EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2019-121 LHCb-PAPER-2019-019 23 July 2019

Measurement of $C\\!P$ violation in the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$
decay and search for the ${{B}^{0}}\rightarrow\phi\phi$ decay

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

A measurement of the time-dependent $C\\!P$-violating asymmetry in
${{B}^{0}_{s}}\rightarrow\phi\phi$ decays is presented. Using a sample of
proton-proton collision data corresponding to an integrated luminosity of
$5.0\text{\,fb}^{-1}$ collected by the LHCb experiment at centre-of-mass
energies $\sqrt{s}=7$ TeV in 2011, 8  TeV in 2012 and 13  TeV in 2015 and
2016, a signal yield of around 9000 ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$
decays is obtained. The $C\\!P$-violating phase $\phi_{s}^{s\bar{s}s}$ is
measured to be $-0.073\pm 0.115\text{\,(stat)}\pm
0.027\text{\,(syst)}\text{\,rad}$, under the assumption it is independent of
the helicity of the $\phi\phi$ decay. In addition, the $C\\!P$-violating
phases of the transverse polarisations under the assumption of $C\\!P$
conservation of the longitudinal phase are measured. The helicity-independent
direct $C\\!P$-violation parameter is also measured, and is found to be
$|\lambda|=0.99\pm 0.05\text{\,(stat)}\pm 0.01\text{\,(syst)}$. In addition,
$T$-odd triple-product asymmetries are measured. The results obtained are
consistent with the hypothesis of $C\\!P$ conservation in
${\overline{b}}\rightarrow{\overline{s}}{{s}{\overline{s}}}$ transitions.
Finally, a limit on the branching fraction of the
${{B}^{0}}\rightarrow\phi\phi$ decay is determined to be
$\mathcal{B}({{B}^{0}}\rightarrow\phi\phi)<2.7\times 10^{-8}$ at 90 %
confidence level.

Published in JHEP 12 (2019) 155

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

##  1 Introduction

In the Standard Model (SM) the ${{B}^{0}_{s}}\rightarrow\phi\phi$ decay, where
the $\phi(1020)$ is implied throughout this paper, is forbidden at tree level
and proceeds predominantly via a gluonic
${\overline{b}}\rightarrow{\overline{s}}{{s}{\overline{s}}}$ loop (penguin)
process. Hence, this channel provides an excellent probe of new heavy
particles entering the penguin quantum loops [1, 2, 3]. In the SM, $C\\!P$
violation is governed by a single phase in the Cabibbo–Kobayashi–Maskawa quark
mixing matrix [4, *Cabibbo:1963yz]. Interference caused by the resulting weak
phase difference between the ${B}^{0}_{s}$-${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ oscillation and decay
amplitudes leads to a $C\\!P$ asymmetry in the decay-time distributions of
${{B}^{0}_{s}}$ and ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$
mesons. For ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ and
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ decays, which proceed via
${\overline{b}}\rightarrow{\overline{s}}{{c}{\overline{c}}}$ transitions, the
SM prediction of the weak phase is
$-2\arg\left(-V_{ts}V_{tb}^{*}/V_{cs}V_{cb}^{*}\right)=-0.0369^{+0.0010}_{-0.0007}\;\rm
rad$ according to the CKMfitter group [6], and
$-2\arg\left(-V_{ts}V_{tb}^{*}/V_{cs}V_{cb}^{*}\right)=-0.0370\pm 0.0010\;\rm
rad$ according to the UTfit collaboration [7]. The LHCb collaboration has
measured the weak phase in several decay processes:
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$,
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$,
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ for the ${{K}^{+}}{{K}^{-}}$ invariant mass region
above 1.05 $\text{\,Ge\kern-1.00006ptV\\!/}c$,
${{B}^{0}_{s}}\rightarrow\psi(2S)\phi$ and ${{B}^{0}_{s}}\rightarrow
D_{s}^{+}D_{s}^{-}$, corresponding to the combined result of $-0.041\pm
0.025\text{\,rad}$ [8]. These measurements are consistent with the SM
prediction and place stringent constraints on $C\\!P$ violation in
${B}^{0}_{s}$-${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$
oscillations [9]. The $C\\!P$-violating phase,
$\phi_{{s}}^{{{s}{\overline{s}}}{s}}$, in the
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decay is expected to be small in the SM.
Calculations using quantum chromodynamics factorisation (QCDf) provide an
upper limit of $0.02\text{\,rad}$ for its absolute value [1, 2, 3]. The
previous most accurate measurement is
${\phi_{{s}}^{{{s}{\overline{s}}}{s}}}=-0.17\pm 0.15\text{\,(stat)}\pm
0.03\text{\,(syst)}\text{\,rad}$ [10].

$C\\!P$ violation can also be probed by time-integrated triple-product
asymmetries. These are formed from $T$-odd combinations of the momenta of the
final-state particles. These asymmetries complement the decay-time-dependent
measurement [11] and are expected to be close to zero in the SM [12]. Previous
measurements of the triple-product asymmetries in ${B}^{0}_{s}$ decays from
the LHCb and CDF collaborations [10, 13] have shown no significant deviations
from zero.

The ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decay is a $P\rightarrow VV$ decay,
where $P$ denotes a pseudoscalar and $V$ a vector meson. This gives rise to
longitudinal and transverse polarisation of the final states with respect to
their direction of flight in the ${B}^{0}_{s}$ reference frame, the fractions
of which are denoted by $f_{L}$ and $f_{T}$, respectively. In the heavy quark
limit, $f_{L}$ is expected to be close to unity at tree level due to the
vector-axial structure of charged weak currents [2]. This is found to be the
case for tree-level $B$ decays measured at the $B$ Factories [14, 15, 16, 17,
18, 19]. However, the dynamics of penguin transitions are more complicated.
Previously LHCb reported a value of $f_{L}\equiv|A_{0}|^{2}=0.364\pm 0.012$ in
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decays [10]. The measurement is in
agreement with predictions from QCD factorisation [2, 3]. The observed value
of $f_{L}$ is significantly larger than that seen in the
${{B}^{0}_{s}}\rightarrow{{K}^{*0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ decay [20, 21].

In addition to the study of the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decay, a
search for the as yet unobserved decay ${{B}^{0}}\rightarrow\phi\phi$ is made.
In the SM this is an OZI suppressed decay [22, *Iizuka:1966fk], with an
expected branching fraction in the range $(0.1-3.0)\times 10^{-8}$ [24, 25, 2,
1]. However, the branching fraction can be enhanced, up to the $10^{-7}$
level, in extensions to the SM such as supersymmetry with R-parity violation
[25]. The most recent experimental limit was determined to be $2.8\times
10^{-8}$ at 90 % confidence level [26].

Measurements presented in this paper are based on $pp$ collision data
corresponding to an integrated luminosity of $5.0\text{\,fb}^{-1}$, collected
with the LHCb experiment at centre-of-mass energies
$\sqrt{s}=7\text{\,Te\kern-1.00006ptV}$ in 2011, 8 TeV in 2012, and 13 TeV
from 2015 to 2016. This paper reports a time-dependent analysis of
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decays, where the $\phi$ meson is
reconstructed in the ${{K}^{+}}{{K}^{-}}$ final state, that measures the
$C\\!P$-violating phase, $\phi_{s}^{s\bar{s}s}$, and the parameter
$|\lambda|$, that is related to the direct $C\\!P$ violation. Results on
helicity-dependent weak phases are also presented, along with helicity
amplitudes describing the $P\rightarrow VV$ transition and strong phases of
the amplitudes. In addition, triple-product asymmetries for this decay are
presented. The analysis also includes a search for the decay
$B^{0}\rightarrow\phi\phi$. Results presented here supersede the previous
measurements based on data collected in 2011 and 2012 [10].

##  2 Detector description

The LHCb detector [27, 28] 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 [29], a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes [30] placed
downstream of the magnet. The tracking system provides a measurement of the
momentum, $p$, of charged particles with a relative uncertainty that varies
from 0.5% at low momentum to 1.0% at 200$\text{\,Ge\kern-1.00006ptV\\!/}c$.
The minimum distance of a track to a primary vertex (PV), the impact parameter
(IP), is measured with a resolution of
$(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$ is the
component of the momentum transverse to the beam, in
$\text{\,Ge\kern-1.00006ptV\\!/}c$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
[31]. Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
and a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers [27].

The online event selection is performed by a trigger, which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. At
the hardware trigger stage, events are required to contain a muon with high
$p_{\mathrm{T}}$ or a hadron, photon or electron with high transverse energy
in the calorimeters. In the software trigger,
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ candidates are selected either by
identifying events containing a pair of oppositely charged kaons with an
invariant mass within 30$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known
$\phi$ meson mass, $m_{\phi}=1019.5\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ [32],
or by using a topological $b$-hadron trigger. This topological trigger
requires a three-track secondary vertex with a large sum of the
$p_{\mathrm{T}}$ of the charged particles and significant displacement from
the PV. At least one charged particle should have
$p_{\mathrm{T}}>1.7\text{\,Ge\kern-1.00006ptV\\!/}c$ and
$\chi^{2}_{\text{IP}}$ with respect to any primary vertex greater than 16,
where $\chi^{2}_{\text{IP}}$ is defined as the difference in $\chi^{2}$ of a
given PV fitted with and without the considered track. A multivariate
algorithm [33] is used for the identification of secondary vertices consistent
with the decay of a $b$ hadron.

Simulation samples are used to optimise the signal candidate selection, to
derive the angular acceptance and the correction to the decay-time acceptance.
In the simulation, $pp$ collisions are generated using Pythia [34,
*Sjostrand:2007gs] with a specific LHCb configuration [36]. Decays of hadronic
particles are described by EvtGen [37], in which final-state radiation is
generated using Photos [38]. The interaction of the generated particles with
the detector and its response are implemented using the Geant4 toolkit [39,
*Agostinelli:2002hh], as described in Ref. [36].

##  3 Selection and mass model

For decay-time-dependent measurements and the $T$-odd asymmetries presented in
this paper, the previously analysed data collected in 2011 and 2012 [10] is
supplemented with the additional data taken in 2015 and 2016, to which the
selection described below is applied. For the case of the
${{B}^{0}}\rightarrow\phi\phi$ search, a wider invariant-mass window is
required, along with more stringent background rejection requirements.

Events passing the trigger are required to satisfy loose criteria on the fit
quality of the four-kaon vertex, the $\chi^{2}_{\text{IP}}$ of each track, the
transverse momentum of each particle, and the product of the transverse
momenta of the two $\phi$ candidates. In addition, the reconstructed mass of
the $\phi$ candidates is required to be within
25$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known $\phi$ mass [32].

In order to separate further the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ signal
candidates from the background, a multilayer perceptron (MLP) [41] is used. To
train the MLP, simulated ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ candidates
satisfying the same requirements as the data candidates are used as a proxy
for signal, whereas the four-kaon invariant-mass sidebands from data are used
as a proxy for background. The invariant-mass sidebands are defined to be
inside the region
$120<|m_{{{K}^{+}}{{K}^{-}}{{K}^{+}}{{K}^{-}}}-m_{{{B}^{0}_{s}}}|<180\text{\,Me\kern-1.00006ptV\\!/}c^{2}$,
where $m_{{{K}^{+}}{{K}^{-}}{{K}^{+}}{{K}^{-}}}$ is the four-kaon invariant
mass. Separate MLP classifiers are trained for each data taking period. The
variables used in the MLP comprise the minimum and the maximum
$p_{\mathrm{T}}$ and $\eta$ of the kaon and $\phi$ candidates, the
$p_{\mathrm{T}}$ and $\eta$ of the ${B}^{0}_{s}$ candidate, the quality of the
four-kaon vertex fit, and the cosine of the angle between the momentum of the
${B}^{0}_{s}$ and the direction of flight from the PV to the ${B}^{0}_{s}$
decay vertex, where the PV is chosen as that with the smallest impact
parameter $\chi^{2}$ with respect to the ${B}^{0}_{s}$ candidate. For
measurements of $C\\!P$ violation, the requirement on each MLP is chosen to
maximise $N_{\rm S}/\sqrt{N_{\rm S}+N_{\rm B}}$, where ${N_{\rm S}\,(N_{\rm
B})}$ represents the expected signal and background yields in the signal
region, defined as $m_{{{B}^{0}_{s}}}\pm 3\sigma$, where $m_{{{B}^{0}_{s}}}$
is the known ${B}^{0}_{s}$ mass [32]. The signal yield is estimated using
simulation, whereas the number of background candidates is estimated from the
data sidebands. For the search of the ${{B}^{0}}\rightarrow\phi\phi$ decay,
the figure of merit is chosen to maximise $\varepsilon/(a/2+\sqrt{N_{\rm B}})$
[42], where $a=3$ corresponds to the desired significance, and $\varepsilon$
is the signal efficiency, determined from simulation. This figure of merit
does not depend on the unknown ${{B}^{0}}\rightarrow\phi\phi$ decay rate.

