Abstract
We analyze the perturbative quantization of the spectral action in noncommutative geometry and establish its one-loop renormalizability in a generalized sense, while staying within the spectral framework of noncommutative geometry. Our result is based on the perturbative expansion of the spectral action in terms of higher Yang-Mills and Chern-Simons forms. In the spirit of random noncommutative geometries, we consider the path integral over matrix fluctuations around a fixed noncommutative gauge background and show that the corresponding one-loop counterterms are of the same form so that they can be safely subtracted from the spectral action. A crucial role will be played by the appropriate Ward identities, allowing for a fully spectral formulation of the quantum theory at one loop.
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References
N. Alkofer, F. Saueressig and O. Zanusso, Spectral dimensions from the spectral action, Phys. Rev. D 91 (2015) 025025 [arXiv:1410.7999] [INSPIRE].
S. Azarfar and M. Khalkhali, Random Finite Noncommutative Geometries and Topological Recursion, arXiv:1906.09362 [INSPIRE].
J. W. Barrett and L. Glaser, Monte Carlo simulations of random non-commutative geometries, J. Phys. A 49 (2016) 245001 [arXiv:1510.01377] [INSPIRE].
W. Beenakker, T. van den Broek and W. D. van Suijlekom, Supersymmetry and noncommutative geometry, in SpringerBriefs in Mathematical Physics 9, Springer, Cham, Switzerland (2016).
D. Bessis, C. Itzykson and J. B. Zuber, Quantum field theory techniques in graphical enumeration, Adv. Appl. Math. 1 (1980) 109 [INSPIRE].
A. Bochniak and A. Sitarz, Spectral geometry for the standard model without fermion doubling, Phys. Rev. D 101 (2020) 075038 [arXiv:2001.02902] [INSPIRE].
L. Boyle and S. Farnsworth, Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics, New J. Phys. 16 (2014) 123027 [arXiv:1401.5083] [INSPIRE].
T. van den Broek and W. D. van Suijlekom, Supersymmetric QCD from noncommutative geometry, Phys. Lett. B 699 (2011) 119 [INSPIRE].
A. H. Chamseddine and A. Connes, Universal formula for noncommutative geometry actions: Unification of gravity and the standard model, Phys. Rev. Lett. 77 (1996) 4868 [hep-th/9606056] [INSPIRE].
A. H. Chamseddine, A. Connes and M. Marcolli, Gravity and the standard model with neutrino mixing, Adv. Theor. Math. Phys. 11 (2007) 991 [hep-th/0610241] [INSPIRE].
A. H. Chamseddine, A. Connes and V. Mukhanov, Quanta of Geometry: Noncommutative Aspects, Phys. Rev. Lett. 114 (2015) 091302 [arXiv:1409.2471] [INSPIRE].
A. H. Chamseddine, A. Connes and W. D. van Suijlekom, Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification, JHEP 11 (2013) 132 [arXiv:1304.8050] [INSPIRE].
A. H. Chamseddine, A. Connes and W. D. van Suijlekom, Grand Unification in the Spectral Pati-Salam Model, JHEP 11 (2015) 011 [arXiv:1507.08161] [INSPIRE].
A. H. Chamseddine, J. Iliopoulos and W. D. van Suijlekom, Spectral action in matrix form, Eur. Phys. J. C 80 (2020) 1045 [arXiv:2009.03367] [INSPIRE].
A. Connes, Noncommutative Geometry, Academic Press, San Diego, CA, U.S.A. (1994).
A. Connes, Gravity coupled with matter and foundation of noncommutative geometry, Commun. Math. Phys. 182 (1996) 155 [hep-th/9603053] [INSPIRE].
A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives, American Mathematical Society, Providence, RI, U.S.A. (2008).
A. Connes and A. H. Chamseddine, Inner fluctuations of the spectral action, J. Geom. Phys. 57 (2006) 1 [hep-th/0605011] [INSPIRE].
L. Dąbrowski, F. D’Andrea and A. Sitarz, The Standard Model in noncommutative geometry: fundamental fermions as internal forms, Lett. Math. Phys. 108 (2018) 1323 [Erratum ibid. 109 (2019) 2585] [arXiv:1703.05279] [INSPIRE].
