Contribution of pulsars to cosmic-ray positrons in light of recent observation of inverse-Compton halos

The hypothesis that pulsar wind nebulae (PWNe) can significantly contribute to the excess of the positron ($e^+$) cosmic-ray flux has been consolidated after the observation of a $\gamma$-ray emission at TeV energies of a few degree size around Geminga and Monogem PWNe, and at GeV energies for Geminga at a much larger extension. The $\gamma$-ray halos around these PWNe are interpreted as due to electrons ($e^-$) and $e^+$ accelerated and escaped by their PWNe, and inverse Compton scattering low-energy photons of the interstellar radiation fields. The extension of these halos suggests that the diffusion around these PWNe is suppressed by two orders of magnitude with respect to the average in the Galaxy. We implement a two-zone diffusion model for the propagation of $e^+$ accelerated by the Galactic population of PWNe. We consider pulsars from the ATNF catalog and build up simulations of the PWN Galactic population. In both scenarios, we find that within a two-zone diffusion model, the total contribution from PWNe and secondary $e^+$ is at the level of AMS-02 data, for an efficiency of conversion of the pulsar spin down energy in $e^\pm$ of $\eta\sim0.1$. For the simulated PWNe, a $1\sigma$ uncertainty band is determined, which is of at least one order of magnitude from 10 GeV up to few TeV. The hint for a decreasing $e^+$ flux at TeV energies is found, even if it is strongly connected to the chosen value of the radius of the low diffusion bubble around each source.