Reanalysis of the newly observed $\Omega^*$ state in hadronic molecule model

After the discovery of the new $\Omega^{*}$ state, the ratio of the branching fractions of $\Omega(2012)\to \bar{K}\pi\Xi$ relative to $\bar{K}\Xi$ decay channel was investigated by the Belle Collaboration recently. The measured $11.9\%$ up limit on this ratio is in sharp tension with the $S$-wave $\bar{K}\Xi(1530)$ molecule interpretation for $\Omega(2012)$ which indicates the dominant $\bar{K}\pi\Xi$ three-body decay. In the present work, we try to explore the possibility of the $P$-wave molecule assignments for $\Omega(2012)$ (where $\Omega(2012)$ has positive parity). It is found that the latest experimental measurements are compatible with the $1/2^+$ and $3/2^+$ $\bar{K}\Xi(1530)$ molecular pictures, while the $5/2^+$ $\bar{K}\Xi(1530)$ molecule shows the larger $\bar{K}\pi\Xi$ three-body decay compared with the $\bar{K}\Xi$ decay as the case of $S$-wave molecule. Thus, the newly observed $\Omega(2012)$ can be interpreted as the $1/2^+$ or $3/2^+$ $\bar{K}\Xi(1530)$ molecule state according to current experiment data.


I. INTRODUCTION
Last year, the Belle Collaboration reported a new Ω * state in theKΞ invariant mass distribution via the Υ(1S, 2S, 3S) decays, with measured mass M = 2012.4 ± 0.7 (stat) ± 0.6 (syst) MeV and decay width Γ = 6.4 +2. 5 −2.0 (stat) ± 1.6 (syst) MeV [1]. This discovery aroused bright-eyed interest of theoreticians in understanding the nature of the excited Ω state. On the one hand, before the discovery of Ω(2012), only two states, the four-star ground state Ω(1672) and the threestar excited state Ω(2250), are listed in the review of the Particle Data Group (PDG) [2] for the S = −3 baryon spectrum. In addition, two other two-star states with even higher masses are also mentioned in PDG. The ground state Ω(1672) is well established in the wellknown quark model based on the SU (3)-flavor symmetry [3,4]. However, our knowledge on the nature of the Ω(2250) and other higher Ω * states is quite scarce. In particular, the almost 600 MeV mass difference between the ground state Ω(1672) and the first observed excited state Ω(2250) is surprising since the negative-parity orbital excitations of many other baryons are approximately 300 MeV above their respective ground states. And various models, such as quark model [5,6], Skyrme model [7] and lattice gauge theory [8], predicted the masses of the first orbital (1P ) excitations of Ω(1672) with J P = 1/2 − or 3/2 − are around 2000 MeV which is quite close to the observed value. Stimulated by these facts on the Ω baryon spectrum, recent theoretical works interpreted the newly observed Ω(2012) as the 1P orbital excitation of the ground state Ω baryon and investigated its strong decays via the chiral quark model [9], QCD sume rules [10,11], SU (3) flavor symmetry [12] and 3 P 0 model [13]. The measured width ∼ 6 MeV will be almost saturated by the two-bodyKΞ channel if the Ω(2012) is treated as the spin-parity-3/2 − 1P excited state. From the point of view of the compact decuplet baryon interpretation for Ω(2012), the troubling mass gap in the current S = −3 baryon spectrum will disappear naturally and the ∆(1700) resonance with the quantum number of J P = 3/2 − can be assigned as the decuplet partner of Ω(2012) as pointed out in Ref. [12].
