Effective flavor interaction for the charged-lepton mass hierarchy and the Koide relation
This article proposes a dynamical framework to address the charged-lepton mass hierarchy and derive the empirical Koide formula. We postulate a $U(3)_F$ flavor symmetry at the high-energy scale, where a heavy dynamical flavon field couples to the electroweak Higgs sector via an effective operator. As the universe cools, spontaneous symmetry breaking of the flavor group occurs following the pattern $U(3)_F \to U(1)^3_F$. Subsequently, the electroweak symmetry breaking at the Fermi scale generates the charged-lepton mass matrix. The phenomenological values of these masses are determined in the infrared regime, where the effective potential governing the flavon VEV undergoes a dynamical vacuum alignment.The stable minimum of this potential enforces an algebraic equipartition between the flavor-singlet and flavor-octet invariants ($I_{(0)} = I_{(8)}$). This geometric equilibrium yields the Koide mass ratio ($Q=2/3$), successfully identifying the dynamically generated eigenvalues with the physical pole masses of the electron, muon, and tau.