Probing the Geometry of Viable Froggatt-Nielsen-like Flavor Textures

Technical Summary: Probing the Geometry of Viable Froggatt–Nielsen-like Flavor Textures

Problem Statement
The origin of fermion mass hierarchies and flavor mixing patterns remains a central unresolved issue in particle physics. While the Standard Model parametrizes these via Yukawa couplings, it offers no explanation for the observed hierarchies. Froggatt–Nielsen (FN) models address this by generating hierarchical couplings through powers of a small symmetry-breaking parameter, $\epsilon$. However, the landscape of viable FN models exhibits significant redundancy: many distinct charge assignments and texture structures can reproduce experimental data. This redundancy suggests that the problem may not be identifying a single unique model, but rather understanding the global geometric organization of the space of viable theories. The central question addressed is whether the subset of FN textures compatible with experimental observations concentrates around a low-dimensional manifold within the high-dimensional parameter space, or if they remain broadly distributed.

Methodology
The author performs an exploratory analysis of the geometry of viable FN-like flavor textures within a controlled toy landscape. The study avoids imposing specific charge-factorization constraints of particular FN models, instead treating the eighteen integer exponents ($n_{ij}^{(u,d)}$) entering the up- and down-type Yukawa matrices as independent coordinates in an 18-dimensional exponent space.

1. Model Setup:

   • Yukawa matrices are parametrized as $(Y_{u,d})_{ij} = a_{ij}^{(u,d)} \epsilon^{n_{ij}^{(u,d)}}$, with $\epsilon = 0.22$.
   • Coefficients $a_{ij}$ are real $\mathcal{O}(1)$ numbers; exponents are non-negative integers.
   • Quark masses and CKM moduli are derived from real Yukawa matrices (no CP-violating phase is included).

2. Benchmark Ensembles:
   Four benchmark datasets are constructed, each containing 1,000 unique viable textures. Viability is determined by independent multiplicative cuts on observables:

   • Benchmark 1: Quark mass ratios only ($m_u/m_c, m_c/m_t, m_d/m_s, m_s/m_b$) with a tolerance of $f(O) \leq 5$.
   • Benchmark 2: Mass ratios + $|V_{us}|$ (tolerance $f(O) \leq 1.5$).
   • Benchmark 3: Mass ratios + $|V_{us}| + |V_{cb}|$.
   • Benchmark 4: Full set including $|V_{ub}|$.

3. Geometric Probes:
   Two complementary methods are employed to diagnose the dimensionality of the viable point clouds:

   • Principal Component Analysis (PCA): Used to detect linear compression. The study measures the fraction of variance explained by leading components and the number of components ($N_{50}, N_{75}, N_{90}$) required to capture 50%, 75%, and 90% of the total variance.
   • Intrinsic Dimensionality Estimators: Used to detect nonlinear manifold structures.
     • TwoNN: Estimates dimension $d$ from the ratio of distances to the first and second nearest neighbors ($\mu = r_2/r_1$).
     • Levina–Bickel (LB): A maximum-likelihood estimator based on the scaling of distances to the first $k$ nearest neighbors.
   • Robustness Checks: The analysis includes tests for metric dependence (Euclidean vs. standardized Euclidean vs. $L_\infty$), fitting range variations, and finite-sample effects (sub-sampling).

Key Results

• PCA Analysis:

  • The inclusion of CKM constraints increases the anisotropy of the point cloud, raising the variance explained by the first principal component from $\sim9.9\%$ (masses only) to $\sim15.2\%$ (full CKM).
  • However, no strong linear compression is observed. In all four benchmarks, 15 principal components are required to account for 90% of the total variance ($N_{90} = 15$). This indicates that viable textures occupy a large fraction of the original 18-dimensional space.

• Intrinsic Dimensionality (TwoNN):

  • The estimated effective dimensions are consistently high.
  • Masses-only: $d_{\text{TwoNN}} \approx 12.07$.
  • Full CKM: $d_{\text{TwoNN}} \approx 10.83$.
  • The reduction in dimensionality upon adding CKM constraints is visible but moderate, failing to collapse the landscape into a very-low-dimensional structure.

• Intrinsic Dimensionality (Levina–Bickel):

  • Estimates vary with the neighborhood size $k$ but remain high across all scales.
  • For the full CKM benchmark, estimates range from $14.16$($k=5$) to $10.22$($k=30$).
  • The ordering of benchmarks is consistent with PCA and TwoNN: adding mixing constraints organizes the landscape but does not drastically reduce its effective dimension.

• Robustness:

  • The high-dimensional conclusion holds across different distance metrics (though $L_\infty$ induces degeneracy due to the integer lattice), different fitting intervals, and reduced sample sizes (down to $N=300$).
  • Discreteness effects (degenerate nearest-neighbor distances) are present but do not dominate the primary Euclidean analysis.

Significance and Claims
The paper presents a "negative" but informative result: raw Froggatt–Nielsen exponents do not provide natural coordinates for revealing a strongly compressed, low-dimensional geometric structure of viable flavor theories.

• Interpretation: The author explicitly states that this result does not rule out the existence of an underlying geometry in flavor theory space. Rather, it suggests that the geometry is not strongly exposed in the raw exponent coordinates. The viable landscape is organized (as evidenced by the increased anisotropy with CKM constraints), but this organization remains broadly high-dimensional in the current representation.
• Implication: The apparent redundancy of flavor model building may reflect a geometric structure that requires a change of variables, metric, or representation to become manifest. The raw exponents may not be the "natural coordinates" for this geometry.
• Scope: The work is framed as a first step in a broader program to reconstruct and characterize the geometry of viable flavor theories. It establishes a baseline using the simplest available coordinates and identifies the limitations of that description, motivating future investigations into observable-space geometry and alternative representations.