A new method for the Ω_ccb baryon spectroscopy in the nonrelativistic quark model: ansatz approach
Subject Areas : Journal of Theoretical and Applied Physics
1 - Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran.
Keywords:
Abstract :
A New Method for the 
Baryon Spectroscopy in the Nonrelativistic Quark Model: Ansatz Approach
NASRIN SALEHI
Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran
*salehi@shahroodut.ac.ir
Triply heavy 
baryon is considerable theoretical interest in a baryonic analogue of heavy quarkonium because of the color-singlet bound state of three heavy quark (c, b) combination inside. In this paper, we will discuss baryon in the nonrelativistic quark model based on the ansatz approach in Hypercentral Constituent Quark Model. The masses of the ground and excited states of the baryon are computed. The hypercentral potential is regarded as a combination of the color Coulomb plus linear confining term and the six-dimensional harmonic oscillator potential in this work. Also, we added the first order correction and the spin-dependent part to the hypercentral potential. The Regge trajectories has been plotted for this baryon and a detailed comparison with previous theoretical calculations is given. Further, using the computed spectroscopic data, the magnetic moments are determined for the ground state based on the nonrelativistic Hypercentral Constituent Quark Model.
Keywords: Triply Heavy Baryon, Hypercentral Constituent Quark Model, Regge Trajectories, Magnetic Moment.
1. Introduction
In recent years, a large number of heavy baryon states, charmonium-like states and bottomonium-like states have been observed, which have attracted intensive attentions and have revitalized many works on the singly heavy, doubly heavy, triply heavy and quadruply heavy hadron spectroscopy. Many new states of heavy baryons as well as of heavy mesons were observed. For the first time, the LHCb collaboration observed the doubly charmed baryon state 
in the 
mass spectrum [1]. The LHCb collaboration studied the doubly charmed tetraquark, 
, with a quark content 
[2] and the J/ψJ/ψ invariant mass distribution using pp collision data 
and 13 TeV was investigated by the LHCb collaboration in 2020. The narrow resonance structure X(6900) around 6.9 GeV and a broad structure just above the J/ψJ/ψ mass with global significances of more than 5σ was observed [3]. The observation of the and X(6900) provides some crucial experimental inputs on the strong correlation between the two charm quarks, which may shed light on the spectroscopy of the doubly heavy, triply heavy baryon states, doubly heavy, triply heavy, quadruply heavy tetraquark states, and pentaquark states [4]. As we know producing the triply charm/bottom heavy baryons is very difficult and thus no experimental signal for any of them has yet been reported. Baranov et al. believed that ccc-baryons couldn’t be observed in 
collisions and the expectations for triply-bottom baryons would be worse [5]. Bjorken in the 1980s [6] came to this conclusion as well. He offered hadron-induced fixed target experiments as the best strategy to observe the ground-state triply-charmed baryon, 
. On the other hand, estimates of the production cross section of triply heavy baryons in proton-proton [7] and heavy ion [8] collisions indicate that triply charmed 
baryons have good chances to be observed at LHC. The first theoretical study of heavy baryon spectroscopy was carried out by ref. [9] using the QCD motivated bag model. In this article, we discuss one of the triply heavy baryons which is 
. Many theoretical approaches have determined the masses of this baryon. They are, non-relativistic quark model [8], Fadeev approach [9, 10], Sum rules [11], Bag model [12], di-quark model [13], Lattice QCD [14, 15], relativistic quark model [16, 17] and variational cornell [18]. Since the solution of the hypercentral Schrödinger equation with Coulombic-like term plus a linear confining term potential cannot be obtained analytically, we have used the hypercentral constituent quark model (hCQM) with Coulombic-like term plus a linear confining term and the harmonic oscillator potential [19, 20] (the six-dimensional hyper-Coulomb potential is attractive for small separations, while at large separations a hyper-linear term gives rise to quark confinement. The six-dimensions harmonic oscillator potential, which has a two-body character, and turns out to be exactly hypercentral). We also added the first order correction and the spin-dependent part (the spin-spin, spin-orbital and tensor interaction are considered for hyperfine splitting) to the potential and spectra have been generated for the ground states and radial excited states (2S-5S) of
for
and 
. For the orbital states, 1P-5P, 1D-4D and 1F-2F orbital excited states are calculated.
