Precision Measurement of CP Violation in the Penguin-Mediated Decay B 0 s → ϕϕ

A flavor-tagged time-dependent angular analysis of the decay B 0 s → ϕϕ is performed using pp collision data collected by the LHCb experiment at the center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 6 fb − 1 . The CP -violating phase and direct CP -violation parameter are measured to be ϕ s ¯ sss ¼ − 0 . 042 (cid:2) 0 . 075 (cid:2) 0 . 009 rad and j λ j ¼ 1 . 004 (cid:2) 0 . 030 (cid:2) 0 . 009 , respectively, assuming the same values for all polarization states of the ϕϕ system. In these results, the first uncertainties are statistical and the second systematic. These parameters are also determined separately for each polarization state, showing no evidence for polarization dependence. The results are combined with previous LHCb measurements using pp collisions at center-of-mass energies of 7 and 8 TeV, yielding ϕ s ¯ sss ¼ − 0 . 074 (cid:2) 0 . 069 rad and j λ j ¼ 1 . 009 (cid:2) 0

Probing the nature of CP violation is central to understanding how the matter-dominated universe came to existence.Precision studies of time-dependent CP asymmetries in b → ccs decays at the e + e − B factories [1,2] helped establish the CKM mechanism [3,4] as the dominant source of CP violation in the Standard Model (SM).However, the amount of CP violation provided by this mechanism is insufficient to explain the observed matter-antimatter asymmetry [5].Flavor-changing neutral current (FCNC) decays of B mesons are highly sensitive to new physics contributions entering via loop (penguin) processes and provide excellent opportunities to reveal new sources of CP violation.
The penguin-dominated decay B 0 s → ϕϕ, which proceeds via a b → sss transition, is a benchmark channel to study CP violation in FCNC decays at the LHCb experiment 1 .Time-dependent CP violation in this decay, arising from the interference between the direct decay and the decay after B 0 s mixing, is characterized by the CP -violating phase ϕ sss s and the parameter |λ|, which is related to direct CP violation.In the SM, the phase ϕ sss s is expected to be very close to zero due to a cancellation of the mixing and decay weak phases [6][7][8][9][10], and the parameter |λ| is expected to be close to unity, indicating vanishing direct CP asymmetry [7][8][9][10][11].However, new physics contributions in the penguin decay or the B 0 s mixing could significantly alter the values of ϕ sss s and |λ| [12][13][14][15].In addition, the ϕϕ system in this decay is produced in three linear polarization states, and the effects of new physics may be polarization dependent [14,15].This is in contrast to the B 0 s → J/ψϕ decay, where potential new physics would mainly affect the mixing process and the polarization dependence is expected to be less pronounced, since the decay amplitude is dominated by a tree-level b → ccs diagram.With no hint of new physics found in the B 0 s mixing, this study of CP violation in B 0 s → ϕϕ decays aims to probe new physics in b → s penguin processes.
The LHCb collaboration has previously studied CP violation in the decay B 0 s → ϕϕ [16][17][18].The most precise measurements, ϕ sss s = −0.073± 0.115 (stat) ± 0.027 (syst) rad and |λ| = 0.99 ± 0.05 (stat) ± 0.01 (syst) [18], are based on a data sample collected before 2017, corresponding to an integrated luminosity of 5 fb −1 .Due to the limited size of the data sample, a full polarization-dependent analysis was not performed.Instead, the CP -violating phases for the parallel and perpendicular polarization states were measured separately, with the phase of the longitudinal polarization fixed to zero.No evidence for polarization-dependent CP violation was observed.
