The $\varOmega _{ c}$-puzzle solved by means of quark model predictions

Abstract

The observation of five $\varOmega _c$= 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 $\varOmega _{ c}$ baryons and predictions for their $\varXi _{ c}^{+}K^{-}$ and $\varXi _{ c}'^{+}K^{-}$ decay amplitudes within an harmonic oscillator based model, we suggest a possible solution to the $\varOmega _{c}$ quantum number puzzle and we extend our mass and decay width predictions to the $\varOmega _b$ states. Finally, we discuss why the set of $\varOmega _{ c(b)}$ baryons is the most suitable environment to test the validity of three-quark and quark–diquark effective degrees of freedom.

1 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.

Table 1 Measured masses (in MeV) of the six resonances observed in the $\varXi _c^{+}K^{-}$ decay channel (see text) according to the LHCb [1] and the Belle [2] collaborations in pp and $e^{+}e^{-}$ collisions, respectively

In 2017, the LHCb Collaboration announced the observation of five narrow $\varOmega _{ c}$ states in the $\varXi _{c}^{+}K^{-}$ decay channel [1]: $\varOmega _{ c}(3000)$, $\varOmega _{ c}(3050)$, $\varOmega _{ c}(3066)$, $\varOmega _{ c}(3090)$ and $\varOmega _{ c}(3119)$. They also reported the observation of another structure around 3188 MeV, the so-called $\varOmega _{ c}(3188)$, though they did not have enough statistical significance to interpret it as a genuine resonance [1]. Later, Belle observed five resonant states in the $\varXi _c^{+} K^{-}$ invariant mass distribution and unambiguously confirmed four of the states announced by LHCb, $\varOmega _{ c}(3000)$, $\varOmega _{ c}(3050)$, $\varOmega _{ c}(3066)$, and $\varOmega _{ c}(3090)$, but no signal was found for the $\varOmega _{ c}(3119)$ [2]. Belle also measured a signal excess at 3188 MeV, corresponding to the $\varOmega _{ c}(3188)$ 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 $\varOmega _{ c}(3188)$, even if not yet confirmed, was seen both by LHCb and Belle, while, on the contrary, the $\varOmega _{ c}(3119)$ 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, $\varXi _b(6227)^-$, in both $\varLambda _b^0 K^-$ and $\varXi ^0_b \pi$ decay modes [3], and of two bottom resonances, $\varSigma _b(6097)^\pm$, in the $\varLambda _b^0 \pi ^\pm$ channels [4].

However, neither LHCb nor Belle were able to measure the $\varOmega _c$ angular momenta and parities. For this reason, several authors tried to provide different quantum number assignments for these states. The current $\varOmega _{c}$ 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 $\varOmega _{ c}$ experimental state, more than one quantum number assignment was suggested [5]. In particular, the $\varOmega _{ c}(3119)$ was allocated to possibly be a $J^{P}=\frac{1}{2}^{+}$ or a $J^{P}=\frac{3}{2}^{+}$ state [7], while the authors in Ref. [8] proposed a $J^{P}=\frac{5}{2}^{-}$ assignment.

From the previous discussion it comes out that, in the case of the $\varOmega _{ c}(3119)$, 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 $\varOmega _c(3066)$ and $\varOmega _c(3090)$ based on mass and decay width estimates. Moreover, the nature of the $\varOmega _c(3188)$ 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 $\varOmega _{c}$-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 $\varOmega _{c}$ 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 $\lambda$-excitations of the ssc system; we also calculate their $\varXi _{ c}^{+}K^{-}$ and $\varXi _{ c}'^{+}K^{-}$ decay widths (Sect. 2.3). Similarly to Refs. [10,11,12], we suggest a molecular interpretation of the $\varOmega _{ c}(3119)$ state, which was not observed by Belle. Later, we extend our mass and decay width predictions to the $\varOmega _b$ sector, which will be useful for future experimental searches. Finally, we calculate the mass splitting between the $\rho$- and $\lambda$-mode excitations of $\varOmega _{c(b)}$ resonances (see Fig. 1 upper-pannel). This calculation is fundamental to access to inner heavy-light baryon structure, as the presence or absence of $\rho$-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.

