Investigation of the mass spectra of singly heavy baryons $\Sigma_{Q}$, $\Xi^{\prime}_{Q}$ and $\Omega_{Q}$ $(Q=c, b)$ in the Regge trajectory model

Very recently, LHCb Collaboration observed that two new $\Omega_{c}^{0}$ states decay into $\Xi^{+}_{c}K^{-}$ with masses of about $3185$ MeV and $3327$ MeV. However, their spin parity quantum numbers $J^{P}$ have not been determined. In this paper, we exploit the quark-diquark model, the linear Regge trajectory and the perturbation treatment method to analyze the mass spectra of the discovered experimental data for the singly heavy baryons $\Sigma_{c}/\Sigma_{b}$, $\Xi^{\prime}_{c}/\Xi^{\prime}_{b}$ and $\Omega_{c}/\Omega_{b}$. In addition, we further predict the mass spectra of several unobserved $\Sigma_{c}/\Sigma_{b}$, $\Xi^{\prime}_{c}/\Xi^{\prime}_{b}$ and $\Omega_{c}/\Omega_{b}$ baryons. In the case of the $\Omega_c(3185)^{0}$ and $\Omega_c(3327)^{0}$ states, we determine $\Omega_{c}(3185)^{0}$ as $2S$ state and $\Omega_{c}(3327)^{0}$ as $1D$ state with $J^{P}=1/2^{+}$ and $J^{P}=3/2^{+}$, respectively. An overall good agreement of the obtained predictions with available experimental data are found.


I. INTRODUCTION
With the discovery of more and more highly excited strongly interacting particles in experiments, such as LHCb, Belle, BaBar, and CLEO, a deeper understanding of the singly heavy baryons has been gained.In the quark-diquark picture, singly heavy baryons are composed of an anti-color triplet (3 c ) diquark with spin one (S d = 1), formed by two light quarks, and a heavy quark (S Q = 1/2).The latest review of particle physics by PDG can shed new light on the singly heavy baryons Σ c /Σ b , Ξ ′ c /Ξ ′ b and Ω c /Ω b .From PDG [1] in 2022, the establishment of S, P and D-wave excited states are gradually improved providing valuable insights into the fundamental structure and behavior for the Σ c /Σ b , Ξ ′ c /Ξ ′ b and Ω c /Ω b baryons.In the Σ c /Σ b baryons, the Σ c (2455) 0,+,++ and Σ c (2520) 0,+,++ states can be well interpreted as S-wave charmed baryons with J P = 1/2 + , J P = 3/2 + , respectively.The triplet of the excited Σ c (2800) 0,+,++ states decaying to Λ + c π were observed by Belle Collaboration

