Measurement of CP violation in B0 → D∓π± decays

Journal of High Energy Physics, Jun 2018

Abstract A measurement of the CP asymmetries S f and \( {S}_{\overline{f}} \) in B0 → D∓π± decays is reported. The decays are reconstructed in a dataset collected with the LHCb experiment in proton-proton collisions at centre-of-mass energies of 7 and 8 TeV and corresponding to an integrated luminosity of 3.0 fb−1. The CP asymmetries are measured to be S f = 0.058 ± 0.020(stat) ± 0.011(syst) and \( {S}_{\overline{f}}=0.038\pm 0.020\left(\mathrm{stat}\right)\pm 0.007\left(\mathrm{syst}\right) \). These results are in agreement with, and more precise than, previous determinations. They are used to constrain angles of the unitarity triangle, | sin (2β + γ) | and γ, to intervals that are consistent with the current world-average values. Open image in new window

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Measurement of CP violation in B0 → D∓π± decays

Received: May B0 A measurement of the CP asymmetries Sf and Sf in B0 B physics; CKM angle gamma; CP violation; Flavor physics; Hadron-Hadron - HJEP06(218)4 The LHCb collaboration ! D decays is reported. The decays are reconstructed in a dataset collected with the LHCb experiment in proton-proton collisions at centre-of-mass energies of 7 and 8 TeV and corresponding to an integrated luminosity of 3.0 fb 1 . The CP asymmetries are measured to be Sf = 0:058 0:020(stat) 0:011(syst) and S f = 0:038 0:020(stat) 0:007(syst). These results are in agreement with, and more precise than, previous determinations. They are used to constrain angles of the unitarity triangle, j sin (2 + ) j and , to intervals that are consistent with the current world-average values. https://doi.org/10.1007/JHEP06(2018)084 1 Introduction 2 3 4 5 6 7 Decay-time-dependent CP asymmetries in B0 In the Standard Model, the decays B0 ! D + and B0 ! D+ b ! cud and b ! ucd quark transitions, respectively.1 The relative weak phase between proceed through the these two decay amplitudes is arg( VudVub=VcdVcb). The B0 meson can undergo a avour oscillation before the decay. The amplitude of the direct decay and that of a decay preceded by an oscillation have a total relative phase di erence of 2 + , where ments of CP violation in B0 ! D arg( VcdVcb=VtdVtb). The phases and are angles of the unitary triangle. Measuredecays provide information on these angles. analysing the decay rates as a function of the decay time of B0 mesons of known initial ! D decays can be measured by = jA(B0 ! D+ )=A(B0 ! D +) , j avour [1{3]. The ratio of the decay amplitudes, rD is around 2%, and limits the size of the CP asymmetries. Given its small value, this ratio needs to be determined from independent measurements, for example using the branching ratio of B0 ! Ds+ decays under the assumption of SU(3) avour symmetry [4, 5]. The decay rates of initially produced B0 mesons to the nal states f = D + and f = D+ as a function of the B0-meson decay time, t, are given by B0!f (t) / e B0!f (t) / e t [1 + Cf cos( m t) t 1 + Cf cos( m t) Sf sin( m t)] ; Sf sin( m t) ; (1.1) 1Inclusion of charge conjugate modes is implied unless explicitly stated. { 1 { where is the average B0 decay width and m is the B0{B0 oscillation frequency. For an initially produced B0 meson, the same equations hold except for a change of sign of the coe cients in front of the sine and cosine functions. No CP violation in the decay is assumed, i.e. only tree-level processes contribute to the decay amplitudes. It is also assumed that jq=pj = 1, where q and p are the complex coe cients de ning the heavy and light mass eigenstates of the B0 system, and = 0, where is the decay-width di erence between the two mass eigenstates. These assumptions follow from the known values of these quantities [6]. Under these assumptions, the coe cients of the cosine and sine terms of eq. (1.1) are given by Cf = Sf = Sf = 1 1 + rD2 r 2 D = Cf ; 1 + rD2 1 + rD2 2rD sin [ (2 + )] 2rD sin [ + (2 + )] ; ; (1.2) (1.3) (1.4) HJEP06(218)4 xing Cf = C amplitudes. Due to the small value of rD , terms of O(rD2 ) are neglected in this analysis, is the CP -conserving phase di erence between the b ! cud and b ! ucd decay A measurement of the CP asymmetries Sf and Sf can be interpreted in terms of 2 + by using the value of rD as input. Additionally, using the known value of [6], the angle can be evaluated. The determination of from tree-level decays is important because processes beyond the Standard Model are not expected to contribute. Constraints from the analysis of B0 ! D the ultimate sensitivity to this angle [7]. decays can be combined with other measurements to improve Measurements of Sf and S f using B0 ! D( ) and corresponding to an integrated luminosity of 3.0 fb 1 . This is the rst measurement decays reconstructed in a dataset collected with of Sf and Sf at a hadron collider. 