This Letter reports an updated measurement of the CP -violation parameters in B 0 s → ϕϕ decays using the full data sample of pp collisions collected with the LHCb detector at a center-of-mass energy of 13 TeV in 2015-2018 (Run 2), corresponding to an integrated luminosity of 6 fb −1 .The results are then combined with the LHCb result based on the data collected at 7 and 8 TeV in 2011 and 2012 (Run 1) [17], corresponding to an integrated luminosity of 3 fb −1 .Apart from the increased data sample size, this analysis also benefits from recent software improvements including new reconstruction algorithms for the vertex detectors that lead to a better decay-time resolution, and optimized flavor-tagging algorithms that increase the effective tagging efficiency.The flavor-tagged time-dependent angular analysis reported here largely follows the procedure described in Refs.[18].New features related to the calibration of flavor tagging and decay-time resolution, and the modelling of decay-time acceptance are described in detail below.The increased sample size enables the CP -violation parameters in the decay B 0 s → ϕϕ to be measured independently for all polarization states for the first time.
The LHCb detector is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5 and is described in detail in Refs.[19,20].The simulated events used in this analysis are produced with the software described in Refs.[21][22][23][24][25][26][27].The online event selection is performed by a trigger [28], which consists of a hardware trigger followed by a two-stage software trigger.For this analysis, the hardware trigger selects hadron or muon candidates with high transverse momentum (p T ).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 three-body topological b-hadron trigger [29].
In the offline selection, B 0 s → ϕϕ candidates, with the ϕ mesons decaying to a K + K − pair, are reconstructed by combining four high-quality charged tracks compatible with originating from the same vertex and identified as kaons.All tracks are required to have a p T above 0.4 GeV/c.Two K + K − pairs, each with an invariant mass within 20 MeV/c 2 of the known mass of the ϕ meson [30], are combined to form B 0 s → ϕϕ candidates and the product of their p T is required to be above 1.2 GeV 2 /c 2 .The decay time of the B 0 s candidates must be larger than 0.3 ps.The K + K − K + K − invariant mass, m(K + K − K + K − ), must fall in the interval [5000, 5800] MeV/c 2 .Potential contamination from B 0 → ϕK * 0 decays is largely suppressed by removing candidates that have a K + K − invariant mass within 30 MeV/c 2 of the known K * (892) 0 mass [30] when the pion mass is assigned to a kaon candidate.
A multilayer perceptron (MLP) classifier [31,32] is used to further suppress background and improve the signal significance.The vertex fit χ 2 per degree of freedom, p T and η of the B 0 s candidate, the cosine of the angle between the B 0 s momentum and its flight direction, as well as the p T and η of the kaon and ϕ candidates are used as discriminating variables.The MLP classifier is trained using a sample of simulated B 0 s → ϕϕ decays representing the signal and a background data sample consisting of B 0 s → ϕϕ candidates with an invariant mass outside a ±120 MeV/c 2 window around the known B 0 s mass [30].The k-fold method [33] with k = 8 is employed in the training of the MLP classifier.The lower threshold on the MLP output is chosen by maximizing the figure-of-merit defined as N s / √ N s + N b , where N s is the expected signal yield in the signal region [5322, 5412] MeV/c 2 and N b is the expected background yield in this region estimated by interpolating the number of candidates observed in the sideband regions [5100, 5160] MeV/c 2 and [5487, 5547] MeV/c 2 .Roughly 3% of the events contain more than one candidate.In these cases one candidate is chosen at random, and the influence of this choice is found to be negligible.Selected B 0 s → ϕϕ candidates in the mass range [5150, 5600] MeV/c 2 are used in subsequent analysis.The mass distribution of these candidates is shown in Fig. 1(a).Two sources of background are identified: the combinatorial background and a peaking background component from Λ 0 b → ϕK − p decays due to misidentification of a proton as a kaon.A maximum-likelihood fit is performed to the mass distribution of selected B 0 s → ϕϕ candidates.The signal shape is described by the sum of a double-sided Crystal Ball function [34] and a Student's t-function [35] with all parameters fixed to the values estimated in the simulation except the peak position and resolution.The Λ 0 b → ϕK − p background is described by a double-sided Crystal Ball function with the shape parameters fixed to the values determined using the fast simulation package RapidSim [36].The combinatorial background is represented by an exponential function.The yields of the three components, the position and resolution of the signal component and the slope of the background exponential function are allowed to vary in the fit.The B 0 s → ϕϕ signal yield is measured to be 15840 ± 140.Based on the result of the fit to the mass distribution, a signal weight is assigned to each candidate using the sPlot method [37].These signal weights are subsequently used in a maximum-likelihood fit [38] to the decay-time and angular distributions in order to statistically subtract the background contribution.
