从复Yukawa耦合到费米子质量及CKM/PMNS混合的精确解析桥梁
An Exact Analytical Bridge from Complex Yukawa Couplings to Fermion Masses and CKM/PMNS Mixing
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中文总结 AI 辅助
该研究针对费米子质量与味混合起源问题，引入对易假设简化复质量矩阵，构建从Yukawa耦合到费米子质量及CKM/PMNS混合矩阵的精确解析桥梁，为相关物理研究提供了新方法。
中文摘要 AI 辅助
费米子质量与味混合的起源常被标准欧拉角参数化所掩盖，这种参数化遮蔽了物理可观测量与Yukawa耦合之间的内在联系。我们证明，在厄米质量平方矩阵$\boldsymbol{M}^2 = M \cdot M^\boldsymbol{\top}$上引入一个单一、极弱但具有物理依据的对易假设$[\boldsymbol{M}_R^2, \boldsymbol{M}_I^2] = 0$，可系统地将一个具有18个参数的一般复质量矩阵$M$简化为每个 sector 含5个参数的形式，该形式可实现精确解析对角化。所得质量本征值为闭式代数表达式，完全规避了一般厄米矩阵所需的超越性卡尔达诺三角化简；而对角化幺正矩阵$U_f$仅依赖两个参数，这两个参数构成无量纲二维几何味比率向量$\boldsymbol{v}_f=(x,y)$。将两个此类 sector 结合，可直接得到物理的CKM/PMNS混合矩阵$V = U_1^\boldsymbol{\top}(\boldsymbol{v}_1) \cdot U_2(\boldsymbol{v}_2)$，建立了从原始Yukawa耦合到费米子质量及混合矩阵的直接第一性原理解析桥梁，无需中间唯象学输入。最后，我们强调该精确领头阶基线在混合元之间施加了精确的四重模简并，直接指向非对易扩展以获得完全的唯象学精度。
英文摘要
The origin of fermion masses and flavor mixing is often obscured by standard Euler-angle parameterizations, which mask the underlying connection between physical observables and Yukawa couplings. We prove that introducing a single, remarkably weak yet physically grounded commutation hypothesis, $[\mathbf{M}_R^2, \mathbf{M}_I^2] = 0$, on the Hermitian mass-squared matrix $\mathbf{M}^2 = M \cdot M^\dagger$ systematically reduces a general 18-parameter complex mass matrix $M$ down to a 5-parameter form per sector that admits \emph{exact} analytic diagonalization. The resulting mass eigenvalues are closed-form algebraic expressions that completely bypass the transcendental Cardano trigonometric reduction required for generic Hermitian matrices, while the diagonalizing unitary matrix $U_f$ depends solely on two parameters that form a dimensionless 2D geometric flavor-ratio vector $\mathbf{v}_f=(x,y)$. Combining two such sectors yields the physical CKM/PMNS mixing matrix directly as $V = U_1^\dagger(\mathbf{v}_1) \cdot U_2(\mathbf{v}_2)$, establishing a direct, first-principles analytical bridge from raw Yukawa couplings to fermion masses and mixing matrices without intermediate phenomenological inputs. Finally, we highlight that this exact leading-order baseline enforces an exact four-fold moduli degeneracy among mixing elements, pointing directly toward non-commuting extensions for full phenomenological precision.
