Measurement of |Vcb| with Bs0→Ds(*)-μ+νμ decays

The element jVcbj of the Cabibbo-Kobayashi-Maskawa matrix is measured using semileptonic Bs decays produced in proton-proton collision data collected with the LHCb detector at center-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3 fb−1. Rates of Bs → Ds μþνμ and Bs → D − s μþνμ decays are analyzed using hadronic form-factor parametrizations derived either by Caprini, Lellouch and Neubert (CLN) or by Boyd, Grinstein and Lebed (BGL). The measured values of jVcbj are ð41.4 0.6 0.9 1.2Þ × 10−3 and ð42.3 0.8 0.9 1.2Þ × 10−3 in the CLN and BGL parametrization, respectively. The first uncertainty is statistical, the second systematic, and the third is due to the external inputs used in the measurement. These results are in agreement with those obtained from decays of Bþ and B mesons. They are the first determinations of jVcbj at a hadron-collider experiment and the first using Bs meson decays.

μ + ν μ decays R. Aaij et al. * (LHCb Collaboration) (Received 10 January 2020; accepted 9 March 2020; published 20 April 2020) The element jV cb j of the Cabibbo-Kobayashi-Maskawa matrix is measured using semileptonic B 0 s decays produced in proton-proton collision data collected with the LHCb detector at center-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3 fb −1 . Rates of B 0 s → D − s μ þ ν μ and B 0 s → D Ã− s μ þ ν μ decays are analyzed using hadronic form-factor parametrizations derived either by Caprini, Lellouch and Neubert (CLN) or by Boyd, Grinstein and Lebed (BGL). The measured values of jV cb j are ð41.4 AE 0.6 AE 0.9 AE 1.2Þ × 10 −3 and ð42.3 AE 0.8 AE 0.9 AE 1.2Þ × 10 −3 in the CLN and BGL parametrization, respectively. The first uncertainty is statistical, the second systematic, and the third is due to the external inputs used in the measurement. These results are in agreement with those obtained from decays of B þ and B 0 mesons. They are the first determinations of jV cb j at a hadron-collider experiment and the first using B 0 s meson decays. DOI: 10.1103/PhysRevD.101.072004

I. INTRODUCTION
The semileptonic quark-level transitionb →cl þ ν l , where l is an electron or a muon, provides the cleanest way to access the strength of the coupling between the b and c quarks, expressed by the element jV cb j of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. 1 Two complementary methods have been used to determine jV cb j. One measures the decay rate by looking at inclusive b-hadron decays to final states made of a c-flavored hadron and a charged lepton; the other measures the rate of a specific (exclusive) decay, such as B 0 → D Ã ð2010Þ − μ þ ν μ or B 0 → D − μ þ ν μ . The average of the inclusive method yields jV cb j ¼ ð42.19 AE 0.78Þ × 10 −3 , while the exclusive determinations give jV cb j ¼ ð39.25 AE 0.56Þ × 10 −3 [1]. The two values are approximately three standard deviations apart, and this represents a long-standing puzzle in flavor physics.
Exclusive determinations rely on a parametrization of strong-interaction effects in the hadronic current of the quarks bound in mesons, the so-called form factors. These are Lorentz-invariant functions of the squared mass q 2 of the virtual W þ emitted in theb →c transition and are calculated using nonperturbative quantum chromodynamics (QCD) techniques, such as lattice QCD (LQCD) or QCD sum rules. Several parametrizations have been proposed to model the form factors [2][3][4][5][6][7]. The parametrization derived by Caprini, Lellouch and Neubert (CLN) [2] has been the reference model for the exclusive determinations of jV cb j. The approximations adopted in this parametrization have been advocated as a possible explanation for the discrepancy with the inclusive measurement [8][9][10][11]. A more general model by Boyd, Grinstein and Lebed (BGL) [3][4][5] has been used in recent high-precision measurements of jV cb j [12,13] to overcome the CLN limitations. However, no significant difference in the jV cb j values measured with the two parametrizations has been found and the issue remains open [14][15][16][17].
All exclusive measurements of jV cb j performed so far make use of decays of B þ and B 0 mesons. The study of other b-hadron decays, which are potentially subject to different sources of uncertainties, can provide complementary information and may shed light on this puzzle. In particular, semileptonic B 0 s decays, which are abundant at the LHC, have not yet been exploited to measure jV cb j. Exclusive semileptonic B 0 s decays are more advantageous from a theoretical point of view. The larger mass of the valence s quark compared to u or d quarks makes LQCD calculations of the form factors for B 0 s decays less computationally expensive than those for B þ or B 0 decays, thus possibly allowing for a more precise determination of jV cb j [18][19][20][21]. Calculations of the form factor over the full q 2 spectrum are available for B 0 s → D − s l þ ν l decays [22,23] and can be used along with experimental data to measure jV cb j. Exclusive B 0 s → D − s l þ ν l and B 0 s → D Ã− s l þ ν l decays are also experimentally appealing because background contamination from partially reconstructed decays is expected to be less severe than for their B þ=0 counterparts. Indeed, the majority of the excited states of the D − s meson (other than D Ã− s ) are expected to decay dominantly into D ðÃÞ K final states. This paper presents the first determination of jV cb j from the exclusive decays B 0 s → D − s μ þ ν μ and B 0 s → D Ã− s μ þ ν μ . The analysis uses proton-proton collision data collected with the LHCb detector at center-of-mass energies of 7 and 8 TeV and corresponding to an integrated luminosity of 3 fb −1 . In both decays, only the D − s μ þ final state is reconstructed using the Cabibbo-favored mode D − s → ½K þ K − ϕ π − , where the kaon pair is required to have invariant mass in the vicinity of the ϕð1020Þ resonance. The photon or the neutral pion emitted along with the D − s in the D Ã− s decay is not reconstructed. The value of jV cb j is determined from the observed yields of B 0 s decays normalized to those of reference B 0 decays after correcting for the relative reconstruction and selection efficiencies. The reference decays are chosen to be B 0 → D − μ þ ν μ and B 0 → D Ã− μ þ ν μ , where the D − meson is reconstructed in the Cabibbo-suppressed mode D − →½K þ K − ϕ π − . Hereafter the symbol D Ã− refers to the D Ã ð2010Þ − meson. Signal and reference decays thus have identical final states and similar kinematic properties. This choice results in a reference sample of smaller size than that of the signal but allows suppressing systematic uncertainties that affect the calculation of the efficiencies. Using the B 0 decays as a reference, the determination of jV cb j needs in input the measured branching fractions of these decays and the ratio of B 0 s -to B 0 -meson production fractions. The latter is measured by LHCb using an independent sample of semileptonic decays with respect to that exploited in this analysis [24], and it assumes universality of the semileptonic decay width of b hadrons [25]. The ratios of the branching fractions of signal and reference decays, are also determined from the same analysis. From the measured branching fractions of the reference decays, the branching fractions of B 0 s → D − s μ þ ν μ and B 0 s → D Ã− s μ þ ν μ decays are determined for the first time.
This analysis uses either the CLN or the BGL parametrization to model the form factors, with parameters determined by analyzing the decay rates using a novel method: instead of approximating q 2 , which cannot be determined precisely because of the undetected neutrino, a variable that can be reconstructed fully from the final-state particles and that preserves information on the form factors is used. This variable is the component of the D − s momentum perpendicular to the B 0 s flight direction, denoted as p ⊥ ðD − s Þ. The p ⊥ ðD − s Þ variable is highly correlated with the q 2 value of the B 0 s → D − s μ þ ν μ and B 0 s → D Ã− s μ þ ν μ decays and, to a minor extent, with the helicity angles of the B 0 s → D Ã− s μ þ ν μ decay. When used together with the corrected mass m corr , it also helps in determining the sample composition. The corrected mass is calculated from the mass of the reconstructed particles, mðD − s μ þ Þ, and from the momentum of the D − s μ þ system transverse to the Signal and background decays accumulate in wellseparated regions of the two-dimensional space spanned by m corr and p ⊥ ðD − s Þ. A fit to the data distribution in the m corr versus p ⊥ ðD − s Þ plane identifies the B 0 s → D − s μ þ ν μ and B 0 s → D Ã− s μ þ ν μ signal decays and simultaneously provides a measurement of jV cb j and of the form factors.
The paper is structured as follows. The formalism describing the semileptonic B 0 ðsÞ decays and the parametrization of their form factors is outlined in Sec. II. Section III gives a brief description of the LHCb detector and of the simulation software. The selection and the expected composition of the signal and reference samples are presented in Sec. IV. Section V describes the method used to measure jV cb j and the other parameters of interest. The determination of the reference B 0 -decay yields is reported in Sec. VI, and the analysis of the signal B 0 s decays is discussed in Sec. VII. Section VIII describes the systematic uncertainties affecting the measurements and Sec. IX presents the final results, before concluding.

