Constraints on beta functions in field theories

TL;DR

This paper demonstrates that the entire set of beta functions in certain field theories can be reconstructed from a measure zero subset, revealing strong constraints on the renormalization group flow.

Contribution

It introduces a quantum renormalization group algorithm to determine full beta functions from a subspace, with applications to specific models.

Findings

Full beta functions derived from subspace in $O(N)$ vector model

Full beta functions derived from subspace in matrix model

Flow constraints significantly reduce the complexity of beta function determination

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.