Landau singularities and symbology: one- and two-loop MHV amplitudes in SYM theory

  • Tristan Dennen
  • Marcus Spradlin
  • Anastasia Volovich
Open Access
Regular Article - Theoretical Physics


We apply the Landau equations, whose solutions parameterize the locus of possible branch points, to the one- and two-loop Feynman integrals relevant to MHV amplitudes in planar \( \mathcal{N}=4 \) super-Yang-Mills theory. We then identify which of the Landau singularities appear in the symbols of the amplitudes, and which do not. We observe that all of the symbol entries in the two-loop MHV amplitudes are already present as Landau singularities of one-loop pentagon integrals.


Scattering Amplitudes Supersymmetric gauge theory 


  1. [1]
    L.D. Landau, On analytic properties of vertex parts in quantum field theory, Nucl. Phys. 13 (1959) 181 [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    R.J. Eden et al., The analytic S-matrix, Cambridge University Press, Cambridge U.K. (1966).MATHGoogle Scholar
  3. [3]
    G.F. Sterman, An introduction to quantum field theory, Cambridge University Press, Cambridge U.K. (1963).Google Scholar
  4. [4]
    K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    A.B. Goncharov, A simple construction of Grassmannian polylogarithms, arXiv:0908.2238 [INSPIRE].
  6. [6]
    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
  7. [7]
    J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, arXiv:1509.03612 [INSPIRE].
  8. [8]
    S. Abreu, R. Britto, C. Duhr and E. Gardi, From multiple unitarity cuts to the coproduct of Feynman integrals, JHEP 10 (2014) 125 [arXiv:1401.3546] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    S. Abreu, R. Britto and H. Grönqvist, Cuts and coproducts of massive triangle diagrams, JHEP 07 (2015) 111 [arXiv:1504.00206] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    R.K. Ellis and G. Zanderighi, Scalar one-loop integrals for QCD, JHEP 02 (2008) 002 [arXiv:0712.1851] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    N. Arkani-Hamed et al., Scattering amplitudes and the positive grassmannian, arXiv:1212.5605.
  17. [17]
    J.M. Drummond and J.M. Henn, Simple loop integrals and amplitudes in N = 4 SYM, JHEP 05 (2011) 105 [arXiv:1008.2965] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    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
  20. [20]
    J.L. Bourjaily and J. Trnka, Local integrand representations of all two-loop amplitudes in planar SYM, JHEP 08 (2015) 119 [arXiv:1505.05886] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    D.B. Fairlie et al., Singularities of the second type, J. Math. Phys. 3 (1962) 594.ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    D.B. Fairlie et al., Physical sheet properties of second type singularities, Phys. Lett. 3 (1962) 55.ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    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
  24. [24]
    L.F. Alday, J.M. Henn, J. Plefka and T. Schuster, Scattering into the fifth dimension of N = 4 super Yang-Mills, JHEP 01 (2010) 077 [arXiv:0908.0684] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    G. ’t Hooft and M.J.G. Veltman, Scalar one loop integrals, Nucl. Phys. B 153 (1979) 365 [INSPIRE].
  26. [26]
    H.J. Lu and C.A. Perez, Massless one loop scalar three point integral and associated Clausen, Glaisher and L functions, SLAC-PUB-5809 (1992) [INSPIRE].
  27. [27]
    A. Goncharov, Volumes of hyperbolic manifolds and mixed Tate motives, J. AMS 12 (1996) 569 [alg-geom/9601021].MathSciNetMATHGoogle Scholar
  28. [28]
    M. Spradlin and A. Volovich, Symbols of one-loop integrals from mixed Tate motives, JHEP 11 (2011) 084 [arXiv:1105.2024] [INSPIRE].MATHGoogle Scholar
  29. [29]
    L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    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].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar N = 4 Super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    L.J. Dixon, J.M. Drummond, C. Duhr, M. von Hippel and J. Pennington, Bootstrapping six-gluon scattering in planar N = 4 super-Yang-Mills theory, PoS(LL2014)077 [arXiv:1407.4724] [INSPIRE].
  34. [34]
    L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    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
  36. [36]
    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

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Tristan Dennen
    • 1
  • Marcus Spradlin
    • 1
  • Anastasia Volovich
    • 1
  1. 1.Department of PhysicsBrown UniversityProvidenceU.S.A.

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