**4. Radiative Decays of** *X***(3872)**

The first experimental evidence for the radiative decay of the *X*(3872) particle was given in [129] by the Belle experiment. From the measured branching fraction product:

$$\mathcal{B}(B \to XK) \cdot \mathcal{B}(X \to \gamma + J/\psi) = (1.8 \pm 0.6 \text{ (stat)} \pm 0.1 \text{ (syst)}) \times 10^{-6} \tag{36}$$

the partial width ratio was deduced:

$$\frac{\Gamma(X \to \gamma + I/\psi)}{\Gamma(X \to \pi^+ \pi^- \restriction/\psi)} = 0.14 \pm 0.05. \tag{37}$$

This finding was supported by the BaBar observation [137]:

$$\mathcal{B}(\mathcal{B}^+ \to X\mathcal{K}^+) \cdot \mathcal{B}(X \to \gamma + \text{J/}\psi) = (3.3 \pm 1.0 \text{ (stat)} \pm 0.3 \text{ (syst)}) \times 10^{-6} \tag{38}$$

which had a limited significance of 3.4 *σ*. The same experiment reaffirmed the observation in 2009 [138] with smaller errors:

$$\mathcal{B}(\mathcal{B}^{\pm} \to XK^{\pm}) \cdot \mathcal{B}(X \to \gamma + f/\psi) = (2.8 \pm 0.8 \text{ (stat)} \pm 0.1 \text{ (syst)}) \times 10^{-6} \tag{39}$$

from which one can deduce [36]:

$$\frac{\Gamma(X \to \gamma + J/\psi)}{\Gamma(X \to \pi^+ \pi^- \restriction/\psi)} = 0.22 \pm 0.06. \tag{40}$$

BaBar also presented a result related to *ψ*(2*s*):

$$\mathcal{B}(\mathcal{B}^\pm \to XK^\pm) \cdot \mathcal{B}(X \to \gamma + \psi(2S)) = (9.5 \pm 2.7 \, (\text{stat}) \pm 0.6 \, (\text{syst})) \times 10^{-6}.\tag{41}$$

In 2011, the Belle collaboration published measurements with *J*/*ψ* and *ψ*(2*s*) in the final state [139]:

$$\mathcal{B}(\mathcal{B}^{\pm} \to XK^{\pm}) \cdot \mathcal{B}(X \to \gamma + J/\psi) \quad = \quad (1.78^{+0.48}\_{-0.44} \text{(stat)} \pm 0.12 \text{ (syst)}) \times 10^{-6} \text{)}$$

$$\mathcal{B}(\mathcal{B}^{\pm} \to XK^{\pm}) \cdot \mathcal{B}(X \to \gamma + \psi(2S)) \quad < \quad 3.45 \times 10^{-6} \text{.} \tag{42}$$

The first result was in good agreement with the previous one from the same experiment (36); however, the second number brought some tension when compared to BaBar and a later LHCb measurement [140]:

$$\frac{\Gamma(X \to \psi(2s) + \gamma)}{\Gamma(X \to J/\psi + \gamma)} = \begin{cases} 3.4 \pm 1.4 & \text{BaBar} \\ \\ < 2.0 \, (90\% \, \text{CL}) & \text{Belle} \\ \\ 2.46 \pm 0.64 \, (\text{stat}) \pm 0.29 \, (\text{sys}) & \text{LHCb} \end{cases} \tag{43}$$

