Measurements of Weak Decay Asymmetries of $\Lambda_c^+\to pK_S^0$, $\Lambda\pi^+$, $\Sigma^+\pi^0$, and $\Sigma^0\pi^+$

TL;DR

This study measures decay asymmetries and transverse polarization of $\Lambda_c^+$ baryons in four decay modes using BESIII data, providing new insights and improved precision, including first-time measurements for some modes.

Contribution

First-time measurement of $\Lambda_c^+$ decay asymmetries and polarization in multiple modes with improved precision, using a full angular analysis of BESIII data.

Findings

Non-zero $\Lambda_c^+$ transverse polarization observed at 2.1$\sigma$ significance.

Decay asymmetry parameters measured with improved precision for $\Lambda o pK_S^0$, $\Lambda o ext{mode}$, $\Sigma^+ o ext{mode}$, and $\Sigma^0 o ext{mode}$.

First measurements of asymmetry parameters for $pK_S^0$ and $\Sigma^0 o ext{mode}$ modes.

Abstract

Using $e^+e^-\to\Lambda_c^+\bar\Lambda_c^-$ production from a 567 pb$^{-1}$ data sample collected by BESIII at 4.6 GeV, a full angular analysis is carried out simultaneously on the four decay modes of $\Lambda_c^+\to pK_S^0$, $\Lambda \pi^+$, $\Sigma^+\pi^0$, and $\Sigma^0\pi^+$. For the first time, the $\Lambda_c^+$ transverse polarization is studied in unpolarized $e^+e^-$ collisions, where a non-zero effect is observed with a statistical significance of 2.1$\sigma$. The decay asymmetry parameters of the $\Lambda_c^+$ weak hadronic decays into $pK_S^0$, $\Lambda\pi^+$, $\Sigma^+\pi^0$ and $\Sigma^0\pi^+$ are measured to be $0.18\pm0.43(\rm{stat})\pm0.14(\rm{syst})$, $-0.80\pm0.11(\rm{stat})\pm0.02(\rm{syst})$, $-0.57\pm0.10(\rm{stat})\pm0.07(\rm{syst})$, and $-0.73\pm0.17(\rm{stat})\pm0.07(\rm{syst})$, respectively. In comparison with previous results, the measurements for the $\Lambda\pi^+$ and $\Sigma^+\pi^0$ modes are consistent but with improved precision, while the parameters for the $pK_S^0$ and $\Sigma^0\pi^+$ modes are measured for the first time.

Tables

Table 1. Table 1: Comparisons between different theoretical calculations and experimental measurements.

${\Lambda }_c^{+}\to$		$p{K}_S^0$		$\Lambda {\pi }^{+}$		${\Sigma }^{+}{\pi }^0$		${\Sigma }^0{\pi }^{+}$
${\alpha }_{BP}^{{\Lambda }_c^{+}}$	Predicted	$-1.0$ Korner:1992wi ,	$\mathrm{\hspace{0.17em}0.51}$ Xu:1992vc	$-0.70$ Korner:1992wi ,	$-0.67$ Xu:1992vc	$\mathrm{\hspace{0.17em}0.71}$ Korner:1992wi ,	$\mathrm{\hspace{0.17em}0.92}$ Xu:1992vc	$\mathrm{\hspace{0.17em}0.70}$ Korner:1992wi ,	$\mathrm{\hspace{0.17em}0.92}$ Xu:1992vc
		$-0.49$ Cheng:1993gf ,	$-0.90$ Cheng:1993gf	$-0.95$ Cheng:1993gf ,	$-0.99$ Cheng:1993gf	$\mathrm{\hspace{0.17em}0.79}$ Cheng:1993gf	$-0.49$ Cheng:1993gf	$\mathrm{\hspace{0.17em}0.78}$ Cheng:1993gf ,	$-0.49$ Cheng:1993gf
		$-0.49$ Cheng:1991sn ,	$-0.97$ Ivanov:1997ra	$-0.96$ Cheng:1991sn ,	$-0.95$ Ivanov:1997ra	$\mathrm{\hspace{0.17em}0.83}$ Cheng:1991sn ,	$\mathrm{\hspace{0.17em}0.43}$ Ivanov:1997ra	$\mathrm{\hspace{0.17em}0.83}$ Cheng:1991sn ,	$\mathrm{\hspace{0.17em}0.43}$ Ivanov:1997ra
		$-0.66$ Zenczykowski:1993jm ,	$-0.90$ Zenczykowski:1993hw	$-0.99$ Zenczykowski:1993jm ,	$-0.86$ Zenczykowski:1993hw	$\mathrm{\hspace{0.17em}0.39}$ Zenczykowski:1993jm ,	$-0.76$ Zenczykowski:1993hw	$\mathrm{\hspace{0.17em}0.39}$ Zenczykowski:1993jm ,	$-0.76$ Zenczykowski:1993hw
		$-0.99$ Sharma:1998rd ,	$-0.91$ Datta:1995mn	$-0.99$ Sharma:1998rd ,	$-0.94$ Datta:1995mn	$-0.31$ Sharma:1998rd ,	$-0.47$ Datta:1995mn	$-0.31$ Sharma:1998rd ,	$-0.47$ Datta:1995mn
	PDG pdg			$-0.91\pm 0.15$		$-0.45\pm 0.32$
	This work	$0.18\pm 0.43\pm 0.14$		$-0.80\pm 0.11\pm 0.02$		$-0.57\pm 0.10\pm 0.07$		$-0.73\pm 0.17\pm 0.07$
${\Delta }_1^{BP}\left(\mathrm{rad}\right)$	This work			$3.0\pm 2.4\pm 1.0$		$4.1\pm 1.1\pm 0.6$		$0.8\pm 1.2\pm 0.2$
${\beta }_{BP}$	This work			${0.06}_{-0.47-0.06}^{+0.58+0.05}$		$-{0.66}_{-0.25-0.02}^{+0.46+0.22}$		${0.48}_{-0.57-0.13}^{+0.35+0.07}$
${\gamma }_{BP}$	This work			$-{0.60}_{-0.05-0.03}^{+0.96+0.17}$		$-{0.48}_{-0.42-0.04}^{+0.45+0.21}$		${0.49}_{-0.56-0.12}^{+0.35+0.07}$

