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

Sphere-level Ramond-Ramond couplings in Ramond-Neveu-Schwarz formalism