Boundaries of amplituhedra and NMHV symbol alphabets at two loops

  • I. Prlina
  • M. Spradlin
  • J. Stankowicz
  • S. Stanojevic
Open Access
Regular Article - Theoretical Physics


In this sequel to [3] we classify the boundaries of amplituhedra relevant for determining the branch points of general two-loop amplitudes in planar \( \mathcal{N}=4 \) super-Yang-Mills theory. We explain the connection to on-shell diagrams, which serves as a useful cross-check. We determine the branch points of all two-loop NMHV amplitudes by solving the Landau equations for the relevant configurations and are led thereby to a conjecture for the symbol alphabets of all such amplitudes.


Scattering Amplitudes Supersymmetric Gauge Theory 


Open Access

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  1. [1]
    T. Dennen, M. Spradlin and A. Volovich, Landau singularities and symbology: one- and two-loop MHV amplitudes in SYM theory, JHEP 03 (2016) 069 [arXiv:1512.07909] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    T. Dennen, I. Prlina, M. Spradlin, S. Stanojevic and A. Volovich, Landau singularities from the amplituhedron, JHEP 06 (2017) 152 [arXiv:1612.02708] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    I. Prlina, M. Spradlin, J. Stankowicz, S. Stanojevic and A. Volovich, All-helicity symbol alphabets from unwound amplituhedra, arXiv:1711.11507 [INSPIRE].
  4. [4]
    A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    L.D. Landau, On analytic properties of vertex parts in quantum field theory, Nucl. Phys. 13 (1959) 181 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix, Cambridge University Press, Cmbridge U.K. (1966).zbMATHGoogle Scholar
  8. [8]
    L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP 01 (2016) 053 [arXiv:1509.08127] [INSPIRE].CrossRefGoogle Scholar
  12. [12]
    S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a five-loop amplitude using Steinmann relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    J.M. Drummond, G. Papathanasiou and M. Spradlin, A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon, JHEP 03 (2015) 072 [arXiv:1412.3763] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    L.J. Dixon et al., Heptagons from the Steinmann cluster bootstrap, JHEP 02 (2017) 137 [arXiv:1612.08976] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J.L. Bourjaily et al., The elliptic double-box integral: massless amplitudes beyond polylogarithms, Phys. Rev. Lett. 120 (2018) 121603 [arXiv:1712.02785] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    D. Nandan, M.F. Paulos, M. Spradlin and A. Volovich, Star integrals, convolutions and simplices, JHEP 05 (2013) 105 [arXiv:1301.2500] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    N. Arkani-Hamed and J. Trnka, The amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the amplituhedron in binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    N. Arkani-Hamed and J. Trnka, Into the amplituhedron, JHEP 12 (2014) 182 [arXiv:1312.7878] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    N. Arkani-Hamed et al,, Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge U.K. (2016), arXiv:1212.5605 [INSPIRE].
  25. [25]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    S. Caron-Huot and S. He, Jumpstarting the all-loop S-matrix of planar N = 4 super Yang-Mills, JHEP 07 (2012) 174 [arXiv:1112.1060] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic amplitudes and cluster coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    J. Golden and M. Spradlin, A cluster bootstrap for two-loop MHV amplitudes, JHEP 02 (2015) 002 [arXiv:1411.3289] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    J. Drummond, J. Foster and O. Gurdogan, Cluster adjacency properties of scattering amplitudes, arXiv:1710.10953 [INSPIRE].
  31. [31]
    J. Golden, M.F. Paulos, M. Spradlin and A. Volovich, Cluster polylogarithms for scattering amplitudes, J. Phys. A 47 (2014) 474005 [arXiv:1401.6446] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  32. [32]
    T. Harrington and M. Spradlin, Cluster functions and scattering amplitudes for six and seven points, JHEP 07 (2017) 016 [arXiv:1512.07910] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    J. Golden and M. Spradlin, The differential of all two-loop MHV amplitudes in \( \mathcal{N}=4 \) Yang-Mills theory, JHEP 09 (2013) 111 [arXiv:1306.1833] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Singularity structure of maximally supersymmetric scattering amplitudes, Phys. Rev. Lett. 113 (2014) 261603 [arXiv:1410.0354] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Logarithmic singularities and maximally supersymmetric amplitudes, JHEP 06 (2015) 202 [arXiv:1412.8584] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a nonplanar amplituhedron, JHEP 06 (2016) 098 [arXiv:1512.08591] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of PhysicsBrown UniversityProvidenceU.S.A.
  2. 2.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.
  3. 3.Kavli Institute for Theoretical PhysicsUniversity of CaliforniaSanta BarbaraU.S.A.

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