2 Detector and simulation The LHCb detector [ 12, 13 ] is a single-arm forward spectrometer covering the pseudorapidity range 2{5, designed for the study of particles containing b or c quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the pp interaction region [14], 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 [15] placed downstream of the magnet. The tracking system provides a measurement of the momentum, p, of charged particles with a relative uncertainty that varies from 0:5% at low momentum to 1:0% at 200 GeV=c. The minimum distance of a track to a primary vertex (PV), the impact parameter (IP), is measured with a resolution of (15 + 29=pT) m, where pT is the component of the momentum { 2 { transverse to the beam, in GeV=c. Di erent types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors. Photons, electrons and hadrons are identi ed by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identi ed by a system composed of alternating layers of iron and multiwire proportional chambers. In the simulation, pp collisions are generated using Pythia [16, 17] with a speci c LHCb con guration [18]. Decays of hadronic particles are described by EvtGen [19], in which nal-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, 22] as described in ref. [23]. 3 Candidate selection The online event selection is performed by a trigger, which consists of a hardware stage, using information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. Events containing a muon with high pT or a hadron, photon or electron with high transverse energy in the calorimeters are considered at the hardware trigger stage. Events selected by the trigger using hadrons from the signal decay represent 70% of the sample used in this analysis, the rest being collected using trigger criteria satis ed by other properties of the event. The software trigger requires a two-, three-, or four-track secondary vertex with a signi cant displacement from the primary pp interaction vertices. At least one charged particle must have pT > 1:7 GeV=c and be inconsistent with originating from a PV. A multivariate algorithm is used for the identi cation of secondary vertices consistent with HJEP06(218)4 candidates is performed by reconstructing candidates from charged particle tracks with high momentum and the decay of a b hadron [24]. selection of B0 ! D transverse momentum, and originating from a common displaced vertex. Particle identi cation (PID) information is used to select kaon and pion candidates, and the invariant mass is required to be within 35 MeV=c2 of the known value of the mass [25]. These candidates are combined with a fourth charged particle, referred to as the companion, to form the B0 vertex, which must be displaced from any PV. The PV with respect to which the B0 candidate has the smallest I2P is considered as the production vertex. The I2P is de ned as the di erence in the vertex- t 2 of a given PV reconstructed with and without the B0 candidate. No PID requirement is applied to the companion track at this stage. The B0 candidates are required to match the secondary vertices found in the software trigger, to have a proper decay time larger than 0:2 ps, and to have a momentum vector aligned with the vector formed by joining the PV and the B0 decay vertex. The decay time is determined from a kinematic t in which the B0 candidate is constrained to originate from the PV to improve the decay-time resolution, while the B0-candidate mass is computed assigning the known value [25] to the mass of the D candidate to improve the mass resolution [26]. A combination of PID information and mass-range vetoes is used to { 3 { suppress to a negligible level cross-feed backgrounds such as Bs0 ! Ds (! K K+ ) +, due to the misidenti cation of protons and kaons as pions. 0 b ! c+(! pK +) and A boosted decision tree (BDT) [27, 28] is used to increase the signal purity by suppressing background from random combinations of particles. Candidates reconstructed from simulated B0 ! D data candidates with an invariant mass larger than 5:5 GeV=c2 are used as background. A set of 16 variables are combined into a single response, which is used to categorise the B0 candidates. The most relevant variables entering the BDT are the quality of the t of the B0 vertex and that of the kinematic t to calculate the B0 decay time, the transverse momentum of the D candidate, and the quality of the t of the companion-particle track. The requirement placed on the BDT response is chosen to maximise the expected sensitivity to Sf and Sf as derived from a set of simulated samples of signal plus background that are passed through the entire analysis. The data sample is further required to consist of B0 candidates whose initial avour has been determined by means of the avour tagging decays are used as signal in the training of the BDT, and algorithms described in section 5. 4 Sample composition The data sample after the selection is split into two disjoint subsets according to the PID information of the companion particle: a sample referred to as pion-like consisting mostly of genuine B0 mostly of genuine B0 mass distributions span the range 5090{6000 MeV=c2 and are shown in gure 1 with t projections overlaid. The mass distribution of B0 candidates in the pion-like sample features a peak at the known B0 mass with a width of about 20 MeV=c2, corresponding to B0 signal decays, and is modelled with the sum of a double-sided Hypatia function [29] and a Johnson SU function [30]. The combinatorial background is modelled using the sum of two exponential functions. At values lower than 5:2 GeV=c2, broad structures corresponding to partially reconstructed decays, such as B0 ! D +(! + 0), B ! D 0(! + ) 0 ) + where the additional pion is not reconstructed, are present; decays, due to kaon-to-pion misidenti cation, contaminating the left tail of the signal peak, are described with a double-sided Hypatia function with parameters determined 0K+) decays, where the charged pion is misidenti ed as a kaon and the from simulated decays. nents: