Are the $X$(4160) and $X$(3915) charmonium states?

TL;DR

This paper investigates whether the newly observed X(4160) and X(3915) can be explained as charmonium states by analyzing their mass spectra and decay widths using quark models and the $^3P_0$ decay model.

Contribution

It provides a detailed analysis suggesting X(4160) is likely the charmonium state $ ext{η}_{c2}(2^1D_2)$, while X(3915) is unlikely to be $ ext{χ}_0(2^3P_0)$ based on decay width calculations.

Findings

X(4160) compatible with η_{c2}(2^1D_2) charmonium state

X(3915) unlikely to be χ_0(2^3P_0) due to decay width mismatch

Decay width calculations align with experimental data for X(4160) hypothesis

Abstract

Inspired by the newly observed $X$(4160) and $X$(3915) states, we analyze the mass spectrum of these states in different quark models and calculate their strong decay widths by the $^3P_0$ model. According to the mass spectrum of charmonium states predicted by the potential model, the states $\chi_0(3^3P_0), \chi_1(3^3P_1), \eta_{c2}(2^1D_2), \eta_c(4^1S_0)$ all can be candidates for the $X$(4160). However, only the decay width of the state $\eta_{c2}(2^1D_2)$ in our calculation is in good agreement with the data reported by Belle and the decay of $\eta_{c2}(2^1D_2)\to D\bar{D}$, which is not seen in experiment, is also forbidden. Therefore, it is reasonable to interpret the charmonium state $\eta_{c2}(2^1D_2)$ as the state X(4160). For the state X(3915), although the mass of $\chi_0(2^3P_0)$ is compatible with the experimental value, the calculated strong decay width is much larger than experimental data. Hence, the assignment of X(3915) to charmonium state $\chi_0(2^3P_0)$ is disfavored in our calculation.

Figures

Tables

Table 1. Table 1: Theoretical mass spectrum of the charmonium candidates for the X ​ ( 4160 ) 𝑋 4160 X(4160) and X ​ ( 3915 ) 𝑋 3915 X(3915) . The mass are in units of MeV. The results are taken from Ref. prd79094004 with color screening potential model, and Ref. prd72054026 including Nonrelativistic potential and Godfrey-Isgur relativized potential model.

State	${\chi }_0\left(2^3{P}_0\right)$	${\eta }_c\left(4^1{S}_0\right)$	${\chi }_0\left(3^3{P}_0\right)$	${\chi }_1\left(3^3{P}_1\right)$	${\eta }_{c2}\left(2^1{D}_2\right)$
$J^{PC}$	$0^{++}$	$0^{-+}$	$0^{++}$	$1^{++}$	$2^{-+}$
Ref.prd79094004 SCR	3842	4250	4131	4178	4099
Ref.prd72054026 NR	3852	4384	4202	4271	4158
Ref.prd72054026 GI	3916	4425	4292	4317	4208

Table 2. Table 2: The OZI rule and phase space allowed open-charm strong decay modes of the possible charmonium states for the X(4160) and X(3915).

State	$J^{PC}$	Decay mode	Decay channel
${\eta }_c\left(4^1{S}_0\right)$	$0^{-+}$	$0^{-}+1^{-}$	$D{\overline{D}}^{\ast },D_s^{+}{D}_s^{\ast -}$
		$1^{-}+1^{-}$	$D^{\ast }{\overline{D}}^{\ast }$
${\chi }_0\left(3^3{P}_0\right)$	$0^{++}$	$0^{-}+0^{-}$	$D\overline{D},D_s^{+}{D}_s^{-}$
		$1^{-}+1^{-}$	$D^{\ast }{\overline{D}}^{\ast }$
${\chi }_1\left(3^3{P}_1\right)$	$1^{++}$	$0^{-}+1^{-}$	$D{\overline{D}}^{\ast },D_s^{+}{D}_s^{\ast -}$
		$1^{-}+1^{-}$	$D^{\ast }{\overline{D}}^{\ast }$
${\eta }_{c2}\left(2^1{D}_2\right)$	$2^{-+}$	$0^{-}+1^{-}$	$D{\overline{D}}^{\ast },D_s^{+}{D}_s^{\ast -}$
		$1^{-}+1^{-}$	$D^{\ast }{\overline{D}}^{\ast }$
${\chi }_0\left(2^3{P}_0\right)$	$0^{++}$	$0^{-}+0^{-}$	$D\overline{D}$

Table 3. Table 3: The partial wave amplitude for the strong decays of relevant charmonium state. The element of flavor matrix ⟨ ϕ B 14 ​ ϕ C 32 | ϕ A 12 ​ ϕ 0 34 ⟩ = 1 / 3 inner-product subscript superscript italic-ϕ 14 𝐵 subscript superscript italic-ϕ 32 𝐶 subscript superscript italic-ϕ 12 𝐴 subscript superscript italic-ϕ 34 0 1 3 \langle\phi^{14}_{B}\phi^{32}_{C}|\phi^{12}_{A}\phi^{34}_{0}\rangle=1/\sqrt{3} in present work. We take ℰ = γ ​ E A ​ E B ​ E C ℰ 𝛾 subscript 𝐸 𝐴 subscript 𝐸 𝐵 subscript 𝐸 𝐶 \mathcal{E}=\gamma\sqrt{E_{A}E_{B}E_{C}} in this table.

