# Can the three new states around 2.2 GeV assign to $\omega(3D)$

Ya-Rong Wang1,2<EMAIL_ADDRESS>Yan Ma1,2<EMAIL_ADDRESS>Cheng-
Qun Pang1,2111Corresponding author<EMAIL_ADDRESS>1College of Physics and
Electronic Information Engineering, Qinghai Normal University, Xining 810000,
China
2Joint Research Center for Physics, Lanzhou University and Qinghai Normal
University, Xining 810000, China

###### Abstract

Recently, the BESIII Collaboration reported three resonances: $X(2232)$ with
$M=2232\pm 19\pm 27$ MeV and $\Gamma=93\pm 53\pm 20$ MeV, $X(2200)$ whose mass
$M=2200\pm 11\pm 17$ MeV and width $\Gamma=74\pm 20\pm 24$ MeV as well as
$X(2222)$ which has mass of $2222\pm 7\pm 2$ MeV and the width of $59\pm 30\pm
6$ MeV. The mass spectrum of $\omega$ meson family is studied utilizing the
modified Godfrey-Isgur model, and the two-body strong decays of $X(2232)$,
$X(2200)$ and $X(2222)$ within two different approaches of the ${}^{3}P_{0}$
model. We find that the newly discovered states $X(2232)$, $X(2200)$ and
$X(2222)$ may be the same and are most likely to be the $\omega(3D)$ state.
The discovery could be useful in establishing entire $\omega$ mesons.

###### pacs:

14.40.Be, 12.38.Lg, 13.25.Jx

##  I Introduction

Light flavor physics is a fascinating and significant research field. In the
past few decades, more and more novel hadronic states have been discovered due
to the improvement of particle detector accuracy. Very recently, the BESIII
Collaboration has announced the discovery of two new states (X(2232) and
X(2200)) in the measurement of the $e^{+}e^{-}\to\omega\pi^{+}\pi^{-}$ cross
section at $\sqrt{s}=2.00\to 3.080$ GeV Ablikim  _et al._ (2022a). $X(2232)$
has the mass of $2232\pm 19\pm 27$ MeV and width of $93\pm 53\pm 20$ MeV,
according to the measurement. $X(2200)$ is discovered with a combined
significance of $7.9\sigma$, and its mass and width were determined to be
$M=2200\pm 11\pm 17$ MeV and $\Gamma=74\pm 20\pm 24$ MeV, respectively. The
BESIII Collaboration also announced the observation of the $X(2222)$ state in
the $e^{+}e^{-}\to\omega\pi^{0}\pi^{0}$ process Ablikim  _et al._ (2022b),
which has a statistical significance larger than $5\sigma$. $X(2220)$ has the
mass of $M=2222\pm 7\pm 2$ MeV and the width of $\Gamma=59\pm 30\pm 6$ MeV,
according to the measurements. Additionally, $X(2222)$ may be an $\omega(3D)$
state, according to our earlier work Wang  _et al._ (2022), Lanzhou group
indicates that the enhancement structures around 2.2 GeV existing in
$e^{+}e^{-}\to\omega\eta$ and $e^{+}e^{-}\to\omega\pi^{0}\pi^{0}$ contain the
$\omega(4S)$ and $\omega(3D)$ signals Zhou  _et al._ (2022). The BESIII
Collaboration announced $X(2222)$ could be an excited $\omega$ resonances Li
_et al._ (2022).

In 2002, $\omega(2205)$ state was firstly observed in the
$\bar{p}p\to\omega\eta$ and $\omega\pi^{0}\pi^{0}$ process with $M=2205\pm 30$
MeV and $\Gamma=350\pm 90$ MeV Anisovich  _et al._ (2002). According to the
authors, this state is most likely the radial excitation of $\omega(1650)$ and
it couples strongly to ${}^{3}D_{1}$ Anisovich  _et al._ (2002). Furthermore,
they also stated that this resonance has quite a large ${}^{3}D_{1}$
amplitude, and is consistent with the ${}^{3}D_{1}$ state expected around 2230
MeV. $X(2232)$, $X(2200)$ and $X(2222)$ states have adjacent masses and their
widths are relatively narrow. This phenomenon is so intriguing that it catches
our attention. According to our previous work, there is a great possibility
that these three states can be grouped into $\omega$ meson family. $\omega$
meson plays a vital role in the light meson families. This work helps us
reveal the inner structure of $\omega(3D)$, which is crucial to understanding
the $\omega$ meson family.

