*3.1. Decays X* <sup>→</sup> *<sup>D</sup>*<sup>∗</sup> <sup>0</sup>(<sup>→</sup> *<sup>D</sup>*0*π*0)*D*¯ <sup>0</sup>*, X* <sup>→</sup> *<sup>ρ</sup>*0(<sup>→</sup> *<sup>π</sup>*+*π*−)¯*J*/*ψ, and X* <sup>→</sup> *<sup>ω</sup>*(<sup>→</sup> *<sup>π</sup>*+*π*−*π*0)¯*J*/*<sup>ψ</sup>*

The controversy raised by the discovery of the *X*(3872) state can be best seen in the large number of publications it provoked (with many different interpretations). The proximity of the *D*∗<sup>0</sup> *D*<sup>0</sup> threshold:

$$M\_{X(3872)} - (M\_{D^{\*0}} + M\_{D^{0}}) = -0.30 \pm 0.40 \,\text{MeV} \tag{20}$$

naturally suggests the idea of a loosely bound charm meson molecule. This idea was studied in several texts: implications of the molecular hypothesis for interference and binding effects were discussed in [89]; the authors of [90] found support for the molecular interpretation within a non-relativistic quark model; a published text [91] analyzed the molecular assumption in an effective field theory approach; and further works [92–94] based their analyses on an effective field theory with pion exchange, Monte Carlo simulations, and heavy quark spin symmetry. A rather strong support for the molecular picture was given in [95] (line shapes study) and [96] (potential model). The lattice study [97] found an explanation for *X*(3872) in both the molecular and tetraquark scenario. An important group of analyses focused on charmonium [98–100] or mixed charmonium [101–104] explanations. Further arguments in favor of a charmonium structure followed from the Flatté analysis performed in [105], and both molecular and charmonium hypotheses were discussed in [106]. Several works [107–110] disfavored the molecular description. The authors of [107] based their conclusion on a non-relativistic quark model with the pion exchange, and the analysis presented in [108] favored the charmonium picture instead, while the conclusions in [109] were based on the pion and sigma exchange model. More rare were approaches based on the glueball picture [111] and chromomagnetic interaction [112]. The authors of [113] put in question the existence of a bound state at all. A hybrid hypothesis was considered in [114] and [115] (here, together with the molecular and charmonium one). Lattice computations in relation to *X*(3872) were used in [116,117], QCD sum rules in [118,119], and the coupled channel approach in [120–122]. One should also mention the studies based on quark models [56,123,124] and other strategies [125–127].

The description of the *X*(3872) state by the CCQM was presented in [78]. There, one assumed a tetraquark structure, and within this assumption, decays *<sup>X</sup>* <sup>→</sup> *<sup>J</sup>*/*<sup>ψ</sup>* <sup>+</sup> <sup>2</sup>*π*(3*π*) and *<sup>X</sup>* <sup>→</sup> *<sup>D</sup>*¯ <sup>0</sup> <sup>+</sup> *<sup>D</sup>*<sup>0</sup> <sup>+</sup>

*π*0, proceeding through the off-shell *ρ*/*ω* and *D*<sup>∗</sup> states respectively, were computed. In addition, possible implications of the *X*(3872) dominance in the *s*-channel dissociation of *J*/*ψ* were discussed.

When describing the *X*(3872) state, one follows the suggestions of [123,128], where a symmetric spin distribution of this *JPC* = 1++ state was proposed:

$$[cq]\_{S=0}[\bar{c}\bar{q}]\_{S=1} + [cq]\_{S=1}[\bar{c}\bar{q}]\_{S=0} \quad (q=u,d). \tag{21}$$

A non-local generalization of this diquark-antidiquark current is written as:

$$J\_{\mathbf{X}\_{q}}^{\mu}(\mathbf{x}) = \int d\mathbf{x}\_{1} \dots \int d\mathbf{x}\_{4} \,\delta\left(\mathbf{x} - \sum\_{i=1}^{4} w\_{i}\mathbf{x}\_{i}\right) \Phi\_{\mathbf{X}}\left(\sum\_{i
$$J\_{4q}^{\mu} = \frac{1}{\sqrt{2}} \varepsilon\_{abc} \varepsilon\_{dcc} \left\{ \left[ q\_{d}(\mathbf{x}\_{4}) \mathrm{C} \gamma^{5} c\_{b}(\mathbf{x}\_{1}) \right] \left[ q\_{d}(\mathbf{x}\_{3}) \gamma^{\mu} \mathrm{C} \mathcal{E}\_{\varepsilon}(\mathbf{x}\_{2}) \right] + (\gamma^{5} \leftrightarrow \gamma^{\mu}) \right\} \,\, ^{\prime} \tag{22}$$
$$

