Abstract
The Asymptotic Safety hypothesis states that the high-energy completion of gravity is provided by an interacting renormalization group fixed point. This implies non-trivial quantum corrections to the scaling dimensions of operators and correlation functions which are characteristic for the corresponding universality class. We use the composite operator formalism for the effective average action to derive an analytic expression for the scaling dimension of an infinite family of geometric operators \( \int {d}^dx\sqrt{g}{R}^n \). We demonstrate that the anomalous dimensions interpolate continuously between their fixed point value and zero when evaluated along renormalization group trajectories approximating classical general relativity at low energy. Thus classical geometry emerges when quantum fluctuations are integrated out. We also compare our results to the stability properties of the interacting renormalization group fixed point projected to f (R)-gravity, showing that the composite operator formalism in the single-operator approximation cannot be used to reliably determine the number of relevant parameters of the theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Weinberg, Critical phenomena for field theorists, in Understanding the fundamental constituents of matter, A. Zichichi ed., Springer, Germany (1976).
S. Weinberg, Ultraviolet divergences in quantum theories of gravitation, in General relativity: an Einstein centenary survey, S.W. Hawking and W. Israel eds., Cambridge University Press, Cambridge U.K. (1979).
R. Percacci, An introduction to covariant quantum gravity and asymptotic safety, in 100 years of general relativity, volume 3, W.T. Ni ed., World Scientific, Singapore (2017).
M. Reuter and F. Saueressig, Quantum gravity and the functional renormalization group, Cambridge University Press, Cambridge U.K. (2019).
M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev.D 57 (1998) 971 [hep-th/9605030] [INSPIRE].
C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett.B 301 (1993) 90 [arXiv:1710.05815] [INSPIRE].
T.R. Morris, The exact renormalization group and approximate solutions, Int. J. Mod. Phys.A 9 (1994) 2411 [hep-ph/9308265] [INSPIRE].
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].
W. Souma, Nontrivial ultraviolet fixed point in quantum gravity, Prog. Theor. Phys.102 (1999) 181 [hep-th/9907027] [INSPIRE].
S. Falkenberg and S.D. Odintsov, Gauge dependence of the effective average action in Einstein gravity, Int. J. Mod. Phys.A 13 (1998) 607 [hep-th/9612019] [INSPIRE].
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].
D.F. Litim, Fixed points of quantum gravity, Phys. Rev. Lett.92 (2004) 201301 [hep-th/0312114] [INSPIRE].
A. Bonanno and M. Reuter, Proper time flow equation for gravity, JHEP02 (2005) 035 [hep-th/0410191] [INSPIRE].
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].
E. Manrique and M. Reuter, Bimetric truncations for quantum Einstein gravity and asymptotic safety, Annals Phys.325 (2010) 785 [arXiv:0907.2617] [INSPIRE].
A. Eichhorn and H. Gies, Ghost anomalous dimension in asymptotically safe quantum gravity, Phys. Rev.D 81 (2010) 104010 [arXiv:1001.5033] [INSPIRE].
K. Groh and F. Saueressig, Ghost wave-function renormalization in asymptotically safe quantum gravity, J. Phys.A 43 (2010) 365403 [arXiv:1001.5032] [INSPIRE].
E. Manrique, M. Reuter and F. Saueressig, Bimetric renormalization group flows in quantum Einstein gravity, Annals Phys.326 (2011) 463 [arXiv:1006.0099] [INSPIRE].
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].
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].
D. Benedetti, On the number of relevant operators in asymptotically safe gravity, EPL102 (2013) 20007 [arXiv:1301.4422] [INSPIRE].
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].
K. Falls, Renormalization of Newton’s constant, Phys. Rev.D 92 (2015) 124057 [arXiv:1501.05331] [INSPIRE].
H. Gies, B. Knorr and S. Lippoldt, Generalized parametrization dependence in quantum gravity, Phys. Rev.D 92 (2015) 084020 [arXiv:1507.08859] [INSPIRE].
