Toward discovering the excited $\Omega$ baryons through nonleptonic weak decays of $\Omega_{c}$

A unique feature of $\Omega_{c}\to\Omega^{(*)}(X)\pi^{+}$ is that this decay proceeds only via external $W$-emission diagram Cheng:2021qpd , which is displayed in Fig. 1. We consider the simple quark-level transition $c\to su\bar{d}$, which is relevant for the Cabibbo-favored decay process of $\Omega_{c}\to\Omega(X)^{-}\pi^{+}$. The effective Hamiltonian for $c\to su\bar{d}$ can be given by Buchalla:1995vs

$\displaystyle H_{W}$

$\displaystyle=$

$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cs}^{*}V_{ud}(C_{1}\mathcal{O}_{1}+C_{2}\mathcal{O}_{2}),$

(1)

where $G_{F}=1.1663787\times 10^{-5}$ GeV^-2 is the Fermi constant ParticleDataGroup:2020ssz , and $C_{1}=1.26$ and $C_{2}=-0.51$ are the Wilson coefficients taken at the $m_{c}$ scale Buchalla:1995vs . The Cabibbo-Kobayashi-Maskawa matrix elements $V_{cs}=0.987$ and $V_{ud}=0.974$ are taken from the Review of Particle Physics (RPP) ParticleDataGroup:2020ssz , and the current-current operators are

	$\displaystyle\mathcal{O}_{1}$	$\displaystyle=$	$\displaystyle\bar{\psi}_{\bar{s}_{a}}\gamma_{\mu}(1-\gamma_{5})\psi_{c_{a}}\bar{\psi}_{\bar{u}_{b}}\gamma^{\mu}(1-\gamma_{5})\psi_{d_{b}},$		(2)
	$\displaystyle\mathcal{O}_{2}$	$\displaystyle=$	$\displaystyle\bar{\psi}_{\bar{s}_{a}}\gamma_{\mu}(1-\gamma_{5})\psi_{c_{b}}\bar{\psi}_{\bar{u}_{b}}\gamma^{\mu}(1-\gamma_{5})\psi_{d_{a}},$		(3)

with $\psi_{j_{\delta}}$ ($j=u/d/s/c$, $\delta=a/b$) representing the $j$th quark field in a meson or baryon, and $a$ and $b$ being color indices.

According to its parity behavior, $H_{W}$ can be separated into a parity-conserving part ($H_{W}^{PC}$) and a parity-violating part ($H_{W}^{PV}$) Niu:2020gjw ,

$\displaystyle H_{W}=H_{W}^{PC}+H_{W}^{PV}.$

(4)

With a non-relativistic expansion, the two operators can be approximately expressed as Niu:2020gjw

