The 1-loop vacuum polarization for a graphene-like medium in an external magnetic field; corrections to the Coulomb potential
Résumé
I calculate the 1-loop vacuum polarization
$\Pi_{\mu\nu}(k,B,a)$
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 $\Pi_{\mu\nu}$ factorize into a quantum, ``reduced''
$T_{\mu\nu}(\hat k,B)$
and a transmittance function $V(k,a)$, in which
the geometry of the sample and the resulting confinement of the
$\gamma\,e^+\,e^-$ 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 $\Pi_{\mu\nu}$ and to fix
the corresponding counterterms. Their connection with the
transversality of $\Pi_{\mu\nu}$ is investigated.
The corrections to the Coulomb potential and their dependence on $B$
strongly differ from QED$_{3+1}$.