the B0 signal region, with a long tail towards the high-mass region; the shape of this distribution, a double-sided Hypatia function, is taken from simulation. The yields of all components are oating parameters of the t. The yield of the cross-feed decays in the pion-like sample is constrained to that of the signal decays in the kaon-like sample using the kaon-to-pion misidenti cation probability and the kaon identi cation e ciency of the PID requirement on the companion particle. In a similar manner, the yield of the B0 the kaon-like sample is constrained to that of B0 mined from a large sample of D + ! D0(! K +) + decays in which the charged tracks are weighted in momentum and pseudorapidity to match those of the companion particle cross-feed decays in signal decays in the pion-like in B0 ! D decays [31]. An unbinned maximum-likelihood t to the B0-mass distribution of the pion-like sample is performed to determine sWeights [32], which are used to statistically subtract the background in the decay-time analysis of section 6. This unbinned t contains the same components as the binned t, but applied in a smaller mass window, 5220{5600 MeV=c2, to suppress the background contamination. All backgrounds entering this mass region are combined to form a single shape according to the fractions found in the previous t. The shape parameters of the signal and background components are also xed to the values A combination of tagging algorithms is used to determine the avour of the B0 candidates at production. Each algorithm provides a decision (tag), d, which determines the avour, { 5 { found in the preceding t. The B0 that of the background to be 34 400 300. and an estimate, , of the probability that the decision is incorrect (mistag probability). The decision takes the value of d = 1 for a candidate tagged as a B0, and d = 1 for a candidate tagged as B0. The mistag probability is de ned only between 0 and 0.5, since > 0:5 corresponds to an opposite tag with a mistag probability of (1 ). Two classes of avour tagging algorithms are used: opposite-side, OS, and same-side, SS, taggers. The OS tagger exploits the dominant production mechanism of b hadrons, the incoherent production of bb pairs, by identifying signatures of the b hadron produced together with the signal B0 meson. The time evolution of the signal B0 meson is independent from that of the accompanying b hadron. The OS tagger uses the charge of the electron or muon from semileptonic b-hadron decays, the charge of the kaon from a b ! c ! s decay chain, the charge of a reconstructed secondary charm hadron, and the charge of particles associated with a secondary vertex distinct from the signal decay; further details are given in refs. [33, 34]. The SS tagger selects pions and protons related to the hadronisation process of the signal B0 meson by means of BDT classi ers that determine the tag decision and mistag probability, as described in ref. [35]. Unlike ref. [35], where B0 f = 0, the BDT classi ers of the SS algorithm exploited in this analysis are trained on a control sample of avour-speci c B0 ! J= K 0 decays, whose distributions of pT, pseudorapidity, azimuthal angle of the B0 candidate, as well as number of tracks and PVs in the event, are weighted to match those of the B0 ! D signal decay. Around 37% of the B0 candidates are tagged by the OS tagger, 79% by the SS tagger, and 31% by both algorithms. About 15% of the B0 candidates are not tagged by either of the algorithms and are discarded. Each tagging decision is weighted by the estimated mistag probability , which dilutes the sensitivity to the CP asymmetry. To correct for potential biases in , a function ! ( ) is used to calibrate the mistag probability which provides an unbiased estimate of the mistag fraction ! (!), i.e. the fraction of incorrectly tagged candidates for a B0 (B0) meson, for any value of . Charged particles used for avour tagging, such as the kaons from the b ! c ! s decay chain exploited in the OS tagger, can have di erent interaction cross-sections with the detector material and therefore di erent reconstruction e ciencies. This can result in di erent tagging e ciencies and mistag probabilities for initial B0 and B0 mesons. Asymmetries in the tagging e ciency are found to be consistent with zero in simulation and data for both taggers and are therefore neglected in the baseline t, but considered as a source of systematic uncertainty. This is not the case for the asymmetries of the mistag probability, which can bias the determination of the CP asymmetries and must be corrected for. Therefore, the calibration functions depend on the initial avour of the B0 candidate: !( ) for d = +1 and !( ) for d = 1. They are expressed as generalised linear models (GLMs) of the form where pi and tion [36]. ( ) ! ( ) = g h( ) = g g 1( ) + X fi( ) ; (5.1) N i=1 pi + ( ) pi 2 pi are free parameters, fi are the basis functions , and g is the link func{ 6 { g(x) = 12 (1 + ex) 1, is used as the link function. To account for the tagging decision and mistag probability, the following substitutions occur in eq. (1.1): Sf ! ( Cf ! ( +)Sf ; +)Cf : (5.2) Similar equations hold for Sf and Cf . The calibration functions enter the coe cients along with the tagging e ciencies "OS and "SS of the OS and SS taggers, according to for candidates tagged by the OS algorithm and not by the SS algorithm (and vice-versa, exchanging the OS and SS indexes), and = 1 2 "OS 1 1 2 "OS 1 = " 1 4 "OS"SS 1 + X dj 1 j=OS;SS " 1 4 "OS"SS 1 + X dj 1 j=OS;SS "SS + dOS 1 "SS 2!