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.
There are many theories that cannot be checked. But one thing you can check: they will come to a miserable end.
- 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.
Named after Pontecorvo, Maki, Nakagawa and Sakata [42].
- 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.
These are the Sakharov [61] conditions.
- 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.
For earlier work see [51].
- 8.
- 9.
This is easily justified for infinitesimal transformations x′ = x + δx.
- 10.
The reduced Planck mass MP compares to the Fermi scale as v∕MP = 10−16.
- 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.
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.
In absence of torsion.
- 14.
For a physical interpretation see ’t Hooft [68].
References
Fock, V., 1926, Über die invariante Form der Wellen- und der Bewegungsgleichungen für einen geladenen Massenpunkt, Zeit. für Physik 39, 226–232.
Weyl, H., 1929, Elektron und Gravitation, Zeit. für Physik 56, 330–352.
Weyl,H.,1950, “A remark on the coupling of gravitation and electron, Phys.Rev.77,699.
Yang, C. N., Mills, R. (1954). “Conservation of Isotopic Spin and Isotopic Gauge Invariance”, Phys. Rev. 96 (1): 191–195.
Cartan, E., 1922, Sur une generalisation de la notion de courbure de Riemann et les espaces de torsion, C. R. Acad. Sci. (Paris)174 593 595. Cartan E. 1923, Sur la connexion affine et la theorie de la relativite generalisee, Part I Ann. Ec. Norm. 40: 325–412 and 41 1–25; Part II: 42 17–88.
R. Utiyama, Phys. Rev.101 (1956) 1597. Kibble T. W. B. 1961, “Lorentz Invariance and the Gravitational Field”, JMP2 212–221. D. W. Sciama, “On the analogy between charge and spin in General Relativity”, Festschrift for Infeld pg 415, Pergamon Press, Oxford 1962. R. Utiyama, Phys. Rev., 101, 1597 (1956).
Hehl, Friedrich W., von der Heyde Paul, Kerlick, G. David, and Nester. James M., 1976a, “General relativity with spin and torsion: Foundations and prospects”, Rev. Mod. Phys. 48, 393–416.
I. L. Shapiro, Physical Aspects of the Space-Time Torsion, Phys.Rept.357:113,2002, arXiv:hep-th/0103093.
Cosserat, E and F (1909).Théorie des corps déformables. Paris: Hermann.
Atom-Interferometric Test of the Equivalence Principle at the 10−12 Level Peter Asenbaum, Chris Overstreet, Minjeong Kim, Joseph Curti, and Mark A. Kasevich Phys. Rev. Lett. 125, 191101.
Eötvös R.V., Pekar D., Fekete E., Ann. Physik, 68(1922) 11–66. Roll P.G., Krotkov R., Dicke R.H., Ann. Phys., 26 (1964)442–517; Baessler S., Heckel B.R., Adelberger E.G., Gundlach J.H., Schmidt U. and Swanson H.E., Phys. Rev. Lett.83 (1999) 3585–3588.
S. Weinberg, Phys.Rev. B 135 (1964) 1049.
S. Bludman, Nuovo Cim. 9 (1958) 433. S. Glashow, Nucl.Phys. 22 (1961) 579.
Adler, S. L. Axial-vector vertex in spinor electrodynamics. Phys. Rev. 177, 2426 (1969); Bell, J. S., Jackiw, R.: A PCAC puzzle in the sigma-model. Nuovo Cim. 60A, 47–61 (1969).
Nielsen, H. B., Ninomiya, M. The Adler-Bell-Jackiw anomaly and Weyl Fermions in a crystal. Phys. Lett. B 130, 389–396 (1983).
Rabi, I.I. Das freie Elektron im homogenen Magnetfeld nach der Diracschen Theorie. Z. Physik 49, 507–511 (1928).
M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422–433.
Lee, T.D.; Yang, C.N. (1956). “Question of Parity Conservation in Weak Interactions”. Physical Review. 104 (1): 254–258.