The theoretical study of radiative *X*(3872) decays includes several different approaches. Such decays were analyzed in [98] in the charmonium picture. The authors studied excited 1D and 2P states and their decays in relation with the electric dipole radiation and provided implications for quantum number assignments. The molecular hypothesis was considered in [106]. There, the authors argued that the validity of the molecular picture could be determined from the study of several *X*(3872) decay channels (including some with the photon emission). The work in [141] was dedicated to radiative decays with two *D* mesons in the final state. It was claimed that the discrimination

between the molecular and charmonium picture could be obtained via analysis of the photon spectrum. Several decay modes, which also included *J*/*ψ* + *γ*, were examined in [142] within a phenomenological Lagrangian approach. The predicted value of the radiative decay width depended on the model parameters and varied from 125 KeV to 250 KeV. In [143], *X*(3872) was described as a mixture of charmonium and exotic molecular states and treated using QCD sum rules. The predicted radiative decay width ratio <sup>Γ</sup>*X*(*J*/*ψγ*)/Γ*X*(*J*/*ψπ*+*π*−) = 0.19 <sup>±</sup> 0.13 was in agreement with experimental measurements. The excited charmonium hypothesis and study of E1 decay widths within the relativistic Salpeter method was presented in [144]. A description based on a charmonium-like picture with high spin 2−<sup>+</sup> using a light front quark model was proposed in [145]. Later works [146–149] were mostly interested in the puzzling Γ*X*(*ψ*(2*s*)*γ*)/Γ*X*(*J*/*ψγ*) ratio (43) and analyzed it with different approaches (quark potential model, single-channel approximation, coupled-channel approach, charmonium-molecule hybrid model, and an effective theory framework).

Here, we focus on the *J*/*ψ* decay channel, which was studied using the CCQM in [79]. The non-local quark current for the *X*(3872) hadron was given in the previous section; see Equation (22). The *J*/*ψ* quark current is written as:

$$\|f\|\_{l/\Psi}^{\mu}(y) = \int dy\_1 \int dy\_2 \,\delta\left(y - \frac{1}{2}(y\_1 + y\_2)\right) \times \Phi\_{l/\Psi}\left((y\_1 - y\_2)^2\right) \bar{c}\_a(y\_1)\gamma^\mu c\_a(y\_2). \tag{44}$$

The related size parameter was established in earlier works and has the value of Λ*J*/*<sup>ψ</sup>* = 1.738 GeV. The knowledge of the quark currents enables us to give more details concerning the interaction with photons, addressed before in Section 2.4. The second part of the electromagnetic interaction Lagrangian stands:

$$\begin{array}{rclclcl}\mathcal{L}\_{\text{int}}^{\text{EM}(2)}(\mathbf{x})&=&\operatorname{g}\_{X}\mathcal{X}\_{q,\mu}(\mathbf{x})\cdot l\_{\text{X}\_{q}-\text{em}}^{\mu}(\mathbf{x})+\operatorname{g}\_{I/\forall}\mathcal{I}/\psi\mu\_{\mu}(\mathbf{x})\cdot l\_{\text{J}/\forall}^{\mu}\psi\_{\text{em}}(\mathbf{x}),&(q=\mu,d)\end{array}$$

$$\begin{array}{rclclcl}\mathcal{J}\_{\text{X}\_{q}-\text{em}}^{\mu}&=&\int d\tilde{\rho}\,\Phi\_{X}(\tilde{\rho}^{2})\,l\_{\text{J}\_{\text{Q}}}^{\mu}(\mathbf{x}\_{1},\ldots,\mathbf{x}\_{4})\left\{ie\_{q}\left[I\_{\text{x}}^{\text{x}}-I\_{\text{x}}^{\text{x}\_{4}}\right]+ie\_{c}\left[I\_{\text{x}}^{\text{x}\_{2}}-I\_{\text{x}}^{\text{x}\_{1}}\right]\right\},\\\mathcal{J}\_{\text{J}/\Psi\text{-cm}}^{\mu}&=&\int d\rho\,\Phi\_{\text{J}/\Psi}(\rho^{2})\,l\_{\text{J}\_{\text{Q}}}^{\mu}(\mathbf{x}\_{1},\mathbf{x}\_{2})\,ie\_{c}\left[I\_{\text{x}}^{\text{x}\_{1}}-I\_{\text{x}}^{\text{x}\_{2}}\right],&I\_{\text{x}}^{\text{x}\_{i}}\equiv\operatorname{I}(\mathbf{x}\_{i},\mathbf{x},\mathbf{P}).\end{array}$$

