Rethinking Connes' approach to the standard model of particle physics via non-commutative geometry

Abstract

Connes' non-commutative geometry (NCG) is a generalization of Riemannian geometry that is particularly apt for expressing the standard model of particle physics coupled to Einstein gravity. In a previous paper, we suggested a reformulation of this framework that is: (i) simpler and more unified in its axioms, and (ii) allows the Lagrangian for the standard model of particle physics (coupled to Einstein gravity) to be specified in a way that is tighter and more explanatory than the traditional algorithm based on effective field theory. Here we explain how this same reformulation yields a new perspective on the symmetries of a given NCG. Applying this perspective to the NCG traditionally used to describe the standard model we find, instead, an extension of the standard model by an extra $U(1{)}_{B-L}$ gauge symmetry, and a single extra complex scalar field $\sigma$, which is a singlet under $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y$, but has $B-L=2$. This field has cosmological implications, and offers a new solution to the discrepancy between the observed Higgs mass and the NCG prediction.

arXiv commentv2: 5 pages, no figures, minor changes matching NJP published version

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• The standard model, the Pati-Salam model, and "Jordan geometry"
• The graded product of real spectral triples
• Spectral Noncommutative Geometry, Standard Model and all that
• The geometry of physical observables
• A $U(1{)}_{B-L}$-extension of the Standard Model from Noncommutative Geometry
• A new algebraic structure in the standard model of particle physics
• Fermionic spectral action and the origin of nonzero neutrino masses
• Algebraic backgrounds: a framework for noncommutative Kaluza-Klein theory
• Non-Associative Geometry and the Spectral Action Principle
• Particle models from special Jordan backgrounds and spectral triples
• Topics in Noncommutative Geometry
• Type-II two-Higgs-doublet model in noncommutative geometry