On the other hand, the reported mass of the newly observed Ω(2012) state lies quite close to the threshold of KΞ(1530) channel with the binding energy ∼ 15 MeV. The closeness to the thresholds leads naturally to the interpretation of hadronic molecule composed ofKΞ(1530) for Ω(2012). After the first successful attempt of the hadronic molecule picture on uncovering the composite nature of deuteron performed by Weinberg [14,15], many exotic states which are discovered during last decades can be described well with the molecule scenarios, such as the D * s0 (2317) as a DK molecule, X(3872) as D * D molecule and newly observed series of P c pentaquark-like states asD ( * ) Σ ( * ) c molecules. The systematic discussions on hadronic molecule can be found in the recent reviews [16][17][18]. The possibility of Ω(2012) as the 3/2 − -KΞ(1530) hadronic molecule is investigated in Refs. [12,[19][20][21][22]. It is remarkable that the three-bodyKπΞ channel contributes sizable width to Ω(2012) in the S-wave hadronic molecule scenario while it is difficult to happen in the compact decuplet baryon assignment. Inspired by this significant difference, the Belle Collaboration tried to distinguish these two interpretations of the Ω(2012) by searching for its three-body decay toKπΞ [23]. They did not observe the significant Ω(2012) signals in thē KπΞ channel and drew their conclusion with the 90% credibility level upper limits on the ratios of the branching fractions of three-bodyKπΞ toKΞ two-body decay, B(Ω(2012) →KπΞ)/B(Ω(2012) →KΞ) < 11.9%. It is in sharp tension with the prediction of the S-wavē KΞ(1530) molecule assignment for Ω(2012).
However, it should be noted that the quantum num-ber of the Ω(2012) cannot be determined in experiments at present. Refocusing the processes considered by the Belle Collaboration, the samples of the Ω(2012) was collected in the final states of the Υ decay processes. The parity of the Ω(2012) prefers to be positive if the Ω(2012) is measured via the reaction Υ →ΩΩ * . In that way, the Υ decays in S wave which indicates that more events of Ω(2012) can be observed in experiments. The positive-parity Ω(2012) should be assigned as the P -wavē KΞ(1530) bound state in the molecular scenario. Such the existence of the P -wave and even higher partial wave bound states in the hidden charm sector was already suggested with the unitary coupled-channel approaches in Ref. [24]. In the present work, we would like to explore the possibility of the P -wave molecule assignments for Ω(2012). Similar to our previous work, we investigate the strong decays of Ω(2012) with the Effective lagragian approach by treating it as the P -wave molecular states with J P = 1/2 + , 3/2 + and 5/2 + , respectively. This work is organized as follows: In Sec. II, we introduce formalism and some details about the theoretical tools used to calculate the decay modes of exotic hadronic molecular states. In Sec. III, the numerical results and discussion are presented. The last section is devoted to the summary of the present work.

II. FORMALISM
In the hadronic molecule picture, the two-bodyKΞ decay happens via the triangle mechanism where the vector-meson-exchange potential is adopted for the interaction between theKΞ(1530) molecular component and KΞ final state. And theKπΞ three-body decay through the decay of the intermediate Ξ(1530) happens in the tree level. The decay diagrams of the Ω(2012) molecules are shown in the Fig. 1. The partial decay widths of these diagrams can be calculated with the effective Lagrangian approach. As we did previously, the Lorentz covariant L-S scheme proposed in Ref. [25] is used to describe the first vertex that Ω(2012) couples to theKΞ(1530) component in the P wave. The Lagrangians for the different spin parities of the Ω(2012) are presented in the following, LK Ξ * Ω * (5/2 + ) = g withg µν defined as (g µν − p µ p ν /p 2 ), where p denotes the momentum of initial Ω * state andp = p/m Ω * . The effective couplings g 1/2 + KΞ * Ω * , g 3/2 + KΞ * Ω * and g 5/2 + KΞ * Ω * are estimated with the compositeness criterion which states the relation between the derivative of self-energy operator of hadron resonance and its compositeness [14,15]. In our molecule scenario where the Ω(2012) is assumed to be the pureKΞ(1530) molecular state, the compositeness of Ω(2012) equals to one, that is χ ≡ 1 − Z = 1. In general, the compositeness criterion is used to estimate the effective coupling only for the S-wave state. It is because only the self-energy loop of the S-wave composite state can be evaluated model-independently while for the higher partial wave state, the loop integral is definitely divergent, see Ref. [16] for the detail statements. Consequently, additional scale parameters which are usually the cutoffs in some regulators need to be introduced to cope with the UV divergence in that case. In the present work, we use the compositeness criterion to estimate the P -wave couplings between the Ω(2012) andKΞ(1530) channel by including a Gaussian form factor in the evaluation of the self-energy operator of Ω(2012). The cutoff dependence of these couplings will be also given when we present our numerical results. The left vertices in the decay diagrams are the same as the S-wave case and we take the same convention with our previous calculation [20].
