A Massive Scalar Field under the Effects of the Lorentz Symmetry Violation by a CPT-Odd Nonminimal Coupling

Vitória, R. L. L.; Belich, H.

doi:https://doi.org/10.1155/2019/8462973

Advances in High Energy Physics

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 8462973 | https://doi.org/10.1155/2019/8462973

A Massive Scalar Field under the Effects of the Lorentz Symmetry Violation by a CPT-Odd Nonminimal Coupling

R. L. L. Vitória¹and H. Belich¹

Academic Editor: Osvaldo Civitarese

Received07 May 2019

Accepted12 Aug 2019

Published22 Dec 2019

Abstract

In this paper, based on the Standard Model Extended gauge sector, we made a nonminimal coupling in the Klein–Gordon equation which characterizes the Lorentz symmetry violation and, through this nonminimal CPT-odd coupling, we investigate the effects of possible scenarios of Lorentz symmetry violation by electrical and magnetic field configurations on a massive scalar field in this background, where, analytically, we determine solutions of bound states.

1. Introduction

Recently, physical experiments have found supposed evidence that some particles detected in the Antarctic, despised as anomalies because they do not fit into theories, are real and must be taken into account [1, 2]. In addition, in order to know better the atomic nucleus, researchers placed a muon in place of an orbital electron and showed that the proton radius is somewhat different from the theoretical predictions [3]. Two decades ago, physicists have shown that one type of neutrino can transform into another, and for such a change to occur, it is necessary for the particle to have mass [4] and that the fine structure constant, [5, 6]. It is important to note that all these differences are in contrast to the predictions of the Standard Model (SM), which represents the best quantum field theory (QFT) that describes the subatomic world.

Due to the existence of conflicts between experimental/observational data and the theoretical predictions of the SM, the scientific community is induced to seek theories that best explain possible physical effects underlying the SM. In this context, the Lorentz symmetry violation (LSV) has been one of the alternative theories in an attempt to seek answers about the discrepancies between the experimental/observational results and the theoretical predictions of the SM. The maturation of the LSV culminated in a QFT that goes beyond the SM, which became known in the literature as Standard Model Extended (SME) [7, 8]. The LSV has been studied in various branches of physics [9–56]. It is noteworthy that the LSV is also related to other theories, for example, the principle of generalized uncertainty [57, 58] and rainbow gravity [59, 60], where both imply a deformed Lorentz symmetry such that the energy-moment relations are modified in the plane spacetime by the corrections in the Plank scale. Recently, these two approaches, which modify the Klein–Gordon equation, have been investigated in black hole thermodynamics [61, 62].

The LSV has been investigated in nonrelativistic quantum mechanics, for example, on the influence of a Coulomb-like potential induced by the Lorentz symmetry breaking effects on the harmonic oscillator [63], on geometric phases for a Dirac neutral particle [64, 65], in a Rashba coupling induced by LSV effects [66], in quantum holonomies, on a Dirac neutral particle inside a two-dimensional quantum ring [67], in a Landau-type quantization [68], in a spin-orbit coupling for a neutral particle [69], in a Rashba-type coupling induced by Lorentz-violating effects on a Landau system for a neutral particle [70] and on an Aharonov-Bohm effect induced by the LSV [71]. In relativistic quantum mechanics, the LSV has been studied in a relativistic Anandan quantum phase [72], in a relativistic Landau-Aharonov-Casher quantization [73], in a relativistic Landau-He-McKellar-Wilkens quantization and relativistic bound states solutions for a Coulomb-like potential induced by the Lorentz symmetry breaking effects [74], on geometric quantum phases from Lorentz symmetry breaking effects in the cosmic string spacetime [64], on relativistic EPR correlations [76], on the relativistic Anandan quantum phase and the Aharonov-Casher effect under Lorentz symmetry breaking effects in the cosmic string spacetime [77] and in a relativistic quantum scattering yielded by Lorentz symmetry breaking effects [78]. All of these examples are investigations of a spin- fermionic field. In addition, these studies were possible through nonminimal couplings in the Dirac equation.

Nonminimum couplings, based on the SME gauge sector, which carry information that the background is characterized by the LSV are classified in the literature as CPT-even and CPT-odd. The first conserves the CPT symmetry, while the second violates the CPT symmetry [79]. Recently, the calibre sector CPT-even coupling has been investigated on a scalar field in solutions of bound states, for example, on a relativistic scalar particle subject to a Coulomb-type potential given by Lorentz symmetry breaking effects [80], on the harmonic-type and linear-type confinement of a relativistic scalar particle yielded by Lorentz symmetry breaking effects [81], in a relativistic scalar particle subject to a confining potential and Lorentz symmetry breaking effects in the cosmic string spacetime [82], on the effects of the Lorentz symmetry violation yielded by a tensor field on the interaction of a scalar particle and a Coulomb-type field [83, 84] and on the Klein–Gordon oscillator [85, 86]. However, a point that has not yet been raised in the literature, in the context of relativistic quantum mechanics, is the CPT-odd nonminimal coupling, based on the SME gauge sector, in the Klein–Gordon equation. Therefore, following Refs. [87, 88], we insert a background vector field, which governs the LSV, into the Klein–Gordon equation through a nonminimal coupling. Next, we consider distributions of electric and magnetic fields to which they characterize possible scenarios of LSV where it is possible to obtain solutions of bound states for such backgrounds. In addition, we consider one of these backgrounds, the most general, and we analyze a massive scalar field with position-dependent mass.