The decay of a B 0 s meson to the K + K − K + K − final state can proceed via the ϕϕ, ϕf 0 and f 0 f 0 intermediate states.Due to the small phase space of the decay f 0 → K + K − and the narrow K + K − mass window used to select the ϕ candidates, the latter two contributions are highly suppressed and found to be negligible from an angular fit that accounts for these contributions.Thus in the subsequent analysis, only the B 0 s → ϕϕ decay is considered.The differential decay rate is written as the sum of six terms, corresponding to contributions from the three polarization states and their interferences, where t is the decay time of the B 0 s meson, and ⃗ Ω = (θ 1 , θ 2 , χ) denotes the helicity angles of the two K + mesons in the corresponding ϕ rest frame (θ 1 , θ 2 ) and the angle between the two ϕ → K + K − decay planes (χ).The angular functions f k ( ⃗ Ω) are defined in Ref. [18].
The time-dependent functions h k (t) are given by [39] h Here Q equals +1 (−1) for an initial B 0 s (B 0 s ) state, ∆m s is the mass difference between the heavy and light B 0 s mass eigenstates, ∆Γ s is the decay width difference between the light and heavy mass eigenstates, and Γ s is the average decay width.Ignoring CP violation in the B 0 s mixing, in line with experimental measurements [40], the coefficients N k , a k , b k , c k and d k are defined [18] in terms of the magnitudes |A i |, phases δ i , CP -violating phases ϕ s,i and direct CP -violation parameters |λ i | for the three polarization states of the B 0 s decay at t = 0, with i = 0, ∥, ⊥.The three amplitudes satisfy The parameters ϕ s,i and |λ i | are defined by the equation where η i is the CP eigenvalue of the polarization state i, q and p are complex numbers relating the B 0 s mass eigenstates to the flavor eigenstates.A subset of parameters, chosen here as (ϕ s,i , , can be determined by performing a maximum-likelihood fit to the distributions of t, ⃗ Ω and Q.In the SM-like case or new physics scenarios where CP violation is polarization independent, the set of CP -violation observables can be reduced to ϕ s,i = ϕ sss s and |λ i | = |λ|.In this analysis, the above formalism is used to obtain both polarization-independent and polarization-dependent results, taking into account the experimental effects discussed below.
The detector acceptance and selection requirements lead to a nonuniform efficiency as a function of the angular variables, referred to below as the angular acceptance.This effect is accounted for through the use of normalization factors calculated with simulated signal events subject to the same selection criteria as the data.Weights are assigned to the simulated events to improve the agreement with the data, in the shape of the kaon p T distribution.These weights are determined with an iterative algorithm [18,41].
The reconstruction, trigger and selection requirements result in a decay-time dependent efficiency.A cubic spline function [42], with 7 knots at [0.3, 0.5, 1.0, 1.5, 2.0, 3.0, 8.0] ps and 9 coefficients, is employed to describe the decay-time dependent efficiency function, referred to below as the decay-time acceptance.One coefficient is fixed to unity for normalization, and all the other coefficients are determined in the fit to the data, where the parameters Γ s and ∆Γ s are constrained to the recent measurements by the LHCb collaboration in B 0 s → J/ψϕ decays [43].Compared with the previous analysis in Ref. [18], which used B 0 s → D − s π + and B 0 → J/ψK * 0 decays as control channels to determine the decay-time acceptance, this method with free acceptance parameters simplifies the analysis without loss of precision for the physics parameters.
