Hidden-charmonium decays of Z _c (3900) and Z _c (4025) in intermediate meson loops model

Abstract

The BESIII collaboration reported an observation of two charged charmonium-like structure $Z_{c}^{\pm}(3900)$ and $Z_{c}^{\pm}(4025)$ in e ⁺ e ⁻→(J/ψπ)^± π ^∓ and $e^{+}e^{-} \to(D^{*} \bar{ D}^{*})^{\pm} \pi^{\mp}$ at $\sqrt{s} =4.26$ GeV recently, which could be an analogue of Z _b (10610) and Z _b (10650) claimed by the Belle Collaboration. In this work, we investigate the hidden-charmonium transitions of $Z_{c}^{\pm}(3900)$ and $Z_{c}^{\pm}(4025)$ via intermediate D ^(∗) D ^(∗) meson loops. Reasonable results for the branching ratios by taking appropriate values of α in this model can be obtained, which shows that the intermediate D ^(∗) D ^(∗) meson loops process may be a possible mechanism in these decays. Our results are consistent with the power-counting analysis, and comparable with the calculations in the framework of nonrelativistic effective field theory to some extent. We expect more experimental measurements on these hidden-charmonium decays and search for the decays of $Z_{c}\to D\bar{D}^{*} +c.c$. and $Z_{c}^{\prime}\to D^{*} \bar{D}^{*}$, which will help us investigate the $Z_{c}^{(\prime)}$ decays deeply.

Appendices

Appendix A: The transition amplitude in ELA

In the following, we present the transition amplitudes for the intermediate meson loops listed in Figs. 1 and 2 in the framework of the ELA. Notice that the expressions are similar for the charged and neutral charmed mesons except that different charmed meson masses are applied. We thus only present the amplitudes for those charged charmed meson loops.

(i) $Z_{c}^{(\prime) +} \to J/\psi\pi^{+}$ and ψ′π ⁺.

