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Nonsingular black holes in nonlinear gravity coupled to Euler-Heisenberg electrodynamics

We study static, spherically symmetric black holes supported by Euler-Heisenberg theory of electrodynamics and coupled to two different modified theories of gravity. Such theories are the quadratic $f(R)$ model and Eddington-inspired Born-Infeld gravity, both formulated in metric-affine spaces, where metric and affine connection are independent fields. We find exact solutions of the corresponding field equations in both cases, characterized by mass, charge, the Euler-Heisenberg coupling parameter and the modified gravity one. For each such family of solutions, we characterize its horizon structure and the modifications in the innermost region, finding that some subclasses are geodesically complete. The singularity regularization is achieved under two different mechanisms: either the boundary of the manifold is pushed to an infinite affine distance, not being able to be reached in finite time by any geodesic, or the presence of a wormhole structure allows for the smooth extension of all geodesics overcoming the maximum of the potential barrier.