A new group of doubly charmed molecule with $T$-doublet charmed meson pair

Before deducing the effective potentials, we first construct the flavor and spin-orbital wave functions for these discussed $TT$-type doubly charmed tetraquark systems. The isospin quantum numbers for the $TT$-type doubly charmed tetraquark systems can be either $0$ or $1$, and we collect the relevant flavor wave functions $|I,I_{3}\rangle$ for the $TT$ systems in Table 2 Li:2012ss ; Chen:2021vhg ; Liu:2019stu ; Wang:2021yld .

These allowed quantum numbers for the $S$-wave $D_{1}D_{1}$ and $D_{2}^{\ast}D_{2}^{\ast}$ states include

	$\displaystyle D_{1}D_{1}[I(J^{P})]:1(0^{+}),0(1^{+}),1(2^{+}),$
	$\displaystyle D_{2}^{\ast}D_{2}^{\ast}[I(J^{P})]:1(0^{+}),0(1^{+}),1(2^{+}),0(3^{+}),1(4^{+}).$

The spin-orbital wave functions $|{}^{2S+1}L_{J}\rangle$ for the $D_{1}D_{1}$, $D_{1}{D}_{2}^{\ast}$, and $D_{2}^{\ast}D_{2}^{\ast}$ systems can be expressed as

	$\displaystyle D_{1}D_{1}:$		$\displaystyle|{}^{2S+1}L_{J}\rangle=\sum_{m,m^{\prime},m_{S},m_{L}}C^{S,m_{S}}_{1m,1m^{\prime}}C^{J,M}_{Sm_{S},Lm_{L}}\epsilon_{m}^{\mu}\epsilon_{m^{\prime}}^{\nu}|Y_{L,m_{L}}\rangle,$
	$\displaystyle D_{1}D_{2}^{\ast}:$		$\displaystyle|{}^{2S+1}L_{J}\rangle=\sum_{m,m^{\prime},m_{S},m_{L}}C^{S,m_{S}}_{1m,2m^{\prime}}C^{J,M}_{Sm_{S},Lm_{L}}\epsilon_{m}^{\lambda}\zeta_{m^{\prime}}^{\mu\nu}|Y_{L,m_{L}}\rangle,$
	$\displaystyle D_{2}^{\ast}D_{2}^{\ast}:$		$\displaystyle|{}^{2S+1}L_{J}\rangle=\sum_{m,m^{\prime},m_{S},m_{L}}C^{S,m_{S}}_{2m,2m^{\prime}}C^{J,M}_{Sm_{S},Lm_{L}}\zeta_{m}^{\mu\nu}\zeta_{m^{\prime}}^{\lambda\rho}|Y_{L,m_{L}}\rangle,$

respectively. Here, $C^{e,f}_{ab,cd}$ is the Clebsch-Gordan coefficient, and $|Y_{L,m_{L}}\rangle$ stands for the spherical harmonics function. $\epsilon^{\mu}_{m}\,(m=0,\,\pm 1)$ and $\zeta^{\mu\nu}_{m}\,(m=0,\,\pm 1,\,\pm 2)$ denote the polarization vector and tensor for the axial-vector charmed meson $D_{1}$ and the tensor charmed meson $D_{2}^{*}$, respectively. In the static limit, their explicit expressions can be written as Cheng:2010yd

	$\displaystyle\epsilon_{0}^{\mu}$	$\displaystyle=$	$\displaystyle\left(0,0,0,-1\right),~{}~{}~{}~{}~{}~{}\epsilon_{\pm}^{\mu}=\frac{1}{\sqrt{2}}\left(0,\,\pm 1,\,i,\,0\right),$
	$\displaystyle\zeta^{\mu\nu}_{m}$	$\displaystyle=$	$\displaystyle\sum_{m_{1},m_{2}}C^{2,m}_{1m_{1},1m_{2}}\epsilon^{\mu}_{m_{1}}\epsilon^{\nu}_{m_{2}}.$

