Abstract
It has been recently conjectured that scattering amplitudes in planar \( \mathcal{N}=4 \) super Yang-Mills are given by the volume of the (dual) amplituhedron. In this paper we show some interesting connections between the tree-level amplituhedron and a special class of differential equations. In particular we demonstrate how the amplituhedron volume for NMHV amplitudes is determined by these differential equations. The new formulation allows for a straightforward geometric description, without any reference to triangulations. Finally we discuss possible implications for volumes related to generic NkMHV amplitudes.
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References
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [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, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Scattering Amplitudes and the Positive Grassmannian, arXiv:1212.5605 [INSPIRE].
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in \( \mathcal{N}=4 \) super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].
J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in \( \mathcal{N}=4 \) super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].
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, F. Cachazo and C. Cheung, The Grassmannian Origin Of Dual Superconformal Invariance, JHEP 03 (2010) 036 [arXiv:0909.0483] [INSPIRE].
J.M. Drummond and L. Ferro, Yangians, Grassmannians and T-duality, JHEP 07 (2010) 027 [arXiv:1001.3348] [INSPIRE].
J.M. Drummond and L. Ferro, The Yangian origin of the Grassmannian integral, JHEP 12 (2010) 010 [arXiv:1002.4622] [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].
L.J. Mason and D. Skinner, Dual Superconformal Invariance, Momentum Twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].
N. Arkani-Hamed, A. Hodges and J. Trnka, Positive Amplitudes In The Amplituhedron, JHEP 08 (2015) 030 [arXiv:1412.8478] [INSPIRE].
G.P. Korchemsky and E. Sokatchev, Superconformal invariants for scattering amplitudes in \( \mathcal{N}=4 \) SYM theory, Nucl. Phys. B 839 (2010) 377 [arXiv:1002.4625] [INSPIRE].
L. Ferro, T. Lukowski, C. Meneghelli, J. Plefka and M. Staudacher, Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization, Phys. Rev. Lett. 110 (2013) 121602 [arXiv:1212.0850] [INSPIRE].
L. Ferro, T. Lukowski, C. Meneghelli, J. Plefka and M. Staudacher, Spectral Parameters for Scattering Amplitudes in \( \mathcal{N}=4 \) Super Yang-Mills Theory, JHEP 01 (2014) 094 [arXiv:1308.3494] [INSPIRE].
I.M. Gelfand, General theory of hypergeometric functions, Dokl. Akad. Nauk SSSR 288 (1986) 14.
K. Aomoto, Les équations aux différences linéaires et les intégrales des fonctions multiformes, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975) 271.
M. Kita and K. Aomoto, Theory of hypergeometric functions, Springer-Verlag (2011).
L. Ferro, T. Lukowski and M. Staudacher, \( \mathcal{N}=4 \) scattering amplitudes and the deformed Graßmannian, Nucl. Phys. B 889 (2014) 192 [arXiv:1407.6736] [INSPIRE].
I.M. Gelfand, M.I. Graev and V.S. Retakh, General hypergeometric systems of equations and series of hypergeometric type, Uspekhi Mat. Nauk SSSR 47 (1992) 3.
T. Oshima, Capelli Identities, Degenerate Series and Hypergeometric Functions, in Proceedings of a symposium on Representation Theory at Okinawa (1995), pg. 1-19.
I. Gel’fand and G. Shilov, Generalized Functions: Properties and operations, Academic Press (1964).
N. Arkani-Hamed, The amplituhedron, scattering amplitudes and Ψ U , New geometric structures in scattering amplitudes, http://people.maths.ox.ac.uk/lmason/NGSA14/Films/Nima-Arkani-Hamed.mp4.
T. Bargheer, Y.-t. Huang, F. Loebbert and M. Yamazaki, Integrable Amplitude Deformations for \( \mathcal{N}=4 \) Super Yang-Mills and ABJM Theory, Phys. Rev. D 91 (2015) 026004 [arXiv:1407.4449] [INSPIRE].
R. Frassek, N. Kanning, Y. Ko and M. Staudacher, Bethe Ansatz for Yangian Invariants: Towards Super Yang-Mills Scattering Amplitudes, Nucl. Phys. B 883 (2014) 373 [arXiv:1312.1693] [INSPIRE].
N. Kanning, T. Lukowski and M. Staudacher, A shortcut to general tree-level scattering amplitudes in \( \mathcal{N}=4 \) SYM via integrability, Fortsch. Phys. 62 (2014) 556 [arXiv:1403.3382] [INSPIRE].
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ArXiv ePrint: 1512.04954
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Ferro, L., Lukowski, T., Orta, A. et al. Towards the amplituhedron volume. J. High Energ. Phys. 2016, 14 (2016). https://doi.org/10.1007/JHEP03(2016)014
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DOI: https://doi.org/10.1007/JHEP03(2016)014