One-Loop Calculations and Detailed Analysis of the Localized Non-Commutative $p^{-2}$ $U(1)$ Gauge Model

This paper carries forward a series of articles describing our enterprise to construct a gauge equivalent for the $\theta$-deformed $\frac1{p^2}$ model originally introduced by Gurau et. al. [Comm. Math. Phys. 287 (2009), 275–290]. It is shown that breaking terms of the form used by Vilar [J. Phys. A: Math. Theor. 43 (2010), 135401, 13 pages] and ourselves [Eur. Phys. J. C: Part. Fields 62 (2009), 433–443] 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.