Measurement of the CKM angle γ using B0 → DK *0 with D → K0S π + π − decays

JHEP08(2016)137 Published for SISSA by Springer Received:May 10, 2016 Revised:July 4, 2016 Accepted:August 10, 2016 Published:August 24, 2016 Measurement of the CKM angle γusing B0→DK∗0 with D→K0 Sπ+π−decays The LHCb collaboration E-mail: [email protected] Abstract: A model-dependent amplitude analysis of the decay B0→D(K0 Sπ+π−)K∗0 is performed using proton-proton collision data corresponding to an integrated luminosity of 3.0 fb−1, recorded at √s= 7 and 8 TeV by the LHCb experiment. The CP violation observables x±and y±, sensitive to the CKM angle γ, are measured to be x−=−0.15 ±0.14 ±0.03 ±0.01, y−= 0.25 ±0.15 ±0.06 ±0.01, x+= 0.05 ±0.24 ±0.04 ±0.01, y+=−0.65 +0.24 −0.23 ±0.08 ±0.01, where the first uncertainties are statistical, the second systematic and the third arise from the uncertainty on the D→K0 Sπ+π−amplitude model. These are the most precise measurements of these observables. They correspond to γ= (80+21 −22)◦and rB0= 0.39 ±0.13, where rB0is the magnitude of the ratio of the suppressed and favoured B0→DK+π− decay amplitudes, in a Kπ mass region of ±50 MeV around the K∗(892)0mass and for an absolute value of the cosine of the K∗0decay angle larger than 0.4. Keywords: B physics, CKM angle gamma, CP violation, Flavor physics, Hadron-Hadron scattering (experiments) ArXiv ePrint: 1605.01082 Open Access, Copyright CERN, for the benefit of the LHCb Collaboration. Article funded by SCOAP3. doi:10.1007/JHEP08(2016)137 JHEP08(2016)137 Contents 1 Introduction 1 2 The LHCb detector 4 3 Candidate selection and background sources 5 4 Efficiency across the phase space 6 5 Analysis strategy and fit results 6 5.1 Invariant mass fit of B0→DK∗0candidates 7 5.2 CP fit 8 6 Systematic uncertainties 11 7 Determination of the parameters γ,rB0and δB017 8 Conclusion 17 The LHCb collaboration 25 1 Introduction The Standard Model can be tested by checking the consistency of the Cabibbo-KobayashiMaskawa (CKM) mechanism [1,2], which describes the mixing between weak and mass eigenstates of the quarks. The CKM phase γcan be expressed in terms of the elements of the complex unitary CKM matrix, as γ≡arg [−VudVub∗/VcdVcb∗]. Since γis also the angle of the unitarity triangle least constrained by direct measurements, its precise determination is of considerable interest. Its value can be measured in tree-level processes such as B±→ DK±and B0→DK∗0, where Dis a superposition of the D0and D0flavour eigenstates, and K∗0is the K∗(892)0meson. Since loop corrections to these processes are of higher order, the associated theoretical uncertainty on γis negligible [3]. As such, measurements of γin tree-level decays provide a reference value, allowing searches for potential deviations due to physics beyond the Standard Model in other processes. The combination of measurements by the BaBar [4] and Belle [5] collaborations gives γ= (67 ±11)◦[6], whilst an average value of LHCb determinations in 2014 gave γ= 73+9 −10◦[7]. Global fits of all current CKM measurements by the CKMfitter [8,9] and UTfit [10] collaborations yield indirect estimates of γwith an uncertainty of 2◦. Some of the CKM measurements included in these combinations can be affected by new physics contributions. – 1 – JHEP08(2016)137 Since the phase difference between Vub and Vcb depends on γ, the determination of γin tree-level decays relies on the interference between b→cand b→utransitions. The strategy of using B±→DK±decays to determine γfrom an amplitude analysis of D-meson decays to the three-body final state K0 Sπ+π−was first proposed in refs. [11,12]. The method requires knowledge of the D→K0 Sπ+π−decay amplitude across the phase space, and in particular the variation of its strong phase. This may be obtained either by using a model to describe the D-meson decay amplitude in phase space (model-dependent approach), or by using measurements of the phase behaviour of the amplitude (modelindependent approach). The model-independent strategy, used