EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2019-108 LHCb-PAPER-2019-013 19 March 2020

Updated measurement of time-dependent $C\\!P$-violating observables in
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}$ decays

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

The decay-time-dependent $C\\!P$ asymmetry in
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays is measured using proton-proton collision
data, corresponding to an integrated luminosity of $1.9\mbox{\,fb}^{-1}$,
collected with the LHCb detector at a centre-of-mass energy of
$13\,\mathrm{TeV}$ in 2015 and 2016. Using a sample of approximately 117 000
signal decays with an invariant ${{K}^{+}}{{K}^{-}}$ mass in the vicinity of
the $\phi(1020)$ resonance, the $C\\!P$-violating phase $\phi_{s}$ is
measured, along with the difference in decay widths of the light and heavy
mass eigenstates of the ${{B}^{0}_{s}}$-${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ system,
$\Delta\Gamma_{s}$. The difference of the average ${{B}^{0}_{s}}$ and
${{B}^{0}}$ meson decay widths, $\Gamma_{s}-\Gamma_{d}$, is determined using
in addition a sample of
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$ decays. The values obtained are
$\phi_{s}=-0.083\pm 0.041\pm 0.006\mathrm{\,rad}$, $\Delta\Gamma_{s}=0.077\pm
0.008\pm 0.003{\mathrm{\,ps^{-1}}}$ and $\Gamma_{s}-\Gamma_{d}=-0.0041\pm
0.0024\pm 0.0015{\mathrm{\,ps^{-1}}}$, where the first uncertainty is
statistical and the second systematic. These are the most precise single
measurements of these quantities to date and are consistent with expectations
based on the Standard Model and with a previous LHCb analysis of this decay
using data recorded at centre-of-mass energies 7 and 8 TeV. Finally, the
results are combined with recent results from
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ decays obtained using the same dataset as this
analysis, and with previous independent LHCb results.

Submitted to EPJC

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

##  1 Introduction

The existence of new phenomena beyond those predicted by the Standard Model
(SM), hereafter referred to as New Physics (NP), could introduce sizeable
effects on $C\\!P$-violating observables. In the SM, $C\\!P$ violation
originates from an irreducible complex phase in the Cabibbo–Kobayashi–Maskawa
(CKM) matrix that describes the mixing of the mass and weak interaction
eigenstates of the quarks [1, 2]. In decays of a ${{B}^{0}_{s}}$ meson to a
$C\\!P$ eigenstate, $C\\!P$ violation can originate from the interference of
the amplitude of the decay and that of the adjoint decay preceded by
${{B}^{0}_{s}}$-${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$
oscillation. It manifests itself through a nonzero value of the phase
$\phi_{s}=-\mathrm{arg}\left(\lambda\right)$, where the parameter
$\lambda\equiv\left({q}/{p}\right)\left({\overline{A}}/{A}\right)$ describes
$C\\!P$ violation in the interference between mixing and decay. Here, $A$ and
$\overline{A}$ are the amplitudes for a $B_{s}^{0}$ or a ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ meson to decay to the same
final state and the complex parameters $p=\langle B_{s}^{0}|B_{\rm L}\rangle$
and $q=\langle{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}|B_{\rm L}\rangle$ describe
the relation between the flavour and the mass eigenstates, light, L, and
heavy, H. The two eigenstates have a decay width difference
$\Delta\Gamma_{s}\equiv\Gamma_{\rm L}-\Gamma_{\rm H}$ and a mass difference
$\Delta m_{s}\equiv m_{\rm H}-m_{\rm L}$. In the absence of ${C\\!P}$
violation in the decay and assuming negligible ${C\\!P}$ violation in
${{B}^{0}_{s}}$-${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$
mixing [3], $|\lambda|$ is expected to be unity. In the SM, ignoring
subleading contributions, the phase $\phi_{s}$ can be related to the CKM
matrix elements $V_{ij}$, such that $\phi_{s}\approx-2\beta_{s}$, where
$\beta_{s}\equiv\arg[-(V_{ts}V_{tb}^{*})/(V_{cs}V_{cb}^{*})]$. Global fits to
experimental data, assuming unitarity of the CKM matrix, give a precise
prediction of a small value, namely
$-2\beta_{s}=-0.0369^{+0.0010}_{-0.0007}\mathrm{\,rad}$ according to the
CKMfitter group [4] and $-2\beta_{s}=-0.0370\pm 0.0010\mathrm{\,rad}$
according to the UTfit collaboration [5]. However, many NP models [6, 7]
predict larger values for this phase if non-SM particles were to contribute to
${{B}^{0}_{s}}$-${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$
oscillations, while satisfying all existing constraints. Thus, a measurement
of $\phi_{s}$ different from the SM prediction would provide clear evidence
for NP.

Due to its high yield and clean experimental signature, the most sensitive
decay channel to NP contributions is
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\rightarrow\mu^{+}\mu^{-}){{K}^{+}}{{K}^{-}}$ [8], where the kaon pair
predominantly originates from the decay of a $\phi(1020)$ resonance.111The
inclusion of charge-conjugate processes is implied throughout this paper,
unless otherwise noted. For simplicity, the resonance $\phi(1020)$ is referred
to as $\phi$ in the following. Angular momentum conservation in the decay
implies that the final state is an admixture of $C\\!P$-even and $C\\!P$-odd
components, with orbital angular momentum of 0 or 2, and 1, respectively.
Moreover, along with the three polarisation states of the $\phi$ meson (P-wave
states), there is also a $C\\!P$-odd ${{K}^{+}}{{K}^{-}}$ component in an
S-wave state [9]. The data can therefore be described considering four
polarisation amplitudes $A_{g}=|A_{g}|e^{-i\delta_{g}}$, where the indices
$g\in\\{0,\parallel,\perp,S\\}$ refer to the longitudinal, transverse-parallel
and transverse-perpendicular relative orientations of the linear polarisation
vectors of the $J/\psi$ and $\phi$ mesons and $S$ to the single S-wave
amplitude, respectively. The $C\\!P$-even and $C\\!P$-odd components are
disentangled by a decay-time-dependent angular analysis, where the angular
observables $\cos\theta_{K}$, $\cos\theta_{\mu}$ and $\phi_{h}$ are defined in
the helicity basis as described in Ref. [10]. The polar angle $\theta_{K}$
($\theta_{\mu}$) is the angle between the $K^{+}$ ($\mu^{+}$) momentum and the
direction opposite to the ${B}^{0}_{s}$ momentum in the $K^{+}K^{-}$
($\mu^{+}\mu^{-}$) centre-of-mass system and $\phi_{h}$ is the azimuthal angle
between the $K^{+}K^{-}$ and $\mu^{+}\mu^{-}$ decay planes. The $\phi_{h}$
angle is defined by a rotation from the $K^{-}$ side of the $K^{+}K^{-}$ plane
to the $\mu^{+}$ side of the $\mu^{+}\mu^{-}$ plane. The rotation is positive
in the $\mu^{+}\mu^{-}$ direction in the ${B}^{0}_{s}$ rest frame.

A decay-time-dependent angular analysis also allows the determination of
$\Delta\Gamma_{s}$, and of the average ${{B}^{0}_{s}}$ decay width,
$\Gamma_{s}\equiv\left(\Gamma_{\rm L}+\Gamma_{\rm H}\right)/2$. In the SM,
$\Gamma_{s}$ and $\Delta\Gamma_{s}$ can be calculated within the framework of
the heavy quark expansion (HQE) theory [11, 12, 13, 14, 15, 16, 17], where a
perturbative expansion of the amplitudes in inverse powers of the $b$-quark
mass is used to calculate $b$-hadron observables. The ratio of the average
decay width of ${{B}^{0}_{s}}$ and ${{B}^{0}}$ mesons,
$\Gamma_{s}/\Gamma_{d}$, is usually the preferred observable to compare with
experimental measurements as it allows the suppression of common uncertainties
in the calculation. The predictions are $\Delta\Gamma_{s}=0.088\pm
0.020{\mathrm{\,ps^{-1}}}$ [18] and $\Gamma_{s}/\Gamma_{d}=1.0006\pm 0.0025$
[19]. The high precision of the ratio $\Gamma_{s}/\Gamma_{d}$ makes it an
excellent testing ground for the validity of the HQE [20, 19]. In addition,
$\Delta\Gamma_{s}$ can provide bounds complementary to those from
$\Gamma_{s}/\Gamma_{d}$ on quark-hadron duality violation [21].

Measurements of $\phi_{s}$, $\Delta\Gamma_{s}$ and $\Gamma_{s}$ using
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays, with
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow\mu^{+}\mu^{-}$,
have been previously reported by the D0 [22], CDF [23], ATLAS [24, *ATLAS-
CONF-2019-009], CMS [26] and LHCb [27] collaborations. The LHCb collaboration
has also exploited different decay channels, namely
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ [28], ${{B}^{0}_{s}}\rightarrow\psi(2S)\phi$
[29], ${{B}^{0}_{s}}\rightarrow D_{s}^{+}D_{s}^{-}$ [30] and
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ for the ${{K}^{+}}{{K}^{-}}$ invariant-mass region
above 1.05${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ [31]. The world-average
values, including all of the above mentioned results, are $\phi_{s}=-0.021\pm
0.031\mathrm{\,rad}$, $\Delta\Gamma_{s}=0.085\pm 0.006{\mathrm{\,ps^{-1}}}$
and $\Gamma_{s}/\Gamma_{d}=1.006\pm 0.004$ [32]. They are in agreement with
the abovementioned predictions.

The main parameters of interest in this paper are ${\phi_{{s}}}$, $|\lambda|$,
${\Gamma_{{s}}}-{\Gamma_{{d}}}$, ${\Delta\Gamma_{{s}}}$ and $\Delta m_{s}$
measured in ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays, in the ${{K}^{+}}{{K}^{-}}$ mass region
0.99–1.05${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The new measurement
reported is based on a data sample of proton-proton collisions recorded at a
centre-of-mass energy of $\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}$ in 2015
and 2016 during Run 2 of LHC operation, corresponding to an integrated
luminosity of $0.3\mbox{\,fb}^{-1}$ and $1.6\mbox{\,fb}^{-1}$, respectively.
The decay width difference $\Gamma_{s}-\Gamma_{d}$ is determined using
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$ decays as a reference, reconstructed in the same
data set as the signal. The ${{K}^{+}}{{\pi}^{-}}$ in the final state
originates predominantly from the decay of a $K^{*}(892)^{0}$ resonance. The
analysis procedure gives access to $\Gamma_{s}-\Gamma_{d}$ rather than
$\Gamma_{s}$ due to the dependence of the time efficiency parametrisation on
$\Gamma_{d}$. This allows the determination of $\Gamma_{s}-\Gamma_{d}$ with a
significant reduction of the systematic uncertainty associated with lifetime-
biasing selection requirements compared to the previous measurement. Taking as
an input the precisely known value of $\Gamma_{d}$ [32], the ratio
$\Gamma_{s}/\Gamma_{d}$ may be determined with higher precision with respect
to measuring the two lifetimes independently.

In this analysis, the polarisation-independent $C\\!P$-violating parameter
$\lambda_{r}$, associated with each polarisation state $r$, is defined such
that $\lambda_{r}=\eta_{r}\lambda$, where $\eta_{r}=+1\>\ \text{for
$r\in\\{0,\parallel\\}$}$ and $\eta_{r}=-1\>\text{for $r\in\\{\perp,{\rm
S}\\}$}$. As a consequence, $\phi_{s}=-\arg{\lambda}$. However, this
assumption can be relaxed such that the values of $\phi_{s}^{r}$ and
$|\lambda_{r}|$ are measured separately for each polarisation state. In
addition, the following quantities are measured: the $\phi$ polarization
fractions $|A_{0}|^{2}$ and $|A_{\perp}|^{2}$; the strong-phase differences
$\delta_{\perp}-\delta_{0}$ and $\delta_{\parallel}-\delta_{0}$; the fraction
of S-wave, $F_{\rm S}$, and the phase difference $\delta_{S}-\delta_{\perp}$.
The S-wave parameters are measured in bins of $m({{K}^{+}}{{K}^{-}})$. The sum
$|A_{0}|^{2}+|A_{\perp}|^{2}+|A_{\parallel}|^{2}$ equals unity and by
convention $\delta_{0}$ is zero.

After a brief description of the LHCb detector in Sec. 2, the candidate
selection and the background subtraction using the $sPlot$ technique [33] are
outlined in Sec. 3. The relevant inputs to the analysis, namely the decay-time
resolution, the decay-time efficiency, the angular efficiency and the flavour-
tagging calibration, are described in Secs. 4, 5, 6 and 7, respectively. The
$sFit$ procedure [34], the evaluation of the systematic uncertainties and the
results are discussed in Secs. 8, 9 and 10, respectively. The combination of
the results obtained in this analysis with those measured by the LHCb
collaboration using data collected in 2011 and 2012 and determined using 2015
and 2016 ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ data is presented in Sec. 11. Finally,
conclusions are drawn in Sec. 12.

##  2 Detector and simulation

The LHCb detector [35, 36] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector 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 the momentum, $p$,
of charged particles with a 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 a 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 detectors.
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
and a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers.

Samples of simulated events are used to optimise the signal selection, to
derive the angular efficiency and to correct the decay-time efficiency. In
simulations, $pp$ collisions are generated using Pythia [37,
*Sjostrand:2006za] with a specific LHCb configuration [39]. Decays of hadronic
particles are described by EvtGen [40], in which final-state radiation is
generated using Photos [41]. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [42,
*Agostinelli:2002hh] as described in Ref. [44]. The ${{B}^{0}_{s}}\rightarrow
J/\psi\phi$ simulated sample used in this analysis is generated taking into
account the three possible polarization states of the $\phi$ meson while
S-wave contributions are not included.

