Measurement of CP asymmetry in B0s → D ∓s K± decays

JHEP11(2014)060 Published for SISSA by Springer Received:July 24, 2014 Revised:October 15, 2014 Accepted:October 20, 2014 Published:November 13, 2014 Measurement of CP asymmetry in B0 s→D∓ sK± decays The LHCb collaboration E-mail: [email protected] Abstract: We report on measurements of the time-dependent CP violating observables in B0 s→D∓ sK±decays using a dataset corresponding to 1.0 fb−1of pp collisions recorded with the LHCb detector. We find the CP violating observables Cf= 0.53±0.25±0.04, A∆Γ f= 0.37±0.42±0.20, A∆Γ ¯ f= 0.20±0.41±0.20, Sf=−1.09±0.33±0.08, S¯ f=−0.36 ±0.34 ±0.08, where the uncertainties are statistical and systematic, respectively. Using these observables together with a recent measurement of the B0 smixing phase −2βsleads to the first extraction of the CKM angle γfrom B0 s→D∓ sK±decays, finding γ= (115+28 −43)◦modulo 180◦at 68% CL, where the error contains both statistical and systematic uncertainties. Keywords: CP violation, CKM angle gamma, B physics, Flavor physics, Hadron-Hadron Scattering ArXiv ePrint: 1407.6127 Open Access, Copyright CERN, for the benefit of the LHCb Collaboration. Article funded by SCOAP3. doi:10.1007/JHEP11(2014)060 JHEP11(2014)060 Contents 1 Introduction 1 1.1 Decay rate equations and CP violation observables 2 1.2 Analysis strategy 3 2 Detector and software 4 3 Event selection 5 4 Signal and background shapes 6 4.1 B0 scandidate mass shapes 7 4.2 D− scandidate mass shapes 8 4.3 Companion L(K/π) shapes 8 5 Multivariate fit to B0 s→D∓ sK±and B0 s→D− sπ+8 6 Flavour tagging 9 6.1 Tagging calibration 9 6.2 Combination of OS and SSK taggers 12 6.3 Mistag distributions 12 7 Decay-time resolution and acceptance 13 8 Decay-time fit to B0 s→D∓ sK±14 9 Systematic uncertainties 17 10 Interpretation 19 11 Conclusion 21 The LHCb collaboration 25 1 Introduction Time-dependent analyses of tree-level B0 (s)→D∓ (s)π±, K±decays1are sensitive to the angle γ≡arg(−VudV∗ ub/VcdV∗ cb) of the unitarity triangle of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2] through CP violation in the interference of mixing and decay amplitudes [3–5]. The determination of γfrom such tree-level decays is important because it is not sensitive to potential effects from most models of physics beyond the Standard 1Inclusion of charge conjugate modes is implied except where explicitly stated. – 1 – JHEP11(2014)060 ✁ Vcb ×Vus ≈λ3 B0 s K− D+ s b s s u c s ✁ Vub ×Vcs ≈λ3 B0 s D+ s K− b u, c, t W± W± u, c, t s b s c u s Figure 1. Feynman diagrams for B0 s→D+ sK−without (left) and with (right) B0 smixing. Model (BSM). The value of γhence provides a reference against which other BSM-sensitive measurements can be compared. Due to the interference between mixing and decay amplitudes, the physical CP violating observables in these decays are functions of a combination of γand the relevant mixing phase, namely γ+ 2β(β≡arg(−VcdV∗ cb/VtdV∗ tb)) in the B0and γ−2βs (βs≡arg(−VtsV∗ tb/VcsV∗ cb)) in the B0 ssystem. A measurement of these physical observables can therefore be interpreted in terms of γor β(s)by using an independent measurement of the other parameter as input. Such measurements have been performed by both the BaBar [6,7] and the Belle [8,9] collaborations using B0→D(∗)∓π±decays. In these decays, however, the ratios rD(∗)π= |A(B0→D(∗)−π+)/A(B0→D(∗)+π−)|between the interfering b→uand b→camplitudes are small, rD(∗)π≈0.02, limiting the sensitivity on γ[10]. The leading order Feynman diagrams contributing to the interference of decay and mixing in B0 s→D∓ sK±are shown in figure 1. In contrast to B0→D(∗)∓π±decays, here both the B0 s→D− sK+(b→cs¯u) and B0 s→D+ sK−(b→u¯cs) amplitudes are of the same order in the sine of the Cabibbo angle λ= 0.2252 ±0.0007 [11,12], O(λ3), and the amplitude ratio of the interfering diagrams is approximately |VubVcs/VcbVus| ≈ 0.4. Moreover, the decay width difference in the B0 ssystem, ∆Γs, is nonzero [13], which allows a determination of γ−2βsfrom the sinusoidal and hyperbolic terms in the decay time evolution, up to a two-fold ambiguity. This paper presents the first measurements of the CP violating observables in B0 s→ D∓ sK±decays using a dataset corresponding to 1.0 fb−1of pp collisions recorded with the LHCb detector at √s= 7 TeV, and the first determination of γ−2βsin these decays. 