Strong decays of the hadronic molecule ${\mathbf{\Omega }}^{\mathbf{{\ast}}}$(2012)

Last year the Belle collaboration reported on a new excited isosinglet hyperon Ω^∗− state decaying into Ξ⁰ K⁻ and ${{\Xi}}^{-}{K}_{S}^{0}$ pairs with a mass of 2012.4 ± 0.7 ± 0.6 MeV and a width of ${\Gamma}=6.{4}_{-2.0}^{+2.5}\left(\text{stat}\right){\pm}1.6\left(\text{syst}\right)$ MeV [1]. The spin-parity quantum numbers of the Ω^∗− have been favored to be ${J}^{P}={\frac{3}{2}}^{-}$ based on two arguments: (1) the observed mass value of the Ω^∗− is close to the theoretical predictions for the ${\frac{3}{2}}^{-}$ states, (2) the rather narrow width of the Ω^∗− decaying to a ${\Xi}\bar{K}$ pair via a d wave. Recently, the Belle collaboration searched for the cascade three-body decay ${\Omega}\left(2012\right)\to \bar{K}{\Xi}\left(1530\right)\to \bar{K}\pi {\Xi}$ [2]. They did not observe any significant signal in this channel and derived upper limits for the ratios of the branchings relative to the two-body $\bar{K}{\Xi}$ decay modes. In particular, the most stringent upper limits read [2]:

$$\begin{align}\hfill {R}_{1}& =\frac{B\left({\Omega}\left(2012\right)\to {\Xi}{\left(1530\right)}^{0}\left(\to {{\Xi}}^{-}{\pi }^{+}\right){K}^{-}\right)}{B\left({\Omega}\left(2012\right)\to {{\Xi}}^{-}{\bar{K}}^{0}\right)}{< }9.3\%,\hfill & \hfill \\ \hfill & \hfill & \hfill \\ \hfill {R}_{2}& =\frac{B\left({\Omega}\left(2012\right)\to {\Xi}{\left(1530\right)}^{0}\left(\to {{\Xi}}^{-}{\pi }^{+}\right){K}^{-}\right)}{B\left({\Omega}\left(2012\right)\to {{\Xi}}^{0}{K}^{-}\right)}{< }7.8\%.\hfill & \hfill \end{align} \tag{ 1 }$$

An excited Ω* state with ${J}^{P}={\frac{3}{2}}^{-}$ and a mass of 2020 MeV has been predicted in a quark model with a QCD based potential in reference [3]. Later, different types of quark models [4], large N_c approaches [5], algebraic string-like model [6], Skyrme model [7], and lattice QCD [8] reported on an estimate for the Ω* mass in the region from 1953 to 2120 MeV. Inclusion of sizeable five-quark Fock components and their mixing with three-quark components in the constituent quark models, considered in reference [9], lead to a reduction of the Ω* mass by about 200 MeV. The dynamical generation of the Ω* state using two (Ωη and ${{\Xi}}^{{\ast}}\bar{K}$) coupled channels in the chiral unitary approach has been proposed and developed in references [10–12], where the mass of the Ω* has a strong dependence on the choice of model parameters leading to 2141 MeV in reference [11] and 1800 MeV in reference [12].

The understanding of the structure and decays of the Ω*(2012) state has been of increased interest since the discovery by the Belle collaboration. In references [13–22] possible interpretations of the Ω*(2012) hyperon have been critically discussed. In reference [13] the two-body decays Ω* → Ξ⁰ K⁻ and ${{\Omega}}^{{\ast}}\to {{\Xi}}^{-}{\bar{K}}^{0}$ have been analyzed in the chiral quark model. It was argued that the obtained numerical results are in agreement with spin-parity ${\frac{3}{2}}^{-}$ of the Ω* state, while alternative assignments as ${\frac{1}{2}}^{-}$ and ${\frac{3}{2}}^{+}$ cannot be completely excluded. In reference [14] the possible structure and resulting strong decays of the Ω* were studied using QCD sum rules. From the analysis of the mass and the strong decay properties (coupling constants and widths) it was concluded that the Ω* hyperon is the 1P orbital excitation of the ground state Ω(1670) baryon with ${J}^{P}={\frac{3}{2}}^{-}$. The same conclusion about the nature of the Ω* state has been made from an analysis of its two-body decays in the framework of the ³ P₀ model. In reference [15] the Ω* state has been considered on the basis of the SU(3) flavor picture. It was found that if the Ω* state is the $\bar{K}{\Xi}\left(1530\right)$ molecular state formed in the isospin zero channel, then its main decay mode is the tree-body process ${{\Omega}}^{{\ast}}\to {\Xi}\bar{K}\pi$. In reference [16] a hadronic molecular scenario for the Ω(2012) has been tested in an effective field-theoretical approach. It was found that the partial width of the three-body decay ${{\Omega}}^{{\ast}}\to {\Xi}\bar{K}\pi$ is in the 2–3 MeV interval, while the partial width of the two-body decay ${{\Omega}}^{{\ast}}\to {\Xi}\bar{K}$ is in the 1–11 MeV range. Here and also in references [17–19] the dominance or sizable contribution of the tree-body decays ${{\Omega}}^{{\ast}}\to {\Xi}\bar{K}\pi$ has been based on the description of these processes by a tree-level diagram, while the two-body processes ${{\Omega}}^{{\ast}}\to {\Xi}\bar{K}$ have been described by a loop-diagram generated by the Ω* constituents. In reference [21] the Ω baryon spectrum up to the N = 2 shell has been calculated based on a nonrelativistic constituent quark potential model. In reference [22] the analysis of the Ω(2012) strong decays has been performed using a hadronic molecular model and different spin-parity assignments for these state.

