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On the quantum field theory of the gravitational interactions

  • Damiano Anselmi
Open Access
Regular Article - Theoretical Physics

Abstract

We study the main options for a unitary and renormalizable, local quantum field theory of the gravitational interactions. The first model is a Lee-Wick superrenormalizable higher-derivative gravity, formulated as a nonanalytically Wick rotated Euclidean theory. We show that, under certain conditions, the S matrix is unitary when the cosmological constant vanishes. The model is the simplest of its class. However, infinitely many similar options are allowed, which raises the issue of uniqueness. To deal with this problem, we propose a new quantization prescription, by doubling the unphysical poles of the higher-derivative propagators and turning them into Lee-Wick poles. The Lagrangian of the simplest theory of quantum gravity based on this idea is the linear combination of R, R μν R μν , R 2 and the cosmological term. Only the graviton propagates in the cutting equations and, when the cosmological constant vanishes, the S matrix is unitary. The theory satisfies the locality of counterterms and is renormalizable by power counting. It is unique in the sense that it is the only one with a dimensionless gauge coupling.

Keywords

Models of Quantum Gravity Beyond Standard Model Renormalization Regularization and Renormalons 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    G. ’t Hooft and M. Veltman, One-loop divergences in the theory of gravitation, Ann. Inst. H. Poincaré Phys. Theor. A 20 (1974) 69.Google Scholar
  2. [2]
    P. Van Nieuwenhuizen, On the renormalization of quantum gravitation without matter, Annals Phys. 104 (1977) 197 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    M.H. Goroff and A. Sagnotti, The ultraviolet behavior of Einstein gravity, Nucl. Phys. B 266 (1986) 709 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    A.E.M. van de Ven, Two loop quantum gravity, Nucl. Phys. B 378 (1992) 309 [INSPIRE].ADSMathSciNetGoogle Scholar
  5. [5]
    D. Anselmi, Aspects of perturbative unitarity, Phys. Rev. D 94 (2016) 025028 [http://renormalization.com/16a1/] [arXiv:1606.06348] [INSPIRE].
  6. [6]
    K.S. Stelle, Renormalization of higher derivative quantum gravity, Phys. Rev. D 16 (1977) 953 [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    J. Julve and M. Tonin, Quantum gravity with higher derivative terms, Nuovo Cim. B 46 (1978) 137 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    E.S. Fradkin and A.A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B 201 (1982) 469 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    I.G. Avramidi and A.O. Barvinsky, Asymptotic freedom in higher derivative quantum gravity, Phys. Lett. B 159 (1985) 269 [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    U.G. Aglietti and D. Anselmi, Inconsistency of Minkowski higher-derivative theories, Eur. Phys. J. C 77 (2017) 84 [http://renormalization.com/16a2/] [arXiv:1612.06510] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    D. Anselmi and M. Piva, A new formulation of Lee-Wick quantum field theory, [http://renormalization.com/17a1/] arXiv:1703.04584 [INSPIRE].
  12. [12]
    T.D. Lee and G.C. Wick, Negative metric and the unitarity of the S-matrix, Nucl. Phys. B 9 (1969) 209 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  13. [13]
    T.D. Lee and G.C. Wick, Finite theory of quantum electrodynamics, Phys. Rev. D 2 (1970) 1033 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  14. [14]
    D. Anselmi and M. Piva, Perturbative unitarity of Lee-Wick quantum field theory, http://renormalization.com/17a2/ [arXiv:1703.05563] [INSPIRE].
  15. [15]
    R.E. Cutkosky, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, A non-analytic S matrix, Nucl. Phys. B 12 (1969) 281 [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    B. Grinstein, D. O’Connell and M.B. Wise, Causality as an emergent macroscopic phenomenon: the Lee-Wick O(N) model, Phys. Rev. D 79 (2009) 105019 [arXiv:0805.2156] [INSPIRE].ADSGoogle Scholar
  17. [17]
    B. Grinstein, D. O’Connell and M.B. Wise, The Lee-Wick standard model, Phys. Rev. D 77 (2008) 025012 [arXiv:0704.1845] [INSPIRE].
  18. [18]
    C.D. Carone and R.F. Lebed, Minimal Lee-Wick extension of the standard model, Phys. Lett. B 668 (2008) 221 [arXiv:0806.4555] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    J.R. Espinosa and B. Grinstein, Ultraviolet properties of the Higgs sector in the Lee-Wick standard model, Phys. Rev. D 83 (2011) 075019 [arXiv:1101.5538] [INSPIRE].
  20. [20]
    C.D. Carone and R.F. Lebed, A higher-derivative Lee-Wick standard model, JHEP 01 (2009) 043 [arXiv:0811.4150] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  21. [21]
    B. Grinstein and D. O’Connell, One-Loop Renormalization of Lee-Wick Gauge Theory, Phys. Rev. D 78 (2008) 105005 [arXiv:0801.4034] [INSPIRE].ADSGoogle Scholar
  22. [22]
