Abstract
The effective action in quantum general relativity is strongly dependent on the gauge-fixing and parametrization of the quantum metric. As a consequence, in the effective approach to quantum gravity, there is no possibility to introduce the renormalization-group framework in a consistent way. On the other hand, the version of effective action proposed by Vilkovisky and DeWitt does not depend on the gauge-fixing and parametrization off- shell, opening the way to explore the running of the cosmological and Newton constants as well as the coefficients of the higher-derivative terms of the total action. We argue that in the effective framework the one-loop beta functions for the zero-, two- and four-derivative terms can be regarded as exact, that means, free from corrections coming from the higher loops. In this perspective, the running describes the renormalization group flow between the present-day Hubble scale in the IR and the Planck scale in the UV.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D 50 (1994) 3874 [gr-qc/9405057] [INSPIRE].
G.A. Vilkovisky, The Unique Effective Action in Quantum Field Theory, Nucl. Phys. B 234 (1984) 125 [INSPIRE].
G.A. Vilkovisky, The Gospel according to DeWitt, in Quantum Theory of Gravity, S.M. Christensen ed., Adam Hilger, Bristol, U.K. (1984).
B.S. DeWitt, The effective action, in Quantum Field Theory and Quantum Statistics, essays in honor of the sixtieth birthday of E.S. Fradkin, Vol. 1: Quantum Statistics and methods of Field Theory, C.J. Isham, I.A. Batalin and G.A. Vilkovisky eds., Hilger, Bristol, U.K., (1987).
B.S. DeWitt, The effective action, in Architecture of fundamental interactions at short distances, P. Ramond and R. Stora eds., North-Holland, Amsterdam, The Netherlands (1987).
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].
A.O. Barvinsky and G.A. Vilkovisky, The Generalized Schwinger-DeWitt Technique in Gauge Theories and Quantum Gravity, Phys. Rept. 119 (1985) 1 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, On the New Definition of Off-shell Effective Action, Nucl. Phys. B 234 (1984) 509 [INSPIRE].
G. Kunstatter, Vilkovisky’s unique effective action: an introduction and explicit calculation, in proceedings of NATO Advanced Research Workshop on Superfield Theories, H.C. Lee, V. Elias, G. Kunstatter, R.B. Mann and K.S. Viswanathan eds., NATO ASI Series B, Vol. 160, Plenum Press, New York, U.S.A. (1987).
A. Rebhan, The Vilkovisky-DeWitt Effective Action and its application to Yang-Mills Theories, Nucl. Phys. B 288 (1987) 832 [INSPIRE].
A. Rebhan, Feynman Rules and S Matrix Equivalence of the Vilkovisky-de Witt Effective Action, Nucl. Phys. B 298 (1988) 726 [INSPIRE].
P. Ellicott and D.J. Toms, On the New Effective Action in Quantum Field Theory, Nucl. Phys. B 312 (1989) 700 [INSPIRE].
S.R. Huggins, G. Kunstatter, H.P. Leivo and D.J. Toms, The Vilkovisky-de Witt Effective Action for Quantum Gravity, Nucl. Phys. B 301 (1988) 627 [INSPIRE].
B.L. Giacchini, T. de Paula Netto and I.L. Shapiro, On the Vilkovisky unique effective action in quantum gravity, arXiv:2006.04217 [INSPIRE].
A.T. Banin and I.L. Shapiro, Gauge dependence and new kind of two-dimensional gravity theory with trivial quantum corrections, Phys. Lett. B 327 (1994) 17 [INSPIRE].
T.R. Taylor and G. Veneziano, Quantum Gravity at Large Distances and the Cosmological Constant, Nucl. Phys. B 345 (1990) 210 [INSPIRE].
B. Voronov and I. Tyutin, Formulation of gauge theories of general form. II. Gauge invariant renormalizability and renormalization structure, Theor. Math. Phys. 52 (1982) 628 [INSPIRE].
