Dirac Equation under Scalar , Vector , and Tensor Cornell Interactions

Spin and pseudospin symmetries of Dirac equation are solved under scalar, vector, and tensor interactions for arbitrary quantum number via the analytical ansatz approach. The spectrum of the system is numerically reported for typical values of the potential parameters.


Introduction
No doubt, the Cornell potential alternatively called Funnel in the literature is, if not the best, among the most appealing interactions in particle physics.The Cornell potential contains a confining term besides the Coulomb interaction and has successfully accounted for the particle physics data 1 .Unfortunately, to our best knowledge, the potential does not possess exact solutions under all common equations of quantum mechanics, that is, the nonrelativistic Schr ödinger equation, and relativistic Dirac, Klein-Gordon, Proca, and Duffin-Kemmer-Petiau DKP equations.Here, we focus on the relativistic symmetries of Dirac equation, that is, spin and pseudospin symmetries, which provide a reliable theoretical basis for hadronic and nuclear spectroscopy 2-12 .This has motivated many studies under various interactions within the past two decades 13-18 and many references therein .Nevertheless, none of these papers has investigated the symmetry limits under the Cornell potential.This is definitely due to the complicated nature of the resulting differential equation which cannot be solved by common analytical techniques of quantum mechanics such as the supersymmetry quantum mechanics SUSYQ , Lie groups, Nikiforov-Uvarov NU technique, and point canonical transformations, In our study, we make use of the ansatz approach to deal with this complicated equation.A survey on the application of this technique to other wave equations including Schr ödinger, spinless-Salpeter, Dirac, Klein-Gordon, and DKP equations can be found in 19-27 .We organize the study as follows.In the first step, we review the most essential equations of the symmetry limits.We next propose a physical ansatz solution to the equation and, in a systematic manner, calculate the spectrum of the system for any arbitrary state.To provide a better understanding of the solutions, we provide some numerical data for the spectrum as well.

Dirac Equation Including Tensor Coupling
In spherical coordinates, Dirac equation with both scalar potential S r and vector potential V r is expressed as 2, 3 where E is the relativistic energy of the system; α and β are the 4 × 4 Dirac matrices and p −i ∇ stands for the momentum operator.For a particle in a spherical field, the total angular momentum operator J and spin-orbit matrix operator K σ • L 1 , where σ and L are, respectively, the Pauli matrix and orbital angular momentum, commute with the Dirac Hamiltonian.The eigenvalues of K are κ − j 1/2 for the aligned spin s 1/2 , p 3/2 , etc. and κ j 1/2 for the unaligned spin p 1/2 , d 3/2 , etc. .The complete set of the conservative quantities can be chosen as H, K, J 2 , J z .As shown in 13 , the Dirac spinor is considered as where F nk r and G nk r are the radial wave functions of the upper and lower components, respectively, and Y jm θ, ϕ and Y jm θ, ϕ , respectively, stand for spin and pseudospin spherical harmonics coupled to the angular momentum j. m is the projection of the angular momentum on the z-axis.The orbital angular momentum quantum numbers and refer to the upper and lower components, respectively.The quasidegenerate doublet structure can be expressed in terms of pseudospin angular momentum s 1/2 and pseudoorbital angular momentum , which is defined as 1 for aligned spin j − 1/2 and − 1 for unaligned spin j 1/2.As shown in 2, 3 , substitution of 2.2 into 2.1 yields the following two-coupled differential equations:

2.3
Advances in High Energy Physics 3 which simply give where, as the notation indicates, Δ r V r − S r and Σ r V r S r .

Pseudospin Symmetry Limit
Under the condition of the pseudospin symmetry, dΣ r /dr 0 or equivalently Σ r C gps Const.2, 3 .We choose Δ r as the Cornell potential: Δ r a ps r b ps r .

2.6
For the tensor term, we consider the Cornell potential: Substitution of these two terms in into 2.5 gives where κ − and κ 1 for κ < 0 and κ > 0, respectively.