1.1 Decay rate equations and CP violation observables The time-dependent decay rates of the initially produced flavour eigenstates |B0 s(t= 0)i and |B0 s(t= 0)iare given by dΓB0 s→f(t) dt=1 2|Af|2(1 + |λf|2)e−Γstcosh ∆Γst 2+A∆Γ fsinh ∆Γst 2 +Cfcos (∆mst)−Sfsin (∆mst),(1.1) – 2 – JHEP11(2014)060 dΓB0 s→f(t) dt=1 2|Af|2 p q 2 (1 + |λf|2)e−Γstcosh ∆Γst 2+A∆Γ fsinh ∆Γst 2 −Cfcos (∆mst) + Sfsin (∆mst),(1.2) where λf≡(q/p)(Af/Af) and Af(Af) is the decay amplitude of a B0 sto decay to a final state f(¯ f). Γsis the average B0 sdecay width, and ∆Γsis the positive [14] decay-width difference between the heavy and light mass eigenstates in the B0 ssystem. The complex coefficients pand qrelate the B0 smeson mass eigenstates, |BL,H i, to the flavour eigenstates, |B0 siand |B0 si |BLi=p|B0 si+q|B0 si,(1.3) |BHi=p|B0 si−q|B0 si,(1.4) with |p|2+|q|2= 1. Similar equations can be written for the CP -conjugate decays replacing Cfby C¯ f,Sfby S¯ f, and A∆Γ fby A∆Γ ¯ f. In our convention fis the D− sK+final state and ¯ fis D+ sK−. The CP asymmetry observables Cf,Sf,A∆Γ f,C¯ f,S¯ fand A∆Γ ¯ fare given by Cf=1−|λf|2 1 + |λf|2=−C¯ f=−1−|λ¯ f|2 1 + |λ¯ f|2, Sf=2Im(λf) 1 + |λf|2, A∆Γ f=−2Re(λf) 1 + |λf|2, S¯ f=2Im(λ¯ f) 1 + |λ¯ f|2, A∆Γ ¯ f=−2Re(λ¯ f) 1 + |λ¯ f|2.(1.5) The equality Cf=−C¯ fresults from |q/p|= 1 and |λf|=|1 λ¯ f|, i.e. the assumption of no CP violation in either the decay or mixing amplitudes. The CP observables are related to the magnitude of the amplitude ratio rDsK≡ |λDsK|=|A(B0 s→D− sK+)/A(B0 s→D− sK+)|, the strong phase difference δ, and the weak phase difference γ−2βsby the following equations: Cf=1−r2 DsK 1 + r2 DsK , A∆Γ f=−2rDsKcos(δ−(γ−2βs)) 1 + r2 DsK , A∆Γ ¯ f=−2rDsKcos(δ+ (γ−2βs)) 1 + r2 DsK , Sf=2rDsKsin(δ−(γ−2βs)) 1 + r2 DsK , S ¯ f=−2rDsKsin(δ+ (γ−2βs)) 1 + r2 DsK .(1.6) 1.2 Analysis strategy To measure the CP violating observables defined in section 1.1, it is necessary to perform a fit to the decay-time distribution of the selected B0 s→D∓ sK±candidates. The kinematically similar mode B0 s→D− sπ+is used as control channel which helps in the determination of the time-dependent efficiency and flavour tagging performance. Before a fit to the decay time can be performed, it is necessary to distinguish the signal and background candidates in the selected sample. This analysis uses three variables to maximise sensitivity – 3 – JHEP11(2014)060 when discriminating between signal and background: the B0 smass; the D− smass; and the log-likelihood difference L(K/π) between the pion and kaon hypotheses for the companion particle. In section 4, the signal and background shapes needed for the analysis are obtained in each of the variables. Section 5describes how a simultaneous extended maximum likelihood fit (in the following referred to as multivariate fit) to these three variables is used to determine the yields of signal and background components in the samples of B0 s→D− sπ+ and B0 s→D∓ sK±candidates. Section 6describes how to obtain the flavour at production of the B0 s→D∓ sK±candidates using a combination of flavour-tagging algorithms, whose performance is calibrated with data using flavour-specific control modes. The decay-time resolution and acceptance are determined using a mixture of data control modes and simulated signal events, described in section 7. Finally, section 8describes two approaches to fit the decay-time distribution of the B0 s→D∓ sK±candidates which extract the CP violating observables. The first fit, henceforth referred to as the sFit, uses the results of the multivariate fit to obtain the so-called sWeights [15] which allow the background components to be statistically subtracted [16]. The sFit to the decay-time distribution is therefore performed using only the probability density function (PDF) of the signal component. The second fit, henceforth referred to as the cFit, uses the various shapes and yields of the multivariate fit result for the different signal and background components. The cFit subsequently performs a six-dimensional maximum likelihood fit to these variables, the decay-time distribution and uncertainty, and the probability that the initial B0 sflavour is correctly determined, in which all contributing signal and background components are described with their appropriate PDFs. In section 10, we extract the CKM angle γusing the result of one of the two approaches. 