Study of the $B^0\to \rho (770{)}^0{K}^{\ast }(892{)}^0$ decay with an amplitude analysis of $B^0\to ({\pi }^{+}{\pi }^{-})(K^{+}{\pi }^{-})$ decays

An amplitude analysis of $B^0\to ({\pi }^{+}{\pi }^{-})(K^{+}{\pi }^{-})$ decays is performed in the two-body invariant mass regions $300<m({\pi }^{+}{\pi }^{-})<1100$ MeV/$c^2$, accounting for the ${\rho }^0$, $\omega$, $f_0(500)$, $f_0(980)$ and $f_0(1370)$ resonances, and $750<m(K^{+}{\pi }^{-})<1200$ MeV/$c^2$, which is dominated by the $K^{\ast }(892{)}^0$ meson. The analysis uses 3 fb${}^{-1}$ of proton-proton collision data collected by the LHCb experiment at centre-of-mass energies of 7 and 8 TeV. The $CP$ averages and asymmetries are measured for the magnitudes and phase differences of the contributing amplitudes. The $CP$-averaged longitudinal polarisation fractions of the vector-vector modes are found to be ${\tilde{f}}_{\rho K^{\ast }}^0=0.164\pm 0.015\pm 0.022$ and ${\tilde{f}}_{\omega K^{\ast }}^0=0.68\pm 0.17\pm 0.16$, and their $CP$ asymmetries, ${\mathcal{A}}_{\rho K^{\ast }}^0=-0.62\pm 0.09\pm 0.09$ and ${\mathcal{A}}_{\omega K^{\ast }}^0=-0.13\pm 0.27\pm 0.13$, where the first uncertainty is statistical and the second systematic.