where *J μ* <sup>4</sup>*<sup>q</sup>* and *J μ* <sup>2</sup>*<sup>q</sup>* correspond to the parts of usual currents (22), (44) not containing the vertex function. In order to make use of the definition (15), it is convenient to switch to the Fourier transforms of the vertex functions and quark fields:

$$\begin{aligned} \Phi\_X(\vec{\rho}^2) &= \int \frac{d^4 \vec{\omega}}{(2\pi)^4} \tilde{\Phi}\_X(-\vec{\omega}^2) e^{-i\vec{\rho}\vec{\omega}} = \tilde{\Phi}\_X(\vec{\theta}\_\rho^2) \,\delta^{(4)}(\vec{\rho}),\\ \Phi\_{\mathcal{I}/\Psi}(\rho^2) &= \int \frac{d^4 \vec{\omega}}{(2\pi)^4} \tilde{\Phi}\_{\mathcal{I}/\Psi}(-\omega^2) e^{-i\rho\omega} = \tilde{\Phi}\_{\mathcal{I}/\Psi}(\partial\_\rho^2) \,\delta^{(4)}(\rho),\\ q(\mathbf{x}\_i) &= \int \frac{d^4 p\_i}{(2\pi)^4} e^{-ip\_i \mathbf{x}\_i} \tilde{q}(p\_i), \qquad \vec{q}(\mathbf{x}\_i) = \int \frac{d^4 p\_i}{(2\pi)^4} e^{i p\_i \mathbf{x}\_i} \vec{q}(p\_i) \,\rho \end{aligned}$$

so that the differential operator can be placed in front of the path integrals:

$$\begin{split} \|f\|\_{X\_{q}-\text{emp}}^{\mu} &= \prod\_{i=1}^{4} \int \frac{d^4 p\_i}{(2\pi)^4} \tilde{l}\_{4q}^{\mu}(p\_1,\dots,p\_4) \int d\vec{\rho} \,\delta^{(4)}(\vec{\rho}) \tilde{\Phi}\_X(\vec{\partial}\_{\rho}^2) e^{-i(p\_1x\_1-p\_2x\_2-p\_3x\_3+p\_4x\_4)} \cdot Q \chi^{(4)} \\ &= \prod\_{i=1}^{4} \int \frac{d^4 p\_i}{(2\pi)^4} \tilde{l}\_{4q}^{\mu}(p\_1,\dots,p\_4) e^{-i(p\_1-p\_2-p\_3+p\_4)x} \int d\vec{\rho} \,\delta^{(4)}(\vec{\rho}) e^{-i\vec{\rho}\vec{\omega}} \tilde{\Phi}\_X(\vec{\mathcal{D}}\_{\rho}^2) \cdot Q\_X \\ Q\_X &= \ \mathrm{ic}\_{q} \left[I\_x^{x3} - I\_x^{x4}\right] + \mathrm{ic}\_{\mathbb{C}} \left[I\_x^{x2} - I\_x^{x1}\right]\_{\prime} \end{split}$$

