Recent Progress in ${\mathsf{\Lambda }}_c^{+}$ Decays

Abstract

Studying ${\mathsf{\Lambda }}_c^{+}$ decays provides important and complementary information to help understand the effects of non-perturbative quantum chromodynamics. Recently, many interesting results about ${\mathsf{\Lambda }}_c^{+}$ decays have been reported thanks to the high luminosity in Belle and dedicated operation at the ${\mathsf{\Lambda }}_c^{+}$ pair production threshold in BESIII. This paper reviews the recent progress in experimental measurements, containing sections about the hadronic weak decay, semileptonic decay, and decay asymmetry parameters and CP violation.

1. Introduction

The lowest-lying charmed baryon ${\mathsf{\Lambda }}_c^{+}$ was discovered more than 40 years ago. However, its properties still need much exploration in both experiments and theory. Investigation of the properties of ${\mathsf{\Lambda }}_c^{+}$ is essential for the exploration and measuring of excited and higher-mass charmed baryons, as they are the lightest charmed baryons and the major daughter particle of all the higher-mass ones. In 2017, an LHCb experiment announced the observation of a new resonance in the ${\mathsf{\Lambda }}_c^{+}{K}^{-}{\pi }^{+}{\pi }^{+}$ mass spectrum, which was consistent with the expectations for the doubly charmed baryon ${\mathsf{\Xi }}_{cc}^{++}$ [1]. The measurement of ${\mathsf{\Lambda }}_c^{+}$ decays with the improved precision provided crucial input for the relevant theoretical prediction [2]. Afterward, in 2020, the LHCb reported the observation of three new ${\mathsf{\Xi }}_c$ excited states in the ${\mathsf{\Lambda }}_c^{+}{K}^{-}$ spectrum [3].

Studying the charmed baryon also provides the complementary information to understand the effects of non-perturbative quantum chromodynamics (QCD), and this is a unique area in which to probe the behavior of a light diquark in the environment of a charm quark [4]. Both the experimental and theoretical progress in the decay of charmed baryons were very slow, but this situation reversed in 2014, as there were several breakthroughs in experiments regarding the weak decays of ${\mathsf{\Lambda }}_c^{+}$. The representative one is the improved measurement of hadronic decay ${\mathsf{\Lambda }}_c^{+}\to p{K}^{-}{\pi }^{+}$ from both Belle and BESIII. It was concerned with the absolute branching fraction of ${\mathsf{\Lambda }}_c^{+}\to p{K}^{-}{\pi }^{+}$, because nearly all the branching fractions of the ${\mathsf{\Lambda }}_c^{+}$ decays were measured relative to the $p{K}^{-}{\pi }^{+}$ mode. In 2014, Belle reported a value of $(6.84\pm 0.{24}_{-0.27}^{+0.21})$% from the reconstruction of $D^{\ast }p\pi$ recoiling against the ${\mathsf{\Lambda }}_c^{+}$ production in $e^{+}{e}^{-}$ annihilation [5]. Most importantly, this measurement is model-independent. Soon after, BESIII measured this mode directly with the result $\mathcal{B}({\mathsf{\Lambda }}_c^{+}\to p{K}^{-}{\pi }^{+})=$ $(5.84\pm 0.27\pm 0.23)$% [6]. The ${\mathsf{\Lambda }}_c^{+}$ pair was produced near the threshold region at the BEPCII collider, and this made BESIII also conduct the absolute branching fraction measurement in a model-independent way. Its precision was comparable to Belle’s result. A new average of $(6.3\pm 0.5)$% for this benchmark mode was quoted by the PDG [7]. Thanks to the dedicated data set collected at the ${\mathsf{\Lambda }}_c^{+}$ pair production threshold, BESIII followed up and conducted a series of model-independent measurements on ${\mathsf{\Lambda }}_c^{+}$ decays.

