Order α′ 2 equivalence of the string equations of motion and the σ-model weyl invariance conditions. Dependence on the dilaton field

Abstract

We determine the order α′ 2 terms in the closed string effective actions, when the dilaton field is included, by comparison with the four-point string amplitudes, both in the S-matrix parametrization and in the σ-model parametrization. We find that for the bosonic and heterotic string theories there are terms involving derivatives of the dilaton field which cannot be removed by field redefinitions and whose coefficients are non-zero. Our result is not in agreement with existing σ-model computations.