Triply charmed and bottom baryons in a constituent quark model

In this work, we study the mass spectrum of the $\Omega_{ccc}$ and $\Omega_{bbb}$ baryons up to the $N=2$ shell within a nonrelativistic constituent quark model (NRCQM). The model parameters are adopted from the determinations by fitting the charmonium and bottomonium spectra in our previous works. The masses of the $\Omega_{ccc}$ and $\Omega_{bbb}$ baryon states predicted in present work reasonably agree with the results obtained with the Lattice QCD calculations. Furthermore, to provide more knowledge of the $\Omega_{ccc}$ and $\Omega_{bbb}$ states, we evaluate their radiative decays with the available masses and wave functions from the potential model.

The triply heavy baryons, as a system of fully heavy quarks, may provide a new window for understanding the structure of baryons. The complications of light-quark interaction are absent in the triply heavy baryons, thus, they provide an ideal place for our better understanding the heavy quark dynamics. From the theoretical point view, the potential models might be able to describe triply heavy baryons to a similar level of precision as their success in heavy quarkonia. Just as the quark-antiquark interactions are examined in charmonia and bottomonia, the studies of the triply heavy baryon spectra will probe the quark-quark interactions in the heavy quark sector [9]. In the past years, many studies about the triply heavy baryons can be found in the literature. Most of them focused on the predictions of the masses  and the production [7,8,[39][40][41][42][43][44]. However, only several works have paid attentions to the weak decays [45][46][47], magnetic moments [33,48], M1 decays [48] of triply heavy baryons.
Stimulated by the large discovery potentials of the heavy baryons at the LHC facility, in this work we carry out a systemical study of the triply heavy baryon spectra of Ω ccc and Ω bbb within a nonrelativistic potential model. Recently, with this model we have studied the spectra of the charmonium, bottomonium, B c meson, Ω baryon, and fully-heavy tetraquark states. For there are no measurements of the triply heavy baryons which can be used to constrain the parameters potential model, the model parameters of this work are adopted with the determinations by fitting the charmonium and bottomonium spectra in our previous works [49][50][51]. The masses of the Ω ccc and Ω bbb baryon states predicted in present work reasonably agree with the results obtained with the Lattice QCD calculations [9,15].
Furthermore, to provide more knowledge of the Ω ccc and Ω bbb states, we evaluate their radiative decays with the available wave functions from the potential model. It should be emphasized that the OZI allowed two-body strong decay channels are absence for the low-lying 1P-, 1D-, and 2Swave Ω ccc and Ω bbb states, thus, the radiative transitions become important in their decays. Consequently, the radiative decay processes of the excited Ω ccc or Ω bbb states may be crucial for establishing them if they are produced in experiments. In this work, radiative decays of the Ω ccc and Ω bbb states are calculated within a nonrelativistic constituent quark model developed in our previous study of the heavy quarkonia [49,52]. This model was also successfully extended to deal with the radiative decays of the B c meson states [53], Ω baryon states [54], singly baryon states [55][56][57], and doubly heavy baryon states [58,59]. This paper is organized as follows. In Sec. II, a brief review of the potential model is given and the mass spectra of the Ω ccc and Ω bbb baryons are calculated. Then, in Sec. III, we give a review of the radiative decay model, and calculate radiative decays of the excited Ω ccc and Ω bbb states by using the masses and wave functions obtained from the potential model. In Sec. IV, we give our discussions based on the obtained radiative decay properties and masses of the Ω ccc and Ω bbb resonances. Finally, a summary is given in Sec. V.

