Excited states of $\phi$ meson

TL;DR

This paper investigates the excited states of the $$ meson, analyzing mass spectra and decay behaviors, and proposes assignments for observed states as specific $$ excitations, predicting properties of higher states.

Contribution

It provides a detailed analysis of $$ meson excited states, including recent observations, and offers assignments and predictions for their masses and decay widths.

Findings

$X(2000)$ is a candidate for $(3S)$ state.

$Y(2175)$ is a candidate for $(2D)$ state.

Predicted masses and widths for $(1D)$ and $(4S)$ states.

Abstract

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 550 MeV and 230 MeV, respectively.

Tables

Table 1. Table 1: Parameters and their values in this work, which are determined by fitting the meson experimental data listed in PDG.

Parameter	Value Pang et al. (2019)	Parameter	Value Pang et al. (2019)
$m_u$ (GeV)	0.163	${\sigma }_0$ (GeV)	1.799
$m_d$ (GeV)	0.163	$s$ (GeV)	1.497
$m_s$ (GeV)	0.387	$\mu$ (GeV)	0.0635
$b$ (GeV2)	0.221	$c$ (GeV)	-0.240
${ϵ}_c$	-0.138	${ϵ}_{sov}$	0.157
${ϵ}_{sos}$	0.9726	${ϵ}_t$	0.893

Table 2. Table 2: The decay widths of the ϕ ​ ( 1680 ) ​ ( 2 ​ S ) italic-ϕ 1680 2 𝑆 \phi(1680)(2S) state.

Decay channel	Expe. (MeV)	This work
$Total$	150	150
$\varphi \left(1680\right)\to K{K}^{\ast }$	–	117
$\varphi \left(1680\right)\to \eta \varphi$	–	16.7
$\varphi \left(1680\right)\to KK$	–	15.5
$\gamma =6.57$

Table 3. Table 3: The mass spectrum of ϕ italic-ϕ \phi mesons. ”Expe.” represents experimental value. The unit of the mass is GeV.

Table 4. Table 4: The partial decay widths of the ϕ ​ ( 3 ​ S ) italic-ϕ 3 𝑆 \phi(3S) , s ​ s ¯ 𝑠 ¯ 𝑠 s\bar{s} hybrid and ϕ ​ ( 4 ​ S ) italic-ϕ 4 𝑆 \phi(4S) , the unit of widths is MeV.

Table 5. Table 5: The partial decay widths of the ϕ italic-ϕ \phi (D-wave) mesons, the unit of widths is MeV.

Equations

$$\tilde{H}=\sqrt{m_1^2+{\mathbf{p}}^2}+\sqrt{m_2^2+{\mathbf{p}}^2}+{\tilde{V}}_{\mathrm{eff}}(\mathbf{p},\mathbf{r}),$$

$$V_{\mathrm{eff}}(r)=H^{\mathrm{conf}}+H^{\mathrm{hyp}}+H^{\mathrm{so}},$$

$$H^{\mathrm{conf}}$$

$$=S(r)+G(r),$$

$$H^{\mathrm{hyp}}$$

$$\displaystyle\quad-\bm{S}_{1}\cdot\bm{S}_{2}\Big{)}\Bigg{]}(\bm{F}_{1}\cdot\bm{F}_{2}),$$

$$H^{\mathrm{so}}=$$

$${\alpha }_s(r)=\sum _k\frac{2{\alpha }_k}{\sqrt{\pi }}{\int }_0^{{\gamma }_{k}r}{e}^{-x^2}dx,$$

$$H^{\mathrm{so}(\mathrm{cm})}=\frac{-{\alpha }_s(r)}{r^3}\left(\frac{1}{m_1}+\frac{1}{m_2}\right)\left(\frac{{\mathbf{S}}_1}{m_1}+\frac{{\mathbf{S}}_2}{m_2}\right)\cdot \mathbf{L}({\mathbf{F}}_1\cdot {\mathbf{F}}_2),$$

$$\displaystyle H^{\mathrm{so(tp)}}=-\frac{1}{2r}\frac{\partial H^{\mathrm{conf}}}{\partial r}\Bigg{(}\frac{\bm{S}_{1}}{m^{2}_{1}}+\frac{\bm{S}_{2}}{m^{2}_{2}}\Bigg{)}\cdot\bm{L}.$$

