Graviton emission in Einstein-Hilbert gravity

  • Agustín Sabio Vera
  • Eduardo Serna Campillo
  • Miguel Á. Vázquez-Mozo


The five-point amplitude for the scattering of two distinct scalars with the emission of one graviton in the final state is calculated in exact kinematics for Einstein- Hilbert gravity. The result, which satisfies the Steinmann relations, is expressed in Sudakov variables, finding that it corresponds to the sum of two gauge invariant contributions written in terms of a new two scalar - two graviton effective vertex. A similar calculation is carried out in Quantum Chromodynamics (QCD) for the scattering of two distinct quarks with one extra gluon in the final state. The effective vertices which appear in both cases are then evaluated in the multi-Regge limit reproducing the well-known result obtained by Lipatov where the Einstein-Hilbert graviton emission vertex can be written as the product of two QCD gluon emission vertices, up to corrections to preserve the Steinmann relations.


Models of Quantum Gravity QCD 


  1. [1]
    J.M. Maldacena, The large- \( \mathcal{N} \) limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1133 ] [hep-th/9711200] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  2. [2]
    S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].MathSciNetADSGoogle Scholar
  3. [3]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  4. [4]
    Z. Bern, J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, The complete four-loop four-point amplitude in \( \mathcal{N} = {4} \) super-Yang-Mills theory, Phys. Rev. D 82 (2010) 125040 [arXiv:1008.3327] [INSPIRE].ADSGoogle Scholar
  5. [5]
    Z. Bern, L.J. Dixon, D. Dunbar, M. Perelstein and J. 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] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    Z. Bern, J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, The ultraviolet behavior of \( \mathcal{N} = {8} \) supergravity at four loops,Phys. Rev. Lett. 103(2009) 081301 [arXiv:0905.2326] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    J.F. Donoghue, General relativity as an effective field theory: the leading quantum corrections, Phys. Rev. D 50 (1994) 3874 [gr-qc/9405057] [INSPIRE].ADSGoogle Scholar
  8. [8]
    N. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein, Quantum gravitational corrections to the nonrelativistic scattering potential of two masses, Phys. Rev. D 67 (2003) 084033 [Erratum ibid. D 71 (2005) 069903] [hep-th/0211072] [INSPIRE].MathSciNetADSGoogle Scholar
  9. [9]
    J.F. Donoghue and T. Torma, Infrared behavior of graviton-graviton scattering, Phys. Rev. D 60 (1999) 024003 [hep-th/9901156] [INSPIRE].ADSGoogle Scholar
  10. [10]
    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] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    Z. Bern, D.C. Dunbar and T. Shimada, String based methods in perturbative gravity, Phys. Lett. B 312 (1993) 277 [hep-th/9307001] [INSPIRE].ADSGoogle Scholar
  12. [12]
    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 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    S.-Q. Su, Graviton bremsstrahlung at high energies, Doctoral Thesis, Katholieke Universiteit Leuven, Leuven Belgium (1982).Google Scholar
  14. [14]
    J. Geris and S.-Q. Su, Single bremsstrahlung processes in quantum gravity, Commun. Theor. Phys. 8 (1987) 325 [INSPIRE].Google Scholar
