EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2017-320 LHCb-PAPER-2017-048 22 December 2017

First measurement of the $C\\!P$-violating phase
$\phi_{s}^{{d}{\overline{d}}}$ in
${{{B}^{0}_{s}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})}$
decays

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

A flavour-tagged decay-time-dependent amplitude analysis of
${{{B}^{0}_{s}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})}$
decays is presented in the ${{K}^{\pm}}{{\pi}^{\mp}}$ mass range from 750 to
1600${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The analysis uses $pp$
collision data collected with the LHCb detector at centre-of-mass energies of
$7$ and $8\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an integrated
luminosity of $3.0\mbox{\,fb}^{-1}$. Several quasi-two-body decay modes are
considered, corresponding to ${K}^{\pm}$${\pi}^{\mp}$ combinations with spin
0, 1 and 2, which are dominated by the ${K}_{0}^{*}(800)^{0}$ and
${K}_{0}^{*}(1430)^{0}$, the ${K}^{*}(892)^{0}$ and the
${K}_{2}^{*}(1430)^{0}$ resonances, respectively. The longitudinal
polarisation fraction for the
${{B}^{0}_{s}}\rightarrow{{K}^{*}(892)^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*}(892)^{0}}$ decay is measured as
$f_{\rm L}=0.208\pm 0.032\pm 0.046$, where the first uncertainty is
statistical and the second is systematic. The first measurement of the mixing-
induced $C\\!P$-violating phase, $\phi_{s}^{d\bar{d}}$, in
${b}\rightarrow{d}{\overline{d}}{s}$ transitions is performed, yielding a
value of $\phi_{s}^{d\bar{d}}=-0.10\pm 0.13\mathrm{\,(stat)}\pm
0.14\mathrm{\,(syst)}\mathrm{\,rad}$.

Published in JHEP 03 (2018) 140

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

##  1 Introduction

The $C\\!P$-violating weak phases $\phi_{{s}}$ arise in the interference
between the amplitudes of ${B}^{0}_{s}$ mesons directly decaying to $C\\!P$
eigenstates and those decaying to the same final state after
${B}^{0}_{s}$–${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$
oscillation. The ${{{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}}$ decay,111Throughout this
article, charge conjugation is implied and ${K}^{*0}$ refers to the
${K}^{*}(892)^{0}$ resonance, unless otherwise stated. which in the Standard
Model (SM) is dominated by the gluonic loop diagram shown in Fig. 1, has been
discussed extensively in the literature as a benchmark test for the SM and as
an excellent probe for physics beyond the SM [1, 2, 3, 4, 5, 6, 7]. New heavy
particles entering the loop would introduce additional amplitudes and modify
properties of the decay from their SM values. In general, the weak phase
$\phi_{{s}}$ depends on the ${B}^{0}_{s}$ decay channel under consideration,
and can be different between channels as it depends on the contributions from
tree- and loop-level processes. The notation $\phi_{s}^{{d}{\overline{d}}}$ is
used when referring to the weak phase measured in
${b}\rightarrow{d}{\overline{d}}{s}$ transitions. For
${b}\rightarrow{c}{\overline{c}}{s}$ transitions, e.g.
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ and
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ decays, the weak phase
$\phi_{s}^{{c}{\overline{c}}}$ has been measured by several experiments [8, 9,
10, 11]. The world average reported by HFLAV,
$\phi_{s}^{{c}{\overline{c}}}=-0.021\pm 0.031\mathrm{\,rad}$ [8], is dominated
by the LHCb measurement $\phi_{s}^{{c}{\overline{c}}}=-0.010\pm
0.039\mathrm{\,rad}$ [9]. The LHCb collaboration has also measured the
$\phi_{s}^{{s}{\overline{s}}}$ phase in ${{B}^{0}_{s}}\rightarrow\phi\phi$
transitions [12], reporting a value of $\phi_{s}^{{s}{\overline{s}}}=-0.17\pm
0.15\mathrm{\,rad}$.

Figure 1: Leading-order SM Feynman diagram of the
${{{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}}$ decay.

The decay ${{{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}}$, with ${{K}^{*0}}\rightarrow
K^{+}\pi^{-}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}\rightarrow K^{-}\pi^{+}$, was
first observed by the LHCb collaboration, based on $pp$ collision data
corresponding to an integrated luminosity of $35\mbox{\,pb}^{-1}$ at a centre-
of-mass energy $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ [13]. A branching
fraction and a final-state polarisation analysis were reported. An updated
analysis of the
${{{B}^{0}_{s}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})}$ decay
was performed by LHCb using $1.0$$\mbox{\,fb}^{-1}$ of data at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ [14]. In both analyses, the
invariant mass of the two $K\pi$ pairs222Hereafter the notation $K\pi$ will
stand for both $K^{+}\pi^{-}$ and $K^{-}\pi^{+}$ pairs. was restricted to a
window of $\pm 150$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the known
${K}^{*0}$ mass. This publication reports the first decay-time-dependent
amplitude analysis of
${{{B}^{0}_{s}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})}$
decays using a $K\pi$ mass window that extends from 750 to
1600${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, approximately corresponding to
the region between the $K\pi$ production threshold and the
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ resonance. At the current level of
sensitivity, the assumption of common $C\\!P$-violating parameters for the
contributing amplitudes is appropriate. Consequently, such a wide window
provides a four-fold increase of the signal sample size with respect to the
narrow window of $150{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the
${K}^{*0}$ mass. The analysis uses $pp$ collision data collected by LHCb in
2011 and 2012 at $\sqrt{s}=7$ and $8\mathrm{\,Te\kern-1.00006ptV}$,
corresponding to an integrated luminosity of $3.0\mbox{\,fb}^{-1}$. In this
study, nine different quasi-two-body decay channels are considered,
corresponding to the different possible combinations of $K\pi$ pairs with spin
0, 1 or 2. Additional contributions were studied and found to be negligible in
the phase-space region considered in this analysis. The $K\pi$ spectrum is
dominated by the $K_{0}^{*}(800)^{0}$, ${K}_{0}^{*}(1430)^{0}$,
${K}^{*}(892)^{0}$ and ${K}_{2}^{*}(1430)^{0}$ resonances. Angular momentum
conservation in the decay allows for one single amplitude in modes involving
at least one scalar $K\pi$ pair, three amplitudes for vector-vector or vector-
tensor decays and five amplitudes for a tensor-tensor decay. These
possibilities are listed in Table 1. There is a physical difference between
decay pairs of the form scalar-vector and vector-scalar. Namely, in the used
convention, the spectator quark from the $B_{s}^{0}$ decay (see Fig. 1) always
ends up in the second $K\pi$ pair. The $C\\!P$-averaged fractions of the
contributing amplitudes, $f_{i}$, as well as their strong-phase differences,
$\delta_{i}$, are determined together with the $C\\!P$-violating weak phase
$\phi_{s}^{{d}{\overline{d}}}$ and a parameter that accounts for the amount of
$C\\!P$ violation in decay, $|\lambda|$. This is the first time that the weak
phase in ${b}\rightarrow{d}{\overline{d}}{s}$ transitions has been measured.
It is also the first time that the tensor components in the
$(K^{+}\pi^{-})(K^{-}\pi^{+})$ system have been studied.

Table 1: Quasi-two-body decay channels and corresponding polarisation amplitudes contributing to the ${{{B}^{0}_{s}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})}$ final state in the $K\pi$ mass window from 750 to 1600${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The different contributions are identified by the spin $j_{1}$ ($j_{2}$) of the $K^{+}\pi^{-}$ ($K^{-}\pi^{+}$) pair and the helicity $h$. In cases where more than one amplitude contributes, the polarisations are defined as being longitudinal, parallel, or perpendicular, which are then denoted by $0$, $\parallel$ and $\perp$ respectively, following the definitions given in Ref. [15]. The subscripts 1 and 2 in the parallel and perpendicular helicities of the tensor-tensor component denote different spin states leading to a parallel or a perpendicular configuration, as discussed in Appendix A. Decay | Mode | $j_{1}$ | $j_{2}$ |  Allowed values of $h$ |  Number of amplitudes
---|---|---|---|---|---
${{B}^{0}_{s}}\rightarrow(K^{+}\pi^{-})^{*}_{0}(K^{-}\pi^{+})^{*}_{0}$ | scalar-scalar | 0 | 0 |  0 |  1
${{B}^{0}_{s}}\rightarrow(K^{+}\pi^{-})^{*}_{0}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*}(892)^{0}}$ | scalar-vector | 0 | 1 |  0 |  1
${{B}^{0}_{s}}\rightarrow{{K}^{*}(892)^{0}}(K^{-}\pi^{+})^{*}_{0}$ | vector-scalar | 1 | 0 |  0 |  1
${{B}^{0}_{s}}\rightarrow(K^{+}\pi^{-})^{*}_{0}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}_{2}^{*}(1430)^{0}}$ | scalar-tensor | 0 | 2 |  0 |  1
${{B}^{0}_{s}}\rightarrow{{K}_{2}^{*}(1430)^{0}}(K^{-}\pi^{+})^{*}_{0}$ | tensor-scalar | 2 | 0 |  0 |  1
${{B}^{0}_{s}}\rightarrow{{K}^{*}(892)^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*}(892)^{0}}$ | vector-vector | 1 | 1 |  0, $\parallel$, $\perp$ |  3
${{B}^{0}_{s}}\rightarrow{{K}^{*}(892)^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}_{2}^{*}(1430)^{0}}$ | vector-tensor | 1 | 2 |  0, $\parallel$, $\perp$ |  3
${{B}^{0}_{s}}\rightarrow{{K}_{2}^{*}(1430)^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*}(892)^{0}}$ | tensor-vector | 2 | 1 |  0, $\parallel$, $\perp$ |  3
${{B}^{0}_{s}}\rightarrow{{K}_{2}^{*}(1430)^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}_{2}^{*}(1430)^{0}}$ | tensor-tensor | 2 | 2 |  0, $\parallel_{1}$, $\perp_{1}$, $\parallel_{2}$, $\perp_{2}$ |  5

##  2 Phenomenology

The phenomenon of quark mixing means that a ${B}^{0}_{s}$ meson can oscillate
into its antiparticle equivalent, ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$. Consequently, the physical
states, $B^{0}_{s,\rm{H}}$ (heavy) and $B^{0}_{s,\rm{L}}$ (light), which have
mass and decay width differences defined by $\Delta
m_{s}=m_{B^{0}_{s,\rm{H}}}-m_{B^{0}_{s,\rm{L}}}$ and
$\Delta\Gamma_{s}=\Gamma_{B^{0}_{s,\rm{L}}}-\Gamma_{B^{0}_{s,\rm{H}}}$,
respectively, are admixtures of the flavour eigenstates such that

$B^{0}_{s,\rm{H}}=p{{B}^{0}_{s}}+q{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\;\;\;\;\;\mathrm{and}\;\;\;\;\;B^{0}_{s,\rm{L}}=p{{B}^{0}_{s}}-q{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}},$ (1)

where $p$ and $q$ are complex coefficients that satisfy $|p|^{2}+|q|^{2}=1$.
The time evolution of the initially pure flavour eigenstates at $t=0$,
$|{{B}^{0}_{s}}(0)\rangle$ and $|{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}(0)\rangle$, is described by

$\displaystyle|{{B}^{0}_{s}}(t)\rangle$
$\displaystyle=g_{+}(t)|{{B}^{0}_{s}}(0)\rangle+\frac{q}{p}g_{-}(t)|{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}(0)\rangle,$ (2)
$\displaystyle|{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}(t)\rangle$
$\displaystyle=\frac{p}{q}g_{-}(t)|{{B}^{0}_{s}}(0)\rangle+g_{+}(t)|{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}(0)\rangle,$

where the decay-time-dependent functions $g_{\pm}(t)$ are given by

$g_{\pm}(t)=\frac{1}{2}e^{-im_{s}t}e^{-\frac{\Gamma_{s}}{2}t}\left(e^{i\frac{\Delta
m_{s}}{2}t}e^{-\frac{\Delta\Gamma_{s}}{4}t}\pm e^{-i\frac{\Delta
m_{s}}{2}t}e^{\frac{\Delta\Gamma_{s}}{4}t}\right),$ (3)

with $m_{s}$ and $\Gamma_{s}$ being the average mass and width of the
$B^{0}_{s,\rm{H}}$ and $B^{0}_{s,\rm{L}}$ states. Negligible $C\\!P$ violation
in mixing is assumed in this analysis, leading to the parameterisation
$q/p=e^{-i\phi_{\rm M}}$, where $\phi_{\rm M}$ is the ${B}^{0}_{s}$–${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ mixing phase. The total
decay amplitude of the flavour eigenstates at $t=0$ into the final state
$f=({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$, denoted by $\langle
f|{{B}^{0}_{s}}(0)\rangle$ and $\langle f|{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}(0)\rangle$, is a coherent
sum of scalar-scalar (SS), scalar-vector (SV), vector-scalar (VS), scalar-
tensor (ST), tensor-scalar (TS), vector-vector (VV), vector-tensor (VT),
tensor-vector (TV) and tensor-tensor (TT) contributions. The quantum numbers
used to label the $({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ final states
are the spin $j_{1}$ ($j_{2}$) of the $K^{+}\pi^{-}$ ($K^{-}\pi^{+}$) pair and
the helicity $h$. The vector component is represented in this analysis by the
$K^{*0}$ meson, since this resonance is found to be largely dominant in this
spin configuration. Potential contributions from the $K_{1}^{*}(1410)^{0}$ and
$K_{1}^{*}(1680)^{0}$ resonances are considered as sources of systematic
uncertainty. For the tensor case, only the $K_{2}^{*}(1430)^{0}$ resonance
contributes in the considered $K\pi$ mass window. The scalar component,
denoted in this paper by $(K\pi)_{0}^{*}$ requires a more careful treatment.
It can have contributions from the $K_{0}^{*}(800)^{0}$ and
$K_{0}^{*}(1430)^{0}$ resonances and from a nonresonant $K\pi$ component. The
parameterisation of the $K\pi$ invariant mass spectrum for the scalar
contribution is explained later in this section. All of the considered decay
modes, together with the quantum numbers for the corresponding amplitudes, are
shown in Table 1. In order to separate components with different $C\\!P$
eigenvalues, $\eta_{h}^{j_{1}j_{2}}=\pm 1$, the differential decay rate is
expressed as a function of three angles and the two $K\pi$ invariant masses.
The angles $\theta_{1}$, $\theta_{2}$ and $\varphi$, are written in the
helicity basis and defined according to the diagram shown in Fig. 2. The
invariant mass of the ${K}^{+}$ ${\pi}^{-}$ pair is denoted as $m_{1}$, while
that of the ${K}^{-}$ ${\pi}^{+}$ pair as $m_{2}$. The symbol $\Omega$ is used
to represent all three angles and the two invariant masses,
$\Omega=(m_{1},m_{2},\cos\theta_{1},\cos\theta_{2},\varphi)$.