The structure of this paper is as follows: in Section 2, we rewrite the Klein–Gordon equation under the perspective of a nonminimal coupling that carries the information that the LSV is present by the a background vector field. Next, we analyze the first LSV scenario possible through an electric field configuration which induces a Coulomb-type potential and determine the relativistic energy levels of the system. Continuing, we investigated the second LSV possible scenario where there is a uniform magnetic field, where we determine analytically the energy spectrum of the system; in Section 3, still in the second background, we insert a linear central potential in the Klein–Gordon equation by modifying the mass term and show that the relativistic energy levels of the system are drastically modified; in Section 4, we present our conclusions.

2. CPT-Old Nonminimal Coupling in the Klein–Gordon Equation

The description of a massive scalar field is given by the Klein–Gordon equation

where , with , and is rest mass of the scalar field . By considering the Minkowski spacetime with cylindrical symmetry described by the metric

where , the Klein–Gordon equation is given as follows

which represents the relativistic quantum dynamics of a free scalar field in the Minkowski spacetime, that is, in an isotropic medium. On the other hand, we can leave an isotropic medium to an anisotropic medium through a nonminimal coupling in the Klein–Gordon equation which is characterized by the presence of background vector and tensor fields that governs the LSV [7, 8, 79]. Then, based on the Refs. [87, 88], let us consider the CPT-old nonminimum coupling , where is a coupling constant, is dual electromagnetic tensor, is the Maxwell tensor and is the background constant vector field that governs the LSV. In the way, equation (1) becomes

or

It is worth remembering that gives the homogenous Maxwell equations. Moreover, since , we neglect the penultimate term of equation (5) such that we obtain

Hence, in a spacetime described by the metric given in equation (2), equation (6) is given in the form

Equation (7) represents the Klein–Gordon equation in a spacetime of cylindrical symmetry affected by the LSV, which is governed by the presence of a background vector field. Note that equation (7) supports several LSV scenarios through possible electric and magnetic field configurations. We will investigate two particular cases from now on.

2.1. First Background

Let us consider a background of the LSV determined by the field configuration:

where and is a constant associated with a linear distribution of electric charges on the -axis. Note that, with this field configuration, we have . Then, for this possible scenario, equation (7) becomes

The solution to equation (9) is given in form

where and are the eigenvalues of the angular momentum and linear operators, respectively. Then, by substituting equation (10) into equation (9), we obtain

Now, let us assume the case . In the way, equation (11) is rewrite in the form

where we define the parameter

By making the change of variable into equation (12), we have

with

The radial wave function must be analytic at the origin and at the infinity. Then, by analyzing the asymptotic behavior of equation (14) at and , we obtain the following solution in terms of a function :

By substituting equation (16) into equation (14), we have

Equation (17) is the well known confluent hypergeometric equation [89, 90] and is the confluent hypergeometric power series: . It is well known that the confluent hypergeometric power series becomes a polynomial of degree when we have the condition . Then, with this condition, we obtain

which represents the relativistic energy levels of a massive scalar field in a LSV background governed for a constant vector field. This energy spectrum arises through a Coulomb-type central potential induced by the LSV, by the electric and magnetic field configuration given in equation (8). Hence, the energy spectrum is influenced by the LSV through parameters , and . We can note that for the particular case where , that is, the scale field is in the -plane, we obtain the rest energy of the scalar field, in contrast to Ref. [81], where the scalar field, which is subjected to the CPT-even nonminimal coupling of the SME gauge sector, remains confined in the -plane. This effect may be associated with the nonminimal CPT-odd coupling. We can also notice that, for , we recover the energy corresponding to a free scalar field in the Minkowski spacetime.

2.2. Second Background

In this section, we establish another LSV scenario possible. This LSV scenario possible is determined by a field configuration defined as

where is a constant. Note that . In the way, equation (7) becomes

By following the steps from equation (10) to equation (12), we have

with the parameters

By making the change of variable into equation (13), we obtain

where we define the new parameter

We can note that equation (23) is analogous to equation (14). Then, by analyzing the asymptotic behavior of equation (23) at and at , we obtain a general solution in terms of a confluent hypergeometric function: . As we saw in the previous case, in order to obtain relativistic bound state solutions, we must impose the condition , with , which gives us

Equation (25) represents the energy spectrum of massive scalar field in a LSV background governed for a constant vector field. We can note that the new LSV background determined by the configuration given in equation (19) modifies relativistic energy levels, compared to equation (18). This modification can be seen by the presence of parameters and in equation (25). Again, we can also note that, for the particular case where , the massive scalar field is not confined in the -plane, as in Ref. [81]. In addition, in this particular case, in contrast to equation (18), the massive scalar field gains a relativistic energy term determined by the parameters associated with the LSV background on the rest energy, that is, , where . By making in equation (25), we recover the energy corresponding to a free massive scalar field in the Minkowski space.

