Skip to main content
SpringerLink
Log in

Augmented recursion for one-loop gravity amplitudes

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We present a semi-recursive method for calculating the rational parts of one-loop gravity amplitudes which utilises axial gauge diagrams to determine the nonfactorising pieces of the amplitude. This method is used to compute the amplitudes M 1-loop(1−, 2+, 3+, 4+, 5+) and M 1-loop(1−, 2+, 3+, 4+, 5+, 6+).

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. 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 

  2. 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 

  3. 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 

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

  5. K. Risager, A direct proof of the CSW rules, JHEP 12 (2005) 003 [hep-th/0508206] [SPIRES].

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  7. 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 

  8. 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] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  9. 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] [SPIRES].

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  11. D.C. Dunbar, W.B. Perkins and E. Warrick, The unitarity method using a canonical basis approach, JHEP 06 (2009) 056 [arXiv:0903.1751] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  12. Z. Bern, N.E.J. Bjerrum-Bohr, D.C. Dunbar and H. Ita, Recursive calculation of one-loop QCD integral coefficients, JHEP 11 (2005) 027 [hep-ph/0507019] [SPIRES].

    Article  ADS  Google Scholar 

  13. C.F. Berger, Z. Bern, L.J. Dixon, D. Forde and D.A. Kosower, Bootstrapping one-loop QCD amplitudes with general helicities, Phys. Rev. D 74 (2006) 036009 [hep-ph/0604195] [SPIRES].

    ADS  Google Scholar 

  14. C.F. Berger, Z. Bern, L.J. Dixon, D. Forde and D.A. Kosower, All one-loop maximally helicity violating gluonic amplitudes in QCD, Phys. Rev. D 75 (2007) 016006 [hep-ph/0607014] [SPIRES].

    ADS  Google Scholar 

  15. W.L. van Neerven, Dimensional regularization of mass and infrared singularities in two loop on-shell vertex functions, Nucl. Phys. B 268 (1986) 453 [SPIRES].

    Article  ADS  Google Scholar 

  16. Z. Bern and A.G. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467 (1996) 479 [hep-ph/9511336] [SPIRES].

    Article  ADS  Google Scholar 

  17. A. Brandhuber, S. McNamara, B.J. Spence and G. Travaglini, Loop amplitudes in pure Yang-Mills from generalised unitarity, JHEP 10 (2005) 011 [hep-th/0506068] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  18. 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] [SPIRES].

    ADS  Google Scholar 

  19. 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, JHEP 03 (2007) 111 [hep-ph/0612277] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  20. R. Britto and B. Feng, Unitarity cuts with massive propagators and algebraic expressions for coefficients, Phys. Rev. D 75 (2007) 105006 [hep-ph/0612089] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  21. W.T. Giele, Z. Kunszt and K. Melnikov, Full one-loop amplitudes from tree amplitudes, JHEP 04 (2008) 049 [arXiv:0801.2237] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  22. R.K. Ellis, W.T. Giele, Z. Kunszt and K. Melnikov, Masses, fermions and generalized D-dimensional unitarity, Nucl. Phys. B 822 (2009) 270 [arXiv:0806.3467] [SPIRES].

    Article  ADS  Google Scholar 

  23. C.F. Berger et al., An automated implementation of on-shell methods for one-loop amplitudes, Phys. Rev. D 78 (2008) 036003 [arXiv:0803.4180] [SPIRES].

    ADS  Google Scholar 

  24. 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 

  25. 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 

  26. Z. Xiao, G. Yang and C.-J. Zhu, The rational parts of one-loop QCD amplitudes III: the six-gluon case, Nucl. Phys. B 758 (2006) 53 [hep-ph/0607017] [SPIRES].

    Article  ADS  Google Scholar 

  27. T. Binoth, J.P. Guillet and G. Heinrich, Algebraic evaluation of rational polynomials in one-loop amplitudes, JHEP 02 (2007) 013 [hep-ph/0609054] [SPIRES].

    Article  ADS  Google Scholar 

  28. S.D. Badger, Direct extraction of one loop rational terms, JHEP 01 (2009) 049 [arXiv:0806.4600] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  29. G. Ossola, C.G. Papadopoulos and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys. B 763 (2007) 147 [hep-ph/0609007] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  30. G. Ossola, C.G. Papadopoulos and R. Pittau, CutTools: a program implementing the OPP reduction method to compute one-loop amplitudes, JHEP 03 (2008) 042 [arXiv:0711.3596] [SPIRES].

    Article  ADS  Google Scholar 

  31. Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One-loop self-dual and N = 4 super Yang-Mills, Phys. Lett. B 394 (1997) 105 [hep-th/9611127] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  32. C.F. Berger, Z. Bern, L.J. Dixon, D. Forde and D.A. Kosower, All one-loop maximally helicity violating gluonic amplitudes in QCD, Phys. Rev. D 75 (2007) 016006 [hep-ph/0607014] [SPIRES].

    ADS  Google Scholar 

  33. C.F. Berger, Z. Bern, L.J. Dixon, D. Forde and D.A. Kosower, Bootstrapping one-loop QCD amplitudes with general helicities, Phys. Rev. D 74 (2006) 036009 [hep-ph/0604195] [SPIRES].