Fig. 1

Comparison between three-quark and quark–diquark baryon effective degrees of freedom. Upper panel: three-quark picture with two excitation modes. Lower panel: quark–diquark picture with one excitation mode

2 Results

2.1 S- and P-wave ssQ states.

The three-quark system (ssQ,  where $Q = c\, \mathrm{or}\, b$) Hamiltonian can be written in terms of two coordinates [13], $\varvec{\rho }$ and $\varvec{\lambda }$, which encode the system spatial degrees of freedom (see Fig. 1). Let $m_{\rho }=m_s$ and $m_{\lambda }= \frac{3 m_s m_{Q}}{2m_s+m_{Q}}$ be the ssQ system reduced masses; then, the $\rho$- and $\lambda$-mode frequencies are $\omega _{\rho ,\lambda }=\sqrt{\frac{3 K_Q}{m_{\rho ,\lambda }}}$, where $K_Q$ is the spring constant, which implies that in three equal-mass-quark baryons, in which $m_{\rho } = m_{\lambda }$, the $\lambda$- and $\rho$-orbital excitation modes are completely mixed together. By contrast, in heavy-light baryons, in which $m_{\rho } \ll m_{\lambda }$, 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 $\varOmega _c$ states. A single quark is described by its spin, flavor and color quantum numbers. As a fermion, its spin is $S=\frac{1}{2}$, its flavor, spin-flavor and color representations are $\mathbf{{3}}_{\mathrm{f}}$, $\mathbf{6}_{\mathrm{sf}}$, and $\mathbf{{3}}_{\mathrm{c}}$, respectively. An ssQ state, $\left| ssQ, S_{\rho },S_{\mathrm{tot}}, l_{\rho }, l_{\lambda }, J \right\rangle$, is characterized by total angular momentum $\mathbf{J} = \mathbf{l}_{\rho }+\mathbf{l}_{\lambda }+ \mathbf{S}_{\mathrm{tot}}$, where $\mathbf{S}_{\mathrm{tot}}=\mathbf{S}_{\rho }+\frac{\mathbf{1}}{\mathbf{2}}$. In order to construct an ssQ color singlet state, the light quarks must transform under $\hbox {SU}_{\mathrm{c}}$(3) as the anti-symmetric ${\bar{\mathbf{3}}}_{\mathrm{c}}$ 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 P-wave. Two equal flavour quarks are necessarily in the $\mathbf{{6}}_{\mathrm{f}}$ flavor-symmetric state. Thus, they are in an S-wave symmetric spin-triplet state, $S_{\rho }=1$, or in a P-wave antisymmetric spin-singlet state, $S_{\rho }=0$.

If $l_{\rho }=l_{\lambda }=0$, then $S_{\rho }=1$, and we find the two ground states, $\varOmega _Q$ and $\varOmega _{Q}^{*}$: $\left| ssQ,1,S_{\mathrm{tot}},0_{\rho },0_{\lambda },J \right\rangle$ with $J=S_{\mathrm{tot}}=\frac{1}{2}$ and $\frac{3}{2}$, respectively. If $l_{\rho }=0$ and $l_{\lambda }=1$, then $S_{\rho }=1$ and, by coupling the spin and orbital angular momentum, we find five excited states: $\left| ssQ,1,S_{\mathrm{tot}},0_{\rho },1_{\lambda },J \right\rangle$ with $J=\frac{1}{2}$, $\frac{3}{2}$ for $S_{\mathrm{tot}}=\frac{1}{2}$, and $J=\frac{1}{2}$, $\frac{3}{2}$, $\frac{5}{2}$ for $S_\mathrm{tot}=\frac{3}{2}$, which we interpret as $\lambda$-mode excitations of the ssQ system. On the other hand, if $l_{\rho }=1$ and $l_{\lambda }=0$, then $S_{\rho }=0$, and we find two excited states $\left| ssQ,0, \frac{1}{2},1_{\rho },0_{\lambda },J \right\rangle$ with $J=\frac{1}{2}$, $\frac{3}{2}$ which we interpret as $\rho$- mode excitations of the ssQ system.¹

2.2 Mass spectra of $\varOmega _{Q}$ states

We make use of a three-dimensional harmonic oscillator hamiltonian (h.o.) plus a perturbation term given by spin–orbit, isospin and flavour dependent contributions:

$$\begin{aligned} H = H_{\mathrm{h.o.}}+A\; \mathbf{S }^2 + B \; \mathbf{S} \cdot \mathbf{L} +E\; \varvec{I}^2+G \; {\mathbf{C}}_{\mathbf{2}}(\text{ SU(3) }_{\mathrm{f}}) ; \end{aligned}$$