II. FORMALISM
The formalism used to describe the decay rate of a B meson into a semileptonic final state with a pseudoscalar or a vector D meson is outlined here. In this section, the notation B → D ðÃÞ μν is used to identify both B 0 → D ðÃÞ− μ þ ν μ and B 0 s → D ðÃÞ− s μ þ ν μ decays, clarifying when the distinction is relevant.
The B → D Ã μν differential decay rate can be expressed in terms of one recoil variable, w, and three helicity angles, θ μ , θ D and χ, as where G F is the Fermi constant and the coefficient η EW ≈ 1.0066 accounts for the leading-order electroweak correction [26]. The recoil variable is defined as the scalar product of the four-velocities of the B and D Ã mesons, with m BðD Ã Þ being the mass of the B (D Ã ) meson. The minimum value w ¼ 1 corresponds to zero recoil of the D Ã meson in the B rest frame, i.e., the largest kinematically allowed value of q 2 . The helicity angles (represented in Fig. 1) are θ μ , the angle between the direction of the muon in the W rest frame and the direction of the W boson in the B rest frame; θ D , the angle between the direction of the D in the D Ã rest frame and the direction of the D Ã in the B rest frame; and χ, the angle between the plane formed by the D Ã decay products and that formed by the two leptons. In the limit of massless leptons, the decay amplitude A can be decomposed in terms of three amplitudes H AE=0 ðwÞ, corresponding to the three possible helicity states of the D Ã meson, and its squared modulus is written as with the H i and k i terms defined in Table I. The helicity amplitudes are expressed by three form factors, h A 1 ðwÞ, R 1 ðwÞ, and R 2 ðwÞ, as with r ¼ m D Ã =m B and The CLN parametrization uses dispersion relations and reinforced unitarity bounds based on heavy quark effective theory to derive simplified expressions for the form factors [2]. For the B → D Ã μν case, the three form factors are written as [2] where the same numerical coefficients, originally computed for B 0 decays, are considered also for B 0 s decays, and where the conformal variable z is defined as The form factors depend only on four parameters: ρ 2 , R 1 ð1Þ, R 2 ð1Þ and h A 1 ð1Þ. The BGL parametrization follows from more general arguments based on dispersion relations, analyticity, and crossing symmetry [3][4][5]. In the case of B → D Ã μν decays, the form factors are written in terms of three functions, fðwÞ, gðwÞ and F 1 ðwÞ, as follows: These functions are expanded as convergent power series of z as n¼0 a n z n ; ð17Þ Here, the P 1 AE ðzÞ functions are known as Blaschke factors for the J P ¼ 1 AE resonances, and ϕ f;g;F 1 ðzÞ are the so-called outer functions. Adopting the formalism of Ref. [27], the Blaschke factors take the form where t AE ¼ ðm B AE m D Ã Þ 2 , and m k denotes the pole masses of the kth excited B þ c states that are below the BD Ã threshold and have the appropriate J P quantum numbers. The constants C 1 AE are scale factors calculated to use in B 0 s decays the same Blaschke factor derived for B 0 decays. The outer functions are defined as where n I ¼ 2.6 is the number of spectator quarks (three), corrected for SUð3Þ-breaking effects [8]. The B þ c resonances used in the computation of the Blaschke factors, thẽ χ 1 AE ð0Þ coefficients of the outer functions, and the constants C 1 AE are reported in Table II. The coefficients of the series in Eqs. (16)-(18) are bound by the unitarity constraints The first coefficient of fðzÞ, b 0 , is related to h A 1 ð1Þ by the expression while c 0 is fixed from b 0 through In the B → Dμν case, the decay rate only depends upon the recoil variable w ¼ v B · v D . In the limit of negligible lepton masses, the differential decay rate can be written as [28] dΓðB → DμνÞ In the CLN parametrization, using the conformal variable zðwÞ defined in Eq. (12), the form factor GðzÞ is expressed in terms of its value at zero recoil, Gð0Þ, and a slope parameter, ρ 2 , as [2] In the BGL parametrization, it is expressed as [3][4][5] with r ¼ m D =m B and The outer function ϕðzÞ is defined as The coefficients of the series in Eq. (30) are bound by unitarity, with the coefficient d 0 being related to Gð0Þ through

