An Exact Analytical Bridge from Complex Yukawa Couplings to Fermion Masses and CKM/PMNS Mixing

Abstract

The origin of fermion masses and flavor mixing is often obscured by standard Euler-angle parameterizations, which mask the underlying connection between physical observables and Yukawa couplings. We prove that introducing a single, remarkably weak yet physically grounded commutation hypothesis, $[{\mathbf{M}}_R^2,{\mathbf{M}}_I^2]=0$, on the Hermitian mass-squared matrix ${\mathbf{M}}^2=M\cdot M^{†}$ systematically reduces a general 18-parameter complex mass matrix $M$ down to a 5-parameter form per sector that admits \emph{exact} analytic diagonalization. The resulting mass eigenvalues are closed-form algebraic expressions that completely bypass the transcendental Cardano trigonometric reduction required for generic Hermitian matrices, while the diagonalizing unitary matrix $U_f$ depends solely on two parameters that form a dimensionless 2D geometric flavor-ratio vector ${\mathbf{v}}_f=(x,y)$. Combining two such sectors yields the physical CKM/PMNS mixing matrix directly as $V=U_1^{†}({\mathbf{v}}_1)\cdot U_2({\mathbf{v}}_2)$, establishing a direct, first-principles analytical bridge from raw Yukawa couplings to fermion masses and mixing matrices without intermediate phenomenological inputs. Finally, we highlight that this exact leading-order baseline enforces an exact four-fold moduli degeneracy among mixing elements, pointing directly toward non-commuting extensions for full phenomenological precision.

Comments: 4pages, no figure