*Symmetry* **2020**, *12*, 884

$$\begin{split} \|I\_{1/\varphi\text{-}cm}^{\mu}\|\_{\mathcal{I}/\Psi\text{-}cm} &= \prod\_{i=1}^{2} \int \frac{d^4 p\_i}{(2\pi)^4} \tilde{l}\_{2q}^{\mu}(p\_1, p\_2) \int d\rho \,\delta^{(4)}(\rho) \tilde{\Phi}\_{\mathcal{I}/\Psi}(\hat{\sigma}\_{\rho}^2) \, e^{i(p\_1 x\_1 - p\_2 x\_2)} \cdot Q\_{\mathcal{I}/\Psi} \\ &= \prod\_{i=1}^{2} \int \frac{d^4 p\_i}{(2\pi)^4} \tilde{l}\_{2q}^{\mu}(p\_1, p\_2) e^{i(p\_1 - p\_2)x} \int d\rho \,\delta^{(4)}(\rho) e^{i\text{pp}} \tilde{\Phi}\_{\mathcal{I}/\Psi}(D\_{\rho}^2) \cdot Q\_{\mathcal{I}/\Psi} \\ Q\_{\mathcal{I}/\Psi} &= \text{i} c\_{\mathbb{C}} \left[l\_{\mathbb{x}}^{\text{x1}} - l\_{\mathbb{x}}^{\text{x2}}\right]\_{\text{\textquotedblleft}} \end{split}$$

where the long derivatives are defined as *D<sup>μ</sup> <sup>ρ</sup><sup>i</sup>* = *∂ μ <sup>ρ</sup><sup>i</sup>* <sup>−</sup> *<sup>i</sup>ω<sup>μ</sup> <sup>i</sup>* and *<sup>D</sup><sup>μ</sup> <sup>ρ</sup>* = *∂ μ <sup>ρ</sup>* + *ipμ*, *p* = <sup>1</sup> <sup>2</sup> (*p*<sup>1</sup> + *p*2) with *ω<sup>i</sup>* being combinations of the integration four-vectors *pi* and mass parameters *wq* and *wc*. Next, the identity involving the operator function action on the path integral [150] is applied:

$$F(D\_{\rho\_{\rangle}}^2)I\_x^{\chi\_{\rangle}} = \int\_0^1 d\tau F'(\tau D\_{\rho\_{\rangle}}^2 - (1-\tau)\omega\_{\rangle}^2) \, w\_{l\rangle} \cdot \left(\partial\_{\rho\_{\rangle}}^{\nu} A\_{\nu}(\mathbf{x}\_l) - 2i \,\omega\_{\rangle}^{\nu} A\_{\nu}(\mathbf{x}\_l)\right) + F(-\omega\_{\rangle}^2)I\_x^{\chi\_{\rangle}}.\tag{45}$$

Its validity extends to all functions *F* analytic at zero. The result for *X*(3872) reads:

*J μ Xq*−em(*x*) = 4 ∏*i*=1 - *d*<sup>4</sup>*xi* - *d*4*y J<sup>μ</sup>* <sup>4</sup>*q*(*x*1,..., *<sup>x</sup>*4) *<sup>A</sup>ρ*(*y*) · *<sup>E</sup><sup>ρ</sup> <sup>X</sup>*(*x*; *x*1,..., *x*4, *y*), (46) *Eρ <sup>X</sup>*(*x*; *x*1,..., *x*4, *y*) = 4 ∏*i*=1 *d*<sup>4</sup> *pi* (2*π*)<sup>4</sup> *d*4*r* (2*π*)<sup>4</sup> *<sup>e</sup>* −*ip*1(*x*−*x*1)+*ip*2(*x*−*x*2)+*ip*3(*x*−*x*3)−*ip*4(*x*−*x*4)−*ir*(*x*−*y*) *Eρ <sup>X</sup>*(*p*1,..., *p*4,*r*), *Eρ <sup>X</sup>*(*p*1,..., *<sup>p</sup>*4,*r*) = -1 0 *dτ* 3 ∑ *j*=1 *ec* −Φ- *<sup>X</sup>*(−*z*1*j*)*l ρ* <sup>1</sup>*<sup>j</sup>* + Φ- *<sup>X</sup>*(−*z*2*j*)*l ρ* 2*j* +*eq* −Φ- *<sup>X</sup>*(−*z*4*j*)*l ρ* <sup>4</sup>*<sup>j</sup>* + Φ- *<sup>X</sup>*(−*z*3*j*)*l ρ* 3*j* , *lij* = *wij* (*wijr* + 2 *ωj*), (*i* = 1, . . . , 4; *j* = 1, . . . , 3), *zi*<sup>1</sup> <sup>=</sup> *<sup>τ</sup>* (*wi*1*<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>*1)<sup>2</sup> + (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*) *<sup>ω</sup>*<sup>2</sup> <sup>1</sup> + *<sup>ω</sup>*<sup>2</sup> <sup>2</sup> + *<sup>ω</sup>*<sup>2</sup> 3, *zi*<sup>2</sup> = (*wi*1*<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>*1)<sup>2</sup> <sup>+</sup> *<sup>τ</sup>* (*wi*2*<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>*2)<sup>2</sup> + (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*) *<sup>ω</sup>*<sup>2</sup> <sup>2</sup> + *<sup>ω</sup>*<sup>2</sup> 3, *zi*<sup>3</sup> = (*wi*1*<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>*1)<sup>2</sup> + (*wi*2*<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>*2)<sup>2</sup> <sup>+</sup> *<sup>τ</sup>* (*wi*3*<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>*3)<sup>2</sup> + (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*) *<sup>ω</sup>*<sup>2</sup> 3.

