Description of the newly observed $\Omega^{*}_{c}$ states as molecular states

After the discovery of the five $\Omega^{*}_{c}$ states in one observation simultaneously back in 2017 LHCb:2017uwr , the latest experimental results of the LHCb Collaboration reported the existence of an additional two $\Omega_{c}$ states in the $\Xi_{c}^{+}K^{-}$ invariant mass spectrum, originating from $pp$ collisions LHCb:2023rtu . The measured parameters of these newly states are as follows:

	$\displaystyle M_{\Omega_{c}(3185)}$	$\displaystyle=3185.1\pm{}1.7^{+7.4}_{-0.9}\pm{}0.2~{}~{}~{}~{}~{}{\rm MeV},$
	$\displaystyle\Gamma_{\Omega_{c}(3185)}$	$\displaystyle=50\pm 7^{+10}_{-20}~{}~{}~{}~{}~{}{\rm MeV},$		(1)
	$\displaystyle M_{\Omega_{c}(3327)}$	$\displaystyle=3327.1\pm{}1.2^{+0.1}_{-1.3}\pm{}0.2~{}~{}~{}~{}~{}{\rm MeV},$
	$\displaystyle\Gamma_{\Omega_{c}(3327)}$	$\displaystyle=20\pm 5^{+13}_{-1}~{}~{}~{}~{}~{}{\rm MeV}.$

Compared to the earlier discovery of two ground states $\Omega^{*}_{c}(2695)$ and $\Omega^{*}_{c}(2770)$ Workman:2022ynf , the findings presented in not only expand our understanding of the charmed baryon with quantum numbers $C=1$ and $S=2$, which is composed of one charm quark and two strange quarks, but also help us to strange quarks, but also help us to understand the formation mechanism of exotic hadron states. This is mainly because the newly observed $\Omega^{*}_{c}$ states may display a more complex internal structure than the ground states $\Omega^{*}_{c}(2695)$ and $\Omega^{*}_{c}(2770)$, which are made up of three quarks.

Indeed, the $\Omega^{*}_{c}(3050)$ and $\Omega_{c}^{*}(3119)$ were suggested to be the exotic pentaquarks in the chiral quark-soliton model Kim:2017jpx ; Kim:2017khv . Similar qualitative results can also be found in Refs. Yang:2017rpg ; An:2017lwg . In Refs. Debastiani:2017ewu ; Montana:2017kjw , based on the analysis of the mass spectrum, the $\Omega^{*}_{c}(3050)$ was identified as a meson-baryon molecule with $J^{P}=1/2^{-}$. This is the same with the results in Ref. Huang:2018wgr that the total decay widths of the $\Omega^{*}_{c}(3050)$ can be well reproduced with the assumption that $\Omega^{*}_{c}(3050)$ as $S$-wave $\Xi{}D$ bound state with $J^{P}=1/2^{-}$. By solving a coupled-channel Bethe Salpeter equation using a $SU(6)_{lsf}\times$ HQSS extended WT interaction as a kernel, three meson-baryon molecular states were obtained Nieves:2017jjx that can be associated to the experimental states $\Omega^{*}_{c}(3000)$, $\Omega^{*}_{c}(3050)$, and $\Omega^{*}_{c}(31119)$ or $\Omega^{*}_{c}(3090)$.

There also exist some clues support the newly observed $\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3227)$ as molecular states. It was suggested in Ref. Yu:2023bxn that $\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3327)$ were proposed to be the $2S(3/2^{+})$ and $1D(3/2^{+})$ states, respectively. However, a completely different conclusion was drawn from Ref. luo:2023bxn that the $\Omega^{*}_{c}(3327)$ is a good candidate of $\Omega^{*}_{c}(1D)$ state with $J^{P}=5/2^{+}$. There exists spin-parity puzzle for $\Omega^{*}_{c}(3327)$ indicate it difficult to put $\Omega^{*}_{c}(3327)$ into the conventional $css$ states. Since the mass of $\Omega^{*}_{c}(3327)$ is very close to the threshold of $D^{*}\Xi$, the hadronic molecular configuration for $\Omega^{*}_{c}(3327)$ is possible. Although the $\Omega^{*}_{c}(3185)$ can be considered as a conventional charmed baryon Yu:2023bxn , the hadronic molecule interpretations cannot be excluded due to the mass of $\Omega^{*}_{c}(3185)$ is about 6.37 MeV below the threshold of $D\Xi$.

