Measurement of the absolute branching fractions of $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ and $\Sigma(1385)^{+}\eta$

We study the decays $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ and $\Sigma(1385)^{+}\eta$ based on $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pairs produced in $e^+e^-$ collisions at a center-of-mass energy of $\sqrt{s}=4.6~\mbox{GeV}$, corresponding to an integrated luminosity of $567\;\mbox{pb$^{-1}$}$. The data sample was accumulated with the BESIII detector at the BEPCII collider. The branching fractions are measured to be $\mathcal{B}(\Lambda_{c}^{+} \to \Lambda\eta\pi^{+})=(1.84\pm0.21({\rm{stat.}})\pm0.15({\rm{syst.}}))\%$ and $\mathcal{B}(\Lambda_{c}^{+} \to \Sigma(1385)^{+}\eta)=(0.91\pm0.18({\rm{stat.}})\pm0.09({\rm{syst.}}))\%$, constituting the most precise measurements to date.


I. INTRODUCTION
Since the charmed baryon ground state Λ + c was first observed at the Mark II experiment in 1979 [1], progress in the studies of charmed baryon decays was relatively slow both theoretically and experimentally due to the limits of the factorization approach in complicated three quark systems [2] and the lack of experimental data, respectively. Therefore, more efforts in studying hadronic decays of the Λ + c are useful to understand the internal dynamics of charmed baryons.
In this paper, we present an improved measurement of the absolute BFs of the Λ + c → Ληπ + and study the intermediate state Σ * + in the three-body decay. The measurements are based on a Λ + cΛ − c pair data sample produced in e + e − collisions at a center-of-mass energy √ s = 4.6 GeV [9], corresponding to an integrated luminosity of 567 pb −1 [10]. The sample was collected by the BESIII detector [11] at the Beijing Electron Positron Collider (BEPCII) [12]. The collision energy is just above the 1 For simplicity, we use the symbol Σ * + to represent Σ(1385) + resonance throughout this paper.
mass threshold for the production of Λ + cΛ − c pairs, providing a very clean environment without the production of additional hadrons. Taking advantage of this and the excellent performance of the BESIII detector, a singletag method (i.e. only one Λ c of the Λ + cΛ − c pair is reconstructed in each event and the otherΛ c is assumed in the recoil side) is used in the analysis, in order to improve the detection efficiency and acquire more Λ + c candidates. The single-tag method is valid under the condition that Λ + c andΛ − c are always produced in pairs. In this paper, CP violation will be neglected which is reasonable from the studies on the current statistics-limited data set; thus the charge conjugate states are always implied unless mentioned explicitly.

II. BESIII EXPERIMENT AND MONTE CARLO SIMULATION
The BESIII detector is a magnetic spectrometer located at the BEPCII collider. The cylindrical core of the BESIII detector 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 identifier modules interleaved with steel. The acceptance of charged particles and photons is 93% over 4π solid angle. The chargedparticle momentum resolution at 1 GeV/c is 0.5%, and the dE/dx resolution is 6% for the 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 of the TOF barrel part is 68 ps, while that of the end cap part is 110 ps. More detailed descriptions can be found in Refs. [11,12].
Simulated samples produced with the geant4based [13] Monte Carlo (MC) package which includes the geometric description of the BESIII detector [14,15] and the detector response, are used to determine the detection efficiency and to estimate the backgrounds. The simulation includes the beam energy spread and initial state radiation (ISR) in the e + e − annihilations modelled with the generator kkmc [16]. The inclusive MC samples consist of the production of open charm processes, the ISR production of vector charmonium(-like) states, and the continuum processes incorporated in kkmc [16]. The known decay modes are modelled with evtgen [17] using branching fractions taken from the Particle Data Group [8], and the remaining unknown decays from the charmonium states with lundcharm [18]. The final state radiations (FSR) from charged final state particles are incorporated with the photos package [19]. For the production of e + e − → Λ + cΛ − c signal MC samples, which are used to estimate the detection efficiencies, the observed cross sections [20] are taken into account in simulating ISR, and the observed kinematic behavior is considered when simulating Λ + c decays.

