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We have combined perturbative unitarity and renormalisation group equationarguments in order to find a dynamical way to constrain the space of the gaugecouplings ($g'_1$, $\widetilde{g}$) of the so-called "Minimal $Z'$ Models". Wehave analysed the role of the gauge couplings evolution in the perturbativestability of the two-to-two body scattering amplitudes of the vector and scalarsectors of these models and we have shown that perturbative unitarity imposesan upper bound that is generally stronger than the triviality constraint. Wehave also demonstrated how this method quantitatively refines the usualtriviality bound in the case of benchmark scenarios such as the $U(1)_\chi$,the $U(1)_R$ or the "pure" $U(1)_{B-L}$ extension of the Standard Model.Finally, a description of the underlying model structure in Feynman gauge isprovided.

Theoretical constraints on the couplings of non-exotic minimal $Z'$ bosons