The excited states of $\phi$ meson

In this paper, the excited states of $\phi$ meson, especially containing the newly observed $X(2000)$ with $I(J^P)=0(1^-)$ by the BESIII Collaboration, is studied. In addition, $Y(2175)$ as a $\phi$ {\color{black}{meson excited state}} is investigated. The mass spectrum and strong decay behaviors of $\phi$ {\color{black}{meson excited states}} are analyzed, which indicates that $X(2000)$ and $Y(2175)$ are the candidates of $\phi(3S)$ and $\phi(2D)$ states with $I(J^P)=0(1^-)$, respectively. In addition, {\color{black}{$\phi(1D)$ and $\phi(4S)$ are}} predicted to have the mass of 1.87 GeV and 2.5 GeV and width of 440 MeV and 940 MeV, respectively.

In this paper, the excited states of φ meson, especially containing the newly observed X (2000) with I(J P ) = 0(1 − ) by the BESIII Collaboration, is studied. In addition, Y(2175) as a φ meson excited state is investigated. The mass spectrum and strong decay behaviors of φ meson excited states are analyzed, which indicates that X (2000) and Y(2175) are the candidates of φ(3S ) and φ(2D) states with I(J P ) = 0(1 − ), respectively. In addition, φ(1D) and φ(4S ) are predicted to have the mass of 1.87 GeV and 2.5 GeV and width of 440 MeV and 940 MeV, respectively.
Naturally, one can note that the resonance parameters with the first assumption are close to the observed Y(2175) [21][22][23]. Y(2175) has been studied in various theoretical explanations [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. X(2000) has also been studied by recent work [24][25][26]. Ref. [16] studied Y(2175) as a φ(2 3 D 1 ) state. Ref. [24] treated X(2000) as a h 1 (3 1 P 1 ) state with ss component under J P = 1 + assignment, Cui et al. [25] argued that the X(2000) is the partner of the tetraquark state Y(2175) with J P = 1 + , and Ref. [26] assigned X(2000) to be a new ssss tetraquark state with the same J P . As another possibility, i.e., X(2000) has the resonance parameters M=2002.1 ± 27.5 ± 15.0 MeV and Γ = 129 ± 17 ± 7 MeV with J P = 1 − , has not been theoretically studied. In addition, a hybrid with the same quantum numbers and a similar mass and width are predicted by the flux-tube model [27][28][29][30]. Identifying whether X(2000) is ss or ss hybrid is a difficult, * Electronic address: xuehua45@163.com interesting, and urgent research issue. In J PC = 1 −− assignment, X(2000) is the candidate of an excited state of φ meson in the conventional ss meson framework. In fact, Refs. [31,32] predicted a φ(3S ) state with the mass of 2050 MeV and 1900-1960 MeV, Refs. [31] also predicted that the width of φ(3S ) will be 380 MeV. If this X state is considered as the conventional mesons under the J P = 1 − assignment, what is the relation between X(2000) and Y(2175)? Is X(2000) a φ(3S ) state? These questions should be clarified. In addition, the angular excited state of φ(1S ), the mass and the width of φ(1D) are unclear. A systemic study of excited states of φ meson represents an intriguing and important research topic. This paper is aimed to give a systemic study of excited states of φ meson. By using modified Godfrey-Isgur (GI) model and quark pair creation model, the mass spectrum and strong decay behavior of excited states of φ meson are analyzed, which indicates that X(2000) is a candidate of the φ(3S ) meson with I(J P ) = 0(1 − ) and Y(2175) is a candidate of the φ(2D) state. At the same time, the mass and the width of φ(1D), φ(3D), and φ(4S ) are predicted.
In this work, the spectra of the φ meson family are studied using the modified Godfrey-Isgur (MGI) model [33][34][35][36], which contains the screening effect. At higher excited states of φ meson, the screening effect should be considered for the larger average distance between the quark pair. The former studies [33][34][35][37][38][39][40][41] show that the GI model works well for describing hadron spectroscopy. Then, for further studying the properties of φ mesons, their Okubo-Zweig-Iizuka (OZI)allowed two-body strong decays are studied, taking input with the spatial wave functions obtained from the mass spectrum by numerical calculation. Their partial and total decay widths are calculated by using a quark pair creation (QPC) model that was proposed by Micu [42] and extensively applied to studies of strong decay of other hadrons [13,16,35,36,39,. This paper also gives a comparison of X(2000) ′ s two-body decay information between that of φ(3S ) and ss hybrid [29]. The effort will be helpful to uncover the structure of X (2000) and Y(2175) and establish φ meson families. This paper is organized as follows. In Section II, the models employed in this work are briefly reviewed. The mass spectrum and decay behavior phenomenological analysis of φ mesons will be performed in Section III. The paper ends with a conclusion in Section IV.

