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

We calculate in detail 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 α^′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