Measurements of the branching fractions of $\Xi_c^0 \to \Lambda K_S^0$, $\Xi_c^0 \to \Sigma^0 K_S^0$, and $\Xi_c^0 \to \Sigma^+ K^-$ decays at Belle

Using the entire data sample of $980\mathrm{~fb}^{-1}$ collected with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider, we present measurements of the branching fractions of the Cabibbo-favored decays $\Xi_c^0 \to \Lambda K_S^0$, $\Xi_c^0 \to \Sigma^0 K_S^0$, and $\Xi_c^0 \to \Sigma^+ K^-$. Taking the decay $\Xi_c^0 \to \Xi^- \pip$ as the normalization mode, we measure the branching fraction ratio ${\cal B}(\Xi_c^0 \to \Lambda K_S^0)/{\cal B}(\Xi_c^0 \to \Xi^- \pi^+) = 0.229\pm0.008\pm0.012$ with improved precision, and measure the branching fraction ratios ${\cal B}(\Xi_c^0 \to \Sigma^0 K_S^0)/{\cal B}(\Xi_c^0 \to \Xi^- \pi^+) = 0.038\pm0.006\pm0.004$ and ${\cal B}(\Xi_c^0 \to \Sigma^+ K^-)/{\cal B}(\Xi_c^0 \to \Xi^- \pi^+) = 0.123\pm0.007\pm0.010$ for the first time. Taking into account the branching fraction of the normalization mode, the absolute branching fractions are determined to be ${\cal B}(\Xi_c^0 \to \Lambda K_S^0) = (3.27\pm0.11\pm0.17\pm0.73) \times 10^{-3}$, ${\cal B}(\Xi_c^0 \to \Sigma^0 K_S^0) = (0.54\pm 0.09\pm 0.06\pm 0.12) \times 10^{-3}$, and ${\cal B}(\Xi_c^0 \to \Sigma^+ K^-) = (1.76\pm 0.10\pm0.14\pm 0.39) \times 10^{-3}$. The first and second uncertainties above are statistical and systematic, respectively, while the third ones arise from the uncertainty of the branching fraction of $\Xi_c^0 \to \Xi^- \pi^+$.

Theoretical calculations for the two-body hadronic weak decays Ξ 0 c → B + P have been performed using dynamical models [4] and SU (3) F flavor symmetry methods [5,6], where B and P represent light baryons and pseudoscalar mesons.In hadronic weak decays of charmed baryons, nonfactorizable contributions from inner W -emission and W -exchange topological diagrams play an essential role and cannot be neglected, in contrast with their negligible effects in heavy meson decays [7]. Figure 1 shows the Feynman diagrams from internal W -emission for Ξ 0 c → Λ K0 /Σ 0 K0 decays and W -exchange for Ξ 0 c → Λ K0 /Σ 0 K0 /Σ + K − decays as examples.In Ref. [4], the authors found that the factorizable and nonfactorizable terms in both the S-and P -wave amplitudes of the decay Ξ 0 c → Σ 0 K0 interfere destructively, resulting in a small branching fraction.On the other hand, the interference in the decay Ξ 0 c → Λ K0 is found to be constructive.The decay Ξ 0 c → Σ + K − proceeds only through purely nonfactorizable diagrams, and it allows us to check the importance of such decay diagrams.The branching fractions of Ξ 0 c → Λ K0 , Ξ 0 c → Σ 0 K0 , and Ξ 0 c → Σ + K − decays predicted by different theoretical models are listed in Table I.The ratio of the branching fraction of Ξ 0 c → ΛK 0 S relative to that of Ξ 0 c → Ξ − π + has been measured to be 0.21 ± 0.02 ± 0.02 by Belle using a 140 fb −1 data sample [8].In this paper, we measure the branching fraction ratio B(Ξ 0 c → ΛK 0 S )/B(Ξ 0 c → Ξ − π + ) to improve the precision, and present the first measurements of the branching fraction ratios ) using the entire data sample of 980 fb −1 collected with the Belle detector.Charge-conjugate modes are also implied unless otherwise stated throughout this paper.
Monte Carlo (MC) simulated signal events are generated using EvtGen [13] to optimize the signal selection criteria and calculate the reconstruction efficiencies.Events for the e + e − → cc production are generated using PYTHIA [14] with a specific Belle configuration, where one of the two charm quarks hadronizes into a Ξ 0 c baryon.The Ξ 0 c → ΛK 0 S /Σ 0 K 0 S /Σ + K − decays are generated using a phase space model.
The simulated events are processed with a detector simulation based on GEANT3 [15].Inclusive MC samples of Υ(1S, 2S, 3S) decays, Υ(4S) , and e + e − → q q (q = u, d, s, c) at center-of-mass (C.M.) energies of 9.460, 10.024, 10.355, 10.520, 10.580, and 10.867 GeV corresponding to the total integrated luminosity of data are used to check possible peaking backgrounds and to verify the event selection criteria.

