Observation of excited $\Omega_c^0$ baryons in $\Omega_b^- \to \Xi_c^+ K^-\pi^-$ decays

The first observation of the $\Omega_b^- \to \Xi_c^+ K^- \pi^-$ decay is reported using $pp$ collision data at centre-of-mass energies of 7, 8 and 13\,$\mathrm{TeV}$ collected by the LHCb experiment, corresponding to an integrated luminosity of 9\,$\mathrm{fb}^{-1}$. Four excited $\Omega_c^0$ baryons are observed in the $\Xi_c^+ K^-$ mass projection of the $\Omega_b^- \to \Xi_c^+ K^- \pi^-$ decays with the significance of each exceeding five standard deviations. They coincide with the states previously observed in prompt \proton\proton and $e^+e^-$ production. Relative production rates, masses and natural widths of the states are measured, and a test of spin hypotheses is performed. Moreover, the branching ratio of $\Omega_b^- \to \Xi_c^+ K^- \pi^-$ is measured relative to the $\Omega_b^- \to \Omega_c^0 \pi^-$ decay mode and a precise measurement of the $\Omega_b^-$ mass of \mbox{$6044.3 \pm 1.2 \pm 1.1^{\,+0.19}_{\,-0.22}\,\mathrm{MeV}$} is obtained.


Introduction
The spectrum of the baryons with a single heavy quark Qqq (Q = b or c and q, q = u, d or s) is well classified using the heavy quark-diquark degrees of freedom. Heavy quark effective theory [1][2][3][4][5][6][7][8] provides the basis for factoring out the heavy-quark dynamics up to corrections of the first order of 1/m Q , where m Q is the heavy quark mass. Therefore, the observation of new baryons and measurements of their properties provide information about the role played by diquarks in baryons, and can also help to tune tetraquark and pentaquark models.
This letter presents the first observation of the Ω * * 0 c states produced in exclusive Ω − b decays. These are studied in the previously unobserved Ω − b → Ξ + c K − π − decays [44,45], where the Ξ + c baryons are reconstructed in the pK − π + final state. The mass of the Ω − b baryon has been measured in decays to the Ω 0 c π − and Ω − J/ψ final states. The new decay mode Ω − b → Ξ + c K − π − is a prominent reaction to measure also the Ω − b mass due to a multi-particle final state and smaller phase space with respect to the Ω 0 c π − mode. 1 The analysis is based on samples of pp collision data at centre-of-mass energies of √ s = 7, 8 and 13 TeV, corresponding to an integrated luminosity of 9 fb −1 .

Detector and simulation
The LHCb detector [46,47] 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 siliconstrip 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 together with straw drift tubes placed down-stream of the magnet. Simulation is necessary to train a multivariate algorithm used to suppress background, model shapes of mass distributions, and calculate efficiencies. In the simulation, pp collisions are generated using Pythia [48] with a specific LHCb configuration [49]. Decays of unstable particles are described by EvtGen [50]. The interaction of the generated particles with the detector is implemented using the Geant4 toolkit [51] as described in Ref [52].
The Ξ + c candidates are formed by combining three tracks that are detached from any primary pp interaction vertex (PV) in the event. A good-quality vertex fit is required to select tracks originating from the same secondary vertex. The Ω − b candidates are selected by combining the Ξ + c candidate with two tracks identified as a K − and a π − meson. Loose particle identification (PID) requirements are applied to all five final-state tracks in order to reduce background. The Ω − b candidates are required to have a transverse momentum p T > 3.5 GeV and are constrained to originate from the PV by requiring a small χ 2 IP , where χ 2 IP is defined as the difference in the vertex-fit χ 2 of a given PV reconstructed with and without the candidate under consideration. The Ω − b decay time is required to be larger than 0.2 ps, making the overlap with the prompt sample analysed in Ref. [16] negligible.
