Effects of heavy meson loops on heavy quarkonium radiative transitions

Abstract

Radiative transitions between charmonium states offer an insight into the internal structure of heavy quark bound states within QCD. In this work, we systematically investigate the coupled-channel effects of intermediate charmed mesons to charmonium radiative transitions utilizing an effective Lagrangian approach. Our results show that the coupled-channel effects of these decays under the open charm thresholds are relatively weak, while the coupled-channel effects of the excited P-wave states close to the charmed pairs thresholds are relatively important.

Appendix: The transition amplitude

In the following, we present the transition amplitudes for the intermediate meson loops listed in Table 1 in the framework of the ELA. Notice that the expressions are similar for the charged, neutral, and charmed-strange mesons except that different charmed meson masses are applied. We thus only present the amplitudes for those charged charmed meson loops. G ₁, G ₂, and G ₃ are vertex couplings corresponding the initial charmonium, final charmonium and final photon vertex for each loop, and the explicit expression can be found in Eq. (16). [p ₁, p ₃, p ₂] are the four-vector momenta for the intermediate mesons [M1, M3, M2], respectively. p _i (p _f ), ε _i (ε _f ), and ϕ _i (ϕ _f ) are the initial (final) charmonium four-vector momentum, polarization vector, and polarization tensor, respectively. p _γ and ε _γ are the final photon four-vector momentum and polarization vector, respectively.

(i) 1⁻⁻→γ0⁺⁺ (ψ′→γχ _c0, ψ(3770)→γχ _c0)

$$\begin{aligned} \mathcal{M}_{[DDD]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi)^4} \bigl[G_1 \varepsilon _i^\mu(p_{1\mu}-p_{3\mu}) \bigr] [iG_2] \bigl[e\varepsilon_\gamma^{*\theta}(p_{1\theta}+p_{2\theta}) \bigr] \\ &{}\times\frac{i}{p_1^2-m_1^2} \frac{i}{p_2^2-m_2^2} \frac{i}{ p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[DD^*D^*]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi )^4} \bigl[G_1\varepsilon_{\mu\nu\alpha\beta} p_i^\mu \varepsilon_i^\nu \bigl(p_3^\beta-p_1^\beta \bigr)\bigr] [iG_2] \bigl[G_3\varepsilon_{\theta\phi\kappa\lambda} p_\gamma^\theta\varepsilon_\gamma^{*\phi} p_2^\kappa\bigr] \\ &{}\times \frac{i}{p_1^2-m_1^2} \frac{i(-g^{\rho\lambda}+p_2^\rho p_2^\lambda/m_2^2)}{p_2^2-m_2^2} \frac {i(-g^\alpha_\rho+p_3^\alpha p_{3\rho}/m_3^2)}{p_3^2-m_3^2}\mathcal {F} \bigl(m_2,p_2^2\bigr) \\ \mathcal{M}_{[D^*DD]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi )^4} \bigl[-G_1\varepsilon_{\mu\nu\alpha\beta} p_i^\mu \varepsilon_i^\nu \bigl(p_3^\beta-p_1^\beta \bigr)\bigr] [iG_2] \bigl[G_3\varepsilon_{\theta\phi\kappa\lambda} p_\gamma^\theta\varepsilon_\gamma^{*\phi} p_1^\kappa\bigr] \\ &{}\times \frac{i(-g^{\alpha\lambda}+p_1^\alpha p_1^\lambda/m_1^2)}{ p_1^2-m_1^2} \frac{i}{p_2^2-m_2^2} \frac{i}{p_3^2-m_3^2}\prod _i\mathcal {F}_i\bigl(m_i,p_i^2 \bigr) \\ \mathcal{M}_{[D^*D^*D^*]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi )^4} \bigl[G_1(2p_{1\mu}g_{\alpha\beta}+p_{3\alpha}g_{\beta\mu}-p_{1\beta} g_{\alpha\mu})\varepsilon_i^\mu\bigr] [iG_2] \\ &{}\times\bigl[-e\varepsilon_{\gamma}^{*\theta}\bigl(g_{\phi\kappa}(p_{2\theta }+p_{1\theta})+g_{\kappa\theta} p_{1\phi} - g_{\phi\theta} p_{2\kappa}\bigr) + G_3 \varepsilon_{\gamma}^{*\theta}(g_{\kappa\theta} p_{1\phi} - g_{\phi \theta} p_{2\kappa})\bigr] \\ &{}\times\frac{i(-g^{\alpha\kappa}+p_1^\alpha p_1^\kappa/m_1^2)}{p_1^2-m_1^2} \frac {i(-g^{\rho\phi}+p_2^\rho p_2^\phi/m_2^2)}{p_2^2-m_2^2} \frac {i(-g^{\beta\rho}+p_3^\beta p_3^\rho/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) . \end{aligned}$$

