Regarding the Charmed-Strange Member of the 2³𝑆₁ Meson State

Abstract

By employing the mass relations derived from the mass matrix and Regge trajectory, we investigate the masses of charmed and charmed-strange members of the 2³𝑆₁ meson. The masses are compared with the values predicted by other theoretical approaches and experimental data. The results may be useful for the discovery of the unobserved meson and the determination of the quantum number of the newly discovered states.

1. Introduction

Charmed spectroscopy becomes an active field with many new states observed in the experiment in the last few years [1–8]. The 𝐷_𝑆𝐽(2632) (with the mass 2632.5±1.7 MeV and decay width Γ_tot≤17 MeV) was reported in the two decay modes 𝐷⁺_𝑠𝜂 and 𝐷⁰𝐾⁺ by the Selex collaboration [5]. In [9–11], 𝐷_𝑆𝐽(2632) was interpreted as the first radical excitation of 𝐷^∗_𝑆(2112). However, the narrow and dominated 𝐷⁺_𝑠𝜂 decay mode implies that the assignment is problematic. On the other hand, the mass of the 𝐷_𝑆𝐽(2632) is lower than the value of the tradition potential model predictions. Therefore, the 𝐷_𝑆𝐽(2632) is also interpreted as a tetraquark, a hybrid, a diquark-antiquark bound state, and so forth.

Recently, Belle collaboration observed a new 𝑐𝑠 state 𝐷_𝑆𝐽(2700) with a mass of 2708±9⁺¹¹₋₁₀ MeV and width 108±23⁺³⁶₋₃₁ MeV [8]. Based on its observed decay channel 𝐵⁺→𝐷⁰𝐷_𝑆𝐽→𝐷⁰𝐷⁰𝐾⁺, the 𝐷_𝑆𝐽(2700) resonance is interpreted as a charmed-strange meson state.

Firstly, we reviewed the assignment of the 2³𝑆₁ meson state in the 𝑞𝑞 quark model (see Table 1). According to the new edition of Particle Data Group (PDG) [12], the states 𝜔(1420) and 𝜙(1680) have been well established as the isoscalar member in the 2³𝑆₁ meson state. 𝐾^∗(1410) is assigned as isodoublet member however, it is still based on very weak experimental signals. Till now, the 𝐾^∗(1410) has been reported by two experiments 𝐾⁻𝑝→𝐾⁻𝜋⁺𝑛 (with mass 1380±21±19 MeV and width 176±52±22 MeV) and 𝐾⁻𝑝→𝐾⁰𝜋⁺𝜋⁻𝑛 [12] (with mass 1420±7±10 MeV and width 240±18±12 MeV). The assignment of isodoublet of the 2³𝑆₁ meson state was investigated in our previous work [13]. For the heavy-light meson, the charmed-strange member of the 2³𝑆₁ meson state has attracted more attention, recently. Both the 𝐷_𝑆𝐽(2632) and the 𝐷_𝑆𝐽(2700) are probably interpreted as charmed-strange candidate of the 2³𝑆₁ meson state [9–11, 14]. The nonstrange partner of 𝐷_𝑆𝐽 has not been observed in the experiment.

Table 1 

Assignment of the 2³𝑆₁ meson state in PDG [12].

In this work, based on the isoscalar states can mix to form the physical states in the 𝑞𝑞 quark model and the linear Regge trajectory; we establish the new mass relations which relate the mass spectrum of meson state and constituent quark masses. Inserting the corresponding constituent quark masses and the following well-established states, we reexamine the mass spectrum of the 2³𝑆₁ meson state. The results could be a useful comparison with the experiment data in the new experiment.

