Wilsonian Effective Actions and the IR/UV Mixing in Noncommutative Gauge Theories

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 theta k ->0, where theta 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 divergencies cancel in supersymmetric gauge theories. The logarithmic divergencies disappear in mass-deformed N=4 theories, but not in other finite N=2 theories. We next concentrate on finite N=2 and mass-deformed N=4 supersymmetric U(1) gauge theories with massive hypermultiplets. The Wilsonian running coupling exhibits a non-trivial threshold behaviour at and well below the noncomutativity scale, eventually becoming flat in the extreme infrared in N=4 theories, but not in N=2 theories. This is interpreted as the (non)-existence of a non-singular commutative limit where the theory is described by a commutative N=2 pure U(1) theory. We expect that our analysis of finite theories is exact to all orders in perturbation theory.