Enhanced $K_L\to {\pi }^0\nu \overline{\nu }$ from direct $CP$ violation in $B\to K\pi$ with four generations

Recent $CP$ violation results in $B$ decays suggest that $Z$ penguins may have large weak phase. This can be realized by the four generation (standard) model. Concurrently, $B\to X_s{\ell }^{+}{\ell }^{-}$ and $B_s$ mixing allow for sizable $V_{t^{\prime }s}^{*}{V}_{t^{\prime }b}$ only if it is nearly imaginary. Such large effects in $b\leftrightarrow s$ transitions would affect $s\leftrightarrow d$ transitions, as kaon constraints would demand $V_{t^{\prime }d}\ne 0$. Using $\Gamma (Z\to b\overline{b})$ to bound $|V_{t^{\prime }b}|$, we infer sizable $|V_{t^{\prime }s}|\lesssim |V_{t^{\prime }b}|\lesssim |V_{us}|$. Imposing ${\epsilon }_K$, $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ and ${\epsilon }^{\prime }/\epsilon$ constraints, we find $V_{t^{\prime }d}^{*}{V}_{t^{\prime }s}\sim$ few $\times {10}^{-4}$ with large phase, enhancing $K_L\to {\pi }^0\nu \overline{\nu }$ to $5\times {10}^{-10}$ or even higher. Interestingly, $\Delta m_{B_d}$ and $\sin 2{\Phi }_{B_d}$ are not much affected, as $|V_{t^{\prime }d}^{*}{V}_{t^{\prime }b}|\ll |V_{td}^{*}{V}_{tb}|\sim 0.01$.

Figure 1

Unitarity quadrangles of (a) Eq. (2), with $|V_{us}^{*}{V}_{ub}|$ exaggerated; (b) Eq. (17), where actual scale is $\sim 1/4$ of (a). From Eq. (1), $V_{t^{\prime }s}^{*}{V}_{t^{\prime }b}$ (dashed) drastically changes the invariant phase and $V_{ts}^{*}{V}_{tb}$ from the SM3 triangle (solid), but from Eq. (16), the dashed lines for $V_{td}{V}_{tb}^{*}$ and $V_{t^{\prime }d}{V}_{t^{\prime }b}^{*}$ can hardly be distinguished from SM3 case.

Figure 2

Allowed region from $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ (valley shaped shaded region), ${\epsilon }_K$ (simulated dots) and ${\epsilon }^{\prime }/\epsilon$ (elliptic rings) in $r_{ds}$ and ${\varphi }_{ds}$ plane, as described in text, where $V_{t^{\prime }d}^{*}{V}_{t^{\prime }s}\equiv r_{ds}{e}^{i{\varphi }_{ds}}$. For ${\epsilon }^{\prime }/\epsilon$, the rings on upper right correspond to $R_6=2.2$, and $R_8=0.8$, $1.1$ (bottom to top), and on upper left, $R_6=1.0$, $1.2$ (bottom to top), $R_8=1.2$.

Figure 3

(a) ${\epsilon }_K$, (b) $\mathcal{B}(K^{+}\to {\pi }^{+}\nu \overline{\nu })$, (c) $\mathrm{Re}({\epsilon }^{\prime }/\epsilon )$ and (d) $\mathcal{B}(K_L\to {\pi }^0\nu \overline{\nu })$ vs. ${\varphi }_{ds}$, for $r_{ds}=4$ and $6\times {10}^{-4}$ and $m_{t^{\prime }}=300\mathrm{GeV}$. Larger $r_{ds}$ gives stronger variation, and horizontal bands are current ($1\sigma$) experimental range [8] (the bound for (d) is outside the plot). For (c), solid (dashed) lines are for $R_6=2.2$, $R_8=1.1$ ($R_6=1.1$, $R_8=1.2$).

Figure 4

(a) $\Delta m_{B_d}$ and (b) $\sin 2{\Phi }_{B_d}$ vs. ${\varphi }_{db}$ for $r_{db}=8$ and $12\times {10}^{-4}$, with $V_{t^{\prime }d}^{*}{V}_{t^{\prime }b}\equiv r_{db}{e}^{i{\varphi }_{db}}$. Larger $r_{db}$ gives stronger variation, and horizontal bands are the experimental range [8].