EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2015-224 LHCb-PAPER-2015-034

Measurement of $C\\!P$ violation parameters and polarisation fractions in
$B_{s}^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
4.14793pt\overline{\kern-4.14793ptK}{}^{*0}$ decays

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

##  1 Introduction

The $C\\!P$-violating phase $\phi_{s}$ arises in the interference between the
amplitudes of $B^{0}_{s}$ mesons decaying via $b\to c\bar{c}s$ transitions to
$C\\!P$ eigenstates directly and those decaying after oscillation. The phase
$\phi_{s}$ can be measured using the decay
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$. Within the
Standard Model (SM), and ignoring penguin contributions to the decay,
$\phi_{s}$ is predicted to be $-2\beta_{s}$, with
$\beta_{s}\equiv\arg(-V_{cb}V_{cs}^{*}/V_{tb}V_{ts}^{*})$, where $V_{ij}$ are
elements of the CKM matrix [1]. The phase $\phi_{s}$ is a sensitive probe of
dynamics beyond the SM (BSM) since it has a very small theoretical uncertainty
and BSM processes can contribute to $B^{0}_{s}$-$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing [2, 3, 4, 5]. Global
fits to experimental data, excluding the direct measurements of $\phi_{s}$,
give $-2\beta_{s}=-0.0363\pm 0.0013\rm\,rad$ [6]. The current world average
value is $\phi_{s}=-0.015\pm 0.035\rm\,rad$ [7], dominated by the LHCb
measurement reported in Ref. [8]. In the SM expectation of $\phi_{s}$ [6],
additional contributions to the leading $b\to c\bar{c}s$ tree Feynman diagram,
as shown in Fig. 1, are assumed to be negligible. However, the shift in
$\phi_{s}$ due to these contributions, called hereafter “penguin pollution”,
is difficult to compute due to the non-perturbative nature of the quantum
chromodynamics (QCD) processes involved. This penguin pollution must be
measured or limited before using the $\phi_{s}$ measurement in searches for
BSM effects, since a shift in this phase caused by penguin diagrams is
possible. Various methods to address this problem have been proposed [9, 10,
11, 12, 13, 14], and LHCb has recently published upper limits on the size of
the penguin-induced phase shift using
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$ decays [15].

Tree and penguin diagrams contributing to both
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ and
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays are shown in Fig. 1. In
this paper, the penguin pollution in $\phi_{s}$ is investigated using
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays111Charge conjugation is
implicit throughout this paper, unless otherwise specified., with
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\to\mu^{+}\mu^{-}$ and $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\to K^{-}\pi^{+}$, following the
method first proposed in Ref. [9] for the
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$ decay and
later also discussed for the
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay in Refs. [11, 13]. This
approach requires the measurement of the branching fraction, direct $C\\!P$
asymmetries, and polarisation fractions of the
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay. The measurements use data
from proton-proton ($pp$) collisions recorded with the LHCb detector
corresponding to 3.0$\mbox{\,fb}^{-1}$ of integrated luminosity, of which 1.0
(2.0)$\mbox{\,fb}^{-1}$ was collected in 2011 (2012) at a centre-of-mass
energy of 7 (8)$\mathrm{\,Te\kern-1.00006ptV}$. The LHCb collaboration
previously reported a measurement of the branching fraction and the
polarisation fractions using data corresponding to 0.37$\mbox{\,fb}^{-1}$ of
integrated luminosity [16].

The paper is organised as follows: a description of the LHCb detector,
reconstruction and simulation software is given in Sect. 2, the selection of
the $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ signal candidates and the
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{*0}$ control channel
are presented in Sect. 3 and the treatment of background in Sect. 4. The
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{-}\pi^{+}$ invariant mass
fit is detailed in Sect. 5. The angular analysis and $C\\!P$ asymmetry
measurements, both performed on weighted distributions where the background is
statistically subtracted using the $sP\\!lot$ technique [17], are detailed in
Sect. 6. The measurement of the branching fraction is explained in Sect. 7.
The evaluation of systematic uncertainties is described in Sect. 8 along with
the results, and in Sect. 9 constraints on the penguin pollution are evaluated
and discussed.

Figure 1: Decay topologies contributing to the
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ channel (a,b)
and $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ channel (c,d). The tree diagrams
(a,c) are shown on the left and the penguin diagrams (b,d) on the right.

##  2 Experimental setup

The LHCb detector [18, 19] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream of the
magnet. The tracking system provides a measurement of momentum, $p$, of
charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at 200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum
distance of a track to a primary vertex, the impact parameter, is measured
with a resolution of (15+29$/\mbox{$p_{\rm T}$}$)$\,\rm\mu m$, where $p_{\rm
T}$ is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors.
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.

The online event selection is performed by a trigger, which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. In
this analysis, candidates are first required to pass the hardware trigger,
which selects muons with a transverse momentum $\mbox{$p_{\rm
T}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in the
7$\mathrm{\,Te\kern-1.00006ptV}$ data or $\mbox{$p_{\rm
T}$}>1.76{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in the
8$\mathrm{\,Te\kern-1.00006ptV}$ data. In the subsequent software trigger, at
least one of the final-state particles is required to have both $\mbox{$p_{\rm
T}$}>0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter larger than
100$\,\rm\mu m$ with respect to all of the primary $pp$ interaction vertices
(PVs) in the event. Finally, the tracks of two or more of the final-state
particles are required to form a vertex that is significantly displaced from
any PV. Further selection requirements are applied offline in order to
increase the signal purity.

In the simulation, $pp$ collisions are generated using Pythia [20, 21] with a
specific LHCb configuration [22]. Decays of hadronic particles are described
by EvtGen [23], in which final-state radiation is generated using Photos [24].
The interaction of the generated particles with the detector, and its
response, are implemented using the Geant4 toolkit [25, 26] as described in
Ref. [27].

##  3 Event selection

The selection of $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ candidates consists
of two steps: a preselection consisting of discrete cuts, followed by a
specific requirement on a boosted decision tree with gradient boosting (BDTG)
[28, 29] to suppress combinatorial background. All charged particles are
required to have a transverse momentum in excess of
$0.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and to be positively identified
as muons, kaons or pions. The tracks are fitted to a common vertex which is
required to be of good quality and significantly displaced from any PV in the
event. The flight direction can be described as a vector between the
$B^{0}_{s}$ production and decay vertices; the cosine of the angle between
this vector and the $B^{0}_{s}$ momentum vector is required to be greater than
$0.999$. Reconstructed invariant masses of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ candidates are required to be in
the ranges
$2947<m_{\mu^{+}\mu^{-}}<3247{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$826<m_{K^{-}\pi^{+}}<966{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The
$B^{0}_{s}$ invariant mass is reconstructed by constraining the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate to its nominal mass
[30], and is required to be in the range
$5150<m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}\pi^{+}}<5650{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The training of the BDTG is performed independently for 2011 and 2012 data,
using information from the $B^{0}_{s}$ candidates: time of flight, transverse
momentum, impact parameter with respect to the production vertex and
$\chi^{2}$ of the decay vertex fit. The data sample used to train the BDTG
uses less stringent particle identification requirements. When training the
BDTG, simulated $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ events are used to
represent the signal, while candidates reconstructed from data events with
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{-}\pi^{+}$ invariant mass
above $5401{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are used to represent the
background. The optimal threshold for the BDTG is chosen independently for
2011 and 2012 data and maximises the effective signal yield.

##  4 Treatment of peaking backgrounds

After the suppression of most background with particle identification
criteria, simulations show residual contributions from the backgrounds
$\mathchar 28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}pK^{-}$, $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}$, $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}\pi^{-}$, and $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}\pi^{-}$. The invariant mass distributions of misidentified
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\pi^{+}\pi^{-}$ and
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\pi^{+}\pi^{-}$
events peak near the $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}\pi^{+}$ signal peak due to the effect of a wrong-mass hypothesis,
and the misidentified $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}$ candidates are located in the vicinity of the
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}\pi^{-}$ signal
peak. It is therefore not possible to separate such background from signal
using information based solely on the invariant mass of the
${{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}K^{-}\pi^{+}$ system.
Moreover the shape of the reflected invariant mass distribution is sensitive
to the daughter particles momenta. Due to these correlations it is difficult
to add the
$b{\text{-hadron}}~{}{\text{to}}~{}{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}h^{+}h^{-}$ (where $h$ is either a pion, a kaon or a proton)
misidentified backgrounds as extra modes to the fit to the invariant mass
distribution. Instead, simulated events are added to the data sample with
negative weights in order to cancel the contribution from those peaking
backgrounds, as done previously in Ref. [8]. Simulated
$b{\text{-hadron}}~{}{\text{to}}~{}{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}h^{+}h^{-}$ events are generated using a phase-space model, and then
weighted on an event-by-event basis using the latest amplitude analyses of the
decays $\mathchar
28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}pK^{-}$
[31], $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}K^{-}$
[32], $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}\pi^{-}$ [33], and
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\pi^{+}\pi^{-}$ [34].
The sum of weights of each decay mode is normalised such that the injected
simulated events cancel out the expected yield in data of the specific
background decay mode.

In addition to $\mathchar
28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}pK^{-}$
and $B\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}h^{+}h^{-}$ decays,
background from $\mathchar
28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}p\pi^{-}$ is also expected. However, in Ref. [35] a full amplitude
analysis was not performed. For this reason, as well as the fact that the
$\mathchar 28931\relax^{0}_{b}$ decays have broad mass distributions, the
contribution is explicitly included in the mass fit described in the next
section. Expected yields for both $B\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}h^{+}h^{-}$ and $\mathchar
28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}ph^{-}$
background decays are given in Table 1.

Table 1: Expected yields of each background component in the signal mass range. Background sources | 2011 data | 2012 data
---|---|---
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\pi^{+}\pi^{-}$ | $\phantom{0}51\pm 10\phantom{0}$ | $115\pm 23\phantom{0}$
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\pi^{+}\pi^{-}$ | $\phantom{0}9.3\pm 2.1\phantom{0}$ | $25.0\pm 5.4\phantom{0}$
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}K^{-}$ | $10.1\pm 2.3\phantom{0}$ | $19.2\pm 4.0\phantom{0}$
$\mathchar 28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}pK^{-}$ | $\phantom{0}36\pm 17\phantom{0}$ | $\phantom{0}90\pm 43\phantom{0}$
$\mathchar 28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}p\pi^{-}$ | $13.8\pm 5.3\phantom{0}$ | $27.3\pm 9.0\phantom{0}$

##  5 Fit to the invariant mass distribution

After adding simulated $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}\pi^{-}$, $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}\pi^{-}$, $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}$, and $\mathchar
28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}pK^{-}$
events with negative weights, the remaining sample consists of
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}\pi^{-}$,
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{-}\pi^{+}$,
$\mathchar 28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}p\pi^{-}$ decays, and combinatorial background. These four modes are
statistically disentangled through a fit to the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{-}\pi^{+}$ invariant mass.
The combinatorial background is described by an exponential distribution, the
$\mathchar 28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}p\pi^{-}$ decay by the Amoroso distribution [36] and the $B^{0}$ and
$B^{0}_{s}$ signals by the double-sided Hypatia distribution [37],

$I(m,\mu,\sigma,\lambda,\zeta,\beta,a_{1},a_{2},n_{1},n_{2})\propto\\\
\begin{cases}\frac{A}{\left(B+m-\mu\right)^{n_{1}}}\phantom{,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,-}\text{if
}m-\mu<-a_{1}\sigma\,,\\\
\frac{C}{\left(D+m-\mu\right)^{n_{2}}}\phantom{,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,}\text{if
}m-\mu>a_{2}\sigma\,,\\\
\left((m-\mu)^{2}+\delta^{2}\right)^{\frac{1}{2}\lambda-\frac{1}{4}}e^{\beta(m-\mu)}K_{\lambda-\frac{1}{2}}\left(\alpha\sqrt{(m-\mu)^{2}+\delta^{2}}\right)\phantom{,,,,,,}\text{otherwise}\,,\phantom{,,}\\\
\end{cases}$ (1)

where $K_{\nu}(z)$ is the modified Bessel function of the second kind,
$\delta\equiv\sigma\sqrt{\frac{\zeta\,K_{\lambda}(\zeta)}{K_{\lambda+1}(\zeta)}}$,
$\alpha\equiv\frac{1}{\sigma}\sqrt{\frac{\zeta\,K_{\lambda+1}(\zeta)}{K_{\lambda}(\zeta)}}$,
and $A,B,C,D$ are obtained by imposing continuity and differentiability. This
function is chosen because the event-by-event uncertainty on the mass has a
dependence on the particle momenta. The estimate of the number of
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}\pi^{-}$ decays
lying under the $B^{0}_{s}$ peak is very sensitive to the modelling of the
tails of the $B^{0}$ peak. The fitted fraction is in good agreement with the
estimate from simulation.

