Neutrino Emission in Jet Propagation Process

Relativistic jets are universal in long-duration gamma-ray burst (GRB) models. Before breaking out, they must propagate in the progenitor envelope along with a forward shock and a reverse shock forming at the jet head. Both electrons and protons will be accelerated by the shocks. High energy neutrinos could be produced by these protons interacting with stellar materials and electron-radiating photons. The jet will probably be collimated, which may have a strong effect on the final neutrino flux. Under the assumption of a power-law stellar-envelope density profile $\rho \propto r^{-\alpha}$ with an index $\alpha$, we calculate this neutrino emission flux by these shocks for low-luminosity GRBs (LL-GRBs) and ultra-long GRBs (UL-GRBs) in different collimation regimes, using the jet propagation framework developed by \citet{bro11}. We find that LL-GRBs and UL-GRBs are capable for detectable high energy neutrinos up to $\sim {\rm PeV}$, and obtain the final neutrino spectrum. Besides, we conclude that larger $\alpha$ corresponds to greater neutrino flux at high energy end ($\sim {\rm PeV}$) and higher maximum neutrino energy as well. However, such differences are so small that it is not promising for us to distinguish from observations, given the energy resolution we have now.