Theoretical study of the $\Omega(2012)$ state in the $\Omega_c^0 \to \pi^+ \Omega(2012)^- \to \pi^+ (\bar{K}\Xi)^-$ and $\pi^+ (\bar{K}\Xi\pi)^-$ decays

We report on a theoretical study of the newly observed $\Omega(2012)$ resonance in the nonleptonic weak decays of $\Omega_c^0 \to \pi^+ \bar{K}\Xi^*(1530) (\eta \Omega) \to \pi^+ (\bar{K}\Xi)^-$ and $\pi^+ (\bar{K}\Xi\pi)^-$ via final-state interactions of the $\bar{K}\Xi^*(1530)$ and $\eta \Omega$ pairs. The weak interaction part is assumed to be dominated by the charm quark decay process: $c(ss) \to (s + u + \bar{d})(ss)$, while the hadronization part takes place between the $sss$ cluster from the weak decay and a quark-antiquark pair with the quantum numbers $J^{PC} = 0^{++}$ of the vacuum, produces a pair of $\bar{K}\Xi^*(1530)$ and $\eta \Omega$. Accordingly, the final $\bar{K}\Xi^*(1530)$ and $\eta \Omega$ states are in pure isospin $I= 0$ combinations, and the $\Omega_c^0 \to \pi^+ \bar{K}\Xi^*(1530)(\eta \Omega) \to \pi^+ (\bar{K}\Xi)^-$ decay is an ideal process to study the $\Omega(2012)$ resonance. With the final-state interaction described in the chiral unitary approach, up to an arbitrary normalization, the invariant mass distributions of the final state are calculated, assuming that the $\Omega(2012)$ resonance with spin-parity $J^P = 3/2^-$ is a dynamically generated state from the coupled channels interactions of the $\bar{K}\Xi^*(1530)$ and $\eta \Omega$ in $s$-wave and $\bar{K}\Xi$ in $d$-wave. We also calculate the ratio, $R^{\bar{K}\Xi\pi}_{\bar{K}\Xi} = {\rm Br}[\Omega_c^0 \to \pi^+ \Omega(2012)^- \to \pi^+ (\bar{K}\Xi \pi)^-] / {\rm Br}[\Omega_c^0 \to \pi^+ \Omega(2012)^- \to \pi^+ (\bar{K}\Xi)^-$]. The proposed mechanism can provide valuable information on the nature of the $\Omega(2012)$ and can in principle be tested by future experiments.


I. INTRODUCTION
The study of baryon spectroscopy is one of the most important issues in hadron physics and is an essential tool to analyze baryon structure. Data on baryon masses and decay modes are compiled in the Particle Data Book Review [1]. For the light flavor hyperons with strangeness −3, not much is known about their properties [1]. In 2018, the Belle collaboration reported an Ω exited state, Ω(2012), in the K − Ξ 0 and K 0 S Ξ − invariant mass distributions [2], with the measured mass M = 2012.4 ± 0.7 ± 0.6 MeV and width Γ = 6.4 +2.5 −2.0 ± 1.6 MeV. The Ω(2012) is the first Ω excited state with preferred negative parity [2] and it is a PDG three-star state. Before this new observation, there is only one three-star Ω resonance, Ω(2250), with its spin-parity unknown. Further investigations about the Ω excited states are mostly welcome.
On the theoretical side, there exist several quark model studies of the Ω excited states before the Belle observation. A pioneer work was done in Ref. [3], where an Ω resonance was predicted with mass about 2020 MeV and spin-parity J P = 3/2 − . The excitation spectrum for multistrange baryons were investigated in Ref. [4] in a constituent quark model, where a 3/2 − Ω excited state was obtained with mass about 1953 MeV. Within the extended chiral quark model, theKΞ and ωΩ interactions were studied in Refs. [5,6], in which, the Ω excited states in theKΞ system with J P = 1/2 − and ωΩ system with J P = 3/2 − or 5/2 − , were obtained. These Ω excited states with negative parity have also been studied by using an extended quark model in the five quark picture [7][8][9]. It was found that the lowest 3/2 − state has a mass around 1785 ± 25 MeV, which is lower than the one of the lowest 1/2 − state [9]. This indicates that fivequark components are dominant in the wave functions of those Ω resonances with lower masses [8,9].
After its discovery the Ω(2012) resonance was studied in the framework of QCD sum rules in Refs. [10,11], where the Ω(2012) can be interpreted as a 1P orbital excitation of the ground Ω baryon with J P = 3/2 − . The two body strong decays of Ω(2012) resonance were also studied in Refs. [12][13][14], within a non-relativistic constituent quark potential model, in which it was found that the strong decay of Ω(2012) is predominated byKΞ mode.
Furthermore, the topic of hadronic molecular states, with mesons and baryons bound by strong interactions in s-wave, has been well developed by the combination of the chiral Lagrangians with nonperturbative unitary techniques in coupled channels, which has been a very fruitful scheme to study the nature of many baryon resonances [15][16][17][18]. The analysis of meson-baryon scattering amplitudes shows poles, which can be identified with existing baryon resonances or new ones. In this way the Ω resonances are dynamically generated [19][20][21][22] from the coupled channels interactions of theKΞ * (1530) and ηΩ in s-wave.
Indeed, there is growing evidence that the newly observed Ω(2012) can be interpreted as a hadronic molecular state, with J P = 3/2 − , as discussed in Refs. [23][24][25][26][27][28]. However, the large decay width for Ω(2012) → KΞ * →KπΞ, predicted by the molecular nature [23][24][25], is in disagreement with the very recent Belle measurement [29]. The measured ratio of the three body decay width to the one of the two body decay, RK πΞ KΞ = Γ Ω(2012)→KπΞ /Γ Ω(2012)→KΞ , is less than 11.9% at the 90% confidence level [29]. Based on this recent measurement, Refs. [30,31] claimed a reasonable reproduction of the experimental data of the Belle collaboration [2,29], and concluded that the experimental data on the Ω(2012) are compatible with the molecular picture and the theoretical results are rather stable with different sets of model parameters of natural size.
The paper is organized as follows. In the next section, we present the formalism and ingredients of the decay amplitudes of the three and four-body decays of Ω 0 c . Numerical results are given in Section III, followed by a short summary in the last section.