In this work, we shall adopt the modified Godfrey-Isgur (MGI) model to
estimate the mass spectrum of the $\omega$ states and calculate the OZI-
allowed two-body strong decays of $X(2232)$, $X(2200)$ and $X(2222)$ within
two different approaches of the ${}^{3}P_{0}$ model. Our results show that the
enhancement structures $X(2232)$, $X(2200)$ and $X(2222)$ may be the same
states, they are all the $\omega(3D)$ states.

##  II Mass spectrum and Okubo-Zweig-Iizuka allowed two-body strong decay

###  II.1 A brief review of MGI model

In 1985, Godfrey and Isgur proposed the GI model for describing relativistic
meson spectra with great success Godfrey and Isgur (1985), Q.-T. Song
$et~{}al$. Song  _et al._ (2015a, b) introduced the color screening effect on
the basis of the Cornell potential to modify the GI model (MGI model).

The basic equation of the model is the rest frame Schrödinger-type equation
Godfrey and Isgur (1985):

$H|\psi\rangle=E|\psi\rangle,$ (2.1)

the Hamiltonian of the potential model and can be written as

$\tilde{H}=\sqrt{m_{1}^{2}+\mathbf{p}^{2}}+\sqrt{m_{2}^{2}+\mathbf{p}^{2}}+\tilde{V}_{\mathrm{eff}}(\mathbf{p,r}),$
(2.2)

where $m_{1}$ and $m_{2}$ denote the masses of quarks and antiquarks,
respectively, the effective potential has a familiar format in the
nonrelativistic limit Godfrey and Isgur (1985); Lucha  _et al._ (1991)

$\displaystyle
V_{\mathrm{eff}}(r)=H^{\mathrm{hyp}}+H^{\mathrm{conf}}+H^{\mathrm{so}},$ (2.3)

with

$\displaystyle H^{\mathrm{hyp}}$
$\displaystyle=-\frac{\alpha_{s}(r)}{m_{1}m_{2}}\Bigg{[}\frac{8\pi}{3}\bm{S}_{1}\cdot\bm{S}_{2}\delta^{3}(\bm{r})+\frac{1}{r^{3}}\Big{(}\frac{3\bm{S}_{1}\cdot\bm{r}\bm{S}_{2}\cdot\bm{r}}{r^{2}}$
$\displaystyle\quad-\bm{S}_{1}\cdot\bm{S}_{2}\Big{)}\Bigg{]}(\bm{F}_{1}\cdot\bm{F}_{2}),$
(2.4) $\displaystyle H^{\mathrm{conf}}$
$\displaystyle=\Big{[}-\frac{3}{4}(\frac{b(1-e^{-\mu
r})}{\mu}+c)+\frac{\alpha_{s}(r)}{r}\Big{]}(\bm{F}_{1}\cdot\bm{F}_{2}),$ (2.5)
$\displaystyle H^{\mathrm{so}}=$ $\displaystyle
H^{\mathrm{so(cm)}}+H^{\mathrm{so(tp)}},$ (2.6)

in color space, $\bm{F}$ are related to the Gell-Mann matrices.
$\bm{S}_{1}/\bm{S}_{2}$ indicates the spin of quark/antiquark and $\bm{L}$ is
the orbital momentum. All the parameters involved in the MGI model are listed
in Table 1 given by Ref. Wang  _et al._ (2021), where a systematic study in
performed for the spectroscopy behavior of light unflavored vector mesons with
mass at the range of $2.4-3$ GeV. More details of the MGI model can be seen in
Ref. Wang  _et al._ (2022).

