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The tree formula for MHV graviton amplitudes

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Abstract

We present and prove a formula for the MHV scattering amplitude of n gravitons at tree level. Some of the more interesting features of t he formula, which set it apart as being significantly different from many more familiar formulas, include the absence of any vestigial reference to a cyclic ordering of the gravitons — making it in a sense a truly gravitational formula, rather than a recycled Yang-Mills result, and the fact that it simultaneously manifests both S n − 2 symmetry as well as large-z behavior that is \( \mathcal{O}\left( {{{1} \left/ {{{z^2}}} \right.}} \right) \) term-by-term, without relying on delicate cancellations. The formula is seemingly related to others by an enormous simplification provided by \( \mathcal{O}\left( {{n^n}} \right) \) iterated Schouten identities, but our proof relies on a complex analysis argument rather than such a brute force manipulation. We find that the formula has a very simple link representation in twistor space, where cancellations that are non-obvious in physical space become manifest.

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References

  1. S.J. Parke and T.R. Taylor, An Amplitude for n Gluon Scattering, Phys. Rev. Lett. 56 (1986) 2459 [SPIRES].

    Article  ADS  Google Scholar 

  2. F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [SPIRES].

    Article  ADS  Google Scholar 

  3. H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  4. Z. Bern, Perturbative quantum gravity and its relation to gauge theory, Living Rev. Rel. 5 (2002) 5 [gr-qc/0206071] [SPIRES].

    MathSciNet  Google Scholar 

  5. M.B. Green, J.G. Russo and P. Vanhove, Non-renormalisation conditions in type-II string theory and maximal supergravity, JHEP 02 (2007) 099 [hep-th/0610299] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  6. R. Kallosh, The Effective Action of N = 8 Supergravity, arXiv:0711.2108 [SPIRES].

  7. R. Kallosh and M. Soroush, Explicit Action of E7(7) on N = 8 Supergravity Fields, Nucl. Phys. B 801 (2008) 25 [arXiv:0802.4106] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  8. S.G. Naculich, H. Nastase and H.J. Schnitzer, Two-loop graviton scattering relation and IR behavior in N = 8 supergravity, Nucl. Phys. B 805 (2008) 40 [arXiv:0805.2347] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  9. N.E.J. Bjerrum-Bohr and P. Vanhove, Absence of Triangles in Maximal Supergravity Amplitudes, JHEP 10 (2008) 006 [arXiv:0805.3682] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  10. N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, arXiv:0808.1446 [SPIRES].

  11. S. Badger, N.E.J. Bjerrum-Bohr and P. Vanhove, Simplicity in the Structure of QED and Gravity Amplitudes, JHEP 02 (2009) 038 [arXiv:0811.3405] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  12. R. Kallosh and T. Kugo, The footprint of E7 in amplitudes of N = 8 supergravity, JHEP 01 (2009) 072 [arXiv:0811.3414] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  13. R. Kallosh, C.H. Lee and T. Rube, N= 8 Supergravity 4-point Amplitudes, JHEP 02 (2009) 050 [arXiv:0811.3417] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  14. R. Kallosh, N = 8 Supergravity on the Light Cone, Phys. Rev. D 80 (2009) 105022 [arXiv:0903.4630] [SPIRES].

    ADS  Google Scholar 

  15. Z. Bern, L.J. Dixon and R. Roiban, Is N = 8 Supergravity Ultraviolet Finite?, Phys. Lett. B 644 (2007) 265 [hep-th/0611086] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  16. M.B. Green, J.G. Russo and P. Vanhove, Ultraviolet properties of maximal supergravity, Phys. Rev. Lett. 98 (2007) 131602 [hep-th/0611273] [SPIRES].

    Article  ADS  Google Scholar 

  17. M.B. Green, H. Ooguri and J.H. Schwarz, Decoupling Supergravity from the Superstring, Phys. Rev. Lett. 99 (2007) 041601 [arXiv:0704.0777] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  18. R. Kallosh, On a possibility of a UV finite N = 8 supergravity, arXiv:0808.2310 [SPIRES].

