Dispersion Relations in Non-Linear Electrodynamics and the Kinematics of the Compton Effect in a Magnetic Background
M. J. Neves, Jorge B. de Oliveira, L. P. R. Ospedal, J. A. Helayël-Neto
aa r X i v : . [ h e p - t h ] J a n Dispersion Relations in Non-Linear Electrodynamics and theKinematics of the Compton Effect in a Magnetic Background
M. J. Neves, ∗ Jorge B. de Oliveira, † L. P. R. Ospedal, ‡ and J. A. Helay¨el-Neto § Departamento de F´ısica, Universidade Federal Rural do Rio de Janeiro, BR 465-07, 23890-971, Serop´edica, RJ, Brazil Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro, Brazil, CEP 22290-180 (Dated: January 22, 2021)Non-linear electrodynamic models are re-assessed in this paper to pursue an investigation of thekinematics of the Compton effect in a magnetic background. Before considering specific models, westart off by presenting a general non-linear Lagrangian built up in terms of the most general Lorentz-and gauge-invariant combinations of the electric and magnetic fields. The extended Maxwell-likeequations and the energy-momentum tensor conservation are presented and discussed in their gen-erality. We next expand the fields around a uniform and time-independent electric and magneticbackgrounds up to second order in the propagating wave, and compute dispersion relations whichaccount for the effect of the external fields. We obtain thereby the refraction index and the groupvelocity for the propagating radiation in different situations. In particular, we focus on the kine-matics of the Compton effect in presence of external magnetic fields. This yields constraints thatrelate the derivatives of the general Lagrangian with respect to the field invariants and the magneticbackground under consideration. We carry out our inspection by focusing on some specific non-linear electrodynamic effective models: Hoffmann-Infeld, Euler-Heisenberg, generalized Born-Infeldand Logarithmic.
PACS numbers: 11.15.-q; 11.10.Ef; 11.10.NxKeywords: Non-linear electrodynamics, dispersion relations in a magnetic background, Compton effect.
I. INTRODUCTION
Maxwell Electrodynamics is a highly successful the-ory to describe properties of the electromagnetic interac-tion at both the classical and the quantum length scales.Photon-photon scattering in Quantum Electrodynamics(QED) motivates the study of non-linear extensions ofthe Maxwell Electrodynamics (MED), and phenomenalike vacuum birefringence and vacuum dichroism may bea guide to also inspect the consistency of non-linear ex-tensions of MED [1–7]. Different scenarios of the latterintroduce new effects into the photon-photon scatteringprocess. To be more specific, we quote scenarios such asBorn-Infeld Electrodynamics, models with milli-chargedparticles and models of axion-like particles that interacttopologically with hidden-photons [8, 10–13, 33]. Thenon-linear Born-Infeld Electrodynamics was originally in-troduced to remove the singularity of the electric field ofpoint-like charges on their space position. Nowadays,Born-Infeld effective actions emerge in diverse scenar-ios, like superstring theory, quantum-gravitational mod-els and theories with magnetic monopoles [14–18]. Fur-thermore, a number of new phenomena in Cosmology andblack-hole physics have been reported in connection withnon-linear extensions of electrodynamic systems coupledto gravity [19–24]. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected]
The introduction of external backgrounds in field-theoretic models is an old procedure to reproduce effectsof the vacuum polarization phenomenon. If the exter-nal electromagnetic fields are strong enough (as com-pared to the so-called Schwinger critical electric andmagnetic fields), they can induce the creation of vir-tual particle-antiparticle pairs in the non-trivial quan-tum vacuum, such as, the electron-positron pairs of QEDvacuum [25, 26]. Non-linear extensions of electrodynam-ics in connection with external electromagnetic fields areable to describe effects of the vacuum structure on wavesthat propagate in empty space, through the observationof birefringence and vacuum dichroism phenomena [27–31]. A well-known case of non-linear extension that stemsfrom the vacuum polarization is the Euler-Heisenberg La-grangian [32], which is an effective photonic model withhigher powers in the electric and magnetic fields, attainedupon integration over the quantum effects of the (virtual)electron-positron pairs. We would like to point out herethe interesting paper of Reference [33], where the EHmodel is studied beyond the 1-loop level in a great dealof details. Let us also recall that, in 1961, Franken etal. opened up the field of non-linear optics with the cel-ebrated experiment in which they successfully measuredthe second-harmonic (optical) generation [34].Motivations that strongly justify the renewed interestin non-linear extensions of MED are also coming from therecent high-intensity LASERs, which are the best devicesto test both Classical and Quantum Electrodynamics inthe strong-field regime, whose (field) scales are fixed bythe critical intensity I crit ∼ . × W · cm − . The Sta-tion of Extreme Light (SEL), the Europe’s Extreme LightInfrastructure (ELI Project) and the ExaWatt Center for Rs in Non-Linear Electrodynamics and the Kinematics of the Compton Effect in a Magnetic Background PW, hopefully getting