2026/07/29 by Chilong Lin
Physics and Astronomy · #hep-ph #hep-th
16 pages, no figures
arxiv created 2026/07/31 · arxiv updated 2026/08/03
We present a top-down geometric framework within the CP-violating Standard Model (CPVSM), utilizing dimensionless 2D ratio vectors (x,y) and (x',y') to establish a Cartesian-like parameterization of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. This approach provides an intuitive physical mapping that explicitly disentangles up- and down-type quark sector contributions to electroweak mixing and CP violation. Crucially, we prove an exact algebraic dimensional reduction from the 9 real degrees of freedom of the complex mass-squared matrix M2 down to 5 baseline parameters, and ultimately to 2 invariant ratio coordinates (x,y) in the unitary diagonalization matrix U. This baseline yields exact analytical expressions for the CKM matrix and a closed-form Jarlskog invariant J, achieving robust numerical convergence to |J0| ≈ 3.08 × 10-5. We demonstrate that the exact commutativity condition [M2R, M2I] = 0 is the fundamental origin of an intrinsic fourfold magnitude degeneracy governed by |p| = |p'| = |p^*| = |p'^*|, inducing non-perturbative Wolfenstein scale mixing that signals the necessity of non-commuting quantum corrections ([M2R, M2I] ≠ 0). Remarkably, near critical boundary geometries (xy → 0), the system exhibits a quark mass spurt that reshapes the mass hierarchy while geometrically preserving non-collinear CP violation (xy' - x'y ≠ 0), yielding magnitudes sufficient to account for the cosmological Baryon Asymmetry of the Universe (BAU). Extending this vector mismatch paradigm to the PMNS lepton matrix with Dirac neutrinos (νR), we naturally accommodate large leptonic mixing angles through angular vector offsets, establishing a unified, rephasing-invariant geometric framework for flavor origins across both quark and lepton sectors.