Search for the semi-leptonic decays $\Lambda_c^+ \to \Lambda \pi^+ \pi^- e^+ \nu_e$ and $\Lambda_c^+ \to p K_S^0 \pi^- e^+ \nu_e$

We search for the semi-leptonic decays $\Lambda_c^+ \to \Lambda \pi^+ \pi^- e^+ \nu_e$ and $\Lambda_c^+ \to p K_S^0 \pi^- e^+ \nu_e$ in a sample of 4.5 $\mathrm{fb}^{-1}$ of $e^{+}e^{-}$ annihilation data collected in the center-of-mass energy region between 4.600 GeV and 4.699 GeV by the BESIII detector at the BEPCII. No significant signals are observed, and the upper limits on the decay branching fractions are set to be $\mathcal{B}(\Lambda_c^+ \to \Lambda \pi^+ \pi^- e^+ \nu_e)<3.9\times10^{-4}$ and $\mathcal{B}(\Lambda_c^+ \to p K_S^0 \pi^- e^+ \nu_e)<3.3\times10^{-4}$ at the 90% confidence level, respectively.


Introduction
The study of Λ + c semi-leptonic (SL) decays provides valuable information about weak and strong interactions in baryons containing a heavy quark.According to Fermi's Golden Rule, the decay rate depends on the product of kinematic phase space (PHSP) and dynamic amplitude.The dynamic amplitude in Λ + c SL decays is much simpler than in non-leptonic decays, and can be factorized into a hadronic term, leptonic term and weak quark-mixing Cabibbo-Kobayashi-Maskawa matrix element [1].The hadronic current describes the weak transition of the charm quark to a light quark, and the leptonic current describes the coupling to the charged-lepton-neutrino pair.In principle, the leptonic current can be precisely calculated, in contrast to the hadronic current, which suffers from difficulties due to the strong interaction [2], and can be parameterized by form factors. Recently, the first measurement of Λ + c → Λ form factors from the BESIII Collaboration [3] shows large discrepancies with the lattice Quantum Chromodynamics (LQCD) calculation [4], which attracts wide attention.
Ref. [6] suggests these other decay modes could be from Λ + c decays to excited states such as Λ(1405) and Λ(1520) or continuum Σπ and N K contributions.Recently, BESIII presented the first observation of the SL decay Λ + c → pK − e + ν e [7], in which evidence for Λ + c → Λ(1520)e + ν e is reported with a BF of (1.02±0.52 stat.±0.11 syst.)×10 −3 and a combined statistical and systematic significance of 3.3σ.This result stimulates further research on Λ + c SL decays into various excited Λ * baryons.Over the years, many theoretical calculations concerning Λ + c → Λ * form factors and BFs based on constituent quark model (CQM) [8], nonrelativistic quark model (NRQM) [9], light-front quark model (LFQM) [10] and LQCD [11] have been performed; their results for the BF of Λ + c → Λ * e + ν e are shown in Table 1.Searching for Λ + c SL decays which may contribute to these excited Λ * baryons is important for testing and constraining these theoretical calculations.

BESIII experiment and Monte Carlo simulation
The BESIII detector [16] records symmetric e + e − collisions provided by the BEPCII storage ring [17] in the CMS energy range from 2.0 to 4.95 GeV, with a peak luminosity of 1 × 10 33 cm −2 s −1 achieved at √ s = 3.77 GeV.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 [18].The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identification modules interleaved with steel.The charged-particle momentum resolution at 1 GeV/c is 0.5%, and the specific ionization energy loss dE/dx resolution 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, while that in the end-cap region is 110 ps.The end-cap TOF system was upgraded in 2015 using multi-gap resistive plate chamber technology, providing a time resolution of 60 ps [19].
Simulated data samples are produced with a GEANT4-based [20] Monte Carlo (MC) package, which includes the geometric description of the BESIII detector [21,22] and the detector response.The simulation models the beam-energy spread and initial-state radiation (ISR) in the e + e − annihilations with the generator KKMC [23].The final-state radiation (FSR) from charged final-state particles is incorporated using PHO-TOS [24].
