# $O\left(N\right)$ model in $4<d<6$ : Instantons and complex CFTs

## Abstract

We revisit the scalar O($$\textit{N}$$) model in the dimension range 4 < $$\mathit{d}$$ < 6 and study the effects caused by its metastability. As shown in previous work, this model formally possesses a fixed point where, perturbatively in the 1/$$\textit{N}$$ expansion, the operator scaling dimensions are real and above the unitarity bound. Here, we further show that these scaling dimensions do acquire small imaginary parts due to the instanton effects. In d dimensions and for large $$\textit{N}$$, we find that they are of order $$e^{₋ Nf(d)}$$, where, remarkably, the function $$\textit{f(d)}$$ equals the sphere free energy of a conformal scalar in $$\mathit{d}$$ ₋ 2 dimensions. The non-perturbatively small imaginary parts also appear in other observables, such as the sphere free energy and two and three-point function coefficients, and we present some of their calculations. Therefore, at sufficiently large $$\textit{N}$$, the O($$\textit{N}$$) models in 4 < $$\mathit{d}$$ < 6 may be thought of as complex CFTs. When $$\textit{N}$$ is large enough for the imaginary parts to be numerically negligible, the five-dimensional O($$\textit{N}$$) models may be studied using the techniques of numerical bootstrap.

- Authors:

- Publication Date:

- Research Org.:
- Harvard Univ., Cambridge, MA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC); National Science Foundation (NSF); Simons Foundation; US Army Research Office (ARO)

- OSTI Identifier:
- 1600078

- Alternate Identifier(s):
- OSTI ID: 1801777

- Grant/Contract Number:
- SC0007870; SC0019030; PHY-1620059; PHY-1620542; PHY-1820651; PHY-1914860; 488653; W911NF-14-1-0003

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 101 Journal Issue: 4; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Astronomy & Astrophysics; Physics

### Citation Formats

```
Giombi, Simone, Huang, Richard, Klebanov, Igor R., Pufu, Silviu S., and Tarnopolsky, Grigory. O ( N ) model in 4 < d < 6 : Instantons and complex CFTs. United States: N. p., 2020.
Web. doi:10.1103/PhysRevD.101.045013.
```

```
Giombi, Simone, Huang, Richard, Klebanov, Igor R., Pufu, Silviu S., & Tarnopolsky, Grigory. O ( N ) model in 4 < d < 6 : Instantons and complex CFTs. United States. https://doi.org/10.1103/PhysRevD.101.045013
```

```
Giombi, Simone, Huang, Richard, Klebanov, Igor R., Pufu, Silviu S., and Tarnopolsky, Grigory. Tue .
"O ( N ) model in 4 < d < 6 : Instantons and complex CFTs". United States. https://doi.org/10.1103/PhysRevD.101.045013.
```

```
@article{osti_1600078,
```

title = {O ( N ) model in 4 < d < 6 : Instantons and complex CFTs},

author = {Giombi, Simone and Huang, Richard and Klebanov, Igor R. and Pufu, Silviu S. and Tarnopolsky, Grigory},

abstractNote = {We revisit the scalar O($\textit{N}$) model in the dimension range 4 < $\mathit{d}$ < 6 and study the effects caused by its metastability. As shown in previous work, this model formally possesses a fixed point where, perturbatively in the 1/$\textit{N}$ expansion, the operator scaling dimensions are real and above the unitarity bound. Here, we further show that these scaling dimensions do acquire small imaginary parts due to the instanton effects. In d dimensions and for large $\textit{N}$, we find that they are of order $e^{₋ Nf(d)}$, where, remarkably, the function $\textit{f(d)}$ equals the sphere free energy of a conformal scalar in $\mathit{d}$ ₋ 2 dimensions. The non-perturbatively small imaginary parts also appear in other observables, such as the sphere free energy and two and three-point function coefficients, and we present some of their calculations. Therefore, at sufficiently large $\textit{N}$, the O($\textit{N}$) models in 4 < $\mathit{d}$ < 6 may be thought of as complex CFTs. When $\textit{N}$ is large enough for the imaginary parts to be numerically negligible, the five-dimensional O($\textit{N}$) models may be studied using the techniques of numerical bootstrap.},

doi = {10.1103/PhysRevD.101.045013},

journal = {Physical Review D},

number = 4,

volume = 101,

place = {United States},

year = {2020},

month = {2}

}

https://doi.org/10.1103/PhysRevD.101.045013

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Web of Science

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