# Effects of a linear central potential induced by the Lorentz symmetry violation on the Klein–Gordon oscillator

- 154 Downloads

## Abstract

Inspired by the Standard Model Extension, we have investigated a possible scenario arising from the Lorentz symmetry violation governed by a background tensor field on a scalar field subject to the Klein–Gordon oscillator, where this possible scenario gives rise to a linear central potential. We analyse the behaviour of the relativistic quantum oscillator under the influence of a Coulomb-type scalar potential in this background. Then, we solve the Klein–Gordon equation analytically and discuss the influence of the background which violates the Lorentz symmetry in the relativistic energy levels.

## 1 Introduction

In recent years, in order to observe more accurately the atomic nuclei, some investigations, taken into account the hydrogen atom, have considered to change the orbital electron by a muon, and they have shown that the proton radius is little different [1]. On the other hand, there is a line of research which argue that the fine structure constant \(\left( \alpha =\frac{e^{2}}{\hbar c}\right) \) is slowly changing [2, 3], and this fact can imply the light velocity variation. Such variation can be consequence of the Lorentz symmetry violation (LSV). It has been extensively studied and applied in various branches of physics [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

The development of the spontaneous symmetry breaking of Lorentz (LSV) in quantum field theory has allowed the formulation of a new model that aims to go beyond the Standard Model which has been known in the literature as Standard Model Extension (SME) [16, 17]. In the SME there are the fermionic and gauge sectors which are modified by background tensors that are manifested by expected vacuum of more fundamental fields [17]. In particular, the gauge sector is divided in the sectors CPT-odd and CPT-even [18].

In the context of non-relativistic quantum mechanics, through the nonminimal coupling of the term CPT-odd, inspired by gauge sector of the SME, there are studies in geometric phases [18], holonomies [19], on a Dirac neutral particle inside a two-dimensional quantum ring [20], on Aharonov–Bohm-type effect [21] and Landau-type quantization [22]. Recently, also by nonminimal coupling, the CPT-even term, inspired by gauge sector of the SME, has been applied in the analysis of the relativistic quantum dynamics of a scalar field under effects of central potentials induced by the LSV, for example, Coulomb-type potential [23], harmonic-type and linear-type potentials [24, 25] and the Klein–Gordon oscillator under effects of a Coulomb-type potential induced by the LSV [26]. However, the Klein–Gordon oscillator was not investigated under the effects of a central linear potential induced by the LSV caused by the presence of a background tensor field.

Inspired by the relativistic oscillator model for the spin-\(\frac{1}{2}\) fermionic field known as the Dirac oscillator [27, 28], Bruce and Minning have proposed a relativistic oscillator model for the scalar field which it was known in the literature as the Klein–Gordon oscillator [29] that, in the non-relativistic limit, is reduced to the oscillator described by the Schoröndinger equation [30]. The Klein–Gordon oscillator has been studied by a \(\mathcal {PT}\)-symmetric Hamiltonian [31], in noncommutative space [32, 33], in spacetime with cosmic string [34], in a spacetime with torsion [35], in a Kaluza–Klein theory [36], with noninertial effects [37] and under effects of linear and Coulomb-type central potentials [38, 39, 40].

In this paper, we have investigated the effects of a linear central potential induced by the LSV provided by the presence of a background tensor field on a scalar field subject to the Klein–Gordon oscillator. This analysis is made through the nonminimum coupling into the Klein–Gordon equation. Then, we search for relativistic bound state solutions to the Klein–Gordon equation, where we have shown that analytical solutions can be achieved.

The structure of this paper is as follows: in Sect. 2, we introduce a possible scenario of the LSV defined by the presence of a background tensor field \((K_F)_{\mu \nu \alpha \beta }\) which governs the LSV out of the SME, and thus, the possible scenario of the LSV induces a linear central potential; in Sect. 3, we generalize our analysis by inserting a Coulomb-type central potential by modifying the mass term of the Klein–Gordon equation and determining solutions of bound states for the scalar field subject to the effects of the LSV; in Sect. 4, we present our conclusions.

## 2 Klein–Gordon oscillator under effects of a linear central potential induced by the LSV

*g*is a coupling constant, \(F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }\) is the electromagnetic tensor and \((K_F)_{\mu \nu \alpha \beta }\) is the constant background tensor field that governs the LSV out of the SME [16, 17] and describes the anisotropy of the spacetime by creating preferred directions in the spacetime. Thus, the Klein–Gordon equation is given in the form \((\hbar =c=1)\):

*m*is the rest mass of the scalar field.