2 Detector and software The LHCb detector [17] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, designed for the study of particles containing bor cquarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the pp interaction region [18], 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 [19] placed downstream of the magnet. The tracking system provides a measurement of momentum, p, with a relative uncertainty that varies from 0.4% at low momentum to 0.6% at 100 GeV/c. The minimum distance of a track to a primary pp collision vertex, the impact parameter, is measured with a resolution of (15 + 29/pT)µm, where pTis the component of ptransverse to the beam, in GeV/c. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors [20]. The magnet polarity is reversed regularly to control systematic effects. The trigger [21] consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. The software trigger requires a two-, threeor four-track secondary vertex with a large sum of the transverse momentum of the charged particles and a significant displacement from – 4 – JHEP11(2014)060 the primary pp interaction vertices (PVs). A multivariate algorithm [22] is used for the identification of secondary vertices consistent with the decay of a bhadron. In the simulation, pp collisions are generated using Pythia [23] with a specific LHCb configuration [24]. Decays of hadrons are described by EvtGen [25], in which final state radiation is generated using Photos [26]. The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [27,28] as described in ref. [29]. 3 Event selection The event selection begins by building D− s→K−K+π−,D− s→K−π+π−, and D− s→ π−π+π−candidates from reconstructed charged particles. These D− scandidates are subsequently combined with a fourth particle, referred to as the “companion”, to form B0 s→ D∓ sK±and B0 s→D− sπ+candidates. The flavour-specific Cabibbo-favoured decay mode B0 s→D− sπ+is used as a control channel in the analysis, and is selected identically to B0 s→D∓ sK±except for the PID criteria on the companion particle. The decay-time and B0 smass resolutions are improved by performing a kinematic fit [30] in which the B0 scandidate is constrained to originate from its associated proton-proton interaction, i.e. the one with the smallest IP with respect to the B0 scandidate, and the B0 smass is computed with a constraint on the D− smass. The B0 s→D− sπ+mode is used for the optimisation of the selection and for studying and constraining physics backgrounds to the B0 s→D∓ sK±decay. The B0 s→D∓ sK±and B0 s→D− sπ+candidates are required to be matched to the secondary vertex candidates found in the software trigger. Subsequently, a preselection is applied to the B0 s→D∓ sK± and B0 s→D− sπ+candidates using a similar multivariate displaced vertex algorithm to the trigger selection, but with offline-quality reconstruction. A selection using the gradient boosted decision tree (BDTG) [31] implementation in the Tmva software package [32] further suppresses combinatorial backgrounds. The BDTG is trained on data using the B0 s→D− sπ+,D− s→K−K+π−decay sample, which is purified with respect to the previous preselection exploiting PID information from the Cherenkov detectors. Since all channels in this analysis are kinematically similar, and since no PID information is used as input to the BDTG, the resulting BDTG performs equally well on the other D− sdecay modes. The optimal working point is chosen to maximise the expected sensitivity to the CP violating observables in B0 s→D∓ sK±decays. In addition, the B0 sand D− scandidates are required to be within m(B0 s)∈[5300,5800] MeV/c2and m(D− s)∈[1930,2015] MeV/c2, respectively. Finally, the different final states are distinguished by using PID information. This selection also strongly suppresses cross-feed and peaking backgrounds from other misidentified decays of b-hadrons to c-hadrons. We will refer to such backgrounds as “fully reconstructed” if no particles are missed in the reconstruction, and “partially reconstructed” otherwise. The decay modes B0→D−π+,B0→D− sπ+,Λ0 b→Λ− cπ+,B0 s→D∓ sK±, and B0 s→D∗− sπ+are backgrounds to B0 s→D− sπ+, while B0 s→D− sπ+,B0 s→D∗− sπ+, B0 s→D− sρ+,B0→D− sK+,B0→D−K+,B0→D−π+,Λ0 b→Λ− cK+,Λ0 b→Λ− cπ+, and – 5 – JHEP11(2014)060 Λ0 b→D(∗)− spare backgrounds to B0 s→D∓ sK±. This part of the selection is necessarily different for each D− sdecay mode, as described below. •For D− s→π−π+π−none of the possible misidentified backgrounds fall inside the D− smass window. Loose PID requirements are nevertheless used to identify the