III. DETECTOR AND SIMULATION
The LHCb detector [29,30] 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 a track to a primary vertex, the impact parameter, is measured with a resolution of ð15 þ 29=p T Þ 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. Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillatingpad and preshower detectors and an electromagnetic and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers.
Simulation is required to model the expected sample composition and develop the selection requirements, to calculate the reconstruction and selection efficiencies, and to build templates describing the distributions of signal and background decays used in the fit that determines the parameters of interest. In the simulation, pp collisions are generated using PYTHIA [31] with a specific LHCb configuration [32]. Decays of unstable particles are described by EvtGen [33], in which final-state radiation is generated using PHOTOS [34]. The interaction of the generated particles with the detector, and its response, are implemented using the GEANT4 toolkit [35] as described in Ref. [36]. The simulation is corrected for mismodeling of the reconstruction and selection efficiency, of the response of the particle identification algorithms, and of the kinematic properties of the generated B 0 ðsÞ mesons. The corrections are determined by comparing data and simulation in large samples of control decays, such as Residual small differences between data and the corrected simulation are accounted for in the systematic uncertainties.

IV. SELECTION AND EXPECTED SAMPLE COMPOSITION
The selection of the B 0 ðsÞ → D ðÃÞ− ðsÞ μ þ ν μ candidates closely follows that of Ref. [37]. Online, a trigger [38] selects events containing a high-p T muon candidate associated with one, two, or three charged particles, all with origins displaced from the collision points. In the offline reconstruction, the muon candidate is combined with three charged particles consistent with the topology and kin- mass is restricted to be in the ranges ½1.945; 1.995 GeV=c 2 and ½1.850; 1.890 GeV=c 2 to define the inclusive samples of D − s μ þ signal and D − μ þ reference candidates, respectively. Cross-contamination between signal and reference samples is smaller than 0.1%, as estimated from simulation. The K þ K − mass must be in the range ½1.008; 1.032 GeV=c 2 , to suppress the background under the D − ðsÞ peaks and ensure similar kinematic distributions for signal and reference decays. Same-sign D − ðsÞ μ − candidates are also reconstructed to model combinatorial background from accidental D − ðsÞ μ þ associations. The candidate selection is optimized toward suppressing the background under the charm signals and making same-sign candidates a reliable model for the combinatorial background: track-and vertexquality, vertex-displacement, transverse-momentum, and particle-identification criteria are chosen to minimize shape and yield differences between same-sign and signal candidates in the mðD − ðsÞ μ þ Þ > 5.5 GeV=c 2 region, where genuine b-hadron decays are kinematically excluded and combinatorial background dominates. Mass vetoes suppress to a negligible level background from misreconstructed decays, such as B 0 where the proton is misidentified as a kaon or a pion (and X indicates other possible final-state particles), and B 0 ðsÞ → D − ðsÞ π þ decays where the pion is misidentified as a muon. The requirement p ⊥ ðD − ðsÞ Þ ½GeV=c < 1.5 þ 1.1 × ðm corr ½GeV=c 2 − 4.5Þ is imposed to suppress background from all other partially reconstructed b-hadron decays, as shown in Fig. 2 for B 0 s decays. Tighter and looser variations of this requirement are used in Sec. VIII to estimate the systematic uncertainty due to the residual background contamination.
A total of 2.72 × 10 5 D − s μ þ and 0.82 × 10 5 D − μ þ candidates satisfy the selection criteria. Simulation is used to describe all sources of b-hadron decays contributing to these inclusive samples. Assuming for B 0 s → D − s μ þ ν μ and B 0 s → D Ã− s μ þ ν μ decays the same branching fractions as decays are expected to constitute about 30% and 60% of the inclusive sample of the selected D − s μ þ candidates, while B 0 → D − μ þ ν μ and B 0 → D Ã− μ þ ν μ decays are expected to constitute about 50% and 30% of the D − μ þ sample. The lower expected fraction of semimuonic decays into D Ã ðsÞ mesons for B 0 decays compared to B 0 s decays is due to the branching fraction of 5Þ is drawn as a dashed line; the dot-dashed line shows the tighter requirement, applied on top of the baseline, which is used in Sec. VIII to further suppress background and assess the systematic uncertainty due to the residual contamination.
The decays considered here are those into D Ã s0 ð2317Þ − and D s1 ð2460Þ − mesons, because these states have a mass below the kinematic threshold required to decay strongly into DK or D Ã K final states. Decays into the D s1 ð2536Þ − meson are also considered, even if this state is above the D Ã K threshold, because it has been observed to decay to a D − s meson [39]. Branching fractions for these B 0 s decays are not known, but, based on the yields measured in Ref. [37], they are estimated to be a few percent of the D − s μ þ sample. Background from semileptonic B þ decays into a D − μ þ X final state is expected to be about 9% of the D − μ þ sample, including both semimuonic and semitauonic decays, with τ þ → μ þ ν μντ . Semitauonic B 0 ðsÞ decays are estimated to contribute less than 1% to both the D − s μ þ and D − μ þ samples, comprising all decays into D decaying into a pair of charm hadrons, where one hadron is the fully reconstructed D − ðsÞ candidate and the other decays semileptonically. While this background is expected to be negligible in the D − μ þ sample, it is estimated to be about 2% of the D − s μ þ sample, following Ref. [37]. Such Cross-feed semileptonic B 0 s decays can be neglected in the inclusive D − μ þ sample, whereas those of B 0 and B þ decays to final states with a D − s candidate and an unreconstructed kaon, such as B → D ðÃÞ− s Kμ þ ν μ , must be considered in the D − s μ þ sample. This contamination is estimated to be at most 2%.
Reconstruction and selection efficiencies are determined from simulation. Given that signal decays are measured relative to reference B 0 decays, only efficiency ratios are needed. They are measured to be 1.568 AE 0.008 They depart from unity mainly because of the requirement on mðK þ K − Þ to be around the ϕð1020Þ mass. This requirement reduces systematic uncertainties due to the modeling of trigger and particle-identification criteria. However, its efficiency relies on an accurate description in the simulation of the D − ðsÞ → K þ K − π − amplitude model; a systematic uncertainty is assigned to cover for a possible mismodeling, as discussed in Sec. VIII. An additional difference between the efficiency of signal and reference decays originates from the D − lifetime being about 2 times longer than the D − s lifetime [39]. The trigger selection is more efficient for decays with closely spaced B 0 ðsÞ and D − ðsÞ vertices, favoring smaller D − ðsÞ flight distances and hence decay times [37]. As a consequence, the efficiency for selecting D − s μ þ candidates in the trigger is about 10% larger than that for D − μ þ candidates.

