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.

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.

Current status:

Has been resubmitted

Reports on this Submission

Report #2 by
Anonymous
(Referee 4) on 2021-9-25
(Invited Report)

Cite as: Anonymous, Report on arXiv:2009.11880v4, delivered 2021-09-25, doi: 10.21468/SciPost.Report.3566

We thank the referee for the comments. Below, we provide our responses to the comments.

1) ``the procedure proposed by the authors is a mere rewriting of the Wilsonian path integral which, unless the the theory has a meaningful gravitational dual, does not lead to any improvement of understanding of RG flows''

Besides being an exact reformulation of the Wilsoninan RG, quantum RG (QRG) can be of practical use. First, it can be used to compute the exact quantum effective action of interacting field theories in the large N limit. Recently, the exact quantum effective action has been computed for the O(N) vector model in the large N limit via QRG [arXiv:2107.05654]. The exact effective action was not known before even for that relatively simple model because it includes operators with arbitrarily many fields and derivatives. Second, QRG provides a prescription to derive concrete holographic duals for general quantum field theories. As far as boundary quantum field theories are regularized, the bulk theories that arise from QRG are fully regularized. It is known that the bulk theory includes dynamical gravity [JHEP 1210 (2012) 160; JHEP 01 (2014) 076]. What is not well understood is how those regularized theory of dynamical gravity are related to continuum theories.

2) ``why not choosing just the field bi-linears as a basis also in the example of section IIB (O(N) × O(N) matrix model)?''

One can not choose bi-linears as a basis in the matrix model. This is because the single-trace operators should be singlets of the symmetry, and the set of bi-linear singlet operators of the matrix model do not form a complete basis of the vector space formed by the actions that respect the symmetry.

3) ``I would ask the authors to make a very concrete list of calculable physical quantities or properties of realistic QFTs that can be computed using their method, as well as an estimate of how hard the computation is''

We are grateful to the referee for the suggestion. In the revised manuscript, we added the following paragraph that includes a list of open questions and future directions. We feel that it is natural to place it in the conclusion as these are not directly related to the main content of the present paper.

We conclude with open questions and future directions. First, QRG can be used to compute the exact quantum effective action. The scale dependence of the quantum effective action obeys the exact RG equation[6,8]. In general solving the exact RG equation is challenging because the exact effective action includes operators made of arbitrarily many fields and derivatives. As a result, exact effective actions remain unknown even for relatively simple theories. In QRG, the exact RG equation is mapped to a quantum evolution of a wavefunction for single-trace couplings. Since the set of single-trace operators is far smaller than the set of all possible operators, QRG can be potentially more tractable. For general quantum field theories, it is still difficult to solve the corresponding quantum evolution problem in QRG. However, in the large N limit, quantum fluctuations of the single-trace couplings become weak, and the theory that describes QRG evolution becomes classical. In the large N limit, the solution to the exact RG equation can be obtained from the saddle-point solution. Recently, the exact effective action for the O(N) vector model has been computed from QRG in the large N limit[34]. It would be of great interest to compute exact effective actions for matrix models in the large N limit. Second, QRG provides a concrete prescription for constructing the holographic duals for general quantum field theories[16]. The construction gives a well-defined bulk theory that includes dynamical gravity as far as the boundary theory is regularized[15,30,39]. However, the continuum limit of the bulk theory obtained from regularized boundary theories such as lattice models is not fully understood. It is of great interest to understand how the regularized bulk theory obtained from QRG is related to continuum theories conjectured as holographic duals of known field theories in the semi-classical limit.

Report #1 by
Anonymous
(Referee 3) on 2021-8-22
(Invited Report)

Report

The addition of the new sec. II has considerably clarified and strengthened the paper. I recommend publication in its current form.

## Author: Han Ma on 2021-10-11 [id 1838]

(in reply to Report 2 on 2021-09-25)We thank the referee for the comments. Below, we provide our responses to the comments.

1) ``the procedure proposed by the authors is a mere rewriting of the Wilsonian path integral which, unless the the theory has a meaningful gravitational dual, does not lead to any improvement of understanding of RG flows''

Besides being an exact reformulation of the Wilsoninan RG, quantum RG (QRG) can be of practical use. First, it can be used to compute the exact quantum effective action of interacting field theories in the large N limit. Recently, the exact quantum effective action has been computed for the O(N) vector model in the large N limit via QRG [arXiv:2107.05654]. The exact effective action was not known before even for that relatively simple model because it includes operators with arbitrarily many fields and derivatives. Second, QRG provides a prescription to derive concrete holographic duals for general quantum field theories. As far as boundary quantum field theories are regularized, the bulk theories that arise from QRG are fully regularized. It is known that the bulk theory includes dynamical gravity [JHEP 1210 (2012) 160; JHEP 01 (2014) 076]. What is not well understood is how those regularized theory of dynamical gravity are related to continuum theories.

2) ``why not choosing just the field bi-linears as a basis also in the example of section IIB (O(N) × O(N) matrix model)?''

One can not choose bi-linears as a basis in the matrix model. This is because the single-trace operators should be singlets of the symmetry, and the set of bi-linear singlet operators of the matrix model do not form a complete basis of the vector space formed by the actions that respect the symmetry.

3) ``I would ask the authors to make a very concrete list of calculable physical quantities or properties of realistic QFTs that can be computed using their method, as well as an estimate of how hard the computation is''

We are grateful to the referee for the suggestion. In the revised manuscript, we added the following paragraph that includes a list of open questions and future directions. We feel that it is natural to place it in the conclusion as these are not directly related to the main content of the present paper.

We conclude with open questions and future directions. First, QRG can be used to compute the exact quantum effective action. The scale dependence of the quantum effective action obeys the exact RG equation[6,8]. In general solving the exact RG equation is challenging because the exact effective action includes operators made of arbitrarily many fields and derivatives. As a result, exact effective actions remain unknown even for relatively simple theories. In QRG, the exact RG equation is mapped to a quantum evolution of a wavefunction for single-trace couplings. Since the set of single-trace operators is far smaller than the set of all possible operators, QRG can be potentially more tractable. For general quantum field theories, it is still difficult to solve the corresponding quantum evolution problem in QRG. However, in the large N limit, quantum fluctuations of the single-trace couplings become weak, and the theory that describes QRG evolution becomes classical. In the large N limit, the solution to the exact RG equation can be obtained from the saddle-point solution. Recently, the exact effective action for the O(N) vector model has been computed from QRG in the large N limit[34]. It would be of great interest to compute exact effective actions for matrix models in the large N limit. Second, QRG provides a concrete prescription for constructing the holographic duals for general quantum field theories[16]. The construction gives a well-defined bulk theory that includes dynamical gravity as far as the boundary theory is regularized[15,30,39]. However, the continuum limit of the bulk theory obtained from regularized boundary theories such as lattice models is not fully understood. It is of great interest to understand how the regularized bulk theory obtained from QRG is related to continuum theories conjectured as holographic duals of known field theories in the semi-classical limit.