(1)

here $\mathbf{S}, {\varvec{I}}$ and $\mathbf{C_2}(\text{ SU(3) }_{\mathrm{f}})$ are the spin, the isospin and the $\mathbf{C_2}(\text{ SU(3) }_{\mathrm{f}})$ Casimir operators, and

$$\begin{aligned} H_{\mathrm{h.o.}} =\sum _{i=1}^3m_i + \frac{{\mathbf {p}}_{\rho }^2}{2 m_{\rho }} + \frac{{\mathbf {p}}_{\lambda }^2}{2 m_{\lambda }} +\frac{1}{2} m_{\rho } \omega ^2_{\rho } \varvec{\rho }^2 +\frac{1}{2} m_{\lambda } \omega ^2_{\lambda } \varvec{\lambda }^2\nonumber \\ \end{aligned}$$

(2)

is the three-dimensional harmonic oscillator Hamiltonian written in terms of Jacobi coordinates, $\varvec{\rho }$ and $\varvec{\lambda }$, and conjugated momenta, ${\mathbf {p}}_{\rho }$ and ${\mathbf {p}}_{\lambda }$, whose eigenvalues are $\sum _{i=1}^3m_i + \omega _{\rho } \; n_{\rho } + \omega _{\lambda } n_{\lambda }\;$, where $\omega _{\rho (\lambda )}=\sqrt{\frac{3K_Q}{m_{\rho (\lambda )}}}\;$, $n_{\rho (\lambda )}= 2 k_{\rho (\lambda )}+l_{\rho (\lambda )}\;$, $k_{\rho (\lambda )}=0,1,...$, and $l_{\rho (\lambda )}=0,1,...$

We set the quark masses to reproduce the $\varOmega _c(2695)$, $\varOmega _c^{*}(2765)$, $\varXi _{cc}(3621)$ and $\varSigma _b(5814)$ ground state masses [15]: $m_q=295$ MeV, $m_{s}=450$ MeV, $m_c=1605$ MeV and $m_b=4920$ MeV; the spring constant $K_c$ is set to reproduce the mass difference between $\varXi _c(2790)$, with $J^P=\frac{1}{2}^{-}$, and the $\varXi _c(2469)$ ground state: $K_c=0.0328$ $\hbox {GeV}^{3}$, while $K_b$ is set to reproduce the mass difference between $\varLambda _b(5919)$, with $J^P=\frac{1}{2}^{-}$, and the $\varLambda _b(5619)$ ground state: $K_b=0.0235$ $\hbox {GeV}^{3}$. In order to calculate the mass difference between the $\rho$ and $\lambda$ orbital excitations of ssQ states, we scale the h.o. frequency by the $\rho$ and $\lambda$ oscillator masses. From the definition of $m_{\rho }$ and $m_{\lambda }$, one finds $m_{\rho }=m_s=450$ MeV and $m_{\lambda }= \frac{3 m_s m_{c}}{2m_s+m_{c}}\simeq 865$ MeV for $\varOmega _c$ states, and $m_{\lambda }= \frac{3 m_s m_{b}}{2m_s+m_{b}}\simeq 1141$ MeV for $\varOmega _b$ states; the $\rho$- and $\lambda$-mode frequencies are $\omega _{\rho ,\lambda }=\sqrt{\frac{3K_Q}{m_{\rho ,\lambda }}}$. Finally, the mass splitting parameters, A, B, E and G, calculated in the following, are reported in Table 2.

Table 2 Values of the parameter reported in Eq. (1) with the corresponding uncertainties expressed in MeV

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 $\varOmega _c$ 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 parameter-free procedure the spectrum of the ssQ excited states constructed in the previous section. The predicted masses of the $\lambda$- and $\rho$-orbital excitations of the $\varOmega _c$ and $\varOmega _b$ 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 $\varOmega _{c}$ 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 $\varSigma _{c}^*(2520)$ and $\varSigma _{c}(2453)$. This value ($65 \pm 8$ MeV) agrees with the mass difference between $\varOmega _{c}$ (2695) and $\varOmega _{c}^*$ (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 $S_{\mathrm{tot}}=\frac{1}{2}$ and $S_{\mathrm{tot}}=\frac{3}{2}$, is around 65 MeV plus corrections due the spin–orbit contribution which can be calculated, for example, from the $\varLambda _{ c}(2595)$-$\varLambda _{ c}(2625)$ mass difference. According to the quark model, $\varLambda _{ c}(2595)$ and $\varLambda _{ c}(2625)$ are the charmed counterparts of $\varLambda _{ }(1405)$ and $\varLambda _{ }(1520)$, respectively; their spin-parities are $\frac{1}{2}^{-}$ and $\frac{3}{2}^{-}$, and their mass difference, about 36 MeV, is due to spin–orbit effects.

