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

TL;DR

This paper reports precise measurements of the absolute branching fractions for the decays of the b1_c^+ baryon into b1 b5 b7^+ and b1 b5 b7^+ b5, using data from electron-positron collisions at 4.6 GeV.

Contribution

First precise measurement of these specific b1_c^+ decay branching fractions using BESIII data at 4.6 GeV.

Findings

b1_c^+ b1 b5 b7^+ branching fraction: (1.84 b1 0.21 b1 0.15)%

b1_c^+ b1 b5 b7^+ b5 branching fraction: (0.91 b1 0.18 b1 0.09)%

Most precise measurements to date

Abstract

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.

Tables

Table 1. Table 1: Summary of the signal yields, the detection efficiencies, and the BFs for the different Λ c + superscript subscript Λ 𝑐 \Lambda_{c}^{+} decay modes. In the BFs, the first uncertainties are statistical, and the second are systematic.

	$\Lambda \eta {\pi }^{+}$	${\Sigma }^{\ast +}\eta$
$N_{\mathrm{sig}}$	$154\pm 17$	$54\pm 11$
$\epsilon (\%)$	$15.73\pm 0.01$	$12.84\pm 0.01$
$ℬ(\%)$	$1.84\pm 0.21\pm 0.15$	$0.91\pm 0.18\pm 0.09$

Table 2. Table 2: Summary of the relative systematic uncertainties in percentage. The total values are calculated by summing up all contributions in quadrature.

Source	$\Lambda \eta {\pi }^{+}$	${\Sigma }^{\ast +}\eta$
Tracking	1.0	1.0
PID	1.0	1.0
${\Lambda }_{p{\pi }^{-}}$ reconstruction	3.7	3.7
${\eta }_{\gamma \gamma }$ reconstruction	3.4	3.4
$\Delta E$ requirement	2.3	1.5
Fitting range	0.9	2.7
Background description	1.8	4.8
Signal MC model	2.9	1.4
Peaking background	1.9	1.6
$N_{{\Lambda }_c^{+}{\overline{\Lambda }}_c^{-}}$	4.6	4.6
${ℬ}_{\mathrm{inter}}$	0.9	1.9
Total	8.4	9.5

Equations

$$\mathcal{B}({\Lambda }_c^{+}\to \Lambda \eta {\pi }^{+})=\frac{N_{\mathrm{sig}}}{2\cdot N_{{\Lambda }_c^{+}{\bar{\Lambda }}_c^{-}}\cdot \epsilon \cdot {\mathcal{B}}_{\mathrm{inter}}},$$

$$|A(m){|}^2\propto \frac{q^{2L_b+1}{f}_{L_b}^2(q)\cdot p^{2L_d+1}{f}_{L_d}^2(p)}{(m^2-m_0^2{)}^2+m_0^2{\Gamma }^2(m)},$$

$$\Gamma (m)={\Gamma }_0{\left(\frac{p}{p_0}\right)}^{2L_d+1}\left(\frac{m_0}{m}\right)\frac{f_{L_d}^2(p)}{f_{L_d}^2(p_0)},$$

Full text

I Introduction

Since the charmed baryon ground state $\Lambda_{c}^{+}$ was first observed at the Mark II experiment in 1979 1980GSAbrams , 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 1992HYCheng and the lack of experimental data, respectively. Therefore, more efforts in studying hadronic decays of the $\Lambda_{c}^{+}$ are useful to understand the internal dynamics of charmed baryons.