$$\begin{aligned} &{M_{DD^* [D]}} \\ &{\quad = (i)^3\int\frac{d^4q_2}{(2\pi)^4}[g_{Z_c^{(\prime)} D^*D} \varepsilon_{i\mu}] \bigl[g_{\psi DD} \varepsilon_f^{*\rho} (q_1-q_2)_\rho \bigr]} \\ &{\quad\phantom{=} \times [g_{D^*D\pi} p_{\pi\theta}] \times \frac{i}{q_1^2-m_1^2} \frac{i}{q_2^2-m_2^2}} \\ &{\quad\phantom{=} \times\frac {i(-g^{\mu\theta} +q_3^\mu q_3^\theta/m_3^2)}{q_3^2-m_3^2} \prod _i\mathcal{F}_i \bigl(m_i,q_i^2 \bigr),} \\ &{M_{DD^* [D^*]}} \\ &{\quad= (i)^3\int\frac{d^4q_2}{(2\pi)^4}[g_{Z_c^{(\prime)} D^*D} \varepsilon_{i\mu} ] \bigl[g_{\psi D^*D} \varepsilon_{\rho\sigma\xi\tau }p_f^\rho \varepsilon_f^{*\sigma} q_2^\xi \bigr]} \\ &{\quad\phantom{=} \times \bigl[-g_{D^*D^*\pi} \varepsilon_{\theta\phi\kappa\lambda} p_{\pi}^\kappa q_2^\lambda \bigr]} \\ &{\quad\phantom{=} \times\frac{i}{q_1^2-m_1^2} \frac{i(-g^{\tau\theta} +q_2^\tau q_2^\theta/m_2^2)}{q_2^2-m_2^2}} \\ &{\quad\phantom{=} \times\frac{i(-g^{\mu\phi} +q_3^\mu q_3^\phi/m_3^2)}{q_3^2-m_3^2} \prod _i\mathcal{F}_i \bigl(m_i,q_i^2 \bigr),} \\ &{M_{D^*D [D^*]}} \\ &{\quad = (i)^3\int\frac{d^4q_2}{(2\pi)^4}[g_{Z_c^{(\prime )} D^*D} \varepsilon_{i\mu}] \bigl[g_{\psi D^*D^*} (g_{\rho\sigma} g_{\xi\tau}} \\ &{\quad\phantom{=} {} - g_{\rho\tau} g_{\sigma\xi} + g_{\rho\xi} g_{\sigma\tau}) \varepsilon_f^{*\rho} (q_1+q_2)^\tau \bigr] [-g_{B^*B\pi} p_{\pi\theta}]} \\ &{\quad\phantom{=} {} \times\frac{i(-g^{\mu\xi} +q_1^\mu q_1^\xi/m_1^2)}{ q_1^2-m_1^2} \frac{i(-g^{\sigma\theta} +q_2^\sigma q_2^\theta/m_2^2)}{q_2^2-m_2^2}} \\ &{\quad\phantom{=} {} \times\frac{i}{q_3^2-m_3^2} \prod _i\mathcal {F}_i \bigl(m_i,q_i^2 \bigr),} \\ &{M_{D^*D^* [D]}} \\ &{\quad = (i)^3\int\frac{d^4q_2}{(2\pi)^4} \bigl[g_{Z_c^{(\prime )} D^*D^*} \varepsilon_{\mu\nu\alpha\beta} q_i^\mu \varepsilon_{i}^\nu \bigr]} \\ &{\quad\phantom{=} {} \times \bigl[g_{\psi D^*D} \varepsilon_{\rho\sigma\xi\tau} p_f^\rho\varepsilon_f^{*\sigma} q_1^\xi \bigr] [g_{D^*D\pi} p_{\pi\theta}]} \\ &{\quad\phantom{=} {} \times\frac{i(-g^{\alpha\tau} +q_1^\alpha q_1^\tau/m_1^2)}{ q_1^2-m_1^2}} \\ &{\quad\phantom{=} {} \times\frac{i}{q_2^2-m_2^2} \frac{i(-g^{\beta\theta} +q_3^\beta q_3^\theta/m_3^2)}{q_3^2-m_3^2} \prod _i\mathcal {F}_i \bigl(m_i,q_i^2 \bigr),} \\ &{M_{D^*D^* [D^*]}} \\ &{\quad = (i)^3\int\frac{d^4q_2}{(2\pi)^4} \bigl[g_{Z_c^{(\prime )} D^*D^*} \varepsilon_{\mu\nu\alpha\beta} p_i^\mu \varepsilon_{i}^\nu \bigr]} \\ &{\quad\phantom{=} {} \times \bigl[g_{\psi D^*D^*} (g_{\rho\sigma} g_{\xi\tau} - g_{\rho\tau} g_{\sigma\xi} + g_{\rho\xi} g_{\sigma\tau})} \\ &{\quad\phantom{=} {} \times \varepsilon_f^{*\rho} (q_1+q_2)^\tau \bigr] \bigl[-g_{D^*D^*\pi} \varepsilon_{\theta\phi\kappa\lambda} p_{\pi}^\kappa q_2^\lambda \bigr]} \\ &{\quad\phantom{=} {} \times\frac{i(-g^{\alpha\xi} +q_1^\alpha q_1^\xi/m_1^2)}{q_1^2-m_1^2} \frac {i(-g^{\sigma\theta} +q_2^\sigma q_2^\theta/m_2^2)}{q_2^2-m_2^2}} \\ &{\quad\phantom{=} {} \times \frac{i(-g^{\beta\phi} +q_3^\beta q_3^\phi/m_3^2)}{q_3^2-m_3^2} \prod _i\mathcal{F}_i \bigl(m_i,q_i^2 \bigr) .} \end{aligned}$$

(A.1)

(ii) $Z_{c}^{(\prime) +}\to h_{c} \pi^{+}$.

