Low-lying charmed and charmed-strange baryon states

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 nonrelativistic 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 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 $\Sigma_c(2800)^{0,+,++}$, \emph{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.

Obviously, these new measurements are very useful to understand the nature of these excited charmed baryon states.
Theoretically, the charm and and charm-strange baryons 1 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 could be more helpful for improving our understanding of the mechanism of confinement. The mass spectroscopy and strong decays of charmed baryons have been investigated in various models. So far, the different 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 the Lattice QCD [18,19]. The strong decay behaviors of charmed baryons have been studied by the heavy hadron chiral perturbation theory (HHChPT) [20,21], the chiral quark model [22,23], the 3 P 0 model [24], and a nonrelativistic quark model [25]. The decays of 1P Λ c and Ξ 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].

−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's observation. In this paper, we will denote the signal discovered by BaBar as Σ c (2850) 0 . Ξ c (2930) 0 was only seen by BaBar in the decay mode Λ + c K − [36], and it still needs more confirmation. Ξ c (2980) 0,+ was first reported by Belle in the channels Λ + c K − π + and Λ + c K 0 S π − [37], and was later confirmed by Belle [4,38] and BaBar [39] in the channels Ξ ′ c (2580)π, Ξ c (2645)π and Σ c (2455)K, respectively. However, the decay widths reported by Refs. [4,[37][38][39] were quite different 2 from each other. 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 find them. To study the dynamics of baryons, one crucial question which should be answered is "What are the relevant degrees of freedom in a baryon?" [44]. In a constituent quark model, a baryon system consists of three confined quarks. Then 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 sug- 2 More experimental information about the charmed baryons can be found in Refs. [40][41][42][43].
gests that the couplings between c quark and two light quarks are weak [45]. Therefore, two light quarks in a charmed baryon first couple with each other to form a light diquark, then the diquark couples with a charm quark ,and finally a charmed baryon resonance is formed. Accordingly, the dynamics of three quarks in the baryon system can be separated into two parts by the Jacobi coordinates in the following way. The effective degrees of freedom between two light quarks are denoted as the ρ mode while the effective degrees of freedom between the light diquark and the c quark as the λ mode (see Fig.1). Both ρ and λ modes can in principle be excited as a baryon state. Due to the heavier mass of a c quark, the excitation energies of the ρ and λ modes are different in the charmed baryons. For the ordinary confining potential, such as the linear or harmonic form, the excited energy of the ρ mode is larger than the λ mode [46]. Hence the low excited charmed baryons may be dominated by the λ mode excitations. Recently, the investigation by Yoshida et al. confirmed this point [47]. Furthermore, they find that the ρ and λ modes are well separated for the charmed and bottom baryons, which means the component of the ρ mode can be ignored for the low excited charmed baryons. Interestingly, the works [9][10][11] have also shown that the masses of existing charmed baryons can be explained by the λ mode. So a study of strong decays of the low excited charmed baryons is an important complement to these works [9][10][11]47].
In the present work, both the mass spectra and strong decays will be calculated for the 1P and 2S charmed baryon states. Here the heavy quark-light diquark picture of Refs. [9][10][11] will be used. The concept of diquark in these works should be understood in the narrow sense that the effective degrees of freedom within the light diquark is frozen. In other words, only the states with lower excited energies will be studied in our paper. Fortunately, most of the observed charmed baryons can be appropriately assigned by our results. The paper is arranged as follows. In Sec. II, the mass spectra of low-excited charmed baryons are calculated by the nonrelativistic quark potential model. In Sec. III, the strong decays are investigated by the Eichten-Hill-Quigg (EHQ) decay formula. Finally, the paper ends with the conclusion and outlook. Some detailed calculations and definitions are collected in the Appendixes.

