Chamseddine-Fröhlich-Grandjean metric and localized Higgs coupling in noncommutative geometry
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\[NEWLINE{ \mathcal A}= \left(\begin{matrix} 0 & f(x){v\over \sqrt 2}\\ \widetilde f(x){v \over \sqrt 2} & 0\end{matrix} \right),\quad {\mathcal A}^*= \left(\begin{matrix} 0 & \widetilde f^*(x) {v\over\sqrt 2}\\ f^*(x){v \over\sqrt 2} & 0\end{matrix} \right).NEWLINE\]NEWLINE Then on-shell Higgs couplings are discussed. The detailed calculation for unitarity conditions are given in Appendix A and components of the torsion are given in Appendix B. By these calculations, Higgs couplings are expressed as Wilson-like operations of the connection coefficients and the gauge fields. Differential equations describing torsionless condition are also given. At a quantum level, imposing the torsion-less condition on the ground state \((\partial_\mu|0\rangle=0)\), solutions of these equations are given.
scientific article; zbMATH DE number 1713639