  The scattering amplitudes of gauge bosons in heterotic and open superstring
theories are related by the single-valued projection which yields heterotic
amplitudes by selecting a subset of multiple zeta value coefficients in the
$\alpha'$ (string tension parameter) expansion of open string amplitudes. In
the present work, we argue that this relation holds also at the level of
low-energy expansions (or individual Feynman diagrams) of the respective
effective actions, by investigating the beta functions of two-dimensional sigma
models describing world-sheets of open and heterotic strings. We analyze the
sigma model Feynman diagrams generating identical effective action terms in
both theories and show that the heterotic coefficients are given by the
single-valued projection of the open ones. The single-valued projection appears
as a result of summing over all radial orderings of heterotic vertices on the
complex plane representing string world-sheet.
