The renormalization group invariants and exact results for various supersymmetric theories

TL;DR

The paper demonstrates that for certain $\mathcal{N}=1$ supersymmetric theories, one can construct renormalization group invariants (RGIs) that are exact to all orders in perturbation theory, with the NSVZ beta-function playing a central role. It analyzes HD regularization as a framework where these all-order NSVZ relations hold for bare and, in the HD+MSL scheme, renormalized couplings, while highlighting scheme dependence in schemes like $\overline{DR}$. Concrete all-loop RGIs are derived for ${\cal N}=1$ SQCD+SQED and for the MSSM, with explicit formulas illustrating how gauge, Yukawa, and mu terms combine to yield scale-independent quantities in specific prescriptions. The study further extends to a 6D ${\cal N}=(1,0)$ higher-derivative theory in harmonic superspace, proposing an all-loop NSVZ-like beta function and a corresponding 6D RGI, inviting further multi-loop verification. Overall, the work clarifies how exact RGIs can emerge in tightly constrained SUSY frameworks and under particular regularization schemes, with implications for coupling unification and beyond-4D quantum field theories.

Abstract

Some recent all-loop results on the renormalization of supersymmetric theories are summarized and reviewed. In particular, we discuss how it is possible to construct expressions which do not receive quantum corrections in all orders for certain ${\cal N}=1$ supersymmetric theories. For instance, in ${\cal N}=1$ SQED+SQCD there is a renormalization group invariant combination of two gauge couplings. For the Minimal Supersymmetric Standard Model there are two such independent combinations of the gauge and Yukawa couplings. We investigate the scheme-dependence of these results and verify them by explicit three-loop calculations. We also argue that the all-loop exact $β$-function and the corresponding renormalization group invariant can exist in the $6D$, ${\cal N}=(1,0)$ supersymmetric higher derivative gauge theory interacting with a hypermultiplet in the adjoint representation.