# Renormalization group properties in the conformal sector: towards perturbatively renormalizable quantum gravity

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## Abstract

The Wilsonian renormalization group (RG) requires Euclidean signature. The conformal factor of the metric then has a wrong-sign kinetic term, which has a profound effect on its RG properties. Generically for the conformal sector, complete flows exist only in the reverse direction (i.e. from the infrared to the ultraviolet). The Gaussian fixed point supports infinite sequences of composite eigenoperators of increasing infrared relevancy (increasingly negative mass dimension), which are orthonormal and complete for bare interactions that are square integrable under the appropriate measure. These eigenoperators are non-perturbative in ℏ and evanescent. For ℝ^{4} spacetime, each renormalized physical operator exists but only has support at vanishing field amplitude. In the generic case of infinitely many non-vanishing couplings, if a complete RG flow exists, it is characterised in the infrared by a scale Λ_{p} > 0, beyond which the field amplitude is exponentially suppressed. On other spacetimes, of length scale *L*, the flow ceases to exist once a certain universal measure of inhomogeneity exceeds *O*(1) + 2*πL*^{2}*Λ* _{p} ^{2} . Importantly for cosmology, the minimum size of the universe is thus tied to the degree of inhomogeneity, with space-times of vanishing size being required to be almost homogeneous. We initiate a study of this exotic quantum field theory at the interacting level, and discuss what the full theory of quantum gravity should look like, one which must thus be perturbatively renormalizable in Newton’s constant but non-perturbative in ℏ.

## Keywords

Models of Quantum Gravity Renormalization Group Renormalization Regularization and Renormalons## Notes

