# Constraining Born–Infeld-like nonlinear electrodynamics using hydrogen’s ionization energy

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## Abstract

In this work, the hydrogen’s ionization energy was used to constrain the free parameter *b* of three Born–Infeld-like electrodynamics namely Born–Infeld itself, Logarithmic electrodynamics and Exponential electrodynamics. An analytical methodology capable of calculating the hydrogen ground state energy level correction for a generic nonlinear electrodynamics was developed. Using the experimental uncertainty in the ground state energy of the hydrogen atom, the bound \(b>5.37\times 10^{20}K\frac{V}{m}\), where \(K=2\), \(4\sqrt{2}/3\) and \(\sqrt{\pi }\) for the Born–Infeld, Logarithmic and Exponential electrodynamics respectively, was established. In the particular case of Born–Infeld electrodynamics, the constraint found for *b* was compared with other constraints present in the literature.

## 1 Introduction

Nonlinear electrodynamics (NLED) are extensions of Maxwell’s electromagnetism which arise when self-interaction in field equations is allowed. From the axiomatic point of view, they can be built from a Lagrangian of a vector field that respects three conditions: invariance under the Lorentz group, invariance under the *U*(1) gauge group and the Lagrangian depending only on combinations of the field and its first derivative, i.e. \(\mathcal {L}=\mathcal {L}\left( A_{\mu },\partial _{\nu }A_{\mu }\right) \).

The first two NLED proposals emerged in the 1930s in two very different contexts. In 1934, Born and Infeld proposed the Born–Infeld electrodynamics (BI) in order to deal with the divergence of the self-energy of a point charge [1, 2]. The BI electrodynamics was originally conceived as a fundamental theory for electromagnetism, but later it was found that it was not renormalizable and therefore should be considered as an effective theory.^{1} In 1936, W. Heisenberg and H. Euler showed that, for energies below the electron mass, the self-coupling of the electromagnetic field induced by virtual pairs of electron-positrons can be treated as an effective field theory [4]. This theory is known as Euler–Heisenberg electrodynamics and it provided the first description of the vacuum polarization effect present in the QED [5].

Due to different motivations, other nonlinear electrodynamics were proposed – e.g. Logarithmic and Exponential electrodynamics [6, 7, 8, 9] – and the NLED became a class of electromagnetic theories [10]. This class of theories has applications in several branches of physics being particularly interesting in systems where the NLED are minimally coupled with gravitation as in the cases of charged black holes [11, 12, 13, 14, 15, 16, 17] and cosmology [18, 19, 20, 21, 22].

Nonlinear electrodynamics have some different features with respect to Maxwell’s electrodynamics. Among these features, the most interesting is its non-trivial structure for radiation propagation. Due to nonlinearity of the field equations, the electromagnetic field self-interacts generating deformities in the light cone [23]. Thus, in the NLED context, the introduction of a background field affects the propagation velocity of the electromagnetic waves and generates the birefringence phenomenon. This phenomenon is present in all physically acceptable NLED with the exception of BI electrodynamics [24].

Excluding the Euler–Heisenberg electrodynamics and its variations [5], all other NLED have at least one free parameter which must be experimentally constrained [25]. These constraints can be directly obtained from measurements of atomic transitions [26, 27] and photon-photon scattering [28, 29] associated with self-interaction of NLED. Another possibility occurs in the astrophysical context where bounds to the NLED are imposed through photon splitting process present in magnetars spectra [30]. Moreover, for NLED where the birefringence effect is not negligible, bounds can be established through measurements of vacuum magnetic birefringence generated by the passage of a polarized laser beam through a magnetic dipole field (PVLAS collaboration – see [31] and references therein).

The purpose of this paper is to build a procedure capable of constraining nonlinear electrodynamics and to apply this procedure to three Born–Infeld-like electrodynamics. In Sect. 2, an introduction to the NLED is presented with emphasis on three specific nonlinear electrodynamics: Born–Infeld NLED, Logarithmic NLED and Exponential NLED. The procedure based on the hydrogen’s ionization energy which constrains NLED is developed in Sect. 3. In this section, bounds on the free parameters of each NLED are established and the results obtained are compared with those present in the literature. The final remarks are discussed in Sect. 4.

