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Low-lying charmed and charmed-strange baryon states

  • Bing Chen
  • Ke-Wei Wei
  • Xiang Liu
  • Takayuki Matsuki
Open Access
Regular Article - Theoretical Physics

Abstract

In this work, we systematically study the mass spectra and strong decays of 1P and 2S charmed and charmed-strange baryons in the framework of non-relativistic constituent quark models. With the light quark cluster–heavy quark picture, the masses are simply calculated by a potential model. The strong decays are studied by the Eichten–Hill–Quigg decay formula. Masses and decay properties of the well-established 1S and 1P states can be reproduced by our method. \(\Sigma _c(2800)^{0,+,++}\) can be assigned as a \(\Sigma _{c2}(3/2^-)\) or \(\Sigma _{c2}(5/2^-)\) state. We prefer to interpret the signal \(\Sigma _c(2850)^0\) as a \(2S(1/2^+)\) state although at present we cannot thoroughly exclude the possibility that this is the same state as \(\Sigma _c(2800)^0\). \(\Lambda _c(2765)^+\) or \(\Sigma _c(2765)^+\) could be explained as the \(\Lambda _c^+(2S)\) state or \(\Sigma ^+_{c1}(1/2^-)\) state, respectively. We propose to measure the branching ratio of \(\mathcal {B}(\Sigma _c(2455)\pi )/\mathcal {B}(\Sigma _c(2520)\pi )\) in the future, which may disentangle the puzzle of this state. Our results support \(\Xi _c(2980)^{0,+}\) as the first radial excited state of \(\Xi _c(2470)^{0,+}\) with \(J^P=1/2^+\). The assignment of \(\Xi _c(2930)^0\) is analogous to \(\Sigma _c(2800)^{0,+,++}\), i.e., a \(\Xi ^\prime _{c2}(3/2^-)\) or \(\Xi ^\prime _{c2}(5/2^-)\) state. In addition, we predict some typical ratios among partial decay widths, which are valuable for experimental search for these missing charmed and charmed-strange baryons.

1 Introduction

At present, the particle data group (PDG) lists nine charmed and ten charmed-strange baryons [1]. They are \(\Lambda _c(2286)^+\), \(\Lambda _c(2595)^+\), \(\Lambda _c(2625)^+\), \(\Lambda _c(2765)^+\) (or \(\Sigma _c(2765)^+\)), \(\Lambda _c(2880)^+\), \(\Lambda _c(2940)^+\), \(\Sigma _c(2800)^{0,+,++}\), \(\Sigma _c(2455)^{0,+,++}\), \(\Sigma _c(2520)^{0,+,++}\), \(\Xi _c(2470)^{0,+}\), \(\Xi ^{\prime }_c(2580)^{0,+}\), \(\Xi ^\prime _c(2645)^{0,+}\), \(\Xi _c(2790)^{0,+}\), \(\Xi _c(2815)^{0,+}\), \(\Xi ^{(\prime )}_c(2930)^0\), \(\Xi _c(2980)^{0,+}\), \(\Xi _c(3055)^{0,+}\), \(\Xi _c(3080)^{0,+}\), and \(\Xi ^{(\prime )}_c(3123)^+\). For brevity, we call these charmed and charmed-strange baryons just charmed baryons in the following. Among these observed states, some new measurements have been performed by experiments in the past several years. The masses and widths of \(\Sigma _c(2455)^{0,+,++}\), \(\Sigma _c(2520)^{0,+,++}\), \(\Lambda _c(2595)^+\), and \(\Lambda _c(2625)^+\) have been measured with significantly small uncertainties with the efforts of CDF [2] and Belle [3]. Very recently, the Belle Collaboration updated the measurements of \(\Xi ^{\prime }_c(2580)^{0,+}\), \(\Xi ^\prime _c(2645)^{0,+}\), \(\Xi _c(2790)^{0,+}\), \(\Xi _c(2815)^{0,+}\), and \(\Xi _c(2980)^{0,+}\) [4]. On the other hand, new decay modes for the higher excited charmed baryon states have been found by experiment. For instance, the decay channel of \(\Lambda D^+\) was first found for \(\Xi _c(3055)^+\) and \(\Xi _c(3080)^+\), and the following ratios of branching fractions were first reported by Belle several months ago [5]:
$$\begin{aligned} \frac{\mathcal {B}(\Xi _c(3055)^+\rightarrow \Lambda D^+)}{\mathcal {B}(\Xi _c(3055)^+\rightarrow \Sigma _c(2455)^{++}K^-)}=5.09\pm 1.01\pm 0.76, \end{aligned}$$
$$\begin{aligned} \frac{\mathcal {B}(\Xi _c(3080)^+\rightarrow \Lambda D^+)}{\mathcal {B}(\Xi _c(3080)^+\rightarrow \Sigma _c(2455)^{++}K^-)}=1.29\pm 0.30\pm 0.15, \end{aligned}$$
and
$$\begin{aligned} \frac{\mathcal {B}(\Xi _c(3080)^+\rightarrow \Sigma _c(2520)^{++}K^-)}{\mathcal {B}(\Xi _c(3055)^+\rightarrow \Sigma _c(2455)^{++}K^-)}=1.07\pm 0.27\pm 0.01, \end{aligned}$$
where the uncertainties are statistical and systematic. Obviously, these new measurements are very useful to understand the nature of these excited charmed baryon states.

Theoretically, the charmed baryons which contain one heavy quark and two light quarks occupy a particular position in the baryon physics. Since the chiral symmetry and heavy quark symmetry (HQS) can provide some qualitative insight into the dynamics of charmed baryons, the investigation of charmed baryons should be more helpful for improving our understanding of the confinement mechanism. The spectroscopy of charmed baryons has been investigated in various models. So far, the several kinds of quark potential models [6, 7, 8, 9, 10], the relativistic flux tube (RFT) model [11, 12], the coupled channel model [13], the QCD sum rule [14, 15, 16], and the Regge phenomenology [17] have been applied to study the mass spectra of excited charmed baryons, and so did lattice QCD [18, 19]. The strong decay behaviors of charmed baryons have been studied by several methods, such as the heavy hadron chiral perturbation theory (HHChPT) [20, 21], the chiral quark model [22, 23], the \(^3P_0\) model [24], and a non-relativistic quark model [25]. The decays of 1P \(\Lambda _c\) and \(\Xi _c\) baryons have also been investigated by a light front quark model [26, 27], a relativistic three-quark model [28], and the QCD sum rule [29].

Although many experimental and theoretical efforts have been made for the research of charmed baryons, most of the 1P and 2S charmed baryons are not yet established. Several candidates, including \(\Lambda _c(2765)^+\), \(\Sigma _c(2800)^{0,+,++}\), \(\Xi _c(2930)^0\), and \(\Xi _c(2980)^{0,+}\) are still in controversy. \(\Lambda _c(2765)^+\) was first observed by the CLEO Collaboration in the decay channel of \(\Lambda _c(2765)^+\rightarrow \Lambda _c^+\pi ^+\pi ^-\) [30], and confirmed by Belle in the \(\Sigma _c(2455)\pi \) mode [31, 32]. Because both \(\Lambda _c^+\) and \(\Sigma _c^+\) excitations can decay through \(\Lambda _c^+\pi ^+\pi ^-\) and \(\Sigma _c(2455)\pi \), we even do not know whether the observed charmed baryon signal around 2765 MeV is the \(\Lambda _c^+\) or \(\Sigma _c^+\) state, or their overlapping structure [33]. In the \(e^+e^-\) annihilation process, an isotriplet state, \(\Sigma _c(2800)^{0,+,++}\), was observed by Belle in the channel of \(\Lambda _c^+\pi \), and it was tentatively identified as the \(\Sigma _{c2}\) state with \(J^P=3/2^-\) [34]. Interestingly, another neutral resonance was later found by BaBar in the process of \(B^-\rightarrow \Sigma _c^{*0}\bar{p}\rightarrow \Lambda _c^+\pi ^-\bar{p}\) with the mass \(2846\pm 8\pm 10\) MeV and decay width \(86^{+33}_{-22}\) MeV [35]. The higher mass and the weak evidence of \(J=1/2\) indicate that the signal observed by BaBar might be different from the Belle observation. In this paper, we denote the signal discovered by BaBar by \(\Sigma _c(2850)^0\). \(\Xi _c(2930)^0\), which was only seen by BaBar in the decay mode \(\Lambda ^+_cK^-\) [36], still needs more confirmation. \(\Xi _c(2980)^{0,+}\) was first reported by Belle in the channels \(\Lambda _c^+K^-\pi ^+\) and \(\Lambda _c^+K_S^0\pi ^-\) [37], and was later confirmed by Belle [4, 38] and BaBar [39] in the channels \(\Xi ^\prime _c(2580)\pi \), \(\Xi _c(2645)\pi \) and \(\Sigma _c(2455)K\), respectively. However, the decay widths reported by Refs. [4, 37, 38, 39] were quite different from each other. More experimental information as regards the charmed baryons can be found in Refs. [40, 41, 42, 43].

Obviously, a systematic study of masses and decays is required for these unestablished charmed baryons. More importantly, most of 2S and 1P charmed baryons have not yet been detected by any experiments. Such a research can also help the future experiments to find them. In the present work, we will focus on both the mass spectra and the strong decays of low-lying 1P and 2S charmed baryons. We pay attention to only the charmed baryons inside of which degrees of freedom of two light quarks are frozen. It means that two light quarks are not considered to be excited, neither radially nor orbitally. As illustrated in Ref. [44], these kinds of charmed excitations carry lower-excited energies, which means these excited charmed baryons may more likely be detected by experiments. Fortunately, our results indicate that most of the observed charmed baryons can be accommodated in this way.

The paper is arranged as follows. In Sect. 2, the masses of low-excited charmed baryons are calculated by the non-relativistic quark potential model. In Sect. 3, the Eichten–Hill–Quigg (EHQ) decay formula, which is employed to study the strong decays of excited charmed baryons, is introduced. The properties of low-lying charmed baryon states are fully discussed in Sect. 4. Finally, the paper ends with a conclusion and an outlook. Some detailed calculations and definitions are collected in the appendices.

