Angular analysis of B0→D∗−D∗+s with D∗+s→D+sγ decays

JHEP06(2021)177 Published for SISSA by Springer Received:May 7, 2021 Accepted:June 3, 2021 Published:June 29, 2021 Angular analysis of B0→D∗−D∗+ swith D∗+ s→D+ sγ decays The LHCb collaboration E-mail: [email protected] Abstract: The first full angular analysis of the B0→D∗−D∗+ sdecay is performed using 6 fb−1of pp collision data collected with the LHCb experiment at a centre-of-mass energy of 13 TeV. The D∗+ s→D+ sγand D∗− →D0π−vector meson decays are used with the subsequent D+ s→K+K−π+and D0→K+π−decays. All helicity amplitudes and phases are measured, and the longitudinal polarisation fraction is determined to be fL= 0.578±0.010±0.011 with world-best precision, where the first uncertainty is statistical and the second is systematic. The pattern of helicity amplitude magnitudes is found to align with expectations from quark-helicity conservation in Bdecays. The ratio of branching fractions [B(B0→D∗−D∗+ s)×B(D∗+ s→D+ sγ)]/B(B0→D∗−D+ s)is measured to be 2.045 ±0.022 ±0.071 with world-best precision. In addition, the first observation of the Cabibbo-suppressed Bs→D∗−D+ sdecay is made with a significance of seven standard deviations. The branching fraction ratio B(Bs→D∗−D+ s)/B(B0→D∗−D+ s)is measured to be 0.049 ±0.006 ±0.003 ±0.002, where the third uncertainty is due to limited knowledge of the ratio of fragmentation fractions. Keywords: Bphysics, Branching fraction, Hadron-Hadron scattering (experiments), Polarization ArXiv ePrint: 2105.02596 Open Access, Copyright CERN, for the benefit of the LHCb Collaboration. Article funded by SCOAP3. https://doi.org/10.1007/JHEP06(2021)177 JHEP06(2021)177 Contents 1 Introduction 1 2 Angular decay rate formalism 3 3 LHCb detector and simulation 3 4 Event selection 5 5 Measurement of fLand branching fraction ratios 6 5.1 Fit components 7 5.2 Results 10 6 Invariant-mass fit to B0→D∗−D∗+ sdecays 11 7 Angular acceptance functions 12 7.1 Acceptance functions for cos θDand cos θX13 7.2 Acceptance function for χ14 8 Angular fit to data 15 9 Systematic uncertainties 16 10 Results and conclusion 18 A Relationship between m(D∗−D+ s)and cos θX20 The LHCb collaboration 24 1 Introduction The B0→D∗−D∗+ sdecay involves the production of two vector charm mesons from a pseudoscalar B0parent. This process exhibits a polarisation structure, where three complex helicity amplitudes H0,H+, and H−contribute to the total decay rate. These amplitudes correspond to the relative orientation of the linear polarisation vectors of the two vector mesons. Parity-even (k) and parity-odd (⊥) transversity amplitudes can also be defined in terms of H+and H−, namely Ak,⊥= (H+±H−)/√2. The helicity amplitudes can interfere, with interference governed by the strong phases of the transverse components, φ+and φ−, relative to the phase of the longitudinal component, φ0, which is conventionally taken to be equal to zero. Therefore, five parameters in total determine the decay rate: •|H0|, the magnitude of the longitudinal amplitude; •|H+|and |H−|, the magnitudes of the two transverse amplitudes; •φ+and φ−, the phases of the transverse amplitudes relative to H0. – 1 – JHEP06(2021)177 In order to normalise the total decay rate, |H0|2+|H+|2+|H−|2=fL+fT= 1 is required, where fL≡ |H0|2is the longitudinal polarisation fraction and fT≡ |H+|2+|H−|2is the transverse polarisation fraction. The current world average for fLis 0.52 ±0.05 [1,2], while theoretical predictions cover a similar range [3–6]; the transverse helicity amplitudes have not been measured previously. The normalisation condition reduces the total number of independent observables to four, where the additional observable is absorbed into the absolute branching fraction of the decay which is not measured. Measuring the relative magnitudes of the helicity amplitudes offers a test of quark-helicity conservation in this tree-level decay involving a b→cquark transition. In such decays, a |H0|>|H+|>|H−| hierarchy is expected [7], where the V−Anature of the weak interaction causes the longitudinal component to dominate. The B0→D∗−D∗+ sdecay has a large branching fraction, B(B0→D∗−D∗+ s) = (1.77 ±0.14)% [2], and is thus a prominent background in B0→D∗−τ+ντanalyses that exploit the hadronic three-prong τ+→π+π+π−¯ντmode in order to measure the ratio R(D∗)≡ B(B0→D∗−τ+ντ)/B(B0→D∗−`+ν`)[8] or the angular coefficients of the B0→D∗−τ+ντdecay [9]. Such a background arises when the neutral particle produced in the D∗+ sdecay is not reconstructed, and the D+ smeson decays to three pions plus additional non-reconstructed particles. Using data corresponding to an integrated luminosity of 6 fb−1collected at a centre-of- mass energy of 13 TeV with the LHCb experiment between 2015 and 2018, B0→D∗−D∗+ s with D∗+ s→D+ sγdecays are reconstructed via the D∗− →(D0→K+π−)π−and D+ s→K+K−π+channels; the inclusion of charge-conjugate processes is implied throughout. Partially reconstructed decays, where the photon is not considered in the invariantmass calculation, are used in a fit to the m(D∗−D+ s)distribution to measure fL. Fully reconstructed decays are then considered in a subsequent angular analysis to measure the remaining helicity observables. Measurements are performed under the assumption that both the D0π−and D+ sγsystems are pure vector, as no evidence for a scalar contribution is found