Abstract
Weyl conformal geometry is a gauge theory of scale invariance that naturally brings together the Standard Model (SM) and Einstein gravity. The SM embedding in this geometry is possible without new degrees of freedom beyond SM and Weyl geometry, while Einstein gravity is generated by the broken phase of this symmetry. This follows a Stueckelberg breaking mechanism in which the Weyl gauge boson becomes massive and decouples, as discussed in the past [1,2,3]. However, Weyl anomaly could break explicitly this gauge symmetry, hence we study it in Weyl geometry. We first note that in Weyl geometry metricity can be restored with respect to a new differential operator (\( \hat{\nabla} \)) that also enforces simultaneously a Weyl-covariant formulation. This leads to a metric-like Weyl gauge invariant formalism that enables one to do quantum calculations directly in Weyl geometry, rather than use a Riemannian (metric) geometry picture. The result is the Weyl-covariance in d dimensions of all geometric operators (\( \hat{R} \), etc) and of their derivatives (\( \hat{\nabla} \)μ\( \hat{R} \), etc), including the Euler-Gauss-Bonnet term. A natural (geometric) Weyl-invariant dimensional regularisation of quantum corrections exists and Weyl gauge symmetry is then maintained and manifest at the quantum level. This is related to a non-trivial current of this symmetry, the divergence of which cancels the trace of the energy-momentum tensor. The “usual” Weyl anomaly and Riemannian geometry are recovered in the (spontaneously) broken phase. The relation to holographic Weyl anomaly is discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D.M. Ghilencea, Spontaneous breaking of Weyl quadratic gravity to Einstein action and Higgs potential, JHEP 03 (2019) 049 [arXiv:1812.08613] [INSPIRE].
D.M. Ghilencea, Stueckelberg breaking of Weyl conformal geometry and applications to gravity, Phys. Rev. D 101 (2020) 045010 [arXiv:1904.06596] [INSPIRE].
D.M. Ghilencea, Standard Model in Weyl conformal geometry, Eur. Phys. J. C 82 (2022) 23 [arXiv:2104.15118] [INSPIRE].
W.A. Bardeen, On naturalness in the standard model, FERMILAB-CONF-95-391-T, Fermilab, Batavia, IL, U.S.A. (1995).
H. Weyl, Gravitation und Elektrizität (in German), Königlich Preussischen Akademie der Wissenschaften, Berlin, Germany (1918), p. 465 [https://doi.org/10.1007/978-3-663-19510-8_11].
H. Weyl, A new extension of relativity theory, Annalen Phys. 59 (1919) 101 [INSPIRE].
H. Weyl, Raum, Zeit, Materie (in German), Springer, Berlin, Heidelberg, Germany (1921) [https://doi.org/10.1007/978-3-662-02044-9].
D.M. Ghilencea and C.T. Hill, Standard Model in conformal geometry: local vs gauged scale invariance, arXiv:2303.02515 [INSPIRE].
A.D.I. Latorre, G.J. Olmo and M. Ronco, Observable traces of non-metricity: new constraints on metric-affine gravity, Phys. Lett. B 780 (2018) 294 [arXiv:1709.04249] [INSPIRE].
D.M. Ghilencea, Non-metric geometry as the origin of mass in gauge theories of scale invariance, Eur. Phys. J. C 83 (2023) 176 [arXiv:2203.05381] [INSPIRE].
G. ’t Hooft, Local conformal symmetry: the missing symmetry component for space and time, arXiv:1410.6675 [INSPIRE].
G. ’t Hooft, Local conformal symmetry in black holes, Standard Model, and quantum gravity, in 14th Marcel Grossmann meeting on recent developments in theoretical and experimental general relativity, astrophysics, and relativistic field theories, World Scientific, Singapore (2017), p. 3 [INSPIRE].
P.G. Ferreira, C.T. Hill, J. Noller and G.G. Ross, Scale-independent R2 inflation, Phys. Rev. D 100 (2019) 123516 [arXiv:1906.03415] [INSPIRE].