even to the Exa-W scale in acouple of years. We bring to the reader’s attention thathigh-intensity LASER experiments designed to inspectnon-linearity effects in QED are presented and discussedin [35–37].Other scenarios that strengthen the ever increasing re-search interest in modified dispersion relations are theextensions of the Standard Model (SM), such as LorentzSymmetry Violation (LSV), Lee-Wick-type (LW) modelsand proposals of physical scenarios based on a fundamen-tal length scale. LSV introduces background vectors asa consequence of space-time anisotropies that may arisefrom tensor field condensation. QED and the electroweakphenomenology impose bounds on the background vec-tor components [38–40]. LW-type electrodynamics in-troduces a new heavy boson in association with higherderivatives, keeping the theory gauge invariant despitethe presence of mass in the spin-1 sector [41, 42]. Lateron, the LW proposal was extended to include the SM[43, 44]. Noncommutative theories are one of the mostpopular approaches to introduce a fundamental lengthscale. In this context, the coordinates of the space-time are promoted to quantum observables that do notcommute among themselves [45]. The consequence ofthe noncommutativity is the introduction of a lengthscale and quantum gravity effects must be taken into ac-count [46]. The noncommutative formalism introducesthe Moyal-Weyl product of fields and depends on thenew length scale and higher-order derivatives of the fields.The main motivation to study noncommutative theoriesis that the algebra involving the spatial coordinates is notcommutative as it follows from a string theory embeddedin a magnetic field background [47, 48]. It is also impor-tant to recall that non-linear electrodynamic effects natu-rally emerge in the framework of abelian gauge models asan outcome of the non-commutativity of the space-timecoordinates [49].Non-linearity means corrections to the Maxwell elec-trodynamics that, in general, depend on the two Lorentz-and gauge-invariant quantities, F = − F µν / G = − F µν e F µν /
4. In many examples, the Lagrangian densityof the non-linear model depends exclusively on powers of F [50, 51]; in other situations, there may be dependenceon even powers of G (even powers avoid charge-parity(CP) symmetry violation) [52–55]. In our present con-tribution, we start off with a general Lagrangian that isa function of these two invariants to obtain the corre-sponding field equations and the energy-momentum ten-sor. Next, we expand the field-strength tensor aroundan electromagnetic background, initially considered non-uniform and time-dependent. We keep the terms of theexpansion in the non-linear Lagrangian up to second or-der in the propagating excitation, and the correspond- ing field equations in presence of an external electromag-netic field are written down. The expansion displays co-efficients that depend on the external background fields.The components of the energy-momentum tensor are cal-culated in the case of general space-time-dependent back-grounds. In the absence of external sources, plane wavesolutions are used to calculate the allowed frequencies,the dispersion relations and the refraction index of thephoton in a uniform magnetic background. The disper-sion relations, and consequently, the refraction indicesboth depend on the relative direction of the wave vectorwith respect to the external magnetic field.The group velocity of the electromagnetic wave is de-termined from the solutions of the frequency dictatedby the dispersion relations . The variation of the photonwavelength in the Compton effect is investigated in termsof the new dispersion relations and depends, of course, onthe magnetic background field. We apply these resultsin some particular cases of non-linear electrodynamics: Hoffmann-Infeld, generalized Born-Infeld, Logarithmelectrodynamics and the Euler-Heisenberg effective La-grangian in the regime of weak electromagnetic fields.More recently, the use of astrophysical sources hasshown to be a very fruitful procedure to constrain mod-ified dispersion relations (MDRs) for photons propa-gating in the vacuum [56]. We should also recallthat, early in 2019, the Major Atmospheric GammaImaging Cherenkov (MAGIC) telescopes identified theGRB190114C above 0.2TeV. This corresponds to pho-tons with the highest frequencies detected so far inGamma-Ray Bursts. Photons at this energy scale maywell probe the quantum structure of the vacuum, so thatnon-linear electrodynamic effects should be taken intoaccount. These observations motivate the growing of in-terest in the activity of photonic MDRs. Non-linearityin association with the presence of strong backgroundmagnetic fields yields a rich class of MDRs which may,in turn, unveil effects of new physics beyond the Stan-dard Model (SM) of Particles and Fundamental Inter-actions [57]. Moreover, let us recall that, in the SM,parity violation is verified in the weak-interaction sec-tor. However, physics beyond the SM may well be sensi-tive to parity transformations. In this context, we couldseek for evidence of parity-violating physics by inspect-ing MDRs in a special class on non-linear extensions ofelectrodynamic models, namely, the ones which explic-itly depend on the special Lorentz- and gauge-invariantquantity G which, appearing with an odd power in a