Euler-Heisenberg waves propagating in a magnetic background

We derive the Euler-Heisenberg solutions that describe electromagnetic waves propagating through very intense uniform magnetic or electric backgrounds. We first explore the case of a magnetic background: as a result of the interaction between the wave and the background the wave presents an extra longitudinal electric field component and the phase velocity is determined by the intensity of the external magnetic field; the phase velocity can be slowed down on request and even would be frozen if the required strong magnetic background was reached. We as well considered the situation when the background is in movement, modeling then a magnetized flowing medium, and we determined how this motion affects the speed of propagation of the electromagnetic wave. In this case the phase velocity of the wave depends on both, the magnetic background and the velocity of the medium. In case the fluid moves opposite to the wave propagation, the latter could be frozen with a less intense magnetic background. The analogous study is done when the wave propagates through an intense uniform electric field.


Introduction
In the presence of intense electromagnetic fields quantum electrodynamics (QED) predicts that vacuum has properties of a material medium as a consequence of the electromagnetic field self-interactions. These effects become significant when the electromagnetic field strengths approach E cr ≈ m 2 e c 3 /(e ) ≈ 10 18 Volt/m or B cr ≈ 10 9 Tesla; B cr represents the field at which the cyclotron energy equals m e c 2 , and defines the field scale at which the impact of the external field on quantum processes becomes significant.
The EulerHeisenberg (EH) Lagrangian was derived from QED principles by W. Heisenberg and H. Euler in 1936 [1], for a nice discussion on the history of the Euler-Heisenberg approach see [2] and a pedagogical review can be found in [3]. By treating the vacuum as a medium, EH effective action predicts rates of nonlinear light interaction processes since it takes into account vacuum polarization to one loop, and is valid for electromagnetic fields that change slowly compared to the inverse electron mass. The EH LagrangianL EH (F, G) depends in nonlinear way of the two Lorentz and gauge invariants of the Faraday tensor F µλ , F = F µλ F µλ = 2(B 2 − E 2 ) and G = −F * µλ F µλ = 4 B · E, e −m 2 s (es) 2 Re[cosh(es 2(F + iG))] Im[cosh(es 2(F + iG))] − 2 3 (es) 2 F − 1 ds s 3 . (1) From this Lagrangian new nonlinear interactions can be derived, which do not occur in the tree level Maxwell action; among them are light-light interaction and pair production from vacuum excited by an electric field. We shall derive solutions in the weak field limit of Lagrangian (1), where b is the parameter of the EH theory that in terms of the fine structure constant, α, is b = 2α 2 45m 4 e ; ( 3) or in terms of the critical fields, is of the order b ∼ α/B 2 cr . There is a formal similarity to the work of Born and Infeld (1934) [4] who obtained similar nonlinear corrections to Maxwell theory but from a classical perspective. See [5] for a generalized Born-Infeld electrodynamics. The linear electromagnetic Maxwell theory is recovered if b = 0, L Maxwell (F ) = −F/4.
The situation of strong magnetic fields has astrophysical interest as well, neutron stars can possess magnetic fields in the range of 10 6 − 10 9 Tesla, then processes like photon splitting and pair conversion are expected to occur in their vicinity [6].
Efforts are currently in progress for measuring some of these nonlinear effects, we mention just a few of them: Light by light interactions can be studied using heavy-ion collisions; the electromagnetic (EM) field strengths produced, for example by a Pb nucleus would be up to 10 25 Vm −1 , those intense EM fields can be treated as a beam of quasi-real photons, and it has been measured light by light scattering in P b + P b collisions at the Large Hadron Collider [7]. Other experimental evidences include the measurement of photon splitting in strong magnetic fields [8]; the search for vacuum polarization with laser beams crossing magnetic fields or the detection of vacuum birefringence with intense laser pulses [9]. There is also the detection of QED vacuum nonlinearities using waveguides [10]. Vacuum pair production, known as the Sauter-Schwinger effect [11], was a prediction in the 1936 EH paper however the necessary electric field strengths are not reached yet, since the corresponding critical laser intensity is about I cr = 4.3 × 10 29 W/cm 2 [12].
The phase velocity of an electromagnetic wave traveling through intense EM fields will be altered due to vacuum polarization. In order to attempt the detection of nonlinear electromagnetic effects it is crucial to determine the velocity of propagation of the electromagnetic wave in the intense EM background, and this is our aim in this paper. It is organized as follows: In the next section we derive the solution to the EH equations, derived from Lagrangian (2), that represents an electromagnetic wave propagating through an intense magnetic uniform background. In Section 3 by means of the effective metric approach we determine the phase velocity of the propagating wave and explore its limits when the magnetic background is near the critical value B cr . In Section 4, by performing a Lorentz boost, we consider the background in movement and we investigate the changes that this situation introduces in the phase velocity of the propagation. In Section 6 we present the EH solution for a wave propagating through an intense electric field, and perform an analogous study of the phase velocity. Finally, conclusions are presented in the last section.

