## The four-loop six-gluon NMHV ratio function

## Abstract

We use the hexagon function bootstrap to compute the ratio function which characterizes the next-to-maximally-helicity-violating (NMHV) six-point amplitude in planar N=4 super-Yang-Mills theory at four loops. A powerful constraint comes from dual superconformal invariance, in the form of a Q¯ differential equation, which heavily constrains the first derivatives of the transcendental functions entering the ratio function. At four loops, it leaves only a 34-parameter space of functions. Constraints from the collinear limits, and from the multi-Regge limit at the leading-logarithmic (LL) and next-to-leading-logarithmic (NLL) order, suffice to fix these parameters and obtain a unique result. We test the result against multi-Regge predictions at NNLL and N ^{3}LL, and against predictions from the operator product expansion involving one and two flux-tube excitations; all cross-checks are satisfied. We study the analytical and numerical behavior of the parity-even and parity-odd parts on various lines and surfaces traversing the three-dimensional space of cross ratios. As part of this program, we characterize all irreducible hexagon functions through weight eight in terms of their coproduct. As a result, we also provide representations of the ratio function in particular kinematic regions in terms of multiple polylogarithms.

- Authors:

- SLAC National Accelerator Lab., Menlo Park, CA (United States); California Inst. of Technology (CalTech), Pasadena, CA (United States)
- Perimeter Institute for Theoretical Physics, Waterloo, ON (Canada)
- SLAC National Accelerator Lab., Menlo Park, CA (United States)

- Publication Date:

- Research Org.:
- SLAC National Accelerator Lab., Menlo Park, CA (United States)

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

- OSTI Identifier:
- 1239131

- Grant/Contract Number:
- AC03-76SF00515

- 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: 2016; Journal Issue: 1; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; scattering amplitudes; supersymmetric gauge theory; 1/N expansion

### Citation Formats

```
Dixon, Lance J., von Hippel, Matt, and McLeod, Andrew J.. The four-loop six-gluon NMHV ratio function. United States: N. p., 2016.
Web. doi:10.1007/JHEP01(2016)053.
```

```
Dixon, Lance J., von Hippel, Matt, & McLeod, Andrew J.. The four-loop six-gluon NMHV ratio function. United States. doi:10.1007/JHEP01(2016)053.
```

```
Dixon, Lance J., von Hippel, Matt, and McLeod, Andrew J.. Mon .
"The four-loop six-gluon NMHV ratio function". United States. doi:10.1007/JHEP01(2016)053. https://www.osti.gov/servlets/purl/1239131.
```

```
@article{osti_1239131,
```

title = {The four-loop six-gluon NMHV ratio function},

author = {Dixon, Lance J. and von Hippel, Matt and McLeod, Andrew J.},

abstractNote = {We use the hexagon function bootstrap to compute the ratio function which characterizes the next-to-maximally-helicity-violating (NMHV) six-point amplitude in planar N=4 super-Yang-Mills theory at four loops. A powerful constraint comes from dual superconformal invariance, in the form of a Q¯ differential equation, which heavily constrains the first derivatives of the transcendental functions entering the ratio function. At four loops, it leaves only a 34-parameter space of functions. Constraints from the collinear limits, and from the multi-Regge limit at the leading-logarithmic (LL) and next-to-leading-logarithmic (NLL) order, suffice to fix these parameters and obtain a unique result. We test the result against multi-Regge predictions at NNLL and N3LL, and against predictions from the operator product expansion involving one and two flux-tube excitations; all cross-checks are satisfied. We study the analytical and numerical behavior of the parity-even and parity-odd parts on various lines and surfaces traversing the three-dimensional space of cross ratios. As part of this program, we characterize all irreducible hexagon functions through weight eight in terms of their coproduct. As a result, we also provide representations of the ratio function in particular kinematic regions in terms of multiple polylogarithms.},

doi = {10.1007/JHEP01(2016)053},

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

number = 1,

volume = 2016,

place = {United States},

year = {2016},

month = {1}

}

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