Emergent unitarity from the amplituhedron

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We present a proof of perturbative unitarity for planar \( \mathcal{N} \) = 4 SYM, following from the geometry of the amplituhedron. This proof is valid for amplitudes of arbitrary multiplicity n, loop order L and MHV degree k.

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  1. [1]

    Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Manifest ultraviolet behavior for the three-loop four-point amplitude of N = 8 supergravity, Phys. Rev.D 78 (2008) 105019 [arXiv:0808.4112] [INSPIRE].

  2. [2]

    C. Anastasiou, R. Britto, B. Feng, Z. Kunszt and P. Mastrolia, D-dimensional unitarity cut method, Phys. Lett.B 645 (2007) 213 [hep-ph/0609191] [INSPIRE].

  3. [3]

    R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys.D 725 (2005) 275 [hep-th/0412103] [INSPIRE].

  4. [4]

    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys.B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].

  5. [5]

    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n-point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys.B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].

  6. [6]

    R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys.D 715 (2005) 499 [hep-th/0412308] [INSPIRE].

  7. [7]

    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, JHEP01 (2011) 041 [arXiv:1008.2958] [INSPIRE].

  8. [8]

    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge, U.K. (2016) [arXiv:1212.5605] [INSPIRE].

  9. [9]

    N. Arkani-Hamed and J. Trnka, The amplituhedron, JHEP10 (2014) 030 [arXiv:1312.2007] [INSPIRE].

  10. [10]

    N. Arkani-Hamed and J. Trnka, Into the amplituhedron, JHEP12 (2014) 182 [arXiv:1312.7878] [INSPIRE].

  11. [11]

    A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP05 (2013) 135 [arXiv:0905.1473] [INSPIRE].

  12. [12]

    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A. Hodges and J. Trnka, A note on polytopes for scattering amplitudes, JHEP04 (2012) 081 [arXiv:1012.6030] [INSPIRE].

  13. [13]

    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP03 (2010) 020 [arXiv:0907.5418] [INSPIRE].

  14. [14]

    L.J. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and Grassmannians, JHEP11 (2009) 045 [arXiv:0909.0250] [INSPIRE].

  15. [15]

    A. Postnikov, Total positivity, Grassmannians and networks, math.CO/0609764 [INSPIRE].

  16. [16]

    N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the amplituhedron in binary, JHEP01 (2018) 016 [arXiv:1704.05069] [INSPIRE].

  17. [17]

    J.L. Bourjaily, J. Trnka, A. Volovich and C. Wen, The Grassmannian and the twistor string: connecting all trees in N = 4 SYM, JHEP01 (2011) 038 [arXiv:1006.1899] [INSPIRE].

  18. [18]

    N. Arkani-Hamed, F. Cachazo and C. Cheung, The Grassmannian origin of dual superconformal invariance, JHEP03 (2010) 036 [arXiv:0909.0483] [INSPIRE].

  19. [19]

    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local integrals for planar scattering amplitudes, JHEP06 (2012) 125 [arXiv:1012.6032] [INSPIRE].

  20. [20]

    C. Langer and A. Yelleshpur Srikant, All-loop cuts from the amplituhedron, JHEP04 (2019) 105 [arXiv:1902.05951] [INSPIRE].

  21. [21]

    N. Arkani-Hamed, C. Langer, A. Yelleshpur Srikant and J. Trnka, Deep into the amplituhedron: amplitude singularities at all loops and legs, Phys. Rev. Lett.122 (2019) 051601 [arXiv:1810.08208] [INSPIRE].

  22. [22]

    S. Caron-Huot, Loops and trees, JHEP05 (2011) 080 [arXiv:1007.3224] [INSPIRE].

  23. [23]

    N. Arkani-Hamed, Y. Bai and T. Lam, Positive geometries and canonical forms, JHEP11 (2017) 039 [arXiv:1703.04541] [INSPIRE].

  24. [24]

    G. Salvatori, 1-loop amplitudes from the halohedron, JHEP12 (2019) 074 [arXiv:1806.01842] [INSPIRE].

  25. [25]

    G. Salvatori and S.L. Cacciatori, Hyperbolic geometry and amplituhedra in 1 + 2 dimensions, JHEP08 (2018) 167 [arXiv:1803.05809] [INSPIRE].

  26. [26]

    S. He, G. Yan, C. Zhang and Y. Zhang, Scattering forms, worldsheet forms and amplitudes from subspaces, JHEP08 (2018) 040 [arXiv:1803.11302] [INSPIRE].

  27. [27]

    N. Arkani-Hamed, Y. Bai, S. He and G. Yan, Scattering forms and the positive geometry of kinematics, color and the worldsheet, JHEP05 (2018) 096 [arXiv:1711.09102] [INSPIRE].

  28. [28]

    S. He and C. Zhang, Notes on scattering amplitudes as differential forms, JHEP10 (2018) 054 [arXiv:1807.11051] [INSPIRE].

  29. [29]

    D. Damgaard, L. Ferro, T. Lukowski and M. Parisi, The momentum amplituhedron, JHEP08 (2019) 042 [arXiv:1905.04216] [INSPIRE].

  30. [30]

    C. Anastasiou, R. Britto, B. Feng, Z. Kunszt and P. Mastrolia, Unitarity cuts and reduction to master integrals in d dimensions for one-loop amplitudes, JHEP03 (2007) 111 [hep-ph/0612277] [INSPIRE].

  31. [31]

    R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys.1 (1960) 429 [INSPIRE].

  32. [32]

    R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix. Cambridge Univ. Press, Cambridge, U.K. (1966) [INSPIRE].

  33. [33]

    T. Lukowski, On the boundaries of the m = 2 amplituhedron, arXiv:1908.00386 [INSPIRE].

  34. [34]

    S. Franco, D. Galloni, A. Mariotti and J. Trnka, Anatomy of the amplituhedron, JHEP03 (2015) 128 [arXiv:1408.3410] [INSPIRE].

  35. [35]

    D. Galloni, Positivity sectors and the amplituhedron, arXiv:1601.02639 [INSPIRE].

  36. [36]

    I. Prlina, M. Spradlin, J. Stankowicz and S. Stanojevic, Boundaries of amplituhedra and NMHV symbol alphabets at two loops, JHEP04 (2018) 049 [arXiv:1712.08049] [INSPIRE].

  37. [37]

    I. Prlina, M. Spradlin, J. Stankowicz, S. Stanojevic and A. Volovich, All-helicity symbol alphabets from unwound amplituhedra, JHEP05 (2018) 159 [arXiv:1711.11507] [INSPIRE].

  38. [38]

    T. Dennen, I. Prlina, M. Spradlin, S. Stanojevic and A. Volovich, Landau singularities from the amplituhedron, JHEP06 (2017) 152 [arXiv:1612.02708] [INSPIRE].

  39. [39]

    T. Dennen, M. Spradlin and A. Volovich, Landau singularities and symbology: one- and two-loop MHV amplitudes in SYM theory, JHEP03 (2016) 069 [arXiv:1512.07909] [INSPIRE].

  40. [40]

    I. Prlina, M. Spradlin and S. Stanojevic, All-loop singularities of scattering amplitudes in massless planar theories, Phys. Rev. Lett.121 (2018) 081601 [arXiv:1805.11617] [INSPIRE].

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Correspondence to Akshay Yelleshpur Srikant.

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ArXiv ePrint: 1906.10700

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Srikant, A.Y. Emergent unitarity from the amplituhedron. J. High Energ. Phys. 2020, 69 (2020) doi:10.1007/JHEP01(2020)069

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  • Scattering Amplitudes
  • Supersymmetric Gauge Theory