( OS) 1 + "SS "SS + dOS 1 "SS 2!( OS) 1 + "SS ; (5.3) 2!( j ) + dOSdSS 1 2!( j ) + 2!( OS)!( SS) 2!( j ) + dOSdSS 1 2!( j ) + 2!( OS)!( SS) ; (5.4) # # for candidates tagged by both algorithms. The form of the coe cients and of the substitutions of eq. (5.2) is convenient to also account for other spurious asymmetries considered in section 6. The seven pairs of calibration parameters (pi; pi) are left free in the t from which the Sf and Sf observables are extracted. This is possible because the Cf and C f coe cients are xed parameters, so that the cosine terms of the decay rates permit the calibration parameters to be measured. This procedure has been validated with pseudoexperiments and possible deviations of Cf and Cf from unity are taken into account in the systematic uncertainties. To account for possible mismodelling of the calibration functions, systematic uncertainties are assigned to Sf and Sf . The calibration functions obtained in the data are shown in gure 2, where the measured mistag fraction is presented as a function of the predicted mistag probability of the tagger. Considering only candidates retained for the analysis, i.e. those with a avour tag, the statistical uncertainties of Sf and Sf are inversely proportional to p the average of the squared dilution of the signal, calculated as hD2i. Here, hD2i is 1 Ntag PiN=t1ag wi [1 2!( i)] , 2 where Ntag is the number of candidates, wi is the sW eight of the candidate i determined in the t of the sample composition, and Ntag = PiN=t1ag wi. The total dilution squared of the sample is found to be (6:554 0:017)%. Considering also the number of discarded candidates because no tagging decision is determined by either tagger, Nuntag and Nuntag = PiN=u1ntag wi, the tagging e ciency "tag Ntag=(Ntag + Nuntag) is found to be (85:23 0:05)%. Hence, { 7 { g0.45 a t is0.4 LHCb g0.45 a t is0.4 (left) OS and (right) SS taggers as determined in signal decays with the t described in section 6. The black histograms are the distributions of the mistag probabilities in arbitrary units. The shaded areas correspond to the 68% and 95% con dence-level regions of the calibration functions and do not include systematic uncertainties on the parameters. The calibration functions and the distributions of mistag probabilities are shown summing over candidates tagged as either B0 or B0. the e ective tagging e ciency of the initial sample is "taghD2i = (5:59 0:01)%. All quoted uncertainties are statistical only. The e ective tagging e ciency is similar to that of the measurement of CP violation in Bs0 ! Ds K decays [38]. 6 Decay-time t The CP asymmetries Sf and S f are determined from a multidimensional maximumlikelihood t to the unbinned distributions of the signal candidates weighted with the sWeights. The probability density function (PDF) describing the signal decay to a nal state F equal to f or f , at the reconstructed decay time t, and given the tags d~ = (dOS; dSS) and mistag probabilities ~ = ( OS; SS), is P (t; F; d~ j ~) / (t) P(t0; F; d~ j ~) R(t0 t) ; where P(t0; F; d~j~) is the function describing the distribution of true decay times t0, R(t0 t) is the decay time resolution, and (t) describes the decay-time-dependent e ciency of reconstructing and selecting the signal decays. The function P(t0; F; d~ j ~) corresponds to one of the decay rates of eq. (1.1), according to the nal state F , and with the substitutions A production asymmetry, AP, and a nal-state detection asymmetry, AD, must also (6.1) (6.2) of eq. (5.2) to include the avour tagging. be taken into account. These are de ned as AP = (B0) (B0) + (B0) ; (B0) AD = "(f ) "(f ) "(f ) + "(f ) ; where " is the decay-time-integrated e ciency in reconstructing and selecting the nal state f or f , and is the production cross-section of the given B0 or B0 meson. The asymmetry AP arises from the di erent production cross-sections of B0 and B0 mesons in proton-proton collisions and is measured to be at the percent level at LHC energies [39]. The detection asymmetry is also measured to be at the percent level and to be independent of the decay time. Therefore, eq. (5.2) is further modi ed as follows: ( ( +)Sf ! ( +)Cf ! ( AP AP +)(1 + AD)Sf ; +)(1 + AD)Cf ; where Cf is xed to 1. Similar equations hold for Sf and Cf ( xed to 1) with AD ! The decay-time resolution is determined from a sample of fake B0 candidates formed from a genuine D meson and a charged track originating from the same PV and consistent with being a pion of opposite charge. These candidates are subjected to a selection similar to that of the signal decays except for all decay-time biasing requirements, which are removed. The decay-time distribution of these candidates is therefore expected to peak at zero with a Gaussian shape given by the resolution function. Its width is determined in bins of the uncertainty on the decay time provided by the kinematic t of the decay chain. A second-order polynomial is used to describe the measured width as a function of the decay-time uncertainty. The average resolution of (54:9 0:4) fs is used as the width of the Gaussian resolution function R(t0 t). The e ciency function (t) is modelled by segments of cubic b-splines [40] with nine free parameters in total. The free parameters of the t are the Sf and Sf coe cients, the detection and production asymmetries AD and AP, the seven pairs of parameters (pi; pi) for the calibration functions of the OS and SS taggers, their e ciencies "OS and "SS, and the nine parameters of (t). The average B0 decay width, in eq. (1.1), is constrained by means of a Gaussian function whose mean is the world average value and whose width is the uncertainty [6]. Similarly, the B0{B0 mixing frequency, m, is constrained to the value measured in ref. [ 41 ]. The t determines Sf = 0:058 0:021 and S include the contributions from the constraints on the decay width and mixing frequency. When the t is repeated by xing m and to the central values used in the constraints, the central values for Sf and Sf do not change and their uncertainties decrease to 0:020. This is considered as the statistical uncertainty for both Sf and Sf . The statistical correlation between Sf and Sf is 60%. This correlation is introduced by the avour tagging and by the production asymmetry. The distribution of the decay time with the overlaid projection of the t is shown in gure 3. The values reported for Sf and Sf result in a signi cance of 2:7 for the CP -violation hypothesis, according to Wilks' theorem. Figure 4 reports the decay-time-dependent signalyield asymmetries between candidates tagged as B0 and B0, for the decays split according to the favoured (F) b ! cud and the suppressed (S) b ! ucd transitions (6.3) B0!f (t) B0!f (t) B0!f (t) B0!f (t) : { 9 { (6.4) (6.5) 2 4 6 8 10 12 Decay time [ps] curve is the projection of the signal PDF. The red dotted curve indicates the e ciency function " (t) in arbitrary units. LHCb LHCb LHCb Data Fit Efficiency suppressed decays. The signal-yield asymmetries are de ned in eq. (6.4) and eq. (6.5). The blue solid curve is the projection of the signal PDF, the red dotted curve indicates the projection of the t when CP conservation is imposed. The t projections are overlaid to the asymmetries of the data, along with the curves expected when Sf = Sf is imposed, i.e. in the hypothesis of no CP violation. Several consistency checks are made by performing the t on subsets of the data sample split according to di erent data-taking conditions, tagging algorithms, number of tracks in the event, and trigger requirements. These ts show good agreement with the result presented here. The stability of the result is also analysed in bins of the transverse momentum of the B0 meson and in bins of the di erence of pseudorapidity between the D candidate and the companion pion. The production asymmetry and the detection asymmetry are compared with results of independent LHCb measurements. The values found in this analysis are AP = ( 0:64 0:28)% and AD = (0:86 0:19)%, where the uncertainties are statistical, in agreement with those derived from ref. [39], when accounting for the di erent kinematics of the signals. The values of the avour-tagging parameters are also determined in control samples. The B+ ! D0 + decay is used for the OS tagger. As the quarks that accompany the b quark in B+ and B0 mesons di er, the SS calibration function is studied with B0 decays from a sample that is disjoint to that used in the training of the BDT classi ers. In both cases, distributions of pT and pseudorapidity of the B0 candidate, number of tracks and PVs in the event, and the composition of software trigger decisions are weighted to match those of the B0 ! D signal sample. In the case of the B+ ! D0 + mode, the azimuthal angle of the B0 is weighted to match that of the B0 decay-time distribution of the B+ and D0 mesons are also weighted to match those of the B0 and D mesons of the signal decays, while in the case of the B0 ! J= K 0 decay the signal sample. ! J= K 0 ! D The charged pion produced in B+ ! D0 + decays directly identi es the B+ avour at production. Therefore, the calibration of the OS tagger is achieved by counting the number of correctly and incorrectly tagged signal candidates. In contrast, the SS tagger calibration with B0 ! J= K 0 decays requires the B0{B0 avour oscillations to be resolved by using the decay time as an additional observable, since the amplitude of the observed oscillation is related to the mistag fraction [35]. The values of the calibration parameters found in the control decays are in agreement with those determined in the t to the signal, with the largest deviation being of 2 standard deviations for two of the pi parameters. 7 Systematic uncertainties Systematic uncertainties due to external measurements used in the t are accounted for through Gaussian constraints in the likelihood function. These parameters are the mixing frequency, m, and the B0 decay width, . In order to disentangle these contributions from the statistical uncertainty of Sf and Sf , the t is repeated by xing m and to the central values used in the constraints. The systematic uncertainty due to the constraint on is found to be negligible, and that due to m is 0:0073 and 0:0061 for Sf and Sf , respectively. These are the largest systematic uncertainties of Sf and Sf and are found to be fully anticorrelated. The correlation of m with Sf is 34% and that with Sf is 29%. Validation of the entire analysis using ensembles of simulated signal candidates shows that the values of Sf and S f are biased up to 0:0068 and 0:0018, respectively. The size of these potential biases are small and so are taken as a systematic uncertainty. The correlation of these systematic uncertainties is 40%. Variation of the t to the D + invariant-mass distribution used to calculate the sWeights for