Download PDF

\( \mathcal{N} \) = 7 On-shell diagrams and supergravity amplitudes in momentum twistor space

Journal of High Energy Physics volume 2021, Article number: 181 (2021) Cite this article

Abstract

We derive an on-shell diagram recursion for tree-level scattering amplitudes in \( \mathcal{N} \) = 7 supergravity. The diagrams are evaluated in terms of Grassmannian integrals and momentum twistors, generalising previous results of Hodges in momentum twistor space to non-MHV amplitudes. In particular, we recast five and six-point NMHV amplitudes in terms of \( \mathcal{N} \) = 7 R-invariants analogous to those of \( \mathcal{N} \) = 4 super-Yang-Mills, which makes cancellation of spurious poles more transparent. Above 5-points, this requires defining momentum twistors with respect to different orderings of the external momenta.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  2. [2]

    N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the Amplituhedron in Binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  3. [3]

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  4. [4]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (2016), https://doi.org/10.1017/CBO9781316091548 [arXiv:1212.5605] [INSPIRE].

  6. [6]

    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].

    ADS  MathSciNet  Article  Google Scholar 

  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, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  8. [8]

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  9. [9]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  10. [10]

    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].

    ADS  Article  Google Scholar 

  11. [11]

    P. Benincasa, On-shell diagrammatics and the perturbative structure of planar gauge theories, arXiv:1510.03642 [INSPIRE].

  12. [12]

    P. Benincasa and D. Gordo, On-shell diagrams and the geometry of planar \( \mathcal{N} \) < 4 SYM theories, JHEP 11 (2017) 192 [arXiv:1609.01923] [INSPIRE].

  13. [13]

    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A. Postnikov and J. Trnka, On-Shell Structures of MHV Amplitudes Beyond the Planar Limit, JHEP 06 (2015) 179 [arXiv:1412.8475] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  14. [14]

    S. Franco, D. Galloni, B. Penante and C. Wen, Non-Planar On-Shell Diagrams, JHEP 06 (2015) 199 [arXiv:1502.02034] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  15. [15]

    J.L. Bourjaily, S. Franco, D. Galloni and C. Wen, Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams, JHEP 10 (2016) 003 [arXiv:1607.01781] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  16. [16]

    L. Ferro and T. Lukowski, Amplituhedra, and Beyond, J. Phys. A 54 (2021) 033001 [arXiv:2007.04342] [INSPIRE].

  17. [17]

    J. Bedford, A. Brandhuber, B.J. Spence and G. Travaglini, A Recursion relation for gravity amplitudes, Nucl. Phys. B 721 (2005) 98 [hep-th/0502146] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  18. [18]

    F. Cachazo and P. Svrček, Tree level recursion relations in general relativity, hep-th/0502160 [INSPIRE].

  19. [19]

    P. Heslop and A.E. Lipstein, On-shell diagrams for \( \mathcal{N} \) = 8 supergravity amplitudes, JHEP 06 (2016) 069 [arXiv:1604.03046] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  20. [20]

    E. Herrmann and J. Trnka, Gravity On-shell Diagrams, JHEP 11 (2016) 136 [arXiv:1604.03479] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  21. [21]

    A. Hodges, New expressions for gravitational scattering amplitudes, JHEP 07 (2013) 075 [arXiv:1108.2227] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  22. [22]

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

  23. [23]

    F.A. Berends, W.T. Giele and H. Kuijf, On relations between multi - gluon and multigraviton scattering, Phys. Lett. B 211 (1988) 91 [INSPIRE].

    ADS  Article  Google Scholar 

  24. [24]

    D. Skinner, Twistor strings for \( \mathcal{N} \) = 8 supergravity, JHEP 04 (2020) 047 [arXiv:1301.0868] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  25. [25]

    Y. Geyer, A.E. Lipstein and L.J. Mason, Ambitwistor Strings in Four Dimensions, Phys. Rev. Lett. 113 (2014) 081602 [arXiv:1404.6219] [INSPIRE].

  26. [26]

    J.A. Farrow and A.E. Lipstein, From 4d Ambitwistor Strings to On Shell Diagrams and Back, JHEP 07 (2017) 114 [arXiv:1705.07087] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  27. [27]

    N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  28. [28]

    J.-Y. Liu and E. Shih, Bonus scaling and BCFW in \( \mathcal{N} \) = 7 supergravity, Phys. Lett. B 740 (2015) 151 [arXiv:1409.1710] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  29. [29]

    J.A. Farrow, Y. Geyer, A.E. Lipstein, R. Monteiro and R. Stark-Muchão, Propagators, BCFW recursion and new scattering equations at one loop, JHEP 10 (2020) 074 [arXiv:2007.00623] [INSPIRE].

    ADS  Article  Google Scholar 

  30. [30]

    S. He, D. Nandan and C. Wen, Note on Bonus Relations for N = 8 Supergravity Tree Amplitudes, JHEP 02 (2011) 005 [arXiv:1011.4287] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  31. [31]

    S. Lal and S. Raju, The Next-to-Simplest Quantum Field Theories, Phys. Rev. D 81 (2010) 105002 [arXiv:0910.0930] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  32. [32]

    H. Elvang, Y.-t. Huang and C. Peng, On-shell superamplitudes in N < 4 SYM, JHEP 09 (2011) 031 [arXiv:1102.4843] [INSPIRE].

  33. [33]

    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].