II. MASS SPECTRA
Generally the light diquark in the baryons can be classified into two kinds: one is flavor symmetric and another is flavor antisymmetric. Constrained by the Pauli's exclusion principle, the total wave function of the light diquark should be antisymmetric in exchange of two quarks. Since the spatial and color parts of a diquark are always symmetric and antisymmetric, respectively, the function, |flavor × |spin , should be symmetric. Therefore, the scalar diquark [qq] (S = 0) is always flavor antisymmetric, and the axial-vector diquark {qq} (S = 1) is flavor symmetric. In terms of the Jaffe's terminology, the "scalar" and "vector" diquarks are named as the "good" and "bad" diquarks [48]. The color-spin interaction between two light quarks gives the good diquark in the Λ + c baryons the mass about 200 MeV lighter than the bad diquark in the Σ 0,+,++ c baryons. By considering the flavor SU(3) symmetry, the diquark formed by s and q (q=u or d quark) in the charm-strange baryons can also be classified in the similar way. Henceforth, we will call the Λ + c and Ξ 0,+ c baryons the G-type baryons, and Σ 0,+,++ c and Ξ ′0,+ c the B-type baryons for convenience.
In the heavy quark-light diquark picture, a color singlet baryon system should be formed as Fig.1). The abbreviation "di" here denotes the diquark system. Since the degrees of freedom of two light quarks are frozen, the light diquark can be treated as a block with the antitriplet color structure and peculiar size. In this way, a heavy baryon could be treated as a quasi two-body system. If this is true, the similarity of dynamics should exist between baryons with a single heavy flavor and heavy-light mesons [49]. More importantly, the scenario of a heavy quarklight diquark picture is not contradictory to the HQS. If the color-magnetic interaction is the principal effects of quark correlations, two light quarks in the heavy-baryon system are expected to strongly correlate since the color-spin interaction is proportional to the inverse of the quark masses. Thus, they may develop into a diquark [48] 3 . In this way, the dynamics of a heavy quark in a heavy baryon should finally couple to the light degrees of freedom, i.e., diquark.
Because of similarity of the dynamics, the quark potential models which have been applied to calculate the meson spectra may be extended to investigate the low excited charmed baryons. Here, a nonrelativistic quark potential model is employed to calculate the mass spectra of excited charmed baryons. In this model, the Hamiltonian is written as The last three terms represent the spin-dependent forces which are taken from Ref. [52]. The H con f Q−di term contains a Coulomb term from one gluon exchange approximation at short distances and a phenomenological confining term, which is written as [53] H con f where the constant C is the renormalization constant which is used to fit the total energy to the physical mass spectrum. The static potential of Eq. (2) is slightly different from the Cornell potential which has the form −a/r + br + C [54]. The color contact interaction is usually given by the following form 3 In fact, the existence of diquark correlations was partly confirmed by lattice QCD [50] and the Bethe-Salpeter equation [51].
where S Q and S di refer to the heavy quark and diquark spins. A Gaussian-smeared function (σ/ √ π) 3 e −σ 2 r 2 is normally used forδ σ (r) [55]. If the SU(3) flavor symmetry is kept intact for the charmed baryons, we may modify the color contact interaction as where the mass of light diquark is just assumed as a unit. This assumption is supported by the mass differences of the 1S B−type charmed baryons, 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 diquark masses. The color tensor interaction is Finally, H S O Q−di denotes the spin-orbit interaction which contains two terms. One is the color magnetic interaction which arises from one-gluon exchange where S denotes the spin of a baryon, S = S Q + S di . Another spin-orbit interaction is the Thomas-precession term To reflect the importance of the heavy quark symmetry, we rewrite the spin-dependent interactions as The degrees of freedom of the diquark are characterized by its total angular momentum j di , i.e., j di = S di + L. Obviously, the orbital angular momentum L of a charmed baryon in the diquark picture is defined by the angular momentum between diquark and c quark, i.e., L = L λ . The tensor operator is defined asŜ 12 With the confining term of Eq. (2), the coefficients V ss , V 1 , V 1 , and V t in Eq. (8) are defined as In our calculation, the following Schrödinger equation is solved for the s-wave states: 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 considered for convenience to extract the mass matrix elements.