(A.1)

(ii) 1⁻⁻→γ1⁺⁺ (ψ′→γχ _c1, ψ(3770)→γχ _c1)

$$\begin{aligned} \mathcal{M}_{[DDD^*]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi)^4} \bigl[G_1 \varepsilon_i^\mu(p_{1\mu}-p_{3\mu}) \bigr] \bigl[G_2\varepsilon_{f\rho }^*\bigr] \bigl[G_3\varepsilon_{\theta\phi\kappa\lambda} p_\gamma^\theta \varepsilon _\gamma^{*\phi} p_2^\kappa\bigr] \\ &{}\times \frac{i}{p_1^2-m_1^2} \frac{i(-g^{\rho\lambda}+p_2^\rho p_2^\lambda/m_2^2)}{p_2^2-m_2^2} \frac {i}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) \\ \mathcal{M}_{[DD^*D]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi )^4} \bigl[G_1\varepsilon_{\mu\nu\alpha\beta} p_i^\mu \varepsilon_i^\nu \bigl(p_3^\beta-p_1^\beta \bigr)\bigr] \bigl[G_2\varepsilon_{f \rho}^*\bigr] \bigl[e \varepsilon_\gamma ^{*\theta}(p_{1\theta}+p_{2\theta}) \bigr] \\ &{}\times\frac{i}{p_1^2-m_1^2}\frac{i}{p_2^2-m_2^2} \frac {i(-g^{\alpha\rho}+p_3^\alpha p_3^\rho/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) \\ \mathcal{M}_{[D^*DD^*]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{ (2\pi)^4} \bigl[-G_1\varepsilon_{\mu\nu\alpha\beta} p_i^\mu \varepsilon_i^\nu \bigl(p_3^\beta-p_1^\beta \bigr)\bigr] \bigl[G_2\varepsilon_{f \rho}^*\bigr] \\ &{}\times\bigl[-e\varepsilon_{\gamma}^{*\theta}\bigl(g_{\phi\kappa}(p_{2\theta }+p_{1\theta})+g_{\kappa\theta} p_{1\phi} - g_{\phi\theta} p_{2\kappa}\bigr) + G_3 \varepsilon_{\gamma}^{*\theta}(g_{\kappa\theta} p_{1\phi} - g_{\phi \theta} p_{2\kappa})\bigr] \\ &{}\times\frac{i(-g^{\alpha\kappa}+p_1^\alpha p_1^\kappa/m_1^2)}{ p_1^2-m_1^2} \frac{i(-g^{\rho\phi}+p_2^\rho p_2^\phi/m_2^2)}{p_2^2-m_2^2} \frac{i}{p_3^2-m_3^2}\prod _i\mathcal {F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[D^*D^*D]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{ (2\pi)^4} \bigl[G_1(2p_{1\mu}g_{\alpha\beta}+p_{3\alpha}g_{\beta\mu }-p_{1\beta} g_{\alpha\mu})\varepsilon_i^\mu\bigr] \bigl[G_2\varepsilon_{f\rho }^*\bigr] \\ &{}\times\bigl[-e\varepsilon_{\gamma}^{*\theta}\bigl(g_{\phi\kappa}(p_{2\theta }+p_{1\theta})+g_{\kappa\theta} p_{1\phi} - g_{\phi\theta} p_{2\kappa}\bigr) + G_3 \varepsilon_{\gamma}^{*\theta}(g_{\kappa\theta} p_{1\phi} - g_{\phi \theta} p_{2\kappa})\bigr] \\ &{}\times \frac{i(-g^{\alpha\lambda}+p_1^\alpha p_1^\lambda/m_1^2)}{p_1^2-m_1^2} \frac {i}{p_2^2-m_2^2} \frac {i(-g^{\beta\rho}+p_3^\beta p_3^\rho/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) . \end{aligned}$$