2 The deduction of mass spectra

2.1 Treating charmed baryon system as a two-body problem

To study the baryon dynamics, one crucial question which should be answered is “What are the relevant degrees of freedom in a baryon?” [45]. In constituent quark models, a baryon system consists of three confined quarks. Thus the dynamics of a baryon resonance is surely more complex than a meson. Due to the HQS, however, the dynamics of charmed baryons could be greatly simplified. The HQS suggests that the couplings between a c quark and two light quarks are weak [46]. Therefore, two light quarks in a charmed baryon could first couple with each other to form a light quark cluster.1 Then the light quark cluster couples with a charm quark, and a charmed baryon resonance forms. With this assumption, two light quarks have the same status to a c quark, including the average distances to a c quark. In the light cluster–heavy quark picture, the dynamics of heavy baryon can be simplified. In the non-relativistic constituent quark model, the spin-independent part of the Hamiltonian is
$$\begin{aligned} H = \sum \limits _{i=1}^{3}\left( \frac{p_i^2}{2m_i}+m_i\right) + \sum \limits _{i<j}\left( -\frac{2\alpha }{3r_{ij}}+\frac{b}{2}r_{ij}\right) , \end{aligned}$$
(1)
where the Cornell potential [47] is used as a phenomenological confining term. The Jacobi coordinates are usually taken to deal with the three-body problem for baryons. \(\rho \), \(\lambda \), and R are related to the quark positions by
$$\begin{aligned} \vec {\rho }= & {} \vec {r}_1-\vec {r}_2,~~~~\\ \vec {\lambda }= & {} \frac{m_1\vec {r}_1+m_2\vec {r}_2}{m_1+m_2}-\vec {r}_Q,~~~~\\ \vec {R}= & {} \frac{m_1\vec {r}_1+m_2\vec {r}_2+m_Q\vec {r}_Q}{m_1+m_2+m_Q} , \end{aligned}$$
where the indices, 1, 2, and Q, are for two light quarks and a heavy quark, respectively. The momenta \(\vec {p}_\rho \), \(\vec {p}_\lambda \) and \(\vec {p}_R\), which are conjugate to the Jacobi coordinates above, can be defined easily. Now the spin-independent Hamiltonian becomes
$$\begin{aligned} H= & {} \frac{p^2_\rho }{2m_\rho } + \frac{p^2_\lambda }{2m_\lambda } + \frac{p^2_R}{2M} + M + \left( -\frac{2\alpha }{3r_{12}}+\frac{b}{2}r_{12}\right) \nonumber \\&+ \left( -\frac{2\alpha }{3r_{1Q}}+\frac{b}{2}r_{1Q}\right) + \left( -\frac{2\alpha }{3r_{2Q}}+\frac{b}{2}r_{2Q}\right) , \end{aligned}$$
(2)
where \(m_\rho =m_1m_2/(m_1+m_2)\), \(m_\lambda =(m_1+m_2)m_Q/M\), and \(M=m_1+m_2+m_Q\). According to the definitions above, the relative motion between two light quarks is usually called the \(\rho \)-mode, while the one between the center of mass of the two light quarks and the heavy quark is called the \(\lambda \)-mode.
Fig. 1

A single heavy baryon in the light cluster–heavy quark picture. With the SU(3) flavor symmetry, the relation of \(\tau =\sqrt{\lambda ^2+\rho ^2/4}\) is obvious

As emphasized above, the average distances to c quark should be equal for two light quarks in a cluster, i.e., \(\tau \equiv r_{1Q}=r_{2Q}\) (see Fig. 1). In practice, we should solve the following Schrödinger equation for the mass of a heavy baryon:
$$\begin{aligned}&\left[ -\frac{\nabla _\rho ^2}{2m_\rho }-\frac{\nabla _\lambda ^2}{2m_\lambda }+\left( \frac{b\rho }{2}-\frac{2\alpha }{3\rho }\right) +\left( b\tau -\frac{4\alpha }{3\tau }\right) +C\right] \nonumber \\&\quad \times \quad \psi = E\psi . \end{aligned}$$
(3)
Since the excited mode between two light quarks is not considered in this paper, the light quark cluster can be treated as a block with the antitriplet color structure and peculiar size. Specifically, a color singlet baryon system should be formed as \(3_{q_1}\otimes 3_{q_2}\otimes 3_Q\ni \bar{3}_{\mathrm{cl.}}\otimes 3_Q\ni \bar{1}_{\mathrm{Q-cl.}}\). In this way, a heavy baryon could be treated as a quasi-two-body system. In the light quark cluster–heavy quark picture, Isgur has discussed the similarity of dynamics between heavy baryons and heavy–light mesons [48]. It should be stressed that the scenario of a light cluster–heavy quark picture is not contradictory to the one-gluon exchange interaction. Since the color–spin interaction is proportional to the inverse of the quark masses, two light quarks in the heavy-baryon system are expected to strongly couple to each other [49]. Thus, they may develop into a quark cluster. In fact, the existence of a light quark cluster correlations was partly confirmed by the lattice QCD [50] and the Bethe–Salpeter equation [51].
Thus, we expect the light quark cluster to be an effective degree of freedom for a charmed baryon. Since the \(\rho \) mode of a cluster is not considered here, the Schrödinger equation (3) is simplified as
$$\begin{aligned} \left( -\frac{\nabla _\lambda ^2}{2m_\lambda }-\frac{4\alpha }{3\tau }+b\tau +C \right) \psi = E\psi . \end{aligned}$$
(4)

2.2 Adopted effective potentials

As a whole, the light quark cluster which occupies an antitriplet color structure interacts with the c quark. Then we would like to substitute \(\lambda \) (distance between light cluster and c quark) for \(\tau \) (distance between light quark and c quark). To this end, the effective potential [52]
$$\begin{aligned} H_{\mathrm{Q-cl.}}^{\mathrm{conf}}(\lambda )=-\frac{4}{3}\frac{\alpha _s}{\lambda }+b\lambda ^\nu -C_{Qqq'} \end{aligned}$$
(5)
describes the interaction between the cluster and c quark, where \(\nu \) is an adjustable parameter. This approximation can greatly decrease the computational complexity. As shown in Tables 2 and 3, the mass spectra given in this way are reasonable for the low-lying excited charmed baryons.
As a two-body problem, we treat the masses of different kinds of clusters as parameters and first fix them before calculating the masses of the low-lying charmed baryons. According to the flavor and spin, the light cluster can be classified into two kinds: one is the “scalar” cluster, and another is the “vector” cluster. Constrained by the Pauli exclusion principle, the total wave function of the light quark cluster should be antisymmetric in the exchange of two quarks. Because the spatial and color parts of this light quark cluster are always symmetric and antisymmetric, respectively, the function \(|\mathrm {flavor}\rangle \times |\mathrm {spin}\rangle \) should be symmetric. Therefore, the scalar light quark cluster \([qq]~(S =0)\) is always flavor antisymmetric, and the axial-vector light quark cluster \(\{qq\}~(S =1)\) is flavor symmetric. In Jaffe’s terminology [49], the “scalar” and “vector” quark clusters are named the “good” and “bad” quark clusters, respectively. The masses of the “good” light quark cluster [qq] and [qs] are taken from our previous work where \(m_{[qq]}\) and \(m_{[qs]}\) were fixed as 450 and 630 MeV by the RFT model [11], respectively. Following Jaffe’s method [49], the bad light quark cluster masses can be evaluated by the following relationships:
$$\begin{aligned} \frac{4\times \Sigma _c(2520)+2\times \Sigma _c(2455)}{6}-\Lambda _c(2286)\approx & {} 210~\mathrm {MeV},\\ \frac{4\times \Xi ^*_c(2645)+2\times \Xi '_c(2580)}{6}-\Xi _c(2470)\approx & {} 150~\mathrm {MeV}. \end{aligned}$$
Evidently, the masses of \(\{qq\}\) and \(\{qs\}\) are about 660 MeV and 780 MeV, respectively. Henceforth, we will call the \(\Lambda _c^+\) and \(\Xi _c^{0,+}\) baryons the \(\mathcal {G}\)-type baryons, and \(\Sigma _c^{0,+,++}\) and \(\Xi _c^{\prime 0,+}\) the \(\mathcal {B}\)-type baryons for convenience.
Due to an antitriplet color structure, the spin-dependent interactions between light cluster and c quark are expected to be the same as the meson systems. In a constituent quark model [53], the spin-dependent interaction is written as
$$\begin{aligned} H_S = H_{\mathrm{Q-cl.}}^{\mathrm{cont}}+H_{\mathrm{Q-cl.}}^{\mathrm{ten}}+H_{\mathrm{Q-cl.}}^{SO}. \end{aligned}$$
(6)
The color contact interaction \(H_{\mathrm{Q-cl.}}^{\mathrm{cont}}\) is usually given by the following form:
$$\begin{aligned} H_{\mathrm{Q-cl.}}^{\mathrm{cont}}=\frac{32\pi }{9}\frac{\alpha _s}{m_Qm_{\mathrm{cl.}}}\tilde{\delta }_{\sigma }(\lambda )\vec {S}_Q\cdot \vec {S}_{\mathrm{cl.}}, \end{aligned}$$
(7)
where \(\vec {S}_Q\) and \(\vec {S}_{\mathrm{cl.}}\) refer to the heavy quark and light cluster spins. A Gaussian-smeared function \(({\sigma }/\sqrt{\pi })^3e^{-{\sigma }^2\lambda ^2}\) is normally used for \(\tilde{\delta }_{\sigma }(\lambda )\) [54]. If the SU(3) flavor symmetry is kept for the charmed baryons, we may modify the color contact interaction as
$$\begin{aligned} H_{\mathrm{Q-cl.}}^{\mathrm{cont}}=\frac{32}{9\sqrt{\pi }}\frac{\alpha _s\sigma ^3}{m_Q}e^{-\sigma ^2\lambda ^2}\vec {S}_Q\cdot \vec {S}_{\mathrm{cl.}}, \end{aligned}$$
(8)
where the mass of a light quark cluster is just replaced by a unit. This assumption is supported by the mass differences of the 1S \(\mathcal {B}\)-type charmed baryons,
$$\begin{aligned}&\Sigma _c(2520)^{++}-\Sigma _c(2455)^{++}=64.44^{+0.25}_{-0.24}~\mathrm {MeV},~~~~\\&\Xi _c(2645)^{+}~-\Xi '_c(2580)^{+}=70.2\pm 3.0~\mathrm {MeV},~~~~\\&\Omega _c(2770)^0~-\Omega _c(2695)^0=70.7\pm 2.6~\mathrm {MeV}. \end{aligned}$$
The mass differences shown above are mainly due to the color contact interaction in the quark potential model. Clearly, these values are almost independent of the light quark cluster masses. The color tensor interaction in Eq. (6) is
$$\begin{aligned} H_{\mathrm{Q-cl.}}^{\mathrm{ten}}=\frac{4}{3}\frac{\alpha _s}{m_Qm_{\mathrm{cl.}}}\frac{1}{\lambda ^3}\left( \frac{3\left( \vec {S}_Q\cdot \vec {\lambda }\right) \left( \vec {S}_{\mathrm{cl.}}\cdot \vec {\lambda }\right) }{\lambda ^2}-\vec {S}_Q\cdot \vec {S}_{\mathrm{cl.}}\right) .\nonumber \\ \end{aligned}$$
(9)
Finally, \(H_{\mathrm{Q-cl.}}^{\mathrm{SO}}\) denotes the spin–orbit interaction which contains two terms. One is the color magnetic interaction which arises from one-gluon exchange,
$$\begin{aligned} H_{\mathrm{Q-cl.}}^{\mathrm{SO(cm)}}=\frac{4}{3}\frac{\alpha _s}{\lambda ^3}\left( \frac{\vec {S}\cdot \vec {L}}{m_Qm_{\mathrm{cl.}}}+\frac{\vec {S}_Q\cdot \vec {L}}{m_Q^2}+\frac{\vec {S}_{\mathrm{cl.}}\cdot \vec {L}}{m_{\mathrm{cl.}}^2}\right) , \end{aligned}$$
(10)
where \(\vec {S}\) denotes the spin of a baryon, \(\vec {S}=\vec {S}_Q+\vec {S}_{\mathrm{cl.}}\). Another spin–orbit interaction is the Thomas-precession term,
$$\begin{aligned} H_{\mathrm{Q-cl.}}^{\mathrm{SO(tp)}}=-\frac{1}{2\lambda }\frac{\partial H_{\mathrm{Q-cl.}}^{\mathrm{conf}}}{\partial \lambda }\left( \frac{\vec {S}_Q\cdot \vec {L}}{m_Q^2}+\frac{\vec {S}_{\mathrm{cl.}}\cdot \vec {L}}{m_{\mathrm{cl.}}^2}\right) . \end{aligned}$$
(11)
To reflect the importance of the heavy quark symmetry, we rewrite the spin-dependent interactions as
$$\begin{aligned} H_S=V_{ss}~\vec {S}_Q\cdot \vec {S}_{\mathrm{cl.}}+V_1~\vec {S}_{\mathrm{cl.}}\cdot \vec {L}+V_2~\vec {S}_Q\cdot \vec {j}_{\mathrm{cl.}}+V_t~\hat{S}_{12}.\nonumber \\ \end{aligned}$$
(12)
The degrees of freedom of the light quark cluster are characterized by its total angular momentum \(\vec {j}_{\mathrm{cl.}}\), i.e., \(\vec {j}_{\mathrm{cl.}}=\vec {S}_{\mathrm{cl.}}+\vec {L}\). Obviously, the orbital angular momentum \(\vec {L}\) of a charmed baryon in the present picture is defined by the angular momentum between light quark cluster and c quark, i.e., \(\vec L=\vec L_\lambda \). The tensor operator is defined as \(\hat{S}_{12}=3\left( \vec {S}_Q\cdot \vec {\lambda }\right) \left( \vec {S}_{\mathrm{cl.}}\cdot \vec {\lambda }\right) /\lambda ^2-\vec {S}_Q\cdot \vec {S}_{\mathrm{cl.}}\).
With the confining term of Eq. (5), the coefficients \(V_{ss}\), \(V_1\), \(V_2\), and \(V_t\) in Eq. (12) are defined by
$$\begin{aligned} V_{ss}= & {} \frac{1}{m_Q}\left[ \frac{32\alpha _s}{9\sqrt{\pi }}\sigma ^3e^{-\sigma ^2\lambda ^2}-\frac{1}{m_Q}\left( \frac{2\alpha _s}{3\lambda ^3}-\frac{b\nu }{2}\lambda ^{\nu -2}\right) \right. \nonumber \\&\left. -\frac{4\alpha _s}{3\lambda ^3}\frac{1}{m_{\mathrm{cl.}}}\right] ,~~~~\nonumber \\ V_1= & {} \frac{1}{m_{\mathrm{cl.}}}\left[ \frac{1}{m_{\mathrm{cl.}}}\left( \frac{2\alpha _s}{3\lambda ^3}-\frac{b\nu }{2}\lambda ^{\nu -2}\right) +\frac{4\alpha _s}{3\lambda ^3}\frac{1}{m_{Q}}\right] ,~~~~\nonumber \\ V_2= & {} \frac{1}{m_Q}\left[ \frac{1}{m_Q}\left( \frac{2\alpha _s}{3\lambda ^3}-\frac{b\nu }{2}\lambda ^{\nu -2}\right) +\frac{4\alpha _s}{3\lambda ^3}\frac{1}{m_{\mathrm{cl.}}}\right] ,~~~~\nonumber \\ V_t= & {} \frac{4\alpha _s}{3\lambda ^3}\frac{1}{m_Qm_{\mathrm{cl.}}}. \end{aligned}$$
(13)

