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Title: Multi-loop positivity of the planar $$ \mathcal{N} $$ = 4 SYM six-point amplitude

Abstract

We study the six-point 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 six-gluon amplitude. While the remainder function ceases to be positive at four loops, the BDS-like normalized MHV amplitude appears to be positive through five loops.

Authors:
ORCiD logo [1];  [2];  [1];  [3]
  1. SLAC National Accelerator Lab., Menlo Park, CA (United States)
  2. Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
  3. Univ. of California, Davis, CA (United States). Center for Quantum Mathematics and Physics (QMAP)
Publication Date:
Research Org.:
SLAC National Accelerator Lab., Menlo Park, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP)
OSTI Identifier:
1334238
Report Number(s):
SLAC-PUB-16873
Journal ID: ISSN 1029-8479; arXiv:1611.08325
Grant/Contract Number:  
AC02-76SF00515
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Volume: 2017; Journal Issue: 2; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; HEPTH, MATH

Citation Formats

Dixon, Lance J., von Hippel, Matt, McLeod, Andrew J., and Trnka, Jaroslav. Multi-loop positivity of the planar $ \mathcal{N} $ = 4 SYM six-point amplitude. United States: N. p., 2017. Web. doi:10.1007/JHEP02(2017)112.
Dixon, Lance J., von Hippel, Matt, McLeod, Andrew J., & Trnka, Jaroslav. Multi-loop positivity of the planar $ \mathcal{N} $ = 4 SYM six-point amplitude. United States. https://doi.org/10.1007/JHEP02(2017)112
Dixon, Lance J., von Hippel, Matt, McLeod, Andrew J., and Trnka, Jaroslav. 2017. "Multi-loop positivity of the planar $ \mathcal{N} $ = 4 SYM six-point amplitude". United States. https://doi.org/10.1007/JHEP02(2017)112. https://www.osti.gov/servlets/purl/1334238.
@article{osti_1334238,
title = {Multi-loop positivity of the planar $ \mathcal{N} $ = 4 SYM six-point amplitude},
author = {Dixon, Lance J. and von Hippel, Matt and McLeod, Andrew J. and Trnka, Jaroslav},
abstractNote = {We study the six-point 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 six-gluon amplitude. While the remainder function ceases to be positive at four loops, the BDS-like normalized MHV amplitude appears to be positive through five loops.},
doi = {10.1007/JHEP02(2017)112},
url = {https://www.osti.gov/biblio/1334238}, journal = {Journal of High Energy Physics (Online)},
issn = {1029-8479},
number = 2,
volume = 2017,
place = {United States},
year = {Wed Feb 22 00:00:00 EST 2017},
month = {Wed Feb 22 00:00:00 EST 2017}
}

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Works referenced in this record:

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Works referencing / citing this record:

Yangian invariants and cluster adjacency in $$ \mathcal{N} $$ = 4 Yang-Mills
journal, October 2019


The cosmic Galois group and extended Steinmann relations for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes
text, January 2019


The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N} = 4$ SYM Amplitudes
text, January 2019


Rationalizing Loop Integration
text, January 2018


The Sklyanin Bracket and Cluster Adjacency at All Multiplicity
text, January 2019


PolyLogTools - Polylogs for the masses
text, January 2019