The presence of peaking backgrounds is studied using simulation. The decay
modes considered include ${{B}^{0}}\rightarrow\phi{{K}^{*0}}$, ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow\phi{p}{{K}^{-}}$,
${{B}^{0}}\rightarrow\phi{{\pi}^{+}}{{\pi}^{-}}$ and
${{{B}^{+}}}\rightarrow\phi{{K}^{+}}$, where the last decay mode could
contribute if an extra kaon track is added. The
${{B}^{0}}\rightarrow\phi{{\pi}^{+}}{{\pi}^{-}}$ and
${{{B}^{+}}}\rightarrow\phi{{K}^{+}}$ decays do not contribute significantly.
The ${{B}^{0}}\rightarrow\phi{{K}^{*0}}$ decay, resulting from a
misidentification of a pion as a kaon, is vetoed by rejecting candidates which
simultaneously have
${{K}^{+}}{{\pi}^{-}}$(${{K}^{+}}{{K}^{-}}{{K}^{+}}{{\pi}^{-}}$) invariant
masses within 50 (30)$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known
${K}^{*0}$ (${B}^{0}$) masses. The ${{K}^{+}}{{\pi}^{-}}$ and
${{K}^{+}}{{K}^{-}}{{K}^{+}}{{\pi}^{-}}$ invariant masses are computed by
taking the kaon with the highest probability of being misidentified as a pion
and assigning it the pion mass. These vetoes reduce the number of
${{B}^{0}}\rightarrow\phi{{K}^{*0}}$ candidates to a negligible level.
Similarly, the number of ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow\phi{p}{{K}^{-}}$ decays, resulting from a
misidentification of a proton as a kaon, is estimated from data by assigning
the proton mass to the final-state particle that has the largest probability
to be a misidentified proton based on the particle-identification information.
This method yields $241\pm 30$ ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow\phi{p}{{K}^{-}}$ decays in the total data
set.

In order to determine the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ yield in the
final data sample, the four-kaon invariant-mass distributions are fitted with
the sum of the following components: a ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$
signal model, which comprises the sum of a Crystal Ball [43] and a Student’s
t-function; the peaking background contribution modelled by a Crystal Ball
function, with the shape parameters fixed to the values obtained from a fit to
simulated events, and the combinatorial background component, described using
an exponential function. The yield of the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow\phi{p}{{K}^{-}}$ peaking background
contribution is fixed to the number previously stated. Once the MLP
requirements are imposed, an unbinned extended maximum-likelihood fit to the
four-kaon invariant mass gives a total yield of $8843\pm 102$
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decays and $2813\pm 67$ combinatorial
background candidates in the total data set. The fits to the four-kaon
invariant-mass distributions, after the selection optimised for the
$C\\!P$-violation measurement, separately for each data taking year, are shown
in Fig. 1.

Figure 1: A fit to the four-kaon mass for the (top left) 2011, (top right)
2012, (bottom left) 2015 and (bottom right) 2016 data sets, which are
represented by the black points. Also shown are the results of the total fit
(blue solid line), with the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ (red
dashed), the ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow\phi{p}{{K}^{-}}$
(magenta long dashed), and the combinatorial (blue short dashed) fit
components.

##  4 Formalism

The final state of the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decay comprises a
mixture of $C\\!P$ eigenstates, which are disentangled by means of an angular
analysis in the helicity basis. In this basis, the decay is described by three
angles, $\theta_{1}$, $\theta_{2}$ and $\phi$, defined in Fig. 2.

$\hat{n}_{V_{1}}$$\hat{n}_{V_{2}}$$B^{0}_{s}$$\Phi$$\theta_{2}$$\theta_{1}$$K^{-}$$K^{+}$$K^{-}$$K^{+}$$\phi_{1}$$\phi_{2}$
Figure 2: Decay angles for the ${{B}^{0}_{s}}\rightarrow\phi\phi$ decay, where
$\theta_{1,2}$ is the angle between the $K^{+}$ momentum in the $\phi_{1,2}$
meson rest frame and the $\phi_{1,2}$ momentum in the ${B}^{0}_{s}$ rest
frame, $\Phi$ is the angle between the two $\phi$ meson decay planes and
$\hat{n}_{V_{1,2}}$ is the unit vector normal to the decay plane of the
$\phi_{1,2}$ meson.

###  4.1 Decay-time-dependent model

As discussed in Sec. 1, the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decay is a
$P\rightarrow VV$ decay. However, due to the proximity of the $\phi$ resonance
to the scalar $f_{0}(980)$ resonance, there are irreducible contributions to
the four-kaon mass spectrum from $P\rightarrow V\kern-1.00006ptS$ ($S$-wave)
and $P\rightarrow S\kern-1.00006ptS$ (double $S$-wave) processes, where $S$
denotes a scalar meson, or a nonresonant pair of kaons. Thus, the total
amplitude is a coherent sum of $P$-, $S$-, and double $S$-wave processes, and
is modelled by making use of the different dependence on the helicity angles
associated with these terms, where the helicity angles are defined in Fig. 2.
A randomised choice is made for which $\phi$ meson is used to determine
$\theta_{1}$ and which is used to determine $\theta_{2}$. The total amplitude
($\mathcal{A}$) containing the $P$-, $S$-, and double $S$-wave components as a
function of time, $t$, can be written as [44]

$\displaystyle\mathcal{A}(t,\theta_{1},\theta_{2},\Phi)$
$\displaystyle=A_{0}(t)\cos\theta_{1}\cos\theta_{2}+\frac{A_{\parallel}(t)}{\sqrt{2}}\sin\theta_{1}\sin\theta_{2}\cos\Phi$
$\displaystyle+i\frac{A_{\perp}(t)}{\sqrt{2}}\sin\theta_{1}\sin\theta_{2}\sin\Phi+\frac{A_{S}(t)}{\sqrt{3}}(\cos\theta_{1}+\cos\theta_{2})+\frac{A_{SS}(t)}{3},$
(1)

where $A_{0}$, $A_{\parallel}$, and $A_{\perp}$ are the $C\\!P$-even
longitudinal, $C\\!P$-even parallel, and $C\\!P$-odd perpendicular
polarisations of the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decay. The
$P\rightarrow V\kern-1.00006ptS$ and $P\rightarrow S\kern-1.00006ptS$
processes are described by the $A_{S}$ and $A_{SS}$ amplitudes, respectively,
where $P\rightarrow V\kern-1.00006ptS$ is $C\\!P$-odd and $P\rightarrow
S\kern-1.00006ptS$ is $C\\!P$-even. The resulting differential decay rate is
proportional to the square of the total amplitude and consists of 15 terms
[44]

$\displaystyle\frac{\mathrm{d}\Gamma}{\mathrm{d}t\,\mathrm{d}\cos\theta_{1}\,\mathrm{d}\cos\theta_{2}\,\mathrm{d}\Phi}\propto|\mathcal{A}(t,\theta_{1},\theta_{2},\Phi)|^{2}=\frac{1}{4}\sum_{i=1}^{15}K_{i}(t)f_{i}(\theta_{1},\theta_{2},\Phi),$
(2)

where the $f_{i}$ terms are functions of the angular variables and the time-
dependence is contained in

$\displaystyle
K_{i}(t)=N_{i}e^{-{\Gamma_{{s}}}t}\left[a_{i}\cosh\left(\frac{1}{2}{\Delta\Gamma_{{s}}}t\right)+b_{i}\sinh\left(\frac{1}{2}{\Delta\Gamma_{{s}}}t\right)+c_{i}\cos({\Delta
m_{{s}}}t)+d_{i}\sin({\Delta m_{{s}}}t)\right]\kern-2.5pt.$ (3)

The coefficients $N_{i},a_{i},b_{i},c_{i}$ and $d_{i}$, which are functions of
the $C\\!P$ observables, are defined in Appendix A.
${\Delta\Gamma_{{s}}}\equiv\Gamma_{\rm L}-\Gamma_{\rm H}$ is the decay-width
difference between the light and heavy ${B}^{0}_{s}$ mass eigenstates,
${\Gamma_{{s}}}\equiv(\Gamma_{\rm L}+\Gamma_{\rm H})/2$ is the average decay
width, and ${\Delta m_{{s}}}$ is the ${B}^{0}_{s}$-${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ oscillation frequency. The
differential decay rate for a ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ meson produced at $t=0$ is
obtained by changing the sign of the $c_{i}$ and $d_{i}$ coefficients. The
amplitudes of helicity state $k$ are expressed as

$A_{k}(t)=|A_{k}|e^{i\delta_{k}}\left(g_{+}(t)+\eta_{k}|\lambda_{k}|e^{-i\phi_{s,k}}g_{-}(t)\right),$
(4)

where $g_{+}(t)$ and $g_{-}(t)$ describe the time evolution of ${B}^{0}_{s}$
and ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ mesons,
respectively. $C\\!P$ violation is parameterised through

$\frac{q}{p}\frac{\bar{A}_{k}}{A_{k}}=\eta_{k}|\lambda_{k}|e^{-i\phi_{s,k}}.$
(5)

where, $q$ and $p$ relate the light and heavy mass eigenstates to the flavour
eigenstates and $\eta_{k}$ is the $C\\!P$ eigenvalue of the polarisation being
considered. Defining the amplitude in this way leads to the forms of
$N_{i},a_{i},b_{i},c_{i}$ and $d_{i}$, listed in Table 7 (Appendix A). The
$C\\!P$-violating asymmetry in ${B}^{0}_{s}$ mixing, which can be
characterised by the semileptonic asymmetry, $a_{\mathrm{sl}}^{s}$ is small
[45]. Thus, to good approximation $|q/p|=1$, and $|\lambda_{k}|$ quantifies
the level of $C\\!P$ violation in the decay. Two different fit configurations
are performed, one in which the $C\\!P$-violation parameters are assumed to be
helicity independent and the other in which $C\\!P$-violation parameters are
allowed to differ as a function of helicity. The helicity independent fit
assumes one $C\\!P$-violating phase, $\phi_{{s}}^{{{s}{\overline{s}}}{s}}$,
which takes the place of all $\phi_{s,k}$ contained in the coefficients of
Appendix A, and likewise one parameter that describes direct $C\\!P$
violation, $|\lambda|$, which takes the place of all $\lambda_{k}$
coefficients. Due to the small sample size, the number of degrees of freedom
is reduced for the case of the helicity-dependent $C\\!P$-violation fit. This
involves assuming $C\\!P$ conservation for the case of the direct
$C\\!P$-violation parameters, $\lambda=1$, and also for the phase of the
longitudinal polarisation, $\phi_{s,0}^{s\bar{s}s}=0$. The longitudinal
polarisation has been theoretically calculated as close to zero in the
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decay [1].

The $\phi_{{s}}^{{{s}{\overline{s}}}{s}}$ and $|\lambda|$ parameters are
measured with respect to contributions with the same flavour content as the
$\phi$ meson, i.e. ${s}{\overline{s}}$. Regarding the $S$-wave and double
$S$-wave terms, the impact of the non-${s}{\overline{s}}$ component of the
$\phi$ wavefunction is negligible in this analysis.

###  4.2 Triple-product asymmetries

Scalar triple products of three-momentum or spin vectors are odd under time
reversal, $T$. Nonzero asymmetries for these observables can either be due to
a CP-violating phase or from CP-conserving strong final-state interactions.
Four-body final states give rise to three independent momentum vectors in the
rest frame of the decaying ${B}^{0}_{s}$ meson. For a detailed review of the
phenomenology the reader is referred to Ref. [11].

Two triple products can be defined:

$\displaystyle\sin\Phi=(\hat{n}_{V_{1}}\times\hat{n}_{V_{2}})\cdot\hat{p}_{V_{1}},$
(6) $\displaystyle\sin
2\Phi=2(\hat{n}_{V_{1}}\cdot\hat{n}_{V_{2}})(\hat{n}_{V_{1}}\times\hat{n}_{V_{2}})\cdot\hat{p}_{V_{1}},$
(7)

where $\hat{n}_{V_{i}}$ ($i=1,2$) is a unit vector perpendicular to the vector
meson ($V_{i}$) decay plane and $\hat{p}_{V_{1}}$ is a unit vector in the
direction of $V_{1}$ in the ${B}^{0}_{s}$ rest frame, defined in Fig. 2. This
then provides a method of probing $C\\!P$ violation without the need to
measure the decay time or the initial flavour of the ${B}^{0}_{s}$ meson. It
should be noted, that while the observation of nonzero triple-product
asymmetries implies $C\\!P$ violation or final-state interactions (in the case
of ${B}^{0}_{s}$ meson decays), measurements of triple-product asymmetries
consistent with zero do not rule out the presence of $C\\!P$-violating
effects, as the size of the asymmetry also depends on the differences between
the strong phases [11].

In the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decay, two triple products are
defined as $U\equiv\sin\Phi\cos\Phi$ and $V\equiv\sin(\pm\Phi)$ where the
positive sign is taken if $\cos\theta_{1}\cos\theta_{2}\geq 0$ and the
negative sign otherwise [11]. The $T$-odd asymmetry corresponding to the $U$
observable, $A_{U}$, is defined as the normalised difference between the
number of decays with positive and negative values of $\sin\Phi\cos\Phi$,

$\displaystyle
A_{U}\equiv\frac{\Gamma(U>0)-\Gamma(U<0)}{\Gamma(U>0)+\Gamma(U<0)}\propto\int^{\infty}_{0}\Im\left(A_{\perp}(t)A_{\parallel}^{*}(t)+\bar{A}_{\perp}(t)\bar{A}_{\parallel}^{*}(t)\right)\mathrm{d}t.$
(8)

Similarly, $A_{V}$ is defined as

$\displaystyle
A_{V}\equiv\frac{\Gamma(V>0)-\Gamma(V<0)}{\Gamma(V>0)+\Gamma(V<0)}\propto\int^{\infty}_{0}{\Im}\left(A_{\perp}(t)A_{0}^{*}(t)+\bar{A}_{\perp}(t)\bar{A}_{0}^{*}(t)\right)\mathrm{d}t.$
(9)

Here, $A_{\perp}$, $A_{\parallel}$ and $A_{0}$ correspond to the three
transversity amplitudes. The determination of the triple-product asymmetries
is then reduced to a simple counting experiment. Comparing these formulae with
Eq. 3 and Appendix A it can be seen that the triple products are related to
the $K_{4}(t)$ and $K_{6}(t)$ terms in the decay amplitude.

