We explicitly extract the structure of higher-derivative curvature-squared terms at genus 0 and 1 in the d = 4 heterotic string effective action compactified on symmetric orbifolds by computing on-shell S-matrix superstring amplitudes. In particular, this is done within the context of calculating the graviton 4-point amplitude. We also discuss the modulidependent gravitational threshold corrections to the coupling associated with the CP even quadratic curvature terms.
Förger, K., Ovrut, B. A., Theisen, S. J., & Waldram, D. (1996). Higher-derivative gravity in string theory. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 388(3), 512–520. https://doi.org/10.1016/S0370-2693(96)01175-6