We are considering the double radiative decays of the \(\psi \) meson,
$$\begin{aligned} \psi (p,m_\psi ) \rightarrow \gamma \;R \rightarrow \gamma (q_1, m_\gamma )\; \gamma (q_2, m^\prime _\gamma )\; V(p_V, m_V),\nonumber \\ \end{aligned}$$
(1)
where p, \(q_1\), \(q_2\) and \(p_V\) are the four momenta of the \(\psi \), two photons and the vector particle \(V(\rho , \phi , \omega )\), respectively; \(m_\psi \), \(m_\gamma \), \(m^\prime _\gamma \) and \(m_V\) denote the third components of each particle’s spin. The transition amplitude can be written as follows by using the polarization four-vectors of the initial and final state particles,
$$\begin{aligned} A= & {} \psi _\mu (p,m_\psi )\;\varepsilon ^*_\nu (q_1,m_\gamma )\;\varepsilon ^*_\alpha (q_2,m^\prime _\gamma ) \;A^{\mu \nu \alpha }\;\nonumber \\= & {} \psi _\mu (p,m_\psi )\;\varepsilon ^*_\nu (q_1,m_\gamma )\;\varepsilon ^*_\alpha (q_2,m^\prime _\gamma ) \;\sum _i\Lambda _i A_i^{\mu \nu \alpha }\;, \end{aligned}$$
(2)
where \(\psi (p,m_\psi )\) is the polarization four vector of the \(\psi \)-mesons; \(\varepsilon _\nu (q_i,m_\gamma )\), \(i=1,2\) are the polarization vectors of the two photons. The sum over the two physical polarization states of a photon is given by,
$$\begin{aligned}&\sum ^2_{m_\gamma =1}\varepsilon ^*_\mu (q_i,m_\gamma )\; \varepsilon _\nu (q_i,m_\gamma ) = -g_{\mu \nu }\nonumber \\&\qquad + \frac{q_{i\mu } \; (p-q_i)_\nu + (p-q_i)_\mu \; q_{i\nu }}{q_i\cdot p}\nonumber \\&\qquad -\frac{(p-q_i)^2}{(q_i\cdot p)^2} \;q_{i\mu } \; q_{i\nu }\nonumber \\&\quad = -g_{\mu \nu } + \frac{q_{i\mu } \; p_\nu + p_\mu \; q_{i\nu }}{q_i\cdot p} -\frac{p^2}{(q_i\cdot p)^2} \;q_{i\mu }\; q_{i\nu } \nonumber \\&\quad \equiv -g^{(\perp )}_{\mu \nu }(q_i)\; \end{aligned}$$
(3)
where the relations \(q^\nu _i \;\varepsilon _\nu (q_i,m_\gamma )=0\) and \(p^\nu \;\varepsilon _\nu (q_i,m_\gamma ) =0\) hold.
For \(\psi \) production from \(e^+ e^-\) annihilation, the electrons are highly relativistic, with the result that \(J_z = \pm 1\). If we take the beam direction to be the z-axis, this limits \(m_\psi \) to 1 and 2, i.e., components along x and y. Note that for \(\psi \) at rest system with \(p=(M_\psi , 0, 0, 0)\)
$$\begin{aligned} \sum ^2_{m_\psi =1}\psi _{\mu }(p,m_\psi )\psi ^*_{\mu '}(p,m_\psi ) =\delta _{\mu \mu '}(\delta _{\mu 1}+\delta _{\mu 2}). \end{aligned}$$
(4)
Then the differential decay width for the \(\psi \) radiative decay to an n-body final state is:
