Renormalization Group Invariance of Exact Results in Supersymmetric Gauge Theories

TL;DR

The paper clarifies how Wilsonian renormalization group invariance operates in supersymmetric gauge theories by showing that the dynamical scale Λ is not invariant under changing the ultraviolet cutoff when bare fields are kept canonically normalized. The authors derive a precise Λ-shift formula, Λ' = Λ ∏_i Z_i(M,M')^{-T^i_F/b0}, and confirm it both via holomorphic-one-loop arguments and the NSVZ β-function, aligning with known exact results. They apply this framework to quantum modified moduli spaces, matching equations, and nonperturbative superpotentials, demonstrating RG invariance of these exact results under cutoff changes. In particular, they use the formalism to analyze inverted-hierarchy questions in SP(N) models, concluding that the inverted hierarchy does not arise when RG improvements are implemented correctly, thereby clarifying the low-energy vacuum structure in these theories.

Abstract

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) = Λ^{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 Λas RG invariant. The resolution is that Λis 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 Λmust 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 Λwe 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.