Skip to main content
SpringerLink
Log in

Gravity, Twistors and the MHV Formalism

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We give a self-contained proof of the formula for the MHV amplitudes for gravity conjectured by Berends, Giele & Kuijf and use the associated twistor generating function to define a twistor action for the MHV diagram approach to gravity. Starting from a background field calculation on a spacetime with anti-self-dual curvature, we obtain a simple spacetime formula for the scattering of a single, positive helicity linearized graviton into one of negative helicity. Re-expressing our integral in terms of twistor data allows us to consider a spacetime that is asymptotic to a superposition of plane waves. Expanding these out perturbatively yields the gravitational MHV amplitudes of Berends, Giele & Kuijf. We go on to take the twistor generating function off-shell at the perturbative level. Combining this with a twistor action for the anti-self-dual background, the generating function provides the MHV vertices for the MHV diagram approach to perturbative gravity. We finish by extending these results to supergravity, in particular \({\mathcal {N} = 4}\) and \({\mathcal {N} = 8}\) .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bern Z., Dixon L.J., Perelstein M., Rozowsky J.S.: Multi-leg one-loop gravity amplitudes from Gauge theory. Nucl. Phys. B 546, 423 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Bern Z., Bjerrum-Bohr N.E.J., Dunbar D.C.: Inherited twistor-space structure of gravity loop amplitudes. JHEP 0505, 056 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  3. Bern Z., Carrasco J.J., Dixon L.J., Johansson H., Kosower D.A., Roiban R.: Three-loop superfiniteness of N = 8 supergravity. Phys. Rev. Lett. 98, 161303 (2007)

    Article  ADS  Google Scholar 

  4. Bern Z., Carrasco J.J., Forde D., Ita H., Johansson H.: Unexpected cancellations in gravity theories. Phys. Rev. D 77, 025010 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  5. Bedford J., Brandhuber A., Spence B., Travaglini G.: A recursion relation for gravity amplitudes. Nucl. Phys. B 721, 98 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Cachazo, F., Svrcek, P.: Tree level recursion relations in general relativity. http://arXiv.org/abs/hep-th/0502160v3, 2005

  7. Benincasa P., Boucher-Veronneau C., Cachazo F.: Taming tree amplitudes in general relativity. JHEP 0711, 057 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  8. Cachazo, F., Skinner, D.: On the structure of scattering amplitudes in N = 4 super Yang-Mills and N = 8 supergravity. http://arXiv.org/abs/0801.4574v2[hep-th], 2008

  9. Arkani-Hamed, N., Cachazo, F., Kaplan, J.: What is the simplest quantum field theory?. http://arXiv.org/abs/0808.1446v2[hep-th], 2008

  10. Bjerrum-Bohr N.E.J., Vanhove P.: Explicit cancellation of triangles in one-loop gravity amplitudes. JHEP 0804, 065 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  11. Bjerrum-Bohr, N.E.J., Vanhove, P.: Absence of triangles in maximal supergravity amplitudes. http://arXiv.org/abs/0805.3682v2[hep-th], 2008

  12. Bjerrum-Bohr N.E.J., Dunbar D.C., Ita H., Perkins W.B., Risager K.: The no-triangle hypothesis for N = 8 supergravity. JHEP 0612, 072 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  13. Bjerrum-Bohr N.E.J., Dunbar D.C., Ita H., Perkins W.B., Risager K.: MHV-vertices for gravity amplitudes. JHEP 0601, 009 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  14. Nasti A., Travaglini G.: One-loop N = 8 supergravity amplitudes from MHV diagrams. Class. Quant. Grav. 24, 6071 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Bianchi M., Elvang H., Freedman D.Z.: Generating tree amplitudes in N = 4 SYM and N = 8 SG. JHEP 0809, 063 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  16. Elvang H., Freedman D.Z.: Note on graviton MHV amplitudes. JHEP 0809, 063 (2008)

    MathSciNet  Google Scholar 

  17. Parke S.J., Taylor T.R.: An amplitude for n gluon scattering. Phys. Rev. Lett. 56, 2459 (1986)

    Article  ADS  Google Scholar 

  18. Berends F.A., Giele W.T.: Recursive calculations for processes with n gluons. Nucl. Phys. B 306, 759 (1988)

    Article  ADS  Google Scholar 

  19. Berends F.A., Giele W.T., Kuijf H.: On relations between Multi-Gluon and Multi-Graviton scattering. Phys. Lett. B 211, 91 (1988)