For *J*/*ψ*, one obtains:

$$\begin{split} I^{\mathbb{P}}\_{\mathbb{I}/\varphi\text{-cm}}(y) &= \int d^4 y\_1 \int d^4 y\_2 \int d^4 z \,\tilde{\mu}^{\mathbb{P}}\_{\mathbb{I}/\varphi}(y\_1, y\_2) \, A\_{\rho}(z) \, \tilde{E}^{\rho}\_{\mathbb{I}/\varphi\text{-}\mathfrak{z}}\_{\mathbb{I}}(y; y\_1, y\_2, z), \\ E^{\mathbb{P}}\_{\mathbb{I}/\varphi\text{-}\mathfrak{z}}(y; y\_1, y\_2, z) &= \int \frac{d^4 p\_1}{(2\pi)^4} \int \frac{d^4 p\_2}{(2\pi)^4} \int \frac{d^4 q}{(2\pi)^4} e^{-ip\_1(y\_1 - y) + ip\_2(y\_2 - y) + iq(z - y)} \tilde{E}^{\rho}\_{\mathbb{I}/\varphi\text{-}\mathfrak{z}}\_{\mathbb{I}/\varphi\text{-}\mathfrak{z}}(p\_1, p\_2, q), \\ \tilde{E}^{\rho}\_{\mathbb{I}/\varphi\text{-}\mathfrak{z}}(p\_1, p\_2, q) &= \, \_6 \epsilon \int d\tau \left\{ -\tilde{\Phi}^{\prime}\_{\mathbb{I}/\varphi\text{-}\mathfrak{z}}(-z\_-) \, l^{\rho}\_{-} - \tilde{\Phi}^{\prime}\_{\mathbb{I}/\varphi\text{-}\mathfrak{z}}(-z\_+) \, l^{\rho}\_{+} \right\}, \\ z\_{\!\!\!z} &= \, \ \tau\left(p \mp \frac{1}{2}q\right) - \left(1 - \tau\right)p^2, \qquad l\_{\!\!z} = p \mp \frac{1}{4}q, \qquad p = \frac{1}{2}\left(p\_1 + p\_2\right). \end{split}$$

The amplitude evaluation requires evaluation of four Feynman diagrams displayed in Figure 5.

The corresponding expression stands:

$$M(X\_{\emptyset}(p)\to I/\psi(q\_1)\,\gamma(q\_2)) = i(2\pi)^4 \delta^{(4)}(p-q\_1-q\_2)\,\varepsilon\_X^{\mu}\,\varepsilon\_{\uparrow}^{\rho}\,\varepsilon\_{\downarrow/\psi}^{\nu}\,T\_{\mathbb{H}^{\mu\nu}}(q\_1,q\_2)\,,\tag{48}$$

where *Tμρν*(*q*1, *q*2) can be expanded in terms of appropriate Lorentz structures. Using the on-mass shell condition, gauge invariance, and Schouten identities [151], one can show that only two independent structures remain:

$$T\_{\mu\rho\upsilon} = \mathcal{W}\_A \varepsilon\_{q\_1 q\_2 \mu\rho} q\_{2\upsilon} + \mathcal{W}\_B \varepsilon\_{q\_1 q\_2 \upsilon\rho} q\_{1\mu} \,. \tag{49}$$

