Mass Spectrum of Dirac Equation with Local Parabolic Potential

In this paper, we solve the eigen solutions to the Dirac equation with local parabolic potential which is approximately equal to the short distance potential generated by spinor itself. The energy spectrum is quite different from that with Coulomb potential. The mass spectrum of some baryons is similar to this one. The angular momentum-mass relation is quite similar to the Regge trajectories. The parabolic potential has a structure of asymptotic freedom near the center and confinement at a large distance. So, the results imply that, the local parabolic potential may be more suitable for describing the nuclear potential. The procedure of solving can also be used for some other cases of Dirac equation with complicated potential.


Introduction
For hadrons, the relation between mass m and quantum numbers ( ) where ( ) 0 , , a b m are constants for the exited states of the same kind particle. In many cases, the coefficients satisfy 2 b a b ≤ ≤ [3] [4]. The Regge trajectory is an important tool widely used to analyze the spectroscopy of mesons and baryons.
In many models, the total potential between quarks is given by Cornell potential with some hyperfine terms of correction, and the mass spectrum is solved in relative Jacobi coordinates [8] [10] [13] [30] [31] [35] [49]. In [49], by semi classical approximation and Bohr-Sommerfeld quantization, the Regge-like relation Recently, a number of experimental data for highly exited resonances were reported [50]- [58]. These data provide opportunity to check the previous calculations and develop more effective models. As pointed out in [58], a better understanding of the nucleon as a bound state of quarks and gluons as well as the spectrum and internal structure of excited baryons remains a fundamental challenge and goal in hadronic physics. In particular, the mapping of the nucleon excitations provides access to strong interactions in the domain of quark confinement. While the peculiar phenomenon of confinement is experimentally well established and believed to be true, it remains analytically unproven and the connection to quantum chromodynamics (QCD)-the fundamental theory of the strong interactions-is only poorly understood. In the early years of the 20th century, the study of the hydrogen spectrum has established without question that the understanding of the structure of a bound state and of its excitation spectrum needs to be addressed simultaneously. The spectroscopy of excited baryon resonances and the study of their properties are thus complementary to understanding the structure of the nucleon in deep inelastic scattering experiments that provide access to the properties of its constituents in the ground state.
The quark models employ multiplets of spinors and nonlinear interactive vectors with gauge symmetries, which are too complicated to get exact solutions and an overview for the properties. In this paper we examine the following simple and closed Dirac equation with short range self-generating vector potential in which γ φ γφ + =  . (1.2) has plentiful spectra. By the Regge trajectories we find the excited states may be relevant to some of baryons.

Equations and Simplification
At first, we introduce some notations. Denote the Minkowski metric by Define 4 4 × Hermitian matrices as follows x ct = and For the eigen states of φ , only the magnetic quantum number z m and the sipn s are conserved. So the eigen solution takes the following form where the index "T" stands for transpose, u v k = are real functions of r and θ . However, the exact solution of (2.5) does not exist, and we have to solve it by effective algorithm [59] [60]. Since the numerical solutions are also unhelpful to understand the global structure of the mass spectrum, we seek for the approximate analytic solutions in this paper.
Different from the case of an electron, a proton has a hard core with charge distribution, and the radius of the distribution is about 1 × 10 −15 m. The following calculation shows the local parabolic potential is approximately equal to µ Φ near the center, then we have in which w is the strength factor, η is a parameter to adjust the depth of confinement to fit the true confining potential. c ρ µ =  is the theoretical Compton wave length, which is used for nondimensionalization of the Dirac equation.
In order to simplify (1.2), we make transformation [60] Substituting (2.5), (2.6) and (2.7) into (1.2) we get Lagrangian as In (2.9), ε is relative mass defect defined by and ρ is used as length unit, κ is a constant to let For (2.9), the rigorous eigen solutions take the following form [60] By variation of (2.9), we get in which 1, 2, K = ± ±  corresponding to orbital angular momentum, , P Q are associated Legendre functions. The radial functions satisfy

Eigen Solutions to the Equation
For (2.16), we have the solution ( ) where 1 J n L − is associated Laguerre polynomials, n is radial quantum number, and J is angular momentum quantum number; 1 0 C = corresponds to 0 K < In (3.5), we have 3 constants ( ) , , w η µ for the same series of particles to be determined by empirical data. Although the form of (3.5) or (3.6) is quite different from (1.1), the following calculation shows that the curves of (3.5) in the effective domain are quite near straight lines (see Figure 1).  Considering energy degeneracy, we only need to calculate the energy spectrums   particles. This means a particle with local parabolic potential or short distance potential α Φ has a very hard core. This phenomenon is quite different from the case of Coulomb potential, where we have 2 r n ∝ .
For convenience, we take J as row index and n as column index, then the mass spectrums of the eigen states are listed in Table 1.
We find the masses of many baryons are near the spectra. Obviously each excited state should correspond to an observable particle. This means some baryons can be regarded as excited resonances of a proton. How to exactly identify the quantum numbers for each particle observed in experiments is an important but fallible problem.
As the 0th order approximation with only 3 free coefficients, the result is satisfactory. To get more accurate solutions of (1.2), we can expand φ as series of the eigen functions of (2.9) and then solve mass spectra of (1.2) [60]. However, in this case, we have only numerical results without an overview on the spectra.

Effectiveness of the Parabolic Potential
Now we check the effectiveness of the parabolic potential for nuclear potential. It is well known the global parabolic potential cannot be used as confining potential of Dirac equation. However, the following calculations show the local parabolic potential is effective to describe nuclear potential approximately.

Discussion and Conclusion
As the 0th order approximation, the above calculation provides some important (see Figure 3 and Figure 4).
To get more accurate results, we should directly calculate the coupling system of (2.4) and (2.3), and expand the radial functions ( ) , U V upon the bases , , , K n K n U V of the representation space [60]. However, in this case, we have not explicit analytic expression (3.3) for mass spectra.
As the alternative models for fundamental particles, some simple and closed systems such as the following one are worth to be carefully studied, Some deep secrets may be concealed under the nonlinear potential F and short distance potentials, because the spinor equation is a magic equation.
If we denote In (5.2), ( ) , , ψ ψ ψ − may be easily interpreted as quarks with fraction electric charge and confinement, and the cross terms G may be interpreted as gauge fields. For any complicated mathematical models, a little vigilance should be remained, because Nature only uses simple but best mathematics, and the complicated equations easily lead to inconsistence and singularity.
On the other hand, the regression analysis for empirical data to derive mass function with single integer variable ( ) ( ) N an bJ m = + + is relatively easy. This procedure needs not to concern the physical meanings of N at first and gets rid of the fallible and misleading task to identify the quantum numbers n and J for each particle at the beginning. If we can arrange the masses of similar particles from small to large at each horizontal integer coordinates N to get smooth curves, the regressive function

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.