##  3 Selection and mass fit

Events are first required to pass an online event selection performed by a
trigger [45], which consists of a hardware stage, based on information from
calorimeters and muon systems, followed by a software stage, which applies a
full event reconstruction. At the hardware stage, events are required to have
a muon with high $p_{\mathrm{T}}$ or a hadron, photon or electron with high
transverse-energy deposit in the calorimeters. A difference with respect to
the previous analysis is that all the events passing any of the hardware
trigger requirements are accepted. This increases the signal yield by 13% in
2015 and by 7% in 2016 with respect to using the muon system information only.
The different signal gain in the two data taking years is due to tighter L0
trigger thresholds employed in the 2015 data. The subsequent software trigger
consists of two separate stages. In the first stage, the events can be divided
into two categories. In the first category, they are required to have two
well-identified oppositely charged muons with invariant mass larger than
2700${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. This trigger has an almost
uniform efficiency as a function of ${{B}^{0}_{s}}$ decay time and will be
referred to as unbiased. In the second category, events are retained if there
is at least one muon with transverse momentum larger than about
1${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and with a large impact-parameter
significance with respect to all PVs in the event. The latter is defined as
the difference in the vertex-fit $\chi^{2}$ of the PV fitted with and without
the considered track. Events are also included in the second category if they
pass the selection by a multivariate algorithm that identifies a two-track
good-quality secondary vertex with a large scalar sum of the $p_{\mathrm{T}}$
of the associated charged particles and a significant displacement from the
PVs. These triggers, whose selection thresholds changed slightly between 2015
and 2016 data taking, introduce a nontrivial dependence of the efficiency on
the ${{B}^{0}_{s}}$ decay time and will be referred to as biased. In the
second stage of the trigger, events containing a ${\mu^{+}}\mu^{-}$ pair with
invariant mass within 120${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ mass [46] and which form a
vertex that is significantly displaced from the PV are selected, introducing
another small decay-time bias.

In the offline selection, the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
meson candidates are formed from two oppositely charged particles, originating
from a common vertex, which are identified as muons and which have
$p_{\mathrm{T}}$ larger than 500${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. The
invariant mass of the ${\mu^{+}}\mu^{-}$ pair, $m({\mu^{+}}\mu^{-})$, must be
in the range 3020–3170${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ meson candidates are combined
with ${{K}^{+}}{{K}^{-}}$ candidates formed from two oppositely charged
particles that are identified as kaons and that originate from a common
vertex. The ${{K}^{+}}{{K}^{-}}$ pair is required to have $p_{\mathrm{T}}$
larger than 500${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. The invariant mass of
the ${{K}^{+}}{{K}^{-}}$ pair, $m({{K}^{+}}{{K}^{-}})$, must be in the range
990–1050${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The ${{B}^{0}_{s}}$
candidates are reconstructed by combining the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ candidate with the
${{K}^{+}}{{K}^{-}}$ pair, requiring that they form a good vertex and have an
invariant mass, $m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}})$, in the range
5200–5550${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The ${B}^{0}_{s}$ origin
vertex is defined as the PV in the interaction, or if multiple PVs are
reconstructed the PV with the minimum value of the ${{B}^{0}_{s}}$ impact
parameter significance is associated with the candidate. The invariant mass is
calculated from a kinematic fit that constrains the ${{B}^{0}_{s}}$ candidate
to originate from its origin vertex and constrains $m({\mu^{+}}\mu^{-})$ to
the known ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ mass [46]. When
deriving the decay time, $t$, and the helicity angles of the ${{B}^{0}_{s}}$
candidate the origin vertex constraint is also applied. In addition, $t$ is
required to be in the range 0.3–15.0${\mathrm{\,ps}}$, which suppresses a
large fraction of prompt combinatorial background whilst having a negligible
effect on the sensitivity to $\phi_{s}$. The kinematic fit also estimates a
per-candidate decay-time uncertainty, $\delta_{t}$.

The selection is optimised with respect to the previous analysis [27] by means
of a gradient-boosted decision tree (BDT) [47, 48], which is used to further
suppress combinatorial background. To train the BDT, simulated
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
candidates are used as a signal sample and data candidates with
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}})$ in the
range 5450–5550${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are used as a sample
of combinatorial background. The simulation is corrected to match the
distributions observed in data of particle identification variables, the
${{B}^{0}_{s}}$ transverse momentum and pseudorapidity, the quality of the
muon and kaon track fits and the number of tracks in an event with
measurements both in the VELO and the tracking stations. Various input
quantities are used in the BDT to exploit the features of the signal decay in
order to distinguish it from background, namely the track-fit $\chi^{2}$ of
the final-state particles, the particle identification probability as provided
mainly from the RICH and muon systems, the quality of the candidate
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and ${{B}^{0}_{s}}$ decay
vertices, the $p_{\mathrm{T}}$ of the ${{B}^{0}_{s}}$ candidate and of the
${{K}^{+}}{{K}^{-}}$ combination and the ${{B}^{0}_{s}}$ IP with respect to
its origin vertex. The selection requirement on the BDT output is chosen to
maximise the effective signal sample size approximated by the square of the
sum of sWeights divided by sum of squared sWeights.

In addition to combinatorial background, studies of the data in sidebands of
the $m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}})$
spectrum show contributions from approximately 5200 ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p{{K}^{-}}$ (350
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$) decays where the proton (pion) is misidentified
as a kaon. These backgrounds lie around the ${{B}^{0}_{s}}$ signal peak in the
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}})$
distribution, as shown in Fig. 1.

Figure 1: Distribution of the invariant mass of ${B}^{0}_{s}$ candidates,
selected from simulated
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ (green filled area), ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p{{K}^{-}}$ (solid red line) and
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$ (dotted blue line) decays. The distributions are
weighted to correct differences in the kinematics and the resonance content
between simulation and data.

These contributions are suppressed using more stringent kaon identification
requirements if the $m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}})$ mass, with the kaon interpreted as a proton
(pion), lies within $15{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the
${{\mathchar 28931\relax}^{0}_{b}}$ (${{B}^{0}}$) known mass [46]. This
reduces the ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$ peaking background contribution to approximately
120 decays. This background is neglected and a systematic uncertainty is
assigned to account for this approximation. The contribution due to the
${{\mathchar 28931\relax}^{0}_{b}}$ background is $1600\pm 160$, where the
uncertainty includes statistical and systematic sources. The ${{\mathchar
28931\relax}^{0}_{b}}$ background is statistically subtracted by inserting
simulated ${{\mathchar 28931\relax}^{0}_{b}}$ decays into the data sample with
negative weights. This is done prior to the sPlot procedure, in which the
combinatorial background is subtracted in a fit to
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}})$.
Correlations between the candidate mass and the angular variables are
preserved and the simulated candidates are weighted such that the
distributions of the kinematic variables used in the fit, and their
correlations, match those of data.

(a)(b)

Figure 2: (a) Distribution of the invariant mass of selected
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays. The signal component is shown by the long-
dashed red line, the background component by the dashed green line and the
total fit function by the solid blue line. The background contribution due to
${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p{{K}^{-}}$ decays is statistically subtracted. The contribution from
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$decays is not shown separately due to its small
size. (b) Distribution of ${{K}^{+}}{{K}^{-}}$ invariant mass from selected
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays. The background is subtracted using the
$sPlot$ method. The dashed blue lines define the boundaries of the six
$m(K^{+}K^{-})$ bins that are used in the analysis.

Figure 2(a) shows the $m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}})$ distribution and the result of an unbinned
maximum-likelihood fit to the sample in the range
5200–5550${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The sample is divided
into 24 independent subsamples, corresponding to six bins in
$m({{K}^{+}}{{K}^{-}})$ with boundaries at 990, 1008, 1016, 1020, 1024, 1032,
1050${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, to the biased and the unbiased
trigger categories, and to the year of data taking. The probability density
function (PDF) used for the fit is independent for each of these subsamples
and is composed of a single double-sided Crystal Ball (CB) [49] function for
the signal and an exponential function for the combinatorial background. The
CB tail parameters are fixed to those obtained from simulation.

The sPlot technique relies on the variable used for background subtraction to
be uncorrelated with the variables to which the sWeights are applied. However,
a correlation between the signal mass shape with $\cos\theta_{\mu}$ is
observed, due to the dependence of the mass resolution on the transverse
momentum of the muons. The per-candidate mass uncertainty, $\sigma_{m}$,
obtained in the vertex and kinematic fit used to obtain
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}})$, is
found to represent a good proxy of $\cos\theta_{\mu}$ due to its correlation
with the ${B}^{0}_{s}$ candidate mass resolution. Therefore, the signal
function uses $\sigma_{m}$ as a conditional observable. The width parameter
$\sigma_{\mathrm{CB}}$ of the double-sided CB function is parametrised as a
quadratic function of the per-candidate mass uncertainty such that
$\sigma_{\mathrm{CB}}=a_{1}\sigma_{m}+a_{2}\sigma_{m}^{2}$, $a_{1}$ and
$a_{2}$ are free parameters determined from the data. The quadratic dependence
is motivated by simulation studies.

A small contribution from
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ background candidates is observed at the known
${{B}^{0}}$ mass [46]. This contribution is included in the PDF and is
modelled with a Gaussian distribution, where the mean is fixed to the fitted
${{B}^{0}_{s}}$ mass minus the difference between ${{B}^{0}_{s}}$ and
${{B}^{0}}$ masses [46] and the resolution is fixed to
7${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which is determined from a fit to
the ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$ data control channel. Figure 2(b) shows the
background-subtracted invariant-mass distributions of the ${{K}^{+}}{{K}^{-}}$
system in the selected
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ candidates. After the trigger and full offline
selection requirements, the signal yield totals approximately 15 000 and 102
000 ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays in the 2015 and 2016 data sets,
respectively.

The fraction of events containing more than one ${B}^{0}_{s}$ candidate within
the $m(J/\psi K^{+}K^{-})$ range
$5340$–$5400{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is 0.3%. All candidates
are retained in the subsequent stages of the analysis and a systematic
uncertainty on the impact of allowing multiple candidates per event to be
present in the analysis is assigned.

##  4 Decay-time resolution

The value and the uncertainty of the decay-time resolution strongly affects
the relative precision on $\phi_{s}$, thus the knowledge of the decay-time
resolution calibration is pivotal. The resolution function is modelled with a
Gaussian distribution with a mean of zero and a width $\sigma_{\rm eff}$,
where $\sigma_{\rm eff}$ is determined using a sample of candidates
constructed from combinations of ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$, ${{K}^{+}}$ and ${{K}^{-}}$ candidates that originate predominantly
in the primary interaction (prompt component). This sample is referred to as
the prompt ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}}$
sample. It is selected as described in Sec. 3 for
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays except for the lower limit requirement for
the decay time, by making use of a different trigger line which is heavily
prescaled.

The prompt component has zero decay time and is used to calibrate the detector
resolution by studying the shape of the decay-time distribution around zero.
This distribution is modelled by a delta function. In addition to the prompt
component, there is a contribution at later decay times originating from
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ mesons produced in $b$-hadron
decays, and a small fraction of a background due to candidates that have a
decay time computed with respect to a wrong PV (wrong-PV component). The
$b$-hadron component contributes to a tail at positive decay times and is
described by two exponential functions. The shape of the wrong-PV component is
determined from a data control sample in which the decay-time distribution of
candidates is constructed by computing their decay time with respect to an
independent PV from the following event. This contribution is found to be
approximately $0.5\%$ of the prompt sample.

The sum of the prompt and $b$-hadron components is convolved with a triple-
Gaussian resolution function

${\cal
R}(t)=\sum_{i=1}^{3}f_{i}\frac{1}{\sqrt{2\pi}\sigma_{i}}\exp\left[-\frac{(t-\mu)^{2}}{2\sigma_{i}^{2}}\right],$
(1)

where $\sum_{i}f_{i}=1$, $\mu$ is a parameter that describes a bias in the
decay time measurement and $\sigma_{i}$ are the individual widths. The bias,
$\mu$, is assumed to be zero and a systematic uncertainty is assigned studying
a possible deviation from this value. The rest of the parameters are
determined from the fit.

(a)(b)

Figure 3: (a) Decay-time distribution of the prompt
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}}$ calibration
sample with the result of an unbinned maximum-likelihood fit overlaid in blue.
The overall triple-Gaussian resolution is represented by the dashed red line,
while the two long-lived and the wrong-PV components are shown by the long-
dashed-dotted and dashed-multiple-dotted brown and pink lines and the long-
dashed purple line, respectively. (b) Variation of the effective single-
Gaussian decay-time resolution, $\sigma_{\rm eff}$, as a function of the
estimated per-candidate decay-time uncertainty, $\delta_{t}$, obtained from
the prompt ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}}$
sample. The red line shows the result of a linear fit. The data points are
positioned at the barycentre of each $\delta_{t}$ bin. The shaded histogram
(see right $y$ axis) shows the distribution of $\delta_{t}$ in the background-
subtracted ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ sample.

The calibration sample is split into eleven subsets according to the per-
candidate decay-time uncertainty, $\delta_{t}$. The model is fit to the decay-
time distribution in order to extract the parameters governing the decay-time
resolution of Eq. (1) as shown in Fig. 3(a). The dilution of the amplitude of
the ${{B}^{0}_{s}}$-${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ oscillation due to the
calibrated resolution is determined in each bin of $\delta_{t}$ as

$\mathcal{D}=\sum_{i=1}^{3}f_{i}\exp\left[-\sigma_{i}^{2}\Delta
m_{s}^{2}/2\right],$ (2)

and is then used to evaluate an effective single-Gaussian width given by

$\sigma_{\rm eff}=\sqrt{(-2/\Delta m_{s}^{2})\ln\mathcal{D}}.$ (3)

This effective single-Gaussian resolution of width $\sigma_{\rm eff}$ gives
the same damping effect on the magnitude of the ${{B}^{0}_{s}}$ meson
oscillation as the triple-Gaussian model. Figure 3(b) shows the variation of
$\sigma_{\rm eff}$ as a function of $\delta_{t}$. The variation is fit with a
linear function $\sigma_{\mathrm{eff}}(\delta_{t})=b_{0}+b_{1}\delta_{t}$ to
determine the calibration parameters $b_{0}=12.97\pm 0.22\mathrm{\,fs}$ and
$b_{1}=0.846\pm 0.006$, where the uncertainties are statistical only. A
quadratic dependence is also evaluated and used as an alternate model to
compute a systematic uncertainty. The calibration procedure is validated using
simulated signal and prompt samples. The difference between the effective
resolutions obtained in these simulated samples is approximately
0.8$\mathrm{\,fs}$ and is treated as a source of systematic uncertainty.

