{$K^0$}-{$\overline{K}^0$} mixing and the CKM parameters $(ρ,η)$ from the Laplace sum rules

Using the Laplace sum rule (LSR) approach, which is less affected by the contribution of the higher mass hadronic states than the Finite Energy Sum Rule (FESR), we test the reliability of the existing estimate of the $K^0$-$\overline {K}^0$ mixing parameter from the four-quark two-point correlator. We obtain, for the renormalization group invariant $B$-parameter $\Big[ f_K/(1.2f_\pi)\Big]^2 \hat {B}_K$, the upper bound: 0.83 and the $best$ estimate: $0.55 \pm 0.09$. Combining the previous estimate with the updated value of $f_B\sqrt{B_B}=(1.49\pm 0.14)f_\pi$ obtained from the same LSR method, one can deduce the $best$ fitted values $(\rho,\eta)\approx (0.41,0.09)$ of the CKM parameters.