# Mass Spectrum of1P11 Meson State and Mixing Angle of Strange Axial-Vector Mesons

Authors: Xue-Chao Feng; Ke-Wei Wei; Jie Wu; Yun-Qiang Zhang; Ming-Yang Wu; Feng-Chun Jiang
Journal: Advances in High Energy Physics
 (2013)

---

## Abstract

In the presence of theb1(1235), h1(1170), D1(2420), and Ds1(2536)± being the members of the 1P11 meson multiplet, we investigate the mass spectrum of the 1P11 meson state and the mixing angle of strange axial-vector mesons in the framework of meson-meson mass mixing and Regge phenomenology. Also, in the glueball dominance picture, the kaon is phenomenologically determined to have a mass of about 1356 MeV. The results are compared with the values from different phenomenological models and may be useful for the assignment of the 1P11 meson multiplet in the future.


---

## Body

## 1. Introduction

Quantum Chromodynamics (QCD) is widely accepted as a successful theory of the strong interactions in particle physics. However, the understanding of the strong interactions is far from complete, and it is difficult to interpret the particle properties for the experimental data from the first principles. To be able to determine the nature of a new experimental resonance, it is necessary to create a template to compare the observed states with theoretical predictions. Therefore, different phenomenological models, such as quenched lattice gauge theory [1], the Dyson-Schwinger formalism [2], and constituent quark model [3], are built to describe the properties of experimental results [4].In the recent issue of Review of Particle Physics (PDG), theb1(1235), h1(1170), D1(2420), and Ds1(2536)± have been observed in the experiment and assigned as the members of the 11P1 meson multiplet [5]. However, the assignment for the strange member ss- state is problematic. On the one hand, based on the decay results in the 3P0 model, Barnes et al. suggest that the h1(1380) should be a candidate of the 11P1 meson [6]. On the other hand, the mass of h1(1380) is inconsistent with the prediction in the potential model, which is about 100 MeV higher than the measured masses of the h1(1380) [7]. Accordingly, the mass and decays of the h1(1380) need to be further examined experimentally.Apart from thess- member, the mass of the K1B also remains interesting. Due to the states K1(1P1) and K1(3P1) having no definite C-parity and the SU(3) broken, the K1(1P1) and K1(3P1) can mix via the spin-orbit interaction to form the physical states K1(1270) and K1(1400). In order to compare the theoretical prediction with experimental data, the mixing angle of K1A and K1B has been discussed with different theoretical approaches [8–12]. In addition to the light meson states, the charmonium is one of the most exciting areas of high energy physics. In the 11P1 meson state, the hc(1P) has been observed in the decay ψ(2s)→π0hc with hc→γηc by CLEO [13] and in the reaction pp-→hc→γηc→γγγ [14]. The experimental signals about hc(1P) are still very weak so far.In allusion to the problems mentioned earlier, the purpose of this work is to investigate the mass spectrum of11P1 meson state and mixing angle of strange axial-vector mesons. The results should be useful for exploring the nature and the assignment of 11P1 meson state. The organization of this paper is as follows. In Section 2, the brief review of the mass mixing matrix of isoscalar states is given (for the detailed review, see, e.g., [15–17]). In Section 3, the mass spectrum of 11P1 meson state is presented in the framework of Regge phenomenology and glueball-meson mass relation. The mixing angle of K1A and K1B is predicted in Section 4, and a summary and conclusion are given in Section 5.


## 2. The Review of Mass Mixing Matrix of Isoscalar States

In the quark model, the two isoscalar states with the sameJPC quantum number will mix to form the physical states. One can obtain the mass-squared matrix in the S=ss- and N=(uu-+dd-)/2 basis [18] as follows:
(1)M2=(MN2+2Ann2Ans2AnsMss-2+Ass),
where MN and Mss- are the masses of bare states N and S, respectively; Ann, Ans, and Ass are the mixing parameters which describe the qq-↔q′q′- transition amplitudes. In order to reduce the number of parameters, the following expressions are widely used in [7, 19]:
(2)Ans=AnnX,Ass=AnnX2,
where X is a phenomenological parameter describing the SU(3) broken ratio of the nonstrange and strange quark propagators via the constituent quark mass ratio. In the present work, we take X=0.63, which is determined within the nonrelativistic constituent quark model [17, 20].In the meson nonet, the physical isoscalar statesφ and φ′ are the eigenstates of mass-squared matrix M2, and the masses square Mφ2 and Mφ′2 are the eigenvalues, respectively. The physical states φ and φ′ can be related to the S=ss- and N=(uu-+dd-)/2 by
(3)(|φ〉|φ′〉)=U(|N〉|S〉),
and the unitary matrix U can be described as
(4)UM2U†=(Mφ200Mφ′2).The mix of the physical isoscalar states can also be described as(5)(|φ〉|φ′〉)=(cosθ-sinθsinθcosθ)(φ8φ1)
with
(6)φ1=uu-+dd-+ss-3,φ8=uu-+dd--2ss-6,
where θ is the nonet mixing angle.From the relations (3), (4), and (5), one will obtain
(7)U=(cosθ-sinθsinθcosθ)(13-232313).Based on the relations (4) and (7), the following relations are obtained:
(8)MN2+2A=(13cosθ-23sinθ)2Mφ2z+(23cosθ+13sinθ)2Mφ′2,Mss-2+AX2=(13cosθ-23sinθ)2Mφ′2z+(23cosθ+13sinθ)2Mφ2.
Considering the fact that N is the orthogonal partner of the isovector state of a meson nonet, one can expect that N degenerates with isovector state MN=MI=1. Applying the relations (8) for the ground pseudoscalar meson, we obtain the pure ss- member mass of the ground pseudoscalar meson nonet Mss-(11S0)=685MeV [17]. Here and in the following, all the masses used as input for our calculation are taken from the PDG [5].


