Fermion masses, mass-mixing and the almost commutative geometry of the Standard Model

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.

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References

  1. [1]

    A. Connes, Noncommutative geometry and reality, J. Math. Phys. 36 (1995) 6194 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    A. Connes, Gravity coupled with matter and foundation of noncommutative geometry, Commun. Math. Phys. 182 (1996) 155 [hep-th/9603053] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  3. [3]

    A. Connes and M. Marcolli, Colloquium Publications. Vol. 55: Noncommutative geometry, quantum fields and motives, AMS Press, New York U.S.A. (2007).

  4. [4]

    A.H. Chamseddine and A. Connes, Resilience of the Spectral Standard Model, JHEP 09 (2012) 104 [arXiv:1208.1030] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    A. Devastato, F. Lizzi and P. Martinetti, Higgs mass in Noncommutative Geometry, Fortsch. Phys. 62 (2014) 863 [arXiv:1403.7567] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    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].

    ADS  MathSciNet  Google Scholar 

  7. [7]

    A. Bochniak and A. Sitarz, Finite pseudo-Riemannian spectral triples and the standard model, Phys. Rev. D 97 (2018) 115029 [arXiv:1804.09482] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  8. [8]

    F. D’Andrea and L. Dabrowski, The Standard Model in Noncommutative Geometry and Morita equivalence, arXiv:1501.00156 [INSPIRE].

  9. [9]

    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].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    A. Connes, On the spectral characterization of manifolds, J. Noncommut. Geom. 7 (2013) 1 [arXiv:0810.2088] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    S. Lord, A. Rennie and J.C. Várilly, Riemannian manifolds in noncommutative geometry, J. Geom. Phys. 62 (2012) 1611 [arXiv:1109.2196] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    R.J. Plymen, Strong Morita equivalence, spinors and symplectic spinors, J. Operat. Theor. 16 (1986) 305.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    M. Paschke and A. Sitarz, Discrete sprectral triples and their symmetries, J. Math. Phys. 39 (1998) 6191 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    W.D. van Suijlekom, Noncommutative Geometry and Particle Physics, Springer, Heidelberg Germany (2015).

    Google Scholar 

  15. [15]

    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].

    ADS  MathSciNet  Article  Google Scholar 

  16. [16]

    M. Paschke, F. Scheck and A. Sitarz, Can (noncommutative) geometry accommodate leptoquarks?, Phys. Rev. D 59 (1999) 035003 [hep-th/9709009] [INSPIRE].

    ADS  Google Scholar 

  17. [17]

    W. Burnside, On the condition of reducibility of any group of linear substitutions, Proc. London Math. Soc. 3 (1905) 430.

    MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    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].

    ADS  Article  Google Scholar 

  19. [19]

    Particle Data Group, Review of Particle Physics, Chin. Phys. C 40 (2016) 100001.

  20. [20]

    T. Ohlsson and S. Zhou, Renormalization group running of neutrino parameters, Nature Commun. 5 (2014) 5153 [arXiv:1311.3846] [INSPIRE].

    ADS  Article  Google Scholar 

  21. [21]

    A. Denner and T. Sack, Renormalization of the Quark Mixing Matrix, Nucl. Phys. B 347 (1990) 203 [INSPIRE].

    ADS  Article  Google Scholar 

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Correspondence to Andrzej Sitarz.

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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)

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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

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Keywords

  • Non-Commutative Geometry
  • Quark Masses and SM Parameters