$$\rho \left(\mathbf{r}-{\mathbf{r}}^{\prime }\right)=\frac{{\sigma }^3}{{\pi }^{3/2}}{e}^{-{\sigma }^2{\left(\mathbf{r}-{\mathbf{r}}^{\prime }\right)}^2},$$

$$\tilde{f}(r)=\int d^3{r}^{\prime }\rho (\mathbf{r}-{\mathbf{r}}^{\prime })f(r^{\prime }),$$

$$\displaystyle\sigma_{12}^{2}=\sigma_{0}^{2}\Bigg{[}\frac{1}{2}+\frac{1}{2}\left(\frac{4m_{1}m_{2}}{(m_{1}+m_{2})^{2}}\right)^{4}\Bigg{]}+s^{2}\left(\frac{2m_{1}m_{2}}{m_{1}+m_{2}}\right)^{2},$$

$$\tilde{G}(r)\to {\left(1+\frac{p^2}{E_1{E}_2}\right)}^{1/2}\tilde{G}(r){\left(1+\frac{p^2}{E_1{E}_2}\right)}^{1/2},$$

$$\frac{{\tilde{V}}_i(r)}{m_1{m}_2}\to {\left(\frac{m_1{m}_2}{E_1{E}_2}\right)}^{1/2+{ϵ}_i}\frac{{\tilde{V}}_i(r)}{m_1{m}_2}{\left(\frac{m_1{m}_2}{E_1{E}_2}\right)}^{1/2+{ϵ}_i},$$

$${\Psi }_{nL{M}_L}(\mathbf{r})=R_{nL}(r,\beta )Y_{L{M}_L}({\Omega }_r),$$

$${\Psi }_{nL{M}_L}(\mathbf{p})=R_{nL}(p,\beta )Y_{L{M}_L}({\Omega }_p),$$

$$R_{nL}(r,\beta )={\beta }^{3/2}\sqrt{\frac{2n!}{\Gamma (n+L+3/2)}}(\beta r{)}^L{e}^{\frac{-r^2{\beta }^2}{2}}$$

$$\times L_n^{L+1/2}({\beta }^2{r}^2),$$

$$R_{nL}(p,\beta )=\frac{(-1{)}^n(-i{)}^L}{{\beta }^{3/2}}{e}^{-\frac{p^2}{2{\beta }^2}}\sqrt{\frac{2n!}{\Gamma (n+L+3/2)}}{(\frac{p}{\beta })}^L$$

$$\times L_n^{L+1/2}(\frac{p^2}{{\beta }^2}),$$

$$⟨BC|\mathcal{T}|A⟩={\delta }^3({\mathbf{P}}_B+{\mathbf{P}}_C){\mathcal{M}}^{M_{J_A}{M}_{J_B}{M}_{J_C}},$$

$$\mathcal{T}$$

$$\Gamma$$

$${\mathcal{M}}^{JL}(\mathbf{P})$$

Full text

I Introduction

In light meson spectroscopy, there exist two systems: $q\bar{q}$ ($q$ defines the $u,d$ or $s$ quark) mesons and exotic states (glueballs, hybrids, and multiquark states). Exotic states have exotic quantum numbers (such as $\pi_{1}(1400)$) or the same quantum numbers as the conventional meson system. In the latter case, i.e., if exotic states have the same quantum numbers as conventional meson system, it is difficult but intriguing to identify the exotica from light meson spectroscopy Li et al. (2006); Ding and Yan (2005); Wang (2011); Li (2006); Chao (2006); Bicudo et al. (2007); Klempt and Zaitsev (2007); Wang (2007); Ding and Yan (2007a); Martinez Torres et al. (2008); Wang et al. (2017); Huang and Zhu (2006); Yu et al. (2011); Liu et al. (2010a); Li and Ma (2008); Wang et al. (2012); Ding and Yan (2007b). Thus, some states may be the conventional $q\bar{q}$ mesons or exotic states, which are related to $X(1835)$ Ablikim et al. (2005, 2011), $X(1860)$ Bai et al. (2003), $X(1812)$, $Y(2175)$ Shen et al. (2009); Aubert et al. (2006); Ablikim et al. (2019a) and so on.

