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In the past twenty years, many charmonium-like $ XYZ $ states have been discovered in particle experiments [1]. All of these are good multiquark candidates, and their relevant experimental and theoretical studies have significantly improved our understanding of the strong interaction at the low energy region. In particular, in 2013, BESIII reported the $ Z_c(3900)^+ $ in the $ Y(4260) \to J/\psi\pi^+\pi^- $ process [2], which was later confirmed by Belle [3] and CLEO [4]. Since it couples strongly to the charmonium and yet it is charged, the $ Z_c(3900)^+ $ is not a conventional charmonium state and contains at least four quarks. It is quite interesting to understand how it is composed of these four quarks, and there have been various models developed to explain this, such as a compact tetraquark state composed of a diquark and an antidiquark [5, 6], a loosely-bound hadronic molecular state composed of two charmed mesons [7-13], a hadro-quarkonium [8, 14, 15], or due to the kinematical threshold effect [16-19], etc. We refer to reviews [20-24] for detailed discussions.
The charged charmonium-like state $ Z_c(3900) $ of $ J^{PC} = 1^{+-} $ [25] has been observed in the $ J/\psi \pi $ and $ D \bar D^* $ channels [2, 3, 26, 27], and there were some events in the $ h_c \pi $ channel [28]. In a recent BESIII experiment [29], evidence for the $ Z_c(3900) \rightarrow \eta_c\rho $ decay was reported with a statistical significance of $ 3.9\sigma $ at $ \sqrt{s} = 4.226 $ GeV, and the relative branching ratio

$ {\cal{R}}_{Z_c} \equiv {{\cal{B}}(Z_c(3900) \rightarrow \eta_c\rho) \over {\cal{B}}(Z_c(3900) \rightarrow J/\psi\pi)} \, , $

(1)

was evaluated to be $ 2.2 \pm 0.9 $ at the same center-of-mass energy. This ratio has been studied by many theoretical methods/models [30-39], and was suggested in Ref. [40] to be useful to discriminate between the compact tetraquark and hadronic molecule scenarios. As summarized in Table 1, this ratio was calculated in many molecular models, but the extracted values are highly model dependent. Hence, it would be useful to derive a model-independent result, and it would be even better to do so within the same framework for both the tetraquark and molecule scenarios.

interpretations	${\cal{R}}_{Z_c}$	methods/models
compact tetraquark	$\left( 2.3^{+3.3}_{-1.4} \right) \times 10^2$	Type-I diquark-antidiquark model [40]
	$0.27^{+0.40}_{-0.17}$	Type-II diquark-antidiquark model [40]
	$0.95$	QCD sum rules [30]
	$0.57$	QCD sum rules [31]
	$1.1$	QCD sum rules [32]
	$1.28$	covariant quark model [33]
hadronic molecule	$\left( 4.6^{+2.5}_{-1.7} \right) \times 10^{-2}$	Non-Relativistic effective field theory [40]
	$0.12$	light front model [34]
	$0.68 \times 10^{-2}$	effective field theory [35]
	$1.78$	covariant quark model [33]

Table1.The relative branching ratio ${\cal{R}}_{Z_c} \equiv {\cal{B}}(Z_c(3900) \rightarrow \eta_c\rho) / {\cal{B}}(Z_c(3900) \rightarrow J/\psi\pi)$, calculated by various theoretical methods/models.

In this paper we study the decay properties of the $ Z_c(3900) $ under both the compact tetraquark and hadronic molecule interpretations. This study is based on our previous finding that the diquark-antidiquark currents ($ [qq][\bar q \bar q] $) and the meson-meson currents ($ [\bar q q][\bar q q] $) are related to each other through the Fierz rearrangement of the Dirac and color indices [41-51]. More studies on light baryon operators can be found in Refs. [52-54]. In the present case the $ Z_c(3900) $ contains four quarks: the c, $ \bar c $, q, $ \bar q $ quarks ($ q = u/d $). Thus, there are three configurations:

$ [cq][\bar c \bar q]\, , \; \; [\bar c q][\bar q c]\, , \; \; {\rm{and}} \; \; [\bar c c][\bar q q] \, . $

Again, the Fierz rearrangement can be applied to relate them. Based on these relations, we shall extract some decay properties of the $ Z_c(3900) $ in this paper.
There are eight independent $ [cq][\bar c \bar q] $ currents of $ J^{PC} = 1^{+-} $, which have been systematically constructed in Ref. [55]. Here, we choose one of them,

$ \eta^{{\cal{Z}}}_{\mu} = \epsilon^{abe} \epsilon^{cde}\; q_a^T{\mathbb{C}}\gamma_{\mu} c_b \; \bar q_c \gamma_5{\mathbb{C}}\bar c_d^T - \{ \gamma_{\mu} \leftrightarrow \gamma_5 \} \, , $

(2)

where $ {\mathbb{C}} $ is the charge-conjugation matrix, the subscripts $ a \cdots e $ are the color indices, and the sum over repeated indices is taken. This current would strongly couple to the $ Z_c(3900) $, if it has the same internal structure (internal symmetry) as that state.
The above current is useful from the viewpoints of both effective field theory and QCD sum rules. Note that there are various quark-based effective field theories, which have been successfully applied to describe the meson and baryon systems, such as the Non-Relativistic QCD for the heavy quarkonium system [56, 57]:

$ \begin{split} {\cal{L}}_{\rm{NRQCD}} =& \psi^{\dagger} \left\{ {\rm i} D_0 + \cdots \right\} \psi + \chi^{\dagger} \left\{ {\rm i} D_0 + \cdots \right\} \chi \\&+ {f_1(^1S_0) \over m_1 m_2} \psi^{\dagger} \chi \chi^{\dagger} \psi + {f_1(^3S_0) \over m_1 m_2} \psi^{\dagger} {{\sigma}} \chi \chi^{\dagger} {{\sigma}} \psi \\ &+ {f_8(^1S_0) \over m_1 m_2} \psi^{\dagger} T^a\chi \chi^{\dagger} T^a \psi \\ &+ {f_8(^3S_0) \over m_1 m_2} \psi^{\dagger} T^a {{\sigma}} \chi \chi^{\dagger} T^a {{\sigma}} \psi + \cdots \, . \end{split} $

(3)

We refer to Ref. [58] for a detailed review of this method. The above Lagrangian contains four four-fermion operators, which can be used to study the annihilation width of a heavy quarkonium into light particles. In this method the Fierz rearrangement is applied to decouple the Dirac and color indices that connect the short-distance part to the long-distance part [57].
Compared with this, the quark-based effective field theory for the multiquark system is much more difficult [24]. Let us attempt to do this for the $ Z_c(3900) $. Based on Eq. (2), we can add an eight-quark operator (the same argument applies for other Lagrangians containing $ \eta^{{\cal{Z}}}_{\mu} $):

$ \begin{split} {\cal{L}} = & c_0 \times \eta^{{\cal{Z}}}_{\mu} \times \left(\eta^{{{\cal{Z}}},\mu}\right)^{\dagger} \\ = &c_0 \times \left( \epsilon^{abe} \epsilon^{cde}\; q_a^T{\mathbb{C}}\gamma_{\mu} c_b \; \bar q_c \gamma_5{\mathbb{C}}\bar c_d^T - \{ \gamma_{\mu} \leftrightarrow \gamma_5 \} \right) \\& \times \left( \epsilon^{a^{\prime} b^{\prime} e^{\prime} } \epsilon^{c^{\prime} d^{\prime} e^{\prime} }\; \bar c_{b^{\prime}} \gamma^\mu {\mathbb{C}} \bar q_{a^{\prime}}^T \; c_{d^{\prime}}^T {\mathbb{C}} \gamma_5 q_{c^{\prime}} - \{ \gamma_{\mu} \leftrightarrow \gamma_5 \} \right) \, , \end{split} $

(4)

where $ c_0 $ is a constant. Next, we can use the Fierz rearrangement to transform it to

$ \begin{split} {\cal{L}} =& c_0 \times \Big( + {1\over3} \; \bar c_{a} \gamma_5 c_a \; \bar q_{b} \gamma_{\mu} q_b - {1\over3} \; \bar c_{a} \gamma_{\mu} c_a \; \bar q_{b} \gamma_5 q_b \\ & + {{\rm i}\over3} \; \bar c_{a} \gamma^\nu \gamma_5 c_a \; \bar q_{b} \sigma_{\mu\nu} q_b - {{\rm i}\over3} \; \bar c_{a} \sigma_{\mu\nu} c_a \; \bar q_{b} \gamma^\nu \gamma_5 q_b \\ & - {1\over4} \; {\lambda^n_{ab}}{\lambda^n_{cd}} \; \bar c_{a} \gamma_5 c_b \; \bar q_{c} \gamma_{\mu} q_d \\ & + {1\over4} \; {\lambda^n_{ab}}{\lambda^n_{cd}} \; \bar c_{a} \gamma_{\mu} c_b \; \bar q_{c} \gamma_5 q_d \\ & - {{\rm i}\over4} \; {\lambda^n_{ab}}{\lambda^n_{cd}} \; \bar c_{a} \gamma^\nu \gamma_5 c_b \; \bar q_{c} \sigma_{\mu\nu} q_d \\ & + {{\rm i}\over4} \; {\lambda^n_{ab}}{\lambda^n_{cd}} \; \bar c_{a} \sigma_{\mu\nu} c_b \; \bar q_{c} \gamma^\nu \gamma_5 q_d \Big) \\ & \times \left( \epsilon^{a^{\prime} b^{\prime} e^{\prime} } \epsilon^{c^{\prime} d^{\prime} e^{\prime} }\; \bar c_{b^{\prime}} \gamma^\mu {\mathbb{C}} \bar q_{a^{\prime}}^T \; c_{d^{\prime}}^T {\mathbb{C}} \gamma_5 q_{c^{\prime}} - \{ \gamma_{\mu} \leftrightarrow \gamma_5 \} \right) \, . \end{split} $

(5)

Detailed discussions on this transformation will be given below.
Considering that the meson operators, $ \bar q \gamma_5 q $, $ \bar q \gamma_{\mu} q $, $ \bar c \gamma_5 c $, and $ \bar c \gamma_{\mu} c $ couple to the $ \pi $, $ \rho $, $ \eta_c $, and $ J/\psi $ mesons (see Table 2 at Sec. 3), the above eight-quark operator can describe the fall-apart decays of the $ Z_c(3900) $ into the $ \eta_c \rho $ and $ J/\psi \pi $ final states simultaneously, together with some other possible decay channels. In order to extract the widths of these decays, one still needs to do further calculations, which we shall not examine further. However, their relative branching ratios can be extracted much more easily, which are useful and important for understanding the nature of the $ Z_c(3900) $ [59].

