EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2020-214 LHCb-PAPER-2020-030 April 12, 2021

Measurement of the CKM angle $\gamma$

and ${B}^{0}_{s}$-${\kern 3.73305pt\overline{\kern-3.73305ptB}}{}^{0}_{s}$
mixing frequency

with
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}h^{\pm}{{\pi}^{\pm}}{{\pi}^{\mp}}$
decays

LHCb collaboration†††Authors are listed at the end of this paper.

The CKM angle $\gamma$ is measured for the first time from mixing-induced
$C\\!P$ violation between
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
and ${{\kern
1.61993pt\overline{\kern-1.61993ptB}}{}^{0}_{s}}\\!\rightarrow{{D}^{\pm}_{s}}{{K}^{\mp}}{{\pi}^{\mp}}{{\pi}^{\pm}}$
decays reconstructed in proton-proton collision data corresponding to an
integrated luminosity of $9\text{\,fb}^{-1}$ recorded with the LHCb detector.
A time-dependent amplitude analysis is performed to extract the
$C\\!P$-violating weak phase $\gamma-2\beta_{s}$ and, subsequently, $\gamma$
by taking the ${{B}^{0}_{s}}-{{\kern
1.61993pt\overline{\kern-1.61993ptB}}{}^{0}_{s}}$ 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 model-independent measurement, integrating over the five-
dimensional phase space of the decay, yields
$\gamma=(44^{\,+\,20}_{\,-\,13})^{\circ}$ modulo $180^{\circ}$. Moreover, the
${{B}^{0}_{s}}-{{\kern 1.61993pt\overline{\kern-1.61993ptB}}{}^{0}_{s}}$
oscillation frequency is measured from the flavour-specific control channel
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ to
be $\Delta m_{s}=(17.757\pm 0.007\text{\,(stat)}\pm
0.008\text{\,(syst)})\text{\,ps}^{-1}$, consistent with and more precise than
the current world-average value.

Published in JHEP 03 (2021) 137

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.

##  1 Introduction

Within the Standard Model of particle physics, the charge-parity ($C\\!P$)
symmetry between quarks and antiquarks is broken by a single complex phase in
the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix [1,
*Kobayashi:1973fv]. The unitarity of this matrix leads to the condition
$V_{ud}^{\phantom{*}}V^{*}_{ub}+V_{cd}^{\phantom{*}}V^{*}_{cb}+V_{td}^{\phantom{*}}V^{*}_{tb}=0$,
where $V_{ij}$ are the complex elements of the CKM matrix. This equation can
be visualised as a triangle in the complex plane with angles $\alpha$, $\beta$
and $\gamma$. It is a key test of the Standard Model to verify this unitarity
condition by over-constraining the CKM matrix with independent measurements
sensitive to various distinct combinations of matrix elements. In particular,
measurements of
$\gamma\equiv\text{arg}[-(V_{ud}^{\phantom{*}}V_{ub}^{*})/(V_{cd}^{\phantom{*}}V_{cb}^{*})]$
in tree-level decays provide an important benchmark of the Standard Model, to
be compared with loop-level measurements of $\gamma$ and other CKM parameters.
The world-average value,
$\gamma=\left(72.1^{\,+\,4.1}_{\,-\,4.5}\right)^{\circ}$ [3, 4], is dominated
by a combination of LHCb measurements obtained from analyses of beauty meson
decays to open-charm final states [5]. Improved direct measurements are needed
to set a conclusive comparison to the indirect predictions from global CKM
fits, $\gamma=\left(65.7^{\,+\,0.9}_{\,-\,2.7}\right)^{\circ}$ [6] or
$\gamma=\left(65.8\pm 2.2\right)^{\circ}$ [7].

This paper presents the first measurement of the CKM angle $\gamma$ with
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
decays.111Inclusion of charge-conjugate modes is implied throughout except
where explicitly stated. The data set is collected with the LHCb experiment in
proton-proton ($pp$) collisions at centre-of-mass energies222Natural units
with $\hbar=c=1$ are used throughout the paper. of $7,8$ and
$13\text{\,Te\kern-1.00006ptV}$, corresponding to an integrated luminosity of
$9\text{\,fb}^{-1}$. In these decays, the interference between
${b}\rightarrow{c}$ and ${b}\rightarrow{u}$ quark-level transitions achieved
through ${{B}^{0}_{s}}-{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$ mixing provides sensitivity
to the $C\\!P$-violating weak phase $\gamma-2\beta_{s}$ [8, 9]. The mixing
phase, $\beta_{s}$, is well constrained from
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}K^{+}K^{-}$ [10,
11] and related decays [12, 13, 14, 15] and is taken as an external input. The
leading-order Feynman diagrams for ${{B}^{0}_{s}}\rightarrow
D_{s}^{-}K^{+}{{\pi}^{+}}{{\pi}^{-}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow
D_{s}^{-}K^{+}{{\pi}^{+}}{{\pi}^{-}}$ decays are shown in Fig. 1. The
amplitudes for both processes are of the same order in the Wolfenstein
parameter $\lambda$[16], $\mathcal{O}(\lambda^{3})$, so that interference
effects are expected to be large. To account for the strong-phase variation
across the phase space of the decay, a time-dependent amplitude analysis is
performed. An alternative, model-independent approach analyses the phase-space
integrated decay-time spectrum and is pursued as well; this method is
conceptually similar to the analysis of
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ decays [17, 18].
However, a coherence factor needs to be introduced as an additional hadronic
parameter, which dilutes the observable $C\\!P$ asymmetry since constructive
and destructive interference effects cancel when integrated over the entire
phase space. The topologically similar and flavour-specific decay
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ is
used to calibrate detector-induced effects. This mode is also employed to make
a precise measurement of the ${{B}^{0}_{s}}-{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$ mixing frequency, which can
be related to one side of the unitarity triangle. The relative branching
fraction of these decay modes was measured by LHCb to be
$\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}})/\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}})=\left(5.2\pm
0.5(\text{stat})\pm 0.3(\text{syst})\right)\%$ [19].

The paper is structured as follows. After introducing the amplitude analysis
formalism and the differential decay rates in Sec. 2, the LHCb detector, the
event reconstruction and candidate selection are described in Sec. 3. Section
4 presents the measurement of the ${{B}^{0}_{s}}$ mixing frequency, followed
by the analysis of the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
signal channel in Sec. 5. Experimental and model-dependent systematic
uncertainties are evaluated in Sec. 6, the results are discussed in Sec. 7,
and our conclusions are given in Sec. 8.

$s$$s$$s$$\bar{b}$$\bar{c}$$W^{+}$$\bar{s}$$u$$K^{+}\pi^{+}\pi^{-}$$D_{s}^{-}$${B}^{0}_{s}$$\bar{s}$$\bar{s}$$b$$u$$W^{-}$$\bar{c}$$s$$D_{s}^{-}$$K^{+}\pi^{+}\pi^{-}$${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$

Figure 1: Leading-order Feynman diagrams for (left) ${B}^{0}_{s}$ and (right)
${\kern 1.61993pt\overline{\kern-1.61993ptB}}{}^{0}_{s}$ decays to the
$D_{s}^{-}K^{+}{{\pi}^{+}}{{\pi}^{-}}$ final state, where the
${{\pi}^{+}}{{\pi}^{-}}$ subsystem exemplarily hadronises in conjunction with
the kaon.

##  2 Phenomenology of the decay

Assuming tree-level processes are dominant, transitions from ${B}^{0}_{s}$ and
${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ flavour eigenstates
to the final state $f=D_{s}^{-}K^{+}{{\pi}^{+}}{{\pi}^{-}}$ are described by
decay amplitudes

$\displaystyle\langle f|{{B}^{0}_{s}}\rangle$ $\displaystyle\equiv
A^{c}(\mathbf{x}),$ $\displaystyle\langle f|{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rangle$ $\displaystyle\equiv
r\,e^{i(\delta-\gamma)}\,A^{u}(\mathbf{x}),$ (1)

with a relative magnitude $r$ and (constant) strong- and weak-phase
differences $\delta$ and $\gamma$, respectively. A set of five independent
kinematic observables (e.g. invariant-mass combinations of the final state
particles or helicity angles) fully describes the phase space $\mathbf{x}$ of
the decay. The hadronic amplitudes $A^{c}(\mathbf{x})$ and
$A^{u}(\mathbf{x})$, where the superscript $c$ ($u$) refers to a $b\rightarrow
c$ ($b\rightarrow u$) quark-level transition, contain the strong-interaction
dynamics and are given by a coherent sum over intermediate-state amplitudes,
$A_{i}(\mathbf{x})$:

$\displaystyle A^{c}(\mathbf{x})=$
$\displaystyle\sum_{i}a_{i}^{c}\,A_{i}(\mathbf{x}),$ $\displaystyle
A^{u}(\mathbf{x})$ $\displaystyle=\sum_{i}a_{i}^{u}\,A_{i}(\mathbf{x}).$ (2)

The complex amplitude coefficients $a_{i}^{c}$ and $a_{i}^{u}$ need to be
determined from data. Since the hadronisation process is different for
${{B}^{0}_{s}}\rightarrow f$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow f$ decays, their
respective amplitude coefficients are distinct ($a_{i}^{c}\neq a_{i}^{u}$). To
ensure that the parameters $r$ and $\delta$ do not depend on the convention
employed for the amplitude coefficients, the magnitude squared of the hadronic
amplitudes is normalised to unity when integrated over the phase space (with
four-body phase-space element $\text{d}\Phi_{4}$) and the overall strong-phase
difference between $A^{c}(\mathbf{x})$ and $A^{u}(\mathbf{x})$ is set to zero,
i.e.
$\int|A^{c}(\mathbf{x})|^{2}\,\text{d}\Phi_{4}(\mathbf{x})=\int|A^{u}(\mathbf{x})|^{2}\,\text{d}\Phi_{4}(\mathbf{x})=1$
and $\arg\left(\int
A^{c}(\mathbf{x})^{*}\,A^{u}(\mathbf{x})\,\text{d}\Phi_{4}(\mathbf{x})\right)=0$.
Within this convention, the decay fractions and interference fractions for
$b\rightarrow c$ ($b\rightarrow u$) transitions are defined as

$\displaystyle F^{c(u)}_{i}$
$\displaystyle\equiv\int\left|a_{i}^{c(u)}\,A_{i}(\mathbf{x})\right|^{2}\,\text{d}\Phi_{4},$
$\displaystyle I_{ij}^{c(u)}\equiv\int
2\,\text{Re}[a_{i}^{c(u)}\,a^{c(u)*}_{j}\,A_{i}(\mathbf{x})A_{j}^{*}(\mathbf{x})]\,\text{d}\Phi_{4}.$
(3)

###  2.1 Amplitude formalism

The isobar model is used to construct the intermediate-state amplitudes
$A_{i}(\mathbf{x})$[20, 21, 22]. Within this model, the four-body decay
${{B}^{0}_{s}}\rightarrow h_{1}\,h_{2}\,h_{3}\,h_{4}$ proceeds via two isobar
states $R_{1}$ and $R_{2}$ (typically associated to intermediate resonances),
which gives rise to two distinct decay topologies; quasi-two-body decays
${{B}^{0}_{s}}\rightarrow(R_{1}\rightarrow h_{1}\,h_{2})\,(R_{2}\rightarrow
h_{3}\,h_{4})$ or cascade decays ${{B}^{0}_{s}}\rightarrow
h_{1}\,\left[R_{1}\rightarrow h_{2}\,(R_{2}\rightarrow h_{3}\,h_{4})\right]$.
The intermediate-state amplitude for a given decay channel $i$ can be
parameterised as

$A_{i}(\mathbf{x})=B_{L_{{{B}^{0}_{s}}}}(\mathbf{x})\,[B_{L_{R_{1}}}(\mathbf{x})\,T_{R_{1}}(\mathbf{x})]\,[B_{L_{R_{2}}}(\mathbf{x})\,T_{R_{2}}(\mathbf{x})]\,S_{i}(\mathbf{x})\,,$
(4)

where the form factors $B_{L}$ account for deviations from point-like
interactions and the propagator $T_{R}$ describes the lineshape of resonance
$R$. The angular correlation of the final state particles subject to total
angular momentum conservation is encoded in the spin factor $S$. The
Blatt–Weisskopf penetration factors [23, 24] are used as form factors. They
depend on the effective interaction radius $r_{\rm BW}$, the break-up momentum
$b$ and the orbital angular momentum $L$ between the final-state particles.
The explicit expressions for $L=\\{0,1,2\\}$ are

$\displaystyle B_{0}(b)$ $\displaystyle=1,$ $\displaystyle B_{1}(b)$
$\displaystyle=1/\sqrt{{1+(b\,r_{\rm BW})^{2}}}\,\,\,\text{and}$
$\displaystyle B_{2}(b)$ $\displaystyle=1/\sqrt{9+3\,(b\,r_{\rm
BW})^{2}+(b\,r_{\rm BW})^{4}}.$ (5)

Resonance lineshapes are described by Breit–Wigner propagators,

$T_{R}(s)=(m_{0}^{2}-s-i\,\sqrt{s}\,\Gamma(s))^{-1},$ (6)

where $s$ is the square of the centre-of-mass energy. The energy-dependent
decay width, $\Gamma(s)$, is normalised to give the nominal width,
$\Gamma_{0}$, when evaluated at the nominal mass, $m_{0}$. For a decay into
two stable particles $R\rightarrow AB$, the energy dependence of the decay
width can be described by

$\sqrt{s}\,\Gamma_{R\rightarrow
AB}(s)=m_{0}\,\Gamma_{0}\,\frac{m_{0}}{\sqrt{s}}\,\left(\frac{b}{b_{0}}\right)^{2L+1}\,\frac{B_{L}(b)^{2}}{B_{L}(b_{0})^{2}}\,,$
(7)

where $b_{0}$ is the value of the break-up momentum at the resonance pole
[25]. Specialised lineshape parameterisations are used for the
$f_{0}(500)^{0}$ (Bugg [26]), $K_{0}^{*}(1430)^{0}$ (LASS [27, 28]) and
$\rho(770)^{0}$ (Gounaris–Sakurai [29]) resonances. The lineshapes for non-
resonant states are set to a constant.

