Correlations among newCPviolating effects inΔF=2observables

We point out that the observed $CP$ violation in $B_d-{\overline{B}}_d$ mixing, taking into account the measured ratio $\Delta M_d/\Delta M_s$, the recently decreased lattice value of the nonperturbative parameter ${\widehat{B}}_K$ and an additional effective suppression factor ${\kappa }_{ϵ}\simeq 0.92$ in ${ϵ}_K$ neglected so far in most analyses, may be insufficient to describe the measured value of ${ϵ}_K$ within the standard model (SM), thus hinting at new $CP$ violating contributions to the $K-\overline{K}$ and/or $B_d-{\overline{B}}_d$ systems. Furthermore, assuming that $\Delta M_d/\Delta M_s$ is SM-like, the signs and the magnitudes of new-physics effects in ${ϵ}_K$ and in the $CP$ asymmetries $S_{\psi K_s}$ and $S_{\psi \varphi }$ may turn out to be correlated. For example, in a scenario with new $CP$ phases in $B_d$ and $B_s$ mixings being approximately equal and negative, a common new phase $\approx -5°$ could remove the tension between ${ϵ}_K$ and $S_{\psi K_s}$ present in the SM and simultaneously accommodate, at least partly, the recent claim of $S_{\psi \varphi }$ being much larger than the SM expectation. We emphasize the importance of precise determinations of $V_{cb}$, ${\widehat{B}}_K$, $F_K$, and ${\xi }_s$, to which the parameter ${ϵ}_K$ and its correlation with the $CP$ violation in the $B_d-{\overline{B}}_d$ system are very sensitive.