1 Introduction

It is well-known that the Standard Model of particle physics relies on Lorentz invariance, an exact symmetry governing the interactions among the smallest building blocks of matter [1]. The SM has been remarkably successful in describing the phenomena that are known to us today. Interestingly enough, despite its experimental success, the observation of Lorentz symmetry violation (LSV) would indicate the existence of new physics. In this form, the possibility that Lorentz and CPT symmetries be spontaneously broken at a very fundamental scale, such as in string theories, has motivated a very intensive research activity. More precisely, the necessity of a new scenario has been proposed to overcome theoretical difficulties in the quantum gravity framework [2,3,4,5]. It is worth recalling that theories with LSV are to be considered as effective theories and the analysis of its physical consequences at low energies may provide information and impose constraints on a more fundamental theory. It is to be specially noted that a suitable framework for testing the low-energy manifestations of LSV is the effective approach referred to as the Standard Model Extension (SME), where it is possible to realize spontaneous LSV.

On the other hand, quantum vacuum nonlinearities and their physical consequences such as vacuum birefringence and vacuum dichroism have been of great interest since its earliest days [6,7,8,9,10,11]. The work due to Euler and Heisenberg [6] is the key example, who computed an effective nonlinear electromagnetic theory in vacuum emerging from the interaction of photons with virtual electron-positron pairs. Nevertheless, these amazing quantum characteristics of light have generated a growing interest on the experimental side [12,13,14]. For example, the PVLAS Collaboration [15], and more recently the ATLAS Collaboration has reported on the direct detection of the light-by-light scattering in LHC Pb–Pb collisions [16, 17]. The advent of laser facilities has given rise to various proposals to probe quantum vacuum nonlinearities [18, 19].

From the preceding considerations, and given the ongoing experiments related to photon–photon interaction physics, we further investigate how nonlinear electromagnetic effects relate to the parameters that indicate LSV. To be more precise, we are interested in studying birefringence, electromagnetic radiation, and the static potential along the lines [20,21,22,23]. These topics were previously explored in a study that began in [24]. It is worth noting that an electromagnetic theory coupled to parameters that signal LSV has been considered from different points of view [25,26,27].

To carry out such studies we consider a non-linear electrodynamic coupled to an LSV term (actually, CFJ). We begin by computing the photonic dispersion relations to get how the parameters associated with the non-linearity interfere with the CFJ external vector in constitutive properties of the vacuum. Furthermore, we compute the refractive indices illustrating that the new medium is birefringent as well as dispersive. Based on these results, let us discuss a more physically motivated aspect of the subject. To this end, we consider the electromagnetic radiation produced by a moving charged particle interacting with this new quantum medium. To do this, we will work out the radiated energy following the usual approach of calculating the Poynting vector. Our analysis reveals the vital role played by vacuum electromagnetic nonlinearities with the CFJ term in triggering the radiated energy.

Our next objective is to study another aspect of the model under consideration, namely, the effect of the non-linearities with a CFJ term on a physical observable. To this end, we shall compute the static potential for this theory by using the gauge-invariant but path-dependent variables formalism. One important advantage of this approach is that it provides a physical alternative to the Wilson loop approach. As a result, we obtain a logarithmic correction to the usual static Coulomb potential.

The paper is organized as follows: in Sect. 1, we introduce the model we are considering and provide the dispersion relations and the refractive indices. Section 2 discusses electromagnetic radiation and shows that the radiation obtained is similar to that of the Cherenkov effect. In Sect. 3, we focus on calculating the interaction energy for a pair of static probe charges within the framework of the gauge-invariant variables formalism. Specifically, we shall be interested in the parameters’ dependence arising from the nonlinearities with the CFJ term. Finally, in Sect. 4, we provide some closing remarks.

In our conventions, the signature of the metric is (\(+1,-1, -1,-1\)), and natural units \(\hbar = c = 1\) are adopted.

2 Model under consideration

2.1 General aspects

We begin by providing a brief overview of the model under consideration. The model is defined by the following Lagrangian density:

$$\begin{aligned} {{\mathcal {L}}} = {{{\mathcal {L}}}_{NL}}\left( {{{\mathcal {F}}};{{\mathcal {G}}}} \right) + {{{\mathcal {L}}}_{CFJ}}, \end{aligned}$$
(1)

where the electromagnetic invariants (\({{\mathcal {F}}}\); \({{\mathcal {G}}}\)) are given by \({{\mathcal {F}}} \equiv - \frac{1}{4}{F_{\mu \nu }}{F^{\mu \nu }} = {\textstyle {1 \over 2}}\left( {{\textbf{E}^2} - {\textbf{B}^2}} \right) \) and \({{\mathcal {G}}} \equiv - \frac{1}{4}{F_{\mu \nu }}{\tilde{F}^{\mu \nu }} = \textbf{E} \cdot \textbf{B}\). Here, \({{{\mathcal {L}}}_{NL}}\), describes the nonlinear part, whereas the term Carroll, Field, Jackiw term, \({{{\mathcal {L}}}_{CFJ}}\), is given by

$$\begin{aligned} {{\mathcal {L}}}_{CFJ} = \frac{1}{4}{\varepsilon ^{\mu \nu \kappa \lambda }}{v_\mu }{A_\nu }{F_{\kappa \lambda }}. \end{aligned}$$
(2)

Here, \({v_\mu }\), is an arbitrary four-vector that selects a preferred direction in space-time, which has mass dimensions in natural units.

Next, we split the \(A_{\mu }\)-field into a classical background \(A_{B\mu }\) and a photonic field \(a_{\mu }\), following our earlier procedure. In this case, the tensor \(F_{\mu \nu }\) can be expressed as \(F_{\mu \nu }=f_{\mu \nu }+F_{B\mu \nu }\), where \(f^{\mu \nu }=\partial ^{\mu }a^{\nu }-\partial ^{\nu }a^{\mu }=\left( \, -e^{i} \,, \, -\epsilon ^{ijk} \, b^{k} \, \right) \), while \(F_{B}^{\;\,\mu \nu }=\partial ^{\mu }A_{B}^{\;\,\nu }-\partial ^{\nu }A_{B}^{\;\,\mu } =\left( \, -E_{B}^{\,\,i} \,, \, -\epsilon ^{ijk} \, B_{B}^{\,\,k} \, \right) \). Accordingly, the corresponding equations of motion read

$$\begin{aligned}{} & {} \nabla \cdot \textbf{d} - \textbf{v} \cdot \textbf{b} = 0, \nonumber \\{} & {} \nabla \times \textbf{e} = - \frac{{\partial \,\textbf{b}}}{{\partial \, t}}, \nonumber \\{} & {} \nabla \cdot \textbf{b} = 0, \nonumber \\{} & {} \nabla \times \textbf{h} - {v^0}\, \textbf{b} + \textbf{v} \times \textbf{e} = \frac{{\partial \,\textbf{d}}}{{\partial \,t}}, \end{aligned}$$
(3)

where the \(\textbf{d}\) and \(\textbf{h}\) fields are given by

(4)
(5)

in this case the tensors, and , are given by \({\zeta _{ij}} = - \frac{{{D_3}}}{{{C_1}}}{B_{Bi}}{B_{Bj}}\) and \({\eta _{ij}} = \frac{{{D_3}}}{{\,{C_1}}}{B_{Bi}}{B_{Bj}}\).

