Nature of $X(3872)$ from its radiative decay

We take the covariant light-front quark model used in Jaus:1991cy ; Jaus:1989au ; Jaus:1996np ; Jaus:1999zv ; Cheng:2003sm ; Choi:2007se ; Shi:2016cef ; Chang:2019obq to calculate the width of radiative decay. In Ref. Ke:2011jf , the radiative decay of $X(3872)$ is investigated for the case of $2^{-+}$. In the light-front framework, a momentum $p$ is expressed as

	$\displaystyle p^{\mu}$	$\displaystyle=$	$\displaystyle(p^{-},p^{+},p_{\bot}),$
	$\displaystyle p_{\bot}$	$\displaystyle=$	$\displaystyle(p^{1},p^{2}),\,\,p^{-}=p^{0}-p^{3},\,\,p^{+}=p^{0}+p^{3},$		(1)

and thus

$$p^{2}=p^{+}p^{-}-p_{\bot}^{2},\,\,\,p\cdot q=\frac{1}{2}(p^{-}q^{+}+p^{+}q^{-})-p_{\bot}q_{\bot}.$$

(2)

The incoming (outgoing) meson has momentum $P^{\prime}(^{\prime\prime})=p_{1}^{\prime}(^{\prime\prime})+p_{2}$, where $p_{1}^{\prime}(^{\prime\prime})$ and $p_{2}$ are the momenta of the off-shell active quark and spectator quark, respectively, with masses $m^{\prime}_{1}(^{\prime\prime})$ and $m_{2}$. We defined $P=P^{\prime}+P^{\prime\prime}$ and $q=P^{\prime}-P^{\prime\prime}$. An illustration is shown in Fig. 1 ¹¹1In principle, there are diagrams of the photon that couples directly to the quark-quark-vector meson vertex. However, as shown in Ref. Ganbold:2021nvj those contributions from the bubble diagrams are small and not more than 10% of the result. Thus they are omitted without loss of our accuracy for a conclusion. These momenta can be expressed in terms of the internal variables ($x_{i},p^{\prime}_{\bot}$),

	$\displaystyle p_{1}^{\prime}(^{\prime\prime})^{+}=x_{1}P^{{}^{\prime}(^{\prime\prime})+},\,\,\,p_{2}^{\prime}(^{\prime\prime})^{+}=x_{2}P^{\prime}(^{\prime\prime})^{+}$
	$\displaystyle p^{\prime}(^{\prime\prime})_{1\bot}=x_{1}P^{\prime}(^{\prime\prime})_{\bot}+p^{\prime}(^{\prime\prime})_{\bot},$
	$\displaystyle p^{\prime}(^{\prime\prime})_{2\bot}=x_{2}P^{{}^{\prime}(^{\prime\prime})}_{\bot}-p^{\prime}(^{\prime\prime})_{\bot},$
	$\displaystyle p^{\prime\prime}_{\bot}=p^{\prime}_{\bot}-x_{2}q_{\bot},$		(3)

with $x_{1}+x_{2}=1$. The following physical quantities are defined for subsequent derivations:

	$\displaystyle M_{0}^{\prime 2}=(e_{1}^{\prime}+e_{2})^{2}=\frac{p_{\bot}^{\prime 2}+m_{1}^{\prime 2}}{x_{1}}+\frac{p_{\bot}^{\prime 2}+m_{2}^{2}}{1-x_{1}},$
	$\displaystyle e_{1}^{\prime}=\sqrt{p_{\bot}^{\prime 2}+p_{z}^{\prime 2}+m_{1}^{\prime 2}},\,\,e_{2}=\sqrt{p_{\bot}^{\prime 2}+p_{z}^{\prime 2}+m_{2}^{2}},$
	$\displaystyle p_{z}^{\prime}=\frac{(1-x_{1})M_{0}^{\prime}}{2}-\frac{p_{\bot}^{\prime 2}+m_{2}^{2}}{2(1-x_{1})M_{0}^{\prime}},$
	$\displaystyle M_{0}^{\prime\prime 2}=\frac{p_{\bot}^{\prime\prime 2}+m_{1}^{\prime\prime 2}}{x_{1}}+\frac{p_{\bot}^{\prime\prime 2}+m_{2}^{2}}{1-x_{1}},$
	$\displaystyle e_{1}^{\prime\prime}=\sqrt{p_{\bot}^{\prime\prime 2}+p_{z}^{\prime\prime 2}+m_{1}^{\prime\prime 2}},$
	$\displaystyle p_{z}^{\prime\prime}=\frac{(1-x_{1})M_{0}^{\prime\prime}}{2}-\frac{p_{\bot}^{\prime\prime 2}+m_{2}^{2}}{2(1-x_{1})M_{0}^{\prime\prime}}.$		(4)

