Chern-Simons couplings at order $O({\alpha }^{\prime 2})$

Using the explicit string scattering calculation and the linear T-dual ward identity, we evaluate the string theory disc amplitude of one Ramond-Ramond field $C^{(p+1)}$ and two Neveu-Schwarz B-fields in the presence of a single $D_p$-brane in type $IIB$ string theory. From this amplitude we extract the $O({\alpha }^{\prime 2})$ (or equivalently four-derivative) part of the $D_p$-brane couplings involving these fields.