by Belle [13] and LHCb [14, 15], incorporates measurements from CLEO [16] of the Ddecay strong phase in bins across the phase space. The present paper reports a new unbinned model-dependent measurement, following the method used by the BaBar [17–19], Belle [20–22] and LHCb [23] collaborations in their analyses of B±→D(∗)K(∗)±decays. This method allows the statistical power of the data to be fully exploited. The sensitivity to γdepends both on the yield of the sample analysed and on the magnitude of the ratio rBof the suppressed and favoured decay amplitudes in the relevant region of phase space. Due to colour suppression, the branching fraction B(B0→D0K∗0) = (4.2± 0.6)×10−5is an order of magnitude smaller than that of the corresponding charged B-meson decay mode, B(B+→D0K+) = (3.70 ±0.17) ×10−4[24]. However, this is partially compensated by an enhancement in rB0, which was measured to be rB0= 0.240+0.055 −0.048 in B0→ DK∗0decays in which the Dis reconstructed in two-body final states [25]; the charged decays have an average value of rB= 0.097 ±0.006 [8,9]. Model-dependent and independent determinations of γusing B0→D(K0 Sπ+π−)K∗0decays have already been performed by the BaBar [26] and Belle [27] collaborations, respectively. The model-independent approach has also been employed recently by LHCb [28]. For these decays a time-independent CP analysis is performed, as the K∗0is reconstructed in the self-tagging mode K+π−, where the charge of the kaon provides the flavour of the decaying neutral Bmeson. The K∗0meson is one of several possible states of the (K+π−) system. Letting X0 s represent any such state, the B-meson decay amplitude to DK+π−may be expressed as a superposition of favoured b→cand suppressed b→ucontributions: A(B0→DX0 s)∝ |Ac|Af+|Au|ei(δB0−γ)¯ Af, A(B0→DX0 s)∝ |Ac|¯ Af+|Au|ei(δB0+γ)Af,(1.1) where |Ac,u|are the magnitudes of the favoured and suppressed B-meson decay amplitudes, δB0is the strong phase difference between them, and γis the CP-violating weak phase. The quantities Ac,u and δB0depend on the position in the B0→DK+π−phase space. The amplitudes of the D0and D0mesons decaying into the common final state f, Af≡fHD0and ¯ Af≡fHD0, are functions of the K0 Sπ+π−final state, which can be completely specified by two squared invariant masses of pairs of the three final-state particles, chosen to be m2 +≡m2 K0 Sπ+and m2 −≡m2 K0 Sπ−. The other squared invariant mass is m2 0≡m2 π+π−. Making the assumption of no CP violation in the D-meson decay, the amplitudes Afand ¯ Afare related by ¯ Af(m2 +, m2 −) = Af(m2 −, m2 +). – 2 – JHEP08(2016)137 The amplitudes in eq. (1.1) give rise to distributions of the form dΓB0∝ |Ac|2|Af|2+|Au|2|¯ Af|2+ 2|Ac||Au| RehA? f¯ Afei(δB0−γ)i, dΓB0∝ |Ac|2|¯ Af|2+|Au|2|Af|2+ 2|Ac||Au| RehAf¯ A? fei(δB0+γ)i, (1.2) which are functions of the position in the B0→DK+π−phase space. Integrating only over the region φK∗0of the B0→DK+π−phase space in which the K∗0resonance is dominant, r2 B0≡RφK∗0dφ|Au|2 RφK∗0dφ|Ac|2.(1.3) The functional P(A, z, κ) = A 2+|z|2¯ A 2+ 2κRezA?¯ A,(1.4) describes the distribution within the phase space of the D-meson decay, PB0(m2 −, m2 +)∝ P(Af, z−, κ), PB0(m2 −, m2 +)∝ P(¯ Af, z+, κ),(1.5) where the coherence factor κis a real constant (0 ≤κ≤1) [29] measured in ref. [30], parameterising the fraction of the region φK∗0that is occupied by the K∗0resonance, and the complex parameters z±are z±=rB0ei(δB0±γ).