Chirality invariance and the universal Fermi interaction, E.C.G. Sudarshan, R. E. Marshak, Phys.Rev. 109 (1958) 1860. R. P. Feynman and M. Gell-Mann, Phys. Rev. 109 (1958) 193.
M. Goldhaber, L.Grodzins and A.W. Sunyar, Phys.Rev. 109 (1958) 1015–1017.
Y. Nambu and G. Jona-Lasinio (1961), Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I, Phys. Rev. 122, 345–358.
P.W.Higgs, Phys. Rev. Lett. 13: 508–9 (1964). Englert, F. and Brout, R. Broken Symmetry and the Mass of Gauge Vector Mesons. Phys. Rev. Lett. 13: 321 (1964).
LHCb collaboration, arXiv:2103.11769.
’t Hooft, G (1971). Renormalization of massless Yang-Mills fields. Nucl. Phys. B33: 173. ’t Hooft, G (1971). Renormalizable Lagrangians for massive Yang-Mills fields. Nucl. Phys. B35: 167. ’t Hooft, G and Veltman, M (1972). Regularization and renormalization of gauge fields. Nucl. Phys. B44: 189.
D.J. Gross; F. Wilczek, Ultraviolet behavior of non-abelian gauge theories, Physical Review Letters. 30 (26): 1343–1346. H.D. Politzer, Reliable perturbative results for strong interactions”. Physical Review Letters. 30 (26): 1346–1349.
C.P. Korthals Altes and M. Perrottet, Phys Lett. B 39 (1972), 546.
B. Abi et al. (Muon g-2 Collaboration) Phys. Rev. Lett. 126, 141801.
The Standard Model Higgs boson as the inflaton, Fedor L. Bezrukov, Mikhail Shaposhnikov, Phys.Lett.B 659 (2008) 703–706, 0710.3755 [hep-th]
C. Germani, A. Kehagias, Phys.Rev.Lett.105.011302; arXiv:1003.2635v2 and references therein.
A.A. Belavin; A.M. Polyakov; A.S. Schwartz; Yu.S.Tyupkin (1975). “Pseudoparticle solutions of the Yang-Mills equations”. Phys. Lett. B. 59 (1): 85–87.
G.’t Hooft, Phys. Rev. Lett.37, 8 (1976); Phys. Rev.D14, 3432 (1976).
Super Kamiokande collaboration, Phys.Rev.Lett.86: 5656–5660, 2001; arXiv:hep-ex/0103033.
Q.R. Ahmad et al., Phys.Rev. Lett.89, 011302, 2002, nucl-ex/0204008. For a review see A.B. McDonald, Invited Paper for Nobel Symposium 129, August 19–24, 2004, Enköping, Sweden, Phys.Scripta T121 (2005) 29–32; arXiv:hep-ex/0412060v1.
Atlas collaboration, Phys.Lett. B716 (2012) 1–29; arXiv:1207.7214 [hep-ex].
A Model of Leptons. Steven Weinberg, Phys.Rev.Lett. 19 (1967) 1264–1266.
Bouchiat, C., Iliopoulos, J., Meyer, Ph. (1972), An anomaly-free version of Weinberg’s model, Physics Letters B 38 (7): 519–523, (1972).
H. Georgi and S. L. Glashow, “Gauge theories without anomalies,” Phys. Rev. D 6, 429 (1972). D. J. Gross and R. Jackiw, “Effect of anomalies on quasi-renormalizable theories,” Phys. Rev. D6, 477 (1972). L. Alvarez-Gaume and E. Witten, “Gravitational Anomalies,” Nucl. Phys. B 234, 269 (1984).
Fritzsch, H.; Gell-Mann, M.; Leutwyler, H. (1973). “Advantages of the color octet gluon picture”. Physics Letters. 47B (4): 365–368.
S. Weinberg, Baryon and lepton non-conserving processes, Phys. Rev. Lett. 43, 1566, 1979.
Kobayashi, M.; Maskawa, T. (1973). “CP-violation in the renormalizable theory of weak interaction”. Progress of Theoretical Physics. 49 (2): 652–657.
