# Constraints on beta functions in field theories

### Submission summary

 As Contributors: Han Ma Arxiv Link: https://arxiv.org/abs/2009.11880v4 (pdf) Date submitted: 2021-07-18 20:54 Submitted by: Ma, Han Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical

### Abstract

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 group-based 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 single-trace couplings.

###### Current status:
Editor-in-charge assigned

### List of changes

The main change of the manuscript is that we added a new section (Sec. II) in which our result is applied to two realistic field theories: the O(N) vector model and the O(N)*O(N) matrix model. In the revised manuscript, we explicitly construct the well-defined bulk theories that govern the quantum RG flow of these realistic models. We also demonstrates the main claim of our paper by constructing all beta functions entirely from the beta functions defined in the subspace of single-trace couplings in those theories.