  We investigate the quantum effects of the nonlocal gauge invariant operator
$\frac{1}{{}{D}^{2}}{F}_{\mu \nu}\ast \frac{1}{{}{D}^{2}}{F}^{\mu \nu}$ in the
noncommutative U(1) action and its consequences to the infrared sector of the
theory. Nonlocal operators of such kind were proposed to solve the infrared
problem of the noncommutative gauge theories evading the questions on the
explicit breaking of the Lorentz invariance. More recently, a first step in the
localization of this operator was accomplished by means of the introduction of
an extra tensorial matter field, and the first loop analysis was carried out
$(Eur.Phys.J.\textbf{C62}:433-443,2009)$. We will complete this localization
avoiding the introduction of new degrees of freedom beyond those of the
original action by using only BRST doublets. This will allow us to make a
complete BRST algebraic study of the renormalizability of the theory, following
Zwanziger's method of localization of nonlocal operators in QFT.

High Energy Physics - Theory