Probing the interference between non-linear, axionic and space-time-anisotropy effects in the QED vacuum

We pursue the investigation of a generic non-linear extension of axionic electrodynamics in a Carroll-Field-Jackiw (CFJ) scenario that implements Lorentz-symmetry violation (LSV). The model we inspect consists of an arbitrary non-linear electrodynamic action coupled to the axion field in presence of an anisotropy four-vector that realizes the breaking of Lorentz symmetry under the particle point of view. The non-linear electromagnetic field is expanded around a constant and uniform magnetic background up to second order in the propagating photon field. The focus of our attention is the study of the material properties of the vacuum in the particular case of a space-like CFJ $4$-vector. The dispersion relations associated to the plane wave solutions are explicitly worked out in two situations: the magnetic background perpendicular and parallel to the wave direction. We extend these results to consider the analysis of the birefringence phenomenon in presence of non-linearity, the axion and the LSV manifested through the spatial anisotropy. Three specific proposals of non-linear electrodynamics are contemplated: Euler-Heisenberg, Born-Infeld and the Modified Maxwell electrodynamics. Throughout the paper, we shall justify why we follow the unusual path of connecting, in a single Lagrangian density, three pieces of physics beyond the Standard Model, namely, non-linearity, axions and LSV. Our true goal is to actually inspect and describe how axionic, non-linear and LSV effects interfere with one another whenever physical entities like group velocity, refraction indices, birefringence and effective masses of physical excitations are computed in presence of an external constant and homogeneous magnetic field.


I. INTRODUCTION
The strong CP problem is still an intriguing question in the Standard Model (SM) of elementary particles.Certainly, the mechanism proposed by Peccei and Quinn is the most popular and elegant approach to solve this issue by introducing the axions [1,2].We should point out here that there are, however, other resolutions to solve the strong CP problem without introducing new particles beyond the Standard Model, as one may find in the articles of Refs.[3][4][5][6].The Axion-like Particles (ALPs) has been the subject of investigation in diverse branches of the high energy physics.A good motivation is that such particles are strong candidates for the dark matter content [7][8][9].Furthermore, ALPs naturally arise in string theories [10].
Over the past decades, a considerable effort has been made for the detection of ALPs, both in astrophysical experiments [11][12][13][14][15] and in particle accelerators [16][17][18][19].The challenge is that the ALPs couple very weakly to the SM matter, so the bounds obtained have a stringent parameter space.For example, the CAST experiment that searches for ALPs produced in the solar core provides a well-established limit for the ALP-photon interaction with coupling constant g aγ ≃ 0.66 × 10 −10 GeV −1 and ALP mass restricted to m a < 0.02 eV at 95% CL [14].We also highlight that ALPs can be produced by ALP-photon conversion in the presence of an intense magnetic background field, as described by the Primakoff process.There is also a great deal of interest in looking for ALPs with very small masses, the so-called ultra-light Axions (ULAs).They are particles with mass in the broad range of 10 −33 eV m a 10 −18 eV.The search for ULAs involves cosmological observations from the cosmic microwave background (CMB) and the large-scale structure (LSS).A thorough review is provided in ref. [20].
In presence of intense magnetic fields close to the Schwinger's critical magnetic field, i.e., |B| S = m 2 e /q e = 4.41 × 10 9 T, non-linear effects acquire relevance [21].In the work [22], a general approach has been followed to investigate ALPs in non-linear electrodynamic scenarios, where some optical properties of the vacuum have been investigated, such as the vacuum magnetic birefringence (VMB) and Kerr effect.Furthermore, it has been shown that the presence of the axion generates dispersion relations that depend on the wavelength, so that dispersive refractive indices show up that would not be present if only non-linearity were considered.In a seminal work [23], the authors connected axionic physics with the conversion, as well as the axion-graviton conversion in the vicinity of stars with an intense magnetic fields.
It is well-known that the ALP-photon conversion in a magnetic background changes the optical properties of the vacuum.Therefore, the measure of the VMB can provide bounds on the axion mass and coupling constant g aγ [24][25][26].Although VMB is an effect predicted by quantum electrodynamics (QED), there is still no experimental evidence of its existence produced in laboratories.The PVLAS Collaboration was one of the most notable projects in this search, having ended its activity in 2017 after 25 years of efforts to measure the birefringence and vacuum dichroism phenomena, providing very reliable limits for such quantities [27][28][29].Even so, indirect evidence of vacuum birefringence was found from measurement of optical polarization of the neutron star RX J1856.5-3754[30].
