Constraints on Cosmic Rays Acceleration in Bright Gamma-ray Bursts with Observations of Fermi

Gamma-ray bursts (GRBs) are widely suggested as potential sources of ultrahigh-energy cosmic rays (UHECRs). The kinetic energy of the jets dissipates, leading to the production of an enormous amount of $\gamma$-ray photons and possibly also the acceleration of protons. The accelerated protons will interact with the radiation of the GRB via the photomeson and Bethe-Heitler processes, which can initiate electromagnetic cascades. This process can give rise to broadband radiation up to the GeV-TeV $\gamma$-ray regime. The expected $\gamma$-ray flux from cascades depends on properties of the GRB jet, such as the dissipation radius $R_{\rm diss}$, the bulk Lorentz factor $\Gamma$, and the baryon loading factor $\eta_p$. Therefore, observations of Fermi-LAT can impose constraints on these important parameters. In this study, we select 12 GRBs of high keV-MeV fluence and constrain the baryon loading factor, under different combinations of the bulk Lorentz factor and the dissipation radius based on Fermi-LAT's measurements. Our findings indicate a strong constraint of $\eta_p<10$ for most selected GRBs over a large parameter space except for large dissipation radii ($\gtrsim 10^{15}\rm cm$) and high bulk Lorentz factors ($\gtrsim 600$). The constraint is comparable to, and in some GRBs even stronger than, that from high-energy neutrinos for stacked GRBs. Our results suggest that for typical bulk Lorentz factor of several hundreds, the dissipation radii of GRBs need be large to avoid overshooting the GeV gamma-ray flux during the prompt emission phase of GRBs, which can be used to constrain GRBs.