Towards the full Heisenberg-Euler effective action at large N

A bstract We study the Heisenberg-Euler effective action in constant electromagnetic fields F F ¯ for QED with N charged particle flavors of the same mass and charge e in the large N limit characterized by sending N → ∞ while keeping Ne 2 ∼ eF e F ¯ ∼ N 0 fixed. This immediately implies that contributions that scale with inverse powers of N can be neglected and the resulting effective action scales linearly with N. Interestingly, due to the presence of one-particle reducible diagrams, even in this limit the Heisenberg-Euler effective action receives contributions of arbitrary loop order. In particular for the special cases of electric- and magnetic-like field configurations we construct an explicit expression for the associated effective Lagrangian that, upon extremization for two constant scalar coefficients, allows to evaluate its full, all-order result at arbitrarily large field strengths. We demonstrate that our manifestly nonperturbative expression correctly reproduces the known results for the Heisenberg-Euler effective action at large N, namely its all-loop strong field limit and its low-order perturbative expansion in powers of the fine-structure constant.