Triply heavy baryons in the constituent quark model

The constituent quark model is used to compute the ground and excited state masses of QQQbaryons containing either c or bquarks. The quark model parameters previously used to describe the properties of charmonium and bottomonium states were used in this analysis. The non-relativistic three-body bound state problem is solved by means of the Gaussian expansion method which provides sufficient accuracy and simplifies the subsequent evaluation of the matrix elements. Several low-lying states with quantum numbers and are reported. We compare the results with those obtained by the other theoretical formalisms. There is a general agreement for the mass of the ground state in each sector of triply heavy baryons. However, the situation is more puzzling for the excited states, and appropriate comments about the most relevant features of our comparison are given.


I. INTRODUCTION
Quantum chromodynamics (QCD) is the stronginteraction part of the Standard Model of Particle Physics and solving it presents a fundamental problem that is unique in the history of science. Never before have we been confronted by a theory whose elementary excitations (quarks and gluons) are not those degreesof-freedom readily accessible via experiment; i.e., whose elementary excitations are confined inside hadrons. This complexity makes hadron spectroscopy, the collection of readily accessible states constituted from gluons and quarks, the starting point for all further investigations. A very successful classification scheme for hadrons in terms of their valence quarks and antiquarks was independently proposed by Murray Gell-Mann [1] and George Zweig [2] in 1964. This classification was called the quark model which basically separates hadrons in two big families: quark-antiquark (mesons) and three-quark (baryons) bound-states. The quark model received experimental verification beginning in the late 1960s and, * ygz0788a@sina.com † jlping@njnu.edu.cn ‡ pgortega@usal.es § jsegovia@upo.es despite extensive experimental searches, no unambiguous candidates for exotic quark-gluon configurations were identified. Until 2003 when the Belle Collaboration discovered an unexpected enhancement at 3872 MeV in the π + π − J/ψ invariant mass spectrum while studying the reaction B + → K + π + π − J/ψ [3]. The X(3872) state was later studied by the CDF, D0, and BaBar collaborations confirming that its quantum numbers, mass and decay patterns make it an unlikely conventional charmanti-charm (charmonium) candidate.
Mesons containing only heavy valence quarks, either cc (charmonium) or bb (bottomonium), have contributed to the understanding of QCD due to their approximately non-relativistic nature and the clean spectrum of narrow states, at least below open-flavor threshold. In fact, many precise experimental results are available for heavy quarkonium mesons, and their analysis has contributed significantly to the understanding of the quark-antiquark forces (see, for instance, Refs. [4,5] for nice reviews). On the other hand, almost two dozen charmonium-and bottomonium-like XYZ states [6] have been identified since the discovery of the first one: X(3872). Their analysis and new determinations will continue with the upgrade of experiments such as BES III [7], Belle II [8] and HL-and HE-LHC [9]. This will provide a sustained progress in the field as well as the breadth and depth necessary for a vibrant heavy quarkonium research environment.
In the same way that heavy mesons are useful to examine the quark-antiquark interaction in its non-relativistic limit, the triply-charmed and -bottom baryons may provide a new window for understanding QCD's properties without taking into account the usual light-quark complications. Moreover, as in the heavy quarkonium sector, there is no restriction of finding exotic structures in the triply-heavy baryon spectra and thus a reliable prediction of conventional QQQ-baryons 1 is interesting by itself in order to provide a template from which compare the future experimental findings.
The production of triply-heavy baryons is extremely difficult and thus no experimental signal for any of them has been reported yet. Baranov et al. estimated that triply-charmed baryons may not be observed in e + e − collisions and the expectations for bbb-baryons would be even worse [10]. With the advent of high-luminosity and -energy periods of the LHC accelerator facility, the LHCb experiment could be the best device to establish triplyheavy baryons in the near future [11][12][13]. Meanwhile, the Ω ccc is most probable to be discovered at RHIC by investigating the production of multi-charmed baryons in high energy nuclear collisions [14,15]. Finally, authors of Ref. [16] have suggested to look for triply-heavy baryons through their semi-leptonic and non-leptonic decays.
From a theoretical point of view, there are three promising approaches to the triply-heavy baryon problem which are non-relativistic effective field theories [17][18][19][20], lattice gauge theories [21][22][23][24][25][26], and the continuum approach to QCD based on Dyson-Schwinger and Faddeev equations [27,28]. All of them have been able to compute ground state masses of QQQ-baryons but still much work is needed in other to have a complete excited spectrum. In order to assist on this issue, one should expect that potential models work well on describing the triplyheavy baryons, at least with a similar level of precision as their success in heavy quarkonia. Partial results on the spectra for triply-heavy baryons have been reported by the non-relativistic [29][30][31][32][33] and relativistic [34,35] quark models, the Faddeev formalism using a non-relativistic reduction of the quark-quark interaction [36], the frontform formulation of an effective QCD Hamiltonian [37], the bag model approach [38], and applying QCD sum rules [39][40][41][42].
