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LHCb-CONF-2018-001April 19, 2018

Measurement of CP violation in theB0s→ φφ decay and search for the

B0 → φφ decay

LHCb collaboration†

Abstract

A measurement of the time-dependent CP -violating asymmetry in B0s → φφ de-

cays is presented, along with measurements of the T -odd triple-product asym-metries. Using a sample of proton-proton collision data corresponding to anintegrated luminosity of 5.0 fb−1 collected with the LHCb detector, a signalyield of approximately 8500 B0

s → φφ decays is obtained. The CP -violatingphase is measured to be φssss = −0.07± 0.13 (stat)± 0.03 (syst) rad. The triple-product asymmetries are measured to be AU = 0.000± 0.012 (stat)± 0.004 (syst)and AV = −0.003± 0.012 (stat)± 0.004 (syst). The results obtained are consis-tent with the hypothesis of CP conservation in b → sss transitions. In addi-tion, an improved limit on the branching fraction of the B0 → φφ decay is set,B(B0 → φφ) < 2.4× 10−8 at 90 % confidence level.

c© 2018 CERN for the benefit of the LHCb collaboration. CC-BY-4.0 licence.

†Conference report prepared for the 53rd Rencontres de Moriond, La Thuile, Italy, 10–17 March 2018.Contact authors: Sean Benson, sean.benson@nikhef.nl and Emmy Gabriel, emmy.gabriel@cern.ch.

1 Introduction

The B0s → φ(→ K+K−)φ(→ K+K−) decay is forbidden at tree level in the Standard

Model (SM) and proceeds predominantly via a gluonic b→ sss loop (penguin) process.Hence, this channel provides an excellent probe of new heavy particles entering thepenguin quantum loops [1–3]. In the SM, CP violation is governed by a single phasein the Cabibbo-Kobayashi-Maskawa quark mixing matrix [4]. Interference between theB0s -B

0s oscillation and decay amplitudes leads to a CP asymmetry in the decay-time

distributions of B0s and B0

s mesons. The CP asymmetry is characterised by a CP -violatingweak phase. Due to different decay amplitudes, the actual value of the weak phasedepends on the B0

s decay channel. For B0s → J/ψK+K− and B0

s → J/ψπ+π− decays,which proceed via b → scc transitions, the SM prediction of the weak phase is givenby −2 arg (−VtsV ∗tb/VcsV ∗cb) = −0.0364 ± 0.0016 rad [5]. The LHCb collaboration hasmeasured the weak phase in the combination of B0

s → J/ψK+K− and B0s → J/ψπ+π−

decays to be −0.010 ± 0.039 rad [6]. These measurements are consistent with the SMprediction and place stringent constraints on CP violation in B0

s -B0s oscillations [7]. The

CP -violating phase, φssss , in the B0s → φφ decay is expected to be small in the SM.

Calculations using quantum chromodynamics factorisation (QCDf) provide an upper limitof 0.02 rad for |φssss | [1–3]. The previous most accurate measurement of the CP -violatingphase is −0.17± 0.15 (stat)± 0.03 (syst) [8].

Triple-product asymmetries are formed from T -odd combinations of the momenta ofthe final-state particles. Such asymmetries provide a method of measuring CP violation ina time-integrated manner that complements the time-dependent measurement [9]. Theseasymmetries expressed in terms of the angular observables and are expected to be closeto zero in the SM [10]. Previous measurements of the triple-product asymmetries in B0

s

decays from the LHCb and CDF collaborations [8,11] have shown no significant deviationsfrom zero [8].

The B0s → φφ decay is a P → V V decay, where P denotes a pseudoscalar and V

a vector meson. This gives rise to longitudinal and transverse polarisation states, thefractions of which are denoted by fL and fT , respectively. In the heavy quark limit, fL isexpected to be close to unity due to the V –A (vector-axial) structure of charged weakcurrents [2]. This is found to be the case for tree-level B decays measured at the Bfactories [12–17]. However, the dynamics of penguin transitions are more complicated. Inthe context of QCDf, fL is predicted to be 0.36+0.23

−0.18 for the B0s→ φφ decay [3].

A search for the as yet unobserved decay B0 → φ(→ K+K−)φ(→ K+K−) is made, inaddition to the study of the B0

s→ φφ decay. This decay is suppressed in the StandardModel by the OZI rule [18], with an expected branching fraction in the range (0.1 −3.0)× 10−8 [1, 2, 19,20]. However, the branching fraction can be enhanced, up to the 10−7

level, in models such as supersymmetry with R-parity violation [20]. The most recentexperimental limit was determined to be 2.8× 10−8 at 90 % confidence level [21].

Measurements presented in this note are based on pp collision data corresponding toan integrated luminosity of 5.0 fb−1 collected by the LHCb experiment at centre-of-massenergies

√s = 7 TeV in 2011, 8 TeV in 2012, and 13 TeV from 2015 to 2016. Results

presented supersede the previous measurements based on data collected in 2011 and2012 [8].

1

2 Detector description

The LHCb detector [22, 23] is a single-arm forward spectrometer covering thepseudorapidity range 2 < η < 5, designed for the study of particles containing b orc quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the pp interaction region, a large-area silicon-stripdetector located upstream of a dipole magnet with a bending power of about 4 Tm,and three stations of silicon-strip detectors and straw drift tubes placed downstream ofthe magnet. The tracking system provides a measurement of momentum, p, of chargedparticles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at200 GeV/c. The impact parameter, i.e. the minimum distance of a track to a primaryvertex (PV), is measured with a resolution of (15 + 29/pT)µm, where pT is the componentof the momentum transverse to the beam, in GeV/c. Photons, electrons and hadrons areidentified by a calorimeter system consisting of scintillating-pad and preshower detectors,an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by asystem composed of alternating layers of iron and multiwire proportional chambers. Theonline event selection is performed by a trigger, which consists of a hardware stage, basedon information from the calorimeter and muon systems, followed by a software stage, whichapplies a full event reconstruction. At the hardware trigger stage, events are required tocontain a muon with high pT or a hadron, photon or electron with high transverse energyin the calorimeters. In the software trigger, B0

s→ φφ candidates are selected either byidentifying events containing a pair of oppositely charged kaons with an invariant masswithin 30 MeV/c2 of the known φ meson mass [24], or by using a topological b-hadrontrigger. It requires a three-track secondary vertex with a large sum of the pT of thecharged particles and significant displacement from the PV. At least one charged particleshould have pT > 1.7 GeV/c and χ2

IP with respect to any primary interaction greater than16, where χ2

IP is defined as the difference in χ2 of a given PV fitted with and without theconsidered track. A multivariate algorithm [25] is used for the identification of secondaryvertices consistent with the decay of a b hadron.

