Bottomonium spectrum in the relativistic flux tube model

The bottomonium spectrum is far from being established. The structures of higher vector states, including the Υ(10580), Υ(10860), and Υ(11020) states, are still in dispute. In addition, whether the Υ(10750) signal which was recently observed by the Belle Collaboration is a normal bb̄ state or not should be examined. Faced with such situation, we carried out a systematic investigation of the bottomonium spectrum in the scheme of the relativistic flux tube (RFT) model. A Chew-Frautschi like formula was derived analytically for the spin average mass of bottomonium states. We further incorporated the spin-dependent interactions and obtained a complete bottomonium spectrum. We found that the most established bottomonium states can be explained in the RFT scheme. The Υ(10750), Υ(10860), and Υ(11020) could be predominantly the 3D1, 5 S 1, and 4 D1 states, respectively. Our predicted masses of 1F and 1G bb̄ states are in agreement with the results given by the method of lattice QCD, which can be tested by experiments in future.


I. INTRODUCTION
The toponium system (tt) can hardly exist in the nature due the very short lifetime of top quark (≈ 0.5×10 −24 s) [1]. Then the bottomonium is the heaviest meson system which have been researched by experiments for many years. This fact makes the bottomonium family occupy an important position in the hadron zoo and play a special role in the study of the strong interactions. A prominent feature of the bottomonium spectrum is that many excited states are below the threshold BB, which provides a good platform to test the different kinds of effective theories and phenomenological models.
Comparing with the theoretical expectations, however, the complete bottomonium spectrum is far from being established. The first three bottomonium states, namely Υ(1S ), Υ(2S ), and Υ(3S ), were observed by the E288 Collaboration at Fermilab in 1977 [2,3]. Since then nearly twenty bottomonium states have been established [4]. The experimental history of the bb states has been reviewed in Ref. [5]. Here, we just briefly review some important measurements of bottomonium in the past fifteen years.
Obviously, it is not an easy task to establish the bottomonium spectrum completely because even many bb states below the BB threshold have not been discovered. However, the situation may be changed especially because of the running of Belle II [26]. It is expected that more excited bottomonium states will be detected in the near future. So it is time to investigate the spectrum of bb by different approaches which incorporate the spirits of QCD.
In this work, we will explore bottomonium spectrum in the scheme of the RFT model which can be rigorously derived 2 Belle also measured R ≡ σ(h b (np)π + π − ) σ(Υ(2S )π + π − ) (n = 1, 2) and the result indicated that the Υ(5S ) → h b (np)π + π − and Υ(5S ) → Υ(2S )π + π − processes have the similar production ratios [19]. This interesting result not only implied the complicated structure of high excited Υ states [20], but also provided a new route to search the unknown bb states. from the Wilson area law in QCD [53]. The investigation of bb spectrum here by the RFT model could be regraded as an extension of our previous work [54]. There we have shown that the RFT model can describe the masses of single heavy baryons well. Especially, the predicted massed of 1D Λ + c and Λ 0 b states in Ref. [54] are in good agreement with the later measurements by the LHCb Collaboration [55,56].
The manuscript is organized as follows. The RFT model is introduced in Sec. II where a spin average mass formula of the heavy quarkonia is derived. In Sec. III, we test the mass formula by the well measured bb states. In Sec. IV, the spin-dependent interactions are incorporated and the complete bottomonium spectrum is presented. Finally, the paper ends with the discussion and conclusion. The idea of RFT model stemmed from the Nambu-Goto QCD string model [57][58][59]. Different aspects of the RFT model have been investigated by Olsson and the collaborators [60][61][62][63][64][65]. The deep relationship between the RFT model and QCD has been verified in Refs. [53,66]. The basic assumption of the RFT model is that the gluon field connecting the largely separated quarks in the QCD dynamical ground state could be regarded as a rigid straight tube-like color flux configuration. Thus the angular momentum of gluon field is taken into account by the RFT model, which is qualitatively different from the usual quark potential models. The RFT model has been applied to study the masses of heavy-light mesons [67][68][69], charmonium states [70], single heavy baryons [54,71], glueballs [72], and other exotic hadrons [73].
As shown in Fig. 2, the Lagrangian of a q 1q2 meson in the RFT model is written as [74] where m i and r i denote the mass of i (i = 1, 2) quark and its distance from the center of gravity (see Fig. 2). τ represents the string (flux tube) tension. Here, we only consider the transverse velocity of the quark and antiquark, i.e.,ṙ i = 0.
Then the total orbital angular momentum L is defined by The Hamiltonian of q 1q2 meson is given by When we set u i = r iθ = r i ω, the energy and orbital angular momentum can be written as and We have set c = 1 in natural unit for simplicity. Eqs. (5) and (6) have also been obtained by the Wilson area law [53]. With Eqs. (5) and (6), a mass formula for the heavy-light hadrons has been derived analytically in our previous work [54]. For the bottomonium system, the masses of b andb quarks are denoted as m. Then Eqs. (5) and (6) become as and Combing with the following relationship in the RFT model we have and Since the Eqs. (7) and (8) can be derived from the QCD [53], the m in the above equations could be regarded as the "current quark masses" of bottom quark. In practice, the constituent quark mass is more suitable for the phenomenological analysis. To this end, we assume From Eqs. (7)−(11), we have In above equations, we set the following functions and Since m b has included the relativistic effect, we may treat it as the constituent quark mass of b quark. In this way, the value of m b can be fixed by the experimental data, directly. The treatment of m b which includes the relativistic effect is different from the work [70] where the RFT model has been applied to investigate the assignment of X(3872). As shown later, the velocity of bottom quark in the bb meson is no more than 0.50 c. The Eqs. (13) can be expanded as If we ignore the higher order of u, the following relationship can be obtained 3 We replace the string tension τ by the parameter σ with the relationship σ ≡ 2πτ. As done in Ref. [54], we further extend Eq. (17) to include the radial excited bb states, The coefficient λ will also be determined by the experimental data. Eq. (18) is a Chew-Frautschi like formula of the mass of bb states. When the distance between the b andb quarks in a bb meson is denoted as r, we have the relationship: r = 2u/ω. Combing with Eq. (9), we get In the region of u ∈ 0.3c ∼ 0.6c, we find (13) and (18), we obtain the expression of r as In next Section, we shall test the Eq. (18) by the measured masses of bb states. In Section IV, we will incorporate the spin-dependent interactions and present a complete bottomonium spectrum.