II. MODELS EMPLOYED IN THE WORK
In this work, the modified GI quark model and quark pair creation (QPC) model are utilized to calculate the mass spectrum and the two-body strong decays of the meson family, respectively. In the following, these models will be illustrated briefly.

A. The modified GI model
In 1985, Godfrey and Isgur proposed the GI model for describing relativistic meson spectra with great success, specifically for low-lying mesons [37]. Regarding the excited states, the screening potential should be taken into account for its coupled-channel effect [33][34][35][36].
The internal interaction of mesons is depicted by the Hamiltonian of the potential model and can be written as where m 1 and m 2 denote the mass of quark and antiquark, respectively, the relation betweenṼ eff (p, r) and V eff (p, r) will be illustrated later, and the effective potential has a familiar format in the nonrelativistic limit [37,68] where S 1 /S 2 indicates the spin of quark/antiquark and L is the orbital momentum. F are related to the Gell-Mann matrices in color space. For a meson, F 1 · F 2 = −4/3, the running coupling constant α s (r) has following form: where k is from 1 to 3 and the corresponding α k and γ k are constant, α 1,2,3 = 0.25, 0.15, 0.2 and γ 1,2,3 = 1 2 , √ 10 2 , √ 1000 2 [37]. H conf consists of two pieces, the spin-independent linear confinement piece S (r) and Coulomb-like potential G(r). H hyp is the color-hyperfine interaction and also includes two parts, tensor and contact terms; H SO denotes the spin-orbit interaction with the color magnetic term due to one-gluon exchange and the Thomas precession term, which can be written as In the light meson system, relativistic effects in effective potential must be considered; the GI model introduces these relativistic effects in two ways.
First, the GI model introducs a smearing function for a qq meson, which includes nonlocal interactions and new r dependence.
then, S (r) and G(r) become smeared potentialsS (r) andG(r), respectively, by the following procedure: where the values of σ 0 and s are defined in Table I [36].
Second, to make up for the loss of relativistic effects in the nonrelativistic limit, a general potential relying on the the center-of-mass of interacting quarks and momentum are applied as whereṼ i (r) delegates the contact, tensor, vector spin-orbit and scalar spin-orbit terms, and ǫ i represents the relevant modification parameters as shown in Table I. After the above revision in two points,Ṽ eff (p, r) is replaced by V eff (p, r).
Diagonalizing and solving the Hamiltonian in Eq.(2.1) by exploiting a simple harmonic oscillator (SHO) basis, the mass spectrum and wave functions will be obtained. In configuration and momentum space, SHO wave functions have explicit forms: with where Y LM L (Ω) is spherical harmonic function, L L+1/2 n−1 (x) is the associated Laguerre polynomial, and β = 0.4 GeV for the calculation.
After diagonalization of the Hamiltonian matrix, the mass and wave function of the meson that is available to undergo the strong decay process can be obtained.
For the process A → B + C, 1m; 1 − m|00 dp 3 dp 4 δ 3 (p 3 + p 4 ) where the quark and antiquark are denoted by indices 3 and 4, respectively, and γ depicts the strength of the creation of qq from vacuum. In this work, γ = 6.57, which is obtained by fitting the decay width of φ(1680)(2S ) state as shown in Table  II and is independent of the decay channels′ branch ratios.
Y ℓm (p) = |p| ℓ Y ℓm (p) are the solid harmonics. χ, φ, and ω denote the spin, flavor, and color wave functions, respectively, which can be separately treated. Subindices i and j denote the color of a qq pair. The decay width reads where m A is the mass of an initial state A and the two decay amplitudes can related to the Jacob-Wick formula as [78] M JL (P) = √ 4π(2L + 1) In the calculation, the spatial wave functions of the discussed mesons can be numerically obtained by the MGI model.

