Fermion mass relations in one-parameter modular models

Technical Summary: Fermion Mass Relations in One-Parameter Modular Models

Problem Statement
The Standard Model (SM) successfully describes fundamental interactions via gauge symmetry but fails to explain the "flavour puzzle": the origin of fermion masses, mixing patterns, and CP violation. The Yukawa sector contains numerous free parameters, exhibiting marked mass hierarchies and contrasting mixing patterns (large for leptons, small for quarks). Traditional non-Abelian flavour symmetries address this by unifying generations into multiplets, broken by flavon vacuum expectation values (VEVs). Modular flavour symmetries offer a refinement, replacing generic couplings with modular forms $Y(\tau)$ controlled by a single complex modulus $\tau$. While previous models utilized a limited number of parameters, this paper investigates the extreme predictive limit: One-Parameter Modular Models (OPMs). In an OPM, each charged-fermion mass matrix is fixed by a single modular invariant contraction, determined solely by the modulus $\tau$ and an overall coefficient. The central challenge is determining whether such highly constrained models can reproduce the observed charged-fermion mass hierarchies, given the rigid structure of the modular group $SL(2, \mathbb{Z})$.

Methodology
The authors develop a systematic framework to construct and test OPMs within global $N=1$ supersymmetry:

1. Theoretical Constraints: The authors impose strict conditions for an OPM:

   • Matter fields ($\psi, \psi^c$) must transform as triplets of a finite modular group to ensure all three generations acquire mass via a single contraction.
   • The superpotential must contain only one modular form of a specific weight $k$ that contracts with the matter triplets to form a trivial singlet.
   • The search is restricted to the lowest non-trivial modular weights ($k=1, 2$) and groups admitting triplet irreducible representations (irreps).

2. Hierarchical Expansion: To generate hierarchical masses, the modulus $\tau$ is expanded around fixed points ($\tau_{sym} = i, \omega, i\infty$) where residual symmetries exist. The authors focus on the limit $\tau \to i\infty$ (large $\text{Im}\,\tau$), where the expansion parameter is $\epsilon \sim |q|^{1/N}$ (with $q = e^{2\pi i \tau}$). This "modular proximity-induced hierarchy" mechanism naturally suppresses lighter generations.

3. Group Search: A systematic scan of finite modular groups with order $< 100$ is performed. The analysis identifies $\Delta(96)$ and its subgroup $\Delta(384)$ as the only viable candidates capable of producing the required triplet contractions with non-vanishing determinants in the symmetric limit.

4. Phenomenological Analysis:

   • High-Scale Relations: The mass matrices for charged leptons and down-quarks are constructed using specific triplet assignments in $\Delta(384)$. This yields exact mass relations at the flavour scale ($M_F \sim 2 \times 10^{16}$ GeV).
   • Renormalization Group (RG) Evolution: The high-scale predictions are evolved down to the electroweak scale ($M_Z$) using SM and MSSM beta functions, accounting for the SUSY scale ($M_{SUSY}$) and $\tan\beta$.
   • Threshold Corrections: Finite supersymmetric threshold corrections at $M_{SUSY}$ are included to reconcile high-scale predictions with low-energy data, specifically testing if selective corrections can resolve discrepancies.

Key Contributions and Results

• Classification of Hierarchies: The study identifies five distinct hierarchical patterns (H1–H5) allowed by OPMs. These patterns are characterized by specific power-law dependencies on $\epsilon$ and fixed numerical coefficients derived from Clebsch-Gordan coefficients.

  • H2: $m_1 : m_2 : m_3 \propto \epsilon^6 : \epsilon^2 : 1$
  • H5: $m_1 : m_3 : m_3 \propto \epsilon^5 : \epsilon^3 : 1$
  • A "meta-relation" is derived for all OPMs: $(m_2/m_3)^{p/q} = \sqrt{2} (m_1/m_3)$, where $p/q$ characterizes the hierarchy.

• Double OPM Realization: The authors construct a concrete "double OPM" based on $\Delta(384)$ where:

  • The charged-lepton sector follows the H2 hierarchy.
  • The down-quark sector follows the H5 hierarchy.
  • Both sectors share a common modulus $\tau$ and expansion parameter $\epsilon$.

• Exact High-Scale Mass Relations: This construction leads to three exact mass relations at the flavour scale:

  1. $m_s^5 = 2\sqrt{2} m_d^3 m_b^2$ (Down sector)
  2. $m_\mu^3 = \sqrt{2} m_e m_\tau^2$ (Lepton sector)
  3. $m_s^2 m_\tau = \sqrt{2} m_e m_b^2$ (Inter-sector)

• Phenomenological Viability:

  • RG Running: When evolved to $M_Z$ with $M_{SUSY} \sim 10^9$ GeV and $\tan\beta = 50$, the down-quark and electron-to-tau ratios align well with experimental data.
  • Muon Discrepancy: The model initially predicts a muon mass $\sim 20\%$ higher than observed.
  • Threshold Resolution: The authors demonstrate that a selective one-loop SUSY threshold correction, specifically modifying the muon-to-tau ratio while preserving the electron-to-tau and down-quark ratios, can resolve this discrepancy. A benchmark soft spectrum ($x_\mu \approx 0.5, x_{e\tau} \approx 0.012$) successfully reproduces the data.

• Up-Quark Speculation: The paper notes that the observed up-quark hierarchy ($m_u : m_c : m_t$) aligns with a hypothetical pattern H6 ($p/q = 2$), which would imply $m_c^2 = \sqrt{2} m_u m_t$. If this pattern holds, a unified mass relation involving all nine charged fermions is satisfied at the $M_Z$ scale with high precision ($1.05 \pm 0.05$).

Significance and Claims
The paper claims to provide a "working proof of principle" for One-Parameter Modular Models. It demonstrates that the extreme constraint of a single modular invariant contraction per sector is not inherently incompatible with observed fermion data, provided the model is based on specific finite modular groups ($\Delta(384)$) and includes appropriate RG evolution and selective threshold effects.

The authors emphasize that OPMs do not offer a flexible parametrization but rather a discrete set of predictive curves in the hierarchy plane. The fact that the H2 and H5 patterns naturally match the charged-lepton and down-quark sectors, respectively, is presented as a non-trivial alignment rather than a result of fine-tuning. The work suggests a potential route to solving the flavour puzzle through highly constrained constructions where mass hierarchies and mixing angles (via a common $\epsilon \sim \theta_C$) share a modular origin. However, the authors remain modest regarding the up-quark sector, labeling the H6 connection as "speculative" pending an explicit model construction and a full derivation of the CKM matrix.