Abstract
The origin of the positron excess is one of the most intriguing mysteries in
astroparticle physics. The recent discovery of extended $\gamma$-ray halos
around the pulsars Geminga, Monogem and PSR J0621+3755 have brought indirect
evidence that pulsar wind nebulae accelerate $e^\pm$ up to very-high-energy.
While the precision of previous data does not permit precise evaluation of the
parameters for the pulsars, we are able to find the more precise shape of the
injection spectrum using new data released by HAWC and LHAASO in 2020 and 2021.
We find that this is well fitten by a power-law with an exponential cutoff. The
spectral index is quite hard with values around 1 while the cutoff energy is
roughly 100 TeV. We also derive the strength of the diffusion coefficient
around the pulsars finding that it is two orders of magnitude lower than the
average of the Galaxy. Finally, we use the above mentioned results to estimate
the contribution of Geminga to the positron excess. This source alone can
contribute to the entire positron excess at around 1 TeV.
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