The formalism of quantum field theory for treatment of bound states based on the Weinberg–Salam (WS) compositeness condition. It was formulated in references [23–26] and widely applied to the study of hadronic molecules (HM) in references [23, 27–30]. Application of the WS condition for description of composite structure of conventional hadrons [23–26] and exotic states [23, 27–30] is based on the following steps: (1) first, we derive a phenomenological Lagrangian, which is manifestly Lorenz covariant and gauge invariant and describes the interaction of the bound state and its constituents; (2) second, the coupling constant of the HM with the constituents is fixed by solving by the WS condition Z_HM = 1 − Π_HM' = 0 [23–30], where Π_HM' is the derivative of the HM mass operator induced by interaction Lagrangian of the HM with its constituents. The WS condition is understood as the probability to find the HM as the bare state is always equal to zero; (3) third, we construct the S-matrix operator, which consistently generates matrix elements of physical processes involving HM. By construction we are able to consider hadronic molecules as an admixture of two or more hadronic constituent channels (Fock states). This procedure corresponds to the consideration of two- or multiple channel couplings in potential approaches. Of course, the use of a mixture of hadronic constituents leads to additional parameters, which reduce the predictive power of the approach. Therefore, one can try to reduce parameters by using additional physical arguments, like, e.g. the hadronic components whose sum of the masses is far away from the mass of the hadronic molecule have strongly reduced contributions. We therefore restrict to the leading components.

The main our goal is to present a self-consistent picture of strong two- and three-body decays of the Ω*(2012) state in the hadronic molecular approach formulated and developed in [27–30]. In the recent paper by the Belle collaboration [2] it is claimed that the strong three-body decay modes of the Ω(2012) state are suppressed in comparison with the two-body one, which raises doubts on a molecular interpretation of this state. We find that due to a possible mixture of the hadronic components ${{\Xi}}^{{\ast}}\bar{K}$ and Ωη in the structure of Ω the three-body decays ${\Omega}\left(2012\right)\to {\Xi}\pi \bar{K}$ are suppressed when the Ωη hadronic component dominates. Note that applications of our approach to hadrons composed of two and more hadronic configurations have been performed in references [29–31] with respect to the X(3872), Z(4430), and Λ(2940) states.

Following the conjecture of the Belle collaboration [1] and assignments of most of the theoretical approaches we accept that Ω*(2012) has ${J}^{P}={\frac{3}{2}}^{-}$ spin-parity. We consider the Ω*(2012) as a weakly bound hadronic molecule, which involves a superposition of two hadronic components—(Ω[1670]η) and $\left({{\Xi}}^{{\ast}}\left[1530\right]\bar{K}\right)$:

$$\begin{equation}\vert {{\Omega}}^{{\ast}}\left(2012\right)\rangle =\mathrm{cos}\theta \enspace \frac{\vert {{\Xi}}^{{\ast}0}{K}^{-}\rangle \enspace +\enspace \vert {{\Xi}}^{{\ast}-}{\bar{K}}^{0}\rangle }{\sqrt{2}}-\mathrm{sin}\enspace \theta \vert {{\Omega}}^{-}\eta \rangle .\end{equation} \tag{ 2 }$$

A mixing angle θ is introduced between the two components, which later on will play an important role in explaining a possible suppression of the three-body decays ${\Omega}\left(2012\right)\to {\Xi}\pi \bar{K}$ recently observed by the Belle collaboration [2]. Such angle can be constrained from data on the Ω*(2012) decays. In particular, in this paper we show that a possible scenario for the dominance of the two-body decay rates over three-body ones recently noticed by the Belle collaboration could occur due to the dominant admixture of the Ωη hadronic component in the Ω*(2012) state.

For convenience, in table 1 we present the quantum numbers (isospin I, spin-parity J^P ) and values of masses (current central values from the particle data group [33]) of the hadrons, which will be used in our calculations. Note that our approach is able to describe the molecular states as mixing of different Fock components and before it was successfully applied to the problem of the X(3872) state [30].

Table 1. Quantum numbers and masses of relevant hadrons.

Hadron	I	JP	Mass (MeV)
π±	1	0−	139.57061
π0	1	0−	134.977
${\bar{K}}^{0}$	$\frac{1}{2}$	0−	493.677
K−	$\frac{1}{2}$	0−	497.611
η	0	0−	547.862
${\bar{K}}^{{\ast}-}$	$\frac{1}{2}$	1−	891.76
K∗0	$\frac{1}{2}$	1−	895.55
Ξ0	$\frac{1}{2}$	${\frac{1}{2}}^{+}$	1314.86
Ξ−	$\frac{1}{2}$	${\frac{1}{2}}^{+}$	1321.71
Ξ∗0	$\frac{1}{2}$	${\frac{3}{2}}^{+}$	1531.8
Ξ∗−	$\frac{1}{2}$	${\frac{3}{2}}^{+}$	1535.0
Ω−	0	${\frac{3}{2}}^{+}$	1672.45
Ω∗−	0	${\frac{3}{2}}^{-}$	2012.4