In the following, we adopt the effective Lagrangian approach to deduce the OBE effective potentials for the $D_{1}D_{1}$, $D_{1}{D}_{2}^{\ast}$, and $D_{2}^{\ast}D_{2}^{\ast}$ systems. Based on the heavy quark symmetry, the chiral symmetry, and the hidden gauge symmetry Casalbuoni:1992gi ; Casalbuoni:1996pg ; Yan:1992gz ; Harada:2003jx ; Bando:1987br , the compact form of the relevant effective Lagrangians can be constructed as Ding:2008gr

	$\displaystyle\mathcal{L}_{TT\sigma}$	$\displaystyle=$	$\displaystyle g^{\prime\prime}_{\sigma}\langle T^{(Q)\mu}_{a}\sigma\overline{T}^{\,({Q})}_{a\mu}\rangle,$		(2.1)
	$\displaystyle\mathcal{L}_{TT\mathbb{P}}$	$\displaystyle=$	$\displaystyle ik\langle T^{(Q)\mu}_{b}{\cal A}\!\!\!/_{ba}\gamma_{5}\overline{T}^{\,(Q)}_{a\mu}\rangle,$		(2.2)
	$\displaystyle\mathcal{L}_{TT\mathbb{V}}$	$\displaystyle=$	$\displaystyle i\beta^{\prime\prime}\langle T^{(Q)\lambda}_{b}v^{\mu}({\cal V}_{\mu}-\rho_{\mu})_{ba}\overline{T}^{\,(Q)}_{a\lambda}\rangle$		(2.3)
			$\displaystyle+i\lambda^{\prime\prime}\langle T^{(Q)\lambda}_{b}\sigma^{\mu\nu}F_{\mu\nu}(\rho)_{ba}\overline{T}^{\,(Q)}_{a\lambda}\rangle,$

where the four velocity is $v=(1,\bm{0})$ in the static approximation. Surperfield $T^{(Q)\mu}_{a}$ can be expressed as a linear combination of the axial-vector heavy flavor meson $P^{(Q)}_{1}$ with $I(J^{P})=1/2(1^{+})$ and the tensor heavy flavor meson $P_{2}^{*(Q)}$ with $I(J^{P})=1/2(2^{+})$, which reads as Ding:2008gr

$\displaystyle T^{(Q)\mu}_{a}$

$\displaystyle=$

$\displaystyle\frac{1+{v}\!\!\!/}{2}\left[P^{*(Q)\mu\nu}_{2a}\gamma_{\nu}-\sqrt{\frac{3}{2}}P^{(Q)}_{1a\nu}\gamma_{5}\left(g^{\mu\nu}-\frac{\gamma^{\nu}(\gamma^{\mu}-v^{\mu})}{3}\right)\right],$

where $P_{1}^{(c)}=(D_{1}^{0},\,D_{1}^{+})$, $P_{2}^{*(c)}=(D_{2}^{*0},\,D_{2}^{*+})$, and the normalization relations for these discussed charmed mesons are $\langle 0|D_{1}^{\mu}|c\bar{q}(1^{+})\rangle=\epsilon^{\mu}\sqrt{m_{D_{1}}}$ and $\langle 0|D_{2}^{*\mu\nu}|c\bar{q}(2^{+})\rangle=\xi^{\mu\nu}\sqrt{m_{D_{2}^{\ast}}}$. Its conjugate field $\overline{T}_{a}^{(Q)\mu}$ is written as $\overline{T}_{a}^{(Q)\mu}=\gamma^{0}T_{a}^{(Q)\mu{\dagger}}\gamma^{0}$.

In Eqs. (2.1)-(2.3), we define the axial current $\mathcal{A}_{\mu}$, the vector current ${\cal V}_{\mu}$, the vector meson field $\rho_{\mu}$, and the vector meson field strength tensor $F_{\mu\nu}(\rho)$, i.e.,

