Assignments of the $Y(2040)$, $\rho(1900)$, and $\rho(2150)$ in the quark model

Recently, the BESIII Collaboration reported a resonance $Y(2040)$ with $M=2034\pm13\pm9$ MeV and $\Gamma=234\pm30\pm25$ MeV in the process of $e^+e^-\rightarrow\omega\pi^0$. In addition, new measurements with much higher precision for the $\rho(1900)$ and $\rho(2150)$ states are obtained by the BESIII and BABAR Collaborations. In this work, we perform a systematic study on the mass spectrum of the excited $\rho$ resonances using the modified Godfrey-Isgur model, and the strong decays of $Y(2040)$, $\rho(1900)$, and $\rho(2150)$ within the $^3P_0$ model.We find that $Y(2040)$, $\rho(1900)$, and $\rho(2150)$ can be interpreted as the $\rho(2^3D_1)$, $\rho(3^3S_1)$, and $\rho(4^3S_1)$ states, respectively. Meanwhile, the mass and strong decays of the $\rho(3^3D_1)$ state are predicted as well, which could be helpful to search for this state in future.


I. INTRODUCTION
The study of low-energy properties of mesons such as mass spectra and decay modes is important to understand the nonperturbative behaviors of QCD. Recently, there are abundant experimental data on the excited ρ mesons, for example, the resonance Y (2040) with M = 2034 ± 13 ± 9 MeV and Γ = 234 ± 30 ± 25 MeV was observed in the process of e + e − → ωπ 0 by the BESIII Collaboration [1]. Y (2040) could be the same state as ρ(2000) which is omitted from the summary table of the latest version of Particle Data Group (PDG) [2], because they have similar masses and decay widths [3][4][5][6]. In addition, many new experimental measurements with much higher precision for ρ(1900) [7][8][9][10] and ρ(2150) [11][12][13] are reported, especially recent measurements show that the decay width of ρ(2150) is around 100 MeV [11][12][13] while a rather large width 200 − 400 MeV was obtained in previous measurements [3,[14][15][16]. Thus, one can investigate the possible quark-model assignments for those states using these accurate experimental data.
In 1985, Godfrey and Isgur proposed the relativized quark model (GI model) [17] which is widely used to investigate the mass spectrum of mesons, however, there appears a discrepancy between theoretical predictions and experimental values as more and more excited states were observed in the past decades. As discussed in Ref. [18], the linear confining potential br will be screened and softened by the vacuum polarization effects at large distance, thus the linear confining potential in a meson will be modified by the screening effects at large distance, which are induced by the quark-antiquark creation. In Refs. [19,20] the screening effects are introduced into the GI model, as known the modified GI (MGI) model, which gives a better description of the mass spectrum * Electronic address: lidm@zzu.edu.cn † Electronic address: wangen@zzu.edu.cn ‡ Electronic address: yanwc@zzu.edu.cn § Electronic address: songqintao@zzu.edu.cn for excited meson states. In this work, we shall adopt the MGI model to estimate the mass spectrum of the excited ρ mesons, based on which we give assignments for ρ(1900), Y (2040), and ρ(2150). Furthermore, in order to check these assignments, we calculate the two-body OZIallowed strong decays for ρ(1900), Y (2040), and ρ(2150) within the 3 P 0 model by employing the meson wave functions obtained from the MGI model.
The organization of this paper is as follows. In Sec. II, a brief review of theoretical works and experiments is presented for ρ(1900), Y (2040), and ρ(2150). We introduce the MGI model and the 3 P 0 model in Secs. III and IV, respectively. The numerical results of mass spectrum and strong decays are shown in Sec. V. In Sec. VI we give a summary of our results.
Based on the measurements with large uncertainties of the Crystal Barrel Collaboration [6], ρ(2000) was assigned as ρ(2 3 D 1 ) in Ref. [27] by studying its two-body OZI-allowed strong decay behaviors. Thus, those accumulated abundant data with better accuracy on the resonance parameters provide an ideal lab to restudy the possible assignments of ρ(2000)/Y (2040).
The ρ(2150) is a good candidate of ρ(4 3 S 1 ) according to the analysis of mass spectrum [14,26,39], and the results from the effective Lagrangian approach also support this assignment [28]. In Ref. [27] the authors studied the two-body strong decay behavior of ρ(2150) by using simple harmonic oscillator (SHO) functions as the wave functions of mesons, and a good description of ρ(2150) can be obtained by considering it as ρ(4 3 S 1 ), however, a large Γ ρ(2150) = 230±50 MeV [3] was adopted in the analysis. In this work we investigate the quark-model assignment for ρ(2150) basing on more accurate measurements of the BESIII [11] and BABAR [12] Collaborations.

