# Nature of the \(\Omega (2012)\) through its strong decays

## Abstract

We extend our previous analysis on the mass of the recently discovered \(\Omega (2012)\) state by investigation of its strong decays and calculation of its width employing the method of light cone QCD sum rule. Considering two possibilities for the quantum numbers of \(\Omega (2012)\) state, namely 1*P* orbital excitation with \(J^P=\frac{3}{2}^-\) and 2*S* radial excitation with \(J^P=\frac{3}{2}^+\), we obtain the strong coupling constants defining the \(\Omega (1P/2S)\rightarrow \Xi K\) decays. The results of the coupling constants are then used to calculate the decay width corresponding to each possibility. Comparison of the obtained results on the total widths in this work with the experimental value and taking into account the results of our previous mass prediction on the \(\Omega (2012)\) state, we conclude that this state is 1*P* orbital excitation of the ground state \(\Omega \) baryon, whose quantum numbers are \(J^P=\frac{3}{2}^-\).

## 1 Introduction

The study of the different parameters of the hadrons allows us gain deeper understanding on the properties of the observed particles as well as provide insight into the future experiments searching for the new states. Up to now many baryons have been observed and their properties have been examined extensively. However we are still in need of much work to obtain comprehensive information even for the light baryons. We need more information especially on their excited states. For these baryons some excited states predicted by the quark model have not been observed yet experimentally. Therefore investigations on these experimentally not yet observed states, providing information to the experimental researches, are essential.

Among these states are the excited states of the \(\Omega \) with three strange quark content. So far the list provided by the Particle Data Group (PDG) [1] contains a few of the \(\Omega \) states. Except for the ground state \(\Omega (1672)\), our knowledge is limited on the nature of these baryons. Therefore the recent observation of the Belle Collaboration reporting \(\Omega (2012)\) state with mass \(2012.4 \pm 0.7\) (stat) \(\pm 0.6\)(syst) MeV and width \(6.4^{+2.5}_{-2.0}\) (stat) \(\pm 1.6\) (syst) MeV [2] has triggered the attentions of the researchers on this particle. The Belle Collaboration observed this particle in the \(\Omega ^{*-}\rightarrow \Xi ^0K^-\) and \(\Omega ^{*-}\rightarrow \Xi ^{-}K_S^0\) decays and reported its being an excited \(\Omega ^-\) state with probable quantum numbers \(J^P=\frac{3}{2}^-\). This decision was made based on the comparison of the observation with the present theoretical works on the masses of excited \(\Omega \) states with different models. The mass predictions of these models for \(J^P=\frac{3}{2}^-\) state are close to the observed value. Among these models are the quark model [3, 4, 5, 6, 7, 8, 9, 10, 11], lattice gauge theory [12, 13] and Skyrme model [14]. The predictions of these models put support behind the possibility of \(J^P=\frac{3}{2}^-\) assignment for the observed state. However besides the mass prediction, the investigation of other properties of the considered state would be helpful to identify the state more reliably. The magnetic dipole moment and Radiative decays are among these properties which may help us gain information about the properties of these particles. Such investigations for negative parity baryons exist in the literature [15, 16, 17]. The study of the strong decay of the particle is also helpful in this respect. With this motivation, to provide a possible interpretation for the observed \(\Omega (2012)\) particle, using the chiral quark model an analysis on its strong decay was carried out in Ref. [18]. The result of this work suggested the possibility of \(\Omega (2012)\) being 1*P* state with \(J^P=\frac{3}{2}^-\) without completely excluding the other possibilities such as 1*P* state with \(J^P=\frac{1}{2}^-\) and 2*S* state with \(J^P=\frac{3}{2}^+\) quantum numbers. The possibility of the \(\Omega (2012)\) being a hadronic molecular state also discussed in Refs. [19, 20, 21].