Fig. 2

$\varOmega _c$ mass spectra and tentative quantum number assignments. The theoretical predictions (red dots) are compared with the experimental results by LHCb [1] (blue line), Belle [2] (violet line) and Particle Data Group (black lines) [15]. Except the $\varOmega _c(3188)$ case, the experimental error for the other states is too small to be appreciated in this energy scale. The spin-$\frac{1}{2}$ and -$\frac{3}{2}$ ground-state masses, $\varOmega _c(2695)$ and $\varOmega _c^{*}(2770)$ are indicated with $\dagger$ because are inputs while all the others are predictions

Table 3 Our ssc state quantum number assignments (first column), predicted masses (second column) and open-flavor strong decay widths into $\varXi _c^{+}K^{-}$ and $\varXi _c'^{+}K^{-}$ channels (fourth column) are compared with the experimental masses (third column) and total decay widths (fifth column) [1, 15]. An ssc state, $\left| ssc, S_{\rho }, S_{\mathrm{tot}}, l_{\rho }, l_{\lambda }, J \right\rangle$, is characterized by total angular momentum $\mathbf{J} = \mathbf{l}_{\rho }+\mathbf{l}_{\lambda } + \mathbf{S}_{\mathrm{tot}}$, where ${\mathbf{S}}_{\mathrm{tot}} = {\mathbf{S}}_{\rho }+\frac{1}{2}$. 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 $\dagger$ are used as inputs while all the others are predictions; partial decay widths denoted with $\dagger \dagger$ and with $\dagger \dagger \dagger$ are zero for phase space and for selection rules, respectively

In conclusion, by taking into account the spin-spin and spin–orbit contributions, the mass difference between the lowest $\varOmega _c$ excitation, $\left| ssc,1, \frac{1}{2},0_{\rho },1_{\lambda },\frac{1}{2} \right\rangle \equiv \varOmega _c(3000)$ and $\left| ssc,1, \frac{3}{2},0_{\rho },1_{\lambda },\frac{1}{2} \right\rangle$, is about $65-36\simeq 30$ MeV. So, we identify the $\left| ssc,1, \frac{3}{2},0_{\rho },1_{\lambda },\frac{1}{2} \right\rangle$ with the observed $\varOmega _c(3050)$ (see Fig.  2 and Table 3). In the bottom sector, the energy splitting due to the spin–spin interaction through the (isospin-averaged) mass difference between $\varSigma _{b}^*$ and $\varSigma _{b}$ is $20 \pm 7$ MeV. In such a way, we expect a mass difference between the two S-wave ground states, $\varOmega _{b}^{*}$ and $\varOmega _{b}^{}$, close to $20 \pm 7$ MeV. Hence, we suggest the experimentalists to look for a $\varOmega _{b}^{*}$ resonance with a mass of about 6082 MeV, as we can see in Fig. 3 and Table 4.

We estimate that the mass of $\left| ssc,1, \frac{1}{2},0_{\rho },1_{\lambda }, \frac{3}{2} \right\rangle$ is related to the previous spin–orbit splitting. We obtain a value of $3052 \pm 15$ MeV, which is compatible with the mass of the $\varOmega _{ c}^{} (3066)$ within the experimental error. Thus, we identify the $\left| ssc,1, \frac{1}{2},0_{\rho },1_{\lambda }, \frac{3}{2} \right\rangle$ state with the $\varOmega _{ c}(3066)$ resonance. Through the estimation of orbital, spin–spin and spin–orbit interactions, we estimate the $\left| ssc,1, \frac{3}{2},0_{\rho },1_{\lambda }, \frac{3}{2} \right\rangle$ and $\left| ssc,1, \frac{3}{2},0_{\rho },1_{\lambda }, \frac{5}{2} \right\rangle$ mass values as 3080 ±13 MeV and 3140 ±14, respectively. Thence, we propose the following assignments: $\left| ssc,1, \frac{3}{2},0_{\rho },1_{\lambda }, \frac{3}{2} \right\rangle$ $\rightarrow$ $\varOmega _c$(3090) and $\left| ssc,1, \frac{3}{2},0_{\rho },1_{\lambda },\right.$ $\left. \frac{5}{2} \right\rangle$ $\rightarrow$ $\varOmega _c$(3188).

