[Paper Review] Interaction of Electromagnetic Radiation with Supercritical Magnetic Field

A. E. Shabad|ArXiv.org|Jul 22, 2003

Crystallography and Radiation Phenomena4 references3 citations

TL;DR

This paper investigates the interaction of electromagnetic radiation with supercritical magnetic fields ($B \gg B_{\text{cr}} = 4.4 \times 10^{13}$ G) in the vacuum, showing that one polarization mode exhibits a refraction index that grows linearly with $B$, while the other saturates. The effect arises from the one-loop polarization operator's asymptotic expansion, leading to strong, direction-dependent refraction for soft photons across X-ray, optical, and radio bands.

ABSTRACT

It is pointed, that effects of refraction of electromagnetic radiation in the medium, formed by the magnetized vacuum, become essential already for relatively soft photons, not hard enough to create an electron-positron pair, including those belonging to soft gamma-, X-ray, optic and radio- range, if the magnetic field B exceeds the critical value of Bcr=m^2/e=4.4 10^13 Gauss. Three leading terms in the asymptotic expansion of the one-loop polarization operator in a constant magnetic field are found for B>>Bcr, and the corresponding refraction index is shown to depend only on the propagation direction of the photon relative to the external field. It is established, that the refraction index for one of polarization modes unlimitedly grows with the field, while the other is saturated at a moderate level. The photon capture effect is extended to soft photons. The results may be essential in studying reflection, refraction and splitting of X-rays, light and radio waves by magnetic fields of magnetars, as well as in considering emission of such waves by charged particles .

Motivation & Objective

• To analyze the propagation of electromagnetic radiation in supercritical magnetic fields ($B \gg B_{\text{cr}}$) beyond the standard $̲\gamma$-ray regime.
• To determine how the one-loop polarization operator's asymptotic behavior in strong fields alters photon refraction and dispersion.
• To extend the photon capture effect to soft photons (X-ray, optical, radio) by identifying a new, field-strength-dependent refraction mechanism.
• To establish that refraction becomes significant even for low-energy photons when $B \gg B_{\text{cr}}$, particularly in magnetar magnetospheres.

Proposed method

• Derives the asymptotic expansion of the one-loop polarization operator eigenvalues in a constant magnetic field for $B \gg B_{\text{cr}}$.
• Identifies three leading terms in the expansion of the polarization operator, focusing on the behavior of the eigenvalue $\kappa_2$.
• Applies exact field-theoretic constraints—relativistic invariance, gauge invariance, and Onsager's theorem—without approximations.
• Evaluates the dispersion relation using the polarization operator to compute the refractive index as a function of photon propagation direction relative to $\mathbf{B}$.
• Performs asymptotic integration over proper time and momentum variables, isolating the dominant $B$-dependent contributions.
• Neglects subleading terms (scaling as $z_1/eB$, $z_2/eB$) to focus on the linear $B$-dependence in $\kappa_2$, which governs the refractive index.

Experimental results

Research questions

• RQ1 How does the refraction index of electromagnetic radiation in a supercritical magnetic field depend on the photon's propagation direction relative to $\mathbf{B}$?
• RQ2 What is the asymptotic behavior of the one-loop polarization operator eigenvalues for $B \gg B_{\text{cr}}$?
• RQ3 Can the photon capture effect—previously restricted to high-energy $\gamma$-rays—be extended to soft photons (X-ray, optical, radio) in supercritical fields?
• RQ4 Why does one polarization mode exhibit a divergent refraction index with increasing $B$, while the other saturates?
• RQ5 What is the role of the $\kappa_2$ eigenvalue in enabling strong, direction-sensitive refraction for low-energy photons?

Key findings

• For $B \gg B_{\text{cr}}$, one polarization mode's refractive index grows linearly with $B$, while the other saturates at a moderate value.
• The dominant contribution to the polarization operator's eigenvalue $\kappa_2$ scales as $B/B_{\text{cr}}$, leading to a direction-dependent, large refraction index.
• Refraction becomes significant for soft photons (X-ray, optical, radio) due to the $B$-linear term in $\kappa_2$, even when photons are too soft to create $e^+e^-$ pairs.
• The photon capture effect is extended to soft photons because the modified dispersion relation allows for strong interaction with the magnetized vacuum.
• The refraction index depends only on the angle between the photon momentum and the magnetic field direction, not on photon energy in the kinematic domain far from thresholds.
• The asymptotic expansion of the polarization operator yields a leading-order term proportional to $eB/\pi$, which dominates over logarithmic and constant terms in the $B \to \infty$ limit.