Chamseddine-Fröhlich-Grandjean metric and localized Higgs coupling in noncommutative geometry
type: Item
instance of: scholarly article
zbMATH Open document ID: 1012.81048
DOI: 10.1143/PTP.105.161
author: Akio Sugamoto
author: Rie Kuriki
published in: Progress of Theoretical Physics
publication date: 2003-06-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/hep-th/0004127
Mathematics Subject Classification ID: 81T75
Mathematics Subject Classification ID: 81V22
Mathematics Subject Classification ID: 46L87
zbMATH DE Number: 1713639
zbMATH Keywords: standard model
zbMATH Keywords: gravitational sector
zbMATH Keywords: Connes-Lott formulation
zbMATH Keywords: Wilson-like operators
MaRDI profile type: Publication
arXiv ID: hep-th/0004127
title: Chamseddine-Fröhlich-Grandjean metric and localized Higgs coupling in noncommutative geometry
review text: A geometrized Higgs mechanism on the gravitational sector in the Connes-Lott formulation of the Standard Model, which was constructed by \textit{A. H. Chamseddine}, \textit{J. Fröhlich} and \textit{O. Grandjean}, J. Math. Phys. 155, 6255-6275 (1995; Zbl 0846.58006)], is proposed. The authors' proposal is to promote the Higgs coupling from a constant to one of the components of the vielbein on noncommutative geometry, \(M_4\times Z_2\), and make the Higgs coupling depend on the local coordinates of \(M_4\). As a result of the on-shell condition (the unitarity condition and the torsion-less condition) in the discrete space, Higgs couplings can be expressed as Wilson-like operators of the connection coefficients and the gauge fields, which provides geometric conditions in which the Higgs couplings become zero. To describe the model, Chamseddine-Fröhlich-Grandjean metric is reviewed in Sect. 2. The underlying noncommutative space of the authors' model is NEWLINE\[NEWLINE{\mathcal A}= ({\mathcal A}_1 \oplus {\mathcal A}_2\oplus {\mathcal A}_3)\otimes C^\infty(M_4), \quad {\mathcal A}_1= {\mathcal A}_2= \mathbb{C},\;{\mathcal A}_3= M_2(\mathbb{C}).NEWLINE\]NEWLINE The elements of \({\mathcal A}\) are written as diagonal form (3,3)-matrices. The Hilbert space is the space of spinors \(u_R\), \(d_R\), which are regarded as composing an \(SU(2)\) singlet, and \(u_L\), \(d_L\), which are regarded as composing an \(SU(2)\) doublet. \(R\) and \(L\) indicate the two kinds of chiralities that are defined by \(\gamma^5\). After explicitly giving generators of the noncommutative one-forms, introducing the localized Higgs couplings \(f(x)\) and \(\widetilde f(x)\), the discrete space generators of the one-form on curved \(M\) are presented as NEWLINE\[NEWLINE{\mathcal E}^r= \gamma^5 \left(\begin{matrix} 0 & {\mathcal A}^*e_r\\ -e^*_r{\mathcal A} & 0\end{matrix} \right),NEWLINE\]NEWLINE NEWLINE\[NEWLINE{ \mathcal A}= \left(\begin{matrix} 0 & f(x){v\over \sqrt 2}\\ \widetilde f(x){v \…
reviewed by: Akira Asada
description: scientific article; zbMATH DE number 1713639