The shadow and gamma-ray bursts of a Schwarzschild black hole in asymptotic safety

We research on the neutrino pair annihilation $\nu+\overline{\nu}\longrightarrow e^{-}+e^{+}$ around a massive source in asymptotic safety. Since neutrinos and photons have the same geodesic equation around black holes, we can estimate the radius where the neutrinos will be released by obtaining a series of trajectory curves with various correction values $\xi$. The black hole shadow radius is influenced by the correction parameter $\xi$. The black hole shadow radius decreases with increasing the $\xi$. The ratio $\dfrac{\dot{Q}}{\dot{Q}_{Newt}}$ corresponding to the energy deposition per unit time over that in the Newtonian case is derived and calculated. We find that the quantum corrections to the black hole spacetime affect the emitted energy rate ratio for the annihilation. It is interesting that the more considerable quantum effect reduces the ratio value slightly. Although the energy conversion is damped because of the quantum correction, the energy deposition rate is enough during the neutrino-antineutrino annihilation. The corrected annihilation process can become a source of gamma ray burst. We also investigate the derivative $\dfrac{\mathrm{d}\dot{Q}}{\mathrm{d}r}$ relating to the star's radius $r$ to show that the quantum effect for the black hole will drop the ratio. The more manifest quantum gravity influence leads the weaker neutrino pair annihilation.