with simplified weights resulting from only two quark flavors being present:

$$w\_1 = w\_2 = w\_c = \frac{m\_c}{2(m\_q + m\_c)}, \qquad w\_3 = w\_4 = w\_q = \frac{m\_q}{2(m\_q + m\_c)}.\tag{23}$$

The strong isospin violation observed by comparing the *ρ* and *ω* vector meson mediated decays:

$$\frac{\mathcal{B}(X \to \mathcal{J}/\psi\pi^{+}\pi^{-}\pi^{0})}{\mathcal{B}(X \to \mathcal{J}/\psi\pi^{+}\pi^{-})} = 1.0 \pm 0.4(\text{stat}) \pm 0.3(\text{syst})\tag{24}$$

experimentally established by Belle [129] suggested a mixed nature of the physical states *Xl*, *Xh*:

$$\begin{aligned} X\_l &\equiv X\_{\text{lcm}} &= & X\_{\text{ll}} \cos \theta + X\_d \sin \theta, \\ X\_{\text{h}} &\equiv X\_{\text{high}} &= & -X\_{\text{u}} \sin \theta + X\_d \cos \theta, \end{aligned}$$

where *θ* is the mixing angle. The state *Xu* breaks the isospin symmetry maximally:

$$X\_{\mathfrak{u}} = \frac{1}{\sqrt{2}} \left\{ \underbrace{\frac{X\_{\mathfrak{u}} + X\_{\mathfrak{d}}}{\sqrt{2}}}\_{I=0} + \underbrace{\frac{X\_{\mathfrak{u}} - X\_{\mathfrak{d}}}{\sqrt{2}}}\_{I=1} \right\}.$$

The mixing angle is to be adjusted to fit the branching fraction ratio (24).

The first step in our calculation is to determine the coupling constant *gX* by using the so-called compositeness condition discussed before. The derivative of the tetraquark mass operator needed for this can be written as:

Π- *<sup>X</sup>*(*p*2) = <sup>1</sup> <sup>2</sup>*p*<sup>2</sup> *<sup>p</sup><sup>α</sup> <sup>∂</sup> <sup>∂</sup>p<sup>α</sup>* <sup>Π</sup>*X*(*p*2) (25) <sup>=</sup> <sup>2</sup> *<sup>g</sup>*<sup>2</sup> *X* 3 *p*<sup>2</sup> *<sup>g</sup>μν* <sup>−</sup> *<sup>p</sup><sup>μ</sup> <sup>p</sup><sup>ν</sup> p*2 3 ∏*i*=1 *d*<sup>4</sup>*ki* (2*π*)4*i* Φ2 *X* <sup>−</sup> *<sup>K</sup>*<sup>2</sup> × <sup>−</sup>*wc*tr *<sup>S</sup>*[12] *<sup>c</sup>* ✁ *pS*[12] *<sup>c</sup> <sup>γ</sup>*5*S*[2] *<sup>q</sup> γ*<sup>5</sup> tr *S*[3] *<sup>c</sup> <sup>γ</sup>μS*[13] *<sup>q</sup> <sup>γ</sup><sup>ν</sup>* <sup>+</sup> *wq*tr *<sup>S</sup>*[12] *<sup>c</sup> <sup>γ</sup>*5*S*[2] *<sup>q</sup>* ✁ *pS*[2] *<sup>q</sup> γ*<sup>5</sup> tr *S*[3] *<sup>c</sup> <sup>γ</sup>μS*[13] *<sup>q</sup> <sup>γ</sup><sup>ν</sup>* <sup>−</sup>*wc*tr *<sup>S</sup>*[12] *<sup>c</sup> <sup>γ</sup>*5*S*[2] *<sup>q</sup> γ*<sup>5</sup> tr *S*[3] *<sup>c</sup>* ✁ *pS*[3] *<sup>c</sup> <sup>γ</sup>μS*[13] *<sup>q</sup> <sup>γ</sup><sup>ν</sup>* <sup>+</sup> *wq*tr *<sup>S</sup>*[12] *<sup>c</sup> <sup>γ</sup>*5*S*[2] *<sup>q</sup> γ*<sup>5</sup> tr *S*[3] *<sup>c</sup> <sup>γ</sup>μS*[13] *<sup>q</sup>* ✁ *pS*[13] *<sup>q</sup> <sup>γ</sup><sup>ν</sup>* ,

