T-duality of O-plane action at order ${\alpha }^{\mathrm{\prime }2}$

We use compatibility of O-plane action with nonlinear T-duality as a guiding principle to find all NS-NS couplings in the O-plane action at order ${\alpha }^{\mathrm{\prime }2}$. We find that the dilaton couplings appear in the string frame only via the transformation $\hat{R}{\mu\nu}\rightarrow \hat{R}{\mu\nu}+\nabla_{\mu}\nabla_{\nu}\phi$.