One is the eigenstates |S di , L, j di , S Q , J ( j − j coupling) and another is |S di , S Q , S , L, J (L − S coupling). The relationship between these two bases is given by Due to S di = 0 for a good diquark, only V 2 S Q · j di survives for the Λ + c and Ξ 0,+ c baryons . With a bad diquark, however, Σ 0,+,++ c , Ξ ′0,+ c and Ω 0 c have more complicated splitting structures. Within the framework of the heavy quark effective theory, the spin of an axial-vector diquark, j di , first couples with the orbital angular momentum L. As illustrated in Fig. (2), in the heavy quark limit m c → ∞, there are only three states which are characterized by j di for 1P charmed baryons. When the heavy quark spin S Q couples with j di , the degeneracy is resolved and the five states appear. They are two J ′ = 1/2 − , two J ′ = 3/2 − , and one J ′ = 5/2 − states. Lastly, the states with the same J P mix with each other by the interactions of V ss S Q · S di and V tŜ 12 , and the physical states are formed.
For 1P states with J P = 1/2 − , the mass matrix is given by in the |S di , L, j di , S Q , J basis which, in the following, is represented as | j di , J P for brevity. Similarly, for two states with FIG. 2: Schematic diagram for the splittings of p-wave Σ c and Ξ ′ c . Here j l = L + j di and subindices ℓ and h of the last column mean low and high states in mass after includingṼ ss andṼ t interactions.
For the J P = 5/2 − state, The notations | Φ 1/2 and | Φ 3/2 appearing above are defined by The mass matrix of 1D states can also be obtained by the similar procedure. As shown above, there are seven parameters in the nonrelativistic quark potential model. They are m Q , m di , b, α, γ, ν, and C Qqq ′ . All values of parameters are listed in Table I. If the SU(3) flavor symmetry is taken into account for the charm and charm-strange baryons, the dynamics of Λ + c states should be like Ξ c . The case of Σ c and Ξ ′ c is alike. Thereby, we select the same value of γ for Λ + c and Ξ c , so do the case of Σ c and Ξ ′ c .
The masses of the good diquarks [qq] and [qs] have been fixed as 450 MeV and 630 MeV by the RFT model [11]. The bad diquark masses may be evaluated by the following relationships . We have taken the typical values in the quark potential models for m c , b, α, and ν (see in Table I). It is an effective method to investigate charmed baryons in a diquark picture. We do not expect the value of ν to be the same both for G-type and B-type baryons. Here, ν of Λ + c /Ξ c is is slightly lager than Σ c /Ξ ′ c . The predicted masses of low excited charmed baryons are collected in Tables II and  III. As mentioned earlier, the nonzero off-diagonal elements in mass matrices of Φ 1/2 | H S | Φ 1/2 and Φ 3/2 | H S | Φ 3/2 cause the mixing between two states with the same J P but different j di . However, the mechanism of mixing effects in hadron physics is still unclear. In principle, a physical hadron state with a specific J P comprise 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 between charmonium and molecular state with J PC = 1 ++ [56].
Here we take the | j di , J P basis to describe the mixing for the B−type baryons. Then two physical states characterized by different masses can be denoted as For example, two 1P Σ c states with J P = 1/2 − can be represented as Here we have denoted the physical states by their masses (see Table II). The mixing angles for other states in Tables II and  III with the same J P are listed in Table IV. Our results of mixing angles in Table IV indicate that the  heavier 1 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 the heavier masses have a dominant component of smaller j di .