However, no work has been conducted to study whether the newly observed $\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3327)$ can be explained as $D\Xi$ and $D^{*}\Xi$ molecular state, respectively. In this work, we estimate possible strong decay modes by assuming $\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3327)$ as molecular states. By comparing the calculated decay widths with experimental data, we can evaluate the validity of the proposed molecular explanations for the structures of $\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3327)$.

This paper is organized as follows. In Sec. II, we will present the theoretical formalism. In Sec. III, the numerical result will be given, followed by discussions and conclusions in last section.

With assuming the newly observed $\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3227)$ as $S$-wave molecular states, the decay of $\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3227)$ into $\bar{K}\Xi_{c}$ and $\bar{K}\Xi_{c}^{{}^{\prime}}$ are allowed. The corresponding Feynman diagrams are illustrated in Fig. 1, which includes the $t$-channel $\Sigma$, $\Lambda$ baryons exchange and $D_{s}$, $D_{s}^{*}$ mesons exchange. To evaluate these diagrams, the Lagrangian for the coupling between the mesons and the baryons are obtained using the $SU(4)$ invariant interaction Lagrangians Liu:2001ce

	$\displaystyle\mathcal{L}_{PBB}$	$\displaystyle=ig_{p}(a\phi^{*\alpha\nu\nu}\gamma^{5}P^{\beta}_{\alpha}\phi_{\beta\mu\nu}+b\phi^{*\alpha\mu\nu}\gamma^{5}P^{\beta}_{\alpha}\phi_{\beta\mu\nu}),$		(2)
	$\displaystyle\mathcal{L}_{VBB}$	$\displaystyle=ig_{v}(c\phi^{*\alpha\nu\nu}\gamma P^{\beta}_{\alpha}\phi_{\beta\mu\nu}+d\phi^{*\alpha\nu\nu}\gamma P^{\beta}_{\alpha}\phi_{\beta\mu\nu}),$

where $V_{\mu}$ and $P$ are the $SU(4)$ vector meson and pseudoscalar meson matrix, respectively. The meson matrices are

$\displaystyle P=\begin{pmatrix}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}+\frac{\eta_{c}}{\sqrt{12}}&\pi^{+}&K^{+}&\bar{D}^{0}\\ \pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}+\frac{\eta_{c}}{\sqrt{12}}&K^{0}&D^{-}\\ K^{-}&\bar{K}^{0}&-\frac{2\eta}{\sqrt{6}}+\frac{\eta_{c}}{\sqrt{12}}&D_{s}^{-}\\ D^{0}&D^{+}&D_{s}^{+}&-\frac{3\eta_{c}}{\sqrt{12}}\end{pmatrix}$

(3)

and

$\displaystyle V=\begin{pmatrix}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega_{8}}{\sqrt{6}}+\frac{J/\Psi}{\sqrt{12}}&\rho^{+}&K^{*+}&\bar{D}^{*0}\\ \rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega_{8}}{\sqrt{6}}+\frac{J/\Psi}{\sqrt{12}}&K^{*0}&D^{*-}\\ K^{*-}&\bar{K}^{*0}&-\frac{2\omega_{8}}{\sqrt{6}}+\frac{J/\Psi}{\sqrt{12}}&D_{s}^{*-}\\ D^{*0}&D^{*+}&D_{s}^{*+}&-\frac{3J/\Psi}{\sqrt{12}}\end{pmatrix}.$

(4)

with $\omega_{8}=\omega\cos\theta+\phi\sin\theta$ and $\sin\theta=-0.761$. The tensors $\phi_{\beta\mu\nu}$ in above equation represents 20-plet of the proton Liu:2001ce ,

