# A New Model for Calculating the Ground and Excited States Masses Spectra of Doubly HeavyΞ Baryons

Authors: Neda Mohajery; Nasrin Salehi; Hassan Hassanabadi
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

Since the doubly heavy baryons masses are experimentally unknown (exceptΞcc+ and Ξcc++), we present the ground state masses and the positive and negative parity excited state masses of doubly heavy Ξ baryons. For this purpose, we have solved the six-dimensional hyperradial Schrödinger equation analytically for three particles under the hypercentral potential by using the ansatz approach. In this paper, the hypercentral potential is regarded as a combination of the color Coulomb plus linear confining term and the six-dimensional harmonic oscillator potential. We also added the first-order correction and the spin-dependent part contains three types of interaction terms (the spin-spin term, spin-orbit term, and tensor term) to the hypercentral potential. Our obtained masses for the radial excited states and orbital excited states of Ξccd, Ξccu, Ξbbd, Ξbbu, Ξbcd, and Ξbcu systems are compared with other theoretical reports, which could be a beneficial tool for the interpretation of experimentally unknown doubly heavy baryons spectrum.


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## Body

## 1. Introduction

The doubly heavy baryons have two heavy quarks (c and b) with a light quark (d or u or s). The doubly heavyΞ baryons family have up or down quarks but Ω family has a light strange quark and their masses spectra have been predicted in the quark model [1]. The SELEX collaboration announced only the experimental mass for the ground state of Ξcc+ baryon and LHCb has determined the ground state of Ξcc++ baryon mass while no triply heavy baryons have been observed yet [2]. Recently experiments and theoretical outcomes have been used in studying the heavy baryons. A lot of new experimental results have been reported by various experimental facilities like CLEO, Belle, BaBar, LHCb, and so forth [3, 4] on ground states and many new excited states of heavy flavor baryons. Bottom baryons are investigated at LHC and Lattice QCD whereas charm baryons are announced at the B-factories [5, 6]. On the other hand, the theoretical works are providing new results for doubly heavy baryons like the Hamiltonian model [7], relativistic quark model [8], the chiral unitary model [9], QCD sum rule [10, 11], and many more. Single- and double- heavy baryons in the constituent quark model were studied by Yoshida et al. They used a model in which there were two exceptions, a color Coulomb term depending on quark masses and an antisymmetric L.S force. They studied the low-lying negative parity states and structures within the framework of a constituent quark model [7]. In [12], the authors calculated the masses of baryons with the quadratic mass relations for ground and orbitally excited states. Wei et al. estimated the masses of singly, doubly, and triply bottom baryons in [13]. Then they studied the linear mass relations and quadratic mass relations.The light flavor dependence of the singly and doubly charmed states is investigated by Rubio et al. They focused on searching the masses of charmed baryons with positive and negative parity [5]. In [14], the authors used lattice QCD for baryons containing one, two, or three heavy quarks. They applied nonrelativistic QCD for the bottom quarks and relativistic heavy-quark action for the charm quarks. Padmanath et al. determined the ground and excited state spectra of doubly charmed baryons from lattice QCD with dynamical quark fields [15]. The mass of the heavy baryons with two heavy b or c quarks for spin 1/2 in the framework of QCD sum rules is estimated by Aliev et al. They use the most general form of the interpolating current in its symmetric and antisymmetric forms with respect to the exchange of heavy quarks, to calculate the two point correlation functions describing the baryons under consideration [16]. The authors calculated the masses and residues of the spin 3/2 doubly heavy baryons within the QCD sum rules method. In [17], Eakins et al. ignored all spin-dependent interactions and assume a flavor independent potential, working in the limit where the two heavy quarks are massive enough that their motion can be treated as essentially nonrelativistic, and QCD interactions can be well described by an adiabatic potential [18]. The three-quark problem was solved by Valcarce et al. by means of the Faddeev method in momentum space [19].The masses of the ground and excited states of the doubly heavy baryons were calculated by Ebert et al. baryons on the basis of the quark-diquark approximation in the framework of the relativistic quark model [20]. In [21], the authors, in the model with the quark-diquark factorization of wave functions, estimated the spectroscopic characteristics of baryons containing two heavy quarks. Albertus et al. used five different quark-quark potentials that include a confining term plus Coulomb and hyperfine terms coming from one–gluon exchange. They solved the three-body problem by means of a variational ansatz made possible by heavy-quark spin symmetry constraints [22].In this study, we have used the hypercentral constituent quark model (hCQM) with Coulombic-like term plus a linear confining term and the harmonic oscillator potential [23]. We also added the first-order correction and the spin-dependent part to the potential and calculation has been performed by solving six-dimensional hyperradial Schrödinger equations by using the ansatz method. We have obtained the mass spectra of radial excited states up to 5S and orbital excited states for 1P-5P, 1D-4D, and 1F-2F states.This paper is organized as follows: we briefly present the hypercentral constituent quark model and introduce the interaction potentials between three quarks in doubly heavy baryons in Section2. In Section 3, we present the exact analytical solution of the hyperradial Schrödinger equation for our proposed potential. In Section 4, our masses spectra results for ground, radial, and orbital excited states of baryon family with six members are given and compared with other predictions. We present the conclusions in Section 5.