Here, $M_{0}^{\prime 2}$ ($M_{0}^{\prime\prime 2}$) can be interpreted as the kinetic invariant mass squared of the incoming (outgoing) $q\overline{q}$ system, and $e_{i}$ is the energy of Quark $i$.

The transition amplitude for $X(3872)\to J/\psi\,\gamma,\psi^{\prime}\gamma$ can be written as

	$\displaystyle A(X(3872)\rightarrow\psi\gamma)=\epsilon^{*\alpha}(q)\epsilon^{\prime\mu}(P^{\prime})\epsilon^{\prime\prime*\nu}(P^{\prime\prime})\mathcal{A}_{\alpha\mu\nu}$
	$\displaystyle\mathcal{A}_{\alpha\mu\nu}=\varepsilon_{\alpha\nu\beta\eta}P^{\beta}q^{\eta}[P_{\mu}f_{1}(q^{2})+q_{\mu}f_{2}(q^{2})]$
	$\displaystyle\qquad\quad+\varepsilon_{\alpha\mu\beta\eta}P^{\beta}q^{\eta}[P_{\nu}f_{3}(q^{2})+q_{\nu}f_{4}(q^{2})]$
	$\displaystyle\qquad\quad+\varepsilon_{\alpha\mu\nu\rho}[P^{\rho}f_{5}(q^{2})+q^{\rho}f_{6}(q^{2})]$
	$\displaystyle\qquad\quad+\varepsilon_{\mu\nu\beta\eta}P^{\beta}q^{\eta}[P_{\alpha}f_{7}(q^{2})+q_{\alpha}f_{8}(q^{2})],$		(5)

where $\epsilon^{\prime}$ denotes the polarization of $X(3872)$, $\epsilon^{\prime\prime}$ denotes the polarization of $J/\psi$ or $\psi^{\prime}$, and $\epsilon$ denotes the polarization of the photon. $f_{i}(q^{2})$ with $i=1\cdots 8$ is the $q^{2}$-dependent form factor that encodes the underlying dynamics. By considering the properties of the polarization vectors, the following can be obtained:

	$\displaystyle\epsilon^{*\alpha}(q)\cdot q_{\alpha}=0$
	$\displaystyle\epsilon^{\prime\mu}(P^{\prime})\cdot P^{\prime}_{\mu}=\epsilon^{\prime\mu}(P^{\prime})\cdot\frac{1}{2}(P+q)_{\mu}=0$
	$\displaystyle\epsilon^{\prime\prime*\nu}(P^{\prime\prime})\cdot P^{\prime\prime}_{\nu}=\epsilon^{\prime\prime*\nu}(P^{\prime\prime})\cdot\frac{1}{2}(P-q)_{\nu}=0,$		(6)

$\mathcal{A}_{\alpha\mu\nu}$ will be reduced to, and only the nonvanishing terms are maintained as follows:

	$\displaystyle\mathcal{A}_{\alpha\mu\nu}$	$\displaystyle=$	$\displaystyle\varepsilon_{\alpha\nu\beta\eta}P^{\beta}q^{\eta}P_{\mu}f_{m}(q^{2})+\varepsilon_{\alpha\mu\beta\eta}P^{\beta}q^{\eta}P_{\nu}f_{p}(q^{2})$		(7)
			$\displaystyle+\varepsilon_{\alpha\mu\nu\rho}[P^{\rho}f_{5}(q^{2})+q^{\rho}f_{6}(q^{2})]$
			$\displaystyle+\varepsilon_{\mu\nu\beta\eta}P^{\beta}q^{\eta}P_{\alpha}f_{7}(q^{2}),$

where $f_{m}(q^{2})=f_{1}(q^{2})-f_{2}(q^{2})$, $f_{p}(q^{2})=f_{3}(q^{2})+f_{4}(q^{2})$. Imposing the gauge invariance $q^{\alpha}A_{\alpha\mu\nu}=0$, we further obtained

