T-duality invariant effective actions at orders ${\alpha }^{\prime }$, ${{\alpha }^{\prime }}^2$

We use compatibility of the $D$-dimensional effective actions for diagonal metric and for dilaton with the T-duality when theory is compactified on a circle, to find the $D$-dimensional couplings of curvatures and dilaton as well as the higher derivative corrections to the $(D-1)$-dimensional Buscher rules at orders ${\alpha }^{\prime }$ and ${{\alpha }^{\prime }}^2$. We observe that the T-duality constraint on the effective actions fixes the covariant effective actions at each order of ${\alpha }^{\prime }$ up to field redefinitions and up to an overall factor. Inspired by these results, we speculate that the $D$-dimensional effective actions at any order of ${\alpha }^{\prime }$ must be consistent with the standard Buscher rules provided that one uses covariant field redefinitions in the corresponding reduced $(D-1)$-dimensional effective actions. This constraint may be used to find effective actions at all higher orders of ${\alpha }^{\prime }$.