X(3872) in an unquenched quark model

TL;DR

This study models the X(3872) meson as a mixed state of charmonium and meson-meson components using an unquenched quark model with a modified pair creation operator, providing insights into its structure.

Contribution

It introduces a modified quark pair creation operator considering energy and separation effects, and applies it to describe X(3872) as a mixed state.

Findings

X(3872) is approximately 70% charmonium and 30% meson-meson components.

The modified pair creation operator improves the modeling of meson structures.

The wavefunctions are obtained via solving Schrödinger equations with Gaussian expansion.

Abstract

In this paper, we calculate mass and probability fractions of meson-meson components of $X(3872)$ in an unquenched quark model. Different from most of other unquenched quark models, the quark pairs creation operator from $^3P_0$ is modified by considering effects of the created quark pair energy and separation between the created quark pair and the valence quark-pair. In the calculation, all the wavefunctions of mesons and the relative motion between two mesons are obtained by solving the corresponding Schr\"{o}dinger equation with the help of gaussian expansion method. The multi-channel coupling of quark-antiquark state with possible meson-meson states are performed. The results show that the $X(3872)$ can be described as a mixing state of dominant charmonium state (70\%) and meson-meson components (30\%).

Tables

Table 1. Table 1: Quark Model Parameters.

Quark masses	$m_u=m_d$(MeV)	313
	$m_s$(MeV)	536
	$m_c$(MeV)	1728
	$m_b$(MeV)	5112
Goldstone bosons	$m_{\pi }\left(f{m}^{-1}\right)$	0.70
	$m_{\sigma }\left(f{m}^{-1}\right)$	3.42
	$m_{\eta }\left(f{m}^{-1}\right)$	2.77
	$m_K\left(f{m}^{-1}\right)$	2.51
	${\Lambda }_{\pi }={\Lambda }_{\sigma }\left(f{m}^{-1}\right)$	4.2
	${\Lambda }_{\eta }={\Lambda }_K\left(f{m}^{-1}\right)$	5.2
	$g_{ch}^2/\left(4\pi \right)$	0.54
	${\theta }_p{(}^{\circ })$	-15
Confinement	$a_c$(MeV)	101
	$\Delta$(MeV)	-78.3
OGE	${\alpha }_0$	3.67
	${\Lambda }_0\left(f{m}^{-1}\right)$	0.033
	${\mu }_0$(MeV)	36.976
	${\hat{r}}_0$(MeV)	28.17

Table 2. Table 2: The mass shifts and probability fractions of four-quark components in accumulating approach and the three-channel coupling calculation. ’S’ in the parentheses denotes the relative motion between two clusters is in S 𝑆 S -wave.

	${\eta }_c$		${\chi }_{c1}\left(2P\right)$
bare mass	2986.28		3889.62
meson-meson	$\Delta m_i$ (MeV)	$P_i$	$\Delta m_i$ (MeV)	$P_i$
$D{\overline{D}}^{\ast }\left(S\right)$	$-189.75$	4.02%	$-73.65$	43.09%
$D_s^{\ast }{\overline{D}}_s\left(S\right)$	$-76.50$	1.75%	$-15.13$	1.26%
Total	$-266.25$	5.77%	$-88.78$	44.35%
$D{\overline{D}}^{\ast }$+$D_s^{\ast }{\overline{D}}_s\left(S\right)$	$-266.71$	5.20%	$-83.39$	35.53%
$D{\overline{D}}^{\ast }\left(S\right)$	$-189.75$	4.02%	$-73.65$	43.09%
$D^{\ast }\overline{D}\left(S\right)$	$-189.75$	4.02%	$-73.65$	43.09%
Total	$-379.50$	8.04%	$-147.30$	86.18%
$D{\overline{D}}^{\ast }$+$D^{\ast }\overline{D}\left(S\right)$	$-369.43$	6.4%	$-117.11$	33.38%

Table 3. Table 3: The mass shift of η c ​ ( 1 ​ S ) subscript 𝜂 𝑐 1 𝑆 \eta_{c}(1S) in the different calculations. (unit: MeV)

meson-meson	Ref.18Kalashnikova:2005	Ref.19Ferretti:2013	Ref.31Barnes:2008	This work
$D{\overline{D}}^{\ast }\left(D^{\ast }\overline{D}\right)$	$-59$	$-34$	$-114$	$-189.75$
$D^{\ast }{\overline{D}}^{\ast }$	$-55$	$-31$	$-105$	$-354.48$
$D_s{\overline{D}}_s^{\ast }\left(D_s^{\ast }{\overline{D}}_s\right)$	$-26$	$-8$	$-106$	$-76.50$
$D_s^{\ast }{\overline{D}}_s^{\ast }$	$-35$	$-8$	$-98$	$-147.14$

Table 4. Table 4: The mass shifts and probability fraction of four-quark components of χ c ​ 1 ​ ( 2 ​ P ) subscript 𝜒 𝑐 1 2 𝑃 \chi_{c1}(2P) . ’ S 𝑆 S ’ and ’ D 𝐷 D ’ in the parentheses denote the relative motion between two clusters is in S 𝑆 S -wave and D 𝐷 D -wave. (unit: MeV)

