Is Z _c (3900) a molecular state?

Abstract

Assuming the newly observed Z _c (3900) to be a molecular state of $D\bar{D}^{*}(D^{*} \bar{D})$, we calculate the partial widths of Z _c (3900)→J/ψ+π; ψ′+π; η _c +ρ and $D\bar{D}^{*}$ within the light-front model (LFM). Z _c (3900)→J/ψ+π is the channel by which Z _c (3900) was observed, our calculation indicates that it is indeed one of the dominant modes whose width can be in the range of a few MeV depending on the model parameters. Similar to Z _b and $Z_{b}'$, Voloshin suggested that there should be a resonance $Z_{c}'$ at 4030 MeV, which can be a molecular state of $D^{*}\bar{D}^{*}$. Then we go on calculating its decay rates to all the aforementioned final states and the $D^{*}\bar{D}^{*}$ as well. It is found that if Z _c (3900) is a molecular state of ${1\over\sqrt{2}}(D\bar{D}^{*}+D^{*}\bar{D})$, the partial width of $Z_{c}(3900)\to D\bar{D}^{*}$ is rather small, but the rate of Z _c (3900)→ψ(2s)π is even larger than Z _c (3900)→J/ψπ. The implications are discussed and it is indicated that with the luminosity of BES and BELLE, the experiments may finally determine if Z _c (3900) is a molecular state or a tetraquark.

Appendices

Appendix A: The vertex function of molecular state

Supposing Z _c (3900) and Z _c (4030) are molecular states which consists of D and $\bar{D^{*}}$ and D ^∗ and $\bar{D^{*}}$, respectively. If the orbital angular momentum between the two components is zero, i.e. l=0, the total spin S should be 0, 1, and 2 and total angular momentum J also is 0, 1, and 2.

Similar to our previous works on baryons [18–20], we construct the vertex function of molecular in the same model. The wavefunction of a molecular with total spin J and momentum P is

$$\begin{aligned} & \bigl|X(P,J,J_z)\bigr\rangle \\ &\quad =\int\bigl\{d^3 \tilde{p}_1\bigr\} \bigl\{d^3\tilde{p}_2\bigr\} \, 2(2\pi)^3\delta^3(\tilde{P}-\tilde{p_1}- \tilde{p_2}) \\ &\qquad {}\times\sum_{\lambda_1}\varPsi^{SS_z}( \tilde{p}_1,\tilde {p}_2,\lambda_1, \lambda_2) \\ &\qquad {}\times\mathcal{F}\bigl \vert D^{(*)}(p_1, \lambda_1) \bar{D}^*(p_2,\lambda_2)\bigr\rangle , \end{aligned}$$

(A.1)

with

$$\begin{aligned} & \varPsi^{SS_z}(\tilde{p}_1,\tilde{p}_2, \lambda_1,\lambda_2)\\ &\quad = \langle \lambda_1 \vert \mathcal{R}^{\dagger}_M(x_1,k_{1\perp},m_1) \vert s_1 \rangle \langle \lambda_2 \vert \mathcal {R}^{\dagger}_M(x_2,k_{2\perp},m_2) \vert s_2 \rangle \\ &\qquad {}\times \langle 1s_1 ;1 s_2\vert S S_z \rangle \langle S S_z ;0 0\vert J J_z \rangle \varphi(x,k_{\perp}), \end{aligned}$$

where 〈1s ₁;1s ₂|SS _z 〉〈SS _z ;00|JJ _z 〉 is the C-G coefficients and s ₁,s ₂ are the spin projections of the constituents. These C-G coefficients can be rewritten as