The interaction of the Ω*(2012) state and their constituents is described by the phenomenological Lagrangian:

$$\begin{align}\hfill {\mathcal{L}}_{{{\Omega}}^{{\ast}}}\left(x\right)& ={g}_{{{\Omega}}^{{\ast}}}\enspace {\bar{{\Omega}}}_{\mu }^{{\ast}}\left(x\right)\enspace \int {\mathrm{d}}^{4}y\enspace {\Phi}\left({y}^{2}\right)\enspace \left[\frac{\mathrm{cos}\enspace \theta }{\sqrt{2}}\enspace {{\Xi}}_{i}^{{\ast}}\left(x+{\omega }_{K}y\right){\bar{K}}_{i}\left(x-{\omega }_{{{\Xi}}^{{\ast}}}y\right)\right.\hfill \\ \hfill & \left.\quad ,-\mathrm{sin}\enspace \theta \enspace {\Omega}\left(x+{\omega }_{\eta }y\right)\eta \left(x-{\omega }_{{\Omega}}y\right)\right]+\enspace \text{H.c.}\hfill \end{align} \tag{ 3 }$$

Here ${{\Xi}}_{i}^{{\ast}}=\left({{\Xi}}^{{\ast}0},{{\Xi}}^{{\ast}-}\right)$ and ${\bar{K}}_{i}=\left({K}^{-},{\bar{K}}^{0}\right)$ are the doublets of Ξ* hyperons and $\bar{K}$ mesons, Φ(y²) is the vertex form factor modeling the distribution of $\left({{\Xi}}^{{\ast}}\bar{K}\right)$ and (Ωη) constituents in the Ω* state, ${\omega }_{{{\Xi}}^{{\ast}}}={M}_{{{\Xi}}^{{\ast}}}/\left({M}_{{{\Xi}}^{{\ast}}}+{M}_{K}\right)$, ${\omega }_{K}={M}_{K}/\left({M}_{{{\Xi}}^{{\ast}}}+{M}_{K}\right)$, ω_Ω = M_Ω/(M_Ω + M_η ), and ω_η = M_η /(M_Ω + M_η ) are the fractions of the constituent masses obeying the conditions ${\omega }_{{{\Xi}}^{{\ast}}}+{\omega }_{K}=1$ and ω_Ω + ω_η = 1. For the Fourier transform of the Φ(y²) we use $\tilde {{\Phi}}\left({p}_{\mathrm{E}}^{2}\right)\doteq \mathrm{exp}\left(-{p}_{\mathrm{E}}^{2}/{{\Lambda}}^{2}\right)$, where p_E is the relative Jacobi momentum in Euclidean space and Λ is the dimensional parameter, which has a value of about 1 GeV. The cutoff parameter Λ characterizes distribution of the bound state Ω*(2012). Hence, Λ has a universal value independent of the parameter decay process.

Note that the |Ω*(2012)> state could also contain an admixture of a three s-quark component. In such a case the Fock state and the corresponding phenomenological Lagrangian describing the coupling of the Ω*(2012) to the constituents are modified as

$$\begin{equation}\vert {{\Omega}}^{{\ast}}\left(2012\right)\rangle =\mathrm{cos}\enspace \phi \enspace \left[\mathrm{cos}\enspace \theta \enspace \frac{\vert {{\Xi}}^{{\ast}0}{K}^{-}\rangle \enspace +\enspace \vert {{\Xi}}^{{\ast}-}{\bar{K}}^{0}\rangle }{\sqrt{2}}-\mathrm{sin}\enspace \theta \enspace \vert {{\Omega}}^{-}\eta \rangle \right]+\mathrm{sin}\enspace \phi \enspace \vert sss\rangle \end{equation} \tag{ 4 }$$

and

$$\begin{align}\hfill {\mathcal{L}}_{{{\Omega}}^{{\ast}}}\left(x\right)& ={g}_{{{\Omega}}^{{\ast}}}\enspace {\bar{{\Omega}}}_{\mu }^{{\ast}}\left(x\right)\enspace \int {\mathrm{d}}^{4}y\left\{{\Phi},\left({y}^{2}\right),\enspace ,\mathrm{cos}\enspace \phi \left[\frac{\mathrm{cos}\enspace \theta }{\sqrt{2}}\enspace {{\Xi}}_{i}^{{\ast}}\left(x+{\omega }_{K}y\right){\bar{K}}_{i}\left(x-{\omega }_{{{\Xi}}^{{\ast}}}y\right)\right.\right.\hfill \\ \hfill & \left.\quad ,-\mathrm{sin}\theta \enspace {\Omega}\left(x+{\omega }_{\eta }y\right)\eta \left(x-{\omega }_{{\Omega}}y\right)\right]+\enspace \int {\mathrm{d}}^{4}y\enspace {\Phi}\left({y}^{2}+{z}^{2}\right)\enspace \mathrm{sin}\enspace \phi \enspace {\varepsilon }^{abc}\hfill \\ \hfill & \quad \left.{\times},{s}^{a},\left(x+{\xi }_{+}\right),{s}^{b},\left(x+{\xi }_{-}\right),C,{\gamma }^{\mu },{s}^{b},\left(x-{\xi }_{+}-{\xi }_{-}\right)\right\}+\text{H.c.},\quad {\xi }_{{\pm}}=\frac{y}{3\sqrt{2}}{\pm}\frac{z}{\sqrt{6}}.\hfill \end{align} \tag{ 5 }$$

Now ϕ is an additional parameter, which defines the mixing of hadronic and quark components in the Ω*(2012). In this paper we proceed with a hadronic molecular scenario. The possible mixing of the hadronic molecular and quark components will be considered in an extended work. One should stress that our approach was already successfully applied to the mixing of hadronic and quark components in exotic bound states. In particular, in a series of papers [30] the exotic state X(3872) was considered in the mixing of a charm-anticharm component and a set of hadronic components (DD*), (Jψρ), and (Jψω), while the decay properties were successfully described.