Finally, two form factors are also included in the loop integrals of the triangle diagrams. The first one is the Gaussian form factor. As discussed in the Ref. [26], there are two different Gaussian formulas are used commonly in the phenomenological analysis, the four dimensional Euclidean formula [27][28][29][30][31] and the three dimensional nonrelativistic formula [16,32,33]. They are defined as and respectively. p E is the four dimensional Euclidean Jacobi momentum defined as mKp Ξ * /(mK + m Ξ * ) − m Ξ * pK/(mK + m Ξ * ) and p is the spatial part of the momentums ofK and Ξ * in the rest frame of Ω(2012) state. Comparing these two equations, we can find that the form factor f 1 includes an additional constraint on the energy of molecular components, which demands that the center of mass energy is divided as the mass distribution of compounding particles inside the molecular states as happening usually for the bound states in quantum mechanics. The difference between these two kinds of Gaussian form factors will be presented in the next section.
The second form factor is chosen to be the multipolar formula as shown in Eq. (6). It is introduced to suppress the off-shell contributions of the exchanged mesons in our triangle diagrams.
where m and q is the mass and momentum of the exchanged particle. The cutoffs Λ 0 and Λ 1 are the free parameters in our calculation and we vary both of them The decay mechanisms of the Ω(2012) molecules. The left diagram stands for theKΞ two-body decay and the right one is theKπΞ three-body decay. Ω * and Ξ * denote the Ω(2012) and Ξ(1530), respectively.
in the range of 0.6-1.4 GeV to scrutinize how the decay behaviors undergo changes as the cutoffs are varied. A specific set of values for Λ 0 and Λ 1 is chosen to give the decay patterns of Ω(2012) molecules by fitting to the measured total widths.

III. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we will present our numerical results on the strong decays of the Ω(2012) in the P -waveKΞ * molecule scenarios. Firstly, the short discussion on the estimation of the P -wave effective couplings g Ω * K Ξ * will be given. After that, we will show the numerical decay patterns of these P -waveKΞ * molecules and also the parameter dependence of our results. The following is the comparison with the latest experimental data and our conclusion. At the end, we try to estimate the partial widths of the three-bodyKπΞ decays which are generated from the rescattering ofKΞ channel for the Ω(2012) states with various quantum numbers.

A. Couplings
As introduced previously, the effective couplings between the Ω(2012) andKΞ * channel are estimated with the compositeness condition. We include the form factor f 1 (Eq. (4)) in the calculations of the self-energy operators for the P -wave Ω(2012) molecules to get rid of the UV divergence. And for consistently, the same form factor is also included in the estimation of the S-wave coupling. The dependence of these effective couplings for various quantum numbers on cutoff Λ 0 is presented in Fig. 2. It does not escape attention that the P -wave effective couplings g 1/2 + Ω * K Ξ * , g 3/2 + Ω * K Ξ * and g 5/2 + Ω * K Ξ * are more sensitive to the cutoff Λ 0 than the S-wave coupling g 3/2 − Ω * K Ξ * as expected. And all the couplings decrease when the cutoff gets larger. The 5/2 + Ω(2012) has the largest coupling with theKΞ * channel, 1/2 + and 3/2 + are next, and the 3/2 − is smallest. In particular, g 1/2 + Ω * K Ξ * is quite similar with g 3/2 + Ω * K Ξ * . In spite of some model-dependence The cutoff dependence of the effective coupling constants g Ω * K Ξ * for the different quantum numbers of Ω * . The black-dotted, red-square, blue-diamond and orange-triangular points denote the cases of the J P = 3/2 − S-wave, 1/2 + Pwave, 3/2 + P -wave and 5/2 + P -wave Ω * molecules, respectively.
existence, the uncertainty of the determination of P -wave effective couplings from the compositeness condition is still under control. As shown in Fig. 2, the largest magnitude of the coupling decrease is less than the half of its value in the whole range of Λ 0 . It is the authors' opinion that the P -wave effective couplings obtained with the compositeness condition are available for the estimation of the decay widths of the molecular states, especially to estimate the relative ratios of branch fractions among various decay channels.

