A possible link between the GeV excess and the 511 keV emission line in the Galactic Centre

The morphology and characteristics of the so-called GeV gamma-ray excess detected in the Milky Way lead us to speculate about a possible common origin with the 511 keV line mapped by the SPI experiment about ten years ago. In the previous version of our paper, we assumed 30 GeV dark matter particles annihilating into $b \bar{b}$ and obtained both a morphology and a 511 keV flux (\phi_{511 keV} ~ 10^{-3} ph/cm^2/s) in agreement with SPI observation. However our estimates assumed a negligible number density of electrons in the bulge which lead to an artificial increase in the flux (mostly due to negligible Coulomb losses in this configuration). Assuming a number density greater than $n_e > 10^{-3} cm^{-3}$, we now obtain a flux of 511 keV photons that is smaller than \phi_{511 keV} ~ 10^{-6} ph/cm^2/s and is essentially in agreement with the 511 keV flux that one can infer from the total number of positrons injected by dark matter annihilations into $b \bar{b}$. We thus conclude that -- even if 30 GeV dark matter particles were to exist-- it is impossible to establish a connexion between the two types of signals, even though they are located within the same 10 deg region in the galactic centre.