Theoretically, in Ref. 2016JJXie , the decay $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ was pointed out as an ideal process to study the $a_{0}(980)$ and $\Lambda(1670)$, because the final states $\eta\pi^{+}$ and $\Lambda\eta$ are in pure isospin $I=1$ and $I=0$ combinations. Also in Ref. 2015Kenta , resonances $\Lambda(1405)$ and $\Lambda(1670)$ have been studied in $\Lambda\eta$ combinations, and in Ref. 2017JJXie , several $\Sigma^{*}$ states including possible pentaquark state $\Sigma_{1/2^{-}}(1380)$ and resonance $\Sigma(1385)$ have been studied in $\Lambda\pi^{+}$ combinations. Experimentally, the decays $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ and $\Sigma(1385)^{+}\eta$ 111For simplicity, we use the symbol $\Sigma^{*+}$ to represent $\Sigma(1385)^{+}$ resonance throughout this paper. have been studied at the CLEO experiment in 1995 1995RAmmar and 2003 2003DCronin . The branching fractions (BFs) for both channels are measured relative to $\mathcal{B}(\Lambda_{c}^{+}\to pK^{-}\pi^{+})$. After scaling with the average $\mathcal{B}(\Lambda_{c}^{+}\to pK^{-}\pi^{+})$ given by the Particle Data Group (PDG) 2016PDG , the absolute BFs are estimated as $\mathcal{B}(\Lambda_{c}^{+}\to\Lambda\eta\pi^{+})=(2.2\pm 0.5)\%$ and $\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{*+}\eta)=(1.22\pm 0.37)\%$, with large uncertainties at the 20% and 30% level, respectively.

In this paper, we present an improved measurement of the absolute BFs of the $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ and study the intermediate state $\Sigma^{*+}$ in the three-body decay. The measurements are based on a $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pair data sample produced in $e^{+}e^{-}$ collisions at a center-of-mass energy $\sqrt{s}=4.6\;\mbox{GeV}$ 2016Energy , corresponding to an integrated luminosity of 567 pb*-1* 2015Lumi . The sample was collected by the BESIII detector 2009Detector at the Beijing Electron Positron Collider (BEPCII) Yu:IPAC2016-TUYA01 . The collision energy is just above the mass threshold for the production of $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ 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 single-tag method ($i.e.$ only one $\Lambda_{c}$ of the $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pair is reconstructed in each event and the other $\bar{\Lambda}_{c}$ is assumed in the recoil side) is used in the analysis, in order to improve the detection efficiency and acquire more $\Lambda_{c}^{+}$ candidates. The single-tag method is valid under the condition that $\Lambda_{c}^{+}$ and $\bar{\Lambda}_{c}^{-}$ 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 $4\pi$ solid angle. The charged-particle momentum resolution at $1~{}{\rm GeV}/c$ is $0.5\%$, and the $\mbox{d}E/\mbox{d}x$ 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. 2009Detector ; Yu:IPAC2016-TUYA01 .

Simulated samples produced with the geant4-based geant4 Monte Carlo (MC) package which includes the geometric description of the BESIII detector GDMLMethod ; BesGDML 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 ref:kkmc . 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 ref:kkmc . The known decay modes are modelled with evtgen ref:evtgen using branching fractions taken from the Particle Data Group 2016PDG , and the remaining unknown decays from the charmonium states with lundcharm ref:lundcharm . The final state radiations (FSR) from charged final state particles are incorporated with the photos package photos . For the production of $e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ signal MC samples, which are used to estimate the detection efficiencies, the observed cross sections 2018CrossSection are taken into account in simulating ISR, and the observed kinematic behavior is considered when simulating $\Lambda_{c}^{+}$ decays.

III Event Selection

Charged particle tracks are reconstructed from hits in the MDC, and are required to have a polar angle $\theta$ with respect to the beam direction satisfying $|\cos\theta|<$ 0.93 and a distance of closest approach to the interaction point (IP) of less than $10\,\mbox{cm}$ along the beam axis ($V_{z}$) and less than $1\,\mbox{cm}$ in the plane perpendicular to the beam axis, except for those used to reconstruct the $\Lambda\to p\pi^{-}$ decay. Particle identification (PID) for charged particle tracks combines the information from the flight time in the TOF and measurements of ionization energy loss ($\mbox{d}E/\mbox{d}x$) to form a likelihood $\mathcal{L}(h)$ ($h=\pi,K,p$) for each hadron ($h$) hypothesis. Tracks will be identified as protons when this hypothesis is determined to have the largest PID likelihood ($\mathcal{L}(p)>\mathcal{L}(K)$ and $\mathcal{L}(p)>\mathcal{L}(\pi)$), while charged pions are differentiated from kaons by the likelihood requirement $\mathcal{L}(\pi)>\mathcal{L}(K)$.