This paper will mainly review the recent experimental progress in ${\mathsf{\Lambda }}_c^{+}$ decays at Belle and BESIII, containing sections about hadronic weak decay, semileptonic decay, and decay asymmetry parameters and CP polarization. The study of the ${\mathsf{\Lambda }}_c^{+}$ at Belle is based on data samples taken at or near the $\mathrm{Y}\left(1S\right)$, $\mathrm{Y}\left(2S\right)$, $\mathrm{Y}\left(3S\right)$, $\mathrm{Y}\left(4S\right)$, and $\mathrm{Y}\left(5S\right)$ resonances, which were collected with the Belle detector at the KEKB asymmetric energy $e^{+}{e}^{-}$ collider [8,9]. The integrated luminosity of the data samples was 980.6 fb${}^{-1}$. BESIII [10] runs at the symmetric energy of the BEPCII $e^{+}{e}^{-}$ collider [11], covering a center-of-mass energy between 2 and 5 GeV.In 2014 and 2020, BESIII operated at the dedicated energy region of ${\mathsf{\Lambda }}_c^{+}$ pair production that is between 4.6 and 4.7 GeV. The integrated luminosity was 567 pb${}^{-1}$ at $\sqrt{s}=4.6$ GeV [12] and 4.5 fb${}^{-1}$ between 4.61 and 4.7 GeV [13]. Belle has larger data sets where searching activities for rare hadronic weak decays of ${\mathsf{\Lambda }}_c^{+}$ are always carried out, as well as precise measurements of the decay asymmetry parameters. BESIII has the threshold effect of ${\mathsf{\Lambda }}_c^{+}$ pair production, although the luminosity is smaller. Hence, the missing energy reconstruction for neutrinos and neutrons could be conducted, which led to results of semi-leptonic decay and decay with neutrons in ${\mathsf{\Lambda }}_c^{+}$.

2. Hadronic Weak Decays

Today, we still do not have a good phenomenological model to describe the complicated physics of charmed baryon hadronic weak decays as the quantum chromodynamics (QCD)-inspired approach is no longer applicable to charmed baryons, although it has been applied successfully to heavy mesons. From the theoretical point of view, in contrast to two quarks in mesons, the baryons being made out of three quarks brings along several essential complications. The hadronic decay amplitude of the charmed baryon generally consists of factorizable and nonfactorizable contributions [14], and the size of a nonfactorizable contribution can be comparable to that of a factorizable one, as W exchange and inner W emission are no longer subject to helicity and color suppression. Although the study of nonfactorizable effects could rely on the pole model, the evaluation of pole diagrams is still far more difficult than the factorizable term, and the experimental measurements are the crucial input for reducing the uncertainty.

Following the milestone work of measuring the absolute branching fraction of ${\mathsf{\Lambda }}_c^{+}\to p{K}^{-}{\pi }^{+}$, BESIII measured the absolute branching fractions for more than a dozen Cabibbo-favored (CF) decay modes directly for the first time [6]. Not only were the central values substantially different from the PDG ones (versions before 2016), but the uncertainties were also significantly improved. Several of the ${\mathsf{\Lambda }}_c^{+}$ decays, such as ${\mathsf{\Sigma }}^0{\pi }^{+}$, ${\mathsf{\Sigma }}^{+}{\pi }^0$, and ${\mathsf{\Sigma }}^{+}\varphi$, did not receive any factorizable contributions, and the measurements of them implied the importance of the nonfactorizable terms of W exchange and inner W-emission. Other interesting decays are ${\mathsf{\Lambda }}_c^{+}\to {\mathsf{\Sigma }}^{+}\eta$ and ${\mathsf{\Sigma }}^{+}{\eta }^{\prime }$. The branching fraction of ${\mathsf{\Lambda }}_c^{+}\to {\mathsf{\Sigma }}^{+}\eta$ is expected to be comparable to or larger than that of ${\mathsf{\Lambda }}_c^{+}\to {\mathsf{\Sigma }}^{+}{\eta }^{\prime }$. However, the first measurement by BESIII for ${\mathsf{\Lambda }}_c^{+}\to {\mathsf{\Sigma }}^{+}{\eta }^{\prime }$ indicated that the opposite was true [15], and $\mathcal{B}({\mathsf{\Lambda }}_c^{+}\to {\mathsf{\Sigma }}^{+}{\eta }^{\prime })/\mathcal{B}({\mathsf{\Lambda }}_c^{+}\to {\mathsf{\Sigma }}^{+}\eta )=3.5\pm 2.1\pm 0.4$. Both the experiments and theory need improved precision to confirm this in the future.

