Generalized Riemann curvature corrections to type II supergravity

We observe that the replacement of the Riemann curvature with the generalized Riemann curvature in the corrections to type-II supergravity at order ${\alpha }^{\prime 3}$, which are in terms of the contractions of four Riemann curvatures $R^4$, is not fully consistent with the S-matrix elements in the superstring theory. In particular, they produce nonzero S-matrix elements for an odd number of B-field strengths which are not consistent with the string theory results. Using the consistency of the couplings with the linear T duality as a guiding principle, we consider all T-duality-invariant couplings and fix their coefficients by requiring them to be consistent with the S-matrix elements. The new Lagrangian density is then equivalent to the placement of the generalized Riemann curvature into the expression $t_8{t}_8{R}^4$.