Revisiting the $\Omega(2012)$ as a hadronic molecule and its strong decays

Recently, Belle collaboration measured the ratios of the branching fractions of the newly observed $\Omega(2012)$ excited state. They did not observe significant signals for the $\Omega(2012) \to \bar{K} \Xi^*(1530) \to \bar{K} \pi \Xi$ decay, and reported an upper limit for the ratio of the three body decay to the two body decay mode of $\Omega(2012) \to \bar{K} \Xi$. In this work, we revisit the newly observed $\Omega(2012)$ from the molecular perspective where this resonance appears to be a dynamically generated state with spin-parity $3/2^-$ from the coupled channels interactions of the $\bar{K} \Xi^*(1530)$ and $\eta \Omega$ in $s$-wave and $\bar{K} \Xi$ in $d$-wave. With the model parameters for the $d$-wave interaction, we show that the ratio of these decay fractions reported recently by the Belle collaboration can be easily accommodated.


I. INTRODUCTION
In 2018, the Belle collaboration reported an Ω * state in theKΞ invariant mass distributions [1]. The measured mass and width of the Ω * state are M = 2012.4±0.7±0. 6 MeV and Γ = 6.4 +2. 5 −2.0 ± 1.6 MeV. Such kind of Ω excited states have been studied before Belle collaboration publishes their results. In Refs. [2][3][4] using the chiral unitary approach where the coupled channels interactions of thē KΞ * (1530) and ηΩ were taken into account, the Ω excited states were investigated. An Ω excited state with spin-parity J P = 3/2 − and mass around 2012 MeV can be dynamically generated with a reasonable value of the subtraction constant [4]. Using a spin-flavor-SU (6) extended Weinberg-Tomozawa meson-baryon interaction, the Ω resonances with J P = 1/2 − , 3/2 − and 5/2 − were studied in Ref. [5]. On the other hand, the Ω excited states were also investigated in classical quark models [6][7][8][9] and in the five-quark picture [10][11][12], in which, however, their predicted masses are always much different from the mass observed by the Belle collaboration. In Ref. [13], baryon states with strangeness −3 were predicted employing a quark model with ingredients suggested by QCD, and the mass of one predicted state with J P = 3/2 − is about 2020 MeV.
After the observation of the above mentioned Ω(2012) by the Belle collaboration [1], there were many theoretical studies on its mass, width, quantum numbers and decay modes. In Refs. [14,15], the mass and the two-body strong decays of the Ω(2012) state were studied by the QCD sum rule method and it was found that the Ω(2012) can be interpreted as a 1P orbital excitation of the ground state Ω baryon with quantum * Electronic address: xiejujun@impcas.ac.cn † Electronic address: lisheng.geng@buaa.edu.cn numbers J P = 3/2 − . In Refs. [16][17][18], the Ω excited spectrum and their two body strong decays were evaluated within a non-relativistic constituent quark potential model, and it was found that the Ω(2012) resonance is most likely to be a 1P state with J P = 3/2 − . In Ref. [19], the authors performed a SU (3) flavor analysis of the Ω(2012) state and discussed itsKΞ * (1530) molecular picture. They concluded that the preferred quantum numbers of Ω(2012) are also 3/2 − . On the other hand, the mass of the Ω(2012) is just a few MeV below thē KΞ * (1530) mass threshold, which indicates that it could be a possibleKΞ * (1530) molecule state [20]. Indeed, the hadronic molecule nature of the Ω(2012) were investigated in Refs. [21][22][23][24], and these calculations 1 predicted a large decay width for Ω(2012) →KΞ * (1530) →KπΞ. However, in a very recent measurement of the Belle collaboration [25], it was found that there is no significant signals for the Ω(2012) →KΞ * (1530) →KπΞ decay, and an upper limit was obtaining, at the 90% credibility level, for the ratio of the three body decay to the two body decay mode of Ω(2012) →KΞ, R = Br[Ω(2012) → KπΞ]/Br[Ω(2012) →KΞ], which is only 11.9%. There are also other experimental results for the ratios of different final charged decay modes [25], but because of large background for those decay channels these values are obtained without including the constraints of the isospin symmetry 2 . Later on, based on the new measurements by the Belle collaboration [25], the strong decays of the Ω(2012) were restudied in Refs. [26,27] within the hadronic molecular approach. In Ref. [26] it concluded that the Ω(2012) can be interpreted as the p-wavē KΞ * (1530) molecule state with J P = 1/2 + or 3/2 + , while in Ref. [27], it was pointed out that the Ω(2012) state contains mixedKΞ * (1530) and ηΩ hadronic components and the sizable ηΩ hadronic component leads to a suppression of theKπΞ decay mode.
The Ω(2012) state was investigated within a coupled channel approach in Ref. [24], in which, in addition to the interaction ofKΞ * (1530) and ηΩ in s-wave, theKΞ in d-wave interaction was also taken into account. The pole position of the Ω(2012) was well reproduced in the scattering amplitude. However, the predicted value of R is about 90% [24], which is much larger than the experimental measurements [25]. Based on the new measurements of Ref. [25], we follow Ref. [24] and restudy the Ω(2012) state from the molecular perspective in which the resonance is dynamically generated from the interactions ofKΞ * (1530), ηΩ andKΞ in coupled channels, withKΞ * (1530) and ηΩ in s-wave andKΞ in d-wave.
In this work, we determine the unknown parameters α and β introduced in Ref. [24], fitting to the experimental data, and calculate the partial decay widths of the two and three body strong decays of Ω(2012), with the strong couplings obtained at the pole position of the state.
The paper is organized as follows. In Section II, we present the formalism and ingredients of the chiral unitary approach for the treatment of the Ω(2012) as a dynamically generated hadronic state from the interactions ofKΞ * (1530), ηΩ andKΞ in coupled channels. Numerical results for the two and three body strong decays of the Ω(2012) state and discussions are given in Section III, followed by a short summary in the last section.