La-grangian density, signals parity-symmetry breaking. Thisdirectly addresses us to the set of Planck 2018 Polar-ization Data, which could become a rich laboratory toconstrain parity-violating non-linear extensions of elec-tromagnetism, which may, in turn, be opening up trendsto see for new physics beyond the SM [58].We organize our paper as follows. In Section II, we givehighlights of the non-linear electrodynamics framework,the corresponding field equations and energy-momentumtensor in the presence of general electric and magnetic Rs in Non-Linear Electrodynamics and the Kinematics of the Compton Effect in a Magnetic Background F -invariant. In Section V, wediscuss the results by exploiting other cases of non-linearED known in the literature that depend on both the F -and G -invariants. Our Conclusions and Final Remarksare cast in Section VI.The convention for the metric we adopt is η µν =diag (+1 , − , − , − ~ = c = 1 and 4 πǫ = 1. In this unit system, theelectric and magnetic fields have squared-energy dimen-sion. The conversion of Volt/m and Tesla (T) to the nat-ural system is as follows: 1 Volt/m = 2 . × − GeV and 1 T = 6 . × − GeV , respectively. II. A QUICK GLANCE AT A GENERALNON-LINEAR ELECTRODYNAMIC MODEL
We start off the description of the non-linear electrody-namics through the most general Lagrangian, L , writtenas a function of the Lorentz- and gauge-invariant bilin-ears, F and G , defined, respectively, as follows below: F = − F µν = 12 (cid:0) E − B (cid:1) , (1a) G = − F µν e F µν = E · B , (1b)where F µν = ∂ µ A ν − ∂ ν A µ = (cid:0) − E i , − ǫ ijk B k (cid:1) is the skew-symmetric field-strength tensor, e F µν = ǫ µναβ F αβ / (cid:0) − B i , ǫ ijk E k (cid:1) its corresponding dualtensor, which satisfies the Bianchi identity ∂ µ e F µν = 0.The Euler-Lagrange field equations in presence of an ex-ternal source, J µ are written as ∂ µ (cid:18) ∂ L ∂ F F µν + ∂ L ∂ G e F µν (cid:19) = J ν . (2)The external current is conserved, i. e. , ∂ µ J µ = 0, asconsequence of (2). The conserved energy-momentumtensor of the general theory can be obtained by contract-ing F νρ with both sides of (2) and making use of theBianchi identity and the field equations themselves . Af-ter algebraic manipulations, we arrive at the well-knownform: ∂ µ Θ µρ = J ν F νρ , (3)where Θ µρ stands for the energy-momentum tensor,Θ µρ = ∂ L ∂ F F µν F ρν + η µρ (cid:18) ∂ L ∂ G G − L (cid:19) . (4)This tensor comes out symmetric and gauge invariant.In the absence of external charges and currents, J ν = 0; and, as usually, the Θ - and Θ i -components denotethe conserved energy and momentum transported by thefield. Replacing the components of F µν , the energy andmomentum densities read as below :Θ = ∂ L ∂ F E + ∂ L ∂ G G − L , (5a)Θ i = ∂ L ∂ F ( E × B ) i . (5b)We decompose the tensor F µν according to F µν = f µν + F Bµν , in which f µν = ∂ µ a ν − ∂ ν a µ = (cid:0) − e i , − ǫ ijk b k (cid:1) is the electromagnetic field-strength tensor of the prop-agating excitation, whereas F µνB = ∂ µ A νB − ∂ ν A µB = (cid:0) − E i , − ǫ ijk B k (cid:1) corresponds to the field-strength asso-ciated with the electric and magnetic background fields.These fields, in general, depend on the space-time coordi-nates. Therefore, by expanding the Lagrangian L ( F , G )around the background fields and keeping terms up tothe second-order in the propagating field, we get L (2) = − c f µν − c f µν e f µν − f µν G µνB + 18 Q µνκλB f µν f κλ − J µ a µ , (6)where the background tensors are defined by G µνB = c F µνB + c e F µνB ,Q µνκλB = d F µνB F κλB + d e F µνB e F κλB ++ d F µνB e F κλB + d e F µνB F κλB . (7)By construction, G µνB = − G νµB and the tensor Q µνκλB is antisymmetric under the exchange µ ↔ ν or κ ↔ λ ,and symmetric in µν ↔ κλ . The coefficients c , c , d , d and d are evaluated at the background fields E , B : c = ∂ L ∂ F (cid:12)(cid:12)(cid:12)(cid:12) E , B , c = ∂ L ∂ G (cid:12)(cid:12)(cid:12)(cid:12) E , B ,d = ∂ L ∂ F (cid:12)(cid:12)(cid:12)(cid:12) E , B , d = ∂ L ∂ G (cid:12)(cid:12)(cid:12)(cid:12) E , B , d = ∂ L ∂ F ∂ G (cid:12)(cid:12)(cid:12)(cid:12) E , B , (8)that, in a general situation, are space-time-dependent.Using the second-order expanded Lagrangian (6), wegive below the energy-momentum, up to the second orderin the photon field strength, f µν , for a general space-time-dependent background, F Bµν . Let us consider again theLagrangian (6) whose corresponding field equations areas follows: ∂ µ G µν = − ∂ µ G µνB + J ν , (9)where G µν is defined by G µν = c f µν + c e f µν − Q µνκλB f κλ . (10)The dual tensor e f µν satisfies the Bianchi identity : ∂ µ e f µν = 0. We contract the field equations with f να , Rs in Non-Linear Electrodynamics and the Kinematics of the Compton Effect in a Magnetic Background f να , we obtain thecontinuity equation ∂ µ Θ µαph = h α , (11)where the energy-momentum tensor of the photon fieldis given byΘ µαph = c f µν f αν − Q µνκλB f κλ f αν + η µα (cid:18) c f ρσ − Q ρσωτB f ρσ f ωτ (cid:19) , (12)and the vector h α is h α = J ν f να − ( ∂ µ G Bµν ) f να + 14 ( ∂ α c ) f µν + 14 ( ∂ α c ) e f µν f µν − (cid:16) ∂ α Q µνκλB (cid:17) f µν f κλ . (13)Notice that the topological term, the one in c , is can-celled in the expression for (12); as expected, it does notcontribute to the stress-tensor by virtue of its topolog-ical nature. In the