Measurement of CP observables in B ± → D ( ∗ ) K ± and B ± →

Measurements of CP observables in B → D(∗)K± and B → D(∗)π± decays are presented, where D(∗) indicates a neutral D or D meson that is an admixture of D(∗)0 and D̄(∗)0 states. Decays of the D meson to the Dπ0 and Dγ final states are partially reconstructed without inclusion of the neutral pion or photon, resulting in distinctive shapes in the B candidate invariant mass distribution. Decays of the D meson are fully reconstructed in the Kπ, K+K− and π+π− final states. The analysis uses a sample of charged B mesons produced in pp collisions collected by the LHCb experiment, corresponding to an integrated luminosity of 2.0, 1.0 and 2.0 fb taken at centre-of-mass energies of √ s = 7, 8 and 13TeV, respectively. The study of B → DK and B → Dπ decays using a partial reconstruction method is the first of its kind, while the measurement of B → DK and B → Dπ decays is an update of previous LHCb measurements. The B → DK results are the most precise to date. Published in Phys. Lett. B777 (2018) 16−30 c © CERN on behalf of the LHCb collaboration, licence CC-BY-4.0. Authors are listed at the end of this Letter.


Introduction
Overconstraining the Unitarity Triangle (UT) derived from the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix is central to testing the Standard Model (SM) description of CP violation [1]. The least well known angle of the UT is γ ≡ arg(−V ud V * ub /V cd V * cb ), which has been determined with a precision of about 7 • from a combination of measurements [2, 3] (cf . 3 • and < 1 • on the angles α and β [4,5]). Among the UT angles, γ is unique in that it does not depend on any top-quark coupling, and can thus be measured in decays that are dominated by tree-level contributions. In such decays, the interpretation of physical observables (rates and CP asymmetries) in terms of the underlying UT parameters is subject to small theoretical uncertainties [6]. Any disagreement between these measurements of γ and the value inferred from global CKM fits performed without any γ information would invalidate the SM description of CP violation.
The most powerful method for determining γ in decays dominated by tree-level contributions is through the measurement of relative partial widths in B − → D K − decays, where D represents an admixture of the D 0 and D 0 states. 1 The amplitude for the B − → D 0 K − decay, which at the quark level proceeds via a b → cūs transition, is proportional to V cb . The corresponding amplitude for the B − → D 0 K − decay, which proceeds via a b → ucs 1 The inclusion of charge-conjugate processes is implied except in any discussion of asymmetries.
transition, is proportional to V ub . By studying hadronic D decays accessible to both D 0 and D 0 mesons, phase information can be extracted from the interference between these two amplitudes. The degree of the resulting CP violation is governed by the size of ≈ 0.10 [3] in B − → D K − decays allows the determination of the relative phase of the two interfering amplitudes. This relative phase has both CP -violating (γ ) and CP -conserving (δ D K B ) contributions; a measurement of the decay rates for both B + and B − gives sensitivity to γ . Similar interference effects also occur in B − → Dπ − decays, albeit with lower sensitivity to the phases. The reduced sensitivity is the result of additional Cabibbo suppression factors, which decrease the ratio of amplitudes relative to B − → D K − decays by around a factor of 20.
