Strong decay modes $\bar{K}\Xi$ and $\bar{K}\Xi\pi$ of the $\Omega(2012)$ in the $\bar{K}\Xi(1530)$ and $\eta\Omega$ molecular scenario

We study the $\bar{K} \Xi$ decay mode of the newly observed $\Omega(2012)$ assuming that the $\Omega(2012)$ is a dynamically generated state with spin-parity $J^P = 3/2^-$ from the coupled channel $S$-wave interactions of $\bar{K}\Xi(1530)$ and $\eta \Omega$. In addition we also calculate its $K\pi\Xi$ three-body decay mode. It is shown that the so-obtained total decay width is in fair agreement with the experimental data. We compare our results with those of other recent studies and highlight differences among them.

After the observation of the Ω(2012), its two-body strong decays were studied within the chiral quark model in Ref. [14], where it was shown that the newly observed Ω(2012) could be assigned to the J P = 3/2 − three quark state. In Ref. [15] the mass and residue of the Ω(2012) were calculated by employing the QCD sum rule method with the conclusion that the Ω(2012) could be a 1P orbitally excited Ω state with J P = 3/2 − . The analysis of Ref. [15] was extended in Ref. [16] to study the Ω(2012) → K − Ξ 0 decay. In Ref. [17], the authors performed a flavor SU(3) analysis and concluded that the preferred J P for the Ω(2012) is also 3/2 − . In Refs. [18,19], its strong decay modes were studied assuming that the Ω(2012) is aKΞ(1530) hadronic molecule.
In this work, we take the chiral unitary approach and assume that the Ω(2012) is a dynamically generated state from theKΞ(1530) and ηΩ interactions. The coupling of the Ω(2012) toKΞ(1530) is then obtained from the residule at the pole position. We then calculate its decay into KΞ via a triangle diagram. We also calculate the three-body partial decay width of the Ω(2012) into KΞπ. The total decay width compares favorably with the experimental data [1] and agrees with other theoretical approaches.
This work is organized as follows. In Section II, we briefly explain the chiral unitary approach and calculate the two and three body decays of the Ω(2012). Results and discussions are shown in Section III, followed by a short summary in Section IV.

II. FORMALISM
A. Ω(2012) as aKΞ(1530) and ηΩ molecula The mass of the Ω(2012) is slightly below theKΞ(1530) threshold. It is natural to treat it as aKΞ(1530) molecular state, dynamically generated from the interaction of the coupled channelsKΞ(1530) and ηΩ in the isospin I = 0 sector. However, the possibility to be an I = 1 molecule cannot be excluded [17]. Within the chiral unitary approach, the interaction of the coupled channelsKΞ(1530) and ηΩ in the strangeness −3 and isospin 0 was first studied in Ref. [12], where a pole at (2141 − i38) MeV was found with a natural subtraction constant a µ = −2 and a renormalization scale µ = 700 MeV. Later, it was explicitly shown in Ref. [13] that the pole position of the 3/2 − Ω state can shift by varying a µ . If we take a µ = −2.5 and µ = 700 MeV, we obtain a pole at z R = (2012.7, i0) MeV, which can be associated to the newly observed Ω(2012) state [1]. In the cutoff regularization scheme, the corresponding momentum cutoff is Λ =726 MeV, which seems to quite natural as well.
The couplings of the bound state to the coupled channels,KΞ(1530) (channel 1) and ηΩ (channel 2), can be obtained from the residue of the scattering amplitude at the pole position z R , where g ii is the coupling of the state to the i-th channel. One finds Then one can write down the effective interaction of Ω(2012)KΞ(1530) (≡ Ω * K Ξ * ) and v It is worth to mention that the g Ω * (2012)KΞ(1530) obtained in Ref. [19] with the assumption that the Ω * (2012) is a pure S-waveKΞ(1530) hadronic molecule with spin-parity 3/2 − is about 1.97, which is different from ours by a mere 16%, and the difference is only 2.6% in terms of g 2 Ω * (2012)KΞ(1530) . For the Ω * (2012) →KπΞ three-body decay, since only thē KΞ(1530) component contributes at tree level and the partial decay width is proportional to g 2 Ω * (2012)KΞ(1530) , our three-body decay width is almost the same as that of Ref. [19].
For the three-body final states K − Ξ 0 π 0 andK 0 Ξ − π 0 , the isospin factors are f I = 1 and −1, respectively. The isospin factor is f I = √ 2 for all the other three-body final states. To take into account the finite size of hadrons, for each vertex, we multiply a form factor F (k 2 1 ) of the following form [19] where m is the mass of the exchanged particle and k 1 is its momentum, with the cutoff Λ varying from 0.8 GeV to 2.0 GeV.
The partial decay width of the Ω * →KΞ and Ω * →KΞπ in its rest frame are given by dΓ where M is the mass of the Ω(2012), while p is the module of the Ξ (orK) three-momentum in the rest frame of the Ω(2012). The (| p * 3 |, Ω * p 3 ) is the momentum and angle of the particle Ξ in the rest frame of Ξ and π, and Ω k 2 is the angle of theK in the rest frame of the decaying particle. The m πΞ is the invariant mass for π and Ξ and m π + m Ξ ≤ m πΞ ≤ M − mK. The averaged squared amplitude is then where

