Background independent exact renormalization group for conformally reduced gravity

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Regular Article - Theoretical Physics


Within the conformally reduced gravity model, where the metric is parametrised by a function f (ϕ) of the conformal factor ϕ, we keep dependence on both the background and fluctuation fields, to local potential approximation and \( \mathcal{O}\left({\partial}^2\right) \) respectively, making no other approximation. Explicit appearances of the background metric are then dictated by realising a remnant diffeomorphism invariance. The standard non-perturbative Renormalization Group (RG) scale k is inherently background dependent, which we show in general forbids the existence of RG fixed points with respect to k. By utilising transformations that follow from combining the flow equations with the modified split Ward identity, we uncover a unique background independent notion of RG scale, \( \widehat{k} \). The corresponding RG flow equations are then not only explicitly background independent along the entire RG flow but also explicitly independent of the form of f. In general f (ϕ) is forced to be scale dependent and needs to be renormalised, but if this is avoided then k-fixed points are allowed and furthermore they coincide with \( \widehat{k} \)-fixed points.


Models of Quantum Gravity Nonperturbative Effects Renormalization Group 


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  1. [1]
    C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301 (1993) 90 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    T.R. Morris, The exact renormalization group and approximate solutions, Int. J. Mod. Phys. A 9 (1994) 2411 [hep-ph/9308265] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    J. Berges, N. Tetradis and C. Wetterich, Nonperturbative renormalization flow in quantum field theory and statistical physics, Phys. Rept. 363 (2002) 223 [hep-ph/0005122] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  4. [4]
    H. Gies, Introduction to the functional RG and applications to gauge theories, Lect. Notes Phys. 852 (2012) 287 [hep-ph/0611146] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    J.M. Pawlowski, Aspects of the functional renormalisation group, Annals Phys. 322 (2007) 2831 [hep-th/0512261] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    T.R. Morris, Elements of the continuous renormalization group, Prog. Theor. Phys. Suppl. 131 (1998) 395 [hep-th/9802039] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev. D 57 (1998) 971 [hep-th/9605030] [INSPIRE].ADSMathSciNetGoogle Scholar
  8. [8]
    M. Reuter and F. Saueressig, Quantum Einstein gravity, New J. Phys. 14 (2012) 055022 [arXiv:1202.2274] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    R. Percacci, A short introduction to asymptotic safety, arXiv:1110.6389 [INSPIRE].
  10. [10]
    M. Niedermaier and M. Reuter, The Asymptotic Safety Scenario in Quantum Gravity, Living Rev. Rel. 9 (2006) 5.CrossRefMATHGoogle Scholar
  11. [11]
    S. Nagy, Lectures on renormalization and asymptotic safety, Annals Phys. 350 (2014) 310 [arXiv:1211.4151] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    D.F. Litim, Renormalisation group and the Planck scale, Phil. Trans. Roy. Soc. Lond. A 369 (2011) 2759 [arXiv:1102.4624] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    S. Weinberg, Ultraviolet Divergences In Quantum Theories Of Gravitation, in General Relativity, S.W. Hawking and W. Israel eds., Cambridge University Press, Cambridge U.K. (1980), pg. 790.Google Scholar
  14. [14]
    J.A. Dietz and T.R. Morris, Background independent conformally reduced gravity and asymptotic safety, to appear.Google Scholar
  15. [15]
    M. Reuter and H. Weyer, Background Independence and Asymptotic Safety in Conformally Reduced Gravity, Phys. Rev. D 79 (2009) 105005 [arXiv:0801.3287] [INSPIRE].ADSGoogle Scholar
  16. [16]
    M. Reuter and H. Weyer, Conformal sector of Quantum Einstein Gravity in the local potential approximation: non-Gaussian fixed point and a phase of unbroken diffeomorphism invariance, Phys. Rev. D 80 (2009) 025001 [arXiv:0804.1475] [INSPIRE].ADSGoogle Scholar
  17. [17]
