Study of the $B^{-} \to \Lambda_{c}^{+} \bar{\Lambda}_{c}^{-} K^{-}$ decay

The decay $B^{-} \to \Lambda_{c}^{+} \bar{\Lambda}_{c}^{-} K^{-}$ is studied in proton-proton collisions at a center-of-mass energy of $\sqrt{s}=13$ TeV using data corresponding to an integrated luminosity of 5 $\mathrm{fb}^{-1}$ collected by the LHCb experiment. In the $\Lambda_{c}^+ K^{-}$ system, the $\Xi_{c}(2930)^{0}$ state observed at the BaBar and Belle experiments is resolved into two narrower states, $\Xi_{c}(2923)^{0}$ and $\Xi_{c}(2939)^{0}$, whose masses and widths are measured to be $$ m(\Xi_{c}(2923)^{0}) = 2924.5 \pm 0.4 \pm 1.1 \,\mathrm{MeV}, \\ m(\Xi_{c}(2939)^{0}) = 2938.5 \pm 0.9 \pm 2.3 \,\mathrm{MeV}, \\ \Gamma(\Xi_{c}(2923)^{0}) = \phantom{000}4.8 \pm 0.9 \pm 1.5 \,\mathrm{MeV},\\ \Gamma(\Xi_{c}(2939)^{0}) = \phantom{00}11.0 \pm 1.9 \pm 7.5 \,\mathrm{MeV}, $$ where the first uncertainties are statistical and the second systematic. The results are consistent with a previous LHCb measurement using a prompt $\Lambda_{c}^{+} K^{-}$ sample. Evidence of a new $\Xi_{c}(2880)^{0}$ state is found with a local significance of $3.8\,\sigma$, whose mass and width are measured to be $2881.8 \pm 3.1 \pm 8.5\,\mathrm{MeV}$ and $12.4 \pm 5.3 \pm 5.8 \,\mathrm{MeV}$, respectively. In addition, evidence of a new decay mode $\Xi_{c}(2790)^{0} \to \Lambda_{c}^{+} K^{-}$ is found with a significance of $3.7\,\sigma$. The relative branching fraction of $B^{-} \to \Lambda_{c}^{+} \bar{\Lambda}_{c}^{-} K^{-}$ with respect to the $B^{-} \to D^{+} D^{-} K^{-}$ decay is measured to be $2.36 \pm 0.11 \pm 0.22 \pm 0.25$, where the first uncertainty is statistical, the second systematic and the third originates from the branching fractions of charm hadron decays.


