About one year ago, it was speculated that decaying or annihilating Light Dark Matter (LDM) particles could explain the flux and extension of the 511 keV line emission in the galactic centre. Here we present a thorough comparison between theoretical expectations of the galactic positron distribution within the LDM scenario and observational data from INTEGRAL/SPI. Unlike previous analyses, there is now enough statistical evidence to put tight constraints on the shape of the dark matter halo of our galaxy, if the galactic positrons originate from dark matter. For annihilating candidates, the best fit to the observed 511 keV emission is provided by a radial density profile with inner logarithmic slope gamma=1.03+-0.04. In contrast, decaying dark matter requires a much steeper density profile, gamma>1.5, rather disfavoured by both observations and numerical simulations. Within the annihilating LDM scenario, a velocity-independent cross-section would be consistent with the observational data while a cross-section purely proportional to v^2 can be rejected at a high confidence level. Assuming the most simplistic model where the galactic positrons are produced as primaries, we show that the LDM candidate should be a scalar rather than a spin-1/2 particle and obtain a very stringent constraint on the value of the positron production cross-section to explain the 511 keV emission. One consequence is that the value of the fine structure constant should differ from that recommended in the CODATA. This is a very strong test for the LDM scenario and an additional motivation in favour of experiments measuring alpha directly. Our results finally indicate that an accurate measurement of the shape of the dark halo profile could have a tremendous impact on the determination of the origin of the 511 keV line and vice versa.
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