On RR couplings on D-branes at order $O(\alpha'^2)$

Recently, it has been found that there are couplings of the RR field strength $F^{(p)}$ and the B-field strength $H$ on the world volume of D$_p$-branes at order ${\cal O}(\alpha'^2)$. These couplings which have both world-volume and transverse indices, are invariant under the linear T-duality transformations. Consistency with the nonlinear T-duality indicates that the RR field strength $F^{(p)}$ in these couplings should be replaced by ${\cal F}^{(p)}=d{\cal C}^{(p-1)}$ where ${\cal C}=e^{B}C$. This replacement, however, reproduces some non-gauge invariant terms. On the other hand, the nonlinear terms are invariant under the linear T-duality transformations at the level of two B-fields. This allows one to remove some of the nonlinear terms in ${\cal F}^{(p)}$. We fix this by comparing the nonlinear couplings with the S-matrix element of one RR and two NSNS vertex operators. Our results indicate that in the expansion of ${\cal F}^{(p)}$ one should keep only the B-field gauge invariant terms, e.g. $B\wedge dC^{(p-3)}$ where both indices of B-field lie along the brane. Moreover, in this case one should replace $B$ with $B+2\pi\alpha'f$ to have the $B$-field gauge invariance.