Charmless hadronicBdecays involving scalar mesons: Implications on the nature of light scalar mesons

Abstract

The hadronic charmless $B$ decays into a scalar meson and a pseudoscalar meson are studied within the framework of QCD factorization. Based on the QCD sum rule method, we have derived the leading-twist light-cone distribution amplitudes of scalar mesons and their decay constants. Although the light scalar mesons ${f}_{0}(980)$ and ${a}_{0}(980)$ are widely perceived as primarily the four-quark bound states, in practice it is difficult to make quantitative predictions based on the four-quark picture for light scalars. Hence, predictions are made in the 2-quark model for the scalar mesons. The short-distance approach suffices to explain the observed large rates of ${f}_{0}(980){K}^{\ensuremath{-}}$ and ${f}_{0}(980){\overline{K}}^{0}$ that receive major penguin contributions from the $b\ensuremath{\rightarrow}ss\overline{s}$ process. When ${f}_{0}(980)$ is assigned as a four-quark bound state, there exist extra diagrams contributing to $B\ensuremath{\rightarrow}{f}_{0}(980)K$. Therefore, a priori the ${f}_{0}(980)K$ rate is not necessarily suppressed for a four-quark state ${f}_{0}(980)$. The predicted ${\overline{B}}^{0}\ensuremath{\rightarrow}{a}_{0}^{\ifmmode\pm\else\textpm\fi{}}(980){\ensuremath{\pi}}^{\ensuremath{\mp}}$ and ${a}_{0}^{+}(980){K}^{\ensuremath{-}}$ rates exceed the current experimental limits, favoring a four-quark nature for ${a}_{0}(980)$. The penguin-dominated modes ${a}_{0}(980)K$ and ${a}_{0}(1450)K$ receive predominant weak annihilation contributions. There exists a twofold experimental ambiguity in extracting the branching ratio of ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\overline{K}}_{0}^{*0}(1430){\ensuremath{\pi}}^{\ensuremath{-}}$, which can be resolved by measuring other ${K}_{0}^{*}(1430)\ensuremath{\pi}$ modes in conjunction with the isospin symmetry consideration. Large weak annihilation contributions are needed to explain the ${K}_{0}^{*}(1430)\ensuremath{\pi}$ data. The decay ${\overline{B}}^{0}\ensuremath{\rightarrow}{\ensuremath{\kappa}}^{+}{K}^{\ensuremath{-}}$ provides a nice ground for testing the 4-quark and 2-quark nature of the $\ensuremath{\kappa}$ meson. It can proceed through $W$-exchange and hence is quite suppressed if $\ensuremath{\kappa}$ is made of two quarks, while it receives a tree contribution if $\ensuremath{\kappa}$ is predominately a four-quark state. Hence, an observation of this channel at the level of $\ensuremath{\gtrsim}{10}^{\ensuremath{-}7}$ may imply a four-quark assignment for the $\ensuremath{\kappa}$. Mixing-induced $CP$ asymmetries in penguin-dominated modes are studied and their deviations from $\mathrm{sin}2\ensuremath{\beta}$ are found to be tiny.