$$ \begin{aligned} &\langle 1s_1;0 0|S S_z\rangle \langle S S_z ;0 0|1 J_z\rangle =A_{10} \epsilon_{1}(s_1)\cdot \epsilon(J_z), \\ &\langle 0 0;1s_2|S S_z\rangle \langle S S_z ;0 0|1 J_z\rangle =A_{01} \epsilon_{2}(s_2)\cdot \epsilon(J_z), \\ &\langle 1s_1 ;1 s_2|S S_z\rangle \langle S S_z ;0 0|0 0\rangle =A_0 \epsilon_{1}(s_1) \cdot \epsilon_{2}(s_2), \\ &\langle 1s_1 ;1 s_2|S S_z\rangle \langle S S_z ;0 0|1 J_z\rangle\\ &\quad =A_1 \varepsilon^{\mu\nu\alpha\beta} \epsilon_{1\mu}(s_1) \epsilon_{2\nu}(s_2)\epsilon_{\alpha}(J_z)P_\beta, \\ &\langle 1s_1 ;1 s_2|S S_z\rangle \langle S S_z ;0 0|2 J_z\rangle\\ &\quad =A_2 \epsilon_{1\mu}(s_1) \epsilon_{2\nu}(s_2) \epsilon^{\mu\nu}(J_z), \end{aligned} $$

(A.2)

with

$$\begin{aligned} &A_{01}=\frac{\sqrt{3}m_1}{\sqrt{e_1^2+2m_1^2}}, \\ & A_{10}=\frac{\sqrt{3}m_2}{\sqrt{e_2^2+2m_2^2}}, \\ &A_0=\frac{2 {m_1} {m_2}}{\sqrt{{M_0'}^4-2 {M_0'}^2 ({m_1}^2+{m_2}^2)+{m_1}^4+10 {m_1}^2 {m_2}^2+{m_2}^4}}, \\ & A_1=\frac{2\sqrt{3} {m_1} {m_2}}{\sqrt{{M'}^2 [4 {e_1}^2 {m_2}^2-4 {e_1} {e_2} (-{M_0'}^2+{m_1}^2+{m_2}^2)+4 {e_2}^2 {m_1}^2+10 {m_1}^2 {m_2}^2-C_A]}} , \\ &A_2=\frac{\sqrt{120} {m_1} {m_2}}{\sqrt{4 {e_1}^2 (4 {e_2}^2+7 {m_2}^2)+4 {e_1} {e_2} (-{M_0'}^2+{m_1}^2+{m_2}^2)+28 {e_2}^2 {m_1}^2+54 {m_1}^2 {m_2}^2+C_A}}, \\ &C_A={M_0'}^4-2 {M_0'}^2 \bigl({m_1}^2+{m_2}^2 \bigr)+{m_1}^4+{m_2}^4. \end{aligned}$$

A Melosh transformation brings the matrix elements from the spin-projection-on-fixed-axes representation into the helicity representation and is explicitly written as

$$\langle \lambda_2\vert \mathcal{R}^{\dagger}_M(x_2,k_{2\perp},m_2) \vert s_2 \rangle =\xi^*(\lambda_2,m_2) \cdot\xi(s_2,m_2), $$

and

$$\langle \lambda_1\vert \mathcal{R}^{\dagger}_M(x_1,k_{1\perp},m_1) \vert s_1 \rangle =\xi^*(\lambda_1,m_1) \cdot\xi(s_1,m_1). $$

Following Refs. [13–15, 27], the Melosh transformed matrix can be expressed as

$$\begin{aligned} & \langle \lambda_1|\mathcal{R}^{\dagger}_M(x_1,k_{1\perp},m_1) |s_1\rangle \langle \lambda_2| \mathcal{R}^{\dagger }_M(x_2,k_{2\perp},m_2) |s_2\rangle \\ &\qquad {}\times\langle 1s_1 ;1 s_2|S S_z\rangle \langle S S_z ;0 0|J J_z\rangle \\ &\quad =A_1 \varepsilon^{\mu\nu\alpha\beta} \epsilon_{1\mu}( \lambda_1) \epsilon_{2\nu}(\lambda_2) \epsilon_{\alpha}(J_z)P_\beta, \end{aligned}$$

(A.3)

so the wavefunction of 1⁺ molecular state of $D^{(*)}\bar{D}^{*}$ is

$$\begin{aligned} & \varPsi^{SS_z}(\tilde{p}_1,\tilde{p}_2, \lambda_1,\lambda_2) \\ &\quad = A_1 \varphi(x,k_{\perp}) \varepsilon^{\mu\nu\alpha\beta} \epsilon_{1\mu}( \lambda_1) \epsilon_{2\nu}(\lambda_2) \epsilon_{\alpha}(J_z)P_\beta \\ &\quad =h_{A_1}' \varepsilon^{\mu\nu\alpha\beta} \epsilon_{1\mu}(\lambda_1) \epsilon_{2\nu}( \lambda_2)\epsilon_{\alpha}(J_z)P_\beta, \end{aligned}$$

(A.4)

with $\varphi=4(\frac{\pi}{\beta^{2}})^{3/4}\frac{e_{1}e_{2}}{x_{1}x_{2}M_{0}}{\rm exp}(\frac{-\mathbf{k}^{2}}{2\beta^{2}})$.