The result of the calibration leads to an effective single-Gaussian resolution
function averaged over the $\delta_{t}$ bins with $\sigma_{\rm eff}=45.54\pm
0.04\pm 0.05\mathrm{\,fs}$, where the first uncertainty is statistical, and
the second contribution comes from the uncertainties on the calibration
parameters. This corresponds to a dilution ${\cal D}=0.721\pm 0.001$ assuming
$\Delta m_{s}=17.757\pm 0.021$${\mathrm{\,ps^{-1}}}$ [32].

##  5 Decay-time efficiency

The selection and reconstruction efficiency depends on the $B^{0}_{s}$ decay
time due to displacement requirements made on the signal tracks and a decrease
in reconstruction efficiency for tracks with large impact parameter with
respect to the beam line [50]. The efficiency as a function of the decay time
is determined using a new technique with respect to Ref. [27], exploiting the
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$ decay, with
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow\mu^{+}\mu^{-}$, as
a control sample. This control mode is kinematically similar to the signal
decay. Since the decay-width difference between the two mass eigenstates in
the ${{B}^{0}}$ system is measured to be consistent with zero [46],
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$ candidates are assumed to have a purely
exponential decay-time distribution with lifetime
$\tau^{{{B}^{0}}}_{\mathrm{data}}=1.520{\mathrm{\,ps}}$ [32]. The
${B}^{0}_{s}$ efficiency is determined via a simultaneous fit to background-
subtracted data and simulated samples through the relation

$\varepsilon^{{{B}^{0}_{s}}}_{\mathrm{data}}(t)=\varepsilon^{{{B}^{0}}}_{\mathrm{data}}(t)\times\frac{\varepsilon^{{{B}^{0}_{s}}}_{\mathrm{sim}}(t)}{\varepsilon^{{{B}^{0}}}_{\mathrm{sim}}(t)}\,,$
(4)

where $\varepsilon^{{{B}^{0}}}_{\mathrm{data}}(t)$ is the efficiency of the
control channel and
$\varepsilon^{{{B}^{0}_{s}}}_{\mathrm{sim}}(t)/\varepsilon^{{{B}^{0}}}_{\mathrm{sim}}(t)$
is the ratio of efficiencies of simulated signal and reference decays after
reconstruction and selection. Residual differences between either signal and
control mode or data and simulation are automatically corrected for in the
ratio of Eq. (4). In order to correct first-order differences between the
${{B}^{0}_{s}}$ and ${{B}^{0}}$ data samples, the latter is weighted to match
the $p$ and $p_{\mathrm{T}}$ distribution of ${{B}^{0}_{s}}$ data. In
addition, both ${{B}^{0}_{s}}$ and ${{B}^{0}}$ simulated samples are weighted
to match the $p_{\mathrm{T}}$ distribution of the ${{B}^{0}_{s}}$ data sample.
The simulated samples are further corrected according to the ratio of the PDF
used to generate them and the PDF obtained with the parameters measured in
data [51, 27]. Together with an additional weighting to match the
$m({{K}^{+}}{{\pi}^{-}})$ and $m({{K}^{+}}{{K}^{-}})$ distributions in data,
this procedure reproduces the correct mixture of P- and S-waves in the
${{K}^{+}}{{\pi}^{-}}$ and ${{K}^{+}}{{K}^{-}}$ final state. The decay-time
efficiency is obtained separately for the data-taking periods 2015 and 2016
and the two trigger categories.

The ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$ candidates are selected using trigger and
preselection requirements similar to those of the
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}$ channel. The main difference is an additional selection on
the pion-identification requirement, in order to reduce the probability of
reconstructing two different ${{B}^{0}}$ candidates by swapping the kaon and
pion mass hypotheses. In addition, the $p_{\mathrm{T}}$ of the pion is
required to be larger than $250{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ to reduce
the number of multiple candidates per event to 0.5$\%$ in the
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}\pi^{-})$ region
$5260$–$5300{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The invariant mass of
the kaon-pion pair is required to be in the range
$826$–$966{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The BDT as trained and
optimised on the signal channel is used, applying the same selection
requirement. Several potential peaking backgrounds arising from the
misidentification of particles are considered but they are all found to be
negligible. A small contribution from
$B^{0}_{s}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}\pi^{-}$ decays is removed by selecting candidates with
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}\pi^{-})<5350{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

(a)(b)

Figure 4: Distribution of the invariant mass of selected (a)
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}\pi^{-}$ and (b)
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}$ decays
used for the calibration and validation of the decay-time efficiency. The
signal component is shown by the long-dashed red line, the background
component by the dashed green line and the total fit function by the solid
blue line.

Figure 4(a) shows the $m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}\pi^{-})$ distribution and corresponding result of an unbinned
maximum-likelihood fit to the sample. The model used for the fit is the same
for the 2015 and 2016 data-taking periods and the two trigger categories but
with independently fitted parameters. It is composed of a Hypatia [52]
function for the signal, where the parameters describing the tails are fixed
to the values obtained from simulation, and an exponential function for the
combinatorial background. In total, $75\,000$ and $480\,000$ ${{B}^{0}}$
mesons are found in 2015 and 2016, respectively. The result of this fit is
used to statistically subtract the background when determining the decay-time
efficiency in data, by using weights computed with the sPlot technique.

The PDF used to describe the decay-time distribution of the ${{B}^{0}}$ data,
and of the ${{B}^{0}_{s}}$ and ${{B}^{0}}$ simulated samples is composed of
the product of the efficiency function and a single exponential function,
convolved with a single Gaussian resolution function centred at zero. For the
${{B}^{0}}$ candidates, the width of the resolution function is set to
39$\mathrm{\,fs}$ and 42$\mathrm{\,fs}$ for the simulated and data samples,
respectively. The first value is obtained from simulation, and the second
value is obtained by scaling the ${{B}^{0}_{s}}$ resolution obtained in data,
as described in Sec. 4, by the ratio seen between the ${{B}^{0}}$ and
${{B}^{0}_{s}}$ resolutions in simulated samples. A ${{B}^{0}_{s}}$ simulated
sample is generated with $\Delta\Gamma_{s}=0{\mathrm{\,ps^{-1}}}$ and thus a
single exponential function is used to determine
$\varepsilon_{\mathrm{sim}}^{{{B}^{0}_{s}}}$. As a cross-check, the decay-time
efficiency is also derived from the nominal
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
simulated sample, weighted to have $\Delta\Gamma_{s}=0{\mathrm{\,ps^{-1}}}$
such that the same fitting strategy can be used as defined above. The
difference between these two strategies is considered as a source of
systematic uncertainty.

The efficiency functions are parametrised using cubic splines with nodes at
0.3, 0.58, 0.91, 1.35, 1.96, 3.01, 7.00${\mathrm{\,ps}}$ and the first
coefficient fixed to unity. The node positions are defined as to create six
uniformly populated bins in the interval $0.3$–$15{\mathrm{\,ps}}$, assuming
an exponential distribution with $\Gamma=0.66{\mathrm{\,ps}}^{-1}$. The
position of the last node is chosen due to the lack of candidates at large
decay times in the 2015 data control sample. The final decay-time
efficiencies, $\varepsilon^{{{B}^{0}_{s}}}_{\mathrm{data}}(t)$, are shown in
Fig. 5. The structure around 1${\mathrm{\,ps}}$ visible in Fig. 5(a) and (c)
is due to the different definition of the origin vertex used in the trigger
and in the offline selection.

(a)(b)
(c)(d)

Figure 5: Decay-time efficiency for the (a) 2015 unbiased, (b) 2015 biased,
(c) 2016 unbiased and (d) 2016 biased
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
sample. The cubic-spline function described in the text is shown by the blue
line. For comparison, the black points show the efficiency when computed using
histograms for each of the input component efficiencies.

The full procedure is validated in data using two approaches where the
${B}^{0}_{s}$ samples are replaced with alternative $B$ meson samples of known
lifetime. First, a sample of approximately 1.6 million
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\rightarrow{\mu^{+}}\mu^{-})K^{+}$ candidates is reconstructed in the
same data set as the
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ candidates and selected using similar selection
requirements. The mass distribution of these candidates is shown in Fig. 4(b).
This sample is used to measure the difference of the ${{B}^{+}}$ and
${{B}^{0}}$ decay widths, $\Gamma_{u}-\Gamma_{d}$, with the same methods used
for the measurement of $\Gamma_{s}-\Gamma_{d}$. A simulated sample of
${{B}^{+}}$ decays is used in the calculation of the numerator of Eq. (4) and
this sample is corrected such that the particle-identification, event-
multiplicity and other kinematic and selection variables match those in data.
The measured difference of decay widths is $\Gamma_{u}-\Gamma_{d}=-0.0478\pm
0.0013{\mathrm{\,ps^{-1}}}$, where the uncertainty is statistical only. This
is in agreement with the world average value, $-0.0474\pm
0.0023{\mathrm{\,ps^{-1}}}$ [46], and validates the measurement of
$\Gamma_{s}-\Gamma_{d}$ with a precision of 0.003${\mathrm{\,ps^{-1}}}$.

A similar test is done using the
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$ decays both as the signal and the reference to
measure a null decay-width difference. The sample is split into two
independent sets according to different selection criteria, where one is used
to evaluate the decay-time efficiency with the procedure defined above, and
the other is used as the signal sample. In all cases, the measured decay-width
difference is found to be consistent with zero with a precision around
0.003${\mathrm{\,ps^{-1}}}$.

##  6 Angular efficiency

The LHCb detector geometry and the selection requirements give rise to
efficiencies that vary as a function of the helicity angles $\theta_{K}$,
$\theta_{\mu}$ and $\phi_{h}$. The three-dimensional angular-efficiency
correction is determined from simulated signal events to which the same
trigger and selection criteria as in the data are applied. The efficiency is
evaluated separately for the different years of data taking and for the two
trigger categories. Two sets of corrections are applied to the simulated
events such that they match the data. First, the simulated samples are
weighted, using a boosted decision tree method [53], to match the
$p_{\mathrm{T}}$, $p$ and $m({{K}^{+}}{{K}^{-}})$ distributions of the
${B}^{0}_{s}$ signal. A second procedure is performed to correct the
differences observed in the kinematic distributions of the final-state
particles and the fact that the simulated events do not include
${{K}^{+}}{{K}^{-}}$ pairs in an S-wave configuration. This correction is
implemented as an iterative procedure that gradually modifies the simulation
such that the S-wave fraction matches the value measured in the data. As a
result, the agreement of the kaon momentum and $p_{\mathrm{T}}$ distributions
between the simulation and the data is improved. The efficiencies as a
function of the three helicity angles are shown for illustration in Fig. 6.
The angular efficiency correction is introduced in the analysis through
normalisation weights in the PDF describing the signal decays in the fit of
Sec. 8, following the procedure described in Ref. [54]. The weights are
calculated using simulated candidates and their statistical uncertainties are
propagated to the parameters of interest as a systematic uncertainty.

(a)(b)(c)

Figure 6: Normalised angular efficiency as a function of (a) $\cos\theta_{K}$,
(b) $\cos\theta_{\mu}$ and (c) $\phi_{h}$, where in all cases the efficiency
is integrated over the other two angles. The efficiency is evaluated using
simulated ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ decays that have been weighted to match the kinematics and
physics of ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays in data, as described in the text. The
points are obtained by dividing the angular distribution in the simulated
sample by the distribution expected without any efficiency effect and the
curves represent an even fourth-order polynomial parameterisation of each one-
dimensional efficiency. The figure is for illustration only as the angular
efficiency is accounted for by normalisation weights in the signal PDF.

A cross-check of the angular efficiency procedure is made using the
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$ data and
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{*}(892)^{0}(\rightarrow K^{+}\pi^{-})$ simulated samples. The
simulation contains $K^{+}\pi^{-}$ systems in P-wave only and is corrected to
match the kinematic distributions of the data using the iterative method
defined above and the angular-efficiency weights are determined. The P- and
S-wave ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}\pi^{-}$ polarisation amplitudes are measured by means of an
unbinned fit to the distribution of helicity angles of the final-state
particles and found to be consistent with those in Ref. [55].

Another high-precision test of the angular-efficiency correction is made by
using the large sample of
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$
decays presented in Sec. 5. In
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$,
the helicity angle $\theta_{\mu}$ distribution follows a
$1-\cos^{2}\theta_{\mu}$ dependence. The ${{B}^{+}}$ data sample is split into
nine disjoint subsets according to the pseudorapidity of the ${{B}^{+}}$
meson, to check the large efficiency variation as a function of this quantity.
In each subset, background is subtracted with the sPlot technique using the
${{B}^{+}}$ candidate mass as a discriminating variable. Prior to any angular
efficiency correction, the $\theta_{\mu}$ distribution presents up to a 30%
deviation from the expected shape, three times larger than in
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays. However, when the
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$
simulation is used to correct the data with the same method used for this
analysis, a fit of the background-subtracted and efficiency-corrected data
demonstrates that the expected distribution is fully recovered in each bin,
with an overall precision of about 0.1$\%$. The test is stable against
variation of the binning of the ${{B}^{+}}$ sample and choice of different
variables used to correct the simulation to match the data with respect to the
baseline strategy.

