  Two years ago, the AMS collaboration released the most precise measurement of
the cosmic ray positron flux. It confirms that pure secondary predictions fall
below the data above 10 GeV, suggesting the presence of a primary component,
e.g. annihilations of WIMPs dark matter. Most analyses have focused on the
high-energy part of the spectrum, disregarding the GeV energy region where
cosmic ray transport is harder to model and solar modulation comes into play.
Given the high quality of AMS measurements, it is timely to re-examine the
positron anomaly over the entire energy range, taking into account transport
processes so far neglected, e.g. convection or diffusive re-acceleration. We
devise a new semi-analytical method to take into account transport processes so
far neglected, but important below a few GeV. It is based on the pinching of
inverse Compton and synchrotron energy losses inside the Galactic disc. It
allows to carry out extensive scans over the cosmic ray propagation parameters,
which we strongly constrain by requiring that the secondary component does not
overshoot the AMS measurements. Only models with large diffusion coefficients
survive this test. The positron flux is a powerful and independent probe of
cosmic ray propagation, complementary to the boron-to-carbon ratio. We then
scan over WIMP mass to fit the annihilation cross section and branching ratios,
exploring both direct annihilations into standard model particles or through
light mediators. In the former case, the best fit yields a $p$-value of 0.4%
for a mass of 264 GeV, a value that does not allow to reproduce the highest
energy data points. Worse quality fits are found in the latter case. The
interpretation of the positron excess in terms of single DM species
annihilations is strongly disfavored. This conclusion is based solely on the
positron data, and no other observation needs to be invoked.