The energy-dependent width for a three-body decay $R\rightarrow ABC$ is
computed by integrating the squared transition amplitude over the phase space,

$\Gamma_{R\rightarrow ABC}(s)=\frac{1}{2\,\sqrt{s}}\,\int|A_{R\rightarrow
ABC}|^{2}\,\text{d}\Phi_{3},$ (8)

as described in Ref. [30]. For the
$K_{1}(1270)^{+}\rightarrow\rho(770)^{0}K^{+}$ cascade decay chain, mixing
between the $\rho(770)^{0}$ and $\omega(782)^{0}$ states is included [31],
with relative magnitude and phase fixed to the values determined in Ref. [32].
More details are given in Appendix A. The spin factors are constructed in the
covariant Zemach (Rarita–Schwinger) tensor formalism [33, 34, 35, 36, 37]. The
explicit expressions for the decay topologies relevant for this analysis are
taken from Refs. [30, 38].

###  2.2 Decay rates

Since ${B}^{0}_{s}$ mesons can convert into ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ and vice versa, the flavour
eigenstates are an admixture of the physical mass eigenstates $B_{L}$ and
$B_{H}$,

$\displaystyle|B_{L}\rangle$ $\displaystyle=p|{{B}^{0}_{s}}\rangle+q|{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rangle,$
$\displaystyle|B_{H}\rangle$ $\displaystyle=p|{{B}^{0}_{s}}\rangle-q|{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rangle,$ (9)

where the complex coefficients are normalised such that $|p|^{2}+|q|^{2}=1$.
The light, $|B_{L}\rangle$, and heavy, $|B_{H}\rangle$, mass eigenstates have
distinct masses, $m_{L}$ and $m_{H}$, and decay widths, $\Gamma_{L}$ and
$\Gamma_{H}$. Their arithmetic means (differences) are denoted as $m_{s}$ and
$\Gamma_{s}$ ($\Delta m_{s}=m_{H}-m_{L}$ and
$\Delta\Gamma_{s}=\Gamma_{L}-\Gamma_{H}$). The time evolution of the flavour
and mass eigenstates can be described by an effective Schrödinger equation
[39, 40] resulting in the following differential decay rates for initially
produced ${{B}^{0}_{s}}$ or ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$ mesons [41, 40, 42]:

$\footnotesize\begin{split}\frac{\text{d}\Gamma({{B}^{0}_{s}}\rightarrow
f)}{\text{d}t\,\text{d}\Phi_{4}(\mathbf{x})}=|\langle
f|{{B}^{0}_{s}}(t)\rangle|^{2}\propto
e^{-\Gamma_{s}\,t}&\Biggl{[}\left(|A^{c}(\mathbf{x})|^{2}+r^{2}\,|A^{u}(\mathbf{x})|^{2}\right)\,\text{cosh}\left(\frac{\Delta\Gamma_{s}\,t}{2}\right)\\\
&+\left(|A^{c}(\mathbf{x})|^{2}-r^{2}\,|A^{u}(\mathbf{x})|^{2}\right)\,\text{cos}\left(\Delta
m_{s}\,t\right)\\\
&-2\,r\,\text{Re}\left(A^{c}(\mathbf{x})^{*}\,A^{u}(\mathbf{x})\,e^{i\left(\delta-(\gamma-2\beta_{s})\right)}\right)\,\text{sinh}\left(\frac{\Delta\Gamma_{s}\,t}{2}\right)\\\
&-2\,r\,\text{Im}\left(A^{c}(\mathbf{x})^{*}\,A^{u}(\mathbf{x})\,e^{i\left(\delta-(\gamma-2\beta_{s})\right)}\right)\,\text{sin}\left(\Delta
m_{s}\,t\right)\Biggr{]},\\\ \frac{\text{d}\Gamma({{\kern
1.43994pt\overline{\kern-1.43994ptB}}{}^{0}_{s}}\rightarrow
f)}{\text{d}t\,\text{d}\Phi_{4}(\mathbf{x})}=|\langle f|{{\kern
1.43994pt\overline{\kern-1.43994ptB}}{}^{0}_{s}}(t)\rangle|^{2}\propto
e^{-\Gamma_{s}\,t}&\Biggl{[}\left(|A^{c}(\mathbf{x})|^{2}+r^{2}\,|A^{u}(\mathbf{x})|^{2}\right)\,\text{cosh}\left(\frac{\Delta\Gamma_{s}\,t}{2}\right)\\\
&-\left(|A^{c}(\mathbf{x})|^{2}-r^{2}\,|A^{u}(\mathbf{x})|^{2}\right)\,\text{cos}\left(\Delta
m_{s}\,t\right)\\\
&-2\,r\,\text{Re}\left(A^{c}(\mathbf{x})^{*}\,A^{u}(\mathbf{x})\,e^{i\left(\delta-(\gamma-2\beta_{s})\right)}\right)\,\text{sinh}\left(\frac{\Delta\Gamma_{s}\,t}{2}\right)\\\
&+2\,r\,\text{Im}\left(A^{c}(\mathbf{x})^{*}\,A^{u}(\mathbf{x})\,e^{i\left(\delta-(\gamma-2\beta_{s})\right)}\right)\,\text{sin}\left(\Delta
m_{s}\,t\right)\Biggr{]}.\\\ \end{split}$ (10)

Here, the magnitude of $q/p$ is set to unity (i.e. no $C\\!P$ violation in
mixing is assumed [4, 43, 44]) and the phase between $q$ and $p$ can be
related to the mixing phase $\beta_{s}$, $\arg(q/p)\approx-2\beta_{s}$ [41,
42]. The decay rates to the $C\\!P$-conjugate final state
$\bar{f}=D_{s}^{+}K^{-}{{\pi}^{-}}{{\pi}^{+}}$ (with phase-space point
$\overline{\mathbf{x}}\equiv CP\,\mathbf{x}$), $\text{d}\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow\bar{f})$ and
$\text{d}\Gamma({{B}^{0}_{s}}\rightarrow\bar{f})$, follow from the expressions
for $\text{d}\Gamma({{B}^{0}_{s}}\rightarrow f)$ and $\text{d}\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow f)$ in Eq. (10) by
replacing $A^{c}(\mathbf{x})\rightarrow A^{c}(\overline{\mathbf{x}})$,
$A^{u}(\mathbf{x})\rightarrow A^{u}(\overline{\mathbf{x}})$ and
$-(\gamma-2\beta_{s})\rightarrow+(\gamma-2\beta_{s})$. This assumes no $C\\!P$
violation in the hadronic decay amplitudes (i.e., $a_{i}^{c}=\bar{a}_{i}^{c}$
and $a_{i}^{u}=\bar{a}_{i}^{u}$), as expected for tree-level-dominated decays.

It is also instructive to examine the decay rates as functions of the decay
time only, by marginalising the phase space

$\small\begin{split}\frac{\text{d}\Gamma({{B}^{0}_{s}}\rightarrow
f)}{\text{d}t}&\propto\Biggl{[}\text{cosh}\left(\frac{\Delta\Gamma_{s}\,t}{2}\right)+\,C_{f}\,\text{cos}\left(\Delta
m_{s}\,t\right)\\\
&\phantom{\propto[}+A^{\Delta\Gamma}_{f}\,\text{sinh}\left(\frac{\Delta\Gamma_{s}\,t}{2}\right)-\,\,S_{f}\,\text{sin}\left(\Delta
m_{s}\,t\right)\Biggr{]}e^{-\Gamma_{s}\,t},\\\ \frac{\text{d}\Gamma({{\kern
1.61993pt\overline{\kern-1.61993ptB}}{}^{0}_{s}}\rightarrow
f)}{\text{d}t}&\propto\Biggl{[}\text{cosh}\left(\frac{\Delta\Gamma_{s}\,t}{2}\right)-\,C_{f}\,\text{cos}\left(\Delta
m_{s}\,t\right)\\\
&\phantom{\propto[}+A^{\Delta\Gamma}_{f}\,\text{sinh}\left(\frac{\Delta\Gamma_{s}\,t}{2}\right)+\,\,S_{f}\,\text{sin}\left(\Delta
m_{s}\,t\right)\Biggr{]}e^{-\Gamma_{s}\,t}.\\\ \end{split}$ (11)

Analogous expressions for the $C\\!P$-conjugate processes can be written by
replacing $A^{\Delta\Gamma}_{f}$ with $A^{\Delta\Gamma}_{\bar{f}}$ and $S_{f}$
with $-S_{\bar{f}}$, where the $C\\!P$ coefficients are defined as

$\displaystyle C_{f}=\frac{1-r^{2}}{1+r^{2}},$ $\displaystyle
A^{\Delta\Gamma}_{f}=-\frac{2\,r\,\kappa\,\text{cos}\left(\delta-\,(\gamma-2\beta_{s})\right)}{1+r^{2}},$
$\displaystyle\phantom{\quad}A^{\Delta\Gamma}_{\bar{f}}=-\frac{2\,r\,\kappa\,\text{cos}\left(\delta+\,(\gamma-2\beta_{s})\right)}{1+r^{2}},$
$\displaystyle
S_{f}^{\phantom{\Delta\Gamma}}=\,+\,\frac{2\,r\,\kappa\,\text{sin}\left(\delta-\,(\gamma-2\beta_{s})\right)}{1+r^{2}},$
$\displaystyle\phantom{\quad}S_{\bar{f}}^{\phantom{\Delta\Gamma}}=\,-\,\frac{2\,r\,\kappa\,\text{sin}\left(\delta+\,(\gamma-2\beta_{s})\right)}{1+r^{2}}.$
(12)

The coherence factor, $\kappa$, results from the integration over the
interfering amplitudes across the phase space, $\kappa\equiv\int
A^{c}(\mathbf{x})^{*}\,A^{u}(\mathbf{x})\,\mathrm{d}\Phi_{4}$. It is bounded
between zero and unity, and dilutes the sensitivity to the weak phase. For the
two-body decay ${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ the
coherence factor is $\kappa=1$ [18]. Measured values of
$A^{\Delta\Gamma}_{f}\neq A^{\Delta\Gamma}_{\bar{f}}$ or $S_{f}\neq-
S_{\bar{f}}$ signify time-dependent $C\\!P$ violation and lead to different
mixing asymmetries for decays into the $f$ or $\bar{f}$ final states. These
mixing asymmetries are defined as [8, 9]

$\displaystyle A_{\text{mix}}^{f}(t)$
$\displaystyle=\frac{N_{f}(t)-\bar{N}_{f}(t)}{N_{f}(t)+\bar{N}_{f}(t)}=\frac{C_{f}\cos\left(\Delta
m_{s}\,t\right)-S_{f}\sin\left(\Delta
m_{s}\,t\right)}{\cosh\left(\frac{\Delta\Gamma_{s}\,t}{2}\right)+A^{\Delta\Gamma}_{f}\sinh\left(\frac{\Delta\Gamma_{s}\,t}{2}\right)},$
$\displaystyle A_{\text{mix}}^{\bar{f}}(t)$
$\displaystyle=\frac{\bar{N}_{\bar{f}}(t)-N_{\bar{f}}(t)}{\bar{N}_{\bar{f}}(t)+N_{\bar{f}}(t)}=\frac{C_{f}\cos\left(\Delta
m_{s}\,t\right)+S_{\bar{f}}\sin\left(\Delta
m_{s}\,t\right)}{\cosh\left(\frac{\Delta\Gamma_{s}\,t}{2}\right)+A^{\Delta\Gamma}_{\bar{f}}\sinh\left(\frac{\Delta\Gamma_{s}\,t}{2}\right)},$
(13)

where $N_{f}(t)$ ($\bar{N}_{f}(t)$) and $N_{\bar{f}}(t)$
($\bar{N}_{\bar{f}}(t)$) denote the number of initially produced
${{B}^{0}_{s}}$ (${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$)
mesons decaying at proper time $t$ to the final states $f$ and $\bar{f}$,
respectively. Flavour-specific decay modes such as
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
have $r=0$ and consequently $C_{f}=1$ as well as
$A^{\Delta\Gamma}_{f}=A^{\Delta\Gamma}_{\bar{f}}=S_{f}=S_{\bar{f}}=0$.

##  3 Event reconstruction

The LHCb detector [45, 46] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector (VELO) surrounding the
$pp$ interaction region [47], a large-area silicon-strip detector located
upstream of a dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$,
and three stations of silicon-strip detectors and straw drift tubes [48, 49]
placed downstream of the magnet. The polarity of the dipole magnet is reversed
periodically throughout the data-taking process to control systematic
asymmetries. The tracking system provides a measurement of the momentum, $p$,
of charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at 200 GeV. The minimum distance of a track to a primary
vertex (PV), the impact parameter (IP), is measured with a resolution of
$(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$ is the
component of the momentum transverse to the beam, in  GeV. Different types of
charged hadrons are distinguished using information from two ring-imaging
Cherenkov detectors [50]. The online event selection is performed by a trigger
[51], which consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which applies a
full event reconstruction. At the hardware trigger stage, events are required
to have a muon with high $p_{\mathrm{T}}$ or a hadron, photon or electron with
high transverse energy in the calorimeters. For hadrons, the transverse energy
threshold is 3.5 GeV. The software trigger requires a two-, three- or four-
track secondary vertex with a significant displacement from any primary $pp$
interaction vertex. At least one charged particle must have a transverse
momentum $p_{\mathrm{T}}>1.6\text{\,Ge\kern-1.00006ptV}$ and be inconsistent
with originating from a PV. A multivariate algorithm [52] is used for the
identification of secondary vertices consistent with the decay of a $b$
hadron. The data-taking period from 2011 to 2012 with centre-of-mass energies
of 7 and 8  TeV (2015 to 2018 with centre-of-mass energy of 13  TeV) is
referred to as Run 1 (Run 2) throughout the paper.

Simulated events are used to study the detector acceptance and specific
background contributions. In the simulation, $pp$ collisions are generated
using Pythia [53, *Sjostrand:2007gs] with a specific LHCb configuration [55].
Decays of hadrons are described by EvtGen [56], in which final-state radiation
is generated using Photos [57]. The simulated signal decays are generated
according to a simplified amplitude model with an additional pure phase-space
component. The interaction of the generated particles with the detector, and
its response, are implemented using the Geant4 toolkit [58,
*Agostinelli:2002hh] as described in Ref. [60].