State	decay channel	Decay amplitude
${\eta }_c\left(4^1{S}_0\right)$	$0^{-}+1^{-}$	${ℳ}^{11}=\frac{\sqrt{2}}{3}ℰ{ℐ}^{00}$
	$1^{-}+1^{-}$	${ℳ}^{11}=\frac{2}{3}ℰ{ℐ}^{00}$
${\chi }_0\left(3^3{P}_0\right)$	$0^{-}+0^{-}$	${ℳ}^{00}=\frac{\sqrt{2}}{3\sqrt{3}}ℰ\left({ℐ}^{00}-2{ℐ}^{\pm }\right)$
	$1^{-}+1^{-}$	${ℳ}^{00}=\frac{\sqrt{2}}{9}ℰ\left({ℐ}^{00}-2{ℐ}^{\pm }\right)$
		${ℳ}^{22}=\frac{4}{9}ℰ\left({ℐ}^{00}+{ℐ}^{\pm }\right)$
${\chi }_1\left(3^3{P}_1\right)$	$0^{-}+1^{-}$	${ℳ}^{10}=\frac{2}{9}ℰ\left({ℐ}^{00}-2{ℐ}^{\pm }\right)$
		${ℳ}^{12}=\frac{\sqrt{2}}{9}ℰ\left({ℐ}^{00}+{ℐ}^{\pm }\right)$
	$1^{-}+1^{-}$	${ℳ}^{22}=\frac{2}{3\sqrt{3}}ℰ\left({ℐ}^{00}+{ℐ}^{\pm }\right)$
${\eta }_{c2}\left(2^1{D}_2\right)$	$0^{-}+1^{-}$	${ℳ}^{11}=\frac{2}{15}ℰ\left(\sqrt{3}{ℐ}^{\pm }-{ℐ}^{00}\right)$
	$1^{-}+1^{-}$	${ℳ}^{11}=\frac{2\sqrt{2}}{15}ℰ\left(\sqrt{3}{ℐ}^{\pm }-{ℐ}^{00}\right)$
${\chi }_0\left(2^3{P}_0\right)$	$0^{-}+0^{-}$	${ℳ}^{00}=\frac{\sqrt{2}}{3\sqrt{3}}ℰ\left({ℐ}^{00}-2{ℐ}^{\pm }\right)$

Table 4. Table 4: The parameters relevant to the two-body strong decays of the charmonium state in the P 0 3 superscript subscript 𝑃 0 3 {}^{3}P_{0} model.

State	Mass (MeV) PDG	$R$ (GeV-1) Godfrey
$D$	$1869.62(\pm )$  1864.84(0)	1.52
$D^{\ast }$	$2021.27(\pm )$  2006.97(0)	1.85
$D_s$	$1968.49(\pm )$	1.41
$D_s^{\ast }$	$2112.3(\pm )$	1.69

Equations

$${\mathcal{B}}_{D\bar{D}}(X(4160))/{\mathcal{B}}_{D^{\ast }{\bar{D}}^{\ast }}(X(4160))<0.09,$$

$${\mathcal{B}}_{D^{\ast }\bar{D}}(X(4160))/{\mathcal{B}}_{D^{\ast }{\bar{D}}^{\ast }}(X(4160))<\mathrm{0.22.}$$

$$M=3914\pm 4\pm 2\text{MeV},\Gamma =28\pm 12_{-8}^{+2}\text{MeV}.$$

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

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

$$|A(n_A{}^{2S_A+1}{L}_A\text{mbox}{}_{J_A{M}_{J_A}})({\mathbf{P}}_A)⟩$$

$$⟨A(n_A{}^{2S_A+1}{L}_A\text{mbox}{}_{J_A{M}_{J_A}})({\mathbf{P}}_A)|A(n_A{}^{2S_A+1}{L}_A\text{mbox}{}_{J_A{M}_{J_A}})({{\mathbf{P}}^{\prime }}_A)⟩=2E_A{\delta }^3({\mathbf{P}}_A-{{\mathbf{P}}^{\prime }}_A).$$

$$⟨BC|S|A⟩=I-2\pi i\delta (E_A-E_B-E_C)⟨BC|T|A⟩,$$

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

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

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

$$⟨{\chi }_{S_B{M}_{S_B}}^{14}{\chi }_{S_C{M}_{S_C}}^{32}|{\chi }_{S_A{M}_{S_A}}^{12}{\chi }_{1-m}^{34}⟩=$$

$$\sum _{S,M_S}⟨S_B{M}_{S_B}{S}_C{M}_{S_C}|S{M}_S⟩⟨S_A{M}_{S_A}1-m|S{M}_S⟩$$

$$\times (-1{)}^{S_C+1}\sqrt{3(2S_A+1)(2S_B+1)(2S_C+1)}$$

$$\displaystyle\times\left\{\begin{array}[]{ccc}\frac{1}{2}&\frac{1}{2}&S_{A}\\ \frac{1}{2}&\frac{1}{2}&1\\ S_{B}&S_{C}&S\end{array}\right\}.$$

$${\Psi }_{nL{M}_L}(\mathbf{p})=(-1{)}^n(-i{)}^L{R}^{L+\frac{3}{2}}\sqrt{\frac{2n!}{\Gamma (n+L+\frac{3}{2})}}$$

$$\times \exp \left(-\frac{R^2{p}^2}{2}\right)L_n^{L+\frac{1}{2}}\left(R^2{p}^2\right){\mathcal{Y}}_{L{M}_L}(\mathbf{p}),$$

$$\displaystyle\Gamma=\pi^{2}\frac{|\mathbf{P}|^{2}}{M_{A}^{2}}\frac{1}{2J_{A}+1}\sum_{\begin{array}[t]{l}\scriptstyle M_{J_{M_{A}}},M_{J_{M_{B}}},\\ \scriptstyle\quad M_{J_{M_{C}}}\end{array}}\Big{|}\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}\Big{|}^{2}\,.$$