$$\begin{aligned}&\frac{d\Gamma }{d\Phi _n} = \frac{(2\pi )^4}{2M_\psi }\cdot \frac{1}{2}\cdot \frac{1}{2}\sum ^2_{m_\psi =1}\sum ^2_{m'_\gamma ,m_\gamma =1}\nonumber \\&\qquad \times |\psi _{\mu }(p,m_\psi )\; \varepsilon ^*_{\nu }(q_1,m_\gamma )\;\varepsilon ^*_{\alpha }(q_2,m'_\gamma )A^{\mu \nu \alpha }|^2 \nonumber \\&\quad = \frac{(2\pi )^4}{2M_\psi }\cdot \frac{1}{4}\sum ^2_{m_\psi =1} \psi _{\mu }(p,m_\psi )\;\psi ^*_{\mu '}(p,m_\psi )\; g^{(\perp )}_{\nu \nu '}(q_1)\;g^{(\perp )}_{\alpha \alpha '}(q_2)\; A^{\mu \nu \alpha }\;A^{*\mu '\nu '\alpha '}\nonumber \\&\quad = \frac{(2\pi )^4}{2M_\psi }\cdot \frac{1}{4}\sum ^2_{\mu =1}A^{\mu \nu \alpha }\; g^{(\perp )}_{\nu \nu '}(q_1)\;g^{(\perp )}_{\alpha \alpha '}(q_2)\;A^{*\mu \nu '\alpha '}\nonumber \\&\quad = \frac{(2\pi )^4}{2M_\psi }\cdot \frac{1}{4}\sum _{i,j}\Lambda _i\;\Lambda _j^*\sum ^2_{\mu =1} U_i^{\mu \nu \alpha }\;g^{(\perp )}_{\nu \nu '}(q_1)\;g^{(\perp )}_{\alpha \alpha '}(q_2)\;\nonumber \\&\qquad \times U_j^{*\mu \nu '\alpha '} \equiv \sum _{i,j}P_{ij}\cdot F_{ij}\;, \end{aligned}$$
(5)
where
$$\begin{aligned} P_{ij}= & {} P^*_{ji} = \frac{(2\pi )^4}{2M_\psi }\Lambda _i\;\Lambda ^*_j, \end{aligned}$$
(6)
$$\begin{aligned} F_{ij}= & {} F^*_{ji} = \frac{1}{4}\sum ^2_{\mu =1} U_i^{\mu \nu \alpha }\;g^{(\perp )}_{\nu \nu '}(q_1)\;g^{(\perp )}_{\alpha \alpha '}(q_2)\; U_j^{*\mu \nu '\alpha '}\;. \end{aligned}$$
(7)
\(d\Phi _n\) is the standard element of n-body phase space given by
$$\begin{aligned} d\Phi _n(p; p_1, \ldots p_n)=\delta ^4(p-\sum ^n_{i=1}p_i) \prod ^n_{i=1}\frac{d^3\mathbf{p}_i}{(2\pi )^32E_i}\;. \end{aligned}$$
(8)
2.1 4.1 Amplitudes for the doubly radiative decay \(\psi \rightarrow \gamma \gamma V(\rho , \omega , \phi )\)
This is a three step process: \(\psi \rightarrow \gamma R\) with \(R\rightarrow \gamma V(\rho , \omega , \phi )\) and \(\rho \rightarrow \pi ^+\pi ^-\), \(\omega \rightarrow \pi ^0\pi ^+\pi ^-\), \(\phi \rightarrow K^+ K^-\). The intermediate resonance state R that may appear in the process with \(J^{PC}\) values are \(0^{++}, 0^{-+}\), \(1^{++}\), \(1^{-+}\), \(2^{++}\), \(2^{-+}\), etc. Here J, P, C are the intrinsic spin, parity and C-parity of the R particle, respectively. For \(\psi \rightarrow \gamma R\), we denote the spin-orbital angular momenta between the photon and \(\psi \) by S and L, respectively. The tensor describing the first and the second steps will be denoted by \(\tilde{T}^{(L)}_{\mu _1\ldots \mu _L}\) and \(\tilde{t}^{(L_1)}_{\mu _1\ldots \mu _{L_1}}\), respectively. The vector describing the third step will be denoted by \(V_\mu \), where \(V(\rho ,\phi )_\mu = p_{1\mu } - p_{2\mu }\), here we use the fact that \(\pi ^+\) and \(\pi ^-\) (or \(K^+\) and \(K^-\)) have equal masses, and
