Measurement of the branching fractions of the singly Cabibbo-suppressed decays Λ + c → pη and

: Based on 4.5 fb − 1 e + e − collision data collected with BESIII detector at seven energy points between 4.600 and 4.699 GeV, the branching fractions for Λ + c → pη and Λ + c → pω were measured by means of single-tag method. The branching fractions of Λ + c → pη and Λ + c → pω are determined to be (1 . 58 ± 0 . 11 stat ± 0 . 04 syst ) × 10 − 3 and (1 . 11 ± 0 . 20 stat ± 0 . 05 syst ) × 10 − 3 , with a statistical signiﬁcance of greater than 10 σ and 5.7 σ , respectively. These results are consistent with the previous measurements by BESIII, LHCb and Belle, and the result of Λ + c → pη is the most precise to date.


Introduction
Studies on weak decays of charmed baryons provide crucial information on the strong and weak interactions in the charm sector.Decay amplitudes of charmed hadrons can be divided into factorizable terms, where the strong and weak parts can be treated separately, and non-factorizable terms, where the two parts are entangled and hard to calculate [1][2][3].In the case of charmed mesons, non-factorizable terms are negligible compared to factorizable ones.However, for charmed baryon decays, the non-factorizable terms, such as inner exchange and emission of W bosons, are no longer negligible and increase the complexity of calculation.For example, W -exchange manifesting a pole diagram is no longer subject to helicity and color suppression [4].This makes theoretical calculations of the charmed baryon decays more challenging than those of charmed mesons.There are many theoretical models and approaches dealing with charmed baryon decays, including the covariant confined quark model [5][6][7], pole model [4,[8][9][10][11][12], current algebra [4,12,13], and SU(3) flavor symmetry [14].To discriminate between these different models, experimental measurements are necessary.
There has been much progress in measurements of decay rates [15,16] of the charmed baryons.Using a data set at √ s = 4.600 GeV, which is just above the threshold of Λ + Table 1: Measurements and predictions of the branching fractions of Λ + c → pη and Λ + c → pω (in units of 10 −3 ) from different experiments and theoretical calculations.The superscript a(b) denotes the assumption of P -wave amplitude of Λ + c → Ξ 0 K + is positive(negative).The superscript c(d) denotes SU(3) flavor symmetry is conserved(broken).