Figure 2: Graphical definition of the angles in the helicity basis. Taking the
example of a ${{B}^{0}_{s}}\rightarrow Q_{1}Q_{2}$ decay (this analysis uses
${{B}^{0}_{s}}\rightarrow SS$, ${{B}^{0}_{s}}\rightarrow SV$,
${{B}^{0}_{s}}\rightarrow VS$, ${{B}^{0}_{s}}\rightarrow VV$,
${{B}^{0}_{s}}\rightarrow ST$, ${{B}^{0}_{s}}\rightarrow TS$,
${{B}^{0}_{s}}\rightarrow VT$, ${{B}^{0}_{s}}\rightarrow TV$ and
${{B}^{0}_{s}}\rightarrow TT$), with each final-state quasi-two-body meson
decaying to pseudoscalars ($Q_{1}\rightarrow K^{+}\pi^{-}$ and
$Q_{2}\rightarrow K^{-}\pi^{+}$), $\theta_{1}$ ($\theta_{2}$) is defined as
the angle between the directions of motion of $K^{+}$ ($K^{-}$) in the $Q_{1}$
($Q_{2}$) rest frame and $Q_{1}$ ($Q_{2}$) in the ${B}^{0}_{s}$ rest frame,
and $\varphi$ as the angle between the plane defined by $K^{+}\pi^{-}$ and the
plane defined by $K^{-}\pi^{+}$ in the ${B}^{0}_{s}$ rest frame.

Summing over the possible states and using the partial wave formalism, the
decay amplitudes at $t=0$ can be written as

$\displaystyle\langle f|{{B}^{0}_{s}}(0)\rangle(\Omega)$
$\displaystyle=\sum_{j_{1},j_{2},h}\mathcal{A}_{h}^{j_{1}j_{2}}\Theta^{j_{1}j_{2}}_{h}(\cos{\theta_{1}},\cos{\theta_{2}},\varphi)\mathcal{H}_{h}^{j_{1}j_{2}}(m_{1},m_{2}),$
(4) $\displaystyle\langle f|{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}(0)\rangle(\Omega)$
$\displaystyle=\sum_{j_{1},j_{2},h}\eta_{h}^{j_{1}j_{2}}\overline{\mathcal{A}}_{h}^{j_{1}j_{2}}\Theta^{j_{1}j_{2}}_{h}(\cos{\theta_{1}},\cos{\theta_{2}},\varphi)\mathcal{H}_{h}^{j_{1}j_{2}}(m_{1},m_{2}).$

The complex parameters $\mathcal{A}_{h}^{j_{1}j_{2}}$ and
$\overline{\mathcal{A}}_{h}^{j_{1}j_{2}}$ contain the physics of the decays to
the final states with $j_{1}$, $j_{2}$ and $h$ as defined in Table 1. The
angular terms, $\Theta_{h}^{j_{1}j_{2}}$, are built from combinations of
spherical harmonics as shown in Appendix A. The $\eta_{h}^{j_{1}j_{2}}$ factor
is equal to $(-1)^{j_{1}+j_{2}}\,\eta_{h}$, where $\eta_{h}=1$ for
$h\in\\{0,\parallel,\parallel_{1},\parallel_{2}\\}$ and $\eta_{h}=-1$ for
$h\in\\{\perp,\perp_{1},\perp_{2}\\}$. The mass-dependent terms are
parameterised as

$\mathcal{H}_{h}^{j_{1}j_{2}}(m_{1},m_{2})=\mathcal{F}_{h}^{j_{1}j_{2}}(m_{1},m_{2})\mathcal{M}_{j_{1}}(m_{1})\mathcal{M}_{j_{2}}(m_{2}),$
(5)

where $\mathcal{F}_{h}^{j_{1}j_{2}}(m_{1},m_{2})$ is the Blatt–Weisskopf
angular-momentum centrifugal-barrier factor [16] and $\mathcal{M}_{j}$
describes the shape of the $K\pi$ invariant mass of a $K\pi$ pair with spin
$j$. Relativistic Breit–Wigner functions of spin 1 and 2, parameterising the
${K}^{*0}$ and the ${K}_{2}^{*}(1430)^{0}$ resonances, are used for
$\mathcal{M}_{1}$ and $\mathcal{M}_{2}$, respectively. The parameterisation of
$\mathcal{M}_{0}$ is based on the phenomenological $S$-wave scattering
amplitude of isospin $1/2$ presented in Ref. [17]. Since only the phase
evolution of $\mathcal{M}_{0}$ is linked to that of the scattering amplitude
(by virtue of Watson’s theorem [18]), its modulus is parameterised with a
fourth-order polynomial whose coefficients are determined in the final fit to
data. Details of this parameterisation can be found in Appendix B. The
normalisation condition for the mass-dependent terms is

$\int dm_{1}\int
dm_{2}\;|\mathcal{H}_{h}^{j_{1}j_{2}}(m_{1},m_{2})|^{2}\Phi_{4}(m_{1},m_{2})=1,$
(6)

where $\Phi_{4}$ is the four-body phase-space factor. The phase of
$\mathcal{H}_{h}^{j_{1}j_{2}}(m_{1},m_{2})$ is set to 0 at
$m_{1}=m_{2}=M({{K}^{*0}})$, where $M({{K}^{*0}})$ is the mass of the
${{K}^{*0}}$ state [15], in order to normalise the relative global phases of
the $K\pi$ mass-dependent amplitudes. The $C\\!P$-violating effects are
assumed to be the same for all of the modes under study. Consequently, the
value of $\phi_{s}^{{d}{\overline{d}}}$ and $|\lambda|$ determined in this
article is effectively an average over the various channels considered in
Table 1. Within this approach, the physical amplitudes
$\mathcal{A}_{h}^{j_{1}j_{2}}$ and $\overline{\mathcal{A}}_{h}^{j_{1}j_{2}}$
in Eq. (4) can be separated into a $C\\!P$-averaged complex amplitude,
$A_{h}^{j_{1}j_{2}}$, a direct $C\\!P$ asymmetry, $\Delta^{{C\\!P}}_{\rm
dir}=(|\overline{\mathcal{A}_{h}^{j_{1}j_{2}}}|^{2}-|\mathcal{A}_{h}^{j_{1}j_{2}}|^{2})/(|\overline{\mathcal{A}_{h}^{j_{1}j_{2}}}|^{2}+|\mathcal{A}_{h}^{j_{1}j_{2}}|^{2})$,333The
direct $C\\!P$ asymmetry is often notated elsewhere as $A_{{C\\!P}}$. and a
$C\\!P$-violating weak phase in the decay, $\phi_{\rm D}$, as

$\displaystyle\mathcal{A}_{h}^{j_{1}j_{2}}$
$\displaystyle=\sqrt{1-\Delta^{{C\\!P}}_{\rm dir}}e^{-i\phi_{\rm
D}}A_{h}^{j_{1}j_{2}},$ (7)
$\displaystyle\overline{\mathcal{A}}_{h}^{j_{2}j_{1}}=\overline{\mathcal{A}_{h}^{j_{1}j_{2}}}$
$\displaystyle=\sqrt{1+\Delta^{{C\\!P}}_{\rm dir}}e^{i\phi_{\rm
D}}A_{h}^{j_{1}j_{2}}.$

In the expressions above the $C\\!P$ transformation also changes $j_{1}j_{2}$
to $j_{2}j_{1}$. The total $C\\!P$-violating phase associated to the
interference between mixing and decay is given by
$\phi_{s}^{{d}{\overline{d}}}=\phi_{\rm M}-2\phi_{\rm D}$ and its
determination is the main goal of this analysis. In the SM the size of
$\phi_{s}^{{d}{\overline{d}}}$ is expected to be small due to an almost exact
cancellation in the values of $\phi_{\rm M}$ and $2\phi_{\rm D}$ [5]. The
parameter $|\lambda|$ is defined in terms of the direct $C\\!P$ asymmetry by

$|\lambda|=\frac{\sqrt{1+\Delta^{{C\\!P}}_{\rm
dir}}}{\sqrt{1-\Delta^{{C\\!P}}_{\rm dir}}}.$ (8)

##  3 Detector and simulation

The LHCb detector [19, 20] is a single-arm forward spectrometer covering the
pseudorapidity range between $2$ and $5$, 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 surrounding the $pp$
interaction region, 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 placed downstream of
the magnet. The tracking system provides a measurement of momentum, $p$, of
charged particles with relative uncertainty that varies from 0.5% at low
momentum to 1.0% at 200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum
distance of a track to a primary vertex (PV), the impact parameter (IP), is
measured with resolution of
$(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where $p_{\mathrm{T}}$
is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov (RICH)
detectors. Photons, electrons and hadrons are identified by a calorimeter
system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers. The online event selection is performed by a trigger, 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 contain
a muon with high $p_{\mathrm{T}}$ or a hadron, photon or electron with high
transverse energy in the calorimeters. The software trigger requires a two-,
three- or four-track secondary vertex with significant displacement from the
primary $pp$ interaction vertices. At least one charged particle must have
transverse momentum
$\mbox{$p_{\mathrm{T}}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and be
inconsistent with originating from a PV. A multivariate algorithm [21] is used
for the identification of secondary vertices consistent with the decay of a
$b$ hadron. Simulated samples of resonant
${{{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}}$,
${{{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}_{0}^{*}(1430)^{0}}}$ and
${{{B}^{0}_{s}}\rightarrow{{K}_{0}^{*}(1430)^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}_{0}^{*}(1430)^{0}}}$ decays, as well
as phase-space
${{{B}^{0}_{s}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{K}^{-}}{{\pi}^{+}}}$ decays,
are used to study the signal. Simulated samples of
${{{B}^{0}}\rightarrow{{K}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}}$,
${{{B}^{0}}\rightarrow{{K}^{*0}}{{\rho}^{0}}}$,
${{{B}^{0}}\rightarrow{{K}^{*0}}\phi}$ and ${{{\mathchar
28931\relax}^{0}_{b}}\rightarrow({p}{{K}^{-}})({{\pi}^{+}}{{\pi}^{-}})}$ are
created to study peaking backgrounds. In the simulation, $pp$ collisions are
generated using Pythia [22] with a specific LHCb configuration [23]. Decays of
particles are described by EvtGen [24], in which final-state radiation is
generated using Photos [25]. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [26,
*Agostinelli:2002hh] as described in Ref. [28].

##  4 Signal candidate selection

Events passing the trigger are required to satisfy requirements on the fit
quality of the ${B}^{0}_{s}$ decay vertex as well as the $p_{\mathrm{T}}$ and
$\chi^{2}_{\text{IP}}$ of each track, where $\chi^{2}_{\text{IP}}$ is defined
as the difference between the $\chi^{2}$ of the secondary vertex reconstructed
with and without the track under consideration. The tracks are assigned as
kaon or pion candidates using particle identification information from the
RICH detectors by requiring that the likelihood for the kaon hypothesis is
larger than that for the pion hypothesis and vice versa. In addition, the
$p_{\mathrm{T}}$ of each $K\pi$ pair is required to be larger than
$500$${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, the reconstructed mass of each
$K\pi$ pair is required to be within the range $750\leq m(K\pi)\leq
1600$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and the reconstructed mass of
the ${B}^{0}_{s}$ candidate is required to be within the range $5000\leq
m({{K}^{+}}{{\pi}^{-}}{{K}^{-}}{{\pi}^{+}})\leq
5800$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. A boosted decision tree (BDT)
algorithm [29, 30] is trained to reject combinatorial background, where at
least one of the final-state tracks originates from a different decay or
directly from the PV. The signal is represented in the BDT training with
simulated ${{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ candidates, satisfying the
same requirements as the data, while selected data candidates in the four-body
invariant mass sideband, $5600\leq
m({{K}^{+}}{{\pi}^{-}}{{K}^{-}}{{\pi}^{+}})\leq
5800$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, are used to represent the
background. The input variables employed in the training are kinematic and
geometric quantities associated with the four final-state tracks, the two
$K\pi$ candidates and the ${B}^{0}_{s}$ candidate. The features used to train
the BDT response are chosen to minimise any correlation with the ${B}^{0}_{s}$
and two $K\pi$ pair invariant masses. Separate trainings are performed for the
data samples collected in 2011 and 2012, due to the different data-taking
conditions. The $k$-fold cross-validation method [31], with $k=4$, is used to
increase the training statistics while reducing the risk of overtraining. The
requirement on the BDT response is optimized by maximising the metric $N_{\rm
S}/\sqrt{N_{\rm S}+N_{\rm B}}$, where $N_{\rm S}$ is the estimated number of
signal candidates after selection and $N_{\rm B}$ is the estimated number of
combinatorial background candidates within $\pm
60{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known ${B}^{0}_{s}$ mass
[15]. The BDT requirement is 95% efficient for simulated signal candidates and
rejects 70% of the combinatorial background. After applying the BDT
requirement, specific background contributions containing two real oppositely
charged kaons and two real oppositely charged pions are removed by mass vetoes
on the two- and three-body invariant masses. Candidates are removed if they
fulfill either
$m({{K}^{+}}{{K}^{-}}{{\pi}^{\pm}})<2100$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
or $m({{K}^{+}}{{K}^{-}})$ within 30${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
of the known ${D}^{0}$ mass [15]. Sources of peaking background in which one
of the final-state tracks is misidentified are suppressed by introducing
further particle identification requirements. The particle identification
quantities make use of information from the RICH detectors and are calibrated
using ${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ decays in data. These
requirements significantly reduce contributions from
${{B}^{0}}\rightarrow{{\rho}^{0}}{{K}^{*0}}$,
${{B}^{0}}\rightarrow\phi{{K}^{*0}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{p}{{\pi}^{-}}{{K}^{-}}{{\pi}^{+}}$ in which a
pion or proton is misidentified as a kaon, or a kaon is misidentified as a
pion. In addition, there are specific extra particle identification
requirements for candidates whose reconstructed mass falls within $\pm
30$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known ${B}^{0}$ or
${\mathchar 28931\relax}^{0}_{b}$ mass under the relevant mass-hypothesis
change ($K\rightarrow{p}$, $K\rightarrow\pi$ or $\pi\rightarrow K$). These
requirements remove 40% of the simulated signal but almost all of the
simulated background: 80% of ${{B}^{0}}\rightarrow\phi{{K}^{*0}}$, 96% of
${{B}^{0}}\rightarrow\rho{{K}^{*0}}$ and 88% of ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{p}{{\pi}^{-}}{{K}^{-}}{{\pi}^{+}}$ events.
Subsequently, each of these background components is found to have a small
effect on the signal determination. After all of the selection criteria have
been imposed, 1.4% of selected events contain multiple candidates, from which
one is randomly selected. A fit to the four-body invariant mass distribution
is performed in order to determine a set of signal weights, obtained using the
sPlot procedure [32], which allows the decay-time-dependent $C\\!P$ fit to be
performed on a sample that represents only the signal. For the invariant mass
fit the ${{B}^{0}_{s}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$
signal and the peaking background components
${{B}^{0}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$,
$B^{0}_{(s)}\rightarrow({{K}^{+}}{{K}^{-}})({{K}^{-}}{{\pi}^{+}})$,
${{B}^{0}}\rightarrow({{\pi}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ and
${{\mathchar
28931\relax}^{0}_{b}}\rightarrow({p}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ are
modelled as Ipatia functions [33] in which the tail parameters are fixed to
values obtained from fits to the simulated samples. The mass difference
between the ${B}^{0}_{s}$ and ${B}^{0}$ mesons is fixed to its known value
whilst the mean of the ${B}^{0}_{s}$ component, as well as the width of both
the ${B}^{0}$ and ${B}^{0}_{s}$ components, are allowed to vary freely. The
yields of the
${{{{B}^{0}_{s}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})}}$,
${{{B}^{0}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})}$ and
${{{B}^{0}}\rightarrow({{K}^{+}}{{K}^{-}})({{K}^{-}}{{\pi}^{+}})}$ components
are allowed to freely vary, whilst the yields of the other components are
Gaussian constrained to values relative to the known
${{B}^{0}}\rightarrow({{K}^{+}}{{K}^{-}})({{K}^{-}}{{\pi}^{+}})$ branching
fraction taking into account the relevant production fractions [34] and
reconstruction efficiencies. There is an additional background contribution in
the low-mass region from partially reconstructed $b$-hadron decays in which a
pion is missed in the final state. This component is modelled as an ARGUS
function [35] convolved with a Gaussian mass resolution function. The ARGUS
cutoff parameter is fixed to the fitted ${B}^{0}_{s}$ mass minus the neutral
pion mass, with the other parameters and yield allowed to vary. The
combinatorial background is modelled as an exponential function whose shape
parameter and yield are allowed to vary. The result of the four-body invariant
mass fit, which is used to obtain the sPlot signal weights, is shown in Fig.
3. The two $K\pi$ pair invariant masses, with the signal weights applied, are
shown in Fig. 4. The resulting yields of the various fit components are shown
in Table 2.