In the fit to data, the mean and resolution parameters of both the $B^{0}_{s}$
and $B^{0}$ Hypatia functions are free to vary. All the remaining parameters,
namely $\lambda$, $a_{1}$, $n_{1}$, $a_{2}$ and $n_{2}$, are fixed to values
determined from fits to $B^{0}_{s}$ and $B^{0}$ simulated events. All the
$\mathchar 28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}p\pi^{-}$ shape parameters are fixed to values obtained from fits to
simulated $\mathchar
28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}p\pi^{-}$ events, while the exponent of the combinatorial background
is free to vary.

Due to the small expected yield of $\mathchar
28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}p\pi^{-}$ decays compared to those of the other modes determined in
the fit to data, and to the broad distribution of $\mathchar
28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}p\pi^{-}$ decays across the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}\pi^{+}$ invariant mass spectrum, its yield is included in the fit
as a Gaussian constraint using the expected number of events and its
uncertainties, as shown in Table 1.

---
Figure 2: The ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{-}\pi^{+}$
invariant mass distribution with the sum of the fit projections in the 20
$m_{K^{-}\pi^{+}}$ and $\cos(\theta_{\mu})$ bins. Points with error bars show
the data. The projection of the fit result is represented by the solid blue
line, and the contributions from the different components are detailed in the
legend. At this scale the contribution of the $\mathchar
28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}p\pi^{-}$ is barely visible. All the other peaking background
components are subtracted as described in the text.

From studies of simulated (MC) samples, it is found that the resolution of
$B^{0}_{s}$ and $B^{0}$ mass peaks depends on both $m_{K^{-}\pi^{+}}$ and
$\cos(\theta_{\mu})$, where $\theta_{\mu}$ is one of the helicity angles used
in the angular analysis as defined in Sect. 6. The fit to the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{-}\pi^{+}$ invariant mass
spectrum, including the evaluation of the $sW\\!eights$, is performed
separately in twenty bins, corresponding to four $m_{K^{-}\pi^{+}}$ bins of
$35{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ width, and five equal bins in
$\cos(\theta_{\mu})$. The overall $B^{0}_{s}$ and $B^{0}$ yields are obtained
from the sum of yields in the twenty bins, giving

$\displaystyle N_{B^{0}}$ $\displaystyle=208656\pm
462\>(\text{stat})^{+78}_{-76}\>(\text{syst})\,,$ (2) $\displaystyle
N_{B^{0}_{s}}$ $\displaystyle=\phantom{00}1808\pm
51\>(\text{stat})^{+38}_{-33}\>(\text{syst})\,,$ (3)

where the statistical uncertainties are obtained from the quadratic sum of the
uncertainties determined in each of the individual fits. Systematic
uncertainties are discussed in Sect. 8. The correlation between the $B^{0}$
and $B^{0}_{s}$ yields in each bin are found to be smaller than $4\%$. The
ratio of the $B^{0}_{s}$ and $B^{0}$ yields is found to be
$N_{B^{0}_{s}}/N_{B^{0}}=(8.66\pm
0.24\>(\text{stat})^{+0.18}_{-0.16}\>(\text{syst}))\times 10^{-3}$. Figure 2
shows the sum of the fit results for each bin, overlaid on the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{-}\pi^{+}$ mass spectrum for
the selected data sample.

##  6 Angular analysis

###  6.1 Angular formalism

This analysis uses the decay angles defined in the helicity basis. The
helicity angles are denoted by $(\theta_{K},\theta_{\mu},\varphi_{h})$, as
shown in Fig. 3. The polar angle $\theta_{K}$ ($\theta_{\mu}$) is the angle
between the kaon ($\mu^{+}$) momentum and the direction opposite to the
$B^{0}_{s}$ momentum in the $K^{-}\pi^{+}$ ($\mu^{+}\mu^{-}$) centre-of-mass
system. The azimuthal angle between the $K^{-}\pi^{+}$ and $\mu^{+}\mu^{-}$
decay planes is $\varphi_{h}$. The definitions are the same for $B^{0}_{s}$ or
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays. They are also
the same for $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{*0}$
decays.

Figure 3: Representation of helicity angles as discussed in the text.

The shape of the angular distribution of
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays is given by Ref. [38],

$\frac{\text{d}\Gamma(\theta_{K},\theta_{\mu},\varphi_{h})}{\text{d}\Omega}\propto\sum_{\alpha_{\mu}=\pm
1}\Bigg{|}\sum_{\lambda,J}^{|\lambda|<J}\sqrt{\frac{2J+1}{4\pi}}\mathcal{H}_{\lambda}^{J}e^{-\text{i}\lambda\varphi_{h}}d_{\lambda,\alpha_{\mu}}^{1}(\theta_{\mu})d_{-\lambda,0}^{1}(\theta_{K})\Bigg{|}^{2}\>,$
(4)

where $\lambda=0,\pm 1$ is the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ helicity, $\alpha_{\mu}=\pm 1$ is the helicity difference between the
muons, $J$ is the spin of the $K^{-}\pi^{+}$ system, $\mathcal{H}$ are the
helicity amplitudes, and $d$ are the small Wigner matrices.

The helicity amplitudes are rotated into transversity amplitudes, which
correspond to final $P$ eigenstates,

$\displaystyle A_{\rm S}$ $\displaystyle=\mathcal{H}_{0}^{0}\,,$ (5)
$\displaystyle A_{0}$ $\displaystyle=\mathcal{H}_{0}^{1}\,,$ (6)
$\displaystyle A_{\parallel}$
$\displaystyle=\frac{1}{\sqrt{2}}({\mathcal{H}}_{+}^{1}+{\mathcal{H}}_{-}^{1})\,,$
(7) $\displaystyle A_{\perp}$
$\displaystyle=\frac{1}{\sqrt{2}}({\mathcal{H}}_{+}^{1}-{\mathcal{H}}_{-}^{1})\,.$
(8)

The distribution in Eq. 4 can be written as the sum of ten angular terms, four
corresponding to the square of the transversity amplitude of each final state
polarisation, and six corresponding to the cross terms describing interference
among the final polarisations.

The modulus of a given transversity amplitude, $A_{x}$, is written as
$|A_{x}|$, and its phase as $\delta_{x}$. The convention $\delta_{0}=0$ is
used in this paper. The P–wave polarisation fractions are
$f_{i}=|A_{i}|^{2}/(|A_{0}|^{2}+|A_{\parallel}|^{2}+|A_{\perp}|^{2})$, with
$i=0,\parallel,\perp$ and the S–wave fraction is defined as $F_{\rm S}=|A_{\rm
S}|^{2}/(|A_{0}|^{2}+|A_{\parallel}|^{2}+|A_{\perp}|^{2}+|A_{\rm S}|^{2})$.
The distribution of the $C\\!P$-conjugate decay is obtained by flipping the
sign of the interference terms which contain $|A_{\perp}|$. For the
$C\\!P$-conjugate case, the amplitudes are denoted as $\overline{A}_{i}$. Each
$A_{i}$ and the corresponding $\overline{A}_{i}$ are related through the
$C\\!P$ asymmetries, as described in Sect. 6.3.

###  6.2 Partial-wave interference factors

In the general case, the transversity amplitudes of the angular model depend
on the $K^{-}\pi^{+}$ mass ($m_{K^{-}\pi^{+}}$). This variable is limited to
be inside a window of $\pm$70${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around
the $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ mass. Figure 4 shows
the efficiency-corrected $m_{K^{-}\pi^{+}}$ spectra for $B^{0}_{s}$ and
$B^{0}$ using the nominal sets of $sW\\!eights$.

In order to account for the $m_{K^{-}\pi^{+}}$ dependence while keeping the
framework of an angular-only analysis, a fit is performed simultaneously in
the same four $m_{K^{-}\pi^{+}}$ bins defined in Sect. 5. Different values of
the parameters $|A_{\rm S}|^{2}$ and $\delta_{\rm S}$ are allowed for each
bin, but the angular distribution still contains mass-dependent terms
associated with the interference between partial-waves. If only the S–wave and
P–wave are considered, such interference terms correspond to the following
complex integrals,

$\frac{\int_{m_{K\pi}^{L}}^{m_{K\pi}^{H}}{\mathcal{P}\times\mathcal{S}^{*}\>{\Phi}\>\varepsilon_{m}(m_{K\pi})}\>\>\text{d}m_{K\pi}}{\sqrt{\int_{m_{K\pi}^{L}}^{m_{K\pi}^{H}}{|\mathcal{P}|^{2}\>{\Phi}\>\varepsilon_{m}(m_{K\pi})}\>\>\text{d}m_{K\pi}\int_{m_{K\pi}^{L}}^{m_{K\pi}^{H}}{|\mathcal{S}|^{2}\>{\Phi}\>\varepsilon_{m}(m_{K\pi})}\>\>\text{d}m_{K\pi}}}=C_{\rm
SP}e^{-\text{i}\theta_{\rm SP}}\,,$ (9)

where $m_{K\pi}^{L(H)}$ is the lower (higher) limit of the bin,
$\varepsilon_{m}(m_{K\pi})$ is the acceptance for a $K^{-}\pi^{+}$ candidate
with mass $m_{K\pi}$ (see Appendix A for a discussion on the angular
acceptance), $\Phi$ stands for the phase space, and $\mathcal{P}$
($\mathcal{S}$) is the P–wave (S–wave) propagator. The phase space term is
computed as

$\Phi=\frac{p\>q}{m_{K\pi}^{2}},$ (10)

where $p$ denotes the $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
momentum in the $B_{s}^{0}$ rest frame and $q$ refers to the $K^{-}$ momentum
in the $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ rest frame.

The phase $\theta_{\rm SP}$ is included in the definition of $\delta_{\rm S}$
but the $C_{\rm SP}$ factors, corresponding to real numbers in the interval
$[0,1]$, have to be computed and input to the angular fit. The contribution of
D–wave ($J=2$) in the $m_{K^{-}\pi^{+}}$ range considered is expected to be
negligible. Therefore the nominal model only includes S–wave and P–wave. To
determine the systematic uncertainty due to possible D–wave contributions,
$C_{\rm SD}$ and $C_{\rm PD}$ factors are also computed, using analogous
expressions to that given in Eq. 9. The $C_{ij}$ factors are calculated by
evaluating numerically the integrals using the propagators outlined below, and
are included as fixed parameters in the fit. A systematic uncertainty
associated to the different possible choices of the propagator models is
afterwards evaluated.

Figure 4: Efficiency corrected $m_{K^{-}\pi^{+}}$ distribution for $B^{0}_{s}$
shown in squares (red) and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$
shown in circles (black) using $sW\\!eights$ computed from the maximum
likelihood fit to the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}\pi^{+}$ invariant mass spectrum.

The S–wave propagator is constructed using the LASS parametrisation [39],
consisting of a linear combination of the $K^{*}_{0}(1430)^{0}$ resonance with
a non-resonant term, coming from elastic scattering. The P–wave is described
by a combination of the $K^{*}(892)^{0}$ and $K^{*}_{1}(1410)^{0}$ resonances
using the isobar model [40], and the D–wave is assumed to come from the
$K^{*}_{2}(1430)^{0}$ contribution. Relativistic Breit-Wigner functions,
multiplied by angular momentum barrier factors, are used to parametrise the
different resonances. Table 2 contains the computed $C_{\rm SP}$, $C_{\rm SD}$
and $C_{\rm PD}$ factors.