D− s decay products as pions in order to suppress combinatorial background. •For D− s→K−π+π−, the relevant peaking backgrounds are Λ− c→pπ+π−in which the antiproton is misidentified, and D−→K+π−π−in which both the kaon and a pion are misidentified. As this is the smallest branching fraction D− sdecay mode used, and hence that most affected by background, all D− sdecay products are required to pass tight PID requirements. •The D− s→K−K+π−mode is split into three submodes. We distinguish between the resonant D− s→φπ−and D− s→K∗0K−decays, and the remaining decays. Candidates in which the K+K−pair falls within 20 MeV/c2of the φmass are identified as a D− s→φπ−decay. This requirement suppresses most of the cross-feed and combinatorial background, and only loose PID requirements are needed. Candidates within a 50 MeV/c2window around the K∗0mass are identified as a D− s→K∗0K−decay; it is kinematically impossible for a candidate to satisfy both this and the φrequirement. In this case there is non-negligible background from misidentified D−→K+π−π− and Λ− c→pπ−K+decays which are suppressed through tight PID requirements on the D− skaon with the same charge as the D− spion. The remaining candidates, referred to as nonresonant decays, are subject to tight PID requirements on all decay products to suppress cross-feed backgrounds. Figure 2shows the relevant mass distributions for candidates passing and failing this PID selection. Finally a loose PID requirement is made on the companion track. After all selection requirements, fewer than 2% of retained events contain more than one signal candidate. All candidates are used in the subsequent analysis. 4 Signal and background shapes The signal and background shapes are obtained using a mixture of data-driven approaches and simulation. The simulated events need to be corrected for kinematic differences between simulation and data, as well as for the kinematics-dependent efficiency of the PID selection requirements. In order to obtain kinematic distributions in data for this weighting, we use the decay mode B0→D−π+, which can be selected with very high purity without the use of any PID requirements and is kinematically very similar to the B0 ssignals. The PID efficiencies are measured as a function of particle momentum and event occupancy using prompt D∗+→D0(K−π+)π+decays which provide pure samples of pions and kaons [33], henceforth called D∗+calibration sample. – 6 – JHEP11(2014)060 ] 2 c) [MeV/ ± π - K + m(K 1900 1950 2000 2050 ) 2 cCandidates/(1.0 MeV/ 0 2000 4000 6000 8000 10000 12000 LHCb Failing Selection Passing Selection ] 2 c) [MeV/ ± π - π + πm( 1900 1950 2000 2050 ) 2 cCandidates/(1.0 MeV/ 0 200 400 600 800 1000 1200 1400 1600 1800 2000 LHCb Failing Selection Passing Selection ] 2 c) [MeV/ - π + π ± m(K 1900 1950 2000 2050 ) 2 cCandidates/(1.0 MeV/ 0 2000 4000 6000 8000 10000 12000 LHCb Failing Selection Passing Selection Figure 2. Mass distributions for D− scandidates passing (black, open circles) and failing (red, crosses) the PID selection criteria. In reading order: D− s→K−K+π−,D− s→π−π+π−, and D− s→K−π+π−. 4.1 B0 scandidate mass shapes In order to model radiative and reconstruction effects, the signal shape in the B0 smass is the sum of two Crystal Ball [34] functions with common mean and oppositely oriented tails. The signal shapes are determined separately for B0 s→D∓ sK±and B0 s→D− sπ+from simulated candidates. The shapes are subsequently fixed in the multivariate fit except for the common mean of the Crystal Ball functions which floats for both the B0 s→D− sπ+and B0 s→D∓ sK±channel. The functional form of the combinatorial background is taken from the upper B0 s sideband, with its parameters left free to vary in the subsequent multivariate fit. Each D− s mode is considered independently and parameterised by either an exponential function or by a combination of an exponential and a constant function. The shapes of the fully or partially reconstructed backgrounds are fixed from simulated events using a non-parametric kernel estimation method (KEYS, [35]). Exceptions to this are the B0→D−π+background in the B0 s→D− sπ+fit and the B0 s→D− sπ+background in the B0 s→D∓ sK±fit, which are obtained from data. The latter two backgrounds are – 7 – JHEP11(2014)060 reconstructed with the “wrong” mass hypothesis but without PID requirements, which would suppress them. The resulting shapes are then weighted to account