Quark Mixing from a Lattice Flavon Model: A Four-Magnitude Parameterization

Abstract

We present the quark weak-mixing component of a Froggatt--Nielsen program, with one flavon and three messengers, in which a single hierarchy parameter $B$ (with $ε\equiv 1/B$) and a rational-exponent ``$B$-lattice'' organize fermion Yukawa textures. Building on companion mass-fit work, we translate the lattice into sharp predictions for quark mixing. The four-magnitude parameterization serves as a practical interface between the flavon Yukawa textures and quark weak mixing observables, yielding coefficient-free ratio tests of the lattice structure.

Figures (3)

• Figure 1: Comparison of the sharp FX ratio predictions in Eq. \ref{['eq:ratio-preds']} with the PDG global-fit magnitudes of the CKM matrix. The plotted points show the PDG central values divided by our Eq. \ref{['eq:ratio-preds']} predictions; error bars are obtained by propagating the PDG (one-sigma) uncertainties in Eq. (12.27) of the PDG CKM review PDGCKM2024.
• Figure 2: $B$ inferred from CKM magnitudes using the PDG Standard-Model global-fit values (unitarity imposed), with $1\sigma$ error bars from standard propagation. The vertical reference line is drawn at $B=75/14\simeq 5.357$.
• Figure 3: $\chi^2$ for the four $B$-power CKM magnitude predictions in Table \ref{['tab:CKM_Bpowers']} as a function of $B$, using PDG global-fit values PDGCKM2024. The minimum $\chi^2\simeq 0.13$ occurs at $B\simeq 5.359$, consistent with $B=75/14\simeq 5.357$ (dashed red line). The $\Delta\chi^2=1$ interval yields $B=5.345$--$5.373$; the $95\%$ CL interval ($\Delta\chi^2=3.84$) gives $B=5.330$--$5.388$. The steep parabolic profile demonstrates that the four CKM magnitudes tightly constrain $B$ to within $\pm 0.3\%$ at $1\sigma$.