III. MASS SPECTRUM
In this section, we will give a brief introduction of the GI model which was proposed by Godfrey and Isgur in 1985 [17]. The GI model plays an important role in studying the mass spectrum of mesons, especially for the low-lying states. For high excited states, it is necessary to introduce the screening effects to the GI model, because the linear confining potential br will be screened and softened by the vacuum polarization effects at large distance, as discussed in Refs. [18,40,41], The MGI model turns out to be able to give a better description of the mass spectra for the higher radial and orbital excitations [19,20,[42][43][44][45][46].

A. The GI model
In GI model [17], the Hamiltonian of a meson includes the kinetic energy term and the effective potential term, where m 1 and m 2 denote the masses of quark and antiquark, respectively,Ṽ eff (p, r) is the effective potential between quark and antiquark. In the nonrelativistic limit, the effective potential can be simplified as with where F 1 · F 2 = −4/3 for a meson. The running coupling constant α s (Q 2 ) depends on the energy scale Q which is related to the relative momentum between quark and antiquark as Q = | p 1 − p 2 |. Since α s (Q 2 ) is divergent at low Q region, the authors of Ref. [17] assume that α s (Q 2 ) saturates as α s (Q 2 = 0) = α critital s , and α critital s is a parameter which is determined by fitting to the mass spectrum. α s (r) is obtained from α s (Q 2 ) by using Fourier transform, where r is the relative distance between quark and antiquark. H conf reflects the spin-independent interaction, and it can be divided into two parts S(r) and G(r). The linear confining potential S(r) = br + c plays an important role at large r, and the Coulomb-type potential G(r) = −4α s (r)/(3r) is dominant at small r. H hyp denotes the color-hyperfine interaction which can cause the mixing of different angular momenta, namely 3 L J and 3 L ′ J . H so is the spinorbit interaction, which contains the color-magnetic term H so(cm) and the Thomas-precession term H so(tp) . The spin-orbit interaction will give rise to the mixing between spin singlet 1 L J and spin triplet 3 L J if m 1 = m 2 , and the specific expression of H so is given in Ref. [17].
The GI model is a relativized quark model, and the relativistic effects are introduced by two main ways. First, in the quark-antiquark scattering, the interactions should depend on both quark momentum p 1 and antiquark momentum p 2 , or a linear combination of them as p 1 − p 2 and p 1 + p 2 , so they must be nonlocal interaction potentials as pointed out in Ref. [17]. In order to take this effect into account, a smearing function ρ 12 (r − r ′ ) is used to transform the basic potentials G(r) and S(r) into the smeared onesG(r) andS(r), and the smearing function is defined as For a heavy-heavy QQ meson system, ρ 12 (r − r ′ ) will turn into delta function δ 3 (r − r ′ ) as one increases the quark mass m Q . In this case, one can obtainf (r) = f (r), which indicates that the relativistic effects can be neglected for a heavy-heavy QQ meson. However, the relativistic effects are important for heavy-light mesons and light mesons, so it is necessary to adopt a relativized quark model such as the GI model to study the excited ρ mesons in this work.
Second, the momentum-dependent factors are introduced to modify the effective potentials, whereG(r) is the Coulomb-type potential, andṼ i (r) represents the contact, tensor, vector spin-orbit, and scalar spin-orbit terms as explained in Ref. [17]. In the nonrelativistic limit, those momentum-dependent factors will become unity.