	$\displaystyle H_{W}^{PC}$	$\displaystyle\simeq$	$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cs}^{*}V_{ud}\frac{C_{i}\phi_{c}^{i}\gamma}{(2\pi)^{3}}\delta^{3}(\textbf{p}_{3}-\textbf{p}_{3}^{\prime}-\textbf{p}_{4}-\textbf{p}_{5})\{\langle s_{3}^{\prime}|I|s_{3}\rangle$		(5)
			$\displaystyle\langle s_{5}\bar{s}_{4}|\mbox{\boldmath$\sigma$\unboldmath}|0\rangle\left(\frac{\textbf{p}_{5}}{2m_{5}}+\frac{\textbf{p}_{4}}{2m_{4}}\right)-\Big{[}\left(\frac{\textbf{p}_{3}^{\prime}}{2m_{3}^{\prime}}+\frac{\textbf{p}_{3}}{2m_{3}}\right)\langle s_{3}^{\prime}|I|s_{3}\rangle$
			$\displaystyle-i\langle s_{3}^{\prime}|\mbox{\boldmath$\sigma$\unboldmath}|s_{3}\rangle\times\left(\frac{\textbf{p}_{3}}{2m_{3}}-\frac{\textbf{p}_{3}^{\prime}}{2m_{3}^{\prime}}\right)\Big{]}\langle s_{5}\bar{s}_{4}|\mbox{\boldmath$\sigma$\unboldmath}|0\rangle$
			$\displaystyle-\langle s_{3}^{\prime}|I|s_{3}\rangle\Big{[}\left(\frac{\textbf{p}_{5}}{2m_{5}}+\frac{\textbf{p}_{4}}{2m_{4}}\right)\langle s_{5}\bar{s}_{4}|\mbox{\boldmath$\sigma$\unboldmath}|0\rangle-i\langle s_{5}\bar{s}_{4}|\mbox{\boldmath$\sigma$\unboldmath}|0\rangle$
			$\displaystyle\times\left(\frac{\textbf{p}_{4}}{2m_{4}}-\frac{\textbf{p}_{5}}{2m_{5}}\right)\Big{]}+\langle s_{3}^{\prime}|I|s_{3}\rangle\left(\frac{\textbf{p}_{3}^{\prime}}{2m_{3}^{\prime}}+\frac{\textbf{p}_{3}}{2m_{3}}\right)$
			$\displaystyle\langle s_{5}\bar{s}_{4}|I|0\rangle\}\hat{\alpha}_{3}^{-},$
	$\displaystyle H_{W}^{PV}$	$\displaystyle\simeq$	$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cs}^{*}V_{ud}\frac{C_{i}\phi_{c}^{i}\gamma}{(2\pi)^{3}}\delta^{3}(\textbf{p}_{3}-\textbf{p}_{3}^{\prime}-\textbf{p}_{4}-\textbf{p}_{5})\{-\langle s_{3}^{\prime}|I|s_{3}\rangle$		(6)
			$\displaystyle\langle s_{5}\bar{s}_{4}|I|0\rangle-\langle s_{3}^{\prime}|\mbox{\boldmath$\sigma$\unboldmath}|s_{3}\rangle\langle s_{5}\bar{s}_{4}|\mbox{\boldmath$\sigma$\unboldmath}|0\rangle)\hat{\alpha}_{3}^{-}.$

In the above equations, $\textbf{p}_{j}$ and $m_{j}$ stand for the momentum and mass of the $j$th quark, respectively, as shown in Fig. 1. The $\phi_{c}^{i}$ ($i=1,2$ and $\phi_{c}^{1}=1$, $\phi_{c}^{2}=\frac{1}{3}$) are color factors, $I$ is the dimension-two unit matrix, and $\hat{\alpha}_{3}^{-}$ is the flavor operator which transforms $c$ quark to $s$ quark. The $s_{j}$ and $\bar{s}_{4}$ stand for the spin of the $j$th quark and the fourth antiquark, respectively. $\gamma$ is a symmetry factor and equals to one for a direct pion emission process considering in present work.

In order to evaluate the spin matrix element $\langle s_{5}\bar{s}_{4}|I|0\rangle$ and $\langle s_{5}\bar{s}_{4}|\mbox{\boldmath$\sigma$\unboldmath}|0\rangle$ including an antiquark, the particle-hole conjugation Racah:1942gsc should be employed. Within the particle-hole conjugation relation:

$\displaystyle|j,-m\rangle\rightarrow(-1)^{j+m}|j,m\rangle\,,$

(7)

the antiquark spin transforms as follows: $|\bar{\uparrow}\rangle\rightarrow|\downarrow\rangle$ and $|\bar{\downarrow}\rangle\rightarrow-|\uparrow\rangle$. For instance:

	$\displaystyle\langle\frac{1}{\sqrt{2}}(\uparrow_{5}\bar{\downarrow}_{4}-\downarrow_{5}\bar{\uparrow}_{4})|I|0\rangle=\frac{1}{\sqrt{2}}\left(\langle\uparrow_{5}|I|-\uparrow_{4}\rangle-\langle\downarrow_{5}|I|\downarrow_{4}\rangle\right)$
	$\displaystyle=-\sqrt{2}.$		(8)

For a given decay process $A\to BC$, the transition amplitude $\mathcal{M}$ is calculated by