## 2. Theoretical Framework: The HCQM Model and Hypercentral Potential

The hypercentral model has been applied to solve bound states and scattering problems in many various fields of physics. In this model, we consider baryons as three-body systems of constituent quarks. In the center of mass frame, the internal quark motion is described by the Jacobi coordinates (ρ and λ) [37] and the respective reduced masses are given by(1)mρ=2m1m2m1+m2,mλ=2m3m12+m22+m1m2m1+m2m1+m2+m3Here m1, m2, and m3 are the current quark masses. In order to describe three-quark dynamics, we define hyperradius x=ρ2+λ2 and hyperangle ξ=arctanρ/λ [38]. In present work, the confining three-body potential is regarded as a combination of three hypercentral interacting potentials. First, the six-dimensional hyper-Coulomb potential Vhyc(x)=τ/x, which is attractive for small separations [39–41], while at large separations a hyper-linear term, Vcon=βx, gives rise to quark confinement [42], where β corresponds to the string tension of the confinement [43]. Third, the six-dimension harmonic oscillator potential Vh.o.=px2, which has a two-body character and turns out to be exactly hypercentral [44], where p is constant. The solution of the hypercentral Schrödinger equation with Coulombic-like term plus a linear confining term potential cannot be obtained analytically [45]; therefore, Giannini et al. used the dynamic symmetry O(7) of the hyper-Coulomb problem to obtain the hyper-Coulomb Hamiltonian and eigenfunctions analytically and they regarded the linear term as a perturbation. Combination of the color Coulomb plus linear confining term and the six-dimensional harmonic oscillator potential has interesting properties since it can be solved analytically, with a good correspondence to physical results. The first-order correction V(1)(x) can be written as [44–47] (2)V1x=-CFCAαS24x2The parameters CF=2/3 and CA=3 are the Casimir charges of the fundamental and adjoint representation. The hyper-Coulomb strength τ=-(2/3)αS, 2/3 is the color factor for the baryon. αs is the strong running coupling constant, which is written as (3)αS=αSμ01+33-2nf/12παSμ0ln⁡m1+m2+m3/μ0The spin-dependent part VSD(x) is given as(4)VSDx=VSSxS→ρ.S→λ+VγSxγ→.S→+VTxS2-3S→.x→S→.x→x2The spin-dependent potential,VSD(x), contains three types of the interaction terms [48], such as the spin-spin term VSS(x), the spin-orbit term VγS(x), and tensor term VT(x) described as [35]. Here S=Sρ+Sλ, where Sρ and Sλ are the spin vectors associated with the ρ and λ variables, respectively. The coefficient of these spin-dependent terms of the above equation can be written in terms of the vector, VV(x)=τ/x, and scalar, VS(x)=βx+px2 parts of the static potential as [38](5)Vγs=12mρmλx3dVVdx-dVSdx(6)VTx=16mρmλ3d2VVd2x-1xdVVdx(7)VSSx=13mρmλ∇2VVIn our model, the hypercentral interaction potential is assumed as follows [48]: (8)Vx=V0x+1mρ+1mλV1x+VSDxwhere V(0)(x) is given by (9)V0x=Vhycx+Vconx+Vh.o.x=τx+βx+px2The baryons masses are determined by the sum of the model quark masses plus kinetic energy, potential energy, and the spin-dependent interaction as MB=∑mi+H [49]. First, we have solved the hyperradial Schrödinger equation exactly and find eigenvalue under the proposed potential by using the ansatz approach.


