On the relationship between string low-energy effective actions and O(α'3) σ-model β-functions

We examine in detail the connection between low-energy string effective actions and the σ-model β-functions up to O(α'3) in the torsion-free case. In particular we show that the relationship between, on the one hand, the equations of motion derived from the O(α'3) string effective action, and, on the other hand, the O(α'3) σ-model β-functions necessarily involves derivative operators acting on the β-functions. (orig.)