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 α′s≫1, fixed scattering angle, and very weak coupling gs≪1/(α′s), the four-graviton amplitude exhibits the usual exponential suppression. As the energy approaches 1/gs, 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 α′s≪1/gs, the pure D-instanton contribution to the scattering amplitude is found to produce a factor A4D-inst≅exp(cgs3/2e−2πgss3). At energies 1/gs≪α′s≪1/gs, the D-instanton factor becomes A4D-inst≅exp(2e−2πgs+πgss2), α′=4. At higher energies α′s≫1/gs 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.