Exotic Vector-Like Quark Phenomenology in the Minimal Linear $\sigma$ Model

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 $SO(5)/SO(4)$ composite Higgs model in a specific limit, we consider the phenomenology of vector-like quarks when a scalar singlet $\sigma$ is present. This new particle may be produced in the decays $T \to t \sigma$, $B \to b \sigma$, where $T$ and $B$ are vector-like quarks of charges $2/3$ and $-1/3$, respectively, with subsequent decay $\sigma \to W^+ W^-, ZZ, hh$. By scanning over the allowed parameter space we find that these decays may be dominant. In addition, we find that the presence of several new particles allows for single $T$ production cross sections larger than those expected in minimal models. We discuss the observability of these new signatures in existing searches.