Abstract
The quantum dynamics of the gravitational field non-minimally coupled to an (also dynamical) scalar field is studied in the broken phase. For a particular value of the coupling the system is classically conformal, and can actually be understood as the group averaging of Einstein-Hilbert’s action under conformal transformations. Conformal invariance implies a simple Ward identity asserting that the trace of the equation of motion for the graviton is the equation of motion of the scalar field. We perform an explicit one-loop computation to show that the DeWitt effective action is not UV divergent on shell and to find that the Weyl symmetry Ward identity is preserved on shell at that level. We also discuss the fate of this Ward identity at the two-loop level — under the assumption that the two-loop UV divergent part of the effective action can be retrieved from the Goroff-Sagnotti counterterm — and show that its preservation in the renormalized theory requires the introduction of counterterms which exhibit a logarithmic dependence on the dilaton field.
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References
E. Alvarez and A.F. Faedo, Unimodular cosmology and the weight of energy, Phys. Rev. D 76 (2007) 064013 [hep-th/0702184] [INSPIRE].
E. Alvarez and M. Herrero-Valea, No Conformal Anomaly in Unimodular Gravity, Phys. Rev. D 87 (2013) 084054 [arXiv:1301.5130] [INSPIRE].
E. Alvarez and M. Herrero-Valea, Some Comments on Dilaton Gravity, arXiv:1307.2060 [INSPIRE].
E. Alvarez and A.F. Faedo, Renormalized Kaluza-Klein theories, JHEP 05 (2006) 046 [hep-th/0602150] [INSPIRE].
E. Alvarez, A.F. Faedo and J.J. Lopez-Villarejo, Ultraviolet behavior of transverse gravity, JHEP 10 (2008) 023 [arXiv:0807.1293] [INSPIRE].
A.O. Barvinsky, A.Y. Kamenshchik and I.P. Karmazin, The renormalization group for nonrenormalizable theories: Einstein gravity with a scalar field, Phys. Rev. D 48 (1993) 3677 [gr-qc/9302007] [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].
L. Bonora, P. Cotta-Ramusino and C. Reina, Conformal Anomaly and Cohomology, Phys. Lett. B 126 (1983) 305 [INSPIRE].
P. Breitenlohner and D. Maison, Dimensional Renormalization and the Action Principle, Commun. Math. Phys. 52 (1977) 11 [INSPIRE].
S.M. Christensen and M.J. Duff, Quantizing Gravity with a Cosmological Constant, Nucl. Phys. B 170 (1980) 480 [INSPIRE].
Y. Decanini and A. Folacci, Off-diagonal coefficients of the Dewitt-Schwinger and Hadamard representations of the Feynman propagator, Phys. Rev. D 73 (2006) 044027 [gr-qc/0511115] [INSPIRE].
P.A.M. Dirac, Long range forces and broken symmetries, Proc. Roy. Soc. Lond. A 333 (1973) 403 [INSPIRE].
M.J. Duff, Ultraviolet divergences in extended supergravity, arXiv:1201.0386 [INSPIRE].
M.J. Duff, Twenty years of the Weyl anomaly, Class. Quant. Grav. 11 (1994) 1387 [hep-th/9308075] [INSPIRE].
F. Englert, C. Truffin and R. Gastmans, Conformal Invariance in Quantum Gravity, Nucl. Phys. B 117 (1976) 407 [INSPIRE].
C. Fefferman and C.R. Graham, The ambient metric, arXiv:0710.0919 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Conformal supergravity, Phys. Rept. 119 (1985) 233 [INSPIRE].
E.S. Fradkin and G.A. Vilkovisky, Conformal Off Mass Shell Extension and Elimination of Conformal Anomalies in Quantum Gravity, Phys. Lett. B 73 (1978) 209 [INSPIRE].
E.S. Fradkin and G.A. Vilkovisky, Conformal Invariance and Asymptotic Freedom in Quantum Gravity, Phys. Lett. B 77 (1978) 262 [INSPIRE].
P.B. Gilkey, The spectral geometry of a Riemannian manifold, J. Diff. Geom. 10 (1975) 601 [INSPIRE].
M.H. Goroff and A. Sagnotti, The Ultraviolet Behavior of Einstein Gravity, Nucl. Phys. B 266 (1986) 709 [INSPIRE].
M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Background Field Method Versus Normal Field Theory in Explicit Examples: One Loop Divergences in S Matrix and Green’s Functions for Yang-Mills and Gravitational Fields, Phys. Rev. D 12 (1975) 3203 [INSPIRE].
T. Henz, J.M. Pawlowski, A. Rodigast and C. Wetterich, Dilaton Quantum Gravity, Phys. Lett. B 727 (2013) 298 [arXiv:1304.7743] [INSPIRE].
G. Narain and R. Percacci, Renormalization Group Flow in Scalar-Tensor Theories. I, Class. Quant. Grav. 27 (2010) 075001 [arXiv:0911.0386] [INSPIRE].
G. ’t Hooft, Probing the small distance structure of canonical quantum gravity using the conformal group, arXiv:1009.0669 [INSPIRE].
G. ’t Hooft, A class of elementary particle models without any adjustable real parameters, Found. Phys. 41 (2011) 1829 [arXiv:1104.4543] [INSPIRE].
I. Jack and H. Osborn, Background Field Calculations in Curved Space-time. 1. General Formalism and Application to Scalar Fields, Nucl. Phys. B 234 (1984) 331 [INSPIRE].
R. Jackiw, C. Núñez and S.-Y. Pi, Quantum relaxation of the cosmological constant, Phys. Lett. A 347 (2005) 47 [hep-th/0502215] [INSPIRE].
G. Giavarini, C.P. Martin and F. Ruiz Ruiz, Chern-Simons theory as the large mass limit of topologically massive Yang-Mills theory, Nucl. Phys. B 381 (1992) 222 [hep-th/9206007] [INSPIRE].
J.M. Martın-García et al., xAct: Efficient tensor computer algebra for Mathematica, 2002-2014, http://xact.es.
L. Parker and D. Toms, Quantum Field Theory in curved Spacetime, Cambridge University Press, (2009).
R.D. Pisarski, Trying to Zero the Cosmological Constant with Conformal Symmetry, FERMILAB-PUB-85-030-T [INSPIRE].
I.L. Shapiro and H. Takata, One loop renormalization of the four-dimensional theory for quantum dilaton gravity, Phys. Rev. D 52 (1995) 2162 [hep-th/9502111] [INSPIRE].
C.F. Steinwachs and A.Y. Kamenshchik, One-loop divergences for gravity non-minimally coupled to a multiplet of scalar fields: calculation in the Jordan frame. I. The main results, Phys. Rev. D 84 (2011) 024026 [arXiv:1101.5047] [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, One loop divergencies in the theory of gravitation, Annales Poincare Phys. Theor. A 20 (1974) 69 [INSPIRE].
P. Van Nieuwenhuizen, On the Renormalization of Quantum Gravitation Without Matter, Annals Phys. 104 (1977) 197 [INSPIRE].
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Alvarez, E., Herrero-Valea, M. & Martín, C.P. Conformal and non conformal dilaton gravity. J. High Energ. Phys. 2014, 115 (2014). https://doi.org/10.1007/JHEP10(2014)115
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DOI: https://doi.org/10.1007/JHEP10(2014)115