In the bottom sector, the mass splitting due to the spin–orbit interaction between $\varLambda _{b}(5912)$ and $\varLambda _{b}(5920)$ is 8 MeV and we estimated previously that the spin-spin splitting is $20 \pm 7$ MeV. Thus, we interpret the predicted $\varOmega _b (6305)$, $\varOmega _b (6313)$, $\varOmega _b (6317)$, $\varOmega _b (6325)$ and $\varOmega _b (6338)$ states, reported in Table 4, as the bottom counterparts of the $\varOmega _{ c}(3000)$, $\varOmega _c(3066)$, $\varOmega _c(3050)$, $\varOmega _c(3090)$ and $\varOmega _c(3188)$, respectively. We observe that, unlike the charm sector, in the bottom sector the state $\left| ssb,1, \frac{3}{2},0_{\rho },1_{\lambda }, \frac{1}{2} \right\rangle$ is heavier than the state $\left| ssb,\right.$ $\left. 1, \frac{1}{2},0_{\rho },1_{\lambda }, \frac{3}{2} \right\rangle$: this is due to the fact that in the charm sector the spin–orbit contribution is lesser than the spin-spin one, while in the bottom sector the situation is reversed (see Table 2).

Fig. 3

$\varOmega _b$ mass spectrum predictions (red dots) and $\varOmega _b$ ground-state experimental mass (black line) [15]. The experimental error on the $\varOmega _b(6046)$ state, 2 MeV, is too small to be appreciated in this energy scale

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, $\left| ssb, S_{\rho }, S_{\mathrm{tot}}, l_{\rho }, l_{\lambda }, J \right\rangle$, is characterized by total angular momentum $\mathbf{J} = \mathbf{l}_{\rho }+\mathbf{l}_{\lambda } + \mathbf{S}_{\mathrm{tot}}$, where $\mathbf{S}_{\mathrm{tot}} = \mathbf{S}_{\rho }+\frac{1}{2}$. Partial decay widths denoted with $\dagger \dagger$ and with $\dagger \dagger \dagger$ are zero for phase space and for selection rules, respectively

In the charm sector, the mass splitting due to the flavor-dependent interaction can be estimated from the mass difference between $\varXi _c$ and $\varXi _c^{'}$, 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 $\varXi _c$ and $\varXi _c^{'}$ are $\varXi _b$ and $\varXi _b^{'}$, with masses 5793.2 MeV and 5935.02 MeV, respectively. Therefore, in the bottom sector the flavor-dependent 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, $\varSigma _c$ and $\varLambda _c$, is related to the different isospin and flavor structures of the light quark multiplets: $\varLambda _c$ is an isospin-singlet state belonging to an SU(3)$_{\mathrm{f}}$ flavor anti-triplet, while $\varSigma _c$ is an isospin-triplet state belonging to an SU(3)$_{\mathrm{f}}$ flavor sextet. In the bottom sector, the isospin-flavor contribution to the baryon masses can be calculated from the mass difference between $\varSigma _b$ and $\varLambda _b$.

We summarize all our proposed quantum number assignments for both $\varOmega _c$ and $\varOmega _b$ 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, $\varOmega _c(3000)$ and $\varOmega _c(3188)$, respectively, also the absolute mass predictions are in agreement within the experimental error, which is very small (less than 1 MeV).

2.3 Decay widths of ssQ states

In the following, we compute the strong decays of ssQ baryons in $sqQ - K$ ($q = u,d$) final states by means of the $^3P_0$ model [20,21,22,23] (see Appendix A).

In the $^3P_0$ model, the parameters depend on the harmonic oscillator frequency of the initial and final states. For charmed baryons, we expect the parameters $\alpha _\rho$ and $\alpha _\lambda$ to lie in the range 0.4–0.7 GeV. In principle, the values of the $\alpha _\rho$ and $\alpha _\lambda$ 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 $\alpha _\rho$ and $\alpha _\lambda$. We also prefer not to take $\alpha _\rho$ and $\alpha _\lambda$ as free parameters, but to express them in terms of the baryon $\rho$- and $\lambda$-mode frequencies, $\omega _{\rho ,\lambda }=\sqrt{3K_Q/m_{\rho ,\lambda }}$, using the relation $\alpha ^2_{\rho ,\lambda }=\omega _{\rho ,\lambda }m_{\rho ,\lambda }$ for both initial- and final-state baryon resonances; see Appendix B. In light of this, the only free parameter is the pair creation strength, $\gamma _0 =9.2$, which is fitted to the reproduction of the $\varOmega _c(3066)$ width. The frequency of K meson is set to be $\omega _{c}=0.46$ GeV [25].