Table 1: Parameters and their values in this work.  Parameter | Value Wang  _et al._ (2021) | Parameter | Value Wang  _et al._ (2021)
---|---|---|---
$m_{u}$ (GeV) | 0.22 |  $\sigma_{0}$ (GeV) | 1.8
$m_{d}$ (GeV) | 0.22 |  $s$ (GeV) | 3.88
$m_{s}$ (GeV) | 0.424 |  $\mu$ (GeV) | 0.081
$b$ (GeV2) | 0.229 |  $c$ (GeV) | -0.3
$\epsilon_{c}$ | -0.164 | $\epsilon_{sov}$ | 0.262
$\epsilon_{sos}$ | 0.9728 | $\epsilon_{t}$ | 1.993

We calculate the mass spectrum of $\omega$ meson family and obtain the masses
of $\omega(3S)$, $\omega(4S)$, $\omega(5S)$, $\omega(1D)$, $\omega(2D)$ and
$\omega(3D)$ are 1861.6 MeV, 2181.6 MeV, 2429.0 MeV, 2633.1 MeV, 2003.0 MeV
and 2284.1 MeV, respectively. The new observed states $X(2200)$, $X(2232)$ and
$X(2222)$ have approximate masses $2200\pm 11\pm 17$ MeV, $2232\pm 19\pm 27$
MeV and $2222\pm 7\pm 2$ MeV, respectively. According to their mass spectrum
we obtained, these three states may be $\omega(3D)$ states.

###  II.2 A brief review of QPC model

The ${}^{3}P_{0}$ model is widely applied in 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._ (2010); 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).

In our previous treatment of the two-body decays $A\to BC$, the decay widths
were obtained from the amplitudes read

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{\pi}{4}\frac{|\mathbf{P}|}{m_{A}^{2}}\sum_{J,L}|\mathcal{M}^{JL}(\mathbf{P})|^{2},$
(2.7)

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

$\displaystyle\mathcal{M}^{JL}(\mathbf{P})=$
$\displaystyle\frac{\sqrt{4\pi(2L+1)}}{2J_{A}+1}\sum_{M_{J_{B}}M_{J_{C}}}\langle
L0;JM_{J_{A}}|J_{A}M_{J_{A}}\rangle$ (2.8) $\displaystyle\times\langle
J_{B}M_{J_{B}};J_{C}M_{J_{C}}|{J_{A}}M_{J_{A}}\rangle\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}.$

The QPC model’s input parameter $\gamma$, has two different values from method
A and method B. In method A, $\gamma$ is taken to be 8.7 for
$u\bar{u}/d\bar{d}$ pair creation, while the strength of the $s\bar{s}$ pair
creation satisfies $\gamma=8.7/\sqrt{3}$ Le Yaouanc  _et al._ (1977); Ye  _et
al._ (2012); He  _et al._ (2013). In method B, we adopt $\gamma=8.1$, which
was obtained by fitting the decay width of $\omega(1420)$ Zyla  _et al._
(2020) state.

We use two methods to verify the conclusion. The parameter $R$ of the final
channel in the simple harmonic oscillator wave function is determined by
reproducing the root mean square radius by solving the Schrödinger equation
with the effective potential Close and Swanson (2005). We may set a $R$ range
between the $R$ values determined by the potential model in Refs. Close and
Swanson (2005); Godfrey and Isgur (1985) for these three states for method A.
For method B, We use a simple harmonic oscillator wave function $\psi_{n\ell
m}({\bf k})=R_{n\ell}(R,{\bf k})\mathcal{Y}_{n\ell m}(\bf{k})$ to describe the
meson wave function involved in the decays when calculating the spatial
integral of the decay amplitude.