A boosted decision tree (BDT) classifier, implemented using the TMVA toolkit [53], is used to further reduce the background. Variables found to provide good discrimination between signal and background are: the PID information and p T of the final-state tracks, the Ξ + c p T , the Ξ + c and Ω − b χ 2 IP , the Ξ + c and Ω − b vertex-fit χ 2 , the Ω − b flight-distance significance, defined as the measured flight distance divided by its uncertainty, and the cosine of the Ξ + c and Ω − b direction angles. The direction angle is defined as the angle between the Ξ + c (Ω − b ) momentum and the vector joining the PV and the Ξ + c (Ω − b ) decay vertex. The training of the BDT classifier is performed using simulated samples as signal and data as background separately for Run 1 and Run 2 data samples. The candidates used for the background sample are in the 6200-6300 MeV range of the Ξ + c K − π − mass spectrum, which is not populated by partially reconstructed Ω − b decays. The optimal selection criterion on the BDT response is found by maximising the figure of merit /(5/2 + √ B P ) [54], where is the signal efficiency in simulation, and B P is the number of Ξ + c K − π − candidates in the mass region 6200 < m(Ξ + c K − π − ) < 6256 MeV, roughly matching the expected number of background events in the Ω − b mass window. Roughly 4% of selected events contain more than one candidate and are removed. Finally, a kinematic fit [55] is applied to the Ω − b decays to improve the mass resolution where the Ξ + c candidate mass is constrained to its known value [56], and the Ω − b candidate is constrained to originate from its associated PV, defined as the PV to which the impact parameter of the combination of two-track and Ξ + c candidate is the smallest. The resulting Ξ + c K − π − mass spectrum is shown in Fig. 1 (left) and an extended unbinned maximum-likelihood fit is performed. The signal shape is modelled by the combination of two Gaussian functions with a common mean, where the ratios of the resolutions and yields between the functions are fixed according to the simulation. The main sources of background are due to the partially reconstructed decays with Ω 0 c → pK − K − π + for all candidates passing the selection requirements. The black symbols show the data. The result of a fit is overlaid (solid red line). The missing particles in partially reconstructed decays are indicated in grey in the legends.
background shape is fixed according to a wrong-sign sample, consisting of Ξ + c K − π + combinations processed in the same way as the right-sign Ξ + c K − π − combinations. The shape of the partially reconstructed decays is taken from simulated samples generated using the RapidSim package [57]. The shape of misidentified decays 8% based on |V us | 2 /|V ud | 2 ≈ 5% corrected by the difference in reconstruction efficiency and the phase space. The fit returns a combined mass resolution of 17.9 ± 1.3 MeV, a yield of N Ξ + c K − π − = 240 ± 17 and an Ω − b mass, m(Ω − b ) = 6044.3 ± 1.2 MeV, where the uncertainty is statistical only (see Table 1). The Dalitz plot distribution of the candidates, with a mass within two standard deviations of the Ω − b peak, is shown in Fig. 2. Excited Ω 0 c baryons appear in the Ξ + c K − projection while no excited Ξ 0 c states are clearly visible in the Ξ + c π − system. The branching fraction of The selections of the two decay modes differ in the requirements applied to the invariant mass of the pK − π + and pK − K − π + systems to select Ξ + c and Ω 0 c candidates, respectively. A kinematic fit is applied to the Ω − b decay where the Ω 0 c candidate mass is constrained to its known value [56]. The two largest background components are due to the partially reconstructed The result of an unbinned maximum-likelihood fit is overlaid to the data in Fig. 1 (right). All decays are modelled in the same way as for the The combinatorial background shape is fixed according to the projection of the Ω 0 c sidebands in the Ω 0 c π − mass spectrum, where the Ω 0 c sidebands are defined as the 2650-2670 and 2720-2740 MeV ranges in the pK − K − π + invariant mass distribution. The yield of reconstructed Ω − b candidates is N Ω 0 c π − = 174 ± 14, and the mass resolution is 18.4 ± 1.5 MeV. The ratio of branching fractions is obtained as which is calculated from the ratio of efficiency-corrected yields, where the error is statistical only (see Table 1).