$$\begin{aligned} V(\omega )_\mu= & {} \epsilon ^{\mu }_{\;\;\nu \lambda \sigma } p_1^\nu p_2^\lambda p_0^\sigma [B_1(Q_{\omega \rho 0})f^{(\rho )}_{(12)}B_1(Q_{\rho 12}) \\&+ B_1(Q_{\omega \rho 2})f^{(\rho )}_{(01)}B_1(Q_{\rho 10}) \\&+ B_1(Q_{\omega \rho 1})f^{(\rho )}_{(02)}B_1(Q_{\rho 20})]. \end{aligned}$$
Now we write the decay amplitude of the \(\psi \) into two photons and a vector in a general and compact form using the covariant tensor formalism. When writing the covariant tensor amplitude we have to keep in mind that there are two identical particles (photons) in the final state, due to Bose statistics decay amplitude is symmetric with respect to the exchange of photons. There is one independent covariant tensor amplitude for \(\psi \rightarrow \gamma 0^{++} \rightarrow \gamma \gamma V(\rho , \omega , \phi )\)
$$\begin{aligned} U^{\mu \nu \alpha } = g^{\mu \nu }\; \Big (f^{(R)}_{V\gamma (q_1)} + f^{(R)}_{V\gamma (q_2)}\Big )\; f^{(V)}\;V^\alpha , \end{aligned}$$
(9)
where \(f^{(V)}\) represents either \(f^{(\rho ,\phi )}_{(12)}\) or \(f^{(\omega )}_{(012)}\),
$$\begin{aligned} f^{(R)}_{V\gamma (q_i)}= & {} \frac{M_R\; \Gamma _R}{(p_V+q_i)^2 - M^2_R + iM_R\;\Gamma _R}\;,\nonumber \\ f^{(V)}= & {} \frac{M_V \;\Gamma _V}{p_V^2 - M^2_V + iM_V\;\Gamma _V}\;, \end{aligned}$$
(10)
here \(M_R\), \(M_V\) and \(\Gamma _R\), \(\Gamma _V\) are the mass and width of each resonance.
There is also one independent covariant tensor amplitude for \(\psi \rightarrow \gamma 0^{-+} \rightarrow \;\gamma \; \gamma \;V(\rho ,\omega , \phi )\)
$$\begin{aligned} U^{\mu \nu \alpha }_{12}&=\epsilon ^{\mu \nu \lambda \sigma } \; \epsilon ^{\alpha \beta \rho \delta } \; p_\lambda \nonumber \\&\quad \times \; \Big ({\tilde{T}}^{(1)}_\sigma (q_1) \; (p-q_1)_\rho \; t^{(1)}_{\beta }(q_2) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + \; {\tilde{T}}^{(1)}_\sigma (q_2) \; (p-q_2)_\rho \; t^{(1)}_{\beta }(q_1) \; f^{(R)}_{V\gamma (q_1)} \Big )V_\delta \; f^{(V)}\;,\nonumber \\ \end{aligned}$$
(11)
where
$$\begin{aligned} {\tilde{T}}^{(1)}_\mu (q_i)= & {} {\tilde{g}}_{\mu \nu }(p)\; (q_i -p_R)^\nu \; B_1(Q_{\psi \gamma (q_i) R}) \;,\nonumber \\ {\tilde{t}}^{(1)}_\mu (q_i)= & {} {\tilde{g}}_{\mu \nu }(p_R)\; (q_i -p_V)^\nu \; B_1(Q_{R \gamma (q_i) V}) \;. \end{aligned}$$