In the simulation, pp collisions are generated using Pythia [26] with a specific LHCbconfiguration [27]. Decays of hadronic particles are described by EvtGen [28], in whichfinal-state radiation is generated using Photos [29]. The interaction of the generatedparticles with the detector and its response are implemented using the Geant4 toolkit [30],as described in Ref. [27].

3 Selection and mass model

For time-dependent measurements, the previously analysed dataset using data taken in2011 and 2012 [8] is supplemented with the additional data taken in 2015 and 2016, towhich the selection described below is applied. However, for the case of the B0 → φφsearch, a wider invariant mass window is required than that used in Ref. [8]. Therefore,the selection used for this measurement is wholly as described in this section, and isapplied to the entire 5.0 fb−1 dataset.

Events passing the trigger are initially required to pass loose requirements on the fitquality of the four-kaon vertex fit, the χ2

IP of each track, the transverse momentum ofeach particle, and the product of the transverse momenta of the two φ candidates. In

2

addition, the reconstructed mass of the φ candidates is required to be within 25 MeV/c2

of the known φ mass [24].In order to further separate the B0

s → φφ signal candidates from the background,a multilayer perceptron (MLP) is used [31]. To train the MLP, simulated B0

s → φφcandidates passing the same requirements as the data candidates are used as a proxy forsignal, whereas the four-kaon invariant-mass sidebands from data are used as a proxy forbackground. The signal mass region is defined to be less than 120 MeV/c2 from the knownB0s mass, mB0

s[24]. The invariant-mass sidebands are defined to be inside the region

120 < |mK+K−K+K− −mB0s| < 180 MeV/c2, where mK+K−K+K− is the four-kaon invariant

mass. Separate MLP classifiers are trained for the data of each year. Variables used inthe MLP consist of the minimum and maximum kaon pT and η, the minimum and themaximum pT and η of the φ candidates, the pT and η of the B0

s candidate, the quality ofthe four-kaon vertex fit, and the cosine of the angle between the momentum of the B0

s

and the direction of flight from the best PV to the B0s decay vertex, where the best PV is

chosen as that with the smallest χ2IP with respect to the decay vertex. For the case of CP

violation measurements, the optimum requirement on each MLP is chosen to maximiseNS/√NS +NB, where NS (NB) represents the expected number of signal (background)

candidates in the signal region. The number of signal candidates is estimated usingsimulated events, whereas the number of background candidates is estimated from thedata sidebands. However, for the case of the B0 search, the figure of merit is chosen tomaximise ε/(3/2 +

√NB) [32], where ε is the signal efficiency, determined from simulated

events. The different figure of merit is chosen for the B0 search in order to have thegreatest sensitivity to the unobserved decay.

The presence of peaking backgrounds is studied using simulated samples. The decaymodes considered include B+ → φK+, B0 → φπ+π−, B0 → φK∗0, and Λ0

b → φpK−,of which only the last two are found to contribute significantly, and are the result of amisidentification of a pion or proton as a kaon, respectively. The B0 → φK∗0 decay isvetoed by rejecting candidates which simultaneously have K+π− invariant mass within50 MeV/c2 and K+K−K+π− invariant mass within 30 MeV/c2 of the nominal K∗0 andB0 masses, respectively. The number of Λ0

b → φpK− decays is estimated from data bychanging the mass hypothesis of the final-state particle most likely to have the mass ofthe proton. The misidentified particle is chosen to be the one with the highest probabilityof having the true mass of the species in question using particle identification information.This method yields 193± 30 Λ0

b → φpK− decays.In order to correctly determine the B0

s → φφ yield in the final data sample, thefour-kaon invariant mass distributions are fitted with a B0

s → φφ signal model, whichconsists of the sum of a Crystal Ball [33] and Student’s t function; and the combinatorialbackground component, described using an exponential function. The peaking backgroundcontribution is modelled by a Crystal Ball function, with the shape parameters fixedto the values obtained from a fit to simulated events. The yield of the Λ0

b → φpK−

peaking background contribution is fixed to the number previously stated. Once theMLP requirements are imposed, an unbinned extended maximum likelihood fit to thefour-kaon invariant mass gives a total yield of 8481± 101 B0

s→ φφ decays and 3405± 73combinatorial background decays. The fits to the four-kaon invariant mass are shown inFig. 1.

3

-4-2024

M(K+K-K+K-) [MeV/c2]5200 5300 5400 5500 560010-2

10-1

1

10

102

Candidates/

(4.5MeV

/c2 )

Pull

LHCb Preliminary2011

-4-2024

M(K+K-K+K-) [MeV/c2]5200 5300 5400 5500 560010-2

10-1

1

10

102

Candidates/

(4.5MeV

/c2 )

Pull

LHCb Preliminary2012

-4-2024

M(K+K-K+K-) [MeV/c2]5200 5300 5400 5500 560010-2

10-1

1

10

102LHCb Preliminary

2015

Candidates/

(4.5MeV

/c2 )

Pull

-4-2024

M(K+K-K+K-) [MeV/c2]5200 5300 5400 5500 560010-2

10-1

1

10

102

Candidates/

(4.5MeV

/c2 )

Pull

LHCb Preliminary2016

Figure 1: A fit to the 2011 (top left), 2012 (top right), 2015 (bottom left) and 2016 (bottomright) data, represented by the black points. Also shown are the results of the total fit (bluesolid line), with the B0

s→ φφ (red dashed), the Λ0b → φpK− (magenta long dashed), and the

combinatorial (blue short dashed) fit components.

4 Formalism

The B0s→ φφ decay is composed of a mixture of CP eigenstates, that are disentangled by

means of an angular analysis in the helicity basis. This basis is defined in Fig. 2.