Table 2: The $C_{\rm SP}$, $C_{\rm SD}$ and $C_{\rm PD}$ factors calculated in each of the four $m_{K^{-}\pi^{+}}$ bins around the $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ peak. Bin |  $m_{K^{-}\pi^{+}}$ range (${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) | $C_{\rm SP}$ | $C_{\rm SD}$ | $C_{\rm PD}$
---|---|---|---|---
0 | [826, 861] | 0.968 $\pm$ 0.017 | 0.9968 $\pm$ 0.0030 | 0.9827 $\pm$ 0.0048
1 | [861, 896] | 0.931 $\pm$ 0.012 | 0.9978 $\pm$ 0.0021 | 0.9402 $\pm$ 0.0048
2 | [896, 931] | 0.952 $\pm$ 0.012 | 0.9983 $\pm$ 0.0016 | 0.9421 $\pm$ 0.0056
3 | [931, 966] | 0.988 $\pm$ 0.011 | 0.9986 $\pm$ 0.0012 | 0.9802 $\pm$ 0.0066

###  6.3 $C\\!P$ asymmetries

The direct $C\\!P$ violation asymmetry in the $B^{0}_{(s)}$ decay rate to the
final state $f_{(s)\,i}$, with $f_{s,i}={{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(K^{-}\pi^{+})_{i}$ and $f_{i}={{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(K^{+}\pi^{-})_{i}$, is defined as

$A^{C\\!P}_{i}(B^{0}_{(s)}\to
f_{(s)\,i})=\frac{|\overline{A}_{(s)\,i}|^{2}-|A_{(s)\,i}|^{2}}{|\overline{A}_{(s)\,i}|^{2}+|A_{(s)\,i}|^{2}}\,,$
(11)

where $A_{(s)\,i}$ are the transversity amplitudes defined in Sect. 6.1 and
the additional index $s$ is used to distinguish the $B^{0}_{s}$ and the
$B^{0}$-meson. The index $i$ refers to the polarisation of the final state
($i=0,\parallel,\perp,{\rm S}$) and is dropped in the rest of this section,
for clarity.

The raw $C\\!P$ asymmetry is expressed in terms of the number of observed
candidates by

$A^{C\\!P}_{\rm raw}(B^{0}_{(s)}\to f_{(s)})=\frac{N^{\rm
obs}(\overline{f}_{(s)})-N^{\rm obs}(f_{(s)})}{N^{\rm
obs}(\overline{f}_{(s)})+N^{\rm obs}(f_{(s)})}\,.$ (12)

Both asymmetries in Eq. 11 and Eq. 12 are related by [41]

$A^{C\\!P}(B^{0}_{(s)}\to f_{(s)})\simeq A^{C\\!P}_{\rm raw}(B^{0}_{(s)}\to
f_{(s)})-\zeta_{(s)}A_{\rm D}(f)-\kappa_{(s)}A_{\rm P}(B^{0}_{(s)})\,,$ (13)

where $A_{\rm D}(f)$ is the detection asymmetry, defined as in Eq. (16),
$A_{\rm P}(B^{0}_{(s)})$ is the $B^{0}_{(s)}$$-$$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$ production asymmetry,
defined as in Eq. (15), $\zeta_{(s)}=+1(-1)$ and $\kappa_{(s)}$ accounts for
the dilution due to $B^{0}_{(s)}$$-$$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$ oscillations [42]. The
$\kappa_{(s)}$ factor is evaluated by

$\kappa_{(s)}\\!=\\!\frac{\int_{0}^{\infty}\\!e^{-\Gamma_{(s)}t}\\!\cos\\!\left(\Delta
m_{(s)}t\right)\\!\varepsilon(t)\mathrm{d}t}{\int_{0}^{\infty}\\!e^{-\Gamma_{(s)}t}\\!\cosh\\!\left(\frac{\Delta\Gamma_{(s)}}{2}t\right)\\!\varepsilon(t)\mathrm{d}t}\,,$
(14)

where $\varepsilon(t)$ is the time-dependent acceptance function, assumed to
be identical for the $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{*0}$ decays. The
symbols $\Gamma_{(s)}$ and $\Delta m_{(s)}$ denote the decay width and mass
differences between the $B^{0}_{(s)}$ mass eigenstates.

The $B^{0}_{(s)}$$-$$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$
production asymmetry is defined as

$A_{\text{P}}\left(B_{(s)}^{0}\right)\equiv\frac{\sigma\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}_{(s)}^{0}\right)-\sigma\left(B_{(s)}^{0}\right)}{\sigma\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}_{(s)}^{0}\right)+\sigma\left(B_{(s)}^{0}\right)}\>,$
(15)

where $\sigma$ is the $B_{(s)}^{0}$ production cross-section within the LHCb
acceptance. The production asymmetries reported in Ref. [43] are reweighted in
bins of $B_{(s)}^{0}$ transverse momentum to obtain

$\displaystyle A_{\text{P}}(B^{0})=$ $\displaystyle\left(-1.04\pm
0.48\>(\text{stat})\pm 0.14\>(\text{syst})\right)\%\>,$ $\displaystyle
A_{\text{P}}(B^{0}_{s})=$ $\displaystyle\left(-1.64\pm 2.28\>(\text{stat})\pm
0.55\>(\text{syst})\right)\%\,.$

The $\kappa_{(s)}$ factor in Eq. 14 is determined by fixing
$\Delta\Gamma_{(s)}$, $\Delta m_{(s)}$ and $\Gamma_{(s)}$ to their world
average values [30] and by fitting the decay time acceptance $\varepsilon(t)$
to the nominal data sample after applying the $B^{0}$ $sW\\!eights$, in a
similar way to Ref. [44]. It is equal to 0.06$\%$ for $B^{0}_{s}$ decays, and
41$\%$ for $B^{0}$. This reduces the effect of the production asymmetries to
the level of $10^{-5}$ for $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $10^{-3}$ for
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{*0}$ decays.

Other sources of asymmetries arise from the different final-state particle
interactions with the detector, event reconstruction and detector acceptance.
The detection asymmetry, $A_{\text{D}}(f)$, is defined in terms of the
detection efficiencies of the final states, $\varepsilon^{\rm det}$, as

$A_{\text{D}}(f)\equiv\frac{\varepsilon^{\rm
det}(\overline{f})-\varepsilon^{\rm det}(f)}{\varepsilon^{\rm
det}(\overline{f})+\varepsilon^{\rm det}(f)}\,.$ (16)

The detection asymmetry, measured in bins of the $K^{+}$ momentum in Ref.
[45], is weighted with the momentum distribution of the kaon from the
$B^{0}_{(s)}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{*0}(\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})$ decays to give

$\displaystyle A_{\text{D}}(B^{0})$ $\displaystyle=$
$\displaystyle\left(\phantom{-}1.12\pm 0.55\>(\text{stat})\right)\%\,,$
$\displaystyle A_{\text{D}}(B^{0}_{s})$ $\displaystyle=$
$\displaystyle\left({-}1.09\pm 0.53\>(\text{stat})\right)\%\,.$

##  7 Measurement of ${\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})$

The branching fraction ${\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})$ is obtained by normalising to
two different channels, $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ and $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{*0}$, and then averaging the results. The expression

$\frac{{\cal B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0})\times{\cal
B}(K^{*0}\to K^{+}\pi^{-})}{{\cal
B}(B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}X)\times{\cal B}(X\to
h^{+}h^{-})}=\frac{N_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}{N_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}X}}\times\frac{\varepsilon_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}X}}{\varepsilon_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}\times\frac{f_{q}}{f_{s}}\,,$
(17)

is used for the normalisation to a given
$B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}X$ decay, where $N$
refers to the yield of the given decay, $\varepsilon$ corresponds to the total
(reconstruction, trigger and selection) efficiency, and $f_{q}=f_{s}(f_{d})$
are the $B_{s}^{0}$($B^{0}$)-meson hadronisation fractions.

The event selection of $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ candidates consists of the same requirements as those for
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ candidates (see Sect. 3), with
the exception that $\phi$ candidates are reconstructed in the $K^{+}K^{-}$
state so there are no pions among the final state particles. In addition to
the other requirements, reconstructed $\phi$ candidates are required to have
mass in the range
$1000<m_{K^{-}K^{+}}<1040{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and to have
a transverse momentum in excess of
$1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

###  7.1 Efficiencies obtained in simulation

A first estimate of the efficiency ratios is taken from simulated events,
where the particle identification variables are calibrated using $D^{*\pm}$
decays. The efficiency ratios estimated from simulation, for 2011 (2012) data,
are $\varepsilon_{B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{*0}}^{\rm
MC}/\varepsilon_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}^{\rm MC}=0.929\pm
0.012~{}(0.927\pm 0.012)$ and
$\varepsilon_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}^{\rm
MC}/\varepsilon_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}^{\rm MC}=1.991\pm
0.025~{}(1.986\pm 0.027)$.

###  7.2 Correction factors for yields and efficiencies

The signal and normalisation channel yields obtained from a mass fit are
affected by the presence of a non-resonant S–wave background as well as
interference between S–wave and P–wave components. Such interference would
integrate to zero for a flat angular acceptance, but not for experimental data
that are subject to an angle-dependent acceptance. In addition, the
efficiencies determined in simulation correspond to events generated with an
angular distribution different from that in data; therefore the angular
integrated efficiency also needs to be modified with respect to simulation
estimates. These effects are taken into account using a correction factor
$\omega$, which is the product of the correction factor to the angular-
integrated efficiency and the correction factor to the P–wave yield:

$\frac{N_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}{N_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}X}}\times\frac{\varepsilon_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}X}}{\varepsilon_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}=\frac{N_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}{N_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}X}}\times\frac{\varepsilon_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}X}^{\rm
MC}}{\varepsilon_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}^{\rm
MC}}\times\frac{\omega_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}X}}{\omega_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}\,,$ (18)

where $N_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$,
$N_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}X}$ are the yields
obtained from the mass fits,
$\varepsilon_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}X}^{\rm
MC},\varepsilon_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}^{\rm MC}$ are the
efficiencies obtained in simulation, and $\omega$ is calculated as

$\omega_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}X}=\frac{F_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}X}^{X}}{c_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}X}}\,,$ (19)

where $F_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}X}^{X}$ is the
fraction of the P–wave $X$ resonance in a given
$B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}X$ decay (related to
the presence of S–wave and its interference with the P–wave), and
$c_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}X}$ is a correction
to $\varepsilon_{B_{q}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}X}^{\rm
MC}$ due to the fact that the simulated values of the decay parameters differ
slightly from those measured. The values obtained for the $\omega$ correction
factors are

$\displaystyle\omega_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$
$\displaystyle=1.149\pm 0.044\,{\rm(stat)}\pm 0.018\,{\rm(syst)}\,,$
$\displaystyle\omega_{B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{*0}}$ $\displaystyle=1.107\pm 0.003\,{\rm(stat)}\pm
0.038\,{\rm(syst)}\,,$
$\displaystyle\omega_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}$ $\displaystyle=1.013\pm 0.002\,{\rm(stat)}\pm
0.007\,{\rm(syst)}\,.$

###  7.3 Normalisation to $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$

The study of penguin pollution requires the calculation of ratios of absolute
amplitudes between $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$. Thus,
normalising ${\cal B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0})$ to ${\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi)$ is very
useful. This normalisation is given by

$\small\frac{{\cal B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.79997pt\overline{\kern-1.79997ptK}{}^{*0})}{{\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)}=\frac{N_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}\pi^{+}}}{N_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}}}\times\frac{\varepsilon_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}^{\rm
MC}}{\varepsilon_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.25998pt\overline{\kern-1.25998ptK}{}^{*0}}^{\rm
MC}}\times\frac{\omega_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}}{\omega_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.25998pt\overline{\kern-1.25998ptK}{}^{*0}}}\times\frac{{\cal
B}(\phi\to K^{+}K^{-})}{{\cal B}(\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}\to K^{-}\pi^{+})}\,,$ (20)

where ${\cal B}(\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\to
K^{-}\pi^{+})=2/3$ and ${\cal B}(\phi\to K^{+}K^{-})=(49.5\pm 0.5)\%$ [30].
Using $N_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}\pi^{+}}$ as given in Eq. 3, and
$N_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}}=58\,091~{}\pm~{}243~{}({\rm stat})\pm 319~{}({\rm syst})$
as obtained from a fit to the invariant mass of selected
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ candidates,
where the signal is described by a double-sided Hypatia distribution and the
combinatorial background is described by an exponential distribution, a value
of

$\frac{{\cal B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0})}{{\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)}=\big{(}4.05\pm 0.19({\rm stat})\pm 0.13({\rm syst})\big{)}\%\\\
$

is obtained.