	$\displaystyle\mathcal{A}_{\alpha\mu\nu}$	$\displaystyle=$	$\displaystyle\varepsilon_{\alpha\nu\beta\eta}P^{\beta}q^{\eta}P_{\mu}f_{m}(q^{2})+\varepsilon_{\alpha\mu\beta\eta}P^{\beta}q^{\eta}P_{\nu}f_{p}(q^{2})$		(8)
			$\displaystyle+\varepsilon_{\alpha\mu\nu\rho}q^{\rho}f_{6}(q^{2}).$

For the calculation of radiative decay, we employed the covariant light-front quark model based on the assumption that $X(3872)$ is regarded as an axial vector ($A$) meson of $c\overline{c}$, while $J/\psi$ ($\psi^{\prime}$) is a vector ($V$) meson. The vertex functions for the axial vector meson and vector meson states are written as Cheng:2003sm

	$\displaystyle-iH_{A}^{\prime}\left(\gamma_{\mu}+\frac{(p_{1}^{\prime}-p_{2})_{\mu}}{W_{A}}\right)\gamma_{5},$
	$\displaystyle iH_{V}^{\prime\prime}\left(\gamma_{\mu}-\frac{(p_{1}^{\prime\prime}-p_{2})_{\mu}}{W_{V}}\right),$		(9)

where $H_{A}^{\prime}$ and $H_{V}^{\prime\prime}$ are the wave functions of the axial vector meson and vector meson that will be given below. The Feynman amplitude corresponding to Fig. 1 can be written as

	$\displaystyle A_{\alpha\mu\nu}$	$\displaystyle=$	$\displaystyle\frac{N_{c}}{(2\pi)^{4}}\int d^{4}p_{1}^{\prime}H_{A}^{\prime}H_{V}^{\prime\prime}$		(10)
			$\displaystyle\times\left(\frac{2e}{3}\frac{S_{\alpha\mu\nu}^{a}}{N_{1}^{\prime}N_{1}^{\prime\prime}N_{2}}+\frac{2e}{3}\frac{S_{\alpha\mu\nu}^{b}}{{N_{2}^{\prime}N_{2}^{\prime\prime}N_{1}}}\right),$

where $N_{c}=3$, $2e/3$ is the electric charge of the charm quark, $N_{i}=p_{i}^{2}-m_{i}^{2}+i0^{+}$, $N^{\prime}_{i}=p^{\prime 2}_{i}-m^{\prime 2}_{i}+i0^{+}$, and $N^{\prime\prime}_{i}=p^{\prime\prime 2}_{i}-m^{\prime\prime 2}_{i}+i0^{+}$. The terms $S_{\alpha\mu\nu}^{a}$ and $S_{\alpha\mu\nu}^{b}$ are the traces corresponding to Fig. 1(a) and Fig. 1(b), respectively. They are shown as follows:

	$\displaystyle S_{\alpha\mu\nu}^{a}$	$\displaystyle=$	$\displaystyle\text{Tr}\Big{[}(\gamma_{\mu}+\frac{(p_{1}^{\prime}-p_{2})_{\mu}}{W_{A}^{\prime}})\gamma_{5}(-p_{2}\mkern-10.5mu/+m_{2})(\gamma_{\nu}-\frac{(p_{1}^{\prime\prime}-p_{2})_{\nu}}{W_{V}^{\prime\prime}})(p_{1}^{\prime\prime}\mkern-10.5mu/+m_{1}^{\prime\prime})\gamma_{\alpha}(p_{1}^{\prime}\mkern-10.5mu/+m_{1}^{\prime})\Big{]},$
	$\displaystyle S_{\alpha\mu\nu}^{b}$	$\displaystyle=$	$\displaystyle\text{Tr}\Big{[}(\gamma_{\mu}+\frac{(p_{1}-p_{2}^{\prime})_{\mu}}{W_{A}^{\prime}})\gamma_{5}(-p_{2}^{\prime}\mkern-10.5mu/+m_{2}^{\prime})\gamma_{\alpha}(-p_{2}^{\prime\prime}\mkern-10.5mu/+m_{2}^{\prime\prime})(\gamma_{\nu}-\frac{(p_{1}-p_{2}^{\prime\prime})_{\nu}}{W_{V}^{\prime\prime}})(p_{1}\mkern-10.5mu/+m_{1})\Big{]}.$		(11)

