  Using Dimensional Regularization (DR) for some two-point functions of a
prototype Non-Comutative (NC) \phi^4 scalar theory in 4-dimensions, we
explictly analyze, to one-loop, the IR and UV divergences of non-planar
diagrams having quadratic divergences and compare to the case of the
Pauli-Villars cut-off regularization (PVR). We also note that the IR structure
1/p0p obtained from DR is reproduced by PVR in the limit where the UV cut-off
\Lambda is set to infinity. We study the phenomenological implications of this
result by rederiving bounds from low-energy data on the violation of Lorentz
invariance based on the existence of the quadratic divergence. The most
stringent (and regularization independent) bound on Lorentz violation from
low-energy data is 1/\sqrt{\theta}\approx \nu\geq 10^{15} GeV for NCQCD and
10^{10} GeV for NCQED, which comes from the absence of sidereal variations
between the Cs and Hg atomic clocks.