general case, the background fieldsare non-homogeneous over space and time-dependent.Whenever J ν = 0, the term of h α is not zero and, asconsequence, the components of the energy-momentumtensor are not conserved if the background fields are nei-ther uniform nor constant in time. If we consider thebackground fields to be constant and uniform, h α is van-ishing and, in this case, the energy-momentum tensor(12) satisfies a continuity equation with the conservedcomponents given in what follows below:Θ ph = 12 c (cid:0) e + b (cid:1) + 12 d ( e · E ) + 12 d ( e · B ) − d ( b · B ) − d ( b · E ) + d ( e · E ) ( e · B )+ d ( b · E ) ( b · B ) , (14a)Θ iph = c ( e × b ) i − d ( e · E ) ( E × b ) i − d ( b · B ) ( E × b ) i − d ( e · B ) ( B × b ) i + d ( b · E ) ( B × b ) i − d ( e · B ) ( E × b ) i − d ( e · E ) ( B × b ) i − d ( b · B ) ( B × b ) i + d ( b · E ) ( E × b ) i , (14b)where all the coefficients depend on the external fields, E and B . We recover the Maxwell limit by turning offthe background fields and by taking c = 1.The energy density can be written asΘ ph = 12 K ij e i e j + 12 Λ ij b i b j , (15)where K ij and Λ ij are, respectively, defined by K ij = c δ ij + d E i E j + d B i B j ++ d ( E i B j + E j B i ) , Λ ij = c δ ij − d B i B j − d E i E j ++ d ( E i B j + E j B i ) . (16) The energy density (15) is positive-definite whenever theeigenvalues of the symmetric matrices K ij and Λ ij arenon-negative. Let us contemplate the case d = 0 andassume a purely magnetic background, i. e. , E i = 0.These conditions shall actually be the ones we are goingto work with in the forthcoming Sections. Therefore,with these assumptions, the eigenvalues of K ij are c , c + ( d − | d | ) B / c + ( d + | d | ) B /
2, and theeigenvalues of Λ ij are c , c and c − d B , respectively.If d < d >
0, to ensure positive eigenvalues, thefollowing conditions should be fulfilled: c > , c − d B > c + d B > . (17)To elaborate more on the positivity of the energy den-sity, let us recall that a general non-linear Lagrangiancan be expanded as an asymptotic series in powers of F and G : L = a ij F i G j , i, j = 0 , , , ... . (18)MED corresponds to i = 1 , j = 0. However, exceptfor a = 1, all the coefficients of the expansion aboveare small, for they describe tiny non-linear effects, evenfor strong external fields, such as, for example, magneticfields in the neighborhood of magnetized astrophysicalobjects. So, in the energy density (14a) and in (16), c = 1 + δ , with δ ≪
1, since the latter stems fromthe coefficients a ij that extend the Maxwellian version.We are then arguing that, in (14a) and (16), δ and thecoefficients d i correspond all to tiny corrections, so thatit is expected that the eigenvalues above are, in a widerange of situations, but not generally, are all positive. Inthese cases, the energy density (14a) is consequently non-negative, once the contribution given by the Maxwellianterm, e + b , dominates over the other terms. Nev-ertheless, we shall consider, further on, specific cases ofnon-linear models and the argument above may not workif the external fields become stronger than some criticalvalue. In all situations, we are going to point out criticalvalues of the external magnetic fields above which theaverage energy density of plane waves become negative,which is, to our sense, void of physical meaning. There-fore, for all models we shall discuss, we will be boundto consider external fields below the critical values weare going to derive, so as to undertake that the averageenergy density of the radiation be positive.After the previous considerations, in the incomingSections IV and V, we are going to apply these re-sults and conditions to the particular cases of Hoffman-Infeld, the generalized Born-Infeld, Logarithm and Euler-Heisenberg electrodynamics. III. PHOTON DISPERSION RELATIONS ANDTHE KINEMATICS OF THE COMPTON EFFECT
In this Section, we consider the expansion, up to sec-ond order in f µν , to compute the dispersion relations Rs in Non-Linear Electrodynamics and the Kinematics of the Compton Effect in a Magnetic Background e , b ), and the background given by( E , B ), the field equations (9) read as follows below: ∇ · e + χ · ∇ F + λ · ∇ G = −∇ · E + ρc , (19a) ∇ × e + ∂ t b = , ∇ · b = 0 , (19b) ∇ × b + ξ × ∇ F + µ × ∇ G = ∂ t e ++ χ ∂ t F + λ ∂ t G − ∇ × B + J c , (19c)where we have defined the scalar functions F and G , andthe background vectors χ , λ , ξ and µ as F = E · e − B · b , G = B · e + E · b , χ = d c E + d c B , λ = d c B + d c E , ξ = − d c B + d c E , µ = d c E − d c B . (20)Notice that F