Similarly the wavefunction of 1⁺ molecular state of $D\bar{D}^{*}$

$$\begin{aligned} \varPsi^{SS_z}(\tilde{p}_1,\tilde{p}_2, \lambda_1,\lambda_2) &= A\varphi(x,k_{\perp}) \epsilon_{1\mu}(\lambda_1) \cdot\epsilon_{\alpha}(J_z) \\ & =h_{A_1}' \epsilon_{1\mu}( \lambda_1) \cdot\epsilon_{\alpha}(J_z) \end{aligned}$$

(A.5)

and normalization of the state is |X(P,J,J _z )〉,

$$\begin{aligned} & \bigl\langle X\bigl(P',J',J_z' \bigr)\big|X(P,J,J_z)\bigr\rangle \\ &\quad =2(2\pi)^3P^+ \delta^3\bigl(\tilde{P}'-\tilde{P}\bigr) \delta_{J'J}\delta_{J'_zJ_z}. \end{aligned}$$

(A.6)

All other notations can be found in Refs. [18–20].

Appendix B: The effective vertices

The effective vertices can be found in [33–37],

$$\begin{aligned} &\mathcal{L}_{\pi DD^*}=ig_{_{\pi DD^*}} \bigl(D^{*\mu}\partial_\mu\pi \bar{D} - \partial^\mu D \pi\bar{D}^{*}_\mu+h.c.\bigr) , \end{aligned}$$

(B.1)

$$\begin{aligned} &\mathcal{L}_{\pi D^*D^*}=-g_{_{\pi D^*D^*}}\varepsilon^{\mu\nu \alpha\beta} \partial^\mu\bar{D}^*_\nu\pi\partial_\alpha D^{*\beta}, \end{aligned}$$

(B.2)

$$\begin{aligned} &\mathcal{L}_{\psi DD}=ig_{_{\psi DD}}\psi_\mu\bigl(\partial ^\mu D\bar{D} - D\partial^\mu \bar{D}\bigr) , \end{aligned}$$

(B.3)

$$\begin{aligned} &\mathcal{L}_{\psi DD^*}=-g_{_{\psi DD^*}}\varepsilon^{\mu\nu\alpha \beta} \partial^\mu\psi_\nu \bigl[{\partial}_\beta D^*_\alpha\bar{D}+D{ \partial}_\beta\bar{D}^*_\alpha \bigr], \end{aligned}$$

(B.4)

$$\begin{aligned} &\mathcal{L}_{\psi D^*D^*}=ig_{_{\psi D^*D^*}} \bigl[-\psi^\mu D^{*\nu}(\overrightarrow{\partial}-\overleftarrow {\partial})_\mu D_\nu^{*\dagger} \\ &\hphantom{\mathcal{L}_{\psi D^*D^*}=}{}+ \psi^\mu D^{*\nu}{ \partial}_\nu D_\mu^{*\dagger}-\psi^\mu { \partial}_\nu D^{*\mu} D_\nu^{*\dagger}\bigr], \end{aligned}$$

(B.5)

$$\begin{aligned} & \mathcal{L}_{\sigma DD}=-g_{_{\sigma DD}}\sigma D\bar{D}, \end{aligned}$$

(B.6)

$$\begin{aligned} &\mathcal{L}_{\sigma D^*D^*}=g_{_{\sigma D^*D^*}}\sigma D^* \cdot \bar{D}^*. \end{aligned}$$

(B.7)

The effective vertices η _c DD ^∗ and η _c D ^∗ D ^∗ are similar to those in Eq. (B.1) and Eq. (B.2) and the effective vertices ρDD, ρDD ^∗, and ρD ^∗ D ^∗ can be obtained by replacing the ψ by ρ in Eq. (B.3) and Eq. (B.4).