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

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

$$\displaystyle\Gamma=\pi^{2}\frac{{|\textbf{P}|}}{M_{A}^{2}}\sum_{JL}\Big{|}\mathcal{M}^{JL}\Big{|}^{2},$$

$$|\text{P}|=\frac{\sqrt{[M_A^2-(M_B+M_C{)}^2][M_A^2-(M_B-M_C{)}^2]}}{2M_A},$$

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

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

$$\times \exp \left(-\frac{R^{-2}{r}^2}{2}\right)r^L{L}_n^{L+\frac{1}{2}}\left(R^{-2}{r}^2\right).$$

$$\frac{\mathcal{B}({\eta }_c(4^1{S}_0)\to D{\bar{D}}^{\ast })}{\mathcal{B}({\eta }_c(4^1{S}_0)\to D^{\ast }{\bar{D}}^{\ast })}=\mathrm{1.25.}$$

$$\frac{\mathcal{B}({\eta }_{c2}(2^1{D}_2)\to D{\bar{D}}^{\ast })}{\mathcal{B}({\eta }_{c2}(2^1{D}_2))\to D^{\ast }{\bar{D}}^{\ast })}=1.4\sim 0.76$$

$$I^{\pm }=I^{1-1}=I^{-11}$$

$$=i\frac{\sqrt{6}}{\sqrt{5}{\pi }^{5/4}{\Delta }^7}\left(R_A^{5/2}{R}_B^{3/2}{R}_C^{3/2}\right)\exp \left(-\frac{1}{2}{\zeta }^2{\text{P}}^2\right)(10R_A^2+{\Delta }^2(-5+2{\text{P}}^2{R}_A^2(1+\lambda {)}^2))$$

$$I^{00}=-i\frac{\sqrt{6}}{\sqrt{5}{\pi }^{5/4}{\Delta }^7}\left(R_A^{5/2}{R}_B^{3/2}{R}_C^{3/2}\right)\exp \left(-\frac{1}{2}{\zeta }^2{\text{P}}^2\right)$$

$$(10R_A^2+{\Delta }^2(-5+{\text{P}}^2(1+\lambda )(-5{\Delta }^2\lambda +2R_A^2(3+\lambda (8+{\Delta }^2{\text{P}}^2(1+\lambda {)}^2))))).$$

$$I^{\pm }=I^{1-1}=I^{-11}$$

$$=\frac{2\sqrt{3}}{\sqrt{7}{\pi }^{5/4}{\Delta }^7}\left(R_A^{7/2}{R}_B^{3/2}{R}_C^{3/2}\right)\exp \left(-\frac{1}{2}{\zeta }^2{\text{P}}^2\right)\text{P}(1+\lambda )(14R_A^2+{\Delta }^2(-7+2{\text{P}}^2{R}_A^2(1+\lambda {)}^2))$$

$$I^{00}=-\frac{2}{\sqrt{7}{\pi }^{5/4}{\Delta }^7}\left(R_A^{7/2}{R}_B^{3/2}{R}_C^{3/2}\right)\exp \left(-\frac{1}{2}{\zeta }^2{\text{P}}^2\right)\text{P}(1+\lambda )(28R_A^2+{\Delta }^2(-14+{\text{P}}^2(1+\lambda )$$

$$(-7{\Delta }^2\lambda +2R_A^2(4+\lambda (11+{\Delta }^2{\text{P}}^2(1+\lambda {)}^2))))).$$

$$I^{\pm }=I^{1-1}=I^{-11}$$

$$=i\frac{\sqrt{3}}{\sqrt{70}{\pi }^{5/4}{\Delta }^9}\left(R_A^{5/2}{R}_B^{3/2}{R}_C^{3/2}\right)\exp \left(-\frac{1}{2}{\zeta }^2{\text{P}}^2\right)(140R_A^4+28{\Delta }^2{R}_A^2(-5+2{\mathbf{P}}^2{R}_A^2(1+\lambda {)}^2)$$

$$+{\Delta }^4(35-28{\mathbf{P}}^2{R}_A^2(1+\lambda {)}^2+4{\mathbf{P}}^4{R}_A^4(1+\lambda {)}^4))$$

Full text

I Introduction

Many new charmonium like states, the so-called $XYZ$ mesons, have been reported by Belle and BaBar collaborations in recent years. Some of these states can be understood as conventional mesons that are comprised of only pure $c\bar{c}$ quark pair. However, most of the $XYZ$ states do not match well the mass spectrum of $c\bar{c}$ predicted by the QCD-motivated potential models. By considering the effects of virtual mesons loop Eichtenprd17 ; Eichtenprd73 ; Barnesprc77 ; chaoprd80 and color screening prd79094004 , the masses of some excited charmonium states are smaller than it calculated by conventional quark model. Therefore, some $XYZ$ states Eichtenprd73 may be still compatible with the mass spectrum of charmonium. However, the state $X(3872)$ Eichtenprd73 ; chaoprd80 ; Suzuki72 is probably the most robust of all the charmonium like objects.

