**5. Nature of** *Zc***(3900)**

As stated in the Introduction, the detected *Zc*(3900) decays include both the *π*<sup>±</sup> *J*/*ψ* and *D*∗*D* final states (assuming *Zc*(3885) and *Zc*(3900) are the same particle). The ratio of the decay widths of these two channels was measured by BESIII [27]:

$$\frac{\Gamma(Z\_{\varepsilon} \to D\hat{D}^\*)}{\Gamma(Z\_{\varepsilon} \to \pi I/\psi)} = 6.2 \pm 1.1(\text{stat}) \pm 2.7(\text{syst})\tag{55}$$

and represents a quantitative observation to be explained by the theorists. There are many different theoretical approaches that are trying to understand the nature of this state.

The tetraquark interpretation was intensively discussed within QCD sum rules [152–154] and also in the color flux-tube model [155]. The molecular scenario seems to be more abundant in the literature and is discussed or preferred in several theoretical frameworks. A light front theory description was presented in [156]; an effective field theory description was proposed in [157]; and QCD sum rules were used in [158,159]. The molecular interpretation was also supported by the quark model developed in [160]. The authors of [161] made a proposal for BESIII and forthcoming Belle II measurements by using also the molecular scenario. Further molecular picture oriented works can be found in [162] (constituent quark model, coupled channels) and in [163] (quark interchange model). It is interesting to note that most of the lattice QCD based studies obtained different results from previous ones: some did not see (within the approach they used) a bound state at all [164–167], invoked a threshold cusp explanation [168,169], or indicated that the understanding of *Zc* within the lattice QCD was only approaching [170]. For completeness, one can mention the charmonium hybrid interpretation studied in [171], the hadro charmonium picture presented in [172] with the tetraquark and molecular interpretation and the color magnetic interaction [173]. Further ideas can be found in [174–184].

The description of *Zc*(3900) in the framework of the CCQM was presented in [80]. Two options were tested: the molecular interpretation and the tetraquark hypothesis. For each option, the strong decays into *J*/*ψπ*+, *ηcρ*+, *D*¯ <sup>0</sup>*D*∗+, and *D*¯ <sup>∗</sup>0*D*<sup>+</sup> were computed and compared to available experimental data. First, we investigate the tetraquark hypothesis. In this scenario, the non-local *Zc* current is written as:

$$J\_{Z\_{\mathbf{c}}}^{\mu}(\mathbf{x}) = \int d\mathbf{x}\_{1} \dots \int d\mathbf{x}\_{4} \,\delta\left(\mathbf{x} - \sum\_{i=1}^{4} w\_{i}\mathbf{x}\_{i}\right) \cdot \Phi\_{\mathbf{Z}\_{\mathbf{c}}}\left(\sum\_{i 
$$J\_{4\eta}^{\mu} = \frac{i}{\sqrt{2}} \varepsilon\_{abc} \varepsilon\_{dcc} \left\{ \left[ u\_{d}(\mathbf{x}\_{4}) \mathbb{C} \gamma\_{5} c\_{b}(\mathbf{x}\_{1}) \right] \left[ d\_{d}(\mathbf{x}\_{3}) \gamma^{\mu} \mathbf{C} \vec{c}\_{\mathbf{c}}(\mathbf{x}\_{2}) \right] - (\gamma\_{5} \leftrightarrow \gamma^{\mu}) \right\}.$$
$$

The tetraquark mass operator looks like:

$$\begin{split} \Pi\_{Z\_{\mathbb{C}}}^{\mu\nu}(p) &= \, \, \, \, \, \, \, \prod\_{i=1}^{3} \int \frac{d^4 k\_i}{(2\pi)^4 i} \, \tilde{\Phi}\_{Z\_{\mathbb{C}}}^2 \left( -\vec{\omega}^2 \right) \\ &\quad \times \, \, \left\{ \, \, \text{tr} \left( S\_4(\hat{k}\_4) \gamma\_5 S\_1(\hat{k}\_1) \gamma\_5 \right) \text{tr} \left( S\_3(\hat{k}\_3) \gamma^\mu S\_2(\hat{k}\_2) \gamma^\nu \right) \\ &\quad \, \, \, \, \text{tr} \left( S\_4(\hat{k}\_4) \gamma^\nu S\_2(\hat{k}\_2) \gamma^\mu \right) \text{tr} \left( S\_3(\hat{k}\_3) \gamma\_5 S\_1^\n (\hat{k}\_1) \gamma\_5 \right) \right\}, \end{split} \tag{57}$$

