Two-over-Two Lattice Flavor from a Single Flavon with Three Messenger Chains

Abstract

Flavor hierarchies are organized by a single parameter $B\simeq 5.357$ in a single-flavon Froggatt--Nielsen (FN) framework, in which each effective Yukawa entry arises from the sum of \emph{three} unit-magnitude messenger chains. We present benchmark complex $O(1)$ Yukawa matrices that reproduce quark and charged-lepton masses at $M_Z$ as powers of $ε\equiv 1/B$. The organizing principle is a two-over-two (2/2) lattice of quadrilateral mass ratios, which maps directly to a rational lattice of FN exponents. Sequential dominance preserves the leading-power exponent matrices, while subleading messenger chains generate entry-dependent complex $O(1)$ coefficients and provide a UV-friendly origin for CP violation. Neutrino masses are discussed at the level of eigenvalues within the same $B^n$ counting.

Figures (3)

• Figure 1: Schematic three-messenger realization of a single Yukawa entry $(Y_f)_{ij}$ in the FN framework Froggatt:1978ntLeurer:1992qjLeurer:1993gy. The external legs are the left-handed quark doublet $Q_i$, the right-handed singlet $f^c_j$ ($f=u,d$), and the Higgs doublet $H$. Heavy vector-like messenger fermions (doublets and singlets) run in the internal lines. Each messenger chain carries a certain number of flavon insertions $\Phi$, generating a power of $\epsilon = \langle \Phi\rangle/\Lambda = 1/B$. Summing the three chains yields an effective suppression $\epsilon^{p^f_{ij}}$ times an $\mathcal{O}(1)$ complex coefficient $C^f_{ij}$.
• Figure 2: Two-over-two relations as points in the integer $(x,y)$ plane. Each ratio $R_{abcd}$ lies on a line of constant $n_{abcd}$ given by Eq. \ref{['eq:n-from-k']}. The straight lines (from bottom to top at fixed $X$) correspond to $n=0,1,2,3,5,6,7,8,9$. All lattice points are plotted; the $(x,y)$ coordinates are listed in Table \ref{['tab:kXY']}. The horizontal axis is $x\equiv 9\,(n_b+n_c)$ and the vertical axis is $y\equiv 9\,(n_a+n_d)$. The axes are linear in the integers $(x,y)$ (i.e. not log--scaled), but they encode logarithmic ratio information: if $R_{abcd}\sim B^{-n_{abcd}}$ with $n_{abcd}=(y-x)/9$, then $n_{abcd}=-\log_B R_{abcd}$.
• Figure 3: Quark flavor points in the $(x^{\prime},y^{\prime})$ plane constructed from the FN charges in Table \ref{['tab:fncharges']} via Eq. \ref{['eq:k6k7-quarks']}. Up-type quarks are shown as squares and down-type quarks as triangles. The origin is chosen so that the third generation $(t,b)$ sits at $(0,0)$, which is invariant under uniform shifts of all charges.