B. The MGI model with screening effects
The following replacement is often employed to modify the linear confining potential br in the quark model [40,41], If r is small enough, we can have V scr (r) = V (r). Therefore, this replacement will not affect the low-lying meson states. The parameter µ is related to the strength of the screening effects, and one can roughly understand that the screening effects begin to work from r ∼ 1/µ. The value of µ can be determined by fitting to the experimental measurements. Furthermore, the smeared potential [19,20] can be obtained by using Eq. (7), where σ = σ 12 defined in Eq. (8). The MGI model is widely used in Refs. [19,20,[42][43][44][45][46], and it gives a better description of the mass spectrum for both heavy-light mesons and light mesons.
In this work, all the parameters involved in the MGI model are listed in Table II followed Ref. [45], where a systematic study is performed for the mass spectrum of the light mesons. Here, we need to emphasize that there exist differences of parameters between relativistic quark models and nonrelativistic ones. For example, the mass parameters of quarks are roughly consistent as m u = m d ∼ 0.2 GeV and m s ∼ 0.4 GeV in relativistic quark models [17,45,47], while m u = m d ∼ 0.3 GeV and m s ∼ 0.6 GeV are often adopted by the nonrelativistic quark models [48,49]. The mass spectrum and wave functions of the mesons can be obtained by solving the Schrödinger equation with the Hamiltonian in Eq.
(1). Furthermore, the meson wave functions are used as inputs to investigate the subsequent strong decays for mesons.  [45] Parameter value Parameter value mu

IV. TWO-BODY OZI-ALLOWED STRONG DECAYS
In addition to the mass spectrum, the decay widths are crucial to identify the assignments for mesons. Here, we give a brief introduction of the 3 P 0 model which is widely used in studying two-body OZI-allowed strong decays of mesons [50][51][52][53][54][55][56][57][58].
The 3 P 0 model was originally proposed by Micu [59] in 1968, and later it was further developed by Le Yaouanc et al. [60]. Hitherto, 3 P 0 model has been considered as an effective tool to study two-body strong decays of hadrons, namely, A → BC. In the decay process, a flavor-singlet and color-singlet quark-antiquark pair with J P C = 0 ++ is created from the vacuum first, then, the created antiquark (quark) combines with the quark (antiquark) in meson A to form meson B (C) as shown in the left diagram of Fig. 1, besides, another similar decay mode is depicted in the right diagram of Fig. 1. The transition operator T of the decay A → BC in the 3 P 0 model is given by [61] (13) where p 3 (p 4 ) is the momentum of the created quark (antiquark). γ is a dimensionless parameter which stands for the strength of the quark-antiquark q 3q4 pair created from the vacuum, and it is often determined by fitting to the experimental data. χ 34 1,−m , φ 34 0 , and ω 34 0 are spin, flavor, and color wave functions of the created quarkantiquark pair, respectively.
The helicity amplitude M MJ A MJ B MJ C (P ) of the decay process is defined with the help of the transition operator T , (14) where |A , |B , and |C denote the mock meson states defined in Ref. [62], and P is the momentum of meson B in the center of mass frame.
Furthermore, one can express the partial wave amplitude M LS (P ) with helicity amplitude for the decay A → BC [63], Finally, with the relativistic phase space, the total width Γ(A → BC) can be expressed in terms of the partial wave amplitude squared [61], where , M A , M B , and M C are the masses of the mesons A, B, and C, respectively.