Tables 3 and 4 report our $\varOmega _c \rightarrow \varXi _c^{+}K^{-},\varXi _c^{\prime +} K^-$ and $\varOmega _b\rightarrow \varXi _b^{0}K^{-}$ predicted decay widths. The $\varXi _c^{+}K^{-}$ decay channel is where the $\varOmega _c$ states were observed by LHCb and Belle; we also consider the $\varXi _c^{\prime +} K^-$ channel, which contributes to the $\varOmega _{\mathrm{c}}(3090)$ and $\varOmega _{\mathrm{c}}(3188)$ open-flavor decay widths. Both the $\varXi _c^{+}K^{-}$ branching ratios and the quantum numbers of the $\varOmega _c$’s are unknown; we only have experimental informations on their total widths, $\varGamma _{\mathrm{tot}}$. Thus, our predictions have to satisfy the constraint: $\varGamma (\varOmega _c \rightarrow \varXi _c^{+}K^{-})\le \varGamma _{\mathrm{tot}}$. 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 $\lambda$-mode decay widths of the $\varOmega _c$ states are in the order 1 MeV, while the $\varXi _{\mathrm{c}}^{+}K^{-}$ decay mode of the two $\rho$-excitations, $\left| ssc,0, \frac{1}{2},1_{\rho },0_{\lambda }, \frac{1}{2} \right\rangle$ and $\left| ssc,0, \frac{1}{2},1_{\rho },0_{\lambda }, \frac{3}{2} \right\rangle$, is forbidden by spin conservation. Similar considerations can be applied to the decay widths of $\rho$-mode $\varOmega _b$ 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: I) We used a single set of values for the $\alpha _\rho$ and $\alpha _\lambda$ h.o. parameters. Those values were extracted from a fit to the spectrum and not fitted to the reproduction of the $\varOmega _c$’s decay widths; II) There is not a single model which is capable of providing a completely satisfactory description of baryon open-flavor strong decay widths [26].

Fig. 4

Adapted from Fig. 2 of Ref. [1], APS copyright. Proposed spin- and parity-assignments for the $\varOmega _{ c} = css$ excited states reported by the LHCb Collaboration and later confirmed by Belle: $\varOmega _{ c}^{}(3000)$, $\varOmega _{ c}^{}(3050)$, $\varOmega _{ c}^{}(3066)$, $\varOmega _{ c}^{}(3090)$, and $\varOmega _{ c}^{}(3188)$. We interpret $\varOmega _{ c}(3119)$ as a $\varXi _c^{*}K$ molecule

In conclusion, in addition to our mass estimates, also the $^3P_0$ model results suggest that the five $\varOmega _{ c}$ resonances, $\varOmega _{ c}^{}(3000)$, $\varOmega _{ c}^{}(3050)$, $\varOmega _{ c}^{}(3066)$, $\varOmega _{ c}^{}(3090)$, and $\varOmega _{ c}^{}(3188)$, could be interpreted as ssc ground-state P-wave $\lambda$-excitations. In principle, both the $\varOmega _{ c}^{}(3090)$ and $\varOmega _{ c}^{}(3119)$ resonances observed by LHCb are compatible with the properties (mass and decay width) of the $\left| ssc, \frac{3}{2},1_{\lambda },\frac{3}{2} \right\rangle$ theoretical state. As Belle could neither confirm nor deny the existence of the $\varOmega _{ c}^{}(3119)$, given the low significance of its results for the previous state ($0.4 \sigma$), we prefer to: 1) Assign $\left| ssc, \frac{3}{2},1_{\lambda },\frac{3}{2} \right\rangle$ to the $\varOmega _{ c}^{}(3090)$; 2) Interpret the $\varOmega _{ c}^{}(3119)$ as a $\varXi _c^{*}K$ bound state [10,11,12], the $\varOmega _{ c}^{}(3119)$ lying 22 MeV below the $\varXi _c^{*}K$ threshold. See Fig. 4. Additionally, in Table 5, we present a comparison of different quantum number assignments for the $\varOmega _c$ states.

3 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, $\rho$ and $\lambda$, have approximately the same frequency, $\omega _{\rho }\simeq \omega _{\lambda }$; therefore, the $\rho$- and $\lambda$-excitations are degenerate. By contrast, in the case of heavy-light baryons $m_{\rho } \ll m_{\lambda }$; thus, the two excitation modes are decoupled from one another; specifically $\omega _{\rho }-\omega _{\lambda }\simeq 130$ MeV for $\varOmega _c$ states and $\omega _{\rho }-\omega _{\lambda }\simeq 150$ MeV for $\varOmega _b$ 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 $\rho$-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 $\rho$-excitations, $\varOmega _c(3146)$, $\varOmega _c(3182)$, $\varOmega _b(6452)$, and $\varOmega _b(6460)$, are not observed, then the other $\varOmega _c$ states will be characterized by a quark–diquark structure.