The inclusive MC sample includes the production of Λ + c Λ− c pairs, and open-charm mesons, ISR production of vector charmonium(-like) states, and continuum processes which are incorporated in KKMC [23,25].Known decay modes are modeled with EVT-GEN [26,27] using the BFs taken from the Particle Data Group (PDG) [28].The signal decay modes Λ + c → Λπ + π − e + ν e and Λ + c → pK 0 S π − e + ν e are not includ-ed in the inclusive MC sample.The remaining unknown charmonium decays are modeled with LUND-CHARM [29,30].The inclusive MC sample is used to study background contributions and to optimize event selection criteria.The e + e − → Λ + c Λ− c signal MC sample are generated to estimate the detection efficiencies, in which the Λ + c decays through signal modes while the other Λ− c decays through 12 single-tag (ST) modes as described below.In the baseline analysis, all Λπ + π − combinations are assumed to be from the Λ(1520) resonance.The SL decay of Λ + c → Λ(1520)e + ν e is simulated based on the heavy-quark effective-theory (HQET) model [9], while the decay of Λ(1520) → Λπ + π − is simulated with the uniformly distributed PHSP model.The SL decay Λ + c → pK 0 S π − e + ν e is simulated by using the PHSP model.For two-body Λ− c ST modes, the angular distributions are described with the transverse polarization and decay asymmetry parameters of the Λ + c and its daughter baryons [31].For three-body and fourbody Λ− c ST modes, the intermediate states are modeled according to individual internal partial-wave analysis models.

Event selection
At the CMS energy region between √ s = 4.600 GeV and 4.699 GeV, the Λ + c Λ− c pair is produced in the electron-positron annihilation without additional hadron companions.Since the neutrinos in the signal decays can not be detected by the BESIII detector, we use the double tag (DT) technique which was first applied by the Mark III Collaboration [32].
The ST sample consists of events in which the Λ− c baryon is reconstructed with any of the following 12 exclusive hadronic decay modes: which are the same as those considered in Ref. [33].The DT sample is formed of those events in the ST sample that also contain candidates for the ) is determined by is the average efficiency of detecting a Λ + c decaying into the signal-mode s in the system recoiling against the ST Λ− c .
The selection criteria of the ST Λ− c candidate events follow the previous BESIII analysis [33].The beam-energy-constrained mass , where E beam is the average value of the e + and e − beam energies, p is the total measured Λ− c momentum in the CMS of the e + e − collision.To improve the signal purity, the energy difference ∆E ≡ E − E beam for the Λ− c candidate is required to fulfill a mode-dependent ∆E requirement.Here, E is the total reconstructed energy of the Λ− c candidate in the CMS of the e + e − collision.For each ST mode, if more than one candidate satisfies the above requirements, the one with the minimal |∆E| is retained.The total yield of the 12 ST modes is N ST = 123509 ± 461, where the uncertainty is statistical.Full information about the ∆E requirements, M BC distributions, signal regions, ST yields, and efficiencies for the various ST modes at each energy point is detailed in Ref. [33].
The signal candidate events for Λ + c → Λπ + π − e + ν e and Λ + c → pK 0 S π − e + ν e are selected with those tracks in the event that are not used to form the ST Λ− c candidates.The Λ and K 0 S candidates are reconstructed from pπ − and π + π − combinations, respectively.Charged tracks are reconstructed in the MDC, and are required to have a polar angle θ with respect to the zaxis, defined as the symmetry axis of the MDC, satisfying |cos θ| < 0.93.The distance of closest approach to the interaction point (IP) is required to be less than 10 cm along the z-axis (V z ) and less than 1 cm in the perpendicular plane (V r ), which are denoted as tight track requirements.Tracks originating from K 0 S and Λ decays are not subjected to these distance requirements.Instead, they are subjected to the loose track requirements of |V z | < 20 cm and no restriction on V r .To suppress background events, it is required that there are only five charged tracks to be reconstructed in the signal side, which must satisfy loose track requirements.
