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

We study photon propagation in a strong magnetic field $B\sim B_{\mathrm{c}\mathrm{r}}$, where $B_{\mathrm{c}\mathrm{r}}=\frac{m^2}{e}\simeq 4.4\times {10}^{13}$ 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 $0\le B\le 30B_{\mathrm{c}\mathrm{r}}$. We find that the anomalous magnetic moment ${\mu }_{\gamma }$ of a photon for $B=30B_{\mathrm{c}\mathrm{r}}$ is $\sim 8\mathrm{/}3$ of the anomalous magnetic moment of a photon for $B=1\mathrm{/}2B_{\mathrm{c}\mathrm{r}}$. 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.