Table 2. Table 2: Summary of the systematic uncertainties. A 𝐴 A , B 𝐵 B , C 𝐶 C and D 𝐷 D stand for the modes of p ​ K S 0 𝑝 superscript subscript 𝐾 𝑆 0 pK_{S}^{0} , Λ ​ π + Λ superscript 𝜋 \Lambda\pi^{+} , Σ + ​ π 0 superscript Σ superscript 𝜋 0 \Sigma^{+}\pi^{0} , and Σ 0 ​ π + superscript Σ 0 superscript 𝜋 \Sigma^{0}\pi^{+} , respectively.

Source	${\alpha }_A^{+}$	${\alpha }_B^{+}$	${\alpha }_C^{+}$	${\alpha }_D^{+}$	$\sin {\Delta }_0$	${\Delta }_1^B$	${\Delta }_1^C$	${\Delta }_1^D$
Reconstruction	0.00	0.00	0.00	0.01	0.00	0.8	0.0	0.0
${\pi }^0{\pi }^0$ veto	0.01	0.00	0.01	0.00	0.00	0.0	0.2	0.0
$\Delta E$ signal region	0.07	0.01	0.02	0.05	0.02	0.3	0.1	0.1
$M_{\mathrm{BC}}$ signal region	0.12	0.01	0.05	0.02	0.02	0.5	0.4	0.1
Bkg subtraction	0.03	0.01	0.05	0.04	0.02	0.3	0.3	0.0
Total	0.14	0.02	0.07	0.07	0.03	1.0	0.6	0.2

Equations

$${\mathcal{P}}_T(\cos {\theta }_0)\equiv \sqrt{1-{\alpha }_0^2}\cos {\theta }_0\sin {\theta }_0\sin {\Delta }_0,$$

$$\begin{array}[]{ll}\beta_{BP}&=\sqrt{1-(\alpha^{+}_{BP})^{2}}\sin\Delta_{1}^{BP},\\ \gamma_{BP}&=\sqrt{1-(\alpha^{+}_{BP}})^{2}\cos\Delta_{1}^{BP}.\\ \end{array}$$

$${\mathcal{L}}_{\mathrm{data}}=\prod _{i=1}^{N_{\mathrm{data}}}{f}_S(\vec{\xi }).$$

$$f_S(\vec{\xi })=\frac{ϵ(\vec{\xi })|M(\vec{\xi };\vec{\eta }){|}^2}{\int ϵ(\vec{\xi })|M(\vec{\xi };\vec{\eta }){|}^2\mathrm{d}\vec{\xi }},$$

