Hidden-charm pentaquarks with triple strangeness due to the $\Omega_{c}^{(*)}\bar{D}_{s}^{(*)}$ interactions

In the present work, we study the interactions between an $S$-wave charmed baryon $\Omega_{c}^{(*)}$ and an $S$-wave anticharmed-strange meson $\bar{D}_{s}^{(*)}$. Here, we adopt the OBE model Chen:2016qju ; Liu:2019zoy , and consider the effective potentials from the $f_{0}(980)$, $\eta$, and $\phi$ exchanges. In particular, we need to emphasize that the light scalar meson $f_{0}(980)$ exchange provides effective interaction for these investigated systems, and we do not consider the $\sigma$ and $a_{0}(980)$ exchanges in our calculation, since the $\sigma$ is usually considered as a meson with only up and down quarks and the $a_{0}(980)$ is the light isovector scalar meson. In this subsection, we construct the relevant wave functions and effective Lagrangians, and deduce the OBE effective potentials in the coordinate space for all of the investigated systems.

Firstly, we introduce the flavor and spin-orbital wave functions involved in our calculation. For the $\Omega_{c}^{(*)}\bar{D}_{s}^{(*)}$ systems, the flavor wave function $|I,I_{3}\rangle$ is quite simple and reads as $|0,0\rangle=|\Omega_{c}^{(*)0}{D}_{s}^{(*)-}\rangle$, where $I$ and $I_{3}$ are the isospin and its third component of the discussed system. In addition, the spin-orbital wave functions $|{}^{2S+1}L_{J}\rangle$ for these investigated $\Omega_{c}^{(*)}\bar{D}_{s}^{(*)}$ systems are explicitly written as

	$\displaystyle\left|\Omega_{c}\bar{D}_{s}\left({}^{2S+1}L_{J}\right)\right\rangle$	$\displaystyle=$	$\displaystyle\sum_{m,m_{L}}C^{J,M}_{\frac{1}{2}m,Lm_{L}}\chi_{\frac{1}{2}m}\left|Y_{L,m_{L}}\right\rangle,$
	$\displaystyle\left|\Omega_{c}^{*}\bar{D}_{s}\left({}^{2S+1}L_{J}\right)\right\rangle$	$\displaystyle=$	$\displaystyle\sum_{m,m_{L}}C^{J,M}_{\frac{3}{2}m,Lm_{L}}\Phi_{\frac{3}{2}m}\left|Y_{L,m_{L}}\right\rangle,$
	$\displaystyle\left|\Omega_{c}\bar{D}_{s}^{*}\left({}^{2S+1}L_{J}\right)\right\rangle$	$\displaystyle=$	$\displaystyle\sum_{m,m^{\prime},m_{S},m_{L}}C^{S,m_{S}}_{\frac{1}{2}m,1m^{\prime}}C^{J,M}_{Sm_{S},Lm_{L}}\chi_{\frac{1}{2}m}\epsilon_{m^{\prime}}^{\mu}\left|Y_{L,m_{L}}\right\rangle,$
	$\displaystyle\left|\Omega_{c}^{*}\bar{D}_{s}^{*}\left({}^{2S+1}L_{J}\right)\right\rangle$	$\displaystyle=$	$\displaystyle\sum_{m,m^{\prime},m_{S},m_{L}}C^{S,m_{S}}_{\frac{3}{2}m,1m^{\prime}}C^{J,M}_{Sm_{S},Lm_{L}}\Phi_{\frac{3}{2}m}\epsilon_{m^{\prime}}^{\mu}\left|Y_{L,m_{L}}\right\rangle.$

In the above expressions, $S$, $L$, and $J$ denote the spin, orbit angular momentum, and total angular momentum for the discussed system, respectively. The constant $C^{e,f}_{ab,cd}$ is the Clebsch-Gordan coefficient, and $|Y_{L,m_{L}}\rangle$ is the spherical harmonics function. In the static limit, the polarization vector $\epsilon_{m}^{\mu}\,(m=0,\,\pm 1)$ with the spin-1 field can be expressed as $\epsilon_{0}^{\mu}=\left(0,0,0,-1\right)$ and $\epsilon_{\pm}^{\mu}=\left(0,\,\pm 1,\,i,\,0\right)/\sqrt{2}$. $\chi_{\frac{1}{2}m}$ stands for the spin wave function of the charmed baryon with spin $S={1}/{2}$, and the polarization tensor $\Phi_{\frac{3}{2}m}$ of the charmed baryon with spin quantum number $S={3}/{2}$ can be written in a general form, i.e.,