operators	$J^{PC}$	mesons	$J^{PC}$	couplings	decay constants
$J^{S} = \bar d u$	$0^{++}$	–	$0^{++}$	–	–
$J^{P} = \bar d i\gamma_5 u$	$0^{-+}$	$\pi^+$	$0^{-+}$	$\langle 0 | J^{P} | \pi^+ \rangle = \lambda_\pi$	$\lambda_\pi = \dfrac{f_\pi m_\pi^2}{m_u + m_d}$
$J^{V}_{\mu} = \bar d \gamma_{\mu} u$	$1^{–}$	$\rho^+$	$1^{–}$	$\langle 0 | J^{V}_{\mu} | \rho^+ \rangle = m_\rho f_{\rho^+} \epsilon_{\mu}$	$f_{\rho^+} = 208$ MeV [82]
$J^{A}_{\mu} = \bar d \gamma_{\mu} \gamma_5 u$	$1^{++}$	$\pi^+$	$0^{-+}$	$\langle 0 | J^{A}_{\mu} | \pi^+\rangle = {\rm i} p_{\mu} f_{\pi^+}$	$f_{\pi^+} = 130.2$ MeV [1]
		$a_1(1260)$	$1^{++}$	$\langle 0 | J^{A}_{\mu} | a_1 \rangle = m_{a_1} f_{a_1} \epsilon_{\mu} $	$f_{a_1} = 254$ MeV [87]
$J^{T}_{\mu\nu} = \bar d \sigma_{\mu\nu} u$	$1^{\pm-}$	$\rho^+$	$1^{–}$	$\langle 0 | J^{T}_{\mu\nu} | \rho^+ \rangle = {\rm i} f^T_{\rho} (p_{\mu}\epsilon_\nu - p_\nu\epsilon_{\mu})$	$f_{\rho}^T = 159$ MeV [82]
		$b_1(1235)$	$1^{+-}$	$\langle 0 | J^{T}_{\mu\nu} | b_1 \rangle = {\rm i} f^T_{b_1} \epsilon_{\mu\nu\alpha\beta} \epsilon^\alpha p^\beta$	$f_{b_1}^T = 180$ MeV [95]
$I^{S} = \bar c c$	$0^{++}$	$\chi_{c0}(1P)$	$0^{++}$	$\langle 0 | I^S | \chi_{c0} \rangle = m_{\chi_{c0}} f_{\chi_{c0}}$	$f_{\chi_{c0}} = 343$ MeV [76]
$I^{P} = \bar c i\gamma_5 c$	$0^{-+}$	$\eta_c$	$0^{-+}$	$\langle 0 | I^{P} | \eta_c \rangle = \lambda_{\eta_c}$	$\lambda_{\eta_c} = \dfrac {f_{\eta_c} m_{\eta_c}^2}{2 m_c}$
$I^{V}_{\mu} = \bar c \gamma_{\mu} c$	$1^{–}$	$J/\psi$	$1^{–}$	$\langle 0 | I^{V}_{\mu} | J/\psi \rangle = m_{J/\psi} f_{J/\psi} \epsilon_{\mu}$	$f_{J/\psi} = 418$ MeV [83]
$I^{A}_{\mu} = \bar c \gamma_{\mu} \gamma_5 c$	$1^{++}$	$\eta_c$	$0^{-+}$	$\langle 0 | I^{A}_{\mu} | \eta_c \rangle = {\rm i} p_{\mu} f_{\eta_c}$	$f_{\eta_c} = 387$ MeV [83]
		$\chi_{c1}(1P)$	$1^{++}$	$\langle 0 | I^{A}_{\mu} | \chi_{c1} \rangle = m_{\chi_{c1}} f_{\chi_{c1}} \epsilon_{\mu} $	$f_{\chi_{c1}} = 335$ MeV [77]
$I^{T}_{\mu\nu} = \bar c \sigma_{\mu\nu} c$	$1^{\pm-}$	$J/\psi$	$1^{–}$	$\langle 0 | I^{T}_{\mu\nu} | J/\psi \rangle = {\rm i} f^T_{J/\psi} (p_{\mu}\epsilon_\nu - p_\nu\epsilon_{\mu})$	$f_{J/\psi}^T = 410$ MeV [83]
		$h_c(1P)$	$1^{+-}$	$\langle 0 | I^{T}_{\mu\nu} | h_c \rangle = {\rm i} f^T_{h_c} \epsilon_{\mu\nu\alpha\beta} \epsilon^\alpha p^\beta$	$f_{h_c}^T = 235$ MeV [83]
$O^{S} = \bar d c$	$0^{+}$	$D_0^{*+}$	$0^{+}$	$\langle 0 | O^{S} | D_0^{*+} \rangle = m_{D_0^{*}} f_{D_0^{*}}$	$f_{D_0^{*}} = 410$ MeV [108]
$O^{P} = \bar d i\gamma_5 c$	$0^{-}$	$D^+$	$0^{-}$	$\langle 0 | O^{P} | D^+ \rangle = \lambda_D$	$\lambda_D = \dfrac{f_D m_D^2 }{{m_c + m_d}}$
$O^{V}_{\mu} = \bar c \gamma_{\mu} u$	$1^{-}$	$\bar D^{*0}$	$1^{-}$	$\langle 0 | O^{V}_{\mu} | \bar D^{*0} \rangle = m_{D^*} f_{D^*} \epsilon_{\mu}$	$f_{D^*} = 253$ MeV [105]
$O^{A}_{\mu} = \bar c \gamma_{\mu} \gamma_5 u$	$1^{+}$	$\bar D^{0}$	$0^{-}$	$\langle 0 | O^{A}_{\mu} | \bar D^{0} \rangle = {\rm i} p_{\mu} f_{D}$	$f_{D} = 211.9$ MeV [1]
		$D_1$	$1^{+}$	$\langle 0 | O^{A}_{\mu} | D_1 \rangle = m_{D_1} f_{D_1} \epsilon_{\mu} $	$f_{D_1} = 356$ MeV [108]
$O^{T}_{\mu\nu} = \bar d \sigma_{\mu\nu} c$	$1^{\pm}$	$\bar D^{*+}$	$1^{-}$	$\langle 0 | O^{T}_{\mu\nu} | D^{*+} \rangle = {\rm i} f_{D^*}^T (p_{\mu}\epsilon_\nu - p_\nu\epsilon_{\mu})$	$f_{D^*}^T \approx 220$ MeV
		–	$1^{+}$	–	–

Table2.Couplings of meson operators to meson states. Color indices are omitted for simplicity.

The current $ \eta^{{\cal{Z}}}_{\mu} $ can also be investigated from the viewpoint of QCD sum rules [60, 61]. We assume it couples to the $ Z_c(3900) $ through

$ \langle 0 | \eta^{{\cal{Z}}}_{\mu} | Z_c \rangle = f_{Z_c} \epsilon_{\mu} \, . $

(6)

After the Fierz rearrangement, $ \eta^{{\cal{Z}}}_{\mu} $ transforms to the long expression inside Eq. (5). Through the first and second terms, it couples to the $ \eta_c \rho $ and $ J/\psi \pi $ channels simultaneously:

$ \begin{split} & \langle 0 | \eta^{{\cal{Z}}}_{\mu} | \eta_c \rho \rangle = {1\over3} \langle 0 | \bar c_{a} \gamma_5 c_a | \eta_c \rangle \langle 0 | \bar q_{b} \gamma_{\mu} q_b | \rho \rangle + \cdots \, , \\ & \langle 0 | \eta^{{\cal{Z}}}_{\mu} | J/\psi \pi \rangle = - {1\over3} \langle 0 | \bar c_{a} \gamma_{\mu} c_a | J/\psi \rangle \langle 0 | \bar q_{b} \gamma_5 q_b | \pi \rangle + \cdots \, . \end{split} $

(7)

Again, these two equations can be easily used to calculate the relative branching ratio $ {\cal{R}}_{Z_c} $. Detailed discussions on this will be given below.
In the above equations, we have worked within the naive factorization scheme, so our uncertainty is significantly larger than the well-developed QCD factorization method [62-64], which has been widely and successfully applied to study weak and radiative decay properties of conventional (heavy) hadrons, e.g., see Refs. [65, 66]. However, given that we still do not fully understand the internal structure of the $ Z_c(3900) $ (as well as all the other exotic hadrons), the naive factorization scheme at this moment can be useful. Besides, the tetraquark decay constant $ f_{Z_c} $ is removed when calculating relative branching ratios, which significantly reduces our uncertainty.
In this study, we shall examine the strong decay properties of the $ Z_c(3900) $ under the naive factorization scheme. To do this we just need to replace the weak-interaction Lagrangian by some interpolating current, and apply the similar technics here together with the Fierz arrangement. Note that a similar arrangement of the spin and color indices in the nonrelativistic case was used to study strong decay properties of the $ Z_c(3900) $ in Refs. [8, 67, 68].
This paper is organized as follows. In Sec. 2 we systematically construct all the tetraquark currents of $ J^{PC} = 1^{+-} $ with the quark content $ c \bar c q \bar q $. There are three configurations, $ [cq][\bar c \bar q] $, $ [\bar c q][\bar q c] $, and $ [\bar c c][\bar q q] $, and their relations are also derived in this section by using the Fierz rearrangement of the Dirac and color indices. In Sec. 3 we discuss the couplings of meson operators to meson states and list those which are needed in the present study. In Sec. 4 and Sec. 5 we extract some decay properties of the $ Z_c(3900) $, separately for the compact tetraquark interpretation and the hadronic molecule interpretation. The obtained results are discussed and summarized in Sec. 6.

By using the c, $ \bar c $, q, $ \bar q $ quarks ($ q = u/d $), one can construct three types of tetraquark currents, as illustrated in Fig. 1:
Figure1. (color online) Three types of tetraquark currents. Quarks are shown in red/green/blue color, and antiquarks are shown in cyan/magenta/yellow color.

$ \begin{split}& \eta(x,y) = [q^T_a(x)\; {\mathbb{C}} \Gamma_1\; c_b(x)] \times [\bar q_c(y)\; \Gamma_2 {\mathbb{C}}\; \bar c_d^T(y)] \, , \\ & \xi(x,y) = [\bar c_a(x)\; \Gamma_3\; q_b(x)] \times [\bar q_c(y)\; \Gamma_4\; c_d(y)] \, , \\ & \theta(x,y) = [\bar c_a(x)\; \Gamma_5\; c_b(x)] \times [\bar q_c(y)\; \Gamma_6\; q_d(y)] \, , \end{split} $

(8)

where $ \Gamma_i $ are the Dirac matrices, $ {\mathbb{C}} $ is the charge-conjugation matrix, the subscripts $ a, b, c, d $ are color indices, and the sum over repeated indices is taken. One typically calls $ \eta(x,y) $ the diquark-antidiquark current, and $ \xi(x,y) $ and $ \theta(x,y) $ the mesonic-mesonic currents. We separately construct them as follows:
2

2.1.$ [qc][\bar q \bar c] $ currents $ \eta_{\mu}^i(x,y) $

$ \begin{split} & \eta^1_{\mu} = q_a^T{\mathbb{C}}\gamma_{\mu} c_b \; \bar q_{a} \gamma_5{\mathbb{C}}\bar c_{b}^T - q_a^T{\mathbb{C}}\gamma_5 c_b \; \bar q_{a} \gamma_{\mu}{\mathbb{C}}\bar c_{b}^T \, , \\ & \eta^2_{\mu} = q_a^T{\mathbb{C}}\gamma_{\mu} c_b \; \bar q_{b} \gamma_5{\mathbb{C}}\bar c_{a}^T - q_a^T{\mathbb{C}}\gamma_5 c_b \; \bar q_{b} \gamma_{\mu}{\mathbb{C}}\bar c_{a}^T \, , \\ & \eta^3_{\mu} = q_a^T{\mathbb{C}}\gamma^\nu c_b \; \bar q_{a} \sigma_{\mu\nu} \gamma_5{\mathbb{C}}\bar c_{b}^T - q_a^T{\mathbb{C}}\sigma_{\mu\nu} \gamma_5 c_b \; \bar q_{a} \gamma^\nu{\mathbb{C}}\bar c_{b}^T \, , \\ & \eta^4_{\mu} = q_a^T{\mathbb{C}}\gamma^\nu c_b \; \bar q_{b} \sigma_{\mu\nu} \gamma_5{\mathbb{C}}\bar c_{a}^T - q_a^T{\mathbb{C}}\sigma_{\mu\nu} \gamma_5 c_b \; \bar q_{b} \gamma^\nu{\mathbb{C}}\bar c_{a}^T \, , \\ & \eta^5_{\mu} = q_a^T{\mathbb{C}}\gamma_{\mu} \gamma_5 c_b \; \bar q_{a}{\mathbb{C}}\bar c_{b}^T - q_a^T{\mathbb{C}}c_b \; \bar q_{a} \gamma_{\mu} \gamma_5{\mathbb{C}}\bar c_{b}^T \, , \\ & \eta^6_{\mu} = q_a^T{\mathbb{C}}\gamma_{\mu} \gamma_5 c_b \; \bar q_{b}{\mathbb{C}}\bar c_{a}^T - q_a^T{\mathbb{C}}c_b \; \bar q_{b} \gamma_{\mu} \gamma_5{\mathbb{C}}\bar c_{a}^T \, , \\ & \eta^7_{\mu} = q_a^T{\mathbb{C}}\gamma^\nu \gamma_5 c_b \; \bar q_{a} \sigma_{\mu\nu}{\mathbb{C}}\bar c_{b}^T - q_a^T{\mathbb{C}}\sigma_{\mu\nu} c_b \; \bar q_{a} \gamma^\nu \gamma_5{\mathbb{C}}\bar c_{b}^T \, , \\ & \eta^8_{\mu} = q_a^T{\mathbb{C}}\gamma^\nu \gamma_5 c_b \; \bar q_{b} \sigma_{\mu\nu}{\mathbb{C}}\bar c_{a}^T - q_a^T{\mathbb{C}}\sigma_{\mu\nu} c_b \; \bar q_{b} \gamma^\nu \gamma_5{\mathbb{C}}\bar c_{a}^T \, . \end{split} $

