Abstract
Using background field perturbation theory we study
wilsonian effective actions of noncommutative gauge theories with an
arbitrary matter content. We determine the wilsonian coupling constant
and the gauge boson polarization tensor as functions of the momentum
scale k at the one-loop level and study their short-distance
behaviour as θ · k→0, where θ is the
noncommutativity parameter. The mixing between the short-distance and
the long-distance degrees of freedom characteristic of noncommutative
field theories violates the universality of the wilsonian action and
leads to IR-singularities. We find, in agreement with known results,
that the quadratic IR divergences cancel in supersymmetric gauge
theories. The logarithmic divergences disappear in mass-deformed
= 4 theories, but not in other finite = 2 theories.
We next concentrate on finite = 2 and mass-deformed = 4 supersymmetric U(1) gauge theories with massive
hypermultiplets. The wilsonian running coupling exhibits a non-trivial
threshold behaviour at and well below the noncommutativity scale
1/√θ, eventually becoming flat in the extreme infrared in
= 4 theories, but not in = 2 theories. This is
interpreted as the (non)-existence of a non-singular commutative limit
where the theory is described by a commutative = 2 pure
U(1) theory. We expect that our analysis of finite theories is
exact to all orders in perturbation theory.