Measurement of Branching Fractions of Hadronic Decays of the $\Omega_c^0$ Baryon

Using a data sample of 980 ${\rm fb}^{-1}$ of $e^+e^-$ annihilation data taken with the Belle detector operating at the KEKB asymmetric-energy $e^+e^-$ collider, we report the results of a study of the decays of the $\Omega_c^0$ charmed baryon into hadronic final states. We report the most precise measurements to date of the relative branching fractions of the $\Omega_c^0$ into $\Omega^-\pi^+\pi^0$, $\Omega^-\pi^+\pi^-\pi^+$, $\Xi^-K^-\pi^+\pi^+$, and $\Xi^0K^-\pi^+$, as well as the first measurements of the branching fractions of the $\Omega_c^0$ into $\Xi^-\bar{K}^0\pi^+$, $\Xi^0\bar{K}^0$, and $\Lambda \bar{K}^0\bar{K}^0$, all with respect to the $\Omega^-\pi^+$ decay. In addition, we investigate the resonant substructure of these modes. Finally, we present a limit on the branching fraction for the decay $\Omega_c^0\to\Sigma^+K^-K^-\pi^+$.

Using a data sample of 980 fb −1 of e + e − annihilation data taken with the Belle detector operating at the KEKB asymmetric-energy e + e − collider, we report the results of a study of the decays of the Ω 0 c charmed baryon into hadronic final states. We report the most precise measurements to date of the relative branching fractions of the Ω 0 c into Ω − π + π 0 , Ω − π + π − π + , Ξ − K − π + π + , and

INTRODUCTION
The Ω 0 c comprises the combination of a charm quark and two strange quarks [1]. The ground-state Ω 0 c has the ss diquark in a J P = 1 + configuration, and decays weakly. There are no measurements of the absolute branching fractions of the Ω 0 c , but some measurements of the branching ratios of modes with respect to the normalizing mode Ω − π + have been made [2][3][4]. However, because the production cross section of the Ω 0 c is lower than the other singly charmed baryons, and because it typically decays to more complicated final states, there is less information on its hadronic decays than there is for the other weakly decaying charmed baryons (Λ + c , Ξ 0 c , and Ξ + c ) or for the charmed mesons. In this paper, we present the most precise measurements of the branching fractions of Ω 0 c decays into the four decay modes (Ω − π + π 0 , Ω − π + π − π + , Ξ − K − π + π + , Ξ −K 0 π + ). These modes have previously been measured by the CLEO [2] and/or BaBar [4] Collaborations. We also present the measurement of three previously unreported decays (Ξ −K 0 π + , Ξ 0K 0 and ΛK 0K 0 ) and a search for one other decay, Σ + K − K − π + , that was reported by the E687 Collaboration [5]. All branching fractions are measured relative to the decay Ω 0 c → Ω − π + . In addition, we investigate the resonant substructure of the decays we observe. The choice of decay modes was guided by previous observations, analogy with other charmed baryon decay modes, and consideration of the detector capabilities.
The four ground-state charmed baryons all decay predominantly through the weak decay c → sW + , but each has its own features. Uniquely among the four, the two spectator quarks of the Ω 0 c have the same flavor, and this leads to many decay diagrams producing the same final states. Constructive interference among these diagrams is thought to explain the short lifetime, despite the fact that, unlike the Λ + c and Ξ 0 c , the Ω 0 c cannot decay via a Cabibbo-favored W-exchange diagram [6]. Measuring the branching fractions of all the charmed hadrons helps disentangle the various processes involved and adds to our knowledge of the dynamics of charmed baryon decays.
This analysis uses a data sample of e + e − annihilations recorded by the Belle detector [7] operating at the KEKB asymmetric-energy e + e − collider [8]. It corresponds to an integrated luminosity of 980 fb −1 . The majority of these data were taken with the accelerator energy tuned for production of the Υ(4S) resonance, as this is optimum for investigation of B decays. However, the Ω 0 c particles in this analysis are produced in continuum charm pro-duction and are of higher momentum than those that are decay products of B mesons, so the dataset used in this analysis also includes the Belle data taken at beam energies corresponding to the other Υ resonances and the nearby continuum (e + e − → qq, where q ∈ {u, d, s, c}).

