  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.
