The Supersymmetric Effective Action of the Heterotic String in Ten   Dimensions
  We construct the supersymmetric completion of quartic $R+R^4$-actions in the ten-dimensional effective action of the heterotic string. Two invariants, of which the bosonic parts are known from one-loop string amplitude calculations, are obtained. One of these invariants can be generalized to an $R+F^2+F^4$-invariant for supersymmetric Yang-Mills theory coupled to supergravity.   Supersymmetry requires the presence of $B\wedge R\wedge R\wedge   R\wedge R$-terms, ($B\wedge F\wedge F\wedge F\wedge F$ for   Yang-Mills) which correspond to counterterms in the Green-Schwarz anomaly cancellation. Within the context of our calculation the $\zeta(3)R^4$-term from the tree-level string effective action does not allow supersymmetrization. 
M. de Roo, H. Suelmann and A. Wiedemann