(A.2)

(iii) 1⁻⁻→γ2⁺⁺ (ψ′→γχ _c2, ψ(3770)→γχ _c2)

$$\begin{aligned} \mathcal{M}_{[DD^*D^*]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi )^4} \bigl[G_1\varepsilon_{\mu\nu\alpha\beta} p_i^\mu \varepsilon_i^\nu \bigl(p_3^\beta-p_1^\beta \bigr)\bigr] \bigl[iG_2\phi_{f\rho\sigma}^*\bigr] \bigl[G_3\varepsilon_{\theta \phi\kappa\lambda} p_\gamma^\theta \varepsilon_\gamma^{*\phi} p_2^\kappa \bigr] \\ &{}\times\frac{i}{p_1^2-m_1^2} \frac {i(-g^{\rho\lambda}+p_2^\rho p_2^\lambda/m_2^2)}{p_2^2-m_2^2} \frac{i(-g^{\alpha\sigma}+p_3^\alpha p_3^\sigma/m_3^2)}{ p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{D^*D^*D^*} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi )^4} \bigl[G_1(2p_{1\mu}g_{\alpha\beta}+p_{3\alpha}g_{\beta\mu}-p_{1\beta} g_{\alpha\mu})\varepsilon_i^\mu\bigr] [iG_2\phi_{f\rho\sigma}] \\ &{}\times\bigl[-e\varepsilon_{\gamma}^{*\theta}\bigl(g_{\phi\kappa}(p_{2\theta }+p_{1\theta})+g_{\kappa\theta} p_{1\phi} - g_{\phi\theta} p_{2\kappa}\bigr) + G_3 \varepsilon_{\gamma}^{*\theta}(g_{\kappa\theta} p_{1\phi} - g_{\phi \theta} p_{2\kappa})\bigr] \\ &{}\times \frac{i(-g^{\alpha\kappa}+p_1^\alpha p_1^\kappa/m_1^2)}{ p_1^2-m_1^2}\frac {i(-g^{\rho\phi}+p_2^\rho p_2^\phi/m_2^2)}{p_2^2-m_2^2} \frac {i(-g^{\beta\sigma}+p_3^\beta p_3^\sigma/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) . \end{aligned}$$

(A.3)

(iv) 0⁻⁺→γ1⁺⁻ (η′→γh _c )