Last year, Belle collaborations reported a new charmonium like state, the $X(4160)$ X4160 , in the processes $e^{+}e^{-}\to J/\psi D^{(*)}\bar{D}^{(*)}$ with a significance of 5.1$\sigma$. It has the mass $M=4156^{+25}_{-20}\pm 15$ MeV, and width $\Gamma=139^{+111}_{-61}\pm 21$ MeV. Based on the the processes $e^{+}e^{-}\to J/\psi D\bar{D}$, $e^{+}e^{-}\to J/\psi D^{*}\bar{D}$, and $e^{+}e^{-}\to J/\psi D^{*}\bar{D}^{*}$, The upper limits of the branch ratios of $X(4160)$ are given as,

[TABLE]

The $X(4160)$ has possible charge parity $C=+$ mostly, since the photon $\gamma$ and $J/\psi$ have $J^{PC}=1^{--}$, and $e^{+}e^{-}\to\gamma\to J/\psi X(4160)$ is a main process. Hence the $X(4160)$ can have $J^{PC}=0^{-+}$, $0^{++}$, $1^{-+}$, $2^{-+}$, $1^{++},~{}2^{++},\ldots$. In Ref.chaox4160 , Chao discussed the possible interpretation of the $X(4160)$ in view of production rate in $e^{+}e^{-}\to J/\psi X(4160)$. He believes that the charmonium states $4^{1}S_{0},~{}3^{3}P_{0}$ may be assigned to the state $X(4160)$ by analogy with the cross section of $e^{+}e^{-}\to J/\psi\eta_{c}(1S)(\eta_{c}(2S)\chi_{c0}(1P))$, while the $2^{1}D_{2}$ chao77014002 can not be rule out. According to the mass spectrum prd79094004 predicted by the potential model with color screening, Li and Chao also give some arguments about the $\chi_{0}(3^{3}P_{0})$ as an assignment for the $X(4160)$.

Using the vector-vector interaction within the framework of the hidden gauge formalism, Molina and Oset Oset0907 suggested that the $X(4160)$ is a molecular state of $D_{s}^{*}\bar{D}_{s}^{*}$ with $J^{PC}=2^{++}$.

Very recently, Refs.Olsen0909 ; CZYuan0910 ; Zupanc0910 ; Gordfrey0910 reported the newest charmonium like state, the $X(3915)$, which is observed by Belle in $\gamma\gamma\to\omega J/\psi$ with a statistical significance of 7.5$\sigma$. It has the mass and width

[TABLE]

Belle collaborations determine the $X(3915)$ production rate $\Gamma_{\gamma\gamma}(X(3915))$ $\mathcal{B}(X(3915)\to\omega J/\psi)=69\pm 16^{+7}_{-18}$ eV and $\Gamma_{\gamma\gamma}(X(3915))$ $\mathcal{B}(X(3915)\to\omega J/\psi)=21\pm 4^{+2}_{-5}$ eV for $J^{P}=0^{+}$ or $2^{+}$, respectively. Because the partial width of this state to $\gamma\gamma$ or $\omega J/\psi$ is too large, it is very unlikely to be a charmonium state analyzed by Yuan CZYuan0910 .

The $X(3915)$ also has the charge parity $C=+$, because it is observed in the process of $\gamma\gamma\to\omega J/\psi$. In Ref.XiangLiu , Liu et al. argued that the $\chi_{0}(2^{3}P_{0})$ can be assigned to the $X(3915)$ if taking $R=1.8\sim 1.85$ GeV*-1* in the SHO (the simple harmonic oscillator wave functions).

Up to now, the interpretation of the $X(4160)$ and $X(3915)$ is still unclear. The states $\chi_{0}(3^{3}P_{0})$, $\chi_{1}(3^{3}P_{1})$, $\eta_{c2}(2^{1}D_{2})$ listed in Table 1 all can be interpreted as the $X(4160)$ just on mass level. Which charmonium state is an assignment for the $X(4160)$? One can answer this question in different ways. We study the $X(4160)$ and $X(3915)$ via strong decay by the ${}^{3}P_{0}$ model NPB10521 ; Yaouanc ; PRD546811 ; Roberts1 in this work. In following discussion, we take the $\chi_{0}(3^{3}P_{0}),~{}\chi_{1}(3^{3}P_{1}),~{}\eta_{c2}(2^{1}D_{2}),~{}\eta_{c}(4^{1}S_{0})$ and $\chi_{0}(2^{3}P_{0})$ as candidates of the $X(4160)$ and $X(3915)$, respectively.

The paper is organized as follows. In the next section we take a review of the ${}^{3}P_{0}$ model. Sect. III devotes to discuss the possible strong decay channels and gives the corresponding amplitudes of the candidates for the $X(4160)$ and $X(3915)$. In Sect. IV we present and analyze the results obtained by the ${}^{3}P_{0}$ model. Finally, the summary of the present work is given in the last section.

II A review of the ${}^{3}P_{0}$ model of meson decay

Fig.1 The two possible diagrams contributing to $A\rightarrow B+C$ in the ${}^{3}P_{0}$ model.

The ${}^{3}P_{0}$ decay model, also known as the Quark-Pair Creation model (QPC), was originally introduced by MicuNPB10521 and further developed by Le Yaouanc, Ackleh, Roberts et al.Yaouanc ; PRD546811 ; Roberts1 . It is applicable to OZI (Okubo, Zweig and Iizuka) rule allowed strong decays of a hadron into two other hadrons, which are expected to be the dominant decay modes of a hadron. Due to the ${}^{3}P_{0}$ model gives a good description of many observed partial widths of the hadrons, it has been widely used to evaluate the strong decays of mesons and baryons composed of $u,~{}d,~{}s,~{}c,~{}b$ quarks Close ; XiangLiu ; Swanson ; zhusl ; XiangLiu2 ; prd342809 ; prd506855 ; Blundell ; prd35907 ; prd554157 ; prd68054014 ; prd62014011 ; ldm . The ${}^{3}P_{0}$ model of strong decays assumes that quark-antiquark pair are created with vacuum quantum number $J^{PC}=0^{++}$NPB10521 . The diagrams of all possible decay process $A\rightarrow B+C$ of meson are shown in Fig.1. In many cases only one of them contributes to the strong decay of meson.