V. ANALYSIS METHOD
Signal and reference yields can be precisely measured through a fit to the corrected mass distribution following the method of Ref. [37]. To be able to access the form factors, yields are measured as a function of the recoil variable w and of the helicity angles, as discussed in Sec. II. However, these quantities cannot be computed precisely because of the undetected neutrino and the inability to resolve the b-hadron kinematic properties by balancing it against the accompanyingb hadron produced in the event, as done in e þ e − collisions.
Approximate methods, based on geometric and kinematic constraints and on the assumption that only the neutrino is undetected, allow the determination of these quantities up to a twofold ambiguity in the neutrino momentum component parallel to the b-hadron flight direction [40][41][42][43]. Such an ambiguity can be resolved, e.g., by using multivariate regression algorithms [44] or by imposing additional constraints on the b-hadron production [45]. These approximate methods have already been successfully used by several LHCb analyses of semileptonic b-hadron decays [46][47][48][49]. However, Oð20%Þ inefficiencies are introduced because, due to resolution effects, the second-order equation responsible for the twofold ambiguity does not always have real solutions. The inability to use candidates for which no real solutions are found also restricts the candidate m corr values to be smaller than the nominal B 0 ðsÞ mass, thus reducing the discriminating power between the different sample components.
To overcome such problems, a novel approach is adopted in this analysis. In B 0 ðsÞ → D − ðsÞ μ þ ν μ decays the component of the D − ðsÞ momentum perpendicular to the B 0 ðsÞ flight direction, p ⊥ ðD − ðsÞ Þ, is opposite and equal in magnitude to the component of the W þ momentum vector that is perpendicular to the B 0 ðsÞ flight direction. Therefore, p ⊥ ðD − ðsÞ Þ is highly correlated with w, as shown in the top-left distribution of Fig. 3 for B 0 s decays. In B 0 ðsÞ → D Ã− ðsÞ μ þ ν μ decays the correlation is kept, as shown in the top-right distribution of Fig. 3, because the unreconstructed photon or pion from the D Ã− ðsÞ decay carries very little energy, which only leads to a small dilution. In these decays, the p ⊥ ðD − ðsÞ Þ variable is also correlated, albeit to a lesser extent, with the helicity angles θ μ and θ D , as shown in the bottom distributions of Fig. 3 for B 0 s decays. Through such correlations, the distribution of p ⊥ ðD − ðsÞ Þ has a strong dependence on the form factors, particularly on GðwÞ for the scalar case and on h A 1 ðwÞ for the vector case. Therefore, the form factors can be accessed by analysing the shape of the p ⊥ ðD − ðsÞ Þ distribution of the signal decays, with no need to estimate the momentum of the unreconstructed particles. The p ⊥ ðD − ðsÞ Þ variable has the experimental advantage of being reconstructed fully from the tracks of the D − ðsÞ decay products and from the wellmeasured origin and decay vertices of the B 0 ðsÞ meson. It is also correlated with m corr , and the two variables together provide very efficient discrimination between signal and background decays, which accumulate in different regions of the two-dimensional space spanned by m corr and p ⊥ ðD − ðsÞ Þ, as already shown in Fig. 2 for B 0 s decays. A least-squares fit to the m corr -p ⊥ ðD − ðsÞ Þ distribution of the selected inclusive samples of D − ðsÞ μ þ candidates is used to simultaneously determine the form factors and (signal) reference yields that are needed for the measurement of jV cb j or of the ratios of branching fractions R ðÃÞ . In the fit, the data are described by several fit components, which will be detailed later, separately for the B 0 and B 0 s cases. The shape of each component in the m corr -p ⊥ ðD − ðsÞ Þ space is modeled with two-dimensional histogram templates derived either from simulation (for signal, reference and all physics background decays) or from same-sign data candidates (for combinatorial background). The binning of the histograms is chosen such that there are at least 15 entries per bin (for both data and templates distributions), to guarantee unbiased estimates of the least-squares fit. A few bins at the edges of the m corr -p ⊥ ðD − ðsÞ Þ space have a smaller number of entries, but studies performed on pseudoexperiments show that they do not introduce biases in the fit results.
Signal templates are built using a per-candidate weight calculated as the ratio between the differential decay rate featuring a given set of form-factor parameters and that with the parameters used in the generation of the simulated samples. The set of parameters of the differential decay rate at the numerator is varied in the least-squares minimization. The differential decay rates are given in Eq. form-factor parameters probed at that iteration. With this weighting procedure, all efficiency and resolution effects are accounted for, making the templates independent of the form-factor values assumed in the generation of the simulated candidates.
In the fit, the yield of each component is a free parameter. To determine jV cb j, the signal yields N ðÃÞ sig are expressed as the integral of the differential decay rates multiplied by the B 0 s lifetime τ. The signal yields are normalized to the yields N ðÃÞ ref and to the measured branching fractions of the reference B 0 modes, correcting for the efficiency ratios between signal and reference decays, ξ ðÃÞ . The full expression for the signal yields is where the integral is performed over ζ with f s =f d being the ratio of B 0 s -to B 0 -meson production fractions. The dependence on jV cb j in Eq. (34) is enclosed in the differential decay rate of Eqs. (4) and (27). The other parameters entering the differential decay rate are either left free to float in the fit, together with jV cb j, or constrained to external determinations by a penalty term in the least-squares function, as detailed in the following sections. A similar fit is performed to determine the ratios of branching fractions, with the difference that the expression of the signal yields simplifies to and R and R Ã become free parameters instead of jV cb j.
In the fit to the reference sample, the yields are free parameters, not expressed in terms of jV cb j. Their histogram templates are functions of the form factors and are allowed to float in the fit.