A. Mass spectrum analysis
Applying the MGI model and the parameters in Table I, the mass spectrum of the φ family can be obtained, as shown in Table III. In addition, the mass spectrum of mesons with J P = 1 − was calculated by the GI model, Ref. [79] also gave a spectrum for the φ meson. The mass spectrum of these φ states can be obtained by the MGI model which is listed in Table III. The numerical results are compared with the GI model [37], Ref. [79] and experiments in Table III.

The spectrum of φ meson excitations
The spectrum of φ meson excitations is calculated, and the values are listed in Table III. The third radial excited state of φ(1S ) has a mass of 2.5 GeV, which is smaller than the result of GI model and close to that reported in Ref. [79]. For the ground state of a D-wave φ meson (φ(1D)), its first and second radial excited states (φ(2D) and φ(3D)) have the mass of 1.869 GeV, 2.276 GeV and 2.6 GeV, respectively, which are also smaller than those reported in Ref. [37].

Y(2175) and X(2000)
According to Table III, one can note that Y(2175) tends to be the candidate of φ(3S ) rather than φ(2D) state; Ref. [79] and the MGI model mass spectrum show that Y(2175) could be the φ(3S ) or φ(2D) state because the mass of Y(2175) is between their masses. The position of Y(2175) in the φ family needs further discussion based on the decay behavior, which will be given in the next section.
As shown in Table III, the mass spectrum of Ref. [79], the GI model and MGI model all indicate that the newly observed state X(2000) [23] may be φ(1D) or φ(3S ) state. In fact, Ref. [31] estimated the mass of φ(3S ) to be 2050 MeV, which is smaller than the mass obtained with the GI model, Ref. [79] and MGI model. Further discussion based on the decay behaviors on the assignment of X(2000) will be given below.
The above discussions are only from the point of view of the mass spectra. In the next section, their strong decays will be studied.

B. Decay behavior analysis
Applying the QPC model, one can obtain the OZI-allowed two-body strong decay of vector light family, which is shown in Tables IV and V.

The radial excited states of S-wave φ meson
In this section, the radial excited states of S-wave φ meson will be discussed. φ(1680) has been established as a φ(2S ) state [31]. As presented in Table II, the branch ratio Γ KK /Γ KK * is approximately 0.13, which is closer to the experimental value 0.07±0.01 [80] than the theoretical result of Ref. [31]. The ratio Γ ηφ /Γ KK * is predicted to be 0.14, which is close to the value (0.18) of Ref. [31].
The decay widths of φ(3S ) state with the mass of 2188(Y(2175)) and 2002(X(2000)) are compared in Table IV. Ref. [31] also estimated the mass and the width of φ(3S ) to be 2050 MeV and 380 MeV, respectively. If Y(2175) is the second excited state of φ(1S ), its total width is 217 MeV, which does not agree with the experimental value [22]. According to the mass spectrum analysis section and Ref. [31], the mass and the width of Y(2175) will be larger than the theoretical result when it is treated as φ(3S ).
According to Table IV, when X(2000) is treated as the φ(3S ) state, the width (133 MeV) is in very good agreement with the experimental value [23] and smaller than the theoretical result of Ref. [31]. Unfortunately, the width of the corresponding ss hybrid is in the range of 100 − 150 MeV in flux tube model [29,30], which makes it difficult to determine the internal structure of this X state. Table IV gives a comparison of the two-body decay information between φ(3S ) and ss hybrid [29]. Under the φ(3S ) assignment, X(2000) → KK * (1410) → KKππ will be the main decay mode with the branch ratio Γ KK * (1410) Γ T otal ≈ 0.34, which is smaller than that of the ss hybrid assignment. KK * , KK ′ 1 and KK 1 are predicted to be its important decay channels, which have the ratios of 0.2, 0.16 and 0.14, respectively. When treated as a ss hybrid [29], X(2000) dominantly decays to KK ′ 1 , with the branch ratio 1 and KK 1 can decay to KKππ, which indicates that KKππ will be the dominant final states of X(2000) as the candidate of second excitation of φ. We suggest that experimentalists focus on this final channel. KK, KK * 2 (1430), K * K * , and ηφ are the sizable decay modes as well. To summarize, the branch ratios of KK * (1410), KK ′ 1 and KK differ greatly when X(2000) is treated as φ(4S ) and ss hybrid. These predictions of the branch ratios can help reveal the internal structure of this X state.

