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.
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