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

The CKM angle $\gamma$ 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 ${\bar{B}}_s^0\to D_s^{\pm }{K}^{\mp }{\pi }^{\mp }{\pi }^{\pm }$ decays reconstructed in proton-proton collision data corresponding to an integrated luminosity of 9 ${\mathrm{f}\mathrm{b}}^{-1}$ recorded with the LHCb detector. A time-dependent amplitude analysis is performed to extract the $CP$-violating weak phase $\gamma -2{\beta }_s$ and, subsequently, $\gamma$ by taking the $B_s^0$-${\bar{B}}_s^0$ mixing phase ${\beta }_s$ as an external input. The measurement yields $\gamma =(44\pm 12{)}^{\circ }$ modulo ${180}^{\circ }$, where statistical and systematic uncertainties are combined. An alternative model-independent measurement, integrating over the five-dimensional phase space of the decay, yields $\gamma =(44_{-13}^{+20}{)}^{\circ }$ modulo ${180}^{\circ }$. Moreover, the $B_s^0$-${\bar{B}}_s^0$ oscillation frequency is measured from the flavour-specific control channel $B^0_s \rightarrow D_s^- \pi^+ \pi^+ \pi^-$tobe$\Delta m_s = (17.757 \pm 0.007 ,({\rm stat.}) \pm 0.008 ,({\rm syst.})) $\text{ps}$^{-1}$, consistent with and more precise than the current world-average value.