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 ($|\cos\theta|<$ 0.80) or 50 MeV in the end-cap region (0.86 $<|\cos\theta|<$ 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 $\eta$ meson candidates are reconstructed from photon pairs using an invariant mass requirement of 505<M(\gamma\gamma)<575\;\mbox{MeV/c^{2}}. The invariant mass spectrum of $\gamma\gamma$ pairs in data is shown in Fig. 1. To improve the momentum resolution, a kinematic fit constraining the invariant mass to the $\eta$ nominal mass 2016PDG is applied to the photon pairs and the resultant energy and fitted momentum of the $\eta$ meson are used for further analysis.

Candidate $\Lambda$ baryons are reconstructed by combining two oppositely charged tracks for any pairs of $p\pi^{-}$. Those tracks are required to satisfy the polar angle requirement $|\cos\theta|<0.93$ and $V_{z}<20\;\mbox{cm}$ 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 $\pi^{-}$ tracks are constrained to originate from a common decay vertex by requiring the $\chi^{2}$ of a vertex fit to be less than 100. Furthermore, the reconstructed momentum of the $\Lambda$ 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 $\Lambda$ peak appears in the invariant mass spectrum of $p\pi^{-}$ in data, as shown in Fig. 1. The $p\pi^{-}$ pairs satisfying the mass requirement 1.111<M(p\pi^{-})<1.121\;\mbox{GeV/c^{2}} are chosen as the final $\Lambda$ candidates. This requirement is chosen corresponding to $\pm\,3$ standard deviations of the reconstruction resolution around the $\Lambda$ nominal mass 2016PDG .

The $\Lambda_{c}^{+}$ baryon candidates are reconstructed using all combinations of the selected $\Lambda$, $\eta$ and $\pi^{+}$ candidates. To differentiate $\Lambda_{c}^{+}$ from background, two kinematic variables calculated in the center-of-mass system, the beam constrained mass $M_{\rm{BC}}\equiv\sqrt{E_{\rm{beam}}^{2}/c^{4}-|\overrightarrow{p}_{\Lambda_{c}^{+}}|^{2}/c^{2}}$, and the energy difference $\Delta E\equiv E_{\Lambda_{c}^{+}}-E_{\rm{beam}}$ are used, where $E_{\Lambda_{c}^{+}}$ and $\overrightarrow{p}_{\Lambda_{c}^{+}}$ are the energy and momentum of the reconstructed $\Lambda_{c}^{+}$ candidate respectively, and $E_{\rm{beam}}$ is the average value of the electron and positron beam energies. For a well reconstructed $\Lambda_{c}^{+}$ candidate, $M_{\rm{BC}}$ and $\Delta E$ are expected to be consistent with the $\Lambda_{c}^{+}$ nominal mass and zero, respectively. Candidates are rejected when they fail the requirement of $-0.03<\Delta E<0.03\;\mbox{GeV}$, which corresponds to $\pm\,3$ standard deviations of the signal $\Delta E$ distribution. The $\Delta E$ distribution in data is shown in Fig. 1. If more than one candidate satisfies the above requirements, we select the one with the minimal $|\Delta E|$.

IV Signal Yield and branching fraction

To extract the signal yield for the $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ decay, an unbinned extended maximum likelihood fit is performed to the $M_{\rm{BC}}$ distribution in data with fitting range 2.25<M_{\rm{BC}}<2.30\;\mbox{GeV/c^{2}}, as illustrated in Fig. 2. In the fit, the signal shape is derived from the kernel-estimated non-parametric shape keyspdf 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 1990argus with fixed end-point $E_{\rm{beam}}$. The obtained signal yield and the corresponding detection efficiency are listed in Table 1. The validity of the ARGUS function to describe the background shape in the $M_{\rm{BC}}$ spectrum is checked using the inclusive MC samples. No obvious peaking background from the decay $\Lambda_{c}^{+}\to pK_{S}^{0}\eta$ with $K_{S}^{0}\to\pi^{+}\pi^{-}$ is observed and the influence of cross feed is neglected. The BF is calculated using

