Finite-Field QED Corrections to Vacuum Birefringence and Magnetar Polarization Transport

Abstract

{We study low-energy photon propagation in a constant magnetic field within the one-loop Heisenberg--Euler theory, retaining the refractive-index normalization $γ_s$ without expansion. Here ``finite-field'' denotes exact dependence on $B/B_{\rm cr}$ within the one-loop, constant-field approximation. The resulting birefringence is propagated into magnetar polarization transport. In a centered-dipole model, the polarization-limiting radius is unchanged to better than $10^{-12}$ because mode decoupling occurs at $\sim10^2R_{\rm NS}$, where $B\ll B_{\rm cr}$. Near the surface, however, the weak-field Cotton--Mouton expression overestimates the accumulated birefringent phase by up to a factor $2.9$ at $10^{15}$~G. At the plasma--vacuum resonance, finite-field corrections reduce the resonance density by $32\%$ and raise the adiabatic conversion energy by $14\%$ for 1E~1547.0$-$5408; the corresponding changes are factors $2.6$ and $1.37$ for 1RXS~J1708$-$4009, and factors $9.7$ and $2.13$ for SGR~1806$-$20, the latter controlled by the strong-field asymptote. The resummed one-loop parallel-mode magnetic response remains positive and develops a broad maximum near $17B_{\rm cr}$. The strictly truncated $\mathcal O(α)$ response is monotonic; therefore the maximum is a structural prediction of the resummed one-loop constitutive model, while its detailed profile and precise location require higher-loop validation. These results identify vacuum-resonance observables as the most sensitive channel for testing finite-field QED in magnetars.