Constraints on beta functions in field theories

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.

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

1. “The ideas rest on formal manipulations. It is far from clear that these manipulations are well defined outside the simple models considered.” The bulk theory that describes the quantum RG evolution of a boundary field theory is well defined as far as the boundary field theory is regularized. The regularization of the bulk theory follows from the regularization of the boundary field theory. For example, the bulk theory is a finite quantum mechanical theory of lattice variables for a boundary theory defined on a lattice. For example, well-defined bulk theories for lattice gauge theories and lattice vector models can be found in Nucl. Phys. B 862, 781 (2012) and JHEP (2016) 2016: 44, respectively. While it is generally hard to solve those bulk theories analytically, they can be solved in a computer with enough computational power as they are well defined quantum mechanics problems. In section II of the revised manuscript, we explicitly construct well-defined bulk theories for two realistic field theories that are regularized on a lattice.

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; 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 (a finite number of marginally relevant) couplings — what the authors actually do is weaker” What is new in this paper is the constraints that beta functions satisfy non-perturbatively in general field theories. This result is beyond the known constraint of beta functions present in perturbative field theories defined in the continuum. To understand this, let us consider a field theory defined non-perturbatively with a finite UV cutoff. Examples include field theories regularized on a lattice. In such theories, there are infinitely many couplings that can be turned on independently at UV. If the energy scale of an effective theory is much smaller than the UV cutoff, the RG flow of all couplings is indeed fixed by a small number of marginal and relevant couplings. At high energies, however, irrelevant couplings are not fixed by the marginal and relevant couplings through that constraint that emerges only in the low-energy limit, and one has to keep track of the flow of infinitely many couplings. The constraint discussed in our paper is new as it applies to the exact beta functions at all energy scales even at scales that are comparable to the UV cutoff.

3. “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)” It is true that the bulk theory obtained form quantum RG is generally hard to solve because the bulk theory becomes semi-classical only in the large N limit. However, the main purpose of the present paper is not to provide a general way of solving strongly interacting bulk theories. Rather, our goal is to uncover non-perturbative constraints that beta functions satisfy through quantum RG. These constraints hold for any N, irrespective of whether the bulk theory is in the semi-classical limit.

4. “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.” The quantum RG gives well-defined bulk theories as far as the boundary field theories are regularized. Furthermore, the constraints of beta functions uncovered in our paper are valid for general theories beyond the toy models. In order to demonstrate this, in the revised manuscript, we added a new section (Sec. II) in which well-defined bulk theories are explicitly derived for two realistic field theories regularized on a lattice. We also derive the full beta functions of these theories solely from the beta functions defined in the subspace of single-trace couplings. The salient point of our paper is not the exact beta-functions itself but the fact that the entire beta-functions are fully characterized by a small set of data defined in the subspace of single-trace couplings.