D-wave φ mesons
The decay information of the D-wave φ mesons is listed in Table V.
As shown in the second column of Table V, the strong decay of φ(1D) is predicted, which is still unobserved. φ(1D) has the total width of 442 MeV. KK 1 is its dominant decay channel, which is consistent with Ref. [16]. KK * and KK are the important final states. ηφ and K * K * have the same ratio of 3%. If X(2000) is treated as the φ(1D) state, its total width will be larger than 440 MeV, which does not agree with the experimental value [22]. Thus, it can be basically ruled out that X(2000) is the candidate of φ(1D). When treated as the φ(2D) state, Y(2175) has a total width of 186 MeV, which is consistent with that in Ref. [21]. Under this assignment, Y(2175)→ KK 1 will be the dominant decay mode. In the calculation, K * K * and KK are the important decay channels. However, K * K * and KK modes are not observed in recent experiments [23,81]. If Y(2175) is the φ(2D) state, this puzzle should be explained in theory or experiment.
The decay information of the φ(3D) state is also predicted in this work. The total width of φ(3D) is approximately 230 MeV, with a mass of 2.6 GeV. The channels KK 1 , K * K * (1410) and K * K * have the branch ratios of 0.27, 0.2 and 0.18, respectively, which are the main decay modes. K * K 1 and KK are its important decay channels. Their branch ratios are approximately 0.12 and 0.08, respectively. This work suggests that experimentalists should search for this missing state in KK or KKππ final states. Otherwise, KK * and KK * 2 (1430) have sizable contributions to the total width of φ(3D).

IV. CONCLUSION
This paper presents an analysis of mass spectra of the excitations of φ meson, in particular the newly observed X(2000) state, using the modified Godfrey-Isgur quark model, and the structure information of these excitations of the φ meson is obtained. After comparing our theoretical results of the two-body strong decays with the experimental data, we can reach the following conclusions under the conventional meson framework.

Mass and strong decay behavior analysis indicates that
the newly observed state X(2000) [23] may be the φ(3S ) state, and KK * (1410) will be the dominant decay mode.
3. φ(4S ) is predicted to have a mass of 2.5 GeV and a width of 942 MeV. The ground state, φ(1D) and second radial excited state φ(3D) have the mass of 1.869 GeV and 2.6 GeV and the widths of 442 MeV and 229 MeV, respectively.
According to the comparison of the two-body strong decays under the φ(3S ) assignment with that of the ss hybrid, it is apparent that the study of the branch ratios of KK * (1410), KK ′ 1 and KK in experiment will be very valuable for identifying the nature of X (2000).
This study is crucial not only to establish the φ meson family and future search for the missing excitations but also to help us reveal the structure information of the newly observed X(2000) state. Thus, more experimental measurements of the resonance parameters should be conducted by the BESIII and other experiments, which can help us to identify the nature of X(2000) and establish the φ meson family in the future.