The CKM angle $\gamma$ is measured for the first time from mixinginduced $CP$ violation between $B^0_s \rightarrow D_s^\mp K^\pm \pi^\pm \pi^\mp$ and $\bar{B}^0_s \rightarrow D_s^\pm K^\mp \pi^\mp \pi^\pm$ decays reconstructed in protonproton collision data corresponding to an integrated luminosity of 9 ${\rm fb}^{1}$ recorded with the LHCb detector. A timedependent amplitude analysis is performed to extract the $CP$violating weak phase $\gamma2\beta_s$ and, subsequently, $\gamma$ by taking the $B^0_s$$\bar{B}^0_s$ mixing phase $\beta_{s}$ as an external input. The measurement yields $\gamma = (44 \pm 12)^\circ$ modulo $180^\circ$, where statistical and systematic uncertainties are combined. An alternative modelindependent measurement, integrating over the fivedimensional phase space of the decay, yields $\gamma = (44^{ + 20}_{  13})^\circ$ modulo $180^\circ$. Moreover, the $B^0_s$$\bar{B}^0_s$ oscillation frequency is measured from the flavourspecific control channel $B^0_s \rightarrow D_s^ \pi^+ \pi^+ \pi^$ to be $\Delta m_s = (17.757 \pm 0.007 ({\rm stat.}) \pm 0.008 ({\rm syst.})) \text{ps}^{1}$, consistent with and more precise than the current worldaverage value.
Leadingorder Feynman diagrams for (left) $ B ^0_ s $ and (right) $\overline{ B } {}^0_ s $ decays to the $D_s^ K^+ \pi ^+ \pi ^ $ final state, where the $\pi ^+ \pi ^ $ subsystem is generically drawn in conjunction with the kaon. 
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Invariant mass distribution of selected (left) $ B ^0_ s \rightarrow D ^_ s \pi ^+ \pi ^+ \pi ^ $ and (right) $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ candidates with fit projections overlaid. 
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Backgroundsubtracted decaytime distribution of (top) all and (bottom left) tagged $ B ^0_ s \rightarrow D ^_ s \pi ^+ \pi ^+ \pi ^ $ candidates as well as (bottom right) the dilutionweighted mixing asymmetry folded into one oscillation period along with the fit projections (solid lines). The decaytime acceptance (top) is overlaid in an arbitrary scale (dashed line). 
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Decaytime distribution of (left) backgroundsubtracted $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ candidates and (right) dilutionweighted mixing asymmetry along with the modelindependent fit projections (solid lines). The decaytime acceptance (left) is overlaid in an arbitrary scale (dashed line). 
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Decaytime distribution of (left) backgroundsubtracted $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ candidates and (right) dilutionweighted mixing asymmetry along with the modeldependent fit projections (solid lines). The decaytime acceptance (left) is overlaid in an arbitrary scale (dashed line). 
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Invariantmass distribution of backgroundsubtracted $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ candidates (data points) and fit projections (blue solid line). Contributions from $b\rightarrow c$ and $b\rightarrow u$ decay amplitudes are overlaid. 
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Invariantmass distribution of backgroundsubtracted $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ candidates (data points) and fit projections (blue solid line). Incoherent contributions from intermediatestate components are overlaid. 
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The 1$$CL contours for the physical observables $r,\kappa,\delta$ and $\gamma2\beta_s$ obtained with the modelindependent fit. 
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Running width distributions of the threebody resonances included in the baseline model for $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ decays: (top left) $K_1(1270)^+$, (top right) $K_1(1400)^+$, (bottom left) $K^*(1410)^+$ and (bottom right) $K(1460)^+$. 
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Invariantmass and angular distributions of backgroundsubtracted $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ candidates (data points) and fit projections (blue solid line). Contributions from $b\rightarrow c$ and $b\rightarrow u$ decay amplitudes are overlaid, colour coded as in Fig. ???. 
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Invariantmass and angular distributions of backgroundsubtracted $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ candidates (data points) and fit projections (blue solid line). Incoherent contributions from intermediatestate components are overlaid, colour coded as in Fig. ???. 