  15. [15]
    J.F. Donoghue, Introduction to the effective field theory description of gravity, gr-qc/9512024 [INSPIRE].
  16. [16]
    L. Lipatov, Effective action for the Regge processes in gravity, arXiv:1105.3127 [INSPIRE].
  17. [17]
    M.T. Grisaru, P. van Nieuwenhuizen and C. Wu, Reggeization and the question of higher loop renormalizability of gravitation, Phys. Rev. D 12 (1975) 1563 [INSPIRE].ADSGoogle Scholar
  18. [18]
    M.T. Grisaru and H.J. Schnitzer, Dynamical calculation of bound state supermultiplets in \( \mathcal{N} = {8} \) supergravity, Phys. Lett. B 107 (1981) 196 [INSPIRE].ADSGoogle Scholar
  19. [19]
    L. Lipatov, Graviton reggeization, Phys. Lett. B 116 (1982) 411 [INSPIRE].ADSGoogle Scholar
  20. [20]
    L. Lipatov, Multi-Regge processes in gravitation, Sov. Phys. JETP 55 (1982) 582 [Zh. Eksp. Teor. Fiz. 82 (1982) 991] [INSPIRE].Google Scholar
  21. [21]
    L. Lipatov, High-energy scattering in QCD and in quantum gravity and two-dimensional field theories, Nucl. Phys. B 365 (1991) 614 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    L. Lipatov, Reggeization of the vector meson and the vacuum singularity in non-Abelian gauge theories, Sov. J. Nucl. Phys. 23 (1976) 338 [INSPIRE].Google Scholar
  23. [23]
    V.S. Fadin, E. Kuraev and L. Lipatov, On the Pomeranchuk singularity in asymptotically free theories, Phys. Lett. B 60 (1975) 50 [INSPIRE].ADSGoogle Scholar
  24. [24]
    E. Kuraev, L. Lipatov and V.S. Fadin, Multi-reggeon processes in the Yang-Mills theory, Sov. Phys. JETP 44 (1976) 443 [INSPIRE].ADSGoogle Scholar
  25. [25]
    E. Kuraev, L. Lipatov and V.S. Fadin, The Pomeranchuk singularity in non-Abelian gauge theories, Sov. Phys. JETP 45 (1977) 199 [INSPIRE].MathSciNetADSGoogle Scholar
  26. [26]
    I. Balitsky and L. Lipatov, The Pomeranchuk singularity in quantum chromodynamics, Sov. J. Nucl. Phys. 28 (1978) 822 [INSPIRE].Google Scholar
  27. [27]
    J.M. Martın-García, xPerm: fast index canonicalization for tensor computer algebra, Comput. Phys. Commun. 179 (2008) 597 [arXiv:0803.0862].ADSMATHCrossRefGoogle Scholar
  28. [28]
    Z. Bern, Perturbative quantum gravity and its relation to gauge theory, Living Rev. Rel. 5 (2002)5 [gr-qc/0206071] [INSPIRE].MathSciNetGoogle Scholar
  29. [29]
    J.J.M. Carrasco and H. Johansson, Generic multiloop methods and application to \( \mathcal{N} = {4} \) super-Yang-Mills, J. Phys. A 44 (2011) 454004 [INSPIRE].MathSciNetADSGoogle Scholar
  30. [30]
    R. Gastmans and T.T. Wu, The ubiquitous photon: helicity method for QED and QCD, Clarendon, Oxford U.K. (1990) [INSPIRE].
  31. [31]
    Z. Xu, D.-H. Zhang and L. Chang, Helicity amplitudes for multiple bremsstrahlung in massless non-Abelian gauge theories, Nucl. Phys. B 291 (1987) 392 [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    O. Steinmann, Über den Zusammenhang zwischen den Wightmanfunktionen und der retardierten Kommutatoren (in German), Helv. Phys. Acta 33 (1960) 257.MathSciNetMATHGoogle Scholar
  33. [33]
    O. Steinmann, Wightman-Funktionen und retardierten Kommutatoren. II (in German), Helv. Phys. Acta 33 (1960) 347.MathSciNetMATHGoogle Scholar
  34. [34]
    L. Lipatov, High-energy asymptotics of multicolor QCD and two-dimensional conformal field theories, DESY-93-055, DESY, Zeuthen Germany April 1993 [Phys. Lett. B 309 (1993) 394 ] [INSPIRE].