Due to the limited statistics, the Cabibbo-suppressed (CS) decays of ${\mathsf{\Lambda }}_c^{+}$ could not be studied with high precision. Recently, high luminosity has been achieved by both Belle and BESIII experiments, and this investigation became feasible. The activities of measuring the two-body CS decays were conducted for the first time because of the simplest topology and their higher detection efficiencies. This was also highly motivated by various theoretical predictions. The significant experimental progress in CS processes was in the observation of decay ${\mathsf{\Lambda }}_c^{+}\to n{\pi }^{+}$ [16], and its branching fraction was determined to be $(6.6\pm 1.2\pm 0.4)\times {10}^{-4}$. The observed distribution is presented on the left-hand side of Figure 1, where the neutron signals are evidently seen. This is essential experimental progress toward pinning down the puzzle of the SU(3) flavor relation between ${\mathsf{\Lambda }}_c^{+}\to n{\pi }^{+}$ and ${\mathsf{\Lambda }}_c^{+}\to p{\pi }^0$.

In the very beginning, SU(3) flavor symmetry provided predictions of the two-body decays, where it fit to the data with both CF and CS decays and made some plausible assumptions to reduce the parameters. One would have the relation $M({\mathsf{\Lambda }}_c^{+}\to n{\pi }^{+})=\sqrt{2}M({\mathsf{\Lambda }}_c^{+}\to p{\pi }^0)$, and it predicted $\mathcal{B}({\mathsf{\Lambda }}_c^{+}\to n{\pi }^{+})$ to be $\sim 0.97\times {10}^{-3}$. Hence, $\mathcal{B}({\mathsf{\Lambda }}_c^{+}\to p{\pi }^0)\sim 5\times {10}^{-4}$. However, this leads to a contradiction when compared with the observed branching fractions $\mathcal{B}({\mathsf{\Lambda }}_c^{+}\to p{\pi }^0)$ for BESIII (<$2.7\times {10}^{-4}$) [17] and Belle (<$0.8\times {10}^{-4}$) [18]. Later, in [19], the SU(3) flavor symmetry considered the contribution of $\mathcal{O}\left(\overline{15}\right)$, and the predictions were close to the current measurements for both the ${\mathsf{\Lambda }}_c^{+}\to p{\pi }^0$ and ${\mathsf{\Lambda }}_c^{+}\to n{\pi }^{+}$ decays. The alternative theoretical prediction in [20], with current-algebra approach and input of ${\mathsf{\Lambda }}_c^{+}\to p\varphi$ that was measured by BESIII, studied couples of two-body CS processes and proposed that there exists large destructive interference between the factorizable and nonfactorizable amplitudes for both S and P waves in ${\mathsf{\Lambda }}_c^{+}\to p{\pi }^0$, which leads to a relatively small decay rate. Therefore, the study of ${\mathsf{\Lambda }}_c^{+}\to p{\pi }^0$ needs to be revisited with higher statistics at Belle-II and BESIII.

