# Low-lying charmed and charmed-strange baryon states

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## Abstract

In this work, we systematically study the mass spectra and strong decays of 1*P* and 2*S* 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 1*S* and 1*P* 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

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 1*P* \(\Lambda _c\) 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 1*P* and 2*S* 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 2*S* and 1*P* 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 1*P* and 2*S* 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

*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

*R*are related to the quark positions by

*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

*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:

### 2.2 Adopted effective potentials

*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]

*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.

*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:

*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

*S*\(\mathcal {B}\)-type charmed baryons,

*c*quark, i.e., \(\vec L=\vec L_\lambda \). The tensor operator is defined as \(\hat{S}_{12}=3\left( \vec {S}_Q\cdot \vec {\lambda }\right) \left( \vec {S}_{\mathrm{cl.}}\cdot \vec {\lambda }\right) /\lambda ^2-\vec {S}_Q\cdot \vec {S}_{\mathrm{cl.}}\).

### 2.3 Getting masses of charmed baryons

*nS*state:

*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

*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 1

*P*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.

*jj*coupling scheme. For 1

*P*states with \(J^P=1/2^-\), the mass matrix is given by

*D*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.

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 | | 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 |

*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.

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 |

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 |

*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

*P*\(\Sigma _c\) states with \(J^P=1/2^-\) can be represented as

*P*states is similar to the 1

*P*states. For the 1

*D*states, one notices that both \(3/2^+\) and \(5/2^+\), with heavier masses, are dominated by smaller \(j_{\mathrm{cl.}}\) components.

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

*P*and 2

*S*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.

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 |

*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.

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)

[ | \(\{qq\}\) | [ | \(\{qs\}\) | \(\pi \) | \(\rho \) | | \(K^*\) | | \(D^*\) | | \(\Lambda \) |
---|---|---|---|---|---|---|---|---|---|---|---|

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

*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 1

*D*\(\Lambda _c\) and \(\Xi _c\) states [11].

*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:

*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 1*S* and 1*P* states

\(\Sigma _c(2455)^{++}\) | 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 1*P* \(\mathcal {G}\)-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 1*S* 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}

*P*\(\mathcal {G}\)-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 1

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

*P*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 1

*P*charmed states, more experiments are required in the future.

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

1 | |||||||
---|---|---|---|---|---|---|---|

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

\(\Sigma _c(2455)^{++}\) | \(\Xi '_c(2580)^+\) | \(\Sigma _c(2520)^{++}\) | \(\Xi '_c(2645)^+\) | ||||

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

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

1 | |||||||
---|---|---|---|---|---|---|---|

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

\(\Lambda _c(2595)^+\) | \(\Xi _c(2790)^+\) | \(\Lambda _c(2625)^+\) | \(\Xi _c(2815)^+\) | ||||

\(\Sigma _c\pi \) | 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 1*P*\(\Sigma _c^{~0,+,++}\) states

The partial and total decay widths of 1*P* \(\Sigma _c\) 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 1*P* \(\Sigma _c\) 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 1*P* \(\Sigma _c\) 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 \(\Sigma _c(2765)^+\).

^{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.

The partial and total decay widths of 1*P* \(\Xi ^\prime _c\) 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] |

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

As shown in Table 3, the predicted mass of 1*P* \(\Xi _c^\prime \) 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 1*P* \(\Xi _c^\prime \) 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 1*P* state with \(J^P=1/2^-\).

*P*\(\Xi _c^\prime \) 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 1

*P*\(\Xi ^\prime _c\) state might be excluded. As shown in the next subsection, \(\Xi _c(2980)^{0,+}\) could be a good 2

*S*\(\Xi _c\) 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).

The partial and total decay widths of 2*S* \(\Lambda _c^+\) 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 2*S*\(\Lambda _c^+\) 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 (2*S*) 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.

*S*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 2

*S*excitation of \(\Xi _c(2470)\), the branching ratio

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

*S*\(\Sigma _c(1/2^+,3/2^+)\) 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 2

*S*\(\Sigma _c\) 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:

- 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.
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.

*S*\(1/2^+\) 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:

^{4}Besides the masses and decay widths, the following branching ratios may also be valuable for future experiments:

## 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 1*P* and 2*S* 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 1*P* and 2*S* 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 1*P* \(\mathcal {G}\)-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 1*P* and 2*S* 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 2*S* 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 2*S* and 1*P* 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.
In some work, the light quark cluster may also be named a light diquark.

- 2.
- 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.
If \(\Xi _c(2980)\) is the first radial excited state of \(\Xi _c(2470)\). Then our predicted masses for 2

*S*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 2*S*\(\Xi _c^\prime \) 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.

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