##  5 Decay-time resolution

The sensitivity to $\phi_{{s}}^{{{s}{\overline{s}}}{s}}$ is affected by the
accuracy of the measured decay time. In order to resolve the fast
${B}^{0}_{s}$-${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$
oscillations, it is necessary to have a decay-time resolution that is much
smaller than the oscillation period. To account for the resolution of the
measured decay-time distribution, all decay-time-dependent terms are convolved
with a Gaussian function, with width $\sigma^{t}_{i}$ that is estimated for
each candidate, $i$, based upon the uncertainty obtained from the vertex and
kinematic fit [46].

In order to apply a candidate-dependent resolution model during fitting, the
estimated per-event decay time uncertainty is needed. This is calibrated using
the fact the decay time resolution for the
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ mode is dominated by the secondary
vertex resolution. A sample of good-quality tracks, which originate from the
primary interaction vertex is selected. Due to the small opening angle of the
kaons in the decay of a $\phi$ meson, it is sufficient to use a single prompt
track and assign it the mass of a $\phi$ meson. When combining this with
another pair of tracks, the invariant mass of the three-body combination is
required to be within 250 $\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known
${B}^{0}_{s}$ mass. That the decay-time resolution of the signal ${B}^{0}_{s}$
decays can be described well using three tracks has been validated using
simulation.

A linear function is then fitted to the distribution of $\sigma^{t}_{i}$
versus $\sigma_{\rm true}^{t}$, with parameters $q_{0}$ and $q_{1}$. Here,
$\sigma_{\rm true}^{t}$ denotes the difference between reconstructed decay
time and the exact decay time of simulated signal. The per-event decay-time
uncertainty used in the decay-time-dependent fit is then calculated as
$\sigma^{\rm cal}_{i}=q_{0}+q_{1}\sigma^{t}_{i}$. Gaussian constraints are
used to account for the uncertainties on the calibration parameters in the
decay-time-dependent fit. The effective single-Gaussian decay-time resolution
is found to be between 41 and 44 fs, depending on the data-taking year, in
agreement with the expectation from the simulation.

##  6 Acceptances

The ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ differential decay rate depends on
the decay time and three helicity angles as shown in Eq. 2. Good understanding
of the efficiencies in these variables is required. The decay-time and angular
acceptances are assumed to factorise. Control channels show this assumption
has a negligible systematic uncertainty on the physics parameters.

###  6.1 Angular acceptance

The geometry of the LHCb detector and the momentum requirements imposed on the
final-state particles introduce distortions of the helicity angles, giving
rise to acceptance effects. Simulated signal events, selected with the same
criteria as those applied to data are used to determine these efficiency
corrections. The angular acceptances as a function of the three helicity
angles are shown in Fig. 3.

Figure 3:  Angular acceptance normalised to the average obtained using
simulated ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decays (top-left) integrated
over $\cos\theta_{2}$ and $\Phi$ as a function of $\cos\theta_{1}$, (top-
right) integrated over $\cos\theta_{1}$ and $\Phi$ as a function of
$\cos\theta_{2}$, and (bottom) integrated over $\cos\theta_{1}$ and
$\cos\theta_{2}$ as a function of $\Phi$. Each figure includes the resulting
fit curve.

The efficiency is parameterised in terms of the decay angles as

$\epsilon(\Omega)=\sum_{i,j,k}c_{ijk}P_{i}(\cos\theta_{1})Y_{jk}(\cos\theta_{2},\Phi),$
(10)

where $\Omega$ depends on the decay angles, $\cos\theta_{1}$, $\cos\theta_{2}$
and $\phi$, the $c_{ijk}$ are coefficients, $P_{i}(\cos\theta_{1})$ are
Legendre polynomials, and $Y_{jk}(\cos\theta_{2},\Phi)$ are spherical
harmonics. The procedure followed to calculate the coefficients is described
in detail in Ref. [47] and exploits the orthogonality of Legendre polynomials.
The coefficients are given by

$\displaystyle c_{ijk}\equiv(j+1/2)\int d\Omega
P_{i}(\cos\theta_{1})Y_{jk}(\cos\theta_{2},\Phi)\epsilon(\Omega).$ (11)

This integral is calculated by means of a Monte Carlo technique, which reduces
the integral to a sum over the number of accepted simulated events ($N^{\rm
obs}$)

$\displaystyle c_{ijk}\propto(j+1/2)\frac{1}{N^{\rm obs}}\sum_{e=1}^{N^{\rm
obs}}\frac{P_{i}(\cos\theta_{1,e})Y_{jk}(\cos\theta_{2,e},\Phi_{e})}{P^{\rm
gen}(\Omega_{e}|t_{e})},$ (12)

where $P^{\rm gen}$ is the probability density function (PDF) without
acceptance where the parameters are set to values used in the Monte Carlo
generation. In order to easily incorporate the angular acceptance, it is
convenient to write angular functions of Eq. 2 in the same basis as the
efficiency parameterisation, i.e.

$\displaystyle
f_{a}(\cos\theta_{1},\cos\theta_{2},\Phi)=\sum_{ijkl}\kappa_{ijkl,a}P_{ij}(\cos\theta_{1})Y_{kl}(\cos\theta_{2},\Phi),$
(13)

where $P_{ij}(\cos\theta_{1})$ are the associated Legendre polynomials,
$\kappa_{ijkl,a}$ are coefficients and $a$ numerates the 15 terms outlined
earlier. The parameterisation for each angular function is given in Table 1.

The normalisation of the angular component in the decay-time dependent fit
occurs through the 15 integrals
$\zeta_{k}=\int\epsilon(\Omega)f_{k}(\Omega)\mathrm{d}\Omega$, where
$\epsilon(\Omega)$ is the efficiency as a function of the helicity angles as
shown in Eq. 10 and $f_{k}(\Omega)$ are the angular functions as defined in
Eq. 13.

Table 1: Angular coefficients, written in the same basis as the efficiency
parameterisation.

$\scriptsize\begin{array}[]{c|c|c}i&f_{i}\,\mathrm{in}P_{i\,j}Y_{k\,l}\,{\rm
basis}&f_{i}\\\ \hline\cr
1&8/9P_{00}Y_{00}+16/9/\sqrt{5}P_{00}Y_{20}+16/9P_{20}Y_{00}+32/(9\sqrt{5})P_{20}Y_{20}&4\cos^{2}\theta_{1}\cos^{2}\theta_{2}\\\
\hline\cr
2&8/9P_{00}Y_{00}-8/9P_{00}Y_{20}-8/9\sqrt{5}P_{20}Y_{00}+8/(9\sqrt{5})P_{20}Y_{20}+(2/9)\sqrt{12/5}P_{22}Y_{22}&\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1{+}\cos
2\Phi)\\\ \hline\cr
3&8/9P_{00}Y_{00}-8/9P_{00}Y_{20}-8/9\sqrt{5}P_{20}Y_{00}+8/(9\sqrt{5})P_{20}Y_{20}-(2/9)\sqrt{12/5}P_{22}Y_{22}&\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1{-}\cos
2\Phi)\\\ \hline\cr
4&-8/9\sqrt{3/5}P_{2,2}Y_{2,-2}&-2\sin^{2}\theta_{1}\sin^{2}\theta_{2}\sin
2\Phi\\\ \hline\cr 5&8/9\sqrt{6/5}P_{2,1}Y_{2,1}&\sqrt{2}\sin 2\theta_{1}\sin
2\theta_{2}\cos\Phi\\\ \hline\cr 6&-8/9\sqrt{6/5}P_{2,1}Y_{2,1}&-\sqrt{2}\sin
2\theta_{1}\sin 2\theta_{2}\sin\Phi\\\ \hline\cr
7&(8/9)P_{00}Y_{00}&\frac{4}{9}\\\ \hline\cr
8&16/9P_{00}Y_{00}+16/9/\sqrt{5}P_{00}Y_{20}+16/9P_{20}Y{00}+16\sqrt{3}/9P_{10}Y_{10}&\frac{4}{3}(\cos\theta_{1}+\cos\theta_{2})^{2}\\\
\hline\cr
9&16\sqrt{3}/2P_{10}Y_{00}+16/9P_{00}Y_{10}&\frac{8}{3\sqrt{3}}(\cos\theta_{1}+\cos\theta_{2})\\\
\hline\cr
10&16/(3\sqrt{3})P_{10}Y_{10}&\frac{8}{3}\cos\theta_{1}\cos\theta_{2}\\\
\hline\cr
11&(8/9)\sqrt{6}P_{11}Y_{11}&\frac{4\sqrt{2}}{3}\sin\theta_{1}\sin\theta_{2}\cos\Phi\\\
\hline\cr
12&(8/9)\sqrt{6}P_{11}Y_{1-1}&-\frac{4\sqrt{2}}{3}\sin\theta_{1}\sin\theta_{2}\sin\Phi\\\
\hline\cr
13&16\sqrt{3}/9P_{10}Y_{00}+16/9P_{00}Y_{10}+32/9P_{20}Y{10}+32/(9\sqrt{5})P_{20}Y_{20}&\begin{array}[]{c}\frac{8}{\sqrt{3}}\cos\theta_{1}\cos\theta_{2}\\\
\times(\cos\theta_{1}+\cos\theta_{2})\end{array}\\\ \hline\cr
14&(8/9)\sqrt{2/3}P_{21}Y_{11}+(24/9)\sqrt{2/15}P_{11}Y_{21}&\begin{array}[]{c}\frac{4\sqrt{2}}{\sqrt{3}}\sin\theta_{1}\sin\theta_{2}\\\
\times(\cos\theta_{1}+\cos\theta_{2})\cos\Phi\end{array}\\\ \hline\cr
15&-(8/9)\sqrt{2/3}P_{21}Y_{1-1}-(24/9)\sqrt{2/15}P_{11}Y_{2-1}&\begin{array}[]{c}-\frac{4\sqrt{2}}{3}\sin\theta_{1}\sin\theta_{2}\\\
\times(\cos\theta_{1}+\cos\theta_{2})\sin\Phi\end{array}\\\ \end{array}$

The angular acceptance is calculated correcting for the differences in
kinematic variables between data and simulation. This includes differences in
the MLP training variables that can affect acceptance corrections through
correlations with the helicity angles.

The fit to determine the triple-product asymmetries assumes that the $U$ and
$V$ observables are symmetric in the acceptance corrections. Simulation is
used to assign a systematic uncertainty related to this assumption.

###  6.2 Decay-time acceptance

The impact-parameter requirements on the final-state particles efficiently
suppress the background from the numerous pions and kaons originating from the
PV, but introduce a decay-time dependence in the selection efficiency.

The efficiency as a function of the decay time is taken from the
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}(\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{-}}){{\pi}^{+}}$
decay, in the case of data taken between 2011 and 2012, and from the
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\rightarrow{\mu^{+}}{\mu^{-}}){{K}^{*0}}(\rightarrow{{K}^{+}}{{\pi}^{-}})$
decay in the case of data taken between 2015 and 2016. The reason for the
change in control channel is related to changes to the software-trigger
selection between the two data-taking periods. The decay-time acceptances of
the control modes are weighted by a multivariate algorithm based on simulated
kinematic and topological information, in order to match more closely those of
the signal ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decay.

Cubic splines are used to model the acceptance as a function of decay time in
the PDF. The PDF can then be computed analytically with the inclusion of the
decay-time acceptance following Ref. [48]. Example decay-time acceptances are
shown for the case of the ${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$
and ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ decays in Fig. 4.

Figure 4:  Decay-time acceptances calculated from (left)
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ decays to match Run 1 data
and (right) ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$ decays to match Run 2 data. Superimposed is a
parameterisation using cubic splines..

To simplify the measurement of the triple-product asymmetries, the decay-time
acceptance is not applied in the fit to determine the triple-product
asymmetries. The time acceptance correction has an impact on the asymmetry of
0.3$\%$ and is treated as a source of systematic uncertainty, as further
described in Sec. 9.3.

##  7 Flavour tagging

To obtain sensitivity to $\phi_{{s}}^{{{s}{\overline{s}}}{s}}$, the flavour of
the ${B}^{0}_{s}$ meson at production must be determined. At LHCb, tagging is
achieved through the use of various algorithms described in Refs.[49, 50].
With these algorithms, the flavour-tagging power, defined as $\epsilon_{\rm
tag}\mathcal{D}^{2}$ can be evaluated. Here, $\epsilon_{\rm tag}$ is the
flavour-tagging efficiency defined as the fraction of candidates with a
flavour tag with respect to the total, and $\mathcal{D}\equiv(1-2\omega)$ is
the dilution, where $\omega$ is the average fraction of candidates with an
incorrect flavour assignment. This analysis uses opposite-side (OS) and same-
side kaon (SSK) flavour taggers.

The OS flavour-tagging algorithm [49] makes use of the $\overline{b}$ ($b$)
hadron produced in association with the signal $b$ ($\overline{b}$) hadron. In
this analysis, the predicted probability of an incorrect flavour assignment,
$\omega$, is determined for each candidate by a neural network that is
calibrated using ${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$, ${{B}^{+}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{\pi}^{+}}$,
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*0}}$, ${{B}^{0}}\rightarrow{{D}^{*-}}{\mu^{+}}{{\nu}_{\mu}}$,
and ${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ data as control modes.
Details of the calibration procedure can be found in Ref. [51].