Finally, we observe that in the case of a quark–diquark-picture 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 $\rho$ coordinate:²

$$\begin{aligned} H_{\mathrm{h.o.}}= & {} m_{D}+m_Q + \frac{{\mathbf {p}}_{\lambda }^2}{2 m_{\lambda }} +\frac{1}{2} m_{\lambda } \omega ^2_{\lambda } \varvec{\lambda }^2. \end{aligned}$$

(3)

Here $m_D=2m_s$ is the diquark mass. Indeed, the mass spectrum predicted with this definition of $H_{\mathrm{h.o.}}$ is the same as that reported in Figs.  2 and 3, but without the frozen $\rho$ excitations. We observe also that, if the quark–diquark scenario turns out to be the correct one, the suppression of the spin–spin 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.

Table 5 $J^P$ quantum number assignments of $\varOmega _c$ resonances from previous studies. $^*$ Ref. [5] provided two different sets of $J^P$ assignments

4 Discussion

We calculated the $\varOmega _{c(b)}$’s masses and $\varXi _{ c(b)}^{+}K^{-}$ strong decay amplitudes. By means of these mass and decay width predictions, we proposed an univocal assignment to the five $\varOmega _{c}$ states observed both by LHCb [1] and Belle [2]:

$$\begin{aligned}&\left| ssc, 1, {\scriptstyle \frac{1}{2}},0_{\rho },1_{\lambda },{\scriptstyle \frac{1}{2}} \right\rangle \rightarrow \varOmega _{ c}(3000)\,, \end{aligned}$$

(4)

$$\begin{aligned}&\left| ssc,1, {\scriptstyle \frac{3}{2}},0_{\rho },1_{\lambda },{\scriptstyle \frac{1}{2}} \right\rangle \rightarrow \varOmega _{ c}(3050)\,, \end{aligned}$$

(5)

$$\begin{aligned}&\left| ssc, 1,{\scriptstyle \frac{1}{2}},0_{\rho },1_{\lambda },{\scriptstyle \frac{3}{2}} \right\rangle \rightarrow \varOmega _{ c}(3066)\,,\end{aligned}$$

(6)

$$\begin{aligned}&\left| ssc, 1, {\scriptstyle \frac{3}{2}}, 0_{\rho },1_{\lambda },{\scriptstyle \frac{3}{2}} \right\rangle \rightarrow \varOmega _{ c}(3090)\,,\end{aligned}$$

(7)

$$\begin{aligned}&\left| ssc, 1, {\scriptstyle \frac{3}{2}}, 0_{\rho },1_{\lambda },{\scriptstyle \frac{5}{2}} \right\rangle \rightarrow \varOmega _{ c}(3188)\,. \end{aligned}$$

(8)

The latter was completely ignored in other studies [5,6,7,8,9]. In principle, both the $\varOmega _{ c}(3119)$ and $\varOmega _{ c}(3090)$ could be assigned to the $|ssc,1,\frac{3}{2},0_{\rho },1_{\lambda },\frac{3}{2}\rangle$ state. However, as Belle could neither confirm nor deny the existence of the $\varOmega _{ c}(3119)$, we preferred the $\varOmega _{ c}(3119)$ interpretation as a $\varXi _c^{*} K$ meson-baryon molecule and assigned the $\varOmega _{ c}(3090)$ to the $\left| ssc, 1, \frac{3}{2},\right.$ $\left. 0_{\rho },1_{\lambda },\frac{3}{2} \right\rangle \rightarrow \varOmega _{ c}(3090)$ state, providing a consistent solution to the $\varOmega _{ c}$ puzzle.

We calculated the mass splitting between the $\rho$- and $\lambda$-mode excitations of the $\varOmega _{c(b)}$ 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 $\rho$-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 heavy-quark–light-diquark baryon sector, opening the way to new future theoretical applications.

Appendices

Appendix A: $^3P_0$ Decay model

The $^3P_0$ is an effective model to compute the open-flavor strong decays of hadrons in the quark model formalism [20,21,22,23]. In this model, a hadron decay takes place in its rest frame and proceeds via the creation of an additional $q {\bar{q}}$ pair with vacuum quantum numbers, i.e. $J^{PC} = 0^{++}$. We label the initial baryon- and final baryon- and meson-states as A, B and C, respectively. The final baryon–meson state BC is characterized by a relative orbital angular momentum $\ell$ between B and C and a total angular momentum $\mathbf{J} = \mathbf{J}_B + \mathbf{J}_C + {\varvec{\ell }}$. The decay widths can be calculated as [20, 21, 27]

$$\begin{aligned} \varGamma = \frac{2 \pi \gamma _0^2}{2J_{A}+1} \varPhi _{A \rightarrow BC}(q_0)\sum _{M_{J_A},M_{J_B}} \big |{\mathcal {M}}^{M_{J_A},M_{J_B}}\big |^2 . \end{aligned}$$