where the momenta are defined by:

$$\begin{array}{cccc}\hat{k}\_{1} &=& k\_{1} - w\_{1}p\_{\prime} & \hat{k}\_{2} = k\_{2} - w\_{2}p\_{\prime} & \hat{k}\_{3} = k\_{3} + w\_{3}p\_{\prime} & \hat{k}\_{4} = k\_{1} + k\_{2} - k\_{3} + w\_{4}p\_{\prime} \\ \left(\vec{\omega}^{2}\right)^{2} &=& 1/2\left(k\_{1}^{2} + k\_{2}^{2} + k\_{3}^{2} + k\_{1}k\_{2} - k\_{1}k\_{3} - k\_{2}k\_{3}\right). \end{array}$$

The matrix elements of the decays *Z*<sup>+</sup> *<sup>c</sup>* <sup>→</sup> *<sup>J</sup>*/*<sup>ψ</sup>* <sup>+</sup> *<sup>π</sup>*<sup>+</sup> and *<sup>Z</sup>*<sup>+</sup> *<sup>c</sup>* <sup>→</sup> *<sup>η</sup><sup>c</sup>* <sup>+</sup> *<sup>ρ</sup>*<sup>+</sup> are written down:

$$\begin{split} &\mathcal{M}^{\mu\nu}\left(Z\_{\mathcal{L}}(p,\epsilon\_{p}^{\mu}) \rightarrow \mathbb{I}/\psi(q\_{1},\epsilon\_{q\_{1}}^{\nu}) + \pi^{+}(q\_{2})\right) = \frac{6}{\sqrt{2}} \,\mathcal{GL}\_{\mathcal{S}}\mathbb{I}/\psi\mathcal{S}\pi \\ &\times \quad \int \frac{d^{4}k\_{1}}{(2\pi)^{4}i} \int \frac{d^{4}k\_{2}}{(2\pi)^{4}i} \,\tilde{\mathcal{O}}\_{\mathcal{Z}\_{\mathcal{L}}}\left(-\vec{\eta}^{2}\right)^{2} \tilde{\mathcal{O}}\_{\mathcal{I}/\psi\mathcal{I}}\left(-(k\_{1} + \upsilon\_{2}q\_{1})^{2}\right) \tilde{\mathcal{O}}\_{\mathcal{R}}\left(-(k\_{2} + \imath\_{4}q\_{2})^{2}\right) \\ &\times \quad \left\{ \text{tr}\left(\gamma\_{5}\mathcal{S}\_{4}(k\_{2})\gamma\_{5}\mathcal{S}\_{3}(k\_{2} + q\_{2})\gamma^{\mu}\mathcal{S}\_{2}(k\_{1})\gamma^{\nu}\mathcal{S}\_{1}(k\_{1} + q\_{1})\right) + (\gamma\_{5} \leftrightarrow \gamma^{\mu}) \right\} \\ &= \quad A\_{\mathcal{I}/\psi\pi}\,\mathcal{g}^{\mu\nu} + B\_{\mathcal{I}/\psi\pi}\,q\_{1}^{\mu}q\_{2}^{\nu}, \end{split} \tag{58}$$