At this stage, it is worthy mentioning that the axionic interaction term can be generated via radiative corrections in a theory with Lorentz symmetry violation (LSV) [31].Although Lorentz invariance is a fundamental invariance principle in elementary particle physics and in Einstein's classical General Relativity, it is known that, for a quantum theory of gravity, such invariance may not hold any longer.For example, in string theories [32][33][34][35][36], it is estimated that there should be small violations of Lorentz symmetry next to the Planck energy scale, namely, E P l = 10 19 GeV.This would occur in the early universe.In this perspective, we are motivated to study which influence a LSV background would have on an axionic theory.To generalize our approach, we consider that the electrodynamics model may have non-linear contributions.Although, in specific cases, one can easily reduce to the usual Maxwell's theory.Moreover, we point out that LSV theories introduce an anisotropy in space-time, such that is reasonable to obtain a birefringence effect [37,38].This characteristic added to the non-linear ALP-photon mixing model [22] generates a very rich effective model with implications in the optical properties of the vacuum.In particular, for the LSV term, we adopt the Carroll-Field-Jackiw (CFJ) electrodynamics [39], which is a generalization of a Chern-Simons term for (3 + 1) dimensions.For the consistency of the model, a quadrivector is introduced that guarantees the gauge symmetry of the theory, but does not preserve the Lorentz and CPT symmetry.The CFJ term appears in the CPT-odd gauge sector of the Standard Model Extension (SME).This model developed by Colladay and Kosteletsky describes a general action with terms that violate Lorentz and CPT symmetry [40,41].
There is a rich literature on the CFJ electrodynamic model.In the work [42], limits were obtained for the CFJ Lorentz-breaking parameter in the time-like case through laboratory experiments such as quantum corrections to the spectrum of the hydrogen atom, electric dipole moment, as well as the interparticle potential between fermions.Studies on the possible effects of contributions of the CFJ model for the cosmic microwave background (CMB) were carried out in ref. [43].Recently, in the supersymmetry scenario, the gauge boson-gaugino mixing was investigated by taking into account the effects of the LSV due to a CFJ term [44].We also highlight that there is a connection between the axionic theory and Lorentz and CPT-violation.For example, in ref. [45], the author establishes this connection by embedding Carroll-Field-Jackiw (CFJ) electrodynamics in a premetric framework.The fact is used that in CFJ electrodynamics the constraint ∂ µ v ν − ∂ ν v µ = 0 allows writing the Lorentz-breaking vector as the gradient of a scalar v µ = ∂ µ φ; so, in performing this redefinition in the pre-metric Lagrangian, the axionic interaction term naturally arises.However, this approach does not provide a dynamics for the axion.Also, the corresponding energy-momentum tensor does not depend on the axion field.Furthermore, the author discussed the relation between the birefringence phenomenon with Lorentz and CPT symmetry violation.It is possible to associate the non-observation of birefringence with the preservation of these symmetries.For more details on LSV, we indicate the review [46] and references therein.
Before going on and starting to work out the developments of our paper, we would like to call into question our motivation to bring together three different physical scenarios beyond the Standard Model in a single Lagrangian, namely: axions, non-linear electrodynamic extensions and Lorentz-symmetry violating physics (LSV is here realized by means of the Carroll-Field-Jackiw term).The usual procedure is to consider each of these physical situations separately, once we expect that their respective individual effects correspond to tiny corrections to current physics.Connecting these three diverse physics in a single action might appear as a waste of efforts or, simply, an exercise to mix up different effects.Nevertheless, what we truly wish by coupling axions to non-linear electrodynamics and LSV physics is to show how the parameters associated to the axion and LSV sectors couple to external electric and magnetic fields whenever non-linearity is considered.Actually, the main effort we endeavor is to inspect how the magnetic background field may broaden the effects of the tiny axionic and LSV parameters on physical properties such as birefringence, refractive indices, dichroism and group velocity.This is investigated with the help of the dispersion relations we shall derive in different situations characterized by particular configurations of external fields.And to enforce our claim to consider the simultaneous presence of these three sorts of effects, we gather some works in Refs.[47][48][49][50].In these papers, nonlinear quantum electrodynamics, axion electrodynamics and LSV are studied in Condensed Matter scenarios such as Dirac and Weyl semimetals and topological magnetic materials.
