The Renormalization Group flow of unimodular f(R) gravity

Open Access
Regular Article - Theoretical Physics

Abstract

Unimodular gravity is classically equivalent to General Relativity. This equivalence extends to actions which are functions of the curvature scalar. At the quantum level, the dynamics could differ. Most importantly, the cosmological constant is not a coupling in the unimodular action, providing a new vantage point from which to address the cosmological constant fine-tuning problem. Here, a quantum theory based on the asymptotic safety scenario is studied, and evidence for an interacting fixed point in unimodular f (R) gravity is found. We study the fixed point and its properties, and also discuss the compatibility of unimodular asymptotic safety with dynamical matter, finding evidence for its compatibility with the matter degrees of freedom of the Standard Model.

Keywords

Models of Quantum Gravity Renormalization Group 

Notes

Open Access

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References

  1. [1]
    S. Weinberg, The Cosmological Constant Problem, Rev. Mod. Phys. 61 (1989) 1 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    A. Einstein, Do gravitational fields play an essential part in the structure of the elementary particles of matter? (in German), Sitzungsber. Preuss. Akad. Wiss. Berlin (1919) 433 [INSPIRE].
  3. [3]
    W.G. Unruh, A Unimodular Theory of Canonical Quantum Gravity, Phys. Rev. D 40 (1989) 1048 [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    Y.J. Ng and H. van Dam, Unimodular Theory of Gravity and the Cosmological Constant, J. Math. Phys. 32 (1991) 1337 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    D.R. Finkelstein, A.A. Galiautdinov and J.E. Baugh, Unimodular relativity and cosmological constant, J. Math. Phys. 42 (2001) 340 [gr-qc/0009099] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    E. Alvarez and A.F. Faedo, Unimodular cosmology and the weight of energy, Phys. Rev. D 76 (2007) 064013 [hep-th/0702184] [INSPIRE].ADSGoogle Scholar
  7. [7]
    B. Fiol and J. Garriga, Semiclassical Unimodular Gravity, JCAP 08 (2010) 015 [arXiv:0809.1371] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    M. Shaposhnikov and D. Zenhausern, Scale invariance, unimodular gravity and dark energy, Phys. Lett. B 671 (2009) 187 [arXiv:0809.3395] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    E. Alvarez and M. Herrero-Valea, No Conformal Anomaly in Unimodular Gravity, Phys. Rev. D 87 (2013) 084054 [arXiv:1301.5130] [INSPIRE].ADSGoogle Scholar
  10. [10]
    G.F.R. Ellis, The Trace-Free Einstein Equations and inflation, Gen. Rel. Grav. 46 (2014) 1619 [arXiv:1306.3021] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    C. Barceló, R. Carballo-Rubio and L.J. Garay, Unimodular gravity and general relativity from graviton self-interactions, Phys. Rev. D 89 (2014) 124019 [arXiv:1401.2941] [INSPIRE].ADSGoogle Scholar
  12. [12]
    E. Alvarez, Can one tell Einsteins unimodular theory from Einsteins general relativity?, JHEP 03 (2005) 002 [hep-th/0501146] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    J.J. van der Bij, H. van Dam and Y.J. Ng, The Exchange Of Massless Spin Two Particles, Physica A 116 (1982) 307.ADSMathSciNetGoogle Scholar
  14. [14]
    M. Henneaux and C. Teitelboim, The Cosmological Constant and General Covariance, Phys. Lett. B 222 (1989) 195 [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    L. Smolin, The quantization of unimodular gravity and the cosmological constant problems, Phys. Rev. D 80 (2009) 084003 [arXiv:0904.4841] [INSPIRE].ADSMathSciNetGoogle Scholar
  16. [16]
    L. Smolin, Unimodular loop quantum gravity and the problems of time, Phys. Rev. D 84 (2011) 044047 [arXiv:1008.1759] [INSPIRE].ADSGoogle Scholar
  17. [17]
    A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Rel. 13 (2010) 3 [arXiv:1002.4928] [INSPIRE].CrossRefMATHGoogle Scholar
  18. [18]
    S. Weinberg, Ultraviolet Divergences In Quantum Theories Of Gravitation, [INSPIRE].
  19. [19]