(12)
For the production reaction \(\psi \rightarrow \gamma 1^{++}\) there are two independent covariant tensor amplitudes; there are also two amplitudes for the decay reaction \(1^{++} \rightarrow \gamma V(\rho , \omega , \phi )\), all in all we have four amplitudes,
$$\begin{aligned} U^{\mu \nu \alpha }_ 1= & {} \epsilon ^{\mu \nu \lambda \sigma }\; \epsilon ^{\alpha \beta \rho }_{\sigma }\;p_\lambda \;\Big ( (p - q_1)_\rho \;f^{(R)}_{V\gamma (q_2)} \nonumber \\&+ (p - q_2)_\rho \; f^{(R)}_{V\gamma (q_1)} \Big ) V_\beta \; f^{(V)}\;, \end{aligned}$$
(13)
$$\begin{aligned} U^{\mu \nu \alpha }_ 2= & {} \epsilon ^{\mu \nu \lambda \sigma } \; \epsilon ^{\alpha \beta \rho \delta }\; p_\lambda \nonumber \\&\times \Big ( {\tilde{T}}^{(2)}_{\sigma \zeta }(q_1)\; (p -q_1)_\rho \; {\tilde{t}}^{(2)\zeta }_\delta (q_2)\;f^{(R)}_{V\gamma (q_2)} \nonumber \\&+ {\tilde{T}}^{(2)}_{\sigma \zeta }(q_2)\; (p -q_2)_\rho \; {\tilde{t}}^{(2)\zeta }_\delta (q_1)\;f^{(R)}_{V\gamma (q_1)} \Big )V_\beta \; f^{(V)}\;, \end{aligned}$$
(14)
$$\begin{aligned} U^{\mu \nu \alpha }_3= & {} \epsilon ^{\mu \nu \lambda \sigma } \; \epsilon ^{\alpha \beta \rho \delta }\;p_\lambda \Big ( (p-q_1)_\rho \; {\tilde{t}}^{(2)}_{\sigma \delta }(q_2) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&+ (p-q_2)_\rho \; {\tilde{t}}^{(2)}_{\sigma \delta }(q_1) \; f^{(R)}_{V\gamma (q_1)} \Big ) V_\beta \; f^{(V)}\;, \end{aligned}$$
(15)
$$\begin{aligned} U^{\mu \nu \alpha }_4= & {} \epsilon ^{\mu \nu \lambda \sigma }\; \epsilon ^{\alpha \beta \rho \delta } \; p_\lambda \; \Big ( {\tilde{T}}^{(2)}_{\sigma \delta }(q_1) \; (p -q_1)_\rho \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&+ {\tilde{T}}^{(2)}_{\sigma \delta }(q_2) \; (p -q_2)_\rho \; f^{(R)}_{V\gamma (q_1)} \Big ) V_\beta \; f^{(V)}\;, \end{aligned}$$
(16)
$$\begin{aligned} {\tilde{T}}^{(2)}_{\sigma \gamma }(q_i)= & {} [{\tilde{r}}_\sigma \;{\tilde{r}}_\gamma -{1\over 3}({\tilde{r}}\cdot {\tilde{r}}) \; {\tilde{g}}_{\sigma \gamma }(p_\psi )]B_2(Q_{\psi \gamma (q_i)R})\;, \end{aligned}$$
(17)
$$\begin{aligned} {\tilde{t}}^{(2)}_{\sigma \gamma }(q_i)= & {} [{\tilde{r}}_\sigma \; {\tilde{r}}_\gamma -{1\over 3}({\tilde{r}}\cdot {\tilde{r}}) \; {\tilde{g}}_{\sigma \gamma }(p_R)]B_2(Q_{R\gamma (q_i)V})\;, \end{aligned}$$
(18)
where \(r=p_b-p_c\) is the relative four momentum of the two decay products in the parent particle rest frame.