D.M. Ghilencea, Weyl R2 inflation with an emergent Planck scale, JHEP 10 (2019) 209 [arXiv:1906.11572] [INSPIRE].
D.M. Ghilencea, Gauging scale symmetry and inflation: Weyl versus Palatini gravity, Eur. Phys. J. C 81 (2021) 510 [arXiv:2007.14733] [INSPIRE].
A.A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91 (1980) 99 [INSPIRE].
M.J. Duff, Twenty years of the Weyl anomaly, Class. Quant. Grav. 11 (1994) 1387 [hep-th/9308075] [INSPIRE].
M.J. Duff, Observations on conformal anomalies, Nucl. Phys. B 125 (1977) 334 [INSPIRE].
D.M. Capper, M.J. Duff and L. Halpern, Photon corrections to the graviton propagator, Phys. Rev. D 10 (1974) 461 [INSPIRE].
D.M. Capper and M.J. Duff, Trace anomalies in dimensional regularization, Nuovo Cim. A 23 (1974) 173 [INSPIRE].
S. Deser, M.J. Duff and C.J. Isham, Nonlocal conformal anomalies, Nucl. Phys. B 111 (1976) 45 [INSPIRE].
F. Englert, C. Truffin and R. Gastmans, Conformal invariance in quantum gravity, Nucl. Phys. B 117 (1976) 407 [INSPIRE].
M. Shaposhnikov and A. Tokareva, Exact quantum conformal symmetry, its spontaneous breakdown, and gravitational Weyl anomaly, Phys. Rev. D 107 (2023) 065015 [arXiv:2212.09770] [INSPIRE].
M. Shaposhnikov and A. Tokareva, Anomaly-free scale symmetry and gravity, Phys. Lett. B 840 (2023) 137898 [arXiv:2201.09232] [INSPIRE].
L. Ciambelli and R.G. Leigh, Weyl connections and their role in holography, Phys. Rev. D 101 (2020) 086020 [arXiv:1905.04339] [INSPIRE].
W. Jia, M. Karydas and R.G. Leigh, Weyl-ambient geometries, Nucl. Phys. B 991 (2023) 116224 [arXiv:2301.06628] [INSPIRE].
W. Jia and M. Karydas, Obstruction tensors in Weyl geometry and holographic Weyl anomaly, Phys. Rev. D 104 (2021) 126031 [arXiv:2109.14014] [INSPIRE].
L. Smolin, Towards a theory of space-time structure at very short distances, Nucl. Phys. B 160 (1979) 253 [INSPIRE].
K. Hayashi and T. Kugo, Everything about Weyl’s gauge field, Prog. Theor. Phys. 61 (1979) 334 [INSPIRE].
D.S. Gorbunov and V.A. Rubakov, Introduction to the theory of the early universe, World Scientific, Singapore (2011) [https://doi.org/10.1142/7874].
I.L. Buchbinder and I. Shapiro, Introduction to quantum field theory with applications to quantum gravity, Oxford University Press, Oxford, U.K. (2021) [https://doi.org/10.1093/oso/9780198838319.001.0001].
M. Asorey, E.V. Gorbar and I.L. Shapiro, Universality and ambiguities of the conformal anomaly, Class. Quant. Grav. 21 (2003) 163 [hep-th/0307187] [INSPIRE].
J.F. Donoghue, M.M. Ivanov and A. Shkerin, EPFL lectures on general relativity as a quantum field theory, arXiv:1702.00319 [INSPIRE].
S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [INSPIRE].
M. Shaposhnikov and D. Zenhausern, Quantum scale invariance, cosmological constant and hierarchy problem, Phys. Lett. B 671 (2009) 162 [arXiv:0809.3406] [INSPIRE].
R. Armillis, A. Monin and M. Shaposhnikov, Spontaneously broken conformal symmetry: dealing with the trace anomaly, JHEP 10 (2013) 030 [arXiv:1302.5619] [INSPIRE].