###  3.1 Candidate selection

The selection of
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
and
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
candidates is performed by first reconstructing
${{D}^{-}_{s}}\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{-}}$,
${{D}^{-}_{s}}\rightarrow{{K}^{-}}{{\pi}^{-}}{{\pi}^{+}}$ and
${{D}^{-}_{s}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ candidates from
charged particle tracks with high momentum and transverse momentum originating
from a common displaced vertex. Particle identification (PID) information is
used to assign a kaon or pion hypothesis to the tracks. Candidate
${D}^{-}_{s}$ mesons with a reconstructed invariant mass within
$25\text{\,Me\kern-1.00006ptV}$ of the known ${D}^{-}_{s}$ mass [3] are
combined with three additional charged tracks to form a ${B}^{0}_{s}$ vertex,
which must be displaced from any PV. The PV with respect to which the
${B}^{0}_{s}$ candidate has the smallest impact parameter significance is
considered as the production vertex. The reconstructed invariant mass of the
${B}^{0}_{s}$ candidate is required to be between
$5200\text{\,Me\kern-1.00006ptV}$ and $5700\text{\,Me\kern-1.00006ptV}$. The
mass resolution is improved by performing a kinematic fit [61] where the
${B}^{0}_{s}$ candidate is constrained to originate from the PV and the
reconstructed ${{D}^{-}_{s}}$ mass is constrained to the world-average
${{D}^{-}_{s}}$ mass [3]. When deriving the decay time, $t$, of the
${B}^{0}_{s}$ candidate and the phase-space observables, $\mathbf{x}$, the
reconstructed ${B}^{0}_{s}$ mass is constrained to its known value [3]. The
${B}^{0}_{s}$ proper time is required to be larger than $0.4\text{\,ps}$ to
suppress most of the prompt combinatorial background. The considered phase-
space region is limited to
$m(K^{+}\pi^{+}\pi^{-})<1950\text{\,Me\kern-1.00006ptV}$,
$m(K^{+}\pi^{-})<1200\text{\,Me\kern-1.00006ptV}$ and
$m(\pi^{+}\pi^{-})<1200\text{\,Me\kern-1.00006ptV}$ since the decay proceeds
predominantly through the low-mass axial-vector states $K_{1}(1270)^{+}$ and
$K_{1}(1400)^{+}$ [19], while the combinatorial background is concentrated at
high $K^{+}\pi^{+}\pi^{-}$, $K^{+}\pi^{-}$ and $\pi^{+}\pi^{-}$ invariant
masses. A combination of PID information and kinematic requirements is used to
veto charmed meson or baryon decays reconstructed as ${D}^{-}_{s}$ candidates
due to the misidentification of protons or pions as kaons.

A boosted decision tree (BDT) [62, 63] with gradient boosting is used to
suppress background from random combinations of charged particles. The
multivariate classifier is trained using a background-subtracted
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
data sample as signal proxy, while
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
candidates with invariant mass greater than $5500\text{\,Me\kern-1.00006ptV}$
are used as background proxy. The features used in the BDT are topological
variables related to the vertex separation, such as the impact parameters of
the ${B}^{0}_{s}$ candidate and final-state particles, the flight distance of
the ${D}^{-}_{s}$ candidate with respect to the secondary vertex, as well as
several criteria on the track quality and vertex reconstruction and estimators
of the isolation of the ${B}^{0}_{s}$ candidate from other tracks in the
event. The working point of the BDT classifier is chosen to optimise the
significance of the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
signal.

###  3.2 Data sample composition

Irreducible background contributions to the selected
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
and
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
data samples are disentangled from signal decays on a statistical basis by
means of an extended maximum likelihood fit to the reconstructed
$m(D_{s}^{\mp}h^{\pm}\pi^{\pm}\pi^{\mp})$ invariant mass, where $h$ is either
a pion or a kaon. A Johnson’s $S_{\text{U}}$ function [64] is used as
probability density function (PDF) for the signal component. The shape
parameters are initially determined from simulation. To account for small
differences between simulation and data, scale factors for the mean and
standard deviation of the signal PDF are introduced. These are determined from
a fit to the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
calibration sample and thereafter fixed when fitting
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
candidates. Background decays of ${{B}^{0}}$ mesons are described by the same
PDF shifted by the known mass difference between ${{B}^{0}_{s}}$ and
${{B}^{0}}$ mesons [3]. The combinatorial background is modelled with a
second-order polynomial function. The shapes for partially reconstructed
${{B}^{0}_{s}}\rightarrow{{D}^{*-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$,
${{B}^{0}_{s}}\rightarrow{{D}^{*-}_{s}}{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}}\rightarrow{{D}^{*-}_{s}}{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decays,
where the ${{D}^{*-}_{s}}$ meson decays to ${{D}^{-}_{s}}\gamma$ or
${{D}^{-}_{s}}{{\pi}^{0}}$, are derived from simulated decays. The same
applies to the shape for misidentified
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
and ${{B}^{0}_{s}}\rightarrow{{D}^{*-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
decays contributing to the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
sample. The expected yields of these cross-feed background contributions are
estimated by determining the probability of a pion to pass the PID requirement
imposed on the kaon candidate from a control sample of
$D^{*+}\rightarrow\left(D^{0}\rightarrow K^{-}{{\pi}^{+}}\right){{\pi}^{+}}$
decays [65]. All other yields are determined from the fit.

Figure 2 displays the invariant mass distributions of
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
and
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
candidates with fit projections overlaid. A signal yield of $148\,000\pm 400$
($7500\pm 100$) is obtained for
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
(${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$)
decays. The results are used to assign weights to the candidates to
statistically subtract the background with the sPlot technique [66]. Here, the
$m(D_{s}^{\mp}h^{\pm}\pi^{\pm}\pi^{\mp})$ invariant mass is used as
discriminating variable when performing fits to the decay-time and phase-space
distributions [67].

Figure 2: 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.

##  4 Measurement of the ${B}^{0}_{s}$ mixing frequency

A likelihood fit to the background-subtracted decay-time spectrum of the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
control channel is performed in order to calibrate detector-induced effects
and flavour-tagging algorithms as well as to measure the mixing frequency
$\Delta m_{s}$.

###  4.1 Decay-time resolution

Excellent decay-time resolution is essential in order to resolve the fast
${{B}^{0}_{s}}-{{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$
mixing. The global kinematic fit to the decay topology provides an estimate of
the decay-time resolution for each candidate. The per-candidate decay-time
uncertainty, $\delta_{t}$, is calibrated by reconstructing ${{B}^{0}_{s}}$
candidates from particles originating directly from the PV. These prompt
${{B}^{0}_{s}}$ candidates have a known true decay time of zero, such that the
width of the decay-time distribution is a measure of the true resolution,
$\sigma_{t}$. It is determined in equally populated slices of $\delta_{t}$. A
linear calibration function is used to map the per-candidate decay-time
uncertainty to the actual resolution. On average, the resolution amounts to
$\langle\sigma_{t}\rangle=36.6\pm 0.5\text{\,fs}$, where the uncertainty is
statistical only. A decay-time bias, $\mu_{t}$, of approximately
$-2\text{\,fs}$ is observed which results from the applied selection
requirements and the precision of the detector alignment. The implementation
of the decay-time bias and decay-time resolution in the fit are discussed in
Sec. 4.3.

###  4.2 Flavour tagging

Two complementary methods are used to determine the flavour of the
${{B}^{0}_{s}}$ candidates at production. The opposite-side (OS) tagger [68,
69] exploits the fact that $b$ quarks are predominantly produced in quark-
antiquark pairs, which leads to a second $b$ hadron alongside the signal
${{B}^{0}_{s}}$ meson. The flavour of the non-signal $b$ hadron is determined
using the charge of the lepton ($\mu,e$) produced in semileptonic decays, the
charge of a reconstructed secondary charm hadron, the charge of the kaon from
the $b\rightarrow c\rightarrow s$ decay chain, and/or the charge of the
inclusive secondary vertex reconstructed from the $b$ hadron decay products.
The same-side (SS) tagger [70] determines the flavour of the signal candidate
by identifying the charge of the kaon produced together with the
${{B}^{0}_{s}}$ meson in the fragmentation process.

Each tagging algorithm provides a flavour-tagging decision, $d$, which takes
the value $d=+1$ ($d=-1$) for a candidate tagged as a ${B}^{0}_{s}$ (${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$) meson and $d=0$ if no
decision can be made (untagged). The tagging efficiency
$\epsilon_{\text{tag}}$ is defined as the fraction of selected candidates with
non-zero tag decision. The tagging algorithms also provide an estimate,
$\eta$, of the probability that the decision is incorrect. The tagging
decision and mistag estimate are obtained using flavour specific, self-
tagging, decays. Multivariate classifiers are employed combining various
inputs such as kinematic variables of tagging particles and of the signal
candidate. These are trained on simulated samples of
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ decays for the SS tagger
and on data samples of
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ decays for the
$\mathrm{OS}$ tagger. The mistag estimate does not necessarily represent the
true mistag probability, $\omega$, since the algorithms might perform
differently on the signal decay than on the decay modes used for the training.
Therefore, it is calibrated for each tagger using the flavour-specific decay
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$.

###  4.3 Decay-time fit

The signal PDF describing the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
proper-time spectrum is based on the theoretical decay rate in Eq. (11) taking
several experimental effects into account [17, 18, 71, 72]

$\displaystyle\mathcal{P}(t,d_{\mathrm{OS}},d_{\mathrm{SS}},q_{f}|\delta_{t},\eta_{\mathrm{OS}},\eta_{\mathrm{SS}})\propto$
$\displaystyle\left[(1-q_{f}A_{D})\,p(t^{\prime},d_{\mathrm{OS}},d_{\mathrm{SS}},q_{f}|\delta_{t},\eta_{\mathrm{OS}},\eta_{\mathrm{SS}})\,e^{-\Gamma_{s}\,t^{\prime}}\otimes
R(t-t^{\prime}|\mu_{t},\sigma_{t}(\delta_{t}))\right]\,\epsilon(t),$

where

$\displaystyle
p(t^{\prime},d_{\mathrm{OS}},d_{\mathrm{SS}},q_{f}|\delta_{t},\eta_{\mathrm{OS}},\eta_{\mathrm{SS}})\equiv$
$\displaystyle\Biggl{[}(1-A_{P})h(d_{\mathrm{OS}},\eta_{\mathrm{OS}})h(d_{\mathrm{SS}},\eta_{\mathrm{SS}})\biggl{(}\text{cosh}\left(\frac{\Delta\Gamma_{s}\,t^{\prime}}{2}\right)+\,q_{f}\,\text{cos}\left(\Delta
m_{s}\,t^{\prime}\right)\biggr{)}\Biggr{.}$
$\displaystyle\Biggl{.}+(1+A_{P})\bar{h}(q_{\mathrm{OS}},\eta_{\mathrm{OS}})\bar{h}(d_{\mathrm{SS}},\eta_{\mathrm{SS}})\biggl{(}\text{cosh}\left(\frac{\Delta\Gamma_{s}\,t^{\prime}}{2}\right)-\,q_{f}\,\text{cos}\left(\Delta
m_{s}\,t^{\prime}\right)\biggr{)}\Biggr{]},$ $\displaystyle
h(d,\eta)\equiv|d|\Biggl{[}1+d\,\biggl{(}1-2\,w(\eta)\biggr{)}\Biggr{]}\,\epsilon_{\text{tag}}+2\,(1-|d|)\,(1-\epsilon_{\text{tag}}),$
$\displaystyle\bar{h}(d,\eta)\equiv|d|\Biggl{[}1-d\,\biggl{(}1-2\,\bar{w}(\eta)\biggr{)}\Biggr{]}\,\bar{\epsilon}_{\text{tag}}+2\,(1-|d|)\,(1-\bar{\epsilon}_{\text{tag}}).$
(14)

This PDF depends on the tagging decisions of the $\mathrm{OS}$ and
$\mathrm{SS}$ taggers, $d_{\mathrm{OS}}$ and $d_{\mathrm{SS}}$; on the final
state, $q_{f}=+1(-1)$ for
$f=D_{s}^{-}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}(\bar{f}=D_{s}^{+}{{\pi}^{-}}{{\pi}^{-}}{{\pi}^{+}}$);
and is conditional on the per-event observables [73]
$\delta_{t},\eta_{\mathrm{OS}}$ and $\eta_{\mathrm{SS}}$, describing the
estimated decay-time error and the estimated mistag rates of the OS and SS
taggers, respectively. The parameters of the Gaussian resolution model,
$R(t-t^{\prime}|\mu_{t},\sigma_{t}(\delta_{t}))$, are fixed to the values
determined from the prompt candidate data sample. The decay-time-dependent
efficiency, $\epsilon(t)$, of reconstructing and selecting signal decays is
modelled by a B-spline curve [74, 75], whose cubic polynomials are uniquely
defined by a set of knots. These are placed across the considered decay-time
range to account for local variations [76]. Six knots are chosen such that
there is an approximately equal amount of data in-between two consecutive
knots. By fixing the decay width $\Gamma_{s}=(0.6624\pm 0.0018)\textrm{
ps}^{-1}$ and decay width difference $\Delta\Gamma_{s}=(0.090\pm
0.005)\textrm{ ps}^{-1}$ to their world-average values [4], the spline
coefficients can be directly determined in the fit to the data. The
correlation $\rho(\Gamma_{s},\,\Delta\Gamma_{s})=-0.080$ is taken into account
when evaluating the systematic uncertainties.

Both taggers are simultaneously calibrated during the fit as described by the
functions $h(d,\eta)$ and $\bar{h}(d,\eta)$. These also take into account a
small dependence of the tagging performance on the initial ${{B}^{0}_{s}}$
flavour by introducing linear calibration functions for initial ${B}^{0}_{s}$
and ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ mesons denoted as
$\omega(\eta)$ and $\bar{\omega}(\eta)$, respectively. Similarly,
$\epsilon_{\text{tag}}$ ($\bar{\epsilon}_{\text{tag}}$) denotes the tagging
efficiency for an initial ${B}^{0}_{s}$ (${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$) meson. The calibrated
responses of the OS and SS taggers are then explicitly combined in the PDF.
Since the tagging algorithms have been retuned for the Run 2 data-taking
period [77] to account for the changed conditions, separate calibrations for
the two data-taking periods are performed.