${\chi }_{c1}\left(2P\right)$
states	$\Delta m_i$	$P_i$
$D{\overline{D}}^{\ast }+D^{\ast }\overline{D}\left(S+D\right)$	$-76.37$	27.76%
$D_s{\overline{D}}_s^{\ast }+D_s^{\ast }{\overline{D}}_s\left(S+D\right)$	$-8.84$	1.98%
$D^{\ast }{\overline{D}}^{\ast }\left(D\right)$	$-34.15$	6.42%
$D_s^{\ast }{\overline{D}}_s^{\ast }\left(D\right)$	$-5.61$	0.72%
Total	$-125.37$	36.88%
Unquenched mass	3862.80

Table 5. Table 5: Adjusted quark model parameters.

parameter		ChQM	Ref.29chen:2018	This work
Quark masses	$m_c$ (MeV)	1728	1728	1710
Confinement	$a_c$ (MeV fm-2)	101	101	105.3
	$\Delta$ (MeV)	$-78.3$	$-58.3$	$-57.4$
${\alpha }_s$	${\alpha }_{qq}$	0.57	0.60	0.60
	${\alpha }_{qs}$	0.54	0.56	0.56
	${\alpha }_{qc}$	0.49	0.52	0.52
	${\alpha }_{sc}$	0.44	0.46	0.46
	${\alpha }_{cc}$	0.38	0.39	0.39

Table 6. Table 6: The mass shifts and probability fractions of four-quark components in UQM. ’ S 𝑆 S ’, ’ P 𝑃 P ’ and ’ D 𝐷 D ’ in the parentheses denotes the relative motion between two clusters are in S 𝑆 S -, P 𝑃 P - and D 𝐷 D -wave.

State	${\eta }_c$		$J/\psi$		${\chi }_{c1}\left(2P\right)$
Bare mass	3047.0		3169.7		3986.1
Meson-meson state	$\Delta m_i$	$P_i$	$\Delta m_i$	$P_i$	$\Delta m_i$	$P_i$
$D{\overline{D}}^{\ast }+D^{\ast }\overline{D}\left(P\right)$	$-26.3$	2.5%	$-20.5$	2.2%	-	-
$D_s{\overline{D}}_s^{\ast }+D_s^{\ast }{\overline{D}}_s\left(P\right)$	$-6.0$	0.5%	$-4.6$	0.4%	-	-
$D^{\ast }{\overline{D}}^{\ast }\left(P\right)$	$-24.8$	2.2%	-	-	-	-
$D_s^{\ast }{\overline{D}}_s^{\ast }\left(P\right)$	$-5.9$	0.5%	-	-	-	-
$D\overline{D}+D^{\ast }{\overline{D}}^{\ast }\left(P\right)$	-	-	$-39.4$	4.0%	-	-
$D_s{\overline{D}}_s+D_s^{\ast }{\overline{D}}_s^{\ast }\left(P\right)$	-	-	$-9.0$	0.8%	-	-
$D{\overline{D}}^{\ast }+D^{\ast }\overline{D}\left(S+D\right)$	-	-	-	-	$-68.2$	22.5%
$D_s{\overline{D}}_s^{\ast }+D_s^{\ast }{\overline{D}}_s\left(S+D\right)$	-	-	-	-	$-8.2$	1.7%
$D^{\ast }{\overline{D}}^{\ast }\left(D\right)$	-	-	-	-	$-32.6$	5.6%
$D_s^{\ast }{\overline{D}}_s^{\ast }\left(D\right)$	-	-	-	-	$-5.4$	0.7%
Total	$-63.0$	5.7%	$-73.5$	7.4%	$-114.4$	30.5%
Unquenched mass	2984.0	94.3%	3096.2	92.6%	3871.7	69.5%

Equations

$$H{\Psi }_{I{M}_I}^{J{M}_J}(1,2)=E^{IJ}{\Psi }_{I{M}_I}^{J{M}_J}(1,2).$$

$${\Psi }_{I{M}_I}^{J{M}_J}(1,2)=\sum _{\alpha }{\left[{\psi }_l(\mathbf{r}){\chi }_s(1,2)\right]}^{J{M}_J}{\omega }^c(1,2){\varphi }_{I{M}_I}(1,2),$$

$${\psi }_{lm}(\mathbf{r})$$

$${\psi }_{nlm}^G(\mathbf{r})$$

$${\nu }_n=\frac{1}{r_n^2},r_n=r_1{a}^{n-1},a={\left(\frac{r_{n_{max}}}{r_1}\right)}^{\frac{1}{n_{max}-1}}.$$