7→ − G , G F , χ ξ and λ µ ,whenever we perform the replacements E → − B and B → E . Since the Bianchi identity remains valid innon-linear electrodynamics, the divergent of b and therotational of e keep like in Maxwell electrodynamics.The particular case of a Lagrangian that dependsonly on the invariant F in the presence of the back-ground fields E and B , we have c = 0, d = 0 and c = d = d = 0. The simplest case is Maxwell elec-trodynamics, where the Lagrangian is given by F , thefirst coefficient is c = 1, and the other coefficients ofthe expansion are all vanishing. We shall consider in thispaper the case of a purely magnetic background, i. e. , E = 0. Whenever the non-linear model also exhibits de-pendence on G , this dependence must be quadratic (or aneven power) in G to insure the charge-parity symmetry.This fact happens in non-linear electrodynamics such asthe generalized Born-Infeld, Logarithm and ArcSinh the-ories. If there is no electric background, we obtain c = 0and the CP-symmetry is recovered. We work with a uni-form external magnetic field, B , and, as consequence, thecoefficients are also uniform and constant in time. Otherimportant fact is that whenever the electric backgroundfield is not present, d = 0 for all the examples of non-linear electrodynamics in the literature. Thereby, usingthat d = 0, the equations (19b) and (19c) with no clas-sical sources read as below : ∇ · e + d c B · ∇ ( B · e ) = 0 , (21a) ∇ × e + ∂ t b = , ∇ · b = 0 , (21b) ∇ × b + d c B × ∇ ( B · b ) = ∂ t e ++ d c B ∂ t ( B · e ) − ∇ × B . (21c) The usual Maxwell equations are obtained for d = d =0 and c = 1, which is equivalent to taking | B | → e ( x , t ) = e e i ( k · x − ωt ) and b ( x , t ) = b e i ( k · x − ωt ) in (21a),(21b) and (21c), the relation between the frequency, ω ,and the wave vector, k , can be written in a matrix form: M ij e j = 0 , (22)where e j ( j = 1 , ,
3) are the components of the ampli-tude of the electric field, e . The matrix elements M ij take the form M ij = α δ ij + u i v j + w i t j , (23)where the coefficients are defined by α = ω − k , u = d c B × k , v = B × k , w = d c ω B − d c ( B · k ) k , t = B . (24)The matrix equation (22) has non-trivial solutions onlyif the M -matrix is singular. It can be cast in the formdet M = α [( α + u · v ) ( α + w · t ) − ( u · t ) ( v · w )] , (25)and the condition det M = 0 leads to the usual photondispersion relation ω = | k | as one of the solutions. Thisis so by virtue of gauge invariance, which, in a particlescenario, corresponds to the presence of the genuine (zeromass) photon. Along with this possibility, there appearother solution as the zeroes of the polynomial equationthat follows: P ω + Q ω + R = 0 , (26)where P = 1 + d c B ,Q = − k + d c ( B × k ) − d c h B k + ( B · k ) i + d d c B ( B × k ) ,R = k − d c k ( B × k ) + d c k ( B · k ) − d d c ( B · k ) ( B × k ) . (27)The roots of (26) are ω ( ± )1 = ± ω ( k ) and ω ( ± )2 = ± ω ( k ), whose frequencies are shown below: ω ( k ) = | k | r − d c ( B × ˆ k ) , (28a) ω ( k ) = | k | s − d ( B × ˆ k ) c + d B . (28b) Rs in Non-Linear Electrodynamics and the Kinematics of the Compton Effect in a Magnetic Background d → d →
0, or, equivalently, whenever | B | →
0. Notice that, if the non-linear theory only depends onthe F -invariant, d = 0 and the second solution recoversthe usual dispersion relation. The frequencies (28a) and(28b) are real if c > d ( B × ˆ k ) and c + d ( B · ˆ k ) > ω i / | k | ( i =1 ,
2) : n − = r − d c ( B × ˆ k ) , (29a) n − = s − d ( B × ˆ k ) c + d B . (29b)From the De Broglie duality correspondence, the energy-momentum relations for the propagating excitations readas below: E = p (cid:20) − d c ( B × ˆ p ) (cid:21) , (30a) E = p (cid:20) − d ( B × ˆ p ) c + d B (cid:21) . (30b)The group velocity associated with the previous fre-quencies can be read off from the equations cast in whatfollows: v g | ω = ω = c ˆ k + d B × ( B × ˆ k ) c q − d c ( B × ˆ k ) , (31a) v g | ω = ω = ˆ k c + d B ( B · ˆ k )( c + d B ) q − d ( B × ˆ k ) c + d B . (31b) The group velocity vectors have components in the di-rections of ˆ k and ˆ B . Both results go to v g = ˆ k ω/ | k | , inthe limit | B | →
0, or if we consider the Maxwellian limit.If the magnetic background, B , is perpendicular to thedirection of propagation, ˆ k , the solutions also reduce to v g = (1 − d B /c ) / ˆ k and v g = (1 + d B /c ) − / ˆ k ,respectively.The Compton effect is a scattering process in whichthe dispersion relations (30a) and (30b) can be appliedto study the increasing of the photon wavelength afterbeing scattered by the electron, taken as the target, inpresence of a magnetic background. The photon initialstate has wavelength λ = 1 / | p | , with energy E , where therelation between E and the momentum, p , must now fol-low from (30a) or (30b). After the collision, the photontrajectory is deviated by an angle θ c , with wavelength λ ′ = 1 / | p ′ | and energy E ′ . From the energy and lin-ear momentum conservation, the variation of the photonwavelength, after the collision process, turns out to be λ ′ i − λ = 2 λ e sin (cid:18) θ c (cid:19) + a i " λ ′ i − λ + λ e λ ′ i − λ ) λ ′ i λ , (32)where λ e = m − e = 2 (MeV) − is the Compton wave-length of the electron, a i ( i = 1 ,
2) means a := | B × ˆ p | d /c and a := d | B × ˆ p | / (cid:0) c + d B (cid:1) forboth the cases of (30a) and (30b), respectively, and λ ′ i arethe wavelengths for both cases i = 1 , λ = 1 . × (cid:0) − (cid:1) MeV − . Thecorresponding solutions to (32) yielding the final wave-length of the photon are given by λ ′ ( ± ) i = λ + 2 λ e sin ( θ c / − a i ( λ + λ e )2 − a i − a i λ e /λ ± q(cid:0) λ + 2 λ e sin ( θ c / − a i λ − a i λ e (cid:1) + a i λ e (2 λ − a i λ − a i λ e )2 − a i − a i λ e /λ . (33)If we assume λ ≫ λ e in (33) , the real and positive so-lutions lead us to the two wavelength variations below:∆ λ ≃ λ e sin (cid:18) θ c (cid:19) (cid:20) − d c | B × ˆ p | (cid:21) − , (34a)∆ λ ≃ λ e sin (cid:18) θ c (cid:19) (cid:20) − d | B × ˆ p | c + d B (cid:21) − , (34b)that are positive if c > d | B × ˆ p | and c + d ( B · ˆ p ) >