C. Pagani and M. Reuter, Composite operators in asymptotic safety, Phys. Rev.D 95 (2017) 066002 [arXiv:1611.06522] [INSPIRE].
K. Falls, Physical renormalization schemes and asymptotic safety in quantum gravity, Phys. Rev.D 96 (2017) 126016 [arXiv:1702.03577] [INSPIRE].
B. Knorr and S. Lippoldt, Correlation functions on a curved background, Phys. Rev.D 96 (2017) 065020 [arXiv:1707.01397] [INSPIRE].
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].
M. Reuter and F. Saueressig, A class of nonlocal truncations in quantum Einstein gravity and its renormalization group behavior, Phys. Rev.D 66 (2002) 125001 [hep-th/0206145] [INSPIRE].
A. Codello and R. Percacci, Fixed points of higher derivative gravity, Phys. Rev. Lett.97 (2006) 221301 [hep-th/0607128] [INSPIRE].
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].
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].
M.R. Niedermaier, Gravitational fixed points from perturbation theory, Phys. Rev. Lett.103 (2009) 101303 [INSPIRE].
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].
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].
D. Benedetti, P.F. Machado and F. Saueressig, Four-derivative interactions in asymptotically safe gravity, AIP Conf. Proc.1196 (2009) 44 [arXiv:0909.3265] [INSPIRE].
D. Benedetti, K. Groh, P.F. Machado and F. Saueressig, The universal RG machine, JHEP06 (2011) 079 [arXiv:1012.3081] [INSPIRE].
S. Rechenberger and F. Saueressig, The R2phase-diagram of QEG and its spectral dimension, Phys. Rev.D 86 (2012) 024018 [arXiv:1206.0657] [INSPIRE].
N. Ohta and R. Percacci, Higher derivative gravity and asymptotic safety in diverse dimensions, Class. Quant. Grav.31 (2014) 015024 [arXiv:1308.3398] [INSPIRE].
K. Falls, D.F. Litim, K. Nikolakopoulos and C. Rahmede, A bootstrap towards asymptotic safety, arXiv:1301.4191 [INSPIRE].
K. Falls, D.F. Litim, K. Nikolakopoulos and C. Rahmede, Further evidence for asymptotic safety of quantum gravity, Phys. Rev.D 93 (2016) 104022 [arXiv:1410.4815] [INSPIRE].
A. Eichhorn, The renormalization group flow of unimodular f (R) gravity, JHEP04 (2015) 096 [arXiv:1501.05848] [INSPIRE].
N. Ohta, R. Percacci and G.P. Vacca, Flow equation for f (R) gravity and some of its exact solutions, Phys. Rev.D 92 (2015) 061501 [arXiv:1507.00968] [INSPIRE].
K. Falls, D.F. Litim, K. Nikolakopoulos and C. Rahmede, On de Sitter solutions in asymptotically safe f (R) theories, Class. Quant. Grav.35 (2018) 135006 [arXiv:1607.04962] [INSPIRE].
K. Falls and N. Ohta, Renormalization group equation for f (R) gravity on hyperbolic spaces, Phys. Rev.D 94 (2016) 084005 [arXiv:1607.08460] [INSPIRE].
N. Christiansen, Four-derivative quantum gravity beyond perturbation theory, arXiv:1612.06223 [INSPIRE].
S. Gonzalez-Martin, T.R. Morris and Z.H. Slade, Asymptotic solutions in asymptotic safety, Phys. Rev.D 95 (2017) 106010 [arXiv:1704.08873] [INSPIRE].
D. Becker, C. Ripken and F. Saueressig, On avoiding Ostrogradski instabilities within asymptotic safety, JHEP12 (2017) 121 [arXiv:1709.09098] [INSPIRE].
