Measurement of the Decays $\Lambda_c\to \Sigma\pi\pi$ at Belle

We report measurements of the branching fractions of the decays $\Lambda^+_c\to\Sigma^+\pi^-\pi^+$, $\Lambda^+_c\to\Sigma^0\pi^+\pi^0$ and $\Lambda^+_c\to\Sigma^+\pi^0\pi^0$ relative to the reference channel $\Lambda^+_c\to pK^-\pi^+$. The analysis is based on the full data sample collected at and close to $\Upsilon(4S)$ resonance by the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider corresponding to the integrated luminosity of 711 fb$^{-1}$. We measure ${\cal B}(\Lambda^+_c\rightarrow\Sigma^+\pi^-\pi^+)/{\cal B}(\Lambda^+_c\to pK^-\pi^+) = 0.706~\pm 0.003~\pm 0.029$, ${\cal B}(\Lambda^+_c\rightarrow\Sigma^0\pi^+\pi^0)/{\cal B}(\Lambda^+_c\to pK^-\pi^+) = 0.491~\pm 0.005~\pm 0.028$ and ${\cal B}(\Lambda^+_c\rightarrow\Sigma^+\pi^0\pi^0)/{\cal B}(\Lambda^+_c\to pK^-\pi^+) = 0.198~\pm 0.006~\pm 0.016$. The listed uncertainties are statistical and systematic, respectively.


I. INTRODUCTION
Charmed baryon decays provide crucial information for the study of both strong and weak interactions. The Λ c , which is the lightest charmed baryon and has a udc quark configuration, plays a key role. As most Λ 0 b decays include a Λ + c [1,2] in their decay products, improved measurements of Λ + c hadronic branching fractions help constrain fragmentation functions of bottom, as well as charm, quarks through the measurement of inclusive heavy-flavor baryon production [3] [4]. The recent model-independent measurements of the normalization mode Λ c → pKπ by Belle [5] and BESIII [6] improve the accuracy of Λ + c branching fractions measured relative to this mode and similarly advance other related measurements [7]. The decay Λ + c → Σππ is particularly interesting as it has been proposed as a possible avenue to extract the Σ-π scattering length [8], and this measurement would provide crucial information in the study of the Λ(1405) resonance [9].
This analysis is based on the full Belle data sample taken at the Υ(4S) resonance. In principle, it would be desirable to also measure Λ + c → Σ − π + π + . However Σ − decays almost completely into nπ − , a mode that cannot be reconstructed at Belle. Belle's inability to measure neutrons also limits us to the decay modes Σ + → pπ 0 and Λ → pπ − when reconstructing hyperons. While the Λ + c → Σ + π − π + and Λ + c → Σ 0 π + π 0 modes have been studied previously by BESIII [6] and by CLEO [11], respectively, we present here the first measurement of the Λ + c → Σ + π 0 π 0 channel.

A. Data sample
This analysis is based on the 711 fb −1 data sample collected with the Belle detector at the KEKB asymmetric-energy e + e − collider [12] operating at an energy at or near the Υ(4S) resonance. Belle is a largesolid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of timeof-flight scintillation counters (TOF), and an electromagnetic calorimeter comprised of 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 (KLM). Two inner detector configurations were used. A 2.0 cm radius beampipe and a 3-layer silicon vertex detector were used for the first sample of 140 fb −1 , while a 1.5 cm radius beampipe, a 4-layer silicon detector and a small-cell inner drift chamber were used to record the remaining 571 fb −1 [13]. The detector is described in detail elsewhere [14].
In addition, we use Monte Carlo (MC) simulated events, which are created with the JETSET [15] and EVTGEN [16] generators. A full detector simulation based on GEANT3 [17] is applied to MC events to model the response of the detector and its acceptance. Final-state radiation is taken into account using the PHOTOS [18] package. MC-simulated data samples are equivalent to at least six times the data luminosity.