$\displaystyle\Phi_{\frac{3}{2}m}=\sum_{m_{1},m_{2}}C^{\frac{3}{2},m}_{\frac{1}{2}m_{1},1m_{2}}\chi_{\frac{1}{2}m_{1}}\epsilon_{m_{2}}^{\mu}.$

(2.2)

In order to write out the relevant scattering amplitudes quantitatively, we usually adopt the effective Lagrangian approach. To be convenient, we construct two types of super-fields $\mathcal{S}_{\mu}$ and $H^{(\overline{Q})}_{a}$ via the heavy quark limit Wise:1992hn . The superfield $\mathcal{S}_{\mu}$ is expressed as a combination of the charmed baryons $\mathcal{B}_{6}$ with $J^{P}=1/2^{+}$ and $\mathcal{B}^{*}_{6}$ with $J^{P}=3/2^{+}$ in the $6_{F}$ flavor representation Chen:2017xat , and the superfield $H^{(\overline{Q})}_{a}$ includes the anticharmed-strange vector meson $\bar{D}^{*}_{s}$ with $J^{P}=1^{-}$ and the pseudoscalar meson $\bar{D}_{s}$ with $J^{P}=0^{-}$ Ding:2008gr . The general expressions of the super-fields $\mathcal{S}_{\mu}$ and $H^{(\overline{Q})}_{a}$ can be given by

	$\displaystyle\mathcal{S}_{\mu}$	$\displaystyle=$	$\displaystyle-\sqrt{\frac{1}{3}}(\gamma_{\mu}+v_{\mu})\gamma^{5}\mathcal{B}_{6}+\mathcal{B}_{6\mu}^{*},$
	$\displaystyle H^{(\overline{Q})}_{a}$	$\displaystyle=$	$\displaystyle\left(\bar{D}^{*(\overline{Q})\mu}_{a}\gamma_{\mu}-\bar{D}^{(\overline{Q})}_{a}\gamma_{5}\right)\frac{1-/\!\!\!v}{2}.$		(2.3)

Here, $v_{\mu}=(1,\bm{0})$ is the four velocity under the nonrelativistic approximation.

With the above preparation, we construct the relevant effective Lagrangians to describe the interactions among the heavy hadrons $\mathcal{B}_{6}^{(*)}/\bar{D}_{s}^{(*)}$ and the light scalar, pseudoscalar, or vector mesons as Ding:2008gr ; Chen:2017xat

	$\displaystyle\mathcal{L}_{\mathcal{B}^{(*)}_{6}}$	$\displaystyle=$	$\displaystyle l_{S}\langle\bar{\mathcal{S}}_{\mu}f_{0}\mathcal{S}^{\mu}\rangle-\frac{3}{2}g_{1}\varepsilon^{\mu\nu\lambda\kappa}v_{\kappa}\langle\bar{\mathcal{S}}_{\mu}{\mathcal{A}}_{\nu}\mathcal{S}_{\lambda}\rangle$
			$\displaystyle+i\beta_{S}\langle\bar{\mathcal{S}}_{\mu}v_{\alpha}\left(\mathcal{V}^{\alpha}-\rho^{\alpha}\right)\mathcal{S}^{\mu}\rangle+\lambda_{S}\langle\bar{\mathcal{S}}_{\mu}F^{\mu\nu}(\rho)\mathcal{S}_{\nu}\rangle,$
	$\displaystyle\mathcal{L}_{H}$	$\displaystyle=$	$\displaystyle g_{S}\langle\bar{H}^{(\overline{Q})}_{a}f_{0}H^{(\overline{Q})}_{a}\rangle+ig\langle\bar{H}^{(\overline{Q})}_{a}\gamma_{\mu}{\mathcal{A}}_{ab}^{\mu}\gamma_{5}H^{(\overline{Q})}_{b}\rangle$		(2.4)
			$\displaystyle-i\beta\langle\bar{H}^{(\overline{Q})}_{a}v_{\mu}\left(\mathcal{V}^{\mu}-\rho^{\mu}\right)_{ab}H^{(\overline{Q})}_{b}\rangle$
			$\displaystyle+i\lambda\langle\bar{H}^{(\overline{Q})}_{a}\sigma_{\mu\nu}F^{\mu\nu}(\rho)_{ab}H^{(\overline{Q})}_{b}\rangle,$

