Wilsonian renormalization group flows are ordinary distributions

In nonperturbative formulation of quantum field theory (QFT), the vacuum state is characterized by the Wilsonian renormalization group (RG) flow of Feynman type field correlators. Such a flow is a parametric family of ultraviolet (UV) regularized field correlators, the parameter being the strength of the UV regularization, and the instances with different strength of UV regularizations are linked by the renormalization group equation (RGE). For renormalizable QFTs, the flow is meaningful at any UV regularization strengths. In this paper it is shown that for these flows a natural, mathematically rigorous generally covariant definition can be given, and that they form a well-behaved generalized function space. The main theorem proved in the paper is that the running of Wilsonian RG flows of renormalizable QFTs, for bosonic fields over flat (affine) spacetime, factorize in a rather simple manner: they always originate from a regularization-independent distributional correlator, and its running is described by an algebraic ansatz, independent of the underlying QFT model details.

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