Precision measurement of the $\Xi_{cc}^{++}$ mass

A measurement of the $\Xi_{cc}^{++}$ mass is performed using data collected by the LHCb experiment between 2016 and 2018 in $pp$ collisions at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 5.6 $\mathrm{fb}^{-1}$. The $\Xi_{cc}^{++}$ candidates are reconstructed via the decay modes $\Xi_{cc}^{++}\to\Lambda_c^+K^-\pi^+\pi^+$ and $\Xi_{cc}^{++}\to\Xi_c^+\pi^+$. The result, $3621.55 \pm 0.23{\rm\,(stat)\,} \pm 0.30 {\rm\,(syst)\,}{\rm MeV}/c^2$, is the most precise measurement of the $\Xi_{cc}^{++}$ mass to date.

At present, the experimental uncertainty on the Ξ ++ cc mass is still large compared to that of the singly charmed baryons. This paper presents an updated measurement of the Ξ ++ cc mass using the decay modes Ξ ++ cc → Λ + c (→ pK − π + )K − π + π + and Ξ ++ cc → Ξ + c (→ pK − π + )π + . The analysis uses a data sample corresponding to an integrated luminosity of 5.6 fb −1 , collected by the LHCb experiment during 2016-2018 in pp collisions at a centre-of-mass energy of 13 TeV. This measurement supersedes the results reported on the Ξ ++ cc mass in Refs. [3,4], which only use pp collision data at 13 TeV taken in 2016, corresponding to an integrated luminosity of 1.7 fb −1 .

Detector and simulation
The LHCb detector [22,23] 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 [24], 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 [25,26] placed downstream of the magnet. The tracking system provides a measurement of the momentum 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 [27,28]. The relative accuracy of this procedure is estimated to be 3 × 10 −4 using samples of other fully reconstructed b-hadron, Υ and K 0 S decays. The minimum distance of a track to a primary pp collision vertex (PV), the impact parameter (IP), is measured with a resolution of (15 + 29/(p T / GeV/c)) µm, where p T is the momentum component transverse to the beam axis. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors [29]. 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 [30].
The online event selection is performed by a trigger [31], 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. In between the two software stages, an alignment and calibration of the detector is performed in near real-time [32]. This process allows the reconstruction of the Ξ ++ cc decays to be performed entirely in the software trigger, whose output is used as input to the present analysis.
Simulated samples are used to model the effects of the detector acceptance, optimise selections and verify the validity of the methods used in the measurement. In the simulation, pp collisions are generated using Pythia 8 [33] with a LHCb specific configuration [34]. The production of doubly charmed Ξ ++ cc baryons is simulated using the dedicated generator GenXicc2.0 [35]. Decays of hadrons are described by EvtGen [36], in which final-state radiation is generated using Photos [37]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [38] as described in Ref. [39]. Sources of background, such as those from Ξ ++ cc → Ξ + c (→ Ξ + c γ)π + and Ξ ++ cc → Ξ + c ρ + (→ π + π 0 ), are studied using the fast simulation package RapidSim [40].

Event selection
The reconstruction of Ξ ++ cc → Λ + c (→ pK − π + )K − π + π + and Ξ ++ cc → Ξ + c (→ pK − π + )π + decays is similar to that used in previous LHCb analyses [3,4]. Candidate Ξ + c (Λ + c ) → pK − π + decays are reconstructed from three charged particles identified as a p, K − and π + using information from the RICH detectors [29]. The charged particles are required to form a good-quality vertex and be inconsistent with originating from any PV. The Ξ + c (Λ + c ) candidate is then combined with one (three) additional charged particle(s) to form a Ξ ++ cc → Ξ + c π + (Ξ ++ cc → Λ + c K − π + π + ) decay candidate. These additional particles must form a good-quality vertex with the Ξ + c (Λ + c ) candidate, which is required to be upstream of the Ξ + c (Λ + c ) decay vertex. Each Ξ ++ cc candidate is required to have p T > 2 GeV/c and to be consistent with originating from its associated PV. The associated PV is that with respect to which the Ξ ++ cc candidate 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. To avoid candidates including duplicated tracks, each track pair is required to have an opening angle larger than 0.5 mrad or a momentum difference larger than 5% of the minimum momentum of the track pair.
