Weak disorder corrections of the scattering and transport mean free path

We present a calculation of the weak localisation correction to the scattering and to the transport mean free path, for waves propagating in a $delta$-correlated random potential, going beyond the usual diffusion approximation for the loops in the interfering path amplitudes. We find that not only the transport mean free path, but also the scattering mean free path is affected by the interference contributions. We also find the dependence of the leading order contribution to both, the scattering and the transport mean free path, on the disorder parameter $1\mathrm{/}kl$ to be linear. This is in contrast to the result obtained from the diffusion approximation according to which the scattering mean free path remains unaffected when changing the disorder parameter, whereas the correction of the transport mean free path scales like $1\mathrm{/}(kl{)}^2$.