$\displaystyle\left.\begin{array}[]{ll}{\mathcal{A}}_{\mu}=\dfrac{1}{2}(\xi^{\dagger}\partial_{\mu}\xi-\xi\partial_{\mu}\xi^{\dagger}),&{\mathcal{V}}_{\mu}=\dfrac{1}{2}(\xi^{\dagger}\partial_{\mu}\xi+\xi\partial_{\mu}\xi^{\dagger}),\\ \rho_{\mu}=\dfrac{ig_{V}}{\sqrt{2}}\mathbb{V}_{\mu},&F_{\mu\nu}(\rho)=\partial_{\mu}\rho_{\nu}-\partial_{\nu}\rho_{\mu}+[\rho_{\mu},\rho_{\nu}],\end{array}\right.$

with $\xi=\exp(i\mathbb{P}/f_{\pi})$. Here, $\mathbb{P}$ and $\mathbb{V}_{\mu}$ stand for the pseudoscalar meson and vector meson matrices, respectively, which have the forms of

$\displaystyle\left.\begin{array}[]{l}{\mathbb{P}}={\left(\begin{array}[]{ccc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}&K^{+}\\ \pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&K^{0}\\ K^{-}&\bar{K}^{0}&-\sqrt{\frac{2}{3}}\eta\end{array}\right)},\\ {\mathbb{V}}_{\mu}={\left(\begin{array}[]{ccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}\\ \rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{*0}\\ K^{*-}&\bar{K}^{*0}&\phi\end{array}\right)}_{\mu},\end{array}\right.$

respectively. Once expanding the effective Lagrangians in Eqs. (2.1)-(2.3), we can further obtain the concrete effective Lagrangians between the charmed mesons in the $T$-doublet and the light mesons, which can be explicitly expressed as