H. Gies, B. Knorr, S. Lippoldt and F. Saueressig, Gravitational two-loop counterterm is asymptotically safe, Phys. Rev. Lett.116 (2016) 211302 [arXiv:1601.01800] [INSPIRE].
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].
D. Benedetti and F. Caravelli, The local potential approximation in quantum gravity, JHEP06 (2012) 017 [Erratum ibid.10 (2012) 157] [arXiv:1204.3541] [INSPIRE].
M. Demmel, F. Saueressig and O. Zanusso, Fixed-functionals of three-dimensional quantum Einstein gravity, JHEP11 (2012) 131 [arXiv:1208.2038] [INSPIRE].
J.A. Dietz and T.R. Morris, Asymptotic safety in the f (R) approximation, JHEP01 (2013) 108 [arXiv:1211.0955] [INSPIRE].
I.H. Bridle, J.A. Dietz and T.R. Morris, The local potential approximation in the background field formalism, JHEP03 (2014) 093 [arXiv:1312.2846] [INSPIRE].
J.A. Dietz and T.R. Morris, Redundant operators in the exact renormalisation group and in the f (R) approximation to asymptotic safety, JHEP07 (2013) 064 [arXiv:1306.1223] [INSPIRE].
M. Demmel, F. Saueressig and O. Zanusso, RG flows of quantum einstein gravity on maximally symmetric spaces, JHEP06 (2014) 026 [arXiv:1401.5495] [INSPIRE].
M. Demmel, F. Saueressig and O. Zanusso, RG flows of quantum einstein gravity in the linear-geometric approximation, Annals Phys.359 (2015) 141 [arXiv:1412.7207] [INSPIRE].
M. Demmel, F. Saueressig and O. Zanusso, A proper fixed functional for four-dimensional quantum Einstein gravity, JHEP08 (2015) 113 [arXiv:1504.07656] [INSPIRE].
N. Ohta, R. Percacci and G.P. Vacca, Renormalization group equation and scaling solutions for f (R) gravity in exponential parametrization, Eur. Phys. J.C 76 (2016) 46 [arXiv:1511.09393] [INSPIRE].
P. Labus, T.R. Morris and Z.H. Slade, Background independence in a background dependent renormalization group, Phys. Rev.D 94 (2016) 024007 [arXiv:1603.04772] [INSPIRE].
J.A. Dietz, T.R. Morris and Z.H. Slade, Fixed point structure of the conformal factor field in quantum gravity, Phys. Rev.D 94 (2016) 124014 [arXiv:1605.07636] [INSPIRE].
B. Knorr, Infinite order quantum-gravitational correlations, Class. Quant. Grav.35 (2018) 115005 [arXiv:1710.07055] [INSPIRE].
K. Falls et al., Asymptotic safety of quantum gravity beyond Ricci scalars, Phys. Rev.D 97 (2018) 086006 [arXiv:1801.00162] [INSPIRE].
N. Christiansen, B. Knorr, J.M. Pawlowski and A. Rodigast, Global flows in quantum gravity, Phys. Rev.D 93 (2016) 044036 [arXiv:1403.1232] [INSPIRE].
J. Meibohm, J.M. Pawlowski and M. Reichert, Asymptotic safety of gravity-matter systems, Phys. Rev.D 93 (2016) 084035 [arXiv:1510.07018] [INSPIRE].
N. Christiansen et al., Local quantum gravity, Phys. Rev.D 92 (2015) 121501 [arXiv:1506.07016] [INSPIRE].
T. Denz, J.M. Pawlowski and M. Reichert, Towards apparent convergence in asymptotically safe quantum gravity, Eur. Phys. J.C 78 (2018) 336 [arXiv:1612.07315] [INSPIRE].
N. Christiansen, D.F. Litim, J.M. Pawlowski and M. Reichert, Asymptotic safety of gravity with matter, Phys. Rev.D 97 (2018) 106012 [arXiv:1710.04669] [INSPIRE].
