Precision measurement of the $B^{0}_{s}$-$\bar{B}^{0}_{s}$ oscillation frequency with the decay $B^{0}_{s}\rightarrowD^{-}_{s}\pi^{+}$

A key ingredient to searches for physics beyond the Standard Model in $B^{0}_{s}$ mixing phenomena is the measurement of the $B^{0}_{s}$-$\bar{B}^{0}_{s}$ oscillation frequency, which is equivalent to the mass difference $\Delta m_{s}$ of the $B^{0}_{s}$ mass eigenstates. Using the world's largest $B^{0}_{s}$ meson sample accumulated in a dataset, corresponding to an integrated luminosity of 1.0 fb$^{-1}$, collected by the LHCb experiment at the CERN LHC in 2011, a measurement of $\Delta m_{s}$ is presented. A total of about 34,000 $B^{0}_{s}\rightarrowD^{-}_{s}\pi^{+}$ signal decays are reconstructed, with an average decay time resolution of 44 fs. The oscillation frequency is measured to be \mbox{$\Delta m_{s}$ = 17.768 $\pm$ 0.023 (stat) $\pm$ 0.006 (syst) ps$^{-1}$}, which is the most precise measurement to date.

The LHCb collaboration † Abstract A key ingredient to searches for physics beyond the Standard Model in B 0 s mixing phenomena is the measurement of the B 0 s -B 0 s oscillation frequency, which is equivalent to the mass difference ∆m s of the B 0 s mass eigenstates. Using the world's largest B 0 s meson sample accumulated in a dataset, corresponding to an integrated luminosity of 1.0 fb −1 , collected by the LHCb experiment at the CERN LHC in 2011, a measurement of ∆m s is presented. A total of about 34,000 B 0 s → D − s π + signal decays are reconstructed, with an average decay time resolution of 44 fs. The oscillation frequency is measured to be ∆m s = 17.768 ± 0.023 (stat) ± 0.006 (syst) ps −1 , which is the most precise measurement to date.

Introduction
The Standard Model (SM) of particle physics, despite its great success in describing experimental data, is considered an effective theory only valid at low energies, below the TeV scale. At higher energies, new physics phenomena are predicted to emerge. For analyses looking for physics beyond the SM (BSM) there are two conceptually different approaches: direct and indirect searches. Direct searches are performed at the highest available energies and aim at producing and detecting new heavy particles. Indirect searches focus on precision measurements of quantum-loop induced processes. Accurate theoretical predictions are available for the heavy quark sector in the SM. It is therefore an excellent place to search for new phenomena [1,2], since any deviation from these predictions can be attributed to contributions from BSM.
In the SM, transitions between quark families (flavours) are possible via the charged current weak interaction. Flavour changing neutral currents (FCNC) are forbidden at lowest order, but are allowed in higher order processes. Since new particles can contribute to these loop diagrams, such processes are highly sensitive to contributions from BSM. An example FCNC transition is neutral meson mixing, where neutral mesons can transform into their antiparticles. Particle-antiparticle oscillations have been observed in the K 0 -K 0 system [3], the B 0 -B 0 [4] system, the B 0 s -B 0 s system [5,6] and the D 0 -D 0 system [7][8][9][10]. The frequency of B 0 s -B 0 s oscillations is the highest. On average, a B 0 s meson changes its flavour nine times between production and decay. This poses a challenge to the detector for the measurement of the decay time. Another key ingredient of this measurement is the determination of the flavour of the B 0 s meson at production, which relies heavily on good particle identification and the separation of tracks from the primary interaction point.
The observed particle and antiparticle states B 0 s and B 0 s are linear combinations of the mass eigenstates B H and B L with masses m H and m L and decay widths Γ H and Γ L , respectively [11]. The B 0 s oscillation frequency is equivalent to the mass difference ∆m s = m H − m L . The parameter ∆m s is an essential ingredient for all studies of timedependent matter-antimatter asymmetries involving B 0 s mesons, such as the B 0 s mixing phase φ s in the decay B 0 s → J/ψ φ [12]. It was first observed by the CDF experiment [6]. The LHCb experiment published a measurement of this frequency using a dataset, corresponding to an integrated luminosity of 37 pb −1 , taken in 2010 [13]. This analysis complements the previous result and is obtained in a similar way, using a data sample, corresponding to an integrated luminosity of 1.0 fb −1 , collected by LHCb in 2011.