The production asymmetries between ${B}^{0}_{s}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ mesons at centre-of-mass
energies of $7\text{\,Te\kern-1.00006ptV}$ and $8\text{\,Te\kern-1.00006ptV}$
are taken from an LHCb measurement using
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ decays [78]. After
correcting for kinematic differences between
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
decays, the effective production asymmetry, $A_{P}=(N({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}})-N({{B}^{0}_{s}}))/(N({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}})+N({{B}^{0}_{s}}))$, for Run
1 data amounts to $A_{P}=(-0.1\pm 1.0)\%$. The production asymmetry at a
centre-of-mass energy of $13\text{\,Te\kern-1.00006ptV}$ is determined in the
fit. A detection asymmetry between the final states is caused by the different
interaction cross-sections of positively and negatively charged kaons with the
detector material. The detection asymmetry is defined as
$A_{D}=({\varepsilon}(\bar{f})-{\varepsilon}(f))/({\varepsilon}(\bar{f})+{\varepsilon}(f))$,
where ${\varepsilon}(\bar{f})$ $({\varepsilon}(f))$ denotes the detection
efficiency of final state $\bar{f}$ $(f)$. It is computed by comparing the
charge asymmetries in $D^{\pm}\rightarrow K^{\mp}\pi^{\pm}\pi^{\pm}$ and
$D^{\pm}\rightarrow{{K}^{0}_{\mathrm{S}}}\pi^{\pm}$ calibration samples [79],
weighted to match the kinematics of the signal kaon. Only the decay mode
${{D}^{-}_{s}}\rightarrow{{K}^{-}}{{\pi}^{-}}{{\pi}^{+}}$ is a possible source
of detection asymmetry for
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
decays resulting in an average detection asymmetry of $A_{D}=(-0.07\pm
0.15)\%$ for Run 1 and $A_{D}=(-0.08\pm 0.21)\%$ for Run 2 data. A
sufficiently large subsample of the Run 2 data set is used to reconstruct the
calibration modes for this study.

Figure 3:  Background-subtracted decay-time distribution of (top) all and
(bottom left) tagged
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
candidates as well as (bottom right) the dilution-weighted mixing asymmetry
folded into one oscillation period along with the fit projections (solid
lines). The decay-time acceptance (top) is overlaid in an arbitrary scale
(dashed line).

Figure 3 displays the decay-time distribution and the mixing asymmetry for
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
signal candidates. The latter is weighted by the per-candidate time resolution
and flavour-tagging dilution to enhance the visible asymmetry. All features
are well reproduced by the fit projections which are overlaid. The
${{B}^{0}_{s}}-{{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$
mixing frequency is determined to be

$\Delta m_{s}=(17.757\pm 0.007\pm 0.008)\text{\,ps}^{-1},$

where the first uncertainty is statistical and the second systematic. The
systematic studies are discussed in Sec. 6. Within uncertainties, the measured
production asymmetry for Run 2 data, $A_{P}=(0.1\pm 0.5\pm 0.1)\%$, is
consistent with zero. The calibrated per-candidate mistag probabilities,
$\omega_{i}$, are used to compute the effective tagging power as
$\epsilon_{\text{eff}}=\frac{1}{N}\sum_{i}(1-2\omega_{i})^{2}$, where $N$ is
the total number of signal candidates and a value of $\omega_{i}=0.5$ is
assigned to untagged candidates. Table 1(b) reports the observed tagging
performance for Run 1 and Run 2 data considering three mutually exclusive
categories: tagged by the $\mathrm{OS}$ combination algorithm only, tagged by
the $\mathrm{SS}$ kaon algorithm only and tagged by both $\mathrm{OS}$ and
$\mathrm{SS}$ algorithms. While the flavour taggers suffer from the higher
track multiplicity during the Run 2 data-taking period, they profit from the
harder momentum spectrum of the produced ${b}\bar{b}$ quark pair. Combined,
this results in a net relative improvement of $14\%$ in effective tagging
power.

Table 1: The flavour-tagging performance for only OS-tagged, only SS-tagged
and both OS- and SS-tagged
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
signal candidates.

(a) Run 1 data. | $\epsilon_{\text{tag}}[\%]$ | $\langle\omega\rangle[\%]$ | $\epsilon_{\text{eff}}[\%]$
---|---|---|---
Only OS |  14.74 $\pm$ 0.11 |  39.09 $\pm$ 0.80 |  1.25 $\pm$ 0.16
Only SS |  35.38 $\pm$ 0.18 |  44.26 $\pm$ 0.62 |  1.05 $\pm$ 0.18
Both OS-SS |  33.04 $\pm$ 0.30 |  37.33 $\pm$ 0.73 |  3.41 $\pm$ 0.33
Combined |  83.16 $\pm$ 0.37 |  40.59 $\pm$ 0.70 |  5.71 $\pm$ 0.40

(b) Run 2 data. | $\epsilon_{\text{tag}}[\%]$ | $\langle\omega\rangle[\%]$ | $\epsilon_{\text{eff}}[\%]$
---|---|---|---
Only OS |  11.91 $\pm$ 0.04 |  37.33 $\pm$ 0.41 |  1.11 $\pm$ 0.05
Only SS |  40.95 $\pm$ 0.08 |  42.41 $\pm$ 0.29 |  1.81 $\pm$ 0.10
Both OS-SS |  28.96 $\pm$ 0.12 |  35.51 $\pm$ 0.32 |  3.61 $\pm$ 0.13
Combined |  81.82 $\pm$ 0.15 |  39.23 $\pm$ 0.32 |  6.52 $\pm$ 0.17

##  5 Measurement of the CKM angle $\gamma$

This section first describes the phase-space-integrated decay-time analysis of
the signal channel
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
that allows the determination of the CKM angle $\gamma$ in a model-independent
way. Afterwards, the resonance spectrum in
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
decays is studied and a full time-dependent amplitude analysis is performed
for a model-dependent determination of the CKM angle $\gamma$.

###  5.1 Model-independent analysis

The decay-time fit to the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
candidates uses a signal PDF based on Eq. (11) with modifications accounting
for the experimental effects described in Sec. 4.3. The ${{B}^{0}_{s}}$
production asymmetry for Run 2 data and the ${{B}^{0}_{s}}-{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$ mixing frequency are fixed
to the values obtained from the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
data sample, whereas the tagging calibration parameters are allowed to vary
within Gaussian constraints taking into account their correlation. The kaon
detection asymmetry for
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
decays is determined in a similar manner as for
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
decays and amounts to $A_{D}=(-1.02\pm 0.15)\%$ for Run 1 and $A_{D}=(-0.91\pm
0.22)\%$ for Run 2 data. The decay-time acceptance is also fixed to the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
result, corrected by a decay-time dependent correction factor derived from
simulation to account for small differences in the selection and decay
kinematics between the decay modes. Otherwise, the fit strategy is identical
to that discussed in the previous section. Figure 4 shows the decay-time
distribution and mixing asymmetries together with the fit projections. The
mixing asymmetries for $D_{s}^{-}K^{+}{{\pi}^{+}}{{\pi}^{-}}$ and
$D_{s}^{+}K^{-}{{\pi}^{-}}{{\pi}^{+}}$ final states are shifted with respect
to each other indicating mixing-induced $C\\!P$ violation, cf. Eq. (13). The
$C\\!P$ coefficients
$C_{f},A^{\Delta\Gamma}_{f},A^{\Delta\Gamma}_{\bar{f}},S_{f}$ and
$S_{\bar{f}}$ determined from the fit are reported in Table 2. They are
converted to the parameters of interest $r,\kappa,\delta$ and
$\gamma-2\beta_{s}$ in Sec. 7.

Figure 4:  Decay-time distribution of (left) background-subtracted ${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$ candidates and (right) dilution-weighted mixing asymmetry along with the model-independent fit projections (lines). The decay-time acceptance (left) is overlaid in an arbitrary scale (dashed line).  Table 2: $C\\!P$ coefficients determined from the phase-space fit to the ${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$ decay-time distribution. The uncertainties are statistical and systematic (discussed in Sec. 6). Fit parameter | Value
---|---
$C_{f}$ | 0.631 $\pm$ 0.096 $\pm$ 0.032
$A^{\Delta\Gamma}_{f}$ |  $-0.334$ $\pm$ 0.232 $\pm$ 0.097
$A^{\Delta\Gamma}_{\bar{f}}$ |  $-0.695$ $\pm$ 0.215 $\pm$ 0.081
$S_{f}$ |  $-0.424$ $\pm$ 0.135 $\pm$ 0.033
$S_{\bar{f}}$ |  $-0.463$ $\pm$ 0.134 $\pm$ 0.031

###  5.2 Time-dependent amplitude analysis

To perform the time-dependent amplitude fit, a signal PDF is employed which
replaces the phase-space integrated decay rate with the full decay rate given
in Eq. (10), but is otherwise identical to the PDF used in Sec. 5.1.
Variations of the selection efficiency over the phase space are incorporated
by evaluating the likelihood normalisation integrals with the Monte Carlo (MC)
integration technique using fully simulated decays [30, 80, 81, 82].

The light meson spectrum comprises a large number of resonances potentially
contributing to the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
decay in various decay topologies and angular momentum configurations. The
full list of considered intermediate-state amplitudes can be found in Table
B.1. A significant complication arises from the fact that two (quasi-
independent) amplitude models need to be developed simultaneously: one
amplitude describes decays via $b\rightarrow c$ ($A^{c}(\mathbf{x})$), the
other decays via $b\rightarrow u$ ($A^{u}(\mathbf{x})$) quark-level
transitions. A model building procedure is applied to obtain a good
description of the observed phase-space distribution while keeping the number
of included amplitudes as small as possible. This is accomplished in two
stages. The first stage identifies the set of intermediate-state amplitudes
contributing at a significant level to either decays via $b\rightarrow c$ or
$b\rightarrow u$ quark-level transitions or to both. To that end, the time-
integrated and flavour-averaged phase-space distribution is examined. A single
total amplitude,
$A^{\text{eff}}(\mathbf{x})=\sum_{i}a^{\text{eff}}_{i}A_{i}(\mathbf{x}),$ is
sufficient in this case, which effectively describes the incoherent
superposition of the $b\rightarrow c$ and $b\rightarrow u$ amplitudes,
$|A^{\text{eff}}(\mathbf{x})|^{2}=|A^{c}(\mathbf{x})|^{2}+r^{2}\,|A^{u}(\mathbf{x})|^{2}$.
This significantly simplified fitting procedure allows the initial inclusion
of the whole pool of considered intermediate-state amplitudes, limiting the
model complexity with the LASSO technique[83, 84, 30]. This method adds a
penalty term to the likelihood function,

$-2\,\log\mathcal{L}\rightarrow-2\,\log\mathcal{L}+\lambda\,\sum_{i}\sqrt{\int|a^{\text{eff}}_{i}A_{i}(\mathbf{x})|^{2}\,\text{d}\Phi_{4}},$
(15)

which shrinks the amplitude coefficients towards zero. The optimal value for
the LASSO parameter $\lambda$, which controls the model complexity, is found
by minimising the Bayesian information criterion
$\text{BIC}(\lambda)=-2\,\log\mathcal{L}+k\,\log N_{\rm Sig}$ [85], where
$N_{\rm Sig}$ is the signal yield and $k$ is the number of amplitudes with a
decay fraction above the threshold of $0.5\%$. The amplitudes with a decay
fraction above the threshold at the minimum $\text{BIC}(\lambda)$ value are
selected. The second stage of the model selection performs a full time-
dependent amplitude fit. The components selected by the first stage are
included for both $b\rightarrow c$ and $b\rightarrow u$ transitions and a
LASSO penalty term for each is added to the likelihood function. As the strong
interaction is $C\\!P$ symmetric, the subdecay modes of three-body resonances
and their conjugates are constrained to be the same. The final set of
$b\rightarrow c$ and $b\rightarrow u$ amplitudes is henceforth referred to as
the baseline model. The LASSO penalty term is only used to select the model
and discarded in the final fit to avoid biasing the parameter uncertainties.

Figure 5:  Decay-time distribution of (left) background-subtracted
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
candidates and (right) dilution-weighted mixing asymmetry along with the
model-dependent fit projections (lines). The decay-time acceptance (left) is
overlaid in an arbitrary scale (dashed line).

Table B.2 and B.3 list the moduli and phases of the complex amplitude
coefficients obtained by fitting the baseline model to the background-
subtracted
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
signal candidates. The corresponding decay fractions for the $b\rightarrow c$
and $b\rightarrow u$ amplitudes are given in Table 3. The decay-time
projection and mixing asymmetries shown in Fig. 5 are consistent with those of
the phase-space integrated fit in Fig. 4. Invariant-mass projections are shown
in Figs. 6 and 7 indicating that the model provides a reasonable description
of the data. Decays via $b\rightarrow c$ quark-level transitions are found to
be dominated by the axial-vector states $K_{1}(1270)^{+}$ and
$K_{1}(1400)^{+}$. These resonances are produced by the external weak current
(see Fig. 1). The sub-leading contribution comes from the vector resonance
$K^{*}(1410)^{+}$. In $b\rightarrow u$ quark-level transitions, the excited
kaon states are produced by the spectator-quark interaction. Here, no clear
hierarchy is observed. There are sizeable contributions from the axial-vector
resonances but also from the pseudoscalar state $K(1460)^{+}$ and from the
quasi-two-body process
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$,
where $({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}$ denotes a non-resonant two-
particle system in a $P$-wave ($L=1$) configuration. Interference fractions of
the $b\rightarrow c$ and $b\rightarrow u$ intermediate-state amplitudes are
given in Tables B.6 and B.7. Sizeable interference effects between the decay
modes ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow
K^{*}(892)^{0}\,{{\pi}^{\pm}})$,
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow
K^{*}(892)^{0}\,{{\pi}^{\pm}})$ and
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$
are observed since the overlap of their phase-space distributions is
significant. A net constructive (destructive) interference effect of all
amplitude components of around $+26\%$ ($-12\%$) remains for $b\rightarrow c$
($b\rightarrow u$) quark-level transitions when integrated over the phase
space.

Table 3:  Decay fractions of the intermediate-state amplitudes contributing to
decays via $b\rightarrow c$ and $b\rightarrow u$ quark-level transitions. The
uncertainties are statistical, systematic and due to alternative amplitude
models considered.

