EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2020-215 LHCb-PAPER-2020-029 March 8, 2021

Observation of $C\\!P$ violation

in two-body ${B}_{({s})}^{0}$-meson decays

to charged pions and kaons

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

The time-dependent $C\\!P$ asymmetries of
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays are measured using a
data sample of $p$ $p$ collisions corresponding to an integrated luminosity of
1.9 $\text{\,fb}^{-1}$, collected with the LHCb detector at a centre-of-mass
energy of 13  TeV. The results are

$\displaystyle{C}_{{\pi}{\pi}}$ $\displaystyle=$ $\displaystyle-0.311\pm
0.045\pm 0.015,$ $\displaystyle{S}_{{\pi}{\pi}}$ $\displaystyle=$
$\displaystyle-0.706\pm 0.042\pm 0.013,$ $\displaystyle{C}_{{K}{K}}$
$\displaystyle=$ $\displaystyle\phantom{-}0.164\pm 0.034\pm 0.014,$
$\displaystyle{S}_{{K}{K}}$ $\displaystyle=$ $\displaystyle\phantom{-}0.123\pm
0.034\pm 0.015,$ $\displaystyle{\cal A}^{\Delta\Gamma}_{{K}{K}}$
$\displaystyle=$ $\displaystyle-0.83\phantom{0}\pm 0.05\phantom{0}\pm 0.09,$

where the first uncertainties are statistical and the second systematic. The
same data sample is used to measure the time-integrated $C\\!P$ asymmetries of
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays and the results are

$\displaystyle A_{{C\\!P}}^{{{B}^{0}}}$ $\displaystyle=$
$\displaystyle-0.0824\pm 0.0033\pm 0.0033,$ $\displaystyle
A_{{C\\!P}}^{{{B}^{0}_{s}}}$ $\displaystyle=$
$\displaystyle\phantom{-}0.236\phantom{0}\pm 0.013\phantom{0}\pm 0.011.$

All results are consistent with earlier measurements. A combination of LHCb
measurements provides the first observation of time-dependent $C\\!P$
violation in ${B}^{0}_{s}$ decays.

Published in JHEP 03 (2021) 075.

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

##  1 Introduction

Charge-parity ($C\\!P$) asymmetries of charmless ${B}_{({s})}^{0}$-meson
decays to two-body charged final states are important inputs to the validation
of the Cabibbo-Kobayashi-Maskawa (CKM) mechanism [1, 2], which models $C\\!P$
violation in charged-current quark transitions. Deviations from Standard Model
(SM) predictions may reveal the presence of phenomena not included in the SM,
manifested as modifications to the amplitudes of these decays. [3, 4, 5, 6, 7,
8, 9]. The $C\\!P$ asymmetry in the
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decay is a fundamental input
to the isospin analysis of ${B}\\!\rightarrow{\pi}{\pi}$ decays that allows
the determination of the CKM angle $\alpha$ [10, 11, 12]. The analysis can be
extended by exploiting the approximate U-spin symmetry [13] that relates the
hadronic parameters entering the decay amplitudes of the
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays.111Unless stated
otherwise, the inclusion of charge-conjugate decay modes is implied throughout
this paper. It has been shown that, by incorporating the $C\\!P$ asymmetry and
branching fraction of the ${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$
decay into the standard isospin analysis, stringent constraints on the CKM
angle $\gamma$ and on the $C\\!P$-violating phase $-2\beta_{s}$ can be set,
even when allowing for U-spin breaking effects [14, 15]. Furthermore, a
substantial reduction of uncertainties on the determination of $-2\beta_{s}$
can be achieved by combining the $C\\!P$ asymmetries of the
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays with information
provided by the semileptonic decays
${{B}^{0}}\\!\rightarrow{{\pi}^{-}}{\ell^{+}}{\nu}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{\ell^{+}}{\nu}$ [16, 17]. The $C\\!P$
asymmetries and branching fractions of the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ provides the test of the SM,
assuming U-spin symmetry, proposed in Ref. [7]. The $C\\!P$ asymmetry of the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decay is also a key input to the
long-standing ${B}\\!\rightarrow{K}{\pi}$ puzzle [18, 19, 20]. Strategies have
been proposed to combine information from several decays of the
${B}\\!\rightarrow{\pi}{\pi}$ and ${B}\\!\rightarrow{K}{\pi}$ systems in order
to investigate the presence of physics beyond the SM [21, 22, 23].

This paper presents measurements of time-dependent $C\\!P$ asymmetries in
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays and of time-integrated
$C\\!P$ asymmetries in ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays. The analysis is
based on a data sample of $p$ $p$ collisions corresponding to an integrated
luminosity of 1.9$\text{\,fb}^{-1}$, collected with the LHCb detector at a
centre-of-mass energy of 13 TeV during 2015 and 2016. These results are
combined with previous LHCb results, published in Ref. [24], based on a sample
corresponding to 3.0$\text{\,fb}^{-1}$, collected at 7 and 8 TeV in the Run 1
data taking.

In decays of ${B}_{({s})}^{0}$ mesons to a final state $f$, where $f$ is a
$C\\!P$ eigenstate ($f=\overline{f}$), $C\\!P$ violation originates from the
interference between the decay and ${B}_{({s})}^{0}$-${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}_{({s})}^{0}$ mixing. The latter can be
modelled by an effective Hamiltonian whose mass eigenstates are linear
combinations of the two flavour eigenstates, $p|{{B}_{({s})}^{0}}\rangle\pm
q|{{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}_{({s})}^{0}}\rangle$, where
$p$ and $q$ are complex parameters, normalised such that $|p|^{2}+|q|^{2}=1$.
The $C\\!P$ asymmetry as a function of decay time for
${{B}_{({s})}^{0}}\\!\rightarrow f$ decays is given by

$A_{{C\\!P}}(t)=\frac{\Gamma_{{{\kern
1.25995pt\overline{\kern-1.25995ptB}}{}_{({s})}^{0}}\rightarrow
f}(t)-\Gamma_{{{B}_{({s})}^{0}}\rightarrow f}(t)}{\Gamma_{{{\kern
1.25995pt\overline{\kern-1.25995ptB}}{}_{({s})}^{0}}\rightarrow
f}(t)+\Gamma_{{{B}_{({s})}^{0}}\rightarrow f}(t)}=\frac{-C_{f}\cos(\Delta
m_{{d}({s})}t)+S_{f}\sin(\Delta
m_{{d}({s})}t)}{\cosh\left(\frac{\Delta\Gamma_{{d}({s})}}{2}t\right)+A^{\Delta\Gamma}_{f}\sinh\left(\frac{\Delta\Gamma_{{d}({s})}}{2}t\right)},$
(1)

where $\Delta m_{{d}({s})}$ and $\Delta\Gamma_{{d}({s})}$ are the mass and
width differences of the mass eigenstates of the ${B}_{({s})}^{0}$ system. In
accordance with current experimental knowledge, the value of
$\Delta\Gamma_{{d}}$ is assumed to be negligible. The quantities $C_{f}$,
$S_{f}$ and $A^{\Delta\Gamma}_{f}$ are defined as

$C_{f}\equiv\frac{1-|\lambda_{f}|^{2}}{1+|\lambda_{f}|^{2}},\qquad
S_{f}\equiv\frac{2{\rm Im}\lambda_{f}}{1+|\lambda_{f}|^{2}},\qquad
A^{\Delta\Gamma}_{f}\equiv-\frac{2{\rm Re}\lambda_{f}}{1+|\lambda_{f}|^{2}},$
(2)

where $\lambda_{f}$ is given by

$\lambda_{f}\equiv\frac{q}{p}\frac{\overline{A}_{f}}{A_{f}}$ (3)

and $A_{f}$ ($\overline{A}_{f}$) is the decay amplitude for the
${B}_{({s})}^{0}$ (${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}_{({s})}^{0}$) $\rightarrow f$
transition. As current experimental determinations [25, 26, 27] confirm the SM
expectation [28, 29] of negligible $C\\!P$ violation in the
${B}_{({s})}^{0}$-${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}_{({s})}^{0}$
mixing (implying $\left|q/p\right|=1$), a nonzero value of $C_{f}$ and $S_{f}$
indicates the presence of $C\\!P$ violation in the decay and in the
interference between mixing and decay, respectively. The quantities $C_{f}$,
$S_{f}$ and $A_{f}^{{\Delta\Gamma}}$ are related through the unitary condition
$\left(C_{f}\right)^{2}+\left(S_{f}\right)^{2}+\left(A_{f}^{{\Delta\Gamma}}\right)^{2}=1$.
This constraint is not imposed in this analysis and is instead used as a
cross-check of the consistency of the results. Previous determinations of
${C}_{{\pi}{\pi}}$ and ${S}_{{\pi}{\pi}}$ were performed by BaBar [30], Belle
[31] and LHCb [24] experiments, while only LHCb has measured ${C}_{{K}{K}}$,
${S}_{{K}{K}}$ and ${\cal A}^{\Delta\Gamma}_{{K}{K}}$ [24].

The time-integrated $C\\!P$ asymmetry for a ${B}_{({s})}^{0}$ decay to a
flavour-specific final state $f$, such as
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$, is defined as

$A_{{C\\!P}}=\frac{\left|\overline{A}_{\overline{f}}\right|^{2}-\left|A_{f}\right|^{2}}{\left|\overline{A}_{\overline{f}}\right|^{2}+\left|A_{f}\right|^{2}}.$
(4)

Measurements of $A_{{C\\!P}}$ for the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decay
($A_{{C\\!P}}^{{{B}^{0}}}$) were carried out by BaBar [30], Belle [32], CDF
[33] and LHCb [24], while $A_{{C\\!P}}$ for the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay
($A_{{C\\!P}}^{{{B}^{0}_{s}}}$) was measured only by CDF [33] and LHCb [24].

This paper is organised as follows. The LHCb detector, its trigger system and
the simulation process are briefly introduced in Sec. 2, while the sample
selection is described in Sec. 3. The $C\\!P$ asymmetries are determined by
means of unbinned maximum-likelihood fits to the invariant-mass and decay-time
distributions of ${B}_{({s})}^{0}$ candidates reconstructed in the ${\pi}^{+}$
${\pi}^{-}$, ${{K}^{+}}\\!{{K}^{-}}$ and ${K}^{\pm}$ ${\pi}^{\mp}$ final
states. In order to measure the time-dependent $C\\!P$ asymmetries, it is
necessary to determine the flavour of the ${B}_{({s})}^{0}$ meson at its
production. In addition, a precise determination of the ${B}_{({s})}^{0}$
decay time is important, in particular for the ${B}^{0}_{s}$ meson, due to its
fast oscillation frequency. The flavour-tagging algorithms and their
calibration are presented in Sec. 4, while the determination of the decay-time
resolution is discussed in Sec. 5. The models used in the fits are described
in Sec. 6. Two measurements of the $C\\!P$-violating parameters for the
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays are performed with
different experimental techniques. The first method, referred to as the
simultaneous method, fits all the signal decays simultaneously and uses a fit
model similar to that described in Ref. [24]. The second method, referred to
as the per-candidate method, describes the selection efficiency as a function
of the decay time of the ${B}_{({s})}^{0}$ meson on a per-candidate basis
using the swimming technique [34, 35, 36, 37]. The determination of the
detection asymmetry between the ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$
and ${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays and their charge-
conjugate final states, necessary to measure $A_{{C\\!P}}$, is discussed in
Sec. 7. The results are given in Sec. 8 and the assessment of systematic
uncertainties is presented in Sec. 9. The statistical and systematic
uncertainties on the simultaneous method are found to be, in general, smaller
than those for the per-candidate method. The results from the simultaneous
method are therefore given as the main results of this paper. The final
results and their combination with previous LHCb measurements from Ref. [24]
are presented in Sec. 10, while considerations on the combined measurements
are reported in Sec. 11.

##  2 Detector, trigger and simulation

The LHCb detector [38, 39] is a single-arm forward spectrometer covering the
pseudorapidity in the range between 2 and 5, designed for the study of
particles containing $b$ or $c$ quarks. The detector includes a high-precision
tracking system consisting of a silicon-strip vertex detector surrounding the
$p$ $p$-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 [40, 41]. The tracking system provides a measurement
of momentum, $p$, of charged particles with a relative uncertainty that varies
from 0.5% at low momentum to 1.0% at 200$\text{\,Ge\kern-1.00006ptV\\!/}c$.
The minimum distance of a track to a primary $p$ $p$-collision vertex (PV),
the impact parameter (IP), is measured with a resolution of
$(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$ is the
component of the momentum transverse to the beam, in
$\text{\,Ge\kern-1.00006ptV\\!/}c$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov (RICH)
detectors [42]. Photons, electrons and hadrons are identified by a calorimeter
system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers. The online event selection is performed by a trigger [43], which
consists of a hardware stage, based on information from the calorimeter and
muon systems, followed by a software stage, which applies a full event
reconstruction.

At the hardware trigger stage, events are required to have a muon with high
$p_{\mathrm{T}}$, or a hadron, photon or electron with high transverse energy
in the calorimeters. For hadrons, the transverse energy threshold is 3.5 GeV.
The software trigger requires the presence in the event of at least one
charged particle with $p_{\mathrm{T}}>1.6\text{\,Ge\kern-1.00006ptV\\!/}c$ and
inconsistent with originating from any PV. The tracks identified at this stage
are used by a trigger selection dedicated for two-body $b$-hadron decays. The
selection algorithm imposes requirements on the quality of the reconstructed
tracks, their $p_{\mathrm{T}}$ and minimum $\chi^{2}_{\text{IP}}$ with respect
to every PV in the event, where the $\chi^{2}_{\text{IP}}$ is defined as the
difference in the vertex-fit $\chi^{2}$ of a given PV reconstructed with and
without the track under consideration. Pairs of oppositely charged tracks must
have a small distance of closest approach and a large scalar sum of their
$p_{\mathrm{T}}$ in order to be eligible to form a ${B}_{({s})}^{0}$
candidate. Finally, the ${B}_{({s})}^{0}$ candidates are required to pass
criteria based on their $p_{\mathrm{T}}$, $\chi^{2}_{\text{IP}}$, flight
distance with respect to their associated PV, and angle between the direction
of the ${B}_{({s})}^{0}$ candidate momentum and the direction defined by its
decay vertex and associated PV. Candidates are associated with the PV that is
most consistent with their flight direction.

Simulation is used to study the discrimination between signal and background
candidates, and to assess differences between signal and calibration decays.
The $pp$ collisions are generated using Pythia [44, 45] with a specific LHCb
configuration [46]. Decays of hadronic particles are described by EvtGen [47],
in which final-state radiation is generated using Photos [48]. The interaction
of the generated particles with the detector, and its response, are
implemented using the Geant4 toolkit [49, *Agostinelli:2002hh] as described in
Ref. [51].

##  3 Selection

The ${B}_{({s})}^{0}$ candidates selected by the dedicated software trigger
are further filtered, requiring that either the decay products or particles
from the rest of the event are responsible for the positive decision of the
hadronic hardware trigger. Candidates are then classified into mutually
exclusive samples of different final states (${\pi}^{+}$ ${\pi}^{-}$,
${{K}^{+}}\\!{{K}^{-}}$ and ${K}^{\pm}$ ${\pi}^{\mp}$) using particle
identification (PID) information. Finally, a boosted decision tree (BDT)
algorithm [52, 53] is used to separate signal candidates from combinatorial
background candidates for each of the final states.

Four types of background contributions are considered: two-body $b$-hadron
decays with misidentified pions, kaons or protons in the final state (cross-
feed background); pairs of randomly associated and oppositely charged tracks
(combinatorial background); pairs of oppositely charged tracks from partially
reconstructed three-body decays of $b$ hadrons (three-body background);
${B}_{({s})}^{0}$ mesons produced in ${B}_{c}^{+}$ decays rather than at a PV,
whose measured decay time is biased due to the finite lifetime of the
${B}_{c}^{+}$ meson. Given the small production rate of ${B}_{c}^{+}$ mesons
[54], this background contribution is neglected in the analysis and a
systematic uncertainty is assessed in Sec. 9. Since the three-body background
candidates give rise to ${B}_{({s})}^{0}$ candidates with invariant-mass
values well separated from the mass peak, the candidate selection is
customised to reject mainly the cross-feed and combinatorial background
candidates, as they affect the invariant-mass region around the ${B}^{0}$ and
${B}^{0}_{s}$ nominal masses.

The requirements imposed on the PID variables, used to identify the
${\pi}^{+}$ ${\pi}^{-}$ and ${K}^{+}$ ${K}^{-}$ samples, are optimised using
pseudoexperiments that take into account the different background
contributions. First the PID efficiencies and misidentification probabilities
for kaons and pions are determined, for different requirements, using samples
of
${{D}^{*+}}\\!\rightarrow{{D}^{0}}(\rightarrow{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$
decays[55] and are used to estimate the cross-feed background yields in each
of the final states. The results of the PID calibration and the fitting model
described in Sec. 6 are used to generate pseudoexperiments that are fitted
with the same model. The results of the fits are used to find the
configuration of PID requirements giving the best trade-off between the
statistical sensitivity to the $C\\!P$-violation parameters of the
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays and the systematic
effects due to large contributions of cross-feed background candidates. The
PID selection used to identify the ${K}^{\pm}$ ${\pi}^{\mp}$ samples is,
instead, optimised to reduce the amount of the
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ cross-feed background yields
to approximately 10% of the ${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$
yield.