	$\displaystyle\mathcal{M}_{J_{f},J_{f}^{z};J_{i},J_{i}^{z}}$	$\displaystyle=$	$\displaystyle\langle C(\textbf{P}_{f};J_{f},J_{f}^{z})B(\textbf{q})|H_{W}|A(\textbf{P}_{i};J_{i},J_{i}^{z})\rangle,$		(9)
		$\displaystyle=$	$\displaystyle\langle C(\textbf{P}_{f};J_{f},J_{f}^{z})B(\textbf{q})|H_{W}^{PC}|A(\textbf{P}_{i};J_{i},J_{i}^{z})\rangle$
			$\displaystyle+\langle C(\textbf{P}_{f};J_{f},J_{f}^{z})B(\textbf{q})|H_{W}^{PV}|A(\textbf{P}_{i};J_{i},J_{i}^{z})\rangle.$
		$\displaystyle=$	$\displaystyle\mathcal{M}_{J_{f}^{z},J_{i}^{z}}^{PC}+\mathcal{M}_{J_{f}^{z},J_{i}^{z}}^{PV},$

where $A(\textbf{P}_{i};J_{i},J_{i}^{z})$, $B(\textbf{q})$ and $C(\textbf{P}_{f};J_{f},J_{f}^{z})$ stands for the wave functions of the initial baryon $A$, final meson $B$ and final baryon $C$, respectively. $(\mathbf{P}_{i},\mathbf{P}_{f})$, $(J_{i},J_{f})$, and $(J_{i}^{z},J_{f}^{z})$ are the momentum, the total angular momentum and the third component of the total angular momentum of the initial baryon $A$ and the final baryon $C$, respectively. q is the three-momentum of the final state meson in the initial state rest frame.

Then, the partial decay width for a given decay process $A\to BC$ can be expressed as

$$\Gamma=\frac{\Phi(ABC)}{2J_{A}+1}\sum_{\texttt{spins}}|\mathcal{M}|^{2},$$

(10)

where $\Phi(ABC)$ is the phase-space factor for the decay.

The choice of phase space is not clear. For the phase space factor $\Phi(ABC)$, there are three typical options adopted in the literature Kokoski:1985is ; Kumano:1988ga ; Geiger:1994kr ; Capstick:2000qj . The usual option is the relativistic phase-space factor (RPF)

$$\Phi(ABC)=8\pi^{2}\frac{|\textbf{q}|E_{B}E_{C}}{M_{A}},$$

(11)

where $M_{A}$ is the mass of the initial hadron $A$, while $E_{B}$ and $E_{C}$ stand for the energies of final hadrons $B$ and $C$, respectively.

To match the transition matrix element calculated non-relativistically, a fully nonrelativistic phase-space factor (NRPF) is used, that is

$$\Phi(ABC)=8\pi^{2}\frac{|\textbf{q}|M_{B}M_{C}}{M_{A}},$$

(12)

where $M_{B}$ and $M_{C}$ is the mass of the final hadron $B$ and $C$, respectively.

However, in many cases the momenta of the final hadrons are quite large so that the relativistic phase space is significantly different from the nonrelativistic limit. In Ref. Kokoski:1985is , Kokoski and Isgur suggested a “mock-hadron” phase-space factor (MHPF),

$$\Phi(ABC)=8\pi^{2}\frac{|\textbf{q}|\tilde{M}_{B}\tilde{M}_{C}}{\tilde{M}_{A}},$$

(13)

in their calculation of meson decay widths. The $\tilde{M}_{A}$, $\tilde{M}_{B}$ and $\tilde{M}_{C}$ are effective hadron masses of hadron $A$, $B$ and $C$, respectively. They are evaluated with a spin-independent inter-quark interaction. In the weak-binding limit, the mass of $\pi$ meson is degenerate with that of $\rho$ meson.

To work out the decay amplitude $\mathcal{M}$, we need the wave functions of the initial and final states. Here, the initial state is the ground $\Omega_{c}$ baryon, the final states are the $\pi^{+}$ meson and the $\Omega^{(*)}(X)$ states. These wave functions are constructed within the non-relativistic constituent quark model. For simplicity, the spatial wave functions of the baryons and mesons are adopted the harmonic oscillator form in our calculations.

