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Recently, it has been observed that in a T-duality invariant world-volumetheory in flat spacetime, all orders of gauge field strength and all orders ofthe D-brane velocity appear in two specific matrices. Using these two matrices,we construct the world-volume couplings of two massless NSNS states at order$\alpha'^2$ and all orders of the velocity and the gauge field strength, byrequiring them to be invariant under the linear T-duality. The standardextension $F\rightarrow F+P[B]$, then produces all orders of the pull-back ofB-field into the action. We compare the resulting couplings for zero velocityand gauge field strength, with the $\alpha'^2$ terms of the disk-level S-matrixelement of two massless NSNS vertex operators in the presence of a constantbackground B-field. We have found an exact agreement.

On T-duality of $R^2$-corrections to DBI action at all orders of gauge field