First observation of $CP$ violation and measurement of polarization in $B^{+}\to \rho (770{)}^0{K}^{\ast }(892{)}^{+}$ decays

An amplitude analysis of the $B^{+}\to ({\pi }^{+}{\pi }^{-})(K_{\mathrm{S}}^0{\pi }^{+})$ decay is performed in the mass regions $0.30<m_{{\pi }^{+}{\pi }^{-}}<1.10\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}{c}^2$ and $0.75<m_{K_{\mathrm{S}}^0{\pi }^{+}}<1.20\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}{c}^2$, using $pp$ collision data recorded with the LHCb detector corresponding to an integrated luminosity of $9{\mathrm{f}\mathrm{b}}^{-1}$. The polarization fractions and $CP$ asymmetries for $B^{+}\to \rho (770{)}^0{K}^{\ast }(892{)}^{+}$ decays are measured. Violation of the $CP$ symmetry in the decay $B^{+}\to \rho (770{)}^0{K}^{\ast }(892{)}^{+}$ is observed for the first time, with a significance exceeding nine standard deviations. The $CP$ asymmetry is measured to be ${\mathcal{A}}_{CP}=0.507\pm 0.062\text{ (stat)}\pm 0.024\text{ (syst)}$ and the $CP$-averaged longitudinal polarization fraction of $f_L=0.720\pm 0.028\text{ (stat)}\pm 0.009\text{ (syst)}$. The measurements help to shed light on the polarization puzzle of $B$ mesons decaying to two vector mesons.