How negative can null energy be in large N CFTs?

My main comment is about the motivation for the question that is addressed in the paper. I sympathize with the motivation from general theoretical curiosity, but are there any physical questions that hinge on the "strong" conjecture posted by the authors? For example, the averaged null energy condition (ANEC) became a hot topic after Hoffman and Maldacena pointed out that it could be used to prove interesting general statements about conformal field theories in general spacetime dimensions. I understand that this is an exploratory work, but I believe that a serious attempt should be made to grapple with this and convey the motivation to the reader. To be frank, this paper reads like it is addressed to an in-group that has already agreed that any result about energy inequalities in quantum field theory is interesting.

The general motivation that is conventionally given for studying energy inequalities in quantum field theory (and indeed it is mentioned in this paper) comes from the fact that classical general relativity requires various energy positivity constraints to avoid pathological solutions. However, the path from quantum energy inequalities to any kind of statement about macroscopic general relativity is not at all clear, even in broad outline. The case of a non-minimally coupled scalar is a good example of this. As the authors know well from their own previous work (Ref. 17) a 4D scalar field with a non-minimal coupling to gravity violates energy conditions even classically. However, it does not give rise to any pathological solution because we can simply go to Einstein frame, where the non-minimal coupling to gravity is replaced by a scalar potential. While this is certainly a special example (a theory with a scalar operator of dimension 2, allowing a marginal coupling to gravity), it illustrates the point that the consistency of a theory coupled to gravity will necessarily involve the mixing of the gravitational and non-gravitational degrees of freedom, and that this need not be dominated by short-distance quantum effects. In fact, back-reaction is generically important in pathological solutions of the kind that often result when energy inequalities are violated. (For example, it is often the stability rather than the existence of these solutions that is problematic.)

I get that these are difficult questions to answer, and I am not asking the
authors to provide a general solution in this paper. However, as someone not working in this subject, I would like to see a little more discussion of these points.

A somewhat related comment is that the first page and a half of the introduction focuses on the authors' own work, rather than giving a broad overview of the field. I was surprised that there was no mention of the ANEC, since it is both a limiting case of the energy condition studied by the authors, and a perfect example of a "success story" for this kind of work. Namely, the ANEC was shown to be related to interesting constraints on CFTs, and this led to further theoretical work proving it in general CFTs.

The rest of the paper is well written and appears to be technically sound. I would be more cautious about saying that the results in the paper "suggest" various more general results, but the statements in the paper are clear and the reader can make up their own mind. I believe that overall this paper is above threshold for publication.

1. Make a serious attempt to address the motivation issues in the report above. I suggest that the audience that the authors keep in mind is a wider audience of theorists interested in structural questions in quantum field theory.

Ask for minor revision

This paper investigates the possibility of having states with negative smeared null energy, particularly in the large N limit of CFTs. This is an interesting question, since little is understood about energy conditions in QFT in general, especially interacting theories in higher dimensions. This paper is written well, but I have some suggestions that would make it clearer and some clarifications I would like to be made before it can be published.

1) The authors make use of a smearing function g(x) in their calculations. They mention some conditions (smooth, real, compact support) that were used for the 2d condition in equation (2) of the introduction, and it later sections it seems as though the same conditions are used in the higher dimension calculations, but for clarity it would be useful to have the general conditions on g(x) in higher dimensions stated explicitly in the introduction.

2) In line (159) there is a grammatical error, I suspect "diverges" should be "divergences".

3) In line 290, for clarity this should be changed to: "there is one operator guaranteed to exist in every \emph{local} CFT: the stress-energy tensor..."

4) There are two different definitions of the state $|\psi\rangle$, one in equation (44) and one in equation (51). The notation for one of these should be changed.

5) In line 477 the state $|h^p_∆\rangle$ is referenced, but upon initial reading it was not clear to me that this was the first introduction of this definition of the state. I think changing the language to be something like “Alternatively we could define another state $|h^p_∆\rangle$...” or “Alternatively we could define a different state $|h^p_∆\rangle$...” would make this clearer and distinguish it from $|\mathcal{O}_h^p\rangle$.

6) Can the authors comment on the contributions of contact terms in general for their analysis? Are there assumptions on the smearing functions that ensure that contact terms do not appear? And is it clear they would not appear after analytic continuation (as they do in e.g. 2309.14409)?

I recommend the paper for publication after the above points have been addressed.

Ask for minor revision