    Article  ADS  Google Scholar 

  20. Bialynicki-Birula I., Newman E.T., Porter J., Winicour J., Lukacs B., Perjes Z., Sebestyen A.: A note on helicity. J. Math. Phys. 22, 2530 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  21. Ashtekar A.: A note on helicity and selfduality. J. Math. Phys. 27, 824 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  22. Penrose R.: Non-linear gravitons and curved twistor theory. Gen. Rel. Grav. 7, 31 (1976)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Hansen R.O., Newman E.T., Penrose R., Tod K.P.: The metric and curvature properties of H space. Proc. Roy. Soc. Lond. A 363, 445 (1978)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Plebanski J.F.: On the separation of Einsteinian substructures. J. Math. Phys. 18, 2511 (1977)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. Capovilla R., Jacobson T., Dell J., Mason L.: Selfdual two forms and gravity. Class. Quant. Grav. 8, 41 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Mason, L., Frauendiener, J.: The Sparling 3-Form, Ashtekar Variables and Quasi-Local Mass. Lond. Math. Soc. Lect. Notes 156, Cambridge: Cambridge University Press, 1990, p. 189

  27. Ashtekar A.: New variables for classical and quantum gravity. Phys. Rev. Lett. 57, 2244 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  28. Penrose R., MacCallum M.A.H.: Twistor theory: An approach to the quantization of fields and spacetime. Phys. Rept. 6, 241 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  29. Ashtekar A., Jacobson T., Smolin L.: A new characterization of half flat solutions to einstein’s equation. Commun. Math. Phys. 115, 631 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. Abou-Zeid M., Hull C.M.: A chiral perturbation expansion for gravity. JHEP 0602, 057 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  31. Rosly, A.A., Selivanov, K.G.: Gravitational SD perturbiner. http://arXiv.org/abs/hep-th/9710196v1, 1997

  32. Newman E.T.: Heaven and its Properties. Gen. Rel. Grav. 7, 107 (1976)

    Article  ADS  Google Scholar 

  33. Mason L.J.: Twistor actions for non-self-dual fields: a derivation of twistor-string theory. JHEP 0510, 009 (2005)

    Article  ADS  Google Scholar 

  34. Boels R., Mason L., Skinner D.: Supersymmetric Gauge theories in twistor space. JHEP 0702, 014 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  35. Berkovits N., Witten E.: Conformal supergravity in twistor-string theory. JHEP 0408, 009 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  36. Mason L., Skinner D.: Heterotic twistor-string theory. Nucl. Phys. B 795, 105 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  37. Abou-Zeid M., Hull C., Mason L.: Einstein supergravity and new twistor string theories. Commun. Math. Phys. 282, 519–573 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  38. Nair V.P.: A note on graviton amplitudes for new twistor string theories. Phys. Rev. D 78, 041501 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  39. Mason L.J., Wolf M.: A twistor action for N = 8 self-dual supergravity. Commun. Math. Phys. 288, 97–123 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  40. Witten E.: Perturbative Gauge theory as a string theory in twistor space. Commun. Math. Phys. 252, 189 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  41. Boels R., Mason L., Skinner D.: From twistor actions to MHV diagrams. Phys. Lett. B 648, 90 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  42. Kawai H., Lewellen D.C., Tye S.H.H.: A relation between tree amplitudes of closed and open strings. Nucl. Phys. B 269, 1 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  43. Britto R., Cachazo F., Feng B., Witten E.: Direct proof of tree-level recursion relation in Yang-Mills theory. Phys. Rev. Lett. 94, 181602 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  44. Ananth S., Theisen S.: KLT relations from the Einstein-Hilbert Lagrangian. Phys. Lett. B 652, 128 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  45. Mansfield P.: The Lagrangian origin of MHV rules. JHEP 0603, 037 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  46. Ettle J.H., Morris T.R.: Structure of the MHV-rules Lagrangian. JHEP 0608, 003 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  47. Nair V.P.: A Note on MHV amplitudes for gravitons. Phys. Rev. D 71, 121701 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  48. Rosly A.A., Selivanov K.G.: On amplitudes in self-dual sector of Yang-Mills theory. Phys. Lett. B 399, 135 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  49. Penrose, R., Rindler, W.: Spinors and Spacetime 1 & 2. Cambridge Monographs on Math. Phys., Cambridge: CUP, 1984 & 1986