II. THE REGGE TRAJECTORY AND THE SPIN-AVERAGE MASSES
In the QCD rotating string model [32,33], the strong interaction binds the heavy and light quark inside the hadron, where one end of the string is a heavy quark and the other is a light antiquark or light diquark moving around the heavy quark.Based on this model, it is interesting to investigate the Regge trajectory behavior of the hadronic system.
For the orbital excitations of the baryons, we obtain the spin-average mass M and angular momentum L following the equations given by Refs.[34,35] M = m curQ where α is the QCD string tension coefficient, and v Q , v d the velocity of the string end tied to between the heavy quark Q and light diquark d.We define the velocity where ω and r i are the angular velocity and the position from the centre of mass, respectively.For simplicity, we have chosen the velocity of light c = 1.The light diquark is ultrarelativistic, we take the velocity of light diquark v d ≈ 1 for approximation.Then m curQ and m curd can be regarded as current mass of the heavy quark and light diquark, respectively.Including relativistic effects, one can obtain the constituent quark masses Eqs. ( 1) and ( 2) can be integrated to give where for the string ending at the heavy quark we use the boundary condition Substituting Eq. ( 6) into Eqs.( 4) and ( 5) eliminating the angular velocity ω gives the spin-averaged mass formula [36][37][38][39] for the orbital excited states, here, the intercept factor a 0 = (m d + M Q v 2 Q ) 2 depends on the diquark mass m d and the nonrelativistic 3-kinematic energy M Q v 2 Q = P 2 Q /M Q for the heavy quark.Note that the non-relativistic kinematic 3-momentum P Q is conserved in the heavy quark limit, which has been associated with both M Q and v Q .Using a variant of Eq. ( 3), the velocity v Q is is the orbital angular momentum of the baryon systems (L = 0, 1, 2, • • •).Accordingly, the current masses, the constituent quark masses and the string tension are applied in Eq. ( 9) as listed in Table I, which were previously determined in Refs.[30,39] via matching the measured mass spectra of the singly heavy baryons.To obtain the spin-average masses of the orbital and radial excited states Σ c /Σ b , Ξ ′ c /Ξ ′ b and Ω c /Ω b , we re-examine the Regge-like mass relation Eq. (9).By an analysis of the experimental data given by PDG [1] we suggest that the slope ratio of the Regge trajectory between the radial and angular momentum is 1.37 : 1. Accordingly, παL in Eq. ( 9) is replaced by πα(L + 1.37n), where n is a radial quantum number (n = 0, 1, 2, •••).We use Eq. ( 10) to calculate the spin-average masses of the Σ c /Σ b , Ξ ′ c /Ξ ′ b and Ω c /Ω b baryons.The results are listed in Table II.Accordingly, the squared mass difference (M − M ) 2 of the heavy-light hadronic system is related to L and n by The squared mass difference (M − M ) 2 for the charm baryons is calculated and plotted against L in Fig. 1

