# Constraints on beta functions in field theories

### Submission summary

 As Contributors: Sung-Sik Lee · Han Ma Arxiv Link: https://arxiv.org/abs/2009.11880v5 (pdf) Date accepted: 2021-11-02 Date submitted: 2021-10-11 21:57 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:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

### List of changes

We have added a paragraph that lists open questions and future directions in the conclusion of the revised manuscript.