Embedding dependence of fermion mass hierarchies in S3-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 S3 assignment V=1⊕2. Since Sym2V=2 1⊕2 2, three-dimensional invariant Yukawa subspaces of type 1⊕2 form a continuous ℂℙ1×ℂℙ1 family, denoted ℒt,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, ℒt,κ contains a nonzero rank-one matrix if and only if 2tκ2=1; equivalently, these are precisely the finite-chart embeddings for which σ3/σ2 is unbounded. In a conventional complex-symmetric S3 three-Higgs sector, the same condition becomes y2y52=√2 y1y32, 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 ℒ2,0, whose singular values obey m1+m2≤ m3≤ m1+3m2. 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 S3 representation content alone.
Related
- Fermion Hierarchies from Fricke Modular Symmetry
- Fine structure of fermion mixing and quark-mass hierarchies
- Embedding Paired Free-Fermion Gaussian States into Gutzwiller-Projected Bardeen--Cooper--Schrieffer Wave Functions
- Fractional θ-dependence from the sum over all instanton sectors
- Symmetric mass generation of interacting chiral fermions on a one-dimensional lattice without fermion doubling
BibTeX
@misc{zhou2026embedding,
title = {{Embedding dependence of fermion mass hierarchies in $S_3$-symmetric Yukawa sectors}},
author = {Ye Zhou},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29259},
eprint = {2608.29259},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29259},
}
RIS
TY - UNPB
TI - Embedding dependence of fermion mass hierarchies in S_3-symmetric Yukawa sectors
AU - Ye Zhou
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29259
UR - https://arxiv.org/abs/2608.29259
AB - 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⊕2. Since Sym^2V=2 1⊕2 2, three-dimensional invariant Yukawa subspaces of type 1⊕2 form a continuous ℂℙ^1×ℂℙ^1 family, denoted 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, 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=√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 ℒ_2,0, whose singular values obey m_1+m_2≤ m_3≤ 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.
ER -