Study of light-meson resonances decaying to $K_{\mathrm{S}}^{0}K\pi$ in the $B\to (K_{\mathrm{S}}^{0}K\pi )K$ channels

A study is presented of $B^{+}\to K_{\mathrm{S}}^0{K}^{-}{\pi }^{+}{K}^{-}$ and $B^{+}\to K_{\mathrm{S}}^0{K}^{+}{\pi }^{-}{K}^{+}$ decays based on the analysis of proton-proton collision data collected with the LHCb detector at centre-of-mass energies of 7, 8 and 13 TeV, corresponding to an integrated luminosity of $9f{b}^{-1}$. The $K_{\mathrm{S}}^{0}K\pi$ invariant-mass distributions of both $B^{+}$ decay modes show, in the $m(K_{\mathrm{S}}^{0}K\pi )<1.85$ GeV mass region, large activity which is resolved using an amplitude analysis. A simple model, were $J^{PC}$ amplitudes are described by multiple Breit–Wigner functions with appropriate angular distributions provides a good description of the experimental data. In this approach a complex mixture of $J^{PC}=0^{-+},1^{++}$ and $1^{+-}$ amplitudes is observed that is dominated by $\eta (1405)$, $\eta (1470)$, $\eta (1760)$, $f_1(1285)$, $f_1(1420)$ and $h_1(1405)$ resonances. The $K_{\mathrm{S}}^{0}K\pi$ Dalitz plots are dominated by asymmetric crossing $K^{\ast }\bar{K}$ bands which are different for the two $B^{+}$ decay modes. This is due to a different interference pattern between the $1^{++}$ and $1^{+-}$ amplitudes in the two channels. Branching fractions are measured for each resonant contribution.