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The observation of five Ωc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega _c $$\end{document}= ssc states by LHCb [Aaij et al. Phys. Rev. Lett. 118, 182001 (2017)] and the confirmation of four of them by Belle [Yelton et al. Phys. Rev. D 97, 051102 (2018)], may represent an important milestone in our understanding of the quark organization inside hadrons. By providing results for the spectrum of Ωc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega _{ c}$$\end{document} baryons and predictions for their Ξc+K-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varXi _{ c}^{+}K^{-}$$\end{document} and Ξc′+K-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varXi _{ c}'^{+}K^{-}$$\end{document} decay amplitudes within an harmonic oscillator based model, we suggest a possible solution to the Ωc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega _{c}$$\end{document} quantum number puzzle and we extend our mass and decay width predictions to the Ωb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega _b$$\end{document} states. Finally, we discuss why the set of Ωc(b)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega _{ c(b)}$$\end{document} baryons is the most suitable environment to test the validity of three-quark and quark–diquark effective degrees of freedom.


Introduction
The discovery of new resonances always enriches the present experimental knowledge of the hadron zoo, and it also provides essential information to explain the fundamental forces that govern nature. As the hadron mass patterns carry information on the way the quarks interact one another, they provide a means of gaining insight into the fundamental binding mechanism of matter at an elementary level.
In 2017, the LHCb Collaboration announced the observation of five narrow Ω c states in the Ξ + c K − decay channel [1]: Ω c (3000), Ω c (3050), Ω c (3066), Ω c (3090) and Ω c (3119). They also reported the observation of another structure around 3188 MeV, the so-called Ω c (3188), though they did not have enough statistical significance to interpret it as a genuine resonance [1]. Later, Belle observed five resonant states in the Ξ + c K − invariant mass distribution and unambiguously a e-mail: santopinto@ge.infn.it confirmed four of the states announced by LHCb, Ω c (3000), Ω c (3050), Ω c (3066), and Ω c (3090), but no signal was found for the Ω c (3119) [2]. Belle also measured a signal excess at 3188 MeV, corresponding to the Ω c (3188) state reported by LHCb [2]. A comparison between the results reported by the two collaborations is displayed in Table 1. Here, it is shown that the Ω c (3188), even if not yet confirmed, was seen both by LHCb and Belle, while, on the contrary, the Ω c (3119) was not observed by Belle. It is also worth to mention that the LHCb collaboration has just announced the observation of a new bottom baryon, Ξ b (6227) − , in both Λ 0 b K − and Ξ 0 b π decay modes [3], and of two bottom resonances, Σ b (6097) ± , in the Λ 0 b π ± channels [4]. However, neither LHCb nor Belle were able to measure the Ω c angular momenta and parities. For this reason, several authors tried to provide different quantum number assignments for these states. The current Ω c puzzle consists in the discrepancy between the experimental results, reported by LHCb [1] and Belle [2], and the existing theoretical predictions [5][6][7][8][9]. Indeed, for a given Ω c experimental state, more than one quantum number assignment was suggested [5]. In particular, the Ω c (3119) was allocated to possibly be a J P = 1 2 + or a J P = 3 2 + state [7], while the authors in Ref. [8] proposed a J P = 5 2 − assignment.
From the previous discussion it comes out that, in the case of the Ω c (3119), not only the quantum number assignments are not univocal, but also the quark structure of this baryon is still unclear. The issues we have to deal with are not restricted to the contrasts between the different interpretations provided in the previous studies, they are also related to the discrepancies on the quantum number assignments within a given study. For example, in Ref. [9] the authors provided different J P assignments for the Ω c (3066) and Ω c (3090) based on mass and decay width estimates. Moreover, the nature of the Ω c (3188) state is not addressed in these studies [5][6][7][8][9]. These divergences between the theoretical interpretations created a puzzle which needs to be addressed urgently.
In the present article, we first study the Ω c -mass spectra by estimating the contributions due to spin-orbit interactions, spin-, isospin-and flavour-dependent interaction from the well-established charmed baryon mass spectrum. We reproduce quantitatively the spectrum of the Ω c states within a harmonic oscillator hamiltonian plus a perturbation term given by spin-orbit, isospin and flavour dependent contributions (Sects. 2.1 and 2.2). Based on our results, we describe these five states as P-wave λ-excitations of the ssc system; we also calculate their Ξ + c K − and Ξ + c K − decay widths (Sect. 2.3). Similarly to Refs. [10][11][12], we suggest a molecular interpretation of the Ω c (3119) state, which was not observed by Belle. Later, we extend our mass and decay width predictions to the Ω b sector, which will be useful for future experimental searches. Finally, we calculate the mass splitting between the ρand λ-mode excitations of Ω c(b) resonances (see Fig. 1 upper-pannel). This calculation is fundamental to access to inner heavy-light baryon structure, as the presence or absence of ρ-mode excitations in the experimental spectrum will be the key to discriminate between the three-quark (see Fig. 1 upper-pannel) and the quark-diquark structures (see Fig. 1 lower-pannel), as it will be discussed in Sect. 3.