  19. G. Bossard, P.S. Howe and K.S. Stelle, The ultra-violet question in maximally supersymmetric field theories, Gen. Rel. Grav. 41 (2009) 919 [arXiv:0901.4661] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. R. Kallosh, On UV Finiteness of the Four Loop N = 8 Supergravity, JHEP 09 (2009) 116 [arXiv:0906.3495] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

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

    ADS  Google Scholar 

  22. L. Mason and D. Skinner, Gravity, Twistors and the MHV Formalism, Commun. Math. Phys. 294 (2010) 827 [arXiv:0808.3907] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  23. 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] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  24. H. Elvang and D.Z. Freedman, Note on graviton MHV amplitudes, JHEP 05 (2008) 096 [arXiv:0710.1270] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

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

  26. N.E.J. Bjerrum-Bohr, D.C. Dunbar, H. Ita, W.B. Perkins and K. Risager, MHV-vertices for gravity amplitudes, JHEP 01 (2006) 009 [hep-th/0509016] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  27. P. Benincasa, C. Boucher-Veronneau and F. Cachazo, Taming tree amplitudes in general relativity, JHEP 11 (2007) 057 [hep-th/0702032] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  28. Z. Bern, J.J. Carrasco, D. Forde, H. Ita and H. Johansson, Unexpected Cancellations in Gravity Theories, Phys. Rev. D 77 (2008) 025010 [arXiv:0707.1035] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  29. N. Arkani-Hamed and J. Kaplan, On Tree Amplitudes in Gauge Theory and Gravity, JHEP 04 (2008) 076 [arXiv:0801.2385] [SPIRES].

    Article  MathSciNet  Google Scholar 

  30. M. Bianchi, H. Elvang and D.Z. Freedman, Generating Tree Amplitudes in N = 4 SYM and N = 8 SG, JHEP 09 (2008) 063 [arXiv:0805.0757] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  31. 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] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  32. Z. Bern et al., Three-Loop Superfiniteness of N = 8 Supergravity, Phys. Rev. Lett. 98 (2007) 161303 [hep-th/0702112] [SPIRES].

    Article  ADS  Google Scholar 

  33. 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] [SPIRES].

    ADS  Google Scholar 

  34. Z. Bern, J.J.M. Carrasco and H. Johansson, Progress on Ultraviolet Finiteness of Supergravity, arXiv:0902.3765 [SPIRES].

  35. Z. Bern, J.J. Carrasco, L.J. Dixon, H. Johansson and R. R oiban, The Ultraviolet Behavior of N = 8 Supergravity at Four Loops, arXiv:0905.2326 [SPIRES].

  36. M. Spradlin, A. Volovich and C. Wen, Three Applications of a Bonus Relation for Gravity Amplitudes, Phys. Lett. B 674 (2009) 69 [arXiv:0812.4767] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  37. S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  38. N. Arkani-Hamed, Holography and the S-Matrix, talk at Amplitudes 09, http://www.ippp.dur.ac.uk/Workshops/09/Amplitudes/.

  39. L. Mason and D. Skinner, Scattering Amplitudes and BCFW Recursion in Twistor Space, arXiv:0903.2083 [SPIRES].

  40. N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, The S-matrix in Twistor Space, JHEP 03 (2010) 110 [arXiv:0903.2110] [SPIRES].

    Article  Google Scholar 

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

  42. A.P. Hodges, Twistor diagrams for all tree amplitudes in gauge theory: A helicity-independent formalism, hep-th/0512336 [SPIRES].

  43. A.P. Hodges, Scattering amplitudes for eight gauge fields, hep-th/0603101 [SPIRES].

  44. R. Britto, F. Cachazo and B. Feng, New Recursion Relations for Tree Amplitudes of Gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  45. A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, arXiv:0905.1473 [SPIRES].

  46. N. Arkani-Hamed, Holography and the S-Matrix, talk at Strings 09, http://strings2009.roma2.infn.it/ .

  47. N. Arkani-Hamed, Holography in Flat Space: Algebraic Geometry and the S-Matrix, talk at Integrability in Gauge and String Theory, http://int09.aei.mpg.de/ .