(9)

$ |0_{qc}1_{\bar q \bar c}; 1^{+-} \rangle = {1\over\sqrt2} \left(| 0_{qc}, 1_{\bar q \bar c} \rangle_{J = 1} - |1_{qc}, 0_{\bar q \bar c} \rangle_{J = 1} \right) \, . $

(10)

$ \begin{split} \eta^{{\cal{Z}}}_{\mu}(x,y) & = \eta^1_{\mu}([uc][\bar d \bar c]) - \eta^2_{\mu}([uc][\bar d \bar c]) \\ & = u_a^T(x){\mathbb{C}}\gamma_{\mu} c_b(x) \\&\times\left( \bar d_{a}(y) \gamma_5{\mathbb{C}}\bar c_{b}^T(y) - \{ a \leftrightarrow b \} \right) \\& - \; \{ \gamma_{\mu} \leftrightarrow \gamma_5 \} \, . \end{split} $

(11)

2.2.$ [\bar c q][\bar q c] $ currents $ \xi_{\mu}^i(x,y) $

$ \begin{split} & \xi^1_{\mu} = \bar c_{a} \gamma_{\mu} q_a \; \bar q_{b} \gamma_5 c_b + \bar c_{a} \gamma_5 q_a \; \bar q_{b} \gamma_{\mu} c_b \, , \\& \xi^2_{\mu} = \bar c_{a} \gamma^\nu q_a \; \bar q_{b} \sigma_{\mu\nu} \gamma_5 c_b - \bar c_{a} \sigma_{\mu\nu} \gamma_5 q_a \; \bar q_{b} \gamma^\nu c_b \, , \\ & \xi^3_{\mu} = \bar c_{a} \gamma_{\mu} \gamma_5 q_a \; \bar q_{b} c_b - \bar c_{a} q_a \; \bar q_{b} \gamma_{\mu} \gamma_5 c_b \, , \\& \xi^4_{\mu} = \bar c_{a} \gamma^\nu \gamma_5 q_a \; \bar q_{b} \sigma_{\mu\nu} c_b + \bar c_{a} \sigma_{\mu\nu} q_a \; \bar q_{b} \gamma^\nu \gamma_5 c_b \, , \\& \xi^5_{\mu} = {\lambda^n_{ab}}{\lambda^n_{cd}} \; \left( \bar c_{a} \gamma_{\mu} q_b \; \bar q_{c} \gamma_5 c_d + \bar c_{a} \gamma_5 q_b \; \bar q_{c} \gamma_{\mu} c_d \right) \, , \end{split} $

$ \begin{split} & \xi^6_{\mu} = {\lambda^n_{ab}}{\lambda^n_{cd}} \; \left( \bar c_{a} \gamma^\nu q_b \; \bar q_{c} \sigma_{\mu\nu} \gamma_5 c_d - \bar c_{a} \sigma_{\mu\nu} \gamma_5 q_b \; \bar q_{c} \gamma^\nu c_d \right) \, , \\ & \xi^7_{\mu} = {\lambda^n_{ab}}{\lambda^n_{cd}} \; \left( \bar c_{a} \gamma_{\mu} \gamma_5 q_b \; \bar q_{c} c_d - \bar c_{a} q_b \; \bar q_{c} \gamma_{\mu} \gamma_5 c_d \right) \, , \\& \xi^8_{\mu} = {\lambda^n_{ab}}{\lambda^n_{cd}} \; \left( \bar c_{a} \gamma^\nu \gamma_5 q_b \; \bar q_{c} \sigma_{\mu\nu} c_d + \bar c_{a} \sigma_{\mu\nu} q_b \; \bar q_{c} \gamma^\nu \gamma_5 c_d \right) \, . \end{split} $

(12)

$ | D \bar D^*; 1^{+-} \rangle = {1\over\sqrt2} \left(| D \bar D^* \rangle_{J = 1} - | \bar D D^* \rangle_{J = 1} \right) \, , $

(13)

$ \begin{split} \xi^{{\cal{Z}}}_{\mu}(x,y) & = \xi^1_{\mu}([\bar c u][\bar d c]) \\& = \bar c_{a}(x) \gamma_{\mu} u_a(x) \; \bar d_{b}(y) \gamma_5 c_b(y) + \{ \gamma_{\mu} \leftrightarrow \gamma_5 \} \, . \end{split} $

(14)

2.3.$ [\bar c c][\bar q q] $ currents $ \theta_{\mu}^i(x,y) $

$ \begin{split} & \theta^1_{\mu}(x,y) = \bar c_{a}(x) \gamma_5 c_a(x) \; \bar q_{b}(y) \gamma_{\mu} q_b(y) \, , \\ & \theta^2_{\mu}(x,y) = \bar c_{a}(x) \gamma_{\mu} c_a(x) \; \bar q_{b}(y) \gamma_5 q_b(y) \, , \\ & \theta^3_{\mu}(x,y) = \bar c_{a}(x) \gamma^\nu \gamma_5 c_a(x) \; \bar q_{b}(y) \sigma_{\mu\nu} q_b(y) \, , \\ & \theta^4_{\mu}(x,y) = \bar c_{a}(x) \sigma_{\mu\nu} c_a(x) \; \bar q_{b}(y) \gamma^\nu \gamma_5 q_b(y) \, , \\ & \theta^5_{\mu}(x,y) = {\lambda^n_{ab}}{\lambda^n_{cd}} \; \bar c_{a}(x) \gamma_5 c_b(x) \; \bar q_{c}(y) \gamma_{\mu} q_d(y) \, , \\& \theta^6_{\mu}(x,y) = {\lambda^n_{ab}}{\lambda^n_{cd}} \; \bar c_{a}(x) \gamma_{\mu} c_b(x) \; \bar q_{c}(y) \gamma_5 q_d(y) \, , \\ & \theta^7_{\mu}(x,y) = {\lambda^n_{ab}}{\lambda^n_{cd}} \; \bar c_{a}(x) \gamma^\nu \gamma_5 c_b(x) \; \bar q_{c}(y) \sigma_{\mu\nu} q_d(y) \, , \\ & \theta^8_{\mu}(x,y) = {\lambda^n_{ab}}{\lambda^n_{cd}} \; \bar c_{a}(x) \sigma_{\mu\nu} c_b(x) \; \bar q_{c}(y) \gamma^\nu \gamma_5 q_d(y) \, . \end{split} $

(15)

$ \left( {\begin{array}{*{20}{c}}{\eta _\mu ^1}\\{\eta _\mu ^2}\\{\eta _\mu ^3}\\{\eta _\mu ^4}\\{\eta _\mu ^5}\\{\eta _\mu ^6}\\{\eta _\mu ^7}\\{\eta _\mu ^8}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}{1/2}&{ - 1/2}&{i/2}&{ - i/2}&0&0&0&0\\{1/6}&{ - 1/6}&{i/6}&{ - i/6}&{1/4}&{ - 1/4}&{i/4}&{ - i/4}\\{3i/2}&{3i/2}&{ - 1/2}&{ - 1/2}&0&0&0&0\\{i/2}&{i/2}&{ - 1/6}&{ - 1/6}&{3i/4}&{3i/4}&{ - 1/4}&{ - 1/4}\\{1/2}&{1/2}&{ - i/2}&{ - i/2}&0&0&0&0\\{1/6}&{1/6}&{ - i/6}&{ - i/6}&{1/4}&{1/4}&{ - i/4}&{ - i/4}\\{3i/2}&{ - 3i/2}&{1/2}&{ - 1/2}&0&0&0&0\\{i/2}&{ - i/2}&{1/6}&{ - 1/6}&{3i/4}&{ - 3i/4}&{1/4}&{ - 1/4}\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}{\theta _\mu ^1}\\{\theta _\mu ^2}\\{\theta _\mu ^3}\\{\theta _\mu ^4}\\{\theta _\mu ^5}\\{\theta _\mu ^6}\\{\theta _\mu ^7}\\{\theta _\mu ^8}\end{array}} \right){\mkern 1mu} , $

(16)

$ \left( {\begin{array}{*{20}{c}}{\eta _\mu ^1}\\{\eta _\mu ^2}\\{\eta _\mu ^3}\\{\eta _\mu ^4}\\{\eta _\mu ^5}\\{\eta _\mu ^6}\\{\eta _\mu ^7}\\{\eta _\mu ^8}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}0&{i/6}&{ - 1/6}&0&0&{i/4}&{ - 1/4}&0\\0&{i/2}&{ - 1/2}&0&0&0&0&0\\{ - i/2}&0&0&{1/6}&{ - 3i/4}&0&0&{1/4}\\{ - 3i/2}&0&0&{1/2}&0&0&0&0\\{1/6}&0&0&{ - i/6}&{1/4}&0&0&{ - i/4}\\{1/2}&0&0&{ - i/2}&0&0&0&0\\0&{ - 1/6}&{i/2}&0&0&{ - 1/4}&{3i/4}&0\\0&{ - 1/2}&{3i/2}&0&0&0&0&0\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}{\xi _\mu ^1}\\{\xi _\mu ^2}\\{\xi _\mu ^3}\\{\xi _\mu ^4}\\{\xi _\mu ^5}\\{\xi _\mu ^6}\\{\xi _\mu ^7}\\{\xi _\mu ^8}\end{array}} \right){\mkern 1mu} , $

(17)

$ \left( {\begin{array}{*{20}{c}}{\eta _\mu ^1}\\{\eta _\mu ^2}\\{\eta _\mu ^3}\\{\eta _\mu ^4}\\{\eta _\mu ^5}\\{\eta _\mu ^6}\\{\eta _\mu ^7}\\{\eta _\mu ^8}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}0&0&0&0&{1/2}&{ - 1/2}&{i/2}&{ - i/2}\\0&{i/2}&{ - 1/2}&0&0&0&0&0\\0&0&0&0&{3i/2}&{3i/2}&{ - 1/2}&{ - 1/2}\\{ - 3i/2}&0&0&{1/2}&0&0&0&0\\0&0&0&0&{1/2}&{1/2}&{ - i/2}&{ - i/2}\\{1/2}&0&0&{ - i/2}&0&0&0&0\\0&0&0&0&{3i/2}&{ - 3i/2}&{1/2}&{ - 1/2}\\0&{ - 1/2}&{3i/2}&0&0&0&0&0\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}{\xi _\mu ^1}\\{\xi _\mu ^2}\\{\xi _\mu ^3}\\{\xi _\mu ^4}\\{\theta _\mu ^1}\\{\theta _\mu ^2}\\{\theta _\mu ^3}\\{\theta _\mu ^4}\end{array}} \right){\mkern 1mu} , $

(18)

