# Perturbative and Nonperturbative Studies of CFTs with MN Global Symmetry

### Submission summary

 As Contributors: Johan Henriksson · Andreas Stergiou Arxiv Link: https://arxiv.org/abs/2101.08788v2 (pdf) Date accepted: 2021-07-06 Date submitted: 2021-06-03 17:54 Submitted by: Henriksson, Johan Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical

### Abstract

Fixed points in three dimensions described by conformal field theories with $MN_{m,n}= O(m)^n\rtimes S_n$ global symmetry have extensive applications in critical phenomena. Associated experimental data for $m=n=2$ suggest the existence of two non-trivial fixed points, while the $\varepsilon$ expansion predicts only one, resulting in a puzzling state of affairs. A recent numerical conformal bootstrap study has found two kinks for small values of the parameters $m$ and $n$, with critical exponents in good agreement with experimental determinations in the $m=n=2$ case. In this paper we investigate the fate of the corresponding fixed points as we vary the parameters $m$ and $n$. We find that one family of kinks approaches a perturbative limit as $m$ increases, and using large spin perturbation theory we construct a large $m$ expansion that fits well with the numerical data. This new expansion, akin to the large $N$ expansion of critical $O(N)$ models, is compatible with the fixed point found in the $\varepsilon$ expansion. For the other family of kinks, we find that it persists only for $n=2$, where for large $m$ it approaches a non-perturbative limit with $\Delta_\phi\approx 0.75$. We investigate the spectrum in the case $MN_{100,2}$ and find consistency with expectations from the lightcone bootstrap.

Published as SciPost Phys. 11, 015 (2021)

We thank both reviewers for their carefully written reports and their useful suggestions for improvements of the draft. We have made some changes following their reports.

### List of changes

Introduction: Added some further clarifications and references (including the new footnote 3) concerning the literature discussion of non-perturbative fixed-points, and added some literature values for these. Clarified some formulations relating to the numerical bootstrap (regarding what a kink signals and numerical precision).

Table 2. Added a dashed line to separate experimental and MC data from kink 1 results. Added a new reference with additional MC result.

Figure 4. Added precision for qboot computations.

Section 4. Fixed typo. Clarified what we mean by a simplifying limit and that the cartoons in figure 8 need to be confirmed or disproved by further studies.

### Submission & Refereeing History

Resubmission 2101.08788v2 on 3 June 2021
Submission 2101.08788v1 on 8 March 2021

## Reports on this Submission

### Anonymous Report 1 on 2021-6-18 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2101.08788v2, delivered 2021-06-18, doi: 10.21468/SciPost.Report.3033

### Report

I thank the authors for their careful replies, and I appreciate the authors being conservative in the estimates even though the shown plots would suggest quite smaller error bars. The authors have addressed all my points, and I am happy to recommend the paper for publication. It can be published as is, although I have one minor final clarification to ask. Indeed eqs. (3.6-3.8) do not suggest a large spin expansion, the cause of confusion is the statement above (3.3) where it seems to say the dDisc is computed in the limit $v \ll 0$, and since the section is called large spin perturbation theory it was also not clear if non-perturbative finite spin effects were taken into account. (I am not sure [29,30] were what the authors meant to refer to in their reply.) For the paper to be self-contained, could the authors just briefly comment down to which spin they expect the inversion formula to give the correct answer, and thus their results to hold? In a generic CFT it would be for $\ell>1$, however the correlator being inverted here is an expansion for large $m$ - is the Regge growth expected to be the same for the $\mathcal{O}(m^{-1})$ piece of the correlator, or assumed to be? Or is it even better, since below (3.8) the results are evaluated for $\ell=1$? This issue is commented upon when discussing the $\ell=0$ evaluation of the authors' results but I believe it would be beneficial to comment a little bit earlier.

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