where the short notations for the quark propagators and loop momenta are:

$$\begin{array}{rcl} S\_{\varepsilon}^{[12]} &=& S\_{\varepsilon}(k\_1 + k\_2 - w\_{\varepsilon}p)\_{\prime} \\ S\_{q}^{[2]} &=& S\_{q}(k\_2 + w\_{q}p)\_{\prime} \\ \end{array} \qquad \begin{array}{rcl} S\_{\varepsilon}^{[3]} &= S\_{\varepsilon}(k\_3 - w\_{\varepsilon}p)\_{\prime} \\ S\_{q}^{[13]} &= S\_{q}(k\_1 + k\_3 + w\_{q}p)\_{\prime} \\ \end{array}$$
 
$$\begin{array}{rcl} \mathcal{K}^2 &=& \frac{1}{2} \sum\_{i \le j} k\_i k\_j. \end{array}$$

The evaluation of this expression is related to the determination of the size parameter Λ*<sup>X</sup>* value and allows us to study the Λ*<sup>X</sup>* dependence of the results.

Because the *X*(3872) mass lies close to the studied thresholds:

$$\begin{array}{rcl} m\_X - (m\_{f/\Psi} + m\_\rho) &=& -0.90 \pm 0.41 \,\mathrm{MeV},\\ m\_X - (m\_{D^0} + m\_{D^\*0}) &=& -0.30 \pm 0.34 \,\mathrm{MeV},\end{array}$$

the off-mass-shell character of the *ρ*, *ω*, and *D*∗ vector mesons has to be taken into account when evaluating the transition amplitudes *<sup>X</sup>* <sup>→</sup> *<sup>J</sup>*/*<sup>ψ</sup>* <sup>+</sup> *<sup>ρ</sup>*(*ω*) and *<sup>X</sup>* <sup>→</sup> *<sup>D</sup>*∗0*D*¯ 0. The Feynman diagrams to be considered within the CCQM are depicted in Figure 2.

**Figure 2.** Feynman diagrams describing the decays *<sup>X</sup>* <sup>→</sup> *<sup>J</sup>*/*<sup>ψ</sup>* <sup>+</sup> *<sup>ρ</sup>*(*ω*) and *<sup>X</sup>* <sup>→</sup> *<sup>D</sup>* <sup>+</sup> *<sup>D</sup>*¯ <sup>∗</sup>.

In what follows, we use the notation for the light vector mesons *v*<sup>0</sup> = *ρ*, *ω*. The amplitude of the decay *Xu* <sup>→</sup> *<sup>D</sup>*¯ <sup>+</sup> *<sup>D</sup>*<sup>∗</sup> is written as:

$$M^{\mu\nu}\left(X\_{\mathfrak{u}}(p,\mu)\to\bar{D}(q\_{1})+D^{\*}(q\_{2},\nu)\right)=3\sqrt{2}\,\mathcal{G}X\,\mathcal{G}D\,\mathcal{G}^{\nu}\int\frac{d^{4}k\_{1}}{(2\pi)^{4}i}\int\frac{d^{4}k\_{2}}{(2\pi)^{4}i}\,\widetilde{\mathcal{G}}\nu\left(-k\_{2}^{2}\right)$$

$$\times\quad\ddot{\mathcal{G}}\_{D}\left(-(k\_{1}+w\_{\mathfrak{e}}q\_{1})^{2}\right)\ddot{\mathcal{G}}\_{D^{\*}}\left(-(k\_{2}+w\_{\mathfrak{e}}q\_{2})^{2}\right)$$

$$\times\quad\text{tr}\left[\gamma^{5}S\_{\mathfrak{e}}(k\_{1})\gamma^{5}S\_{\mathfrak{u}}(k\_{1}+q\_{1})\gamma^{\mu}S\_{\mathfrak{e}}(k\_{2})\gamma^{\nu}S\_{\mathfrak{u}}(k\_{2}+q\_{2})\right]+(m\_{\mathfrak{u}}\leftrightarrow m\_{\mathfrak{e}},m\_{\mathfrak{u}}\leftrightarrow m\_{\mathfrak{e}})$$

