From Yukawa Couplings to Flavor Observables: A Top-Down, Analytically Solvable Geometric Framework for CKM/PMNS Mixing and CP Violatio

Abstract

Standard Euler-angle parameterizations obscure the origin of flavor mixing. We present an explicit algebraic framework for the Standard Model Yukawa sector expressing fermion masses, CKM/PMNS matrices, and CP invariants $J$ directly via Yukawa couplings. Starting from a general $3\times3$ complex mass matrix $M$ with 18 parameters, $M^2$ reduces independent parameters to 9. Introducing $[M^2_R, M^2_I] = 0$ further reduces the single-sector space to 5 parameters: scale parameters $(A,B,C)$ and a flavor ratio vector $v_q = (x,y)$. Crucially, $M^2$ admits exact analytical diagonalization without ad hoc texture zeros. The CKM matrix emerges directly as the geometric misalignment between flavor vectors, $V_{\text{CKM}} = U_u^\dagger(v_u)U_d(v_d)$. We derive closed-form expressions for $J(v_u, v_d)$, identifying $v_u \times v_d = 0$ and generation permutations as explicit CP-violation control switches. Substituting $J$ into the CP measure $\Delta_{\text{CP}}$, we show that the enhancement ratio $R_\Delta \equiv \Delta_{\text{CP}} / \Delta^{(0)}_{\text{CP}} > 10^{10}$ is fully accessible, resolving concerns that low-energy CP violation cannot account for the Baryon Asymmetry of the Universe. Finally, four-fold moduli degeneracy establishes our analytical 5-parameter model as an essential leading-order baseline, highlighting the physical necessity of non-commuting extensions ($[M^2_R, M^2_I] \neq 0$) across CKM and PMNS sectors.