In order to increase the selection efficiency of the Ω * * 0 c states, an additional BDT classifier is deployed for the study of the Ξ + c K − spectrum, where a sample of simulated GeV, is used as the signal sample. For the background, the upper region of the Ξ + c K − π − mass distribution is used, as in the previous BDT classifier. After the optimization of the BDT response, the Ω − b candidates with a mass within two standard deviations of the Ω − b peak are selected. Figure 3 shows the distribution of the mass difference is the invariant mass of the Ξ + c K − system, and m Ξ + c and m K − are the world averages of the Ξ + c and K − masses, respectively [56]. Four narrow peaking structures are clearly visible close to the Ξ + c K − kinematic threshold.
An extended maximum-likelihood fit is performed to the ∆M distribution, where each signal is modelled by an S-wave relativistic Breit-Wigner (BW) function multiplied by the phase space function and convolved with a Gaussian function to describe the mass resolution. The widths and masses of the relativistic BW functions vary freely. The background consists of two components: the combinatorial background under the Ω − b signal peak ( Fig. 1 (left)) and the nonresonant Ξ and Ω c (3090) 0 baryons. The distribution shows an enhancement at the threshold, as seen in the previous analysis [16]. The total fit is overlaid in red. The background distribution (grey shaded area) is the combination of the combinatorial and nonresonant Ξ + c K − backgrounds.
is modelled by projecting the Ω − b sideband into the Ξ + c K − invariant mass distribution and the latter (nonresonant Ξ + c K − ) according to phase space. While the shapes of the two contributions and the yield of the combinatorial component are fixed, the yield of the nonresonant background can vary freely. The Ξ + c K − spectrum also features an excess at the Ξ + c K − mass threshold which is modelled by an S-wave BW component. Fit results superimposed to the data are shown in Fig. 3. The yields attributed to the four peaks are 24 ± 7, 33 ± 6, 51 ± 8, and 41 ± 9 respectively. The resulting BW parameters of the four signals, which are listed in Table 1, are consistent with those of the previously observed Ω c (3000) 0 , Ω c (3050) 0 , Ω c (3065) 0 and Ω c (3090) 0 baryons [16]. The natural width of the Ω c (3050) 0 is consistent with zero, therefore an upper limit is set. In order to determine the significance of the peaking structures, another fit is performed by fixing the masses and widths of the Ω * * 0 c states to the previously measured values [16]. Therefore, the statistical significance of each peak is calculated using 2∆(NLL), where ∆(NLL) is the variation of the fit log-likelihood when the corresponding BW function is excluded from the reference fit model. The local significance exceeds six standard deviations (6 σ) for each of the four main states. For the threshold structure, the null hypothesis of the background fluctuation is tested using the likelihood ratio of two fits. The p-value expressed in standard deviations using the one-sided convention corresponds to 4.3 σ after systematic uncertainties are accounted for. Finally, the production rate of the Ω * * 0 The rate is measured for the Ω c (3000) 0 , Ω c (3050) 0 , Ω c (3065) 0 and Ω c (3090) 0 , and an upper limit on the production of the Ω c (3120) 0 state is set. The results are reported in Table 1 with the statistical error computed using the binomial distribution.