4.1 Time-dependent model

As mentioned in Sec. 1, the B0s → φφ decay is a P → V V decay, where P denotes a

pseudoscalar and V a vector meson. However, due to the proximity of the φ resonanceto the f0(980) resonance, there will also be contributions from S-wave (P → VS) anddouble S-wave (P → SS) processes, where S denotes a spin-0 meson, or a nonresonantpair of kaons. Thus, the total amplitude is a coherent sum of P -, S-, and double S-waveprocesses, and is modelled by making use of the different functions of the helicity anglesassociated with these terms, where the helicity angles are defined in Fig. 2. A randomisedchoice of which φ meson is used to determine θ1 and which is used to determine θ2 ismade. The total amplitude (A) containing the P -, S-, and double S-wave components as

4

nV1 nV2

B0s

Φ

θ2θ1

K−

K+

K−

K+

φ1 φ2

Figure 2: Decay angles for the B0s → φφ decay, where θ1,2 is the angle between the K+ track

momentum in the φ1,2 meson rest frame and the parent φ1,2 momentum in the B0s rest frame, Φ

is the angle between the two φ meson decay planes and nV1,2 is the unit vector normal to thedecay plane of the φ1,2 meson.

a function of time, t, can be written as [34]

A(t, θ1, θ2,Φ) = A0(t) cos θ1 cos θ2 +A‖(t)√

2sin θ1 sin θ2 cos Φ

+ iA⊥(t)√

2sin θ1 sin θ2 sin Φ +

AS(t)√3

(cos θ1 + cos θ2) +ASS(t)

3, (1)

where A0, A‖, and A⊥ are the CP -even longitudinal, CP -even parallel, and CP -oddperpendicular polarisations of the B0

s→ φφ decay. The P → VS and P → SS processesare described by the AS and ASS amplitudes, respectively. The differential decay ratemay be found through the square of the total amplitude leading to the 15 terms [34]

dΓ

dt d cos θ1 d cos θ2 dΦ∝ 4|A(t, θ1, θ2,Φ)|2 =

15∑i=1

Ki(t)fi(θ1, θ2,Φ). (2)

The Ki(t) term can be written as

Ki(t) = Nie−Γst

[ai cosh

(1

2∆Γst

)+ bi sinh

(1

2∆Γst

)+ ci cos(∆mst) + di sin(∆mst)

],

(3)

where the coefficients ai, bi, ci and di are shown in Table 1, ∆Γs ≡ ΓL − ΓH is the decaywidth difference between the light and heavy B0

s mass eigenstates, Γs ≡ (ΓL + ΓH)/2 isthe average decay width, and ∆ms is the B0

s -B0s oscillation frequency. The differential

decay rate for a B0s meson produced at t = 0 is obtained by changing the sign of the ci

and di coefficients.

5

Table 1: Coefficients of the time-dependent terms and angular functions used in Eq. 2. Ampli-tudes are defined at t = 0.

i Ni ai bi ci di fi

1 |A0|2 1 D C −S 4 cos2 θ1 cos2 θ2

2 |A‖|2 1 D C −S sin2 θ1 sin2 θ2(1+ cos 2Φ)

3 |A⊥|2 1 −D C S sin2 θ1 sin2 θ2(1− cos 2Φ)

4 |A‖||A⊥| C sin δ1 S cos δ1 sin δ1 D cos δ1 −2 sin2 θ1 sin2 θ2 sin 2Φ

5 |A‖||A0| cos(δ2,1) D cos(δ2,1) C cos δ2,1 −S cos(δ2,1)√

2 sin 2θ1 sin 2θ2 cos Φ

6 |A0||A⊥| C sin δ2 S cos δ2 sin δ2 D cos δ2 −√

2 sin 2θ1 sin 2θ2 sin Φ

7 |ASS |2 1 D C −S 49

8 |AS |2 1 −D C S 43 (cos θ1 + cos θ2)2

9 |AS ||ASS | C cos(δS − δSS) S sin(δS−δSS) cos(δSS−δS) D sin(δSS−δS) 83√3(cos θ1 + cos θ2)

10 |A0||ASS | cos δSS D cos δSS C cos δSS −S cos δSS83 cos θ1 cos θ2

11 |A‖||ASS | cos(δ2,1−δSS) D cos(δ2,1−δSS) C cos(δ2,1−δSS) −S cos(δ2,1−δSS) 4√2

3 sin θ1 sin θ2 cos Φ

12 |A⊥||ASS | C sin(δ2 − δSS) S cos(δ2 − δSS) sin(δ2−δSS) D cos(δ2−δSS) − 4√2

3 sin θ1 sin θ2 sin Φ

13 |A0||AS | C cos δS −S sin δS cos δS −D sin δS

8√3

cos θ1 cos θ2

×(cos θ1 + cos θ2)

14 |A‖||AS | C cos(δ2,1 − δS) S sin(δ2,1 − δS) cos(δ2,1−δS) D sin(δ2,1 − δS)4√2√3

sin θ1 sin θ2

×(cos θ1 + cos θ2) cos Φ

15 |A⊥||AS | sin(δ2 − δS) −D sin(δ2 − δS) C sin(δ2 − δS) S sin(δ2 − δS)− 4

√2√3

sin θ1 sin θ2

×(cos θ1 + cos θ2) sin Φ

The three CP -violating terms introduced in Table 1 are defined as

C ≡ 1− |λ|2

1 + |λ|2, (4)

S ≡ −2|λ| sinφssss1 + |λ|2

, (5)

D ≡ −2|λ| cosφssss1 + |λ|2

, (6)

where φssss measures CP violation in the interference between the direct decay amplitudeand those from mixing, λ ≡ (q/p)(A/A), q and p are the complex parameters relatingthe B0

s flavour and mass eigenstates, and A (A) is the B0s meson decay amplitude (CP

conjugate decay amplitude). Under the assumption that |q/p| = 1, |λ| measures CPviolation in decay. The CP violation parameters are assumed to be helicity independent.The association of φssss and |λ| with S-wave and double S-wave terms implies that these

6

consist solely of contributions with the same flavour content as the φ meson, i.e.ss. Theimpact of the non-ss component of the φ wavefunction is negligible in this analysis.