Decay channel | $F^{c}_{i}[\%]$ | $F^{u}_{i}[\%]$
---|---|---
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 13.0$\,\pm\,$ | 2.4 $\,\pm\,$ | 2.7 $\,\pm$ | 3.4 | 4.1$\,\pm\,$ | 2.2 $\,\pm\,$ | 2.9 $\,\pm$ | 2.6
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ | 16.0$\,\pm\,$ | 1.4 $\,\pm\,$ | 1.8 $\,\pm$ | 2.1 | 5.1$\,\pm\,$ | 2.2 $\,\pm\,$ | 3.5 $\,\pm$ | 2.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)^{0}\,{{\pi}^{\pm}})$ | 3.4$\,\pm\,$ | 0.5 $\,\pm\,$ | 1.0 $\,\pm$ | 0.4 | 1.1$\,\pm\,$ | 0.5 $\,\pm\,$ | 0.6 $\,\pm$ | 0.5
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 63.9$\,\pm\,$ | 5.1 $\,\pm\,$ | 7.4 $\,\pm$ | 13.5 | 19.3$\,\pm\,$ | 5.2 $\,\pm\,$ | 8.3 $\,\pm$ | 7.8
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 12.8$\,\pm\,$ | 0.8 $\,\pm\,$ | 1.5 $\,\pm$ | 3.2 | 12.6$\,\pm\,$ | 2.0 $\,\pm\,$ | 2.6 $\,\pm$ | 4.1
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ | 5.6$\,\pm\,$ | 0.4 $\,\pm\,$ | 0.6 $\,\pm$ | 0.7 | 5.6$\,\pm\,$ | 1.0 $\,\pm\,$ | 1.2 $\,\pm$ | 1.8
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  | 11.9$\,\pm\,$ | 2.5 $\,\pm\,$ | 2.9 $\,\pm$ | 3.1
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ | 10.2$\,\pm\,$ | 1.6 $\,\pm\,$ | 1.8 $\,\pm$ | 4.5 | 28.4$\,\pm\,$ | 5.6 $\,\pm\,$ | 6.4 $\,\pm$ | 15.3
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{P}\,\,\rho(770)^{0}$ | 0.9$\,\pm\,$ | 0.4 $\,\pm\,$ | 0.5 $\,\pm$ | 1.0 |
Sum | 125.7$\,\pm\,$ | 6.4 $\,\pm\,$ | 6.9 $\,\pm$ | 19.9 | 88.1$\,\pm\,$ | 7.0 $\,\pm\,$ | 10.0 $\,\pm$ | 20.9

The mass and width of the $K_{1}(1400)^{+}$ and $K^{*}(1410)^{+}$ resonances
are determined from the fit to be

$\displaystyle m_{K_{1}(1400)}$ $\displaystyle=(1406\pm 7\pm 6\pm
11)\text{\,Me\kern-1.00006ptV},$ $\displaystyle\Gamma_{K_{1}(1400)}$
$\displaystyle=(195\pm 11\pm 12\pm 16)\text{\,Me\kern-1.00006ptV},$
$\displaystyle m_{K^{*}(1410)}$ $\displaystyle=(1433\pm 10\pm 23\pm
8)\text{\,Me\kern-1.00006ptV},$ $\displaystyle\Gamma_{K^{*}(1410)}$
$\displaystyle=(402\pm 24\pm 47\pm 22)\text{\,Me\kern-1.00006ptV},$

in good agreement with the PDG average values [3]. The uncertainties are
statistical, systematic and due to alternative models considered as detailed
in Sec. 6. The ratio of the $B_{s}^{0}\rightarrow
D_{s}^{-}K^{+}{{\pi}^{+}}{{\pi}^{-}}$ and $\bar{B}_{s}^{0}\rightarrow
D_{s}^{-}K^{+}{{\pi}^{+}}{{\pi}^{-}}$ decay amplitudes as well as their
strong- and weak-phase difference are measured to be

$\displaystyle r$ $\displaystyle=0.56\pm 0.05\pm 0.04\pm 0.07,$
$\displaystyle\delta$ $\displaystyle=(-14\pm 10\pm 4\pm 5)^{\circ},$
$\displaystyle\gamma-2\beta_{s}$ $\displaystyle=(\phantom{-}42\pm 10\pm 4\pm
5)^{\circ},$

where the angles are given modulo $180^{\circ}$. The coherence factor is
computed by numerically integrating over the phase space using the baseline
model resulting in

$\displaystyle\kappa=0.72\pm 0.04\text{\,(stat)}\pm 0.06\text{\,(syst)}\pm
0.04\,(\text{model}).$

Figure 6: Invariant-mass distribution of background-subtracted
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
candidates (data points) and fit projections (blue solid line). Contributions
from $b\rightarrow c$ and $b\rightarrow u$ decay amplitudes are overlaid.

Figure 7: Invariant-mass distribution of background-subtracted
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
candidates (data points) and fit projections (blue solid line). Incoherent
contributions from intermediate-state components are overlaid.

##  6 Systematic uncertainties

The systematic uncertainties on the measured observables are summarised in
Table 4 for the decay-time fits to
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
and
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
decays and in Tables 5 and B.4 for the time-dependent amplitude fit to
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
decays. The various sources of systematic uncertainties are described in the
following.

The overall fit procedure is tested by generating pseudoexperiments from the
default fit model using the measured values and subsequently fitting them with
the same model. For each pseudoexperiment and fit parameter, a pull is
calculated by dividing the difference between the fitted and generated values
by the statistical uncertainty. The means of the pull distributions are
assigned as systematic uncertainties due to an intrinsic fit bias. A closure
test using a large sample of fully simulated signal candidates shows a non-
significant bias for the determination of $\Delta m_{s}$, which is assigned as
a systematic uncertainty.

The statistical subtraction of the residual background relies on the correct
description of the reconstructed $m(D_{s}^{\mp}h^{\pm}\pi^{\pm}\pi^{\mp})$
invariant mass distribution. Alternative parameterisations are tested for the
signal and each background component. The yields of the cross-feed
contributions to the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
candidates estimated from a combination of simulated data and control modes
are fixed to zero or doubled. The standard deviations of the obtained
differences to the default fit values are assigned as a systematic uncertainty
due to the background subtraction. The background subtraction technique relies
on the assumption of independence between the reconstructed
$m(D_{s}^{\mp}h^{\pm}\pi^{\pm}\pi^{\mp})$ invariant mass and the observables
that are used in the final fit. The impact of ignoring the small correlation
between reconstructed mass and decay time observed for the combinatorial
background is determined with pseudoexperiments in which the correlation is
included at generation and neglected in the fit.

Table 4: 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 phase-space 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. Systematic | $\Delta m_{s}$ | $C_{f}$ | $A^{\Delta\Gamma}_{f}$ | $A^{\Delta\Gamma}_{\bar{f}}$ | $S_{f}$ | $S_{\bar{f}}$
---|---|---|---|---|---|---
Fit bias ($+$ closure test) | 0.39 | 0.09 | 0.00 | 0.01 | 0.02 | 0.11
Background subtraction | 0.18 | 0.17 | 0.19 | 0.18 | 0.11 | 0.06
Correlations | 0.09 | 0.11 | 0.32 | 0.24 | 0.04 | 0.06
Acceptance | 0.02 | 0.03 | 0.19 | 0.22 | 0.02 | 0.02
Resolution | 0.16 | 0.24 | 0.01 | 0.03 | 0.12 | 0.10
Decay-time bias | 1.00 | 0.06 | 0.00 | 0.00 | 0.10 | 0.08
Nuisance asymmetries | 0.02 | 0.06 | 0.06 | 0.06 | 0.06 | 0.05
VELO $z$ scale | 0.26 |  |  |  |  |
VELO alignment | 0.44 |  |  |  |  |
$\Delta m_{s}$ |  | 0.02 | 0.01 | 0.01 | 0.12 | 0.12
Total | 1.21 | 0.34 | 0.42 | 0.38 | 0.24 | 0.23

The systematic uncertainties related to the decay-time acceptance as well as
due to the limited experimental knowledge of $\Gamma_{s}$ and
$\Delta\Gamma_{s}$ are studied simultaneously. Pseudoexperiments are generated
assuming the default configuration and subsequently fitted under both this
default and an alternative configuration in which the acceptance parameters
together with $\Gamma_{s}$ and $\Delta\Gamma_{s}$ are randomised within their
uncertainties (taking their correlation into account). The bias of the mean of
the resulting pull distribution is added in quadrature to the pull width in
order to arrive at the final systematic uncertainty.

Systematic effects originating from the calibration of the decay-time
uncertainty estimate are studied with two alternative parameterisations which
either slightly overestimate or underestimate the decay-time resolution. Due
to the high correlation between the decay-time resolution and the calibration
of the flavour taggers, their systematic uncertainty is studied
simultaneously. As a first step, the decay-time fit to the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
candidates is repeated using the alternative decay-time error calibration
functions. New tagging calibration parameters are obtained which are then used
(together with the respective decay-time error calibration function) for the
fits to the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
candidates. The largest deviations of the central values from their default
values are assigned as a systematic uncertainty for each fit parameter. A
systematic uncertainty due to the limited knowledge of the decay-time bias
which is fixed in the fit is evaluated by randomising the value within its
uncertainty.

The systematic uncertainty from the production and detection asymmetries and
$\Delta m_{s}$ (in case of
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
decays) which are fixed in the fit are evaluated by means of
pseudoexperiments, analogously to the procedure performed for the decay-time
acceptance.

The precision with which the ${{B}^{0}_{s}}$ flight distance can be determined
is limited by the knowledge of the overall length of the VELO detector (VELO
$z$ scale) and the position of the individual VELO modules (VELO alignment).
This VELO-reconstruction uncertainty translates into a relative uncertainty on
$\Delta m_{s}$ of $0.02\%$ [71] with other parameters being unaffected. In the
fit to the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
candidates, the VELO-reconstruction uncertainty is then implicitly included in
the systematic error due to the $\Delta m_{s}$ uncertainty described above.

Table 5: Systematic uncertainties on the physical observables and resonance
parameters determined from the full time-dependent 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 B.4.

Systematic | $m_{K_{1}(1400)}$ | $\Gamma_{K_{1}(1400)}$ | $m_{K^{*}(1410)}$ | $\Gamma_{K^{*}(1410)}$ | $r$ | $\delta$ | $\gamma-2\beta_{s}$
---|---|---|---|---|---|---|---
Fit bias | 0.00 | 0.14 | 0.14 | 0.42 | 0.06 | 0.13 | 0.13
Background subtraction | 0.15 | 0.28 | 0.15 | 0.41 | 0.15 | 0.15 | 0.22
Correlations | 0.23 | 0.27 | 0.18 | 0.07 | 0.13 | 0.05 | 0.08
Time acceptance | 0.01 | 0.01 | 0.00 | 0.00 | 0.06 | 0.04 | 0.07
Resolution | 0.11 | 0.33 | 0.09 | 0.03 | 0.30 | 0.27 | 0.26
Decay-time bias | 0.00 | 0.02 | 0.01 | 0.00 | 0.02 | 0.12 | 0.01
Nuisance asymmetries | 0.00 | 0.01 | 0.00 | 0.00 | 0.01 | 0.07 | 0.03
$\Delta m_{s}$ | 0.00 | 0.01 | 0.00 | 0.00 | 0.03 | 0.06 | 0.03
Phase-space acceptance | 0.17 | 0.34 | 0.32 | 0.21 | 0.35 | 0.08 | 0.06
Acceptance factorisation | 0.25 | 0.41 | 0.76 | 0.36 | 0.49 | 0.08 | 0.05
Lineshapes | 0.52 | 0.53 | 0.43 | 0.29 | 0.34 | 0.14 | 0.13
Resonances $m,\Gamma$ | 0.07 | 0.06 | 0.02 | 0.02 | 0.02 | 0.01 | 0.01
Form factors | 0.51 | 0.60 | 2.03 | 1.74 | 0.12 | 0.13 | 0.07
Amplitude model | 1.59 | 1.50 | 0.77 | 0.88 | 1.60 | 0.51 | 0.47
Total | 1.80 | 1.86 | 2.38 | 2.10 | 1.79 | 0.67 | 0.63

The treatment of the phase-space acceptance relies on simulated data. The
integration error due to the limited size of the simulated sample used to
normalise the signal PDF is below $0.2\%$ and thus negligibly small. To assess
the uncertainty due to possible data-simulation differences, alternative
phase-space acceptances are derived by varying the selection requirements (for
the simulated sample only) on quantities that are expected not to be well
described by the simulation. It is assumed that the phase-space acceptance is
independent of the decay time. This assumption is tested by using only
simulated candidates within specific decay-time intervals to calculate the MC
normalisation integrals. Four approximately equally populated decay-time
intervals are chosen and the sample variance of the fitted values is assigned
as a systematic uncertainty.

The lineshape parameterisations for the $\rho(770)^{0}$ and
$K_{0}^{*}(1430)^{0}$ resonances are replaced by a relativistic Breit–Wigner
propagator given by Eq. (7) as part of the systematic studies. Moreover,
energy-dependent decay widths of three-body resonances are recomputed taking
only the dominant ${{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$ decay mode into
account. For each alteration, a time-dependent amplitude fit is performed and
the standard deviation of the obtained fit results is assigned as a systematic
uncertainty. Systematic uncertainties due to fixed resonance masses and widths
are computed with the same procedure used for the other fixed parameters
mentioned above. Similarly, pseudoexperiments are performed in which the
Blatt–Weisskopf radial parameter $r_{\mathrm{BW}}$ (set to
$1.5\text{\,Ge\kern-1.00006ptV}^{-1}$ by default) is varied uniformly within
the interval $[0,3]\text{\,Ge\kern-1.00006ptV}^{-1}$ to assign a systematic
uncertainty due to the form-factor modelling.

Several modifications to the baseline model are tested to assign an additional
uncertainty due to the choice of amplitude components: all amplitudes selected
by the first stage of the model selection are included for both $b\rightarrow
c$ and $b\rightarrow u$ transitions, the amplitudes with the smallest decay
fraction are removed, additional sub-decay modes of selected three-body
resonances are considered, higher orbital angular momentum states are included
where applicable, the orbital angular momentum state of non-resonant two-body
states is set to other allowed values, additional cascade and quasi-two-body
amplitudes (which were removed by the first stage of the model selection) are
considered. In total, twelve amplitude models with similar fit quality as the
baseline model are identified. The fit results for those are summarised in
Tables B.8 and B.9. The largest deviations from the baseline values for
$r,\delta$ and $\gamma-2\beta$ are $0.19,10^{\circ}$ and $12^{\circ}$ observed
for alternative models 8, 1 and 11, respectively. The standard deviation of
the twelve fit results is taken as the model uncertainty. No model uncertainty
is assigned to the amplitude coefficients since they are, by definition,
parameters of a given model.