III. NUMERICAL RESULTS AND DISCUSSIONS
As explained in the previous section, the triangle diagrams are regularized with a sharp momentum cutoff Λ, which is taken to be the same as those appearing in the form factor, F (k 2 1 ). Because the triangle diagrams are ultraviolet divergent, our two-body decay width will depend on the value of the cutoff. Therefore, it is important to check whether one can obtain a decay width consistent with the experimental data using a reasonable value for the cutoff.
In Fig. 2 we show the total decay width of Ω(2012) →KΞ as a function of the cutoff parameter Λ. Note that the Ω(2012) →KπΞ three-body decay does not depend on the cut-off parameter Λ, but it depends weakly on the parameter Λ 0 as in Ref. [19]. We can see that theKΞ(1530) component provides the dominant contribution to the partial decay width of theKΞ two-body channel. The ηΩ contribution to theKΞ two-body channel is very small. However, the interference between them is still sizable and increases with the cutoff parameter Λ.
In Refs [17,19], the three-body decay was emphasized, while our result shows that two-bodyKΞ decay width can become larger than theKπΞ three-body decay width when Λ is larger than 1.65 GeV. We note that the hyperon exchange and ηΩ component contribution are not considered in Ref. [19]. More specifically, our three-body partial decay width, ∼ 3.0 MeV, is close to the estimate of Ref. [19], but smaller than that of Ref. [17], about10 MeV.
We note that our total decay width and that of Ref. [18] agree with the experimental data.
The difference is that Ref. [18] assumes that the Ω * (2012) is a pure Ξ(1530)K molecule and invokes some power counting arguments to calculate its two-body decay width. Indeed, our study shows that the contribution from the ηΩ component is small. Note that the molecule picture is different from the qqq picture of the chiral quark model [14] and light cone QCD sum rules [15,16]. The contribution of theKΞ(1530) component includes two parts: i) Σ and Λ exchanges ( fig. 1 (a) and (b)); ii) ρ, φ, and ω meson exchanges ( fig. 1 (c) and (d)). The relative importance of these two mechanisms to the Ω * →KΞ decay is shown in Fig. 3. One can see that the contribution from Σ and Λ exchanges is larger than those from ρ, φ, and ω meson exchanges for the cutoff range studied. It was shown that the calculated total decay width of the Ω * (2012) is in fair agreement with the experimental data, thus supporting the assignment of its spin-parity as 3/2 − , In addition, we showed that the ηΩ channel plays a less relevant role.
The present work should be viewed as a natural extension of the works of Refs. [12,13], where the chiral unitary approach was employed to dynamically generate such an exited Ω * .
The present work showed indeed that the chiral unitary approach can provide a satisfactory explanation of not only the mass but also the decay width of the Ω(2012), consistent with the experimental data.