  35. [35]
    L. Lipatov, The bare Pomeron in quantum chromodynamics, Sov. Phys. JETP 63 (1986) 904 [Zh. Eksp. Teor. Fiz. 90 (1986) 1536] [INSPIRE].Google Scholar
  36. [36]
    J. Bartels, High-energy behavior in a non-Abelian gauge theory. 2. First corrections to T(nm) beyond the leading LNS approximation, Nucl. Phys. B 175 (1980) 365 [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    J. Kwiecinski and M. Praszalowicz, Three gluon integral equation and odd c singlet Regge singularities in QCD, Phys. Lett. B 94 (1980) 413 [INSPIRE].ADSGoogle Scholar
  38. [38]
    L. Lipatov, Duality symmetry of Reggeon interactions in multicolor QCD, Nucl. Phys. B 548 (1999) 328 [hep-ph/9812336] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    L. Lipatov, High-energy asymptotics of multicolor QCD and exactly solvable lattice models, Padua preprint DFPD-93-TH-70, unpublished, University of Padua, Padua Italy October 1993 [hep-th/9311037] [INSPIRE].
  40. [40]
    L. Lipatov, Asymptotic behavior of multicolor QCD at high energies in connection with exactly solvable spin models, JETP Lett. 59 (1994) 596 [Pisma Zh. Eksp. Teor. Fiz. 59 (1994)571] [INSPIRE].ADSGoogle Scholar
  41. [41]
    L. Faddeev and G. Korchemsky, High-energy QCD as a completely integrable model, Phys. Lett. B 342 (1995) 311 [hep-th/9404173] [INSPIRE].ADSGoogle Scholar
  42. [42]
    L. Lipatov, Integrability of scattering amplitudes in \( \mathcal{N} = {4} \) SUSY, J. Phys. A 42 (2009) 304020 [arXiv:0902.1444] [INSPIRE].MathSciNetGoogle Scholar
  43. [43]
    J. Bartels, L. Lipatov and A. Prygarin, Integrable spin chains and scattering amplitudes, J. Phys. A 44 (2011) 454013 [arXiv:1104.0816] [INSPIRE].MathSciNetADSGoogle Scholar
  44. [44]
    J. Bartels, L. Lipatov and A. Sabio Vera, BFKL Pomeron, reggeized gluons and Bern-Dixon-Smirnov amplitudes, Phys. Rev. D 80 (2009) 045002 [arXiv:0802.2065] [INSPIRE].ADSGoogle Scholar
  45. [45]
    J. Bartels, L. Lipatov and A. Sabio Vera, N = 4 supersymmetric Yang-Mills scattering amplitudes at high energies: the Regge cut contribution, Eur. Phys. J. C 65 (2010) 587 [arXiv:0807.0894] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    A. Romagnoni and A. Sabio Vera, A hidden \( BFKL/XX{X_{{ - \frac{1}{2}}}} \) spin chain mapping, arXiv:1111.4553 [INSPIRE].
  47. [47]
    Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    Z. Bern, T. Dennen, Y.-T. Huang and M. Kiermaier, Gravity as the square of gauge theory, Phys. Rev. D 82 (2010) 065003 [arXiv:1004.0693] [INSPIRE].ADSGoogle Scholar
  49. [49]
    Z. Bern and T. Dennen, A color dual form for gauge-theory amplitudes, arXiv:1103.0312 [INSPIRE].
  50. [50]
    L.D. Landau and E.M. Lifshitz, The classical theory of fields, 3rd revised edition, Pergamon, London U.K. (1971).Google Scholar
  51. [51]
    B.S. DeWitt, Quantum theory of gravity. 3. Applications of the covariant theory, Phys. Rev. 162 (1967) 1239 [INSPIRE]. ADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Agustín Sabio Vera
    • 1
  • Eduardo Serna Campillo
    • 2
  • Miguel Á. Vázquez-Mozo
    • 2
  1. 1.Instituto de Física Teórica UAM/CSIC & Universidad Autónoma de MadridMadridSpain
  2. 2.Departamento de F´ısica Fundamental & IUFFyMUniversidad de SalamancaSalamancaSpain

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