S-and P-wave ss Q states.
The three-quark system (ss Q, where Q = c or b) Hamiltonian can be written in terms of two coordinates [13], ρ and λ, which encode the system spatial degrees of freedom (see Fig. 1). Let m ρ = m s and m λ = 3m s m Q 2m s +m Q be the ss Q system reduced masses; then, the ρand λ-mode frequencies are ω ρ,λ = 3K Q m ρ,λ , where K Q is the spring constant, which implies that in three equal-mass-quark baryons, in which m ρ = m λ , the λand ρ-orbital excitation modes are completely mixed together. By contrast, in heavy-light baryons, in which m ρ m λ , the two excitation modes can be decoupled from each other as long as the light-heavy quark mass difference increases. First of all, we construct the ssc and ssb ground and excited states to establish the quantum numbers of the five confirmed Ω c states. A single quark is described by its spin, flavor and color quantum numbers. As a fermion, its spin is S = 1 2 , its flavor, spin-flavor and color representations are 3 f , 6 sf , and 3 c , respectively. An ss Q state, ss Q, S ρ , S tot , l ρ , l λ , J , is characterized by total angular momentum J = l ρ + l λ + S tot , where S tot = S ρ + 1 2 . In order to construct an ss Q color singlet state, the light quarks must transform under SU c (3) as the anti-symmetric3 c representation. The Pauli principle postulates that the wave function of identical fermions must be anti-symmetric for particle exchange. Thus, the ss spin-flavor and orbital wave functions have the same permutation symmetry: symmetric spin-flavor in S-wave, or antisymmetric spin-flavor in antisymmetric Pwave. Two equal flavour quarks are necessarily in the 6 f flavor-symmetric state. Thus, they are in an S-wave symmet-ric spin-triplet state, S ρ = 1, or in a P-wave antisymmetric spin-singlet state, S ρ = 0.

Mass spectra of Ω Q states
We make use of a three-dimensional harmonic oscillator hamiltonian (h.o.) plus a perturbation term given by spinorbit, isospin and flavour dependent contributions: here S, I and C 2 (SU(3) f ) are the spin, the isospin and the is the three-dimensional harmonic oscillator Hamiltonian written in terms of Jacobi coordinates, ρ and λ, and conjugated momenta, p ρ and p λ , whose eigenvalues are 3  Table 2. We estimate the mass splittings due to the spin-orbit, spin-, isospin-and flavor-dependent interactions from the well established charmed (bottom) baryon mass spectrum. The spin-orbit interaction, which is mysteriously small in light baryons [16][17][18], turns out to be fundamental to describe the heavy-light baryon mass patterns, as it is clear from those of the recently observed Ω c states. The spin-, isospin-, and flavour-dependent interactions are necessary to reproduce the masses of charmed baryon ground states, as observed in Ref. [19]. By means of these estimates, we predict in a parameterfree procedure the spectrum of the ss Q excited states constructed in the previous section. The predicted masses of the λand ρ-orbital excitations of the Ω c and Ω b baryons are reported in Tables 3 and 4, respectively. In particular, Table 3 shows that we are able to reproduce quantitatively the mass spectra of the Ω c states observed both by LHCb and Belle; the latter are reported in Table 1.
We estimate the energy splitting due to the spin-spin interaction from the (isospin-averaged) mass difference between Σ * c (2520) and Σ c (2453). This value (65 ± 8 MeV) agrees with the mass difference between Ω c (2695) and Ω * c (2770), a value close to 71 MeV. As a consequence, the spin-spin mass splitting between two orbitally excited states characterized by the same flavor configuration but different spins, specifically S tot = 1 2 and S tot = 3 2 , is around 65 MeV plus corrections due the spin-orbit contribution which can be calculated, for example, from the Λ c (2595)-Λ c (2625) mass difference. According to the quark model, Λ c (2595) and Λ c (2625) are the charmed counterparts of Λ(1405) and Λ(1520), respectively; their spin-parities are 1 2 − and 3 2 − , and their mass difference, about 36 MeV, is due to spin-orbit effects.
