Skip to main content
SpringerLink
Log in

Eikonal methods applied to gravitational scattering amplitudes

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

Abstract

We apply factorization and eikonal methods from gauge theories to scattering amplitudes in gravity. We hypothesize that these amplitudes factor into an IR-divergent soft function and an IR-finite hard function, with the former given by the expectation value of a product of gravitational Wilson line operators. Using this approach, we show that the IR-divergent part of the n-graviton scattering amplitude is given by the exponential of the one-loop IR divergence, as originally discovered by Weinberg, with no additional subleading IR-divergent contributions in dimensional regularization.

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. F. Bloch and A. Nordsieck, Note on the Radiation Field of the electron, Phys. Rev. 52 (1937) 54 [SPIRES].

    Article  ADS  MATH  Google Scholar 

  2. T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [SPIRES].

    Article  ADS  MATH  Google Scholar 

  3. T.D. Lee and M. Nauenberg, Degenerate systems and mass singularities, Phys. Rev. B 33 (1964) 1549 [SPIRES].

    Article  MathSciNet  Google Scholar 

  4. S. Weinberg, Infrared photons and gravitons, Phys. Rev. B 140 (1965) 516 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  5. J.F. Donoghue and T. Torma, Infrared behavior of graviton-graviton scattering, Phys. Rev. D 60 (1999) 024003 [hep-th/9901156] [SPIRES].

    ADS  Google Scholar 

  6. D.C. Dunbar and P.S. Norridge, Infinities within graviton scattering amplitudes, Class. Quant. Grav. 14 (1997) 351 [hep-th/9512084] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. D.R. Yennie, S.C. Frautschi and H. Suura, The infrared divergence phenomena and high-energy processes, Ann. Phys. 13 (1961) 379 [SPIRES].

    Article  ADS  Google Scholar 

  8. A.H. Mueller, On the asymptotic behavior of the Sudakov form-factor, Phys. Rev. D 20 (1979) 2037 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  9. J.C. Collins, Algorithm to compute corrections to the Sudakov form-factor, Phys. Rev. D 22 (1980) 1478 [SPIRES].

    ADS  Google Scholar 

  10. A. Sen, Asymptotic Behavior of the Sudakov Form-Factor in QCD, Phys. Rev. D 24 (1981) 3281 [SPIRES].

    ADS  Google Scholar 

  11. A. Sen, Asymptotic Behavior of the Wide Angle On-Shell Quark Scattering Amplitudes in Nonabelian Gauge Theories, Phys. Rev. D 28 (1983) 860 [SPIRES].

    ADS  Google Scholar 

  12. G.P. Korchemsky and A.V. Radyushkin, Loop space formalism and renormalization group for the infrared asymptotics of QCD, Phys. Lett. B 171 (1986) 459 [SPIRES].

    ADS  Google Scholar 

  13. J.C. Collins, D.E. Soper and G.F. Sterman, Factorization of hard processes in QCD, Adv. Ser. Direct. High Energy Phys. 5 (1988) 1 [hep-ph/0409313] [SPIRES].

    Google Scholar 

  14. G.P. Korchemsky, Double logarithmic asymptotics in QCD, Phys. Lett. B 217 (1989) 330 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  15. G.P. Korchemsky, Sudakov form-factor in QCD, Phys. Lett. B 220 (1989) 629 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  16. J.C. Collins, Sudakov form factors, Adv. Ser. Direct. High Energy Phys. 5 (1989) 573 [hep-ph/0312336] [SPIRES].

    Google Scholar 

  17. L. Magnea and G.F. Sterman, Analytic continuation of the Sudakov form-factor in QCD, Phys. Rev. D 42 (1990) 4222 [SPIRES].

    ADS  Google Scholar 

  18. S. Catani, The singular behaviour of QCD amplitudes at two-loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [SPIRES].

    ADS  Google Scholar 

  19. N. Kidonakis, G. Oderda and G.F. Sterman, Evolution of color exchange in QCD hard scattering, Nucl. Phys. B 531 (1998) 365 [hep-ph/9803241] [SPIRES].