BESIII Experiment and Monte Carlo Simulation
The BESIII detector [34] records symmetric e + e − collisions provided by the BEPCII storage ring [37] in the center-of-mass 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 BESIII detector has collected large data samples in this energy region [38].The cylindrical core of the BESIII detector covers 93% of the full solid angle and comprises 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 [39].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 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 [40].
Simulated data samples generated with geant4-based [41] Monte Carlo (MC) software, containing the geometric description of the BESIII detector and the detector response [42,43], 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.The inclusive MC samples include the production of photons from anti-neutrons: the lateral moment [51] should be less than 0.4 and E 3×3 /E 5×5 should be greater than 0.85, where E 3×3 (E 5×5 ) is the deposited energy summed over 3 × 3 (5 × 5) crystals around the center of the shower.
For π 0 (η γγ ) candidates, the invariant mass of the photon pair is required to be within the interval 0.115 GeV/c 2 < M γγ < 0.150 GeV/c 2 (0.510 GeV/c 2 < M γγ < 0.570 GeV/c 2 ).A one-constraint (1C) kinematic fit (KF) is performed by constraining the invariant mass of the photon pair to the nominal π 0 (η) mass [46] to improve the momentum resolution.The χ 2 of the 1C KF is required to be less than 50 (20) for π 0 (η γγ ) candidates.The momenta after the 1C KF are used in the subsequent analysis.To eliminate miscombinations that accumulate at large | cos θ decay |, we further require | cos θ decay | < 0.9 for η γγ candidates, where θ decay is the helicity angle of one photon candidate in the rest frame of the η γγ candidate.For η 3π (ω) candidates, the invariant mass of the three pion combinations is required to be in the region 0.536 GeV/c 2 < M π + π − π 0 < 0.560 GeV/c 2 (0.750 GeV/c 2 < M π + π − π 0 < 0.810 GeV/c 2 ).
For Λ + c → pη 3π candidates, events with the invariant mass M pπ 0 within the region (1.17, 1.20) GeV/c 2 are rejected to suppress the intermediate Σ + contributions from Λ + c → Σ + (→ pπ 0 )π + π − .For the Λ + c → pω candidates, we apply a vertex fit to the proton and two charged pions, and the resultant momenta are used in further analysis.In addition, we veto events with the invariant mass of M pπ 0 , M pπ − , and M π + π − in the regions of (1.17, 1.20) GeV/c 2 , (1.10, 1.12) GeV/c 2 , and (0.47, 0.51) GeV/c 2 , respectively, to remove the contributions from Λ + c → Σ + π + π − , Λ + c → Λπ + π 0 , and Λ + c → pK 0 S π 0 .The amplitude for ω → π + π − π 0 is conventionally expanded as a polynomial around the center of the Dalitz plot, as shown in Figure 2, in terms of symmetrized coordinates X and Y [52], which are defined as where T + , T − , and T 0 are the kinetic energies of π + , π − , and π 0 in the rest frame of ω.Q is defined as , where m π + , m π 0 , and m π − are the nominal masses from the PDG [46], and M ω denotes the invariant mass of ω meson reconstructed with three pions M ω = M π + π − π 0 .The Dalitz plot with symmetrized coordinates has advantage in deriving a variable of normalized distance R, which is defined as Here, (X bound , Y bound ) is the intersection of a line through the origin (0, 0) and (X, Y ) with the boundary curve, as shown in Figure 2. Hence, R represents the scaled distance between (0, 0) and (X, Y ). Figure 2a shows the ω signal and the sideband regions.The signal region is defined as 0.750 GeV/c 2 < M π + π − π 0 < 0.810 GeV/c 2 , while the sideband regions as 0.620 GeV/c 2 < M π + π − π 0 < 0.720 GeV/c 2 and 0.840 Figures 2c and 2d show the Dalitz plots of the signal MC sample and non=ω background events, respectively, where ω signals concentrate at the origin, while non-ω background events distribute uniformly.Therefore, we require the R value to be less than 0.9 to further suppress the backgrounds from non-ω contribution.To further identify signal candidates, the beam constrained mass M BC and the energy difference ∆E variables are used, defined as ) where E beam is the beam energy, and ⃗ p Λ + c and E Λ + c are momentum and energy of the Λ + c candidate in the rest frame of e + e − , respectively.For a correctly reconstructed Λ +

Branching Fraction Measurement
The branching fractions of signal modes are determined by a simultaneous maximum likelihood fit to the data sets at seven energy points.The branching fractions of Λ + c → pη and Λ + c → pω at each energy point (i indicates the i-th energy point, from 1 to 7) is determined by where Λ− c pairs, ε i is the detection efficiency estimated according to the signal MC sample, and B inter represents the branching fractions of intermediate states, including η → γγ, η → π + π − π 0 , π 0 → γγ, and ω → π + π − π 0 , from the PDG [46].The numbers of Λ + c Λ− c pairs for each energy point, which are listed in Table 3, are calculated by where L i and σ i denote the luminosity [35] and cross section [53], respectively.4.1 Λ + c → pη To determine the signal yield, a 2D unbinned maximum likelihood fit is performed on the 2D distribution of M BC versus ∆E for each energy point.Figure 3a shows 2D distributions from all data sets.The signal shape obtained from the signal MC sample is convolved with a 2D Gaussian function to account for the resolution difference between data and MC simulation.The signal shapes among seven energy points are obtained separately.The parameters of the 2D Gaussian are fixed in the fit to the ones obtained from the control sample of D + → ηπ + taken at 3.773 GeV, because of the similar final states as the signal mode.The background shape is modeled as a product of an ARGUS function [54], denoted as f ARGUS , and a second order Chebyshev polynomial function, denoted as P Cheb .The parameters of f ARGUS , P Cheb , and B are left floating in the simultaneous fit.The last one is shared by all energy points while the others are allowed to be different.Figures 3b and 3c    pη γγ and Λ + c → pη 3π , respectively.The branching fractions obtained via the two decay modes are consistent and are found to be (1.55 ± 0.13 stat ) × 10 −3 and (1.64 ± 0.21 stat ) × 10 −3 , with a statistical significance greater than 10σ and greater than 5σ, respectively.The statistical significance is evaluated by −2 ln L 0 /L max , where L max is the maximum likelihood obtained from the nominal fit and L 0 is the maximum likelihood of the fit without including the signal component.The efficiencies and signal yields from simultaneous and separate fits are given in Table 3, where the yields from separate fits are in agreement with the yields from simultaneous fits.