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Visualisation of how the $ C P$ coefficients contribute towards the overall constraint on the weak phase, $\gamma  2\beta_s$. The difference between the phase of $(A_f^{\Delta\Gamma},S_f)$ and $(A_{\bar{f}}^{\Delta\Gamma},S_{\bar{f}})$ is proportional to the strong phase $\delta$, which is close to $0 ^{\circ} $ and thus not indicated in the figure. 
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Animated gif made out of all figures. 
PAPER2020030.gif Thumbnail 
The flavourtagging performance for only OStagged, only SStagged and both OS and SStagged $ B ^0_ s \rightarrow D ^_ s \pi ^+ \pi ^+ \pi ^ $ signal candidates. 
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$ C P$ coefficients determined from the phasespace fit to the $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ decaytime distribution. The uncertainties are statistical and systematic (discussed in Sec. ???). 
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Decay fractions of the intermediatestate amplitudes contributing to decays via $b \rightarrow c$ and $b \rightarrow u$ quarklevel transitions. The uncertainties are statistical, systematic and due to alternative amplitude models considered. 
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Systematic uncertainties on the $ B ^0_ s $ mixing frequency determined from the fit to $ B ^0_ s \rightarrow D ^_ s \pi ^+ \pi ^+ \pi ^ $ signal candidates and on the fit parameters of the phasespace integrated fit to $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ signal candidates in units of the statistical standard deviations. 
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Systematic uncertainties on the physical observables and resonance parameters determined from the full timedependent amplitude fit to $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ data in units of the statistical standard deviations. The systematic uncertainties for the amplitude coefficients are given in Table ???. 
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Parameters determined from the modelindependent and modeldependent fits to the $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ signal candidates. The uncertainties are statistical, systematic and (if applicable) due to alternative amplitude models considered. The angles are given modulo $180 ^{\circ} $. 
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Parameters of the resonances included in the $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ baseline model. 
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Intermediatestate components considered for the $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ LASSO model building procedure. The letters in square brackets and subscripts refer to the relative orbital angular momentum of the decay products in spectroscopic notation. If no angular momentum is specified, the lowest angular momentum state compatible with angular momentum conservation and, where appropriate, parity conservation, is used. 
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Moduli and phases of the amplitude coefficients for decays via $b \rightarrow c$ and $b \rightarrow u$ quarklevel transitions. The uncertainties are statistical and systematic. 
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Moduli and phases of the amplitude coefficients for cascade decays. The amplitude coefficients are defined relative to the respective threebody production amplitude coefficients in Table ??? and are shared among $b \rightarrow c$ and $b \rightarrow u$ transitions. The uncertainties are statistical and systematic. 
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Systematic uncertainties on the fit parameters of the full timedependent amplitude fit to $ B ^0_ s \rightarrow D ^{\mp}_ s K ^\pm \pi ^\pm \pi ^\mp $ data in units of the statistical standard deviations. The different contributions are: 1) fit bias, 2) background subtraction, 3) correlation of observables, 4) time acceptance, 5) resolution, 6) decaytime bias, 7) nuisance asymmetries, 8) $\Delta m_s$, 9) phasespace acceptance, 10) acceptance factorisation, 11) lineshape models, 12) masses and widths of resonances, 13) form factor. 
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Amplitude ratio and strongphase difference for a given decay channel. The uncertainties are statistical and systematic. 
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Interference fractions (ordered by magnitude) of the $b\rightarrow c$ intermediatestate amplitudes included in the baseline model. Only the statistical uncertainties are given. 
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Interference fractions (ordered by magnitude) of the $b\rightarrow u$ intermediatestate amplitudes included in the baseline model. Only the statistical uncertainties are given. 
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Decay fractions in percent for several alternative amplitude models (Alt. 1  Alt. 6). Resonance parameters and the observables $r, \kappa, \delta, \gamma  2 \beta_s$ are also given. 
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Decay fractions in percent for several alternative amplitude models (Alt. 7  Alt. 12). Resonance parameters and the observables $r, \kappa, \delta, \gamma  2 \beta_s$ are also given. 
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Statistical correlation of the $ C P$ coefficients. 
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Supplementary material full pdf 
supple[..].pdf [121 KiB] 

This ZIP file contains supplemetary material for the publication LHCbPAPER2020030. The files are: Supplementary.pdf : An overview of the extra figures *.pdf, *.png, *.eps, *.C : The figures in variuous formats 
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Created on 22 January 2021.