$$=g^{\mu\nu}M^{(1)}\_{\text{XDD}^{\*}}+\,q\_{1}^{\mu}q\_{1}^{\nu}M^{(2)}\_{\text{XDD}^{\*}}+\,q\_{1}^{\mu}q\_{2}^{\nu}M^{(3)}\_{\text{XDD}^{\*}}+\,q\_{2}^{\mu}q\_{1}^{\nu}M^{(4)}\_{\text{XDD}^{\*}}+q\_{2}^{\mu}q\_{2}^{\nu}M^{(5)}\_{\text{XDD}^{\*}}$$

where the argument of the X-vertex function is equal to:

$$K\_2^2 = \frac{1}{8}(k\_1 - k\_2)^2 + \frac{1}{8}(k\_1 - k\_2 + q\_1 - q\_2)^2 + \frac{1}{4}(k\_1 + k\_2 + w\_c p)^2 .$$

The amplitude of the decay *Xu* <sup>→</sup> *<sup>J</sup>*/*<sup>ψ</sup>* <sup>+</sup> *<sup>v</sup>*<sup>0</sup> is written as:

*Mμνρ Xu*(*p*, *<sup>μ</sup>*) <sup>→</sup> *<sup>J</sup>*/*ψ*(*q*1, *<sup>ν</sup>*) + *<sup>v</sup>*0(*q*2, *<sup>ρ</sup>*) = 6 *gX gJ*/*<sup>ψ</sup> gv*<sup>0</sup> *d*4*k*<sup>1</sup> (2*π*)4*i d*4*k*<sup>2</sup> (2*π*)4*i* Φ *<sup>X</sup>* <sup>−</sup> *<sup>K</sup>*<sup>2</sup> 1 × Φ *<sup>J</sup>*/*<sup>ψ</sup>* <sup>−</sup> (*k*<sup>1</sup> <sup>+</sup> <sup>1</sup> <sup>2</sup> *<sup>q</sup>*1)<sup>2</sup> Φ*v*<sup>0</sup> <sup>−</sup> (*k*<sup>2</sup> <sup>+</sup> <sup>1</sup> <sup>2</sup> *<sup>q</sup>*2)<sup>2</sup> <sup>×</sup> tr *iγ*5*Sc*(*k*1)*γνSc*(*k*<sup>1</sup> + *q*1)*γμSu*(*k*2)*γρSu*(*k*<sup>2</sup> + *q*2) = *ε <sup>q</sup>*1*q*2*μνq ρ* <sup>1</sup> *<sup>M</sup>*(1) *XJv* + *ε <sup>q</sup>*1*q*2*μνq ρ* <sup>2</sup> *<sup>M</sup>*(2) *XJv* + *ε <sup>q</sup>*1*q*2*μρq<sup>ν</sup>* <sup>2</sup> *<sup>M</sup>*(3) *XJv* + *ε <sup>q</sup>*1*q*2*νρq μ* <sup>1</sup> *<sup>M</sup>*(4) *XJv* + *ε <sup>q</sup>*1*μνρ M*(5) *XJv* + *ε <sup>q</sup>*2*μνρ M*(6) *XJv* + *ε <sup>q</sup>*1*q*2*μρq<sup>ν</sup>* <sup>1</sup> *<sup>M</sup>*(7) *XJv* + *ε <sup>q</sup>*1*q*2*νρq μ* <sup>2</sup> *<sup>M</sup>*(8) *XJv* (27)

where the argument of the X-vertex function is equal to:

$$K\_1^2 = \frac{1}{2}(k\_1 + \frac{1}{2}q\_1)^2 + \frac{1}{2}(k\_2 + \frac{1}{2}q\_2)^2 + \frac{1}{4}(w\_{\text{\tiny u}}q\_1 - w\_{\text{\tiny c}}q\_2)^2.$$

In the latter expression, the number of Lorentz structures is reduced to six when *X* and *J*/*ψ* are on the mass-shell because, in that case, one has *μ*(*q μ* <sup>1</sup> + *q μ* <sup>2</sup> ) = 0 and *νq<sup>ν</sup>* <sup>1</sup> = 0.