III. COMMON EVENT SELECTION CRITERIA
The selection of the photon candidates as well as the particle identifications (PID) of kaon, pion, and proton are performed using the same methods as in Ref. [3].Furthermore, the impact parameters of kaons with respect to the interaction point (IP) are required to be less than 0.2 cm and 1.0 cm perpendicular to, and along the beam direction, respectively.The K 0 S candidates are first reconstructed from pairs of oppositely charged tracks, which are treated as pions, with a production vertex significantly separated from the IP, and then selected using an artificial neural network [17,18].The Λ candidates are reconstructed via Λ → pπ − decays.The invariant masses of the K 0 S and Λ candidates are required to be within 9.5 MeV/c 2 and 3.5 MeV/c 2 of the corresponding nominal masses [19] (> 95% signal events are retained), respectively.
For the Σ 0 → Λγ reconstruction, the selected Λ candidate is combined with a photon to form a Σ 0 candidate.The energy of the photon is required to exceed 130 MeV in the laboratory frame to suppress combinatorial backgrounds.This criterion is optimized by maximizing the figure-of-merit N sig / N sig + N bkg , where N sig is the number of expected signal events of Ξ 0 c → Σ 0 K 0 S decay, and N bkg is the number of background events in the normalized Ξ 0 c sidebands in data.N sig is obtained from the following formula [19], and B(Ξ − → Λπ − ) = (99.887±0.035)%[19].The optimized selection criterion is the same using the assumed branching fractions from the above-mentioned theoretical predictions.
The Σ + → pπ 0 reconstruction is performed as follows [20].Photon pairs are kept as π 0 candidates.The reconstructed invariant mass of the π 0 candidates is required to be within 15 MeV/c 2 of the π 0 nominal mass [19], corresponding to approximately twice the resolution.To reduce the combinatorial backgrounds, the momentum of the π 0 in the e + e − C.M. frame is required to exceed 0.3 GeV/c, which is optimized using the same method that was used for the energy of photon from the Σ 0 decay [21].Combinations of π 0 candidates and protons are made using those protons with a significantly large (> 1 mm) distance of closest approach to the IP.Then, taking the IP as the point of origin of the Σ + , the sum of the proton and π 0 momenta is taken as the momentum vector of the Σ + candidate.The intersection of this trajectory with the reconstructed proton trajectory is then found and this position is taken as the decay location of the Σ + baryon.The π 0 is then refit using this location as its point of origin.Only those combinations with the decay location of the Σ + indicating a positive Σ + pathlength are retained.
The ΛK 0 S , Σ 0 K 0 S , or Σ + K − combinations are made to form Ξ 0 c candidates with their daughter tracks fitted to a common vertex.The helicity angle of Ξ 0 c candidates is required to be |cosθ(Ξ 0 c )| < 0.75 to suppress the combinatorial background, where θ(Ξ 0 c ) is the angle between the Λ/Σ 0 /Σ + momentum vector and the boost direction from the laboratory frame in the Ξ 0 c rest frame.To reduce combinatorial backgrounds, especially from B-meson decays, the scaled momentum x p = p * Ξ 0 c /p max is required to be larger than 0.55.Here, p * Ξ 0 c is the momentum of Ξ 0 c candidates in the e + e − frame, and p max = 1 For the reference channel Ξ 0 c → Ξ − (→ Λπ − )π + , except for the scaled momentum x p and the Λ selection criteria, all other selection criteria are similar to those used in Ref. [2].The required x p value and the Λ selection criteria of the reference channel are the same as those of the