The B − → D * K − decay, in which the vector D * meson 2 decays to either the Dπ 0 or Dγ final state, also exhibits CP -violating effects when hadronic D decays accessible to both D 0 and D 0 mesons are studied. In this decay, the exact strong phase difference of π between D * → Dπ 0 and D * → Dγ decays can be exploited to measure CP observables for states with opposite CP eigenvalues [7]. The degree of CP violation observed in B − → D * K − decays is set by the magnitude of the ratio r D * K B ≈ 0.12 [3], and 2  The study of B − → D ( * ) K − decays for measurements of γ was first suggested for CP eigenstates of the D decay, for example the CP -even D → K + K − and D → π + π − decays, labelled here as GLW modes [8,9]. In this work, the GLW decays D → K + K − and D → π + π − are considered along with the Cabibbo-favoured D → K − π + decay, where the latter decay is used for normalisation purposes and to define shape parameters in the fit to data (see Sec. 4).
2 and h − can each represent either a charged kaon or pion and the D-meson decay products are denoted inside square brackets, have been studied at the B factories [10,11] and at LHCb [12]. This Letter reports updated and improved results using a sample of charged B mesons from pp collisions collected by the LHCb experiment, corresponding to an integrated luminosity of 2. ] D h − modes gains approximately a factor of two in signal yield relative to Ref. [12].
where the D * -meson decay products are denoted in parentheses, have also been studied by the B factories [13,14], while this work presents the first analysis of these decays at LHCb.
The small D * − D mass difference and the conservation of angular momentum in D * → Dπ 0 and D * → Dγ decays results in distinctive signatures for the B − → D * K − signal in the D K − invariant mass, allowing yields to be obtained with a partial reconstruction technique. Since the reconstruction efficiency for low momentum neutral pions and photons is relatively low in LHCb [15], the partial reconstruction method provides significantly larger yields compared to full reconstruction, but the statistical sensitivity per signal decay is reduced due to the need to distinguish several signal and background components in the same region of D K − invariant mass.
A total of 19 measurements of CP observables are reported, eight of which correspond to the fully reconstructed In the latter case, the neutral pion or photon produced in the decay of the D * vector meson is not reconstructed in the final state. A summary of all measured CP observables is provided in Table 1. In addition, the branching fractions B(B − → D * 0 π − ) and B(D * 0 → D 0 π 0 ), along with the ratio of branching fractions All of the charge asymmetry measurements are affected by an asymmetry in the B ± production cross-section and any charge asymmetry arising from the LHCb detector efficiency, together denoted as σ . This effective production asymmetry, defined as , is measured from the charge asymmetry of the most abundant B − → [K − π + ] D π − mode. In this mode, the CP asymmetry is fixed to have the value A K π π = (+0.09 ± 0.05)%, which is determined using knowledge of γ and r D K B from Ref. [2], where A K π π was not used as an input observable. This uncertainty is smaller than that of previous measurements of the B ± production asymmetry measured at √ s = 7 and 8 TeV [16,17], and reduces the systematic uncertainties of the asymmetries listed in Table 1. The value of A eff B ± is applied as a correction to all other charge asymmetries. The remaining detection asymmetries, most notably due to different numbers of K + and K − mesons appearing in each final state, are corrected for using independent calibration Table 1 Summary table of the 19 measured CP observables, defined in terms of B meson decay widths. Where indicated, CP represents an average of the D → K + K − and D → π + π − modes. The R observables represent partial width ratios and double ratios, where R K π ,π 0 /γ K /π is an average over the D * → Dπ 0 and D * → Dγ modes.
The A observables represent CP asymmetries.