The present manuscript is arranged as follows. We describe briefly in Sec. II the constituent quark model, the triply-heavy baryon wave-function and the computational formalism based of Gaußian expansion method. Section III is devoted to the analysis and discussion of the obtained results. We summarize and give some prospects in Sec. IV.

II. THEORETICAL FRAMEWORK
The Hamiltonian which describes the triply-heavy baryon bound-state system can be written as [61]: where T CM is the center-of-mass kinetic energy and, since chiral symmetry is explicitly broken in the heavy quark sector, the two-body potential can be deduced from the one-gluon exchange and confining interactions. The onegluon exchange potential is given by where m i is the quark mass, λ c are the SU (3)-color Gell-Mann matrices, and the Pauli matrices are denoted by σ. The contact term of the central potential has been regularized as with r 0 (µ ij ) =r 0 µ nn /µ ij a regulator that depends on the reduced mass of the quark-quark pair, µ ij ; being µ nn = (313/2) MeV the one corresponding to the lightest quark-quark couple. The wide energy range needed to provide a consistent description of light, strange and heavy mesons requires an effective scale-dependent strong coupling constant. We use the frozen coupling constant [43] in which α 0 , µ 0 and Λ 0 are parameters of the model determined by a global fit to the meson spectra. Color confinement should be encoded in the non-Abelian character of QCD. Studies on a lattice have demonstrated that multi-gluon exchanges produce an attractive linearly rising potential proportional to the distance between infinitely heavy quarks [62]. However, the spontaneous creation of light-quark pairs from the QCD vacuum may give rise at the same scale to a breakup of the created color flux-tube [62]. We have tried to mimic these two phenomenological observations by the expression: where a c and µ c are model parameters. One can see in Eq. (5) that the potential is linear at short interquark distances with an effective confinement strength σ = −a c µ c ( λ c i · λ c j ), while it becomes constant at large distances.
The quark model parameters relevant for this work are shown in Table I. They, among others, were fitted over all meson spectra and the interested reader is referred to Refs. [43,46,63] for further details.
Triply-heavy baryon wave function is constructed from a product of four terms: color, flavor, spin and space wave functions. Concerning the color one, it can be written as usually for a baryon: The spin wave-function of a 3-quark system has been worked out in e.g. Ref. [61] and, for clarity purposes, we repeat our expressions herein: being α = 1 2 1 2 and β = 1 2 − 1 2 the spin states of the constituent quarks inside the baryon. The flavor wavefunction of a fully heavy quark baryon is quite simple and are given by Finally, the spatial wave function of the 3-body system can be written as where r and R are the internal Jacobi coordinates Among the different methods to solve the Schrödinger equation in order to find the triply-heavy baryon bound states, we replace the orbital wave functions, φ's in Eq. (19), by a superposition of infinitesimally-shifted Gaussians [64]: where the limit ε → 0 must be carried out after the matrix elements have been calculated analytically. This new set of basis functions makes the calculation of 3body matrix elements easier without the laborious Racah algebra. Following Ref. [64], the Gaußian ranges ν n are taken in a geometric progression. This enables their optimization employing a small number of free parameters. Moreover, the geometric progression is dense at short distances, so that it allows the description of the dynamics mediated by short range potentials. The fast damping of the Gaußian tail is not a problem, since we can choose the maximal range much longer than the hadronic size. Finally, in order to fulfill the Pauli principle, the complete antisymmetric wave-function is written as where the antisymmetric operator A is the same for Ω ccc and Ω bbb , i.e. A = 1 − (13) − (23) in a system with three identical particles. However, A = 1 for Ω ccb and A = 1 − (23) for Ω cbb . This is needed because we have constructed an antisymmetric wave function for only two quarks of the 3-quark cluster, the remaining quark of the system has been added to the wave function by simply considering the appropriate Clebsch-Gordan coefficient.
III. RESULTS Table II shows the total spin and parity of the triplyheavy baryons whose masses shall be calculated herein. In a non-relativistic approximation, the total angular momentum, L, and spin, S, are good quantum numbers and they couple to total spin, J. The total angular momentum is reached by the coupling of the two possible excitations along the Jacobi coordinates, i.e. the l 1 and l 2 of Eq. (19) couple to L with l's never greater than L in this analysis. Since a baryon is a 3-quark bound-state system, its total spin can only take values 1/2 and 3/2 and its parity is given by (−1) L since the parity of a quark is positive by convention.
