# Analytic and Numerical Bootstrap of CFTs with $O(m)\times O(n)$ Global Symmetry in 3D

### Submission summary

 As Contributors: Andreas Stergiou Arxiv Link: https://arxiv.org/abs/2004.14388v3 (pdf) Date accepted: 2020-09-04 Date submitted: 2020-08-26 20:43 Submitted by: Stergiou, Andreas Submitted to: SciPost Physics Discipline: Physics Subject area: High-Energy Physics - Theory Approaches: Theoretical, Computational

### Abstract

Motivated by applications to critical phenomena and open theoretical questions, we study conformal field theories with $O(m)\times O(n)$ global symmetry in $d=3$ spacetime dimensions. We use both analytic and numerical bootstrap techniques. Using the analytic bootstrap, we calculate anomalous dimensions and OPE coefficients as power series in $\varepsilon=4-d$ and in $1/n$, with a method that generalizes to arbitrary global symmetry. Whenever comparison is possible, our results agree with earlier results obtained with diagrammatic methods in the literature. Using the numerical bootstrap, we obtain a wide variety of operator dimension bounds, and we find several islands (isolated allowed regions) in parameter space for $O(2)\times O(n)$ theories for various values of $n$. Some of these islands can be attributed to fixed points predicted by perturbative methods like the $\varepsilon$ and large-$n$ expansions, while others appear to arise due to fixed points that have been claimed to exist in resummations of perturbative beta functions.

### Ontology / Topics

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Published as SciPost Phys. 9, 035 (2020)

We would like to thank both referees for their comments on our manuscript. A list of changes to address the referees' comments can be found below.

### List of changes

1. We added tables with results for critical exponents in our conclusion section, as suggested in Report 1. Hopefully this also ameliorates the indicated weakness in Report 2.
2. In response to Requested Change 1 in Report 2, we emphasize that our expansion parameter is 1/n, not m/n. This is clearly stated in the paragraph before subsection 1.2.
3. We added footnote 1 to address Requested Change 2 in Report 2.
4. We added footnote 17 to address Requested Change 3 in Report 2.