    ADS  MathSciNet  Article  Google Scholar 

  34. [34]

    A. Brandhuber, P. Heslop and G. Travaglini, A Note on dual superconformal symmetry of the N = 4 super Yang-Mills S-matrix, Phys. Rev. D 78 (2008) 125005 [arXiv:0807.4097] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  35. [35]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  36. [36]

    T. Bargheer, F. Loebbert and C. Meneghelli, Symmetries of Tree-level Scattering Amplitudes in N = 6 Superconformal Chern-Simons Theory, Phys. Rev. D 82 (2010) 045016 [arXiv:1003.6120] [INSPIRE].

  37. [37]

    Y.-t. Huang and A.E. Lipstein, Dual Superconformal Symmetry of N = 6 Chern-Simons Theory, JHEP 11 (2010) 076 [arXiv:1008.0041] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  38. [38]

    D. Gang, Y.-t. Huang, E. Koh, S. Lee and A.E. Lipstein, Tree-level Recursion Relation and Dual Superconformal Symmetry of the ABJM Theory, JHEP 03 (2011) 116 [arXiv:1012.5032] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  39. [39]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  40. [40]

    L.J. Mason and D. Skinner, Dual Superconformal Invariance, Momentum Twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  41. [41]

    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  42. [42]

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  43. [43]

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

    ADS  MATH  Article  Google Scholar 

  44. [44]

    J.M. Drummond, M. Spradlin, A. Volovich and C. Wen, Tree-Level Amplitudes in N = 8 Supergravity, Phys. Rev. D 79 (2009) 105018 [arXiv:0901.2363] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  45. [45]

    P. Benincasa and M. Parisi, Positive geometries and differential forms with non-logarithmic singularities. Part I, JHEP 08 (2020) 023 [arXiv:2005.03612] [INSPIRE].

  46. [46]

    J. Trnka, private communication.

  47. [47]

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

    ADS  MathSciNet  Article  Google Scholar 

  48. [48]

    A. Brandhuber, P. Heslop and G. Travaglini, MHV amplitudes in N = 4 super Yang-Mills and Wilson loops, Nucl. Phys. B 794 (2008) 231 [arXiv:0707.1153] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  49. [49]

    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  50. [50]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  51. [51]

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  52. [52]

    A. Brandhuber, P. Heslop, A. Nasti, B. Spence and G. Travaglini, Four-point Amplitudes in N = 8 Supergravity and Wilson Loops, Nucl. Phys. B 807 (2009) 290 [arXiv:0805.2763] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  53. [53]

    E. Herrmann, C. Langer, J. Trnka and M. Zheng, Positive geometry, local triangulations, and the dual of the Amplituhedron, JHEP 01 (2021) 035 [arXiv:2009.05607] [INSPIRE].

    ADS  Article  Google Scholar 

  54. [54]

    J.L. Bourjaily, E. Herrmann, C. Langer, A.J. McLeod and J. Trnka, Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops, JHEP 12 (2019) 073 [arXiv:1909.09131] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  55. [55]

    N. Berkovits and E. Witten, Conformal supergravity in twistor-string theory, JHEP 08 (2004) 009 [hep-th/0406051] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  56. [56]

    T. Adamo and L. Mason, Einstein supergravity amplitudes from twistor-string theory, Class. Quant. Grav. 29 (2012) 145010 [arXiv:1203.1026] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  57. [57]

    L. Dolan and J.N. Ihry, Conformal Supergravity Tree Amplitudes from Open Twistor String Theory, Nucl. Phys. B 819 (2009) 375 [arXiv:0811.1341] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  58. [58]

    T. Adamo and L. Mason, Conformal and Einstein gravity from twistor actions, Class. Quant. Grav. 31 (2014) 045014 [arXiv:1307.5043] [INSPIRE].

  59. [59]

    J.A. Farrow and A.E. Lipstein, New Worldsheet Formulae for Conformal Supergravity Amplitudes, JHEP 07 (2018) 074 [arXiv:1805.04504] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  60. [60]

    Z. Bern, D. Kosower and J. Parra-Martinez, Two-loop n-point anomalous amplitudes in N = 4 supergravity, Proc. Roy. Soc. Lond. A 476 (2020) 20190722 [arXiv:1905.05151] [INSPIRE].

    MathSciNet  Google Scholar 

  61. [61]

    H. Johansson and J. Nohle, Conformal Gravity from Gauge Theory, arXiv:1707.02965 [INSPIRE].

  62. [62]

    H. Johansson, G. Mogull and F. Teng, Unraveling conformal gravity amplitudes, JHEP 09 (2018) 080 [arXiv:1806.05124] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  63. [63]

    J.A. Farrow, A Monte Carlo Approach to the 4D Scattering Equations, JHEP 08 (2018) 085 [arXiv:1806.02732] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  64. [64]

    J.L. Bourjaily, Positroids, Plabic Graphs, and Scattering Amplitudes in Mathematica, arXiv:1212.6974 [INSPIRE].

Download references

Author information

Affiliations

  1. Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, United Kingdom

    Connor Armstrong, Joseph A. Farrow & Arthur E. Lipstein

Authors
  1. Connor Armstrong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joseph A. Farrow
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Arthur E. Lipstein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arthur E. Lipstein.

Additional information

ArXiv ePrint: 2010.11813

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

ESM 1

(TGZ 37 kb)

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Armstrong, C., Farrow, J.A. & Lipstein, A.E. \( \mathcal{N} \) = 7 On-shell diagrams and supergravity amplitudes in momentum twistor space. J. High Energ. Phys. 2021, 181 (2021). https://doi.org/10.1007/JHEP01(2021)181

Download citation

Keywords

  • Scattering Amplitudes
  • Supergravity Models