AI 中文总结
本文研究S₃对称汤川 sector 的嵌入数据对费米子质量谱的影响，发现改变嵌入可容纳任意正三代质量谱，而克利福德代数构造的汤川张量嵌入无法解释观测到的带电轻子层级。
AI 中文摘要
当不可约表示以重数出现时，家族对称性会确定汤川 sector 的表示内容，但不一定确定其在代空间张量中的嵌入方式。我们研究了这种额外嵌入数据对具有S₃赋值V=1⊕2的复对称三代张量的谱学影响。由于Sym²V=2 1⊕2 2，类型为1⊕2的三维不变汤川子空间构成连续的ℂℙ¹×ℂℙ¹族，在有限仿射图上记为ℒ_{t,κ}。在κ=0分支上，我们求解了逆奇异值问题，得到了任意规定的有序正质量谱的充要条件：固定单重态嵌入会施加有限的层级界限，而改变嵌入则可容纳所有正三代谱。对于一般嵌入，ℒ_{t,κ}包含非零秩1矩阵当且仅当2tκ²=1；等价地，这些恰好是σ₃/σ₂无界的有限图嵌入。在常规复对称S₃三代希格斯 sector 中，相同条件变为y₂y₅²=√2 y₁y₃²，因此量级为1的汤川耦合比可支撑任意大的质量层级。随后我们将该结果应用于近期的克利福德代数构造，表明其家族分辨的汤川张量张成ℒ_{2,0}，其奇异值满足m₁+m₂≤m₃≤m₁+3m₂。这排除了现有希格斯方向内任意复真空取向下观测到的带电轻子层级，因此产生的阻碍与固定重数嵌入相关，而非仅与S₃表示内容相关。
英文摘要
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 $\mathcal L_{t,κ}$ on a finite affine chart. On the $κ=0$ 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, $\mathcal L_{t,κ}$ contains a nonzero rank-one matrix if and only if $2tκ^2=1$; equivalently, these are precisely the finite-chart embeddings for which $σ_3/σ_2$ 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.
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