The uncertainty may exist for 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 low mass of D s (2317) ± with respect to the expectations [52] can be well explained by the corrected spindependent potential [60,61]. If we use this 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 [62], which may contribute to the mixing phenomenon in hadron physics. Anyway, we expect that the mixing angles in Table  IV reflect main features of the mixing states. Due to the uncertainties of the mixing angles, we ignore the mixing effects as the first step to study the decays of charmed excitations in the next Subsection. Obviously, it is a good approximation only when the mixing effects are not large. Luckily, this crude procedure is partially supported by the analysis of charmed mesons [63][64][65]. If the decay properties obtained in this way describe principal characteristics of the mixing states, the angles obtained by the potential model may be overestimated. In the next Section, we will also discuss the mixing effects for the decays of the relevant states.
In the next Section, the two-body strong decays will be calculated for the 1P and 2S charmed baryons where the simple harmonic oscillator (SHO) wave functions are used to evaluate the transition factors via the 3 P 0 model. Following the method of Ref. [66], all values of the SHO wave function scale, denoted as β in the following, are calculated (see Table V). The values of β reflect the distances between the diquark and c quark.
In our calculation of strong decays, we will consider the structures of light diquarks. What is more 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., π, K, D, p, and Λ. For the β of these hadrons, the following potential will be used Here, the parameters α s and b are also given by 0.45 and 0.145 GeV as in Table I, respectively. To reproduce the masses of light diquarks in Table I, the masses of u/d, s are fixed as 0.195 GeV and 0.380 GeV. While σ and C are treated as adjustable parameters, the masses of π/ρ, K/K * , D/D * , p/∆, and Λ families are fitted with experimental data. Meanwhile, the values of β for the corresponding states are also obtained, which are collected in Table VI.

III. STRONG DECAYS
In this section, we will use the formula provided by Eichten, Hill, and Quigg (EHQ) [67] 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 di but different J shall have the 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 [45]. Some typical ratios of excited charmed baryons with negative-parity were predicted by this law [45]. Later, a more concise formula (the EHQ formula) was proposed for the widths of heavy-light mesons [67]. The EHQ formula has been applied systematically to the decays of excited open-charm mesons [63][64][65]. Recently, the   EHQ formula has been extended to study the decay properties of 1D Λ c and Ξ c states [11].
For the charmed baryons, the EHQ formula can be written as where ξ is the flavor factor given in Table XIII in Appendix B. q = | 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 concrete expression ofβ is given in Eq. (A11) in Appendix A. In addition, C where j C ≡ s C + ℓ. The symbols s C and ℓ represent the spin of the light hadron C and the orbital angular momentum relative to B, respectively. The transition factors M j A , j B j C ,ℓ (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 [68] and the 3 P 0 model [63,65,69]. In our work, we will employ the 3 P 0 model [70][71][72] to obtain the transition factors. More details for an estimate of the transition factors are given in Appendix A.

A. Experimentally well established 1S and 1P states
At present, all the ground states and 1P 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 Σ c (2455) and Σ c (2520) have been largely improved [2,3] (see Table IX). In our calculation, the mass and decay width of Σ c (2520) ++ measured by CDF will be taken as input data to fix the constant γ peculiar to the 3 P 0 model. With the transition factor of the process Σ c (2520) → Λ c (2286) + π (see Eq. (A12) in the Appendix A), the value of γ is fixed as 1.296. 4 As shown in Tables VII and VIII, the predicted widths of other 1S charmed baryons are well consistent with experiments. Our results of mass spectra and decay widths indicate that Λ c (2595) + , Λ c (2625) + , Ξ c (2790) 0,+ , and Ξ c (2815) 0,+ can be accommodated with the 1P G−type charmed baryons. Λ c (2595) + and Ξ c (2790) 0,+ can be grouped as the 1/2 − states while Λ c (2625) + and Ξ c (2815) 0,+ as 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 Λ c and Ξ c baryons, respectively, which are also consistent with the experiments. The assignments of Λ c (2595) + , Λ c (2625) + , Ξ c (2790) 0,+ , and Ξ c (2815) 0,+ are also supported by other works [9][10][11][12] in which the diquark 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,[57][58][59]. 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 Λ c (2595) + and Λ c (2625) + form the heavy dou-bletΛ c1 (1/2 − , 3/2 − ) (the same assignments as the case of Ξ c (2790) 0,+ and Ξ c (2815) 0,+ ) [16], which are 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 future.  Table X is about 63.52 MeV which is also in agreement with the measurements [30][31][32]. Furthermore, the signal of Σ c (2765) + has been observed in the Σ c (2455)π intermediate states while there is no clear evidence for the decay of Σ c (2765) + through Σ c (2520)π [31,32]. This is also consistent with our results of the |1P, 1/2 − h . Based on the combined analysis of the mass spectrum and strong decays, we, therefore, conclude that Σ c (2765) + could be regarded as a good Σ c1 (1/2 − ) candidate. Considering uncertainties of the quark potential models, the masses obtained by Refs [9,10,57] are not contradictory to our assignment to Σ c (2765) + here.