(1.6) A direct determination of rB0,δB0and γcan lead to bias, when rB0gets close to zero [17]. The Cartesian CP violation observables, x±=Re(z±) and y±=Im(z±), are therefore used instead. This paper reports model-dependent Cartesian measurements of z±made using B0→ D(K0 Sπ+π−)K∗0decays selected from pp collision data, corresponding to an integrated luminosity of 3 fb−1, recorded by LHCb at centre-of-mass energies of 7 TeV in 2011 and 8 TeV in 2012. The measured values of z±place constraints on the CKM angle γ. Throughout the paper, inclusion of charge conjugate processes is implied, unless specified otherwise. Section 2describes the LHCb detector used to record the data, and the methods used to produce a realistic simulation of the data. Section 3outlines the procedure used to select candidate B0→D(K0 Sπ+π−)K∗0decays, and section 4describes the determination of the selection efficiency across the phase space of the D-meson decay. Section 5details the fitting procedure used to determine the values of the Cartesian CP violation observables and section 6describes the systematic uncertainties on these results. Section 7presents the interpretation of the measured Cartesian CP violation observables in terms of central values and confidence intervals for rB0,δB0and γ, before section 8concludes with a summary of the results obtained. – 3 – JHEP08(2016)137 2 The LHCb detector The LHCb detector [31,32] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, designed for the study of particles containing bor c quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the pp interaction region, a large-area silicon-strip detector located upstream of a dipole magnet of reversible polarity with a bending power of about 4 Tm, and three stations of silicon-strip detectors and straw drift tubes placed downstream of the magnet. The tracking system provides a measurement of the momentum pof charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV. The minimum distance of a track to a primary vertex (PV), the impact parameter (IP), is measured with a resolution of (15 + 29/pT)µm, where pTis the component of the momentum transverse to the beam, in GeV. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors. Photons, electrons and hadrons 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 trigger consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, in which all charged particles with pT> 500 (300) MeV are reconstructed for 2011 (2012) data. The software trigger requires a two-, threeor four-track secondary vertex with a large sum of the transverse momentum, pT, of the tracks and a significant displacement from the primary pp interaction vertices. At least one track should have pT>1.7 GeVand χ2 IP with respect to any primary interaction greater than 16, where χ2 IP is defined as the difference in χ2of a given PV reconstructed with and without the considered track. A multivariate algorithm [33] is used for the identification of secondary vertices consistent with the decay of a bhadron. In the offline selection, trigger signals are associated with reconstructed particles. Selection requirements can therefore be made on the trigger selection itself and on whether the decision was due to the signal candidate, other particles produced in the pp collision, or a combination of both. Decays of K0 S→π+π−are reconstructed in two different categories: the first involving K0 Smesons that decay early enough for the daughter pions to be reconstructed in the vertex detector, and the second containing K0 Sthat decay later such that track segments of the pions cannot be formed in the vertex detector. These categories are referred to as long and downstream, respectively. The long category has better mass, momentum and vertex resolution than the downstream category. Large samples of simulated B0 (s)→D() K∗0decays and various background decays are used in this study. In the simulation, pp collisions are generated using Pythia [34, 35] with a specific LHCb configuration [36]. Decays of hadronic particles are described by EvtGen [37], in which final-state radiation is generated using Photos [38]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [39,40], as described in ref. [41]. – 4 – JHEP08(2016)137 3 Candidate selection and background sources In addition to the hardware and software trigger requirements, after a kinematic fit [42] to constrain the B0candidate to point towards the PV and the Dcandidate to have its nominal mass, the invariant mass of the K0 Scandidates must lie within ±14.4 MeV (±19.9 MeV) of the known value [24] for long (downstream) categories. Likewise, after