Nonlocal quantum effective actions in Weyl-Flat spacetimes

  • Teresa Bautista
  • André Benevides
  • Atish Dabholkar
Open Access
Regular Article - Theoretical Physics


Virtual massless particles in quantum loops lead to nonlocal effects which can have interesting consequences, for example, for primordial magnetogenesis in cosmology or for computing finite N corrections in holography. We describe how the quantum effective actions summarizing these effects can be computed efficiently for Weyl-flat metrics by integrating the Weyl anomaly or, equivalently, the local renormalization group equation. This method relies only on the local Schwinger-DeWitt expansion of the heat kernel and allows for a re-summation of the anomalous leading large logarithms of the scale factor, log a(x), in situations where the Weyl factor changes by several e-foldings. As an illustration, we obtain the quantum effective action for the Yang-Mills field coupled to massless matter, and the self-interacting massless scalar field. Our action reduces to the nonlocal action obtained using the Barvinsky-Vilkovisky covariant perturbation theory in the regime R2 ≪ ∇2R for a typical curvature scale R, but has a greater range of validity effectively re-summing the covariant perturbation theory to all orders in curvatures. In particular, it is applicable also in the opposite regime R2 ≫ ∇2R, which is often of interest in cosmology.


Anomalies in Field and String Theories Effective Field Theories Renormalization Group Models of Quantum Gravity 


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.


