Measurement of the $CP$ asymmetry in $B_s^0-\overline{B}{}_s^0$ mixing

The $CP$ asymmetry in the mixing of $B_s^0$ and $\overline{B}{}_s^0$ mesons is measured in proton-proton collision data corresponding to an integrated luminosity of 3.0 fb$^{-1}$, recorded by the LHCb experiment at centre-of-mass energies of 7 and 8 TeV. Semileptonic $B_s^0$ and $\overline{B}{}_s^0$ decays are studied in the inclusive mode $D_s^{\mp} \mu^{\pm} \nu_{\mu} X$ with the $D_s^{\mp}$ mesons reconstructed in the $K^+ K^- \pi^{\mp}$ final state. Correcting the observed charge asymmetry for detection and background effects, the $CP$ asymmetry is found to be $a_{\text{sl}}^s = (0.39 \pm 0.26 \pm 0.20) \% $, where the first uncertainty is statistical and the second systematic. This is the most precise measurement of $a_{\text{sl}}^s$ to date. It is consistent with the prediction from the Standard Model and will constrain new models of particle physics.

When neutral B mesons evolve in time they can change into their own antiparticles. This quantum-mechanical phenomenon is known as mixing and occurs in both neutral B meson systems, B 0 and B 0 s , where B is used to refer to either system. In this mixing process, the CP (charge-parity) symmetry is broken if the probability for a B meson to change into a B meson is different from the probability for the reverse process. This effect can be measured by studying decays into flavour-specific final states, B → f , such that B → f transitions can only occur through the mixing process B → B → f . Such processes include semileptonic B decays, as the charge of the lepton identifies the flavour of the B meson at the time of its decay. The magnitude of the CP -violating asymmetry in B mixing can be characterized by the semileptonic asymmetry a sl . This is defined in terms of the partial decay rates, Γ, to semileptonic final states as where ∆m (∆Γ) is the difference in mass (decay width) between the mass eigenstates of the B system and φ 12 is a CP -violating phase [1]. In the Standard Model (SM), the asymmetry is predicted to be as small as a d sl = (−4.7 ± 0.6) × 10 −4 in the B 0 system and a s sl = (2.22 ± 0.27) × 10 −5 in the B 0 s system [1,2]. However, these values may be enhanced by non-SM contributions to the mixing process [3].
Measurements of a sl have led to an inconclusive picture. In 2010, the D0 collaboration reported an anomalous charge asymmetry in the inclusive production rates of like-sign dimuons [4], which is sensitive to a combination of a d sl and a s sl . Their most recent study shows a discrepancy with SM predictions of about three standard deviations [5]. The current experimental world averages, excluding the anomalous D0 result, are a d sl = (0.01 ± 0.20)% and a s sl = (−0.48 ± 0.48)% [6], compatible with both the SM predictions and the D0 measurement. The measurement of a s sl presented in this letter is based on data recorded by LHCb in 2011 and 2012, corresponding to an integrated luminosity of 3.0 fb −1 . It supersedes the previous LHCb measurement [7], which used the 1.0 fb −1 data sample taken in 2011. Semileptonic decays B 0 s → D − s µ + ν µ X, where X represents any number of particles, are reconstructed inclusively in D − s µ + . Charge-conjugate modes are implied throughout, except in the definitions of charge asymmetry. The D − s meson is reconstructed in the K + K − π − final state. This analysis extends the previous LHCb measurement, which considered only D − s → φπ − decays, by including all possible D − s decays to the K + K − π − final state.
Starting from a sample with equal numbers of B 0 s and B 0 s mesons, a s sl can be measured without determining (tagging) the initial flavour. The raw asymmetry of observed D − s µ + and D + s µ − candidates, integrated over B 0 s decay time, is . (2) The high oscillation frequency ∆m s reduces the effect of the small asymmetry in the production rates between B 0 s and B 0 s mesons in pp collisions by a factor 10 −3 [7,8].