The Euler-Heisenberg propagating wave
In this section we derive the solution to the EH equations that represents an electromagnetic wave propagating through an intense uniform magnetic field. The nonlinear electrodynamics (NLED) field equations derived from Lagrangian (2), are where L X denotes the derivative of L with respect to the invariant X, dL/dX. We shall consider an electromagnetic wave propagating in theẑ direction, and a uniform magnetic field given by B = B xx + B yŷ + B zẑ (see Fig. 1). The solution will be a function of ξ = (z − βt), where β is the phase velocity of the propagation in theẑ direction; considering that the electric and magnetic fields of the propagating wave are, respectively, E w (ξ) = −E(ξ)x + κE(ξ)ẑ, and B w (ξ) = −E(ξ)ŷ, the corresponding Faraday tensor is of the form, where the phase velocity β and the constant κ should be determined by solving the nonlinear electrodynamics (NLED) field Eqs. (4). Note that κ is a longitudinal electric component that arises due to the interaction between the wave and the background; this component lifts the wavefront from the plane defined by ( E, B), and the ray direction differs from the propagation direction, as if the wave were in an anisotropic medium. In this case κ turns out to be κ depends on the EH parameter, b, on the background field, B 2 = B 2 x +B 2 y +B 2 z , and on the phase velocity of the propagation β; for a plane wave κ = 0. In [13] it was presented a wave solution in a Born-Infeld background, there also arises an electric component in the propagating direction.
We need to calculate β to completely determine the solution (5) for the EH wave propagating in the uniform magnetic field background. In the next section we determine the phase velocity β using the NLED effective metric approach.

Effective metric and phase velocity of light rays
It is well known that the intense EM fields can resemble a curved spacetime, in the sense that light trajectories are not straight lines but the trajectories suffer deflection. Deviations from the straight trajectories in vacuum are described in NLED by the null trajectories of an effective metric. According to this approach [14], [15], the magnetic background distorts the spacetime where the EM wave propagates and considering the propagation as a perturbation, (i. e. the EM fields of the propagating wave are much smaller than the background fields) the effective metric is derived from the analysis of the characteristic surfaces (wavefronts) [14], [15]. If k µ is a null vector normal to the characteristic surface of the wave, the effective metric g µν eff is given by The plus minus subscript corresponds to the two metrics that can arise in NLED, where the phenomenon of birefringence can occur. See [16] for a study on the Fresnel equation in nonlinear electrodynamics and [17] for a classification of the effective metrics. The effective metric corresponding to a NLED Lagrangian L(F, G) that depends on the electromagnetic invariants F and G, is given by where η µν = diag[−1, 1, 1, 1] is the Minkowski metric and Ω ± is a function of the first and second derivatives of L w. r. t. the invariants F, G, [15], [17]. In the Maxwell case L F = −F/4, the effective metric g µν eff is conformal to η µν . Assuming that the electric background field is zero, then the invariant G vanishes (G = 4 B · E = 0) and the previous expression simplifies to Another consequence of the vanishing of the invariant G = 0 is that there is no birefringence.
To determine the phase velocity of the EM wave β, we calculate the null geodesics of the effective metric, by making zero the line element, where we have located the label eff as a superscript and we shall use the notation for contravariant and covariant effective metric, respectively, as g µν eff and g eff µν . Considering Cartesian coordinates (t, x, y, z) and a light trajectory for fixed x and y, (ẋ = 0 =ẏ), we obtain from Eq. (10) a quadratic equation for the phase velocity along the z-direction, β = dz/dt =ż/ṫ; then solving for β we obtain the phase velocity in terms of the metric components of the effective metric, In order to calculate β from Eq. (11) we have to determine the covariant effective metric [15], for the EH Lagrangian (2) the effective metric, in terms of the background field B and the EH parameter b, is given by (13) Then the phase velocity β of the propagating wave, from Eq. (11), is .
(14) As a consequence of the nonlinear interaction wave-background there is a retarding term, 8bB 2 , that slows down the velocity of the propagation. In order that β be a real number, a lower bound arises for the magnetic field B, bB 2 ≤ 1/12, that in terms of the critical magnetic field B cr amounts to B ≤ 3.38B cr (recall that bB 2 ≈ α(B/B cr ) 2 ). This bound also guarantees that the denominator in (14) does not become zero. Moreover, the wave would be frozen, β ≈ 0, if the magnetic background field reached the critical value bB 2 = 1/12. If b = 0, that means the absence of vacuum polarization, the light velocity in vacuum, β = 1, is recovered. Note as well that if the magnetic background has only one component in the propagation direction, B = B z , then the velocity is the one in vacuum, i.e. the retarding effect on the wave is in inverse proportion to the B z component of the background field.