Introduction
Table 1: Definitions of the sideband regions, where m(B − ) and m(D + ) represent the known values from Ref [25].The same sideband region is defined for the pK + π − combination as for the pK − π + combination.The values are in MeV.
) [ a wide structure is evident around 2880 MeV.A similar structure was observed in the prompt Λ + c K − study [7], but background due to feed-down from higher Ξ 0 c states, Ξ c (3055) 0 and Ξ c (3080) 0 , with a π 0 or π + missing in the reconstruction, could not be discounted.Such background will not affect the Λ + c K − system from B − decays, since there is not enough phase space for B − → Λ − c Ξ c (3055/3080) 0 and such feed-down component with a missing particle will not peak at the B − mass.To study the effect of partially reconstructed Ξ c (3055/3080) 0 from other b-hadron decays, a sample of Λ 0 b decays is generated with the decay chain Λ 0 b → Ξ c (3055 The selection criteria remove them completely.Therefore, the wide structure cannot be due to feed-down, and is considered as a resonant state in the fit.Both Ξ c (2790) 0 and Ξ c (2815) 0 states are expected at the lower end of the M (Λ + c K − ) spectrum, and their masses, widths and spin-parity J P are fixed to known values [25].The Ξ c (2790) 0 state has J P = (1/2) − and Ξ c (2815) 0 has J P = (3/2) − .However, the significance of the Ξ c (2815) 0 state is only 2.1 σ, and so it is not included in the nominal fit.The yield of the resonant states is not enough to determine their quantum numbers with sufficient significance.The spin-parity J P of both Ξ c (2923) 0 and Ξ c (2939) 0 states is fixed to be (3/2) − and that of The invariant mass distributions of the 4 Masses and widths of the excited Ξ 0 c states The masses and widths of the excited Ξ 0 c states are obtained from an unbinned maximum likelihood fit to the M (Λ + c K − ) spectrum.The total fitting function is constructed as   where M stands for M (Λ + c K − ).The function describing the excited Ξ 0 c states, f Ξ 0 c , which will be discussed in detail later, is convolved with a Gaussian function to account for the detector resolution.The Gaussian function has a mean value of zero and width σ res in the range of 0-2 MeV dependent on M (Λ + c K − ).The resolution σ res is parameterised as ) are lower and upper thresholds of the M (Λ + c K − ) spectrum, respectively, and a, b, c, d are determined from simulation.The contribution from the non-resonant component f phsp is modelled with simulation of pure phase-space decay.The background contribution f bkg is extrapolated from the sideband regions as defined in Fig. 2. The expected number of background events N bkg is also fixed using extrapolation from the sideband regions.The total number of candidates within the fitting range, N evt , is 1494.All fitting functions are normalised to unity.
As discussed in Sec. 3, four excited states are considered: Ξ c (2790) 0 , Ξ c (2880) 0 , Ξ c (2923) 0 and Ξ c (2939) 0 .The interference between the Ξ c (2923) 0 and Ξ c (2939) 0 states is important and cannot be neglected to correctly describe the data.The significance of the two-state hypothesis with respect to the hypothesis of a single Ξ c (2930) 0 state is over 11 σ.The function used for excited Ξ 0 c states is expressed as where The quantities q and p represent the breakup momenta of the Ξ 0 c and B − decays, expressed as The subscript α runs through all four resonant states considered in the fit.The amplitude of each state is described by a relativistic Breit-Wigner function, where m α and Γ α are the mass and width of the state, c α is a complex coefficient, and q α is the breakup momentum q computed at the mass m α .The quantities F l (q) and F L (p) are Blatt-Weisskopf barrier functions, defined as where r is the effective radius of the resonant state, fixed to 3.0 GeV −1 [27], l is the orbital angular momentum between the Λ − c and K + particles, and L is the smallest possible orbital angular momentum between the Λ + c and Ξ 0 c states.As mentioned before, the spin-parity of the Ξ c (2790) 0 state is known to be (1/2) − [25].The 1P multiplet contains 5 states with the quantum numbers (1/2) − , (1/2) − , (3/2) − , (3/2) − and (5/2) − .The quantum numbers of the Ξ c (2880) 0 state are assumed to be (1/2) − as well [8].The spin-parity of the Ξ c (2923) 0 and Ξ c (2939) 0 states is assumed to be (3/2) − .The values of l are set to respect momentum and parity conservation, namely l = 0 for the Ξ c (2790) 0 and Ξ c (2880) 0 states and l = 2 for the Ξ c (2923) 0 and Ξ c (2939) 0 states.Alternative assumptions for the quantum numbers of the states are considered in the determination of the systematic
To study the possible bias on the measured mass and width of the Ξ c (2923) 0 and Ξ c (2939) 0 states, 3 000 pseudoexperiments are performed where all other parameters, except the masses and widths, are fixed.A fit is performed for each pseudoexperiment, and the pull of each mass or width parameter is calculated with respect to the input.The pull is defined as the difference between the fitted value and the input value, divided by the uncertainty obtained from the fit.The pull distributions are then fitted with Gaussian functions.The deviation of the Gaussian mean from zero is used to correct the fitted mass values.The correction values are smaller than the statistical uncertainties and will be considered in the systematic uncertainty determination.
The fitted M (Λ + c K − ) distribution is shown in Fig. 3, and the measured masses and widths are listed in Table 2.The significance of the Ξ c (2790) 0 and Ξ c (2880) 0 states is calculated by studying 30 000 pseudoexperiments.Each is generated with a null hypothesis, then fitted both with and without the excited Ξ 0 c state of interest.The test statistic t 0 , defined as twice the difference in log-likelihood with and without the state, 2 log(L 1 /L 0 ), is expected to follow a χ 2 distribution.The t 0 values from the pseudoexperiments are fitted with a χ 2 distribution, and the p-value of the observed yield corresponds to the fraction of integrated area above the t 0 value measured in real data divided by the total integrated area.The significance of both the Ξ c (2790) 0 and Ξ c (2880) 0 states is estimated to be 3.9 σ.The significance of the Ξ c (2880) 0 state is stable even assuming the absence of the Ξ c (2790) 0 state.
Systematic uncertainties on the mass and width measurements from various sources are studied.Multiple alternative assumptions on the fixed parameters are tested.The spin-parity of the Ξ c (2923) 0 and Ξ c (2939) 0 states is set to (1/2) − , (1/2) + or (3/2) − , and J P of the Ξ c (2880) 0 is set to (1/2) + , (3/2) − or (3/2) + .In these tests, the states with the same spin-parity are always added coherently.The effective radius r is set to either 2.0 or 4.0 GeV −1 .The mass and width of the Ξ c (2790) 0 state are varied within their uncertainty, and the hypothesis without the Ξ c (2790) 0 state is tested.A different coefficient k α is assigned to each group of Ξ 0 c states with the same spin-parity.An additional state around 2970 MeV with orbital angular momentum of 0, 1 or 2 is added.The fit including the Ξ c (2815) 0 state is considered.The potential interference with non-resonant decays is considered by adding a constant term in the Λ + c K − mass distribution.The maximum variation in the fit results is obtained for each alternative assumption if multiple values are considered, and the total systematic uncertainty due to model assumptions is the The Ξ c (2923) 0 and Ξ c (2939) 0 lineshape is described alternatively using a K-matrix formalism [28], which preserves unitarity.The variation in the results is considered as a systematic uncertainty.The pull distributions from pseudoexperiments are used to correct a possible bias in the masses and widths of the Ξ c (2923) 0 and Ξ c (2939) 0 states, where the resulting corrections are smaller than statistical uncertainties.It is found that the fitted width σ pull of the pull distribution is slightly wider than unity, indicating a potential under-estimation of the statistical uncertainty.Therefore, the term σ 2 pull − 1 • σ stat is assigned as a systematic uncertainty.A relative uncertainty of 3 × 10 −4 on the charged particle momentum scale [7] is propagated through the simulation.The resolution σ res is varied within uncertainty and the resulting difference on the fitted parameters of the resolution function is then propagated to the masses and natural widths of the excited Ξ 0 c states, which is found to be negligible.The magnitude of the energy-loss correction for charged particles is known to 10% accuracy [29].In the study of Ref. [14], a correction of less than 0.01 MeV per track is estimated.A conservative estimation of 0.01 MeV per track is taken as the systematic uncertainty, which is also found to be negligible.
Mass constraints on the B − and Λ + c candidates are applied when calculating M (Λ + c K − ).The maximum change after varying the B − or Λ + c mass by one standard deviation is assigned as a systematic uncertainty.The systematic uncertainty due to the background is estimated by generating and fitting 300 pseudoexperiments, randomly varying the background yield according to a Poisson distribution while fixing the Ξ 0 c yield, the parameters of the Ξ 0 c model and the non-resonant yield.The average of the fitted values is calculated, and its difference from the nominal result is assigned as a systematic uncertainty.
A summary of the systematic uncertainties are listed in Table 3.The uncertainties from different sources are uncorrelated and are added in quadrature to give the total systematic uncertainty.The significance of the Ξ c (2880) 0 and Ξ c (2790) 0 states is 3.8 σ and 3.7 σ, respectively, after taking the systematic uncertainties into account.
All selected events are triggered due to B − candidates.The legend is the same as in Fig. 1, except replacing Λ

