Measurement of the absolute branching fraction of the three-body decay Λ + c → Ξ 0 K + π 0 and search for Λ + c → nK + π 0 , Σ 0 K + π 0 and Λ K + π 0

The Cabbibo-favored decay Λ + c → Ξ 0 K + π 0 is studied for the first time using 6.1 fb − 1 of e + e − collision data at center-of-mass energies between 4.600 and 4.840 GeV, collected with the BESIII detector at the BEPCII collider. With a double-tag method, the branching fraction of the three-body decay Λ + c → Ξ 0 K + π 0 is measured to be (7 . 79 ± 1 . 46 ± 0 . 95) × 10 − 3 , where the first and second uncertainties are statistical and systematic, respectively. The branching fraction of the two-body decay Λ + c → Ξ(1530) 0 K + is (5 . 99 ± 1 . 04 ± 0 . 32) × 10 − 3 , which is consistent with the previous result of (5 . 02 ± 0 . 99 ± 0 . 31) × 10 − 3 . In addition, the upper limit on the branching fraction of the doubly Cabbibo-suppressed decay Λ + c → nK + π 0 is 7 . 1 × 10 − 4 at the 90% confidence level. The upper limits on the branching fractions of Λ + c → Σ 0 K + π 0 and Λ K + π 0 are also determined to be 1 . 8 × 10 − 3 and 2 . 0 × 10 − 3 , respectively.


I. INTRODUCTION
Hadronic decays of charmed baryons provide an ideal laboratory to understand the interplay of weak and strong interactions in the charm system [1][2][3][4][5][6][7][8][9], and the measured branching fractions of charmed baryons play an important role in constraining models of charmed hadronic decays [10,11].However, no reliable phenomenological model of charmed baryon decays currently exists.The Λ + c baryon is the lightest charmed baryon, thus it is often a decay product in many charmed baryon decays.Hence, an investigation of the Λ + c decay is essential for understanding excited charmed baryons [12].In charmed baryon decays, the non-factorizable contributions from W-exchange diagrams play an essential role.
In contrast, these effects are negligible in heavy meson decays [13].Therefore, measurements of the absolute branching fractions of Λ + c decays are important to understanding the internal dynamics of charmed baryon decays [14].
suppressed (DCS) decays of Λ + c have not yet been studied.Based on the data sets at BESIII, measurements of the branching fractions of Λ + c decays of currently known decay modes could potentially reach a sensitivity of 10 −4 .In particular, the branching fraction of Λ + c → nπ + was recently measured to be (6.6 ± 1.2 ± 0.4) × 10 −4 [17].In addition, the data allows for a search of many rare Λ + c decays.
The three-body CF decay Λ + c → Ξ 0 K + π 0 is expected to have a large decay rate.Figure 1(a) presents a typical Feynman diagram of this decay.Some phenomenological models have predicted different branching fractions for the Λ + c → Ξ 0 K + π 0 decay, which are (4.5±0.8)×10−2 [23] and (3.2 ± 0.6) × 10 −2 [24] by assuming SU(3) flavor symmetry and isospin asymmetry, respectively.Experimentally, there are only a few studies of Λ + c decays with a Ξ 0 baryon in the final state, and the three-body decay Λ + c → Ξ 0 K + π 0 has not yet been studied.Measurement of the branching fraction will help us further understand the underlying dynamics of Λ + c decays and distinguish among the different theoretical models [23].

