Abstract
Grand symmetry models in noncommutative geometry, characterized by a non-trivial action of functions on spinors, have been introduced to generate minimally (i.e. without adding new fermions) and in agreement with the first order condition an extra scalar field beyond the standard model, which both stabilizes the electroweak vacuum and makes the computation of the mass of the Higgs compatible with its experimental value. In this paper, we use a twist in the sense of Connes-Moscovici to cure a technical problem due to the non-trivial action on spinors, that is the appearance together with the extra scalar field of unbounded vectorial terms. The twist makes these terms bounded and - thanks to a twisted version of the first-order condition that we introduce here - also permits to understand the breaking to the standard model as a dynamical process induced by the spectral action, as conjectured in [24]. This is a spontaneous breaking from a pre-geometric Pati-Salam model to the almost-commutativegeometryofthestandardmodel,withtwoHiggs-likefields: scalar and vector.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Andrianov, A.A., Kurkov, M.A., Lizzi, F.: Spectral action, Weyl anomaly and the higgs-dilaton potential. JHEP 1110(001) (2011)
Andrianov, A.A., Lizzi, F.: Bosonic spectral action induced from anomaly cancelation. JHEP 05(057) (2010)
Barrett, J.W.: A Lorentzian version of the non-commutative geometry of the standard model of particle physics. J. Math. Phys. 48, 012303 (2007)
Farnsworth, S., Boyle, L.: Rethinking Connes’ approach to the standard model of particle physics via non-commutative geometry NJP (2014)
Buttazzo, D., Degrassi, G., Giardino, P.P., Giudice, G.F., Sala, F., Salvio, A.: Investigating the near-criticality of the Higgs boson, arXiv:1307.3536[hep-ph]
Chamseddine, A.H., Connes, A.: The spectral action principle. Commun. Math. Phys. 186, 737–750 (1996)
Chamseddine, A.H., Connes, A.: Why the standard model. J. Geom. Phys. 58, 38–47 (2008)
Chamseddine, A.H., Connes, A.: Resilience of the spectral standard model. JHEP 09, 104 (2012)
Chamseddine, A.H., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991–1089 (2007)
Chamseddine, A.H., Connes, A., Mukhanov, V.: Quanta of geometry, arXiv:1409.2471; Geometry and the quantum: basics, 1409.2471; Geometry and the quantum: basics, arXiv:1411.0977 (2014)
Chamseddine, A.H., Connes, A., Van Suijlekom, W.D: Beyond the spectral standard model: emergence of Pati-Salam unification. JHEP 11, 132 (2013)
Chamseddine, A.H., Connes, A., Van Suijlekom, W.D.: Inner fluctuations in noncommutative geometry without first order condition. J. Geom. Phys. 73, 222–234 (2013)
Chen, C.-S., Tang, Y.: Vacuum stability, neutrinos, and dark matter. JHEP 1204(019) (2012)
Connes, A.: Gravity coupled with matter and the foundations of noncommutative geometry. Commun. Math. Phys. 182, 155–176 (1996)
Connes, A., Lott, J.: Particle models and noncommtative geometry. Nuclear Phys. B Proc. Suppl. 18B, 29–47 (1990)
Connes, A., Marcolli, M.: Noncommutative geometry, quantum fields and motives AMS (2008)
Connes, A., Moscovici, H.: Type III and spectral triples, Traces in number theory, geo. and quantum fields. Aspects Math. E38 (2008). no. Friedt. Vieweg, Wiesbaden, 57–71
Connes, A.: Noncommutative geometry. Academic Press (1994)
Connes, A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194–6231 (1995)
Connes, A.: On the spectral characterization of manifolds. J. Noncom. Geom. 7(1), 1–82 (2013)
Brzezinski, T., Ciccoli, N., Dabrowski, L., Sitarz, A.: Twisted reality condition for Dirac operators. Math. Phys. Anal. Geo., 19–16 (2016)
Devastato, A.: Spectral action and gravitational effects at the Planck scale, arXiv:1309.5973
Devastato, A., Lizzi, F., Flores, C.V., Vassilevich, D.: Unification of coupling constants, dimension six operators and the spectral action, arXiv:1410.6624
Devastato, A., Lizzi, F., Martinetti, P.: Grand Symmetry, Spectral Action and the Higgs mass. JHEP 01, 042 (2014)
Agostino D., Lizzi, F., Martinetti, P.: Higgs mass in noncommutative geometry. Fortschr. Phys. 62(9-10), 863–868 (2014)
Elias-Miro, J., Espinosa, J.R., Giudice, G.F., Min Lee, H., Strumia, A.: Stabilization of the electroweak vacuum by a scalar threshold effect. JHEP 06(031) (2012)
Espinosa, J.R.: Vacuum stability and the Higgs boson, PoS (LATTICE 2013) 010
Espinosa, J.R., Giudice, G.F., Riotto, A.: Cosmological implications of the Higgs mass measurement. JCAP 0805(002) (2008)
Fathizadeh, F., Khalkhali, M.: Twisted spectral triples and Connes’ character formula. Fields Inst. Commun. Ser., 61 (2011)
Gilkey, P.B.: Invariance theory, the heat equation and the atiya-singer theorem, Publish or Perish (1984)
Greenfield, M., Marcolli, M., Teh, K.: Twisted spectral triples and quantum statistical mechanical systems, P-Adic Numbers. Ultrametric Anal. Appl. 6(2), 81–104 (2014)
Gracia-Bondia, B., Iochum J. M., Schücker, T.: The standard model in noncommutative geometry and fermion doubling. Phys. Lett. B 416, 123 (1998)
Iochum, B., Masson, T.: Heat trace for Laplacian type operators with non-scalar symbols (2016)
Landi, G., Martinetti, P.: On twisting real spectral triples by algebra automorphisms, Letters Mathematics Physics (2016)
Lizzi, F., Mangano, G., Miele, G., Sparano, G.: Fermion Hilbert space and fermion doubling in the noncommutative geo. approach to gauge theories. Phys. Rev. D 55, 6357 (1997)
Lizzi, F., Mangano, G., Miele, G., Sparano, G.: Mirror fermions in noncommutative geometry. Mod. Phys. Lett. A 13, 231 (1998)
Stephan, C.A.: New scalar fields in noncommutative geo. Phys. Rev. D 79, 065013 (2009)
Stephan, C.A.: Noncommutative geometry in the LHC-era (2013)
Van Suijlekom, W.: Noncommutative geometry and particle physics. Springer (2015)
Vassilevich, D.V.: Heat kernel expansion: user’s manual. Phys. Rep. 388, 279–360 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Devastato, A., Martinetti, P. Twisted Spectral Triple for the Standard Model and Spontaneous Breaking of the Grand Symmetry. Math Phys Anal Geom 20, 2 (2017). https://doi.org/10.1007/s11040-016-9228-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-016-9228-7