The local potential approximation in quantum gravity

Article

Abstract

Within the context of the functional renormalization group flow of gravity, we suggest that a generic f (R) ansatz (i.e. not truncated to any specific form, polynomial or not) for the effective action plays a role analogous to the local potential approximation (LPA) in scalar field theory. In the same spirit of the LPA, we derive and study an ordinary differential equation for f (R) to be satisfied by a fixed point of the renormalization group flow. As a first step in trying to assess the existence of global solutions (i.e. true fixed point) for such equation, we investigate here the properties of its solutions by a comparison of various series expansions and numerical integrations. In particular, we study the analyticity conditions required because of the presence of fixed singularities in the equation, and we develop an expansion of the solutions for large R up to order N = 29. Studying the convergence of the fixed points of the truncated solutions with respect to N, we find a characteristic pattern for the location of the fixed points in the complex plane, with one point stemming out for its stability. Finally, we establish that if a non-Gaussian fixed point exists within the full f (R) approximation, it corresponds to an R 2 theory.

Keywords

Models of Quantum Gravity Renormalization Group 

References

  1. [1]
    S. Weinberg, Ultraviolet Divergences in Quantum Theories of Gravitation, in General Relativity S.W. Hawking and W. Israel eds., Cambridge University Press (1979).Google Scholar
  2. [2]
    M. Niedermaier and M. Reuter, The Asymptotic Safety Scenario in Quantum Gravity, Living Rev. Rel. 9 (2006) 5.Google Scholar
  3. [3]
    R. Percacci, Asymptotic Safety, arXiv:0709.3851 [INSPIRE].
  4. [4]
    D.F. Litim, Fixed Points of Quantum Gravity and the Renormalisation Group, arXiv:0810.3675 [INSPIRE].
  5. [5]
    M. Reuter and F. Saueressig, Quantum Einstein Gravity, New J. Phys. 14 (2012) 055022 [arXiv:1202.2274] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301 (1993) 90 [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    T.R. Morris, Elements of the continuous renormalization group, Prog. Theor. Phys. Suppl. 131 (1998) 395 [hep-th/9802039] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    C. Bagnuls and C. Bervillier, Exact renormalization group equations. An Introductory review, Phys. Rept. 348 (2001) 91 [hep-th/0002034] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  9. [9]
    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].MathSciNetADSCrossRefMATHGoogle Scholar
  10. [10]
    J.M. Pawlowski, Aspects of the functional renormalisation group, Annals Phys. 322 (2007) 2831 [hep-th/0512261] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  11. [11]
    H. Gies, Introduction to the functional RG and applications to gauge theories, hep-ph/0611146 [INSPIRE].
  12. [12]
    B. Delamotte, An introduction to the nonperturbative renormalization group, cond-mat/0702365 [INSPIRE].
  13. [13]
    D.F. Litim and D. Zappala, Ising exponents from the functional renormalisation group, Phys. Rev. D 83 (2011) 085009 [arXiv:1009.1948] [INSPIRE].ADSGoogle Scholar
  14. [14]
    T.R. Morris, On truncations of the exact renormalization group, Phys. Lett. B 334 (1994) 355 [hep-th/9405190] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    L. Canet, B. Delamotte, D. Mouhanna and J. Vidal, Nonperturbative renormalization group approach to the Ising model: A Derivative expansion at order partial**4, Phys. Rev. B 68 (2003) 064421 [hep-th/0302227] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev. D 57 (1998) 971 [hep-th/9605030] [INSPIRE].MathSciNetADSGoogle Scholar
  17. [17]
    D. Dou and R. Percacci, The running gravitational couplings, Class. Quant. Grav. 15 (1998) 3449 [hep-th/9707239] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  18. [18]
    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].MathSciNetADSGoogle Scholar
  19. [19]
    D.F. Litim, Fixed points of quantum gravity, Phys. Rev. Lett. 92 (2004) 201301 [hep-th/0312114] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    L. Granda and S. Odintsov, Effective average action and nonperturbative renormalization group equation in higher derivative quantum gravity, Grav. Cosmol. 4 (1998) 85 [gr-qc/9801026] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  21. [21]
    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].MathSciNetADSGoogle Scholar