Obvious relations:

$$M(\mathbf{X}\_d \rightarrow \mathbf{J}/\boldsymbol{\psi} + \boldsymbol{\rho}) = -M(\mathbf{X}\_u \rightarrow \mathbf{J}/\boldsymbol{\psi} + \boldsymbol{\rho}), \qquad M(\mathbf{X}\_d \rightarrow \mathbf{J}/\boldsymbol{\psi} + \boldsymbol{\omega}) = M(\mathbf{X}\_u \rightarrow \mathbf{J}/\boldsymbol{\psi} + \boldsymbol{\omega}) \tag{1}$$

allow expressing all amplitudes of physical states transitions in terms of the *Xu* ones:

$$\begin{array}{rcl} M(\mathbf{X}\_{\ell/\hbar} \to \mathbf{J}/\Psi + \omega) & = & (\cos \theta \pm \sin \theta) \, M(\mathbf{X}\_{\mu} \to \mathbf{J}/\Psi + \omega), \\ M(\mathbf{X}\_{\ell/\hbar} \to \mathbf{J}/\Psi + \rho) & = & (\pm \cos \theta - \sin \theta) \, M(\mathbf{X}\_{\mu} \to \mathbf{J}/\Psi + \rho). \end{array}$$

The differential decay rate in the narrow-width approximation is written as [130]:

$$\frac{d\Gamma(X \to f/\psi + n\pi)}{d\eta^2} = \frac{1}{8\,m\_X^2\,\pi} \cdot \frac{1}{3} |M\_{X\backslash\psi}|^2 \frac{\Gamma\_{\upsilon^0} m\_{\upsilon^0}}{\pi} \frac{p^\*(q^2)}{(m\_{\upsilon^0}^2 - q^2)^2 + \Gamma\_{\upsilon^0}^2 m\_{\upsilon^0}^2} \mathcal{B}(\upsilon^0 \to n\pi), \quad \text{(28)}$$

$$\frac{1}{3} |M\_{X\backslash\upsilon}|^2 \quad = \frac{1}{3} \sum\_{\text{pol}} |\varepsilon\_X^\mu \,\varepsilon\_{f/\varphi}^\nu \,\varepsilon\_{\psi^0}^\rho |M\_{\mu\upsilon\rho}|^2,$$

where *p*∗(*q*2) = *λ*1/2(*m*<sup>2</sup> *<sup>X</sup>*, *<sup>m</sup>*<sup>2</sup> *<sup>J</sup>*/*ψ*, *<sup>q</sup>*2)/2*mX* is the momentum of the *<sup>v</sup>*<sup>0</sup> in the *<sup>X</sup>* rest frame. The allowed kinematic range is given by:

$$(n\,m\_{\pi})^2 \le q^2 \le (m\_X - m\_{I/\Phi})^2,$$

where *n* = 2 for the *ρ* meson and *n* = 3 for the *ω* meson. The masses, decay widths, and branching fractions appearing in (28) were taken from PDG [13]. In addition to the model parameter values presented in Table 1, further model parameters are needed, namely the size parameters of the appearing mesons. Their values have been settled earlier and are presented in Table 2.

**Table 2.** Size parameters for selected mesons in GeV.


Two adjustable parameters remain, the size parameter Λ*<sup>X</sup>* and the mixing angle *θ*. It was found out that the dependence of the branching fraction:

$$\frac{\Gamma\left(X\_{\mu} \to J/\psi + 3\,\,\pi\right)}{\Gamma\left(X\_{\mu} \to J/\psi + 2\,\,\pi\right)} \approx 0.25\tag{29}$$

on the size parameter Λ*<sup>x</sup>* is in the CCQM small and close to 1/4. Using this observation and the central value of the experimental ratio in Equation (24), one can deduce the mixing angle from:

$$\frac{\Gamma(X\_{l,h} \to l/\psi + 3\,\pi)}{\Gamma(X\_{l,h} \to l/\psi + 2\,\pi)} \approx 0.25 \cdot \left(\frac{1 \pm \tan\theta}{1 \mp \tan\theta}\right)^2 \approx 1.\tag{30}$$

The latter equation yields *θ* ≈ 18.4◦ for *Xl* and *θ* ≈ −18.4◦ for *Xh*. When not considering the ratio, the sensitivity of the decay widths on the size parameter is more important. One may expect the size parameter value to be close to those of the charmonia Λ*J*/*<sup>ψ</sup>* and Λ*η<sup>c</sup>* , i.e., to be in the range 3 GeV < Λ*<sup>X</sup>* < 4 GeV. This range was scanned, and the behavior of the decay width is depicted in Figure 3.