In the charm sector, the mass splitting due to the flavordependent interaction can be estimated from the mass difference between Ξ c and Ξ c , whose isospin-averaged masses are 2469.37 MeV and 2578.1 MeV, respectively; this leads to a value of 109 MeV, approximately. The bottom partner of Ξ c and Ξ c are Ξ b and Ξ b , with masses 5793.2 MeV and 5935.02 MeV, respectively. Therefore, in the bottom sector the flavordependent interaction gives a contribution of about 142 MeV, which is more than 30% larger than in the charm sector. The mass difference between the lightest charmed ground states, Σ c and Λ c , is related to the different isospin and flavor structures of the light quark multiplets: Λ c is an isospin-singlet state belonging to an SU(3) f flavor anti-triplet, while Σ c is an isospin-triplet state belonging to an SU(3) f flavor sextet. In the bottom sector, the isospin-flavor contribution to the Table 3 Our ssc state quantum number assignments (first column), predicted masses (second column) and open-flavor strong decay widths into Ξ + c K − and Ξ + c K − channels (fourth column) are compared with the experimental masses (third column) and total decay widths (fifth column) [1,15]. An ssc state, ssc, S ρ , S tot , l ρ , l λ , J , is characterized by total angular momentum J = l ρ + l λ + S tot , where S tot = S ρ + 1 2 .
Our results are compatible with the experimental data, the predicted partial decay widths being lower than the total measured decay widths. Masses of states denoted with † are used as inputs while all the others are predictions; partial decay widths denoted with † † and with † † † are zero for phase space and for selection rules, respectively  We summarize all our proposed quantum number assignments for both Ω c and Ω b states in Figs. 2 and 3, respectively. In the charm sector, we find a good agreement between the mass pattern predicted for the spectrum and the experimental data: in particular, with the exception of the lightest and the heaviest resonant states, Ω c (3000) and Ω c (3188), respectively, also the absolute mass predictions are in agreement within the experimental error, which is very small (less than 1 MeV).

Decay widths of ss Q states
In the following, we compute the strong decays of ss Q baryons in sq Q − K (q = u, d) final states by means of the 3 P 0 model [20][21][22][23] (see Appendix A).
In the 3 P 0 model, the parameters depend on the harmonic oscillator frequency of the initial and final states. For charmed baryons, we expect the parameters α ρ and α λ to lie in the range 0.4-0.7 GeV. In principle, the values of the α ρ and α λ h.o. parameters of lower-and higher-lying resonances should be different; see e.g. Ref. [24]. However, as widely discussed in the literature, it is customary to use constant values for α ρ and α λ . We also prefer not to take α ρ and α λ as free parameters, but to express them in terms of the baryon ρand λ-mode frequencies, ω ρ,λ = 3K Q /m ρ,λ , using the relation α 2 ρ,λ = ω ρ,λ m ρ,λ for both initial-and finalstate baryon resonances; see Appendix B. In light of this, the only free parameter is the pair creation strength, γ 0 = 9.2, which is fitted to the reproduction of the Ω c (3066) width. The frequency of K meson is set to be ω c = 0.46 GeV [25].
Tables 3 and 4 report our The Ξ + c K − decay channel is where the Ω c states were observed by LHCb and Belle; we also consider the Ξ + c K − channel, which contributes to the Ω c (3090) and Ω c (3188) open-flavor decay widths. Both the Ξ + c K − branching ratios and the quantum numbers of the Ω c 's are unknown; we only have experimental informations on their total widths, Γ tot . Thus, our predictions have to satisfy the constraint: In light of this, we state that our strong decay width results, based both on our mass estimates and quantum number assignments, are compatible with the present experimental data. In particular, the λ-mode decay widths of the Ω c states are in the order 1 MeV, while the Ξ + c K − decay mode of the two ρ-excitations, ssc, 0, 1 2 , 1 ρ , 0 λ , 1 2 and ssc, 0, 1 2 , 1 ρ , 0 λ , 3 2 , is forbidden by spin conservation. Similar considerations can be applied to the decay widths of ρmode Ω b states. The presence of inconsistencies between our predictions for the mass spectrum and the open-flavor strong decay widths of Table 3 can have two possible explanations: Table 4 Our ssb state quantum number assignments (first column), predicted masses (second column) and open-flavor strong decay widths (fourth column) are compared with the experimental masses (third column) and total decay widths (fifth column) [15]. An ssb state, ssb, S ρ , S tot , l ρ , l λ , J , is characterized by total angular momentum J = l ρ + l λ + S tot , where S tot = S ρ + 1 2 . Partial decay widths denoted with † † and with † † † are zero for phase space and for selection rules, respectively
In conclusion, in addition to our mass estimates, also the 3 P 0 model results suggest that the five Ω c resonances, Ω c (3000), Ω c (3050), Ω c (3066), Ω c (3090), and Ω c (3188), could be interpreted as ssc ground-state P-wave λ-excitations. In principle, both the Ω c (3090) and Ω c (3119) resonances observed by LHCb are compatible with the properties (mass and decay width) of the ssc, 3 2 , 1 λ , 3 2 theoretical state. As Belle could neither confirm nor deny the existence of the Ω c (3119), given the low significance of its results for the previous state (0.4σ ), we prefer to: 1) Assign ssc, 3 2 , 1 λ , 3 2 to the Ω c (3090); 2) Interpret the Ω c (3119) as a Ξ * c K bound state [10][11][12], the Ω c (3119) lying 22 MeV below the Ξ * c K threshold. See Fig. 4. Additionally, in Table 5, we present a comparison of different quantum number assignments for the Ω c states.