B. Event selection
Charged particles are reconstructed in the tracking system consisting of the SVD and CDC detectors. Particle identification is based on the specific ionization in the CDC, the Cherenkov light yield in the ACC, and the time-of-flight information in the TOF. For each track, the normalized likelihood ratio for distinct hypotheses i ∈ {p, π, K} and j = i is defined as L(i : j) = L(i)/(L(i) + L(j)). For a track to be identified as a proton (pion), the corresponding likelihood ratios must exceed 0.6. For pK − π + alone, the more stringent requirement of L(p : K) > 0.9 and L(p : π) > 0.9 for proton candidates is adopted. These selection criteria are about 90 % efficient for detected kaons, 98 % for pions and 80 % (90 %) for protons coming directly from Λ c (from hyperons). For all charged particles except the protons and pions from the Σ + and Λ decays, we require the distance of closest approach |dz| (dr) to the interaction point (IP) along the beam axis (in the transverse plane) to be smaller than 4 cm (2 cm).
Photons are reconstructed from clusters in the ECL are not matched to a CDC track trajectory. We require a minimum cluster energy of 40 MeV. A neutral π 0 candidate is formed by combining two photons selected in a M(γγ) window of [120, 150] MeV/c 2 (about ±3σ around the nominal π 0 mass). The reconstructed π 0 momentum must ex- ceed 100 MeV/c in the laboratory frame.
A Λ candidate is reconstructed by combining a proton and a pion with an invariant-mass M(pπ) between 1.1130 and 1.1180 GeV/c 2 (about ±3σ around the nominal Λ mass). In Belle analyses, additional criteria may be applied, based on the distance along the beam axis of the two daughter tracks at their closest approach (z dist ), the minimum dr of each track, the angular difference in the transverse plane between the Λ flight direction and the vector between the IP and the decay vertex (dφ), and the flight length in the transverse plane of the Λ candidate (|ℓ f |). Two levels of Λ candidate purity are commonly used in Belle, based on the selection criteria for these four parameters [19][20] [21]. Level 1 (2) is determined by optimizing these Λ-selection criteria on MC samples after (without) selections on the charged particle likelihood ratios. The threshold values for each parameter are given in Table I for the two levels. However, at this point we make no selection based on the purity level.
A Σ 0 candidate is formed by combining a Λ candidate with a photon, with M(Λγ) required to lie between 1.18 and 1.206 GeV/c 2 (about ±3σ). Similarly, a Σ + candidate is formed from the combination of a proton with a π 0 , with M(pπ 0 ) lying between 1.159 and 1.219 GeV/c 2 (about ±2.5σ). The Σ + → pπ 0 reconstruction relies on the long hyperon lifetime: we require the proton's dr to exceed 0.3 mm. Then, the Σ + trajectory is approximated by a straight line from the IP in the direction of the reconstructed Σ + three-momentum and intersected with the proton path. This point is taken as an estimate of the Σ + decay vertex and used to refit the π 0 candidate, assuming that the γγ pair originates from this vertex rather than from the IP. Only Σ + candidates with a positive flight length from the IP to the decay vertex are retained.
Finally, the Σ baryon candidate is combined with two pions. To reduce combinatorial background, the scaled momentum x = p/p max is required to be larger than 0.5. Here, p is the magnitude of the Λ + c threemomentum and p max is its maximum value assuming only a pair of Λ + c baryons is produced in the event. As a consequence of this requirement, all Λ + c candidates from B decays are completely eliminated and only candidates originating directly from the e + e − → cc continuum are retained. Charged daughter particles are fitted to a common decay vertex; the χ 2 of this fit is required to be compatible with the daughters being produced by a common parent.

C. Boosted decision tree selector
To further increase the purity of the reconstructed signal, we combine several discriminant variables into a single Boosted Decision Tree (BDT) output, based on the Ad-aBoost [22] algorithm.
The input variables to the BDT are: the scaled momenta of the Λ + c candidate and the hyperon, all final-state charged-particle and π 0 candidate momenta in the center-of-mass (c.m.) frame, the cluster energy and direction of detected photons in the ECL, the cosine of the angle between the two photons from all π 0 particles in the laboratory frame, the χ 2 of the vertex fit (described above) in modes with several charged daughters, the distances of closest approach to the interaction point (dr, |dz|) of all charged trajectories, the Λ-candidate purity level (described earlier), and a purity flag for each π 0 candidate. This binary flag is assigned by (1) forming π 0 candidates from all possible twophoton combinations, starting from the most energetic photons, then (2) processing this ordered list to assign a value of one for the first combination with an invariant-mass in the range of ±15 MeV/c 2 of the nominal π 0 mass and zero for all other combinations using the same photons. This requirement ensures that only the most likely γγ combinations are used and avoids double counting.
The classifier is trained on MC event samples corresponding to the same integrated luminosity as the real data sample except in the case of the Σ + π 0 π 0 decay mode, where six times the real data luminosity is used. If there are multiple candidates in one event, the one with the highest-ranking BDT classifier is selected. The selection threshold applied to the BDT output is optimized by maximizing a figure of merit defined as S/ √ S + B, where S represents the number of signal events and B the number of background events that pass the selection criteria, as estimated from MC samples introduced earlier. For the Σ + π 0 π 0 channel, where no previous measurement is available, a branching fraction of 1.8 % is assumed from isospin considerations.

