Renormalization Group Running of the Koide Parameter: Exact Equation, Two-Loop Doublet Structure, and Predicted Magnitude

We derive the exact renormalization group (RG) equation for the Koide parameter K(µ) = mi(µ)/( Pp mi(µ))2 in terms of the diagonal anomalous dimensions γi of the charged lepton masses. The result has the form dK/dt = K ∆⟨γ⟩, where ∆⟨γ⟩ is the difference between the m-weighted and the √ m-weighted averages of γi . This vanishes identically when γi is uncorre- lated with mi , in which case K is RG-invariant. We then argue—at the structural rather than diagrammatic level—that the non-trivial running of K is supported on the S3 doublet representation of the flavor space, and arises at two loops through commutators of Y † ℓ Yℓ and Y † ν Yν. We provide the numerical setup needed to test this prediction and estimate |∆K| ∼ 10−6–10−4 between the electroweak and Planck scales depending on the neutrino sector. This is intended as a companion to [1], where K = 2/3 is shown to be the unique fixed point of a Z2 self-duality on the lepton flavor space.