2.3 Getting masses of charmed baryons

In our calculation, the following Schrödinger equation is solved for the nS state:
$$\begin{aligned} \left( -\frac{\nabla _\lambda ^2}{2m_\lambda }+H_{\mathrm{Q-cl.}}^{\mathrm{conf}}+H_{\mathrm{Q-cl.}}^{\mathrm{cont}}\right) \Psi =E\Psi . \end{aligned}$$
(14)
The confining and contact terms have been given by Eqs. (5) and (8). For the orbital excitations, all spin-dependent interactions are treated as the leading-order perturbations. Our calculation indicates that the color contact interaction can be ignored for the orbital excitations.
Two bases are employed to extract the mass matrix elements. One is defined by the eigenstates \(|S_{\mathrm{cl.}}, L, j_{\mathrm{cl.}}, S_Q, J\rangle \) (jj coupling scheme) and the other is by \(|S_{\mathrm{cl.}}, S_Q, S, L, J\rangle \) (LS coupling scheme). The relation between these two bases is
$$\begin{aligned} \begin{array}{ll} |[S_{\mathrm{cl.}}, L]_{j_{\mathrm{cl.}}}, S_Q\rangle _J&{}=\sum \limits _S(-1)^{S_{\mathrm{cl.}}+S_Q+L+J}\\ &{}\quad \times \sqrt{(2J_{\mathrm{cl.}}+1)(2S+1)}\\ &{}\quad \times \left\{ \begin{array}{ccc} s_{\mathrm{cl.}} &{} L &{} j_{\mathrm{cl.}}\\ J &{} S_Q &{} S\\ \end{array} \right\} |[S_{\mathrm{cl.}}, S_Q]_S, L\rangle _J. \end{array} \end{aligned}$$
(15)
Due to \(S_{\mathrm{cl.}}=0\), only \(V_2~\vec {S}_Q\cdot \vec {j}_{\mathrm{cl.}}\) contributes to the masses of \(\mathcal {G}\)-type charmed baryons. With a bad light quark cluster, however, \(\mathcal {B}\)-type charmed baryons have more complicated splitting structures. Within the framework of the heavy quark effective theory, the spin of an axial-vector light quark cluster, \(S_{\mathrm{cl.}}\), first couples with the orbital angular momentum L. As illustrated in Fig. 2, in the heavy quark limit \(m_c\rightarrow \infty \), there are only three states which are characterized by \(j_{\mathrm{cl.}}\) for 1P charmed baryons. When the heavy quark spin \(S_Q\) couples with \(j_{\mathrm{cl.}}\), the degeneracy is resolved and the five states appear. These are two \(J^P=1/2^-\), two \(J^P=3/2^-\), and one \(J^P=5/2^-\) states. Lastly, the states with the same \(J^P\) mix with each other by the interactions of \(V_{ss}~\vec {S}_Q\cdot \vec {S}_{\mathrm{cl.}}\) and \(V_t~\hat{S}_{12}\), and physical states are formed.
Fig. 2

A schematic diagram for the splittings of p-wave \(\Sigma _c\) and \(\Xi '_c\). Here \(\vec j_l=\vec L+\vec j_{\mathrm{cl.}}\) and subindices \(\ell \) and h of the last column mean low and high states in mass after including \(\tilde{V}_{ss}\) and \(\tilde{V}_t\) interactions

We now turn to a calculation of the mass matrix in the jj coupling scheme. For 1P states with \(J^P=1/2^-\), the mass matrix is given by
$$\begin{aligned} \langle \Phi _{1/2}\mid H_S\mid \Phi _{1/2}\rangle =\left( \begin{array}{ccc} -2V_1-4V_t &{}\quad \frac{V_{ss}-4V_t}{\sqrt{2}}\\ \frac{V_{ss}-4V_t}{\sqrt{2}} &{}\quad -V_1-V_2-\frac{V_{ss}}{2}-2V_t \\ \end{array} \right) . \end{aligned}$$
Similarly, for two states with \(J^P=3/2^-\),
$$\begin{aligned} \begin{array}{ll} &{}\langle \Phi _{3/2} \mid H_S\mid \Phi _{3/2}\rangle \\ &{}\quad =\left( \begin{array}{ccc} -V_1+\frac{V_2}{2}+\frac{V_{ss}}{4}+4V_t &{}\quad \frac{5V_{ss}+16V_t}{4\sqrt{5}}\\ \frac{5V_{ss}+16V_t}{4\sqrt{5}} &{}\quad V_1-\frac{3V_2}{2}-\frac{3V_{ss}}{4}+\frac{4V_t}{5} \\ \end{array} \right) . \end{array} \end{aligned}$$
For the \(J^P=5/2^-\) state,
$$\begin{aligned}&\langle j_l=2, J^P=5/2^- \mid H_S\mid j_l=2, J^P=5/2^-\rangle \\&\quad =V_1+V_2+\frac{1}{2}V_{ss}-\frac{6}{5}V_{t}. \end{aligned}$$
In the following, we denote \(|S_{\mathrm{cl.}}, L, j_{\mathrm{cl.}}, S_Q, J\rangle \) as \(|j_{\mathrm{cl.}}, J^P\rangle \) for brevity. Then the notations \(\mid \Phi _{1/2}\rangle \) and \(\mid \Phi _{3/2}\rangle \) appearing above are defined by
$$\begin{aligned} \begin{array}{l} \mid \Phi _J\rangle =\left( \begin{array}{cc} \mid j_l=J-1/2, J^P\rangle \\ \mid j_l=J+1/2, J^P\rangle \\ \end{array} \right) . \end{array} \end{aligned}$$
The mass matrix of 1D states can also be obtained by the similar procedure. As shown above, there are seven parameters in the non-relativistic quark potential model, which are \(m_Q\), \(m_{\mathrm{cl.}}\), b, \(\alpha \), \(\gamma \), \(\nu \), and \(C_{Qqq'}\). All values of parameters are listed in Table 1. If the SU(3) flavor symmetry is taken into account for the charmed and charmed-strange baryons, the dynamics of \(\Lambda _c^+\) states should be like \(\Xi _c\). The case of \(\Sigma _c\) and \(\Xi '_c\) is alike. Accordingly, the same value of \(\gamma \) is selected for the \(\mathcal {G}\)-type charmed baryons, as well as the case of \(\mathcal {B}\)-type.
Table 1

Values of the parameters of the non-relativistic quark potential model. The unit of b is GeV\(^{\nu +1}\) which varies depending on each value of \(\nu \)

\(m_c\)