D.M. Ghilencea, Manifestly scale-invariant regularization and quantum effective operators, Phys. Rev. D 93 (2016) 105006 [arXiv:1508.00595] [INSPIRE].
D.M. Ghilencea, Z. Lalak and P. Olszewski, Standard Model with spontaneously broken quantum scale invariance, Phys. Rev. D 96 (2017) 055034 [arXiv:1612.09120] [INSPIRE].
D.M. Ghilencea, Z. Lalak and P. Olszewski, Two-loop scale-invariant scalar potential and quantum effective operators, Eur. Phys. J. C 76 (2016) 656 [arXiv:1608.05336] [INSPIRE].
D.M. Ghilencea, Quantum implications of a scale invariant regularization, Phys. Rev. D 97 (2018) 075015 [arXiv:1712.06024] [INSPIRE].
F. Gretsch and A. Monin, Perturbative conformal symmetry and dilaton, Phys. Rev. D 92 (2015) 045036 [arXiv:1308.3863] [INSPIRE].
C. Tamarit, Running couplings with a vanishing scale anomaly, JHEP 12 (2013) 098 [arXiv:1309.0913] [INSPIRE].
R. Bach, Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs (in German), Math. Z. 9 (1921) 110.
W. Drechsler and H. Tann, Broken Weyl invariance and the origin of mass, Found. Phys. 29 (1999) 1023 [gr-qc/9802044] [INSPIRE].
J.T. Wheeler, Weyl gravity as general relativity, Phys. Rev. D 90 (2014) 025027 [arXiv:1310.0526] [INSPIRE].
M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Gauge theory of the conformal and superconformal group, Phys. Lett. B 69 (1977) 304 [INSPIRE].
R. Jackiw and S.-Y. Pi, Fake conformal symmetry in conformal cosmological models, Phys. Rev. D 91 (2015) 067501 [arXiv:1407.8545] [INSPIRE].
R. Jackiw and S.Y. Pi, New setting for spontaneous gauge symmetry breaking?, Fundam. Theor. Phys. 183 (2016) 159 [arXiv:1511.00994] [INSPIRE].
P.G. Ferreira, C.T. Hill and G.G. Ross, Scale-independent inflation and hierarchy generation, Phys. Lett. B 763 (2016) 174 [arXiv:1603.05983] [INSPIRE].
P.G. Ferreira, C.T. Hill and G.G. Ross, No fifth force in a scale invariant universe, Phys. Rev. D 95 (2017) 064038 [arXiv:1612.03157] [INSPIRE].
P.G. Ferreira, C.T. Hill and G.G. Ross, Inertial spontaneous symmetry breaking and quantum scale invariance, Phys. Rev. D 98 (2018) 116012 [arXiv:1801.07676] [INSPIRE].
P.G. Ferreira, C.T. Hill and G.G. Ross, Weyl current, scale-invariant inflation and Planck scale generation, Phys. Rev. D 95 (2017) 043507 [arXiv:1610.09243] [INSPIRE].
J. Garcia-Bellido, J. Rubio, M. Shaposhnikov and D. Zenhausern, Higgs-dilaton cosmology: from the early to the late universe, Phys. Rev. D 84 (2011) 123504 [arXiv:1107.2163] [INSPIRE].
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].
Acknowledgments
The author thanks Luca Ciambelli (Perimeter Institute), Cezar Condeescu (IFIN Bucharest), Christopher T. Hill (Fermilab), Weizhen Jia (University of Illinois, Urbana), Andrei Micu (IFIN Bucharest), Mikhail Shaposhnikov (University of Lausanne) for discussions on Weyl conformal geometry. This work was supported by a grant of Ministry of Education and Research (Romania), CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2020-2255.
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: 2309.11372
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
Ghilencea, D.M. Weyl conformal geometry vs Weyl anomaly. J. High Energ. Phys. 2023, 113 (2023). https://doi.org/10.1007/JHEP10(2023)113
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2023)113