    E. Manrique, M. Reuter and F. Saueressig, Bimetric Renormalization Group Flows in Quantum Einstein Gravity, Annals Phys. 326 (2011) 463 [arXiv:1006.0099] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    E. Manrique and M. Reuter, Bimetric Truncations for Quantum Einstein Gravity and Asymptotic Safety, Annals Phys. 325 (2010) 785 [arXiv:0907.2617] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    M. Demmel, F. Saueressig and O. Zanusso, RG flows of Quantum Einstein Gravity on maximally symmetric spaces, JHEP 06 (2014) 026 [arXiv:1401.5495] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    J.A. Dietz and T.R. Morris, Redundant operators in the exact renormalisation group and in the f(R) approximation to asymptotic safety, JHEP 07 (2013) 064 [arXiv:1306.1223] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    I.H. Bridle, J.A. Dietz and T.R. Morris, The local potential approximation in the background field formalism, JHEP 03 (2014) 093 [arXiv:1312.2846] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    D. Becker and M. Reuter, En route to Background Independence: Broken split-symmetry and how to restore it with bi-metric average actions, Annals Phys. 350 (2014) 225 [arXiv:1404.4537] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    E. Manrique, M. Reuter and F. Saueressig, Matter Induced Bimetric Actions for Gravity, Annals Phys. 326 (2011) 440 [arXiv:1003.5129] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    A. Codello, G. D’Odorico and C. Pagani, Consistent closure of renormalization group flow equations in quantum gravity, Phys. Rev. D 89 (2014) 081701 [arXiv:1304.4777] [INSPIRE].ADSGoogle Scholar
  25. [25]
    N. Christiansen, B. Knorr, J.M. Pawlowski and A. Rodigast, Global Flows in Quantum Gravity, arXiv:1403.1232 [INSPIRE].
  26. [26]
    K. Groh and F. Saueressig, Ghost wave-function renormalization in Asymptotically Safe Quantum Gravity, J. Phys. A 43 (2010) 365403 [arXiv:1001.5032] [INSPIRE].MATHGoogle Scholar
  27. [27]
    A. Eichhorn and H. Gies, Ghost anomalous dimension in asymptotically safe quantum gravity, Phys. Rev. D 81 (2010) 104010 [arXiv:1001.5033] [INSPIRE].ADSGoogle Scholar
  28. [28]
    P. Donà, A. Eichhorn and R. Percacci, Matter matters in asymptotically safe quantum gravity, Phys. Rev. D 89 (2014) 084035 [arXiv:1311.2898] [INSPIRE].ADSGoogle Scholar
  29. [29]
    P. Donà, A. Eichhorn and R. Percacci, Consistency of matter models with asymptotically safe quantum gravity, arXiv:1410.4411 [INSPIRE].
  30. [30]
    D. Becker and M. Reuter, Propagating gravitons vs.dark matterin asymptotically safe quantum gravity, JHEP 12 (2014) 025 [arXiv:1407.5848] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    K. Falls, On the renormalisation of Newtons constant, arXiv:1501.05331 [INSPIRE].
  32. [32]
    P.F. Machado and R. Percacci, Conformally reduced quantum gravity revisited, Phys. Rev. D 80 (2009) 024020 [arXiv:0904.2510] [INSPIRE].ADSGoogle Scholar
  33. [33]
    A. Bonanno and F. Guarnieri, Universality and Symmetry Breaking in Conformally Reduced Quantum Gravity, Phys. Rev. D 86 (2012) 105027 [arXiv:1206.6531] [INSPIRE].ADSGoogle Scholar
  34. [34]
    G.W. Gibbons, S.W. Hawking and M.J. Perry, Path Integrals and the Indefiniteness of the Gravitational Action, Nucl. Phys. B 138 (1978) 141 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    M. Reuter and H. Weyer, The Role of Background Independence for Asymptotic Safety in Quantum Einstein Gravity, Gen. Rel. Grav. 41 (2009) 983 [arXiv:0903.2971] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    D.F. Litim and J.M. Pawlowski, Wilsonian flows and background fields, Phys. Lett. B 546 (2002) 279 [hep-th/0208216] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    M. Reuter and C. Wetterich, Gluon condensation in nonperturbative flow equations, Phys. Rev. D 56 (1997) 7893 [hep-th/9708051] [INSPIRE].ADSGoogle Scholar
  38. [38]
    D.F. Litim and J.M. Pawlowski, On gauge invariant Wilsonian flows, hep-th/9901063 [INSPIRE].
  39. [39]
    D.F. Litim and J.M. Pawlowski, Renormalization group flows for gauge theories in axial gauges, JHEP 09 (2002) 049 [hep-th/0203005] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    M. Demmel, F. Saueressig and O. Zanusso, RG flows of Quantum Einstein Gravity in the linear-geometric approximation, arXiv:1412.7207 [INSPIRE].