$$\begin{split} &M^{\mu\mu}\left(Z\_{\varepsilon}(p,\epsilon\_{p}^{\mu}) \rightarrow \eta\_{\varepsilon}(q\_{1}) + \rho(q\_{2},\epsilon\_{q\_{2}}^{a})\right) = \frac{6}{\sqrt{2}} \,\mathfrak{S} \,\mathrm{z}\_{\varepsilon} \,\mathfrak{S} \,\eta\_{\varepsilon} \,\mathrm{g} \\ &\times \quad \times \int \frac{d^{4}k\_{1}}{(2\pi)^{4}i} \int \frac{d^{4}k\_{2}}{(2\pi)^{4}i} \,\breve{\mathfrak{d}}\_{\mathrm{Z}\_{\varepsilon}}\left(-\vec{\eta}^{2}\right)^{2} \breve{\Phi}\_{\eta\_{\varepsilon}}\left(-(k\_{1} + \upsilon\_{2}q\_{1})^{2}\right) \,\breve{\Phi}\_{\rho}\left(-(k\_{2} + \iota\_{4}q\_{2})^{2}\right) \\ &\times \quad \left\{ \,\mathrm{tr}\left[\gamma\_{5}S\_{4}(k\_{2})\gamma^{\mu}S\_{3}(k\_{2} + q\_{2})\gamma^{\mu}S\_{2}(k\_{1})\gamma\_{5}S\_{1}(k\_{1} + q\_{1})\right] + (\gamma\_{5} \leftrightarrow \gamma^{\mu}) \right\} \\ &=: \quad A\_{\eta,\rho}\,\mathfrak{g}^{\mu\alpha} - B\_{\eta,\rho}\,\eta\_{2}^{\mu}q\_{1}^{\alpha}. \tag{59} \end{split}$$

where the argument of the *Zc*-vertex function is given by:

$$\begin{array}{rcl} \eta\_1 &=& \frac{1}{2\sqrt{2}} \left( 2k\_1 + (1 - w\_1 + w\_2)q\_1 - (w\_1 - w\_2)q\_2 \right), \\ \eta\_2 &=& \frac{1}{2\sqrt{2}} \left( 2k\_2 - (w\_3 - w\_4)q\_1 + (1 - w\_3 + w\_4)q\_2 \right), \\ \eta\_3 &=& \frac{1}{2} \left( (w\_3 + w\_4)q\_1 - (w\_1 + w\_2)q\_2 \right), & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \vec{\eta}^2 = \eta\_1^2 + \eta\_2^2 + \eta\_3^2. \end{array}$$

The notations used are as follows: *m*<sup>1</sup> = *m*<sup>2</sup> = *mc*, *m*<sup>3</sup> = *m*<sup>4</sup> = *md* = *mu*, *v*<sup>1</sup> = *m*1/(*m*<sup>1</sup> + *m*2), *v*<sup>2</sup> = *m*2/(*m*<sup>1</sup> + *m*2), *u*<sup>3</sup> = *m*3/(*m*<sup>3</sup> + *m*4), and *u*<sup>4</sup> = *m*4/(*m*<sup>3</sup> + *m*4).

The amplitudes of the *Z*<sup>+</sup> *<sup>c</sup>* <sup>→</sup> *<sup>D</sup>*¯ <sup>0</sup> <sup>+</sup> *<sup>D</sup>*<sup>∗</sup> <sup>+</sup> and *<sup>Z</sup>*<sup>+</sup> *<sup>c</sup>* <sup>→</sup> *<sup>D</sup>*¯ <sup>∗</sup> <sup>0</sup> <sup>+</sup> *<sup>D</sup>*<sup>+</sup> decays are:

$$M^{\mu\nu}\left(\mathcal{Z}\_c(p,\epsilon\_p^{\mu}) \to \bar{D}^0(q\_1) + D^{\*+}(q\_2,\epsilon\_{q\_2}^{\nu})\right) = \frac{6}{\sqrt{2}} g \mathcal{Z}\_c \mathfrak{g} \mathcal{D} \mathfrak{g} D^\*$$

$$\times \quad \times \quad \int \frac{d^4 k\_1}{(2\pi)^4 i} \int \frac{d^4 k\_2}{(2\pi)^4 i} \tilde{\Phi}\_{\mathcal{Z}\_c} \left(-\vec{\mathcal{S}}^2\right) \tilde{\Phi}\_D \left(-(k\_2 + \upsilon\_2 q\_2)^2\right) \tilde{\Phi}\_{\mathcal{D}^\*} \left(-(k\_1 + \iota\_1 q\_2)^2\right)$$