In the meantime, by using the large statistics in the B factory, Belle reported the first observation of ${\mathsf{\Lambda }}_c^{+}\to p{\eta }^{\prime }$ [21] and improved measurement of $p\eta$, $p\omega$ [18,22], with branching fractions which were determined relative to ${\mathsf{\Lambda }}_c^{+}\to p{K}^{-}{\pi }^{+}$. The distribution of the ${\mathsf{\Lambda }}_c^{+}\to p{\eta }^{\prime }$ signals is shown on the right-hand side of Figure 1. All of the recent two-body experimental results are listed in

Table 1. The recent experimental measurements of two-body Cabibbo-suppressed decays of ${\mathsf{\Lambda }}_c^{+}$ (${10}^{-4}$) compared to various theoretical predictions.

CS Mode	Exp.	Geng et al.	Cheng et al.	Zhao et al.
${\mathsf{\Lambda }}_c^{+}\to n{\pi }^{+}$	$6.6\pm 1.2\pm 0.4$	$6.1\pm 2.1$	2.7	$7.7\pm 2.0$
${\mathsf{\Lambda }}_c^{+}\to p{\pi }^0$	<0.8	$1.3\pm 0.7$	0.8	$0.8\pm 0.8$
${\mathsf{\Lambda }}_c^{+}\to p\eta$	$14.2\pm 0.5\pm 01.1$	$13.0\pm 1.0$	12.8	$11.4\pm 3.5$
${\mathsf{\Lambda }}_c^{+}\to p{\eta }^{\prime }$	$4.73\pm 0.82\pm 0.52$	⋯	⋯	$7.1\pm 1.4$
${\mathsf{\Lambda }}_c^{+}\to p\omega$	$8.27\pm 0.75\pm 0.75$	$6.3\pm 3.4$	⋯	⋯

and compared to the predictions of various models.

With the threshold advantage at BESIII, the double-tag approach could be conducted, and the probe of ${\mathsf{\Lambda }}_c^{+}$ decays with a neutron in the final state became applicable, where the neutron signals were reconstructed as the missing energy under energy–momentum conservation. Aside from ${\mathsf{\Lambda }}_c^{+}\to n{\pi }^{+}$, BESIII reported the observations of ${\mathsf{\Lambda }}_c^{+}\to n{\pi }^{+}{\pi }^0$, $n{\pi }^{+}{\pi }^{+}{\pi }^{-}$, $n{K}^{-}{\pi }^{+}{\pi }^{+}$ [23] and ${\mathsf{\Lambda }}_c^{+}\to {\mathsf{\Sigma }}^{-}2{\pi }^{+}$, ${\mathsf{\Sigma }}^{-}{\pi }^{0}2{\pi }^{+}$ [24], where ${\mathsf{\Sigma }}^{-}$ was reconstructed with its dominant decay mode ${\mathsf{\Sigma }}^{-}\to n{\pi }^{-}$. All the above measurements on CF and CS modes are useful for testing SU(3) flavor symmetry and understanding the dynamics.

3. Semileptonic Decays

The decay rates of semileptonic (SL) decays ${\mathsf{\Lambda }}_c^{+}\to \mathsf{\Lambda }\ell \nu$ depend on the weak quark mixing a Cabibbo–Kobayashi–Maskawa (CKM) matrix element and the strong interaction effects parameterized by form factors describing the hadronic transition between the initial and the final baryons. Studies on SL decays are valuable for the determination of standard model parameters and non-perturbative effects. Measurements of the absolute branching fractions of SL decays ${\mathsf{\Lambda }}_c^{+}\to \mathsf{\Lambda }{e}^{+}{\nu }_e$ and ${\mathsf{\Lambda }}_c^{+}\to \mathsf{\Lambda }{\mu }^{+}{\nu }_{\mu }$ were carried out by BESIII in 2015 for the first time. With the larger data sets collected in 2020, BESIII improved the precision of the branching fraction of ${\mathsf{\Lambda }}_c^{+}\to \mathsf{\Lambda }{e}^{+}{\nu }_e$ to $(3.56\pm 0.11\pm 0.07)$%, and most importantly, the form factors between the transitions of ${\mathsf{\Lambda }}_c^{+}$ and $\mathsf{\Lambda }$ were extracted for the first time [25], which was motivated by the lattice QCD calculation [26]. Figure 2 presents a comparison between the measured decay rate of ${\mathsf{\Lambda }}_c^{+}\to \mathsf{\Lambda }\ell \nu$ and the LQCD-predicted one as function of the $e^{+}{\nu }_e$ mass squared ($q^2$). The measurement provides important inputs in understanding the SL decays of charmed baryons and helps calibrate the calculation of SL decays of other charmed baryons as well as the bottomed baryons.

