Study of the B ⁰ → ρ(770)°K ^* (892) ⁰ decay with an amplitude analysis of B ⁰ → (π ⁺ π ⁻ )(K ⁺ π ⁻ ) 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 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.[Figure not available: see fulltext.].