= Solution
First, the displayed equations are only the one-loop <sigma-model beta functions>. Higher worldsheet loops generate additional covariant terms with more curvatures, H-fields and derivatives, each accompanied by higher powers of $\alpha'$. Second, string scattering amplitudes contain momentum corrections from the finite string length and from integrating out the infinite tower of massive string states. Their low-energy expansion produces the same higher-derivative spacetime operators, such as curvature-squared and higher-curvature terms. Both arguments require $\alpha'$ corrections even though local field redefinitions can move individual correction terms between equations.
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