The coupling ${g}_{{{\Omega}}^{{\ast}}}$ is fixed determined using the compositeness condition [23–30]

$$\begin{equation}{Z}_{{{\Omega}}^{{\ast}}}=1-{{\Sigma}}_{{{\Omega}}^{{\ast}}}^{T\prime }\left({M}_{{{\Omega}}^{{\ast}}}\right)\equiv 0,\end{equation} \tag{ 6 }$$

where ${{\Sigma}}_{{{\Omega}}^{{\ast}}}^{T\prime }$ is the derivative of the transversal contribution to the Ω*(2012) mass operator:

$$\begin{equation}{{\Sigma}}_{{{\Omega}}^{{\ast}}}^{\mu \nu }\left(p\right)={g}_{\perp }^{\mu \nu }{{\Sigma}}_{{{\Omega}}^{{\ast}}}^{T}\left(p\right)+\frac{{p}^{\mu }{p}^{\nu }}{{p}^{2}}{{\Sigma}}_{{{\Omega}}^{{\ast}}}^{L}\left(p\right),\end{equation} \tag{ 7 }$$

with ${g}_{\perp }^{\mu \nu }={g}^{\mu \nu }-{p}^{\mu }{p}^{\nu }/{p}^{2}$. Note, the longitudinal part of the mass operator vanishes due to the Lorenz transversality condition p^μ _μ (p) = 0, where _μ (p) is the polarization vector of the Ω(2012) state. The corresponding Feynman diagrams contributing to the mass operator ${{\Sigma}}_{{{\Omega}}^{{\ast}}}$, which are generated by the loops of the (Ξ^∗0 K⁻), $\left({{\Xi}}^{{\ast}-}{\bar{K}}^{0}\right)$, and (Ω⁻ η) constituents, are displayed in figure 1. In addition we have a free parameter θ, which is the mixing angle between the ${{\Xi}}^{{\ast}}\bar{K}$ and the Ωη hadronic molecular components of the Ω* state. One should stress that the mass operator of the Ω*(2012) state is represented by only two loop diagrams in terms of two types of the constituents ${{\Xi}}^{{\ast}}\bar{K}$ and Ωη. It is the diagonalized representation of the compositeness condition, which is fully equivalent to the formalism used in references [12, 18, 19] where the bound state is generated by scattering processes of its constituents, which includes both diagonal ${{\Xi}}^{{\ast}}\bar{K}\to {{\Xi}}^{{\ast}}\bar{K}$ and Ωη → Ωη and non-diagonal $\bar{K}{{\Xi}}^{{\ast}}\to \eta {\Omega}$ transitions.

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The expression for the mass operator ${{\Sigma}}_{{{\Omega}}^{{\ast}}}$ is given by

$$\begin{equation}\begin{gathered}{{\Sigma}}_{{{\Omega}}^{{\ast}}}^{\mu \nu }\left(p\right)=\frac{{\mathrm{cos}}^{2}\enspace \theta }{2}\enspace \left({{\Sigma}}_{{{\Omega}}^{{\ast}}}^{\mu \nu ,{{\Xi}}^{0}{K}^{-}}\left(p\right)+{{\Sigma}}_{{{\Omega}}^{{\ast}}}^{\mu \nu ,{{\Xi}}^{-}{\bar{K}}^{0}}\left(p\right)\right)\enspace +\enspace {\mathrm{sin}}^{2}\enspace \theta \enspace {{\Sigma}}_{{{\Omega}}^{{\ast}}}^{\mu \nu ,{\Omega}\eta }\left(p\right),\\ {{\Sigma}}_{{{\Omega}}^{{\ast}}}^{\mu \nu ,{H}_{1}{H}_{2}}\left(p\right)={g}_{{{\Omega}}^{{\ast}}}^{2}\enspace \int \frac{{\mathrm{d}}^{4}k}{{\left(2\pi \right)}^{4}i}{\tilde {{\Phi}}}^{2}\left(-{\left(k+p{\omega }_{2}\right)}^{2}\right)\enspace {S}_{{H}_{1}}^{\mu \nu }\left(k+p\right)\enspace {S}_{{H}_{2}}\left(k\right),\end{gathered}\end{equation} \tag{ 8 }$$

where

$$\begin{equation}{S}_{{H}_{1}}^{\mu \nu }\left(k\right)=\frac{1}{{M}_{{H}_{1}}- k /}\enspace \left[-{g}^{\mu \nu }+\frac{{\gamma }^{\mu }{\gamma }^{\nu }}{3}+\frac{2{k}^{\mu }{k}^{\nu }}{3{M}_{{H}_{1}}^{2}}+\frac{{\gamma }^{\mu }{k}^{\nu }-{\gamma }^{\nu }{k}^{\mu }}{3{M}_{{H}_{1}}}\right],\end{equation} \tag{ 9 }$$

$$\begin{equation}{S}_{{H}_{2}}\left(k\right)=\frac{1}{{M}_{{H}_{2}}^{2}-{k}^{2}}\end{equation} \tag{ 10 }$$

are the propagators of baryon H₁ with spin $\frac{3}{2}$ and of meson H₂ with spin 0. Here H₁ = Ξ^∗−, Ξ^∗0, Ω⁻ and ${H}_{2}={K}^{-},{\bar{K}}^{0},\eta$.

