Spectral energy-loss bump and $\gamma$-ray pulsar halos

LHAASO J0248$+$6021, a possible $\gamma$-ray pulsar halo associated with PSR J0248$+$6021 (J0248), exhibits a highly curved spectrum as revealed by LHAASO and Fermi-LAT measurements. We propose a direct interpretation of this large curvature: the energy-loss bump in the parent electron spectrum has not yet significantly departed from the high-energy cutoff. This requires either that the ambient magnetic field strength $B$ around J0248 be lower than the typical value in the interstellar medium, or that the electron injection age be significantly shorter than the pulsar characteristic age. For the much older Geminga pulsar, the expected energy-loss bump in its $\gamma$-ray halo spectrum has shifted below $100\ \text{GeV}$, in excellent agreement with Fermi-LAT measurements. Thus, the broadband spectra of young and old pulsar halos find a unified interpretation in the picture of a time-dependent energy-loss bump. Meanwhile, the spectral measurements of LHAASO J0248$+$6021 only constrain the combination of $B$ and electron injection age. The uncertainty in $B$ leads to an order-of-magnitude variation in the fitted diffusion coefficient. Future X-ray observations are expected to break the degeneracies.