No statistically significant effect on the results is observed when repeating
the analysis on subsets of the data, splitting by data-taking period or
tagging category (OS-tagged, only SS-tagged or both OS- and SS-tagged).

##  7 Results

To interpret the parameters determined in the model-independent fit,
$P_{\text{obs}}\equiv(C_{f},A^{\Delta\Gamma}_{f},A^{\Delta\Gamma}_{\bar{f}},S_{f},S_{\bar{f}})$,
in terms of the physical observables
$\Lambda\equiv(r,\kappa,\delta,\gamma-2\beta_{s})$, the equations for the
$C\\!P$ coefficients in terms of these physical variables reported in Eq.
(2.2), $P(\Lambda)$, need to be inverted. This is accomplished by minimising
the likelihood function [86, *LHCb-PAPER-2016-032]

$-2\mathcal{L}(\Lambda)=-2\,\exp\left(-\frac{1}{2}(P(\Lambda)-P_{\text{obs}})^{T}\,V^{-1}\,(P(\Lambda)-P_{\text{obs}})\right).$
(16)

Here, $V$ denotes the experimental (statistical and systematic) covariance
matrix of the measured observables, see Appendix C. Figure 8 displays the
confidence levels (CL) for the physical parameters $\Lambda$ obtained from the
model-independent method, where the physical boundary of the coherence factor
is enforced. The $1-\text{CL}=68.3\%$ ($1\sigma$) confidence intervals are
given in Table 6 together with the results of the full time-dependent
amplitude analysis. Considering the difference in statistical sensitivity of
the two methods and that the model-dependent uncertainty only affects the full
time-dependent amplitude fit, a good agreement between the measurements is
observed. As a cross-check, pseudoexperiments are performed to study the
distribution of the test statistic
$Q^{2}=\sum_{i}(\Lambda_{i}^{\text{MI}}-\Lambda_{i}^{\text{MD}})^{2}/(\sigma_{\text{stat}}(\Lambda_{i}^{\text{MI}})^{2}-\sigma_{\text{stat}}(\Lambda_{i}^{\text{MD}})^{2})$,
where $\Lambda_{i}^{\text{MI}}$ ($\Lambda_{i}^{\text{MD}}$) and
$\sigma_{\text{stat}}(\Lambda_{i}^{\text{MI}})$
($\sigma_{\text{stat}}(\Lambda_{i}^{\text{MD}})$) denote the measured value of
the physical observable $\Lambda_{i}$ and its statistical uncertainty obtained
with the model-independent (model-dependent) method. It is found that $p=33\%$
of the pseudoexperiments have a larger $Q^{2}$ value than observed on data,
considering only the statistical uncertainty. The p-value increases to
$p=49\%$ when the uncertainty due to the amplitude modelling is included.

The measured ratio of the $b\rightarrow u$ and $b\rightarrow c$ decay
amplitudes is qualitatively consistent with the naive expectation based on the
involved CKM elements ($r\approx 0.4$). Note that the parameters $r,\kappa$
and $\delta$ are determined in a limited phase-space region (cf. Sec. 3.1) and
might differ when the full phase space is considered.

Figure 8: The 1$-$CL contours for the physical observables $r,\kappa,\delta$ and $\gamma-2\beta_{s}$ obtained with the model-independent fit.  Table 6: Parameters determined from the model-independent and model-dependent 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 $180^{\circ}$.  Parameter | Model-independent | Model-dependent
---|---|---
$r$ | $0.47$ | ${}^{\,+\,0.08}_{\,-\,0.08}$ | ${}^{\,+\,0.02}_{\,-\,0.03}$ | 0.56$\,\pm\,$ | 0.05 $\,\pm\,$ | 0.04 $\,\pm\,$ | 0.07
$\kappa$ | $0.88$ | ${}^{\,+\,0.12}_{\,-\,0.19}$ | ${}^{\,+\,0.04}_{\,-\,0.07}$ | 0.72$\,\pm\,$ | 0.04 $\,\pm\,$ | 0.06 $\,\pm\,$ | 0.04
$\delta$ [∘] | $-6$ | ${}^{\,+\,10}_{\,-\,12}$ | ${}^{\,+\,2}_{\,-\,4}$ |  $-14$$\,\pm\,$ | 10 $\,\pm\,$ | 4 $\,\pm\,$ | 5
$\gamma-2\beta_{s}$ [∘] | $42$ | ${}^{\,+\,19}_{\,-\,13}$ | ${}^{\,+\,6}_{\,-\,2}$ | 42$\,\pm\,$ | 10 $\,\pm\,$ | 4 $\,\pm\,$ | 5

##  8 Conclusion

Mixing-induced $C\\!P$ violation in
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
decays is studied for the first time using $9\text{\,fb}^{-1}$ of proton-
proton collision data recorded by the LHCb detector. A time-dependent
amplitude analysis is performed, disentangling the various intermediate-state
components contributing via $b\rightarrow c$ or $b\rightarrow u$ quark-level
transitions. The $C\\!P$-violating weak phase $\gamma-2\beta_{s}$ is measured
and used to determine the CKM angle $\gamma$ by taking the mixing phase,
$-2\beta_{s}=\phi_{s}=(-2.35\pm 1.43)^{\circ}$ [11], as an external input.
This results in a measured value of $\gamma=(44\pm 12)^{\circ}$ modulo 180∘. A
model-independent fit to the phase-space integrated decay-time spectrum yields
a compatible, but statistically less precise, value of
$\gamma=(44^{\,+\,20}_{\,-\,13})^{\circ}$ modulo 180∘. These results
correspond to $4.4\sigma$ and $4.6\sigma$ evidence for mixing-induced $C\\!P$
violation and agree with the world-average value of the CKM angle $\gamma$ [3,
4] within $2.2\sigma$ and $1.4\sigma$ for the model-dependent and model-
independent methods, respectively. The ${{B}^{0}_{s}}-{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$ oscillation frequency is
measured from flavour-specific
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
decays to be $\Delta m_{s}=(17.757\pm 0.007\pm 0.008)\text{\,ps}^{-1}$. This
is the most precise measurement of this quantity and consistent with the
world-average value [3] and theoretical predictions [88]. Both the measurement
of the CKM angle $\gamma$ and of the ${{B}^{0}_{s}}-{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$ mixing frequency are vital
inputs for global fits of the CKM matrix.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MICINN (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and OSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); A*MIDEX, ANR, Labex P2IO
and OCEVU, and Région Auvergne-Rhône-Alpes (France); Key Research Program of
Frontier Sciences of CAS, CAS PIFI, Thousand Talents Program, and Sci. & Tech.
Program of Guangzhou (China); RFBR, RSF and Yandex LLC (Russia); GVA, XuntaGal
and GENCAT (Spain); the Royal Society and the Leverhulme Trust (United
Kingdom).

## Appendices

##  Appendix A Lineshapes

Lineshape parameters for resonances contributing to
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
decays are fixed to the values given in Table A.1. The running-width
distributions of the three-body resonances are shown in Figure 9.

Table A.1: Parameters of the resonances included in the ${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$ baseline model. Resonance |  $m_{0}$ $[\text{\,Me\kern-1.00006ptV}]$ |  $\Gamma_{0}$ $[\text{\,Me\kern-1.00006ptV}]$ | $J^{P}$ | Source
---|---|---|---|---
$\rho(770)^{0}$ |  $775.26$$\,\pm\,$ | $0.25$ |  $149.1$$\,\pm\,$ | $0.8$ | $1^{-}$ |  [3]
$K^{*}(892)^{0}$ |  $895.55$$\,\pm\,$ | $0.20$ |  $47.3$$\,\pm\,$ | $0.5$ | $1^{-}$ |  [3]
$K_{1}(1270)^{+}$ |  $1289.81$$\,\pm\,$ | $1.75$ |  $116.11$$\,\pm\,$ | $3.4$ | $1^{+}$ |  [89]
$K(1460)^{+}$ |  $1482.4$$\,\pm\,$ | $15.6$ |  $335.6$$\,\pm\,$ | $10.6$ | $0^{-}$ |  [89]
$K_{0}^{*}(1430)^{0}$ |  $1425$$\,\pm\,$ | $50$ |  $270$$\,\pm\,$ | $80$ | $0^{+}$ |  [3]

Figure 9: Running width distributions of the three-body 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) $K(1460)^{+}$.

##  Appendix B Amplitude model

Table B.1: Intermediate-state 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.

(a) Cascade decays ${{B}^{0}_{s}}\rightarrow
D_{s}^{\mp}\,[K_{1}(1270)^{\pm}[S,D]\rightarrow\pi^{\pm}\,K^{*}(892)^{0}]$
---
${{B}^{0}_{s}}\rightarrow
D_{s}^{\mp}\,[K_{1}(1270)^{\pm}\rightarrow\pi^{\pm}\,K_{0}^{*}(1430)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K_{1}(1270)^{\pm}\rightarrow
K^{\pm}\,f_{0}(500)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K_{1}(1270)^{\pm}\rightarrow
K^{\pm}\,f_{0}(980)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K_{1}(1270)^{\pm}[S,D]\rightarrow
K^{\pm}\,\rho(770)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K_{1}(1270)^{\pm}[S,D]\rightarrow
K^{\pm}\,\rho(1450)^{0}]$
${{B}^{0}_{s}}\rightarrow
D_{s}^{\mp}\,[K_{1}(1400)^{\pm}[S,D]\rightarrow\pi^{\pm}\,K^{*}(892)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K_{1}(1400)^{\pm}[S,D]\rightarrow
K^{\pm}\,\rho(770)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K(1460)^{\pm}\rightarrow
K^{\pm}\,f_{0}(500)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K(1460)^{\pm}\rightarrow
K^{\pm}\,\rho(770)^{0}]$
${{B}^{0}_{s}}\rightarrow
D_{s}^{\mp}\,[K(1460)^{\pm}\rightarrow\pi^{\pm}\,K^{*}(892)^{0}]$
${{B}^{0}_{s}}\rightarrow
D_{s}^{\mp}\,[K(1460)^{\pm}\rightarrow\pi^{\pm}\,K^{*}(1430)^{0}]$
${{B}^{0}_{s}}\rightarrow
D_{s}^{\mp}\,[K^{*}(1410)^{\pm}\rightarrow\pi^{\pm}\,K^{*}(892)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K^{*}(1410)^{\pm}\rightarrow
K^{\pm}\,\rho(770)^{0}]$
${{B}^{0}_{s}}\rightarrow
D_{s}^{\mp}\,[K_{2}^{*}(1430)^{\pm}\rightarrow\pi^{\pm}\,K^{*}(892)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K_{2}^{*}(1430)^{\pm}\rightarrow
K^{\pm}\,\rho(770)^{0}]$
${{B}^{0}_{s}}\rightarrow
D_{s}^{\mp}\,[K_{1}(1650)^{\pm}\rightarrow\pi^{\pm}\,K^{*}(892)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K_{1}(1650)^{\pm}\rightarrow
K^{\pm}\,\rho(770)^{0}]$
${{B}^{0}_{s}}\rightarrow
D_{s}^{\mp}\,[K^{*}(1680)^{\pm}\rightarrow\pi^{\pm}\,K^{*}(892)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K^{*}(1680)^{\pm}\rightarrow
K^{\pm}\,\rho(770)^{0}]$
${{B}^{0}_{s}}\rightarrow
D_{s}^{\mp}\,[K_{2}(1770)^{\pm}\rightarrow\pi^{\pm}\,K^{*}(892)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K_{2}(1770)^{\pm}\rightarrow
K^{\pm}\,\rho(770)^{0}]$
${{B}^{0}_{s}}\rightarrow
D_{s}^{\mp}\,[K_{2}(1770)^{\pm}\rightarrow\pi^{\pm}\,K_{2}^{*}(1430)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K_{2}(1770)^{\pm}\rightarrow
K^{\pm}\,f_{2}(1270)^{0}]$
${{B}^{0}_{s}}\rightarrow
D_{s}^{\mp}\,[K_{2}^{*}(1980)^{\pm}\rightarrow\pi^{\pm}\,K^{*}(892)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K_{2}^{*}(1980)^{\pm}\rightarrow
K^{\pm}\,\rho(770)^{0}]$
${{B}^{0}_{s}}\rightarrow
D_{s}^{\mp}\,[K_{2}^{*}(1980)^{\pm}\rightarrow\pi^{\pm}\,K_{2}^{*}(1430)^{0}]$
${{B}^{0}_{s}}\rightarrow D_{s}^{\mp}\,[K_{2}^{*}(1980)^{\pm}\rightarrow
K^{\pm}\,f_{2}(1270)]$
${{B}^{0}_{s}}\rightarrow K^{\pm}[D_{s1}(2536)^{\mp}\rightarrow
D_{s}^{\mp}\,\,\rho(770)^{0}]$
${{B}^{0}_{s}}\rightarrow K^{\pm}[D_{s1}(2536)^{\mp}\rightarrow
D_{s}^{\mp}\,\,f_{0}(500)^{0}]$

(b) Quasi-two-body decays ${{B}^{0}_{s}}\rightarrow
f_{0}(500)^{0}(D_{s}^{\mp}K^{\pm})_{S}$
---
${{B}^{0}_{s}}\rightarrow f_{0}(500)^{0}(D_{s}^{\mp}K^{\pm})_{P}$
${{B}^{0}_{s}}\rightarrow f_{0}(980)^{0}(D_{s}^{\mp}K^{\pm})_{S}$
${{B}^{0}_{s}}\rightarrow f_{0}(980)^{0}(D_{s}^{\mp}K^{\pm})_{P}$
${{B}^{0}_{s}}\rightarrow\rho(770)^{0}(D_{s}^{\mp}K^{\pm})_{S}$
${{B}^{0}_{s}}[S,P,D]\rightarrow\rho(770)^{0}\,(D_{s}^{\mp}K^{\pm})_{P}$
${{B}^{0}_{s}}\rightarrow f_{2}(1270)^{0}(D_{s}^{\mp}K^{\pm})_{S}$
${{B}^{0}_{s}}\rightarrow K^{*}(892)^{0}\,(D_{s}^{\mp}\pi^{\pm})_{S}$
${{B}^{0}_{s}}[S,P,D]\rightarrow K^{*}(892)^{0}\,(D_{s}^{\mp}\pi^{\pm})_{P}$
${{B}^{0}_{s}}\rightarrow K_{0}^{*}(1430)^{0}\,(D_{s}^{\mp}\pi^{\pm})_{S}$
${{B}^{0}_{s}}\rightarrow K_{2}^{*}(1430)^{0}\,(D_{s}^{\mp}\pi^{\pm})_{S}$
${{B}^{0}_{s}}\rightarrow(D_{s}^{\mp}K^{\pm})_{S}\,(\pi^{\pm}\pi^{\mp})_{S}$

The full list of intermediate-state amplitudes considered for the model
building procedure is given in Table B.1. In addition to those, the presence
of unknown states decaying into $D_{s}^{-}\pi^{+}\pi^{-}$,
$D_{s}^{-}K^{+}\pi^{-}$, $D_{s}^{-}\pi^{+}$ or $D_{s}^{-}K^{+}$ is
investigated. These states are typically fitted with huge decay widths
(effectively indistinguishable from a non-resonant state) and small decay
fractions. Therefore, such states (if existent in the relevant mass range) are
not resolvable with the current data sample.