    Article  ADS  Google Scholar 

  20. G. Sterman and M.E. Tejeda-Yeomans, Multi-loop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130] [SPIRES].

    ADS  Google Scholar 

  21. S.M. Aybat, L.J. Dixon and G.F. Sterman, The two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole, Phys. Rev. D 74 (2006) 074004 [hep-ph/0607309] [SPIRES].

    ADS  Google Scholar 

  22. L.J. Dixon, L. Magnea and G.F. Sterman, Universal structure of subleading infrared poles in gauge theory amplitudes, JHEP 08 (2008) 022 [arXiv:0805.3515] [SPIRES].

    Article  ADS  Google Scholar 

  23. S.G. Naculich, H. Nastase and H.J. Schnitzer, Subleading-color contributions to gluon-gluon scattering in N = 4 SYM theory and relations to N = 8 supergravity, JHEP 11 (2008) 018 [arXiv:0809.0376] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  24. T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [arXiv:0901.0722] [SPIRES].

    Article  ADS  Google Scholar 

  25. E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [SPIRES].

    Article  ADS  Google Scholar 

  26. T. Becher and M. Neubert, On the Structure of Infrared Singularities of Gauge-Theory Amplitudes, JHEP 06 (2009) 081 [arXiv:0903.1126] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  27. T. Becher and M. Neubert, Infrared singularities of QCD amplitudes with massive partons, Phys. Rev. D 79 (2009) 125004 [arXiv:0904.1021] [SPIRES].

    ADS  Google Scholar 

  28. S.G. Naculich and H.J. Schnitzer, IR divergences and Regge limits of subleading-color contributions to the four-gluon amplitude in N = 4 SYM Theory, JHEP 10 (2009) 048 [arXiv:0907.1895] [SPIRES].

    Article  ADS  Google Scholar 

  29. E. Gardi and L. Magnea, Infrared singularities in QCD amplitudes, Nuovo Cim. C32N 5-6 (2009) 137 [arXiv:0908.3273] [SPIRES].

    Google Scholar 

  30. L.J. Dixon, E. Gardi and L. Magnea, On soft singularities at three loops and beyond, JHEP 02 (2010) 081 [arXiv:0910.3653] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  32. N. Beisert et al., E 7(7) constraints on counterterms in N = 8 supergravity, Phys. Lett. B 694 (2010) 265 [arXiv:1009.1643] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  33. Z. Bern, L.J. Dixon and R. Roiban, Is \( \mathcal{N} = 8 \) supergravity ultraviolet finite?, Phys. Lett. B 644 (2007) 265 [hep-th/0611086] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  34. Z. Bern, J.J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, The Ultraviolet Behavior of N = 8 Supergravity at Four Loops, Phys. Rev. Lett. 103 (2009) 081301 [arXiv:0905.2326] [SPIRES].

    Article  ADS  Google Scholar 

  35. J. Broedel and L.J. Dixon, R 4 counterterm and E 7(7) symmetry in maximal supergravity, JHEP 05 (2010) 003 [arXiv:0911.5704] [SPIRES].

    Article  ADS  Google Scholar 

  36. H. Elvang, D.Z. Freedman and M. Kiermaier, A simple approach to counterterms in N = 8 supergravity, JHEP 11 (2010) 016 [arXiv:1003.5018] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  37. G. Bossard, P.S. Howe and K.S. Stelle, On duality symmetries of supergravity invariants, JHEP 01 (2011) 020 [arXiv:1009.0743] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  38. R. Kallosh, The Ultraviolet Finiteness of N = 8 Supergravity, JHEP 12 (2010) 009 [arXiv:1009.1135] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  39. H. Elvang, D.Z. Freedman and M. Kiermaier, SUSY Ward identities, Superamplitudes and Counterterms, arXiv:1012.3401 [SPIRES].