We are then motivated to assume that topological materials appear as a natural laboratory that justify the inspection of how the effects of non-linearity, axions and LSV interfere with each other.Cosmology provides another viable scenario that may justify efforts in the quest for the interference between the three effects we are here discussing.In Refs.[20,51,52], we cast reference works that support our proposal.Finally, knowing that non-linearity, axions and LSV are issues currently investigated in connection with astrophysical structures [53][54][55], we can also elect Astrophysics as another field of interest to study the concomitant presence of these three issues and how they affect each other in the study of the propagation of electromagnetic waves in the QED vacuum.
In this contribution, we investigate the propagation effects of a general axionic nonlinear ED in presence of a CFJ term.As mentioned, the CFJ introduces the 4-vector that breaks the Lorentz symmetry, and the isotropy of the space-time.We introduce a uniform magnetic field expanding the propagating field of the model up to second order around this background field.The properties of the medium are discussed in presence of the magnetic background.We obtain the dispersion relations of the linearized theory in terms of the magnetic background, the CFJ 4-vector, and the axion coupling constant.The case of a space-like quadrivector is analysed, such that the plane wave frequencies are functions of the wave vector (k), the magnetic background (B), and the CFJ background vector (v).
Thereby, we consider two cases : (a) when k, B and v are perpendiculars, and (b) when k is parallel to B, but both vectors remain perpendicular to v. The solutions of these cases define the perpendicular and parallel frequencies, respectively.Using these dispersion relations, we calculate the birefringence through the perpendicular and parallel refractive indices.We apply our results to the non-linear electrodynamics of Euler-Heisenberg [56], Born-Infeld [57], and Modified Maxwell (ModMax) [58][59][60].
This paper is organized according to the following outline: In Section (II), the axionic non-linear theory is presented with the CFJ term in an electromagnetic background field.
In Section (III), we consider a purely magnetic background field and obtain the permittivity and permeability tensors, as well as the dispersion relations associated with the plane wave solutions.Next, in Section (IV), the birefringence phenomenon is discussed in the framework of Euler-Heisenberg, Born-Infeld, and ModMax electrodynamics.Finally, the Conclusions and Perspectives are cast in Section (V).

II. THE NON-LINEAR AXION-PHOTON ELECTRODYNAMICS INCLUDING THE CARROLL-FIELD-JACKIW TERM
We initiate with the description of the model whose Lagrangian density reads as follows : where L nl (F 0 , G 0 ) denotes the most general Lagrangian of a non-linear electrodynamics that is function of the Lorentz-and gauge-invariant bilinears : These definitions introduce the antisymmetric field strength tensor as , and the correspondent dual tensor is The CFJ term introduces the background 4-vector v µ = (v 0 , v) whose components do not depend on the space-time coordinates.It has mass dimension in natural units and is responsible for the Lorentz symmetry violation in the gauge sector of the model.In addition, φ is the axion scalar field with mass m, and g is the non-minimal coupling constant (with length dimension) of the axion with the electromagnetic field, i.e., the usual coupling with the G 0 -invariant in the axion-photon model.There are many investigations and experiments to constraint the possible regions in the space of the parameters g and m, which still remains with a wide range of values, depending on the phenomenological scale in analysis.
We expand the abelian gauge field as A µ 0 = a µ + A µ B , in which a µ is the photon 4potential, and A µ B denotes a background potential.In this conjecture, the tensor F µν 0 is also written as the combination is the EM field strength tensor that propagates in the space-time, and corresponds to the EM background field.The notation of the 4-vector and tensors with index (B) indicates that it is associated with the background.
