On NS-NS couplings at order ${\alpha }^{\mathrm{\prime }3}$

Recently, imposing the gauge-symmetry and the T-duality constraints on the string frame effective actions of type II superstring theories, the NS-NS cuplings have been found in a particular scheme which has 445 couplings in 15 different structures. In this paper, using a field redefinition, we write them in terms of 251 couplings which appear in 9 different structures. The 9 structures involve only Riemann curvature, $H_{\alpha \beta \mu }$ and ${\mathrm{\nabla }}_{\alpha }{H}_{\beta \mu \nu }$. The number of couplings in the structures $R^4$, $H^8$, $H^{6}R$, $H^4{R}^2$, $H^2{R}^3$ are $2,2,1,7,22$, respectively, and the number of couplings in the structures $(\mathrm{\nabla }H{)}^4$, $(\mathrm{\nabla }H{)}^2{R}^2$, $(\mathrm{\nabla }H{)}^2{H}^4$, $(\mathrm{\nabla }H{)}^2{H}^{2}R$ are $12,, 22,, 77,, 106$, respectively. The new couplings are fully consistent with the sphere-level S-matrix element of four NS-NS vertex operators.