III. TESTING EQ. (18) BY THE MEASURED MASSES OF BOTTOMONIUM STATES
Three parameters in Eq. (18), namely the mass of bottom quark m b , the string tension σ, and the dimensionless λ, should be fixed by the experimental data. We will used the spin average masses of the 1S , 2S , and 1P bb states to fix the m b , σ, and λ. The spin average mass of 1S bb is and the averaged mass of 2S bb is Here, the masses of 1S and 2S bb states are taken from the latest "Review of Particle Physics" (RPP) [4] by the Particle Data Group (PDG). Since the average mass of 1 3 P 0 , 1 3 P 1 , and 1 3 P 2 bb states is quite close to the 1 1 P 1 state (see Ref.  The mass of h b (2P) is predicted to be 10262 MeV which is consistent with the experimental result. The η b (3S ) state has not been discovered by experiment. Nevertheless, the spin average mass of the 2S bottomonium states is about 6 MeV below the Υ(2S ) state (see Eq. (22)). So one could reasonably expect the average mass of 3S states to be about 10350 MeV which is also close to our prediction. As argued in Ref. [15], two D-wave bb states, namely the Υ(10152) and Υ 3 (10173), may have been detected in the experimental data by the CLEO [8] and BABAR [9] Collaborations. Although the measured masses of these two states need more confirmations, the average mass of the Υ(10152), Υ 2 (10164), and is quite consistent with our result (see Table II).
As shown above, the predicted average masses of Υ(3S ), h b (2P) and Υ 2 (1D) multiplets are well comparable with the experimental results. For completeness, we will incorporate the spin-dependent interactions and give a whole bottomonium spectrum in the next section.

IV. THE COMPLETE BOTTOMONIUM SPECTRUM BY INCORPORATING THE SPIN-DEPENDENT INTERACTIONS
For simplicity, we consider the color hyperfine interaction which arises from the one gluon exchange (OGE) forces, and the following spin-orbit term which includes the OGE spin-orbit and the longer-ranged inverted spin-orbit terms. This type of spin-dependent interactions has been used to study the mass spectrum of cc spectrum [76]. TheŜ bb denotes the tensor operator. The "δ 3 (r)" function which comes from a contact hyperfine interaction can be simulated by the different smearing functions [33,77]. In our calculations, we take the following smearing function to reproduce the mass splitting of nS (n ≥ 2). 4 Here, we take the r 0 as 0.94 GeV −1 . Due to the heavy masses, the distance between b andb quarks in the low-lying bottomonium states is much small. Therefore, one should treat the running coupling constant α s in Eqs. (25) and (26) seriously. We use the following to simulate the running coupling constant, where the Erf[· · · ] refers to the error function. In our calculations, the running coupling constant is assumed to saturate at 0.68, i.e., α 0 = 0.68. To reduce the free parameters, we take the value of b in Eq. (26) as the string tension τ in the RFT model, i.e., b = σ/2π = 0.471 GeV 2 . With Eqs. (20), (25), (27), and (28), the splitting masses of n 3 S 1 and n 1 S 0 states (n ≥ 2) are presented in Table III. Obviously, our results are comparable with these from Refs. [27,34]. As shown later, the masses of observed excited bb states will be well reproduced though our method is quite phenomenological.