a kinematic fit to constrain the B0candidate to point towards the PV and the K0 Scandidate to have the K0 Smass, the reconstructed D-meson candidate must lie within ±30 MeV of the D0mass. To reconstruct the B0mass, a third kinematic fit of the whole decay chain is used, constraining the B0candidate to point towards the PV and the Dand K0 Sto have their nominal masses. The χ2of this fit is used in the multivariate classifier described below. This fit improves the resolution of the m2 ±invariant masses and ensures that the reconstructed Dcandidates are constrained to lie within the kinematic boundaries of the phase space. The K∗0candidate must have a mass within ±50 MeV of the world average value and |cos θ∗|>0.4, where the decay angle θ∗is defined in the K∗0rest frame as the angle between the momentum of the kaon daughter of the K∗0, and the direction opposite to the B0momentum. The criteria placed on the K∗0candidate are identical to those used in the analysis of B0→DK∗0with two-body Ddecays [25]. A multivariate classifier is then used to improve the signal purity. A boosted decision tree (BDT) [43,44] is trained on simulated signal events and background candidates lying in the high B0mass sideband [5500,6000] MeV in data. This mass range partially overlaps with the range of the invariant mass fit described below. To avoid a potential fit bias, the candidates are randomly split into two disjoint subsamples, A and B, and two independent BDTs (BDTA and BDTB) are trained with them. These classifiers are then applied to the complementary samples. The BDTs are based on 16 discriminating variables: the B0 meson χ2 IP, the sum of the χ2 IP of the K0 Sdaughter pions, the sum of the χ2 IP of the final state particles except the K0 Sdaughters, the B0and Ddecay vertex χ2, the values of the flight distance significance with respect to the PV for the B0,Dand K0 Smesons, the D (K0 S) flight distance significance with respect to the B0(D) decay vertex, the transverse momenta of the B0,Dand K∗0, the cosine of the angle between the momentum direction of the B0and the displacement vector from the PV to the B0decay vertex, the decay angle of the K∗0and the χ2of the kinematic fit of the whole decay chain. Since some of the variables have different distributions for long or downstream candidates, the two event categories have separate BDTs, giving a total of four independent BDTs. The optimal cut value of each BDT classifier is chosen from pseudoexperiments to minimise the uncertainties on z±. Particle identification (PID) requirements are applied to the daughters of the K∗0to select kaon-pion pairs and reduce background coming from B0→Dρ0decays. A specific veto is also applied to remove contributions from B±→DK±decays: B0→DK∗0candidates with a DK invariant mass lying in a ±50 MeV window around the B±-meson mass are removed. To reject background from D0→ππππ decays, the decay vertex of each long K0 Scandidate is required to be significantly displaced from the Ddecay vertex along the beam direction. – 5 – JHEP08(2016)137 The decay B0 s→DK∗0has a similar topology to B0→DK∗0, but exhibits much less CP violation [30], since the decay B0 s→D0K∗0is doubly-Cabbibo suppressed compared to B0 s→D0K∗0. These decays are used as a control channel in the invariant mass fit. Background from partially reconstructed B0 (s)→D∗( ) K∗0decays, where D∗stands for either D∗0or D∗0, are difficult to exclude since they have a topology very similar to the signal. The D∗0→D0γand D∗0→D0π0decays where the photon or the neutral pion is not reconstructed lead to B0 (s)→D( ) K∗0candidates with a lower invariant mass than the B0 (s)mass. 4 Efficiency across the phase space The variation of the detection efficiency across the phase space is due to detector acceptance, trigger and selection criteria and PID effects. To evaluate this variation, a simulated sample generated uniformly over the D→K0 Sπ+π−phase space is used, after applying corrections for known differences between data