$$\begin{aligned} \mathcal{M}_{[DD^*D]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi)^4} \bigl[-G_1 (p_{i\mu }+p_{1\mu})\bigr] [G_2\varepsilon_{f\rho}] \bigl[e\varepsilon_\gamma^{*\theta} (p_{1\theta}+p_{2\theta})\bigr] \\ &{}\times\frac{i}{p_1^2-m_1^2} \frac {i}{p_2^2-m_2^2} \frac{i(-g^{\mu\rho}+p_3^\mu p_3^\rho/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[DD^*D^*]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi)^4} \bigl[-G_1 (p_{i\mu}+p_{1\mu})\bigr] \bigl[-G_2\varepsilon_{\rho\sigma\xi\tau} p_{f}^\sigma \varepsilon_{f}^\xi\bigr] \bigl[G_3 \varepsilon_{\theta\phi\kappa\lambda} p_\gamma ^\theta\varepsilon_\gamma^{*\phi} p_2^\kappa\bigr] \\ &{}\times\frac{i}{p_1^2-m_1^2} \frac {i(-g^{\rho\lambda}+p_2^\rho p_2^\lambda/m_2^2)}{p_2^2-m_2^2} \frac{i(-g^{\mu\tau}+p_3^\mu p_3^\tau/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[D^*DD^*]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi)^4} \bigl[-G_1(p_{i \mu}+p_{3\mu})\bigr] [G_2 \varepsilon_{f\rho}] \\ &{}\times\bigl[-e\varepsilon_{\gamma}^{*\theta}\bigl(g_{\phi\kappa}(p_{2\theta }+p_{1\theta})+g_{\kappa\theta} p_{1\phi} - g_{\phi\theta} p_{2\kappa}\bigr) + G_3 \varepsilon_{\gamma}^{*\theta}(g_{\kappa\theta} p_{1\phi} - g_{\phi \theta} p_{2\kappa})\bigr] \\ &{}\times\frac{i(-g^{\mu\kappa}+p_1^\mu p_1^\kappa/m_1^2)}{ p_1^2-m_1^2} \frac{i(-g^{\rho\phi}+p_2^\rho p_2^\phi/m_2^2)}{p_2^2-m_2^2} \frac{i}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[D^*D^*D]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi )^4} \bigl[G_1\varepsilon_{\mu\nu\alpha\beta} p_1^\mu p_{i}^\beta\bigr] [G_2\varepsilon_{f\rho}] \bigl[G_3\varepsilon_{\theta\phi\kappa\lambda} p_\gamma^\theta \varepsilon_\gamma^{*\phi} p_1^\kappa\bigr] \\ &{}\times\frac{i(-g^{\nu\lambda}+p_1^\nu p_1^\lambda/m_1^2)}{ p_1^2-m_1^2} \frac{i}{p_2^2-m_2^2} \frac{i(-g^{\alpha\rho}+p_3^\alpha p_3^\rho/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[D^*D^*D^*]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi )^4} \bigl[G_1\varepsilon_{\mu\nu\alpha\beta} p_1^\mu p_{i}^\beta\bigr] \bigl[-G_2 \varepsilon_{\rho\sigma\xi\tau} p_{f}^\sigma\varepsilon_{f}^\xi \bigr] \\ &{}\times\bigl[-e\varepsilon_{\gamma}^{*\theta}\bigl(g_{\phi\kappa}(p_{2\theta }+p_{1\theta})+g_{\kappa\theta} p_{1\phi} - g_{\phi\theta} p_{2\kappa}\bigr) + G_3 \varepsilon_{\gamma}^{*\theta}(g_{\kappa\theta} p_{1\phi} - g_{\phi \theta} p_{2\kappa})\bigr] \\ &{}\times \frac{i(-g^{\nu\kappa}+p_1^\nu p_1^\kappa/m_1^2)}{ p_1^2-m_1^2} \frac{i(-g^{\rho\phi}+p_2^\rho p_2^\phi/m_2^2)}{p_2^2-m_2^2} \frac{i(-g^{\alpha\tau}+p_3^\alpha p_3^\tau/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) . \end{aligned}$$

(A.4)

(v) 0⁺⁺→γ1⁻⁻ (χ _c0→γJ/ψ, $\chi _{c0}' \to\gamma J/\psi, \gamma\psi, \gamma\psi(3770)$)

$$\begin{aligned} \mathcal{M}_{[DDD]} =&\bigl(i^3\bigr)\int \frac{d^4p_2}{(2\pi)^4}[iG_1] \bigl[-G_2 \varepsilon_f^{*\rho}(p_{2\rho}-p_{3\rho}) \bigr] \bigl[e\varepsilon_\gamma ^{*\theta} (p_{1\theta}+p_{2\theta}) \bigr] \\ &{}\times\frac{i}{p_1^2-m_1^2} \frac{i}{p_2^2-m_2^2} \frac{i}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[DDD^*]} =&\bigl(i^3\bigr)\int \frac{d^4p_2}{(2\pi )^4}[iG_1] \bigl[G_2\varepsilon_{\rho\sigma\xi\tau} p_f^\rho\varepsilon _f^{*\sigma} \bigl(p_3^\tau-p_2^\tau\bigr)\bigr] \bigl[-eG_3\varepsilon_{\theta\phi\kappa \lambda} p_\gamma^\theta \varepsilon_\gamma^{*\phi} p_2^\kappa\bigr] \\ &{}\times\frac{i}{p_1^2-m_1^2} \frac{i(-g^{\xi\lambda}+p_2^\xi p_2^\lambda/m_2^2)}{p_2^2-m_2^2} \frac{i}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[D^*D^*D]} =&\bigl(i^3\bigr)\int \frac{d^4p_2}{(2\pi )^4}[iG_1] \bigl[G_2\varepsilon_{\rho\sigma\xi\tau} p_f^\rho\varepsilon _f^{*\sigma} \bigl(p_3^\tau-p_2^\tau\bigr)\bigr] \bigl[-e G_3\varepsilon_{\theta\phi\kappa \lambda} p_\gamma^\theta \varepsilon_\gamma^{*\phi} p_1^\kappa\bigr] \\ &{}\times\frac{i(-g^{\mu\lambda}+p_1^\mu p_1^\lambda/m_1^2)}{ p_1^2-m_1^2} \frac{i}{p_2^2-m_2^2} \frac{i(-g^{\mu\xi}+p_3^\mu p_3^\xi/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[D^*D^*D^*]} =&\bigl(i^3\bigr)\int \frac{d^4p_2}{(2\pi )^4}[iG_1] \bigl[G_2(2p_{3\rho}g_{\sigma\xi}+p_{2\sigma}g_{\xi\rho}-p_{3\xi} g_{\sigma\rho})\varepsilon_f^{*\rho}\bigr] \\ &{}\times\bigl[-e\varepsilon_{\gamma}^{*\theta}\bigl(g_{\phi\kappa}(p_{2\theta }+p_{1\theta})+g_{\kappa\theta} p_{1\phi} - g_{\phi\theta} p_{2\kappa}\bigr) + G_3 \varepsilon_{\gamma}^{*\theta}(g_{\kappa\theta} p_{1\phi} - g_{\phi \theta} p_{2\kappa})\bigr] \\ &{}\times \frac{i(-g^{\mu\kappa}+p_1^\mu p_1^\kappa/m_1^2)}{ p_1^2-m_1^2} \frac{i(-g^{\xi\phi}+p_2^\xi p_2^\phi/m_2^2)}{p_2^2-m_2^2} \frac{i(-g^{\mu\sigma}+p_3^\mu p_3^\sigma/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) . \end{aligned}$$