According to the predicted masses in Table III,    1P Λ c and Ξ c  Σ c (2800) 0,+,++ could be assigned to either |1P, 3/2 − l , or |1P, 3/2 − h , or |1P, 5/2 − states. When we consider the decay properties of these three states (see Table X), the possibility of assignment to the |1P, 3/2 − l state can be excluded since the Belle Collaboration observed this state in the Λ + c π mode. 5 At present, the Belle Collaboration tentatively identified Σ c (2800) 0,+,++ as members of the Σ c2 (3/2 − ) isospin triplet, which agrees with our results of both mass spectrum and strong decays. When the measured mass of Σ c (2800) 0 is used for the Σ 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 Σ c (2800) 0,+,++ has not been measured yet. Then the possibility of this state as the Σ c2 (5/2 − ) candidate can not be excluded by our results since the decay mode of Λ + c π is dominant for this state. In addition, the predicted mass and total width of Σ c2 (5/2 − ) state are also possible for the Σ c (2800) 0,+,++ baryon. Therefore, we would like to point out that the signal of Σ c (2800) 0,+,++ found by Belle might be their overlapping structure. The future experiments may measure the following branching ratios to disentangle this state: For the Σ c2 (3/2 − ) state, For the Σ c2 (5/2 − ) state, As mentioned earlier, the signal Σ c (2850) 0 discovered by the BaBar collaboration may be a J = 1/2 state. If Σ c (2850) 0 is the 1/2 + (2S ) state, the corresponding ratios (see Subsection IV E) are different from Eqs. (18∼21). So the measurements of these ratios of branching fractions can help us understand the nature of Σ c (2800) 0,+,++ and Σ c (2850) 0 .
D. 2S Λ + c and Ξ 0,+ c states According to the mass spectrum, Λ c /Σ c (2765) + can also be regarded as the first radial (2S) excitation of the Λ c (2286) + with J P = 1/2 + . Interestingly, the results of strong decays in Table XII do not contradict with this assignment. Our calculation indicates that the decay channel Σ c (2455)π is a dominant decay channel for the Λ + c (2S ) state. This is in line with the observations by Belle [31,32]. At present, both 1/2 + (2S ) Λ + c and 1/2 − (1P) Σ + c are possible for the assignment of Λ c /Σ c (2765) + . However, there is a very important discriminator for experiments to distinguish these two assignments in future. Specifically, we suggest to search Λ c /Σ c (2765) + in the channel of Σ c (2520)π. As shown in Table XII, the channel Σ c (2520)π is large enough to find the the Λ + c (2S ) state. Differently, this mode seems too small to be detected for the Σ c1 (1/2 − )(see Table X). Explaining the criteria concretely, we give the following branching ratios for these two states, For the Λ c (2S ) state, For the Σ c1 (1/2 − ) state, The branching ratio of B(Σ c (2520)π)/B(Σ c (2455)π) for the Σ c1 (1/2 − ) state is roughly an order of magnitude smaller than Λ c (2S ). If Λ c (2765) is the 2S excitation, Ξ c (2980) could be a good candidate as its charm-strange analog [21] as seen in Table XII. The mass difference between Λ c (2765) and Ξ c (2980) is about 200 MeV which nearly equals the mass difference between Λ c (2287) and Ξ c (2470). The predicted width of Ξ c (2980) is 27.44 MeV which is in good agreement with the experimental data [4,39]. As the 2S excitation of Ξ c (2470), the branching ratio below is predicted for Ξ c (2980), which can be tested by the future experiments. Recently, the following ratio of branching fraction (26) has been estimated by Belle Collaboration [4]. Combining with the predicted partial widths of Ξ c (2815) and Ξ c (2645) in Table VII and VIII, the branching fraction B(Ξ c (2980) + → Ξ ′ c (2580) 0 π + ) is evaluated about 40% which is consistent with our result of 41.8%.  