Non-orthogonality and $\kappa$-dependence eccentricity of polarized electromagnetic waves in CPT-even Lorentz violation

We discuss the modified Maxwell action of a $K_{F}$-type Lorentz symmetry breaking theory and present a solution of Maxwell equations derived in the cases of linear and elliptically polarized electromagnetic waves in the vacuum of CPT-even Lorentz violation. We show in this case the Lorentz violation has the effect of changing the amplitude of one component of the magnetic field, while leaving the electric field unchanged, leading to non-orthogonal propagation of eletromagnetic fields and dependence of the eccentricity on $\kappa$-term. Further, we exhibit numerically the consequences of this effect in the cases of linear and elliptical polarization, in particular, the regimes of non-orthogonality of the electromagnetic wave fields and the eccentricity of the elliptical polarization of the magnetic field with dependence on the $\kappa$-term.


I. INTRODUCTION
The recent discovery of the Higgs boson at the LHC established a research program that aims to explain the physics of fundamental interactions, like excitations manifested from fundamental fields, ending a program of tremendous experimental predictions -the Standard Model (SM) of particle physics. This model has the merit of unifying the weak force and electromagnetism by the Anderson-Higgs mechanism. Nevertheless, it is known that the description of the SM without neutrino mass is unsatisfactory. Recent experiments indicate that the electrons have a bound for the electric dipole moment [1] and the SM predicts that the electron is a punctual particle. By these experimental results we understand the necessity of investigating physics beyond the Standard Model.
Kostelecky and Samuel [6] showed that interactions in the context of the string field theory could lead to spontaneous breaking of the Lorentz symmetry. Such a scenario cannot exist in the regime of SM, but can be the key for explaining regimes where quantum gravity and the early stages of the universe bring remanecent breaking effects in the limits of validity of SM. In these other regimes, structure of spacetime and the corresponding particle physics are affected in their symmetries. A spontaneous breaking of Lorentz symmetry can then be extented beyond the known scenarios of QED and QCD [7][8][9][10][11][12].
Lorentz violation theories are in the framework of Standard Model Extension (SME) [13,14]. From the SM perspective, Lorentz symmetry breaking induces an assymmetry in the structure of spacetime, implying for instance that vacuum Maxwell equations has to be changed by non-trivial contributions. One important feature of LVT in the domain of electrodynamics is the possibility of violating Lorentz symmetry without breaking gauge symmetry. One of the first models in this sense was car-ried out by the Carroll-Field-Jackiw model [15] where breaking of the Lorentz invariance is realized by the inclusion of an additional term in the Maxwell lagrangian, while preserving the gauge symmetry.
General case of κ-electrodynamics was introduced in [16], determining the energy momentum tensor and basic properties of the plane wave solutions of κelectrodynamics. A more in-depth study of the model and its plane wave solutions was performed in [17], where equations of motion in Maxwell form are derived. A special case of the κ-parameter was discussed in [18] that also constructs the plane-wave solutions including with polarization vectors. The idea that the special case of the κ-parameter acts like an effective metric has been used widely in the literature [19]. The presence of terms that violate the Lorentz symmetry imposes at least one privileged direction in the spacetime [20]. Nowadays, studies in relativistic quantum effects [21] that stem from a non minimal coupling with Lorentz symmetry breaking [22] has opened the possibility of investigating new implications in quantum mechanics that this violator background can promote.
In this work, we discuss free electromagnetic fields with a particular type of Lorentz symmetry violation: the vacuum reduction of a K F -type Lorentz symmetry breaking theory [23] in the absence of complex scalar field ϕ and λ-parameter associated to a λ|φ 4 | are zero. The modified dispersion laws of electromagnetic waves are obtained in [5] with two four-vectors as violating background, i. e., they are interpreted as preferred directions in spacetime. Here we investigate the scenario of the the modified Maxwell equations resulting from this CPT-even scenario of Lorentz violation. In this case, we investigate the modified Maxwell equations in the vacuum of a K F -type Lorentz symmetry breaking where the Lorentz violation is manifested only by the presence of a κ µνρσ tensor. We then consider a solution of Maxwell equations derived in the cases of linear and elliptical polarization electromagnetic waves in the vacuum of CPT-even Lorentz viola-tion scenario and investigate the effect of the amplitude changing of the amplitudes electromagnetic wave fields and regimes of non-orthogonality of the electromagnetic wave fields and the eccentricity of the elliptical polarization with dependence on the κ-term.
The structure of this paper is the following: in section II we discuss the modified Maxwell action by the presence of the tensor κ abcd ; in section III, the discuss the lagrangian; in section IV, the effective metric; in section V, the Momentum Energy Tensor for the modified Maxwell action and in the section VI, the Maxwell equations from the modified Maxwell lagrangian. In section VII, we present our conclusions.

II. THE TENSOR κ ABCD
We starting off our analysis with the Maxwell action in the presence of the Lorentz violating term where the part responsible by the Lorentz violation is given by The tensor κ µνσρ is CPT-even, that is, it does not violate the CPT-symmetry. Although the violation of the CPT-symmetry implies that the Lorentz invariance is violated [25], the reverse is not necessarily true. The action with the presence of κ abcd breaks the Lorentz symmetry in the sense that the tensor κ abcd has a non-null vacuum expectation value. Besides, the tensor κ abcd has the same properties of the Riemann tensor, as well as an additional double-traceless condition. This tensor possesses the following symmetries: By following Refs. [26][27][28], we can write the tensor κ abcd in terms of a traceless and symmetric matrixκ ab as By defining a normalized parameter four-vector ξ a , which satisfies the conditions: ξ a ξ a = 1 for the timelike case and ξ a ξ a = −1 for the spacelike case; thus, we can decompose the tensorκ ab as where We deal with a Lorentz symmetry violating tensor k µνκλ in such a way that the CPT symmetry is preserved. It has been shown in Ref. [23] that a particular decomposition [29,30] of the tensor κ µνρσ produces a modification on the equations of motion of the electromagnetic waves due to the presence of vacuum anisotropies, which gives rise to the modified Maxwell equations. As a consequence, the anisotropy can be a source of the electric field and, then, Gauss's law is modified. Besides, the Ampère-Maxwell law is modified by the presence of anisotropies and it has a particular interest in the analysis of vortices solutions since it generates the dependence of the vortex core size on the intensity of the anisotropy.

