Measurement of Branching Fractions of $\Lambda_{c}^{+} \rightarrow \eta\Lambda\pi^{+}$, $\eta \Sigma^{0} \pi^{+}$, $\Lambda(1670) \pi^{+}$, and $\eta \Sigma(1385)^{+}$

We report branching fraction measurements of four decay modes of the $\Lambda_{c}^{+}$ baryon, each of which includes an $\eta$ meson and a $\Lambda$ baryon in the final state, and all of which are measured relative to the $\Lambda_{c}^{+} \rightarrow p K^{-} pi^{+}$ decay mode. The results are based on a $980~\mathrm{fb^{-1}}$ data sample collected by the Belle detector at the KEKB asymmetric-energy $e^{+}e^{-}$ collider. Two decays, $\eta \Sigma^{0} \pi^{+}$ and $\Lambda(1670) \pi^{+}$, are observed for the first time, while the measurements of the other decay modes, $\Lambda_{c}^{+} \rightarrow \eta\Lambda\pi^{+}$ and $\eta\Sigma(1385)^{+}$, are more precise than those made previously. We obtain $\mathcal{B}(\Lambda_{c}^{+} \rightarrow \eta \Lambda \pi^{+})/\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+})$ = $0.293 \pm 0.003 \pm 0.014$, $\mathcal{B}(\Lambda_{c}^{+} \rightarrow \eta \Sigma^{0} \pi^{+})/\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+})$ = $0.120 \pm 0.006 \pm 0.006$, $\mathcal{B}(\Lambda_{c}^{+} \rightarrow \Lambda(1670) \pi^{+}) \times \mathcal{B}(\Lambda(1670) \rightarrow \eta \Lambda)/\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+})$ = $(5.54 \pm 0.29 \pm 0.73 ) \times 10^{-2}$, and $\mathcal{B}(\Lambda_{c}^{+} \rightarrow \eta \Sigma(1385)^{+})/\mathcal{B}(\Lambda_{c}^{+} \rightarrow p K^{-} \pi^{+})$ = $0.192 \pm 0.006 \pm 0.016$. The mass and width of the $\Lambda(1670)$ are also precisely determined to be $1674.3 \pm 0.8 \pm 4.9~{\rm MeV}/c^{2}$ and $36.1 \pm 2.4 \pm 4.8~{\rm MeV}$, respectively, where the uncertainties are statistical and systematic, respectively.


I. INTRODUCTION
The branching fractions of weakly decaying charmed baryons provide a way to study both strong and weak interactions. Although there are theoretical models that estimate the branching fractions, for example constituent quark models and Heavy Quark Effective Theories (HQET) [1,2], the lack of experimental measurements of branching fractions of charmed baryons makes it difficult to test the models. Therefore, branching fraction measurements of new decay modes of the Λ + c or known decay modes with higher statistics are crucial. Modelindependent measurements of the branching fraction of Λ + c → pK − π + by Belle [3] and BESIII [4] now enable branching ratios measured relative to the Λ + c → pK − π + mode to be converted to absolute branching fraction measurements with high precision [5]. The Λ + c → ηΛπ + decay mode is especially interesting since it has been suggested [6] that it is an ideal decay mode to study the Λ(1670) and a 0 (980) because, for any combination of two particles in the final state, the isospin is fixed.
Two different models have been proposed to explain the structure of the Λ(1670). One is based on a quark model and assigns it to be the SU(3) octet partner of the N (1535) [7]. The other describes the Λ(1670) as a KΞ bound state using a meson-baryon model that has also been used to describe the Λ(1405) as aKN bound state [8]. There have been few experimental efforts to confirm the structure of the Λ(1670); and the interpretation of partial-wave analyses ofKN scattering data depends on theoretical models [9,10]. Here we investigate the production and decays of the Λ(1670) in the resonant substructure of the Λ + c → ηΛπ + decay, in order to elucidate * Corresponding author. sbyang@korea.ac.kr † now at Hiroshima University the nature of this particle.