Very recently, the BESIII Collaboration observed a structure ($X(2000)$) appearing in the $\phi\eta^{\prime}$ invariant mass spectrum of the $J/\psi\to\phi\eta\eta^{\prime}$ process Ablikim et al. (2019a). With assumption that the spin-parity quantum number $J^{P}=1^{-}$, its measured resonance parameters are M=$2002.1\pm 27.5\pm 15.0\,{\text{MeV}}$ and $\Gamma=129\pm 17\pm 7\,{\text{MeV}}$ with significance of 5.3$\sigma$. Assuming the spin-parity quantum number $J^{P}=1^{+}$, the resonance parameters are M=$2062.8\pm 13.1\pm 4.2\,{\text{MeV}}$ $\Gamma=177\pm 36\pm 20\,{\text{MeV}}$ with significance of 4.9$\sigma$.

Naturally, one can note that the resonance parameters with the first assumption are close to the observed $Y(2175)$ Shen et al. (2009); Aubert et al. (2006); Ablikim et al. (2019a). $Y(2175)$ has been studied in various theoretical explanations Li et al. (2006); Ding and Yan (2005); Wang (2011); Li (2006); Chao (2006); Bicudo et al. (2007); Klempt and Zaitsev (2007); Wang (2007); Ding and Yan (2007a); Martinez Torres et al. (2008); Wang et al. (2017); Huang and Zhu (2006); Yu et al. (2011); Liu et al. (2010a); Li and Ma (2008); Wang et al. (2012); Ding and Yan (2007b). $X(2000)$ has also been studied by recent work Wang et al. (2019); Cui et al. (2019); Wang (2019). Reference Wang et al. (2012) studied $Y(2175)$ as a $\phi(2^{3}D_{1})$ state. Reference Wang et al. (2019) treated $X(2000)$ as a $h_{1}(3^{1}P_{1})$ state with $s\bar{s}$ component under $J^{P}=1^{+}$ assignment, Cui ${\it et~{}al}$. Cui et al. (2019) argued that the $X(2000)$ is the partner of the tetraquark state $Y(2175)$ with $J^{P}=1^{+}$, and Ref. Wang (2019) assigned $X(2000)$ to be a new $ss\bar{s}\bar{s}$ tetraquark state with the same $J^{P}$. As another possibility, i.e., $X(2000)$ has the resonance parameters M=$2002.1\pm 27.5\pm 15.0\,{\text{MeV}}$ and $\Gamma=129\pm 17\pm 7\,{\text{MeV}}$ 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 Isgur et al. (1985); Barnes et al. (1995); Page et al. (1999); Close and Page (1995). Identifying whether $X(2000)$ is $s\bar{s}$ or $s\bar{s}$ hybrid is a difficult, interesting, and urgent research issue. In $J^{PC}=1^{--}$ assignment, $X(2000)$ is the candidate of an excited state of $\phi$ meson in the conventional $s\bar{s}$ meson framework. In fact, Refs. Barnes et al. (2003); Afonin and Pusenkov (2014) predicted a $\phi(3S)$ state with the mass of 2050 MeV and 1900-1960 MeV, Refs. Barnes et al. (2003) also predicted that the width of $\phi(3S)$ 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 $\phi(3S)$ state? These questions should be clarified. In addition, the angular excited state of $\phi(1S)$, the mass and the width of $\phi(1D)$ are unclear. A systemic study of excited states of $\phi$ meson represents an intriguing and important research topic.

This paper is aimed to give a systemic study of excited states of $\phi$ meson. By using modified Godfrey-Isgur (GI) model and quark pair creation model, the mass spectrum and strong decay behavior of excited states of $\phi$ meson are analyzed, which indicates that $X(2000)$ is a candidate of the $\phi(3S)$ meson with $I(J^{P})=0(1^{-})$ and $Y(2175)$ is a candidate of the $\phi(2D)$ state. At the same time, the mass and the width of $\phi(1D)$, $\phi(3D)$, and $\phi(4S)$ are predicted.