$$\begin{aligned} &{M_{DD^* [D^*]}} \\ &{\quad = (i)^3\int\frac{d^4q_2}{(2\pi)^4}[g_{Z_c^{(\prime )} D^*D} \varepsilon_{i\mu} ] \bigl[g_{h_c D^*D} \varepsilon_{f\rho}^{*} \bigr]} \\ &{\quad\phantom{=} \times \bigl[-g_{D^*D^*\pi} \varepsilon_{\theta\phi\kappa\lambda} p_{\pi}^\kappa q_2^\lambda \bigr]} \\ &{\quad\phantom{=} \times\frac{i}{q_1^2-m_1^2} \frac{i(-g^{\rho\theta} +q_2^\rho q_2^\theta/m_2^2)}{q_2^2-m_2^2}} \\ &{\quad\phantom{=} \times\frac{i(-g^{\mu\phi} +q_3^\mu q_3^\phi /m_3^2)}{q_3^2-m_3^2} \prod _i\mathcal{F}_i \bigl(m_i,q_i^2 \bigr),} \\ &{M_{D^*D [D^*]}} \\ &{\quad = (i)^3\int\frac{d^4q_2}{(2\pi)^4}[g_{Z_c^{(\prime )} D^*D} \varepsilon_{i\mu}]} \\ &{\quad\phantom{=}\times \bigl[g_{h_c D^*D^*} \varepsilon_{\rho\sigma\xi \tau} p_f^\rho\varepsilon_f^{*\sigma} \bigr] [-g_{D^*D\pi} p_{\pi\theta} ]} \\ &{\quad\phantom{=}\times\frac{i(-g^{\mu\xi} +q_1^\mu q_1^\xi/m_1^2)}{q_1^2-m_1^2} \frac{i(-g^{\tau\theta} +q_2^\tau q_2^\theta/m_2^2)}{q_2^2-m_2^2}} \\ &{\quad\phantom{=} \times\frac{i}{q_3^2-m_3^2} \prod _i\mathcal{F}_i \bigl(m_i,q_i^2 \bigr),} \\ &{M_{D^*D^* [D]}} \\ &{\quad = (i)^3\int\frac{d^4q_2}{(2\pi)^4} \bigl[g_{Z_c^{(\prime )} D^*D^*} \varepsilon_{\mu\nu\alpha\beta} p_i^\mu \varepsilon_{0}^\nu \bigr] \bigl[g_{h_c D^*D} \varepsilon_{f\rho}^{*} \bigr]} \\ &{\quad\phantom{=}\times [g_{D^*D\pi} p_{\pi\theta}] \frac{i(-g^{\alpha\rho} +q_1^\alpha q_1^\rho/m_1^2)}{ q_1^2-m_1^2} \frac{i}{q_2^2-m_2^2}} \\ &{\quad\phantom{=} \times\frac{i(-g^{\beta\theta} +q_3^\beta q_3^\theta/m_3^2)}{q_3^2-m_3^2} \prod _i\mathcal {F}_i \bigl(m_i,q_i^2 \bigr),} \\ &{M_{D^*D^* [D^*]}} \\ &{\quad = (i)^3\int\frac{d^4q_2}{(2\pi)^4} \bigl[g_{Z_c^{(\prime )} D^*D^*} \varepsilon_{\mu\nu\alpha\beta} p_i^\mu \varepsilon_{i}^\nu \bigr]} \\ &{\quad\phantom{=}\times \bigl[g_{h_c D^*D^*} \varepsilon_{\rho\sigma\xi\tau} p_f^\rho\varepsilon_f^{*\sigma} \bigr] \bigl[-g_{D^*D^*\pi} \varepsilon_{\theta\phi\kappa\lambda} p_{\pi}^{\kappa} q_2^\lambda \bigr]} \\ &{\quad\phantom{=} \times\frac{i(-g^{\alpha\xi} +q_1^\alpha q_1^\xi/m_1^2)}{ q_1^2-m_1^2} \frac{i(-g^{\tau\theta} +q_2^\tau q_2^\theta/m_2^2)}{ q_2^2-m_2^2}} \\ &{\quad\phantom{=}\times \frac{i(-g^{\beta\phi} +q_3^\beta q_3^\phi/m_3^2)}{ q_3^2-m_3^2} \prod _i\mathcal{F}_i \bigl(m_i,q_i^2 \bigr),} \end{aligned}$$

(A.2)

where p _i , p _f , p _π are the four-vector momenta of the initial $Z_{c}^{(\prime)}$, final state charmonium and pion, respectively, and q ₁, q ₂, and q ₃ are the four-vector momenta of the intermediate charmed mesons as defined in Figs. 1 and 2.

Appendix B: Amplitudes in NREFT approach

The basic three-point loop function worked out using dimensional regularization in d=4 is

$$\begin{aligned} &{I(q,m_1,m_2,m_3)} \\ &{\quad = \frac{-i}{8} \int\frac{d^d l}{(2\pi)^d} \frac{1}{ [l^0 - \frac{\vec{l}^2}{m_1}+i\epsilon]}} \\ &{\quad\phantom{=}\times \frac{1}{[l^0-b_{12}+ \frac{\vec{l}^2}{m_2}-i\epsilon]} \frac{1}{[l^0+b_{12}-b_{23}- \frac{(\vec{l}-\vec{q})^2}{m_2}+i\epsilon]}} \\ &{\quad= \frac{\mu_{12} \mu_{23}}{16\pi} \frac{1}{{\sqrt{2}}} \biggl[ \tan^{-1} \biggl( \frac{c^\prime-c}{2\sqrt{a(c-i\epsilon)}} \biggr)} \\ &{\quad\phantom{=} {} + \tan^{-1} \biggl(\frac {2a+c-c^\prime}{2\sqrt{a(c^\prime-a -i\epsilon)}} \biggr) \biggr],} \end{aligned}$$

(B.1)

where m _i (i=1,2,3) are the masses of the particles in the loop; μ _ij =m _i m _j /(m _i +m _j ) are the reduced masses; b ₁₂=m ₁+m ₂−M and b ₂₃=m ₂+m ₃+q ⁰−M with M being the mass of the initial particle; and

$$ \begin{aligned} &a= \biggl(\frac{\mu_{23}}{m_3}\biggr)^2 \vec{q}^2, \qquad c=2\mu_{12}b_{12},\\ & c^\prime= 2 \mu_{23} b_{23}+\frac{\mu_{23}}{m_3} \vec{q}^2. \end{aligned} $$

