Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Lin, Chilong
Institution: National Museum of Natural Science
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171838274