Particle identification (PID) [34] for charged tracks is implemented using combined information from the flight time measured in the TOF and the dE/dx measured in the MDC.Charged tracks are identified as pro-tons when they satisfy L(p) > L(K), L(p) > L(π) and L(p) > 0, where L(h) is the PID probability for each particle (h) hypothesis with h = p, π, K. Charged tracks are identified as pions when they satisfy L(π) > L(K) and L(π) > 0. The energy deposited in the EMC is also considered when constructing the PID probability for the positron hypothesis, L(e).Charged tracks are identified as positrons when they satisfy L(e) > 0.001 and a requirement on the PID probability ratio which is L(e)/(L(e) + L(π) + L(K)) > 0.99(0.98)for ).The energy loss due to FSR and bremsstrahlung photon(s) of the positron candidates is partially recovered by adding the showers that are within a 5 • cone relative to the track momentum.
Long-lived Λ (K 0 S ) candidates are reconstructed by combining pπ − (π + π − ) pairs.Here, a PID requirement is imposed on the proton candidate, but not on the π candidates.A vertex fit is applied to pairs of charged tracks, constraining them to originate from a common decay vertex, and the χ 2 of this vertex fit is required to be less than 100.The invariant mass of the pπ − (π + π − ) pair must satisfy 1.09 < M pπ − < 1.14 GeV/c 2 (0.490 < M π + π − < 0.504 GeV/c 2 ).Here, M pπ − and M π + π − are the invariant masses of pπ − and π + π − pairs, calculated with the common decay-vertex constraint imposed.To further suppress background, we require a positive value of the decay length.These selection criteria are optimized by using the Punzi figureof-merit (FOM) [35].The definition of the Punzi FOM is ε/(3/2+ √ B), where ε is the detection efficiency and B is the number of background events in the inclusive MC sample.If there are more than one Λ (K 0 S ) or e + candidates in an event, the Λ(K 0 S ) with the largest L/σ L or the e + with the largest L(e) is retained to avoid double counting of the DT events.
The missing energy and missing momentum carried by undetected neutrinos are denoted by E miss and p miss , which are calculated from E miss = E beam − E SL and p miss = p Λ + c − p SL in the initial e + e − rest frame.Here, E SL and p SL are the measured energy and momentum of SL decay products, which are determined as With the help of a generic event type analysis tool, TopoAna [36], the inclusive MC sample is used to study background events after applying the primary selection criteria described above.In the events of the ST modes Λ− c → pπ + π − and Λ− c → Σ− π + π − , to suppress contamination due to misidentification of pions and positrons, only positrons with a detected energy deposit in the EMC are retained.The background levels of these two modes are the highest among the 12 tag modes due to the random combination of final state particles, and this additional selection allows to enhance the discrimination of the PID.To suppress γ-conversion background events, cos θ e,π is required to be less than 0.88(0.92)for where θ e,π is the opening angle between the oppositely charged pion and positron.To suppress contamination from , where e(π) + denotes that the e + mass is replaced by that of the π + in the calculation, is required to be less than 2.20 (2.28) GeV/c 2 .The remaining dominant background events are from In these events, the charged pions misidentified with positrons and the electromagnetic showers due to π 0 /γ are not detected.To suppress this category of background, cos θ Pmiss,γ is required to be less than 0.82(0.90)for Λ + c → Λπ + π − e + ν e (Λ + c → pK 0 S π − e + ν e ), where θ Pmiss,γ is the opening angle between the missing momentum P miss and the most energetic shower.After applying these requirements, which have been optimized through the Punzi FOM, the level of background is greatly suppressed.The resulting U miss distributions of candidates in data and MC samples are shown in Figure 1, where no significant excess over the expected backgrounds is observed.The total number of observed events, N obs , is counted in the U miss signal region and listed in Table 2.The average efficiencies ε sig Λππ and ε sig pK 0 S π are determined to be (9.69 ± 0.03)% and (13.58 ± 0.02)%, respectively, where the uncertainties are statistical.The two dimensional efficiency maps of M (Λπ + π − ) or M (pK 0 S π − ) versus q 2 are shown in Figure 2, where q 2 = (p e +p νe ) 2 .