III. EVENT SELECTION
Charged particle tracks are reconstructed from hits in the MDC, and are required to have a polar angle θ with respect to the beam direction satisfying | cos θ| < 0.93 and a distance of closest approach to the interaction point (IP) of less than 10 cm along the beam axis (V z ) and less than 1 cm in the plane perpendicular to the beam axis, except for those used to reconstruct the Λ → pπ − decay. Particle identification (PID) for charged particle tracks combines the information from the flight time in the TOF and measurements of ionization energy loss (dE/dx) to form a likelihood L(h) (h = π, K, p) for each hadron (h) hypothesis. Tracks will be identified as protons when this hypothesis is determined to have the largest PID likelihood (L(p) > L(K) and L(p) > L(π)), while charged pions are differentiated from kaons by the likelihood requirement L(π) > L(K).
Clusters with no association to a charged particle track in the EMC crystals are identified as photon candidates when satisfying the following requirements: The deposited energy is required to be larger than 25 MeV in the barrel region (| cos θ| < 0.80) or 50 MeV in the end-cap region (0.86 < | cos θ| < 0.92). To suppress background from electronic noise and showers unrelated to the events, the measured EMC time is required to be within 0 and 700 ns of the event start time. Additionally, in order to eliminate showers related to charged particle tracks, showers are required to be separated by more than 10 • from charged particle tracks. The η meson candidates are reconstructed from photon pairs using an invariant mass requirement of 505 < M (γγ) < 575 MeV/c 2 . The invariant mass spectrum of γγ pairs in data is shown in Fig. 1. To improve the momentum resolution, a kinematic fit constraining the invariant mass to the η nominal mass [8] is applied to the photon pairs and the resultant energy and fitted momentum of the η meson are used for further analysis.
Candidate Λ baryons are reconstructed by combining two oppositely charged tracks for any pairs of pπ − . Those tracks are required to satisfy the polar angle requirement | cos θ| < 0.93 and V z < 20 cm for the distance of closest approach to the IP along the beam axis. No distance constraint is applied in the plane perpendicular to the beam axis. Proton PID is required to improve the signal purity while no PID requirement is applied to the charged pion candidates. The p and π − tracks are constrained to originate from a common decay vertex by requiring the χ 2 of a vertex fit to be less than 100. Furthermore, the reconstructed momentum of the Λ candidate is constrained to be aligned with the line joining the IP and the decay vertex, and the resultant flight distance is required to be larger than twice the fitted resolution. A clear Λ peak appears in the invariant mass spectrum of pπ − in data, as shown in Fig. 1. The pπ − pairs satisfying the mass requirement 1.111 < M (pπ − ) < 1.121 GeV/c 2 are chosen as the final Λ candidates. This requirement is chosen corresponding to ± 3 standard deviations of the reconstruction resolution around the Λ nominal mass [8].
The Λ + c baryon candidates are reconstructed using all combinations of the selected Λ, η and π + candidates. To differentiate Λ + c from background, two kinematic variables calculated in the center-of-mass system, the beam constrained mass c are the energy and momentum of the reconstructed Λ + c candidate respectively, and E beam is the average value of the electron and positron beam energies. For a well reconstructed Λ + c candidate, M BC and ∆E are expected to be consistent with the Λ + c nominal mass and zero, respectively. Candidates are rejected when they fail the requirement of −0.03 < ∆E < 0.03 GeV, which corresponds to ± 3 standard deviations of the signal ∆E distribution. The ∆E distribution in data is shown in Fig. 1. If more than one candidate satisfies the above requirements, we select the one with the minimal |∆E|.