B. Partial decay widths of the Ω(2012) molecules
With the couplings g Ω * K Ξ * obtained, the partial widths of the Ω * molecules can be calculated straightforwardly. The results with the cutoff Λ 0 = 1.0 GeV and Λ 1 = 0.8 GeV which are fitted to the measured width of Ω(2012) are displayed in Table I for the form factor set (f 1 , f 3 ) and Table II for the (f 2 , f 3 ). It is intriguing that the 3/2 − S-wave and 5/2 + P -wave Ω(2012) molecules  S-wave P -wave have quite similar decay patterns except the highly suppressed widths of 5/2 + state and the decay pattern of the 1/2 + state is almost same with that of the 3/2 + state. And the remarkable difference between these two set of states is that the dominant decay channel of 1/2 + and 3/2 + P -wave states is the two-bodyKΞ channel which is compatible with the experimental observations, while it is the three-bodyKπΞ channel for the 3/2 − S-wave and 5/2 + P -wave states which is in sharp tension with the experimental data. It also can be noticed that the form factor f 2 gives much larger decay width for theKΞ channel than f 1 . It is similar with the case discussed in Ref. [26], the exchanged vector mesons (with the energy about 0.2 GeV) are off the mass shell when the component particlesK and Ξ(1530) are confined on nearly their mass shells by the form factor f 1 and its contribution will be suppressed by the form factor f 3 . The cutoff dependence of the partial decay widths of Ω(2012) states with various quantum numbers are given in the Fig. 3 for the form factor set (f 1 , f 3 ) and Fig. 4 for the (f 2 , f 3 ) case. Also the widths of P -wave states are more sensitive to both cutoff Λ 0 and Λ 1 than that of the S-wave state. The partial width of the three-bodȳ KπΞ channel only depends slightly on the cutoff Λ 0 due to the cutoff dependent coupling constant g 3/2 − Ω * K Ξ . The partial widths of the two-bodyKΞ channel obtained with the form factor set (f 1 , f 3 ) depend a lot on the Λ 1 while keep almost steady as the Λ 0 is varied. In the case where the form factor set (f 2 , f 3 ) is used, however, the two-bodyKΞ decay widths depend heavily on the Λ 0 and its dependence on Λ 1 is relatively modest but not such slight as the Λ 0 dependence in the case of (f 1 , f 3 ).
Besides the decay widths, the dependence of the relative ratios between the partial decay widths ofKπΞ andKΞ channels are also considered and the results are displayed in Fig. 5. Although the decay widths of Ω(2012) molecules are cutoff dependent, the characteristic behaviors on the ratios of the branching fractions ofKπΞ channel relative to theKΞ channel of theKΞ * molecules with different spin parities are deserving of special attention. As shown in Fig. 5, these ratios of all the Ω(2012) molecules does not change significantly in the whole range of cutoffs relative to the experimental upper limit. The green bands in the these plots denote the allowed region of the ratios by experimental data. The spin-parity 1/2 + and 3/2 +K Ξ(1530) molecular assignments for the Ω(2012) are consistent with the experiments when the cut off is larger than 0.7 GeV. While the 3/2 − and 5/2 +K Ξ(1530) molecular assumptions strongly disagree with the experiments in the range of cutoffs we considered. Then it can pave another way to understand the nature of Ω(2012) state besides the 1P orbital excitation of the ground Ω state. Our results suggest the Ω(2012) might be the P -waveKΞ(1530) molecule state with J P = 1/2 + or 3/2 + . Since the decay behaviors of the 1/2 + and 3/2 +K Ξ(1530) molecules are quite similar with each other, it is difficult to distinguish these two possible quantum numbers in the current hadronic molecular framework. The experimental determination of the quantum number for the Ω(2012) state is critical to uncover the mystery on its inner structure in future.