When a signal ${B}^{0}_{s}$ meson is formed, there is an associated
$\overline{s}$ quark formed in the first branches of the fragmentation that
about 50 % of the time forms a charged kaon, which is likely to originate
close to the ${B}^{0}_{s}$ meson production point. The kaon charge therefore
allows for the identification of the flavour of the signal ${B}^{0}_{s}$
meson. This principle is exploited by the SSK flavour-tagging algorithm [50],
which is calibrated with the
${{B}^{0}_{s}}\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ decay mode. A neural
network is used to select fragmentation particles, improving the flavour-
tagging power quoted in the previous decay-time-dependent measurement [10].

Table 2 shows the tagging power for the candidates tagged by only one of the
algorithms and those tagged by both. Uncertainties due to the calibration of
the flavour tagging algorithms are applied as Gaussian constraints in the
decay-time-dependent fit. The initial flavour of the ${B}^{0}_{s}$ meson
established from flavour tagging is accounted for during fitting.

Table 2:  Tagging performance of the opposite-side (OS) and same-side kaon (SSK) flavour taggers for the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decay. Category |  $\varepsilon$ (%) | $\mathcal{D}^{2}$ |  $\varepsilon\mathcal{D}^{2}$ (%)
---|---|---|---
OS-only | $12.5$ | $0.10$ | $1.24\pm 0.10$
SSK-only | $41.0$ | $0.04$ | $1.74\pm 0.36$
OS&SSK | $23.3$ | $0.12$ | $2.76\pm 0.20$
Total | $76.8$ | $0.08$ | $5.74\pm 0.43$

##  8 Decay-time-dependent measurement

###  8.1 Likelihood fit

The fit parameters in the polarisation-independent fit are the $C\\!P$
violation parameters, ${\phi_{{s}}^{{{s}{\overline{s}}}{s}}}$ and $|\lambda|$,
the squared amplitudes, $|A_{0}|^{2}$, $|A_{\perp}|^{2}$, $|A_{S}|^{2}$, and
$|A_{SS}|^{2}$, and the strong phases, $\delta_{\perp}$, $\delta_{\parallel}$,
$\delta_{0}$, $\delta_{S}$, and $\delta_{SS}$, as defined in Sec. 4.1. The
$P$-wave amplitudes are defined such that
$|A_{0}|^{2}+|A_{\perp}|^{2}+|A_{\parallel}|^{2}=1$, hence only two of the
three amplitudes are free parameters. This normalisation effectively means the
$S$ and $SS$ components are measured relative to the $P$-wave. The
polarisation-dependent fit allows for a perpendicular, parallel and
longitudinal component of $\phi_{{s}}^{{{s}{\overline{s}}}{s}}$ and
$|\lambda|$.

The measurement of the parameters of interest is performed through an unbinned
negative log likelihood minimisation. The log-likelihood, $\mathcal{L}$, of
each candidate is weighted using the sPlot method [52, 53], to remove partly
reconstructed and combinatorial background. The negative log-likelihood then
takes the form

$\displaystyle-\ln\mathcal{L}=-\alpha\sum_{e\in{\rm
candidates}}W_{e}\ln(S^{e}_{\rm TD}),$ (14)

where $W_{e}$ are the signal sPlot weights calculated using the four-kaon
invariant mass as the discriminating variable. The correlations between the
angular and decay-time variables used in the fit with the four-kaon mass are
small enough for this technique to be appropriate. The factor
$\alpha=\sum_{e}W_{e}/\sum_{e}W_{e}^{2}$ accounts for the sPlot weights in the
determination of the statistical uncertainties. The parameter $S^{e}_{\rm TD}$
is the differential decay rate of Eq. 2, modified to the effects of decay-time
and angular acceptance, in addition to the probability of an incorrect flavour
tag. Explicitly, this can be written as

$\displaystyle S^{e}_{\rm
TD}=\frac{\sum_{i}s^{e}_{i}(t_{e})f_{i}(\Omega_{e})\epsilon(t_{e})}{\sum_{k}\zeta_{k}\int
s_{k}(t)f_{k}(\Omega)\epsilon(t)\mathrm{d}t\,\mathrm{d}\Omega},$ (15)

where $\zeta_{k}$ are the normalisation integrals used to describe the angular
acceptance described in Sec. 6.1 and

$\displaystyle s^{e}_{i}(t)$
$\displaystyle=N_{i}e^{-{\Gamma_{{s}}}t_{e}}\left[c_{i}q_{e}(1-2\omega_{e})\cos({\Delta
m_{{s}}}t_{e})+d_{i}q_{e}(1-2\omega_{e})\sin({\Delta m_{{s}}}t_{e})\right.$
$\displaystyle\qquad\left.+a_{i}\cosh\left(\frac{1}{2}{\Delta\Gamma_{{s}}}t_{e}\right)+b_{i}\sinh\left(\frac{1}{2}{\Delta\Gamma_{{s}}}t_{e}\right)\right]\otimes
R(\sigma^{\rm cal}_{e},t_{e}).$ (16)

The calibrated probability of an incorrect flavour assignment is given by
$\omega_{e}$, $R$ denotes the Gaussian time-resolution function, and the
$\otimes$ denotes a convolution operation. In Eq. 16, $q_{e}=1$ $(-1)$ for a
${B}^{0}_{s}$ (${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$) meson
at $t=0$ or $q_{e}=0$ where no flavour-tagging information is assigned. The
data samples corresponding to the different years of data taking are assigned
independent signal weights, decay-time and angular acceptances, and separate
Gaussian constraints are applied to the decay-time resolution parameters, as
defined in Sec. 5. The ${B}^{0}_{s}$-${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ oscillation frequency is
constrained to the value measured by LHCb of ${\Delta m_{{s}}}=17.768\pm
0.023\text{\,(stat)}\pm 0.006\text{\,(syst)}\text{\,ps}^{-1}$ [54], with the
assumption that the systematic uncertainties are uncorrelated with those of
the current measurement. The values of the decay width and decay-width
difference are constrained to the current best known values of
$\Gamma_{s}=0.6646\pm 0.0020$$\text{\,ps}^{-1}$ and $\Delta\Gamma_{s}=0.086\pm
0.006$$\text{\,ps}^{-1}$ [55].

Correction factors must be applied to each of the $S$-wave and double $S$-wave
interference terms in the differential decay width. These factors modulate the
sizes of the contributions of the interference terms in the angular PDF due to
the different line-shapes of kaon pairs originating from spin-1 and spin-0
configurations. This takes the form of a multiplicative factor for each time a
$S$-wave pair of kaons interferes with a $P$-wave pair. Their $K^{+}K^{-}$
invariant-mass parameterisations are denoted by $g(m_{{{K}^{+}}{{K}^{-}}})$
and $h(m_{{{K}^{+}}{{K}^{-}}})$, respectively. The $P$-wave configuration is
described by a Breit–Wigner function and the $S$-wave configuration is assumed
to be uniform. The correction factors, denoted by $C_{SP}$, are defined in
Ref. [51]

$\displaystyle C_{SP}=\int^{m_{\rm h}}_{m_{\rm
l}}g^{*}(m_{{{K}^{+}}{{K}^{-}}})h(m_{{{K}^{+}}{{K}^{-}}})\mathrm{d}m_{{{K}^{+}}{{K}^{-}}},$
(17)

where $m_{\rm h}$ and $m_{\rm l}$ are the upper and lower edges of the
$m_{{{K}^{+}}{{K}^{-}}}$ window and the phase of $C_{SP}$ is absorbed in the
measurements of $\delta_{S}-\delta_{\perp}$. The factor $|C_{SP}|$, is
calculated to be 0.36. In order to determine systematic uncertainties due to
the model dependence of the S-wave, $C_{SP}$ factors are recalculated based on
the $S$-wave originating from an $f_{0}(980)$ resonance and incorporating the
effects of the $m_{K^{+}K^{-}}$ resolution. These alternative assumptions on
the $P$-wave and $S$-wave lineshapes yield a $|C_{SP}|$ value of 0.34, which
is found to have a negligible effect on the parameter estimation.

###  8.2 Results

The resulting parameters are given in Table 3. A polarisation-independent fit
is performed to calculate values for ${\phi_{{s}}^{{{s}{\overline{s}}}{s}}}$
and $|\lambda|$. A negligible fraction of $S$-wave and double $S$-wave is
observed.

Table 3:  Results of the decay-time-dependent, polarisation-independent fit for the $C\\!P$-violation fit. Uncertainties shown do not include systematic contributions. Parameter | Fit result
---|---
$\phi_{{s}}^{{{s}{\overline{s}}}{s}}$ [rad] | $-0.073\pm 0.115~{}~{}$
$|\lambda|$ | $0.99\pm 0.05$
$|A_{0}|^{2}$ | $0.381\pm 0.007$
$|A_{\perp}|^{2}$ | $0.290\pm 0.008$
$\delta_{\perp}$ [rad] | $2.82\pm 0.18$
$\delta_{\parallel}$ [rad] | $2.56\pm 0.05$

In addition, the $C\\!P$-violating phases are also determined in a
polarisation-dependent manner. Due to limited size of data samples, the phases
$\phi_{s,\parallel}$ and $\phi_{s,\perp}$ are measured under the assumptions
that the longitudinal weak phase is $C\\!P$-conserving and that there is no
direct $C\\!P$ violation. In addition, all $S$-wave and double $S$-wave
components of the fit are set to zero. The results of the polarisation
dependent fit are shown in Table 4. The results for $|A_{0}|^{2}$,
$|A_{\perp}|^{2}$, $\delta_{\perp}$ and $\delta_{\parallel}$ are not shown but
are in agreement with the results reported in Table 3.

Table 4:  Results of the polarisation-dependent fit for the $C\\!P$ violation fit. Uncertainties shown do not include systematic contributions.  Parameter | Fit result
---|---
$\phi_{s,\parallel}$ [rad] | $0.014\pm 0.055$
$\phi_{s,\perp}$ [rad] | $0.044\pm 0.059$

The correlation matrices for the two fit configurations are provided in
Appendix B. Correlations with such decay-time-dependent measurements depend on
the central values of the parameters. No large correlation is expected between
the $C\\!P$-violating parameters when the central values are consistent with
$C\\!P$ conservation. The largest correlations are found to be between the
different decay amplitudes. Cross-checks are performed on simulated data sets
generated with the same yield as observed in data, and with the same physics
parameters, to establish that the generated values are recovered without
biases.

Figure 5 shows the distributions of the ${B}^{0}_{s}$ decay time and the three
helicity angles. Superimposed are the projections of the fit result. The
projections include corrections for acceptance effects. Pseudoexperiments were
used to confirm that the deviation of the data around $\cos\theta_{2}=\pm 0.5$
from the resulting distribution of the fit is compatible with a statistical
fluctuation.

Figure 5: One-dimensional projections of the
${{B}^{0}_{s}}\rightarrow\phi\phi$ fit for (top-left) decay time with binned
acceptance, (top-right) helicity angle $\Phi$ and (bottom-left and bottom-
right) cosine of the helicity angles $\theta_{1}$ and $\theta_{2}$. The
background-subtracted data are marked as black points, while the blue solid
lines represent the projections of the fit. The $C\\!P$-even $P$-wave, the
$C\\!P$-odd $P$-wave and the combined $S$-wave and double $S$-wave components
are shown by the red long dashed, green short dashed and purple dot-dashed
lines, respectively. Fitted components are plotted taking into account
efficiencies in the time and angular observables.

###  8.3 Systematic uncertainties

Various sources of systematic uncertainty are considered in addition to those
applied as Gaussian constraints in the fit. These arise from the angular and
decay-time acceptances, the mass model used to describe the mass distribution,
the determination of the time resolution calibration, and the fit bias. A
summary of the systematic uncertainties is given in Table 5.

An uncertainty due to the angular acceptance arises from the choice of
weighting scheme described in Sec. 6. This is accounted for by performing
multiple alternative weighting schemes for the weighting procedure, based on
different kinematic variables in the decay. The largest variation is then
assigned as the uncertainty. Further checks are performed to verify that the
angular acceptance does not depend on the way in which the event was
triggered.

Two sources of systematic uncertainty are considered concerning the decay-time
acceptance. These are the statistical uncertainty from the spline
coefficients, and also the residual disagreement between the weighted control
mode and the signal decay acceptances (see Sec. 6.2). The former is evaluated
through fitting the signal data set with 30 different spline functions, whose
coefficients are varied according to the corresponding uncertainties. This
study is performed with the three different choices of the knot points. The
RMS of the fitted parameters is then assigned as the uncertainty. The residual
disagreement between the control mode and the signal mode is accounted for
with a simulation-based correction. Simplified simulation is used with the
corrected acceptance and then fitted with the nominal acceptance. This process
is repeated and the resulting bias on the fitted parameters is used as an
estimate of the systematic uncertainty.

The uncertainty on the mass model is found by refitting the data with various
alternative signal models, consisting of the sum of two Crystal Ball models,
the sum of a double-sided Crystal Ball and a Gaussian model. In addition, a
Chebyshev polynomial is used to describe the combinatorial background. The
signal weights are recalculated and the largest deviation from the nominal fit
results is used as the corresponding uncertainty.