3. Effects of a Linear Central Potential

The Ref. [91] presents a procedure to insert central potentials in the Klein–Gordon equation different from the standard procedure that is through the minimum coupling , where is the electric charge of the scalar field. This other procedure of inserting central potentials is done by modifying the mass term of the Klein–Gordon equation by the transformation , where is the scalar central potential. Then, through this procedure, let us insert a central linear potential in the Klein–Gordon equation in the form:

where is a constant that characterizes the linear central potential. In the context of nonrelativistic quantum mechanical, the linear central potential has been studied on the effects of a screw dislocation on the harmonic oscillator [92], on an atom with a magnetic quadrupole moment [93], in atomic and molecular physics [94–99] and on a quantum particle subject to the uniform force field [100, 101]; in the context of relativistic quantum mechanics, the liner central potential has been studied in atomic physics [102], in the Klein–Gordon equation in the presence of a dyon and magnetic flux in the spacetime of gravitational defects [103], in the relativistic quantum dynamics of a charged particle in cosmic string spacetime in the presence of magnetic field [104], on Klein–Gordon oscillator [105, 106], on a scalar field in the spacetime with torsion [107–109], on a Majorana fermion [110], on a scalar field in the rotating cosmic string spacetime [111, 112] and on a scalar particle in a Gödel-type spacetime [113, 114].

By substituting equation (26) into equation (7) and by considering the background given in Section 2.2, we have the radial wave equation

where

From now on, let us consider the change of variable , such that equation (27) becomes

where we define the new parameters

Again, we are interested in analytical solutions for and . In this sense, by analyzing the asymptotic behavior of equation (29) for and , we can write a general solution in terms of a unknown function in the form [104]:

By substituting equation (31) into equation (29), we obtain

where

Equation (32) is the biconfluent Heun equation [104, 115] and is the biconfluent Heun function: . Equation (32) has two singular points: and . represents the regular singular point, while represents the irregular singular point [104]. In this case, equation (32) has at least one solution around the origin given by the power series [89, 104]

By substituting equation (34) into equation (32), we obtain recurrence relation

with the coefficients

By following Refs. [103, 104, 114, 116], it is possible to obtain solutions of bound states and, consequently, a polynomial of degree of equation (34) imposing the following conditions:

Let us consider the lowest energy state of the system, that is, the radial mode in the condition such that we obtain . For this equality to give us a physical meaning, an adjustment parameter is needed, not only for the radial mode , but for any value of . Then, we choose the parameter that characterizes the linear central potential as the adjustment parameter, which, from the , gives the following expression:

Equation (38) represents the allowed values of the parameter associated to the linear central potential for the radial mode . We can observe that the parameter is influenced by the LSV through the presence of the parameters , , and in equation (38). In addition, the specific values of the parameter depend on the quantum numbers and for this reason, we have labelled in equation (38). We can note that, for the particular case , equation (38) becomes , which represents the allowed values of the parameter for radial mode , that is, the allowed values of for the radial mode continue to be influenced by the LSV through the parameters and . We can also note that, for , we recover the expression given in Ref. [107] without topological defect.

For our analysis to become complete, it is necessary to analyze the condition for radial mode . Then, the condition becomes , which gives us the expression

Then, by substituting equation (38) into equation (39), we obtain

Equation (40) (or equation (39)) represents the allowed values of relativistic energy of a massive scalar field to the lowest energy state subject to a linear central potential in a LSV possible scenario governed by a background vector field. We can note that the allowed values of relativistic energy for the state of lower energy of the system are influenced by the LSV through the parameters associated with the background considered . In contrast to equations (18) and (25), it is not possible to obtain a closed equation representing the relativistic energy spectrum, but rather separate expressions representing the allowed values of relativistic energy for the radial modes. This effect is due to the presence of the linear central potential in the system, which has its parameter dependent on the quantum numbers. We can also note that, for the particular case , in contrast to equations (18) and (25), the massive scalar field remains confined in the plane and influenced by the LSV through the parameters , and . Again, this effect is due to the presence of the linear central potential. By making , we recover the allowed values of relativistic energy for radial mode obtained in Ref. [107] without torsion.

4. Conclusion

We have investigated a massive scalar field subject to the effects of the LSV through a nonminimal coupling based on the SME gauge sector. We have shown that through this minimum coupling the Klein–Gordn equation is modified and, consequently, provides several LSV backgrounds to be investigated. In this sense, we investigated two backgrounds, where we can observe that in both a Coulomb-type central potential is induced by the LSV and, therefore, we determine the relativistic energy spectra, which are influenced by the LSV.

In addition, we introduce a linear central potential in the Klein–Gordon equation by modifying the mass term, where we analyze the effects of this central potential on the second background. We can note that the presence of this potential modifies the relativistic energy levels of the system. This modification is characterized by the impossibility of determining a closed expression representing the relativistic energy spectrum of the system. Given this, we have shown that it is possible to determine allowed values of relativistic energy for each radial mode of the system separately, where these allowed values are influenced by the LSV. As an example, we determine the allowed values of relativistic energy for the lower energy state, that is, radial mode , instead of . Another interesting quantum effect is that the parameter associated with the linear central potential is determined by the LSV and on the quantum numbers of the system.

Data Availability

The data used to support the findings of this study are available from the corresponding upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brazil). Ricardo L. L. Vitória was supported by the CNPq project No. 150538/2018-9.

