Constraints on beta functions in field theories

Abstract

The \betaβ-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 \betaβ-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 \betaβ-functions defined in a measure zero subspace of couplings completely determine the \betaβ-functions in the entire space of couplings. We provide a quantum renormalization group-based algorithm for reconstructing the full \betaβ-functions from the \betaβ-functions defined in the subspace. As examples, we derive the full \betaβ-functions for the O(N)O(N) vector model and the O_L(N) \times O_R(N)OL(N)×OR(N) matrix model entirely from the \betaβ-functions defined in the subspace of single-trace couplings.

Exact effective action for the O(N) vector model in the large N limit

“…The β-functions are then replaced with an action that governs the dynamics of the z-dependent hopping field defined in a (D + 1)-dimensional bulk [27,28,31], where the extra direction corresponds to the length scale ln z. The fact that the β-functions for all higher order couplings can be encoded in the dynamics of a much smaller subset of couplings also implies that the original β-functions are highly constrained even in the presence of arbitrary irrelevant deformations [30]. In the phase space path integral representation, |S z * 1 〉 can be written as…”

“…We note that most of the results obtained in our paper for the O(N ) vector model through the quantum RG can be in principle obtained through alternative methods that use collective fields. However, the quantum RG can be readily extended to more complicated theories such as matrix models [27,30] for which the method of the collective field used for the vector model is not applicable.…”