THE BELLE DETECTOR AND PARTICLE RECONSTRUCTION
The Belle detector is a large-solid-angle spectrometer comprising six sub-detectors: the Silicon Vertex Detector (SVD), the 50-layer Central Drift Chamber (CDC), the Aerogel Cherenkov Counter (ACC), the Time-of-Flight scintillation counter (TOF), the electromagnetic calorimeter, and the K L and muon detector. A superconducting solenoid produces a 1.5 T magnetic field throughout the first five of these sub-detectors. The detector is described in detail elsewhere [7]. Two inner detector configurations were used. The first comprised a 2.0 cm radius beampipe and a 3-layer silicon vertex detector, and the second a 1.5 cm radius beampipe and a 4-layer silicon detector and a small-cell inner drift chamber.
Final-state charged particles, π ± , K − , and p, are selected using the likelihood information from the tracking (SVD, CDC) and charged-hadron identification (CDC, ACC, TOF) systems, L(h1 : h2) = L h1 /(L h1 + L h2 ), where h 1 and h 2 are p, K, and π as appropriate. In general, we require proton candidates to have L(p : K) > 0.6 and L(p : π) > 0.6 (≈ 96% efficient); kaon candidates to have L(K : p) > 0.6 and L(K : π) > 0.6 (≈ 94% efficient); and pions to have the less restrictive requirements of L(π : K) > 0.2 and L(π : p) > 0.2 (≈ 99% efficient). The π 0 candidates used in hyperon reconstruction are formed from two clusters unassociated with a charged track, each consistent with being due to a photon, and each of energy above 50 MeV in the laboratory frame. The invariant mass of the photon pair is required to be within 3 standard deviations (σ) of the π 0 mass [9]. Because of the large combinatorial background, the π 0 candidates used for Ω 0 c → Ω − π + π 0 reconstruction have more restrictive requirements of at least 100 MeV energy per photon, at least 300 MeV/c π 0 momentum, and an invariant mass within 2σ of the π 0 nominal mass.
The Λ (K 0 S ) candidates are reconstructed from pπ − (π + π − ) pairs with a production vertex significantly separated from the nominal interaction point (IP) in the r − φ plane (perpendicular to the beam axis). For the case of the proton from the Λ, the particle identification (PID) is loosened to L(p : K) > 0.2 and L(p : π) > 0.2. The Λ candidates used as immediate daughters of Ξ c can-didates are required to have trajectories consistent with origination at the IP, but those that are daughters of Ξ − , Ξ 0 or Ω − candidates do not have this requirement.
The Ξ − and Ω − candidates are reconstructed from the Λ candidates detailed above, together with a π − or K − candidate. The vertex formed from the Λ and π/K is required to be at a smaller radial distance from the IP than the Λ decay vertex.
The Ξ 0 and Σ + reconstruction is complicated by the fact that the parent hyperon decays with a π 0 (which has negligible vertex position information) as one of its daughters. In the case of the Σ + → pπ 0 reconstruction, combinations of π 0 candidates and protons are made using those protons with a significantly large (> 1 mm) distance of closest approach (DOCA) to the IP. Then, taking the IP as the point of origin of the Σ + , the point of intersection of the Σ + trjectory and the reconstructed proton trajectory is found. This position is taken as the decay location of the Σ + hyperon, and the π 0 is then re-fit using this as its point of origin. Only those combinations with the decay location of the Σ + indicating a positive Σ + pathlength are retained. The Ξ 0 is reconstructed in a similar manner, but it is not necessary to require a large DOCA with respect to the IP.
Mass requirements are placed on all the hyperons reconstructed, based on the nominal masses of these particles [9]. The half-widths of the allowed ranges of these mass requirements, all corresponding to approximately two standard deviations of the resolution, are 8.0, 5.0, 3.5, 3.5, and 3.5 MeV/c 2 for Σ + , Ξ 0 , Ξ − , Ω − , and Λ, respectively. The particles are then kinematically constrained to the expected masses for further analysis. Baryons and mesons detailed above are combined to reconstruct Ω 0 c candidates. Once the daughter particles of a Ω c candidate are selected, the Ω c candidate itself is made by kinematically fitting the daughters to a common decay vertex. The IP is not included in this vertex, as the small decay length associated with the Ω c decays, though very short compared with the Ξ − , Ξ 0 , Ω − , and Σ + decay lengths, is not negligible. The χ 2 of this vertex fit is required to be consistent with all the daughters being produced by a common parent. To reduce combinatorial background, we require a scaled momentum of x p > 0.6, where x p = p * c/ (s/4 − m 2 c 2 ), p * is the momentum of the Ω c candidate in the e + e − center-of-mass frame, s is the total center-of-mass energy squared, and m is the reconstructed mass. Charmed baryons are known to have a hard fragmentation function, and this requirement produces a good signal-to-noise ratio while retaining high signal efficiency. Figure 1 shows the invariant mass distribution for the normalizing mode Ω 0 c → Ω − π + . A double-Gaussian signal function together with a first-order polynomial function to represent the background are fit to this distri- bution. For this and all similar distributions in this analysis, the resolution function is obtained by studying Monte Carlo (MC) events generated using EvtGen [10], and having the Belle detector response simulated using GEANT3 [11]. Taking the measure of each width to be the weighted average of the widths of the two Gaussian functions of the resolution function, the ratio of the width found by fitting the data in this channel to that found by fitting the MC is 1.035 ± 0.045. This confirms that the MC simulation predicts the resolution well. Figure 2 shows the invariant mass distributions for the other eight Ω 0 c decay modes under consideration. A fit is made to each distribution comprising the sum of a double-Gaussian signal function, as obtained from MC, and a Chebyshev polynomial background function whose order is the lowest that allows a satisfactory fit. An exception is the case of the Ω − π + π 0 final state, for which the resolution function is a bifurcated Gaussian to account for the asymmetry in the mass distribution found in MC. With the exception of the mode Ω 0 c → Σ + K − K − π + , the masses in the fits are free parameters; nevertheless, the resultant masses are consistent with the world-average [9], which is dominated by the measurement in a previous Belle analysis using a subset of the data presented here [12]. In all cases, the resolution functions are fixed from the MC simulation, but should their widths be allowed to float, each would have a width within two standard deviations of the MC values.
The yields and statistical uncertainties for each mode are listed in Table I  clude all branching fractions of the subsequent decays [9]. In the cases where significant substructure is observed (as described in the next section), the MC is generated with this substructure included. This last effect does not change the efficiency of any mode by more than 3% of its nominal value.