Table 2: Yields of the signal decay and the various background components considered in the four-body invariant mass fit. The uncertainties are statistical only. The signal region is defined as $\pm 60$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ from the known ${B}^{0}_{s}$ meson mass [15]. Channel | Yield | Yield in Signal Region
---|---|---
${{B}^{0}_{s}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ |  6080 | $\pm$ |  83 | 6004
${{B}^{0}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ |  1013 | $\pm$ |  49 | 103
${{B}^{0}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{K}^{+}})$ |  281 | $\pm$ |  47 | 1
${{B}^{0}_{s}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{K}^{+}})$ |  8 | $\pm$ |  3 | 4
${{B}^{0}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{\pi}^{-}}{{\pi}^{+}})$ |  57 | $\pm$ |  13 | 33
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow({p}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ |  44 | $\pm$ |  10 | 13
Partially reconstructed |  2580 | $\pm$ |  151 | 0
Combinatorial |  2810 | $\pm$ |  214 | 372

Figure 3: Four-body invariant mass distribution on a (left) linear and (right)
logarithmic scale superimposed with the mass fit model. Figure 4: Distribution
of the two $(K\pi)$ pair invariant masses, with the signal weights applied,
after all of the selection requirements.

##  5 Flavour tagging

At the LHC, $b$ quarks are predominantly produced in ${b}{\overline{b}}$
pairs. This analysis focuses on events where one of the quarks hadronises to
produce the ${B}^{0}_{s}$ meson while the other quark hadronises and decays
independently. Taking advantage of this effect, two types of tagging
algorithms aimed at identifying the $b$-quark flavour at production time are
used in this analysis: same-side (SS) taggers, based on information from
accompanying particles associated with the signal ${B}^{0}_{s}$ hadronisation
process; and opposite-side (OS) taggers, based on particles produced in the
decay of the other $b$ quark. This analysis uses the neural-network-based SS-
kaon tagging algorithm presented in Ref. [36]; and the combination of OS
tagging algorithms explained in Ref. [37], based on information from
$b$-hadron decays to electrons, muons or kaons and the total charge of tracks
that form a vertex. Both the SS and OS tagging algorithms provide for each
event a tagging decision, $\mathfrak{q}$, and an estimated mistag probability,
$\eta_{\rm tag}$. The tagging decision takes the value $1$ for ${B}^{0}_{s}$,
$-1$ for ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ and $0$
for untagged. To obtain the calibrated mistag probability for a ${B}^{0}_{s}$
(${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$) meson, $\omega$
($\bar{\omega}$), the estimated probability is calibrated on several flavour-
specific control channels. The following linear functions are used in the
calibration

$\displaystyle\omega^{\rm X}(\eta_{\rm tag}^{\rm X})$
$\displaystyle=\left(p_{0}^{\rm X}+\frac{\Delta p_{0}^{\rm
X}}{2}\right)+\left(p_{1}^{\rm X}+\frac{\Delta p_{1}^{\rm
X}}{2}\right)(\eta_{\rm tag}^{\rm X}-\langle\eta_{\rm tag}^{\rm X}\rangle),$
(9) $\displaystyle\bar{\omega}^{\rm X}(\eta_{\rm tag}^{\rm X})$
$\displaystyle=\left(p_{0}^{\rm X}-\frac{\Delta p_{0}^{\rm
X}}{2}\right)+\left(p_{1}^{\rm X}-\frac{\Delta p_{1}^{\rm
X}}{2}\right)(\eta_{\rm tag}^{\rm X}-\langle\eta_{\rm tag}^{\rm X}\rangle),$

where $\rm X\in\\{OS,SS\\}$, $\langle\eta_{\rm tag}^{\rm X}\rangle$ is the
mean $\eta_{\rm tag}^{\rm X}$ of the sample, $p_{0,1}^{\rm X}$ correspond to
calibration parameters averaged over ${B}^{0}_{s}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$, and $\Delta p_{0,1}^{\rm
X}$ account for ${B}^{0}_{s}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ asymmetries in the
calibration. Among other modes, the portability of the SS tagger calibration
was checked on $B_{s}^{0}\rightarrow\phi\phi$ decays [36], which are
kinematically similar to the considered signal mode. The tagging efficiency,
$\epsilon_{\rm tag}$, denotes the fraction of candidates with a nonzero
tagging decision. The tagging power of the sample, $\epsilon_{\rm
eff}=\epsilon_{\rm tag}(1-2\langle\omega\rangle)^{2}$, characterises the
tagging performance. Information from the SS and OS algorithms is combined on
a per-event basis (see Eq. (13)) for the decay-time-dependent amplitude fit
discussed in Sec. 7. The overall effective tagging power is found to be
$(5.15\pm 0.14)\%$. The flavour-tagging performance is shown in Table 3. When
separating the ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ and
${B}^{0}_{s}$ components at $t=0$, the value of the production asymmetry
$A_{p}=[\sigma({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}})-\sigma({{B}^{0}_{s}})]/[\sigma({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}})+\sigma({{B}^{0}_{s}})]$,
where $\sigma({{B}^{0}_{s}})$ ($\sigma({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}})$) is the production cross-
section for the ${B}^{0}_{s}$ (${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$) meson, also has to be
incorporated in the model. This asymmetry was measured by LHCb in $pp$
collisions at $\sqrt{s}=7$ TeV by means of a decay-time-dependent analysis of
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ decays [38]. To correct for
the different kinematics of ${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$
and ${{{B}^{0}_{s}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})}$
decays, a weighting in bins of ${B}^{0}_{s}$ transverse momentum and
pseudorapidity is performed, yielding a value of $A_{p}=-0.005\pm 0.019$. No
detection asymmetry need be considered in this analysis since the final state
under consideration is charge symmetric.

Table 3: The flavour-tagging performance of the SS and OS tagging algorithms, as well as the combination of both, for the signal data sample used in the analysis. The quoted uncertainty includes both statistical and systematic contributions. Tagging algorithm |  $\epsilon_{\rm tag}$ [%] |  $\epsilon_{\rm eff}$ [%]
---|---|---
SS | $62.0\pm 0.7$ | $1.63\pm 0.21$
OS | $37.1\pm 0.7$ | $3.70\pm 0.21$
Combination | $75.6\pm 0.6$ | $5.15\pm 0.14$

##  6 Acceptance and resolution effects

The LHCb geometrical coverage and selection procedure induce acceptance
effects that depend on the three decay angles, the $K\pi$ two-body invariant
masses and the decay time. In addition, imperfect reconstruction gives rise to
resolution effects. Any deviations caused by imperfect angular and mass
resolution are small and are accounted for within the evaluation of systematic
uncertainties (see Sec. 8). However, knowledge of the decay-time resolution is
of key importance in the determination of $\phi_{s}^{{d}{\overline{d}}}$ and
is consequently included in the decay-time-dependent fit. In this analysis,
both acceptance and resolution effects are studied using samples of simulated
events which have been weighted to match the data distributions in several
important kinematic variables. In the description of the acceptance, the
decay-time-dependent part is factorised with respect to the part that depends
on the kinematic quantities, since they are found to be only $5\%$ correlated.
The acceptance and the decay-time resolutions are determined from simulated
events that contain an appropriate combination of the vector-vector
${{{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}}$ component with a sample of
${{B}^{0}_{s}}\rightarrow K^{+}\pi^{-}K^{-}\pi^{+}$ decays generated according
to a phase-space distribution. This combination sufficiently populates the
phase-space regions to represent the signal decay. To obtain the acceptance
function, the simulated events are weighted by the inverse of the probability
density function (PDF) used for generation (defined in terms of angles, masses
and decay time). The decay-time acceptance is treated analytically and
parameterised using cubic spline functions, following the procedure outlined
in Ref. [39], with the number of knots chosen to be six. The effect of this
choice is addressed as a systematic uncertainty in Sec. 8. The decay-time
acceptance is shown in Fig. 5 (bottom right). The five-dimensional kinematic
acceptance in angles and masses is included by using normalisation weights in
the denominator of the PDF used in the fit to the data, following the
procedure described in Ref. [40]. When visualising the fit results (see Fig.
7), the simulated events are weighted using the matrix element of the
amplitude fit model. For illustrative purposes, some projections of the
kinematic acceptance are shown in Fig. 5. In order to obtain the best possible
sensitivity for the measurement of the $\phi_{s}^{{d}{\overline{d}}}$ phase,
the time resolution is evaluated event by event, using the estimated decay-
time uncertainty, $\delta_{t}$, obtained in the track reconstruction process.
This variable is calibrated using the simulation sample described above to
provide the per-event decay-time resolution, $\sigma_{t}$, using a linear
relationship

$\sigma_{t}(\delta_{t})=p_{0}^{\sigma_{t}}+p_{1}^{\sigma_{t}}(\delta_{t}-\langle\delta_{t}\rangle),$
(10)

where $\langle\delta_{t}\rangle$ is the mean $\delta_{t}$ of the sample and
$p_{(0,1)}^{\sigma_{t}}$ are the calibration parameters. During fitting,
$\sigma_{t}$ is taken to be the width of a Gaussian resolution function which
convolves the decay-time-dependent part of the total amplitude model. Figure 6
shows the relationship between the estimated decay-time uncertainty,
$\delta_{t}$, and the calibrated per-event decay-time resolution,
$\sigma_{t}$.

Figure 5: Kinematic acceptance and decay-time distributions evaluated with
simulated vector-vector ${{{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}}$ and pure phase-space
${{{B}^{0}_{s}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})}$
candidates scaled by the mean acceptance. In the bottom right plot the decay-
time acceptance obtained from the simulated sample is shown as the black
points and the parametric form of the acceptance obtained with cubic splines
is shown as the red curve. In the other three plots the black points show the
acceptance distribution for the masses and angles. The two $\cos\theta$
variables and the two $m(K\pi)$ masses have been averaged for the purpose of
illustration. In the fit, the kinematic acceptance enters via the
normalisation weights. Figure 6: Per-event decay-time resolution,
$\sigma_{t}$, versus the estimated per-event decay-time uncertainty,
$\delta_{t}$, obtained from simulated samples containing both vector-vector
resonant ${{{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}}$ and phase-space
${{{B}^{0}_{s}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{K}^{-}}{{\pi}^{+}}}$ events.

