Mass Spectrum of 1¹𝑃₁ Meson State and Mixing Angle of Strange Axial-Vector Mesons

Abstract

In the presence of the 𝑏₁(1235), ℎ₁(1170), 𝐷₁(2420), and 𝐷_𝑠1(2536)^± being the members of the 1¹𝑃₁ meson multiplet, we investigate the mass spectrum of the 1¹𝑃₁ 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 1¹𝑃₁ meson multiplet in the future.

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), the 𝑏₁(1235), ℎ₁(1170), 𝐷₁(2420), and 𝐷_𝑠1(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 the 𝑠𝑠 member, the mass of the 𝐾_1𝐵 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 𝐾_1𝐴 and 𝐾_1𝐵 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 ℎ_𝑐(1𝑃) has been observed in the decay 𝜓(2𝑠)→𝜋⁰ℎ_𝑐 with ℎ_𝑐→𝛾𝜂_𝑐 by CLEO [13] and in the reaction 𝑝𝑝→ℎ_𝑐→𝛾𝜂_𝑐→𝛾𝛾𝛾 [14]. The experimental signals about ℎ_𝑐(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 of 1¹𝑃₁ 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 𝐾_1𝐴 and 𝐾_1𝐵 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 same 𝐽^𝑃𝐶 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: 𝑀²=𝑀²_𝑁+2𝐴_𝑛𝑛√2𝐴_𝑛𝑠√2𝐴_𝑛𝑠𝑀²_𝑠𝑠+𝐴_𝑠𝑠, (1) where 𝑀_𝑁 and 𝑀_𝑠𝑠 are the masses of bare states 𝑁 and 𝑆, respectively; 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, the following expressions are widely used in [7, 19]: 𝐴_𝑛𝑠=𝐴_𝑛𝑛𝑋,𝐴_𝑠𝑠=𝐴_𝑛𝑛𝑋², (2) 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 states 𝜑 and 𝜑^′ are the eigenstates of mass-squared matrix 𝑀², and the masses square 𝑀²_𝜑 and 𝑀²_𝜑′ are the eigenvalues, respectively. The physical states 𝜑 and 𝜑^′ can be related to the 𝑆=𝑠𝑠 and 𝑁=(𝑢𝑢+𝑑𝑑)/√2 by 𝜑𝜑^′|𝑁⟩|𝑆⟩=𝑈, (3) and the unitary matrix 𝑈 can be described as 𝑈𝑀²𝑈^†=𝑀²_𝜑00𝑀²_𝜑′. (4)

The mix of the physical isoscalar states can also be described as 𝜑𝜑^′=cos𝜃−sin𝜃𝜑sin𝜃cos𝜃𝜑⁸₁ (5) with 𝜑₁=𝑢𝑢+𝑑𝑑+𝑠𝑠√3,𝜑₈=𝑢𝑢+𝑑𝑑−2𝑠𝑠√6, (6) where 𝜃 is the nonet mixing angle.

From the relations (3), (4), and (5), one will obtain 𝑈=cos𝜃−sin𝜃sin𝜃cos𝜃13−232313. (7)