The BDT algorithm exploits the following properties of the ${B}_{({s})}^{0}$
decay products: the $p_{\mathrm{T}}$ of the two tracks; the
$\chi^{2}_{\text{IP}}$ of each track with respect to their associated PV; the
distance of closest approach between the two tracks, and the quality of their
common vertex. The BDT classifier also uses properties of the reconstructed
${B}_{({s})}^{0}$ candidate, particularly the $p_{\mathrm{T}}$, the
$\chi^{2}_{\text{IP}}$ and the $\chi^{2}$ of the flight distance with respect
to the associated PV. Separate BDT algorithms are trained and optimised for
the selection of the ${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays. Simulated events of
the two decay modes are used to model the signal candidates, while data from
their high-mass sidebands (from 5.6$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ to
6.2$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$) are used to model the combinatorial
background candidates. The optimal threshold on the response of the BDT
algorithm is chosen to maximise $S/\sqrt{S+B}$, where $S$ and $B$ represent
the estimated yield of signal and combinatorial background candidates within
$\pm 60\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ (corresponding to about $\pm 3$
times the invariant-mass resolution) around the known ${B}_{({s})}^{0}$ mass.
The ${K}^{\pm}$ ${\pi}^{\mp}$ samples are selected using the BDT classifier
optimised for the ${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decay.222A
BDT classifier optimised for ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$
decays was found to have a comparable performance to that optimised for
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays and applied to the
${K}^{\pm}$ ${\pi}^{\mp}$ sample. Multiple candidates are present in less than
0.06% of the events satisfying the offline selections. Only one candidate is
accepted at random from each event.

The optimisation of the selection criteria preferentially rejects short-lived
candidates over longer lived ones. This introduces a distorted decay-time
efficiency that must be corrected for. The selection criteria present in the
analysis that produce this efficiency are the requirements on the
$\chi^{2}_{\text{IP}}$ of all particles, the $\chi^{2}$ of the
${B}_{({s})}^{0}$ flight distance, the direction defined by its decay vertex
and associated PV, and the outputs of the BDT algorithm. In addition, there
are also decay-time biasing selection criteria due to the geometry of the
detector. These are the limit on the radial flight distance of the
${B}_{({s})}^{0}$, which is required to avoid secondary interactions with the
vertex detector material, and the minimal number of the vertex-detector
sensors required to have track hits, which is imposed by the software
triggers. The bias introduced by the radial flight distance is only present in
the per-candidate method.

##  4 Flavour tagging

Tagging of the initial flavour of the ${B}_{({s})}^{0}$ meson plays a crucial
role in measuring the time-dependent $C\\!P$ asymmetries of decays to $C\\!P$
eigenstates, since the sensitivity to the $C_{f}$ and $S_{f}$ coefficients,
defined in Eq. (1), is related to the tagging performance. The flavour of the
${B}_{({s})}^{0}$ candidates is inferred by two classes of the flavour-tagging
algorithms called opposite-side (OS) and same-side (SS) taggers. The OS
taggers [56] exploit the fact that in $p$ $p$ collisions beauty quarks are
almost exclusively produced in ${b}{\overline{b}}$ pairs. Thus the flavour of
the decaying signal ${B}_{({s})}^{0}$ meson can be determined by looking at
the decay products of the other $b$ hadron in the event, for example, the
charge of the lepton originating from semileptonic decays, the charge of the
kaon from the ${b}\rightarrow{c}\rightarrow{s}$ transition, or the charge of a
charm hadron. An additional OS tagger is based on the inclusive reconstruction
of the opposite $b$-hadron decay vertex by computing the
$p_{\mathrm{T}}$-weighted average of the charges of all tracks associated to
that vertex. The SS taggers are based on the identification of the particles
produced in the hadronisation of the signal beauty quarks. In contrast to the
OS taggers, which to a very good approximation act equally on ${B}^{0}$ and
${B}^{0}_{s}$ mesons, SS taggers are specific to the light quark of the
${B}_{({s})}^{0}$ meson under study. Additional $\overline{d}$ ($d$) or
$\overline{s}$ ($s$) quarks produced in association with a ${B}^{0}$ (${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$) or a ${B}^{0}_{s}$ (${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$) meson, respectively, can
form charged pions and protons, in the down-quark case, or charged kaons, in
the strange-quark case. The so-called SS$\pi$ and SS$p$ taggers [57] are used
to determine the initial flavour of ${B}^{0}$ mesons, while the SS$K$ tagger
[58] is used for ${B}^{0}_{s}$ mesons.

For each tagger, the probability of misidentifying the flavour of the
${B}_{({s})}^{0}$ meson at production, the mistag probability, $\eta$, is
estimated by means of a multivariate classifier, and is defined in the range
$0\leq\eta\leq 0.5$. The flavour-tagging performance of each tagger can be
quantified by means of the tagging power, defined as

$\displaystyle\varepsilon_{\rm
eff}=\frac{1}{N}\sum\limits_{i}{|\xi_{i}|\,(1-2\eta_{i})^{2}},$ (5)

where $\xi_{i}$ and $\eta_{i}$ are the tagging decision and the probability of
misidentifying the flavour of the $i$-th out of $N$ ${B}_{({s})}^{0}$
candidates, respectively. The tagging decision $\xi_{i}$ takes the value of
$+1$ when the candidate is tagged as ${B}_{({s})}^{0}$, $-1$ when the
candidate is tagged as ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}_{({s})}^{0}$, and zero for untagged
candidates. Multivariate algorithms are used to determine the values of $\eta$
for the OS and SS taggers, denoted as $\eta_{\rm OS}$ and $\eta_{\rm SS}$.
These are trained using specific $B$-meson decay channels and selections. The
differences between the training samples and the selected signal
${B}_{({s})}^{0}$ candidates can lead to an imperfect determination of the
mistag probability. Hence, a more accurate estimate, denoted as $\omega$
hereafter, is obtained by means of a calibration procedure that takes into
account the specific kinematics of selected signal ${B}_{({s})}^{0}$ mesons.
The relation between $\eta$ and $\omega$ is calibrated using
${{B}^{+}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{\pi}^{+}}$,
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ decays for the OS, SS$K$, and
SS$\pi$ and SS$p$ taggers, respectively. The flavour for the ${{B}^{+}}$ meson
is tagged by the charge of the pion in the final state. For the ${B}^{0}$ and
${B}^{0}_{s}$ modes, which decay into flavour-specific final states, the
amplitude of the tagged time-dependent asymmetry is proportional to
$1-2\omega$. When the response of more than one OS tagger is available per
candidate, the different decisions and associated calibrated mistag
probabilities are combined into a unique decision $\xi_{\rm OS}$ and a single
$\eta_{\rm OS}$. A similar combination is also performed between the SS$\pi$
and SS$p$ taggers to create a combined same-side tagger, SSc, where a combined
tagging decision $\xi_{\rm SSc}$ and mistag probability $\eta_{\rm SSc}$ is
evaluated, as discussed in App. A.2.

In the simultaneous method, the OS and SSc combinations are recalibrated in
the final fit, discussed in Sec. 6, using the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays in order to correct for
possible correlations between the individual algorithms not taken into account
in the combination procedure. For the SS$K$ case, since the small yield of
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays is insufficient for a
reliable recalibration, the original calibration is kept and a systematic
uncertainty is assigned. In the per-candidate method, the OS and SS
combinations are further combined into a unique tagging decision and mistag
probability using the calibrations determined by the simultaneous method. This
combination is again recalibrated with the calibration samples. The
description of the implementation of the flavour tagging into the fit models
is presented in Sec. 6.

##  5 Decay-time resolution

The decay-time resolution is modelled with a Gaussian function, whose mean and
width are calibrated with a sample of
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ decays
produced directly in $p$ $p$ collisions. The background contribution in the
${J\mskip-3.0mu/\mskip-2.0mu\psi}$ sample is subtracted using the sPlot
technique [59] with the dimuon invariant mass acting as a discriminating
variable. The background-subtracted sample is separated in intervals of decay-
time uncertainty, $\delta_{t}$, which is determined for each candidate from
the kinematic fit used to measure the decay time. The decay-time distribution
in each bin of $\delta_{t}$ is fitted with a model comprising three Gaussian
functions with shared mean and independent widths. According to Ref. [60] the
parameters obtained from the fits are combined into an effective resolution,
$\sigma_{\rm eff}$, such that a single-Gaussian resolution model with width
$\sigma_{\rm eff}$ gives the same dilution effect on the amplitude of the
time-dependent asymmetry as the triple-Gaussian model. The value of
$\sigma_{\rm eff}$ is calibrated assuming all the signal decays have the same
mixing frequency as the ${B}^{0}_{s}$ meson. This assumption does not impact
the analysis for ${B}^{0}$ mesons, since for them the effect of the decay-time
resolution is negligible. Figure 1 shows the dependence of $\sigma_{\rm eff}$
on $\delta_{t}$ and is found to be well modelled with a linear function with
an intercept $q_{0}$ and slope $q_{1}$. The fit is repeated for different
numbers of bins of $\delta_{t}$, and the obtained mean values of the slope and
intercept are found to be 0.94$\,\pm\,$0.02 and 1.64$\,\pm\,$1.09 fs,
respectively. Differences in the decay-time resolution between
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ and two-
body $b$-hadron decays are studied using samples of fully simulated
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays. The calibrated decay-
time resolution as a function of $\delta_{t}$ is

$\sigma_{t}(\delta_{t})=\sigma_{\rm
eff}(\delta_{t})\frac{\sigma^{{{K}^{+}}{{K}^{-}}}_{\rm
eff}(\delta_{t})}{\sigma^{{\mu^{+}}{\mu^{-}}}_{\rm eff}(\delta_{t})},$ (6)

where $\sigma^{{{K}^{+}}{{K}^{-}}}_{\rm eff}(\delta_{t})$ and
$\sigma^{{\mu^{+}}{\mu^{-}}}_{\rm eff}(\delta_{t})$ are the effective
resolution widths for the simulated
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ and
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ decays,
respectively.

For the per-candidate method, the calibrated resolution in Eq. 6 is applied to
each candidate in the fit to the ${B}^{0}_{s}$ $\rightarrow$ ${K}^{+}$
${K}^{-}$ decay-time spectrum.333A calibrated per-candidate resolution is not
required for ${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays as the
${B}^{0}$ oscillation is significantly slower than that of the ${B}^{0}_{s}$
meson. For the simultaneous method, the decay-time resolution is not used on a
per-candidate basis, but an average model is used instead. The consequence of
using the average model is a small loss in the statistical precision for
${C}_{{K}{K}}$ and ${S}_{{K}{K}}$, corresponding to a relative 1% difference
on the final uncertainties, while the effect on the other $C\\!P$-violation
parameters is negligible. The loss is compensated by a significant
simplification of the fit model, as will be discussed in detail in Sec. 6.3.
To obtain the average resolution, $\sigma_{t}(\delta_{t})$ in Eq. (6) is
integrated over the distribution of $\delta_{t}$ from background-subtracted
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays, and an averaged
resolution of $\hat{\sigma}_{t}=42.9\pm 0.1\text{\,fs}$ is obtained. A
dependence of the resolution on the decaying particle mass is found when
repeating the procedure using a sample of ${\mathchar
28935\relax{(1S)}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ decays instead of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ sample,
which yields $\hat{\sigma}_{t}=44.1\pm 0.1\text{\,fs}$. The average between
the two calibrations, $\hat{\sigma}_{t}=43.5\text{\,fs}$, is used in the fit
to data with the simultaneous method, and the difference between them is
considered in the determination of the related systematic uncertainty.

In the fit to the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ data
sample, an offset of the mean of the triple-Gaussian model is observed and
attributed to a misalignment in the vertex detector. The size of the bias,
$\mu_{t}=-6.5\text{\,fs}$, is used as mean value in the resolution model in
both fit methods.

Figure 1: Dependence of the effective decay-time resolution, $\sigma_{\rm
eff}$, on the estimated decay-time uncertainty, $\delta_{t}$, for the
background-subtracted data sample of
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ decays.
The result of a linear fit is superimposed.

##  6 Fitting methods

Two independent methods, called simultaneous and per-candidate, are used to
determine the $C\\!P$-violation parameters in the
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays, while the simultaneous
method also determines the direct $C\\!P$-asymmetries in
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$. A comparison of their
respective results serves as validation of the measurements. The common
aspects of the two methods are described in Sec. 6.1 and 6.2, while the
specific details of each one are discussed in Sec. 6.3 and 6.4.

###  6.1 Components of the fit models

For each component, the distributions of the final-state invariant mass, decay
time and flavour-tagging assignment with the associated mistag probability are
modelled for ${B}_{({s})}^{0}$ candidates. Signal components are
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays in the ${K}^{\pm}$
${\pi}^{\mp}$ samples, the ${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$
decay in the ${\pi}^{+}$ ${\pi}^{-}$ sample, and the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decay in the
${{K}^{+}}\\!{{K}^{-}}$ sample. In the ${\pi}^{+}$ ${\pi}^{-}$ and
${{K}^{+}}\\!{{K}^{-}}$ samples, a small contribution from
${{B}^{0}_{s}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays is present and must be
taken into account. Cross-feed, combinatorial and three-body background
contributions are described by the model. Apart from $B$-meson decays, the
only relevant source of cross-feed background is the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow p{{K}^{-}}$ decay with the proton
misidentified as a kaon in the ${{K}^{+}}\\!{{K}^{-}}$ sample. Considering the
PID efficiencies, the branching fractions and the relative hadronisation
probabilities [25], the contribution of this background component is expected
to be about 2.5% relative to the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decay and is included in the
fit. Components describing partially reconstructed three-body
${B}_{({s})}^{0}$-meson decays and combinatorial background candidates are
necessary in all of the three final states.

###  6.2 Decay-time model for two-body ${B}_{({s})}^{0}$ decays

The time-dependent decay rate of a flavour-specific ${B}\\!\rightarrow f$
decay and of its $C\\!P$ conjugate ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\\!\rightarrow\overline{f}$, as for the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays, is given by the
probability density function (PDF)

$\begin{split}T_{\rm FS}\left(t,\,\psi,\,\vec{\xi},\,\vec{\eta}\right)=&K_{\rm
FS}\left(1-\psi A_{{C\\!P}}\right)\left(1-\psi A_{\rm D}\right)\times\\\
&\left\\{\left[\left(1\\!-\\!A_{\rm P}\right)\\!\Omega_{\rm
sig}(t,\vec{\xi},\vec{\eta})\\!+\\!\left(1\\!+\\!A_{\rm
P}\right)\\!\overline{\Omega}_{\rm
sig}(t,\vec{\xi},\vec{\eta})\right]\\!H_{+}\left(t\right)\\!+\\!\right.\\\
&\left.\psi\\!\left[\left(1\\!-\\!A_{\rm P}\right)\\!\Omega_{\rm
sig}(t,\vec{\xi},\vec{\eta})\\!-\\!\left(1\\!+\\!A_{\rm
P}\right)\\!\overline{\Omega}_{\rm
sig}(t,\vec{\xi},\vec{\eta})\right]\\!H_{-}\left(t\right)\\!\right\\},\end{split}$
(7)

where $K_{\rm FS}$ is a normalisation factor and the discrete variable $\psi$
assumes the value $+1$ for the final state $f$ and $-1$ for the final state
$\overline{f}$. The functions $H_{\pm}$, $\Omega_{\rm sig}$ and
$\overline{\Omega}_{\rm sig}$ are defined below. The direct $C\\!P$ asymmetry,
$A_{{C\\!P}}$, is defined in Eq. (4), while the final-state detection
asymmetry, $A_{\rm D}$, and the ${B}_{({s})}^{0}$-meson production asymmetry,
$A_{\rm P}$, are defined as

$A_{\rm D}=\frac{\varepsilon_{\rm
tot}\left(\overline{f}\right)-\varepsilon_{\rm
tot}\left(f\right)}{\varepsilon_{\rm
tot}\left(\overline{f}\right)+\varepsilon_{\rm tot}\left(f\right)},\qquad
A_{\rm P}=\frac{\sigma_{{{\kern
1.25995pt\overline{\kern-1.25995ptB}}{}_{({s})}^{0}}}-\sigma_{{{B}_{({s})}^{0}}}}{\sigma_{{{\kern
1.25995pt\overline{\kern-1.25995ptB}}{}_{({s})}^{0}}}+\sigma_{{{B}_{({s})}^{0}}}},$
(8)

where $\varepsilon_{\rm tot}$ is the time-integrated efficiency in
reconstructing and selecting the final state $f$ or $\overline{f}$, and
$\sigma_{{{B}_{({s})}^{0}}}$ ($\sigma_{{{\kern
1.25995pt\overline{\kern-1.25995ptB}}{}_{({s})}^{0}}}$) is the production
cross-section of the given ${B}_{({s})}^{0}$ (${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}_{({s})}^{0}$) meson. The asymmetry
$A_{\rm P}$ arises because production rates of ${B}_{({s})}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}_{({s})}^{0}$ mesons are not identical
in $p$ $p$ collisions. It is measured to be of the order of one percent at LHC
energies [61]. From the time-dependent fit it is possible to determine
simultaneously $A_{\rm P}$ and the sum $A_{{C\\!P}}+A_{\rm D}$. The
contribution of $A_{\rm D}$ is subtracted a posteriori as described in Sec. 7.