$$\int ϵ(\vec{\xi })|M(\vec{\xi };\vec{\eta }){|}^2\mathrm{d}\vec{\xi }=\frac{1}{N_{\mathrm{gen}}}\sum _{k_{\mathrm{MC}}}^{N_{\mathrm{MC}}}|M({\vec{\xi }}_k;\vec{\eta }){|}^2,$$

$$\frac{dN}{d\cos {\theta }_2}\propto 1+{\alpha }_{\Lambda {\pi }^{+}({\Sigma }^{+}{\pi }^0)}^{+}{\alpha }_{\Lambda ({\Sigma }^{+})}\cos {\theta }_2.$$

$$⟨\cos {\theta }_i⟩=-\frac{1}{6}{\alpha }_{{\Sigma }^0{\pi }^{+}}^{+}{\alpha }_{\Lambda }\cos {\theta }_j,$$

$$\mathcal{W}\propto 1+{\alpha }_0{\cos }^2{\theta }_0+{\mathcal{P}}_T{\alpha }_{BP}^{+}\sin {\theta }_1\sin {\varphi }_1.$$

$$⟨\sin {\theta }_1\sin {\varphi }_1⟩=\frac{{\int }_0^{2\pi }{\int }_{-1}^1\sin {\theta }_1\sin {\varphi }_1\mathcal{W}\mathrm{d}\cos {\theta }_1\mathrm{d}{\varphi }_1}{{\int }_0^{2\pi }{\int }_{-1}^1\mathcal{W}\mathrm{d}\cos {\theta }_1\mathrm{d}{\varphi }_1}$$

Full text

The study of the lightest charmed baryon $\Lambda_{c}^{+}$ is important for the understanding of the whole charmed baryon sector. In recent years, there has been significant progress in studying the $\Lambda_{c}^{+}$, both experimentally and theoretically Amhis:2016xyh ; pdg . This provides crucial information in detailed explorations of the singly charmed baryons ($\Sigma_{c}$, $\Xi_{c}$ and $\Omega_{c}$) Lu:2016ogy ; Geng:2017mxn , and further searches or discoveries of the doubly charmed baryons ($\Xi_{cc}$ and $\Omega_{cc}$) Yu:2017zst ; Aaij:2017ueg . Moreover, as the charmed baryon is the favored weak decay final state of $b$-baryons and its properties are inputs to study $b$-baryons, improved knowledge in the charm sector can contribute substantially to understanding the properties of $b$-baryons.

Some QCD-inspired charmed baryon models that have been developed Cheng:2015iom are the flavor symmetry model Savage:1989qr , factorization model Bjorken:1988ya , pole model Cheng:1993gf , and current algebra framework  Xu:1992vc . As shown in Refs. Cheng:2015iom ; pdg , many of these models calculate $\Lambda_{c}^{+}$ decay rates in good agreement with experimental results. But the decay asymmetries predicted by these models for $\Lambda_{c}^{+}$ two-body hadronic weak decays do not agree very well.

The decay asymmetry parameter, $\alpha^{+}_{BP}$, in a weak decay $\Lambda_{c}^{+}\to BP$ ($B$ denotes a $J^{P}=\frac{1}{2}^{+}$ baryon and $P$ denotes a $J^{P}=0^{-}$ pseudoscalar meson) is defined as $\alpha^{+}_{BP}\equiv\frac{2{\rm Re}(s\cdot p)}{|s|^{2}+|p|^{2}}$, where $s$ and $p$ stand for the parity-violating $s$-wave and parity-conserving $p$-wave amplitudes in the decay, respectively. Model calculations of $\alpha^{+}_{BP}$ in $\Lambda_{c}^{+}\to pK_{S}^{0}$, $\Lambda\pi^{+}$, $\Sigma^{+}\pi^{0}$, and $\Sigma^{0}\pi^{+}$ are listed in Table 1, which shows large variations among the different models. As predictions of $\alpha^{+}_{BP}$ rely on the relative phase between the two amplitudes, the experimental measurements of the decay asymmetry parameters serve as very sensitive probes to test different theoretical models.

