Phase diagram of the U(2)×U(2) σ model and its implications for chiral hierarchies

Motivated by the issue of whether it is possible to construct phenomenologically viable models where the electroweak symmetry breaking is triggered by new physics at a scale $\Lambda \gg 4\pi v$, where $v$ is the order parameter of the transition $(v\sim 250$ GeV) and $\Lambda$ is the scale of new physics, we have studied the phase diagram of the $\mathrm{U}\mathrm{}\left(2\right)\mathrm{}\times \mathrm{U}\mathrm{}\left(2\right)\mathrm{}$ model. This is the relevant low energy effective theory for a class of models which will be discussed below. We find that the phase transition in these models is first order in most of parameter space. The order parameter cannot be made much smaller than the cutoff and, consequently, a large hierarchy does not appear sustainable. In the relatively small region in the space of parameters where the phase transition is very weakly first order or second order the model effectively reduces to the O(8) theory for which the triviality considerations should apply.