Minimal $SU(3)\times SU(3)$ symmetry breaking patterns

We study the vacua of an $SU(3)\times SU(3)$-symmetric model with a bifundamental scalar. Structures of this type appear in various gauge theories such as the Renormalizable Coloron Model, which is an extension of QCD, or the Trinification extension of the electroweak group. In other contexts, such as chiral symmetry, $SU(3)\times SU(3)$ is a global symmetry. As opposed to more general $SU(N)\times SU(N)$ symmetric models, the $N=3$ case is special due to the presence of a trilinear scalar term in the potential. We find that the most general tree-level potential has only three types of minima: one that preserves the diagonal $SU(3)$ subgroup, one that is $SU(2)\times SU(2)\times U(1)$ symmetric, and a trivial one where the full symmetry remains unbroken. The phase diagram is complicated, with some regions where there is a unique minimum, and other regions where two minima coexist.