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Based on models of confinement of quarks, we analyse a relativistic scalar particle subject to a scalar potential proportional to the inverse of the radial distance and under the effects of the violation of the Lorentz symmetry. We show that the effects of the Lorentz symmetry breaking can induce a harmonic-type potential. Then, we solve the Klein-Gordon equation analytically and discuss the influence of the background of the violation of the Lorentz symmetry on the relativistic energy levels.

The search for physics beyond the Standard Model (SM) has been increased with the need for understanding new phenomena, such as the unbalance between matter-antimatter and the dark matter. With respect to the dark matter, it is expected that the dark sector can interact with the visible sector, and thus, it can induce the detection of a weak fifth force in such a way that it can be investigated in decay of an excited state of

From the studies of quantum chromodynamics (QCD), the size of the proton radius can be estimated through the quark interaction mediated by virtual gluons. Further, experiments with electrons have shown that the value of the size of the proton radius is in agreement with that yielded by the existing theories. However, recent researchers have considered a muon in orbit around a proton and shown that the radius of the proton is different [

In this paper, we consider a relativistic scalar particle subject to a scalar potential proportional to the inverse of the radial distance. We also consider this particle to be under the effects of the violation of the Lorentz symmetry. The violation of the Lorentz symmetry is established by a tensor field. Then, we chose a particular background of the violation of the Lorentz symmetry that yields a harmonic-type potential. Thereby, we show that the Klein-Gordon equation can be solved analytically and then discuss the influence of the background of the violation of the Lorentz symmetry on the relativistic energy levels.

The structure of this paper is as follows: in Section

In this section, we consider a relativistic scalar particle subject to a scalar potential proportional to the inverse of the radial distance, and thus, we investigate the effects of a harmonic-type potential produced by an anisotropic environment generated by a violating term of the Lorentz symmetry on this system. We deal with a model that goes for energy scales beyond the Standard Model. In recent years, two of us [

On the other hand, as discussed in [

Now, let us consider a background of the Lorentz symmetry violation determined by the presence of the electric field

Observe that the last term of the right-hand side of the Klein-Gordon equation (

We proceed with a change of variables given by

Next, by substituting (

Let us write

We go further in search of bound state solutions; thus, we must impose that the biconfluent Heun series terminates. This occurs, from (

Furthermore, in order that the biconfluent Heun series terminates, we also need to analyse the condition

In this way, both conditions established in (

Finally, let us use the label

From (

We have investigated relativistic effects on a scalar particle under the influence of a scalar potential proportional to the inverse of the radial distance and the violation of the Lorentz symmetry. We have considered a background of the violation of the Lorentz symmetry that yields a harmonic-type potential on the Klein-Gordon equation. Then, in search of polynomial solutions to the function

The authors declare that they have no conflicts of interest.

The authors would like to thank the Brazilian agencies CNPq and CAPES for financial support.

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