4
Advances in High Energy Physics

Spin Symmetry Limit
In the spin symmetry limit dΔ r /dr 0 or Δ r C gs const.2, 3 .As the previous section, we consider

Solution of the Pseudospin Symmetry Limit
In the previous section, we obtained a Schr ödinger-like equation of the form where

3.2
Equation 3.1 fails to admit exact analytical solutions.Therefore, we follow the ansatz approach with the starting square: where By substitution of f n r and g κ r into 3. 2β ps δ ps 0 , where Actually, to have well-behaved solutions of the radial wave function at boundaries, namely, the origin and the infinity, we need to take δ from 3.8 as Advances in High Energy Physics Form 3.2 , 3.8 , the ground-state energy satisfies which is more compactly written as where the parameter a ps of potential 2.6 from 3.8 should satisfy the following restriction: .

3.12
From 3.3 , 3.4 , and 3.8 , the upper and lower components of the wave function are

3.13c
For the first node n 1 , using and g κ r from 3.5 , we arrive at .

3.14
Here, the consequent relations between the potential parameters and the coefficients α ps , β ps , δ ps , and α .

3.16b
The energy eigenvalues therefore are

3.18
For the upper and lower components of the wave function we thus have

3.19b
Following the analytic iteration procedures for the second node n 2 with and g κ r as defined in 3.6 , the relations between the potential parameters and the coefficients α ps , β ps , δ ps , α

3.20
The coefficients α Advances in High Energy Physics 9 Therefore, the energy eigenvalue in this case is 3.24

Solution of the Spin Symmetry Limit
In this case, our ordinary differential equation is

3.26
which cannot be solved by our common exact analytical techniques.Let us propose the ansatz solution: where α s r 2 − β s r δ s ln r, α s > 0, β s > 0.

3.29
By substitution of f n r and g κ r into 3.27 , we find F s nκ r .

3.30
For the case of n 0, from 3.26 -3.29 , we find

3.31
By comparing the corresponding powers of 3.27 and 3.33 , we have

3.32
where k s 1 2κ 1 2 − 4d s 1 .To have physically acceptable solutions, we pick up the value

3.33
By considering 3.26 , 3.32 , the first node eigenvalue satisfies

3.35
Advances in High Energy Physics 11 where the parameter a s of potential 2.9a should satisfy the restriction

3.36
From 3.27 , 3.28 , and 3.32 , the upper and lower components of the wave function are 37

3.39
Secondly, for the first node n 1 , using and g κ r from 3.28 -3.30 , our resulting equation is .

3.40
The relations between the potential parameters and the coefficients α s , β s , δ s , and α

3.41
where c s 1 and α 1 1 are found from 3.41 as 20, 27-29 which determine the corresponding energy as

3.44
The upper and lower components of the wave function are then simply found to be

3.45b
Advances in High Energy Physics 13 For the second node n 2 , we choose and g κ r as defined in 3.28 and 3.29 .The relations between the potential parameters and the coefficients α s , β s , δ s , α 3.48 The energy eigenvalue therefore is

3.49
For the upper component of the wave function, we have

3.50
We have given some numerical values of the energy eigenvalues in Tables 1, 2, 3, 4, 5, and 6 for various states.For the final point, we wish to emphasize on the degeneracy-removing role of the Cornell tensor potential.As we already know, for vanishing tensor interaction A B 0 , the pseudospin doublets, that is, states with quantum numbers n, l, j l 1/2 and n − 1, l 2, j l 3/2 are degenerate.The degenerate states in the spin doublets are those with quantum numbers n, , j 1/2 and n, , j − 1/2 , where n, l,and j are the radial, the orbital, and the total angular momentum quantum numbers, respectively see Tables 1 and 2 .Our numerical data reveals that, in the pseudospin symmetry limit, the degenerate states for A B 0 are n s

Conclusion
Because of the established roles of the Cornell potential and spin, and pseudospin symmetries in nuclear and hadrons spectroscopy, we solved the Dirac equation under these symmetry limits for vector, scalar, and tensor interactions of Cornell-type.In our calculations, on the one hand, due to the failure of other common analytical techniques, and, on the other hand, the better insight which analytical techniques provide us in comparison with their numerical counterparts, we used the quasianalytical ansatz approach.By proposing novel physical solutions and after lengthy calculations, we could find the arbitrary-state solutions.Our results clearly show the degeneracy-removing role of the tensor term and provide the requisite understanding of the solutions for possible further studies.Both the

2 1 and α 2 2
are found from the constraint relations 20, 27-29 : component of the wave function is

Table 6 :
Energies in the spin symmetry limit for A 0.5, B 0.2, b 1, C s 0 fm −1 .