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Euler-Heisenberg Lagrangian to all orders in the magnetic field and the Chiral Magnetic Effect

In high energy heavy ion collisions as well as in astrophysical objects like magnetars extreme magnetic field strengths are reached. Thus, there exists a need to calculate divers QED processes to all orders in the magnetic field. We calculate the vacuum polarization graph in second order of the electric field and all orders of the magnetic field resulting in a generalization of the Euler-Heisenberg Lagrangian. We perform the calculation in the effective Lagrangian approach of J. Schwinger as well as using modified Feynman rules. We find that both approaches give the same results provided that the different finite renormalization terms are taken into account. Our results imply that any quantitative explanation of the recently proposed Chiral Magnetic Effect has to take 'Strong QED' effects into account, because these corrections are huge.