Spin test for the excited Ω 0 c baryons
In order to probe the spin of the Ω * * 0 c baryons, the distribution of the helicity angle in the Ω − b → Ω * * 0 c (→ Ξ + c K − )π − decay is studied. The helicity angle θ is defined as the angle between the p K − and the − p π − directions in the Ξ + c K − rest frame, where p is the momentum of the meson. The spin projection of the Ω * * 0 c baryon in the direction of the π − meson is limited to 1/2 as it is produced in the Ω − b → Ω * * 0 c π − decay. Additionally, it cannot exceed 1/2 in the direction of either decay product, Ξ + c or K − , due to their spins. Therefore, the angular distribution for a Ω * * 0 c state with spin J is given as The expectations for the angular density function, D J (cos θ), shown by the colored lines in Fig. 4, are calculated as a sum of the signal PDF and the two background components (combinatorial and nonresonant Ξ + c K − ) by where f s and f b are the fractions of the signal and the combinatorial background fixed according to the result of the mass fit. The angular distribution for the combinatorial background, B 1 (cos θ), is fixed by selecting candidates in the Ξ + c K − π − mass range above the Ω − b peak. A flat distribution is assumed for nonresonant background, B 2 (cos θ). The efficiency, (cos θ), is calculated separately for each signal region using simulation. The efficiency maps are combined according to the fraction of the signal candidates in the corresponding data-taking periods. The efficiency for the helicity angle is calculated by convolving the efficiency map with the Ω * * 0 c line-shape profile. The fall of the curves at cos θ = 1 indicates the smaller reconstruction efficiency for candidates with a low momentum K − in the Ω − b rest frame. Discrimination of different spin hypotheses is based on the likelihood-ratio test statistic, where H J and H J are the compared hypotheses for the state to have spin J and J , respectively, N is the number of candidates in the mass region around the peak. The test statistic t (data) = (t  the number of signal and background events obtained from data are simulated for each spin hypothesis and for every Ω * * 0 c state. The two-dimensional distribution of t is well described by the multivariate normal distribution from which we extract the covariance matrix and the two-dimensional mean, t (mean) . The p-value in the double-tailed convention is calculated by exp(−r 2 /2), where r is the Mahalanobis distance [58] between t (data) and t (mean) . All results are summarized in Table 1. The significance of the rejection of the J = 1/2 hypothesis for Ω c (3050) 0 and Ω c (3065) 0 is 2.2 σ and 3.6 σ respectively, including systematic effects listed in the next section. The combined hypothesis of the four peaks to have quantum numbers in the order 1/2, 1/2, 3/2, 3/2 is tested and rejected with a significance of 3.5 σ.

Systematic uncertainties
Various systematic uncertainties for each observable are considered, where the largest deviation from the default model on every source is used. A summary of the systematic uncertainties is provided in the supplemental material [59]. The uncertainties from different sources are combined in quadrature. A source of systematic uncertainty is determined from varying components of the Ω − b fit model. The helicity couplings of the partially reconstructed decays in the Ω 0 c π − invariant mass spectrum are modified as well as the shape used to describe the signal peaks. The uncertainty in the yield of misidentified decays is quantified by varying the fractional contribution by ±40% relative to the default value. In simulation, the Ξ + c →pK − π + Dalitz plot is generated according to phase space and a binned weighting is performed to match the data. A systematic uncertainty is found by varying the binning scheme.
The uncertainty in the mass measurements due to momentum calibration is determined following Ref. [60] as ±0.03% of the energy released in the decay. The PID variables in simulation are corrected in order to match the PID performance in data. To calculate an uncertainty, a modified weighting is applied to the PID variables. For the uncertainty in the Ω − b kinematics, the p T and η of the Ω − b candidates, as well as the track multiplicity in simulation, are weighted according to data. Several alternative models are considered for Ξ + c K − fit. Firstly, the resolution of each Gaussian function is varied by ±10%. In addition, different orbital angular momenta (L = 1, 2) are tested along with the variation of the Blatt-Weisskopf factors [61,62] from 1.5 to 5 GeV −1 . A constant-width BW approximation and the scattering-length approximation are probed for the threshold structure. Lastly, for each signal peak, interference with neighbours and the nonresonant Ξ + c K − background is tested. The full list of results including systematic uncertainties are listed in Table 1.

Summary and conclusion
In summary, data collected by the LHCb experiment at centre-of-mass energies 7, 8 and 13 TeV corresponding to an integrated luminosity of 9 fb −1 are used to observe the new decay mode Ω − b → Ξ + c K − π − and to measure its branching fraction relative to the Ω − b → Ω 0 c π − decay mode. A precise measurement of the Ω − b mass, m(Ω − b ) = 6044.3±1.2±1.1 +0.19 −0.22 MeV, is obtained where the first uncertainty is statistical, the second is systematic, and the third asymmetric error is due to the uncertainty in the Ξ + c mass. Averaging with the previous LHCb measurements [63,64], taking correlated systematic uncertainties into account, gives a mass of m(Ω − b ) = 6044.8 ± 1.3 MeV, which is the most precise to date. The investigation of the Ξ + c K − mass spectrum has revealed four excited Ω 0 c baryons, Ω c (3000) 0 , Ω c (3050) 0 , Ω c (3065) 0 , Ω c (3090) 0 , and a threshold enhancement as also seen in Ref. [16]. The Ω c (3120) 0 state is not observed, therefore an upper limit on its production rate is set by scanning the likelihood profile, P Ωc(3120) 0 < 0.03 at 95% confidence level (CL). Measurements of the Ω * * 0 c masses and widths, together with an upper limit of Γ Ωc(3050) 0 < 1.6 MeV at 95% CL are reported. Their spin assignments are tested based on the distribution of the helicity angle in the decay chain Ω − b → Ω * * 0 c π − , Ω * * 0 c → Ξ + c K − . Significance values of excluding the J = 1/2 spin hypothesis for Ω c (3050) 0 and Ω c (3065) 0 are 2.2 σ and 3.6 σ, respectively. All results are summarised in Table 1. The combined hypothesis on the spin of the four peaks in the order J = 1/2, 1/2, 3/2, 3/2, as proposed in several works [20,31,36], is rejected with a p-value corresponding to 3.5 standard deviations once systematic uncertainties are taken into account.