The variable $\vec{\xi}=\left(\xi_{\rm OS},\,\xi_{\rm SS}\right)$ is the pair
of flavour-tagging assignments of the OS and SS algorithms used to identify
the ${B}_{({s})}^{0}$-meson flavour at production, and
$\vec{\eta}=\left(\eta_{\rm OS},\,\eta_{\rm SS}\right)$ is the pair of
associated mistag probabilities defined in Sec. 4. The functions $\Omega_{\rm
sig}(t,\vec{\xi},\vec{\eta})$ and $\overline{\Omega}_{\rm
sig}(t,\vec{\xi},\vec{\eta})$ describe how the flavour tagging modifies the
time-dependent decay rate. The functions $H_{+}\left(t\right)$ and
$H_{-}\left(t\right)$ are defined as

$\displaystyle H_{+}\left(t\right)$ $\displaystyle=$
$\displaystyle\left[e^{-\Gamma_{{d}({s})}t^{\prime}}\cosh{\left(\frac{\Delta\Gamma_{{d}({s})}}{2}t^{\prime}\right)}\right]\otimes
R\left(t-t^{\prime}\right),$ (9) $\displaystyle H_{-}\left(t\right)$
$\displaystyle=$
$\displaystyle\left[e^{-\Gamma_{{d}({s})}t^{\prime}}\cos{\left(\Delta
m_{{d}({s})}t^{\prime}\right)}\right]\otimes R\left(t-t^{\prime}\right),$

where $\Gamma_{d}$ and $\Gamma_{s}$ are the ${B}^{0}$ and ${B}^{0}_{s}$ mean
decay widths, respectively, $R\left(t-t^{\prime}\right)$ is the decay-time
resolution model described in Sec. 5 and $\otimes$ denotes the convolution
product.

In the case of a decay to a $C\\!P$ eigenstate $f$, as it is for the
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays, the decay-time PDF is
given by

$\begin{split}T_{{C\\!P}}\left(t,\,\vec{\xi},\vec{\eta}\right)=K_{{C\\!P}}&\left\\{\left[\left(1-A_{\rm
P}\right)\Omega_{\rm sig}\left(t,\vec{\xi},\vec{\eta}\right)+\left(1+A_{\rm
P}\right)\overline{\Omega}_{\rm
sig}\left(t,\vec{\xi},\vec{\eta}\right)\right]I_{+}\left(t\right)+\right.\\\
&\left.~{}\left[\left(1-A_{\rm P}\right)\Omega_{\rm
sig}\left(t,\vec{\xi},\vec{\eta}\right)-\left(1+A_{\rm
P}\right)\overline{\Omega}_{\rm
sig}\left(t,\vec{\xi},\vec{\eta}\right)\right]I_{-}\left(t\right)\right\\},\end{split}$
(10)

where $K_{{C\\!P}}$ is a normalisation factor and the functions
$I_{+}\left(t\right)$ and $I_{-}\left(t\right)$ are

$\displaystyle I_{+}\left(t\right)$ $\displaystyle=$
$\displaystyle\left\\{e^{-\Gamma_{{d}({s})}t^{\prime}}\left[\cosh{\left(\frac{\Delta\Gamma_{{d}({s})}}{2}t^{\prime}\right)}+A_{f}^{\Delta\Gamma}\sinh{\left(\frac{\Delta\Gamma_{{d}({s})}}{2}t^{\prime}\right)}\right]\right\\}\otimes
R\left(t-t^{\prime}\right),$ (11) $\displaystyle I_{-}\left(t\right)$
$\displaystyle=$
$\displaystyle\left\\{e^{-\Gamma_{{s}({s})}t^{\prime}}\left[C_{f}\cos{\left(\Delta
m_{{d}({s})}t^{\prime}\right)}-S_{f}\sin{\left(\Delta
m_{{d}({s})}t^{\prime}\right)}\right]\right\\}\otimes
R\left(t-t^{\prime}\right).$

In this case $f$ is equal to $\overline{f}$, hence the final-state detection
asymmetry $A_{\rm D}$ is zero. The parameters $\Delta m_{{d}({s})}$,
$\Gamma_{{d}({s})}$, and $\Delta\Gamma_{{d}({s})}$ are fixed in the fit to
data to the values reported in Table 1.

Table 1:  Values of the parameters $\Delta m_{{d}}$, $\Delta m_{{s}}$, $\Gamma_{{d}}$ [25], $\Gamma_{{s}}$ and $\Delta\Gamma_{{s}}$ [60] used in the two methods. For $\Gamma_{{s}}$ and $\Delta\Gamma_{{s}}$ the correlation factor, $\rho$, between the two quantities is also reported. The decay width difference $\Delta\Gamma_{{d}}$ is fixed to zero. Parameter | Value
---|---
$\Delta m_{{d}}$ | $\phantom{0}0.5065\pm 0.0019\text{\,ps}^{-1}$
$\Gamma_{{d}}$ | $\phantom{0}0.6579\pm 0.0017\text{\,ps}^{-1}$
$\Delta\Gamma_{{d}}$ | $\hskip 5.69046pt0\text{\,ps}^{-1}$
$\Delta m_{{s}}$ | $17.757\phantom{0}\pm 0.021\text{\,ps}^{-1}\phantom{0}$
$\Gamma_{{s}}$ | $\phantom{0}0.6562\pm 0.0021\text{\,ps}^{-1}$
$\Delta\Gamma_{{s}}$ | $\phantom{0}0.082\pm 0.005\text{\,ps}^{-1}$
$\rho({\Gamma_{{s}}},{\Delta\Gamma_{{s}}})$ | $-0.170$

###  6.3 Simultaneous fit method

The simultaneous method relies on a concurrent fit to all the final-state
samples (${\pi}^{+}$ ${\pi}^{-}$, ${K}^{+}$ ${K}^{-}$ and ${K}^{\pm}$
${\pi}^{\mp}$), modelling the multidimensional space defined by the final-
state invariant mass, ${B}_{({s})}^{0}$ decay time, flavour-tagging decision
and associated mistag probability for the signal and background components.
The models used in the fit are a modification of those described in Ref. [24].

The model describing the invariant-mass shape of the signal components
comprises a sum of two Gaussian functions and a Johnson function [62], while
the model for cross-feed background is based on a kernel estimation (KDE)
method [63] and tuned with simulated decays. The normalisation of each cross-
feed background component is determined by rescaling the yields of the decay
reconstructed with the correct mass hypothesis by the ratio between the
misidentification probability and the PID efficiency for the wrong and correct
mass hypotheses.

The decay-time model of the signal components is also used for the cross-feed
background components originating from the signal decays reconstructed with
the wrong mass hypothesis. This is valid under the assumption that the decay-
time calculated under the wrong mass hypothesis is equal to that calculated
using the correct hypothesis, and is verified using samples of simulated
decays. The flavour-tagging assignments and related mistag probabilities for
OS and SS taggers enter the time-dependent decay rates of Eqs. (7) and (10)
through the functions $\Omega_{\rm sig}(t,\vec{\xi},\vec{\eta})$ and
$\overline{\Omega}_{\rm sig}(t,\vec{\xi},\vec{\eta})$. These functions are the
same as already used in Ref. [24] with the only difference being that they now
depend on the decay time, as do the efficiencies of the SS taggers. This
dependence is accommodated using separate efficiencies: one independent of the
SS-tagger decision and one specific for the candidates tagged by the SS
taggers. More details are reported in App. A.

The decay-time efficiency, $\varepsilon_{\rm sig}\left(t\right)$, is sculpted
by the selection criteria presented in Sec. 3. It is parameterised using an
empirical function determined using the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ calibration decay, whose time-
dependent decay rate is independent of the flavour-tagging decision and
described by an exponential distribution with ${\Gamma_{{d}}}=0.6588\pm
0.0017\text{\,ps}^{-1}$ [25]. A sample of background-subtracted
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ candidates is obtained from the
${K}^{\pm}$ ${\pi}^{\mp}$ sample in the invariant-mass window
$5.23<m({{K}^{\pm}}{{\pi}^{\mp}})<5.32~{}\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$.
The contributions of the combinatorial background, the only non-negligible
background in this region, is subtracted by injecting, with negative weights,
candidates from the sideband
$m({{K}^{\pm}}{{\pi}^{\mp}})>5.6\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. As
explained above, the procedure is repeated for the subsample with $\xi_{\rm
SS}\neq 0$, in order to model the time dependence of the SS-tagging
efficiency. For the ${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays, a small correction is
applied to the efficiency in order to take into account the differences
between signal and calibration modes. The correction for a given mode is a
product of the efficiency determined from the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ data and the ratio between the
efficiencies of this mode and of the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decay, as determined from
simulation.

The final difference with respect to the model used in Ref. [24] is that the
decay-time resolution is no longer modelled on a per-candidate basis. This
change is made since a correlation between the distributions of the decay-time
and decay-time error is observed for the combinatorial background candidates.
A full description of this correlation would imply a considerable complication
of the fitting model that outweighs the small loss in statistical power that
the use of an average decay-time resolution implies. A systematic uncertainty
is established in order to cover for possible biases coming from using an
average rather than per-candidate decay-time resolution.

The invariant-mass model for the combinatorial background components for each
decays is an exponential function, with its slope depending on the decay time,
in order to take into account a slight correlation between invariant mass and
decay time observed in the high-mass sideband. The time dependence of the
slope is studied using a two-dimensional unbinned maximum-likelihood fit to
the invariant mass and decay time of the sample in the high-mass sideband
above 5.6$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$, where only combinatorial
background candidates contribute. The obtained time-dependent mass slope is
used for the combinatorial background model in the entire invariant-mass
window, going from 5.0 to 6.2$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. The
relative normalisation of each candidate in the sideband is scaled to
reproduce that in the total invariant-mass window. A KDE method is applied to
the weighted candidates and the output is used to model the decay-time shape
of the combinatorial-background component. A dependence of the decay-time
shape of combinatorial background candidates on the tagging assignment of the
OS- and SS-taggers is also observed. Hence the time dependence of the mass
slope is studied separately for the subsamples corresponding to the tagging
decision $(|\xi_{\rm OS}|,\,|\xi_{\rm
SS}|)=\left\\{(1,1),(1,0),(0,1),(0,0)\right\\}$. Different weights are
determined for each subsample, and also the KDE method is applied separately
to each of them. The weighting procedure is the same as employed for the
background subtraction used to study the decay-time efficiency for
${B}_{({s})}^{0}$ decays. The functions taking into account the flavour-
tagging assignment and mistag probabilities are the same used in Ref [24], but
are generalised to consider all the possible combinations of $(|\xi_{\rm
OS}|,\,|\xi_{\rm SS}|)$. Finally, in the case of the ${K}^{\pm}$ ${\pi}^{\mp}$
samples, possible asymmetries in the flavour-tagging or reconstruction
efficiencies for the two charge-conjugate final states are taken into account.

The invariant-mass model of partially reconstructed ${B}_{({s})}^{0}$ decays
is the same as that used in Ref. [24], comprising the sum of two Gaussian
functions, which are defined using the same parameters as in the signal model
and are convolved with ARGUS functions [64]. For the ${K}^{\pm}$ ${\pi}^{\mp}$
sample two three-body background components are used: one describing three-
body ${B}^{0}$ and ${{B}^{+}}$ decays and another describing three-body
${B}^{0}_{s}$ decays. For the ${\pi}^{+}$ ${\pi}^{-}$ and
${{K}^{+}}\\!{{K}^{-}}$ samples a single ARGUS component is found to be
sufficient to describe the invariant-mass shape in the low-mass region. The
shape of the decay-time distribution is obtained by applying a KDE method to
the candidates in the low-mass sideband below
$5.2\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$, after subtracting the combinatorial
background contribution, as explained above. This is repeated separately for
the candidates with $|\xi_{\rm SS}|=0$ and $|\xi_{\rm SS}|\neq 0$, since a
difference in the decay-time shape is observed in data for the two subsamples.
The functions used to take into account the flavour-tagging information are
the same as used for the combinatorial background model, but with independent
parameters. Also for this component possible differences in flavour-tagging
and reconstruction efficiencies between the ${K}^{+}$ ${\pi}^{-}$ and
${\pi}^{+}$ ${K}^{-}$ final states are taken into account in the same way as
used for the combinatorial background model.

###  6.4 Per-candidate fit method

The per-candidate method relies on independent fits to the ${\pi}^{+}$
${\pi}^{-}$ and ${K}^{+}$ ${K}^{-}$ samples with all background components
statistically subtracted using the sFit technique [59, 65] with the
${\pi}^{+}$ ${\pi}^{-}$ and ${K}^{+}$ ${K}^{-}$ invariant mass as the
discriminating variable. Hence only the decay-time distributions are modelled
for the signal modes ${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$.

The invariant-mass distributions of the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ and
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ signal components are modelled
with the sum of two Crystal Ball functions [66] where the tail parameters are
fixed to the values obtained from the simulation. The mean and width of the
Gaussian core are allowed to vary in the fit for the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ and
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ signal modes, while these
parameters are constrained for the ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$
and ${{B}^{0}_{s}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ signal components
using the known mass difference between ${B}^{0}$ and ${B}^{0}_{s}$ and the
ratio of resolutions obtained from simulations, respectively. The decay-time
model for the signal components is the function described in Sect. 6.2,
multiplied on a per-candidate basis with the acceptance functions described
below.

The invariant-mass distributions of the misidentified background candidates
from other two-body ${B}_{({s})}^{0}$ decays are modelled with templates from
simulations and their yields are constrained using efficiencies measured in
data calibration samples. The three-body background components, which are the
same as in the simultaneous method, are modelled using an exponentially
modified Gaussian PDF.

The decay-time resolution consists of a single Gaussian function with its
width varying candidate by candidate, depending on the decay-time error
$\delta_{t}$ for each candidate and calibrated as presented in Sec. 5. The
per-candidate acceptance function is determined with the swimming method [34,
35, 36, 37] by artificially changing the decay time of the ${B}_{({s})}^{0}$
meson and re-evaluating whether the candidate would have been accepted by the
selection requirements that are known to bias the decay-time measurement. The
decay time is changed by moving the position of every PV in the event along
the direction of the ${B}_{({s})}^{0}$ momentum vector. For decay times for
which the candidate is accepted the efficiency is 1, otherwise the efficiency
is 0. By scanning a range of hypothetical decay times, a series of top-hat
functions are constructed for each candidate as it changes from being
rejected, to being accepted, finally to being rejected again.444A series of
top hat functions are produced as each event can have more than one primary
vertex. The procedure is re-evaluated in steps of 50$\,\upmu\text{m}$ along
the ${B}_{({s})}^{0}$ momentum vector and, when the selection decision
changes, the position at which this change occurs is determined with a finer
granularity, giving an overall resolution of 0.5$\,\upmu\text{m}$ on the
decay-time efficiency. The effective lifetime measured on the fully simulated
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ events, assuming an
exponential decay-time model and using only the swimming-based efficiency for
this simulation, is found to be 1.416  ps. Compared to a generated effective
lifetime of 1.394  ps it exhibits a bias of $1.5\%$. This arises from effects
that are not fully modelled in the swimming method and can result in an
incorrect measurement of the parameter ${\cal A}^{\Delta\Gamma}_{{K}{K}}$, for
which high precision is expected. To correct for this, an additional decay-
time efficiency weight is applied by comparing the decay-time efficiency
extracted using the swimming method for the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ data with the decay-time
efficiency determined from the ratio of background-subtracted
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ events and the unbiased decay-
time PDF. The unbiased decay-time PDF consists of an exponential function,
whose decay time is fixed to the known ${B}^{0}$ lifetime, convolved with a
Gaussian function to account for the intrinsic decay-time resolution. The
width of the Gaussian is fixed to the effective decay-time resolution as
detailed in Sec. 5. The ratio of these efficiencies is modelled with an
empirical function

$f(t)=p_{0}(1+\tanh[p_{1}(t-p_{2})])+p_{3}t.$ (12)

where $t$ is the decay time of the candidate and $p_{\\{0,1,2,3\\}}$ are free
parameters measured in the fit to the ratio. Applying this weight to the
swimming-based efficiency allows to correctly recover the effective lifetime
of the simulated ${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays and
the mean lifetime of ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays
extracted from the ${K}^{\pm}$ ${\pi}^{\mp}$ samples.

##  7 Detection asymmetry between ${K}^{-}$ ${\pi}^{+}$ and ${K}^{+}$
${\pi}^{-}$ final states

In order to extract the $C\\!P$ asymmetries $A_{{C\\!P}}^{{{B}^{0}}}$ and
$A_{{C\\!P}}^{{{B}^{0}_{s}}}$ from the asymmetries measured through the
simultaneous fit, an estimation of the nuisance experimental detection
asymmetry is required as indicated in Eq. 7. This asymmetry is a consequence
of the different efficiency for selecting the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays and their charge-
conjugate final states. To an excellent approximation, it can be expressed as
the sum of two contributions

$A_{\rm D}=A^{{K}{\pi}}_{\rm det}+A^{{K}{\pi}}_{\rm PID},$ (13)

where $A^{{K}{\pi}}_{\rm det}$ is the asymmetry between the selection
efficiencies without the application of the PID requirements and
$A^{{K}{\pi}}_{\rm PID}$ is the asymmetry between the efficiencies of the PID
requirements selecting the two final states. The convention used in the
following to determine $A^{{K}{\pi}}_{\rm det}$ and $A^{{K}{\pi}}_{\rm PID}$
is such that a positive value of the asymmetry means a larger efficiency for
the ${K}^{-}$ ${\pi}^{+}$ pair with respect to the ${K}^{+}$ ${\pi}^{-}$ pair.
As a consequence of this convention, the values reported below for the
${B}^{0}$ and ${B}^{0}_{s}$ asymmetries must be used with an inverted sign for
the ${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay.