1.68 GeV

b

0.145

\(C_{\Lambda _C}\)

0.233 GeV

\(m_{[qq]}\)

0.45 GeV

\(\alpha \)

0.45

\(C_{\Sigma _C}\)

0.100 GeV

\(m_{[qs]}\)

0.63 GeV

\(\nu _{[\Lambda _c,\Xi _c]}\)

0.84

\(C_{\Xi _C}\)

0.156 GeV

\(m_{\{qq\}}\)

0.66 GeV

\(\nu _{[\Sigma _c,\Xi '_c]}\)

0.70

\(C_{\Xi '_C}\)

0.060 GeV

\(m_{\{qs\}}\)

0.78 GeV

\(\sigma \)

1.00 GeV

  
We have adopted the typical values in the quark potential models for \(m_c\), b, \(\alpha \), and \(\nu \) (see Table 1). It is an effective method to investigate charmed baryons in the heavy quark–light quark cluster picture. We do not expect the values of \(\nu \) to be the same for \(\mathcal {G}\)-type and \(\mathcal {B}\)-type baryons. Here, \(\nu \) of \(\Lambda _c^+/\Xi _c\) is slightly larger than \(\Sigma _c/\Xi '_c\). The predicted masses of the low-excited charmed baryons are collected in Tables 2 and 3.
Table 2

Predicted masses for \(\Lambda _c^+\) and \(\Xi _c\) states of ours and other approaches in Refs. [10, 11, 56, 57] compared to experimental data [1] (in MeV)

States

\(\Lambda _c^+\) baryons

\(\Xi _c\) baryons

 

PDG [1]

Prediction

Ref. [10]

Ref. [11]

Refs. [56]

PDG [1]

Prediction

Ref. [10]

Ref. [11]

Ref. [57]

\(\mid 1S, 1/2^+\rangle \)

2286.86

2286

2286

2286

2265

2470.88

2470

2476

2467

2466

\(\mid 2S, 1/2^+\rangle \)

2766.6

2772

2769

2766

2775

2968.0

2940

2959

2959

2924

\(\mid 3S, 1/2^+\rangle \)

 

3116

3130

3112

3170

 

3265

3323

3325

 

\(\mid 1P, 1/2^-\rangle \)

2592.3

2614

2598

2591

2630

2791.8

2793

2792

2779

2773

\(\mid 1P, 3/2^-\rangle \)

2628.1

2639

2627

2629

2640

2819.6

2820

2819

2814

2783

\(\mid 1D, 3/2^+\rangle \)

 

2843

2874

2857

2910

3054.2

3033

3059

3055

3012

\(\mid 1D, 5/2^+\rangle \)

2881.53

2851

2880

2879

2910

3079.9

3040

3076

3076

3004

\(\mid 2P, 1/2^-\rangle \)

2939.3

2980

2983

2989

3030

3122.9

3140

3179

3195

 

\(\mid 2P, 3/2^-\rangle \)

 

3004

3005

3000

3035

 

3164

3201

3204

 
Table 3

Predicted masses for \(\Sigma _c\) and \(\Xi '_c\) states of ours and other approaches in Refs. [9, 10, 56, 58] compared to experimental data [1] (in MeV)

States

\(\Sigma _c\) baryons

\(\Xi '_c\) baryons

 

PDG [1]

Prediction

Ref. [9]

Ref. [10]

Refs. [56]

Ref. [58]

PDG [1]

Prediction

Ref. [9]

Ref. [10]

\(\mid 1S, 1/2^+\rangle \)

2452.9

2456

2439

2443

2440

2452

2575.6

2579

2579

2579

\(\mid 1S, 3/2^+\rangle \)

2517.5

2515

2518

2519

2495

2501

2645.9

2649

2654

2649

\(\mid 2S, 1/2^+\rangle \)

2846\(^\mathrm{a}\)

2850

2864

2901

2890

2961

 

2977

2984

2983

\(\mid 2S, 3/2^+\rangle \)

 

2876

2912

2936

2985

2996

 

3007

3035

3026

\(\mid 3S, 1/2^+\rangle \)

 

3091

 

3271

3035

3381

 

3215

 

3323

\(\mid 3S, 3/2^+\rangle \)

 

3109

 

3293

3200

3403

 

3236

 

3396

\(\mid 1P, 1/2^-\rangle _l\)

 

2702

2795

2713

2765

2832

 

2839

2928

2854

\(\mid 1P, 1/2^-\rangle _h\)

2766.6

2765

2805

2799

2770

2841

 

2900

2934

2936

\(\mid 1P, 3/2^-\rangle _l\)

 

2785

2761

2773

2770

2812

2931

2921

2900

2912

\(\mid 1P, 3/2^-\rangle _h\)

2801

2798

2798

2798

2805

2822

 

2932

2931

2935

\(\mid 1P, 5/2^-\rangle \)

 

2790

2799

2789

2815

2796

 

2927

2921

2929

\(\mid 1D, 1/2^+\rangle \)

 

2949

3014

3041

3005

  

3075

3132

3163

\(\mid 1D, 3/2^+\rangle _l\)

 

2952

3005

3040

3060

  

3089

3127

3160

\(\mid 1D, 3/2^+\rangle _h\)

 

2964

3010

3043

3065

  

3081

3131

3167

\(\mid 1D, 5/2^+\rangle _l\)

 

2942

2960

3023

3065

  

3091

3087

3153

\(\mid 1D, 5/2^+\rangle _h\)

 

2962

3001

3038

3080

  

3077

3123

3166

\(\mid 1D, 7/2^+\rangle \)

 

2943

3015

3013

3090

  

3078

3136

3147

\(\mid 2P, 1/2^-\rangle _l\)

 

2971

3176

3125

3185

3245

 

3094

3294

3267

\(\mid 2P, 1/2^-\rangle _h\)

 

3018

3186

3172

3195

3256

 

3144

3300

3313

\(\mid 2P, 3/2^-\rangle _l\)

 

3036

3147

3151

3195

3223

 

3172

3269

3293

\(\mid 2P, 3/2^-\rangle _h\)

 

3044

3180

3172

3210

3233

 

3165

3296

3311

\(\mid 2P, 5/2^-\rangle \)

 

3040

3167

3161

3220

3203

 

3170

3282

3303

\(^\mathrm{a}\) The mass value for the \(\mid 2S, 1/2^+\rangle \) state is taken from the measurement of BaBar [35]

As mentioned earlier, the nonzero off-diagonal elements in mass matrices of \(\langle \Phi _{1/2}\mid H_S\mid \Phi _{1/2}\rangle \) and \(\langle \Phi _{3/2}\mid H_S\mid \Phi _{3/2}\rangle \) cause the mixing between two states with the same \(J^P\) but different \(j_{cl.}\). However, the mechanism of mixing effects in hadron physics is still unclear. In principle, a physical hadron state with a specific \(J^P\) comprises all possible Fock states with the same total spin and parity. As the most famous member of the XYZ family, X(3872) may be explained as a mixture of charmonium and the molecular state with \(J^{PC}=1^{++}\) [55]. Here we take the \(|j_{cl.}, J^P\rangle \) basis to describe the mixing for the \(\mathcal {B}\)-type baryons. Then two physical states characterized by different masses can be denoted
$$\begin{aligned} \begin{array}{ll} \left( \begin{array}{c} |\mathrm{High}, J^P\rangle \\ |\mathrm{Low}, J^P\rangle \\ \end{array} \right) &{}=\left( \begin{array}{cc} ~\cos \phi &{}\quad \sin \phi \\ -\sin \phi &{}\quad \cos \phi \\ \end{array} \right) \left( \begin{array}{c} |J-1/2, J^P\rangle \\ |J+1/2, J^P\rangle \\ \end{array} \right) . \end{array}\nonumber \\ \end{aligned}$$
(16)
For example, two 1P \(\Sigma _c\) states with \(J^P=1/2^-\) can be represented as
$$\begin{aligned} \begin{array}{ll} \left( \begin{array}{c} \Sigma _c(2765)\\ \Sigma _c(2702)\\ \end{array} \right) &{}=\left( \begin{array}{cc} ~\cos 125.4^\circ &{}\quad \sin 125.4^\circ \\ -\sin 125.4^\circ &{}\quad \cos 125.4^\circ \\ \end{array} \right) \\ &{}\quad \times \left( \begin{array}{c} |0, 1/2^-\rangle \\ |1, 1/2^-\rangle \\ \end{array} \right) . \end{array} \end{aligned}$$
(17)
Here we have denoted the physical states by their masses (see Table 3). The mixing angles for other states in Table 3 with the same \(J^P\) are listed in Table 4.
Our results of the mixing angles in Table 4 indicate that the heavier \(1/2^-\) state, \(\Sigma _c(2765)\), is dominated by a \(|1, 1/2^-\rangle \) component, while \(\Sigma _c(2702)\) is dominated by a \(|0, 1/2^-\rangle \) component. For two \(3/2^-\) states, the light \(\Sigma _c(2785)\) is dominated by \(|2, 1/2^-\rangle \), and the heavy \(\Sigma _c(2798)\) by \(|1, 1/2^-\rangle \). The mixing of 2P states is similar to the 1P states. For the 1D states, one notices that both \(3/2^+\) and \(5/2^+\), with heavier masses, are dominated by smaller \(j_{\mathrm{cl.}}\) components.
Table 4

The mixing angles for the 1P, 2P, and 1D \(\Sigma _c/\Xi '_c\) states

 

\(1P(1/2^-)\)

\(1P(3/2^-)\)

\(2P(1/2^-)\)

\(2P(3/2^-)\)

\(1D(3/2^+)\)

\(1D(5/2^+)\)

\(\Sigma _c\)

\(125.4^\circ \)

\(-156.8^\circ \)

\(124.8^\circ \)

\(-151.4^\circ \)

\(172.2^\circ \)

\(-175.6^\circ \)

\(\Xi '_c\)

\(125.0^\circ \)

\(-153.6^\circ \)

\(124.3^\circ \)

\(-145.1^\circ \)

\(168.9^\circ \)

\(-173.8^\circ \)

The uncertainty may exist in the mixing angles. Firstly, the loop corrections to the spin-dependent one-gluon-exchange potential may be important for the heavy–light hadrons. As an example, the lower mass of \(D_s(2317)^\pm \) compared with the old calculations [53] can be well explained by the corrected spin-dependent potential [59, 60]. If we use this type of potential in our calculation, of course, the mixing angle will change. Secondly, the mixing angles depend on the parameters. Thirdly, there are other mechanisms, e.g., hadron loop effects [61], which may contribute to the mixing phenomenon in hadron physics. Anyway, we expect that the mixing angles in Table 4 reflect the main features of the mixing states. Due to the uncertainties of the mixing angles, however, we ignore the mixing effects as the first step to the study of the decays of charmed excitations in the next subsection. Obviously, it is a good approximation only when the mixing effects are not large. Fortunately, this crude procedure is partially supported by the former analysis of charmed mesons [62, 63, 64]. If the decay properties obtained in this way describe the principal characteristics of the mixing states, the angles obtained by the potential model may be overestimated.