	$\displaystyle p=\phi_{112},n=\phi_{221},\Lambda=\sqrt{\frac{2}{3}}(\phi_{321}-\phi_{312}),$
	$\displaystyle\Sigma^{+}=\phi_{113},\Sigma^{0}=\sqrt{2}\phi_{123},\Sigma^{-}=\phi_{233},$
	$\displaystyle\Xi^{0}=\phi_{311},\Xi^{-}=\phi_{332},\Sigma^{++}_{c}=\phi_{114},$
	$\displaystyle\Sigma^{+}_{c}=\phi_{124},\Sigma^{0}_{c}=\phi_{224},\Xi_{c}^{+}=\phi_{134},$
	$\displaystyle\Xi_{c}^{0}=\phi_{234},\Xi_{c}^{+^{\prime}}=\sqrt{\frac{2}{3}}(\phi_{413}-\phi_{431}),\Xi_{c}^{0^{\prime}}=\sqrt{\frac{2}{3}}(\phi_{423}-\phi_{432})$
	$\displaystyle\Lambda_{c}^{+}=\sqrt{\frac{2}{3}}(\phi_{421}-\phi_{412}),\Omega_{c}^{0}=\phi_{334},\Xi^{++}_{cc}=\phi_{441}$
	$\displaystyle\Xi_{cc}^{+}=\phi_{442},\Omega_{cc}^{+}=\phi_{443},$		(5)

where the indices $\beta$, $\mu$, $\nu$ denote the quark content of the baryon fields with the identification $1\leftrightarrow{}u$, $2\leftrightarrow{}d$, $3\leftrightarrow{}s$, and $4\leftrightarrow{}c$. Hence, the tensors $\phi_{\beta\mu\nu}$ satisfy following condition

$\displaystyle\phi_{\mu\nu\lambda}+\phi_{\nu\lambda\mu}+\phi_{\lambda\mu\nu}=0,\phi_{\mu\nu\lambda}=\phi_{\nu\mu\lambda}.$

(6)

By using the exact form of the matrix and the above relationship, the interaction vertices between baryons and pseudoscalar mesons can be estimated by expanding the SU(4) invariant interaction Lagrangians. The values of the coupling constants adopted in this work can be computed by comparing with the coefficients of the interaction Lagrangians ${\cal{L}}_{\pi{}NN}$ and ${\cal{L}}_{\rho{}NN}$ and determining the constants $g_{V}$, $g_{P}$, $a$, $b$, $c$, and $d$ in terms of $g_{\pi{}NN}=13.5$ and $g_{\rho{}NN}=3.25$ Liu:2001ce . Then the values of the coupling constants can be compute and are listed in Tab. 1.

In addition to the Lagrangian shown in Eq.2, the effective Lagrangians of relevant interaction vertices are also needed Yang:2021pio

	$\displaystyle\mathcal{L}_{PPV}$	$\displaystyle=\frac{iG}{2\sqrt{2}}\langle\partial^{\mu}P(PV_{\mu}-V_{\mu}P)\rangle,$		(7)
	$\displaystyle\mathcal{L}_{VVP}$	$\displaystyle=\frac{G^{\prime}}{\sqrt{2}}\epsilon^{\mu\nu\alpha\beta}\langle\partial_{\mu}V_{\nu}\partial_{\alpha}V_{\beta}P\rangle,$		(8)
	$\displaystyle\mathcal{L}_{VVV}$	$\displaystyle=\frac{iG}{2\sqrt{2}}\langle\partial^{\mu}V^{\nu}(V_{\mu}V_{\nu}-V_{\nu}V_{\mu})\rangle,$		(9)

where the coupling constants $G=12.0$ and $G^{{}^{\prime}}=55.51$ Yang:2021pio .

Since the $\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3227)$ states are considered as $S$-wave bound states of $D^{(*)}\Xi$, the couplings of $\Omega^{*}_{c}(3185)$ and $\Omega^{*}_{c}(3227)$ to their components are written as

