# Bootstrapping the Three-Loop Hexagon

## Abstract

We consider the hexagonal Wilson loop dual to the six-point MHV amplitude in planar N = 4 super Yang-Mills theory. We apply constraints from the operator product expansion in the near-collinear limit to the symbol of the remainder function at three loops. Using these constraints, and assuming a natural ansatz for the symbol's entries, we determine the symbol up to just two undetermined constants. In the multi-Regge limit, both constants drop out from the symbol, enabling us to make a non-trivial confirmation of the BFKL prediction for the leading-log approximation. This result provides a strong consistency check of both our ansatz for the symbol and the duality between Wilson loops and MHV amplitudes. Furthermore, we predict the form of the full three-loop remainder function in the multi-Regge limit, beyond the leading-log approximation, up to a few constants representing terms not detected by the symbol. Our results confirm an all-loop prediction for the real part of the remainder function in multi-Regge 3 {yields} 3 scattering. In the multi-Regge limit, our result for the remainder function can be expressed entirely in terms of classical polylogarithms. For generic six-point kinematics other functions are required.

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1029137

- Report Number(s):
- SLAC-PUB-14528

Journal ID: ISSN 1029-8479; arXiv:1108.4461; TRN: US1105623

- DOE Contract Number:
- AC02-76SF00515

- Resource Type:
- Journal Article

- Journal Name:
- JHEP 1111:023,2011

- Additional Journal Information:
- Journal Volume: 2011; Journal Issue: 11; Journal ID: ISSN 1029-8479

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AMPLITUDES; DUALITY; FORECASTING; OPERATOR PRODUCT EXPANSION; SCATTERING; SUPERSYMMETRY; SYMMETRY; WILSON LOOP; YANG-MILLS THEORY; Theory-HEP,HEPTH

### Citation Formats

```
Dixon, Lance J, /CERN /SLAC, Drummond, James M, /CERN /Annecy, LAPTH, Henn, Johannes M, and /Humboldt U., Berlin /Santa Barbara, KITP.
```*Bootstrapping the Three-Loop Hexagon*. United States: N. p., 2011.
Web. doi:10.1007/JHEP11(2011)023.

```
Dixon, Lance J, /CERN /SLAC, Drummond, James M, /CERN /Annecy, LAPTH, Henn, Johannes M, & /Humboldt U., Berlin /Santa Barbara, KITP.
```*Bootstrapping the Three-Loop Hexagon*. United States. doi:10.1007/JHEP11(2011)023.

```
Dixon, Lance J, /CERN /SLAC, Drummond, James M, /CERN /Annecy, LAPTH, Henn, Johannes M, and /Humboldt U., Berlin /Santa Barbara, KITP. Tue .
"Bootstrapping the Three-Loop Hexagon". United States. doi:10.1007/JHEP11(2011)023. https://www.osti.gov/servlets/purl/1029137.
```

```
@article{osti_1029137,
```

title = {Bootstrapping the Three-Loop Hexagon},

author = {Dixon, Lance J and /CERN /SLAC and Drummond, James M and /CERN /Annecy, LAPTH and Henn, Johannes M and /Humboldt U., Berlin /Santa Barbara, KITP},

abstractNote = {We consider the hexagonal Wilson loop dual to the six-point MHV amplitude in planar N = 4 super Yang-Mills theory. We apply constraints from the operator product expansion in the near-collinear limit to the symbol of the remainder function at three loops. Using these constraints, and assuming a natural ansatz for the symbol's entries, we determine the symbol up to just two undetermined constants. In the multi-Regge limit, both constants drop out from the symbol, enabling us to make a non-trivial confirmation of the BFKL prediction for the leading-log approximation. This result provides a strong consistency check of both our ansatz for the symbol and the duality between Wilson loops and MHV amplitudes. Furthermore, we predict the form of the full three-loop remainder function in the multi-Regge limit, beyond the leading-log approximation, up to a few constants representing terms not detected by the symbol. Our results confirm an all-loop prediction for the real part of the remainder function in multi-Regge 3 {yields} 3 scattering. In the multi-Regge limit, our result for the remainder function can be expressed entirely in terms of classical polylogarithms. For generic six-point kinematics other functions are required.},

doi = {10.1007/JHEP11(2011)023},

journal = {JHEP 1111:023,2011},

issn = {1029-8479},

number = 11,

volume = 2011,

place = {United States},

year = {2011},

month = {11}

}