发表机构
National Museum of Natural Science（国立自然科学博物馆）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在标准模型中提出几何CPVSM框架，通过参数化CKM矩阵实现维度约化，解释夸克质量激增与CP破坏，还将范式扩展至轻子 sector，为味起源提供统一几何框架。
AI 中文摘要
我们在标准模型（SM）中提出了一种自上而下的几何框架，利用无量纲二维比值向量$(x,y)$和$(x',y')$对卡比博-小林-益川（CKM）矩阵建立类笛卡尔参数化。该方法提供了直观的物理映射，明确区分了上型和下型夸克区对电弱混合及CP破坏的贡献。关键是，我们证明了复质量平方矩阵$\boldsymbol{M}^2$的9个实自由度可精确代数约化至5个基线参数，最终约化至幺正对角化矩阵$U$中的2个不变比值坐标$(x,y)$。此基线给出了CKM矩阵的精确解析表达式及闭式雅尔斯戈不变量$J$，实现了对$|J_0| \times 3.08 \times 10^{-5}$的稳健数值收敛。我们证明，精确对易条件$[\boldsymbol{M}^2_R, \boldsymbol{M}^2_I] = 0$是固有四重大小简并的根本起源，该简并由$|p| = |p'| = |p^*| = |p'^*|$控制，诱导出非微扰的沃尔芬斯坦标度混合，这表明非对易量子修正（$[\boldsymbol{M}^2_R, \boldsymbol{M}^2_I] \neq 0$）是必要的。值得注意的是，在近临界边界几何（$xy \to 0$）处，系统会表现出“夸克质量激增”，该激增在几何上保持非共线CP破坏（$xy' - x'y \neq 0$）的同时重塑质量等级，产生的大小足以解释宇宙学重子不对称性（BAU）。将此矢量失配范式扩展至具有狄拉克中微子（$\nu_R$）的PMNS轻子矩阵，我们通过角矢量偏移自然容纳了大的轻子混合角，建立了跨夸克和轻子区的统一、重相位不变的味起源几何框架。
英文摘要
Flavor mixing and CP violation in the Standard Model are conventionally parameterized by Euler angles that entangle the physical contributions of the up- and down-type quark sectors, obscuring the origin of CP violation itself. We present a non-perturbative geometric framework that resolves this entanglement through the commutativity condition $[\mathbf{M}_R^2,\mathbf{M}_I^2]=0$ for the Hermitian components of the mass-squared matrices, reducing the 18 real degrees of freedom of a general $3\times3$ Yukawa matrix to five independent parameters and yielding an exact Cartesian-like parameterization. Explicit diagonalization directly gives fermion masses, CKM matrix elements, and the CP phase $δ$ in closed form, cleanly separating the distinct physical contributions of the up- and down-type sectors. Mapping flavor space onto a two-dimensional vector geometry, we derive the Jarlskog invariant $J$ in exact closed form and show that CP violation is governed by a discrete topological selection rule ($η_{\text{relative}}\in\{0,\pm1\}$) together with the vector cross-product $\mathbf{v}_u\times\mathbf{v}_d=xy'-x'y$. Across all 36 $S_3\times S_3$ flavor topologies, this yields an exact $18/18$ classification into CP-violating ($J \neq 0$) and CP-conserving ($J\equiv 0$) cases, associated with two distinct CP-suppression mechanisms: geometric alignment through a vanishing vector cross-product and algebraic cancellation of imaginary contributions under permutation. At tree level, fitting to experimental CKM elements yields agreement at the $\mathcal{O}(10^{-2})$ level, establishing this framework as a transparent analytical baseline for future non-commutative flavor extensions.
Comments20 pages, no figures