II. BESIII DETECTOR AND MONTE CARLO
The BESIII detector [29] records symmetric e + e − collisions provided by the BEPCII [30] storage ring, which operates in the CM energy range from 2.0 to 4.95 GeV, with a peak luminosity of 1.1 × 10 33 cm −2 s −1 achieved at √ s = 3.77 GeV [31].The cylindrical core of the BESIII detector covers 93% of the full solid angle and consists of a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic field.The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identification modules interleaved with steel [32].The charged-particle momentum resolution at 1 GeV/c is 0.5%, and resolution of the ionization energy loss in the MDC (dE/dx) is 6% for electrons from Bhabha scattering.The EMC measures photon energies with a resolution of 2.5% (5%) at 1 GeV in the barrel (end-cap) region.The time resolution in the TOF barrel region is 68 ps.The end-cap TOF system was upgraded in 2015 using multi-gap resistive plate chamber technology, providing a time resolution of 60 ps [33].
Simulated samples produced with geant4-based [34] Monte Carlo software, which includes the geometric description [35,36] of the BESIII detector and the detector response, are used to determine the detection effi-ciency and to estimate backgrounds.Exclusive simulation samples of e + e − → Λ + c Λ− c are produced with Λ− c decaying into ten specific tag modes and Λ + c decaying into Ξ 0 K + π 0 and nK + π 0 .The resonances are modeled with Breit-Wigner functions, of which masses and widths are taken from the Particle Data Group (PDG) [42].The simulation includes the beam-energy spread and initialstate radiation (ISR) in the e + e − annihilations with the specific tag modes [37] modeled with the generator kkmc [38,39].The signal events are modeled with a phase-space generator.The e + e − → Λ + c Λ− c line-shape implements the description from Ref. [18].The inclusive simulation sample, which consists of Λ + c Λ− c events, D (s) production, ISR return to lower-mass ψ states, and continuum processes (e + e − → uū, d d and ss) is used to estimate the potential background.All the known decay modes of charmed hadrons and charmonia are modeled with evtgen [40,41] using branching fractions either taken from the PDG [42], when available, or otherwise estimated with lundcharm [43,44].Final state radiation from charged final state particles is incorporated using photos [45].
The branching fraction is expressed as: where N sig is the signal yield of the DT candidates and B inter is the product of branching fractions of intermediate decays on the signal side from the PDG [42].N ST i denotes the yields of the ST candidates observed in data, and ϵ ST i and ϵ DT i are the ST and DT efficiencies, respectively.

