IndirectCPviolation in the neutral kaon system beyond leading logarithms

We have calculated the short distance QCD coefficient ${\mathrm{\eta }}_3$ of the effective ‖ΔS‖=2 Hamiltonian in the next-to-leading order of renormalization group improved perturbation theory. Since now all coefficients ${\mathrm{\eta }}_1$, ${\mathrm{\eta }}_2$, and ${\mathrm{\eta }}_3$ are known beyond the leading log approximation, one can achieve a much higher precision in the theoretical analysis of ${\mathrm{\epsilon }}_{\mathit{K}}$, the parameter of indirect CP violation in ${\mathit{K}}^0$-${\mathit{K}}^0$¯ mixing. The measured value for ${\mathrm{\epsilon }}_{\mathit{K}}$ yields a lower bound on each of ‖${\mathit{V}}_{\mathit{c}\mathit{b}}$‖,${\mathit{V}}_{\mathit{u}\mathit{b}}$/${\mathit{V}}_{\mathit{c}\mathit{b}}$‖, the top quark mass ${\mathit{m}}_{\mathit{t}}$, and the nonperturbative parameter ${\mathit{B}}_{\mathit{K}}$ as a function of the remaining three quantities. For example, ${\mathit{m}}_{\mathit{t}}^{\mathrm{pole}}$=176 GeV, ‖${\mathit{V}}_{\mathit{c}\mathit{b}}$‖=0.040, and ${\mathit{B}}_{\mathit{K}}$=0.75 implies ‖${\mathit{V}}_{\mathit{u}\mathit{b}}$/${\mathit{V}}_{\mathit{c}\mathit{b}}$‖≥0.0778, if the measured value for ${\mathrm{\epsilon }}_{\mathit{K}}$ is attributed solely to standard model physics. We further discuss the implications on the CKM phase δ, ‖${\mathit{V}}_{\mathit{t}\mathit{d}}$‖, and the key quantity for all CP-violating processes, Im${\lambda }_{\mathit{t}}$=Im[${\mathit{V}}_{\mathit{t}\mathit{s}}^{\mathrm{*}}$ ${\mathit{V}}_{\mathit{t}\mathit{d}}$]. These quantities and the improved Wolfenstein parameters ρ¯ and η¯ are tabulated and the shape of the unitarity triangle is discussed.