  We consider an external gauge potential minimally coupled to a renormalisable
scalar theory on 4-dimensional Moyal space and compute in position space the
one-loop Yang-Mills-type effective theory generated from the integration over
the scalar field. We find that the gauge invariant effective action involves,
beyond the expected noncommutative version of the pure Yang-Mills action,
additional terms that may be interpreted as the gauge theory counterpart of the
harmonic oscillator term, which for the noncommutative $\phi^4$-theory on Moyal
space ensures renormalisability. The expression of a possible candidate for a
renormalisable action for a gauge theory defined on Moyal space is conjectured
and discussed.
