  A radiation-magnetohydrodynamic simulation for the black hole-torus system is
performed in the framework of full general relativity for the first time. A
truncated moment formalism is employed for a general relativistic neutrino
radiation transport. Several systems in which the black hole mass is $M_{\rm
BH}=3$ or $6M_{\odot}$, the black hole spin is zero, and the torus mass is
$\approx 0.14$--$0.38M_{\odot}$ are evolved as models of the remnant formed
after the merger of binary neutron stars or black hole-neutron star binaries.
The equation of state and microphysics for the high-density and
high-temperature matter are phenomenologically taken into account in a
semi-quantitative manner. It is found that the temperature in the inner region
of the torus reaches $\agt 10$ MeV which enhances a high luminosity of
neutrinos $\sim 10^{51}$ ergs/s for $M_{\rm BH}=6M_{\odot}$ and $\sim 10^{52}$
ergs/s for $M_{\rm BH}=3M_{\odot}$. It is shown that neutrinos are likely to be
emitted primarily toward the outward direction in the vicinity of the
rotational axis and their energy density may be high enough to launch a
low-energy short gamma-ray burst via the neutrino-antineutrino
pair-annihilation process with the total energy deposition $\sim
10^{47}$--$10^{49}$ ergs. It is also shown in our model that for $M_{\rm
BH}=3M_{\odot}$, the neutrino luminosity is larger than the electromagnetic
luminosity while for $M_{\rm BH}=6M_{\odot}$, the neutrino luminosity is
comparable to or slightly smaller than the electromagnetic luminosity.
