📝 SummarXiv

Abstract

The origin of the three fermion generations and their highly hierarchical mass spectra remains one of the most profound puzzles in particle physics. We show that the complexified exceptional Jordan algebra \(J_{3}(\mathbb{O}_{\mathbb{C}})\), the natural mathematical framework for the exceptional Lie group \(E_{6}\), provides a unified explanation for both. In this approach, the three generations arise from the three canonical eigenvalues of Jordan elements, while their mass hierarchies are determined by a minimal, universal ladder in the \(\mathrm{Sym}^{3}(\mathbf{3})\) representation of a flavor \(SU(3)\) subgroup. The construction yields several parameter-free predictions. The universal Jordan eigenvalue spectrum \((q-\delta, q, q+\delta)\) is fixed by the algebra to have a spread \(\delta^{2}=3/8\). The requirement of a minimal, consistent ladder uniquely selects fixed Clebsch-Gordan factors \((2,1,1)\) for state mixing. Together, these inputs derive closed-form expressions for the square-root mass ratios of all charged fermions (quarks and leptons) from a single underlying structure. A Dynkin \(\mathbb{Z}_{2}\) automorphism of \(E_{6}\) (a ``swap'') naturally relates the down-quark and charged-lepton sectors, predicting \(\sqrt{m_{\tau}/m_{\mu}} = \sqrt{m_{s}/m_{d}}\). The first-generation mass scale is set by a trace split \(\operatorname{Tr}X_{\ell}:\operatorname{Tr}X_{u}:\operatorname{Tr}X_{d}=1:2:3\), leading to the relation \(\sqrt{m_{e}}:\sqrt{m_{u}}:\sqrt{m_{d}}=1:2:3\).