Theoretical interpretation of $\Xi_c(2970)$

The open charm strong decay widths and certain ratio of branching fractions of a charmed strange baryon $\Xi_c(2970)$ are calculated in a $^3P_0$ model. The results are compatible with the latest experimental data. The theoretical ratio of decay branching fractions $R = \mathcal{B} [ \Xi_{c}(2970)^+ \rightarrow \Xi_c(2645)^{0}\pi^{+} ] / \mathcal{B} [ \Xi_{c}(2970)^+ \rightarrow \Xi_c^{\prime0}\pi^{+}] \approx 1.0$. The spin-parity $J^P = 1/2^+$ and $3/2^+$ for different assignments are analyzed. From the results of our calculation, $\Xi_{c}(2970)$ can be interpreted as a $2S$-wave state with $J^P(s_l) = 1/2^+(0)$. The distinguishing between the 2S-wave $n_\rho$- and $n_\lambda$-excitation states and between states with $s_l = 0$ and $s_l = 1$ and between states with total spin $1/2$ and $3/2$($s_l = 1$) are also discussed.

The open charm strong decay widths and certain ratio of branching fractions of a charmed strange baryon Ξc(2970) are calculated in a 3 P0 model. The results are compatible with the latest experimental data. The theoretical ratio of decay branching fractions R = B[Ξc(2970) + → Ξc(2645) 0 π + ]/B[Ξc(2970) + → Ξ 0 c π + ] ≈ 1.0. The spin-parity J P = 1/2 + and 3/2 + for different assignments are analyzed. From the results of our calculation, Ξc(2970) can be interpreted as a 2Swave state with J P (s l ) = 1/2 + (0). The distinguishing between the 2S-wave nρ-and n λ -excitation states and between states with s l = 0 and s l = 1 and between states with total spin 1/2 and 3/2(s l = 1) are also discussed.

I. INTRODUCTION
In the convention of quark model [1,2], Ξ c baryon is consisted of with one u or d, one strange and one charmed quark, which is also known as the charmedstrange baryon. The low lying charmed-strange baryons are confirmed with their J P numbers measured experimentally. However, with the increasing energy, more and more highly excited charmed strange baryons have been observed. In Ξ c sector, Ξ c (2970)(was Ξ c (2980)) was first observed by the Belle Collaboration in the Λ + c K − π + channel [3], and then confirmed by the BABAR Collaboration in the Ξ c (2645) 0 π + decay channel [4]. With its quantum numbers unmeasured, calculations and debates of Ξ c (2970) have been carried out.
Recently, the Belle Collaboration reported the results from a study of the spin and parity of Ξ c (2970) + [5]. The angular distributions strongly favors Ξ c (2970) to be spin J = 1/2. And they also measured the ratio of decay branching fractions, where the IS means the uncertainty due to possible isospin-symmetry-breaking effects. This R value favors the spin-parity J P = 1/2 + with the spin of the lightquark degrees of freedom s l = 0.
In experiments, J P quantum numbers for most of the observed excited charmed baryons have not yet been measured so far. How to identify the observed baryons is an important topic in baryon spectroscopy. The spectroscopy of charmed baryons has been studied in many models. Ebert et al calculated the mass spectra of heavy baryons in the heavy-quark-light-quark picture in the * zhaoze@mail.itp.ac.cn QCD-motivated relativistic quark model and they suggested that Ξ c (2970) be assigned as the 2S excitation with J P = 1/2 + [6]. Bing Chen etal investigated the Λ c and Ξ c in the heavy quark-light diquark picture and they also concluded that Ξ c (2970) to be assigned to the first radial excitations with J P = 1/2 + [7]. For more spectral study, one can see literature [2] and references therein.
Hadronic decays of Ξ c (2970) have been studied in a heavy hadron chiral perturbation theory [8], where Ξ c (2970) is suggested a positive-parity excitation of Ξ c . It has also been studied in a chiral quark model [9], in which Ξ c (2970) is suggested as one of the orbital excitations of Ξ c . There is never certain conclusions about these excited baryons before their quantum numbers be measured experimentally. Since the report of the spinparity of Ξ c (2970) + , one of the remained questions is to ascertain its radial quantum number, that is to decide whether it is a 2S-wave or 1D-wave state. One of the purposes of this article is to solve this question theoretically by calculating the hadronic decay of Ξ c (2970) + under different assignments, and the other is to predict the properties of other Ξ c baryons that may carry similar masses or quantum numbers to be observed.
There are a lot of theoretical approaches to study the properties of hadrons. 3 P 0 model is one of the phenomenological methods to calculate the OZI-allowed hadronic decays of hadrons. In addition to mesons, it is employed successfully to explain the hadronic decays of baryons [10][11][12][13][14][15]. In this paper, we will study the hadronic decays of Ξ c (2970) + in the framework of 3 P 0 model. This work is organized as follows. In Sec. II, we give a brief review of the 3 P 0 model while in Sec. III we present our numerical results . In the last section, we give our conclusions and discussions. ated from the vacuum and thus with quantum numbers J P C = 0 ++ ( 2S+1 L J = 3 P 0 ). It was first proposed to calculate the open strong decays of two-body mesons. Furthermore, the model has been subsequently employed and developed to study the OZI-allowed hadronic decays of three-body baryons by many authors not cited here.
In the model, the created quark-anti-quark qq then regroup with the quarks from the initial hadron A to form two daughter hadrons B and C. The interaction Hamiltonian for the creating process has the form [20][21][22] where ψ f is a Dirac quark field with flavor f . m f is the constituent quark mass. The strength of the quark pair creation is represented by the dimensionless parameter γ.
For meson decays, the created quark regroup with the anti-quark of the initial meson, the created anti-quark regroup with the quark of the initial meson, and two mesons appear in the final states. For baryon decays, one quark of the initial baryon regroups with the created antiquark to form a meson, and the rest two quarks regroup with the created quark to form a daughter baryon. The process of a baryon decay is shown in Fig. 1.
In the 3 P 0 model, the hadronic decay width Γ of a process A → B + C is as follows [19], In the equation, p is the momentum of the daughter baryon in A's center of mass frame, In the equation above, ϕ 1,4,3 is the flavor matrix, to calculate the flavor matrix element, equations from Ref. [19] is employed, and taking into account that I P = 0(isospin of the created quark pair) we have, where f takes the value of ( 2 3 ) 1/2 or −( 1 3 ) 1/2 according to the isospin 1 2 or 0 of the created quarks. I A , I B and I M represent the isospins of the initial baryon, the final baryon and the final meson. I 12 , I 3 , I 4 are the isospins of relevant quarks, respectively.
The core of the calculation lies in the space integral in Eq. (4), Simple harmonic oscillator (SHO) wave functions are employed to model the baryon wave functions [10,12,13] where N represents a normalization coefficient of the total wave function and denotes the Laguerre polynomial function, Y LM L (Ω p ) is a spherical harmonic function. The relation between the solid harmonica polynomial y LM ( p) and