Explicitly, the following can be obtained:

	$\displaystyle S_{\alpha\mu\nu}^{a}$	$\displaystyle=$	$\displaystyle\varepsilon_{\alpha\mu\nu\rho}\Big{[}-2p_{1}^{\prime\rho}(N_{1}^{\prime\prime}-2m_{2}m_{1}^{\prime\prime}+m_{1}^{\prime\prime 2}-2m_{1}^{\prime\prime}m_{1}^{\prime}+N_{1}^{\prime}-q^{2})+q^{\rho}(N_{1}^{\prime\prime}+4m_{2}m_{1}^{\prime}$		(12)
			$\displaystyle+m_{1}^{\prime\prime 2}-2m_{1}^{\prime\prime}m_{1}^{\prime}+m_{1}^{\prime 2}+N_{1}^{\prime}-q^{2})+P^{\rho}(N_{1}^{\prime\prime}+m_{1}^{\prime\prime 2}-2m_{1}^{\prime\prime}m_{1}^{\prime}+m_{1}^{\prime 2}+N_{1}^{\prime}-q^{2})\Big{]}$
			$\displaystyle+2\varepsilon_{\alpha\mu\beta\eta}q^{\beta}p_{1}^{\prime\eta}\Big{(}P_{\nu}+q_{\nu}-2p_{1\nu}^{\prime}\Big{)}+2\varepsilon_{\alpha\nu\beta\eta}q^{\beta}p_{1}^{\prime\eta}\Big{(}P_{\mu}+q_{\mu}-2p_{1\mu}^{\prime}\Big{)}$
			$\displaystyle+\varepsilon_{\mu\nu\beta\eta}\Big{[}2P^{\beta}p_{1}^{\prime\eta}(2p_{1\alpha}^{\prime}-q^{\alpha})-2P^{\beta}q^{\eta}p_{1\alpha}^{\prime}-2q^{\beta}p_{1}^{\prime\eta}q_{\alpha}\Big{]}$
			$\displaystyle+\frac{\varepsilon_{\alpha\nu\beta\eta}}{W_{A}^{\prime}}(P_{\mu}+q_{\mu}-4p_{1\mu}^{\prime})\Big{[}p_{1}^{\prime\eta}q^{\beta}(m_{1}^{\prime}+m_{1}^{\prime\prime}-2m_{2})+P^{\beta}p_{1}^{\prime\eta}(m_{1}^{\prime\prime}+m_{1}^{\prime})+P^{\beta}q^{\eta}m_{1}^{\prime}\Big{]}$
			$\displaystyle+\frac{\varepsilon_{\alpha\mu\beta\eta}}{W_{V}^{\prime\prime}}(P_{\nu}+3q_{\nu}-4p_{1\nu}^{\prime})\Big{[}-p_{1}^{\prime\eta}q^{\beta}(m_{1}^{\prime}+m_{1}^{\prime\prime}+2m_{2})+P^{\beta}p_{1}^{\prime\eta}(+m_{1}^{\prime}-m_{1}^{\prime\prime})-P^{\beta}q^{\eta}m_{1}^{\prime}\Big{]}$
			$\displaystyle+\frac{\varepsilon_{\alpha\beta\eta\rho}P^{\beta}q^{\eta}p_{1}^{\prime\rho}(P_{\nu}+3q_{\nu}-4p_{1\nu}^{\prime})(P_{\mu}+q_{\mu}-4p_{1\mu}^{\prime})}{2W_{A}^{\prime}W_{V}^{\prime\prime}}.$

In the light-front formalism, the integration measure is

$$d^{4}p^{\prime}_{1}=\frac{1}{2}P^{\prime+}dp_{1}^{\prime-}dx_{1}d^{2}p^{\prime}_{\bot}.$$

(13)

As discussed in Ref. Jaus:1999zv , if it is assumed that the vertex function ($H_{A}^{\prime}$, $H_{V}^{\prime\prime}$) has no pole in the upper complex $p_{1}^{\prime-}$ plane, the CLFQM used here and the standard light-front formulism where the covariance property is not satisfied will lead to identical results at the one-loop level. We assume such a situation holds here. Then, the integration over $dp_{1}^{\prime-}$ picks up a residue at $p_{2}=\hat{p}_{2}$, where $\hat{p}_{2}^{2}=m_{2}^{2}$. Thus, the following replacements will be obtained:

	$\displaystyle N_{1}^{\prime}\rightarrow\hat{N}_{1}^{\prime}=x_{1}(M^{\prime 2}-M_{0}^{\prime 2}),$
	$\displaystyle N_{1}^{\prime\prime}\rightarrow\hat{N}_{1}^{\prime\prime}=x_{1}(M^{\prime\prime 2}-M_{0}^{\prime\prime 2}),$
	$\displaystyle H_{A}^{\prime}\rightarrow\hat{H}_{A}^{\prime}\equiv h_{A}^{\prime},\quad H_{V}^{\prime\prime}\rightarrow\hat{H}_{V}^{\prime\prime}\equiv h_{V}^{\prime\prime},$
	$\displaystyle W_{A}^{\prime}\rightarrow\hat{W}_{A}^{\prime}\equiv w_{A}^{\prime},\quad W_{V}^{\prime\prime}\rightarrow\hat{W}_{V}^{\prime\prime}\equiv w_{V}^{\prime\prime},$
	$\displaystyle\int d^{4}p_{1}^{\prime}\frac{H_{A}^{\prime}H_{V}^{\prime\prime}}{N_{1}^{\prime}N_{1}^{\prime\prime}N_{2}}S_{\alpha\mu\nu}^{a}$
	$\displaystyle\qquad\rightarrow-i\pi\int dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{A}^{\prime}h_{V}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}\hat{S}_{\alpha\mu\nu}^{a},$		(14)

with Cheng:2003sm

	$\displaystyle h_{A}^{\prime}=\sqrt{\frac{3}{2}}(M^{\prime 2}-M_{0}^{\prime 2})\sqrt{\frac{x_{1}x_{2}}{N_{c}}}\frac{1}{\sqrt{2}\widetilde{M}_{0}^{\prime}}\frac{\widetilde{M}_{0}^{\prime 2}}{2\sqrt{3}M_{0}^{\prime}}\varphi^{\prime}(nP),$
	$\displaystyle h_{V}^{\prime\prime}=(M^{\prime\prime 2}-M_{0}^{\prime\prime 2})\sqrt{\frac{x_{1}x_{2}}{N_{c}}}\frac{1}{\sqrt{2}\widetilde{M}_{0}^{\prime\prime}}\varphi^{\prime\prime}(nS),$
	$\displaystyle w_{A}^{\prime}=\frac{\widetilde{M}_{0}^{\prime 2}}{m_{1}^{\prime}-m_{2}},\quad w_{V}^{\prime\prime}=M_{0}^{\prime\prime}+m_{1}^{\prime\prime}+m_{2},$
	$\displaystyle\widetilde{M}_{0}^{\prime}(^{\prime\prime})=\sqrt{M_{0}^{\prime}(^{\prime\prime})^{2}-(m_{1}^{\prime}(^{\prime\prime})-m_{2})^{2}}.$		(15)

Considering $X(3872)$ as a $2P$ charmonium, we need the following wave functions ($1S$ for $J/\psi$ and $2S$ for $\psi^{\prime}$) Hwang:2008qi

	$\displaystyle\varphi(1S)$	$\displaystyle=$	$\displaystyle 4\left(\frac{\pi}{\beta^{2}}\right)^{\frac{3}{4}}\sqrt{\frac{dp_{z}}{dx_{2}}}\exp\left(-\frac{p_{z}^{2}+p_{\bot}^{2}}{2\beta^{2}}\right),$
	$\displaystyle\varphi(2S)$	$\displaystyle=$	$\displaystyle 4\sqrt{\frac{8}{3}}\left(\frac{\pi}{\beta^{2}}\right)^{\frac{3}{4}}\sqrt{\frac{dp_{z}}{dx_{2}}}\left(\frac{p_{z}^{2}+p_{\bot}^{2}}{2\beta^{2}}-\frac{3}{4}\right)$
			$\displaystyle\times\exp\left(-\frac{p_{z}^{2}+p_{\bot}^{2}}{2\beta^{2}}\right),$
	$\displaystyle\varphi(2P)$	$\displaystyle=$	$\displaystyle 4\left(\frac{\pi}{\beta^{2}}\right)^{\frac{3}{4}}\sqrt{\frac{dp_{z}}{dx_{2}}}\sqrt{\frac{5}{\beta^{2}}}\left(1-\frac{2}{5}\frac{p_{z}^{2}+p_{\bot}^{2}}{2\beta^{2}}\right)$		(16)
			$\displaystyle\times\exp\left(-\frac{p_{z}^{2}+p_{\bot}^{2}}{2\beta^{2}}\right).$