Neglecting corrections, the untagged, time-integrated asymmetry is A raw = a s sl /2, where the factor two reduction compared to the tagged asymmetry in Eq. 1 comes from the summation over mixed and unmixed decays. The tagged asymmetry would actually suffer from a larger reduction because of the tagging efficiency [9,10]. The unmixed decays have zero asymmetry due to CP T symmetry. The raw asymmetry is still affected by possible differences in detection efficiency for the two charge-conjugate final states and by backgrounds from other b-hadron decays to D − s µ + X. Hence, a s sl is calculated as where A det is the detection asymmetry, which is assessed from data using calibration samples, f bkg is the fraction of b-hadron background and A bkg the background asymmetry. The LHCb detector is a single-arm forward spectrometer designed for the study of particles containing b or c quarks [11,12]. A high-precision tracking system with a dipole magnet measures the momentum (p) and impact parameter (IP) of charged particles. The IP is defined as the distance of closest approach between the track and any primary proton-proton interaction and is used to distinguish between D − s mesons from B decays and D − s mesons promptly produced in the primary interaction. The regular reversal of the magnet polarity allows a quantitative assessment of detector-induced charge asymmetries. Different types of charged particles are distinguished using particle identification (PID) information from two ring-imaging Cherenkov detectors, an electromagnetic calorimeter, a hadronic calorimeter and a muon system. Online event selection is performed by a two-stage trigger. For this analysis, the first (hardware) stage selects muons in the muon system; the second (software) stage applies a full event reconstruction. Here the events are first selected by the presence of the muon or one of the hadrons from the D − s decay, after which a combination of the decay products is required to be consistent with the topological signature of a b-hadron decay. Simulated events are produced using the software described in Refs. [13][14][15][16][17].
Different intermediate states, clearly visible in the Dalitz plot shown in Fig. 1, contribute to the three-body D − s → K + K − π − decays. Three disjoint regions are defined, which have different levels of background. The φπ region is the cleanest and is selected by requiring the reconstructed K + K − mass to be within ±20 MeV/c 2 of the known φ mass. The K * K region is selected by requiring the reconstructed K + π − mass to be within ±90 MeV/c 2 of the known K * (892) 0 mass. The remaining D − s candidates are included in the non-resonant (NR) region, which also covers other intermediate states [18].
The D − s candidates are reconstructed from three charged tracks, and then a muon track with opposite charge is added. All four tracks are required to have a good quality track fit and significant IP. The contribution from prompt D − s background is suppressed to a negligible level by imposing a lower bound on the IP of the D − s candidates. To ensure a good overlap with the calibration samples, minimum momenta of 2, 5 and 6 GeV/c and minimum transverse momenta, p T , of 300, 400 and 1200 MeV/c are required for the pions, kaons and muons, respectively. To suppress background, kaon and pion candidates are required to be positively identified by the PID system. Candidates are selected by requiring a good quality of the D − s and B 0 s decay vertices. A source of background arises from D − s candidates where one of the three decay particles is misidentified. The main contributions are from Λ − c → K + pπ − , D − → K + π − π − , J/ψ X, and misidentified or partially reconstructed multibody D decays, all originating from semileptonic b-hadron decays. They are suppressed to a negligible level by specific vetoes, which apply tight PID requirements in a small window of invariant mass of the corresponding particle combination. These vetoes are optimized separately for each Dalitz plot region. To check that this does not introduce additional asymmetries, these selections are applied to control samples of promptly produced D − s mesons. The asymmetries are found to be consistent between the Dalitz regions.
The D − s µ + signal yields are obtained from fits to the K + K − π − invariant mass distributions. These yields contain contributions from backgrounds that also peak at the D − s mass, originating from other b-hadron decays into D − s mesons and muons. Simulation studies indicate that these peaking backgrounds are mainly composed of b-hadron decays to D − s X c X, where the D − s meson originates from a b → ccs transition, and X c is a charmed hadron decaying semileptonically.
