Measurement of branching fractions of $\Lambda_{c}^{+}$ decays to $\Sigma^{+}K^{+}K^{-}$, $\Sigma^{+}\phi$ and $\Sigma^{+}K^{+}\pi^{-}(\pi^{0})$

Weak decays of charmed baryons provide useful information for understanding the interplay of weak and strong interactions, complementary to the information obtained from charmed mesons Klein:1989tj ; Asner:2008nq . The lightest charmed baryon $\Lambda_{c}^{+}$ is the cornerstone of all charmed baryon spectroscopy, and the measurement of the properties of the $\Lambda_{c}^{+}$ provides essential inputs for studying other heavy-quark baryons, such as the singly- and doubly-charmed baryons Yu:2017zst and $b$-baryons Rosner:2012gj . Contrary to the significant progress made in the studies of the heavy meson decay, advancement in the arena of heavy baryons, on both theoretical and experimental sides, has been very slow Cheng:2015iom . Since 2014, much progress has been made by BESIII, Belle and LHCb experiments in studying $\Lambda_{c}^{+}$ decays. This includes measurements of absolute branching fractions (BFs) and observation of new $\Lambda_{c}^{+}$ decay modes  BESIII:lmdenv ; BESIII:hadron ; BESIII:phh ; BESIII:lmdmunv ; BESIII:sigmapipi ; BESIII:nkspi ; BESIII:LmdX ; BESIII:eX ; BESIII:xik ; BESIII:Lmdpieta ; BESIII:KsX ; BESIII:pkseta ; BESIII:npi ; Belle:pkpi ; Belle:pkpi_dcs ; Belle:sigmapipi ; Belle:ppi0peta ; Belle:pomega ; Li:2021iwf , measurements of weak decay asymmetries of $\Lambda_{c}^{+}$ decays  BESIII:2019odb ; Belle:2022uod , determination of $\Lambda_{c}^{+}$ spin BESIII:2020kap , lifetime Belle-II:2022ggx ; LHCb:2019ldj and search for the rare $\Lambda_{c}^{+}$ decays LHCb:pomega . All of this supplies rich data for improving theoretical models Cheng:2021qpd . The external $W$-emission process is factorizable, but internal $W$-emission and $W$-exchange processes are not, and non-factorizable contributions have sizeable theoretical uncertainties. The $W$-exchange process is particularly relevant for charmed baryons, where it is not suppressed by helicity and color as in the meson case. Therefore, more precise measurements of the $W$-exchange decays of the $\Lambda_{c}^{+}$ play an important role in the identification of the non-factorizable contribution in different theoretical calculations Cheng:2021qpd ; Belle:2001hyr . The two decay modes $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\phi$ and $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}K^{-}$ are expected to proceed entirely through non-factorizable $W$-exchange diagrams Cheng:1993gf , as shown in Figures 1 and  1.

Complementary studies of charmed baryons, such as the problem of the $\Xi_{c}^{+}$ lifetime Guberina:2002fz , are considered in the framework of Heavy-Quark Expansion Khoze:1983yp , which can be expressed in terms of measurable inclusive quantities of the other two charmed baryons belonging to the same SU(3) flavor multiplet in a model-independent way. In such a treatment, the inclusive decay rates of singly Cabibbo-suppressed decay modes play a prominent role. Two such modes, $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}$ and $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}\pi^{0}$, are expected to proceed entirely through internal and external $W$-emission diagrams as shown in Figures 1 and 1, respectively. Measurements of the $\Lambda_{c}^{+}$ singly Cabibbo-suppressed decay rates can improve our understanding of the lifetime hierarchy LHCb:2018nfa ; LHCb:2021vll ; Cheng:2021vca . The BFs of both $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}$ and $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}K^{-}$ are predicted to be $(0.025\pm 0.03)\%$ Cen:2019ims , and $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\phi$ is predicted to be $(0.39\pm 0.06)\%$ Hsiao:2019yur .

The ratios of the BFs (RBFs) of $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\phi$ and $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}K^{-}$ relative to $\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$ were measured by CLEO to be $0.069\pm 0.023\pm 0.016$ and $0.070\pm 0.011\pm 0.011$ respectively CLEO:1993fhs . In 2002, Belle reported the RBFs of $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}K^{-}$, $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\phi$ and $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}$ relative to $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\pi^{+}\pi^{-}$, which are $0.076\pm 0.007\pm 0.009$, $0.085\pm 0.012\pm 0.012$ and $0.047\pm 0.011\pm 0.008$, respectively Belle:2001hyr .

