  We clarify the notion of Wilsonian renormalization group (RG) invariance in
supersymmetric gauge theories, which states that the low-energy physics can be
kept fixed when one changes the ultraviolet cutoff, provided appropriate
changes are made to the bare coupling constants in the Lagrangian. We first
pose a puzzle on how a quantum modified constraint (such as Pf(Q^i Q^j) =
\Lambda^{2(N+1)} in SP(N) theories with N+1 flavors) can be RG invariant, since
the bare fields Q^i receive wave function renormalization when one changes the
ultraviolet cutoff, while we naively regard the scale \Lambda as RG invariant.
The resolution is that \Lambda is not RG invariant if one sticks to canonical
normalization for the bare fields as is conventionally done in field theory. We
derive a formula for how \Lambda must be changed when one changes the
ultraviolet cutoff. We then compare our formula to known exact results and show
that their consistency requires the change in \Lambda we have found. Finally,
we apply our result to models of supersymmetry breaking due to quantum modified
constraints. The RG invariance helps us to determine the effective potential
along the classical flat directions found in these theories. In particular, the
inverted hierarchy mechanism does not occur in the original version of these
models.
