# Vacuum structure of Yang-Mills theory as a function of θ

## Abstract

It is believed that in SU(N) Yang-Mills theory observables are N -branched functions of the topological θ angle. This is supposed to be due to the existence of a set of locally-stable candidate vacua, which compete for global stability as a function of θ. We study the number of θ vacua, their interpretation, and their stability properties using systematic semiclassical analysis in the context of adiabatic circle compactification on $$ \mathbb{R}$$ ^{3} × S ^{1}. We find that while observables are indeed N-branched functions of θ, there are only ≈ N/2 locally-stable candidate vacua for any given θ. We point out that the different θ vacua are distinguished by the expectation values of certain magnetic line operators that carry non-zero GNO charge but zero ’t Hooft charge. Finally, we show that in the regime of validity of our analysis YM theory has spinodal points as a function of θ, and gather evidence for the conjecture that these spinodal points are present even in the $$ \mathbb{R}$$ ^{4} limit.

- Authors:

- Univ. of Washington, Seattle, WA (United States). Dept. of Physics
- Univ. of Washington, Seattle, WA (United States). Inst. for Nuclear Theory
- North Carolina State Univ., Raleigh, NC (United States). Dept. of Physics

- Publication Date:

- Research Org.:
- Univ. of Washington, Seattle, WA (United States); North Carolina State Univ., Raleigh, NC (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1507783

- Grant/Contract Number:
- FG02-00ER41132; FG02-03ER41260; SC0011637

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2018; Journal Issue: 9; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Discrete Symmetries, Global Symmetries, Spontaneous Symmetry Breaking, Wilson, 't Hooft and Polyakov loops

### Citation Formats

```
Aitken, Kyle, Cherman, Aleksey, and Ünsal, Mithat. Vacuum structure of Yang-Mills theory as a function of θ. United States: N. p., 2018.
Web. doi:10.1007/jhep09(2018)030.
```

```
Aitken, Kyle, Cherman, Aleksey, & Ünsal, Mithat. Vacuum structure of Yang-Mills theory as a function of θ. United States. doi:10.1007/jhep09(2018)030.
```

```
Aitken, Kyle, Cherman, Aleksey, and Ünsal, Mithat. Thu .
"Vacuum structure of Yang-Mills theory as a function of θ". United States. doi:10.1007/jhep09(2018)030. https://www.osti.gov/servlets/purl/1507783.
```

```
@article{osti_1507783,
```

title = {Vacuum structure of Yang-Mills theory as a function of θ},

author = {Aitken, Kyle and Cherman, Aleksey and Ünsal, Mithat},

abstractNote = {It is believed that in SU(N) Yang-Mills theory observables are N -branched functions of the topological θ angle. This is supposed to be due to the existence of a set of locally-stable candidate vacua, which compete for global stability as a function of θ. We study the number of θ vacua, their interpretation, and their stability properties using systematic semiclassical analysis in the context of adiabatic circle compactification on $ \mathbb{R}$3 × S1. We find that while observables are indeed N-branched functions of θ, there are only ≈ N/2 locally-stable candidate vacua for any given θ. We point out that the different θ vacua are distinguished by the expectation values of certain magnetic line operators that carry non-zero GNO charge but zero ’t Hooft charge. Finally, we show that in the regime of validity of our analysis YM theory has spinodal points as a function of θ, and gather evidence for the conjecture that these spinodal points are present even in the $ \mathbb{R}$4 limit.},

doi = {10.1007/jhep09(2018)030},

journal = {Journal of High Energy Physics (Online)},

number = 9,

volume = 2018,

place = {United States},

year = {2018},

month = {9}

}

*Citation information provided by*

Web of Science

Web of Science

#### Figures / Tables:

*N*) YM theory for

*N*= 4 as a function of

*θ*. The vertical axis depicts the vacuum energy density. The gray dashed curves indicate the energy density associated with the four distinct

*θ*-extrema. The colored bold curves indicatemore »

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Figures / Tables found in this record:

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*