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

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A Massive Scalar Field Under the Effects of the Lorentz Symmetry Violation by a CPT-Odd Nonminimal Coupling Hindawi Advances in High Energy Physics Volume 2019, Article ID 8462973, 9 pages https://doi.org/10.1155/2019/8462973 Research Article A Massive Scalar Field under the Effects of the Lorentz Symmetry Violation by a CPT-Odd Nonminimal Coupling R. L. L. Vitória and H. Belich Departamento de Física e Química, Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 514, Goiabeiras, 29060-900 Vitória, ES, Brazil Correspondence should be addressed to R. L. L. Vitória; [email protected] Received 7 May 2019; Accepted 12 August 2019; Published 22 December 2019 Academic Editor: Osvaldo Civitarese Copyright © 2019 R. L. L. Vitória and H. Belich. is 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. 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 eects of possible scenarios of Lorentz symmetry violation by electrical and magnetic eld congurations on a massive scalar eld in this background, where, analytically, we determine solutions of bound states. 1. Introduction Model Extended (SME) [7, 8]. e LSV has been studied in various branches of physics [9–56]. It is noteworthy that the Recently, physical experiments have found supposed evidence LSV is also related to other theories, for example, the prin- that some particles detected in the Antarctic, despised as ciple of generalized uncertainty [57, 58] and rainbow gravity anomalies because they do not t into theories, are real and [59, 60], where both imply a deformed Lorentz symmetry must be taken into account [1, 2]. In addition, in order to know such that the energy-moment relations are modied in the better the atomic nucleus, researchers placed a muon in place plane spacetime by the corrections in the Plank scale. of an orbital electron and showed that the proton radius is Recently, these two approaches, which modify the Klein– somewhat dierent from the theoretical predictions [3]. Two Gordon equation, have been investigated in black hole ther- decades ago, physicists have shown that one type of neutrino modynamics [61, 62]. can transform into another, and for such a change to occur, it e LSV has been investigated in nonrelativistic quan- is necessary for the particle to have mass [4] and that the ne tum mechanics, for example, on the in¤uence of a Coulomb- structure constant, 2 [5, 6]. It is important to note that like potential induced by the Lorentz symmetry breaking all these dierences= are in/ contrast to the predictions of the eects on the harmonic oscillator [63], on geometric phases Standard Model (SM), which represents the best quantum eld for a Dirac neutral particle [64, 65], in a Rashba coupling theory (QFT) that describes the subatomic world. induced by LSV eects [66], in quantum holonomies, on a Due to the existence of con¤icts between experimental/ Dirac neutral particle inside a two-dimensional quantum observational data and the theoretical predictions of the SM, ring [67], in a Landau-type quantization [68], in a spin-orbit the scientic community is induced to seek theories that best coupling for a neutral particle [69], in a Rashba-type cou- explain possible physical eects underlying the SM. In this pling induced by Lorentz-violating eects on a Landau sys- context, the Lorentz symmetry violation (LSV) has been one tem for a neutral particle [70] and on an Aharonov-Bohm of the alternative theories in an attempt to seek answers eect induced by the LSV [71]. In relativistic quantum about the discrepancies between the experimental/observa- mechanics, the LSV has been studied in a relativistic tional results and the theoretical predictions of the SM. e Anandan quantum phase [72], in a relativistic Landau- maturation of the LSV culminated in a QFT that goes beyond Aharonov-Casher quantization [73], in a relativistic the SM, which became known in the literature as Standard Landau-He-McKellar-Wilkens quantization and relativistic 2 Advances in High Energy Physics bound states solutions for a Coulomb-like potential induced 2. CPT-Old Nonminimal Coupling in the by the Lorentz symmetry breaking eects [74], on geometric Klein–Gordon Equation quantum phases from Lorentz symmetry breaking eects in the cosmic string spacetime [64], on relativistic EPR cor- e description of a massive scalar eld is given by the Klein– relations [76], on the relativistic Anandan quantum phase Gordon equation 2 and the Aharonov-Casher eect under Lorentz symmetry 2 (1) breaking eects in the cosmic string spacetime [77] and in − = 0, 2 a relativistic quantum scattering yielded by Lorentz sym- where , with , and is rest mass of metry breaking eects [78]. All of these examples are inves - the scalar= eld . By considering = 0, = 1, the 2, 3 Minkowski spacetime tigations of a spin- fermionic eld. In addition, these with cylindrical symmetry described by the metric studies were possible1/2 through nonminimal couplings in the 2 2 2 2 2 2 (= =1) (2) Dirac equation. = − + + + , Nonminimum couplings, based on the SME gauge sec- where 2 2, the Klein–Gordon equation is given as tor, which carry information that the background is char- follows= + acterized by the LSV are classied in the literature as 2 2 2 2 CPT-even and CPT-odd. e rst conserves the CPT sym- 2 metry, while the second violates the CPT symmetry [79]. 