##  7 Tagging the ${B}^{0}_{s}$ meson flavour at production

The determination of the initial flavour of the ${B}^{0}_{s}$ meson, called
tagging, is a fundamental component for measuring $C\\!P$ asymmetries in the
decays of ${B}^{0}_{s}$ mesons to $C\\!P$ eigenstates. Two classes of
algorithms are used. The opposite side (OS) tagger exploits the fact that $b$
and $\overline{b}$ quarks are almost exclusively produced in pairs in $pp$
collisions, allowing the flavour of the signal ${B}^{0}_{s}$ candidate to be
inferred from the flavour of the other $b$ hadron in the event. The OS tagger
combines information on the charge of the muon or electron from semileptonic
$b$ decays, the charge of the kaon from the $b\rightarrow c\rightarrow s$
decay chain, the charge of a reconstructed secondary charm hadron and the
charges of the tracks that form the secondary vertex of the other $b$-hadron
decay, combined into a weighted average, with weights depending on the
transverse momenta of the tracks. The same-side kaon (SSK) tagger exploits the
additional correlated kaon that tends to be produced during the hadronisation
of the $\overline{b}$ ($b$) quark that forms the signal ${B}^{0}_{s}$ (${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$) candidate, with its
initial flavour identified by the kaon charge. These flavour tagging
algorithms have been revisited and optimised using Run 2 data [56], obtaining
significantly higher combined tagging performances with respect to Run 1.
Further details on the OS and SSK taggers can be found in Refs. [57, 58, 59].

The tagging algorithms each provide a flavour-tagging decision,
$\mathfrak{q}$, and an estimate, $\eta$, of the probability that the decision
is incorrect (mistag) for each reconstructed ${B}^{0}_{s}$ candidate. The
tagging decision takes the value $+1$ ($-1$) for each tagged ${B}^{0}_{s}$
(${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$) candidate and 0
if the taggers cannot make a decision (untagged). The mistag probability is
defined in the range from 0 to 0.5, since $\eta>0.5$ corresponds to the
opposite decision with a mistag of $(1-\eta)$. For untagged events $\eta$ is
0.5.

Each tagging algorithm is implemented as a BDT that is trained and optimised
using large samples of simulated $b$-hadron decays for the SSK tagger and a
large data sample of
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$
decays for the OS tagger. The mistag probability for each tagger is given by
the output of the BDT, which is calibrated using dedicated data control
channels to relate $\eta$ to the true mistag probability, $\omega$, as
described in the following sections. Each tagger has a corresponding tagging
power given by $\epsilon_{\rm tag}D^{2}$, where $\epsilon_{\rm tag}$ is the
fraction of tagged candidates and $D=1-2\omega$ is the dilution induced on the
amplitude of the ${B}^{0}_{s}$ oscillation. The tagging power represents the
effective reduction in statistical power due to imperfect tagging.

###  7.1 Opposite-side tagging

The OS tagging algorithm is calibrated using the sample of
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$
decays (Sec. 5), whose flavour is determined by the kaon charge. This sample
of ${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decays is independent of that used to train and optimise the
BDT of the tagging algorithm. The result of the fit to the distribution of
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})$ shown in Fig.
4(b) is used to compute sWeights, which are applied in subsequent stages of
the analysis to subtract the background. The
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$
sample is further weighted to match the background-subtracted
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
sample in the distributions of charged-track and PV multiplicities and the
$p_{\mathrm{T}}$ and rapidity of the ${B}$ meson.

Figure 7: Calibration of the OS tagger using
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$
decays. The black points show the average measured mistag probability,
$\omega$, in bins of predicted mistag, $\eta$, the red line shows the
calibration as described in the text and the yellow area the calibration
uncertainty within one standard deviation. The shaded histogram shows the
distribution, with arbitrary normalisation, of $\eta$ in the background
subtracted ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ sample, summing over candidates tagged as ${{B}^{0}_{s}}$ or
${{\kern 1.61993pt\overline{\kern-1.61993ptB}{}}{}^{0}_{s}}$.

The calibration between true and estimated mistag for each tagging algorithm
is given by an empirically determined linear relationship,

$\displaystyle\omega(\eta)$ $\displaystyle=$
$\displaystyle\left(p_{0}+\frac{\Delta
p_{0}}{2}\right)+\left(p_{1}+\frac{\Delta
p_{1}}{2}\right)\left(\eta-\langle\eta\rangle\right)$ (5)
$\displaystyle\overline{\omega}(\eta)$ $\displaystyle=$
$\displaystyle\left(p_{0}-\frac{\Delta
p_{0}}{2}\right)+\left(p_{1}-\frac{\Delta
p_{1}}{2}\right)\left(\eta-\langle\eta\rangle\right),$ (6)

where $\omega(\eta)$ and $\overline{\omega}(\eta)$ are the calibrated mistag
probabilities for $B^{+}$ and $B^{-}$ mesons, respectively, $\Delta p_{0,1}$
are mistag asymmetries and $\langle\eta\rangle$ is the average estimated
mistag of the ${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ sample. The calibration parameters are determined from an
unbinned maximum-likelihood fit to the $\eta$ distribution of the probability

$\mathcal{P}(a|\eta)=(1-a)\overset{\textbf{(---)}}{\omega}(\eta)+a(1-\overset{\textbf{(---)}}{\omega}(\eta)),$
(7)

for an initial flavour of the $B^{+}$ ($B^{-}$) meson. The discrete variable
$a$ has the value 0 or 1 for an incorrect or correct tagging decision,
respectively, based upon comparing the decision $\mathfrak{q}$ to the kaon
charge. Figure 7 shows the relation between the flavour-averaged value of
$\omega$ and $\eta$ determined by the fit and the values of the measured
mistag in bins of estimated mistag, which supports the use of a linear
calibration function. The final calibration parameters are given in Table 1
and the overall tagging power for candidates with an OS tag only can be found
in Table 2. Differences of the tagging efficiency are expected to be
negligible as their effects are washed out by the fast
${{B}^{0}_{s}}$-${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$
oscillations. The applicability of the calibration from
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$
to ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ decays is tested using simulated samples and observed differences
between the calibration parameters are treated as a source of systematic
uncertainty. Variations in the parameters caused by the use of a different
model for the combinatorial background in the fit to the
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})$ distribution are
found to be negligible.

Table 1: Calibration parameters for the OS and SSK taggers. Where given, the first uncertainty is statistical and the second is systematic. Tagger | OS | SSK
---|---|---
$p_{0}$ | $0.3890\pm 0.0007\pm 0.0028$ | $0.4325\pm 0.0108\pm 0.0030$
$p_{1}$ | $0.849\pm 0.006\pm 0.027$ | $0.92\pm 0.13\pm 0.02$
$\Delta p_{0}$ | $0.0090\pm 0.0014$ | $0.00\pm 0.03$
$\Delta p_{1}$ | $0.014\pm 0.012$ | $0.00\pm 0.03$
$\langle\eta\rangle$ | $0.360$ | $0.417$
Table 2:  Overall tagging performance for ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}K^{-}$. The uncertainty on $\epsilon_{\rm tag}D^{2}$ is obtained by varying the tagging calibration parameters within their statistical and systematic uncertainties summed in quadrature. Category |  $\epsilon_{\rm tag}$(%) | $D^{2}$ | $\epsilon_{\rm tag}D^{2}(\%)$
---|---|---|---
OS only | 11.4 | 0.078 | $0.88\pm 0.04$
SSK only | 42.6 | 0.032 | $1.38\pm 0.30$
OS & SSK | 23.8 | 0.104 | $2.47\pm 0.15$
Total | 77.8 | 0.061 | $4.73\pm 0.34$

###  7.2 Same-side tagging

Figure 8: Distribution of the invariant mass of selected
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}\pi^{+}$ candidates (black points). The
total fit function is shown as the solid blue line. The signal component is
shown by the red long-dashed line, the combinatorial background by the light-
blue short-dashed line and other small background components are also shown as
specified in the legend. Only the dominant backgrounds are shown.

The SSK tagger is calibrated by resolving the ${{B}^{0}_{s}}$-${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ flavour oscillations in a
sample of flavour-specfic ${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$
decays. The amplitude of this oscillation is related to the averaged
${B}^{0}_{s}$-${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$
mistag probability, $\tilde{\omega}$, via the PDF of the decay-time
distribution of flavour-tagged
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ decays, given by

$\displaystyle\mathcal{P}(t)$ $\displaystyle=\epsilon(t)\left[\Gamma(t)\otimes
R(t-t^{\prime})\right],$ (8) $\displaystyle\Gamma(t)$
$\displaystyle=\Gamma_{s}e^{-\Gamma_{s}t}\left[\cosh(\Delta\Gamma_{s}t/2)+q^{\rm
mix}(1-2\tilde{\omega}(\eta))\cos(\Delta m_{s}t)\right],$

where $t^{\prime}$ and $t$ are the true and reconstructed decay time of the
${{B}^{0}_{s}}$ meson, respectively, and $\Gamma(t)$ is the ${{B}^{0}_{s}}$
decay rate. The decay time and the decay-time uncertainty are estimated from a
kinematic fit [60] in which the ${{D}^{-}_{s}}\pi^{+}$ candidate is
constrained to originate from the PV. The decay-time efficiency is empirically
parameterised as $\epsilon(t)=$ $1-1/(1+(at)^{n}+b)$, and $R(t-t^{\prime})$ is
the decay-time resolution model. Here $q^{\rm mix}=+1$ $(-1)$ if the
${{B}^{0}_{s}}$ meson has (has not) changed flavour between its production and
decay, determined by comparing the flavour-tagging decision and charge of the
pion. A linear relationship between the true and estimated mistag
probabilities is assumed, as given in Eq. (5).

Approximately $70\,000$ same-side flavour-tagged
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}\pi^{+}$ decays, with
${{D}^{-}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}\pi^{-}$, are selected with similar
requirements as in Ref. [59]. Due to trigger requirements, only candidates
with $\mbox{$p_{\mathrm{T}}$}({{B}^{0}_{s}})$ larger than
$2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are used to perform the
calibration. Figure 8 shows the distribution of $m({{D}^{-}_{s}}\pi^{+})$ for
the selected sample. Superimposed is the result of a fit with a model composed
of a signal contribution described by a Hypatia with tail parameters fixed to
those from simulation and a combinatorial background component modelled by an
exponential function. In addition, template shapes for several peaking
backgrounds (${{B}^{0}_{s}}\rightarrow{{D}^{\pm}_{s}}K^{\mp}$,
${{B}^{0}}\rightarrow{{D}^{-}_{s}}\pi^{+}$, ${{\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}}\rightarrow{{\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}}{}^{-}_{c}}\pi^{+}$,
${{B}^{0}_{s}}\rightarrow{{D}^{*-}_{s}}\pi^{+}$ and
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}\rho^{+}$) are evaluated from simulation
and included in the fit model. The yield of the peaking backgrounds is
determined from a fit to $m({{D}^{-}_{s}}\pi^{+})$ in the mass range
$5100$–$5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Using the fit results,
the yield is extrapolated to the narrower region
$5300$–$5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and fixed in the
subsequent $m({{D}^{-}_{s}}\pi^{+})$ fit, which is used to compute sWeights
for background subtraction as in the OS calibration. The
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}\pi^{+}$ sample is also weighted to
match the background-subtracted
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
sample in the distributions of charged-track and PV multiplicities and the
$p_{\mathrm{T}}$ and rapidity of the ${{B}^{0}_{s}}$ meson.

Figure 9: Variation of the effective single-Gaussian decay-time resolution,
$\sigma_{\rm eff}$, as a function of the estimated per-event decay-time
uncertainty, $\delta_{t}$, obtained from the prompt ${{D}^{-}_{s}}\pi^{+}$
sample. The red line shows the result of a linear fit to the data and the
yellow band its uncertainty within one standard deviation.

To calibrate the decay-time resolution in Eq. (8), a sample of promptly
produced ${{D}^{-}_{s}}\pi^{+}$ candidates is selected following the
requirements defined in Ref. [61]. The procedure to obtain the calibration for
the decay-time resolution is similar to that described in Sec. 4. An unbinned
maximum-likelihood fit is made to the ${{D}^{-}_{s}}$ candidate invariant-mass
distribution in 18 bins of $\delta_{t}$. The model consists of a single
Gaussian component for the signal and a second-order polynomial for the
combinatorial background. From this fit, sWeights are computed that are used
to subtract the background contribution in an unbinned fit to the decay-time
distribution in each $\delta_{t}$ bin. The model for this fit is composed of
two Gaussian functions with a common mean and different widths. Only
candidates with reconstructed decay time in the range from $-1.0$ to
$0.1{\mathrm{\,ps}}$ are fitted. At such low values the longer-lived
background components can be neglected. The effective single-Gaussian
resolution is calculated from the double-Gaussian model using Eqs. (2) and
(3). The variation of the effective resolution with the average value of
$\delta_{t}$ in each bin is shown in Fig. 9. From a binned fit using a linear
calibration function, $\sigma_{\rm eff}(\delta_{t})=c_{0}+c_{1}\delta_{t}$,
the calibration constants are determined to be $c_{0}=18.8\pm
1.0$$\mathrm{\,fs}$ and $c_{1}=1.03\pm 0.02$, where the uncertainties are
statistical only. Applying a similar procedure to a sample of simulated
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}\pi^{+}$ decays indicates a difference
in the calibration parameters between prompt ${{D}^{-}_{s}}\pi^{+}$ candidates
and ${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}\pi^{+}$ decays. A systematic
uncertainty of $0.1$ is assigned to $c_{1}$ to account for this difference.