C. TheKπΞ decays via the rescattering ofKΞ
We also investigate the three-bodyKπΞ decays of Ω(2012) by considering theKπΞ are generated from the rescattering ofKΞ. The Ω(2012) states are suspected to decay intoKΞ firstly. And next theKΞ rescatters to thē KΞ(1530) channel which can be described by the vector meson dominance model. Finally, the three-bodyKπΞ final state is generated through the decay of the intermediate Ξ(1530). The decay mechanism is shown in Fig. 6.
Here the quantum numbers of Ω(2012) are considered to be 1/2 − , 1/2 + , 3/2 − , 3/2 + , 5/2 − and 5/2 + . We expect that these various quantum numbers of Ω ( 2012) can be distinguished by such three-bodyKπΞ decays. Similarly, the partial widths of these diagrams in Fig. 6 are calculated with the effective Lagrangian approach. The  (f1, f3). The left upper, left lower, right upper and right lower plots are the results of the Ω * molecule with J P = 3/2 − , 3/2 + , 1/2 + and 5/2 + , respectively. The red-solid, green-solid, blue-solid, black-solid and black-dashed lines denote the Λ0 dependence ofKΞ channel, the Λ1 dependence ofKΞ channel, the Λ0 dependence ofKπΞ channel, the Λ0 dependence of total width and the Λ1 dependence of total width, respectively. Note that the Λ0 is fixed at 1.0 GeV when Λ1 is varied and the Λ1 is fixed at 0.8 GeV when Λ0 is varied.
The corresponding partial widths are displayed in Ta The red-solid, green-solid, blue-solid, black-solid and black-dashed lines denote the Λ0 dependence ofKΞ channel, the Λ1 dependence ofKΞ channel, the Λ0 dependence ofKπΞ channel, the Λ0 dependence of total width and the Λ1 dependence of total width, respectively. Note that the Λ0 is fixed at 1.0 GeV when Λ1 is varied and the Λ1 is fixed at 0.8 GeV when Λ0 is varied.

IV. SUMMARY
The latest observation on the ratio of the branching fractions of Ω(2012) →KπΞ relative to theKΞ channel reported by the Belle Collaboration strongly disfavors the S-waveKΞ(1530) molecule interpretation for the Ω(2012). It seems to indicate that the Ω(2012) can only be considered as the 1P orbital excitation of the ground Ω baryon with J P = 3/2 − . In fact, there is no definite conclusion on the quantum number of the newly observed Ω(2012) state so far. In the present work, we explore the possibility of the Ω(2012) being a P -waveKΞ(1530) molecular state. Analogous to our previous work, we investigate the strong decays of the P -wave Ω(2012) molecules with the quantum numbers of Non-Relativistic FIG. 5. The ratio of the branching fractions of three-bodyKπΞ channel relative to the two-bodyKΞ channel varying with the cutoff Λ0 and Λ1. The upper panels are the results of the Ω * molecule with J P = 1/2 + and 3/2 + , where the left one is obtained with the form factor set of (f1, f3) and the right is obtained with the form factor set of (f2, f3). The lower panels are for the J P = 3/2 − and 5/2 + cases. And the green band in the plot denotes the region allowed by the latest Belle measurements. Note that the Λ0 is fixed at 1.0 GeV when Λ1 is varied and the Λ1 is fixed at 0.8 GeV when Λ0 is varied.  1/2 + , 3/2 + and 5/2 + by using the effective lagrangian approach. It is found that the decay behaviors of the 1/2 + and 3/2 + Ω(2012) molecules are compatible with the current experimental data. Then it suggests that Ω(2012) might be the P -waveKΞ(1530) molecules with J P = 1/2 + or 3/2 + besides being the J P = 3/2 − orbital excitation of the ground Ω baryon. The determination of the quantum number of Ω(2012) would be a landmark experimental feat on understanding its nature.