##  7 Decay-time-dependent amplitude fit

The model used to fit the data is built by taking the squared moduli of the
amplitudes $\langle f|{{B}^{0}_{s}}(t)\rangle$ and $\langle f|{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}(t)\rangle$ introduced in
Sec. 2, multiplying them by the four-body phase-space factor, incorporating
the relevant flavour-tagging and production-asymmetry parameters, and
including the acceptance and resolution factors obtained in Sec. 6. The
observables $\eta_{\rm tag}^{\rm SS}$, $\eta_{\rm tag}^{\rm OS}$ and
$\delta_{t}$ (introduced in Sec. 5 and Sec. 6) are treated as conditional
variables. The effective444In the PDF used for fitting, the marginal PDFs on
the conditional variables as well as the acceptance function in the numerator
are factored out (see Ref. [40] for details on the acceptance treatment used
in this analysis). normalised PDF can be written as

$\text{PDF}(t,\Omega)=\frac{\sum_{\alpha=1}^{19}\sum_{\beta\leq\alpha}\Re
e[K_{\alpha\beta}(t)F_{\alpha\beta}(\Omega)]}{\sum_{\alpha^{\prime}=1}^{19}\sum_{\beta^{\prime}\leq\alpha^{\prime}}\Re
e[(\int dt^{\prime}\,K^{\rm
untag}_{\alpha^{\prime}\beta^{\prime}}(t^{\prime})\epsilon_{t}(t^{\prime}))\xi_{\alpha^{\prime}\beta^{\prime}}]},$
(11)

where the subscript $\alpha$ ($\beta$) represents the state labels
$\\{j_{1},j_{2},h\\}$ ($\\{j_{1}^{\prime},j_{2}^{\prime},h^{\prime}\\}$),
$K_{\alpha\beta}(t)$ parameterises the decay-time dependence and is defined in
Eq. (12), and $F_{\alpha\beta}(\Omega)$ are terms that parameterise the
angular and mass dependence. Both the numerator and the denominator of Eq.
(11) are constructed as a sum over 190 real terms, which arise when squaring
the amplitudes decomposed in the combination of the nineteen contributing
polarisation states. The decay-time-dependent factors are constructed as

$\displaystyle K_{\alpha\beta}(t)$
$\displaystyle=R(t,\delta_{t})\otimes\bigg{\\{}e^{-\Gamma_{s}t}\bigg{[}\zeta_{+}\bigg{(}a_{\alpha\beta}\cosh\bigg{(}\frac{1}{2}\Delta\Gamma_{s}t\bigg{)}+b_{\alpha\beta}\sinh\bigg{(}\frac{1}{2}\Delta\Gamma_{s}t\bigg{)}\bigg{)}$
(12) $\displaystyle+\zeta_{-}\bigg{(}c_{\alpha\beta}\cos\left(\Delta
m_{s}t\right)+d_{\alpha\beta}\sin\left(\Delta
m_{s}t\right)\bigg{)}\bigg{]}\bigg{\\}},$

where $R(t,\delta_{t})$ is the decay-time resolution function and the factors
$\zeta_{\pm}$ contain the flavour-tagging and production-asymmetry
information. These factors are

$\zeta_{\pm}=\frac{(1+A_{p})}{2}P^{\rm OS}(\mathfrak{q}^{\rm OS})\,P^{\rm
SS}(\mathfrak{q}^{\rm SS})\;\pm\;\frac{(1-A_{p})}{2}\bar{P}^{\rm
OS}(\mathfrak{q}^{\rm OS})\,\bar{P}^{\rm SS}(\mathfrak{q}^{\rm SS}),$ (13)

where

$\displaystyle P^{\rm X}(\mathfrak{q}^{\rm X})$
$\displaystyle=\begin{cases}1-\omega^{\rm X}(\eta^{\rm X})&\text{ for
}\mathfrak{q}^{\rm X}=\phantom{-}1,\\\ 1&\text{ for }\mathfrak{q}^{\rm
X}=\phantom{-}0,\\\ \omega^{\rm X}(\eta^{\rm X})&\text{ for }\mathfrak{q}^{\rm
X}=-1,\end{cases}$ (14) $\displaystyle\bar{P}^{\rm X}(\mathfrak{q}^{\rm X})$
$\displaystyle=\begin{cases}\bar{\omega}^{\rm X}(\eta^{\rm X})&\text{ for
}\mathfrak{q}^{\rm X}=\phantom{-}1,\\\ 1&\text{ for }\mathfrak{q}^{\rm
X}=\phantom{-}0,\\\ 1-\bar{\omega}^{\rm X}(\eta^{\rm X})&\text{ for
}\mathfrak{q}^{\rm X}=-1,\end{cases}$

with $\rm X\in\\{OS,SS\\}$. The complex quantities $a_{\alpha\beta}$,
$b_{\alpha\beta}$, $c_{\alpha\beta}$ and $d_{\alpha\beta}$ are defined in
terms of the $C\\!P$-averaged amplitudes, the $C\\!P$-violating parameters and
the $\eta_{h}^{j_{1}j_{2}}$ factors, as

$\displaystyle a_{\alpha\beta}$
$\displaystyle=\frac{2}{1+|\lambda|^{2}}\left(A_{\alpha}A_{\beta}^{*}+\eta_{\alpha}\eta_{\beta}|\lambda|^{2}A_{\bar{{\alpha}}}A_{\bar{{\beta}}}^{*}\right),$
(15) $\displaystyle b_{\alpha\beta}$
$\displaystyle=\frac{-2|\lambda|}{1+|\lambda|^{2}}\left(\eta_{\beta}e^{i\phi_{s}^{{d}{\overline{d}}}}A_{\alpha}A_{\bar{{\beta}}}^{*}+\eta_{\alpha}e^{-i\phi_{s}^{{d}{\overline{d}}}}A_{\bar{{\alpha}}}A_{\beta}^{*}\right),$
$\displaystyle c_{\alpha\beta}$
$\displaystyle=\frac{2}{1+|\lambda|^{2}}\left(A_{\alpha}A_{\beta}^{*}-\eta_{\alpha}\eta_{\beta}|\lambda|^{2}A_{\bar{{\alpha}}}A_{\bar{{\beta}}}^{*}\right),$
$\displaystyle d_{\alpha\beta}$
$\displaystyle=\frac{-2|\lambda|i}{1+|\lambda|^{2}}\left(\eta_{\beta}e^{i\phi_{s}^{{d}{\overline{d}}}}A_{\alpha}A_{\bar{{\beta}}}^{*}-\eta_{\alpha}e^{-i\phi_{s}^{{d}{\overline{d}}}}A_{\bar{{\alpha}}}A_{\beta}^{*}\right),$

where the bars on the amplitude indices $\alpha$ and $\beta$ denote the
$C\\!P$ transformation of the considered final state, i.e. the change of
quantum numbers $j_{1}j_{2}\rightarrow j_{2}j_{1}$. The functions $K^{\rm
untag}_{\alpha\beta}$ are obtained by summing $K_{\alpha\beta}$ over the
tagging decisions. The angular- and mass-dependent terms are constructed as

$\displaystyle F_{\alpha\beta}(\Omega)$
$\displaystyle=(2-\delta_{\alpha\beta})\Theta^{j_{1}j_{2}}_{h}(\cos{\theta_{1}},\cos{\theta_{2}},\varphi)[\Theta^{j_{1}^{\prime}j_{2}^{\prime}}_{h^{\prime}}(\cos{\theta_{1}},\cos{\theta_{2}},\varphi)]^{*}$
(16)
$\displaystyle\times\mathcal{M}_{j_{1}}(m_{1})\mathcal{M}_{j_{2}}(m_{2})\mathcal{M}^{*}_{j_{1}^{\prime}}(m_{1})\mathcal{M}^{*}_{j_{2}^{\prime}}(m_{2})$
$\displaystyle\times\mathcal{F}_{h}^{j_{1}j_{2}}(m_{1},m_{2})\mathcal{F}_{h^{\prime}}^{j_{1}^{\prime}j_{2}^{\prime}}(m_{1},m_{2})\Phi_{4}(m_{1},m_{2}),$

where $\delta_{\alpha\beta}$ is the Kronecker delta and the other terms have
been introduced in Sec. 2. The decay-time acceptance function,
$\epsilon_{t}(t)$, and the normalisation weights, $\xi_{\alpha\beta}$, are
included in the denominator of Eq. (11). The normalisation weights correspond
to angular and mass integrals that involve the five-dimensional kinematic
acceptance, $\epsilon_{\Omega}(\Omega)$, and are obtained by summing over the
events in the simulated sample

$\xi_{\alpha\beta}\equiv\int{d\Omega\;F_{\alpha\beta}(\Omega)\;\epsilon_{\Omega}(\Omega)}\propto\sum_{i}^{N_{\rm
events}}\frac{F_{\alpha\beta}(\Omega_{i})}{G(\Omega_{i})},$ (17)

where $G(\Omega)$ is the model used for generation. The $C\\!P$-conserving
amplitudes, $A_{h}^{j_{1},j_{2}}$, the direct $C\\!P$-asymmetry parameter,
$|\lambda|$, and the mixing induced $C\\!P$-violating phase,
$\phi_{s}^{{d}{\overline{d}}}$, are allowed to vary during the fit. Gaussian
constraints are applied to $\Delta m_{s}$, $\Gamma_{s}$ and $\Delta\Gamma_{s}$
from their known values [8], and to the flavour-tagging and decay-time
resolution calibration parameters, introduced in Sec. 5 and Sec. 6. The
$C\\!P$-averaged amplitudes are characterised in the fit by wave fractions,
$f^{w}$, polarisation fractions, $f^{w}_{h}$, and strong phases,
$\delta^{w}_{h}$, given by

$\displaystyle f^{w}$
$\displaystyle=\frac{\sum_{h}|A^{w}_{h}|^{2}}{\sum_{w^{\prime}}\sum_{h^{\prime}}|A^{w^{\prime}}_{h^{\prime}}|^{2}},$
(18) $\displaystyle f^{w}_{h}$
$\displaystyle=\frac{|A^{w}_{h}|^{2}}{\sum_{h^{\prime}}|A^{w}_{h^{\prime}}|^{2}},$
$\displaystyle\delta^{w}_{h}$ $\displaystyle=\arg{(A^{w}_{h})},$

with $w$ running over the nine decays under study and $h$ running over the
available helicities for each channel. With these definitions it follows that

$\sum_{w}f^{w}=1,\hskip 42.67912pt\sum_{h}f_{h}^{w}=1,\;\forall\,w,$ (19)

so not all the fractions are independent of each other, for example
$f_{\perp}^{VV}=1-f_{\rm L}^{VV}-f_{\parallel}^{VV}$. The phase of the
longitudinal polarisation amplitude of the vector-vector component is set to
zero to serve as a reference.

##  8 Systematic uncertainties

The decay-time-dependent amplitude model and the fit procedure are cross-
checked in several independent ways: using purely simulated decays, fitting in
a narrow window around the dominant $K^{*0}$ resonance, fitting only in the
high-mass region above the $K^{*0}$ resonance, considering higher-spin
contributions (whose effect is found to be negligible), ensuring that there is
no bias when repeating the fit procedure on ensembles of pseudo-experiments
and by repeating the fit on subsamples of the data set split by the year of
data taking, the magnet polarity and using a different mass range. These
checks give compatible results. Several sources of systematic uncertainty are
considered for each of the physical observables extracted in the decay-time-
dependent fit. These are described in this section. A summary of the
systematic uncertainties is given in Table 4.

###  8.1 Fit to the four-body invariant mass distribution

The uncertainty on the yield of each of the partially reconstructed components
used in the four-body invariant mass fit is propagated to the decay-time-
dependent amplitude fit by recalculating the sPlot signal weights after
varying each of the yields by one standard deviation. Sources of systematic
uncertainty which arise from mismodelling the shapes of both the background
and signal components are calculated by performing the full fit procedure
using alternative parameterisations. The signal is replaced with a double-
sided Crystal Ball function [41] instead of the nominal Ipatia shape described
in Sec. 4 and the combinatorial-background shape is replaced with a first-
order polynomial instead of the nominal exponential function.

###  8.2 Weights derived from the sPlot procedure

The sPlot procedure assumes that there is no correlation between the fit
variable used to determine the weights, in this case the four-body invariant
mass, $m({{K}^{+}}{{\pi}^{-}}{{K}^{-}}{{\pi}^{+}})$, and the projected
variables in which the signal distribution is unfolded, in this case the three
angles and two masses, $\Omega$. This is checked to be valid to a close
approximation for signal decays. In order to assess the impact of any residual
correlations in the signal weights, the four-body mass fit is performed by
splitting the data into different bins of $\cos\theta$ for each $(K\pi)$ pair.
For each subcategory the four-body fit is repeated and the resulting model is
used to compute a new set of signal weights for the full sample. The largest
difference between each subcategory value and the nominal fit value is taken
as the systematic uncertainty.

###  8.3 Decay-time-dependent fit procedure

An ensemble of pseudoexperiments is generated to estimate the bias on the
parameters of the decay-time-dependent fit. For each experiment, a sample with
a similar size to the selected signal is generated using the matrix element of
the nominal model (employing the measured amplitudes) and then refitted to
determine the deviation induced in the fit parameters. The systematic
uncertainty is calculated as the mean of the deviation over the ensemble.

###  8.4 Decay-time-dependent fit parameterisation

Several sources of systematic uncertainty originating from the decay-time-
dependent fit model have been studied. These include the parameterisations of
the angular momentum centrifugal-barrier factors, the mean and width of the
Breit–Wigner functions and the model for the S-wave propagator. An alternative
model-independent approach is used, as described in Appendix B. The systematic
uncertainties are obtained for each of these cases by comparing the fitted
parameter values of the alternative model with the fitted values from the
nominal model. Additional contributions from higher mass $(K\pi)$ vector
resonances, namely the $K_{1}^{*}(1410)^{0}$ and the $K_{1}^{*}(1680)^{0}$
states, are also considered. In this case, the size of these components is
first estimated on data through a simplified fit. Afterwards, an ensemble of
pseudoexperiments is generated including these resonances in the model and
then refitting with the nominal PDF. The total systematic uncertainty for the
decay-time-dependent fit model is taken as the sum in quadrature of these
alternatives.

###  8.5 Acceptance normalisation weights

The kinematic acceptance weights, explained in Sec. 7, are computed from
simulated samples of limited size, which induces an uncertainty. This
systematic uncertainty is calculated using an ensemble of pseudoexperiments in
which the acceptance weights are randomly varied according to their covariance
matrix (evaluated on the simulated sample). The root-mean-square of the
distribution of the differences between the nominal fitted value and the value
obtained in each pseudoexperiment is taken as the size of the systematic
uncertainty. This effect is found to be the largest systematic uncertainty
impacting the measurement of the $\phi_{s}^{{d}{\overline{d}}}$ phase.

###  8.6 Other acceptance and resolution effects

Various other acceptance and resolution effects for the decay angles, the two
$K\pi$ pair masses and the decay-time are accounted for. Most of these
quantities are nominally computed in the decay-time-dependent fit using
simulation samples. Any differences between data and simulation are accounted
for by the systematic uncertainties described in this section. Furthermore,
various other effects originating from mismodelling of the decay-time
acceptance and decay-time resolution functions are considered. Each of these
effects are summed in quadrature to provide the value listed in Table 4. The
kinematic and decay-time acceptances, shown in Fig. 5, are computed from
samples of simulated signal events. Small systematic effects can arise due to
differences between the data and the simulated samples. In particular,
mismodelling of the ${B}^{0}_{s}$ and the four-track momentum distributions
can impact the acceptance in $\cos\theta$. This effect is checked by producing
a data-driven correction for the simulation in several relevant physical
quantities.555The variables used to correct the distributions of the
simulation are the momentum and pseudorapidity of the kaons and pions, the
transverse momentum of the ${B}^{0}_{s}$ and the number of tracks in the
event. This correction is produced using an iterative procedure that removes
any effects arising from differences between the model used in the event
generation and the actual decay kinematics of
${{{B}^{0}_{s}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})}$
decays. The systematic uncertainty is computed as the difference in the fit
parameters before and after the iterative correction has been applied.
Systematic effects due to the possible mismodelling of the decay-time-
dependent acceptance are studied by generating ensembles of pseudoexperiments
in two different configurations: one in which the decay-time acceptance spline
coefficients are randomised and one in which the configuration of the decay-
time acceptance knots is varied. The nominal decay-time-dependent fit
procedure is repeated for each pseudoexperiment and the systematic uncertainty
for each of these two effects is computed as the average deviation of the fit
parameters from their generated values over each ensemble. Sources of
systematic uncertainty which affect the decay-time resolution are studied by
modifying the calibration function in Eq. (10) that is used to obtain the per-
event decay-time resolution. First, the nominal function is substituted by an
alternative quadratic form, to asses the effect of nonlinearity in the
calibration. Second, the nominal function is multiplied by a scale factor that
accounts for possible remaining differences between data and simulation. This
scale factor is taken from the analysis of $B_{s}^{0}\rightarrow J/\psi\phi$
decays performed by LHCb in Ref. [42]. In the both cases, the systematic
uncertainties are obtained by comparing the values resulting from the
alternative configurations with the nominal values. The effect of the
resolution on the masses and angles is studied by generating ensembles of
pseudoexperiments for which the masses and angles are smeared using a multi-
dimensional Gaussian resolution function, obtained from simulation. The
systematic uncertainty is computed as the mean deviation between the fitted
and generated values.