II. DATA SAMPLE AND MONTE CARLO SIMULATION
This measurement is based on data recorded at or near the Υ(1S), Υ(2S), Υ(3S), Υ(4S), and Υ(5S) resonances by the Belle detector at the KEKB asymmetric-energy e + e − collider [13]. The total data sample corresponds to an integrated luminosity of 980 fb −1 . The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprising CsI(Tl) crystals (ECL) located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented to detect K 0 L mesons and to identify muons. The detector is described in detail elsewhere [14].
Two inner detector configurations were used. A 2.0-cm radius beampipe and a three-layer silicon vertex detector were used for the first sample of 156 fb −1 , while a 1.5-cm radius beampipe, a four-layer silicon detector and a small-cell inner drift chamber were used to record the remaining 824 fb −1 .
Monte Carlo (MC) simulation events are generated with PYTHIA [15] and EvtGen [16] and propagated by GEANT3 [17]. The effect of final-state radiation is taken into account in the simulation using the PHOTOS [18] package. A generic MC simulation sample, having the same integrated luminosity as real data, is used to optimize selection criteria for Λ + c → ηΛπ + signal events. We also generate several signal MC simulation samples of specific Λ + c decays in order to study particle reconstruction efficiencies and the detector performance; the signal MC events follow a uniform distribution in phase space.

III. EVENT SELECTION
We reconstruct Λ + c candidates via Λ + c → ηΛπ + decays with the η and Λ in η → γγ and Λ → pπ − decays. Starting from selection criteria typically used in other charmed-hadron analyses at Belle [19,20], our final criteria are determined through a figure-of-merit (FoM) study based on the generic MC sample. We optimize the FoM, defined as n sig / √ n sig + n bkg , where n sig is the number of reconstructed Λ + c signal events while n bkg is the number of background events. The yields n sig and n bkg are counted in the M (ηΛπ + ) range from 2.2755 GeV/c 2 to 2.2959 GeV/c 2 .
The η meson candidates are reconstructed from photon pairs in which M (γγ) is in the range 0.50-0.58 GeV/c 2 corresponding to an efficiency of about 79%. A massconstrained fit is performed to improve the momentum resolution of η candidates, and the fitted momentum and energy are used for the subsequent steps of analysis. In addition, we require η candidates to have momenta greater than 0.4 GeV/c and an energy asymmetry, defined as |(E(γ 1 ) − E(γ 2 ))/(E(γ 1 ) + E(γ 2 ))|, less than 0.8. For the selection of photons, the energy deposited in the ECL is required to be greater than 50 MeV for the barrel region and greater than 100 MeV for the endcap region [14]. In order to reject neutral hadrons, the ratio between energy deposited in the 3 × 3 array of crystals centered on the crystal with the highest energy, to that deposited in the corresponding 5 × 5 array of crystals, is required to be greater than 0.85. To reduce the background in the η signal region due to photons from π 0 decays, the photons used to reconstruct the η candidates are not allowed to be a part of a reconstructed π 0 with mass between 0.12 GeV/c 2 and 0.15 GeV/c 2 .
Charged π + candidates are selected using requirements on a distance-of-closest-approach (DOCA) to the interaction point (IP) of less than 2.0 cm in the beam direction (z) and less than 0.2 cm in the transverse (r) direction. Measurements from CDC, TOF, and ACC are combined to form particle identification (PID) likelihoods L(h) (h = p ± , K ± , or π ± ), and the L(h : h ′ ), defined , is the ratio of likelihoods for h and h ′ . For the selection of π + , L(π : K) > 0.2 and L(π : p) > 0.4 are required. Furthermore, the electron likelihood ratio R(e), derived from ACC, CDC, and ECL measurements [21], is required to be less than 0.7.