	$\displaystyle{\cal{L}}^{1/2^{-}}_{\Omega^{*}_{c}(3185)}$	$\displaystyle=\sum_{j=\Xi^{-}D^{+},\Xi^{0}D^{0}}C_{j}g^{1/2^{-}}_{\Omega_{c}(3185)\Xi D}\Omega_{c}(3185)(x)$
		$\displaystyle\times\int dy\Phi(y^{2})\Xi(x+\omega_{D}y)D(x-\omega_{\Xi}y),$
	$\displaystyle{\cal{L}}^{1/2^{-}}_{\Omega^{*}_{c}(3327)}=$	$\displaystyle\sum_{j=\Xi^{-}D^{*+},\Xi^{0}D^{*0}}C_{j}g^{1/2^{-}}_{\Omega^{*}_{c}(3327)\Xi D^{*}}\Omega^{*}_{c}(3327)(x)\gamma^{\mu}\gamma^{5}$
		$\displaystyle\times\int dy\Phi(y^{2})\Xi(x+\omega_{D^{*}}y)D^{*}_{\mu}(x-\omega_{\Xi}y),$
	$\displaystyle{\cal{L}}^{3/2^{-}}_{\Omega^{*}_{c}(3327)}=$	$\displaystyle-i\sum_{j=\Xi^{-}D^{*+},\Xi^{0}D^{*0}}C_{j}g^{3/2^{-}}_{\Omega_{c}(3327)\Xi D^{*}}\Omega_{c}(3327)^{\mu}(x)$
		$\displaystyle\int dy\Phi(y^{2})\Xi(x+\omega_{D^{*}}y)D^{*}_{\mu}(x-\omega_{\Xi}y),$		(10)

where $\omega_{\Xi}=m_{\Xi}/(m_{\Xi}+m_{D^{(*)}})$ and $\omega_{D^{(*)}}=m_{D^{(*)}}/(m_{\Xi}+m_{D^{(*)}})$ with $m_{\Xi}$ and $m_{D^{(*)}}$ are the masses of the $\Xi$ and $D^{(*)}$, respectively. $C_{j}=1/\sqrt{2}$ is the isospin coefficient, which is calculated from the following isospin assignments for the $\Xi$ and $D^{(*)}$

$$\begin{pmatrix}\Xi^{-}\\ \Xi^{0}\end{pmatrix}\sim\begin{pmatrix}-|\frac{1}{2},-\frac{1}{2}\rangle\\ |\frac{1}{2},+\frac{1}{2}\rangle\end{pmatrix};~{}~{}~{}\begin{pmatrix}D^{(*)+}\\ D^{(*)0}\end{pmatrix}\sim\begin{pmatrix}|\frac{1}{2},+\frac{1}{2}\rangle\\ |\frac{1}{2},-\frac{1}{2}\rangle\end{pmatrix}.$$

In the above equation, $\Phi(y^{2})$ is the effective correlation function and serves the following two roles: 1) it shows the distribution of the components in the hadronic molecule, 2) it has the same role with the form factor that can avoid the Feynman diagram’s ultraviolet divergence. Usually, the correlation function can vanish quickly in the ultraviolet region. Here we choose the Fourier transformation of the correlation function to have a Gaussian form

$\displaystyle\Phi(-p_{E}^{2})\doteq{}\exp(-p_{E}^{2}/\Lambda^{2}),$

(11)

where $p_{E}$ is the Euclidean Jacobi momentum and the parameter $\Lambda$ is taken as a parameter, which will be discussed later. The coupling constants of $g^{1/2^{-}}_{\Omega^{*}_{c}(3185)\Xi D}$, $g^{1/2^{-}}_{\Omega^{*}_{c}(3327)\Xi D^{*}}$, $g^{3/2^{-}}_{\Omega^{*}_{c}(3327)\Xi{}D^{*}}$ appearing in Eq. (10) are given in Eq. (12)