N. Christiansen, K. Falls, J.M. Pawlowski and M. Reichert, Curvature dependence of quantum gravity, Phys. Rev.D 97 (2018) 046007 [arXiv:1711.09259] [INSPIRE].
A. Eichhorn, P. Labus, J.M. Pawlowski and M. Reichert, Effective universality in quantum gravity, SciPost Phys.5 (2018) 031 [arXiv:1804.00012] [INSPIRE].
A. Eichhorn et al., How perturbative is quantum gravity?, Phys. Lett.B 792 (2019) 310 [arXiv:1810.02828] [INSPIRE].
L. Bosma, B. Knorr and F. Saueressig, Resolving spacetime singularities within asymptotic safety, Phys. Rev. Lett.123 (2019) 101301 [arXiv:1904.04845] [INSPIRE].
B. Knorr, C. Ripken and F. Saueressig, Form factors in asymptotic safety: conceptual ideas and computational toolbox, Class. Quant. Grav.36 (2019) 234001 [arXiv:1907.02903] [INSPIRE].
P. Donà, A. Eichhorn and R. Percacci, Matter matters in asymptotically safe quantum gravity, Phys. Rev.D 89 (2014) 084035 [arXiv:1311.2898] [INSPIRE].
J. Biemans, A. Platania and F. Saueressig, Renormalization group fixed points of foliated gravity-matter systems, JHEP05 (2017) 093 [arXiv:1702.06539] [INSPIRE].
A. Eichhorn and A. Held, Mass difference for charged quarks from asymptotically safe quantum gravity, Phys. Rev. Lett.121 (2018) 151302 [arXiv:1803.04027] [INSPIRE].
N. Alkofer and F. Saueressig, Asymptotically safe f (R)-gravity coupled to matter I: the polynomial case, Annals Phys.396 (2018) 173 [arXiv:1802.00498] [INSPIRE].
J.M. Pawlowski, M. Reichert, C. Wetterich and M. Yamada, Higgs scalar potential in asymptotically safe quantum gravity, Phys. Rev.D 99 (2019) 086010 [arXiv:1811.11706] [INSPIRE].
G.P. De Brito, Y. Hamada, A.D. Pereira and M. Yamada, On the impact of Majorana masses in gravity-matter systems, JHEP08 (2019) 142 [arXiv:1905.11114] [INSPIRE].
B. Bürger, J.M. Pawlowski, M. Reichert and B.-J. Schaefer, Curvature dependence of quantum gravity with scalars, arXiv:1912.01624 [INSPIRE].
A. Eichhorn, An asymptotically safe guide to quantum gravity and matter, Front. Astron. Space Sci.5 (2019) 47 [arXiv:1810.07615] [INSPIRE].
T.R. Morris and R. Percacci, Trace anomaly and infrared cutoffs, Phys. Rev.D 99 (2019) 105007 [arXiv:1810.09824] [INSPIRE].
A. Baldazzi, R. Percacci and V. Skrinjar, Wicked metrics, Class. Quant. Grav.36 (2019) 105008 [arXiv:1811.03369] [INSPIRE].
A. Baldazzi, R. Percacci and V. Skrinjar, Quantum fields without Wick rotation, Symmetry11 (2019) 373 [arXiv:1901.01891] [INSPIRE].
A. Maas, The Fröhlich-Morchio-Strocchi mechanism and quantum gravity, arXiv:1908.02140 [INSPIRE].
J. Ambjørn, A. Görlich, J. Jurkiewicz and R. Loll, Nonperturbative quantum gravity, Phys. Rept.519 (2012) 127 [arXiv:1203.3591] [INSPIRE].
R. Loll, Quantum gravity from causal dynamical triangulations: a review, Class. Quant. Grav.37 (2020) 013002 [arXiv:1905.08669] [INSPIRE].
