The exact column texture: tree-level Yukawa universality in heterotic $Z_3 \times Z_3$ orbifolds

TL;DR

This paper proves that in certain heterotic orbifold models, the leading-order Yukawa couplings have an exact column texture with universal coefficients, and explores the geometric and symmetry reasons behind this property.

Contribution

The authors demonstrate that the Froggatt-Nielsen column texture is an exact property of the leading-order string amplitude in specific heterotic orbifold models, supported by multiple lines of evidence.

Findings

The geometric worldsheet instanton analysis yields identical areas for all relevant triangles.

All three generations are gauge-identical due to trivial Wilson lines.

The Froggatt-Nielsen column texture is confirmed as an exact property, not an approximation.

Abstract

On $T^6/(Z_3 \times Z_3)$ heterotic orbifolds where three quark generations arise from $Z_3$ fixed-point triplication, we prove that the leading-order tree-level Yukawa amplitude -- the three-point coupling among massless string states -- has an exact column texture: $Y_{\rm lead}(i,j) = c\,\varepsilon^{q_R[j]}$, with the $O(1)$ coefficient $c$ universal across all left-handed generations $i$. Five independent lines of evidence are given: (1) the worldsheet instanton geometry on the $SU(3)$ root lattice gives identical areas for all non-degenerate triangles, making the geometric $O(1)$ coefficient exactly $1$; (2) the generation direction necessarily has trivial Wilson line, rendering all three generations gauge-identical, as verified across all 77 MSSM-like models in the Mini-Landscape classification; (3) an extension to two-Wilson-line models, verified on the complete Parr-Vaudrevange-Wimmer classification of 3,337 $Z_3 \times Z_3$ MSSM models, confirms that no Wilson line configuration can break gauge blindness; (4) the K\"ahler metric is generation-universal by $\Delta(54)$ representation theory; (5) the full Froggatt-Nielsen chain computation with 534 trilinear superpotential couplings and vacuum-aligned singlet VEVs produces left-circulant Yukawa matrices whose eigenstructure is generation-universal. The Froggatt-Nielsen column texture is therefore not an approximation but an exact property of the leading-order string amplitude. Non-trivial $O(1)$ coefficients, which are required for CKM mixing angles beyond the Wolfenstein hierarchy, must originate from beyond-leading-order contributions: integrated-out heavy messenger propagators (tree-level in the low-energy effective theory), vacuum-alignment effects, multi-instanton corrections, or loop corrections.