At this stage, we consider the general case in which the background depends on the spacetime coordinates.Under this prescription, we also expand the Lagrangian (1) around the background up to second order in the propagating field a µ to yield the expression where Jν = J ν −∂ µ (H Bµν )−v µ F Bµν represent an effective external current that couples to the photon field; it includes an eventual matter current and the contributions that stem from the background electromagnetic fields.The tensors associated with this electromagnetic background are defined in what follows: The axion field is shifted as φ → φ + φ 0 in order to eliminate the g φ G B term that would appear in the Lagrangian (2).The coefficients c 1 , c 2 , d 1 , d 2 and d 3 are evaluated at E and B, as follows : that depend on the EM field magnitude and may also be functions of the space-time coordinates.Following the previous definitions, the background tensors satisfy the properties H Bµν = −H Bνµ , whereas Q Bµνκλ is symmetric under exchange µν ↔ κλ, and antisymmetric under µ ↔ ν and κ ↔ λ.Note that the current J µ couples to the external potential A µ B , but this term and L nl (F B , G B ) are irrelevant for the field equations in which we are interested.
Using the minimal action principle by varying a µ , the Lagrangian (2) yields the EM field equations with source Jµ and the Bianchi identity remains the same one for the photon field, namely, ∂ µ f µν = 0.The action principle in relation to φ in (2) yields the axion field equation evaluated at the EM background : Since we consider a uniform magnetic background field, the all the cases of non-linear ED mentioned previously.

III. THE DISPERSION RELATIONS IN PRESENCE OF A UNIFORM MAG-NETIC FIELD
In this Section, we obtain the dispersion relations of the axion and photon fields in a uniform magnetic background.Thus, we can take E = 0 in the equations of the section II.Thereby, from now on, all the coefficients defined in (4) are not space-time dependent; they actually depend only on the magnetic vector B. We start the description of the field propagating with the equations written in terms of e and b, in the presence of constant and uniform magnetic background field.For the analysis of the free wave propagation, we just consider the linear terms in e, b and φ, as well as, the equations with no source, J = 0 and ρ = 0.Under these conditions, the electrodynamics equations in terms of the propagating vector field are read below : where the vectors D and H are, respectively, given by The scalar field equation ( 6) in terms of the magnetic background field leads to We substitute the plane wave solutions of e, b and φ in the field equations ( 7)(a-d) and (9).Eliminating conveniently the amplitudes of b and φ in terms of the electric field amplitude, the wave equation in the momentum space is read below : where e j 0 (j = 1, 2, 3) are the components of the electric amplitude, and the matrix elements M ij are given by (11) whose the coefficients a, b, c are defined by in which d := d 1 /c 1 , f := d 2 /c 1 and g a := g 2 /c 1 for simplicity in the equations.Thus, the non-linearity evaluated on the magnetic background is manifested in the parameters d and f , and the coupling constant g a corrects the axion coupling constant with the coefficient c 1 .
Notice that the b-coefficient depends only on the magnetic background, but the others one depends on the ω-frequency and on the k-wave vector.
Back to the expressions of D and H in (8a)-(8b) with the plane wave solutions, the components of D and H in terms of the electric and magnetic amplitudes can be written as where ǫ ij and (µ ij ) −1 are the permittivity and permeability (inverse) tensors, respectively, The permeability tensor is obtained by computing the inverse of (14b) Notice that the electric permittivity depends on the ω-frequency and the k-wave vector due to the axion coupling g = 0. Also, the definition of these tensors do not include the components of the CFJ 4-vector v µ .Thereby, this LSV scenario does not contribute with the physical properties of the tensors.
According to the works of refs.[61][62][63], the v 0 -component may induce contributions that violate the causality and stability.For this reason, we shall adopt a space-like CFJ 4-vector, i.e., v 0 = 0 in the matrix element M ij from eq. ( 11).The dispersion relations come from the non-trivial solutions to the wave equation (11).The condition for non-trivial solutions is det M ij = 0; for the space-like case of the CFJ background, it is reduced to an ω-polynomial equation: . Finally, the LSV appears represented by the background vector, v.It is worthy to remark that no term in the dispersion relation (16) couples the three effects together.We actually mean that non-linearity, axionic and LSV effects interfere with one another only in pairs.No term is present in which parameters of the three different effects appear grouped together in a product.However, below, in discussing the effective masses of axion-photon coupled system, it will become clear that three effects mix up to give these effective masses.
