F⁵ contributions to the nonabelian Born Infeld action from a supersymmetric Yang-Mills five-point function

We consider the N=4 supersymmetric Yang-Mills theory in four dimensions. We compute the one-loop contributions to the effective action with five external vector fields and compare them with corresponding results in open superstring theory. Our calculation determines the structure of the F^5 terms that appear in the nonabelian generalization of the Born Infeld action. The trace operation on the gauge group indices receives contributions from the symmetric as well as the antisymmetric part. We find that in order to study corrections to the symmetrized trace prescription one has to consistently take into account derivative contributions not only with antisymmetrized products \nabla_{[\mu}\nabla_{\nu]} but also with symmetrized ones \nabla_{(\mu}\nabla_{\nu)}.