P. Minkowski, Physics Letters B67 (1977), 421–428; Gell-Mann, M.; Ramond, P.; Slansky, R. (1979). Freedman, D.; Van Nieuwenhuizen, P. (eds.). Supergravity. Amsterdam: North Holland. pp. 315–321.
Maki, Z.; Nakagawa, M.; Sakata, S. (1962). “Remarks on the unified model of elementary particles”. Progress of Theoretical Physics. 28 (5): 870.
A. Boyarsky, J. Franse, D. Iakubovskyi, and O. Ruchayskiy, Phys. Rev. Lett. 115, 161301 (2015), 1408.2503. Deep XMM Observations of Draco rule out at the 99% Confidence Level a Dark Matter Decay Origin for the 3.5 keV Line; Tesla E. Jeltema, Stefano Profumo, Mon.Not.Roy.Astron.Soc. 458 (2016) 4, 3592–3596, 1512.01239 [astro-ph.HE].
Takehiko Asaka and Mikhail Shaposhnikov, The νMSM, Dark Matter and Baryon Asymmetry of the Universe, Phys.Lett.B620:17–26,2005;arXiv:hep-ph/0505013. Takehiko Asaka, Steve Blanchet, and Mikhail Shaposhnikov, Phys.Lett.B631:151–156,2005, arXiv:hep-ph/0503065v1. Uniting Low-Scale Leptogenesis Mechanisms, Juraj Klarić, Mikhail Shaposhnikov(EPFL, Lausanne, LPPC and Ecole Polytechnique, Lausanne), Inar Timiryasov, Phys.Rev.Lett. 127 (2021) 11, 111802 ⋅ e-Print: 2008.13771 [hep-ph]
P.F. De Salas, S. Gariazzo, O. Mena, C.A. Ternes, M. Tórtola, Front.Astron.Space Sci. 5 (2018) 36; e-Print: 1806.11051 [hep-ph].
see e.g. Relic neutrino decoupling with flavour oscillations revisited, Pablo F. de Salas, Sergio Pastor, JCAP 07 (2016) 051 ⋅ e-Print: 1606.06986 [hep-ph].
Matching conditions and Higgs mass upper bounds revisited, Thomas Hambye, Kurt Riesselmann, Phys.Rev.D 55 (1997) 7255–7262; hep-ph/9610272 [hep-ph]. For a review: F. Jegerlehner, Acta Phys.Polon.B 52 (2021) 6–7, 575–605; 2106.00862 [hep-ph].
M. Laine, M.Meyer, JCAP 1507 (2015) 035; arXiv:1503.04935v2 [hep-ph].
Michela D’Onofrio, Kari Rummukainen, Phys. Rev. D 93, 025003 (2016); arXiv:1508.07161v1 [hep-ph] and references therein.
J. Ghiglieri and M. Laine, Gravitational wave background from Standard Model physics: Qualitative features, JCAP 07 (2015) 022 [1504.02569]. Gravitational wave background from non-Abelian reheating after axion-like inflation, P. Klose, M. Laine, S. Procacci; 2201.02317 [hep-ph]
C.Q. Geng and R.E. Marshak, Phys. Rev. D 39. 693 (1989); J.A. Minahan, Pierre Ramond, R.C. Warner, Phys.Rev.D 41 (1990) 715; A. Font, L. Ibanez and F. Quevedo, Phys. Lett. 228B, 79 (1989); R. Foot, G. C. Joshi, H. Lew and R. R. Volkas, Mod. Phys. Lett. A5, 95 (1990); K.S. Babu and R. N. Mohapatra, Phys. Rev. Lett. 63, 938 (1989); J.Minahan, P. Ramond and R. Warner, Phys. Rev. D 41, 715 (1990); S. Rudaz, Phys. Rev. D41, 2619 (1990). E. Golowich and P. B. Pal, Phys. Rev.D 41, 3537 (1990); P.H. Frampton, R.N. Mohapatra, Phys.Rev.D 50 (1994) 3569–3571; hep-ph/9312230 [hep-ph].
Nakarin Lohitsiri, David Tong, SciPost Phys. 8 (2020) 1, 009; 1907.00514 [hep-th]. .