  1. [1]
    J.S. Schwinger, The Theory of quantized fields. 1., Phys. Rev. 82 (1951) 914 [INSPIRE].
  2. [2]
    B.S. DeWitt, Dynamical theory of groups and fields, North Carolina University, Chapel Hill U.S.A. (1963).Google Scholar
  3. [3]
    A.O. Barvinsky and G.A. Vilkovisky, The generalized Schwinger-Dewitt technique and the unique effective action in quantum gravity, Phys. Lett. B 131 (1983) 313 [INSPIRE].
  4. [4]
    A.O. Barvinsky and G.A. Vilkovisky, The Generalized Schwinger-Dewitt Technique in Gauge Theories and Quantum Gravity, Phys. Rept. 119 (1985) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    A.O. Barvinsky and G.A. Vilkovisky, The Effective Action in Quantum Field Theory: Two Loop Approximation, Quantum Field Theory and Quantum Statistics 1 (1988) 245.Google Scholar
  6. [6]
    A.O. Barvinsky, Yu. V. Gusev, G.A. Vilkovisky and V.V. Zhytnikov, The Basis of nonlocal curvature invariants in quantum gravity theory. (Third order.), J. Math. Phys. 35 (1994) 3525 [gr-qc/9404061] [INSPIRE].
  7. [7]
    A.O. Barvinsky, Yu. V. Gusev, G.A. Vilkovisky and V.V. Zhytnikov, The One loop effective action and trace anomaly in four-dimensions, Nucl. Phys. B 439 (1995) 561 [hep-th/9404187] [INSPIRE].
  8. [8]
    A.O. Barvinsky, A.G. Mirzabekian and V.V. Zhytnikov, Conformal decomposition of the effective action and covariant curvature expansion, in Proceedings of 6th Seminar on Quantum gravity, Moscow Russia (1995) [gr-qc/9510037] [INSPIRE].
  9. [9]
    R.J. Riegert, A Nonlocal Action for the Trace Anomaly, Phys. Lett. 134B (1984) 56 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    S. Deser, Conformal anomalies: Recent progress, Helv. Phys. Acta 69 (1996) 570 [hep-th/9609138] [INSPIRE].zbMATHGoogle Scholar
  11. [11]
    J. Erdmenger and H. Osborn, Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions, Nucl. Phys. B 483 (1997) 431 [hep-th/9605009] [INSPIRE].
  12. [12]
    J. Erdmenger, Conformally covariant differential operators: Properties and applications, Class. Quant. Grav. 14 (1997) 2061 [hep-th/9704108] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    S. Deser, Conformal anomalies revisited: Closed form effective actions in D ≥ 4, Nucl. Phys. Proc. Suppl. 88 (2000) 204 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    A. Dabholkar, J. Gomes and S. Murthy, Localization & Exact Holography, JHEP 04 (2013) 062 [arXiv:1111.1161] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    A. Dabholkar, J. Gomes and S. Murthy, Nonperturbative black hole entropy and Kloosterman sums, JHEP 03 (2015) 074 [arXiv:1404.0033] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    A. Sen, Logarithmic Corrections to N = 2 Black Hole Entropy: An Infrared Window into the Microstates, Gen. Rel. Grav. 44 (2012) 1207 [arXiv:1108.3842] [INSPIRE].
  17. [17]
    S. Banerjee, R.K. Gupta, I. Mandal and A. Sen, Logarithmic Corrections to N = 4 and N = 8 Black Hole Entropy: A One Loop Test of Quantum Gravity, JHEP 11(2011) 143 [arXiv:1106.0080] [INSPIRE].
  18. [18]
    M.S. Turner and L.M. Widrow, Inflation Produced, Large Scale Magnetic Fields, Phys. Rev. D 37 (1988) 2743 [INSPIRE].
  19. [19]
    B. Ratra, Cosmologicalseedmagnetic field from inflation, Astrophys. J. 391 (1992) L1 [INSPIRE].
  20. [20]
    V. Demozzi, V. Mukhanov and H. Rubinstein, Magnetic fields from inflation?, JCAP 08 (2009) 025 [arXiv:0907.1030] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    B.K. El-Menoufi, Inflationary magnetogenesis and non-local actions: The conformal anomaly, JCAP 02 (2016) 055 [arXiv:1511.02876] [INSPIRE].CrossRefGoogle Scholar
  22. [22]
    A. Benevides, A. Dabholkar and T. Kobayashi, Weyl Anomalies and Primordial Magnetogenesis, to appear (2018).Google Scholar
  23. [23]
    A. Dolgov, Breaking of conformal invariance and electromagnetic field generation in the universe, Phys. Rev. D 48 (1993) 2499 [hep-ph/9301280] [INSPIRE].
  24. [24]
    A. Dabholkar, Quantum Weyl Invariance and Cosmology, Phys. Lett. B 760 (2016) 31 [arXiv:1511.05342] [INSPIRE].
  25. [25]
    T. Bautista and A. Dabholkar, Quantum Cosmology Near Two Dimensions, Phys. Rev. D 94 (2016) 044017 [arXiv:1511.07450] [INSPIRE].
  26. [26]
    T. Bautista, A. Benevides, A. Dabholkar and A. Goel, Quantum Cosmology in Four Dimensions, arXiv:1512.03275 [INSPIRE].
  27. [27]
    E. Mottola and R. Vaulin, Macroscopic Effects of the Quantum Trace Anomaly, Phys. Rev. D 74 (2006) 064004 [gr-qc/0604051] [INSPIRE].
  28. [28]
    S. Deser and R.P. Woodard, Nonlocal Cosmology, Phys. Rev. Lett. 99 (2007) 111301 [arXiv:0706.2151] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    S. Nojiri and S.D. Odintsov, Modified non-local-F (R) gravity as the key for the inflation and dark energy, Phys. Lett. B 659 (2008) 821 [arXiv:0708.0924] [INSPIRE].
  30. [30]
    S. Jhingan, S. Nojiri, S.D. Odintsov, M. Sami, I. Thongkool and S. Zerbini, Phantom and non-phantom dark energy: The Cosmological relevance of non-locally corrected gravity, Phys. Lett. B 663 (2008) 424 [arXiv:0803.2613] [INSPIRE].
  31. [31]
    S. Park and S. Dodelson, Structure formation in a nonlocally modified gravity model, Phys. Rev. D 87 (2013) 024003 [arXiv:1209.0836] [INSPIRE].
  32. [32]
    R.P. Woodard, Nonlocal Models of Cosmic Acceleration, Found. Phys. 44 (2014) 213 [arXiv:1401.0254] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    J.F. Donoghue and B.K. El-Menoufi, Nonlocal quantum effects in cosmology: Quantum memory, nonlocal FLRW equations and singularity avoidance, Phys. Rev. D 89 (2014) 104062 [arXiv:1402.3252] [INSPIRE].
  34. [34]
