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The Physics of $K^0 -\bar K^0$ Mixing: $\widehat{B}_K$ and $ΔM_{LS}$ in the Chiral Quark Model

We compute the $\widehat B_K$ parameter and the mass difference $ΔM_{LS}$ of the $K^0-\bar K^0$ system by means of the chiral quark model. The chiral coefficients of the relevant $ΔS=2$ and $ΔS=1$ chiral lagrangians are computed via quark-loop integration. We include the relevant effects of one-loop corrections in chiral perturbation theory. The final result is very sensitive to non-factorizable corrections of $O(α_S N)$ coming from gluon condensation. The size of the gluon condensate is determined by fitting the experimental value of the amplitude $K^+ \to π^+π^0$. By varying all the relevant parameters we obtain $\widehat{B}_K= 0.87 \pm 0.33\ .$ We evaluate within the model the long-distance contributions to $ΔM_{LS}$ induced by the double insertion of the $ΔS = 1$ chiral lagrangian and study the interplay between short- and long-distance amplitudes. By varying all parameters we obtain $ΔM_{LS}^{th}/ΔM_{LS}^{exp} = 0.76 ^{+0.64}_{-0.34} .$ Finally, we investigate the phenomenological constraints on the Kobayashi-Maskawa parameter Im $λ_t$ entering the determination of $ε'/ε$.