Observable
Definition samples. These corrections transform the measured charge asymmetries into CP asymmetries.

Detector and simulation
The LHCb detector [15,18] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 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, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of silicon-strip detectors and straw drift tubes placed downstream of the magnet.
The tracking system provides a measurement of 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/p T ) μm, where p T is the component of the momentum transverse to the beam, in GeV/c. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors (RICH) [19,20]. 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 p T > 500 (300) MeV are reconstructed for 2011 (2012) data, and p T > 70 MeV for 2015 and 2016 data. At the hardware trigger stage, events are required to contain a muon with high p T or a hadron, photon or electron with high transverse energy in the calorimeters. For hadrons, the transverse energy threshold varied between 3 and 4 GeV between 2011 and 2016. The software trigger requires a two-, three-or four-track secondary vertex with significant displacement from all primary pp interaction vertices. A multivariate algorithm [21,22] is used for the identification of secondary vertices consistent with the decay of a b hadron.
In the simulation, pp collisions are generated using Pythia8 [23] with a specific LHCb configuration [24]. Decays of hadronic particles 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] as described in Ref. [28].

Event selection
After reconstruction of the D-meson candidate from two oppositely charged particles, the same event selection is applied to all B − → D ( * ) h − channels. Since the neutral pion or photon from the vector D * decay is not reconstructed, partially reconstructed B − → D * h − decays and fully reconstructed B − → Dh − decays contain the same reconstructed particles, and thus appear in the same sample. These decays are distinguished according to the reconstructed invariant mass m(Dh), as described in Sec. 4.
The reconstructed D-meson candidate mass is required to be within ±25 MeV/c 2 of the known D 0 mass [29], which corresponds to approximately three times the mass resolution. The kaon or pion originating from the B − decay, subsequently referred to as the companion particle, is required to have p T in the range 0.5-10 GeV/c and p in the range 5-100 GeV/c. These requirements ensure that the track is within the kinematic coverage of the RICH detectors, which are used to provide particle identification (PID) information. Details of the PID calibration procedure are given in Sec. 4. A kinematic fit is performed to each decay chain, with vertex constraints applied to both the B − and D decay products, and the D candidate constrained to its known mass [30]. Events are required to have been triggered by either the decay products of the signal candidate, or by particles produced elsewhere in the pp collision. Each B − candidate is associated to the primary vertex (PV) to which it has the smallest χ 2 IP , which is quantified as the difference in the vertex fit χ 2 of a given PV reconstructed with and without the considered particle. The B − meson candidates with invariant masses in the interval 4900-5900 MeV/c 2 are retained. This range is wider than that considered in Ref. [12], in order to include the partially re- A pair of boosted decision tree (BDT) classifiers, implementing the gradient boost algorithm [31], is employed to achieve further background suppression. The BDTs are trained using simulated B − → [K − π + ] D K − decays and a background sample of K − π + K − combinations in data with invariant mass in the range 5900-7200 MeV/c 2 ; the training was also repeated The input to both BDTs is a set of features that characterise the signal decay. These features can be divided into two categories: (1) properties of any particle and (2) properties of composite particles only (the D and B − candidates). Specifically: 1. p, p T and χ 2 IP ; 2. decay time, flight distance, decay vertex quality, radial distance between the decay vertex and the PV, and the angle between the particle's momentum vector and the line connecting the production and decay vertices.
In addition, a feature that estimates the imbalance of p T around the B − candidate momentum vector is also used in both BDTs. It is defined as where the sum is taken over tracks inconsistent with originating from the PV which lie within a cone around the B − candidate, excluding tracks used to make the signal candidate. The cone is defined by a circle with a radius of 1.5 units in the plane of pseudorapidity and azimuthal angle (expressed in radians). Including the I p T feature in the BDT training gives preference to B − candidates that are either isolated from the rest of the event, or consistent with a recoil against another b hadron. Since no PID information is used in the BDT classifier, the effi- insignificant variations arising from small differences in the decay kinematics. The criteria applied to the two BDT responses are optimised by minimising the expected statistical uncertainty on R CP ,π 0 and R CP ,γ , as measured with the method described below. The purity of the sample is further improved by requiring that all kaons and pions in the D decay are positively identified by the RICH. This PID selection used to separate the Dπ and D K samples has an efficiency of about 85% per final-state particle. Peaking background contributions from charmless decays that result in the same final state as the signal are suppressed by requiring that the flight distance of the D candidate from the B − decay vertex is larger than two times its uncertainty. After the above selections, multiple candidates exist in 0.1% of the events in the sample. When more than one candidate is selected, only the candidate with the best B − vertex quality is retained. The overall effect of the multiple-candidate selection is negligible.

Fit to data
The values of the CP observables are determined using a binned extended maximum likelihood fit to the data. Distinguishing between B + and B − candidates, companion particle hypotheses, and the three D decay product final states, yields 12 independent samples which are fitted simultaneously. The total probability density function (PDF) is built from six signal functions, one for each of the In addition, there are functions which describe the combinatorial background components, background contributions from B decays to charmless final states and background contributions from partially reconstructed decays. All functions are identical for B + and B − decays.
which has a peak position μ and core width σ c , where α L (m < μ) and α R (m > μ) parameterise the tails. The μ and α parameters are shared across all samples but the core width parameter varies independently for each D final state. The additional Gaussian function, with a small fractional contribution, is necessary to model satisfactorily the tails of the peak.
The B − → Dπ − decays misidentified as B − → D K − are displaced to higher mass in the D K − subsamples. These misidentified candidates are modelled by the sum of two Gaussian functions with a common mean, modified to include tail components as in Eq. (2). The mean, widths and α R are left to vary freely, while α L is fixed to the value found in simulation.

