Effective action of heterotic string theory at order ${\alpha }^{\mathrm{\prime }2}$

Upon examining the effective action of the heterotic string theory at order ${\alpha }^{\mathrm{\prime }2}$, an inconsistency between the Chern-Simons coupling ${\mathrm{\Omega }}^2$ and T-duality has been discovered. To address this issue, we introduce 60 parity-even independent geometrical couplings involving the $B$-field, metric, and dilation at the same order, each with arbitrary coefficients. To ensure the invariance of these couplings under T-duality, we consider the Miessner action for the couplings at order ${\alpha }^{\mathrm{\prime }}$ and rigorously determine the coefficients of the 60 couplings in terms of the Chern-Simons coupling. Notably, it is found that the coefficients of the Riemann cubed terms must be zero, which aligns with the results of S-matrix calculations. Additionally, the parity odd couplings at order ${\alpha }^{\mathrm{\prime }2}$ are obtained through T-duality. Overall, our analysis successfully resolves the inconsistency between the Chern-Simons coupling and T-duality, providing a more comprehensive understanding of the behavior of the heterotic string theory at order ${\alpha }^{\mathrm{\prime }2}$.