Renormalization Group Invariance of Exact Results in Supersymmetric Gauge Theories

TL;DR

This paper clarifies how Wilsonian renormalization group invariance applies to supersymmetric gauge theories, resolving puzzles about scale invariance of quantum constraints and deriving formulas for scale changes under cutoff variations.

Contribution

It provides a formula for how the dynamical scale mbda must change with the UV cutoff, ensuring RG invariance of exact results in supersymmetric gauge theories.

Findings

Derived a formula for mbda change with cutoff adjustments.

Confirmed the formula's consistency with known exact results.

Applied the results to supersymmetry breaking models, clarifying effective potential behavior.

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) = \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.