[TABLE]

where $N_{\rm{sig}}$ is the signal yield obtained from the $M_{\rm{BC}}$ fit, $N_{\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}}=(105.9\pm 4.8(\rm{stat.})\pm 0.5(\rm{syst.}))\times 10^{3}$ is the number of $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pairs in the data sample 2015MAblikimLambdac , $\varepsilon$ is the detection efficiency estimated using the signal MC simulation sample, and $\mathcal{B}_{\rm{inter}}=\mathcal{B}(\Lambda\to p\pi^{-})\cdot\mathcal{B}(\eta\to\gamma\gamma)$ is taken from the PDG 2016PDG . The factor of 2 in the denominator takes into account the charge conjugate decay mode of the $\Lambda_{c}^{+}$ baryon. The resultant BF and corresponding statistical uncertainty are listed in Table 1.

To check the possible intermediate states fore-mentioned in the theoretical calculations 2016JJXie ; 2015Kenta ; 2017JJXie , the two-dimensional Dalitz distributions of $M^{2}(\Lambda\eta)$ versus $M^{2}(\Lambda\pi^{+})$ for selected $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ candidates in the $M_{\rm{BC}}$ signal region 2.282<M_{\rm{BC}}<2.291\;\mbox{GeV/c^{2}} and the sideband region 2.250<M_{\rm{BC}}<2.270\;\mbox{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 $M(\Lambda\pi^{+})$ spectrum, an obvious peak of the $\Sigma^{*+}$ resonance is seen, which has been studied at CLEO 1995RAmmar , while other potential states are not evident in these projections. Hence, under the current statistics, we only measure the decay rate of $\Lambda_{c}^{+}\to\Sigma^{*+}\eta$.

To extract the signal yield of the cascade decay $\Lambda_{c}^{+}\to\Sigma^{*+}\eta$, $\Sigma^{*+}\to\Lambda\pi^{+}$, an unbinned extended maximum likelihood fit is performed to the invariant mass spectrum of $M(\Lambda\pi^{+})$ for the events within the $M_{\rm{BC}}$ signal region. The fitting range is 1.25<M(\Lambda\pi^{+})<1.56\;\mbox{GeV/c^{2}} as illustrated in Fig. 4. In the fit, the signal shape is derived from the kernel-estimated non-parametric shape keyspdf 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 $\Sigma^{*+}$ is generated according the following formula

[TABLE]

using the mass-dependent width $\Gamma(m)$ with the expression

[TABLE]

where $m=M(\Lambda\pi^{+})$, $m_{0}$ and $\Gamma_{0}$ are the $\Sigma^{*+}$ nominal mass and width, respectively, $q$ and $p$ ($p_{0}$) are the daughter momenta of $\Lambda_{c}^{+}$ and $\Sigma^{*+}$ (when $\Sigma^{*+}$ is at its nominal mass $m_{0}$) at their rest frame, respectively, and $L_{b}=1$($L_{d}=1$) is angular momentum between the two-body decay products in the $\Lambda_{c}^{+}$($\Sigma^{*+}$) rest frame. $f(p)$ are Blatt-Weisskopf barrier factors which have been detailed in Ref. BW . Possible interference between $\Sigma^{*+}$ and non-$\Sigma^{*+}$ amplitudes is neglected. The random combinatorial background is also modeled with kernel-estimated non-parametric shape keyspdf based on data in the $M_{\rm{BC}}$ sideband region. The non-$\Sigma^{*+}$ background is described with a smooth background function $f_{\rm bkg}\left(M\left(\Lambda\pi^{+}\right)\right)\propto(M\left(\Lambda\pi^{+}\right)-1.25)^{c}\cdot(1.75-M\left(\Lambda\pi^{+}\right))^{d}$, where the parameters $c$ and $d$ are obtained from MC-simulated non-$\Sigma^{*+}$ backgrounds and fixed in the fit. Only the integral of the signal shape in the signal region 1.32<M(\Lambda\pi^{+})<1.45\;\mbox{GeV/c^{2}} is counted as signal yield. The signal yield and the corresponding detection efficiency are listed in Table 1. The corresponding BF is calculated using Eq. (1), where $\varepsilon$ is the corresponding detection efficiency and $\mathcal{B}_{\rm{inter}}=\mathcal{B}(\Sigma^{*+}\to\Lambda\pi^{+})\cdot\mathcal{B}(\Lambda\to p\pi^{-})\cdot\mathcal{B}(\eta\to\gamma\gamma)$ taken from the PDG 2016PDG . The resultant BF and the corresponding statistical uncertainty are also listed in Table 1.