## 3. The Exact Analytical Solution of the Hyperradial Schrödinger Equation under the Hypercentral Potential

The Hamiltonian of three bodies’ baryonic system in the hypercentral constituent quark model is expressed as [50] (10)H=Pρ22m+Pλ22m+Vxand the hyperradial wave function ψνγ(x) is determined by the hypercentral Schrödinger equation. The hyperradial Schrödinger equation corresponding to the above Hamiltonian can be written as [51](11)d2dx2+5xddx-γγ+4x2ψνγx=-2mE-Vxψνγxwhere γ is the grand angular quantum number and given by γ=2n+lρ+lλ, n=0,1,…;lρ and lλ are the angular momenta associated with the  ρ→ and λ→ variable and ν denotes the number of nodes of the space three-quark wave function [36]. In (11), m is the reduced mass which is defined as m=2mρmλ/mρ+mλ [32]. By regarding ψνγ(x)=x-5/2φνγ [20, 35], (11) reduces to the following form: (12)φνγ′′x+ε-r1x2-r2x-r3x-r4x2-r5x3+r6x5+r7-2γ+32γ+54x2φνγx=0The hyperradial wave function φνγ(x) is a solution of the reduced Schrödinger equation for each of the three identical particles with the mass m and interacting potential (8), where(13)ε=2mE,r1=2mp,r2=2mβ,r3=2mτ,r4=2m1mρ+1mλ-CfCAαs24,r5=2m2τ3mρmλSρ·Sλ-3τ2mρmλγ→·s→+7τ6mρmλs2,r6=2m21τ6mρmλs→·x→s→·x→,r7=2mβ+2p2mρmλγ→·s→.We suppose the φνγ=h(x)eg(x) form for the wave function. Now we make use of the ansatz for h(x) and g(x)[33, 34]:(14)hx=Πx-aiνν=1,2,…,hx=1ν=0gx=aln⁡x+qx2+cx+dxwhere a, q, c, and d are positive. From (14), we obtain(15)φ′′x=g′′x+g′2x+h′′x+2h′xg′xhxφxComparing (12) and (15), it can be found that(16)r1x2+r2x+r3x+r4x2+r5x3-r6x5-r7+2γ+32γ+54x2-ε=g′′x+g′2x+h′′x+2h′xg′xhxBy substituting (14) into (16), we obtained the following equation:(17)-ε+r1x2+r2x+r3x+r4x2+r5x3-r6x5-r7+2γ+32γ+54x2=4q2x2+4cqx+2ac-4dqx+a2-a-2cdx2+2d1-ax3+d2x4+c2+2q+4acBy equating the corresponding powers of x on both sides of (17), we can obtain (18)a=2τβmp2,c=mβ22mp,q=mp2,ε=-mβ22p+2mp2+4mpτβ+2mβ+2p2mρmλγ→·s→Since p=mω2/2, we have a=2mω/2β, c=β/ω, q=mω/2. The energy eigenvalues for the mode ν=0 and grand angular momentum γ from (13) and (18) are given as follows: (19)E=-β22mω+ω2+mω2τβ+β+mω22mρmλγ→·s→At last for the best doubly heavy baryons masses (Ξccd, Ξccu, Ξbbd, Ξbbu, Ξbcd, Ξbcu) predictions, the values of mu, md, mc, mb, αS, ω, and β (which are listed in Table 1) are selected using genetic algorithm. The cost function of a genetic algorithm is the minimum difference between our calculated baryon mass and the reported baryons mass of other works.Table 1
The quark mass (in GeV) and the fitted values of the parameters used in our calculations.
m b m c m d m u α S C F C A β ω 4.750 1.348 0.35 0.34 0.340 2 3 3 0.02 0.11 fm−1


## 4. Results and Discussions: Mass Spectrum

The ground and excited states of doubly heavyΞ baryons are unclear to us experimentally (except Ξcc+ and Ξcc++). Hence, we have obtained the ground and excited state masses of Ξcc+, Ξcc++, Ξbb-, Ξbb0, Ξbc0, and Ξbc+ (see Tables 2, 3, 4, 5, and 6, respectively). These mass spectra are estimated by using the hypercentral potential equation (8) in the hypercentral constituent quark model. We begin with the ground state 1S; the masses are computed for both parities JP=1/2+and JP=3/2+. Our predicted ground state masses of doubly heavy Ξ baryons are compared with other predictions in Table 2.Table 2
The outcomes ground state masses ofΞ are listed with other theoretical predictions (in GeV). Standard devotion of the result is 0.350.
Baryon Ξ c c d / Ξccu Ξ b b d / Ξbbu Ξ b c d/ Ξbcu J P 1 2 + 3 2 + 1 2 + 3 2 + 1 2 + 3 2 + Our Calc 3.522 / 3.515 3.696 / 3.689 9.716 / 9.711 9.894 / 9.889 6.628 / 6.622 6.688 / 6.682 Ref.[1] 3.520 / 3.511 3.695 / 3.687 10.317 / 10.312 10.340 / 10.335 6.920 / 6.914 6.986 / 6.980 Ref.[24] 3.519 Ref.[7] 3.685 3.754 10.314 Ref.[12, 13] 3.520 3.695 10.199 10.316 Ref.[5] 3.610 3.694 Ref.[14] 3.610 3.692 10.143 10.178 6.943 6.985 Ref.[25] 3.561 3.642 Ref.[17] 3.720  9.960  6.720 Ref.[18] 3.687 3.752 10.322 10.352 7.014 7.064 Ref.[26] 3.676 3.753 10.340 10.367 7.011 7.074 Ref.[27] 3.547 3.719 10.185 10.216 6.904 6.936 Ref.[19] 3.579 3.656 10.189 10.218 Ref.[20] 3.620 3.727 10.202 10.237 6.933 6.980 Ref.[21] 3.478 3.610 10.093 10.133 6.820 6.900 Ref.[28] 3.627 3.690 10.162 10.184 6.914 Ref.[29] 3.519 3.620 9.800 9.980 6.650 6.690 Ref.[22] 3.612 3.706 10.197 10.136 6.919 6.986 Ref.[30] 3.510 3.548 10.130 10.144 6.792 6.827 Ref.[31] 3.570 3.610 10.170 10.220Table 3
The masses of radial excited states for doubly heavyΞ baryons (in GeV). Standard devotions of the result are 0.435 and 0.434.