The transition operator of this model takes

[TABLE]

where $\gamma$, which is a dimensionless parameter, represents the probability of the quark-antiquark pair created from the vacuum and can be extracted by fitting observed experimental data. $\phi^{34}_{0}=(u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}$,   $\omega^{34}_{0}=(R\bar{R}+G\bar{G}+B\bar{B})/\sqrt{3}$ are flavor and color singlet state, respectively. $\chi_{{1,-m}}^{34}$ is a spin-triplet state. ${\cal{Y}}^{m}_{l}(\mathbf{p})\equiv|p|^{l}Y^{m}_{l}(\theta_{p},\phi_{p})$ is the $l$th solid harmonic polynomial that reflects the momentum-space distribution of the created quark-antiquark pair. $b^{\dagger}_{3}(\mathbf{p}_{3})$, $d^{\dagger}_{4}(\mathbf{p}_{4})$ are the creation operators of the quark and antiquark, respectively.

In general, the mock state is adopted to describe the meson with the spatial wave function $\psi_{n_{A}L_{A}M_{L_{A}}}(\mathbf{p}_{1},\mathbf{p}_{2})$ in the momentum representation mock .

[TABLE]

with the normalization conditions

[TABLE]

where $n_{A}$ represent the radial quantum number of the meson $A$ composed of $q_{1},~{}\bar{q}_{2}$ with momentum $\mathbf{p_{1}}$ and $\mathbf{p_{2}}$. $E_{A}$ is the total energy, $\mathbf{P}_{A}$ is the momentum of the meson $A$ and $\mathbf{p}_{A}=(m_{1}\mathbf{p}_{1}-m_{1}\mathbf{p}_{2})/(m_{1}+m_{2})$ is the relative momentum between quark and antiquark. $\mathbf{S}_{A}=\mathbf{s}_{q_{1}}+\mathbf{s}_{q_{2}}$, $\mathbf{J}_{A}=\mathbf{L}_{A}+\mathbf{S}_{A}$ stand for the total spin and total angular momentum, respectively. $\mathbf{L}_{A}$ is the relative orbital angular momentum between $q_{1}$ and $\bar{q}_{2}$. $\langle L_{A}M_{L_{A}}S_{A}M_{S_{A}}|J_{A}M_{J_{A}}\rangle$ denotes a Clebsch-Gordan coefficient, and $\chi^{12}_{S_{A}M_{S_{A}}}$, $\phi^{12}_{A}$ and $\omega^{12}_{A}$ are the spin, flavor and color wave functions, respectively.

The $S$-matrix of the process $A\rightarrow B+C$ is defined by

[TABLE]

with

[TABLE]

where ${\cal{M}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}$ is the helicity amplitude of $A\rightarrow B+C$. In the center of mass frame of meson $A$, $\mathbf{P}_{A}=0$, and ${\cal{M}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}$ can be written as

[TABLE]

with the momentum space integral,

[TABLE]

where $\mathbf{P}=\mathbf{P}_{B}=-\mathbf{P}_{C}$, $\mathbf{p}=\mathbf{p}_{3}$, $m_{3}$ is the mass of the created quark $q_{3}$; $\langle\chi^{14}_{S_{B}M_{S_{B}}}\chi^{32}_{S_{C}M_{S_{C}}}|\chi^{12}_{S_{A}M_{S_{A}}}\chi^{34}_{1-m}\rangle$ and $\langle\phi^{14}_{B}\phi^{32}_{C}|\phi^{12}_{A}\phi^{34}_{0}\rangle$ are the overlap of spin and flavor wave function, respectively.

The spin overlap in terms of Winger’s $9j$ symbol can be given by

[TABLE]

Generally, one takes the simple harmonic oscillator (SHO) approximation for the meson space wave functions in Eq. (11). In momentum-space, the SHO wave function reads

[TABLE]

with $\mathcal{Y}_{LM_{L}}(\mathbf{p})=|\mathbf{p}|^{L}Y_{LM_{L}}(\Omega_{p})$. Here $R$ denotes the SHO wave function scale parameter; $\mathbf{p}$ represents the relative momentum between the quark and the antiquark within a meson; $L^{L+\frac{1}{2}}_{n}\left(R^{2}p^{2}\right)$ is an associated Laguerre polynomial.

The decay width for the process $A\to B+C$ in terms of the helicity amplitude is

[TABLE]

For comparing with experiments, ${\cal{M}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}(\mathbf{P})$ can be converted into the partial amplitude via the Jacob-Wick formula Jacob

[TABLE]

where $\mathbf{J}=\mathbf{J}_{B}+\mathbf{J}_{C}$, $\mathbf{J}_{A}=\mathbf{J}_{B}+\mathbf{J}_{C}+\mathbf{L}$ ,and $M_{J_{A}}=M_{J_{B}}+M_{J_{C}}$. Then the decay width in terms of the partial wave amplitude is taken as,

[TABLE]

where $|\textbf{P}|$, as mentioned above, is the three momentum of the outgoing meson in the rest frame of meson $A$. According to the calculation of 2-body phase space, one can get

[TABLE]

where $M_{A}$, $M_{B}$, and $M_{C}$ are the masses of the meson $A$, $B$, and $C$, respectively.