Experimentally, only $\alpha^{+}_{\Lambda\pi^{+}}$ and $\alpha^{+}_{\Sigma^{+}\pi^{0}}$ have been measured previously Link:2005ft ; Bishai:1995gp ; Albrecht:1991vs ; Avery:1990ya ; Bishai:1995gp . The measured value for $\alpha^{+}_{\Sigma^{+}\pi^{0}}$ is $-0.45\pm 0.32$, in contradiction with the predicted values in many theoretical models Cheng:1993gf ; Xu:1992vc ; Korner:1992wi ; Cheng:1991sn ; Ivanov:1997ra ; Zenczykowski:1993jm . Therefore, it is important to carry out independent measurements of $\alpha^{+}_{\Sigma^{+}\pi^{0}}$ to confirm the sign of $\alpha^{+}_{\Sigma^{+}\pi^{0}}$ and test these models. Moreover, $\alpha^{+}_{\Sigma^{+}\pi^{0}}$ and $\alpha^{+}_{\Sigma^{0}\pi^{+}}$ should have the same value according to hyperon isospin symmetry Sharma:1998rd , and any deviation from this expectation provides critical information on final state interactions in $\Lambda_{c}^{+}$ hadronic decays. All the models predict $\alpha^{+}_{\Lambda\pi^{+}}$ consistent with the measured values, and it is necessary to further improve the experimental precision to discriminate between them.

In previous experiments, $\Lambda_{c}^{+}$ was assumed to be unpolarized, and the decay asymmetry parameter $\alpha^{+}_{BP}$ was obtained by analyzing the longitudinal polarization from the weak two-body decay of the produced baryon $B$, such as $\Lambda\to p\pi^{-}$ and $\Sigma^{+}\to p\pi^{0}$ for $\alpha^{+}_{\Lambda\pi^{+}}$ and $\alpha^{+}_{\Sigma^{+}\pi^{0}}$, respectively. However, the hypothesis of unpolarized $\Lambda_{c}^{+}$ may not be valid. There have been observations of transverse $\Lambda$ polarization in inclusive $\Lambda$ production in $e^{+}e^{-}$ collisions at 10.58 GeV Guan:2018ckx and in $e^{+}e^{-}\to\Lambda\bar{\Lambda}$ at $J/\psi$ mass position Lamtrans , and it has been postulated that the produced $\Lambda_{c}^{+}$ could be polarized Faldt:2017yqt . Further, as the polarization of the proton in the decay $\Lambda_{c}^{+}\to pK_{S}^{0}$ is not accessible with the above method, a non-zero transverse polarization of the $\Lambda_{c}^{+}$ provides an alternative way to measure $\alpha^{+}_{pK_{S}^{0}}$ supple .

In this Letter, we investigate for the first time the transverse polarization of the $\Lambda_{c}^{+}$ baryon in unpolarized $e^{+}e^{-}$ annihilations. We present for the first time measurements of the decay asymmetry parameters in $\Lambda_{c}^{+}$ decays into $pK_{S}^{0}$, $\Lambda\pi^{+}$, $\Sigma^{+}\pi^{0}$, and $\Sigma^{0}\pi^{+}$based on a multi-dimensional angular analysis of the cascade-decay final states, which greatly improves the resulting precision. Data sample used in this analysis corresponds to an integrated luminosity of $567\,\rm{pb}^{-1}$ collected with the BESIII detector at BEPCII at center-of-mass (CM) energy of 4.6 GeV.

Since the close proximity of the CM energy to the $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ mass threshold does not allow an additional hadron to be produced, $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ are always generated in pairs, which provides a clean environment to study their decays. When one $\Lambda_{c}^{+}$ is detected, another $\bar{\Lambda}_{c}^{-}$ partner is inferred. Hence, to increase signal yields, we adopt a partial reconstruction method, in which only one $\Lambda_{c}^{+}$ is reconstructed out of all the final-state particles in an event. The charge conjugation modes are always implied in the context, unless otherwise stated explicitly.

Details of the BESIII apparatus, the software framework and the Monte Carlo (MC) simulation sample have been given in Ref. lipr . The $\Lambda_{c}^{+}$ signal candidates are reconstructed through the decays into $pK_{S}^{0}$, $\Lambda\pi^{+}$, $\Sigma^{+}\pi^{0}$ and $\Sigma^{0}\pi^{+}$. Here, the intermediate particles $K_{S}^{0}$, $\Lambda$, $\Sigma^{+}$, $\Sigma^{0}$ and $\pi^{0}$ are reconstructed via the decays $K_{S}^{0}\to\pi^{+}\pi^{-}$, $\Lambda\to p\pi^{-}$, $\Sigma^{+}\to p\pi^{0}$, $\Sigma^{0}\to\gamma\Lambda$, and $\pi^{0}\to\gamma\gamma$. The event selection criteria follow those described in Ref. lipr , unless otherwise stated explicitly. To suppress the $\Lambda_{c}^{+}\to pK_{S}^{0}$, $K_{S}^{0}\to\pi^{0}\pi^{0}$ events in the $\Sigma^{+}\pi^{0}$ candidate samples, the invariant mass of the $\pi^{0}\pi^{0}$ system is required to be outside the range $[400,550]\,\mathrm{MeV}/c^{2}$.