For the production reaction \(\psi \rightarrow \gamma 1^{-+}\) there are two independent covariant tensor amplitudes; there are also two amplitudes for the decay reaction \(1^{-+} \rightarrow \gamma V(\rho , \omega , \phi )\), all in all we have four amplitudes,
$$\begin{aligned} U^{\mu \nu \alpha }_1= & {} \, g^{\mu \nu } \Big ({\tilde{T}}^{(1)}_\beta (q_1) \; {\tilde{t}}^{(1)\beta } (q_2) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&+ \, T^{(1)}_\beta (q_2) \; {\tilde{t}}^{(1)\beta } (q_1) \; f^{(R)}_{V\gamma (q_1)} \Big ) V^\alpha f^{(V)} \;, \end{aligned}$$
(19)
$$\begin{aligned} U^{\mu \nu \alpha }_2= & {} \Big ({\tilde{T}}^{(1)\mu }(q_1) \; {\tilde{t}}^{(1)\nu }(q_2) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&+ \, {\tilde{T}}^{(1)\mu }(q_2) \; {\tilde{t}}^{(1)\nu }(q_1) \; f^{(R)}_{V\gamma (q_1)}\Big ) V^\alpha \; f^{(V)} \;, \end{aligned}$$
(20)
$$\begin{aligned} U^{\mu \nu \alpha }_3= & {} \, g^{\mu \nu } \Big ( {\tilde{T}}^{(1)\alpha }(q_1) \; {\tilde{t}}^{(1)\beta } (q_2) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&+ \, {\tilde{T}}^{(1)\alpha }(q_2) \; {\tilde{t}}^{(1)\beta } (q_1) \; f^{(R)}_{V\gamma (q_1)} \Big )V_\beta \; f^{(V)}, \end{aligned}$$
(21)
$$\begin{aligned} U^{\mu \nu \alpha }_4= & {} \, g^{\nu \alpha } \Big ( {\tilde{T}}^{(1)\mu }(q_1) \; {\tilde{t}}^{(1)\beta }(q_2) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&+ \, {\tilde{T}}^{(1)\mu }(q_2) \; {\tilde{t}}^{(1)\beta }(q_1) \; f^{(R)}_{V\gamma (q_1)} \Big ) V_\beta \; f^{(V)}\;. \end{aligned}$$
(22)
For the production reaction \(\psi \rightarrow \gamma 2^{++}\) there are three independent covariant tensor amplitudes; there are also three amplitudes for the decay reaction \(2^{++} \rightarrow \gamma V(\rho , \omega , \phi )\), all in all we have nine amplitudes,
$$\begin{aligned} U^{\mu \nu \alpha }_ 1&= \Big ( P^{(2)\mu \nu \alpha \beta }(p-q_1) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + P^{(2)\mu \nu \alpha \beta }(p-q_2) \; f^{(R)}_{V\gamma (q_1)} \Big ) V_\beta \; f^{(V)}\;, \end{aligned}$$
(23)
$$\begin{aligned} U^{\mu \nu \alpha }_ 2&= g^{\mu \nu }\Big ( P^{(2)\lambda \sigma \rho \delta }(p-q_1) \; {\tilde{T}}^{(2)}_{\lambda \sigma }(q_1) \; {\tilde{t}}^{(2)}_{\rho \delta }(q_2) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + P^{(2)\lambda \sigma \rho \delta }(p-q_2) \; \tilde{T}^{(2)}_{\lambda \sigma }(q_2) \; {\tilde{t}}^{(2)}_{\rho \delta }(q_1) \; f^{(R)}_{V\gamma (q_1)} \Big ) V^\alpha \; f^{(V)},\nonumber \\ \end{aligned}$$
(24)
$$\begin{aligned} U^{\mu \nu \alpha }_3&= \Big ( P^{(2)\nu \sigma \alpha \lambda }(p-q_1) \; {\tilde{T}}^{(2)\mu }_{\sigma } (q_1) \; {\tilde{t}}^{(2)}_{\lambda \beta }(q_2) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + P^{(2)\nu \sigma \alpha \lambda }(p-q_2) \; {\tilde{T}}^{(2)\mu }_{\sigma } (q_2) \; {\tilde{t}}^{(2)}_{\lambda \beta }(q_1) \; f^{(R)}_{V\gamma (q_1)} \Big ) V^\beta \; f^{(V)},\nonumber \\ \end{aligned}$$