    J. Henson, The causal set approach to quantum gravity, gr-qc/0601121 [INSPIRE].
  20. [20]
    P. Wallden, Causal Sets: Quantum Gravity from a Fundamentally Discrete Spacetime, J. Phys. Conf. Ser. 222 (2010) 012053 [arXiv:1001.4041] [INSPIRE].CrossRefGoogle Scholar
  21. [21]
    J. Ambjørn, A. Görlich, J. Jurkiewicz, A. Kreienbuehl and R. Loll, Renormalization Group Flow in CDT, Class. Quant. Grav. 31 (2014) 165003 [arXiv:1405.4585] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  22. [22]
    J.F. Donoghue, The effective field theory treatment of quantum gravity, AIP Conf. Proc. 1483 (2012) 73 [arXiv:1209.3511] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    J.F. Donoghue, Leading quantum correction to the Newtonian potential, Phys. Rev. Lett. 72 (1994) 2996 [gr-qc/9310024] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    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
  25. [25]
    N.E. 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].
  26. [26]
    G. ’t Hooft and M.J.G. Veltman, One loop divergencies in the theory of gravitation, Annales Poincare Phys. Theor. A 20 (1974) 69 [INSPIRE].ADSMathSciNetGoogle Scholar
  27. [27]
    M.H. Goroff and A. Sagnotti, Quantum gravity at two loops, Phys. Lett. B 160 (1985) 81 [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    A.E.M. van de Ven, Two loop quantum gravity, Nucl. Phys. B 378 (1992) 309 [INSPIRE].ADSMathSciNetGoogle Scholar
  29. [29]
    A. Codello, R. Percacci and C. Rahmede, Ultraviolet properties of f(R)-gravity, Int. J. Mod. Phys. A 23 (2008) 143 [arXiv:0705.1769] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  30. [30]
    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
  31. [31]
    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
  32. [32]
    D. Benedetti and F. Caravelli, The Local potential approximation in quantum gravity, JHEP 06 (2012) 017 [Erratum ibid. 1210 (2012) 157] [arXiv:1204.3541] [INSPIRE].
  33. [33]
    J.A. Dietz and T.R. Morris, Asymptotic safety in the f(R) approximation, JHEP 01 (2013) 108 [arXiv:1211.0955] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    K. Falls, D.F. Litim, K. Nikolakopoulos and C. Rahmede, A bootstrap towards asymptotic safety, arXiv:1301.4191 [INSPIRE].
  35. [35]
    K. Falls, D.F. Litim, K. Nikolakopoulos and C. Rahmede, Further evidence for asymptotic safety of quantum gravity, arXiv:1410.4815 [INSPIRE].
  36. [36]
    M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev. D 57 (1998) 971 [hep-th/9605030] [INSPIRE].ADSMathSciNetGoogle Scholar
  37. [37]
    D. Dou and R. Percacci, The running gravitational couplings, Class. Quant. Grav. 15 (1998) 3449 [hep-th/9707239] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    M. Reuter and F. Saueressig, Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation, Phys. Rev. D 65 (2002) 065016 [hep-th/0110054] [INSPIRE].ADSMathSciNetGoogle Scholar
  39. [39]
    O. Lauscher and M. Reuter, Flow equation of quantum Einstein gravity in a higher derivative truncation, Phys. Rev. D 66 (2002) 025026 [hep-th/0205062] [INSPIRE].ADSMathSciNetGoogle Scholar
  40. [40]
    D.F. Litim, Fixed points of quantum gravity, Phys. Rev. Lett. 92 (2004) 201301 [hep-th/0312114] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    P. Fischer and D.F. Litim, Fixed points of quantum gravity in extra dimensions, Phys. Lett. B 638 (2006) 497 [hep-th/0602203] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    A. Eichhorn, H. Gies and M.M. Scherer, Asymptotically free scalar curvature-ghost coupling in Quantum Einstein Gravity, Phys. Rev. D 80 (2009) 104003 [arXiv:0907.1828] [INSPIRE].ADSGoogle Scholar
  43. [43]
    A. Codello and R. Percacci, Fixed points of higher derivative gravity, Phys. Rev. Lett. 97 (2006) 221301 [hep-th/0607128] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    D. Benedetti, P.F. Machado and F. Saueressig, Asymptotic safety in higher-derivative gravity, Mod. Phys. Lett. A 24 (2009) 2233 [arXiv:0901.2984] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  45. [45]
    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
