Understanding the newly observed $\Xi_c^0$ states through their decays

Inspired by the newly observed $\Xi_c^0$ states by the LHCb Collaboration, we investigate the OZI-allowed two-body strong decays of the $\lambda$-mode $1P$ wave $\Xi'_c$ states within the chiral quark model. Our results indicate that: (i) the newly observed states $\Xi_c(2923)^0$ and $\Xi_c(2939)^0$ are good candidates of the $\lambda$-mode $1P$ wave $\Xi'_c$ states with the spin-parity $J^P=3/2^-$, namely $|^4P_{\lambda}3/2^-\rangle$ and $|^2P_{\lambda}3/2^-\rangle$, respectively. (ii) The another newly observed state $\Xi_c(2965)^0$ mostly corresponds to the $\lambda$-mode $1P$-wave $\Xi'_c$ state with the spin-parity $J^P=5/2^-$, namely $|^4P_{\lambda}5/2^-\rangle$. (iii) For the two $\lambda$-mode $J^P=1/2^-$ mixed states, the $|P_{\lambda}~1/2^-\rangle_1$ is a narrow state with a width of $\Gamma\sim15$ MeV and mainly decays into $\Xi'_c\pi$; while the $|P_{\lambda}~1/2^-\rangle_2$ state has a width of $\Gamma\sim52$ MeV and dominantly decays into $\Xi_c\pi$ and $\Lambda_cK$ channels. If the broad structure around $2880$ MeV observed at LHCb arises from the new $\Xi^0_c$ state, this state is very likely to be the $|P_{\lambda}~1/2^-\rangle_2$ state.


I. INTRODUCTION
As an important kind of singly heavy baryons, the charmedstrange baryon Ξ ′ c spectrum belonging to the flavor sextet 6 F plays a crucial role in perfecting baryon spectra. Although there are some discussions for the Ξ ′ c states in theory during the past several decades [1][2][3][4], no obvious progress of the observations for the Ξ ′ c state has been achieved in experiments [5]. In the Ξ ′ c spectrum, only the two ground states Ξ ′ c with J P = 1/2 + and Ξ * c (2645) with J P = 3/2 + (1S wave) have been established. So far, the low-lying P-wave Ξ ′ c states predicted in the quark model are still missing. It should be mentioned that in 2007 a structure Ξ c (2930) 0 was observed by BaBar in the Λ + c K − mass spectrum in B − → K − Λ + cΛ − c process [6]. Later, the Ξ c (2930) 0 was confirmed by the Belle Collaboration in the same decay process [7], while its charged partner Ξ c (2930) + was also observed inK 0 Λ + c final state in the reactionB 0 →K 0 Λ + cΛ − c [8]. In addition, another structure Ξ c (2970) 0 was first observed by BaBar in the Σ c (2455) 0 K 0 S decay mode [9]. This structure was also observed in both Ξ ′+ c π − [10] and Ξ c (2645) + π − [11] final states at Belle. The structures Ξ c (2930) 0 and Ξ c (2970) 0 may be some signals of the missing P-wave Ξ ′ c states first observed in experiments. Very recently, the LHCb Collaboration observed three new states Ξ c (2923) 0 , Ξ c (2939) 0 , and Ξ c (2965) 0 in the Λ + c K − mass spectrum with a large significance [12], and their masses and natural widths are determined to be m[Ξ c (2923) 0 ] = 2923.04 ± 0.59 MeV, Γ[Ξ c (2923) 0 ] = 7.1 ± 2.6 MeV, c process [6,7] might be due to the overlap of the two new narrower states, Ξ c (2923) 0 and Ξ c (2939) 0 . Stimulated by these three newly observed states, recently some groups have discussed their nature. In Ref. [13], these three newly observed states Ξ c (2923) 0 , Ξ c (2939) 0 , and Ξ c (2965) 0 are suggested to be assigned as the P-wave Ξ ′ c states. In Ref. [14], Ξ c (2923) 0 and Ξ c (2939) 0 may be good candidates of the Pwave states with J P = 3/2 − and 5/2 − states and Ξ c (2965) 0 can be assigned as Ξ ′ c (2S ) state. To understand the nature of these newly observed states and clarify whether they can be identified as the Ξ ′ c (1P) and Ξ ′ c (2S ) state or not, more theoretical analysis is urgently needed.
