measured with high precision by the AMS-02 experiment demonstrating that an excess above 10 GeV, with respect to the secondary production, is present. We study in this paper the possible DM origin of the positron excess finding the values of the DM mass $M$ and annihilation cross section $\langle \sigma v \rangle$ that are needed to fit high-energy positron data. In particular, we find that for DM annihilating into $b\bar{b}$ it is required to have $M=43$ TeV and $\langle \sigma v \rangle = 10^{-21}$ cm$^3$/s while for $\tau^+\tau^-$ $M=2$ TeV and $\langle \sigma v \rangle = 3\times 10^{-23}$ cm$^3$/s. If DM produce positrons, they are expected to generate gamma rays from the center of the Milky Way and around dwarf galaxy satellites of the Galaxy. We thus combine the values for the DM mass and annihilation cross section obtained with the fit to AMS-02 positron data with the upper limits derived with the non-detection of $\gamma$ rays with HESS in the direction of the Galactic center and Fermi-LAT for the combined analysis of dwarf galaxies. The main result of the paper is that only DM annihilating into $\mu^+ \mu^-$ with a mass around 500 GeV and $\langle \sigma v \rangle = 4\times 10^{-24}$ cm$^3$/s can fit AMS-02 data and be compatible with the upper limits found with $\gamma$ rays. As for the $\tau^+ \tau^-$ ($b\bar{b}$) channel, DM can contribute at most at a few tens $\%$ (a few \%) level.