0. Whenever d , d →
0, the standard variation ofthe photon wavelength in the Compton effect is recov-ered. The non-linear contribution depends on the exter-nal magnetic field, B , and the parameters c , d and d .These parameters, in turn, also depend on the external magnetic field and the specific dependence is governedby the non-linear electrodynamics under consideration. IV. EXAMPLE OF AN F -DEPENDENTELECTRODYNAMIC MODEL The Hoffmann-Infeld (HI) model is an example of non-linear electrodynamics with interesting application in thestudy of special black-hole solutions [24]. The Lagrangianis given by L HI ( F ) = β − η ( F ) − ln η ( F ) ] , (35) Rs in Non-Linear Electrodynamics and the Kinematics of the Compton Effect in a Magnetic Background θ n H I | B | = | B | = | B | = FIG. 1: The vacuum refraction index of the HI ED appearsas a function of the θ -angle between B and the direction ofˆ k . We choose β = 10 . , for the values | B | = 1 . , | B | = 4 . and | B | = 6 . . where η ( F ) is defined by η ( F ) = 4 F β − β p β − F . (36)The parameter β is introduced to guarantee a finite elec-trostatic field configuration in the case the particle-likecharges. Maxwell Electrodynamics is recovered when β → ∞ . In this Section, since the model depends exclu-sively on the invariant F , the coefficient vanishes, d = 0.Thereby, the non-trivial dispersion relation in a magneticbackground corresponds to (30a), which depends on d and c . The non-trivial coefficients, c and d , are givenby c HI (cid:12)(cid:12) E =0 , B = β B − β + 2 B + β p β + 4 B p β + 4 B , (37a) d HI (cid:12)(cid:12) E =0 , B = β B B − β + 2 β p β + 4 B ( β + 4 B ) / − β B β − p β + 4 B ( β + 4 B ) / . (37b)The corresponding refraction index as function of theangle θ between the magnetic background and the di-rection of propagation, ˆ k , is shown in figure (1). Wechoose the following values for the magnetic background: | B | = 1 . (black line), | B | = 4 . (blue line)and | B | = 8 . (red line). For a strong magneticbackground, i. e. , | B | ≫ β , the refraction index of theHI model is n HI = | sec θ | ; this refraction index vanishesif B is perpendicular to the direction ˆ k , and n HI = 1 if B is parallel to ˆ k .The components of the group velocity (31b) for theHI ED as functions of θ -angle are plotted in the figure(2). We choose β = 50 . and | B | = 10 . inthis case. The black line stands for the component inthe direction of ˆ k , whereas the red line represents the - - θ v g H I | B | = β = FIG. 2: The components of the group velocity as functions of θ in the HI ED. The black line is the velocity component in thedirection of ˆ k , while the red line stands for the ˆ B component. component in the direction of magnetic field ˆ B . Thiscomponent is negative for the range π/ ≤ θ ≤ π . When | B | ≫ β , the group velocity has the behaviour v gHI ≃ − β | B | | sec θ | ˆ k + sgn(cos θ ) ˆ B . (38)The next analysis refers to the energy density of theHI ED in the presence of the background, B . We use theresult (14a) with the coefficients (37a) and (37b). Un-der the conditions (17), the energy density in the HI EDis positive if | B | < . β , or | B | > √ β . To illus-trate the energy density (14a), we consider the case inwhich the plane wave for the electric and magnetic fields e ( x , t ) = e e i ( k · x − ωt ) and b ( x , t ) = b e i ( k · x − ωt ) , re-spectively, propagate in a medium with an uniform andconstant magnetic background. We obtain thereby thetime average of the energy density of the HI ED per unitof the squared electric field : h Θ ph i HI e = 14 c HI (cid:20) k ω − d HI c HI k ω (cid:16) ˆ e · (ˆ k × B ) (cid:17) (cid:21) . (39)Notice that this result depends on the frequency solutionsof ω , that in the HI case just (28a) depend on the externalmagnetic field. We consider the vectors ˆ k , ˆ e and B perpendicular to each other in this analysis, and we alsochoose β = 50MeV . Under these conditions, the result(39) is plotted as function of the magnetic background inthe figure (3). In this case, the energy density is positivefor all values of the B magnitude, and goes to zero if | B | → ∞ . The limit | B | → V. SOME F - AND G -DEPENDENTELECTRODYNAMIC MODELS Many examples of non-linear electrodynamics with de-pendence on both F and G are discussed in the litera- Rs in Non-Linear Electrodynamics and the Kinematics of the Compton Effect in a Magnetic Background ( MeV ) Θ p h ω H I e FIG. 3: The average (in time) of the energy density per unit ofthe squared amplitude of the electric field versus the externalmagnetic field in the HI ED. We choose β = 50 MeV , andthe vectors ˆ k , ˆ e and B are perpendicular to one another. ture. In these cases, the coefficient d = 0 and the twofrequency solutions, (28a) and (28b), must be considered.The models, in general, depend on G (simplest case) oran even power of G , so as to ensure CP-invariance.Let us start off by contemplating the well-known caseof the Euler-Heisenberg (EH) electrodynamics, describedby the effective Lagrangian L EH ( F , G ) = F − π Z ∞ dss e − m s ×× " ( es ) G ℜ cosh (cid:0) es √−F + i G (cid:1) ℑ cosh (cid:0) es √−F + i G (cid:1) + 23 ( es ) F − , (40)where ℜ and ℑ stand for the real and imaginary parts,respectively, and m = 0 . L EH ( F , G ) ≃ F + 2 α m (cid:0) F + 7 G (cid:1) , (41)in which α = e = (137) − = 0 . c , d and d of