VI. FIT TO THE REFERENCE SAMPLE
The reference yields N ðÃÞ ref are determined by fitting the m corr -p ⊥ ðD − Þ distribution of the inclusive D − μ þ sample using the following four components: the two reference decays B 0 → D − μ þ ν μ and B 0 → D Ã− μ þ ν μ ; physics background due to the sum of semileptonic B 0 feed-down and B þ → D − μ þ X decays; and combinatorial background. The B 0 → D Ã− μ þ ν μ template is generated assuming a fraction of approximately 5% for D Ã− → D − γ decays and 95% for D Ã− → D − π 0 decays, according to the measured D Ã− branching fractions [39]. The physics background components are grouped together into a single template because their m corr -p ⊥ ðD − Þ distributions are too similar to be discriminated by the fit. A contribution from semitauonic decays is neglected because its yield is found to be consistent with zero in an alternate fit in which this component is included, and no significant change of the reference yields is observed. The fit parameters are N ðÃÞ ref , the yields of the background components and the B 0 → D ðÃÞ− μ þ ν μ form-factor parameters expressed in the CLN parametrization: ρ 2 ðD − Þ, ρ 2 ðD Ã− Þ, R 1 ð1Þ and R 2 ð1Þ. Given the limited size of the D − μ þ samples, the CLN parametrization is preferred over BGL because of its reduced number of free parameters.
The reference yields are determined to be N ref ¼ ð36.4 AE 1.6Þ × 10 3 and N Ã ref ¼ ð27.8 AE 1.2Þ × 10 3 with a correlation of −70.3%. These results do not depend significantly on the choice of the form-factor parametrization. The onedimensional projections of the fit on the m corr and p ⊥ ðD − Þ variables are shown in Fig. 4. The fit describes the data well with a minimum χ 2 =ndf of 76=70, corresponding to a p value of 29%. The form-factor parameters are measured to be in agreement with their world-average values [1], with relative uncertainties ranging from 20% to 50% depending on the parameter.

VII. FIT TO THE SIGNAL SAMPLE
The fit function for the D − s μ þ sample features five components: the two signal decays B 0 s → D − s μ þ ν μ and B 0 s → D Ã− s μ þ ν μ ; a background component made by the sum of semimuonic B 0 s feed-down decays and b-hadron decays to a doubly charmed final state; a background component made by the sum of cross-feed semileptonic B 0 decays and semitauonic B 0 s decays; and combinatorial background. The B 0 s → D Ã− s μ þ ν μ template is generated assuming a fraction of approximately 94% for D Ã− s → D − s γ decays and 6% for D Ã− s → D − s π 0 decays, according to the measured D Ã− s branching fractions [39]. The physics background components that are merged together in the two templates have very similar shapes in the m corr versus p ⊥ ðD − s Þ plane and cannot be discriminated by the fit when considered as separate components. They are therefore merged according to the expected approximate fractions.
The yields of the five components are free parameters in the fit, with the signal yields expressed in terms of the parameters of interest according to Eq. (34), when determining jV cb j, or Eq. (38), when determining R ðÃÞ . The measurement relies on the external inputs reported in Tables III and IV. Correlations between external inputs, e.g., between N ref and N Ã ref or between the LQCD inputs, are accounted for in the fit. The value of f s =f d is derived from the measurement of Ref. [24], which is the most precise available. It is obtained using an independent sample of semileptonic B 0 ðsÞ decays collected with the LHCb detector in pp collisions at the center-of-mass energy of 13 TeV. This measurement uses the branching fraction of the D − s → K þ K − π − decay and the B 0 s lifetime as external inputs [39]. To properly account for all correlations, the value of the product f s =f d × BðD − s → K − K þ π − Þ × τ is derived directly from Ref. [24]. The measured dependence of f s =f d on the collision energy [50] is also accounted for in the computation, by scaling the 13 TeV measurement to the value at 7 and 8 TeV needed in this analysis. All other branching fractions and the particle masses are taken from Ref. [39]. The external inputs listed in Table IV are based exclusively on theory calculations: η EW and h A 1 ð1Þ are constrained to the values reported in Refs. [26,18], respectively; the constraints on the B 0 s → D − s μ þ ν μ form factors are based on the LQCD calculations of Ref. [23], which provide the form factor f þ ðzÞ over the full q 2 spectrum using the parametrization proposed by Bourrely, Caprini and Lellouch (BCL) [6]. In Appendix A, the corresponding CLN and BGL parameters reported in Table IV are derived.