Baryon State J P Our Calc Our Calc [1] [1] [7] [26] [27] [19] [20] [18] Ξ c c d  andΞccu 2S 1 2 + 3.905 3.901 3.925 3.920 4.079 4.029 4.183 3.976 3.910 4.030 3S 4.185 4.118 4.233 4.159 4.206  4.640  4.154 4S 4.430 4.429 4.502 4.501 5S 4.653 4.653 4.748 4.748 2S 3 2 + 3.962 3.958 3.988 3.983 4.114 4.042 4.282 4.025 4.027 4.078 3S 4.213 4.211 4.264 4.261 4.131  4.719 4S 4.446 4.445 4.520 4.519 5S 4.663 4.663 4.759 4.759 Ξ b b d  andΞbbu 2S 1 2 + 9.984 9.981 10.612 10.609 10.571 10.576 10.751 10.482 10.441 10.551 3S 10.211 10.211 10.862 10.862 10.612  11.170  10.630 4S 10.417 10.418 11.088 11.090     10.812 5S 10.606 10.610 11.297 11.301 2S 3 2 + 9.990 9.988 10.619 10.617 10.592 10.578 10.770 10.501 10.482 10.574 3S 10.205 10.233 10.855 10.866 10.593  11.184  10.673 4S 10.418 10.420 11.090 11.092     10.856 5S 10.607 10.611 11.298 11.302 Ξ b c d  andΞbcu 2S 1 2 + 6.922 6.919 7.244 7.240   7.478   7.321 3S 7.163 7.161 7.509 7.507   7.904 4S 7.379 7.377 7.746 7.744 5S 7.576 7.581 7.963 7.964 2S 3 2 + 6.943 6.939 7.267 7.263   7.495   7.353 3S 7.174 7.171 7.521 7.518   7.917 4S 7.384 7.384 7.752 7.752 5S 7.580 7.581 7.968 7.969Table 4
The masses of orbital excited states forΞcc baryon (in GeV). Standard devotions of the result are 0.072 and 0.068.
(a) State Our cal  Ξcc+ Our Cal  Ξcc++ [1]   Ξcc+ [1]   Ξcc++ [7] [26] [19] [20] [12] [21] [18] [5] ( 1 2   P 1 / 2 ) 3.851 3.847 3.865 3.861 3.947 3.910 3.880 3.838   4.073 3.892 ( 1 2   P 3 / 2 ) 3.834 3.830 3.847 3.842 3.949 3.921  3.959 3.786 3.834 4.079 3.989 ( 1 4   P 1 / 2 ) 3.860 3.856 3.875 3.871 ( 1 4   P 3 / 2 ) 3.842 3.838 3.856 3.851 ( 1 4   P 5 / 2 ) 3.873 3.872 3.890 3.888 4.163 4.092  4.155 3.949 4.047 4.089 ( 2 2   P 1 / 2 ) 4.120 4.101 4.161 4.140 4.135 4.074 4.018 4.085 ( 2 2   P 3 / 2 ) 4.104 4.101 4.144 4.140 4.137 4.078 4.197 ( 2 4   P 1 / 2 ) 4.127 4.125 4.169 4.167 ( 2 4   P 3 / 2 ) 4.111 4.109 4.152 4.149 ( 2 4   P 5 / 2 ) 4.140 4.138 4.183 4.181 4.488 ( 3 2   P 1 / 2 ) 4.361 4.345 4.426 4.409 4.149 ( 3 2   P 3 / 2 ) 4.347 4.345 4.411 4.409 4.159 ( 3 4   P 1 / 2 ) 4.367 4.366 4.433 4.432 ( 3 4   P 3 / 2 ) 4.354 4.352 4.419 4.417 ( 3 4   P 5 / 2 ) 4.336 4.333 4.399 4.396 4.534 ( 4 2   P 1 / 2 ) 4.583 4.583 4.671 4.671 ( 4 2   P 3 / 2 ) 4.571 4.571 