In Table 1, δS and δSS are the strong phases of the P → VS and P → SS processes,respectively. The P -wave strong phases are defined to be δ1 ≡ δ⊥ − δ‖ and δ2 ≡ δ⊥ − δ0,with the notation δ2,1 ≡ δ2 − δ1 = δ‖ − δ0.

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 final-state interactions that arise from long distance strong interaction effects,which conserve C, P and CP . Four-body final states give rise to three independentmomentum vectors in the rest frame of the decaying B0

s meson. For a detailed review ofthe phenomenology the reader is referred to Ref. [9].

The two independent terms in the time-dependent decay rate that contribute to aT -odd asymmetry are the K4(t) and K6(t) terms, defined in Eq. 3. The triple productsthat allow access to these terms are

sin Φ = (nV1 × nV2) · pV1 , (7)

sin 2Φ = 2(nV1 · nV2)(nV1 × nV2) · pV1 , (8)

where nVi (i = 1, 2) is a unit vector perpendicular to the vector meson, Vi, decay planeand pV1 is a unit vector in the direction of V1 in the B0

s rest frame, defined in Fig. 2. Thisthen provides a method of probing CP violation without the need to measure the decaytime or the initial flavour of the B0

s meson. It should be noted that while the observationof nonzero triple-product asymmetries implies CP violation or final-state interactions (inthe case of B0

s meson decays), measurements of triple-product asymmetries consistentwith zero do not rule out the presence of CP -violating effects, as differences between thestrong phases can cause suppression [9].

In the B0s → φφ decay, two triple products are defined as U ≡ sin Φ cos Φ and

V ≡ sin(±Φ) where the positive sign is taken if cos θ1 cos θ2 ≥ 0 and a negative signotherwise [9]. The T -odd asymmetry corresponding to the U observable, AU , is definedas the normalised difference between the number of decays with positive and negativevalues of sin Φ cos Φ,

AU ≡Γ(U > 0)− Γ(U < 0)

Γ(U > 0) + Γ(U < 0)∝∫ ∞

0

=(A⊥(t)A∗‖(t) + A⊥(t)A∗‖(t)

)dt. (9)

Similarly AV is defined as

AV ≡Γ(V > 0)− Γ(V < 0)

Γ(V > 0) + Γ(V < 0)∝∫ ∞

0

=(A⊥(t)A∗0(t) + A⊥(t)A∗0(t)

)dt. (10)

Determination of the triple-product asymmetries is then reduced to a simple countingexperiment.

5 Decay-time resolution

The sensitivity to φssss is affected by the accuracy of the measured decay time. In order toresolve the fast B0

s -B0s oscillation period, it is necessary to have a decay-time resolution

7

that is much smaller than this period. To account for decay-time resolution, all time-dependent terms are convolved with a Gaussian function, with width σti that is estimatedfor each candidate, i, based upon the uncertainty obtained from the vertex and kinematicfit. In order to apply a candidate-dependent resolution model during fitting, the estimatedper-event decay time uncertainty must be calibrated.

The calibration of the per-event decay-time uncertainty is performed using samplesof prompt data, consisting of good quality tracks, which originate from the primaryinteraction vertex. Due to the small opening angle of the kaons in the decay of a φmeson, it is sufficient to use a single prompt track and assign the mass of a φ meson.When combining this with another pair of tracks, the invariant mass of the three bodycombination is required to be within 250 MeV/c2 of the nominal B0

s mass. Such anapproach has been validated using simulated prompt events and signal B0

s decays.A first-order polynomial is then fitted to the distribution of σti versus σttrue, with

parameters denoted by q0 and q1. The calibrated per-event decay-time uncertainty usedin the time-dependent fit is then calculated as σcal

i = q0 + q1σti . Gaussian constraints are

used to account for the uncertainties on the calibration parameters in the time-dependentfit. The effective single Gaussian resolution is found to be between 41–44 fs, depending onthe data-taking year.

6 Acceptances

The four observables used to analyse B0s→ φφ decays consist of the decay time and the

three helicity angles. A good understanding of efficiencies in these variables is required.It is assumed that the decay-time and angular acceptances can be factorised.

6.1 Angular acceptance

The geometry of the LHCb detector and the momentum requirements imposed on thefinal-state particles introduce efficiencies that depend on the helicity angles. Simulatedsignal events, selected with the same criteria as those applied to data candidates are usedto determine these efficiency corrections. The angular acceptances as a function of thethree helicity angles are shown in Fig. 3.

In order to account for the efficiency as a function of the decay angles, a parameterisa-tion is used, defined as

ε(Ω) =∑i,j,k

cijkPi(cos θ1)Yjk(cos θ2,Φ), (11)

where cijk are coefficients, Pi(cos θ1) are Legendre polynomials, and Yjk(cos θ2,Φ) arespherical harmonics. The procedure followed to calculate the coefficients is outlined indetail in Ref. [35]. It uses the orthogonal nature of Legendre polynomials. This results inthe evaluation of the equation

cijk ≡ (j + 1/2)

∫dΩPi(cos θ1)Yjk(cos θ2,Φ)ε(Ω). (12)

8

cos θ1-1 -0.5 0 0.5 1

A.U.

00.20.40.60.81

1.21.41.61.82

LHCb simulation

cos θ2-1 -0.5 0 0.5 1

A.U.

00.20.40.60.81

1.21.41.61.82

LHCb simulation

[rad]Φ-1 -0.5 0 0.5 1

A.U.

00.20.40.60.81

1.21.41.61.82

LHCb simulation

Figure 3: Angular acceptance found from simulated B0s→ φφ decays (top-left) integrated over

cos θ2 and Φ as a function of cos θ1, (top-right) integrated over cos θ1 and Φ as a function ofcos θ2, and (bottom) integrated over cos θ1 and cos θ2 as a function of Φ.

