Relativistic scalar ﬁeld under the eﬀects of Lorentz Symmetry Violation subject to Cornell-type potential

We investigate a scalar particle under Lorentz symmetry breaking eﬀects determined by a tensor out of the Standard Model Extension (SME) in the presence of a Cornell-type potential by modifying the mass term M → M + S in the KG-equation. The ﬁeld conﬁguration is such that a Coulomb-type radial electric ﬁeld and a constant magnetic ﬁeld can be induced by Lorentz symmetry violation, and analyze the behaviour of a scalar particle. One can see that the bound states solution to the KG-equation under the consider eﬀects can be achieved, and a quantum eﬀect characterized by the dependence of charge den-sity distribution parameter on the quantum numbers of the system is observed.


Introduction
We study the behaviour of a scalar particle by solving the Klein-Gordon equation subject to a Cornell-type scalar potential in a possible scenario of anisotropy generated by Lorentz symmetry breaking effects defined by a tensor (K F ) µναβ that governs the Lorentz symmetry violation out of the Standard Model Extension [1,2]. We investigate the effects of a radial electric field and a uniform magnetic field induced by Lorentz symmetry violation by showing that the bound states solutions to the Klein-Gordon equation can be obtained. The Standard Model extension (SME) is an effective field theory that incorporates known physics and also the possibility of Lorentz violation. The gauge sector of the SME model has been extensively studied in several works by several authors [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18], with many interesting results.
The quantum dynamics of a scalar particle under the effects of Lorentz symmetry violation (LSV) [1,2,19,20,21,22,23,24] subject to a scalar potential is given by where α is a constant, F µν (x) = ∂ µ A ν − ∂ ν A µ is the electromagnetic tensor, (K F ) µναβ corresponds to a tensor that governs the Lorentz symmetry violation out of the Standard Model Extension and S is the scalar potential.
2 Relativistic scalar particle subject to Cornelltype potential under LSV We consider the Minkowski flat space-time where the ranges of the cylindrical coordinates are −∞ < (t, z) < ∞, r ≥ 0 For the geometry (2), the KG-equation under the effects of the Lorentz symmetry violation using (1) and finally using the properties of tensor (K F ) µναβ [22,23,24] become Let us consider a possible scenario of the Lorentz symmetry violation determined by (κ DE ) 11 = const, (κ HB ) 33 = const and (κ DB ) 13 = const and the field configuration given by [22,23,24]: where B 0 > 0,ẑ is a unit vector in the z-direction, λ is a constant associated with a linear distribution of electric charge along the axial direction, andr is the unit vectors in the radial direction.
Hence, equation (3) using the configuration (4) becomes Since the metric is independent of time and symmetrical by translations along the z-axis, as well by rotations. It is reasonable to write the solution to Eq. (6) as where E is the energy of the particle, l = 0, ±1, ±2, .... are the eigenvalues of the z-component of the angular momentum operator, and k is a constant. We have chosen a Cornell-type potential in cylindrical system that has been used to obtained bound states of hadrons [25,26], and the ground state of three quarks [27] in particle physics. This type of potential is given by [28,29,30] where η L > 0, η c > 0 are arbitrary constants.
Therefore using the function (6) and using the potential (7), we obtain the radial wave-equation for ψ(r): where Transforming x = √ η L r in the above equation (8), we have where Suppose the possible solution to the Eq. (10) is Substituting the solution (12) into the Eq. (10), we obtain the following equation where Equation (13) is the biconfluent Heun's differential equation [28,29,30,31,32] with H(x) is the Heun polynomials function.
The above equation (13) can be solved by the Frobenius method. Writing the solution as a power series expansion around the origin [33]: Substituting the power series solution into the Eq. (15), we obtain the following recurrence relation With few coefficients are The power series expansion H(x) becomes a polynomial of degree n by imposing the following two conditions [28,29,30] Θ = 2 n, (n = 1, 2, ...) Note that for the above conditions imposed simultaneously, one can show that the radial wave-function ψ(x) is finite both at the origin x → 0 as well as at x → ∞.
By analyzing the first condition, we obtain following equation of the energy eigenvalue E n,l : Note that Eq. (19) is not the general expression of the relativistic energy eigenvalues of the relativistic scalar particle. One can obtain the individual energy levels and eigenfunction one by one by imposing the additional recurrence condition d n+1 = 0 on the eigenvalue problem.
The corresponding wave-functions are given by Now, we evaluate the individual energy levels and eigenfunctions one by one as in [28,29,30]. For example, n = 1, we have Θ = 2 and d 2 = 0 which a constraint on the potential parameter η L 1,l . We can see, from Eq. (21), that the allowed values of this potential parameter depends on quantum numbers {n, l} of the system, and the Lorentz symmetry breaking parameter (α λ B 0 ).
Similarly, one can find another relation of the potential parameter η L 2,l for the radial mode n = 2 and so on. We can see that for the allowed values of η L given by (21) is defined for the radial mode n = 1 which gives us a first degree polynomial function of H(x).
Thus, the ground state energy level for the radial mode n = 1 using (19) is given by And the ground state eigenfunction is where we have chosen d 0 = 1 and We can see that the lowest energy state (22) plus the ground state wavefunction (23)- (24) with the restriction on the potential parameter η L given by Eq. (21) is defined for the radial mode n = 1.
We can see that the presence of the tensor field (K F ) µναβ that governs the Lorentz symmetry breaking effects and the Cornell-type scalar S(r) potential modified the energy spectrum and the wave-function of a relativistic scalar particle. Furthermore, we can see that the energy levels for each radial mode is symmetrical on either side about E = 0, and are equally spaced.
For zero Lorentz symmetry parameter (κ HB ) 33 = 0, the energy eigenvalues (21) becomes Therefore, the ground state energy level for the radial mode n = 1 is given by And the wave-function is given by Eqs.  (21) imposed on the potential parameter η L for the radial mode n = 1. This effect arises due to a Cornell-type scalar potential, and the Lorentz symmetry breaking parameters present in the quantum system. Furthermore, we have seen a quantum effect characterized by the dependence of the parameter η L on the quantum numbers {n, l} of the system.

Conflict of Interest
There is no conflict of interest regarding publication this paper.

Data Availability
No data has been used to prepare this paper.