V Systematic uncertainty

Different sources of systematic uncertainties are considered in the BF measurement, including charged particle tracking, PID, reconstruction of intermediate states, the $\Delta E$ requirement, the fitting range, the background description, the signal MC model, peaking backgrounds and intermediate BFs.

Tracking and PID for $\pi^{+}$ particle. By studying a set of control samples of $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ events based on data collected at energies above $\sqrt{s}=4.0\;\mbox{GeV}$, which are the same as used in Ref. 2015MAblikimLambdac , the tracking and PID efficiencies are estimated in data and MC simulations. After weighting these efficiencies to the $\pi^{+}$ kinematics in the signal samples, the uncertainties associated with $\pi^{+}$ tracking and PID efficiencies are derived out to be 1.0% for each decay mode.

Reconstruction for $\Lambda$ particle. The efficiencies for $\Lambda$ reconstruction in data and MC simulations are measured with control samples of $J/\psi\to\bar{p}K^{+}\Lambda$ and $J/\psi\to\Lambda\bar{\Lambda}$ events, which are the same as studied by Ref. RecLmd . The uncertainties of $\Lambda$ reconstruction efficencies are estimated to be 3.7% for each decay mode, according to the $\Lambda$ momentum and angular distributions in the signal samples.

Reconstruction for $\eta$ particle. We use a control sample of $\pi^{0}$ from $D$ meson decays RecEta to evaluate the $\eta$ 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 $\gamma\gamma$ reconstruction efficiencies are obtained in data and MC simulations, and an uncertainty of 3.4% is assigned by weighting these efficiencies to the $\eta$ momentum distribution in the signal samples.

Requirement for $\Delta E$. To estimate the systematic uncertainty arising from $\Delta E$ requirement, we repeat the measurement procedure by varying the boundaries of the $\Delta E$ signal ranges with $\pm\,1\;\mbox{MeV}$. The largest changes in the resultant BFs, 2.3% and 1.5% for the decays $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ and $\Lambda_{c}^{+}\to\Sigma^{*+}\eta$, 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 $M_{\rm{BC}}$ fitting ranges of 2.26<M_{\rm{BC}}<2.30\;\mbox{GeV/c^{2}} for the decay $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ and of 1.25<M(\Lambda\pi^{+})<1.55\;\mbox{GeV/c^{2}} for the decay $\Lambda_{c}^{+}\to\Sigma^{*+}\eta$. The changes in resultant BFs, 0.9% and 2.7% for the decays $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ and $\Lambda_{c}^{+}\to\Sigma^{*+}\eta$, respectively, are considered as the systematic uncertainties.

Background description. For the $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ decay, we repeat the measurement by varying the $M_{\rm{BC}}$ end-point (2.3\;\mbox{GeV/c^{2}}) in the ARGUS function by \pm\,0.5\;\mbox{MeV/c^{2}}, by adding a Gaussian function to model the affection rising from the possible peak around 2.26\;\mbox{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 $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ decay. For $\Lambda_{c}^{+}\to\Sigma^{*+}\eta$ decay, we let the parameters of non-$\Sigma^{*+}$ 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 $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ decay, we consider the difference of angular and momentum distributions of final states $\Lambda$, $\eta$ and $\pi^{+}$ particles between data and signal MC samples and calculate weight factors using $w^{i}=\frac{n^{i}_{\mathrm{Data}}}{n^{i}_{\mathrm{MC}}}$, 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 $\Lambda_{c}^{+}\to\Sigma^{*+}\eta$ decay, we calculate the polar angle $\theta_{\Sigma^{*+}}$ of the momentum of the $\Sigma^{*+}$ with respect to that of the $\Lambda_{c}^{+}$ in the rest frame of the $\Lambda_{c}^{+}$. We model the signal process according to the distribution of $1+\alpha\cdot\cos^{2}\theta_{\Sigma^{*+}}$ in the range of $-1\leq\alpha\leq 1$. The maximum change on the MC-determined efficiency is 1.3%. Furthermore, we vary the nominal mass and width of the $\Sigma^{*+}$ within uncertainties in PDG 2016PDG , 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 $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ decay rate, which is estimated to be 1.9%. For the studies of the $\Lambda_{c}^{+}\to\Sigma^{*+}\eta$ decay rate, we incorporate complex components from non-$\Sigma^{*+}$ intermediate processes in the MC simulations of the $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ decays, and analyze the amplitude of the peaking background contribution beneath the $\Sigma^{*+}$ peak. The relative peaking background rate is evaluated to be 1.6%.