## 3. Mass Spectrum of11P1 Meson State in Regge Phenomenology and Glueball Dominance Picture

Regge theory is cornered with the particle spectrum, the forces between particles, and the high energy behavior of scattering amplitudes. In [21], Li et al. investigate the masses of different meson multiplets in the framework of Regge phenomenology and suggest that the quasilinear Regge trajectories could provide a reasonable description for the meson mass spectrum. Khruschov predicts the masses spectrum of excited meson states using the phenomenology formulas derived from the Regge trajectories [22]. Anisovich et al. also suggest that the mass of meson states can fit to the quasilinear Regge trajectories with good accuracy [23]. In the last decade, Regge trajectories are reanalysed to make predictions for the masses of states not yet to be discovered in experiment and to determine the quantum numbers of the newly discovered states [21–28].Based on the assumption that the hadrons with identicalJPC quantum numbers obey quasilinear form of Regge trajectories, one has the following relations [21]:
(9)l=αii′-(0)+αii′-′Mii′-2,l=αij′-(0)+αij′-′Mij′-2,l=αjj′-(0)+αjj′-′Mjj′-2,
where ii′- denotes the quark and antiquark flavors, and l and Mii′- are the orbital momentum and mass of the ii′- meson. The parameters αii′-′ and αii′-(0) are, respectively, the slope and intercept of the Regge trajectory. The intercepts can be described as the following relation by [21, 24]:
(10)αii-(0)+αjj-(0)=2αij-(0).
The relation (10) is satisfied in two-dimensional QCD [29] and the quark bremsstrahlung model [30].With the aid of the parity partners trajectories having the same slopes, we can apply relations (9)-(10) to the 11S0 and 11P1 meson multiplets to predict the following meson masses:
(11)αnn-′M2(11S0)nn-+αss-′M2(11S0)ss-=2αns-′M2(11S0)ns-,αnn-′M2(11S0)nn-+αcc-′M2(11S0)cc-=2αcn-′M2(11S0)cn-,αss-′M2(11S0)ss-+αcc-′M2(11S0)cc-=2αcs-′M2(11S0)cs-,αnn-′M2(11P1)nn-+αss-′M2(11P1)ss-=2αns-′M2(11P1)ns-,αnn-′M2(11P1)nn-+αcc-′M2(11P1)cc-=2αcn-′M2(11P1)cn-,αss-′M2(11P1)ss-+αcc-′M2(11P1)cc-=2αcs-′M2(11P1)cs-.
In order to eliminate the Regge slopes from the previous formulas, we adopt the following relation [24]:
(12)1αii′-′+1αjj′-′=2αij′-′.
For the 11S0 meson multiplet, all members are established well. For the 11P1 meson multiplet, only the b1(1235) and h1(1170) are established well, and the strange meson K1B and the charmed mesons D1(2420) and Ds1(2536)± are the mixtures of spin-singlet 11P1 and spin-triplet 13P1 states. In the present work, based on the small mass splitting between charmed mesons, we conclude that the effect of mixing on the charmed meson masses will be small. Therefore, inserting the masses of the corresponding mesons states and M(11P1)cs-=MDs1(2536) as well as M(11P1)cn-=MD1(2420) into relations (11)-(12), the masses of the ss- member, K1B, and hc(1P) for the 11P1 meson multiplet are obtained, and the results are shown in Table 1.Table 1
Comparison of predictions for the masses of the11P1 meson states with other references. (All in MeV.)
Present work
[5]
[15]
[7]
[31]
[32]
[33]
M
s
s
-
1499
1386
1495.18
1470
M
n
s
-
1375
1370.03
1340
1368
1356
M
c
c
-
3509
3525.93
3520
3526The mass of theK1B is important for the calculation of mixing angle of the K1A and K1B states. In order to check consistence of the results in Table 1, we investigate the mass of kaon in the glueball dominance picture in the next section [18]. And the result is in good agreement with the prediction in the Regge phenomenology.In the mass matrix, the quark mixing amplitudes in (1) can be expressed as
(13)Aqq′=∑k‍〈qq-|Hp.c.q′q′-|k〉〈k|Hp.c.qq-|q′q′-〉Mq′q′-2-Mk2,
where Hp.c.qq- is the quark pair creation operator for the flavor q, and |k〉 is a complete set of the intermediate states. Based on the assumption that there is no direct quarkonium-quarkonium mixing and the qq-↔q′q′- transitions are dominated by the ground glueball with the corresponding quantum numbers, we obtain the following relations:
(14)Ann=fnn-G2Mnn2-MG2,Ass=fss-G2Mss2-MG2,
where fqq-G≡〈qq-|Hqq-|G〉|pμPμ=Mqq-2, and q refers to the n and s quarks. The parameters Mnn2=MI=12 and Mss2=2MI=1/22-MI=12; MI=1 and MI=1/2 are the masses of the isovector and isodoublet states of a meson nonet. Considering the functions fqq-G, Brisudova et al. indicate the product of fnn-G and fss-G in QCD to be a constant approximately independent of the quantum numbers of a meson nonet; namely [18],
(15)fnn-Gfss-G≈Const.
Following the relations (1), (4), and (14)-(15), we have
(16)(Mb1(1235)2-Mpse2)(2MK1B2-Mb1(1235)2-Mpse2)(Ma2(1320)2-Mten2)(2MK2*2-Ma2(1320)2-Mten2)=A13P2A11P1
with
(17)A11P1=Mh1(1170)2+Mh1(1380)2-2MK1B222+X2,A13P2=Mf2(1270)2+Mf2′(1525)2-2MK2*22+X2,
where Mten and Mpse are the masses of ground tensor and pseudovector glueball, respectively. In order to extract the mass of K1B by using relation (16), we should determine the approximate masses of ground tensor and pseudovector glueballs. Some predictions on the masses of the glueball given by different theoretical models are shown in Table 2. In the present work, taking the masses of corresponding mesons in the relation (16) and the average values of glueball masses in Table 2 as input, we obtain the mass of the K1B. Our predicted result 1356 MeV is in agreement with our previous work.Table 2
Masses (in GeV) of the ground tensor and pseudovector glueball from different theoretical models.
Mass
[34]
[35]
[36]
[37]
[38]
[39]
[40]
Average
M
ten
2.59
2.40
2.39
2.15
2.26
2.4
2.417
2.37
M
pse
2.94
2.98
2.67
2.71
2.83