A. nS (n ≥ 2) states
With the predicted splitting masses in Table III, the masses of n 1 S 0 and n 3 S 1 bottomonium states (n ≥ 2) are predicted in Table IV where the experimental data [4] and the results from other works [29,34,41] are also listed for comparison.
The masses of the Υ(4S ), Υ(5S ), and Υ(6S ) obtained by the RFT model are quite close to the results given by the Godfrey-Isgur model [34]. Our results favor the Υ(10860) as a predominantly 5 3 S 1 state. Interestingly, a recent work based on the lattice QCD also suggested the Υ(10860) as a 5 3 S 1 state [78]. The mass of Υ(4S ) predicted by the RFT model is about 60 MeV higher than the measured mass of Υ(10580) (see Table IV). The mass of Υ(4S ) state predicted in Refs. [5, 27-29, 34, 41] was also larger than the Υ(10580) state. In the quark potential models, the mass gap between the 3 3 S 1 and 4 3 S 1 bb states is expected to be larger than the gap between the 4 3 S 1 and 5 3 S 1 states. However, the experimental measurement is contrary to the expectation, i.e., ∆M(Υ(10580) − Υ(10355)) ≈ 224.2 MeV, (29) which is smaller than ∆M(Υ(10860) − Υ(10580)) ≈ 310.5 MeV. (30) It indicates that the mass of Υ(4S ) shifts down about 40∼50 MeV due to a particular mechanism. This anomalously mass gaps of "Υ(4S ) − Υ(3S )" and "Υ(5S ) − Υ(4S )" can not be simply solved by the näive quark model. Törnqvist proposed a solution to this puzzle. Specifically, it may be disentangled by considering the coupled-channel effects [40]. More importantly, the masses of Υ(5S ) and Υ(6S ) were well predicted in the scheme of coupled-channel model [40] before the observations of candidates Υ(10860) and Υ(11020) [6,7]. The scheme suggested by Törnqvist was supported by the recent work [42]. As a pure 6 3 S 1 bb state, the measured mass of Υ(11020) is about 100∼200 MeV lower than the predictions by the RFT model and other methods [29,34,41,79]. These results indicate that the Υ(11020) is not a pure 6S upsilon resonance. This conclusion is partially supported by the analysis of its dielectron widths [79] (see subsection IV C).

B. nP states
The masses of nP (n = 1 ∼ 5) states which are predicted by the RFT model are listed in Table V with the experimental data [4] and other theoretical results from Refs. [29,34,36]. Up to now, the 1P and 2P bottomonium states are well established [4]. Obviously, the masses of these states are well reproduced by the RFT model.
The candidates of 3P bottomonium states have been detected by the ATLAS [21], D0 [22], and LHCb [23,24] collaborations (see Table I). The masses of the χ b1 (3P) and χ b2 (3P) collected by the PDG are listed in Table V. The experimental results are about 20∼40 MeV smaller than the theoretical results. One notices that the predicted masses 3P bb states are about 30∼100 MeV above the thresholds of BB, BB * + B * B , and B * B * decay channels. So the coupled-channel channel effect may affect the properties of 3P bottomonium states including their masses. 5 More theoretical and experimental efforts are desirable for the 3P bb states in future.
The 4P and 5P bottomonium states are predicted around 10800 MeV and 11050 MeV, respectively, which means these states locate above the open-bottom thresholds. Then the Okubo-Zweig-Iizuka (OZI) allowed decays are probable for these states. In Ref. [34], the investigation of strong decays by the 3 P 0 model indicated that the χ b0 (4P) state mainly decays through the BB and B * B * channels while the BB * +B * B is the largest decay channel for the χ b1 (4P), χ b2 (4P), and h b (4P) states. Different from the 4P bottomonium states, the largest decay channel of 5P states is the B * B * . The total decay widths of 4P and 5P bottomonium states were predicted to be 30∼70 MeV. The decays predicted in Ref. [35] were slight different from these results in Ref. [34]. Of course, discovery of these high P-wave bottomonium states is a great challenge for present experiments.

