*-Trek:: the one-loop N=4 noncommutative SYM action

We investigate N = 4 noncommutative super-Yang-Mills (SYM) theory. We compute the one-loop four gauge boson scattering amplitude on parallel Dp-branes, and find the corresponding contribution to the noncommutative SYM one-loop action in a momentum expansion. The result is somewhat surprising. We find that while the planar diagram can be written using the usual product, the contributions from nonplanar diagrams in general involve additional structure beyond the *-product, arising from the nontrivial worldsheet correlations surviving the field theory limit. To each nonplanar diagram, depending on the number n of external vertex operator insertions on each boundary, there is a corresponding *(n) n-ary operation. We further find that it is no longer possible to write down an off-shell gauge invariant one-loop effective action using the noncommutative field strength defined at tree-level. (C) 2001 Published by Elsevier Science B.V.