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Standard Model, and Its Standard Problems

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Theoretical Physics, Wavelets, Analysis, Genomics

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

Since fifty years the standard model of particle physics reigns supreme. Not only does it explain the huge amount of accelerator data amassed with great precision. In cosmology it does have a say and may explain dark matter and absence of anti-matter. The model obtains by gauging its symmetries to generate the electroweak, strong and gravitational force. Symmetry breaking effects on the quantum level cause the standard model to obey a set of consistency conditions in order to be a viable quantum theory. These conditions need gravity to fix the hypercharges. The inclusion of gravity poses one of the standard problems, the unbearably heavy weight of the vacuum. Another standard problem discussed is the colossal spread in masses of the particles, more than 1013 decades.

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Notes

  1. 1.

    There are many theories that cannot be checked. But one thing you can check: they will come to a miserable end.

  2. 2.

    This result, the equivalence principle, is to be expected for elementary particles if you are ready to accept that the gravitational field couples to the particles through a massless spin two particle. Furthermore that relativistic field theory governs elementary particles. The argument is sixty years old and due to Weinberg [12].

  3. 3.

    Named after Pontecorvo, Maki, Nakagawa and Sakata [42].

  4. 4.

    It is amusing to note that the only other known scale on the horizontal axis between 10−3 − 10−2 eV is the scale of the energy density of the cosmological constant.

  5. 5.

    These are the Sakharov [61] conditions.

  6. 6.

    The effective coupling depends also on the occupation number, \(g_{eff}^2=g^2(T)/\left (\exp (p/T)-1\right )\). So for long wave length modes p ∼ g2(T)T the effective coupling is \({\mathcal O}(1)\), no matter how small g(T)!

  7. 7.

    For earlier work see [51].

  8. 8.

    Inversely, (3.16), (3.10), and (3.9) imply the gravitational constraint (3.19). Had there been a U(1) vector field coupling to lepton “colour” then the corresponding anomaly cancellation would have given (3.16).

  9. 9.

    This is easily justified for infinitesimal transformations x′ = x + δx.

  10. 10.

    The reduced Planck mass MP compares to the Fermi scale as v∕MP = 10−16.

  11. 11.

    In the original Einstein theory any given configuration with \(\sqrt {-g}\neq 1\) can be brought by an Einstein transformation to the uniform configuration \(\sqrt {-g}=1\), a fact already known to and used by Schwarzschild in his original black hole solution.

  12. 12.

    Note the factor 1∕2 compared to the anomaly of the axial current (3.7). It is due to the absence of the righthanded neutrino.

  13. 13.

    In absence of torsion.

  14. 14.

    For a physical interpretation see ’t Hooft [68].

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Acknowledgements

Thanks are due to Jan Smit, members of the NIKHEF theory group, Mikko Laine and Gerard ’t Hooft for discussions. Hospitality of the NIKHEF theory group and patience of Thierry Paul are gratefully acknowledged.

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  1. NIKHEF, Theory Group, Amsterdam, The Netherlands

    Chris P. Korthals Altes

  2. Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

    Chris P. Korthals Altes

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  1. Laboratoire de Physique, Université Lyon, ENS de Lyon, Université Claude Bernard, CNRS, Lyon, France

    Patrick Flandrin

  2. Mathematics, Université Paris Est Créteil, Créteil, France

    Stéphane Jaffard

  3. Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL), Paris, France

    Thierry Paul

  4. Mathematics, Université d’Aix-Marseille, Marseille, France

    Bruno Torresani

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Korthals Altes, C.P. (2023). Standard Model, and Its Standard Problems. In: Flandrin, P., Jaffard, S., Paul, T., Torresani, B. (eds) Theoretical Physics, Wavelets, Analysis, Genomics. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-45847-8_10

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