Table III, masses of the 2S Σ c (1/2 + , 3/2 + ) states are predicted as 2850 MeV and 2876 MeV, respectively. The neutral Σ c (2850) 0 which was found by the BaBar Collaboration in the decay channel B − → Σ c (2850) 0p → Λ + c π −p [35] can be regarded as the 2S Σ c state with J P = 1/2 + . The mass and width of the neutral Σ c (2800) 0 and Σ c (2850) 0 are collected below. For lack of experimental information, at present, PDG treated Σ c (2850) 0 and Σ 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 Σ * c states. The main reasons are listed as follows: 1. Although the widths of Σ c (2800) 0,+,++ and Σ c (2850) 0 are consistent with each other, their masses are 3σ apart.
Our results also indicate that Σ c (2800) 0,+,++ and Σ c (2850) 0 are the different Σ c excited states. One notices that the predicted mass of 1/2 + (2S ) Σ c state in this work and in Ref. [9] are around 2850 MeV. Even the results in Refs. [10,57] are only about 50 MeV larger than the measurements. Due to the intrinsic uncertainties of the quark potential model, it is appropriate to assign Σ c (2850) 0 as a 2S 1/2 + state. More importantly, the predicted decay width of Σ c (1/2 + , 2S ) state is 118.36 MeV which is comparable with the measurement by BaBar [35]. The partial width of Λ c π is 35.11 MeV, which can explain why Σ c (2850) 0 was first found in this channel. We find that the decay modes of Σ c (2455)π and Σ c (2520)π are also large. Finally, we give the following branching ratio, and which can be tested by the future experiments. If Σ 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 as Σ c (2880)). According to the predicted decay widths in Table XII, this state might be also broad. Λ + c π, Σ c (2455)π, and Σ c (2520)π are also dominant for Σ c (2880). The ratio of Γ(Σ c (2455)π)/Γ(Σ c (2520)π) for Σ c (2880) is different from Σ c (2850) and its numerical value is given by, Even though the strange partners of Σ c (2850) and Σ c (2880) have not been found by any experiments, their decay properties are calculated and presented in Table XII. Our results indicate that Λ + c K, Ξ ′ c (2580)π, and Σ c (2455)K are the dominant decay modes of the Ξ ′ c (3000) state with J P = 1/2 + , while Λ + c K and Ξ ′ c (2645)π of the Ξ ′ c (3030). 6 Besides the masses and decay widths, the following branching ratios may also be valuable for the future experiments: and

V. SUMMARY AND OUTLOOK
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 a systematical research of the mass spectra and strong decays for the 1P and 2S charmed baryon states in the framework of nonrelativistic constituent quark model. The masses have been calculated in the potential model where the charmed baryons are treated simply as a quasi two body system in a diquark picture. The strong decays are computed by the the EHQ decay formula [67] where the transition factors are determined by the 3 P 0 model. When calculating the decays, the inner structure of a diquark has also been considered. Except for the unique parameter γ 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 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 the predictions for other 1P and 2S states. Our main conclusions are given as follows: The broad state Λ c (2765) + (or Σ c (2765) + ) which is still ambiguous could be assigned to the 1/2 + (2S ) Λ + c , or the 1/2 − (1P) Σ + c1 state. The branching ratio   [35] B(Σ c (2455)π)/B(Σ c (2520)π) is found to be different for these two assignments, which may help us understand the nature of this state.