There was the breakthrough of searching for new SL processes. BESIII reported the observation of ${\mathsf{\Lambda }}_c^{+}\to p{K}^{-}{e}^{+}{\nu }_e$ with a significance of 8.2$\sigma$ [27], and the measured branching fraction with respect to the inclusive decay ${\mathsf{\Lambda }}_c^{+}\to X{e}^{+}{\nu }_e$ was $\mathcal{B}({\mathsf{\Lambda }}_c^{+}\to p{K}^{-}{e}^{+}{\nu }_e)/\mathcal{B}({\mathsf{\Lambda }}_c^{+}\to X{e}^{+}{\nu }_e)=(2.1\pm 0.4\pm 0.2)$%, which was direct confirmation that the SL ${\mathsf{\Lambda }}_c^{+}$ decays were not saturated by the final state of $\mathsf{\Lambda }\ell \nu$.

4. Decay Asymmetry Parameters and CP Violation

The polarization of the daughter particle in the two-body decay of ${\mathsf{\Lambda }}_c^{+}$ is very useful information, which can be reflected by the decay asymmetry parameter $\alpha$. The $\alpha$ is given as

$$\alpha =\frac{2\mathrm{Re}\left(A^{\ast }B\right)}{{\left|A\right|}^2+{\left|B\right|}^2},$$

(1)

The amplitudes A and B correspond to the parity-violating S wave and parity-conserving P wave amplitudes, respectively. The decay asymmetry parameters of ${\mathsf{\Lambda }}_c^{+}\to \mathsf{\Lambda }{\pi }^{+}$, ${\mathsf{\Sigma }}^0{\pi }^{+}$, ${\mathsf{\Sigma }}^{+}{\pi }^0$, and $p{K}_S^0$ were measured by BESIII [28], where $\alpha ({\mathsf{\Lambda }}_c^{+}\to {\mathsf{\Sigma }}^{+}{\pi }^0)=-0.57\pm 0.12$ was obtained and confirmed the results of CLEO [29]. However, it was always predicted to be positive in the pole model. For the decay ${\mathsf{\Lambda }}_c^{+}\to p{K}_S^0$, all the predictions except the pole mode gave a large and negative value, while the current $\alpha ({\mathsf{\Lambda }}_c^{+}\to p{K}_S^0)=0.18\pm 0.45$. This result is limited by statistical uncertainty, and it needs to be resolved with high precision in the future. With the data sets collected in 2020, BESIII carried out partial wave analysis with the three-body decay ${\mathsf{\Lambda }}_c^{+}\to \mathsf{\Lambda }{\pi }^{+}{\pi }^0$ to disentangle the information of excited hyperons [30]. Not only were the branching fractions of two-body modes ${\mathsf{\Lambda }}_c^{+}\to \mathsf{\Sigma }{\left(1385\right)}^0{\pi }^{+}$, $\mathsf{\Sigma }{\left(1385\right)}^{+}{\pi }^0$, and $\mathsf{\Lambda }{\rho }^{+}$ determined, but the decay asymmetry parameters were also measured to be $-0.789\pm 0.113$, $-0.917\pm 0.083$, and $-0.763\pm 0.066$, respectively.