II. FORMALISM AND INGREDIENTS
Following Refs. [34,35], the Cabibbo favored process of Ω 0 c into a π + plus a pair of ground state pseudoscalar mesons and decuplet baryons (MB) is as follows: in the first step the charmed quark in Ω 0 c turns into a strange quark with a π + (ud pair) by the weak decay as shown in Fig. 1. Then the sss cluster hadronizes with a new qq pair with the quantum numbers J P C = 0 ++ of the vacuum, to form a pair of meson and baryon. Finally, the final-state interactions of the MB leads to the dynamical generation of the Ω(2012). 1 It is worth to mention that, in this work, the Ω(2012) is a dynamically generated state with J P = 3/2 − from the coupled channels interactions of theKΞ * (1530) and ηΩ in s-wave and KΞ in d-wave [30,31].ū c → π + MB decay.

A. Decay amplitudes
As in Refs. [34,35], one can easily obtain the final meson-baryon states as |MB =|s(ūu +dd + ss)ss where the last step is obtained in the isospin basis using the convention of Ref. [42]: |K − = −| 1 2 − 1 2 , and the flavor states of the baryons and η meson are as follows [23,34]: After the production ofKΞ * (1530) and ηΩ pairs, the final-state interactions between the mesons (K, η) and the baryons [Ξ * (1530), Ω] take place, which can be parameterized by the re-scattering shown in Fig. 2 at the hadronic level for the production of Ω(2012), and then it decays intoKΞ in d-wave.
According to Eq. (1), we can write down the Ω 0 c → π +K Ξ decay amplitude of Fig. 2 as, with MK Ξ the invariant mass ofKΞ. Similarly, one can obtain the decay amplitude for Ω 0 c → π +K Ξ * (1530) as, where the factor V P is assumed to be constant in the relevant energy region [34,43,44], and its actual value should be determined from the experimental measurements for a certain decay channel. The loop functions GK Ξ * and G ηΩ depend on the invariant mass, M inv , of the finalKΞ orKΞ * (1530) system. The two body scattering amplitudes tK Ξ * →KΞ(Ξ * ) and t ηΩ→KΞ(Ξ * ) are those obtained in the chiral unitary approach, which depend also on M inv , and we take them as, where these coupling constants, the mass and width of the Ω(2012) are obtained in Ref. [30] with different sets of parameters as shown in Table I  The decay amplitude of Ω 0 c → π +K Ξπ decay can be written as where M Ξπ is the invariant mass of Ξπ system, and Γ Ξ * is energy dependent, and its explicit form is given by 2 wherē On the other hand, the coupling constant g Ξ * Ξπ = 4.4 × 10 −3 MeV −1 can be easily obtained from Eq. (13) with the values of m π = 138.04, M Ξ * = 1533.4, M Ξ = 1318.29 MeV, and Γ on Ξ * = 9.5 MeV.