Fit biases can arise in maximum-likelihood fits where the number of candidates
is small compared to the number of free parameters. The effect of such a bias
is taken as a systematic uncertainty which is evaluated by generating and
fitting simulated data sets and taking the resulting bias as the uncertainty.

The uncertainties of the effective flavour-tagging power are included in the
statistical uncertainty through Gaussian constraints on the calibration
parameters, and amount to 10 % of the statistical uncertainty on the
$C\\!P$-violating phases.

Table 5: Summary of systematic uncertainties (in units of $10^{-2}$) for parameters of interest in the decay-time-dependent measurement. Parameter | Mass | Angular | Decay-time | Time | Fit | Total
---|---|---|---|---|---|---
| model | acceptance | acceptance | resolution | bias |
$|A_{0}|^{2}$ | 0.4 | 1.1 | 0.1 | - | 0.2 | 1.2
$|A_{\perp}|^{2}$ | - | 0.5 | - | - | 0.1 | 0.5
$\delta_{\parallel}$ [rad] | 2.7 | 0.2 | 0.5 | 0.1 | 1.7 | 3.3
$\delta_{\perp}$ [rad] | 3.8 | 0.3 | 0.8 | 1.4 | 6.0 | 7.3
$\phi_{{s}}^{{{s}{\overline{s}}}{s}}$ [rad] | 1.2 | 0.5 | 0.6 | 2.0 | 1.1 | 2.7
$\lambda$ | 0.5 | 0.5 | 0.2 | 0.3 | 0.9 | 1.2
$\phi_{s,\parallel}$ [rad] | 0.2 | 0.2 | 0.4 | 0.2 | 1.0 | 1.1
$\phi_{s,\perp}$ [rad] | 1.4 | 0.3 | 0.4 | 0.3 | 0.4 | 1.9

##  9 Triple-product asymmetries

###  9.1 Likelihood

To determine the triple-product asymmetries, the data sets are divided
according to the sign of the observables $U$ and $V$. Simultaneous likelihood
fits to the four-kaon mass distributions are preformed for the $U$ and $V$
variables separately. The set of free parameters in the fits to determine the
$U$ and $V$ observables consists of their total yields and the asymmetries
$A_{U(V)}$. The mass model is the same as that described in Sec. 3. The total
PDF, $D_{\rm TP}$, is then of the form

$D_{\rm
TP}=\sum_{i\in\\{+,-\\}}\left(f_{i}^{S}G^{S}(m_{{{K}^{+}}{{K}^{-}}{{K}^{+}}{{K}^{-}}})+\sum_{k}f^{k}_{i}P^{k}(m_{{{K}^{+}}{{K}^{-}}{{K}^{+}}{{K}^{-}}})\right),$
(18)

where $k$ indicates the sum over the background components with corresponding
PDFs, $P^{j}$, and $G^{S}$ is the signal PDF, as described in Sec. 3. The
parameters $f^{S}_{i}$ found in Eq. 18 are related to the asymmetry,
$A_{U(V)}^{S}$, through

$\displaystyle f_{U(V),+}^{S}=\frac{1}{2}(1+A^{S}_{U(V)}),$ (19)
$\displaystyle f_{U(V),-}^{S}=\frac{1}{2}(1-A^{S}_{U(V)}),$ (20)

where $S$ denotes the signal component of the four-kaon mass fit, as described
in Sec. 3. Peaking backgrounds are assumed to be symmetric in $U$ and $V$.

###  9.2 Results

The triple-product asymmetries found from the simultaneous fit described in
Sec. 9.1 are measured separately for the 2015 and 2016 data. The results are
combined by performing likelihood scans of the asymmetry parameters and
summing the two years. This gives

$A_{U}$ $~{}=~{}$ |  $-$0.003$\,\,\pm\,$ | 0.015 ,
---|---|---
$A_{V}$ $~{}=~{}$ |  $-$0.012$\,\,\pm\,$ | 0.015 ,

where the uncertainties are statistical only.

###  9.3 Systematic uncertainties

As for the case of the decay-time-dependent fit, significant contributions to
the systematic uncertainty arise from the decay-time and angular acceptances.
Minor uncertainties also result from the knowledge of the mass model of the
signal and the composition of peaking backgrounds.

The effect of the decay-time acceptance is determined through the generation
of simulated samples including the decay-time acceptance and fitted with the
method described in Sec. 9.1. The resulting deviation from the nominal fit
results is used to assign a systematic uncertainty. The effect of the angular
acceptance is evaluated by generating simulated data sets with and without the
inclusion of the angular acceptance. The difference between the nominal fit
results and the results obtained using the simulated samples including the
decay-time acceptance is then used as a systematic uncertainty.

Uncertainties related to the mass model are evaluated using a similar approach
to that described in Sec. 8.3. Additional uncertainties arise from the
assumption that the peaking background is symmetric in $U$ and $V$. The
deviation observed without this assumption is then added as a systematic
uncertainty. Similarly, the assumption that the combinatorial background has
no asymmetry yields an identical uncertainty. The systematic uncertainties are
summarised in Table 6.

Table 6: Summary of systematic uncertainties on $A_{U}$ and $A_{V}$. Source | Uncertainty
---|---
Time acceptance | 0.003
Angular acceptance | 0.003
Mass model | 0.001
Combinatorial background | 0.001
Peaking background | 0.001
Total | 0.005

###  9.4 Combination of Run 1 and Run 2 results

The Run 2 (2015–2016) values for the triple product asymmetries are

$A_{U}$ $~{}=~{}$ |  $-$0.003 $\,\,\pm\,$ | 0.015 (stat) $\pm\,$ | 0.005 (syst) | ,
---|---|---|---|---
$A_{V}$ $~{}=~{}$ |  $-$0.012 $\,\,\pm\,$ | 0.015 (stat) $\pm\,$ | 0.005 (syst) | ,

whilst the Run 1 (2011–2012) values from Ref. [10] are

$A_{U}$ $~{}=~{}$ |  $-$0.003 $\,\,\pm\,$ | 0.017 (stat) $\pm\,$ | 0.006 (syst) | ,
---|---|---|---|---
$A_{V}$ $~{}=~{}$ |  $-$0.017 $\,\,\pm\,$ | 0.017 (stat) $\pm\,$ | 0.006 (syst) | .

The Run 1 and Run 2 results are combined by calculating a weighted average. In
this procedure the decay-time and angular acceptance systematic uncertainties
and peaking backgrounds are assumed to be fully correlated. All other
systematic uncertainties are assumed to be uncorrelated. This gives a final
result of

$A_{U}$ $~{}=~{}$ |  $-$0.003$\,\,\pm\,$ | 0.011 (stat)$\,\,\pm\,$ | 0.004 (syst),
---|---|---|---
$A_{V}$ $~{}=~{}$ |  $-$0.014$\,\,\pm\,$ | 0.011 (stat)$\,\,\pm\,$ | 0.004 (syst).

The Run 1 and Run 2 results are compatible with each other, and the
asymmetries are consistent with zero. No evidence for $C\\!P$ violation is
found.

##  10 Search for the $B^{0}\rightarrow\phi\phi$ decay

The selection criteria for the $B^{0}\rightarrow\phi\phi$ mode are based on
the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ selection, with some modifications.
The Punzi figure of merit [42] is used for the ${{B}^{0}}\rightarrow\phi\phi$
search, resulting in a more stringent MLP requirement. Furthermore, the
uncertainty on the four-kaon mass is required to be less than
25$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, corresponding to roughly $3\sigma$
separation between the ${{B}^{0}_{s}}$ and $B^{0}$ mass peaks. The
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decay is used as normalisation decay
mode. The signal PDF for the mass of the ${B}^{0}$ meson is assumed to be the
same as that of the ${B}^{0}_{s}$ decay, with the modification of the
resolution according to a scaling factor, which is defined as

$\alpha=\frac{m_{B^{0}}-4m_{K}}{m_{B^{0}_{s}}-4m_{K}}=0.974,$ (21)

where $m_{K}$ is the known ${K}^{+}$ mass.

Figure 6 shows the fit to the full data set. The ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow\phi pK$ contribution is fixed to 109
candidates, following the same method described in Sec. 3. The fit returns a
yield of $4.9\pm 9.2$ ${{B}^{0}}\rightarrow\phi\phi$ decays.

Figure 6: Fit to the four-kaon invariant mass. The total PDF as described in
the text is shown as a blue solid line, ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$
as a red dashed line, ${{B}^{0}}\rightarrow\phi\phi$ as a green dotted line,
the ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow\phi pK$ contribution as a
magenta long-dashed line and the combinatorial background as a blue short-
dashed line.

The Confidence Levels (${\rm CL}_{s}$) method [56] is used to set a limit on
the $B^{0}\rightarrow\phi\phi$ branching fraction. A total of 10,000
pseudoexperiments are used to calculate each point of the scan.

Figure 7: Results of the ${\rm CL}_{s}$ scan as a function of the
${{B}^{0}}\rightarrow\phi\phi$ yield. The solid black line shows the observed
${\rm CL}_{s}$ distribution, while the dotted black line indicates the
expected distribution. The green (yellow) band marks the $1\sigma$ ($2\sigma$)
confidence region on the expected ${\rm CL}_{s}$. The $90\,\%$ CL limit is
shown as a red line.

Figure 7 shows the results of the ${\rm CL}_{s}$ scan. At $90\,\%$ CL,
$N_{{{B}^{0}}}<23.7$. These limits are converted to a branching fraction using

$\mathcal{B}(B^{0}\rightarrow\phi\phi)=N_{B^{0}}\times\frac{\epsilon_{B^{0}\rightarrow\phi\phi}}{\epsilon_{{{B}^{0}_{s}}\\!\rightarrow\phi\phi}}\times\frac{\mathcal{B}(B_{s}^{0}\rightarrow\phi\phi)\times
f_{s}/f_{d}}{N_{B_{s}^{0}\rightarrow\phi\phi}},$ (22)

where $N_{{{B}^{0}}}$ is the limit on the $B^{0}\rightarrow\phi\phi$ yield,
and $N_{{{B}^{0}_{s}}\\!\rightarrow\phi\phi}$ is the $B_{s}^{0}$ yield from
the fit displayed in Fig. 6. The relative reconstruction and selection
efficiency of the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ and
$B^{0}\rightarrow\phi\phi$ decays,
$\epsilon_{B^{0}\rightarrow\phi\phi}/\epsilon_{{{B}^{0}_{s}}\\!\rightarrow\phi\phi}$,
is determined to be 0.986 using simulation. The ratio of the fragmentation
functions has been measured at 7 and 8  TeV to be $f_{s}/f_{d}=0.259\pm 0.015$
within the LHCb acceptance [57]. The production fraction at 13  TeV has been
shown to be consistent with that of the 7 and 8  TeV data [58]. The
$\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow\phi\phi)=(1.84\pm
0.05\text{\,(stat)}\pm 0.07\text{\,(syst)}\pm 0.11(f_{s}/f_{d})\pm 0.12\,(\rm
norm))\times 10^{-5}$ branching fraction is an external input taken from Ref.
[26]. To set the limit, the uncertainties on the
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ branching fraction are propagated to the
limit, where the uncertainty on the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$
branching fraction arising from $f_{s}/f_{d}$ is already included in the
uncertainty on the normalisation mode, ${{B}^{0}}\rightarrow\phi{{K}^{*}}$.
The maximum value of $\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow\phi\phi)$
including the systematic contribution is found to be $1.99\times 10^{-5}$ and
is used in Eq. 22. This therefore translates to a limit of

$\displaystyle\mathcal{B}(B^{0}\rightarrow\phi\phi)<2.7\,(3.0)\times
10^{-8}\,\mathrm{at}\,90\,\%\,(95\,\%)\,{\rm CL},$

which supersedes the previous best limit.

##  11 Summary and conclusions

Measurements of $C\\!P$ violation in the ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$
decay are presented, based on a sample of proton-proton collision data
corresponding to an integrated luminosity of 5.0$\text{\,fb}^{-1}$ collected
with the LHCb detector. The $C\\!P$-violating phase,
$\phi_{{s}}^{{{s}{\overline{s}}}{s}}$, and $C\\!P$ violation parameter,
$|\lambda|$, are determined in a helicity-independent manner to be

$\phi_{{s}}^{{{s}{\overline{s}}}{s}}$ $~{}=~{}$ |  $-$0.073$\,\,\pm\,$ | 0.115 (stat)$\,\,\pm\,$ | 0.027 (syst) $\text{\,rad},$
---|---|---|---
$|\lambda|$ $~{}=~{}$ | 0.99$\,\,\pm\,$ | 0.05 (stat)$\,\,\pm\,$ | 0.01 (syst) .

The $C\\!P$-violating phases are also measured in a polarisation-dependent
manner, with the assumption that the longitudinal weak phase is
$C\\!P$-conserving ($\phi_{s,0}=0$) and that no direct $C\\!P$ violation is
present ($|\lambda|=1$). The $C\\!P$ phases corresponding to the parallel,
$\phi_{s,\parallel}$, and perpendicular, $\phi_{s,\perp}$, polarisations are
determined to be

$\phi_{s,\parallel}$ $~{}=~{}$ | 0.014$\,\,\pm\,$ | 0.055 (stat)$\,\,\pm\,$ | 0.011 (syst) $\text{\,rad},$
---|---|---|---
$\phi_{s,\perp}$ $~{}=~{}$ | 0.044$\,\,\pm\,$ | 0.059 (stat)$\,\,\pm\,$ | 0.019 (syst) $\text{\,rad}.$

The results are in agreement with SM predictions [1, 2, 3]. The uncertainties
have been validated with simulation. When compared with the $C\\!P$-violating
phase measured in
${{{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}}$ and
${{{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}}$ decays [51], these results show that no
significant $C\\!P$ violation is present either in ${B}^{0}_{s}$-${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ mixing or in the
${\overline{b}}\rightarrow{\overline{s}}{{s}{\overline{s}}}$ decay amplitude,
though the increased precision of the measurement presented in Ref. [51] leads
to more stringent constraints on $C\\!P$ violation in ${B}^{0}_{s}$-${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ mixing.