    ADS  Google Scholar 

  34. Z. Bern, L.J. Dixon and D.A. Kosower, On-shell recurrence relations for one-loop QCD amplitudes, Phys. Rev. D 71 (2005) 105013 [hep-th/0501240] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  35. D. Vaman and Y.-P. Yao, The space-cone gauge, Lorentz invariance and on-shell recursion for one-loop Yang-Mills amplitudes, arXiv:0805.2645 [SPIRES].

  36. A. Brandhuber, S. McNamara, B. Spence and G. Travaglini, Recursion relations for one-loop gravity amplitudes, JHEP 03 (2007) 029 [hep-th/0701187] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  37. Z. Bern and D.A. Kosower, The computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  38. Z. Bern and D.C. Dunbar, A mapping between Feynman and string motivated one loop rules in gauge theories, Nucl. Phys. B 379 (1992) 562 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  39. Z. Bern, D.C. Dunbar and T. Shimada, String based methods in perturbative gravity, Phys. Lett. B 312 (1993) 277 [hep-th/9307001] [SPIRES].

    ADS  Google Scholar 

  40. D.C. Dunbar and P.S. Norridge, Calculation of graviton scattering amplitudes using string based methods, Nucl. Phys. B 433 (1995) 181 [hep-th/9408014] [SPIRES].

    Article  ADS  Google Scholar 

  41. Z. Bern and G. Chalmers, Factorization in one loop gauge theory, Nucl. Phys. B 447 (1995) 465 [hep-ph/9503236] [SPIRES].

    Article  ADS  Google Scholar 

  42. Z. Bern, L.J. Dixon and D.A. Kosower, One loop corrections to five gluon amplitudes, Phys. Rev. Lett. 70 (1993) 2677 [hep-ph/9302280] [SPIRES].

    Article  ADS  Google Scholar 

  43. C. Schwinn and S. Weinzierl, Scalar diagrammatic rules for Born amplitudes in QCD, JHEP 05 (2005) 006 [hep-th/0503015] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  44. 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 

  45. S. Alston, D.C. Dunbar, J.H. Ettle and W.B. Perkins, One-loop amplitudes using augmented recursion for gauge and gravity theories, work in progress.

  46. G. Passarino and M.J.G. Veltman, One loop corrections for e + e − annihilation into μ + μ − in the Weinberg model, Nucl. Phys. B 160 (1979) 151 [SPIRES].

    Article  ADS  Google Scholar 

  47. R.G. Stuart, Algebraic reduction of one loop Feynman diagrams to scalar integrals, Comput. Phys. Commun. 48 (1988) 367 [SPIRES].

    Article  ADS  Google Scholar 

  48. G.J. van Oldenborgh and J.A.M. Vermaseren, New algorithms for one loop integrals, Z. Phys. C 46 (1990) 425 [SPIRES].

    Google Scholar 

  49. M.T. Grisaru and J. Zak, One loop scalar field contributions to graviton-graviton scattering and helicity nonconservation in quantum gravity, Phys. Lett. B 90 (1980) 237 [SPIRES].

    ADS  Google Scholar 

  50. D.C. Dunbar and N.W.P. Turner, Gravity and form scattering and renormalisation of gravity in six and eight dimensions, Class. Quant. Grav. 20 (2003) 2293 [hep-th/0212160] [SPIRES].

    Article  MATH  MathSciNet  Google Scholar 

  51. D.C. Dunbar, B. Julia, D. Seminara and M. Trigiante, Counterterms in type-I supergravities, JHEP 01 (2000) 046 [hep-th/9911158] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  52. M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, Supergravity and the S matrix, Phys. Rev. D 15 (1977) 996 [SPIRES].

    ADS  Google Scholar 

  53. M.T. Grisaru and H.N. Pendleton, Some properties of scattering amplitudes in supersymmetric theories, Nucl. Phys. B 124 (1977) 81 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  54. 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 

  55. N.E.J. Bjerrum-Bohr, D.C. Dunbar and H. Ita, Six-point one-loop N = 8 supergravity NMHV amplitudes and their IR behaviour, Phys. Lett. B 621 (2005) 183 [hep-th/0503102] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  56. Z. Bern, N.E.J. Bjerrum-Bohr and D.C. Dunbar, Inherited twistor-space structure of gravity loop amplitudes, JHEP 05 (2005) 056 [hep-th/0501137] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  57. N.E.J. Bjerrum-Bohr, D.C. Dunbar, H. Ita, W.B. Perkins and K. Risager, The no-triangle hypothesis for N = 8 supergravity, JHEP 12 (2006) 072 [hep-th/0610043] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  58. Z. Bern, A compact representation of the one loop N gluon amplitude, Phys. Lett. B 296 (1992) 85 [SPIRES].

    ADS  Google Scholar 

  59. Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Swansea University, Swansea, SA2 8PP, U.K.

    David C. Dunbar, James H. Ettle & Warren B. Perkins

Authors
  1. David C. Dunbar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James H. Ettle
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Warren B. Perkins
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David C. Dunbar.

Additional information

ArXiv ePrint: 1003.3398

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dunbar, D.C., Ettle, J.H. & Perkins, W.B. Augmented recursion for one-loop gravity amplitudes. J. High Energ. Phys. 2010, 27 (2010). https://doi.org/10.1007/JHEP06(2010)027

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP06(2010)027

Keywords

Search

Navigation