## 2 Nonlinear electrodynamics

### 2.1 Born–Infeld-like electrodynamics

*F*and

*G*. In this case, the Lagrangian can be written as a series of the invariants

*G*can be neglected because of Bianchi identity. The main NLED (Born–Infeld, Euler–Heisenberg, etc) have this structure. For instance, the first coefficients for Born–Infeld electrodynamics [1] are

#### 2.1.1 Born–Infeld electrodynamics

*e*is the electron charge and

*Z*is the atomic number. The substitution of this expression into (10), with \(\vec {B}=0\), results in the electric field given by

#### 2.1.2 Logarithmic and Exponential electrodynamics

The Logarithmic and Exponential electrodynamics belong to a special class, called Born–Infeld-like NLED, which was proposed in order to study topics such as inflation [6] and exact solutions of spherically symmetric static black holes [7, 8]. These electrodynamics are characterized by having a finite self-energy solution for a point-like charge but, unlike Born–Infeld NLED, they predict a birefringence effect in the presence of an electromagnetic background field.

*x*and \(\varepsilon \) are defined as in (14). The function \(W\left( z\right) \) is the Lambert function

^{2}defined as the inverse function of \(z\left( W\right) =We^{W}\). When \(\varepsilon \rightarrow 0\), both electric fields reduce to the Maxwell case. Besides, \(\vec {E}_{Ex}\) diverges at the origin but slower than Maxwell, and \(\vec {E}_{Lg}\) is bounded from above in a similar way such as Born–Infeld field.

## 3 Testing NLED using hydrogen’s ionization energy

The theory about the energy levels of a hydrogen-like atom is described by the quantization of Dirac equation and subject to several correction factors such as the relativistic-recoil of the nucleus, electron self-energy, vacuum polarization due to the creation of virtual electron-positron pairs, etc (for details see [34, 35] and references therein). This theoretical structure in the context of Maxwell electrodynamics establishes a theoretical experimental agreement for the hydrogen’s ionization (HI) energy of 2 parts per \(10^{10}\) [36]. Thus, any correction to HI energy generated by modifications in the Maxwell potential must be a small correction and it can be treated perturbatively.

### 3.1 Hydrogen’s ionization energy for Born–Infeld electrodynamics

^{3}

Results of \(\frac{a_{0}}{\left( Ze\right) ^{2}}E_{HI_{1}} ^{BI}\) for \(\varepsilon =10^{-1}\), \(10^{-3}\) and \(10^{-5}\). The second column shows the numerical result calculated from (22), the third column shows the first order correction given by the first term in (23) and the last column shows the relative error between the two approaches

\(\varepsilon \) | Numerical | First correction | Relative error (%) |
---|---|---|---|

\(10^{-1}\) | \(5.66\times 10^{-3}\) | \(6.67\times 10^{-3}\) | 17.758 |

\(10^{-3}\) | \(6.65\times 10^{-7}\) | \(6.67\times 10^{-7}\) | 0.185 |

\(10^{-5}\) | \(6.67\times 10^{-11}\) | \(6.67\times 10^{-11}\) | 0.0019 |

### 3.2 Hydrogen’s ionization energy for Logarithmic electrodynamics

Results of \(\frac{a_{0}}{\left( Ze\right) ^{2}}E_{HI_{1}} ^{Lg}\) for \(\varepsilon =10^{-1}\), \(10^{-3}\) and \(10^{-5}\). The second column shows the numerical result calculated from (24), the third column shows the first order correction given by (25) and the last column shows the relative error between the two approaches

\(\varepsilon \) | Numerical | First correction | Relative error (%) |
---|---|---|---|

\(10^{-1}\) | \(5.325\times 10^{-3}\) | \(6.285\times 10^{-3}\) | 18.044 |

\(10^{-3}\) | \(6.274\times 10^{-7}\) | \(6.285\times 10^{-7}\) | 0.188 |

\(10^{-5}\) | \(6.285\times 10^{-11}\) | \(6.285\times 10^{-11}\) | 0.0019 |

### 3.3 Hydrogen’s ionization energy for Exponential electrodynamics

*W*after the change of variable \(ue^{u}=\frac{\varepsilon ^{4}}{x^{4}}\). The other three integrals do not have analytical solutions. However, approximated solutions can be achieved following the steps described in Appendix A. The leading order correction for HI energy due the Exponential electrodynamics is obtained substituting (27), (A2), (A3) and (A4) into (26):

Results of \(\frac{a_{0}}{\left( Ze\right) ^{2}}E_{HI_{1}} ^{Ex}\) for \(\varepsilon =10^{-1}\), \(10^{-3}\) and \(10^{-5}\). The second column shows the numerical result calculated from (26), the third column shows the first order correction given by (28) and the last column shows the relative error between the two approaches