Radial excitation of the decuplet baryons were investigated in Refs. [3] and [22] and mass value for radial excitation of \(\Omega \) state was predicted as 2065 MeV and \(2176\pm 219\) MeV, respectively. In Ref. [23] the mass of the newly observed \(\Omega (2012)\) state was extracted using QCD sum rule approach. Giving very consistent mass value with that of the experimentally observed \(\Omega (2012)\) state, this result allowed us to interpret this state as the orbital excitation of the ground state \(\Omega \) baryon. To understand the nature of \(\Omega (2012)\), the investigations on other aspects, namely the strong decay of the \(\Omega (2012)\) state into \(\Xi ^0\) and \(K^-\), is necessary. Considering this, in this work we extend our analysis for \(\Omega (2012)\) presented in Ref. [23] using the spectroscopic parameters obtained in [23] as input parameters. We investigate the \(\Omega (2012)\rightarrow \Xi ^0K^-\) transition and calculate corresponding coupling constant. While doing so, we consider two possibilities for the \(\Omega (2012)\) taking it as being 1*P* or 2*S* state with spin-parity quantum numbers \(J^P=\frac{3}{2}^-\) and \(J^P=\frac{3}{2}^+\), in what follows represented by \(\widetilde{\Omega }\) and \(\Omega ^{\prime }\) respectively, and calculate the decay widths for these possible configurations. Then we compare the results of these analysis with the experimental value of the width. For the calculations we apply the light cone QCD sum rule (LCSR) approach [24, 25, 26, 27, 28] which is an extension of the traditional QCD sum rule method. This method has been extensively and successfully used to study the various properties of the hadrons such as form factors, strong coupling constants etc.

The arrangement of the remaining part of the paper is as follows: Section II presents the calculations of the coupling constants for \(\widetilde{\Omega }\) and \(\Omega ^{\prime }\) transitions to \(\Xi ^0K^-\) in detail. Section III is devoted to the numerical analysis of the results and decay width calculations employing the results obtained for coupling constants. Final section is separated for the conclusion. In the last section the experimental width is also compared with the results of the decay widths obtained for different \(J^P\) scenarios assigned to the \(\Omega (2012)\) state.

## 2 \(\widetilde{\Omega }\) and \(\Omega ^{\prime }\) transitions to \(\Xi ^0K^-\)

*a*,

*b*and

*c*are used to represent the color indices, \(\beta \) is an arbitrary parameter and

*C*is the charge conjugation operator.

The correlation function can be calculated either in terms of hadronic degrees of freedom or in terms of QCD degrees of freedom. In QCD sum rule formalism firstly we get these two representations of the correlator. Its representation acquired in terms of hadronic parameters such as mass, residue, coupling constant of considered hadrons is called as the physical or phenomenological side. The second representation including QCD parameters such as masses of quarks, quark-gluon condensates is called QCD or theoretical side. By matching of the coefficients of the same Lorentz structures obtained in both sides we obtain the QCD sum rules for the physical parameters in question.

*p*and

*q*are the momenta of \(\widetilde{\Omega }\), \(\Xi \) and

*K*states, respectively. The result comes up with the matrix elements which can be written in terms of physical parameters as

*K*-meson. \(M_1^2\) and \(M_2^2\) represents the Borel parameters. As is seen from the above equation there are many structures entering the calculations. Among these structures we need only the ones that are independent and give contributions only to the spin-3/2 states at initial channel. The current \(J_{\Omega \mu } \) in Eq. (2) couples not only to the spin-3/2 states but also to the possible spin-1/2 states with the same quark contents. Hence we need to remove the spin-1/2 pollution to get pure spin-3/2 contributions. The procedure for removing these unwanted contributions is presented for instance in Ref. [29] in details. After applying this procedure there remain only four structures, Open image in new window , Open image in new window , Open image in new window and \( q_{\mu }\) that give contributions to the pure spin-3/2 states at initial channel. We will use these structures and match their coefficients in physical side to the ones obtained in the QCD side of the calculations.

The QCD side of the correlation function is made with the same correlation function given in Eq. (1) using the explicit form of interpolating currents. After inserting the currents to the correlation function we make the contraction via Wick’s theorem considering all possible contractions between the quark fields. The contractions give us the result in terms of the light quark propagators in the presence of background field which contain the perturbative and non-perturbative terms. The propagators are used explicitly in the coordinate space and via Fourier transformation the expression is converted to the momentum space. In addition to the propagators, in the calculations there appear matrix elements of non-local operators between *K*-meson and vacuum states which have the common form \(\langle K(q)|\bar{q}(x)\Gamma q(y)|0\rangle \) or \(\langle K(q)|\bar{q}(x)\Gamma G_{\mu \nu } q(y)|0\rangle \). These matrix elements are parameterized in terms of *K*-meson distribution amplitudes (DAs). The \(\Gamma \) in these matrix elements is the full set of Dirac matrices and \(G_{\mu \nu }\) is the gluon field strength tensor. These matrix elements are used as inputs of LCSR to get the nonperturbative contributions. The DAs for *K*-meson are derived in Refs. [30, 31, 32] and their expressions can be found there.

