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

We consider renormalization groups of transformations composed of a Gaussianconvolution and a field dilatation. As an example, we consider perturbations ofa single component real Euclidean free field $\phi$ with covariance$(-\bigtriangleup)^{-1+\frac{\epsilon}{2}}$. We show that the renormalizationgroup admits two equivalent formulations called massless picture and massivepicture respectively. We then show in the massive picture that therenormalization group has a symmetry. The symmetry consists of global scaletransformations composed with certain Gaussian convolutions. We translate thesymmetry back to the massless picture. The relation between the symmetry andthe notion of an anomalous dimension is briefly discussed.Comment: 17 pages LaTeX2