and simulation that arise for the hardware trigger and PID requirements. The trigger corrections are determined separately for two independent event categories. In the first category, events have at least one energy deposit in the hadronic calorimeter, associated with the signal decay, which passes the hardware trigger. In the second category, events are triggered only by particles present in the rest of the event, excluding the signal decay. The probability that a given energy deposit in the hadronic calorimeter passes the hardware trigger is evaluated with calibration samples, which are produced for kaons and pions separately, and give the trigger efficiency as a function of the dipole magnet polarity, the transverse energy and the hit position in the calorimeter. The efficiency functions obtained for the two categories are combined according to their proportions in data. The PID corrections are calculated with calibration samples of D∗+→D0π+,D0→ K−π+decays. After background subtraction, the PID efficiencies for kaon and pion candidates are obtained as functions of momentum and pseudorapidity. The product of the kaon and pion efficiencies, taking into account their correlation, gives the total PID efficiency. The various efficiency functions are combined to make two separate global efficiency functions, one for long candidates and one for downstream candidates, which are used as inputs to the fit to obtain the Cartesian observables z±. To smooth out statistical fluctuations, an interpolation with a two-dimensional cubic spline function is performed to give a continuous description of the efficiency ε(m2 +, m2 −), as shown in figure 1. 5 Analysis strategy and fit results To determine the CP observables z±defined in eq. (1.6), an unbinned extended maximum likelihood fit is performed in three variables: the B0candidate reconstructed invariant mass mB0and the Dalitz variables m2 +and m2 −. This fit is performed in two steps. First, the signal and background yields and some parameters of the invariant mass PDFs are determined with a fit to the reconstructed B0invariant mass distribution, described in section 5.1. An amplitude fit over the phase space of the D-meson decay is then performed to measure z±, using only candidates lying in a ±25 MeV window around the fitted B0 – 6 – JHEP08(2016)137 ) 2 (GeV − 2 m 1 2 3 ) 2 (GeV + 2 m 0.5 1 1.5 2 2.5 3 arbitrary units 0.4 0.6 0.8 1 Simulation LHCb ) 2 (GeV − 2 m 1 2 3 ) 2 (GeV + 2 m 0.5 1 1.5 2 2.5 3 arbitrary units 0.4 0.6 0.8 1 Simulation LHCb Figure 1. Variation of signal efficiency across the phase space for (left) long and (right) downstream candidates. mass, and taking the results of the invariant mass fit as inputs, as explained in section 5.2. The cfit [45] library has been used to perform these fits. Candidate events are divided into four subsamples, according to K0 Stype (long or downstream), and whether the candidate is identified as a B0or B0-meson decay. In the B-candidate invariant mass fit, the B0 and B0samples are combined, since identical distributions are expected for this variable, whilst in the CP violation observables fit (CP fit) they are kept separate. 5.1 Invariant mass fit of B0→DK∗0candidates An unbinned extended maximum likelihood fit to the reconstructed invariant mass distributions of the B0candidates in the range [4900,5800] MeV determines the signal and background yields. The long and downstream subsamples are fitted simultaneously. The total PDF includes several components: the B0→DK∗0signal PDF, background PDFs for B0 s→DK∗0decays, combinatorial background, partially reconstructed B0 (s)→D∗( ) K∗0 decays and misidentified B0→Dρ0decays, as illustrated in figure 2. The fit model is similar to that used in the analysis of B0→DK∗0decays with D-meson decays to two-body final states [25]. The B0→DK∗0and B0 s→DK∗0components are each described as the sum of two Crystal Ball functions [46] sharing the same central value, with the relative yields of the two functions and the tail