  46. [46]
    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
  47. [47]
    A. Nink and M. Reuter, On the physical mechanism underlying Asymptotic Safety, JHEP 01 (2013) 062 [arXiv:1208.0031] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    S. Rechenberger and F. Saueressig, A functional renormalization group equation for foliated spacetimes, JHEP 03 (2013) 010 [arXiv:1212.5114] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  49. [49]
    A. Eichhorn, Faddeev-Popov ghosts in quantum gravity beyond perturbation theory, Phys. Rev. D 87 (2013) 124016 [arXiv:1301.0632] [INSPIRE].ADSGoogle Scholar
  50. [50]
    M. Demmel, F. Saueressig and O. Zanusso, RG flows of Quantum Einstein Gravity in the linear-geometric approximation, arXiv:1412.7207 [INSPIRE].
  51. [51]
    K. Falls, On the renormalisation of Newtons constant, arXiv:1501.05331 [INSPIRE].
  52. [52]
    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].ADSMathSciNetGoogle Scholar
  53. [53]
    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
  54. [54]
    E. Manrique, M. Reuter and F. Saueressig, Matter Induced Bimetric Actions for Gravity, Annals Phys. 326 (2011) 440 [arXiv:1003.5129] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    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
  56. [56]
    I. Donkin and J.M. Pawlowski, The phase diagram of quantum gravity from diffeomorphism-invariant RG-flows, arXiv:1203.4207 [INSPIRE].
  57. [57]
    N. Christiansen, D.F. Litim, J.M. Pawlowski and A. Rodigast, Fixed points and infrared completion of quantum gravity, Phys. Lett. B 728 (2014) 114 [arXiv:1209.4038] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  58. [58]
    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
  59. [59]
    N. Christiansen, B. Knorr, J.M. Pawlowski and A. Rodigast, Global Flows in Quantum Gravity, arXiv:1403.1232 [INSPIRE].
  60. [60]
    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
  61. [61]
    A. Eichhorn, On unimodular quantum gravity, Class. Quant. Grav. 30 (2013) 115016 [arXiv:1301.0879] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  62. [62]
    C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301 (1993) 90 [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    T.R. Morris, The exact renormalization group and approximate solutions, Int. J. Mod. Phys. A 9 (1994) 2411 [hep-ph/9308265] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  64. [64]
    R. Percacci and G.P. Vacca, Search of scaling solutions in scalar-tensor gravity, arXiv:1501.00888 [INSPIRE].
  65. [65]
    D.F. Litim, Optimized renormalization group flows, Phys. Rev. D 64 (2001) 105007 [hep-th/0103195] [INSPIRE].ADSGoogle Scholar
  66. [66]
    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
  67. [67]
    K. Aoki, Introduction to the nonperturbative renormalization group and its recent applications, Int. J. Mod. Phys. B 14 (2000) 1249 [INSPIRE].ADSMATHGoogle Scholar
  68. [68]
    J.M. Pawlowski, Aspects of the functional renormalisation group, Annals Phys. 322 (2007) 2831 [hep-th/0512261] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  69. [69]
    H. Gies, Introduction to the functional RG and applications to gauge theories, Lect. Notes Phys. 852 (2012) 287 [hep-ph/0611146] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  70. [70]
    J. Braun, Fermion Interactions and Universal Behavior in Strongly Interacting Theories, J. Phys. G 39 (2012) 033001 [arXiv:1108.4449] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    M. Niedermaier and M. Reuter, The Asymptotic Safety Scenario in Quantum Gravity, Living Rev. Rel. 9 (2006) 5.CrossRefMATHGoogle Scholar
  72. [72]
    M. Niedermaier, The Asymptotic safety scenario in quantum gravity: An Introduction, Class. Quant. Grav. 24 (2007) R171 [gr-qc/0610018] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  73. [73]
    R. Percacci, Asymptotic Safety, arXiv:0709.3851 [INSPIRE].
  74. [74]
    D.F. Litim, Fixed Points of Quantum Gravity and the Renormalisation Group, PoS(QG-Ph)024 [arXiv:0810.3675] [INSPIRE].