for the effect of the momentum-dependent efficiency of the PID requirements from the D∗+calibration samples, and KEYS templates are extracted for use in the multivariate fit. 4.2 D− scandidate mass shapes The signal shape in the D− smass is again a sum of two Crystal Ball functions with common mean and oppositely oriented tails. The signal shapes are extracted separately for each D− sdecay mode from simulated events that have the full selection chain applied to them. The shapes are subsequently fixed in the multivariate fit except for the common mean of the Crystal Ball functions, which floats independently for each D− sdecay mode. The combinatorial background consists of both random combinations of tracks which do not peak in the D− smass, and, in some D− sdecay modes, backgrounds that contain a true D− s, and a random companion track. It is parameterised separately for each D− sdecay mode either by an exponential function or by a combination of an exponential function and the signal D− sshape. The fully and partially reconstructed backgrounds which contain a correctly reconstructed D− scandidate (B0 s→D∓ sK±and B0→D− sπ+as backgrounds in the B0 s→D− sπ+ fit; B0→D− sK+and B0 s→D− sπ+as backgrounds in the B0 s→D∓ sK±fit) are assumed to have the same mass distribution as the signal. For other backgrounds, the shapes are KEYS templates taken from simulated events, as in the B0 smass. 4.3 Companion L(K/π) shapes We obtain the PDFs describing the L(K/π) distributions of pions and kaons from dedicated D∗+calibration samples. We obtain the PDF describing the protons using a calibration sample of Λ+ c→pK−π+decays. These samples are weighted to match the signal kinematic and event occupancy distributions in the same way as the simulated events. The weighting is done separately for each signal and background component, as well as for each magnet polarity. The shapes for each magnet polarity are subsequently combined according to the integrated luminosity in each sample. The signal companion L(K/π) shape is obtained separately for each D− sdecay mode to account for small kinematic differences between them. The combinatorial background companion L(K/π) shape is taken to consist of a mixture of pions, protons, and kaons, and its normalisation is left floating in the multivariate fit. The companion L(K/π) shape for fully or partially reconstructed backgrounds is obtained by weighting the PID calibration samples to match the event distributions of simulated events, for each background type. 5 Multivariate fit to B0 s→D∓ sK±and B0 s→D− sπ+ The total PDF for the multivariate fit is built from the product of the signal and background PDFs, since correlations between the fitting variables are measured to be small in simulation. These product PDFs are then added for each D− sdecay mode, and almost all background yields are left free to float. The only exceptions are those backgrounds whose – 8 – JHEP11(2014)060 The signal production asymmetry is fixed to zero because the fast B0 soscillations wash out any initial asymmetry and make its effect on the CP observables negligible. The signal detection asymmetry is fixed to (1.0±0.5)%, with the sign convention in which positive detection asymmetries correspond to a higher efficiency to reconstruct positive kaons [41,42]. The background production and detection asymmetries are floated within constraints of ±1% for B0 sand B0decays, and ±3% for Λ0 bdecays. The signal and background mistag and decay-time uncertainty distributions, including k-factors, are modelled by kernel templates as described in section 6and 7. The tagging calibration parameters are constrained to the values obtained from the control channels for all B0 sdecay modes, except for B0and Λ0 bdecays where the calibration parameters of the SSK tagger are fixed to p0= 0.5, p1= 0. All modes use the same spline-based decay-time acceptance function described in section 7. The backgrounds from B0 sdecay modes are all flavour-specific, and are modelled by the decay-time PDF used for B0 s→D− sπ+decays convolved with the appropriate decay-time resolution and k-factors model for the given background. The backgrounds from Λ0 bdecay modes are all described by a single exponential convolved with the appropriate decay-time resolution and k-factor models. The B0→D−K+background is flavour specific and is described with the same PDF as B0 s→D− sπ+, except with ∆mdinstead of ∆msin the