The polarisation amplitudes and strong phases are measured independently of
polarisation to be

$|A_{0}|^{2}$ $~{}=~{}$ |  $0.381$ ± | 0.007 (stat) $\,\,\pm\,$ | 0.012 (syst) ,
---|---|---|---
$|A_{\perp}|^{2}$ $~{}=~{}$ |  $0.290$ ± | 0.008 (stat) $\,\,\pm\,$ | 0.007 (syst) ,
$\delta_{\perp}$ $~{}=~{}$ |  $2.818$ ± | 0.178 (stat) $\,\,\pm\,$ | 0.073 (syst) $\text{\,rad},$
$\delta_{\parallel}$ $~{}=~{}$ |  $2.559$ ± | 0.045 (stat) $\,\,\pm\,$ | 0.033 (syst) $\text{\,rad}.$

The polarisation amplitudes and strong phases measured in the polarisation-
dependent fit are in agreement with the results listed here. In addition,
values of the polarisation amplitudes are found to agree well with previous
measurements [59, 60, 13, 10] and with predictions from QCD factorisation [2,
3].

The most precise measurements to date of the triple-product asymmetries are
determined from a separate time-integrated fit to be

$A_{U}$ $~{}=~{}$ |  $-$0.003$\,\,\pm\,$ | 0.011 (stat)$\,\,\pm\,$ | 0.004 (syst) ,
---|---|---|---
$A_{V}$ $~{}=~{}$ |  $-$0.014$\,\,\pm\,$ | 0.011 (stat)$\,\,\pm\,$ | 0.004 (syst) ,

in agreement with previous measurements [59, 13, 10]. The measured values of
the $C\\!P$-violating phase and triple-product asymmetries are consistent with
the hypothesis of $C\\!P$ conservation in
${\overline{b}}\rightarrow{\overline{s}}{{s}{\overline{s}}}$ transitions.

In addition, the most stringent limit on the branching fraction of the
${{B}^{0}}\rightarrow\phi\phi$ decay is presented and it is found to be

$\mathcal{B}(B^{0}\rightarrow\phi\phi)<2.7\times
10^{-8}\,\mathrm{at}\,90\,\%\,{\rm CL}.$

## Acknowledgements

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

##  Appendix A Time-dependent terms

In Table 7, $\delta_{S}$ and $\delta_{SS}$ are the strong phases of the
$P\rightarrow V\kern-1.00006ptS$ and $P\rightarrow S\kern-1.00006ptS$
processes, respectively. The $P$-wave strong phases are dependent on
$\delta_{\parallel}$ and $\delta_{0}$.

Table 7: Coefficients of the time-dependent terms and angular functions used
in Eq. 2. Amplitudes are defined at $t=0$.

$i$ | $N_{i}$ | $a_{i}$ | $b_{i}$ | $c_{i}$ | $d_{i}$ | $f_{i}$
---|---|---|---|---|---|---
$1$ | $|A_{0}|^{2}$ | $1+|\lambda_{0}|^{2}$ | $-2|\lambda_{0}|\cos(\phi)$ | $1-|\lambda_{0}|^{2}$ | $2|\lambda_{0}|\sin(\phi)$ | $4\cos^{2}\theta_{1}\cos^{2}\theta_{2}$
$2$ | $|A_{\parallel}|^{2}$ | $1+|\lambda_{\parallel}|^{2}$ | $-2|\lambda_{\parallel}|\cos(\phi_{s,\parallel})$ | $1-|\lambda_{\parallel}|^{2}$ | $2|\lambda_{\parallel}|\sin(\phi_{s,\parallel})$ | $\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1{+}\cos 2\Phi)$
$3$ | $|A_{\perp}|^{2}$ | $1+|\lambda_{\perp}|^{2}$ | $2|\lambda_{\perp}|\cos(\phi_{s,\perp})$ | $1-|\lambda_{\perp}|^{2}$ | $-2|\lambda_{\parallel}|\sin(\phi_{s,\perp})$ | $\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1{-}\cos 2\Phi)$
$4$ | $\frac{|A_{\parallel}||A_{\perp}|}{2}$ | $\begin{array}[]{c}\sin(\delta_{\parallel}-\delta_{\perp})-|\lambda_{\parallel}||\lambda_{\perp}|\cdot\\\ \sin(\delta_{\parallel}-\delta_{\perp}-\phi_{s,\parallel}+\phi_{s,\perp})\end{array}$ | $\begin{array}[]{c}-|\lambda_{\parallel}|\sin(\delta_{\parallel}-\delta_{\perp}-\phi_{s,\parallel})\\\ +|\lambda_{\perp}|\sin(\delta_{\parallel}-\delta_{\perp}+\phi_{s,\perp})\end{array}$ | $\begin{array}[]{c}\sin(\delta_{\parallel}-\delta_{\perp})+|\lambda_{\parallel}||\lambda_{\perp}|\cdot\\\ \sin(\delta_{\parallel}-\delta_{\perp}-\phi_{s,\parallel}+\phi_{s,\perp})\end{array}$ | $\begin{array}[]{c}|\lambda_{\parallel}|\cos(\delta_{\parallel}-\delta_{\perp}-\phi_{s,\parallel})\\\ +|\lambda_{\perp}|\cos(\delta_{\parallel}-\delta_{\perp}+\phi_{s,\perp})\end{array}$ | $-2\sin^{2}\theta_{1}\sin^{2}\theta_{2}\sin 2\Phi$
$5$ | $\frac{|A_{\parallel}||A_{0}|}{2}$ | $\begin{array}[]{c}\cos(\delta_{\parallel}-\delta_{0})+|\lambda_{\parallel}||\lambda_{0}|\cdot\\\ \cos(\delta_{\parallel}-\delta_{0}-\phi_{s,\parallel}+\phi)\end{array}$ | $\begin{array}[]{c}-|\lambda_{\parallel}|\cos(\delta_{\parallel}-\delta_{0}-\phi_{s,\parallel})\\\ +|\lambda_{0}|\cos(\delta_{\parallel}-\delta_{0}+\phi)\end{array}$ | $\begin{array}[]{c}\cos(\delta_{\parallel}-\delta_{0})-|\lambda_{\parallel}||\lambda_{0}|\cdot\\\ \sin(\delta_{\parallel}-\delta_{0}-\phi_{s,\parallel}+\phi)\end{array}$ | $\begin{array}[]{c}-|\lambda_{\parallel}|\sin(\delta_{\parallel}-\delta_{0}-\phi_{s,\parallel})\\\ +|\lambda_{0}|\sin(\delta_{\parallel}-\delta_{0}+\phi)\end{array}$ | $\sqrt{2}\sin 2\theta_{1}\sin 2\theta_{2}\cos\Phi$
$6$ | $\frac{|A_{0}||A_{\perp}|}{2}$ | $\begin{array}[]{c}\sin(\delta_{0}-\delta_{\perp})-|\lambda_{0}||\lambda_{\perp}|\cdot\\\ \sin(\delta_{0}-\delta_{\perp}-\phi+\phi_{s,\perp})\end{array}$ | $\begin{array}[]{c}-|\lambda_{0}|\sin(\delta_{0}-\delta_{\perp}-\phi)\\\ +|\lambda_{\perp}|\sin(\delta_{0}-\delta_{\perp}+\phi_{s,\perp})\end{array}$ | $\begin{array}[]{c}\sin(\delta_{0}-\delta_{\perp})+|\lambda_{0}||\lambda_{\perp}|\cdot\\\ \sin(\delta_{0}-\delta_{\perp}-\phi+\phi_{s,\perp})\end{array}$ | $\begin{array}[]{c}|\lambda_{0}|\cos(\delta_{0}-\delta_{\perp}-\phi)\\\ +|\lambda_{\perp}|\cos(\delta_{0}-\delta_{\perp}+\phi_{s,\perp})\end{array}$ | $-\sqrt{2}\sin 2\theta_{1}\sin 2\theta_{2}\sin\Phi$
$7$ | $|A_{SS}|^{2}$ | $1+|\lambda_{ss}|^{2}$ | $-2|\lambda_{ss}|\cos(\phi_{s,ss})$ | $1-|\lambda_{ss}|^{2}$ | $2|\lambda_{ss}|\sin(\phi_{s,ss})$ | $\frac{4}{9}$
$8$ | $|A_{S}|^{2}$ | $1+|\lambda_{s}|^{2}$ | $2|\lambda_{s}|\cos(\phi_{s,s})$ | $1-|\lambda_{s}|^{2}$ | $-2|\lambda_{s}|\sin(\phi_{s,s})$ | $\frac{4}{3}(\cos\theta_{1}+\cos\theta_{2})^{2}$
$9$ | $\frac{|A_{S}||A_{SS}|}{2}$ | $\begin{array}[]{c}\cos(\delta_{s}-\delta_{ss})-|\lambda_{s}||\lambda_{ss}|\cdot\\\ \cos(\delta_{s}-\delta_{ss}-\phi_{s,s}+\phi_{s,ss})\end{array}$ | $\begin{array}[]{c}|\lambda_{s}|\cos(\delta_{s}-\delta_{ss}-\phi_{s,s})\\\ +|\lambda_{ss}|\cos(\delta_{s}-\delta_{ss}+\phi_{s,ss})\end{array}$ | $\begin{array}[]{c}\cos(\delta_{s}-\delta_{ss})+|\lambda_{s}||\lambda_{ss}|\cdot\\\ \sin(\delta_{s}-\delta_{ss}-\phi_{s,s}+\phi_{s,ss})\end{array}$ | $\begin{array}[]{c}|\lambda_{s}|\sin(\delta_{s}-\delta_{ss}-\phi_{s,s})\\\ +|\lambda_{ss}|\sin(\delta_{s}-\delta_{ss}+\phi_{s,ss})\end{array}$ | $\frac{8}{3\sqrt{3}}(\cos\theta_{1}+\cos\theta_{2})$
$10$ | $\frac{|A_{0}||A_{SS}|}{2}$ | $\begin{array}[]{c}\cos(\delta_{0}-\delta_{ss})+|\lambda_{0}||\lambda_{ss}|\cdot\\\ \cos(\delta_{0}-\delta_{ss}-\phi+\phi_{s,ss})\end{array}$ | $\begin{array}[]{c}-|\lambda_{0}|\cos(\delta_{0}-\delta_{ss}-\phi)\\\ +|\lambda_{ss}|\cos(\delta_{0}-\delta_{ss}+\phi_{s,ss})\end{array}$ | $\begin{array}[]{c}\cos(\delta_{0}-\delta_{ss})-|\lambda_{0}||\lambda_{ss}|\cdot\\\ \sin(\delta_{0}-\delta_{ss}-\phi+\phi_{s,ss})\end{array}$ | $\begin{array}[]{c}-|\lambda_{0}|\sin(\delta_{0}-\delta_{ss}-\phi)\\\ +|\lambda_{ss}|\sin(\delta_{0}-\delta_{ss}+\phi_{s,ss})\end{array}$ | $\frac{8}{3}\cos\theta_{1}\cos\theta_{2}$
$11$ | $\frac{|A_{\parallel}||A_{SS}|}{2}$ | $\begin{array}[]{c}\cos(\delta_{\parallel}-\delta_{ss})+|\lambda_{\parallel}||\lambda_{ss}|\cdot\\\ \cos(\delta_{\parallel}-\delta_{ss}-\phi_{s,\parallel}+\phi_{s,ss})\end{array}$ | $\begin{array}[]{c}-|\lambda_{\parallel}|\cos(\delta_{\parallel}-\delta_{ss}-\phi_{s,\parallel})\\\ +|\lambda_{ss}|\cos(\delta_{\parallel}-\delta_{ss}+\phi_{s,ss})\end{array}$ | $\begin{array}[]{c}\cos(\delta_{\parallel}-\delta_{ss})-|\lambda_{\parallel}||\lambda_{ss}|\cdot\\\ \sin(\delta_{\parallel}-\delta_{ss}-\phi_{s,\parallel}+\phi_{s,ss})\end{array}$ | $\begin{array}[]{c}-|\lambda_{0}|\sin(\delta_{\parallel}-\delta_{ss}-\phi_{s,\parallel})\\\ +|\lambda_{ss}|\sin(\delta_{\parallel}-\delta_{ss}+\phi_{s,ss})\end{array}$ | $\frac{4\sqrt{2}}{3}\sin\theta_{1}\sin\theta_{2}\cos\Phi$
$12$ | $\frac{|A_{\perp}||A_{SS}|}{2}$ | $\begin{array}[]{c}\sin(\delta_{\perp}-\delta_{ss})-|\lambda_{\perp}||\lambda_{ss}|\cdot\\\ \sin(\delta_{\perp}-\delta_{ss}-\phi_{s,\perp}+\phi_{s,ss})\end{array}$ | $\begin{array}[]{c}|\lambda_{\perp}|\sin(\delta_{\perp}-\delta_{ss}-\phi_{s,\perp})\\\ -|\lambda_{ss}|\sin(\delta_{\perp}-\delta_{ss}+\phi_{s,ss})\end{array}$ | $\begin{array}[]{c}\sin(\delta_{\perp}-\delta_{ss})+|\lambda_{\perp}||\lambda_{ss}|\cdot\\\ \sin(\delta_{\perp}-\delta_{ss}-\phi_{s,\perp}+\phi_{s,ss})\end{array}$ | $\begin{array}[]{c}-|\lambda_{\perp}|\cos(\delta_{\perp}-\delta_{ss}-\phi_{s,\perp})\\\ -|\lambda_{ss}|\cos(\delta_{\perp}-\delta_{ss}+\phi_{s,ss})\end{array}$ | $-\frac{4\sqrt{2}}{3}\sin\theta_{1}\sin\theta_{2}\sin\Phi$
$13$ | $\frac{|A_{0}||A_{S}|}{2}$ | $\begin{array}[]{c}\cos(\delta_{0}-\delta_{s})-|\lambda_{0}||\lambda_{s}|\cdot\\\ \cos(\delta_{0}-\delta_{s}-\phi+\phi_{s,s})\end{array}$ | $\begin{array}[]{c}-|\lambda_{0}|\cos(\delta_{0}-\delta_{s}-\phi)\\\ -|\lambda_{s}|\cos(\delta_{0}-\delta_{s}+\phi_{s,s})\end{array}$ | $\begin{array}[]{c}\cos(\delta_{0}-\delta_{s})+|\lambda_{0}||\lambda_{s}|\cdot\\\ \sin(\delta_{0}-\delta_{s}-\phi+\phi_{s,s})\end{array}$ | $\begin{array}[]{c}-|\lambda_{0}|\sin(\delta_{0}-\delta_{s}-\phi)\\\ -|\lambda_{s}|\sin(\delta_{0}-\delta_{ss}+\phi_{s,s})\end{array}$ | $\begin{array}[]{c}\frac{8}{\sqrt{3}}\cos\theta_{1}\cos\theta_{2}\\\ \times(\cos\theta_{1}+\cos\theta_{2})\end{array}$
$14$ | $\frac{|A_{\parallel}||A_{S}|}{2}$ | $\begin{array}[]{c}\cos(\delta_{\parallel}-\delta_{s})-|\lambda_{\parallel}||\lambda_{s}|\cdot\\\ \cos(\delta_{\parallel}-\delta_{s}-\phi_{s,\parallel}+\phi_{s,s})\end{array}$ | $\begin{array}[]{c}-|\lambda_{\parallel}|\cos(\delta_{\parallel}-\delta_{s}-\phi_{s,\parallel})\\\ -|\lambda_{s}|\cos(\delta_{\parallel}-\delta_{s}+\phi_{s,s})\end{array}$ | $\begin{array}[]{c}\cos(\delta_{\parallel}-\delta_{s})+|\lambda_{\parallel}||\lambda_{s}|\cdot\\\ \sin(\delta_{\parallel}-\delta_{s}-\phi_{s,\parallel}+\phi_{s,s})\end{array}$ | $\begin{array}[]{c}-|\lambda_{\parallel}|\sin(\delta_{\parallel}-\delta_{s}-\phi_{s,\parallel})\\\ -|\lambda_{s}|\sin(\delta_{\parallel}-\delta_{ss}+\phi_{s,s})\end{array}$ | $\begin{array}[]{c}\frac{4\sqrt{2}}{\sqrt{3}}\sin\theta_{1}\sin\theta_{2}\\\ \times(\cos\theta_{1}+\cos\theta_{2})\cos\Phi\end{array}$
$15$ | $\frac{|A_{\perp}||A_{S}|}{2}$ | $\begin{array}[]{c}\sin(\delta_{\perp}-\delta_{s})+|\lambda_{\perp}||\lambda_{s}|\cdot\\\ \sin(\delta_{\perp}-\delta_{s}-\phi_{s,\perp}+\phi_{s,s})\end{array}$ | $\begin{array}[]{c}|\lambda_{\perp}|\sin(\delta_{\perp}-\delta_{s}-\phi_{s,\perp})\\\ +|\lambda_{s}|\sin(\delta_{\perp}-\delta_{s}+\phi_{s,s})\end{array}$ | $\begin{array}[]{c}\sin(\delta_{\perp}-\delta_{s})-|\lambda_{\perp}||\lambda_{s}|\cdot\\\ \sin(\delta_{\perp}-\delta_{s}-\phi_{s,\perp}+\phi_{s,s})\end{array}$ | $\begin{array}[]{c}-|\lambda_{\perp}|\cos(\delta_{\perp}-\delta_{s}-\phi_{s,\perp})\\\ +|\lambda_{s}|\cos(\delta_{\perp}-\delta_{s}+\phi_{s,s})\end{array}$ | $\begin{array}[]{c}-\frac{4\sqrt{2}}{3}\sin\theta_{1}\sin\theta_{2}\\\ \times(\cos\theta_{1}+\cos\theta_{2})\sin\Phi\end{array}$