	$\displaystyle\mathcal{L}_{D_{1}D_{1}\sigma}$	$\displaystyle=$	$\displaystyle-2g_{\sigma}^{\prime\prime}D_{1a\mu}D^{\mu\dagger}_{1a}\sigma,$		(2.8)
	$\displaystyle\mathcal{L}_{D_{1}D_{1}\mathbb{P}}$	$\displaystyle=$	$\displaystyle-\frac{5ik}{3f_{\pi}}~{}\epsilon^{\mu\nu\rho\tau}v_{\tau}D_{1b\nu}D^{\dagger}_{1a\mu}\partial_{\rho}\mathbb{P}_{ba},$		(2.9)
	$\displaystyle\mathcal{L}_{D_{1}D_{1}\mathbb{V}}$	$\displaystyle=$	$\displaystyle-\sqrt{2}\beta^{\prime\prime}g_{V}\left(v\cdot\mathbb{V}_{ba}\right)D_{1b\mu}D^{\mu\dagger}_{1a}$		(2.10)
			$\displaystyle+\frac{5\sqrt{2}i\lambda^{\prime\prime}g_{V}}{3}\left(D^{\mu}_{1b}D^{\nu\dagger}_{1a}-D^{\nu}_{1b}D^{\mu\dagger}_{1a}\right)\partial_{\mu}\mathbb{V}_{ba\nu},$
	$\displaystyle\mathcal{L}_{D^{\ast}_{2}D^{\ast}_{2}\sigma}$	$\displaystyle=$	$\displaystyle 2g_{\sigma}^{\prime\prime}D^{*\mu\nu}_{2a}D^{*\dagger}_{2a\mu\nu}\sigma,$		(2.11)
	$\displaystyle\mathcal{L}_{D^{\ast}_{2}D^{\ast}_{2}\mathbb{P}}$	$\displaystyle=$	$\displaystyle\frac{2ik}{f_{\pi}}~{}\epsilon^{\mu\nu\rho\tau}v_{\nu}D^{*}_{2b\alpha\tau}D^{*\alpha\dagger}_{2a\rho}\partial_{\mu}\mathbb{P}_{ba},$		(2.12)
	$\displaystyle\mathcal{L}_{D^{\ast}_{2}D^{\ast}_{2}\mathbb{V}}$	$\displaystyle=$	$\displaystyle\sqrt{2}\beta^{\prime\prime}g_{V}\left(v\cdot\mathbb{V}_{ba}\right)D_{2b}^{*\lambda\nu}D^{*\dagger}_{2a{\lambda\nu}}+2\sqrt{2}i\lambda^{\prime\prime}g_{V}$		(2.13)
			$\displaystyle\times\left(D^{*\lambda\nu}_{2b}D^{*\mu\dagger}_{2a\lambda}-D^{*\mu}_{2b\lambda}D^{*\lambda\nu\dagger}_{2a}\right)\partial_{\mu}\mathbb{V}_{ba\nu},$
	$\displaystyle\mathcal{L}_{D_{1}D^{\ast}_{2}\sigma}$	$\displaystyle=$	$\displaystyle\sqrt{\frac{2}{3}}ig_{\sigma}^{\prime\prime}\epsilon^{\mu\nu\rho\tau}v_{\rho}\left(D^{*}_{2a\mu\tau}D^{\dagger}_{1a\nu}-D_{1a\nu}D^{*\dagger}_{2a\mu\tau}\right)\sigma,$		(2.14)
	$\displaystyle\mathcal{L}_{D_{1}D^{\ast}_{2}\mathbb{P}}$	$\displaystyle=$	$\displaystyle-\sqrt{\frac{2}{3}}\frac{k}{f_{\pi}}\left(D^{*\mu\lambda}_{2b}D^{\dagger}_{1a\mu}+D_{1b\mu}D^{*\mu\lambda\dagger}_{2a}\right)\partial_{\lambda}\mathbb{P}_{ba},$		(2.15)
	$\displaystyle\mathcal{L}_{D_{1}D^{\ast}_{2}\mathbb{V}}$	$\displaystyle=$	$\displaystyle\frac{i\beta^{\prime\prime}g_{V}}{\sqrt{3}}\epsilon^{\lambda\alpha\rho\tau}v_{\rho}\left(v\cdot\mathbb{V}_{ba}\right)\left(D^{*}_{2b\lambda\tau}D^{\dagger}_{1a\alpha}-D_{1b\alpha}D^{*\dagger}_{2a\lambda\tau}\right)$		(2.16)
			$\displaystyle+\frac{2\lambda^{\prime\prime}g_{V}}{\sqrt{3}}\left[3\epsilon^{\mu\lambda\nu\tau}v_{\lambda}\left(D^{*}_{2b\alpha\tau}D^{\alpha\dagger}_{1a}+D^{\alpha}_{1b}D^{*\dagger}_{2a\alpha\tau}\right)\partial_{\mu}\mathbb{V}_{ba\nu}\right.$
			$\displaystyle+2\epsilon^{\lambda\alpha\rho\nu}v_{\rho}\left(D^{*\mu}_{2b\lambda}D^{\dagger}_{1a\alpha}+D_{1b\alpha}D^{*\mu\dagger}_{2a\lambda}\right)$
			$\displaystyle\left.\times\left(\partial_{\mu}\mathbb{V}_{ba\nu}-\partial_{\nu}\mathbb{V}_{ba\mu}\right)\right].$

In this work, we estimate all the coupling constants in the quark model (see Refs. Wang:2019nwt ; Liu:2011xc ; Wang:2019aoc ; Riska:2000gd for more details). The values for the involved coupling constants are $g^{\prime\prime}_{\sigma}=0.76$, $k=0.59$, $f_{\pi}=132~{}\rm{MeV}$, $\beta^{\prime\prime}=0.90$, $\lambda^{\prime\prime}=0.56~{}\rm{GeV}^{-1}$, and $g_{V}=5.83$ Wang:2019nwt . The masses of these involved hadrons are $m_{\sigma}=600.00~{}\rm{MeV}$, $m_{\pi}=137.27~{}\rm{MeV}$, $m_{\eta}=547.86~{}\rm{MeV}$, $m_{\rho}=775.26~{}\rm{MeV}$, $m_{\omega}=782.66~{}\rm{MeV}$, $m_{D_{1}}=2422.00~{}\rm{MeV}$, and $m_{D_{2}^{*}}=2463.05~{}\rm{MeV}$ Zyla:2020zbs . As the important input parameters within the OBE model, the information about the coupling constants is crucial when studying the existence possibility of the hadronic molecular states. However, the coupling constants with the charmed meson in $T$-doublet cannot be fixed based on the experimental data at present, and we estimate these coupling constants by the quark model in this work Riska:2000gd , which is often adopted to determine the coupling constants. Because of the uncertainties of the coupling constants, we should be cautious when studying the existence possibility of the hadronic molecular states. Here, we need to indicate our results of the existence possibility of the hadronic molecular states at the qualitative level, but the uncertainties of the coupling constants do not significantly affect our qualitative conclusions.