Branching fraction
The relative branching fraction R B of the B − → Λ + c Λ − c K − decay with respect to the B − → D + D − K − channel is calculated as a ratio of efficiency-corrected yield divided by the corresponding charm decay branching fractions, expressed as The yield of the signal and normalisation channels is obtained from the 3D fit described in Sec. 3. The charm decay branching fractions B(D + → K − π + π + ) [(9.38 ± 0.16)%] and B(Λ + c → pK − π + ) [(6.28 ± 0.32)%] are obtained from Ref. [25].
In addition to the selection described in Sec. 2, it is required that all selected signal events are triggered on the B − candidate decay products for both signal and normalisation modes so that the trigger efficiencies can be estimated accurately.The projections of the 3D fit to the B − → Λ + c Λ − c K − spectra with these additional requirements are shown in Fig. 5  (a-c) and the measured yield is 977 ± 36.For the normalisation decay B − → D + D − K − , a similar 3D fit is performed to the mass distributions of the B − , D + and D − candidates with the same fitting functions except replacing Λ c with D wherever applicable.The fitted distributions are shown in Fig. 5 (d-f), and the yield of The total efficiency ε tot is the product of the efficiencies due to the detector acceptance, trigger, reconstruction, offline preselection and multivariate selection.The efficiencies from different sources are estimated with a combination of simulation and data.Weights are assigned to the simulated events such that the track multiplicity distribution agrees with the data.The tracking efficiency for charged tracks is estimated using data, and the difference between simulation and data is obtained from a correction table as a function of p T , η and number of charged tracks [30].The total efficiencies of the signal and normalisation decays are (3.41 ± 0.02) × 10 −4 and (8.15 ± 0.11) × 10 −4 , respectively, where the uncertainties come from the statistics of the simulation samples.An alternative set of selections of the B − → Λ + c Λ − c K − signal with tighter criteria on p T and looser criteria on B − decay time is used as a cross-check, and the efficiency-corrected yield is found to be consistent with the baseline selections.
The systematic uncertainty on the relative branching fraction comes from uncertainties in signal yield determination and efficiency estimation.The working point of the BDT output is chosen to maximise the signal significance.The choice of BDT thresholds are varied and no significant bias is found.The average of the efficiency-corrected yield agrees with the nominal result within statistical uncertainties, and the maximum change in the central value is taken as a systematic uncertainty.Alternative assumptions are made when performing the 3D fits, including varying the fitting range, using fixed shapes from simulation for the signal in the fit, and varying the mass resolution for B decay backgrounds.The shifts in yield are taken as systematic uncertainties.The simulation sample is weighted to match the track multiplicity distribution of data, and the uncertainties of the weights are propagated to the efficiencies.The efficiency dependence on the decay phase space is studied in the plane for the signal (normalisation) decay.The efficiencies are calculated alternatively according to each candidate's decay phase space and the shift in central values is taken as a systematic uncertainty.The tracking efficiency has a negligible effect on the final results.The PID values in simulation are sampled to better agree with data using a dedicated tool [24].An alternative template is used for the sampling and the shift in the efficiency-corrected yield is taken as a systematic uncertainty.The parameterisation of the VELO materials, which influences IP-related variables, is tuned on simulation.The χ 2 IP distribution in simulation is scaled with parameters given by the Λ 0 b → Λ + c π − sample, and the shift in R B using the scaled simulation is assigned as a systematic uncertainty.The trigger efficiencies are estimated using the simulation as the baseline.They are also estimated using data, taking the events triggered independent of the B − candidates.The estimated efficiencies between the two methods agree within statistical uncertainties, and the difference between efficiencies calculated using data and the nominal value is assigned as a systematic uncertainty.The uncertainty due to the limited simulation sample size is propagated to the final result.The systematic uncertainty on R B is summarised in Table 4, and all the terms mentioned above are added in quadrature.
The ratio of branching fractions is measured to be where the uncertainties are statistical, systematic and that due to uncertainties on the Λ + c and D + branching fractions.This measurement is significantly improved with respect to R B = 2.23 ± 0.78 calculated using values from Ref. [25].

