2018ApJ...857...24W

The high-energy (>100 MeV) emission observed by the Fermi Large Area Telescope during the prompt phase of some luminous gamma-ray bursts (GRBs) could arise from the cascade induced by interactions between accelerated protons and the radiation field of GRBs. The photomeson process, which is usually suggested to operate in such a hadronic explanation, requires a rather high proton energy (>10¹⁷ eV) for an efficient interaction. However, whether GRBs can accelerate protons to such a high energy is far from guaranteed, although they have been suggested as the candidate source for ultrahigh-energy cosmic rays. In this work, we revisit the hadronic model for the prompt high-energy emission of GRBs with a smaller maximum proton energy than the usually adopted value estimated from the Bohm condition. In this case, the Bethe-Heitler pair production process becomes comparably important or even dominates over the photomeson process. We show that with a relatively low maximum proton energy with a Lorentz factor of 10⁵ in the comoving frame, the cascade emission can still reproduce various types of high-energy spectra of GRBs. For most GRBs without high-energy emission detected, the maximum proton energy could be even lower and relax the constraints on the parameters of the GRB jet resulting from the nondetection of GRB neutrinos by IceCube.