In this last line, we have used

$$\begin{aligned}{} & {} C_{1}=\left. \frac{\partial \mathcal{L}_{NL}}{\partial \mathcal{F}}\right| _{\textbf{E}_{B},\textbf{B}_{B}} \,, \, \left. C_{2}=\frac{\partial \mathcal{L}_{NL}}{\partial \mathcal{G}}\right| _{\textbf{E}_{B},\textbf{B}_{B}} \,, \, \nonumber \\{} & {} \left. D_{1}=\frac{\partial ^2\mathcal{L}_{NL}}{\partial \mathcal{F}^2}\right| _{\textbf{E}_{B},\textbf{B}_{B}} \,, \, \left. D_{2}=\frac{\partial ^2\mathcal{L}_{NL}}{\partial \mathcal{G}^2}\right| _{\textbf{E}_{B},\textbf{B}_{B}}, \nonumber \\{} & {} \left. D_{3}=\frac{\partial ^2\mathcal{L}_{NL}}{\partial \mathcal{F}\partial \mathcal{G}}\right| _{\textbf{E}_{B},\textbf{B}_{B}}. \end{aligned}$$
(6)

It should be emphasized again that the vacuum electromagnetic properties are characterized by the following expressions for the vacuum permittivity and the vacuum permeability:

$$\begin{aligned}{} & {} {\varepsilon _{ij}} \equiv {\delta _{ij}} + {\alpha _i}E_{Bj} + {\beta _i}B_{Bj}, \end{aligned}$$
(7)
$$\begin{aligned}{} & {} {\mu ^{ - 1}}_{ij} \equiv {\delta _{ij}} - B_{Bi}{\gamma _j} - E_{Bi}{\Delta _j}, \end{aligned}$$
(8)

here we have simplified our notation by writing

$$\begin{aligned}{} & {} \varvec{\alpha } \equiv \frac{1}{{{C_1}}}\left( {{D_1}\,\textbf{E}_{B} + {D_3}\,\textbf{B}_{B}} \right) ,\nonumber \\{} & {} \varvec{\beta } \equiv \frac{1}{{{C_1}}}\left( {{D_2}\, \textbf{B}_{B} + {D_3} \,\textbf{E}_{B}} \right) , \end{aligned}$$
(9)
$$\begin{aligned}{} & {} \varvec{\gamma } \equiv \frac{1}{{{C_1}}}\left( {{D_1}\, \textbf{B}_{B} - {D_3} \,\textbf{E}_{B}} \right) ,\nonumber \\{} & {} \varvec{\Delta } \equiv \frac{1}{{{C_1}}}\left( { - {D_3}\, \textbf{B}_{B} + {D_2} \,\textbf{E}_{B}} \right) . \end{aligned}$$
(10)

Furthermore, restricting our considerations to the \(\textbf{E}_{B}=0\) case, we have that \(D_{3}=0\). Making use of this result one encounters that Eq. (7) becomes \({\varepsilon _{ij}} = {\delta _{ij}} + \frac{{{D_2}}}{{{C_1}}}{B_{Bi}}{B_{Bj}}\), which have two eigenvalues \(\varepsilon =1\) and \(\varepsilon =1 + \frac{{{D_2}}}{{{C_1}}}{} \textbf{B}_B^2\). Similarly, from Eq. (8) we have \(\mu _{ij}^{ - 1} = {\delta _{ij}} - \frac{{{D_1}}}{{{C_1}}}{B_{Bi}}{B_{Bj}}\). In this case the eigenvalues of \(\mu _{ij}\) are given by \(\mu = 1\) and \(\mu = \frac{1}{{\left( {1 - \frac{{{D_1}}}{{{C_1}}}{} \textbf{B}_B^2} \right) }}\).

At this point, it is important to discuss another aspect related to vacuum electromagnetic properties. Specifically, we need to understand how the parameters and external fields from the nonlinear (NL) sector are connected to LSV parameters, such as \(v^{\mu }\) for CFJ. In the following sections, we will explore this question in detail.

2.2 Dispersion relations

To adequately deal with this issue, one must inspect the dispersion relations (DRs) for an electromagnetic wave propagating in the external electromagnetic background. To this purpose, we introduce a decomposition into plane waves for the fields \(\textbf{e}\) and \(\textbf{b}\) in the form:

$$\begin{aligned} \textbf{e} = {\textbf{e}_{0R}}\cos \left( {\textbf{k} \cdot \textbf{x} - \omega t} \right) - {\textbf{e}_{0I}}\sin \left( {\textbf{k} \cdot \textbf{x} - \omega t} \right) , \end{aligned}$$
(11)

and

$$\begin{aligned} \textbf{b} = {\textbf{b}_{0R}}\cos \left( {\textbf{k} \cdot \textbf{x} - \omega t} \right) - {\textbf{b}_{0I}}\sin \left( {\textbf{k} \cdot \textbf{x} - \omega t} \right) . \end{aligned}$$
(12)

Next, without restricting generality we take the z-axis as the direction of propagation of the external field, that is, \(\textbf{B}=B_{0}\,\pmb {{{\hat{e}}}_z}\). We further consider that \(\textbf{k}\) and \(\textbf{v}\) are given by \(\textbf{k} = 0\,\pmb {{{\hat{e}}}_x} + {k_y}\,\pmb {{{\hat{e}}}_y} + {k_z}\,\pmb {{{\hat{e}}}_z}\) and \(\textbf{v} = 0\, \pmb {{{\hat{e}}}_x} + {v_y}\, \pmb {{{\hat{e}}}_y} + {v_z}\, \pmb {{{\hat{e}}}_z}\).

Our general approach throughout the present analysis will be to consider that the electromagnetic wave propagating is polarized as follows below:

$$\begin{aligned} \textbf{e}= & {} {e_{0Rx}}\cos \left( {\textbf{k} \cdot \textbf{x} - \omega \, t} \right) \pmb {{{\hat{e}}}_x} - {e_{0Iy}}\sin \left( {\textbf{k} \cdot \textbf{x} - \omega \, t} \right) \pmb {{{\hat{e}}}_y} \nonumber \\{} & {} - \,{e_{0Iz}}\sin \left( {\textbf{k} \cdot \textbf{x} - \omega \, t} \right) \pmb {{{\hat{e}}}_z}, \end{aligned}$$
(13)

and

$$\begin{aligned} \textbf{b}= & {} - {b_{0Ix}}\sin \left( {\textbf{k} \cdot \textbf{x} - \omega \, t} \right) \pmb {{{\hat{e}}}_x} + {b_{0Ry}}\cos \left( {\textbf{k} \cdot \textbf{x} - \omega \, t} \right) \pmb {{{\hat{e}}}_y}\nonumber \\{} & {} +\, {b_{0Rz}}\cos \left( {\textbf{k} \cdot \textbf{x} - \omega \, t} \right) \pmb {{{\hat{e}}}_z}. \end{aligned}$$
(14)

With the preceding expressions, we find that Eq. (3) reduce to

$$\begin{aligned}{} & {} - {k_y}\,{e_{0Iy}} + {\varepsilon _{33}}\,{k_z}\,{e_{0Iz}} = {v_y}\,{b_{0Ry}} + {v_z}\,{b_{0Rz}}, \end{aligned}$$
(15)
$$\begin{aligned}{} & {} \quad - {k_y}\,{e_{0Iz}} + {k_z}\,{e_{0Iy}} = - \omega \, {b_{0Ix}},\,\,\,\ \nonumber \\{} & {} {k_z}\,{e_{oRx}} = \omega \, {b_{0Ry}},\,\,\,\, {k_y}\,{e_{oRx}} = - \omega \,{b_{0Ry}}, \end{aligned}$$
(16)
$$\begin{aligned}{} & {} {k_y}\,{b_{oRy}} + {k_z}\,{b_{0Rz}} = 0, \end{aligned}$$
(17)
$$\begin{aligned}{} & {} - \mu _{33}^{ - 1}\,{k_y}\,{b_{0Rz}} + {k_z}\,{b_{0Ry}} = \omega \,{e_{0Rx}} + {v_y}\,{e_{0Iz}} - {v_z}\,{e_{0Iy}}, \nonumber \\{} & {} - {k_z}\,{b_{0Ix}} = \omega \,{e_{0Iy}} - {v_z}\,{e_{0Rx}}, \nonumber \\{} & {} {k_y}\,{b_{0Ix}} = \omega \,{\varepsilon _{33}}{e_{0Iz}} + {v_y}\,{e_{0Rx}}. \end{aligned}$$
(18)