	$\displaystyle\Sigma^{1/2^{-}}_{\Omega^{*}_{c}(3185)}(k_{0})=$	$\displaystyle(g^{1/2^{-}}_{\Omega_{c}\Xi D})^{2}\int_{0}^{\infty}d\alpha\int_{0}^{\infty}d\beta\sum_{j}C_{j}{\cal{Y}}(\omega_{D},m_{D})$
		$\displaystyle\times{\cal{Z}}(\omega_{D},m_{D}),$
	$\displaystyle\Sigma^{1/2^{-}}_{\Omega^{*}_{c}(3327)}(k_{0})=$	$\displaystyle(g^{1/2^{-}}_{\Omega_{c}\Xi D^{*}})^{2}\int_{0}^{\infty}d\alpha\int_{0}^{\infty}d\beta\sum_{j}C_{j}{\cal{Y}}(\omega_{D^{*}},m_{D^{*}})$
		$\displaystyle\times{}[2{\cal{Z}}(\omega_{D^{*}},m_{D^{*}})+\frac{k^{3}_{0}(-4\omega_{D^{*}}-2\beta)^{3}}{8m^{2}_{D^{*}}z^{3}}$
		$\displaystyle-\frac{3k_{0}\Lambda^{2}(-4\omega_{D^{*}}-2\beta)}{2m^{2}_{D^{*}}z^{2}}+\frac{k_{0}^{3}(-4\omega_{D^{*}}-2\beta)}{4m^{2}_{D^{*}}z^{2}}$
		$\displaystyle-\frac{k^{2}_{0}(-4\omega_{D^{*}}-2\beta)^{2}m_{\Xi}}{4m^{2}_{D^{*}}z^{2}}+\frac{k_{0}\Lambda^{2}}{m^{2}_{D^{*}}z}+\frac{2\Lambda^{2}m_{\Xi}}{m^{2}_{D^{*}}z}],$

	$\displaystyle\Sigma^{T3/2^{-}}_{\Omega^{*}_{c}(3327)}(k_{0})=$	$\displaystyle g_{\Omega_{c}\Xi D^{*}}^{2}\int_{0}^{\infty}d\alpha\int_{0}^{\infty}d\beta\sum_{j}C_{j}{\cal{Y}}(\omega_{D^{*}},m_{D^{*}})$
		$\displaystyle\times[{\cal{Z}}(\omega_{D^{*}},m_{D^{*}})+\frac{k_{0}\Lambda^{2}(-4\omega_{D^{*}}-2\beta)}{4m^{2}_{D^{*}}z^{2}}$
		$\displaystyle+\frac{k_{0}\Lambda^{2}}{2m^{2}_{D^{*}}z}+\frac{\Lambda^{2}m_{\Xi}}{2m^{2}_{D^{*}}z}],$		(12)

with

	$\displaystyle{\cal{Y}}(\omega_{D^{(*)}},m_{D^{(*)}})$	$\displaystyle=\frac{1}{16\pi^{2}z^{2}}\exp\{-\frac{1}{\Lambda^{2}}[-2k_{0}^{2}\omega_{D}^{2}+\alpha m_{D}^{2}$
		$\displaystyle+\beta(-k_{0}^{2}+m^{2}_{\Xi})+\frac{(-4\omega_{D}-2\beta)^{2}k_{0}^{2}}{4z}]\}$
	$\displaystyle{\cal{Z}}(\omega_{D^{(*)}},m_{D^{(*)}})$	$\displaystyle=k_{0}+m_{\Xi}+\frac{k_{0}(-4\omega_{D}-2\beta)}{2z},$		(13)

where $Z=2+\alpha+\beta$ and $k_{0}^{2}=m^{2}_{\Omega^{*}_{c}}$ with $k_{0}$, $m_{\Omega^{*}_{c}}$ are the four momenta and the mass of the newly observed $\Omega^{*}_{c}$, respectively. If the $\Omega_{c}^{*}$ is a $D^{(*)}\Xi$ molecular states with $J^{P}=1/2^{-}$, the $\Sigma^{1/2^{-}}$ is the self-energy operator of the hadronic molecule $\Omega_{c}^{*}$. However, $\Sigma^{T3/2^{-}}$ is the transverse part of the self-energy operator $\Sigma^{\mu\nu}_{\Omega_{c}^{*}}$ when we assuming $\Omega_{c}^{*}$ as $D^{(*)}\Xi$ molecular states with $J^{P}=3/2^{-}$. Once the self-energy operator and its transverse part are obtained, the coupling constants of the hadronic molecule $\Omega_{c}^{*}$ to its constituents $D^{(*)}\Xi$ can be determined using the compositeness condition based on the work in Ref. Weinberg:1962hj . This condition requires that the renormalization constant of the hadronic molecular wave function is equal to zero