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## References

- [1]G. ’t Hooft and M.J.G. Veltman,
*One loop divergencies in the theory of gravitation*,*Ann. Inst. H. Poincare Phys. Theor.***A20**(1974) 69 [INSPIRE]. - [2]M.H. Goroff and A. Sagnotti,
*Quantum gravity at two loops*,*Phys. Lett.***B 160**(1985) 81 [INSPIRE].ADSCrossRefGoogle Scholar - [3]M.H. Goroff and A. Sagnotti,
*The Ultraviolet Behavior of Einstein Gravity*,*Nucl. Phys.***B 266**(1986) 709 [INSPIRE].ADSCrossRefGoogle Scholar - [4]A.E.M. van de Ven,
*Two loop quantum gravity*,*Nucl. Phys.***B 378**(1992) 309 [INSPIRE].ADSMathSciNetGoogle Scholar - [5]S. Weinberg,
*Ultraviolet Divergences In Quantum Theories Of Gravitation*, in*General Relativity*, S.W. Hawking and W. Israel eds., Cambridge University Press (1980), pp. 790-831.Google Scholar - [6]M. Reuter,
*Nonperturbative evolution equation for quantum gravity*,*Phys. Rev.***D 57**(1998) 971 [hep-th/9605030] [INSPIRE].ADSMathSciNetGoogle Scholar - [7]K.G. Wilson and J.B. Kogut,
*The Renormalization group and the ϵ-expansion*,*Phys. Rept.***12**(1974) 75 [INSPIRE].ADSCrossRefGoogle Scholar - [8]T.R. Morris,
*Elements of the continuous renormalization group*,*Prog. Theor. Phys. Suppl.***131**(1998) 395 [hep-th/9802039] [INSPIRE].ADSCrossRefGoogle Scholar - [9]G.W. Gibbons, S.W. Hawking and M.J. Perry,
*Path Integrals and the Indefiniteness of the Gravitational Action*,*Nucl. Phys.***B 138**(1978) 141 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [10]T.R. Morris,
*On the fixed point structure of scalar fields*,*Phys. Rev. Lett.***77**(1996) 1658 [hep-th/9601128] [INSPIRE].ADSCrossRefGoogle Scholar - [11]T.R. Morris,
*Three-dimensional massive scalar field theory and the derivative expansion of the renormalization group*,*Nucl. Phys.***B 495**(1997) 477 [hep-th/9612117] [INSPIRE].ADSCrossRefGoogle Scholar - [12]I. Hamzaan Bridle and T.R. Morris,
*Fate of nonpolynomial interactions in scalar field theory*,*Phys. Rev.***D 94**(2016) 065040 [arXiv:1605.06075] [INSPIRE].ADSMathSciNetGoogle Scholar - [13]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].ADSMathSciNetGoogle Scholar - [14]A. Bonanno and F. Guarnieri,
*Universality and Symmetry Breaking in Conformally Reduced Quantum Gravity*,*Phys. Rev.***D 86**(2012) 105027 [arXiv:1206.6531] [INSPIRE].ADSGoogle Scholar - [15]C.G. Bollini and J.J. Giambiagi,
*Evanescent couplings and compensation of adler anomaly*,*Acta Phys. Austriaca***38**(1973) 211 [INSPIRE].Google Scholar - [16]T.R. Morris,
*The Exact renormalization group and approximate solutions*,*Int. J. Mod. Phys.***A 9**(1994) 2411 [hep-ph/9308265] [INSPIRE]. - [17]J. Polchinski,
*Renormalization and Effective Lagrangians*,*Nucl. Phys.***B 231**(1984) 269 [INSPIRE].ADSCrossRefGoogle Scholar - [18]T.R. Morris and Z.H. Slade,
*Solutions to the reconstruction problem in asymptotic safety*,*JHEP***11**(2015) 094 [arXiv:1507.08657] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [19]J.F. Nicoll and T.S. Chang,
*An Exact One Particle Irreducible Renormalization Group Generator for Critical Phenomena*,*Phys. Lett.***A 62**(1977) 287 [INSPIRE].ADSCrossRefGoogle Scholar - [20]M. Bonini, M. D’Attanasio and G. Marchesini,
*Perturbative renormalization and infrared finiteness in the Wilson renormalization group: The Massless scalar case*,*Nucl. Phys.***B 409**(1993) 441 [hep-th/9301114] [INSPIRE].ADSCrossRefGoogle Scholar - [21]C. Wetterich,
*Exact evolution equation for the effective potential*,*Phys. Lett.***B 301**(1993) 90 [arXiv:1710.05815] [INSPIRE].ADSCrossRefGoogle Scholar - [22]G. Keller, C. Kopper and M. Salmhofer,
*Perturbative renormalization and effective Lagrangians in ϕ*^{4}*in four-dimensions*,*Helv. Phys. Acta***65**(1992) 32 [INSPIRE].MathSciNetGoogle Scholar - [23]K. Halpern and K. Huang,
*Fixed point structure of scalar fields*,*Phys. Rev. Lett.***74**(1995) 3526 [hep-th/9406199] [INSPIRE].ADSCrossRefGoogle Scholar - [24]A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkin,
*Pseudoparticle Solutions of the Yang-Mills Equations*,*Phys. Lett.***B 59**(1975) 85 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [25]G. ’t Hooft,
*Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle*,*Phys. Rev.***D 14**(1976) 3432 [*Erratum ibid.***D 18**(1978) 2199] [INSPIRE]. - [26]G. ’t Hooft,
*Can We Make Sense Out of Quantum Chromodynamics?*,*Subnucl. Ser.***15**(1979) 943 [INSPIRE]. - [27]R. Jackiw,