B.L. Voronov and I.V. Tyutin, On renormalization of R2 gravitation (in Russian), Yad. Fiz. 39 (1984) 998 [INSPIRE].
B.L. Voronov, P.M. Lavrov and I.V. Tyutin, Canonical transformations and the gauge dependence in general gauge theories (in Russian), Yad. Fiz. 36 (1982) 498 [INSPIRE].
J. Honerkamp, Chiral multiloops, Nucl. Phys. B 36 (1972) 130 [INSPIRE].
B.S. DeWitt, Dynamical theory of groups and fields, Gordon and Breach, New York, U.S.A. (1965).
J.L. Synge, Relativity: the general theory, North-Holland, Amsterdam, The Netherlands (1960).
B.S. DeWitt, Quantum Theory of Gravity. 1. The Canonical Theory, Phys. Rev. 160 (1967) 1113 [INSPIRE].
S.D. Odintsov, Does the Vilkovisky-De Witt effective action in quantum gravity depend on the configuration space metric?, Phys. Lett. B 262 (1991) 394 [INSPIRE].
J.F. Barbero G. and J. Pérez-Mercader, Superspace dependence of the Vilkovisky-DeWitt effective action for quantum gravity, Phys. Rev. D 48 (1993) 3663 [INSPIRE].
C.P. Burgess, Quantum gravity in everyday life: General relativity as an effective field theory, Living Rev. Rel. 7 (2004) 5 [gr-qc/0311082] [INSPIRE].
G. ’t Hooft and M. Veltman, One loop divergencies in the theory of gravitation, Ann. Inst. H. Poincare Phys. Theor. A 20 (1974) 69 [INSPIRE].
S.M. Christensen and M.J. Duff, Quantizing Gravity with a Cosmological Constant, Nucl. Phys. B 170 (1980) 480 [INSPIRE].
J.D. Gonçalves, T. de Paula Netto and I.L. Shapiro, Gauge and parametrization ambiguity in quantum gravity, Phys. Rev. D 97 (2018) 026015 [arXiv:1712.03338] [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B 201 (1982) 469 [INSPIRE].
E.V. Gorbar and I.L. Shapiro, Renormalization group and decoupling in curved space, JHEP 02 (2003) 021 [hep-ph/0210388] [INSPIRE].
E.V. Gorbar and I.L. Shapiro, Renormalization group and decoupling in curved space. 2. The standard model and beyond, JHEP 06 (2003) 004 [hep-ph/0303124] [INSPIRE].
S.A. Franchino-Viñas, T. de Paula Netto, I.L. Shapiro and O. Zanusso, Form factors and decoupling of matter fields in four-dimensional gravity, Phys. Lett. B 790 (2019) 229 [arXiv:1812.00460] [INSPIRE].
G. de Berredo-Peixoto and I.L. Shapiro, Higher derivative quantum gravity with Gauss-Bonnet term, Phys. Rev. D 71 (2005) 064005 [hep-th/0412249] [INSPIRE].
L. Modesto, L. Rachwa·l and I.L. Shapiro, Renormalization group in super-renormalizable quantum gravity, Eur. Phys. J. C 78 (2018) 555 [arXiv:1704.03988] [INSPIRE].
P.d.M. Teixeira, I.L. Shapiro and T.G. Ribeiro, One-loop effective action: nonlocal form factors and renormalization group, Grav. Cosmol. 26 (2020) 185 [arXiv:2003.04503] [INSPIRE].
Supernova Cosmology Project collaboration, Discovery of a supernova explosion at half the age of the Universe and its cosmological implications, Nature 391 (1998) 51 [astro-ph/9712212] [INSPIRE].
Supernova Search Team collaboration, Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998) 1009 [astro-ph/9805201] [INSPIRE].