III. NUMERICAL RESULTS
In this section, we show our theoretical predictions for the production of Ω(2012) in different final states in the Ω 0 c decay. Note that the physical masses of the involved particles are taken from PDG [1] and we take the isospin averaged values for m K = 495.64, m η = 547.86, 3 Since the transition of Ω 0 c (1/2 + ) → π + (0 − )Ω(2012) − (3/2 − ) is weak decay, in general, the d-wave term also gives contribution, which is neglected in this work, because it is suppressed at the low energy region.
m Ω = 1672.45, and M Ω 0 c = 2695.2 MeV. In addition, the following numerical results are obtained with V P = 1.
We first show in Figs. 4 and 5 the theoretical predictions for the invariant mass distribution dΓ/dMK Ξ and dΓ/dMK Ξ * (1530) . From Fig. 4 one can see clearly the shape of the Ω(2012), while from the invariant mass distribution ofKΞ * (1530) as shown in Fig. 5, there is no signal of the Ω(2012) resonance, this is because its mass is below theKΞ * mass threshold, and its width is narrow. The interference between the tree level contribution and the final state interactions ofKΞ * (1530) and ηΩ makes the production of Ω(2012) in theKΞ * (1530) channel even worse. In other words, the first term in Eq. (7) contributes at the tree level to the Ω 0 c → π +K Ξ * (1530) decay, but does not contribute to the production of Ω(2012), followed by decaying intoKΞ * (1530).   Next we consider the Ω 0 c → π +K Ξ * (1530) → π +K Ξπ decay. The invariant mass distribution dΓ/dMK Ξπ is shown in Fig. 6, where one can see that the tree level contributions provide a very big background, and, hence, the signal of the Ω(2012) is rather weak. However, the Ω(2012) may give significant contribution to the invariant mass distributions ofKΞπ close to threshold, as shown in the sub-figure of Fig. 6, especially for the Models I, II and III. In general we cannot fix the value of V P , which should be determined by experimental measurements. Therefore, it would be interesting to remove the uncertainties arising from the introduction of V P by investigating the ratios between different partial decay widths, where the effect of the V P factor is canceled, and which only reflect the production of Ω(2012) through the final state interactions ofKΞ * (1530) and ηΩ. For such a purpose, we define As discussed before, the main contribution to the Ω 0 c → π +K Ξπ decay comes from the tree level of the production ofKΞ * (1530). In order to consider the contribution from the Ω(2012) resonance, we need to switch off the first term in Eq. (7), which means that we have to change the M Ω 0 c →πKΞ * in the following form Furthermore, as shown in Eq. (17), to get the partial decay widths of Γ[Ω 0 c → π + Ω(2012) − → π +K Ξπ] and Γ[Ω 0 c → π + Ω(2012) − → π +K Ξπ], we have integrated dMK Ξπ and dMK Ξ over the range of [M Ω * −2Γ Ω * , M Ω * + 2Γ Ω * ], in which the contributions from Ω(2012) resonance are covered.
The numerical results for the ratio RK Ξπ KΞ are listed in Table II. On can see that these predictions are in agreement with the measurements reported by the Belle collaboration [29], as expected. This ratio is relevant because it is obtained with no free parameters (all the model parameters are fixed by previous works) and, thus, it is a prediction of the model. We expect that these numerical results could be tested by future experimental measurements.

IV. SUMMARY
In summary, we have investigated the nonleptonic weak decays of Ω 0 c → π +K Ξ * (1530)(ηΩ) → π + (KΞ) − as a tool to study the newly observed Ω(2012) resonance via the final-state interactions of theKΞ * (1530) and ηΩ pairs. We assume that the weak interaction part is dominated by the Cabibbo favored charm quark decay process: c(ss) → (s + u +d)(ss), then the sss cluster and a quark-antiquark pair from the vacuum hadronize into the intermediate meson-baryon states, in this case, thē KΞ * (1530) and ηΩ. Accordingly, the finalKΞ * (1530) and ηΩ states are in pure isospin I = 0 combinations, and the final state interaction ofKΞ * (1530) and ηΩ can produce the Ω(2012) state by using the chiral unitary approach. After the Ω(2012) is dynamically generated in the above process, it will decay intoKΞ andKΞπ, and shows a peak or bump structure in theKΞ andKΞπ invariant mass distributions. Thus, we have calculated the invariant mass distributions of dΓ Ω 0 c →π +K Ξ /dMK Ξ and dΓ Ω 0 c →π +K Ξπ /dMK Ξπ . We have seen that the Ω 0 c → π +K Ξπ decay is not well suited to study the Ω(2012) resonance because the dominant contribution is from the Ω 0 c → π +K Ξ * (1530) decay at tree level, which will not contribute to the production of Ω(2012). However, the Ω(2012) peak can be clearly seen in theKΞ invariant mass distribution of the Ω 0 c → π +K Ξ decay. We have also calculated the ratio, RK Ξπ KΞ = Br[Ω 0 c → π + Ω(2012) − → π + (KΞπ) − ]/Br[Ω 0 c → π + Ω(2012) − → π + (KΞ) − ]. The numerical results are in agreement with the Belle measurements [29]. The good agreement with experimental data of the chiral unitary approach as shown by us here, provides extra support to the picture of the Ω(2012) as a dynamically generated resonance.
Finally, we would like to stress that the predictions here are very qualitative, since the contributions from other resonances are neglected. We hope that the theoretical calculations presented in this work may stimulate experimental interest in exploring the Ω(2012) resonance or other Ω excited states through the Ω 0 c decays. The proposed mechanism can provide valuable information on the nature of the Ω(2012) and can in principle be tested by future experiments.