(A.5)

(vi) 1⁺⁺→γ1⁻⁻ (χ _c1→γJ/ψ, $\chi _{c1}' \to\gamma J/\psi, \gamma\psi, \gamma\psi(3770)$)

$$\begin{aligned} \mathcal{M}_{[DD^*D]} =&\bigl(i^3\bigr)\int \frac{d^4p_2}{(2\pi)^4}[G_1\varepsilon _{i \mu}] \bigl[G_2\varepsilon_{\rho\sigma\xi\tau} p_f^\rho \varepsilon _f^{*\sigma} \bigl(p_3^\tau-p_2^\tau \bigr)\bigr] \bigl[e\varepsilon_\gamma^{*\theta} (p_{1\theta}+p_{2\theta})\bigr] \\ &{}\times\frac{i}{p_1^2-m_1^2} \frac {i}{p_2^2-m_2^2} \frac{i(-g^{\mu\xi}+p_3^\mu p_3^\xi/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[DD^*D^*]} =&\bigl(i^3\bigr)\int \frac{d^4p_2}{(2\pi )^4}[G_1\varepsilon_{i\mu}] \bigl[G_2(2p_{3\rho}g_{\sigma\xi}+p_{2\sigma }g_{\xi\rho}-p_{3\xi} g_{\sigma\rho})\varepsilon_f^{*\rho}\bigr] \bigl[-e G_3\varepsilon_{\theta\phi\kappa\lambda} p_\gamma^\theta \varepsilon _\gamma^{*\phi} p_2^\kappa\bigr] \\ &{}\times\frac{i}{p_1^2-m_1^2} \frac {i(-g^{\xi\lambda}+p_2^\xi p_2^\lambda/m_2^2)}{p_2^2-m_2^2} \frac{i(-g^{\mu\sigma}+p_3^\mu p_3^\sigma/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[D^*DD]} =&\bigl(i^3\bigr)\int \frac{d^4p_2}{(2\pi)^4}[G_1\varepsilon _{i\mu}] \bigl[-G_2 \varepsilon_f^\rho(p_{2\rho}-p_{3\rho}) \bigr] \bigl[G_3\varepsilon_{\theta\phi\kappa\lambda} p_\gamma^\theta \varepsilon _\gamma^{*\phi} p_1^\kappa\bigr] \\ &{}\times\frac{i(-g^{\mu\lambda}+p_1^\mu p_1^\lambda/m_1^2)}{ p_1^2-m_1^2} \frac {i}{p_2^2-m_2^2} \frac{i}{p_3^2-m_3^2}\prod _i\mathcal {F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[D^*DD^*]} =&\bigl(i^3\bigr)\int \frac{d^4p_2}{(2\pi )^4}[G_1\varepsilon_{i\mu}] \bigl[G_2\varepsilon_{\rho\sigma\xi\tau} p_f^\rho \varepsilon_f^{*\sigma} \bigl(p_3^\tau-p_2^\tau \bigr)\bigr] \\ &{}\times\bigl[-e\varepsilon_{\gamma}^{*\theta}\bigl(g_{\phi\kappa}(p_{2\theta }+p_{1\theta})+g_{\kappa\theta} p_{1\phi} - g_{\phi\theta} p_{2\kappa}\bigr) + G_3 \varepsilon_{\gamma}^{*\theta}(g_{\kappa\theta} p_{1\phi} - g_{\phi \theta} p_{2\kappa})\bigr] \\ &{}\times\frac{i(-g^{\mu\kappa}+p_1^\mu p_1^\kappa/m_1^2)}{ p_1^2-m_1^2} \frac {i(-g^{\xi\phi}+p_2^\xi p_2^\phi/m_2^2)}{p_2^2-m_2^2} \frac{i}{ p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) . \end{aligned}$$