with

$$\frac{dp_{z}^{\prime}(^{\prime\prime})}{dx_{2}}=\frac{e_{1}^{\prime}(^{\prime\prime})e_{2}}{x_{1}x_{2}M_{0}^{\prime}(^{\prime\prime})}.$$

(17)

For the practical calculation, the symbols $p_{z}$ and $p_{\bot}$ in Eq. (II.2) should be associated with the superscript ^′ for the initial meson and ^′′ for the final meson.

In a concrete calculation in the light front quark model, one usually works in the $q^{+}=0$ frame, where one only needs to consider the valence quark part, and all other non-valence contributions e.g., the pair creations from the vacuum (the so-called Z diagram) vanish. By making the contour integral over $p_{1}^{\prime-}$ for the four-dimensional integration, one gets the above hat quantities. These matrix elements will acquire a spurious $\omega$ dependence, with $\omega=(2,0,0_{\perp})$ as a constant. Thus the covariance is lost. The reason is due to the missing contribution in the $p_{1}^{\prime-}=p_{1}^{\prime\prime-}=0$ region. Once such contribution is correctly included, the covariance will be restored. W. Jaus finds an effective way for inclusion of such zero mode contribution as the necessary condition for the covariance, and the net results are Jaus:1999zv :

	$\displaystyle\hat{p}^{\prime}_{1\mu}\rightarrow$		$\displaystyle P_{\mu}A_{1}^{(1)}+q_{\mu}A_{2}^{(1)}$
	$\displaystyle\hat{p}^{\prime}_{1\mu}\hat{p}^{\prime}_{1\nu}\rightarrow$		$\displaystyle g_{\mu\nu}A_{1}^{(2)}+P_{\mu}P_{\nu}A_{2}^{(2)}+(P_{\mu}q_{\nu}+q_{\mu}P_{\nu})A_{3}^{(2)}$
			$\displaystyle+q_{\mu}q_{\nu}A_{4}^{(2)}$
	$\displaystyle\hat{p}^{\prime}_{1\mu}\hat{p}^{\prime}_{1\nu}\hat{p}^{\prime}_{1\alpha}\rightarrow$		$\displaystyle(g_{\mu\nu}P_{\alpha}+g_{\mu\alpha}P_{\nu}+g_{\nu\alpha}P_{\mu})A_{1}^{(3)}$		(18)
			$\displaystyle+(g_{\mu\nu}q_{\alpha}+g_{\mu\alpha}q_{\nu}+g_{\nu\alpha}q_{\mu})A_{2}^{(3)}$
			$\displaystyle+P_{\mu}P_{\nu}P_{\alpha}A_{3}^{(3)}$
			$\displaystyle+(P_{\mu}P_{\nu}q_{\alpha}+P_{\mu}q_{\nu}P_{\alpha}+q_{\mu}P_{\nu}P_{\alpha})A_{4}^{(3)}$
			$\displaystyle+(q_{\mu}q_{\nu}P_{\alpha}+q_{\mu}P_{\nu}q_{\alpha}+P_{\mu}q_{\nu}q_{\alpha})A_{5}^{(3)}$
			$\displaystyle+q_{\mu}q_{\nu}q_{\alpha}A_{6}^{(3)},$

