Charmless three-body decays of $B_s$ mesons

Branching fractions and $CP$-violating asymmetries of charmless ${\overline{B}}_s^0\to PPP$ decays with $P=K$, $\pi$ are studied using a simple model based on the factorization approach. The penguin-dominated modes ${\overline{B}}_s^0\to {\overline{K}}^0{K}^{+}{\pi }^{-}$ and $K^0{K}^{-}{\pi }^{+}$ have the largest rates among the three-body $B_s$ decays, and they are dominated by the $K_0^{*}(1430)$ resonances and nonresonant contributions. The branching fraction of ${\overline{B}}_s^0\to K^0{K}^{+}{K}^{-}$ is predicted to be of order $1.4\times {10}^{-6}$, which is in the middle of the measured result $(0.2–3.4)\times {10}^{-6}$ obtained by LHCb. We find sizable $CP$ violation in $K^0{\pi }^{+}{\pi }^{-}$, $K^0{\pi }^0{\pi }^0$, $K^0{K}^{+}{K}^{-}$, and $K_S{K}_S{K}_S$ channels. Just as the $B_u$ sector, the $CP$ asymmetries of ${\overline{B}}_s^0\to K^0{\pi }^{+}{\pi }^{-}$ and ${\overline{B}}_s^0\to K^0{K}^{+}{K}^{-}$ have similar magnitudes but are opposite in sign. Several model-independent relations based on $U$-spin symmetry for $CP$ asymmetries are derived. Although they cannot be tested by the present available data, they can be checked by the dynamical calculations. Since the $U$-spin symmetry that relates various matrix elements of scalar densities, for example, $⟨K^{0}K|\overline{d}q|0⟩$ and $⟨{\overline{K}}^0\pi |\overline{s}q|0⟩$, is badly broken, the derived $U$-spin relations are generally not well respected.

Figure 1

Dalitz-plot distributions of penguin dominated ${\overline{B}}_s^0$ to (a) $K^0{K}^{-}{\pi }^{+}$, (b) $K^{+}{K}^{-}{\pi }^0$, and (c) $K^0{\overline{K}}^0{\pi }^0$ decays. The $CP$ averaged differential rates are in units of ${10}^{-6}{\mathrm{GeV}}^{-4}$. Note that the contour lines are separated logarithmically.

Figure 2

Same as Fig. 1, but for tree-dominated ${\overline{B}}_s^0$ to (a) $K^0{\pi }^{+}{\pi }^{-}$, (b) $K^{+}{\pi }^{-}{\pi }^0$, (c) $K^0{\pi }^0{\pi }^0$ and (d) $K^0{K}^{+}{K}^{-}$ decays.