An example of such a background is All of these peaking backgrounds have more missing particles than the B 0 s → D − s µ + ν µ X signal decay. Their contribution is reduced by requiring the corrected B 0 s mass, defined as m corr ≡ m 2 + p T 2 + p T , to be larger than 4200 MeV/c 2 , where m is the D − s µ + invariant mass and p T the D − s µ + momentum transverse to the line connecting the primary and B 0 s decay vertices.
The estimates of f bkg and A bkg are based on known branching fractions [18], selection efficiencies and background asymmetries, using a similar approach as in the previous measurement [7]. The reconstruction and selection efficiencies of the backgrounds relative to the signal efficiency are determined from simulation. The total background asymmetry is given by the sum of all contributions as The background asymmetries mainly originate from the production asymmetries of b hadrons. The production asymmetry between B + and B − mesons is A bkg (B + ) = (−0.6 ± 0.6)%, obtained from the observed asymmetry in B + → J/ψ K + decays [19], after correcting for the kaon detection asymmetry and the direct CP asymmetry [18]. For the B 0 background, there are contributions from the production asymmetry and from a d sl [20]. Both asymmetries are diluted when integrating over the B 0 decay time, resulting in A bkg (B 0 ) = (−0.18 ± 0.13)%. The production asymmetry in the Λ 0 b backgrounds is estimated based on the combined CP and production asymmetry measured in . The direct CP asymmetry in this decay mode is estimated to be (−0.6 ± 0.3)%, using the measurements in Ref.
[22] and the method proposed in Ref. [23]. Subtracting this from the combined The overall peaking background fraction is f bkg = (18.4 ± 6.0)% and the correction for the background asymmetry is f bkg A bkg = (−0.023 ± 0.031)%.
The K + K − π ∓ mass distributions are shown in Fig. 2, with the fit results superimposed. The D ∓ s µ ± yields are found to be 899 × 10 3 in the φπ region, 413 × 10 3 in the K * K region, and 280 × 10 3 in the NR region. Extended maximum likelihood fits are made separately for the three Dalitz regions, for the two magnet polarities, and the two datataking periods (2011 and 2012). To accurately determine the background shape from random combinations of K + K − π − candidates, a wide mass window between 1800 and 2047 MeV/c 2 is used, which includes the Cabibbo-suppressed D − → K + K − π − decay. Both peaks are modelled with a double-sided Hypatia function [24]. The tail parameters of this function are determined for each Dalitz region by a fit to the combined data sets for all magnet polarities and data-taking periods, and subsequently fixed in the twelve individual mass fits. A systematic uncertainty is assigned to account for fixing these parameters. The combinatorial background is modelled with a second-order polynomial. A simultaneous fit to the m(K + K − π − ) and m(K + K − π + ) distributions is performed. All signal parameters except the mean masses and signal yields are shared between the D − s and D + s candidates. All background parameters vary independently in the fit to allow for any asymmetry in the combinatorial background. Possible biases from the fit model are studied by generating invariant mass distributions with the signal component described by a double Gaussian function with power-law tails on both sides, and subsequently applying the fit with the default Hypatia shape. The change in the value of A raw is assigned as a systematic uncertainty.