*Functional evaluation of the effective potential*,*Phys. Rev.***D 9**(1974) 1686 [INSPIRE].ADSGoogle Scholar - [28]N.K. Nielsen,
*On the Gauge Dependence of Spontaneous Symmetry Breaking in Gauge Theories*,*Nucl. Phys.***B 101**(1975) 173 [INSPIRE].ADSCrossRefGoogle Scholar - [29]I.S. Gradshteyn and I.M. Ryzhik,
*Tables of integrals, series and products*, 4th edition, Academic Press Inc., New York (1980).MATHGoogle Scholar - [30]S. Arnone, Y.A. Kubyshin, T.R. Morris and J.F. Tighe,
*Gauge invariant regularization via*SU(*N*|*N*),*Int. J. Mod. Phys.***A 17**(2002) 2283 [hep-th/0106258] [INSPIRE].ADSCrossRefMATHGoogle Scholar - [31]M.P. Kellett and T.R. Morris,
*Renormalization group properties of the conformal mode of a torus*,*Class. Quant. Grav.***35**(2018) 175002 [arXiv:1803.00859] [INSPIRE].ADSCrossRefGoogle Scholar - [32]S. Hollands and R.M. Wald,
*An Alternative to inflation*,*Gen. Rel. Grav.***34**(2002) 2043 [gr-qc/0205058] [INSPIRE]. - [33]L. Kofman, A.D. Linde and V.F. Mukhanov,
*Inflationary theory and alternative cosmology*,*JHEP***10**(2002) 057 [hep-th/0206088] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [34]S. Hollands and R.M. Wald,
*Comment on inflation and alternative cosmology*, hep-th/0210001 [INSPIRE]. - [35]
- [36]P. Hasenfratz and H. Leutwyler,
*Goldstone Boson Related Finite Size Effects in Field Theory and Critical Phenomena With O*(*N*)*Symmetry*,*Nucl. Phys.***B 343**(1990) 241 [INSPIRE].ADSCrossRefGoogle Scholar - [37]M.B. Green, J.H. Schwarz and L. Brink,
*N*= 4*Yang-Mills and N*= 8*Supergravity as Limits of String Theories*,*Nucl. Phys.***B 198**(1982) 474 [INSPIRE].ADSCrossRefGoogle Scholar - [38]K. Kikkawa and M. Yamasaki,
*Casimir Effects in Superstring Theories*,*Phys. Lett.***B 149**(1984) 357 [INSPIRE].ADSCrossRefGoogle Scholar - [39]N. Sakai and I. Senda,
*Vacuum Energies of String Compactified on Torus*,*Prog. Theor. Phys.***75**(1986) 692 [*Erratum ibid.***77**(1987) 773] [INSPIRE]. - [40]J. Feldbrugge, J.-L. Lehners and N. Turok,
*No rescue for the no boundary proposal: Pointers to the future of quantum cosmology*,*Phys. Rev.***D 97**(2018) 023509 [arXiv:1708.05104] [INSPIRE].ADSGoogle Scholar - [41]S.N. Gupta,
*Gravitation and Electromagnetism*,*Phys. Rev.***96**(1954) 1683 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [42]R.H. Kraichnan,
*Special-Relativistic Derivation of Generally Covariant Gravitation Theory*,*Phys. Rev.***98**(1955) 1118 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [43]R.P. Feynman,
*Feynman lectures on gravitation*, (1996) [INSPIRE]. - [44]S. Weinberg,
*Photons and gravitons in perturbation theory: Derivation of Maxwell’s and Einstein’s equations*,*Phys. Rev.***138**(1965) B988 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [45]V.I. Ogievetsky and I.V. Polubarinov,
*Interacting field of spin 2 and the einstein equations*,*Annals Phys.***35**(1965) 167.ADSCrossRefGoogle Scholar - [46]
- [47]S. Deser,
*Selfinteraction and gauge invariance*,*Gen. Rel. Grav.***1**(1970) 9 [gr-qc/0411023] [INSPIRE]. - [48]D.G. Boulware and S. Deser,
*Classical General Relativity Derived from Quantum Gravity*,*Annals Phys.***89**(1975) 193 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [49]J. Fang and C. Fronsdal,
*Deformation of Gauge Groups. Gravitation*,*J. Math. Phys.***20**(1979) 2264 [INSPIRE]. - [50]R.M. Wald,
*Spin-2 Fields and General Covariance*,*Phys. Rev.***D 33**(1986) 3613 [INSPIRE].ADSMathSciNetGoogle Scholar - [51]N. Boulanger, T. Damour, L. Gualtieri and M. Henneaux,
*Inconsistency of interacting, multigraviton theories*,*Nucl. Phys.***B 597**(2001) 127 [hep-th/0007220] [INSPIRE].ADSCrossRefMATHGoogle Scholar - [52]P.J. Mohr, D.B. Newell and B.N. Taylor,
*CODATA Recommended Values of the Fundamental Physical Constants: 2014*,*Rev. Mod. Phys.***88**(2016) 035009 [arXiv:1507.07956] [INSPIRE].ADSCrossRefGoogle Scholar - [53]T.R. Morris and A.W.H. Preston,
*Manifestly diffeomorphism invariant classical Exact Renormalization Group*,*JHEP***06**(2016) 012 [arXiv:1602.08993] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [54]K.S. Stelle,
*Renormalization of Higher Derivative Quantum Gravity*,*Phys. Rev.***D 16**(1977) 953 [INSPIRE].ADSMathSciNetGoogle Scholar - [55]M.C. Bergere and Y.-M.P. Lam,
*Equivalence Theorem and Faddeev-Popov Ghosts*,*Phys. Rev.***D 13**(1976) 3247 [INSPIRE].ADSGoogle Scholar - [56]C. Itzykson and J.B. Zuber,
*Quantum Field Theory*, International Series In Pure and Applied Physics, McGraw-Hill, New York (1980).Google Scholar