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

Abstract

We show that quark weak mixing follows from the same one-flavon, three-messenger Froggatt-Nielsen construction, with a single hierarchy parameter $B$ ($\epsilon \equiv 1/B$) and a rational-exponent "$B$-lattice", that organizes the fermion mass spectrum in companion work. The lattice exponents, together with a four-phase messenger coefficient model, account for the four measured CKM magnitudes and, through the Fritzsch-Xing reconstruction, the full $3\times 3$ CKM matrix and the Jarlskog invariant $J$, providing a consistency test of the single-$B$ hypothesis. A benchmark Yukawa pair on the lattice reproduces, under direct diagonalization, the light-quark masses to better than 5%, all nine CKM magnitudes to better than 1.2%, and $J$ within $0.3\sigma$ of the PDG global fit, with order-unity coefficients; a $10^7$-configuration phase scan shows the CKM hierarchy to be generic once the quark masses are fit, while percent-level agreement across all four magnitudes is not. The near-maximal CP phase admits a geometric interpretation: equal-weight messenger interference fixes the phase at exactly $\pi/2$ for any individual phase values, and the fitted $\phi_{\rm FX} \simeq 93^\circ$ measures the departure from this limit. The same lattice parameter fixes the charged-lepton mass ratio $m_\mu/m_\tau = c_\mu \epsilon^{5/3}$, whose exponent is the second-generation lepton charge sum; this yields $|V_{ub}| \simeq (m_\mu/m_\tau)^2$, agreeing with the measured value at about the 6% level, and the equivalent form $|V_{ub}| \simeq \sin^3\theta_{13}$ ties the smallest CKM magnitude to the smallest PMNS magnitude, the two routes furnishing complementary determinations of the same ninths-lattice quantity from the quark and lepton sectors.