in the m(D0π−)distribution in data and no scalar component is permitted in m(D+ sγ)due to the photon angular momentum. The analysis includes an improved measurement of fLand first measurements of the transverse helicity amplitude magnitudes and phases. The data sample is also used to measure the ratio of branching fractions R ≡ [B(B0→D∗−D∗+ s)×B(D∗+ s→D+ sγ)]/B(B0→D∗−D+ s), where the current value of R= 2.07±0.33 is calculated using world-average branching fractions taken from ref. [2]. In addition, a measurement of the previously unobserved Cabibbo-suppressed B0 s→D∗−D+ s decay is performed and the ratio of branching fractions B(B0 s→D∗−D+ s)/B(B0→D∗−D+ s) determined. The formalism adopted is described in section 2, essential details of the LHCb detector and simulation are given in section 3, and the event selection is outlined in section 4. The longitudinal polarisation fraction and ratios of branching fractions are measured in section 5, and the remaining helicity observables are measured in sections 6–8. Systematic uncertainties are determined in section 9, and final results and conclusions are presented in section 10. – 2 – JHEP06(2021)177 2 Angular decay rate formalism The B0→D∗−D∗+ sdecay rate is a function of three decay angles, θD,θX, and χ, where θDis the angle between the D0meson and the direction opposite the B0momentum vector in the D∗− rest frame, θXis the angle between the D+ smeson and the direction opposite the B0momentum vector in the D∗+ srest frame, and χis the angle between the two decay planes as defined in the B0rest frame. The angles are illustrated in figure 1, and are explicitly defined as follows cos θD=ˆp(D∗− ) D0·ˆp(B0) D∗− =ˆp(D∗− ) D0·−ˆp(D∗− ) B0, cos θX=ˆp(D∗+ s) D+ s·ˆp(B0) D∗+ s=ˆp(D∗+ s) D+ s·−ˆp(D∗+ s) B0, cos χ=ˆp(B0) D+ s׈p(B0) γ·ˆp(B0) D0׈p(B0) π−,(2.1) sin χB0=−hˆp(B0) D+ s׈p(B0) γ×ˆp(B0) D0׈p(B0) π−i·ˆp(B0) D∗− , sin χB0= +hˆp(B0) D− s׈p(B0) γ×ˆp(B0) D0׈p(B0) π−i·ˆp(B0) D∗+, where the ˆp(Y) Xare unit vectors describing the direction of a particle Xin the rest frame of the system Y. In the B0rest frame, the angular definition for the B0decay is a chargeparity (CP) transformation of that for the B0decay. The sign of sin χis negative for B0 candidates and positive for B0candidates, where the B-meson flavour is tagged by the D∗-meson charge. This formalism is the same as that adopted in other LHCb angular analyses such as that of B→K∗µ+µ−decays [10,11]. The full three-dimensional differential decay rate expressed in terms of the helicity amplitudes is given by [3] d3Γ dcos θDdcos θXdχ ∝9 8cos2θDsin2θX|H0|2+1 4sin2θD1 + cos2θX|H+|2+|H−|2 −1 2sin2θDsin2θXcos 2χRe H+H∗ −−sin 2χIm H+H∗ − (2.2) −1 4sin 2θDsin 2θX[cos χRe (H+H∗ 0+H−H∗ 0)−sin χIm (H+H∗ 0−H−H∗ 0)] . 3 LHCb detector and simulation The LHCb detector [12,13] 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 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 – 3 – JHEP06(2021)177 D*−D*+ sB0 χ ¯ D0 ¯ D0 π−D+ s γ D+ s γ π−  pB0  pB0 θX θD rest frame rest frame rest frame Figure 1. Illustration of the B0→D∗−D∗+ sdecay angles. a track to a primary pp collision 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/c. 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 and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers. The online event selection is performed by a trigger, which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. At the hardware trigger stage, events are required to have a muon with high pTor a hadron, photon or electron with high transverse energy in the calorimeters. For hadrons, the transverse energy threshold is 3.5 GeV. The software trigger requires a two-, three- or four-track secondary vertex with a significant displacement from any primary pp interaction vertex. At least one charged particle must have a transverse momentum pT>1.6GeV/cand be inconsistent with originating from any PV. A multivariate algorithm is used for the identification of secondary vertices consistent with the decay of a bhadron. In the offline selection, trigger information is 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 an overlap of both. Simulation is required to model the effects of the detector acceptance and the imposed selection requirements. In the simulation, pp collisions are generated using Pythia [14] with a specific LHCb configuration [16]. Decays of unstable particles are described by EvtGen [17], in which final-state radiation is generated using Photos [18]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [19] as described in ref. [21]. The underlying pp interaction is reused multiple times, with an independently generated signal decay for each [22]. In addition, the m(D∗−D+ s)distributions of pure longitudinal and transverse polarised B0→D∗−D∗+ s – 4 – JHEP06(2021)177 decays are studied using fast-simulated samples generated with the RapidSim package [23], where an LHCb momentum resolution configuration is used to smear the generated fourmomenta. The same tool is used to study the m(D∗−D+ s)distributions of various background contributions from decays involving higher-excited charm mesons. 