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 $μγ$ 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.