## 3 Numerical analysis

*K*-meson distribution amplitudes given in Refs. [30, 31, 32] we need the values of various input parameters, such as quark condensates, masses and residues of the considered hadrons, which are presented in Table 1.

Some input parameters

Parameters | Values |
---|---|

\(m_{\Xi }\) | \(1314.86\pm 0.20~\mathrm {MeV}\) [1] |

\(m_{\widetilde{\Omega }}\) | \(2019^{+17}_{-29}~\mathrm {MeV}\) [23] |

\(m_{\Omega ^{\prime }}\) | \(2176\pm 219~\mathrm {MeV}\) [22] |

\(\lambda _{\Xi }(1\text{ GeV })\) | \(0.017\pm 0.003~\mathrm {GeV}^3\) [33] |

\(\lambda _{\widetilde{\Omega }}(1\text{ GeV })\) | \(0.108^{+0.004}_{-0.005}~\mathrm {GeV}^3\) [23] |

\(\lambda _{\Omega ^{\prime }}(1\text{ GeV })\) | \(0.129\pm 0.039~\mathrm {GeV}^3\) [22] |

\(f_{K}(1\text{ GeV })\) | \(160~\mathrm {MeV}\) [32] |

\(m_{s}(1\text{ GeV })\) | \(128^{+12}_{-4}~\mathrm {MeV}\) [1] |

\(\langle \bar{q}q \rangle (1\text{ GeV })\) | |

\(\langle \bar{s}s \rangle (1\text{ GeV })\) | |

\(m_{0}^2 (1\text{ GeV })\) | |

\(\langle g_s^2 G^2 \rangle \) | |

\( \Lambda (1\text{ GeV })\) | \( (0.5\pm 0.1) \)\(\mathrm {GeV} \) [38] |

Sum rules predictions for the strong coupling constants and widths corresponding to the decays of the orbitally and radially excited \(\Omega \) states to \(\Xi \) and *K* using different Lorentz structures

Structure | Decay | g \((\text{ GeV }^{-1})\) | \(\Gamma \) (MeV) |
---|---|---|---|

\(\widetilde{\Omega }\rightarrow \Xi K\) | \(11.13^{+1.14}_{-1.31}\) | \(6.98\pm 1.75\) | |

\(\Omega ^{\prime } \rightarrow \Xi K\) | \(7.76^{+0.71}_{-0.87}\) | \(355.02\pm 88.68\) | |

\(\widetilde{\Omega }\rightarrow \Xi K\) | \(11.57^{+1.42}_{-1.53}\) | \(7.57 \pm 1.98\) | |

\(\Omega ^{\prime } \rightarrow \Xi K\) | \(7.53^{+0.69}_{-0.85}\) | \(333.64\pm 83.79\) | |

\(\widetilde{\Omega }\rightarrow \Xi K\) | \(10.64^{+0.98}_{-1.14}\) | \(6.41\pm 1.57\) | |

\(\Omega ^{\prime } \rightarrow \Xi K\) | \(7.56^{+0.68}_{-0.76}\) | \(336.36\pm 86.79\) | |

\( q_{\mu }\) | \(\widetilde{\Omega }\rightarrow \Xi K\) | \(11.54^{+1.39}_{-1.47}\) | \(7.53\pm 1.87\) |

\( q_{\mu }\) | \(\Omega ^{\prime } \rightarrow \Xi K\) | \(7.95^{+0.81}_{-0.92}\) | \(371.99\pm 92.85\) |

Total widths corresponding to the orbitally and radially excited \(\Omega \) states