4.658 4.657 ( 4 4   P 1 / 2 ) 4.590 4.590 4.678 4.678 ( 4 4   P 3 / 2 ) 4.577 4.577 4.664 4.664 ( 4 4   P 5 / 2 ) 4.561 4.561 4.646 4.646 ( 5 2   P 1 / 2 ) 4.792 4.793 4.901 4.902 ( 5 2   P 3 / 2 ) 4.781 4.781 4.889 4.889 ( 5 4   P 1 / 2 ) 4.799 4.800 4.908 4.909 ( 5 4   P 3 / 2 ) 4.705 4.788 4.895 4.896 ( 5 4   P 5 / 2 ) 4.771 4.772 4.878 4.879 ( 1 4   D 1 / 2) 4.043 4.038 4.077 4.071 ( 1 2   D 3 / 2 ) 4.018 4.013 4.049 4.044 ( 1 4   D 3 / 2 ) 4.026 4.022 4.058 4.053 ( 1 2   D 5 / 2 ) 3.995 3.991 4.024 4.019 4.043 4.115 4.047  4.391 4.034 4.050 4.388 ( 1 4   D 5 / 2 ) 4.003 4.000 4.033 4.029 4.027 4.052  4.187 4.089 4.393 ( 1 4   D 7 / 2 ) 3.975 3.972 4.002 3.998 4.097 ( 2 4   D 1 / 2) 4.287 4.284 4.345 4.342 ( 2 2   D 3 / 2 ) 4.265 4.262 4.321 4.318 ( 2 4   D 3 / 2 ) 4.272 4.270 4.329 4.326 ( 2 2   D 5 / 2 ) 4.245 4.243 4.299 4.297 4.164 4.091 ( 2 4   D 5 / 2 ) 4.252 4.251 4.307 4.305 ( 2 4   D 7 / 2 ) 4.228 4.226 4.280 4.278 4.394  (b) State Our Cal  Ξcc+ Our Cal  Ξcc++ [1]  Ξcc+ [1]  Ξcc++ [7] [32] [33] [34] [35] [33] [36] [5] ( 3 4   D 1 / 2) 4.511 4.511 4.592 4.592 4.511 4.511 4.592 4.592 4.511 4.511 4.592 4.592 ( 3 2   D 3 / 2 ) 4.492 4.491 4.571 4.570 4.492 4.491 4.571 4.570 4.492 4.491 4.571 4.570 ( 3 4   D 3 / 2 ) 4.499 4.499 4.578 4.578 ( 3 2   D 5 / 2 ) 4.475 4.474 4.552 4.551 4.348 ( 3 4   D 5 / 2 ) 4.481 4.481 4.559 4.558 ( 3 4   D 7 / 2 ) 4.460 4.459 4.535 4.534 ( 4 4   D 1 / 2) 4.723 4.724 4.825 4.826 ( 4 2   D 3 / 2 ) 4.706 4.706 4.806 4.806 ( 4 4   D 3 / 2 ) 4.711 4.712 4.812 4.813 ( 4 2   D 5 / 2 ) 4.690 4.690 4.788 4.788 ( 4 4   D 5 / 2 ) 4.696 4.696 4.795 4.795 ( 4 4   D 7 / 2 ) 4.675 4.675 4.772 4.772 ( 1 4   F 3 / 2 ) 4.198 4.193 4.247 4.242 ( 1 2   F 5 / 2 ) 4.169 4.164 4.215 4.210 ( 1 4   F 5 / 2 ) 4.142 4.172 4.186 4.219 ( 1 4   F 7 / 2 ) 4.150 4.147 4.194 4.191 ( 1 2   F 7 / 2 ) 4.178 4.139 4.225 4.182     4.267 ( 1 4   F 9 / 2 ) 4.118 4.115 4.159 4.156     4.413 ( 2 4   F 3 / 2 ) 4.422 4.425 4.494 4.497 ( 2 2   F 5 / 2 ) 4.399 4.399 4.468 4.468 ( 2 4   F 5 / 2 ) 4.405 4.406 4.475 4.476 ( 2 4   F 7 / 2 ) 4.378 4.382 4.445 4.450 ( 2 2   F 7 / 2 ) 4.384 4.376 4.452 4.443 ( 2 4   F 9 / 2 ) 4.359 4.355 4.424 4.420Table 5
The masses of orbital excited states forΞbb baryon (in GeV).