Based on the above preparation, we can further deduce the effective potentials to judge whether these discussed doubly-charmed molecular tetraquark states exist or not. As indicated in Ref. Wang:2019nwt , there exists the standard strategy to judge the possibility of the existence of the investigated hadronic molecular state within the OBE model, which is given in Fig. 1.

For extracting the effective potential in the coordinate space, which is shown in the upper part of the Fig. 1, we will give a brief introduction in the following. First, we can write down the scattering amplitude $\mathcal{M}^{h_{1}h_{2}\to h_{3}h_{4}}(\bm{q})$ of the scattering process $h_{1}h_{2}\to h_{3}h_{4}$ by exchanging the light mesons under the effective Lagrangian approach. The relevant Feynman diagram is given in Fig. 2, and the scattering amplitude $\mathcal{M}^{h_{1}h_{2}\to h_{3}h_{4}}(\bm{q})$ can be written as

$\displaystyle i\mathcal{M}^{h_{1}h_{2}\to h_{3}h_{4}}(\bm{q})=\sum_{m=\sigma,\,\mathbb{P},\,\mathbb{V}}i\Gamma^{h_{1}h_{3}m}_{(\mu)}P_{m}^{(\mu\nu)}i\Gamma^{h_{2}h_{4}m}_{(\nu)},$

(2.17)

where the interaction vertices $\Gamma^{h_{1}h_{3}m}_{(\mu)}$ and $\Gamma^{h_{2}h_{4}m}_{(\nu)}$ can be extracted from the former effective Lagrangians.

And then, we adopt the Breit approximation Breit:1929zz ; Breit:1930zza and the nonrelativistic normalizations to obtain the relations between the effective potential in the momentum space $\mathcal{V}^{h_{1}h_{2}\to h_{3}h_{4}}_{E}(\bm{q})$ and the corresponding scattering amplitude $\mathcal{M}^{h_{1}h_{2}\to h_{3}h_{4}}(\bm{q})$, i.e.,

$\displaystyle\mathcal{V}_{E}^{h_{1}h_{2}\to h_{3}h_{4}}(\bm{q})=-\frac{\mathcal{M}^{h_{1}h_{2}\to h_{3}h_{4}}(\bm{q})}{\sqrt{\prod_{i}2m_{i}\prod_{f}2m_{f}}},$

(2.18)

where $m_{i(f)}$ are the masses of the initial (final) states. Finally, we can perform the Fourier transformation to deduce the effective potential in the coordinate space $\mathcal{V}^{h_{1}h_{2}\to h_{3}h_{4}}_{E}(\bm{r})$, i.e.,

$\displaystyle\mathcal{V}^{h_{1}h_{2}\to h_{3}h_{4}}_{E}(\bm{r})=\int\frac{d^{3}\bm{q}}{(2\pi)^{3}}e^{i\bm{q}\cdot\bm{r}}\mathcal{V}^{h_{1}h_{2}\to h_{3}h_{4}}_{E}(\bm{q})\mathcal{F}^{2}(q^{2},m_{E}^{2}).$

In order to compensate the off-shell effect of the exchanged particles, we introduce the form factor $\mathcal{F}(q^{2},m_{E}^{2})$ at every interaction vertex, $\mathcal{F}(q^{2},m_{E}^{2})=(\Lambda^{2}-m_{E}^{2})/(\Lambda^{2}-q^{2})$. Here, $\Lambda$, $q$, and $m_{E}$ are the cutoff parameter, the four-momentum, and the mass of the exchanged light mesons, respectively. According to the experience of studying the deuteron, a reasonable cutoff value is taken around 1.00 GeV Tornqvist:1993ng ; Tornqvist:1993vu , in this cutoff region, we can reproduce the masses of the three $P_{c}$ states Wu:2010jy ; Karliner:2015ina ; Wang:2011rga ; Yang:2011wz ; Wu:2012md ; Chen:2015loa and the $T_{cc}$ state Li:2021zbw ; Chen:2021vhg ; Agaev:2021vur ; Ren:2021dsi ; Xin:2021wcr ; Chen:2021tnn ; Albaladejo:2021vln ; Ding:2020dio ; Li:2012ss ; Xu:2017tsr ; Liu:2019stu ; Dong:2021bvy in the hadronic molecular picture.

