First measurement of the $CP$-violating phase $\phi_s^{d\overline{d}}$ in $B_s^0\to(K^+\pi^-)(K^-\pi^+)$ 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/c$^{2}$. 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 $ {B}_s^0\to {K}^{\ast }(892){}^{\circ}{\overline{K}}^{\ast }{(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, $ {\phi}_s^{d\overline{d}} $ , in $ b\to d\overline{d}s $ transitions is performed, yielding a value of $ {\phi}_s^{d\overline{d}}=-0.10\pm 0.13\left(\mathrm{stat}\right)\pm 0.14 $ (syst) rad.