which satisfy the requirement of the heavy quark symmetry, the chiral symmetry, and the hidden local symmetry Casalbuoni:1992gi ; Casalbuoni:1996pg ; Yan:1992gz ; Harada:2003jx ; Bando:1987br . The axial current $\mathcal{A}_{\mu}$ and the vector current ${\cal V}_{\mu}$ can be defined as ${\mathcal{A}}_{\mu}=\left(\xi^{\dagger}\partial_{\mu}\xi-\xi\partial_{\mu}\xi^{\dagger}\right)/2$ and ${\mathcal{V}}_{\mu}=\left(\xi^{\dagger}\partial_{\mu}\xi+\xi\partial_{\mu}\xi^{\dagger}\right)/2$, respectively. Here, the pseudo-Goldstone field can be written as $\xi=\exp(i\mathbb{P}/f_{\pi})$, where $f_{\pi}$ is the pion decay constant. In the above formulas, the vector meson field $\rho_{\mu}$ and its strength tensor $F_{\mu\nu}(\rho)$ are $\rho_{\mu}=i{g_{V}}\mathbb{V}_{\mu}/{\sqrt{2}}$ and $F_{\mu\nu}(\rho)=\partial_{\mu}\rho_{\nu}-\partial_{\nu}\rho_{\mu}+[\rho_{\mu},\rho_{\nu}]$, respectively. Here, $\mathcal{B}_{6}^{(*)}$, $\mathbb{V}_{\mu}$, and ${\mathbb{P}}$ are the matrices of the charmed baryon in the $6_{F}$ flavor representation, light vector meson, and light pseudoscalar meson, which can be written as

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

(2.17)

respectively. By expanding the compact effective Lagrangians to the leading order of the pseudo-Goldstone field $\xi$, we can further obtain the concrete effective Lagrangians. The effective Lagrangians for $\mathcal{B}_{6}^{(*)}$ and the light mesons are expressed as

	$\displaystyle\mathcal{L}_{\mathcal{B}_{6}^{(*)}\mathcal{B}_{6}^{(*)}f_{0}}$	$\displaystyle=$	$\displaystyle-l_{S}\langle\bar{\mathcal{B}}_{6}f_{0}\mathcal{B}_{6}\rangle+l_{S}\langle\bar{\mathcal{B}}_{6\mu}^{*}f_{0}\mathcal{B}_{6}^{*\mu}\rangle$		(2.18)
			$\displaystyle-\frac{l_{S}}{\sqrt{3}}\langle\bar{\mathcal{B}}_{6\mu}^{*}f_{0}\left(\gamma^{\mu}+v^{\mu}\right)\gamma^{5}\mathcal{B}_{6}\rangle+h.c.,$
	$\displaystyle\mathcal{L}_{\mathcal{B}_{6}^{(*)}\mathcal{B}_{6}^{(*)}\mathbb{P}}$	$\displaystyle=$	$\displaystyle i\frac{g_{1}}{2f_{\pi}}\varepsilon^{\mu\nu\lambda\kappa}v_{\kappa}\langle\bar{\mathcal{B}}_{6}\gamma_{\mu}\gamma_{\lambda}\partial_{\nu}\mathbb{P}\mathcal{B}_{6}\rangle$		(2.19)
			$\displaystyle-i\frac{3g_{1}}{2f_{\pi}}\varepsilon^{\mu\nu\lambda\kappa}v_{\kappa}\langle\bar{\mathcal{B}}_{6\mu}^{*}\partial_{\nu}\mathbb{P}\mathcal{B}_{6\lambda}^{*}\rangle$
			$\displaystyle+i\frac{\sqrt{3}g_{1}}{2f_{\pi}}v_{\kappa}\varepsilon^{\mu\nu\lambda\kappa}\langle\bar{\mathcal{B}}_{6\mu}^{*}\partial_{\nu}\mathbb{P}{\gamma_{\lambda}\gamma^{5}}\mathcal{B}_{6}\rangle+h.c.,$
	$\displaystyle\mathcal{L}_{\mathcal{B}_{6}^{(*)}\mathcal{B}_{6}^{(*)}\mathbb{V}}$	$\displaystyle=$	$\displaystyle-\frac{\beta_{S}g_{V}}{\sqrt{2}}\langle\bar{\mathcal{B}}_{6}v\cdot\mathbb{V}\mathcal{B}_{6}\rangle$		(2.20)
			$\displaystyle-i\frac{\lambda_{S}g_{V}}{3\sqrt{2}}\langle\bar{\mathcal{B}}_{6}\gamma_{\mu}\gamma_{\nu}\left(\partial^{\mu}\mathbb{V}^{\nu}-\partial^{\nu}\mathbb{V}^{\mu}\right)\mathcal{B}_{6}\rangle$
			$\displaystyle-\frac{\beta_{S}g_{V}}{\sqrt{6}}\langle\bar{\mathcal{B}}_{6\mu}^{*}v\cdot\mathbb{V}\left(\gamma^{\mu}+v^{\mu}\right)\gamma^{5}\mathcal{B}_{6}\rangle$
			$\displaystyle-i\frac{\lambda_{S}g_{V}}{\sqrt{6}}\langle\bar{\mathcal{B}}_{6\mu}^{*}\left(\partial^{\mu}\mathbb{V}^{\nu}-\partial^{\nu}\mathbb{V}^{\mu}\right)\left(\gamma_{\nu}+v_{\nu}\right)\gamma^{5}\mathcal{B}_{6}\rangle$
			$\displaystyle+\frac{\beta_{S}g_{V}}{\sqrt{2}}\langle\bar{\mathcal{B}}_{6\mu}^{*}v\cdot{V}\mathcal{B}_{6}^{*\mu}\rangle$
			$\displaystyle+i\frac{\lambda_{S}g_{V}}{\sqrt{2}}\langle\bar{\mathcal{B}}_{6\mu}^{*}\left(\partial^{\mu}\mathbb{V}^{\nu}-\partial^{\nu}\mathbb{V}^{\mu}\right)\mathcal{B}_{6\nu}^{*}\rangle+h.c.,$