This integral is calculated by means of Monte Carlo integration, which reduces the integralto a sum over the number of accepted simulated events (Nobs)

cijk ∝ (j + 1/2)1

Nobs

Nobs∑e=1

Pi(cos θ1,e)Yjk(cos θ2,e,Φe)

P gen(Ωe|te), (13)

where P gen is the probability density function (PDF) without acceptance with parametersset to the generation value of the Monte Carlo simulation. In order to have this processfully optimised, the angular functions themselves are written in the same basis as theefficiency parameterisation, i.e.

fa(cos θ1, cos θ2,Φ) =∑ijkl

κijkl,aPij(cos θ1)Ykl(cos θ2,Φ), (14)

where Pij(cos θ1) are the associated Legendre polynomials. The acceptance can then becombined with the angular functions to again recover functions in the same basis, allowingfor fast analytic computation and normalisation of the PDF. The parameterisation foreach angular function is given in Table 2.

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

The fit to determine the triple-product asymmetries assumes that the U and Vobservables are symmetric in the acceptance corrections. Simulation is used to assign asystematic uncertainty related to this assumption.

9

Table 2: Angular coefficients, written in the same basis as the efficiency parameterisation.

i fi Pi,jYk,l basis fi1 8/9P00Y00 + 16/9/

√5P00Y20 + 16/9P20Y 00 + 32/(9

√5)P20Y20 4 cos2 θ1 cos2 θ2

2 8/9P00Y00 − 8/9P00Y20 − 8/9√

5P20Y00 + 8/(9√

5)P20Y20 + (2/9)√

12/5P22Y22 sin2 θ1 sin2 θ2(1+ cos 2Φ)

3 8/9P00Y00 − 8/9P00Y20 − 8/9√

5P20Y00 + 8/(9√

5)P20Y20 − (2/9)√

12/5P22Y22 sin2 θ1 sin2 θ2(1− cos 2Φ)

4 −8/9√

3/5P2,2Y2,−2 −2 sin2 θ1 sin2 θ2 sin 2Φ

5 8/9√

6/5P2,1Y2,1√

2 sin 2θ1 sin 2θ2 cos Φ

6 −8/9√

6/5P2,1Y2,1 −√

2 sin 2θ1 sin 2θ2 sin Φ

7 (8/9)P00Y0049

8 16/9P00Y00 + 16/9/√

5P00Y20 + 16/9P20Y 00 + 16√

3/9P10Y1043

(cos θ1 + cos θ2)2

9 16√

3/2P10Y00 + 16/9P00Y108

3√3

(cos θ1 + cos θ2)

10 16/(3√

3)P10Y1083

cos θ1 cos θ2

11 (8/9)√

6P11Y114√2

3sin θ1 sin θ2 cos Φ

12 (8/9)√

6P11Y1−1 − 4√2

3sin θ1 sin θ2 sin Φ

13 16√

3/9P10Y00 + 16/9P00Y10 + 32/9P20Y 10 + 32/(9√

5)P20Y208√3

cos θ1 cos θ2

×(cos θ1 + cos θ2)

14 (8/9)√

2/3P21Y11 + (24/9)√

2/15P11Y214√2√3

sin θ1 sin θ2

×(cos θ1 + cos θ2) cos Φ

15 −(8/9)√

2/3P21Y1−1 − (24/9)√

2/15P11Y2−1− 4√2

3sin θ1 sin θ2

×(cos θ1 + cos θ2) sin Φ

6.2 Decay-time acceptance

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

The efficiency as a function of the decay time is taken from the B0s → D−s (→

K+K−π−)π+ decay, in the case of data taken between 2011 and 2012, and from theB0 → J/ψ (→ µ+µ−)K∗0(→ K+π−) decay in the case of data taken between 2015 and2016. The reason for this is related to changes to the software trigger selection between thetwo data-taking periods. The decay-time acceptances of the control modes are weightedby a multivariate algorithm based on simulated kinematic and topological information, inorder to more closely match the signal B0

s→ φφ decay.Cubic splines are used to construct the acceptance as a function of decay time in the

PDF. The PDF can then be computed analytically with the inclusion of the decay-timeacceptance following Ref. [36]. The use of cubic splines allows the acceptance to beconstructed as

ε(t) =6∑i=0

cibi(t), (15)

where ci are coefficients and bi(t) are third-order polynomial functions defined over distincttime ranges. The cubic spline acceptance description is chosen to have 9 subintervals, thususing 10 knots. These are chosen to be at: 0.31, 0.4, 0.7, 0.9, 1.5, 2.0, 3.0, 5.0, 7.0, 9.0ps. With these knots, the coefficients are calculated as described in Ref. [36]. Exampledecay-time acceptances are shown for the case of the B0

s → D−s π+ and B0 → J/ψK∗0

decays in Fig. 4.In the fit to determine the triple-product asymmetries, the decay-time acceptance is

treated only as a systematic uncertainty.

10

Time [ps]1 2 3 4 5 6 7 8 9 10

Acc

epta

nce

(A. U

.)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

LHCb Preliminary+π-s D→sB

Time [ps]1 2 3 4 5 6 7 8 9 10

Acc

epta

nce

(A. U

.)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

LHCb Preliminary*

Kψ J/→0B

Figure 4: Decay-time acceptances calculated from (left) B0s → D−s π

+ and (right) B0 → J/ψK∗0

decays. Superimposed are calculated analytical splines.

7 Flavour tagging

To ensure the sensitivity to φssss , the initial flavour of the B0s meson must be determined.

The terms in the differential decay rate with the largest sensitivity to φssss require theidentification (tagging) of the flavour at production. At LHCb, tagging is achievedthrough the use of various algorithms described in Refs. [6,37]. With these algorithms, theflavour-tagging power, defined as εtagD2 can be evaluated. Here, εtag is the flavour-taggingefficiency and D ≡ (1− 2ω) is the dilution.

This analysis uses opposite side (OS) and same side kaon (SSK) flavour taggers. TheOS flavour-tagging algorithm [37] makes use of the b(b) hadron produced in associationwith the signal b(b)-hadron. In this analysis, the predicted probability of an incorrectflavour assignment, ω, is determined for each candidate by a neural network that iscalibrated using B+ → J/ψK+, B+ → D0π+, B0 → J/ψK∗0, B0 → D∗−µ+νµ, andB0s → D−s π

+ data as control modes. Details of the calibration procedure can be found inRef. [6].