4 Event selection Candidate B0→D∗−D+ sdecays are reconstructed through the D∗− →(D0→K+π−)π− and D+ s→K+K−π+channels. The tracks of the final-state particles are required to have a good quality, fulfil loose particle identification (PID) criteria, and have a high χ2 IP value with respect to any PV, where χ2 IP is defined as the difference in the vertex-fit χ2of a given PV reconstructed with and without the particle being considered. The reconstructed masses of the D0and D+ scandidates are required to lie inside mass windows of ±20 MeV/c2 around their known values [2]. The D∗− candidate mass is required to be within ±40 MeV/c2 of the known value [2], while the difference in mass between the D∗− and D0candidates is required to be in the range 140–150 MeV/c2. In combination with the track PID cuts, these narrow mass windows reduce potential backgrounds from misidentified decays such as B0→D∗−D+to negligible levels. The B0candidate is reconstructed by combining the D∗− and D+ scandidates to form a common vertex. If multiple PVs are reconstructed in the same event, the PV for which the B0candidate has the lowest χ2 IP is assigned as the associated PV. The pTof the B0candidate is required to be larger than 5 GeV/c, and the χ2 IP of the B0candidate for the associated PV is required to be small. To suppress combinatorial background and background from decays involving the production of a D∗− and three prompt tracks, the flight distance of the D+ scandidate along the beam axis is required to be different from zero by more than one standard deviation, considering both the origin and decayvertex uncertainties of the D+ scandidate. To suppress combinatorial background from combinations of tracks originating from the PV, the decay time of the B0candidate is required to be larger than 0.2 ps. To improve the invariant-mass resolution, a kinematic fit is performed to the decay chain [24], the B0candidate is constrained to originate from the PV and the D+ sand D0masses are constrained to their known values. Candidates are retained if the resulting invariant mass of the D∗−D+ scombination falls within the 4900–5500 MeV/c2 range, which includes the region occupied by partially reconstructed B0→D∗−D∗+ sdecays when the neutral particle produced in the D∗+ sdecay is not reconstructed. This sample is considered in section 5, where a fit to the m(D∗−D+ s)distribution of candidates is used to measure fL. A subsample of fully reconstructed B0→D∗−D∗+ scandidates is selected by combining D+ scandidates from the above dataset with photons. The difference between the D∗+ s and D+ scandidate masses is required to be in the range 120–180 MeV/c2, and the photon is required to have a pTlarger than 500 MeV/c. Each D∗+ scandidate is then recombined with the corresponding D∗− candidate from the above dataset to form a B0candidate, where candidates in the invariant-mass range 5150–5500 MeV/c2are retained. Fully reconstructed candidates with m(D∗−D+ s)values greater than 5240 MeV/c2are vetoed to remove – 5 – JHEP06(2021)177 B0→D∗−D+ sdecays where a random photon is combined with the D+ scandidate. This dataset is used in section 8to measure the remaining helicity observables in an angular analysis. 5 Measurement of fLand branching fraction ratios The longitudinal polarisation fraction, fL, determines the fractional contribution of the H0 helicity amplitude to the total B0→D∗−D∗+ sdecay rate. The longitudinal and transverse amplitudes contribute to the one-dimensional differential decay rate in cos θXas follows, dΓ dcos θX∝3 4|H0|2(1 −cos2θX) + 1 2(|H+|2+|H−|2)(1 + cos2θX) =3 4fL(1 −cos2θX) + (1 −fL) 2(1 + cos2θX),(5.1) which is obtained from eq. (2.2) via a definite integral over cos θDand χ. Experimentally, the integral over cos θDand χmust also include the acceptance in these angles. However, the acceptance is predominantly linear for both angles, as shown in figures 4and 5, such that no significant residual dependence remains after the integration. Due to a common dependence on photon kinematics, the angle cos θXand the invariant mass of the D∗−D+ s system are strongly negatively correlated, as illustrated in appendix Ain figure 7. More positive values of cos θXcorrespond to higher momentum photons and thus lower values of m(D∗−D+ s). As a result, the different cos θXshapes for longitudinal and transverse polarised B0→D∗−D∗+ sdecays manifest in corresponding m(D∗−D+ s)distributions with different parabolic forms, as shown in appendix Ain figure 8. This feature enables fLto be measured using a binned maximum-likelihood fit to the m(D∗−D+ s)distribution in data, where the total B0→D∗−D∗+ scontribution is modelled by the sum of probability density functions (PDFs) for the longitudinal and transverse components with relative fractions fLand 1−fL. Determining fLvia an m(D∗−D+ s)fit enables partially reconstructed B0→D∗−D∗+ sdecays to be used, which increases the sample size by avoiding efficiency losses due to the limited photon reconstruction efficiency of the LHCb detector. Due to the presence of B0→D∗−D+ sdecays in the same sample, a measurement of the branching fraction ratio R ≡ B(B0→D∗−D∗+ s)×B(D∗+ s→D+ sγ) B(B0→D∗−D+ s)(5.2) can also be made. Experimentally, this quantity is defined as R=N(B0→D∗−(D∗+ s→D+ sγ)) N(B0→D∗−D+ s)×(B0→D∗−D+ s) (B0→D∗−(D∗+ s→D+ sγ)) =N(B0→D∗−(D∗+ s→D+ sγ)) N(B0→D∗−D+ s)×ξ , (5.3) where Ndenotes the yields for each decay mode, and ξis the ratio of their total reconstruction and selection efficiencies. In the case of B0→D∗−D∗+ sdecays, the yields and – 6 – JHEP06(2021)177 efficiencies correspond to those of partially