Λ +
c → pω A 1D unbinned maximum likelihood fit is simultaneously performed to the M BC distributions for data selected in the ω signal and sideband regions at seven energy points, as shown in Figure 4, to estimate the branching fraction of Λ + c → pω.The 2D fit is not used because the q q background would be too high to control if the restriction on the ∆E is released.The signal shape is obtained from signal MC simulations convolved with a Gaussian function.The signal shapes among seven energy points are obtained separately.The parameters of the Gaussian function are fixed according to the control sample D + → ωπ + taken at 3.773 GeV, because of the similar final states as the signal mode.The background is composed of two parts: the combinatorial part and the remaining non-ω peaking background.The combinatorial background is modeled by an ARGUS function, while the non-ω peaking background is modeled with the shape derived from the Λ + c → pπ + π − π 0 MC sample, where its yields are constrained in the simultaneous fit to the Λ + c yields in the ω sideband regions in Figure 4b.Here, according to studies on MC simulations, other non-ω backgrounds apart from Λ + c → pπ + π − π 0 are negligible after the requirements described in Sec. 3.According to the fit to the M π + π − π 0 spectrum, as shown in Figure 2a, the ratio between the number of non-ω peaking background events in Figure 4a and the corresponding number of events in ω sideband regions in Figure 4b is fixed to 0.468±0.003.Therefore, the fitted branching fraction of Λ + c → pω is given as (1.11 ± 0.20 stat ) × 10 −3 , with a statistical significance of 5.7σ.The efficiencies and signal yields from simultaneous and separate fits are given in Table 3, where the yields from separate fits are in agreement with the yields from simultaneous fits.A potential peak is seen around 2.26 GeV in the M BC distribution in Figure 4a, which is not observed in MC simulations and the significance of the peak is merely 2.4σ, so we treat it as statistical fluctuation in the nominal model.The systematic uncertainty caused by the peak will be discussed in Sec. 5.