  75. [75]
    D.F. Litim, Renormalisation group and the Planck scale, Phil. Trans. Roy. Soc. Lond. A 369 (2011) 2759 [arXiv:1102.4624] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  76. [76]
    R. Percacci, A Short introduction to asymptotic safety, arXiv:1110.6389 [INSPIRE].
  77. [77]
    M. Reuter and F. Saueressig, Quantum Einstein Gravity, New J. Phys. 14 (2012) 055022 [arXiv:1202.2274] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  78. [78]
    M. Reuter and F. Saueressig, Asymptotic Safety, Fractals and Cosmology, Lect. Notes Phys. 863 (2013) 185 [arXiv:1205.5431] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  79. [79]
    S. Nagy, Lectures on renormalization and asymptotic safety, Annals Phys. 350 (2014) 310 [arXiv:1211.4151] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  80. [80]
    A. Ashtekar, M. Reuter and C. Rovelli, From General Relativity to Quantum Gravity, arXiv:1408.4336 [INSPIRE].
  81. [81]
    L.F. Abbott, The Background Field Method Beyond One Loop, Nucl. Phys. B 185 (1981) 189 [INSPIRE].ADSCrossRefGoogle Scholar
  82. [82]
    A. Nink, Field Parametrization Dependence in Asymptotically Safe Quantum Gravity, Phys. Rev. D 91 (2015) 044030 [arXiv:1410.7816] [INSPIRE].ADSMathSciNetGoogle Scholar
  83. [83]
    A. Codello and G. D’Odorico, Scaling and Renormalization in two dimensional Quantum Gravity, arXiv:1412.6837 [INSPIRE].
  84. [84]
    E. Alvarez, D. Blas, J. Garriga and E. Verdaguer, Transverse Fierz-Pauli symmetry, Nucl. Phys. B 756 (2006) 148 [hep-th/0606019] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  85. [85]
    D. Benedetti, K. Groh, P.F. Machado and F. Saueressig, The Universal RG Machine, JHEP 06 (2011) 079 [arXiv:1012.3081] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  86. [86]
    E. Alvarez, A.F. Faedo and J.J. Lopez-Villarejo, Ultraviolet behavior of transverse gravity, JHEP 10 (2008) 023 [arXiv:0807.1293] [INSPIRE].ADSCrossRefGoogle Scholar
  87. [87]
    D.F. Litim and J.M. Pawlowski, Completeness and consistency of renormalisation group flows, Phys. Rev. D 66 (2002) 025030 [hep-th/0202188] [INSPIRE].ADSGoogle Scholar
  88. [88]
    H. Gies, Running coupling in Yang-Mills theory: A flow equation study, Phys. Rev. D 66 (2002) 025006 [hep-th/0202207] [INSPIRE].ADSMathSciNetGoogle Scholar
  89. [89]
    M.A. Rubin and C.R. Ordonez, Symmetric Tensor Eigen Spectrum of the Laplacian on n Spheres, J. Math. Phys. 26 (1985) 65 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  90. [90]
    I.D. Saltas, UV structure of quantum unimodular gravity, Phys. Rev. D 90 (2014) 124052 [arXiv:1410.6163] [INSPIRE].ADSGoogle Scholar
  91. [91]
    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
  92. [92]
    A. Eichhorn and H. Gies, Light fermions in quantum gravity, New J. Phys. 13 (2011) 125012 [arXiv:1104.5366] [INSPIRE].ADSCrossRefGoogle Scholar
  93. [93]
    A. Eichhorn, Quantum-gravity-induced matter self-interactions in the asymptotic-safety scenario, Phys. Rev. D 86 (2012) 105021 [arXiv:1204.0965] [INSPIRE].ADSGoogle Scholar
  94. [94]
    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
  95. [95]
    P. Donà, A. Eichhorn and R. Percacci, Consistency of matter models with asymptotically safe quantum gravity, arXiv:1410.4411 [INSPIRE].
  96. [96]
    P. Donà and R. Percacci, Functional renormalization with fermions and tetrads, Phys. Rev. D 87 (2013) 045002 [arXiv:1209.3649] [INSPIRE].ADSGoogle Scholar
  97. [97]
    F. Synatschke, G. Bergner, H. Gies and A. Wipf, Flow Equation for Supersymmetric Quantum Mechanics, JHEP 03 (2009) 028 [arXiv:0809.4396] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  98. [98]
    M. Shaposhnikov and C. Wetterich, Asymptotic safety of gravity and the Higgs boson mass, Phys. Lett. B 683 (2010) 196 [arXiv:0912.0208] [INSPIRE].ADSCrossRefGoogle Scholar

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© The Author(s) 2015

Authors and Affiliations

  1. 1.Blackett Laboratory, Imperial CollegeLondonUnited Kingdom

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