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The Minimal Linear $sigma$ Model is a useful theoretical laboratory. One can investigate in a perturbative renormalisable model the properties of the Higgs boson as a pseudo-Goldstone boson, the phenomenological effects of the radial mode of the field $mathtt{s}$ which spontaneously breaks the global $SO(5)$ symmetry and the validity of conclusions based on the Effective Field Theory approach with the field $mathtt{s}$ in the spectrum, after the decoupling of heavy degrees of freedom. In this paper all those issues are discussed in the framework of the Minimal Linear $sigma$ Model with CP violating phases leading to pseudoscalar components in the effective Standard Model Yukawa couplings. Also the character of the electroweak phase transition in the presence of the field $mathtt{s}$ is investigated.
Extensions of the Standard Model that include vector-like quarks commonly also include additional particles that may mediate new production or decay modes. Using as example the minimal linear $sigma$ model, that reduces to the minimal $SO(5)/SO(4)$ c
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