Total $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ number and intermediate BFs. In Ref. 2015MAblikimLambdac , absolute BFs of the twelve $\Lambda_{c}^{+}$ decay modes were measured and the total number of $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pairs was calculated using the absolute BFs and corresponding single-tag yields. The total number is $N_{\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}}=(105.9\pm 4.8(\rm{stat.})\pm 0.5(\rm{syst.}))\times 10^{3}$ and corresponding uncertainty is calculated to be 4.6% for each decay mode by adding both the statistical and systematic uncertainties in quadrature. The uncertainties of the intermediate BFs quoted from the PDG 2016PDG are $\mathcal{B}(\Sigma^{*+}\to\Lambda\pi^{+})=(87.0\pm 1.5)\%$, $\mathcal{B}(\Lambda\to p\pi^{-})=(63.9\pm 0.5)\%$ and $\mathcal{B}(\eta\to\gamma\gamma)=(39.41\pm 0.20)\%$, and corresponding uncertainties are calculated to be 0.9% and 1.9% for $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ and $\Lambda_{c}^{+}\to\Sigma^{*+}\eta$, respectively.

All these systematic uncertainties are summarized in Table 2, and the total systematic uncertainties are evaluated to be 8.4% and 9.5% for the $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ and $\Lambda_{c}^{+}\to\Sigma^{*+}\eta$ decays, respectively, by summing up all contributions in quadrature.

VI Summary

In summary, the absolute branching fractions of the two processes $\Lambda_{c}^{+}\to\Lambda\eta\pi^{+}$ and $\Sigma^{*+}\eta$ are measured using a single-tag method on a data sample produced in $e^{+}e^{-}$ collisions at $\sqrt{s}=4.6\;\mbox{GeV}$ collected with the BESIII detector. The results are $\mathcal{B}(\Lambda_{c}^{+}\to\Lambda\eta\pi^{+})=(1.84\pm 0.21\pm 0.15)\%$ and $\mathcal{B}(\Lambda_{c}^{+}\to\Sigma^{*+}\eta)=(0.91\pm 0.18\pm 0.09)\%$, 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 1995RAmmar ; 2003DCronin , but with improved precisions. Under the current statistics, no other potential intermediate states are concluded. Future $\Lambda_{c}^{+}$ data samples with larger statistics will allow for detailed studies of the intermediate states proposed in Refs. 2016JJXie ; 2015Kenta ; 2017JJXie .

Acknowledgements.

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. This work is supported in part by National Key Basic Research Program of China under Contract No. 2015CB856700; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11335008, 11425524, 11625523, 11635010, 11675275, 11735014; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos. U1532257, U1532258, U1732263; CAS Key Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; the Recruitment Program of Global Experts in China; 100 Talents Program of CAS; INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; German Research Foundation DFG under Contracts Nos. Collaborative Research Center CRC 1044, FOR 2359; Istituto Nazionale di Fisica Nucleare, Italy; Koninklijke Nederlandse Akademie van Wetenschappen (KNAW) under Contract No. 530-4CDP03; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Science and Technology fund; The Swedish Research Council; U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374, DE-SC-0010118, DE-SC-0010504, DE-SC-0012069; University of Groningen (RuG) and the Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt; the undergraduate research program of Sun Yat-sen University.