$${\Psi }_{I{M}_I}^{JM}(1,2)$$

$$\sum _{n^{\prime },{\alpha }^{\prime }}(H_{n\alpha ,n^{\prime }{\alpha }^{\prime }}^{IJ}-E^{IJ}{N}_{n\alpha ,n^{\prime }{\alpha }^{\prime }}^{IJ})C_{n^{\prime }{\alpha }^{\prime }}^{IJ}=0,$$

$$H_{n\alpha ,n^{\prime }{\alpha }^{\prime }}^{IJ}=⟨{\Phi }_{I{M}_I,n\alpha }^{JM}|H|{\Phi }_{I{M}_I,n^{\prime }{\alpha }^{\prime }}^{JM}⟩,$$

$$N_{n\alpha ,n^{\prime }{\alpha }^{\prime }}^{IJ}=⟨{\Phi }_{I{M}_I,n\alpha }^{JM}|{\Phi }_{I{M}_I,n^{\prime }{\alpha }^{\prime }}^{JM}⟩.$$

$$H{\Psi }_{I{M}_I}^{J{M}_J}(1234)=E^{IJ}{\Psi }_{I{M}_I}^{J{M}_J}(1234),$$

$${\Psi }_{I_1{M}_{I_1}}^{J_1{M}_{J_1}}(12)={\left[{\psi }_{l_1}({\mathbf{r}}_{12}){\chi }_{s_1}(12)\right]}^{J_1{M}_{J_1}}{\omega }^{c_1}(12){\varphi }_{I_1{M}_{I_1}}(12),$$

$${\Psi }_{I_2{M}_{I_2}}^{J_2{M}_{J_2}}(34)={\left[{\psi }_{l_2}({\mathbf{r}}_{34}){\chi }_{s_2}(34)\right]}^{J_2{M}_{J_2}}{\omega }^{c_2}(34){\varphi }_{I_2{M}_{I_2}}(34),$$

$${\Psi }_{I{M}_I}^{J{M}_J}(1234)=\mathcal{A}[{\left[{\psi }_{l_1}({\mathbf{r}}_{12}){\chi }_{s_1}(12)\right]}^{J_1}$$

$${{\left[{\psi }_{l_2}({\mathbf{r}}_{34}){\chi }_{s_2}(34)\right]}^{J_2}{\psi }_{L_r}({\mathbf{r}}_{1234})]}^{J{M}_J}$$

$${\left[{\omega }^{c_1}(12){\omega }^{c_2}(34)\right]}^{[222]}{\left[{\varphi }_{I_1}(12){\varphi }_{I_2}(34)\right]}_{I{M}_I},$$

$$\mathcal{A}=\frac{1}{2}(1-P_{13}-P_{24}+P_{13}{P}_{24}).$$

$${\psi }_{nlm}^G(\mathbf{r})$$

$$H$$

$$V_{ij}^G$$

$$V_{ij}^C$$

$$V_{ij}^{\pi }$$

$$V_{ij}^K$$

$$V_{ij}^{\eta }$$

$$V_{ij}^{\sigma }$$

$$v_{ij}^{\chi }$$

$${\alpha }_s({\mu }_{ij})=\frac{{\alpha }_0}{\ln \left[({\mu }_{ij}^2+{\mu }_0^2)/{\Lambda }_0^2\right]}$$

$$T_1=-3\gamma \sum _m⟨1m1-m|00⟩\int d{\mathbf{p}}_{3}d{\mathbf{p}}_4{\delta }^3({\mathbf{p}}_3+{\mathbf{p}}_4)$$

$$\times {\mathcal{Y}}_1^m(\frac{{\mathbf{p}}_3-{\mathbf{p}}_4}{2}){\chi }_{1-m}^{34}{\varphi }_0^{34}{\omega }_0^{34}{b}_3^{†}({\mathbf{p}}_3)d_4^{†}({\mathbf{p}}_4),$$

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

$${\mathcal{M}}^{M_{J_A}{M}_{J_B}{M}_{J_C}}(\mathbf{P})$$

$${\mathcal{I}}_{M_{L_B},M_{L_C}}^{M_{L_A},m}(\mathbf{P},m_1,m_2,m_3)=\int d\mathbf{p}\text{mbox}{\psi }_{n_B{L}_B{M}_{L_B}}^{\ast }(\frac{m_3}{m_1+m_3}\mathbf{P}+\mathbf{p}){\psi }_{n_C{L}_C{M}_{L_C}}^{\ast }(\frac{m_3}{m_2+m_3}\mathbf{P}+\mathbf{p}){\psi }_{n_A{L}_A{M}_{L_A}}(\mathbf{P}+\mathbf{p}){\mathcal{Y}}_1^m(\mathbf{p}),$$

$${\mathcal{M}}^{JL}(A\to BC)=\frac{\sqrt{2L+1}}{2J_A+1}\sum _{M_{J_B},M_{J_C}}⟨L0J{M}_{J_A}|J_A{M}_{J_A}⟩$$

$$\times ⟨J_B{M}_{J_B}{J}_C{M}_{J_C}|J{M}_{J_A}⟩{\mathcal{M}}^{M_{J_A}{M}_{J_B}{M}_{J_C}}(\text{P}).$$

$$T_2$$

$${\mathbf{R}}_A$$

$${\mathbf{R}}_V$$

$$\gamma =32.2f=0.5\text{mbox}fm{R}_0=1.0\text{mbox}fm.$$

$${\alpha }_0=3.85,\Delta =-58.5$$

Full text

I Introduction

The nonrelativistic quark model has been successfully describing the properties of heavy mesons. However, since the well-known exotic state, $X(3872)$, was discovered, more and more exotic particles which cannot be fitted well into meson spectrum, are reported by experimental collaborations. These exotic states put forward great challenge on quark model.