A. Determination of jV cb j with the CLN parametrization
The analysis in the CLN parametrization uses the form factors defined in Eqs. (9)-(11), for B 0 s → D Ã− s μ þ ν μ decays, and in Eq. (28), for B 0 s → D − s μ þ ν μ decays. The form-factor parameters ρ 2 ðD Ã− s Þ, R 1 ð1Þ, and R 2 ð1Þ are free to float in the fit, while h A 1 ð1Þ, Gð0Þ and ρ 2 ðD − s Þ are constrained. One-dimensional projections of the fit results on m corr and p ⊥ ðD − s Þ are shown in Fig. 5. The fit has a minimum χ 2 =ndf of 279=285, corresponding to a p value of 58%. The results for the parameters of interest are reported in Table V. In addition to jV cb j, these include the form-factor parameters that are determined exclusively by the data,

Parameter
Value Reference 0.0191 AE 0.0008 [24,50] [23] such as ρ 2 ðD Ã− s Þ, R 1 ð1Þ and R 2 ð1Þ, and those for which the precision improves compared to the external constraints, such as Gð0Þ and ρ 2 ðD − s Þ. Detailed fit results for all parameters, including their correlations, are reported in Appendix B. The uncertainties returned by the fit include the statistical contribution arising from the limited size of the data and simulation samples (stat) and the contribution due to the external inputs (ext). The calculation of this latter contribution is detailed in Sec. VIII. The value of jV cb j, ð41.4 AE 0.6ðstatÞ AE 1.2ðextÞÞ × 10 −3 , agrees with the exclusive determination from B þ and B 0 decays [1]. When only Gð0Þ is constrained and ρ 2 ðD − s Þ is left free, the fit returns ρ 2 ðD − s Þ ¼ 1.30 AE 0.06ðstatÞ, in agreement with the LQCD estimation, and jV cb j ¼ ð41.8 AE 0.8ðstatÞ AE 1.2ðextÞÞ × 10 −3 . Including the constraint on ρ 2 ðD − s Þ improves the statistical precision on jV cb j by about 20% and also that on Gð0Þ by 10%, because of the large correlation between Gð0Þ and ρ 2 ðD − s Þ.

B. Determination of jV cb j with the BGL parametrization
The BGL form-factor functions are given by Eqs. (13) (25) and (26). The coefficients b 1 , a 0 , a 1 , and c 1 are free parameters. For B 0 s → D − s μ þ ν μ decays, the expansion of the f þ ðzÞ form factor is truncated after the second order in z and the coefficients d 0 , d 1 and d 2 are constrained to the values obtained in Appendix A using Ref. [23], with d 0 expressed in terms of the parameter Gð0Þ using Eq. (33). No constraints from the unitarity bounds of Eqs. (24) and (32) are imposed, to avoid potential biases on the parameters or fit instabilities due to convergence at the boundary of the parameter space.
The fit has minimum χ 2 =ndf of 276=284, corresponding to a p value of 63%. Figure 6 shows a comparison of the p ⊥ ðD − s Þ background-subtracted distributions obtained with the CLN and BGL fits. No significant differences are found between the two fits for both B 0 s → D − s μ þ ν μ and B 0 s → D Ã− s μ þ ν μ decays. The fit results for the parameters of interest are reported in Table VI. Detailed fit results for all parameters, including their correlations, are reported in Appendix B. The values found for the form-factor coefficients satisfy the unitarity bounds of Eqs. (24) and (32). The value of jV cb j is found to be ð42.3 AE 0.8ðstatÞ AE 1.2ðextÞÞ × 10 −3 , in agreement with the CLN analysis. The correlation between the BGL and CLN results is 34.0%. When only Gð0Þ is constrained and d 1 and d 2 are left free, jV cb j is found to be ð42.2 AE 1.5ðstatÞ AE 1.2ðextÞÞ × 10 −3 . The constraints on d 1 and d 2 improve the statistical precision on jV cb j by about 50% and that on Gð0Þ by 10%. Without such constraints, the fit returns d 1 ¼ 0.02 AE 0.05ðstatÞ and d 2 ¼ −0.9 AE 0.8ðstatÞ, both in agreement with the LQCD estimations and within the unitarity bound of Eq. (32).
Variations of the orders of the form-factor expansions have been probed for the B 0 s → D Ã− s μ þ ν μ decay, while for  the B 0 s → D − s μ þ ν μ decay the expansion is kept at order z 2 to exploit the constraints on d 1 and d 2 . A first alternative fit, where only the order zero of the g series is considered by fixing a 1 to zero, returns a p value of 62% and jV cb j ¼ ð41.7 AE 0.6ðextÞ AE 1.2ðextÞÞ × 10 −3 , in agreement with the nominal result of Table VI. The shift in the central value of jV cb j is consistent with that observed in pseudoexperiments where data are generated by using the nominal truncation and fit with the zero-order expansion of g. In a second alternative fit, g is kept at order zero and f is expanded at order z 2 , by adding the coefficient b 2 as a free parameter. The fit has a p value of 64% and returns jV cb j ¼ ð42.2 AE 0.8ðstatÞAE1.2ðextÞÞ × 10 −3 and b 2 ¼ 1.9AE 1.4ðstatÞ. Configurations at lower order than those considered for f and F 1 lead to poor fit quality and are discarded. Higher orders than those discussed here are not considered because they result in fit instabilities and degrade the sensitivity to jV cb j and to the form-factor coefficients.