##  Appendix B Correlation matrices for the time-dependent fits

Table 8:  Statistical correlation matrix of the time-dependent fit.  | $\delta_{\parallel}$ | $|A_{\perp}|^{2}$ | $\delta_{\perp}$ | $|A_{0}|^{2}$ | $|\lambda|$ | $\phi_{{s}}^{{{s}{\overline{s}}}{s}}$
---|---|---|---|---|---|---
$\delta_{\parallel}$ | 1.00 | 0.14 | 0.13 | $-0.03$ | $0.02$ | $0.01$
$|A_{\perp}|^{2}$ |  | 1.00 | 0.01 | $-0.45$ | $0.00$ | $-0.03$
$\delta_{\perp}$ |  |  | 1.00 | $0.00$ | $-0.26$ | $-0.15$
$|A_{0}|^{2}$ |  |  |  | $1.00$ | $-0.01$ | $0.01$
$|\lambda|$ |  |  |  |  | $1.00$ | $-0.05$
${\phi_{{s}}^{{{s}{\overline{s}}}{s}}}$ |  |  |  |  |  | $1.00$
Table 9: Statistical correlation matrix of the time-dependent fit in which $C\\!P$ violation is polarisation dependent. | $\delta_{\parallel}$ | $|A_{\perp}|^{2}$ | $\delta_{\perp}$ | $|A_{0}|^{2}$ | $\phi_{s,\parallel}$ | $\phi_{s,\perp}$
---|---|---|---|---|---|---
$\delta_{\parallel}$ | 1.00 | 0.13 | 0.13 | $-0.02$ | $0.58$ | $0.41$
$|A_{\perp}|^{2}$ |  | 1.00 | 0.03 | $-0.45$ | $0.00$ | $0.01$
$\delta_{\perp}$ |  |  | 1.00 | $0.00$ | $0.08$ | $0.13$
$|A_{0}|^{2}$ |  |  |  | $1.00$ | $0.00$ | $0.01$
$|\lambda|$ |  |  |  |  | $1.00$ | $0.71$
${\phi_{{s}}^{{{s}{\overline{s}}}{s}}}$ |  |  |  |  |  | $1.00$