In order to improve the signal purity, multivariate classifiers are trained to separate signal from background. The choice of classifier algorithms is based on their performance for each decay mode. A classifier based on the Boosted Decision Tree (BDT) algorithm [41,42] implemented in the TMVA toolkit [43] is used for the the Ξ ++ cc → Λ + c K − π + π + mode, while a Multilayer Perceptron (MLP) algorithm [43] is used for the Ξ ++ cc → Ξ + c π + mode. The BDT for the Ξ ++ cc → Λ + c K − π + π + decay is trained with simulated signal events as a signal proxy and wrong-sign Λ + c K − π + π − combinations (3525-3725 MeV/c 2 ) in data, where the two final-state pions have opposite charges, as a background proxy. Both the signal and background proxies are required to pass the selection described above. Variables associated with the Ξ ++ cc candidates used in the training include the vertex-fit quality, the χ 2 IP , the angle between the momentum and the displacement vector, the flight-distance χ 2 between the PV and the decay vertex. The flight-distance χ 2 is defined as the χ 2 of the hypothesis that the decay vertex of the candidate coincides with its associated PV. Variables associated with the decay products of the Ξ ++ cc candidates (the Λ + c , K − and π + ) used in the training include their p T and χ 2 IP , the vertex-fit quality of the Λ + c candidates and the smallest p T among the Λ + c decay products (p, K − and π + ). Particle-identification information for the final-state particles is also used.
The threshold applied to the classifier response is determined by maximising the signal significance S/ √ S + B, where S is the expected signal yield estimated using simulation, and B is the background yield evaluated in the upper sideband of data (3800-3900 MeV/c 2 ) extrapolated to the signal region (3607-3635 MeV/c 2 ). The multivariate classifier for the Ξ ++ cc → Ξ + c π + decay is developed following the same strategy as that for the Ξ ++ cc → Λ + c K − π + π + decay. After the full selection, an event may still contain more than one Ξ ++ cc candidate. According to studies on simulated decays and the wrong-sign control sample, multiple candidates in the same event may form a peaking structure in the mass distribution of the Ξ ++ cc candidates if they are obtained from the same final-state tracks, but via swapping two final state tracks (e.g. the K − from the Λ + c decay and the K − from the Ξ ++ cc decay). In this case, one candidate is chosen randomly. Other kinds of multiple candidates, which account for 8% (< 1%) of the Ξ ++ cc → Λ + c K − π + π + (Ξ ++ cc → Ξ + c π + ) signal events, are not removed since they do not form a peaking background.
The m cand (Ξ ++ cc ) mass distributions of the selected Ξ ++ cc candidates are shown in Fig. 1 for the Ξ ++ cc → Λ + c K − π + π + and Ξ ++ cc → Ξ + c π + decay modes. The Ξ ++ cc mass is determined by performing unbinned extended maximum-likelihood fits to the two m cand (Ξ ++ cc ) mass distributions. The signal components are described by a modified Gaussian function with a power-law tail on the left-hand side of the distribution [47], parameterised as The peak position, x, and width, σ, of the function are allowed to vary in the fit. The power-law tail parameters, N and α, are fixed from simulation. The background from randomly associated tracks is modelled using an exponential function. Background contributions from the partially reconstructed decays Ξ ++ cc → Ξ + c (→ Ξ + c γ)π + and Ξ ++ cc → Ξ + c ρ + (→ π + π 0 ), where photons and neutral π 0 mesons are not reconstructed, can contribute to the Ξ ++ cc → Ξ + c π + decay mode. The mass line shapes of these partially reconstructed backgrounds are determined from simulation. The fits return signal yields of 1598 ± 64 and 616 ± 47 for the Ξ ++ cc → Λ + c K − π + π + and Ξ ++ cc → Ξ + c π + decay modes, respectively. The peak positions are determined with the Ξ ++ cc → Λ + c K − π + π + and Ξ ++ cc → Ξ + c π + decay modes to be 3622.08 ± 0.24 MeV/c 2 and 3622.37 ± 0.60 MeV/c 2 , respectively, where the uncertainty is statistical only.