III The possible strong decay channels and amplitudes of the candidates for the $X(4160)$ and $X(3915)$

As analyzed in section I, we consider the $\eta_{c}(4^{1}S_{0})$, $\chi_{0}(3^{3}P_{0})$, $\chi_{1}(3^{3}P_{1})$, $\eta_{c2}(2^{1}D_{2})$ as the possible candidates of the $X(4160)$, and assume that the upper limit of the mass is 4156 MeV observed by Belle. For the X(3915), one chooses charmonium state $\chi_{0}(2^{3}P_{0})$ with mass 3916 MeV. According to the ${}^{3}P_{0}$ model discussed in the above section, the OZI rule allows open-charm strong decay and corresponding amplitudes of possible charmonium states are listed in Tables 2 and 3. We replace $\mathcal{I}^{+1-1}_{0,0},\mathcal{I}^{-1+1}_{0,0}$ with $\mathcal{I}^{\pm}$ and $\mathcal{I}^{0,0}_{0,0}$ with $\mathcal{I}^{0,0}$ in Table 3, respectively. The details of the spatial integral about $\mathcal{I}^{\pm}(\mathbf{P})$ and $\mathcal{I}^{0,0}(\mathbf{P})$ are given in the Appendix.

IV Numerical results and discussion

There are several parameters should be input to calculate the strong decay in the ${}^{3}P_{0}$ model. In the present work, the masses of constituent quarks are taken as $m_{u}=m_{d}=0.22$ GeV, $m_{s}=0.419$ GeV, $m_{c}=1.6$ GeV Godfrey . The strength of quark pair creation $\gamma=6.95$ has been adopted by many literatures Blundell ; zhusl , which is fitted by strong decay of light-, charmonium-, open charmed-mesons and baryons observed by experiments. The value of $\gamma$ is higher than that used in Ref. isgur2 by a factor of $\sqrt{96\pi}$ due to different field theory conventions. The strength of $s\bar{s}$ creation satisfies $\gamma_{s}=\gamma/\sqrt{3}$ Yaouanc2 . Refs.XiangLiu ; zhusl ; XiangLiu2 also take this value to study the strong decay of charmonium, heavy-light meson and heavy baryons. In this work, we take these parameters for calculation as well. The $R$ values of $D,~{}D^{*},~{}D_{s},~{}D_{s}^{*}$ in the SHO are shown in Table 4, which are obtained by the calculation of the nonrelativistic quark model with Coulomb item, linear confinement and smeared hyperfine interactions.

First of all, we study the strong decay of the $\chi_{0}(3^{3}P_{0})$ which is discussed by Chao and Li in Refs.chaox4160 ; prd79094004 from the production process of $e^{+}e^{-}\to J/\psi+X(4160)$ and the mass spectrum is obtained by the potential model with color screening. Using the method of Numerov algorithm Koonin , we also obtain the mass 4149 MeV by the same potential and parameters in Ref. prd79094004 . Usually, the width of strong decay is sensitive Close ; XiangLiu ; Swanson ; zhusl ; Blundell ; ldm to the $R$ value in the SHO. Here the reasonable value of $R$ is obtained by fitting the wave function obtained by solving the schrödinger equation prd79094004 .

Through the Fourier transform, the Eq. (16) turns into

[TABLE]

with the radial wave function

[TABLE]

The wave function $u(r)=r~{}R_{nL}(r)$ of charmonium state $3P$ is shown in Fig.2. Using Eq.(21) to fit the wave function got by Numerov algorithm method (the wave function is denoted as ’NAWF’ in the following), we can get the $R=2.5\sim 2.98$ GeV*-1*.

Fig.2 The wave function of charmonium state $3P$.

The $\chi_{0}(3^{3}P_{0})$ has decay channels of $0^{++}\to 0^{-}+0^{-}$ with $S$-wave and $0^{++}\to 1^{-}+1^{-}$ with $S$-,$~{}D$-wave, while the $0^{++}\to 0^{-}+1^{-}$ is forbidden. Therefore, it can decay into $D\bar{D},~{}D_{s}D_{s},~{}D^{*}\bar{D}^{*}$, which are allowed by the phase space. In Fig.3, we show the dependence of the partial widths of the strong decay of the $\chi_{0}(3^{3}P_{0})$ on the $R_{A}$. Taking $R_{A}=2.5\sim 2.98$ GeV*-1* discussed above, the total width ranges from 105 to 143 MeV which falls in the range of experimental data. However, the dominate contribution comes from the $\chi_{0}(3^{3}P_{0})\to DD$ which is inconsistent with the experimental result. So the assignment of the charmonium state $\chi_{0}(3^{3}P_{0})$ to the $X(4160)$ is disfavored.

Fig.3 The possible strong decay of the $\chi_{0}(3^{3}P_{0})$.

The $\eta_{c}(4^{1}S_{0})$ is mostly like the $X(4160)$ for it has high production cross sections in the process of $e^{+}e^{-}\to J/\psi+X(4160)$ discussed by Chao chaox4160 . However, it is difficult to understand why the predicted mass 4250 MeV prd79094004 , 4384, 4425 MeV prd72054026 are much higher than 4156 MeV. By considering the effect of the meson loops our , the mass may be lower than that of Refs.prd79094004 ; prd72054026 . Here, we assume the mass of the $\eta_{c}(4^{1}S_{0})$ is 4156 MeV. The main decay channels of the $\eta_{c}(4^{1}S_{0})$ are $0^{-+}\to 0^{-}+1^{-}$ and $0^{-+}\to 1^{-}+1^{-}$ with $P$-wave between outgoing mesons. Obviously, the $0^{-+}\to 0^{-}+0^{-}$ is forbidden. The decay width of main decay channels are shown in Fig.4. The total width can only reach up to about 25 MeV with $R_{A}$ around 2.9 GeV, which is obtained by fitting to NAWF of the $\eta_{c}(4^{1}S_{0})$. It is about 3 times smaller than the lower limit of the experimental result of the $X(4160)$. Since the results of some hadron states predicted by the ${}^{3}P_{0}$ model may be a factor of $2\sim 3$ off the experimental width due to inherent uncertainties of this model NPB10521 ; Yaouanc ; PRD546811 ; Roberts1 ; Blundell , the assignment of the X(4160) to the $\eta_{c}(4^{1}S_{0})$ cannot be excluded. The ratio of main decay channel $D\bar{D}^{*},~{}D^{*}\bar{D}^{*}$ is

