Higher derivative corrections to the Wess-Zumino action

In the first part of this paper, we calculate the disk-level $S$-matrix elements of one Ramond-Ramond (RR), one Neveu-Schwarz-Neveu-Schwarz (NSNS), and one Neveu-Schwarz (NS) vertex operator, and show that they are consistent with the amplitudes that have been recently found by applying various Ward identities. We show that the massless poles of the amplitude at low energy are fully consistent with the known $D$-brane couplings at order ${\alpha }^{\prime 2}$ that involve one RR or NSNS and two NS fields. Subtracting the massless poles, we then find the contact terms of one RR, one NSNS and one NS fields at order ${\alpha }^{\prime 2}$. Some of these terms are reproduced by the Taylor expansion and the pullback of two closed string couplings; some other couplings are reproduced by the linear graviton in the second fundamental form and by the $B$-field in the gauge field extension $F\to F+B$, in one closed and two open string couplings. In the second part, we write all of the independent covariant contractions of one RR, one NSNS, and one NS fields with unknown coefficients. We then constrain the couplings to be consistent with the linear $T$ duality and with the above contact terms. Interestingly, we have found that up to total derivative terms and Bianchi identities, these constraints uniquely fix all the unknown coefficients.

Figure 1

One RR, one NSNS, and one NS Feynman diagram.