Data-based analysis of Forward-Backward Asymmetry in $B^\pm \to K^\pm K^\mp K^\pm$

An analysis of the Forward-Backward Asymmetry (FBA) in the decay $B^\pm \to K^\pm K^\mp K^\pm$ is carried out based on the LHCb data. It is found that the large FBA observed for the invariant mass of the $K^+ K^-$ pair around 1.5 GeV can be explained by the interference of the amplitudes between the resonances with even and odd spins, where the former can be the spin-0 $f_0(1500)$ resonance plus a non-resonance $s$-wave, while the latter is a spin-1 resonance which is probably $\rho^0 (1450)$. The analysis shows the existence of the decay channel $B^\pm\to \rho^0(1450) K^\pm$, with $C\!P$ asymmetry of $A_{CP}(B^\pm\to \rho^0(1450) K^\pm)=(-3.4\pm3.0)\%$. This is in contradiction with the conclusion of BaBar in Phys. Rev. D 85, 112010, according to which the analysis showed no signal of the spin-1 resonance $\rho^0(1450)$. We suggest our experimental colleagues to perform a closer analysis to this channel. We also suggest to perform the measurements of the FBAs (as well as the FB-$C\!P$As) in other three-body decay channels of beauty and charmed mesons, as it is helpful for resonance analysis.