Vacuum Birefringence, Ellipticity, and the Anomalous Magnetic Moment of a Photon

Abstract

We study photon propagation in a strong magnetic field $B\sim B_{\rm{cr}}$, where $B_{\rm cr}= \frac{m^2}{e} \simeq 4.4 \times 10^{13}$ Gauss is the Schwinger critical field. We show that the expected value of the Hamiltonian of a quantized photon for a perpendicular mode is a convex function of the magnetic field $B$. We find that the anomalous magnetic moment of a photon in the one-loop approximation is a non-decreasing function of the magnetic field $B$ in the range $0\leq B \leq 30 \, B_{\rm cr}$. We find that the anomalous magnetic moment $μ_γ$ of a photon for $B=30\, B_{\rm cr}$ is $\sim 8/3$ of the anomalous magnetic moment of a photon for $B = 1/2 ~ B_{\rm cr}$. We establish new connections between $μ_γ$, vacuum birefringence, and directly measurable polarization observables. Based on recent experimental observations -- including the ATLAS detection of light-by-light scattering at $8.2σ$ significance, IXPE X-ray polarimetry of magnetars revealing polarization degrees up to 80\%, and continuing PVLAS measurements approaching QED sensitivity -- we provide predictions for ellipticity and polarization degree as important observables for future experiments. Numerical verification of our analytical results confirms the theoretical predictions with high precision.

Figures (2)

• Figure 1: Ellipticity $\chi/(\alpha |\mathbf{k}|\ell\sin^2\theta)$ as a function of $h = B_{\rm cr}/(2B)$. The rapid growth at small $h$ (strong-field regime) reflects the logarithmic structure of the Heisenberg--Euler Lagrangian.
• Figure 2: Numerical verification of theoretical predictions and comparison with experiments. Top left: Vacuum magnetic birefringence $|\Delta n|$ as a function of $B/B_{\rm cr}$, showing the $\Delta n \propto (B/B_{\rm cr})^2$ scaling. Vertical dashed lines indicate the field strengths probed by PVLAS (laboratory), VLT J1856 (weak field), and magnetars (strong field). Top right: PVLAS measurement compared with QED prediction. The blue curve shows the QED prediction $\Delta n = k_{\rm CM} B^2$; the green line and shaded region show the PVLAS measured value and $1\sigma$ uncertainty. The QED prediction (blue dotted) at $B = 2.5$ T lies within a factor of $\sim 5$ of current sensitivity. Bottom left: IXPE magnetar observations compared with theoretical expectations. Magnetars probe fields of $B = 3$--$45\,B_{\rm cr}$, in the strong-field regime where polarization effects are large. The high observed polarization degrees (15--65%) provide strong evidence for vacuum birefringence. Bottom right: Normalized photon magnetic moment $m\mu_\gamma/(\alpha|\mathbf{k}|\sin^2\theta)$ as a function of $B/B_{\rm cr}$, confirming the monotonic increase toward the asymptotic value of $2/3$ (dashed line), consistent with Eq. (\ref{['eq:mu_gamma_exact']}).