Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 4 / iv / 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 ${\alpha }^{\prime }$. 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 ${\alpha }^{\prime }$ corrections even though local field redefinitions can move individual correction terms between equations.