The coupling constant ${g}_{{{\Omega}}^{{\ast}}}$ deduced from the compositeness condition (6) reads

$$\begin{equation}\frac{{g}_{{{\Omega}}^{{\ast}}}^{2}}{16{\pi }^{2}}=\frac{1}{{I}_{{{\Omega}}^{{\ast}}}},\end{equation} \tag{ 11 }$$

where ${I}_{{{\Omega}}^{{\ast}}}$ is the structure integral

$$\begin{equation}\begin{gathered}{I}_{{{\Omega}}^{{\ast}}}=\frac{{\mathrm{cos}}^{2}\enspace \theta }{2}\enspace \left({I}_{{{\Omega}}^{{\ast}}}^{{{\Xi}}^{{\ast}0}{K}^{-}}+{I}_{{{\Omega}}^{{\ast}}}^{{{\Xi}}^{{\ast}-}{\bar{K}}^{0}}\right)\enspace +\enspace {\mathrm{sin}}^{2}\enspace \theta \enspace {I}_{{{\Omega}}^{{\ast}}}^{{\Omega}\eta },\\ {I}_{{{\Omega}}^{{\ast}}}^{{H}_{1}{H}_{2}}=\underset{0}{\overset{\infty }{\int }}\enspace \frac{\mathrm{d}{\alpha }_{1}\mathrm{d}{\alpha }_{2}}{{{\Delta}}^{3}}\enspace {R}_{{{\Omega}}^{{\ast}}}\enspace {\mathrm{e}}^{-{w}_{{{\Omega}}^{{\ast}}}}\enspace ,\quad {\Delta}=2+{\alpha }_{1}+{\alpha }_{2}\end{gathered}\end{equation} \tag{ 12 }$$

and

$$\begin{equation}\begin{gathered}{R}_{{{\Omega}}^{{\ast}}}=\left(1+\frac{1}{3{\mu }_{{H}_{1}^{2}}{\Delta}}\right)\enspace \left[{\alpha }_{2}+2{\omega }_{1}+2{\mu }_{{{\Omega}}^{{\ast}}}\left({\mu }_{{H}_{1}}+{\mu }_{{{\Omega}}^{{\ast}}}\frac{{\alpha }_{2}+2{\omega }_{1}}{{\Delta}}\right)\enspace z\left({\alpha }_{1},{\alpha }_{2}\right)\right],\\ {w}_{{{\Omega}}^{{\ast}}}=\frac{{\left({\mu }_{{H}_{1}}{\alpha }_{1}-{\mu }_{{H}_{2}}{\alpha }_{2}\right)}^{2}}{{\Delta}}+\frac{{\left({\mu }_{{H}_{1}}+{\mu }_{{H}_{2}}\right)}^{2}-{\mu }_{{{\Omega}}^{{\ast}}}^{2}}{{\Delta}}\enspace z\left({\alpha }_{1},{\alpha }_{2}\right),\\ {\mu }_{i}=\frac{{M}_{i}}{{\Lambda}}\enspace ,\quad z\left({\alpha }_{1},{\alpha }_{2}\right)={\alpha }_{1}{\alpha }_{2}+2{\alpha }_{1}{\omega }_{1}^{2}+2{\alpha }_{2}{\omega }_{2}^{2}.\end{gathered}\end{equation} \tag{ 13 }$$

The leading diagrams contributing to the strong decays of the Ω* state are shown in figure 2: (a) two-body decay ${{\Omega}}^{{\ast}}\to {\Xi}\bar{K}$, (b) three-body decay ${{\Omega}}^{{\ast}}\to {\Xi}\pi \bar{K}$. Note that in agreement with references [15–19] the two-body decay ${{\Omega}}^{{\ast}}\to {\Xi}\bar{K}$ proceeds via the hadronic loops $\left({{\Xi}}^{{\ast}}\bar{K}\right)$ and (Ωη) involving the constituents of the Ω* state. The three-body decay ${{\Omega}}^{{\ast}}\to {\Xi}\pi \bar{K}$ is described by the two-cascade tree-level diagram, where the Ω* first couples to the constituents Ξ* and $\bar{K}$ and then Ξ* decays via the dominant mode into Ξ and π. For calculation of the diagrams in figure 2 we built the Lagrangian including the interaction of the Ω* with its hadronic constituents ${{\Xi}}^{{\ast}}\bar{K}$ and Ωη, and two additional terms describing the couplings Ξ*ΞKK (ΞΩηK) and Ξ*Ξπ.