IV. SIGNAL YIELD AND BRANCHING FRACTION
To extract the signal yield for the Λ + c → Ληπ + decay, an unbinned extended maximum likelihood fit is performed to the M BC distribution in data with fitting range 2.25 < M BC < 2.30 GeV/c 2 , as illustrated in Fig. 2. In the fit, the signal shape is derived from the kernelestimated non-parametric shape [21] based on signal MC samples convolved with a Gaussian function to account for the difference between data and the MC simulation caused by imperfect modeling of the detector resolution and beam-energy spread. The high mass tail in that signal shape reflects ISR effects. The parameters of the Gaussian function are free in the fit. The background shape is modeled with an ARGUS function [22] with fixed end-point E beam . The obtained signal yield and the corresponding detection efficiency are listed in Table I. The validity of the ARGUS function to describe the background shape in the M BC spectrum is checked using the inclusive MC samples. No obvious peaking background from the decay Λ + c → pK 0 S η with K 0 S → π + π − is observed and the influence of cross feed is neglected. The BF is calculated using where N sig is the signal yield obtained from the M BC fit, N Λ + cΛ − c = (105.9 ± 4.8(stat.) ± 0.5(syst.)) × 10 3 is the number of Λ + cΛ − c pairs in the data sample [23], ε is the detection efficiency estimated using the signal MC simulation sample, and B inter = B(Λ → pπ − ) · B(η → γγ) is taken from the PDG [8]. The factor of 2 in the denominator takes into account the charge conjugate decay mode of the Λ + c baryon. The resultant BF and corresponding statistical uncertainty are listed in Table I.  2. Fit to the MBC distribution for the Λ + c → Ληπ + decay. The dots with error bars are data, the (black) solid curve is the fit function which is the sum of the signal shape (red dashed curve) and the background shape (blue dashdotted curve). A test of goodness-of-fit with χ 2 divided by the degrees of freedom is shown.
To check the possible intermediate states forementioned in the theoretical calculations [3][4][5], the two-dimensional Dalitz distributions of M 2 (Λη) versus M 2 (Λπ + ) for selected Λ + c → Ληπ + candidates in the M BC signal region 2.282 < M BC < 2.291 GeV/c 2 and the sideband region 2.250 < M BC < 2.270 GeV/c 2 are shown in Fig. 3(a) and (b), respectively. In addition, the corresponding one-dimensional projections are presented in Fig. 3(c)-(e). In the M (Λπ + ) spectrum, an obvious peak of the Σ * + resonance is seen, which has been studied at CLEO [6], while other potential states are not evident in these projections. Hence, under the current statistics, we only measure the decay rate of Λ + c → Σ * + η. Σ * + η, Σ * + → Λπ + , an unbinned extended maximum likelihood fit is performed to the invariant mass spectrum of M (Λπ + ) for the events within the M BC signal region. The fitting range is 1.25 < M (Λπ + ) < 1.56 GeV/c 2 as illustrated in Fig. 4. In the fit, the signal shape is derived from the kernel-estimated nonparametric shape [21] based on signal MC samples convolved with a Gaussian function. In the Gaussian function, their parameters are allowed to vary in the fit. The signal lineshape of the Σ * + is generated according the following formula using the mass-dependent width Γ(m) with the expression where m = M (Λπ + ), m 0 and Γ 0 are the Σ * + nominal mass and width, respectively, q and p (p 0 ) are the daughter momenta of Λ + c and Σ * + (when Σ * + is at its nominal mass m 0 ) at their rest frame, respectively, and L b = 1(L d = 1) is angular momentum between the twobody decay products in the Λ + c (Σ * + ) rest frame. f (p) are Blatt-Weisskopf barrier factors which have been detailed in Ref. [24]. Possible interference between Σ * + and non-Σ * + amplitudes is neglected. The random combinatorial background is also modeled with kernel-estimated non-parametric shape [21] based on data in the M BC sideband region. The non-Σ * + background is described with a smooth background function f bkg (M (Λπ + )) ∝ (M (Λπ + ) − 1.25) c · (1.75 − M (Λπ + )) d , where the parameters c and d are obtained from MC-simulated non-Σ * + backgrounds and fixed in the fit. Only the integral of the signal shape in the signal region 1.32 < M (Λπ + ) < 1.45 GeV/c 2 is counted as signal yield. The signal yield and the corresponding detection efficiency are listed in Table I. The corresponding BF is calculated using Eq. (1), where ε is the corresponding detection efficiency and B inter = B(Σ * + → Λπ + ) · B(Λ → pπ − ) · B(η → γγ) taken from the PDG [8]. The resultant BF and the corresponding statistical uncertainty are also listed in Table I.

