Phenomenological Dual Koide Relation for Down-Type Quarks

Koide's mass formula and Brannen's phase-parametrized extension provide a compact empirical description of the charged-lepton spectrum. In this paper we examine whether a structurally similar ansatz can organize the down-type quark sector. The construction is presented as an empirical benchmark parametrization rather than a derived common-scale mass law: the down-type pattern is read as a \emph{Brannen-type geometric skeleton}---a phase-parametrized square-root pattern---together with \emph{QCD renormalization-group dressing}. Adopting \(\phi_d=\tfrac{1}{3}\) and \(\cos(\delta_d)=\tfrac{1}{2}\cos(\tfrac{3}{8}\pi)\) yields percent-level consistency with the PDG~2024 central values, comparable to the current input uncertainties.
With \(m_d, m_s\) quoted at \(\mu=2\,\mathrm{GeV}\) and \(m_b\) at its own reference scale \(\mu=m_b\), the benchmark is a mixed-scale skeleton; light-quark entries acquire a multiplicative dressing under transport to a common scale, with four-loop \(\overline{\mathrm{MS}}\) running giving a nearly common factor \(\approx 0.9191\) for \(\sqrt{m_d}\) and \(\sqrt{m_s}\) between these scales. At a common scale, the same three-mass tuple can be recast in a Brannen-type form with negligible numerical residual; we treat this as a diagnostic parametrization rather than as an independent prediction, and the residual fit parameters \(T_d^{\mathrm{fit}}\) and \(\phi_d^{\mathrm{fit}}\) are kept approximately scale-independent. In this form, the inverse-Koide quantity \(K_{\mathrm{inv}}\)---the Koide combination with each mass replaced by its reciprocal---remains close to \(\tfrac{2}{3}\); since QCD running is flavour-universal for these masses in the approximation used here, the shape parameters of the inverse square-root tuple are kinematically preserved, and the empirical content is the proximity of the corresponding ratio parameter to \(\tfrac{1}{\sqrt{2}}\). Following correspondence with Rivero, we further parametrize the inverse square-root tuple in the same Brannen-type form and observe that the corresponding fit parameter \(T_{d\,\mathrm{inv}}^{\mathrm{fit}}\) is exactly scale-independent within the QCD-only, flavour-universal running approximation adopted in this work and lies within \(\sim 8\times 10^{-4}\) of the symmetric value \(\tfrac{1}{\sqrt{2}}\). Whether full Standard-Model running further reduces this residual is left as an external open question and is not assumed in the present analysis. The whole construction is presented as a compact empirical organization of the down-type spectrum, not as a first-principles derivation, together with a structurally compact statement of the dual Koide relation.