GeV observations of the extended pulsar wind nebulae constrain the pulsar interpretations of the cosmic-ray positron excess

It has long been suggested that nearby pulsars within $\sim 1 \,{\rm kpc}$ are the leading candidate of the 10-500 GeV cosmic-ray positron excess measured by PAMELA and other experiments. The recent measurement of surface brightness profile of TeV nebulae surrounding Geminga and PSR~B0656+14 by the High-Altitude Water Cherenkov Observatory (HAWC) suggests inefficient diffusion of particles from the sources, giving rise to a debate on the pulsar interpretation of the cosmic-ray positron excess. Here we argue that GeV observations provide more direct constraints on the positron density in the TeV nebulae in the energy range of 10-500 GeV and hence on the origin of the observed positron excess. Motivated by this, we search for GeV emission from the TeV nebulae with the \textsl{Fermi} Large Area Telescope (LAT). No spatially-extended GeV emission is detected from these two TeV nebulae in the framework of two-zone diffusion spatial templates, suggesting a relatively low density of GeV electrons/positrons in the TeV nebulae. A joint modelling of the data from HAWC and \textsl{Fermi}-LAT disfavors Geminga and PSR~B0656+14 as the dominant source of the positron excess at $\sim 50-500$ GeV for the usual Kolmogorov-type diffusion, while for an energy-independent diffusion, a dominant part of the positron excess contributed by them cannot be ruled out by the current data.