The dilution effect of the decay-time resolution on the B 0 s oscillation is modelled by a Gaussian with a per-candidate width σ t , which is related to the per-candidate decay-time uncertainty, δ t , through a linear calibration function σ t = q 0 + q 1 × δ t .The parameters q 0 and q 1 are obtained using fictitious candidates formed of four prompt tracks from pp interactions, which have a decay time centered around 0. These prompt candidates are weighted to match the momentum and p T distributions of the signal candidates and split into ten δ t intervals.For each interval, the effective time resolution σ t,i is estimated by fitting the sum of three Gaussian functions with a common mean to the decay-time distribution of the prompt candidates and converting the resultant triple Gaussian into a single Gaussian that has the same damping effect on the observed B 0 s oscillation amplitude [43].A linear fit to the data points (δ t,i , σ t,i ) for all intervals provides the estimates of q 0 and q 1 .The reliability of this calibration method is verified using simulated samples of signal decays and prompt candidates.The effective decay-time resolution is found to be between 42 and 44 fs , depending on the data-taking year.
The initial flavor of a B 0 s meson is inferred using both opposite-side (OS) [44] and same-side (SS) [45] tagging algorithms.Each of the two methods yields a tagging decision Q with a mistag probability κ for each B 0 s candidate, where Q = +1, −1 or 0, if the candidate is classified as B 0 s , B 0 s or untagged, respectively.The mistag probability is calibrated using a linear function ω = p 0 + p 1 × κ for the OS tagging and a quadratic function ω = p 0 + p 1 × κ + p 2 × κ 2 for the SS tagging, where ω is the corrected mistag probability.The calibration of the OS mistag probability uses B + → J/ψK + decays, for which the value of ω in an interval of κ can be obtained from the numbers of correct and wrong decisions.The calibration of the SS mistag probability uses flavor-specific B 0 s → D − s π + decays, for which the value of ω in an interval of κ is estimated by fitting the decay-time distribution.Detailed descriptions of the calibration procedures can be found in Refs.[18,41].The effective tagging efficiency is estimated to be (5.7 ± 0.5)%, (6.1 ± 0.7)% and (6.3 ± 0.5)% for the 2015-2016, 2017 and 2018 data samples, respectively.Mistag asymmetries between B 0 s and B 0 s decays are evaluated using B ± → J/ψK ± decays for the OS tagging and prompt D ± s decays as a proxy for B 0 s and B 0 s decays for the SS tagging, and accounted for in the subsequent signal fit.
A weighted maximum-likelihood fit [38] is simultaneously performed to the three subsamples of data recorded in 2015-2016, 2017 and 2018.The probability density function for each period is based on Eq. 1 and takes into account the effects of angular acceptance, decay-time acceptance, decay-time resolution and mistag probability.The parameter ∆m s is constrained to the measurement by the LHCb collaboration, ∆m s = 17.766 ± 0.006 ps −1 [46].
The background-subtracted data distributions of the decay time and angular variables with projections of the polarization-independent fit are shown in Fig. 1(b-d).The fit results are summarized in Table 1, which include both statistical and systematic uncertainties.The correlation matrix is given in the Supplemental Material [47], as well as the observed time-dependent asymmetry between flavor-tagged B 0 s and B 0 s decays.A summary of the systematic uncertainties for the polarization-independent fit is reported in Table 2.The total systematic uncertainty on ϕ sss s , which is the sum in quadrature of the different contributions, is 0.009 rad, significantly smaller than the statistical uncertainty of 0.075 rad.This is also the case for the other physics parameters.
Inaccuracies in the determination of the calibration parameters of the decay-time resolution (q 0 , q 1 ) and the flavor tagging (p 0 , p 1 , p 2 ) lead to systematic effects on the CP -violation observables.These parameters are fixed in the baseline fit, and their combined statistical and systematic uncertainties are propagated to the fit parameters as systematic uncertainties.The compatibility of the calibration parameter values in control and signal channels are checked using simulated samples, and the observed differences are treated as sources of systematic uncertainties.These uncertainties are taken to be fully correlated between different data-taking periods.A bias of roughly −5 fs on the reconstructed decay time is observed in the calibration of the decay-time resolution.Since this bias is much 2.769 ± 0.105 ± 0.011 smaller than the effective resolution, neglecting it in the analysis of the signal decays results in negligible effects on the physics parameters.
