Models of Electroweak Interactions in Non-Commutative Geometry: A Comparison

TL;DR

This paper compares two non-commutative geometric models of electroweak interactions, analyzing their differences in Higgs potential and fermion mass terms, and clarifies their relationship at the differential algebra level.

Contribution

It provides a detailed comparison of Connes' and Marseille-Mainz approaches to non-commutative geometry models of electroweak interactions, clarifying their differences.

Findings

Connes' model avoids gamma_5 factors in fermion mass terms.

The Higgs potential differs between the two models.

Both models can be formulated within the same differential algebra framework.

Abstract

Alain Connes' construction of the standard model is based on a generalized Dirac-Yukawa operator and the K-cycle $(\HD ,D)$, with $\HD$ a fermionic Hilbert space. If this construction is reformulated at the level of the differential algebra then a direct comparison with the alternative approach by the Marseille-Mainz group becomes possible. We do this for the case of the toy model based on the structure group $U(1)\times U(1)$ and for the $SU(2)\times U(1)$ of electroweak interactions. Connes' results are recovered without the somewhat disturbing $\gamma_{5}$-factors in the fermion mass terms and Yukawa couplings. We discuss both constructions in the same framework and, in particular, pinpoint the origin of the difference in the Higgs potential obtained by them.