II. FORMALISM AND INGREDIENTS
In this section, we briefly review the coupled channel approach to study the Ω(2012) state involving the s-wave interaction ofKΞ * (1530), ηΩ and d-wave interaction of KΞ, although these interactions have been detailed in Refs. [4,23,24].
A. Scattering amplitude and the Ω(2012) Follow Ref. [24], we denoteKΞ * (1530), ηΩ, andKΞ channels by 1, 2, and 3, respectively, and then the tree level transition amplitudes, V ij (i, j = 1, 2, 3), between each of the two channels are given by where we take the pion decay constant f π = 93 MeV. The k 0 1 and k 0 2 are the energies of theK meson in channel 1 and η meson in channel 2, respectively, which are, with √ s the invariant mass of the meson-baryon system. In addition, q 3 is the on-shell momentum of theK meson in channel 3, which reads, Then we solve the Bethe-Salpeter equation with the V ij given above, and obtain the unitarized scattering amplitude T : where G is the loop function for each channel and it is a diagonal matrix containing the meson and baryon propagators. Explicitly where G ii ( √ s) can be regularized with a cutoff prescription and the explicit results are 3 : where E i and ω i are the baryon and meson energies for each channel. In general, Λ 1 , Λ 2 and Λ 3 are different. Yet, to minimize the model parameters, Λ 1 = Λ 2 = 726 MeV are used in Ref. [23] and Λ 1 = Λ 2 = Λ 3 = 735 MeV were used in Ref. [24]. In this work, we will determine them with the experimental data of the Belle collaboration [1,25], and discuss them in the following. However, since the Ξ * (1530) resonance has a sizable decay width and theKΞ * (1530) mass threshold is close to the mass of Ω(2012), the width of Ξ * (1530) should be considered. For this purpose, we need to perform a convolution with the spectral function [30] Note that the range of (M Ξ * − 6Γ Ξ * , M Ξ * + 6Γ Ξ * ) includes most of the distribution. Here, theΓ Ξ * is energy dependent, and its explicit form is given bỹ with In this work, the physical masses and spin-parities of the involved particles are taken from PDG [31], and tabulated in Table I. Note that we take the isospin averaged values for m K , M Ξ * , M Ξ and Γ Ξ * , where we take Γ Ξ * = 9.5 MeV.

