Measurement of branching fractions of Λ + c decays to Σ + K + K − , Σ + ϕ and Σ + K + π − ( π 0 )

: Based on 4.5 fb − 1 data taken at seven center-of-mass energies ranging from 4.600 to 4.699 GeV with the BESIII detector at the BEPCII collider, we measure the branching fractions of Λ + c → Σ + + hadrons relative to Λ + c → Σ + π + π − . Combining with the world average branching fraction of Λ + c → Σ + π + π − , their branching fractions are measured to be (0 . 377 ± 0 . 042 ± 0 . 020 ± 0 . 021)% for Λ + c → Σ + K + K − , (0 . 200 ± 0 . 023 ± 0 . 011 ± 0 . 011)% for Λ + c → Σ + K + π − , (0 . 414 ± 0 . 080 ± 0 . 030 ± 0 . 023)% for Λ + c → Σ + ϕ and (0 . 197 ± 0 . 036 ± 0 . 009 ± 0 . 011)% for Λ + c → Σ + K + K − (non-ϕ ). In all the above results, the first uncertainties are statistical, the second are systematic and the third are from external input of the branching fraction of Λ + c → Σ + π + π − . Since no signal for Λ + c → Σ + K + π − π 0 is observed, the upper limit of its branching fraction is determined to be 0.13% at the 90 % confidence level.


BESIII DETECTOR AND MONTE CARLO SIMULATION
The BESIII detector [43] records symmetric e + e − collisions provided by the BEPCII storage ring [44], which operates with a peak luminosity of 1 × 10 33 cm −2 s −1 in the centerof-mass energy range from 2.0 to 4.95 GeV.BESIII has collected large data samples in this energy region [45].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 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 identification modules interleaved with the steel.
The charged-particle momentum resolution at 1 GeV/c is 0.5%, and the 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 [46][47][48].About 87% of the data used were collected after this upgrade.
Simulated data samples produced with the geant4-based [49] MC package BOOST [50], which includes the geometric and material description of the BESIII detector [51,52] and the detector responses, are used to determine detection efficiencies and to estimate backgrounds.The simulation models the beam energy spread and initial state radiation (ISR) in the e + e − annihilations with the generator kkmc [53,54].The inclusive MC sample includes the production of open charm processes, the ISR production of vector charmonium(-like) states, and the continuum processes.All particle decays are generated with evtgen [55] using BFs either taken from the Particle Data Group (PDG) [56], when available, or otherwise estimated with lundcharm [57,58].Final state radiation (FSR) from charged final state particles is incorporated using the photos package [59].For the MC production of the e + e − → Λ + c Λ − c events, the cross section line-shape from BESIII measurements is taken into account.The signal decay processes include an incoherent sum of intermediate state resonances, while for the reference mode Λ + c → Σ + π + π − , a partial-wave analysis is performed to obtain the amplitude model.

EVENT SELECTION AND DATA ANALYSIS
In this work, both the signal and reference Λ + c decays are fully reconstructed.Charged tracks detected in the MDC are required to be within a polar angle (θ) range of |cosθ| < 0.93, where θ is defined with respect to the z axis, which is the symmetry axis of the MDC.The distance of closest approach to the interaction point (IP) must be less than 10 cm along the z axis and less than 1 cm in the transverse plane.
Particle identification (PID) for charged tracks combines measurements of the energy deposited in the MDC (dE/dx) and the flight time in the TOF to form likelihoods L(h) (h = p, K, π) for each hadron h hypothesis.Tracks are identified as protons when the proton hypothesis has the greatest likelihood (L(p) > L(K) and L(p) > L(π)), while charged kaons and pions are identified by comparing the likelihoods for the kaon and pion hypotheses, L(K) > L(π) and L(π) > L(K), respectively.
Photon candidates are reconstructed using showers in the EMC.The deposited energy of each shower must be more than 25 MeV in the barrel region (| cos θ| < 0.80) and more than 50 MeV in the end cap region (0.86 < | cos θ| < 0.92).To exclude showers that originate from charged tracks, the angle subtended by the EMC shower and the position of the closest charged track at the EMC must be greater than 10 degrees as measured from the IP.To suppress electronic noise and showers unrelated to the event, the difference between the EMC time and the event start time is required to be within [0, 700] ns.
All γγ combinations are considered as π 0 candidates, and the reconstructed mass M (γγ) is required to fall in the range of 0.115 GeV/c 2 < M (γγ) < 0.150 GeV/c 2 .A kinematic fit is performed to constrain the γγ invariant mass to the known π 0 mass [56], and candidates with the fit quality χ 2 < 200 are retained.
To improve the signal purity, the energy difference ∆E = E cand − E beam for Λ + c candidates are required to satisfy a mode-dependent ∆E requirement shown in Table 2.These ranges are obtained by optimizing signal yield significance with the inclusive MC sample.Here E cand is the total reconstructed energy of the Λ + c candidate and E beam is the beam energy.Only one candidate with the minimal |∆E| is accepted.The Λ + c signal is identified using the beam constrained mass M BC = E 2 beam /c 4 − p 2 /c 2 , where p is the measured Λ + c momentum in the center-of-mass system of the e + e − collision.After the event selection, there is no obvious peaking background in the M BC distribution for each tag mode, as shown in Figure 2.