A. Notations of baryons and relevant parameters
For the hadronic decays of Ξ c (2970), the quantum numbers of the initial state baryons and final state baryons are presented in Table I and Table II, respectively. Here n ρ , L ρ and S ρ denote the nodal, the orbital and the spin of the two light quarks, while n λ , L λ and S λ correspond to the nodal, the orbital and the spin between the heavy quark and the light quark system. J l is the total angular momentum of S ρ and the total orbital angular momentum L. Finally, J equals to the total spin of the baryon. Notations for the excited D-wave Ξ c baryons are the same as those in Ref. [15]. For the first radially excited Ξ c , there are two kinds of excitations, (n ρ , n λ ) = (1, 0) and (0, 1), which we call them as "n ρ -excitation" and "n λ -excitation", respectively. We useΞ ( , * ) cJ l andΞ ( , * ) cJ l to represent them.

Masses of relevant mesons and baryons involved in the calculation are listed in
Parameters γ and β are taken as those in Refs. [15,23]. In general, uū, dd and ss could be created from the vacuum. However, there exists no experimental signal for the decay mode with a ss creation. On the other hand, according to the decaying threshold the decay mode with a ss creation does not open for Ξ c (2970) 0 . Thus, we consider that the OZI-allowed channels are all assumed from   Table IV, along with the decays of a few other candidate-like states are also calculated and listed. The vanish modes the table indicate forbidden channels due to conservation of some quantum numbers. Ξ c (2970) was first observed by the Belle Collaboration in the Λ + c K − π + channel with Γ = 43.5 ± 7.5 ± 7.0M eV and confirmed by the BABAR Collaboration in the intermediate resonant mode Σ c (2455) + K − . In Ref. [24], some new measurements were reported. The branching fraction ratio of B(Ξ c (2970) Ξ c (2970) decays significantly into Ξ c π + . The decay of Ξ c (2970) into Λ c K or Ξ c π channel has never been observed in experiments. The latest experimental data shows that the width of Ξ c (2970) + is Γ = 20.9 +2.4 −3.5 M eV From Table IV  also has the ratio R ≈ 1 and has a even more larger total decay width, which is about seven times as the experimental measurement. On the other hand, the n λ -excited stateΞ + c ( 1 2 + ) has a relatively smaller total decay width 15.5M eV , which is near the lower threshold of the experimental data. The R value is also ≈ 1.
On the other hand, as mentioned in the experimental article [5], heavy-quark spin symmetry(HQSS) predicts R = 1.06(0.26) for a 1/2 + state with the spin of the light-quark degrees of freedom s l = 0(1) as calculated in Ref [8]. The R value in this work are in consistent with their result, which is R ≈ 1.
As we all know, the results of phenomenological models depend heavily on the parameters. Ref. [25] calculated the decay widths of Ξ c baryons by using different groups of parameters β. The results in their Table IV shows how the decay width and relative branching fractions vary according the different sets of parameters.
Further more, we also calculated the hadronic decays of the 2S-wave Ξ c baryons with quantum numbers s l = 1, the n ρ -excitationsΞ that is, the n ρ -excited states are much more broader than the n λ -excited states. The total decay widths will be the distinguishing between them. Finally, the branching ratiosΞ This may be of help to distinguish the 2S-wave states with total spin 1 2 and 3 2 .

IV. CONCLUSIONS AND DISCUSSIONS
In this work, the hadronic decays of Ξ c (2970) + are studied in a 3 P 0 model. We calculate the decay widths and some ratios of branching fractions of Ξ c (2970) + as possible assignments related to recent Belle experiment. In comparison with experiments, we make an identification of this charmed-strange baryon. Our theoretical predictions are consistent with experiments. Ξ c (2970) was first observed by the Belle Collaboration in the Λ + c K − π + channel, the latest experimental data shows that its decay width is of about 20 MeV. Our results of branching fraction ratios are in consistent with the results that considered the HQSS effects. Our conclusion of Ξ c (2970) + is that it is a 2S-wave n λ -excitation stateΞ + c ( 1 2 + ), whose J P (s l ) is 1/2 + (0