An exploratory study of charmless 3-body decays of $B$ mesons is presented using a simple model based on the framework of the factorization approach. The nonresonant contributions arising from $B\to P_1{P}_2$ transitions are evaluated using heavy meson chiral perturbation theory (HMChPT). The momentum dependence of nonresonant amplitudes is assumed to be in the exponential form $e^{-{\alpha }_{\mathrm{NR}}{p}_B·(p_i+p_j)}$ so that the HMChPT results are recovered in the soft meson limit $p_i,p_j\to 0$. In addition, we have identified another large source of the nonresonant signal in the matrix elements of scalar densities, e.g. $⟨K\overline{K}|\overline{s}s|0⟩$, which can be constrained from the decay ${\overline{B}}^0\to K_S{K}_S{K}_S$ or $B^{-}\to K^{-}{K}_S{K}_S$. The intermediate vector-meson contributions to 3-body decays are identified through the vector current, while the scalar meson resonances are mainly associated with the scalar density. Their effects are described in terms of the Breit-Wigner formalism. Our main results are: (i) All $KKK$ modes are dominated by the nonresonant background. The predicted branching ratios of $K^{+}{K}^{-}{K}_{S(L)}$, $K^{+}{K}^{-}{K}^{-}$ and $K^{-}{K}_S{K}_S$ modes are consistent with the data within errors. (ii) Although the penguin-dominated $B^0\to K^{+}{K}^{-}{K}_S$ decay is subject to a potentially significant tree pollution, its effective $\sin 2\beta$ is very similar to that of the $K_S{K}_S{K}_S$ mode. However, direct $CP$ asymmetry of the former, being of order $-4\%$, is more prominent than the latter. (iii) For $B\to K\pi \pi$ decays, we found sizable nonresonant contributions in $K^{-}{\pi }^{+}{\pi }^{-}$ and ${\overline{K}}^0{\pi }^{+}{\pi }^{-}$ modes, in agreement with the Belle measurements but larger than the BABAR result. (iv) Time-dependent $CP$ asymmetries in $K_S{\pi }^0{\pi }^0$, a purely $CP$-even state, and $K_S{\pi }^{+}{\pi }^{-}$, an admixture of $CP$-even and $CP$-odd components, are studied. (v) The ${\pi }^{+}{\pi }^{-}{\pi }^0$ mode is found to have a rate larger than ${\pi }^{+}{\pi }^{-}{\pi }^{-}$ even though the former involves a ${\pi }^0$ in the final state. They are both dominated by resonant $\rho$ contributions. (vi) We have computed the resonant contributions to 3-body decays and determined the rates for the quasi-two-body decays $B\to VP$ and $B\to SP$. The predicted $\rho \pi$, $f_0(980)K$ and $f_0(980)\pi$ rates are in agreement with the data, while the calculated $\varphi K$, $K^{*}\pi$, $\rho K$ and $K_0^{*}(1430)\pi$ are in general too small compared to experiment. (vii) Sizable direct $CP$ asymmetry is found in $K^{+}{K}^{-}{K}^{-}$ and $K^{+}{K}^{-}{\pi }^{-}$ modes.

• Received 10 April 2007

DOI:https://doi.org/10.1103/PhysRevD.76.094006