When a signal B0s meson is formed, there is an associated s quark formed in the first

branches of the fragmentation that about 50 % of the time forms a charged kaon, which islikely to originate close to the B0

s meson production point. The kaon charge therefore allowsfor the identification of the flavour of the signal B0

s meson. This principle is exploitedby the SSK flavour-tagging algorithm [37]. The SSK tagging algorithm is calibratedwith the B0

s → D+s π− decay mode. A neural network is used to select fragmentation

particles, improving the flavour-tagging power quoted in the previous time-dependentmeasurement [8]. Table 3 shows the tagging power for the candidates tagged by only oneof the algorithms and those tagged by both. Uncertainties due to the calibration of theflavour tagging algorithms are applied as Gaussian constraints in the time-dependent fit.The dependence of the initial flavour of the B0

s meson established from flavour tagging isaccounted for during fitting.

11

Table 3: Tagging performance of the opposite side (OS) and same side kaon (SSK) flavourtaggers.

Category Fraction(%) ε(%) D2 εD2(%)

OS-only 16.293 12.507 0.099 1.24± 0.10

SSK-only 53.393 40.986 0.043 1.74± 0.36

OS&SSK 30.314 23.270 0.118 2.76± 0.20

Total 100 76.763 0.075 5.74 ±0.43

8 Decay-time dependent measurement

8.1 Likelihood

The parameters of interest are the CP violation parameters (φssss and |λ|), the polarisationamplitudes (|A0|2, |A⊥|2, |AS|2, and |ASS|2), and the strong phases (δ1, δ2, δS, and δSS), asdefined in Sec. 4.1. The P -wave amplitudes are defined such that |A0|2 + |A⊥|2 + |A‖|2 = 1,hence only two of the three amplitudes are free parameters.

Parameter estimation is achieved by minimising of the negative log-likelihood. Thelikelihood, L, is weighted using the sPlot method [38,39]. The log-likelihood then takesthe form

− lnL = −α∑

e∈candidates

We ln(SeTD), (16)

where We are the signal sPlot weights calculated using the four-kaon invariant mass asthe discriminating variable, α =

∑eWe/

∑eW

2e is used to account for the weights in

the determination of the statistical uncertainties, and SeTD is the signal model of Eq. 2,accounting also for the effects of decay-time and angular acceptance, in addition to theprobability of an incorrect flavour tag. Explicitly, this can be written as

SeTD =

∑i sei (te)fi(Ωe)ε(te)∑

k ζk∫sk(t)fk(Ω)ε(t)dt dΩ

, (17)

where ζk are the normalisation integrals used to describe the angular acceptance describedin Sec. 6.1 and

sei (t) = Nie−Γste

[ciqe(1− 2ωe) cos(∆mste) + diqe(1− 2ωe) sin(∆mste) + ai cosh

(1

2∆Γste

)+bi sinh

(1

2∆Γste

)]⊗R(σcal

e , te), (18)

where ωe is the calibrated probability of an incorrect flavour assignment, and R denotesthe Gaussian time resolution function. In Eq. 18, qe = 1(−1) for a B0

s (B0s) meson at

t = 0 or qe = 0 if no flavour-tagging information exists. The data samples correspondingto the different years of data taking are assigned independent signal weights, decay-time

12

Table 4: Results of the time-dependent fit. Uncertainties shown do not include systematiccontributions.

Parameter Resultφssss ( rad) −0.07± 0.13|λ| 1.02± 0.05|A⊥|2 0.287± 0.008

|A0|2 0.382± 0.008δ⊥ ( rad) 2.81± 0.21δ‖ ( rad) 2.52± 0.05

and angular acceptances, and separate Gaussian constraints are applied to the decay-time resolution parameters, as defined in Sec. 5. The value of the B0

s -B0s oscillation

frequency is constrained to the LHCb measured value of ∆ms = 17.768± 0.023 (stat)±0.006 (syst) ps−1 [40], with the assumption that the uncertainties are uncorrelated. Thevalues of the decay width and decay-width difference are constrained to the current bestknown values [41].

Correction factors must be applied to each of the S-wave and double S-wave inter-ference terms in the differential cross-section. These factors modulate the sizes of thecontributions of the interference terms in the angular PDF due to the different line-shapesof kaon pairs originating from spin-1 and spin-0 configurations. This takes the form ofa multiplicative factor for each time a spin-0 pair of kaons interferes with a spin-1 pair.Their parameterisations are denoted by g(mK+K−) and h(mK+K−), respectively. Thespin-1 configuration is described by a Breit-Wigner function and the spin-0 configurationis assumed to be approximately uniform. The correction factors, denoted by CSP , aredefined in Ref. [6]

CSP eiθSP =

∫ mh

ml

g∗(mK+K−)h(mK+K−)dmK+K− , (19)

where mh and ml are the upper and lower edges of a given mK+K− bin and θSP is absorbedin the measurements of δS − δ⊥. Alternative assumptions on the P -wave and S-wavelineshapes are found to have a negligible effect on the parameter estimation. The correctionfactor, CSP , is calculated to be 0.36.

8.2 Results

The results of the fit of the parameters of interest are given in Table 4. A negligiblefraction of S-wave is observed. The correlation matrix is shown in Table 5. Correlationswith such time-dependent measurements depend on the central values of parameters. Nolarge correlation is expected to be observed between the CP -violating parameters when thecentral values are consistent with CP conservation. As expected, the largest correlationsare found to be between the different decay amplitudes. The observed correlations havebeen verified with simulated datasets. Cross-checks are performed on simulated datasetsgenerated with the same yield as observed in data, and with the same physics parameters,to establish that the generated values are recovered with negligible biases.

13

Table 5: Correlation matrix associated with the result of the time-dependent fit and relate tothe statistical uncertainties.

δ‖ |A⊥|2 δ⊥ |A0|2 |λ| φssssδ‖ 1.00 0.14 0.13 −0.03 0.02 0.01|A⊥|2 1.00 0.01 −0.45 0.00 −0.03δ⊥ 1.00 0.00 −0.26 −0.15|A0|2 1.00 −0.01 0.01|λ| 1.00 −0.05φssss 1.00

Figure 5: One-dimensional projections of the B0s → φφ fit for (top-left) decay time with binned

acceptance, (top-right) helicity angle Φ and (bottom-left and bottom-right) cosine of the helicityangles θ1 and θ2. The background-subtracted data are marked as black points, while the bluesolid lines represent the projections of the best fit. The CP -even P -wave, the CP -odd P -waveand S-wave combined with double S-wave components are shown by the red long dashed, greenshort dashed and purple dot-dashed lines, respectively. Fitted components are plotted taking into account efficiencies in the time and angular observables.