Based on the relations (4) and (7), the following relations are obtained: 𝑀²_𝑁+2𝐴=13cos𝜃−23sin𝜃²𝑀²_𝜑+23cos𝜃+13sin𝜃²𝑀²_𝜑′,𝑀²_𝑠𝑠+𝐴𝑋²=13cos𝜃−23sin𝜃²𝑀²_𝜑′+23cos𝜃+13sin𝜃²𝑀²_𝜑. (8) 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 𝑀_𝑁=𝑀_𝐼=1. Applying the relations (8) for the ground pseudoscalar meson, we obtain the pure 𝑠𝑠 member mass of the ground pseudoscalar meson nonet 𝑀_𝑠𝑠(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 of 1¹𝑃₁ 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 identical 𝐽^𝑃𝐶 quantum numbers obey quasilinear form of Regge trajectories, one has the following relations [21]: 𝑙=𝛼_𝑖𝑖′(0)+𝛼^′_𝑖𝑖′𝑀²_𝑖𝑖′,𝑙=𝛼_𝑖𝑗′(0)+𝛼^′_𝑖𝑗′𝑀²_𝑖𝑗′,𝑙=𝛼_𝑗𝑗′(0)+𝛼^′_𝑗𝑗′𝑀²_𝑗𝑗′, (9) where 𝑖𝑖^′ denotes the quark and antiquark flavors, and 𝑙 and 𝑀_𝑖𝑖′ are the orbital momentum and mass of the 𝑖𝑖^′ meson. The parameters 𝛼^′_𝑖𝑖′ and 𝛼_𝑖𝑖′(0) are, respectively, the slope and intercept of the Regge trajectory. The intercepts can be described as the following relation by [21, 24]: 𝛼_𝑖𝑖(0)+𝛼_𝑗𝑗(0)=2𝛼_𝑖𝑗(0). (10) 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: 𝛼^′_𝑛𝑛𝑀²1¹𝑆_0𝑛𝑛+𝛼^′_𝑠𝑠𝑀²1¹𝑆_0𝑠𝑠=2𝛼^′_𝑛𝑠𝑀²1¹𝑆_0𝑛𝑠,𝛼^′_𝑛𝑛𝑀²1¹𝑆_0𝑛𝑛+𝛼^′_𝑐𝑐𝑀²1¹𝑆_0𝑐𝑐=2𝛼^′_𝑐𝑛𝑀²1¹𝑆_0𝑐𝑛,𝛼^′_𝑠𝑠𝑀²1¹𝑆_0𝑠𝑠+𝛼^′_𝑐𝑐𝑀²1¹𝑆_0𝑐𝑐=2𝛼^′_𝑐𝑠𝑀²1¹𝑆_0𝑐𝑠,𝛼^′_𝑛𝑛𝑀²1¹𝑃_1𝑛𝑛+𝛼^′_𝑠𝑠𝑀²1¹𝑃_1𝑠𝑠=2𝛼^′_𝑛𝑠𝑀²1¹𝑃_1𝑛𝑠,𝛼^′_𝑛𝑛𝑀²1¹𝑃_1𝑛𝑛+𝛼^′_𝑐𝑐𝑀²1¹𝑃_1𝑐𝑐=2𝛼^′_𝑐𝑛𝑀²1¹𝑃_1𝑐𝑛,𝛼^′_𝑠𝑠𝑀²1¹𝑃_1𝑠𝑠+𝛼^′_𝑐𝑐𝑀²1¹𝑃_1𝑐𝑐=2𝛼^′_𝑐𝑠𝑀²1¹𝑃_1𝑐𝑠. (11) In order to eliminate the Regge slopes from the previous formulas, we adopt the following relation [24]: 1𝛼^′_𝑖𝑖′+1𝛼^′_𝑗𝑗′=2𝛼^′_𝑖𝑗′. (12) 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 𝐾_1𝐵 and the charmed mesons 𝐷₁(2420) and 𝐷_𝑠1(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=𝑀_{𝐷𝑠1(2536)} as well as 𝑀(1¹𝑃₁)_𝑐𝑛=𝑀_𝐷1(2420) into relations (11)-(12), the masses of the 𝑠𝑠 member, 𝐾_1𝐵, and ℎ_𝑐(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 the 1¹𝑃₁ meson states with other references. (All in MeV.)