For each signal decay mode, the yields are obtained from a fit to the beam-constrained mass ($M_{\rm{BC}}$) distribution, $M_{\rm{BC}}\equiv\sqrt{E^{2}_{\rm{beam}}-p_{\Lambda_{c}^{+}}^{2}}$, where $E_{\rm beam}$ is the average beam energy and $p_{\Lambda_{c}^{+}}$ is the measured $\Lambda_{c}^{+}$ momentum in the CM system of the $e^{+}e^{-}$ collisions. If more than one candidate is reconstructed in the event, the one with the smallest energy difference ($|\Delta E|$) is kept, where $\Delta E\equiv E_{\Lambda_{c}^{+}}-E_{\rm{beam}}$, and $E_{\Lambda_{c}^{+}}$ is the measured total energy of the $\Lambda_{c}^{+}$ candidate.

Figure 1 shows the $M_{\rm{BC}}$ distributions for the signal candidates, where the $\Lambda_{c}^{+}$ signal peak is evident at the nominal $\Lambda_{c}^{+}$ mass. The backgrounds can be classified into two types. The Type-I backgrounds are from the true $\Lambda_{c}^{+}$ signal decays, where at least one of the final state particle candidates is wrongly assigned in reconstruction. The Type-II backgrounds correspond to combinatorial backgrounds mostly from $e^{+}e^{-}\to q\bar{q}~{}(q=u,d,s)$ processes. To evaluate the Type-I and Type-II background level, unbinned maximum likelihood fits (shown in Fig. 1) are applied to the $M_{\rm{BC}}$ spectra. The signal and Type-I background shapes, as well as the ratio of their yields, are derived from the signal MC simulation samples. These two shapes are convolved with a common Gaussian function, whose width is left free and represents the difference in resolution between data and MC simulations. The Type-II background shape is modeled by an ARGUS function Albrecht:1990am . The $\Lambda_{c}^{+}$ signal and sideband regions are chosen as $[2.278,2.294]\,\rm{GeV}/\emph{c}^{2}$ and $[2.250,2.270]\,\rm{GeV}/\emph{c}^{2}$, respectively.

The decay asymmetry parameters are determined by analyzing the multi-dimensional angular distributions, where the full cascade decay chains are considered. The full angular dependence formulae (4), (6), and (10) in Ref. supple , constructed under the helicity basis, are used in the fit. To illustrate the helicity system defined in this analysis, we take as an example the two-level cascade decay process $\Lambda_{c}^{+}\to\Lambda\pi^{+},\Lambda\to p\pi^{-}$ following the level-0 process $e^{+}e^{-}\to\gamma^{*}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$. An analogous formalism is applied to the other $\Lambda_{c}^{+}\to BP$ decays.

Figure 2 illustrates the definitions of the full system of helicity angles for the $\Lambda_{c}^{+}\to\Lambda\pi^{+}$ mode. In the helicity frame of $e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$, $\theta_{0}$ is the polar angle of the $\Lambda_{c}^{+}$ with respect to the $e^{+}$ beam axis in the $e^{+}e^{-}$ CM system. For the helicity angles of the $\Lambda_{c}^{+}\to\Lambda\pi^{+}$ decay, $\phi_{1}$ is the angle between the $e^{+}\Lambda_{c}^{+}$ and $\Lambda\pi^{+}$ planes, and $\theta_{1}$ is the polar angle of the $\Lambda$ momentum in the rest frame of the $\Lambda_{c}^{+}$ with respect to the $\Lambda_{c}^{+}$ momentum in the CM frame. The angle subscript represents the level numbering of the cascade signal decays. For the helicity angles describing the $\Lambda\to p\pi^{+}$ decay, $\phi_{2}$ is the angle between the $\Lambda\pi^{+}$ plane and $p\pi^{-}$ plane and $\theta_{2}$ is the polar angle of the proton momentum with respect to opposite direction of $\pi^{+}$ momentum in the rest frame of $\Lambda$. For the three-level cascade decays $\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+},~{}\Sigma^{0}\to\Lambda\gamma,~{}\Lambda\to p\pi^{-}$ process, $\phi_{3}$ is the angle between the $\Lambda\gamma$ and $p\pi^{-}$ planes, while $\theta_{3}$ is the polar angle of the proton with respect to the opposite direction of the photon momentum (from $\Sigma^{0}\to\Lambda\gamma$) in the rest frame of $\Lambda$.