In our previous works [28,35], the decay properties of the 1P wave Ξ c states were estimated with a chiral quark model, the mostly predicted decay widths of the λ-mode 1P wave Ξ ′ c states were about a few dozen MeV, which were roughly consistent with the LHCb's measurement [12]. In the present work, by combining the newest data we further analyze the strong decay properties of the λ-mode 1P wave Ξ ′ c states with the chiral quark model, and attempt to put forward views on the inner structures of the three Ξ 0 c states observed by the LHCb Collaboration [12].
This paper is organized as follows. In Sec. II we give a brief introduction of the strong decay model. We discuss the strong decays of the low-lying λ-mode 1P wave Ξ ′ c states in Sec. III and summarize our results in Sec. IV.

II. THE MODEL
In this work we apply the chiral quark model [37] to study the strong decay properties. Within the chiral quark model, the effective low energy quark-pseudoscalar-meson coupling in the SU(3) flavor basis at tree level is adopted as [37] where f m is the pseudoscalar meson decay constant; ψ j represents the jth quark field in a baryon, and φ m denotes the pseudoscalar meson octet.
To match the nonrelativistic harmonic oscillator spatial wave function in our calculations, one should adopt a nonrelativistic form of the quark-pseudoscalar coupling [38][39][40] where (ω m , q) denote the energy and three-vector momentum of the final light pseudoscalar meson; (E i , M i , P i ) and (E f , M f , P f ) are the energy, mass and three-vector momentum of the initial and final baryons, respectively. The p ′ j (= p j − (m j /M)P c.m. ) stands for the internal momentum of the jth quark in the baryon rest frame; σ j sands for the Pauli spin vector on the jth quark; µ q represents the reduced mass expressed as 1/µ q = 1/m j + 1/m ′ j . The isospin operator I j associated with π and K mesons is given by Here, a † j (u, d) and a j (u, d, s) are the creation and annihilation operator for the u, d and u, d, s quarks on jth quark, respectively.
Then the partial decay width for the emission of a light pseudoscalar meson in a hadron strong decay can be calculated with [41,42] where δ is a global parameter accounting for the strength of the quark-meson couplings; J iz and J f z are the third components of the total angular momenta of the initial and final baryons, respectively; M J iz ,J f z denotes the transition amplitude.
With momentum q of the final light pseudoscalar meson increasing, the relativistic effect should be significant [43]. To partly remedy the inadequacy of the nonrelativistic wave function as the momentum q increases, a commonly used Lorentz boost factor γ f is introduced into the decay amplitudes [42,[44][45][46] where In most decays, the corrections from the Lorentz boost are not drastic and the nonrelativistic prescription is reasonable. The model parameters have been well determined in previous works [28,35], and we collect them in Table II. In the calculations the masses of the final baryons and mesons are taken from the PDG [5] and collected in Table II as well. The harmonic oscillator space-wave functions Ψ n lm = R nl Y lm are adopted to describe the spatial wave function of the initial and final baryons, and the harmonic oscillator parameter α ρ in the wave functions for ds/us system is taken as α ρ = 420 MeV. Another harmonic oscillator parameter α λ can be related to α ρ with the relation α λ = [3m c /(2m q + m c )] 1/4 α ρ , where m q denotes the light quark mass.

III. RESULTS AND ANALYSIS
The masses of the three Ξ 0 c states newly observed by the LHCb Collaboration [12] are in the predicted mass region of the λ-mode 1P wave Ξ ′ c states(see Table I). To clarify the possibility and further investigate their inner structures, we conduct a systematic study of the strong decay properties for the λ-mode 1P wave Ξ ′ c states within the framework of a chiral quark model. Our results and theoretical predictions are presented as follows.
