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We characterize CP violation in the SU(2) × U(1) model due to an extra vectorlike quark or sequential family, giving special emphasis to the chiral limit m u,d,s = 0. In this limit, CP is conserved in the three generation Standard Model (SM), thus implying that all CP violation is due to the two new CP violating phases whose effects may manifest either at high energy in processes involving the new quark or as deviations from SM unitarity equalities among imaginary parts of invariant quartets (or, equivalently, areas of unitarity triangles). In our analysis we use an invariant formulation, independent of the choice of weak quark basis or the phase convention in the generalized Cabibbo-Kobayashi-Maskawa matrix. We identify the three weakbasis invariants, as well as the three imaginary parts of quartets B 1−3 which, in the chiral limit, give the strength of CP violation beyond the SM. We find that for an extra vector-like quark |B i | ≤ 10 −4 , whereas for an extra sequential family |B i | ≤ 10 −2 .

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 $$4\times 3$$ 4 × 3 quark mixing matrix which contains the three angles of the $$3\times 3$$ 3 × 3 CKM matrix plus three new angles denoted $$\theta _{14}$$ θ 14 , $$\theta _{24}$$ θ 24 , $$\theta _{34}$$ θ 34 . It is assumed that the mixing of T with standard quarks is dominated by $$\theta _{14}$$ θ 14 . Imposing a recently derived, and much more restrictive, upper-bound on the New Physics contributions to $$\epsilon _{K}$$ ϵ K , we find, in the limit of exact $$\theta _{14}$$ θ 14 dominance where the other extra angles vanish, that $$\epsilon _{K}^{\text {NP}}$$ ϵ K NP is too large. However, if one relaxes the exact $$\theta _{14}$$ θ 14 dominance limit, there exists a parameter region, where one may obtain $$\epsilon _{K}^{\text {NP}}$$ ϵ K 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 $$K_L\rightarrow \pi ^0 \nu {\overline{\nu }}$$ K L → π 0 ν ν ¯ .

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