S-duality constraint on higher-derivative couplings

The Riemann curvature correction to the type II supergravity at eight-derivative level in string frame is given as e^−2ϕ(t₈t₈R⁴+(1/4)ϵ₈ϵ₈R⁴). For constant dilaton, it has been extended in the literature to the S-duality invariant form by extending the dilaton factor in the Einstein frame to the non-holomorphic Eisenstein series. For non-constant dilaton, however, there are various couplings in the Einstein frame which are not consistent with the S-duality. By constructing the tensors t_2n from Born-Infeld action, we include the appropriate Ricci and scalar curvatures as well as the dilaton couplings to make the above action to be consistent with the S-duality