The final-state detection asymmetry is determined using
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and
${{D}^{+}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{0}}{{\pi}^{+}}$ control modes, with
the neutral kaon decaying to ${\pi}^{+}$ ${\pi}^{-}$, following the strategy
used in Ref. [24]. Assuming negligible $C\\!P$ violation in these Cabibbo-
favoured $D$-meson decays, the raw asymmetries between the measured yields of
${D}^{+}$ and ${D}^{-}$ decays can be written as

$\displaystyle A_{\rm RAW}^{{K}{\pi}{\pi}}$ $\displaystyle=$ $\displaystyle
A_{\rm P}^{{{D}^{+}}}+A_{\rm det}^{{K}{\pi}}+A_{\rm det}^{{\pi}},$ (14)
$\displaystyle A_{\rm RAW}^{{{\kern
1.25995pt\overline{\kern-1.25995ptK}}{}^{0}}{\pi}}$ $\displaystyle=$
$\displaystyle A_{\rm P}^{{{D}^{+}}}+A_{\rm det}^{{\pi}}-A_{\rm
det}^{{{K}^{0}}},$ (15)

where $A_{\rm P}^{{{D}^{+}}}$ is the asymmetry between the production cross-
sections of ${D}^{+}$ and ${D}^{-}$ mesons, and $A^{{\pi}}_{\rm det}$
($A^{{K}^{0}}_{\rm det}$) is the asymmetry between the detection efficiencies
of ${\pi}^{+}$ (${K}^{0}$) and ${\pi}^{-}$ (${\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{0}$) mesons. The difference between
Eqs. (14) and (15) leads to

$A_{\rm det}^{{K}{\pi}}=A_{\rm RAW}^{{K}{\pi}{\pi}}-A_{\rm RAW}^{{{\kern
1.25995pt\overline{\kern-1.25995ptK}}{}^{0}}{\pi}}-A_{\rm det}^{{{K}^{0}}}.$
(16)

The asymmetry $A_{\rm det}^{{{K}^{0}}}$ includes the effects from the kaon
mixing and $C\\!P$ violation, and was estimated to be $\left(0.054\pm
0.014\right)\%$ [67]. The asymmetries $A_{\rm P}^{{{D}^{+}}}$ and $A_{\rm
det}^{{\pi}}$ can depend on the kinematics of the ${D}^{+}$ and ${\pi}^{+}$
mesons. To obtain a better cancellation of these nuisance asymmetries in Eq.
(16), the momentum and the transverse momentum of the ${D}^{+}$ and
${\pi}^{+}$ mesons from the
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ sample are
simultaneously weighted to match the corresponding distributions in the
${{D}^{+}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{0}}{{\pi}^{+}}$ sample. The $A_{\rm
det}^{{K}{\pi}}$ is determined in intervals of the kaon momentum, to account
for the kinematic-dependent variation of the interaction cross-sections of
positive and negative kaons with the detector material. This binned asymmetry
is averaged over the momentum distribution of the kaon in the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays, giving no difference
between the absolute values of the corrections for the two modes.

Figure 2: Distributions of (top left) ${K}^{\pm}$ ${\pi}^{\mp}$ invariant
mass, (top right) ${B}_{({s})}^{0}$ decay time, mistag fractions (bottom left)
$\eta_{\rm OS}$ and (bottom right) $\eta_{\rm SSc}$ for ${K}^{\pm}$
${\pi}^{\mp}$ candidates. The result of the simultaneous fit is overlaid. The
various components contributing to the fit model are drawn as stacked
histograms.

The final-state detection asymmetry values for the 2015 and 2016 data samples
are

$\displaystyle A_{\rm det}^{{K}{\pi}}(2015)$ $\displaystyle=$
$\displaystyle\left(-0.96\pm 0.32\right)\%,$ (17) $\displaystyle A_{\rm
det}^{{K}{\pi}}(2016)$ $\displaystyle=$ $\displaystyle\left(-1.05\pm
0.13\right)\%.$

The asymmetry between the PID efficiencies is computed in intervals of
momentum, pseudorapidity and azimuthal angle of the two final-state particles,
using the ${{D}^{*+}}\\!\rightarrow{{D}^{0}}({{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$
calibration samples, as discussed in Sec. 3. The computation is repeated using
several binning schemes, and then the average and standard deviation of the
PID asymmetries determined in each scheme are used as the central value and
associated uncertainty for $A^{{K}{\pi}}_{\rm PID}$, respectively.

Figure 3: Distributions of (top left) ${\pi}^{+}$ ${\pi}^{-}$ invariant mass,
(top right) ${B}_{({s})}^{0}$ decay time, mistag fractions (bottom left)
$\eta_{\rm OS}$ and (bottom) $\eta_{\rm SSc}$ for ${\pi}^{+}$ ${\pi}^{-}$
candidates. The result of the simultaneous fit is overlaid. The various
components contributing to the fit model are drawn as stacked histograms.

The PID asymmetry is calculated taking into account the differences in the
running conditions of the two years of data taking and the numerical results
are:

$\displaystyle A^{{K}{\pi}}_{\rm PID}\left(2015\right)$ $\displaystyle=$
$\displaystyle\left(-1.2\pm 0.7\right)\%,$ (18) $\displaystyle
A^{{K}{\pi}}_{\rm PID}\left(2016\right)$ $\displaystyle=$
$\displaystyle\left(0.5\pm 0.3\right)\%.$

##  8 Fit results

The results obtained from unbinned maximum likelihood fits to data of the
models described in Secs. 6 are presented in the following. Their comparison
is also discussed.

###  8.1 Simultaneous method

Figure 4: Distributions of (top left) ${{K}^{+}}\\!{{K}^{-}}$ invariant mass,
(top right) ${B}_{({s})}^{0}$ decay time, mistag fractions (bottom left)
$\eta_{\rm OS}$ and (bottom right) $\eta_{{\rm SS}{K}}$ for
${{K}^{+}}\\!{{K}^{-}}$ candidates. The result of the simultaneous fit is
overlaid. The various components contributing to the fit model are drawn as
stacked histograms.

The simultaneous fit to the final-state invariant mass, the ${B}_{({s})}^{0}$
decay time, and the tagging decisions and their associated mistag
probabilities of the ${\pi}^{+}$ ${\pi}^{-}$, ${{K}^{+}}\\!{{K}^{-}}$ and
${K}^{\pm}$ ${\pi}^{\mp}$ samples determines the coefficients
${C}_{{\pi}{\pi}}$, ${S}_{{\pi}{\pi}}$, ${C}_{{K}{K}}$, ${S}_{{K}{K}}$, ${\cal
A}^{\Delta\Gamma}_{{K}{K}}$ and the $C\\!P$ asymmetries
$A_{{C\\!P}}^{{{B}^{0}}}$ and $A_{{C\\!P}}^{{{B}^{0}_{s}}}$. The signal yields
are $N({{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}})=45\hskip 1.42262pt620\pm
260$, $N({{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}})=70\hskip
1.42262pt310\pm 320$,
$N({{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}})=140\hskip 1.42262pt340\pm 420$
and $N({{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}})=10\hskip
1.42262pt580\pm 150$, where uncertainties are statistical only.

Figure 5: Time-dependent asymmetries for ${K}^{\pm}$ ${\pi}^{\mp}$ candidates
with
$5.20<m({{K}^{\pm}}{{\pi}^{\mp}})<5.32\text{\,Ge\kern-0.90005ptV\\!/}c^{2}$:
(left) using the OS-tagging decision and (right) the SSc-tagging decision. The
result of the simultaneous fit is overlaid.

The distributions of the mass and decay time of the selected candidates are
shown in Figs. 2, 3 and 4, for the ${K}^{\pm}$ ${\pi}^{\mp}$, ${\pi}^{+}$
${\pi}^{-}$ and ${K}^{+}$ ${K}^{-}$ samples, respectively. The time-dependent
asymmetries, obtained separately by using the OS or the SS tagging decisions,
for the ${B}_{({s})}^{0}$ candidates in the region
$5.20<m({{K}^{\pm}}{{\pi}^{\mp}})<5.32\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$,
dominated by the ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decay, are
shown in Fig. 5.

Figure 6: Time-dependent asymmetries for (top) ${\pi}^{+}$ ${\pi}^{-}$ and
(bottom) ${{K}^{+}}\\!{{K}^{-}}$ candidates with
$5.20<m({{\pi}^{+}}{{\pi}^{-}})<5.35\text{\,Ge\kern-0.90005ptV\\!/}c^{2}$ and
$5.30<m({{K}^{+}}{{K}^{-}})<5.44\text{\,Ge\kern-0.90005ptV\\!/}c^{2}$,
respectively: (left) using the OS-tagging decision and (right) using either
the SSc-tagging decision (for the ${\pi}^{+}$ ${\pi}^{-}$ candidates) or the
SS$K$-tagging decision (for the ${{K}^{+}}\\!{{K}^{-}}$ candidates). The
result of the simultaneous fit is overlaid. The asymmetry for the
${{K}^{+}}\\!{{K}^{-}}$ candidates is folded into one mixing period
$2\pi/{\Delta m_{{s}}}$ and the parameter $t_{0}=0.2\text{\,ps}$ corresponds
to the minimum value of the decay-time used in the fit.

The production asymmetries for the ${B}^{0}$ and ${B}^{0}_{s}$ mesons are
determined to be $(-0.60\pm 0.49)\%$ and $(-1.2\pm 1.5)\%$, respectively,
where uncertainties are statistical only. They are consistent with the
expectations from Ref. [61]. The time-dependent asymmetries for the
${\pi}^{+}$ ${\pi}^{-}$ candidates with
$5.20<m({{\pi}^{+}}{{\pi}^{-}})<5.35\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$, and
for the ${{K}^{+}}\\!{{K}^{-}}$ candidates with
$5.30<m({{K}^{+}}{{K}^{-}})<5.45\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$,
dominated by the corresponding ${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$
and ${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ signal components, are
shown in Fig. 6, again separately for the OS and SS tagging decision. The
effective tagging powers for the
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays are $\left(4.5\pm
0.2\right)\%$ and $\left(5.1\pm 0.2\right)\%$, respectively. The results for
the $C\\!P$-violating quantities are

$\displaystyle{C}_{{\pi}{\pi}}$ $\displaystyle=$
$\displaystyle-0.311\phantom{0}\pm 0.045,$ $\displaystyle{S}_{{\pi}{\pi}}$
$\displaystyle=$ $\displaystyle-0.706\phantom{0}\pm 0.042,$ $\displaystyle
A_{{C\\!P}}^{{{B}^{0}}}$ $\displaystyle=$ $\displaystyle-0.0824\pm 0.0033,$
$\displaystyle A_{{C\\!P}}^{{{B}^{0}_{s}}}$ $\displaystyle=$
$\displaystyle\phantom{-}0.236\phantom{0}\pm 0.013,$ (19)
$\displaystyle{C}_{{K}{K}}$ $\displaystyle=$
$\displaystyle\phantom{-}0.164\phantom{0}\pm 0.034,$
$\displaystyle{S}_{{K}{K}}$ $\displaystyle=$
$\displaystyle\phantom{-}0.123\phantom{0}\pm 0.034,$ $\displaystyle{\cal
A}^{\Delta\Gamma}_{{K}{K}}$ $\displaystyle=$
$\displaystyle-0.833\phantom{0}\pm 0.054,$

where the uncertainties are statistical, and the central values of
$A_{{C\\!P}}^{{{B}^{0}}}$ and $A_{{C\\!P}}^{{{B}^{0}_{s}}}$ are corrected for
the ${K}^{+}$ ${\pi}^{-}$ detection and PID asymmetry.

###  8.2 Per-candidate method

The signal yields in the ${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays, used to determine the
$C\\!P$-violating parameters with the per-candidate method, are in agreement
with those of the simultaneous method. The parameters $\Delta m_{{d}({s})}$,
$\Gamma_{{d}({s})}$, and $\Delta\Gamma_{{d}({s})}$ are fixed to the values
reported in Table 1. The value of the production asymmetry is fixed to that
measured by the simultaneous method. The fits to the ${\pi}^{+}$ ${\pi}^{-}$
and ${K}^{+}$ ${K}^{-}$ invariant-mass spectra are shown in Figs. 7 and 8
along with the decay-time fits to the ${B}_{({s})}^{0}$ mesons having their
flavours tagged.

The results for the $C\\!P$-violating parameters using the per-candidate
method are

$\displaystyle{C}_{{\pi}{\pi}}$ $\displaystyle=$ $\displaystyle-0.338\pm
0.048,$ $\displaystyle{S}_{{\pi}{\pi}}$ $\displaystyle=$
$\displaystyle-0.673\pm 0.043,$ $\displaystyle{C}_{{K}{K}}$ $\displaystyle=$
$\displaystyle\phantom{-}0.173\pm 0.042,$ $\displaystyle{S}_{{K}{K}}$
$\displaystyle=$ $\displaystyle\phantom{-}0.166\pm 0.042,$ $\displaystyle{\cal
A}^{\Delta\Gamma}_{{K}{K}}$ $\displaystyle=$ $\displaystyle-0.973\pm 0.071,$

where the uncertainties are statistical only.

Figure 7: Distributions of the (top left) ${\pi}^{+}$ ${\pi}^{-}$ invariant
mass, (top right) decay time for tagged ${B}^{0}$ mesons, (bottom left) decay
time for untagged ${B}^{0}$ mesons and (bottom right) asymmetry for the
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays. The individual
components are shown for the invariant-mass spectrum while only tagged
background-subtracted candidates are shown in the decay-time spectrum. The fit
results to the different distributions are overlaid. The various components
contributing to the invariant mass model are drawn as stacked histograms.

Figure 8: Distributions of the ${K}^{+}$ ${K}^{-}$ invariant mass (top left),
decay time for tagged ${B}^{0}_{s}$ mesons (top right), decay time for
untagged ${B}^{0}_{s}$ candidates (bottom left) and asymmetry (bottom right)
for the ${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays. The individual
components are shown for the invariant mass spectrum while only background-
subtracted candidates are shown in the decay time spectrum. The fit results to
the different distributions are overlaid. The various components contributing
to the invariant mass model are drawn as stacked histograms. The asymmetry for
the ${{K}^{+}}\\!{{K}^{-}}$ candidates is folded into one mixing period
$2\pi/{\Delta m_{{s}}}$ and the parameter $t_{0}=0.2\text{\,ps}$ corresponds
to the minimum value of the decay-time used in the fit.

###  8.3 Comparison

To evaluate the compatibility of the results from the two methods, their
statistical correlation is determined from 500 simulated pseudoexperiments.
The correlation is found to be approximately 84% for all $C\\!P$-violating
parameters. This is used to determine the uncorrelated statistical uncertainty
on the difference between the results of the two methods. The
pseudoexperiments also confirm the smaller total uncertainty observed by the
simultaneous method. A sizeable difference between the two results is observed
for ${\cal A}^{\Delta\Gamma}_{{K}{K}}$. This difference is reduced to
approximately 1.5 standard deviations when taking into account the systematic
uncertainties due to the determination of the decay-time efficiency (see Tab.
2), which are completely uncorrelated between the two methods. Adding in
quadrature the uncorrelated statistical and systematic uncertainties, the
results are found to be compatible within one standard deviation. The
resulting contour plots from measuring ${C}_{{\pi}{\pi}}$, ${S}_{{\pi}{\pi}}$,
${C}_{{K}{K}}$ and ${S}_{{K}{K}}$ are given in Fig. 9.

Given the large correlation between the two determinations, the values
obtained from the simultaneous method are quoted as the LHCb results. They are
chosen due to the slightly smaller total uncertainty and the fact that the
simultaneous method gives also the direct $C\\!P$ asymmetries allowing for a
complete combination with the results published in Ref. [24].

Figure 9: Two-dimensional 68% and 95% confidence-level regions for the
measured $C\\!P$-violating parameters of the
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ (left) and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ (right) decays from the two
methods. The simultaneous method is shown in purple while the per-candidate
method in green. Previous measurements of these parameters are also shown,
with the LHCb Run 1 result in blue, the Belle result in yellow and the BaBar
result in red. The confidence-level regions are calculated using only the
statistical uncertainties of all the measurements. The correlation is found to
be approximately 84% for all $C\\!P$-violating parameters between the
simultaneous and per-event methods.

##  9 Systematic uncertainties

The systematic uncertainties are evaluated for both the simultaneous and the
per-candidate methods, and the total systematic uncertainties for both results
are given in Table 2. A full description of the systematic uncertainties is
only given for the simultaneous method since it is used as the LHCb result and
for combination with the Run 1 measurement. Hence the description given in
this section and the breakdown of the individual components in Table 2 refers
to that method. The main differences in systematic uncertainties between the
two methods are briefly discussed at the end of this section.