We reconstruct Λ candidates via Λ → pπ − decays in the mass range, 1.108 GeV/c 2 < M (pπ − ) < 1.124 GeV/c 2 , and selected using Λ-momentumdependent criteria based on four parameters: the distance between two daughter tracks along the z direction at their closest approach; the minimum distance between daughter tracks and the IP in the transverse plane; the angular difference between the Λ flight direction and the direction pointing from the IP to the Λ decay vertex in the transverse plane; and the flight length of Λ in the transverse plane. We require L(p : π) > 0.6 for the proton from the Λ decay.
Finally, η, Λ, and π + candidates are combined to form a Λ + c with its daughter tracks fitted to a common vertex. The χ 2 value from the vertex fit is required to be less than 40, with an efficiency of about 87%. To reduce combinatorial background, especially from B meson decays, the scaled momentum x p = p * /p max is required to be greater than 0.51; here, p * is the momentum of Λ + c in the center-of-mass frame and p max is the maximum possible momentum.
Since the branching fractions are determined relative to B(Λ + c → pK − π + ), Λ + c candidates from Λ + c → pK − π + decays are also reconstructed using the same selection criteria in Ref. [19] except for the scaled momentum requirement of the Λ + c , which is chosen to be the same as that used for the Λ + c → ηΛπ + channel. All charged tracks in the Λ + c → pK − π + decay are required to have their DOCA less than 2.0 cm and 0.1 cm in the z and r directions, respectively, and at least one SVD hit in both the z and r directions. The PID requirements are L(p : K) > 0.9 and L(p : π) > 0.9 for p, L(K : p) > 0.4 and L(K : π) > 0.9 for K, and L(π : p) > 0.4 and L(π : K) > 0.4 for π. In addition, R(e) < 0.9 is required for all tracks. The charged tracks from the Λ + c decay are fitted to a common vertex and the χ 2 value from the vertex fit must be less than 40.
The branching fractions of the Λ + c → ηΛπ + and ηΣ 0 π + decays are calculated relative to that of the Λ + c → pK − π + decay using the efficiency-corrected event yields via the following equation,  where Decay Mode is either Λ + c → ηΛπ + or Λ + c → ηΣ 0 π + , and y(Decay Mode) refers to the efficiencycorrected yield of the corresponding decay mode. Here B PDG denotes subdecay branching fractions of the η, Λ, and Σ 0 ; we use B(η → γγ) = (39.41 ± 0.20)%, B(Λ → pπ − ) = (63.9 ± 0.5)%, and B(Σ 0 → Λγ) = 100% from Ref. [22]. Figure 1 shows the M (ηΛπ + ) spectrum after the event selection described in the previous section. In the spectrum, we find a peaking structure from the Λ + c → ηΛπ + channel at 2.286 GeV/c 2 . The enhancement to the left of the peak corresponds to the Λ + c → ηΣ 0 π + channel with a missing photon from the Σ 0 → Λγ decay. First, we perform a binned-χ 2 fit to the M (ηΛπ + ) distribution to extract the Λ + c → ηΣ 0 π + signal yield. The probability density functions (PDFs) of the signals are modeled empirically based on MC samples as the sum of a Gaussian and two bifurcated Gaussian functions with a common mean for Λ + c → ηΛπ + , and a histogram PDF for the feed-down of the Λ + c → ηΣ 0 π + decay. The latter PDF is derived from Λ + c → ηΣ 0 π + ; Σ 0 → Λγ decays where the photon decaying from the Σ 0 is not reconstructed. The PDF of the combinatorial backgrounds used for the fit is a third-order polynomial function. The signal yield for the feed-down from the Λ + c → ηΣ 0 π + channel shown in Fig. 1 is 17058 ± 871. This yield is then corrected for the reconstruction efficiency obtained from MC to give an efficiency-corrected yield of (3.05 ± 0.16) × 10 5 , where the uncertainty is statistical only.