We would like to point out that, if only non-linearity is considered, the dispersion relations do not exhibit momentum dependence, so no dispersive effect shows up.By introducing the axion sector, the profile changes and dispersive effects emerges.Now, if LSV is also considered in addition to non-linearity and axionic physics, not only dispersion is enforced, but another fact should be highlighted: photonic dispersion relations show that the modulus of the LSV background vector, |v| = v, along with the axion mass endows the photon with an effective mass, m ef f .For instance, let us consider the rest frame (k = 0 with the identification ω 2 = m 2 ef f ), as well as B = B ẑ and v = v ŷ in eq. ( 16).Therefore, it is possible to show that the effective masses for the photon and the axion correspond to the roots of the equation The solutions of this quartic equation are In the limit g a → 0, the first solution (18a) is reduced to whereas, when v → 0, the effective mass is null.Consequently, in the uncoupled limit (g a → 0), the CFJ parameter (v) gives an effective mass with the correction of the nonlinearity evaluated at the magnetic background field.Considering the same limit of v → 0, the second solution (18b) is reduced to the expression where the axion mass, and the coupling constant have a fundamental role for the effective mass.Thus, the axion mass is corrected by the coupling constant and by the magnetic field.In the uncoupled limit, g a → 0, the second root (18b) yields the axion mass, i.e., Now let us return to eq. ( 16).The general solution is quite involved in view of the coefficients (12a)-(12d).For simplicity, we consider the two cases below : (a) The case of the vectors B, k and v perpendiculars among themselves : Considering this condition, the equation ( 16) is reduced to : where we denote the perpendicular frequency ω ⊥ , and B, k and v are the magnitudes of the previous vectors.The first solution is ω ⊥ = 0, and the non-trivial solutions from ( 21) are given by The analysis of the limits to establish comparisons with the results in the literature is immediate.The limits f → 0 and c 1 = 1 yield the dispersion relations of the axionic ED coupled to CFJ term in the presence of an external magnetic field.Furthermore, if we also take g a → 0, the dispersion relations are reduce to Note that in ω 2⊥ (k), occurs the characteristic effect of CFJ, where the Lorentz-breaking parameter gives a small mass for the photon.These results confirm the roots of the eq.( 17) in the rest frame (k = 0).The usual Maxwell limit reduces all the frequencies to : The refractive (perpendicular) index are defined by (b) The second case consists in considering v orthogonal to both B and k, but B parallel In this case, the equation ( 16) is where ω is now the frequency for B parallel to k.The trivial solution is ω = 0, and the others solutions are read below : The first solution (25a) is the usual photon DR due to B ×k = 0 in the a-parameter in (12a).The limits of f → 0 and c 1 = 1 also recover the DRs of the axionic ED coupled to CFJ term in the presence of the external magnetic field B. The limit g a → 0, when the axion is decoupled from the CFJ ED, the DRs are reduced to the results : This confirm the same results recovered in the case (a).The correspondent refractive (parallel) index are defined by where we must substitute the DRs (25a)-(25c).Notice that, in both the cases (a) and (b), the refractive index of the medium depends on the modulus k, so, consequently, it depends on the wavelength, as λ = 2π/|k|.
To close this section, in possess of the set of dispersion relations ( 22) and ( 25 ModMax will be shown in the next section, namely, by equations ( 33),( 38) and ( 44), respectively.

IV. THE BIREFRINGENCE PHENOMENON
Birefringence is an optical property of an anisotropic medium expressed by the dependence of the refractive index on the polarization and direction of propagation of an electromagnetic wave.Just to recall, the polarization conventionally refers to the configuration of the electric field of the wave.However, in the previous Section, we have worked out refraction indices associated to the propagation of the waves in two situations: perpendicular and parallel to the background magnetic field: k • B = 0 and k • B = |k||B|, respectively, with no reference to the polarization established by the electric field.Eqs. ( 23) and ( 26) explicitly show how the non-linearity -manifested by the external magnetic field -the axion parameters and the LSV vector interfere with one another in the expressions for the perpendicular and parallel refraction indices.And we would like to stress that we are here adopting the point of view that the phenomenon of birefringence manifests itself by the difference between the refractive indices of eqs.( 23) and ( 26) , as defined below, where we are contemplating the cases in which i = j and i = j; in general, ∆n ij = 0, and it depends on the wavelength, which characterizes dispersive propagation.Notice also that ∆n ij = ∆n ji according to the definition (27).The difference between the refraction indices in these situations is exclusively due to the choice of the wave propagation direction with respect to the external B-field.