III. THE SPIN-DEPENDENT POTENTIAL AND THE SCALING RELATIONS
Even though the baryon is a three-body system under the strong interaction, it is helpful to understand the measured mass data of the excited baryons using a simple heavy quark-diquark picture.To estimate the mass splitting for the singly heavy baryons, we consider the spin-dependent Hamiltonian H SD [11,40] between the heavy quark (Q) and the spin-1 diquark (d) as where a 1 , a 2 , b 1 , c 1 are the spin coupling parameters.The first two terms are spin-orbit interactions, the third is the tensor energy, and the last is the contact interaction between the heavy quark spin S Q and the diquark spin S d .For the particular choice L = 0 for the S-wave baryons in appendix A, the first three terms of Eq. ( 12) can be eliminated and only the last term survives, see Eq.
(A1).Here, [27] with L = 1 and L = 2 can be given by Combined with the experimental data [9] of the Ω c (css), we used the Regge trajectory Eq. ( 9) to fit the constituent quark masses of the charm quark (c) and two strange quarks (ss) in Ref. [30], the results are M c = 1.44 GeV and m ss = 0.991 GeV.In the case of doubly strange Qss baryons with the mass of the diquark ss comparable with the mass of the heavy quark Q (m ss ≈ M c ), the finite mass effect of the heavy quark may become important and makes it appropriate to go beyond the jj coupling.Therefore, in contrast to the scheme used in Ref. [40], we proposed a new scheme of state classification named the JLS coupling [30].The first three terms are treated as operators defining representations and the last term 12) as a perturbation.
The operator H SD 1 is given by    Using the bases |J, j in terms of eigenvalues J, j of the total angular momentum J and total light-quark angular momentum j, respectively, in order to diagonalize the mass operators H SD 1 and H SD 2 , we can obtain the mass shifts ∆M of P -wave in Eq. (B6) and D-wave in Eq. (C4) for the singly heavy baryons, see appendix B and C. In this scheme, the P -wave states of the baryons may be classified as 2S+1 P J = 2 P 1/2 , 4 P 1/2 , 2 P 3/2 , 4 P 3/2 , 4 P 5/2 and the D-wave states as Next, it is necessary to estimate the four spin coupling parameters a 1 , a 2 , b 1 , c 1 in the heavylight quark system.If Eq. ( 12) is taken as a spin-relevant relativistic correction, the parameters a 1 , a 2 , b 1 , c 1 are related to the magnetic moment S Q /M Q of the heavy quark.Therefore, these parameters can be considered roughly inversely proportional to the heavy quark mass (M Q ).In Ref. [40], the authors calculated the parameters of the partner in baryons using the scaling relations with the constituent quark masses (M c , M b ) of the heavy quark in baryons.The parameter c 1 is expected to be negligible, because it should be very small in the P -wave states of the baryons.
In order to calculate the mass splitting of all excited states, we utilize the scaling relations based on the similarity between a baryon and its the partner baryons in the color configurations to study the spin coupling parameters.In this subsection, we need to generalize Eq. ( 16) and consider the parameter c 1 which should include the effect of the principal quantum number N together with the radial quantum number n and orbital quantum number L [11,[41][42][43][44]. The parameters a 1 , a 2 , b 1 are obtained by following the scaling rules: Here, 1/r = 1/((n + L + 1) 2 a B ), 1/r 3 = 1/(L(L + 1/2)(L + 1)(n + L + 1) 3 a 3 B ) and a B is the Bohr radius.According to the scaling rules, a 2 can be of the same order as a 1 with the same n, L in the excited states, while the parameter b 1 should be smaller than the a 1 , a 2 , as b 1 scales with In order to obtain the parameter c 1 in Eq. ( 12), we need a scaling rule similar to (i)-(iii).
Considering that c 1 becomes dominant in determining the mass splitting Eq. (A5), we can estimate c 1 based on the hyperfine structure term given by [45,46] where ∇ 2 is the Laplace operator and δ 3 (r) is the three-dimensional delta distribution.The derivative of the Coulomb potential V gives ▽ 2 V = 4πα s δ 3 (r) with the strong coupling α s .By taking the average δ 3 (r) = |ψ(0)| 2 established for the hydrogen-like atoms wave function ψ(r) of S-wave (L = 0) [47], Eq. ( 17) becomes with N = n + L + 1.To extend Eq. ( 18) further to the excited states of the baryons, we introduce a parameter λ as follows, Based on the systematic analysis of experimental values, we find the parameter λ = 3.3.Analyzing the coefficient in Eq. ( 19), the parameter Thus, the scaling rule of c 1 can be determined as follows: (iv) The parameter c 1 is proportional to 1 Eventually, the scaling relations of the spin coupling parameters in Eq. ( 12) for the baryon system are where and n ′ , respectively.The prime denotes the quantities of the baryon B ′ a obtained from experiments, distinguishing them from that of an unobserved baryon B a .

IV. THE BARYONS Ω c AND Ω b
For Ω c baryon family, it was a pleasant surprise that the LHCb Collaboration recently discovered Later, the Belle Collaboration confirmed the existence of these states [48].In Ref. [30], the authors employ the quark model to analyze the narrow Ω c states, and suggested that the parity was negative for all of five states.These can be interpreted as 1P -wave charmed baryons candidates.
Correspondingly, the masses which can give good results for the Ω c states, and are consistent with the experimental data of the LHCb Collaboration.At the same time, by fitting, the spin coupling parameters a 1 , a 2 , b 1 , c 1 are also obtained in Ref. [30], These results are the same as those in both of Refs.[27,49].For more information of the Ω c baryons, we recommend interested readers to see Refs.[11-22, 24, 50].
To elaborate on the mass shifts ∆M (J, j) for the entire baryon systems, we utilize the parameters (22) of the 1P -wave Ω c states as the object of the scaling relations in Eq. ( 20) to calculate the parameters of the other states.Adding the spin-average mass M , the baryon mass becomes M (J, j) = M + ∆M (J, j), where details of calculating ∆M (J, j) and M (J, j) are presented in the Appendix.Therefore, the mass spectra of the singly heavy baryons can be predicted.
The mixed state Ω c (3327) 0 has been speculated as a 2S state in Ref. [51] and as a 1D state in Refs.[52][53][54].However, we still need more observable objects to get clarity about the internal structure.In addition, in Ref. [24] the authors suggested that Ω c (3185) 0 may be regarded as a 2S state with J P = 1/2 + or J P = 3/2 + , or their overlapping structure, and Ω c (3185) 0 is interpreted as a P -wave state in Ref. [55].Very recently, the Ω c (3185) 0 and Ω c (3327) 0 states of Ω c baryons were observed by LHCb Collaboration [31] with masses 3185.1 MeV and 3327.1 MeV, respectively.The quantum numbers of these states remain to be determined.According to our model, the calculation of the spin-averaged mass and the parameter for the Ω c states in 2S-wave (L = 0, n = 1) are obtained by using Eqs.(10), (20) and (22), Hence, the masses of the 2S-wave Ω c states are 3185.20MeV and 3193.09MeV as listed in Table V with J P = 1/2 + and J P = 3/2 + , respectively.The Ω c (3185) 0 can be grouped into the 2S state.