Making use of the previous equations and after a long algebraic work, we find that the corresponding dispersion relation becomes

$$\begin{aligned}{} & {} {\varepsilon _{33}}\,{\omega ^4} - \left( {2\,{\varepsilon _{33}}\,k_z^2 + \left[ {1 + \mu _{33}^{ - 1}\,{\varepsilon _{33}}} \right] k_y^2 + v_y^2 + {\varepsilon _{33}}\,v_z^2} \right) {\omega ^2} \nonumber \\{} & {} \quad +\left( \mu _{33}^{ - 1}\,{\varepsilon _{33}}\,k_y^2\,k_z^2 + \mu _{33}^{ - 1}\,k_y^4 \nonumber \right. \\{} & {} \quad \left. + {\varepsilon _{33}}\,k_z^4 + k_y^2\,k_z^2 + v_y^2\,k_z^2 + v_z^2\,k_y^2 - 2\,{v_y}\,{v_z}\,{k_y}\,{k_z} \right) = 0, \nonumber \\ \end{aligned}$$
(19)

where \({\varepsilon _{33}} = 1 + \frac{{{D_2}}}{{{C_1}}}{} \textbf{B}_B^2\) and \(\mu _{33}^{ - 1} = 1 - \frac{{{D_1}}}{{{C_1}}}{} \textbf{B}_B^2\). Then equation (19) can be written alternatively in the form

$$\begin{aligned}{} & {} {\varepsilon _{33}}\,{\omega ^4} - \left[ 2|\textbf{k}{|^2} + |\textbf{v}{|^2} + \frac{{\textbf{B}_B^2}}{{{C_1}}}\left( {{D_2} - {D_1} - \frac{{{D_1}{D_2}}}{{{C_1}}}{} \textbf{B}_B^2} \right) k_y^2 \nonumber \right. \\{} & {} \quad \left. + \frac{{{D_2}}}{{{C_1}}}{} \textbf{B}_B^2\left( {2k_z^2 + v_z^2} \right) \right] {\omega ^2} +\left[ |\textbf{k}{|^4} + {{\left( {\textbf{v} \times \textbf{k}} \right) }^2} +\frac{{\textbf{B}_B^2}}{{{C_1}}}\right. \nonumber \\{} & {} \quad \left. \left( { - {D_1}\,k_y^2\,|\textbf{k}{|^2} + {D_2}\,k_z^2\,|\textbf{k}{|^2} - \frac{{{D_1}{D_2}}}{{{C_1}}}\textbf{B}_B^2\,k_y^2\,k_z^2} \right) \right] = 0. \nonumber \\ \end{aligned}$$
(20)

We are now in a position to check the internal consistency of our procedure. To be more precise, if we remove the nonlinearity, that is, we consider \(D_{1}=0\) and \(D_{2}=0\), Eq. (20) reduces to

$$\begin{aligned} {\omega ^4} - \left( {2\,|\textbf{k}{|^2} + |\textbf{v}{|^2}} \right) {\omega ^2} + |\textbf{k}{|^4} + |\textbf{v}{|^2}|\textbf{k}{|^2} - {\left( {\textbf{v} \cdot \textbf{k}} \right) ^2} = 0. \nonumber \\ \end{aligned}$$
(21)

One immediately sees that the above relation dispersion is identical to that encountered in the CFJ model, which is given by

$$\begin{aligned} {k^4} + {v^2}{k^2} - {\left( {v \cdot k} \right) ^2} = 0, \end{aligned}$$
(22)

with \({v^\mu } = \left( {0,\textbf{v}} \right) \) and \({v^2} = - |\textbf{v}{|^2}\).

The refractive indices can also be found with the help of our dispersion relation. This will be considered in the next subsection.

2.3 Refractive indices

As stated previously, our immediate objective is to compute the refractive indices. We begin our discussion by noting that if \(\theta \) is the angle between \(\textbf{k}\) and the z-axis, and \(\alpha \) is the angle between \(\textbf{k}\) and \(\textbf{v}\), Eq. (20) can be brought to the form

$$\begin{aligned}{} & {} {\varepsilon _{33}}\,{\omega ^4} - \left[ |\textbf{v}{|^2} + 2|\textbf{k}{|^2} + \frac{{\textbf{B} _B^2}}{{{C_1}}}\left( {{D_2} - {D_1} - \frac{{{D_1}{D_2}}}{{{C_1}}}{} \textbf{B}_B^2} \right) \nonumber \right. \\{} & {} \quad \left. {{\sin }^2}\theta |\textbf{k}{|^2} + \frac{{{D_2}}}{{{C_1}}}{} \textbf{B}_B^2\left( {v_z^2 + 2\,{{\cos }^2}\theta |\,\textbf{k}{|^2}} \right) \right] \!{\omega ^2} \nonumber \\{} & {} \quad +\left[ |\textbf{k}{|^4} + |\textbf{v}{|^2}|\textbf{k}{|^2}\,{{\sin }^2}\alpha + \frac{{\textbf{B}_B^2}}{{{C_1}}}\left( - {D_1}\,{{\sin }^2}\theta + {D_2}\,{{\cos }^2}\theta \nonumber \right. \right. \\{} & {} \quad \left. \left. - \frac{{{D_1}{D_2}}}{{{C_1}}} \textbf{B}_B^2\,{{\sin }^2}\theta \, {{\cos }^2}\theta \right) |\textbf{k}{|^4} \right] =0. \end{aligned}$$
(23)

Let us, for later convenience, observe that if \(\textbf{B}_B\) and \(\textbf{v}\) are along the z-axis, and using \(\frac{{{D_1}}}{{{C_1}}}\textbf{B}_B^2 = 1 - \mu _{33}^{ - 1}\), \(\frac{{{D_2}}}{{{C_1}}}{} \textbf{B}_B^2 = - \left( {1 - {\varepsilon _{33}}} \right) \), \(\left( {\frac{{{D_1}}}{{{C_1}}} - \frac{{{D_2}}}{{{C_1}}}} \right) \textbf{B}_B^2 = 2 - {\varepsilon _{33}} - \mu _{33}^{ - 1}\), \(\frac{{{D_1}{D_2}}}{{C_1^2}}{} \textbf{B}_B^4 = - \left( {1 - {\varepsilon _{33}}} \right) \left( {1 - \mu _{33}^{ - 1}} \right) \), it follows that

$$\begin{aligned}{} & {} {\textbf{k}^4} - \left( {{\omega ^2} + {\varepsilon _{33}}\mu _{33}^{ - 1}{\omega ^2} - {\textbf{v}^2} + \mu _{33}^{ - 1}{\textbf{v}^2}{{\cos }^2}\theta } \right) {\textbf{k}^2} \nonumber \\{} & {} \quad + \left( {{\varepsilon _{33}}{\omega ^4} - {\varepsilon _{33}}{\textbf{v}^2}{\omega ^2}} \right) = 0. \end{aligned}$$
(24)

In this manner, we obtain the following refractive indices

$$\begin{aligned} {n^2}\left( \omega \right)= & {} \frac{1}{{2\mu _{33}^{ - 1}}} \left( {1 + {\varepsilon _{33}}\mu _{33}^{ - 1} - \frac{{{\textbf{v}^2}}}{{{\omega ^2}}} + \mu _{33}^{ - 1}\frac{{{\textbf{v}^2}}}{{{\omega ^2}}}{{\cos }^2}\theta } \right) \nonumber \\{} & {} \pm \frac{1}{{2\mu _{33}^{ - 1}{\omega ^2}}}\sqrt{{{\left( {{\omega ^2} + {\varepsilon _{33}}\mu _{33}^{ - 1}{\omega ^2} - {\textbf{v}^2} + \mu _{33}^{ - 1}{v^2}{{\cos }^2}\theta } \right) }^2} - 4{\varepsilon _{33}}\mu _{33}^{ - 1}{\omega ^2}\left( {{\omega ^2} - {\textbf{v}^2}} \right) }. \end{aligned}$$
(25)

This implies that the preceding electromagnetic vacuum behaves like a birefringent medium that has two refractive indices that are determined by the polarization of the incoming electromagnetic waves. An important observation from Eq. (25) is that this new vacuum is both dispersive and quite anisotropic.

