Skip to main content

Advertisement

SpringerLink
The Amplituhedron
Download PDF
Download PDF
  • Open Access
  • Published: 06 October 2014

The Amplituhedron

  • Nima Arkani-Hamed1 &
  • Jaroslav Trnka2 

Journal of High Energy Physics volume 2014, Article number: 30 (2014) Cite this article

  • 3009 Accesses

  • 275 Citations

  • 222 Altmetric

  • Metrics details

A preprint version of the article is available at arXiv.

Abstract

Perturbative scattering amplitudes in gauge theories have remarkable simplicity and hidden infinite dimensional symmetries that are completely obscured in the conventional formulation of field theory using Feynman diagrams. This suggests the existence of a new understanding for scattering amplitudes where locality and unitarity do not play a central role but are derived consequences from a different starting point. In this note we provide such an understanding for \( \mathcal{N}=4 \) SYM scattering amplitudes in the planar limit, which we identify as “the volume” of a new mathematical object — the Amplituhedron — generalizing the positive Grassmannian. Locality and unitarity emerge hand-in-hand from positive geometry.

Download to read the full article text

Working on a manuscript?

Avoid the most common mistakes and prepare your manuscript for journal editors.

Learn more

References

  1. S.J. Parke and T.R. Taylor, An amplitude for n gluon scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].

    Article  ADS  Google Scholar 

  2. 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. 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].

    Article  ADS  Google Scholar 

  4. E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. F. Cachazo, P. Svrček and E. Witten, MHV vertices and tree amplitudes in gauge theory, JHEP 09 (2004) 006 [hep-th/0403047] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. 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].

    Article  ADS  MathSciNet  Google Scholar 

  8. L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  9. 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].

    Article  ADS  MathSciNet  Google Scholar 

  10. S. Caron-Huot, Notes on the scattering amplitude/Wilson loop duality, JHEP 07 (2011) 058 [arXiv:1010.1167] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. L.J. Mason and D. Skinner, The complete planar S-matrix of N = 4 SYM as a Wilson loop in twistor space, JHEP 12 (2010) 018 [arXiv:1009.2225] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. L.F. Alday, B. Eden, G.P. Korchemsky, J. Maldacena and E. Sokatchev, From correlation functions to Wilson loops, JHEP 09 (2011) 123 [arXiv:1007.3243] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  14. N. Arkani-Hamed and J. Kaplan, On tree amplitudes in gauge theory and gravity, JHEP 04 (2008) 076 [arXiv:0801.2385] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, The S-matrix in twistor space, JHEP 03 (2010) 110 [arXiv:0903.2110] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S-matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. N. Arkani-Hamed et al., Scattering amplitudes and the positive Grassmannian, arXiv:1212.5605 [INSPIRE].

  20. R.H. Boels, On BCFW shifts of integrands and integrals, JHEP 11 (2010) 113 [arXiv:1008.3101] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. A.P. Hodges and S. Huggett, Twistor diagrams, Surveys High Energ. Phys. 1 (1980) 333 [INSPIRE].

    Article  ADS  Google Scholar 

  22. A.P. Hodges, Twistor diagram recursion for all gauge-theoretic tree amplitudes, hep-th/0503060 [INSPIRE].

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

  24. A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. N. Arkani-Hamed and J. Trnka, Into the Amplituhedron, arXiv:1312.7878 [INSPIRE].

  27. N. Arkani-Hamed, A. Hodges and J. Trnka, Three views of the Amplituhedron, to appear.

  28. N. Arkani-Hamed and J. Trnka, Scattering amplitudes from positive geometry, in preparation.

  29. H. Elvang and Y.-T. Huang, Scattering amplitudes, arXiv:1308.1697 [INSPIRE].

  30. A. Hodges, The box integrals in momentum-twistor geometry, JHEP 08 (2013) 051 [arXiv:1004.3323] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. L. Mason and D. Skinner, Amplitudes at weak coupling as polytopes in AdS 5, J. Phys. A 44 (2011) 135401 [arXiv:1004.3498] [INSPIRE].

    ADS  MATH  Google Scholar 

  32. 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Cyclic polytope — wikipedia webpage, http://en.wikipedia.org/wiki/Cyclic_polytope.

  34. L.J. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  35. S. Caron-Huot, Loops and trees, JHEP 05 (2011) 080 [arXiv:1007.3224] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. 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].

    ADS  MathSciNet  Google Scholar 

  37. 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].

    ADS  MathSciNet  Google Scholar 

  38. 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].

    ADS  MathSciNet  Google Scholar 

  39. 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  42. E. Witten, Quantum gravity in de Sitter space, hep-th/0106109 [INSPIRE].

  43. V.V. Fock and A.B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. École Norm. Sup. 42 (2009) 865 [math.AG/0311245] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  44. R. Roiban, M. Spradlin and A. Volovich, Dissolving N = 4 loop amplitudes into QCD tree amplitudes, Phys. Rev. Lett. 94 (2005) 102002 [hep-th/0412265] [INSPIRE].

