# Regarding the Charmed-Strange Member of the23S1  Meson State

Authors: Xue-Chao Feng; Jing Chen
Journal: The Scientific World Journal
 (2013)

---

## 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 the23S1 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.


---

## Body

## 1. Introduction

Charmed spectroscopy becomes an active field with many new states observed in the experiment in the last few years [1–8]. The DSJ(2632) (with the mass 2632.5±1.7 MeV and decay width Γtot≤17 MeV) was reported in the two decay modes Ds+η and D0K+ by the Selex collaboration [5]. In [9–11], DSJ(2632) was interpreted as the first radical excitation of DS*(2112). However, the narrow and dominated Ds+η decay mode implies that the assignment is problematic. On the other hand, the mass of the DSJ(2632) is lower than the value of  the tradition potential model predictions. Therefore, the DSJ(2632) is also interpreted as a tetraquark, a hybrid, a diquark-antiquark bound state, and so forth.Recently, Belle collaboration observed a newcs- state DSJ(2700) with a mass of 2708±9-10+11 MeV and width 108±23-31+36 MeV [8]. Based on its observed decay channel B+→D-0DSJ→D-0D0K+, the DSJ(2700) resonance is interpreted as a charmed-strange meson state.Firstly, we reviewed the assignment of the23S1 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→K-0π+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 DSJ(2632) and the  DSJ(2700) are probably interpreted as charmed-strange candidate of the 23S1 meson state [9–11, 14]. The nonstrange partner of DSJ has not been observed in the experiment.Table 1
Assignment of the23S1 meson state in PDG [12].
2
S
+
1
L
J
n
n
-
n
n
-
(
s
s
-
)
n
s
-
c
c
-
c
n
-
c
s
-
2
3
S
1
ρ
(
1450
)
ω
(
1420
)
ϕ
(
1680
)
K
*
(
1410
)
ψ
(
2
S
)
D
D
sIn this work, based on the isoscalar states can mix to form the physical states in theqq- 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 sameJPC 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:
(1)M2=(Mρ(1450)2+2Ann2Ans2Ans2MK*(1410)2-Mρ(1450)2+Ass),
where Mρ(1450) and MK*(1410) are the masses of isovector and isodoublet states of the 23S1 meson nonet, respectively and  Ann, Ans, and Ass 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:
(2)Ann=Λmnmn,Ass=Λmsms,Ans=Λmnms,
where Λ is a phenomenological parameter. Based on the isospin symmetry, we have mu=mu-=md=mu-, ms=ms- (mn,ms denote the mass of light quark u, d, and s).In the23S1 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 Mω(1420)2 and Mϕ(1680)2 are the eigenvalues, respectively. The physical states ω(1420) and ϕ(1680) can be related to the ss- and N=(uu-+dd-)/2 by
(3)(|ω(1420)〉|ϕ(1680)〉)=U(|N〉|S〉),
and the unitary matrix U can be described as
(4)UM2U†=(Mω(1420)200Mϕ(1680)2).According to relations (1), (2), (3), and (4), we will obtain
(5)2Λmn2+2MK*(1410)2+Λms2=Mω(1420)2+Mϕ(1680)2,(Mρ(1450)2+2Λmn2)(2MK*(1410)2-Mρ(1450)2+Λms2)-2Λms2mn2=Mω(1420)2*Mϕ(1680)2.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]:
(6)J=αii′-(0)+αii′-′Mii′-2,
where ii′- refers to the quark (antiquark) flavor, J and Mii′- are, respectively, the spin and mass of the ii′- meson. The parameters αii′-′ and αii′-(0) are, respectively, the slope and intercept of the trajectory. The intercepts and slopes can be described by [15, 29]
(7)αii-(0)+αjj-(0)=2αij-(0),(8)1αii′-′+1αjj′-′=2αij′-′.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:
(9)π4αji′+π4α′mi+mj2αji′=α′,
where mi and mj 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:
(10)Mρ(1450)2α′2+2α′mn+Mψ(2S)2α′2+2α′mc=2MD2α′2+α′(mn+mc),4MK*(1410)2α′2+α′(ms+mn)+Mψ(2S)2α′1+α′mc=Mρ(1450)2α′2+α′mn+4MDs2α′2+α′(ms+mc).
Using relations (5) and (10), we can obtain the masses of K*(1410), D, and Ds 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.
Mass
Ref [15]
Ref [16]
Ref [17]
Ref [18]
Ref [19]
Average
m
n
(
n
=
u
,
d
)
290
360
337.5
311
310
321
m
s
460
540
486
487
483
491.2
m
c
1650
1710
1550
1592
1625.5Table 3
Masses (MeV) ofK*(1410), ϕ(1680), and ψ(2S) in the 23S1 meson state.
Mass
M
K
*
(
1410
)
M
D
M
D
s
Our results
1579.82
2593.2
2700
Ref [12]
1414
±
15
Ref [14]
1567
±
22.6
Ref [20]
1573
2588
Ref [21]
2757.8
Ref [22]
2629
2716
Ref [23]
2640
2730
Ref [24]
2620
2730IfMK*(1410)=1579.82 MeV and Mω(1420) and Mϕ(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
(11)U=(Xω(1420)Yω(1420)Xϕ(1680)Yϕ(1680)).Inserting the masses of isoscalar states, we obtain the quarkonia content ofω(1420) and ϕ(1680) as follows:
(12)|ω(1420)〉=-0.997|N〉-0.074|S〉,|ϕ(1680)〉=0.074|N〉-0.997|S〉.


## 3. The Results and Conclusions

In summary, based on the mass matrix and Regge trajectory, we investigate the masses ofK*(1410), D, and Ds 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 DSJ+(2632) and in agreement with the DSJ(2700). Moreover, on the basis of unusual decay modes and narrow width of the DSJ+(2632), we suggest that the DSJ(2700) should be the first radial excitation of the DS*(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.



---

*Source: 658105-2013-10-22.xml*