\(\varepsilon \) | Numerical | First correction | Relative error (%) |
---|---|---|---|

\(10^{-1}\) | \(4.989\times 10^{-3}\) | \(5.908\times 10^{-3}\) | 18.424 |

\(10^{-3}\) | \(5.897\times 10^{-7}\) | \(5.908\times 10^{-7}\) | 0.193 |

\(10^{-5}\) | \(5.908\times 10^{-11}\) | \(5.908\times 10^{-11}\) | 0.0019 |

### 3.4 Constraining parameter *b*

*b*(with \(Z=1\)) is constrained by the expression

## 4 Final remarks

In this work the ground state energy level correction \(E_{HI_{1}}\) for the hydrogen atom generated by three Born–Infeld-like electrodynamics was obtained. More specifically, a general expression for the correction \(E_{HI_{1}}\) was derived through a perturbative approach, then this correction was calculated for the Born–Infeld, Logarithmic and Exponential electrodynamics. Using the experimental uncertainty for HI energy, the free parameters *b*’s of each of these NLED were lower bounded, and for the particular Born–Infeld case the result found was compared with other constraints present in the literature. It is worth mentioning that the method developed here based on the expression (21) and the techniques of the Appendix A can easily be extended to constrain other nonlinear electrodynamics.

An important point in the derivation of \(E_{HI_{1}}\) concerns the need to know the electric field exactly (see discussion below Eq. (20)). This point can be observed by the distinct values obtained for \(E_{HI_{1}}^{BI}\), \(E_{HI_{1}}^{Lg}\) and \(E_{HI_{1}}^{Ex}\). Although different, these values are similar and we can wonder if the expression (30) could be used to constrain a more general class of NLED. The necessary and sufficient condition to apply the result (30) to others NLED is related to the behavior of the electric field. Observing Fig. 1 and the values of *K* (\(K_{BI}=2\), \(K_{Lg}=4\sqrt{2}/3\) and \(K_{Ex}=\sqrt{\pi }\)) we see that the greater is the difference between the Maxwell and Born–Infeld-like NLED electric fields the higher is the *K* value. Thus, we can state that any NLED whose electric field absolute value \(E_{NLED}\) fulfills the condition \(E_{BI}<E_{NLED}<E_{Ex}\) will have \(b_{Ex}<b_{NLED}<b_{BI}\). Also, since *K* slightly varies from \(K_{Ex}\) to \(K_{BI}\) we can estimate that any NLED which has an \(E_{NLED}\) near to \(E_{BI}\) or \(E_{Ex}\) will have its free parameter bounded by \(b_{NLED} \gtrsim 10^{21}V/m\). Thus, we can impose limits on a broad class of NLED only by knowing the behavior of its electric field.

Finally, it is important to discuss the possibility of application involving the electrodynamics of Euler–Heisenberg (EH) [4, 5]. EH electrodynamics is an effective description of the self-interaction process due the electron-positron virtual pairs present in QED (vacuum polarization). Thus, starting from EH NLED one could think of using the procedure developed in this work to obtain, in an alternative way, the vacuum polarization correction for the hydrogen’s ionization energy [34, 43]. The problem with this approach is that the EH electrodynamics is built assuming a slowly varying electromagnetic field in distances of the order of the electron Compton wavelength \(\lambda _{e}\), and this requirement is not satisfied in the calculation of \(E_{HI_{1}}\). The essential part of the integral \(E_{HI_{1}}\) is in the range [0,1[ (see Appendix A), and within this range the EH electric field rapidly varies at distances of order \(\lambda _{e} \). Therefore, the vacuum polarization effect associated with the hydrogen atom can not be described by the Euler–Heisenberg effective electrodynamics [43].

## Footnotes

- 1.
This kind of approach was explicitly performed when Fradkin and Tseytlin showed that BI electrodynamics appears as an effective theory of low energies in open string theories [3].

- 2.
For \(z\in \mathbb {R} \) and \(z\ge 0\), \(W\left( z\right) \ge 0\) and it is monotonically increasing. Besides, \(\lim _{z\rightarrow 0}\frac{W\left( z\right) }{z}=1\) and \(\lim _{z\rightarrow \infty }W\left( z\right) =\infty \).

- 3.
The \(\gamma _{E}\) is the Euler–Mascheroni constant.

## Notes

### Acknowledgements

The authors acknowledge A. E. Kramida and R. R. Cuzinatto for their useful comments. They are also grateful to CNPq-Brazil for financial support.

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