  We calculate in details the sphere-level scattering amplitude of two
Ramond-Ramond (RR) and two Neveu-Schwarz-Neveu-Schwarz (NSNS) vertex operators
in type II superstring theories in Ramond-Neveu-Schwarz (RNS) formalism. We
then compare the expansion of this amplitude at order $\alpha'^3$ with the
eight-derivative couplings of the gravity and B-field that have been recently
found based on S-dual and T-dual Ward identities. We find exact agreement.
Moreover, using the above S-matrix element, we have found various couplings
involving the dilaton field, and shown that they are also fully consistent with
these Ward identities.