IV. EVENT SELECTION
The selection criteria in this analysis follow that of Ref. [17].Charged tracks are required to have a polar angle (θ) within |cos θ| < 0.93, where θ is defined with respect to the z-axis, which is the symmetry axis of the MDC.Tracks, except for those from K 0 S and Λ decays, are required to originate from an interaction region defined by |V xy | < 1 cm and |V z | < 10 cm (referred as a tight track hereafter), where |V xy | and |V z | refer to the distances of closest approach of the reconstructed track to the interaction point (IP) in the xy plane and the z direction, respectively.
Particle identification (PID) for charged tracks combines measurements of the energy deposited in the MDC (dE/dx) and the flight time in the TOF to form likelihoods L(h) (h = p, K, π) for each hadron h hypothesis.Tracks are identified as protons when the proton hypothesis has the greatest likelihood (L(p) > L(K) and L(p) > L(π)), while charged kaons and pions are identified by comparing the likelihoods for the kaon and pion hypotheses, L(K) > L(π) and L(π) > L(K), respectively.
The selection of tracks from K 0 S and Λ is different from those of other tracks.Candidates of K 0 S and Λ hadrons are reconstructed from their decays to π + π − and pπ + , respectively, where the charged tracks must satisfy |V z | < 20 cm (referred as a loose track hereafter).The PID selection criteria are imposed on the antiproton candidate, while the charged pion is not subject to any PID requirement.A secondary vertex fit is performed for each K 0 S or Λ candidate, and the momentum obtained from the fit is used in the subsequent analysis.The K 0 S or Λ candidate is retained if the χ 2 of the secondary vertex fit is less than 100.Furthermore, the decay vertex is required to be separated from the IP by a distance of at least twice the vertex resolution.For K 0 S and Λ, the invariant masses of π + π − and pπ + pairs are required to be within (0.487, 0.511) GeV/c 2 and (1.111, 1.121) GeV/c 2 , respectively.The π + π − and pπ − invariant mass resolutions determined through simulations, are 2.9 MeV/c 2 and 1.2 MeV/c 2 , respectively.Similarly, the Σ0 and Σ− candidates are reconstructed from the γ Λ and pπ 0 final states with invariant masses within the ranges of (1.179, 1.203) GeV/c 2 and (1.176, 1.200) GeV/c 2 , respectively.The mass resolutions for Σ0 and Σ− are found, using simulation, to be 3.6 MeV/c 2 and 4.3 MeV/c 2 , respectively.
Photon candidates are identified using showers in the EMC.The deposited energy of each shower must be more than 25 MeV in the barrel region (|cos θ| ≤ 0.80) or more than 50 MeV in the end-cap region (0.86 ≤ |cos θ| ≤ 0.92).To suppress electronic noise and showers unrelated to the event, the difference between the EMC time and the event start time is required to be within (0, 700) ns.
A π 0 candidate is reconstructed with a photon pair, and their invariant mass is required to be within the range (0.115, 0.150) GeV/c 2 .To improve the resolution, a kinematic fit is performed, where the diphoton invariant mass is constrained to the known π 0 mass [42], and the χ 2 of the kinematic fit is required to be less than 200.The momenta obtained from the kinematic fit are used in the subsequent analysis.
To distinguish the ST Λ− c baryons from combinatorial backgrounds, the distributions of the energy difference ∆E are used, defined as where E beam is the beam energy, and E Λ− c is the total energy of the ST candidate, calculated in the e + e − rest frame.The signal events are expected to peak around zero in the ∆E distribution.If an event has multiple Λ− c candidates, the one with the smallest |∆E| is retained, and the ∆E requirements are listed in Table 2.
The beam-constrained mass M BC of the selected ST candidates is defined as where ⃗ p Λ− c is the total momentum of the ST candidate and M BC peaks at the Λ− c mass.For each tag mode, the ST yield is determined by fitting the M BC distribution.In the fit, the Λ− c signal is described by the MC simulated shape convolved with a Gaussian function to compensate for the resolution difference between data and MC.The parameters of the Gaussian function are free in the fit and different for each energy and tag mode.The combinatorial background is described by an ARGUS function [46], with the end-point parameter fixed to the nominal beam energy.The fits to the M BC distributions of the various tag modes for the 4.600 -4.700 datasets are the same as Ref. [37].The fits to the M BC distributions for the 4.740, 4.750, 4.780, and 4.840 datasets are shown in Fig. 2. Candidates in the M BC signal region, (2.275, 2.310) GeV/c 2 , are kept for further analysis.The ST yields in data and the ST efficiencies for individual tags are listed in Table 2.The total ST yield, obtained by summing the ST yield of all tag modes and all energies, is found to be N tot ST = 130439 ± 425, where the uncertainty is statistical.
After the ST selection, we want to determine the number of Λ + c → Ξ 0 K + π 0 , nK + π 0 , Σ 0 K + π 0 and ΛK + π 0 candidates on the opposite side of the ST Λ− c .The K + and π 0 are reconstructed, as described above, from the remaining showers and tracks.Using total four-momentum conservation, the Ξ 0 signal is obtained from the distribution of recoil mass, where E K + , E π 0 , ⃗ p K + and ⃗ p π 0 are the energies and momenta of K + and π 0 candidates, respectively, is the magnitude of the Λ + c momentum, constrained by the beam energy, and Similarly, the yield of the two-body decay Λ + c → Ξ(1530) 0 K + is obtained using the recoil mass.
The Λ baryon has two main decay modes, which are studied in this analysis: the neutral decay mode Λ → nπ 0 , which will be referred to as Cat-1, and the charged decay mode Λ → pπ − , which will be referred to as Cat-2.The Cat-1 and Cat-2 decays are shown schematically in Fig. 3. Four variables M Cat−1 recoil Λ ST c K + are defined by Eqs. 4 and 5.For Cat-1, a single tight track is reconstructed and identified as a K + meson.In Cat-2, three charged tracks are reconstructed as a K + , p, and π − .The K + is a tight track, while the other two are loose tracks from the Λ → pπ − decay.The p and π − are constrained to originate from a common vertex to form the Λ candidates by a vertex fit [47].The invariant mass of the Λ candidate is required to fall inside the range (1.111, 1.121) GeV/c 2 .
The π 0 candidate is reconstructed via the process π 0 → γγ.If there are multiple π 0 candidates, the one with the minimum kinematic-fit χ 2 is assumed to originate from the Λ + c baryon.To suppress sources of background due to neutral hyperon decays from the decays Λ + c → ΛK + , Σ 0 K + and Ξ 0 K + , the events falling inside the corresponding intervals of these three decays of the variable M Cat−1 recoil Λ ST c K + are rejected.This requirement suppresses 95% of these backgrounds.Furthermore, electron mis-PID backgrounds are suppressed using PID and EMC information.To suppress contaminations from long-lived particles in the final state, any event with additional tracks, either loose or tight, is rejected.The DT efficiencies of each signal decay for Cat-1 and Cat-2 are summarized in Tables 3-7.For the Cat-1 q q background, the data events in the M BC sideband region, defined as M BC ∈ (2.200, 2.265) GeV/c 2 , are used.Since Λ + c Λ− c background is present in the sideband region, it is estimated using inclusive simulation samples and subtracted from the total sideband yield to determine the q q background in the sideband.The q q background in the signal region is extrapolated from the yield in the sideband using information from    the simulation.In Cat-2, the simulated q q samples are used.
There exists a contamination from mismatched π 0 candidates (denoted as π 0 mismatch), which consists of three sources: the first is due to the selection of an incorrect π 0 candidate in a signal event with multiple π 0 candidates; the second is where the π 0 candidate is reconstructed using photons that originated from two different π 0 decays; the third is caused by noise in the EMC, which creates a fake photon candidate that is used in the π 0 reconstruction.It has a wide shape and will not contribute to the narrow resonances in the distributions of The π 0 mismatch background is estimated using simulation samples.Events containing showers where the angular separation between the true and reconstructed π 0 directions is greater than 20 • are considered π 0 mismatch background, and the background is normalized according to the branching fractions of Λ + c → Ξ(1530