$ \left( {\begin{array}{*{20}{c}}{\xi _\mu ^1}\\{\xi _\mu ^2}\\{\xi _\mu ^3}\\{\xi _\mu ^4}\\{\xi _\mu ^5}\\{\xi _\mu ^6}\\{\xi _\mu ^7}\\{\xi _\mu ^8}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}{ - 1/6}&{ - 1/6}&{ - i/6}&{ - i/6}&{ - 1/4}&{ - 1/4}&{ - i/4}&{ - i/4}\\{ - i/2}&{i/2}&{1/6}&{ - 1/6}&{ - 3i/4}&{3i/4}&{1/4}&{ - 1/4}\\{1/6}&{ - 1/6}&{ - i/6}&{i/6}&{1/4}&{ - 1/4}&{ - i/4}&{i/4}\\{i/2}&{i/2}&{1/6}&{1/6}&{3i/4}&{3i/4}&{1/4}&{1/4}\\{ - 8/9}&{ - 8/9}&{ - 8i/9}&{ - 8i/9}&{1/6}&{1/6}&{i/6}&{i/6}\\{ - 8i/3}&{8i/3}&{8/9}&{ - 8/9}&{i/2}&{ - i/2}&{ - 1/6}&{1/6}\\{8/9}&{ - 8/9}&{ - 8i/9}&{8i/9}&{ - 1/6}&{1/6}&{i/6}&{ - i/6}\\{8i/3}&{8i/3}&{8/9}&{8/9}&{ - i/2}&{ - i/2}&{ - 1/6}&{ - 1/6}\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}{\theta _\mu ^1}\\{\theta _\mu ^2}\\{\theta _\mu ^3}\\{\theta _\mu ^4}\\{\theta _\mu ^5}\\{\theta _\mu ^6}\\{\theta _\mu ^7}\\{\theta _\mu ^8}\end{array}} \right){\mkern 1mu} . $

(19)

There are altogether six types of meson operators: $ \bar q_a q_a $, $ \bar q_a \gamma_5 q_a $, $ \bar q_a \gamma_{\mu} q_a $, $ \bar q_a \gamma_{\mu} \gamma_5 q_a $, $ \bar q_a \sigma_{\mu\nu} q_a $, and $ \bar q_a \sigma_{\mu\nu} \gamma_5 q_a $. The last two can be related to each other through

$ \sigma_{\mu\nu} \gamma_5 = {{\rm i}\over2} \epsilon_{\mu\nu\rho\sigma} \sigma^{\rho\sigma} \, . $

(20)

The couplings of these operators to meson states are already well understood, i.e., some of them have been measured in particle experiments, and some of them have been studied and calculated by various theoretical methods, such as Lattice QCD and QCD sum rules, etc.
In this study, we require the following couplings, as summarized in Table 2:
1. The scalar operators $ J^{S} = \bar q_a q_a $ and $ I^{S} = \bar c_a c_a $ of $ J^{PC} = 0^{++} $ couple to scalar mesons. In Ref. [76] the authors used the method of QCD sum rules and extracted the coupling of $ I^{S} $ to $ \chi_{c0}(1P) $ to be

$ \langle 0 | \bar c_a c_a | \chi_{c0}(p) \rangle = m_{\chi_{c0}} f_{\chi_{c0}} \, , $

(21)

where

$ f_{\chi_{c0}} = 343\; {\rm{MeV}} \, . $

(22)

See also discussions in Refs. [77-79]. The light scalar mesons have a complicated nature [80], so we shall not investigate their relevant decay channels in this study.
2. The pseudoscalar operators $J^{P} = \bar q_a {\rm i}\gamma_5 q_a$ and $I^{P} = \bar c_a {\rm i}\gamma_5 c_a$ of $ J^{PC} = 0^{-+} $ couple to the pseudoscalar mesons $ \pi $ and $ \eta_c $, respectively. We can evaluate them through [81]:

$ \begin{split} & \langle 0 | \bar d_a {\rm i}\gamma_5 u_a | \pi^+(p) \rangle = \lambda_{\pi} = {f_{\pi^+} m_{\pi^+}^2 \over m_u + m_d} \, , \\ & \;\;\;\;\langle 0 | \bar c_a {\rm i}\gamma_5 c_a | \eta_c(p) \rangle = \lambda_{\eta_c} = {f_{\eta_c} m_{\eta_c}^2 \over 2 m_c} \, . \end{split} $

(23)

3. The vector operators $ J^{V}_{\mu} = \bar q_a \gamma_{\mu} q_a $ and $ I^{V}_{\mu} = \bar c_a \gamma_{\mu} c_a $ of $ J^{PC} = 1^{–} $ couple to the vector mesons $ \rho $ and $ J/\psi $, respectively. In Refs. [82, 83] the authors used the method of Lattice QCD to obtain

$ \begin{split} & \langle 0 | \bar d_a \gamma_{\mu} u_a | \rho^+(p, \epsilon) \rangle = m_{\rho} f_{\rho^+} \epsilon_{\mu} \, , \\ & \langle 0 | \bar c_a \gamma_{\mu} c_a | J/\psi(p, \epsilon) \rangle = m_{J/\psi} f_{J/\psi} \epsilon_{\mu} \, , \end{split} $

(24)

where

$ \begin{array}{l} \;\;\;f_{\rho^+} = 208\; {\rm{MeV}} \, , \;\;\; f_{J/\psi} = 418\; {\rm{MeV}} \, . \end{array} $

(25)

See also discussions in Refs. [84-86].
4. The axialvector operators $ J^{A}_{\mu} = \bar q_a \gamma_{\mu} \gamma_5 q_a $ and $ I^{A}_{\mu} = \bar c_a \gamma_{\mu} \gamma_5 c_a $ of $ J^{PC} = 1^{++} $ couple to both the pseudoscalar mesons ($ \pi $ and $ \eta_c $ of $ J^{PC} = 0^{-+} $) and axialvector mesons ($ a_1(1260) $ and $ \chi_{c1}(1P) $ of $ J^{PC} = 1^{++} $). The coupling of $ J^{A}_{\mu} $ to $ \pi $ has been well measured in particle experiments [1]:

$ \langle 0 | \bar d_a \gamma_{\mu} \gamma_5 u_a | \pi^+(p) \rangle = {\rm i} p_{\mu} f_{\pi^+} \, , $

(26)

while its coupling to $ a_1(1260) $ was evaluated by using Lattice QCD [87]:

$ \langle 0 | \bar d_a \gamma_{\mu} \gamma_5 u_a | a_1(p, \epsilon) \rangle = m_{a_1} f_{a_1} \epsilon_{\mu} \, , $

(27)

where

$ \begin{array}{l} f_{\pi^+} = 130.2\; {\rm{MeV}} \, , \;\;\; f_{a_1} = 254\; {\rm{MeV}} \, . \end{array} $

(28)

The coupling of $ I^{A}_{\mu} $ to $ \eta_c $ and $ \chi_{c1}(1P) $ was evaluated by using Lattice QCD [83] and QCD sum rules [77]:

$ \begin{split} & \langle 0 | \bar c_a \gamma_{\mu} \gamma_5 c_a | \eta_c(p) \rangle = {\rm i} p_{\mu} f_{\eta_c} \, , \\ & \langle 0 | \bar c_a \gamma_{\mu} \gamma_5 c_a | \chi_{c1}(p, \epsilon) \rangle = m_{\chi_{c1}} f_{\chi_{c1}} \epsilon_{\mu} \, , \end{split} $

(29)

where

$ \begin{array}{l} f_{\eta_c} = 387\; {\rm{MeV}} \, , \;\;\; f_{\chi_{c1}} = 335\; {\rm{MeV}} \, . \end{array} $

(30)

See also discussions in Refs. [77, 85, 88-94].
5. The tensor operators $ J^{T}_{\mu\nu} = \bar q_a \sigma_{\mu\nu} q_a $ and $ I^{T}_{\mu\nu} = \bar c_a \sigma_{\mu\nu} c_a $ of $ J^{PC} = 1^{\pm-} $ couple to both the vector mesons ($ \rho $ and $ J/\psi $ of $ J^{PC} = 1^{–} $) and the axialvector mesons ($ b_1(1235) $ and $ h_{c}(1P) $ of $ J^{PC} = 1^{+-} $). The coupling of $ J^{T}_{\mu\nu} $ to $ \rho $ and $ b_1(1235) $ was calculated through Lattice QCD [82] and QCD sum rules [95]:

$ \begin{split} & \langle 0 | \bar d_a \sigma_{\mu\nu} u_a | \rho^+(p, \epsilon) \rangle = {\rm i} f^T_{\rho} (p_{\mu}\epsilon_\nu - p_\nu\epsilon_{\mu}) \, , \\& \langle 0 | \bar d_a \sigma_{\mu\nu} u_a | b_1(p, \epsilon) \rangle = {\rm i} f^T_{b_1} \epsilon_{\mu\nu\alpha\beta} \epsilon^\alpha p^\beta \, , \end{split} $

(31)

where

$ \begin{array}{l} f_{\rho}^T = 159\; {\rm{MeV}} \, , \;\;\; f_{b_1}^T = 180\; {\rm{MeV}} \, . \end{array} $

(32)

The coupling of $ I^{T}_{\mu\nu} $ to $ J/\psi $ and $ h_{c}(1P) $ was calculated through Lattice QCD [83]:

$ \begin{split}& \langle 0 | \bar c_a \sigma_{\mu\nu} c_a | J/\psi(p, \epsilon) \rangle = {\rm i} f^T_{J/\psi} (p_{\mu}\epsilon_\nu - p_\nu\epsilon_{\mu}) \, , \\ & \langle 0 | \bar c_a \sigma_{\mu\nu} c_a | h_c(p, \epsilon) \rangle = {\rm i} f^T_{h_c} \epsilon_{\mu\nu\alpha\beta} \epsilon^\alpha p^\beta \, , \end{split} $

(33)

where

$ \begin{array}{l} f_{J/\psi}^T = 410\; {\rm{MeV}} \, , \;\;\; f_{h_c}^T = 235\; {\rm{MeV}} \, . \end{array} $

(34)

See also discussions in Refs. [96-104].
6. The $ Z_c(3900) $ is above the $ D \bar D^* $ threshold; thus, we need the couplings of $O^{P} = \bar q_a {\rm i}\gamma_5 c_a$ and $ O^{A}_{\mu} = \bar c_a \gamma_{\mu} \gamma_5 q_a $ to the D meson [1]:

$ \begin{split} & \langle 0 | \bar d_a {\rm i}\gamma_5 c_a | D^+(p) \rangle = \lambda_D \, , \\ & \langle 0 | \bar c_a \gamma_{\mu} \gamma_5 u_a | \bar D^0(p) \rangle = {\rm i} p_{\mu} f_{D} \, , \end{split} $

(35)

and the couplings of $ O^{V}_{\mu} = \bar c_a \gamma_{\mu} q_a $ and $ O^{T}_{\mu\nu} = \bar q_a \sigma_{\mu\nu} c_a $ to the $ D^* $ meson [105]:

$ \begin{split} & \langle 0 | \bar c_a \gamma_{\mu} u_a | \bar D^{*0}(p, \epsilon) \rangle = m_{D^*} f_{D^*} \epsilon_{\mu} \, , \\ & \langle 0 | \bar d_a \sigma_{\mu\nu} c_a | D^{*+}(p, \epsilon) \rangle = {\rm i} f^T_{D^*} (p_{\mu}\epsilon_\nu - p_\nu\epsilon_{\mu}) \, , \end{split} $

(36)

where

$ \begin{array}{l} \lambda_D = \dfrac{{f_{D} m_{D^+}^2}}{m_c + m_d} \, , \;\; f_{D} = 211.9\; {\rm{MeV}} \, , \;\; f_{D^*} = 253\; {\rm{MeV}} \, . \end{array} $

(37)

We found no theoretical study on the transverse decay constant $ f^T_{D^*} $; thus, we simply fit it among the decay constants, $ f_{\pi^+} $-$ f_{\rho^+} $-$ f_{\rho}^T $, $ f_{\eta_c} $-$ f_{J/\psi} $-$ f_{J/\psi}^T $, and $ f_{D} $-$ f_{D^*} $-$ f_{D^*}^T $, to obtain

$ f_{D^*}^T \approx 220\; {\rm{MeV}} \, . $

(38)

See also discussions in Refs. [106, 107].
7. The $ Z_c(3900) \rightarrow D \bar D^*_0 \rightarrow D \bar D \pi $ decay is kinematically allowed, so we need the coupling of $ O^{S} = \bar q_a c_a $ to the $ D_0^* $ meson [108]:

$ \langle 0 | \bar d_a c_a | D_0^{*+}(p) \rangle = m_{D_0^{*}} f_{D_0^{*}} \, , $

(39)

where

$ f_{D_0^{*}} = 410\; {\rm{MeV}} \, . $

(40)

See also discussions in Refs. [109, 110].