A. 1P states with J P = 1/2 − In the Ξ ′ c family, there are two λ-mode J P = 1/2 − states |1 2 P λ 1/2 − and |1 4 P λ 1/2 − . Their masses are predicted to be about M = (2830 − 2940) MeV (see Table I), and their OZI-allowed two-body strong decay channels are Ξ c π, Ξ ′ c π, I: The mass spectrum of Ξ ′ c belonging to 6 F up to the 1P-wave states in various models and effective theories. The Ξ ′ c states are denoted by |N 2S +1 L σ J P in the LS coupling scheme. The unit of mass is MeV in the table.
For the state Ξ ′ c |1 2 P λ 1/2 − , the total decay width varies in the region of Γ ∼ (23 − 27) MeV, and the dominant decay modes are Ξ c π, Ξ ′ c π and Λ c K. The branching fraction for each of the dominant decay mode is about 30%. For the other MeV with mass varied in the region what we considered. It mainly decays into Ξ c π and Λ c K. Meanwhile, the partial decay width of the Ξ ′ c π mode is significant with a branching ratio of ∼10%.
We notice that for the singly heavy flavour quark systems, proper consideration of the heavy quark symmetry is necessary [47]. Namely, the physical states with the spinparity J P = 1/2 − are very likely to be mixed states between |1 2 P λ 1/2 − and |1 4 P λ 1/2 − by the following mixing scheme, The mixing angle φ may rang from φ = 0 • to that of heavy quark symmetry limit (φ = 35 • ). In our previous work [28,48], we obtained that the state Ω c (3000) could be explained as the mixed state |1P λ 1/2 − 1 with a mixing angle φ ≃ 24 • . As the same flavour multiplet, the mixing angle in the Ξ ′ c family should be comparable with that in the Ω c family. Meanwhile, according to the equal spacing rule [26,27], the mass of the mixed state |1P λ 1/2 − 1 in the Ξ ′ c family is lighter about ∼120 MeV than that of state in the Ω c family, namely M Ξ ′ c |1P λ 1/2 − 1 ≃ 2880 MeV. Thus according to the mixing scheme defined in Eq. (6), in Fig. 2 we plot the strong decay width of the |1P λ 1/2 − 1 state as a function of the mixing angle φ in the range of (0 • , 35 • ) by fixing the mass at M = 2880 MeV.
It is found that with the mixing angle increasing, the partial decay widths of the Λ c K and Ξ c π modes for Ξ ′ c |1P λ 1/2 − 1 are rapidly suppressed and the Ξ ′ c π decay channel almost saturates The decay rate of Ξ ′ c |1P λ 1 2 − 1 into Λ c K strongly depends on the mixing angle. If taking a slightly larger mixing φ ≃ 28 • than that (φ ≃ 24 • ) determined by Ω c (3000), one finds the decay rate into the Λ c K channel is nearly zero. Thus, the Ξ ′ c |1P λ 1 2 − 1 may be hardly observed in the Λ c K final state. Furthermore, the predicted decay width of the mixed state Ξ ′ c |1P λ 1/2 − 1 seems to be comparable with the decay width of the three Ξ 0 c states observed by the LHCb Collaboration [12]. However, it should be kept in mind that the mass of Ξ ′ c |1P λ 1/2 − 1 should be around ∼ 2880 MeV considering a reasonable explanation for the properties of Ω c (3000). Thus, most likely the three Ξ 0 c states as the state Ξ ′ c |1P λ 1/2 − 1 is excluded. Considering the predicted width of Ξ ′ c |1P λ 1/2 − 1 being narrow, this state might be observed in the Ξ ′ c π channel when enough data are accumulated in experiments.