Figure 5 shows the distributions of the B0s decay time and the three helicity angles.

Superimposed are the projections of the fit result. The projections are weighted to yieldthe signal distribution and include corrections for acceptance effects.

14

Table 6: Summary of systematic uncertainties for physics parameters in the time-dependentmeasurement, where AA denotes angular acceptance, TA denotes time acceptance, and TR timeresolution.

Parameter Mass model AA TA TR Fit bias Total|A0|2 0.0035 0.0098 0.0008 0.0001 0.0018 0.0106|A⊥|2 0.0021 0.0046 0.0007 0.0002 0.0012 0.0052δ‖ (rad) 0.0128 0.0653 0.0049 0.0031 0.0179 0.0692δ⊥ (rad) 0.0640 0.0100 0.0085 0.0064 0.0701 0.0960φssss (rad) 0.0119 0.0072 0.0077 0.0035 0.0126 0.0206|λ| 0.0063 0.0217 0.0023 0.0053 0.0097 0.0253

8.3 Systematic uncertainties

Various sources of systematic uncertainty are considered in addition to those applied asGaussian constraints in the nominal fit. These arise from the angular and decay-timeacceptances, the mass model used to construct the weights, the determination of the timeresolution calibration, and the fit bias. A summary of the systematic uncertainties isgiven in Table 6.

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

Two sources of systematic uncertainty are considered concerning the decay-timeacceptance. These are the statistical uncertainty from the spline coefficients, and also theresidual disagreement between the weighted control mode and the signal decay, acceptances(see Sec. 6.2). The former is evaluated through fitting the signal dataset with 20 differentspline functions, whose coefficients are varied according to the corresponding uncertainties.The RMS of the fitted parameters is then assigned as the uncertainty. The residualdisagreement between the control mode and the signal mode is accounted for with asimulation based correction. Simplified simulations are used with the corrected acceptanceand then fitted with the nominal acceptance. This process is repeated and the resultingbias 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 variousalternative signal models, consisting of a double Crystal Ball model, the sum of a double-sided Crystal Ball and a Gaussian model. In addition, a Chebyshev polynomial is used todescribe the background. The signal weights are recalculated and the largest deviationfrom the nominal fit results is used as the corresponding uncertainty. Fit bias arises inlikelihood fits when the number of candidates used to determine the free parameters isnot sufficient to achieve the Gaussian limit. This uncertainty is evaluated by generatingand fitting simulated datasets and taking the resulting bias as the uncertainty.

Uncertainties due to flavour tagging are included in the statistical uncertainty throughGaussian constraints on the calibration parameters, and amount to 10 % of the statisticaluncertainty on the CP -violating phase.

15

9 Triple-product asymmetries

9.1 Likelihood

In order to determine the triple-product asymmetries, a separate likelihood fit is performed.The fit is based around the simultaneous fitting of separate datasets to the four-kaoninvariant mass, which are split according to the sign of U and V observables. Simultaneousmass fits are performed for the U and V observables separately. The set of free parametersin the fits to determine the U and V observables consist of the asymmetries (AU(V )) alongwith their associated total yields. The mass model is the same as that described in Sec. 3.The total PDF, STP, is then of the form

STP =∑

i∈+,−

(fSi G

S(mK+K−K+K−) +∑j

f ji Pj(mK+K−K+K−)

), (20)

where j indicates the sum over the background components with corresponding PDFs,P j, and GS is the signal PDF, as described in Sec. 3. The parameters fki found in Eq. 20are related to the asymmetry, AkU(V ), through

fkU(V ),+ =1

2(AkU(V ) + 1), (21)

fkU(V ),− =1

2(1− AkU(V )), (22)

where k denotes a four-kaon mass fit component, as described in Sec. 3. Peaking back-grounds 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 aremeasured separately for the case of 2015 and 2016 data. The results are combined bymaking likelihood scans of the asymmetry parameters and summing the two years. Thisgives

AU = 0.003 ± 0.016 ,AV = 0.010 ± 0.016 .

9.3 Systematic uncertainties

As for the case of the time-dependent fit, significant contributions to the systematicuncertainty arise from the decay-time and angular acceptances. Minor uncertainties alsoresult from the mass model and peaking background knowledge.

The effect of the decay-time acceptance is determined through the generation ofsimulated samples including the decay-time acceptance and fitted with the methoddescribed in Sec. 9.1. The resulting bias is used to assign a systematic uncertainty. Theeffect of the angular acceptance is evaluated by generating simulated datasets with andwithout the inclusion of the angular acceptance. The resulting bias found on the fit resultsof the triple-product asymmetries is then used as a systematic uncertainty.

16

Uncertainties related to the mass model are evaluated using a similar approach tothat described in Sec. 8.3. Additional uncertainties arise from the assumption that thepeaking background is symmetric in U and V . The deviation observed without thisassumption is then added as a systematic uncertainty. Similarly, the assumption thatthe combinatorial background has no asymmetry yields an identical uncertainty. Thesystematic uncertainties are summarised in Table 7

Table 7: Summary of systematic uncertainties on AU and AV .

Quantity Assigned UncertaintyTime acceptance 0.003

Angular acceptance 0.003Mass model 0.001

Combinatorial background 0.001Peaking background 0.001

Quadrature sum 0.005

9.4 Combination of Run 1 and Run 2 results

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

AU = 0.003 ± 0.016 (stat) ± 0.005, (syst)AV = 0.010 ± 0.016 (stat) ± 0.005, (syst)

whilst the Run 1 (2011-2012) values from Ref. [8] are

AU = −0.003 ± 0.017 (stat) ± 0.006, (syst)AV = −0.017 ± 0.017 (stat) ± 0.006. (syst)

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

AU = 0.000 ± 0.012 (stat) ± 0.004 (syst)AV = −0.003 ± 0.012 (stat) ± 0.004 (syst)

The Run 1 and Run 2 results are compatible with each other, and the asymmetriesare in agreement with zero. No evidence for CP violation is found.

