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Advances in High Energy Physics
Volume 2018, Article ID 1326438, 11 pages
https://doi.org/10.1155/2018/1326438
Research Article

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

1Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran
2Faculty of Physics, Shahrood University of Technology, Shahrood, Iran

Correspondence should be addressed to Nasrin Salehi; ri.ca.tudoorhahs@ihelas

Received 9 March 2018; Revised 31 May 2018; Accepted 11 June 2018; Published 26 July 2018

Academic Editor: Chun-Sheng Jia

Copyright © 2018 Neda Mohajery et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Abstract

Since the doubly heavy baryons masses are experimentally unknown (except and ), we present the ground state masses and the positive and negative parity excited state masses of doubly heavy 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 , , , , , and systems are compared with other theoretical reports, which could be a beneficial tool for the interpretation of experimentally unknown doubly heavy baryons spectrum.

1. Introduction

The doubly heavy baryons have two heavy quarks (c and b) with a light quark (d or u or s). The doubly heavy baryons family have up or down quarks but 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 baryon and LHCb has determined the ground state of 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 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 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 Section 2. 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 ( and ) [37] and the respective reduced masses are given byHere , , and are the current quark masses. In order to describe three-quark dynamics, we define hyperradius and hyperangle [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 , which is attractive for small separations [3941], while at large separations a hyper-linear term, , gives rise to quark confinement [42], where corresponds to the string tension of the confinement [43]. Third, the six-dimension harmonic oscillator potential , which has a two-body character and turns out to be exactly hypercentral [44], where 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 can be written as [4447] The parameters and are the Casimir charges of the fundamental and adjoint representation. The hyper-Coulomb strength , is the color factor for the baryon. is the strong running coupling constant, which is written as The spin-dependent part is given as

The spin-dependent potential, , contains three types of the interaction terms [48], such as the spin-spin term , the spin-orbit term , and tensor term described as [35]. Here , where and are the spin vectors associated with the and variables, respectively. The coefficient of these spin-dependent terms of the above equation can be written in terms of the vector, , and scalar, parts of the static potential as [38]In our model, the hypercentral interaction potential is assumed as follows [48]: where is given by The baryons masses are determined by the sum of the model quark masses plus kinetic energy, potential energy, and the spin-dependent interaction as [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] and the hyperradial wave function is determined by the hypercentral Schrödinger equation. The hyperradial Schrödinger equation corresponding to the above Hamiltonian can be written as [51]where is the grand angular quantum number and given by , and are the angular momenta associated with the   and variable and denotes the number of nodes of the space three-quark wave function [36]. In (11), is the reduced mass which is defined as [32]. By regarding [20, 35], (11) reduces 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 (8), whereWe suppose the form for the wave function. Now we make use of the ansatz for and [33, 34]:where , , , and are positive. From (14), we obtainComparing (12) and (15), it can be found thatBy substituting (14) into (16), we obtained the following equation:By equating the corresponding powers of on both sides of (17), we can obtain Since , we have , , . The energy eigenvalues for the mode and grand angular momentum from (13) and (18) are given as follows: At last for the best doubly heavy baryons masses (, , , , , ) predictions, the values of , , , , , , and (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.

4. Results and Discussions: Mass Spectrum

The ground and excited states of doubly heavy baryons are unclear to us experimentally (except and ). Hence, we have obtained the ground and excited state masses of , , , , , and (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 and . Our predicted ground state masses of doubly heavy baryons are compared with other predictions in Table 2.

Table 2: The outcomes ground state masses of are listed with other theoretical predictions (in GeV). Standard devotion of the result is 0.350.
Table 3: The masses of radial excited states for doubly heavy baryons (in GeV). Standard devotions of the result are 0.435 and 0.434.
Table 4: The masses of orbital excited states for baryon (in GeV). Standard devotions of the result are 0.072 and 0.068.
Table 5: The masses of orbital excited states for baryon (in GeV).
Table 6: The masses of orbital excited states for baryon (in GeV).

We can observe that, in the case of baryon, for 2S states and , our predictions are close to [34] and [1], respectively. Our outcomes for 3S state of baryon show 21 MeV (with [7]) and shows 51 MeV (with [1]) difference. Analyzing the 2S and 3S states masses for and baryons (with both parities) shows that our masses have a difference in the range of 0.5 GeV with [1, 7, 20, 3234, 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 spin and total angular momentum . Our outcomes and the comparison of masses with other approaches are also tabulated in Tables 4, 5, and 6.

Our obtained orbital excited masses for , 1P state show a difference of 14 MeV (with [1]), 29 MeV (with [33]), 13 MeV (with [34]), and 41 MeV (with [5]), while 1P state shows 14 MeV (with [1]), 48 MeV (with [35]), and 0 MeV (with [33]). Our 2P state shows a difference of 15 MeV (with [7]), 35 MeV (with [34]), and 41 MeV (with [1]), while 2P state shows 26 MeV (with [32]), 33 MeV (with [7]), and 40 MeV (with [1]). Results for 3P states and 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 is quite equal to the predictions of [7, 32, 33, 35]. For the ground and excited states of doubly heavy baryons (), 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].

For and 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 and , respectively.

5. Conclusion

In this study, we have computed the mass spectra of ground and excited states for doubly heavy 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 and quarks have 10 MeV difference mass, so there is a very small mass difference between and , and , and . Our model has succeeded to assign the values to the exited states of doubly heavy baryons (, , , , , and ). 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 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].

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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