Here, ${\mathcal {M}}^{M_{J_A},M_{J_B}}$ is the $A \rightarrow BC$ amplitude which, for simplicity, is usually expressed in terms of hadron harmonic-oscillator wave functions, $\gamma _0$ is the dimensionless pair-creation strength. $q_0$ is the relative momentum between B and C, and the coefficient $\varPhi _{A \rightarrow BC}(q_0)$ is the relativistic phase space factor [27],

$$\begin{aligned} \varPhi _{A \rightarrow BC}(q_0)= & {} 4 \pi q_0 \frac{E_B(q_0) E_C(q_0)}{M_A} , \\ \text{ with }\;\;E_{B,C}= & {} \sqrt{M_{B,C}^2 + q_0^2}. \end{aligned}$$

Appendix B: Baryon wave functions

Differently from light baryon phenomenology, where the $\rho$- and $\lambda$-modes of the mixed-symmetry spatial wave function are degenerate in energy, in the heavy-light sector the previous modes decouple; so, they can be distinguished through an analysis of the heavy-light baryon mass spectra. This happens because frequency of the $\rho$- and $\lambda$-modes are different,

$$\begin{aligned} \omega _{\rho }=\sqrt{\frac{3 K_Q}{m_{\rho }}} \;\; \text {and} \;\;\omega _{\lambda }=\sqrt{\frac{3 K_Q}{m_{\lambda }}}, \end{aligned}$$

(B.1)

where $m_{\rho }$ and $m_{\lambda }$ are defined in Sect. 2.1 . We write the baryon wave functions in terms of $\omega _{\rho }$ and $\omega _{\lambda }$ by using the relation $\alpha ^2_{\rho ,\lambda }=\omega _{\rho ,\lambda }m_{\rho ,\lambda }$ .

For the S-wave charmed baryon,

$$\begin{aligned} \psi (0,0,0,0)= & {} 3^{3/4}\;\Bigg (\frac{1}{\pi \omega _{\rho }m_{\rho }}\Bigg )^{\frac{3}{4}}\,\Bigg (\frac{1}{\pi \omega _{\lambda }m_{\lambda }}\Bigg )^{\frac{3}{4}}\nonumber \\&\times \,\exp \Bigg [{-\frac{ {{\mathbf {P}}}^2_{\rho }}{2\omega _{\rho }m_{\rho }} -\frac{ {{\mathbf {P}}}^2_{\lambda }}{2\omega _{\lambda }m_{\lambda }}}\Bigg ]. \end{aligned}$$

(B.2)

For the P-wave charmed baryon,

$$\begin{aligned} \psi _{\rho }(1,m,0,0)= & {} -i\,\Bigg (\frac{8}{3\sqrt{\pi }}\Bigg )^{{1}/{2}}\Bigg (\frac{1}{ \omega _{\rho }m_{\rho }}\Bigg )^{ {5}/{4}}\,\mathcal{Y}^m_1({{\mathbf {P}}}_{\rho })\nonumber \\&\times \Bigg (\frac{3}{\pi \omega _{\lambda }m_{\lambda }}\Bigg )^{{3}/{4}}\,\exp \Bigg [{-\frac{{{\mathbf {P}}}^2_{\rho }}{2\omega _{\rho }m_{\rho }} -\frac{{{\mathbf {P}}}^2_{\lambda }}{2\omega _{\lambda }m_{\lambda }}}\Bigg ], \end{aligned}$$

(B.3)

$$\begin{aligned} \psi _\lambda (0,0,1,m)= & {} -i\;\,\Bigg (\frac{8}{3\sqrt{\pi }}\Bigg )^{{1}/{2}}\Bigg (\frac{1}{ \omega _{\lambda }m_{\lambda }}\Bigg )^{{5}/{4}}\,\mathcal{Y}^m_1({{\mathbf {P}}}_{\lambda })\nonumber \\&\times \Bigg (\frac{3}{\pi \omega _{\rho }m_{\rho }}\Bigg )^{{3}/{4}}\,\exp \Bigg [{-\frac{{{\mathbf {P}}}^2_{\rho }}{2\omega _{\rho }m_{\rho }} -\frac{{{\mathbf {P}}}^2_{\lambda }}{2\omega _{\lambda }m_{\lambda }}}\Bigg ].\nonumber \\ \end{aligned}$$

(B.4)