J. Ambjørn, S. Jordan, J. Jurkiewicz and R. Loll, A second-order phase transition in CDT, Phys. Rev. Lett.107 (2011) 211303 [arXiv:1108.3932] [INSPIRE].
J. Ambjørn, S. Jordan, J. Jurkiewicz and R. Loll, Second- and first-order phase transitions in CDT, Phys. Rev.D 85 (2012) 124044 [arXiv:1205.1229] [INSPIRE].
J. Ambjørn et al., Critical phenomena in causal dynamical triangulations, Class. Quant. Grav.36 (2019) 224001 [arXiv:1904.05755] [INSPIRE].
J. Ambjørn et al., Towards an UV fixed point in CDT gravity, JHEP07 (2019) 166 [arXiv:1906.04557] [INSPIRE].
E. Manrique, S. Rechenberger and F. Saueressig, Asymptotically safe Lorentzian gravity, Phys. Rev. Lett.106 (2011) 251302 [arXiv:1102.5012] [INSPIRE].
S. Rechenberger and F. Saueressig, A functional renormalization group equation for foliated spacetimes, JHEP03 (2013) 010 [arXiv:1212.5114] [INSPIRE].
J. Biemans, A. Platania and F. Saueressig, Quantum gravity on foliated spacetimes: asymptotically safe and sound, Phys. Rev.D 95 (2017) 086013 [arXiv:1609.04813] [INSPIRE].
W.B. Houthoff, A. Kurov and F. Saueressig, Impact of topology in foliated quantum Einstein gravity, Eur. Phys. J.C 77 (2017) 491 [arXiv:1705.01848] [INSPIRE].
B. Knorr, Lorentz symmetry is relevant, Phys. Lett.B 792 (2019) 142 [arXiv:1810.07971] [INSPIRE].
A. Eichhorn, A. Platania and M. Schiffer, Lorentz invariance violations in the interplay of quantum gravity with matter, arXiv:1911.10066 [INSPIRE].
J. Ambjørn, J. Jurkiewicz and R. Loll, Spectral dimension of the universe, Phys. Rev. Lett.95 (2005) 171301 [hep-th/0505113] [INSPIRE].
O. Lauscher and M. Reuter, Fractal spacetime structure in asymptotically safe gravity, JHEP10 (2005) 050 [hep-th/0508202] [INSPIRE].
M. Reuter and F. Saueressig, Asymptotic safety, fractals, and cosmology, Lect. Notes Phys.863 (2013) 185 [arXiv:1205.5431].
S. Carlip, Dimension and dimensional reduction in quantum gravity, Universe5 (2019) 83 [arXiv:1904.04379] [INSPIRE].
C. Pagani and H. Sonoda, Products of composite operators in the exact renormalization group formalism, PTEP2018 (2018) 023B02 [arXiv:1707.09138] [INSPIRE].
M. Becker and C. Pagani, Geometric operators in the asymptotic safety scenario for quantum gravity, Phys. Rev.D 99 (2019) 066002 [arXiv:1810.11816] [INSPIRE].
M. Becker and C. Pagani, Geometric operators in the Einstein-Hilbert truncation, Universe5 (2019) 75.
M. Becker, C. Pagani and O. Zanusso, Fractal geometry of higher derivative gravity, arXiv:1911.02415 [INSPIRE].
N. Klitgaard and R. Loll, Introducing quantum Ricci curvature, Phys. Rev.D 97 (2018) 046008 [arXiv:1712.08847] [INSPIRE].
N. Klitgaard and R. Loll, Implementing quantum Ricci curvature, Phys. Rev.D 97 (2018) 106017 [arXiv:1802.10524] [INSPIRE].
M. Reuter and C. Wetterich, Effective average action for gauge theories and exact evolution equations, Nucl. Phys.B 417 (1994) 181 [INSPIRE].
M. Demmel, F. Saueressig and O. Zanusso, Fixed functionals in asymptotically safe gravity, in proceedings of the 13thMarcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories (MG13), July 1–7, Stockholm, Sweden (2015), arXiv:1302.1312 [INSPIRE].
