Heterotic string couplings at order $\alpha'^3$ in NS-NS sector
We utilize the standard T-duality procedure to derive the classical effective action of heterotic string theory at the eight-derivative order within the NS-NS sector, which comprises the metric, the \(B\)-field, and the dilaton. Starting from the minimal basis at this order, consisting of 872 even-parity and 477 odd-parity couplings, we perform a dimensional reduction on a circle and impose invariance under T-duality transformations, specifically the Buscher rules supplemented by higher-derivative corrections. This invariance uniquely fixes a subset of the couplings to match those of type II theory up to an overall factor, while the remaining couplings are determined in terms of the coefficients at order \(\alpha'\). For these latter couplings, we adopt both the Metsaev-Tseytlin and Meissner schemes. Subsequently, through field redefinitions, we recast the action into a canonical form in which the dilaton appears solely through the overall factor \(e^{-2\Phi}\). In the Meissner scheme, the pure gravity sector precisely reproduces the known S-matrix results. In both schemes, the pure gravity terms can be expressed as the double trace \((\Tr(R^2))^2\), and when combined with the corresponding Yang-Mills terms \((\Tr(F^2))^2\), they take the unified form \((\Tr(R^2-F^2))^2\), as anticipated in the literature.