oscillating terms, Γdinstead of Γsand the appropriate decay-time resolution and k-factor KEYS templates. The B0→D−π+background, on the other hand, is not a flavour specific decay, and is itself sensitive to CP violation as discussed in section 1. Its decay-time PDF therefore includes nonzero Sfand S¯ fterms which are constrained to their world-average values [12]. The decay-time PDF of the combinatorial background used in the cFit is a double exponential function split by the tagging category of the event, whose parameters are measured using events in the B0 smass sidebands. All decay-time PDFs include the effects of flavour tagging, are convolved with a single Gaussian representing the per-candidate decay-time resolution, and are multiplied by the decay-time acceptance described in section 7. Once the decay-time PDFs are constructed, the sFit proceeds by fitting the signal PDF to the sWeighted B0 s→D∓ sK±candidates. The cFit, on the other hand, performs a six-dimensional fit to the decay time, decay-time error, predicted mistag, and the three variables used in the multivariate fit. The B0 smass range is restricted to m(B0 s)∈[5320,5420] MeV/c2, and the yields of the different signal and background components are fixed to those found in this fit range in the multivariate fit. The decay-time range of the fit is τ(B0 s)∈[0.4,15.0] ps in both cases. The results of the cFit and sFit for the CP violating observables are given in table 3, and their correlations in table 4. The fits to the decay-time distribution are shown in figure 6together with the folded asymmetry plots for D+ sK−and D− sK+final states. The folded asymmetry plots show the difference in the rates of B0 sand B0 stagged D+ sK−and D− sK+candidates, plotted in slices of 2π/∆ms, where the sWeights obtained with the multivariate fit have been used to subtract background events. The plotted asymmetry function is drawn using the sFit central values of the CP observables, and is normalised using the expected dilution due to mistag and time resolution. – 15 – JHEP11(2014)060 Parameter sFit fitted value cFit fitted value Cf0.52 ±0.25 ±0.04 0.53 ±0.25 ±0.04 A∆Γ f0.29 ±0.42 ±0.17 0.37 ±0.42 ±0.20 A∆Γ ¯ f0.14 ±0.41 ±0.18 0.20 ±0.41 ±0.20 Sf−0.90 ±0.31 ±0.06 −1.09 ±0.33 ±0.08 S¯ f−0.36 ±0.34 ±0.06 −0.36 ±0.34 ±0.08 Table 3. Fitted values of the CP observables to the B0 s→D∓ sK±time distribution for (left) sFit and (right) cFit, where the first uncertainty is statistical, the second is systematic. All parameters other than the CP observables are constrained in the fit. lab0_LifetimeFit_ctau 2 4 6 8 10 12 14 Candidates / ( 0.1 ps) 1 10 2 10 LHCb Data ± K ± s D→ 0 s B ) [ps] ± K ± s D→ 0 s (Bτ 2 4 6 8 10 12 14 -2 0 2 ) [ps] s m∆/π modulo (2τ 00.1 0.2 0.3 ) − K + s A(D -0.6 -0.4 -0.2 0 0.2 0.4 0.6 LHCb ) [ps] s m∆/π modulo (2τ 00.1 0.2 0.3 ) + K − s A(D -0.6 -0.4 -0.2 0 0.2 0.4 0.6 LHCb Figure 6. Result of the decay-time (top left) sFit and (top right) cFit to the B0 s→D∓ sK± candidates; the cFit plot groups B0 s→D∗− sπ+and B0 s→D− sρ+, and also groups B0→D−K+, B0→D−π+,Λ0 b→Λ− cK+,Λ0 b→Λ− cπ+,Λ0 b→D− sp,Λ0 b→D∗− sp, and B0→D− sK+together for the sake of clarity. The folded asymmetry plots for (bottom left) D+ sK−, and (bottom right) D− sK+are also shown. – 16 – JHEP11(2014)060 Parameter CfA∆Γ fA∆Γ ¯ fSfS¯ f sFit Cf1.000 0.071 0.097 0.117 −0.042 A∆Γ f1.000 0.500 −0.044 −0.003 A∆Γ ¯ f1.000 −0.013 −0.005 Sf1.000 0.007 S¯ f1.000 cFit Cf1.000 0.084 0.103 −0.008 −0.045 A∆Γ f1.000 0.544 −0.117 −0.022 A∆Γ ¯ f1.000 −0.067 −0.032 Sf1.000 0.002 S¯ f1.000 Table 4. Statistical correlation matrix of the B0 s→D∓ sK±(top) sFit and (bottom) cFit CP parameters. Other fit parameters have negligible correlations with the CP parameters and are omitted for brevity. 9 Systematic uncertainties Systematic uncertainties arise from the fixed parameters ∆ms, Γs, and ∆Γs, and from the limited knowledge of the decay time resolution and acceptance. These uncertainties are estimated using large sets of simulated pseudoexperiments, in which the relevant parameters are varied. The pseudoexperiments are generated with the average of the cFit and sFit central values reported in section 8. They are subsequently processed by the full data fitting procedure: first the multivariate fit to obtain the sWeights, and then the decay time fits. The fitted values of the observables are compared between the nominal fit, where all fixed parameters are kept at their