The other mixed state |1P λ 1/2 − 2 should be a relatively broad state with a width much larger than the three newly observed Ξ 0 c states. At this moment we investigate its width range with M=2880 MeV in Fig. 2 as well. From the figure, the total decay width is about Γ ∼ (38 − 50) MeV with the mixing angle varying in range of (0 • , 35 • ). Taking a possible mixing angle φ = 24 • constrained by the Ω c (3000) state, we find that the mixed state Ξ ′ c |1P λ 1/2 − 2 has a width of Γ ∼ 52 MeV, and dominantly decays into the Ξ c π and Λ c K final states with comparable branching fractions Both the mass and decay width of this mixed state are inconsistent with the observations of the three newly observed Ξ 0 c states, thus, the three Ξ 0 c states as the state Ξ ′ c |1P λ 1/2 − 2 should be excluded. Since Ξ ′ c |1P λ 1/2 − 2 is not a very broad state, it might be observed in the Ξ c π and Λ c K channels.
It should be remarked that according to the LHCb' measurement [12], the Λ c K mass spectrum shows a broad structure around 2880 MeV, which might be due to the presence of additional new Ξ 0 c states. Combining the predicted decay properties of Ξ ′ c |1P λ 1/2 − 2 in this work, if the broad structure in the region around M ∼ 2880 MeV arises from a new Ξ 0 c state, this state is very likely to be the Ξ ′ c |1P λ 1/2 − 2 state. Moreover, the possibility of the broad structure arising from the overlapping of Ξ ′ c |1P λ 1/2 − 1 and Ξ ′ c |1P λ 1/2 − 2 cannot be ruled out as well.
Fixing the mass of Ξ ′ c |1 4 P λ 3/2 − with M = 2923 MeV, we The decay properties of the P-wave Ξ ′ c states compared with the observations. Γ th total presents the total decay width calculated in the present work, while Γ exp total presents the total width obtained from the LHCb experiment [12].
The predicted branching fraction of the dominant decay mode Meanwhile, the decay rates of Ξ ′ c |1 4 P λ 3/2 − into Ξ c π and Λ c K are considerable, and the predicted branching fractions are The sizeable branching fraction for Ξ ′ c |1 4 P λ 3/2 − into Λ c K is consistent with the nature of Ξ c (2923) 0 , which is observed in the Λ c K channel. If Ξ c (2923) 0 corresponds to the state Ξ ′ c |1 4 P λ 3/2 − indeed, the Ξ c π and Ξ * c π also may be measured in future experiments due to their large branching fractions.
In the same way, we fix the mass of Ξ ′ c |1 2 P λ 3/2 − with M = 2939 MeV. Then, the total decay width is predicted to be This state mainly decays into Ξ c π channel with the branching fraction The partial decay widths of the other three decay channels Ξ ′ c π, Ξ * c π, and Λ c K are comparable. The partial decay ratios are If Ξ c (2939) 0 observed in the Λ c K channel corresponds to the state Ξ ′ c |1 2 P λ 3/2 − indeed, the comparable partial decay widths indicate this state may be established in the Ξ c π, Ξ ′ c π and Ξ * c π decay channels as well in future experiments. Moreover, assigning the Ξ c (2923) 0 and Ξ c (2939) 0 to Ξ ′ c |1 4 P λ 3/2 − and Ξ ′ c |1 2 P λ 3/2 − , respectively, we notice that the predicted total decay width ratio highly agrees with the observed central value R There is only one λ-mode J P = 5/2 − state |1 4 P λ 5/2 − . The predicted mass of this state is listed in Table I. In terms of the predicted mass, the possibility of the three newly observed Ξ 0 c states taken as the state Ξ ′ c |1 4 P λ 5/2 − cannot be excluded. To investigate the effects of the uncertainties of the mass on the decay properties of the Ξ ′ c |1 4 P λ 5/2 − state, we show the variation of the partial decay width with the change of mass in Fig. 1. From the figure, the variation curves between the partial decay width and the mass for this state is similar to that for Ξ ′ c |1 2 P λ 3/2 − . The dominant decay mode for Ξ ′ c |1 4 P λ 5/2 − is Ξ c π. Meanwhile, the partial decay width of the Λ c K mode is sizeable, and becomes more and more significant with the mass increasing in the region of (2860 − 2970) MeV.