reconstructed signal. The efficiency ratio is determined using simulated samples of B0→D∗−D∗+ sand B0→D∗−D+ sdecays, and is found to be ξ= 1.142 ±0.034, where the uncertainty quoted accounts only for the use of finite simulated samples and potential variation in the efficiency across data-taking years. This uncertainty is considered as a source of systematic uncertainty on R. A contribution from Cabibbo-suppressed B0 s→D∗−D+ sdecays is also considered in the m(D∗−D+ s)fit, enabling a measurement of the branching fraction ratio r(B0 s)≡B(B0 s→D∗−D+ s) B(B0→D∗−D+ s)(5.4) to be made. Experimentally, r(B0 s)is defined as r(B0 s) = fd fs×N(B0 s→D∗−D+ s) N(B0→D∗−D+ s)×(B0→D∗−D+ s) (B0 s→D∗−D+ s) =fd fs×N(B0 s→D∗−D+ s) N(B0→D∗−D+ s)×ξ(B0 s),(5.5) where Ndenotes the yields for each decay mode, and fs/fd= 0.2539 ±0.0079 is the ratio of fragmentation fractions at √s= 13 TeV as measured inside the LHCb acceptance [25]. The relative efficiency ξ(B0 s)is assumed to be unity, with a 5% relative systematic uncertainty assigned to account for potential variation in efficiency due to mass and lifetime differences. 5.1 Fit components The m(D∗−D+ s)distribution of selected candidates is shown in figure 2, and is dominated by the narrow signal due to fully reconstructed B0→D∗−D+ sdecays and a broad structure due to B0→D∗−D∗+ sdecays with missing a photon or π0from the D∗+ sdecay. The distribution is modelled as a sum of several components which are described below. B0→D∗−D+ sdecays. Fully reconstructed B0→D∗−D+ sdecays are modelled using the sum of two Crystal Ball PDFs [26] with a freely varying common mean and width, and a relative yield fraction that is Gaussian-constrained according to simulation. The component PDF tails are modelled on opposite sides, and the tail parameters are Gaussianconstrained from simulation. The branching fraction ratio Ris measured directly in the fit, such that the yield of the B0→D∗−D+ scomponent is related to the yield of the B0→D∗−(D∗+ s→D+ sγ)component via a freely varying parameter Rand the fixed relative efficiency ratio ξ. B0→D∗−(D∗+ s→D+ sγ)decays. The partially reconstructed B0→D∗−(D∗+ s→D+ sγ) signal is modelled using the sum of a longitudinal component and a transverse component, where a freely varying parameter fLdetermines the relative proportion of the longitudinal component. To derive invariant-mass PDFs for each component, fits are performed to simulated samples of pure longitudinal and transverse polarised decays as shown in appendix A in figure 9. The m(D∗−D+ s)distributions are modelled with parabolas convolved with Gaussian resolution functions, where the parabolas are based on the cos θXdependence in – 7 – JHEP06(2021)177 4900 5000 5100 5200 5300 5400 m(D∗−D+ s) [MeV/c2] 500 1000 1500 2000 2500 Candidates / (2.0 MeV/c2) LHCb 6 fb−1 Data Total fit B0→D∗−(D∗+ s→D+ sγ), L B0→D∗−(D∗+ s→D+ sγ), T B0→D∗−(D∗+ s→D+ sπ0), L B0→D∗−(D∗+ s→D+ sπ0), T B0→D∗−D+ s B0 s→D∗−D(∗)+ s B→D∗(∗)D∗(∗) s Combinatorial 5325 5350 5375 5400 5425 5450 5475 m(D∗−D+ s) [MeV/c2] 10 20 30 40 50 Candidates / (2.0 MeV/c2) LHCb 6 fb−1 Figure 2. (Top) Distribution of m(D∗−D+ s)for selected candidates in data, with the fit overlaid. Where indicated, L(T) represents longitudinally (transverse) polarised decays. (Bottom) Restricted to region for candidates with m(D∗−D+ s)>5325 MeV/c2, where the Cabibbo-suppressed B0 s→D∗−D+ scontribution is visible. eq. (5.1). This approach closely follows the method used in refs. [27] and [28] for CP violation studies of partially reconstructed B−→D∗0h−with D∗0→Dγ/π0decays, where h− is a pion or a kaon and the neutral particle produced in the D∗0decay is not reconstructed. The total yield of the B0→D∗−(D∗+ s→D+ sγ)component, N(B0→D∗−(D∗+ s→D+ sγ)), varies freely and is used along with Rand ξto set the B0→D∗−D+ scomponent yield. All PDF parameters for the B0→D∗−(D∗+ s→D+ sγ)component are fixed in the data fit, and are varied within their uncertainties to determine the systematic uncertainties on fL, R, and r(B0 s). – 8 – JHEP06(2021)177 −π−π 20π 2π χ[rad] 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Density / (0.63) LHCb Reco. LHCb simulation Generated sample −π−π 20π 2π χ[rad] 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 Acceptance LHCb Reco. LHCb simulation / Generated ratio Polynomial fit Figure 5. Comparison of reconstructed χdistribution in a fully-simulated B0→D∗−D∗+ ssample and the generated χdistribution in a RapidSim sample produced with the same helicity amplitude model (left). The ratio is fitted with a second-order polynomial to determine the acceptance function for use in the data fit (right). their uncertainties to determine the systematic uncertainties on the helicity parameters. In this procedure, the correlations between the polynomial coefficients are accounted for using the acceptance fit covariance matrix. 