DETERMINATION OF THE BRANCHING FRACTIONS
The RBF between the signal and reference modes is calculated with where ε i and ε j are the detection efficiencies, N i and N j are the signal yields of the signal mode i and the reference mode j, respectively.The B inter is the product of the BFs of the intermediate states (Σ + → pπ 0 , π 0 → γγ, and ϕ → K + K − for Λ + c → Σ + ϕ), and the charge-conjugate channel is included in the simulation.The observables in Equation (4.1) are determined as follows.
• Reference mode Λ + c → Σ + π + π − To obtain the yield of the reference mode, an unbinned maximum likelihood (UML) fit is performed on the M BC distribution separately at each energy point for data.In each fit, the signal shape is described by the MC simulated shape convolved with a Gaussian function with floating mean and width, in order to take into account the resolution difference between data and MC simulation.The distribution of the combinatorial backgrounds is modeled with an ARGUS function [60] where the parameter a is free in the fit.The fitted yields of the reference mode at different energy points are given in Table 3.In addition, the detection efficiencies are estimated according to MC simulations, as listed in Table 4.The total fit to data summing over all the energy points is shown in Figure 3(a).
• Signal mode Λ + c → Σ + K + π − The simultaneous UML fit is performed using seven energy point data samples for signal mode, to obtain a more precise result.For this purpose, a common RBF is fitted Other plots (c) and (d) show the distributions of M BC and M K + K − , respectively, for the projections of the 2-D fit used to separate the Λ + c → Σ + ϕ and Λ + c → Σ + K + K − (non-ϕ) processes.The points with error bars are combined from data at all energy points, the red curves are the overall fit result, the blue and green dashed curves are the signal shapes, the black and pink dashed curves are the background shapes.
at various energy points, where the signal yields can be derived from Equation 4.1 with the input yields of the reference mode in Table 3 and the detection efficiencies in Table 4.The uncertainties of the input values are taken into account in systematic uncertainties, as listed in Table 5.In the fit, the signal shape is extracted from the corresponding signal MC sample and convolved with a Gaussian function with floating mean and width.The background shape is described by an ARGUS function in Equation 4.2.The summed fit to data from all of the energy points is shown in Figure 3(b).The RBF results obtained from the fit are given in Table 1, and the signal yields at different energy points are calculated, as listed in Table 3.
• Signal mode Λ + c → Σ + K + K − To separate the ϕ contribution for Σ + K + K − mode, a two-dimensional simultaneous UML fit is performed on the M BC vs the M K + K − distributions, in which common RBF values are estimated.Four components (Σ + ϕ, Σ + K + K − (non-ϕ), ϕ background and non-ϕ background) are considered in this fit, as shown in Figures 3(c) and 3(d).
All the signal shapes are extracted from MC samples and convolved with a Gaussian resolution function with floating mean and width.In the M BC distribution, the combinatorial background is described using the ARGUS function.In the M K + K − distribution, we use a second-order Chebyshev function to describe the background of the non-ϕ process and use the ϕ line-shape distribution extracted from the Λ + c → Σ + ϕ signal MC sample to describe the ϕ background.
The simultaneous UML fit is also performed for Λ + c → Σ + K + π − π 0 , as shown in Figure 3(e).Since there is no significant signal observed, the upper limit on the BF of this decay is estimated with a likelihood scan method which takes into account the systematic uncertainties as follows Here, L(BF) is the likelihood expression of BF, L stat is the statistical likelihood given by the data without considering the systematic uncertainties when taking the nominal BF obtained, ∆ is the relative deviation of the estimated BF from the nominal value and σ syst is the total systematic uncertainty given in Table 5.The likelihood curve calculated according to Equation 4.3 is shown in Figure 4.The upper limit on the BF of Λ + c → Σ + π − π + π 0 mode at the 90% confidence level (CL) is estimated to be 0.11% by integrating the likelihood curve in the physical region and considering the BF of the reference mode and systematic uncertainties.

