Constraints on Tree-Level Higher Order Gravitational Couplings in Superstring Theory

We consider the scattering amplitudes of five and six gravitons at tree level in superstring theory. Their power series expansions in the Regge slope α^' are analyzed through the order α^'8 showing some interesting constraints on higher order gravitational couplings in the effective superstring action such as the absence of R⁵ terms. Furthermore, some transcendentality constraints on the coefficients of the nonvanishing couplings are observed: the absence of zeta values of even weight through the order α^'8 like the absence of ζ(2)ζ(3)R⁶ terms. Our analysis is valid for any superstring background in any space-time dimension, which allows for a conformal field theory description.