The backgrounds can be separated into two categories: events with a wrong ST candidate denoted as bkg1 which is dominantly from non-Λ c decay process, and events with a correct ST but wrong signal candidate denoted as bkg2 which is dominantly from Λ c decay process.The size of the bkg1 component can be estimated with the surviving events in the ST sideband (SB) region of M tag BC , which is defined as (2.25, 2.27) GeV/c 2 .The corresponding number of bkg1 events, N bkg1 , is estimated from the number of events in the SB region (N SB bkg1 ) normalized by a scale factor r, which is the ratio of the integrated numbers of background events in the signal region and SB region.The scale factor r is found to be 1.533 ± 0.004, where the uncertainty is statistical only.The number of events in the SB region, N SB bkg1 , is expected to follow a Poisson (P) distribution with central value of N bkg1 ×  Table 2: The total number of observed events N obs in the signal region, the average efficiency ε sig , the number of events in the SB region N SB bkg1 , the number of bkg2 events estimated by MC simulation N MC bkg2 , the corresponding statistical uncertainty σ MC bkg2 for Λ + c → Λπ + π − e + νe and Λ + c → pK 0 S π − e + νe and the upper limit on the DT signal yield N DT at the 90% confidence level.The uncertainties are statistical.

Systematic uncertainties
With the DT technique, the systematic uncertainties in the BF measurements due to the detection and reconstruction of the ST Λ− c baryons mostly cancel out.For the signal side, the signal yield, N sig , which is N DT Λππ or N DT pK 0 S π , is calculated by N obs − N bkg1 − N bkg2 , where N obs is the total number of observed events obtained from counting, without any uncertainty assigned, N bkg1 and N bkg2 are the numbers of bkg1 and bkg2 events, the statistical uncertainties of which are assigned assuming Poisson and Gaussian distributions, respectively.All sources of systematic uncertainties are summarized in Table 3 and discussed below.It should be noted that the systematic uncertainties due to the M Λπ + π − e(π) + or M pK 0 S π − e(π) + requirement are negligible.(IV) p tracking/PID.The proton (anti-proton) tracking/PID efficiency is studied with a J/ψ → ppπ + π − control sample.The detection efficiency of Λ + c → pK 0 S π − e + ν e is recalculated after re-weighting the signal MC sample on an event-by-event basis according to the momentum-and polar angle-dependent efficiency differences between data and MC simulation.The relative differences between the baseline and corrected efficiencies, 0.4% and 0.2%, are taken as the systematic uncertainties due to p tracking and PID efficiencies for B pK 0 S π , respectively.The systematic uncertainties due to p tracking and PID efficiencies for B Λππ are included in the Λ reconstruction, as described below.
(V) π tracking/PID.The charged pion tracking/PID efficiency is also studied with the J/ψ → ppπ + π − control sample.The detection efficiencies of Λ + c → Λπ + π − e + ν e and Λ + c → pK 0 S π − e + ν e are re-weighted in the same way as in the p tracking study.The resultant data-MC differences are assigned as the systematic uncertainties, which are 2.6% and 0.4% in π tracking, and 0.7% and 0.3% in π PID for B Λππ and B pK 0 S π , respectively.
