  UV/IR mixing is one of the most important features of noncommutative field
theories. As a consequence of this coupling of the UV and IR sectors, the
configuration of fields at the zero momentum limit in these theories is a very
singular configuration. We show that the renormalization conditions set at a
particular momentum configuration with a fixed number of zero momenta,
renormalizes the Green's functions for any general momenta only when this
configuration has same set of zero momenta. Therefore only when renormalization
conditions are set at a point where all the external momenta are nonzero, the
quantum theory is renormalizable for all values of nonzero momentum. This
arises as a result of different scaling behaviors of Green's functions with
respect to the UV cutoff ($\Lambda$) for configurations containing different
set of zero momenta. We study this in the noncommutative $\phi^4$ theory and
analyse similar results for the Gross-Neveu model at one loop level. We next
show this general feature using Wilsonian RG of Polchinski in the globally O(N)
symmetric scalar theory and prove the renormalizability of the theory to all
orders with an infrared cutoff. In the context of spontaneous symmetry breaking
(SSB) in noncommutative scalar theory, it is essential to note the different
scaling behaviors of Green's functions with respect to $\Lambda$ for different
set of zero momenta configurations. We show that in the broken phase of the
theory the Ward identities are satisfied to all orders only when one keeps an
infrared regulator by shifting to a nonconstant vacuum.
