Study of the decay $B^0 \rightarrow \rho^0K^*(892)^0$

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 $K^{*}(892)^0$ 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 fraction of the $B^0\rightarrow \rho K^{*}$ mode is found to be $\tilde{f}^0_{\rho K^{*}} = 0.164 \pm 0.015 \pm 0.022$ and its $CP$ asymmetry $\mathcal{A}^0_{\rho K^{*}} = -0.62 \pm 0.09 \pm 0.09$, where the first uncertainty is statistical and the second systematic.