Constraints on the Mass and Mixing of the 4th Generation Quark From Direct CP Violation$\epsilon^{\prime}/\epsilon$ and Rare K Decays

Abstract

We investigate the $\epsilon^{\prime} /\epsilon$ for $K\to \pi\pi$ in a sequential fourth generation model. By giving the basic formulae for $\epsilon^{\prime}/\epsilon$ in this model, we analyze the numerical results which are dependent of $m_{t^{\prime}}$ and imaginary part of the fourth CKM factor, ${Im}V^{*}_{t^{'}s}V_{t^{'}d}$ (or $V^{*}_{t^{'}s}V_{t^{'}d}$ and the fourth generation CKM matrix phase $\theta$). We find that, unlike the SM, when taking the central values of all parameters for $\epsilon^{\prime}/\epsilon$, the values of $\epsilon^{\prime}/ \epsilon$ can easily fit to the current experimental data for all values of hadronic matrix elements estimated from various approaches. Also, we show that the experimental values of $\epsilon^{\prime}/\epsilon$ and rare K decays can provide a strong constraint on both mass and mixing of the fourth generation quark. When taking the values of hadronic matrix elements from the lattice or 1/N expansion calculations, a large region of the up-type quark mass $m_{t^{\prime}}$ is excluded.