$$\times \quad \times \quad \left\{ \text{tr}\left(\gamma\_5 S\_4 (k\_2 + q\_1) \gamma\_5 S\_1 (k\_1) \gamma^\nu S\_3 (k\_1 + q\_2) \gamma^\mu S\_2 (k\_2)\right) - (\gamma\_5 \leftrightarrow \gamma^\mu) \right\}$$

$$= A\_{D\mathcal{D}^\*} \, g^{\mu\nu} - B\_{D\mathcal{D}^\*} \, q\_2^\mu q\_1^\nu,\tag{60}$$

$$M^{\mu\mu} \left( Z\_{\mathbf{c}} (p, \epsilon\_p^{\mu}) \to D^{\*0} (q\_1, \epsilon\_{q\_1}^{\mu}) + D^{+} (q\_{2^\*}) \right) = \frac{6}{\sqrt{2}} \mathcal{G}\_{\mathbf{c}} \mathcal{G} D^{\*} \mathcal{G} D$$

$$\times \quad \times \quad \int \frac{d^4 k\_1}{(2\pi)^4 i} \int \frac{d^4 k\_2}{(2\pi)^4 i} \tilde{\Phi}\_{\mathbf{Z}\mathbf{c}} \left( -\tilde{\delta}^2 \right) \tilde{\Phi}\_{D^\*} \left( -(k\_1 + \vartheta\_1 q\_1)^2 \right) \tilde{\Phi}\_D \left( -(k\_2 + \vartheta\_2 q\_2)^2 \right)$$

$$\times \quad \times \quad \left\{ \text{tr} \left( S\_4 (k\_2 + q\_1) \gamma\_5 S\_1 (k\_1) \gamma\_5 S\_3 (k\_1 + q\_2) \gamma^\mu S\_2 (k\_2) \gamma^\mu \right) - (\gamma\_5 \leftrightarrow \gamma^\mu) \right\}$$

$$= \quad A\_{D'D} \mathcal{g}^{\mu\nu} + B\_{D'D} q\_1^\mu q\_2^\nu \tag{61}$$

with the argument of the *Zc*-vertex function being:

$$\begin{array}{rcl}\delta\_1 &=& -\frac{1}{2\sqrt{2}}\left(k\_1 - k\_2 + (w\_1 - w\_2)(q\_1 + q\_2)\right),\\\delta\_2 &=& +\frac{1}{2\sqrt{2}}\left(k\_1 - k\_2 - (1 + w\_3 - w\_4)q\_1 + (1 - w\_3 + w\_4)q\_2\right),\\\delta\_3 &=& -\frac{1}{2}\left(k\_1 + k\_2 + (w\_1 + w\_2)(q\_1 + q\_2)\right),\end{array} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\delta\_1^2 = \delta\_1^2 + \delta\_2^2 + \delta\_3^2.\tag{62}$$

Now, the notation used is *m*<sup>1</sup> = *m*<sup>2</sup> = *mc*, *m*<sup>3</sup> = *m*<sup>4</sup> = *md* = *mu*, *v*ˆ2 = *m*2/(*m*<sup>2</sup> + *m*4), *v*ˆ4 = *m*4/(*m*<sup>2</sup> + *m*4), *u*ˆ1 = *m*1/(*m*<sup>1</sup> + *m*3), and *u*ˆ3 = *m*3/(*m*<sup>1</sup> + *m*3).

The decay width for the 1+(*p*) <sup>→</sup> <sup>1</sup>−(*qv*) + <sup>0</sup>−(*qs*) transition is given by:

$$\Gamma = \frac{1}{8\pi} \frac{1}{2s+1} \frac{|\mathbf{q}\_\mathbf{v}|}{m^2} (|H\_{+1}\mathbf{i}\_\mathbf{l}|^2 + |H\_{-1}\mathbf{i}\_\mathbf{l}|^2 + |H\_{00}\mathbf{i}|^2),\tag{63}$$

where *H* denotes the helicity amplitudes and **qv** is the three-momentum of the final state vector particle *q μ <sup>v</sup>* = (*Ev*, 0, 0, |**qv**|). The helicity amplitudes can be related to the invariant amplitudes A<sup>1</sup> and A2, which parametrize the matrix element in terms of the Lorentz structures:

$$
\mathcal{M} = \mathcal{A}\_1 \, m \, g^{\mu \rho} + \mathcal{A}\_2 \, \frac{1}{m} q\_1^{\mu} q\_2^{\rho} \tag{64}
$$

by means of the relations:

$$H\_{00} = -\frac{m}{m\_1} E\_\mathbf{v} \mathcal{A}\_1 - \frac{1}{m\_1} |\mathbf{q}\_\mathbf{v}|^2 \mathcal{A}\_2, \qquad H\_{+1+1} = H\_{-1-1} = -m \mathcal{A}\_1.$$

