We present our first numerical results of axisymmetric magnetohydrodynamic simulations for neutrino-cooled accretion tori around rotating black holes in general relativity. We consider tori of mass $sim 0.1$--0.4$M_{odot}$ around a black hole of mass $M=4M_{odot}$ and spin $a=0$--$0.9M$; such systems are candidates for the central engines of gamma-ray bursts (GRBs) formed after the collapse of massive rotating stellar cores and the merger of a black hole and a neutron star. In this paper, we consider the short-term evolution of a torus for a duration of $approx 60$ ms, focusing on short-hard GRBs. Simulations were performed with a plausible microphysical equation of state that takes into account neutronization, the nuclear statistical equilibrium of a gas of free nucleons and $alpha$-particles, black body radiation, and a relativistic Fermi gas (neutrinos, electrons, and positrons). Neutrino-emission processes, such as $e^{pm}$ capture onto free nucleons, $e^{pm}$ pair annihilation, plasmon decay, and nucleon-nucleon bremsstrahlung are taken into account as cooling processes. Magnetic braking and the magnetorotational instability in the accretion tori play a role in angular momentum redistribution, which causes turbulent motion, resultant shock heating, and mass accretion onto the black hole. The mass accretion rate is found to be $dot M_* sim 1$--$10 M_{odot}$/s, and the shock heating increases the temperature to $sim 10^{11}$ K. This results in a maximum neutrino emission rate of $L_{ u}=$ several $times 10^{53}$ ergs/s and a conversion efficiency $L_{ u}/dot M_* c^2$ on the order of a few percent for tori with mass $M_{rm t} approx 0.1$--0.4$M_{odot}$ and for moderately high black hole spins.
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