Systematic Uncertainties
Table 4: Systematic uncertainties from multiple sources of three signal channels in percentage.Items in bold are treated as fully correlated between Λ + c → pη γγ and Λ + c → pη 3π .The systematic uncertainties for Λ + c → pη and Λ + c → pω are summarized in Table 4 and are described below.I. Proton tracking and PID.We select a control sample of J/ψ → ppπ + π − to study the proton tracking and PID efficiencies as functions of proton momenta based on tag-and-based method.The relative difference between data and MC simulations is estimated and used to reweight the nominal efficiencies in the dimensions of the signal momenta to obtain alternative efficiency.The relative difference between the nominal efficiency and the alternative one is taken as the systematic uncertainty.This method of estimating systematic uncertainty has been used in Refs.[17,55].The systematic uncertainties for proton tracking are evaluated to be 0.4% for the pη γγ and pη 3π modes, and 0.7% for the pω mode; the PID uncertainties are 0.1% for the pη γγ and pη 3π modes, and 0.2% for the pω mode.
II. Proton V r requirement.A control sample of Λ + c → pK − π + is used to study the systematic uncertainty of the V r requirement for proton candidates.The difference of the estimated efficiency-corrected signal yields with and without the V r requirement is found to be 1.2%, which is taken as the systematic uncertainty.III.Charged π tracking and PID.The uncertainties due to charged π tracking and PID are studied based on a control sample J/ψ → π + π − π 0 .We adopt the same method of estimating the systematic uncertainty as for proton tracking and PID.Accordingly, the relevant systematic uncertainties are assigned as 0.6% (0.2%), and 0.6% (0.3%) for tracking (PID) efficiencies in the pη 3π and pω modes, respectively.
V. Further η and ω requirements.The photon candidates from η γγ and π 0 are chosen with additional shower requirements on the lateral moment and E 3×3 /E 5×5 .We further apply the helicity angle requirement, mass window requirement and the R value selection on η γγ , η 3π and ω candidates, respectively.The efficiencies on these additional requirements are studied by control samples of D + → η γγ π + , D + → η 3π π + , and D 0 → K 0 S ω.The differences between efficiency-corrected signal yields of the control samples with and without these requirements are taken as systematic uncertainties.The systematic uncertainty due to shower requirements and cos θ decay criteria for pη γγ is estimated to be 0.8% in total, and that due to shower requirements and M π + π − π 0 mass requirement for pη 3π is 0.2%.The systematic uncertainties due to shower requirements, R value requirement and M π + π − π 0 mass window for pω are given as 0.9%, 0.5% and 1.1%, respectively.VI.Vetoing Σ + , K 0 S , and Λ.We use control samples Λ + c → Σ + π + π − , Λ + c → pK 0 S π 0 , and Λ + c → Λπ + π 0 to study the data and MC difference in the M pπ 0 , M π + π − and M pπ − spectra.The difference is described by a Gaussian function which is used to correct the signal MC sample.The relative changes on the detection efficiencies before and after correction are assigned as systematic uncertainties, which are 1.8% and 0.7% for pη 3π and pω, respectively.VII.∆E requirement.Using the control sample of D + → ωπ + , the resolution in the ∆E distribution in the pω mode is studied.The relative change of the detection efficiencies before and after correcting the MC-simulated resolution effect is assigned as the systematic uncertainty, which is 0.1%.
VIII.Non-η and non-ω contribution.For the pη 3π mode, a potential non-η contribution from Λ + c → pπ + π − π 0 is found to be negligible, according to a control sample in the η 3π sideband region.For the pω mode, the ratio of the sizes of Λ + c → pπ + π − π 0 in the ω signal and sideband regions is varied to understand the potential bias on the estimation of non-ω contributions.An alternative method to calculate the ratio is implemented by counting the surviving numbers of events in the π + π − π 0 signal and sideband regions of the Λ + c → pπ + π − π 0 MC sample.The resultant difference of the branching fractions with the new ratio from the nominal result is assigned as systematic uncertainty, which is 0.7%.
c are 1.6%, which include the uncertainties on the luminosity and cross section [35,53] of each data set.XI MC statistics.The statistical uncertainties of the estimated efficiencies from the signal MC samples are both 0.1%.XII Signal model.In the nominal analysis, the efficiencies are calculated by using a PHSP MC sample.The alternative efficiencies are estimated with the weighted MC samples according to the joint angular distributions for pη and pω, which are written as and respectively.Here, α 0 and ∆ 0 denote the parameters related to the polarization of e + e − → Λ + c Λ+ c .The parameters θ 0 and θ 1 denote the helicity angles of the corresponding process e + e − → Λ + The maximum differences are considered as systematic uncertainties, which are 0.2%, 0.1%, and 0.7% for pη γγ , pη 3π and pω modes, respectively.XIII Fitting model.The systematic uncertainty from the fitting model results from the signal and background shapes.To estimate the potential effects, we vary the smearing Gaussian parameters within the uncertainties from the control samples, and vary the parameter E beam of the ARGUS function by ±0.2 MeV.Six thousand pseudo data sets are generated randomly, where for each pseudo data set these parameters are varied randomly.The pull distribution of the fitted branching fractions in the pseudo data sets indicates a relative shift of 0.2%, 0.5%, and 2.2% on the results of pη γγ , pη 3π and pω modes, respectively, which are assigned as systematic uncertainties.For Λ + c → pω, the peak around 2.26 GeV in the M BC distribution in Figure 4a is insignificant in data and not reproduced in MC simulations.However, the peak could be caused by some unknown process in truth.To estimate the potential systematic uncertainty in the nominal fit, an alternative fitting model is considered, which assumes the peak around 2.26 GeV with a Gaussian function.Two thousands toy data samples are generated based on the alternative fitting model.We fit the toy data samples with the nominal model, and the deviation of the pull distribution is taken as systematic uncertainty, which is 4.7%.
We combine the measured branching fractions of Λ + c → pη γγ and Λ + c → pη 3π considering correlations of systematic uncertainties utilizing BLUE (Best Linear Unbiased Estimate) [59].We assume the uncertainties from proton tracking, PID, V r requirement and total number of Λ + c Λ− c pairs are 100% correlated, and the other sources of systematic uncertainties are uncorrelated.The branching fraction of Λ + c → pη is estimated to be (1.57± 0.11 stat ± 0.04 syst ) × 10 −3 .For Λ + c → pω, we add the systematic uncertainties in quadrature, and the branching fraction is calculated to be (1.11± 0.20 stat ± 0.07 syst ) × 10 −3 .