From the comparison of Equation (64) with Equations (58)–(61), one can express A1,2 as a function of *Axy*, *Bxy*. The results are importantly influenced by the fact that the amplitudes *ADD*¯ <sup>∗</sup> and *AD*∗*<sup>D</sup>* (Formulas (60) and (61)) vanish exactly within the CCQM description *ADD*¯ <sup>∗</sup> = *AD*∗*<sup>D</sup>* = 0, and the contributions from the non-zero *B* amplitudes are strongly suppressed by the |**qv**| <sup>5</sup> factor. Before arriving at the numerical predictions, the size parameters need to be specified, and a strategy with respect to the choice of Λ*Zc* value has to be settled. The numerical values of the size parameters were in [80] (i.e., the herein presented *Zc* analysis) re-adjusted with respect to those in [78] and are shown in Table 3.

**Table 3.** The size parameters for selected mesons in GeV used in the *Zc*(3900) analysis.


As concerns the Λ*Zc* parameter, first, it is taken as Λ*Zc* = 2.24 ± 0.10 GeV to make the predicted value of the decay width Γ(*Z*<sup>+</sup> *<sup>c</sup>* <sup>→</sup> *<sup>J</sup>*/*<sup>ψ</sup>* <sup>+</sup> *<sup>π</sup>*+) close to the one from [152,176]. One obtains:

$$
\Gamma(\mathbf{Z}\_{\varepsilon}^{+} \rightarrow \mathbb{J}/\Psi + \pi^{+}) \quad \quad = \quad \text{(27.9)}\\\quad \text{MeV}, \qquad \Gamma(\mathbf{Z}\_{\varepsilon}^{+} \rightarrow \bar{\mathbf{D}}^{0} + \mathbf{D}^{\*+}) \approx 10^{-8} \,\text{MeV},
$$

$$
\Gamma(\mathbf{Z}\_{\varepsilon}^{+} \rightarrow \eta\_{\varepsilon} + \rho^{+}) \quad \quad = \quad \text{(35.7}\\\,^{+6.3}\_{-5.2}) \,\text{MeV}, \qquad \Gamma(\mathbf{Z}\_{\varepsilon}^{+} \rightarrow \bar{\mathbf{D}}^{\*0} + \mathbf{D}^{+}) \approx 10^{-8} \,\text{MeV}.\tag{65}
$$

These outputs contradict the experimental number (see Equation (55)), which indicates a larger coupling to *DD*<sup>∗</sup> than to the *J*/*ψπ* mode. If trying to adjust the Λ*Zc* parameter to a more realistic value, the results do not become any better. Assuming Λ*Zc* = 3.3 ± 1.1 GeV, one gets:

$$
\Gamma(\mathbf{Z}\_{\varepsilon}^{+} \rightarrow \mathbb{J}/\Psi + \pi^{+}) \quad \quad = \quad (4.3^{+0.7}\_{-0.6})\,\mathrm{MeV}, \qquad \Gamma(\mathbf{Z}\_{\varepsilon}^{+} \rightarrow \mathbb{D}^{0} + \mathcal{D}^{\*+}) \propto 10^{-9} \,\mathrm{MeV},
$$

$$
\Gamma(\mathbf{Z}\_{\varepsilon}^{+} \rightarrow \eta\_{\varepsilon} + \rho^{+}) \quad \quad = \quad (8.0^{+1.2}\_{-1.0})\,\mathrm{MeV}, \qquad \Gamma(\mathbf{Z}\_{\varepsilon}^{+} \rightarrow \mathcal{D}^{\*0} + \mathcal{D}^{+}) \propto 10^{-9} \,\mathrm{MeV}.\tag{66}
$$

These predictions suggest that the tetraquark picture is not appropriate for the *Zc*(3900) state.