A. Hamiltonian
To calculate the spectrum of the Ω ccc and Ω bbb baryons, the following nonrelativistic Hamiltonian is adopted in this work where m i and T i stand for the constituent quark mass and kinetic energy of the i-th quark, respectively; T G stands for the center-of-mass (c.m.) kinetic energy of the baryon system; r i j ≡ |r i − r j | is the distance between the i-th quark and j-th quark; and V i j (r i j ) stands for the effective potential between the i-th and j-th quark. In this work, we adopt a widely used potential form for V i j (r i j ) [49,52,[60][61][62][63][64][65][66][67], i.e.
where V con f i j stands for the potential for confinement, and is adopted the standard Cornell form: while V sd i j (r i j ) stands for the spin-dependent interaction, which is the sum of the spin-spin contact hyperfine potential V S S i j , the tensor term V T i j , and the spin-orbit term V LS i j : The spin-spin potential V S S i j and the tensor term V T i j are adopted the often used forms: In this work, a simplified phenomenological spin-orbit potential is adopted as that suggested in the literature [54,68,69], i.e., In the above equations, the S i , S and L are the spin operator of the i-th quark, the total spin of the baryon and the total orbital angular momentum of the baryon, respectively; the parameter b, α i j , and α S O denote the strength of confinement potential, strong coupling, and spin-orbit potential, respectively. The seven parameters m c , m b , α cc , α bb , σ cc , σ bb , and b have been determined by fitting the charmonium and bottomonium spectra in our previous works [49][50][51]. In this work, we use the same value of parameter α S O as in Ref. [54]. The quark model parameters adopted in present work are collected in Table I. The Ω ccc and Ω bbb spectra should satisfy the requirements of the SU(6)×O(3) symmetry. The states in the SU(6)×O(3) representation up to the N = 2 shell are given in Table II. We denote the baryon states as |N 6 , 2S +1 N 3 , N, L, J P , where N 6 stands for the irreducible representation of spin-flavor SU (6) group, N 3 stands for the irreducible representation of flavor SU(3) group, and N, S , L, and J P stand for the principal, spin, total orbital angular momentum, and spin-parity quantum numbers, respectively. The SU(6)×O(3) wave functions, which correspond to the |N 6 , 2S +1 N 3 , N, L, J P states, are also listed in Table II. The ψ σ NLM L (ρ, λ) and χ σ M S are the spatial and spin wave functions, respectively, where σ(= s, ρ, λ, a) denotes the representation of the S 3 group. In the spatial wave functions, ρ and λ are the internal Jacobi coordinates. The explicit forms of the ψ σ NLM L (ρ, λ) and χ σ M S have been given in the Ref. [54,70].

C. Numerical calculation
The key problem of our numerical calculations is how to deal with the spatial wave functions. To work out the spatial wave functions, in this work we expand them in terms of Gaussian basis functions. The spatial wave function ψ σ NLM L (ρ, λ) may be expressed as [54] ψ σ The coefficients C n ρ l ρ m ρ n λ l λ m λ in the spatial wave function ψ σ NLM L (ρ, λ) up to the N = 2 shell have been given in our previous work [54]. In the above equation, ψ n ρ l ρ m ρ (ρ) and ψ n λ l λ m λ (λ) stand for the spatial wave functions of the ρand λ-mode excitations, respectively.
The radial wave functions of the ρand λ-mode excitations, R n ξ l ξ (ξ) (ξ = ρ, λ), are expanded by a series of Gaussian basis functions [54]: The F −n ξ , l ξ + 3 2 , ξ d ξℓ 2 is the confluent hypergeometric function. The parameter d ξℓ can be related to the harmonic oscillator frequency ω ξℓ with 1/d 2 ξℓ = M ξ ω ξℓ . The reduced masses M ρ,λ are defined by M ρ ≡ 2m 1 m 2 (m 1 +m 2 ) , M λ ≡ 3(m 1 +m 2 )m 3 2(m 1 +m 2 +m 3 ) . On the other hand, the harmonic oscillator frequency ω ξℓ can be related to the harmonic oscillator stiffness factor K ℓ with ω ξℓ = 3K ℓ /M ξ [70]. For the identical quark system, one has d ρℓ = d λℓ = d ℓ = (3m Q K ℓ ) −1/4 , where m Q stands for the constituent mass of the charm or bottom quark. With this relation, the spatial wave function ψ σ NLM L (ρ, λ) can be simply expanded as where ψ σ NLM L (d ℓ , ρ, λ) stands for the trial harmonic oscillator functions, To solve the Schrödinger equation, the variation principle is adopted in this work. Following the method used in Refs. [50,71], the oscillator length d ℓ is set to be where n is the number of Gaussian functions, and a is the ratio coefficient. There are three parameters {d 1 , d n , n} to be determined through variation method. It is found that when we take parameters {0.068fm, 2.711fm, 15} and {0.050fm, 2.016fm, 15} for Ω ccc baryons and Ω bbb baryons, respectively, we will obtain stable solutions for the Ω ccc and Ω bbb baryons. Finally, the problem of solving the Schrödinger equation become a problem of solving the generalized matrix eigenvalues of the following equation where The calculations of matrix elements H ℓℓ ′ and N ℓℓ ′ have been detailed discussed in Ref. [54]. The physical state corresponds to the solution with a minimum energy E m . By solving this generalized matrix eigenvalue problem, the masses and spacial wave functions of the Ω ccc and Ω bbb baryons can be determined.
The predicted masses of the Ω ccc and Ω bbb baryons up to N = 2 shell have been given in Table III and also shown in

