Beta functions and S-matrix in string theory

The relationship between sigma model β-functions and string theory scattering amplitudes is proved. We derive our results for the closed bosonic string using the weak field expansion around the flat space. The equations of motion for all the background fields, including the heavy fields are considered. We show that the effective equations for the light fields are obtained by integrating out the heavy fields. It is shown that the contributions to the β-functions come from the boundary of moduli space on a punctured Riemann surface. String loop corrections to the equations of motion are also studied.