###  8.7 Production asymmetry

The uncertainty of the production asymmetry for the ${B}^{0}_{s}$ meson is
studied by computing the maximum difference between the nominal conditions and
when the production asymmetry is shifted to $\pm 1\sigma$ of its nominal
value.

Table 4: Summary of the systematic uncertainties on the two $C\\!P$
parameters, the $C\\!P$-averaged fractions and the strong phase differences
(in radians) for each of the components listed in Table 1.

Parameter |  $\phi_{s}^{{d}{\overline{d}}}$ [rad] | $|\lambda|$ | $f^{VV}$ | $f_{L}^{VV}$ | $f_{\parallel}^{VV}$ | $\delta_{\parallel}^{VV}$ | $\delta_{\perp}^{VV}$ | $f^{SV}$ | $f^{VS}$ | $\delta^{SV}$ | $\delta^{VS}$ | $f^{SS}$ | $\delta^{SS}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Yield and shape of mass model | $0.012$ | $0.001$ | $0.001$ | $0.004$ | $0.004$ | $0.011$ | $0.020$ | $0.002$ | $0.003$ | $0.023$ | $0.023$ | $0.004$ | $0.012$
Signal weights of mass model | $0.012$ | $0.007$ | $0.002$ | $0.006$ | $0.005$ | $0.024$ | $0.112$ | $0.004$ | $0.005$ | $0.049$ | $0.022$ | $0.005$ | $0.047$
Decay-time-dependent fit procedure | $0.006$ | $0.002$ | $0.001$ | $0.006$ | $0.002$ | $0.007$ | $0.017$ | $0.003$ | $0.002$ | $0.007$ | $0.027$ | $0.001$ | $0.009$
Decay-time-dependent fit parameterisation | $0.049$ | $0.013$ | $0.021$ | $0.025$ | $0.026$ | $0.187$ | $0.202$ | $0.042$ | $0.029$ | $0.159$ | $0.234$ | $0.064$ | $0.227$
Acceptance weights (simulated sample size) | $0.106$ | $0.078$ | $0.004$ | $0.031$ | $0.029$ | $0.236$ | $0.564$ | $0.037$ | $0.039$ | $0.250$ | $0.290$ | $0.015$ | $0.256$
Other acceptance and resolution effects | $0.063$ | $0.008$ | $0.005$ | $0.018$ | $0.005$ | $0.136$ | $0.149$ | $0.006$ | $0.004$ | $0.167$ | $0.124$ | $0.017$ | $0.194$
Production asymmetry | $0.002$ | $0.002$ | $0.000$ | $0.000$ | $0.000$ | $0.001$ | $0.017$ | $0.002$ | $0.002$ | $0.002$ | $0.008$ | $0.000$ | $0.002$
Total | $0.141$ | $0.089$ | $0.024$ | $0.046$ | $0.042$ | $0.333$ | $0.641$ | $0.071$ | $0.065$ | $0.346$ | $0.405$ | $0.069$ | $0.399$

Parameter | $f^{ST}$ | $f^{TS}$ | $\delta^{ST}$ | $\delta^{TS}$ | $f^{VT}$ | $f_{L}^{VT}$ | $f_{\parallel}^{VT}$ | $f^{TV}$ | $f_{L}^{TV}$ | $f_{\parallel}^{TV}$ | $\delta_{0}^{VT}$ | $\delta_{\parallel}^{VT}$ | $\delta_{\perp}^{VT}$ | $\delta_{0}^{TV}$ | $\delta_{\parallel}^{TV}$ | $\delta_{\perp}^{TV}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Yield and shape of mass model | $0.002$ | $0.004$ | $0.111$ | $0.023$ | $0.001$ | $0.003$ | $0.001$ | $0.001$ | $0.043$ | $0.025$ | $0.023$ | $0.055$ | $0.110$ | $0.053$ | $0.018$ | $0.065$
Signal weights of mass model | $0.004$ | $0.006$ | $0.151$ | $0.105$ | $0.002$ | $0.003$ | $0.001$ | $0.001$ | $0.043$ | $0.029$ | $0.025$ | $0.131$ | $0.126$ | $0.080$ | $0.073$ | $0.150$
Decay-time-dependent fit procedure | $0.001$ | $0.002$ | $0.248$ | $0.017$ | $0.002$ | $0.004$ | $0.002$ | $0.002$ | $0.008$ | $0.005$ | $0.012$ | $0.069$ | $0.025$ | $0.062$ | $0.017$ | $0.030$
Decay-time-dependent fit parameterisation | $0.006$ | $0.017$ | $0.736$ | $0.247$ | $0.011$ | $0.053$ | $0.019$ | $0.008$ | $0.080$ | $0.048$ | $0.286$ | $0.308$ | $0.260$ | $0.260$ | $0.228$ | $0.405$
Acceptance weights (simulated sample size) | $0.014$ | $0.015$ | $1.463$ | $0.719$ | $0.026$ | $0.145$ | $0.054$ | $0.027$ | $0.199$ | $0.102$ | $1.117$ | $1.080$ | $0.888$ | $0.712$ | $0.417$ | $0.947$
Other acceptance and resolution effects | $0.002$ | $0.003$ | $0.184$ | $0.226$ | $0.015$ | $0.024$ | $0.004$ | $0.005$ | $0.045$ | $0.017$ | $0.163$ | $0.168$ | $0.191$ | $0.229$ | $0.246$ | $0.171$
Production asymmetry | $0.001$ | $0.001$ | $0.037$ | $0.026$ | $0.001$ | $0.003$ | $0.001$ | $0.002$ | $0.012$ | $0.006$ | $0.015$ | $0.030$ | $0.018$ | $0.003$ | $0.007$ | $0.041$
Total | $0.031$ | $0.033$ | $1.688$ | $0.817$ | $0.049$ | $0.165$ | $0.063$ | $0.048$ | $0.252$ | $0.143$ | $1.171$ | $1.159$ | $0.970$ | $0.802$ | $0.546$ | $1.076$

Parameter | $f^{TT}$ | $f_{L}^{TT}$ | $f_{\parallel_{1}}^{TT}$ | $f_{\perp_{1}}^{TT}$ | $f_{\parallel_{2}}^{TT}$ | $\delta_{0}^{TT}$ | $\delta_{\parallel_{1}}^{TT}$ | $\delta_{\perp_{1}}^{TT}$ | $\delta_{\parallel_{2}}^{TT}$ | $\delta_{\perp_{2}}^{TT}$
---|---|---|---|---|---|---|---|---|---|---
Yield and shape of mass model | $0.000$ | $0.045$ | $0.019$ | $0.037$ | $0.002$ | $0.038$ | $0.027$ | $0.009$ | $0.079$ | $0.114$
Signal weights of mass model | $0.000$ | $0.066$ | $0.025$ | $0.024$ | $0.002$ | $0.147$ | $0.046$ | $0.112$ | $0.123$ | $0.215$
Decay-time-dependent fit procedure | $0.001$ | $0.022$ | $0.022$ | $0.014$ | $0.004$ | $0.127$ | $0.036$ | $0.068$ | $0.058$ | $0.040$
Decay-time-dependent fit parameterisation | $0.005$ | $0.051$ | $0.071$ | $0.113$ | $0.038$ | $1.213$ | $0.199$ | $0.685$ | $0.820$ | $0.476$
Acceptance weights (simulated sample size) | $0.003$ | $0.135$ | $0.110$ | $0.127$ | $0.077$ | $1.328$ | $0.454$ | $1.348$ | $1.443$ | $1.161$
Other acceptance and resolution effects | $0.002$ | $0.031$ | $0.028$ | $0.056$ | $0.024$ | $0.226$ | $0.275$ | $0.156$ | $0.343$ | $0.301$
Production asymmetry | $0.000$ | $0.002$ | $0.001$ | $0.008$ | $0.003$ | $0.005$ | $0.002$ | $0.062$ | $0.015$ | $0.043$
Total | $0.007$ | $0.176$ | $0.142$ | $0.205$ | $0.107$ | $1.825$ | $0.573$ | $1.546$ | $1.706$ | $1.330$

##  9 Fit results

An unbinned maximum likelihood fit is applied to the background-subtracted
data using the PDF defined in Eq. (11). The large computational load due to
the complexity of the fit motivates the parallelisation of the process on a
Graphics Processing Unit (GPU), for which the Ipanema software package [43,
44] is used. The one-dimensional projections of the results in the six
analysis variables are shown in Fig. 7 along with the separate components from
the contributing decay modes listed in Table 1. The resulting fit values for
the common $C\\!P$ observables, $\phi_{s}^{{d}{\overline{d}}}$ and
$|\lambda|$, as well as the $C\\!P$-averaged fractions, $f_{i}$, and
polarisation strong-phase differences, $\delta_{i}$, for each component are
given in Table 5. The central values are given along with the statistical
uncertainties obtained from the fit and the systematic uncertainties, which
are discussed in Sec. 8. These are the first measurements in a
${b}\rightarrow{d}{\overline{d}}{s}$ transition of the $C\\!P$-violation
parameter $|\lambda|=1.035\pm 0.034\pm 0.089$ and the $C\\!P$-violating weak
phase $\phi_{s}^{{d}{\overline{d}}}=-0.10\pm 0.13\pm 0.14\mathrm{\,rad}$. Both
are consistent with no $C\\!P$ violation and with the SM predictions. In the
region of phase space considered, the
${{{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}}$ vector-vector component has
a relatively small fraction, of $f^{VV}=0.067\pm 0.004\pm 0.024$, mainly due
to the large scalar $K\pi$ contributions. Indeed, a relatively large
contribution from the scalar-scalar double $S$-wave fraction is determined to
be $f^{SS}=0.225\pm 0.010\pm 0.069$. The tensor-tensor double $D$-wave
fraction is measured to be $f^{TT}=0.011\pm 0.003\pm 0.007$. The cross-term
contributions from the scalar with the vector combination (single $S$-wave)
and the vector with the tensor combination (single $D$-wave) are also found to
be large, $f^{SV}=0.329\pm 0.015\pm 0.071$, $f^{VS}=0.133\pm 0.013\pm 0.065$,
$f^{VT}=0.160\pm 0.016\pm 0.049$ and $f^{TV}=0.036\pm 0.014\pm 0.048$, while a
small contribution from the scalar with the tensor combination is found,
$f^{TS}=0.025\pm 0.007\pm 0.033$ and $f^{ST}=0.014\pm 0.006\pm 0.031$. The
values of the longitudinal polarisation fractions of the vector-vector and
tensor-tensor components are found to be small, $f_{\rm L}^{TT}=0.25\pm
0.14\pm 0.18$ and $f_{\rm L}^{VV}=0.208\pm 0.032\pm 0.046$, while the
longitudinal polarisation fractions of the vector with the tensor components
are measured to be large, $f_{\rm L}^{VT}=0.911\pm 0.020\pm 0.165$ and $f_{\rm
L}^{TV}=0.62\pm 0.16\pm 0.25$.

Figure 7: One-dimensional projections of the decay-time-dependent, flavour-
tagged fit to (black points) the sPlot weighted data for (top row) the two
$(K\pi)$ invariant masses, (middle row) the two $(K\pi)$ decay plane angles,
(bottom left) the angle between the two $(K,\pi)$ decay planes and (bottom
right) the decay-time. The solid gray line represents the total fit model
along with the $C\\!P$-averaged components for each contributing decay. Table
5: Results of the decay-time-dependent amplitude fit to data. The first
uncertainty is statistical and the second uncertainty is systematic.