  40. Z. Bern, L.J. Dixon, D.C. Dunbar, M. Perelstein and J.S. Rozowsky, On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys. B 530 (1998) 401 [hep-th/9802162] [SPIRES].

    Article  ADS  Google Scholar 

  41. V.A. Smirnov, Analytical result for dimensionally regularized massless on-shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323] [SPIRES].

    ADS  Google Scholar 

  42. J.B. Tausk, Non-planar massless two-loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  43. 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 

  44. 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] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  45. C.D. White, Factorization Properties of Soft Graviton Amplitudes, JHEP 05 (2011) 060 [arXiv:1103.2981] [SPIRES].

    Article  ADS  Google Scholar 

  46. E. Laenen, G. Stavenga and C.D. White, Path integral approach to eikonal and next-to-eikonal exponentiation, JHEP 03 (2009) 054 [arXiv:0811.2067] [SPIRES].

    Article  ADS  Google Scholar 

  47. 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 

  48. G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys. B 283 (1987) 342 [SPIRES].

    Article  ADS  Google Scholar 

  49. D.R. Green, Worldlines as Wilson Lines, Phys. Rev. D 78 (2008) 064066 [arXiv:0804.4450] [SPIRES].

    ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  51. G.P. Korchemsky, J.M. Drummond and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  52. 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] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  53. 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] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  54. J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, Nucl. Phys. B 826 (2010) 337 [arXiv:0712.1223] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  55. J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude, Phys. Lett. B 662 (2008) 456 [arXiv:0712.4138] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  56. J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude, Nucl. Phys. B 815 (2009) 142 [arXiv:0803.1466] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  57. A. Mitov, G.F. Sterman and I. Sung, Computation of the Soft Anomalous Dimension Matrix in Coordinate Space, Phys. Rev. D 82 (2010) 034020 [arXiv:1005.4646] [SPIRES].

    ADS  Google Scholar 

  58. M.B. Green, J.H. Schwarz and L. Brink, N = 4 Yang-Mills and N = 8 Supergravity as Limits of String Theories, Nucl. Phys. B 198 (1982) 474 [SPIRES].

    Article  ADS  Google Scholar 

  59. Z. Bern and A.K. Grant, Perturbative gravity from QCD amplitudes, Phys. Lett. B 457 (1999) 23 [hep-th/9904026] [SPIRES].

    ADS  Google Scholar 

  60. D.M. Capper, G. Leibbrandt and M. Ramon Medrano, Calculation of the graviton selfenergy using dimensional regularization, Phys. Rev. D 8 (1973) 4320 [SPIRES].

    ADS  Google Scholar 

  61. J.G.M. Gatheral, Exponentiation of eikonal cross-sections in nonabelian gauge theories, Phys. Lett. B 133 (1983) 90 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  62. J. Frenkel and J.C. Taylor, Nonabelian eikonal exponentiation, Nucl. Phys. B 246 (1984) 231 [SPIRES].

    Article  ADS  Google Scholar 

  63. S.B. Giddings, M. Schmidt-Sommerfeld and J.R. Andersen, High energy scattering in gravity and supergravity, Phys. Rev. D 82 (2010) 104022 [arXiv:1005.5408] [SPIRES].

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Bowdoin College, Brunswick, ME, 04011, U.S.A.

    Stephen G. Naculich

  2. Theoretical Physics Group, Martin Fisher School of Physics, Brandeis University, Waltham, MA, 02454, U.S.A.

    Howard J. Schnitzer

Authors
  1. Stephen G. Naculich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Howard J. Schnitzer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephen G. Naculich.

Additional information

Research supported in part by the NSF under grant PHY-0756518. (Stephen G. Naculich)

Research supported in part by the DOE under grant DE–FG02–92ER40706. (Howard J. Schnitzer)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Naculich, S.G., Schnitzer, H.J. Eikonal methods applied to gravitational scattering amplitudes. J. High Energ. Phys. 2011, 87 (2011). https://doi.org/10.1007/JHEP05(2011)087

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP05(2011)087

Keywords

Search

Navigation