Tables III, V, VII and IX show, respectively, the spectrum of Ω ccc , Ω ccb , Ω cbb and Ω bbb baryon sectors computed within our formalism and following the pattern of quantum numbers discussed above and shown in Table II. Since there is no experimental data related with the triply-heavy baryons, we compare our results with those recently predicted by lattice QCD when available. It is worth to note here that lattice-regularized computations need to deal better with excited states and thus our level of agreement should be judged mostly through the comparison with ground state predictions. Moreover, continuum and chiral extrapolations are still needed in the available lattice data. For this reason, we shall add for each sector a table which shows predicted masses by other theoretical approaches and our results (Tables IV, VI, VIII, and X).
The Ω ccc baryon sector.  [23]. We report herein S-, P -and Dwave ground and radial-excited states for all channels mentioned. One can realize looking at Table III that we are predicting a mass of 4.80 GeV for the ground state of the spectrum which has quantum numbers nL (J P ) = 1S ( 3 2 + ). The same state is predicted by lattice-QCD to be the lowest and its mass, 4.76 GeV, compares well with ours. In fact, the agreement between lattice and our calculation is repeated for each ground state of the channels reported in Table III. There is some mismatch between our calculation and that using lattice-regularized QCD for excited states. The origin of some discrepancies is clear. For instance, the extra state predicted by lattice that appear around the energy range of our 1D ( 1 2 + ) state must correspond to a state with quantum numbers S = 1/2 and L = 0 with l 1 = l 2 = 1. As mentioned at the beginning of this section, such case is not considered in our computation. For other positive parity baryons, where the mentioned ambiguity does not take place at low-energy spectrum, we obtain good agreement with lattice results. It is the case of the J P = 3 2 + channel in which the absolute value of the 2S ( 3 2 + ) state predicted by lattice, 5.31 GeV, compares well with our prediction: 5.29 GeV. It is also interesting to observe that the excited states predicted by lattice in the negative parity channels have energies which slightly disagree with ours, 0.1 GeV. Table IV compares the masses computed by other theoretical approaches with ours. Only Ref. [37] provides a spectrum which is as complete as ours; we shall see later that this feature is repeated in other triply-heavy baryon sectors with a very few exceptions. As shown by Table IV This work 4798 5286 5376 5129 5129 [18] 4760 ± 60 ---- [20] 4979 ± 271 ---- [24] 4789 ± 6 ± 21 ---- [25] 4796 ± 8 ± 18 ---- [27] 4760 5150 --5027 [28] 5000 ---- [30] 4632 -4915 ± 283 4808 ± 176 - [31] 4812 ± 85 ---- [34] 4803 ---- [35] 4773 -5216 5109 5014 [36] 4801 ± 5 ---- [39] 4670 ± 150 ---- [37] 4797 5309 5358 5103 5103 [40] 4990 ± 140 ---5110 ± 150 [42] 4720 ± 120 ---4900 ± 100 is quite sparse with big uncertainties in some cases, making difficult to perform a quantitative comparison. However, we can state that our prediction is in reasonable agreement with the one which is available, even if the theoretical approach we compare differ from one state to another.
The Ω ccb baryon sector. Table V [26] is remarkable. We predict the ground state for their corresponding parity partners at around 8.3 GeV. As clearly shown in Fig 2, there is a rich spectrum of low-lying states which concentrate in an energy region of 1 GeV, i.e. the masses go from 8 to 9 GeV which should be an accessible energy region for current experimental facilities such as the LHCb at CERN.
We collect in Table VIII the computed masses for Ω cbb by other approaches and compare them with our results. The reported mass of Ref. [18], which is a model independent prediction based on the application of a non-relativistic effective field theory, agrees nicely with our result. Let us highlight here that the same level of agreement with the predictions of Ref. [18] exists in all Ω QQQ sectors, with Q either a c-or b-quark. The general trend in most of the theoretical results collected in Table VIII is to predict a ground state of Ω cbb with quantum numbers 1S ( 1 2 + ) and mass around 11.2 GeV. As we already mentioned above, its partner with quantum numbers 1S ( 3 2 + ) is quasi-degenerated with a mass only 20 MeV higher, a similar pattern is observed in other formalisms. There are only two computations, Refs. [37] and [27], that provide results for all the states shown in Table VIII; our results agree nicely with those predicted by the two approaches.