(VI) e tracking.The positron tracking efficiency is studied with a e + e − → γe + e − control sample.The detection efficiencies of Λ + c → Λπ + π − e + ν e and Λ + c → pK 0 S π − e + ν e are re-weighted in the same way as in the p tracking study.The resultant data-MC differences, 0.5% and 0.1%, are assigned as the systematic uncertainties for B Λππ and B pK 0 S π , respectively.(VII) e PID.The positron PID efficiency with the requirement of L(e) > 0.001 and L(e)/(L(e) + L(π) + L(K)) > 0.99(0.98) is studied with control samples of e + e − → γe + e − and D 0 → K0 π − e + ν e decays.The differences in the acceptance efficiencies between data and MC simulation are assigned as the corresponding systematic uncertainties, which are 2.8% and 3.6% for B Λππ and B pK 0 S π , respectively.(VIII) Λ reconstruction.The Λ reconstruction efficiency is studied by using the control samples of J/ψ → ΛpK − and J/ψ → Λ Λ decays.Using the same procedure as in the p tracking study, the systematic uncertainty due to Λ reconstruction is assigned to be 2.2% for B Λππ .
(IX) K 0 S reconstruction.The K 0 S reconstruction efficiency of the selections 0.490 < M π + π − < 0.504 GeV/c 2 and L/σ L > 0 is studied by using control samples of J/ψ → K * (892) ∓ K ± , K * (892) ∓ → K 0 S π ∓ , J/ψ → φK 0 S K ± π ∓ and D 0 → K0 π − e + ν e decays.The difference in the acceptance efficiencies between data and MC simulation is assigned as the corresponding systematic uncertainty, which is 3.2% for B pK 0 S π .(X) cos θ e,π requirement/cos θ Pmiss,γ requirement/FSR recovery.The systematic uncertainties due to the cos θ e,π requirement, cos θ Pmiss,γ requirement and FSR recovery are assigned to be 1.5%, 0.1%, and 0.2% for the two signal modes, respectively, from measuring the differences in the acceptance efficiencies between data and MC simulation with the control sample of D 0 → K0 π − e + ν e decays.
(XI) Signal model.To evaluate the systematic uncertainty due to signal model, additional Λ * resonance contributions are considered.For the two signal modes, MC events of Λ + c → Λ(1820)/Λ(1890)e + ν e are generated in both the PHSP model and the HQET model [9].The contribution from the transition through the Λ(1600) is also considered for Λ + c → Λπ + π − e + ν e .The largest changes in the detection efficiencies are assigned as the associated systematic uncertainties, which are 2.2% and 5.6% for B Λππ and B pK 0 S π , respectively.

BF upper limits
To calculate the upper limits on the BFs of the signal decays, we use a maximum likelihood estimator extended from the profile likelihood method [37].According to Eq. ( 1), the effective signal yield is defined to be N eff which follows a Gaussian distribution with mean B inter •N ST •ε sig , and width B inter •N ST •ε sig •σ, where σ is the relative uncertainty of N eff including both statistical and systematic components.From error propagation it follows that σ is equal to the relative systematic uncertainty of B, as given in Table 3.Therefore, the joint likelihood is Based on the Bayesian statistics, B is priorly assumed to be the uniform distribution, and the likelihood L maximized by the variation of the other parameters N eff , N bkg1 and N bkg2 , is the posterior probability of B. By scanning B, the likelihood distribution as a function of B is obtained.The resultant distributions of likelihoods plotted as a function of the individual BFs of Λ + c → Λπ + π − e + ν e and Λ + c → pK 0 S π − e + ν e are shown in Figure 3.The upper limits on the signal BFs at the 90% confidence level (CL) are estimated by integrating the likelihood curves in the physical region of B ≥ 0 [38].The upper limits on the BFs of Λ + c → Λπ + π − e + ν e and Λ + c → pK 0 S π − e + ν e are determined to be 3.9 × 10 −4 and 3.3 × 10 −4 , respectively, and the related upper limit on the DT signal yield is listed in Table 2. Assuming that all the Λππ combinations come from Λ + c → Λ(1520)e + ν e , which is expected to be the dominant decay, the upper limit of B(Λ + c → Λ(1520)e + ν e ) is determined to be 4.3 × 10 −3 at 90% CL after considering B(Λ(1520) → Λπ + π − ) = (10 ± 1)% [28].Assuming that all the Λππ combinations come from Λ + c → Λ(1600)e + ν e , the upper limit of B(Λ + c → Λ(1600)e + ν e ) is determined to be 9.0 × 10 −3 at 90% CL after taking into account B(Λ(1600) → Σ(1385)π) = (9 ± 4)% and B(Σ(1385) → Λπ) = (87.5 ± 1.5)% [28].Our result is consistent with B(Λ + c → Λ(1520)e + ν e ) = (1.02± 0.52 stat.± 0.11 syst. ) × 10 −3 , as measured via Λ(1520) → pK − by BESIII [7].