The molecular description of *Zc*(3900) appears as a natural alternative. In such a scenario, the non-local interpolation quark current is written as [53]:

$$J\_{4q}^{\mu} = \frac{1}{\sqrt{2}} \left\{ (\bar{d}(\mathbf{x}\_3)\gamma\_5\mathbf{c}(\mathbf{x}\_1))(\bar{\varepsilon}(\mathbf{x}\_2)\gamma^\mu\mathbf{u}(\mathbf{x}\_4)) + (\bar{d}(\mathbf{x}\_3)\gamma^\mu\mathbf{c}(\mathbf{x}\_1))(\bar{\varepsilon}(\mathbf{x}\_2)\gamma\_5\mathbf{u}(\mathbf{x}\_4)) \right\}.\tag{67}$$

By using similar steps as in the tetraquark analysis, one writes down the Fourier transformed *Zc* mass operator in the form:

$$\begin{split} \Pi\_{Z\_{\epsilon}}^{\mu\nu}(p) &= \ \frac{9}{2} \prod\_{i=1}^{3} \int \frac{d^4 k\_i}{(2\pi)^4 i} \Phi\_{\mathbf{Z}\_{\epsilon}}^2 \left(-\vec{\omega}^2\right) \\ &\quad \times \ \left\{ \text{tr}\left[\gamma\_5 \mathbf{S}\_1(\hat{k}\_1) \gamma\_5 \mathbf{S}\_3(\hat{k}\_3)\right] \cdot \text{tr}\left[\gamma^\mu \mathbf{S}\_4(\hat{k}\_4) \gamma^\nu \mathbf{S}\_2(\hat{k}\_2)\right] \right. \\ &\left. + \text{tr}\left[\gamma^\mu \mathbf{S}\_1(\hat{k}\_1) \gamma^\nu \mathbf{S}\_3(\hat{k}\_3)\right] \cdot \text{tr}\left[\gamma\_5 \mathbf{S}\_4(\hat{k}\_4) \gamma\_5 \mathbf{S}\_2(\hat{k}\_2)\right] \right\} \end{split} (68)$$

in order to pin down the Λ*Zc* dependence of the coupling *gZc* . Next, the transition amplitudes are constructed:

$$M^{\mu\upsilon} \left( Z\_c(\boldsymbol{\rho}, \boldsymbol{\epsilon}\_p^{\mu}) \to \boldsymbol{l}/\psi(q\_1, \boldsymbol{\epsilon}\_{q\_1}^{\upsilon}) + \boldsymbol{\pi}^+(q\_2) \right) = \frac{3}{\sqrt{2}} \operatorname{\mathcal{Z}\boldsymbol{z}\mathcal{S}} \boldsymbol{l}/\psi \boldsymbol{\mathcal{S}} \pi$$

$$\times \quad \times \quad \int \frac{d^4 k\_1}{(2\pi)^4 i} \int \frac{d^4 k\_2}{(2\pi)^4 i} \tilde{\boldsymbol{\Theta}} \mathcal{Z}\_{\boldsymbol{\epsilon}} \left( -\vec{\eta}^2 \right) \tilde{\boldsymbol{\Theta}}\_{\boldsymbol{l}/\psi} \left( -(k\_1 + v\_1 q\_1)^2 \right) \tilde{\boldsymbol{\Theta}}\_{\boldsymbol{\pi}} \left( -(k\_2 + u\_4 q\_2)^2 \right)$$

$$\times \quad \times \quad \left\{ \text{tr} \left( \gamma\_5 \boldsymbol{S}\_1(k\_1) \gamma^\nu \boldsymbol{S}\_2(k\_1 + q\_1) \gamma^\mu \boldsymbol{S}\_4(k\_2) \gamma\_5 \boldsymbol{S}\_3(k\_2 + q\_2) \right) + (\gamma\_5 \leftrightarrow \gamma^\mu) \right\}$$