Comparison between the three-quark and quark-diquark structures
In the light sector, the quark model reproduces successfully the baryon spectrum by assuming that the constituent u, d and s quarks have roughly the same mass. This implies that the two oscillators, ρ and λ, have approximately the same frequency, ω ρ ω λ ; therefore, the ρand λ-excitations are degenerate. By contrast, in the case of heavy-light baryons m ρ m λ ; thus, the two excitation modes are decoupled from one another; specifically ω ρ − ω λ 130 MeV for Ω c states and ω ρ − ω λ 150 MeV for Ω b states. Thus, the heavy-light baryon sector is the most suitable environment to test what are the correct effective spatial degrees of freedom for reproducing the mass spectra, as the presence or absence of ρ-mode excitations in the spectrum will be the key to discriminate between the three-quark and the quark-diquark structures (see Fig. 1). Specifically, if the predicted four ρ-excitations, Ω c (3146), Ω c (3182), Ω b (6452), and Ω b (6460), are not observed, then the other Ω c states will be characterized by a quark-diquark structure.
Finally, we observe that in the case of a quark-diquarkpicture experimental confirmation, the model Hamiltonian employed, see Eq. (1), still holds because the quark-diquark h.o. Hamiltonian is the limit of the three-quark h.o. Hamiltonian, Eq. (2), when we freeze the ρ coordinate: 2 Here m D = 2m s is the diquark mass. Indeed, the mass spectrum predicted with this definition of H h.o. is the same as that reported in Figs. 2 and 3, but without the frozen ρ excitations. We observe also that, if the quark-diquark scenario turns out to be the correct one, the suppression of the spinspin interaction that we found going from the charmed to the bottom sector is consistent with the heavy quark symmetry, this suppression being an indication that heavy quark effective theory, HQET, still holds also in the heavy-light baryon sector.
The latter was completely ignored in other studies [5][6][7][8][9]. In principle, both the Ω c (3119) and Ω c (3090) could 2 In this limit, in fact, we recover the expression of the well known two-body h.o. Hamiltonian in the quark-diquark centre of mass frame, whose modulus is the quark and the diquark momentum in the centre of mass frame, in which p cm = p D + p Q = 0, and r = r D − r Q is the quark-diquark relative position vector. Ω c (3000) Molecule 5/2 − 3/2 + 5/2 − Ω c (3188) 5/2 − · · · · · · 1/2 − (3/2 − ) be assigned to the |ssc, 1, 3 2 , 0 ρ , 1 λ , 3 2 state. However, as Belle could neither confirm nor deny the existence of the Ω c (3119), we preferred the Ω c (3119) interpretation as a Ξ * c K meson-baryon molecule and assigned the Ω c (3090) to the ssc, 1, 3 2 , 0 ρ , 1 λ , 3 2 → Ω c (3090) state, providing a consistent solution to the Ω c puzzle. We calculated the mass splitting between the ρand λ-mode excitations of the Ω c(b) resonances. This large mass splitting, that we predicted to be larger than 150 MeV, is fundamental to access to the inner heavy-light baryon structure. If the ρ-excitations in the predicted mass region are not observed in the future, then the three-quark model effective degrees of freedom for the heavy-light baryons will be ruled out, supporting the Heavy Quark Effective Theory (HQET) picture of the heavy-light baryons described as heavy quark-light diquark systems. If the HQET is valid for the heavy-light baryons, the heavy quark symmetry, predicted by the HQET in the heavy-light meson sector, can be extended to the heavyquark-light-diquark baryon sector, opening the way to new future theoretical applications.