2.4 Simple harmonic oscillator (SHO) \(\beta \) values

In the next section, the Okubo–Zweig–Iizuka (OZI) allowed decays will be calculated for the 1P and 2S charmed baryons where the SHO wave functions are used to evaluate the transition factors via the \(^3P_0\) model. We will also discuss the mixing effects for the decays of the relevant states. Following the method of Ref. [65], all values of the SHO wave function scale, denoted \(\beta \) in the following, are calculated (see Table 5). The values of \(\beta \) reflect the distances between the light quark cluster and the c quark.
Table 5

The meson effective \(\beta \) values in GeV

States

\(\Lambda _c^+\)

\(\Xi _c\)

\(\Sigma _c\)

\(\Xi '_c\)

1S

    

   \(1/2^+\)

0.291

0.331

0.335

0.362

   \(3/2^+\)

  

0.296

0.315

2S

0.145

0.162

0.144

0.152

3S

0.102

0.113

0.098

0.103

1P

0.184

0.205

0.182

0.192

2P

0.117

0.130

0.112

0.118

1D

0.142

0.156

0.136

0.143

In our calculation of strong decays, we will consider the structures of light diquarks. What is more important is that the possible final states of an excited charmed baryon may contain a light flavor meson, a charmed meson, a light flavor baryon, e.g., \(\pi \), K, D, p, and \(\Lambda \). For the \(\beta \) of these hadrons, the following potential will be used:whereHere, the parameters \(\alpha _s\) and b are taken as 0.45 and 0.145 GeV\(^{\nu +1}\) as in Table 1, respectively. To reproduce the masses of the light quark clusters in Table 1, the masses of u / d, s are fixed as 0.195 and 0.380 GeV. While \(\sigma \) and C are treated as adjustable parameters, the masses of the \(\pi /\rho \), \(K/K^*\), \(D/D^*\), \(p/\Delta \), and \(\Lambda \) families are fitted with experimental data. Meanwhile, the values of \(\beta \) for the corresponding states are also obtained; they are collected in Table 6.
Table 6

The effective \(\beta \) values in GeV for the light quark cluster and various hadrons (the second row). The values of \(\sigma \) and C are given in the square brackets for various hadron structures (the third row)

[qq]

\(\{qq\}\)

[qs]

\(\{qs\}\)

\(\pi \)

\(\rho \)

K

\(K^*\)

D

\(D^*\)

p

\(\Lambda \)

0.201

0.143

0.207

0.159

0.298

0.179

0.291

0.201

0.250

0.230

0.189

0.226

[1.17, 0.39]

[1.57, 0.38]

[0.73, 0.63]

[0.83, 0.48]

[1.20, 0.63]

[−, 0.38]

[−, 0.26]

Before ending this section, we briefly summarize the complicated deduction presented here. Firstly, the dynamics of heavy baryon is simplified as a two-body system when the symmetric configuration is considered. Secondly, the mass matrices were calculated in the jj coupling scheme. By solving the Schrödinger equation, we obtain the mass spectra and mixing angles for the relevant states. For estimating the two-body strong decays in the next section, finally, we also present the values of the SHO wave function scale for all initial and final states.

3 Strong decays

In this section, we will use the formula provided by Eichten, Hill, and Quigg (EHQ) [66] to extract the decay widths of excited charmed baryons. Since the dynamical behavior of the heavy–light hadrons is governed by the light degrees of freedom in the limit of heavy quark symmetry, a doublet formed by two states with the same \({j}_{\mathrm{cl.}}\) but different J shall have similar decay properties. More specifically, the transitions between two doublets should be determined by a single amplitude which is proportional to the products of four Clebsch–Gordan coefficients [46]. Some typical ratios of excited charmed baryons with negative parity were predicted by this law [46]. Later, a more concise formula (the EHQ formula) was proposed for the widths of heavy–light mesons [66]. The EHQ formula has been applied systematically to the decays of excited open-charm mesons [62, 63, 64]. Recently, the EHQ formula has been extended to the study of the decay properties of 1D \(\Lambda _c\) and \(\Xi _c\) states [11].
Fig. 3

The two topological diagrams for an excited charmed baryon A decaying into the final states B and C. The brown line 3 denotes a charm quark

For the charmed baryons, the EHQ formula can be written as
$$\begin{aligned} \Gamma ^{A\rightarrow BC}_{j_C,\ell } = \xi \,\left( \mathcal {C}^{s_Q,j_B,J_B}_{j_C,j_A,J_A}\right) ^2\left| \mathcal {M}^{j_A,j_B}_{j_C,\ell }(q)\right| ^2 \,q^{2\ell +1}\, e^{-q^2/\tilde{\beta }^2},\nonumber \\ \end{aligned}$$
(20)
where \(\xi \) is the flavor factor given in Table 13 in Appendix B. \(q=|\vec q|\) denotes the three-momentum of a final state in the rest frame of an initial state. A and B represent the initial and final heavy–light hadrons, respectively. C denotes the light flavor hadron (see Fig. 3). The explicit expression of \(\tilde{\beta }\) is given in Eq. (A.11) in Appendix A. In addition, \(\mathcal {C}^{s_Q,j_B,J_B}_{j_C,j_A,J_A}\) is a normalized coefficient given by the following equation:
$$\begin{aligned} \begin{array}{ll} \mathcal {C}^{s_Q,j_B,J_B}_{j_C,j_A,J_A}&{}=(-1)^{J_A+j_B+j_C+s_Q}~\sqrt{(2j_A+1)(2J_B+1)}\\ &{}\quad \times \left\{ \begin{array}{ccc} s_Q &{} j_B &{} J_B\\ j_C &{} J_A &{} j_A\\ \end{array} \right\} , \end{array} \end{aligned}$$
(21)
where \(\vec {j}_C \equiv \vec {s}_C + \vec {\ell }\). The symbols \(s_C\) and \(\ell \) represent the spin of the light hadron C and the orbital angular momentum relative to B, respectively. The transition factors \(\mathcal {M}^{j_A,j_B}_{j_C,\ell }(q)\) involved in the concrete dynamics can only be calculated by various phenomenological models. For the decays of heavy–light mesons, transition factors have been calculated by the relativistic chiral quark model [67] and the \(^3P_0\) model [62, 64, 68]. In our work, we will employ the \(^3P_0\) model [69, 70, 71] to obtain the transition factors. More details of an estimate of the transition factors are given in Appendix A.

4 Discussion

4.1 Experimentally well-established 1S and 1P states

Table 7

The masses and widths (in units of MeV) of \(\Sigma _c(2455)^{++}\) and \(\Sigma _c(2520)^{++}\) measured by CDF [2] and Belle [3]

\(\Sigma _c(2455)^{++}\)

2453.90 ± 0.13 ± 0.14

2.34 ± 0.47

CDF

 

2453.97 ± 0.01 ± 0.02 ± 0.14

1.84 ± 0.04\(^{+0.07}_{-0.20}\)

Belle

\(\Sigma _c(2520)^{++}\)

2517.19 ± 0.46 ± 0.14

15.03 ± 2.52

CDF

 

2518.45 ± 0.10 ± 0.02 ± 0.14

14.77 ± 0.25\(^{+0.18}_{-0.30}\)

Belle

At present, all the ground states and 1P \(\mathcal {G}\)-type charmed states have been experimentally established [1]. These states have been observed, at least, by two different collaborations, and their properties including masses and decays have been well determined. With good precision, the strong decays of these states provide a crucial test of our method.

Among the 1S charmed baryons, the measurements of \(\Sigma _c(2455)\) and \(\Sigma _c(2520)\) have been largely improved [2, 3] (see Table 7). In our calculation, the mass and decay width of \(\Sigma _c(2520)^{++}\) measured by CDF will be taken as input data to fix the constant \(\gamma \) peculiar to the \(^3P_0\) model. With the transition factor for the process \(\Sigma _c(2520)\rightarrow \Lambda _c(2286)+\pi \) [see Eq. (A.12) in Appendix A], the value of \(\gamma \) is fixed as 1.296.2

As shown in Tables 8 and 9, the predicted widths of other ground and \(1P\, \mathcal {G}\)-type charmed baryons are well consistent with experiments. Our results of mass spectra and decay widths indicate that \(\Lambda _c(2595)^+\), \(\Lambda _c(2625)^+\), \(\Xi _c(2790)^{0,+}\), and \(\Xi _c(2815)^{0,+}\) can be accommodated with the 1P \(\mathcal {G}\)-type charmed baryons. \(\Lambda _c(2595)^+\) and \(\Xi _c(2790)^{0,+}\) can be classified into the \(1/2^-\) states while \(\Lambda _c(2625)^+\) and \(\Xi _c(2815)^{0,+}\) into the \(3/2^-\) states. The predicted mass splittings between the 1P \(1/2^-\) and \(3/2^-\) states are 25 MeV and 27 MeV for the \(\Lambda _c\) and \(\Xi _c\) baryons, respectively, which are also consistent with the experiments. The assignments of \(\Lambda _c(2595)^+\), \(\Lambda _c(2625)^+\), \(\Xi _c(2790)^{0,+}\), and \(\Xi _c(2815)^{0,+}\) are also supported by other work [9, 10, 11, 12] in which the light quark cluster scenario was also employed. In addition, the mass spectra obtained by different types of the quark potential models in the three-body picture also support these assignments [7, 8, 56, 57, 58]. However, the investigations by QCD sum rules indicate that these 1P candidates may have more complicated structures [14, 15, 16]. Especially, the work by Chen et al. suggested that \(\Lambda _c(2595)^+\) and \(\Lambda _c(2625)^+\) form the heavy doublet \(\tilde{\Lambda }_{c1}(1/2^-, 3/2^-)\) (the same assignments as the case of \(\Xi _c(2790)^{0,+}\) and \(\Xi _c(2815)^{0,+}\)) [16], which is different from our conclusion. Since the quantum numbers of \(J^P\) have not yet been determined for these 1P charmed states, more experiments are required in the future.
Table 8

Open-flavor strong decay widths of 1S \(\Sigma _c\) and \(\Xi '_c\) in MeV

1S \(\Sigma _c\) and \(\Xi '_c\)

\(1/2^+\)

\(3/2^+\)

\(\Sigma _c(2455)^{++}\)

\(\Xi '_c(2580)^+\)

\(\Sigma _c(2520)^{++}\)

\(\Xi '_c(2645)^+\)

\(\Lambda _c^+\pi ^+\)

1.53

  

\(\Lambda _c^+\pi ^+\)

Input

\(\Xi _c^0\pi ^+\)