8 Angular fit to data To measure |H−|,φ−, and φ+, an unbinned maximum-likelihood fit to the three-dimensional angular distribution of signal-weighted data is performed using zfit [30]. For the fit, the B0→D∗−(D∗+ s→D+ sγ)candidates from the m(D∗−D∗+ s)fit in section 6are used with per-candidate signal weights assigned. The longitudinal polarisation amplitude, H0, is assigned a fixed magnitude |H0|using the value of fLmeasured in section 5, and its phase is set to the arbitrary value φ0= 0. The parameter |H+|is fully determined by the normalisation of the helicity amplitudes to unity. The signal density at each point in angular phase space is described using eq. (2.2) multiplied by acceptance functions in each of the decay angles. To determine the statistical uncertainties of the observables, the fit applies an asymptotic correction to the covariance matrix as detailed in ref. [31], which correctly accounts for the use of signal-weighted data. The distributions for each decay angle are shown in figure 6, with the one-dimensional fit projections overlaid. Studies with pseudoexperiments are performed to determine the level of bias present in the results, where pull distributions of mean µx Pand width σx Pare constructed for each observable x. The pull distributions for each helicity observable are found to follow Gaussian distributions closely, where σ|H−| Pis consistent with unity. However, σφ+ P= 1.14 ±0.02 and σφ− P= 1.12 ±0.02, indicating that the default fit uncertainties for these observables are underestimated. The mean values of the pulls for the transverse phases are consistent with zero, but µ|H−| P=−0.14 ±0.02. These biases are traced to the finite size of the fitted – 15 – JHEP06(2021)177 −1.0−0.5 0.0 0.5 1.0 cos θD 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Candidate density / (0.07) LHCb 6 fb−1 −0.5 0.0 0.5 1.0 cos θX 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Candidate density / (0.06) LHCb 6 fb−1 −π−π 20π 2π χ[rad] 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Candidate density / (0.21 rad) LHCb 6 fb−1 Figure 6. Decay-angle distributions of signal-weighted B0→D∗−D∗+ scandidates in data, with the one-dimensional angular fit projections overlaid. dataset, and are found to resolve when pseudoexperiment datasets containing more events than are present in data are generated. The values of µx Pand σx Pare used to correct the default fit results x±σxas follows xc=x−µx P×σx(8.1) σc x=σx P×σx(8.2) where xc±σc xare the corrected fit results. In section 10, the results for |H−|,φ+, and φ− are quoted after this correction procedure. 9 Systematic uncertainties The values of R,r(B0 s), and fLmeasured in section 5are subject to systematic uncertainties due to limited knowledge of the shape parameters, branching fractions, and relative efficiency corrections used in the fit. To determine these systematic uncertainties, the – 16 – JHEP06(2021)177 Systematic uncertainty R R(B0 s)fL Fixed PDF shape parameters 0.030 0.00197 0.0074 Fixed branching fractions 0.016 0.00004 0.0080 Efficiency corrections 0.062 0.00253 0.0001 Total 0.071 0.00320 0.0109 Table 1. Systematic uncertainties on the branching fraction ratios and fLas measured in the m(D∗−D+ s)fit. Systematic uncertainty |H−|φ+φ− Fixed fLin angular fit and cos(θX/D)acceptance 0.0005 0.0007 0.005 Use of sWeighted data 0.0003 0.0011 0.002 Statistical uncertainty of acceptance functions 0.0034 0.0132 0.044 m(D∗D∗ s)fit background model 0.0319 0.0156 0.025 Total 0.0321 0.0205 0.051 Table 2. Systematic uncertainties on the helicity parameters measured in the unbinned angular fit. m(D∗−D+ s)fit to data is performed many times with the parameters randomly varied within their prescribed uncertainties according to Gaussian distributions. This procedure is performed separately for the shape parameters, branching fractions, and efficiency corrections, and the total systematic uncertainties calculated as the sum in quadrature of these contributions. The systematic uncertainties are summarised in table 1. The observables |H−|,φ+, and φ−measured in the angular fit are subject to several systematic uncertainties. Firstly, the angular analysis is performed at a fixed value of fL, which is used as input in the cos θDand cos θXacceptance fits and also to set the value of |H0|in the angular fit. To determine the systematic uncertainty, the angular analysis is repeated many times with fLvaried within its total uncertainty; the standard deviations of the helicity observable results are taken as the systematic uncertainties. In this procedure, the varied fLvalue used in the acceptance fits is shared with the angular fit to ensure consistency. A small systematic uncertainty is also assigned for the use of signalweighted data, where the angular fit is run many times while varying the signal weights within the signal yield uncertainties from the m(D∗−D∗+ s)fit. To determine the systematic uncertainty from the use of finite samples to obtain the acceptance functions, the acceptance coefficients are varied within their uncertainties according to the acceptance fit covariance matrices. Finally, the angular analysis is repeated with an alternative background model in the m(D∗−D∗+ s)fit, and the differences in central value for each helicity observable are assigned as a systematic uncertainty. The contributing systematic uncertainties are summarised in table 2. – 17 – JHEP06(2021)177 10 Results and conclusion Using a fit to the m(D∗−D+ s)distribution to determine the properties of partially reconstructed B0→D∗−(D∗+ s→D+ sγ)decays, the longitudinal polarisation fraction is measured to be fL= 0.578 ±0.010 ±0.011, where the first uncertainty is statistical and the second is systematic. The corresponding magnitude of the longitudinal helicity amplitude, given by |H0|=√fL, is |H0|= 0.760 ±0.007 ±0.007. This information is used to