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With this formalism and the former ingredients, one can easily obtain the scattering matrix T . Then one can also look for the poles of the scattering amplitude T ij on the complex plane of √ s. The poles, z R , on the second Riemann sheet could be associated with the Ω(2012) resonance. The real part of z R is associated with the mass (M ) of the state, and the minus imaginary part of z R is associated with one half of its width (Γ). Close to the pole at z R = M R − iΓ R /2, T ij can be written as where g kk is the coupling constant of the resonance to channel k. Thus, by determining the residues of the scattering amplitude T at the pole, one can obtain the couplings of the resonance to different channels, which are complex in general. Since we consider the s-wave interactions of thē KΞ * (1530) and the ηΩ channels, the quantum numbers of the Ω(2012) should be J P = 3/2 − , and it decays intoKΞ in d-wave as shown in Fig. 1, where the effective interactions are obtained from the s-wave Ω(2012)KΞ * (1530) and Ω(2012)ηΩ decays and the rescattering of theKΞ * (1530) and ηΩ pairs, which proceed as shown in Fig. 2. Then the partial decay width of the Ω(2012) →KΞ is easily obtained as 4 where g Ω * K Ξ is the effective coupling constant of Ω(2012)KΞ vertex obtained as explained above, and M is the mass of the obtained Ω(2012) state, and For the Ω(2012) →KπΞ decay, it can proceed via Ω(2012) →KΞ * (1530) →KπΞ. The decay diagram is shown in Fig. 3. And the partial decay width can be calculated using whereΓ Ξ * is dependent on the invariant mass of π and Ξ system, M πΞ . And . With all the formulae above, one can easily work out the Γ Ω(2012)→KΞ * →KπΞ performing the integration over M πΞ from M Ξ + m π to M − mK.

III. NUMERICAL RESULTS
To calculate the scattering amplitude T , we have to fix the unknown parameters α, β, and the cutoffs Λ k . Since there are very limited experimental data: the mass and the width of the Ω(2012) and the upper limit of the ratio R, we will take the same value for Λ 1 = Λ 2 = Λ 3 = q max . Even so, we still have three free parameters, and there are only two experimental data plus one more constraint, the upper limit of the ratio R < 11.9%. Varying the unknown model parameters of α, β and q max , we find that one can reproduce the mass and width of Ω(2012) and the upper limit R < 11.9% with the following range of the model parameters 5 : To minimize the number of the free parameters, we fix q max = 735 (Set I), 750 (Set II), 800 (Set III), 850 (Set IV), and 900 MeV (Set V), and determine α and β by fitting them to the experimental data. Since we only know the upper limit of R, it is difficult to perform a χ 2 fit to it. Technically, one can define where M th and Γ th are evaluated at the pole position of T , and we take M exp = 2012.4 MeV, ∆M exp = 0.9 MeV, Γ exp = 6.4 MeV, and ∆Γ exp = 3.0 MeV as measured by the Belle collaboration [1]. Then we vary firstly the values of α and β in the range as in Eq. (19). If the obtained mass and width of Ω(2012), and R are in agreement with the experimental values within errors, we call that a best fit. In this way, we obtain sets of the fitted parameters (α, β) with different best χ 2 best . The fitted parameters corresponding to the minimum χ 2 min that we get from the best fits are:  Table II. We will take these values as the central values of parameters α and β. In addition, with all the fitted parameters of α and β with the χ 2 best fit, we search for the minimal values of the ratio R, which are 9%, 8%, 7%, 5% and 4% for sets I, II, III, IV and V, respectively. It is worth to mention that, in Ref. [27], the minimal value of R could be zero.
Next, we collect these sets of the fitted parameters, such that the corresponding χ 2 best are below χ 2 min + 1. With these collected best fitted parameters, we obtain the standard deviations of parameters α and β, which are quoted in Table II Table II. In addition, with the coupling constants obtained from the best fit, we calculate the partial decay widths of Ω(2012) →KπΞ and Ω(2012) →KΞ, and also their ratio R. We show these results in Table III. From these results, one can easily find that the sum of the branching fractions of Ω(2012) →KπΞ and Ω(2012) →KΞ is more than 95%, which indicates that the other decay modes and other strong decay mechanisms of Ω(2012) are small, such as those of the triangle mechanisms of Refs. [23,26].
Finally, we pay attention to the πΞ invariant mass distributions of the Ω(2012) →KΞ * →KπΞ decay. The theoretical calculations with the parameters of Set I are shown in Fig. 4. On can see that, because of the phase space limitations, dΓ Ω(2012)→KπΞ /dM πΞ peaks around M πΞ = 1515 MeV, which is lower than the mass of Ξ * (1530).

IV. SUMMARY
Based on the recent measurements by the Belle collaboration [25], where they did not observe significant signals for the Ω(2012) →KΞ * (1530) →KπΞ decay, we revisit the Ω(2012) state from the molecular perspective in which this resonance appears to be dynamically generated from the coupled channels interactions of thē KΞ * (1530) and ηΩ in s-wave andKΞ in d-wave. In such a scenario, the Ω(2012) is interpreted as a 3/2 − molecule state. We studied the two and three body strong decays of Ω(2012), within the model parameters for the d-wave interaction, it is shown that the experimental proper- II: Used or determined values of the unknown parameters in this work. We also give the pole positions (MR, ΓR) of the Ω(2012) and the couplings to different channel obtained with the central values of these fitted parameters.