In this work, we present the measurements of the RBFs of the signal decays $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}K^{-}$, $\Sigma^{+}K^{+}\pi^{-}$, $\Sigma^{+}\phi$, $\Sigma^{+}K^{+}K^{-}$(non-$\phi$) and $\Sigma^{+}K^{+}\pi^{-}\pi^{0}$, relative to the reference decay $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\pi^{+}\pi^{-}$, by analyzing 4.5 fb^-1 data taken at the center-of-mass energies $\sqrt{s}=4.600$, 4.612, 4.628, 4.641, 4.661, 4.682 and 4.699 GeV BESIII:2022ulv with the BESIII detector at the BEPCII collider. Throughout this paper, charge-conjugate modes are implicitly assumed. In Sect. 2, the BESIII detector and the data samples used in this analysis are described. The event selection is introduced in Sect. 3 and the determination of the BF is presented in Sect. 4. The systematic uncertainties are discussed in Sect. 5. Finally, Sect. 6 summarizes the results.

The BESIII detector Ablikim:2009aa records symmetric $e^{+}e^{-}$ collisions provided by the BEPCII storage ring Yu:IPAC2016-TUYA01 , which operates with a peak luminosity of $1\times 10^{33}$ cm^-2s^-1 in the center-of-mass energy range from 2.0 to 4.95 GeV. BESIII has collected large data samples in this energy region Ablikim:2019hff . 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 $1~{}{\rm GeV}/c$ 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 etof ; etof2 ; etof3 . About 87% of the data used were collected after this upgrade.

Simulated data samples produced with the geant4-based geant4 MC package BOOST bes:boost , which includes the geometric and material description of the BESIII detector geo2 ; detvis 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 ref:kkmc ; ref:kkmc2 . 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 ref:evtgen using BFs either taken from the Particle Data Group (PDG) pdg:2020 , when available, or otherwise estimated with lundcharm ref:lundcharm ; ref:lundcharm2 . Final state radiation (FSR) from charged final state particles is incorporated using the photos package photos . For the MC production of the $e^{+}e^{-}\rightarrow\Lambda_{c}^{+}\Lambda_{c}^{-}$ 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 $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\pi^{+}\pi^{-}$, a partial-wave analysis is performed to obtain the amplitude model.

In this work, both the signal and reference $\Lambda_{c}^{+}$ decays are fully reconstructed. Charged tracks detected in the MDC are required to be within a polar angle ($\theta$) range of $|\rm{cos\theta}|<0.93$, where $\theta$ 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 (d$E$/d$x$) and the flight time in the TOF to form likelihoods $\mathcal{L}(h)~{}(h=p,K,\pi)$ for each hadron $h$ hypothesis. Tracks are identified as protons when the proton hypothesis has the greatest likelihood ($\mathcal{L}(p)>\mathcal{L}(K)$ and $\mathcal{L}(p)>\mathcal{L}(\pi)$), while charged kaons and pions are identified by comparing the likelihoods for the kaon and pion hypotheses, $\mathcal{L}(K)>\mathcal{L}(\pi)$ and $\mathcal{L}(\pi)>\mathcal{L}(K)$, 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 ($|\cos\theta|<0.80$) and more than 50 MeV in the end cap region ($0.86<|\cos\theta|<0.92$). 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 $\gamma\gamma$ combinations are considered as $\pi^{0}$ candidates, and the reconstructed mass $M(\gamma\gamma)$ is required to fall in the range of $0.115$ GeV/$c^{2}<M(\gamma\gamma)<0.150$ GeV/$c^{2}$. A kinematic fit is performed to constrain the $\gamma\gamma$ invariant mass to the known $\pi^{0}$ mass pdg:2020 , and candidates with the fit quality $\chi^{2}<200$ are retained.