	$\displaystyle 1-\frac{d\Sigma_{\Omega^{*}_{c}(3185/3327)}}{dk_{0}}=0,~{}~{}~{}~{}~{}J=\frac{1}{2}$
	$\displaystyle 1-\frac{d\Sigma^{T}_{\Omega^{*}_{c}(3327)}}{dk_{0}}=0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}J=\frac{3}{2}$		(14)

With the above prepared, we can obtain the general expressions of the amplitudes corresponding to the Feynman diagrams Fig. 1

	$\displaystyle\mathcal{M}_{a}^{1/2^{-}}$	$\displaystyle=\mu(p_{2})[\frac{1}{\sqrt{2}}g_{\Omega 11}g_{\Xi^{(^{\prime})+}_{c}D^{+}\Sigma^{0}}g_{\Xi^{-}K^{-}\Sigma^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
		$\displaystyle\times\Phi[(p\omega_{D^{+}}-q\omega_{\Xi^{-}})^{2}]\gamma^{5}\frac{i(k\!\!\!/_{1}+m_{\Sigma^{0}})}{k_{1}^{2}-m_{\Sigma^{0}}^{2}}\gamma^{5}$
		$\displaystyle\times\frac{i(p\!\!\!/+m_{\Xi^{-}})}{p^{2}-m_{\Xi^{-}}^{2}}\frac{i}{q^{2}-m_{D^{+}}^{2}}+\frac{1}{\sqrt{2}}g_{\Omega 11}g_{\Xi^{(^{\prime})+}_{c}D^{+}\Lambda^{0}}$
		$\displaystyle\times{}g_{\Xi^{-}K^{-}\Lambda^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\Phi[(p\omega_{D^{+}}-q\omega_{\Xi^{-}})^{2}]\gamma^{5}$
		$\displaystyle\times\frac{i(k\!\!\!/_{1}+m_{\Lambda^{0}})}{k_{1}^{2}-m_{\Lambda^{0}}^{2}}\gamma^{5}\frac{i(p\!\!\!/+m_{\Xi^{-}})}{p^{2}-m_{\Xi^{-}}^{2}}\frac{i}{q^{2}-m_{D^{+}}^{2}}]\mu(k_{0}),$		(15)
	$\displaystyle\mathcal{M}_{b}^{1/2^{-}}$	$\displaystyle=\frac{1}{\sqrt{2}}g_{\Omega 12}g_{D^{0}\Xi_{c}^{(^{\prime})+}\Sigma^{+}}g_{\Xi^{0}K^{-}\Sigma^{+}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
		$\displaystyle\times\phi[(p\omega_{D^{0}}-q\omega_{\Xi^{0}})^{2}]\mu(p_{2})\gamma^{5}\frac{i(k\!\!\!/_{1}+m_{\Sigma^{+}})}{k_{1}^{2}-m_{\Sigma^{+}}^{2}}\gamma^{5}$
		$\displaystyle\times\frac{i(p\!\!\!/+m_{\Xi^{0}})}{p^{2}-m_{\Xi^{0}}^{2}}\mu(k_{0})\frac{i}{q^{2}-m_{D^{0}}^{2}},$		(16)