the background subtraction leads to systematic uncertainties on Sf and S f of 0:0042 and 0:0023, respectively. Their correlation is 70%. The remaining systematic uncertainties are much smaller than those reported above. Hence, the correlation between the systematic uncertainty of Sf and Sf for the sources that follow are neglected. The systematic uncertainties associated with the PID e ciencies used in the t to the D + invariant mass are also propagated by means of Gaussian constraints. These uncertainties take into account the size of the calibration samples and the dependence of the results on the binning scheme adopted for weighting the kinematic distributions of the particles of the control decays to match those of the companion tracks. They contribute an uncertainty of 0:0008 to both Sf and Sf . To test the impact of the choice of the calibration models, pseudoexperiments are HJEP06(218)4 The other sources of systematic uncertainty are calculated by means of pseudoexperiments, where samples of the same size as the data are generated by sampling the PDF with parameters xed to the value found in data. In the generation of the pseudoexperiments the PDF is modi ed to consider alternative models according to the source of systematic uncertainty under investigation. The generated sample is then t with the nominal model. For each parameter, the mean of the distribution of the residuals is considered, (Sigen Si t), from 1000 pseudoexperiments as the systematic uncertainty. If the mean di ers from zero by less than one standard deviation, the error on the mean is taken as the systematic uncertainty. generated using for the SS calibration the nominal model, while for the OS the degree of the polynomial used in the model is reduced by one unit compared to the nominal model. In the t for both taggers the degrees of the calibration models are increased by one degree compared to that used to generate the pseudoexperiments. The systematic uncertainties are determined to be 0:0008 and 0:0016 for Sf and Sf , respectively. Assuming values for the avour-tagging e ciency asymmetries di erent from zero, based on what is found in simulation, leads to systematic uncertainties of 0:0012 and 0:0015 for Sf and Sf , respectively. A di erent decay-time acceptance model is used in generation by considering new boundaries of the subranges of the spline functions. This results in a systematic uncertainty of 0:0007 for both Sf and Sf . Mismodelling of the decay-time resolution is also considered by increasing and decreasing the nominal resolution by 20 fs. The largest residuals are considered as the systematic uncertainties, and are 0:0012 and 0:0008 for Sf and Sf , respectively. A value for Cf = C f di erent from 1, based on the value of rD from refs. [4, 5] is assumed, resulting in a variation of 0:0006 for both Sf and Sf . By assigning to a value di erent from zero and equal to the world-average value plus its uncertainty [6] leads to a systematic uncertainty of 0:0007 on both Sf and Sf . The sources of systematic uncertainties are summarised in table 1. They total 0:011 and 0:007 for Sf and Sf , respectively, with a correlation of 41%. 8 Interpretation of the CP asymmetries The values of Sf and Sf are interpreted in terms of the angle 2 + , the ratio of amplitudes rD , and the strong phase , using the statistical method described in ref. [7]. By taking external measurements of rD , con dence intervals for j sin(2 are derived. The ratio rD is calculated from the branching fraction of B0 decays, assuming SU(3) symmetry, following the same relation used in refs. [4, 5]: + )j and 0:00032 is the tangent of the Cabibbo angle from 0:003 is the ratio of decay constants [43{45], and (8.1) avour-tagging e ciency asymmetries Source uncertainty of m t biases background subtraction PID e ciencies avour-tagging models (t) model assumption on decay-time resolution assumption on C total statistical uncertainty 1 L C 1−0.8 0.6 0.4 0.2 0 0 68.3% 95.5% 0.2 0:0073 0:0068 0:0042 0:0008 0:0011 0:0012 0:0007 0:0007 0:0012 0:0006 f 0:0061 0:0018 0:0023 0:0008 0:0015 0:0015 0:0007 0:0007 0:0008 0:0006 sum in quadrature of the individual contributions. branching fractions taken from ref. [25]. We determine rD where the second uncertainty accounts for possible nonfactorizable SU(3)-breaking e ects, considered to be 20% of the value of rD as suggested in ref. [46]. In addition, using the known value of = (22:2 0:7) [6], con dence intervals for are determined. The con dence intervals are j sin(2 + )j 2 [0:77; 1:0] ; 2 [5; 86] [ [185; 266] ; 2 [ 41; 41] [ [140; 220] ; all at the 68% con dence level (CL). The uncertainties on rD and have a negligible impact on these values. The intervals are illustrated in gures 5 and 6. C 1−0.8 200 100 0 0 100 200 300 100 200 and (right) con dence regions for and . The con dence regions hold the 39% and 87% CL. Points denote the preferred values. 9 A measurement of the CP asymmetries Sf and Sf in the decay B0 ! D The decay candidates are reconstructed in a data set collected with the LHCb experiment is reported. at centre-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3:0 fb 1. We measure with a correlation of 60% ( 41%) between the statistical (systematic) uncertainties. These values are in agreement with, and more precise than, measurements from the Belle and BaBar collaborations [9, 10]. This measurement, in combination with the external inputs to be in the interval [5; 86] [ [185; 266] at the of rD and , constrains the CKM angle 68% con dence level. 