L. Dąbrowski and A. Sitarz, Fermion masses, mass-mixing and the almost commutative geometry of the Standard Model, JHEP 02 (2019) 068 [arXiv:1806.07282] [INSPIRE].
A. Devastato, F. Lizzi and P. Martinetti, Higgs mass in Noncommutative Geometry, Fortsch. Phys. 62 (2014) 863 [arXiv:1403.7567] [INSPIRE].
A. Devastato, F. Lizzi and P. Martinetti, Grand Symmetry, Spectral Action, and the Higgs mass, JHEP 01 (2014) 042 [arXiv:1304.0415] [INSPIRE].
A. Devastato and P. Martinetti, Twisted spectral triple for the Standard Model and spontaneous breaking of the Grand Symmetry, Math. Phys. Anal. Geom. 20 (2017) 2.
E. Gesteau, Renormalizing Yukawa interactions in the standard model with matrices and noncommutative geometry, J. Phys. A 54 (2020) 035203.
L. Glaser and A. Stern, Understanding truncated non-commutative geometries through computer simulations, J. Math. Phys. 61 (2020) 033507 [arXiv:1909.08054] [INSPIRE].
L. Glaser and A. B. Stern, Reconstructing manifolds from truncations of spectral triples, J. Geom. Phys. 159 (2021) 103921.
J. Gomis and S. Weinberg, Are nonrenormalizable gauge theories renormalizable?, Nucl. Phys. B 469 (1996) 473 [hep-th/9510087] [INSPIRE].
H. Grosse and R. Wulkenhaar, Power counting theorem for nonlocal matrix models and renormalization, Commun. Math. Phys. 254 (2005) 91 [hep-th/0305066] [INSPIRE].
H. Grosse and R. Wulkenhaar, Renormalization of ϕ4 theory on noncommutative ℝ4 in the matrix base, Commun. Math. Phys. 256 (2005) 305 [hep-th/0401128] [INSPIRE].
B. Iochum, C. Levy and D. Vassilevich, Spectral action beyond the weak-field approximation, Commun. Math. Phys. 316 (2012) 595 [arXiv:1108.3749] [INSPIRE].
R. A. Iseppi, The BV formalism: theory and application to a matrix model, Rev. Math. Phys. 31 (2019) 1950035.
R. A. Iseppi and W. D. van Suijlekom, Noncommutative geometry and the BV formalism: application to a matrix model, J. Geom. Phys. 120 (2017) 129 [arXiv:1604.00046] [INSPIRE].
M. Khalkhali and N. Pagliaroli, Phase Transition in Random Noncommutative Geometries, J. Phys. A 54 (2021) 035202 [arXiv:2006.02891] [INSPIRE].
M. A. Kurkov, F. Lizzi and D. Vassilevich, High energy bosons do not propagate, Phys. Lett. B 731 (2014) 311 [arXiv:1312.2235] [INSPIRE].
M. Nakahara, Geometry, Topology and Physics, IOP Publishing, Bristol, U.K. (1990).
T. D. H. van Nuland and W. D. van Suijlekom, Cyclic cocycles in the spectral action, to appear in J. Noncommut. Geom., arXiv:2104.09899 [INSPIRE].
A. Sitarz, Towards the signs of new physics through the spectral action, Int. J. Geom. Meth. Mod. Phys. 17 (2020) 2040008 [INSPIRE].
A. Skripka, Asymptotic expansions for trace functionals, J. Funct. Anal. 266 (2014) 2845.
W. D. van Suijlekom, Perturbations and operator trace functions, J. Funct. Anal. 260 (2011) 2483.
W. D. van Suijlekom, Renormalization of the spectral action for the Yang-Mills system, JHEP 03 (2011) 146 [arXiv:1101.4804] [INSPIRE].
W. D. van Suijlekom, Noncommutative Geometry and Particle Physics, in Mathematical Physics Studies, Springer, Dordrecht, The Netherlands (2015).
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van Nuland, T.D.H., van Suijlekom, W.D. One-loop corrections to the spectral action. J. High Energ. Phys. 2022, 78 (2022). https://doi.org/10.1007/JHEP05(2022)078
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DOI: https://doi.org/10.1007/JHEP05(2022)078