Summary
The B − → Λ + c Λ − c K − decay is studied for the first time in pp collisions using data collected by the LHCb experiment, corresponding to an integrated luminosity of 5 fb −1 .In the

Source
No resonance structure is found at higher masses.Evidence for the Ξ c (2880) 0 state is found with a significance of 3.8 σ.In addition, evidence of a new Ξ c (2790) 0 → Λ + c K − decay is found with a significance of 3.7 σ.
The relative branching fraction of B − → Λ + c Λ − c K − is measured with respect to B − → D + D − K − as R B = 2.36 ± 0.11 ± 0.22 ± 0.25, where the uncertainties are statistical, systematic and that due to the charm decay branching fraction.This is the most precise measurement to date of this ratio.q Scuola Normale Superiore, Pisa, Italy r Università di Pisa, Pisa, Italy s Università della Basilicata, Potenza, Italy t Università di Roma Tor Vergata, Roma, Italy u Università di Siena, Siena, Italy v Università di Urbino, Urbino, Italy w Universidad de Alcalá, Alcalá de Henares , Spain † Deceased

Figure 1 :Figure 2 :
Figure 1: Distributions of (a) M(Λ + c Λ − c K − ), (b) M (pK + π − ) and (c) M (pK − π + ) of selected B − → Λ + c Λ − c K − candidates.The data points with error bars are shown along with the total fitted shape, which is composed of signal and background components, as shown in the legend.
are shown in Fig. 4. No significant structure is seen.The distributions of M (Λ − c K − ) and M (Λ + c Λ − c ) are also produced after rejecting candidates with 2900 < M (Λ + c K − ) < 2970 MeV, to remove contributions from the Ξ c (2923) 0 and Ξ c (2939) 0 states.No significant structure is seen.

Figure 3 :
Figure 3: Mass spectrum of the Λ + c K − pair from the B − → Λ + c Λ − c K − decays.The data points with error bars are shown along with the total fitted shape in blue solid line, which is composed of the components, as shown in the legend.

Table 2 :
Measured masses, widths and significance of excited Ξ 0 c states.

Table 4 :
Relative systematic uncertainties of the branching fractions of signal, normalisation channel, and their ratio R B (in percent).Correlation between the two channels are considered.
− invariant mass spectrum, two neutral excited charm baryon states, Ξ c (2923) 0 and Ξ c (2939) 0 , are observed.Their masses and widths are in agreement with those of the states observed in a prompt Λ + c K − measurement [7].These new measurements confirm that the Ξ c (2930) 0 state observed in B