We develop a numerical scheme and code for estimating the energy and momentum transfer via neutrino pair annihilation ( \nu + { \bar { \nu } } \rightarrow e ^ { - } + e ^ { + } ) , bearing in mind the application to the collapsar models of gamma-ray bursts ( GRBs ) .
To calculate the neutrino flux illuminated from the accretion disk , we perform a ray-tracing calculation in the framework of special relativity .
The numerical accuracy of the developed code is certificated by several tests , in which we show comparisons with the corresponding analytical solutions .
Using hydrodynamical data in our collapsar simulation , we estimate the annihilation rates in a post-processing manner .
We show that the neutrino energy deposition and momentum transfers are strongest near the inner edge of the accretion disk .
The beaming effects of special relativity are found to change the annihilation rates by several factors in the polar funnel region .
After the accretion disk settles into a stationary state ( typically later than \sim 9 s from the onset of gravitational collapse ) , we find that the neutrino-heating timescale in the vicinity of the polar funnel ( \lesssim 80 km ) can become shorter than the hydrodynamical timescale , indicating that the neutrino-heated outflows can be launched there .
We point out that the momentum transfer can play as important role as the energy deposition for the efficient acceleration of neutrino-driven outflows .
Our results suggest that the neutrino pair annihilation has a potential importance equal to the conventional magnetohydrodynamic mechanism for igniting the GRB fireballs .