Relativistic Symmetries of Hulthén Potential Incorporated with Generalized Tensor Interactions

Spin and pseudospin symmetries of theDirac equation for aHulthén potential with a novel tensor interaction, that is, a combination of the Coulomb and Yukawa potentials, are investigated using the Nikiforov-Uvarov method. The bound-state energy spectra and the radial wave functions are approximately obtained in the case of spin and pseudospin symmetries. The tensor interactions and the degeneracy-removing role are presented in details.


Introduction
The exact solutions of wave equations are still an interesting problem in fundamental quantum mechanics.Unfortunately, there are only a few potentials for which the Schrödinger, Dirac, Klein-Gordon, and Duffin-Kemmer-Petiau (DKP) equations can be exactly solved.Several potential models have been introduced to explore the relativistic and nonrelativistic energy spectra and the corresponding wave functions [1][2][3][4][5].Jia et al. [6] have derived the bound-state solution of the Klein-Gordon equation under unequal scalar and vector kink-like potentials.By using the series expansion method, the authors in [7] have obtained the analytical solutions of the two-dimensional Schrödinger equation with the Morse potential.Pseudospin symmetry solution of Dirac equation for the modified Rosen-Morse potential is investigated in [8].The DKP equation under a scalar Coulomb interaction is solved in [9] where the authors have used the ansatz approach.Ginocchio et al. [10][11][12][13][14] established the relationship between the pseudospin and the Dirac equation by recognizing that pseudo-orbital angular momentum l =  + 1 is nothing but the usual orbital angular momentum  of the lower component of the Dirac spinor.They went further to show that, within the framework of the Dirac theory, the spin symmetry occurs when the difference of the potential between the repulsive Lorentz vector potential () and the attractive Lorentz scalar potential () is a constant; that is, Δ() = ()−() = const.On the other hand, the pseudospin symmetry arises when the sum of the potentials is a constant; that is, Σ() = () + () = const.Many researchers in the field have investigated the symmetries in the presence of various interactions.The list includes Manning-Rosen [15], Eckart [16], Hyllerass [17], Deng-Fan [18], Mobious square [19], Tietz [20], and Hyperbolical potentials [21].The analysis of the symmetries has been done by using various methods such as Nikiforov-Uvarov (NU) technique [22], supersymmetric quantum mechanics (SUSYQM) [23], and many others [24, and references therein].Due to the mathematical structure of the problem, the tensor interaction is often considered as a Coulomb or Cornell interaction [25].Hassanabadi et al. were the first who introduced the Yukawa tensor interaction [26] besides the Coulomb-like term.
Here, we intend to report the solution of Dirac equation for the Hulthén potential under the generalized Coulomblike and Yukawa-like potentials as tensor interactions.The Hulthén potential is defined as [27,28] Advances in High Energy Physics  where  0 ,  are two positive real parameters and represent the strength and the screening range of the potential, respectively.

Pseudospin Symmetry Limit
The pseudospin symmetry limit occurs in Dirac equation when Σ()/ = 0 or Σ() =  ps = const.In this limit, we take Δ() as the Hulthén potential: In addition, we propose a novel generalized tensor interaction of the form where   () and   () are the Coulomb-like and Yukawalike potentials defined as with where   is the Coulomb radius and   and   denote the charges of the projectile a and the target nuclei , respectively.Also,  1 is the depth of the potential.Substituting ( 14) into (13), we can write the tensor potential as Substitution of ( 16) and ( 29) into (9) gives It is well known that the above equation cannot be exactly solved due to the centrifugal term  −2 .In order to get rid of the centrifugal term, we make use of the following approximation [27,29] (see Figure 1): where  = − l and  = l + 1 for  < 0 and  > 0, respectively.
Using a new variable of the form  =  −2 and introducing where (17) is transformed to Comparing ( 23) with (A.1), we find Substitution of the values of the parameters given by ( 25) into (A.2) and (A.3) gives us the following relations for the eigenfunctions and energy eigenvalues: where On the other hand, the upper component can be found by using the following relation:

Spin Symmetry Limit
In this section, we consider the spin symmetry limit where Δ()/ = 0 or Δ() =  ps = const.As in the previous section, we consider Substitution of ( 16) and ( 22) into (10) gives with  2  = (   +  −   )( −    ),   = (   +  −   ) 0 , and  = ℓ and  = −ℓ − 1 for  < 0 and  > 0, respectively.Substituting ( 18) into (30), we have Introducing a new transformation of the form  =  −2 , we arrive at the Schrödinger-like equation: where Using the same approach we used to solve (23), the energy equation is obtained as    where and the corresponding upper and lower radial wave functions are

Discussion and Numerical Results
We obtained the energy eigenvalues in the absence ( = 0,  1 = 0) and the presence ( = 0.5,  1 = 0.5) of the Coulomblike plus Yukawa tensor interaction for various values of the quantum numbers  and .The results are reported in Tables 1 and 2 under the condition of the pseudospin and spin symmetries, respectively, where we can see the way the tensor interaction affects the degeneracy of the system.If we set  =  1 = 0, the potential reduces into the Hulthén potential and our result is consistent with that of Aydogdua et al. [27].In Tables 1 and 2, we have compared our results with [27].We represent the effects of the -parameter on the bound states for  =  1 = 0.5 in Figure 2. It is seen that if   the -parameter increases, the bound states become more bounded in both for symmetry limits.Figures 3 and 4 present the effect of the tensor interaction on the bound states.We have plotted the energy versus  in Figure 5.
Figure 6 shows the effects of  ps and   on the boundstates.In Figures 7 and 8, the wave functions are plotted for pseudospin and spin symmetry limits with and without a tensor interaction, respectively.It is seen in Figures 7 and  8 that the tensor interaction affects only the shape of the wave functions and does not change the node structure of the radial upper and lower components of the Dirac spinors.

Conclusion
In this paper, we obtained the approximate analytical solutions of the Dirac equation for the Hulthén potential with a novel generalized tensor interaction consisting of the Coulomb and Yukawa interactions within the framework of pseudospin and spin symmetry limits using the NU technique.We have obtained the energy levels in a closed form and the corresponding wave functions in terms of the Jacobi polynomials.We also included some numerical results to investigate the way the combination of Coulomb and Yukawa potentials.Finally, the results of our work find many applications in both nuclear and Hadron physics and therefore provide more general solutions compared to other previous works performed in [30,31].