Due to multiple scattering, the opening angle between the Ξ ++ cc decay products can be increased or decreased. This can bias both the resulting Ξ ++ cc mass and the measured decay length. Since the selection is more efficient for candidates with larger reconstructed decay lengths, and the decay length is correlated with the mass by the effect of the multiple scattering, this can bias the Ξ ++ cc mass measurement. This effect was studied with charmed hadrons, D + , D 0 , D + s , Λ + c , and was found to be well reproduced by simulation [3]. Corresponding corrections of −0.61 ± 0.09 MeV/c 2 for Ξ ++ cc → Λ + c K − π + π + and −0.45 ± 0.09 MeV/c 2 for Ξ ++ cc → Ξ + c π + are determined using simulated candidates by comparing the fitted mass with signal candidates before and after applying the event selection. These corrections are applied to the fitted mass values. The uncertainties are due to the limited size of simulated samples, and are taken as the systematic uncertainties from the selection-induced bias on the Ξ ++ cc mass. The difference of the kinematic distributions in simulation and data is considered as a systematic uncertainty and is discussed in Sec. 5.
Low-momentum photons emitted by the final-state particles are not reconstructed. This distorts the reconstructed mass distribution and can bias the fitted mass value. This effect is studied with the simulation. To disentangle this effect from those due to reconstruction, the mass of the Ξ ++ cc candidates calculated with the true momenta of the final-state particles is smeared with different resolution values. The difference between the fitted and input mass values is studied as a function of the mass resolution, and the difference corresponding to the mass resolution in data is taken as a correction. Alternative signal models are also considered. The largest difference of the fitted mass with final-state radiation corrections between the nominal and the alternative is quoted as the uncertainty. Following the procedure described above, the corrections due to the final-state radiation are determined as 0.06 ± 0.05 MeV/c 2 and 0.03 ± 0.16 MeV/c 2 for the Ξ ++ cc → Λ + c K − π + π + and Ξ ++ cc → Ξ + c π + decay modes, respectively. The uncertainties on the corrections are due to the limited size of the simulated samples, and the difference between the corrections with different signal models.

Systematic uncertainties
The dominant source of systematic uncertainty on the mass measurement is due to the momentum-scale calibration [27,28]. It amounts to 0.21 MeV/c 2 for the Ξ ++ cc → Λ + c K − π + π + decay, and 0.34 MeV/c 2 for the Ξ ++ cc → Ξ + c π + decay due to larger Q-value. A further uncertainty arises from the correction for energy loss in the spectrometer, which is known with 10% accuracy [23]. This uncertainty was studied in Ref. [28], and amounts to 0.03 MeV/c 2 for D 0 →K + K − π + π − decays. The uncertainties on the Ξ ++ cc mass are scaled from that of the D 0 decay by the number of final-state particles, and are determined to be 0.05 MeV/c 2 and 0.03 MeV/c 2 for the Ξ ++ cc → Λ + c K − π + π + and Ξ ++ cc → Ξ + c π + decays, respectively.