###  7.4 Normalisation to $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{*0}$

The normalisation to $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{*0}$ is given by

$\frac{{\cal B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0})}{{\cal
B}(B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{*0})}=\frac{N_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}\pi^{+}}}{N_{B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}\pi^{-}}}\times\frac{f_{d}}{f_{s}}\times\frac{\varepsilon_{B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{*0}}^{\rm
MC}}{\varepsilon_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}^{\rm
MC}}\times\frac{\omega_{B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{*0}}}{\omega_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}\,,$ (21)

where $N_{B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}\pi^{-}}$
and $N_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}\pi^{+}}$ are given in Eq. 2 and Eq. 3, respectively, and

$\frac{\omega_{B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{*0}}}{\omega_{B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}=0.963\pm
0.036\,({\rm stat})\pm 0.031\,({\rm syst})\,,\\\ $

resulting in a value of

$\frac{{\cal B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0})}{{\cal
B}(B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{*0})}=\left(2.99\pm 0.14\,({\rm stat})\pm 0.12\,({\rm syst})\pm
0.17\,(f_{d}/f_{s})\right)\%\,,\\\ $ (22)

where the third uncertainty comes from the hadronisation fraction ratio
$f_{d}/f_{s}=3.86\pm 0.22$ [7].

###  7.5 Computation of ${\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})$

By multiplying the fraction given in Eq. 22 by the branching fraction of the
decay $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{*0}$ measured
at Belle222The result from Belle was chosen rather than the PDG average, since
it is the only ${\cal B}(B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{*0})$ measurement that subtracts S–wave contributions., $(1.29\pm
0.05\,({\rm stat})\pm 0.13\,({\rm syst)})\times 10^{-3}$ [46], and taking into
account the difference in production rates for the $B^{+}$$B^{-}$ and
$B^{0}$$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ pairs at the
$\Upsilon(4S)$ resonance, i.e. $\Gamma(B^{+}B^{-})/\Gamma(B^{0}\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0})=1.058\pm 0.024$ [7], the value

$\displaystyle{\cal B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0})_{d}$
$\displaystyle=(3.95$ $\displaystyle\pm 0.18\,({\rm stat})\pm 0.16\,({\rm
syst})\pm 0.23\,(f_{d}/f_{s})$ $\displaystyle\pm 0.43\,({\cal
B}(B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{*0})))\times
10^{-5}\,$

is obtained, where the fourth uncertainty arises from ${\cal
B}(B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{*0})$. A second
estimate of this quantity is found via the normalisation to ${\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi)$ [32],
updated with the value of $f_{d}/f_{s}$ from Ref. [7] to give ${\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi)=(1.038\pm
0.013\>(\text{stat})\pm 0.063\>(\text{syst})\pm 0.060\>(f_{d}/f_{s}))\times
10^{-3}$, resulting in a value of

${\cal B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})_{\phi}=\left(4.20\pm
0.20\,\text{(stat)}\pm 0.13\,\text{(syst)}\pm 0.36\,({\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi))\right)\times 10^{-5}\,,$

where the third uncertainty comes from ${\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi)$. Both
values are compatible within uncorrelated systematic uncertainties and are
combined, taking account of correlations, to give

${\cal B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=\left(4.14\pm
0.18\,\text{(stat)}\pm 0.26\,\text{(syst)}\pm 0.24\,(f_{d}/f_{s})\right)\times
10^{-5}\,,$

which is in good agreement with the previous LHCb measurement [16], of
$(4.4_{-0.4}^{+0.5}\pm 0.8)\times 10^{-5}$.

##  8 Results and systematic uncertainties

Section 8.1 presents the results of the angular fit as well as the procedure
used to estimate the systematic uncertainties, while in Sect. 8.2 the results
of the branching fraction measurements and the corresponding estimated
systematic uncertainties are discussed.

###  8.1 Angular parameters and $C\\!P$ asymmetries

The results obtained from the angular fit to the
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ events are given in Table 3 and
Table 4 for the P–wave and S–wave parameters, respectively. For comparison,
the previous LHCb measurements [16] of $f_{0}$ and $f_{\|}$ were $0.50\pm
0.08\pm 0.02$ and $0.19^{+0.10}_{-0.08}\pm 0.02$, respectively. The angular
distribution of the signal and the projection of the fitted distribution are
shown in Fig. 5. The statistical-only correlation matrix as obtained from the
fit to data is given in Appendix B. The polarisation-dependent $C\\!P$
asymmetries are compatible with zero, as expected in the SM. The polarisation
fractions are in good agreement with the previous measurements [16] performed
on the same decay mode by the LHCb collaboration using data corresponding to
an integrated luminosity of 0.37$\mbox{\,fb}^{-1}$.

Figure 5: Fitted signal distributions compared with the weighted angular
distributions with $B^{0}_{s}$ $sW\\!eights$. Points with error bars show the
data. The projection of the fit result is represented by the solid black line,
and the contributions from the different amplitude components are described in
the legend.

Various sources of systematic uncertainties on the parameters of the angular
fit are studied, as summarised in Table 3 and Table 4 for the P–wave and
S–wave parameters. Two classes of systematic uncertainties are defined, one
from the angular fit model and another from the mass fit model. Since the
angular fit is performed on the data weighted using the signal $sW\\!eights$
calculated from the fit to the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}\pi^{+}$ invariant mass, biases on the mass fit results may be
propagated to the $sW\\!eights$ and thus to the angular parameters. Overall,
two sources of systematic uncertainties dominate: the angular acceptance and
the correlation between the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}\pi^{+}$ invariant mass and $\theta_{\mu}$.

Table 3: Summary of the measured $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ P–wave properties and their statistical and systematic uncertainties. When no value is given, it means an uncertainty below $5\times 10^{-4}$, except for the two phases, $\delta_{\parallel}$ (rad) and $\delta_{\perp}$ (rad), in which case the uncertainty is below $5\times 10^{-3}$. | $f_{0}$ | $f_{\|}$ | $\delta_{\parallel}$ | $\delta_{\perp}$ | $A^{C\\!P}_{0}$ | $A^{C\\!P}_{\|}$ | $A^{C\\!P}_{\perp}$ |
---|---|---|---|---|---|---|---|---
Fitted value | $0.497$ | $0.179$ | $-2.70$ | $0.01$ | $-0.048$ | $0.171$ | $-0.049$ |
Statistical uncertainties | $0.025$ | $0.027$ | $0.16$ | $0.11$ | $0.057$ | $0.152$ | $0.096$ |
Angular acceptance | $0.018$ | $0.008$ | $0.02$ | $0.01$ | $0.009$ | $0.017$ | $0.008$ |
(simulation statistics) |
Angular acceptance | $0.015$ | $0.007$ | $0.17$ | $0.10$ | $0.007$ | — | $0.015$ |
(data–simulation differences) |
$C_{\rm SP}$ factors | — | $0.001$ | — | — | $0.001$ | $0.002$ | $0.002$ |
D–wave contribution | $0.004$ | $0.003$ | — | — | $0.002$ | $0.015$ | $0.002$ |
Background | $0.004$ | $0.002$ | $0.02$ | $0.01$ | $0.004$ | ${}^{+0.012}_{-0.004}$ | $0.002$ |
angular model |
Mass parameters and | — | — | — | — | $0.001$ | $0.001$ | — |
$B^{0}$ contamination |
Mass–$\cos(\theta_{\mu})$ | $0.007$ | $0.006$ | $0.07$ | ${}^{+0.02}_{-0.04}$ | $0.014$ | ${}^{+0.009}_{-0.012}$ | $0.016$ |
correlations |
Fit bias | — | $0.001$ | $0.01$ | $0.07$ | $0.003$ | $0.002$ | $0.005$ |
Detection | — | — | — | — | $0.005$ | $0.005$ | $0.006$ |
asymmetry |
Production | — | — | — | — | — | — | — |
asymmetry |
Quadratic sum of | $0.025$ | $0.013$ | $0.19$ | ${}^{+0.012}_{-0.013}$ | $0.020$ | ${}^{+0.028}_{-0.027}$ | $0.025$ |
systematic uncertainties |
Total uncertainties | $0.035$ | $0.030$ | $0.25$ | ${}^{+0.016}_{-0.017}$ | $0.060$ | $0.154$ | $0.099$ |
Table 4: Summary of the measured $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ S–wave properties and their statistical and systematic uncertainties. When no value is given, it means an uncertainty below $5\times 10^{-4}$, except for the four phases related to the S–wave component, $\delta_{\rm S}$ (rad), in which case the uncertainty is below $5\times 10^{-3}$. The $m_{K^{-}\pi^{+}}$ binning definition is identical to the one given in Table 2. | $A^{C\\!P}_{\rm S}$ | $m_{K^{-}\pi^{+}}^{\rm bin0}$ | $m_{K^{-}\pi^{+}}^{\rm bin1}$ | $m_{K^{-}\pi^{+}}^{\rm bin2}$ | $m_{K^{-}\pi^{+}}^{\rm bin3}$
---|---|---|---|---|---
$F_{\rm S}$ | $\delta_{\rm S}$ | $F_{\rm S}$ | $\delta_{\rm S}$ | $F_{\rm S}$ | $\delta_{\rm S}$ | $F_{\rm S}$ | $\delta_{\rm S}$
Fitted value | $0.167$ | $0.475$ | $0.54$ | $0.080$ | $-0.53$ | $0.044$ | $-1.46$ | $0.523$ | $-1.76$
Statistical uncertainties | $0.114$ | ${}^{+0.108}_{-0.112}$ | $0.16$ | ${}^{+0.031}_{-0.025}$ | ${}^{+0.25}_{-0.21}$ | ${}^{+0.042}_{-0.029}$ | ${}^{+0.22}_{-0.19}$ | ${}^{+0.109}_{-0.112}$ | ${}^{+0.13}_{-0.14}$
Angular acceptance | $0.028$ | $0.039$ | $0.03$ | $0.012$ | $0.065$ | $0.015$ | $0.10$ | $0.065$ | $0.06$
(simulation statistics)
Angular acceptance | $0.015$ | $0.058$ | $0.08$ | $0.019$ | $0.18$ | $0.027$ | $0.27$ | $0.006$ | $0.04$
(data–simulation differences)
$C_{\rm SP}$ factors | — | $0.002$ | $0.01$ | $0.001$ | — | $0.002$ | — | $0.001$ | $0.01$
D–wave contribution | $0.008$ | $0.010$ | $0.02$ | $0.005$ | $0.03$ | $0.008$ | $0.08$ | $0.002$ | $0.04$
Background | $0.001$ | $0.002$ | $0.01$ | ${}^{+0.000}_{-0.001}$ | $0.01$ | — | $0.03$ | ${}^{+0.002}_{-0.000}$ | ${}^{+0.07}_{-0.04}$
angular model
Mass parameters and | $0.001$ | $0.001$ | $0.01$ | — | — | — | — | — | —
$B^{0}$ contamination
Mass–$\cos(\theta_{\mu})$ | ${}^{+0.023}_{-0.029}$ | ${}^{+0.040}_{-0.028}$ | $0.05$ | $0.003$ | $0.04$ | ${}^{+0.006}_{-0.016}$ | $0.02$ | ${}^{+0.009}_{-0.011}$ | $0.03$
correlations
Fit bias | $0.004$ | $0.005$ | $0.01$ | $0.003$ | $0.02$ | $0.007$ | $0.032$ | $0.015$ | $0.01$
Detection | $0.005$ | — | — | — | — | — | — | — | —
asymmetry
Production | — | — | — | — | — | — | — | — | —
asymmetry
Quadratic sum of | ${}^{+0.041}_{-0.044}$ | ${}^{+0.081}_{-0.076}$ | $0.10$ | $0.023$ | $0.20$ | ${}^{+0.033}_{-0.036}$ | $0.30$ | $0.068$ | ${}^{+0.11}_{-0.09}$
systematic uncertainties
Total uncertainties | ${}^{+0.120}_{-0.122}$ | $0.135$ | $0.19$ | ${}^{+0.039}_{-0.034}$ | ${}^{+0.32}_{-0.29}$ | ${}^{+0.054}_{-0.047}$ | ${}^{+0.37}_{-0.35}$ | ${}^{+0.128}_{-0.131}$ | $0.17$