1.54

      

\(\Xi _c^+\pi ^0\)

1.01

1.53

Input

2.55

\(1.89^{+0.09}_{-0.18}\) [1]

\(14.9\pm 1.5\) [1]

\(2.6\pm 0.6\) [72]

Table 9

Open-flavor strong decay widths of 1P \(\Lambda _c\) and \(\Xi _c\) in MeV

1P \(\Lambda _c\) and \(\Xi _c\)

\(1/2^-\)

\(3/2^-\)

\(\Lambda _c(2595)^+\)

\(\Xi _c(2790)^+\)

\(\Lambda _c(2625)^+\)

\(\Xi _c(2815)^+\)

\(\Sigma _c\pi \)

2.78

\(\Xi '_c\pi \)

6.01

\(\Sigma _c\pi \)

0.04

\(\Xi '_c\pi \)

0.15

      

\(\Xi ^*_c\pi \)

4.09

2.78

6.01

0.04

4.24

\(2.6\pm 0.6\) [1]

\(8.9\pm 1.4\) [4]

<0.97 [1]

\(2.43\pm 0.37\) [4]

4.2 1P \(\Sigma _c^{~0,+,++}\) states

Table 10

The partial and total decay widths of 1P \(\Sigma _c\) states in MeV

Decay modes

\(1/2^-~(1P)\)

\(3/2^-~(1P)\)

\(5/2^-~(1P)\)

 

\(\Sigma _{c0}(2702)\)

\(\Sigma _{c1}(2765)\)

\(\Sigma _{c1}(2798)\)

\(\Sigma _{c2}(2785)\)

\(\Sigma _{c2}(2790)\)

\(\Lambda _c\pi \)

3.64

\(\times \)

\(\times \)

24.06

24.63

\(\Sigma _c(2455)\pi \)

\(\times \)

58.94

3.48

5.22

2.50

\(\Sigma _c(2520)\pi \)

\(\times \)

1.70

63.72

2.47

4.34

\(\Lambda _c(2595)\pi \)

 

2.88

2.31

1.93

0.03

\(\Lambda _c(2625)\pi \)

  

3.12

0.07

0.63

Theory

3.64

63.52

72.63

33.75

32.13

Expt. [1]

 

\(\approx \)50

 

\(72^{+22}_{-15}\)

 

As shown in Tables 2 and 3, the masses of 1P \(\Sigma _c\) states are predicted in the range of 2700–2800 MeV. Then \(\Sigma _c(2765)^+\) and \(\Sigma _c(2800)^{0,+,++}\) can be grouped into the candidates of 1P \(\Sigma _c\) family. The predicted mass of the \(|1P,1/2^-\rangle _h\) state is about 2765 MeV, which is in good agreement with the measured mass of \(\Sigma _c(2765)^+\). In addition, the theoretical result for the decay width of the \(\Sigma _{c1}(1/2^-)\) state in Table 10 is about 63.52 MeV, which is also in agreement with the measurements [30, 31, 32]. Furthermore, the signal of \(\Sigma _c(2765)^+\) has been observed in the \(\Sigma _c(2455)\pi \) intermediate state, while there is no clear evidence for the decay of \(\Sigma _c(2765)^+\) through \(\Sigma _c(2520)\pi \) [31, 32]. This is also consistent with our results of the \(|1P,1/2^-\rangle _h\). Based on the combined analysis of the mass spectrum and strong decays, we therefore conclude that \(\Sigma _c(2765)^+\) could be regarded as a good candidate of \(\Sigma _{c1}(1/2^-)\). Considering the uncertainties of the quark potential models, the masses obtained by Refs. [9, 10, 56] are not contradictory to our assignment to \(\Sigma _c(2765)^+\).

According to the predicted masses in Table 3, \(\Sigma _c(2800)^{0,+,++}\) could be assigned to either the \(|1P,3/2^-\rangle _l\), or the \(|1P,3/2^-\rangle _h\), or the \(|1P,5/2^-\rangle \) states. When we consider the decay properties of these three states (see Table 10), the possibility of assignment to the \(|1P,3/2^-\rangle _l\) state can be excluded since the Belle Collaboration observed this state in the \(\Lambda _c^+\pi \) mode.3 At present, the Belle Collaboration tentatively identified \(\Sigma _c(2800)^{0,+,++}\) as members of the \(\Sigma _{c2}(3/2^-)\) isospin triplet, which agrees with our results of both mass spectrum and strong decays. When the measured mass of \(\Sigma _c(2800)^0\) (2806 MeV) is used for the \(\Sigma _{c2}(3/2^-)\) state, the predicted width is about 40.1 MeV, which is comparable with the experiment [34]. However, we notice that the quantum number \(J^P\) of \(\Sigma _c(2800)^{0,+,++}\) has not yet been measured. Then the possibility of this state as the \(\Sigma _{c2}(5/2^-)\) candidate cannot be excluded by our results since the decay mode of \(\Lambda _c^+\pi \) is dominant for this state. In addition, the predicted mass and total width of the \(\Sigma _{c2}(5/2^-)\) state are also compatible with experimental data of the \(\Sigma _c(2800)^{0,+,++}\) baryon. Therefore, we would like to point out that the signal of \(\Sigma _c(2800)^{0,+,++}\) found by Belle might be their overlapping structure. We hope the future experiments measure the following branching ratios to disentangle this state.
Table 11

The partial and total decay widths of 1P \(\Xi ^\prime _c\) states in MeV

Decay modes

\(1/2^-~(1P)\)

\(3/2^-~(1P)\)

\(5/2^-~(1P)\)

 

\(\Xi '_{c0}(2839)\)

\(\Xi '_{c1}(2900)\)

\(\Xi '_{c1}(2932)\)

\(\Xi '_{c2}(2921)\)

\(\Xi '_{c2}(2927)\)

\(\Lambda _cK\)

46.59

\(\times \)

\(\times \)

11.59

12.43

\(\Xi _c\pi \)

4.39

\(\times \)

\(\times \)

7.42

7.75

\(\Xi '_c(2580)\pi \)

\(\times \)

9.44

0.76

1.20

0.57

\(\Xi '_c(2645)\pi \)

\(\times \)

0.52

3.23

0.75

1.31

\(\Xi _c(2790)\pi \)

  

0.01

  

Theory

50.98

9.96

4.00

20.96

22.06

Expt.

   

\(36\pm 7\pm 11\) [36]

For the \(\Sigma _{c2}(3/2^-)\) state,
$$\begin{aligned} \frac{\mathcal {B}(\Sigma _{c2}(3/2^-)\rightarrow \Sigma _c(2455)~\pi )}{\mathcal {B}(\Sigma _{c2}(3/2^-)\rightarrow \Sigma _c(2520)~\pi )}~=~1.90; \end{aligned}$$
(22)
$$\begin{aligned} \frac{\mathcal {B}(\Sigma _{c2}(3/2^-)\rightarrow \Lambda _c(2286)~\pi )}{\mathcal {B}(\Sigma _{c2}(3/2^-)\rightarrow \Sigma _c(2455)~\pi )}~=~4.07. \end{aligned}$$
(23)
For the \(\Sigma _{c2}(5/2^-)\) state,
$$\begin{aligned} \frac{\mathcal {B}(\Sigma _{c2}(5/2^-)\rightarrow \Sigma _c(2455)~\pi )}{\mathcal {B}(\Sigma _{c2}(5/2^-)\rightarrow \Sigma _c(2520)~\pi )}~=~0.58; \end{aligned}$$
(24)
$$\begin{aligned} \frac{\mathcal {B}(\Sigma _{c2}(5/2^-)\rightarrow \Lambda _c(2286)~\pi )}{\mathcal {B}(\Sigma _{c2}(5/2^-)\rightarrow \Sigma _c(2455)~\pi )}~=~9.85. \end{aligned}$$
(25)
As mentioned earlier, the signal \(\Sigma _c(2850)^0\) discovered by the BaBar Collaboration may be a \(J=1/2\) state. If \(\Sigma _c(2850)^0\) is the \(1/2^+(2S)\) state, the corresponding ratios (see Sect. 4.5) are different from Eqs. (22)–(25). So the measurements of these ratios of branching fractions can help us understand the nature of \(\Sigma _c(2800)^{0,+,++}\) and \(\Sigma _c(2850)^0\).
Although, at present, the \(\Sigma _{c0}(1/2^-)\) and \(\Sigma _{c1}(3/2^-)\) states are still missing in the experiments, our results indicate that the \(\Sigma _{c0}(1/2^-)\) state may be a narrow resonance and its predominant decay channel \(\Lambda _c^+\pi \) is only about 3.64 MeV (see Table 10). Since the decay mode of \(\Sigma _c(2520)~\pi \) is the largest for the \(\Sigma _{c1}(3/2^-)\) state, we suggest to search this channel for this state in future experiments. In the heavy quark limit, the branching ratio for the \(\Sigma _{c1}(3/2^-)\) state,
$$\begin{aligned} \frac{\mathcal {B}(\Sigma _{c1}(3/2^-)\rightarrow \Sigma _c(2455)~\pi )}{\mathcal {B}(\Sigma _{c1}(3/2^-)\rightarrow \Sigma _c(2520)~\pi )}~=~0.04, \end{aligned}$$
(26)
is much smaller than \(\Sigma _{c2}(3/2^-)\) (Eq. 22).

4.3 1P \(\Xi _c^{\prime ~0,+}\) states

As shown in Table 3, the predicted mass of 1P \(\Xi _c^\prime \) is in the range from 2840 to 2930 MeV. Then the resonance structure observed by BaBar [36] in the decay channel \(B^-\rightarrow \Xi _c^{\prime }(2930)^0\bar{\Lambda }_c^-\rightarrow \Lambda ^+_cK^-\bar{\Lambda }_c^-\) with an invariant mass of 2.93 GeV could be a good candidate of 1P \(\Xi _c^\prime \) members. The results of the decays in Table 11 favor \(\Xi _c^{\prime }(2930)^0\) as the \(\Xi '_{c2}(3/2^-)\) or \(\Xi '_{c2}(5/2^-)\) state. Then \(\Xi _c^{\prime }(2930)^0\) might be regarded as the strange partner of \(\Sigma _c(2800)^{0,+,++}\) by our results. Interestingly, the mass difference between \(\Xi _c^{\prime }(2930)^0\) and \(\Sigma _c(2800)^{0,+,++}\) is about 130 MeV, which is comparable with the mass differences among sextet states of the ground charmed baryons [21]. With a chiral quark model, Liu et al. also analyzed the \(\Xi _c^{\prime }(2930)^0\) by the two-body strong decays [23]. Their results support \(\Xi _c^{\prime }(2930)^0\) as the \(|\Xi _c^{\prime 2}P_\lambda ,~1/2^-\rangle \) or \(|\Xi _c^{\prime 4}P_\lambda ,~1/2^-\rangle \) state. Since the heavy quark symmetry was not considered in Ref. [23], the notations of charm-strange baryons in Ref [23] are different from our \(\Xi '_{c0}(1/2^-)\) and \(\Xi '_{c1}(1/2^-)\). Although the results in Table 11 indicate that the \(\Lambda _c^+K\) decay mode dominates the decay of \(\Xi ^\prime _{c0}(1/2^-)\) state, the mass of this state is predicted to be about 2840 MeV, which is much smaller than \(\Xi _c^{\prime 0}(2930)\). In addition, the \(\Lambda _c^+K\) decay mode is forbidden for the \(\Xi ^\prime _{c1}(1/2^-)\) state. Thus, according to our results, \(\Xi _c^{\prime 0}(2930)\) is unlikely to be a 1P state with \(J^P=1/2^-\).