Finally, it is straightforward to obtain the refractive indices for the CFJ model. If \(\theta \) is the angle between \(\textbf{k}\) and \(\textbf{v}\), from Eq. (22), we readily obtain

$$\begin{aligned} {\left( {{\omega ^2} - {\textbf{k}^2}} \right) ^2} - {\textbf{v}^2}\left( {{\omega ^2} - {\textbf{k}^2}} \right) - {\textbf{v}^2}{\textbf{k}^2}{\cos ^2}\theta = 0. \end{aligned}$$
(26)

We thus find that the refractive indices are as follows

$$\begin{aligned} {n^2}\left( \omega \right)= & {} 1 - \frac{{{\textbf{v}^2}}}{{2{\omega ^2}}}{\sin ^2}\theta \pm \frac{{|\textbf{v}|}}{2}\nonumber \\{} & {} \sqrt{\frac{4}{{{\omega ^2}}} - \left( {\frac{2}{{{\omega ^2}}} - \frac{{{\textbf{v}^2}}}{{{\omega ^4}}}{{\sin }^2}\theta } \right) {{\sin }^2}\theta }. \end{aligned}$$
(27)

3 Cherenkov radiation

Based on these observations, the purpose of this section is to further elaborate on the physical content of the model being discussed. Our focus here is to examine the problem of obtaining the electromagnetic radiation generated by a moving charged particle when it interacts with this new medium. To this end, we consider the Maxwell equations

$$\begin{aligned}{} & {} \nabla \cdot \textbf{d} - {\textbf{v} \cdot \textbf{b}} = {4\pi }{\rho _{ext}}, \nonumber \\{} & {} \nabla \cdot \textbf{b} = 0, \nonumber \\{} & {} \nabla \times \textbf{e} = - \frac{{\partial \,\textbf{b}}}{{\partial \, t}}, \nonumber \\{} & {} \nabla \times \textbf{h} + \textbf{v} \times \textbf{e} = {4\pi }{\textbf{J}_{ext}} + \frac{{\partial \, \textbf{d}}}{{\partial \, t}}. \end{aligned}$$
(28)

Here, \({\rho _{ext}}\) and \({\textbf{J}_{ext}}\) denote the external charge and current densities, which are given by: \({\rho _{ext}}\left( {t,\textbf{x}} \right) = Q\,\delta \left( x \right) \delta \left( y \right) \delta \left( {z - {{\bar{u}}}t} \right) \) and \(\textbf{J}_{ext}\left( {t,\textbf{x}} \right) = Q\,{{\bar{u}}}\,\delta \left( x \right) \delta \left( y \right) \delta \left( {z - {{\bar{u}}}t} \right) {\hat{\textbf{e}}_z}\). We mention in passing that, for simplicity, we are considering the z-axis as the direction of the moving charged particle. In addition \({{\bar{u}}}\) is the velocity of the particle.

Following our earlier procedure [23], we first observe that the previous equations can be brought to the form

(29a)
(29b)

where the tensors, and , are given by Eqs. (7) and (8).

From the above-coupled equations we observe that are quite complicated in a general setup. However, since the Lorentz- symmetry violating parameter is very small, we can solve the previous equations perturbatively. This assumption is based on current estimates in the literature [28] which suggests that the parameter \(v \equiv |\textbf{v}|\) must be upper-bounded \({v} < {10}^{{-}{42}}\) GeV. The main idea is to explore the impact of the Lorentz-symmetry violating parameter on the field equations. To do this, we shall first solve Eqs. (29) and (29b) to zero order in v, and then use these solutions to calculate the electric and magnetic fields to first order in v \(({\mathcal {O}}(\textbf{v}))\).

As explained in [23], after transformation to momentum space, the aforementioned equations are used to implement the previous approximation, that is,

$$\begin{aligned} F(t,\textbf{x}) = \int {\frac{{d \omega \,{d^3}{} \textbf{k}}}{{{{\left( {2\pi } \right) }^4}}}} {e^{ - i\, \omega \, t \,+\, i\,\textbf{k} \cdot \textbf{x}}}F\left( {\omega ,\textbf{k}} \right) , \end{aligned}$$
(30)

where F stands for the electric and magnetic fields.

We thus find that Eq. (29) becomes

(31)

where

(32)

Next, for simplicity, we assume that \(\textbf{v}=(0,0,v_{z})\). As a result, the inverse matrix elements of \(M_{ij}\) are given by: \(M_{11}^{ - 1}{=}\frac{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }}{{\left[ {{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }^2} - {\Omega ^2}v_z^2} \right] }}\), \(M_{12}^{ - 1}{=}- i\frac{{\Omega {v_z}}}{{\left[ {{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }^2} - {\Omega ^2}v_z^2} \right] }}\), \(M_{13}^{ - 1} = 0\), \(M_{21}^{ - 1} = i\frac{{\Omega {v_z}}}{{\left[ {{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }^2} - {\Omega ^2}v_z^2} \right] }}\), \(M_{22}^{ - 1} = \frac{{\left( {{\textbf{k}^2} - v_z^2} \right) }}{{\left[ {{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }^2} - {\Omega ^2}v_z^2} \right] }}\), \(M_{23}^{ - 1} = 0\), \(M_{31}^{ - 1} = 0\), \(M_{32}^{ - 1} = 0\), and \(M_{33}^{ - 1} = \frac{1}{{\left( {{\textbf{k}^2} - \xi {\Omega ^2}} \right) }}\). In the above we have defined \(\Omega \equiv {\omega }\) and \(\xi \equiv {\varepsilon _{33}}{\mu _{33}}\).

Similarly, we see that Eq. (29b) becomes

(33)

where

(34)

In this case the inverse matrix elements of \(\tilde{M}_{ij}\) are given by: \(\tilde{M}_{11}^{ - 1} = \frac{1}{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }}\), \(\tilde{M}_{12}^{ - 1}=0\), \(\tilde{M}_{13}^{ - 1}=0\), \(\tilde{M}_{21}^{ - 1}=0\), \(\tilde{M}_{22}^{ - 1} = \frac{1}{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }}\), \(\tilde{M}_{23}^{ - 1}=0\), \(\tilde{M}_{31}^{ - 1}=0\), \(\tilde{M}_{32}^{ - 1}=0\) and \(\tilde{M}_{33}^{ - 1} = \frac{1}{{\left( {{\textbf{k}^2} - \xi {\Omega ^2}} \right) }}\).

At first order in \(\textbf{v}\), the electric and magnetic fields are found to take the form

$$\begin{aligned} \textbf{e}\left( {\omega ,\textbf{k}} \right)= & {} \frac{1}{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }} \left( { - i4\pi {k_x}{\rho _{ext}} - i4\pi \frac{{{v_z}{k_y}{\rho _{ext}}}}{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }}} \right) \pmb {{{\hat{e}}}_x} \nonumber \\{} & {} + \frac{1}{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }}\left( { - i4\pi {k_y}{\rho _{ext}} - 4\pi {\omega }\frac{{{v_z}{k_x}{\rho _{ext}}}}{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }}} \right) \pmb {{{\hat{e}}}_y} \nonumber \\{} & {} + \frac{1}{{\left( {{\textbf{k}^2} - \xi {\Omega ^2}} \right) }}\left( { - i4\pi \varepsilon _{33}^{ - 1}{k_z}{\rho _{ext}} + i{4\pi } \omega {\mu _{33}}{J_z}} \right) \pmb {{{\hat{e}}}_z}, \nonumber \\ \end{aligned}$$
(35)

and

$$\begin{aligned} \textbf{h}\left( {\omega ,\textbf{k}} \right)= & {} \frac{1}{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }}\left( {i {4\pi }{k_y}{J_z} + 4\pi \frac{{{k_z}{k_x}{v_z}{\rho _{ext}}}}{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }}} \right) \pmb {{{\hat{e}}}_x} \nonumber \\{} & {} +\frac{1}{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }}\left( { - i{4\pi }{k_x}{J_z} + 4\pi \frac{{{k_z}{k_y}{v_z}{\rho _{ext}}}}{{\left( {{\textbf{k}^2} - {\Omega ^2}} \right) }}} \right) \pmb {{{\hat{e}}}_y} \nonumber \\{} & {} +\frac{1}{{\left( {{\textbf{k}^2} - \xi \,{\Omega ^2}} \right) }}\Bigg ( - 4\pi \varepsilon _{33}^{ - 1}{v_z}{\rho _{ext}} + 4\pi \nonumber \\{} & {} \quad \frac{{\varepsilon _{33}^{ - 1}k_z^2{v_z}}}{{\left( {{\textbf{k}^2} - \xi \,{\Omega ^2}} \right) }}{\rho _{ext}} - 4\pi \frac{{{k_z}{v_z}\omega {\mu _{33}}{J_z}}}{{\left( {{\textbf{k}^2} - \xi \,{\Omega ^2}} \right) }} \Bigg ) \pmb {{{\hat{e}}}_z}. \end{aligned}$$
(36)