The phase velocity of the propagation through the magnetic background, Eq. (14), as a function of bB 2 is illustrated in Fig. 2; bB 2 is dimensionless. The calculation of the phase velocity β allows us to determine the constant κ of the longitudinal electric field component as and it is displayed in Fig. 3. Either if B x = 0 or B z = 0, the longitudinal component vanishes. Therefore the Faraday tensor Eq.(5) along with the Eqs. (14) and (15)   Another interesting situation is when the background is not still but moving with constant velocity; it can be considered as a plasma model, for instance. This situation is obtained by performing a Lorentz transformation on the NLED effective metric. Performing a Lorentz boost of velocity β L in the z−direction Λ z , on the effective metric, Λ z g eff Λ T z , we obtain the components of the transformed metric, that we denote with a prime g ′eff µν , where γ −2 = 1−β 2 L . By comparing effective metrics (13) and (16) it can be seen that the Lorentz transformation mixes the components of the effective metric (13), such that g ′eff tz does not vanish, and velocities in the x and y directions arise in addition to the z component; we shall consider only the propagation in the z direction. We recover the effective metric (13) when β L = 0, γ = 1. Writing the phase velocity from Eq. (11) we have when β L = 0, γ = 1 Eq. (14) is recovered. The phase velocity of the propagation is illustrated in Fig. 4. The direction of the Lorentz velocity is important: when β L is in the same direction than the propagation, the braking of the wave is less effective than if β L is opposite to the propagation. In the latter case the wave slows down with a magnetic background less intense than for a still medium. As an example to illustrate the magnitude of the magnetic background field, we set the values B x = B y = 10 6 T that is equivalent to the field generated by a neutron star [6]. To obtain a phase velocity of β ′ = 0.5, then we need to do a boost of β L = 0.5 and apply a background magnetic field in the z direction .38 × 10 9 T. As in the previous case the wave can be frozen (β ′ = 0) and with the cooperation of the medium moving opposite to the wave, this can occur with a less intense magnetic background. The condition for having β ′ = 0 with β 2 L ≤ 1 is given by, where we assumed for simplicity that B z = 0. In Fig. 5 are shown the values of β L versus bB 2 for which β ′ = 0 and there occurs an optical horizon; note that in this case the magnetic background does not need to reach the critical value B = 1/12 to freeze the wave. The interpretation of the effective metric under the Lorentz boost as a propagating medium can be seen clearly by writing the effective metric g ′eff µν in the form of the Painlevé-Lemaitre-Gullstrand (PLG) metric, that in Cartesian coordinates (t,x,ỹ,z) forx =constỹ = const, is given by where V (t,z) represents the velocity of the propagating medium and c(t,z) the velocity of the perturbation propagating through such a medium [18]. The velocity of the medium V is not the same than the one of the Lorentz transformation β L . Taking advantage of the constant curvature R = 0 of the effective metric, by making a scale transformation on the (t, z) coordinates we can write the effective metric in the PLG form (19). By re-scaling as the effective metric in the (t,z) coordinates acquires the form, comparing with (19), we identify the velocity of the perturbation as c = β ′ and we determine the velocity of the medium V as that in our case amounts to .
(23) An optical horizon occurs if the propagation of the medium equals the one of the propagating wave, V (z, t) = β ′ (z, t); this situation happens if g ′eff tt = 0 or g ′eff zz = 0. Assuming for simplicity that B z = 0, we obtain that the condition V (z, t) = β ′ (z, t) (shown in Fig. 5) requires the fulfilment of Eq. (18), in agreement with our previous analysis of the occurrence of the optical horizon. In Fig. 6 is shown the behavior of the velocity of the medium V , Eq. (23), as a function of bB 2 . Knowing the velocity of the propagation, β ′ , we determine κ ′ from Eq. (6) and in Fig. 7 is illustrated for several values of bB 2 for β L both positive and negative.

