O-plane couplings at order $\alpha'^2$: one R-R field strength

It is known that the anomalous Chern-Simons (CS) coupling of O$_p$-plane is not consistent with the T-duality transformations. Compatibility of this coupling with the T-duality requires the inclusion of couplings involving one R-R field strength. In this paper we find such couplings at order $\alpha'^2$. By requiring the R-R and NS-NS gauge invariances, we first find all independent couplings at order $\alpha'^2$. There are $1,\, 6,\,28,\,20,\, 19,\, 2$ couplings corresponding to the R-R field strengths $F^{(p-4)}$, $\,F^{(p-2)}$, $\,F^{(p)}$, $\,F^{(p+2)}$, $\,F^{(p+4)}$ and $F^{(p+6)}$, respectively. We then impose the T-duality constraint on these couplings and on the CS coupling $C^{(p-3)}\wedge R\wedge R$ at order $\alpha'^2$ to fix their corresponding coefficients. The T-duality constraint fixes all coefficients in terms of the CS coefficient. They are fully consistent with the partial couplings that have been already found in the literature by the S-matrix method.