String Amplitudes from Field-Theory Amplitudes and Vice Versa.

We present an integration-by-parts reduction of any massless tree-level string correlator to an equivalence class of logarithmic functions, which can be used to define a field-theory amplitude via a Cachazo-He-Yuan (CHY) formula. The string amplitude is then shown to be the double copy of the field-theory one and a special disk or sphere integral. The construction is generic as it applies to any correlator that is a rational function of correct SL(2) weight. By applying the reduction to open bosonic or heterotic strings, we get a closed-form CHY integrand for the (DF)^{2}+Yang-Mills+ϕ^{3} theory.