  We calculate the relative permittivity of a cold atomic gas under weak probe
illumination, up to second order in the density. Within the framework of a
diagrammatic representation method, we identify all the second order diagrams
that enter into the description of the relative permittivity for coherent light
transmission. These diagrams originate from pairwise position correlation and
recurrent scattering. Using coupled dipole equations, we numerically simulate
the coherent transmission with scalar and vector waves, and find good agreement
with the perturbative calculations. We applied this perturbative expansion
approach to a classical gas at rest, but the method is extendable to thermal
gas with finite atomic motion and to quantum gases where non-trivial pair
correlations can be naturally included.
