Abstract
We investigate whether the Standard Model, within the accuracy of current experimental measurements, satisfies the regularity in the form of Hodge duality condition introduced and studied in [9]. We show that the neutrino and quark mass-mixing and the difference of fermion masses are necessary for this property. We demonstrate that the current data supports this new geometric feature of the Standard Model, Hodge duality, provided that all neutrinos are massive.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Connes, Noncommutative geometry and reality, J. Math. Phys. 36 (1995) 6194 [INSPIRE].
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, Colloquium Publications. Vol. 55: Noncommutative geometry, quantum fields and motives, AMS Press, New York U.S.A. (2007).
A.H. Chamseddine and A. Connes, Resilience of the Spectral Standard Model, JHEP 09 (2012) 104 [arXiv:1208.1030] [INSPIRE].
A. Devastato, F. Lizzi and P. Martinetti, Higgs mass in Noncommutative Geometry, Fortsch. Phys. 62 (2014) 863 [arXiv:1403.7567] [INSPIRE].
F. D’Andrea, M.A. Kurkov and F. Lizzi, Wick Rotation and Fermion Doubling in Noncommutative Geometry, Phys. Rev. D 94 (2016) 025030 [arXiv:1605.03231] [INSPIRE].
A. Bochniak and A. Sitarz, Finite pseudo-Riemannian spectral triples and the standard model, Phys. Rev. D 97 (2018) 115029 [arXiv:1804.09482] [INSPIRE].
F. D’Andrea and L. Dabrowski, The Standard Model in Noncommutative Geometry and Morita equivalence, arXiv:1501.00156 [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 [arXiv:1703.05279] [INSPIRE].
A. Connes, On the spectral characterization of manifolds, J. Noncommut. Geom. 7 (2013) 1 [arXiv:0810.2088] [INSPIRE].
S. Lord, A. Rennie and J.C. Várilly, Riemannian manifolds in noncommutative geometry, J. Geom. Phys. 62 (2012) 1611 [arXiv:1109.2196] [INSPIRE].
R.J. Plymen, Strong Morita equivalence, spinors and symplectic spinors, J. Operat. Theor. 16 (1986) 305.
M. Paschke and A. Sitarz, Discrete sprectral triples and their symmetries, J. Math. Phys. 39 (1998) 6191 [INSPIRE].
W.D. van Suijlekom, Noncommutative Geometry and Particle Physics, Springer, Heidelberg Germany (2015).
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].
M. Paschke, F. Scheck and A. Sitarz, Can (noncommutative) geometry accommodate leptoquarks?, Phys. Rev. D 59 (1999) 035003 [hep-th/9709009] [INSPIRE].
W. Burnside, On the condition of reducibility of any group of linear substitutions, Proc. London Math. Soc. 3 (1905) 430.
F. Capozzi, E. Lisi, A. Marrone and A. Palazzo, Current unknowns in the three neutrino framework, Prog. Part. Nucl. Phys. 102 (2018) 48 [arXiv:1804.09678] [INSPIRE].
Particle Data Group, Review of Particle Physics, Chin. Phys. C 40 (2016) 100001.
T. Ohlsson and S. Zhou, Renormalization group running of neutrino parameters, Nature Commun. 5 (2014) 5153 [arXiv:1311.3846] [INSPIRE].
A. Denner and T. Sack, Renormalization of the Quark Mixing Matrix, Nucl. Phys. B 347 (1990) 203 [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: 1806.07282
Partially supported through H2020-MSCA-RISE-2015-691246-QUANTUM DYNAMICS and through Polish support grant for the international cooperation project 3542/H2020/2016/2 and 328941/PnH/2016. (Ludwik Dąbrowski)
This work was supported through National Science Centre grant OPUS 2016/21/B/ST1/02438. (Andrzej Sitarz)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Dąbrowski, L., Sitarz, A. Fermion masses, mass-mixing and the almost commutative geometry of the Standard Model. J. High Energ. Phys. 2019, 68 (2019). https://doi.org/10.1007/JHEP02(2019)068
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2019)068