(25)
$$\begin{aligned} U^{\mu \nu \alpha }_4&= \Big ( P^{(2)\mu \nu \lambda \sigma }(p-q_1) \; \tilde{t}^{(2)}_{\lambda \sigma }(q_2) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + P^{(2)\mu \nu \lambda \sigma }(p-q_2) \; \tilde{t}^{(2)}_{\lambda \sigma }(q_1) \; f^{(R)}_{V\gamma (q_1)} \Big )V^\alpha \; f^{(V)}, \end{aligned}$$
(26)
$$\begin{aligned} U^{\mu \nu \alpha }_5&= \Big ( P^{(2)\mu \nu \alpha \lambda }(p-q_1) \; {\tilde{t}}^{(2)}_{\beta \lambda }(q_2) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + P^{(2)\mu \nu \alpha \lambda }(p-q_2) \; {\tilde{t}}^{(2)}_{\beta \lambda }(q_1) \; f^{(R)}_{V\gamma (q_1)} \Big ) V^\beta \; f^{(V)}, \end{aligned}$$
(27)
$$\begin{aligned} U^{\mu \nu \alpha }_6&= g^{\mu \nu } \Big (P^{(2)\lambda \sigma \alpha \beta }(p-q_1) \; \tilde{T}^{(2)}_{\lambda \sigma }(q_1) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + P^{(2)\lambda \sigma \alpha \beta }(p-q_2) \; \tilde{T}^{(2)}_{\lambda \sigma }(q_2) \; f^{(R)}_{V\gamma (q_1)} \Big ) V_\beta \; f^{(V)}, \end{aligned}$$
(28)
$$\begin{aligned} U^{\mu \nu \alpha }_7&= g^{\mu \nu } \Big ( P^{(2)\lambda \sigma \alpha \delta }(p-q_1) \; \tilde{T}^{(2)}_{\lambda \sigma }(q_1) \; {\tilde{t}}^{(2)}_{\beta \delta } (q_2) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + P^{(2)\lambda \sigma \alpha \delta }(p-q_2) \; \tilde{T}^{(2)}_{\lambda \sigma }(q_2) \; {\tilde{t}}^{(2)}_{\beta \delta } (q_1) \; f^{(R)}_{V\gamma (q_1)} \Big ) V^\beta \; f^{(V)}\;,\nonumber \\ \end{aligned}$$
(29)
$$\begin{aligned} U^{\mu \nu \alpha }_8&= \Big (P^{(2)\nu \lambda \alpha \beta }(p-q_1) \; \tilde{T}^{(2)\mu }_{\lambda }(q_1) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + P^{(2)\nu \lambda \alpha \beta }(p-q_2) \; \tilde{T}^{(2)\mu }_{\lambda }(q_2) \; f^{(R)}_{V\gamma (q_1)} \Big )V_\beta \; f^{(V)}, \end{aligned}$$
(30)
$$\begin{aligned} U^{\mu \nu \alpha }_9&= \Big ( P^{(2)\nu \delta \lambda \sigma }(p-q_1) \; {\tilde{T}}^{(2)\mu }_\delta (q_1) \; {\tilde{t}}^{(2)}_{\lambda \sigma }(q_2) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + P^{(2)\nu \delta \lambda \sigma }(p-q_2) \; {\tilde{T}}^{(2)\mu }_\delta (q_2) \; {\tilde{t}}^{(2)}_{\lambda \sigma }(q_1) \; f^{(R)}_{V\gamma (q_1)} \Big ) V^\alpha \; f^{(V)} \;.\nonumber \\ \end{aligned}$$
(31)
For the production reaction \(\psi \rightarrow \gamma 2^{-+}\) there are three independent covariant tensor amplitudes; there are also three amplitudes for the decay reaction \(2^{-+} \rightarrow \gamma V(\rho , \omega , \phi )\), all in all we have nine amplitudes,