parameters fixed from simulation. The separation between the central values of the B0→DK∗0and B0 s→DK∗0PDFs is fixed to the known B0-B0 smass difference. The ratio of the B0→DK∗0and B0 s→DK∗0 yields is constrained to be the same in both the long and downstream subsamples. The combinatorial background is described with an exponential PDF. Partially reconstructed B0 (s)→D∗( ) K∗0decays are described with non-parametric functions obtained by applying kernel density estimation [47] to distributions of simulated events. These distributions depend on the helicity state of the D∗0meson. Due to parity conservation in D∗0→D0γand D∗0→D0π0decays, two of the three helicity amplitudes have the same invariant mass distribution. The B0 s→D∗K∗0PDF is therefore a linear combination of two non-parametric – 7 – JHEP08(2016)137 m(DK*) (MeV) 5000 5200 5400 5600 5800 Candidates / [18 MeV] 0 20 40 60 80 100 LHCb 0 DK*→ 0 B 0 *K D→ 0 s B Combinatorial 0 D*K*→ 0 B 0 *K D*→ 0 s B 0 ρ D→ 0 B Figure 2. Invariant mass distribution for B0→DK∗0long and downstream candidates. The fit result, including signal and background components, is superimposed (solid blue). The points are data, and the different fit components are given in the legend. The two vertical lines represent the signal region in which the CP fit is performed. functions, with the fraction of the longitudinal polarisation in the B0 s→D∗K∗0decays unknown and accounted for with a free parameter in the fit. Each of the two functions describing the different helicity states is a weighted sum of non-parametric functions obtained from simulated B0 s→D∗(D0γ)K∗0and B0 s→D∗(D0π0)K∗0decays, taking into account the known D∗0→D0π0and D∗0→D0γbranching fractions [48] and the appropriate efficiencies. The PDF for B0→D∗K∗0decays is obtained from that for B0 s→D∗K∗0decays, by applying a shift corresponding to the known B0-B0 smass difference. In the nominal fit, the polarisation fraction is assumed to be the same for B0→D∗K∗0and B0 s→D∗K∗0 decays. The effect of this assumption is taken into account in the systematic uncertainties. The B0→Dρ0component is also described with a non-parametric function obtained from the simulation, using a data-driven calibration to describe the pion-kaon misidentification efficiency. This component has a very low yield and, to improve the stability of the fit, a Gaussian constraint is applied, requiring the ratio of yields of B0→Dρ0and B0 s→DK∗0 to be consistent with its expected value. The fitted distribution is shown in figure 2. The resulting signal and background yields in a ±25 MeV range around the B0mass are given in table 1. This range corresponds to the signal region over which the CP fit is performed. 5.2 CP fit A simultaneous unbinned maximum likelihood fit to the four subsamples is performed to determine the CP violation observables z±. The value of the coherence factor is fixed to the – 8 – JHEP08(2016)137 To evaluate the systematic uncertainty due to the choice of amplitude model for D→K0 Sπ+π−, one million B0→DK∗0and one million B0 s→DK∗0decays are simulated according to the nominal decay model, with the Cartesian observables fixed to the nominal fit result. These simulated decays are fitted with alternative models, each of which includes a single modification with respect to the nominal model, as described in the next paragraph. Each of these alternative models is first used to fit the simulated B0 s→DK∗0decays to determine values for the resonance coefficients of the model. Those coefficients are then fixed in a second fit, to the simulated B0→DK∗0decays, to obtain z±. The systematic uncertainties are taken to be the signed differences in the values of z±from the nominal results. The following changes, labelled (a)-(u), are applied in the alternative models, leading to the uncertainties shown in table 3: −ππ S-wave: the F-vector model is changed to use two other solutions of the K-matrix (from a total of three) determined from fits to scattering data [53] (a), (b). The slowly varying part