Parameter | Value
---|---
Common parameters
$\phi_{s}^{{d}{\overline{d}}}$ [rad] | $-0.10\phantom{0}\pm 0.13\phantom{0}\pm 0.14\phantom{0}$
$|\lambda|$ | $\phantom{-}1.035\pm 0.034\pm 0.089$
Vector/Vector (VV)
$f^{VV}$ | $\phantom{-}0.067\pm 0.004\pm 0.024$
$f_{\rm L}^{VV}$ | $\phantom{-}0.208\pm 0.032\pm 0.046$
$f_{\parallel}^{VV}$ | $\phantom{-}0.297\pm 0.029\pm 0.042$
$\delta^{VV}_{\parallel}$ [rad] | $\phantom{-}2.40\phantom{0}\pm 0.11\phantom{0}\pm 0.33\phantom{0}$
$\delta^{VV}_{\perp}$ [rad] | $\phantom{-}2.62\phantom{0}\pm 0.26\phantom{0}\pm 0.64\phantom{0}$
Scalar/Vector (SV and VS)
$f^{SV}$ | $\phantom{-}0.329\pm 0.015\pm 0.071$
$f^{VS}$ | $\phantom{-}0.133\pm 0.013\pm 0.065$
$\delta^{SV}$ [rad] | $-1.31\phantom{0}\pm 0.10\phantom{0}\pm 0.35\phantom{0}$
$\delta^{VS}$ [rad] | $\phantom{-}1.86\phantom{0}\pm 0.11\phantom{0}\pm 0.41\phantom{0}$
Scalar/Scalar (SS)
$f^{SS}$ | $\phantom{-}0.225\pm 0.010\pm 0.069$
$\delta^{SS}$ [rad] | $\phantom{-}1.07\phantom{0}\pm 0.10\phantom{0}\pm 0.40\phantom{0}$
Scalar/Tensor (ST and TS)
$f^{ST}$ | $\phantom{-}0.014\pm 0.006\pm 0.031$
$f^{TS}$ | $\phantom{-}0.025\pm 0.007\pm 0.033$
$\delta^{ST}$ [rad] | $-2.3\phantom{00}\pm 0.4\phantom{00}\pm 1.7\phantom{00}$
$\delta^{TS}$ [rad] | $-0.10\phantom{0}\pm 0.26\phantom{0}\pm 0.82\phantom{0}$

Parameter | Value
---|---
Vector/Tensor (VT and TV)
$f^{VT}$ | $\phantom{-}0.160\pm 0.016\pm 0.049$
$f_{\rm L}^{VT}$ | $\phantom{-}0.911\pm 0.020\pm 0.165$
$f_{\parallel}^{VT}$ | $\phantom{-}0.012\pm 0.008\pm 0.053$
$f^{TV}$ | $\phantom{-}0.036\pm 0.014\pm 0.048$
$f_{\rm L}^{TV}$ | $\phantom{-}0.62\phantom{0}\pm 0.16\phantom{0}\pm 0.25\phantom{0}$
$f_{\parallel}^{TV}$ | $\phantom{-}0.24\phantom{0}\pm 0.10\phantom{0}\pm 0.14\phantom{0}$
$\delta^{VT}_{0}$ [rad] | $-2.06\phantom{0}\pm 0.19\phantom{0}\pm 1.17\phantom{0}$
$\delta^{VT}_{\parallel}$ [rad] | $-1.8\phantom{00}\pm 0.4\phantom{00}\pm 1.0\phantom{00}$
$\delta^{VT}_{\perp}$ [rad] | $-3.2\phantom{00}\pm 0.3\phantom{00}\pm 1.2\phantom{00}$
$\delta^{TV}_{0}$ [rad] | $\phantom{-}1.91\phantom{0}\pm 0.30\phantom{0}\pm 0.80\phantom{0}$
$\delta^{TV}_{\parallel}$ [rad] | $\phantom{-}1.09\phantom{0}\pm 0.19\phantom{0}\pm 0.55\phantom{0}$
$\delta^{TV}_{\perp}$ [rad] | $\phantom{-}0.2\phantom{00}\pm 0.4\phantom{00}\pm 1.1\phantom{00}$
Tensor/Tensor (TT)
$f^{TT}$ | $\phantom{-}0.011\pm 0.003\pm 0.007$
$f_{\rm L}^{TT}$ | $\phantom{-}0.25\phantom{0}\pm 0.14\phantom{0}\pm 0.18\phantom{0}$
$f_{\parallel_{1}}^{TT}$ | $\phantom{-}0.17\phantom{0}\pm 0.11\phantom{0}\pm 0.14\phantom{0}$
$f_{\perp_{1}}^{TT}$ | $\phantom{-}0.30\phantom{0}\pm 0.18\phantom{0}\pm 0.21\phantom{0}$
$f_{\parallel_{2}}^{TT}$ | $\phantom{-}0.015\pm 0.033\pm 0.107$
$\delta^{TT}_{0}$ [rad] | $\phantom{-}1.3\phantom{00}\pm 0.5\phantom{00}\pm 1.8\phantom{00}$
$\delta^{TT}_{\parallel_{1}}$ [rad] | $\phantom{-}3.00\phantom{0}\pm 0.29\phantom{0}\pm 0.57\phantom{0}$
$\delta^{TT}_{\perp_{1}}$ [rad] | $\phantom{-}2.6\phantom{00}\pm 0.4\phantom{00}\pm 1.5\phantom{00}$
$\delta^{TT}_{\parallel_{2}}$ [rad] | $\phantom{-}2.3\phantom{00}\pm 0.8\phantom{00}\pm 1.7\phantom{00}$
$\delta^{TT}_{\perp_{2}}$ [rad] | $\phantom{-}0.7\phantom{00}\pm 0.6\phantom{00}\pm 1.3\phantom{00}$

##  10 Summary

A flavour-tagged decay-time-dependent amplitude analysis of the
${{B}^{0}_{s}}\rightarrow({{K}^{+}}{{\pi}^{-}})({{K}^{-}}{{\pi}^{+}})$ decay,
for $({{K}^{\pm}}{{\pi}^{\mp}})$ invariant masses in the range from 750 to
1600${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, is performed on a data set
corresponding to an integrated luminosity of $3.0\mbox{\,fb}^{-1}$ obtained by
the LHCb experiment with $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ and
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$. Several quasi-two-body decay
components are considered, corresponding to $({{K}^{\pm}}{{\pi}^{\mp}})$
combinations with spins of 0, 1 and 2. The longitudinal polarisation fraction
for the ${{{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}}$ vector-vector decay is
determined to be $f_{\rm L}^{VV}=0.208\pm 0.032\pm 0.046$, where the first
uncertainty is statistical and the second one systematic. This confirms, with
improved precision, the relatively low value reported previously by LHCb [14].
The first determination of the $C\\!P$ asymmetry of the
$(K^{+}\pi^{-})(K^{-}\pi^{+})$ final state and the best, sometimes the first,
measurements of 19 $C\\!P$-averaged amplitude parameters corresponding to
scalar, vector and tensor final states, are also reported. This analysis
determines for the first time the mixing-induced $C\\!P$-violating phase
$\phi_{s}$ using a ${b}\rightarrow{d}{\overline{d}}{s}$ transition. The value
of this phase is measured to be $\phi_{s}^{{d}{\overline{d}}}=-0.10\pm 0.13\pm
0.14\mathrm{\,rad}$, which is consistent with both the SM expectation [7] and
the corresponding LHCb result of $\phi_{s}^{{s}{\overline{s}}}=-0.17\pm
0.15\pm 0.03\mathrm{\,rad}$ measured using ${{B}^{0}_{s}}\rightarrow\phi\phi$
decays [12]. The statistical uncertainty of the two measurements is at a
similar level although the systematic uncertainty of this measurement is
larger, which is mainly due to the treatment of the multi-dimensional
acceptance. It is expected that this can be reduced by increasing the size of
the simulation sample used to determine the acceptance effects. Most other
sources of systematic uncertainty are expected to scale with larger data
samples.

## 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 (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania);
MinES and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU
(Ukraine); STFC (United Kingdom); NSF (USA). We acknowledge the computing
resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany),
INFN (Italy), SURF (The 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),
ANR, Labex P2IO and OCEVU, and Région Auvergne-Rhône-Alpes (France), RFBR, RSF
and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund,
the Royal Society, the English-Speaking Union and the Leverhulme Trust (United
Kingdom).

Appendices

##  Appendix A Angular distributions

The angular dependence of the decay amplitudes introduced in Eq. (4) is shown
in Table 6.

Table 6: Functions containing the angular dependence of the amplitudes, as introduced in Eq. (4). For a discussion on some of the angular terms see Ref. [7]. $j_{1}$ | $j_{2}$ | $h$ | $Y_{\ell_{1}}^{m_{1}}(\theta_{1},-\varphi)Y_{\ell_{2}}^{m_{2}}(\pi-\theta_{2},0)$ | $\Theta^{j_{1}j_{2}}_{h}(\cos\theta_{1},\cos\theta_{2},\varphi)$
---|---|---|---|---
0 | 0 | ${0}$ | $\sqrt{\pi}Y_{0}^{0}Y_{0}^{0}$ | $\frac{1}{2\sqrt{2\pi}}$
0 | 1 | ${0}$ | $\sqrt{\pi}Y_{0}^{0}Y_{1}^{0}$ | $-\frac{\sqrt{3}}{2\sqrt{2\pi}}\cos\theta_{2}$
1 | 0 | ${0}$ | $\sqrt{\pi}Y_{1}^{0}Y_{0}^{0}$ | $\frac{\sqrt{3}}{2\sqrt{2\pi}}\cos\theta_{1}$
0 | 2 | ${0}$ | $\sqrt{\pi}Y_{0}^{0}Y_{2}^{0}$ | $\frac{\sqrt{5}}{4\sqrt{2\pi}}(3\cos^{2}\theta_{2}-1)$
2 | 0 | ${0}$ | $\sqrt{\pi}Y_{2}^{0}Y_{0}^{0}$ | $\frac{\sqrt{5}}{4\sqrt{2\pi}}(3\cos^{2}\theta_{1}-1)$
1 | 1 | ${0}$ | $\sqrt{\pi}Y_{1}^{0}Y_{1}^{0}$ | $-\frac{3}{2\sqrt{2\pi}}\cos\theta_{1}\cos\theta_{2}$
1 | 1 | ${\parallel}$ | $\frac{\sqrt{\pi}}{\sqrt{2}}(Y_{1}^{-1}Y_{1}^{+1}+Y_{1}^{+1}Y_{1}^{-1})$ | $-\frac{3}{4\sqrt{\pi}}\sin\theta_{1}\sin\theta_{2}\cos\varphi$
1 | 1 | ${\perp}$ | $\frac{\sqrt{\pi}}{\sqrt{2}}(Y_{1}^{-1}Y_{1}^{+1}-Y_{1}^{+1}Y_{1}^{-1})$ | $-i\frac{3}{4\sqrt{\pi}}\sin\theta_{1}\sin\theta_{2}\sin\varphi$
1 | 2 | ${0}$ | $\sqrt{\pi}Y_{1}^{0}Y_{2}^{0}$ | $\frac{\sqrt{15}}{4\sqrt{2\pi}}\cos\theta_{1}(3\cos^{2}\theta_{2}-1)$
1 | 2 | ${\parallel}$ | $\frac{\sqrt{\pi}}{\sqrt{2}}(Y_{1}^{-1}Y_{2}^{+1}+Y_{1}^{+1}Y_{2}^{-1})$ | $\frac{3\sqrt{5}}{4\sqrt{\pi}}\sin\theta_{1}\sin\theta_{2}\cos\theta_{2}\cos\varphi$
1 | 2 | ${\perp}$ | $\frac{\sqrt{\pi}}{\sqrt{2}}(Y_{1}^{-1}Y_{2}^{+1}-Y_{1}^{+1}Y_{2}^{-1})$ | $i\frac{3\sqrt{5}}{4\sqrt{\pi}}\sin\theta_{1}\sin\theta_{2}\cos\theta_{2}\sin\varphi$
2 | 1 | ${0}$ | $\sqrt{\pi}Y_{2}^{0}Y_{1}^{0}$ | $-\frac{\sqrt{15}}{4\sqrt{2\pi}}(3\cos^{2}\theta_{1}-1)\cos\theta_{2}$
2 | 1 | ${\parallel}$ | $\frac{\sqrt{\pi}}{\sqrt{2}}(Y_{2}^{-1}Y_{1}^{+1}+Y_{2}^{+1}Y_{1}^{-1})$ | $-\frac{3\sqrt{5}}{4\sqrt{\pi}}\sin\theta_{1}\cos\theta_{1}\sin\theta_{2}\cos\varphi$
2 | 1 | ${\perp}$ | $\frac{\sqrt{\pi}}{\sqrt{2}}(Y_{2}^{-1}Y_{1}^{+1}-Y_{2}^{+1}Y_{1}^{-1})$ | $-i\frac{3\sqrt{5}}{4\sqrt{\pi}}\sin\theta_{1}\cos\theta_{1}\sin\theta_{2}\sin\varphi$
2 | 2 | ${0}$ | $\sqrt{\pi}Y_{2}^{0}Y_{2}^{0}$ | $\frac{5}{8\sqrt{2\pi}}(3\cos^{2}\theta_{1}-1)(3\cos^{2}\theta_{2}-1)$
2 | 2 | ${\parallel_{1}}$ | $\frac{\sqrt{\pi}}{\sqrt{2}}(Y_{2}^{-1}Y_{2}^{+1}+Y_{2}^{+1}Y_{2}^{-1})$ | $\frac{15}{4\sqrt{\pi}}\sin\theta_{1}\cos\theta_{1}\sin\theta_{2}\cos\theta_{2}\cos\varphi$
2 | 2 | ${\perp_{1}}$ | $\frac{\sqrt{\pi}}{\sqrt{2}}(Y_{2}^{-1}Y_{2}^{+1}-Y_{2}^{+1}Y_{2}^{-1})$ | $i\frac{15}{4\sqrt{\pi}}\sin\theta_{1}\cos\theta_{1}\sin\theta_{2}\cos\theta_{2}\sin\varphi$
2 | 2 | ${\parallel_{2}}$ | $\frac{\sqrt{\pi}}{\sqrt{2}}(Y_{2}^{-2}Y_{2}^{+2}+Y_{2}^{+2}Y_{2}^{-2})$ | $\frac{15}{16\sqrt{\pi}}\sin^{2}\theta_{1}\sin^{2}\theta_{2}\cos(2\varphi)$
2 | 2 | ${\perp_{2}}$ | $\frac{\sqrt{\pi}}{\sqrt{2}}(Y_{2}^{-2}Y_{2}^{+2}-Y_{2}^{+2}Y_{2}^{-2})$ | $i\frac{15}{16\sqrt{\pi}}\sin^{2}\theta_{1}\sin^{2}\theta_{2}\sin(2\varphi)$

##  Appendix B Scalar $K\pi$ mass-dependent amplitude

The variation of the phase with $m(K\pi)$ in the nominal model used for the
scalar $K\pi$ mass-dependent amplitude is taken from Ref. [17]. The modulus
line-shape is parameterised with a polynomial expansion as follows

$|\mathcal{M}_{0}(m)|=1+\sum_{i=1}^{4}c_{i}\,T_{i}(X(m)),$ (20)

where $X(m)=(m-1175\,{\rm MeV}/c^{2})/425\,{\rm MeV}/c^{2}$ $X(m)\in[-1,1]$,
and $T_{i}(x)$ are the Chebyshev polynomials defined as

$T_{0}(x)=1$, |  $T_{1}(x)=x$, |  $T_{2}(x)=2x^{2}-1$,
---|---|---
$T_{3}(x)=4x^{3}-3x$, |  $T_{4}(x)=8x^{4}-8x^{2}+1$. |
(21)

This parameterisation is chosen to minimise parameter correlations. The values
of the $c_{i}$ coefficients retrieved from the decay-time-dependent fit are
given in Table 7. The coefficients decrease with the order of the polynomial
term. The expansion is truncated at fourth order since adding an extra term
would not significantly affect the result and the size of the fifth
coefficient is of the order of its statistical uncertainty.