Substituting the results from the previous section, the variation of refractive index in the case of i = j are read where we have considered that g a is very weak in comparison with the squared inverse of the magnetic background (g 2 a B ≪ 1).The birefringence effects for i = j are read below : Turning off the magnetic background (B → 0), birefringence emerges in all the results with lim B→0 ∆n ij = 0, for i = j, and lim B→0 ∆n ij = 0 for i = j.Therefore, in this limit of B → 0, the birefringence phenomenon appears only due to the axion mass and the CFJ parameter.
Only the expression (28a) does not depend on the wavelength.For the usual Maxwell ED coupled to the axion and the CFJ term, the limits of c 1 → 1, d → 0 and f → 0 yield the results below : where now the parameters g, B and the v-CFJ have a fundamental rule for the birefringence phenomenon.It is worth to highlight that if we consider a massless axion, the birefringence is null only in (30g).The limit g a → 0, for which we have a non-linear ED coupled to the CFJ term, the results (28a)-(29f) are reduced to In this case, the non-linearity plays a key role in the birefringence phenomenon.We shall discuss ahead birefringence by contemplating three non-linear electrodynamic models : Euler-Heisenberg, Born-Infeld and ModMax ED.
(a) The Euler-Heisenberg ED is described by the Lagrangian : where α = e 2 = (137) −1 = 0.00729 is the fine structure constant, and m e = 0.5 MeV is the electron mass.Taking this Lagrangian and applying the expansion presented in Section (II), the coefficients read as below: for a weak magnetic field.Substituting these coefficients in (28a)-(28c), we obtain ∆n ∆n ∆n ∆n where we have neglected terms with α 2 g 2 .Using the parameters previously defined, the solution (34a) yields the numeric value that is of the same order of the result presented by the PVLAS-FE experiment for vacuum magnetic birefringence, i.e., ∆n P V LAS−F E /B 2 = (19 ± 27) × 10 −24 T −2 [27].
The result (34b) contains the contribution of g 2 , and also of the v-parameter, whereas the variation ∆n is proportional to g 2 B 2 and it does not depend on α.Turning off the magnetic background, we obtain ∆n (EH) ii = 0.In this limit, the CFJ v-parameter contributes to the birefringence in ∆n = −∆n These conditions constrain the CFJ parameter to depend on the range of the axion mass.This is the case of a ultra-light axion (ULA),candidate to DM, with mass lower-bounded according to m 10 −22 eV (2σ) from non-linear clustering [20], to be compared with v 10 −23 − 10 −25 GeV [42].Notice that the limit B → 0 is equivalent to the cases in which the non-linearity is absent (α ≃ 0), and also when the axion coupling constant disappears(g → 0).Thus, the only parameters that remain are v and the axion mass.The mass for the free axion is not a sufficient parameter to guarantee birefringence, whereas if we have only the v-parameter, the birefringence of the CFJ ED is recovered [39].
(b) The Born-Infeld ED is governed by the Lagrangian : where β is the critical field of this model and has squared mass dimension.The usual Maxwell ED is recovered whenever β → ∞.This is the well motivated non-linear theory in which a point-like charge exhibits a finite electric field at the origin.A maximum electric field produces a finite self-energy for the electron that fixes the BI-parameter at β = 1.187 × 10 20 V/m, that in MeV scale is √ β = 16 MeV [57].As an example in accelerators, the ATLAS collaboration constraints the β-parameter in the stringent bound of √ β 100 GeV through the light-by-light scattering in Pb-Pb collisions [64].
Since our analysis is associated with the propagation effects in the linearized BI ED, we consider the BI β-parameter in the low-energy scale, √ β = 16 MeV.We understand that, upon the inclusion of radiative corrections, the renormalization group equations must exhibit a running of the β-parameter with energy and the external magnetic field, since the latter is present in the propagators and interactions vertices.This then means that the β-parameter is not a fixed universal parameter, but should run with both the energy and the external magnetic field whenever we go beyond the tree-level.
Moreover, in a BI ED coupled to the matter sector, as in the electroweak case with a non-linear realization of the hypercharge U(1)-factor, non-perturbative effects must be included such that the choice √ β 100 GeV may be justified.We would to finally point out that in the work of ref. [65], the reader may find the microscopic origin of the Born-Infeld action as described in the framework of string theory.The microscopic origin of the CFJ term, whose effects we are inspecting in presence of the Born-Infeld Lagrangian, can also backed in a string scenario [32].
