An excess of $\gamma$ rays in the data measured by the {\it Fermi} Large Area Telescope in the direction of the Galactic center (GCE) has been reported in several publications. The characteristics of the GCE, recently measured with unprecedented precision, are all compatible with dark matter particles (DM) annihilating in the main halo of our Galaxy. We investigate the DM candidates that fit the observed GCE spectrum assuming a simple scenario with DM annihilating into a single channel but we inspect also more complicated models with two and three channels.
We perform a search for a $\gamma$-ray flux from a list of 48 Milky Way dwarf spheroidal galaxies (dSphs) and since we do not find any significant signal from the dSphs, we put upper limits on the annihilation cross section that result to be compatible with the DM candidate that fits the GCE.
However, we find that the GCE DM signal is excluded at the $95\%$ confidence level by the AMS-02 $\bar{p}$ flux data for all purely hadronic (semi-hadronic) channels unless the diffusive halo size $L$ is smaller than 1.7 kpc (2.6 kpc). Such a small diffusion halo is at the $2\sigma$ significance lower limit for the results inferred from fluxes of radioactive cosmic rays and radio and $\gamma$-ray data. Furthermore, AMS-02 $e^+$ data rule out the GCE DM interpretation with pure or partial annihilation into $e^+ e^-$.
The only DM candidate that fits the GCE spectrum and fulfills all constraints obtained with the combined dSphs analysis and the AMS-02 $\bar{p}$ and $e^+$ data annihilates purely (or very dominantly) into $\mu^+\mu^-$, has a mass of $\sim$60 GeV and roughly a thermal cross section.