measure the remaining helicity observables in an angular fit to fully reconstructed B0→D∗−(D∗+ s→D+ sγ)decays, obtaining |H−|= 0.195 ±0.022 ±0.032, |H+|= 0.620 ±0.011 ±0.013, φ+=−0.046 ±0.102 ±0.020, φ−= 0.108 ±0.170 ±0.051, where the quoted value and uncertainties for |H+|are fully determined by the normalisation of the three helicity amplitudes to unity. The measurement of fLis consistent with and more precise than the current world average, fL= 0.52 ±0.05 [1,2]. The transverse amplitude magnitudes and phases are measured for the first time, where both phases are consistent with zero but the magnitudes differ from each other at the level of nine standard deviations. It is noted that |H0|>|H+|>|H−|, which is expected from quark-helicity conservation in Bdecays involving a b→cquark transition. In such decays, the V−Anature of the weak interaction causes the longitudinal component to dominate. The inequality is stronger for decays involving light vector mesons [7], but also appears to be satisfied in B0→D∗−D∗+ s decays where two vector charm mesons are produced. This helicity hierarchy is not observed in decays dominated by penguin amplitudes such as B0→φK∗0, where the longitudinal and transverse components are found to have roughly equal amplitudes [32–35]. The branching fraction ratio of B0→D∗−(D∗+ s→D+ sγ)decays relative to B0→D∗−D+ s decays is measured to be R= 2.045 ±0.022 ±0.071, where the first uncertainty is statistical and the second is systematic. This result is in agreement with, but considerably more precise than, the current world-average value R= 2.07 ±0.33 [2]. The branching fraction ratio of the Cabibbo-suppressed B0 s→D∗−D+ s decay relative to the B0→D∗−D+ sdecay is measured to be r(B0 s)=0.049 ±0.006 ±0.003 ±0.002, where the first uncertainty is statistical, the second is systematic, and the third accounts for the use of an external value of fs/fd[25]. This measurement constitutes the first – 18 – JHEP06(2021)177 observation of the Cabibbo-suppressed B0 s→D∗−D+ sdecay with a significance of seven standard deviations. In conclusion, an angular analysis of B0→D∗−D∗+ swith D∗+ s→D+ sγdecays is performed using 6 fb−1of data collected with the LHCb experiment at √s= 13 TeV in order to measure a complete set of helicity amplitude observables. Partially reconstructed candidates are used in a fit to the m(D∗−D+ s)distribution to measure the longitudinal polarisation fraction fL=|H0|2. This knowledge is then used in a subsequent angular fit to fully reconstructed data in order to measure the remaining helicity observables. The measurement of fLis consistent with and more precise than the current world-average value, while the magnitudes and phases of the transverse helicity amplitudes are measured for the first time. The pattern of helicity amplitude magnitudes is found to align with expectations from quark-helicity conservation for tree-level Bdecays involving a b→c transition. The B0→D∗−D∗+ sdecay is a large background in B0→D∗−τ+ντanalyses, particularly when the τ+decays hadronically. Analyses aiming to measure angular observables in B0→D∗−τ+ντdecays must control the angular distributions of prominent hadronic backgrounds such as B0→D∗−D∗+ s, and the results presented herein will help to significantly reduce background model uncertainties in future measurements. 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); MOST and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE (Russia); MICINN (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (U.K.); DOE NP and NSF (U.S.A.). We acknowledge the computing resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF (Netherlands), PIC (Spain), GridPP (U.K.), RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID (Poland) and NERSC (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 ARC and ARDC (Australia); AvH Foundation (Germany); EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union); A*MIDEX, ANR, IPhU and Labex P2IO, and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier Sciences of CAS, CAS PIFI, CAS CCEPP, Fundamental Research Funds for the Central Universities, and Sci. & Tech. Program of Guangzhou (China); RFBR, RSF and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Leverhulme Trust, the Royal Society and UKRI (U.K.). – 19 – JHEP06(2021)177 4950 5000 5050 5100 5150 5200 5250 m(D∗−D+ s) [MeV/c2] −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00 cos θX LHCb simulation 0 20 40 60 80 100 120 140 Figure 7. Relationship between m(D∗−D+ s)and cos θXin a sample of fully reconstructed B0→D∗−(D∗+ s→D+ sγ)simulated decays. The colour scale indicates the number of candidates in each bin. A Relationship between m(D∗−D+ s)and cos θX In figure 7, the relationship between m(D∗−D+ s)and cos θXis shown for fully reconstructed B0→D∗−(D∗+ s→D+ sγ)simulated decays. A strong negative correlation is evident, due to a common dependence on the kinematics of the photon produced in the D∗+ sdecay. The one-dimensional decay rate as a function of cos θXis given by eq. (5.1), where separate transverse and longitudinal components contribute; these components are illustrated in figure 8. Due to the co-dependence of m(D∗−D+ s)and cos θX, the different angular forms for transverse and longitudinal signal give rise to different m(D∗−D+ s)distributions. This is illustrated in figure 9, where RapidSim