Difference in $B^{+}$ and $B^0$ Direct $CP$ Asymmetry as an Effect of a Fourth Generation

Direct $CP$ violation in $B^0\to K^{+}{\pi }^{-}$ decay has emerged at the $-10\%$ level, but the asymmetry in $B^{+}\to K^{+}{\pi }^0$ mode is consistent with zero. This difference points towards possible new physics in the electroweak penguin operator. We point out that a sequential fourth generation, with sizable $V_{t^{\prime }s}^{*}{V}_{t^{\prime }b}$ and near maximal phase, could be a natural cause. We use the perturbative QCD factorization approach for $B\to K\pi$ amplitudes. While the $B^0\to K^{+}{\pi }^{-}$ mode is insensitive to $t^{\prime }$, we critically compare $t^{\prime }$ effects on direct $CP$ violation in $B^{+}\to K^{+}{\pi }^0$ with $b\to s{\ell }^{+}{\ell }^{-}$ and $B_s$ mixing. If the $K^{+}{\pi }^0-K^{+}{\pi }^{-}$ asymmetry difference persists, we predict $\sin 2{\Phi }_{B_s}$ to be negative.

Figure 1

The $t^{\prime }$ correction $-\Delta C_i$ normalized to the strong penguin coefficient $|C_4^t|$ (both at $m_b$ scale) vs $m_{t^{\prime }}$.

Figure 2

Direct CPV asymmetries ${\mathcal{A}}_{K\pi }$ and ${\mathcal{A}}_{K{\pi }^0}$ vs ${\varphi }_s\equiv \arg V_{t^{\prime }s}^{*}{V}_{t^{\prime }b}$. In (a) the solid and dashed curves are for $m_{t^{\prime }}=300$ and 350 GeV, respectively, and for $r_s\equiv |V_{t^{\prime }s}^{*}{V}_{t^{\prime }b}|=0.01$ and 0.03. All curves for ${\mathcal{A}}_{K\pi }$ coalesce, but for ${\mathcal{A}}_{K{\pi }^0}$, the $r_s=0,03$ curves are steeper. For (b) the strong penguin absorptive phase $\delta$ is varied from 155° (dot-dashed line), 156° (solid line) to 160° (dashed line) for $m_{t^{\prime }}=300\mathrm{GeV}$ and $r_s=0.03$.

Figure 3

(a) $\mathcal{B}(b\to s{\ell }^{+}{\ell }^{-})$, (b) $\Delta m_{B_s}$, (c) $A_{CP}(b\to s\gamma )$, and (d) $\sin 2{\Phi }_{B_s}$ vs ${\varphi }_s=\arg V_{t^{\prime }s}^{*}{V}_{t^{\prime }b}$. Notation is the same as Fig. 2a, with effect strongest for larger $r_s$ and $m_{t^{\prime }}$. Horizontal solid band in (a) corresponds to $1\sigma$ experimental range, and solid line in (b) is the lower limit, both from Ref. 6. The experimental range for (c) is outside the plot.