References

P. W. Gorham, B. Rotter, P. Allison et al., “Observation of an unusual upward-going cosmic-ray-like event in the third flight of ANITA,” Physical Review Letters, vol. 121, no. 16, p. 161102, 2018.
View at: Publisher Site | Google Scholar
D. B. Fox, S. Sigurdsson, S. Shandera et al., “The ANITA anomalous events as signatures of a beyond standard model particle, and supporting observations from iceCube,” High Energy Astrophysical Phenomena, 2018, https://arxiv.org/abs/1809.09615.
View at: Google Scholar
R. Pohl, F. Nez, L. M. Fernandes et al., “Laser spectroscopy of muonic deuterium,” Science, vol. 353, no. 6300, pp. 669–673, 2016.
View at: Publisher Site | Google Scholar
Y. Fukuda, “Evidence for oscillation of atmospheric neutrinos,” Physical Review Letters, vol. 81, no. 8, 1998.
View at: Publisher Site | Google Scholar
A. Songaila and L. L. Cowie, “Astronomy: fine-structure variable?” Nature, vol. 398, no. 6729, pp. 667-668, 1999.
View at: Publisher Site | Google Scholar
A. Songaila and L. L. Cowie, “Astrophysics: the inconstant constant?” Nature, vol. 428, no. 6979, pp. 132-133, 2004.
View at: Publisher Site | Google Scholar
D. Colladay and V. A. Kostelecký, “CPT violation and the standard model,” Physical Review D, vol. 55, no. 11, pp. 6760–6774, 1997.
View at: Publisher Site | Google Scholar
D. Colladay and V. A. Kostelecký, “Lorentz-violating extension of the standard model,” Physical Review D, vol. 58, no. 11, p. 116002, 1998.
View at: Publisher Site | Google Scholar
D. Colladay and V. A. Kostelecký, “Cross sections and lorentz violation,” Physics Letters B, vol. 511, no. 2-4, pp. 209–217, 2001.
View at: Publisher Site | Google Scholar
R. Lehnert, “Threshold analyses and Lorentz violation,” Physical Review D, vol. 68, no. 8, p. 085003, 2003.
View at: Publisher Site | Google Scholar
R. Lehnert, Journal of Mathematical Physics, vol. 45, p. 3399, 2004.
B. Altschul, “Compton scattering in the presence of Lorentz and CPT violation,” Physical Review D, vol. 70, no. 5, p. 056005, 2004.
View at: Publisher Site | Google Scholar
G. M. Shore, “Strong equivalence, Lorentz and CPT violation, anti-hydrogen spectroscopy and gamma-ray burst polarimetry,” Nuclear Physics B, vol. 717, no. 1-2, pp. 86–118, 2005.
View at: Publisher Site | Google Scholar
S. Aghababaei, M. Haghighat, and I. Motie, “Muon anomalous magnetic moment in the standard model extension,” Physical Review D, vol. 96, no. 11, p. 115028, 2017.
View at: Publisher Site | Google Scholar
R. Bluhm, V. A. Kostelecký, and C. D. Lane, “CPT and Lorentz tests with Muons,” Physical Review Letters, vol. 84, no. 6, pp. 1098–1101, 2000.
View at: Publisher Site | Google Scholar
R. Bluhm, V. A. Kostelecký, C. D. Lane, and N. Russell, “Clock-comparison tests of Lorentz and CPT symmetry in space,” Physical Review Letters, vol. 88, no. 9, p. 090801, 2002.
View at: Publisher Site | Google Scholar
S. M. Carroll, G. B. Field, and R. Jackiw, “Limits on a Lorentz- and parity-violating modification of electrodynamics,” Physical Review D, vol. 41, no. 4, pp. 1231–1240, 1990.
View at: Publisher Site | Google Scholar
A. A. Andrianov, D. Espriu, P. Giacconi, and R. Soldati, “Anomalous positron excess from Lorentz-violating QED,” Journal of High Energy Physics, vol. 2009, no. 09, p. 057, 2009.
View at: Publisher Site | Google Scholar
J. Alfaro, A. A. Andrianov, M. Cambiaso, P. Giacconi, and R. Soldati, “Bare and induced Lorentz and CPT invariance violations in QED,” International Journal of Modern Physics A, vol. 25, no. 16, pp. 3271–3306, 2010.
View at: Publisher Site | Google Scholar
Y. M. P. Gomes and P. C. Malta, “Laboratory-based limits on the Carroll-Field-Jackiw Lorentz-violating electrodynamics,” Physical Review D, vol. 94, no. 2, p. 025031, 2016.
View at: Publisher Site | Google Scholar
A. Martn-Ruiz and C. A. Escobar, “Equivalence between bumblebee models and electrodynamics in a nonlinear gauge,” Physical Review D, vol. 95, no. 9, p. 036011, 2017.
View at: Publisher Site | Google Scholar
R. Lehnert and R. Potting, “Vacuum Čerenkov Radiation,” Physical Review Letters, vol. 93, no. 11, p. 110402, 2004.
View at: Publisher Site | Google Scholar
R. Lehnert and R. Potting, “Čerenkov effect in Lorentz-violating vacua,” Physical Review D, vol. 70, no. 12, p. 125010, 2004.
View at: Publisher Site | Google Scholar
C. Kaufhold and F. R. Klinkhamer, “Vacuum Cherenkov radiation and photon triple-splitting in a Lorentz-noninvariant extension of quantum electrodynamics,” Nuclear Physics B, vol. 734, no. 1-2, pp. 1–23, 2006.
View at: Publisher Site | Google Scholar
B. Altschul, “Infinitesimally nonlocal Lorentz violation,” Physical Review D, vol. 75, no. 2, p. 105003, 2007.
View at: Publisher Site | Google Scholar