Table 2: The total and partial decay widths of the $X(2222)$, $X(2200)$ and
$X(2232)$ when treated them as $\omega(3D)$ states in method B, the unit of
widths is MeV. The $\gamma$ value is 8.1. Exp. means the experiment width.
$\begin{array}[]{c|c|c|c}\hline\cr\hline\cr\hline\cr\hline\cr&X(2222)&X(2200)&X(2232)\\\
\hline\cr\text{Exp.}&59\pm 30.59&74\pm 31.24&93\pm 56.65\\\
\text{Total}&{73.1}&{65.0}&{67.3}\\\ {b_{1}}\pi&{35.9}&{31.2}&{27.2}\\\
\pi\rho(1450)&{10.6}&{9.46}&{9.58}\\\ \rho\pi&{5.90}&{2.99}&{4.05}\\\
\rho{a_{1}}&{4.40}&{4.66}&{7.87}\\\ \rho{a_{2}}&{3.47}&{3.11}&{5.41}\\\
\eta{h_{1}}&{3.02}&{2.35}&{1.53}\\\ \pi\rho_{3}&{1.72}&{1.47}&{2.84}\\\
\omega{f_{2}}&{1.72}&{1.82}&{2.84}\\\ \omega{f_{1}}&{1.13}&{1.93}&{2.12}\\\
\pi\rho(1700)&{1.03}&{0.971}&{1.16}\\\ KK(1460)&{1.11}&{0.882}&{0.000297}\\\
KK&{1.02}&{0.945}&{0.312}\\\ \eta\omega(1420)&{0.438}&{0.277}&{0.202}\\\
K{K_{1}^{\prime}}&{0.291}&{0.0999}&{0.112}\\\
\omega\eta&{0.364}&{0.289}&{0.117}\\\
\rho\pi(1300)&{0.264}&{0.619}&{0.0640}\\\
KK^{*}(1410)&{0.358}&{0.0490}&{0.0412}\\\
\omega{f_{0}(1370)}&0.135&0.0735&0.160\\\
\eta^{\prime}{h_{1}}&{0.0901}&{0.178}&{0.318}\\\
\omega\eta(1295)&{0.0424}&{0.116}&{0.0137}\\\
K{K_{1}}&{0.0607}&{1.35}&{1.14}\\\
\omega\eta^{\prime}&{0.00357}&{0.121}&{0.103}\\\
KK^{*}&{0.00342}&{0.121}&{0.106}\\\
\hline\cr\hline\cr\hline\cr\hline\cr\end{array}$

###  II.3 Decay modes and widths

It should be more emphasis on the fact that the mass alone is insufficient to
identify this assignment, and decay behaviors need to be analyzed. We adopt
two methods to show the widths of $X(2232)$, $X(2200)$ and $X(2222)$, when
they are treated as $\omega(3D)$. According to our results of the QPC model,
$X(2232)$, $X(2200)$ and $X(2222)$ can decay into $b_{1}\pi$, $\pi\rho(1450)$,
$\rho{a_{1}}$, $\eta\omega(1420)$, $KK$, $KK(1460)$, $\omega\eta$,
$\pi\rho_{3}$, $\eta{h_{1}}$, $\rho\pi$, $\rho{a_{2}}$, $\omega{f_{2}}$,
$\rho\pi(1300)$, $\omega{f_{1}}$, $KK_{1}(1400)$, $KK_{1}(1270)$,
$\pi\rho(1700)$, $KK^{*}$, $KK^{*}(1410)$, $\omega\eta^{\prime}$,
$\eta^{\prime}{h_{1}}$, $\omega\eta(1295)$ and so on. The decay modes
${b_{1}}\pi$, $\omega{f_{0}(1370)}$ and $\omega{f_{2}}$ are also proved in the
experiment Ablikim  _et al._ (2022a).

####  II.3.1 Results of method A

Figure 1: The $R$ dependence of the calculated total decay widths of
$X(2222)$, $X(2200)$ and $X(2232)$ in method A. The colored band is the
experiment width of $X(2222)$, $X(2200)$ and $X(2232)$ from BESIII BESIII
Collaboration (2022); Ablikim  _et al._ (2022b).