S. Weinberg, The Cosmological Constant Problem, Rev. Mod. Phys. 61 (1989) 1 [INSPIRE].
I.L. Shapiro and J. Solà, Scaling behavior of the cosmological constant: Interface between quantum field theory and cosmology, JHEP 02 (2002) 006 [hep-th/0012227] [INSPIRE].
M. Asorey, J.L. López and I.L. Shapiro, Some remarks on high derivative quantum gravity, Int. J. Mod. Phys. A 12 (1997) 5711 [hep-th/9610006] [INSPIRE].
L. Modesto, Super-renormalizable Quantum Gravity, Phys. Rev. D 86 (2012) 044005 [arXiv:1107.2403] [INSPIRE].
L. Modesto and L. Rachwa·l, Super-renormalizable and finite gravitational theories, Nucl. Phys. B 889 (2014) 228 [arXiv:1407.8036] [INSPIRE].
L. Modesto and L. Rachwa·l, Nonlocal quantum gravity: A review, Int. J. Mod. Phys. D 26 (2017) 1730020 [INSPIRE].
I.L. Shapiro, Counting ghosts in the “ghost-free” non-local gravity., Phys. Lett. B 744 (2015) 67 [arXiv:1502.00106] [INSPIRE].
L. Modesto and I.L. Shapiro, Superrenormalizable quantum gravity with complex ghosts, Phys. Lett. B 755 (2016) 279 [arXiv:1512.07600] [INSPIRE].
A. Accioly, B.L. Giacchini and I.L. Shapiro, On the gravitational seesaw in higher-derivative gravity, Eur. Phys. J. C 77 (2017) 540 [arXiv:1604.07348] [INSPIRE].
K.S. Stelle, Renormalization of Higher Derivative Quantum Gravity, Phys. Rev. D 16 (1977) 953 [INSPIRE].
B.L. Nelson and P. Panangaden, Scaling behavior of interacting quantum fields in curved space-time, Phys. Rev. D 25 (1982) 1019 [INSPIRE].
I.L. Buchbinder, S.D. Odintsov and I.L. Shapiro, Effective action in quantum gravity, IOP Publishing, Bristol, U.K. (1992).
I.L. Buchbinder and I.L. Shapiro, Introduction to Quantum Field Theory with Applications to Quantum Gravity, Oxford University Press, to be published.
I.G. Avramidi and A.O. Barvinsky, Asymptotic freedom in higher derivative quantum gravity, Phys. Lett. B 159 (1985) 269 [INSPIRE].
G. Cusin, F. de O.Salles and I.L. Shapiro, Tensor instabilities at the end of the ΛCDM universe, Phys. Rev. D 93 (2016) 044039 [arXiv:1503.08059] [INSPIRE].
A.O. Barvinsky and G.A. Vilkovisky, Covariant perturbation theory. 2: Second order in the curvature. General algorithms, Nucl. Phys. B 333 (1990) 471 [INSPIRE].
I. Antoniadis, P.O. Mazur and E. Mottola, Conformal symmetry and central charges in four-dimensions, Nucl. Phys. B 388 (1992) 627 [hep-th/9205015] [INSPIRE].
G. de Berredo-Peixoto and I.L. Shapiro, Conformal quantum gravity with the Gauss-Bonnet term, Phys. Rev. D 70 (2004) 044024 [hep-th/0307030] [INSPIRE].
M. Niedermaier, The asymptotic safety scenario in quantum gravity: An introduction, Class. Quant. Grav. 24 (2007) R171 [gr-qc/0610018] [INSPIRE].
R. Percacci, Asymptotic Safety, in Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, pp. 111–128, D. Oriti ed., Cambridge University Press, Cambridge, U.K. (2007), [arXiv:0709.3851] [INSPIRE].
A. Bonanno et al., Critical reflections on asymptotically safe gravity, Front. in Phys. 8 (2020) 269 [arXiv:2004.06810] [INSPIRE].
