# Finite field-energy and interparticle potential in logarithmic electrodynamics

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## Abstract

We pursue an investigation of logarithmic electrodynamics, for which the field energy of a point-like charge is finite, as happens in the case of the usual Born–Infeld electrodynamics. We also show that, contrary to the latter, logarithmic electrodynamics exhibits the feature of birefringence. Next, we analyze the lowest-order modifications for both logarithmic electrodynamics and for its non-commutative version, within the framework of the gauge-invariant path-dependent variables formalism. The calculation shows a long-range correction (\(1/r^5\)-type) to the Coulomb potential for logarithmic electrodynamics. Interestingly enough, for its non-commutative version, the interaction energy is ultraviolet finite. We highlight the role played by the new quantum of length in our analysis.

## Keywords

Electrostatic Field Lagrangian Density Star Product Nonlinear Electrodynamic Electromagnetic Field Strength## 1 Introduction

The photon–photon scattering of Quantum Electrodynamics (QED) and its physical consequences such as vacuum birefringence and vacuum dichroism have been of great interest since its earliest days [1, 2, 3, 4, 5, 6, 7]. Even though this subject has had a revival after recent results of the PVLAS collaboration [8, 9], the issue remains as relevant as ever. We also point out alternative scenarios such as Born–Infeld theory [10], millicharged particles [11] or axion-like particles [12, 13, 14] in order to account for the results reported by the PVLAS collaboration.

We further note that recently considerable attention has been paid to the study of nonlinear electrodynamics due to its natural emergence from D-brane physics, where the Born–Infeld theory plays a prominent role. In addition to the string interest, nonlinear electrodynamics has also been investigated in the context of gravitational physics. In fact, Hoffman [15] was the one who first considered the connection between gravity and nonlinear electrodynamics (Born–Infeld theory). In passing we recall that these nonlinear gauge theories are endowed with interesting features, like a finite electron self-energy and a regular point charge electric field at the origin. Very recently, in addition to Born–Infeld theory, other types of nonlinear electrodynamics have been studied in the context of black hole physics [16, 17, 18, 19].

Let us also mention here that Lagrangian densities of nonlinear extensions of electrodynamics with a logarithmic function of the electromagnetic field strengths are a typical characteristic of QED effective actions. In the classical work by Euler and Heisenberg [20], in which the authors studied electrons in a background set up by a uniform electromagnetic field, a logarithmic term of the field strength came out as an exact 1-loop correction to the vacuum polarization. Furthermore, some years ago, Volovik [21] has worked out the action for an electromagnetic field that emerges as a collective field in superfluid \({}^3\hbox {He}-A\); this 4-dimensional action exhibits a logarithmic factor whose argument is a function of the electromagnetic field strengths [22].

On the other hand, it is worth recalling here that the study of extensions of the Standard Model (SM) such as Lorentz invariance violation and fundamental length, have attracted much attention in the past years [25, 26, 27, 28, 29]. As is well known, this is mainly so because the SM does not include a quantum theory of gravitation. In fact, the necessity of a new scenario has been suggested to overcome difficulties theoretical in the quantum gravity research. Among these new scenarios, probably the most studied framework are quantum field theories allowing non-commuting position operators [30, 31, 32, 33, 34, 35], where this non-commutativity is an intrinsic property of spacetime. We call attention to the fact that these studies have been achieved by using a star product (Moyal product). More recently, a novel way to formulate non-commutative field theory (or quantum field theory in the presence of a minimal length) has been proposed in [36, 37, 38]. Later, it has been shown that this approach can be summarized through the introduction of a new multiplication rule which is known as a Voros star product. Evidently, physics will turn out be independent from the choice of the type of product [39]. With these ideas in mind, in previous studies [40, 41], we have considered the effect of the spacetime non-commutativity on a physical observable. In fact, we have computed the static potential for axionic electrodynamics both in \((3+1)\) and \((2+1)\) spacetime dimensions, in the presence of a minimal length. The point we wish to emphasize, however, is that our analysis leads to a well-defined non-commutative interaction energy. Indeed, in both cases we have obtained a fully ultraviolet finite static potential. Later, we have extended our analysis for both Yang–Mills theory and gluodynamics in curved spacetime, where we have obtained a finite string tension [42].

Given the ongoing experiments related to photon–photon interaction physics [43, 44, 45], it is desirable to have some additional understanding of the physical consequences presented by a particular nonlinear electrodynamics, that is, logarithmic electrodynamics. Of special interest will be the study of aspects of birefringence as well as the computation of physical observables. In particular, we mention the static potential between two charges, using the gauge-invariant but path-dependent variables formalism, which is an alternative to the Wilson loop approach. We further note that the model under consideration satisfies the criteria of causality and unitarity as studied in [46].

Our work is organized according to the following outline: in Sect. 2, we present general aspects of logarithmic electrodynamics, show that it yields birefringence and compute the finite electrostatic field energy of a point-like charge. In Sect. 3, we analyze the interaction energy for a fermion–antifermion pair in the usual logarithmic electrodynamics and its version in the presence of a minimal length. Finally, in Sect. 4, we make final remarks.

## 2 The model under consideration

At this point, we would like to draw the reader’s attention to the recent work by Costa et al. [49], where these authors investigate a nonlinear gauge-invariant extension of classical electrodynamics, quartic in the field strength (they consider an \(F^2\)-term) and also attain a finite value for the field energy of a point-like charge.

## 3 Interaction energy

## 4 Final remarks

In summary, within the gauge-invariant but path-dependent variables formalism, we have considered the confinement versus screening issue for logarithmic electrodynamics. Once again, a correct identification of the physical degrees of freedom has been fundamental for understanding the physics hidden in gauge theories. We should highlight the different behaviors of the potentials associated to each of the models. In the logarithmic electrodynamics case, the static potential profile is similar to that encountered in Born–Infeld electrodynamics. Interestingly enough, its non-commutative version displays an ultraviolet finite static potential. The above analysis reveals the key role played by the new quantum of length in our analysis. In a general perspective, the benefit of considering the present approach is to provide unifications among different models, as well as exploiting the equivalence in explicit calculations, as we have illustrated in the course of this work.

## Notes

### Acknowledgments

This work was partially supported by Fondecyt (Chile) Grant 1130426.

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