The LHCb experiment
The LHCb experiment is designed for precision measurements in the beauty and charm hadron systems. At a center-of-mass energy of √ s = 7 TeV, about 3 · 10 11 bb pairs were produced in 2011. The LHCb detector [14] is a single-arm forward spectrometer covering the pseudorapidity range from two to five. The excellent decay time resolution necessary to resolve the fast B 0 s -B 0 s oscillation is provided by a silicon-strip vertex detector surrounding the pp interaction region. At nominal position the sensitive region of the vertex detector is only 8 mm away from the beam. Impact parameter (IP) resolution of 20 µm for tracks with high transverse momentum (p T ) is achieved.
Charged particle momenta are measured with the LHCb tracking system consisting of the aforementioned vertex dector, a large-area silicon-strip detector 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. The combined tracking system has momentum resolution ∆p/p that varies from 0.4% at 5 GeV/c to 0.6% at 100 GeV/c.
Since this analysis is performed with decays involving only hadrons in the final state, excellent particle identification is crucial to suppress background. Charged hadrons are identified using two ring-imaging Cherenkov detectors [15]. Photon, electron, and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter, and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers.
The first stage of the trigger [16] is implemented in hardware, based on information from the calorimeter and muon systems, and selects events that contain candidates with large transverse energy and transverse momentum. This is followed by a software stage which applies a full event reconstruction. The software trigger used in this analysis requires a two-, three-or four-track secondary vertex with a significant displacement from the primary interaction, a large sum of p T of the tracks, and at least one track with p T > 1.7 GeV/c. In addition an IP χ 2 with respect to the primary interaction greater than 16 and a track fit χ 2 per degree of freedom < 2 is required. The IP χ 2 is defined as the difference between the χ 2 of the primary vertex reconstructed with and without the considered track. A multivariate algorithm is used for the identification of the secondary vertices.
For the simulation, pp collisions are generated using Pythia 6.4 [17] with a specific LHCb configuration [18]. Decays of hadronic particles are described by EvtGen [19], in which final state radiation is generated using Photos [20]. The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [21], as described in Ref. [22].