$$= \quad A\_{I/\varphi\pi} \mathcal{g}^{\mu\upsilon} + B\_{I/\varphi\pi} q\_1^\mu q\_2^\upsilon. \tag{69}$$

$$\begin{split} &M^{\mu\iota} \left( Z\_{\varepsilon}(p, \epsilon\_{p}^{\mu}) \rightarrow \eta\_{\varepsilon}(q\_{1}) + \rho(q\_{2}, \epsilon\_{q\_{2}}^{\mu}) \right) = \frac{3}{\sqrt{2}} \operatorname{\mathcal{S}} \mathcal{Z}\_{\varepsilon} \mathcal{S}\_{\mathbb{R}} \mathcal{S}\_{\mathbb{R}} \\ &\times \quad \int \frac{d^{4}k\_{1}}{(2\pi)^{4}i} \int \frac{d^{4}k\_{2}}{(2\pi)^{4}i} \tilde{\operatorname{\mathcal{S}}}\_{\mathbb{Z}\_{\varepsilon}} \left( -\vec{\eta}^{2} \right)^{2} \tilde{\operatorname{\mathcal{S}}}\_{\mathbb{R}\_{\mathbb{R}}} \left( -(k\_{1} + v\_{1}q\_{1})^{2} \right) \tilde{\operatorname{\mathcal{S}}}\_{\mathbb{R}} \left( -(k\_{2} + u\_{4}q\_{2})^{2} \right) \\ &\times \quad \left\{ \operatorname{tr} \left( \gamma\_{5} S\_{1}(k\_{1}) \gamma\_{5} S\_{2}(k\_{1} + q\_{1}) \gamma^{\mu} S\_{4}(k\_{2}) \gamma^{\mu} S\_{3}(k\_{2} + q\_{2}) \right) + (\gamma\_{5} \leftrightarrow \gamma^{\mu}) \right\} \\ &= \quad A\_{\eta,\rho} g^{\mu\nu} - B\_{\eta,\rho} q\_{2}^{\mu} q\_{1}^{\mu} . \end{split} \tag{70}$$

$$\begin{split} &\mathcal{M}^{\mu\nu}\left(Z\_{\mathbf{c}}(p,\epsilon\_{p}^{\mu}) \to D^{0}(q\_{1}) + D^{\*+}(q\_{2},\epsilon\_{q\_{2}}^{\nu})\right) = \frac{9}{\sqrt{2}}\mathcal{g}\mathcal{Z}\_{\mathbf{c}}\mathcal{g}D\mathcal{J}^{\mu} \\ &\times \quad \int \frac{d^{4}k\_{1}}{(2\pi)^{4}i} \int \frac{d^{4}k\_{2}}{(2\pi)^{4}i} \,\widetilde{\mathcal{Q}}\_{\mathbf{Z}\_{\mathbf{c}}}\left(-\widetilde{\delta}^{2}\right) \widetilde{\mathcal{Q}}\_{D}\left(-(k\_{2} + v\_{4}q\_{1})^{2}\right) \widetilde{\mathcal{Q}}\_{\mathcal{D}^{\mu}}\left(-(k\_{1} + u\_{1}q\_{2})^{2}\right) \\ &\times \quad \text{tr}\left(\gamma^{\mu}\mathcal{S}\_{1}(k\_{1})\gamma^{\nu}\mathcal{S}\_{3}(k\_{1} + q\_{2})\right) \cdot \text{tr}\left(\gamma\_{5}\mathcal{S}\_{4}(k\_{2})\gamma\_{5}\mathcal{S}\_{2}(k\_{2} + q\_{1})\right) \\ &= \quad A\_{\mathcal{D}D^{\mu}}\,\mathcal{g}^{\mu\nu} - B\_{\mathcal{D}D^{\mu}}\,q\_{2}^{\mu}q\_{1}^{\nu}. \end{split} \tag{71}$$

