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 $V=1\oplus 2$. Since ${\mathrm{Sym}}^{2}V=21\oplus 22$, three-dimensional invariant Yukawa subspaces of type $1\oplus 2$ 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 $y_2{y}_5^2=\sqrt{2}{y}_1{y}_3^2$, 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 $m_1+m_2\le m_3\le m_1+3m_2$. 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.

Comments: 16 pages