####  8.1.1 Systematic uncertainties related to the mass fit model

To determine the systematic uncertainty arising from the fixed parameters in
the description of the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}\pi^{+}$ invariant mass, these parameters are varied inside their
uncertainties, as determined from fits to simulated events. The fit is then
repeated and the widths of the $B^{0}_{s}$ and $B^{0}$ yield distributions are
taken as systematic uncertainties on the branching fractions. Correlations
among the parameters obtained from simulation are taken into account in this
procedure. For each new fit to the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}\pi^{+}$ invariant mass, the corresponding set of $sW\\!eights$ is
calculated and the fit to the weighted angular distributions is repeated. The
widths of the distributions are taken as systematic uncertainties on the
angular parameters. In addition, a systematic uncertainty is added to account
for imperfections in the modelling of the upper tail of the $B^{0}$ and
$B^{0}_{s}$ peaks. Indeed, in the Hypatia distribution model, the parameters
$a_{2}$ and $n_{2}$ take into account effects such as decays in flight of the
hadron, that affect the lineshape of the upper tail and could modify the
$B^{0}$ leakage into the $B^{0}_{s}$ peak. The estimate of this leakage is
recalculated for extreme values of those parameters, and the maximum spread is
conservatively added as a systematic uncertainty.

Systematic uncertainties due to the fixed yields of the
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}K^{-}$,
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\pi^{+}\pi^{-}$,
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\pi^{+}\pi^{-}$, and
$\mathchar 28931\relax^{0}_{b}\to{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}pK^{-}$ peaking backgrounds,333The yields of the subtracted
backgrounds can be considered as fixed, since the sum of negative weights used
to subtract them is constant in the nominal fit. are evaluated by repeating
the fit to the invariant mass varying the normalisation of all background
sources by either plus or minus one standard deviation of its estimated yield.
For each of the new mass fits, the angular fit is repeated using the
corresponding new sets of $sW\\!eights$. The deviations on each of the angular
parameters are then added in quadrature.

Correlations between the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}\pi^{+}$ invariant mass and the cosine of the helicity angle
$\theta_{\mu}$ are taken into account in the nominal fit model, where the mass
fit is performed in five bins of $\cos(\theta_{\mu})$. In order to evaluate
systematic uncertainties due to these correlations, the mass fit is repeated
with the full range of $\cos(\theta_{\mu})$ divided into four or six equal
bins. For each new mass fit, the angular fit is repeated using the
corresponding set of $sW\\!eights$. The deviations from the nominal result for
each of the variations are summed quadratically and taken as the systematic
uncertainty.

####  8.1.2 Systematic uncertainties related to the angular fit model

In order to account for systematic uncertainties due to the angular
acceptance, two distinct effects are considered, as in Ref. [8]. The first is
due to the limited size of the simulation sample used in the acceptance
estimation. It is estimated by varying the normalisation weights 200 times
following a Gaussian distribution within a five standard deviation range
taking into account their correlations. For each of these sets of
normalisation weights, the angular fit is repeated, resulting in a
distribution for each fitted parameter. The width of the resulting parameter
distribution is taken as the systematic uncertainty. Note that in this
procedure, the normalisation weights are varied independently in each
$m_{K^{-}\pi^{+}}$ bin. The second effect, labelled as data-simulation
corrections in the tables, accounts for differences between the data and the
simulation, using normalisation weights that are determined assuming the
amplitudes measured in Ref. [47]. The difference with respect to the nominal
fit is assigned as a systematic uncertainty. The uncertainties due to the
choice of model for the $C_{\rm SP}$ factors are evaluated as the maximum
differences observed in the measured parameters when computing the $C_{\rm
SP}$ factors with all of the alternative models, as discussed below. Instead
of the nominal propagator for the S–wave, a combination of the
$K_{0}^{*}(800)^{0}$ and $K_{0}^{*}(1430)^{0}$ resonances with a non-resonant
term using the isobar model is considered, as well as a K-matrix [48] version.
A pure phase space term is also used, in order to account for the simplest
possible parametrisation. For the P–wave, the alternative propagators
considered are the $K^{*}(892)^{0}$ alone and a combination of this
contribution with the $K_{1}^{*}(1410)^{0}$ and the $K_{1}^{*}(1430)^{0}$
using the isobar model.

In order to account for the absence of D–wave terms in the nominal fit model a
new fit is performed, including a D–wave component, where the related
parameters are fixed to the values measured in the $K^{*}_{2}(1430)^{0}$
region. The differences in the measured parameters between the results
obtained with and without a D–wave component are taken as the corresponding
systematic uncertainty.

The presence of biases in the fit model itself is studied using parametric
simulation. For this study, 1000 pseudoexperiments were generated and fitted
using the nominal shapes, where the generated parameter values correspond to
the ones obtained in the fit to data. The difference between the generated
value and the mean of the distribution of fitted parameter values are treated
as a source of systematic uncertainty.

Finally, the systematic uncertainties due to the fixed values of the detection
and production asymmetries are estimated by varying their values by $\pm 1$
standard deviation and repeating the fit.

###  8.2 Branching fraction

Several sources of systematic uncertainties on the branching fraction
measurements are studied, summarised along with the results in Table 5:
systematic uncertainties due to the external parameter $f_{d}/f_{s}$ and due
to the branching fraction ${\cal B}(\phi\to K^{+}K^{-})$; systematic
uncertainties due to the ratio of efficiencies obtained from simulation and
due to the angular parameters, propagated into the $\omega$ factors (see Sect.
8.1); and systematic uncertainties affecting the
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{*0}$ yields, which
are determined from the fit to the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}\pi^{-}$ invariant mass and described in Sect. 8.1. Finally, a
systematic uncertainty due to the
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ yield
determined from the fit to the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}K^{-}$ invariant mass distribution, described in Sect. 7.3, is
also taken into account, where only the effect due to the modelling of the
upper tail of the $B^{0}_{s}$ peak is considered (see Sect. 8.1.1). For the
computation of the absolute branching fraction ${\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})$ (see Sect. 7.5), two additional
systematic sources are taken into account, the uncertainties in the external
parameters ${\cal B}(B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{*0})$ and ${\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi)$.

Table 5: Summary of the measured values for the relative branching fractions and their statistical and systematic uncertainties. Relative branching fraction |  $\frac{{\cal B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern 1.67996pt\overline{\kern-1.67996ptK}{}^{*0})}{{\cal B}(B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{*0})}$ (%) |  $\frac{{\cal B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern 1.67996pt\overline{\kern-1.67996ptK}{}^{*0})}{{\cal B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi)}$ (%)
---|---|---
Nominal value | 2.99 | 4.05
Statistical uncertainties | 0.14 | 0.19
Efficiency ratio | 0.04 | 0.05
Angular correction ($\omega$) | 0.09 | 0.07
Mass model (effect on the yield) | 0.06 | 0.08
$f_{d}/f_{s}$ | 0.17 | —
${\cal B}(\phi\to K^{+}K^{-})$ | — | 0.04
Quadratic sum (excluding $f_{d}/f_{s}$) | 0.12 | 0.13
Total uncertainties | 0.25 | 0.23

##  9 Penguin pollution in $\phi_{s}$

###  9.1 Information from $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$

Following the strategy proposed in Refs. [9, 11, 13], the measured branching
fraction, polarisation fractions and $C\\!P$ asymmetries can be used to
quantify the contributions originating from the penguin topologies in
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. To that end, the transition
amplitude for the $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay is written in
the general form

$A\left(B^{0}_{s}\to({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})_{i}\right)=-\lambda\mathcal{A}_{i}\left[1-a_{i}e^{\text{i}\theta_{i}}e^{\text{i}\gamma}\right]\>,$
(23)

where $\lambda=|V_{us}|=0.22548^{+0.00068}_{-0.00034}$ [6] and $i$ labels the
different polarisation states. In the above expression, $\mathcal{A}_{i}$ is a
$C\\!P$-conserving hadronic matrix element that represents the tree topology,
and $a_{i}$ parametrises the relative contribution from the penguin
topologies. The $C\\!P$-conserving phase difference between the two terms is
parametrised by $\theta_{i}$, whereas their weak phase difference is given by
the angle $\gamma$ of the Unitarity Triangle.

Both the branching fraction and the $C\\!P$ asymmetries depend on the penguin
parameters $a_{i}$ and $\theta_{i}$. The dependence of $A^{C\\!P}_{i}$ is
given by [9]

$A^{C\\!P}_{i}=-\frac{2a_{i}\sin\theta_{i}\sin\gamma}{1-2a_{i}\cos\theta_{i}\cos\gamma+a_{i}^{2}}\>.$
(24)

To use the branching fraction information an observable is constructed [9]:

$\displaystyle H_{i}$
$\displaystyle\equiv\frac{1}{\epsilon}\left|\frac{\mathcal{A}^{\prime}_{i}}{\mathcal{A}_{i}}\right|^{2}\frac{\Phi\left(\frac{m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}{m_{B^{0}_{s}}},\frac{m_{\phi}}{m_{B^{0}_{s}}}\right)}{\Phi\left(\frac{m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}{m_{B^{0}_{s}}},\frac{m_{\kern
0.99998pt\overline{\kern-0.99998ptK}{}^{*0}}}{m_{B^{0}_{s}}}\right)}\frac{{\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})_{\text{theo}}}{{\cal
B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)_{\text{theo}}}\frac{f_{i}}{f^{\prime}_{i}}\>,$ (25)
$\displaystyle=\frac{1-2a_{i}\cos\theta_{i}\cos\gamma+a_{i}^{2}}{1+2\epsilon
a^{\prime}_{i}\cos\theta^{\prime}_{i}\cos\gamma+\epsilon^{2}a_{i}^{\prime
2}}\>,$

where $f_{i}^{(\prime)}$ represents the polarisation fraction,

$\epsilon\equiv\frac{\lambda^{2}}{1-\lambda^{2}}=0.0536\pm
0.0003\;\mbox{\cite[cite]{[\@@bibref{}{Charles:2015gya}{}{}]}}\>,$ (26)

and $\Phi(x,y)=\sqrt{(1-(x-y)^{2})(1-(x+y)^{2})}$ is the standard two-body
phase-space function. The primed quantities refer to the
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ channel,
while the non-primed ones refer to
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. The penguin parameters
$a^{\prime}_{i}$ and $\theta^{\prime}_{i}$ are defined in analogy to $a_{i}$
and $\theta_{i}$, and parametrise the transition amplitude of the
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ decay as

$A\left(B^{0}_{s}\to({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{}\phi)_{i}\right)=\left(1-\frac{\lambda^{2}}{2}\right)\mathcal{A}^{\prime}_{i}\left[1+\epsilon
a^{\prime}_{i}e^{\text{i}\theta^{\prime}_{i}}e^{\text{i}\gamma}\right]\>.$
(27)

Assuming $\mathrm{SU}(3)$ flavour symmetry, and neglecting contributions from
exchange and penguin-annihilation topologies, 444We follow the decomposition
introduced in Ref. [49]. which are present in
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ but have no
counterpart in $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, we can identify

$a^{\prime}_{i}=a_{i}\>,\qquad\theta^{\prime}_{i}=\theta_{i}\>.$ (28)

The contributions from the additional decay topologies in
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ can be probed
using the decay $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{}\phi$
[13]. The current upper limit on its branching fraction is ${\cal
B}(B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{}\phi)<1.9\times
10^{-7}$ at 90% confidence level (C.L.) [50], which implies that the size of
these additional contributions is small compared to those associated with the
penguin topologies.

