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