On the other hand, the Λ + c → ηΛπ + and pK − π + channels have sufficiently large statistics to perform the yield extractions in individual bins of the Dalitz plot, in order to take into account the bin-to-bin variations of the efficiencies. Figure 2 shows the binning and the efficien-  for the Λ + c → ηΛπ + channel (top) and of M 2 (K − π + ) vs M 2 (pK − ) for the Λ + c → pK − π + channel (bottom). The red lines indicate the Dalitz plot boundaries. The fits in the three sample bins of (a), (b) and (c) are shown in Fig. 3. for the Λ + c → ηΛπ + channel and in Fig. 4 for the Λ + c → pK − π + channel. cies over the Dalitz plots for Λ + c → ηΛπ + and pK − π + , respectively. For the fit to each bin of the Λ + c → ηΛπ + Dalitz plot, we use PDFs of the same form described above. In the pK − π + channel, two Gaussian functions sharing a common mean value and a third-order polynomial function are used to represent the pK − π + signals and combinatorial backgrounds, respectively. For the signal PDFs in both Λ + c → ηΛπ + and pK − π + fits, all parameters except for normalizations are fixed for each bin. The fixed parameters are first obtained for each bin according to an MC simulation and later corrected by taking into account the difference of the fit results between data and MC samples over the entire region of the Dalitz plot. For the fit to Λ + c → ηΛπ + , all the parameters for the PDF attributed to the feed-down from the Λ + c → ηΣ 0 π + decay with one photon missing are fixed,   Fig. 2) of the Λ + c → ηΛπ + channel. The curves indicate the fit results: the total PDF (solid red), signal from the Λ + c → ηΣ 0 π + channel with a missing photon from the Σ 0 decay (dotted dark green), signal from the Λ + c → ηΛπ + decay (dashed blue) and combinatorial backgrounds (long-dashed green).
including the normalization based on the measured yield in this analysis. The polynomial functions for the combinatorial backgrounds are floated for both Λ + c → ηΛπ + and pK − π + decays. Figures 3 and 4 show examples of fits for three Dalitz plot bins. For the Λ + c → ηΛπ + and pK − π + channels, the extracted yields are efficiencycorrected in each bin and summed up over the Dalitz plots. The results for the total efficiency-corrected signal yields are summarized in Table I.
Finally, we calculate the branching fractions using the efficiency-corrected signal yields and Eq. (1). The branching fractions are summarized in Table II Fig. 2) of the Λ + c → pK − π + channel. The curves indicate the fit results: the total PDF (solid red), signal from the Λ + c → pK − π + decays (dashed blue) and combinatorial backgrounds (longdashed green).
In order to extract yields for the Λ + c → Λ(1670)π + and Λ + c → ηΣ(1385) + contributions to inclusive Λ + c → ηΛπ + decays, we fit the M (ηΛπ + ) mass distributions, and extract Λ + c signal yields, for every 2 MeV/c 2 bin of the M (ηΛ) and M (Λπ + ) distributions. The same form of PDF described in Sec. IV is used to fit the M (ηΛπ + ) mass spectrum, and the PDF parameters for each mass bin are obtained in the same way for the fit of each Dalitz plot bin in Sec. IV. The Λ + c yields as a function of M (ηΛ) and M (Λπ + ) are shown in Fig. 6. The Λ(1670) and Σ(1385) + resonances are clearly seen in Fig. 6(top) and (bottom), respectively. This is the first observation of the Λ(1670) in Λ + c → ηΛπ + decays. To extract the signal yields for the two resonant decay modes, binned least-χ 2 fits are performed to the M (ηΛ) and M (Λπ + ) spectra shown in Fig. 6. For the signal modeling, we use an S-wave relativistic partial width Breit-Wigner (BW) for the Λ(1670) and a corresponding P -wave BW for the Σ(1385) + : where m, m 0 and L are the invariant mass, the nominal mass and the decay angular momentum, respectively, and q and q 0 are the center-of-mass momenta corresponding to m and m 0 , respectively. Here Γ(m) is the partial width for Λ(1670) → ηΛ or Σ(1385) + → Λπ + and Γ 0 = Γ(m 0 ) is a floating parameter in the fit. The contribution Γ others , which indicates the sum of the partial widths for the other decay modes, is fixed to 25 MeV for Λ(1670) and 5 MeV