The mass of the 𝐾_1𝐵 is important for the calculation of mixing angle of the 𝐾_1𝐴 and 𝐾_1𝐵 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 𝐴_𝑞𝑞′=_𝑘𝑞𝑞𝐻^{𝑞′𝑞′}_𝑝.𝑐.𝑘𝑘𝐻^𝑞𝑞_𝑝.𝑐.𝑞^′𝑞^′𝑀²_{𝑞′𝑞′}−𝑀²_𝑘, (13) where 𝐻^𝑞𝑞_𝑝.𝑐. 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: 𝐴_𝑛𝑛=𝑓²_𝑛𝑛𝐺𝑀²_𝑛𝑛−𝑀²_𝐺,𝐴_𝑠𝑠=𝑓²_𝑠𝑠𝐺𝑀²_𝑠𝑠−𝑀²_𝐺, (14) where 𝑓_𝑞𝑞𝐺≡⟨𝑞𝑞|𝐻^𝑞𝑞|𝐺⟩|_{𝑝𝜇𝑃𝜇=𝑀}²_𝑞𝑞, and 𝑞 refers to the 𝑛 and 𝑠 quarks. The parameters 𝑀²_𝑛𝑛=𝑀²_𝐼=1 and 𝑀²_𝑠𝑠=2𝑀²_𝐼=1/2−𝑀²_𝐼=1; 𝑀_𝐼=1 and 𝑀_𝐼=1/2 are the masses of the isovector and isodoublet states of a meson nonet. Considering the functions 𝑓_𝑞𝑞𝐺, Brisudova et al. indicate the product of 𝑓_𝑛𝑛𝐺 and 𝑓_𝑠𝑠𝐺 in QCD to be a constant approximately independent of the quantum numbers of a meson nonet; namely [18], 𝑓_𝑛𝑛𝐺𝑓_𝑠𝑠𝐺≈Const. (15) Following the relations (1), (4), and (14)-(15), we have 𝑀²_𝑏1()1235−𝑀²_pse2𝑀²_𝐾1𝐵−𝑀²_𝑏1()1235−𝑀²_pse𝑀²_𝑎2()1320−𝑀²_ten2𝑀²_𝐾∗2−𝑀²_𝑎2()1320−𝑀²_ten=𝐴₁³_𝑃2𝐴₁¹_𝑃1 (16) with 𝐴₁¹_𝑃1=𝑀²_ℎ1()1170+𝑀²_ℎ1()1380−2𝑀²_𝐾21𝐵2+𝑋²,𝐴₁³_𝑃2=𝑀²_𝑓2()1270+𝑀²_{𝑓′2()1525}−2𝑀²_𝐾^∗₂2+𝑋², (17) where 𝑀_ten and 𝑀_pse are the masses of ground tensor and pseudovector glueball, respectively. In order to extract the mass of 𝐾_1𝐵 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 𝐾_1𝐵. 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.

4. The Mixing Angle of 𝐾_1𝐴 and 𝐾_1𝐵

According to the introduction in Section 1, the 𝐾_1𝐴 and 𝐾_1𝐵 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] 𝐾₁(1270)=sin𝜙1³𝑃₁+cos𝜙1¹𝑃₁,𝐾₁(1400)=cos𝜙1³𝑃₁−sin𝜙1¹𝑃₁. (18) In this work, 𝜙 is defined as the 𝐾_1𝐴 and 𝐾_1𝐵 mixing angle. To obtain the mixing angle 𝜙, we use the formula [31] tan²=𝑀2𝜙^2𝐾1𝐴−𝑀²_𝐾1𝐵𝑀²_𝐾1()1400−𝑀²_𝐾1()1270²−1. (19) The relation (19) can also be rewritten as =𝑀cos2𝜙^2𝐾1()1400−𝑀²_𝐾1()1270𝑀²_𝐾1𝐴−𝑀²_𝐾1𝐵. (20) Moreover, using the following relation which is independent of the mixing angle, the mass of 𝐾_1𝐴 is determined to be 1303 MeV. Consider 𝑀²_𝐾1𝐴+𝑀²_𝐾1𝐵=𝑀²_𝐾1()1400+𝑀²_𝐾1()1270. (21) With the help of the masses of 𝐾_1𝐴, 𝐾_1𝐵, 𝐾₁(1270), and 𝐾₁(1400), the mixing angle is determined to be ±61.6^∘.

5. Results and Conclusions

Based on the fact that the 𝑏₁(1235), ℎ₁(1170), 𝐷₁(2420), and 𝐷_𝑠1(2536)^± are assigned as the 1¹𝑃₁ meson multiplet, we obtain the masses of the 𝐾_1𝐵, ℎ₁(1380), and ℎ_𝑐(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 of 𝐾_1𝐵 is useful for determining the mixing angle of the 𝐾₁(1270) and 𝐾₁(1400). In the present work, the mass for 𝐾_1𝐵 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 angle 𝜙 of the 𝐾₁(1270) and 𝐾₁(1400).

Acknowledgments

This project is supported by the Zhengzhou University of Light Industry Foundation of China (Grant nos. 2009XJJ011 and 2012XJJ008), in part by the National Natural Science Foundation of China (Grant nos. 10975018 and 11147197), and the Key Project of Scientific and Technological Research of the Education Department of Henan Province (Grant nos. 12B140001 and 13B140332).