B − → D K −
In the D ( * )0 K − samples, Eq. Misidentified B − → D K − candidates appearing in the D ( * )0 π − subsamples are described by a fixed shape obtained from simulation, which is later varied to determine a systematic uncertainty associated with this choice.
In partially reconstructed decays involving a vector meson, the Dh − invariant mass distribution depends upon the spin and mass of the missing particle. In the case of B − → (Dπ 0 ) D * π − decays, the missing neutral pion has spin-parity 0 − . The distribution is parameterised by an upward-open parabola, whose range is defined by the kinematic endpoints of the decay. It is convolved with a Gaussian resolution function, resulting in The resulting distribution has a characteristic double-peaked shape, visible in Figs   Partially reconstructed B − → (Dπ 0 ) D * π − decays, where the companion pion is misidentified as a kaon, are parameterised with a semiempirical function, formed from the sum of Gaussian and error functions. The parameters of this function are fixed to the values found in fits to simulated events, and are varied to determine the associated systematic uncertainty. (3) is also used to describe partially reconstructed samples is related to the Dπ − width by a freely varying ratio r σ , which is shared across all functions describing partially reconstructed decays. All other shape parameters are shared with the Partially reconstructed B − → (Dπ 0 ) D * K − decays, where the companion kaon is misidentified as a pion, are parameterised with a semiempirical function, formed from the sum of Gaussian and error functions. The parameters of this function are fixed to the values found in fits to simulated events, and are varied to determine the associated systematic uncertainty.

B − → (Dγ ) D * π −
Partially reconstructed B − → (Dγ ) D * π − decays involve a missing particle of zero mass and spin-parity 1 − . The Dπ − invariant mass distribution is described by a parabola exhibiting a maximum, convolved with a Gaussian resolution function. The functional form of this component is

Combinatorial background
An exponential function is used to describe the combinatorial background. The exponential function is widely used to describe combinatorial backgrounds to B − decays in LHCb, and has been validated for numerous different decay modes. Independent and freely varying exponential parameters and yields are used to model this component in each subsample, with the constraint that the B + and B − yields are required to be equal. The systematic uncertainty associated with this constraint is negligible. Partially reconstructed charmless decays of the type

Charmless background
where X is a charged pion, neutral pion or photon that has not been reconstructed, contribute at low invariant mass. Their contributions are fixed to the fully reconstructed charmless components scaled by relative branching fractions [29] and efficiencies determined from simulated samples. A parabola with negative curvature convolved with a Gaussian resolution function is used to model this component, with shape parameter values taken from simulation [32].
The charmless contribution is interpolated from fits to the B − mass spectrum in both the lower and upper D-mass sidebands, without the kinematic fit of the decay chain. The charmless yields are determined independently for B + and B − candidates and are then fixed in the analysis. Their uncertainties contribute to the systematic uncertainties of the final results. The largest charmless contribution is in the B − → [π + π − ] D K − mode, which has a yield corresponding to 7% of the measured signal yield.

Partially reconstructed background
Several additional partially reconstructed b-hadron decays contribute at low invariant mass values. The dominant contributions are from B − → Dh − π 0 and B 0 → (Dπ + ) D * + π − decays, where a neutral pion or positively charged pion is missed in the reconstruction. 3 The invariant mass distribution of these sources de- 3 When considering partially reconstructed background contributions, the assumption is made that the production fractions f u and f d are equal.
pends upon the spin and mass of the missing particle, as with the B − → D * h − signals. In both cases, the missing particle has spin-parity 0 − , such that the Dh − distribution is parameterised using Eq. (3), with shape parameter values taken from simulation.
The Dalitz structure of B − → Dh − π 0 decays is modelled using Laura++ [33]. Decays in which a particle is missed and a companion pion is misidentified as a kaon are parameterised with a semiempirical function, formed from the sum of Gaussian and error functions. The parameters of each partially reconstructed function are fixed to the values found in fits to simulated events, and are varied to determine the associated systematic uncertainty. The yields of the B − → Dπ − π 0 and B − → D K − π 0 contributions vary independently in each subsample, with a CP asymmetry that is fixed to zero in the case of the favoured mode but allowed to vary freely The increase in relative production rate at 13 TeV is small [44], and so the 7 TeV value is used to describe all data in the analysis.