VI. BRANCHING FRACTION MEASUREMENT
For the two-body decay Λ + c → Ξ(1530) 0 K + , a simultaneous fit is performed by combining the Cat-1 and Cat-2 distributions of M Cat−1 recoil The signal shapes are modeled with the simulation shapes convolved with a Gaussian function representing the resolution difference between data and simulation samples.The background shapes are described by third-order polynomial functions, and the background yields are floating.The fitted curves with distributions of M Cat−1 recoil Λ ST c K + and M Cat−2 recoil Λ ST c K + are shown in Fig. 4. The signal yields are 30±6 and 20±5 events for Cat-1 and Cat-2, respectively.The DT efficiencies in Cat-1 and Cat-2 are summarized in Table 3. From Eq. 1, the branching fraction is determined to be (5.99 ± 1.04) × 10 −3 , where the uncertainty is statistical.It is consistent with the previous result [49].The significance, at 6.9σ, is determined by evaluating the difference in likelihood values between fits that include and exclude the signal component, while considering the change in the number of degrees of freedom.
The DT efficiencies of Cat-1 and Cat-2 for Λ + c → Ξ 0 K + π 0 are summarized in Table 4.The simulation samples use a flat phase space model, but intermediate resonances could affect the momentum and angular distributions of the final state particles, which would affect DT efficiencies.Therefore, the simulation samples are reweighted to match data.The two variables, M K + π 0 and M recoil Λ ST c π 0 , are selected for the reweighting procedure using a 6×6 binning scheme with uniformly spaced bins.The overall efficiency is 3.44% before the procedure and 3.61% after.
Using the results of fits to M Cat−1 recoil Λ ST c K + and M Cat−2 recoil Λ ST c K + and correcting for the differences in efficiency, the contributions to the three-body decays Λ + c → Ξ 0 K + π 0 are calculated.After subtracting them, the yields of the three-body decay Λ + c → Ξ 0 K + π 0 are 48±9 and 31±7 for Cat-1 and Cat-2, respectively.Using Eq. 1 and the corrected efficiency, the branching fraction is determined to be (7.79 ± 1.46) × 10 −3 , where the uncertainty is statistical, and the statistical significance is 8.6σ.
For the three-body decays Λ + c → nK + π 0 , Λ + c → Σ 0 K + π 0 and ΛK + π 0 , simultaneous fits are made to the Λ ST c K + π 0 distributions as shown in Figs. 6, 7 and 8. Since no significant signals are observed, the frequentist method is used to determine the upper limits on the branching fractions of these decays [48].The upper limits, at a 90% confidence level, are determined by integrating the likelihood curves, which are obtained by scanning over the branching fraction.