B. Altschul, “Bound on the photon charge from the phase coherence of extragalactic radiation,” Physical Review Letters, vol. 98, no. 26, p. 041603, 2007.
View at: Publisher Site | Google Scholar
C. Kaufhold and F. R. Klinkhamer, “Vacuum Cherenkov radiation in spacelike Maxwell-Chern-Simons theory,” Physical Review D, vol. 76, no. 2, p. 025024, 2007.
View at: Publisher Site | Google Scholar
B. Altschul, Nuclear Physics B, vol. 796, p. 262, 2008.
C. A. Escobar and M. A. G. Garcia, “FullCPT-even photon sector of the standard model extension at finite temperature,” Physical Review D, vol. 92, no. 2, p. 025034, 2015.
View at: Publisher Site | Google Scholar
A. Martn-Ruiz and C. A. Escobar, “Casimir effect between ponderable media as modeled by the standard model extension,” Physical Review D, vol. 94, no. 7, p. 076010, 2016.
View at: Publisher Site | Google Scholar
B. Altschul, “Bound on the photon charge from the phase coherence of extragalactic radiation,” Physical Review Letters, vol. 98, no. 26, p. 041603, 2007.
View at: Publisher Site | Google Scholar
C. Kaufhold and F. R. Klinkhamer, “Vacuum Cherenkov radiation in spacelike Maxwell-Chern-Simons theory,” Physical Review D, vol. 76, no. 2, p. 025024, 2007.
View at: Publisher Site | Google Scholar
F. R. Klinkhamer and M. Risse, “Ultrahigh-energy cosmic-ray bounds on nonbirefringent modified Maxwell theory,” Physical Review D, vol. 77, no. 1, p. 016002, 2008.
View at: Publisher Site | Google Scholar
F. R. Klinkhamer and M. Risse, “Ultrahigh-energy cosmic-ray bounds on nonbirefringent modified Maxwell theory,” Physical Review D, vol. 77, no. 1, p. 117901, 2008.
View at: Publisher Site | Google Scholar
F. R. Klinkhamer and M. Schreck, “New two-sided bound on the isotropic Lorentz-violating parameter of modified Maxwell theory,” Physical Review D, vol. 78, no. 8, p. 085026, 2008.
View at: Publisher Site | Google Scholar
A. Moyotl, H. Novales-Sánchez, J. J. Toscano, and E. S. Tututi, “One-loop nonbirefringent effects on the electromagnetic vertex in the Standard Model Extension,” International Journal of Modern Physics A, vol. 29, no. 22, p. 1450039, 2014.
View at: Publisher Site | Google Scholar
M. Schreck, “Analysis of the consistency of parity-odd nonbirefringent modified Maxwell theory,” Physical Review D, vol. 86, no. 6, p. 065038, 2012.
View at: Publisher Site | Google Scholar
B. Agostini, F. A. Barone, F. E. Barone, P. Gaete, and J. A. Helayël-Neto, “Consequences of vacuum polarization on electromagnetic waves in a Lorentz-symmetry breaking scenario,” Physics Letters B, vol. 708, no. 1-2, pp. 212–215, 2012.
View at: Publisher Site | Google Scholar
L. C. T. Brito, H. G. Fargnoli, and A. P. Baêta Scarpelli, “Aspects of quantum corrections in a Lorentz-violating extension of the Abelian higgs model,” Physical Review D, vol. 87, no. 12, p. 125023, 2013.
View at: Publisher Site | Google Scholar
T. Mariz, J. R. Nascimento, E. Passos, R. F. Ribeiro, and F. A. Brito, “A remark on Lorentz violation at finite temperature,” Journal of High Energy Physics, vol. 2005, no. 10, p. 019, 2005.
View at: Publisher Site | Google Scholar
J. R. Nascimento, E. Passos, A Yu Petrov, and F. A. Brito, “Lorentz-CPT violation, radiative corrections and finite temperature,” Journal of High Energy Physics, vol. 2007, no. 06, p. 016, 2007.
View at: Publisher Site | Google Scholar
A. P. B. Scarpelli, M. Sampaio, M. C. Nemes, and B. Hiller, “Gauge invariance and the CPT and Lorentz violating induced Chern–Simons-like term in extended QED,” The European Physical Journal C, vol. 56, no. 4, pp. 571–578, 2008.
View at: Publisher Site | Google Scholar
F. A. Brito, J. R. Nascimento, E. Passos, and A Yu Petrov, “The ambiguity-free four-dimensional Lorentz-breaking Chern–Simons action,” Physics Letters B, vol. 664, no. 1-2, pp. 112–115, 2008.
View at: Publisher Site | Google Scholar
F. A. Brito, L. S. Grigorio, M. S. Guimaraes, E. Passos, and C. Wotzasek, “Induced Chern-Simons-like action in Lorentz-violating massless QED,” Physical Review D, vol. 78, no. 12, p. 125023, 2008.
View at: Publisher Site | Google Scholar
O. M. Del Cima, J. M. Fonseca, D. H. T. Franco, and O. Piguet, “Lorentz and CPT violation in QED revisited: a missing analysis,” Physics Letters B, vol. 688, no. 2-3, pp. 258–262, 2010.
View at: Publisher Site | Google Scholar
V. A. Kostelecký and M. Mewes, “Electrodynamics with Lorentz-violating operators of arbitrary dimension,” Physical Review D, vol. 80, no. 1, p. 015020, 2009.
View at: Publisher Site | Google Scholar
M. Mewes, Physical Review D, vol. 85, p. 116012, 2012.
M. Schreck, Physical Review D, vol. 89, p. 105019, 2014.
V. A. Kostelecký and M. Mewes, “Fermions with Lorentz-violating operators of arbitrary dimension,” Physical Review D, vol. 88, no. 9, p. 096006, 2013.