Signal selection and analysis strategy
The analysis uses B 0 s candidates reconstructed in the flavour-specific decay mode To avoid double counting, events that contain a candidate passing the selection criteria of one mode, are not considered for the subsequent modes, using the order listed above. All reconstructed decays are flavour-specific final states, thus the flavour of the B 0 s candidate at the time of its decay is given by the charges of the final state particles. A combination of tagging algorithms is used to identify the B 0 s flavour at production. The algorithms provide for each candidate a tagging decision as well as an estimate of the probability that this decision is wrong (mistag probability). These algorithms have been optimized using large event samples of flavour-specific decays [23,24].
To be able to study the effect of selection criteria that influence the decay time spectrum, we restrict the analysis to those events in which the signal candidate passed the requirements of the software trigger algorithm used in this analysis. Specific features, such as the masses of the intermediate φ and K * 0 resonances or the Dalitz structure of the D − s → π − π + π − decay mode, are exploited for the five decay modes. The most powerful quantity to separate signal from background common to all decay modes is the output of a boosted decision tree (BDT) [25]. The BDT exploits the long B 0 s lifetime by using as input the IP χ 2 of the daughter tracks, the angle of the reconstructed B 0 s momentum relative to the line between the reconstructed primary vertex, and the B 0 s vertex and the radial flight distance in the transverse plane of both the B 0 s and the D − s meson. Additional requirements are applied on the sum of the p T of the B 0 s candidate's decay products as well as on particle identification variables, and on track and vertex quality. The reconstructed D − s mass is required to be consistent with the known value [26]. After this selection, a total of about 47,800 candidates remain in the B 0 s → D − s π + invariant mass window of 5.32 -5.98 GeV/c 2 .
An unbinned likelihood method is employed to simultaneously fit the B 0 s invariant mass and decay time distributions of the five decay modes. The probability density functions (PDFs) for signal and background in each of the five modes can be written as where m is the reconstructed invariant mass of the B 0 s candidate, t is its reconstructed decay time and σ t is an event-by-event estimate of the decay time resolution. The tagging decision q can be 0 if no tag is found, −1 for events with different flavour at production and decay (mixed) or +1 for events with the same flavour at production and decay (unmixed). The predicted event-by-event mistag probability η can take values between 0 and 0.5. The functions P m and P t describe the invariant mass and the decay time probability distributions, respectively. P t is a conditional probability depending on σ t and η. The functions P σt and P η are required to ensure the proper relative normalization of P t for signal and background [27]. The functions P σt and P η are determined from data, using the measured distribution in the upper B 0 s invariant mass sideband for the background PDF and the sideband subtracted distribution in the invariant mass signal region for the signal PDF. This measurement has been performed "blinded", meaning that during the analysis process the fitted value of ∆m s was shifted by an unknown value, which was removed after the analysis procedure had been finalized.
s → K − π + π − , and e) D − s → π − π + π − . The fits and the various background components are described in the text. Misidentified backgrounds refer to background from B 0 and Λ 0 b decays with one misidentified daughter particle.

Invariant mass description
The invariant mass of each B 0 s candidate is determined in a vertex fit constraining the D − s invariant mass to its known value [26]. The invariant mass spectra for the five decay modes after all the selection criteria are applied are shown in Fig. 1. The fit to the five distributions takes into account contributions from signal, combinatorial background and b-hadron decay backgrounds. The signal components are described by the sum of two Crystal Ball (CB) functions [28], which are constrained to have the same peak parameter. The parameters of the CB function describing the tails are fixed to values obtained from simulation, whereas the mean and the two widths are allowed to vary. These are constrained to be the same for all five decay modes. It has been checked on data that the

Decay mode (D
0.599 ± 0.008 0.019 ±0.010 Total 47781 0.714 ± 0.004 0.019 ±0.010 mass resolution is compatible among all modes. The b-hadron decay background includes B 0 and Λ 0 b decays with one misidentified daughter particle. Their mass shapes are derived from simulated samples. The yields for the different b-hadron decay backgrounds are allowed to vary individually for each of the five decay modes. Another component originates from B 0 s → D ∓ s K ± decays, in which the kaon is misidentified as a pion. This contribution is treated as signal in the decay time analysis.
The requirement that the invariant mass be larger than 5.32 GeV/c 2 rejects background candidates from B 0 s decays with additional particles in the decay not reconstructed, such The fitted number of signal candidates does not change with respect to a fit in a larger mass window. The high mass sideband region 5.55 -5.98 GeV/c 2 provides a sample of mainly combinatorial background candidates. The mass distribution is described by an exponential function, whose parameters are allowed to vary individually for the five decay modes. By including this region in the fit, we are able to determine the decay time distribution as well as the tagging behaviour of the combinatorial background.
The number of used candidates, along with the signal fractions extracted from the two dimensional fit in mass and decay time, are reported in Table 1. One complication arises from the fact that the shape of the invariant mass distribution of the B 0 s → D ∓ s K ± events is very similar to that of the B 0 background. Therefore the fraction of B 0 s → D ∓ s K ± candidates has been determined in a fit to the D − s → π − π + π − mode only, in which no B 0 background is present. Subsequently this value is used for all other modes.

