SciPost Phys. 15, 111 (2023) ·
published 22 September 2023

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We present the Wilsonian effective action as a solution of the exact RG equation for the critical $O(N)$ vector model in the large $N$ limit. Below four dimensions, the exact effective action can be expressed in a closed form as a transcendental function of two leading scaling operators with infinitely many derivatives. From the exact solution that describes the RG flow from a UV theory to the fixed point theory in the IR, we obtain the mapping between UV operators and IR scaling operators. It is shown that IR scaling operators are given by sums of infinitely many UV operators with infinitely many derivatives.
SciPost Phys. 12, 046 (2022) ·
published 1 February 2022

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The $\beta$functions describe how couplings run under the renormalization group flow in field theories. In general, all couplings that respect the symmetry and locality are generated under the renormalization group flow, and the exact renormalization group flow is characterized by the $\beta$functions defined in the infinite dimensional space of couplings. In this paper, we show that the renormalization group flow is highly constrained so that the $\beta$functions defined in a measure zero subspace of couplings completely determine the $\beta$functions in the entire space of couplings. We provide a quantum renormalization groupbased algorithm for reconstructing the full $\beta$functions from the $\beta$functions defined in the subspace. As examples, we derive the full $\beta$functions for the $O(N)$ vector model and the $O_L(N) \times O_R(N)$ matrix model entirely from the $\beta$functions defined in the subspace of singletrace couplings.