Geminga contribution to the cosmic-ray positron excess according to the gamma-ray observations

We attempt to interpret the cosmic-ray positron excess by injection from the nearby pulsar Geminga, assuming a two-zone diffusion scenario and an injection spectrum with a low energy cutoff. Since the high energy positrons and electrons from Geminga can induce $\gamma$ rays via inverse Compton scattering, we take into account the extended $\gamma$-ray observations around Geminga from HAWC for $\sim 10$ TeV and from Fermi-LAT for $\mathcal{O}(10)$ GeV. According to the extended $\gamma$-ray observation claimed by an analysis of Fermi-LAT data, we find that Geminga could explain the positron excess for a $30\%$ energy conversion efficiency into positrons and electrons. However, based on the constraint on the extended $\gamma$ rays given by another Fermi-LAT analysis, positrons from Geminga would be insufficient to account for the positron excess. A further robust analysis of Fermi-LAT data for the extended $\gamma$ rays would be crucial to determine whether Geminga can explain the positron excess or not.