CP violation in the charm sector is the most important and fundamental issue. The first observation in the charm meson was announced by LHCb, but it has never been confirmed in charmed baryons. The exploration was conducted in ${\mathsf{\Lambda }}_c^{+}$. Belle performed the measurements of the decay asymmetry parameters $\alpha ({\mathsf{\Lambda }}_c^{+}\to \mathsf{\Lambda }{\pi }^{+})=-0.755\pm 0.006$ and $\alpha ({\mathsf{\Lambda }}_c^{+}\to {\mathsf{\Sigma }}^0{\pi }^{+})=-0.463\pm 0.018$ with improved precision and reported those of the CS processes $\alpha ({\mathsf{\Lambda }}_c^{+}\to \mathsf{\Lambda }{K}^{+})=-0.585\pm 0.052$ and $\alpha ({\mathsf{\Lambda }}_c^{+}\to {\mathsf{\Sigma }}^0{K}^{+})=-0.55\pm 0.20$ for the first time [31]. In the meantime, $\mathcal{A}({\mathsf{\Lambda }}_c^{+}\to \mathsf{\Lambda }{K}^{+})=-0.023\pm 0.112$ and $\mathcal{A}({\mathsf{\Lambda }}_c^{+}\to {\mathsf{\Sigma }}^0{K}^{+})=0.08\pm 0.38$ were derived. The resulting CP-violating decay parameter asymmetry $\mathcal{A}=(\alpha -\overline{\alpha })/(\alpha +\overline{\alpha })$ was consistently zero within the large uncertainty, as expected. Owing to the advantage of extremely high luminosity, the CP violation in the multi-hadron final state of ${\mathsf{\Lambda }}_c^{+}$ decays has been investigated in the next-generation Super Tau-Charm Facility (STCF) program, where the decays ${\mathsf{\Lambda }}_c^{+}\to p{K}^{-}{\pi }^{+}{\pi }^0$, $\mathsf{\Lambda }{\pi }^{+}{\pi }^{+}{\pi }^{-}$, and $p{K}_S^0{\pi }^{+}{\pi }^{-}$ have been probed [32]. By using the 1 ab${}^{-1}$ fast Monte Carlo-simulated data of $e^{+}{e}^{-}$ collision at $\sqrt{s}=4.64$ GeV, an indicated sensitivity at the level of (0.25–0.5)% was accessible.

5. Summary

This paper reviewed the recent experimental progress for ${\mathsf{\Lambda }}_c^{+}$ decays. It mainly contains the results of hadronic weak decay, semileptonic decay, and decay asymmetry parameters and CP violation. The significant progress was in the measurements of several two-body Cabibbo-suppressed decays ${\mathsf{\Lambda }}_c^{+}\to n{\pi }^{+}$, $p\eta$, and $p{\eta }^{\prime }$, among others, as well as the form factor extraction with the semileptonic decay ${\mathsf{\Lambda }}_c^{+}\to \mathsf{\Lambda }{e}^{+}{\nu }_e$. Thanks to the high luminosity of Belle and BESIII, we have such opportunities to yield these measurements with high precision.

Very recently, by using the double tag approach and anti-neutron effects with materials, BESIII reported the absolute branching fraction of an inclusive decay ${\mathsf{\Lambda }}_c^{+}\to n+X$ to be $(33.5\pm 0.7\pm 1.2)$% [33], which directly demonstrates that the decay with a neutron in the final state is only one half of that with a proton. This shows us how large remaining decays are still missing in both the neutron and proton channels and also provides an important constraint for theoretical predictions.

In the near future, fruitful results for ${\mathsf{\Lambda }}_c^{+}$ decays are expected as a benefit of the higher statistics from Belle-II and BESIII upgrades. We are eager to see the results of a single SC process of ${\mathsf{\Lambda }}_c^{+}\to p{\pi }^0$, double SC process of ${\mathsf{\Lambda }}_c^{+}\to n{K}^{+}$, and improved measurements of the decay asymmetry parameters and, hopefully, the study of CP violation in a charmed baryon.