In Ref. supple , we define $\Delta_{0}$ as the phase angle difference between two individual helicity amplitudes, $H_{\lambda_{1},\lambda_{2}}$, for the $\Lambda_{c}^{+}$ production process $\gamma^{*}\to\Lambda_{c}^{+}(\lambda_{1})\bar{\Lambda}_{c}^{-}(\lambda_{2})$ with total helicities $|\lambda_{1}-\lambda_{2}|=0$ and $1$, respectively. In the case where one-photon exchange dominates the production process, $\Delta_{0}$ is also the phase between the electric and magnetic form factors of the $\Lambda_{c}^{+}$ Ablikim:2017lct ; Faldt:2017yqt . The transverse polarization observable of the produced $\Lambda_{c}^{+}$ can be defined as

[TABLE]

whose magnitude varies as a function of $\cos\theta_{0}$. Similarly, two parameters, $\alpha^{+}_{BP}$ and $\Delta_{1}^{BP}$, describe the level-1 decays $\Lambda_{c}^{+}\to\Lambda\pi^{+},~{}\Sigma^{+}\pi^{0}$, and $\Sigma^{0}\pi^{+}$, where $\Delta_{1}^{BP}$ is the phase angle difference between the two helicity amplitudes in the $BP$ mode. The Lee-Yang parameters Lee:1956qn ; supple can be obtained with the relations

[TABLE]

In the angular analysis, the free parameters describing the angular distributions for the four data sets are determined from a simultaneous unbinned maximum likelihood fit, as $\alpha_{0}$ and $\Delta_{0}$ are common. The likelihood function is constructed from the probability density function (PDF) jointly by

[TABLE]

Here, $f_{S}(\vec{\xi})$ is the PDF of the signal process, $N_{\rm data}$ is the number of the events in data and $i$ is event index. Signal PDF $f_{S}(\vec{\xi})$ is formulated as

[TABLE]

where the variable $\vec{\xi}$ denotes the kinematic angular observables, and $\vec{\eta}$ denotes the free parameters to be determined. $M(\vec{\xi})$ is the total decay amplitude supple and $\epsilon(\vec{\xi})$ is the detection efficiency parameterized in terms of the kinematic variables $\vec{\xi}$. The background contribution to the joint likelihood is subtracted according to the calculated likelihoods for the Type-I background based on inclusive MC simulations and for the Type-II background according to the $M_{\rm BC}$ sideband. The MC-integration technique is adopted to compute the normalization factor as follows

[TABLE]

where $N_{\rm gen}$ is the total number of MC-simulated signal events. $N_{\rm MC}$ is the number of the MC signal events survived from the full selection criteria and $k_{\rm MC}$ is its event index.

Minimization of the negative logarithmic likelihood with background subtraction over all the four signal processes is carried out using the MINUIT package James:1975dr . Here, $\alpha_{0}$ is fixed to the known value $-0.20$ Ablikim:2017lct . For the charge-conjugation $\bar{\Lambda}_{c}^{-}$ decays, under the assumption of $CP$ conservation, $\bar{\Delta}_{0}=\Delta_{0}$, $\alpha^{+}_{BP}=-\alpha^{-}_{\bar{B}\bar{P}}$, and $\bar{\Delta}_{1}^{\bar{B}\bar{P}}=-\Delta_{1}^{BP}$. The decay asymmetry parameter $\alpha_{\Lambda}$ for $\Lambda\to p\pi^{-}$ is taken from the recent BESIII measurement Lamtrans and $\alpha_{\Sigma^{+}}$ for $\Sigma^{+}\to p\pi^{0}$ from the Particle Data Group (PDG) pdg . From the fit, we obtain $\sin\Delta_{0}=-0.28\pm 0.13({\rm stat.})$ which differs from zero with a statistical significance of 2.1$\sigma$ according to a likelihood ratio test. This indicates that transverse polarization $\mathcal{P}_{T}$ of the $\Lambda_{c}^{+}$ is non-zero when $\sin(2\theta_{0})\neq 0$. The numerical fit results are given in Table 1, together with the calculated $\gamma_{BP}$ and $\beta_{BP}$.