Structure | State | \(\Gamma _{tot}\) (MeV) |
---|---|---|

\(\widetilde{\Omega }\) | \(12.80\pm 3.21\) | |

\(\Omega ^{\prime } \) | \(650.75\pm 162.55\) | |

\(\widetilde{\Omega }\) | \(13.87 \pm 3.62\) | |

\(\Omega ^{\prime } \) | \(611.56\pm 153.58\) | |

\(\widetilde{\Omega }\) | \(11.74\pm 2.87\) | |

\(\Omega ^{\prime }\) | \(616.54\pm 159.08\) | |

\( q_{\mu }\) | \(\widetilde{\Omega }\) | \(13.80\pm 3.42\) |

\( q_{\mu }\) | \(\Omega ^{\prime } \) | \(681.85\pm 170.19\) |

Comparing the results obtained for the total widths of the considered states with the experimental value of width, \(6.4^{+2.5}_{-2.0}\) (stat) \(\pm 1.6\) (syst) MeV [2], we see that the results for orbitally excited states are consistent with the experimental width within the errors for all structures. The total widths obtained for the radially excited states, however, are very far from the experimental value of the width for all structures. With these information, we conclude that the newly observed \(\Omega (2012)\) is 1*P* excitation of the ground state \(\Omega \) and this assignment is structure independent. Among all Lorentz structures the Open image in new window , containing maximum number of momenta, gives closer value to that of the experimental width compared to other structures.

## 4 Conclusion

Using the LCSR method we made a calculation on the coupling constant and decay width of the \(\Omega (2012)\) transition to \(\Xi K\) considering it as either orbital or radial excitation in \( \Omega \) channel. In a previous work, [23], we extracted the mass of the \(\Omega (2012)\) taking it as an orbital excitation of the ground state \( \Omega \) baryon and got a result in a good consistency with the experimentally measured value. In Ref. [22] the corresponding mass was calculated for the radial excitation of the state under consideration and obtained as \(2176\pm 219\) MeV. Comparison of these results with the experimentally obtained mass allowed us to reach a conclusion on the quantum numbers of the observed \(\Omega (2012)\) state as \(J^P=\frac{3}{2}^-\). The present work has been done to gain more new information on this issue. Hence, the decay widths were calculated for the channels \(\widetilde{\Omega }\rightarrow \Xi ^0K^-\) and \(\Omega ^{\prime }\rightarrow \Xi ^0K^-\) with the assumption of \(\Omega (2012)\) being an orbital or radial excitation of the ground state \(\Omega \). Considering the \(\Omega ^{*-}\rightarrow \Xi ^0K^-\) and \(\Omega ^{*-}\rightarrow \Xi ^{-}\bar{K}^0\) as dominant modes of \(\Omega (2012)\) state and the measured ratio \( \mathcal{R}=\frac{\mathcal{B}(\Omega ^{*-}\rightarrow \Xi ^0K^-)}{\mathcal{B}(\Omega ^{*-}\rightarrow \Xi ^-\bar{K}^0)} =1.2 \pm 0.3\) [2] we estimated the total widths of the considered states for all the Lorentz structures. Comparing the obtained results for the total widths with the observed width value reported by the Belle Collaboration, \(6.4^{+2.5}_{-2.0}\) (stat) \(\pm 1.6\) (syst) MeV [2], we concluded that \(\Omega (2012)\) is 1*P* excitation of the ground state \(\Omega \). This conclusion, which is structure independent, supports our previous assignments for the nature of \(\Omega (2012)\) resonance using mass sum rules. Hence, the \(\Omega (2012)\) state is first orbital excitation in \( \Omega \) channel with spin-parity \(J^P=\frac{3}{2}^-\). This assignment for the spin-parity of \(\Omega (2012)\) is consistent with those of Refs. [19, 20, 21], which consider this particle as the \( \Xi ^* K\), \( \Xi (1530) K\) or \( \Xi (1530) \bar{K} \) molecular state. In the molecular picture considered in Refs. [19, 20, 21] the three-body decays take place at tree level, but the two-particle decays appear only at loop level. Therefore, in this picture, the three-body decays dominate over the two-body modes. In our case that we consider \(\Omega (2012)\) as a usual three-quark state, however, as we also noted before any estimation on the three-body decays of \(\Omega (2012)\) is problematic regarding the general philosophy of the QCD sum rule method and even if it becomes possible it should be suppressed via phase volume. Therefore, more measurements on the possible two- and three-body decay modes of \(\Omega (2012)\) state can play decisive roles on choosing the “right” picture for the internal structure of this particle.

## Notes

### Acknowledgements

H. S. thanks Kocaeli University for the partial financial support through the Grant BAP 2018/070.

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