and the effective Lagrangians to describe the $S$-wave anticharmed-strange mesons $\bar{D}_{s}^{(*)}$ and the light scalar, pseudoscalar, or vector mesons are

	$\displaystyle\mathcal{L}_{{\bar{D}}^{(*)}{\bar{D}}^{(*)}f_{0}}$	$\displaystyle=$	$\displaystyle-2g_{S}{\bar{D}}_{a}{\bar{D}}_{a}^{{\dagger}}f_{0}+2g_{S}{\bar{D}}_{a\mu}^{*}{\bar{D}}_{a}^{*\mu{\dagger}}f_{0},$		(2.21)
	$\displaystyle\mathcal{L}_{{\bar{D}}^{(*)}{\bar{D}}^{(*)}\mathbb{P}}$	$\displaystyle=$	$\displaystyle\frac{2ig}{f_{\pi}}v^{\alpha}\varepsilon_{\alpha\mu\nu\lambda}{\bar{D}}_{a}^{*\mu{\dagger}}{\bar{D}}_{b}^{*\lambda}\partial^{\nu}{\mathbb{P}}_{ab}$		(2.22)
			$\displaystyle+\frac{2g}{f_{\pi}}\left({\bar{D}}_{a}^{*\mu{\dagger}}{\bar{D}}_{b}+{\bar{D}}_{a}^{{\dagger}}{\bar{D}}_{b}^{*\mu}\right)\partial_{\mu}{\mathbb{P}}_{ab},$
	$\displaystyle\mathcal{L}_{{\bar{D}}^{(*)}{\bar{D}}^{(*)}\mathbb{V}}$	$\displaystyle=$	$\displaystyle\sqrt{2}\beta g_{V}{\bar{D}}_{a}{\bar{D}}_{b}^{{\dagger}}v\cdot\mathbb{V}_{ab}-\sqrt{2}\beta g_{V}{\bar{D}}_{a\mu}^{*}{\bar{D}}_{b}^{*\mu{\dagger}}v\cdot\mathbb{V}_{ab}$
			$\displaystyle-2\sqrt{2}i\lambda g_{V}{\bar{D}}_{a}^{*\mu{\dagger}}{\bar{D}}_{b}^{*\nu}\left(\partial_{\mu}\mathbb{V}_{\nu}-\partial_{\nu}\mathbb{V}_{\mu}\right)_{ab}$
			$\displaystyle-2\sqrt{2}\lambda g_{V}v^{\lambda}\varepsilon_{\lambda\mu\alpha\beta}\left({\bar{D}}_{a}^{*\mu{\dagger}}{\bar{D}}_{b}+{\bar{D}}_{a}^{{\dagger}}{\bar{D}}_{b}^{*\mu}\right)\partial^{\alpha}\mathbb{V}^{\beta}_{ab}.$