4.1.$ \eta^{{\cal{Z}}}_{\mu}\big([uc][\bar d \bar c]\big) \rightarrow \theta_{\mu}^i \big([\bar c c] + [\bar d u]\big) $

$ \begin{split} [u(x) c(x)]\; [\bar d(y) \bar c(y)] \Longrightarrow & [u(x \to y^{\prime})\; c(x \to x^{\prime})]\\&\times [\bar d(y \to y^{\prime})\; \bar c(y \to x^{\prime})] \\ \Longrightarrow & [\bar c(x^{\prime}) c(x^{\prime})] + [\bar d(y^{\prime}) u(y^{\prime})] \, .\end{split} $

(41)

$ \begin{split} \eta^{{\cal{Z}}}_{\mu}(x,y) & \Longrightarrow + {1\over3}\; \theta_{\mu}^1(x^{\prime},y^{\prime}) - {1\over3}\; \theta_{\mu}^2(x^{\prime},y^{\prime}) \\& \;\;\;\;\;\;\;\; + {{\rm i}\over3}\; \theta_{\mu}^3(x^{\prime},y^{\prime}) - {{\rm i}\over3}\; \theta_{\mu}^4(x^{\prime},y^{\prime}) + \cdots \\& \;\;\;\;= - {{\rm i}\over3}\; I^{P}(x^{\prime}) \; J^{V}_{\mu}(y^{\prime}) + {{\rm i}\over3}\; I^{V}_{\mu}(x^{\prime}) \; J^{P}(y^{\prime}) \\& \;\;\;\;\;\;\;\; + {{\rm i}\over3}\; I^{A,\nu}(x^{\prime}) \; J^{T}_{\mu\nu}(y^{\prime}) - {{\rm i}\over3}\; I^{T}_{\mu\nu}(x^{\prime}) \; J^{A,\nu}(y^{\prime}) + \cdots \, , \end{split} $

(42)

$ \begin{split}& \langle Z_c^+(p,\epsilon) | \eta_c(p_1)\; \rho^+(p_2,\epsilon_2) \rangle \\\approx & - {{\rm i} c_1\over3}\; \lambda_{\eta_c} m_\rho f_{\rho^+}\; \epsilon \cdot \epsilon_2 \\& - {{\rm i} c_1\over3}\; f_{\eta_c} f^T_{\rho}\; (\epsilon \cdot p_2 \; \epsilon_2 \cdot p_1 - p_1 \cdot p_2\; \epsilon \cdot \epsilon_2) \\\equiv & g^S_{\eta_c \rho}\; \epsilon \cdot \epsilon_2 + g^D_{\eta_c \rho}\; (\epsilon \cdot p_2 \; \epsilon_2 \cdot p_1 - p_1 \cdot p_2\; \epsilon \cdot \epsilon_2) \, , \end{split} $

(43)

$ {\cal{L}}^S_{\eta_c \rho} = g^S_{\eta_c \rho}\; Z_{c}^{+,\mu}\; \eta_c\; \rho^{-}_{\mu} + \cdots \, , $

(44)

$ \begin{split}\quad\quad {\cal{L}}^D_{\eta_c \rho} =& g^D_{\eta_c \rho} \times \left( g^{\mu\sigma}g^{\nu\rho} - g^{\mu\nu}g^{\rho\sigma} \right) \\&\times Z_{c,\mu}^{+}\; \partial_\rho \eta_c\; \partial_\sigma \rho^{-}_{\nu} + \cdots \, . \end{split} $

(45)

$ \begin{split}& \langle Z_c^+(p,\epsilon) | J/\psi(p_1,\epsilon_1)\; \pi^+(p_2) \rangle \\ \approx & {{\rm i} c_1 \over3}\; \lambda_{\pi} m_{J/\psi} f_{J/\psi}\; \epsilon \cdot \epsilon_1 \\&+ {{\rm i} c_1 \over3}\; f_{\pi^+} f^T_{J/\psi}\; (\epsilon \cdot p_1 \; \epsilon_1 \cdot p_2 - p_1 \cdot p_2\; \epsilon \cdot \epsilon_1) \\\equiv & g^S_{\psi \pi}\; \epsilon \cdot \epsilon_1 + g^D_{\psi \pi}\; (\epsilon \cdot p_1 \; \epsilon_1 \cdot p_2 - p_1 \cdot p_2\; \epsilon \cdot \epsilon_1) \, . \end{split} $

(46)

$ {\cal{L}}^S_{\psi \pi} = g^S_{\psi \pi}\; Z_{c}^{+,\mu}\; \psi_{\mu}\; \pi^- + \cdots \, , $

(47)

$ \begin{split}{\cal{L}}^D_{\psi \pi} = g^D_{\psi \pi} \times \left( g^{\mu\rho}g^{\nu\sigma} - g^{\mu\nu}g^{\rho\sigma} \right) \times Z_{c,\mu}^{+}\; \partial_\rho \psi_\nu\; \partial_\sigma \pi^- + \cdots \, .\quad\quad \end{split} $

(48)

$ \begin{split}\langle Z_c^+(p,\epsilon) | \eta_c(p_1)\; b_1^+(p_2,\epsilon_2) \rangle &\approx - {{\rm i} c_1 \over3}\; f_{\eta_c} f^T_{b_1}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu p_1^\nu \epsilon_2^\alpha p_2^\beta \\&\equiv g_{\eta_c b_1}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu p_1^\nu \epsilon_2^\alpha p_2^\beta \, . \end{split} $

(49)

$ \begin{split}&\;\;\;\;\;\; \langle Z_c^+(p,\epsilon) | \chi_{c1}(p_1,\epsilon_1)\; \rho^+(p_2,\epsilon_2) \rangle \\&\approx - {c_1\over3}\; m_{\chi_{c1}} f_{\chi_{c1}} f^T_{\rho}\; (\epsilon_1 \cdot \epsilon_2\; \epsilon \cdot p_2 - \epsilon_1 \cdot p_2\; \epsilon \cdot \epsilon_2) \\&\equiv g_{\chi_{c1} \rho}\; (\epsilon_1 \cdot \epsilon_2\; \epsilon \cdot p_2 - \epsilon_1 \cdot p_2\; \epsilon \cdot \epsilon_2) \, . \end{split} $

(50)

$ \begin{split} &\;\;\;\;\;\; \langle Z_c^+(p,\epsilon) | \chi_{c1}(p_1,\epsilon_1)\; b_1^+(p_2,\epsilon_2) \rangle \\&\approx - {c_1\over3}\; m_{\chi_{c1}} f_{\chi_{c1}} f^T_{b_1}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu \epsilon_1^\nu \epsilon_2^\alpha p_2^\beta \\&\equiv g_{\chi_{c1} b_1}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu \epsilon_1^\nu \epsilon_2^\alpha p_2^\beta \, . \end{split} $

(51)

$ \begin{split}&\;\;\;\;\;\; \langle Z_c^+(p,\epsilon) | h_c(p_1,\epsilon_1)\; \pi^+(p_2) \rangle \\&\approx {{\rm i} c_1\over3}\; f_{\pi^+} f^T_{h_c}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu p_2^\nu \epsilon_1^\alpha p_1^\beta \\&\equiv g_{h_c \pi}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu p_2^\nu \epsilon_1^\alpha p_1^\beta \, . \end{split} $

(52)

$ \begin{split}&\;\;\;\;\; \langle Z_c^+(p,\epsilon) | J/\psi(p_1,\epsilon_1)\; a_1^+(p_2,\epsilon_2) \rangle \\&\approx {c_1\over3}\; f^T_{J/\psi} m_{a_1} f_{a_1}\; (\epsilon_1 \cdot \epsilon_2\; \epsilon \cdot p_1 - \epsilon_2 \cdot p_1\; \epsilon \cdot \epsilon_1) \\&\equiv g_{\psi a_1}\; (\epsilon_1 \cdot \epsilon_2\; \epsilon \cdot p_1 - \epsilon_2 \cdot p_1\; \epsilon \cdot \epsilon_1) \, . \end{split} $

(53)

$ \begin{split}&\;\;\;\;\;\langle Z_c^+(p,\epsilon) | h_c(p_1,\epsilon_1)\; a_1^+(p_2,\epsilon_2) \rangle \\&\approx {c_1\over3}\; f^T_{h_c} m_{a_1} f_{a_1}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu \epsilon_2^\nu \epsilon_1^\alpha p_1^\beta \\&\equiv g_{h_c a_1}\; \epsilon_{\mu\nu\alpha\beta} \epsilon^\mu \epsilon_2^\nu \epsilon_1^\alpha p_1^\beta \, . \end{split} $

(54)

$ \begin{split}& g^S_{\eta_c \rho} = - {\rm i} c_1\; 7.29 \times 10^{10}\; {\rm{MeV}}^4 \, , \\ & g^D_{\eta_c \rho} = - {\rm i} c_1\; 2.05 \times 10^{4}\; {\rm{MeV}}^2 \, , \\ & g^S_{\psi \pi} = {\rm i} c_1\; 11.87 \times 10^{10}\; {\rm{MeV}}^4 \, , \\ & g^D_{\psi \pi} = {\rm i} c_1\; 1.78 \times 10^{4}\; {\rm{MeV}}^2 \, , \\ & g_{\eta_{c} b_1} = - {\rm i} c_1\; 2.32 \times 10^{4}\; {\rm{MeV}}^2 \, , \\ & g_{\chi_{c1} \rho} = - c_1\; 6.23 \times 10^{7}\; {\rm{MeV}}^3 \, , \\ & g_{\chi_{c1} b_1} = - c_1\; 7.06 \times 10^{7}\; {\rm{MeV}}^3 \, , \\ & g_{h_c \pi} = {\rm i} c_1\; 1.02 \times 10^{4}\; {\rm{MeV}}^2 \, , \\ & g_{\psi a_1} = c_1\; 4.27 \times 10^{7}\; {\rm{MeV}}^3 \, , \\ & g_{h_c a_1} = c_1\; 2.45 \times 10^{7}\; {\rm{MeV}}^3 \, . \end{split} $

(55)

$ \begin{split} &{{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.059 \, , \\ & {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow h_c\pi) \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.0088 \, , \\& {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \chi_{c1}\rho \rightarrow \chi_{c1}\pi \pi) \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 1.4 \times 10^{-6} \, . \end{split} $

(56)

$ \begin{split}& |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c b_1 \rightarrow \eta_c \omega \pi \rightarrow \eta_c + 4 \pi \, , \\ & |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi a_1 \rightarrow J/\psi \rho \pi \rightarrow J/\psi + 3 \pi \, . \end{split} $

(57)

4.2.$ \eta^{{\cal{Z}}}_{\mu}\big([uc][\bar d \bar c]\big) \rightarrow \xi_{\mu}^i \big([\bar c u] + [\bar d c]\big) $

$ \eta^{{\cal{Z}}}_{\mu}(x,y) \Longrightarrow - {{\rm i}\over3}\; \xi_{\mu}^2(x^{\prime},y^{\prime}) + {1\over3}\; \xi_{\mu}^3(x^{\prime},y^{\prime}) + \cdots \, . $

(58)

$ \begin{split} \langle Z_c^+(p,\epsilon) | \bar D^0(p_1)D_0^{*+}(p_2) \rangle \approx & \dfrac{{\rm i} c_2}{3}\; f_D m_{D_0^*} f_{D_0^*}\; \epsilon \cdot p_1 \\ \equiv & g_{D \bar D_0^*}\; \epsilon \cdot p_1 \, , \end{split} $

(59)