View at: Publisher Site | Google Scholar
M. Schreck, “Quantum field theoretic properties of Lorentz-violating operators of nonrenormalizable dimension in the fermion sector,” Physical Review D, vol. 90, no. 8, p. 085025, 2014.
View at: Publisher Site | Google Scholar
R. C. Myers and M. Pospelov, “Ultraviolet modifications of dispersion relations in effective field theory,” Physical Review Letters, vol. 90, no. 21, p. 211601, 2003.
View at: Publisher Site | Google Scholar
C. M. Reyes, L. F. Urrutia, and J. D. Vergara, “Quantization of the Myers-Pospelov model: the photon sector interacting with standard fermions as a perturbation of QED,” Physical Review D, vol. 78, no. 12, p. 125011, 2008.
View at: Publisher Site | Google Scholar
J. Lopez-Sarrion and C. M. Reyes, “Microcausality and quantization of the fermionic Myers–Pospelov model,” The European Physical Journal C, vol. 72, no. 9, p. 2150, 2012.
View at: Publisher Site | Google Scholar
C. M. Reyes, L. F. Urrutia, and J. D. Vergara, “The photon sector in the quantum Myers–Pospelov model: an improved description,” Physics Letters B, vol. 675, no. 3-4, pp. 336–339, 2009.
View at: Publisher Site | Google Scholar
C. M. Reyes, Physical Review D, vol. 82, p. 125036, 2010.
C. M. Reyes, Physical Review D, vol. 87, p. 125028, 2013.
E. V. B. Leite and H. Belich, “An effective theory with Lorentz violation and minimum length,” International Journal of Modern Physics D, vol. 27, no. 11, p. 1843006, 2018.
View at: Publisher Site | Google Scholar
B. Khosropour, “Statistical aspects of the Klein–Gordon oscillator in the frame work of GUP,” Indian Journal of Physics, vol. 92, no. 1, pp. 43–47, 2018.
View at: Publisher Site | Google Scholar
J. Magueijo and L. Smolin, “Gravity’s rainbow,” Classical and Quantum Gravity, vol. 21, no. 7, pp. 1725–1736, 2004.
View at: Publisher Site | Google Scholar
V. B. Bezerra, H. R. Christiansen, M. S. Cunha, and C. R. Muniz, “Exact solutions and phenomenological constraints from massive scalars in a gravity’s rainbow spacetime,” Physical Review D, vol. 96, no. 2, p. 024018, 2017.
View at: Publisher Site | Google Scholar
Z. W. Feng, H. L. Li, X. T. Zu, and S. Z. Yang, “Quantum corrections to the thermodynamics of Schwarzschild–Tangherlini black hole and the generalized uncertainty principle,” The European Physical Journal C, vol. 76, no. 4, p. 212, 2016.
View at: Publisher Site | Google Scholar
Z. W. Feng and S. Z. Yang, Physics Letter B, vol. 772, p. 737, 2017.
K. Bakke and H. Belich, “On the influence of a Coulomb-like potential induced by the Lorentz symmetry breaking effects on the harmonic oscillator,” The European Physical Journal Plus, vol. 127, no. 9, p. 102, 2012.
View at: Publisher Site | Google Scholar
K. Bakke and H. Belich, “Abelian geometric phase for a Dirac neutral particle in a Lorentz symmetry violation environment,” Journal of Physics G: Nuclear and Particle Physics, vol. 39, no. 8, p. 085001, 2012.
View at: Publisher Site | Google Scholar
K. Bakke, E. O. Silva, and H. Belich, Journal of Physics G: Nuclear and Particle Physics, vol. 39, p. 055004, 2012.
K. Bakke and H. Belich, “Quantum holonomies based on the Lorentz-violating tensor background,” Annalen der Physik (Leipzig), vol. 526, p. 1, 2013.
View at: Google Scholar
K. Bakke and H. Belich, “Quantum holonomies based on the Lorentz-violating tensor background,” Journal of Physics G: Nuclear and Particle Physics, vol. 40, no. 6, Article ID 065002, 2013.
View at: Publisher Site | Google Scholar
K. Bakke and H. Belich, “A Landau-type quantization from a Lorentz symmetry violation background with crossed electric and magnetic fields,” Journal of Physics G: Nuclear and Particle Physics, vol. 42, no. 9, p. 09500, 2015.
View at: Publisher Site | Google Scholar
H. Belich and K. Bakke, “A spin-orbit coupling for a neutral particle from Lorentz symmetry breaking effects in the CPT-odd sector of the Standard Model Extension,” International Journal of Modern Physics A, vol. 30, no. 22, p. 1550136, 2015.
View at: Publisher Site | Google Scholar
K. Bakke and H. Belich, “On the influence of a Rashba-type coupling induced by Lorentz-violating effects on a Landau system for a neutral particle,” Annals of Physics, vol. 354, pp. 1–9, 2015.
View at: Publisher Site | Google Scholar
H. Belich, E. O. Silva, M. M. Ferreira Jr., and M. T. D. Orlando, “Aharonov-Bohm-Casher problem with a nonminimal Lorentz-violating coupling,” Physical Review D, vol. 83, no. 12, p. 125025, 2011.
View at: Publisher Site | Google Scholar
K. Bakke, H. Belich, and E. O. Silva, “Relativistic Anandan quantum phase in the Lorentz violation background,” Annalen der Physik (Leipzig), vol. 523, no. 11, pp. 910–918, 2011.
View at: Publisher Site | Google Scholar