RESONANT SUBSTRUCTURE
Many of the modes under consideration may have resonant substructure that can help reveal their decay mechanisms. Figure 3(a) shows the π + π 0 invariant mass for the combinations within 22 MeV/c 2 (≈ 90% efficient) of the Ω 0 c peak in the Ω 0 c → Ω − π + π 0 mass distribution. This distribution has been background-subtracted using events from scaled sidebands between 32 and 76 MeV/c 2 from the peak. A fit is made to this distribution using the sum of a ρ + signal shape and a nonresonant shape flat in phase space. The very small efficiency difference between these two distributions is taken into account to calculate that (83 ± 10%) of the Ω − π + π 0 mode proceeds via the ρ + . This result is consistent with the saturation of the Ωπ + π 0 decay by the pseudo-two-body Ω − ρ + channel. We calculate a lower limit for the Ω − ρ + fraction by integrating the likelihood function obtained from the fit, and finding the value of the fraction for which the integral contains 90% of the total area. This 90% confidence-level lower limit value on the Ω − ρ + fraction of Ω − π + π 0 is 71%.
For the mode Ω 0 c → Ξ − K − π + π + , we define signal candidates as those within 7 MeV/c 2 of the Ω 0 c mass; sidebands of 12 − 26 MeV/c 2 from the Ω 0 c peak value; and present the scaled sideband-subtracted Ξ − π + and K − π + invariant mass distributions in Figs. 3(b) and 3(c). Each distribution has two entries per Ω 0 c candidate. Polynomial nonresonant functions are fit to these distributions to find the yield of Ξ 0 (1530) andK * 0 (892), respectively. Clear signals of 74 ± 20 events and 136 ± 39 events are found, where these uncertainties are statistical. These correspond to (33 ± 9)% and (55 ± 16)% of the Ξ − K − π + π + decays proceeding through Ξ 0 (1530) andK * 0 (892), respectively. There are indications that the signals include pseudo-two-body decays of the type Ω 0 c → Ξ 0 (1530)K * 0 (892), but the signal-to-noise ratio is not sufficient to allow for the measurement of this process. Interference effects are expected to be small and are not taken into consideration.
For the mode Ω 0 c → Ξ 0 K − π + , we select signal events within 11 MeV/c 2 of the Ω 0 c peak value, and use sidebands of 22 to 44 MeV/c 2 . We then plot the sidebandsubtracted K − π + invariant mass distribution and observe a clear peak due to theK * 0 (892) meson. The sum of aK * 0 (892) signal shape and a polynomial nonresonant shape is fit to this distribution and shown in Fig. 3(d).