of the nonresonant term of the P-vector is removed (c). −Kπ S-wave: the generalised LASS parametrisation used to describe the K∗ 0(1430)± resonance, is replaced by a relativistic Breit-Wigner propagator with parameters taken from ref. [54] (d). −ππ P-wave: the Gounaris-Sakurai propagator is replaced by a relativistic BreitWigner propagator [19,49] (e). −Kπ P-wave: the mass and width of the K∗(1680)−resonance are varied by their uncertainties from ref. [50] (f)−(i). −ππ D-wave: the mass and width of the f2(1270) resonance are varied by their uncertainties from ref. [24] (j)−(m). −Kπ D-wave: the mass and width of the K∗ 2(1430)±resonance are varied by their uncertainties from ref. [55] (n)−(q). −The radius of the Blatt-Weisskopf centrifugal barrier factors, rBW, is changed from 1.5 GeV−1to 0.0 GeV−1(r) and 3.0 GeV−1(s). −Two further resonances, K∗(1410)0and ρ(1450), parametrised with relativistic BreitWigner propagators, are included in the model [19,49] (t). −The Zemach formalism used for the angular distribution of the decay products is replaced by the helicity formalism [19,49] (u). It results in total systematic uncertainties arising from the choice of amplitude model of δx−= 8 ×10−3, δy−= 7 ×10−3, δx+= 10 ×10−3, δy+= 5 ×10−3. The different systematic uncertainties are combined, assuming that they are independent to obtain the total experimental uncertainties. Depending on the (x±, y±) parameters, the leading systematic uncertainties arise from the invariant mass fit, the description of the non-Dbackground and the fit biases. A larger data sample is expected to reduce all three of – 15 – JHEP08(2016)137 Description δx−δy−δx+δy+ (a) K-matrix 1st solution −2 0.921 (b) K-matrix 2nd solution 0.3 0.3 0.0−0.5 (c) Remove slowly varying −0.7 0.2 0.5 0.6 part in P-vector (d) Generalised LASS 2 3 −1 3 →relativistic Breit-Wigner (e) Gounaris-Sakurai 0.7 0.0−0.1 0.8 →relativistic Breit-Wigner (f) K∗(1680) m+δm −0.0 0.6 0.1 0.5 (g) m−δm −0.2−0.5 0.2−0.9 (h) Γ + δΓ−0.2 0.2 0.0−0.2 (i) Γ −δΓ 0.2−0.1 0.5−0.2 (j) f2(1270) m+δm −0.1 0.0 0.3−0.2 (k) m−δm −0.0 0.1 0.2−0.2 (l) Γ + δΓ−0.0 0.0 0.2−0.2 (m) Γ −δΓ−0.1 0.0 0.2−0.2 (n) K∗ 2(1430) m+δm 0.3 0.2 0.2−0.2 (o) m−δm −0.4−0.2 0.3−0.1 (p) Γ + δΓ−0.2 0.2 0.1−0.2 (q) Γ −δΓ 0.1−0.1 0.3−0.2 (r) rBW = 0.0 GeV−1−2 0.7−1−0.3 (s) rBW = 3.0 GeV−14−242 (t) Add K∗(1410) and ρ(1450) −0.2−0.2 0.3−0.3 (u) Helicity formalism −6 6 −8 2 Total model related 8 7 10 5 Table 3. Model related systematic uncertainties for each alternative model, in units of (10−3). The relative signs indicate full correlation or anti-correlation. these uncertainties. Whilst not intrinsically statistical in nature, the systematic uncertainty due to the description of the non-Dbackground is presently evaluated using a conservative approach due to lack of statistics. The total systematic uncertainties, including the model-related uncertainties, are significantly smaller than the statistical uncertainties. – 16 – JHEP08(2016)137 7 Determination of the parameters γ,rB0and δB0 To determine the physics parameters rB0,δB0and γfrom the fitted Cartesian observables z±, the relations x±=rB0cos(δB0±γ), y±=rB0sin(δB0±γ),(7.1) must be inverted. This is done using the GammaCombo package, originally developed for the frequentist combination of γmeasurements by the LHCb collaboration [7,56]. A global likelihood function is built, which gives the probability of observing a set of z±values given the true values (rB0, δB0, γ), L(x−, y−, x+, y+|rB0, δB0, γ).