  This is a technical work about how to evaluate loop integrals appearing in
one loop nonplanar (NP) diagrams in noncommutative (NC) field theory. The
conventional wisdom says that, barring the ultraviolet/infrared (UV/IR) mixing
problem, NP diagrams whose planar counterparts are UV divergent are rendered
finite by NC phases that couple the loop momentum to the external NC momentum
\rho^{\mu}=\theta^{\mu\nu}p_{\nu}. We show that this is generally not the case.
We find that subtleties arise already on Euclidean spacetime. The situation is
even worse in Minkowski spacetime due to its indefinite metric. We compare
different prescriptions that may be used to evaluate loop integrals in ordinary
theory. They are equivalent in the sense that they always yield identical
results. However, in NC theory there is no a priori reason that these
prescriptions, except for the defining one built in Feynman propagator, are
physically justified. Employing them can lead to ambiguous results. For
\rho^2>0, the NC phase can worsen the UV property of loop integrals instead of
always improving it in high dimensions. We explain how this surprising
phenomenon comes about from the indefinite metric. For \rho^2<0, the NC phase
improves the UV property and softens the quadratic UV divergence in ordinary
theory to a bounded but indefinite UV oscillation. We employ a cut-off method
to quantify the new UV non-regular terms. For \rho^2>0, these terms are
generally complex and thus also harm unitarity. As the new terms are not
available in the Lagrangian, our result casts doubts on previous demonstrations
of one loop renormalizability based exclusively upon analysis of planar
diagrams, especially in theories with quadratic divergences.
