Precision measurement of the $B_{c}^{+}$ meson mass

A precision measurement of the $B_{c}^{+}$ meson mass is performed using proton-proton collision data collected with the LHCb experiment at centre-of-mass energies of $7, 8$ and $13$ TeV, corresponding to a total integrated luminosity of $9.0 \,{\rm fb}^{-1}$. The $B_{c}^{+}$ mesons are reconstructed via the decays $B_{c}^{+} \rightarrow J\mskip -3mu/\mskip -2mu\psi\mskip 2mu \pi^+$, $B_{c}^{+} \rightarrow J\mskip -3mu/\mskip -2mu\psi\mskip 2mu \pi^+ \pi^- \pi^+$, $B_{c}^{+} \rightarrow J\mskip -3mu/\mskip -2mu\psi\mskip 2mu p \bar{p} \pi^+$, $B_{c}^{+} \rightarrow J\mskip -3mu/\mskip -2mu\psi\mskip 2mu D_{s}^{+}$, $B_{c}^{+} \rightarrow J\mskip -3mu/\mskip -2mu\psi\mskip 2mu D^{0} K^{+}$ and $B_{c}^{+} \rightarrow B_{s}^{0} \pi^{+}$. Combining the results of the individual decay channels, the $B_{c}^{+}$ mass is measured to be $6274.47 \pm 0.27 \,({\rm stat}) \pm 0.17 \,({\rm syst}) \mathrm{\,Me\kern -0.1em V}/c^{2}$. This is the most precise measurement of the $B_{c}^{+}$ mass to date. The difference between the $B_{c}^{+}$ and $B_{s}^{0}$ meson masses is measured to be $907.75 \pm 0.37 \,({\rm stat}) \pm 0.27 \,({\rm syst}) \mathrm{\,Me\kern -0.1em V}/c^{2}$.


Introduction
The B c meson family is unique in the Standard Model as its states contain two different heavy-flavour quarks, ab and a c quark. Quantum Chromodynamics (QCD) predicts that theb and c quarks are tightly bound in a compact system, with a rich spectroscopy of excited states. Studies of the B c mass spectrum can reveal information on heavy-quark dynamics and improve our understanding of the strong interaction. Due to the presence of two heavy-flavour quarks the mass spectrum of the B c states can be predicted with much better precision than many other hadronic systems. The mass spectrum of the B c family has been calculated with nonrelativistic quark potential models [1][2][3][4][5][6][7][8], nonperturbative phenomenological models [9], perturbative QCD [10,11], relativistic quark models [12][13][14][15], and lattice QCD [16][17][18][19][20][21]. The ground state of the B c meson family, denoted hereafter as B + c , decays only through the weak interaction, with a relatively long lifetime. The most accurate prediction of the B + c mass, M (B + c ) = 6278 ± 6 ± 4 MeV/c 2 [20], is obtained with unquenched lattice QCD.
This paper presents a precision measurement of the B + c mass using the decay modes B + c → J/ψ π + , B + c → J/ψ π + π − π + , B + c → J/ψ ppπ + , B + c → J/ψ D + s , B + c → J/ψ D 0 K + and B + c → B 0 s π + 1 . The first two decays are chosen for their large signal yield, while the others have a low energy release. The data sample corresponds to an integrated luminosity of 9.0 fb −1 , collected with the LHCb experiment in pp collisions at centre-of-mass energies of 7, 8 and 13 TeV. This measurement supersedes results reported in Refs. [23, 28-30].

Detector and simulation
This LHCb detector [45,46] 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 [47], 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 [48,49] 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 momentum scale is calibrated using samples of B + → J/ψ K + and J/ψ → µ + µ − decays collected concurrently with the data sample used for this analysis [50,51]. The relative accuracy of this procedure is determined to be 3 × 10 −4 using samples of other fully reconstructed B, Υ and K 0 S -meson decays. The minimum distance of a track to a primary vertex (PV), the impact parameter (IP), is measured with a resolution of (15 + 29/p T ) µm, where p T is the component of the momentum transverse to the beam, in GeV/c. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors [52]. Photons, electrons and hadrons are identified by a calorimeter system consisting of a 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 [53]. The online event selection is performed by a trigger [54], which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which performs a full event reconstruction.
Simulated samples are used to model the effects of the detector acceptance, optimise signal selection and validate the analysis technique. In simulation, pp collisions are generated using Pythia 8 [55] with an LHCb specific configuration [56]. The production of B + c mesons is simulated using the dedicated generator BcVegPy [57]. Decays of hadrons are described by EvtGen [58], in which final-state radiation is generated using Photos 3 [59]. The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [60] as described in Ref. [61].

Event selection
The B + c candidates are reconstructed in the following decay modes: A pair of oppositely charged muons form J/ψ candidates. The D + s candidates are reconstructed via the D + s → K + K − π + and D + s → π + π − π + decays, while the D 0 is reconstructed using the D 0 → K − π + decay. The B 0 s candidates are reconstructed in the decay modes B 0 A boosted decision tree [62][63][64] implemented within the TMVA [65] package optimises separation of the signal from combinatorial background for each decay mode. The classifiers are trained with simulated signal samples and a background proxy obtained from the upper mass sideband of the data, in the range [6.6, 7.0] GeV/c 2 . Kinematic variables that generically separate b-hadron decays from background are used in the training of the classifiers. The variables include the decay time, transverse momenta, vertex-fit quality of the B + c candidate, as well as variables related to the fact that the B + c meson is produced at the PV. The requirement on the classifiers is determined by maximising the signal significance S/ √ S + B, where S is the expected signal yield estimated using simulation, and B is the expected background yield evaluated in the upper sideband in data and extrapolated to the signal region.

