---
title: Minimal Modular Quark Flavour Models
url: https://www.emergentmind.com/topics/minimal-modular-quark-flavour-models
type: topic
---

# Minimal Modular Quark Flavour Models

Minimal Modular Quark Flavour Models postulate that the hierarchical structure of quark masses, mixing angles, and CP violation originates from the field-theoretic consequences of modular symmetry, with the Standard Model quark sector realized as a minimal and highly predictive construction—often with one or two complex moduli and typically without flavon fields or fine-tuned input parameters. These models employ modular groups such as $A_4$, $S_4'$, $2O$, $\Gamma_6$, or binary dihedral groups (e.g., $2D_3$), constraining Yukawa textures using properties of modular forms evaluated near special fixed points (cusps) in moduli space. The modular assignment of quark multiplets and modular weights, together with the modular forms of appropriate levels and weights, uniquely structure the mass matrices, leading to mass hierarchies and mixing patterns in quantitative agreement with experimental data [2307.14410], [2601.04529], [2209.08796], [2306.05730], [2311.10136], [2409.15823], [2012.13390], [2301.03737], [2306.14467].

## 1. Modular Symmetry and Field Assignments

Minimal modular quark flavour models assign the left-handed quark doublets $Q$ and right-handed up-type ($u^c$) and down-type ($d^c$) quarks to irreducible representations of a finite modular group $\Gamma_N$ (such as $A_4$, $S_4'$, $2O$, $\Gamma_6$, or $2D_3$), each associated with a modular weight $k$ [2307.14410], [2601.04529], [2409.15823], [2209.08796], [2311.10136]. The modulus $\tau$ parametrizes the shape of an underlying toroidal compactification, and modular transformations enforce that the chiral superfields $\psi(\tau)$ obey $\psi(\gamma\tau) = (c\tau+d)^{-k}\,\rho(\gamma)\,\psi(\tau)$, where $\rho$ is a group representation and $\gamma\in SL(2,\mathbb{Z})$ [2307.14410], [2601.04529]. 

Typical representations include:

- $A_4$: $Q$ as triplet $3$, $u^c$, $d^c$ as singlets $1,1',1''$ [2212.13336], [2306.05730]
- $S'_4$: $Q$ as $3$ or $2\oplus1$, $u^c,d^c$ as singlets or doublets [2307.14410], [2006.10722], [2601.04529]
- $2O$: $Q$ as doublet and singlet, $u^c,d^c$ as doublets/singlets of $2O$ [2409.15823]
- $2D_3$: "2+1" assignments, e.g., $Q=(Q_1,Q_2)\sim 2$, $Q_3\sim 1$ [2311.10136]
- $\Gamma_6$: only singlet modular forms, enabling Z$_6$ residual symmetry near the cusp [2301.03737]

The modular weights are chosen such that Yukawa couplings can be constructed from modular forms of the appropriate weight to ensure overall modular invariance.

## 2. Modular Forms, Fixed Points, and Mass Matrix Texture

The flavour structures—i.e., the entries and hierarchies of the quark mass matrices—arise from holomorphic modular forms $Y^{(k)}_r(\tau)$ selected according to the modular weights and representations [2307.14410], [2212.13336], [2306.05730], [2601.04529], [2012.13390]. These modular forms exhibit a hierarchical structure when expanded near special points (cusps) of moduli space corresponding to residual symmetry, such as:

- $\tau=i\infty$: associated with residual $Z_N$ symmetry, e.g., $N=4$ for $S_4'$ or $N=6$ for $\Gamma_6$ [2601.04529], [2301.03737], [2409.15823]
- $\tau=\omega\equiv e^{2\pi i/3}$: the left $Z_3$ cusp, important in $A_4$ and $S_4'$ models [2307.14410], [2212.13336]

The modular forms themselves possess $q$-expansions in the vicinity of these fixed points (with $q=e^{2\pi i\tau/N}$), producing leading hierarchies among elements of the mass matrices according to their modular transformation properties and the specific Clebsch-Gordan structure of the group representations [2307.14410], [2601.04529], [2301.03737], [2212.13336], [2311.10136]. 

For example, in $S_4'$, the leading behaviour is $M_u\sim\begin{pmatrix}\epsilon^3 & \epsilon^2 & 1\\ \epsilon^2 & \epsilon & \epsilon^3\\ \epsilon^3 & \epsilon^2 & 1\end{pmatrix}$ for an expansion parameter $\epsilon=|q_4|=e^{-\pi\Im\tau/2}\ll 1$ [2601.04529]. In $A_4$ models near $\tau=\omega$, the mass ratios are set by powers of $|\epsilon|=|\tau-\omega|$, e.g., $m_2/m_3 \sim |\epsilon|$, $m_1/m_3\sim|\epsilon|^2$ [2307.14410], [2212.13336].

## 3. Superpotential Construction and Parameter Counting

The holomorphic Yukawa superpotential is built as a sum of terms involving matter multiplets, modular forms, and MSSM Higgs doublets $H_{u,d}$, each term a modular invariant singlet of total weight zero [2307.14410], [2012.13390], [2212.13336], [2306.14467]. No flavon fields are required in the minimal approach; the entire hierarchy arises from the value of $\tau$. 

The number of free real parameters is minimized by the following mechanisms:

- Modular symmetry drastically reduces the number of allowed Yukawa couplings.
- Typically, all superpotential coefficients are taken to be real (or absolute values $O(1)$), with CP violation arising from phases in either $\tau$ or specific couplings.
- The minimal number of input parameters is set by: the real and imaginary parts of $\tau$ (or two moduli if CPV is engineered this way), and a fixed set of Yukawa coupling magnitudes/phases (ranging from 8 to 13 real parameters total for 10 observables) [2307.14410], [2601.04529], [2012.13390], [2212.13336], [2301.03737], [2311.10136], [2409.15823].