with those coefficients $A_{i}^{(j)}$ given by

$$\begin{split}&A_{1}^{(1)}=\frac{x_{1}}{2},\,\,A_{2}^{(1)}=\frac{x_{1}}{2}-\frac{p^{\prime}_{\bot}\cdot q_{\bot}}{q^{2}}\\ &A_{1}^{(2)}=-p^{\prime 2}_{\bot}-\frac{(p^{\prime}_{\bot}\cdot q_{\bot})^{2}}{q^{2}},\,\,A_{2}^{(2)}=(A_{1}^{(1)})^{2}\\ &A_{3}^{(2)}=A_{1}^{(1)}A_{2}^{(1)},\,\,A_{4}^{(2)}=(A_{2}^{(1)})^{2}-\frac{A_{1}^{(2)}}{q^{2}}\\ &A_{1}^{(3)}=A_{1}^{(1)}A_{1}^{(2)},\,\,A_{2}^{(3)}=A_{1}^{(2)}A_{2}^{(1)}\\ &A_{3}^{(3)}=A_{1}^{(1)}A_{2}^{(2)},\,\,A_{4}^{(3)}=A_{2}^{(1)}A_{2}^{(2)}\\ &A_{5}^{(3)}=A_{1}^{(1)}A_{4}^{(2)},\,\,A_{6}^{(3)}=A_{2}^{(1)}A_{4}^{(2)}-\frac{2}{q^{2}}A_{2}^{(1)}A_{1}^{(2)}\\ \end{split}$$

(19)

By matching with Eq. (8), the expressions of the form factors can be obtained as follows:

	$\displaystyle f_{m}^{a}(q^{2})$	$\displaystyle=$	$\displaystyle\frac{2e}{3}\frac{N_{c}}{16\pi^{3}}\int dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{A}^{\prime}h_{V}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}(-4)\left\{\frac{1}{w_{A}^{\prime}}\Big{[}(m_{1}^{\prime}-m_{1}^{\prime\prime})(A_{3}^{(2)}-A_{4}^{(2)})\right.$
			$\displaystyle+(m_{1}^{\prime}+m_{1}^{\prime\prime}-2m_{2})\times(A_{2}^{(2)}-A_{3}^{(2)})+m_{1}^{\prime}(A_{2}^{(1)}-A_{1}^{(1)})\Big{]}+A_{2}^{(2)}-A_{3}^{(2)}$
			$\displaystyle\left.-\frac{1}{w_{A}^{\prime}w_{V}^{\prime\prime}}(2A_{2}^{(3)}-2A_{1}^{(3)})\right\},$
	$\displaystyle f_{p}^{a}(q^{2})$	$\displaystyle=$	$\displaystyle\frac{2e}{3}\frac{N_{c}}{16\pi^{3}}\int dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{A}^{\prime}h_{V}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}(-4)\left\{\frac{1}{w_{V}^{\prime\prime}}\Big{[}(m_{1}^{\prime}-m_{1}^{\prime\prime})(A_{3}^{(2)}+A_{4}^{(2)}\right.$
			$\displaystyle-A_{2}^{(1)})+(m_{1}^{\prime}+m_{1}^{\prime\prime}+2m_{2})\times(A_{2}^{(2)}+A_{3}^{(2)}-A_{1}^{(1)})-m_{1}^{\prime}(A_{1}^{(1)}+A_{2}^{(1)}$
			$\displaystyle\left.-1)\Big{]}+A_{1}^{(1)}-A_{2}^{(2)}-A_{3}^{(2)}-\frac{1}{w_{A}^{\prime}w_{V}^{\prime\prime}}(2A_{1}^{(3)}+2A_{2}^{(3)}-2A_{1}^{(2)})\right\},$
	$\displaystyle f_{6}^{a}(q^{2})$	$\displaystyle=$	$\displaystyle\frac{2e}{3}\frac{N_{c}}{16\pi^{3}}\int dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{A}^{\prime}h_{V}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}(-4)\left\{\frac{1}{w_{A}^{\prime}}(m_{1}^{\prime}+m_{1}^{\prime\prime}-2m_{2})A_{1}^{(2)}\right.$		(20)
			$\displaystyle+\frac{1}{w_{V}^{\prime\prime}}(m_{1}^{\prime}+m_{1}^{\prime\prime}+2m_{2})A_{1}^{(2)}-\frac{1}{4}(1-2A_{2}^{(1)})\Big{[}-q^{2}+\hat{N}_{1}^{\prime}+\hat{N}_{1}^{\prime\prime}$
			$\displaystyle\left.+(m_{1}^{\prime}-m_{1}^{\prime\prime})^{2}\Big{]}-A_{2}^{(1)}(m_{1}^{\prime\prime}m_{2}-m_{1}^{\prime}m_{2})-m_{1}^{\prime}m_{2}\phantom{\frac{1}{w_{A}^{\prime}}}\right\}.$

