Higher-derivative corrections to the non-Abelian Born-Infeld action

We determine higher-derivative terms in the open superstring effective action with U(N) gauge group up to and including order α'4 as can be extracted from 4 boson, 2 boson-2 fermion and 4 fermion string scattering amplitudes. This yields corrections to the non-Abelian Born-Infeld action involving higher derivatives as is relevant for studying D-branes beyond the slowly varying field approximation. While at order α'2 the action has recently been shown to be a symmetrised trace, this no longer is true at order α'3 or α'4. We insist that these terms including higher derivatives are as important in a low-energy expansion as, e.g., the much-discussed α'4F6 terms. In particular, a computation of the fluctuation spectra at order α'4 has to take into account these nonsymmetrised trace higher-derivative terms computed here