    Article  ADS  Google Scholar 

  45. L. Dolan and P. Goddard, Gluon tree amplitudes in open twistor string theory, JHEP 12 (2009) 032 [arXiv:0909.0499] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  46. D. Nandan, A. Volovich and C. Wen, A Grassmannian étude in NMHV minors, JHEP 07 (2010) 061 [arXiv:0912.3705] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  47. N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Unification of residues and Grassmannian dualities, JHEP 01 (2011) 049 [arXiv:0912.4912] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. J.L. Bourjaily, J. Trnka, A. Volovich and C. Wen, The Grassmannian and the twistor string: connecting all trees in N = 4 SYM, JHEP 01 (2011) 038 [arXiv:1006.1899] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. A.E. Lipstein and L. Mason, From the holomorphic Wilson loop to ‘d log’ loop-integrands for super-Yang-Mills amplitudes, JHEP 05 (2013) 106 [arXiv:1212.6228] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. A.E. Lipstein and L. Mason, From d logs to dilogs the super Yang-Mills MHV amplitude revisited, JHEP 01 (2014) 169 [arXiv:1307.1443] [INSPIRE].

    Article  ADS  Google Scholar 

  51. 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].

    Article  ADS  Google Scholar 

  52. 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].

    Article  ADS  MathSciNet  Google Scholar 

  53. B. Basso, A. Sever and P. Vieira, Spacetime and flux tube S-matrices at finite coupling for N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 111 (2013) 091602 [arXiv:1303.1396] [INSPIRE].

    Article  ADS  Google Scholar 

  54. B. Basso, A. Sever and P. Vieira, Space-time S-matrix and flux tube S-matrix II. Extracting and matching data, JHEP 01 (2014) 008 [arXiv:1306.2058] [INSPIRE].

    Article  ADS  Google Scholar 

  55. 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. N. Beisert and M. Staudacher, The N = 4 SYM integrable super spin chain, Nucl. Phys. B 670 (2003) 439 [hep-th/0307042] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 01 (2007) P01021 [hep-th/0610251] [INSPIRE].

    Google Scholar 

  58. B. Eden and M. Staudacher, Integrability and transcendentality, J. Stat. Mech. 11 (2006) P11014 [hep-th/0603157] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  59. L. Ferro, T. Lukowski, 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].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. 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].

    Article  ADS  MATH  Google Scholar 

  61. Y.-T. Huang and C. Wen, ABJM amplitudes and the positive orthogonal Grassmannian, JHEP 02 (2014) 104 [arXiv:1309.3252] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  62. A. Hodges, A simple formula for gravitational MHV amplitudes, arXiv:1204.1930 [INSPIRE].

  63. F. Cachazo and Y. Geyer, A ‘twistor string’ inspired formula for tree-level scattering amplitudes in N = 8 SUGRA, arXiv:1206.6511 [INSPIRE].

  64. F. Cachazo and D. Skinner, Gravity from rational curves in twistor space, Phys. Rev. Lett. 110 (2013) 161301 [arXiv:1207.0741] [INSPIRE].

    Article  ADS  Google Scholar 

  65. D. Skinner, Twistor strings for N = 8 supergravity, arXiv:1301.0868 [INSPIRE].

  66. F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles in arbitrary dimension, arXiv:1307.2199 [INSPIRE].

  67. F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles: scalars, gluons and gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].

    Article  ADS  Google Scholar 

  68. Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  69. Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  70. N. Berkovits and J. Maldacena, Fermionic T-duality, dual superconformal symmetry and the amplitude/Wilson loop connection, JHEP 09 (2008) 062 [arXiv:0807.3196] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. School of Natural Sciences, Institute for Advanced Study, 1 Einstein Dr., Princeton, NJ, 08540, United Kingdom

    Nima Arkani-Hamed

  2. Walter Burke Institute for Theoretical Physics, California Institute of Technology, 1200 E Colorado Blvd., Pasadena, CA, 91125, United Kingdom

    Jaroslav Trnka

Authors
  1. Nima Arkani-Hamed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaroslav Trnka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jaroslav Trnka.

Additional information

ArXiv ePrint: 1312.2007

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Arkani-Hamed, N., Trnka, J. The Amplituhedron. J. High Energ. Phys. 2014, 30 (2014). https://doi.org/10.1007/JHEP10(2014)030

Download citation

  • Received: 04 September 2014

  • Accepted: 08 September 2014

  • Published: 06 October 2014

  • DOI: https://doi.org/10.1007/JHEP10(2014)030

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Supersymmetric gauge theory
  • Scattering Amplitudes
Download PDF

Working on a manuscript?

Avoid the most common mistakes and prepare your manuscript for journal editors.

Learn more

Advertisement

Over 10 million scientific documents at your fingertips

Switch Edition
  • Academic Edition
  • Corporate Edition
  • Home
  • Impressum
  • Legal information
  • Privacy statement
  • California Privacy Statement
  • How we use cookies
  • Manage cookies/Do not sell my data
  • Accessibility
  • FAQ
  • Contact us
  • Affiliate program

Not affiliated

Springer Nature

© 2023 Springer Nature Switzerland AG. Part of Springer Nature.