The systematic uncertainties on the $C\\!P$-violating parameters are
determined following two approaches. In the first case the fit to data is
repeated a large number of times, each time modifying the values of the input
parameters. This approach is used to account for the knowledge of external
inputs whose values are fixed in the fit. In the second case,
pseudoexperiments are performed according to the default model and both the
default model and modified models are used to fit the generated data. This
strategy is used to account for the systematic uncertainties due to the
assumptions on the fitting model. In both cases the difference between the
default and alternative results for the $C\\!P$ asymmetries is measured, and
the mean and width of the obtained distribution is used to assign a systematic
uncertainty.

Three sources of systematic uncertainty are considered on the invariant-mass
model. First, the systematic uncertainty due to a possibly imperfect
description of the mass-resolution function, used for both signal and cross-
feed background components, is determined by replacing the double Gaussian
function with a single Gaussian model. Second, the systematic uncertainty
associated to the combinatorial background model is assessed using an
alternative model with no correlation between decay time and invariant mass.
Finally, a systematic uncertainty associated with the model adopted for the
three-body background components is determined by fitting a set of
pseudoexperiments, after removing the candidates with an invariant mass below
5.2$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ and ignoring the components
describing this background contributions in the model.

The PID efficiencies and misidentification probabilities govern the amount of
cross-feed background components. A systematic uncertainty related to their
calibration is determined by repeating the fit to data changing those values
according to their uncertainties estimated from the calibration samples.

The effect of ignoring the small fraction of ${B}^{0}_{s}$ candidates
originating from decays of the ${B}_{c}^{+}$ meson is studied by injecting
simulated ${{B}_{c}^{+}}\\!\rightarrow{{B}^{0}_{s}}X$ decays (where $X$ stands
for any additional particle in the final state) into the pseudoexperiments,
where the relative ${B}_{c}^{+}$ yield is determined from Ref. [54]. No
systematic uncertainty is assigned for the ${B}^{0}$ $C\\!P$-violating
parameters since the ${{B}_{c}^{+}}\\!\rightarrow{{B}^{0}}X$ decay is Cabibbo
suppressed.

Systematic uncertainties associated with the calibration of the OS and SSc
flavour-tagging responses are determined using an alternative relation between
$\eta_{\rm OS(SS)}$ and the calibrated mistag probability $\omega_{\rm
OS(SS)}$. The linear relation connecting the two quantities in Eq. (24) is
replaced with a second-order polynomial. A similar approach is also used for
the SS$K$ tagger, but the values of the parameters of the alternative
relations are first determined from the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ sample and then used in
the fit to data. For the SS$K$ tagger an additional systematic uncertainty
associated with the calibration of the flavour-tagging response is determined
by varying the calibration parameters according to their uncertainties and
correlations.

Regarding the decay-time model, a systematic uncertainty associated with the
uncertainties on the parameters reported in Table 1 is determined by repeating
the simultaneous fit using different fixed values, generated according to
their uncertainties and correlations. The systematic effect due to the decay-
time resolution can be decomposed into three contributions: one due to the
calibration of the resolution width, another one due to the calibration of the
bias in the determination of the decay time, and the last due to the usage of
an average decay-time resolution instead of a per-candidate value. The first
effect is estimated varying the value of the averaged decay-time resolution
width according to a Gaussian distribution with mean equal to the default
value, reported in Section 5, and with a width equal to the difference between
the decay-time resolution for the fully simulated
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays. The second effect is
determined varying the mean of the decay-time resolution model according to a
Gaussian centered at the default value and with the width of 2  fs. The last
contribution, due to the usage of an average decay-time resolution instead of
a per-candidate value, is evaluated by fitting a set of pseudoexperiments with
both the decay-time resolution models.

Three sources of systematic uncertainty related to the knowledge of the decay-
time efficiency are identified. A systematic uncertainty on the chosen model
is assessed by replacing the effective function with a cubic-spline polynomial
in an alternative model. Second, a systematic uncertainty arising from the
limited calibration-sample size is computed by varying the parameters
governing the decay-time acceptance according to their uncertainties and
correlations. An additional systematic uncertainty due to the imperfect
description of the ratios between the decay-time efficiency of the various
signal and calibration modes, determined from fully simulated samples, is
estimated. In this case, the alternative model is created assuming that all
the decay-time efficiencies are equal to that of the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decay.

To determine the systematic uncertainty associated with the choice of the
decay-time model for the cross-feed background component, an alternative model
is created by disabling the oscillating component of the cross-feed background
model. This means assuming no $C\\!P$ violation for both, the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ component in the ${\pi}^{+}$
${\pi}^{-}$ and ${{K}^{+}}{{K}^{-}}$ samples, and for the
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ components in the ${K}^{\pm}$
${\pi}^{\mp}$ sample. Finally, the uncertainties associated with the detection
and PID asymmetries reported in Eqs. (17) and (18) are accounted for as
systematic uncertainties on $A_{{C\\!P}}^{{{B}^{0}}}$ and
$A_{{C\\!P}}^{{{B}^{0}_{s}}}$.

The total systematic uncertainties are obtained as the quadratic sum of the
individual contributions, and they are smaller than the corresponding
statistical uncertainties for all the $C\\!P$-violating parameters apart from
${\cal A}^{\Delta\Gamma}_{{K}{K}}$. The dominating systematic uncertainty for
this parameter is related to the knowledge of how the efficiency varies with
the decay time, whose knowledge is limited by the size of the calibration
sample of ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays.

Most of the sources of systematic uncertainties related to the per-candidate
method are the same as those on the simultaneous method. The systematic
uncertainties are also similar in size. The main difference is a smaller
uncertainty related to the decay-time acceptance in the per-candidate method.
This uncertainty, which is uncorrelated between the two methods due to their
different strategies, mainly affects ${\cal A}^{\Delta\Gamma}_{{K}{K}}$ and
largely cancels in the other parameters. The second most important difference
is due to systematic uncertainties related to flavour tagging, where the
uncertainties are larger in the per-candidate method, which arises from the
different approaches of incorporating this information in the two fits. These
differences in systematic uncertainties illustrate the strength of validating
the result with two different methods.

Table 2: Systematic uncertainties on the $C\\!P$-violating parameters. The
values given for each individual contribution to the systematic uncertainty
are those for the simultaneous method. The total systematic uncertainties are
given both for the simultaneous and the per-candidate methods. The dash
indicates that the uncertainty is not applicable.

Source | ${C}_{{\pi}{\pi}}$ | ${S}_{{\pi}{\pi}}$ | $A_{{C\\!P}}^{{{B}^{0}}}$ | $A_{{C\\!P}}^{{{B}^{0}_{s}}}$ | ${C}_{{K}{K}}$ | ${S}_{{K}{K}}$ | ${\cal A}^{\Delta\Gamma}_{{K}{K}}$
---|---|---|---|---|---|---|---
Time acceptance |  |  |  |  |  |  |
| Model | $0.005$ | $0.003$ | $0.0005$ | $0.001$ | $0.003$ | $0.003$ | $0.045$
| Calibration channel | $0.003$ | $0.001$ | $0.0003$ | $0.006$ | $0.001$ | $0.001$ | $0.047$
| Ratios between modes | $0.004$ | $0.002$ | $0.0010$ | $0.000$ | $0.001$ | $0.001$ | $0.047$
Time resolution |  |  |  |  |  |  |
| Width | $0.002$ | $0.003$ | $0.0001$ | $0.000$ | $0.0009$ | $0.010$ | $0.000$
| Bias | $0.000$ | $0.000$ | $0.0000$ | $0.000$ | $0.004$ | $0.003$ | $0.000$
| Average | $0.000$ | $0.001$ | $0.0000$ | $0.000$ | $0.004$ | $0.004$ | $0.004$
Input parameters | $0.003$ | $0.002$ | $0.0001$ | $0.000$ | $0.006$ | $0.007$ | $0.047$
${B}^{0}_{s}$ from ${B}_{c}^{+}$ | $-$ | $-$ | $-$ | $-$ | $0.004$ | $0.003$ | $0.004$
Flavour tagging |  |  |  |  |  |  |
| Calibration model | $0.001$ | $0.001$ | $0.0000$ | $0.000$ | $0.004$ | $0.003$ | $0.001$
|  ${\rm SS}{K}$ calibration | $-$ | $-$ | $-$ | $-$ | $0.003$ | $0.004$ | $0.000$
PDF modeling |  |  |  |  |  |  |
| Signal mass | $0.007$ | $0.008$ | $0.0004$ | $0.007$ | $0.002$ | $0.002$ | $0.006$
| Cross-feed bkg. | $0.008$ | $0.004$ | $0.0001$ | $0.000$ | $0.001$ | $0.000$ | $0.002$
| Combinatorial bkg | $0.006$ | $0.003$ | $0.0001$ | $0.002$ | $0.001$ | $0.001$ | $0.006$
| 3-body bkg. | $0.004$ | $0.006$ | $0.0005$ | $0.004$ | $0.001$ | $0.001$ | $0.011$
PID in fit model | $0.002$ | $0.003$ | $0.0002$ | $0.002$ | $0.000$ | $0.001$ | $0.001$
PID asymmetry | $-$ | $-$ | $0.0028$ | $0.003$ | $-$ | $-$ | $-$
Det. asymmetry | $-$ | $-$ | $0.0012$ | $0.001$ | $-$ | $-$ | $-$
Total (simultaneous) | $0.015$ | $0.013$ | $0.0033$ | $0.011$ | $0.014$ | $0.015$ | $0.094$
Total (per-candidate) | $0.018$ | $0.016$ | $-$ | $-$ | $0.021$ | $0.012$ | $0.067$

##  10 Results

The final results for the time-dependent $C\\!P$ violation in
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays, and of the $C\\!P$
asymmetries in ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays are

$\displaystyle{C}_{{\pi}{\pi}}$ $\displaystyle=$
$\displaystyle-0.311\phantom{0}\pm 0.045\phantom{0}\pm 0.015,$
$\displaystyle{S}_{{\pi}{\pi}}$ $\displaystyle=$
$\displaystyle-0.706\phantom{0}\pm 0.042\phantom{0}\pm 0.013,$ $\displaystyle
A_{{C\\!P}}^{{{B}^{0}}}$ $\displaystyle=$ $\displaystyle-0.0824\pm 0.0033\pm
0.0033,$ $\displaystyle A_{{C\\!P}}^{{{B}^{0}_{s}}}$ $\displaystyle=$
$\displaystyle\phantom{-}0.236\phantom{0}\pm 0.013\phantom{0}\pm 0.011,$
$\displaystyle{C}_{{K}{K}}$ $\displaystyle=$
$\displaystyle\phantom{-}0.164\phantom{0}\pm 0.034\phantom{0}\pm 0.014,$
$\displaystyle{S}_{{K}{K}}$ $\displaystyle=$
$\displaystyle\phantom{-}0.123\phantom{0}\pm 0.034\phantom{0}\pm 0.015,$
$\displaystyle{\cal A}^{\Delta\Gamma}_{{K}{K}}$ $\displaystyle=$
$\displaystyle-0.83\phantom{00}\pm 0.05\phantom{00}\pm 0.09,$

where the first uncertainties are statistical and the second systematic. The
corresponding statistical and systematic correlation matrices are reported in
Tables 3 and 4, respectively. The results are compatible with the previous
LHCb measurement in Ref. [24].

Table 3: Correlations of statistical uncertainties among the $C\\!P$-violating parameters. | ${C}_{{\pi}{\pi}}$ | ${S}_{{\pi}{\pi}}$ | $A_{{C\\!P}}^{{{B}^{0}}}$ | $A_{{C\\!P}}^{{{B}^{0}_{s}}}$ | ${C}_{{K}{K}}$ | ${S}_{{K}{K}}$ |  ${\cal A}^{\Delta\Gamma}_{{K}{K}}$
---|---|---|---|---|---|---|---
${C}_{{\pi}{\pi}}$ | $1$ |  |  |  |  |  |
${S}_{{\pi}{\pi}}$ | $\phantom{-}0.394$ | $1$ |  |  |  |  |
$A_{{C\\!P}}^{{{B}^{0}}}$ | $-0.035$ | $\phantom{-}0.011$ | $1$ |  |  |  |
$A_{{C\\!P}}^{{{B}^{0}_{s}}}$ | $\phantom{-}0.000$ | $\phantom{-}0.000$ | $\phantom{-}0.052$ | $1$ |  |  |
${C}_{{K}{K}}$ | $-0.008$ | $-0.029$ | $\phantom{-}0.002$ | $\phantom{-}0.001$ | $1$ |  |
${S}_{{K}{K}}$ | $-0.008$ | $\phantom{-}0.005$ | $-0.006$ | $\phantom{-}0.001$ | $-0.010$ | $1$ |
${\cal A}^{\Delta\Gamma}_{{K}{K}}$ | $\phantom{-}0.000$ | $\phantom{-}0.000$ | $\phantom{-}0.001$ | $\phantom{-}0.000$ | $\phantom{-}0.025$ | $\phantom{-}0.023$ | $1$
Table 4: Correlations of systematic uncertainties among the $C\\!P$-violating parameters. | ${C}_{{\pi}{\pi}}$ | ${S}_{{\pi}{\pi}}$ | $A_{{C\\!P}}^{{{B}^{0}}}$ | $A_{{C\\!P}}^{{{B}^{0}_{s}}}$ | ${C}_{{K}{K}}$ | ${S}_{{K}{K}}$ |  ${\cal A}^{\Delta\Gamma}_{{K}{K}}$
---|---|---|---|---|---|---|---
${C}_{{\pi}{\pi}}$ | $1$ |  |  |  |  |  |
${S}_{{\pi}{\pi}}$ | $\phantom{-}0.306$ | $1$ |  |  |  |  |
$A_{{C\\!P}}^{{{B}^{0}}}$ | $-0.044$ | $-0.024$ | $1$ |  |  |  |
$A_{{C\\!P}}^{{{B}^{0}_{s}}}$ | $\phantom{-}0.075$ | $\phantom{-}0.010$ | $-0.238$ | $1$ |  |  |
${C}_{{K}{K}}$ | $-0.050$ | $-0.022$ | $\phantom{-}0.028$ | $-0.009$ | $1$ |  |
${S}_{{K}{K}}$ | $\phantom{-}0.053$ | $\phantom{-}0.045$ | $-0.025$ | $\phantom{-}0.011$ | $\phantom{-}0.197$ | $1$ |
${\cal A}^{\Delta\Gamma}_{{K}{K}}$ | $-0.117$ | $-0.090$ | $\phantom{-}0.050$ | $-0.006$ | $\phantom{-}0.082$ | $\phantom{-}0.018$ | $1$

A combination is performed between the results in this paper and those based
on the Run 1 data sample reported in Ref. [24]. Since the values of
$\Gamma_{{s}}$ and $\Delta\Gamma_{{s}}$ used as input to the fit have changed
with respect to Ref. [24], the Run 1 analysis is updated to account for the
new values. The main variation is observed for the central value of ${\cal
A}^{\Delta\Gamma}_{{K}{K}}$ that changes from $-0.79\pm 0.07$ to $-0.97\pm
0.07$. The large variation of ${\cal A}^{\Delta\Gamma}_{{K}{K}}$ is expected,
given its correlation of 0.91 with $\Gamma_{{s}}$, and the significant change
in the value of $\Gamma_{{s}}$ from $0.6654\pm 0.0022\text{\,ps}^{-1}$ to
$0.6563\pm 0.0021\text{\,ps}^{-1}$ [60]. The only other variation is for
${C}_{{K}{K}}$, moving from $0.20\pm 0.06$ to $0.19\pm 0.06$. The
compatibility between the updated Run 1 result and the numbers reported in Eq.
(19) is computed by means of $\chi^{2}$ test statistic, finding the two sets
of results in agreement with a $p$-value of 0.68.

The full statistical and systematic covariance matrices of the two results are
taken into account in the combination. The only relevant correlation between
the two results is related to the values of the input parameters in Table 1,
hence the corresponding systematic uncertainties are removed from the
covariance matrices of the two results, before combining them. The systematic
uncertainty due to these input parameters is included again by summing the
corresponding covariance matrix to the covariance matrix of the combination.
The results of the combination are:

$\displaystyle{C}_{{\pi}{\pi}}$ $\displaystyle=$
$\displaystyle-0.320\phantom{1}\pm 0.038,$ $\displaystyle{S}_{{\pi}{\pi}}$
$\displaystyle=$ $\displaystyle-0.672\phantom{1}\pm 0.034,$ $\displaystyle
A_{{C\\!P}}^{{{B}^{0}}}$ $\displaystyle=$ $\displaystyle-0.0831\pm 0.0034,$
$\displaystyle A_{{C\\!P}}^{{{B}^{0}_{s}}}$ $\displaystyle=$
$\displaystyle\phantom{-}0.225\phantom{1}\pm 0.012,$
$\displaystyle{C}_{{K}{K}}$ $\displaystyle=$
$\displaystyle\phantom{-}0.172\phantom{1}\pm 0.031,$
$\displaystyle{S}_{{K}{K}}$ $\displaystyle=$
$\displaystyle\phantom{-}0.139\phantom{1}\pm 0.032,$ $\displaystyle{\cal
A}^{\Delta\Gamma}_{{K}{K}}$ $\displaystyle=$
$\displaystyle-0.897\phantom{1}\pm 0.087$

and their correlation matrix is reported in Table 5.