In the above effective Lagrangians, the coupling constants can be either extracted from the experimental data or calculated by the theoretical models, and the signs of these coupling constants are fixed via the quark model Riska:2000gd . The values of these coupling constants are $l_{S}=6.20$, $g_{S}=0.76$,²²2In this work, we consider the contribution from light scalar meson $f_{0}(980)$ exchange. Here, the corresponding coupling constant involved in effective Lagrangians [Eq. (2.18) and Eq. (2.21)] is approximately taken as the same as that for the case of light scalar $\sigma$. $g_{1}=0.94$, $g=0.59$, $f_{\pi}=132~{}\rm{MeV}$, $\beta_{S}g_{V}=10.14$, $\beta g_{V}=-5.25$, $\lambda_{S}g_{V}=19.2~{}\rm{GeV}^{-1}$, and $\lambda g_{V}=-3.27~{}\rm{GeV}^{-1}$ Chen:2019asm , which are widely used to discuss the hadronic molecular states Wang:2020bjt ; Chen:2017xat ; Wang:2019nwt ; Chen:2019asm ; He:2015cea ; He:2019ify ; Chen:2018pzd . In particular, we need to emphasize that these input coupling constants can well reproduce the masses of the $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$ Aaij:2019vzc under the meson-baryon molecular picture when adopting the OBE model Chen:2019asm ; He:2019ify .

We follow the standard strategy to deduce the effective potentials in the coordinate space in Refs. Wang:2020dya ; Wang:2019nwt ; Wang:2019aoc , which is a lengthy and tedious calculation. In Fig. 1, we present the relevant Feynman diagram for the $\Omega_{c}^{(*)}\bar{D}_{s}^{(*)}\to\Omega_{c}^{(*)}\bar{D}_{s}^{(*)}$ scattering processes.

At the hadronic level, we firstly write out the scattering amplitude $\mathcal{M}(h_{1}h_{2}\to h_{3}h_{4})$ of the scattering process $h_{1}h_{2}\to h_{3}h_{4}$ by considering the effective Lagrangian approach. And then, the effective potential in momentum space $\mathcal{V}^{h_{1}h_{2}\to h_{3}h_{4}}(\bm{q})$ can be related to the scattering amplitude $\mathcal{M}(h_{1}h_{2}\to h_{3}h_{4})$ with the help of the Breit approximation Breit:1929zz ; Breit:1930zza and the nonrelativistic normalization, i.e.,

$\displaystyle\mathcal{V}_{E}^{h_{1}h_{2}\to h_{3}h_{4}}(\bm{q})$

$\displaystyle=$

$\displaystyle-\frac{\mathcal{M}(h_{1}h_{2}\to h_{3}h_{4})}{\sqrt{\prod_{i}2m_{i}\prod_{f}2m_{f}}},$

(2.24)

where $m_{i}$ and $m_{f}$ are the masses of the initial states $(h_{1},\,h_{2})$ and final states $(h_{3},\,h_{4})$, respectively. By performing the Fourier transformation, the effective potential in the coordinate space $\mathcal{V}^{h_{1}h_{2}\to h_{3}h_{4}}(\bm{r})$ can be deduced

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

In order to reflect the finite size effect of the discussed hadrons and compensate the off-shell effect of the exchanged light mesons Wang:2020dya , we need to introduce the monopole form factor $\mathcal{F}(q^{2},m_{E}^{2})=(\Lambda^{2}-m_{E}^{2})/(\Lambda^{2}-q^{2})$ in the interaction vertex Tornqvist:1993ng ; Tornqvist:1993vu . Here, $\Lambda$, $m_{E}$, and $q$ are the cutoff parameter, the mass, and the four momentum of the exchanged light meson, respectively.

In addition, we also need a series of normalization relations for the heavy hadrons $D_{s}$, $D_{s}^{*}$, $\Omega_{c}$, and $\Omega_{c}^{*}$, i.e.,

