Effective Action for Noncommutative U(1) Gauge Theory with Higher Dimensional Terms

In this paper we apply the assumption of our recent work in noncommutative scalar models to the noncommutative U(1) gauge theories. This assumption is that the noncommutative effects start to be visible continuously from a scale λ_NC and that below this scale the theory is a commutative one. Based on thisassumption and using background field method and loop calculations, an effective action is derived for noncommutative U(1) gauge theory. It will be shown that the corresponding low energy effective theory is asymptotically free and that under this condition the noncommutative quadratic IR divergences will not appear. The effective theory contains higher dimensional terms, which become more important at high energies. These terms predict an elastic photon-photon scattering due to the noncommutativity of space. Thecoefficients of these higher dimensional terms also satisfy a positivity constraint indicating that in this theory the related diseases of superluminal signal propagating and bad analytic properties of S-matrix do not exist. In the last section, we will apply our method to the noncommutative extra dimension theories.