VII. SYSTEMATIC UNCERTAINTIES
The systematic uncertainties for the branching fraction measurements include those associated with the ST yields (N ST i ), detection efficiencies of the ST Λ− c (ϵ ST i ), detection efficiencies of the DT events (ϵ DT i ) and signal yield (N sig ).Since Eq. 1 contains a ratio of ST and DT efficiencies, any systematic uncertainty on the tag side is canceled to first order.Each of them is evaluated relative to the measured branching fraction.
The numbers of charged tracks are required to be only one (Cat-1) or three (Cat-2) without any extra charged tracks.This difference between data and MC simulation for this selection is studied with control sample of Λ + c → pK − π + (the other Λ− c goes to the 10 tag modes).The uncertainty on the requirement of the number of charged tracks is assigned to be 2.2%, which is denoted as "No extra charged track" in Table 8.
The tracking efficiencies for K + as a function of transverse momentum has been studied with the process J/ψ → K 0 S K + π − → K + π + π − π − .The efficiency differences between data and MC simulation are both 1% for K + tracking and PID efficiencies, which are taken as the systematic uncertainties [50].The uncertainty associated with the π 0 reconstruction is assigned to be 1.0%, studied with control sample of J/ψ → π + π − π 0 [51].
The uncertainties in the total ST yields are 0.5% [52], which arise from the statistical uncertainty and fitting strategy of extracting ST yields.
The uncertainties of the background and fits of the distributions of M Cat−1 recoil Λ ST c K + π 0 and  M Cat−2 recoil Λ ST c K + π 0 arise from the sideband range and background estimation.First, the uncertainty for the data sideband and q q background estimation is derived from the statistical uncertainty of the background estimate in the M BC sideband region and the variation of the sideband range.Second, the background shape is changed from a polynomial function to an ARGUS function.The differences between the new and nominal results are taken as the systematic uncertainties.
The uncertainty due to the Λ reconstruction efficiency, which is estimated by a weighted root-mean-square of the statistical uncertainties for different (p, cos θ) intervals, is assigned to be 0.6% using a sample of J/ψ → pK + Λ decays [53].
The uncertainty due to the π 0 selection combines two sources.The first source is originated from π 0 selection with the minimum kinematic-fit χ 2 , which is studied with control sample of Λ + c → Σ + ω and Λ + c → Σ + π 0 .The difference between data and MC simulation is 1%.The second one is due to the π 0 mismatch component, which is evaluated using simulation samples with an alternative π 0 matching approach, where the π 0 candidate closest to the true π 0 is chosen to be the reconstructed π 0 candidate.The fit is performed on the simulation samples, us-  ing both the nominal and alternative π 0 matching, and the difference in the fitted branching fraction between them is taken as the systematic uncertainty.The uncertainty in reweighting the Λ + c → Ξ 0 K + π 0 simulation samples is derived using different 2D variables and different choices of 2D binning, for instance 6 bins × 6 bins to 7 bins × 7 bins.The uncertainty is determined in a similar manner as the uncertainty from the background shape.and they are treated without any correlation in the combination.The different systematic sources are listed in Table 8, where the two categories have been separated explicitly.

