One-Loop Calculations and Detailed Analysis of the Localized Non-Commutative 1/p**2 U(1) Gauge Model

This paper carries forward a series of articles describing our enterprise to construct a gauge equivalent for the $\theta$-deformed non-commutative $p^{-2}$ model originally introduced by Gurau et al. arXiv:0802.0791. It is shown that breaking terms of the form used by Vilar et al. arXiv:0902.2956 and ourselves arXiv:0901.1681 to localize the BRST covariant operator $(D^2\theta^2D^2)^{-1}$ lead to difficulties concerning renormalization. The reason is that this dimensionless operator is invariant with respect to any symmetry of the model, and can be inserted to arbitrary power. In the present article we discuss explicit one-loop calculations, and analyze the mechanism the mentioned problems originate from.