    C.D. Carone, Higher-Derivative Lee-Wick Unification, Phys. Lett. B 677 (2009) 306 [arXiv:0904.2359] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    E. Tomboulis, 1/N Expansion and Renormalization in Quantum Gravity, Phys. Lett. B 70 (1977) 361 [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    E. Tomboulis, Renormalizability and Asymptotic Freedom in Quantum Gravity, Phys. Lett. B 97 (1980) 77 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    L. Modesto and I.L. Shapiro, Superrenormalizable quantum gravity with complex ghosts, Phys. Lett. B 755 (2016) 279 [arXiv:1512.07600] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    L. Modesto, Super-renormalizable or finite Lee-Wick quantum gravity, Nucl. Phys. B 909 (2016) 584 [arXiv:1602.02421] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    L. Modesto, L. Rachwal and I.L. Shapiro, Renormalization group in super-renormalizable quantum gravity, arXiv:1704.03988 [INSPIRE].
  28. [28]
    S. Weinberg, Ultraviolet divergences in quantum theories of gravitation, in An Einstein centenary survey, S. Hawking and W. Israel eds., Cambridge University Press, Cambridge U.K. (1979), pg. 790.Google Scholar
  29. [29]
    O. Lauscher and M. Reuter, Ultraviolet fixed point and generalized flow equation of quantum gravity, Phys. Rev. D 65 (2002) 025013 [hep-th/0108040] [INSPIRE].
  30. [30]
    R. Percacci and D. Perini, Asymptotic safety of gravity coupled to matter, Phys. Rev. D 68 (2003) 044018 [hep-th/0304222] [INSPIRE].
  31. [31]
    D.F. Litim, Fixed points of quantum gravity, Phys. Rev. Lett. 92 (2004) 201301 [hep-th/0312114] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    R. Percacci, A Short introduction to asymptotic safety, in Proceedings of “Time and matter” conference, Budva Montenegro (2010) [arXiv:1110.6389] [INSPIRE].
  33. [33]
    Yu.V. Kuz’min, Finite nonlocal gravity, Sov. J. Nucl. Phys. 50 (1989) 6 [Yad. Fiz. 50 (1989) 1630].Google Scholar
  34. [34]
    E.T. Tomboulis, Superrenormalizable gauge and gravitational theories, hep-th/9702146 [INSPIRE].
  35. [35]
    L. Modesto, Super-renormalizable quantum gravity, Phys. Rev. D 86 (2012) 044005 [arXiv:1107.2403] [INSPIRE].
  36. [36]
    L. Modesto, Finite quantum gravity, arXiv:1305.6741 [INSPIRE].
  37. [37]
    F. Briscese, L. Modesto and S. Tsujikawa, Super-renormalizable or finite completion of the Starobinsky theory, Phys. Rev. D 89 (2014) 024029 [arXiv:1308.1413] [INSPIRE].
  38. [38]
    T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar, Towards singularity and ghost free theories of gravity, Phys. Rev. Lett. 108 (2012) 031101 [arXiv:1110.5249] [INSPIRE].
  39. [39]
    D. Chialva and A. Mazumdar, Cosmological implications of quantum corrections and higher-derivative extension, Mod. Phys. Lett. A 30 (2015) 1540008 [arXiv:1405.0513] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys. 1 (1960) 429 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    M.J.G. Veltman, Unitarity and causality in a renormalizable field theory with unstable particles, Physica 29 (1963) 186 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    G. ’t Hooft, Renormalization of massless Yang-Mills fields, Nucl. Phys. B 33 (1971) 173 [INSPIRE].
  43. [43]
    G. ’t Hooft, Renormalizable Lagrangians for massive Yang-Mills fields, Nucl. Phys. B 35 (1971) 167 [INSPIRE].
  44. [44]
    G. Narain and R. Anishetty, Short distance freedom of quantum gravity, Phys. Lett. B 711 (2012) 128 [arXiv:1109.3981] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    G. Narain and R. Anishetty, Unitary and renormalizable theory of higher derivative gravity, J. Phys. Conf. Ser. 405 (2012) 012024 [arXiv:1210.0513] [INSPIRE].
  46. [46]
    D. Anselmi, Properties of the classical action of quantum gravity, JHEP 05 (2013) 028 [http://renormalization.com/13a2/] [arXiv:1302.7100] [INSPIRE].
  47. [47]
    S.B. Giddings, The Boundary S matrix and the AdS to CFT dictionary, Phys. Rev. Lett. 83 (1999) 2707 [hep-th/9903048] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    V. Balasubramanian, S.B. Giddings and A.E. Lawrence, What do CFTs tell us about Anti-de Sitter space-times?, JHEP 03 (1999) 001 [hep-th/9902052] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  49. [49]
    S.J. Hathrell, Trace anomalies and λ \( \phi \) 4 theory in curved space, Annals Phys. 139 (1982) 136 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    S.J. Hathrell, Trace anomalies and QED in curved space, Annals Phys. 142 (1982) 34 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    M.D. Freeman, The renormalization of non-Abelian gauge theories in curved space-time, Annals Phys. 153 (1984) 339 [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    P.S. Howe, K.S. Stelle and P.C. West, A class of finite four-dimensional supersymmetric field theories, Phys. Lett. 124B (1983) 55 [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    A. Salvio and A. Strumia, Agravity, JHEP 06 (2014) 080 [arXiv:1403.4226] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Dipartimento di Fisica “Enrico Fermi”Università di PisaPisaItaly
  2. 2.INFN, Sezione di PisaPisaItaly

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