High energy scattering amplitude in the linearized gravitational theory

The asymptotic behavior of the elastic scattering amplitude by the exchange of graviton between two scalar particles at high energies and fixed momentum transfers is reconsidered in the Logunov-Tavkhelidze equation in the linearized gravitational theory. The corrections to the eikonal approximation in the quasi-potential approach of relative order $1\mathrm{/}p$ is developed with the principal contributions at high energy. The eikonal expression of scattering amplitude and the formal first correction are derived. The Yukawa potential is applied to discuss the results.