Vacuum Birefringence in a Supercritical Magnetic Field and a Subcritical Electric Field

Abstract: Recent ultra-intense lasers of subcritical fields and proposed observations of the x-rays polarization from highly magnetized neutron stars of supercritical fields have attracted attention to vacuum birefringence, a unique feature of nonlinear electrodynamics. We propose a formulation of vacuum birefringence that incorporates the effects of the weaker electric field added to the extremely strong magnetic field. To do so, we first derive a closed analytical expression for the one-loop effective Lagrangian for the combined magnetic and electric fields by using an explicit formula of the one-loop effective Lagrangian for an arbitrarily strong magnetic field. We then employ the expression to derive the polarization and magnetization of the vacuum, from which the permittivity and permeability for weak probe fields are obtained. Finally, we find the refractive indices and the associated polarization vectors for the case of parallel magnetic and electric fields. The proposed formulation predicts that an electric field along the magnetic field reduces the birefringence and rotates the polarization vectors. Such effects should be taken into account for accurate polarimetry of the x-rays from magnetized neutron stars, which will prove the fundamental aspect of the strong field quantum electrodynamics (QED) and explore the extreme fields of astrophysical bodies.