  22. [22]
    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].ADSCrossRefGoogle Scholar
  23. [23]
    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].MathSciNetADSGoogle Scholar
  24. [24]
    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].MathSciNetADSCrossRefMATHGoogle Scholar
  25. [25]
    A. Bonanno, A. Contillo and R. Percacci, Inflationary solutions in asymptotically safe f(R) theories, Class. Quant. Grav. 28 (2011) 145026 [arXiv:1006.0192] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    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].ADSCrossRefGoogle Scholar
  27. [27]
    R. Percacci and D. Perini, Asymptotic safety of gravity coupled to matter, Phys. Rev. D 68 (2003) 044018 [hep-th/0304222] [INSPIRE].ADSGoogle Scholar
  28. [28]
    D. Benedetti, P.F. Machado and F. Saueressig, Taming perturbative divergences in asymptotically safe gravity, Nucl. Phys. B 824 (2010) 168 [arXiv:0902.4630] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    K. Groh and F. Saueressig, Ghost wave-function renormalization in Asymptotically Safe Quantum Gravity, J. Phys. A 43 (2010) 365403 [arXiv:1001.5032] [INSPIRE].MathSciNetGoogle Scholar
  30. [30]
    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
  31. [31]
    E. Manrique and M. Reuter, Bimetric Truncations for Quantum Einstein Gravity and Asymptotic Safety, Annals Phys. 325 (2010) 785 [arXiv:0907.2617] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  32. [32]
    E. Manrique, M. Reuter and F. Saueressig, Bimetric Renormalization Group Flows in Quantum Einstein Gravity, Annals Phys. 326 (2011) 463 [arXiv:1006.0099] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  33. [33]
    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
  34. [34]
    T.P. Sotiriou and V. Faraoni, f(R) Theories Of Gravity, Rev. Mod. Phys. 82 (2010) 451 [arXiv:0805.1726] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  35. [35]
    A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Rel. 13 (2010) 3 [arXiv:1002.4928] [INSPIRE].Google Scholar
  36. [36]
    A. Hasenfratz and P. Hasenfratz, Renormalization Group Study of Scalar Field Theories, Nucl. Phys. B 270 (1986) 687 [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    G. Felder, Renormalization Group in the Local Potential Approximation, Commun. Math. Phys. 111 (1987) 101.MathSciNetADSCrossRefMATHGoogle Scholar
  38. [38]
    G. ’t Hooft and M. Veltman, One loop divergencies in the theory of gravitation, Annales Poincaré Phys. Theor. A 20 (1974) 69 [INSPIRE].MathSciNetADSGoogle Scholar
  39. [39]
    M.H. Goroff and A. Sagnotti, The Ultraviolet Behavior of Einstein Gravity, Nucl. Phys. B 266 (1986) 709 [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    D. Benedetti, Asymptotic safety goes on shell, New J. Phys. 14 (2012) 015005 [arXiv:1107.3110] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    J. Comellas and A. Travesset, O (N) models within the local potential approximation, Nucl. Phys. B 498 (1997) 539 [hep-th/9701028] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    A. Bonanno, An effective action for asymptotically safe gravity, Phys. Rev. D 85 (2012) 081503 [arXiv:1203.1962] [INSPIRE].ADSGoogle Scholar
  43. [43]
    M. Hindmarsh and I.D. Saltas, f(R) Gravity from the renormalisation group, arXiv:1203.3957 [INSPIRE].
  44. [44]
    S. Domazet and H. Stefancic, Renormalization group scale-setting from the action: A Road to modified gravity theories, arXiv:1204.1483 [INSPIRE].
  45. [45]
    M.A. Rubin and C.R. Ordonez, Eigenvalues and degeneracies for n-dimensional tensor spherical harmonics, J. Math. Phys. 25 (1975) 2888.MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    D.F. Litim, Optimized renormalization group flows, Phys. Rev. D 64 (2001) 105007 [hep-th/0103195] [INSPIRE].ADSGoogle Scholar
  47. [47]
    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
  48. [48]
    W.-S. Dai and M. Xie, The number of eigenstates: counting function and heat kernel, JHEP 02 (2009) 033 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    G. Cognola, E. Elizalde, S. Nojiri, S.D. Odintsov and S. Zerbini, One-loop f(R) gravity in de Sitter universe, JCAP 02 (2005) 010 [hep-th/0501096] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Max Planck Institute for Gravitational Physics (Albert Einstein Institute)GolmGermany
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  3. 3.University of WaterlooWaterlooCanada

Personalised recommendations