In Fig. 3, the fit results are illustrated using several projection variables. The real data are compared with the MC generated events re-weighted according to the fit.

For the $\Lambda_{c}^{+}\to\Lambda\pi^{+}$ and $\Sigma^{+}\pi^{0}$ decays, if all angles are integrated over except for the angle $\theta_{2}$, the decay rate becomes wangdan

[TABLE]

Equation (6) shows a characteristically longitudinal polarization of the produced $\Lambda$($\Sigma^{+}$) from the $\Lambda_{c}^{+}$ decays, and the asymmetry of $\cos\theta_{2}$ distribution reflects the product of the decay asymmetries $\alpha^{+}_{\Lambda\pi^{+}}\alpha_{\Lambda}$($\alpha^{+}_{\Sigma^{+}\pi^{0}}\alpha_{\Sigma^{+}}$) Asner:2008nq . The distributions of $\cos\theta_{2}$ in the $\Lambda_{c}^{+}\to\Lambda\pi^{+}$ and $\Sigma^{+}\pi^{0}$ modes are shown in Figs. 3(a) and (b), respectively. The drop at the right side in Fig. 3(b) is due to the $K_{S}^{0}\to\pi^{0}\pi^{0}$ veto.

For the $\Lambda_{c}^{+}\to\Sigma^{0}\pi^{+}$ decay, the correlations of $\cos\theta_{2}$ and $\cos\theta_{3}$ in the subsequent level-2 decay $\Sigma^{0}\to\gamma\Lambda$ and level-3 decay $\Lambda\to p\pi^{-}$, are shown in Figs. 3(c) and (d), respectively. The correlation of the average value of $\cos\theta_{i}$ satisfies the relation

[TABLE]

with $(i,j)$=(2, 3) or (3, 2).

If the full expressions for the joint angular distributions (Ref. supple ) are integrated over the angles of the level 2 and 3 decay products, the remaining partial decay rate $\mathcal{W}$ is

[TABLE]

Therefore, in a given $\cos\theta_{0}$ interval,

[TABLE]

is directly proportional to $\alpha_{BP}P_{T}(\cos\theta_{0})/(1+\alpha_{0}\cos^{2}\theta_{0})$ for the acceptance corrected data. In Fig. 3(e), the effect of the transverse polarization $P_{T}(\cos\theta_{0})$ is illustrated by plotting the average value $\langle{\rm{sign}}(\alpha_{BP})\sin\theta_{1}\sin\phi_{1}\rangle$ from all four decay modes and including both particles and antiparticles. The sign function of the measured decay asymmetry parameter, ${\rm sign}(\alpha_{BP})$, is used to avoid the cancellation of contributions from the opposite charge modes.

The systematic uncertainties arise mainly from the reconstruction of final state tracks, $K_{S}^{0}\to\pi^{0}\pi^{0}$ veto, $\Delta E$ requirement, signal $M_{\rm{BC}}$ selections and background subtraction. The contributions are summarized in Table 2. The uncertainty of the input $\alpha_{0}$ is found to be negligible, after considering the experimental uncertainty Ablikim:2017lct . Systematic uncertainties from different sources are combined in quadrature to obtain the total systematic uncertainties.

To understand the reconstruction efficiencies in data and MC simulations, a series of control samples are used for different final states. The proton and charged pion are studied based on the channel $J/\psi\to p\bar{p}\pi^{+}\pi^{-}$, photon on $e^{+}e^{-}\to\gamma\mu^{+}\mu^{-}$ Prasad:2016wxl , $\pi^{0}$ on $\psi(3686)\to\pi^{0}\pi^{0}J/\psi$ and $e^{+}e^{-}\to\omega\pi^{0}$, $\Lambda$ on $J/\psi\to\bar{p}K^{+}\Lambda$ and $J/\psi\to\Lambda\bar{\Lambda}$ Ablikim:2018jfs , and $K_{S}^{0}$ on $J/\psi\to K^{*}(892)^{+}K^{-},K^{*}(892)^{+}\to K_{S}^{0}\pi^{+}$ and $J/\psi\to\phi K_{S}^{0}K^{+}\pi^{-}$ Ablikim:2015qgt . The efficiency differences between data and MC simulations are used to reweight the summed likelihood values. The changes of the fit results after likelihood minimization are taken as systematic uncertainties. The uncertainties due to the $K_{S}^{0}\to\pi^{0}\pi^{0}$ veto in $\Sigma^{+}\pi^{0}$ candidate events are evaluated by taking the maximum changes with respect to the nominal results when varying the $\pi^{0}\pi^{0}$ veto range. A similar method is applied when estimating the systematic uncertainties from the signal $\Delta E$ and $M_{\rm{BC}}$ selection criteria. In the likelihood construction, the subtraction of the background contributions are modeled with the sideband control samples and the inclusive MC samples. The associated uncertainties are studied by varying the sideband range and adjusting the scaling factors of the two background components. The altered scaling factors are obtained by changing the background lineshapes within their 1$\sigma$ uncertainties from the fits to the $M_{\rm{BC}}$ distribution. The resultant maximum changes of the fit results are taken as corresponding systematic uncertainties.