	$\displaystyle\mathcal{M}_{c}^{1/2^{-}}$	$\displaystyle=i\frac{1}{2}g_{\Omega 12}g_{D^{0}D^{*-}_{s}K^{-}}g_{\Xi^{0}\Xi_{c}^{(^{\prime})+}D^{*-}_{s}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
		$\displaystyle\times\Phi[(p\omega_{D^{0}}-q\omega_{\Xi^{0}})^{2}]\mu({p_{2}})\gamma^{\nu}\frac{i(-g^{\mu\nu}+\frac{k_{1}^{\mu}k_{1}^{\nu}}{m_{D^{*-}_{s}}^{2}})}{k^{2}_{1}-m^{2}_{D^{*-}_{s}}}$
		$\displaystyle\times(iq^{\mu}-ip_{1}^{\mu})\frac{i}{q^{2}-m_{D^{0}}^{2}}\mu(k_{0})\frac{i(p\!\!\!/+m_{\Xi^{0}})}{p^{2}-m^{2}_{\Xi^{0}}},$		(17)
	$\displaystyle\mathcal{M}_{d}^{1/2^{-}}$	$\displaystyle=\mu(p_{2})[\frac{1}{\sqrt{2}}g_{\Omega 21}g_{\Xi^{(^{\prime})+}_{c}D^{*+}\Sigma^{0}}g_{\Xi^{-}K^{-}\Sigma^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
		$\displaystyle\times\Phi[(p\omega_{D^{*+}}-q\omega_{\Xi^{-}})^{2}]\gamma^{\mu}\frac{i(k\!\!\!/_{1}+m_{\Sigma^{0}})}{k_{1}^{2}-m_{\Sigma^{0}}^{2}}\gamma^{5}$
		$\displaystyle\times\frac{i(p\!\!\!/+m_{\Xi^{-}})}{p^{2}-m_{\Xi^{-}}^{2}}\gamma^{\nu}\gamma^{5}\frac{i(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{m^{2}_{D^{*+}}})}{q^{2}-m_{D^{*+}}^{2}}$
		$\displaystyle+\frac{1}{\sqrt{2}}g_{\Omega 21}g_{\Xi^{(^{\prime})+}_{c}D^{*+}\Lambda^{0}}g_{\Xi^{-}K^{-}\Lambda^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
		$\displaystyle\times\Phi[(p\omega_{D^{*+}}-q\omega_{\Xi^{-}})^{2}]\gamma^{\alpha}\frac{i(k\!\!\!/_{1}+m_{\Lambda^{0}})}{k_{1}^{2}-m_{\Lambda^{0}}^{2}}\gamma^{5}$
		$\displaystyle\times\frac{i(p\!\!\!/+m_{\Xi^{-}})}{p^{2}-m_{\Xi^{-}}^{2}}\gamma^{\beta}\gamma^{5}\frac{i(-g^{\alpha\beta}+\frac{q^{\alpha}q^{\beta}}{m^{2}_{D^{*+}}})}{q^{2}-m_{D^{*+}}^{2}}]\mu(k_{0}),$		(18)
	$\displaystyle\mathcal{M}_{d}^{3/2^{-}}$	$\displaystyle=\mu(p_{2})[-i\frac{1}{\sqrt{2}}g_{\Omega 31}g_{\Xi^{(^{\prime})+}_{c}D^{*+}\Sigma^{0}}g_{\Xi^{-}K^{-}\Sigma^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
		$\displaystyle\times\Phi[(p\omega_{D^{*+}}-q\omega_{\Xi^{-}})^{2}]\gamma^{\mu}\frac{i({k\!\!\!/_{1}}+m_{\Sigma^{0}})}{k_{1}^{2}-m_{\Sigma^{0}}^{2}}\gamma^{5}$
		$\displaystyle\times\frac{i({p\!\!\!/}+m_{\Xi^{-}})}{p^{2}-m_{\Xi^{-}}^{2}}g^{\nu\lambda}\frac{i(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{m^{2}_{D^{*+}}})}{q^{2}-m_{D^{*+}}^{2}}$
		$\displaystyle-i\frac{1}{\sqrt{2}}g_{\Omega 31}g_{\Xi^{(^{\prime})+}_{c}D^{*+}\Lambda^{0}}g_{\Xi^{-}K^{-}\Lambda^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
		$\displaystyle\times\Phi[(p\omega_{D^{*+}}-q\omega_{\Xi^{-}})^{2}]\gamma^{\alpha}\frac{i({k\!\!\!/_{1}}+m_{\Lambda^{0}})}{k_{1}^{2}-m_{\Lambda^{0}}^{2}}\gamma^{5}$
		$\displaystyle\times\frac{i({p\!\!\!/}+m_{\Xi^{-}})}{p^{2}-m_{\Xi^{-}}^{2}}g^{\beta\lambda}\frac{i(-g^{\alpha\beta}+\frac{q^{\alpha}q^{\beta}}{m^{2}_{D^{*+}}})}{q^{2}-m_{D^{*+}}^{2}}]\mu_{\lambda}(k_{0}),$		(19)
	$\displaystyle\mathcal{M}_{e}^{1/2^{-}}$	$\displaystyle=\frac{1}{\sqrt{2}}g_{\Omega 22}g_{\Xi^{(^{\prime})+}_{c}D^{*0}\Sigma^{0}}g_{\Xi^{0}K^{-}\Sigma^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
		$\displaystyle\times\Phi[(p\omega_{D^{*0}}-q\omega_{\Xi^{0}})^{2}]\mu(p_{2})\gamma^{\mu}\frac{i({k\!\!\!/_{1}}+m_{\Sigma^{+}})}{k_{1}^{2}-m_{\Sigma^{+}}^{2}}\gamma^{5}$
		$\displaystyle\times\frac{i({p\!\!\!/}+m_{\Xi^{0}})}{p^{2}-m_{\Xi^{0}}^{2}}\mu(k_{0})\gamma^{\nu}\gamma^{5}\frac{i(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{m^{2}_{D^{*0}}})}{q^{2}-m_{D^{*0}}^{2}},$		(20)
	$\displaystyle\mathcal{M}_{e}^{3/2^{-}}$	$\displaystyle=-\frac{i}{\sqrt{2}}g_{\Omega 22}g_{\Xi^{(^{\prime})+}_{c}D^{*0}\Sigma^{0}}g_{\Xi^{0}K^{-}\Sigma^{0}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
		$\displaystyle\times\Phi[(p\omega_{D^{*0}}-q\omega_{\Xi^{0}})^{2}]\mu(p_{2})\gamma^{\mu}\frac{i({k\!\!\!/_{1}}+m_{\Sigma^{+}})}{k_{1}^{2}-m_{\Sigma^{+}}^{2}}\gamma^{5}$
		$\displaystyle\times\frac{i({p\!\!\!/}+m_{\Xi^{0}})}{p^{2}-m_{\Xi^{0}}^{2}}\mu^{\nu}(k_{0})\frac{i(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{m^{2}_{D^{*0}}})}{q^{2}-m_{D^{*0}}^{2}},$		(21)