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LHCb collaboration

R. Aaij30, C. Abellán Beteta47, B. Adeva44, M. Adinolfi51, C.A. Aidala78, Z.
Ajaltouni8, S. Akar62, P. Albicocco21, J. Albrecht13, F. Alessio45, M.
Alexander56, A. Alfonso Albero43, G. Alkhazov36, P. Alvarez Cartelle58, A.A.
Alves Jr44, S. Amato2, Y. Amhis10, L. An20, L. Anderlini20, G. Andreassi46, M.
Andreotti19, J.E. Andrews63, F. Archilli21, J. Arnau Romeu9, A. Artamonov42,
M. Artuso65, K. Arzymatov40, E. Aslanides9, M. Atzeni47, B. Audurier25, S.
Bachmann15, J.J. Back53, S. Baker58, V. Balagura10,b, W. Baldini19,45, A.
Baranov40, R.J. Barlow59, S. Barsuk10, W. Barter58, M. Bartolini22, F.
Baryshnikov74, V. Batozskaya34, B. Batsukh65, A. Battig13, V. Battista46, A.
Bay46, F. Bedeschi27, I. Bediaga1, A. Beiter65, L.J. Bel30, V. Belavin40, S.
Belin25, N. Beliy4, V. Bellee46, K. Belous42, I. Belyaev37, G. Bencivenni21,
E. Ben-Haim11, S. Benson30, S. Beranek12, A. Berezhnoy38, R. Bernet47, D.
Berninghoff15, E. Bertholet11, A. Bertolin26, C. Betancourt47, F. Betti18,e,
M.O. Bettler52, Ia. Bezshyiko47, S. Bhasin51, J. Bhom32, M.S. Bieker13, S.
Bifani50, P. Billoir11, A. Birnkraut13, A. Bizzeti20,u, M. Bjørn60, M.P.
Blago45, T. Blake53, F. Blanc46, S. Blusk65, D. Bobulska56, V. Bocci29, O.
Boente Garcia44, T. Boettcher61, A. Boldyrev75, A. Bondar41,w, N. Bondar36, S.
Borghi59,45, M. Borisyak40, M. Borsato15, M. Boubdir12, T.J.V. Bowcock57, C.
Bozzi19,45, S. Braun15, A. Brea Rodriguez44, M. Brodski45, J. Brodzicka32, A.
Brossa Gonzalo53, D. Brundu25,45, E. Buchanan51, A. Buonaura47, C. Burr59, A.
Bursche25, J.S. Butter30, J. Buytaert45, W. Byczynski45, S. Cadeddu25, H.
Cai69, R. Calabrese19,g, S. Cali21, R. Calladine50, M. Calvi23,i, M. Calvo
Gomez43,m, A. Camboni43,m, P. Campana21, D.H. Campora Perez45, L.
Capriotti18,e, A. Carbone18,e, G. Carboni28, R. Cardinale22, A. Cardini25, P.
Carniti23,i, K. Carvalho Akiba2, A. Casais Vidal44, G. Casse57, M. Cattaneo45,
G. Cavallero22, R. Cenci27,p, M.G. Chapman51, M. Charles11,45, Ph.
Charpentier45, G. Chatzikonstantinidis50, M. Chefdeville7, V. Chekalina40, C.
Chen3, S. Chen25, S.-G. Chitic45, V. Chobanova44, M. Chrzaszcz45, A.
Chubykin36, P. Ciambrone21, X. Cid Vidal44, G. Ciezarek45, F. Cindolo18,
P.E.L. Clarke55, M. Clemencic45, H.V. Cliff52, J. Closier45, J.L.
Cobbledick59, V. Coco45, J.A.B. Coelho10, J. Cogan9, E. Cogneras8, L.
Cojocariu35, P. Collins45, T. Colombo45, A. Comerma-Montells15, A. Contu25, G.
Coombs45, S. Coquereau43, G. Corti45, C.M. Costa Sobral53, B. Couturier45,
G.A. Cowan55, D.C. Craik61, A. Crocombe53, M. Cruz Torres1, R. Currie55, C.L.
Da Silva64, E. Dall’Occo30, J. Dalseno44,51, C. D’Ambrosio45, A. Danilina37,
P. d’Argent15, A. Davis59, O. De Aguiar Francisco45, K. De Bruyn45, S. De
Capua59, M. De Cian46, J.M. De Miranda1, L. De Paula2, M. De Serio17,d, P. De
Simone21, J.A. de Vries30, C.T. Dean56, W. Dean78, D. Decamp7, L. Del Buono11,
B. Delaney52, H.-P. Dembinski14, M. Demmer13, A. Dendek33, D. Derkach75, O.
Deschamps8, F. Desse10, F. Dettori25, B. Dey6, A. Di Canto45, P. Di Nezza21,
S. Didenko74, H. Dijkstra45, F. Dordei25, M. Dorigo27,x, A.C. dos Reis1, A.
Dosil Suárez44, L. Douglas56, A. Dovbnya48, K. Dreimanis57, L. Dufour45, G.
Dujany11, P. Durante45, J.M. Durham64, D. Dutta59, R. Dzhelyadin42,†, M.
Dziewiecki15, A. Dziurda32, A. Dzyuba36, S. Easo54, U. Egede58, V.
Egorychev37, S. Eidelman41,w, S. Eisenhardt55, U. Eitschberger13, R.
Ekelhof13, S. Ek-In46, L. Eklund56, S. Ely65, A. Ene35, S. Escher12, S.
Esen30, T. Evans62, A. Falabella18, C. Färber45, N. Farley50, S. Farry57, D.
Fazzini10, M. Féo45, P. Fernandez Declara45, A. Fernandez Prieto44, F.
Ferrari18,e, L. Ferreira Lopes46, F. Ferreira Rodrigues2, S. Ferreres Sole30,
M. Ferro-Luzzi45, S. Filippov39, R.A. Fini17, M. Fiorini19,g, M. Firlej33, C.
Fitzpatrick45, T. Fiutowski33, F. Fleuret10,b, M. Fontana45, F.
Fontanelli22,h, R. Forty45, V. Franco Lima57, M. Franco Sevilla63, M. Frank45,
C. Frei45, J. Fu24,q, W. Funk45, E. Gabriel55, A. Gallas Torreira44, D.
Galli18,e, S. Gallorini26, S. Gambetta55, Y. Gan3, M. Gandelman2, P.
Gandini24, Y. Gao3, L.M. Garcia Martin77, J. García Pardiñas47, B. Garcia
Plana44, J. Garra Tico52, L. Garrido43, D. Gascon43, C. Gaspar45, G. Gazzoni8,
D. Gerick15, E. Gersabeck59, M. Gersabeck59, T. Gershon53, D. Gerstel9, Ph.
Ghez7, V. Gibson52, A. Gioventù44, O.G. Girard46, P. Gironella Gironell43, L.
Giubega35, K. Gizdov55, V.V. Gligorov11, C. Göbel67, D. Golubkov37, A.
Golutvin58,74, A. Gomes1,a, I.V. Gorelov38, C. Gotti23,i, E. Govorkova30, J.P.
Grabowski15, R. Graciani Diaz43, L.A. Granado Cardoso45, E. Graugés43, E.
Graverini46, G. Graziani20, A. Grecu35, R. Greim30, P. Griffith25, L.
Grillo59, L. Gruber45, B.R. Gruberg Cazon60, C. Gu3, E. Gushchin39, A. Guth12,
Yu. Guz42,45, T. Gys45, T. Hadavizadeh60, C. Hadjivasiliou8, G. Haefeli46, C.
Haen45, S.C. Haines52, P.M. Hamilton63, Q. Han6, X. Han15, T.H. Hancock60, S.
Hansmann-Menzemer15, N. Harnew60, T. Harrison57, C. Hasse45, M. Hatch45, J.
He4, M. Hecker58, K. Heijhoff30, K. Heinicke13, A. Heister13, K. Hennessy57,
L. Henry77, M. Heß71, J. Heuel12, A. Hicheur66, R. Hidalgo Charman59, D.
Hill60, M. Hilton59, P.H. Hopchev46, J. Hu15, W. Hu6, W. Huang4, Z.C. Huard62,
W. Hulsbergen30, T. Humair58, M. Hushchyn75, D. Hutchcroft57, D. Hynds30, P.
Ibis13, M. Idzik33, P. Ilten50, A. Inglessi36, A. Inyakin42, K. Ivshin36, R.
Jacobsson45, S. Jakobsen45, J. Jalocha60, E. Jans30, B.K. Jashal77, A.
Jawahery63, F. Jiang3, M. John60, D. Johnson45, C.R. Jones52, C. Joram45, B.
Jost45, N. Jurik60, S. Kandybei48, M. Karacson45, J.M. Kariuki51, S.
Karodia56, N. Kazeev75, M. Kecke15, F. Keizer52, M. Kelsey65, M. Kenzie52, T.
Ketel31, B. Khanji45, A. Kharisova76, C. Khurewathanakul46, K.E. Kim65, T.
Kirn12, V.S. Kirsebom46, S. Klaver21, K. Klimaszewski34, S. Koliiev49, M.
Kolpin15, A. Kondybayeva74, A. Konoplyannikov37, P. Kopciewicz33, R.
Kopecna15, P. Koppenburg30, I. Kostiuk30,49, O. Kot49, S. Kotriakhova36, M.
Kozeiha8, L. Kravchuk39, M. Kreps53, F. Kress58, S. Kretzschmar12, P.
Krokovny41,w, W. Krupa33, W. Krzemien34, W. Kucewicz32,l, M. Kucharczyk32, V.
Kudryavtsev41,w, G.J. Kunde64, A.K. Kuonen46, T. Kvaratskheliya37, D.
Lacarrere45, G. Lafferty59, A. Lai25, D. Lancierini47, G. Lanfranchi21, C.
Langenbruch12, T. Latham53, C. Lazzeroni50, R. Le Gac9, R. Lefèvre8, A.
Leflat38, F. Lemaitre45, O. Leroy9, T. Lesiak32, B. Leverington15, H. Li68,
P.-R. Li4,aa, X. Li64, Y. Li5, Z. Li65, X. Liang65, T. Likhomanenko73, R.
Lindner45, F. Lionetto47, V. Lisovskyi10, G. Liu68, X. Liu3, D. Loh53, A.
Loi25, J. Lomba Castro44, I. Longstaff56, J.H. Lopes2, G. Loustau47, G.H.
Lovell52, D. Lucchesi26,o, M. Lucio Martinez44, Y. Luo3, A. Lupato26, E.
Luppi19,g, O. Lupton53, A. Lusiani27, X. Lyu4, F. Machefert10, F. Maciuc35, V.
Macko46, P. Mackowiak13, S. Maddrell-Mander51, O. Maev36,45, A. Maevskiy75, K.
Maguire59, D. Maisuzenko36, M.W. Majewski33, S. Malde60, B. Malecki45, A.
Malinin73, T. Maltsev41,w, H. Malygina15, G. Manca25,f, G. Mancinelli9, D.
Marangotto24,q, J. Maratas8,v, J.F. Marchand7, U. Marconi18, C. Marin
Benito10, M. Marinangeli46, P. Marino46, J. Marks15, P.J. Marshall57, G.
Martellotti29, L. Martinazzoli45, M. Martinelli45,23,i, D. Martinez Santos44,
F. Martinez Vidal77, A. Massafferri1, M. Materok12, R. Matev45, A. Mathad47,
Z. Mathe45, V. Matiunin37, C. Matteuzzi23, K.R. Mattioli78, A. Mauri47, E.
Maurice10,b, B. Maurin46, M. McCann58,45, L. Mcconnell16, A. McNab59, R.
McNulty16, J.V. Mead57, B. Meadows62, C. Meaux9, N. Meinert71, D. Melnychuk34,
M. Merk30, A. Merli24,q, E. Michielin26, D.A. Milanes70, E. Millard53, M.-N.
Minard7, O. Mineev37, L. Minzoni19,g, D.S. Mitzel15, A. Mödden13, A. Mogini11,
R.D. Moise58, T. Mombächer13, I.A. Monroy70, S. Monteil8, M. Morandin26, G.
Morello21, M.J. Morello27,t, J. Moron33, A.B. Morris9, R. Mountain65, H. Mu3,
F. Muheim55, M. Mukherjee6, M. Mulder30, D. Müller45, J. Müller13, K.
Müller47, V. Müller13, C.H. Murphy60, D. Murray59, P. Naik51, T. Nakada46, R.
Nandakumar54, A. Nandi60, T. Nanut46, I. Nasteva2, M. Needham55, N. Neri24,q,
S. Neubert15, N. Neufeld45, R. Newcombe58, T.D. Nguyen46, C. Nguyen-Mau46,n,
S. Nieswand12, R. Niet13, N. Nikitin38, N.S. Nolte45, A. Oblakowska-Mucha33,
V. Obraztsov42, S. Ogilvy56, D.P. O’Hanlon18, R. Oldeman25,f, C.J.G.
Onderwater72, J. D. Osborn78, A. Ossowska32, J.M. Otalora Goicochea2, T.
Ovsiannikova37, P. Owen47, A. Oyanguren77, P.R. Pais46, T. Pajero27,t, A.
Palano17, M. Palutan21, G. Panshin76, A. Papanestis54, M. Pappagallo55, L.L.
Pappalardo19,g, W. Parker63, C. Parkes59,45, G. Passaleva20,45, A. Pastore17,
M. Patel58, C. Patrignani18,e, A. Pearce45, A. Pellegrino30, G. Penso29, M.
Pepe Altarelli45, S. Perazzini18, D. Pereima37, P. Perret8, L. Pescatore46, K.
Petridis51, A. Petrolini22,h, A. Petrov73, S. Petrucci55, M. Petruzzo24,q, B.
Pietrzyk7, G. Pietrzyk46, M. Pikies32, M. Pili60, D. Pinci29, J. Pinzino45, F.
Pisani45, A. Piucci15, V. Placinta35, S. Playfer55, J. Plews50, M. Plo
Casasus44, F. Polci11, M. Poli Lener21, M. Poliakova65, A. Poluektov9, N.
Polukhina74,c, I. Polyakov65, E. Polycarpo2, G.J. Pomery51, S. Ponce45, A.
Popov42, D. Popov50, S. Poslavskii42, K. Prasanth32, E. Price51, C. Prouve44,
V. Pugatch49, A. Puig Navarro47, H. Pullen60, G. Punzi27,p, W. Qian4, J. Qin4,
R. Quagliani11, B. Quintana8, N.V. Raab16, B. Rachwal33, J.H. Rademacker51, M.
Rama27, M. Ramos Pernas44, M.S. Rangel2, F. Ratnikov40,75, G. Raven31, M.
Ravonel Salzgeber45, M. Reboud7, F. Redi46, S. Reichert13, F. Reiss11, C.
Remon Alepuz77, Z. Ren3, V. Renaudin60, S. Ricciardi54, S. Richards51, K.
Rinnert57, P. Robbe10, A. Robert11, A.B. Rodrigues46, E. Rodrigues62, J.A.
Rodriguez Lopez70, M. Roehrken45, S. Roiser45, A. Rollings60, V.
Romanovskiy42, A. Romero Vidal44, J.D. Roth78, M. Rotondo21, M.S. Rudolph65,
T. Ruf45, J. Ruiz Vidal77, J.J. Saborido Silva44, N. Sagidova36, B.
Saitta25,f, V. Salustino Guimaraes67, C. Sanchez Gras30, C. Sanchez
Mayordomo77, B. Sanmartin Sedes44, R. Santacesaria29, C. Santamarina Rios44,
M. Santimaria21,45, E. Santovetti28,j, G. Sarpis59, A. Sarti21,k, C.
Satriano29,s, A. Satta28, M. Saur4, D. Savrina37,38, S. Schael12, M.
Schellenberg13, M. Schiller56, H. Schindler45, M. Schmelling14, T.
Schmelzer13, B. Schmidt45, O. Schneider46, A. Schopper45, H.F. Schreiner62, M.
Schubiger30, S. Schulte46, M.H. Schune10, R. Schwemmer45, B. Sciascia21, A.
Sciubba29,k, A. Semennikov37, E.S. Sepulveda11, A. Sergi50,45, N. Serra47, J.
Serrano9, L. Sestini26, A. Seuthe13, P. Seyfert45, M. Shapkin42, T. Shears57,
L. Shekhtman41,w, V. Shevchenko73, E. Shmanin74, B.G. Siddi19, R. Silva
Coutinho47, L. Silva de Oliveira2, G. Simi26,o, S. Simone17,d, I. Skiba19, N.
Skidmore15, T. Skwarnicki65, M.W. Slater50, J.G. Smeaton52, E. Smith12, I.T.
Smith55, M. Smith58, M. Soares18, l. Soares Lavra1, M.D. Sokoloff62, F.J.P.
Soler56, B. Souza De Paula2, B. Spaan13, E. Spadaro Norella24,q, P.
Spradlin56, F. Stagni45, M. Stahl15, S. Stahl45, P. Stefko46, S. Stefkova58,
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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
4University of Chinese Academy of Sciences, Beijing, China
5Institute Of High Energy Physics (ihep), Beijing, China
6Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
7Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
8Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
9Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
10LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
11LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
12I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
13Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
14Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
15Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
16School of Physics, University College Dublin, Dublin, Ireland
17INFN Sezione di Bari, Bari, Italy
18INFN Sezione di Bologna, Bologna, Italy
19INFN Sezione di Ferrara, Ferrara, Italy
20INFN Sezione di Firenze, Firenze, Italy
21INFN Laboratori Nazionali di Frascati, Frascati, Italy
22INFN Sezione di Genova, Genova, Italy
23INFN Sezione di Milano-Bicocca, Milano, Italy
24INFN Sezione di Milano, Milano, Italy
25INFN Sezione di Cagliari, Monserrato, Italy
26INFN Sezione di Padova, Padova, Italy
27INFN Sezione di Pisa, Pisa, Italy
28INFN Sezione di Roma Tor Vergata, Roma, Italy
29INFN Sezione di Roma La Sapienza, Roma, Italy
30Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
31Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
32Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
33AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
34National Center for Nuclear Research (NCBJ), Warsaw, Poland
35Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
36Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
37Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia, Moscow, Russia
38Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
39Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
40Yandex School of Data Analysis, Moscow, Russia
41Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
42Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
43ICCUB, Universitat de Barcelona, Barcelona, Spain
44Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
45European Organization for Nuclear Research (CERN), Geneva, Switzerland
46Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
47Physik-Institut, Universität Zürich, Zürich, Switzerland
48NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
49Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
50University of Birmingham, Birmingham, United Kingdom
51H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
52Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
53Department of Physics, University of Warwick, Coventry, United Kingdom
54STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
55School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
56School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
57Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
58Imperial College London, London, United Kingdom
59School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
60Department of Physics, University of Oxford, Oxford, United Kingdom
61Massachusetts Institute of Technology, Cambridge, MA, United States
62University of Cincinnati, Cincinnati, OH, United States
63University of Maryland, College Park, MD, United States
64Los Alamos National Laboratory (LANL), Los Alamos, United States
65Syracuse University, Syracuse, NY, United States
66Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
67Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
68South China Normal University, Guangzhou, China, associated to 3
69School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
70Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 11
71Institut für Physik, Universität Rostock, Rostock, Germany, associated to 15
72Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 30
73National Research Centre Kurchatov Institute, Moscow, Russia, associated to
37
74National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 37
75National Research University Higher School of Economics, Moscow, Russia,
associated to 40
76National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
37
77Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 43
78University of Michigan, Ann Arbor, United States, associated to 65

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
xSezione INFN di Trieste, Trieste, Italy
ySchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
zPhysics and Micro Electronic College, Hunan University, Changsha City, China
aaLanzhou University, Lanzhou, China
†Deceased