Table 5: Correlation matrix for the $C\\!P$ violation parameters obtained from the combination with Run-1 results. | ${C}_{{\pi}{\pi}}$ | ${S}_{{\pi}{\pi}}$ | $A_{{C\\!P}}^{{{B}^{0}}}$ | $A_{{C\\!P}}^{{{B}^{0}_{s}}}$ | ${C}_{{K}{K}}$ | ${S}_{{K}{K}}$ |  ${\cal A}^{\Delta\Gamma}_{{K}{K}}$
---|---|---|---|---|---|---|---
${C}_{{\pi}{\pi}}$ | $1$ |  |  |  |  |  |
${S}_{{\pi}{\pi}}$ | $\phantom{-}0.405$ | $1$ |  |  |  |  |
$A_{{C\\!P}}^{{{B}^{0}}}$ | $-0.019$ | $\phantom{-}0.001$ | $1$ |  |  |  |
$A_{{C\\!P}}^{{{B}^{0}_{s}}}$ | $\phantom{-}0.014$ | $-0.002$ | $-0.063$ | $1$ |  |  |
${C}_{{K}{K}}$ | $-0.009$ | $-0.032$ | $\phantom{-}0.008$ | $\phantom{-}0.000$ | $1$ |  |
${S}_{{K}{K}}$ | $-0.004$ | $\phantom{-}0.004$ | $-0.007$ | $\phantom{-}0.002$ | $\phantom{-}0.007$ | $1$ |
${\cal A}^{\Delta\Gamma}_{{K}{K}}$ | $-0.019$ | $-0.014$ | $\phantom{-}0.019$ | $-0.003$ | $\phantom{-}0.027$ | $\phantom{-}0.043$ | $1$

##  11 Concluding remarks

The time-dependent $C\\!P$ asymmetries of
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays and the time-integrated
$C\\!P$ asymmetries in ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays are measured using a
data sample of $p$ $p$ collisions corresponding to an integrated luminosity of
1.9$\text{\,fb}^{-1}$, collected with the LHCb detector at a centre-of-mass
energy of 13 TeV. The measurements are compatible with previous LHCb
determinations of the same quantities obtained with Run 1 data [24] and are
combined with them. The measurements of ${C}_{{\pi}{\pi}}$,
${S}_{{\pi}{\pi}}$, $A_{{C\\!P}}^{{{B}^{0}}}$ and
$A_{{C\\!P}}^{{{B}^{0}_{s}}}$ are in good agreement with previous results from
other experiments [30, 31, 32, 33] and are the most precise from a single
experiment to date.

A $\chi^{2}$ test statistic is used to determine the significance for
$({C}_{{K}{K}},\,{S}_{{K}{K}},\,{\cal A}^{\Delta\Gamma}_{{K}{K}})$ to differ
from $(0,\,0,\,-1)$ and for $({C}_{{K}{K}},\,{S}_{{K}{K}})$ to differ from
$(0,\,0)$. The significance for the combined LHCb results is found to be of
$6.5$ and $6.7$ standard deviations, respectively. This constitutes the first
observation of time-dependent $C\\!P$ violation in decays of the ${B}^{0}_{s}$
meson.

The unitary relation among ${C}_{{K}{K}}$, ${S}_{{K}{K}}$ and ${\cal
A}^{\Delta\Gamma}_{{K}{K}}$ is tested, giving
$\sqrt{\left({C}_{{K}{K}}\right)^{2}+\left({S}_{{K}{K}}\right)^{2}+\left({\cal
A}^{\Delta\Gamma}_{{K}{K}}\right)^{2}}=0.93\pm 0.08$. This is compatible with
unity within one standard deviation.

According to the test of the SM proposed in Ref. [7], the following sum must
be satisfied

$\Delta\equiv\frac{A_{{C\\!P}}^{{{B}^{0}}}}{A_{{C\\!P}}^{{{B}^{0}_{s}}}}+\frac{\mathcal{B}\left({{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}\right)}{\mathcal{B}\left({{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}\right)}\frac{{\Gamma_{{s}}}}{{\Gamma_{{d}}}}=0,$
(20)

where $\mathcal{B}\left({{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}\right)$
and $\mathcal{B}\left({{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}\right)$
are $C\\!P$-averaged branching fractions. The LHCb measurements of the
relative fragmentation-fraction ratio between ${B}^{0}_{s}$ and ${B}^{0}$
mesons $f_{s}/f_{d}=0.259\pm 0.015$ [68],
$f_{s}/f_{d}\times\mathcal{B}\left({{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}\right)/\mathcal{B}\left({{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}\right)$
[69] and ${\Gamma_{{s}}}/{\Gamma_{{d}}}$ [60] are used in this test along with
the measurements of $A_{{C\\!P}}^{{{B}^{0}}}$ and
$A_{{C\\!P}}^{{{B}^{0}_{s}}}$. The value $\Delta=-0.085\pm 0.025\pm 0.035$ is
obtained, where the first uncertainty is from the measurements of the $C\\!P$
asymmetries and the second is from the other inputs in Eq. (20). With the
present experimental precision, $\Delta$ is in agreement with zero within two
standard deviations.

Owing to the measurements reported in this paper, improved constraints on the
CKM angles and ${B}^{0}_{s}$ mixing phase can be obtained, as outlined in
Refs. [9, 14, 15]. The comparison of these precises determinations, based on
decays receiving sizeable loop-level contributions, with those provided by the
study of the decays dominated by tree-level amplitudes, will constitute a
stringent test of the SM hypothesis.

## Acknowledgements

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

Appendix

##  Appendix A Additional information on flavour-tagging

###  A.1 Formalism

The functions $\Omega_{\rm sig}(t,\vec{\xi},\,\vec{\eta})$ and
$\overline{\Omega}_{\rm sig}(t,\vec{\xi},\,\vec{\eta})$ in Eqs. (7) and (10)
are

$\displaystyle\Omega_{\rm sig}(t,\xi_{\rm OS},\xi_{\rm SS},\eta_{\rm
OS},\eta_{\rm SS})=\Omega_{\rm OS}^{\rm sig}(\xi_{\rm OS},\eta_{\rm
OS})\Omega^{\rm SS}_{\rm sig}(t,\xi_{\rm SS},\eta_{\rm SS}),$ (21)
$\displaystyle\overline{\Omega}_{\rm sig}(t,\xi_{\rm OS},\xi_{\rm
SS},\eta_{\rm OS},\eta_{\rm SS})=\overline{\Omega}_{\rm OS}^{\rm sig}(\xi_{\rm
OS},\eta_{\rm OS})\overline{\Omega}^{\rm SS}_{\rm sig}(t,\xi_{\rm
SS},\eta_{\rm SS}),$

where $\eta$ is the mistag probability computed by the flavour-tagging
algorithms and discussed in Sec. 4, $\Omega^{\rm sig}_{\rm OS}$ and
$\overline{\Omega}^{\rm sig}_{\rm OS}$ are the same functions used in Ref.
[24], i.e.

$\displaystyle\begin{split}\Omega^{\rm sig}_{\rm OS}(\xi_{\rm OS},\,\eta_{\rm
OS})=&\left\\{\delta_{\xi_{\rm OS},\,+1}\,\varepsilon^{\rm sig}_{\rm
OS}\,\left[1-\omega^{\rm sig}_{\rm OS}(\eta_{\rm OS})\right]+\delta_{\xi_{\rm
OS},\,-1}\,\varepsilon^{\rm sig}_{\rm OS}\,\omega^{\rm sig}_{\rm OS}(\eta_{\rm
OS})\right\\}\,h^{\rm sig}_{\rm OS}(\eta_{\rm OS})\,+\\\ &\delta_{\xi_{\rm
OS},\,0}\,(1-\varepsilon^{\rm sig}_{\rm OS})\,U(\eta_{\rm OS}),\end{split}$
(22) $\displaystyle\begin{split}\overline{\Omega}^{\rm sig}_{\rm OS}(\xi_{\rm
OS},\,\eta_{\rm OS})=&\left\\{\delta_{\xi_{\rm
OS},\,-1}\,\bar{\varepsilon}^{\rm sig}_{\rm
OS}\,\left[1-\overline{\Omega}^{\rm sig}_{\rm OS}(\eta_{\rm
OS})\right]+\delta_{\xi_{\rm OS},\,+1}\,\bar{\varepsilon}^{\rm sig}_{\rm
OS}\,\overline{\Omega}^{\rm sig}_{\rm OS}(\eta_{\rm OS})\right\\}h^{\rm
sig}_{\rm OS}(\eta_{\rm OS})\,+\\\ &\delta_{\xi_{\rm
OS},\,0}\,(1-\bar{\varepsilon}^{\rm sig}_{\rm OS})\,U(\eta_{\rm
OS}),\end{split}$

while $\Omega^{\rm sig}_{\rm SS}$ and $\overline{\Omega}^{\rm sig}_{\rm SS}$
are

$\displaystyle\begin{split}\Omega^{\rm sig}_{\rm SS}(t,\xi_{\rm
SS},\,\eta_{\rm SS})=&\left\\{\delta_{\xi_{\rm SS},\,+1}\,\varepsilon^{\rm
sig}_{\rm SS}(t)\,\left[1-\omega^{\rm sig}_{\rm SS}(\eta_{\rm
SS})\right]+\delta_{\xi_{\rm SS},\,-1}\,\varepsilon^{\rm sig}_{\rm
SS}(t)\,\omega^{\rm sig}_{\rm SS}(\eta_{\rm SS})\right\\}h^{\rm sig}_{\rm
SS}(\eta_{\rm SS})\,+\\\ &\delta_{\xi_{\rm
SS},\,0}\,\left[\varepsilon(t)-\varepsilon^{\rm sig}_{\rm
SS}(t)\right]\,U(\eta_{\rm SS}),\end{split}$ (23)
$\displaystyle\begin{split}\overline{\Omega}^{\rm sig}_{\rm SS}(\xi_{\rm
SS},\,\eta_{\rm SS})=&\left\\{\delta_{\xi_{\rm
SS},\,-1}\,\bar{\varepsilon}^{\rm sig}_{\rm
SS}(t)\,\left[1-\overline{\Omega}^{\rm sig}_{\rm SS}(\eta_{\rm
SS})\right]+\delta_{\xi_{\rm SS},\,1}\,\bar{\varepsilon}^{\rm sig}_{\rm
SS}(t)\,\overline{\Omega}^{\rm sig}_{\rm SS}(\eta_{\rm SS})\right\\}h^{\rm
sig}_{\rm SS}(\eta_{\rm SS})\,+\\\ &\delta_{\xi_{\rm
SS},\,0}\,\left[\varepsilon(t)-\bar{\varepsilon}^{\rm sig}_{\rm
SS}(t)\right]\,U(\eta_{\rm SS}).\end{split}$

Here, $\varepsilon^{\rm sig}_{\rm tag}$ ($\bar{\varepsilon}^{\rm sig}_{\rm
tag}$) is the probability that the flavour of a ${B}_{({s})}^{0}$ (${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}_{({s})}^{0}$) meson is tagged, which
in the case of the SS tagger depends on the decay time555From now on, in order
to simplify the notation, the dependency of SS-tagger efficiency on the decay-
time is omitted.; $\varepsilon(t)$ is the decay-time efficiency independent
from the decision of the SS-tagger, such that $\varepsilon(t)-\varepsilon^{\rm
sig}_{\rm SS}(t)$ is the decay-time efficiency for candidates that have
$\xi_{\rm SS}=0$; $\omega^{\rm sig}_{\rm tag}(\eta_{\rm tag})$ and
$\overline{\Omega}^{\rm sig}_{\rm tag}(\eta_{\rm tag})$ are the calibrated
mistag probabilities as a function of $\eta_{\rm tag}$ for ${B}_{({s})}^{0}$
and ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}_{({s})}^{0}$ mesons;
$h^{\rm tag}_{\rm sig}(\eta_{\rm tag})$ is the PDF describing the distribution
of $\eta_{\rm tag}$ for tagged candidates, and $U(\eta_{\rm tag})$ is a
uniform distribution of $\eta_{\rm tag}$. It is empirically observed that, to
a good approximation, $\eta_{\rm tag}$ and $\omega_{\rm tag}$ are related by a
linear function, i.e.

$\displaystyle\omega^{\rm sig}_{\rm tag}(\eta_{\rm tag})$ $\displaystyle=$
$\displaystyle p^{\rm tag}_{0}\,+\,p^{\rm tag}_{1}\,(\eta_{\rm
tag}\,-\,\hat{\eta}_{\rm tag}),$ (24) $\displaystyle\overline{\omega}^{\rm
sig}_{\rm tag}(\eta_{\rm tag})$ $\displaystyle=$ $\displaystyle\bar{p}^{\rm
tag}_{0}\,+\,\bar{p}^{\rm tag}_{1}\,(\eta_{\rm tag}\,-\,\hat{\eta}_{\rm
tag}),$

where $\hat{\eta}_{\rm tag}$ is a fixed value, chosen to be equal to the mean
value of the $\eta_{\rm tag}$ distribution to minimise the correlation among
the parameters. To reduce the correlation among $\varepsilon^{\rm sig}_{\rm
tag}$ and $\bar{\varepsilon}^{\rm sig}_{\rm tag}$, and $p^{\rm tag}_{0}$,
$\bar{p}^{\rm tag}_{0}$, $p^{\rm tag}_{1}$, and $\bar{p}^{\rm tag}_{1}$, these
variables are conveniently parameterised as

$\displaystyle\varepsilon^{\rm sig}$ $\displaystyle=$
$\displaystyle\hat{\varepsilon}^{\rm sig}_{\rm tag}(1+\Delta\varepsilon^{\rm
sig}_{\rm tag}),$ (25) $\displaystyle\bar{\varepsilon}^{\rm sig}$
$\displaystyle=$ $\displaystyle\hat{\varepsilon}^{\rm sig}_{\rm
tag}(1-\Delta\varepsilon^{\rm sig}_{\rm tag}),$ $\displaystyle p^{\rm
tag}_{0}$ $\displaystyle=$ $\displaystyle\hat{p}^{\rm tag}_{0}(1+\Delta p^{\rm
tag}_{0}),$ $\displaystyle\bar{p}^{\rm tag}_{0}$ $\displaystyle=$
$\displaystyle\hat{p}^{\rm tag}_{0}(1-\Delta p^{\rm tag}_{0}),$ $\displaystyle
p^{\rm tag}_{1}$ $\displaystyle=$ $\displaystyle\hat{p}^{\rm tag}_{1}(1+\Delta
p^{\rm tag}_{1}),$ $\displaystyle\bar{p}^{\rm tag}_{1}$ $\displaystyle=$
$\displaystyle\hat{p}^{\rm tag}_{1}(1-\Delta p^{\rm tag}_{1}),$

where $\hat{p}^{\rm tag}_{0,1}$ and $\Delta p^{\rm tag}_{0,1}$ are the average
and the asymmetry between $p^{\rm tag}_{0,1}$ and $\bar{p}^{\rm tag}_{0,1}$,
and $\hat{\varepsilon}^{\rm sig}_{\rm tag}$ and $\Delta\varepsilon^{\rm
sig}_{\rm tag}$ are the average and the asymmetry between $\varepsilon^{\rm
sig}_{\rm tag}$ and $\bar{\varepsilon}^{\rm sig}_{\rm tag}$. The dependence on
the decay-time is considered only for the averaged efficiency
$\hat{\varepsilon}^{\rm sig}_{\rm SS}$ and not for the asymmetry
$\Delta\varepsilon^{\rm sig}_{\rm tag}$. The strategy used to determine the
decay-time efficiencies $\varepsilon(t)$ and $\hat{\varepsilon}^{\rm sig}_{\rm
SS}(t)$ is reported in Sec. 6.3. The description of $h^{\rm SS}_{\rm
sig}(\eta)$ for the SS taggers is presented in Secs. A.2 and A.3,
respectively.

