Effective action of bosonic string theory at order ${\alpha }^{\mathrm{\prime }3}$

In this work, we derive the classical effective action of bosonic string theory at order ${\alpha }^{\mathrm{\prime }3}$ for the metric, Kalb-Ramond field, and dilaton by imposing a higher-derivative extension of the Buscher rules on the circular reduction of the minimal basis at this order, in the schemes where their corresponding actions at order ${\alpha }^{\mathrm{\prime }}$ are the Meissner and the Metsaev-Tseytlin schemes. We find that T-duality fixes all coupling constants in terms of the known overall factor at order ${\alpha }^{\mathrm{\prime }}$ and a single remaining parameter. This final parameter is determined by matching the single-trace term $\text{Tr}(\epsilon \epsilon \epsilon \epsilon )$ in the four-graviton S-matrix element which lacks a massless pole, with the corresponding string theory amplitude. Our results for the Riemann quartic terms are in full agreement with those obtained from the nonlinear sigma-model approach.