The functions *WA*/*<sup>B</sup>* are to be extracted from the expression following from the CCQM computation:

$$T\_{\mu\mu\nu}(q\_1, q\_2) = \sum\_{i=a,b,c,d} T\_{\mu\mu\nu}^{(i)}(q\_1, q\_2) \,, \tag{50}$$

where the separate contributions are written down:

$$\begin{split} T^{(a)}\_{\mu\rho\upsilon} &= \, \, \, \mathfrak{d}\sqrt{2} \, \mathcal{G}\_{X} \mathcal{G}\_{l/\varphi} \, \varepsilon\_{q} \int \frac{d^4 k\_1}{(2\pi)^4 i} \int \frac{d^4 k\_2}{(2\pi)^4 i} \, \widetilde{\mathfrak{d}}\_X \left( -\mathcal{K}\_a^2 \right) \widetilde{\mathfrak{d}}\_{l/\varphi} \left( -(k\_1 + \frac{1}{2} q\_1)^2 \right) \\ & \times \, \, \, \, \, \, \, \, \frac{1}{2} \, \, \text{tr} \left[ \gamma\_5 \mathcal{S}\_c(k\_1) \gamma\_\upsilon \mathcal{S}\_c(k\_1 + q\_1) \gamma\_\mu \mathcal{S}\_q(k\_2) \gamma\_\rho \mathcal{S}\_q(k\_2 + q\_2) - (\gamma\_5 \leftrightarrow \gamma\_\mu) \right], \\ \mathcal{K}\_a^2 &= \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, q\_\mu \, \, \, \, \mathcal{G}\_{\mu} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,$$

$$\begin{split} T^{(b)}\_{\mu\mu\nu} &= \, \, \, \delta \sqrt{2} \, \mathcal{g}\_X \mathcal{g}\_{J/\Psi} \int \frac{d^4 k\_1}{(2\pi)^4 i} \int \frac{d^4 k\_2}{(2\pi)^4 i} \, \tilde{\mathcal{O}}\_{J/\Psi} \left( -(k\_2 + \frac{1}{2} q\_1)^2 \right) \tilde{E}\_{X,\rho}(p\_1, \dots, p\_4, r), \\ &\quad \times \quad \, \, \, \, \, \, \, \, \, \frac{1}{2} \, \text{tr} \left[ \gamma\_5 S\_q(k\_1) \gamma\_\mu S\_\varepsilon(k\_2) \gamma\_\nu S\_\varepsilon(k\_2 + q\_1) - (\gamma\_5 \leftrightarrow \gamma\_\mu) \right], \\ p\_1 &= \, \, k\_2, \qquad p\_2 = k\_2 + q\_1, \qquad p\_3 = p\_4 = -k\_1, \qquad r = -q\_2. \end{split}$$

$$\begin{split} T^{(c)}\_{\mu\rho\upsilon} &= \, \left\| \sqrt{2} \, \mathcal{G}\_{X} \mathcal{G}\_{l} \right\| \cdot \psi\_{c} \left( \frac{d^{4}k\_{1}}{(2\pi)^{4}i} \int \frac{d^{4}k\_{2}}{(2\pi)^{4}i} \tilde{\Phi}\_{X} \left( -k\_{c}^{2} \right) \tilde{\Phi}\_{l} \left( -(k\_{2} + q\_{2} + \frac{1}{2}q\_{1})^{2} \right) \right) \\ & \times \, \left\| \begin{array}{ll} \frac{1}{2} \, \text{tr} \left[ \gamma\_{5} S\_{q}(k\_{1}) \gamma\_{\mu} S\_{c}(k\_{2}) \gamma\_{\mu} S\_{c}(k\_{2} + q\_{2}) \gamma\_{\nu} S\_{c}(k\_{2} + p) - (\gamma\_{5} \leftrightarrow \gamma\_{\mu}) \right] \\\\ \frac{1}{2} k\_{1}^{2} + \frac{1}{2} \, (k\_{2} + \frac{1}{2}p)^{2} + \frac{1}{4} \, w\_{q}^{2} p^{2} \end{array} \right. \end{split}$$