A. Mass spectrum analysis
Here, we adopt both the MGI and GI models to calculate the mass spectrum for excited ρ mesons. The parameters involved in the GI model are taken from Ref. [17]. In Table III, the numerical results are shown together with recent experimental measurements. Here, we need to mention that, in 2008, the BABAR Collaboration announced an enhancement around 1.57 GeV [8] which is named as ρ(1570) shown in Fig. 2, however, this state has not been confirmed by other experiments since 2008. Therefore, we will not discuss it in this work. The mass differences between the MGI model and the GI model become larger for higher excited ρ mesons as shown in Fig. 2. For example, the mass differences are as large as 100 MeV for ρ(3 3 S 1 ), ρ(2 3 D 1 ), and ρ(4 3 S 1 ). Comparing the MGI model predictions with the measured masses, one can regard ρ(1900), Y (2040), and ρ(2150) as possible candidates of ρ(3 3 S 1 ), ρ(2 3 D 1 ), and ρ(4 3 S 1 ), respectively.

B. Decay behavior analysis
It should be emphasized that the mass alone is insufficient to identify those assignments, and the decay behaviors need to be analyzed. In this work, we employ the 3 P 0 model with the realistic meson wave functions obtained from the MGI model to evaluate the decay widths of ρ(1900), Y (2040), and ρ(2150). In this case, only the parameter γ is unknown in the 3 P 0 model. Since ρ(1900), Y (2040), and ρ(2150) are light mesons, we can assume they share the same γ with other excited light mesons with J P C = 1 −− . We obtain γ = 6.57 for uū/dd pair creation by fitting to the total widths of ρ(1700), K * (1680), ω(1650), ρ(1450), K * (1410), φ(1680), and ω(1420) mesons. As for the ss pair creation in the decay process, the γ value is multiplied by a factor m u /m s .
We also predict the decay behavior for ρ(3 3 D 1 ) state with the initial mass M = 2365 MeV obtained in the MGI model, as listed in Table VI. The ρ(3 3 D 1 ) width is predicted to be 204.89 MeV, and the main decay modes are ρρ, πh 1 (1170), ππ 2 (1670), and πa 1 (1640). It should be stressed that the state ρ(2270) with a mass of 2265 ± 40 MeV and a width of 325 ± 80 MeV was reported by the Crystal Barrel Collaboration [3,6] , but is not further confirmed by other experiments. In Ref. [27], the ρ(2270) was regarded as the candidate of ρ(3 3 D 1 ). Our predictions on the mass and width for the ρ(3 3 D 1 ) do not favor the assignment of ρ(2270) as ρ(3 3 D 1 ).

VI. SUMMARY AND CONCLUSION
In this work, we perform a systematic study on the mass spectrum and decay properties for Y (2040), ρ(1900), and ρ(2150). The mass spectrum of the excited ρ mesons is predicted within the MGI model where the screening effects are taken into account. Moreover, the decay behaviors are calculated in the 3 P 0 model with the meson wave functions obtained from the MGI model. We draw the following conclusions by comparing the recent precise experimental measurements with our theoretical results.  3. The Y (2040), ρ(1900), and ρ(2150) can be interpreted as the ρ(2 3 D 1 ), ρ(3 3 S 1 ), and ρ(4 3 S 1 ) states, respectively.
This work is helpful for us to reveal the inner structure of Y (2040), ρ(1900), and ρ(2150), which is crucial to understand the ρ meson family. In addition, we expect that more and more experimental measurements will be released in future, especially the decay width of ρ(1900) needs to be measured precisely to pin down the ρ(3 3 S 1 ) assignment.
Apart from the phenomenological models employed in this work, much progress has been made on mass spectrum and strong decays of the ρ mesons by using lattice QCD in the past decade. For example, masses are estimated in Ref. [64] for a few excited ρ mesons, and the authors of Ref. [65] also compute the decay widths πρ and πω for ρ(1450) and ρ(1700). We expect that more and more results will be released by lattice QCD in the near future, and the predictions of this work can be checked for the excited ρ mesons.