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.
