Measurement of the CKM angle $\gamma$ and $B^0_s$-$\bar{B}^0_s$ mixing frequency with $B^0_s \rightarrow D_s^\mp h^\pm \pi^\pm \pi^\mp$ decays

The CKM angle γ is measured for the first time from mixing-induced CP violation between $ {B}_s^0\to {D}_s^{\mp }{K}^{\pm }{\pi}^{\pm }{\pi}^{\mp } $ and $ {\overline{B}}_s^0\to {D}_s^{\pm }{K}^{\mp }{\pi}^{\mp }{\pi}^{\pm } $ decays reconstructed in proton-proton collision data corresponding to an integrated luminosity of 9 fb$^{−1}$ recorded with the LHCb detector. A time-dependent amplitude analysis is performed to extract the CP-violating weak phase γ − 2β$_{s}$ and, subsequently, γ by taking the $ {B}_s^0\hbox{-} {\overline{B}}_s^0 $ mixing phase β$_{s}$ as an external input. The measurement yields γ = (44 ± 12)° modulo 180°, where statistical and systematic uncertainties are combined. An alternative model-independent measurement, integrating over the five-dimensional phase space of the decay, yields $ \gamma =\left({44}_{-13}^{+20}\right){}^{\circ} $ modulo 180°. Moreover, the $ {B}_s^0\hbox{-} {\overline{B}}_s^0 $ oscillation frequency is measured from the flavour-specific control channel $ {B}_s^0\to {D}_s^{-}{\pi}^{+}{\pi}^{+}{\pi}^{-} $ to be ∆m$_{s}$ = (17.757 ± 0.007(stat) ± 0.008(syst)) ps$^{−1}$, consistent with and more precise than the current world-average value.[graphic not available: see fulltext]