Decay time description
The decay time of a particle is measured as where L is the distance between the production vertex and the decay vertex of the particle, m its reconstructed invariant mass, and p its reconstructed momentum. We use the decay time calculated without the D − s mass constraint to avoid a systematic dependence of the B 0 s decay time on the reconstructed invariant mass. The theoretical distribution of the decay time, t, ignoring the oscillation and any detector resolution, is where Γ s is the B 0 s decay width and ∆Γ s the decay width difference between the light and heavy mass eigenstate. 2 The value for ∆Γ s is fixed to the latest value measured by LHCb [12] ∆Γ s = 0.106 ± 0.011 ± 0.007 ps −1 . It is varied within its uncertainties to assess the systematic effect on the measurement of ∆m s . The Heaviside step function θ(t) restricts the PDF to positive decay times.
To account for detector resolution effects, the decay time PDF is convolved with a Gaussian distribution. The width σ t is taken from an event-by-event estimate returned by the fitting algorithm that reconstructs the B 0 s decay vertex. Due to tracking detector resolution effects σ t needs to be calibrated. A data-driven method, combining prompt D − s mesons from the primary interaction with random π + mesons, forms fake B 0 s candidates. The decay time distribution of these candidates, each divided by its event-by-event σ t , is fitted with a Gaussian function. The width provides a scale factor S σt = 1.37, by which each σ t is multiplied, such that it represents the correct resolution. By inspecting different regions of phase space of the fake B 0 s candidates, the uncertainty range on this number is found to be 1.25 < S σt < 1.45. The variation is taken into account as part of the ∆m s systematic studies. The resulting average decay time resolution is S σt × σ t = 44 fs.
Some of the selection criteria influence the shape of the decay time distribution, e.g. the requirement of a large IP for B 0 s daughter tracks. Thus a decay time acceptance function E t (t) has to be taken into account. Its parametrization is determined from simulated data and the parameter describing its shape is allowed to vary in the fit to the data, while Γ s is fixed to the nominal value [26]. Taking into account resolution and decay time acceptance, the PDF given in Eq.(3) is modified to with G(t; 0, S σt σ t ) being the resolution function determined by the method mentioned above. The decay time PDFs for the B 0 and Λ 0 b backgrounds are identical to the signal PDF, except for ∆Γ being zero, and Γ s being replaced by their respective decay widths [26]. The shape of the decay time distribution of the combinatorial background is determined with high mass sideband data. It is parametrized by the sum of two exponential functions multiplied by a second order polynomial distribution. The exponential and polynomial parameters are allowed to vary in the fit and are constrained to be the same for the five decay modes.

Flavour tagging
To determine the flavour of the B 0 s meson at production, both opposite-side (OST) and same-side (SST) tagging algorithms are used. The OST exploits the fact that b quarks at the LHC are predominantly produced in quark-antiquark pairs. By partially reconstructing the second b hadron in the event, conclusions on the flavour at production of the signal B 0 s candidate can be drawn. The OST have been optimized on large samples of B + → J/ψ K + , B → µ + D * − X, and B 0 → D − π + decays [23].
The SST takes advantage of the fact that the net strangeness of the pp collision is zero. Therefore, the s quark needed for the hadronization of the B 0 s meson must have been produced in association with an s quark, which in about 50% of the cases hadronizes to form a charged kaon. By identifying this kaon, the flavour at production of the signal B 0 s candidate is determined. The optimization of the SST was performed on a data sample of B 0 s → D − s π + decays, which has a large overlap with the sample used in this analysis [24]. However, since the oscillation frequency is not correlated with the parameters describing tagging performance, this does not bias the ∆m s measurement.
The decisions given by both tagging algorithms have a probability ω to be incorrect. Each tagging algorithm provides an estimate for the mistag probability η which is the output of a neural network combining various event properties. The true mistag probability ω can be parametrized as a linear function of the estimate η [23,24] with η being the mean of the distribution of η. This parametrization is chosen to minimize the correlations between p 0 and p 1 . The calibration is performed separately for the OST and SST. The sets of calibration parameters (p 0 , p 1 ) OST and (p 0 , p 1 ) SST are allowed to vary in the fit. The figure of merit of these tagging algorithms is called the effective tagging efficiency ε eff . It gives the factor by which the statistical power of the sample is reduced due to imperfect tagging decisions. In this analysis, ε eff is found to be (2.6 ± 0.4)% for the OST and (1.2 ± 0.3)% for the SST. Uncertainties are statistical only.

