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Measuring the electron diffusion coefficient is the most straightforward taskin the study of gamma-ray pulsar halos. The updated measurements of the spatialmorphology and spectrum of the Geminga halo by the HAWC experiment enable us toconstrain parameters beyond the diffusion coefficient, including the size ofthe slow-diffusion zone and the electron injection spectrum from the pulsarwind nebulae (PWN). Based on the two-zone diffusion model, we find that theslow-diffusion zone size ($r_*$) around Geminga is within the range of$30-70$pc. The lower boundary of this range is determined by the goodness offit of the model to the one-dimensional morphology of the Geminga halo. Theupper limit is derived from fitting the gamma-ray spectrum of the Geminga halo,along with the expectations for the power-law index of the injection spectrumbased on simulations and PWN observations, i.e., $p\gtrsim1$. With $r_*$ set atits lower limit of $30$~pc, we obtain the maximum $p$ permitted by the HAWCspectrum measurement, with an upper limit of $2.17$ at a $3\sigma$significance. Moreover, we find that when $r_*=30$pc and $p=2.17$, thepredicted positron spectrum generated by Geminga at Earth coincides with theAMS-02 measurement in the $50-500$GeV range.

Constraining the slow-diffusion zone size and electron injection spectral index for the Geminga pulsar halo