New results in models with reduced couplings

The reduction of couplings concept consists in searching for renormalization group invariant relations among parameters that hold to all orders in perturbation theory. This technique has been applied to $N = 1$ supersymmetric Grand Unified Theories, some of which can become finite to all loops. We review the basic idea and tools, as well as two theories in which reduction of couplings has been applied: (i) an all-loop finite $N = 1$ $SU(5)$ model and (ii) a reduced version of the Minimal Supersymmetric Standard Model. The finite model exhibits high relic abundance of cold dark matter, while, on the contrary, the second model has underproduction in the early universe. For each model we select three representative benchmark scenarios. The heavy Higgs and supersymmetric spectrum of the finite $SU(5)$ model lies beyond the reach of the 14 TeV HL-LHC, while large parts of the predicted spectrum can be tested in the 100 TeV FCC-hh, although the higher mass regions are beyond its reach. On the other hand, the Reduced Minimal Supersymmetric Standard Model (Reduced MSSM) is found to be ruled out by LHC searches for heavy neutral MSSM Higgs bosons.