State Our cal  Ξbb- Our Cal  Ξbb0 [1]  Ξbb- [1]  Ξbb0 [7] [26] [19] [20] [12] [18] Others ( 1 2   P 1 / 2 ) 9.895 9.892 10.514 10.511 10.476 10.493 10.406 10.368  10.691 ( 1 2   P 3 / 2 ) 9.890 9.887 10.509 10.506 10.476 10.495  10.408 10.474 10.692 10.390 [31] ( 1 4   P 1 / 2 ) 9.897 9.895 10.517 10.514 ( 1 4   P 3 / 2 ) 9.893 9.890 10.512 10.509       10.430 [17] ( 1 4   P 5 / 2 ) 9.901 9.898 10.521 10.518 10.759    10.588 10.695 ( 2 2   P 1 / 2 ) 10.127 10.127 10.77 10.77 10.703 10.710 10612 10.563 ( 2 2   P 3 / 2 ) 10.124 10.120 10.766 10.762 10.704 10.713  10.607 ( 2 4   P 1 / 2 ) 10.129 10.129 10.772 10.772 ( 2 4   P 3 / 2 ) 10.126 10.125 10.768 10.767 ( 2 4   P 5 / 2 ) 10.121 10.133 10.763 10.776 10.973 10.713 ( 3 2   P 1 / 2 ) 10.337 10.338 11.001 11.002 10.740   10.744 ( 3 2   P 3 / 2 ) 10.334 10.335 10.997 10.998 10.742   10.788 ( 3 4   P 1 / 2 ) 10.339 10.340 11.003 11.004 ( 3 4   P 3 / 2 ) 10.336 10.337 10.999 11.000 ( 3 4   P 5 / 2 ) 10.331 10.343 10.994 11.007 11.004 ( 4 2   P 1 / 2 ) 10.531 10.534 11.214 11.217    10.900 ( 4 2   P 3 / 2 ) 10.527 10.530 11.21 11.213 ( 4 4   P 1 / 2 ) 10.533 10.536 11.216 11.219 ( 4 4   P 3 / 2 ) 10.529 10.532 11.212 11.215 ( 4 4   P 5 / 2 ) 10.526 10.538 11.208 11.222 ( 5 2   P 1 / 2 ) 10.712 10.716 11.413 11.418 ( 5 2   P 3 / 2 ) 10.709 10.714 11.41 11.415 ( 5 4   P 1 / 2 ) 10.714 10.718 11.415 11.420 ( 5 4   P 3 / 2 ) 10.711 10.716 11.412 11.417 ( 5 4   P 5 / 2 ) 10.706 10.721 11.407 11.423 ( 1 4   D 1 / 2 ) 10.043 10.041 10.677 10.675 ( 1 2   D 3 / 2 ) 10.037 10.035 10.670 10.668 ( 1 4   D 3 / 2 ) 10.038 10.037 10.672 10.670      11.011 ( 1 2   D 5 / 2 ) 10.030 10.028 10.663 10.661 10.592 10.676   10.742 11.002 ( 1 4   D 5 / 2 ) 10.033 10.031 10.666 10.664 ( 1 4   D 7 / 2 ) 10.026 10.024 10.658 10.656  10.608   10.853 11.011 ( 2 4   D 1 / 2 ) 10.257 10.257 10.913 10.913 ( 2 2   D 3 / 2 ) 10.252 10.252 10.907 10.907 ( 2 4   D 3 / 2 ) 10.254 10.254 10.909 10.909 ( 2 2   D 5 / 2 ) 10.247 10.247 10.901 10.901  10.712 ( 2 4   D 5 / 2 ) 10.248 10.248 10.903 10.903 10.613 ( 2 4   D 7 / 2 ) 10.242 10.242 10.896 10.896  11.057 ( 3 4   D 1 / 2 ) 10.455 10.457 11.13 11.133   4.592 4.592 ( 3 2   D 3 / 2 ) 10.450 10.452 11.125 11.127   4.571 4.570 ( 3 4   D 3 / 2 ) 10.451 10.454 11.126 11.129 ( 3 2   D 5 / 2 ) 10.446 10.447 11.120 11.122 ( 3 4   D 5 / 2 ) 10.447 10.449 11.122 11.124 10.809 ( 3 4   D 7 / 2 ) 10.442 10.444 11.116 11.118 ( 4 4   D 1 / 2 ) 10.639 10.643 11.333 11.337 ( 4 2   D 3 / 2 ) 10.635 10.638 11.328 11.332 ( 4 4   D 3 / 2 ) 10.636 10.640 11.330 11.334 ( 4 2   D 5 / 2 ) 10.631 10.635 11.324 11.328 ( 4 4   D 5 / 2 ) 10.632 10.636 11.325 11.33 ( 4 4   D 7 / 2 ) 10.627 10.631 11.320 11.324 ( 1 4   F 3 / 2 ) 10.173 10.172 10.82 10.819 ( 1 2   F 5 / 2 ) 10.166 10.165 10.812 10.811 ( 1 4   F 5 / 2 ) 10.158 10.167 10.804 10.813 ( 1 4   F 7 / 2 ) 10.167 10.160 10.814 10.806 ( 1 2   F 7 / 2 ) 10.160 10.157 10.806 10.803     11.004 ( 1 4   F 9 / 2 ) 10.152 10.152 10.797 10.797     11.112 ( 2 4   F 3 / 2 ) 10.357 10.376 11.022 11.043 ( 2 2   F 5 / 2 ) 10.368 10.369 11.035 11.036 ( 2 4   F 5 / 2 ) 10.369 10.371 11.036 11.038 ( 2 4   F 7 / 2 ) 10.362 10.365 11.028 11.031 ( 2 2   F 7 / 2 ) 10.364 10.363 11.030 11.029 ( 2 4   F 9 / 2 ) 10.357 10.357 11.022 11.023Table 6
The masses of orbital excited states forΞbc baryon (in GeV).
