The renormalization group invariants and exact results for various supersymmetric theories

TL;DR

This paper reviews recent all-loop results on renormalization invariants in supersymmetric theories, highlighting exact expressions that remain uncorrected across all orders and their scheme-dependence, verified through explicit calculations.

Contribution

It demonstrates the construction of all-loop renormalization group invariants in various supersymmetric theories, including ${ m N}=1$ models and higher-dimensional theories, with explicit three-loop verification.

Findings

Existence of scheme-independent RG invariants in ${ m N}=1$ SQED+SQCD

Two independent RG invariant combinations in the MSSM

Verification of invariants through explicit three-loop calculations

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 $\beta$-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.