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

Neutrino Pair Annihilation in Collapsars; Ray-Tracing Method in Special Relativity