The exact renormalization group and dimensional regularization

The exact renormalization group (ERG) is formulated implementing the decimation of degrees of freedom by means of a particular momentum integration measure. The definition of this measure involves a distribution that links this decimation process with the dimensional regularization technique employed in field theory calculations. Taking the dimension $d=4-ϵ$, the one-loop solutions to the ERG equations for the scalar field theory in this scheme are shown to coincide with the dimensionally regularized perturbative field theory calculation in the limit $ϵ\to 0^{-}$, if a particular relation between the scale parameter $\mu$ and $ϵ$ is employed. In general, it is shown that in this scheme the solutions to the ERG equations for the proper functions coincide when $ϵ\to 0^{-}$ with the complete diagrammatic contributions appearing in field theory for these functions and this theory, provided that exact relations between $\mu$ and $ϵ$ hold. In addition, a nonperturbative approximation is considered. This approximation consists in a truncation of the ERG equations, which by means of a low-momentum expansion leads to reasonable results.