K. Bakke, H. Belich, and E. O. Silva, “Relativistic Landau-Aharonov-Casher quantization based on the Lorentz symmetry violation background,” Journal of Mathematical Physics, vol. 52, no. 6, p. 063505, 2011.
View at: Publisher Site | Google Scholar
K. Bakke and H. Belich, “Relativistic Landau-He-McKellar-Wilkens quantization and relativistic bound states solutions for a Coulomb-like potential induced by the Lorentz symmetry breaking effects,” Annals of Physics, vol. 333, pp. 272–281, 2013.
View at: Publisher Site | Google Scholar
H. Belich and K. Bakke, “Geometric quantum phases from Lorentz symmetry breaking effects in the cosmic string spacetime,” Physical Review D, vol. 90, no. 2, p. 112, 2014.
View at: Publisher Site | Google Scholar
H. Belich, C. Furtado, and K. Bakke, “Lorentz symmetry breaking effects on relativistic EPR correlations,” The European Physical Journal C, vol. 75, no. 9, p. 410, 2015.
View at: Publisher Site | Google Scholar
K. Bakke, C. Furtado, and H. Belich, “Relativistic Anandan quantum phase and the Aharonov–Casher effect under Lorentz symmetry breaking effects in the cosmic string spacetime,” Annals of Physics, vol. 372, pp. 544–552, 2016.
View at: Publisher Site | Google Scholar
H. F. Mota, K. Bakke, and H. Belich, “Relativistic quantum scattering yielded by Lorentz symmetry breaking effects,” International Journal of Modern Physics A, vol. 32, no. 23-24, p. 1750140, 2017.
View at: Publisher Site | Google Scholar
K. Bakke and H. Belich, Spontaneous Lorentz Symmetry Violation and Low Energy Scenarios, LAMBERT Academic Publishing, Saarbrücken, 2015.
K. Bakke and H. Belich, “On a relativistic scalar particle subject to a Coulomb-type potential given by Lorentz symmetry breaking effects,” Annals of Physics, vol. 360, pp. 596–604, 2015.
View at: Publisher Site | Google Scholar
K. Bakke and H. Belich, “On the harmonic-type and linear-type confinement of a relativistic scalar particle yielded by Lorentz symmetry breaking effects,” Annals of Physics, vol. 373, p. 115122, 2016.
View at: Publisher Site | Google Scholar
H. Belich and K. Bakke, “Relativistic scalar particle subject to a confining potential and Lorentz symmetry breaking effects in the cosmic string space–time,” International Journal of Modern Physics A, vol. 31, no. 7, p. 1650026, 2016.
View at: Publisher Site | Google Scholar
R. L. L. Vitória, H. Belich, and K. Bakke, “Coulomb-type interaction under lorentz symmetry breaking effects,” Advances in High Energy Physics, vol. 2017, p. 6893084, 2017.
View at: Publisher Site | Google Scholar
R. L. L. Vitória, K. Bakke, and H. Belich, “On the effects of the Lorentz symmetry violation yielded by a tensor field on the interaction of a scalar particle and a Coulomb-type field,” Annals of Physics, vol. 399, pp. 117–123, 2018.
View at: Publisher Site | Google Scholar
R. L. L. Vitória, H. Belich, and K. Bakke, “A relativistic quantum oscillator subject to a Coulomb-type potential induced by effects of the violation of the Lorentz symmetry,” The European Physical Journal Plus, vol. 132, no. 1, p. 25, 2017.
View at: Publisher Site | Google Scholar
R. L. L. Vitória and H. Belich, “Effects of a linear central potential induced by the Lorentz symmetry violation on the Klein–Gordon oscillator,” The European Physical Journal C, vol. 78, no. 12, p. 999, 2018.
View at: Publisher Site | Google Scholar
H. Belich, T. Costa-Soares, M. M. Ferreira Jr., and J. A. Helayël-Neto, “Nonminimal coupling to a Lorentz-violating background and topological implications,” The European Physical Journal C, vol. 41, pp. 421–426, 2005.
View at: Publisher Site | Google Scholar
H. Belich, L. P. Colatto, T. Costa-Soares, J. A. Helayël-Neto, and M. T. D. Orlando, “Magnetic moment generation from nonminimal couplings in a scenario with Lorentz-symmetry violation,” The European Physical Journal C, vol. 62, no. 2, pp. 425–432, 2009.
View at: Publisher Site | Google Scholar
G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Elsevier Academic Press, New York, 6th edition, 2005.
M. Abramowitz and I. A. Stegum, Handbook of Mathematical Functions, Dover Publications Inc., New York, 1965.
W. Greiner, “Klein’s paradox,” Relativistic Quantum Mechanics: Wave Equations, Springer, Berlin, 3rd edition, 2000.
View at: Publisher Site | Google Scholar
M. J. Bueno, C. Furtado, and K. Bakke, “On the effects of a screw dislocation and a linear potential on the harmonic oscillator,” Physica. B, Condensed Matter (Print), vol. 496, pp. 45–48, 2016.
View at: Publisher Site | Google Scholar
I. C. Fonseca and K. Bakke, “On an atom with a magnetic quadrupole moment subject to harmonic and linear confining potentials,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 471, no. 2184, p. 20150362, 2015.
View at: Publisher Site | Google Scholar