In summary then, Eqs. (35) and (36) form the basis of our Next, following our earlier line of argument [23], we are in a position to calculate \(\textbf{b}\left( {w,\textbf{x}} \right) \) and \(\textbf{e}\left( {w,\textbf{x}} \right) \). We also recall that, \(\textbf{e}\left( {w,\textbf{x}} \right) \) is given by

$$\begin{aligned} \textbf{e}\left( {w,\textbf{x}} \right) = \int {\frac{{{d^3}\textbf{k}}}{{{{\left( {2\pi } \right) }^3}}}} \; {e^{i\textbf{k} \cdot \textbf{x}}}\;\textbf{e}\left( {w,\textbf{k}} \right) . \end{aligned}$$
(37)

Again, as in [23], by using cylindrical coordinates and taking advantage of the axial symmetry of the problem under consideration, we find that the electric field (37) can be brought to the form

$$\begin{aligned} \textbf{e}\left( {\omega ,\textbf{x}} \right)= & {} - i\frac{Q}{{\pi {\bar{u}}}}{e^{i{{\omega z} / {{\bar{u}}}}}}\int _0^\infty {d{k_T}} \frac{{k_T^2}}{{\left( {k_T^2 - {\sigma ^2}} \right) }}\nonumber \\{} & {} \times \int _0^{2\pi } {d\alpha } \cos \alpha {e^{i{k_T}{x_T}\cos \alpha }}\, \pmb {\hat{\rho }} \nonumber \\{} & {} - \frac{{Q\omega }}{{\pi {\bar{u}}}}{v_z}{e^{i{{\omega z} / {{\bar{u}}}}}}\int _0^\infty {d{k_T}} \frac{{k_T^2}}{{{{\left( {k_T^2 - {\sigma ^2}} \right) }^2}}}\nonumber \\{} & {} \times \int _0^{2\pi } {d\alpha } \cos \alpha {e^{i{k_T}{x_T}\cos \alpha }} \,\pmb {\hat{\phi }} \nonumber \\{} & {} - i\frac{{Q\omega }}{\pi }{e^{i{{\omega z} / {{\bar{u}}}}}}\left( {\frac{{\varepsilon _{33}^{ - 1}}}{{{{{\bar{u}}}^2}}} - {\mu _{33}}} \right) \nonumber \\{} & {} \times \int _0^\infty {d{k_T}} \frac{{k_T^2}}{{\left( {k_T^2 - {\sigma ^2}} \right) }}\int _0^{2\pi } {d\alpha } \cos \alpha {e^{i{k_T}{x_T}\cos \alpha }}\,\pmb {{{{\hat{e}}}_z}}. \nonumber \\ \end{aligned}$$
(38)

where \(\pmb {\hat{\rho }}\) and \(\pmb {\hat{\phi }}\) are unit vectors normal and tangential to the cylindrical surface, respectively. While \(\pmb {{{{\hat{e}}}_z}}\) is a unit vector along the z-direction. We also note that the subscript T in \(k_{T}\) indicates transversal to the z-direction.

With the aid of the integrals: \( \int _0^{2\pi } {d\theta } {e^{ix\cos \theta }}\cos \theta = 2\pi i{J_1}\left( x \right) , \int _0^{2\pi } {d\theta } {e^{ix\cos \theta }} = 2\pi {J_0}\left( x \right) \), where \({{J_0}\left( x \right) }\) and \({{J_1}\left( x \right) }\) represent Bessel functions of the first kind, we can simplify (38) as

$$\begin{aligned} \textbf{e}\left( {\omega ,\textbf{x}} \right)= & {} \frac{{2Q}}{{{\bar{u}}}}{e^{i{{\omega z} / {{\bar{u}}}}}}\int _0^\infty {d{k_T}k_T^2} \frac{{{J_1}\left( {{k_T}{x_T}} \right) }}{{\left( {k_T^2 - {\sigma ^2}} \right) }}\,\pmb {\hat{\rho }} \nonumber \\{} & {} - \frac{{2Q\omega }}{{{\bar{u}}}}{v_z}{e^{i{{\omega z} / {{\bar{u}}}}}}\int _0^\infty {d{k_T}k_T^2} \frac{{{J_1}\left( {{k_T}{x_t}} \right) }}{{{{\left( {k_T^2 - {\sigma ^2}} \right) }^2}}}\,\pmb {\hat{\phi }} \nonumber \\{} & {} - i2Q\omega {e^{i{{\omega z} / {{\bar{u}}}}}} \left( {\frac{{\varepsilon _{33}^{ - 1}}}{{{{{\bar{u}}}^2}}} - {\mu _{33}}} \right) \nonumber \\{} & {} \int _0^\infty {d{k_T}} {k_T}\frac{{{J_0}\left( {{k_T}{x_T}} \right) }}{{\left( {k_T^2 - {\sigma ^2}} \right) }} \, \pmb {{{{\hat{e}}}_z}}. \end{aligned}$$
(39)

After integrating over \(k_T\) and performing further manipulations, we finally obtain that Eq. (39) can be rewritten as

$$\begin{aligned} \textbf{e}\left( {\omega ,\textbf{x}} \right)= & {} - i\frac{{\pi Q}}{{{\bar{u}}}}{e^{i{{\omega z} / {{\bar{u}}}}}}\sigma H_1^{\left( 1 \right) }\left( {\sigma {x_T}} \right) \pmb {\hat{\rho }}\nonumber \\{} & {} + \frac{{Q\pi \omega }}{{2{\bar{u}}}}{v_z}{e^{i{{\omega z} / {{\bar{u}}}}}}\rho H_0^{\left( 1 \right) }\left( {\sigma \rho } \right) \pmb {\hat{\phi }} + {Q\pi {\mu _{33}}} {e^{i{{\omega z} / {{\bar{u}}}}}}\omega \nonumber \\{} & {} \times \left( {1 - \frac{{{1}}}{{\xi {{{\bar{u}}}^2}}}} \right) H_0^{\left( 1 \right) }\left( {\chi \rho } \right) \pmb {{{{\hat{e}}}_z}}, \end{aligned}$$
(40)

where \({\chi ^2} = {\omega ^2}\left( {\xi - \frac{1}{{{{{\bar{u}}}^2}}}} \right) \) and \({\sigma ^2} = {\omega ^2}\left( {1 - \frac{1}{{{{{\bar{u}}}^2}}}} \right) \). In the above, we have used \({x_T} = \rho \) (in cylindrical coordinates). We also recall that \({H}_{0}^{(1)}(x)\) and \({H}_{1}^{(1)}(x)\) are Hankel functions of the first kind.

We mention in passing that in our previous work [23] we explained how the Cherenkov angle comes about in our framework. Specifically, we demonstrated that wave planes are radiated if the velocity of the charge is larger than the velocity of light in the medium.