The spatial wave function for a baryon with principal quantum number $N$, total orbital angular momentum quantum numbers $L$, and $M_{L}$ is a product of the $\rho$-oscillator part and the $\lambda$-oscillator part. In momentum space, the baryon spatial wave function is given by Liu:2019wdr

$\displaystyle\Psi^{\sigma}_{NLM_{L}}(\textbf{p}_{\rho},\textbf{p}_{\lambda})\!=\!\!\!\sum_{N,M_{L}}C^{n_{\rho}l_{\rho}m_{\rho}}_{n_{\lambda}l_{\lambda}m_{\lambda}}\left[\psi_{n_{\rho}l_{\rho}m_{\rho}}(\textbf{p}_{\rho})\psi_{n_{\lambda}l_{\lambda}m_{\lambda}}(\textbf{p}_{\lambda})\right]^{\sigma}_{NLM_{L}},$

(14)

with $N=2(n_{\rho}+n_{\lambda})+l_{\rho}+l_{\lambda}$, $M_{L}=m_{\rho}+m_{\lambda}$, and

	$\displaystyle\psi^{\alpha}_{nlm}(\textbf{p})=(i)^{l}(-1)^{n}\left[\frac{2n!}{(n+l+1/2)!}\right]^{1/2}\frac{1}{\alpha^{l+3/2}}$
	$\displaystyle\mathrm{exp}\left(-\frac{\textbf{p}^{2}}{2\alpha^{2}}\right)L_{n}^{l+1/2}(\textbf{p}^{2}/\alpha^{2})\mathcal{Y}_{lm}(\textbf{p}).$		(15)

Here, $\mathcal{Y}_{lm}(\textbf{p})=|\textbf{p}|^{l}Y_{lm}(\mathbf{\hat{p}})$ is $l$th solid harmonic polynomial. $\textbf{p}_{\rho}$ and $\textbf{p}_{\lambda}$ are the internal momenta of the $\rho$- and $\lambda$-oscillator wave functions, respectively. They can be expressed as functions of the quark momenta $\mathbf{p}_{j}$ ($j=1,2,3$):

	$\displaystyle\mathbf{p}_{\rho}$	$\displaystyle=$	$\displaystyle\frac{\sqrt{2}}{2}(\textbf{p}_{1}-\textbf{p}_{2}),$		(16)
	$\displaystyle\mathbf{p}_{\lambda}$	$\displaystyle=$	$\displaystyle\frac{\sqrt{6}}{2}\frac{m_{3}(\textbf{p}_{1}+\textbf{p}_{2})-(m_{1}+m_{2})\textbf{p}_{3}}{m_{1}+m_{2}+m_{3}}.$		(17)

The $n_{\rho}$ and $n_{\lambda}$ are the principal quantum numbers of the $\rho$- and $\lambda$-mode oscillators, respectively. $(l_{\rho},m_{\rho})$ and $(l_{\lambda},m_{\lambda})$ are the orbital angular momentum quantum numbers of the $\rho$- and $\lambda$-mode oscillators, respectively. $\sigma=s,\rho,\lambda,a,...$ stand for different excitation modes with different permutation symmetries. $\alpha_{\rho}$ and $\alpha_{\lambda}$ are two oscillator parameters. For the $\Omega$ baryons, we have $\alpha_{\rho}=\alpha_{\lambda}$, while for the charmed $\Omega_{c}$ baryons, we have

$\displaystyle\alpha_{\lambda}=\left(\frac{3m_{c}}{2m_{s}+m_{c}}\right)^{1/4}\alpha_{\rho},$

(18)

where $m_{s}$ and $m_{c}$ stand for the masses of the strange and charmed quarks, respectively. The flavor and spin wave functions of the $\Omega_{c}$ and $\Omega$ baryons have been given in our previous works Xiao:2013xi ; Wang:2017kfr . The product of spin, flavor, and spatial wave functions of the heavy baryons must be symmetric since the color wave function is antisymmetric. The details about the quark model classifications for the $\Omega_{c}$ spectrum can be found in the works of Wang:2017kfr ; Yao:2018jmc ; Zhong:2007gp , while for the $\Omega$ baryon spectrum can be found in Refs. Xiao:2013xi ; Liu:2019wdr .