III. RADIATIVE DECAYS
In this work the radiative decays of the Ω ccc and Ω bbb baryon states are evaluated within a nonrelativistic constituent quark model developed in our previous study of the heavy quarkonia [49,52]. This model has been extended to deal with the radiative decays of the B c meson states [53], Ω baryon states [54], singly baryon states [55][56][57].  4  In this model, the quark-photon electromagnetic (EM) coupling at the tree level is adopted as where A µ is the photon field with three momentum k, while r j and e j stand for the coordinate and charge of the jth quark field ψ j . In order to match the nonrelativistic wave functions of the baryons, we should adopt the nonrelativistic form of Eq. (16) in the calculations. Including the effects of the binding potential between quarks [72], the nonrelativistic expansion of H e may be written as [73][74][75] where m j and σ j stand for the constituent mass and Pauli spin vector for the jth quark. The vector ǫ is the polarization vector of the photon. This nonrelativistic EM transition operator has between widely applied to meson photoproduction reactions [73,74,[76][77][78][79][80][81][82][83][84][85]. Then, the standard helicity transition amplitude A λ between the initial baryon state |B and the final baryon state |B can be calculated by where ω γ is the photon energy. Finally, we can calculate the EM decay width by where J i is the total angular momentum of an initial meson, J f z and J iz are the components of the total angular momenta along the z axis of initial and final mesons,respectively.
In our calculations, the masses and the wave functions of the Ω ccc and Ω bbb baryon states are adopted by solving the Schrödinger equation in Sec.II. The radiative decay widths of Ω ccc and Ω bbb baryons up to N = 2 are listed in Table IV. For simplicity one can fit the numerical wave functions with a single Gaussian (SG) form by reproducing the root-mean-square radius of the ρ-mode excitations. The determined harmonic oscillator strength parameters, α, for the Ω ccc and Ω bbb baryon states are listed in Table V. With the the SG effective wave functions, we also calculated the radiative decay widths of the Ω ccc and Ω bbb baryon states, these results are listed in Table IV for a comparison. From Table IV, it is find that the partial widths obtained with the SG effective wave functions show less differences with those obtained with the real numerical wave functions.

A. Ground states
For the ground states Ω ccc and Ω bbb , our predicted masses are ∼ 4828 MeV and ∼ 14432 MeV, respectively. There are many predictions of the masses of Ω ccc and Ω bbb in the literature . For a comparison, our results and those of other works are collected in Table VI and also shown in Fig 2. It is found that in most of the studies the masses of the ground states Ω ccc and Ω bbb are predicted to be in the range of ∼ 4800 ± 50 MeV and ∼ 14410 ± 170 MeV, respectively. Our predicted masses are reasonably consistent with the previous studies, although our results are slightly larger most of the other predictions (see Fig 2). Compared with the results of the lattice QCD, it is found that our predicted mass for Ω ccc just lies the upper limit of the predictions in Refs. [10,12,13], while our predicted mass for Ω bbb is about 60 MeV above the predictions in Refs. [10,15].