III. LAGRANGIAN
We start with the following modified Maxwell lagrangian from the CPT-even K F -type Lorentz symmetry breaking theory Using the description of κ µνρσ in tetrad fields and the decomposition in the non-birrefringent sector eq. (4), we can write the modified contribution as Taking into account the relation between metric and tetrad fields we also have Considering the tensor we can rewrite We also have the following relation for tetrad fields that will lead to the following contribution We can also rewrite Identifying the even symmetry in the terms F µν F ρσ in the pairs µσ, νρ, and using the antisymmetry in the second term under the exchange ρ → σ, σ → ρ, we achieve The effective lagrangian now can be written as

IV. EFFECTIVE METRIC
We can simplify the terms in the lagrangian as follows and rewrite F µν F ρσ (g µρ g νσ + 2κg µρ (ξ ν ξ σ − g νσ ξ e ξ e /4)) The lagrangian can now be written as follows We can define the effective metric and its inverse as In order to prove the identity relation, we have to consider we can simplifyg µνg µν = g µν g µν .

V. MOMENTUM ENERGY TENSOR
The momentum energy tensor from the modified action can be written as The energy associated is then given by The modified Maxwell equations in the vacuum of a K F -type Lorentz symmetry breaking theory in the presence of the complex scalar field ϕ, and the λ-parameter associated to a λ|φ 4 |-theory obtained in [23] We write explicitly the modified Maxwell equations, are reduced to the same equation in the absence of the complex scalar field as we start from the modified Maxwell lagrangian in the bilinear gauge action we derived In this case, we can understand the anisotropy generated by the kind of Lorentz violation we are considering. We obtain the generalized equation In terms of the components (ν = 0) and (ν = i) we have These equations take into account the possible dynamical variation of the background represented by the presence of ξ e and κ. In the static background we achive the usual the modified Maxwell equations in the vacuum of a K F -type Lorentz symmetry breaking theory [23], in the absence of the complex scalar field ϕ, and the λparameter associated to a λ|φ 4 |-theory is zero, is reduced to modified Maxwell equations by the presence of the K µνρσ tensor. In this case, these equations are written Notice that these modified Maxwell equations are written in the vacuum by the presence of Lorentz violation in the electromagnetic field dynamics. A feature of this modification is that the two equations (43) and (44) are not changed. Considering a ξ µ point in a specific direction We have κ = 4κ 11 /3 and The only change is in the x component of the eq. (50) Considering the electromagnetic wave propagating in the z direction, we have from the equations (47) and (48) we can then consider for simplicity E z = 0, B z = 0. Taking the x-component of the electric field as monocromatic wave propagation linearly polarized in the z direction We then have from (52) The component of the magnetic field will be given by from which the relation between electric and magnetic components will be Notice that there is a deformation factor in the component of the magnetic field due to the presence of Lorentz violation ( figure 1). For the other component We have then kB x,0 cos(kz − ωt) = ∂E y (z, t) ∂t = −ωE y,0 cos(kz − ωt). (61) and we have We then have the magnetic fields We can calculate the scalar product between the electric and magnetic field and after simplification This result shows that the effect of Lorentz violation term κ is to make the electric and magnetic field nonorthogonals. Instead of be orthogonals for any time they the angle between the eletric and magnetic fields are oscillating on time.
We can also consider the circularly polarized case, taking the electric field given by As before, the component in the direction of propagation is zero. We can also write wherer (kz − ωt) = cos(kz − ωt)x + sin(kz − ωt)ŷ, (70) We then have The components of the magnetic field will be and then We can then write We then have In this case we have or explicitly We note that the dependence on κ makes the electric and magnetic fields non-orthogonal (figure 3) and the elliptical polarization of magnetic field depend on κ ( figure  4). As a consequence, the eccentricity of the ellipse in the magnetic field is κ-dependent as depicted in figure 5.
The restriction of κ to the interval 0 ≤ κ ≤ 2 respects the limits for unitarity found in [23].

VII. CONCLUSION
We investigated the influence of CPT-even Lorentz violating contribution of a K F -type Lorentz symmetry breaking theory tensor in the Maxwell equations derived from the modified Maxwell action of K F -Lorentz violation. We considered the features associated to the derivation of the electromagnetic wave solutions in such a Lorentz violation scenario and considered solutions of the cases of linear and elliptical polarized electromagnetic waves propagating in the vacuum. We showed explicitly that the dependence of κ of the non-orthogonality between electric and magnetic fields and that the exihibiting the explicit dependence on κ.
We showed in this case the Lorentz violation changes the amplitude of the magnetic field with respect on κ, without changing the electric field. As a consequence, we showed that the electromagnetic wave fields have a non-orthogonal propagation in the vacuum and exhibited an explicity dependency of the eccentricity of the magnetic field polarization on κ-term. Furthermore, we exhibited numerically the consequences of this effect in the cases of linear and elliptical polarization, in particular, the regimes of non-orthogonality of the electromagnetic wave fields and the eccentricity of the elliptical polarization of the magnetic field with dependence on the κ-term.