# Constraints on beta functions in field theories

### Submission summary

 As Contributors: Han Ma Arxiv Link: https://arxiv.org/abs/2009.11880v3 (pdf) Date submitted: 2021-01-04 17:42 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 allowed by symmetry and locality are generated under the renormalization group flow, and the exact renormalization group flow takes place 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. The general prescription is applied to two simple examples.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 2009.11880v4 on 18 July 2021

Submission 2009.11880v3 on 4 January 2021

## Reports on this Submission

### Report

I believe the paper requires a significant rewriting. Please see the report attached

### Attachment

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Han Ma  on 2021-07-18

(in reply to Report 2 on 2021-05-17)

We thank the referee for the comments. Here are our responses to the referee’s comments.

1. “I do not see why for a generic strongly coupled QFT there should be a preferred set of single-trace fields” There needs not be a preferred set of single-trace operators. All we need in quantum RG is the existence of a set of single-trace operators. Its existence follows from the fact that the space of theories can be always viewed as a Hilbert space, where an action S[\phi] of fundamental field \phi defines a wavefunction exp( -S[\phi] ) in the Hilbert space. Then, it follows that there exists a set of basis states that span the Hilbert space. In general, there exist multiple ways of choosing a complete set of basis states. Moreover, the basis states do not need to be orthogonal, and an over-complete set is an acceptable choice. Once a complete set of basis is chosen, the wavefunctions of the basis states define a set of actions. The operators that are needed to construct the wavefunctions of the basis states define the set of single-trace operators in general theories. This is explained in details for the O(N) vector model and the O(N)*O(N) matrix model in section II. While it is true that one choice of basis may give a simpler bulk theory and a stronger constraint, the purpose of our paper is to demonstrate the existence of general constraints from one choice of basis.

2. “why it (the set of single-trace fields) should be the same at different energy scales” The Hilbert space associated with the space of theories is independent of scale. What is scale dependent is the effective action and the associated state in the Hilbert space that runs along the RG flow. Therefore, one can always choose a set of basis states in a scale independent way. This is explicitly shown to be the case in the examples included in Sec. II. While it is in principle possible to choose basis states in a scale dependent way, this is not necessary.

3. “the path integral over HS (Hubbard-Stratonovich) fields will generically get strongly coupled and I expect the book keeping of operators used throughout the paper to fail” Indeed, the dynamical single-trace couplings are strongly interacting in theories with small numbers of flavours or colours. Only in the large N limit, the interactions become weak, and one can use a semi-classical approximation. However, the book keeping does not fail because the basis states made of the single-trace operators span the full Hilbert space independent of their dynamics. Furthermore, the constraints of beta functions discussed in this paper does not require that the bulk theory is weakly interacting. In the revised manuscript, we explicitly compute the full beta functions of two realistic models valid for any N entirely from the beta functions defined in the space of single-trace operators.

4. “there is a Hamiltonian constraint HΨ = 0, and Shroedinger equation only appears in a semiclassical expansion of this constraint” In quantum RG, general states defined at a scale are not annihilated by the RG Hamiltonian. This is because the effective action generally changes as a function of RG scale. What is invariant is an overlap between two states associated with a fixed point and a deformation, where the overlap corresponds to the generating function of the boundary theory. This is described in Sec. II and III in details.

5. “in a real QFT this would still be infinitely many operators” It is true that even in quantum RG one needs to include infinitely many single-trace operators in the thermodynamic limit. This is because couplings are in general space dependent. In the vector model, these are position dependent bi-local operators, and in the matrix model, they are loop operators, as is discussed in Sec. II in details. Nonetheless, the exact mapping from the Wilsonian RG to quantum RG is powerful enough to reveal general constraints among beta functions.

6. “Can any of this be achieved using the authors’ method in any interesting QFTs?” In the revised manuscript, we add a new section for two realistic models. The first is the O(N) vector model and the other is the O(N)*O(N) matrix model. It is explicitly shown that the full beta functions can be obtained solely from the beta functions defined in the subspace of single-trace operators in these models.

7. “what prevents one from considering an RG flow with an arbitrary β2(j, j2), without modifying the beta functions at zero j2?” Naively, one might think that the beta functions away from the subspace of j2=0 can be modified without modifying the beta function on the subspace. However, this is impossible due to the constraint that we find in this paper - this is the main point of our paper. Because multi-trace operators are composites of the single-trace operators, the RG flow in the presence of general multi-trace operators are completely fixed by the beta functions defined in the subspace of single-trace couplings. This constraint holds even when multi-trace operators have large anomalous dimensions. This is explicitly demonstrated through the two realistic field theories in the revised manuscript.

### Strengths

1. This explains reasonably well Lee's ideas about a quantum renormalisation group and attempts some advance on these ideas.

### Weaknesses

1. The ideas rest on formal manipulations. It is far from clear that these manipulations are well defined outside the simple models considered.

2. It is not clear to me that this paper adds anything substantive to previous papers the senior author has already released on the subject.

### Report

The central claim of the paper (that the beta-functions for all symmetric operators in a general quantum field theory can be reconstructed from beta-functions in a measure zero subspace of such conjugate couplings) is not surprising, since the continuum limit of such quantum field theories are parametrised by typically a finite dimensional space of marginally relevant couplings. At least within perturbation theory, there are well established procedures for determining the form of the beta-functions for all other operators in terms of these couplings -- crucially after appropriate care is taken to define the regularisation and renormalisation of those operators. What the authors actually do is both weaker (they are typically left to deal with a subspace still containing an infinite number of couplings) and nowhere near as well-defined. To achieve their result they cast the Polchinski version of the exact renormalisation group flow equation into a kind of functional Schrodinger equation where the dependence on sources for all operators outside the subspace, is finessed into a functional generalisation of a Schrodinger wave function. No algorithm is presented for solving this functional Schrodinger equation, or justification for thinking this is easier than working with the original quantum field theory (most likely it is not) and it is far from clear that this reformulation is in general well defined (again most likely it is not). Instead in examples, the authors make Gaussian-like ansatze for the beta functions inside this subspace, equations (93) and (109). Only with these simple ansatze can they proceed to reconstruct' the full theory but in these now over-simplified models.

• validity: low
• significance: low
• originality: good
• clarity: good
• formatting: good
• grammar: reasonable

### Author:  Han Ma  on 2021-07-18

(in reply to Report 1 on 2021-05-11)
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