(a)(b)

Figure 10: (a) Distribution of the decay time for
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}\pi^{+}$ candidates tagged as mixed and
unmixed with the projection of the fit result, which is described in the text.
(b) Calibration of the SSK tagger using
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}\pi^{+}$ decays. The black points show
the average measured mistag probability, $\omega$, in bins of predicted
mistag, $\eta$, the red line shows the calibration obtained from the fit
described in the text, and the yellow area the calibration uncertainty within
one standard deviation. The shaded histogram shows the distribution of $\eta$
in the background subtracted
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
sample.

To determine the SSK tagger calibration parameters from the
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}\pi^{+}$ decay candidates, an unbinned
maximum-likelihood fit, which uses the PDF of Eq. (8), is performed.
Uncertainties due to the use of external measurements [46] of $\Gamma_{s}$,
$\Delta\Gamma_{s}$, $\Delta m_{s}$, and the decay-time resolution parameters
($c_{0}$, $c_{1}$) are accounted for via Gaussian constraints in the
likelihood function. The parameters of the SSK tagger calibration and decay-
time efficiency are free in the fit. Figure 10 shows the result of this fit,
split by decays that are tagged as being mixed or unmixed, and the obtained
relation between $\omega$ and $\eta$. Also shown are the values of the
measured mistag in bins of estimated mistag, which supports the use of a
linear calibration function for $\omega(\eta)$. The final calibration
parameters are given in Table 1. Two sources of systematic uncertainty are
studied in addition to the knowledge of the time resolution, which is
incorporated in the statistical uncertainty. The first and larger one is due
to the applicability of the calibration from
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}\pi^{+}$ to
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
decays. It is tested using simulated events and the observed difference
between the calibrations is assigned as a systematic uncertainty. Variations
in the parameters through the use of a different model for the combinatorial
background in the fit to the ${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}\pi^{+}$
invariant-mass distribution are treated as systematic uncertainties. The
tagging asymmetry parameters, $\Delta p_{0}$ and $\Delta p_{1}$, are both
assumed to be $0.00\pm 0.03$. The uncertainty is estimated by studying the
tagging calibration using a sample of over 3.1 million promptly produced
${{D}^{-}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}\pi^{-}$ decays, with the method
described in Ref. [59]. The overall tagging power for
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}$ candidates with only an SSK tag can be found in Table 2.

###  7.3 Tagger combination

Approximately 31% of the tagged candidates in the
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}$ sample are tagged by both the OS and the SSK algorithms.
Since the algorithms are uncorrelated, as they select mutually exclusive
charged particles, the two tagging results are combined taking into account
both decisions and their corresponding estimate of $\eta$. The combined
estimated mistag probability and the corresponding uncertainties are obtained
by combining the individual calibrations for the OS and SSK tagging and
propagating their uncertainties. The effective tagging power and efficiency
for these both OS and SSK tagged candidates is given in Table 2.

##  8 Maximum-likelihood fit

The maximum-likelihood fitting procedure is similar to that in Ref. [27], the
only major differences being the treatment of the decay-time efficiency and
that the quantity ${\Gamma_{{s}}}-{\Gamma_{{d}}}$ is measured instead of
$\Gamma_{s}$. It has been checked via pseudoexperiments that, given that the
decay-time efficiency is obtained using $\Gamma_{d}$ as an input parameter
(see Sec. 5), the fitted value of ${\Gamma_{{s}}}-{\Gamma_{{d}}}$ and its
uncertainty are independent of the value and uncertainty of $\Gamma_{d}$. This
strategy has the advantage that the measured value of
${\Gamma_{{s}}}-{\Gamma_{{d}}}$ can be combined with the most up-to-date value
of $\Gamma_{d}$ to obtain $\Gamma_{s}$ or $\Gamma_{s}/\Gamma_{d}$.

Each candidate $i$ is given a signal weight $W_{i}$ using the sPlot method
with $m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}})$ as
a discriminating variable and $\sigma_{m}$ as a conditional variable as
explained in Sec. 3. A weighted fit is then performed to the ${B}^{0}_{s}$
decay time and helicity-angle distributions using a PDF that describes only
the signal. The log-likelihood in each of the 24 data subsamples is scaled by
a per-sample factor $\alpha=\sum_{i}W_{i}/\sum_{i}W_{i}^{2}$ to account for
the effect of the weights in the determination of the parameter uncertainties
[34].

The distribution of the decay time and angles for a ${B}^{0}_{s}$ meson
produced at time $t=0$ is described by a sum of ten terms, corresponding to
the four polarisation amplitudes squared and their interference terms. Each of
these is given by the product of a decay-time-dependent function and an
angular function

$\frac{\mathrm{d}^{4}\Gamma({{B}^{0}_{s}}\rightarrow J/\psi
K^{+}K^{-})}{\mathrm{d}t\;\mathrm{d}\Omega}\;\propto\;\sum^{10}_{k=1}\>N_{k}\>h_{k}(t)\>f_{k}(\Omega)\,,$
(9)

with

$\displaystyle h_{k}(t|B^{0}_{s})$ $\displaystyle=\frac{3}{4\pi}e^{-\Gamma
t}\left(a_{k}\cosh\frac{\Delta\Gamma t}{2}+b_{k}\sinh\frac{\Delta\Gamma
t}{2}+c_{k}\cos(\Delta mt)+d_{k}\sin(\Delta mt)\right),$ (10) $\displaystyle
h_{k}(t|\bar{B}^{0}_{s})$ $\displaystyle=\frac{3}{4\pi}e^{-\Gamma
t}\left(a_{k}\cosh\frac{\Delta\Gamma t}{2}+b_{k}\sinh\frac{\Delta\Gamma
t}{2}-c_{k}\cos(\Delta mt)-d_{k}\sin(\Delta mt)\right),$ (11)

where the definition of the parameters $N_{k}$, $a_{k}$, $b_{k}$, $c_{k}$,
$d_{k}$ and of the function $f_{k}(\Omega)$ can be found in Table 3.

Table 3: Angular and time-dependent functions used in the fit to the data.
Abbreviations used include $c_{K}=\cos\theta_{K}$, $s_{K}=\sin\theta_{K}$,
$c_{l}=\cos\theta_{l}$, $s_{l}=\sin\theta_{l}$, $c_{\phi}=\cos\phi$ and
$s_{\phi}=\sin\phi$.

$f_{k}$ | $N_{k}$ | $a_{k}$ | $b_{k}$ | $c_{k}$ | $d_{k}$
---|---|---|---|---|---
$c^{2}_{K}s^{2}_{l}$ | $|A_{0}|^{2}$ | $\frac{1}{2}(1+|\lambda_{0}|^{2})$ | $-|\lambda_{0}|\cos(\phi_{0})$ | $\frac{1}{2}(1-|\lambda_{0}|^{2})$ | $|\lambda_{0}|\sin(\phi_{0})$
$\frac{1}{2}{s^{2}_{K}(1-c_{\phi}^{2}s_{l}^{2})}$ | $|A_{\parallel}|^{2}$ | $\frac{1}{2}(1+|\lambda_{{\parallel}}|^{2})$ | $-|\lambda_{{\parallel}}|\cos(\phi_{{\parallel}})$ | $\frac{1}{2}(1-|\lambda_{{\parallel}}|^{2})$ | $|\lambda_{{\parallel}}|\sin(\phi_{{\parallel}})$
$\frac{1}{2}s^{2}_{K}(1-s_{\phi}^{2}s_{l}^{2})$ | $|A_{\perp}|^{2}$ | $\frac{1}{2}(1+|\lambda_{{\perp}}|^{2})$ | $|\lambda_{{\perp}}|\cos(\phi_{{\perp}})$ | $\frac{1}{2}(1-|\lambda_{{\perp}}|^{2})$ | $-|\lambda_{{\perp}}|\sin(\phi_{{\perp}})$
$s^{2}_{K}s_{l}^{2}s_{\phi}c_{\phi}$ | $|A_{\perp}A_{\parallel}|$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}\sin(\delta_{\perp}-\delta_{\parallel})-|\lambda_{\perp}\lambda_{\parallel}|\\\ \sin(\delta_{\perp}-\delta_{\parallel}-\phi_{\perp}+\phi_{\parallel})\bigg{]}\end{array}$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}|\lambda_{\perp}|\sin(\delta_{\perp}-\delta_{\parallel}-\phi_{\perp})\\\ +|\lambda_{\parallel}|\sin(\delta_{\parallel}-\delta_{\perp}-\phi_{\parallel})\bigg{]}\end{array}$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}\sin(\delta_{\perp}-\delta_{\parallel})+|\lambda_{\perp}\lambda_{\parallel}|\\\ \sin(\delta_{\perp}-\delta_{\parallel}-\phi_{\perp}+\phi_{\parallel})\bigg{]}\end{array}$ | $\begin{array}[]{l}-\frac{1}{2}\bigg{[}|\lambda_{\perp}|\cos(\delta_{\perp}-\delta_{\parallel}-\phi_{\perp})\\\ +|\lambda_{\parallel}|\cos(\delta_{\parallel}-\delta_{\perp}-\phi_{\parallel})\bigg{]}\end{array}$
$\sqrt{2}s_{K}c_{K}s_{l}c_{l}c_{\phi}$ | $|A_{0}A_{\parallel}|$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}\cos(\delta_{0}-\delta_{\parallel})+|\lambda_{0}\lambda_{\parallel}|\\\ \cos(\delta_{0}-\delta_{\parallel}-\phi_{0}+\phi_{\parallel})\bigg{]}\end{array}$ | $\begin{array}[]{l}-\frac{1}{2}\bigg{[}|\lambda_{0}|\cos(\delta_{0}-\delta_{\parallel}-\phi_{0})\\\ +|\lambda_{\parallel}|\cos(\delta_{\parallel}-\delta_{0}-\phi_{\parallel})\bigg{]}\end{array}$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}\cos(\delta_{0}-\delta_{\parallel})-|\lambda_{0}\lambda_{\parallel}|\\\ \cos(\delta_{0}-\delta_{\parallel}-\phi_{0}+\phi_{\parallel})\bigg{]}\end{array}$ | $\begin{array}[]{l}-\frac{1}{2}\bigg{[}|\lambda_{0}|\sin(\delta_{0}-\delta_{\parallel}-\phi_{0})\\\ +|\lambda_{\parallel}|\sin(\delta_{\parallel}-\delta_{0}-\phi_{\parallel})\bigg{]}\end{array}$
$-\sqrt{2}s_{K}c_{K}s_{l}c_{l}s_{\phi}$ | $|A_{0}A_{\perp}|$ | $\begin{array}[]{l}-\frac{1}{2}\bigg{[}\sin(\delta_{0}-\delta_{\perp})-|\lambda_{0}\lambda_{\perp}|\\\ \sin(\delta_{0}-\delta_{\perp}-\phi_{0}+\phi_{\perp})\bigg{]}\end{array}$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}|\lambda_{0}|\sin(\delta_{0}-\delta_{\perp}-\phi_{0})\\\ +|\lambda_{\perp}|\sin(\delta_{\perp}-\delta_{0}-\phi_{\perp})\bigg{]}\end{array}$ | $\begin{array}[]{l}-\frac{1}{2}\bigg{[}\sin(\delta_{0}-\delta_{\perp})+|\lambda_{0}\lambda_{\perp}|\\\ \sin(\delta_{0}-\delta_{\perp}-\phi_{0}+\phi_{\perp})\bigg{]}\end{array}$ | $\begin{array}[]{l}-\frac{1}{2}\bigg{[}|\lambda_{0}|\cos(\delta_{0}-\delta_{\perp}-\phi_{0})\\\ +|\lambda_{\perp}|\cos(\delta_{\perp}-\delta_{0}-\phi_{\perp})\bigg{]}\end{array}$
$\frac{1}{3}s_{l}^{2}$ | $|A_{\rm S}|^{2}$ | $\frac{1}{2}(1+|\lambda_{\rm S}|^{2})$ | $|\lambda_{\rm S}|\cos(\phi_{\rm S})$ | $\frac{1}{2}(1-|\lambda_{\rm S}|^{2})$ | $-|\lambda_{\rm S}|\sin(\phi_{\rm S})$
$\frac{2}{\sqrt{6}}s_{K}s_{l}c_{l}c_{\phi}$ | $|A_{\rm S}A_{\parallel}|$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}\cos(\delta_{S}-\delta_{\parallel})-|\lambda_{S}\lambda_{\parallel}|\\\ \cos(\delta_{S}-\delta_{\parallel}-\phi_{S}+\phi_{\parallel})\bigg{]}\end{array}$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}|\lambda_{S}|\cos(\delta_{S}-\delta_{\parallel}-\phi_{S})\\\ -|\lambda_{\parallel}|\cos(\delta_{\parallel}-\delta_{S}-\phi_{\parallel})\bigg{]}\end{array}$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}\cos(\delta_{S}-\delta_{\parallel})+|\lambda_{S}\lambda_{\parallel}|\\\ \cos(\delta_{S}-\delta_{\parallel}-\phi_{S}+\phi_{\parallel})\bigg{]}\end{array}$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}|\lambda_{S}|\sin(\delta_{S}-\delta_{\parallel}-\phi_{S})\\\ -|\lambda_{\parallel}|\sin(\delta_{\parallel}-\delta_{S}-\phi_{\parallel})\bigg{]}\end{array}$
$-\frac{2}{\sqrt{6}}s_{K}s_{l}c_{l}s_{\phi}$ | $|A_{\rm S}A_{\perp}|$ | $\begin{array}[]{l}-\frac{1}{2}\bigg{[}\sin(\delta_{S}-\delta_{\perp})+|\lambda_{S}\lambda_{\perp}|\\\ \sin(\delta_{S}-\delta_{\perp}-\phi_{S}+\phi_{\perp})\bigg{]}\end{array}$ | $\begin{array}[]{l}-\frac{1}{2}\bigg{[}|\lambda_{S}|\sin(\delta_{S}-\delta_{\perp}-\phi_{S})\\\ -|\lambda_{\perp}|\sin(\delta_{\perp}-\delta_{S}-\phi_{\perp})\bigg{]}\end{array}$ | $\begin{array}[]{l}-\frac{1}{2}\bigg{[}\sin(\delta_{S}-\delta_{\perp})-|\lambda_{S}\lambda_{\perp}|\\\ \sin(\delta_{S}-\delta_{\perp}-\phi_{S}+\phi_{\perp})\bigg{]}\end{array}$ | $\begin{array}[]{l}-\frac{1}{2}\bigg{[}-|\lambda_{S}|\cos(\delta_{S}-\delta_{\perp}-\phi_{S})\\\ +|\lambda_{\perp}|\cos(\delta_{\perp}-\delta_{S}-\phi_{\perp})\bigg{]}\end{array}$
$\frac{2}{\sqrt{3}}c_{K}s^{2}_{l}$ | $|A_{\rm S}A_{0}|$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}\cos(\delta_{S}-\delta_{0})-|\lambda_{S}\lambda_{0}|\\\ \cos(\delta_{S}-\delta_{0}-\phi_{S}+\phi_{0})\bigg{]}\end{array}$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}|\lambda_{S}|\cos(\delta_{S}-\delta_{0}-\phi_{S})\\\ -|\lambda_{0}|\cos(\delta_{0}-\delta_{S}-\phi_{0})\bigg{]}\end{array}$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}\cos(\delta_{S}-\delta_{0})+|\lambda_{S}\lambda_{0}|\\\ \cos(\delta_{S}-\delta_{0}-\phi_{S}+\phi_{0})\bigg{]}\end{array}$ | $\begin{array}[]{l}\frac{1}{2}\bigg{[}|\lambda_{S}|\sin(\delta_{S}-\delta_{0}-\phi_{S})\\\ -|\lambda_{0}|\sin(\delta_{0}-\delta_{S}-\phi_{0})\bigg{]}\end{array}$

