# Navigating through the O(N) archipelago

### Submission summary

 Authors (as Contributors): Benoit Sirois
Submission information
Date accepted: 2022-08-23
Date submitted: 2022-08-16 16:03
Submitted by: Sirois, Benoit
Submitted to: SciPost Physics
Ontological classification
Specialties:
• High-Energy Physics - Theory
• Statistical and Soft Matter Physics
Approaches: Theoretical, Computational

### Abstract

A novel method for finding allowed regions in the space of CFT-data, coined navigator method, was recently proposed in arXiv:2104.09518. Its efficacy was demonstrated in the simplest example possible, i.e. that of the mixed-correlator study of the 3D Ising Model. In this paper, we would like to show that the navigator method may also be applied to the study of the family of $d$-dimensional $O(N)$ models. We will aim to follow these models in the $(d,N)$ plane. We will see that the "sailing" from island to island can be understood in the context of the navigator as a parametric optimization problem, and we will exploit this fact to implement a simple and effective path-following algorithm. By sailing with the navigator through the $(d,N)$ plane, we will provide estimates of the scaling dimensions $(\Delta_{\phi},\Delta_{s},\Delta_{t})$ in the entire range $(d,N) \in [3,4] \times [1,3]$. We will show that to our level of precision, we cannot see the non-unitary nature of the $O(N)$ models due to the fractional values of $d$ or $N$ in this range. We will also study the limit $N \xrightarrow[]{} 1$, and see that we cannot find any solution to the unitary mixed-correlator crossing equations below $N=1$.

Published as SciPost Phys. 13, 081 (2022)

We are very thankful for the attention every referee gave to the paper. We agree with every modification suggested by the referees; you may find all modifications made in the list of changes below.

### List of changes

- Modification of figures in order to make them easier to view, especially when printed. The modifications include: the use of different styles of markers in Figures 1,4-6,7,11,13-15; the use of darker colors or better contrasting colors in Figures 2,8-11,16; the use of different line styles in 5-7; the use of larger fonts in legends and/or axes labels in Figures 1-11,13-16.

-Addition of Appendix A which summarizes the resummation procedure of Section V of [18], along with a comment in the second paragraph of p.4 announcing this Appendix.

-Re-organization of the paper, moving less crucial calculations to appendices. Appendix B now contains a lot of the more technical details about the navigator method, Appendix C contains the calculations which were previously at the beginning of Section V, and Appendix D contains many of the technical details about the limit N -> 1.

- Addition at the end of the introduction of a paragraph describing the layout of the paper, along with a comment on which sections may be skipped by the more physics-inclined readers.

-Addition of a long comment at the beginning of Section VI better describing the theoretical status on the question of non-unitarity, and why it is normal we don't observe this non-unitarity in our setup.

-Addition of a comment on how our results compare in precision with the state-of-the-art in the middle of the first paragraph of Section VI and in footnote 4.

-Addition in footnote 5 of an approximate time that a typical computation in our setup takes.

-Addition of Eq. (9) in the new submission and removal of Eq. (22) of the original submission, along with modification of the surrounding text. The author was made aware by J. Henriksson of new theoretical results concerning the OPE coefficient of Fig. 10, and so a comparison with this theoretical result was added.