The results of the angular analysis together to the absence of the Ω c (3120) 0 state in the Ξ + c K − spectrum in Ω − b decays and in e + e − collisions at Belle [17], suggest that the interpretation of the five peaks observed in Ref. [16] as λ-mode excited states might be invalid. In such a scenario, only the four peaks observed in this analysis would be λ-mode excitations (with quantum numbers J = 1/2, 3/2, 3/2, and 5/2) and a spin 1/2 λ-mode state would be still to be observed. The nonobservation of the Ω c (3120) 0 baryon would be consistent with the state being either one of the 2S doublet, decaying to Ξ + c K − in Pwave [27,31], or a ρ-mode P -wave excitation with spin 3/2 − that requires D-wave between Ξ + c and K − . Finally, the Ξ + c K − spectrum also features an excess at the Ξ + c K − mass threshold. An analogous enhancement was observed in the inclusive Ξ + c K − spectrum [16] and interpreted as the partially reconstructed decay Ω c (3065) 0 → Ξ + c (→ Ξ + c γ)K − with the photon escaping detection. However, such an explanation does not hold here, given that the partially reconstructed decay Ω − b → Ξ + c K − π − does not populate the mass region selected for the exclusive Ω − b → Ξ + c K − π − decay. While the current data do not provide enough sensitivity to determine the parameters of the structure, such as the mass, natural width and spin, future data acquired with the upgraded LHCb detector will provide insights to establish its nature. Table 1: Results on the Ω − b mass, relative branching fraction of the Ξ + c K − π − decay mode, measured mass differences (∆M ), masses (m), natural widths (Γ) and production fraction (P) of Ω * * 0 c baryons where the first uncertainty is statistical and the second systematic. The third asymmetric uncertainty on the Ω − b and Ω * * 0 c masses is due to the uncertainty in the Ξ + c mass. Upper limits are given for the width of the Ω c (3050) 0 state and the production rate of the Ω c (3120) 0 baryon, which are measured to be consistent with zero. The results of the spin analysis are also listed (J rejection).

State
Observable Measurement

Supplemental material for LHCb-PAPER-2021-012
The mass distributions, m(Ξ + c K − π − ) and m(Ω 0 c π − ), are shown in Fig. 5 with a logarithmic scale. Candidates / (11.6 MeV)    LHCb  Figure 7: Measurements of the Ω − b mass from the LHCb experiment, the LHCb average and the PDG average, which includes the two previous LHCb measurements and one measurement from the CDF collaboration [56].
from the LHCb experiment, the LHCb average which is calculated using the Ω − b mass determined in this analysis and the two previous results, and the PDG average.
The efficiency map of the data reconstruction and selection in the [∆M, cos θ] plane is shown in Fig. 8, where the positions of four of the Ω * * 0 c states are demonstrated by the red lines. Figure 9 shows the value of the test statistic observables t J=1/2|J=3/2 and t J=3/2|J=5/2 for each Ω * * 0 c state, where the red point indicates the value determined from data.   Table 1 of the main text. Tables 2 and 3 summarise the systematic uncertainties considered for each observable where the largest deviation from the default model is quoted on every source.