[TABLE]

It is much larger than the $0.22$ reported by Belle. If one takes the $\eta_{c}(4^{1}S_{0})$ as an assignment of X(4160), the precision measurement of the ratio between the width of the $D\bar{D}^{*}$ and $D^{*}\bar{D}^{*}$ is necessary in further experiment.

Fig.4 The possible strong decay of the $\eta_{c}(4^{1}S_{0})$.

Because the $\chi_{1}(3^{3}P_{1})$ has quantum number $J^{PC}=1^{++}$ and mass 4178 MeV, it is also a possible candidate of the $X(4160)$. $1^{++}\to 0^{-}+1^{-}$ and $1^{++}\to 1^{-}+1^{-}$ with $S$- and $D$-wave are the main decay channels of the $\chi_{1}(3^{3}P_{1})$. Fig.5 shows our results in the ${}^{3}P_{0}$ model. Taking $R_{A}=2.5\sim 2.98$ GeV*-1*, the total width is consistent with the range of the $X(4160)$. However, the dominant decay is $\chi_{1}(3^{3}P_{1})\to D\bar{D}^{*}$ while the decay width has only a few MeV for the $\chi_{1}(3^{3}P_{1})\to D^{*}\bar{D}^{*}$ channel, which is inconsistent with the experimental data. Therefore, regarding the $X(4160)$ as the $\chi_{1}(3^{3}P_{1})$ state is impossible.

Fig.5 The possible strong decay of the $\chi_{1}(3^{3}P_{1})$.

The another possible candidate of the $X(4160)$ is the charmonium state $\eta_{c2}(2^{1}D_{2})$. Firstly, it has quantum number $J^{PC}=2^{-+}$ and mass 4099 MeV prd79094004 , 4158 MeV prd72054026 which are compatible with the result of Belle. Secondly, the $\psi(4160)$PDG is known to be the good candidate of the $\psi(2^{3}D_{1})$ with $J^{PC}=1^{--}$, which is discussed in detail by Chao chaox4160 . So the $X(4160)$ may be the D-wave spin-singlet charmonium state ${}^{1}D_{2}(2D)$. Thirdly, $\eta_{c2}(2^{1}D_{2})$ decaying into $D\bar{D}$ is forbidden, and this decay is also not seen by Belle.

For the strong decay of the $\eta_{c2}(2^{1}D_{2})$, it has $2^{-+}\to 0^{-}+1^{-}$ and $2^{-+}\to 1^{-}+1^{-}$ decay channels with $P$-wave between outgoing mesons. In this case, final states $D\bar{D}^{*},~{}D_{s}\bar{D}_{s}^{*}$ and $D^{*}\bar{D}^{*}$ are phase space allowed. In Fig.6, we present the numerical results of main decay channels for the $\eta_{c2}(2^{1}D_{2})$. By fitting the NAWF of the $\eta_{c2}(2^{1}D_{2})$, we get $R_{A}=2.7\sim 3.0$ GeV*-1*. The total decay width of the $\eta_{c2}(2^{1}D_{2})$ falls in the range of the $X(4160)$ released by Belle. Taking the reasonable $R_{A}$ value of the SHO, the ratio of the main decay channel $D\bar{D}^{*},~{}D^{*}\bar{D}^{*}$ is

[TABLE]

and shown in Fig.7. However, the result is somewhat larger than the $\mathcal{B}_{D^{*}\bar{D}}(X(4160))/\mathcal{B}_{D^{*}\bar{D}^{*}}(X(4160))<0.22$ observed by Belle. We believe that to measure this ratio is very important since it is independent on the uncertain strength $\gamma$ of the quark pair creation from vacuum.

To sum up, the $\eta_{c2}(2^{1}D_{2})$ is a better candidate for the $X(4160)$ in the present calculation.

Fig.6 The possible strong decay of the $\eta_{c2}(2^{1}D_{2})$.

Fig.7 The ration of $\frac{\mathcal{B}(\eta_{c2}(2^{1}D_{2})\to D\bar{D}^{*})}{\mathcal{B}(\eta_{c2}(2^{1}D_{2}))\to D^{*}\bar{D}^{*})}$ with $R_{A}$ value of the SHO.

The $X(3915)$, which was observed by Belle in $\gamma\gamma\to\omega J/\psi$ with a statical significance of $7.5\sigma$ Olsen0909 , is the most recent addition to the collection of the $XYZ$ states. According to the Table 1 predicted by potential model, the excited charmonium state $\chi_{0}(2^{3}P_{0})$ is a good candidate for the $X(3915)$, due to it has mass $M=3914\pm 4\pm 2$ MeV and the possible quantum number is $J^{PC}=0^{++}$.