Although both the masses and strong decays have been explained in the heavy quark-light diquark picture for the observed 2S and 1P candidates, it is not the end of the story to study the excited charmed baryon states. Investigation of the ρ 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 future, a more reasonable scheme for studying the properties of heavy baryons will be achieved 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. [79].

Acknowledgement
Bing Chen thanks to 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 Youth Top-notch Talent Support Program ("Thousands-of-Talents Scheme").
Appendix A: Transition factor M j A , j B j C ,ℓ (q) in the QPC model Usually, there are two possible decay processes for an excited charmed baryon state (see Fig. 3). The final states of the left figure contain a charmed baryon and a light meson. The right one contains a charmed meson and a light baryon. If a baryon decays via the so-called 3 P 0 mechanism, a quarkantiquark pair is created from the vacuum and then regroups two outgoing hadrons by a quark rearrangement process. In the non-relativistic limit, the transition operatorT of the 3 P 0 model is given bŷ where the ω (4,5) i, j and ϕ (4,5) 0 are the color and flavor wave functions of the q 4q5 pair created from the vacuum. Thus, ω (4,5) i, j = (RR + GḠ + BB)/ √ 3, ϕ (4,5) 0 = (uū + dd + ss)/ √ 3 are color and flavor singlets. The pair is also assumed to carry the quantum number of 0 ++ , suggesting that they are in a 3 P 0 state. The χ (4,5) 1,−m represents the pair production in a spin triplet state. The solid harmonic polynomial Y m 1 ( k) ≡ | k|Y m 1 (θ k , φ k ) reflects the momentum-space distribution of the q 4q5 . γ is a dimensionless constant which denotes the strength of the quark-antiquark pair created from the vacuum. The value of γ is usually fixed by fitting the well measured partial decay widths.
When the mock state [76] is adopted to describe the spatial wave function of a meson, the helicity amplitude M M J A ,M J B ,M J C (q) can be easily constructed in the L-S basis [72]. The mock state for an A meson is As for the left decay process in Fig. 3, the wave function of a B baryon can be constructed in the same way. The wave function of a C meson is Here, the symbols of i (i = A, B, and C) represent the Clebsch-Gordan coefficients for the initial and final hadrons, which arise from the couplings among the orbital, spin, and total angular momentum and their projection of l z and s z to j z . More specifically, i (i = A, B, and C) are given by A · · · d 3 k 1 · · · d 3 k 5 δ 3 ( k 1 + k 2 + k 3 )δ 3 ( q − k 1 − k 4 )δ 3 ( q + k 2 + k 3 + k 5 ) where, i = A, B, C and j = 1, 2, · · · , 5. The color matrix element ω 235 B ω 14 C |ω 45 0 ω 123 A is a constant which can be absorbed into the parameter γ. The flavor matrix element ξ = ϕ 235 B ϕ 14 C |ϕ 45 0 ϕ 123 A will be presented in the next Subsection. To obtain the analytical amplitudes, the SHO wave functions are employed to describe the spatial wave function of a hadron. In the momentum space, the SHO radial wave func-tion, ψ n Lm (q), is given by With the help of Eq. (A7), the transition amplitude can be simplified. In the following, we take the process Σ c (2520) → Λ c (2280) + π as an example. The wave functions of initial and final states are Based on Eq. (A7), we obtain the amplitude as M 1 1 2 (q) = − 3g 8π 5/4 f 5/2 λ 3/2 β 3/2 A β 3/2 dA β 3/2 B β 3/2 dB β 3/2