Multiloop positivity of the planar $$ \mathcal{N} $$ = 4 SYM sixpoint amplitude
We study the sixpoint NMHV ratio function in planar $$ \mathcal{N} $$ = 4 SYM theory in the context of positive geometry. The Amplituhedron construction of the integrand for the amplitudes provides a kinematical region in which the integrand was observed to be positive. It is natural to conjecture that this property survives integration, i.e. that the final result for the ratio function is also positive in this region. Establishing such a result would imply that preserving positivity is a surprising property of the Minkowski contour of integration and it might indicate some deeper underlying structure. We find that the ratio function is positive everywhere we have tested it, including analytic results for special kinematical regions at one and two loops, as well as robust numerical evidence through five loops. There is also evidence for not just positivity, but monotonicity in a “radial” direction. We also investigate positivity of the MHV sixgluon amplitude. While the remainder function ceases to be positive at four loops, the BDSlike normalized MHV amplitude appears to be positive through five loops.
 Authors:

^{[1]}
;
^{[2]};
^{[1]};
^{[3]}
 SLAC National Accelerator Lab., Menlo Park, CA (United States)
 Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
 Univ. of California, Davis, CA (United States). Center for Quantum Mathematics and Physics (QMAP)
 Publication Date:
 Report Number(s):
 SLACPUB16873
Journal ID: ISSN 10298479; arXiv:1611.08325
 Grant/Contract Number:
 AC0276SF00515
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of High Energy Physics (Online)
 Additional Journal Information:
 Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 2; Journal ID: ISSN 10298479
 Publisher:
 Springer Berlin
 Research Org:
 SLAC National Accelerator Lab., Menlo Park, CA (United States)
 Sponsoring Org:
 USDOE Office of Science (SC), High Energy Physics (HEP) (SC25)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; HEPTH, MATH
 OSTI Identifier:
 1334238
Dixon, Lance J., von Hippel, Matt, McLeod, Andrew J., and Trnka, Jaroslav. Multiloop positivity of the planar $ \mathcal{N} $ = 4 SYM sixpoint amplitude. United States: N. p.,
Web. doi:10.1007/JHEP02(2017)112.
Dixon, Lance J., von Hippel, Matt, McLeod, Andrew J., & Trnka, Jaroslav. Multiloop positivity of the planar $ \mathcal{N} $ = 4 SYM sixpoint amplitude. United States. doi:10.1007/JHEP02(2017)112.
Dixon, Lance J., von Hippel, Matt, McLeod, Andrew J., and Trnka, Jaroslav. 2017.
"Multiloop positivity of the planar $ \mathcal{N} $ = 4 SYM sixpoint amplitude". United States.
doi:10.1007/JHEP02(2017)112. https://www.osti.gov/servlets/purl/1334238.
@article{osti_1334238,
title = {Multiloop positivity of the planar $ \mathcal{N} $ = 4 SYM sixpoint amplitude},
author = {Dixon, Lance J. and von Hippel, Matt and McLeod, Andrew J. and Trnka, Jaroslav},
abstractNote = {We study the sixpoint NMHV ratio function in planar $ \mathcal{N} $ = 4 SYM theory in the context of positive geometry. The Amplituhedron construction of the integrand for the amplitudes provides a kinematical region in which the integrand was observed to be positive. It is natural to conjecture that this property survives integration, i.e. that the final result for the ratio function is also positive in this region. Establishing such a result would imply that preserving positivity is a surprising property of the Minkowski contour of integration and it might indicate some deeper underlying structure. We find that the ratio function is positive everywhere we have tested it, including analytic results for special kinematical regions at one and two loops, as well as robust numerical evidence through five loops. There is also evidence for not just positivity, but monotonicity in a “radial” direction. We also investigate positivity of the MHV sixgluon amplitude. While the remainder function ceases to be positive at four loops, the BDSlike normalized MHV amplitude appears to be positive through five loops.},
doi = {10.1007/JHEP02(2017)112},
journal = {Journal of High Energy Physics (Online)},
number = 2,
volume = 2017,
place = {United States},
year = {2017},
month = {2}
}