$ \begin{split}\langle Z_c^+(p,\epsilon) | D^+(p_1)\bar D_0^{*0}(p_2) \rangle &\approx- \dfrac{{\rm i} c_2}{3}\; f_D m_{D_0^*} f_{D_0^*}\; \epsilon \cdot p_1\\ &\equiv - g_{D \bar D_0^*}\; \epsilon \cdot p_1 \, , \end{split} $

(60)

$ g_{D \bar D_0^*} = {\rm i}c_2\; 6.80 \times 10^{7}\; {\rm{MeV}}^3 \, . $

(61)

$ {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi) \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi+ \eta_c\rho)} = 9.3 \times 10^{-8} \times {c_2^2 \over c_1^2} \, . $

(62)

$ g_{D \bar D^{*}} \approx 0 \, , $

(63)

$ {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow D \bar D^{*} + \bar D D^{*}) \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} \approx 0 \, . $

(64)

4.3.$ \eta^{{\cal{Z}}}_{\mu}\big([uc][\bar d \bar c]\big) \rightarrow \theta_{\mu}^{1,2,3,4}\big([\bar c c] + [\bar d u]\big) + \xi_{\mu}^{1,2,3,4}\big([\bar c u] + [\bar d c]\big) $

$ \begin{split} \eta^{{\cal{Z}}}_{\mu}(x,y) \Longrightarrow & + {1\over2}\; \theta_{\mu}^1(x^{\prime},y^{\prime}) - {1\over2}\; \theta_{\mu}^2(x^{\prime},y^{\prime}) + {{\rm i}\over2}\; \theta_{\mu}^3(x^{\prime},y^{\prime}) \\ & - {{\rm i}\over2}\; \theta_{\mu}^4(x^{\prime},y^{\prime}) - {{\rm i}\over2}\; \xi_{\mu}^2(x^{\prime\prime},y^{\prime\prime}) + {1\over2}\; \xi_{\mu}^3(x^{\prime\prime},y^{\prime\prime}) \, . \end{split} $

(65)

4.4.Mixing with $ |1_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle $

$ |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle = {1\over\sqrt2} \left(| 0_{qc}, 1_{\bar q \bar c} \rangle_{J = 1} - |1_{qc}, 0_{\bar q \bar c} \rangle_{J = 1} \right) \, , $

$ |x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle = \cos \theta_1 \; |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle + \sin \theta_1 \; |1_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \, , $

(66)

$ |1_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle = | 1_{qc}, 1_{\bar q \bar c} \rangle_{J = 1} \, , $

(67)

$ \eta^{{\cal{Z}}^{\prime}}_{\mu}(x,y) = \eta^3_{\mu}([uc][\bar d \bar c]) - \eta^4_{\mu}([uc][\bar d \bar c]) \, , $

(68)

$ \eta^{\rm{mix}}_{\mu}(x,y) = \cos \theta^{\prime}_1\; \eta^{{\cal{Z}}}_{\mu}(x,y) + {\rm i} \sin \theta^{\prime}_1\; \eta^{{\cal{Z}}^{\prime}}_{\mu}(x,y)\, , $

(69)

$ \begin{split} \eta^{\rm{mix}}_{\mu}(x,y) \Longrightarrow & + \left( -{{\rm i}\over3} \cos \theta^{\prime}_1 + {\rm i} \sin \theta^{\prime}_1 \right) \; I^{P}(x^{\prime}) \; J^{V}_{\mu}(y^{\prime}) \\ & + \left( + {{\rm i}\over3} \cos \theta^{\prime}_1 + {\rm i} \sin \theta^{\prime}_1 \right) \; I^{V}_{\mu}(x^{\prime}) \; J^{P}(y^{\prime}) \\ & + \left( + {{\rm i}\over3} \cos \theta^{\prime}_1 - {{\rm i}\over3} \sin \theta^{\prime}_1 \right) \; I^{A,\nu}(x^{\prime}) \; J^{T}_{\mu\nu}(y^{\prime}) \\ & + \left( - {{\rm i}\over3} \cos \theta^{\prime}_1 - {{\rm i}\over3} \sin \theta^{\prime}_1 \right) \; I^{T}_{\mu\nu}(x^{\prime}) \; J^{A,\nu}(y^{\prime}) + \cdots \, . \end{split} $

(70)

$ \theta_1 = f(\theta^{\prime}_1) \, . $

(71)

$ \begin{split} & {\cal{R}}_{\psi \pi} \equiv {\Gamma(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi) \over \Gamma(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} \, , \\ & {\cal{R}}_{\eta_c \rho} \equiv {\Gamma(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c \rho) \over \Gamma(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c \rho)} \, , \\ & {\cal{R}} \equiv {{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) \over {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} \, , \end{split} $

(72)

$ \begin{split} {\cal{R}} \equiv & \dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) }{ {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 2.2 \, , \\& \dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow h_c\pi) }{ {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.052 \, , \\ & \dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \chi_{c1}\rho \rightarrow \chi_{c1} \pi \pi)}{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 1.5 \times 10^{-5} \, . \end{split} $

(73)

$ \begin{split} \eta^{\rm{mix}}_{\mu}(x,y) \Longrightarrow & - {{\rm i}\over3}\; \cos \theta^{\prime}_1\; \xi_{\mu}^2(x^{\prime},y^{\prime}) + {1\over3}\; \cos \theta^{\prime}_1\; \xi_{\mu}^3(x^{\prime},y^{\prime}) \\ & -\; \sin \theta^{\prime}_1\; \xi_{\mu}^1(x^{\prime},y^{\prime}) - {{\rm i}\over3}\; \sin \theta^{\prime}_1\; \xi_{\mu}^4(x^{\prime},y^{\prime}) + \cdots \, , \end{split} $

(74)

$ \begin{split}& \dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow D \bar D^{*} + \bar D D^{*}) }{ {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} = 0.26 \times \dfrac{c_2^2}{ c_1^2} \, , \\&\dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi) }{ {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} = 2.5 \times 10^{-7} \times \dfrac{c_2^2 }{ c_1^2} \, . \end{split} $

(75)

5.1.$ \xi^{{\cal{Z}}}_{\mu}\big([\bar c u][\bar d c]\big) \longrightarrow \theta_{\mu}^{i}\big([\bar c c] + [\bar d u]\big) $

$ \begin{split} \xi^{{\cal{Z}}}_{\mu}(x,y) \Longrightarrow& -\dfrac{1}{6}\; \theta_{\mu}^1(x^{\prime},y^{\prime}) - {1\over6}\; \theta_{\mu}^2(x^{\prime},y^{\prime}) - \dfrac{{\rm i}}{6}\; \theta_{\mu}^3(x^{\prime},y^{\prime})\\ & - {{\rm i}\over6}\; \theta_{\mu}^4(x^{\prime},y^{\prime}) + \cdots = + {{\rm i}\over6}\; I^{P}(x^{\prime}) \; J^{V}_{\mu}(y^{\prime}) \\ &+ {{\rm i}\over6}\; I^{V}_{\mu}(x^{\prime}) \; J^{P}(y^{\prime}) - \dfrac{{\rm i}}{6}\; I^{A,\nu}(x^{\prime}) \; J^{T}_{\mu\nu}(y^{\prime}) \\ &- \dfrac{{\rm i}}{6}\; I^{T}_{\mu\nu}(x^{\prime}) \; J^{A,\nu}(y^{\prime}) + \cdots \, , \end{split} $

(76)

$ \begin{split} & h^S_{\eta_c \rho} = \dfrac{{\rm i} c_4}{ 6} \lambda_{\eta_c} m_\rho f_{\rho^+} = {\rm i} c_4\; 3.65 \times 10^{10}\; {\rm{MeV}}^4 \, , \\[-1pt]& h^D_{\eta_c \rho} = \dfrac{{\rm i} c_4}{ 6} f_{\eta_c} f^T_{\rho} = {\rm i} c_4\; 1.03 \times 10^{4}\; {\rm{MeV}}^2 \, , \\[-1pt]& h^S_{\psi \pi} = \dfrac{{\rm i} c_4}{ 6} \lambda_{\pi} m_{J/\psi} f_{J/\psi} ={\rm i} c_4\; 5.93 \times 10^{10}\; {\rm{MeV}}^4 \, , \\[-1pt]& h^D_{\psi \pi} = \dfrac{{\rm i} c_4 }{ 6} f_{\pi^+} f^T_{J/\psi} = {\rm i} c_4\; 0.89 \times 10^{4}\; {\rm{MeV}}^2 \, , \\[-1pt]& h_{\eta_c b_1} = \dfrac{{\rm i} c_4}{ 6} f_{\eta_c} f^T_{b_1} = {\rm i} c_4\; 1.16 \times 10^{4}\; {\rm{MeV}}^2 \, , \\[-1pt]& h_{\chi_{c1} \rho} = \dfrac{c_4}{ 6} m_{\chi_{c1}} f_{\chi_{c1}} f^T_{\rho} = c_4\; 3.12 \times 10^{7}\; {\rm{MeV}}^3 \, , \\[-1pt] & h_{\chi_{c1} b_1} = \dfrac{c_4}{ 6} m_{\chi_{c1}} f_{\chi_{c1}} f^T_{b_1} = c_4\; 3.53 \times 10^{7}\; {\rm{MeV}}^3 \, , \\[-1pt]& h_{h_c \pi} = \dfrac{{\rm i} c_4}{ 6} f_{\pi^+} f^T_{h_c} = {\rm i} c_4\; 0.51 \times 10^{4}\; {\rm{MeV}}^2 \, , \\[-1pt] & h_{\psi a_1} = \dfrac{c_4}{ 6} f^T_{J/\psi} m_{a_1} f_{a_1} = c_4\; 2.13 \times 10^{7}\; {\rm{MeV}}^3 \, , \\[-1pt] & h_{h_c a_1} = \dfrac{c_4}{ 6} f^T_{h_c} m_{a_1} f_{a_1} = c_4\; 1.22 \times 10^{7}\; {\rm{MeV}}^3 \, . \end{split} $

(77)

$ \begin{split} & {{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho) \over {\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.059 \, , \\& {{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow h_c\pi) \over {\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.0088 \, , \\ & {{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow \chi_{c1}\rho \rightarrow \chi_{c1}\pi \pi) \over {\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 1.4 \times 10^{-6} \, . \end{split} $

(78)

5.2.$ \xi^{{\cal{Z}}}_{\mu}\big([\bar c u][\bar d c]\big) \longrightarrow \xi^i_{\mu}\big([\bar c u] + [\bar d c]\big) $

$ \xi^{{\cal{Z}}}_{\mu}(x,y) \Longrightarrow \xi_{\mu}^1(x^{\prime},y^{\prime}) = -{\rm i}\; O^{V}_{\mu}(x^{\prime}) \; O^{P}(y^{\prime}) + \{ \gamma_{\mu} \leftrightarrow \gamma_5 \} \, . $

(79)

$ \begin{split}\langle Z_c^+(p,\epsilon) | D^+(p_1) \bar D^{*0}(p_2,\epsilon_2) \rangle \approx &-{\rm i}c_5\; \lambda_D m_{D^*} f_{D^*}\; \epsilon \cdot \epsilon_2 \\ & \equiv h_{D \bar D^*}\; \epsilon \cdot \epsilon_2 \, , \end{split} $

(80)

$ \begin{split} \langle Z_c^+(p,\epsilon) | \bar D^0(p_1) D^{*+}(p_2,\epsilon_2) \rangle \approx & -{\rm i}c_5\; \lambda_D m_{D^*} f_{D^*}\; \epsilon \cdot \epsilon_2 \\& \equiv h_{D \bar D^*}\; \epsilon \cdot \epsilon_2 \, , \end{split} $

(81)

$ h_{D \bar D^*} = -{\rm i}c_5\; 2.95 \times 10^{11}\; {\rm{MeV}}^4 \, . $

(82)

$ {{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow D \bar D^{*} + \bar D D^{*}) \over {\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} = 25 \times {c_5^2 \over c_4^2} \, . $

(83)

$ {{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi) \over {\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} \approx 0 \, . $

(84)