Is the Lee constant a cosmological constant? Andrei D. Linde, JETP Lett. 19 (1974) 183, Pisma Zh.Eksp.Teor.Fiz. 19 (1974) 320–322. Cosmology and the Higgs Mechanism, M.Veltman, Phys.Rev. Lett. 34,(1975), 777.
The Exchange of Massless Spin Two Particles, J.J. van der Bij, H. van Dam, Yee Jack Ng, Physica A 116 (1982) 307–320. Zee, A., 1985, in High Energy Physics: Proceedings of the 20th Annual Orbis Scientiae, 1983, edited by S. L. Mintz and A. Perlmutter (Plenum, New York). Buchmuller, W., and N. Dragon, Phys.Lett.B 207 (1988) 292–294, Phys.Lett.B 223 (1989) 313–317. Self-tuning vacuum variable and cosmological constant, F. R. Klinkhamer, G. E. Volovik, Phys.Rev.D 77 (2008) 085015.
S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 61,1 (1989).
Casimir, H.B.G.,1948, K.Ned.Acad. Wet. 51, 635.
Sparnaay, M (1958). “Measurements of attractive forces between flat plates”. Physica. 24 (6–10): 751–764. Mohideen, U.; Roy, Anushree (1998). “Precision Measurement of the Casimir Force from 0.1 to 0.9 μm”. Physical Review Letters. 81 (21): 4549–4552. arXiv:physics/9805038.
E.P. Wigner, Ann.Math 40(1939) 149. Group Theoretical Discussion of Relativistic Wave Equations, V. Bargmann, Eugene P. Wigner, Proc.Nat.Acad.Sci. 34 (1948) 211
Riess, A. G. et al., The Astronomical Journal. 116 (3): 1009–1038. arXiv:astro-ph/9805201. Perlmutter, S. et al., The Astrophysical Journal. 517 (2): 565–586. arXiv:astro-ph/9812133. Schmidt, B.P. et al., The Astrophysical Journal. 507 (1): 46–63. arXiv:astro-ph/9805200. The Planck Collaboration (2020). “Planck 2018 results. VI. Cosmological parameters”. Astronomy and Astrophysics. 641: A6. arXiv:1807.06209.
Topology in the Weinberg-Salam Theory, N.S. Manton, Phys.Rev.D 28 (1983) 2019. A Saddle Point Solution in the Weinberg-Salam Theory, Frans R. Klinkhamer, N.S. Manton, Phys.Rev.D 30 (1984) 2212.
Sakharov, A. D., “Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe”, JETP Letters. 5 (1): 24–26.
V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 155, 36 (1985).
T. Kimura, Prog. Theor. Phys.42, 1191 (1969). R. Delbourgo and A. Salam, Phys. Lett.40B, 381 (1972). T. Eguchi and P. Freund, Phys. Rev. Lett., 1251 (1976).
A. Dobado and A. Maroto, The standard model anomalies in curved space-time with torsion, Phys. Rev.D54(1996) 5185–5194,hep-ph/9509227.
S. W. Hawking, Gravitational instantons, Phys. Lett.60A(1977) 81.
Tohru Eguchi, Andrew J. Hanson, Self-dual Solutions to Euclidean Gravity, Annals Phys. 120 (1979) 82.
G.W. Gibbons, S.W. Hawking, Phys. Lett. 78B (1978),430.
A Physical Interpretation of Gravitational Instantons, Gerard ’t Hooft, Nucl.Phys.B 315 (1989) 517–527.
Page, D.N., Phys. Lett. 78B 249 (1978).
S.W. Hawking and C.N. Pope, Phys. Lett. 73B (1978) 42.
The Positive action conjecture and asymptotically Euclidean Metrics in Quantum Gravity, G.W. Gibbons, C.N. Pope, Commun.Math.Phys.66, 267–290(1979); New gravitational index theorems and super theorems, S.M.Christensen, M.J.Duff, Nuclear Physics B154 (1979).
D.T. Regan, 66th US Secretary of the Treasury, dixit.
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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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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