The $\Sigma^{+}$ candidates are reconstructed from the combinations of $p\pi^{0}$ with an invariant mass in the range of $1.174$ GeV/$c^{2}<M(p\pi^{0})<1.200$ GeV/$c^{2}$. This requirement corresponds to approximately $\pm 3\sigma$ (standard deviations) around the known $\Sigma^{+}$ mass pdg:2020 . To reject the possible backgrounds for $\Lambda^{+}_{c}\to\Sigma^{+}K^{+}\pi^{-}$ and $\Sigma^{+}\pi^{+}\pi^{-}$ modes including $\Lambda\rightarrow p\pi^{-}$ in the final states, $M(p\pi^{-})$ is required to be outside the range (1.11, 1.12) GeV/$c^{2}$. In addition, to remove the $K^{0}_{S}$ decays in the mode $\Lambda_{c}^{+}\to\Sigma^{+}\pi^{+}\pi^{-}$, events with $M(\pi^{+}\pi^{-})$ in the range (0.48, 0.52) GeV/$c^{2}$ are vetoed.

To improve the signal purity, the energy difference $\Delta E=E_{\rm cand}-E_{\rm beam}$ for $\Lambda_{c}^{+}$ candidates are required to satisfy a mode-dependent $\Delta E$ requirement shown in Table 2. These ranges are obtained by optimizing signal yield significance with the inclusive MC sample. Here $E_{\rm cand}$ is the total reconstructed energy of the $\Lambda_{c}^{+}$ candidate and $E_{\rm beam}$ is the beam energy. Only one candidate with the minimal $|\Delta E|$ is accepted. The $\Lambda^{+}_{c}$ signal is identified using the beam constrained mass $M_{\rm BC}=\sqrt{E_{\rm beam}^{2}/c^{4}-p^{2}/c^{2}}$, where $p$ is the measured $\Lambda_{c}^{+}$ momentum in the center-of-mass system of the $e^{+}e^{-}$ collision. After the event selection, there is no obvious peaking background in the $M_{\rm BC}$ distribution for each tag mode, as shown in Figure 2.

The RBF between the signal and reference modes is calculated with

where $\varepsilon_{i}$ and $\varepsilon_{j}$ 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 $\mathcal{B}_{\rm{inter}}$ is the product of the BFs of the intermediate states ($\Sigma^{+}\rightarrow p\pi^{0}$, $\pi^{0}\rightarrow\gamma\gamma$, and $\phi\to K^{+}K^{-}$ for $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\phi$), and the charge-conjugate channel is included in the simulation. The observables in Equation (1) are determined as follows.

• •

  Reference mode $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\pi^{+}\pi^{-}$

  To obtain the yield of the reference mode, an unbinned maximum likelihood (UML) fit is performed on the $M_{\rm BC}$ 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 ARGUS:1990hfq

  $$f^{\mathrm{ARGUS}}\propto M_{\mathrm{BC}}\sqrt{1-\left(\frac{M_{\mathrm{BC}}\cdot c^{2}}{E_{\mathrm{beam}}}\right)^{2}}\,\,e^{a\left(1-\frac{M_{\mathrm{BC}}\cdot c^{2}}{E_{\mathrm{beam}}}\right)^{2}},$$

  (2)

  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.

  Decay mode	4.600 GeV	4.612 GeV	4.628 GeV	4.641 GeV	4.661 GeV	4.682 GeV	4.699 GeV
  $\Sigma^{+}\pi^{+}\pi^{-}$	1123$\pm$47	200$\pm$21	1003$\pm$46	1026$\pm$48	1025$\pm$48	3132$\pm$85	942$\pm$47
  $\Sigma^{+}K^{+}K^{-}$	14.1$\pm$1.5	2.3$\pm$0.2	11.7$\pm$1.3	13.0$\pm$1.4	12.1$\pm$1.4	43.0$\pm$4.5	14.1$\pm$1.4
  $\Sigma^{+}\phi$	14.3$\pm$2.8	2.4$\pm$0.5	12.7$\pm$2.5	13.2$\pm$2.7	14.2$\pm$2.7	47.2$\pm$8.9	14.9$\pm$2.9
  $\Sigma^{+}K^{+}K^{-}$(non-$\phi$)	9.2$\pm$1.7	1.6$\pm$0.3	8.1$\pm$1.5	8.6$\pm$1.6	9.0$\pm$1.6	29.0$\pm$4.0	9.1$\pm$1.6
  $\Sigma^{+}K^{+}\pi^{-}$	30.5$\pm$3.6	5.8$\pm$0.7	29.3$\pm$3.5	29.6$\pm$3.5	26.1$\pm$3.4	80.3$\pm$9.5	22.3$\pm$2.6