D. Signal yield extraction
The signal yields in the Λ + c → pK − π + , Σ 0 π + π 0 , Σ + π − π + , and Σ + π 0 π 0 modes are extracted using an unbinned extended maximum likelihood fit (EML) [23] to the Λ ccandidate invariant-mass distribution. The probability density functions (PDFs) of the signal and background models are typically defined between 2.2 and 2.4 GeV/c 2 ; for the Σ + π 0 π 0 mode, the lower bound is set to 2.14 GeV/c 2 to accommodate the longer signal tail at low invariant-masses. The signal in each channel is modeled by a combination of Gaussian, Breit-Wigner, and Crystal Ball [24] functions, sharing the same mean. Details are given in Table II. The model is chosen empirically on MC samples and the  Λ + c mode PDF Alternative PDF Σ + π + π − G ⊗ BW + G G + G + BW Σ 0 π + π 0 CB + BW CB + G pK − π + G + G + BW G + G + G Σ + π 0 π 0 CB + G CB + BW Λ 0 π + π 0 + γ Bifurcated G + G CB + G width and peak position are in good agreement with data for all Σππ decay channels. For pK − π + , we find the signal shape to be 12 % broader in data. In the Σ 0 π + π 0 decay mode, Λ c → Λπ + π 0 combined with one random photon causes a peak in the invariantmass distribution that overlaps partially with the signal region. This background is included in the fit model. In all modes with a π 0 in the final state, π 0 candidates containing an incorrect photon produce a broad peak centered at the nominal Λ + c mass. These self-cross-feed events, which amount to between 5% and 23% of true signal depending on the mode, are included in the signal component's PDF. For the combinatorial background, polynomials are used: cubic for pKπ and Σ 0 π + π 0 , quadratic for other Σππ combinations. The reconstruction efficiency depends on the presence of intermediate resonances. To extract the signal yields in a model-independent way, the Dalitz distribution of each decay is binned and independent fits are performed in each bin. The binning and the Dalitz-bin efficiencies for Λ + c → pK − π + , Σ 0 π + π 0 , Σ + π + π − , and Σ + π 0 π 0 are shown in Figs. 1, 3, 5 and 7, re- spectively. The PDF parameters in each bin are determined from simulation. In the fit to Σππ real data, only the normalizations of the signal and combinatorial background are floated, except in the Λ + c → Σ 0 π + π 0 channel, where the distinct contribution of the Λπ + π 0 + γ background is also determined bin by bin. For Λ + c → pK − π + , both the background polynomial and the width of the signal component are allowed to float. For Σππ, the width is measured on the full sample and fixed for yield extraction. At the next step, the extracted yields in each bin are efficiency-corrected and summed over the Dalitz plot to give the total yield Here, the index i runs over the Dalitz plot bins shown in Figs. 1, 3, 5 and 7, and y i and ǫ i are the extracted signal yield and the reconstruction efficiency, respectively, for bin i. The result for the total efficiency-corrected signal yield y is given for each mode in Table III.    Final state i y i /ǫ i [×10 3 ] Σ + π − π + 2687 ± 10 Σ 0 π + π 0 2661 ± 24 pK − π + 7249 ± 9 Σ + π 0 π 0 925 ± 22
The following uncertainties are taken into account and listed in Table V. Unless stated otherwise, we assume no correlation in the individual systematic error components and so add them in quadrature. The systematic uncertainty related to the pion and kaon identification efficiency is estimated from kinematically identified D * + → D 0 π + , D 0 → K − π + real-data events. These events are used both to derive a correction to the MC simulation and to determine the systematic uncertainties of pion and kaon identification. All channels except Σ + π 0 π 0 include a charged pion, IV: Branching-fraction values determined by this analysis. The second column gives the branching fractions of the decays Λ + c → Σ + π − π + , Λ + c → Σ 0 π + π 0 , and Λ + c → Σ + π 0 π 0 relative to the branching fraction of the decay Λ + c → pK − π + . The third column lists the absolute branching fractions taking B(Λ + c → pK − π + ) = 6.35 ± 0.33 [25]. Errors are statistical, systematic, and from B(pKπ), respectively. In the final column, the current world average is given.