*z*-direction. It is important to note that the electric and magnetic fields given in the Eq. (6) are external independent background fields. This configuration has been studied in induced electric dipole moment systems [44, 45, 46, 47] and in LSV possible scenarios [18, 24, 48, 49].

*z*-component of the angular momentum \(\hat{L}_z=-i\partial _{\varphi }\) and of the eigenvalues of the

*z*-component of the linear momentum operator \(\hat{p}_{z}=-i\partial _{z}\) as

*z*-component of the linear momentum. Then, substituting the Eqs. (6) and (7) into the Eq. (5), we obtain the ordinary differential equation

*n*, and thus, we have obtained the relation

*a*, \(B_0\),

*g*and \(\lambda \).

## 3 Klein–Gordon oscillator under effects of a linear central potential induced by the violation of the Lorentz symmetry and Coulomb-type interaction

*q*is the charge of the scalar field and \(V(\vec {r})=V(r)\) is a central potential [53]. Another procedure of inserting central potentials into relativistic wave equations is presented in Ref. [53], which is done by modification of the mass term \(m\rightarrow m+S(\vec {r})\), where

*m*is the rest mass and \(S(\vec {r})\) is the non-electromagnetic central potential. This procedure was used to study Landau quantization in cosmic string spacetime [50], quark–antiquark interaction [54], scalar field in spacetime with torsion [55], the Aharonov–Bohm effect in spacetime with global monopole [56], effects of a central linear potential on a scalar field in Gödel-type spacetime [57] and for a Dirac particle [58]. In this work, we consider a scalar potential proportional to the inverse of the radial distance, then, the mass term of the Klein–Gordon equation becomes [38, 53]

*c*is a constant that characterizes the Coulomb-type potential. In the context of quantum mechanics, the Coulomb-type potential was studied in an atom with permanent electric dipole moment [59, 60], in an Aharonov–Casher system [61], in a Kaluza–Klein theory [62] and in electric quadrupole moment systems [63, 64]. Therefore, by considering the background of the LSV in the previous section, the general form of the Klein–Gordon radial equation which describes the interaction of a scalar field subject to the Klein–Gordon oscillator plus Coulomb-type scalar potential (2) is given by

## 4 Conclusion

We have analyzed the effects of LSV on a scalar field subject to the Klein–Gordon oscillator. The background is characterized by the presence of a constant tensor field from LSV, and it has allowed us a possible scenario which induces a linear central potential. The presence of this linear central potential induced by LSV modifies the energy profile of the system and the lower energy state is determined by the radial mode \(n=1\) instead of the quantum number \(n=0\). By restricting the angular frequency values of the Klein–Gordon oscillator, then the angular frequency of the relativistic oscillator has allowed values defined by the quantum numbers of the system and parameters associated with LSV. Then, a polynomial solution to the Heun biconfluent series can be achieved.

In addition, in order to generalize our analysis by modifying the mass term of the Klein–Gordon equation, we insert a Coulomb-type potential where we have shown that the quantum effects from the influence of the linear central potential and a Coulomb-type potential modify the energy profile of the relativistic oscillator. This restrict the values of the angular frequency of the Klein–Gordon oscillator. They are defined by the quantum numbers of the system and parameters associated with LSV in order to obtain a polynomial solution to the Heun biconfluent series. In this case, we have shown that the allowed values of the angular frequency of the relativistic quantum oscillator associated with the radial mode \(n=1\) are determined by a third degree algebraic equation.