2. Mass Matrix and Regge Trajectory

In the quark model, the two isoscalar states with the same 𝐽^𝑃𝐶 will mix to form the physical isoscalar states. We can establish the mass-squared matrix in the 𝑠𝑠 and 𝑁=(𝑢𝑢+𝑑𝑑)/√2 basis [25] as follows: 𝑀²=𝑀²_𝜌()1450+2𝐴_𝑛𝑛√2𝐴_𝑛𝑠√2𝐴_𝑛𝑠2𝑀²_𝐾^∗₍₎₁₄₁₀−𝑀²_𝜌()1450+𝐴_𝑠𝑠, (1) where 𝑀_𝜌(1450) and 𝑀_𝐾^∗₍₁₄₁₀₎ are the masses of isovector and isodoublet states of the 2³𝑆₁ meson nonet, respectively and 𝐴_𝑛𝑛, 𝐴_𝑛𝑠, and 𝐴_𝑠𝑠 are the mixing parameters which describe the 𝑞𝑞↔𝑞^′𝑞^′ transition amplitudes. In order to reduce the number of parameters, we adopt the similar expression of the transition amplitudes in the 𝑞𝑞↔𝑞^′𝑞^′ process which is widely used in [26–28] as follows: 𝐴_𝑛𝑛=Λ𝑚_𝑛𝑚_𝑛,𝐴_𝑠𝑠=Λ𝑚_𝑠𝑚_𝑠,𝐴_𝑛𝑠=Λ𝑚_𝑛𝑚_𝑠, (2) where Λ is a phenomenological parameter. Based on the isospin symmetry, we have 𝑚_𝑢=𝑚_𝑢=𝑚_𝑑=𝑚_𝑢, 𝑚_𝑠=𝑚_𝑠 (𝑚_𝑛,𝑚_𝑠 denote the mass of light quark 𝑢, 𝑑, and 𝑠).

In the 2³𝑆₁ meson nonet, we assume that the physical states 𝜔(1420) and 𝜙(1680) are the eigenstates of mass-squared matrix and the masses square of 𝑀²_𝜔(1420) and 𝑀²_𝜙(1680) are the eigenvalues, respectively. The physical states 𝜔(1420) and 𝜙(1680) can be related to the 𝑠𝑠 and 𝑁=(𝑢𝑢+𝑑𝑑)/√2 by |𝜔(𝜙(|𝑁⟩|𝑆⟩,1420)⟩1680)=𝑈 (3) and the unitary matrix 𝑈 can be described as 𝑈𝑀²𝑈^†=𝑀²_𝜔()142000𝑀²_𝜙()1680. (4)

According to relations (1), (2), (3), and (4), we will obtain 2Λ𝑚²_𝑛+2𝑀²_𝐾^∗₍₎₁₄₁₀+Λ𝑚²_𝑠=𝑀²_𝜔()1420+𝑀²_𝜙()1680,𝑀²_𝜌()1450Λ+2𝑚²_𝑛2𝑀²_𝐾^∗₍₎₁₄₁₀−𝑀²_𝜌()1450+Λ𝑚²_𝑠Λ−2𝑚²_𝑠𝑚²_𝑛=𝑀²_𝜔()1420∗𝑀²_𝜙()1680. (5)

In relation (5), the masses of 𝐾^∗(1410) and 𝜙(1680) are related with constituent quarks mass and phenomenological parameter Λ.

Regge theory is cornered with the particle spectrum, the forces between particles, and the high energy behavior of scattering amplitudes. Because the Regge trajectories can offer an effective way for the assignment and the classification of meson states, it also become an active field with many new particles and resonances being observed in the experiment in the last decade. In [29], the authors investigated the mass of different meson multiplets and suggested that the quasilinear Regge trajectories could describe the meson mass spectrum. Khruschov [30], using the phenomenology formulae deprive from the Regge trajectories, predicted the masses of excited meson states. Anisovich et al. [31] show that meson states can fit to the quasi-linear Regge trajectories with good accuracy.

According to the hadron with a set of given quantum numbers belonging to a quasilinear trajectory, we will have the following relation [7]: 𝐽=𝛼_𝑖𝑖′(0)+𝛼^′_𝑖𝑖′𝑀²_𝑖𝑖′, (6) where 𝑖𝑖^′ refers to the quark (antiquark) flavor, 𝐽 and 𝑀_𝑖𝑖′ are, respectively, the spin and mass of the 𝑖𝑖^′ meson. The parameters 𝛼^′_𝑖𝑖′ and 𝛼_𝑖𝑖′(0) are, respectively, the slope and intercept of the trajectory. The intercepts and slopes can be described by [15, 29] 𝛼_𝑖𝑖(0)+𝛼_𝑗𝑗(0)=2𝛼_𝑖𝑗(0),1𝛼^′_𝑖𝑖′+1𝛼^′_𝑗𝑗′=2𝛼^′_𝑖𝑗′. (7)(8)

Relation (7) is satisfied in two-dimensional QCD [32], the dual-analytic model [33], and the quark bremsstrahlung model [34]. Relation (8) is derived from the topological and the 𝑞𝑞-string picture of hadrons [35]. According to available data of meson states, Burakovsky constructed a slope formula (9) for all quarks flavors [36] follows: 𝜋4𝛼^′_𝑗𝑖+𝜋4𝛼^′𝑚_𝑖+𝑚_𝑗2𝛼^′_𝑗𝑖=𝛼^′, (9) where 𝑚_𝑖 and 𝑚_𝑗 are the corresponding constituent quark masses and the 𝛼^′=0.88 GeV⁻² is the standard Regge slope in the light quark sector.