State Our cal Our Cal [1] [1] [18] Ξ b c 0 Ξ b c + Ξ b c 0 Ξ b c + ( 1 2   P 1 / 2 ) 6.846 6.842 7.16 7.156 7.390 ( 1 2   P 3 / 2 ) 6.836 6.831 7.149 7.144 7.394 ( 1 4   P 1 / 2 ) 6.851 6.847 7.166 7.161 7.399 ( 1 4   P 3 / 2 ) 6.841 6.837 7.155 7.15 ( 1 4   P 5 / 2 ) 6.859 6.856 7.175 7.171 ( 2 2   P 1 / 2 ) 7.087 7.084 7.425 7.422 ( 2 2   P 3 / 2 ) 7.078 7.075 7.415 7.412 ( 2 4   P 1 / 2 ) 7.091 7.088 7.43 7.426 ( 2 4   P 3 / 2 ) 7.082 7.079 7.42 7.417 ( 2 4   P 5 / 2 ) 7.071 7.095 7.408 7.434 ( 3 2   P 1 / 2 ) 7.304 7.302 7.664 7.662 ( 3 2   P 3 / 2 ) 7.296 7.295 7.655 7.654 ( 3 4   P 1 / 2 ) 7.308 7.306 7.668 7.666 ( 3 4   P 3 / 2 ) 7.299 7.299 7.659 7.658 ( 3 4   P 5 / 2 ) 7.289 7.312 7.648 7.673 ( 4 2   P 1 / 2 ) 7.504 7.623 7.884 8.015 ( 4 2   P 3 / 2 ) 7.497 7.498 7.876 7.877 ( 4 4   P 1 / 2 ) 7.508 7.508 7.888 7.888 ( 4 4   P 3 / 2 ) 7.500 7.500 7.88 7.88 ( 4 4   P 5 / 2 ) 7.491 7.514 7.87 7.895 ( 5 2   P 1 / 2 ) 7.692 7.693 8.091 8.092 ( 5 2   P 3 / 2 ) 7.686 7.687 8.084 8.085 ( 5 4   P 1 / 2 ) 7.695 7.697 8.094 8.096 ( 5 4   P 3 / 2 ) 7.689 7.689 8.087 8.088 ( 5 4   P 5 / 2 ) 7.680 7.681 8.078 8.079 ( 1 4   D 1 / 2 ) 7.006 7.004 7.336 7.334 ( 1 2   D 3 / 2 ) 6.992 6.989 7.321 7.318 ( 1 4   D 3 / 2 ) 6.997 6.980 7.326 7.308 7.324 ( 1 2   D 5 / 2 ) 6.980 6.977 7.308 7.304 ( 1 4   D 5 / 2 ) 6.985 6.969 7.313 7.295 7.309 ( 1 4   D 7 / 2 ) 6.969 6.953 7.296 7.278 7.292 ( 2 4   D 1 / 2 ) 7.087 7.227 7.425 7.579 7.579 ( 2 2   D 3 / 2 ) 7.216 7.214 7.567 7.565 ( 2 4   D 3 / 2 ) 7.219 7.219 7.571 7.57 ( 2 2   D 5 / 2 ) 7.205 7.203 7.555 7.553 7.538 ( 2 4   D 5 / 2 ) 7.209 7.208 7.559 7.558 ( 2 4   D 7 / 2 ) 7.196 7.195 7.545 7.544 ( 3 4   D 1 / 2 ) 7.431 7.431 7.804 7.804 ( 3 2   D 3 / 2 ) 7.420 7.420 7.792 7.792 ( 3 4   D 3 / 2 ) 7.411 7.424 7.782 7.796 ( 3 2   D 5 / 2 ) 7.415 7.410 7.786 7.781 ( 3 4   D 5 / 2 ) 7.402 7.414 7.772 7.785 ( 3 4   D 7 / 2 ) 7.402 7.402 7.772 7.772 ( 4 4   D 1 / 2 ) 7.429 7.504 7.801 7.884 7.797 ( 4 2   D 3 / 2 ) 7.611 7.613 8.002 8.004 ( 4 4   D 3 / 2 ) 7.615 7.617 8.006 8.008 ( 4 2   D 5 / 2 ) 7.603 7.604 7.993 7.994 ( 4 4   D 5 / 2 ) 7.606 7.608 7.996 7.998 ( 4 4   D 7 / 2 ) 7.596 7.597 7.985 7.986 ( 1 4   F 3 / 2 ) 7.143 7.141 7.487 7.485 ( 1 2   F 5 / 2 ) 7.127 7.125 7.469 7.467 ( 1 4   F 5 / 2 ) 7.131 7.129 7.474 7.472 ( 1 4   F 7 / 2 ) 7.117 7.114 7.458 7.455 ( 1 2   F 7 / 2 ) 7.112 7.109 7.453 7.45 ( 1 4   F 9 / 2 ) 7.099 7.097 7.439 7.436 ( 2 4   F 3 / 2 ) 7.350 7.350 7.715 7.715 ( 2 2   F 5 / 2 ) 7.337 7.336 7.7 7.699 ( 2 4   F 5 / 2 ) 7.340 7.339 7.704 7.703 ( 2 4   F 7 / 2 ) 7.328 7.327 7.69 7.689 ( 2 2   F 7 / 2 ) 7.324 7.323 7.686 7.685 ( 2 4   F 9 / 2 ) 7.313 7.311 7.674 7.672We can observe that, in the case ofΞcc baryon, for 2S states JP=1/2+ and JP=3/2+, our predictions are close to [34] and [1], respectively. Our outcomes for 3S state JP=1/2+ of Ξcc baryon show 21 MeV (with [7]) and JP=3/2+ shows 51 MeV (with [1]) difference. Analyzing the 2S and 3S states masses for Ξbb and Ξbc baryons (with both parities) shows that our