## Notes

### Acknowledgements

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

## References

- 1.R. Pohl et al., Science
**353**, 669 (2016)ADSCrossRefGoogle Scholar - 2.A. Songaila, L.L. Cowie, Nature
**398**, 667 (1999)ADSCrossRefGoogle Scholar - 3.A. Songaila, L.L. Cowie, Nature
**428**, 132 (2004)ADSCrossRefGoogle Scholar - 4.V.A. Kostelecký, S. Samuel, Phys. Rev. D
**39**, 683 (1989)ADSCrossRefGoogle Scholar - 5.H. Belich et al., Rev. Bras. Ensino Fís.
**29**, 1 (2007)CrossRefGoogle Scholar - 6.H. Belich et al., Phys. Rev. D
**74**, 065009 (2006)ADSMathSciNetCrossRefGoogle Scholar - 7.H. Belich et al., Eur. Phys. J. C
**62**, 425 (2009)ADSCrossRefGoogle Scholar - 8.R. Casana et al., Phys. Lett. B
**726**, 815 (2013)ADSMathSciNetCrossRefGoogle Scholar - 9.R. Casana et al., Eur. Phys. J. C
**74**, 3064 (2014)ADSCrossRefGoogle Scholar - 10.R. Casana, M.M. Ferreira Jr., F.E.P. dos Santos, Phys. Rev. D
**90**, 105025 (2014)ADSCrossRefGoogle Scholar - 11.R. Casana, C.F. Farias, M.M. Ferreira, Phys. Rev. D
**92**, 125024 (2015)ADSMathSciNetCrossRefGoogle Scholar - 12.R. Casana et al., Phys. Lett. B
**746**, 171 (2015)ADSCrossRefGoogle Scholar - 13.G. Gazzola, J. Phys. G Nucl. Part. Phys.
**39**, 035002 (2012)ADSCrossRefGoogle Scholar - 14.M. Gomes et al., Phys. Rev. D
**81**, 045018 (2010)ADSCrossRefGoogle Scholar - 15.M.B. Cruz, E.R. Bezerra de Mello, AYu. Petrov, Phys. Rev. D
**96**, 045019 (2017)ADSMathSciNetCrossRefGoogle Scholar - 16.D. Colladay, V.A. Kostelecký, Phys. Rev. D
**55**, 6760 (1997)ADSCrossRefGoogle Scholar - 17.D. Colladay, V.A. Kostelecký, Phys. Rev. D
**58**, 116002 (1998)ADSCrossRefGoogle Scholar - 18.K. Bakke, H. Belich,
*Spontaneous Lorentz Symmetry Violation and Low Energy Scenarios*(LAMBERT Academic Publishing, Saarbrücken, 2015)zbMATHGoogle Scholar - 19.K. Bakke, H. Belich, J. Phys. G Nucl. Part. Phys.
**40**, 065002 (2013)ADSCrossRefGoogle Scholar - 20.K. Bakke, H. Belich, Eur. Phys. J. Plus
**129**, 147 (2014)CrossRefGoogle Scholar - 21.A.G. de Lima, H. Belich, K. Bakke, Ann. Phys. (Leipzig)
**526**, 514 (2014)ADSCrossRefGoogle Scholar - 22.K. Bakke, H. Belich, J. Phys. G Nucl. Part. Phys.
**42**, 095001 (2015)ADSCrossRefGoogle Scholar - 23.K. Bakke, H. Belich, Ann. Phys. (NY)
**360**, 596 (2015)CrossRefGoogle Scholar - 24.K. Bakke, H. Belich, Ann. Phys. (NY)
**373**, 115 (2016)ADSCrossRefGoogle Scholar - 25.R.L.L. Vitória, H. Belich, K. Bakke, Adv. High Energy Phys.
**2017**, 6893084 (2017)CrossRefGoogle Scholar - 26.R.L.L. Vitória, H. Belich, K. Bakke, Eur. Phys. J. Plus
**132**, 25 (2017)CrossRefGoogle Scholar - 27.M. Moshinsky, A. Szczepaniak, J. Phys. A Math. Gen.
**22**, L817 (1989)ADSCrossRefGoogle Scholar - 28.K. Bakke, H.F. Mota, Eur. Phys. J. Plus
**133**, 409 (2018)CrossRefGoogle Scholar - 29.S. Bruce, P. Minning, Nuovo Cimento A
**106**, 711 (1993)ADSCrossRefGoogle Scholar - 30.N.A. Rao, B.A. Kagali, Phys. Scr.
**77**, 015003 (2008)CrossRefGoogle Scholar - 31.J.-Y. Cheng, Int. J. Theor. Phys.
**50**, 228 (2011)CrossRefGoogle Scholar - 32.B. Mirza, R. Narimani, S. Zare, Commun. Theor. Phys.
**55**, 405 (2011)CrossRefGoogle Scholar - 33.M.