The systematic uncertainties related to the angular acceptance comprise the uncertainties due to the limited size of the simulation sample, evaluated using a bootstrapping [48] procedure, and uncertainties associated with convergence of the iterative correction procedure, estimated by comparing fit results with different numbers of iterations.
The systematic uncertainties related to the decay-time acceptance are found by varying the knot positions of the spline function to achieve similar fit quality of the decay-time distribution to that of the baseline fit.The maximum changes of the parameter estimates are assigned as systematic uncertainties.
The effects of mismodelling the signal and background mass distributions are studied by using alternative shape functions that can achieve similar fit quality to that of the baseline fit.The maximum changes of the parameter estimates are assigned as systematic uncertainties.The sPlot method used to subtract background in the time-dependent angular fit requires that the discriminating variable, m(K + K − K + K − ), is uncorrelated with the decay angles, ⃗ Ω, and decay time, t.To evaluate the effects of possible correlations among these variables, the data sample is split into three subsamples according to the value of cos 2 θ 1 + cos 2 θ 2 , χ or t, respectively.A fit to the m(K + K − K + K − ) distribution of each subsample is performed, and signal weights are calculated accordingly.The three subsamples are then recombined and a new time-dependent angular fit is performed.The maximum changes of the parameter estimates are assigned as systematic uncertainties.
The intrinsic bias from the maximum-likelihood fit with a limited sample size is evaluated by performing pseudoexperiments, which also demonstrate that the parameter uncertainties estimated in the fit are reliable after correcting for the background dilution effect.
Various checks of the fit procedure are performed by splitting the data sample according to magnet polarity, trigger selection, tagging category, data-taking period, and multiple decay-time and B 0 s -meson p T intervals.The effect of tightening the kaon-identification and MLP-output requirements is also studied.The fit results are compatible between different subsamples in all checks.
The polarization-independent measurements of the CP -violation parameters ϕ sss s and |λ| in B 0 s → ϕϕ decays presented here are combined with the LHCb Run 1 measurements,  [43].In the combination, those systematic uncertainties that arise from the same origin are taken to be completely correlated between the Run 1 and Run 2 results.The combined values of the CP -violation parameters are ϕ sss s = −0.074± 0.069 rad and |λ| = 1.009 ± 0.030, with a correlation coefficient of −0.02.This is the most precise measurement of CP violation in B 0 s → ϕϕ decays to date, as is illustrated in Fig. 2.
where the uncertainties are statistical only.No significant difference between different polarization states is observed.
In conclusion, a measurement of the polarization-independent CP -violation observables in B 0 s → ϕϕ decays is performed using data collected with the LHCb detector in 2015-2018, corresponding to a total integrated luminosity of 6 fb −1 .The results are ϕ sss s = −0.042± 0.075 ± 0.009 rad , |λ| = 1.004 ± 0.030 ± 0.009 , where the first uncertainties are statistical and the second systematic.These results are combined with the LHCb measurements based on data taken in 2011 and 2012 to obtain ϕ sss s = −0.074± 0.069 rad and |λ| = 1.009 ± 0.030.This is the most precise measurement of time-dependent CP asymmetry in the decay B 0 s → ϕϕ and in any penguin-dominated B meson decay.The measurement is consistent with and supersedes the measurement in Ref. [18], and agrees with the SM expectation of tiny CP violation.For the first time, the polarization-dependent CP -violation parameters are measured, which show no significant difference between the three polarization states of B 0 s → ϕϕ decays.These results can be used to constrain new physics contributions in b → s transitions [12][13][14].

Figure 1 :
Figure 1: (a) Mass distribution of the B 0 s → ϕϕ candidates, superimposed by the fit projections.(b-d) Background-subtracted distributions of angular variables (cos θ 1 and χ) and decay time, superimposed by the fit projections.

Table 1 :
Measured observables in the polarization-independent fit.The first uncertainties are statistical and the second systematic.