C. Determination of R and R Ã
The ratios of B 0 s to B 0 branching fractions are determined by a fit where the signal yields are expressed using Eq. (38), with R and R Ã as free parameters. In the fit, the constraint on f s =f d × BðD − s → K þ K − π − Þ is obtained by dividing the value of the first row of Table III by the B 0 s lifetime τ [39]. The form factors are expressed in the CLN parametrization and a systematic uncertainty is assigned for this arbitrary choice, as discussed in Sec. VIII. The fit returns R ¼ 1.09 AE 0.05ðstatÞ AE 0.05ðextÞ and R Ã ¼ 1.06 AE 0.05ðstatÞ AE 0.05ðextÞ, with a p value of 59%. Detailed fit results for all fit parameters, including their correlations, are reported in Appendix B.

VIII. SYSTEMATIC UNCERTAINTIES
Systematic uncertainties affecting the measurements can be split into two main categories: those due to external inputs, indicated with (ext), and those due to the experimental methods, indicated with (syst). The individual contributions for each category are discussed in the following and are reported in Table VII, together with the statistical uncertainties.
The uncertainties returned by the fit include the statistical contribution arising from the finite size of the data and simulation samples and the contribution due to the external inputs that constrain some of the fit parameters through penalty terms in the least-squares function. To evaluate the purely statistical component, a second fit is performed with all external parameters fixed to the values determined by the first fit. The contribution due to the external inputs is then obtained by subtracting in quadrature the uncertainties from the two sets of results. The procedure is repeated for each individual input to estimate its contribution to the uncertainty. The results are reported in the upper section of Table VII. Here the uncertainty on f s =f d × BðD − s → K þ K − π − Þð×τÞ comprises also that due to a difference in the distribution of the transverse momentum of the D − ðsÞ μ þ system with respect to Ref. [24], which results in a relative 1% change of the value of f s =f d . The branching fractions   of the B 0 decays taken in input are obtained from averages that assume isospin symmetry in decays of the ϒð4SÞ meson [39]. This symmetry is observed to hold with a precision of 1%-2%, and no uncertainty is assigned. However, it is noted that considering the correction suggested in Ref. [51] increases the value of jV cb j by 0.2 × 10 −3 in both the CLN and BGL parametrizations. The efficiency of the requirement that limits mðK þ K − Þ to be around the ϕð1020Þ mass is evaluated using simulation. Given that the simulated model of the intermediate amplitudes contributing to the D − ðsÞ → K þ K − π − decays may be inaccurate, a systematic uncertainty is estimated by comparing the efficiency of the mðK þ K − Þ requirement derived from simulation with that based on data from an independent control sample of D − ðsÞ → K þ K − π − decays. The efficiency ratios ξ ðÃÞ change by a relative −4% when substituting the simulation-based efficiency of the mðK þ K − Þ requirement with that determined from data. This variation modifies the values of jV cb j, R and R Ã found in the fit, while producing negligible shifts to the formfactor parameters. The differences with respect to the nominal values are assigned as systematic uncertainties.
The knowledge of the physics backgrounds contributing to the inclusive D − s μ þ sample is limited by the lack of experimental measurements of exclusive semileptonic B 0 s decays. These background components are, however, well separated in the m corr versus p ⊥ ðD − s Þ plane and their contribution is reduced to a few percent by the requirement p ⊥ ðD − s Þ ½GeV=c < 1.5 þ 1.1 × ðm corr ½GeV=c 2 − 4.5Þ (dashed line in Fig. 2). To quantify by how much the assumed background composition can affect the determination of the parameters of interest, the fit is repeated by varying the requirements on the m corr versus p ⊥ ðD − ðsÞ Þ plane for both signal and reference samples. In the first variation, the more restrictive requirement p ⊥ ðD − ðsÞ Þ ½GeV=c < 0.7 þ 4.0 × ðm corr ½GeV=c 2 − 4.5Þ is added on top of the baseline selection to further halve the expected background fractions. This requirement is shown as a dot-dashed line in Fig. 2, for the B 0 s case. In the second variation, the baseline requirement is removed to allow maximum background contamination, which doubles with respect to that of the nominal selection. For both variations, the resulting samples are fit accounting for changes in the templates and in the efficiencies. The residuals for each parameter are computed as the difference between the values obtained in the alternative and baseline fits. The root-mean-square deviation of the residuals is taken as systematic uncertainty.
The analysis method is validated using large ensembles of pseudoexperiments, generated by resampling with repetitions (bootstrapping [52]) the samples of simulated signal and background decays and the same-sign data that model the combinatorial background. The relative proportions of signal and background components of the nominal fit to data are reproduced. Signal decays are generated by using both the CLN and BGL parametrizations with the form factors determined in the fit to data. Each sample is fit with the same form-factor parametrization used in the generation, and residuals between the fit and the generation values of each parameter are computed. The residuals that are observed to be at least two standard deviations different from zero are assigned as systematic uncertainties.
The simulated samples are corrected for mismodeling of the reconstruction and selection efficiency, of the response of the particle identification algorithms, and of the kinematic properties of the generated B 0 ðsÞ meson. A systematic uncertainty is assigned by varying the corrections within their uncertainties.
The measurement of R ðÃÞ is performed only in the CLN parametrization, because, as shown in Fig. 6, the signal templates are marginally affected by the choice of the formfactor parametrization. Nevertheless, a systematic uncertainty is assigned as the shift in the R ðÃÞ central values when fitting the data with the BGL parametrization.
The experimental systematic uncertainties are combined together, accounting for their correlations, in the middle section of Table VII. The correlations are reported in Appendix B.
As a consistency test, the fit is repeated by expressing the signal yields of the B 0 s → D − s μ þ ν μ and B 0 s → D Ã− s μ þ ν μ decays in terms of two different jV cb j parameters. The fit returns values of the two parameters in agreement with each other within one standard deviation.
Finally, a data-based null test of the analysis method is performed using a control sample of B 0 → D ðÃÞ− μ þ ν μ decays where the D − decays to the Cabibbo-favored K þ π − π − final state. These decays are normalized to the same B 0 → D ðÃÞ− μ þ ν μ decays, with D − → ½K þ K − ϕ π − , used in the default analysis to measure ratios of branching fractions between control and reference decays consistent with unity. The control sample is selected with criteria very similar to those of the reference sample, but the different D − final state introduces differences between the efficiencies of the control and reference decays that are 40% larger than those between signal and reference decays. The control sample features the same fit components as described in Sec. VI for the reference sample, with signal and background decays modeled with simulation and combinatorial background with same-sign data. External inputs are changed to reflect the replacement of the signal with the control decays. Fits are performed using both the CLN and the BGL parametrizations. In both cases, the ratios of branching fractions between control and reference decays are all measured to be compatible with unity with 5%-6% relative precision.