The form factors corresponding to Fig. 1(b), i.e., the expressions of $f_{m}^{b}(q^{2}),f_{p}^{b}(q^{2}),f_{6}^{b}(q^{2})$, can be obtained from Eq. (II.2) by the interchanges

$$m_{1}^{\prime}\rightarrow m_{2}^{\prime},\,m_{1}^{\prime\prime}\rightarrow m_{2}^{\prime\prime},\,m_{2}\rightarrow m_{1}.$$

(21)

In practice, all these quark masses are the mass of the charm quark, and the contribution of Fig. 1(b) is the same as that of Fig. 1(a). The form factors will finally be

$\displaystyle f_{i}(q^{2})=f_{i}^{a}(q^{2})+f_{i}^{b}(q^{2})=2f_{i}^{a}(q^{2}),$

(22)

with $i$ denoting the indices $m,\,p$, and 6. For the radiative decay considered here, only the form factor values at $q^{2}=0$ are relevant.

The decay width will be conveniently expressed in the helicity basis. We then define the helicity amplitude as $A_{\lambda^{\prime}\lambda^{\prime\prime}\lambda_{\gamma}}$, with $\lambda^{\prime},\,\lambda^{\prime\prime},\,\lambda_{\gamma}$ denoting the helicity of $X(3872)$, $J/\psi$ (or $\psi^{\prime}$), and the photon, respectively. In the rest frame of the initial meson $X(3872)$, we can obtain the explicit representations of the momenta and polarization vectors as follows Dubnicka:2011mm :

	$\displaystyle P^{\prime\mu}=(M^{\prime},0,0,0),\quad P^{\prime\prime\mu}=(E_{1},0,0,|\bm{q}|),$
	$\displaystyle q^{\mu}=(|\bm{q}|,0,0,-|\bm{q}|),$
	$\displaystyle\epsilon^{\prime\mu}_{\pm}=\frac{1}{\sqrt{2}}(0,\mp 1,-i,0),\,\,\epsilon^{\prime\mu}_{0}=(0,0,0,1),$
	$\displaystyle\epsilon^{\prime\prime\mu}_{\pm}=\frac{1}{\sqrt{2}}(0,\mp 1,-i,0),\,\,\epsilon^{\prime\prime\mu}_{0}=\frac{1}{M^{\prime\prime}}(|\bm{q}|,0,0,E_{1}),$
	$\displaystyle\epsilon^{\mu}_{\pm}(\gamma)=\frac{1}{\sqrt{2}}(0,\pm 1,-i,0),$		(23)

where $|{\bm{q}}|=\frac{M^{\prime 2}-M^{\prime\prime 2}}{2M^{\prime}}$ and $E_{1}=\frac{M^{\prime 2}+M^{\prime\prime 2}}{2M^{\prime}}$ is the energy of the vector meson. Due to the conservation of angular momentum, $\lambda^{\prime}=\lambda^{\prime\prime}-\lambda_{\gamma}$, and the nonvanishing amplitudes for $X(3872)\to J/\psi\gamma,\,\psi^{\prime}\gamma$ decay will be

	$\displaystyle A_{+0-}$	$\displaystyle=$	$\displaystyle-i\frac{M^{\prime}|\bm{q}|}{M^{\prime\prime}}(f_{6}+2M^{\prime}|\bm{q}|f_{p}),$
	$\displaystyle A_{0++}$	$\displaystyle=$	$\displaystyle i|\bm{q}|(f_{6}+2M^{\prime}|\bm{q}|f_{m}),$
	$\displaystyle A_{-0+}$	$\displaystyle=$	$\displaystyle-A_{+0-},\quad A_{0--}=-A_{0++}.$		(24)

The last two equations follow from the parity relation and have also been verified by calculation. Obviously, we can also adopt the convention of the polarization vectors used in Refs. Zhang:2020dla ; Ivanov:2019nqd . The difference lies only in the definition of the momentum direction and the resulting change in the polarization vector forms. We verified that they give identical physical results. Stated differently, this transition includes the $E1$ and $M2$ types, which are characterized by the $\bm{q}$ and $\bm{q}^{2}$ for the near-threshold behavior for the amplitude. Of course, both $S$ and $D$ waves are included, and combined together in Eq. (II.3). The helicity relation $\lambda^{\prime}=\lambda^{\prime\prime}-\lambda_{\gamma}$ does not imply $S$ wave solely, since the projection of any orbital angular momentum onto the linear momentum vanishes.