Mass measurement
The B + c meson mass is determined in each decay mode by performing an unbinned maximum likelihood fit to the invariant mass distributions of the B + c candidates. The signal is described by a double-sided Crystal Ball (DSCB) function [66], while the background is described by an exponential function. The DSCB function comprises a Gaussian core with power-law tails to account for radiative effects. Parameters describing the radiative tails are determined from simulation. The invariant mass of the B + c candidates is calculated from a kinematic fit [67], in which the B + c candidate is assumed to originate from its PV and the intermediate-state masses are constrained to their known values [43]. The PV of the B + c candidate is that with respect to which it has the smallest χ 2 IP . The χ 2 IP is defined as the difference in χ 2 of the PV fit with and without the particle in question. For B + c → B 0 s π + decays, the B 0 s mass is constrained to the value of 5366.89 ± 0.21 MeV/c 2 , which is an average of the measurements of the B 0 s mass performed by the LHCb collaboration [68][69][70][71]. The difference between the B + c and B 0 s meson masses, and m(B 0 s ) are the reconstructed masses of B + c and B 0 s candidates. The mass difference ∆m is calculated with a kinematic fit [67], in which the B + c candidate is assumed to originate from the PV with the smallest χ 2 IP and the masses of the intermediate particles are constrained to their known values [43]. Figure 1 shows the invariant mass distributions and fit results for all B + c decay modes. Figure 2 shows the distributions of ∆m and fit results for the The signal yields, mass and resolution values as determined from fits to the individual mass distributions are given in Table 1. For the B + c → B 0 s π + decays, the results of the fits to the ∆m distribution are reported in Table 2. The reconstructed invariant-mass distribution is distorted due to the missing energy from unreconstructed photons (bremsstrahlung) emitted by final-state particles. The resulting bias in the extracted B + c mass is studied with simulated samples for each decay channel, and is used to correct the mass obtained from the fit. Multiple scattering in detector material can decrease the observed opening angles among the B + c decay products, affecting the reconstructed B + c mass and decay length and thereby the selection efficiency. The corresponding bias of the B + c mass measurement was studied with charmed hadrons (D + , D 0 , D + s , Λ + c ), and was found to be well reproduced by simulation [72]. A bias associated with the selection from simulated samples is assigned as a corresponding correction. The measured masses (M ) and mass difference (∆M ) are corrected for this bias (from -0.46 to 0.27 MeV/c 2 ) due to final-state radiation and the selection, and summarised in Table 1

Systematic uncertainties
To evaluate systematic uncertainties, the complete analysis is repeated varying assumed parameters, models and selection requirements. The observed differences in the B + c mass central values between the nominal result and the alternative estimates are considered as one standard-deviation uncertainties.
The systematic uncertainty of the B + c mass comprises uncertainties on the momentumscale calibration, energy loss corrections, signal and background models, the mass of the intermediate states and the uncertainty on the bias caused by the final-state radiation and selection.
The dominant source of systematic uncertainty arises due to the limited precision of the momentum-scale calibration. For each decay, this uncertainty is propagated to the      Due to the limited knowledge of the energy loss in the material of the tracking system, the amount of material traversed by final-state particles is known to 10% accuracy. This translates into a measured mass uncertainty of 0.03 MeV/c 2 for D 0 → K + K − π + π − decays [51]. The uncertainties on the B + c mass are scaled from that of the D 0 decay by the number of final-state particles. The uncertainties due to the limited size of simulated samples are taken as systematic uncertainties from the selection-induced bias on the B + c masses. The uncertainty on the masses of the intermediate states D + s , D 0 , B 0 s are propagated to the B + c mass measurement. The uncertainty related to the signal shape is estimated by using alternative signal models, including the sum of two Gaussian functions, a Hypatia function [73], the sum of a DSCB and a Gaussian function, and the sum of two DSCB functions. The differences of the fitted mass with final-state radiation corrections between the nominal and the alternative models are found to be smaller than 0.1 MeV/c 2 , which is taken as the corresponding systematic uncertainty. The uncertainty related to the background description is evaluated by using a first-order Chebyshev function instead of an exponential function.
The systematic uncertainties considered for the B + c mass and mass difference measurements are summarised in Table 3 and 4, respectively.

Combination of the measurements
The combination of the B + c mass measurements is performed using the Best Linear Unbiased Estimate (BLUE) method [74][75][76]. In the combination, uncertainties arising from the momentum-scale calibration, energy loss corrections, and final-state radiation are assumed to be 100% correlated, while all other sources of systematic uncertainty are assumed to be uncorrelated. The uncertainty on the momentum-scale calibration of the B 0 s mass is assumed to be 100% correlated with that of the B + c mass. The individual mass measurements and the resulting combination are shown in Fig. 3. The individual measurements are consistent with each other. The breakdown of the combined systematic uncertainty is given in Table 5. The weights of individual measurements returned by the BLUE method are listed in Table 6. The weights are computed including all uncertainties. The measurement contributing most to the combination is obtained   from the B + c → J/ψ D + s (K + K − π + ) decay. The negative weight for the B + c → J/ψ π + channel arises from the 100% correlation between the systematic uncertainties due to the momentum-scale calibration. This results in a larger statistical and smaller systematic uncertainty relative to an uncorrelated average.
The combination for the mass difference ∆M is shown in Fig. 4. The breakdown of the combined systematic uncertainty is given in Table 5 and the weights of decay modes in the combination are listed in Table 6. The combined B + c mass is determined to be M (B + c ) = 6274.47 ± 0.27 (stat) ± 0.17 (syst) MeV/c 2 , while the mass difference between the B + c and B 0 s mesons, ∆M , is determined to be ∆M = 907.75 ± 0.37 (stat) ± 0.27 (syst) MeV/c 2 .