Abstract
In the present work, the nonrelativistic quark model is applied to study baryon systems, where the constituent quarks are bound by a suitable hyper central potential. We proposed a new phenomenological form of the interaction potential, digamma-type potential. Using the Jacobi coordinates, the three-body wave equation is solved numerically to calculate the resonance states of the , and baryon systems. The present model contains only two adjustable parameters in addition to the quark masses. Our theoretical calculations are compared to the available experimental data and Cornell potential results. The description of the spectrum shows that the ground states of the considered light and strange baryon spectra are in general well reproduced.
1. Introduction
Hadron spectroscopy is very important to study its structures and the nature of the interaction forces between its constituents. In the previous works [1–5], the heavy meson spectroscopy is studied by using different potential models. Many authors have been studying the baryon spectroscopy. An analytical solution was provided in case of harmonic and anharmonic potentials for a system consists of three identical particles [6, 7]. Some authors have used the Cornell potential (Coulomb-type plus linear term) to study the resonance states of and baryons [8–14]. The harmonic oscillator potential has been used to study the interaction between three identical particles [15]. In the present work, the digamma-function is used as the interaction potential between constituent quarks of baryon systems. The three-body wave equation is solved, numerically, by using the Jacobi method to calculate the resonance state masses of , , , and baryons. These baryons consist of identical or nonidentical quarks. Our theoretical results will be investigated in Section 3.
2. The Used Model
The nonrelativistic Schroedinger equation for a system consists of three particles is given as where , , and are the three-quark masses and is the relative separation vector.
In the quark model, a baryon is a three-body bound state made of quarks. The mathematical description of a three-body system is more complicated than that of a two-body system. Several methods have been used by different authors to solve three-body problems [11, 15]. In order to describe the baryon as a bound state of three constituent quarks, we define the configuration of three particles by the center of mass, , and two Jacobi coordinates and as where ; ; and are constants.
Using natural units and letting the coefficients of and be equal to , therefore, (1) can be separated into the following two equations: where where The wave function can be expressed as [8, 16, 17] where and are the hyper radial part of the wave function and the hyper spherical harmonic function, respectively. The symbol is called the grand angular quantum number and is given by where and are the angular momenta associated with the and variables and is a nonnegative integer number.
The hyperradius is given as The potential can be assumed to depend on the hyperradius and space wave function is factored similar to the central potential [8].
Equation (4) can be rewritten as where represents the dimension of the and is the angular momentum operator whose eigenfunctions are [6, 9, 11] In the present work, the interaction potential is assumed as digamma-type as follows: where is an adjustable parameter and is the gamma-function.
Now, applying the following transformation (10) can be rewritten as With the new boundary conditions, Using
Equation (12) can be rewritten as
To transform the range of from to , I introduce the dimensionless variables, , , and assume that where GeV−1.
Equation (18) can be rewritten as The new boundary conditions are given as To transform (20) into a matrix form, one can divide the range of from into points with equal interval, , labeled by subscript .
The new form of the boundary conditions is given as Using the finite difference approximation [19], the second derivative term in (20) is given as where the term represents the expected error.
Using the symmetry properties of the radial wave function, one can assume that [15] Using (21), (22), and (23) in (18) and taking , one gets This equation is rewritten in the following matrix form: where is symmetric matrix and is unitary diagonal matrix. Using the Jacobi method [20, 21], (26) can be solved numerically and the eigenvalues, , are determined. The nonrelativistic resonance mass, , for each state is given by
3. Results and Discussions
In the present work, the spin-spin interaction is not considered and we deal with the spin averaged stats. The predicted values of the , , , and resonance masses are calculated through solving (26) numerically using the Jacobi method. The present potential form is proposed as the analytical form of the digamma-function; see (12) and (13). The -test is used to determine the parameter values where is the total number of states, is the resonance mass, and is the experimental error in the th State. The parameter values which are used in the present calculations are given in Table 1. The uncertainties in these parameters may be due to relativistic effects and ignoring the spin interaction terms.
The behavior of the digamma-type and Cornell potentials versus the hyperradius, , is shown in Figure 1. From this figure one notices that the suggested potential behaves nearly like Cornell potential at small (one gluon exchange part) and the Cornell potential is more confined than the digamma-type potential at large .
Table 2 contains the present calculations of resonance states using both digamma-function potential and a Coulomb like hyper central potential plus the linear term (Cornell Model [10]) in comparison to the experimental values [18]. From this table, one notices that the observed seven experimental states are calculated using the suggested potential form while the results of Cornell potential are 4 states only and the present calculations give more satisfied agreement with the experimental data.
Table 3 contains the present calculations of baryon states and Cornell potential results [10] in comparison to the available experimental data [18]. From this figure one notices that the observed 5 states are calculated using the present potential form while the calculated states in [10] are three states only. In case of and baryons, one can notice that the calculated resonance masses using the digamma-type potential agree more with the experiment than the Cornell potential model results [10].
The present calculations of and baryon states compared with the experimental values [21] are given in Tables 4 and 5. One notices that the suggested potential model can reproduce the experimental results of the ground states of and baryon systems.
4. Conclusions
A nonrelativistic quark model is used to study the spectra of the considered baryon systems (, , , and baryons). Using the hyper central approach, we have simplified the three-body problem and solved the Schrodinger equation numerically to obtain the ground-state energy eigenvalues and eigenfunctions. In the present work, we proposed a new phenomenological form of the interaction potential, digamma-type potential, between the quark constituents to study baryon systems. The behavior of this analytical function may be the same as the quark-antiquark interaction inside quarkonium systems. The description of the spectrum shows that the light and strange baryon ground states are in general well reproduced. The suggested potential model may be used to describe the energies of the higher excited states and negative-parity resonances through introducing the spin interaction terms as perturbed terms. Finally, this approach is useful because it allows for predictions of baryon states without extended lattice calculations and provides useful information about the short-distance one gluon exchange and the long-distance confinement interactions that can be useful in understanding the nature of the quark-quark force generated by QCD.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.