| Summary: | Abstract Recent ultra-intense lasers of subcritical fields and proposed X-ray polarimetry for highly magnetized neutron stars of supercritical fields have attracted attention to vacuum birefringence, a unique feature of the nonlinear vacuum under strong electromagnetic fields. We propose a formulation of the vacuum birefringence in a strong magnetic field ( $${\textbf{B}}$$ B ) and a weak electric field ( $${\textbf{E}}$$ E ), including the effect of electromagnetic wrench ( $$G \equiv -{\textbf{E}}\cdot {\textbf{B}}\ne 0$$ G ≡ - E · B ≠ 0 ). To do so, we derive a closed expression of the one-loop effective Lagrangian for the combined magnetic and electric fields by using the 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 we obtain the permittivity and permeability for a weak probe field. Specifically, we find the refractive indices and the associated polarization vectors of the probe field for the case of parallel magnetic and electric fields. The proposed formulation reproduces the known results for pure magnetic fields in the proper limit. Finally, we apply the formulation to the Goldreich–Julian pulsar model. Our formulation reveals the importance of the electromagnetic wrench in vacuum birefringence: it can reduce the difference between refractive indices and rotate polarization vectors to a significant degree. Such a quantitative understanding is crucial to the X-ray polarimetry for magnetized neutron stars or magnetars, which will demonstrate the fundamental feature of the strongly-modified quantum vacuum and estimate the extreme fields surrounding those astrophysical bodies.
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