Perturbative approach to the RG $\beta $-function for the 3-d Anderson localization

The $\beta $-function of the conductance for Anderson Metal-Insulator transition in 2+ $\varepsilon $ dimensions is known from the non-linear sigma model. However, the result is valid for small $\varepsilon $ only. Recently, the $\beta $-function for the two-dimensional unitary case up to two-loop order was reproduced within a standard diagrammatic perturbation theory by including contributions from the ballistic regime in a consistent way [P. Ostrovskii (2009), unpublished]. An extension of the method to three dimensions will be discussed. The result in leading order in 1/g (g=dimensionless conductance) is $\beta $(g)=1-a/g, where a is a constant.