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The Ξ*Ξπ interaction is described by the phenomenological Lagrangian

$$\begin{equation}{\mathcal{L}}_{{{\Xi}}^{{\ast}}{\Xi}\pi }=\frac{{g}_{{{\Xi}}^{{\ast}}{\Xi}\pi }}{{M}_{{{\Xi}}^{{\ast}}}}\enspace {\bar{{\Xi}}}_{\mu }^{{\ast}}{\partial }^{\mu } \overrightarrow {\pi } \overrightarrow {\tau }\enspace {\Xi}\enspace +\enspace \text{H.c.},\end{equation} \tag{ 14 }$$

where the dimensionless coupling ${g}_{{{\Xi}}^{{\ast}}{\Xi}\pi }$ is fixed from data on the Ξ* → Ξπ decay width. In particular, the corresponding two-body decay width reads

$$\begin{equation}{\Gamma}\left({{\Xi}}^{{\ast}}\to {\Xi}\pi \right)=\frac{{g}_{{{\Xi}}^{{\ast}}{\Xi}\pi }^{2}}{64{M}_{{{\Xi}}^{{\ast}}}^{7}}\enspace {\lambda }^{3/2}\left({M}_{{{\Xi}}^{{\ast}}}^{2},{M}_{{\Xi}}^{2},{M}_{\pi }^{2}\right)\enspace \left[{\left({M}_{{{\Xi}}^{{\ast}}}+{M}_{{\Xi}}\right)}^{2}-{M}_{\pi }^{2}\right].\end{equation} \tag{ 15 }$$

Using the measured central values of

$$\begin{equation}\begin{gathered}{{\Gamma}}_{\text{total}}\left({{\Xi}}^{{\ast}0}\right)=2{\Gamma}\left({{\Xi}}^{{\ast}0}\to {{\Xi}}^{-}{\pi }^{+}\right)+{\Gamma}\left({{\Xi}}^{{\ast}0}\to {{\Xi}}^{0}{\pi }^{0}\right)=9.14\enspace \text{MeV},\\ {{\Gamma}}_{\text{total}}\left({{\Xi}}^{{\ast}-}\right)=2{\Gamma}\left({{\Xi}}^{{\ast}-}\to {{\Xi}}^{0}{\pi }^{-}\right)+{\Gamma}\left({{\Xi}}^{{\ast}-}\to {{\Xi}}^{-}{\pi }^{0}\right)=9.9\enspace \text{MeV}\end{gathered}\end{equation} \tag{ 16 }$$

one gets ${g}_{{{\Xi}}^{{\ast}}{\Xi}\pi }=6.79$ from the Ξ^∗0 set and ${g}_{{{\Xi}}^{{\ast}}{\Xi}\pi }=6.71$ from Ξ^∗−. In the following we will use the averaged value ${g}_{{{\Xi}}^{{\ast}}{\Xi}\pi }=\left(6.79+6.71\right)/2=6.75$.

The couplings Ξ*ΞKK and ΞΩηK are generated by a phenomenological Lagrangian:

$$\begin{equation}{\mathcal{L}}_{{\Gamma}DB}={g}_{{\Gamma}DB}\enspace {\bar{B}}^{mk}\enspace {\gamma }^{5}\enspace {{\Gamma}}_{lj}^{\mu }\enspace {D}_{\mu }^{\text{ijk}}\enspace {{\epsilon}}^{\text{ilm}}\enspace +\enspace \text{H.c.},\end{equation} \tag{ 17 }$$

where B^mk and D^ijk are the octet and decuplet baryon fields, ^ilm is the rang-3 Levi–Civita tensor. Γ_μ is the chiral connection which in absence of external vector and axial fields is defined as

$$\begin{equation}{{\Gamma}}_{\mu }=\frac{1}{2}\left[{u}^{+},{\partial }_{\mu }u\right]=\frac{1}{4{F}^{2}}\left[\hat{{\Phi}},{\partial }_{\mu }\hat{{\Phi}}\right]+\mathcal{O}\left({\hat{{\Phi}}}^{4}\right),\end{equation} \tag{ 18 }$$

where $\hat{{\Phi}}=\sum _{i=1}^{8}{\phi }_{i}{\lambda }_{i}$ is the octet matrix of pseudoscalar mesons.

The dimensionless coupling g_ΓDB can be fixed using the following arguments. We constraint g_ΓDB based on the use of SU(2) chiral Lagrangians involving pseudoscalar, spin-$\frac{1}{2}$ and spin-$\frac{3}{2}$ fields and apply the extension to the SU(3) case [34–36]. In the SU(2) case, the g_ΓNΔ coupling is obtained by comparing it to three similar couplings describing the coupling of the two nucleon and nucleon-Δ pair to the pion (g_πNN , g_πNΔ = f_πNΔ M_N /M_π ) and of two nucleons with the chiral connection (g_ΓNN = 1). On phenomenological grounds we postulate that all four couplings are related as:

$$\begin{equation}\frac{{g}_{{\Gamma}N{\Delta}}}{{g}_{{\Gamma}NN}}=\frac{{g}_{\pi N{\Delta}}}{{g}_{\pi NN}}.\end{equation} \tag{ 19 }$$

From the ansatz (19) one gets

$$\begin{equation}{g}_{{\Gamma}N{\Delta}}={g}_{{\Gamma}NN}\enspace \frac{{g}_{\pi N{\Delta}}}{{g}_{\pi NN}}=\frac{{f}_{\pi N{\Delta}}}{{g}_{\pi NN}}\enspace \frac{{M}_{N}}{{M}_{\pi }}.\end{equation} \tag{ 20 }$$

Next using following fundamental constants of hadron physics: πNN coupling g_πNN ≃ 13.4 and the πNΔ coupling f_πNΔ ≃ 2 we obtain g_ΓNΔ ≃ 1. Extending our considerations from SU(2) to SU(3) we finally arrive at the coupling g_ΓBD ≃ 1 defining the Lagrangian in equation (17), whose value will be used in our numerical analysis.