[%] Σ + π − π + 0.719 ± 0.003 ± 0.024 4.57 ± 0.02 ± 0.15 ± 0. 24 4.57 ± 0.29 Σ 0 π + π 0 0.575 ± 0.005 ± 0.036 3.65 ± 0.03 ± 0.23 ± 0. 19 2.3 ± 0.9 Σ + π 0 π 0 0.247 ± 0.006 ± 0.019 1.57 ± 0.04 ± 0.12 ± 0.08 -   Fig. 3. The dotted curve is the signal component, the dashed curve the combinatorial background, and the dash-dotted curve the Λπ + π 0 + γ background. The pull distribution of the fit is shown at the bottom of each panel. directly produced in the Λ + c decay. The uncertainty caused by the PID selection of this particle cancels in the ratio. The uncertainty introduced by proton identification is determined from the ratio of yields of the decay Λ → pπ with and without the proton identification requirement. The difference in the ratio between MC and data is used to correct the efficiency; the statistical uncertainty is treated as a systematic error. The systematic uncertainty due to Λ reconstruction is estimated by considering the data-MC difference of tracks displaced from the IP, the Λ proper time, and Λ mass distributions. The weighted average over the momentum range is taken as the total uncertainty. A study of  τ − → π − π 0 ν τ decays described in [26] is used to correct for MC-data discrepancies in the π 0 reconstruction efficiency. We check model uncertainties by varying the PDF parameters fixed from MC within their statistical uncertainties and repeat the fits one thousand times for each bin. The change in the central value plus the width of the distribution, in terms of standard deviation, of fit results is taken as a systematic error in a given bin and the weighted sum is taken as the total sys- tematic error. Furthermore, we use alternate signal PDFs as described in Table II and alternate background PDFs whose polynomial order is increased by one. The residual Dalitz model dependence of our fitting method is checked by repeating the fit with a four times finer binning. The difference in the yields is taken as a systematic error. Limited statistics preclude us from using a finer binning in the case of Σ + π 0 π 0 . Here, we compare the efficiency-corrected signal yield with the fit on the unbinned sample and take the difference as a systematic error. The uncertainty due to tracking is 0.35 % per charged track. We only apply this uncertainty to pK − π + in the ratio with Σ + π 0 π 0 . In the other decay modes, the equal number of charged tracks in the measured and reference modes causes this uncertainty to cancel. For the reconstruction of the photon from the Σ 0 → Λγ decay, we apply half the uncertainty for low-momentum (below 200 MeV/c) π 0 reconstruction. The additional uncertainty compared to general π 0 reconstruction is obtained from a study of B 0 → D * − π + and B + → D * 0 π + decays to determine the data-MC ratio in bins of pion momentum from the D * decay. The overall systematic error is obtained by lin-   ear summation of this uncertainty and the results of the τ − → π − π 0 ν τ study mentioned previously. Possible uncertainties introduced by the BDT selector are studied by loosening the selection as far as possible while maintaining a plausible fit quality. The changes in the efficiency-corrected yields are found to be consistent with zero within the statistical uncertainty.

IV. SUMMARY
We analyze the decays Λ + c → Σ + π − π + , Λ + c → Σ 0 π + π 0 , and Λ + c → Σ + π 0 π 0 using the full Belle data set at or near the Υ(4S) resonance. Using a model-independent approach, we fit the signal yields in separate bins of the decay Dalitz distribution to avoid uncertainties introduced by intermediate resonances.