  41. [41]
    D. Benedetti and F. Caravelli, The Local potential approximation in quantum gravity, JHEP 06 (2012) 017 [arXiv:1204.3541] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    A. Codello, R. Percacci and C. Rahmede, Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation, Annals Phys. 324 (2009) 414 [arXiv:0805.2909] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    P.F. Machado and F. Saueressig, On the renormalization group flow of f(R)-gravity, Phys. Rev. D 77 (2008) 124045 [arXiv:0712.0445] [INSPIRE].ADSMathSciNetGoogle Scholar
  44. [44]
    T.R. Morris, Derivative expansion of the exact renormalization group, Phys. Lett. B 329 (1994) 241 [hep-ph/9403340] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    T.R. Morris and Z. Slade, Solving the Reconstruction Problem in Asymptotic Safety, to appear.Google Scholar
  46. [46]
    L.F. Abbott, The Background Field Method Beyond One Loop, Nucl. Phys. B 185 (1981) 189 [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    L.F. Abbott, Introduction to the Background Field Method, Acta Phys. Polon. B 13 (1982) 33 [INSPIRE].MathSciNetGoogle Scholar
  48. [48]
    D.F. Litim, Optimized renormalization group flows, Phys. Rev. D 64 (2001) 105007 [hep-th/0103195] [INSPIRE].ADSGoogle Scholar
  49. [49]
    T.R. Morris, The renormalization group and two-dimensional multicritical effective scalar field theory, Phys. Lett. B 345 (1995) 139 [hep-th/9410141] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    D.F. Litim, Critical exponents from optimized renormalization group flows, Nucl. Phys. B 631 (2002) 128 [hep-th/0203006] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  51. [51]
    T.R. Morris, Three-dimensional massive scalar field theory and the derivative expansion of the renormalization group, Nucl. Phys. B 495 (1997) 477 [hep-th/9612117] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    C. Bervillier, Revisiting the local potential approximation of the exact renormalization group equation, Nucl. Phys. B 876 (2013) 587 [arXiv:1307.3679] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  53. [53]
    H.W. Hamber and R. Toriumi, Inconsistencies from a Running Cosmological Constant, Int. J. Mod. Phys. D 22 (2013) 1330023 [arXiv:1301.6259] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  54. [54]
    R. Percacci, Renormalization group flow of Weyl invariant dilaton gravity, New J. Phys. 13 (2011) 125013 [arXiv:1110.6758] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    C. Pagani and R. Percacci, Quantization and fixed points of non-integrable Weyl theory, arXiv:1312.7767 [INSPIRE].
  56. [56]
    A. Nink, Field Parametrization Dependence in Asymptotically Safe Quantum Gravity, Phys. Rev. D 91 (2015) 044030 [arXiv:1410.7816] [INSPIRE].ADSMathSciNetGoogle Scholar
  57. [57]
    R. Percacci and G.P. Vacca, Search of scaling solutions in scalar-tensor gravity, arXiv:1501.00888 [INSPIRE].
  58. [58]
    H. Gies and S. Lippoldt, Fermions in gravity with local spin-base invariance, Phys. Rev. D 89 (2014) 064040 [arXiv:1310.2509] [INSPIRE].ADSGoogle Scholar
  59. [59]
    H. Gies and S. Lippoldt, Global surpluses of spin-base invariant fermions, Phys. Lett. B 743 (2015) 415 [arXiv:1502.00918] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  60. [60]
    S. Lippoldt, Spin-base invariance of Fermions in arbitrary dimensions, arXiv:1502.05607 [INSPIRE].
  61. [61]
    J. Distler and H. Kawai, Conformal Field Theory and 2D Quantum Gravity Or Whos Afraid of Joseph Liouville?, Nucl. Phys. B 321 (1989) 509 [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    V. Branchina, K.A. Meissner and G. Veneziano, The Price of an exact, gauge invariant RG flow equation, Phys. Lett. B 574 (2014) 319 [hep-th/0309234] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  63. [63]
    J.M. Pawlowski, Geometrical effective action and Wilsonian flows, hep-th/0310018 [INSPIRE].
  64. [64]
    I. Donkin and J.M. Pawlowski, The phase diagram of quantum gravity from diffeomorphism-invariant RG-flows, arXiv:1203.4207 [INSPIRE].

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.School of Physics and AstronomyUniversity of SouthamptonSouthamptonUnited Kingdom

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