Standard model with four generations: Selected implications for rare $B$ and $K$ decays

We extend our recent work and study implications of the standard model with four generations (SM4) for rare $B$ and $K$ decays. We again take seriously the several $2–3\sigma$ anomalies seen in $B$, $B_s$ decays and interpret them in the context of this simple extension of the SM. SM4 is also of course of considerable interest for its potential relevance to dynamical electroweak symmetry breaking and to baryogenesis. Using experimental information from processes such as $B\to X_s\gamma$, $B_d$ and $B_s$ mixings, indirect $CP$-violation from $K_L\to \pi \pi$ etc. along with oblique corrections, we constrain the relevant parameter space of the SM4, and find $m_{t^{\prime }}$ of about 400–600 GeV with a mixing angle $|V_{t^{\prime }b}^{*}{V}_{t^{\prime }s}|$ in the range of about $0.05–1.4\times {10}^{-2}$ and with an appreciable $CP$-odd associated phase, are favored by the current data. Given the unique role of the $CP$ asymmetry in $B_s\to \psi \varphi$ due to its gold-plated nature, correlation of that with many other interesting observables, including the semileptonic asymmetry ($A_{\mathrm{SL}}$) are studied in SM4. We also identify several processes, such as $B\to X_s\nu \overline{\nu }$, $K_L\to {\pi }^0\nu \overline{\nu }$ etc., that are significantly different in SM4 from the SM. Experimentally the very distinctive process $B_s\to {\mu }^{+}{\mu }^{-}$ is also discussed; the branching ratio can be larger or smaller than in SM, $(3.2\to 4.2)\times {10}^{-9}$, by a factor of $\mathcal{O}(3)$.

Figure 1

(a) Correlation between $S_{\varphi K_s}$ and $S_{\psi \varphi }$ (left panel) and (b) Variation of $S_{\psi \varphi }$ with the phase ${\varphi }_s^{\prime }$ of ${\lambda }_{t^{\prime }}^s$ (right panel), for $m_{t^{\prime }}=300$ (magenta), 400 (red), 500 (green), and 600 (blue) GeV, respectively. The horizontal lines (left panel) represent the experimental $1\sigma$ range for $S_{\varphi K_s}$ whereas the vertical lines (black $1\mathrm{\text{−}}\sigma$ and red $2\mathrm{\text{−}}\sigma$) represent that for $S_{\psi \varphi }$; in the right panel the horizontal lines are for $S_{\psi \varphi }$.

Figure 2

Correlation between $CP$ asymmetry in $B\to X_s\gamma$ and $S_{\psi \varphi }$, the $CP$ asymmetry in $B_s\to J/\psi \varphi$; where the red and blue regions correspond to $m_{t^{\prime }}=400$ and 600 GeV whereas horizontal lines represent the SM limit for $CP$ asymmetry and the vertical lines represent the $2\sigma$ limit for $CP$ asymmetry in $B_s\to J/\psi \varphi$.

Figure 3

Left panel shows the semileptonic $CP$ asymmetry $A_{\mathrm{SL}}$ as a function of $|{\lambda }_{t^{\prime }}^s|$ whereas in the right panel correlation between $A_{\mathrm{SL}}$ and $S_{\psi \varphi }$ is shown; red and blue region corresponds to $m_{t^{\prime }}=400$ and 600 GeV, respectively, the SM value of $A_{\mathrm{SL}}$ (of order ${10}^{-5}$) is too close to zero to be visible in the plot whereas the SM value for $S_{\psi \varphi }$ is $-0.04$.

Figure 4

Correlation between $CP$ asymmetry in $B\to X_s{\ell }^{+}{\ell }^{-}$ (high-$q^2$ region) and $S_{\psi \varphi }$. In the SM both the values are very small and in the plot they correspond to the point $[-0.04,0.0]$. The red and blue regions correspond to $m_{t^{\prime }}=400$ and 600 GeV whereas the vertical lines represent $2\sigma$ experimental range for $S_{\psi \varphi }$.