SYSTEMATIC UNCERTAINTIES
The systematic uncertainties that enter this analysis of the branching fractions are summarized in Table II. To estimate the uncertainty due to the choice of background shape, the order of the Chebyshev polynomial is increased by one and the change in yield taken as the systematic uncertainty. As this always reduces the yield, this is not done for the Ω 0 c → Σ + K − K − π + mode, for which only an upper limit is quoted. The sensitivity to the signal shape is found by repeating the analysis with single, rather than double, Gaussian signal functions both for the normalizing mode and the signal mode.
The MC simulation program is tested using many similar reconstructed signals, and in all cases the extracted resolution values agree with the data within 10%. The systematic uncertainty due to uncertainties in the resolution width are estimated from the change in yield when adjusting the signal widths by 10%.
In addition, there are uncertainties in the simulation of the reconstruction efficiency that are not specific to this analysis. Care is taken to account for the cancelation of uncertainties in the calculation of the branching ratios with respect to the normalizing mode. We assign a relative uncertainty on the track reconstruction varying from 0.35% to 2.5% [13]. The relative uncertainties on the Λ, K 0 S , and π 0 reconstruction are 4.0% [13], 2.8% [14], and 3% [15], respectively. We use studies of Λ → pπ − and D 0 → K − π + decays to assign uncertainties on the PID identification of the kaons and protons of 1.3% per track [13].
Lastly, there is an uncertainty due to changes in the efficiencies when resonant substructure is present. As visible resonant substructure is already taken into account in the efficiency calculations, this effect is small. In the determination of the fractions due to substructure, the statistical uncertainties dominate over the small systematic uncertainties. The small differences in the efficiencies between the resonant and multi-body decays are taken into account in calculating the resonant contribution to these modes.

FINAL RESULTS
The results for the branching fractions are summarized in Table III. In the case of Ω c → Σ + K − K − π + , there is no significant signal. We calculate a 90% confidence upper limit by first combining the statistical and systematic uncertainties, and integrating the resultant likelihood function starting at N signal = 0; the upper limit is set when the integral reaches 90% of the total area. For the cases where substructure is measured, the fraction of the primary mode is given. The results assume a branching fractionK 0 → K 0 S of 50%. Four of the modes presented here have been measured previously [2,4,5]. In all cases, these new measurements are consistent, within two standard deviations, with the previous measurements [9] and provide substantial improvements in precision. It is surprising that we find a restrictive limit on the decay B(Ω c → Σ + K − K − π + )/B(Ω − π + ), even though the E687 experiment, albeit with different relative efficiencies, finds a much larger signal in Σ + K − K − π + than Ω − π + .
There is a paucity of recent predictions on the branching fractions of charmed baryons. However, some patterns in the data of charmed baryon decays are clear. Whereas the other weakly decaying charmed baryons Y c have branching ratios B(Y c → Y π + π − π + )/B(Y c → Y π + ) ≫ 1, it is confirmed that, when Y c is an Ω c , this ratio is considerably less than 1. While multi-body weak decays are difficult to model theoretically, we hope that these new results on pseudo-two-body decays will spur further theoretical work.

ACKNOWLEDGMENTS
We thank the KEKB group for the excellent operation of the accelerator; the KEK cryogenics group for the efficient operation of the solenoid; and the KEK computer group, the National Institute of Informatics, and the PNNL/EMSL computing group for valuable computing and SINET5 network support.
We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the Promotion of Science (JSPS), and the Tau-Lepton Physics Research Center of Nagoya University; the Australian Research Background-subtracted invariant mass distributions for two particle combinations: (a) π + π 0 for Ω 0 c → Ω − π + π 0 decays, (b) Ξ − π + and (c) K − π + for Ω 0 c → Ξ − K − π + π + decays, and (d) K − π + for Ω 0 c → Ξ 0 K − π + decays. The blue dotted lines show the signals, the green dashed lines show the background, and the solid lines the sum of the two. Data are shown with circles.  Figs. 1, 2, and 3. The numbers in parentheses refer to the fraction of the multi-body final state that includes the listed resonance.