PID efficiencies
In the D ( * ) K − subsamples, the B − → D ( * ) π − cross-feed is determined by the fit to data. The B − → D ( * ) K − cross-feed into the D ( * ) π − subsamples is not well separated from background, so the expected yield is determined by a PID calibration procedure using approximately 20 million D * + → [K − π + ] D π + decays.
The reconstruction of this decay is performed using kinematic variables only, and thus provides a pure sample of K ∓ and π ± particles unbiased in the PID variables. The PID efficiency is parameterised as a function of particle momentum and pseudorapidity, as well as the charged-particle multiplicity in the event. The effective PID efficiency of the signal is determined by weighting the Table 2 Signal yields as measured in the fit to the data.
Mode Yield

Production and detection asymmetries
In order to measure CP asymmetries, the detection asymmetries for K ± and π ± mesons must be taken into account. A detection asymmetry of (−0.87 ± 0.17)% is assigned for each kaon in the final state, primarily due to the fact that the nuclear interaction length of K − mesons is shorter than that of K + mesons. It is computed by comparing the charge asymmetries in D − → K + π − π − and D − → K 0 S π − calibration samples, weighted to match the kinematics of the signal kaons. The equivalent asymmetry for pions is smaller (−0.17 ± 0.10)% [16]. The CP asymmetry in the favoured B − → [K − π + ] D π − decay is fixed to (+0.09 ± 0.05)%, calculated from current knowledge of γ and r B in this decay [2], with no assumption made about the strong phase, δ Dπ B . This enables the effective production asymmetry, A eff B ± , to be measured and simul- Table 4 Systematic uncertainties for the CP observables measured in a fully reconstructed manner, quoted as a percentage of the statistical uncertainty on the observable. The Sim uncertainty on R K π K /π is due to the limited size of the simulated samples used to determine the relative efficiency for reconstructing and selecting B − → Dπ − and B − → D K − decays.
[%] taneously subtracted from the charge asymmetry measurements in other modes.

Yields and selection efficiencies
The total yield for each mode is a sum of the number of correctly identified and cross-feed candidates; their values are given in Table 2. The corresponding invariant mass spectra, separated by charge, are shown in Figs. 1-3.
To obtain the observable R K π K /π (R K π ,π 0 /γ K /π ), which is defined in Table 1, the ratio of yields must be corrected by the relative effi- decays are reconstructed and selected. Both ratios are found to be consistent with unity within their assigned uncertainties, which take into account the size of the simulated samples and the imperfect modelling of the relative pion and kaon absorption in the detector material.
To determine the branching fraction B(D * 0 → D 0 π 0 ), the yields of the B − → (Dπ 0 ) D * π − and B − → (Dγ ) D * π − modes are corrected for the relative efficiencies of the neutral pion and photon modes as determined from simulation. As both of these modes are partially reconstructed with identical selection requirements, the relative efficiency is found to be unity within its assigned uncertainty, and is varied to determine the associated systematic uncertainty. In the measurement of B(D * → Dπ 0 ), the assumption is made that B(D * → Dπ 0 ) + B(D * → Dγ ) = 1 [29].
The branching fraction B(B − → D * 0 π − ) is determined from the total B − → D * π − yield, the total B − → Dπ − yield, the relative efficiencies determined from simulation, and the B − → Dπ − branching fraction [29,34]. Both the efficiencies and external input branching fraction are varied to determine the associated systematic uncertainty.

Systematic uncertainties
The 21 observables of interest are free parameters of the fit, and each of them is subject to a set of systematic uncertainties that result from the use of fixed terms in the fit. The systematic uncer- Table 3 Systematic uncertainties for the CP observables measured in a partially reconstructed manner, quoted as a percentage of the statistical uncertainty on the observable.
[%] A K π ,γ K A K π ,γ π A K π ,π 0 K A K π ,π 0 π A C P,γ K A C P,γ π A C P,π 0 K A C P,π 0 π R C P,γ R C P,π 0 R K π ,π 0 /γ  Table 5 Systematic uncertainties for the branching fraction measurements, quoted as a percentage of the statistical uncertainty on the observable.
[%] tainties associated with using these fixed parameters are assessed by repeating the fit many times, varying the value of each external parameter within its uncertainty according to a Gaussian distribution. The resulting spread (RMS) in the value of each observable is taken as the systematic uncertainty on that observable due to the external source. The systematic uncertainties, grouped into six categories, are listed in Tables 3 and 4 for the CP observables measured in a partially reconstructed and fully reconstructed manner, respectively. The systematic uncertainties for the branching fraction measurements are listed in Table 5. Correlations between the categories are negligible, but correlations within categories are accounted for. The total systematic uncertainties are summed in quadrature.
The first systematic category, referred to as PID in Tables 3−5, accounts for the uncertainty due to the use of fixed PID efficiency values in the fit. The second category Bkg rate corresponds to the use of fixed background yields in the fit. For example, the rate of B 0 → D * − π + decays is fixed in the fit using known branching fractions as external inputs. This category also accounts for charmless background contributions, each of which have fixed rates in the fit. The Bkg func and Sig func categories refer to the use of fixed shape parameters in background and signal functions, respectively; each of these parameters is determined using simulated samples. The category Sim accounts for the use of fixed selection efficiencies derived from simulation, for instance the relative efficiency of selecting B − → (Dπ 0 ) D * π − and B − → Dπ − decays. The final category, Asym, refers to the use of fixed asymmetries in the fit. This category accounts for the use of fixed CP asymmetries and detection asymmetries in the fit, as described earlier.