Asymmetries are averaged as follows. For each magnet polarity and data-taking period, the weighted average of the asymmetries of the three Dalitz regions is taken. Then the arithmetic average for the two magnet polarities is taken to minimize possible residual detection asymmetries [7]. Finally, a weighted average is made over the two data-taking periods. The resulting raw asymmetry is A raw = (0.11 ± 0.09)%. The asymmetry A det , arising from the difference in detection efficiencies between the D − s µ + and D + s µ − candidates, is determined using calibration samples. The asymmetry is split up as where the individual contributions are described below. For each calibration sample, event weights are applied to match the three-momentum distributions of the calibration particles to those of the signal decays. The weights are determined in bins of the distributions of momenta and angles. Alternative binning schemes are used to assess the systematic uncertainties due to the weighting procedure. The track reconstruction asymmetry, A track , is split into a contribution, A track (K + K − ), associated with the reconstruction of the K + K − pair and a contribution, A track (π − µ + ), associated with the π − µ + pair. The track reconstruction efficiency for single kaons suffers from a sizeable difference between K + and K − cross-sections with the detector material, which depends on the kaon momentum. This asymmetry largely cancels in A track (K + K − ), due to the similar kinematic distributions of the positive and negative kaons. The kaon asymmetry is calculated using prompt D − → K + π − π − and D − → K 0 S π − decays, similarly to Refs. [20,25]. For pions and muons, the charge asymmetry due to interactions in the detector material is assumed to be negligible, and a systematic uncertainty is assigned for this assumption [20]. Effects from the track reconstruction algorithms and detector acceptance, combined with a difference in kinematic distributions between pions and muons, can result in a charge asymmetry. It is assessed here with two methods. The first method measures the track reconstruction efficiency using samples of partially reconstructed J/ψ → µ + µ − decays as described in Ref. [26]. The second method uses fully and partially reconstructed D * − → D 0 (K + π − π + π − )π − decays as described in Ref. [27]. The final value of A track (π − µ + ) is obtained as the weighted average from the two methods. The systematic uncertainty on this number includes a small effect from differences in the detector acceptance for positive and negative particles. The asymmetry induced by the PID requirements, A PID , is determined using large samples of D * + → D 0 (K − π + )π + and J/ψ → µ + µ − decays. The D * + charge identifies the kaon and the pion of the D 0 decay without the use of PID requirements, which is then used to determine the PID efficiencies and corresponding charge asymmetries.
The asymmetry induced by the trigger, A trig , is split into contributions from the muon hardware trigger and from the software trigger. The first, A trig (hardware), is assessed using samples of J/ψ → µ + µ − decays in data. The second, A trig (software), is mainly caused by the trigger requirements on the muon or one of the hadrons from the D − s decay. The asymmetry from the muon software trigger is determined in a similar fashion to that from the hardware trigger. The asymmetry due to the trigger requirement on the hadrons is determined using samples of prompt D − s → K + K − π − decays that have been triggered by other particles in the event. The combined asymmetry takes into account the overlap between the two triggers.
The measured values of all detection asymmetries with their statistical and systematic uncertainties are shown in Table 1. The overall corrections are small and compatible with zero. In contrast, corrections for separate magnet polarities are more significant (at most 1.1% in 2011 and 0.3% in 2012), as expected for most of the detector-induced charge asymmetries. The corrections for the detection asymmetries are almost fully correlated between the Dalitz regions.
The previous analysis, based on 1.0 fb −1 , used only candidates in the φπ region of the Dalitz plot, with different selection criteria, and used a different fit method to determine The yellow ellipse represents the D0 dimuon measurement with ∆Γ d /Γ d set to its SM expectation value [5]. The error bands and contours correspond to 68% confidence level. the signal yields [7]. A more stringent selection resulted in a cleaner signal sample, but with roughly 30% fewer signal candidates in the φπ region. As a cross check, the approach of the previous analysis is repeated on the full 3.0 fb −1 data sample and the result is compatible within one standard deviation.
The twelve values of a s sl for each Dalitz region, polarity and data-taking period are consistent with each other. The combined result, taking into account all correlations, is where the first uncertainty is statistical, originating from the size of the signal and calibration samples, and the second systematic. There is a small correlation coefficient of +0.13 between this measurement and the LHCb measurement of a d sl [20]. The correlation mainly originates from the muon detection asymmetry and from the effect of a d sl , due to B 0 background, on the measurement of a s sl . Figure 3 displays an overview of the most precise measurements of a d sl and a s sl [5,20,[28][29][30][31][32]. The simple averages of pure a sl measurements, including the present a s sl result and accounting for the small correlation from LHCb, are found to be a d sl = (0.02 ± 0.20)% and a s sl = (0.17 ± 0.30)% with a correlation of +0.07. In combination, these two averages are marginally compatible with the D0 dimuon result (p = 0.5%) shown in Fig. 3. In summary, the determination of a s sl presented in this letter is the most precise to date. It shows no evidence for new physics effects and will serve to restrict models beyond the SM.