Breaking of permutation symmetry and diagonal group action: Nielsen model and the standard model as low-energy limit
In a series of papers \textit{H. B. Nielsen} and collaborators [see e.g. ``Random dynamics, three generators and skewness'', Preprint Niels Bohr Institute NBI-HE-91-04 (1991) and the references cited there] have considered the possibility that the grand unification group is of the form \[ {\mathcal G} = G_ 1 \otimes G_ 1 \otimes \cdots \otimes G_ 2 \otimes \cdots \otimes G_ s = (G_ 1)^{\otimes k_ 1} \otimes (G_ 2)^{\otimes k_ 2} \otimes \cdots \otimes (G)^{\otimes k_ s} \] and that the low-energy limit corresponds to breaking of this symmetry to the diagonal action over the \(G_ i\) spaces, i.e., to \({\mathcal G}_ 1 = G_ 1 \otimes G_ 2 \otimes \cdots \otimes G_ s\). Naturally, the \(G_ i\) factors should then be \(SU(n)\) groups. Our point here is just to point out that, once a symmetry of the kind considered by Nielsen is assumed, solutions breaking it to diagonal action do necessarily exist, as a consequence of a theorem of \textit{L. Michel} [Points critiques des fonctions invariants sur une \(G\)- variété C. R. Acad. Sci., Paris, Sér. A-B 272, A 433--A 436 (1971)] -- actually motivated by \(SU(3)\) models in particle physics -- on the geometry of group actions.
scientific article; zbMATH DE number 465586
scientific article