In 2003, the Belle collaboration first found $X(3872)$ in the $B$ meson decay 01Choi:2003 . Subsequently, CDF 02Acosta:2004 and D0 03Abazov:2004 collaborations confirmed this state in $p\bar{p}$ collision, and BABAR also found $X(3872)$ in the $B$ meson decay 04Aubert:2005 . Different from the ordinary hadrons, the state $X(3872)$ has some strange properties, its mass is very close to the threshold of $D\bar{D}^{*}$, and decay width is very narrow, less than 1.2 MeV. So the dispute on the nature of $X(3872)$ is quite hot. Due to the ambiguity information about the quantum numbers of $X(3872)$ at that time, someone held the view that traditional quark model can still described its properties 05Barnes:2004 ; 09Tornqvist:2004 ; 07Swanson:2004 ; 13Suzuki:2005 . T. Barnes et al. analyzed some states by calculating their radiative transition, and the results turned out that five states, $1^{3}D_{3}$, $1^{3}D_{2}$, $1^{1}D_{2}$, $2^{3}P_{1}$ and $2^{1}P_{1}$, could be the possible candidate of the new exotic state 05Barnes:2004 . After examining the pictures of mesons and meson-meson molecules through different ways, Pakasa and Swanson et al. concluded the meson picture may be more suitable for $X(3872)$ 06Pakasa:2004 ; 07Swanson:2004 . Indeed, the ratio $\frac{B(X(3872)\rightarrow\psi(2s)\gamma)}{B(X(3872)\rightarrow J/\psi\gamma)}=3.4\pm 1.4$ announced by BABAR collaboration appears to support $X(3872)$ as a traditional meson 08Aubert:2009 . N. Achasov argued that the $X(3872)$ resonance was the $c\bar{c}=\chi_{c1}(2P)$ charmonium which “sit on” the $D^{*0}\bar{D}^{0}$ threshold, and its mass was shifted from the prediction of potential model to the threshold of $D^{*}\bar{D}$ by the contribution of the virtual $D^{*}\bar{D}+c.c.$ intermediate states Achasov .

Due to the mass of $X(3872)$ closes to the threshold of $D\bar{D}^{*}$, it’s natural to classify state $X(3872)$ into molecule picture 09Tornqvist:2004 . Törnqvist put forward that, analogous to deuteron, one pion exchange potential could make contribution to bound state of $X(3872)$ 09Tornqvist:2004 , and $X(3872)$ had structure of $\frac{1}{\sqrt{2}}[D\bar{D}^{*}+\bar{D}D^{*}]$. This picture could easily explain the isospin breaking branching ratio $\frac{Br(J/\psi\pi^{-}\pi^{+}\pi^{0})}{Br(J/\psi\pi^{+}\pi^{-})}=1.0\pm 0.4\pm 0.3$ 10Abe:2005 due to the mass difference between neutral $D$ meson and charged $D$ meson. Based on pion exchange potential proposed by Törnqvist, Swanson et al. considered additional mixture of $J/\psi\rho$ and $J/\psi\omega$ to the state, and arrived a conclusion that although the effect of pion exchange might be responsible for bound state, short range quark dynamics were present and assisted in binding the $X(3872)$ via mixing to hidden charm vector, $J/\psi$ states 11Swanson:2004 . However, some people held the different views of points that pion exchange was too weak to bind the $X(3872)$ 12Liu:2008 ; 13Suzuki:2005 ; 14Thomas:2008 . Because $D^{*}-D-\pi=0$, Suzuki thought there was no long rang attraction between $D$ and $D^{*}$, which means $X(3872)$ was not a bound state 13Suzuki:2005 . What’s more, unless the number of coupling constant and cut off were all big, $X(3872)$ could not be a real molecule depending only on pion exchange potential 12Liu:2008 . Tomas et al. added tensor term and factor on the OPE (one pion exchange) model, and they finally got a bound state 14Thomas:2008 .