In this work, the spectra of the $\phi$ meson family are studied using the modified Godfrey-Isgur (MGI) model Song et al. (2015a, b); Pang et al. (2017, 2019), which contains the screening effect. At higher excited states of $\phi$ meson, the screening effect should be considered for the larger average distance between the quark pair. The former studies Godfrey and Isgur (1985); Barnes et al. (2005); Sun et al. (2014); Song et al. (2015a, b); Godfrey and Moats (2015); Godfrey et al. (2016); Pang et al. (2017) show that the GI model works well for describing hadron spectroscopy. Then, for further studying the properties of $\phi$ 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 Micu (1969) and extensively applied to studies of strong decay of other hadrons Le Yaouanc et al. (1973); van Beveren et al. (1980, 1983); Le Yaouanc et al. (1988); Roberts and Silvestre-Brac (1992); Capstick and Roberts (1994); Blundell and Godfrey (1996); Ackleh et al. (1996); Capstick and Keister ; Bonnaz et al. (2002); Close and Swanson (2005); Zhang et al. (2007); Lu et al. (2006); Sun and Liu (2009); Liu et al. (2010b); Sun et al. (2010); Rijken et al. (2010); Yu et al. (2011); Zhou and Xiao (2011); Ye et al. (2012); Wang et al. (2012); Sun et al. (2013); He et al. (2013); Sun et al. (2014); Pang et al. (2014); Wang et al. (2015); Chen et al. (2015); Pang et al. (2017, 2019); Pan et al. (2016). This paper also gives a comparison of $X(2000)^{\prime}$s two-body decay information between that of $\phi(3S)$ and $s\bar{s}$ hybrid Page et al. (1999). The effort will be helpful to uncover the structure of $X(2000)$ and $Y(2175)$ and establish $\phi$ meson families.

This paper is organized as follows. In Sec. II, the models employed in this work are briefly reviewed. The mass spectrum and decay behavior phenomenological analysis of $\phi$ mesons will be performed in Sec.III. The paper ends with a conclusion in Sec. 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.

II.1 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 mesonsGodfrey and Isgur (1985). Regarding the excited states, the screening potential should be taken into account for its coupled-channel effect Song et al. (2015a, b); Pang et al. (2017, 2019).

The internal interaction of mesons is depicted by the Hamiltonian of the potential model and can be written as

[TABLE]

where $m_{1}$ and $m_{2}$ denote the mass of quark and antiquark, respectively, the relation between $\tilde{V}_{\mathrm{eff}}(\mathbf{p,r})$ and $V_{\mathrm{eff}}(\mathbf{p,r})$ will be illustrated later, and the effective potential has a familiar format in the nonrelativistic limit Godfrey and Isgur (1985); Lucha et al. (1991)

[TABLE]

with

[TABLE]

where $\bm{S}_{1}/\bm{S}_{2}$ indicates the spin of quark/antiquark and $\bm{L}$ is the orbital momentum. $\bm{F}$ are related to the Gell-Mann matrices in color space. For a meson, $\langle\bm{F}_{1}\cdot\bm{F}_{2}\rangle=-4/3$, the running coupling constant $\alpha_{s}(r)$ has the following form:

[TABLE]

where $k$ is from 1 to 3 and the corresponding $\alpha_{k}$ and $\gamma_{k}$ are constant, $\alpha_{1,2,3}=0.25,0.15,0.2$ and $\gamma_{1.2.3}=\frac{1}{2},\frac{\sqrt{10}}{2},\frac{\sqrt{1000}}{2}$ Godfrey and Isgur (1985). $H^{\mathrm{conf}}$ consists of two pieces, the spin-independent linear confinement piece $S(r)$ and Coulomb-like potential $G(r)$. $H^{\mathrm{hyp}}$ is the color-hyperfine interaction and also includes two parts, tensor and contact terms; $H^{\mathrm{SO}}$ 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

[TABLE]

[TABLE]

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 introduces a smearing function for a $q\bar{q}$ meson, which includes nonlocal interactions and new $\mathbf{r}$ dependence.

[TABLE]

then, $S(r)$ and $G(r)$ become smeared potentials $\tilde{S}(r)$ and $\tilde{G}(r)$, respectively, by the following procedure:

[TABLE]

with

[TABLE]

where the values of $\sigma_{0}$ and $s$ are defined in Table 1 Pang et al. (2019).