We now calculate the magnetic field using the same method as before. In such a case, we have

$$\begin{aligned} \textbf{h}\left( {\omega ,\textbf{x}} \right)= & {} i\frac{{2Q{v_z}\omega }}{{{{{\bar{u}}}^2}}}{e^{i{{\omega z} / {{\bar{u}}}}}}\int _0^\infty {d{k_T}} k_T^2\frac{{{J_1}\left( {{k_T}{x_T}} \right) }}{{\left( {k_T^2 - {\sigma ^2}} \right) }}\, \pmb {\hat{\rho }} \nonumber \\{} & {} + {{2Q}}{e^{i{{\omega z} / {{\bar{u}}}}}}\int _0^\infty {d{k_T}} k_T^2\frac{{{J_1} \left( {{k_T}{x_T}} \right) }}{{\left( {k_T^2 - {\sigma ^2}} \right) }}\, \pmb {\hat{\phi }} \nonumber \\{} & {} + 2Q{v_z}{e^{i{{\omega z} / {{\bar{u}}}}}}\int _0^\infty {d{k_T}{k_T}} \frac{{{J_0}\left( {{k_T}{x_T}} \right) }}{{\left( {k_T^2 - {\chi ^2}} \right) }}\nonumber \\{} & {} \times \left\{ - \frac{{\varepsilon _{33}^{ - 1}}}{{{\bar{u}}}} + \frac{1}{{\left( {k_T^2 - {\chi ^2}} \right) }} \left[ \frac{{\varepsilon _{33}^{ - 1}}}{{{\bar{u}}}}\frac{{{\omega ^2}}}{{{{{\bar{u}}}^2}}} - \frac{{{\mu _{33}}}}{{{\bar{u}}}}{\omega ^2} \right] \right\} \pmb {{{{\hat{e}}}_z}}. \nonumber \\ \end{aligned}$$
(41)

By using that \({h_\rho } \rightarrow {b_\rho }\), \({h_\phi } \rightarrow {b_\phi }\), \({h_z} \rightarrow \mu _{33}^{ - 1}{b_z}\). Again, essentially following the same steps as in the case of the electric field, we find that the magnetic field becomes

$$\begin{aligned} \textbf{b}\left( {w,\textbf{x}} \right)= & {} - \frac{{\pi Q\omega }}{{2{{{\bar{u}}}^2}}}{v_z}{e^{i{{\omega z} / {{\bar{u}}}}}}\rho H_0^{\left( 1 \right) }\left( {\sigma \rho } \right) \pmb {\hat{\rho }}\nonumber \\{} & {} - i{{\pi Q}}{e^{i{{\omega z} / {{\bar{u}}}}}}\sigma H_1^{\left( 1 \right) }\left( {\sigma \rho } \right) \pmb {\hat{\phi }} \nonumber \\{} & {} + i\frac{{\pi Q\tau }}{{{\bar{u}}}}{v_z}{e^{i{{\omega z} / {{\bar{u}}}}}}\left( {H_0^{\left( 1 \right) }\left( {\chi \rho } \right) + \frac{\chi }{2}\rho H_1^{\left( 1 \right) }\left( {\chi \rho } \right) } \right) \pmb {{{{\hat{e}}}_z}}. \nonumber \\ \end{aligned}$$
(42)

Equations (40) and (42) will then be exploited to compute the radiated energy for the model under consideration.

Before going ahead, it is appropriate to note that, as was explained in [23], the above development leads to the Cherenkov angle. Specifically, we have demonstrated that wave planes are radiated if the velocity of the charge is larger than the velocity of light in the medium. We will not delve into the technical details here, but they can be found in [23]. Given this situation, we will now move on to calculating radiated energy.

We begin by recalling that the power density carried by the radiation fields across the surface S that bounds volume V is the real part of the time-averaged Poynting vector, that is,

$$\begin{aligned} \textbf{S} = { \frac{1}{{2 }}}{\mathop {\textrm{Re}}\nolimits } \left( {\textbf{e} \times {\textbf{b}^ * }} \right) . \end{aligned}$$
(43)

By using this expression, we can calculate the power radiated through the surface S [29] through

$$\begin{aligned} {{\mathcal {E}}} = \int _{ - \infty }^\infty \! {dt} \int \limits _S {d\textbf{a} \cdot \textbf{S}}. \end{aligned}$$
(44)

One can now further observe that we shall consider a cylinder of length h and radius \(\rho \). Hence the total energy radiated through the surface of the cylinder can be reduced to the form

$$\begin{aligned} {\mathcal {E}} = \left( {2\pi \rho h} \right) \int _0^\infty {d\omega }\, {S_\rho }\,\Theta \left( {{\bar{u}} - \frac{1}{n}} \right) , \end{aligned}$$
(45)

where it may be recalled that the presence of the step function means that Cherenkov radiation will be emitted if and only if the velocity of the particle exceeds the velocity of light in the medium (\({{\bar{u}}} > \frac{1}{n}\)).

Using Eqs. (40) and (42) in the radiation zone, we can express the power radiated per unit length (45) as

(46)

From Eq. (25) to first order in v \(({\mathcal {O}}(\textbf{v}))\), we obtain that refractive indices are given by \(n_ + ^2\left( \omega \right) = {\mu _{33}}\) and \(n_ - ^2\left( \omega \right) = {\varepsilon _{33}}\). Accordingly, the previous expression becomes

(47)

It is useful to point out that the above expression is similar to that obtained in the Cherenkov radiation theory.

Let us also mention here that the new vacuum is analogous to the general case of non-linear electrodynamics [22, 23], where the polarizable medium is the QED vacuum, made up of virtual electrons and positrons. In this manner, the produced Cherenkov radiation corresponds to the energy re-emitted by the excited virtual particles, hence we may set an upper bound for the integration over w in (47). As was explained in [22, 23], we argue that the frequency, 2m, acts as a cutoff frequency for the re-emitted photons, which then corresponds to the energy for a pair creation.

With this, Eq. (47) can be written as

(48)

where \(\Gamma = 2m\).

We thus find that Eq. (48) reads

(49)

where \(|p| = |\sqrt{1 - \frac{{{1}}}{{{{{\bar{u}}}^2}}}} |\) and \(q = n\sqrt{1 - \frac{{{1}}}{{{n^2}{{{\bar{u}}}^2}}}}\).

It is interesting to note that the expression above explicitly depends on \(\rho \) (cylindrical coordinates). However, this was expected due to the nonlinear background and its associated anisotropy. It is also worth noting that taking the limit \(\rho \rightarrow 0\) does not present any problems since the previous expression represents the far-field solution. In other words, to obtain the previous expression, we used the asymptotic expressions of the Hankel functions, which in the radiation zone are given by

(50)

and

(51)

Finally, the limit \(\rho \rightarrow \infty \) also does not present problems, since the oscillating functions cos and sin in the average tend to zero at infinity.

4 Interaction energy

We turn our attention to the calculation of the interaction energy between static point-like sources for our model described by Eq. (1), by using the gauge-invariant but path-dependent variables formalism. However, before proceeding with the determination of the interaction energy, we first note that to study the quantum properties of the electromagnetic field in the presence of external electric and magnetic fields, we should split the \(A_{\mu }\)-field into a classical background \(A_{B\mu }\) and a photonic field \(a_{\mu }\), in the same way as we have done in Sect. 1.

In this manner, following our earlier procedure, the nonlinear part of Lagrangian (1) up to quadratic terms in the fluctuations, is also expressed as

$$\begin{aligned} {{\mathcal {L}}}_{NL}^{(2)}= & {} -\frac{1}{4} \, C_{1} \, f_{\mu \nu }^{\, 2} -\frac{1}{4} \, C_{2} \, f_{\mu \nu } \, \widetilde{f}^{\mu \nu } -\frac{1}{2} \, G_{B\mu \nu } \, f^{\mu \nu }\nonumber \\{} & {} \quad + \,\frac{1}{8} \, Q_{B\mu \nu \kappa \lambda } \, f^{\mu \nu } \, f^{\kappa \lambda } \;, \end{aligned}$$
(52)

where \(f^{\mu \nu }\) has been defined previously in Sect. 1, and \(\widetilde{f}^{\mu \nu }=\epsilon ^{\mu \nu \alpha \beta }f_{\alpha \beta }/2\). We further recall that \(G_{B\mu \nu }=C_{1} \, F_{B\mu \nu }+ C_{2} \, \widetilde{F}_{B\mu \nu }\) and \(Q_{B\mu \nu \kappa \lambda }=D_{1} \, F_{B\mu \nu }F_{B\kappa \lambda } +D_{2} \, \widetilde{F}_{B\mu \nu }\widetilde{F}_{B\kappa \lambda } +D_{3} \, F_{B\mu \nu }\widetilde{F}_{B\kappa \lambda } + D_{3} \, \widetilde{F}_{B\mu \nu } F_{B\kappa \lambda }\) are tensors that depends on the components of the electric and magnetic background fields. Whereas, the coefficients \(C_{1}\), \(C_{2}\), \(D_{1}\), \(D_{2}\) and \(D_{3}\) of this expansion are given by Eq. (6).