V. THE BARYONS Σ c AND Σ b
By analyzing the existing experimental data in PDG [1], we explore some patterns of the oddparity Σ Q (Q = c, b) baryons consisting of a light isospin-one nonstrange diquark (nn = uu, ud, dd) in a state of L with respect to the spin-1/2 heavy quark Q.So far, the Σ Q baryons have been observed in experiments, and the data are available from the Particle Data Group, which provides us with more information to study the mass spectra of the Σ Q states.
Ref. [1] cites the two masses M (Σ c , 1/2 + ) = 2452.65MeV, M (Σ c , 3/2 + ) = 2517.4MeV for Σ c (2455) + , Σ c (2520) + with J P = 1/2 + and 3/2 + , respectively, which was discovered and identified as 1S-wave states by the LHCb experiment.Accordingly, by using Particle Data Group masses, the spin-weighted average mass is obtained by [59] M spin-weighted = Σ(2J + 1)M (J) MeV between Σ c (2455) + and Σ c (2520) + is regarded as a good reference for comparing the results of our model.For the Σ c baryons, comparing the measured masses presented in Table IX with our prediction masses, and the parameters as shown in Table VII, it is seen that the masses of all these states are compatible with the experimental values (within few MeV).We employ Eq. (10) to calculate the spin-averaged mass M of 1S-wave with L = 0, n = 0, as well as the following rough estimate for the parameter c 1 by Eq. ( 20), with c 1 (Ω c , 1P ) = 4.04 MeV given in Eq. ( 22).Note that the heavy quark mass M c cancels out for charmed baryons.
The Σ c (2800) observed by the Belle Collaboration [2] might be a good candidate for a 1Pwave state (cf.e.g.Ref. [40]).For comparison with the experiment values, we also compute the parameters and the masses of the Σ Although the Belle Collaboration observed the excited Σ c (2800) state in the decay channel Λ + c π [60] which mass at M (Σ c ) = 2792 MeV, the J P has not been determined, making it difficult to determine its properties.The Σ c (2800) state is calculated by our model to own the mass 2788.31 MeV, which is in agreement with the experiment as show in Table IX.Hence, for Σ c (2800) we should advocate the fourth state | 4 P 3/2 , 3/2 − of 1P -wave.The nature of these states is discussed in Refs.[11,61].b → Λ + c π − and Λ + c → ρκ + c π + decays in Ref. [3].In our calculations, Σ b (6097) can be a good candidate of 1P -wave excitations.Therefore, we assign J P = 5/2 − to the Σ b (6097) state.Finally, the spin-averaged mass, the parameters and the mass splitting are given by Eq. ( 10) and Eq. ( 20) in 1P -wave   In this section, based on our scheme, a similar method can be applied to the excited Ξ ′ Q (csn or bsn) baryon systems in order to analyze their masses and parameters.For the Ξ ′ c baryon system, the (S-wave) ground states with the spin-parity J P = 1/2 + and J P = 3/2 + correspond to Ξ ′0  MeV lower than the mass of the state Ξ c (2923) 0 , and lower than M (Ξ c (2930) 0 ) = 2935.17MeV, compared with experimental values within a reasonable range.For a more detailed analysis the Ξ ′ c baryons see also Refs.[62,64].
In addition, the state Ξ c (3123) was also confirmed by the BaBar Collaboration [65], with a mass M (Ξ c + ) = 3122.9MeV listed in PDG [1].From the analysis of our data in Table XIII we infer that the mass shifts of about 22 MeV in the 1D-wave are relatively small.In the past, the quantum number of Ξ c (3123) was not determined.In our frame, it is possible to determine Ξ c (3123) as the second state with J P = 3/2 + or mixed with the first state, which can be a good candidate for a 1D state of the Ξ ′ c baryons.
The predicted masses are compatible with the experimental values, closer to the second state or mixed with the first state.The same conclusion holds for the masses of the Ξ ′ b baryons as shown in Table XII and Table XIV.The latter can be inquired also for a discussion of Ξ b (6227) in different models [57,58] and the well-matching with the experiment.For analyzing the D-wave system, the diquark spin S d = 1 can be coupled with the heavy quark spin S Q = 1/2 to determine the total spin S = 1/2, 3/2.Coupling of the orbital angular momentum L = give six states with the total spin J = 1/2, 3/2, 5/2 or 3/2, 5/2, 7/2 with positive parity P = +1.The relevant linear combinations of six basis states are , 3, 5 with n = 0, 1, 2, 3 and 4. Similarly, the results of the bottom baryons are shown in Fig. 2, 4, 6.The (red) solid circles correspond to the observed (mean) masses and the empty circles indicate the predicted value in Fig. 1-6.It can be seen that (M − M ) 2 increases with both L and n.

