Threshold effects and radiative electroweak symmetry breaking in $SU(5)$ extensions of the MSSM

We make a complete analysis of radiative symmetry breaking in the MSSM and its $SU(5)$ extensions including low- and high-energy threshold effects in the framework of the two-loop renormalization group. In particular, we consider minimal $SU(5)$, the ``missing-doublet'' $SU(5)$, a Peccei-Quinn invariant version of $SU(5)$ as well as a version with light adjoint remnants. We derive permitted ranges for the parameters of these models in relation to predicted $\alpha_{s}$ and $M_G$ values within the present experimental accuracy. The parameter regions allowed under the constraints of radiative symmetry breaking, perturbativity and proton stability, include the experimentally designated domain for $\alpha_s$. In the case of the minimal $SU(5)$, the values of $\alpha_s$ obtained are somewhat large in comparison with the experimental average. The ``missing-doublet'' $SU(5)$, generally, predicts smaller values of $\alpha_s$. In both versions of the ``missing-doublet'', the high energy threshold effects on $\alpha_s$ operate in the opposite direction than in the case of the minimal model, leading to small values. In the case of the Peccei-Quinn version however the presence of an extra intermediate scale allows to achieve an excellent agreement with the experimental $\alpha_s$ values. Finally, the last considered version, with light remnants, exhibits unification of couplings at string scale at the expense however of rather large $\alpha_s$ values.