for Σ(1385) + [22]. Unlike the Σ(1385) + , the branching fractions for Λ(1670) decays are not well determined [22], we select 25 MeV as the nominal value for Γ others . A systematic uncertainty from the fixed value of Γ others is calculated by changing this  value over a wide range, 15 to 32 MeV. In Eq. (3), the Blatt-Weisskopf centrifugal barrier factor F (q) is 1 for S wave and (1 + R 2 q 2 0 )/(1 + R 2 q 2 ) for P wave, with R = 3.1 GeV −1 [23]. The detector resolution for Λ(1670) is not included in the signal PDF because the detector response function is not a simple Gaussian near threshold. The effect is small and is treated as a systematic uncertainty in the measurement. On the other hand, for Σ(1385) + the relativistic Breit-Wigner function is convolved with a Gaussian with σ = 1.39 MeV/c 2 to form the signal PDF. This σ value is determined from a MC simulation of detector responses. To represent the background to the Λ(1670) signal, we use a function with a threshold: where p 0 and p 1 are free parameters and m Λη is the sum of the masses of Λ and η. In the case of the Σ(1385) + fit, a third-order Chebyshev polynomial function is used to represent background. The χ 2 /ndf of the Λ(1670) and Σ(1385) + fits are 90.3/90 and 194/167, respectively. We calculate the corresponding reconstruction efficiencies of Λ + c → Λ(1670)π + and Λ + c → ηΣ(1385) + decays from a MC simulation. The extracted yields from the fits in Fig. 6 are divided by the reconstruction efficiencies and the results are summarized in Table I. The branching fractions relative to Λ + c → pK − π + decay are summarized in Table II. From the fit results, we also determine masses and widths (Γ tot = Γ 0 + Γ others ) of the Λ(1670) and Σ(1385) + as summarized in Table III. Changes in efficiency over the M (ηΛ) and M (Λπ + ) distributions are not considered be- cause their effect is negligible as described in Sec. VI. The results obtained for the Σ(1385) + are consistent with previous measurements [22]. For the Λ(1670), the mass and width have not been previously measured directly from a peaking structure in the mass distribution. The values that we obtain fall within the range of the partial wave analyses of theKN reaction [9,10].

VI. SYSTEMATIC UNCERTAINTY
The systematic uncertainties for the Λ + c → ηΛπ + , ηΣ 0 π + , and pK − π + efficiency-corrected yields are listed in Table IV. A study is performed based on a D * + → D 0 π + (D 0 → K − π + ) control sample for πK identification and on the Λ → pπ − decay for the proton identification to give corrections for the reconstruction efficiencies and to estimate the systematic uncertainties due to the PID selection. Conservatively, all PID systematic uncertainties are considered to be independent when calculating the relative branching fractions to the Λ + c → pK − π + channel. The systematic uncertainty due to Λ reconstruction is determined from a comparison of yield ratios of B → ΛΛK + with and without the Λ selection cut in data and MC samples. The weighted average of the difference between data and MC samples over the momentum range is assigned as the systematic uncertainty. A 3.0% systematic uncertainty attributed to η reconstruction is assigned by comparing the MC and data ratios of π 0 reconstruction efficiency for η → 3π 0 and η → π + π − π 0 decays [24]. The binning over the Dalitz plots is varied from 10 × 5 to 6 × 4 and the differences in the results are taken as a systematic uncertainty. Unlike the Λ + c → ηΛπ + and Λ + c → pK − π + channels that are analyzed in a modelindependent way, the efficiency of the Λ + c → ηΣ 0 π + decay mode depends on its substructure. To estimate the effect of possible substructures in the Λ + c → ηΣ 0 π + decay, efficiencies of Λ + c → ηΣ(1385) + → ηΣ 0 π + and Λ + c → Σ 0 a 0 (980) + → ηΣ 0 π + modes are compared to that of the nonresonant decay mode of Λ + c → ηΣ 0 π + which is used to correct the yield and the larger difference is taken as systematic uncertainty. The systematic uncertainty due to the background PDF modeling is estimated by changing the polynomial function from third order to fourth order.
