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Constraints on the spectrum of field theories with noninteger $O(N)$ symmetry from quantum evanescence
by Weiguang Cao, Xiaochuan Lu, Tom Melia
Submission summary
Authors (as registered SciPost users):  Weiguang Cao · Tom Melia 
Submission information  

Preprint Link:  scipost_202404_00027v1 (pdf) 
Date submitted:  20240419 03:00 
Submitted by:  Melia, Tom 
Submitted to:  SciPost Physics Core 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We identify constraints in the energy spectra of quantum theories that have a global $O(N)$ symmetry, where $N$ is treated as a continuous parameter. We point out that a class of evanescent states fall out of the spectrum at integer values of $N$ in pairs, via an annihilation mechanism. This forces the energies of the states in such a pair to approach equality as $N$ approaches a certain integer, with both states disappearing at precisely integer $N$ and the point of wouldbe degeneracy. These constraints occur between different irreducible representations of the analytic continuation of $O(N)$ and hold nonperturbatively. We give examples in the spectra of the critical $O(N)$ model.
Author comments upon resubmission
List of changes
In light of the comments of referees and editor, we have made the following major revisions to the manuscript.
1. To address the issue of the usage of the term degeneracy we have opted to instead describe the phenomena as a "constraint" on energy spectra, replacing the usage of 'degeneracy' everywhere it appeared in the original text. We have changed the title to "Constraints on the spectrum of field theories with noninteger O(N) symmetry from quantum evanescence". We have also added/made revisions to what are now the 4th, 5th, and 6th paragraphs, and to the abstract, described in the following:
a) In light of the comment of referee 3, we explicitly emphasize that certain representations are "becoming negative" at integer N, and that we are identifying thus the fact that some (not all) evanescent operators have to drop out pairwise. This is a result we are not aware of in existing literature. It is of course, as we outlined in our response to referee 3, the key ingredient for obtaining the spectrum constraints. We emphasize, in light of what we believe may be a miscommunication between us and referee 2 and 3, that not all evanescent operators drop out pairwise and thus lead to constraints. Phrasing in terms of the viewpoint of referee 2, that operators are becoming linearly related at certain N, we point out such linear relations do not in general lead to the existence of the constraints we identify.
b) We added a paragraph to highlight the difference of this constraint phenomena from usual degeneracy. Namely, that states become degenerate and simultaneously drop out of the spectrum at precisely integer N, coining this behaviour as 'evanescentdegeneracy'.
We hope that this addresses the referee and editors concerns, making clear the difference to the usual usage of degeneracy.
2. To address another of the points of referee 2, we added a clarifying sentence above eq 15 that states the continued partition function is valid for all real values of N, same as the continued characters and the continued tensor product decomposition algebra.
3. We addressed the misuse of coupling constant to describe N as pointed out by referee 1 in the 3rd paragraph.
We hope that these revisions can satisfy the referees' concerns, clear up some potential miscommunications, and result in the publication of the manuscript.
Current status:
Reports on this Submission
Report
Generally I think the paper is mostly acceptable. However while
the word degeneracy is no longer in the title it appears frequently in
the introduction. I thought it was agreed that what washing described was not degeneracy in the usual sense but a necessary consistency condition. In my view the introductory paragraphs should be modified to reflect this
Secondly the results in (14a,b,c) and in Table II are a direct reflection
of the Racah Speiser algorithm. For what it is worth a concise discussion of this is given in an appendix to hepth/0209056. Similar results can be found in maths textbooks.
With some revision this paper would be acceptable.
Recommendation
Ask for minor revision
Report
I find the revisions, along with the authors' reply to the three referee reports, very helpful in better understanding the paper. The sharpened terminology and added comments serve to clarify the underlying philosophy, the role played by the nonintegrality of N, and the aspects of the paper that are novel. Under a strict view that considers only unitary, integerN theories, the relationships between states that form the subject of this paper are imperceptible, but continuation of integervalued physical parameters is a commonplace procedure, which it is worth understanding at the deepest possible level.
Recommendation
Publish (meets expectations and criteria for this Journal)