On Generalized Logarithmic Electrodynamics

The generalized logarithmic electrodynamics with two parameters $\beta$ and $\gamma$ is considered. The indexes of refraction of light in the external magnetic field are calculated. In the case $\beta=\gamma$ we come to results obtained by P. Gaete and J. Helay\"el-Neto [7] (Eur.Phys.J. C\textbf{74}, 2816 (2014)). The bound on the values of $\beta$, $\gamma$ was obtained from the Bir\'efringence Magn\'etique du Vide (BMV) experiment. The symmetrical Belinfante energy-momentum tensor and dilatation current are found.


Introduction
Nonlinear classical electrodynamics in vacuum is of interest because of the one-loop quantum corrections in QED which give non-linear terms [1], [2]. In addition, to solve the problem of singularity of a point-like charge giving an infinite electromagnetic energy, Born and Infeld (BI) [3], [4] introduced a new parameter with the dimension of length. BI non-linear electrodynamics results in a finite electromagnetic energy of point-like particles. Other examples of non-linear electrodynamics were introduced in [5], [6], [7] and [8]. In the vacuum, in the presence of strong external magnetic field, nonlinear effects can be observed in experiments. Thus, PVLAS [9] and BMV [10] experiments can give the bounds on dimensional parameters introduced in non-linear electrodynamics. In this letter we calculate the correct values of indexes of refraction of light in the external magnetic field in generalized logarithmic electrodynamics and estimate the value of the parameter γ from the BMV experiment.

The model of generalized logarithmic electrodynamics
Let us consider the Lagrangian density of non-linear generalized logarithmic electrodynamics where β, γ are dimensional parameters and At γ = β we arrive at logarithmic electrodynamics considered in [7].
We obtain from Eq. (1), and the Euler-Lagrange equations, the equations of motion where From Eq. (1) and the expression for the electric displacement field D = ∂L/∂E (E j = iF j4 ), one finds Defining the tensor of the electric permittivity ε ij by the relation D i = ε ij E j , we obtain Using the definition of the magnetic field H = −∂L/∂B (B j = (1/2)ε jik F ik ) and Eq. (1), one finds Introducing the magnetic induction field B i = µ ij H j , the inverse magnetic permeability tensor (µ −1 ) ij is given by From the field equations (2) and the Bianchi identity ∂ µ F µν = 0, one can write Maxwell's equations as follows: where the electric permittivity ε ij and magnetic permeability µ ij depend on the fields E and B and are given by Eqs. (5),(7).

Vacuum birefringence
Let us consider the presence of the external constant and uniform magnetic induction field B 0 = (B 0 , 0, 0) and the plane electromagnetic wave, (e, b), which propagates in the z-direction. As a result, the electromagnetic fields become E = e, B = b + B 0 . We consider the case when amplitudes of electromagnetic wave e 0 , b 0 are smaller comparing with the strong magnetic induction field, e 0 , b 0 ≪ B 0 . In this approximation, up to O(e 2 0 ), O(b 2 0 ), the Lagrangian density (1) is Defining the fields [11] d i = ∂L/∂e i , h i = −∂L/∂b i and linearizing these equations with respect to the wave fields e and b, we obtain the electric permittivity tensor and magnetic permeability One finds the wave equation from Maxwell's equations (8) If the polarization is parallel to external magnetic field, e = e 0 (1, 0, 0), and one finds from Eq. (12) that µε 11 ω 2 = k 2 . As a result, the index of refraction is given by In the case when the polarization of the electromagnetic wave is perpendicular to external induction magnetic field, e = e 0 (0, 1, 0), and µε 22 ω 2 = k 2 , ε 22 = 1/λ. The index of refraction is Thus, the equation (23) in [7] is not correct. The phase velocity depends on the polarization of the electromagnetic wave and n > n ⊥ in opposite to [7]. It should be noted that in the case of QCD, we have the same relation n > n ⊥ [11].
In the presence of a transverse external magnetic field the phenomenon of birefringence is named the Cotton-Mouton (CM) effect [12]. The difference in indexes of refraction is defined by the relation From the approximation B 2 0 /γ 2 ≪ 1, and Eqs. (13), (14) one finds and CM coefficient becomes The vacuum magnetic linear birefringence by the BMV experiment for a maximum field of B 0 = 6.5 T gives the value [10] k CM = (5.1 ± 6.2) × 10 −21 T −2 .
From Eqs. (17),(18), we estimate the value of the parameter γ: We note that the value obtained from QED, using one loop approximation, is [10] k CM ≈ 4.0 × 10 −24 T −2 which is much less than the experimental value (18).

The energy-momentum tensor and dilatation current
From Eq.(1) using the method of [13], we obtain the symmetrical Belinfante tensor where Λ is given by Eq. (3). The energy density obtained from Eq.(20) is 2 The trace of the energy-momentum tensor (20) is as follows: and the trace of the energy-momentum tensor is not zero contrarily to classical electrodynamics. According to [13], we obtain the dilatation current (the field-virial V µ is zero) and the divergence of dilatation current is As a result, the scale (dilatation) symmetry is broken because the dimensional parameters β, γ were introduced. In BI electrodynamics the dilatation symmetry is also broken [14] in opposite to the linear Maxwell electrodynamics.

Conclusion
We have considered the model of generalized logarithmic electrodynamics with two parameters β and γ. At β = γ we arrive at logarithmic electrodynamics considered in [7]. We have corrected formulas for indexes of refraction of light in the external magnetic field describing the phenomenon of birefringence. The parameter γ was estimated, γ ≈ 10 10 T, from the BMV experiment. We show that the scale symmetry is broken and dilatation current was obtained.