Decays of the Heavy Top and New Insights on $\epsilon_K$ in a one-VLQ Minimal Solution to the CKM Unitarity Problem

We propose a minimal extension of the Standard Model where an up-type vector-like quark, denoted $T$, is introduced and provides a simple solution to the CKM unitarity problem. We adopt the Botella-Chau parametrization in order to extract the $4\times 3$ quark mixing matrix which contains the three angles of the $3\times 3$ CKM matrix plus three new angles denoted $\theta_{14}$, $\theta_{24}$, $\theta_{34}$. It is assumed that the mixing of $T$ with standard quarks is dominated by $\theta_{14}$. Imposing a recently derived, and much more restrictive, upper-bound on the New Physics contributions to $\epsilon_{K}$, we find, in the limit of exact $\theta_{14}$ dominance where the other extra angles vanish, that $\epsilon_{K}^{\text{NP}}$ is too large. However, if one relaxes the exact $\theta_{14}$ dominance limit, there exists a parameter region, where one may obtain $\epsilon_{K}^{\text{NP}}$ in agreement with experiment while maintaining the novel pattern of $T$ decays with the heavy quark decaying predominantly to the light quarks $d$ and $u$. We also find a reduction in the decay rate of $K_L\rightarrow \pi^o \nu\bar{\nu}$.