Another charm-strange baryon, \(\Xi _c(2980)^{0,+}\), is slightly higher than the predicted mass range of 1P \(\Xi _c^\prime \) states. This state has been observed in \(\Sigma _c(2455)K\), \(\Xi ^\prime _c(2580)\pi \), \(\Xi ^\prime _c(2645)\pi \), and nonresonant \(\Lambda ^+_c\bar{K}\pi \) decay channels. However, it was not seen in the decay modes of \(\Lambda ^+_c\bar{K}\) and \(\Xi _c\pi \) [37, 38, 39]. Comparing the mass and decay properties of \(\Xi _c(2980)^{0,+}\) with our results, the possibility of a 1P \(\Xi ^\prime _c\) state might be excluded. As shown in the next subsection, \(\Xi _c(2980)^{0,+}\) could be a good 2S \(\Xi _c\) candidate. Based on our results on strong decays, we find that the \(\Xi ^\prime _{c1}(1/2^-)\) and \(\Xi ^\prime _{c1}(3/2^-)\) are quite narrow (see Table 11).
Table 12

The partial and total decay widths of 2S \(\Lambda _c^+\) and \(\Xi _c^{+,0}\) states in MeV

\(1/2^+~(2S)\)

\(1/2^+~(2S)'\)

\(3/2^+~(2S)'\)

\(\Lambda _c(2765)^+\)

\(\Xi _c(2980)\)

\(\Sigma _c(2850)^0\)

\(\Xi '_c(3000)\)

\(\Sigma _c(2880)^0\)

\(\Xi '_c(3030)\)

\(\Sigma _c(2455)\pi \)

26.23

\(\Sigma _c(2455)K\)

3.14

\(\Lambda _c^+\pi \)

35.11

\(\Lambda _c^+K\)

17.42

\(\Lambda _c^+\pi \)

34.96

\(\Lambda _c^+K\)

18.37

\(\Sigma _c(2520)\pi \)

19.28

\(\Xi '_c(2580)\pi \)

11.47

\(\Sigma _c(2455)\pi \)

57.16

\(\Xi '_c(2580)\pi \)

12.56

\(\Sigma _c(2455)\pi \)

15.98

\(\Xi '_c(2580)\pi \)

3.50

  

\(\Xi '_c(2645)\pi \)

12.83

\(\Sigma _c(2520)\pi \)

17.54

\(\Xi '_c(2645)\pi \)

4.13

\(\Sigma _c(2520)\pi \)

54.52

\(\Xi '_c(2645)\pi \)

12.92

    

\(\Lambda _c(2595)\pi \)

6.92

\(\Xi _c(2790)\pi \)

5.89

\(\Lambda _c(2595)\pi \)

1.07

\(\Xi _c(2790)\pi \)

0.40

    

\(\Lambda _c(2625)\pi \)

1.57

\(\Xi _c(2815)\pi \)

0.13

\(\Lambda _c(2625)\pi \)

7.62

\(\Xi _c(2815)\pi \)

6.22

    

\(D^0n\)

0.03

\(\Sigma _c(2455)K\)

15.34

\(D^0n\)

3.03

\(\Sigma _c(2455)K\)

6.49

          

\(D^0\Lambda \)

0.01

45.51

27.44

118.33

55.47

117.18

47.91

\(\approx \)50 [1]

28.1\(\pm 2.4^{+1.0}_{-5.0}\) [4]

86\(^{+33}_{-22}\) [35]

   

4.4 2S \(\Lambda _c^+\) and \(\Xi _c^{0,+}\) states

According to the mass spectrum (see Table 2), \(\Lambda _c/\Sigma _c(2765)^+\) can also be regarded as the first radial (2S) excitation of the \(\Lambda _c(2286)^+\) with \(J^P=1/2^+\). Interestingly, the results of strong decays in Table 12 do not contradict this assignment. Our calculation indicates that the decay channel \(\Sigma _c(2455)\pi \) is a dominant decay channel for the \(\Lambda _c^+(2S)\) state. This is in line with the observations by Belle [31, 32]. At present, both \(1/2^+(2S)\) \(\Lambda _c^+\) and \(1/2^-(1P)\) \(\Sigma _c^+\) are possible for the assignment of \(\Lambda _c/\Sigma _c(2765)^+\). However, there is the very important feature for experiments to distinguish these two assignments in future. Specifically, we suggest to search \(\Lambda _c/\Sigma _c(2765)^+\) in the channel of \(\Sigma _c(2520)\pi \). As shown in Table 12, the channel \(\Sigma _c(2520)\pi \) is large enough to find the \(\Lambda _c^+(2S)\) state. On the other hand, this mode seems to be too small to be detected for the \(\Sigma _{c1}(1/2^-)\) (see Table 10). Explaining the criteria concretely, we give the following branching ratios for these two states.

For the \(\Lambda _c(2S)\) state,
$$\begin{aligned} \frac{\mathcal {B}(\Lambda _c(2765)\rightarrow \Sigma _c(2520)~\pi )}{\mathcal {B}(\Lambda _c(2765)\rightarrow \Sigma _c(2455)~\pi )}~=~0.74. \end{aligned}$$
(27)
For the \(\Sigma _{c1}(1/2^-)\) state,
$$\begin{aligned} \frac{\mathcal {B}(\Sigma _c(2765)\rightarrow \Sigma _c(2520)~\pi )}{\mathcal {B}(\Sigma _c(2765)\rightarrow \Sigma _c(2455)~\pi )}~=~0.03. \end{aligned}$$
(28)
The branching ratio of \(\mathcal {B}(\Sigma _c(2520)\pi )/\mathcal {B}(\Sigma _c(2455)\pi )\) for the \(\Sigma _{c1}(1/2^-)\) state is roughly an order of magnitude smaller than \(\Lambda _c(2S)\). If \(\Lambda _c(2765)\) is the 2S excitation, \(\Xi _c(2980)\) could be a good candidate as its charm-strange analog [21] as seen in Table 12. The mass difference between \(\Lambda _c(2765)\) and \(\Xi _c(2980)\) is about 200 MeV, which nearly equals the mass difference between \(\Lambda _c(2287)\) and \(\Xi _c(2470)\). The predicted width of \(\Xi _c(2980)\) is 27.44 MeV which is in good agreement with the experimental data [4, 39]. As the 2S excitation of \(\Xi _c(2470)\), the branching ratio
$$\begin{aligned} \frac{\mathcal {B}(\Xi _c(2980)\rightarrow \Xi ^\prime _c(2580)~\pi )}{\mathcal {B}(\Xi _c(2980)\rightarrow \Xi _c(2645)~\pi )}~=~0.89 \end{aligned}$$
(29)
is predicted for \(\Xi _c(2980)\), which can be tested by future experiments. Recently, the ratio of branching fractions
$$\begin{aligned} \frac{\mathcal {B}(\Xi _c(2980)^+\rightarrow \Xi ^\prime _c(2580)^0\pi ^+)}{\mathcal {B}(\Xi _c(2815)^+\rightarrow \Xi _c(2645)^0\pi ^+\rightarrow \Xi _c^+\pi ^-\pi ^+)}~\approx ~75\%\nonumber \\ \end{aligned}$$
(30)
has been estimated by the Belle Collaboration [4]. Combining this with the predicted partial widths of \(\Xi _c(2815)\) and \(\Xi _c(2645)\) in Tables 8 and 9, the branching fraction \(\mathcal {B}(\Xi _c(2980)^+\rightarrow \Xi ^\prime _c(2580)^0\pi ^+)\) is evaluated to be about \(40\%\), which is consistent with our direct result of \(41.8\%\).

4.5 2S \(\Sigma _c^{0,+,++}\) and \(\Xi _c^{\prime ~0,+}\) states

In Table 3, masses of the 2S \(\Sigma _c(1/2^+,3/2^+)\) states are predicted as 2850 and 2876 MeV, respectively. The neutral \(\Sigma _c(2850)^0\) found by the BaBar Collaboration in the decay channel \(B^-\rightarrow \Sigma _c(2850)^0\bar{p}\rightarrow \Lambda _c^+\pi ^-\bar{p}\) [35] can be regarded as the 2S \(\Sigma _c\) state with \(J^P=1/2^+\). The mass and width of the neutral \(\Sigma _c(2800)^0\) and \(\Sigma _c(2850)^0\) are collected here:
$$\begin{aligned} \begin{array}{ll} \Sigma _c(2800)^0:~~~m~&{}=~2806^{+5}_{-7}~\mathrm {MeV},~~~\Gamma ~=~72^{+22}_{-15}~\mathrm {MeV};\\ \Sigma _c(2850)^0:~~~m~&{}=~2846\pm 8\pm 10~\mathrm {MeV},~~~\Gamma ~=~86^{+33}_{-22}~\mathrm {MeV}. \end{array} \end{aligned}$$
For lack of experimental information, at present, PDG treated \(\Sigma _c(2850)^0\) and \(\Sigma _c(2800)^{0,+,++}\) as the same state [1]. As pointed out by the BaBar Collaboration [35], however, there are indications that these two signals detected by Belle [34] and BaBar [35] are two different \(\Sigma _c^*\) states. The main reasons are listed as follows:
  1. 1.

    Although the widths of \(\Sigma _c(2800)^{0,+,++}\) and \(\Sigma _c(2850)^0\) are consistent with each other, their masses are \(3\sigma \) apart.

     
  2. 2.

    The Belle Collaboration tentatively identified the \(\Sigma _c(2800)^{0,+,++}\) as the \(J=3/2\) isospin triple, while the BaBar Collaboration found weak evidence of \(\Sigma _c(2850)^0\) as a \(J=1/2\) state.