The $\chi_{0}(2^{3}P_{0})$ has only the strong decay channel $0^{++}\to 0^{-}+0^{-}$ allowed by phase space. The width of $\chi_{0}(2^{3}P_{0})\to D\bar{D}$ with $R_{A}$ of the SHO is presented in Fig.8. The total width ranges from 132 to 187 MeV with $R_{A}=2.3\sim 2.5$ GeV*-1* fitted to the NAWF of the $\chi_{0}(2^{3}P_{0})$. It is much larger than the $\Gamma=28\pm 12^{+2}_{-8}$ MeV reported by Refs. Olsen0909 ; CZYuan0910 ; Zupanc0910 . Therefore, the $X(3915)$ is unlikely to be the charmonium state $\chi_{0}(2^{3}P_{0})$ although the mass is compatible with the $X(3915)$.

Fig.8 The possible strong decay of the $\chi_{0}(2^{3}P_{0})$.

V Summery

In summary, we have discussed the possible interpretations of the $X(4160)$ observed by Belle collaborations in $e^{+}e^{-}\to J/\psi+X(4160)$ followed by $X(4160)\to D^{*}\bar{D}^{*}$. We also study the newest state $X(3915)$ observed by Belle in the process $\gamma\gamma\to J/\psi\omega$ Olsen0909 .

In quark models, the masses of the charmonium states: $\chi_{0}(3^{3}P_{0})$, $\chi_{1}(3^{3}P_{1})$, $\eta_{c2}(2^{1}D_{2})$ are all around 4156 MeV. By taking the effect of virtual mesons loop our into account, the $\eta_{c}(4^{1}S_{0})$ may also has mass around 4156 MeV. All the four states have charge parity $C=+$ which are compatible with the $X(4160)$ observed by Belle.

For the strong decay of the $\chi_{0}(3^{3}P_{0})$, the dominant strong decay is $\chi_{0}(3^{3}P_{0})\to D\bar{D}$ while $\chi_{0}(3^{3}P_{0})\to D^{*}\bar{D}^{*}$ contributes to the total width only a little in the reasonable $R$ in the SHO. It is contrast to the experimental result. Thus the excited charmonium state $\chi_{0}(3^{3}P_{0})$ disfavor the $X(4160)$.

The $\eta_{c}(4^{1}S_{0})$ can not decay into $D\bar{D}$ and may has high production rate chaox4160 in $e^{+}e^{-}\to J/\psi+\eta_{c}(4S)$ process by analogy with $e^{+}e^{-}\to J/\psi+\eta_{c}(1S)(\eta_{c}(2S)\chi_{c0}(1P))$. However, the total width in present work is lower than the experimental data of the $X(4160)$.

The main strong decay channel of the $\chi_{1}(3^{3}P_{1})$ is $D\bar{D}^{*}$ while $D^{*}\bar{D}^{*}$ is only a few MeV. It is inconsistent with the results of Belle. Therefore, taking the $\chi_{1}(3^{3}P_{1})$ as an assignment for the $X(4160)$ is impossible.

The $\eta_{c2}(2^{1}D_{2})$ can not decay to $D\bar{D}$ which is also not seen in the experiment. The total width of the $\eta_{c2}(2^{1}D_{2})$ match well with the data of the $X(4160)$ in our calculation. So, the $\eta_{c2}(2^{1}D_{2})$ is a good candidate for the $X(4160)$, for it is not only the mass but also the strong decay are well compatible with the results observed by Belle, although the excited charmonium state $\eta_{c}(4^{1}S_{0})$ can not be rule out as an assignment for the $X(4160)$.

We also give the ratio of $\frac{\mathcal{B}(\eta_{c2}(2^{1}D_{2})\to D\bar{D}^{*})}{\mathcal{B}(\eta_{c2}(2^{1}D_{2}))\to D^{*}\bar{D}^{*})}$ which is independent on the parameter $\gamma$ in the ${}^{3}P_{0}$ model. The numerical result is somewhat larger than the experimental data. Therefore, we suggest Belle, BaBar and other experimental collaborations to measure it to confirm this state.

By assuming the $X(3915)$ is the $\chi_{0}(2^{3}P_{0})$, the strong decay of the state is calculated. From our numerical results, we think this assumption is unacceptable. Due to the partial width of the $X(3915)$ to $\gamma\gamma$ or $\omega J/\psi$ is too large, Yuan CZYuan0910 also believes that it is very unlikely to be a charmonium state. Thus, It is necessary to do more study to understand the properties of the $X(3915)$.

Acknowledgements.

You-chang Yang would like to thank Xin Liu for useful discussion. The work is supported partly by the National Science Foundation of China under Contract No.10775072 and the Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20070319007, No. 1243211601028.

Appendix

The spatial overlap $\mathcal{I}_{M_{L_{B}},M_{L_{C}}}^{M_{L_{A}},m}(\textbf{P},m_{1},m_{2},m_{3})$ is simplified as $\mathcal{I}^{n^{\prime}m^{\prime}}(\textbf{P})$ in present work due to $M_{L_{B}}=M_{L_{C}}=0$. According to the Eq. (11), the concrete calculations of the integration are trivial after choosing the direction of P along $z$ axis Jacob . We list all expressions of $I^{\pm},I^{00}$ used in Table 3

In the case of $2P\to 1S+1S$

[TABLE]

For $2D\to 1S+1S$

[TABLE]

For $3P\to 1S+1S$

[TABLE]

For $4S\to 1S+1S$

[TABLE]

Here, The parameters $\Delta$, $\zeta$ and $\eta$ in Eqs. (24), (25), (26), (27) are defined as

[TABLE]

with

[TABLE]

Here $m_{1},m_{2}$ and $m_{3}$ denotes the mass of quark inside parent meson and created from vacuum, respectively.