the expansion corresponding to the truncation givenby (41) : c EH (cid:12)(cid:12) E =0 , B = 1 − α B m ,d EH (cid:12)(cid:12) E =0 , B = 16 α m ,d EH (cid:12)(cid:12) E =0 , B = 28 α m . (42)Using the results (28a) and (28b) with the coefficients(42), the two solutions for the frequencies are ω EH ( k ) ≃ | k | (cid:20) − α m ( B × ˆ k ) (cid:21) ,ω EH ( k ) ≃ | k | (cid:20) − α m ( B × ˆ k ) (cid:21) , (43) and, then, the corresponding vacuum refraction indexfor the EH effective model, in the approximation we areworking, turn out given by n EH ≃ α m ( B × ˆ k ) , (44a) n EH ≃ α m ( B × ˆ k ) . (44b)It is worthy to highlight that these results are in agree-ment with Reference [1] after suitable changes in the unitsystem.The second case of a ED non-linear is the generalizedBorn-Infeld (BI) Lagrangian [54], L BI ( F , G ) = β (cid:20) − (cid:18) − F β − G β (cid:19) p (cid:21) , (45)where β is a scale parameter with dimension of squaredenergy (in natural units), and p is a real parameter thatsatisfies 0 < p <
1. The usual Born-Infeld theory is ob-tained for p = 1 /
2. For β ≫ ( | E | , | B | ), the Lagrangian(45) leads to L BI ≃ p F + 1 β (cid:20) p (1 − p ) F + p G (cid:21) . (46)Here, we recall that Maxwell electrodynamics is recov-ered in the limit β → ∞ , and p = 1 / B = 0, the correspond-ing field equation in presence of charges is ∇ · D = ρ , (47)where ρ is the charge density, and D is defined by D = E (1 − E /β ) − p . (48)In the point-like particle case, ρ ( r ) = e δ ( r ), equation(47) yields D = ˆ r Q/r with Q = e/ π . So, the solutionsof the electric field in (47) are difficult to obtain due tothe polynomial equation with degree (1 − p ) − . The case p = 3 / E = √ β Q ˆ r q Q + p Q + 9 β r . (49)The magnitude of the electrostatic field goes to zerowhenever r → ∞ . In the limit r →
0, the electric field isfinite at the origin, i. e. , E ( r = 0) = p / β sgn( Q ), inwhich sgn( Q ) denotes the signal function. If the chargeis positive, the electric field does not blow up at thecharge position and the maximum value it reaches is E | max = p / β . Otherwise, if the charge is negative,the electric field has a minimum at E | min = − p / β . Rs in Non-Linear Electrodynamics and the Kinematics of the Compton Effect in a Magnetic Background V BI ( r ) = 3 Q / s β Q ( √ π
15 Γ(9 /
4) cos( π/ / (cid:20) sin φ cos φ cos ( φ / (cid:21) / − (cid:20) sin φ cos ( φ / (cid:21) / ×× F (cid:18) , , , tan φ (cid:19)(cid:27) , (50)in which φ = tan − (3 β r /Q ). In addition, this poten-tial reduces to the Maxwellian case for β → ∞ .The derivations from (8) in the Lagrangian (45) yieldthe coefficients c , d and d for the generalized BI the-ory: c BI (cid:12)(cid:12) E =0 , B = 2 p (1 + B /β ) − p ,d BI (cid:12)(cid:12) E =0 , B = p (1 − p ) β (1 + B /β ) − p ,d BI (cid:12)(cid:12) E =0 , B = 2 pβ (1 + B /β ) − p . (51)The dispersion relations in this case are cast below: ω BI ( k ) = | k | s − (1 − p ) ( B × ˆ k ) β , (52a) ω BI ( k ) = | k | s − ( B × ˆ k ) B + β , (52b)that are real if the conditions β + ( B · ˆ k ) > β > (1 − p )( B × ˆ k ) are respectively fulfilled. No-tice that the second frequency is independent of the p -parameter of the generalized BI theory. Whenever themagnetic field is strong, that is | B | ≫ β , the secondfrequency is ω BI ≃ | k || cos θ | , and the refraction indexis n BI ≃ | sec θ | , where θ is the angle between B andthe direction of ˆ k . Thereby, when the medium is un-der a strong magnetic background, the refraction indexchanges with the angle θ . This refraction index as func-tion of angle θ is plotted in the figure (4). The three casesare for | B | = 1 . (black line), | B | = 5 . (blueline) and | B | = 10 MeV (red line), respectively, with β = 5 . . Notice that, in the limit | B | →
0, therefraction index approaches the value one.The correction to the Compton effect (32) for the BItheory is shown in figure (5). We plot the curves for themagnetic field values | B | = 1 . (black line), | B | =5 . (blue line) and | B | = 10 . (red line). Thevariation of the photon wavelength in the Compton effectincreases with the magnetic field magnitude.For the generalized BI ED, the energy density is pos-itive if the magnetic background satisfies the condition | B | < √ β/ √ − p . Using the plane wave for e and b ,the time average of the energy density in generalized BI θ n B I | B | = M(cid:0)(cid:1) | B | = (cid:2)(cid:3)(cid:4) | B | = θ n B I | B | = (cid:12)(cid:13)(cid:14) | B | = | B | = (cid:20)(cid:21)(cid:22) FIG. 4: The refraction index of the Born-Infeld theory asfunction of angle θ (between B and ˆ k ), when β = 5 . .The top panel is associated with (52a) for the values of | B | =1 . , | B | = 4 . and | B | = 8 . . The bottompanel sets (52b) for | B | = 1 . , | B | = 5 . and | B | = 10 . . theory is given by h Θ ph i BI e = 14 c BI (cid:20) k ω − d BI c BI k ω (cid:16) ˆ e · (ˆ k × B ) (cid:17) + d BI c BI (ˆ e · B ) − (cid:18) d BI c BI (cid:19) k ω (cid:16) B · ˆ k (cid:17) ( B · ˆ e ) . (53)In this case, both frequencies (52a) and (52b) dependon the external magnetic field, and just ω BI depends onthe p -parameter. Therefore, the time average of energydensity associated with the frequency ω BI changes withthe values of the p -parameter. We plot the time averageof energy density by unit of e associated with ω BI for p = 0 . p = 0 .