To summarize, based on the 567 pb*-1* data sample collected from $e^{+}e^{-}$ collisions at a CM energy of 4.6 GeV, a simultaneous full angular analysis of four decay modes of $\Lambda_{c}^{+}\to pK_{S}^{0}$, $\Lambda\pi^{+}$, $\Sigma^{+}\pi^{0}$, and $\Sigma^{0}\pi^{+}$ from the $e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ production is carried out. We study the $\Lambda_{c}^{+}$ transverse polarization in unpolarized $e^{+}e^{-}$ collisions for the first time, which gives $\sin\Delta_{0}=-0.28\pm 0.13\pm 0.03$ with a statistical significance of 2.1$\sigma$. This information will help in understanding the production mechanism of the charmed baryons in $e^{+}e^{-}$ annihilations. With availability of the transverse polarization measurement, the decay asymmetry parameter in $\Lambda_{c}^{+}\to pK_{S}^{0}$ becomes accessible experimentally. Moreover, this improves the precision in determining the decay asymmetry parameters in $\Lambda_{c}^{+}\to\Lambda\pi^{+}$, $\Sigma^{+}\pi^{0}$, and $\Sigma^{0}\pi^{+}$, as listed in Table 1.

The parameters $\alpha^{+}_{pK_{S}^{0}}$ and $\alpha^{+}_{\Sigma^{0}\pi^{+}}$ are measured for the first time. The measured $\alpha^{+}_{\Lambda\pi^{+}}$ and $\alpha^{+}_{\Sigma^{+}\pi^{0}}$ parameters are consistent with previous measurements, but with much improved precisions (by a factor of 3 for $\alpha^{+}_{\Sigma^{+}\pi^{0}}$). The negative sign of the $\alpha^{+}_{\Sigma^{+}\pi^{0}}$ parameter is confirmed and differs from the positive predictions Korner:1992wi ; Xu:1992vc ; Cheng:1993gf ; Cheng:1991sn ; Ivanov:1997ra ; Zenczykowski:1993jm by at least 8$\sigma$, which rules out those model calculations. The measured $\alpha^{+}_{\Sigma^{+}\pi^{0}}$ and $\alpha^{+}_{\Sigma^{0}\pi^{+}}$ values agree well, which supports hyperon isospin symmetry in $\Lambda_{c}^{+}$ decay. For the results on $\alpha^{+}_{pK_{S}^{0}}$, $\alpha^{+}_{\Sigma^{+}\pi^{0}}$, and $\alpha^{+}_{\Sigma^{0}\pi^{+}}$ listed in Table 1, at present no model gives predictions fully consistent with all the measurements. These improved results in $\Lambda_{c}^{+}$ decay asymmetries provide essential inputs for the $b$-baryon decay asymmetry measurements to be performed in the future.

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. This work is supported in part by National Key Basic Research Program of China under Contract No. 2015CB856700; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11335008, 11425524, 11625523, 11635010, 11735014; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos. U1532257, U1532258, U1732263; CAS Key Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; German Research Foundation DFG under Contract No. Collaborative Research Center CRC 1044, FOR 2359; Istituto Nazionale di Fisica Nucleare, Italy; Koninklijke Nederlandse Akademie van Wetenschappen (KNAW) under Contract No. 530-4CDP03; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Science and Technology fund; The Swedish Research Council; U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374, DE-SC-0010118, DE-SC-0012069; University of Groningen (RuG); Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt; the Knut and Alice Wallenberg Foundation (Sweden) under Contract No. 2016.0157 and the Royal Society, UK under Contract No. DH160214.