Table 7: Parameters used in the nominal model for the scalar $K\pi$ mass-dependent amplitude. The correlations among them are found to be small, the largest ones been of the order of $50\%$. Parameter | Value
---|---
$c_{1}$ | $-0.287\pm 0.020$
$c_{2}$ | $-0.180\pm 0.020$
$c_{3}$ | $-0.106\pm 0.016$
$c_{4}$ | $-0.066\pm 0.016$

Figure 8: Line-shapes of the (left) modulus and (right) phase of the scalar
$K\pi$ mass-dependent amplitude. The nominal model is shown with a solid blue
line and the model-independent parameterisation, used in systematic studies,
is shown with a dashed red line.

When computing systematic uncertainties, the scalar $K\pi$ mass-dependent
amplitude is parameterised using a model-independent (MI) approach as follows

$\mathcal{M}_{0}^{MI}(m)=\left[1+\sum_{i=1}^{4}\alpha_{i}\,T_{i}(X(m))\right]+i\,\left[\sum_{j=0}^{4}\beta_{j}\,T_{j}(X(m))\right].$
(22)

The coefficients measured in the decay-time-dependent fit for this case are
given in Table 8.

Table 8: Coefficients used in the model-independent parameterisation of the scalar $K\pi$ mass-dependent amplitude. Parameter | Value | Parameter | Value
---|---|---|---
$\alpha_{1}$ | $-0.854\pm 0.038$ | $\beta_{0}$ | $\phantom{-}0.278\pm 0.038$
$\alpha_{2}$ | $-0.381\pm 0.040$ | $\beta_{1}$ | $\phantom{-}0.817\pm 0.079$
$\alpha_{3}$ | $-0.105\pm 0.032$ | $\beta_{2}$ | $-0.206\pm 0.082$
$\alpha_{4}$ | $\phantom{-}0.046\pm 0.027$ | $\beta_{3}$ | $-0.367\pm 0.053$
|  | $\beta_{4}$ | $-0.115\pm 0.040$

The line-shapes of the two scalar mass amplitude models are shown in Fig. 8.
Both approaches are found to be qualitatively compatible with each other.

## References

  * [1] R. Fleischer, _Extracting CKM phases from angular distributions of $B_{d,s}$ decays into admixtures of CP eigenstates_, Phys. Rev. D60 (1999) 073008, arXiv:hep-ph/9903540
  * [2] R. Fleischer and M. Gronau, _Studying new physics amplitudes in charmless $B_{s}$ decays_, Phys. Lett. B660 (2008) 212, arXiv:0709.4013
  * [3] M. Ciuchini, M. Pierini, and L. Silvestrini, _$B_{s}^{0}\rightarrow K^{(*)0}\bar{K}^{(*)0}$ CP asymmetries: Golden channels for new physics searches_, Phys. Rev. Lett. 100 (2008) 031802, arXiv:hep-ph/0703137
  * [4] S. Descotes-Genon, J. Matias, and J. Virto, _Penguin-mediated $B_{d,s}\rightarrow VV$ decays and the ${{B}^{0}_{s}}-{{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ mixing angle_, Phys. Rev. D76 (2007) 074005, Erratum ibid. D84 (2011) 039901, arXiv:0705.0477
  * [5] S. Descotes-Genon, J. Matias, and J. Virto, _Analysis of $B_{d,s}$ mixing angles in presence of new physics and an update of $B_{s}\rightarrow K^{0*}\bar{K}^{0*}$_, Phys. Rev. D85 (2012) 034010, arXiv:1111.4882
  * [6] B. Bhattacharya, A. Datta, M. Imbeault, and D. London, _Searching for new physics with $B_{s}\rightarrow K^{0(*)}\bar{K}^{0(*)}$ – a reappraisal_, Phys. Lett. B717 (2012) 403, arXiv:1203.3435
  * [7] B. Bhattacharya, A. Datta, M. Duraisamy, and D. London, _Searching for new physics with $\bar{b}\rightarrow\bar{s}$ $B_{s}^{0}\rightarrow V_{1}V_{2}$ penguin decays_, Phys. Rev. D88 (2013) 016007, arXiv:1306.1911
  * [8] Heavy Flavor Averaging Group, Y. Amhis et al., _Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of summer 2016_, arXiv:1612.07233, updated results and plots available at http://www.slac.stanford.edu/xorg/hflav/
  * [9] LHCb collaboration, R. Aaij et al., _Precision measurement of ${C\\!P}$ violation in ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}}$ decays_, Phys. Rev. Lett. 114 (2015) 041801, arXiv:1411.3104
  * [10] ATLAS collaboration, G. Aad et al., _Measurement of the CP-violating phase $\phi_{s}$ and the $B^{0}_{s}$ meson decay width difference with $B^{0}_{s}\rightarrow J/\psi\phi$ decays in ATLAS_, JHEP 08 (2016) 147, arXiv:1601.03297
  * [11] CMS collaboration, V. Khachatryan et al., _Measurement of the CP-violating weak phase $\phi_{s}$ and the decay width difference $\Delta\Gamma_{s}$ using the $B_{s}^{0}\rightarrow J/\psi\phi$(1020) decay channel in $pp$ collisions at $\sqrt{s}=$ 8 TeV_, Phys. Lett. B757 (2016) 97, arXiv:1507.07527
  * [12] LHCb collaboration, R. Aaij et al., _Measurement of ${C\\!P}$ violation in ${{B}^{0}_{s}}\rightarrow\phi\phi$ decays_, Phys. Rev. D90 (2014) 052011, arXiv:1407.2222
  * [13] LHCb collaboration, R. Aaij et al., _First observation of the decay ${{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$_, Phys. Lett. B709 (2012) 50, arXiv:1111.4183
  * [14] LHCb collaboration, R. Aaij et al., _Measurement of ${C\\!P}$ asymmetries and polarisation fractions in ${{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decays_, JHEP 07 (2015) 166, arXiv:1503.05362
  * [15] Particle Data Group, C. Patrignani et al., _Review of particle physics_ , Chin. Phys. C40 (2016) 100001
  * [16] F. Von Hippel and C. Quigg, _Centrifugal-barrier effects in resonance partial decay widths, shapes, and production amplitudes_ , Phys. Rev. D5 (1972) 624
  * [17] J. R. Pelaez and A. Rodas, _Pion-kaon scattering amplitude constrained with forward dispersion relations up to 1.6 GeV_ , Phys. Rev. D93 (2016) 074025, arXiv:1602.08404
  * [18] K. M. Watson, _Some general relations between the photoproduction and scattering of $\pi$ mesons_, Phys. Rev. 95 (1954) 228
  * [19] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [20] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [21] V. V. Gligorov and M. Williams, _Efficient, reliable and fast high-level triggering using a bonsai boosted decision tree_ , JINST 8 (2013) P02013, arXiv:1210.6861
  * [22] T. Sjöstrand, S. Mrenna, and P. Skands, _A brief introduction to PYTHIA 8.1_ , Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
  * [23] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [24] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [25] P. Golonka and Z. Was, _PHOTOS Monte Carlo: A precision tool for QED corrections in $Z$ and $W$ decays_, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [26] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [27] Geant4 collaboration, S. Agostinelli et al., _Geant4: A simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [28] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [29] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
  * [30] Y. Freund and R. E. Schapire, _A decision-theoretic generalization of on-line learning and an application to boosting_ , J. Comput. Syst. Sci. 55 (1997) 119
  * [31] A. Blum, A. Kalai, and J. Langford, _Beating the hold-out: Bounds for k-fold and progressive cross-validation_ , in Proceedings of the Twelfth Annual Conference on Computational Learning Theory, COLT ’99, (New York, NY, USA), pp. 203–208, ACM, 1999.  doi: 10.1145/307400.307439
  * [32] M. Pivk and F. R. Le Diberder, _sPlot: A statistical tool to unfold data distributions_ , Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
  * [33] D. Martínez Santos and F. Dupertuis, _Mass distributions marginalized over per-event errors_ , Nucl. Instrum. Meth. A764 (2014) 150, arXiv:1312.5000
  * [34] LHCb collaboration, R. Aaij et al., _Measurement of the fragmentation fraction ratio $f_{s}/f_{d}$ and its dependence on $B$ meson kinematics_, JHEP 04 (2013) 001, arXiv:1301.5286, $f_{s}/f_{d}$ value updated in LHCb-CONF-2013-011
  * [35] ARGUS collaboration, _Search for hadronic $b\rightarrow u$ decays_, Phys. Lett. B241 (1990) 278
  * [36] LHCb collaboration, R. Aaij et al., _Neural-network-based same side kaon tagging algorithm calibrated with ${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ and $B_{s2}^{*}(5840)^{0}\rightarrow{{{B}^{+}}}{{K}^{-}}$ decays_, JINST 11 (2016) P05010, arXiv:1602.07252
  * [37] LHCb collaboration, R. Aaij et al., _Opposite-side flavour tagging of ${B}$ mesons at the LHCb experiment_, Eur. Phys. J. C72 (2012) 2022, arXiv:1202.4979
  * [38] LHCb collaboration, R. Aaij et al., _Measurement of the ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$–${{B}^{0}}$ and ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$–${{B}^{0}_{s}}$ production asymmetries in ${p}{p}$ collisions at $\sqrt{s}=7$$~{}T\kern-0.50003pteV$_, Phys. Lett. B739 (2014) 218, arXiv:1408.0275
  * [39] T. M. Karbach, G. Raven, and M. Schiller, _Decay time integrals in neutral meson mixing and their efficient evaluation_ , arXiv:1407.0748
  * [40] T. du Pree, Search for a strange phase in beautiful oscillations, PhD thesis, Nikhef, Amsterdam, 2010, CERN-THESIS-2010-124
  * [41] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
  * [42] LHCb collaboration, R. Aaij et al., _Measurement of ${C\\!P}$ violation and the ${{B}^{0}_{s}}$ meson decay width difference with ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}}$ and ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ decays_, Phys. Rev. D87 (2013) 112010, arXiv:1304.2600
  * [43] D. Martínez Santos et al., _Ipanema- $\beta$: Tools and examples for HEP analysis on GPU_, arXiv:1706.01420
  * [44] A. Klöckner et al., _PyCUDA and PyOpenCL: A scripting-based approach to GPU run-time code generation_ , Parallel Computing 38 (2012) 157, arXiv:0911.3456