The $H_{i}$ observables are constructed in terms of the theoretical branching
fractions defined at zero decay time, which differ from the measured time-
integrated branching fractions [51] due to the non-zero decay-width difference
$\Delta\Gamma_{s}$ of the $B^{0}_{s}$ meson system [7]. The conversion factor
between the two branching fraction definitions [51] is taken to be

$\frac{{\cal B}(B\to f)_{\text{theo}}}{{\cal B}(B\to
f)}=\frac{1-y_{s}^{2}}{1-y_{s}\eta_{i}\cos(\phi_{s}^{\text{SM}})}\>,$ (29)

where $\eta_{i}$ is the $C\\!P$ eigenvalue of the final state, and
$y_{s}=\Delta\Gamma_{s}/2\Gamma_{s}$. Taking values for $\Gamma_{s}$,
$\Delta\Gamma_{s}$ and $\phi_{s}^{\text{SM}}$ from Refs. [7, 6], the
conversion factor is $1.0608\pm 0.0045$ $(0.9392\pm 0.0045)$ for the
$C\\!P$-even (-odd) states. For the flavour-specific
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay $\eta_{i}=0$, resulting in
a conversion factor of $0.9963\pm 0.0006$. The ratios of hadronic amplitudes
$|\mathcal{A}^{\prime}_{i}/\mathcal{A}_{i}|$ are calculated in Ref. [52]
following the method described in Ref. [53] and using the latest results on
form factors from Light Cone QCD Sum Rules (LCSR) [54]. This leads to

$\displaystyle H_{0}$ $\displaystyle=0.98\pm 0.07\>\text{(stat)}\pm
0.06\>\text{(syst)}\pm 0.26\>(|\mathcal{A}^{\prime}_{i}/\mathcal{A}_{i}|)\>,$
$\displaystyle H_{\parallel}$ $\displaystyle=0.90\pm 0.14\>\text{(stat)}\pm
0.08\>\text{(syst)}\pm 0.21\>(|\mathcal{A}^{\prime}_{i}/\mathcal{A}_{i}|)\>,$
$\displaystyle H_{\perp}$ $\displaystyle=1.46\pm 0.14\>\text{(stat)}\pm
0.11\>\text{(syst)}\pm 0.28\>(|\mathcal{A}^{\prime}_{i}/\mathcal{A}_{i}|)\>.$

Assuming Eq. 28 and external input on the Unitarity Triangle angle
$\gamma=\left(73.2_{-7.0}^{+6.3}\right)^{\circ}$ [6], the penguin parameters
$a_{i}$ and $\theta_{i}$ are obtained from a modified least-squares fit to
$\\{A^{C\\!P}_{i},H_{i}\\}$ in Eq. 24 and Eq. 25. The information on $\gamma$
is included as a Gaussian constraint in the fit. The values obtained for the
penguin parameters are

$\displaystyle a_{0}$ $\displaystyle=0.04^{+0.95}_{-0.04}\>,$
$\displaystyle\theta_{0}$
$\displaystyle=\phantom{-}\left(40^{+140}_{-220}\right)^{\circ}\>,$
$\displaystyle a_{\parallel}$ $\displaystyle=0.32^{+0.57}_{-0.32}\>,$
$\displaystyle\theta_{\parallel}$
$\displaystyle=-\left(15^{+148}_{-14}\right)^{\circ}\>,$ $\displaystyle
a_{\perp}$ $\displaystyle=0.44^{+0.21}_{-0.27}\>,$
$\displaystyle\theta_{\perp}$
$\displaystyle=\phantom{-}\left(175^{+11}_{-10}\right)^{\circ}\>.$

For the longitudinal polarisation state the phase $\theta$ is unconstrained.
Correlations between the experimental inputs are ignored, but the effect of
including them is small. The two-dimensional confidence level contours are
given in Fig. 6. This figure also shows, as different (coloured) bands, the
constraints on the penguin parameters derived from the individual observables
entering the $\chi^{2}$ fit. The thick inner darker line represents the
contour associated with the central value of the input quantity, while the
outer darker lines represent the contours associated with the one standard
deviation changes. For the parallel polarisation the central value of the $H$
observable does not lead to physical solutions in the
$\theta_{\parallel}$–$a_{\parallel}$ plane, and the thick inner line is thus
absent.

Figure 6: Limits on the penguin parameters $a_{i}$ and $\theta_{i}$ obtained
from intersecting contours derived from the $C\\!P$ asymmetries and branching
fraction information in $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. Superimposed are
the confidence level contours obtained from a $\chi^{2}$ fit to the data.
Shown are the longitudinal (top), parallel (middle) and perpendicular (bottom)
polarisation.

When decomposed into its different sources, the angle $\phi_{s}$ takes the
form

$\phi_{s,i}=-2\beta_{s}+\phi_{s}^{\text{BSM}}+\Delta\phi^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{}\phi}_{s,i}(a^{\prime}_{i},\theta^{\prime}_{i})\>,$ (30)

where $-2\beta_{s}$ is the SM contribution, $\phi_{s}^{\text{BSM}}$ is a
possible BSM phase, and $\Delta\phi^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{}\phi}_{s,i}$ is a shift introduced by the presence of penguin
pollution in the decay $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$. In terms of the penguin parameters $a^{\prime}_{i}$ and
$\theta^{\prime}_{i}$, the shift
$\Delta\phi^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{}\phi}_{s,i}$ is
defined as

$\tan(\Delta\phi^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{}\phi}_{s,i})=\frac{2\epsilon
a^{\prime}_{i}\cos\theta^{\prime}_{i}\sin\gamma+\epsilon^{2}a^{\prime
2}_{i}\sin(2\gamma)}{1+2\epsilon
a^{\prime}_{i}\cos\theta^{\prime}_{i}\cos\gamma+\epsilon^{2}a^{\prime
2}_{i}\cos(2\gamma)}\>.$ (31)

Using Eqs. 28 and 31, the fit results on $a_{i}$ and $\theta_{i}$ given above
constrain this phase shift, giving

$\displaystyle\Delta\phi^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{}\phi}_{s,0}$ $\displaystyle=\phantom{-}0.003$ ${}^{+0.084}_{-0.011}$
(stat) ${}^{+0.014}_{-0.009}$ (syst) ${}^{+0.047}_{-0.030}$
$\displaystyle\>(|\mathcal{A}^{\prime}_{i}/\mathcal{A}_{i}|)\>,$
$\displaystyle\Delta\phi^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{}\phi}_{s,\parallel}$ $\displaystyle=\phantom{-}0.031$
${}^{+0.047}_{-0.037}$ (stat) ${}^{+0.010}_{-0.013}$ (syst) $\displaystyle\pm
0.032$ $\displaystyle\>(|\mathcal{A}^{\prime}_{i}/\mathcal{A}_{i}|)\>,$
$\displaystyle\Delta\phi^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{}\phi}_{s,\perp}$ $\displaystyle=-0.045$ $\displaystyle\pm 0.012$
(stat) $\displaystyle\pm 0.008$ (syst) ${}^{+0.017}_{-0.024}$
$\displaystyle\>(|\mathcal{A}^{\prime}_{i}/\mathcal{A}_{i}|)\>,$

which is in good agreement with the values measured in Ref. [15], and with the
predictions given in Refs. [13, 12, 14].

The above results are obtained assuming $\mathrm{SU}(3)$ flavour symmetry and
neglecting contributions from additional decay topologies. Because
$a_{i}e^{\text{i}\theta_{i}}$ represents a ratio of hadronic amplitudes, the
leading factorisable $\mathrm{SU}(3)$-breaking effects cancel, and the
relation between $a_{i}e^{\text{i}\theta_{i}}$ and
$a^{\prime}_{i}e^{\text{i}\theta^{\prime}_{i}}$ is only affected by non-
factorisable $\mathrm{SU}(3)$-breaking. This can be parametrised using two
$\mathrm{SU}(3)$-breaking parameters $\xi$ and $\delta$ as

$a^{\prime}_{i}=\xi\times
a_{i}\>,\qquad\theta^{\prime}_{i}=\theta_{i}+\delta\>.$ (32)

The above quoted results assume $\xi=1$ and $\delta=0$. The dependence of the
uncertainty on $\Delta\phi_{s,i}^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi}$ on the uncertainty on $\xi$ is illustrated in Fig. 7, while the
dependence on the uncertainty on $\delta$ is negligible for the solutions
obtained for $\\{a_{i},\theta_{i}\\}$.

Figure 7: Dependence of the uncertainty on the penguin shift
$\Delta\phi_{s,i}^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi}$ on the
uncertainty on $\xi$. The bands correspond to the 68% C.L. The longitudinal
(top), parallel (middle) and perpendicular (bottom) polarisations are shown.

###  9.2 Combination with $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{}\rho^{0}$

The information on the penguin parameters obtained from
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ can be complemented with similar
information from the $\mathrm{SU}(3)$-related mode
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$ [15]. Both
modes describe a $\bar{b}\to\bar{c}c\bar{d}$ transition, and are related by
exchanging the spectator $s\leftrightarrow d$ quarks. The decay amplitude of
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$ is also
parametrised as

$A\left(B^{0}\to({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{}\rho^{0})_{i}\right)=-\lambda\tilde{\mathcal{A}}_{i}\left[1-\tilde{a}_{i}e^{\text{i}\tilde{\theta}_{i}}e^{\text{i}\gamma}\right]\>,$
(33)

which is the equivalent of Eq. 23. In contrast to
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, however, $\tilde{a}_{i}$ and
$\tilde{\theta}_{i}$ also include contributions from exchange and penguin-
annihilation topologies, which are present in
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$ but have no
counterpart in $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. Assuming
$\mathrm{SU}(3)$ symmetry, and neglecting the contributions from the
additional decay topologies in
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ and
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$, the relation
in Eq. 28 can be extended to

$a^{\prime}_{i}=a_{i}=\tilde{a}_{i}\>,\qquad\theta^{\prime}_{i}=\theta_{i}=\tilde{\theta}_{i}\>,$
(34)

which allows a combined fit to be performed to the $C\\!P$ asymmetries and
branching fraction information in
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$.

The $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$ decay
exhibits decay-time-dependent $C\\!P$ violation, which is described by two
parameters, the direct $C\\!P$ asymmetry $C_{i}$, which in the
$\mathrm{SU}(3)$ limit is related to $A_{i}^{C\\!P}$ as
$C_{i}=-A_{i}^{C\\!P}$, and the mixing-induced $C\\!P$ asymmetry $S_{i}$.
Their dependence on the penguin parameters $\tilde{a}_{i}$ and
$\tilde{\theta}_{i}$ is given by

$\displaystyle C_{i}$
$\displaystyle=\phantom{-\eta_{f}~{}~{}}\frac{2\,\tilde{a}_{i}\sin\tilde{\theta}_{i}\sin\gamma}{1-2\,\tilde{a}_{i}\cos\tilde{\theta}_{i}\cos\gamma+\tilde{a}_{i}^{2}}\>,$
(35) $\displaystyle S_{i}$
$\displaystyle=-\eta_{i}\left[\frac{\sin\phi_{d}-2\,\tilde{a}_{i}\cos\tilde{\theta}_{i}\sin(\phi_{d}+\gamma)+\tilde{a}_{i}^{2}\sin(\phi_{d}+2\gamma)}{1-2\,\tilde{a}_{i}\cos\tilde{\theta}_{i}\cos\gamma+\tilde{a}_{i}^{2}}\right]\>,$
(36)

where $\eta_{i}$ is the polarisation-dependent $C\\!P$ eigenvalue of the
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$ decay, and
$\phi_{d}$ is a $C\\!P$-violating phase arising from the interference between
$B^{0}$–$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing and the
subsequent $B^{0}$ decay. The use of $S_{i}$ to constrain the penguin
parameters $a_{i}$ and $\theta_{i}$ requires external information on the
$C\\!P$ phase $\phi_{d}$. The most precise value of $\phi_{d}$ is determined
from $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{0}$ decays, but
this determination is also affected by penguin pollution. A recent study of
the penguin effects in $B^{+}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}$, $B^{+}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{+}$, $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\pi^{0}$
and $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{0}_{\mathrm{\scriptscriptstyle S}}$ decays is performed in Ref.
[13], with the latest numerical update [52], including the results from Refs.
[55, 56, 6], leading to $\phi_{d}=0.767\pm 0.029\rm\,rad$.