B. 1P-wave states
There are two 1P-wave Ω QQQ (Q = c, b) states with J P = 1/2 − and J P = 3/2 − according to the quark model classification (see Table II). For a comparison, our predicted masses of the 1P-wave Ω ccc and Ω bbb states together with those of other theoretical predictions have been listed in Table VII and shown in Figure 3. Our predictions of the radiative decay properties of the 1P-wave Ω ccc and Ω bbb states are also given in Table IV. 1. Ω ccc (1P) states In our calculations, the masses of the 1P-wave states Ω ccc (1 2 P 1/2 − ) and Ω ccc (1 2 P 3/2 − ) are predicted to be ∼ 5142 MeV and ∼ 5162 MeV, respectively, which are close to the values ∼ 5120(9) MeV ∼ 5124(13) MeV from the Lattice QCD calculation [9]. Our results are also compatible with the other model predictions in Refs. [16,17,22,30,34]. The mass splitting between Ω ccc (1 2 P 1/2 − ) and Ω ccc (1 2 P 3/2 − ) might be small. With a simplified phenomenological spin-orbit potential as adopted in the study of the Ω spectrum in our previous work [54], we predict that the mass splitting between these two 1P-wave states might be ∼ 20 MeV, which is slightly larger than the value of several MeV predicted in the literature [9,16].
The decays of the 1P-wave Ω ccc (1 2 P 1/2 − ) and Ω ccc (1 2 P 3/2 − ) states may be dominated by the radiative transitions into the ground 1S -wave state Ω ccc , for their OZI-allowed two body strong decay processes are absence. We further estimate the radiative decays of the Ω ccc (1 2 P 1/2 − ) and Ω ccc (1 2 P 3/2 − ) states by using the wave functions calculated from the potential model. It is found that both Ω ccc (1 2 P 1/2 − ) and Ω ccc (1 2 P 3/2 − ) have a comparable radiative decay width into the ground 1Swave state Ω ccc , i.e., The radiative transitions Ω ccc (1 2 P 1/2 − , 1 2 P 3/2 − ) → Ω ccc γ may be crucial to established them in future experiments. It should be mentioned that few studies radiative decay properties of the excited Ω ccc are found in the literature. More theoretical analysis is need to better understand these 1P-wave states.
In the following we give a brief discussion of the relations of the mass splitting between the 1P-wave states in the Ω QQQ (Q ∈ {s, c, b}) baryon spectrum. If the mass splitting between two 1P-wave states is due to the spin-orbit interaction, from Eq. (7) one finds that the mass splitting ∆m[Ω QQQ (1P)] ∝ 1 m 2 Q 1 ρ 2 +λ 2 , where m Q is the mass of constituent quark Q. With a simple harmonic oscillator wave function, one can relate the element matrix 1 ρ 2 +λ 2 to the harmonic oscillator strength parameter α. One further finds that 1 ρ 2 +λ 2 ∝ α 2 . Then we obtain an useful relation for the mass splitting: Taking the constituent quark masses and effective harmonic oscillator strength parameters α for the 1P-wave Ω, Ω ccc , and Ω bbb states determined in present work and our previous   work [54], we obtain the following ratios (23) Future experimental measurements of these ratios may provide a crucial test for the spin-orbit interactions adopted in present work.
The radiative decay properties of the Ω bbb (1 2 P 1/2 − ) and Ω bbb (1 2 P 3/2 − ) states are also estimated in present work by using the wave functions calculated from the potential model. The partial widths for the Ω bbb (1 2 P 1/2 − ) and Ω bbb (1 2 P 3/2 − ) Combing the partial widths of the 1P-wave Ω ccc states, one Since the partial widths for the 1P-wave Ω ccc states are about two orders of magnitude smaller than those corresponding processes of the 1P-wave Ω bbb states, the radiative decay process of Ω bbb (1P) → Ω bbb γ may be more difficultly observed than Ω ccc (1P) → Ω ccc γ.

C. 1D-wave states
There are six 1D-wave states |1 4 D 7/2 + , 5/2 + 3/2 + 1/2 + and |1 2 D 5/2 + 3/2 + in Ω QQQ spectrum according to the quark model classification (see Table II). For a comparison, the masses of the 1D-wave Ω ccc and Ω bbb states predicted in present work together with those from other works are listed in Table VIII. Our predictions of the radiative decay properties of the 1Dwave states are also given in Table IV. To our knowledge, no studies of the radiative decay properties of the 1D-wave triply heavy baryons can be available in the literature.

