T-duality constraint on effective Lagrangians

Recent studies have highlighted the significant role of utilizing $O(1,1)$ symmetry in the circular reduction of effective actions to determine NS-NS couplings in the effective action of string theory. However, these calculations often result in residual terms as total derivatives that do not conform to $O(1,1)$ transformations. In this paper, we present explicit calculations at $\alpha'$ order, demonstrating the enforceability of this symmetry on effective Lagrangians to establish the parameters governing covariant couplings in any scheme. Notably, we discover the $O(1,1)$-invariant Lagrangians corresponding to the Metsaev-Tseytlin action and the Miessner action.