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We investigate the behaviour of a massive scalar field under the influence of a Coulomb-type and central linear central potentials inserted in the Klein-Gordon equation by modifying the mass term in the spacetime with Lorentz symmetry violation. We consider the presence of a background constant vector field which characterizes the breaking of the Lorentz symmetry and show that analytical solutions to the Klein-Gordon equation can be achieved.

The Standard Model (SM) presents in a unified way the electromagnetic, weak, and strong nuclear interactions with the exception of gravitational interaction which makes it an incomplete quantum field theory (QFT). In addition, there are some observations, from both a theoretical and observational point of view, about their predictions. Recent experimental data provided measurements of the proton radius which it is different from the value predicted by the SM [

Due to these questions about limitations of the SM, it has arisen in recent years interest in investigating the possibility of a physics beyond the SM. In this context, the Lorentz symmetry violation (LSV) has been extensively explored in QFT since their possible scenarios have provided directions in the search for answers about physical effects of possible underlying physical theories, in which they can not be explained or observed through usual physics. In this sense, based on string theory, Kosteleký and Samuel dealt with the spontaneous breaking of symmetry through nonscalar fields, in which the vacuum expected value is constant and acquires a tensorial nature which violates the Lorentz symmetry spontaneously [

In this paper, we have investigated the relativistic quantum dynamics of a scalar field subject to a hard-wall potential and to the Coulomb-type and linear central potentials in a spacetime with LSV. Such configuration in the spacetime, which characterizes the LSV, is provided by the direct coupling between the derivative of the field with an arbitrary constant vector field in the Klein-Gordon equation, where we analyze its effects on a scalar field. Thus, we show that it is possible to find out analytically solutions of bound states and to determine the relativistic energy levels for the scalar field in a Lorentz violating background for each case.

The structure of this paper is as follows: in Section

Effective theories with LSV have been the focus of increasing interest in various physics contexts nowadays. The symmetry that permeates all high energy physics and Lorentz covariance is the basis of the SM of particle physics construction and so it is natural to ask why the interest of this type of violation. Inspired by [

In this paper, we work in the Minkowski spacetime with cylindrical symmetry (

From now on, let us make a discussion from the theoretical point of view, where a scalar field is subject to the effects of the LSV given by the presence of the background vector field

Let us consider a particular solution to (

Equation (

Let us restrict the motion of the scalar field to a region where a hard-wall potential is present. This kind of confinement is described by the following boundary condition:

Hence, by substituting (

We note that the background that characterizes the LSV caused by the presence of the particular vector field influences the dynamics of the scalar field subject to the hard-wall potential through the presence of the parameters associated with the LSV,

Let us consider a vector field which governs the LSV with the following configuration:

Note that (

Equation (

From now on, let us consider the background vector field

We can follow the steps from (

We can note that (

Equation (

The standard procedure of inserting central potentials into relativistic wave equations, such as the Klein-Gordon and Dirac equations, is through the minimum coupling which is represented by the transformation in the linear momentum operator,

Let us consider a background vector field with the following configuration:

Hence, by introducing the scalar potential by modification of the mass term, we can note, through (

Let us consider an external vector field with the following configuration:

Let us consider a background vector field with the following configuration:

We can note that (

In this section, we analyse the relativistic quantum effects of a linear central potential, through the modification of the mass term [

Let us consider a background vector field with the following configuration:

From now on, let us consider

Let us search for polynomial solutions to (

In search of polynomial solutions to the biconfluent Heun equation (

However, our analysis is not complete, since condition

Let us consider an external vector field with the following configuration:

Let us define

We can note that (

In search of a polynomial solution to the function

Note that the allowed values of

Equation (

Now, let us consider the configuration of the vector field given in the form

Equation (

In this section, let us consider the scalar field in

Let us consider the background vector field

From now on, let us consider

By analysing the asymptotic behaviour of the possible solutions to (

Further, by using the Fröbenius method as in (

As seen in Section

By comparing the expressions for the allowed values of

We can note that the allowed energy values for

Let us consider the external vector field

We can note that (

As we have discussed in the previous section, the parameter

We can observe that, in contrast to Section

We can note the influence of the LSV in (

Let us consider the background vector field

Equation (

We have investigated the effects of the LSV on a scalar field subject to a hard-wall potential and Coulomb-type and linear central potentials. The LSV is governed by the presence of a background constant vector field which modifies the structure of the Klein-Gordon equation (

In our first analysis, we considered the presence of a hard-wall potential, where we have shown that there is the influence of the effects of the LSV on the relativistic energy levels. Then, by modifying the mass term, we inserted the Coulomb-type central potential into the Klein-Gordon equation, where we determined the energy levels of the analyzed systems, in which in turn in all cases we can note the influence of the LSV on the levels of the relativistic energy levels. In the case of the linear central potential, we have calculated the values allowed for the lower energy states of the system and shown that there is also influence of the LSV. Then, we extend our analysis considering the presence of the Coulom-type potential plus the linear potential and show that relativistic energy allowed for lower energy state is affected by the effects of the LSV. In addition, the influence of the linear central potential and of the Coulomb-type plus linear central potential on the scalar field restricts the values of the parameter related to the linear central potential to a set of values that are established by the quantum numbers of the system which allow us to obtain a polynomial solution to the biconfluent Heun series. We also can note that because the symmetry is cylindrical, the scalar field is subject to the effects of the axial central potentials only in the

It is worth mentioning that the background vector field introduced in the Klein-Gordon equation can be considered more general, that is, where all its components are nonzero. It is in our interest as future perspectives to analyze this more general case on the scalar field, not only for the central potentials considered in the present work, but for other interactions and external effects, for example, the Klein-Gordon oscillator [

There is no use of data in this manuscript.

The authors declare that they have no conflicts of interest.

The authors would like to thank the Brazilian agency CNPq for financial support. R. L. L. Vitória was supported by the CNPq Project no. 150538/2018-9.