	$\displaystyle\mathcal{M}_{f}^{1/2^{-}}$	$\displaystyle=\frac{i}{2}g_{\Omega 31}g_{D^{*0}D^{-}_{s}K^{-}}g_{\Xi^{0}D_{s}^{-}\Xi_{c}^{(^{\prime})+}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
		$\displaystyle\times\Phi[(p\omega_{D^{*0}}-q\omega_{\Xi^{0}})^{2}]\mu(p_{2})\gamma^{5}\frac{1}{k_{1}^{2}-m_{D_{s}^{-}}^{2}}$
		$\displaystyle\times(p_{1}^{\mu}-k_{1}^{\mu})\frac{(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{m^{2}_{D^{*0}}})}{q^{2}-m_{D^{*0}}^{2}}\mu(k_{0})\gamma^{\nu}$
		$\displaystyle\times\gamma^{5}\frac{({p\!\!\!/}+m_{\Xi^{0}})}{p^{2}-m_{\Xi^{0}}^{2}},$		(22)
	$\displaystyle\mathcal{M}_{f}^{3/2^{-}}$	$\displaystyle=\frac{1}{2}g_{\Omega 32}g_{D^{*0}D^{-}_{s}K^{-}}g_{\Xi^{0}D_{s}^{-}\Xi_{c}^{(^{\prime})+}}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}$
		$\displaystyle\Phi[(p\omega_{D^{*0}}-q\omega_{\Xi^{0}})^{2}]\mu(p_{2})\gamma^{5}\frac{1}{k_{1}^{2}-m_{D_{s}^{-}}^{2}}$
		$\displaystyle\times(p_{1}^{\mu}-k_{1}^{\mu})\frac{(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{m^{2}_{D^{*0}}})}{q^{2}-m_{D^{*0}}^{2}}\mu^{\nu}(k_{0})\frac{({p\!\!\!/}+m_{\Xi^{0}})}{p^{2}-m_{\Xi^{0}}^{2}},$		(23)

where the $p_{1}$, $p_{2}$, $p$, $q$, and $k_{1}$ are the four momenta of the $K$, $\Xi_{c}^{(^{\prime})}$, $\Xi$, $D^{(*)}$, and $t$-channel exchanged particles, respectively.