Differences between kinematic distributions in simulation and data are treated as sources of systematic uncertainties on the corrections due to the selection procedure. The kinematic variables used in the event selection that are found to affect the corrections are listed below: p T of Ξ ++ cc candidates; the angle between the momentum and the displacement vector from the PV to the decay vertex of the Ξ ++ cc and Λ + c (Ξ + c ) candidates; the χ 2 IP of Ξ ++ cc and Λ + c (Ξ + c ) candidates and their decay products; the BDT (MLP) response; and the particle identification information. The distributions of these variables in simulation are weighted to match those in data where the background is subtracted by means of the sPlot technique [48]. Then, the corrections obtained with the weights are compared to those without weights, and largest variations of the corrections are taken as systematic uncertainties, which are 0.09 MeV/c 2 and 0.05 MeV/c 2 for the Ξ ++ cc → Λ + c K − π + π + and Ξ ++ cc → Ξ + c π + decays, respectively. The uncertainty related to the background description is estimated by repeating the fits with alternative models which include first and second-order polynomial functions. For the Ξ ++ cc → Λ + c K − π + π + decay, the fit with a second-order polynomial function has better fit quality, but returns identical fitted mass. The largest changes on the fitted mass value are found to be 0.01 MeV/c 2 and 0.04 MeV/c 2 for the Ξ ++ cc → Λ + c K − π + π + and Ξ ++ cc → Ξ + c π + decays, respectively, which are assigned as systematic uncertainties. The mass of Ξ ++ cc candidates also depends on the value of the known Λ + c and Ξ + c masses. The uncertainties on the Λ + c mass and on the mass difference between the Ξ + c and Λ + c are propagated to the Ξ ++ cc mass measurement. The corresponding uncertainties on the Ξ ++ cc mass are 0.14 MeV/c 2 and 0.22 MeV/c 2 for the Ξ ++ cc → Λ + c K − π + π + decay and the Ξ ++ cc → Ξ + c π + decay, respectively. The sources of systematic uncertainty considered in this analysis are summarised in Table 1. When computing the total uncertainty, the uncertainty on the momentum-scale calibration of the Ξ + c mass from Ref. [46] is assumed to be fully correlated to that of the Ξ ++ cc mass. The total systematic uncertainty is calculated by summing the individual sources of uncertainty in quadrature.

Results and summary
The resulting values of the Ξ ++ cc mass using the Ξ ++ cc → Λ + c K − π + π + and Ξ ++ cc → Ξ + c π + decay modes are 3621.53 ± 0.24 ± 0.29 MeV/c 2 , and 3621.95 ± 0.60 ± 0.49 MeV/c 2 , respectively, including corrections and systematic uncertainties. The combination of the two measurements is performed using the Best Linear Unbiased Estimator (BLUE) [49,50]. Table 1: Systematic uncertainties on the Ξ ++ cc mass measurements using Ξ ++ cc → Λ + c K − π + π + and Ξ ++ cc → Ξ + c π + decays. The total systematic uncertainty on each mode is calculated by summing the individual sources of uncertainty in quadrature, except for the uncertainty on the momentum-scale calibration of the Ξ + c mass [46], that is added linearly to that of the Ξ ++ cc mass. In the combination, the correlation between the Λ + c and Ξ + c masses [45,46] is taken into account. Uncertainties arising from the momentum-scale calibration, energy-loss corrections, and final-state radiation are assumed to be 100% correlated while other sources of systematic uncertainty are assumed to be uncorrelated. The individual mass measurements and the resulting combination are illustrated in Fig. 2. The averaged mass is dominated by the result for the Ξ ++ cc → Λ + c K − π + π + mode, due to its larger yield and smaller momentum-scale uncertainty relative to that of the Ξ ++ cc → Ξ + c π + decay. This is the most precise measurement of the Ξ ++ cc mass to date, improving upon the previous weighted average mass value of 3621.24 ± 0.65 (stat) ± 0.31 (syst) MeV/c 2 from Ref. [4]. cc → Λ + c K − π + π + and Ξ ++ cc → Ξ + c π + . The combination is performed using the best linear unbiased estimator [49,50]. The inner error bars represent statistical uncertainties and the outer error bars represent the quadratic sum of statistical and systematic uncertainties. The inner and outer green bands correspond to the uncertainties on the averaged value.