We accordingly rewrite the density Lagrangian (1), up to quadratic terms in the photonic field, in the form

$$\begin{aligned} {{\mathcal {L}}}= & {} -\frac{1}{4} \, C_{1} \, f_{\mu \nu }^{\, 2} -\frac{1}{4} \, C_{2} \, f_{\mu \nu } \, \widetilde{f}^{\mu \nu } -\frac{1}{2} \, G_{B\mu \nu } \, f^{\mu \nu } \nonumber \\{} & {} +\!\!\!\frac{1}{8} \, Q_{B\mu \nu \kappa \lambda } \, f^{\mu \nu } \, f^{\kappa \lambda } \; + \frac{1}{4}{\varepsilon ^{\mu \nu \kappa \lambda }}{v_\mu }{a_\nu }{f_{\kappa \lambda }} - a_{0}J^{0}, \nonumber \\ \end{aligned}$$
(53)

where we have introduced an external current \(J^{0}\).

This effective theory provides us with a suitable starting point to study the interaction energy. To obtain the corresponding Hamiltonian, the canonical quantization of this theory from the Hamiltonian analysis point of view is straightforward and follows closely that of Refs. [20, 21, 32]. The canonical momenta read

$$\begin{aligned} {\Pi ^\mu }= & {} - {C_1}{f^{0\mu }} - {C_2}{\tilde{f}^{0\mu }} - G_B^{0\mu } \nonumber \\{} & {} + \frac{1}{2}Q_B^{0\mu \kappa \rho }{f_{\kappa \rho }} + \frac{1}{2}{\varepsilon ^{\kappa \rho 0\mu }}{v_{\kappa \rho }}. \end{aligned}$$
(54)

This yields the usual primary constraint \({\Pi ^0} = 0\). While the remaining nonzero momenta are given by

$$\begin{aligned} {\Pi ^i}= & {} - {C_1}{f^{0i}} - {C_2}{\tilde{f}^{0i}} - G_B^{0i} + \frac{1}{2}Q_B^{0i\kappa \rho }\nonumber \\{} & {} + \frac{1}{2}{\varepsilon ^{\kappa \rho 0i}}{v_\kappa }{a_\rho }. \end{aligned}$$
(55)

The canonical Hamiltonian is now obtained in the usual way

$$\begin{aligned} {H_C}= & {} \int {{d^3}x} \left\{ { - {a_0}\left( {{\partial _i}{\Pi ^i} - \frac{1}{2}{\varepsilon ^{ijk}}{v_i}{\partial _j}{a_k} - {J^0}} \right) } \right\} \nonumber \\{} & {} + \int {{d^3}x} \left\{ {\frac{1}{2}\left( {{C_1}{\delta ^{ki}} + Q_B^{k0i0}} \right) {f_{i0}} - \frac{1}{2}{C_2}{{\tilde{f}}^{k0}}{f_{k0}}} \right\} \nonumber \\{} & {} + \int {{d^3}x} \left\{ {\frac{1}{4}Q_B^{k0ij}{f_{ij}}{f_{k0}} + \frac{1}{4}{C_1}{f_{ij}}{f^{ij}} + \frac{1}{2}{G_{Bij}}{f^{ij}}} \right\} \nonumber \\{} & {} + \int {{d^3}x} \left\{ {\frac{1}{4}{C_2}{f_{ij}}{{\tilde{f}}^{ij}}} \right\} . \end{aligned}$$
(56)

Before we proceed, we would like to make a technical remark. The general Lagrangian (1) contains higher-order derivatives. However, for non-singular systems, the Hamiltonian approach to higher derivatives theories was first developed by Ostrogradsky [30]. His method involves defining one more pair of canonical variables, thus doubling the dimension of the phase space. For singular higher derivatives systems, such as the Lagrangian (1), Dirac’s theory for constrained systems can be generalized to include the Ostrogradsky approach [31]. It is important to note that the system studied here (53) is up to quadratic terms in the photonic field.

Next, in what follows we will consider the case of a pure magnetic background, that is, \({\textbf{E}_B} = 0\) and \({\textbf{B}_B} \ne 0\). We thus find that

$$\begin{aligned} {H_C}= & {} \int {{d^3}x} \left\{ { - {a_0}\left( {{\partial _i}{\Pi ^i} + \frac{1}{2}{} \textbf{v} \cdot \textbf{b} - {J^0}} \right) + \frac{1}{2}{C_1}{\textbf{e}^2}} \right\} \nonumber \\{} & {} + \int {{d^3}x} \left\{ {\frac{1}{2}{D_2}\,{{\left( {{\textbf{B}_B} \cdot \textbf{e}} \right) }^2} + \frac{1}{2}{C_1}{\textbf{b}^2} + {C_1}{\textbf{B}_B} \cdot \textbf{b}} \right\} . \nonumber \\ \end{aligned}$$
(57)

Time conserving the primary constraint \({\Pi }_0\) yields the secondary constraint \({\Gamma _1} \equiv {\partial _i}{\Pi ^i} + \frac{1}{2}{} \textbf{v} \cdot \textbf{b}-J^{0} = 0\) (Gauss’s law). Therefore, in this case, there are two constraints in all, which are first class. Now we recall that the extended Hamiltonian, which generates the time evolution of the dynamical variables, is then written as \(H = H_C + \int {d^3 x} \left( {u_0(x) \Pi _0(x) + u_1(x) \Gamma _1(x) } \right) \). Here \(u_o(x)\) and \(u_1(x)\) are arbitrary Lagrange multipliers reflecting the gauge invariance of the theory. Considering that \(\Pi ^0=0\) always and \({\dot{a_0}}\left( x \right) = \left[ {{a_0}\left( x \right) , H} \right] = {u_0}\left( x \right) \), which is completely arbitrary, we eliminate \(a^0\) and \(\Pi ^0\) because they add nothing to the description of the system.

With this, the extended Hamiltonian is now given by

$$\begin{aligned} {H}= & {} \int {{d^3}x} \left\{ { w(x)\left( {{\partial _i}{\Pi ^i} + \frac{1}{2}{} \textbf{v} \cdot \textbf{b} - {J^0}} \right) + \frac{1}{2}{C_1}{\textbf{e}^2}} \right\} \nonumber \\{} & {} + \int {{d^3}x} \left\{ {\frac{1}{2}{D_2}\,{{\left( {{\textbf{B}_B} \cdot \textbf{e}} \right) }^2} + \frac{1}{2}{C_1}{\textbf{b}^2} + {C_1}{\textbf{B}_B} \cdot \textbf{b}} \right\} , \nonumber \\ \end{aligned}$$
(58)

where we have defined \(w(x) \equiv u_1 (x) - a_0 (x)\).

According to the usual procedure, we now proceed to introduce a gauge condition such that the full set of constraints becomes second class. Hence, we choose the gauge fixing condition as [33]:

$$\begin{aligned} \Gamma _2 \left( x \right) \equiv \int \limits _{C_{\zeta x} } {dz^\nu } a_\nu \left( z \right) \equiv \int \limits _0^1 {d\lambda x^i } a_i \left( { \lambda x } \right) = 0, \end{aligned}$$
(59)

where \(\lambda \) \((0\le \lambda \le 1)\) is the parameter describing the space-like straight path \(x^i = \zeta ^i + \lambda \left( {x - \zeta } \right) ^i \), and \(\zeta \) is a fixed reference point. There is no essential loss of generality if we restrict our considerations to \(\zeta ^i=0\). One thus obtains the only non-vanishing equal-time Dirac bracket:

$$\begin{aligned}{} & {} \left\{ {a_i \left( \textbf{x} \right) ,\Pi ^j \left( \textbf{y} \right) } \right\} ^ * \nonumber \\{} & {} \quad = \delta {_i^j} \,\delta ^{\left( 3 \right) } \left( {\textbf{x} - \textbf{y}} \right) - \partial _i^x \int \limits _0^1 {d\lambda \,x^j } \delta ^{\left( 3 \right) } \left( {\lambda \textbf{x}- \textbf{y}} \right) . \end{aligned}$$
(60)