**Figure 3.** The dependence of the decay widths <sup>Γ</sup>(*Xl* <sup>→</sup> *<sup>D</sup>*¯ <sup>0</sup>*D*0*π*0) and <sup>Γ</sup>(*<sup>X</sup>* <sup>→</sup> *<sup>J</sup>*/*<sup>ψ</sup>* <sup>+</sup> *<sup>n</sup>π*) on the size parameter Λ*X*.

One can conclude that the predicted values in the interval 2.5 <sup>≤</sup> *<sup>q</sup>*<sup>2</sup> <sup>≤</sup> 3.5 GeV lie in the range 0.05 MeV < Γ*X*(3872) < 0.23 MeV, which is in agreement with the upper limit of 1.2 MeV.

The differential rate of the decay *<sup>X</sup>*(3872) <sup>→</sup> *<sup>D</sup>*¯ <sup>0</sup>*D*0*π*<sup>0</sup> in the narrow-width approximation is written as:

$$\frac{d\Gamma(\mathcal{X}\_{\rm{H}}\rightarrow\mathcal{D}^{0}D^{0}\pi^{0})}{dq^{2}}\quad=\frac{1}{2m\_{X}^{2}\pi}\cdot\frac{1}{3}|M\_{\rm{XDD}^{\ast}}|^{2}\cdot\frac{\Gamma\_{\rm{D}^{\ast,0}}m\_{\rm{D}^{\ast,0}}}{\pi}\frac{p^{\ast}(q^{2})\,\mathcal{B}(D^{\ast,0}\rightarrow D^{0}\pi^{0})}{(m\_{D^{\ast,0}}^{2}-q^{2})^{2}+\Gamma\_{\rm{D}^{\ast,0}}^{2}m\_{D^{\ast,0}}^{2}},\quad(31)$$

$$\frac{1}{3}|M\_{\rm{XDD}^{\ast}}|^{2}\quad=\quad\frac{1}{3}\sum\_{\rm{pol}}|\boldsymbol{\varepsilon}\_{\rm{X}}^{\rm{H}}\boldsymbol{\varepsilon}\_{\rm{D}^{\ast,0}}^{\rm{V}}\,M\_{\rm{l}^{\ast}}|^{2},$$

where *p*∗(*q*2) = *λ*1/2(*m*<sup>2</sup> *<sup>X</sup>*, *<sup>m</sup>*<sup>2</sup> *<sup>D</sup>*<sup>0</sup> , *<sup>q</sup>*2)/2*mX* is the momentum of *<sup>D</sup>*<sup>∗</sup> <sup>0</sup> in the *<sup>X</sup>* rest frame. The matrix element *Mμν* was defined above by Equation (26). One has to note that the allowed kinematic range:

$$3.99928\,\text{GeV}^2 \approx (m\_{D^0} + m\_{\pi^0})^2 \le q^2 \le (m\_X - m\_{D^0})^2 \approx 4.02672\,\text{GeV}^2$$

is very narrow. Taking the masses, widths, and branching fractions of appearing *D*∗ mesons from [13,89,131–134], we can calculate the decay width:

$$
\Gamma(X\_I \to D^0 D^0 \pi^0) = \cos^2 \theta \,\Gamma(X\_u \to D^0 D^0 \pi^0),
$$

and study its dependence on the size parameter Λ*X*. This is shown in Figure 3. By using the experimental data from PDG [13] for the ratio:

$$10^5 \mathcal{B}(\mathcal{B}^\pm \to K^\pm X) \cdot \mathcal{B}(X \to J/\psi \pi^+ \pi^-) \quad = \quad 0.95 \pm 0.19,$$

$$10^5 \mathcal{B}(\mathcal{B}^\pm \to K^\pm X) \cdot \mathcal{B}(X \to D^0 D^0 \pi^0) \quad = \quad 10.0 \pm 4.0,\tag{32}$$

one finds:

$$\frac{\Gamma(X \to D^0 \vec{D}^0 \pi^0)}{\Gamma(X \to \int \!/\psi \pi^+ \pi^-)} = 10.5 \pm 4.7. \tag{33}$$

The latter is to be compared to the CCQM prediction:

$$\frac{\Gamma(X \to D^0 \mathcal{D}^0 \pi^0)}{\Gamma(X \to f/\Psi \pi^+ \pi^-)}\Big|\_{\text{CCQM}} = 6.0 \pm 0.2,\tag{34}$$

where the uncertainty of the result reflects the uncertainty on Λ*X*. One can see that the two numbers agree within errors.