IX. FINAL RESULTS AND CONCLUSIONS
A study of the B 0 s → D − s μ þ ν μ and B 0 s → D Ã− s μ þ ν μ decays is performed using proton-proton collision data collected with the LHCb detector at center-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3 fb −1 . A novel analysis method is used to identify the two exclusive decay modes from the inclusive sample of selected D − s μ þ candidates and measure the CKM matrix element jV cb j using B 0 → D − μ þ ν μ and B 0 → D Ã− μ þ ν μ decays as normalization. The analysis is performed with both the CLN [2] and BGL [3][4][5] parametrizations to determine where the first uncertainties are statistical (including contributions from both data and simulation), the second systematic, and the third due to the limited knowledge of the external inputs. The two results are compatible, when accounting for their correlation. These are the first determinations of jV cb j from exclusive decays at a hadron collider and the first using B 0 s decays. The results are in agreement with the exclusive measurements based on B 0 and B þ decays and as well with the inclusive determination [1].
The ratios of the branching fractions of the exclusive B 0 s → D ðÃÞ− s μ þ ν μ decays relative to those of the exclusive B 0 → D ðÃÞ− μ þ ν μ decays are measured to be Taking the measured values of BðB 0 → D − μ þ ν μ Þ and BðB 0 → D Ã− μ þ ν μ Þ as additional inputs [39], the following exclusive branching fractions are determined for the first time: where the third uncertainties also include the contribution due to the limited knowledge of the normalization branching fractions. Finally, the ratio of B 0 s → D − s μ þ ν μ to B 0 s → D Ã− s μ þ ν μ branching fractions is determined to be The novel method employed in this analysis can also be used to measure jV cb j with semileptonic B 0 decays at LHCb. In this case, the uncertainty from the external inputs can be substantially decreased, as the dominant contribution in the current measurement is due to the knowledge of the B 0 s -to B 0 -meson production ratio f s =f d . The limiting factor for B 0 decays stems from the knowledge of the reference decays branching fractions, but these are expected to improve from new measurements at the Belle II experiment [53].

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:  [22,23] report LQCD calculations of the form-factor function over the full q 2 spectrum for B 0 s → D − s μ þ ν μ decays. The calculations differ in the methodology and in the treatment of the sea quarks, with Ref. [22] using ensembles that include 2 þ 1 flavors and Ref. [23] using 2 þ 1 þ 1 flavors. The two calculations agree.
The results reported in Ref. [23] are expressed in the BCL parametrization [6], with the series expanded up to order z 2 (see Appendix A of Ref. [23]). The parameters describing the f þ ðwÞ form factor are reported in Table VIII. To be used in this analysis, they need to be translated into the CLN and BGL parametrizations. For this purpose, 10 3 ensembles, each consisting of 10 7 q 2 values distributed according to f þ ðwÞ, are generated by sampling the BCL parameters within their covariance. Each sample is then fit with the CLN and BGL equations of Sec. II to derive the corresponding set of parameters. Each fit parameter features a Gaussian distribution. The central value and uncertainty of each parameter are defined as the mean and the width of these distributions, respectively. In the CLN parametrization, the derived parameters are Gð0Þ ¼ 1.07 AE 0.04 and ρ 2 ðD − s Þ ¼ 1.23 AE 0.05, with a correlation of 84.2%. Both values are in agreement with the results reported in Ref. [22], Gð0Þ ¼ 1.068 AE 0.040 and ρ 2 ðD − s Þ ¼ 1.244 AE 0.076. (A combination is not attempted because of the unknown correlation between the two LQCD calculations.) In the BGL parametrization, the derived parameters are Gð0Þ ¼ 1.07 AE 0.04, d 1 ¼ −0.012 AE 0.008 and d 2 ¼ −0.24 AE 0.05, with correlation coefficients ϱðGð0Þ; d 1 Þ ¼ −82.4%, ϱðGð0Þ; d 2 Þ ¼ −37.2% and ϱðd 1 ; d 2 Þ ¼ 10.0%.

APPENDIX B: DETAILED FIT RESULTS
Detailed results for the jV cb j fits, in both the CLN and BGL parametrizations, are reported in Table IX. The full correlation matrices are given in Tables X and XI, separately for the CLN and BGL configurations. Detailed results for the R and R Ã fit are given in Table XII, with  correlations in Table XIII.