Higher-derivative corrections to type II supergravity: Four Ramond-Ramond terms

It is known that the sphere-level S-matrix element of four type II superstrings has one kinematic factor. At the low energy limit, this factor produces the kinematic factor of the corresponding Feynman amplitudes in the supergravity. It also produces higher-derivative couplings of four strings. In this paper, we explicitly calculate the kinematic factor of four Ramond-Ramond (RR) states in the supergravity. Using this factor, we then find the eight-derivative P-even and P-odd couplings of four RR fields, including the self-dual RR five-form field strength. We show that the P-even couplings are mapped to the standard ${\overline{R}}^4$ couplings by linear T-duality and S-duality transformations. We also confirm the P-even couplings with direct calculations in type II superstring theories.