First measurement of the CP-violating phase in $B_s^0 \to \phi \phi$ decays

A first flavour-tagged measurement of the time-dependent CP-violating asymmetry in $B_s^0 \to \phi\phi$ decays is presented. In this decay channel, the CP-violating weak phase arises due to CP violation in the interference between $B_s^0$-$\bar{B}_s^0$ mixing and the $b \to s \bar{s} s $ gluonic penguin decay amplitude. Using a sample of $pp$ collision data corresponding to an integrated luminosity of $1.0\; fb^{-1}$ and collected at a centre-of-mass energy of $7 \rm TeV$ with the LHCb detector, $880\ \B_s^0 \to \phi\phi$ signal decays are obtained. The CP-violating phase is measured to be in the interval [-2.46, -0.76] \rm rad$ at 68% confidence level. The p-value of the Standard Model prediction is 16%.

a P.N.Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia b Università di Bari, Bari, Italy c Università di Bologna, Bologna, Italy d Università di Cagliari, Cagliari, Italy e Università di Ferrara, Ferrara, Italy f Università di Firenze, Firenze, Italy g Università di Urbino, Urbino, Italy h Università di Modena e Reggio Emilia, Modena, Italy i Università di Genova, Genova, Italy j Università di Milano Bicocca, Milano, Italy k Università di Roma Tor Vergata, Roma, Italy l Università di Roma La Sapienza, Roma, Italy m Università della Basilicata, Potenza, Italy n LIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain o IFIC, Universitat de Valencia-CSIC, Valencia, Spain p Hanoi University of Science, Hanoi, Viet Nam q Università di Padova, Padova, Italy r Università di Pisa, Pisa, Italy s Scuola Normale Superiore, Pisa, Italy vi The B 0 s → φφ decay is forbidden at tree level in the Standard Model (SM) and proceeds via a gluonic b → sss penguin process.Hence, this channel provides an excellent probe of new heavy particles entering the penguin quantum loops [1][2][3].Generally, CP violation in the SM is governed by a single phase in the Cabibbo-Kobayashi-Maskawa quark mixing matrix [4].The interference between the B 0 s -B 0 s oscillation and decay amplitudes leads to a CP asymmetry in the decay time distributions of B 0 s and B 0 s mesons, which is characterised by a CP -violating weak phase.The SM predicts this phase to be small.Due to different decay amplitudes the actual value is dependent on the B 0 s decay channel.For B 0 s → J/ψφ, which proceeds via a b → ccs transition, the SM prediction of the weak phase is given by −2 arg (−V ts V * tb /V cs V * cb ) = −0.036± 0.002 rad [5].The LHCb collaboration recently measured the weak phase in this decay to be 0.068 ± 0.091(stat) ± 0.011(syst) rad [6], which is consistent with the SM and places stringent constraints on CP violation in B 0 s -B 0 s oscillations [7].In the SM, the phase in the B 0 s → φφ decay, φ s , is expected to be close to zero due to a cancellation of the phases arising from B 0 s -B 0 s oscillations and decay [8].Calculations using QCD factorization provide an upper limit of 0.02 rad for |φ s | [1][2][3].
In this Letter, we present the first measurement of the CP -violating phase in B 0 s → φφ decays.Charge conjugate states are implied.The result is based on pp collision data corresponding to an integrated luminosity of 1.0 fb −1 and collected by the LHCb experiment in 2011 at a centre-of-mass energy of 7 TeV.This data sample was previously used for a time-integrated measurement of the polarisation amplitudes and triple product asymmetries in the same decay mode [9].The analysis reported here improves the selection efficiency, measures the B 0 s decay time and identifies the flavour of the B 0 s meson at production.This allows a study of CP violation in the interference between mixing and decay to be performed.It is necessary to disentangle the CP -even longitudinal (A 0 ), CPeven transverse (A ), and CP -odd transverse (A ⊥ ) polarisations of the φφ final state by measuring the distributions of the helicity angles [9].