	$\displaystyle\langle 0|D_{s}|c\bar{s}\left(0^{-}\right)\rangle$	$\displaystyle=$	$\displaystyle\sqrt{M_{D_{s}}},$		(2.26)
	$\displaystyle\langle 0|D_{s}^{*\mu}|c\bar{s}\left(1^{-}\right)\rangle$	$\displaystyle=$	$\displaystyle\sqrt{M_{D_{s}^{*}}}\epsilon^{\mu},$		(2.27)
	$\displaystyle\langle 0|\Omega_{c}|css\left({1}/{2}^{+}\right)\rangle$	$\displaystyle=$	$\displaystyle\sqrt{2M_{\Omega_{c}}}{\left(\chi_{\frac{1}{2}m},\frac{\bf{\sigma}\cdot\bf{p}}{2M_{\Omega_{c}}}\chi_{\frac{1}{2}m}\right)^{T}},$		(2.28)
	$\displaystyle\langle 0|\Omega_{c}^{*\mu}|css\left({3}/{2}^{+}\right)\rangle$	$\displaystyle=$	$\displaystyle\sum_{m_{1},m_{2}}C_{1/2,m_{1};1,m_{2}}^{3/2,m_{1}+m_{2}}\sqrt{2M_{\Omega_{c}^{*}}}$		(2.29)
			$\displaystyle\times\left(\chi_{\frac{1}{2}m_{1}},\frac{\bf{\sigma}\cdot\bf{p}}{2M_{\Omega_{c}^{*}}}\chi_{\frac{1}{2}m_{1}}\right)^{T}\epsilon^{\mu}_{m_{2}}.$

With the above preparation, we can deduce the OBE effective potentials in the coordinate space for all of the investigated processes, which are collected in the A.

Now, we attempt to find the loosely bound state solutions of these discussed $\Omega_{c}^{(*)}\bar{D}_{s}^{(*)}$ systems by solving the coupled channel Schrödinger equation, i.e.,

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

(2.30)

with $\nabla^{2}=\frac{1}{r^{2}}\frac{\partial}{\partial r}r^{2}\frac{\partial}{\partial r}$, where $\mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}}$ is the reduced mass for the discussed system. The bound state solutions include the binding energy $E$, the root-mean-square radius $r_{\rm RMS}$, and the probability of the individual channel $P_{i}$, which provides us with valuable information to analyze whether the loosely bound state exists. In this work, we are interested in the $S$-wave $\Omega_{c}^{(*)}\bar{D}_{s}^{(*)}$ systems since there exists the repulsive centrifugal potential for the higher partial wave states $\ell\geqslant 1$.

In our calculation, the masses of these involved hadrons are $m_{f_{0}}=990.00$ MeV, $m_{\eta}=547.85$ MeV, $m_{\phi}=1019.46$ MeV, $m_{D_{s}}=1968.34$ MeV, $m_{D_{s}^{*}}=2112.20$ MeV, $m_{\Omega_{c}}=2695.20$ MeV, and $m_{\Omega_{c}^{*}}=2765.90$ MeV, which are taken from the Particle Data Group (PDG) Zyla:2020zbs . As the remaining phenomenological parameter, we take the cutoff value from 1.00 to 4.00 GeV. Usually, a loosely bound state with the cutoff parameter closed to 1.00 GeV can be suggested as the possible hadronic molecular candidate according to the experience of the deuteron Tornqvist:1993ng ; Tornqvist:1993vu ; Wang:2019nwt ; Chen:2017jjn . For an ideal hadronic molecular candidate, the reasonable binding energy should be at most tens of MeV, and the typical size should be larger than the size of all the included component hadrons Chen:2017xat .

In addition, the $S$-$D$ wave mixing effect is considered in this work, which plays an important role to modify the bound state properties of the deuteron Wang:2019nwt . The relevant channels $|{}^{2S+1}L_{J}\rangle$ are summarized in Table 1.

Before performing numerical calculation, we analyze the OBE effective potentials for these discussed $\Omega_{c}^{(*)}\bar{D}_{s}^{(*)}$ systems as below:

• •

  For the $\Omega_{c}\bar{D}_{s}$ and $\Omega_{c}^{*}\bar{D}_{s}$ systems, only the $f_{0}$ and $\phi$ exchange interactions are allowed. Meanwhile, the tensor force from the $S$-$D$ wave mixing effect disappears in the effective potentials, and thus the contribution of the $S$-$D$ wave mixing effect does not affect the bound state properties of the $\Omega_{c}\bar{D}_{s}$ and $\Omega_{c}^{*}\bar{D}_{s}$ systems.

• •

  For the $\Omega_{c}\bar{D}_{s}^{*}$ and $\Omega_{c}^{*}\bar{D}_{s}^{*}$ systems, in addition to the $f_{0}$ and $\phi$ exchange interactions, the $\eta$ exchange interaction and the $S$-$D$ wave mixing effect need to be taken into account.