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

An amplitude analysis of B$^{0}$ → (π$^{+}$π$^{−}$)(K$^{+}$π$^{−}$) decays is performed in the two-body invariant mass regions 300 < m(π$^{+}$π$^{−}$) < 1100 MeV/c$^{2}$, accounting for the ρ$^{0}$, ω, f$_{0}$(500), f$_{0}$(980) and f$_{0}$(1370) resonances, and 750 < m(K$^{+}$π$^{−}$) < 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 con- tributing amplitudes. The CP-averaged longitudinal polarisation fractions of the vector-vector modes are found to be $ {\tilde{f}}_{\rho {K}^{*}}^0 $ = 0.164 ± 0.015 ± 0.022 and $ {\tilde{f}}_{\omega {K}^{*}}^0 $ = 0.68 ± 0.17 ± 0.16, and their CP asymmetries, $ {\mathcal{A}}_{\rho {K}^{*}}^0 $ = −0.62 ± 0.09 ± 0.09 and $ {\mathcal{A}}_{\omega {K}^{*}}^0 $ = −0.13 ± 0.27 ± 0.13, where the first uncertainty is statistical and the second systematic.