10 Search for the B0 → φφ decay

The figure of merit used for the B0 → φφ search, explained in Sec. 3, results in a morestringent MLP requirement. The relative construction and selection efficiency of theB0s→ φφ and B0 → φφ decays is determined to be 0.986 using simulated samples. For

this reason the B0s→ φφ decay is the most relevant normalisation decay mode. The signal

17

LHCb Preliminary-4-2024

M(K+K-K+K-) [MeV/c2]5200 5300 5400 5500 5600

Candidates/

(9MeV

/c2 )

10-2

10-1

1

10

102

103

Pull

Figure 6: Fit to the four-kaon invariant mass. The total PDF as described in the text is shown asa blue solid line, B0

s→ φφ as a red dashed line, B0 → φφ as a green dotted line, the Λ0b → φpK

as a magenta long-dashed line and the combinatorial background as a blue short-dashed line.

PDF for the mass of the B0 meson is assumed to be the same as that of the B0s decay,

with the modification of the resolution according to a scaling factor, which is defined as

α =MB0 − 4MK

MB0s− 4MK

= 0.974, (23)

where MK is the nominal K+ mass.Figure 6 shows a fit to the full Run 1 and Run 2 selected data. The Λ0

b → φpKcontribution is fixed to 109 candidates, as established in Sec. 3. The result of the fitreturns a yield of 4.6± 8.8 B0 decays.

The CLs method [42] is used to set a limit on the B0 → φφ branching fraction. Thetest statistic used in the CLs method is defined as

t = −2logL(fs+b(x))

L(fb(x)), (24)

where L(fs+b(x)) is the likelihood of the signal plus background PDF, fs+b, and L(fb(x)) isthe likelihood calculated from the background only PDF, fb. The set of input parametersto the PDF is denoted by x. A total of 10,000 pseudoexperiments are used to calculateeach point of the scan. Figure 7 shows the results of the CLs scan. At 90 % CL, NB0 < 18.5decays, and at 95 % CL, NB0 < 22.2 decays. These limits are converted to a branchingfraction using

B(B0 → φφ) = NB0 × B(B0s → φφ)× fs/fdNB0

s→φφ, (25)

where NB0 is the limit on the B0 → φφ yield, and NB0s→φφ is the B0

s yield from the fitdisplayed in Fig. 6. The fragmentation ratio, fs/fd = 0.259, and the B(B0

s → φφ) =

18

B0 yield5 10 15 20 25 30

CL s

10-3

10-2

10-1

1LHCb Preliminary

Figure 7: Results of the CLs scan as a function of NB0 . The solid black line shows the observedCLs distribution, while the dotted black line indicates the expected distribution. The green(yellow) band marks the 1σ (2σ) confidence region on the expected CLs. The 90 (95) % CL limitis indicated as a solid (dashed) red line.

(1.84 ± 0.05 (stat) ± 0.07 (syst) ± 0.11(fs/fd) ± 0.12 (norm)) × 10−5 branching fractionare external inputs taken from Refs. [43] and [21], respectively. To set the limit, theuncertainties on the B0

s→ φφ branching fraction are propagated, where the uncertainty onthe B0

s→ φφ branching fraction arising from fs/fd is already included in the uncertaintyon the normalisation mode. The maximum value including the systematic contribution isfound to be 1.99× 10−5 and is used in Eq. 25. This therefore translates to a limit of

B(B0 → φφ) < 2.4× 10−8 (90 % CL),

B(B0 → φφ) < 2.9× 10−8 (95 % CL),

which is more stringent than the previous best limit of 2.8× 10−8 at 90 % CL [21].

11 Summary and conclusions

Measurements of CP violation in the B0s→ φφ decay are presented, based on a sample of

proton-proton collision data corresponding to an integrated luminosity of 5.0 fb−1 collectedwith the LHCb detector. The CP -violating phase, φssss , and CP violation parameter, |λ|,are determined to be

φssss = −0.07 ± 0.13 (stat) ± 0.03 (syst) rad,|λ| = 1.02 ± 0.05 (stat) ± 0.03 (syst).

The results are in agreement with the theoretical SM predictions [1–3]. Uncertaintiesare slightly larger than the naive expectation from the scaling of the previous result [8].

19

The uncertainties have been validated with simulated events and the differences withrespect to the expectation from the previous results are exaggerated through roundingeffects. When compared with the CP -violating phase measured in B0

s → J/ψK+K− andB0s → J/ψπ+π− decays [6], these results show that no significant CP violation is present

either in B0s -B

0s mixing or in the b→ sss decay amplitude, though the increased precision

of the measurement presented in Ref. [6] leads to more stringent constraints of CP violationin B0

s -B0s mixing.

The polarisation amplitudes and strong phases are measured to be

|A0|2 = 0.382 ± 0.008 (stat) ± 0.011 (syst),|A⊥|2 = 0.287 ± 0.008 (stat) ± 0.005 (syst),δ⊥ = 2.81 ± 0.21 (stat) ± 0.10 (syst) rad,δ‖ = 2.52 ± 0.05 (stat) ± 0.07 (syst) rad.

Values of the polarisation amplitudes are found to agree well with the previous mea-surements [8, 11, 44, 45]. Measurements in other B → V V penguin transitions at the Bfactories generally give higher values of fL ≡ |A0|2 [12–17]. The value of fL found in theB0s→ φφ channel is almost equal to that in the B0

s → K∗0K∗0 decay [46]. As reported inRef. [44], the results are in agreement with QCD factorisation predictions [2, 3].

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

AU = 0.000 ± 0.012 (stat) ± 0.004 (syst),AV = −0.003 ± 0.012 (stat) ± 0.004 (syst),

in agreement with previous measurements [8, 11,44].The measured values of the CP -violating phase and triple-product asymmetries are

consistent with the hypothesis of CP conservation in b→ sss transitions.In addition, the most stringent limit on the branching fraction of the B0 → φφ decay

is presented. It is found to be

B(B0 → φφ) < 2.4× 10−8 (90 % CL).

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