     
Our results also indicate that \(\Sigma _c(2800)^{0,+,++}\) and \(\Sigma _c(2850)^0\) are the different \(\Sigma _c\) excited states. One notices that the predicted mass of \(1/2^+(2S)\) \(\Sigma _c\) state in this work and in Ref. [9] are around 2850 MeV. Even the results in Refs. [10, 56] are only about 50 MeV larger than the measurements. Due to the intrinsic uncertainties of the quark potential model, it is appropriate to assign \(\Sigma _c(2850)^0\) as a 2S \(1/2^+\) state. More importantly, the predicted decay width of \(\Sigma _c(1/2^+,~2S)\) state is 118.36 MeV, which is comparable with the measurement by BaBar [35]. The partial width of \(\Lambda _c\pi \) is 35.11 MeV, which can explain why \(\Sigma _c(2850)^0\) was first found in this channel. We find that the decay modes of \(\Sigma _c(2455)\pi \) and \(\Sigma _c(2520)\pi \) are also large. Finally, we give the following branching ratios:
$$\begin{aligned} \frac{\mathcal {B}(\Sigma _c(2850)\rightarrow \Sigma _c(2455)~\pi )}{\mathcal {B}(\Sigma _c(2850)\rightarrow \Sigma _c(2520)~\pi )}~=~3.26, \end{aligned}$$
(31)
$$\begin{aligned} \frac{\mathcal {B}(\Sigma _c(2850)\rightarrow \Sigma _c(2286)~\pi )}{\mathcal {B}(\Sigma _c(2850)\rightarrow \Sigma _c(2455)~\pi )}~=~0.61, \end{aligned}$$
(32)
which can be tested by future experiments. If \(\Sigma _c(2850)^0\) is the \(1/2^+(2S)\) state, the mass of its doublet partner in the heavy quark effective theory is predicted as 2876 MeV (denoted \(\Sigma _c(2880)\)). According to the predicted decay widths in Table 12, this state might also be broad. \(\Lambda _c^+\pi \), \(\Sigma _c(2455)\pi \), and \(\Sigma _c(2520)\pi \) are also dominant for the decay of \(\Sigma _c(2880)\). The ratio of \(\Gamma (\Sigma _c(2455)\pi )/\Gamma (\Sigma _c(2520)\pi )\) for \(\Sigma _c(2880)\) is different from \(\Sigma _c(2850)\), whose numerical value is given by
$$\begin{aligned} \frac{\mathcal {B}(\Sigma _c(2880)\rightarrow \Sigma _c(2455)~\pi )}{\mathcal {B}(\Sigma _c(2880)\rightarrow \Sigma _c(2520)~\pi )}~=~0.29. \end{aligned}$$
(33)
Even though the strange partners of \(\Sigma _c(2850)\) and \(\Sigma _c(2880)\) have not been found by any experiments, their decay properties are calculated and presented in Table 12. Our results indicate that \(\Lambda _c^+K\), \(\Xi _c^\prime (2580)\pi \), and \(\Sigma _c(2455)K\) are the dominant decay modes of the \(\Xi _c^\prime (3000)\) state with \(J^P=1/2^+\), while \(\Lambda _c^+K\) and \(\Xi _c^\prime (2645)\pi \) are those of the \(\Xi _c^\prime (3030)\).4 Besides the masses and decay widths, the following branching ratios may also be valuable for future experiments:
$$\begin{aligned} \frac{\mathcal {B}(\Xi ^\prime _c(3000)\rightarrow \Xi ^\prime _c(2580)~\pi )}{\mathcal {B}(\Xi ^\prime _c(3000)\rightarrow \Xi ^\prime _c(2645)~\pi )}~=~3.04, \end{aligned}$$
(34)
$$\begin{aligned} \frac{\mathcal {B}(\Xi ^\prime _c(3030)\rightarrow \Xi ^\prime _c(2580)~\pi )}{\mathcal {B}(\Xi ^\prime _c(3030)\rightarrow \Xi ^\prime _c(2645)~\pi )}~=~0.27. \end{aligned}$$
(35)

5 Summary and outlook

In principle, both \(\rho \) and \(\lambda \) modes can be excited in a baryon system. For charmed baryons, the excitation energies of the \(\rho \) and \(\lambda \) modes are different due to the heavier mass of a c quark. For the ordinary confining potential, such as the linear or harmonic form, the excited energy of the \(\rho \) mode is larger than the \(\lambda \) mode [44]. Hence the low-excited charmed baryons may be dominated by the \(\lambda \) mode excitations. Recently, the investigation by Yoshida et al. confirmed this point [75]. Furthermore, they find that the \(\rho \) and \(\lambda \) modes are well separated for the charmed and bottom baryons, which means the component of the \(\rho \) mode can be ignored for the low-excited charmed baryons. Interestingly, Refs. [9, 10, 11] have also shown that the masses of existing charmed baryons can be explained by the \(\lambda \) mode. Hence, our study of strong decays of the low-excited charmed baryons is an important complement to this work [9, 10, 11, 75].

Up to now, several candidates of the 1P and 2S charm and charm-strange baryons have been found by experiments, and some of them are still open to debate. To better understand these low-excited charmed baryons, in this paper, we carry out a systematical study of the mass spectra and strong decays for the 1P and 2S charmed baryon states in the framework of the non-relativistic constituent quark model. The masses have been calculated in the potential model where the charmed baryons are simply treated as a quasi-two-body system in a light quark cluster picture. The strong decays are computed by the EHQ decay formula where the transition factors are determined by the \(^3P_0\) model. When calculating the decays, the inner structure of a light quark cluster has also been considered. Except for the unique parameter \(\gamma \) of the QPC model, the parameters in the potential model and in the EHQ decay formula have the same values.

The well-established ground and 1P \(\mathcal {G}\)-type charmed baryons provide a good test to our method. The experimental properties including both masses and widths for these states can be well explained by our results. This success has made us more confident of our predictions for other 1P and 2S states. Our main conclusions are as follows.

The broad state \(\Lambda _c(2765)^+\) (or \(\Sigma _c(2765)^+\)) which is still ambiguous could be assigned to the \(1/2^+(2S)\) \(\Lambda _c^+\), or the \(1/2^-(1P)\) \(\Sigma _{c1}^+\) state. The branching ratio \(\mathcal {B}(\Sigma _c(2455)\pi )/\mathcal {B}(\Sigma _c(2520)\pi )\) is found to be different for these two assignments, which may help us understand the nature of this state.

\(\Sigma _c(2800)^{0,+,++}\) observed by the Belle Collaboration in \(e^+e^-\) annihilation processes [34] can be regarded as a negative-parity state with \(J^P=3/2^-\), or \(5/2^-\), or their overlapping structure. We suggest to measure the \(\mathcal {B}(\Sigma _c(2455)\pi )/\mathcal {B}(\Sigma _c(2520)\pi )\) in the future. Another neutral state, \(\Sigma _c(2850)^0\), which was found in the \(B^-\) meson decay [35] could be a good candidate for the first radial excited state of \(\Sigma _c(2455)\). With the above assignments, the ratios of \(\mathcal {B}(\Lambda _c(2287)\pi )/\mathcal {B}(\Sigma _c(2455)\pi )\) shall be very different for \(\Sigma _c(2800)^{0,+,++}\) and \(\Sigma _c(2850)^0\), i.e., 4.07 for \(\Sigma _c(2800)^{0,+,++}\) and 0.61 for \(\Sigma _c(2850)^0\). The puzzle of \(\Sigma _c(2800)^{0,+,++}\) and \(\Sigma _c(2850)^0\) may be disentangled if these branching ratios are measured in the future. In addition, the ratio of branching fractions \(\mathcal {B}(\Sigma _c(2455)\pi )/\mathcal {B}(\Sigma _c(2520)\pi )\) for \(\Sigma _c(2850)^0\) is predicted to be 3.26.

The analysis of the mass and decay properties supports that \(\Xi _c(2980)^{0,+}\) is the 2S excitation (the first radial excited state of \(\Xi _c(2470)\)). The existence of \(\Xi _c(2930)^0\) is still in dispute. If it exists, the assignments of \(\Xi _{c2}^\prime (3/2^-)\) and \(\Xi _{c2}^\prime (5/2^-)\) are possible. In other words, it could be regarded as a strange partner of \(\Sigma _c(2800)^{0,+,++}\). Some useful ratios of partial decay widths are also presented for \(\Xi _c(2980)^{0,+}\) and \(\Xi _c(2930)^0\).

Although both the masses and the strong decays have been explained in the heavy quark–light quark cluster picture for the observed 2S and 1P candidates, it is not the end of the story of the study of the excited charmed baryon states. Investigation of the \(\rho \) mode excited states with higher energies are also important to identify the effective degrees of freedom of charmed baryons. However, this topic needs much laborious work and is beyond the scope of the present work. In addition, the quark model employed here neglects the effect of virtual hadronic loops. In the future, a more reasonable scheme for studying the properties of heavy baryons will be obtained by the unquenched quark model. Another topic which is left as a future task is to calculate the sum rules among the branching fractions of charmed baryons by applying the technique found in Ref. [76].

Footnotes

  1. 1.

    In some work, the light quark cluster may also be named a light diquark.

  2. 2.

    For the different conventions to extract the color and flavor factors, the value of \(\gamma \) here is different from those in Refs. [24, 73, 74]. The deviation, of course, does not affect the predictions since \(\gamma \) is regarded as an adjustable parameter in the \(^3P_0\) model.

  3. 3.

    Even if the possible mixing between \(\Sigma _{c1}(3/2^-)\) and \(\Sigma _{c2}(3/2^-)\) is considered, the partial width of \(\Lambda _c^+\pi \) is only 3.87 MeV for the \(|1P,3/2^-\rangle _l\) state where the mixing angle obtained in Table 4 has been used.

  4. 4.

    If \(\Xi _c(2980)\) is the first radial excited state of \(\Xi _c(2470)\). Then our predicted masses for 2S charm-strange baryons may be about 20–30 MeV lower than the experiments. To compensate for this difference, we increase by about 25 MeV for the 2S \(\Xi _c^\prime \) states in this case.

Notes

Acknowledgements

Bing Chen thanks Franz F. Schöberl for the package which is very useful to solve the Schrödinger equation. This project is supported by the National Natural Science Foundation of China under Grant Nos. 11305003, 11222547, 11175073, 11447604, and U1204115. Xiang Liu is also supported by the National Program for Support of Top-notch Young Professionals.

References

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Authors and Affiliations

  • Bing Chen
    • 1
    • 3
  • Ke-Wei Wei
    • 1
  • Xiang Liu
    • 2
    • 3
  • Takayuki Matsuki
    • 4
    • 5
  1. 1.Department of PhysicsAnyang Normal UniversityAnyangChina
  2. 2.School of Physical Science and TechnologyLanzhou UniversityLanzhouChina
  3. 3.Research Center for Hadron and CSR PhysicsInstitute of Modern Physics of CAS & Lanzhou UniversityLanzhouChina
  4. 4.Tokyo Kasei UniversityTokyoJapan
  5. 5.Theoretical Research DivisionNishina Center, RIKENSaitamaJapan

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