Minima of Higgs potentials corresponding to non-maximal isotropy subgroups
\textit{L. Michel} [CERN Report, No.TH. 2196-CERN (1979)] conjectured that the minima of a Higgs-Landau potential has isotropy groups which are maximal isotropy subgroups of the full symmetry group of the potential. In the case of finite groups counterexamples were found by \textit{M. V. Jarić} [Phys. Rev. Lett. 51, 2073-2076 (1983)]. Here the Higgs-Landau potential associated with the irreducible 75-dimensional representation of SU(5) is discussed in detail. The geometrical methods of the first author and \textit{G. Sartori} [Ann. Phys. 150, 307-372 (1983; Zbl 0529.58028)] form the basis of the analysis, which involves the use of a computer to generate random directions in \({\mathbb{R}}^{75}\) to check the algebraic relations between various quartic invariants and to express each invariant in terms of generic parameters, and to study the Hessian matrix to determine which extrema are minima. In this way minima were found having four distinct maximal isotropy subalgebras, as well as a minimum with a non-maximal isotropy algebra together with a discrete symmetry. Finally a counterexample to Michel's conjecture was found having a non-maximal isotropy group SU(2)\(\times U(1)\times U(1).\) The same potential has been studied by \textit{C. J. Cummins} and the reviewer [J. Phys. A 19, 161-181 (1986; see the following review)] who identified absolute minima with non-maximal isotropy group SU(2)\(\times SU(2)\times U(1)\) using a rather different method [\textit{J. S. Kim}, Nucl. Phys. B 196, 285-300 (1982)] more suited to the determination of absolute minima but still requiring computer searches to cope with the complicated geometry.
scientific article; zbMATH DE number 3953120
scientific article