Finally, the wave function of the $\pi^{+}$ meson is constructed by

$\displaystyle\varphi(\textbf{p}_{4},\textbf{p}_{5})=\phi_{\pi^{+}}\chi^{a}\psi(\textbf{p}_{4},\textbf{p}_{5}),$

(19)

where the spin wave function $\chi^{a}$ is

$\displaystyle\chi^{a}=\frac{1}{\sqrt{2}}(\uparrow\downarrow-\downarrow\uparrow),$

(20)

and the flavor wave function $\phi_{\pi^{+}}$ is

$\displaystyle\phi_{\pi^{+}}=u\bar{d}.$

(21)

The spatial wave function in the momentum space is adopted the simple harmonic oscillator form

$\displaystyle\psi(\textbf{p}_{4},\textbf{p}_{5})=\frac{1}{\pi^{3/4}\beta^{3/2}}\mathrm{exp}\left[-\frac{(\textbf{p}_{4}-\textbf{p}_{5})^{2}}{8\beta^{2}}\right],$

(22)

where $\beta$ is a size parameter of the meson wave function. The $\textbf{p}_{4}$ and $\textbf{p}_{5}$ stand for the quark momenta of the $\pi^{+}$ meson as shown in Fig. 1.

For self consistency, the quark model parameters are taken the same as those adopted in our previous work Wang:2017kfr . The constituent masses for the $u/d$, $s$ and $c$ quarks are taken to be $m_{u/d}=330$ MeV, $m_{s}=450$ MeV and $m_{c}=1480$ MeV, respectively. For the initial state $\Omega_{c}$, the harmonic oscillator parameter $\alpha_{\rho}$ is taken to be $\alpha_{\rho}=440$ MeV, the other harmonic oscillator parameter $\alpha_{\lambda}$ is related to $\alpha_{\rho}$ by $\alpha_{\lambda}=[3m_{c}/(2m_{s}+m_{c})]^{1/4}\alpha_{\rho}$. For the final state $\Omega^{(*)}(X)$, a unified harmonic oscillator parameter is adopted, i.e., $\alpha_{\lambda}=\alpha_{\rho}=440$ MeV. For the $\pi^{+}$ meson, the size parameter is taken to be $\beta=280$ MeV as that adopted in Ref. Niu:2020gjw . The masses for the $\pi^{+}$, $\Omega$ and $\Omega_{c}$ are taken the RPP average values 140 MeV, 1672 MeV and 2695 MeV, respectively ParticleDataGroup:2020ssz . In the MHPF defined in Eq. (13), we need determine the effective masses of the mock hadrons. For the process $\Omega_{c}\to\Omega^{(*)}\pi$, we adopt $\tilde{M}_{\pi}=0.72$ GeV, consistent with Kokoski and Isgur Kokoski:1985is , and $\tilde{M}_{\Omega_{c}}=M_{\Omega_{c}}$ and $\tilde{M}_{\Omega_{c}^{(*)}}=M_{\Omega_{c}^{(*)}}$.

First, we study the weak decay process $\Omega_{c}^{0}\rightarrow\Omega^{-}\pi^{+}$. This weak decay process, as an important process, has been widely studied by the Belle, BaBar, CLEO, SELEX, FOCUS collaborations Belle:2017szm ; BaBar:2007jdg ; CLEO:2000dhf ; FOCUS:2003ylx ; Solovieva:2008fw ; SELEX:2007bim ; BaBar:2005yiy . With the RPF, the partial decay width of $\Omega_{c}^{0}\rightarrow\Omega^{-}\pi^{+}$ is predicted to be

$\displaystyle\Gamma[\Omega_{c}^{0}\rightarrow\Omega^{-}\pi^{+}]$

$\displaystyle\simeq$

$\displaystyle 2.6\times 10^{-14}~{}\text{GeV}.$

(23)