2 2 1 12 2 2 (3) − + + + + − = 0, Recently, the calibre sector CPT-even coupling has been investigated on a scalar eld in solutions of bound states, which represents the relativistic quantum dynamics of a free for example, on a relativistic scalar particle subject to a scalar eld in the Minkowski spacetime, that is, in an isotropic Coulomb-type potential given by Lorentz symmetry break- medium. On the other hand, we can leave an isotropic medium ing eects [80], on the harmonic-type and linear-type con- to an anisotropic medium through a nonminimal coupling in nement of a relativistic scalar particle yielded by Lorentz the Klein–Gordon equation which is characterized by the symmetry breaking eects [81], in a relativistic scalar par- presence of background vector and tensor elds that governs ticle subject to a conning potential and Lorentz symmetry the LSV [7, 8, 79]. en, based on the Refs. [87, 88], let us breaking eects in the cosmic string spacetime [82], on the consider the CPT-old nonminimum coupling , eects of the Lorentz symmetry violation yielded by a tensor − ̃ where is a coupling constant, is eld on the interaction of a scalar particle and a Coulomb- ̃ dual electromagnetic ≪ 1 tensor, = (1/2) is the Maxwell type eld [83, 84] and on the Klein–Gordon oscillator [85, tensor and is the background constant= − vector eld that gov- 86]. However, a point that has not yet been raised in the erns the LSV. In the way, equation (1) becomes literature, in the context of relativistic quantum mechanics, is the CPT-odd nonminimal coupling, based on the SME 2 (4) gauge sector, in the Klein–Gordon equation. erefore, fol- − ̃ − ̃ − = 0, lowing Refs. [87, 88], we insert a background vector eld, or which governs the LSV, into the Klein–Gordon equation 2 through a nonminimal coupling. Next, we consider distri- − 22 ̃ − 2 ̃ (5) butions of electric and magnetic elds to which they char- acterize possible scenarios of LSV where it is possible to − ̃ ̃ − = 0. obtain solutions of bound states for such backgrounds. In It is worth remembering that gives the homogenous addition, we consider one of these backgrounds, the most ̃ 2 Maxwell equations. Moreover, since= 0 ≪ , we neglect general, and we analyze a massive scalar eld with position- the penultimate term of equation (5) such that= we1 obtain dependent mass. e structure of this paper is as follows: in Section 2, we 2 2 (6) rewrite the Klein–Gordon equation under the perspective of − 2 ̃ − = 0. a nonminimal coupling that carries the information that the Hence, in a spacetime described by the metric given in equa- LSV is present by the a background vector eld. Next, we tion (2), equation (6) is given in the form analyze the rst LSV scenario possible through an electric 2 2 2 2 eld conguration which induces a Coulomb-type potential 1 1 0 and determine the relativistic energy levels of the system. − 2 + 2 + + 2 2 + 2 + 2⋅⃗ ⃗ (7) Continuing, we investigated the second LSV possible scenario 0 2 where there is a uniform magnetic eld, where we determine + 2 ⃗ ⋅ ∇ − 2×⃗ ⋅∇− ⃗ = 0. analytically the energy spectrum of the system; in Section 3, Equation (7) represents the Klein–Gordon equation in a still in the second background, we insert a linear central spacetime of cylindrical symmetry aected by the LSV, which potential in the Klein–Gordon equation by modifying the is governed by the presence of a background vector eld. Note mass term and show that the relativistic energy levels of the that equation (7) supports several LSV scenarios through pos- system are drastically modied; in Section 4, we present our sible electric and magnetic eld congurations. We will inves- conclusions. tigate two particular cases from now on. Advances in High Energy Physics 3 2.1. First Background. Let us consider a background of the Equation (17) is the well known con¤uent hypergeometric LSV determined by the eld conguration: equation [89, 90] and is the con¤uent hypergeometric power series: 1 1() . It is well known that the( )con¤uent= (|| + hypergeometric (1/2) − , 2|| + 1;power ) series = (0, 0, , 0); (8) becomes a polynomial of degree when we have =⃗ ;̂ the condition .
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