Gamma-ray bursts (GRBs) have been suggested as possible sources of the high-energy neutrino flux recently detected by the IceCube telescope. We revisit the fireball emission model and elaborate an analytical prescription to estimate the high-energy neutrino prompt emission from pion and kaon decays, assuming that the leading mechanism for the neutrino production is lepto-hadronic. To this purpose, we include hadronic, radiative and adiabatic cooling effects and discuss their relevance for long- (including high- and low-luminosity) and short-duration GRBs. The expected diffuse neutrino background is derived, by requiring that the GRB high-energy neutrino counterparts follow up-to-date gamma-ray luminosity functions and redshift evolutions of the long and short GRBs. Although dedicated stacking searches have been unsuccessful up to now, we find that GRBs could contribute up to a few % to the observed IceCube high-energy neutrino flux for sub-PeV energies, assuming that the latter has a diffuse origin. Gamma-ray bursts, especially low-luminosity ones, could however be the main sources of the IceCube high-energy neutrino flux in the PeV range. While high-luminosity and low-luminosity GRBs have comparable intensities, the contribution from the short-duration component is significantly smaller. Our findings confirm the most-recent IceCube results on the GRB searches and suggest that larger exposure is mandatory to detect high-energy neutrinos from high-luminosity GRBs in the near future.
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