signal channels.Figure 2 shows the invariant mass distribution of Ξ − π + with x p > 0.55 from data, together with the results of an unbinned extended maximum-likelihood fit.In the fit, the signal shape of Ξ 0 c candidates is parameterized by a double-Gaussian function with different mean values, and the background shape is described by a first-order polynomial.The parameters of signal and background shapes are free.The fit result is displayed in Fig. 2 along with the pull (N data − N fit )/σ data distribution, where σ data is the uncertainty on N data , and the fitted signal yield of Ξ 0 c → Ξ − π + decay in data is 40539 ± 315.After applying the aforementioned event selection criteria, the invariant mass distributions of pπ − , Λγ, and pπ 0 from the decays Ξ 0 There are significant Λ, Σ 0 , and Σ + signals observed in the Ξ 0 c signal region, defined as a window of ±20 MeV/c 2 around the Ξ 0 c nominal mass [19] (∼ 2.5σ).In the fits, the signal shapes of Λ and Σ + candidates are described by double-Gaussian functions with different mean values, and the signal shape of Σ 0 is described by a Crystal-Ball function [23].The backgrounds are parametrized by a first-order polynomial function for the pπ − mass spectrum, and second-order polynomial functions for the Λγ and pπ 0 mass spectra.The blue solid curves show the best-fit results, and the blue dashed curves represent the fitted backgrounds.The reduced χ 2 values of the fits are χ 2 /ndf = 1.20, 1.23, and 0.73 for M (pπ − ), M (Λγ), and M (pπ 0 ) distributions, respectively, where ndf = 63, 106, and 112 are the corresponding numbers of degrees of freedom.The ratios of mass resolutions of Λ, Σ 0 , and Σ + candidates between the MC simulations and data are found to be σ MC /σ data = 93%, 90%, and 92%, respectively.
Figure 5 shows the invariant mass spectra of ΛK 0 S , Σ 0 K 0 S , and Σ + K − from data.The cyan shaded histograms indicate events from the normalized Λ, Σ 0 , and Σ + sidebands, respectively.There are no evident peaking backgrounds found in the normalized sidebands or in the inclusive MC samples.To extract the Ξ 0 c signal yields from the two-body decays Ξ 0 c → ΛK 0 S , Ξ 0 c → Σ 0 K 0 S , and Ξ 0 c → Σ + K − , we perform an unbinned extended maximum-likelihood fit to each distribution.The signal shapes of Ξ 0 c candidates are described by double-Gaussian functions with different mean values, where the parameters are floated for Ξ 0 c → ΛK 0 S and Ξ 0 c → Σ + K − decays and are fixed to those obtained from the fit to the corresponding simulated signal distribution for Ξ 0 c → Σ 0 K 0 S decay.The backgrounds are parametrized by second-order polynomial functions with free parameters.The fit results are displayed in Fig. 5 along with the pull distributions, and the corresponding reduced χ 2 values of the fits are χ 2 /ndf = 1.12, 1.44, and 1.21, respectively, where ndf = 46, 51, and 46 are the corresponding numbers of degrees of freedom.The fitted mean values of Ξ 0 c candidates in Ξ 0 c → ΛK 0 S and Ξ 0 c → Σ + K − decays are consistent with the Ξ 0 c nominal mass [19], and the fitted signal yields of Ξ 0 c → ΛK 0 S , Ξ 0 c → Σ 0 K 0 S , and Ξ 0 c → Σ + K − decays in data are listed in Table II.The statistical significances of Ξ 0 c → ΛK 0 S and Ξ 0 c → Σ + K − decays are greater than 10σ.The statistical significance of Ξ 0 c → Σ 0 K 0 S decay is 8.5σ calculated using −2 ln(L 0 /L max ), where L 0 and L max are the maximized likelihoods without and with a signal component, respectively.
TABLE II: Summary of the fitted signal yields N obs and reconstruction efficiencies ǫ.All the uncertainties here are statistical only.

