The exact renormalization group and dimensional regularization

Abstract

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-\epsilon, 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 \mu\to0, if a particular relation between the scale parameter \mu and \epsilon is employed. In general, it is shown that in this scheme the solutions to the ERG equations for the proper functions coincide when \mu\to0 with the complete diagrammatic contributions appearing in field theory for these functions and this theory, provided that exact relations between \mu and \epsilon hold. In addition a non-perturbative 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.