Actually, since BABAR collaboration reported the branching ratio of $X\rightarrow\gamma J/\psi$, people gradually started to accept the concept $X(3872)$ might be an unquenched hadron state, a mixture of $c\bar{c}$ and $D\bar{D}^{*}$18Kalashnikova:2005 ; 19Ferretti:2013 ; 20Ortega:2010 ; 32cardoso:2014 . Kang and Oller analyzed the experimental data by a near-threshold parameterization method and found the $X(3872)$ compositeness coefficients in $D^{0}\bar{D}^{*0}$ ranges from nearly 0 up to 1 in the different scenarios Kang . In 2005, Kalashnikova first used ${}^{3}P_{0}$ model to study the mass spectrum of $c\bar{c}$ based on nonrelativistic quark model, in which the simple harmonic oscillator (SHO) functions are used to describe the wavefunctions of mesons, and got the mass of $X(3872)$ 3990 MeV 18Kalashnikova:2005 , only 0.3 MeV over threshold. In the following years, the $q\bar{q}\rightarrow qq\bar{q}\bar{q}$ transition operator in the unquenched quark model is often taken as ${}^{3}P_{0}$ operator, which was first proposed by L. Micu 15Micu:1969 , then Yaouanc et al. applied the operator to calculate the strong decay widths of baryons and mesons 16Yaouanc:1973 ; 17Yaouanc:1974 . Santopinto et al. calculated the decay width, components and mass of $X(3872)$ by relativistic quark model and ${}^{3}P_{0}$ model, where SHO functions were also used 19Ferretti:2013 ; 40santopinto:2009 ; 41santopinto:2012 ; 42santopinto:2012 ; 43santopinto:2015 ; 44santopinto:2015 . What’s more, Ortega et al. utilized the unquenched quark model to analyze the decay width and components of $X(3872)$, and their wavefunctions for the relative motion between two mesons were obtained by solving the resonating group method (RGM) equation 20Ortega:2010 ; 37Rupp:2012 ; 38PGO:2012 ; 39salamanca:2012 . In the most of the previous work, the SHO functions are used to describe the meson dynamics and the relative motion between two mesons are described by plane wave functions. The systematic errors due to the approximations are unpredictable for the bound state calculation, although they are not a bad approximation for decay width calculation 29chen:2018 . In this work, we applied the Gaussian expansion method (GEM), which is a powerful method for few-body system, to analyze $X(3872)$ in an unquenched quark model 29chen:2018 , where a modified ${}^{3}P_{0}$ was employed.

The paper is organized as follows. In section II, the chiral quark model and the GEM for solving the $q\bar{q}$ and $q\bar{q}$-$q\bar{q}$ systems are presented. In Sec. III, we briefly introduce the modified ${}^{3}P_{0}$ model. The numerical results are given in Sec. IV. The last section is devoted to the summary of the present work.

II Chiral quark model and GEM

In the chiral quark model 21Vijande:2005 , the meson spectrum is obtained by solving the Schrödinger equation

[TABLE]

The wavefunction $\Psi_{IM_{I}}^{JM_{J}}$ of a meson with quantum numbers $I^{G}J^{PC}$ can be written as

[TABLE]

where $\alpha$ denotes the intermediate quantum numbers $l,s$ and possible flavor indexes (for isospin $I=0$ states, flavor indexes take $u\bar{u},d\bar{d}$ and $s\bar{s}$). ”[ ]” means angular momentum coupling, $\chi_{sm_{s}}(1,2),\omega^{c}(1,2),\phi_{IM_{I}}(1,2)$ are spin, color and flavor wavefunctions (with specific isospin $I$) of meson, $\psi_{lm}(\mathbf{r}),\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$, is the orbital wavefunction. In GEM, the orbital wave function is written as the product of radial one and spherical harmonics, and the radial part of the wave function is expanded by a set of gaussians,

[TABLE]

Gaussian size parameters are taken as the following geometric progression numbers

[TABLE]

This enables the optimization of ranges employing a small number of gaussians. So, the wavefunction takes the form

[TABLE]

Noting that the gaussians are not orthogonal, the Rayleigh-Ritz variational principle for solving the Schrödinger equation leads to a generalized eigenvalue problem

[TABLE]

Extended to $q\bar{q}$-$q\bar{q}$ system, the same Schrödinger equation is employed to obtained the energy of the system,

[TABLE]

where $\Psi^{JM_{J}}_{IM_{I}}$ is the wave function of the four-quark state, which can be constructed as follows. First, we write down the wave functions of two subclusters,

[TABLE]

where $\chi_{s},\omega^{c},\phi^{I}$ are spin, color and flavor wave functions of the quark-antiquark subcluster (the quarks are numbered as 1, 3 and antiquarks 2, 4). The bracket [ ] denotes the angular momentum coupling. Then, the total wave function of the four-quark state is obtained as:

[TABLE]

where $\psi_{L_{r}}(\mathbf{r}_{1234})$ is the wave function for the relative motion between two clusters with orbital angular momentum $L_{r}$. ${\cal A}$ is the antisymmetrization operator. If all quarks (antiquarks) are taken as identical particles, we have

[TABLE]

The radial part of the wave function is also expanded by gaussians as in Eq. (II). Finally, the infinitesimally-shifted Gaussian basis function (ISG) is employed for the orbital wave functions with non-zero orbital angular momentum to simplify the calculation of the matrix elements 30Hiyama:2003 :