There is some mismatch between our calculation and that using lattice-regularized QCD for excited states. The origin of some discrepancies is clear. For instance, the extra state predicted by lattice that appear around the energy range of our 1D ( 1 2 + ) state must correspond to a state with quantum numbers S = 1/2 and L = 0   It is also interesting to observe that the excited states predicted by lattice in the negative parity channels have energies which slightly disagree with ours, 0.03 GeV.
We can turn now our attention to Table X which compares the results predicted by our quark model with the ones reported by other theoretical approaches. There are again few results, mostly reported from QCD sum rules, which scatter the general trend of the ground state mass. If one keeps these results out of the average, one realizes that the 1S ( 3 2 + ) state should have a mass around 14.4 GeV, which agrees with ours. It is nice to observe good agreement between our reported results and the ones computed in Ref. [37]. However, this agreement is lost when comparing the masses predicted by other approaches for the 2S( We do not have a sensible answer to this issue and then we suggest to continue investigating this sector.

IV. SUMMARY
The study of heavy mesons has revealed very useful to examine some relevant QCD's properties and fundamental parameters, without taking into account the usual light-quark complications. Therefore, the triply-charmed and bottom baryons may provide a new window for understanding deeper QCD. Moreover, as in heavy quarkonia, there is no restriction of finding exotic structures in the triply-heavy baryon spectrum and thus a reliable prediction of conventional QQQ-baryons is interesting in order to deliver a template from which compare the future experimental findings.
A phenomenological constituent quark model which has been successfully applied in the last years to the study of spectrum and electromagnetic, strong and weak decays and reactions of heavy quarkonia is used herein to compute ground-and excited-state energies of triplyheavy baryons with quantum numbers J P = 1 2 ± , 3 2 ± , 5 2 ± and 7 2 + .
We solve the non-relativistic 3-body bound state equation by means of a variational method in which the wave function solution is expanded using infinitesimallyshifted Gaußians. This new set of basis functions makes the calculation of 3-body matrix elements easier without the laborious Racah algebra. The Gaußian ranges are taken in a geometric progression enabling its optimization employing a small number of free parameters. Moreover, the geometric progression is dense at short distances, so that it allows the description of the dynamics mediated by short range potentials. The fast damping of the Gaußian tail is not a problem, since we can choose the maximal range much longer than the hadronic size.
There is no experimental data available for the Ω QQQ baryons. Our computed masses for Ω ccc and Ω bbb baryons agree with lattice data for the ground state of all triplyheavy baryons studied. However, some discrepancies are found between our computation of excited states and the recent one reported by lattice-regularized QCD. In these two sectors we have also compared our results with those computed by other theoretical approaches arriving to the conclusion that there is a general agreement about what would be the lowest state and with what mass. The prediction of higher excited states have not been done in a very systematic way by other approaches but our results compare nicely when data exists, even if the comparison must be done taking into account different theoretical formulations.
The Ω ccb and Ω cbb baryon sectors are less explored nowadays but still an array of theoretical predictions is available. Once results for the ground-state's mass predicted by QCD sum rules are discarded, a general trend of agreement among very different formalisms is revealed for this state: nL (J P ) = 1S ( 1 2 + ). This average mass is compatible with our prediction too. Again, few formalisms report a complete spectrum of lowlying excited states; when available, we have compare them with our calculation and the matching is quite reasonable.
It is interesting to remark herein that the spectra of Ω ccc , Ω ccb , Ω cbb and Ω bbb baryons are revealed to be quite reach in just an energy region of 1 GeV above the corresponding ground state. Therefore, we encourage experimentalists to design experiments able to detect this kind of particles because the reward could be high and, as mentioned above, triply-heavy baryons are ideally suited to study QCD as it has been the case for heavy quarkonia.
Finally, after this computation, a possible worthwhile direction would be coupling the naïve triply-heavy baryons presented herein with their closer baryon-meson thresholds using the 3 P 0 decay model as the mechanism connecting both 3-and 5-quark sectors. Mass-shifts, decay widths and all kind of scattering phenomena will then be available to study exotic structures as potential XY Z states. A similar procedure has been followed by us in the heavy quark meson sector with a great level of success.

V. ACKNOWLEDGMENTS
The authors would like to thank L. He, D.R. Entem, F. Fernández, and C.D. Roberts for their support and informative discussions. Work supported by: National Natural Science Foundation of China under Grant nos. 11535005 and 11775118; and by Spanish Ministerio de Economía, Industria y Competitividad under contracts no. FPA2017-86380-P and FPA2016-77177-C2-2-P. P.G.O. acknowledges the financial support from Spanish MINECO's Juan de la Cierva-Incorporación programme, Grant Agreement No. IJCI-2016-28525.