Massless picture, massive picture, and symmetry in the Gaussian renormalization group

We study transformation properties of momentum space renormalization group transformations. The renormalization group transformations are composed of an integral over a high momentum field with a scale transformation of a low momentum field. As an example, we consider perturbations of a massless rea l euclidean free field phi with a smooth momentum space regulator. We show that this renormalization group admits two equivalent formulations called t he massless picture and the massive picture. In the massive picture, the re normalization group is shown to have a symmetry. The symmetry consists of g lobal (space-time independent) scale transformations of the field composed with a certain (gaussian) integral. We then translate the symmetry back to the massless picture. The relation between the symmetry and the notion of a n anomalous dimension is briefly discussed.