A typical table of parameter assignment for the $S_4'$ minimal model is:

| Sector    | Parameters                   |
|-----------|-----------------------------|
| Up        | $\alpha_u, \beta_u, \gamma_u$      |
| Down      | $\alpha_d, \beta_d, \gamma_d$, $g_d$ (complex) |
| Modulus   | $\Im\tau$                    |
|           | Total: 9 real (for 10 observables)  |

[2601.04529]

## 4. Quark Mass Hierarchies, Mixing, and CKM Structure

Diagonalizing the mass matrices leads naturally to strong mass hierarchies without input fine-tuning of Yukawa parameters. The ratios are set by powers of the small expansion parameter associated with proximity to a cusp, e.g., for $\tau$ near $i\infty$ or $\omega$,

- Up-type: $m_t:m_c:m_u\sim1:\epsilon:\epsilon^3$
- Down-type: $m_b:m_s:m_d\sim1:\epsilon:\epsilon^2$
- CKM angles: $\theta_{12}\sim\epsilon$, $\theta_{23}\sim\epsilon$, $\theta_{13}\sim\epsilon^2$

Typical $\epsilon\sim0.03$ reproduces observed patterns for $m_{u}/m_{c}$, $m_{c}/m_{t}$, $m_{d}/m_{s}$, $m_{s}/m_{b}$, with the absolute values of masses depending on the electroweak vev and overall normalization [2307.14410], [2601.04529], [2212.13336], [2306.05730], [2301.03737].

The CKM matrix and the Jarlskog invariant $J_{CP}$ are obtained from the mismatch of the left-diagonalizing unitaries of $M_u$ and $M_d$. The CKM elements $|V_{us}|$, $|V_{cb}|$, $|V_{ub}|$, and the CP phase $\delta_{CP}$ match experimental central values to within a few percent, for $\tau$ values near the modular fixed points and $O(1)$ couplings [2307.14410], [2601.04529], [2012.13390], [2212.13336], [2409.15823]. 

Sample fit results for $S_4'$ minimal model [2601.04529]:
| Observable         | Fit Value    |
|--------------------|-------------|
| $m_s/m_b$          | $1.8\times10^{-2}$ |
| $m_d/m_b$          | $8.8\times10^{-4}$ |
| $m_c/m_t$          | $2.9\times10^{-3}$ |
| $m_u/m_t$          | $5.7\times10^{-6}$ |
| $\sin\theta_{12}$  | $0.226$     |
| $\sin\theta_{23}$  | $0.039$     |
| $\sin\theta_{13}$  | $0.0044$    |
| $\delta_{CP}$      | $65^\circ$  |

## 5. Sources of CP Violation

Minimal modular models, in their most constrained form, often predict a too-small CKM phase if CP violation arises solely from the phase of $\tau$ (spontaneous CP breaking). This is because near the fixed point, $\Re\tau$ is small and thus Jarlskog $J_{CP}$ is suppressed as $|q|^n\sin\Re\tau$ [2307.14410], [2212.13336], [2601.04529].

To reconcile with experiment, explicit breaking of CP is introduced, typically via a complex coupling in the superpotential (e.g., $g_d = |g_d|e^{i\phi}$ in the down sector) [2601.04529], or by assigning distinct moduli $\tau_u$ and $\tau_d$ for up and down sectors with mismatched phases [2307.14410], [2212.13336]. These modifications restore viable $J_{CP}$ of $O(10^{-5})$ and $\delta_{CP} \sim60^\circ$–$70^\circ$, aligning with data.

## 6. Predictive Power and Phenomenology

Because these models contain fewer free parameters than observables, they yield nontrivial predictions and correlations—often sum rules or parameter relations among mass ratios, mixing angles, and CP phases. The residual symmetry at the cusp fixes the hierarchy pattern, and modular forms' orthogonality governs mixing. As a result, the CKM phase and certain mass ratios are sharp outputs of the fit [2307.14410], [2601.04529], [2012.13390], [2311.10136], [2301.03737], [2409.15823]. 

In unified versions combining quark and lepton sectors, a single modulus $\tau$ for both sectors induces correlations between, e.g., the ratio $m_s/m_b$ and neutrino sector parameters—a feature directly testable by precision measurements [2409.15823]. The viability of the minimal modular flavour models is robust across thresholds and RG running, with best-fit points generally remaining close to residual symmetry points even as (supersymmetry breaking) scales vary [2601.04529], [2307.14410], [2209.08796].

## 7. Limitations and Model Extensions

The minimal framework’s predictive power comes at the cost of some rigidity. Notably, if all Yukawa couplings are taken to be real and only one modulus is present, the predicted CP violation can be insufficient [2307.14410], [2212.13336], [2601.04529]. Remedies—such as explicit complex couplings or sector-split moduli—help retain qualitative minimality while matching all observables. Extensions include:

- Adding weight-8 modular forms for improved fits [2212.13336]
- Employing additional modular symmetries or mixed representations ($2O$, $2D_3$, etc.) for detailed spectrum control [2311.10136], [2409.15823]
- Unification with the lepton sector, leading to predictive cross-correlations with neutrino masses and mixings [2409.15823], [2209.08796], [2012.13390]

The overall theme is that minimal modular quark flavour models, by leveraging the mathematical properties of modular groups and forms near fixed points, provide a highly economical, predictive, and structurally motivated explanation for the origin of the observed pattern of quark masses, mixing angles, and CP violation [2307.14410], [2601.04529], [2212.13336], [2012.13390], [2311.10136], [2409.15823], [2301.03737], [2306.05730].

Source: https://www.emergentmind.com/topics/minimal-modular-quark-flavour-models