By using the measured lifetime $\tau=2.68\times 10^{-13}s$ of $\Omega_{c}^{0}$ ParticleDataGroup:2020ssz , we further predict the branching fraction

$\displaystyle\mathcal{B}[\Omega_{c}^{0}\rightarrow\Omega^{-}\pi^{+}]\simeq 1.05\%.$

(24)

If adopting the MHPF, there is a $\sim 20\%$ correction to the results of RMF. However, when adopting the NRPF the results are about a factor of $\sim 6.6$ smaller than that predicted with RMF. From Table 2, it is found that our predicted branching fraction with both RPF and MHPF is close to the predictions in Refs. Cheng:1996cs ; Hsiao:2020gtc . While, if adopting the NRPF, our predicted branching fraction $\mathcal{B}[\Omega_{c}^{0}\rightarrow\Omega^{-}\pi^{+}]\simeq 0.16\%$ is consistent with that from the covariant confined quark model Gutsche:2018utw .

Furthermore, combined the predicted branching fraction of $\mathcal{B}[\Omega_{c}^{0}\rightarrow\Omega^{-}\pi^{+}]$ with the measured relative branching ratios $\frac{\Gamma[\Omega_{c}^{0}\rightarrow\Xi^{0}\bar{K}^{-}\pi^{+}]}{\Gamma[\Omega_{c}^{0}\rightarrow\Omega^{-}\pi^{+}]}=1.20\pm 0.24$ and $\frac{\Gamma[\Omega_{c}^{0}\rightarrow\Xi^{-}\bar{K}^{0}\pi^{+}]}{\Gamma[\Omega_{c}^{0}\rightarrow\Omega^{-}\pi^{+}]}=2.12\pm 0.38$, the branching fractions for the three-body weak decay processes $\Omega_{c}^{0}\rightarrow\Xi^{0}\bar{K}^{-}\pi^{+}/\Xi^{-}\bar{K}^{0}\pi^{+}$ can be obtained easily. With the RPF, we have

	$\displaystyle\mathcal{B}[\Omega_{c}^{0}\rightarrow\Xi^{0}\bar{K}^{-}\pi^{+}]$	$\displaystyle\simeq$	$\displaystyle(1.26\pm 0.27)\times 10^{-2},$		(25)
	$\displaystyle\mathcal{B}[\Omega_{c}^{0}\rightarrow\Xi^{-}\bar{K}^{0}\pi^{+}]$	$\displaystyle\simeq$	$\displaystyle(2.23\pm 0.43)\times 10^{-2}.$		(26)

While when adopting the NRPF, we have small branching fractions

	$\displaystyle\mathcal{B}[\Omega_{c}^{0}\rightarrow\Xi^{0}\bar{K}^{-}\pi^{+}]$	$\displaystyle\simeq$	$\displaystyle(0.19\pm 0.04)\times 10^{-2},$		(27)
	$\displaystyle\mathcal{B}[\Omega_{c}^{0}\rightarrow\Xi^{-}\bar{K}^{0}\pi^{+}]$	$\displaystyle\simeq$	$\displaystyle(0.33\pm 0.06)\times 10^{-2},$		(28)

due to the small nonrelativistic phase space factor.

In the $\Omega$ family, there are two $1P$-wave states $\Omega(1^{2}P_{1/2^{-}})$ and $\Omega(1^{2}P_{3/2^{-}})$ with spin-parity $J^{P}=1/2^{-}$ and $J^{P}=3/2^{-}$, respectively. The newly observed $\Omega(2012)$ resonance may favor the assignment of $\Omega(1^{2}P_{3/2^{-}})$ state, since both the measured mass and width are consistent with the quark model predictions Xiao:2018pwe ; Liu:2019wdr ; Aliev:2018yjo ; Aliev:2018syi ; Polyakov:2018mow . The masses of the unestablished $\Omega^{*}(X)$ states are taken the predictions in Ref. Liu:2019wdr , which have been collected in Table 1. However, the $\Omega(1^{2}P_{1/2^{-}})$ classified in the quark model is still missing.

