Large time-dependent $CP$ violation in the $B_s^0$ system and finite $D^0\mathrm{\text{−}}{\overline{D}}^0$ mass difference in a four generation standard model

Combining the measured $B_s$ mixing with $b\to s{\ell }^{+}{\ell }^{-}$ rate data, we find a sizable 4 generation $t^{\prime }$ quark effect is allowed, for example, with $m_{t^{\prime }}\sim 300\mathrm{GeV}$ and $V_{t^{\prime }s}^{*}{V}_{t^{\prime }b}\sim 0.025e^{\pm i70°}$, which could underlie the new physics indications in $CP$ violation studies of $b\to s\overline{q}q$ transitions. With positive phase, large and negative mixing-dependent $CP$ violation in $B_s$ system is predicted, $\sin 2{\Phi }_{B_s}\sim -0.5$ to $-0.7$. This also can be probed via width difference methods. As a corollary, the short distance generated $D^0\mathrm{\text{−}}{\overline{D}}^0$ mass difference is found to be consistent with, if not slightly higher than, recent $B$ factory measurements, while $CP$ violation is subdued with $\sin 2{\Phi }_D\sim -0.2$.