Study of the B⁰ → ρ(770)°K^*(892)⁰ decay with an amplitude analysis of B⁰ → (π⁺π⁻)(K⁺π⁻) decays

An amplitude analysis of B⁰ → (π⁺π⁻)(K⁺π⁻) decays is performed in the two-body invariant mass regions 300 < m(π⁺π⁻) < 1100 MeV/c², accounting for the ρ⁰, ω, f₀(500), f₀(980) and f₀(1370) resonances, and 750 < m(K⁺π⁻) < 1200 MeV/c², 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 f˜ρK*0 = 0.164 ± 0.015 ± 0.022 and f˜ωK*0 = 0.68 ± 0.17 ± 0.16, and their CP asymmetries, AρK*0 = −0.62 ± 0.09 ± 0.09 and AωK*0 = −0.13 ± 0.27 ± 0.13, where the first uncertainty is statistical and the second systematic.