masses have a difference in the range of ≈0.5 GeV with [1, 7, 20, 32–34, 36].To calculate the orbital excited state masses (1P–5P, 1D– 4D, 1F–2F), we have considered all possible isospin splitting and all combinations of total spinS and total angular momentum J. Our outcomes and the comparison of masses with other approaches are also tabulated in Tables 4, 5, and 6.Our obtained orbital excited masses forΞcc, 1P state JP=1/2- show a difference of 14 MeV (with [1]), 29 MeV (with [33]), 13 MeV (with [34]), and 41 MeV (with [5]), while 1P state JP=3/2- shows 14 MeV (with [1]), 48 MeV (with [35]), and 0 MeV (with [33]). Our 2P state JP=1/2- shows a difference of 15 MeV (with [7]), 35 MeV (with [34]), and 41 MeV (with [1]), while 2P state JP=3/2- shows 26 MeV (with [32]), 33 MeV (with [7]), and 40 MeV (with [1]). Results for 3P states JP=1/2- and JP=3/2- show a difference in the range of ≈ 60 MeV with [1]. We can easily observe that our calculated masses for 4P-5P, 1D-3D, and 1F-2F are matched with [1]. Our outcome for 3D state JP=3/2+ is quite equal to the predictions of [7, 32, 33, 35]. For the ground and excited states of doubly heavy baryons (Ξcc+), the minimum and maximum percentage of relative error values are 0% and 3.53% between our calculations and the masses reported by Shah et al. [1].ForΞbb and Ξbc baryons, the mass difference from our calculations and other references is large.Comparing our findings with the masses reported by Shah et al. [1], the minimum and maximum percentage of relative error values are 1.2% (0.8%) and 10.317% (6.92%) for the ground and excited states of doubly heavy baryons Ξbb and Ξbc, respectively.


## 5. Conclusion

In this study, we have computed the mass spectra of ground and excited states for doubly heavyΞ baryons by using a hypercentral constituent quark model. For this goal, we have analytically solved the hyperradial Schrödinger equation for three identical interacting particles under the effective hypercentral potential by using the ansatz method. Our proposed potential is regarded as a combination of the Coulombic-like term plus a linear confining term and the harmonic oscillator potential. We also added the first-order correction and the spin-dependent part to the potential. In our calculations, the u and d quarks have 10 MeV difference mass, so there is a very small mass difference between Ξccd and Ξccu, Ξbbd and Ξbbu, Ξbcd and Ξbcu. Our model has succeeded to assign the JP values to the exited states of doubly heavy baryons (Ξccd, Ξccu, Ξbbd, Ξbbu, Ξbcd, and Ξbcu). Comparison of the results with other predictions revealed that they are in agreement and our proposed model can be useful to investigate the doubly heavy baryons states masses. For example, for the ground, radial, and orbital excited states masses of doubly heavy Ξ baryons the minimum and the maximum percentage of relative error values are 0% and 6% between our calculations and the masses reported by Shah et al. [1].



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