Updates and New Results in Models with Reduced Couplings

The idea of reduction of couplings consists in searching for renormalization group invariant relations between parameters of a renormalizable theory that hold to all orders of perturbation theory. Based on the principle of the reduction of couplings, one can construct Finite Unified Theories (FUTs) which are $N=1$ supersymmetric Grand Unified Theories that can be made all-order finite. The prediction of the top quark mass well in advance of its experimental discovery and the prediction of the light Higgs boson mass in the range $\sim 121-126$ GeV much earlier than its experimental discovery are among the celebrated successes of such models. Here, after a brief review of the reduction of couplings method and the properties of the resulting finiteness in supersymmetric theories, we analyse four phenomenologically favoured models: a minimal version of the $N=1$ $SU(5)$, a finite $N=1$ $SU(5)$, a $N=1$ finite $SU(3)\otimes SU(3)\otimes SU(3)$ model and a reduced version of the Minimal Supersymmetric Standard Model (MSSM). A relevant update in the phenomenological evaluation has been the improved light Higgs-boson mass prediction as provided by the latest version of $\texttt{FeynHiggs}$. All four models predict relatively heavy supersymmetric spectra that start just below or above the TeV scale, consistent with the non-observation LHC results. Depending on the model, the lighter regions of the spectra could be accessible at CLIC, while the FCC-hh will be able to test large parts of predicted spectrum of each model. The lightest supersymmetric particle (LSP), which is a neutralino, is considered as a cold dark matter candidate and put to test using the latest $\texttt{MicrOMEGAs}$ code.