Tables B.2 and B.3 list the moduli and phases of the complex amplitude
coefficients obtained by fitting the baseline model to the background-
subtracted
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
signal candidates. The systematic uncertainties are summarised in Table B.4.
The fit results are used to derive the amplitude ratio, $r_{i}$, and strong-
phase difference, $\delta_{i}$, for a given decay channel, defined as

$\displaystyle r_{i}$
$\displaystyle=r\,\left|\frac{a_{i}^{u}}{a_{i}^{c}}\right|\,\sqrt{\frac{\int|A^{c}(\mathbf{x})|^{2}\,\text{d}\Phi_{4}(\mathbf{x})}{\int|A^{u}(\mathbf{x})|^{2}\,\text{d}\Phi_{4}(\mathbf{x})}},$
$\displaystyle\delta_{i}$
$\displaystyle=\delta-\arg(a_{i}^{c})+\arg(a_{i}^{u})-\arg\left(\int
A^{c}(\mathbf{x})^{*}\,A^{u}(\mathbf{x})\,\text{d}\Phi_{4}(\mathbf{x})\right).$

The values are given in Table B.5. Tables B.6 ($b\rightarrow c$ amplitudes)
and B.7 ($b\rightarrow u$ amplitudes) list the interference fractions ordered
by magnitude, for the baseline model. Figures 10 and 11 show additional fit
projections of the baseline model. Here, the acoplanarity angle,
${\phi}_{K^{+}\pi^{-},D_{s}^{-}\pi^{+}}$, is defined as the angle between the
two decay planes formed by the $K^{+}\pi^{-}$ system and the
$D_{s}^{-}{{\pi}^{+}}$ system in the ${{B}^{0}_{s}}$ rest frame; boosting into
the rest frames of the two-body systems defining these decay planes, the two
helicity variables $\cos\theta_{K^{+}\pi^{-}}$ and
$\cos\theta_{D_{s}^{-}{{\pi}^{+}}}$ are defined as the cosine of the angle of
the $K^{+}$ or $D_{s}^{-}$ momentum with the ${{B}^{0}_{s}}$ flight direction.

The decay fractions for several alternative models are summarised in Tables
B.8 and B.9.

Table B.2:  Moduli and phases of the amplitude coefficients for decays via
$b\rightarrow c$ and $b\rightarrow u$ quark-level transitions. The
uncertainties are statistical and systematic.

Decay channel | $A^{c}(\mathbf{x})$ | $A^{u}(\mathbf{x})$
---|---|---
| $|a_{i}^{c}|$ | $\arg(a_{i}^{c})[^{\circ}]$ | $|a_{i}^{u}|$ | $\arg(a_{i}^{u})[^{\circ}]$
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K_{1}(1270)^{\pm}$ | 0.50$\,\pm\,$ | 0.05 $\,\pm\,$ | 0.05 | 24$\,\pm\,$ | 4 $\,\pm\,$ | 6 | 0.51$\,\pm\,$ | 0.27 $\,\pm\,$ | 0.25 |  $-129$$\,\pm\,$ | 18 $\,\pm\,$ | 26
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K_{1}(1400)^{\pm}$ | 1.0 | 0.0 | 1.0 | 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K^{*}(1410)^{\pm}$ | 0.45$\,\pm\,$ | 0.05 $\,\pm\,$ | 0.05 | 53$\,\pm\,$ | 6 $\,\pm\,$ | 6 | 0.81$\,\pm\,$ | 0.20 $\,\pm\,$ | 0.15 | 2$\,\pm\,$ | 13 $\,\pm\,$ | 11
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K(1460)^{\pm}$ |  |  | 0.78$\,\pm\,$ | 0.20 $\,\pm\,$ | 0.16 |  $-92$$\,\pm\,$ | 10 $\,\pm\,$ | 15
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,K^{*}(892)^{0}$ | 0.40$\,\pm\,$ | 0.04 $\,\pm\,$ | 0.04 |  $-33$$\,\pm\,$ | 5 $\,\pm\,$ | 4 | 1.21$\,\pm\,$ | 0.26 $\,\pm\,$ | 0.22 |  $-7$$\,\pm\,$ | 8 $\,\pm\,$ | 17
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,K^{\pm})_{P}\,\rho(770)^{0}$ | 0.12$\,\pm\,$ | 0.03 $\,\pm\,$ | 0.03 | 109$\,\pm\,$ | 9 $\,\pm\,$ | 14 |  |

Table B.3:  Moduli and phases of the amplitude coefficients for cascade decays. The amplitude coefficients are defined relative to the respective three-body production amplitude coefficients in Table B.2 and are shared among $b\rightarrow c$ and $b\rightarrow u$ transitions. The uncertainties are statistical and systematic.  Decay channel | $|a_{i}|$ | $\arg(a_{i})[^{\circ}]$
---|---|---
$K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0}$ | 1.0 | 0.0
$K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\pi^{\pm}$ | 0.90$\,\pm\,$ | 0.09 $\,\pm\,$ | 0.10 | 46$\,\pm\,$ | 7 $\,\pm\,$ | 4
$K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)^{0}{{\pi}^{\pm}}$ | 0.46$\,\pm\,$ | 0.05 $\,\pm\,$ | 0.03 | 120$\,\pm\,$ | 5 $\,\pm\,$ | 6
$K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}}$ | 1.0 | 0.0
$K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0}$ | 0.66$\,\pm\,$ | 0.05 $\,\pm\,$ | 0.05 |  $-165$$\,\pm\,$ | 6 $\,\pm\,$ | 5

Table B.4: Systematic uncertainties on the fit parameters of the full time-
dependent 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) decay-time bias, 7)
nuisance asymmetries, 8) $\Delta m_{s}$, 9) phase-space acceptance, 10)
acceptance factorisation, 11) lineshape models, 12) masses and widths of
resonances, 13) form factor.

Parameter | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | Total
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K_{1}(1270)^{\pm}\,|a_{i}^{c}|$ | 0.09 | 0.32 | 0.17 | 0.01 | 0.36 | 0.05 | 0.00 | 0.02 | 0.60 | 0.11 | 0.31 | 0.07 | 0.48 | 0.99
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K_{1}(1270)^{\pm}\,\arg(a_{i}^{c})$ | 0.10 | 0.37 | 0.12 | 0.03 | 0.13 | 0.05 | 0.05 | 0.02 | 0.40 | 0.31 | 0.98 | 0.21 | 0.58 | 1.33
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K_{1}(1270)^{\pm}\,|a_{i}^{u}|$ | 0.17 | 0.26 | 0.14 | 0.04 | 0.41 | 0.07 | 0.01 | 0.03 | 0.35 | 0.18 | 0.60 | 0.02 | 0.26 | 0.94
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K_{1}(1270)^{\pm}\,\arg(a_{i}^{u})$ | 0.12 | 0.77 | 0.21 | 0.08 | 0.39 | 0.05 | 0.11 | 0.04 | 0.66 | 0.55 | 0.52 | 0.07 | 0.54 | 1.46
$K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}}\,|a_{i}|$ | 0.01 | 0.32 | 0.27 | 0.01 | 0.31 | 0.01 | 0.00 | 0.01 | 0.17 | 0.36 | 0.55 | 0.15 | 0.55 | 1.03
$K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}}\,\arg(a_{i})$ | 0.14 | 0.11 | 0.11 | 0.01 | 0.13 | 0.02 | 0.00 | 0.00 | 0.19 | 0.14 | 0.33 | 0.07 | 0.34 | 0.59
$K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)^{0}\,{{\pi}^{\pm}}\,|a_{i}|$ | 0.10 | 0.48 | 0.14 | 0.00 | 0.05 | 0.01 | 0.00 | 0.00 | 0.10 | 0.25 | 0.42 | 0.05 | 0.17 | 0.73
$K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)^{0}\,{{\pi}^{\pm}}\,\arg(a_{i})$ | 0.24 | 0.33 | 0.53 | 0.00 | 0.14 | 0.01 | 0.00 | 0.00 | 0.27 | 0.46 | 0.16 | 0.08 | 0.85 | 1.22
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K^{*}(1410)^{\pm}\,|a_{i}^{c}|$ | 0.27 | 0.35 | 0.14 | 0.02 | 0.29 | 0.04 | 0.00 | 0.01 | 0.87 | 0.32 | 0.09 | 0.10 | 0.20 | 1.10
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K^{*}(1410)^{\pm}\,\arg(a_{i}^{c})$ | 0.32 | 0.29 | 0.10 | 0.01 | 0.40 | 0.02 | 0.00 | 0.01 | 0.23 | 0.21 | 0.46 | 0.06 | 0.64 | 1.04
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K^{*}(1410)^{\pm}\,|a_{i}^{u}|$ | 0.17 | 0.16 | 0.17 | 0.03 | 0.25 | 0.07 | 0.07 | 0.03 | 0.38 | 0.32 | 0.26 | 0.05 | 0.34 | 0.77
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K^{*}(1410)^{\pm}\,\arg(a_{i}^{u})$ | 0.21 | 0.42 | 0.41 | 0.05 | 0.13 | 0.01 | 0.01 | 0.01 | 0.07 | 0.26 | 0.21 | 0.05 | 0.47 | 0.87
$K^{*}(1410)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0}\,|a_{i}|$ | 0.58 | 0.37 | 0.26 | 0.00 | 0.19 | 0.02 | 0.10 | 0.00 | 0.43 | 0.45 | 0.31 | 0.04 | 0.06 | 1.03
$K^{*}(1410)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0}\,\arg(a_{i})$ | 0.05 | 0.42 | 0.24 | 0.00 | 0.19 | 0.04 | 0.06 | 0.00 | 0.58 | 0.03 | 0.22 | 0.05 | 0.19 | 0.84
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K(1460)^{\pm}\,|a_{i}^{u}|$ | 0.02 | 0.22 | 0.28 | 0.05 | 0.13 | 0.04 | 0.01 | 0.02 | 0.22 | 0.18 | 0.63 | 0.04 | 0.19 | 0.82
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K(1460)^{\pm}\,\arg(a_{i}^{u})$ | 0.14 | 0.51 | 0.10 | 0.06 | 0.30 | 0.01 | 0.19 | 0.02 | 0.39 | 0.48 | 0.11 | 0.10 | 1.15 | 1.46
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}\,|a_{i}^{c}|$ | 0.06 | 0.37 | 0.11 | 0.01 | 0.13 | 0.05 | 0.01 | 0.02 | 0.09 | 0.32 | 0.58 | 0.09 | 0.62 | 1.00
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}\,\arg(a_{i}^{c})$ | 0.03 | 0.60 | 0.29 | 0.03 | 0.19 | 0.04 | 0.02 | 0.02 | 0.22 | 0.10 | 0.43 | 0.06 | 0.29 | 0.90
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}\,|a_{i}^{u}|$ | 0.08 | 0.26 | 0.15 | 0.05 | 0.20 | 0.02 | 0.01 | 0.02 | 0.18 | 0.26 | 0.62 | 0.04 | 0.33 | 0.86
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}\,\arg(a_{i}^{u})$ | 0.20 | 0.75 | 0.66 | 0.05 | 0.54 | 0.06 | 0.15 | 0.04 | 0.36 | 0.68 | 0.31 | 0.13 | 1.37 | 1.98
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{P}\,\,\rho(770)^{0}\,|a_{i}^{c}|$ | 0.11 | 0.46 | 0.14 | 0.01 | 0.12 | 0.03 | 0.00 | 0.00 | 0.32 | 0.05 | 0.71 | 0.06 | 0.37 | 1.01
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{P}\,\,\rho(770)^{0}\,\arg(a_{i}^{c})$ | 0.18 | 0.66 | 0.66 | 0.01 | 0.23 | 0.01 | 0.00 | 0.01 | 0.20 | 0.22 | 1.12 | 0.06 | 0.47 | 1.59

Table B.5: Amplitude ratio and strong-phase difference for a given decay channel. The uncertainties are statistical and systematic. Decay channel | $r_{i}$ | $\delta_{i}[^{\circ}]$
---|---|---
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K_{1}(1270)^{\pm}$ | 0.31$\,\pm\,$ | 0.20 $\,\pm\,$ | 0.16 |  $-109$$\,\pm\,$ | 22 $\,\pm\,$ | 28
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K_{1}(1400)^{\pm}$ | 0.30$\,\pm\,$ | 0.03 $\,\pm\,$ | 0.02 | 44$\,\pm\,$ | 10 $\,\pm\,$ | 4
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,K^{*}(1410)^{\pm}$ | 0.54$\,\pm\,$ | 0.18 $\,\pm\,$ | 0.15 |  $-7$$\,\pm\,$ | 17 $\,\pm\,$ | 14
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ | 0.92$\,\pm\,$ | 0.24 $\,\pm\,$ | 0.20 | 69$\,\pm\,$ | 15 $\,\pm\,$ | 18
Table B.6: Interference fractions (ordered by magnitude) of the $b\rightarrow
c$ intermediate-state amplitudes included in the baseline model. Only the
statistical uncertainties are given.