Table VI .
These mass predictions presented in Table V and Table VI for the Ω Q (Q = c, b) baryons will be helpful for future experimental searches.
2 , b 1 , c 1 as shown in Table IV, while our mass results are compared to results of other models in

TABLE III :
The spin coupling parameters (MeV) of the Ω c baryons.

TABLE IV :
The spin coupling parameters (MeV) of the Ω b baryons.

TABLE V :
The mass spectrum (MeV) of Ω c baryons are given and compared with different quark models.

TABLE VI :
The mass spectrum (MeV) of Ω b baryons are given and compared with different quark models.
M (Σ b , 1P ) : 6048.86MeV, 6070.13MeV, 6076.71MeV, 6087.21MeV, 6099.56MeV.(45) Evidently, a 1 , a 2 , b 1 reasonably fulfill (i)-(iii), and c 1 in (iv) becomes a non-vanishing but small value for the highly excited states.We exploit Eq. (B8) to calculate the mass splitting for the Σ b states.The results of the parameters are listed in Table VIII and the masses in Table X.Under the analysis of the model, these results are consistent with the experimental values.

TABLE VII :
The spin coupling parameters (MeV) of the Σ c baryons.

TABLE VIII :
The spin coupling parameters (MeV) of the Σ b baryons.

TABLE IX :
The mass spectrum (MeV) of Σ c baryons are given and compared with different quark models.

TABLE X :
The mass spectrum (MeV) of Σ b baryons are given and compared with different quark models.

TABLE XI :
The spin coupling parameters (MeV) of the Ξ ′

TABLE XII :
The spin coupling parameters (MeV) of the Ξ ′

TABLE XIII :
The mass spectrum (MeV) of Ξ ′ c baryons are given and compared with different quark models.

TABLE XIV :
The mass spectrum (MeV) of Ξ ′ b baryons are given and compared with different quark models.