J.A. Dietz and T.R. Morris, Background independent exact renormalization group for conformally reduced gravity, JHEP04 (2015) 118 [arXiv:1502.07396] [INSPIRE].
G.P. De Brito et al., Asymptotic safety and field parametrization dependence in the f (R) truncation, Phys. Rev.D 98 (2018) 026027 [arXiv:1805.09656] [INSPIRE].
N. Ohta, R. Percacci and A.D. Pereira, \( f\left(R,{R}_{\mu \nu}^2\right) \)at one loop, Phys. Rev.D 97 (2018) 104039 [arXiv:1804.01608] [INSPIRE].
K.G. Falls, D.F. Litim and J. Schröder, Aspects of asymptotic safety for quantum gravity, Phys. Rev.D 99 (2019) 126015 [arXiv:1810.08550] [INSPIRE].
N. Alkofer, Asymptotically safe f (R)-gravity coupled to matter II: Global solutions, Phys. Lett.B 789 (2019) 480 [arXiv:1809.06162] [INSPIRE].
J.M. Pawlowski, Aspects of the functional renormalisation group, Annals Phys.322 (2007) 2831 [hep-th/0512261] [INSPIRE].
Y. Igarashi, K. Itoh and H. Sonoda, Realization of symmetry in the ERG approach to quantum field theory, Prog. Theor. Phys. Suppl.181 (2010) 1 [arXiv:0909.0327] [INSPIRE].
C. Pagani, Note on scaling arguments in the effective average action formalism, Phys. Rev.D 94 (2016) 045001 [arXiv:1603.07250] [INSPIRE].
U. Ellwanger, Flow equations and BRS invariance for Yang-Mills theories, Phys. Lett.B 335 (1994) 364 [hep-th/9402077] [INSPIRE].
M. D’Attanasio and T.R. Morris, Gauge invariance, the quantum action principle and the renormalization group, Phys. Lett.B 378 (1996) 213 [hep-th/9602156] [INSPIRE].
D.F. Litim and J.M. Pawlowski, Flow equations for Yang-Mills theories in general axial gauges, Phys. Lett.B 435 (1998) 181 [hep-th/9802064] [INSPIRE].
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].
D.F. Litim, Optimization of the exact renormalization group, Phys. Lett.B 486 (2000) 92 [hep-th/0005245] [INSPIRE].
D.F. Litim, Optimized renormalization group flows, Phys. Rev.D 64 (2001) 105007 [hep-th/0103195] [INSPIRE].
M. Reuter and F. Saueressig, Fractal space-times under the microscope: a renormalization group view on Monte Carlo data, JHEP12 (2011) 012 [arXiv:1110.5224] [INSPIRE].
J. Ambjørn, A. Görlich, J. Jurkiewicz and R. Loll, The nonperturbative quantum de Sitter universe, Phys. Rev.D 78 (2008) 063544 [arXiv:0807.4481] [INSPIRE].
J. Ambjørn et al., Impact of topology in causal dynamical triangulations quantum gravity, Phys. Rev.D 94 (2016) 044010 [arXiv:1604.08786] [INSPIRE].
B. Knorr and F. Saueressig, Towards reconstructing the quantum effective action of gravity, Phys. Rev. Lett.121 (2018) 161304 [arXiv:1804.03846] [INSPIRE].
D.V. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rept.388 (2003) 279 [hep-th/0306138] [INSPIRE].
J.W. York, Jr., Conformatlly invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial value problem of general relativity, J. Math. Phys.14 (1973) 456 [INSPIRE].
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
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 2002.00256
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Houthoff, W., Kurov, A. & Saueressig, F. On the scaling of composite operators in asymptotic safety. J. High Energ. Phys. 2020, 99 (2020). https://doi.org/10.1007/JHEP04(2020)099
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2020)099