The present manuscript is arranged as follows. We describe briefly in Sec. 2 the theoretical framework and the hypercentral interaction potentials. In Sec. 3, we present the quasi-exact analytical solution of the radial Schrödinger equation for our proposed potential. In Sec. 4, we calculate the masses of the ground, orbitally, and radially excited states of triply 
baryon and compare our results with previous calculations. Also, we draw Regge trajectories for baryon. The magnetic moments of 
baryon has been presented in Sec. 5. Finally, Sec. 6 contains our conclusions.
2. Theoretical Framework and Interaction Potentials
The ground states and excited states of doubly heavy baryons in charm and bottom sector have been distinguished using hypercentral constituent quark model in our past study [19, 21-22]. We would predict that the suggested model will present a same degree of accuracy for triply heavy 
baryon too. The relevant degrees of freedom for the motion of three heavy quarks are related by the Jacobi coordinates 
and
and the respective reduced masses are given by 
and
[19-24]. The configuration of the three-quark system (Q1Q2Q3) is shown in Fig. 1.
Fig. 1. The configuration of the three-quark system (Q1Q2Q3). O is the string-junction point. Q1, Q2 are the heavy quarks b or c. The quark Q3 is treated as a heavy quark b, c in the triply heavy baryon or a light quark q in the doubly heavy baryon.
The constituent quark masses used in our calculations are m1 = m2=1.345 GeV (mass of c quark) and m3 = 4.902 GeV (mass of b quark). To describe three-quark dynamics, we define hyper radius 
and hyper angle 
. In this paper, we consider the confining hypercentral potential as a combination of the color Coulomb [25] plus linear confining term [26] and the six-dimensional harmonic oscillator potential [27, 28] with a first order correction [27] and the spin-dependent interaction [29],
,
(1)
& 
& 
where 
is the hyper-Coulomb strength, 
corresponds to the string tension for baryons and 
is the strong running coupling constant. The parameters 
and 
are the casimir charges of the fundamental and adjoint representation [30]. The spin dependent part 
contains three types of the interaction terms described as [31]. In above equation 
and 
. We can obtain the baryon masses by 
[32]. Now we want to solve the hyperradial Schrödinger equation for the three-body potential interaction Eq. (1).
2. Quasi-Exact Analytical Solution of the Hyperradial Schrödinger Equation
The dynamics of the baryonic system are considered in the wave-function 
which is the solution of hyperradial Schrödinger equation
(2)
where 
is the grand angular quantum number and given by 
, n = 0, 1, . . . ; 
and 
are the angular momenta associated with the 
and 
variables and ν denotes the number of nodes of the space three-quark wave functions. The transformation 
[33] reduces Eq. (2) to the following form
The hyperradial wave function 
is a solution of the reduced Schrödinger equation for each of the three identical particles with the mass m and interacting potential (1), where
, 
, 
, 
, 
, 
We suppose 
for the wave function and we make use of the ansatz for the 
and 
[34, 35],
(5)
By calculating
from ansatz approach and comparing with Eq. (3), we obtain
(6)
We used the ansatz approach, which is a simplified version of the quasi-exact Lie algebra. The Ansatz approach, despite its wide applications in case of many interactions, has its own challenges. To be more precise, the arising set of equations, which is based on fining the solution of an associated Riccati equation, is subjected to many restricting conditions and the parameters cannot be determined independent of each other. Also, the higher states are very hard to calculate. By equating the corresponding powers of 
on both sides of Eq. (6) and regarding 
, we can obtain the energy eigenvalues for the mode ν = 0 and grand angular momentum γ as follows
(7)
In a similar manner we can continue for other modes (ν = 1, 2, 3, . . .).
For calculating the best triply heavy 
baryon mass predictions, the values of 
, ω, 
and β (which are listed in Table 1) are selected using genetic algorithm.
4. Triply Heavy Baryon Spectra and Regge Trajectories
As the first step, we have calculated the masses of the ground state 1S, for 
and 
. Our calculated masses are obtained by using the hypercentral potential Eq. (1) in the hypercentral constituent quark model. Our calculated masses for the ground states of 
baryon are presented and compared with previous theoretical predictions in Table 2. The predicted mass of the ground state
baryon (for ) range from 7.867 to 8.301 GeV and for 
, range from 7.963 to 8.301 GeV. Our predictions for these masses are well inside both ranges.
We compare our predictions with previous calculations for the radial excited states (2S-5S) for both Table 1. The Quark mass (in GeV) and the fitted values of the parameters used in our calculations.and in Table 3.
