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The worldsheet metric is a gauge variable, so quantum consistency requires the Weyl transformation to remain non-anomalous. The background fields are couplings of a two-dimensional quantum field theory, and their sigma-model beta functions multiply the trace of the worldsheet stress tensor. Requiring every beta function to vanish gives, at the first nontrivial order in the derivative or ${\alpha }^{\prime }$ expansion, precisely the stated metric, B-field and dilaton equations. They are also the Euler-Lagrange equations of the leading spacetime string-frame effective action

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.