## The S matrix of 6D super Yang-Mills and maximal supergravity from rational maps

## Abstract

We present new formulas for n-particle tree-level scattering amplitudes of six-dimensional *N* = (1,1) super Yang-Mills (SYM) and *N* = (2,2) supergravity (SUGRA). They are written as integrals over the moduli space of certain rational maps localized on the (n – 3)! solutions of the scattering equations. Due to the properties of spinor-helicity variables in six dimensions, the even- *n* and odd- *n* formulas are quite different and have to be treated separately. We first propose a manifestly supersymmetric expression for the even- *n* amplitudes of *N* = (1,1) SYM theory and perform various consistency checks. By considering soft-gluon limits of the even- *n* amplitudes, we deduce the form of the rational maps and the integrand for *n* odd. The odd- *n* formulas obtained in this way have a new redundancy that is intertwined with the usual SL(2, C) invariance on the Riemann sphere. We also propose an alternative form of the formulas, analogous to the Witten-RSV formulation, and explore its relationship with the symplectic (or Lagrangian) Grassmannian. Since the amplitudes are formulated in a way that manifests double-copy properties, formulas for the six-dimensional *N* = (2,2) SUGRA amplitudes follow. Furthermore, these six-dimensional results allow us to deduce new formulasmore »

- Authors:

- Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
- Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada); Univ. of Waterloo, Waterloo, ON (Canada); Univ. de Concepcion, Concepcion (Chile)
- California Inst. of Technology (CalTech), Pasadena, CA (United States)
- Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada); Univ. of Waterloo, Waterloo, ON (Canada)
- California Inst. of Technology (CalTech), Pasadena, CA (United States); Queen Mary Univ. of London, London (United Kingdom)

- Publication Date:

- Research Org.:
- California Inst. of Technology, Pasadena, CA (United States)

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

- OSTI Identifier:
- 1501468

- Grant/Contract Number:
- SC0011632

- 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:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Scattering Amplitudes; Field Theories in Higher Dimensions; Supersymmetric Gauge Theory

### Citation Formats

```
Cachazo, Freddy, Guevara, Alfredo, Heydeman, Matthew, Mizera, Sebastian, Schwarz, John H., and Wen, Congkao. The S matrix of 6D super Yang-Mills and maximal supergravity from rational maps. United States: N. p., 2018.
Web. doi:10.1007/jhep09(2018)125.
```

```
Cachazo, Freddy, Guevara, Alfredo, Heydeman, Matthew, Mizera, Sebastian, Schwarz, John H., & Wen, Congkao. The S matrix of 6D super Yang-Mills and maximal supergravity from rational maps. United States. doi:10.1007/jhep09(2018)125.
```

```
Cachazo, Freddy, Guevara, Alfredo, Heydeman, Matthew, Mizera, Sebastian, Schwarz, John H., and Wen, Congkao. Thu .
"The S matrix of 6D super Yang-Mills and maximal supergravity from rational maps". United States. doi:10.1007/jhep09(2018)125. https://www.osti.gov/servlets/purl/1501468.
```

```
@article{osti_1501468,
```

title = {The S matrix of 6D super Yang-Mills and maximal supergravity from rational maps},

author = {Cachazo, Freddy and Guevara, Alfredo and Heydeman, Matthew and Mizera, Sebastian and Schwarz, John H. and Wen, Congkao},

abstractNote = {We present new formulas for n-particle tree-level scattering amplitudes of six-dimensional N = (1,1) super Yang-Mills (SYM) and N = (2,2) supergravity (SUGRA). They are written as integrals over the moduli space of certain rational maps localized on the (n – 3)! solutions of the scattering equations. Due to the properties of spinor-helicity variables in six dimensions, the even-n and odd-n formulas are quite different and have to be treated separately. We first propose a manifestly supersymmetric expression for the even-n amplitudes of N = (1,1) SYM theory and perform various consistency checks. By considering soft-gluon limits of the even-n amplitudes, we deduce the form of the rational maps and the integrand for n odd. The odd-n formulas obtained in this way have a new redundancy that is intertwined with the usual SL(2, C) invariance on the Riemann sphere. We also propose an alternative form of the formulas, analogous to the Witten-RSV formulation, and explore its relationship with the symplectic (or Lagrangian) Grassmannian. Since the amplitudes are formulated in a way that manifests double-copy properties, formulas for the six-dimensional N = (2,2) SUGRA amplitudes follow. Furthermore, these six-dimensional results allow us to deduce new formulas for five-dimensional SYM and SUGRA amplitudes, as well as massive amplitudes of four-dimensional N = 4 SYM on the Coulomb branch.},

doi = {10.1007/jhep09(2018)125},

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

number = 9,

volume = 2018,

place = {United States},

year = {2018},

month = {9}

}