(syst)
A ππ π = −0.008 ± 0.006 (stat) ± 0.002 (syst) A ππ K = +0.115 ± 0.025 (stat) ± 0.007 (syst) R K K = 0.988 ± 0.015 (stat) ± 0.011 (syst) R ππ = 0.992 ± 0.027 (stat) ± 0.015 (syst) The results obtained using fully reconstructed B − → Dh − decays supersede those in Ref. [12], while the B − → D * h − results are reported for the first time. The statistical and systematic correlation matrices are given in the appendix. There is a high degree of anticorrelation between partially reconstructed signal and background components in the fit, which all compete for yield in the same invariant mass region. The anticorrelation between the Table 6 of the appendix. The presence of such anticorrelations is a natural consequence of the method of partial reconstruction, and limits the precision with which the CP observables can be measured using this approach.
The value of A K K K has increased with respect to the previous result [12], due to a larger value being measured in the 13 TeV data sets are consistent within 2.6 standard deviations. All other updated measurements are consistent within one standard deviation with those in Ref. [12].
Observables involving D → K + K − and D → π + π − decays can differ due to CP violation in the D decays or acceptance effects. The latest LHCb results [45] show that charm CP -violation effects are negligible for the determination of γ , and that there is also no significant difference in the acceptance for the two modes. is in agreement with, and substantially more precise than, the current world average [29,34,46]. The branching fraction measurements of B(D * 0 → D 0 π 0 ) and B(B − → D * 0 π − ) are found to agree with the current world average values within 0.6 and 1.3 standard deviations, respectively [29,34,47]. A value for the ratio of branching fractions where the uncertainty quoted is dominated by systematic uncertainties, and the statistical and systematic correlations between the input observables are fully taken into account. This value is in agreement with, and improves upon, the current world average. The ratios R K π K /π and R K π ,π 0 /γ K /π can be interpreted as

Conclusion
World-best measurements of CP observables in B − → Dh − decays are obtained with the D meson reconstructed in the K − π + , K + K − and π + π − final states; these supersede earlier work on the GLW modes presented in Ref. [12]. Studies of partially recon- The K − π + final state, which offers higher sensitivity to γ due to larger interference effects [48], has not been considered in this work, due to the presence of a large background contribution from the poorly un- , R CP ,γ and R CP ,π 0 as input, a derivation of the fundamental parameters r D * K B , δ D * K B and γ has been performed using the approach detailed in Ref. [2]. The profile likelihood contours at 1σ , 2σ and 3σ are shown in Fig. 4. The preferred values of r D * K B are lower than the current world average values, owing to the fact that the values of R CP ,γ and R CP ,π 0 measured in this work are below and above unity, respectively, in contrast to the world averages which are both larger than unity [3]. The preferred values of γ and δ D * K B are consistent within 1 standard deviation with the LHCb combination [2] and the world average.

Appendix A. Correlation matrices
The statistical uncertainty correlation matrices are given in Tables 6 and 7 for the CP observables measured using partially reconstructed and fully reconstructed decays, respectively. The correlations between the systematic uncertainties are provided in Tables 8 and 9. Table 6 Statistical correlation matrix for the CP observables measured using partially reconstructed decays.  Table 7 Statistical correlation matrix for the CP observables measured using fully reconstructed decays.   Table 9 Systematic uncertainty correlation matrix for the CP observables measured using fully reconstructed decays.