The interference between the different S- and P-wave contributions is
accounted for via an effective coupling factor, $C_{\rm SP}$. The $C_{\rm SP}$
factors are computed by integrating the interference between the S- and P-wave
contributions in each of the six $m({{K}^{+}}{{K}^{-}})$ bins in which the
analysis is performed, using the same strategy as in the previous analysis.
They are applied by multiplication to the relevant terms in Eq. (9). The
$C_{\rm SP}$ factors are unity for terms involving P-wave and S-wave
amplitudes only ($k<8$). In the determination of the $C_{\rm SP}$ factors, the
$m({{K}^{+}}{{K}^{-}})$ lineshape of the P-wave component is described by a
relativistic Breit–Wigner distribution, while the S-wave is taken as an
$f_{0}(980)$ resonance modelled as a Flatté amplitude with parameters from
Ref. [62]. The $C_{\mathrm{SP}}$ correction factors are calculated to be
$0.8463,0.8756,0.8478,0.8833,0.9415$ and 0.9756 from the lowest to the highest
$m({{K}^{+}}{{K}^{-}})$ bin. Their effect on the fit results is small and is
discussed further in Sec. 9, where three different S-wave lineshapes are
considered to assign a systematic uncertainty. The PDF considers four disjoint
tagging cases: only OS tagged candidates, only SSK-tagged, OS and SSK tagged,
and untagged candidates. Taking into account all detector response effects,
the full PDF is conditional upon the mistag probability and the estimated
decay-time uncertainty.

A simultaneous fit is made to the different subsamples, divided by
$m({{K}^{+}}{{K}^{-}})$ bin, year of data taking and trigger category. The PDF
for each subsample, up to a normalisation constant, is given by

$\displaystyle{\cal P}\left(t,\Omega|\mathfrak{q}^{\rm OS},\mathfrak{q}^{\rm
SSK},\eta^{\rm OS},\eta^{\rm
SSK},\delta_{t}\right)\propto\sum_{k=1}^{10}C^{k}_{\rm
SP}N_{k}f_{k}(\Omega)\varepsilon^{{{B}^{0}_{s}}}_{\rm data}(t)$ (12)
$\displaystyle\cdot\left\\{\left[{\cal Q}\left(\mathfrak{q}^{\rm
OS},\mathfrak{q}^{\rm SSK},\eta^{\rm OS},\eta^{\rm
SSK}\right)h_{k}\left(t|{{B}^{0}_{s}}\right)+\bar{\cal
Q}\left(\mathfrak{q}^{\rm OS},\mathfrak{q}^{\rm SSK},\eta^{\rm OS},\eta^{\rm
SSK}\right)\right.h_{k}\left(t|{{\kern
1.61993pt\overline{\kern-1.61993ptB}{}}{}^{0}_{s}}\right)\right]\otimes\left.{\cal
R}\left(t-t^{\prime}|\delta_{t}\right)\right\\},$

where ${\cal R}$ is the time resolution function defined in Eq. (1) and the
terms

$\displaystyle{\cal Q}\left(\mathfrak{q}^{\rm OS},\mathfrak{q}^{\rm
SSK},\eta^{\rm OS},\eta^{\rm SSK}\right)$
$\displaystyle=\left[1+\mathfrak{q}^{\rm OS}\left(1-2\omega\left(\eta^{\rm
OS}\right)\right)\right]\left[1+\mathfrak{q}^{\rm
SSK}\left(1-2\tilde{\omega}\left(\eta^{\rm SSK}\right)\right)\right],$ (13)
$\displaystyle\bar{{\cal Q}}\left(\mathfrak{q}^{\rm OS},\mathfrak{q}^{\rm
SSK},\eta^{\rm OS},\eta^{\rm SSK}\right)$
$\displaystyle=\left[1-\mathfrak{q}^{\rm
OS}\left(1-2\bar{\omega}\left(\eta^{\rm
OS}\right)\right)\right]\left[1-\mathfrak{q}^{\rm
SSK}\left(1-2\tilde{\omega}\left(\eta^{\rm SSK}\right)\right)\right].$ (14)

account for the measured flavour of the ${B}^{0}_{s}$ candidate. All physics
parameters are free in the fit and are common across the subsamples, except
for the S-wave fraction and the phase difference $\delta_{\rm
S}-\delta_{\perp}$, which are independent parameters for each
$m({{K}^{+}}{{K}^{-}})$ bin.

##  9 Systematic uncertainties

Systematic uncertainties on the measured physics parameters arise from a
variety of sources that are described in the following. They are summarised in
Table 4.

Table 4: Summary of the systematic uncertainties.

Source | ${\phi_{{s}}}$ | $|\lambda|$ | ${\Gamma_{{s}}}-\Gamma_{d}$ | $\Delta\Gamma_{s}$ | ${\Delta m_{{s}}}$ | $|A_{\perp}|^{2}$ | $|A_{0}|^{2}$ | $\delta_{\perp}-\delta_{0}$ | $\delta_{\parallel}-\delta_{0}$
---|---|---|---|---|---|---|---|---|---
| [$\mathrm{\,rad}$] |  | [${\mathrm{\,ps^{-1}}}$] | [${\mathrm{\,ps^{-1}}}$] | [${\mathrm{\,ps^{-1}}}$] |  |  | [$\mathrm{\,rad}$] | [$\mathrm{\,rad}$]
Mass: width parametrisation | - | - | - | 0.0002 | 0.001 | 0.0004 | 0.0006 | - | 0.003
Mass: decay-time $\&$ angles dependence | 0.004 | 0.0037 | 0.0007 | 0.0022 | 0.016 | 0.0005 | 0.0002 | 0.05 | 0.009
Multiple candidates | 0.0011 | 0.0011 | 0.0003 | 0.0001 | 0.001 | 0.0001 | 0.0001 | 0.01 | 0.002
Fit bias | 0.0010 | - | - | 0.0003 | 0.001 | 0.0006 | 0.0001 | 0.02 | 0.033
$C_{\rm SP}$ factors | 0.0010 | 0.0010 | - | 0.0001 | 0.002 | 0.0001 | - | 0.01 | 0.005
Time resolution: model applicability | - | - | - | - | 0.001 | - | - | - | 0.001
Time resolution: $t$ bias | 0.0032 | 0.0010 | 0.0002 | 0.0003 | 0.005 | - | - | 0.08 | 0.001
Time resolution: wrong PV | - | - | - | - | 0.001 | - | - | - | 0.001
Angular efficiency: simulated sample size | 0.0011 | 0.0018 | - | - | 0.001 | 0.0004 | 0.0003 | - | 0.004
Angular efficiency: weighting | 0.0022 | 0.0043 | 0.0001 | 0.0002 | 0.001 | 0.0011 | 0.0020 | 0.01 | 0.008
Angular efficiency: clone candidates | 0.0005 | 0.0014 | 0.0002 | 0.0001 | - | 0.0001 | 0.0002 | - | 0.002
Angular efficiency: $t$ & $\sigma_{t}$ dependence | 0.0012 | 0.0007 | 0.0002 | 0.0010 | 0.003 | 0.0012 | 0.0008 | 0.03 | 0.006
Decay-time efficiency: statistical | - | - | 0.0012 | 0.0008 | - | 0.0003 | 0.0002 | - | -
Decay-time efficiency: kinematic weighting | - | - | 0.0002 | - | - | - | - | - | -
Decay-time efficiency: PDF weighting | - | - | 0.0001 | 0.0001 | - | - | - | - | -
Decay-time efficiency: $\Delta\Gamma_{s}=0$ simulation | - | - | 0.0003 | 0.0005 | - | 0.0002 | 0.0001 | - | -
Length scale | - | - | - | - | 0.004 | - | - | - | -
Quadratic sum of syst. | 0.0061 | 0.0064 | 0.0015 | 0.0026 | 0.018 | 0.0019 | 0.0023 | 0.10 | 0.036

Three systematic effects due to the $m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-})$ model and the sWeights computation are taken into account.
Firstly, the systematic effect due to statistical uncertainties in the
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}K^{-})$ fit model is
estimated. For this the sWeights are recomputed after varying the fit
parameters within their statistical uncertainties. The systematic
uncertainties are obtained from the difference in fit results and are found to
be negligible. Secondly, the average width of the double-sided CB distribution
is parametrised as a linear function of the per-candidate mass uncertainty,
instead of a quadratic one. The differences to the baseline result are
assigned as systematic uncertainties on the mass shape. Thirdly, the
assumption that the $m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-})$ distribution is independent of the decay time and angles
is tested by re-evaluating the sWeights in bins of these observables,
repeating the fit and assigning the differences in fit results as systematic
uncertainties.

The main physics background contribution comes from misidentified ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p{{K}^{-}}$ decays. Possible effects due to the limited knowledge of
the size of this component are estimated by repeating the fit after varying
the amount of this background by one standard deviation of its measured yield.
The maximum difference is found to be negligible and thus no systematic
uncertainties are assigned. A further systematic effect due to the
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ background is evaluated by repeating the
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}K^{-})$ fit while
leaving the mass resolution for this component free. A new set of sWeights are
computed, leading to negligible systematic uncertainties. Finally,
approximately 0.5% of
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ candidates come from the decays of ${{B}_{c}^{+}}$
mesons via the ${{B}_{c}^{+}}\rightarrow{{B}^{0}_{s}}{{\pi}^{+}}$ decay [63,
64]. The effect of ignoring this component in the fit is evaluated using
simulated pseudoexperiments where $0.5\%$ of the candidates are replaced with
${B}^{0}_{s}$-from-${{B}_{c}^{+}}$ decays that are randomly sampled from
simulated
${{B}_{c}^{+}}\rightarrow{{B}^{0}_{s}}(\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\pi^{+}$ decays. This is found to have a negligible effect on all
parameters.

In the baseline strategy, all candidates are retained even if multiple
candidates are present in a single event. A part of these multiple candidates
is found to peak in the $m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-})$ distribution, which introduces a bias in the physics
parameters, mainly on $\Gamma_{s}$. The peaking component is due to so-called
clone candidates originating from final state tracks that are duplicated in
the reconstruction process. Candidates are considered to be clones if they
belong to the same event and their final-state tracks are separated by an
angle smaller than 5 mrad. To assign systematic uncertainties, a single random
candidate is selected among all clone candidates in an event and the fit is
repeated. Approximately 0.35% (0.2%) of all selected ${{B}^{0}_{s}}$
(${{B}^{0}}$) candidates are removed. The maximum resulting variations of the
fit values are assigned as systematic uncertainties.

Possible biases of the fitting procedure are studied by generating and fitting
over eight thousand pseudoexperiments of the same size as the data. The biases
are determined from the resulting pull distributions. The ones that are
significantly different from zero are assigned as systematic uncertainties.

Different models of the S-wave lineshape based on the results in Ref. [65] are
used to evaluate the coupling factors $C_{\rm SP}$ in each of the six
$m({{K}^{+}}{{K}^{-}})$ bins, according to Ref. [31]. This includes an S-wave
parametrisation with a cubic spline function determined from data, a variation
of the $f_{0}(980)$ pole and width parameters used in the baseline model
within their uncertainties, and the variation of the $f_{0}(980)$ parameters
according to the second solution found in the analysis in Ref. [65]. The
maximum resulting variations of the fit values, mostly due to the spline
parametrisation, are assigned as systematic uncertainties.

The tagging parameters are Gaussian-constrained in the fit and therefore their
uncertainties contribute to the statistical uncertainty of each fit value.
This mainly affects the parameter ${\phi_{{s}}}$, with a contribution to the
uncertainty of 15$\mathrm{\,mrad}$. In addition, the calibration of the OS
tagging is re-evaluated using a quadratic function instead of a linear one.
The observed differences when repeating the fit are found to be negligible.

