Measurement of the CKM angle γ and $$ {B}_s^0\hbox{-} {\overline{B}}_s^0 $$ mixing frequency with $$ {B}_s^0\to {D}_s^{\mp }{h}^{\pm }{\pi}^{\pm }{\pi}^{\mp } $$ decays

The CKM angleγis measured for the first time from mixing-inducedCPviolation between$$ {B}_s^0\to {D}_s^{\mp }{K}^{\pm }{\pi}^{\pm }{\pi}^{\mp } $$Bs0→Ds∓K±π±π∓and$$ {\overline{B}}_s^0\to {D}_s^{\pm }{K}^{\mp }{\pi}^{\mp }{\pi}^{\pm } $$B¯s0→Ds±K∓π∓π±decays reconstructed in proton-proton collision data corresponding to an integrated luminosity of 9 fb−1recorded with the LHCb detector. A time-dependent amplitude analysis is performed to extract theCP-violating weak phaseγ− 2βsand, subsequently,γby taking the$$ {B}_s^0\hbox{-} {\overline{B}}_s^0 $$Bs0‐B¯s0mixing phaseβsas an external input. The measurement yieldsγ= (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$$ \gamma =\left({44}_{-13}^{+20}\right){}^{\circ} $$γ=44−13+20°modulo 180°. Moreover, the$$ {B}_s^0\hbox{-} {\overline{B}}_s^0 $$Bs0‐B¯s0oscillation frequency is measured from the flavour-specific control channel$$ {B}_s^0\to {D}_s^{-}{\pi}^{+}{\pi}^{+}{\pi}^{-} $$Bs0→Ds−π+π+π−to be ∆ms= (17.757 ± 0.007(stat) ± 0.008(syst)) ps−1, consistent with and more precise than the current world-average value.