nominal values, and the systematic fit, where each parameter is varied according to its systematic uncertainty. A distribution is formed by normalising the resulting differences to the uncertainties measured in the nominal fit, and the mean and width of this distribution are added in quadrature and conservatively assigned as the systematic uncertainty. The systematic uncertainty on the acceptance is strongly anti-correlated with that due to the fixed value of Γs. This is because the acceptance parameters are determined from the fit to B0 s→D− sπ+data, where Γsdetermines the expected exponential slope, so that the acceptance parameterises any difference between the observed and the expected slope. The systematic pseudoexperiments are also used to compute the systematic covariance matrix due to each source of uncertainty. The total systematic covariance matrix is obtained by adding the individual covariance matrices. The resulting systematic uncertainties are shown in tables 5and 6relative to the corresponding statistical uncertainties. The contributions from Γsand ∆Γsare listed independently for comparison to convey a feeling for their relative importance. For this comparison, Γsand ∆Γsare treated as uncorrelated systematic effects. When computing the total, however, the correlations between these two, as well as between them and the – 17 – JHEP11(2014)060 Parameter CfA∆Γ fA∆Γ ¯ fSfS¯ f sFit ∆ms0.062 0.013 0.013 0.104 0.100 scale factor 0.104 0.004 0.004 0.092 0.096 ∆Γs†0.007 0.261 0.286 0.007 0.007 Γs†0.043 0.384 0.385 0.039 0.038 acceptance, Γs, ∆Γs0.043 0.427 0.437 0.039 0.038 sample splits 0.124 0.000 0.000 0.072 0.071 total 0.179 0.427 0.437 0.161 0.160 cFit ∆ms0.068 0.014 0.011 0.131 0.126 scale factor 0.131 0.004 0.004 0.101 0.103 ∆Γs†0.008 0.265 0.274 0.009 0.008 Γs†0.049 0.395 0.394 0.048 0.042 acceptance, Γs, ∆Γs0.050 0.461 0.464 0.050 0.043 comb. bkg. lifetime 0.016 0.069 0.072 0.015 0.005 sample splits 0.102 0.000 0.000 0.156 0.151 total 0.187 0.466 0.470 0.234 0.226 Table 5. Systematic errors, relative to the statistical error, for (top) sFit and (bottom) cFit. The daggered contributions (Γs, ∆Γs) are given separately for comparison (see text) with the other uncertainties and are not added in quadrature to produce the total. acceptance parameters, are accounted for, and the full systematic uncertainty which enters into the total is listed as “acceptance, Γs, ∆Γs”. The cFit contains fixed parameters describing the decay time of the combinatorial background. These parameters are found to be correlated to the CP parameters, and a systematic uncertainty is assigned. The result is cross-checked by splitting the sample into two subsets according to the two magnet polarities, the hardware trigger decision, and the BDTG response. There is good agreement between the cFit and the sFit in each subsample. However, when the sample is split by BDTG response, the weighted averages of the subsamples show a small discrepancy with the nominal fit for Cf,Sf, and S¯ f, and a corresponding systematic uncertainty is assigned. In addition, fully simulated signal and background events are fitted in order to check for systematic effects due to neglecting correlations between the different variables in the signal and background PDFs. No bias is found. A potential source of systematic uncertainty is the imperfect knowledge on the tagging parameters p0and p1. Their uncertainties are propagated into the nominal fits by means of Gaussian constraints, and are therefore included in the statistical error. A number of other possible systematic effects were studied, but found to be negligible. These include possible production and detection asymmetries, and missing or imperfectly modelled backgrounds. Potential systematic effects due to fixed background yields are evaluated by generating pseudoexperiments with the nominal value for these yields, and fitting back with the yields – 18 – JHEP11(2014)060 Parameter CfA∆Γ fA∆Γ ¯ fSfS¯ f sFit Cf1.00 0.18 0.18 −0.04 −0.04 A∆Γ f1.00 0.95 −0.17 −0.16 A∆Γ ¯ f1.00 −0.17 −0.16 Sf1.00 0.05 S¯ f1.00 cFit Cf1.00 0.22 0.22 −0.04 −0.03 A∆Γ f1.00 0.96 −0.17 −0.14 A∆Γ ¯ f1.00 −0.17 −0.14 Sf1.00 0.09 S¯ f1.00 Table 6. Systematic uncertainty correlations for (top) sFit and (bottom) cFit. fixed to twice their nominal value. No significant bias is observed and no systematic uncertainty assigned. No systematic uncertainty is attributed to the imperfect