Summary
Based on e + e − collision samples with an integrated luminosity of 4.5 fb −1 collected with the BESIII detector at seven energy points between 4.600 and 4.699 GeV, the branching fractions of Λ + c → pη and Λ + c → pω are measured using the single tag method, and they are found to be (1.57± 0.11 stat ± 0.04 syst ) × 10 −3 and (1.11 ± 0.20 stat ± 0.07 syst ) × 10 −3 , with a statistical significance greater than 10σ and 5.7σ, respectively.Our results are consistent with previous measurements, as given in Table 1.The result of Λ + c → pη is the most precise single measurement to date.The results allow more stringent tests of various phenomenological models, as listed in Table 1, where early calculations in Refs.[13,14,33] are confirmed as inconsistent with experimental measurements.

Figure 2 :
Figure 2: (a) Fit to the M π + π − π 0 distribution, where the blue dashed line denotes the ω signal events, and the magenta dashed line are non-ω contributions.The red arrows mark the signal region, and the blue arrows mark the sideband regions.(b) ∆E distribution from data, where the red arrows mark the ∆E window requirement.(c) and (d) Dalitz plots after all requirements except the R requirement in the dimensions of X and Y , as defined in Eqs.(3.1-3.2), of the ω signal MC sample and non-ω background events from the M π + π − π 0 sideband regions, respectively.The orange lines tracing the borders of Dalitz plots are the kinematic limits of ω → π + π − π 0 .The yellow curve marks the position of the R requirement.
show one-dimensional (1D) projections of the fitting results of Λ + c →

Figure 3 :
Figure 3: (a) 2D distributions of data summing over all the energy points.(b, c) 1D projections of 2D simultaneous fits of Λ + c → pη γγ and Λ + c → pη 3π summing over the corresponding distributions at the seven energy points, respectively.Black points with error bars are data, green dashed lines are the signal shapes, blue dashed lines are background, and red solid lines are the total fitting results.

Figure 4 :
Figure 4: Simultaneous fit to the M BC distributions for data in the ω signal (a) and sideband regions (b) as defined in Figure 2a, summing over the corresponding distributions at seven energy points.Black dots with error bars are data, red dashed lines are signal shapes, violet dashed lines represent the non-ω peaking background contributions of Λ + c → pπ + π − π 0 , blue dashed lines are combinatorial background, and green solid lines are the total fitting results.

Table 2 :
The fractions of events with more than one candidate at different energy points in percentage.