5.3.Mixing with the $ | D^* \bar D^*; 1^{+-} \rangle $

$ | D^* \bar D^*; 1^{+-} \rangle = | D^* \bar D^* \rangle_{J = 1} \, , $

(85)

$ \xi^{{\cal{Z}}^{\prime}}_{\mu}(x,y) = \xi^2_{\mu}([\bar c u][\bar d c]) \, , $

(86)

$ \xi^{\rm{mix}}_{\mu}(x,y) = \cos \theta^{\prime}_2\; \xi^{{\cal{Z}}}_{\mu}(x,y) + {\rm i} \sin \theta^{\prime}_2\; \xi^{{\cal{Z}}^{\prime}}_{\mu}(x,y) \, , $

(87)

$ |D^{(*)} \bar D^*; 1^{+-} \rangle = \cos \theta_2 \; |D \bar D^*; 1^{+-} \rangle + \sin \theta_2 \; |D^* \bar D^*; 1^{+-} \rangle \, . $

(88)

$ \begin{split} \xi^{\rm{mix}}_{\mu}(x,y) \Longrightarrow & + \left( + {{\rm i}\over6}\cos \theta^{\prime}_2 - {{\rm i}\over2}\sin \theta^{\prime}_2 \right) \; I^{P}(x^{\prime}) \; J^{V}_{\mu}(y^{\prime}) \\ & + \left( + {{\rm i}\over6}\cos \theta^{\prime}_2 + {{\rm i}\over2} \sin \theta^{\prime}_2 \right) \; I^{V}_{\mu}(x^{\prime}) \; J^{P}(y^{\prime}) \\ &+ \left( - {{\rm i}\over6}\cos \theta^{\prime}_2 + {{\rm i}\over6} \sin \theta^{\prime}_2 \right) \; I^{A,\nu}(x^{\prime}) \; J^{T}_{\mu\nu}(y^{\prime}) \\ &+ \left( - {{\rm i}\over6}\cos \theta^{\prime}_2 - {{\rm i}\over6} \sin \theta^{\prime}_2 \right) \; I^{T}_{\mu\nu}(x^{\prime}) \; J^{A,\nu}(y^{\prime}) + \cdots \, . \end{split} $

(89)

$ \begin{split} {\cal{R}}^{\prime} \equiv & {{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 2.2 \, , \\ & {{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow h_c\pi) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.052 \, , \\ &{{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \chi_{c1}\rho \rightarrow \chi_{c1} \pi \pi) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 1.5 \times 10^{-5} \, , \end{split} $

(90)

$ \begin{split} & {\cal{R}}^{\prime}_{\psi \pi} \equiv {\Gamma(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi) \over \Gamma(|D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} \, , \\ &{\cal{R}}^{\prime}_{\eta_c \rho} \equiv {\Gamma(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \eta_c \rho) \over \Gamma(|D \bar D^*; 1^{+-} \rangle \rightarrow \eta_c \rho)} \, , \\ & {\cal{R}}^{\prime} \equiv {{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \eta_c \rho) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} \, , \end{split} $

(91)

$ \begin{split} & {{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow D \bar D^{*} + \bar D D^{*}) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} = 67 \times {c_5^2 \over c_4^2} \, , \\ &{{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi + \eta_c\rho)} \approx 0 \, , \end{split} $

(92)

In this paper we systematically construct all the tetraquark currents/operators of $ J^{PC} = 1^{+-} $ with the quark content $ c \bar c q \bar q $ ($ q = u/d $). There are three configurations: $ [cq][\bar c \bar q] $, $ [\bar c q][\bar q c] $, and $ [\bar c c][\bar q q] $, and for each configuration we construct eight independent currents. We use the Fierz rearrangement of the Dirac and color indices to derive their relations, through which we study the strong decay properties of the $ Z_c(3900) $:
● Using the transformation of $ [qc][\bar q \bar c] \to [\bar c c][\bar q q] $, we study the decay properties of the $ Z_c(3900) $ as a compact diquark-antidiquark tetraquark state into one charmonium meson and one light meson.
● Using the transformation of $ [qc][\bar q \bar c] \to [\bar c q][\bar q c] $, we study the decay properties of the $ Z_c(3900) $ as a compact diquark-antidiquark tetraquark state into two charmed mesons.
● We use the transformation of the $ [qc][\bar q \bar c] $ currents to the color-singlet-color-singlet $ [\bar c c][\bar q q] $ and $ [\bar c q][\bar q c] $ currents, and obtain the same relative branching ratios as the above results.
● Using the transformation of $ [\bar c q][\bar q c] \to [\bar c c][\bar q q] $, we study the decay properties of the $ Z_c(3900) $ as a hadronic molecular state into one charmonium meson and one light meson.
● Through the $ [\bar c q][\bar q c] $ currents themselves, we study the decay properties of the $ Z_c(3900) $ as a hadronic molecular state into two charmed mesons.
Our results suggest that the possible decay channels of the $ Z_c(3900) $ are: a) the two-body decays $ Z_c(3900) \to J/\psi\pi $, $ Z_c(3900) \to \eta_c\rho $, $ Z_c(3900) \to h_c\pi $, and $ Z_c(3900) \to D \bar D^{*} $, b) the three-body decays $ Z_c(3900) \rightarrow $$\chi_{c1}\rho \rightarrow \chi_{c1}\pi \pi $ and $ Z_c(3900) \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi $, and c) the many-body decay chains $ Z_c(3900) \rightarrow J/\psi a_1 \rightarrow J/\psi \rho \pi $$ \rightarrow J/\psi + 3 \pi $ and $ Z_c(3900) \rightarrow \eta_c b_1 \rightarrow \eta_c \omega \pi \rightarrow \eta_c + 4 \pi $. Their relative branching ratios are summarized in Table 3, where we have investigated the following interpretations of the $ Z_c(3900) $:

channels	$|0_{qc}1_{\bar q \bar c}; 1^{+-} \rangle$	$|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle$($\theta^{\prime}_1 = -8.8^{\rm{o}}$)	$|D \bar D^*; 1^{+-} \rangle$	$|D^{(*)} \bar D^*; 1^{+-} \rangle$($\theta^{\prime}_2 = -8.8^{\rm{o}}$)
$\dfrac {{\cal{B}}(Z_c \rightarrow \eta_c\rho)}{ {\cal{B}}(Z_c \rightarrow J/\psi\pi)}}$	$0.059$	$2.2$ (input)	$0.059$	$2.2$ (input)
$ {{\cal{B}}(Z_c \rightarrow h_c\pi)}{ {\cal{B}}(Z_c \rightarrow J/\psi\pi)}$	$0.0088$	$0.052$	$0.0088$	$0.052$
$\dfrac {{\cal{B}}(Z_c \rightarrow \chi_{c1}\rho \rightarrow \chi_{c1} \pi \pi) }{{\cal{B}}(Z_c \rightarrow J/\psi\pi)}}$	$1.4 \times 10^{-6}$	$1.5 \times 10^{-5}$	$1.4 \times 10^{-6}$	$1.5 \times 10^{-5}$
$\dfrac {{\cal{B}}(Z_c \rightarrow D \bar D^{*} + \bar D D^{*})}{{\cal{B}}(Z_c \rightarrow J/\psi\pi + \eta_c\rho)}}$	$\approx 0$	$0.26 t_1$	$25 t_2$	$67 t_2$
$\dfrac {{\cal{B}}(Z_c \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi)} {{\cal{B}}(Z_c \rightarrow J/\psi\pi + \eta_c\rho)}}$	$9.3 t_1 \times 10^{-8}$	$2.5 t_1 \times 10^{-7}$	$\approx 0$	$\approx 0$

Table3.Relative branching ratios of the $Z_c(3900)$ evaluated through the Fierz rearrangement. $\theta_{1,2}^{\prime}$ are the two mixing angles defined in Eqs. (69) and (87), which are fine-tuned to be $\theta^{\prime}_1 = \theta^{\prime}_2 = -8.8^{\rm{o}}$, so that $\dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) }{ {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = \dfrac{{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho)}{ {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 2.2$ [29]. In this table, we do not take into account the phase angle $\phi$ between S- and D-wave coupling constants.

● In the second and third columns of Table 3, $ |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle $ and $ |x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle $ denote the compact tetraquark states of $ J^{PC} = 1^{+-} $, as defined in Eq. (10) and Eq. (66), respectively. In particular, we have considered the mixing between the compact tetraquarks states

$ | 0_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \oplus | 1_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \rightarrow | x_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \, . $

(93)

Using the mixing angle $ \theta^{\prime}_1 = -8.8^{\rm{o}} $, we obtain

$ \begin{split} {{\cal{B}}\left(| x_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \rightarrow J/\psi\pi \!:\! \eta_c\rho \!:\! h_c\pi \!:\! \chi_{c1}\rho (\rightarrow \pi \pi) \!:\! D \bar D^{*} \!:\! D \bar D_0^{*} (\rightarrow \bar D \pi) \right) \over {\cal{B}}(| x_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \rightarrow J/\psi\pi)} \approx 1 \!:\! 2.2 ({\rm{input}}) \!:\! 0.05 \!:\! 10^{-5} \!:\! 0.82 t_1 : 10^{-6} t_1 . \end{split} $

(94)

● In the fourth and fifth columns of Table 3, $ | D \bar D^*; 1^{+-} \rangle $ and $ |D^{(*)} \bar D^*; 1^{+-} \rangle $ denote the hadronic molecular states of $ J^{PC} = 1^{+-} $, defined in Eq. (13) and Eq. (88), respectively. Especially, we have considered the mixing between the hadronic molecule states

$ | D \bar D^*; 1^{+-} \rangle \oplus |D^{*} \bar D^*; 1^{+-} \rangle \rightarrow |D^{(*)} \bar D^*; 1^{+-} \rangle \, . $

(95)

Using the mixing angle $ \theta^{\prime}_2 = -8.8^{\rm{o}} $, we obtain

$ \begin{split} &{{\cal{B}}\left(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi : \eta_c\rho : h_c\pi \, : \chi_{c1}\rho (\rightarrow \pi \pi) : D \bar D^{*} \right) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} \\ \approx &\,1 \, : 2.2 ({\rm{input}}) : 0.05 : 10^{-5} \, : 210 t_2 \, . \end{split} $

(96)