V. SYSTEMATIC UNCERTAINTY
Different sources of systematic uncertainties are considered in the BF measurement, including charged particle tracking, PID, reconstruction of intermediate states, the ∆E requirement, the fitting range, the background description, the signal MC model, peaking backgrounds and intermediate BFs.
Tracking and PID for π + particle. By studying a set c → Ληπ + candidates. The dots with error bars are the data, the (black) solid curve is fit function, which is the sum of the signal shape (red dashed curve), a smooth background shape describing the background from non Σ * + states (green dotted curve) and the shape of random combinatorial background estimated using the MBC sideband (blue dash-dotted curve). A test of goodness-of-fit with χ 2 divided by the degrees of freedom is shown.
of control samples of e + e − → π + π − π + π − events based on data collected at energies above √ s = 4.0 GeV, which are the same as used in Ref. [23], the tracking and PID efficiencies are estimated in data and MC simulations. After weighting these efficiencies to the π + kinematics in the signal samples, the uncertainties associated with π + tracking and PID efficiencies are derived out to be 1.0% for each decay mode.
Reconstruction for Λ particle. The efficiencies for Λ reconstruction in data and MC simulations are measured with control samples of J/ψ →pK + Λ and J/ψ → ΛΛ events, which are the same as studied by Ref. [25]. The uncertainties of Λ reconstruction efficencies are estimated to be 3.7% for each decay mode, according to the Λ momentum and angular distributions in the signal samples.
Reconstruction for η particle. We use a control sample of π 0 from D meson decays [26] to evaluate the η reconstruction efficiency in the decay to two photons, taking advantage of their close kinematic phase space in the laboratory frame. By studying the control sample, the γγ reconstruction efficiencies are obtained in data and MC simulations, and an uncertainty of 3.4% is assigned by weighting these efficiencies to the η momentum distribution in the signal samples.
Requirement for ∆E. To estimate the systematic uncertainty arising from ∆E requirement, we repeat the measurement procedure by varying the boundaries of the ∆E signal ranges with ± 1 MeV. The largest changes in the resultant BFs, 2.3% and 1.5% for the decays Λ + c → Ληπ + and Λ + c → Σ * + η, respectively, are taken as systematic uncertainties.
Fitting range. To estimate the systematic uncertainty associated with the fitting range, we repeat the measurements by using alternative M BC fitting ranges of 2.26 < M BC < 2.30 GeV/c 2 for the decay Λ + c → Ληπ + and of 1.25 < M (Λπ + ) < 1.55 GeV/c 2 for the decay Λ + c → Σ * + η. The changes in resultant BFs, 0.9% and 2.7% for the decays Λ + c → Ληπ + and Λ + c → Σ * + η, respectively, are considered as the systematic uncertainties.
Background description. For the Λ + c → Ληπ + decay, we repeat the measurement by varying the M BC end-point (2.3 GeV/c 2 ) in the ARGUS function by ± 0.5 MeV/c 2 , by adding a Gaussian function to model the affection rising from the possible peak around 2.26 GeV/c 2 and also by using an alternative background model of a linear combination of the ARGUS function and the MC-simulated background shape. Quadratically summing the changes in resultant BFs for these three sources brings a systematic variation of 1.8% for Λ + c → Ληπ + decay. For Λ + c → Σ * + η decay, we let the parameters of non-Σ * + background function be float and repeat the measurement procedures, which leads to a systematic change of 4.8% on the BF result.
Signal MC model. For the Λ + c → Ληπ + decay, we consider the difference of angular and momentum distributions of final states Λ, η and π + particles between data and signal MC samples and calculate weight factors us- , where i is a specific kinematic interval and n is the number of events that pass the event selections in data or signal MC samples. The change of the re-weighted efficiency from the nominal efficiency is calculated to be 2.9%, which is assigned as the systematic uncertainty. For the Λ + c → Σ * + η decay, we calculate the polar angle θ Σ * + of the momentum of the Σ * + with respect to that of the Λ + c in the rest frame of the Λ + c . We model the signal process according to the distribution of 1 + α · cos 2 θ Σ * + in the range of −1 ≤ α ≤ 1. The maximum change on the MC-determined efficiency is 1.3%. Furthermore, we vary the nominal mass and width of the Σ * + within uncertainties in PDG [8], and the maximum change on the signal yield is 0.5%. By summing up all contributions in quadrature, an uncertainty of 1.4% assigned.
Peaking background. We estimate the sizes of the potentially underestimated peaking backgrounds by detailed background analysis of the inclusive MC samples in measurement of the Λ + c → Ληπ + decay rate, which is estimated to be 1.9%. For the studies of the Λ + c → Σ * + η decay rate, we incorporate complex components from non-Σ * + intermediate processes in the MC simulations of the Λ + c → Ληπ + decays, and analyze the amplitude of the peaking background contribution beneath the Σ * + peak. The relative peaking background rate is evaluated to be 1.6%.

VI. SUMMARY
In summary, the absolute branching fractions of the two processes Λ + c → Ληπ + and Σ * + η are measured using a single-tag method on a data sample produced in e + e − collisions at √ s = 4.6 GeV collected with the BESIII detector. The results are B(Λ + c → Ληπ + ) = (1.84 ± 0.21 ± 0.15)% and B(Λ + c → Σ * + η) = (0.91 ± 0.18 ± 0.09)%, where the first uncertainties are statistical and the second systematic. These are the first absolute measurements of the branching fractions for these two modes, and are consistent with the previous relative measurements [6,7], but with improved precisions. Under the current statistics, no other potential intermediate states are concluded. Future Λ + c data samples with larger statistics will allow for detailed studies of the intermediate states proposed in Refs. [3][4][5].