    H. Godazgar, K.A. Meissner and H. Nicolai, Conformal anomalies and the Einstein Field Equations, JHEP 04 (2017) 165 [arXiv:1612.01296] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    J.F. Donoghue and B.K. El-Menoufi, QED trace anomaly, non-local Lagrangians and quantum Equivalence Principle violations, JHEP 05 (2015) 118 [arXiv:1503.06099] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    J.F. Donoghue and B.K. El-Menoufi, Covariant non-local action for massless QED and the curvature expansion, JHEP 10 (2015) 044 [arXiv:1507.06321] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    A. Benevides and A. Dabholkar, Weyl Anomalies and Quantum Effective Actions in Cosmological Spacetimes, to appear (2017).Google Scholar
  38. [38]
    L.F. Abbott, Introduction to the Background Field Method, Acta Phys. Polon. B 13 (1982) 33 [INSPIRE].MathSciNetGoogle Scholar
  39. [39]
    R. Percacci, 100 Years of General Relativity. Vol. 3: An Introduction to Covariant Quantum Gravity and Asymptotic Safety, World Scientific, Singapore (2017).Google Scholar
  40. [40]
    D.V. Vassilevich, Heat kernel expansion: Users manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].
  41. [41]
    S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    A. Codello, G. D’Odorico, C. Pagani and R. Percacci, The Renormalization Group and Weyl-invariance, Class. Quant. Grav. 30 (2013) 115015 [arXiv:1210.3284] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    D.M. Capper and M.J. Duff, The one loop neutrino contribution to the graviton propagator, Nucl. Phys. B 82 (1974) 147 [INSPIRE].
  44. [44]
    S. Deser, M.J. Duff and C.J. Isham, Nonlocal Conformal Anomalies, Nucl. Phys. B 111 (1976) 45 [INSPIRE].
  45. [45]
    M.J. Duff, Twenty years of the Weyl anomaly, Class. Quant. Grav. 11 (1994) 1387 [hep-th/9308075] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    I.T. Drummond and G.M. Shore, Conformal Anomalies for Interacting Scalar Fields in Curved Space-Time, Phys. Rev. D 19 (1979) 1134 [INSPIRE].
  47. [47]
    H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [INSPIRE].
  48. [48]
    I. Jack and H. Osborn, Analogs for the c Theorem for Four-dimensional Renormalizable Field Theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE].
  49. [49]
    G.M. Shore, A Local Renormalization Group Equation, Diffeomorphisms and Conformal Invariance in σ Models, Nucl. Phys. B 286 (1987) 349 [INSPIRE].
  50. [50]
    G.M. Shore, New methods for the renormalization of composite operator Green functions, Nucl. Phys. B 362 (1991) 85 [INSPIRE].
  51. [51]
    H. Osborn, Renormalization and Composite Operators in Nonlinear σ Models, Nucl. Phys. B 294 (1987) 595 [INSPIRE].
  52. [52]
    H. Osborn, Derivation of a Four-dimensional c Theorem, Phys. Lett. B 222 (1989) 97 [INSPIRE].
  53. [53]
    P.B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, CRC Press, Boca Raton U.S.A.(1984).Google Scholar
  54. [54]
    R.T. Seeley, Complex powers of an elliptic operator, Proc. Symp. Pure Math. 10 (1967) 288.MathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    R. Seeley, The resolvent of an elliptic boundary problem, Am. J. Math. 91 (1969) 889.MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    J. Hadamard, Lectures on Cauchys problem in linear partial differential equations, Courier Corporation, Chelmsford U.S.A. (2014).Google Scholar
  57. [57]
    S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Can. J. Math. 1 (1949) 242 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    S. Minakshisundaram, Eigenfunctions on Riemannian manifolds, J. Indian Math. Soc. 17 (1953) 158.Google Scholar
  59. [59]
    B.S. DeWitt, Dynamical theory of groups and fields, Conf. Proc. C 630701 (1964) 585 [INSPIRE].
  60. [60]
    B.S. DeWitt, Quantum Theory of Gravity. 1. The Canonical Theory, Phys. Rev. 160 (1967) 1113 [INSPIRE].
  61. [61]
    B.S. DeWitt, Quantum Theory of Gravity. 2. The Manifestly Covariant Theory, Phys. Rev. 162 (1967) 1195 [INSPIRE].
  62. [62]
    B.S. DeWitt, Quantum Theory of Gravity. 3. Applications of the Covariant Theory, Phys. Rev. 162 (1967) 1239 [INSPIRE].
  63. [63]
    G. Gibbons, in General Relativity: An Einstein Centenary Survey, S.W. Hawking and W. Israel eds., Cambridge University Press, Cambridge U.K. (1979).Google Scholar
  64. [64]
    A. Barvinsky, Heat kernel expansion in the background field formalism, Scholarpedia 10 (2015) 31644.ADSCrossRefGoogle Scholar
  65. [65]
    L. Bonora, P. Cotta-Ramusino and C. Reina, Conformal Anomaly and Cohomology, Phys. Lett. 126B (1983) 305 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  66. [66]
    P.O. Mazur and E. Mottola, Weyl cohomology and the effective action for conformal anomalies, Phys. Rev. D 64 (2001) 104022 [hep-th/0106151] [INSPIRE].
  67. [67]
    A.O. Barvinsky and V.F. Mukhanov, New nonlocal effective action, Phys. Rev. D 66 (2002) 065007 [hep-th/0203132] [INSPIRE].
  68. [68]
    A.O. Barvinsky, Yu. V. Gusev, V.F. Mukhanov and D.V. Nesterov, Nonperturbative late time asymptotics for heat kernel in gravity theory, Phys. Rev. D 68 (2003) 105003 [hep-th/0306052] [INSPIRE].ADSMathSciNetGoogle Scholar
  69. [69]
    A.G. Mirzabekian, G.A. Vilkovisky and V.V. Zhytnikov, Partial summation of the nonlocal expansion for the gravitational effective action in four-dimensions, Phys. Lett. B 369 (1996) 215 [hep-th/9510205] [INSPIRE].
  70. [70]
    E.S. Fradkin and G.A. Vilkovisky, Conformal Invariance and Asymptotic Freedom in Quantum Gravity, Phys. Lett. B 77 (1978) 262 [INSPIRE].
  71. [71]
    S. Paneitz, A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary), SIGMA 4 (2008) 036 [arXiv:0803.4331].MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Teresa Bautista
    • 1
  • André Benevides
    • 2
    • 3
  • Atish Dabholkar
    • 3
    • 4
    • 5
  1. 1.Max Planck Institute for Gravitational Physics (Albert Einstein Institute)PotsdamGermany
  2. 2.Scuola Internazionale Superiore di Studi Avanzati (SISSA)TriesteItaly
  3. 3.Abdus Salam International Centre for Theoretical PhysicsTriesteItaly
  4. 4.Sorbonne Universités, UPMC Univ Paris 06, UMR 7589, LPTHEParisFrance
  5. 5.CNRS, UMR 7589, LPTHEParisFrance

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