## 4. The Mixing Angle ofK1A and K1B

According to the introduction in Section1, the K1A and K1B in the 13P1 and 11P1 meson states can mix to produce the physical states K1(1270) and K1(1400). In the |11P1〉 and |13P1〉 basis, the physical states K1(1270) and K1(1400) can be parameterized as [11]
(18)K1(1270)=sinϕ|13P1〉+cosϕ|11P1〉,K1(1400)=cosϕ|13P1〉-sinϕ|11P1〉.
In this work, ϕ is defined as the K1A and K1B mixing angle. To obtain the mixing angle ϕ, we use the formula [31]
(19)tan2(2ϕ)=(MK1A2-MK1B2MK1(1400)2-MK1(1270)2)2-1.
The relation (19) can also be rewritten as
(20)cos(2ϕ)=MK1(1400)2-MK1(1270)2MK1A2-MK1B2.
Moreover, using the following relation which is independent of the mixing angle, the mass of K1A is determined to be 1303 MeV. Consider
(21)MK1A2+MK1B2=MK1(1400)2+MK1(1270)2.
With the help of the masses of K1A, K1B, K1(1270), and K1(1400), the mixing angle is determined to be ±61.6°.


## 5. Results and Conclusions

Based on the fact that theb1(1235), h1(1170), D1(2420), and Ds1(2536)± are assigned as the 11P1 meson multiplet, we obtain the masses of the K1B, h1(1380), and hc(1P) in the 11P1 meson state. The mass of the h1(1380) is determined to be 1492 MeV, which is consistent with the value 1470 MeV predicted by Godfrey and Isgur in the potential model [7]. The value is about 100 MeV higher than the measured mass of the h1(1380) in the experiment. Besides, experimental information on the h1(1380) is rather restricted, which has been reported only by two collaborations, LASS Collaboration [45] (with mass 1380±20MeV and width 80±30MeV) and Crystal Barrel Collaboration [46] (with mass 1440±60MeV and width 170±80MeV). Therefore, we suggest that the assignment for this state needs to be further tested in the new experiments in the future.The mass ofK1B is useful for determining the mixing angle of the K1(1270) and K1(1400). In the present work, the mass for K1B is determined to be 1375 MeV, which is in agreement with the value from the meson-glueball mass relation and other theoretical approaches. The mixing angle of K1(1270) and K1(1400) is compared with other different predictions in Table 3. Our predictions may open a window for the assignment of the 11P1 meson multiplet in the new experiment in the future.Table 3
Predicted mixing angleϕ of the K1(1270) and K1(1400).
Refs
Present work
[15]
[12]
[41]
[42]
[43]
[44]
|
ϕ
|
61
.
6
∘
(
59.55
±
2.81
)
∘
59.2
9
∘
54
.
7
∘ or 35.3∘
6
2
∘
55
.
7
∘
6
0
∘



---

*Source: 704529-2013-04-04.xml*