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 sta at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES and FASO (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 (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 Sklodowska-Curie Actions and ERC (European Union), ANR, Labex P2IO and OCEVU, and Region Auvergne-Rh^one-Alpes (France), Key Research Program of Frontier Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China), RFBR, RSF and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, the Royal Society, the English-Speaking Union and the Leverhulme Trust (United Kingdom). Open Access. 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Physikalisches Institut, RWTH Aachen University, Aachen, Germany 10 Fakultat Physik, Technische Universitat Dortmund, Dortmund, Germany 11 Max-Planck-Institut fur Kernphysik (MPIK), Heidelberg, Germany 12 Physikalisches Institut, Ruprecht-Karls-Universitat Heidelberg, Heidelberg, Germany 13 School of Physics, University College Dublin, Dublin, Ireland HJEP06(218)4 15 INFN Sezione di Bologna, Bologna, Italy 16 INFN Sezione di Ferrara, Ferrara, Italy 17 INFN Sezione di Firenze, Firenze, Italy 18 INFN Laboratori Nazionali di Frascati, Frascati, Italy 19 INFN Sezione di Genova, Genova, Italy 20 INFN Sezione di Milano-Bicocca, Milano, Italy 21 INFN Sezione di Milano, Milano, Italy 22 INFN Sezione di Cagliari, Monserrato, Italy 23 INFN Sezione di Padova, Padova, Italy 24 INFN Sezione di Pisa, Pisa, Italy 25 INFN Sezione di Roma Tor Vergata, Roma, Italy 26 INFN Sezione di Roma La Sapienza, Roma, Italy Krakow, Poland Romania 27 Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Krakow, Poland 28 AGH | University of Science and Technology, Faculty of Physics and Applied Computer Science, 29 National Center for Nuclear Research (NCBJ), Warsaw, Poland 30 Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, 31 Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia 32 Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia 33 Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia 34 Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS), Moscow, Russia 35 Yandex School of Data Analysis, Moscow, Russia 36 Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia 37 Institute for High Energy Physics (IHEP), Protvino, Russia 38 ICCUB, Universitat de Barcelona, Barcelona, Spain 39 Instituto Galego de F sica de Altas Enerx as (IGFAE), Universidade de Santiago de Compostela, Santiago de Compostela, Spain 40 European Organization for Nuclear Research (CERN), Geneva, Switzerland 41 Institute of Physics, Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, Switzerland 42 Physik-Institut, Universitat Zurich, Zurich, Switzerland 43 Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands 44 Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, The Netherlands 45 NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine 46 Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine 47 University of Birmingham, Birmingham, United Kingdom 48 H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom 49 Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom 50 Department of Physics, University of Warwick, Coventry, United Kingdom 51 STFC Rutherford Appleton Laboratory, Didcot, United Kingdom 52 School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom 53 School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom 54 Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom 55 Imperial College London, London, United Kingdom 56 School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom 57 Department of Physics, University of Oxford, Oxford, United Kingdom 58 Massachusetts Institute of Technology, Cambridge, MA, United States 59 University of Cincinnati, Cincinnati, OH, United States 60 University of Maryland, College Park, MD, United States 61 Syracuse University, Syracuse, NY, United States 64 School of Physics and Technology, Wuhan University, Wuhan, China, associated to3 65 Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, China, 66 Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia, associated to8 67 Institut fur Physik, Universitat Rostock, Rostock, Germany, associated to12 68 National Research Centre Kurchatov Institute, Moscow, Russia, associated to32 69 National University of Science and Technology \MISIS", Moscow, Russia, associated to32 70 National Research Tomsk Polytechnic University, Tomsk, Russia, associated to32 71 Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia - CSIC, Valencia, Spain, 72 Van Swinderen Institute, University of Groningen, Groningen, The Netherlands, associated to43 73 Los Alamos National Laboratory (LANL), Los Alamos, United States, associated to61 a Universidade Federal do Tria^ngulo Mineiro (UFTM), Uberaba-MG, Brazil b Laboratoire Leprince-Ringuet, Palaiseau, France c P.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia l AGH - University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Krakow, Poland m LIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain v MSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines w Novosibirsk State University, Novosibirsk, Russia x National Research University Higher School of Economics, Moscow, Russia y Escuela Agr cola Panamericana, San Antonio de Oriente, Honduras z School of Physics and Information Technology, Shaanxi Normal University (SNNU), Xi'an, China aa Physics and Micro Electronic College, Hunan University, Changsha City, China y Deceased [13] LHCb collaboration, LHCb detector performance , Int. J. Mod. Phys. A 30 ( 2015 ) 1530022 [14] R. Aaij et al., Performance of the LHCb vertex locator , 2014 JINST 9 P09007 [16] T. Sj ostrand, S. Mrenna and P.Z. Skands , PYTHIA 6 . 