$$\begin{split} &\mathcal{M}^{\mu\mu}\left(Z\_{\mathbf{c}}(p,\epsilon\_{p}^{\mu})\rightarrow D^{\*0}(q\_{1},\epsilon\_{q\_{1}}^{\mu})+D^{+}(q\_{2})\right) = \frac{9}{\sqrt{2}}\operatorname{gz}\_{\mathbf{c}}\mathcal{g}\_{D^{\*}}\mathcal{g}\_{D} \\ &\times \quad \int \frac{d^{4}k\_{1}}{(2\pi)^{4}i} \int \frac{d^{4}k\_{2}}{(2\pi)^{4}i} \widetilde{\Phi}\_{\mathbf{Z}\_{\mathbf{c}}}\left(-\widetilde{\delta}^{2}\right) \widetilde{\Phi}\_{D^{\*}}\left(-(k\_{1}+\hat{v}\_{1}q\_{1})^{2}\right) \widetilde{\Phi}\_{D}\left(-(k\_{2}+\hat{u}\_{4}q\_{2})^{2}\right) \\ &\times \quad \text{tr}\left(\gamma\_{5}S\_{1}(k\_{1})\gamma\_{5}S\_{3}(k\_{1}+q\_{2})\right)\cdot \text{tr}\left(\gamma^{\mu}S\_{4}(k\_{2})\gamma^{\mu}S\_{2}(k\_{2}+q\_{1})\right) \\ &= \quad A\_{D^{\*}D}\mathcal{g}^{\mu\alpha}+B\_{D^{\*}D}q\_{1}^{\mu}q\_{2}^{\mu}. \end{split} \tag{72}$$

where the argument of the function Φ *Zc* is given by:

$$\begin{array}{rcl}\delta\_1 &=& -\frac{1}{2\sqrt{2}}\left(k\_1 + k\_2 + (1 + w\_1 - w\_2)q\_1 + (w\_1 - w\_2)q\_2\right)\right),\\ \delta\_2 &=& +\frac{1}{2\sqrt{2}}\left(k\_1 + k\_2 - (w\_3 - w\_4)q\_1 + (1 - w\_3 + w\_4)q\_2\right),\\ \delta\_3 &=& +\frac{1}{2}\left(-k\_1 + k\_2 + (1 - w\_1 - w\_2)q\_1 - (w\_1 + w\_2)q\_2\right)\right), & \qquad \widetilde{\delta}^2 = \delta\_1^2 + \delta\_2^2 + \delta\_3^2.\end{array} \tag{73}$$

The meaning of all other letters and symbols is the same as was in the previous paragraph dedicated to the tetraquark description. The decay widths are also evaluated in a fully analogous way. However, the parameter Λ*Zc* needs to be adjusted independently. Tuning its value in such a way so as to provide the best description of the BESIII measurement [27], one gets Λ*Zc* = 3.3 ± 1.1 GeV with the following values for the decay widths:

$$\begin{array}{llll} \Gamma(\boldsymbol{Z}\_{\boldsymbol{\varepsilon}}^{+} \rightarrow \boldsymbol{\}/\psi + \boldsymbol{\pi}^{+}) &=& (1.8 \pm 0.3)\,\text{MeV}, & \Gamma(\boldsymbol{Z}\_{\boldsymbol{\varepsilon}}^{+} \rightarrow \boldsymbol{D}^{0} + \boldsymbol{D}^{\*+}) = (10.0^{+1.7}\_{-1.4})\,\text{MeV}, \\\Gamma(\boldsymbol{Z}\_{\boldsymbol{\varepsilon}}^{+} \rightarrow \boldsymbol{\eta}\_{\boldsymbol{\varepsilon}} + \boldsymbol{\rho}^{+}) &=& (3.2^{+0.5}\_{-0.4})\,\text{MeV}, & \Gamma(\boldsymbol{Z}\_{\boldsymbol{\varepsilon}}^{+} \rightarrow \boldsymbol{D}^{\*,0} + \boldsymbol{D}^{+}) = (9.0^{+1.6}\_{-1.3})\,\text{MeV}. \end{array}$$

One can see that the obtained results at this time are in agreement with the experimental observations by showing an enhancement of the *DD*∗ sector and are in agreement with the observed branching fraction ratio in Equation (55) within the errors. One can conclude that the CCQM supports the molecular picture of the *Zc*(3900) state.