The two- and three-body decay widths of the Ω* state are calculated according to the standard expressions:

$$\begin{equation}{\Gamma}\left({{\Omega}}^{{\ast}}\to {\Xi}\bar{K}\right)=\frac{{g}_{{{\Omega}}^{{\ast}}{\Xi}K}^{2}}{192\pi {M}_{{{\Omega}}^{{\ast}}}^{3}}\enspace {\lambda }^{3/2}\left({M}_{{{\Omega}}^{{\ast}}}^{2},{M}_{{\Xi}}^{2},{M}_{K}^{2}\right)\enspace \left[{\left({M}_{{{\Omega}}^{{\ast}}}-{M}_{{\Xi}}\right)}^{2}-{M}_{K}^{2}\right],\end{equation} \tag{ 21 }$$

$$\begin{equation}{\Gamma}\left({{\Omega}}^{{\ast}}\to {\Xi}\pi \bar{K}\right)=\frac{1}{1024{\pi }^{3}{M}_{{{\Omega}}^{{\ast}}}^{3}}\enspace \underset{{\left({M}_{{\Xi}}+{M}_{\pi }\right)}^{2}}{\overset{{\left({M}_{{{\Omega}}^{{\ast}}}-{M}_{K}\right)}^{2}}{\int }}\mathrm{d}{s}_{2}\enspace \underset{{s}_{1}^{-}}{\overset{{s}_{1}^{+}}{\int }}\mathrm{d}{s}_{1}\enspace \sum _{\text{pol}}{\vert {M}_{\text{inv}}\vert }^{2}.\end{equation} \tag{ 22 }$$

M_inv is the invariant matrix element for the ${{\Omega}}^{{\ast}}\to {\Xi}\pi \bar{K}$ transition given by

$$\begin{equation}{M}_{\text{inv}}={g}_{{{\Omega}}^{{\ast}}}\enspace {\Phi}\left(-{\left[{p}_{1}{w}_{{{\Xi}}^{{\ast}}}-q{w}_{K}\right]}^{2}\right)\enspace \frac{\mathrm{cos}\enspace \theta }{\sqrt{2}}\enspace \frac{{g}_{{{\Xi}}^{{\ast}}{\Xi}\pi }}{{M}_{{{\Xi}}^{{\ast}}}}\enspace {c}_{\pi }\enspace {\bar{u}}_{{\Xi}}\left(q\right)\enspace {S}_{{{\Xi}}^{{\ast}};\mu \nu }\left(q\right)\enspace {u}_{{{\Omega}}^{{\ast}}}^{\nu }\left(p\right)\enspace {p}_{3}^{\mu }\end{equation} \tag{ 23 }$$

and the Ω*ΞK coupling ${g}_{{{\Omega}}^{{\ast}}{\Xi}K}$ is evaluated from the diagram in figure 2(a). Here we introduced the following notations. s₁ and s₂ are the invariant Mandelstam variables ${s}_{1}={\left({p}_{1}+{p}_{2}\right)}^{2}={\left(p-{p}_{3}\right)}^{2}$ and ${s}_{2}={\left({p}_{2}+{p}_{3}\right)}^{2}={\left(p-{p}_{1}\right)}^{2}={q}^{2}$ which are related to the momenta p, q, p₁, p₂, and p₃ of the hadrons–Ω*, Ξ*, K, Ξ, and π, respectively. The variable s₁ has the upper/lower limits ${s}_{1}^{{\pm}}$ with:

$$\begin{align}\hfill {s}_{1}^{{\pm}}& ={M}_{K}^{2}+{M}_{{\Xi}}^{2}+\frac{1}{2\enspace s}\left(s-{s}_{2}-{M}_{K}^{2}\right)\left({s}_{2}+{M}_{{\Xi}}^{2}-{M}_{\pi }^{2}\right)\hfill \\ \hfill & \quad {\pm}\frac{1}{2{s}_{2}}\enspace {\lambda }^{1/2}\left(s,{s}_{2},{M}_{K}^{2}\right)\enspace {\lambda }^{1/2}\left({s}_{2},{M}_{{\Xi}}^{2},{M}_{\pi }^{2}\right),\hfill \end{align} \tag{ 24 }$$

where λ(x, y, z) = x² + y² + z² − 2xy − 2yz − 2xz. ${\bar{u}}_{{\Xi}}\left(q\right)$ and ${u}_{{{\Omega}}^{{\ast}}}\left(p\right)$ are the spinors corresponding to Ξ and Ω* baryons, respectively. ${\Phi}\left(-{\left[{p}_{1}{w}_{{{\Xi}}^{{\ast}}}-q{w}_{K}\right]}^{2}\right)$ is the correlation function from the Lagrangian (3), which can be reduced to

$$\begin{equation}{\Phi}\left(-{\left[{p}_{1}{w}_{{{\Xi}}^{{\ast}}}-q{w}_{K}\right]}^{2}\right)=\mathrm{exp}\left[\frac{{M}_{{{\Xi}}^{{\ast}}}{M}_{K}}{{{\Lambda}}^{2}}\enspace \left(1-\frac{{M}_{{{\Omega}}^{{\ast}}}^{2}}{{\left({M}_{{{\Xi}}^{{\ast}}}+{M}_{K}\right)}^{2}}\right)\right].\end{equation} \tag{ 25 }$$

The factor ${c}_{\pi }=\sqrt{2}$ occurs for modes with charged pions π^± and c_π = 1 for modes with the neutral pion π⁰. ${S}_{{{\Xi}}^{{\ast}}}^{\mu \nu }\left(q\right)$ is the Ω* (fermion spin-$\frac{3}{2}$) propagator introduced in equation (8). Note, in our calculations of the three-body decays of the Ω(2012) we use the kinematical formula (24), where the invariant mass squared M²(Ξπ) is defined by the Mandelstam variable s₂. We do not impose the condition 1.49 GeV < M(Ξπ) < 1.53 GeV used in reference [2] for an efficient signal selection.