(B.2)

The vector loop integrals are defined as

$$\begin{aligned} q^iI^{(1)}(q,m_1,m_2,m_3) = \frac{-i}{8} \int\frac{d^d l}{(2\pi)^d} \frac{l^i}{[l^0 - \frac{\vec{l}^2}{m_1}+i\epsilon] [l^0-b_{12}+ \frac{\vec{l}^2}{m_2}-i\epsilon][l^0+b_{12}-b_{23}- \frac{(\vec {l}-\vec{q})^2}{m_2}+i\epsilon]} \end{aligned}$$

(B.3)

and we get

$$\begin{aligned} &{I^{(1)}(q,m_1,m_2,m_3)} \\ &{\quad = \frac{\mu_{23}}{am_3} \biggl[B\bigl(c^\prime-a\bigr) -B(c) + \frac{1}{2} \bigl(c^\prime-c\bigr) I(q)\biggr],} \end{aligned}$$

(B.4)

where the function B(c) is

$$ B(c)= -\frac{\mu_{12} {\mu_{23}} {\sqrt{c-i\epsilon} }}{16\pi}. $$

(B.5)

In terms of the loop functions given above, the transition amplitudes for the intermediate meson loops listed in Figs. 1 and 2 in the framework of NREFT,

(i) $Z_{c}^{(\prime) +} \to J/\psi\pi^{+}$ and ψ′π ⁺.

$$\begin{aligned} &{\mathcal{M} \bigl(Z_c^{(\prime) +} \to \psi\pi^+ \bigr)} \\ &{\quad= -\frac{2{\sqrt{2}} g g_1z^{(\prime)}}{f_\pi} \sqrt{M_{Z_c^{(\prime)}} M_\psi} \bigl\{ \vec{q} \cdot\vec{\varepsilon }(Z_c) \vec{q} \cdot\vec{ \varepsilon}(\psi )} \\ &{\quad\phantom{=}\times \bigl[2I^{(1)}(q,M_{D^*},M_D,M_D)-I(q,M_{D^*},M_D,M_D)} \\ &{\quad\phantom{=}- 2I^{(1)}(q,M_{D^*},M_D,M_{D^*}) + I(q,M_{D^*},M_D,M_{D^*})} \\ &{\quad\phantom{=}+2I^{(1)}(q,M_{D^*},M_{D^*},M_D) -I(q,M_{D^*},M_{D^*},M_D)} \\ &{\quad\phantom{=}-2I^{(1)}(q,M_{D^*},M_{D^*},M_{D^*})} \\ &{\quad\phantom{=}+I(q,M_{D^*},M_{D^*},M_{D^*}) \bigr]} \\ &{\quad\phantom{=} +\vec{q}^2\vec{\varepsilon}(Z_c) \cdot\vec{ \varepsilon}(\psi) \bigl[2I^{(1)}(q,M_{D^*},M_D,M_{D^*})} \\ &{\quad\phantom{=}-I(q,M_{D^*},M_D,M_{D^*})+ 2I^{(1)}(q,M_D,M_{D^*},M_{D^*})} \\ &{\quad\phantom{=}-I(q,M_D,M_{D^*},M_{D^*})-2I^{(1)}(q,M_{D^*},M_{D^*},M_D)} \\ &{\quad\phantom{=} +I(q,M_{D^*},M_{D^*},M_D) - 2I^{(1)}(q,M_{D^*},M_{D^*},M_{D^*})} \\ &{\quad\phantom{=}+I(q,M_{D^*},M_{D^*},M_{D^*}) \bigr] \bigr\} .} \end{aligned}$$

(B.6)

(ii) $Z_{c}^{(\prime) +} \to h_{c}\pi^{+}$.

$$\begin{aligned} &{\mathcal{M} \bigl(Z_c^{(\prime) +}\to h_c\pi^+ \bigr)} \\ &{\quad = \frac{2{\sqrt{2}}g g_1 z^{(\prime) }}{f_\pi} \sqrt{{M_{Z_c^{(\prime)}} {M_{h_c}}}} \epsilon ^{ijk} q^i \varepsilon^j(Z_c) \varepsilon^k(h_c)} \\ &{\quad\phantom{=}\times \bigl[I(q, M_D, M_{D^*},M_{D^*})+ I(q, M_{D^*}, M_D,M_{D^*})} \\ &{\quad\phantom{=} -I(q, M_{D^*}, M_{D^*},M_D)+I(q, M_{D^*}, M_{D^*},M_{D^*}) \bigr].} \end{aligned}$$

(B.7)