With the aid of this result, one can write the Dirac brackets in terms of the magnetic and electric fields as

$$\begin{aligned}{} & {} {\left\{ {{b_i}\left( \textbf{x} \right) ,{b_j}\left( \textbf{y} \right) } \right\} ^ * } = 0, \end{aligned}$$
(61)
$$\begin{aligned}{} & {} {\left\{ {{e_i}\left( \textbf{x} \right) ,{b_j}\left( \textbf{y} \right) } \right\} ^ * } = \frac{1}{{{C_1}}}{\varepsilon _{ijk}}{\partial ^k}{\delta ^{\left( 3 \right) }}\left( {\textbf{x} - \textbf{y}} \right) \nonumber \\{} & {} \qquad \qquad \qquad \qquad \qquad -\frac{{{D_2}}}{{C_1^2\det D}}{B_{Bi}}{B_{Br}}{\varepsilon _{rjk}}{\partial ^k}{\delta ^{\left( 3 \right) }}\left( {\textbf{x} - \textbf{y}} \right) , \nonumber \\ \end{aligned}$$
(62)

and

$$\begin{aligned}{} & {} {\left\{ {{e_i}\left( \textbf{x} \right) ,{e_j}\left( \textbf{y} \right) } \right\} ^ * } = - \frac{1}{{C_1^2}}{\varepsilon _{ijk}}\,{v_k}\,{\delta ^{\left( 3 \right) }}\left( {\textbf{x} - \textbf{y}} \right) \nonumber \\{} & {} \quad + \frac{{{D_2}}}{{C_1^3\det D}}\left( {{B_j}{B_q}{\varepsilon _{iqk}} - {B_i}{B_q}{\varepsilon _{jqk}}} \right) {v_k}\,{\delta ^{\left( 3 \right) }}\left( {\textbf{x} - \textbf{y}} \right) , \nonumber \\ \end{aligned}$$
(63)

where \(det\,D = 1+\frac{D_2}{C_1}{} \textbf{B}_B^2\).

We can now write the equations of motion for the electric and magnetic fields in the form

$$\begin{aligned} {\dot{e}_i}\left( x \right) = - \frac{1}{{{C_1}}}\left( {1 - \frac{{{D_2}}}{{{C_1}}}{} \textbf{B}_B^2} \right) {\varepsilon _{ijk}}{v_k}{e_j} + \frac{1}{{\det D}}{\varepsilon _{ijk}}{\partial ^k}{b_j}, \nonumber \\ \end{aligned}$$
(64)

and

$$\begin{aligned} {\dot{b}_i}\left( x \right) = {\varepsilon _{ijk}}{\partial _{j}}e_{k}. \end{aligned}$$
(65)

Similarly, we write Gauss’s law as

$$\begin{aligned} {C_1}\,\det D\,{\partial _i}{e^i} + \textbf{b} \cdot \textbf{v} - {J^0} = 0. \end{aligned}$$
(66)

In this last line, we used that the external magnetic field, \({\textbf{B}_B}\), has a fixed direction in space.

We shall now consider the static case, that is, Eqs. (64) and (65) must vanish. In this way one encounters

$$\begin{aligned} e_{i} = \partial _{i}\Phi , \end{aligned}$$
(67)

and

$$\begin{aligned} \Phi = \frac{{{\nabla ^2}}}{{\left[ {{\nabla ^2}\left( {{C_1}\det D\,{\nabla ^2} + \xi \,{\textbf{v}^2}} \right) - \xi \, {v^i}{\partial _i}{v^j}{\partial _j}} \right] }}\left( { - {J^0}} \right) , \nonumber \\ \end{aligned}$$
(68)

where \(\xi = \frac{1}{{{C_1}}}\det D\left( {1 - \frac{{{D_2}}}{{{C_1}}}{} \textbf{B}_B^2} \right) \).

Incidentally, the above expression is analogous to that encountered in our previous work [21]. Given this situation, we skip all the technical details and refer to [21] for them. Accordingly, the static potential between static q-charged point-like sources turns out to be

$$\begin{aligned}{} & {} V = - \frac{{{q^2}}}{{4\pi }}\frac{1}{{{C_1}\left( {1 + \frac{{{D_2}}}{{{C_1}}}{} \textbf{B}_B^2} \right) }}\frac{1}{r} \nonumber \\{} & {} + \frac{{{q^2}}}{{4\pi }}\frac{1}{{{C_1}\left( {1 + \frac{{{D_2}}}{{{C_1}}}{} \textbf{B}_B^2} \right) }}\frac{{{\textbf{v}^2}}}{{C_1^2}}\left( {1 - \frac{{{D_2}}}{{{C_1}}}{} \textbf{B}_B^2} \right) z\ln \left( {\frac{{z + r}}{{2z}}} \right) , \nonumber \\ \end{aligned}$$
(69)

where \(r = \sqrt{{\textbf{x}^2} + {\textbf{y}^2}}\).

Finally, in logarithmic electrodynamics [20], we have the following expressions: \({C_1} = \frac{1}{{\left( {1 + \frac{{\textbf{B}_B^2}}{{{2\beta ^2}}}} \right) }}\), \({D_1} = \frac{1}{{{\beta ^2}}}\frac{1}{{{{\left( {1 + \frac{{\textbf{B}_B^2}}{{2{\beta ^2}}}} \right) }^2}}}\) and \({D_2} = \frac{1}{\beta }\frac{1}{{\left( {1 + \frac{{\textbf{B}_B^2}}{{2{\beta ^2}}}} \right) }}\). Hence we see that the static potential profile can be rewritten in the form

$$\begin{aligned} V= & {} - \frac{{{q^2}}}{{4\pi }}\frac{{\left( {1 + {{\textbf{B}_B^2} / {2{\beta ^2}}}} \right) }}{{\left( {1 + {{\textbf{B}_B^2} / {{\beta ^2}}}} \right) }}\frac{1}{r} \nonumber \\{} & {} + \frac{{{q^2}}}{{4\pi }} \, {\textbf{v}^2} \, \frac{{{{\left( {1 + {{\textbf{B}_B^2} / {2{\beta ^2}}}} \right) }^3}\left( {1 - {{\textbf{B}_B^2} / {{\beta ^2}}}} \right) }}{{\left( {1 + {{\textbf{B}_B^2} / {{\beta ^2}}}} \right) }}\,\,z\,\ln \left( {\frac{{z + r}}{{2z}}} \right) . \nonumber \\ \end{aligned}$$
(70)

Of particular concern to us is the effect of the parameter that carries the LSV message on the static potential profile. To be more precise, the logarithmic correction to the usual Coulomb potential.

5 Concluding remarks

In summary, we have explored the physical consequences of the vacuum exposed to non-linear electrodynamics coupled to parameters that signal a violation of Lorentz-symmetry. This new vacuum behaves as a medium with nontrivial refractive indices \(n_ \bot \) and \(n_ \parallel \). Hence, a charged particle interacting with this new medium can emit Cherenkov radiation if the velocity of the charged particle is larger than the velocity of light in the medium. In addition, we have calculated the interaction potential for logarithmic electrodynamics with the term CFJ. Our calculations show that the interaction energy, at leading order in \(\beta ^{2}\), is similar to our previous calculation [21].

Finally, having in mind that several astrophysical phenomena are strongly influenced by magnetic fields present in the cosmo and that many of these phenomena can be systematically described only in terms of Magnetohydrodynamics (MHD) processes [34, 35], we are taking as a step forward the re-assessment of the MHD equations in the presence of the space-time anisotropy parametrized by the CFJ 4-vector, \(v_\mu \), along with non-linear effects. We recall the MHD equations arise by coupling the equations of fluid mechanics with Maxwell equations. New effects introduced in the latter may modify the MHD set-up and induce corrections or even unveil interesting effects in astrophysical plasmas, such as stellar coronae, magnetospheres, regions around pulsars/ magnetars/kilonovas, black hole accretion disks, and AGN jets. More than \(90 \%\) of visible matter is found in the plasma state; this justifies an endeavor to investigate MHD equations in presence of post-Maxwellian terms induced by physics beyond the Standard Model, as the ones we have considered in this contribution.

As a natural path opens, particular wave modes that stem from the MHD equations, such as Alfvén waves and slow/fast magnetosonic waves, should be re-assessed. These waves, that experience no dispersion, are expected to become dispersive by the LSV term in the Maxwell equations. Recognizing the relevance of MHD waves in astrophysical plasmas [36], we intend to concentrate efforts to re-inspect the relevant phenomenon of magnetic reconnection by considering now the modified MHD equations.