Measurement of ∆m s
Adding the information of the flavour tagging algorithms, the decay time PDF for tagged signal candidates is modified to where gives the fraction of candidates with a tagging decision. Signal candidates without a tagging decision are still described by Eq. The information provided by the opposite-side and same-side taggers for the signal is combined to a single tagging decision q and a single mistag probability ω(η OST , η SST ) using their respective calibration parameters p 0 OST/SST and p 1 OST/SST . The individual background components show different tagging characteristics for candidates tagged by the OST or SST. The b hadron backgrounds show the same opposite-side tagging behaviour (q and ω) as the signal, while the combinatorial background shows random tagging behaviour. For same-side tagged events, we assume random tagging behaviour for all background components. We introduce tagging asymmetry parameters to allow for different numbers of candidates being tagged as mixed or unmixed, and other parameters to describe the tagging efficiencies for these backgrounds. As expected, the fitted values of these asymmetry parameters are consistent with zero within uncertainties.
All tagging parameters, as well as the value for ∆m s , are constrained to be the same for the five decay modes. The result is ∆m s = 17.768 ± 0.023 ps −1 (statistical uncertainty only). The likelihood profile was examined and found to have a Gaussian shape up to nine standard deviations. The decay time distributions for candidates tagged as mixed or unmixed are shown in Fig. 2, together with the decay time projections of the PDF distributions resulting from the fit.

Systematic uncertainties
With respect to the first measurement of ∆m s at LHCb [13] all sources of systematic uncertainties have been reevaluated.
The dominant source is related to the knowledge of the absolute value of the decay time. This has two main contributions. First, the imperfect knowledge of the longitudinal (z) scale of the detector contributes to the systematic uncertainty. It is obtained by comparing the track-based alignment and survey data and evaluating the track distribution in the vertex detector. This results in 0.02% uncertainty on the decay time scale and thus an absolute uncertainty of ±0.004 ps −1 on ∆m s .
The second contribution to the uncertainty of the decay time scale comes from the knowledge of the overall momentum scale. This has been evaluated by an independent study using mass measurements of well-known resonances. Deviations from the reference values [26] are measured to be within 0.15%. However, since both the measured invariant mass and momentum enter the calculation of the decay time, this effect cancels to some extent. The resulting systematic on the decay time scale is evaluated from simulation to be 0.02%. This again translates to an absolute uncertainty of ±0.004 ps −1 on ∆m s . The next largest systematic uncertainty is due to a possible bias of the measured decay time given by the track reconstruction and the selection procedure. This is estimated from simulated data to be less than about 0.2 fs, and results in ±0.001 ps −1 systematic uncertainty on ∆m s . Various other sources contributing to the systematic uncertainty have been studied such as the decay time acceptance, decay time resolution, variations of the value of ∆Γ s , different signal models for the invariant mass and the decay time resolution, variations of the signal fraction and the fraction of B 0 s → D ∓ s K ± candidates. They are all found to be negligible. The sources of systematic uncertainty on the measurement of ∆m s are summarized in Table 2.

Conclusion
A measurement of the B 0 s -B 0 s oscillation frequency ∆m s is performed using B 0 s → D − s π + decays in five different D − s decay channels. Using a data sample corresponding to an integrated luminosity of 1.0 fb −1 collected by LHCb in 2011, the oscillation frequency is found to be ∆m s = 17.768 ± 0.023 (stat) ± 0.006 (syst) ps −1 , in good agreement with the first result reported by the LHCb experiment [13] and the current world average, 17.69 ± 0.08 ps −1 [26]. This is the most precise measurement of ∆m s to date, and will be a crucial ingredient in future searches for BSM physics in B 0 s oscillations.