In addition, a second set of $H_{i}$ observables can be constructed by
replacing $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ by
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$ in Eq. 25. To
minimise the theoretical uncertainties associated with the use of these
$H_{i}$ observables, the strategy proposed in Ref. [13] is adopted. That is,
the relation

$\left|\frac{\mathcal{A}^{\prime}_{i}}{\mathcal{A}_{i}}\right|\equiv\left|\frac{\mathcal{A}^{\prime}_{i}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)}{\mathcal{A}_{i}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})}\right|=\left|\frac{\mathcal{A}^{\prime}_{i}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)}{\mathcal{A}_{i}(B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rho^{0})}\right|$ (37)

between the hadronic amplitudes in
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$ is assumed,
and therefore relying on theoretical input from LCSR is no longer needed.
Instead, the ratio $|\mathcal{A^{\prime}}/\mathcal{A}|$ can be determined
directly from the fit, providing experimental information on this quantity.
Effectively, the three $C\\!P$ asymmetry parameters $A_{i}^{C\\!P}$, $C_{i}$
and $S_{i}$ determine the penguin parameters $a_{i}$ and $\theta_{i}$. Thus,
this result for $a_{i}$ and $\theta_{i}$ predicts the values of the two
observables $H_{i}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0})$ and
$H_{i}(B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0})$. By
comparing these two quantities with the branching fraction and polarisation
information on $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$,
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$ and
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$, the hadronic
amplitude ratios $|\mathcal{A}^{\prime}_{i}/\mathcal{A}_{i}|$ can be
determined. The impact of the $H_{i}$ observables on the penguin parameters
$a_{i}$ and $\theta_{i}$ is negligible in the combined fit.

For the combined analysis of
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$ a modified
least-squares fit is performed. External inputs on
$\gamma=\left(73.2_{-7.0}^{+6.3}\right)^{\circ}$ [6] and $\phi_{d}=0.767\pm
0.029\rm\,rad$ [52] are included as Gaussian constraints in the fit. The
values obtained from the fit are

$\displaystyle a_{0}$ $\displaystyle=0.01^{+0.10}_{-0.01}\>,$
$\displaystyle\theta_{0}$
$\displaystyle=-\left(83^{+97}_{-263}\right)^{\circ}\>,$
$\displaystyle\left|\frac{\mathcal{A}^{\prime}_{0}}{\mathcal{A}_{0}}\right|$
$\displaystyle=1.195^{+0.074}_{-0.056}\>,$ $\displaystyle a_{\parallel}$
$\displaystyle=0.07^{+0.11}_{-0.05}\>,$ $\displaystyle\theta_{\parallel}$
$\displaystyle=-\left(85^{+72}_{-63}\right)^{\circ}\>,$
$\displaystyle\left|\frac{\mathcal{A}^{\prime}_{\parallel}}{\mathcal{A}_{\parallel}}\right|$
$\displaystyle=1.238^{+0.104}_{-0.080}\>,$ $\displaystyle a_{\perp}$
$\displaystyle=0.04^{+0.12}_{-0.04}\>,$ $\displaystyle\theta_{\perp}$
$\displaystyle=\phantom{-}\left(38^{+142}_{-218}\right)^{\circ}\>,$
$\displaystyle\left|\frac{\mathcal{A}^{\prime}_{\perp}}{\mathcal{A}_{\perp}}\right|$
$\displaystyle=1.042^{+0.081}_{-0.063}\>,$

with the two-dimensional confidence level contours given in Fig. 8, which also
shows the constraints on the penguin parameters derived from the individual
observables entering the $\chi^{2}$ fit as different bands. Note that the
plotted contours for the two $H$ observables do not include the uncertainty
due to $|\mathcal{A}^{\prime}/\mathcal{A}|$.

Figure 8: Limits on the penguin parameters $a_{i}$ and $\theta_{i}$ obtained
from intersecting contours derived from the $C\\!P$ asymmetries and branching
fraction information in $B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$. Superimposed
are the confidence level contours obtained from a $\chi^{2}$ fit to the data.
The longitudinal (top), parallel (middle) and perpendicular (bottom)
polarisations are shown.

The results on the penguin phase shift derived from the above results on
$a_{i}$ and $\theta_{i}$ are

$\displaystyle\Delta\phi^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{}\phi}_{s,0}$ $\displaystyle=0.000^{+0.009}_{-0.011}\>\text{(stat)}$
${}^{+0.004}_{-0.009}$ $\displaystyle\>\text{(syst)}\rm\,rad\>,$
$\displaystyle\Delta\phi^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{}\phi}_{s,\parallel}$
$\displaystyle=0.001^{+0.010}_{-0.014}\>\text{(stat)}$ $\displaystyle\>\pm
0.008$ $\displaystyle\>\text{(syst)}\rm\,rad\>,$
$\displaystyle\Delta\phi^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{}\phi}_{s,\perp}$
$\displaystyle=0.003^{+0.010}_{-0.014}\>\text{(stat)}$ $\displaystyle\>\pm
0.008$ $\displaystyle\>\text{(syst)}\rm\,rad\>.$

These results are dominated by the input from the $C\\!P$ asymmetries in
$B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$, and show
that the penguin pollution in the determination of $\phi_{s}$ is small.

##  10 Conclusions

Using the full LHCb Run I data sample, the branching fraction, the
polarisation fractions and the direct $C\\!P$ violation parameters in
$B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays have been measured. The
results are

${\cal B}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=\left(4.14\pm
0.18\text{(stat)}\pm 0.26\text{(syst)}\pm 0.24(f_{d}/f_{s})\right)\times
10^{-5}$
$\begin{array}[]{rclllllll}f_{0}&\;=&\phantom{-}0.497&\pm&0.025&\text{(stat)}&\pm&0.025&\text{(syst)}\\\
f_{\|}&\;=&\phantom{-}0.179&\pm&0.027&\text{(stat)}&\pm&0.013&\text{(syst)}\\\
A^{C\\!P}_{0}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})&\;=&-0.048&\pm&0.057&\text{(stat)}&\pm&0.020&\text{(syst)}\\\
A^{C\\!P}_{\|}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})&\;=&\phantom{-}0.171&\pm&0.152&\text{(stat)}&\pm&0.028&\text{(syst)}\\\
A^{C\\!P}_{\perp}(B^{0}_{s}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})&\;=&-0.049&\pm&0.096&\text{(stat)}&\pm&0.025&\text{(syst)\>,}\\\
\end{array}$

which supersede those of Ref. [16], with precision improved by a factor of
$2-3$. The shift on $\phi_{s}$ due to penguin pollution is estimated from a
combination with the $B^{0}\to{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rho^{0}$ channel [15], and is found be to compatible with the result
from the earlier analysis.

## 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); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); INFN (Italy);
FOM and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania);
MinES and FANO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU
(Ukraine); STFC (United Kingdom); NSF (USA). The Tier1 computing centres are
supported by IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and
SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom). We are indebted
to the communities behind the multiple open source software packages on which
we depend. We are also thankful for the computing resources and the access to
software R&D tools provided by Yandex LLC (Russia). Individual groups or
members have received support from EPLANET, Marie Skłodowska-Curie Actions and
ERC (European Union), Conseil général de Haute-Savoie, Labex ENIGMASS and
OCEVU, Région Auvergne (France), RFBR (Russia), XuntaGal and GENCAT (Spain),
Royal Society and Royal Commission for the Exhibition of 1851 (United
Kingdom).

Appendices

##  Appendix A Angular acceptance

To take into account angular acceptance effects, ten normalisation weights,
$\xi_{ij}$, are computed and embedded in the normalization integral of the
angular distribution given in Eq. 4 following the procedure described in Ref.
[57]. Using the transversity amplitude basis, the fitting ${\rm PDF}$ can be
written as

$\frac{\text{d}\Gamma(\theta_{K},\theta_{\mu},\varphi_{h})}{\text{d}\Omega}=\frac{\sum_{i}\sum_{j\leq
i}{\mathcal{R}e{[A_{i}A_{j}^{*}\>F_{ij}(\theta_{K},\theta_{\mu},\varphi_{h})]}}}{\sum_{k}\sum_{l\leq
k}{\mathcal{R}e{[A_{k}A_{l}^{*}\>\int{F_{kl}(\theta_{K}^{\prime},\theta_{\mu}^{\prime},\varphi_{h}^{\prime})\>\epsilon_{\Omega}(\theta_{K}^{\prime},\theta_{\mu}^{\prime},\varphi_{h}^{\prime})\>\text{d}\Omega^{\prime}}}]}},$
(38)

where the real or imaginary angular functions
$F_{ij}(\theta_{K},\theta_{\mu},\varphi_{h})$ are obtained when combining Eq.
4 and Eq. 5-8, and where
$\epsilon_{\Omega}(\theta_{K},\theta_{\mu},\varphi_{h})$ denotes the angular
acceptance. The normalization weights correspond to the integrals

$\xi_{ij}=\\\
\begin{cases}\int{\mathcal{R}e[F_{ij}(\theta_{K},\theta_{\mu},\varphi_{h})]\>\epsilon_{\Omega}(\theta_{K},\theta_{\mu},\varphi_{h})\>\text{d}\Omega},\;\text{if
}F_{ij}\in\mathbb{R}\,,\\\
\int{\mathcal{I}m[F_{ij}(\theta_{K},\theta_{\mu},\varphi_{h})]\>\epsilon_{\Omega}(\theta_{K},\theta_{\mu},\varphi_{h})\>\text{d}\Omega},\;\text{if
}F_{ij}\in\mathbb{I}\,.\\\ \end{cases}$ (39)

In the absence of acceptance effects, the normalisation weights related to the
interference terms are equal to zero by definition, whereas those related to
each polarisation amplitude squared are equal to unity. Eight sets of
normalisation weights are calculated separately, one for each
$m_{K^{-}\pi^{+}}$ bin and kaon charge.

Table 6: Corrected angular acceptance weights for $K^{-}\pi^{+}$ events lying in the first $m_{K^{-}\pi^{+}}$ bin. The $\xi_{ij}$ weights are normalised with respect to the $\xi_{00}$ weight.  $ij$ | $\xi_{ij}/\xi_{00}$
---|---
1 | (00) | $1.000$
2 | ($\parallel\parallel$) | $+1.379\pm 0.029$
3 | ($\perp\perp$) | $+1.388\pm 0.030$
4 | ($\parallel\perp$) | $+0.035\pm 0.019$
5 | (0$\parallel$) | $-0.003\pm 0.012$
6 | (0$\perp$) | $+0.010\pm 0.011$
7 | (SS) | $+1.190\pm 0.019$
8 | (S$\parallel$) | $-0.042\pm 0.017$
9 | (S$\perp$) | $+0.029\pm 0.016$
10 | (S0) | $-0.929\pm 0.024$

In order to correct both for imperfections in the detector simulation and for
the absence of any S–wave component in the simulation sample, the weights are
refined using an iterative procedure where the angular acceptance is re-
evaluated recursively until it does not change significantly. Table 6 gives
one set of normalisation weights after the iterative procedure. The effect of
this correction is below one standard deviation for all the normalisation
weights except for the (S0) weight. This is expected due to the rapid
efficiency drop close to $\cos\theta_{K}=1$ which directly impacts the (S0)
weight. At each step of this procedure the simulation sample is corrected both
for the absence of an S–wave component and for the imperfections in the
detector simulation. For the first correction, the angular fit result to data
is used, whereas for the second the kaon and muon track momentum distributions
of data are used. In both cases the correction is implemented by assigning
weights to each event of the simulation sample.

