QED Fermions in a noisy magnetic field background: The effective action approach

We consider the effects of a noisy magnetic field background over the fermion propagator in QED, as an approximation to the spatial inhomogeneities and time-fluctuations that would naturally arise in certain physical scenarios, such as heavy-ion collisions or the quark-gluon plasma in the early stages of the evolution of the Universe. We considered a classical, finite and uniform average magnetic field background $⟨\mathbf{B}(\mathbf{x}){⟩}_{\mathrm{\Delta }}=\mathbf{B}$, subject to white-noise fluctuations with auto-correlation of magnitude ${\mathrm{\Delta }}_B$. By means of the Schwinger representation of the propagator in the average magnetic field as a reference system, we used the replica formalism to study the effects of the magnetic noise at the mean field level, in terms of a vector order parameter $Q_j=\text{ii}e{\mathrm{\Delta }}_B⟨⟨\bar{\psi }{\gamma }_j\psi ⟩{⟩}_{\mathrm{\Delta }}$ whose magnitude represents the ensemble average (over magnetic noise) of the fermion currents. We identified the region where this order parameter acquires a finite value, thus breaking the $U(1)$-symmetry of the model due to the presence of the magnetic noise.