In addition, the systematic uncertainties from the subdecay mode analysis that are not in common with the Λ + c → ηΛπ + decay channel are summarized in Table V and described below. In order to estimate the systematic uncertainty due to Γ others , its value in the Λ(1670) (Σ(1385) + ) fit is varied from 15 to 32 (2 to 8) MeV and the maximum difference is taken as the systematic uncertainty. The ranges of Γ others conservatively cover the branching fractions of Λ(1670) and Σ(1385) + decays in Ref. [22] and the q dependence of Γ others is negligible compared to this systematic uncertainty. In the M (ηΛ) spectrum, the mass resolution varies from 0 to 2 MeV/c 2 depending on mass; thus, two fits are performed by setting the mass resolution to 1 MeV/c 2 and 2 MeV/c 2 , and the maximum difference is assigned as a systematic uncertainty. For the M (Λπ + ) spectrum, we increase the detector resolution by 20% and the resultant change is taken as a systematic uncertainty. The systematic uncertainties from the background PDF modeling are estimated by fits with fixed shapes of background PDFs, which are determined by MC simulations including known background sources such as Λ + c → a 0 (980) + Λ, nonresonant, and Λ + c → ηΣ(1385) + (Λ + c → Λ(1670)π + ) decays in the M (ηΛ) (M (Λπ + )) spectrum. In order to consider systematic uncertainties related to angular distributions of Λ(1670) and Σ(1385) + , the efficiencies in 10 bins of helicity angle are calculated and the largest efficiency differences between any efficiency in the helicity angle bin and the efficiency used to correct the yields are taken as systematic uncertainties. It is possible that the results for the Λ(1670) and Σ(1385) + can be affected by another resonant channel, Λ + c → a 0 (980) + Λ. To estimate the interference effect with a 0 (980) + , we apply an additional a 0 (980) + veto selection, removing events from 0.95 to 1.02 GeV/c 2 of M (ηπ + ), to the M (ηΛ) and M (Λπ + ) distributions and subsequently repeat the fits. By comparing the fit results with and without the a 0 (980) + requirement, we determine the systematic uncertainties in the masses and widths. For the efficiency-corrected yields, the expected yields calculated on the assumption that there is no interference effect are compared to the nominal values. Since the centrifugal barrier factor [23] is a model-dependent parameter, it has a sizeable uncertainty. Varying the parameter R by ±0.3 GeV −1 , fits are performed to estimate the systematic uncertainty. We also estimate a systematic uncertainty from binning of M (ηΛ) and M (Λπ + ) distributions by changing the bin widths to 1 MeV/c 2 .
The systematic uncertainties for the mass and width measurements are listed in Table VI. In the same way as described above, the systematic uncertainties from the PDFs and the binning of the Λ(1670) and Σ(1385) + fits are estimated. The absolute mass scaling is determined by comparing the measured mass of Λ + c with that in Ref. [22], and it is considered as a systematic uncertainty. To estimate the systematic uncertainty due to the M (ηΛ)-and M (Λπ + )-dependent reconstruction efficiencies, we apply reconstruction efficiency corrections to the M (ηΛ) and M (Λπ + ) spectra. For the corrections, we calculate the mass dependencies of these efficiencies by MC simulation. They are found to vary between 0.068 and 0.070 for M (ηΛ) and between 0.069 and 0.071 for M (Λπ + ), and in both cases the behavior is nearly flat. The mass spectra are divided by these efficiencies. Differences in fit results with and without the efficiency corrections are negligible compared to these other systematic sources as listed in Table VI.