##  Appendix B Correlation matrix

The statistical-only correlation matrix of the angular parameters obtained
from the fit to data, as described in Sect. 8.1, is given in Table 7. Here,
the superscript $l=0,1,2,3$ in $F_{\rm S}^{l}$ and $\delta_{\rm S}^{l}$
represent the number of the $m_{K^{-}\pi^{+}}$ bin as defined in Table 2.

Table 7:  Statistical correlation matrix for the parameters from the angular
fit.

| $A^{C\\!P}_{0}$ | $A^{C\\!P}_{\rm S}$ | $A^{C\\!P}_{\|}$ | $A^{C\\!P}_{\perp}$ | $F_{\rm S}^{0}$ | $F_{\rm S}^{1}$ | $F_{\rm S}^{2}$ | $F_{\rm S}^{3}$ | $\delta_{\parallel}$ | $\delta_{\perp}$ | $\delta_{\rm S}^{0}$ | $\delta_{\rm S}^{1}$ | $\delta_{\rm S}^{2}$ | $\delta_{\rm S}^{3}$ | $f_{0}$ | $f_{\|}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$A^{C\\!P}_{0}$ | $+1.00$ | $-0.12$ | $-0.11$ | $-0.17$ | $-0.13$ | $-0.02$ | $-0.06$ | $-0.01$ | $+0.03$ | $+0.02$ | $+0.10$ | $-0.00$ | $+0.07$ | $+0.01$ | $+0.06$ | $-0.05$
$A^{C\\!P}_{\rm S}$ |  | $+1.00$ | $-0.14$ | $-0.12$ | $+0.16$ | $-0.12$ | $+0.03$ | $-0.10$ | $+0.00$ | $-0.06$ | $+0.02$ | $+0.07$ | $+0.05$ | $+0.07$ | $+0.01$ | $+0.03$
$A^{C\\!P}_{\|}$ |  |  | $+1.00$ | $-0.49$ | $+0.02$ | $+0.09$ | $-0.02$ | $+0.08$ | $+0.09$ | $+0.06$ | $-0.06$ | $-0.04$ | $-0.05$ | $-0.12$ | $-0.04$ | $-0.07$
$A^{C\\!P}_{\perp}$ |  |  |  | $+1.00$ | $-0.00$ | $-0.01$ | $-0.06$ | $-0.07$ | $-0.09$ | $-0.03$ | $-0.03$ | $+0.01$ | $-0.02$ | $+0.07$ | $+0.01$ | $-0.06$
$F_{\rm S}^{0}$ |  |  |  |  | $+1.00$ | $+0.01$ | $-0.01$ | $-0.03$ | $-0.10$ | $-0.24$ | $-0.77$ | $+0.01$ | $+0.04$ | $-0.00$ | $+0.10$ | $-0.09$
$F_{\rm S}^{1}$ |  |  |  |  |  | $+1.00$ | $-0.01$ | $-0.00$ | $-0.02$ | $-0.05$ | $-0.01$ | $-0.25$ | $+0.03$ | $-0.01$ | $+0.15$ | $-0.10$
$F_{\rm S}^{2}$ |  |  |  |  |  |  | $+1.00$ | $+0.01$ | $-0.04$ | $+0.07$ | $+0.01$ | $-0.00$ | $-0.22$ | $+0.00$ | $-0.02$ | $+0.04$
$F_{\rm S}^{3}$ |  |  |  |  |  |  |  | $+1.00$ | $+0.08$ | $+0.08$ | $+0.00$ | $-0.01$ | $-0.03$ | $-0.29$ | $-0.09$ | $+0.04$
$\delta_{\parallel}$ |  |  |  |  |  |  |  |  | $+1.00$ | $+0.62$ | $+0.10$ | $+0.14$ | $+0.03$ | $+0.11$ | $+0.04$ | $-0.03$
$\delta_{\perp}$ |  |  |  |  |  |  |  |  |  | $+1.00$ | $+0.17$ | $+0.13$ | $-0.02$ | $+0.13$ | $+0.05$ | $-0.04$
$\delta_{\rm S}^{0}$ |  |  |  |  |  |  |  |  |  |  | $+1.00$ | $+0.04$ | $+0.03$ | $+0.04$ | $+0.08$ | $+0.04$
$\delta_{\rm S}^{1}$ |  |  |  |  |  |  |  |  |  |  |  | $+1.00$ | $+0.04$ | $+0.04$ | $+0.13$ | $-0.05$
$\delta_{\rm S}^{2}$ |  |  |  |  |  |  |  |  |  |  |  |  | $+1.00$ | $+0.04$ | $+0.27$ | $-0.08$
$\delta_{\rm S}^{3}$ |  |  |  |  |  |  |  |  |  |  |  |  |  | $+1.00$ | $+0.11$ | $+0.00$
$f_{0}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  | $+1.00$ | $-0.34$
$f_{\|}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | $+1.00$

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LHCb collaboration

R. Aaij38, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, S. Akar6,
J. Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P.
Alvarez Cartelle53, A.A. Alves Jr57, S. Amato2, S. Amerio22, Y. Amhis7, L.
An3, L. Anderlini17, J. Anderson40, G. Andreassi39, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, P.
d’Argent11, A. Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,m, M.
Baalouch5, S. Bachmann11, J.J. Back48, A. Badalov36, C. Baesso60, W.
Baldini16,38, R.J. Barlow54, C. Barschel38, S. Barsuk7, W. Barter38, V.
Batozskaya28, V. Battista39, A. Bay39, L. Beaucourt4, J. Beddow51, F.
Bedeschi23, I. Bediaga1, L.J. Bel41, V. Bellee39, N. Belloli20,j, I.
Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson38, J. Benton46, A.
Berezhnoy32, R. Bernet40, A. Bertolin22, M.-O. Bettler38, M. van Beuzekom41,
A. Bien11, S. Bifani45, P. Billoir8, T. Bird54, A. Birnkraut9, A. Bizzeti17,h,
T. Blake48, F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N.
Bondar30,38, W. Bonivento15, S. Borghi54, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, S. Braun11, M. Britsch10, T. Britton59, J. Brodzicka54,
N.H. Brook46, E. Buchanan46, A. Bursche40, J. Buytaert38, S. Cadeddu15, R.
Calabrese16,f, M. Calvi20,j, M. Calvo Gomez36,o, P. Campana18, D. Campora
Perez38, L. Capriotti54, A. Carbone14,d, G. Carboni24,k, R. Cardinale19,i, A.
Cardini15, P. Carniti20,j, L. Carson50, K. Carvalho Akiba2,38, G. Casse52, L.
Cassina20,j, L. Castillo Garcia38, M. Cattaneo38, Ch. Cauet9, G. Cavallero19,
R. Cenci23,s, M. Charles8, Ph. Charpentier38, M. Chefdeville4, S. Chen54,
S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40, X. Cid Vidal38, G.
Ciezarek41, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47, J. Closier38, V.
Coco38, J. Cogan6, E. Cogneras5, V. Cogoni15,e, L. Cojocariu29, G.
Collazuol22, P. Collins38, A. Comerma-Montells11, A. Contu15, A. Cook46, M.
Coombes46, S. Coquereau8, G. Corti38, M. Corvo16,f, B. Couturier38, G.A.
Cowan50, D.C. Craik48, A. Crocombe48, M. Cruz Torres60, S. Cunliffe53, R.
Currie53, C. D’Ambrosio38, E. Dall’Occo41, J. Dalseno46, P.N.Y. David41, A.
Davis57, K. De Bruyn6, S. De Capua54, M. De Cian11, J.M. De Miranda1, L. De
Paula2, P. De Simone18, C.-T. Dean51, D. Decamp4, M. Deckenhoff9, L. Del
Buono8, N. Déléage4, M. Demmer9, D. Derkach65, O. Deschamps5, F. Dettori38, B.
Dey21, A. Di Canto38, F. Di Ruscio24, H. Dijkstra38, S. Donleavy52, F.
Dordei11, M. Dorigo39, A. Dosil Suárez37, D. Dossett48, A. Dovbnya43, K.
Dreimanis52, L. Dufour41, G. Dujany54, F. Dupertuis39, P. Durante38, R.
Dzhelyadin35, A. Dziurda26, A. Dzyuba30, S. Easo49,38, U. Egede53, V.
Egorychev31, S. Eidelman34, S. Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L.
Eklund51, I. El Rifai5, Ch. Elsasser40, S. Ely59, S. Esen11, H.M. Evans47, T.
Evans55, A. Falabella14, C. Färber38, N. Farley45, S. Farry52, R. Fay52, D.
Ferguson50, V. Fernandez Albor37, F. Ferrari14, F. Ferreira Rodrigues1, M.
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L. Giubega29, V.V. Gligorov38, C. Göbel60, D. Golubkov31, A. Golutvin53,31,38,
A. Gomes1,a, C. Gotti20,j, M. Grabalosa Gándara5, R. Graciani Diaz36, L.A.
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Herwijnen38, M. Heß63, A. Hicheur2, D. Hill55, M. Hoballah5, C. Hombach54, W.
Hulsbergen41, T. Humair53, N. Hussain55, D. Hutchcroft52, D. Hynds51, M.
Idzik27, P. Ilten56, R. Jacobsson38, A. Jaeger11, J. Jalocha55, E. Jans41, A.
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Jost38, N. Jurik59, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38,†,
S. Karodia51, M. Kecke11, M. Kelsey59, I.R. Kenyon45, M. Kenzie38, T. Ketel42,
B. Khanji20,38,j, C. Khurewathanakul39, S. Klaver54, K. Klimaszewski28, O.
Kochebina7, M. Kolpin11, I. Komarov39, R.F. Koopman42, P. Koppenburg41,38, M.
Kozeiha5, L. Kravchuk33, K. Kreplin11, M. Kreps48, G. Krocker11, P.
Krokovny34, F. Kruse9, W. Krzemien28, W. Kucewicz26,n, M. Kucharczyk26, V.
Kudryavtsev34, A. K. Kuonen39, K. Kurek28, T. Kvaratskheliya31, D.
Lacarrere38, G. Lafferty54, A. Lai15, D. Lambert50, G. Lanfranchi18, C.
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Leerdam41, J.-P. Lees4, R. Lefèvre5, A. Leflat32,38, J. Lefrançois7, E. Lemos
Cid37, O. Leroy6, T. Lesiak26, B. Leverington11, Y. Li7, T. Likhomanenko65,64,
M. Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, X. Liu3, D.
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Luo50, A. Lupato22, E. Luppi16,f, O. Lupton55, A. Lusiani23, F. Machefert7, F.
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Mancinelli6, P. Manning59, A. Mapelli38, J. Maratas5, J.F. Marchand4, U.
Marconi14, C. Marin Benito36, P. Marino23,38,s, J. Marks11, G. Martellotti25,
M. Martin6, M. Martinelli39, D. Martinez Santos37, F. Martinez Vidal66, D.
Martins Tostes2, A. Massafferri1, R. Matev38, A. Mathad48, Z. Mathe38, C.
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D. Melnychuk28, M. Merk41, E Michielin22, D.A. Milanes62, M.-N. Minard4, D.S.
Mitzel11, J. Molina Rodriguez60, I.A. Monroy62, S. Monteil5, M. Morandin22, P.
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Mussini14, B. Muster39, P. Naik46, T. Nakada39, R. Nandakumar49, A. Nandi55,
I. Nasteva2, M. Needham50, N. Neri21, S. Neubert11, N. Neufeld38, M. Neuner11,
A.D. Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,p, V. Niess5, R. Niet9, N.
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Mucha27, V. Obraztsov35, S. Ogilvy51, O. Okhrimenko44, R. Oldeman15,e, C.J.G.
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Pilař48, D. Pinci25, A. Pistone19, A. Piucci11, S. Playfer50, M. Plo
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Reis1, S. Ricciardi49, S. Richards46, M. Rihl38, K. Rinnert52, V. Rives
Molina36, P. Robbe7,38, A.B. Rodrigues1, E. Rodrigues54, J.A. Rodriguez
Lopez62, P. Rodriguez Perez54, S. Roiser38, V. Romanovsky35, A. Romero
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1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
63Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
64National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
65Yandex School of Data Analysis, Moscow, Russia, associated to 31
66Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
67Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 41

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità degli Studi di Milano, Milano, Italy
†Deceased