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The paradigm that gamma-ray burst (GRB) fireballs are the sources of theultra-high energy cosmic rays (UHECRs) is being probed by neutrinoobservations. Very stringent bounds can be obtained from the cosmic ray(proton)--neutrino connection, assuming that the UHECRs escape as neutrons. Inthis study, we identify three different regimes as a function of the fireballparameters: the standard "one neutrino per cosmic ray" case, the opticallythick (to neutron escape) case, and the case where leakage of protons from theboundaries of the shells (direct escape) dominates. In the optically thickregime, photomeson production is very efficient, and more neutrinos will beemitted per cosmic ray than in the standard case, whereas in the directescape-dominated regime, more cosmic rays than neutrinos will be emitted. Wedemonstrate that, for efficient proton acceleration, which is required todescribe the observed UHECR spectrum, the standard case only applies to a verynarrow region of the fireball parameter space. We illustrate with severalobserved examples that conclusions on the cosmic ray--neutrino connection willdepend on the actual burst parameters. We show that the definition of the pionproduction efficiency currently used by the IceCube collaborationunderestimates the neutrino production in the optically thick case. Finally, wepoint out that the direct escape component leads to a spectral break in thecosmic ray spectrum emitted from a single source. The resulting "two-componentmodel" can be used to even more strongly pronounce the spectral features of theobserved UHECR spectrum than the dip model.

UHECR escape mechanisms for protons and neutrons from GRBs, and the cosmic ray-neutrino connection