The systematic uncertainties associated with decay-time resolution originate
from four different sources. The first is due to the statistical uncertainties
on the calibration parameters and is found to be negligible. The second is
related to the assumption that the resolution model obtained in the
calibration sample applies also to the signal sample. The corresponding
systematic uncertainty is determined by evaluating the ratio of the
calibration effective resolutions, obtained from the simulated samples of the
calibration and signal decays, and using it to scale the effective resolutions
in the prompt data sample. These scaled effective resolutions are then
described by a quadratic function and used to determine the physics
parameters. The differences with respect to the baseline result are assigned
as systematic uncertainties. A third source of uncertainty is due to a
possible bias of the Gaussian resolution mean, which is assumed to be zero in
the baseline model. A quadratic dependence of the mean on the decay-time
uncertainty is observed in the calibration sample, with a maximum deviation of
about $5\mathrm{\,fs}$ from zero. It is modelled in the prompt data sample
after weighting it in order to match the signal data sample. Corresponding
systematic uncertainties are evaluated as the differences between the results
obtained with this bias and the baseline model. Finally, the fourth systematic
effect is estimated by varying the contribution in the fit of candidates with
an associated wrong origin vertex. The fraction of these candidates is varied
between 0 and 1.5%, corresponding to about three times the fraction that is
measured in the calibration sample, the calibration updated and the fit to
data repeated. The maximum deviations from the baseline fit are assigned as
systematic uncertainties.

The angular efficiency is determined from simulated signal, weighted such that
the kinematic distributions of the final-state particles match those in the
data. Systematic uncertainties are assigned to account for the limited size of
the simulated sample by varying the normalisation weights according to their
uncertainties and their covariance matrix and repeating the fit with a new
varied set of weights. The resulting RMS of each fitted observable is taken as
a systematic uncertainty. In addition, the impact of the specific
configuration of the gradient-boost tree method used in the reweighting of the
simulation is studied by testing approximately one hundred alternative
configurations. The maximal deviations from the fit result obtained with the
default angular efficiency are assigned as systematic uncertainties. The
differences between the fit results obtained using angular corrections from
the baseline or alternative weighting procedures of the simulated candidates
are also considered as systematic uncertainties. An imperfect removal of clone
candidates, in simulation, that peak in the ${B}^{0}_{s}$ candidate mass is
tested as follows. The peaking component is separated from the underlying
background via sWeights using all simulated events to determine its shape. As
an alternative, it is modelled according to the distribution of the
corresponding background classification that is available in simulations,
which however is limited by the small sample size and is therefore not used as
the baseline strategy. In addition, the two components are separated by
matching the reconstructed daughter particles to the simulated particles by
comparing their track momentum magnitudes and directions. The angular
efficiency is determined according to these two changes and the larger
differences are assigned as systematic uncertainties. Finally, from a fit to
several simulated samples of the same size as data, uncertainties are
evaluated as the differences between fitted and generated values to account
for correlations between the angular efficiency and the decay time as well as
the decay-time uncertainty. Such correlations are neglected in the baseline
fit.

Several sources of systematic uncertainties related to the determination of
the spline-based decay-time efficiency are studied and found to be small.
First, the effect due to the limited size of the data and simulated samples is
estimated by repeating the fit several times with the spline coefficients
varied according to their covariance matrix and the RMS of the fitted
observable distributions is taken as systematic uncertainties. Two further
contributions are evaluated by taking the difference between the baseline fit
and alternative fits where the time efficiency is determined without applying
either the kinematic or the PDF weighting procedures used to correct the
physics parameters of ${{B}^{0}_{s}}$ and ${{B}^{0}}$ simulated samples. Next,
the number of spline nodes is doubled and found to have a negligible effect on
the result. Another systematic uncertainty source is due to the differences
observed in decay-time efficiency derived from the simulated samples with
$\Delta\Gamma_{s}$ equal to or different from zero. It has been also checked
that varying the decay-time resolutions used in the determination of the
decay-time efficiency by $10\%$ has negligible effects.

The uncertainty on the LHCb length scale is estimated to be 0.022% [66], as
determined from metrology and track-based alignment. This translates directly
into an uncertainty on ${\Gamma_{{s}}}-{\Gamma_{{d}}}$, $\Delta\Gamma_{s}$ and
$\Delta m_{s}$, which is non-negligible only in the case of $\Delta m_{s}$.
Other parameters are unaffected. The precision on the track momentum scale is
$0.03\%$. Its effect largely cancels in the computation of the decay time,
leading to negligible uncertainties on all observables.

Asymmetries between ${B}^{0}_{s}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ production rates are
diluted by the fast oscillation between particle and antiparticle. They are
found to have a negligible effect on the fit parameters.

No statistically significant systematic effect on the results is observed when
repeating the analysis on subsets of the data, splitting by magnet polarity,
trigger conditions, year of data taking, number of primary vertices, bins of
${B}^{0}_{s}$ $p_{\mathrm{T}}$, pseudorapidity and decay-time uncertainty.

##  10 Results

The results of the maximum-likelihood fit described in Sec. 8 are

$\displaystyle\phi_{s}$ $\displaystyle=-0.083\pm 0.041\pm 0.006\mathrm{\,rad}$
$\displaystyle|\lambda|$ $\displaystyle=1.012\pm 0.016\pm 0.006$
$\displaystyle\Gamma_{s}-\Gamma_{d}$ $\displaystyle=-0.0041\pm 0.0024\pm
0.0015{\mathrm{\,ps^{-1}}}$ $\displaystyle\Delta\Gamma_{s}$
$\displaystyle=0.077\pm 0.008\pm 0.003{\mathrm{\,ps^{-1}}}$
$\displaystyle\Delta m_{s}$ $\displaystyle=17.703\pm 0.059\pm
0.018{\mathrm{\,ps^{-1}}}$ $\displaystyle|A_{\perp}|^{2}$
$\displaystyle=0.2456\pm 0.0040\pm 0.0019$ $\displaystyle|A_{0}|^{2}$
$\displaystyle=0.5186\pm 0.0029\pm 0.0023$
$\displaystyle\delta_{\perp}-\delta_{0}$ $\displaystyle=2.64\pm 0.13\pm
0.10\mathrm{\,rad}$ $\displaystyle\delta_{\parallel}-\delta_{0}$
$\displaystyle=3.06^{\>+\>0.08}_{\>-\>0.07}\pm 0.04\mathrm{\,rad}.$ (15)

The S-wave fractions and phase differences with respect to $\delta_{\perp}$ in
each $m(K^{+}K^{-})$ bin are summarized in Appendix A. The background-
subtracted data distributions with fit projections are shown in Fig. 11.

The results are in good agreement with the previous LHCb measurement. The
measurements of $\phi_{s}$, $\Delta\Gamma_{s}$ and
${\Gamma_{{s}}}-{\Gamma_{{d}}}$ are the most precise to date and agree with
the SM expectations [5, 4, 18, 19]. The results also indicate no $C\\!P$
violation in ${{B}^{0}_{s}}\rightarrow J/\psi{{K}^{+}}{{K}^{-}}$ decays. The
value of $\Delta m_{s}$ is in a good agreement with the world average value
[46]. Relaxing the assumption that $\lambda_{r}$ is the same for all
polarisation states and repeating the fit shows no evidence for any
polarisation dependence. The correlation matrix including systematic
uncertainties can be found in Table 5.

Table 5: Correlation matrix including the statistical and systematic correlations between the parameters. | $\phi_{s}$ | $|\lambda|$ | $\Gamma_{s}-\Gamma_{d}$ | $\Delta\Gamma_{s}$ | ${\Delta m_{{s}}}$ | $|A_{\perp}|^{2}$ | $|A_{0}|^{2}$ | $\delta_{\perp}-\delta_{0}$ | $\delta_{\parallel}-\delta_{0}$
---|---|---|---|---|---|---|---|---|---
$\phi_{s}$ | 1.00 |  +0.16 |  $-$0.05 |  +0.01 |  $-$0.02 |  +0.01 |  +0.00 |  +0.03 |  +0.00
$|\lambda|$ |  |  +1.00 |  +0.07 |  $-$0.09 |  +0.06 |  +0.04 |  $-$0.02 |  +0.04 |  +0.01
$\Gamma_{s}-\Gamma_{d}$ |  |  |  +1.00 |  $-$0.46 |  +0.06 |  +0.35 |  $-$0.24 |  $-$0.01 |  +0.03
$\Delta\Gamma_{s}$ |  |  |  |  +1.00 |  $-$0.05 |  $-$0.64 |  +0.46 |  $-$0.02 |  +0.00
${\Delta m_{{s}}}$ |  |  |  |  |  +1.00 |  +0.01 |  +0.01 |  +0.55 |  $-$0.01
$|A_{\perp}|^{2}$ |  |  |  |  |  |  +1.00 |  $-$0.64 |  +0.01 |  +0.07
$|A_{0}|^{2}$ |  |  |  |  |  |  |  +1.00 |  +0.01 |  $-$0.02
$\delta_{\perp}-\delta_{0}$ |  |  |  |  |  |  |  |  +1.00 |  +0.25
$\delta_{\parallel}-\delta_{0}$ |  |  |  |  |  |  |  |  |  +1.00

Figure 11: Decay-time and helicity-angle distributions for background
subtracted ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays (data points) with the one-dimensional
projections of the PDF at the maximum-likelihood point. The solid blue line
shows the total signal contribution, which contains (long-dashed red)
$C\\!P$-even, (short-dashed green) $C\\!P$-odd and (dotted-dashed purple)
S-wave contributions. Data and fit projections for the different samples
considered (data-taking year, trigger and tagging categories, $m(K^{+}K^{-})$
bins) are combined.

##  11 Combination with other results

The results presented in this paper are combined with related Run 1 and Run 2
LHCb measurements, taking into account all statistical correlations, all
systematic errors and their correlations, and correlations between different
run periods.

###  11.1 Combination with Run 1
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$

The measurements presented in this paper are consistent with those obtained
from the analysis of the data collected by LHCb during the LHC Run 1 [27]. The
Run 1 measurements are combined with the results of this analysis taking into
account a covariance matrix that includes the statistical uncertainties with
their correlations, and the systematic uncertainties with their correlations,
both between the parameters in a single run period and between the two run
periods.

The sources of systematic uncertainty that are correlated between the analyses
are the applicability of the time resolution obtained from the prompt control
sample on the signal sample, the $C_{\rm SP}$ factors, the correction of
simulation for the angular efficiency determination, and the length scale. In
the case of the angular efficiency, a correlation matrix is determined from
the RMS distributions of the parameters in Run 2 and the same matrix is taken
to account for correlations between Run 1 and Run 2. For all other sources of
systematic uncertainty no correlation is assumed. For the parameters showing
asymmetric uncertainties, the larger uncertainty has been used in the
combination.

It has been verified that using the average of the two asymmetric
uncertainties does not change the combination, and that completely ignoring
the systematic correlations has a negligible effect. In the Run 1 measurement,
$\Gamma_{s}$ was measured instead of $\Gamma_{s}-\Gamma_{d}$, hence a linear
transformation is taken into account in the combination, constraining
$\Gamma_{d}$ to the known value [32]. The combined results are

$\displaystyle\phi_{s}$ $\displaystyle=-0.081\pm 0.032\mathrm{\,rad}\,,$
$\displaystyle|\lambda|$ $\displaystyle=0.994\pm 0.013\,,$
$\displaystyle\Gamma_{s}$ $\displaystyle=0.6572\pm
0.0023{\mathrm{\,ps^{-1}}}\,,$ $\displaystyle\Delta\Gamma_{s}$
$\displaystyle=0.0777\pm 0.0062{\mathrm{\,ps^{-1}}}\,,$ $\displaystyle\Delta
m_{s}$ $\displaystyle=17.694\pm 0.042{\mathrm{\,ps^{-1}}}\,,$
$\displaystyle|A_{\perp}|^{2}$ $\displaystyle=0.2489\pm 0.0035\,,$
$\displaystyle|A_{0}|^{2}$ $\displaystyle=0.5195\pm 0.0034\,,$
$\displaystyle\delta_{\perp}-\delta_{0}$ $\displaystyle=2.87\pm
0.11\mathrm{\,rad}\,,$ $\displaystyle\delta_{\parallel}-\delta_{0}$
$\displaystyle=3.153\pm 0.079\mathrm{\,rad}.$ (16)

The correlation matrix can be found in Table 6. The correlation between
$\Gamma_{s}$ and $\Gamma_{d}$ is 0.39. The combined value of $\phi_{s}$ is
$2.5$ standard deviations from zero and agrees with expectations based on the
SM [5, 4].

Table 6: Correlation matrix for the results in Eq. (11.1) taking into account correlated systematics between Run 1 and the 2015 and 2016 results. | $\phi_{s}$ | $|\lambda|$ | $\Gamma_{s}$ | $\Delta\Gamma_{s}$ | ${\Delta m_{{s}}}$ | $|A_{\perp}|^{2}$ | $|A_{0}|^{2}$ | $\delta_{\perp}-\delta_{0}$ | $\delta_{\parallel}-\delta_{0}$
---|---|---|---|---|---|---|---|---|---
$\phi_{s}$ | 1.00 |  +0.10 |  $-$0.02 |  $-$0.03 |  +0.02 |  +0.01 |  $-$0.01 |  +0.07 |  +0.00
$|\lambda|$ |  |  +1.00 |  +0.04 |  $-$0.04 |  $-$0.05 |  +0.03 |  $-$0.02 |  $-$0.04 |  +0.03
$\Gamma_{s}$ |  |  |  +1.00 |  $-$0.35 |  +0.04 |  +0.28 |  $-$0.17 |  +0.01 |  +0.01
$\Delta\Gamma_{s}$ |  |  |  |  +1.00 |  $-$0.01 |  $-$0.62 |  +0.40 |  $-$0.05 |  $-$0.01
${\Delta m_{{s}}}$ |  |  |  |  |  +1.00 |  +0.01 |  $-$0.01 |  +0.62 |  +0.02
$|A_{\perp}|^{2}$ |  |  |  |  |  |  +1.00 |  $-$0.67 |  +0.03 |  +0.01
$|A_{0}|^{2}$ |  |  |  |  |  |  |  +1.00 |  $-$0.06 |  $-$0.06
$\delta_{\perp}-\delta_{0}$ |  |  |  |  |  |  |  |  +1.00 |  +0.28
$\delta_{\parallel}-\delta_{0}$ |  |  |  |  |  |  |  |  |  +1.00

###  11.2 Combination with other LHCb $\phi_{s}$ results

The results obtained in the previous section are further combined with the
recent results from
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ [67] decays, and the Run 1 results from
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ [28],
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ for the ${{K}^{+}}{{K}^{-}}$ invariant mass region
above 1.05${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ [31],
${{B}^{0}_{s}}\rightarrow\psi(2S)\phi$ [29] and ${{B}^{0}_{s}}\rightarrow
D^{+}_{s}D^{-}_{s}$ [30] decays.