LHCb collaboration

R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar59, J. Albrecht10,
F. Alessio40, M. Alexander53, A. Alfonso Albero38, S. Ali43, G. Alkhazov31, P.
Alvarez Cartelle55, A.A. Alves Jr59, S. Amato2, S. Amerio23, Y. Amhis7, L.
An3, L. Anderlini18, G. Andreassi41, M. Andreotti17,g, J.E. Andrews60, R.B.
Appleby56, F. Archilli43, P. d’Argent12, J. Arnau Romeu6, A. Artamonov37, M.
Artuso61, E. Aslanides6, M. Atzeni42, G. Auriemma26, M. Baalouch5, I.
Babuschkin56, S. Bachmann12, J.J. Back50, A. Badalov38,m, C. Baesso62, S.
Baker55, V. Balagura7,b, W. Baldini17, A. Baranov35, R.J. Barlow56, C.
Barschel40, S. Barsuk7, W. Barter56, F. Baryshnikov32, V. Batozskaya29, V.
Battista41, A. Bay41, L. Beaucourt4, J. Beddow53, F. Bedeschi24, I. Bediaga1,
A. Beiter61, L.J. Bel43, N. Beliy63, V. Bellee41, N. Belloli21,i, K. Belous37,
I. Belyaev32,40, E. Ben-Haim8, G. Bencivenni19, S. Benson43, S. Beranek9, A.
Berezhnoy33, R. Bernet42, D. Berninghoff12, E. Bertholet8, A. Bertolin23, C.
Betancourt42, F. Betti15, M.O. Bettler40, M. van Beuzekom43, Ia. Bezshyiko42,
S. Bifani47, P. Billoir8, A. Birnkraut10, A. Bizzeti18,u, M. Bjørn57, T.
Blake50, F. Blanc41, S. Blusk61, V. Bocci26, T. Boettcher58, A. Bondar36,w, N.
Bondar31, I. Bordyuzhin32, S. Borghi56,40, M. Borisyak35, M. Borsato39, F.
Bossu7, M. Boubdir9, T.J.V. Bowcock54, E. Bowen42, C. Bozzi17,40, S. Braun12,
J. Brodzicka27, D. Brundu16, E. Buchanan48, C. Burr56, A. Bursche16,f, J.
Buytaert40, W. Byczynski40, S. Cadeddu16, H. Cai64, R. Calabrese17,g, R.
Calladine47, M. Calvi21,i, M. Calvo Gomez38,m, A. Camboni38,m, P. Campana19,
D.H. Campora Perez40, L. Capriotti56, A. Carbone15,e, G. Carboni25,j, R.
Cardinale20,h, A. Cardini16, P. Carniti21,i, L. Carson52, K. Carvalho Akiba2,
G. Casse54, L. Cassina21, M. Cattaneo40, G. Cavallero20,40,h, R. Cenci24,t, D.
Chamont7, M.G. Chapman48, M. Charles8, Ph. Charpentier40, G.
Chatzikonstantinidis47, M. Chefdeville4, S. Chen16, S.F. Cheung57, S.-G.
Chitic40, V. Chobanova39, M. Chrzaszcz42, A. Chubykin31, P. Ciambrone19, X.
Cid Vidal39, G. Ciezarek40, P.E.L. Clarke52, M. Clemencic40, H.V. Cliff49, J.
Closier40, V. Coco40, J. Cogan6, E. Cogneras5, V. Cogoni16,f, L. Cojocariu30,
P. Collins40, T. Colombo40, A. Comerma-Montells12, A. Contu16, G. Coombs40, S.
Coquereau38, G. Corti40, M. Corvo17,g, C.M. Costa Sobral50, B. Couturier40,
G.A. Cowan52, D.C. Craik58, A. Crocombe50, M. Cruz Torres1, R. Currie52, C.
D’Ambrosio40, F. Da Cunha Marinho2, C.L. Da Silva73, E. Dall’Occo43, J.
Dalseno48, A. Davis3, O. De Aguiar Francisco40, K. De Bruyn40, S. De Capua56,
M. De Cian12, J.M. De Miranda1, L. De Paula2, M. De Serio14,d, P. De Simone19,
C.T. Dean53, D. Decamp4, L. Del Buono8, H.-P. Dembinski11, M. Demmer10, A.
Dendek28, D. Derkach35, O. Deschamps5, F. Dettori54, B. Dey65, A. Di Canto40,
P. Di Nezza19, H. Dijkstra40, F. Dordei40, M. Dorigo40, A. Dosil Suárez39, L.
Douglas53, A. Dovbnya45, K. Dreimanis54, L. Dufour43, G. Dujany8, P.
Durante40, J.M. Durham73, D. Dutta56, R. Dzhelyadin37, M. Dziewiecki12, A.
Dziurda40, A. Dzyuba31, S. Easo51, U. Egede55, V. Egorychev32, S.
Eidelman36,w, S. Eisenhardt52, U. Eitschberger10, R. Ekelhof10, L. Eklund53,
S. Ely61, S. Esen12, H.M. Evans49, T. Evans57, A. Falabella15, N. Farley47, S.
Farry54, D. Fazzini21,i, L. Federici25, D. Ferguson52, G. Fernandez38, P.
Fernandez Declara40, A. Fernandez Prieto39, F. Ferrari15, L. Ferreira Lopes41,
F. Ferreira Rodrigues2, M. Ferro-Luzzi40, S. Filippov34, R.A. Fini14, M.
Fiorini17,g, M. Firlej28, C. Fitzpatrick41, T. Fiutowski28, F. Fleuret7,b, M.
Fontana16,40, F. Fontanelli20,h, R. Forty40, V. Franco Lima54, M. Frank40, C.
Frei40, J. Fu22,q, W. Funk40, E. Furfaro25,j, C. Färber40, E. Gabriel52, A.
Gallas Torreira39, D. Galli15,e, S. Gallorini23, S. Gambetta52, M. Gandelman2,
P. Gandini22, Y. Gao3, L.M. Garcia Martin71, J. García Pardiñas39, J. Garra
Tico49, L. Garrido38, D. Gascon38, C. Gaspar40, L. Gavardi10, G. Gazzoni5, D.
Gerick12, E. Gersabeck56, M. Gersabeck56, T. Gershon50, Ph. Ghez4, S. Gianì41,
V. Gibson49, O.G. Girard41, L. Giubega30, K. Gizdov52, V.V. Gligorov8, D.
Golubkov32, A. Golutvin55,69, A. Gomes1,a, I.V. Gorelov33, C. Gotti21,i, E.
Govorkova43, J.P. Grabowski12, R. Graciani Diaz38, L.A. Granado Cardoso40, E.
Graugés38, E. Graverini42, G. Graziani18, A. Grecu30, R. Greim9, P.
Griffith16, L. Grillo56, L. Gruber40, B.R. Gruberg Cazon57, O. Grünberg67, E.
Gushchin34, Yu. Guz37, T. Gys40, C. Göbel62, T. Hadavizadeh57, C.
Hadjivasiliou5, G. Haefeli41, C. Haen40, S.C. Haines49, B. Hamilton60, X.
Han12, T.H. Hancock57, S. Hansmann-Menzemer12, N. Harnew57, S.T. Harnew48, C.
Hasse40, M. Hatch40, J. He63, M. Hecker55, K. Heinicke10, A. Heister9, K.
Hennessy54, P. Henrard5, L. Henry71, E. van Herwijnen40, M. Heß67, A.
Hicheur2, D. Hill57, P.H. Hopchev41, W. Hu65, W. Huang63, Z.C. Huard59, W.
Hulsbergen43, T. Humair55, M. Hushchyn35, D. Hutchcroft54, P. Ibis10, M.
Idzik28, P. Ilten47, R. Jacobsson40, J. Jalocha57, E. Jans43, A. Jawahery60,
F. Jiang3, M. John57, D. Johnson40, C.R. Jones49, C. Joram40, B. Jost40, N.
Jurik57, S. Kandybei45, M. Karacson40, J.M. Kariuki48, S. Karodia53, N.
Kazeev35, M. Kecke12, F. Keizer49, M. Kelsey61, M. Kenzie49, T. Ketel44, E.
Khairullin35, B. Khanji12, C. Khurewathanakul41, K.E. Kim61, T. Kirn9, S.
Klaver19, K. Klimaszewski29, T. Klimkovich11, S. Koliiev46, M. Kolpin12, R.
Kopecna12, P. Koppenburg43, A. Kosmyntseva32, S. Kotriakhova31, M. Kozeiha5,
L. Kravchuk34, M. Kreps50, F. Kress55, P. Krokovny36,w, W. Krzemien29, W.
Kucewicz27,l, M. Kucharczyk27, V. Kudryavtsev36,w, A.K. Kuonen41, T.
Kvaratskheliya32,40, D. Lacarrere40, G. Lafferty56, A. Lai16, G. Lanfranchi19,
C. Langenbruch9, T. Latham50, C. Lazzeroni47, R. Le Gac6, A. Leflat33,40, J.
Lefrançois7, R. Lefèvre5, F. Lemaitre40, E. Lemos Cid39, O. Leroy6, T.
Lesiak27, B. Leverington12, P.-R. Li63, T. Li3, Y. Li7, Z. Li61, X. Liang61,
T. Likhomanenko68, R. Lindner40, F. Lionetto42, V. Lisovskyi7, X. Liu3, D.
Loh50, A. Loi16, I. Longstaff53, J.H. Lopes2, D. Lucchesi23,o, M. Lucio
Martinez39, H. Luo52, A. Lupato23, E. Luppi17,g, O. Lupton40, A. Lusiani24, X.
Lyu63, F. Machefert7, F. Maciuc30, V. Macko41, P. Mackowiak10, S. Maddrell-
Mander48, O. Maev31,40, K. Maguire56, D. Maisuzenko31, M.W. Majewski28, S.
Malde57, B. Malecki27, A. Malinin68, T. Maltsev36,w, G. Manca16,f, G.
Mancinelli6, D. Marangotto22,q, J. Maratas5,v, J.F. Marchand4, U. Marconi15,
C. Marin Benito38, M. Marinangeli41, P. Marino41, J. Marks12, G.
Martellotti26, M. Martin6, M. Martinelli41, D. Martinez Santos39, F. Martinez
Vidal71, A. Massafferri1, R. Matev40, A. Mathad50, Z. Mathe40, C. Matteuzzi21,
A. Mauri42, E. Maurice7,b, B. Maurin41, A. Mazurov47, M. McCann55,40, A.
McNab56, R. McNulty13, J.V. Mead54, B. Meadows59, C. Meaux6, F. Meier10, N.
Meinert67, D. Melnychuk29, M. Merk43, A. Merli22,40,q, E. Michielin23, D.A.
Milanes66, E. Millard50, M.-N. Minard4, L. Minzoni17, D.S. Mitzel12, A.
Mogini8, J. Molina Rodriguez1, T. Mombächer10, I.A. Monroy66, S. Monteil5, M.
Morandin23, M.J. Morello24,t, O. Morgunova68, J. Moron28, A.B. Morris52, R.
Mountain61, F. Muheim52, M. Mulder43, D. Müller56, J. Müller10, K. Müller42,
V. Müller10, P. Naik48, T. Nakada41, R. Nandakumar51, A. Nandi57, I. Nasteva2,
M. Needham52, N. Neri22,40, S. Neubert12, N. Neufeld40, M. Neuner12, T.D.
Nguyen41, C. Nguyen-Mau41,n, S. Nieswand9, R. Niet10, N. Nikitin33, T.
Nikodem12, A. Nogay68, D.P. O’Hanlon50, A. Oblakowska-Mucha28, V. Obraztsov37,
S. Ogilvy19, R. Oldeman16,f, C.J.G. Onderwater72, A. Ossowska27, J.M. Otalora
Goicochea2, P. Owen42, A. Oyanguren71, P.R. Pais41, A. Palano14, M.
Palutan19,40, G. Panshin70, A. Papanestis51, M. Pappagallo52, L.L.
Pappalardo17,g, W. Parker60, C. Parkes56, G. Passaleva18,40, A. Pastore14,d,
M. Patel55, C. Patrignani15,e, A. Pearce40, A. Pellegrino43, G. Penso26, M.
Pepe Altarelli40, S. Perazzini40, D. Pereima32, P. Perret5, L. Pescatore41, K.
Petridis48, A. Petrolini20,h, A. Petrov68, M. Petruzzo22,q, E. Picatoste
Olloqui38, B. Pietrzyk4, G. Pietrzyk41, M. Pikies27, D. Pinci26, F. Pisani40,
A. Pistone20,h, A. Piucci12, V. Placinta30, S. Playfer52, M. Plo Casasus39, F.
Polci8, M. Poli Lener19, A. Poluektov50, I. Polyakov61, E. Polycarpo2, G.J.
Pomery48, S. Ponce40, A. Popov37, D. Popov11,40, S. Poslavskii37, C.
Potterat2, E. Price48, J. Prisciandaro39, C. Prouve48, V. Pugatch46, A. Puig
Navarro42, H. Pullen57, G. Punzi24,p, W. Qian50, J. Qin63, R. Quagliani8, B.
Quintana5, B. Rachwal28, J.H. Rademacker48, M. Rama24, M. Ramos Pernas39, M.S.
Rangel2, I. Raniuk45,†, F. Ratnikov35,x, G. Raven44, M. Ravonel Salzgeber40,
M. Reboud4, F. Redi41, S. Reichert10, A.C. dos Reis1, C. Remon Alepuz71, V.
Renaudin7, S. Ricciardi51, S. Richards48, M. Rihl40, K. Rinnert54, P. Robbe7,
A. Robert8, A.B. Rodrigues41, E. Rodrigues59, J.A. Rodriguez Lopez66, A.
Rogozhnikov35, S. Roiser40, A. Rollings57, V. Romanovskiy37, A. Romero
Vidal39,40, M. Rotondo19, M.S. Rudolph61, T. Ruf40, P. Ruiz Valls71, J. Ruiz
Vidal71, J.J. Saborido Silva39, E. Sadykhov32, N. Sagidova31, B. Saitta16,f,
V. Salustino Guimaraes62, C. Sanchez Mayordomo71, B. Sanmartin Sedes39, R.
Santacesaria26, C. Santamarina Rios39, M. Santimaria19, E. Santovetti25,j, G.
Sarpis56, A. Sarti19,k, C. Satriano26,s, A. Satta25, D.M. Saunders48, D.
Savrina32,33, S. Schael9, M. Schellenberg10, M. Schiller53, H. Schindler40, M.
Schmelling11, T. Schmelzer10, B. Schmidt40, O. Schneider41, A. Schopper40,
H.F. Schreiner59, M. Schubiger41, M.H. Schune7, R. Schwemmer40, B. Sciascia19,
A. Sciubba26,k, A. Semennikov32, E.S. Sepulveda8, A. Sergi47, N. Serra42, J.
Serrano6, L. Sestini23, P. Seyfert40, M. Shapkin37, I. Shapoval45, Y.
Shcheglov31, T. Shears54, L. Shekhtman36,w, V. Shevchenko68, B.G. Siddi17, R.
Silva Coutinho42, L. Silva de Oliveira2, G. Simi23,o, S. Simone14,d, M.
Sirendi49, N. Skidmore48, T. Skwarnicki61, I.T. Smith52, J. Smith49, M.
Smith55, l. Soares Lavra1, M.D. Sokoloff59, F.J.P. Soler53, B. Souza De
Paula2, B. Spaan10, P. Spradlin53, S. Sridharan40, F. Stagni40, M. Stahl12, S.
Stahl40, P. Stefko41, S. Stefkova55, O. Steinkamp42, S. Stemmle12, O.
Stenyakin37, M. Stepanova31, H. Stevens10, S. Stone61, B. Storaci42, S.
Stracka24,p, M.E. Stramaglia41, M. Straticiuc30, U. Straumann42, S. Strokov70,
J. Sun3, L. Sun64, K. Swientek28, V. Syropoulos44, T. Szumlak28, M.
Szymanski63, S. T’Jampens4, A. Tayduganov6, T. Tekampe10, G. Tellarini17,g, F.
Teubert40, E. Thomas40, J. van Tilburg43, M.J. Tilley55, V. Tisserand5, M.
Tobin41, S. Tolk49, L. Tomassetti17,g, D. Tonelli24, R. Tourinho Jadallah
Aoude1, E. Tournefier4, M. Traill53, M.T. Tran41, M. Tresch42, A. Trisovic49,
A. Tsaregorodtsev6, P. Tsopelas43, A. Tully49, N. Tuning43,40, A. Ukleja29, A.
Usachov7, A. Ustyuzhanin35, U. Uwer12, C. Vacca16,f, A. Vagner70, V.
Vagnoni15,40, A. Valassi40, S. Valat40, G. Valenti15, R. Vazquez Gomez40, P.
Vazquez Regueiro39, S. Vecchi17, M. van Veghel43, J.J. Velthuis48, M.
Veltri18,r, G. Veneziano57, A. Venkateswaran61, T.A. Verlage9, M. Vernet5, M.
Vesterinen57, J.V. Viana Barbosa40, D. Vieira63, M. Vieites Diaz39, H.
Viemann67, X. Vilasis-Cardona38,m, M. Vitti49, V. Volkov33, A. Vollhardt42, B.
Voneki40, A. Vorobyev31, V. Vorobyev36,w, C. Voß9, J.A. de Vries43, C. Vázquez
Sierra43, R. Waldi67, J. Walsh24, J. Wang61, Y. Wang65, D.R. Ward49, H.M.
Wark54, N.K. Watson47, D. Websdale55, A. Weiden42, C. Weisser58, M.
Whitehead40, J. Wicht50, G. Wilkinson57, M. Wilkinson61, M. Williams56, M.
Williams58, T. Williams47, F.F. Wilson51,40, J. Wimberley60, M. Winn7, J.
Wishahi10, W. Wislicki29, M. Witek27, G. Wormser7, S.A. Wotton49, K. Wyllie40,
Y. Xie65, M. Xu65, Q. Xu63, Z. Xu3, Z. Xu4, Z. Yang3, Z. Yang60, Y. Yao61, H.
Yin65, J. Yu65, X. Yuan61, O. Yushchenko37, K.A. Zarebski47, M.
Zavertyaev11,c, L. Zhang3, Y. Zhang7, A. Zhelezov12, Y. Zheng63, X. Zhu3, V.
Zhukov9,33, J.B. Zonneveld52, S. Zucchelli15.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
7LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Universita e INFN, Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
35Yandex School of Data Analysis, Moscow, Russia
36Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
37Institute for High Energy Physics (IHEP), Protvino, Russia
38ICCUB, Universitat de Barcelona, Barcelona, Spain
39Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
40European Organization for Nuclear Research (CERN), Geneva, Switzerland
41Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
42Physik-Institut, Universität Zürich, Zürich, Switzerland
43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
44Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
69National University of Science and Technology MISIS, Moscow, Russia,
associated to 32
70National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
32
71Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 38
72Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43
73Los Alamos National Laboratory (LANL), Los Alamos, United States, associated
to 61

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vIligan Institute of Technology (IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
xNational Research University Higher School of Economics, Moscow, Russia
†Deceased