(7.2) All statistical and systematic uncertainties on z±are accounted for, as well as the statistical correlation between z±. Since the precision of the measurement is statistics dominated, correlations between the systematic uncertainties are ignored. Central values for (rB0, δB0, γ) are obtained by performing a scan of these parameters, to find the values that maximise L(xobs −, yobs −, xobs +, yobs +|rB0, δB0, γ), where zobs ±are the measured values of the Cartesian observables. Associated confidence intervals may be obtained either from a simple profilelikelihood method, or using the Feldman-Cousins approach [57] combined with a “plugin” method [58]. Confidence level curves for (rB0, δB0, γ) obtained using the latter method are shown in figures 6,7and 8. The measured values of z±are found to correspond to γ=80+21 −22◦, rB0= 0.39 ±0.13, δB0=197+24 −20◦. Intrinsic to the method used in this analysis [12], there is a two-fold ambiguity in the solution; the Standard Model solution (0 < γ < 180)◦is chosen. Two-dimensional confidence level curves obtained using the profile-likelihood method are shown in figures 9and 10. 8 Conclusion An amplitude analysis of B0→DK∗0decays, employing a model description of the D→ K0 Sπ+π−decay, has been performed using data corresponding to an integrated luminosity of 3 fb−1, recorded by LHCb at a centre-of-mass energy of 7 TeV in 2011 and 8 TeV in 2012. The measured values of the CP violation observables x±=rB0cos (δB0±γ) and y±=rB0sin (δB0±γ) are x−=−0.15 ±0.14 ±0.03 ±0.01, y−= 0.25 ±0.15 ±0.06 ±0.01, x+= 0.05 ±0.24 ±0.04 ±0.01, y+=−0.65 +0.24 −0.23 ±0.08 ±0.01, – 17 – JHEP08(2016)137 ]°[γ CL−1 0 0.2 0.4 0.6 0.8 1 50 100 150 22− +21 80 68.3% 95.5% LHCb Figure 6. Confidence level curve on γ, obtained using the “plugin” method [58]. 0 B r 0.2 0.4 0.6 0.8 CL−1 0 0.2 0.4 0.6 0.8 1 0.13− +0.13 0.39 68.3% 95.5% LHCb Figure 7. Confidence level curve on rB0, obtained using the “plugin” method [58]. where the first uncertainties are statistical, the second are systematic and the third are due to the choice of amplitude model used to describe the D→K0 Sπ+π−decay. These are the most precise measurements of these observables related to the neutral channel B0→DK∗0. They place constraints on the magnitude of the ratio of the interfering Bmeson decay amplitudes, the strong phase difference between them and the CKM angle γ, – 18 – JHEP08(2016)137 ]°[ 0 B δ CL−1 0 0.2 0.4 0.6 0.8 1 100 200 300 20− +24 197 68.3% 95.5% LHCb Figure 8. Confidence level curve on δB0, obtained using the “plugin” method [58]. Only the δB0solution corresponding to 0 < γ < 180◦is highlighted; the other maximum is due to the (δB0, γ)→(δB0+π, γ +π) ambiguity. ]° [γ 0 B r 0 50 100 150 0 0.2 0.4 0.6 0.8 1 LHCb contours hold 68%, 95% CL Figure 9. Two-dimensional confidence level curves in the (γ, rB0) plane, obtained using the profilelikelihood method. – 19 – JHEP08(2016)137 ]° [γ ]° [ 0 B δ 0 50 100 150 150 200 250 300 350 LHCb contours hold 68%, 95% CL Figure 10. Two-dimensional confidence level curves in the (γ, δB0) plane, obtained using the profile-likelihood method. giving the values γ=80+21 −22◦, rB0= 0.39 ±0.13, δB0=197+24 −20◦. Here, rB0and δB0are defined for a Kπ mass region of ±50 MeV around the K∗(892)0 mass and for an absolute value of the cosine of the K∗0decay angle greater than 0.4. These results are consistent with, and have lower total uncertainties than those reported in ref. [28], where a model independent analysis method is used. The two results are based on the same data set and cannot be combined. The consistency shows that at the current level of statistical precision the assumptions used to obtain the present result are justified. 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 and MPG (Germany); 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.). We acknowledge the computing resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF (The Netherlands), PIC – 20 – JHEP08(2016)137 (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID (Poland) and OSC (U.S.A.). We are indebted to the communities behind the multiple open source software packages on which we depend. Individual groups or members have received support from AvH Foundation (Germany), EPLANET, Marie Sk lodowska-Curie Actions and ERC (European Union), Conseil G´en´eral de Haute-Savoie, Labex ENIGMASS and OCEVU, R´egion Auvergne (France), RFBR and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust (United Kingdom). Open Access. 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