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

[TABLE]

and

[TABLE]

where $\tilde{V}_{i}(r)$ delegates the contact, tensor, vector spin-orbit and scalar spin-orbit terms, and $\epsilon_{i}$ represents the relevant modification parameters as shown in Table 1. After the above revision in two points, $\tilde{V}_{\mathrm{eff}}(\mathbf{p,r})$ is replaced by $V_{\mathrm{eff}}(\mathbf{p,r})$.

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:

[TABLE]

with

[TABLE]

where $Y_{LM_{L}}(\mathrm{\Omega})$ is spherical harmonic function, $L_{n-1}^{L+1/2}(x)$ is the associated Laguerre polynomial, and $\beta=0.4~{}\mathrm{GeV}$ 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.

II.2 QPC model

The QPC model is used to obtain the Okubo-Zweig-Iizuka (OZI) allowed hadronic strong decays. This model was first proposed by Micu Micu (1969) and was further developed by Orsay groupLe Yaouanc et al. (1973, 1974, 1975, 1977a, 1977b). The QPC model was widely applied to the OZI-allowed two-body strong decays of hadrons in Refs. van Beveren et al. (1980, 1983); Capstick and Roberts (1994); Page (1995); Titov et al. (1996); Ackleh et al. (1996); Blundell (1996); Bonnaz et al. (2002); Zhou et al. (2005); Lu et al. (2006); Zhang et al. (2007); Luo et al. (2009); Sun and Liu (2009); Liu et al. (2010b); Sun et al. (2010); Rijken et al. (2010); Ye et al. (2012); Wang et al. (2012); He et al. (2013); Sun et al. (2013); Pang et al. (2014); Wang et al. (2015); Chen et al. (2015); Pang et al. (2017, 2019).

For the process $A\to B+C$,

[TABLE]

where $\mathbf{P}_{B(C)}$ is a three-momentum of a meson $B(C)$ in the rest frame of a meson $A$. $M_{J_{i}}\,(i=A,B,C)$ denotes the magnetic quantum number. The transition operator $\mathcal{T}$ describes a quark-antiquark pair creation from vacuum with $J^{PC}=0^{++}$, i.e., $\mathcal{T}$ can be written as

[TABLE]

where the quark and antiquark are denoted by indices $3$ and $4$, respectively, and $\gamma$ depicts the strength of the creation of $q\bar{q}$ from vacuum. In this work, $\gamma=6.57$, which is obtained by fitting the decay width of $\phi(1680)(2S)$ state as shown in Table 2 and is independent of the decay channels$\prime$ branch ratios. $\mathcal{Y}_{\ell m}(\mathbf{p})={|\mathbf{p}|^{\ell}}Y_{\ell m}(\mathbf{p})$ are the solid harmonics. $\chi$, $\phi$, and $\omega$ denote the spin, flavor, and color wave functions, respectively, which can be separately treated. Subindices $i$ and $j$ denote the color of a $q\bar{q}$ pair. The decay width reads

[TABLE]

where $m_{A}$ is the mass of an initial state $A$ and the two decay amplitudes can related to the Jacob-Wick formula as Jacob and Wick (1959)

[TABLE]

In the calculation, the spatial wave functions of the discussed mesons can be numerically obtained by the MGI model.

III Numerical results and phenomenological analysis

III.1 Mass spectrum analysis

Applying the MGI model and the parameters in Table 1, the mass spectrum of the $\phi$ family can be obtained, as shown in Table 3. In addition, the mass spectrum of mesons with $J^{P}=1^{-}$ was calculated by the GI model, Ref. Ebert et al. (2009) also gave a spectrum for the $\phi$ meson. The mass spectrum of these $\phi$ states can be obtained by the MGI model which is listed in Table 3. The numerical results are compared with the GI model Godfrey and Isgur (1985), Ref. Ebert et al. (2009) and experiments in Table 3.

III.1.1 The spectrum of $\phi$ meson excitations

The spectrum of $\phi$ meson excitations is calculated, and the values are listed in Table 3. The third radial excited state of $\phi(1S)$ has a mass of 2.5 GeV, which is smaller than the result of GI model and close to that reported in Ref. Ebert et al. (2009). For the ground state of a D-wave $\phi$ meson ($\phi(1D)$), its first and second radial excited states ($\phi(2D)$ and $\phi(3D)$) have the mass of 1.869 GeV, 2.276 GeV and 2.6 GeV, respectively, which are also smaller than those reported in Ref. Godfrey and Isgur (1985).

