S-duality of D-brane action at order $O({\alpha }^{\mathrm{\prime }2})$

Using the compatibility of the DBI and the Chern-Simons actions with the T-duality transformations, the curvature corrections to these actions have been recently extended to include the quadratic B-field couplings at order $O({\alpha }^{\mathrm{\prime }2})$. In this paper, we use the compatibility of the couplings on D${}_3$-brane with the S-duality to find the nonlinear RR couplings at order $O({\alpha }^{\mathrm{\prime }2})$. We confirm the quadratic RR couplings in the DBI part with the disk-level scattering amplitude. Using the regularized non-holomorphic Eisenstein series $E_1$, we then write the results in $SL(2,Z)$ invariant form.