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

We consider renormalization groups of transformations composed of a Gaussian convolution and a field dilatation. As an example, we consider perturbations of a single component real Euclidean free field $ϕ$ with covariance $(-\bigtriangleup)^{-1+\fracε{2}}$. We show that the renormalization group admits two equivalent formulations called massless picture and massive picture respectively. We then show in the massive picture that the renormalization group has a symmetry. The symmetry consists of global scale transformations composed with certain Gaussian convolutions. We translate the symmetry back to the massless picture. The relation between the symmetry and the notion of an anomalous dimension is briefly discussed.