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

Abstract

We study photon propagation in a strong magnetic field $B\sim B_{\mathrm{cr}}$, where $B_{\mathrm{cr}}=\frac{m^2}{e}\simeq 4.4\times 10^{13}$ 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 $0\le B\le 30B_{\mathrm{cr}}$. We find that the anomalous magnetic moment ${\mu }_{\gamma }$ of a photon for $B=30B_{\mathrm{cr}}$ is $\sim 8/3$ of the anomalous magnetic moment of a photon for $B=1/2B_{\mathrm{cr}}$. We establish new connections between ${\mu }_{\gamma }$, vacuum birefringence, and directly measurable polarization observables. Based on recent experimental observations -- including the ATLAS detection of light-by-light scattering at $8.2\sigma$ 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.

Comments: 15 pages, 2 figures, references added