Embedding dependence of fermion mass hierarchies in $S_3$-symmetric Yukawa sectors

Abstract

When irreducible representations occur with multiplicity, a family symmetry fixes the representation content of a Yukawa sector but need not fix its embedding in the space of generation tensors. We study the spectral consequences of this additional embedding data for complex-symmetric three-family tensors with the $S_3$ assignment $V=\mathbf{1}\oplus\mathbf{2}$. Since $\operatorname{Sym}^2V=2\,\mathbf{1}\oplus2\,\mathbf{2}$, three-dimensional invariant Yukawa subspaces of type $\mathbf{1}\oplus\mathbf{2}$ form a continuous $\mathbb{CP}^1\times\mathbb{CP}^1$ family, denoted $\mathcal L_{t,\kappa}$ on a finite affine chart. On the $\kappa=0$ branch, we solve the inverse singular-value problem and obtain necessary and sufficient conditions for any prescribed ordered positive mass spectrum. A fixed singlet embedding imposes a finite hierarchy bound, whereas varying the embedding accommodates every positive three-family spectrum. For general embeddings, $\mathcal L_{t,\kappa}$ contains a nonzero rank-one matrix if and only if $2t\kappa^2=1$; equivalently, these are precisely the finite-chart embeddings for which $\sigma_3/\sigma_2$ is unbounded. In a conventional complex-symmetric $S_3$ three-Higgs sector, the same condition becomes $y_2y_5^2=\sqrt{2}\,y_1y_3^2$, so order-one Yukawa coupling ratios can support arbitrarily large mass hierarchies. We then apply the result to a recent Clifford-algebraic construction and show that its family-resolved Yukawa tensors span $\mathcal L_{2,0}$, whose singular values obey $m_1+m_2\leq m_3\leq m_1+3m_2$. This excludes the observed charged-lepton hierarchy for arbitrary complex vacuum alignment within the existing Higgs directions. The resulting obstruction is therefore tied to the fixed multiplicity embedding rather than to the $S_3$ representation content alone.