Phenomenological Dual Koide Pattern in Down-Type Quark Masses

We examine whether a Brannen-type square-root mass pattern provides a compact phenomenological organization of the down-type quark spectrum, treated throughout as an empirical benchmark rather than as a first-principles mass prediction.

Using the fixed benchmark pair \(\phi_{d}=\tfrac{1}{3}\), \(\cos(\delta_{d})=\tfrac{1}{2}\cos(\tfrac{3}{8}\pi)\) together with the b-anchored normalization \(A_{d}=\sqrt{m_{b}(m_{b})}/k_{3}\), the Brannen-type formula reproduces the light \(d\)- and \(s\)-quark entries of the PDG~2024 down-type spectrum at the percent level, comparable to current input uncertainties.

Because \(m_{d}\) and \(m_{s}\) are quoted at \(\mu=2\,\mathrm{GeV}\) whereas \(m_{b}\) is quoted at \(\mu=m_{b}(m_{b})\), the construction is a mixed-scale benchmark dressed by QCD running. The empirical observation emphasized here is that the inverse Koide quantity \(K_{\mathrm{inv}}\)---defined by replacing each mass \(m_{i}\) with \(1/m_{i}\) in the Koide ratio---remains numerically close to \(\tfrac{2}{3}\) across the mixed-to-common-scale comparison, whereas the direct Koide ratio \(K\) relative to its benchmark value \(K_{\mathrm{pred}}\) shows a much larger mixed-to-common-scale shift; in inverse-tuple Brannen variables, the common-scale reconstruction places the shape parameter within about \(8\times 10^{-4}\) of \(\tfrac{1}{\sqrt{2}}\). Within the QCD-only, flavour-universal running approximation adopted in this work, both \(K\) and \(K_{\mathrm{inv}}\) are common-scale invariants by homogeneity, not by a \(K_{\mathrm{inv}}\)-specific dynamical fixed-point mechanism; the empirical asymmetry therefore resides in their proximity to \(\tfrac{2}{3}\) and in the size of the mixed-to-common-scale transport, not in their RG laws.

The construction therefore amounts to an empirical parametrization of the down-type spectrum, not to an independent prediction of the quark masses.