[TABLE]

The Hamiltonian of the chiral quark model includes three parts, the rest masses of quarks, the nonrelativistic kinetic energy and the potential energy. The potential energy is composed of color confinement, one-gluon-exchange (OGE) and one Goldstone boson exchange. The detailed form for the four-quark states is shown below 21Vijande:2005

[TABLE]

where $m_{i}$ is the mass of quarks and antiquarks, and $\mu_{ij}$ is their reduced mass, $r_{0}(\mu_{ij})=\hat{r}_{0}/\mu_{ij}$, $\boldsymbol{\sigma}$ are the $SU(2)$ Pauli matrices, $\boldsymbol{\lambda},~{}\boldsymbol{\lambda}^{c}$ are $SU(3)$ flavor, color Gell-Mann matrices, $g^{2}_{ch}/4\pi$ is the chiral coupling constant, determined from $\pi$-nucleon coupling constant. $\alpha_{s}$ is the effective scale-dependent running quark-gluon coupling constant 21Vijande:2005 ,

[TABLE]

All the parameters are determined by fitting the meson spectrum, from light to heavy, taking into account only a quark-antiquark component. They are shown in Table 1.

III ${}^{3}P_{0}$ model

III.1 traditional ${}^{3}P_{0}$ opertaor

The ${}^{3}P_{0}$ model (quark pair creation model) was originally introduced by Micu 15Micu:1969 and further developed by Le Yaouanc, Ackleh and Roberts et al. 16Yaouanc:1973 ; 17Yaouanc:1974 ; 22ACKleh:1996 ; 23Roberts:1992 . It can be applied to the OZI rule allowed two-body strong decays of a hadron 15Micu:1969 ; 22ACKleh:1996 ; 24Capstic:1986 ; 25Capstic:1994 ; 26page:1995 . The transition operator in the model is,

[TABLE]

where $\gamma$ represents the probability of the quark-antiquark pair with momentum $\mathbf{p}_{3}$ and $\mathbf{p}_{4}$ created from the vacuum. Because the intrinsic parity of the antiquark is negative, the created quark-antiquark pair must be in the state ${}^{2S+1}L_{J}={}^{3}P_{0}$. $\phi^{34}_{0}$ and $\omega^{34}_{0}$ are flavor and color singlet states, respectively (the quark and the antiquark in the original meson are indexed by 1 and 2). The S-matrix element for the process $A\rightarrow B+C$ is written as

[TABLE]

where $\mathbf{P}_{B}$ and $\mathbf{P}_{C}$ are the momenta of B and C mesons in the final state, and satisfy $\mathbf{P}_{A}=\mathbf{P}_{B}+\mathbf{P}_{C}=0$ in the center-of-mass frame of meson A. ${\cal{M}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}$ is the helicity amplitude of the process $A\rightarrow B+C$, which can be obtained as

[TABLE]

with the momentum space integral

[TABLE]

where $\mathbf{P}=\mathbf{P}_{B}=-\mathbf{P}_{C}$, and $\mathbf{p}=\mathbf{p}_{3}$, $m_{3}$ is the mass of the created quark $q_{3}$. To analyze the results and to compare the theoretical results with experimental data, the partial wave amplitude ${\mathcal{M}}^{JL}(A\rightarrow BC)$ is often employed. It is related with the helicity amplitude by the Jacob-Wick formula 27Jacob ,

[TABLE]

In evaluating the momentum space integral, Eq. (22), we use the wave functions of mesons $A$, $B$, $C$ obtained in the mass spectrum calculation. Because the wave functions are expanded by a series of gaussians, the integral can be evaluated analytically.

The parameter $\gamma$ is generally determined by an overall fitting of the strong decay width of hadrons. In this way, one obtains $\gamma=6.95$ for $u\bar{u}$ and $d\bar{d}$ pair creation, and $\gamma=6.95/\sqrt{3}$ for $s\bar{s}$ pair creation 28PLB .

III.2 Modified ${}^{3}P_{0}$ opertaor

The modified transition operator $T_{2}$ (in position space) was first proposed by Chen et al. for dealing with the issue that the mass shift of light meson is too large if the traditional transition operator $T_{1}$ is used 29chen:2018 .

[TABLE]

Here, $\mathbf{R}_{AV}=\mathbf{R}_{A}-\mathbf{R}_{V}$ is the relative coordinate between the source particle “A” and the created quark-antiquark pair in the vacuum with

[TABLE]

The convergence factor $e^{-r^{2}/(4f^{2})}$ of the modified operator $T_{2}$ mainly considers the effect of quark-antiquark energy created in the vacuum, that it’s difficult to create the quark-antiquark with high energy. The damp factor $e^{-{R_{AV}}^{2}/{{R_{0}}}^{2}}$ takes into the fact that the created quark-antiquark pair should not be far away from the source particle. With some reasonable arguments, the parameters $f$ and $R_{0}$ were fixed, and the parameter $\gamma$ was determined by fitting the decay width of $\rho\rightarrow\pi\pi$,

[TABLE]

Based on the modified transition operator, $T_{2}$, Chen et al. calculated the mass shift of light meson and adjusted some parameters of quark model to make unquenched light ground state meson masses in agreement with experimental data.