After getting the effective potentials in the coordinate space for the discussed systems by the above three typical steps, we can search for the bound state solutions by numerically solving the coupled channel Schrödinger equation,

$\displaystyle-\frac{1}{2\mu}\left(\nabla^{2}-\frac{\ell(\ell+1)}{r^{2}}\right)\psi(r)+V(r)\psi(r)=E\psi(r),$

(2.20)

where $\nabla^{2}=\frac{1}{r^{2}}\frac{\partial}{\partial r}r^{2}\frac{\partial}{\partial r}$, and $\mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}}$ is the reduced mass for the discussed systems. In our calculation, the obtained bound state solutions mainly include the binding energy $E$ and the radial wave function $\psi(r)$, with which we can further calculate the root-mean-square radius $r_{\rm RMS}$ and the probability of the individual channel $P_{i}$.

The total OBE effective potentials for all the discussed $TT$-type doubly charmed tetraquarks can be expressed as

	$\displaystyle\mathcal{V}_{1}$	$\displaystyle=$	$\displaystyle-g^{\prime\prime 2}_{\sigma}\mathcal{O}_{1}Y_{\sigma}-\frac{25k^{2}}{108f^{2}_{\pi}}\left(\mathcal{O}_{2}\mathcal{Z}_{r}+\mathcal{O}_{3}\mathcal{T}_{r}\right)\left(\frac{\mathcal{G}Y_{\pi}}{2}+\frac{Y_{\eta}}{6}\right)$		(2.21)
			$\displaystyle+\frac{1}{2}\beta^{\prime\prime 2}g^{2}_{V}\mathcal{O}_{1}\left(\frac{\mathcal{G}Y_{\rho}}{2}+\frac{Y_{\omega}}{2}\right)$
			$\displaystyle-\frac{25}{54}\lambda^{\prime\prime 2}g^{2}_{V}\left(2\mathcal{O}_{2}\mathcal{Z}_{r}-\mathcal{O}_{3}\mathcal{T}_{r}\right)\left(\frac{\mathcal{G}Y_{\rho}}{2}+\frac{Y_{\omega}}{2}\right)$

for the ${D}_{1}{D}_{1}\to{D}_{1}{D}_{1}$ process,

	$\displaystyle\mathcal{V}_{2}$	$\displaystyle=$	$\displaystyle-g^{\prime\prime 2}_{\sigma}\frac{\mathcal{O}_{4}+\mathcal{O}_{4}^{\prime}}{2}Y_{\sigma}$
			$\displaystyle-\frac{5k^{2}}{18f^{2}_{\pi}}\left(\frac{\mathcal{O}_{5}+\mathcal{O}_{5}^{\prime}}{2}\mathcal{Z}_{r}+\frac{\mathcal{O}_{6}+\mathcal{O}_{6}^{\prime}}{2}\mathcal{T}_{r}\right)\left(\frac{\mathcal{G}Y_{\pi}}{2}+\frac{Y_{\eta}}{6}\right)$
			$\displaystyle+\frac{1}{2}\beta^{\prime\prime 2}g^{2}_{V}\frac{\mathcal{O}_{4}+\mathcal{O}_{4}^{\prime}}{2}\left(\frac{\mathcal{G}}{2}Y_{\rho}+\frac{1}{2}Y_{\omega}\right)$
			$\displaystyle-\frac{2}{3}\lambda^{\prime\prime 2}g^{2}_{V}\left(2\frac{\mathcal{O}_{5}+\mathcal{O}_{5}^{\prime}}{2}\mathcal{Z}_{r}-\frac{\mathcal{O}_{6}+\mathcal{O}_{6}^{\prime}}{2}\mathcal{T}_{r}\right)\left(\frac{\mathcal{G}Y_{\rho}}{2}+\frac{Y_{\omega}}{2}\right)$

