Charmless three-body decays ofBmesons

Abstract

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 $B\ensuremath{\rightarrow}{P}_{1}{P}_{2}$ transitions are evaluated using heavy meson chiral perturbation theory (HMChPT). The momentum dependence of nonresonant amplitudes is assumed to be in the exponential form ${e}^{\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{NR}}{p}_{B}\ifmmode\cdot\else\textperiodcentered\fi{}({p}_{i}+{p}_{j})}$ so that the HMChPT results are recovered in the soft meson limit ${p}_{i},{p}_{j}\ensuremath{\rightarrow}0$. 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 ${\overline{B}}^{0}\ensuremath{\rightarrow}{K}_{S}{K}_{S}{K}_{S}$ or ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{K}_{S}{K}_{S}$. 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 ${K}^{+}{K}^{\ensuremath{-}}{K}_{S(L)}$, ${K}^{+}{K}^{\ensuremath{-}}{K}^{\ensuremath{-}}$ and ${K}^{\ensuremath{-}}{K}_{S}{K}_{S}$ modes are consistent with the data within errors. (ii) Although the penguin-dominated ${B}^{0}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{K}_{S}$ decay is subject to a potentially significant tree pollution, its effective $\mathrm{sin}2\ensuremath{\beta}$ is very similar to that of the ${K}_{S}{K}_{S}{K}_{S}$ mode. However, direct $CP$ asymmetry of the former, being of order $\ensuremath{-}4%$, is more prominent than the latter. (iii) For $B\ensuremath{\rightarrow}K\ensuremath{\pi}\ensuremath{\pi}$ decays, we found sizable nonresonant contributions in ${K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${\overline{K}}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ modes, in agreement with the Belle measurements but larger than the BABAR result. (iv) Time-dependent $CP$ asymmetries in ${K}_{S}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$, a purely $CP$-even state, and ${K}_{S}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$, an admixture of $CP$-even and $CP$-odd components, are studied. (v) The ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ mode is found to have a rate larger than ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$ even though the former involves a ${\ensuremath{\pi}}^{0}$ in the final state. They are both dominated by resonant $\ensuremath{\rho}$ contributions. (vi) We have computed the resonant contributions to 3-body decays and determined the rates for the quasi-two-body decays $B\ensuremath{\rightarrow}VP$ and $B\ensuremath{\rightarrow}SP$. The predicted $\ensuremath{\rho}\ensuremath{\pi}$, ${f}_{0}(980)K$ and ${f}_{0}(980)\ensuremath{\pi}$ rates are in agreement with the data, while the calculated $\ensuremath{\phi}K$, ${K}^{*}\ensuremath{\pi}$, $\ensuremath{\rho}K$ and ${K}_{0}^{*}(1430)\ensuremath{\pi}$ are in general too small compared to experiment. (vii) Sizable direct $CP$ asymmetry is found in ${K}^{+}{K}^{\ensuremath{-}}{K}^{\ensuremath{-}}$ and ${K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ensuremath{-}}$ modes.