4 physics and manual , JHEP 05 [17] T. 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The LHCb collaboration, R. Aaij, B. Adeva, M. Adinolfi, Z. Ajaltouni, S. Akar, P. Albicocco, J. Albrecht, F. Alessio, M. Alexander, A. Alfonso Albero, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves, S. Amato, S. Amerio, Y. Amhis, L. An, L. Anderlini, G. Andreassi, M. Andreotti, J. E. Andrews, R. B. Appleby, F. Archilli, P. d’Argent, J. Arnau Romeu, A. Artamonov, M. Artuso, E. Aslanides, M. Atzeni, G. Auriemma, S. Bachmann, J. J. Back, S. Baker, V. Balagura, W. Baldini, A. Baranov, R. J. Barlow, S. Barsuk, W. Barter, F. Baryshnikov, V. Batozskaya, V. Battista, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, A. Beiter, L. J. Bel, N. Beliy, V. Bellee, N. Belloli, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, S. Beranek, A. Berezhnoy, R. Bernet, D. Berninghoff, E. Bertholet, A. Bertolin, C. Betancourt, F. Betti, M. O. Bettler, M. van Beuzekom, Ia. Bezshyiko, S. Bifani, P. Billoir, A. Birnkraut, A. Bizzeti, M. Bjørn, T. Blake, F. Blanc, S. Blusk, V. Bocci, O. Boente Garcia, T. Boettcher, A. Bondar, N. Bondar, S. Borghi, M. Borisyak, M. Borsato, F. Bossu, M. Boubdir, T. J. V. Bowcock, E. Bowen, C. Bozzi, S. Braun, M. Brodski, J. Brodzicka, D. Brundu, E. Buchanan, C. Burr, A. Bursche, J. Buytaert, W. Byczynski, S. Cadeddu, H. Cai, R. Calabrese, R. Calladine, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. H. Campora Perez, L. Capriotti, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, P. Carniti, L. Carson, K. Carvalho Akiba, G. Casse, L. Cassina, M. Cattaneo, G. Cavallero, R. Cenci, D. Chamont, M. G. Chapman, M. Charles, Ph. Charpentier, G. Chatzikonstantinidis, M. Chefdeville, S. Chen, S.-G. Chitic, V. Chobanova, M. Chrzaszcz, A. Chubykin, P. Ciambrone, X. Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J. Closier, V. Coco, J. Cogan, E. Cogneras, V. Cogoni, L. Cojocariu, P. Collins, T. Colombo, A. Comerma-Montells, A. Contu, G. Coombs, S. Coquereau, G. Corti, M. Corvo, C. M. Costa Sobral, B. Couturier, G. A. Cowan, D. C. Craik, A. Crocombe, M. Cruz Torres, R. Currie, C. D’Ambrosio, F. Da Cunha Marinho, C. L. Da Silva, E. Dall’Occo, J. Dalseno, A. Danilina, A. Davis, O. De Aguiar Francisco, K. De Bruyn, S. De Capua, M. De Cian, J. M. De Miranda, L. De Paula, M. De Serio, P. De Simone, C. T. Dean, D. Decamp, L. Del Buono, B. Delaney, H.-P. Dembinski, M. Demmer, A. Dendek, D. Derkach, O. Deschamps, F. Dettori, B. Dey, A. Di Canto, P. Di Nezza, S. Didenko, H. Dijkstra, F. Dordei, M. Dorigo, A. Dosil Suárez, L. Douglas, A. Dovbnya, K. Dreimanis, L. Dufour, G. Dujany, P. Durante, J. M. Durham, D. Dutta, R. Dzhelyadin, M. Dziewiecki, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund, S. Ely, A. Ene, S. Escher, S. Esen, H. M. Evans, T. Evans, A. Falabella, N. Farley, S. Farry, D. Fazzini, L. Federici, G. Fernandez, P. Fernandez Declara, A. Fernandez Prieto, F. Ferrari, L. Ferreira Lopes, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, R. A. Fini, M. Fiorini, M. Firlej, C. Fitzpatrick, T. Fiutowski, F. Fleuret, M. Fontana, F. Fontanelli, R. Forty, V. Franco Lima, M. Frank, C. Frei, J. Fu, W. Funk, C. Färber, E. Gabriel, A. Gallas Torreira, D. Galli, S. Gallorini, S. Gambetta, M. Gandelman, P. Gandini, Y. Gao, L. M. Garcia Martin, B. Garcia Plana, J. García Pardiñas, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar, L. Gavardi, G. Gazzoni, D. Gerick, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, S. Gianí, V. Gibson, O. G. Girard, L. Giubega, K. Gizdov, V. V. Gligorov, D. Golubkov, A. Golutvin, A. Gomes, I. V. Gorelov, C. Gotti, E. Govorkova, J. P. Grabowski, R. Graciani Diaz, L. A. Granado Cardoso, E. Graugés, E. Graverini, G. Graziani, A. Grecu, R. Greim, P. Griffith, L. Grillo, L. Gruber, B. R. Gruberg Cazon, O. Grünberg, E. Gushchin, Yu. Guz, T. Gys, C. Göbel, T. Hadavizadeh, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C. Haines, B. Hamilton, X. Han, T. H. Hancock, S. Hansmann-Menzemer, N. Harnew, S. T. Harnew, C. Hasse, M. Hatch, J. He, M. Hecker, K. Heinicke, A. Heister, K. Hennessy, L. Henry, E. van Herwijnen, M. Heß, A. Hicheur, D. Hill, P. H. Hopchev, W. Hu, W. Huang, Z. C. Huard, W. Hulsbergen, T. Humair, M. Hushchyn, D. Hutchcroft, P. Ibis, M. Idzik, P. Ilten, K. Ivshin, R. Jacobsson, J. Jalocha, E. Jans, A. Jawahery, F. Jiang, M. John, D. Johnson, C. R. Jones, C. Joram, B. Jost, N. Jurik, S. Kandybei, M. Karacson, J. M. Kariuki, S. Karodia, N. Kazeev, M. Kecke, F. Keizer, M. Kelsey, M. Kenzie, T. Ketel, E. Khairullin, B. Khanji, C. Khurewathanakul, K. E. Kim, T. Kirn, S. Klaver, K. Klimaszewski, T. Klimkovich, S. Koliiev, M. Kolpin, R. Kopecna, P. Koppenburg, S. Kotriakhova, M. Kozeiha, L. Kravchuk, M. Kreps, F. Kress, P. Krokovny, W. Krupa, W. Krzemien, W. Kucewicz, M. Kucharczyk, V. Kudryavtsev, A. K. Kuonen, T. Kvaratskheliya, D. Lacarrere, G. Lafferty, A. Lai, G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, A. Leflat, J. Lefrançois, R. Lefèvre, F. Lemaitre, O. Leroy. Measurement of CP violation in B0 → D∓π± decays, Journal of High Energy Physics, 2018, 84, DOI: 10.1007/JHEP06(2018)084