for the ${D}_{1}{D}_{2}^{\ast}\to{D}_{1}{D}_{2}^{\ast}$ process,

	$\displaystyle\mathcal{V}_{3}$	$\displaystyle=$	$\displaystyle\frac{k^{2}}{18f^{2}_{\pi}}\left(\frac{\mathcal{O}_{7}+\mathcal{O}_{7}^{\prime}}{2}\mathcal{Z}_{r}+\frac{\mathcal{O}_{8}+\mathcal{O}_{8}^{\prime}}{2}\mathcal{T}_{r}\right)\left(\frac{\mathcal{G}Y_{\pi 0}}{2}+\frac{Y_{\eta 0}}{6}\right)$
			$\displaystyle+\frac{\lambda^{\prime\prime 2}g^{2}_{V}}{9}\left(2\frac{\mathcal{O}_{7}+\mathcal{O}_{7}^{\prime}}{2}\mathcal{Z}_{r}-\frac{\mathcal{O}_{8}+\mathcal{O}_{8}^{\prime}}{2}\mathcal{T}_{r}\right)\left(\frac{\mathcal{G}Y_{\rho 0}}{2}+\frac{Y_{\omega 0}}{2}\right)$

for the ${D}_{1}{D}_{2}^{\ast}\to{D}_{2}^{\ast}{D}_{1}$ process, and

	$\displaystyle\mathcal{V}_{4}$	$\displaystyle=$	$\displaystyle-g^{\prime\prime 2}_{\sigma}\mathcal{O}_{9}Y_{\sigma}-\frac{k^{2}}{3f^{2}_{\pi}}\left(\mathcal{O}_{10}\mathcal{Z}_{r}+\mathcal{O}_{11}\mathcal{T}_{r}\right)\left(\frac{\mathcal{G}Y_{\pi}}{2}+\frac{Y_{\eta}}{6}\right)$		(2.24)
			$\displaystyle+\frac{1}{2}\beta^{\prime\prime 2}g^{2}_{V}\mathcal{O}_{9}\left(\frac{\mathcal{G}Y_{\rho}}{2}+\frac{Y_{\omega}}{2}\right)$
			$\displaystyle-\frac{2}{3}\lambda^{\prime\prime 2}g^{2}_{V}\left(2\mathcal{O}_{10}\mathcal{Z}_{r}-\mathcal{O}_{11}\mathcal{T}_{r}\right)\left(\frac{\mathcal{G}Y_{\rho}}{2}+\frac{Y_{\omega}}{2}\right)$

for the ${D}_{2}^{\ast}{D}_{2}^{\ast}\to{D}_{2}^{\ast}{D}_{2}^{\ast}$ process. Here, the constant $\mathcal{G}$ is taken as $-3$ for the isoscalar system and 1 for the isovector system, respectively. For the convenience, we define the following expressions

	$\displaystyle\mathcal{Z}_{r}$	$\displaystyle=$	$\displaystyle\frac{1}{r^{2}}\frac{\partial}{\partial r}r^{2}\frac{\partial}{\partial r},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{T}_{r}=r\frac{\partial}{\partial r}\frac{1}{r}\frac{\partial}{\partial r},$
	$\displaystyle Y_{i}$	$\displaystyle=$	$\displaystyle\dfrac{e^{-m_{i}r}-e^{-\Lambda_{i}r}}{4\pi r}-\dfrac{\Lambda_{i}^{2}-m_{i}^{2}}{8\pi\Lambda_{i}}e^{-\Lambda_{i}r}.$		(2.25)

Here, $m_{0}=\sqrt{m^{2}-q_{0}^{2}}$ and $\Lambda_{0}=\sqrt{\Lambda^{2}-q_{0}^{2}}$ with $q_{0}=m_{{D}_{2}^{\ast}}-m_{{D}_{1}}$.