$$\begin{aligned} U^{\mu \nu \alpha }_1&= \epsilon ^{\mu \nu \lambda \sigma } \; \epsilon ^{\alpha \beta \rho \xi } \; p_\sigma \Big ( {\tilde{T}}^{(1)\zeta }(q_1) \; (p-q_1)_\xi \; {\tilde{t}}^{(1)\delta }(q_2) \;\nonumber \\&\quad \times P^{(2)}_{\lambda \zeta \rho \delta }(p-q_1) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + {\tilde{T}}^{(1)\zeta }(q_2) \; (p-q_2)_\xi \; {\tilde{t}}^{(1)\delta }(q_1) \;\nonumber \\&\quad \times P^{(2)}_{\lambda \zeta \rho \delta }(p-q_2) \; f^{(R)}_{V\gamma (q_1)}\Big ) V_\beta \; f^{(V)}\;, \end{aligned}$$
(32)
$$\begin{aligned} U^{\mu \nu \alpha }_2&= \epsilon ^{\mu \nu \lambda \sigma } \; \epsilon ^{\alpha \beta \rho \xi } \; p_\sigma \Big ( {\tilde{T}}^{(3)}_{\lambda \zeta \delta }(q_1) \; (p-q_1)_\xi \; {\tilde{t}}^{(3)}_{\rho \zeta '\delta '}(q_2) \;\nonumber \\&\quad \times P^{(2)\zeta \delta \zeta '\delta '}(p-q_1) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + {\tilde{T}}^{(3)}_{\lambda \zeta \delta }(q_2) \; (p-q_2)_\xi \; {\tilde{t}}^{(3)}_{\rho \zeta '\delta '}(q_1) \;\nonumber \\&\quad \times P^{(2)\zeta \delta \zeta '\delta '}(p-q_2) \; f^{(R)}_{V\gamma (q_1)} \Big ) V_\beta \; f^{(V)} \;, \end{aligned}$$
(33)
$$\begin{aligned} U^{\mu \nu \alpha }_ 3&= \epsilon ^{\nu \lambda \sigma \zeta } \; \epsilon ^{\beta \rho \delta \xi } \; p_\zeta \Big ({\tilde{T}}^{(3)\mu \lambda '}_\sigma (q_1) \; (p-q_1)_\xi \; {\tilde{t}}^{(3)\alpha \rho '}_{\delta }(q_2) \;\nonumber \\&\quad \times P^{(2)}_{\lambda \lambda '\rho \rho '}(p-q_1) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + {\tilde{T}}^{(3)\mu \lambda '}_\sigma (q_2) \; (p-q_2)_\xi \; {\tilde{t}}^{(3)\alpha \rho '}_{\delta }(q_1) \;\nonumber \\&\quad \times P^{(2)}_{\lambda \lambda '\rho \rho '}(p-q_2) \; f^{(R)}_{V\gamma (q_1)} \Big ) V_\beta \; f^{(V)}\;, \end{aligned}$$
(34)
$$\begin{aligned} U^{\mu \nu \alpha }_ 4&= \epsilon ^{\mu \nu \lambda \sigma } \; \epsilon ^{\alpha \beta \rho \xi } \; p_\sigma \Big ({\tilde{T}}^{(1)\zeta '}(q_1) \; (p-q_1)_\xi \; {\tilde{t}}^{(3)\delta \zeta }_\rho (q_2) \;\nonumber \\&\quad \times P^{(2)}_{\lambda \zeta '\delta \zeta }(p-q_1) \; f^{(R)}_{V\gamma (q_2)}\nonumber \\&\quad + {\tilde{T}}^{(1)\zeta '}(q_2) \; (p-q_2)_\xi \; {\tilde{t}}^{(3)\delta \zeta }_\rho (q_1) \;\nonumber \\&\quad \times P^{(2)}_{\lambda \zeta '\delta \zeta }(p-q_2) \; f^{(R)}_{V\gamma (q_1)} \Big ) V_\beta \; f^{(V)} \;, \end{aligned}$$
(35)