Figure 5

Forward-backward (FB) asymmetry in $B\to X_s{\ell }^{+}{\ell }^{-}$ has been plotted with $z=\frac{q^2}{m_b^2}$, the red and blue regions correspond to $m_{t^{\prime }}=400$ and 600 GeV, respectively, and the black thick line represents that for SM and the green line represents the zero of $A_{\mathrm{FB}}$.

Figure 6

FB-asymmetry in $B\to K^{*}{\ell }^{+}{\ell }^{-}$ in the low-$q^2$ (left panel) and the high-$q^2$ region (right panel). The red and the blue regions correspond to $m_{t^{\prime }}=400$ and 600 GeV, respectively, and the grey region represents the SM prediction.

Figure 7

Correlation between branching fraction in $B_s\to {\mu }^{+}{\mu }^{-}$ (left panel) and $B_s\to {\tau }^{+}{\tau }^{-}$ (right panel) with $S_{\psi \varphi }$, where the red and blue regions correspond to $m_{t^{\prime }}=400$ and 600 GeV, respectively, the horizontal lines represent the SM limit for $\mathrm{Br}(B_s\to {\ell }^{+}{\ell }^{-})$ whereas the vertical lines represent the $2\sigma$ experimental range for $S_{\psi \varphi }$.

Figure 8

Correlation between branching fraction in $B\to X_s\nu \overline{\nu }$ and $S_{\psi \varphi }$, where the red and blue regions correspond to $m_{t^{\prime }}=400$ and 600 GeV, respectively, the horizontal lines represent the SM limit for $\mathrm{Br}(B\to X_s\nu \overline{\nu })$ whereas the vertical lines represent the $2\sigma$ experimental range for $S_{\psi \varphi }$.

Figure 9

Plot between the branching fraction of $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ with ${\varphi }_{t^{\prime }}^{ds}={\varphi }_{t^{\prime }}^d-{\varphi }_{t^{\prime }}^s$ bounded by the present experimental limit, red and blue region corresponds to $m_{t^{\prime }}=400$ and 600 GeV, respectively, the green and black horizontal lines represent $1\sigma$ limit for SM and experimental value, respectively. Left panel shows only $1\sigma$ range expected in SM4; full range is shown in the right panel.

Figure 10

The branching fraction of $K_L\to {\pi }^0\nu \overline{\nu }$ versus ${\varphi }_{t^{\prime }}^{ds}={\varphi }_{t^{\prime }}^d-{\varphi }_{t^{\prime }}^s$ in SM4, red and blue region corresponds to $m_{t^{\prime }}=400$ and 600 GeV, respectively, the black horizontal lines represent $1\sigma$ SM limit; left panel shows only $1\sigma$ range expected in SM4, full range for SM4 is shown in the right panel.

Figure 11

The allowed range of the $CP$ asymmetry difference ($\Delta A_{CP}$) in the ($\Delta A_{CP}-{\lambda }_t^{\prime }$) plane, where the red and blue regions correspond to $m_{t^{\prime }}=400$ and 600 GeV; grey shaded regions correspond to the uncertainties due to hadronic parameters.

Figure 12

Correlation plots between the mixing induced $CP$ asymmetry $S_{{\pi }^0{K}_s}$ and the direct $CP$ asymmetry $A_{{\pi }^0{K}_s}$ in the SM (left panel) and in the fourth generation model (right panel) where the red and blue regions correspond to $m_{t^{\prime }}=400$ and 600 GeV. The horizontal and vertical lines represent $1\sigma$ experimental allowed ranges.

Figure 13

The correlation plot between the direct $CP$ asymmetry and the $CP$-averaged branching ratio for the $B^0\to {\pi }^0{\pi }^0$ process where the grey region corresponds to the SM result and the red and blue regions correspond to $m_{t^{\prime }}=400$ and 600 GeV, respectively. The horizontal and vertical lines represent the $1\mathrm{\text{−}}\sigma$ experimental range of the corresponding observables.