Modes
N obs ǫ(%) FIG. 5: The invariant mass distributions of (a) ΛK 0 S , (b) Σ 0 K 0 S , and (c) Σ + K − from data.The points with error bars represent the data, the blue solid curves show the best-fit results, and the blue dashed curves show the fitted backgrounds.The cyan histograms represent events from the normalized Λ, Σ 0 , and Σ + sidebands.

V. SYSTEMATIC UNCERTAINTIES
There are several sources of systematic uncertainties for the measurements of branching fractions, including detection-efficiency-related uncertainties, the branching fractions of intermediate states, as well as the overall fit uncertainties.Note that the uncertainties from detection-efficiency-related sources and the branching fractions of intermediate states partially cancel in the ratio to the reference mode.
The detection-efficiency-related uncertainties include those from tracking efficiency, PID efficiency, K 0 S reconstruction efficiency, Λ reconstruction efficiency, photon reconstruction efficiency, π 0 reconstruction efficiency, and the uncertainty related to the required Σ 0 signal region.Based on a study of D * + → π + D 0 (→ K 0 S π + π − ) decay, the tracking efficiency uncertainty is evaluated to be 0.35% per track.Using the D * + → D 0 π + , D 0 → K − π + , and Λ → pπ − control samples, the PID uncertainties are estimated to be 1.6% per kaon and 3.5% per proton.The uncertainties associated with K 0 S , Λ, and π 0 reconstruction efficiencies are found to be 2.23% [24], 3.0% [25], and 2.25% [26], respectively.The efficiency uncertainty in the photon reconstruction is 2.0% per photon, according to a study of radiative Bhabha events.For the reference channel Ξ 0 c → Ξ − (→ Λπ − )π + , the PID efficiency uncertainties of π + from the Ξ 0 c decay and π − from the Ξ − decay are considered separately, because π + has a larger momentum.The PID efficiency ratio between the data and MC simulation of π + is found to be ǫ data /ǫ MC = (95.4± 0.7)%, and then we take 95.4% and 0.7% as an efficiency correction factor and PID uncertainty for π + ; the PID efficiency ratio between the data and MC simulation of π − is found to be ǫ data /ǫ MC = (99.5 ± 0.8)%, and 1.3% is taken as the PID uncertainty of π − .We assume that Ξ 0 c → ΛK 0 S /Σ 0 K 0 S /Σ + K − decays are isotropic in the rest frame of Ξ 0 c , and a phase space model is used to generate signal events.For the Ξ 0 c → Σ 0 K 0 S decay, the M (Σ 0 ) resolution discrepancy between data and MC simulation brings an efficiency correction factor 95.5% and systematic uncertainty 0.5% because of the required Σ 0 signal region.For the Ξ 0 c → ΛK 0 S and Ξ 0 c → Σ + K − decays, the uncertainties of the required Λ and Σ + signal regions are less than 1%.For the measurements of B(Ξ 0 c → ΛK 0 S ) and B(Ξ 0 c → Σ 0 K 0 S ), the uncertainties from tracking and Λ reconstruction efficiencies mostly cancel by the reference channel.Assuming these uncertainties are independent and adding them in quadrature, the final detection-efficiency-related uncertainties are obtained, as listed in Table III.
The systematic uncertainties associated with the background shape, fit range, and mass resolution are considered as follows.The order of the background polynomial is changed from second to first or third, and the average deviation compared to the nominal fit result is taken as the systematic uncertainty related to the background shape, which are 3.26%, 9.11%, and 5.20% for Ξ 0 c → ΛK 0 S , Ξ 0 c → Σ 0 K 0 S , and Ξ 0 c → Σ + K − decays, respectively.The fit range is changed by ±20 MeV/c 2 , and the average deviation compared to the nominal fit result is taken as the systematic uncertainty related to the fit range, which are 1.67%, 2.14%, and 2.31% for Ξ 0 c → ΛK 0 S , Ξ 0 c → Σ 0 K 0 S , and Ξ 0 c → Σ + K − decays, respectively.For Ξ 0 c → Σ 0 K 0 S decay, the signal shape of Ξ 0 c is replaced by a Gaussian function with a free resolution convolved with the fixed signal shape from signal MC simulation: the difference in the number of signal events, 4.32%, is taken as the systematic uncertainty related to the mass resolution.The fit uncertainty of the reference mode is estimated using the same method as was used for the signal modes, and the uncertainties associated with the background shape and fit range are determined to be 1.54% and 0.57%, respectively.For each mode, all the above uncertainties are summed in quadrature to obtain the total systematic uncertainty due to the fit.Finally, the fit uncertainties of signal and reference modes are added in quadrature to give the total fit uncertainty for each signal mode.

FIG. 2 :
FIG.2:The invariant mass distribution of Ξ − π + from data.The points with error bars represent the data, the blue solid curve shows the best-fit result, and the blue dashed curve represents the fitted background.
and Ξ 0 c → Σ + K − in data are shown in Figs.3(a)−3(c), together with the results of unbinned extended maximum-likelihood fits described below.

2 )FIG. 3 :FIG. 4 :
FIG.3:The invariant mass distributions of (a) pπ − , (b) Λγ, and (c) pπ 0 candidates from the decays Ξ 0 c → ΛK 0 S , Ξ 0 c → Σ 0 K 0 S , and Ξ 0 c → Σ + K − in the Ξ 0 c signal region in data.The points with error bars represent the data, the blue solid curves show the best-fit results, and the blue dashed curves are the fitted backgrounds.The vertical solid lines represent the required signal regions, and the vertical dashed lines show the defined sidebands.