Lessons from HAWC PWNe observations: the diffusion constant is not a
  constant; Pulsars remain the likeliest sources of the anomalous positron
  fraction; Cosmic rays are trapped for long periods of time in pockets of
  inefficient diffusion
  Recent TeV observations of nearby pulsars with the HAWC telescope have been
interpreted as evidence that the diffusion of high-energy electrons and
positrons within pulsar wind nebulae is highly inefficient compared to the rest
of the interstellar medium. If the diffusion coefficient well outside the
nebula is close to the value inferred for the region inside the nebula,
high-energy electrons and positrons produced by the two observed pulsars could
not contribute significantly to the local measured cosmic-ray flux. The HAWC
Collaboration thus concluded that, under the assumption of isotropic and
homogeneous diffusion, the two pulsars are ruled out as sources of the
anomalous high-energy positron flux. Here, we argue that since the diffusion
coefficient is likely not spatially homogeneous, the assumption leading to this
conclusion is flawed. We solve the diffusion equation with a radially dependent
diffusion coefficient, and show that the pulsars observed by HAWC produce
potentially perfect matches to the observed high-energy positron fluxes. We
also study the implications of inefficient diffusion within pulsar wind nebulae
on Galactic scales, and show that cosmic rays are likely to have very long
residence times in regions of inefficient diffusion. We describe how this
prediction can be tested with studies of the diffuse Galactic emission.