III.1.2 $Y(2175)$ and $X(2000)$

According to Table 3, one can note that $Y(2175)$ tends to be the candidate of $\phi(3S)$ rather than $\phi(2D)$ state; Ref. Ebert et al. (2009) and the MGI model mass spectrum show that $Y(2175)$ could be the $\phi(3S)$ or $\phi(2D)$ state because the mass of $Y(2175)$ is between their masses. The position of $Y(2175)$ in the $\phi$ family needs further discussion based on the decay behavior, which will be given in the next section.

As shown in Table 3, the mass spectrum of Ref. Ebert et al. (2009), the GI model and MGI model all indicate that the newly observed state $X(2000)$ Ablikim et al. (2019a) may be $\phi(1D)$ or $\phi(3S)$ state. In fact, Ref. Barnes et al. (2003) estimated the mass of $\phi(3S)$ to be 2050 MeV, which is smaller than the mass obtained with the GI model, Ref. Ebert et al. (2009) 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.

III.2 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 4 and 5.

III.2.1 The radial excited states of S-wave $\phi$ meson

In this section, the radial excited states of S-wave $\phi$ meson will be discussed.

$\phi(1680)$ has been established as a $\phi(2S)$ stateBarnes et al. (2003). As presented in Table 2, the branch ratio $\Gamma_{KK}/\Gamma_{KK^{*}}$ is approximately 0.13, which is closer to the experimental value $0.07\pm 0.01$ Mane et al. (1982) than the theoretical result of Ref. Barnes et al. (2003). The ratio $\Gamma_{\eta\phi}/\Gamma_{KK^{*}}$ is predicted to be 0.14, which is close to the value (0.18) of Ref. Barnes et al. (2003).

The decay widths of $\phi(3S)$ state with the mass of 2188($Y(2175)$) and 2002($X(2000)$) are compared in Table 4. Reference Barnes et al. (2003) also estimated the mass and the width of $\phi(3S)$ to be 2050 MeV and 380 MeV, respectively. If $Y(2175)$ is the second excited state of $\phi(1S)$, its total width is 225 MeV, which does not agree with the experimental value Aubert et al. (2006). According to the mass spectrum analysis section and Ref. Barnes et al. (2003), the mass and the width of $Y(2175)$ will be larger than the theoretical result when it is treated as $\phi(3S)$.

According to Table 4, when $X(2000)$ is treated as the $\phi(3S)$ state, the width (139 MeV) is in very good agreement with the experimental value Ablikim et al. (2019a) and smaller than the theoretical result of Ref. Barnes et al. (2003). Unfortunately, the width of the corresponding $s\bar{s}$ hybrid is in the range of $100-150$ MeV in flux tube model Page et al. (1999); Close and Page (1995), which makes it difficult to determine the internal structure of this $X$ state. Table 4 gives a comparison of the two-body decay information between $\phi(3S)$ and $s\bar{s}$ hybrid Page et al. (1999). Under the $\phi(3S)$ assignment, $X(2000)$ $\to KK^{*}(1410)\to KK\pi\pi$ will be the main decay mode with the branch ratio $\frac{\Gamma_{KK^{*}(1410)}}{\Gamma_{Total}}\approx 0.34$, which is smaller than that of the $s\bar{s}$ hybrid assignment. $KK^{*}$, $KK_{1}^{\prime}$ 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 $s\bar{s}$ hybrid Page et al. (1999), $X(2000)$ dominantly decays to $KK_{1}^{\prime}$, with the branch ratio $\frac{\Gamma_{KK_{1}^{\prime}}}{\Gamma_{Total}}\approx 0.5$. $KK^{*}$, $KK_{1}^{\prime}$ and $KK_{1}$ can decay to $KK\pi\pi$, which indicates that $KK\pi\pi$ will be the dominant final states of $X(2000)$ as the candidate of second excitation of $\phi$. We suggest that experimentalists focus on this final channel. $KK$, $KK_{2}^{*}(1430)$, $K^{*}K^{*}$, and $\eta\phi$ are the sizable decay modes as well. To summarize, the branch ratios of $KK^{*}(1410)$, $KK_{1}^{\prime}$ and $KK$ differ greatly when $X(2000)$ is treated as $\phi(3S)$ and $s\bar{s}$ hybrid. These predictions of the branch ratios can help reveal the internal structure of this $X$ state.

