Probing isovector scalar mesons in the charmless three-body $B$ decays

We propose to study the multiparticle configurations of isovector scalar mesons, saying $a_0(980)$ and $a_0(1450)$, in the charmless three-body $B$ decays by considering the width effects. Two scenarios of $a_0$ configurations are assumed, in which the first one take $a_0(980)$ as the lowest-lying $q{\bar q}$ state and $a_0(1450)$ as the first radial excited state, the second one take $a_0(1450)$ as the lowest-lying $q{\bar q}$ state and $a_0(1950)$ as the first radial excited state while $a_0(980)$ is not a $q{\bar q}$ state. Within these two scenarios, we do the PQCD calculation for the quasi-two-body $B \to a_0 \left[\to K{\bar K}/\pi\eta \right] h$ decays and extract the corresponding branching fractions of two-body $B \to a_0 h$ decays under the narrow width approximation. Our predictions show that the first scenario of $a_0(980)$ configuration can not be excluded by the available measurements in $B$ decays, the contributions from $a_0(1450)$ to the branching fractions in most channels are comparable in the first and second scenarios. Several channels are suggested for the forthcoming experimental measurements to reveal the multiparticle configurations of $a_0$, such as the channel $B^0 \to a_0^-(980) \left[\to \pi^-\eta \right] \pi^+$ with the largest predicted branching fraction, the channels $B^0 \to a_0^{\pm}(1450) \left[\to K^\pm{\bar K}^0, \pi^\pm\eta \right] \pi^\mp$ whose branching fractions obtained in the second scenario is about three times larger in magnitude than that obtained in the first scenario, and also the channels $B^+ \to a_0^+(1950) \left[ K^+{\bar K}^0/\pi^+\eta \right] K^0$ whose branching fractions are linear dependent on the partial width $\Gamma_{a_0(1950) \to KK/\pi\eta}$.