$$\begin{aligned} U^{\mu \nu \alpha }_5&= \epsilon ^{\mu \nu \lambda \sigma } \; \epsilon ^{\beta \rho \delta \xi } \; p_\sigma \Big ( {\tilde{T}}^{(1)\zeta '}(q_1) \; (p-q_1)_\xi \; {\tilde{t}}^{(3)\alpha \zeta }_{\delta }(q_2) \;\nonumber \\&\quad \times P^{(2)}_{\lambda \zeta '\rho \zeta } \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + {\tilde{T}}^{(1)\zeta '}(q_2) \; (p-q_2)_\xi \; {\tilde{t}}^{(3)\alpha \zeta }_{\delta }(q_1) \;\nonumber \\&\quad \times P^{(2)}_{\lambda \zeta '\rho \zeta } \; f^{(R)}_{V\gamma (q_1)} \Big ) V_\beta \; f^{(V)}\;, \end{aligned}$$
(36)
$$\begin{aligned} U^{\mu \nu \alpha }_6&= \epsilon ^{\mu \nu \lambda \sigma } \; \epsilon ^{\alpha \beta \rho \xi } \; p_\sigma \Big ( {\tilde{T}}^{(3)\zeta '\delta }_\lambda (q_1) \; (p-q_1)_\xi \; {\tilde{t}}^{(1)\zeta }(q_2) \;\nonumber \\&\quad \times P^{(2)}_{\zeta '\delta \rho \zeta }(p-q_1) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + {\tilde{T}}^{(3)\zeta '\delta }_\lambda (q_2) \; (p-q_2)_\xi \; {\tilde{t}}^{(1)\zeta }(q_1) \;\nonumber \\&\quad \times P^{(2)}_{\zeta '\delta \rho \zeta }(p-q_2) \; f^{(R)}_{V\gamma (q_1)} \Big ) V^\beta \; f^{(V)}\;, \end{aligned}$$
(37)
$$\begin{aligned} U^{\mu \nu \alpha }_7&= \epsilon ^{\mu \nu \lambda \sigma } \; \epsilon ^{\beta \tau \rho \xi } \; p_\sigma \Big ({\tilde{T}}^{(3)\zeta \delta }_\lambda (q_1) \; (p-q_1)_\xi \; {\tilde{t}}^{(3)\alpha \delta '}_\rho (q_2) \;\nonumber \\&\quad \times P^{(2)}_{\zeta \delta \tau \delta '}(p-q_1) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + {\tilde{T}}^{(3)\zeta \delta }_\lambda (q_2) \; (p-q_2)_\xi \; {\tilde{t}}^{(3)\alpha \delta '}_\rho (q_1) \;\nonumber \\&\quad \times P^{(2)}_{\zeta \delta \tau \delta '}(p-q_1)f^{(R)}_{V\gamma (q_1)}\Big ) V_\beta \; f^{(V)}\;, \end{aligned}$$
(38)
$$\begin{aligned} U^{\mu \nu \alpha }_ 8&= \epsilon ^{\nu \lambda \sigma \zeta '} \; \epsilon ^{\alpha \beta \rho \xi } \; p_\zeta ' \Big ({\tilde{T}}^{(3)\mu \zeta }_\sigma (q_1) \; (p-q_1)_\xi \; {\tilde{t}}^{(1)\delta }(q_2) \;\nonumber \\&\quad \times P^{(2)}_{\lambda \zeta \rho \delta }(p-q_1) \; f^{(R)}_{V\gamma (q_2)}\nonumber \\&\quad + {\tilde{T}}^{(3)\mu \zeta }_\sigma (q_2) \; (p-q_2)_\xi \; {\tilde{t}}^{(1)\delta }(q_1) \;\nonumber \\&\quad \times P^{(2)}_{\lambda \zeta \rho \delta }(p-q_2) \; f^{(R)}_{V\gamma (q_1)} \Big ) V_\beta \; f^{(V)}\;, \end{aligned}$$
(39)
$$\begin{aligned} U^{\mu \nu \alpha }_ 9&= \epsilon ^{\nu \lambda \sigma \zeta } \; \epsilon ^{\alpha \beta \rho \xi } \; p_\zeta \Big ({\tilde{T}}^{(3)\mu \delta }_\sigma (q_1) \; (p-q_1)_\xi \; {\tilde{t}}^{(3)\lambda '\delta '}_\rho (q_2) \;\nonumber \\&\quad \times P^{(2)}_{\lambda \delta \lambda '\delta '}(p-q_1) \; f^{(R)}_{V\gamma (q_2)} \nonumber \\&\quad + {\tilde{T}}^{(3)\mu \delta }_\sigma (q_2) \; (p-q_2)_\xi \; {\tilde{t}}^{(3)\lambda '\delta '}_\rho (q_1) \;\nonumber \\&\quad \times P^{(2)}_{\lambda \delta \lambda '\delta '}(p-q_2) \; f^{(R)}_{V\gamma (q_1)} \Big ) V_\beta \; f^{(V)}\;. \end{aligned}$$
(40)
