Renormalization group invariance of exact results in supersymmetric gauge theories

We clarify the notion of Wilsonian renormalization group (RG) invariance in supersymmetric gauge theories, which states that 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 $\mathrm{Pf}{(Q}^i{Q}^j)={\Lambda }^{2(N+1\mathrm{})\mathrm{}}$ in $\mathrm{SP}\mathrm{}\left(N\right)\mathrm{}$ theories with $N+1\mathrm{}$ 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 Λ as RG invariant. The resolution is that Λ 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 Λ 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 Λ 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.