knowledge of the momentum and longitudinal scale of the detector since both effects are taken into account by the systematic uncertainty in ∆ms. Both the cFit and sFit are found to be unbiased through studies of large ensembles of pseudoexperiments generated at the best-fit point in data. In addition, differences between the cFit and sFit are evaluated from the distributions of the per-pseudoexperiment differences of the fitted values. Both fitters return compatible results. Indeed, an important result of this analysis is that the sFit technique has been successfully used in an environment with such a large number of variables, parameters and categories. The sFit technique was able to perform an accurate subtraction of a variety of time-dependent backgrounds in a multidimensional fit, including different oscillation frequencies, different tagging behaviours, and backgrounds with modified decay-time distributions due to misreconstructed particles. 10 Interpretation The measurement of the CP-sensitive parameters is interpreted in terms of γ−2βsand subsequently γ. For this purpose we have arbitrarily chosen the cFit as the nominal fit result. The strategy is to maximise the following likelihood L(~α) = exp −1 2~ A(~α)−~ AobsTV−1~ A(~α)−~ Aobs,(10.1) where ~α = (γ, φs, rDsK, δ) is the vector of the physics parameters, ~ Ais the vector of observables expressed through eqs. (1.6), ~ Aobs is the vector of the measured CP violating observables and Vis the experimental (statistical and systematic) covariance matrix. Confidence intervals are computed by evaluating the test statistic ∆χ2≡χ2(~α0 min)−χ2(~αmin), – 19 – JHEP11(2014)060 ]° [γ 1-CL 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 160 180 43− +28 115 68.3% 95.5% LHCb ]° [γ K s D r 0 20 40 60 80 100 120 140 160 180 0 0.2 0.4 0.6 0.8 1 1.2 1.4 LHCb ]° [γ ]° [ K s D δ 0 50 100 150 200 250 300 350 -50 0 50 100 150 200 250 LHCb Figure 7. Graph showing 1 −CL for γ, together with the central value and the 68.3% CL interval as obtained from the frequentist method described in the text (top). Profile likelihood contours of rDsKvs. γ(bottom left), and δvs. γ(bottom right). The contours are the 1σ(2σ) profile likelihood contours, where ∆χ2= 1 (∆χ2= 4), corresponding to 39% CL (86% CL) in Gaussian approximation. The markers denote the best-fit values. where χ2(~α) = −2 ln L(~α), in a frequentist way following ref. [43]. Here, ~αmin denotes the global maximum of eq. (10.1), and ~α0 min is the conditional maximum when the parameter of interest is fixed to the tested value. The value of βsis constrained to the LHCb measurement from B0 s→J/ψ K+K−and B0 s→J/ψ π+π−decays, φs= 0.01 ±0.07 (stat) ± 0.01 (syst) rad [13]. Neglecting penguin pollution and assuming no BSM contribution in these decays, φs=−2βs. The resulting confidence intervals are, at 68% CL, γ= (115+28 −43)◦, δ= (3+19 −20)◦, rDsK= 0.53+0.17 −0.16 , where the intervals for the angles are expressed modulo 180◦. Figure 7shows the 1−CL curve for γ, and the two-dimensional contours of the profile likelihood L(~α0 min). The systematic contributions to the uncertainty are quoted separately as γ= – 20 – JHEP11(2014)060 115+26 −35 (stat)+8 −25 (syst) ±4 (φs)◦, assuming the central value to be independent from systematic uncertainties and taking the difference in squares of the total and statistical uncertainties. 11 Conclusion The CP violation sensitive parameters which describe the B0 s→D∓ sK±decay rates have been measured using a dataset of 1.0 fb−1of pp collision data. Their values are found to be Cf= 0.53 ±0.25 ±0.04 , A∆Γ f= 0.37 ±0.42 ±0.20 , A∆Γ ¯ f= 0.20 ±0.41 ±0.20 , Sf=−1.09 ±0.33 ±0.08 , S¯ f=−0.36 ±0.34 ±0.08 , where the first uncertainties are statistical and the second are systematic. The results are interpreted in terms of the CKM angle γ, which yields γ= (115+28 −43)◦,δ= (3+19 −20)◦and rDsK= 0.53+0.17 −0.16 (all angles are given modulo 180◦) at the 68% confidence level. This is the first measurement of γperformed in this channel. Acknowledgments We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES and FANO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (United Kingdom); NSF (U.S.A.). The Tier1 computing centres are supported by IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom). We are indebted to the communities behind the multiple open source software packages on which we depend. 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