The Run 1 analysis of
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ decays measured $|\lambda|$ and $\phi_{s}$
assuming a value of ${\Delta m_{{s}}}$ fixed to $17.768\pm
0.024{\mathrm{\,ps^{-1}}}$. Before the combination, this value is updated to
$17.711\pm 0.059{\mathrm{\,ps^{-1}}}$ [27], and the analysis is repeated to
obtain updated values of $|\lambda|$ and $\phi_{s}$. The analysis of 2015 and
2016 data, instead, measured $\phi_{s}$, $|\lambda|$ and $\Gamma_{\rm
H}-\Gamma_{d}$, assuming the $\Delta m_{s}$ value determined in this analysis.
In the combination, $\Gamma_{\rm H}-\Gamma_{d}$ is parametrised as
$\Gamma_{s}-\Gamma_{d}-\Delta\Gamma_{s}/2$, and the value of $\Gamma_{d}$ is
constrained to the known value. The combined values are

$\displaystyle\phi_{s}$ $\displaystyle=-0.042\pm 0.025\mathrm{\,rad}\,,$
$\displaystyle|\lambda|$ $\displaystyle=0.993\pm 0.010\,,$
$\displaystyle\Gamma_{s}$ $\displaystyle=0.6563\pm
0.0021{\mathrm{\,ps^{-1}}}\,,$ $\displaystyle\Delta\Gamma_{s}$
$\displaystyle=0.0813\pm 0.0048{\mathrm{\,ps^{-1}}}\,.$ (17)

The correlation matrix can be found in Table 7. The correlation between
$\Gamma_{s}$ and $\Gamma_{d}$ is 0.48.

Table 7: Correlation matrix for the results in Eq. (11.2) obtained taking into account correlated systematics between the considered analyses. | $\phi_{s}$ | $|\lambda|$ | $\Gamma_{s}$ | $\Delta\Gamma_{s}$
---|---|---|---|---
$\phi_{s}$ | 1.00 |  +0.06 |  $-$0.01 |  $-$0.03
$|\lambda|$ |  |  +1.00 |  +0.03 |  $-$0.02
$\Gamma_{s}$ |  |  |  +1.00 |  $-$0.17
$\Delta\Gamma_{s}$ |  |  |  |  +1.00
Figure 12: Regions of 68% confidence level in the
$\phi_{s}$-$\Delta\Gamma_{s}$ plane for the individual LHCb measurements and a
combined contour (in blue). The
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ (magenta) and
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ [67] (red) contours show the Run 1 and Run 2
combined numbers. The $\phi_{s}$ [4] and $\Delta\Gamma_{s}$ [18] predictions
are indicated by the thin black rectangle.

The values of these parameters are the most precise to date. Figure 12 shows
the 68% confidence level regions in the $\phi_{s}$ vs. $\Delta\Gamma_{s}$
plane for the considered analyses and the LHCb combination. The combined value
of $\phi_{s}$ is consistent with global fits to data. The parameter
$|\lambda|$ agrees with the hypothesis of no $C\\!P$ violation in the decay.
The values of $\Gamma_{s}$ and $\Delta\Gamma_{s}$ are consistent with
expectations from HQE models.

##  12 Conclusions

In summary, a flavour-tagged decay-time-dependent angular analysis of
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays has been performed, using
1.9$\mbox{\,fb}^{-1}$ of $pp$ collision data recorded by the LHCb experiment
during the 2015 and 2016 runs of the LHC. Approximately $117\,000$ signal
decays are selected, with a decay-time resolution of about 45$\mathrm{\,fs}$
and a tagging power of 4.7%. The $C\\!P$-violating phase ${\phi_{{s}}}$ is
measured to be $-0.083\pm 0.041\pm 0.006$$\mathrm{\,rad}$, the decay width
difference of the ${{B}^{0}_{s}}$ mass eigenstates, $\Delta\Gamma_{s}=0.077\pm
0.008\pm 0.003$${\mathrm{\,ps^{-1}}}$, and the difference of the average decay
widths of the ${B}^{0}_{s}$ and ${B}^{0}$ mesons,
$\Gamma_{s}-\Gamma_{d}=-0.0041\pm 0.0024\pm 0.0015$${\mathrm{\,ps^{-1}}}$.
Using the known value for the ${B}^{0}$ meson lifetime $1.520\pm
0.004{\mathrm{\,ps}}$ [32], the ratio of ${B}^{0}_{s}$ and ${B}^{0}$ meson
decay widths is measured to be $\Gamma_{s}/\Gamma_{d}=0.9938\pm 0.0036\pm
0.0023$. All results are shown with first the statistical and second the
systematic uncertainty. These are the single most precise measurements of
these quantities to date. In addition, the mass difference between the
${B}^{0}_{s}$ mass eigenstates is measured to be $\Delta m_{s}=17.703\pm
0.059\pm 0.018{\mathrm{\,ps^{-1}}}$. All results are consistent with
theoretical predictions based on the SM [5, 4]. The $C\\!P$-violating
parameters are also determined assuming that they are not the same for all
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ polarisation states and no polarisation dependence
is observed.

The measurements presented here for the parameters ${\phi_{{s}}}$,
$|\lambda|$, ${\Gamma_{{s}}}-\Gamma_{d}$, ${\Delta\Gamma_{{s}}}$, ${\Delta
m_{{s}}}$, $|A_{\perp}|^{2}$, $|A_{0}|^{2}$, $\delta_{\perp}-\delta_{0}$ and
$\delta_{\parallel}-\delta_{0}$ are consistent with those from
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays obtained using data collected by the LHCb
experiment during Run 1 of the LHC [27]. The two sets of measurements are
combined accounting for the statistical and systematic correlations between
parameters in each and the systematic correlations between the two run
periods. The combined values are $\phi_{s}=-0.081\pm 0.032$$\mathrm{\,rad}$,
$|\lambda|=0.994\pm 0.013$, $\Gamma_{s}=0.6572\pm 0.0023{\mathrm{\,ps^{-1}}}$,
$\Delta\Gamma_{s}=0.0777\pm 0.0062{\mathrm{\,ps^{-1}}}$ and $\Delta
m_{s}=17.694\pm 0.042{\mathrm{\,ps^{-1}}}$. The value of $\phi_{s}$ is 2.5
standard deviations from zero and consistent with theoretical predictions
based on the SM [5, 4].

The results are further combined with the recent results from
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ [67], and the Run 1 results from
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ [28],
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ for the ${{K}^{+}}{{K}^{-}}$ invariant mass region
above 1.05${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ [31],
${{B}^{0}_{s}}\rightarrow\psi(2S)\phi$ [29] and ${{B}^{0}_{s}}\rightarrow
D^{+}_{s}D^{-}_{s}$ [30]. The combined values are $\phi_{s}=-0.042\pm
0.025$$\mathrm{\,rad}$, $|\lambda|=0.993\pm 0.010$, ${\Gamma_{{s}}}=0.6563\pm
0.0021$${\mathrm{\,ps^{-1}}}$ and ${\Delta\Gamma_{{s}}}=0.0813\pm
0.0048$${\mathrm{\,ps^{-1}}}$. These values are consistent with theoretical
predictions based on the SM [5, 4]. In particular, the value of ${\phi_{{s}}}$
is consistent with a non-zero $C\\!P$-violation predicted within the SM and
with no $C\\!P$-violation in the interference of ${B}^{0}_{s}$ meson mixing
and decay. The parameter $|\lambda|$ is consistent with unity, implying no
evidence for direct $C\\!P$-violation in
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays.

## Acknowledgements

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

## Appendix

##  Appendix A S-wave parameters

The results for the S-wave parameters in each of the six $m(K^{+}K^{-})$ bins
are given in Table 8. The main sources of systematic uncertainties are the
$C_{\rm SP}$ factors, the mass factorization and biases of the fitting
procedure.

Table 8: Values of the S-wave parameters in each $m(K^{+}K^{-})$ bin. The first uncertainty is statistical and the second systematic. Parameter | Value
---|---
$F_{S1}$ | $0.492\pm 0.043\pm 0.010$
$F_{S2}$ | $0.041\pm 0.008\pm 0.006$
$F_{S3}$ | $0.0044^{+0.0030}_{-0.0017}\pm 0.0014$
$F_{S4}$ | $0.0069^{+0.0062}_{-0.0045}\pm 0.0016$
$F_{S5}$ | $0.073\pm 0.013\pm 0.004$
$F_{S6}$ | $0.152^{+0.019}_{-0.018}\pm 0.009$
$\delta_{S1}-\delta_{\perp}[\mathrm{\,rad}]$ | $+2.21^{+0.17}_{-0.20}\pm 0.20$
$\delta_{S2}-\delta_{\perp}[\mathrm{\,rad}]$ | $+1.56\pm 0.29\pm 0.05$
$\delta_{S3}-\delta_{\perp}[\mathrm{\,rad}]$ | $+1.09^{+0.47}_{-0.36}\pm 0.10$
$\delta_{S4}-\delta_{\perp}[\mathrm{\,rad}]$ | $-0.28^{+0.16}_{-0.26}\pm 0.12$
$\delta_{S5}-\delta_{\perp}[\mathrm{\,rad}]$ | $-0.54^{+0.09}_{-0.10}\pm 0.02$
$\delta_{S6}-\delta_{\perp}[\mathrm{\,rad}]$ | $-1.10^{+0.13}_{-0.16}\pm 0.11$

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R. Aaij28, C. Abellán Beteta46, T. Ackernley56, B. Adeva43, M. Adinolfi50, H.
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Barsuk8, W. Barter57, M. Bartolini20, F. Baryshnikov74, G. Bassi25, V.
Batozskaya32, B. Batsukh63, A. Battig11, V. Battista45, A. Bay45, M. Becker11,
F. Bedeschi25, I. Bediaga1, A. Beiter63, L.J. Bel28, V. Belavin38, S. Belin23,
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Ferreira Rodrigues2, S. Ferreres Sole28, M. Ferro-Luzzi44, S. Filippov37, R.A.
Fini15, M. Fiorini17,g, M. Firlej31, K.M. Fischer59, C. Fitzpatrick44, T.
Fiutowski31, F. Fleuret8,b, M. Fontana44, F. Fontanelli20,h, R. Forty44, V.
Franco Lima56, M. Franco Sevilla62, M. Frank44, C. Frei44, D.A. Friday55, J.
Fu22,q, W. Funk44, M. Féo44, E. Gabriel54, A. Gallas Torreira43, D. Galli16,e,
S. Gallorini24, S. Gambetta54, Y. Gan3, M. Gandelman2, P. Gandini22, Y. Gao3,
L.M. Garcia Martin76, B. Garcia Plana43, F.A. Garcia Rosales8, J. García
Pardiñas46, J. Garra Tico51, L. Garrido42, D. Gascon42, C. Gaspar44, G.
Gazzoni6, D. Gerick13, E. Gersabeck58, M. Gersabeck58, T. Gershon52, D.
Gerstel7, Ph. Ghez5, V. Gibson51, A. Gioventù43, O.G. Girard45, P. Gironella
Gironell42, L. Giubega33, C. Giugliano17, K. Gizdov54, V.V. Gligorov9, D.
Golubkov35, A. Golutvin57,74, A. Gomes1,a, I.V. Gorelov36, C. Gotti21,i, E.
Govorkova28, J.P. Grabowski13, R. Graciani Diaz42, T. Grammatico9, L.A.
Granado Cardoso44, E. Graugés42, E. Graverini45, G. Graziani18, A. Grecu33, R.
Greim28, P. Griffith17, L. Grillo58, L. Gruber44, B.R. Gruberg Cazon59, C.
Gu3, X. Guo67, E. Gushchin37, A. Guth10, Yu. Guz41,44, T. Gys44, C. Göbel65,
T. Hadavizadeh59, C. Hadjivasiliou6, G. Haefeli45, C. Haen44, S.C. Haines51,
P.M. Hamilton62, Q. Han69, X. Han13, T.H. Hancock59, S. Hansmann-Menzemer13,
N. Harnew59, T. Harrison56, C. Hasse44, M. Hatch44, J. He66, M. Hecker57, K.
Heijhoff28, K. Heinicke11, A. Heister11, A.M. Hennequin44, K. Hennessy56, L.
Henry76, E. van Herwijnen44, J. Heuel10, M. Heß71, A. Hicheur64, R. Hidalgo
Charman58, D. Hill59, M. Hilton58, P.H. Hopchev45, J. Hu13, W. Hu69, W.
Huang66, Z.C. Huard61, W. Hulsbergen28, T. Humair57, R.J. Hunter52, M.
Hushchyn39, D. Hutchcroft56, D. Hynds28, P. Ibis11, M. Idzik31, P. Ilten49, A.
Inglessi34, A. Inyakin41, K. Ivshin34, R. Jacobsson44, S. Jakobsen44, J.