  A one-loop renormalization group (RG) analysis is performed for
noncommutative Landau-Ginsburg theory in an arbitrary dimension. We adopt a
modern version of the Wilsonian RG approach, in which a shell integration in
momentum space bypasses the potential IR singularities due to UV-IR mixing. The
momentum-dependent trigonometric factors in interaction vertices,
characteristic of noncommutative geometry, are marginal under RG
transformations, and their marginality is preserved at one loop. A negative
$\Theta$-dependent anomalous dimension is discovered as a novel effect of the
UV-IR mixing. We also found a noncommutative Wilson-Fisher (NCWF) fixed point
in less than four dimensions. At large noncommutativity, a momentum space
instability is induced by quantum fluctuations, and a consequential first-order
phase transition is identified together with a Lifshitz point in the phase
diagram. In the vicinity of the Lifshitz point, we introduce two critical
exponents $\nu_m$ and $\beta_k$, whose values are determined to be 1/4 and 1/2,
respectively, at mean-field level.