The LHCb detector is a forward spectrometer at the Large Hadron Collider covering the pseudorapidity range 2 < η < 5 and is described in detail in Ref. [10].Events are selected by a hardware trigger, which selects hadron or muon candidates with high transverse energy or momentum (p T ), followed by a two stage software trigger [11].In the software trigger, B 0 s → φφ candidates are selected either by identifying events containing a pair of oppositely charged kaons with an invariant mass close to that of the φ meson or by using a topological b-hadron trigger.In the simulation, pp collisions are generated using Pythia 6.4 [12], with a specific LHCb configuration [13].Decays of hadronic particles are described by EvtGen [14] and the detector response is implemented using the Geant4 toolkit [15] as described in Ref. [16].The B 0 s → φφ decays are reconstructed by combining two φ meson candidates that decay into the K + K − final state.Kaon candidates are required to have p T > 0.5 GeV/c, and an impact parameter (IP) χ 2 larger than 16 with respect to the primary vertex (PV), where the IP χ 2 is defined as the difference between the χ 2 of the PV reconstructed with and without the considered track.Candidates must also be identified as kaons using the ring-imaging Cherenkov detectors [17], by requiring that the difference in the global likelihood between the kaon and pion mass hypotheses (∆ ln L K π ≡ ln L K − ln L π ) be larger than −5.Both φ meson candidates must have a reconstructed mass, m KK , of the kaon pair within 20 MeV/c 2 of the known mass of the φ meson, a transverse momentum (p φ T ) larger than 0.9 GeV/c and a product, p φ1 T p φ2 T > 2 GeV 2 /c 2 .The χ 2 per degree of freedom (ndf) of the vertex fit for both φ meson candidates and the B 0 s candidate is required to be smaller than 25.Using the above criteria, 17 575 candidates are selected in the invariant four-kaon mass range 5100 < m KKKK < 5600 MeV/c 2 .
A boosted decision tree (BDT) [18] is used to separate signal from background.The six observables used as input to the BDT are: p T , η and χ 2 /ndf of the vertex fit for the B 0 s candidate and the cosine of the angle between the B 0 s momentum and the direction of flight from the closest primary vertex to the decay vertex, in addition to the smallest p T and the largest track χ 2 /ndf of the kaon tracks.The BDT is trained using simulated B 0 s → φφ signal events and background from the data where at least one of the φ candidates has invariant mass in the range 20 The sPlot technique [19,20] is used to assign a signal weight to each B 0 s → φφ candidate.Using the four-kaon mass as the discriminating variable, the distributions of the signal components for the B 0 s decay time and helicity angles can be determined in the data sample.The sensitivity to φ s is optimised taking into account the signal purity and the flavour tagging performance.The final selection of B 0 s → φφ candidates based on this optimisation is required to have a BDT output larger than 0.1, ∆ ln L K π > −3 for each kaon and |m KK − m φ | < 15 MeV/c 2 for each φ candidate.
In total, 1182 B 0 s → φφ candidates are selected.Figure 1 shows the four-kaon invariant mass distribution for the selected events.Using an unbinned extended maximum likelihood fit, a signal yield of 880 ± 31 events is obtained.In this fit, the B 0 s → φφ signal component is modelled by two Gaussian functions with a common mean.The width of the first Gaussian component is measured to be 12.9 ± 0.5 MeV/c 2 , in agreement with the expectation from simulation.The relative fraction and width of the second Gaussian component are fixed from simulation to values of 0.785 and 29.5 MeV/c 2 , respectively, in order to ensure a good quality fit.Combinatorial background is modelled using an exponential function which is allowed to vary in the fit.Contributions from specific backgrounds such as B 0 → φK * 0 , where K * 0 → K + π − , are found to be negligible.