IX. ACKNOWLEDGEMENT
The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong sup-Table 9.The comparison between the measurement and theoretical predictions (×10 −3 ).The first and the second uncertainties are statistical and systematic, respectively.
(b) shows a typical Feynman diagram of Λ + c → nK + π 0 .Experimental measurements of DCS decays of Λ + c are limited due to the suppressed branching fractions.Currently, the only study of a DCS Λ + c decay is Λ Figure 1(c) and 1(d) show typical Feynman diagrams of Λ + V. BACKGROUND ANALYSIS Potential sources of background are classified into two categories: those from e + e − → Λ + c Λ− c (denoted as Λ + c Λ− c background) and those directly originating from continuum hadron production in e + e − annihilation (denoted as q q background).The Λ + c Λ− c background is investigated for both categories with the Λ + c Λ− c inclusive MC samples, after removing the signal processes.The Λ + c Λ− c background distributions are the blue-shaded histograms in Figs. 4 and 5, and they have been normalized to the same luminosity as data.

Fig. 2 .
Fig. 2. The MBC distributions of ST channels of Λ− c at (a) 4.740 GeV, (b) 4.750 GeV, (c) 4.780 GeV and (d) 4.840 GeV.The signal shape of the Λ− c is described by the simulated shape convolved with a Gaussian resolution function, and the background is modeled with an ARGUS function.The points with error bars represent data.The (red) solid curves indicate the fit results and the (blue) dashed curves describe the background shapes.The vertical dashed lines show the MBC selection requirements.

Fig. 3 .
Fig. 3.The schematic diagrams of the two analysis strategies, which are denoted (a) Cat-1 and (b) Cat-2 in the text.

Fig. 4 .
Fig. 4. The fit projections of (a) M Cat−1 recoil Λ ST c K + and (b)M Cat−2 recoil Λ ST c K + distributions.The black points with error bars are data.The black line is the sum of all the components in the fit.The blue-shaded histograms are the Λ + c Λ− c backgrounds.The green-shaded histograms are from the data sideband (in Cat-1) or hadron sample (in Cat-2).

Fig. 5 .
Fig. 5. Fit projections of (a) M Cat−1 recoil Λ ST c K + π 0 and (b) M Cat−2 recoil Λ ST c K + π 0 distributions.The black points with error bars are data.The black line is the sum of all the components in the fit.The blue-shaded histograms are the Λ + c Λ− c background.The green-shaded histograms are from the data sideband (in Cat-1) or hadron sample (in Cat-2).The purple line represents the signal shape of the three-body decay Λ + c → Ξ 0 K + π 0 .The dashed blue line represents the one-dimensional histogram PDF that describes the background distributions of M recoil Λ ST c K + π 0 , which are modeled with an polynomial function.
The total systematic uncertainty for the two strategies is obtained by taking the quadratic sum of the individual values.Some systematic uncertainties are the same between two strategies, e.g.tracking and PID of the charged kaon, so they are assumed to be fully correlated in the combination.Other systematic uncertainties are independent between two strategies, e.g.Λ reconstruction, background estimation in the fits on the distributions ofM Cat−1 recoil Λ ST c K + π 0 and M Cat−2 recoil Λ ST c K + π 0 ,

Table 2 .
The ∆E requirements, the ST yields, and the ST detection efficiencies of each tag mode for the data samples between 4.740 and 4.840 GeV.The uncertainties of the ST yields are statistical only.

Table 5 .
The DT detection efficiency(%) of Λ + c → nK + π 0 in Cat-1 for each tag mode and each energy point.