#### *3.2. Implications of X*(3872) *in the Charm Dissociation Process by Light Mesons*

It is interesting to check the significance of *X*(3872) in the reaction of the charm dissociation process *<sup>J</sup>*/*<sup>ψ</sup>* <sup>+</sup> *<sup>ρ</sup>* (*ω*) <sup>→</sup> *<sup>X</sup>*(3872) <sup>→</sup> *DD*¯ <sup>∗</sup>, which plays an important role in heavy ion physics. This state will contribute to the *s* channel of the process. The X-addition to the full cross-section is written as:

$$\sigma(\slash\!/\psi + v^0 \to D(D) + D^\*(D^\*)) = 2\left(\cos\theta \mp \sin\theta\right)^2 \sigma(\slash\!/\psi + v^0 \to X\_\mu \to D + D^\*), \tag{35}$$

$$\sigma(\slash\!/\psi + v^0 \to X\_\mu \to D + D^\*) = \frac{1}{16\pi\,\text{s}} \frac{\lambda^{1/2}(s, m\_{\text{D}}^2, m\_{\text{D\*}}^2)}{\lambda^{1/2}(s, m\_{\text{I}/\psi}^2, m\_{\text{p}}^2)} \cdot \frac{1}{9} \sum\_{\text{pol}} \frac{|A|^2}{(s - m\_X^2)^2 + \Gamma\_X^2 m\_X^2},$$

$$A = \varepsilon\_{I/\varphi}^\nu \varepsilon\_{\psi^0}^\rho M\_{\text{hyp}} \left( -g^{\mu\nu} + \frac{p^\mu p^\nu}{m\_X^2} \right) \varepsilon\_{\text{D\*}}^6 M\_{\text{ap\%}},$$

where *p* = *p*<sup>1</sup> + *p*<sup>2</sup> = *q*<sup>1</sup> + *q*2. The ∓ sign in the first equation is negative for the *ρ* meson and positive for *ω*. A Breit–Wigner propagator is used with Γ*<sup>X</sup>* = 1 MeV, and the size parameter value is fixed to <sup>Λ</sup>*<sup>X</sup>* <sup>=</sup> 3.5 GeV. With this setting, the dependence of the cross-section on the energy *<sup>E</sup>* <sup>=</sup> <sup>√</sup>*<sup>s</sup>* is shown in Figure 4.

**Figure 4.** The cross-sections of the processes *<sup>J</sup>*/*<sup>ψ</sup>* <sup>+</sup> *<sup>v</sup>*<sup>0</sup> <sup>→</sup> *<sup>X</sup>* <sup>→</sup> *<sup>D</sup>* <sup>+</sup> *<sup>D</sup>*∗. Charged D-mesons in left panel; neutral D-mesons in the right panel.

One can compare the predicted behavior to available results for the charged *D*-mesons: At *<sup>E</sup>* <sup>=</sup> 4.0 GeV, a theoretical evaluation [135] predicts *<sup>σ</sup>*(*J*/*<sup>ψ</sup>* <sup>+</sup> *<sup>π</sup>* <sup>→</sup> *<sup>D</sup>* <sup>+</sup> *<sup>D</sup>*¯ <sup>∗</sup>) = 0.9 mb, and the work in [136] predicted *<sup>σ</sup>*(*J*/*<sup>ψ</sup>* <sup>+</sup> *<sup>ρ</sup>* <sup>→</sup> *<sup>D</sup>* <sup>+</sup> *<sup>D</sup>*¯ <sup>∗</sup>) = 2.9 mb at *<sup>E</sup>* <sup>=</sup> 3.9 GeV. In the case of *<sup>X</sup>*(3872), the cross-section reaches the maximum of approximately 0.32 mb at *E* = 3.88 GeV, and one can conclude that the expected contribution of *X*(3872) in the charm dissociation is non-negligible.