-L. Liang, R.-L. Yang, Int. J. Mod. Phys. A
**27**, 1250047 (2012)ADSCrossRefGoogle Scholar - 34.A. Boumali, N. Messai, Can. J. Phys.
**92**, 1 (2014)CrossRefGoogle Scholar - 35.R.L.L. Vitória, K. Bakke, Int. J. Mod. Phys. D
**27**, 1850005 (2018)ADSCrossRefGoogle Scholar - 36.J. Carvalho et al., Eur. Phys. J. C
**76**, 365 (2016)ADSCrossRefGoogle Scholar - 37.L.C.N. Santos, C.C. Barros Jr., Eur. Phys. J. C
**78**, 13 (2018)ADSCrossRefGoogle Scholar - 38.K. Bakke, C. Furtado, Ann. Phys. (NY)
**355**, 48 (2015)ADSCrossRefGoogle Scholar - 39.R.L.L. Vitória, K. Bakke, Eur. Phys. J. Plus
**131**, 36 (2016)CrossRefGoogle Scholar - 40.R.L.L. Vitória, C. Furtado, K. Bakke, Ann. Phys. (NY)
**370**, 128 (2016)ADSCrossRefGoogle Scholar - 41.A.V. Kostelecký, M. Mewes, Phys. Rev. Lett.
**87**, 251304 (2001)ADSCrossRefGoogle Scholar - 42.A.V. Kostelecký, M. Mewes, Phys. Rev. D
**66**, 056005 (2002)ADSCrossRefGoogle Scholar - 43.A.V. Kostelecký, M. Mewes, Phys. Rev. Lett.
**97**, 140401 (2006)ADSCrossRefGoogle Scholar - 44.Y. Aharonov, A. Casher, Phys. Rev. Lett.
**53**, 319 (1984)ADSMathSciNetCrossRefGoogle Scholar - 45.L.R. Ribeiro, C. Furtado, J.R. Nascimento, Phys. Lett. A
**348**, 135 (2006)ADSCrossRefGoogle Scholar - 46.C. Furtado, J.R. Nascimento, L.R. Ribeiro, Phys. Lett. A
**358**, 336 (2006)ADSCrossRefGoogle Scholar - 47.K. Bakke, C. Furtado, Eur. Phys. J. B
**87**, 222 (2014)ADSCrossRefGoogle Scholar - 48.K. Bakke, H. Belich, Ann. Phys. (NY)
**354**, 1 (2015)ADSCrossRefGoogle Scholar - 49.K. Bakke, H. Belich, Eur. Phys. J. Plus
**127**, 102 (2012)CrossRefGoogle Scholar - 50.E.R. Figueiredo Medeiros, E.R. Bezerra de Mello, Eur. Phys. J. C
**72**, 2051 (2012)ADSCrossRefGoogle Scholar - 51.A. Ronveaux,
*Heuns Differential Equations*(Oxford University Press, Oxford, 1995)Google Scholar - 52.G.B. Arfken, H.J. Weber,
*Mathematical Methods for Physicists*, 6th edn. (Elsevier Academic Press, New York, 2005)zbMATHGoogle Scholar - 53.W. Greiner,
*Relativistic Quantum Mechanics: Wave Equations*, 3rd edn. (Springer, Berlin, 2000)CrossRefGoogle Scholar - 54.M.K. Bahar, F. Yasuk, Adv. High Energy Phys.
**2013**, 814985 (2013)CrossRefGoogle Scholar - 55.R.L.L. Vitória, K. Bakke, Gen. Relativ. Gravity
**48**, 161 (2016)ADSCrossRefGoogle Scholar - 56.L. Cavalcanti de Oliveira, E.R. Bezerra de Mello, Class. Quantum Gravity
**23**, 5249 (2006)ADSCrossRefGoogle Scholar - 57.R.L.L. Vitória, C. Furtado, K. Bakke, Eur. Phys. J. C
**78**, 44 (2018)ADSCrossRefGoogle Scholar - 58.G. Soff et al., Z. Naturforsch. A
**28**, 1389 (1973)ADSCrossRefGoogle Scholar - 59.A.B. Oliveira, K. Bakke, Ann. Phys. (NY)
**365**, 66 (2016)ADSCrossRefGoogle Scholar - 60.A.B. Oliveira, K. Bakke, Proc. R. Soc. A
**472**, 20150858 (2016)ADSCrossRefGoogle Scholar - 61.P.M.T. Barboza, K. Bakke, Eur. Phys. J. Plus
**131**, 32 (2016)CrossRefGoogle Scholar - 62.E.V.B. Leite, H. Belich, K. Bakke, Adv. High Energy Phys.
**2015**, 925846 (2015)CrossRefGoogle Scholar - 63.K. Bakke, Ann. Phys. (NY)
**341**, 86 (2014)ADSCrossRefGoogle Scholar - 64.K. Bakke, Int. J. Mod. Phys. A
**29**, 1450117 (2014)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}