An unbinned maximum likelihood fit is performed to the decay time, t, and the three helicity angles, Ω = {θ 1 , θ 2 , Φ}, of the selected B 0 s → φφ candidates, each of which is reassigned a signal sPlot weight based on the four-kaon invariant mass, m KKKK [19,20].The probability density function (PDF) consists of signal components, which include detector resolution and acceptance effects, and are factorised into separate terms for the decay time and the angular observables.The B 0 s decay into the K + K − K + K − final state can proceed via combinations of intermediate vector (φ) and scalar (f 0 (980)) resonances and scalar non-resonant K + K − pairs.Thus the total decay amplitude is a coherent sum of P -wave (vector-vector), S-wave (vector-scalar) and SS-wave (scalar-scalar) contributions.The differential decay rate of the decay time and helicity angles is described by a sum of 15 terms, corresponding to five polarisation amplitudes and their interference terms, The angular functions f i (Ω) for the P -wave terms are derived in Ref. [21] and the helicity angles of the two φ mesons are randomly assigned to θ 1 and θ 2 .The time-dependent functions K i (t) can be written as [21] K where ∆Γ s = Γ L − Γ H is the decay width difference between the light (L) and heavy (H) B 0 s mass eigenstates, Γ s is the average decay width, Γ s = (Γ L + Γ H )/2, and ∆m s is the B 0 s -B 0 s oscillation frequency.The coefficients N i , a i , b i , c i and d i can be expressed in terms of φ s and the magnitudes, |A i |, and phases, δ i , of the five polarisation amplitudes at t = 0.The three P -wave amplitudes, denoted by A 0 , A , A ⊥ , are normalised such that with the strong phases δ 1 and δ 2 defined as δ 1 = δ ⊥ − δ and δ 2 = δ ⊥ − δ 0 .The S and SS-wave amplitudes and their corresponding phases are denoted by A S , A SS and δ S , δ SS , respectively.For a B 0 s meson produced at t = 0, the coefficients in Eq. 2 and the angular functions f i (θ 1 , θ 2 , Φ) are given in Table 1, where δ 2,1 = δ 2 − δ 1 .Assuming that CP violation in mixing and direct CP violation are negligible, the differential distribution for a B 0 s meson is obtained by changing the sign of the coefficients The parameters of a double Gaussian function used to model the decay time resolution are determined from simulation studies.A single Gaussian function with a resolution of 40 fs is found to have a similar effect on physics parameters and is applied to the data fit.
The φ s measurement requires that the meson flavour be tagged as either a B 0 s or B 0 s meson at production.To achieve this, both the opposite side (OS) and same side kaon (SSK) flavour tagging methods are used [23,24].In OS tagging the b-quark hadron produced in association with the signal b-quark is exploited through the charge of a muon or electron produced in semileptonic decays, the charge of a kaon from a subsequent charmed hadron decay, and the momentum-weighted charge of all tracks in an inclusively reconstructed decay vertex.The SSK tagging makes use of kaons formed from the s-quark produced in association with the B 0 s meson.The kaon charge identifies the flavour of the signal B 0 s meson.The event-by-event mistag is the probability that the decision of the tagging algorithm is incorrect and is determined by a neural network trained on simulated events and calibrated with control samples [23].The value of the event-by-event mistag is used in the fit as an observable and the uncertainties on the calibration parameters are propagated to the statistical uncertainties of the physics parameters, following the procedure described in Ref. [6].For events tagged by both the OS and SSK methods, a combined tagging decision is made.The total tagging power is ε tag D 2 = (3.29 ± 0.48)%, with a tagging efficiency of ε tag = (49.7 ± 5.0)% and a dilution D = (1 − 2ω) where ω is the average mistag probability.Untagged events are included in the analysis as they increase the sensitivity to φ s through the b i terms in Eq. 2.
The total S-wave fraction is determined to be (1.6 +2.4 −1.2 )% where the double S-wave contribution A SS is set to zero, since the fit shows little sensitivity to A SS .A fit to the two-dimensional mass, m KK , for both kaon pairs, where background is subtracted using sidebands is performed and yields a consistent S-wave fraction of (2.1 ± 1.2)%.