C. nD states
So far only one D-wave bb state, namely Υ 2 (1D), was listed in the summary table of PDG [4]. Its measured mass, i.e., 10163.7±1.7 MeV, is quite in agreement with our prediction (see Table VI). The visible evidence of the 1 3 D 1 and 1 3 D 3 bottomonium states at 10152 MeV and 10173 MeV [8,9], respectively, was pointed out in Ref. [15]. Our predictions in Table VI are comparable with these preliminary results. Our results are also consistent with the predicted masses of 1D bb states by Lattice QCD [49].
None of the 2D bb states have been announced by any experiments. Nevertheless, Beveren and Rupp found the Υ(2D) signal with 10.7 standard deviations [81] by reanalyzing the BABAR data [82]. There the mass of Υ(2D) was fitted to be 10495 ± 5 MeV, which is a bit larger than the predictions in Table VI.
As mentioned before, a 1 −− structure Υ(10750) which was discovered by the Belle collaboration [25] is still unclear. Since the 3 3 D 1 bb state is expected to has the masse around 10740 MeV, the Υ(10750) could be a good 3D candidate. Due to the significant mixing between the (n + 1) 3 S 1 and n 3 D 1 states (n ≥ 3), the magnitude of dielectron widths of the mixed Υ(n 3 D 1 ) resonances (n = 3, 4, 5) can increase by 2 orders [79]. For theΥ(3D) state, the dielectron width was obtained to be 0.095 +0.028 −0.025 keV. The result shows that the predominantly 3 3 D 1 bb state can be produced in the e + e − annihilation process with the high statistics data. Furthermore, the decay width of the 3 3 D 1 bb state was obtained as 54.1 MeV [35] which is comparable with the measurement by the Belle Collaboration [25] (see Eq. (1)). So the Υ(10750) could be predominantly a 3 3 D 1 bb state in our scheme. However, the other explanations suggested in Refs. [83,84]

States
Expt. [4] Our Ref. [34] Ref. [36] Ref. [29] The masses of the 1F bb states are predicted in the region around 10400 MeV, which is comparable with the results given by the lattice nonrelativistic QCD [50]. The 1G bb masses are predicted around 10590 MeV which are slightly above the BB threshold at 10.56 GeV. Our predicted masses of 1G bb states seem to be larger than the results obtained by the quark potential models [5,[34][35][36], but very close to the results from the lattice QCD [50], where the masses of 4 −+ and 4 −− bb states were predicted as

V. DISCUSSION AND CONCLUSION
We have carried out a systematical study of the bottomonium spectrum for the first time by the relativistic flux tube (RFT) model. We derived a Chew-Frautschi like formula which can give an intuitive description of the spin average mass of the heavy quarkonium systems. With the measured masses of 1S , 2S , and 1P bb states, we fixed the three parameters in the Chew-Frautschi like formula, namely the mass of b quark, the string tension σ, and the dimensionless parameter λ. Then we tested the mass formula by comparing the predicted the spin average masses of 3S , 2P, and 1D states with the experimental results. The comparison implied that the Chew-Frautschi like formula could describe the spin average masses of high excited bb states well.
Inspired by a good description of the spin average mass, we further incorporate the spin-dependent interactions which include the one gluon exchange (OGE) forces and the longerranged inverted spin-orbit term. As shown in the Tables IV and V, the measured masses of the nS (2 ≤ n ≤ 6) and nP (n = 1 and 2) states were well reproduced. The predicted masses of nD and other high bottomonium states in Tables VI and VII could be tested in future.
According to our results, the main conclusions are listed as follows: (1) The Υ(10860) could be explained as a predominant 5S state since its measured mass is very close to the predictions (see Table IV). The Υ(10580) and Υ(11020) can not be regarded as the pure 4S and 6S states, respectively, since the predicted masses are much larger than the measurements.
(2) The newly discovered Υ(10750) could be regarded as a good candidate of the predominant 3 3 D 1 state since the measured mass is in good agreement with our prediction.
(3) The measured masses of 3P bb states seems to be about 20∼30 MeV smaller than the theoretical results.
(4) Our predicted mass of the 1 3 D 2 bb state is consistent with the experimental value. The predicted masses of 1 3 D 1 and 1 3 D 3 states are also comparable with the signals detected by the CLEO [8] and BABAR [9] Collaborations.
In summary, the bottomonium spectrum has been systematically studied by the RFT model, which could be regarded as an important supplement to the available investigations of the bottomonium spectrum. Since the relativistic color flux tube carries both energy and momentum, the RFT model present a different dynamics picture for the heavy quarkonia system. The larger predicted masses of the high orbital excited states by the RFT model can be tested by the experiments in future.