P.M. Lavrov and I.L. Shapiro, On the Functional Renormalization Group approach for Yang-Mills fields, JHEP 06 (2013) 086 [arXiv:1212.2577] [INSPIRE].
V.F. Barra, P.M. Lavrov, E.A. Dos Reis, T. de Paula Netto and I.L. Shapiro, Functional renormalization group approach and gauge dependence in gravity theories, Phys. Rev. D 101 (2020) 065001 [arXiv:1910.06068] [INSPIRE].
I.L. Buchbinder, On renormalization group equations in curved space-time, Theor. Math. Phys. 61 (1984) 393.
D.J. Toms, The Effective Action and the Renormalization Group Equation in Curved Space-time, Phys. Lett. B 126 (1983) 37 [INSPIRE].
M.H. Goroff and A. Sagnotti, Quantum gravity at two loops, Phys. Lett. B 160 (1985) 81 [INSPIRE].
M.H. Goroff and A. Sagnotti, The Ultraviolet Behavior of Einstein Gravity, Nucl. Phys. B 266 (1986) 709 [INSPIRE].
A.E.M. van de Ven, Two loop quantum gravity, Nucl. Phys. B 378 (1992) 309 [INSPIRE].
A. Babic, B. Guberina, R. Horvat and H. Stefancic, Renormalization group running of the cosmological constant and its implication for the Higgs boson mass in the standard model, Phys. Rev. D 65 (2002) 085002 [hep-ph/0111207] [INSPIRE].
B. Guberina, R. Horvat and H. Stefancic, Renormalization group running of the cosmological constant and the fate of the universe, Phys. Rev. D 67 (2003) 083001 [hep-ph/0211184] [INSPIRE].
J. Goldman, J. Pérez-Mercader, F. Cooper and M.M. Nieto, The dark matter problem and quantum gravity, Phys. Lett. B 281 (1992) 219 [INSPIRE].
O. Bertolami, J.M. Mourão and J. Pérez-Mercader, Quantum gravity and the large scale structure of the universe, Phys. Lett. B 311 (1993) 27 [INSPIRE].
I.L. Shapiro, J. Solà and H. Stefancic, Running G and Lambda at low energies from physics at M(X): Possible cosmological and astrophysical implications, JCAP 01 (2005) 012 [hep-ph/0410095] [INSPIRE].
D.C. Rodrigues, P.S. Letelier and I.L. Shapiro, Galaxy rotation curves from General Relativity with Renormalization Group corrections, JCAP 04 (2010) 020 [arXiv:0911.4967] [INSPIRE].
S. Domazet and H. Stefancic, Renormalization group scale-setting in astrophysical systems, Phys. Lett. B 703 (2011) 1 [arXiv:1010.3585] [INSPIRE].
D.C. Rodrigues, B. Chauvineau and O.F. Piattella, Scalar-Tensor gravity with system-dependent potential and its relation with Renormalization Group extended General Relativity, JCAP 09 (2015) 009 [arXiv:1504.05119] [INSPIRE].
N.R. Bertini, W.S. Hipólito-Ricaldi, F. de Melo-Santos and D.C. Rodrigues, Cosmological framework for renormalization group extended gravity at the action level, Eur. Phys. J. C 80 (2020) 479 [Erratum ibid. 80 (2020) 644] [arXiv:1908.03960] [INSPIRE].
A.A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B 91 (1980) 99 [INSPIRE].
A.A. Starobinsky, The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy, Sov. Astron. Lett. 9 (1983) 302 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2009.04122
Ilya L. Shapiro On leave from Tomsk State Pedagogical University.
Rights and permissions
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.
About this article
Cite this article
Giacchini, B.L., de Paula Netto, T. & Shapiro, I.L. On the Vilkovisky-DeWitt approach and renormalization group in effective quantum gravity. J. High Energ. Phys. 2020, 11 (2020). https://doi.org/10.1007/JHEP10(2020)011
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2020)011