From relations (6)–(9), we obtained the following relation: 𝑀²_𝜌()1450𝛼^′2+2𝛼^′𝑚_𝑛+𝑀²_𝜓()2𝑆𝛼^′2+2𝛼^′𝑚_𝑐=2𝑀²_𝐷𝛼^′2+𝛼^′𝑚_𝑛+𝑚_𝑐,4𝑀²_𝐾^∗₍₎₁₄₁₀𝛼^′2+𝛼^′𝑚_𝑠+𝑚_𝑛+𝑀²_𝜓()2𝑆𝛼^′1+𝛼^′𝑚_𝑐=𝑀²_𝜌()1450𝛼^′2+𝛼^′𝑚_𝑛+4𝑀²_𝐷𝑠𝛼^′2+𝛼^′𝑚_𝑠+𝑚_𝑐. (10) Using relations (5) and (10), we can obtain the masses of 𝐾^∗(1410), 𝐷, and 𝐷_𝑠 in the 2³𝑆₁ meson state. In this paper, we use the average values of the constituent quark mass as input parameters (see Table 2). The mass of 𝜔(1420), 𝜙(1680), and 𝜓(2𝑆) used are taken from the new edition of PDG [1]. Our results are presented in Table 3.

Table 2 

Constituent quark masses (in MeV) in different phenological models.

Table 3 

Masses (MeV) of 𝐾^∗(1410), 𝜙(1680), and 𝜓(2𝑆) in the 2³𝑆₁ meson state.

If 𝑀_𝐾^∗₍₁₄₁₀₎=1579.82 MeV and 𝑀_𝜔(1420) and 𝑀_𝜙(1680) are assigned as the member of the 2³𝑆₁ meson nonet, we can investigate the quarkonia content of isoscalar state. Based on the previous assumption that physical states 𝜔(1420) and 𝜙(1680) are the eigenvectors of mass-squared matrix, the unitary matrix 𝑈 can be described as 𝑋𝑈=^𝜔(1420)𝑌_𝜔(1420)𝑋_𝜙(1680)𝑌_𝜙(1680). (11)

Inserting the masses of isoscalar states, we obtain the quarkonia content of 𝜔(1420) and 𝜙(1680) as follows: |𝜔(|𝑁⟩|𝑆⟩,𝜙(|𝑁⟩|𝑆⟩.1420)⟩=−0.997−0.0741680)=0.074−0.997 (12)

3. The Results and Conclusions

In summary, based on the mass matrix and Regge trajectory, we investigate the masses of 𝐾^∗(1410), 𝐷, and 𝐷_𝑠 in the 2³𝑆₁ meson state. The mass of 𝐾^∗(1410) is determined to be 1579.82 MeV; the results is consistent with our previous work. For the heavy-light meson section, we predicted the mass of charmed-strange member to be 2700 MeV. The value is larger than 𝐷⁺_𝑆𝐽(2632) and in agreement with the 𝐷_𝑆𝐽(2700). Moreover, on the basis of unusual decay modes and narrow width of the 𝐷⁺_𝑆𝐽(2632), we suggest that the 𝐷_𝑆𝐽(2700) should be the first radial excitation of the 𝐷^∗_𝑆(2112). As a byproduct, the quarkonia content of 𝜔(1420) and 𝜙(1680) was offered, which implies that 𝜔(1420) is pure 𝑢𝑢+𝑑𝑑 state and 𝜙(1680) is pure 𝑠𝑠 state. Our results should be useful for the assignment and the identification of the member of the 2³𝑆₁ meson state in the new experiment.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This project was supported by Zhengzhou University of Light Industry Foundation of China (Grant nos. 2009XJJ011 and 2012XJJ008) and the Key Project of Scientific and Technological Research of the Education Department of Henan Province (Grant no. 13B140332).