Phenomenology of $\varepsilon_K$ Beyond Leading Logarithms

I present the QCD short distance coefficient $\eta_3$ of the $\Delta S=2$ hamiltonian in the \emph{next-to-leading order} (NLO) of renormalization group improved perturbation theory. Since now all QCD-factors $\eta_1$, $\eta_2$ and $\eta_3$ are known with NLO accuracy, a much higher precision in the analysis of $\varepsilon_K$ can be achieved. The CKM phase $\delta$, $|V_{td}|$ and the mass difference $\Delta m_{B_s}$ in the $\text{B}_s^0-\overline{\text{B}}_s^0$-system are predicted from the measured values of $\varepsilon_K$ and $\Delta m_{B_d}$. Finally I briefly look at the \kkmd. This work has been done in collaboration with Stefan Herrlich.