Advertisement

Journal of High Energy Physics

, 2014:182 | Cite as

Into the amplituhedron

  • Nima Arkani-Hamed
  • Jaroslav TrnkaEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

We initiate an exploration of the physics and geometry of the amplituhedron, starting with the simplest case of the integrand for four-particle scattering in planar \( \mathcal{N}=4 \) SYM. We show how the textbook structure of the unitarity double-cut follows from the positive geometry. We also use the geometry to expose the behavior of the multicollinear limit, providing a direct motivation for studying the logarithm of the amplitude. In addition to computing the two and three-loop integrands, we explore various lower-dimensional faces of the amplituhedron, thereby computing non-trivial cuts of the integrand to all loop orders.

Keywords

Supersymmetric gauge theory Scattering Amplitudes 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    N. Arkani-Hamed and J. Trnka, The amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    A. Postnikov, Total positivity, Grassmannians and networks, math/0609764 [INSPIRE].
  3. [3]
    V.V. Fock and A.B. Goncharov, Cluster ensembles, quantization and the dilogarithm, math/0311245 [INSPIRE].
  4. [4]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    N. Arkani-Hamed et al., Scattering amplitudes and the positive Grassmannian, arXiv:1212.5605 [INSPIRE].
  6. [6]
    A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 75 (2007) 085010 [hep-th/0610248] [INSPIRE].ADSMathSciNetGoogle Scholar
  8. [8]
    Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    Z. Bern, J.J.M. Carrasco, H. Johansson and D.A. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev. D 76 (2007) 125020 [arXiv:0705.1864] [INSPIRE].ADSMathSciNetGoogle Scholar
  10. [10]
    J.L. Bourjaily, A. DiRe, A. Shaikh, M. Spradlin and A. Volovich, The soft-collinear bootstrap: N = 4 Yang-Mills amplitudes at six and seven loops, JHEP 03 (2012) 032 [arXiv:1112.6432] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N = 4 SYM, Nucl. Phys. B 862 (2012) 450 [arXiv:1201.5329] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    S. Caron-Huot, Loops and trees, JHEP 05 (2011) 080 [arXiv:1007.3224] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local integrals for planar scattering amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  15. [15]
    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].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    N. Arkani-Hamed, A. Hodges and J. Trnka, Positive amplitudes in the amplituhedron, arXiv:1412.8478 [INSPIRE].
  17. [17]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    N. Beisert and M. Staudacher, The N = 4 SYM integrable super spin chain, Nucl. Phys. B 670 (2003) 439 [hep-th/0307042] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 01 (2007) P01021 [hep-th/0610251] [INSPIRE].Google Scholar
  20. [20]
    B. Eden and M. Staudacher, Integrability and transcendentality, J. Stat. Mech. 11 (2006) P11014 [hep-th/0603157] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  21. [21]
    L. Ferro, T. Łukowski, C. Meneghelli, J. Plefka and M. Staudacher, Spectral parameters for scattering amplitudes in N = 4 super Yang-Mills theory, JHEP 01 (2014) 094 [arXiv:1308.3494] [INSPIRE].CrossRefGoogle Scholar
  22. [22]
    L. Ferro, T. Łukowski, 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].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.
  2. 2.Walter Burke Institute for Theoretical PhysicsCalifornia Institute of TechnologyPasadenaU.S.A.

Personalised recommendations