(A.6)

(viii) 2⁺⁺→γ1⁻⁻ (χ _c2→γJ/ψ, $\chi _{c2}' \to\gamma J/\psi, \gamma\psi, \gamma\psi(3770)$)

$$\begin{aligned} \mathcal{M}_{[D^*D^*D]} =&\bigl(i^3\bigr)\int \frac{d^4p_2}{(2\pi)^4}[iG_1 \varepsilon_{i\mu\nu}] \bigl[G_2\varepsilon_{\rho\sigma\xi\tau} p_f^\rho \varepsilon_f^{*\sigma} \bigl(p_3^\tau-p_2^\tau \bigr)\bigr] \bigl[G_3\varepsilon_{\theta\phi \kappa\lambda} p_\gamma^\theta \varepsilon_\gamma^{*\phi} p_1^\kappa\bigr] \\ &{}\times\frac{i(-g^{\mu\lambda}+p_1^\mu p_1^\lambda/m_1^2)}{ p_1^2-m_1^2} \frac{i}{p_2^2-m_2^2} \frac{i(-g^{\nu\xi}+p_3^\nu p_3^\xi/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[D^*D^*D^*]} =&\bigl(i^3\bigr)\int \frac{d^4p_2}{ (2\pi)^4}[iG_1\varepsilon_{i \mu\nu}] \bigl[G_2(2p_{3\rho}g_{\sigma\xi }+p_{2\sigma}g_{\xi\rho}-p_{3\xi} g_{\sigma\rho})\varepsilon_f^{*\rho}\bigr] \\ &{}\times\bigl[-e\varepsilon_{\gamma}^{*\theta}\bigl(g_{\phi\kappa}(p_{2\theta }+p_{1\theta})+g_{\kappa\theta} p_{1\phi} - g_{\phi\theta} p_{2\kappa}\bigr) + G_3 \varepsilon_{\gamma}^{*\theta}(g_{\kappa\theta} p_{1\phi} - g_{\phi \theta} p_{2\kappa})\bigr] \\ &{}\times\frac{i(-g^{\mu\kappa}+p_1^\mu p_1^\kappa/m_1^2)}{ p_1^2-m_1^2} \frac {i(-g^{\xi\phi}+p_2^\xi p_2^\phi/m_2^2)}{p_2^2-m_2^2} \frac{i(-g^{\nu\sigma}+p_3^\nu p_3^\sigma/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) . \end{aligned}$$

(A.7)