E. J. Austin, Molecular Physics, vol. 40, p. 893, 1980.
E. R. Vrscay, “Algebraic methods, Bender-Wu formulas, and continued fractions at large order for charmonium,” Physical Review A, vol. 31, no. 4, pp. 2054–2069, 1985.
View at: Publisher Site | Google Scholar
K. Killingbeck, “Quantum-mechanical perturbation theory,” Reports on Progress in Physics, vol. 40, pp. 963–1031, 1977.
View at: Publisher Site | Google Scholar
K. Killingbeck, Physics Letter A, vol. 65, p. 87, 1978.
R. P. Saxena and V. S. Varma, “Polynomial perturbation of a hydrogen atom,” Journal of Physics A: Mathematical and General, vol. 15, no. 4, pp. L149–L153, 1982.
View at: Publisher Site | Google Scholar
E. Castro and P. Martin, “Eigenvalues of the Schrödinger equation with Coulomb potentials plus linear and harmonic radial terms,” Journal of Physics A: Mathematical and General, vol. 33, no. 30, pp. 5321–5334, 2000.
View at: Publisher Site | Google Scholar
L. D. Landau and E. M. Lifshitz, “The basic concepts of quantum mechanics,” in Quantum Mechanics, the Nonrelativistic Theory, pp. 1–24, Pergamon, Oxford, 3rd edition, 1977.
View at: Publisher Site | Google Scholar
L. E. Ballentine, Quantum Mechanics, a Modern Development, World Scientific, Singapore, 1998.
View at: Publisher Site
G. Soff, B. Müller, J. Rafelski, and W. Greiner, “Solution of the dirac equation for scalar potentials and its implications in atomic physics,” Zeitschrift für Naturforschung A, vol. 28, no. 9, p. 1389, 1973.
View at: Publisher Site | Google Scholar
A. L. Cavalcanti de Oliveira and E. R. Bezerra de Mello, Classical and Quantum Gravity, vol. 23, p. 5249, 2006.
E. R. Figueiredo Medeiros and E. R. Bezerra de Mello, “Relativistic quantum dynamics of a charged particle in cosmic string spacetime in the presence of magnetic field and scalar potential,” The European Physical Journal C, vol. 72, no. 6, p. 2051, 2012.
View at: Publisher Site | Google Scholar
R. L. L. Vitória and K. Bakke, “Relativistic quantum effects of confining potentials on the Klein–Gordon oscillator,” The European Physical Journal Plus, vol. 131, no. 2, p. 36, 2016.
View at: Publisher Site | Google Scholar
R. L. L. Vitória, C. Furtado, and K. Bakke, “On a relativistic particle and a relativistic position-dependent mass particle subject to the Klein–Gordon oscillator and the Coulomb potential,” Annals of Physics, vol. 370, pp. 128–136, 2016.
View at: Publisher Site | Google Scholar
R. L. L. Vitória and K. Bakke, “Torsion effects on a relativistic position-dependent mass system,” General Relativity and Gravitation, vol. 48, no. 12, p. 161, 2016.
View at: Publisher Site | Google Scholar
R. L. L. Vitória and K. Bakke, “Aharonov–Bohm effect for bound states in relativistic scalar particle systems in a spacetime with a spacelike dislocation,” International Journal of Modern Physics D, vol. 27, no. 2, p. 1850005, 2018.
View at: Publisher Site | Google Scholar
R. L. L. Vitória and K. Bakke, “On the interaction of the scalar field with a Coulomb-type potential in a spacetime with a screw dislocation and the Aharonov-Bohm effect for bound states,” The European Physical Journal Plus, vol. 133, no. 11, p. 490, 2018.
View at: Publisher Site | Google Scholar
R. F. Ribeiro and K. Bakke, “On the Majorana fermion subject to a linear confinement,” Annals of Physics, vol. 385, pp. 36–39, 2017.
View at: Publisher Site | Google Scholar
M. S. Cunha, C. R. Muniz, H. R. Christiansen, and V. B. Bezerra, “Relativistic Landau levels in the rotating cosmic string spacetime,” The European Physical Journal C, vol. 76, no. 9, p. 512, 2016.
View at: Publisher Site | Google Scholar
Z. Wang, Z. Long, C. Long, and B. Wang, “The study of a spinless relativistic particle in the spinning cosmic string space–time,” Canadian Journal of Physics, vol. 95, no. 4, pp. 331–335, 2017.
View at: Publisher Site | Google Scholar
Z. Wang, Z. Long, C. Long, and M. Wu, “Relativistic quantum dynamics of a spinless particle in the Som-Raychaudhuri spacetime,” The European Physical Journal Plus, vol. 130, no. 3, p. 36, 2015.
View at: Publisher Site | Google Scholar
R. L. L. Vitória, C. Furtado, and K. Bakke, “Linear confinement of a scalar particle in a Gödel-type spacetime,” The European Physical Journal C, vol. 78, no. 1, p. 44, 2018.
View at: Publisher Site | Google Scholar
A. Ronveaux, Heun’s Differential Equations, Oxford University Press, Oxford, 1995.
C. Furtado, B. G. C. da Cunha, F. Moraes, E. R. Bezerra de Mello, and V. B. Bezerra, “Landau levels in the presence of disclinations,” Physics Letters A, vol. 195, no. 1, pp. 90–94, 1994.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2019 R. L. L. Vitória and H. Belich. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

612

Downloads

1037

Citations