  Decay mode	4.600 GeV	4.612 GeV	4.628 GeV	4.641 GeV	4.661 GeV	4.682 GeV	4.699 GeV
  $\Sigma^{+}\pi^{+}\pi^{-}$	26.12$\pm$0.05	25.21$\pm$0.05	24.90$\pm$0.05	24.96$\pm$0.05	24.78$\pm$0.05	24.44$\pm$0.05	24.23$\pm$0.05
  $\Sigma^{+}K^{+}K^{-}$	4.22$\pm$0.02	3.92$\pm$0.02	3.98$\pm$0.02	4.11$\pm$0.02	4.30$\pm$0.02	4.53$\pm$0.02	5.19$\pm$0.02
  $\Sigma^{+}\phi$	3.66$\pm$0.02	3.38$\pm$0.02	3.46$\pm$0.02	3.55$\pm$0.02	3.75$\pm$0.02	4.02$\pm$0.02	4.19$\pm$0.02
  $\Sigma^{+}K^{+}K^{-}$(non-$\phi$)	4.81$\pm$0.02	4.50$\pm$0.02	4.53$\pm$0.02	4.70$\pm$0.02	4.88$\pm$0.02	5.07$\pm$0.02	6.24$\pm$0.02
  $\Sigma^{+}K^{+}\pi^{-}$	16.30$\pm$0.04	15.78$\pm$0.04	15.20$\pm$0.04	15.06$\pm$0.04	14.56$\pm$0.04	14.75$\pm$0.04	14.20$\pm$0.04
  $\Sigma^{+}K^{+}\pi^{-}\pi^{0}$	5.16$\pm$0.02	4.72$\pm$0.02	4.55$\pm$0.02	4.59$\pm$0.02	4.73$\pm$0.02	4.72$\pm$0.02	4.74$\pm$0.02

  

  

  

  

  

• •

  Signal mode $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}$

  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 at various energy points, where the signal yields can be derived from Equation 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 2. The summed fit to data from all of the energy points is shown in Figure 3. 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 $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}K^{-}$

  To separate the $\phi$ contribution for $\Sigma^{+}K^{+}K^{-}$ mode, a two-dimensional simultaneous UML fit is performed on the $M_{\rm BC}$ vs the $M_{K^{+}K^{-}}$ distributions, in which common RBF values are estimated. Four components ($\Sigma^{+}\phi$, $\Sigma^{+}K^{+}K^{-}$(non-$\phi$), $\phi$ background and non-$\phi$ background) are considered in this fit, as shown in Figures 3 and 3. All the signal shapes are extracted from MC samples and convolved with a Gaussian resolution function with floating mean and width. In the $M_{\rm BC}$ 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-$\phi$ process and use the $\phi$ line-shape distribution extracted from the $\Lambda^{+}_{c}\to\Sigma^{+}\phi$ signal MC sample to describe the $\phi$ background.

• •

  Signal mode $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}\pi^{0}$

  The simultaneous UML fit is also performed for $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}\pi^{0}$, as shown in Figure 3. 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

  $$L(\rm{BF})=\int_{-1}^{1}\emph{L}^{\rm{stat}}\,[(1+\Delta)\,\rm{BF}]\,\rm exp(-\frac{\Delta^{2}}{2\sigma^{2}_{syst}})\,d\Delta.$$

  (3)

  Here, $L(\rm{BF})$ is the likelihood expression of BF, $L^{\rm{stat}}$ is the statistical likelihood given by the data without considering the systematic uncertainties when taking the nominal BF obtained, $\Delta$ is the relative deviation of the estimated BF from the nominal value and $\sigma_{\rm{syst}}$ is the total systematic uncertainty given in Table 5. The likelihood curve calculated according to Equation 3 is shown in Figure 4. The upper limit on the BF of $\Lambda^{+}_{c}\to\Sigma^{+}\pi^{-}\pi^{+}\pi^{0}$ 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.

  

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 $\Sigma^{+}$ 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.

• •

  PID and $\pi^{0}$ reconstruction efficiencies. The uncertainties associated with PID efficiencies are estimated to be 1.0% for each charged track by studying the control samples of $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, $K^{+}K^{-}\pi^{+}\pi^{-}$ and $p\bar{p}\pi^{+}\pi^{-}$ based on data taken at energies above $\sqrt{s}=4.0$ GeV. However, the PID uncertainty for the $p(\bar{p})$ common to signal and reference modes cancels, as does that for any common $\pi^{\pm}$. The uncertainty of $\pi^{0}$ reconstruction efficiency is assigned to be 1.0% per $\pi^{0}$ by studying the control sample of $e^{+}e^{-}\rightarrow\omega\pi^{0}$ process.