EM wave propagating in an intense uniform electric background
The propagation of an EM wave through an intense uniform electric field is of interest [19] since there is the prediction of vacuum electron-positron production that has not yet been measured, however it might be feasible in the near future, due to the high power reached lately by lasers [20], [21]. We determined the corresponding solution to the EH electrodynamics equations (4). Being the EM fields of the wave propagating in theẑ direction B w = −E(ξ)ŷ and E w = −β E E(ξ)x + κ E E(ξ)ẑ and considering the uniform electric background E = E xx + E yŷ + E zẑ , then the corresponding Faraday tensor is of the form, where ξ = (z − β E t) and β E , the phase velocity of the propagating wave, and κ E , the longitudinal electric component, should be determined by solving the field Eqs. (4). We have used the subscript E to differentiate from the magnetic background previously analyzed. For the proposed F µν , Eq.(24), we find that κ is given by To determine the phase velocity of propagation β E from Eq. (11), we need the effective metric, that in terms of b and the magnitude of the electric background E 2 is given by The velocity of the propagation through the electric background takes the form In this case it is not possible to freeze the propagation (β E → 0), since the retarding factor [8bE 2 (1 − E 2 z /E 2 )/(1 + 12bE 2 − 8bE 2 z )] does never reach the value 1; as bE 2 increases, the phase velocity tends to a constant value that depends on E 2 z /E 2 . Therefore, the freezing of the propagation does never occur for an electric background field unless the wave propagates in a medium that moves toward the wave, as we shall see next. In Fig. 8 is illustrated β E versus bE 2 . In the case E 2 = E 2 z there is no vacuum polarization effect and β E = 1. Knowing the velocity β E , we can determine κ E from Eq. (25) as and it is illustrated in Fig. 9. In what follows we analyze the situation of a moving electric field background. Performing a Lorentz boost along z on the effective metric, the phase velocity of the propagation, that we denote with a prime β ′ E , Eq. (11), is given by In this case it is possible to freeze the propagation if β L is in the opposite direction to the propagation; the condition for the occurrence of the optical horizon can be derived from demanding that the velocity of the medium V E equals the phase velocity of the propagating wave, V E = β ′ E . In this case from Eq. (22), the velocity of the medium V is given by and demanding V (z, t) = β ′ (z, t) we obtain β L as a function of bE 2 for the occurrence of the optical horizon, as can be seen in Fig. 10, β ′ E approaches zero in cases with β L negative. The velocity of the medium V is illustrated in Fig. 11. E is illustrated as a function of the electric background bE 2 ; β ′ E tends to its value in vacuum as bE 2 → 0 then, as bE 2 grows, it slowly tends to a constant value. To the left β ′ E is shown for positive and negative Lorentz boost velocities: for positive βL (continuous curves) the phase velocity never reaches zero; dashed curves show the phase velocity for negative values of βL, the medium moving in opposite direction to the wave, then β ′ E might reach zero for some negative βL. To the right β ′ E is shown for three different values of Once we have calculated the phase velocity we determine κ ′ E using Eq. (25). Qualitatively the behavior of κ ′ E as a function of bE 2 is very analogous to the one for the magnetic background that is shown in Fig. 9.

Conclusions
We present the Euler-Heisenberg solution that represents an electromagnetic wave propagating through an intense uniform magnetic background; due to the wave-background interaction one electric longitudinal component, κE, arises affecting the polarization of the wave as well as the direction of propagation. The constant κ depends on three parameters: the velocity of the propagation, the magnetic background and the EH parameter κ (β, B, b); if the magnetic field is such that B x = 0 or B z = 0 this effect does not occur and κ = 0.
By means of the NLED effective metric approach [15], [14] we obtain the phase velocity of the propagating wave from the null geodesics of the NLED effective metric. For an intense magnetic background field, of the order of B cr ≈ 10 9 Teslas, the velocity of the propagation significantly slows down, diminishing as B grows; the phase velocity can even be frozen for fields of the order of the critical field, bB 2 ≈ α(B/B cr ) 2 = 1/12 as is illustrated in Fig.  2. We want to stress that we do not find tachyonic phase velocities for the propagating wave, neither birefringence.
By performing a Lorentz boost on the effective metric we model the situation of a flowing medium. By rescaling the coordinates of the effective metric it acquires the form of a PLG metric, where clearly can be identified the velocity of the propagating medium and the velocity of the perturbation through such a medium. In this case also exists an optical horizon that can be reached for a less intense magnetic background if the medium propagates opposite to the wave. This is shown in Fig. 5.
We briefly address the situation of a propagating wave through an intense electric background field. The main difference comparing with the magnetic background is that there is no optical horizon but the phase velocity reaches a constant saturation value. However when we perform a Lorentz boost on the effective metric, then the possibility of an optical horizon exists for a medium moving in opposite direction to the wave, i.e. β L being negative; that is illustrated in Fig. 10; for positive β L an optical horizon does not occur.
In summary we have analyzed the slowing down of an electromagnetic propagating wave under the effect of a very intense electromagnetic field background, in the context of the Euler-Heisenberg theory, that takes into account in an effective way the vacuum polarization phenomenon.