The coefficients of the expansion around the magnetic background are in which both the coefficients go to zero when β → ∞.Substituting these results in (27), the solutions for the birefringence in the BI theory (when g 2 B ≪ 1) are given coupling constant, for ∆n In the limit B → 0, ∆n that is the same order of the PVLAS-FE experiment.Notice also that, in the limit v → 0 and considering B 2 /β 2 ≈ 0, the expressions ∆n where γ is a real and positive parameter of this theory.In the limit γ → 0, the ModMax Lagrangian reduces to the Maxwell ED.This non-linear ED has been well motivated in the literature due to the conformal invariance.Thus, it is the only non-linear ED that preserve the duality and the conformal symmetries in the same Lagrangian [58].
The coefficients of the expansion in the magnetic background, in this case, are ∆n ∆n ∆n ∆n ∆n (45i) The results (30a)-(30i) also are recovered in the limit γ → 0. Notice that, with γ = 0, the birefringence remains in the second solution (45b) when B → 0 : ∆n whereas ∆n (M M ) 33 = 0 in this limit.Again, ∆n is proportional to g 2 B 2 , but in the case of ModMax ED, the result constraints the axion mass and the v-CFJ parameter with the γ-ModMax.The particular result (46) is the case in which the ModMax ED is added to the CFJ term without the presence of the axion.The result (46) shows the birefringence solution that depends directly on the v-CFJ, and on the γ-ModMax parameters.When v → 0, the birefringence of ( 46) is equal to (45d).This is the simplest case of the pure ModMax ED.Turning off the magnetic background, the photon-axion system, and to compute the modified dispersion relations, the corresponding group velocities, refractive indices and birefringence which will become space-dependent as a consequence of the non-uniformity of the background.
c 2 -and d 3 -coefficients of the expansion vanish for most examples of non-linear EDs known in the literature, such as Euler-Heisenberg, Born-Infeld, ModMax, Logarithmic and some others, where the corresponding non-linear Lagrangian densities depend on the square of the G-invariant.These considerations simplify the results that we shall work out in the next Sections ahead.The usual axionic ED coupled to the CFJ-term is recovered whenever d 1 → 0, d 2 → 0 and c 1 → 1 in Back to eqs.(12a)-(12d), notice that the coefficient a takes into account non-linearity bymeans of the piece d (B × k) 2 = d [ B 2 k 2 -(B • k) 2 ], b incorporates non-linearity in the piece d B 2 .The coefficient c, on the other hand, splits into non-linearity (the d-and f -terms) and axionic (the axion mass, m, and the coupling constant, g a ) effects: c = ), we call back one of the motivations to do this work, namely, to keep track of how the three different physical scenarios we bring together in the action of eq.(1) interfere with one another, which is manifested by means of the terms coupling the parameters of the different scenarios.Keeping in mind that the coefficients c 1 , f and d express the non-linearity and that g a incorporates the axion-photon coupling and the coefficient c 1 , the presence of the denominator (1 + f B 2 ), common to all frequency solutions, in combination with the terms in m 2 , v 2 , g a B 2 , f B 2 m 2 and f B 2 m 2 v 2 , as it appears in eqs.(22a)-(22c) and (25a)-(25c), shows in an explicit way how the three different physics mix among themselves to produce tiny effects in optical quantities like phase and group velocities and refraction indices.The explicit forms of the coefficients in terms of the non-linear electrodynamic models of Euler-Heisenberg, Born-Infeld and BI) ii = 0 in the results (39a)-(39c).For the propagation effects in a low energy scale, we use √ β = 16 MeV associated with the electron's self-energy in BI ED.In this case, the solution (40a) has the numeric value |∆n (BI) 11 | B 2 ≃ 3.2 × 10 −24 T −2 , the result obtained in the ref.[22], when k ≫ m ≫ v in the PVLAS-B → 0, the variations of ∆n (BI) ij (i = j) are reduced to (36a)-(36c), that confirms the birefringence depending on the CFJ v-parameter.(c)The modified Maxwell (ModMax) ED is set by the Lagrangian