The total width of $\phi(4S)$ is approximately 230 MeV. According to Table 4, the main decay modes of $\phi(4S)$ are $KK^{*}$, $K^{*}K_{1}^{\prime}$, $KK_{2}^{*}(1430)$, $KK^{*}(1410)$ and $KK_{1}$ which have the branch ratios of 0.15, 0.13, 0.12, 0.12 and 0.10, respectively. $K^{*}K_{0}^{*}(1430)$, $K^{*}K_{1}$ and $K^{*}K^{*}$ are its important decay channels. Considering the the final decay channels of $KK^{*}$, $K^{*}K_{1}^{\prime}$, $KK\pi$ and $KK\pi\pi$ and will be the most important final channels in searching for the $\phi(4S)$ state experimentally. $KK$, $K^{*}K^{*}(1410)$, and $KK_{3}^{*}(1780)$ also have sizable contributions to the total width of $\phi(4S)$. These predictions can help us search for and establish this $\phi(4S)$ state.

III.2.2 D-wave $\phi$ mesons

The decay information of the D-wave $\phi$ mesons is listed in Table 5.

As shown in the second column of Table 5, the strong decay of $\phi(1D)$ is predicted, which is still unobserved. $\phi(1D)$ has the total width of 547 MeV. $KK_{1}$ is its dominant decay channel, which is consistent with Ref. Wang et al. (2012). $KK^{*}$ and $KK$ are the important final states. $\eta\phi$ and $K^{*}K^{*}$ have the same ratio of 3%.

If $X(2000)$ is treated as the $\phi(1D)$ state, its total width will be larger than 550 MeV, which does not agree with the experimental value Aubert et al. (2006). Thus, it can be basically ruled out that $X(2000)$ is the candidate of $\phi(1D)$.

When treated as the $\phi(2D)$ state, $Y(2175)$ has a total width of 205 MeV, which is consistent with that in Ref.Shen et al. (2009). Under this assignment, Y(2175)$$\to 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 experimentsAblikim et al. (2019a, b). If $Y(2175)$ is the $\phi(2D)$ state, this puzzle should be explained in theory or experiment.

The decay information of the $\phi(3D)$ state is also predicted in this work. The total width of $\phi(3D)$ is approximately 245 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 $KK\pi\pi$ final states. Otherwise, $KK^{*}$ and $KK_{2}^{*}(1430)$ have sizable contributions to the total width of $\phi(3D)$.

IV conclusion

This paper presents an analysis of mass spectra of the excitations of $\phi$ 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 $\phi$ 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 $X(2000)$ Ablikim et al. (2019a) may be the $\phi(3S)$ state, and $KK^{*}(1410)$ will be the dominant decay mode. 2. 2.

Mass analysis supports $Y(2175)$ as a candidate of $\phi(3S)$ or $\phi(2D)$. However, strong decay behavior analysis shows that the $Y(2175)$ is preferably a $\phi(2D)$ state. 3. 3.

$\phi(4S)$ is predicted to have a mass of 2.5 GeV and a width of 230 MeV. The ground state, $\phi(1D)$ and second radial excited state $\phi(3D)$ have the mass of 1.869 GeV and 2.6 GeV and the widths of 547 MeV and 245 MeV, respectively.

According to the comparison of the two-body strong decays under the $\phi(3S)$ assignment with that of the $s\bar{s}$ hybrid, it is apparent that the study of the branch ratios of $KK^{*}(1410)$, $KK_{1}^{\prime}$ and $KK$ in experiment will be very valuable for identifying the nature of $X(2000)$.

This study is crucial not only to establish the $\phi$ 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 $\phi$ meson family in the future.

V ACKNOWLEDGMENTS

C.-Q. P. thanks Xiang Liu, Wen-biao Yan for helpful communications and discussions. This work is supported in part by the Nature Science Foundation Projects of Qinghai Office of Science and Technology, No. 2017-ZJ-748, the Chunhui Plan of China’s Ministry of Education, No. Z2017054.