Determine whether dynamical diagonal symmetry breaking occurs

Determine whether the chiral $SU(N)$ or $SU(N)'$ gauge dynamics realize a self-breaking phase that dynamically generates the diagonal symmetry breaking $SU(N)\times SU(N)'\to SU(N)_D$.

Background

The model realizes diagonal breaking with an elementary scalar $\Phi_0$, whose vacuum expectation value breaks $SU(N)\times SU(N)'$ to the diagonal dark-color group $SU(N)_D$. The paper notes that the same breaking might instead be generated dynamically by strong chiral gauge dynamics of either $SU(N)$ or $SU(N)'$. Such a realization would alter the scalar sector because the resulting composite state would not form a complete multiplet under the original ultraviolet gauge group. The existence of the required dynamical phase is therefore unresolved and important for determining whether a more economical, composite realization of the model is possible.