First measurement of the CP-violating phase φ^dd_s in B_s^o → (K+π−)(K−π⁺) decays

A flavour-tagged decay-time-dependent amplitude analysis of B _s ⁰  → (K⁺π⁻)(K⁻π⁺) decays is presented in the K^±π^∓ mass range from 750 to 1600MeV/c². The analysis uses pp collision data collected with the LHCb detector at centre-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3.0 fb⁻¹. Several quasi-two-body decay modes are considered, corresponding to K^±π^∓ combinations with spin 0, 1 and 2, which are dominated by the K ₀ ^* (800)⁰ and K ₀ ^* (1430)⁰, the K^*(892)⁰ and the K ₂ ^* (1430)⁰ resonances, respectively. The longitudinal polarisation fraction for the B0s→K∗(892)∘K¯¯¯¯¯∗(892)0 decay is measured as f_L = 0.208 ± 0.032 ± 0.046, where the first uncertainty is statistical and the second is systematic. The first measurement of the mixing-induced CP-violating phase, ϕdd¯¯¯s, in b→dd¯¯¯s transitions is performed, yielding a value of ϕdd¯¯¯s=−0.10±0.13(stat)±0.14 (syst) rad.