$$\begin{split} T^{(d)}\_{\mu\rho\nu} &= \, \, \, \, \, \delta \sqrt{2} \, \mathcal{G}\_{X} \mathcal{G}\_{I/\Psi} \, \varepsilon\_{c} \int \frac{d^4 k\_1}{(2\pi)^4 i} \int \frac{d^4 k\_2}{(2\pi)^4 i} \, \breve{\Phi}\_X \left( -\mathcal{K}\_c^2 \right) \tilde{E}\_{I/\Psi \rho} (p\_1, p\_2, q) \\ &\times \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \Gamma \Big{ \,} \sim \, \, \, \Gamma \, \zeta \, \, \, \Gamma \, \zeta \, \, \, \zeta \, \, \, \zeta \, \, \, \zeta \\ p\_1 &= \, \, \, \, \, \, \, \, \, \, \, \, \, \, p\_2 = \, \, \, \, \, \, \, \, \, q = \, \, q\_2 \, \, \, \end{split}$$

One evaluates the traces and the loop momenta integrals, and the expression is re-arranged in two terms following the mentioned Lorentz structure. The behavior of coefficient functions *WA*/*<sup>B</sup>* is predicted using a numerical integration over the Schwinger parameters:

$$\mathcal{W}\_{A,B} = \int\_0^\infty dt \int d^3 \mathcal{J} \, F\_{A,B}(t, \beta\_1, \beta\_2, \beta\_3) \,. \tag{51}$$

The decay width is expressed as:

$$
\Gamma(X \to \gamma \, I/\psi) = \frac{1}{12\pi} \frac{|\vec{q}\_2|}{m\_X^2} \left( |H\_L|^2 + |H\_T|^2 \right),
\tag{52}
$$

where *Hi* denote the helicity amplitudes:

$$H\_L = i \frac{m\_X^2}{m\_I/\psi} |\vec{q}\_2|^2 \mathcal{W}\_{A\_{\prime\prime}} \qquad H\_T = -im\_X |\vec{q}\_2|^2 \mathcal{W}\_B \tag{53}$$

with |*q*2| = *m*2 *<sup>X</sup>* <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *J*/*ψ* /(2*mX*). The dependence of the predicted decay width on the size parameter Λ*<sup>X</sup>* is shown in Figure 6.

**Figure 5.** Four Feynman diagrams describing the decay *X* → *γ* + *J*/*ψ*. One with the photon emission form the light quark line (**a**) and three bubble graphs (**b**–**d**).

**Figure 6.** The dependence of the decay widths Γ(*Xl* → *γ* + *J*/*ψ*) and Γ(*Xl* → *J*/*ψ* 2*π*) on the size parameter Λ*X*.

If we follow the approach from the previous section and take Λ*<sup>X</sup>* = 3.0 ± 0.5 GeV, then the model predicts:

$$\frac{\Gamma(X\_l \to \gamma + l/\psi)}{\Gamma(X\_l \to l/\psi + 2\pi)}\Big|\_{\text{CCQM}} = 0.15 \pm 0.03\,,\tag{54}$$

which is to be compared with the experimental results Equation (37) and Equation (40). One may conclude that the bound-tetraquark description of the *X*(3872) state by the CCQM is in an agreement with the experimental observations.