• •

  $\Delta E$ 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 $\Delta E$. The resolution difference is extract by using signal MC shape smeared with the Gaussian function to fit the $\Delta E$ distribution in data.

• •

  $M_{BC}$ fit. The uncertainty in the $M_{\rm BC}$ fit is mainly due to free parameters of the Gaussian and ARGUS functions and the background description. The relevant systematic 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 $\pm 0.005$ GeV/$c^{2}$. The largest result change is assigned as the systematic uncertainty. For $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\phi$ process, due to that a two-dimensional maximum likelihood fit is performed on the $M_{\rm BC}$ and $M_{K^{+}K^{-}}$ distribution, the uncertainty of $M_{K^{+}K^{-}}$ fit is also considered by studying the difference of $\phi$ signal shape convolved with or without a Gaussian resolution function. In addition, the shape of the backgrounds from $\Lambda^{+}_{c}$ inclusive decays can not be different from the ARGUS function. To account for this discrepancy, an additional component of the $\Lambda^{+}_{c}$ decay backgrounds, which is extracted from the $\Lambda^{+}_{c}$ inclusive MC samples, is included in the $M_{\rm{BC}}$ fit. The resultant changes on the RBF are taken as the systematic uncertainties. For $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}\pi^{0}$ process, an alternative fit to data with $M_{\rm{BC}}$ 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 pdg:2020 are included in the signal simulation. Examples include $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\phi$ and $\Lambda_{c}^{+}\rightarrow\Xi^{*0}K^{+}$ ($\Xi^{*0}\rightarrow\Sigma^{+}K^{-}$) processes for $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}K^{-}$ and $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{*}(892)^{0}$ ($K^{*}(892)^{0}\rightarrow K^{+}\pi^{-}$) for $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}$. In the nominal analysis, the BFs that are used in the generator are taken from the PDG pdg:2020 . Their effects on the new BF measurements are estimated by varying the input BF of the intermediate decay by $\pm 1\sigma$ in the generator. The relative efficiency deviation is taken as the uncertainty.

• •

  Reference mode. For the reference mode $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\pi^{+}\pi^{-}$, the TF-PWA PWA 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 $\pi^{0}\rightarrow\gamma\gamma$, $\phi\rightarrow K^{+}K^{-}$ and $\Sigma^{+}\rightarrow p^{+}\pi^{0}$ are used as inputs in the baseline analysis, and their uncertainties pdg:2020 are propagated as the systematic uncertainty.

Based on 4.5 fb^-1 data taken at $\sqrt{s}=4.600$ to 4.699 GeV with the BESIII detector at the BEPCII collider, the non-factorizable $W$-exchange-only processes, $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\phi$, $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}K^{-}$(non-$\phi$) and the non-factorizable $W$-emission processes $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}$ and $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}\pi^{0}$ have been studied. The BF of $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}\pi^{0}$ relative to $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\pi^{+}\pi^{-}$ is measured for the first time. The precisions of BFs for other channels relative to $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\pi^{+}\pi^{-}$ are improved. Combining with the world average $\mathcal{B}(\Lambda_{c}^{+}\rightarrow\Sigma^{+}\pi^{+}\pi^{-})=(4.50\pm 0.25)\%$ pdg:2020 , the BFs of the aforementioned decays are obtained. Table 6 shows the comparison of our results with the PDG values pdg:2020 and the Belle results Belle:2001hyr . The uncertainties of the BFs of $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}K^{-}$ and $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\phi$ are comparable to those in references  Belle:2001hyr ; pdg:2020 . For $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}$, the precision of the BF is improved by a factor of two. The theoretical predictions of the BFs of $\Lambda_{c}^{+}\rightarrow\Sigma^{+}\phi$, $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}K^{-}$ and $\Lambda_{c}^{+}\rightarrow\Sigma^{+}K^{+}\pi^{-}$ in reference Cen:2019ims ; Hsiao:2019yur differ from our results by about $2\sigma$ and our results will be helpful to correct the theoretical model. The combined results are essential to understand the non-factorizable $W$-exchange and $W$-emission contributions to the decays of $\Lambda_{c}^{+}$.