Minimal independent couplings at order ${\alpha }^{\prime 2}$

Using field redefinitions and Bianchi identities on the general form of the effective action for metric, $B$-field and dilaton, we have found that the minimum number of independent couplings at order ${\alpha }^{\prime 2}$ is 60. We write these couplings in two different schemes in the string frame. In the first scheme, each coupling does not include terms with more than two derivatives and it does not include structures $R$, $R_{\mu \nu }$, ${\nabla }_{\mu }{H}^{\mu \alpha \beta }$, ${\nabla }_{\mu }{\nabla }^{\mu }\mathrm{\Phi }$. In this scheme, 20 couplings which are the minimum number of couplings for metric and $B$-field, include dilaton trivially as the overall factor of $e^{-2\mathrm{\Phi }}$, and all other couplings include derivatives of dilaton. In the second scheme, the dilaton appears in all 60 couplings only as the overall factor of $e^{-2\mathrm{\Phi }}$. In this scheme, 20 of the couplings are the same as those in the previous scheme.