$R~{}(\text{GeV}^{-1})$$R~{}(\text{GeV}^{-1})$$R~{}(\text{GeV}^{-1})$

Figure 2: The $R$ dependence of the calculated partial and total decay widths
of $X(2222)$, $X(2200)$ and $X(2232)$ in method A. Here, we do not list the
tiny decay modes. $b_{1}$, $h_{1}$,$a_{1}$, $a_{2}$, $f_{1}$, $f_{2}$, $K_{1}$
and ${K_{1}}^{\prime}$ represent $b_{1}(1235)$, $h_{1}(1170)$, $a_{1}(1260)$,
$a_{2}(1320)$, $f_{1}(1285)$, $f_{2}(1270)$, $K_{1}(1270)$ and
${K_{1}(1400)}$.

We obtained the $R$ dependence of the calculated partial and total decay
widths of $X(2222)$, $X(2200)$ and $X(2232)$ states in the method A, as shown
in Fig. 1 and 2. In Fig. 2, partial decay widths of $X(2222)$, $X(2200)$ and
$X(2232)$ are calculated by us and the colorful band are the experiment widths
of $X(2222)$, $X(2200)$ and $X(2232)$ in Fig. 1. The main decay channel is
${b_{1}}\pi$ and it has been seen in the experiment Ablikim  _et al._ (2022a),
which also proves our assumption. The total widths we obtained of $X(2222)$,
$X(2200)$ and $X(2232)$ are about $68-81$ MeV, $78-89$ MeV and $68-82$ MeV,
respectively, which are in good agreement with the BESIII measurement of
$59\pm 30\pm 6$ MeV, $74\pm 20\pm 24$ MeV and $93\pm 53\pm 20$MeV,
respectively.

####  II.3.2 Results of method B

We employ the QPC model with the realistic meson wave functions obtained from
the MGI model to evaluate the decay widths of $X(2222)$, $X(2200)$ and
$X(2232)$ when they are treated as $\omega(3D)$ states in the method B. The
total widths and partial widths of these three states are listed in Table 2.
The total widths of $X(2222)$, $X(2200)$ and $X(2232)$ are expected to be
about $73$ MeV, $65$ MeV and $67$ MeV, respectively, which agrees well with
the BESIII measurement of $\Gamma_{X(2222)}=59\pm 30.59$ MeV,
$\Gamma_{X(2232)}=93\pm 56.65$ MeV and $\Gamma_{X(2200)}=74\pm 31.24$ MeV.
Furthermore, the crucial decay modes of $X(2222)$, $X(2200)$ and $X(2232)$ are
$b_{1}\pi$, $\pi\rho(1450)$, $\rho\pi$ and $\rho{a_{1}}$. In addition,
$\rho{a_{2}}$, $\eta{h_{1}}$, $\pi\rho_{3}$ and $\omega{f_{2}}$ are the
sizeable decay modes as well. Other decay channels, such as $KK^{*}$,
$\omega\eta^{\prime}$, $K{K_{1}}$, $\omega\eta(1295)$ and so on, are too
narrow for experiment to find. We hope more precise experimental data can be
released in the future to reveal the inner structure of $\omega(3D)$ state.

##  III Summary

In this work, we perform a study on the properties for $X(2232)$, $X(2200)$
and $X(2222)$. The mass spectrum of $\omega$ indicates that these three states
may be $\omega(3D)$ states. Additionally, the decay widths are calculated in
the QPC model with two methods and the result points out when we treated
$X(2232)$, $X(2200)$ and $X(2222)$ as $\omega(3D)$ states, the widths are in
good agreement with experiment values. The most important decay mode
${b_{1}}\pi$ in our theoretical calculation is also proved in the experiment
of BESIII Collaboration. Other decay channels like $\omega{f_{0}(1370)}$ and
$\omega{f_{2}}$ are found both in the theoretical research and experiment. We
expect that more and more results will be released by lattice QCD in the near
future and further experimental progress on the $\omega(3D)$.

##  IV Acknowledgements

This work is supported by the National Natural Science Foundation of China
under Grants No. 11965016 and Department of Qinghai Province No. 2020-ZJ-728.

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