  48. R. Roiban, M. Spradlin and A. Volovich, A googly amplitude from the B-model in twistor space, JHEP 04 (2004) 012 [hep-th/0402016] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  49. R. Roiban and A. Volovich, All googly amplitudes from the B-model in twistor space, Phys. Rev. Lett. 93 (2004) 131602 [hep-th/0402121] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  50. R. Roiban, M. Spradlin and A. Volovich, On the tree-level S-matrix of Yang-Mills theory, Phys. Rev. D 70 (2004) 026009 [hep-th/0403190] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  51. R. Roiban, M. Spradlin and A. Volovich, Yang-Mills amplitudes from twistor string theory, prepared for AMS-IMS-SIAM Summer Research Conference on String Geometr y, Snowbird Utah U.S.A., 5-11 Jun 2004 [ SPIRES].

  52. M. Spradlin, Yang-Mills amplitudes from string theory in twistor space, Int. J. Mod. Phys. A 20 (2005) 3416 [SPIRES].

    MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  54. S. Giombi, R. Ricci, D. Robles-Llana and D. Trancanelli, A note on twistor gravity amplitudes, JHEP 07 (2004) 059 [hep-th/0405086] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  55. N.E.J. Bjerrum-Bohr, D.C. Dunbar and H. Ita, Perturbative gravity and twistor space, Nucl. Phys. Proc. Suppl. 160 (2006) 215 [hep-th/0606268] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  56. M. Abou-Zeid, C.M. Hull and L.J. Mason, Einstein supergravity and new twistor string theories, Commun. Math. Phys. 282 (2008) 519 [hep-th/0606272] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  57. V.P. Nair, A Note on Graviton Amplitudes for New Twistor String Theories, Phys. Rev. D 78 (2008) 041501 [arXiv:0710.4961] [SPIRES].

    ADS  Google Scholar 

  58. V.P. Nair, A Current Algebra For Some Gauge Theory Amplitudes, Phys. Lett. B 214 (1988) 215 [SPIRES].

    ADS  Google Scholar 

  59. V.P. Nair, A note on MHV amplitudes for gravitons, Phys. Rev. D 71 (2005) 121701 [hep-th/0501143] [SPIRES].

    ADS  Google Scholar 

  60. J.M. Drummond and J.M. Henn, All tree-level amplitudes in N = 4 SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  61. 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] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  62. A. Hall, Supersymmetric Yang-Mills and Supergravity Amplitudes at One Loop, Phys. Rev. D 80 (2009) 045004 [arXiv:0906.0204] [SPIRES].

    ADS  Google Scholar 

  63. P. Katsaroumpas, B. Spence and G. Travaglini, One-loop N = 8 supergravity coefficients from N = 4 super Yang-Mills, JHEP 08 (2009) 096 [arXiv:0906.0521] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  64. H. Elvang, D.Z. Freedman and M. Kiermaier, Recursion Relations, Generating Functions and Unitarity Sums in N = 4 SYM Theory, JHEP 04 (2009) 009 [arXiv:0808.1720] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  65. Z. Bern, J.J.M. Carrasco, H. Ita, H. Johansson and R. Roiban, On the Structure of Supersymmetric Sums in Multi-Loop Unitarity Cuts, Phys. Rev. D 80 (2009) 065029 [arXiv:0903.5348] [SPIRES].

    ADS  Google Scholar 

  66. Z. Bern, L.J. Dixon, M. Perelstein and J.S. Rozowsky, Multi-leg one-loop gravity amplitudes from gauge theory, Nucl. Phys. B 546 (1999) 423 [hep-th/9811140] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Brown University, Providence, Rhode Island, 02912, U.S.A.

    Dung Nguyen, Marcus Spradlin, Anastasia Volovich & Congkao Wen

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  1. Dung Nguyen
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  2. Marcus Spradlin
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  3. Anastasia Volovich
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Correspondence to Dung Nguyen or Anastasia Volovich.

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ArXiv ePrint: 0907.2276

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Nguyen, D., Spradlin, M., Volovich, A. et al. The tree formula for MHV graviton amplitudes. J. High Energ. Phys. 2010, 45 (2010). https://doi.org/10.1007/JHEP07(2010)045

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