  50. Risager K.: A direct proof of the CSW rules. JHEP 0512, 003 (2005)

    Article  ADS  Google Scholar 

  51. Elvang H., Freedman D.Z., Kiermaier M.: Recursion relations, generating functions, and unitarity sums in N = 4 SYM theory. JHEP 0904, 009 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  52. Woodhouse N.M.J.: Geometric Quantization Second edition. Oxford Mathematical Monographs. OUP, Oxford (1992)

    Google Scholar 

  53. Ashtekar A., Engle J., Sloan D.: Asymptotics and hamiltonians in a first order formalism. Class. Quant. Grav 25, 095020 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  54. Wardm, R.S., Wells, R.O.: Twistor Geometry and Field Theory. Cambridge Monographs on Math. Phys. CUP, Campridge, 1990

  55. Huggett, S.A., Tod, K.P.: An Introduction To Twistor Theory. London Mathematical Society Student Texts 4, Campridge: CUP, 1985

  56. Eastwood M., Tod P.: Edth - a differential operator on the sphere. Math. Proc. Camb. Phil. Soc. 92, 317 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  57. Porter J.R.: The nonlinear graviton: superposition of plane waves. Gen. Rel. Grav. 14, 1023 (1982)

    Article  MATH  ADS  Google Scholar 

  58. Cachazo F., Svrcek P., Witten E.: MHV vertices and tree amplitudes in gauge theory. JHEP 0409, 006 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  59. Bena I., Bern Z., Kosower D.A.: Twistor-space recursive formulation of Gauge theory amplitudes. Phys. Rev. D 71, 045008 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  60. Bern Z., Dixon L.J., Dunbar D.C., Perelstein M., Rozowsky J.S.: On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences. Nucl. Phys. B 530, 401 (1998)

    Article  ADS  Google Scholar 

  61. Brandhuber A., Spence B., Travaglini G.: From trees to loops and back. JHEP 0601, 142 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  62. Boels R.: A quantization of twistor Yang-Mills theory through the background field method. Phys. Rev. D 76, 105027 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  63. Gorsky A., Rosly A.: From Yang-Mills Lagrangian to MHV diagrams. JHEP 0601, 101 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  64. Ferber A.: Supertwistors and conformal supersymmetry. Nucl. Phys. B 132, 55 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  65. Nair V.P.: A Current algebra for some Gauge theory amplitudes. Phys. Lett. B 214, 215 (1998)

    ADS  Google Scholar 

  66. Wolf M.: Self-dual supergravity and twistor theory. Class. Quant. Grav. 24, 6287 (2007)

    Article  MATH  ADS  Google Scholar 

  67. Chalmers G., Siegel W.: The self-dual Sector of QCD Amplitudes. Phys. Rev. D 54, 7628 (1996)

    Article  ADS  Google Scholar 

  68. Ward R.S.: On self-dual Gauge fields. Phys. Lett. A 61, 81 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  69. Sparling, G.: Dynamically broken symmetry and global yang-Mills in Minkowski space. Sect. 1.4.2 In: Further Advances in Twistor Theory, Mason, L., Hughston L. (eds), Pitman Research Notes in Maths 231, Essex: Longman, Harlow, 1995

Download references

Author information

Authors and Affiliations

  1. The Mathematical Institute, 24-29 St. Giles’, Oxford, OX1 3LP, United Kingdom

    Lionel Mason & David Skinner

Authors
  1. Lionel Mason
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Skinner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Skinner.

Additional information

Communicated by G.W. Gibbons

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mason, L., Skinner, D. Gravity, Twistors and the MHV Formalism. Commun. Math. Phys. 294, 827–862 (2010). https://doi.org/10.1007/s00220-009-0972-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-009-0972-4

Keywords

Search

Navigation