In the above expressions, we have used the recent BESIII measurement $ {\cal{R}}_{Z_c} \equiv \dfrac{{\cal{B}}(Z_c(3900) \rightarrow \eta_c\rho) }{ {\cal{B}}(Z_c(3900) \rightarrow J/\psi\pi)} = 2.2 \pm 0.9 $ [29] as an input to determine the mixing angles $ \theta^{\prime}_1 $ and $ \theta^{\prime}_2 $. The ratio $ t_1 \equiv {c_2^2 / c_1^2} $ is the parameter measuring which process happens more easily, the process depicted in Fig. 2(b) or the process depicted in Fig. 3(b). Generally, the exchange of one light quark with another light quark seems to be easier than the exchange of one light quark with another heavy quark [67, 111]. Thus, it can be the case that $ t_1 \geq 1 $. As discussed in Sec. 5.2, $ c_5 $ is likely larger than $ c_4 $, so that the other ratio $ t_2 \equiv {c_5^2 / c_4^2} \geq 1 $.
The above relative branching ratios calculated in the present study turn out to be very different, which may be one of the reasons why many multiquark states were observed in only a few decay channels [75]. Note that in order to extract the above results, we have only considered the leading-order fall-apart decays described by color-singlet-color-singlet meson-meson currents but neglected the $ {\cal{O}}(\alpha_s) $ corrections described by color-octet-color-octet meson-meson currents. This means that there can be other decay channels.
Based on Table 3 as well as Eqs. (94) and (96), we conclude this paper:
● The relative branching ratios $ \dfrac{{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) }{ {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} $ and $ \dfrac{{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho) }{{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} $ are both around 0.059, significantly smaller than the BESIII measurement $ {\cal{R}}_{Z_c} = 2.2 \pm 0.9 $ at $ \sqrt{s} = 4.226 $ GeV [29]. However, we can add a small $ |1_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle $ component to $ |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle $ to obtain $ \dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) }{ {\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 2.2 $; we can also add a small $ |D^{*} \bar D^*; 1^{+-} \rangle $ component to $ | D \bar D^*; 1^{+-} \rangle $ to obtain $ \dfrac{{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho) }{{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 2.2 $. Note that if the relevant mixing angles change dynamically, the ratio $ {\cal{R}}_{Z_c} $ would also change dynamically.
● The relative branching ratios of the $ | D \bar D^*; 1^{+-} \rangle $ decays into one charmonium meson and one light meson are the same as those of the $ |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle $ decays. After taking proper mixing angles, the relative branching ratios of the $ |D^{(*)} \bar D^*; 1^{+-} \rangle $ decays into one charmonium meson and one light meson are also the same as those of the $ |x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle $ decays. This suggests that one may not discriminate between the compact tetraquark and hadronic molecule scenarios by only investigating relative branching ratios of the $ Z_c(3900) $ decays into one charmonium meson and one light meson.
● $ |0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle $ mainly decays into one charmonium meson and one light meson, but $ |x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle $ might mainly decay into two charmed mesons after taking into account the barrier of the diquark-antidiquark potential (see detailed discussions in Ref. [67] proposing $ c_2 \gg c_1 $). Both $ | D \bar D^*; 1^{+-} \rangle $ and $ |D^{(*)} \bar D^*; 1^{+-} \rangle $ mainly decay into two charmed mesons.
It is useful to generally discuss our uncertainty. In this study, we have worked within the naive factorization scheme; thus, our uncertainty is greater than that of well-developed QCD factorization method [62–64], which is at 5% when being applied to study the weak and radiative decay properties of conventional (heavy) hadrons. On the other hand, the tetraquark decay constant $ f_{Z_c} $ is removed when calculating relative branching ratios. This significantly reduces our uncertainty because this parameter has not yet been accurately determined. Hence, we estimate our uncertainty to be approximately$ X^{+100\%}_{-\; 50\%} $.
Next,let us compare our results with other theoretical calculations. First, we compare them with the QCD sum rule results obtained in Refs. [30, 31], where the $ Z_c(3900) $ is assumed to be a compact diquark-antidiquark tetraquark state. In this study, we find that decays of the $ Z_c(3900) $ into $ J/\psi\pi $ and $ \eta_c\rho $ can happen through both the S-wave and D-wave, and we have calculated these two amplitudes together, as shown in Eqs. (43-48); we can also calculate them individually and obtain

$ \begin{split} & {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho)_{S{-}{\rm{wave}}} \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)_{S{-}{\rm{wave}}}} = 0.24 \, , \\& {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho)_{D{-}{\rm{wave}}} \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)_{D{-}{\rm{wave}}}} = 0.82 \, . \end{split} $

(97)

In the former equation we have only considered the S-wave amplitudes, and in the latter, only the D-wave amplitudes. The QCD sum rule study in Ref. [30] only considers the S-wave amplitudes, where they obtained

$ \begin{split} & \Gamma(Z_c(3900) \rightarrow \eta_c\rho) = 27.5 \pm 8.5\; {\rm{MeV}} \, , \\& \Gamma(Z_c(3900) \rightarrow J/\psi\pi) = 29.1 \pm 8.2\; {\rm{MeV}} \, , \end{split} $

(98)

so that

$ {{\cal{B}}(Z_c(3900) \rightarrow \eta_c\rho)_{S{-}{\rm{wave}}} \over {\cal{B}}(Z_c(3900) \rightarrow J/\psi\pi)_{S{-}{\rm{wave}}}} = 0.95^{+0.47}_{-0.36} \, . $

(99)

The QCD sum rule study in Ref. [31] only considers the D-wave amplitudes, where they obtained

$ \begin{split}&\Gamma(Z_c(3900) \rightarrow \eta_c\rho) = 23.8 \pm 4.9\; {\rm{MeV}} \, , \\&\Gamma(Z_c(3900) \rightarrow J/\psi\pi) = 41.9 \pm 9.4\; {\rm{MeV}} \, , \end{split} $

(100)

so that

$ {{\cal{B}}(Z_c(3900) \rightarrow \eta_c\rho)_{D{-}{\rm{wave}}} \over {\cal{B}}(Z_c(3900) \rightarrow J/\psi\pi)_{D{-}{\rm{wave}}}} = 0.57^{+0.20}_{-0.16} \, . $

(101)

Hence, our results are more or less consistent with the QCD sum rule calculations [30, 31]. Here, we would like to note that the D-wave decay amplitudes are important and cannot be neglected:

$ \begin{split} & {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho)_{D{-}{\rm{wave}}} \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho)_{S{-}{\rm{wave}}}} = 0.51 \, , \\ & {{\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)_{D{-}{\rm{wave}}} \over {\cal{B}}(|0_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)_{S{-}{\rm{wave}}}} = 0.15 \, . \end{split} $

(102)

In fact, there is still one parameter not considered in our calculations: the phase angle $ \phi $ between the S- and D-wave decay amplitudes. For completeness, we shall investigate its relevant uncertainty in Appendix A.
Following this, we compare our results with Ref. [40], where the authors assumed the $ Z_c(3900) $ to be a hadronic molecular state and used the Non-Relativistic Effective Field Theory (a framework based on HQET and NRQCD) to obtain

$ {{\cal{B}}(Z_c(3900) \rightarrow \eta_c\rho) \over {\cal{B}}(Z_c(3900) \rightarrow J/\psi\pi)} = 0.046^{+0.025}_{-0.017} \, . $

(103)

This value is well consistent with our result

$ {{\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho) \over {\cal{B}}(| D \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 0.059 \, . $

(104)

Finally, we propose the BESIII, Belle, Belle-II, and LHCb Collaborations to search for those decay channels not yet observed, in order to better understand the nature of the $ Z_c(3900) $.
We thank Fu-Sheng Yu and Qin Chang for helpful discussions.

There are two different effective Lagrangians for the $ Z_c(3900) $ decay into the $ \eta_c \rho $ final state, as given in Eqs. (44) and (45):

$ {\cal{L}}^S_{\eta_c \rho} = g^S_{\eta_c \rho}\; Z_{c}^{+,\mu}\; \eta_c\; \rho^{-}_{\mu} + \cdots \, ,\tag{A1} $

(A1)

$ {\cal{L}}^D_{\eta_c \rho} = g^D_{\eta_c \rho} \times \left( g^{\mu\sigma}g^{\nu\rho} - g^{\mu\nu}g^{\rho\sigma} \right) Z_{c,\mu}^{+}\; \partial_\rho \eta_c\; \partial_\sigma \rho^{-}_{\nu} + \cdots \, . \tag{A2}$

(A2)

There are also two different effective Lagrangians for the $ Z_c(3900) $ decay into the $ J/\psi \pi $ final state, as given in Eqs. (47) and (48):

$ {\cal{L}}^S_{\psi \pi} = g^S_{\psi \pi}\; Z_{c}^{+,\mu}\; \psi_{\mu}\; \pi^- + \cdots \, , \tag{A3}$

(A3)

$ {\cal{L}}^D_{\psi \pi} = g^D_{\psi \pi} \times \left( g^{\mu\rho}g^{\nu\sigma} - g^{\mu\nu}g^{\rho\sigma} \right) Z_{c,\mu}^{+}\; \partial_\rho \psi_\nu\; \partial_\sigma \pi^- + \cdots \, . \tag{A4} $

(A4)

There can be a phase angle $ \phi $ between $ g^S_{\eta_c \rho} $ and $ g^D_{\eta_c \rho} $, as well as between $ g^S_{\psi \pi} $ and $ g^D_{\psi \pi} $. This parameter is unknown and therfore not fixed, because QCD sum rules dictate that one can only calculate the modular square of the decay constant, such as $ |f_{\eta_c}|^2 $. This might also be the case for Lattice QCD and the light front model. For example, see the different definitions of $ f_{\eta_c} $ in Refs. [83, 91].
We rotate this phase angle between all S- and D-wave coupling constants to be $ \phi = \pi $, and revise the previous calculations. The results are summarized in Table A1. In particular, using the mixing angle $ \theta^{\prime}_1 = \theta^{\prime}_2 = -10.1^{\rm{o}} $, we obtain

channels	$|0_{qc}1_{\bar q \bar c}; 1^{+-} \rangle$	$|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle$($\theta^{\prime}_1 = -10.1^{\rm{o}}$)	$|D \bar D^*; 1^{+-} \rangle$	$|D^{(*)} \bar D^*; 1^{+-} \rangle$($\theta^{\prime}_2 = -10.1^{\rm{o}}$)
$\dfrac{{\cal{B}}(Z_c \rightarrow \eta_c\rho)}{ {\cal{B}}(Z_c \rightarrow J/\psi\pi)}$	$0.36$	$2.2$ (input)	$0.36$	$2.2$ (input)
$\dfrac{{\cal{B}}(Z_c \rightarrow h_c\pi) }{ {\cal{B}}(Z_c \rightarrow J/\psi\pi)}$	$0.0018$	$0.0038$	$0.0018$	$0.0038$
$\dfrac{{\cal{B}}(Z_c \rightarrow \chi_{c1}\rho \rightarrow \chi_{c1} \pi \pi) }{ {\cal{B}}(Z_c \rightarrow J/\psi\pi)}$	$2.8 \times 10^{-7}$	$1.2 \times 10^{-6}$	$2.8 \times 10^{-7}$	$1.2 \times 10^{-6}$
$\dfrac{{\cal{B}}(Z_c \rightarrow D \bar D^{*} + \bar D D^{*}) }{ {\cal{B}}(Z_c \rightarrow J/\psi\pi + \eta_c\rho)}$	$\approx 0$	$0.059 t_1$	$3.9 t_2$	$5.2 t_2$
$\dfrac{{\cal{B}}(Z_c \rightarrow D \bar D_0^{*} + \bar D D_0^{*} \rightarrow D \bar D \pi)}{ {\cal{B}}(Z_c \rightarrow J/\psi\pi + \eta_c\rho)}$	$1.5 t_1 \times 10^{-8}$	$2.0 t_1 \times 10^{-8}$	$\approx 0$	$\approx 0$

TableA1.Relative branching ratios of the $Z_c(3900)$ evaluated through the Fierz rearrangement. The two mixing angles are fine-tuned to be $\theta^{\prime}_1 = \theta^{\prime}_2 = -10.1^{\rm{o}}$, so that $\dfrac{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow \eta_c\rho) }{{\cal{B}}(|x_{qc}1_{\bar q \bar c} ; 1^{+-} \rangle \rightarrow J/\psi\pi)} = \dfrac{{\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow \eta_c\rho)}{ {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} = 2.2$ [29]. In this table, we fix the phase angle $\theta$ between all the S- and D-wave coupling constants to be $\theta = \pi$.

$ \begin{split} {{\cal{B}}\left(| x_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \rightarrow J/\psi\pi\; : \eta_c\rho : h_c\pi : \chi_{c1}\rho (\rightarrow \pi \pi) : D \bar D^{*} : D \bar D_0^{*} (\rightarrow \bar D \pi)\; \right) \over {\cal{B}}(| x_{qc}1_{\bar q \bar c}; 1^{+-} \rangle \rightarrow J/\psi\pi)} \approx \; 1 : \; 2.2\; ({\rm{input}})\; : 0.004 : 10^{-6} \; \, : 0.19\; t_1 : \; 10^{-7}\; t_1\; \, ,\end{split} \tag{{A5}}$

(A5)

$ \begin{split}{{\cal{B}}\left(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi :\; \eta_c\rho :\; h_c\pi :\; \chi_{c1}\rho (\rightarrow \pi \pi)\; :\; D \bar D^{*}\; \right) \over {\cal{B}}(|D^{(*)} \bar D^*; 1^{+-} \rangle \rightarrow J/\psi\pi)} \approx \,1 \, : 2.2 ({\rm{input}})\; : 0.004 : 10^{-6}\; \, : 16 t_2 \, . \end{split} \tag{{A6}}$

(A6)