75 (gen-eralized BI ED in the bottom panel), when β = 50 MeV .In both plots, the time average of the the energy den-sity becomes negative for | B | > | B n | , where | B n | =108 .
41 MeV is a magnetic critical field for the case ofthe usual BI ED (top panel), and | B n | = 148 .
45 MeV isa magnetic critical field for the case of the generalized BIED (bottom panel).Another interesting model is Logarithm ED [53]. The Rs in Non-Linear Electrodynamics and the Kinematics of the Compton Effect in a Magnetic Background θ c Δ λ B I | B | = | B | = | B | = FIG. 5: The Compton effect in the Born-Infeld theory as-sociated with the second frequency (52b). The variation ofthe photon wavelength as function of the Compton angle, θ c , for the values of | B | = 1 . , | B | = 5 . and | B | = 10 . . We choose here β = 5 . . corresponding Lagrangian is given by L ln ( F , G ) = − β ln (cid:20) − F β − G β (cid:21) . (54)As in the previous case, Maxwell electrodynamics is re-covered for β → ∞ . In this model, we obtain the follow-ing coefficients c , d and d : c ln (cid:12)(cid:12) E =0 , B = 2 β β + B ,d ln (cid:12)(cid:12) E =0 , B = 4 β (2 β + B ) ,d ln (cid:12)(cid:12) E =0 , B = 22 β + B . (55)Using these results, the combination of the coefficients c ln , d ln and d ln in (28b) yields the same result (52a)for the second frequency in logarithm theory. The firstsolution (28a) in this case is ω ln ( k ) = | k | s − | B × ˆ k | β + B . (56)For | B | → ∞ , the correspondent refraction index is n ln = p sec(2 θ ). The energy density is positive-definite if themagnetic background satisfies the condition | B | < √ β . VI. CONCLUSIONS AND FINAL REMARKS
Our contribution sets out to pursue a study of gen-eral non-linear models of electrodynamics in presence ofexternal electric and magnetic fields. The energy and lin-ear momentum of the electromagnetic field fulfill a con-tinuity equation if the background fields are uniform and - - - - - - ( MeV ) Θ p h ω B I e - - -
101 B ( MeV ) Θ p h ω B I e FIG. 6: The time average of energy density by unit of e for the usual BI ED p = 0 . p = 0 .
75 (bottom panel) as function of thebackground magnitude. We choose β = 50 MeV in this plot. constant. We have consider only the case of (uniformand constant) magnetic backgrounds for the analysis ofthe non-linear models we have picked out. The energydensity is positive if the (external) field-dependent coef-ficients satisfy the conditions (17). Otherwise, the energydensity may assume negative values for sufficiently strongexternal fields, as our calculations point out. Plane wavesolutions are considered to describe the non-linear pho-ton. Two frequency solutions come out as it happens inthe case of the usual photon; the other two frequenciesexhibit a dependence on the uniform background mag-netic field and the angle between the latter and the di-rection of the wave propagation. As a consequence, therefraction index also changes with the direction of thefield B relative to the direction of propagation of thewave, ˆ k . We have also obtained the contribution of theuniform magnetic background to the kinematics of theCompton effect by employing the modified dispersion re-lations. We have applied these results to four examplesof non-linear electrodynamics : Hoffmann-Infeld, gener-alized Born-Infeld, Logarithm and the Euler-Heisenbergeffective Lagrangian. In all these cases, the refraction in-dex depends on the angle, θ , between B and ˆ k , and onthe magnitude of B as well. For the generalized Born-Infeld and the Logarithm electrodynamics, the refrac- Rs in Non-Linear Electrodynamics and the Kinematics of the Compton Effect in a Magnetic Background n BI ≃ | sec θ | = | ˆ B · ˆ k | − and n ln ≃ p sec(2 θ ), respectively, if the background magneticis extremely strong.Further research has been initiated in a scenario in-volving models of scalar axions in connection with par-ticular non-linear models. The purpose of this particularinvestigation is to try to understand how the present phe-nomenological astrophysical data known for the axionsmay dictate restrictions on the form of non-linear La-grangian densities. On the other hand, we are also par-ticularly interested in contemplating CP-breaking non-linear models in connection with the Planck 2018 Polar-ization Data to eventually use the latter to derive con-straints on CP- violating non-linearities. We intend toreport on the progress of these two lines of investigationin further papers. Acknowledgments
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