Ω bbb (1D) states
As shown in Table VIII, the masses of the 1D-wave states Ω bbb (1D) are predicted to be in the range of ∼ 14.97 − 15.02 GeV in present work. The mass order for the six 1D-wave states is The mass splitting between two adjacent states is about several MeV. It is interesting to find that our prediction of the masses, mass order, and mass splitting for 1D-wave states Ω bbb (1D) are consistent with those of the Lattice QCD [15]. However, the masses of the 1D-wave states predicted in this work are about 100-300 MeV lower than those predicted in Refs. [16,32], while about 100 MeV higher than those predicted in Ref. [17]. Our predicted mass order for the 1D-wave states is also different from those predictions in Refs. [16,32]. As a whole there are large uncertainties in the predictions of the mass spectrum of the 1D-wave states Ω bbb (1D), more theoretical studies are needed.

Ω ccc (2S ) states
Our predicted masses for the 2S -wave Ω ccc states Ω ccc (2 2 S 1/2 + ) and Ω ccc (2 4 S 3/2 + ) are ∼ 5373 MeV and ∼ 5285 MeV, respectively, which are compatible with the Lattice QCD predictions in Ref. [9]. The mass splitting between these two 2S -wave state predicted in present work, ∼ 90 MeV, is also in good agreement with that of the Lattice QCD [9]. It should be mention that there are only a few predictions of masses of the 2S -wave Ω ccc states. The mass range predicted in this work roughly agrees with the other quark model predictions [16,17,32], although the predicted mass splitting between the two 2S -wave Ω ccc states is different with each other.

Ω bbb (2S ) states
Our predicted masses for the 2S -wave Ω bbb states Ω bbb (2 2 S 1/2 + ) and Ω bbb (2 4 S 3/2 + ) are ∼ 14959 MeV and ∼ 14848 MeV, respectively, which are compatible with the Lattice QCD predictions in Ref. [15]. The mass splitting between these two 2S -wave state predicted in present work, ∼ 110 MeV, is also in good agreement with that of the Lattice QCD [15]. Our predicted mass of Ω bbb (2 4 S 3/2 + ) is also close to recent quark model prediction, 14805 MeV, in Ref. [17]. However, our predictions of the masses for these 2S -wave Ω bbb states are about 200-300 MeV lower than the other quark model predictions in Refs. [16,32].

V. SUMMARY
In this work, we calculate the Ω ccc and Ω bbb spectrum up to the N = 2 shell within a potential model. The potentials are determined by fitting the mass spectra of charmonium and bottomonium in our previous works. For the ground states Ω ccc and Ω bbb , our predicted masses are ∼ 4828 MeV and ∼ 14432 MeV, respectively. Compared with the results of the lattice QCD, it is found our predicted mass for Ω ccc just lies the upper limit of the predictions in Refs. [10,12,13], while our predicted mass for Ω bbb is about 60 MeV above the predictions in Refs. [10,15]. Furthermore, our predictions of the mass ranges for the 1P-, 1D-, and 2S -wave excited Ω ccc and Ω bbb states are in good agreement with the Lattice QCD predictions [9,15]. It should be pointed out that mass orders for the spin multiplets in the excited Ω ccc and Ω bbb states predicted in the literature is very different. To clarify the mass order of these spin multiplets, the spin dependent integrations should be further studied in future works.
Moreover, by using the predicted masses and wave functions from the potential model, the radiative transitions for the 1P → 1S , 1D → 1P, and 2S → 1P are evaluated for the first time with a constituent quark model. For the Ω ccc sector, the transition rates for 1P → 1S might be sizeable, the partial widths are about several keV; the transition rates for the E1 dominant decay processes 1 2 D 3/2 + → 1 2 P 1/2 + ,3/2 + , 1 2 D 5/2 + → 1 2 P 3/2 + , 2 2 S 1/2 + → 1 2 P 1/2 + ,3/2 + are relatively large, their partial widths are predicted to be about 10s keV. For the Ω bbb sector, the partial widths of the corresponding transitions mentioned above are about one or two order of magnitude smaller than those for the Ω ccc sector. To better understand the radiative decay properties of the excited Ω ccc and Ω bbb baryon states, more studies are hoped to be carried out in theory.