Higher-derivative terms in one-loop effective action for general trajectories of D-particles in Matrix theory

The one-loop effective action for general trajectories of D-particles in Matrix theory is calculated in the expansion with respect to the number of derivatives up to six, which gives the equation of motion consistently. The result shows that the terms with six derivatives vanish for straight-line trajectories, however, they do not vanish in general. This provides a concrete example that non-renormalization of twelve-fermion terms does not necessarily imply that of six-derivative terms.