[TABLE]

IV Unquenched Quark Model

The mass and the structure of meson in unquenched quark model are obtained by solving the Schrödinger equation

[TABLE]

where, $\Psi_{IM_{I}}^{JM_{J}}$ is the unquenched wave function of the system which contain two- and four-quark components. It can be written as:

[TABLE]

where $\Psi(2q)$ and $\Psi(4q)$ are the wave functions with two- and four-quark components, respectively (the simplified symbols are used to save space), and the $N$ is the total number of four-quark channels.

In the nonrelativistic quark model, the number of particles is conserved. So there is no rigorous way to write down the hamiltonian of the unquenched quark model. Here we only give a prescription of the Hamiltonian $H$ as follows:

[TABLE]

where $H_{2q}$ is stipulated to act on the wave function of quark-antiquark component, $\Psi(2q)$, and $H_{4q}$ only acts on the wave function of four-quark component, $\Psi(4q)$. $T_{24}$ takes the form of the transition operator in the ${}^{3}P_{0}$ model, Eq. (III.1) or Eq. (24), which undertakes the coupling of the two- and four-quark components. So, in this way, the matrix elements of the Hamiltonian can be written as:

[TABLE]

Then, we get a blocked matrix of Hamiltonian and overlap:

[TABLE]

[TABLE]

Where $\langle H_{2q}\rangle$, $\langle H_{24}\rangle_{j}$ and $\langle H_{4q}\rangle_{ij}$ are $\langle\Psi(2q)|H_{2q}|\Psi(2q)\rangle$, $\langle\Psi(2q)|T_{24}|\Psi_{j}(4q)\rangle$, and $\langle\Psi_{i}(4q)|H_{4q}|\Psi_{j}(4q)\rangle$ respectively. The subscript labels the index of the four-quark channel. By solving the following generalized eigen-equation,

[TABLE]

we get the eigen-energy $E_{n}$ and the expansion coefficients $C_{n}$.

V Numerical Results and discussions

In the present calculation, we focus on the charmonium state $\chi_{c1}(2P)$ and try to explain the well-known exotic state $X(3872)$ in the unquenched quark model. To fix the parameters associated with charm quark, two charmonia, $\eta_{c}$ and $J/\psi$ are also investigated in the unquenched quark model. For comparison, two transition operators, traditional one and the modified one, are used to do the calculation.

V.1 Accumulating approach

Generally, the dimension of $H$ in unquenched quark model is very big and matrix construction process is very complex. So most of unquenched quark models adopted an accumulating approach, that is to do two channel coupling calculation, bare state $\Psi(2q)$ and one of four-state $\Psi_{i}(4q)$, to get the mass shift $\Delta m_{i}$ and the probability fraction of four-quark component $P_{i}$, then the total mass shift $\Delta M_{t}$ and the total fraction $P_{t}$ of four-quark component are obtained by accumulating $\Delta m_{i}$ and $P_{i}$.

[TABLE]

In this approach, the dimension of the hamiltonian matrix $H$ needs to be diagonalized is reduced greatly. The approach works well if the cross matrix elements between different four-quark channels are small enough. Unfortunately, this is not always true. If there is coupling between two four-quark channels, the accumulating approach will introduce errors, especially for the case that the energy of the four-quark state is close to the bare mass of the meson. The comparison of the results for states $\eta_{c}$ and $\chi_{c1}(2P)$ between accumulating approach and the full-channel diagonalization is shown in Table 2, where the traditional transition operator $T_{1}$ is used. There is no coupling between $D\bar{D}^{*}$ and $D_{s}\bar{D}_{s}^{*}$, so the mass shifts and the probability fractions of four-quark components for $\eta_{c}$ and $\chi_{c1}(2P)$ from the accumulating approach are almost the same as that of three-channel diagonalization. Whereas there is a coupling between $D\bar{D}^{*}$ and $D^{*}\bar{D}$, then different approaches give different results. For $\eta_{c}$, the accumulating approach over-estimates the mass shift about 10%, and the probability fraction 30%. For $\chi_{c1}(2P)$, the over-estimation are 30% for mass shift and 160% for probability fraction, because $D\bar{D}^{*}$ is open-channel for state $\chi_{c1}(2P)$. So the multi-channel coupling calculation is adopted if there are non-zero matrix elements between different four-quark channels in the present work.