(ix) 1⁺⁻→γ0⁻⁺ (h _c →γη _c , $h_{c}'\to \gamma\eta_{c}$, $h_{c}'\to\gamma\eta_{c}'$)

$$\begin{aligned} \mathcal{M}_{[DD^*D]} =&\bigl(i^3\bigr)\int \frac{d^4p_2}{(2\pi)^4}[G_1\varepsilon _{i\mu}] \bigl[-G_2 (p_{f\rho}+p_{2\rho})\bigr] \bigl[e \varepsilon_\gamma^{*\theta }(p_{1\theta}+p_{2\theta}) \bigr] \\ &{}\times\frac{i}{p_1^2-m_1^2} \frac{i}{p_2^2-m_2^2} \frac{i(-g^{\mu\rho}+p_3^\mu p_3^\rho/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[DD^*D^*]} =&\bigl(i^3\bigr)\int \frac{d^4p_2}{ (2\pi)^4}[G_1\varepsilon_{i\mu}] \bigl[G_2\varepsilon_{\rho\sigma\xi\tau} p_2^\rho p_{f}^\tau\bigr] \bigl[G_3\varepsilon_{\theta\phi\kappa\lambda} p_\gamma ^\theta\varepsilon_\gamma^{*\phi} p_2^\kappa\bigr] \\ &{}\times\frac{i}{p_1^2-m_1^2} \frac {i(-g^{\sigma\lambda}+p_2^\sigma p_2^\lambda/m_2^2)}{p_2^2-m_2^2} \frac{i(-g^{\mu\xi}+p_3^\mu p_3^\xi/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[D^*DD^*]} =&\bigl(i^3\bigr)\int \frac{d^4p_2}{(2\pi )^4}[G_1\varepsilon_{i\mu}] \bigl[-G_2 (p_{f\rho}+p_{3\rho})\bigr] \\ &{}\times\bigl[-e\varepsilon_{\gamma}^{*\theta}\bigl(g_{\phi\kappa}(p_{2\theta }+p_{1\theta})+g_{\kappa\theta} p_{1\phi} - g_{\phi\theta} p_{2\kappa}\bigr) + G_3 \varepsilon_{\gamma}^{*\theta}(g_{\kappa\theta} p_{1\phi} - g_{\phi \theta} p_{2\kappa})\bigr] \\ &{}\times\frac{i(-g^{\mu\kappa}+p_1^\mu p_1^\kappa/m_1^2)}{ p_1^2-m_1^2} \frac{i(-g^{\rho\phi}+p_2^\rho p_2^\phi/m_2^2)}{p_2^2-m_2^2} \frac{i}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[D^*D^*D]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi )^4} \bigl[-G_1\varepsilon_{\mu\nu\alpha\beta} p_{i}^\nu \varepsilon_{i}^\alpha \bigr] \bigl[-G_2(p_{f\rho}+p_{2\rho}) \bigr] \bigl[G_3\varepsilon_{\theta\phi\kappa\lambda} p_\gamma^\theta \varepsilon_\gamma^{*\phi} p_1^\kappa\bigr] \\ &{}\times\frac{i(-g^{\mu\lambda}+p_1^\mu p_1^\lambda/m_1^2)}{ p_1^2-m_1^2} \frac{i}{p_2^2-m_2^2} \frac{i(-g^{\beta\rho}+p_3^\beta p_3^\rho/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) , \\ \mathcal{M}_{[D^*D^*D^*]} =&\bigl(i^3\bigr)\int\frac{d^4p_2}{(2\pi )^4} \bigl[-G_1\varepsilon_{\mu\nu\alpha\beta} p_{i}^\nu \varepsilon_{i}^\alpha \bigr] \bigl[G_2 \varepsilon_{\rho\sigma\xi\tau} p_2^\rho p_{f}^\tau \bigr] \\ &{}\times\bigl[-e\varepsilon_{\gamma}^{*\theta}\bigl(g_{\phi\kappa}(p_{2\theta }+p_{1\theta})+g_{\kappa\theta} p_{1\phi} - g_{\phi\theta} p_{2\kappa}\bigr) + G_3 \varepsilon_{\gamma}^{*\theta}(g_{\kappa\theta} p_{1\phi} - g_{\phi \theta} p_{2\kappa})\bigr] \\ &{}\times\frac{i(-g^{\mu\kappa}+p_1^\mu p_1^\kappa/m_1^2)}{ p_1^2-m_1^2} \frac{i(-g^{\sigma\phi}+p_2^\sigma p_2^\phi/m_2^2)}{p_2^2-m_2^2} \frac{i(-g^{\beta\xi}+p_3^\beta p_3^\xi/m_3^2)}{p_3^2-m_3^2}\prod _i\mathcal{F}_i\bigl(m_i,p_i^2 \bigr) . \end{aligned}$$

(A.8)