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Tree-level S-matrix elements from S-duality

It has been speculated that the S-matrix elements of type IIB superstring theory satisfy the Ward identity associated with the S-duality. This indicates that a group of S-matrix elements at each loop level are invariant under the linear $SL(2,R)$ transformations. If one evaluates one element of such S-dual multiplet, then all other components can be found by the simple use of the linear $SL(2,R)$ transformations. In this paper, we calculate the disk-level S-matrix element of one graviton(dilaton), one B-field and one gauge boson on the word volume of D$_3$-brane. The S-dual multiplet corresponding to the graviton(dilaton) amplitude has three(six) components. In particular, the graviton multiplet has the S-matrix element of one graviton, one R-R two-form and one gauge boson, and the dilaton multiplet has the S-matrix element of one R-R scalar, one B-field and one gauge boson vertex operators. We calculate explicitly these particular components and show that they are precisely the ones predicted by the S-duality. We have also found the low energy contact terms of the dilaton multiplet at order $α'^2$.