The comparison between our results of $\eta_{c}$ and that of other calculations are shown in Table 3, where the same transition operator $T_{1}$ is employed. Clearly our calculation gives larger mass shifts than other work. The reason is that the different wavefunctions are used in the different calculations. In most previous work 18Kalashnikova:2005 ; 19Ferretti:2013 ; 31Barnes:2008 , the simple harmonic oscillator (SHO) wavefunctions are chosen as the radial part of the orbital wavefunctions of mesons, the relative motion wavefunction between two meson clusters is set to the plane wave function. The plane wave approximation is a good one in the decay calculation, it should not be a good one for the bound state calculation. In most case SHO wave function can describe the ground meson well, but different size parameters should be used for different mesons. To simplify the calculation, the same size parameter is used for different mesons in the most previous work. In our calculation, all the wavefunctions are determined by system dynamics. In this a self-consistent calculation is arrived.

V.2 Results

Using the modified operator $T_{2}$, and keeping all the parameters unchanged, we calculate the mass shifts and probability fractions of four-quark components of state $\chi_{c1}(2P)$. The results are listed in Table 4. From the table, we can see that the open channel $D\bar{D}^{*}$ ($S$- and $D$-wave of the relative motion between $D$ and $\bar{D}^{*}$ are all considered) makes the largest contribution, and pushes the bare mass of $\chi_{c1}(2P)$ down 76.37 MeV. The $D$-wave $D^{*}\bar{D}^{*}$ also pushes the bare mass down 34.15 MeV. The mass shifts from the states $D_{s}\bar{D}_{s}^{*}$ and $D^{*}_{s}\bar{D}_{s}^{*}$ are small. At last the four-quark components make mass shifts $-125.37$ MeV to the state $\chi_{c1}(2P)$, and the unquenched mass of the state is 3862.80 MeV, which is close to mass of $X(3872)$. So to identify $X(3872)$ as $\chi_{c1}(2P)$ charmonium state in the unquenched quark model is possible.

If we wang to reproduce the mass of the state $X(3872)$ exactly, we can fine tune the model parameters related to the charm quark, and keep the light meson sector unchanged. To justify the fine-tuning, the charmonia $\eta_{c}$ and $J/\psi$ are also calculated in the unquenched quark model. The adjusted parameters are listed in Table 5 with the original parameters and the parameters adjusted in Ref. 29chen:2018 . From the table, we can see that the adjustment is small, less than 5% except the energy shift $\Delta$.

The results with new parameters for charmonia, $\eta_{c}$, $J/\psi$ and $\chi_{c1}(2P)$ are shown in Table 6. The masses of $\eta_{c}$ and $J/\psi$ are fitted by adjusting the parameters, so the experimental data are reproduced well, The dominant component is $c\bar{c}$, over 90%. In this case the calculated unquenched mass of $\chi_{c1}(2P)$ is 3871.7 MeV, almost the experimental value of $X(3872)$. In our calculation, the state is the mixing one of $c\bar{c}$ and four-quark components. The dominant component is still $c\bar{c}$, $\sim 70\%$, the fraction of $D\bar{D}^{*}+D^{*}\bar{D}$ is around 22.5%, $D^{*}\bar{D}^{*}$, 5.6% and $D_{s}^{(*)}\bar{D}_{s}^{(*)}$, 2.4%. The results are consistent with some previous work Achasov ; 19Ferretti:2013 ; Kang

VI Summary

To describe the ordinary meson and exotic meson in one framework, an unquenched quark model is developed. As a preliminary work, only four-quark components are taken into account, and the four-quark components are limited to meson-meson states. To related the valence part to the high Fock components, the transition operator is needed. Here the transition operator of ${}^{3}P_{0}$ model with modification is employed. The modification consists of two parts. One considers the fact that the creation probability will decrease when the energy of created quark-pair increases. Another requires that the created quark cannot be far away from the valence quark-pair. To minimize the error from the calculation, a powerful method dealing with few-body system, GEM is used to find all the wavefunctions needed. The unquenched quark model has been applied to the light meson spectrum, a reasonable result was obtained. All the mass shifts are around 15% of the bare masses of the states. In this way, the success of valence quark model in describing low-lying spectrum of meson is kept. The present work applies the model to the charmonium states, trying to explain the exotic state $X(3872)$.

By keeping the model parameters related to light meson unchanged, and fine-tuning the parameters related to charm quark, we get the unquenched mass of $\chi_{c1}(2P)$ very close to the experimental value of $X(3872)$. At the same time, the masses of charmonia, $\eta_{c}$ and $J/\psi$ are reproduced well. In our UQM, the high Fock components of ground state charmonium $\eta_{c}$ and $J/\psi$ are small, less than 10%. Similar results have been obtained for the light mesons 29chen:2018 . For the states which can strong decay to two mesons, the probability fractions of two-meson continua will be large, for example, the $\pi\pi$ continua in $\rho$-meson. Here for $X(3872)$, similar results are obtained. The fraction of two-meson continua is around 30%. However, the dominant component of $X(3872)$ is still $c\bar{c}$, 70%.

From our calculation, the unquenched quark model is a promising phenomenological method to unify the description of ordinary mesons and exotic mesons. Of course further improvements are still needed, the four-quark components may be hidden-color states or diquark-antidiquark states. To give a realistic description of all mesons, adjustment of model parameters is expected. These are all our further work.