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The 1-loop vacuum polarization for a graphene-like medium in an external magnetic field; corrections to the Coulomb potential

I calculate the 1-loop vacuum polarization $Π_{μν}(k,B,a)$ for a photon of momentum $k=(\hat k,k_3)$ interacting with the electrons of a thin medium of thickness $2a$ simulating graphene, in the presence of a constant and uniform external magnetic field $B$ orthogonal to it (parallel to $k_3$). Calculations are done with the techniques of Schwinger, adapted to the geometry and Hamiltonian under scrutiny. The situation gets more involved than for the electron self-energy because the photon is now allowed to also propagate outside the medium. This makes $Π_{μν}$ factorize into a quantum, "reduced" $T_{μν}(\hat k,B)$ and a transmittance function $V(k,a)$, in which the geometry of the sample and the resulting confinement of the $γ\,e^+\,e^-$ vertices play major roles. This drags the results away from reduced QED$_{3+1}$ on a 2-brane. The finiteness of $V$ at $k^2=0$ is an essential ingredient to fulfill suitable renormalization condition for $Π_{μν}$ and to fix the corresponding counterterms. Their connection with the transversality of $Π_{μν}$ is investigated. The corrections to the Coulomb potential and their dependence on $B$ strongly differ from QED$_{3+1}$.