samples of transverse and longitudinal signal are shown. The fits used to derive shape parameters for the m(D∗−D+ s)fit are overlaid. – 20 – JHEP06(2021)177 −1.0−0.5 0.0 0.5 1.0 cos θX 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Density LHCb Total Longitudinal Transverse Figure 8. Transverse and longitudinal contributions to the one-dimensional decay rate shown as a function of cos θX. 4900 5000 5100 5200 5300 m(D∗−D+ s) [MeV/c2] 500 1000 1500 Candidates / (2.0 MeV/c2) LHCb Fit RapidSim simulation 4950 5000 5050 5100 5150 5200 m(D∗−D+ s) [MeV/c2] 500 1000 1500 2000 Candidates / (2.0 MeV/c2) LHCb Fit RapidSim simulation Figure 9. Invariant-mass distributions of (left) pure transverse and (right) longitudinal B0→D∗−(D∗+ s→D+ sγ)simulated decays. Fits to the distributions are overlaid, from which shape parameters for use in the m(D∗−D+ s)data fit are derived. – 21 – JHEP06(2021)177 Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References [1] BaBar collaboration, Measurement of B0→D(∗)+ sD∗− branching fractions and B0→D(∗)+ sD∗− polarization with a partial reconstruction technique,Phys. Rev. D 67 (2003) 092003 [hep-ex/0302015] [INSPIRE]. [2] Particle Data collaboration, Review of particle physics,Prog. Theor. Exp. Phys. 2020 (2020) 083C01 [INSPIRE] and online at http://pdg.lbl.gov/. [3] G. Kramer and W.F. Palmer, Branching ratios and CP asymmetries in the decay B→V V , Phys. Rev. D 45 (1992) 193 [INSPIRE]. [4] CLEO collaboration, Measurements of B→D(∗)+ (s)D(∗)+ branching fractions,Phys. Rev. D 62 (2000) 112003 [hep-ex/0008015] [INSPIRE]. [5] CLEO collaboration, Measurement of the form-factors for B0→D∗+`−ν,Phys. Rev. Lett. 76 (1996) 3898 [INSPIRE]. [6] Z. Luo and J.L. Rosner, Factorization in color-favored Bmeson decays to charm,Phys. Rev. D64 (2001) 094001 [hep-ph/0101089] [INSPIRE]. [7] M. Suzuki, Final-state interactions and s−quark helicity conservation in B→J/ψK∗,Phys. Rev. D 64 (2001) 117503 [hep-ph/0106354] [INSPIRE]. [8] LHCb collaboration, Test of Lepton Flavor Universality by the measurement of the B0→D∗−τ+ντbranching fraction using three-prong τdecays,Phys. Rev. D 97 (2018) 072013 [LHCb-PAPER-2017-027] [CERN-EP-2017-256] [arXiv:1711.02505] [INSPIRE]. [9] D. Hill, M. John, W. Ke and A. Poluektov, Model-independent method for measuring the angular coefficients of B0→D∗−τ+ντdecays,JHEP 11 (2019) 133 [arXiv:1908.04643] [INSPIRE]. [10] LHCb collaboration, Angular analysis of the B0→K∗0µ+µ−decay using 3fb−1of integrated luminosity,JHEP 02 (2016) 104 [LHCb-PAPER-2015-051] [CERN-PH-EP-2015-314] [arXiv:1512.04442] [INSPIRE]. [11] LHCb collaboration, Angular Analysis of the B+→K∗+µ+µ−Decay,Phys. Rev. Lett. 126 (2021) 161802 [LHCb-PAPER-2020-041] [CERN-EP-2020-239] [arXiv:2012.13241] [INSPIRE]. [12] LHCb collaboration, The LHCb Detector at the LHC,2008 JINST 3S08005 [INSPIRE]. [13] LHCb collaboration, LHCb Detector Performance,Int. J. Mod. Phys. A 30 (2015) 1530022 [LHCB-DP-2014-002] [CERN-PH-EP-2014-290] [arXiv:1412.6352] [INSPIRE]. [14] T. Sjöstrand, S. Mrenna and P.Z. Skands, A Brief Introduction to PYTHIA 8.1,Comput. Phys. Commun. 178 (2008) 852 [arXiv:0710.3820] [INSPIRE]. [15] T. Sjöstrand, S. Mrenna and P.Z. Skands, PYTHIA 6.4 Physics and Manual,JHEP 05 (2006) 026 [hep-ph/0603175] [INSPIRE]. [16] LHCb collaboration, Handling of the generation of primary events in Gauss, the LHCb simulation framework,J. Phys. Conf. Ser. 331 (2011) 032047 [INSPIRE]. [17] D.J. Lange, The EvtGen particle decay simulation package,Nucl. Instrum. Meth. A 462 (2001) 152 [INSPIRE]. – 22 – JHEP06(2021)177 [18] N. Davidson, T. Przedzinski and Z. Was, PHOTOS interface in C++: Technical and Physics Documentation,Comput. Phys. Commun. 199 (2016) 86 [arXiv:1011.0937] [INSPIRE]. [19] J. Allison et al., Geant4 developments and applications,IEEE Trans. Nucl. Sci. 53 (2006) 270 [INSPIRE]. [20] GEANT4 collaboration, GEANT4 — a simulation toolkit,Nucl. Instrum. Meth. A 506 (2003) 250 [INSPIRE]. [21] LHCb collaboration, The LHCb simulation application, Gauss: Design, evolution and experience,J. Phys. Conf. Ser. 331 (2011) 032023 [INSPIRE]. [22] D. Müller, M. Clemencic, G. Corti and M. Gersabeck, ReDecay: A novel approach to speed up the simulation at LHCb,Eur. Phys. J. C 78 (2018) 1009 [LHCb-DP-2018-004] [arXiv:1810.10362] [INSPIRE]. [23] G.A. Cowan, D.C. Craik and M.D. Needham, RapidSim: an application for the fast simulation of heavy-quark hadron decays,Comput. Phys. Commun. 214 (2017) 239 [arXiv:1612.07489] [INSPIRE]. [24] W.D. Hulsbergen, Decay chain fitting with a Kalman filter,Nucl. Instrum. Meth. A 552 (2005) 566 [physics/0503191] [INSPIRE]. [25] LHCb collaboration, Precise measurement of the fs/fdratio of fragmentation fractions and of B0 sdecay branching fractions,arXiv:2103.06810 [LHCb-PAPER-2020-046] [CERN-EP-2021-027] [INSPIRE]. [26] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, Ph.D. Thesis, Institute of Nuclear Physics, Krakow Poland (1986). [27] LHCb collaboration, Measurement of CP observables in B±→D(∗)K±and B±→D(∗)π± decays,Phys. Lett. B 777 (2018) 16 [LHCb-PAPER-2017-021] [CERN-EP-2017-195] [arXiv:1708.06370] [INSPIRE]. [28] LHCb collaboration, Measurement of CP observables in B±→D(∗)K±and B±→D(∗)π± decays using two-body Dfinal states,JHEP 04 (2021) 081 [LHCb-PAPER-2020-036] [CERN-EP-2020-225] [arXiv:2012.09903] [INSPIRE]. [29] M. Pivk and F.R. Le Diberder, SPlot: A Statistical tool to unfold data distributions,Nucl. Instrum. Meth. A 555 (2005) 356 [physics/0402083] [INSPIRE]. [30] J. Eschle, A. Puig Navarro, R. Silva Coutinho and N. Serra, zfit: scalable pythonic fitting, SoftwareX 11 (2020) 100508 [arXiv:1910.13429] [INSPIRE]. [31] C. Langenbruch, Parameter uncertainties in weighted unbinned maximum likelihood fits, arXiv:1911.01303 [INSPIRE]. [32] LHCb collaboration, Measurement of polarization amplitudes and CP asymmetries in B0→φK∗(892)0,JHEP 05 (2014) 069 [CERN-PH-EP-2014-038] [LHCb-PAPER-2014-005] [arXiv:1403.2888] [INSPIRE]. [33] BaBar collaboration, Measurements of branching fractions, polarizations, and direct CP-violation asymmetries in B+→ρ0K∗+and B+→f0(980)K∗+decays,Phys. Rev. D 83 (2011) 051101 [arXiv:1012.4044] [INSPIRE]. [34] Belle collaboration, Measurements of branching fractions and polarization in B→K∗ρ decays,Phys. Rev. Lett. 95 (2005) 141801 [hep-ex/0408102] [INSPIRE]. [35] LHCb collaboration, First observation of the decay B0 s→K∗0¯ K∗0,Phys. Lett. B 709 (2012) 50 [CERN-PH-EP-2011-183] [LHCb-PAPER-2011-012] [arXiv:1111.4183] [INSPIRE]. – 23 – JHEP06(2021)177 The LHCb collaboration R. Aaij32, C. Abellán Beteta50, T. Ackernley60, B. Adeva46, M. Adinolfi54, H. Afsharnia9, C.A. Aidala86, S. Aiola25, Z. Ajaltouni9, S. Akar65, J. Albrecht15, F. Alessio48, M. Alexander59, A. Alfonso Albero45, Z. Aliouche62, G. Alkhazov38, P. Alvarez Cartelle55, S. Amato2, Y. Amhis11, L. An48, L. Anderlini22, A. Andreianov38, M. Andreotti21, F. Archilli17, A. Artamonov44, M. Artuso68, K. Arzymatov42, E. Aslanides10, M. Atzeni50, B. Audurier12, S. Bachmann17, M. Bachmayer49, J.J. Back56, P. Baladron Rodriguez46, V. Balagura12, W. Baldini21, J. Baptista Leite1, R.J. Barlow62, S. Barsuk11, W. Barter61, M. Bartolini24, F. Baryshnikov83, J.M. Basels14, G. Bassi29, B. Batsukh68, A. Battig15, A. Bay49, M. Becker15, F. Bedeschi29, I. Bediaga1, A. Beiter68, V. Belavin42, S. Belin27, V. Bellee49, K. Belous44, I. Belov40, I. Belyaev41, G. Bencivenni23, E. Ben-Haim13, A. Berezhnoy40, R. Bernet50, D. Berninghoff17, H.C. Bernstein68, C. Bertella48, A. Bertolin28, C. Betancourt50, F. Betti48, Ia. Bezshyiko50, S. Bhasin54, J. Bhom35, L. Bian73, M.S. Bieker15, S. Bifani53, P. Billoir13, M. Birch61, F.C.R. Bishop55, A. Bitadze62, A. Bizzeti22,k, M. Bjørn63, M.P. Blago48, T. Blake56, F. Blanc49, S. Blusk68, D. Bobulska59, J.A. Boelhauve15, O. Boente Garcia46, T. Boettcher65, A. Boldyrev82, A. Bondar43, N. Bondar38,48, S. Borghi62, M. Borisyak42, M. Borsato17, J.T. Borsuk35, S.A. Bouchiba49, T.J.V. Bowcock60, A. Boyer48, C. Bozzi21, M.J. Bradley61, S. Braun66, A. Brea Rodriguez46, M. Brodski48, J. Brodzicka35, A. Brossa Gonzalo56, D. Brundu27, A. Buonaura50, C. Burr48, A. Bursche72, A. Butkevich39, J.S. Butter32, J. Buytaert48, W. Byczynski48, S. Cadeddu27, H. Cai73, R. Calabrese21,f , L. Calefice15,13, L. Calero Diaz23, S. Cali23, R. Calladine53, M. Calvi26,j , M. Calvo Gomez85, P. Camargo Magalhaes54, A. Camboni45,85, P. Campana23, A.F. Campoverde Quezada6, S. Capelli26,j , L. Capriotti20,d, A. Carbone20,d, G. Carboni31, R. Cardinale24, A. Cardini27, I. Carli4, P. Carniti26,j, L. Carus14, K. Carvalho Akiba32, A. Casais Vidal46, G. Casse60, M. Cattaneo48, G. Cavallero48, S. Celani49, J. Cerasoli10, A.J. Chadwick60, M.G. Chapman54, M. Charles13, Ph. Charpentier48, G. Chatzikonstantinidis53, C.A. Chavez Barajas60, M. Chefdeville8, C. Chen3, S. Chen4, A. Chernov35, V. Chobanova46, S. Cholak49, M. Chrzaszcz35, A. Chubykin38, V. Chulikov38, P. Ciambrone23, M.F. Cicala56, X. Cid Vidal46, G. Ciezarek48, P.E.L. Clarke58, M. Clemencic48, H.V. Cliff55, J. Closier48, J.L. Cobbledick62, V. Coco48, J.A.B. Coelho11, J. Cogan10, E. Cogneras9, L. Cojocariu37, P. Collins48, T. Colombo48, L. Congedo19,c, A. Contu27, N. Cooke53, G. Coombs59, G. Corti48, C.M. Costa Sobral56, B. Couturier48, D.C. Craik64, J. Crkovská67, M. Cruz Torres1, R. Currie58, C.L. Da Silva67, E. Dall’Occo15, J. Dalseno46, C. D’Ambrosio48, A. Danilina41, P. d’Argent48, A. Davis62, O. De Aguiar Francisco62, K. De Bruyn79, S. De Capua62, M. De Cian49, J.M. De Miranda1, L. De Paula2, M. De Serio19,c, D. De Simone50, P. De Simone23, J.A. de Vries80, C.T. Dean67, D. Decamp8, L. Del Buono13, B. Delaney55, H.-P. Dembinski15, A. Dendek34, V. Denysenko50, D. Derkach82, O. Deschamps9, F. Desse11, F. Dettori27,e, B. Dey77, P. Di Nezza23, S. Didenko83, L. Dieste Maronas46, H. Dijkstra48, V. Dobishuk52, A.M. Donohoe18, F. Dordei27, A.C. dos Reis1, L. Douglas59, A. Dovbnya51, A.G. Downes8, K. Dreimanis60, M.W. Dudek35, L. Dufour48, V. Duk78, P. Durante48, J.M. Durham67, D. Dutta62, A. Dziurda35, A. Dzyuba38, S. Easo57, U. Egede69, V. Egorychev41, S. Eidelman43,v, S. Eisenhardt58, S. Ek-In49, L. Eklund59,w, S. Ely68, A. Ene37, E. Epple67, S. Escher14, J. Eschle50, S. Esen13, T. Evans48, A. Falabella20, J. Fan3, Y. Fan6, B. Fang73, S. Farry60, D. Fazzini26,j, M. Féo48, A. Fernandez Prieto46, J.M. Fernandez-tenllado Arribas45, A.D. Fernez66, F. Ferrari20,d, L. Ferreira Lopes49, F. Ferreira Rodrigues2, S. Ferreres Sole32, M. Ferrillo50, M. Ferro-Luzzi48, S. Filippov39, R.A. Fini19, M. Fiorini21,f , M. Firlej34, K.M. Fischer63, D.S. Fitzgerald86, C. Fitzpatrick62, T. Fiutowski34, F. Fleuret12, M. Fontana13, F. Fontanelli24,h, R. Forty48, V. Franco Lima60, M. Franco Sevilla66, M. Frank48, E. Franzoso21, – 24 –