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

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