An Exact Geometric Framework for Flavor Mixing and CP Violation in the Standard Model

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 $\delta$ 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 ($\eta_{\text{relative}}\in\{0,\pm1\}$) 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.