One-loop Euler-Heisenberg Lagrangian for U(1)×U(1) dyon QED produces hybrid refractive indices and vacuum birefringence that reduce exactly to ordinary QED when magnetic charge vanishes.
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The Euler-Heisenberg action for a $U(1)\times U(1)$ dyon quantum electrodynamics
One-loop Euler-Heisenberg Lagrangian for U(1)×U(1) dyon QED produces hybrid refractive indices and vacuum birefringence that reduce exactly to ordinary QED when magnetic charge vanishes.