Is $f_X(1500)$ observed in the $B\to \pi (K)KK$ decays ${\rho }^0(1450)$?

Abstract

We suggest that the uncertain state $f_X(1500)$ observed by Belle and BaBar more than a decade ago, which has been viewed as a single scalar or a combination of several even spin resonances, is the vector ${\rho }^0(1450)$ reported recently by LHCb. Adopting the perturbative QCD approach, we determine the di-kaon distribution amplitudes with the ${\rho }^0(1450)$ resonance from the LHCb data for the quasi-two-body decays $B^{\pm }\to {\pi }^{\pm }{\rho }^0(1450)\to {\pi }^{\pm }{K}^{+}{K}^{-}$. It is then shown that the $B^{+}\to K^{+}{K}^{+}{K}^{-}$ decay spectrum around the invariant mass $M(K^{+}{K}^{-})\sim 1.5\mathrm{GeV}$ measured by BaBar can be well described by the resonant contribution from ${\rho }^0(1450)$. The broad structure in the $B^{+}\to K^{+}{K}_S{K}_S$ spectrum around the invariant mass $1.5\mathrm{GeV}$ of a $K_S{K}_S$ pair, which ${\rho }^0(1450)$ cannot decay into because of Bose-Einstein statistics, can be accounted for by a nonresonant $S$-wave contribution alone. The branching fractions and/or the direct $CP$ asymmetries of the $B^{\pm }\to {\pi }^{\pm }{\rho }^0(1450)\to {\pi }^{\pm }{K}^{+}{K}^{-}$, $B^{+}\to K^{+}{\rho }^0(1450)\to K^{+}{K}^{+}{K}^{-}$ and $B^0\to K^0{\rho }^0(1450)\to K^0{K}^{+}{K}^{-}$ modes are predicted, which can be tested at the ongoing LHCb and Belle-II experiments. We encourage experimental colleagues to scrutinize our postulation by analyzing relevant data with higher precision.

Comments: 7 Pages, 3 Figures. Accepted by PRD