The predicted decay properties of Ξ ′ c |1 4 P λ 5/2 − are consistent with the observations of Ξ c (2965) 0 . Meanwhile, considering the equal spacing rule [26,27], Ξ c (2965) 0 most likely corresponds to its flavour mulitiplet Ω c (3090), which was assigned to the |1 4 P λ 5/2 − state according to our previous study [28]. Thus, it is reasonable to assign Ξ c (2965) 0 as the Ξ ′ c |1 4 P λ 5/2 − state. According to our calculations, we get with a mass of M = 2965 MeV (see Table III). The dominant decay mode is Ξ c π with the branching fraction The decay rate of Ξ ′ c |1 4 P λ 5/2 − into the Λ c K channel is significant as well, and the predicted branching ratio is The sizeable branching fraction of Ξ ′ c |1 4 P λ 5/2 − into Λ c K is consistent with the observation of the Ξ c (2965) 0 signal in the Λ c K decay channel.

IV. SUMMARY
In this paper, we carry out a systematic study of the OZI allowed two-body strong decays of the λ-mode 1P wave Ξ ′ c states in the framework of a chiral quark model. Combining our theoretical predictions and the experimental observations, we give possible interpretations for the three new states Ξ c (2923) 0 , Ξ c (2939) 0 , and Ξ c (2965) 0 observed by the LHCb Collaboration.
The another newly observed state Ξ c (2965) 0 may corre-sponds to the λ-mode 1P-wave Ξ ′ c state with spin-parity J P = 5/2 − , namely Ξ ′ c |1 4 P λ 5/2 − . The Ξ c (2965) 0 may be a flavour partner of Ω c (3090) 0 . Besides the Λ c K decay channel, the decay rate of Ξ ′ c |1 4 P λ 5/2 − into Ξ c π is significant as well, and the predicted branching ratio is about 59%. The large branching fraction indicates that this state may be reconstructed in the Ξ c π decay channel as well.
There are strong configuration mixings in the J P = 1/2 − λmode states. The mixed state Ξ ′ c |1P λ 1/2 − 1 might be a flavour partner of Ω c (3000) 0 . This J P = 1/2 − mixed state may have a mass of M ∼ 2880 MeV and a narrow width of Γ ∼ 15 MeV. The dominant decay mode of |1P λ 1/2 − 1 is Ξ ′ c π with a branching fraction of > 85%. The decay rate into Λ c K is strongly suppressed due to the heavy quark symmetry. The Ξ ′ c |1P λ 1/2 − 1 may be observed in the Ξ ′ c π final state. The other J P = 1/2 − mixed state Ξ ′ c |1P λ 1/2 − 2 has a relatively broad width of Γ ∼ 48 MeV, which is about 3 times larger than that of Ξ ′ c |1P λ 1/2 − 1 . The Ξ ′ c |1P λ 1/2 − 2 mainly decays into Ξ c π and Λ c K channels with branching ratios 45% and 54%, respectively. Considering the mass and decay properties, if the broad structure in the Λ c K mass spectrum around M ∼ 2880 MeV observed by the LHCb Collaboration arises from a new Ξ 0 c state, this state is very likely to be the Ξ ′ c |1P λ 1/2 − 2 state. Moreover, the possibility of the broad structure arising from the overlapping of Ξ ′ c |1P λ 1/2 − 1 and Ξ ′ c |1P λ 1/2 − 2 cannot be ruled out as well. Finally it should be mentioned that combining our previous study [49] of the newly observed Ω * b states at LHCb [50], we find that the Ξ c (2923) 0 , Ξ c (2939) 0 , and Ξ c (2965) 0 may be flavour partners of Ω b (6330), Ω b (6340), and Ω b (6350), respectively. The missing mixed state Ξ ′ c |1P λ 1/2 − 1 may be a flavour partner of Ω b (6316).