# Weak separation, positivity and extremal Yangian invariants

## Abstract

We classify all positive $n$-particle N ^{k}MHV Yangian invariants in $N =$ 4 YangMills theory with $n = 5k$, which we call extremal because none exist for $n > 5k$. We show that this problem is equivalent to that of enumerating plane cactus graphs with k pentagons. We use the known solution of that problem to provide an exact expression for the number of cyclic classes of such invariants for any $k$, and a simple rule for writing them down explicitly. We provide an alternative (but equivalent) classification by showing that a product of $k$ five-brackets with disjoint sets of indices is a positive Yangian invariant if and only if the sets are all weakly separated.

- Authors:

- Brown Univ., Providence, RI (United States)

- Publication Date:

- Research Org.:
- Brown Univ., Providence, RI (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1594824

- Report Number(s):
- arXiv:1906.11034

Journal ID: ISSN 1029-8479

- Grant/Contract Number:
- SC0010010

- 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: 2019; 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; Scattering Amplitudes; Supersymmetric Gauge Theory

### Citation Formats

```
Lippstreu, Luke, Mago, Jorge, Spradlin, Marcus, and Volovich, Anastasia. Weak separation, positivity and extremal Yangian invariants. United States: N. p., 2019.
Web. doi:10.1007/JHEP09(2019)093.
```

```
Lippstreu, Luke, Mago, Jorge, Spradlin, Marcus, & Volovich, Anastasia. Weak separation, positivity and extremal Yangian invariants. United States. doi:10.1007/JHEP09(2019)093.
```

```
Lippstreu, Luke, Mago, Jorge, Spradlin, Marcus, and Volovich, Anastasia. Thu .
"Weak separation, positivity and extremal Yangian invariants". United States. doi:10.1007/JHEP09(2019)093. https://www.osti.gov/servlets/purl/1594824.
```

```
@article{osti_1594824,
```

title = {Weak separation, positivity and extremal Yangian invariants},

author = {Lippstreu, Luke and Mago, Jorge and Spradlin, Marcus and Volovich, Anastasia},

abstractNote = {We classify all positive $n$-particle NkMHV Yangian invariants in $N =$ 4 YangMills theory with $n = 5k$, which we call extremal because none exist for $n > 5k$. We show that this problem is equivalent to that of enumerating plane cactus graphs with k pentagons. We use the known solution of that problem to provide an exact expression for the number of cyclic classes of such invariants for any $k$, and a simple rule for writing them down explicitly. We provide an alternative (but equivalent) classification by showing that a product of $k$ five-brackets with disjoint sets of indices is a positive Yangian invariant if and only if the sets are all weakly separated.},

doi = {10.1007/JHEP09(2019)093},

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

number = 9,

volume = 2019,

place = {United States},

year = {2019},

month = {9}

}

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