The PDF $\Omega_{\rm comb}(\vec{\xi},\vec{\eta})$ for the combinatorial
background is empirically parameterised by

$\begin{split}\Omega_{\rm comb}(\vec{\xi},\vec{\eta})=&\left[\delta_{\xi_{\rm
SS},\,+1}\varepsilon^{\rm SS}_{\rm comb}\,+\,\delta_{\xi_{\rm
SS},\,-1}\bar{\varepsilon}^{\rm SS}_{\rm comb}\right]h^{\rm SS}_{\rm
comb}(\eta_{\rm SS})\Omega^{\rm OS,1}_{\rm comb}(\xi_{\rm OS},\eta_{\rm
OS})+\\\ &\delta_{\xi_{\rm SS},\,0}\,(1-\varepsilon^{\rm SS}_{\rm
comb}-\bar{\varepsilon}^{\rm SS}_{\rm comb})\,U(\eta_{\rm SS})\Omega^{\rm
OS,0}_{\rm comb}(\xi_{\rm OS},\eta_{\rm OS}),\end{split}$ (26)

where the functions $\Omega^{{\rm OS},j}_{\rm comb}(\xi_{\rm OS},\eta_{\rm
OS})$ ($j=0,1$) are the PDF for the $\xi_{\rm OS}$ and $\eta_{\rm OS}$,
defined as

$\begin{split}\Omega^{{\rm OS},j}_{\rm comb}(\xi_{\rm OS},\,\eta_{\rm
OS})=&\left[\delta_{\xi_{\rm OS},\,1}\varepsilon^{{\rm OS},j}_{\rm
comb}\,+\,\delta_{\xi_{\rm OS},\,-1}\bar{\varepsilon}^{{\rm OS},j}_{\rm
comb}\right]\,h^{\rm OS}_{\rm comb}(\eta_{\rm OS})\,+\\\ &\delta_{\xi_{\rm
OS},\,0}\,(1-\varepsilon^{{\rm OS},j}_{\rm comb}-\bar{\varepsilon}^{{\rm
OS},j}_{\rm comb})\,U(\eta_{\rm OS}).\end{split}$ (27)

The variables $\varepsilon^{\rm tag}_{\rm comb}$ and $\bar{\varepsilon}^{\rm
tag}_{\rm comb}$ are the probabilities to tag a combinatorial background
candidate as ${B}_{({s})}^{0}$ or ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}_{({s})}^{0}$ and $h^{\rm tag}_{\rm
comb}(\eta_{\rm tag})$ is the distribution of $\eta_{\rm tag}$. The
distribution is described using histograms taken from the right-hand sideband
with invariant-mass range between 5.6$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$
and 6.2$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. The tagging efficiencies are
parameterised by

$\displaystyle\varepsilon^{\rm tag}_{\rm comb}$ $\displaystyle=$
$\displaystyle\frac{\hat{\varepsilon}^{\rm tag}_{\rm
comb}}{2}(1+\Delta\varepsilon^{\rm tag}_{\rm comb}),$ (28)
$\displaystyle\bar{\varepsilon}^{\rm tag}_{\rm comb}$ $\displaystyle=$
$\displaystyle\frac{\hat{\varepsilon}^{\rm tag}_{\rm
comb}}{2}(1-\Delta\varepsilon^{\rm tag}_{\rm comb}),$

such that the fit to data determines the average probability to tag
combinatorial background as ${B}_{({s})}^{0}$ or ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}_{({s})}^{0}$, $\hat{\varepsilon}^{\rm
tag}_{\rm comb}$, and the asymmetry between the two probabilities,
$\Delta\varepsilon^{\rm tag}_{\rm comb}$. For the OS tagger, the distinction
labelled by the index $j=0,1$ is used to differentiate the OS-tagger
probability between cases that have $\xi_{\rm SS}=0$ ($j=0$) and $\xi_{\rm
SS}\neq 0$ ($j=1$). In the case of the ${K}^{+}$ ${\pi}^{-}$ and ${K}^{-}$
${\pi}^{+}$ samples, Eq. (26) is modified in order to include the dependence
on the final-state tag $\psi$

$\Omega_{\rm comb}(\psi,\vec{\xi},\vec{\eta})=\frac{(1-\psi A_{\rm raw}^{\rm
comb})(1-\psi\xi_{\rm OS}A_{\rm OS}^{\rm comb})(1-\psi\xi_{\rm SS}A_{\rm
SS}^{\rm comb})}{\sum_{\psi=-1,1}{(1-\psi A_{\rm raw}^{\rm
comb})(1-\psi\xi_{\rm OS}A_{\rm OS}^{\rm comb})(1-\psi\xi_{\rm SS}A_{\rm
SS}^{\rm comb})}}\times\Omega_{\rm comb}(\vec{\xi},\vec{\eta}),$ (29)

where $A_{\rm raw}^{\rm comb}$ is the total asymmetry between the
combinatorial-background yields in the ${K}^{+}$ ${\pi}^{-}$ and ${K}^{-}$
${\pi}^{+}$ samples, $A_{\rm OS}^{\rm comb}$ and $A_{\rm SS}^{\rm comb}$ are
additional parameters that take into account the possibility that the flavour-
tagging probabilities $\varepsilon^{\rm tag}_{\rm comb}$ and
$\bar{\varepsilon}^{\rm tag}_{\rm comb}$ may depend on the final state.

The PDF $\Omega_{\rm 3-body}(\vec{\xi},\vec{\eta})$ for the partially
reconstructed $B$ decays are empirically parametersied as

$\Omega_{\rm 3-body}(\vec{\xi},\vec{\eta})=\Omega^{\rm OS}_{\rm
3-body}(\vec{\xi},\vec{\eta})\Omega^{\rm SS}_{\rm
3-body}(\vec{\xi},\vec{\eta}),$ (30)

where $\Omega^{\rm OS}_{\rm 3-body}(\vec{\xi},\vec{\eta})$ and $\Omega^{\rm
SS}_{\rm 3-body}(\vec{\xi},\vec{\eta})$ are

$\begin{split}\Omega^{\rm tag}_{\rm 3-body}(\xi_{\rm tag},\,\eta_{\rm
tag})=&\delta_{\xi_{\rm tag},\,+1}\varepsilon^{\rm tag}_{\rm 3-body}\,h^{\rm
tag}_{\rm 3-body}(\eta_{\rm tag})\,+\,\delta_{\xi_{\rm
tag},\,-1}\bar{\varepsilon}^{\rm tag}_{\rm 3-body}\,h^{\rm tag}_{\rm
3-body}(\eta_{\rm tag})\,+\\\ &\delta_{\xi_{\rm tag},\,0}\,(1-\varepsilon^{\rm
tag}_{\rm 3-body}-\bar{\varepsilon}^{\rm tag}_{\rm 3-body})\,U(\eta_{\rm
tag}),\end{split}$ (31)

where $\varepsilon^{\rm tag}_{\rm 3-body}$ and $\bar{\varepsilon}^{\rm
tag}_{\rm 3-body}$ are the probabilities to tag a background candidate as
${B}_{({s})}^{0}$ or ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}_{({s})}^{0}$, and $h^{\rm tag}_{\rm
3-body}(\eta_{\rm tag})$ is the distribution of $\eta_{\rm tag}$. As before,
the tagging efficiencies are parameterised as a function of the total
efficiency ($\hat{\varepsilon}^{\rm tag}_{\rm 3-body}$) and asymmetry
($\Delta\varepsilon^{\rm tag}_{\rm 3-body}$)

$\displaystyle\varepsilon^{\rm tag}_{\rm 3-body}$ $\displaystyle=$
$\displaystyle\frac{\hat{\varepsilon}^{\rm tag}_{\rm
3-body}}{2}(1+\Delta\varepsilon^{\rm tag}_{\rm 3-body}),$ (32)
$\displaystyle\bar{\varepsilon}^{\rm tag}_{\rm 3-body}$ $\displaystyle=$
$\displaystyle\frac{\hat{\varepsilon}^{\rm tag}_{\rm
3-body}}{2}(1-\Delta\varepsilon^{\rm tag}_{\rm 3-body}).$

The PDF $h_{\rm 3-body}^{\rm tag}(\eta_{\rm tag})$ is determined as a
histogram from the low-mass sideband, where the residual contamination of
combinatorial-background candidates is subtracted by injecting candidates with
negative weights. In the case of the ${K}^{+}$ ${\pi}^{-}$ and ${\pi}^{+}$
${K}^{-}$ samples Eq. (31) is modified in order to include the dependence on
the final-state tag $\psi$. Analogously to Eq. (29), the parmeterisation is

$\begin{split}\Omega_{\rm 3-body}(\psi,\vec{\xi},\vec{\eta})=&\frac{(1-\psi
A_{\rm raw}^{\rm 3-body})(1-\psi\xi_{\rm OS}A_{\rm OS}^{\rm
3-body})(1-\psi\xi_{\rm SS}A_{\rm SS}^{\rm 3-body})}{\sum_{\psi=-1,1}{(1-\psi
A_{\rm raw}^{\rm 3-body})(1-\psi\xi_{\rm OS}A_{\rm OS}^{\rm
3-body})(1-\psi\xi_{\rm SS}A_{\rm SS}^{\rm 3-body})}}\\\ &\times\Omega_{\rm
3-body}(\vec{\xi},\vec{\eta}),\end{split}$ (33)

where $A_{\rm raw}^{\rm 3-body}$, $A_{\rm OS}^{\rm 3-body}$ and $A_{\rm
SS}^{\rm 3-body}$ have the same meaning of the corresponding quantities as for
the combinatorial-background component.

The PDFs in Eqs. (21), (26) and (30) are valid if $\eta_{\rm OS}$ and
$\eta_{\rm SS}$ are uncorrelated. This assumption is verified by means of
background-subtracted [59] signal candidates, and of candidates from the high-
and low-mass sidebands for the combinatorial and three-body backgrounds
components, respectively.

###  A.2 Combination of the single SS and OS taggers

The calibration parameters governing the relations in Eqs. (7) are determined
separately for the individual SS and OS taggers by means of a binomial
regression to the tagged decay-time distribution of background-subtracted
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ decays. Then the extracted
calibration parameters of the SS$\pi$ and SS$p$ taggers are used to combine
the two taggers into a unique one (SSc) with decision $\xi_{\rm SSc}$ and
mistag probability $\eta_{\rm SSc}$. The assumption of a linear relation
between $\eta_{\rm tag}$ and $\omega_{\rm tag}$ for each tagger is validated
splitting the sample in bins of $\eta_{{\rm SS}({\pi},{p})}$, estimating the
average mistag fraction in each bin by means of the binomial regression.
Similarly, the various OS taggers are combined together into a unique OS
tagger with decision $\xi_{\rm OS}$ and mistag probability $\eta_{\rm OS}$,
and the same linearity check is performed.

The PDFs $h^{\rm SS}_{\rm sig}(\eta_{\rm SSc})$ describing the $\eta_{\rm
SSc}$ distributions for the signal ${B}^{0}$ mesons are determined using
background-subtracted distributions of
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ decays. As the pion and proton
kinematics are correlated with those of the ${B}^{0}$ meson, the performance
of the SS$\pi$ and SS$p$ taggers also depend on the latter. The differences
between the ${B}^{0}$-meson kinematics and other relevant distributions in
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ and
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays, due to the different
topologies and selection requirements, are taken into account by means of a
weighting procedure to the ${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$
sample. It is empirically observed that the $B$ meson transverse momentum and
the number of hits in the SPD detector distributions need to be equalised.

###  A.3 Calibration of the SS$K$ tagger

The natural control mode to calibrate the response of the SS$K$ tagger would
be the ${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay. However, since
the signal yield of this decay is approximately 8% of that of the
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decay and 20% of that of the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decay, it would not be
possible obtaining a reliable calibration. Furthermore, the calibration
parameters of the SS$K$ tagger would be affected by large uncertainties,
limiting the precision on ${C}_{{K}{K}}$ and ${S}_{{K}{K}}$. Therefore, the
calibration is performed with a large sample of
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ decays. Analogously to
the SS$\pi$ and SS$p$ case, the SS$K$-calibration parameters are determined
using a binomial regression to the tagged decay-time distribution of the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ decays. Also in this
case the regression is performed using the flavour-tagging information on a
per-candidate basis, determining the calibration parameters directly, and a
check of the linear relation between $\eta_{{\rm SS}{K}}$ and $\omega_{{\rm
SS}{K}}$ is performed. As described for the
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ sample in the previous section,
a weighting procedure is applied to the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ sample in order to
equalise the signal distribution of the $B$ meson transverse momentum and the
number of hits in the SPD detector. The PDF $h^{{\rm SS}{K}}_{\rm
sig}(\eta_{{\rm SS}{K}})$ for ${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$
decays is determined using a background-subtracted histogram of the same
weighted sample of ${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$
decays used for the calibration.