In figures 3 and 4 we show our predictions for the respective charged combinations of two- and three-body decay widths of the Ω*(2012) and their sums. Results are indicated for the values of Λ = 1 and 1.5 GeV, respectively, and for the mixing angle θ varied from 0 to 90 grad. In figure 5 we plot our predictions for the ratios R₁ and R₂ of equation (1) of the three- and two-body decays and compare them with the upper limits derived by the Belle collaboration in reference [2]. One can see that an increase in the admixture of the hadronic component Ωη generates the suppression of the three-body decay widths. For the mixing angle bigger 70° (73°) for Λ = 1 GeV and bigger 55° (61°) for Λ = 1.5 GeV our ratios are consistent with Belle results for R₁ and R₂. To get a handle on the parameters θ and Λ further data on the Ω*(2012) decays are needed. The predictions presented here are strongly correlated with the assumption that the Ω*(2012) has a hadronic molecular structure.

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Finally, we discuss our main numerical results. Varying the model scale parameter Λ from 1 to 1.5 GeV we find that the magnitude of the two-body decay rates slightly increase, while the three-body rates decrease. The relative contribution of two to three-body decays is governed by the mixing angle θ—mixing of the ${{\Xi}}^{{\ast}}\bar{K}$ and Ωη hadronic components. An increase of θ generates the suppression of the three-body decay widths in comparison with the two-body ones. For the total two- and three-body decays rates we get the following numerical results

$$\begin{equation}{\Gamma}\left({{\Omega}}^{{\ast}}\to {\Xi}\bar{K}\right)=2.9{\pm}1.6\enspace \text{MeV}\enspace ,\quad {\Gamma}\left({{\Omega}}^{{\ast}}\to {\Xi}\pi \bar{K}\right)=1.5{\pm}1.5\enspace \text{MeV}\end{equation} \tag{ 26 }$$

for Λ = 1 GeV and θ ∈ [0, π/2],

$$\begin{equation}{\Gamma}\left({{\Omega}}^{{\ast}}\to {\Xi}\bar{K}\right)=6{\pm}3.4\enspace \text{MeV}\enspace ,\quad {\Gamma}\left({{\Omega}}^{{\ast}}\to {\Xi}\pi \bar{K}\right)=1.4{\pm}1.4\enspace \text{MeV}\end{equation} \tag{ 27 }$$

for Λ = 1.5 GeV and θ ∈ [0, π/2]. The errors in equations (21) and (22) are mainly due to the variation of the mixing parameter θ. More data on the Ω*(2012) the needed to further constrain of the parameter θ and to reduce the errors in our predictions. The upper limits of equation (1) set by the Belle collaboration [2] for the ratios of three- and two-body decay rates of the Ω(2012) are fulfilled for following values of the θ angle:

• (a)  
  Λ = 1 GeV

  $$\begin{equation}{R}_{1}{< }9.3\%\enspace \text{at}\enspace \theta {\geqslant}70{^{\circ}}\enspace ,\quad {R}_{2}{< }7.8\%\enspace \text{at}\enspace \theta {\geqslant}73{^{\circ}}.\end{equation} \tag{ 28 }$$
• (b)  
  Λ = 1.5 GeV

  $$\begin{equation}{R}_{1}{< }9.3\%\enspace \text{at}\enspace \theta {\geqslant}55{^{\circ}}\enspace ,\quad {R}_{2}{< }7.8\%\enspace \enspace \enspace \enspace \text{at}\enspace \theta {\geqslant}61{^{\circ}}.\end{equation} \tag{ 29 }$$

  We also estimated limits for the ratio of three- to two-body modes

  $$\begin{equation}R=\frac{B\left({\Omega}\left(2012\right)\to {\Xi}\left(1530\right)\left(\to {\Xi}\pi \right)\bar{K}\right)}{B\left({\Omega}\left(2012\right)\to {\Xi}\bar{K}\right)}.\end{equation} \tag{ 30 }$$

  For θ ⩾ 81.3° at Λ = 1 GeV and θ ⩾ 72.8° at Λ = 1.5 GeV we fulfill the limits of the Belle Collaborations R < 11.9%. Hence a sizable Ωη hadronic component in the Ω*(2012) leads to a suppression of the ${\Xi}\pi \bar{K}$ mode relative to the ${\Xi}\bar{K}$ channel. To get a further handle on the parameters θ and Λ in the context of the hadronic structure precise data on the Ω*(2012) decays are needed. The prediction given here can hopefully support a possible structure interpretation of the Ω*(2012).

This work was funded by Bundesministerium für Bildung und Forschung 'Verbundprojekt 05P2018-Ausbau von ALICE am LHC: Jets und partonische Struktur von Kernen' (Förderkennzeichen No. 05P18VTCA1), 'Verbundprojekt 05A2017-CRESST-XENON" (Förderkennzeichen 05A17VTA)', the Carl Zeiss Foundation (Project Gz: 0653-2.8/581/2), ANID PIA/APOYO AFB180002 (Chile), FONDECYT (Chile) Grant No. 1191103, Tomsk State and Tomsk Polytechnic University Competitiveness Enhancement Programs (Russia).