Summary
In summary, based on 4.5 fb −1 of e + e − annihilation data collected in the CMS energy region between 4.600 GeV and 4.699 GeV by the BESIII detector at the BEPCII, we search for the SL decays Λ + c → Λπ + π − e + ν e and Λ + c → pK 0 S π − e + ν e .No significant signal is observed in data.Therefore, the upper limits on the BFs of these two decays are set to be B(Λ + c → Λπ + π − e + ν e ) < 3.9 × 10 −4 and B(Λ + c → pK 0 S π − e + ν e ) < 3.3 × 10 −4 at 90% CL.Assuming that all the Λππ combinations come from Λ(1520) or Λ(1600), the BF upper limits are determined to be B(Λ + c → Λ(1520)e + ν e ) < 4.3 × 10 −3 and B(Λ + c → Λ(1600)e + ν e ) < 9.0 × 10 −3 at 90% CL.Due to the limitation of statistics, our results are consistent with all theoretical calculations listed in Table 1.The result on B(Λ + c → Λ(1520)e + ν e ) is also consistent with that measured via Λ(1520) → pK − in Ref. [7].This result helps to constrain the theoretical calculations of the BFs and form factors of Λ → Λ * e + ν e .The larger data samples that will be collected at BESIII in future [39,40] will allow the sensitivity to these decays to be further improved, and provide a deeper understanding of charmed baryon decays.
where N DT s is the DT signal yield of the signal-mode s, N ST i is the ST event yield of tag-mode i, ε ST i is the efficiency of detecting a tag-mode i candidate, ε DT s,i is the efficiency of simultaneously detecting the tag-mode i and the signal-mode s candidate, B inter Λππ = B(Λ → pπ − ) and B inter pK 0 S π = B(K 0 S → π + π − ) are the BFs of the decays of the intermediate states taken from the PDG[28], N ST = i N ST i is the total yield of all 12 ST modes in data, and ε sig s where ptag is the direction of the momentum of the ST Λ− c and m Λ− c is the known mass of the Λ− c [28].If the ST Λ− c and SL decay products in the signal side are correctly identified, the U miss = E miss − c| p miss | is expected to peak around zero for the signal mode.Signal candidates are required to satisfy U miss ∈ [−0.08, 0.08] GeV, which is about three times the resolution evaluated in the simulation.

Figure 1 :
Figure 1: The U miss distributions of candidates for (a) Λπ + π − e + νe and (b) pK 0 S π − e + νe, where the black points with error bars denote data, the violet hatched histogram the Λ + c decay background, the orange solid histogram the non-Λ + c decay background and the red hollow histogram the signal MC sample, which is scaled to the measured upper limit on the signal yield at 90% confidence level.The region between two arrows indicates the signal region.

Figure 3 :
Figure 3: Normalized likelihood distribution as a function of the signal BFs.The black curve denotes the nominal fit result including systematic uncertainty, the red curve denotes the fit result without incorporating the systematic uncertainties and the arrows point to the positions of the upper limits at the 90% CL.The two curves are totally overlapping.

Table 3 :
Relative systematic uncertainties for the measurements of the BFs of Λ + c → Λπ + π − e + νe and Λ + c → pK 0 S π − e + νe.The total systematic uncertainty is the sum in quadrature of the individual components."--" indicates the cases where there is no uncertainty.