## References

  * [1] N. Cabibbo, _Unitary symmetry and leptonic decays_ , Phys. Rev. Lett. 10 (1963) 531
  * [2] M. Kobayashi and T. Maskawa, _$C\\!P$ -violation in the renormalizable theory of weak interaction_, Prog. Theor. Phys. 49 (1973) 652
  * [3] N. G. Deshpande and X.-G. He, _$C\\!P$ asymmetry relations between ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{\pi}{\pi}$ and ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{\pi}{K}$ rates_, Phys. Rev. Lett. 75 (1995) 1703, arXiv:hep-ph/9412393
  * [4] X.-G. He, _SU(3) analysis of annihilation contributions and $C\\!P$ violating relations in ${B}\\!\rightarrow PP$ decays_, Eur. Phys. J. C9 (1999) 443, arXiv:hep-ph/9810397
  * [5] R. Fleischer, _New strategies to extract $\beta$ and $\gamma$ from $B_{d}\rightarrow\pi^{+}\pi^{-}$ and $B_{s}\rightarrow K^{+}K^{-}$_, Phys. Lett. B459 (1999) 306, arXiv:hep-ph/9903456
  * [6] M. Gronau and J. L. Rosner, _The role of ${{B}^{0}_{s}}\\!\rightarrow{\pi}{K}$ in determining the weak phase $\gamma$_, Phys. Lett. B482 (2000) 71, arXiv:hep-ph/0003119
  * [7] H. J. Lipkin, _Is observed direct $C\\!P$ violation in ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ due to new physics? Check standard model prediction of equal violation in ${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$_, Phys. Lett. B621 (2005) 126, arXiv:hep-ph/0503022
  * [8] R. Fleischer, _$B_{s,d}\rightarrow\pi\pi,\pi K,KK$ : status and prospects_, Eur. Phys. J. C52 (2007) 267, arXiv:0705.1121
  * [9] R. Fleischer and R. Knegjens, _In pursuit of new physics with $B^{0}_{s}\rightarrow K^{+}K^{-}$_, Eur. Phys. J. C71 (2011) 1, arXiv:1011.1096
  * [10] M. Gronau and D. London, _Isospin analysis of $C\\!P$ asymmetries in $B$ decays_, Phys. Rev. Lett. 65 (1990) 3381
  * [11] UTfit collaboration, M. Bona et al., _Improved determination of the CKM angle $\alpha$ from ${B}\\!\rightarrow{\pi}{\pi}$ decays_, Phys. Rev. D76 (2007) 014015, arXiv:hep-ph/0701204
  * [12] J. Charles, O. Deschamps, S. Descotes-Genon, and V. Niess, _Isospin analysis of charmless $B$-meson decays_, Eur. Phys. J. C77 (2017) 574, arXiv:1705.02981
  * [13] M. Gronau, _U spin symmetry in charmless $B$ decays_, Phys. Lett. B492 (2000) 297, arXiv:hep-ph/0008292
  * [14] M. Ciuchini et al., _Testing the standard model and searching for new physics with ${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and ${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays_, JHEP 10 (2012) 29, arXiv:1205.4948
  * [15] LHCb collaboration, R. Aaij et al., _Determination of $\gamma$ and $-2\beta_{s}$ from charmless two-body decays of beauty mesons_, Phys. Lett. B739 (2015) 1, arXiv:1408.4368
  * [16] R. Fleischer, R. Jaarsma, and K. K. Vos, _New strategy to explore $C\\!P$ violation with ${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$_, Phys. Rev. D94 (2016) 113014, arXiv:1608.00901
  * [17] R. Fleischer, R. Jaarsma, and K. K. Vos, _Towards new frontiers in the exploration of charmless non-leptonic $B$ decays_, JHEP 03 (2017) 055, arXiv:1612.07342
  * [18] M. Gronau and J. L. Rosner, _Combining $C\\!P$ asymmetries in ${B}\\!\rightarrow{K}{\pi}$ decays_, Phys. Rev. D59 (1999) 113002, arXiv:hep-ph/9809384
  * [19] A. J. Buras, R. Fleischer, S. Recksiegel, and F. Schwab, _The ${B}\\!\rightarrow{\pi}{K}$ puzzle and its relation to rare $B$ and $K$ decays_, Eur. Phys. J. C32 (2003) 45, arXiv:hep-ph/0309012
  * [20] S. Baek and D. London, _Is There Still a ${B}\\!\rightarrow{\pi}{K}$ Puzzle?_, Phys. Lett. B 653 (2007) 249, arXiv:hep-ph/0701181
  * [21] R. Fleischer, R. Jaarsma, E. Malami, and K. K. Vos, _Exploring ${B}\\!\rightarrow{\pi}{\pi},\,{\pi}{K}$ decays at the high-precision frontier_, Eur. Phys. J. C78 (2018) 943, arXiv:1806.08783
  * [22] A. Crivellin, C. Gross, S. Pokorski, and L. Vernazza, _Correlating $\epsilon^{\prime}/\epsilon$ to hadronic $B$ decays via $U(2)^{3}$ flavour symmetry_, Phys. Rev. D101 (2020) 015022, arXiv:1909.02101
  * [23] L. Calibbi et al., _$Z^{\prime}$ models with less-minimal flavour violation_, Phys. Rev. D101 (2020) 095003, arXiv:1910.00014
  * [24] LHCb collaboration, R. Aaij et al., _Measurement of $C\\!P$ asymmetries in two-body ${B}_{({s})}^{0}$-meson decays to charged pions and kaons_, Phys. Rev. D98 (2018) 032004, arXiv:1805.06759
  * [25] Heavy Flavor Averaging Group, Y. Amhis et al., _Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of 2018_, arXiv:1909.12524, updated results and plots available at https://hflav.web.cern.ch
  * [26] LHCb collaboration, R. Aaij et al., _Measurement of the $C\\!P$ asymmetry in ${B}^{0}_{s}$–${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ mixing_, Phys. Rev. Lett. 117 (2016) 061803, arXiv:1605.09768
  * [27] LHCb collaboration, R. Aaij et al., _Measurement of the semileptonic $C\\!P$ asymmetry in ${B}^{0}$–${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ mixing_, Phys. Rev. Lett. 114 (2015) 041601, arXiv:1409.8586
  * [28] UTfit collaboration, M. Bona et al., _The unitarity triangle fit in the standard model and hadronic parameters from lattice QCD: A reappraisal after the measurements of $\Delta m_{s}$ and $BR(B\rightarrow\tau\nu_{\tau})$_, JHEP 10 (2006) 081, arXiv:hep-ph/0606167, updated results and plots available at http://www.utfit.org/
  * [29] CKMfitter group, J. Charles et al., _Current status of the standard model CKM fit and constraints on $\Delta F=2$ new physics_, Phys. Rev. D91 (2015) 073007, arXiv:1501.05013, updated results and plots available at http://ckmfitter.in2p3.fr/
  * [30] BaBar collaboration, J. P. Lees et al., _Measurement of $C\\!P$ asymmetries and branching fractions in charmless two-body $B$-meson decays to pions and kaons_, Phys. Rev. D87 (2013) 052009, arXiv:1206.3525
  * [31] Belle collaboration, J. Dalseno et al., _Measurement of the $C\\!P$ violation parameters in ${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays_, Phys. Rev. D88 (2013) 092003, arXiv:1302.0551
  * [32] Belle collaboration, Y.-T. Duh et al., _Measurements of branching fractions and direct $C\\!P$ asymmetries for ${B}\\!\rightarrow{K}{\pi}$, ${B}\\!\rightarrow{\pi}{\pi}$ and ${B}\\!\rightarrow{K}{K}$ decays_, Phys. Rev. D87 (2013) 031103, arXiv:1210.1348
  * [33] CDF collaboration, T. A. Aaltonen et al., _Measurements of direct $C\\!P$-violating asymmetries in charmless decays of bottom baryons_, Phys. Rev. Lett. 113 (2014) 242001, arXiv:1403.5586
  * [34] R. Bailey et al., _Measurement of the lifetime of charged and neutral D mesons with high resolution silicon strip detectors_ , Z. Phys. C28 (1985) 357
  * [35] J. Rademacker, _Reduction of statistical power per event due to upper lifetime cuts in lifetime measurements_ , Nucl. Instrum. Meth. A570 (2007) 525, arXiv:hep-ex/0502042
  * [36] V. Gligorov et al., _Swimming: A Data Driven Acceptance Correction Algorithm_ , Journal of Physics: Conference Series 396 (2012) 022016
  * [37] LHCb collaboration, R. Aaij et al., _Measurement of the effective ${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ lifetime_, Phys. Lett. B707 (2012) 349, arXiv:1111.0521
  * [38] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [39] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [40] R. Aaij et al., _Performance of the LHCb Vertex Locator_ , JINST 9 (2014) P09007, arXiv:1405.7808
  * [41] R. Arink et al., _Performance of the LHCb Outer Tracker_ , JINST 9 (2014) P01002, arXiv:1311.3893
  * [42] M. Adinolfi et al., _Performance of the LHCb RICH detector at the LHC_, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
  * [43] R. Aaij et al., _The LHCb trigger and its performance in 2011_, JINST 8 (2013) P04022, arXiv:1211.3055
  * [44] T. Sjöstrand, S. Mrenna, and P. Skands, _A brief introduction to PYTHIA 8.1_ , Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
  * [45] T. Sjöstrand, S. Mrenna, and P. Skands, _PYTHIA 6.4 physics and manual_ , JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [46] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [47] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [48] P. Golonka and Z. Was, _PHOTOS Monte Carlo: A precision tool for QED corrections in $Z$ and $W$ decays_, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [49] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [50] Geant4 collaboration, S. Agostinelli et al., _Geant4: A simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [51] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [52] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
  * [53] B. P. Roe et al., _Boosted decision trees as an alternative to artificial neural networks for particle identification_ , Nucl. Instrum. Meth. A543 (2005) 577, arXiv:physics/0408124
  * [54] LHCb collaboration, R. Aaij et al., _Observation of the decay ${{B}_{c}^{+}}\\!\rightarrow{{B}^{0}_{s}}{{\pi}^{+}}$_, Phys. Rev. Lett. 111 (2013) 181801, arXiv:1308.4544
  * [55] L. Anderlini et al., _The PIDCalib package_ , LHCb-PUB-2016-021, 2016
  * [56] LHCb collaboration, R. Aaij et al., _Opposite-side flavour tagging of $B$ mesons at the LHCb experiment_, Eur. Phys. J. C72 (2012) 2022, arXiv:1202.4979
  * [57] LHCb collaboration, R. Aaij et al., _New algorithms for identifying the flavour of ${B}^{0}$ mesons using pions and protons_, Eur. Phys. J. C77 (2017) 238, arXiv:1610.06019
  * [58] LHCb collaboration, R. Aaij et al., _A new algorithm for identifying the flavour of ${B}^{0}_{s}$ mesons at LHCb_, JINST 11 (2016) P05010, arXiv:1602.07252
  * [59] M. Pivk and F. R. Le Diberder, _sPlot: A statistical tool to unfold data distributions_ , Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
  * [60] LHCb collaboration, R. Aaij et al., _Updated measurement of time-dependent $C\\!P$-violating observables in ${{B}^{0}_{s}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}{{K}^{-}}$ decays_, Eur. Phys. J. C79 (2019) 706, Erratum ibid. C80 (2020) 601, arXiv:1906.08356
  * [61] LHCb collaboration, R. Aaij et al., _Measurement of ${B}^{0}$, ${B}^{0}_{s}$, ${{B}^{+}}$ and ${\mathchar 28931\relax}^{0}_{b}$ production asymmetries in 7 and 8 TeV proton-proton collisions_, Phys. Lett. B774 (2017) 139, arXiv:1703.08464
  * [62] N. L. Johnson, _System of frequency curves generated by methods of translation_ , Biometrika 36 (1949) 149
  * [63] K. S. Cranmer, _Kernel estimation in high-energy physics_ , Comput. Phys. Commun. 136 (2001) 198, arXiv:hep-ex/0011057
  * [64] ARGUS collaboration, H. Albrecht et al., _Measurement of the polarization in the decay $B\rightarrow J/\psi K^{*}$_, Phys. Lett. B340 (1994) 217
  * [65] Y. Xie, _sFit: a method for background subtraction in maximum likelihood fit_ , arXiv:0905.0724
  * [66] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
  * [67] LHCb collaboration, R. Aaij et al., _Measurement of $C\\!P$ asymmetry in ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and ${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays_, JHEP 07 (2014) 041, arXiv:1405.2797
  * [68] LHCb collaboration, R. Aaij et al., _Measurement of the fragmentation fraction ratio $f_{s}/f_{d}$ and its dependence on $B$ meson kinematics_, JHEP 04 (2013) 001, arXiv:1301.5286, $f_{s}/f_{d}$ value updated in LHCb-CONF-2013-011
  * [69] LHCb collaboration, R. Aaij et al., _Measurement of $b$-hadron branching fractions for two-body decays into charmless charged hadrons_, JHEP 10 (2012) 037, arXiv:1206.2794

LHCb collaboration

R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala84, S. Aiola25, Z. Ajaltouni9, S. Akar64, J.
Albrecht14, F. Alessio47, M. Alexander58, A. Alfonso Albero44, Z. Aliouche61,
G. Alkhazov37, P. Alvarez Cartelle47, S. Amato2, Y. Amhis11, L. An21, L.
Anderlini21, A. Andreianov37, M. Andreotti20, F. Archilli16, A. Artamonov43,
M. Artuso67, K. Arzymatov41, E. Aslanides10, M. Atzeni49, B. Audurier11, S.
Bachmann16, M. Bachmayer48, J.J. Back55, S. Baker60, P. Baladron Rodriguez45,
V. Balagura11, W. Baldini20, J. Baptista Leite1, R.J. Barlow61, S. Barsuk11,
W. Barter60, M. Bartolini23,i, F. Baryshnikov80, J.M. Basels13, G. Bassi28, B.
Batsukh67, A. Battig14, A. Bay48, M. Becker14, F. Bedeschi28, I. Bediaga1, A.
Beiter67, V. Belavin41, S. Belin26, V. Bellee48, K. Belous43, I. Belov39, I.
Belyaev38, G. Bencivenni22, E. Ben-Haim12, A. Berezhnoy39, R. Bernet49, D.
Berninghoff16, H.C. Bernstein67, C. Bertella47, E. Bertholet12, A. Bertolin27,
C. Betancourt49, F. Betti19,e, M.O. Bettler54, Ia. Bezshyiko49, S. Bhasin53,
J. Bhom33, L. Bian72, M.S. Bieker14, S. Bifani52, P. Billoir12, M. Birch60,
F.C.R. Bishop54, A. Bizzeti21,s, M. Bjørn62, M.P. Blago47, T. Blake55, F.
Blanc48, S. Blusk67, D. Bobulska58, J.A. Boelhauve14, O. Boente Garcia45, T.
Boettcher63, A. Boldyrev81, A. Bondar42, N. Bondar37, S. Borghi61, M.
Borisyak41, M. Borsato16, J.T. Borsuk33, S.A. Bouchiba48, T.J.V. Bowcock59, A.
Boyer47, C. Bozzi20, M.J. Bradley60, S. Braun65, A. Brea Rodriguez45, M.
Brodski47, J. Brodzicka33, A. Brossa Gonzalo55, D. Brundu26, A. Buonaura49, C.
Burr47, A. Bursche26, A. Butkevich40, J.S. Butter31, J. Buytaert47, W.
Byczynski47, S. Cadeddu26, H. Cai72, R. Calabrese20,g, L. Calefice14,12, L.
Calero Diaz22, S. Cali22, R. Calladine52, M. Calvi24,j, M. Calvo Gomez83, P.
Camargo Magalhaes53, A. Camboni44, P. Campana22, D.H. Campora Perez47, A.F.
Campoverde Quezada5, S. Capelli24,j, L. Capriotti19,e, A. Carbone19,e, G.
Carboni29, R. Cardinale23,i, A. Cardini26, I. Carli6, P. Carniti24,j, L.
Carus13, K. Carvalho Akiba31, A. Casais Vidal45, G. Casse59, M. Cattaneo47, G.
Cavallero47, S. Celani48, J. Cerasoli10, A.J. Chadwick59, M.G. Chapman53, M.
Charles12, Ph. Charpentier47, G. Chatzikonstantinidis52, C.A. Chavez
Barajas59, M. Chefdeville8, C. Chen3, S. Chen26, A. Chernov33, S.-G. Chitic47,
V. Chobanova45, S. Cholak48, M. Chrzaszcz33, A. Chubykin37, V. Chulikov37, P.
Ciambrone22, M.F. Cicala55, X. Cid Vidal45, G. Ciezarek47, P.E.L. Clarke57, M.
Clemencic47, H.V. Cliff54, J. Closier47, J.L. Cobbledick61, V. Coco47, J.A.B.
Coelho11, J. Cogan10, E. Cogneras9, L. Cojocariu36, P. Collins47, T.
Colombo47, L. Congedo18,d, A. Contu26, N. Cooke52, G. Coombs58, G. Corti47,
C.M. Costa Sobral55, B. Couturier47, D.C. Craik63, J. Crkovská66, M. Cruz
Torres1, R. Currie57, C.L. Da Silva66, E. Dall’Occo14, J. Dalseno45, C.
D’Ambrosio47, A. Danilina38, P. d’Argent47, A. Davis61, O. De Aguiar
Francisco61, K. De Bruyn77, S. De Capua61, M. De Cian48, J.M. De Miranda1, L.
De Paula2, M. De Serio18,d, D. De Simone49, P. De Simone22, J.A. de Vries78,
C.T. Dean66, W. Dean84, D. Decamp8, L. Del Buono12, B. Delaney54, H.-P.
Dembinski14, A. Dendek34, V. Denysenko49, D. Derkach81, O. Deschamps9, F.
Desse11, F. Dettori26,f, B. Dey72, P. Di Nezza22, S. Didenko80, L. Dieste
Maronas45, H. Dijkstra47, V. Dobishuk51, A.M. Donohoe17, F. Dordei26, A.C. dos
Reis1, L. Douglas58, A. Dovbnya50, A.G. Downes8, K. Dreimanis59, M.W. Dudek33,
L. Dufour47, V. Duk76, P. Durante47, J.M. Durham66, D. Dutta61, M.
Dziewiecki16, A. Dziurda33, A. Dzyuba37, S. Easo56, U. Egede68, V.
Egorychev38, S. Eidelman42,v, S. Eisenhardt57, S. Ek-In48, L. Eklund58, S.
Ely67, A. Ene36, E. Epple66, S. Escher13, J. Eschle49, S. Esen31, T. Evans47,
A. Falabella19, J. Fan3, Y. Fan5, B. Fang72, N. Farley52, S. Farry59, D.
Fazzini24,j, P. Fedin38, M. Féo47, P. Fernandez Declara47, A. Fernandez
Prieto45, J.M. Fernandez-tenllado Arribas44, F. Ferrari19,e, L. Ferreira
Lopes48, F. Ferreira Rodrigues2, S. Ferreres Sole31, M. Ferrillo49, M. Ferro-
Luzzi47, S. Filippov40, R.A. Fini18, M. Fiorini20,g, M. Firlej34, K.M.
Fischer62, C. Fitzpatrick61, T. Fiutowski34, F. Fleuret11,b, M. Fontana12, F.
Fontanelli23,i, R. Forty47, V. Franco Lima59, M. Franco Sevilla65, M. Frank47,
E. Franzoso20, G. Frau16, C. Frei47, D.A. Friday58, J. Fu25, Q. Fuehring14, W.
Funk47, E. Gabriel31, T. Gaintseva41, A. Gallas Torreira45, D. Galli19,e, S.
Gambetta57,47, Y. Gan3, M. Gandelman2, P. Gandini25, Y. Gao4, M. Garau26, L.M.
Garcia Martin55, P. Garcia Moreno44, J. García Pardiñas49, B. Garcia Plana45,
F.A. Garcia Rosales11, L. Garrido44, C. Gaspar47, R.E. Geertsema31, D.
Gerick16, L.L. Gerken14, E. Gersabeck61, M. Gersabeck61, T. Gershon55, D.
Gerstel10, Ph. Ghez8, V. Gibson54, M. Giovannetti22,k, A. Gioventù45, P.
Gironella Gironell44, L. Giubega36, C. Giugliano20,47,g, K. Gizdov57, E.L.
Gkougkousis47, V.V. Gligorov12, C. Göbel69, E. Golobardes83, D. Golubkov38, A.
Golutvin60,80, A. Gomes1,a, S. Gomez Fernandez44, F. Goncalves Abrantes69, M.
Goncerz33, G. Gong3, P. Gorbounov38, I.V. Gorelov39, C. Gotti24,j, E.
Govorkova31, J.P. Grabowski16, R. Graciani Diaz44, T. Grammatico12, L.A.
Granado Cardoso47, E. Graugés44, E. Graverini48, G. Graziani21, A. Grecu36,
L.M. Greeven31, P. Griffith20, L. Grillo61, S. Gromov80, B.R. Gruberg Cazon62,
C. Gu3, M. Guarise20, P. A. Günther16, E. Gushchin40, A. Guth13, Y. Guz43,47,
T. Gys47, T. Hadavizadeh68, G. Haefeli48, C. Haen47, J. Haimberger47, S.C.
Haines54, T. Halewood-leagas59, P.M. Hamilton65, Q. Han7, X. Han16, T.H.
Hancock62, S. Hansmann-Menzemer16, N. Harnew62, T. Harrison59, C. Hasse47, M.
Hatch47, J. He5, M. Hecker60, K. Heijhoff31, K. Heinicke14, A.M. Hennequin47,
K. Hennessy59, L. Henry25,46, J. Heuel13, A. Hicheur2, D. Hill62, M. Hilton61,
S.E. Hollitt14, J. Hu16, J. Hu71, W. Hu7, W. Huang5, X. Huang72, W.
Hulsbergen31, R.J. Hunter55, M. Hushchyn81, D. Hutchcroft59, D. Hynds31, P.
Ibis14, M. Idzik34, D. Ilin37, P. Ilten64, A. Inglessi37, A. Ishteev80, K.
Ivshin37, R. Jacobsson47, S. Jakobsen47, E. Jans31, B.K. Jashal46, A.
Jawahery65, V. Jevtic14, M. Jezabek33, F. Jiang3, M. John62, D. Johnson47,
C.R. Jones54, T.P. Jones55, B. Jost47, N. Jurik47, S. Kandybei50, Y. Kang3, M.
Karacson47, M. Karpov81, N. Kazeev81, F. Keizer54,47, M. Kenzie55, T. Ketel32,
B. Khanji14, A. Kharisova82, S. Kholodenko43, K.E. Kim67, T. Kirn13, V.S.
Kirsebom48, O. Kitouni63, S. Klaver31, K. Klimaszewski35, S. Koliiev51, A.
Kondybayeva80, A. Konoplyannikov38, P. Kopciewicz34, R. Kopecna16, P.
Koppenburg31, M. Korolev39, I. Kostiuk31,51, O. Kot51, S. Kotriakhova37,30, P.
Kravchenko37, L. Kravchuk40, R.D. Krawczyk47, M. Kreps55, F. Kress60, S.
Kretzschmar13, P. Krokovny42,v, W. Krupa34, W. Krzemien35, W. Kucewicz33,l, M.
Kucharczyk33, V. Kudryavtsev42,v, H.S. Kuindersma31, G.J. Kunde66, T.
Kvaratskheliya38, D. Lacarrere47, G. Lafferty61, A. Lai26, A. Lampis26, D.
Lancierini49, J.J. Lane61, R. Lane53, G. Lanfranchi22, C. Langenbruch13, J.
Langer14, O. Lantwin49,80, T. Latham55, F. Lazzari28,t, R. Le Gac10, S.H.