Abstract
Describing the geometry of the dual amplituhedron without reference to a particular triangulation is an open problem. In this note we introduce a new way of determining the volume of the tree-level NMHV dual amplituhedron. We show that certain contour integrals of logarithms serve as natural building blocks for computing this volume as well as the volumes of general polytopes in any dimension. These building blocks encode the geometry of the underlying polytopes in a triangulation-independent way, and make identities between different representations of the amplitudes manifest.
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References
H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
L.J. Dixon, Calculating scattering amplitudes efficiently, in QCD and beyond. Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics, TASI-95, Boulder, U.S.A., June 4-30, 1995, pp. 539-584, hep-ph/9601359 [INSPIRE].
J.M. Henn and J.C. Plefka, Scattering Amplitudes in Gauge Theories, in Lecture Notes in Physics 883, Springer (2014).
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
N. Arkani-Hamed and J. Trnka, Into the Amplituhedron, JHEP 12 (2014) 182 [arXiv:1312.7878] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press, (2016), [arXiv:1212.5605] [INSPIRE].
A. Postnikov, Total positivity, Grassmannians and networks, math/0609764 [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A. Hodges and J. Trnka, A Note on Polytopes for Scattering Amplitudes, JHEP 04 (2012) 081 [arXiv:1012.6030] [INSPIRE].
N. Arkani-Hamed, A. Hodges and J. Trnka, Positive Amplitudes In The Amplituhedron, JHEP 08 (2015) 030 [arXiv:1412.8478] [INSPIRE].
L. Ferro, T. Lukowski, A. Orta and M. Parisi, Towards the Amplituhedron Volume, JHEP 03 (2016) 014 [arXiv:1512.04954] [INSPIRE].
M. Enciso, Volumes of Polytopes Without Triangulations, JHEP 10 (2017) 071 [arXiv:1408.0932] [INSPIRE].
L.J. Mason and D. Skinner, Dual Superconformal Invariance, Momentum Twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].
N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].
J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].
S. Huggett and K. Tod, An Introduction to Twistor Theory, Cambridge University Press, (1994).
L. Hughston and T. Hurd, A CP 5 Calculus For Space-time Fields, Phys. Rept. 100 (1983) 275.
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
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ArXiv ePrint: 1612.07370
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Enciso, M. Logarithms and volumes of polytopes. J. High Energ. Phys. 2018, 16 (2018). https://doi.org/10.1007/JHEP04(2018)016
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DOI: https://doi.org/10.1007/JHEP04(2018)016