Considering $\Omega(2012)$ as the $\Omega(1^{2}P_{3/2^{-}})$ assignment, we have studied the $\Omega_{c}^{0}\to\Omega^{-}(2012)\pi^{+}$ process, the results are listed in Table 3. It is found that the $\Omega_{c}$ baryon has a fairly large decay rate into $\Omega(2012)^{-}\pi^{+}$, with the RPF or MHPF the branching fraction is predicted to be

$\displaystyle\mathcal{B}[\Omega_{c}^{0}\rightarrow\Omega(2012)^{-}\pi^{+}]$

$\displaystyle\simeq$

$\displaystyle 2.2\times 10^{-3}.$

(29)

Combining it with the branching fraction of $\mathcal{B}[\Omega_{c}^{0}\rightarrow\Omega^{-}\pi^{+}]$ obtained in Eq.(24), we predict the relative ratio

$\displaystyle R^{{\rm Th.}}_{1}=\frac{\mathcal{B}[\Omega_{c}^{0}\rightarrow\Omega(2012)^{-}\pi^{+}]}{\mathcal{B}[\Omega_{c}^{0}\rightarrow\Omega^{-}\pi^{+}]}\simeq 0.22,$

(30)

which is in good agreement with experimental value $R^{\rm Exp.}_{1}=0.220\pm 0.059(\mathrm{stat.})\pm 0.035(\mathrm{syst.})$ that was recently measured by the Belle Collaboration Belle:2021gtf . According to the strong decay properties of $\Omega(2012)$ predicted using the constituent quark model in Refs. Liu:2019wdr ; Xiao:2018pwe , branching fractions of $\Omega(2012)$ decaying into $\Xi^{0}K^{-}$ and $\Xi^{-}\bar{K}^{0}$ are predicted to be $\mathcal{B}[\Omega_{c}(2012)\to\Xi^{0}K^{-}]\simeq 52\%$ and $\mathcal{B}[\Omega_{c}(2012)\to\Xi^{-}\bar{K}^{0}]\simeq 48\%$, respectively. Combining these strong branching fractions of $\Omega(2012)$ with our predicted branching fractions for the weak decay processes $\mathcal{B}[\Omega_{c}^{0}\to\Xi^{0}\bar{K}^{-}\pi^{+}/\Xi^{-}\bar{K}^{0}\pi^{+}/\Omega(2012)^{-}\pi^{+}$] in Eqs. (27)-(29), one can obtain

	$\displaystyle R^{\rm Th.}_{2}$	$\displaystyle=$	$\displaystyle\frac{\mathcal{B}[\Omega_{c}^{0}\rightarrow\Omega(2012)\pi^{+}]\mathcal{B}[\Omega_{c}(2012)\rightarrow\Xi^{0}K^{-}]}{\mathcal{B}[\Omega_{c}^{0}\rightarrow\Xi^{0}\bar{K}^{-}\pi^{+}]}$		(31)
			$\displaystyle\simeq 0.09,$
	$\displaystyle R^{\rm Th.}_{3}$	$\displaystyle=$	$\displaystyle\frac{\mathcal{B}[\Omega_{c}^{0}\rightarrow\Omega(2012)\pi^{+}]\mathcal{B}[\Omega_{c}(2012)\rightarrow\Xi^{-}\bar{K}^{0}]}{\mathcal{B}[\Omega_{c}^{0}\rightarrow\Xi^{-}\bar{K}^{0}\pi^{+}]}$		(32)
			$\displaystyle\simeq 0.05,$

which are also consistent with the experimental values $R_{2}^{\rm Exp.}=0.096\pm 0.032(\mathrm{stat.})\pm 0.018(\mathrm{syst.})$ and $R_{3}^{\rm Exp.}=0.055\pm 0.028(\mathrm{stat.})\pm 0.007(\mathrm{syst.})$ recently measured by the Belle Collaboration Belle:2021gtf , respectively. It should be mentioned that these predicted relative ratios $R^{{\rm Th.}}_{i}$ ($i=1,2,3$) are nearly independent on the options of phase space factor in the calculations.