Decay channel $i$ |  Decay channel $j$ | $I_{ij}[\%]$
---|---|---
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ |  $-15.4$ $\pm$ 3.1
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  $-12.2$ $\pm$ 5.8
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ |  $-10.2$ $\pm$ 1.2
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  5.8 $\pm$ 0.5
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  4.1 $\pm$ 0.3
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,K^{\pm})_{P}\,\,\rho(770)^{0}$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  3.1 $\pm$ 0.5
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,K^{\pm})_{P}\,\,\rho(770)^{0}$ |  1.3 $\pm$ 0.3
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  $-1.3$ $\pm$ 0.9
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,K^{\pm})_{P}\,\,\rho(770)^{0}$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ |  $-0.5$ $\pm$ 0.3
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ | ${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,K^{\pm})_{P}\,\,\rho(770)^{0}$ |  $-0.2$ $\pm$ 0.1
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  $-0.1$ $\pm$ 0.5
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  $-0.1$ $\pm$ 0.1
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ |  $-0.1$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,K^{\pm})_{P}\,\,\rho(770)^{0}$ |  0.0 $\pm$ 0.1
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ |  0.0 $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  0.0 $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  0.0 $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  0.0 $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,K^{\pm})_{P}\,\,\rho(770)^{0}$ |  0.0 $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ | ${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,K^{\pm})_{P}\,\,\rho(770)^{0}$ |  0.0 $\pm$ 0.0

Table B.7: Interference fractions (ordered by magnitude) of the $b\rightarrow
u$ intermediate-state amplitudes included in the baseline model. Only the
statistical uncertainties are given.

Decay channel $i$ |  Decay channel $j$ | $I_{ij}[\%]$
---|---|---
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ |  9.5 $\pm$ 2.6
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  9.2 $\pm$ 3.2
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ |  $-5.5$ $\pm$ 0.9
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  4.1 $\pm$ 0.8
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ |  $-3.7$ $\pm$ 5.9
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  $-1.2$ $\pm$ 0.6
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  $-0.4$ $\pm$ 0.3
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  0.1 $\pm$ 0.3
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ |  $-0.1$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  $0.0$ $\pm$ 0.2
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  0.0 $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  0.0 $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  0.0 $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  0.0 $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ |  0.0 $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ |  $0.0$ $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  0.0 $\pm$ 0.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{\pm}\,\rho(770)^{0})$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)\,{{\pi}^{\pm}})$ |  0.0 $\pm$ 0.0

Figure 10:  Invariant-mass and angular distributions of background-subtracted
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
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. 6.

Figure 11: Invariant-mass and angular distributions of background-subtracted
${{B}^{0}_{s}}\\!\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}{{\pi}^{\pm}}{{\pi}^{\mp}}$
candidates (data points) and fit projections (blue solid line). Incoherent
contributions from intermediate-state components are overlaid, colour coded as
in Fig. 7.

Table B.8: 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.

|  | Alt.1 | Alt.2 | Alt.3 | Alt.4 | Alt.5 | Alt.6
---|---|---|---|---|---|---|---
$b\rightarrow c$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 12.2 | 12.0 | 13.9 | 11.5 | 14.7 | 12.8
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}[D]\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  | 0.4 |  |  |  |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ | 16.2 | 14.6 | 21.7 | 15.6 | 18.6 | 16.5
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(1450)^{0})$ |  | 0.3 |  |  |  |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)^{0}\,{{\pi}^{\pm}})$ | 3.9 | 3.5 | 3.7 | 3.6 | 3.0 | 3.6
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 43.1 | 61.1 | 59.8 | 57.7 | 69.5 | 60.6
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ |  |  | 0.5 |  |  |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 12.9 | 14.0 | 13.6 | 13.8 | 24.2 | 13.5
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ | 4.9 | 5.8 | 4.9 | 5.3 | 3.6 | 5.2
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 1.7 |  |  | 1.3 |  |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ |  |  |  | 0.2 |  |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow{{K}^{\pm}}\,f_{0}(500)^{0})$ |  |  |  | 0.6 |  |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1680)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  |  |  |  | 5.5 |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1680)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ |  |  |  |  | 1.9 |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{2}(1770)\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  |  |  |  |  | 0.4
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ | 8.8 | 9.3 | 9.5 | 7.7 | 10.2 | 9.2
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{P}\,\,\rho(770)^{0}$ | 1.3 | 0.7 | 0.1 | 0.6 | 0.2 | 0.5
$b\rightarrow u$ | Sum | 105.1 | 121.7 | 127.6 | 118.0 | 151.5 | 122.1
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 4.5 | 2.4 | 1.1 | 4.9 | 2.0 | 3.4
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}[D]\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  | 0.1 |  |  |  |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ | 6.0 | 2.9 | 1.6 | 6.6 | 2.6 | 4.4
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(1450)^{0})$ |  | 0.1 |  |  |  |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)^{0}\,{{\pi}^{\pm}})$ | 1.4 | 0.7 | 0.3 | 1.5 | 0.4 | 1.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 23.3 | 18.9 | 26.2 | 10.3 | 28.2 | 23.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ |  |  | 0.2 |  |  |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 14.8 | 13.7 | 12.7 | 7.2 | 2.4 | 11.5
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ | 5.6 | 5.7 | 4.6 | 2.8 | 0.4 | 4.4
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 12.9 | 10.8 | 12.6 | 8.1 | 12.1 | 11.9
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ |  |  |  | 1.0 |  |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow{{K}^{\pm}}\,f_{0}(500)^{0})$ |  |  |  | 3.7 |  |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1680)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  |  |  |  | 5.6 |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1680)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ |  |  |  |  | 2.0 |
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{2}(1770)\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ |  |  |  |  |  | 0.9
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ | 32.5 | 33.4 | 31.3 | 38.3 | 27.4 | 30.7
Sum | 103.9 | 88.7 | 90.7 | 84.4 | 83.0 | 91.4
| $m_{K_{1}(1400)}\,[\text{MeV}]$ | 1401 | 1406 | 1404 | 1401 | 1397 | 1408
| $\Gamma_{K_{1}(1400)}\,[\text{MeV}]$ | 189 | 193 | 192 | 196 | 199 | 193
| $m_{K^{*}(1410)}\,[\text{MeV}]$ | 1434 | 1437 | 1436 | 1442 | 1419 | 1434
| $\Gamma_{K^{*}(1410)}\,[\text{MeV}]$ | 391 | 400 | 386 | 382 | 453 | 395
| $r$ | 0.50 | 0.55 | 0.59 | 0.48 | 0.54 | 0.58
| $\kappa$ | 0.64 | 0.75 | 0.74 | 0.77 | 0.71 | 0.75
| $\delta\,[^{\circ}]$ | $-4$ | $-15$ | $-14$ | $-24$ | $-19$ | $-18$
| $\gamma-2\beta_{s}\,[^{\circ}]$ | 40 | 44 | 42 | 50 | 52 | 51

Table B.9: 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.

|  | Alt.7 | Alt.8 | Alt.9 | Alt.10 | Alt.11 | Alt.12
---|---|---|---|---|---|---|---
$b\rightarrow c$ | ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 13.3 | 12.8 | 15.5 | 24.0 | 11.2 | 13.3
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ | 17.5 | 17.5 | 19.1 | 14.9 | 17.7 | 19.9
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)^{0}\,{{\pi}^{\pm}})$ | 3.3 | 4.0 | 4.3 | 2.8 | 3.2 | 3.7
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 66.0 | 93.1 | 55.5 | 77.9 | 44.7 | 63.5
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 12.5 | 16.5 | 13.3 | 13.7 | 15.1 | 12.7
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ | 5.4 | 5.9 | 6.4 | 4.7 | 5.8 | 5.1
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{S}\,\,K^{*}(892)^{0}$ |  |  |  | 5.1 |  |
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ | 11.4 | 19.8 | 8.5 |  | 5.5 | 10.8
${{B}^{0}_{s}}[P]\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ |  | 1.7 |  |  |  |
${{B}^{0}_{s}}[D]\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ |  | 4.7 |  |  |  |
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{S}\,\,f_{0}(500)^{0}$ |  |  |  |  | 1.8 |
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{S}\,\,f_{0}(980)^{0}$ |  |  |  |  | 1.5 |
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{S}\,\,f_{2}(1270)^{0}$ |  |  |  |  | 0.1 |
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{S}\,\,\rho(770)^{0}$ | 0.3 |  |  |  |  |
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{P}\,\,\rho(770)^{0}$ |  | 1.6 | 3.5 | 1.3 | 0.5 |
${{B}^{0}_{s}}[P]\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{P}\,\,\rho(770)^{0}$ |  |  | 0.2 |  |  |
${{B}^{0}_{s}}[D]\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{P}\,\,\rho(770)^{0}$ |  |  | 0.9 |  |  |
$b\rightarrow u$ | Sum | 129.6 | 177.5 | 127.2 | 144.4 | 107.1 | 129.0
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 3.3 | 5.2 | 1.1 | 10.6 | 2.5 | 2.5
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ | 4.4 | 7.2 | 1.4 | 6.6 | 3.9 | 3.7
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1270)^{\pm}\rightarrow K^{*}_{0}(1430)^{0}\,{{\pi}^{\pm}})$ | 0.8 | 1.6 | 0.3 | 1.2 | 0.7 | 0.7
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K_{1}(1400)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 19.8 | 35.8 | 27.8 | 7.7 | 17.1 | 22.4
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 14.2 | 9.8 | 9.9 | 11.7 | 4.0 | 14.5
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K^{*}(1410)^{\pm}\rightarrow{{K}^{\pm}}\,\rho(770)^{0})$ | 6.1 | 3.5 | 4.8 | 4.0 | 1.5 | 5.8
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}\,(K(1460)^{\pm}\rightarrow K^{*}(892)^{0}\,{{\pi}^{\pm}})$ | 11.7 | 2.6 | 13.2 | 8.8 | 13.6 | 12.3
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{S}\,\,K^{*}(892)^{0}$ |  |  |  | 22.3 |  |
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ | 25.8 | 68.5 | 33.6 |  | 43.1 | 29.3
${{B}^{0}_{s}}[P]\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ |  | 5.8 |  |  |  |
${{B}^{0}_{s}}[D]\rightarrow({{D}^{\mp}_{s}}\,{{\pi}^{\pm}})_{P}\,\,K^{*}(892)^{0}$ |  | 16.3 |  |  |  |
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{S}\,\,f_{0}(500)^{0}$ |  |  |  |  | 0.7 |
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{S}\,\,f_{0}(980)^{0}$ |  |  |  |  | 0.6 |
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{S}\,\,f_{2}(1270)^{0}$ |  |  |  |  | 0.0 |
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{S}\,\,\rho(770)^{0}$ | 0.4 |  |  |  |  |
${{B}^{0}_{s}}\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{P}\,\,\rho(770)^{0}$ |  |  | 4.4 |  |  |
${{B}^{0}_{s}}[P]\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{P}\,\,\rho(770)^{0}$ |  |  | 0.3 |  |  |
| ${{B}^{0}_{s}}[D]\rightarrow({{D}^{\mp}_{s}}\,{{K}^{\pm}})_{P}\,\,\rho(770)^{0}$ |  |  | 1.1 |  |  |
| Sum | 86.5 | 156.3 | 97.9 | 73.0 | 87.8 | 91.0
| $m_{K_{1}(1400)}\,[\text{MeV}]$ | 1405 | 1398 | 1404 | 1365 | 1406 | 1406
| $\Gamma_{K_{1}(1400)}\,[\text{MeV}]$ | 193 | 247 | 188 | 203 | 184 | 190
| $m_{K^{*}(1410)}\,[\text{MeV}]$ | 1430 | 1443 | 1447 | 1440 | 1427 | 1434
| $\Gamma_{K^{*}(1410)}\,[\text{MeV}]$ | 406 | 432 | 419 | 373 | 406 | 399
| $r$ | 0.57 | 0.75 | 0.58 | 0.45 | 0.54 | 0.57
| $\kappa$ | 0.73 | 0.81 | 0.73 | 0.70 | 0.75 | 0.73
| $\delta\,[^{\circ}]$ | $-16$ | $-18$ | $-12$ | $-14$ | $-19$ | $-14$
| $\gamma-2\beta_{s}\,[^{\circ}]$ | 41 | 45 | 44 | 52 | 53 | 41

##  Appendix C Interpretation of the $C\\!P$ coefficients

The statistical and systematic correlation matrices of the $C\\!P$
coefficients obtained from the phase-space integrated fit are given in Table
C.2 and Table C.2.

Figure 12 visualises the measured $C\\!P$ coefficients in the complex plane,
where $\lambda_{f}\equiv(q/p)\,(\int
A^{u}(\mathbf{x})\,\mathrm{d}\Phi_{4}/\int
A^{c}(\mathbf{x})\,\mathrm{d}\Phi_{4})$. The points determined by
$(-A_{f}^{\Delta\Gamma},S_{f})$ and
$(-A_{\bar{f}}^{\Delta\Gamma},S_{\bar{f}})$ are proportional to
$r\,\kappa\,e^{\pm\delta-(\gamma-2\beta_{s})}$, whilst an additional
constraint on $r$ arises from $C_{f}$.

Table C.1: Statistical correlation of the $C\\!P$ coefficients.

| $C_{f}$ | $A^{\Delta\Gamma}_{f}$ | $A^{\Gamma\Delta}_{\bar{f}}$ | $S_{f}$ | $S_{\bar{f}}$
---|---|---|---|---|---
$C_{f}$ | 1.000 | 0.135 | 0.168 | 0.040 | 0.042
$A^{\Delta\Gamma}_{f}$ |  | 1.000 | 0.504 | $-0.082$ | $-0.052$
$A^{\Delta\Gamma}_{\bar{f}}$ |  |  | 1.000 | $-0.036$ | $-0.113$
$S_{f}$ |  |  |  | 1.000 | 0.014
$S_{\bar{f}}$ |  |  |  |  | 1.000

Table C.2: Systematic correlation of the $C\\!P$ coefficients.

| $C_{f}$ | $A^{\Delta\Gamma}_{f}$ | $A^{\Gamma\Delta}_{\bar{f}}$ | $S_{f}$ | $S_{\bar{f}}$
---|---|---|---|---|---
$C_{f}$ | 1.000 | 0.039 | 0.042 | $-0.136$ | $-0.006$
$A^{\Delta\Gamma}_{f}$ |  | 1.000 | 0.208 | $-0.031$ | $-0.023$
$A^{\Delta\Gamma}_{\bar{f}}$ |  |  | 1.000 | $-0.021$ | $-0.035$
$S_{f}$ |  |  |  | 1.000 | $-0.278$
$S_{\bar{f}}$ |  |  |  |  | 1.000

Figure 12:  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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