Effects of D-instantons in string amplitudes

We investigate the different energy regimes in the conjectured SL(2,Z) invariant four graviton scattering amplitude that incorporates D-instanton contributions in 10d type IIB superstring theory. We show that the infinite product over SL(2,Z) rotations is convergent in the whole complex plane s,t. For high energies s>> 1, fixed scattering angle, and very weak coupling g<< 1/s, the four-graviton amplitude exhibits the usual exponential suppression. As the energy approaches 1/g, the suppression gradually diminishes until there appears a strong amplification near a new pole coming from the exchange of a (p,q) string. At energies s<< 1/\sqrt{g}, the pure D instanton contribution to the scattering amplitude is found to produce a factor $A_4^{Dinst}\cong \exp (c g^{3/2}e^{-{2\pi\over g}} s^3)$. At energies $1/\sqrt{g} << s<< 1/g$, the D-instanton factor becomes $A_4^{Dinst}\cong \exp (2 e^{-{2\pi\over g_s}+\pi g_s s^2})$. At higher energies s>> 1/g the D-instanton contribution becomes very important, and one finds an oscillatory behavior which alternates suppression and amplification. This suggests that non-perturbative effects can lead to a high-energy behavior which is significantly different from the perturbative string behavior.