The results of the fit for the main observables are shown in Table 2. Figure 2 shows the distributions for the decay time and helicity angles with the projections for the best fit PDF overlaid.The likelihood profile for the CP -violating weak phase φ s , shown in Fig. 3, is not parabolic.To obtain a confidence level a correction is applied due to a small under-coverage of the likelihood profile using the method described in Ref. [25].Including systematic uncertainties (discussed below) and assuming the values of the polarisation amplitudes and strong phases observed in data, an interval of [−2.46, −0.76] rad at 68% confidence level is obtained for φ s .The polarisation amplitudes and phases, shown in Table 2, differ from those reported in Ref. [9] as φ s is not constrained to zero.
The uncertainties related to the calibration of the tagging and the assumed values of Γ s , ∆Γ s and ∆m s are absorbed into the statistical uncertainty, described above.Systematic uncertainties are determined and the sum in quadrature of all sources is reported in Table 2 for each observable.To check that the background is properly accounted for, an additional fit is performed where the angular and time distributions are parameterised using the B 0 s mass sidebands.This gives results in agreement with those presented here and no further systematic uncertainty is assigned.The uncertainty due to the modelling of the S-wave component is evaluated by allowing the SS-wave component to vary in the fit.The difference between the two fits leads to the dominant uncertainty on φ s of 0.20 rad.The systematic uncertainty due to the decay time acceptance is found by taking the difference in the values of fitted parameters between the nominal fit, using a binned time acceptance, and a fit in which the time acceptance is explicitely paramaterised.This  is found to be 0.09 rad for φ s .Possible differences in the simulated decay time resolution compared to the data are studied by varying the resolution according to the discrepancies observed in the B 0 s → J/ψφ analysis [6].This leads to a systematic uncertainty of 0.01 rad for φ s .The distributions of maximum p T and χ 2 /ndf of the final state tracks and the p T and η of the B 0 s candidate are reweighted to better match the data.From this, the angular acceptance is recalculated, leading to small changes in the results (0.02 rad for φ s ), which are assigned as systematic uncertainty.Biases in the fit method are studied using simulated pseudo-experiments that lead to an uncertainty of 0.02 rad for φ s .Further small systematic uncertainties (0.02 rad for φ s ) are due to the limited number of events in the simulation sample used for the determination of the angular acceptance and to the choice of a single versus a double Gaussian function for the mass PDF, which is used to assign the signal weights.The total systematic uncertainty on φ s is 0.22 rad, significantly smaller than the statistical uncertainty.
In summary, we present the first study of CP violation in the decay time distribution of hadronic B 0 s → φφ decays.The CP -violating phase, φ s , is restricted to the interval of [−2.46, −0.76] rad at 68% C.L. The p-value of the Standard Model prediction [8] is 16%, taking the values of the strong phases and polarisation amplitudes observed in data and assuming that systematic uncertainties are negligible.The precision of the φ s measurement is dominated by the statistical uncertainty and is expected to improve with larger LHCb data sets.

Figure 1 :
Figure 1: Invariant K + K − K + K − mass distribution for selected B 0 s → φφ candidates.The total fit (solid line) consists of a double Gaussian signal component together with an exponential background (dotted line).

Figure 2 :
Figure 2: One-dimensional projections of the B 0 s → φφ fit for (a) decay time, (b) helicity angle Φ and the cosine of the helicity angles (c) θ 1 and (d) θ 2 .The data are marked as points, while the solid lines represent the projections of the best fit.The CP -even P -wave, the CP -odd P -wave and S-wave components are shown by the long dashed, short dashed and dotted lines, respectively.

Figure 3 :
Figure 3: Negative ∆ln likelihood scan of φ s .Only the statistical uncertainty is included.

Table 2 :
Fit results with statistical and systematic uncertainties.A 68% statistical confidence interval is quoted for φ s .Amplitudes are defined at t = 0.