SYSTEMATIC UNCERTAINTY
The sources of systematic uncertainties in the RBF measurements are summarized in Table 5.The systematic uncertainties of MC statistics is 0.1% for each channel and can be neglected.The systematic uncertainties in Σ + reconstruction in both signal and reference modes are canceled in the RBF calculation.The systematic uncertainties of tracking efficiency for charged tracks are also canceled.For each signal decay, the square root of the quadratic sum of all the uncertainties is taken as the total systematic uncertainty.c → Σ + K + π − π 0 after incorporating the systematic uncertainty.The black points are the initial curve, the red curve is the result with the systematic uncertainty, and the blue arrow points to the upper limit on the BF at the 90% CL.
Table 5.Relative systematic uncertainties in the RBF measurements, in %. • PID and π 0 reconstruction efficiencies.The uncertainties associated with PID efficiencies are estimated to be 1.0% for each charged track by studying the control samples of e + e − → π + π − π + π − , K + K − π + π − and ppπ + π − based on data taken at energies above √ s = 4.0 GeV.However, the PID uncertainty for the p(p) common to signal and reference modes cancels, as does that for any common π ± .The uncertainty of π 0 reconstruction efficiency is assigned to be 1.0% per π 0 by studying the control sample of e + e − → ωπ 0 process.
• ∆E requirement.The resolution difference between data and MC simulation need to be considered.Thus a Gaussian function is used to smear MC sample and obtain a new efficiency.The changes in the efficiencies are assigned as the corresponding uncertainties of ∆E.The resolution difference is extract by using signal MC shape smeared with the Gaussian function to fit the ∆E distribution in data.
• M BC fit.The uncertainty in the M BC fit is mainly due to free parameters of the Gaussian and ARGUS functions and the background description.The relevant sys-tematic uncertainty is estimated with an alternative signal shape without smearing the Gaussian resolution function and an alternative background shape which varies the high-end cutoff of the ARGUS function by ±0.005 GeV/c 2 .The largest result change is assigned as the systematic uncertainty.For Λ + c → Σ + ϕ process, due to that a two-dimensional maximum likelihood fit is performed on the M BC and M K + K − distribution, the uncertainty of M K + K − fit is also considered by studying the difference of ϕ signal shape convolved with or without a Gaussian resolution function.In addition, the shape of the backgrounds from Λ + c inclusive decays can not be different from the ARGUS function.To account for this discrepancy, an additional component of the Λ + c decay backgrounds, which is extracted from the Λ + c inclusive MC samples, is included in the M BC fit.The resultant changes on the RBF are taken as the systematic uncertainties.For Λ + c → Σ + K + π − π 0 process, an alternative fit to data with M BC larger than 2.27 GeV is implemented and the relative changes on the fitting results are taken into account as systematic uncertainties.All the above items are summed over in quadrature.
• Signal model.The influence of the assumed signal model on the BF measurement comes from the estimation of the signal efficiency.Decay processes with intermediate resonances listed in the PDG [56] are included in the signal simulation.Examples include In the nominal analysis, the BFs that are used in the generator are taken from the PDG [56].Their effects on the new BF measurements are estimated by varying the input BF of the intermediate decay by ±1σ in the generator.The relative efficiency deviation is taken as the uncertainty.
• Reference mode.For the reference mode Λ + c → Σ + π + π − , the TF-PWA [61] is used to perform the simple Partial Wave Analysis and consider the possible intermediate resonance states, In the estimation of systematic uncertainty, we remove the low significance state.The difference of efficiency (1.1%) is taken as the systematic uncertainty.For the statistical uncertainties of the fitted yields in the reference mode, a relative uncertainty (1.6%) is transferred into the systematic uncertainty of the RBF.In total, a quadrature sum of the systematic uncertainty (1.9%) is assigned.
• BFs of the intermediate states.The BFs of π 0 → γγ, ϕ → K + K − and Σ + → p + π 0 are used as inputs in the baseline analysis, and their uncertainties [56] are propagated as the systematic uncertainty.
Table 6.Comparison of our RBF and BF results with the Belle results [32] and the PDG values [56] (in unit of %).Except for the mode Σ + K + π − π 0 , in all our RBF results, the first uncertainties are statistical, and the second are systematic.The third uncertainties of our BF results are from external input of the branching fraction of Λ + c → Σ + π + π − .

Figure 3 .
Figure 3. Fit results for different decay modes.The plots show theM BC distributions of Λ + c → Σ + π + π − (a), Λ + c → Σ + K + π − (b) and Λ + c → Σ + K + π − π 0 (e) processes.Other plots (c) and (d) show the distributions of M BC and M K + K − , respectively, for the projections of the 2-D fit used to separate the Λ + c → Σ + ϕ and Λ + c → Σ + K + K − (non-ϕ) processes.The points with error bars are combined from data at all energy points, the red curves are the overall fit result, the blue and green dashed curves are the signal shapes, the black and pink dashed curves are the background shapes.

Figure 4 .
Figure 4.The normalized likelihood value versus the BF of Λ +c → Σ + K + π − π 0 after incorporating the systematic uncertainty.The black points are the initial curve, the red curve is the result with the systematic uncertainty, and the blue arrow points to the upper limit on the BF at the 90% CL.

Table 1 .
The center-of-mass energy E cms and the integrated luminosity measured L we used in this analysis.

Table 2 .
Requirements on ∆E for different Λ + c decay modes.

Table 3 .
Signal yields in data for various decay modes, where the uncertainties are statistical only.Λ + c → Σ + K + π − π 0 has no significant signal observed.

Table 4 .
Detection efficiencies (in unit of %) for various decay modes, where the uncertainties are statistical only.The efficiencies do not include the BF of the sequential decay of π 0 , ϕ or Σ + .