Minimal Modular Quark Flavour Models

• Minimal Modular Quark Flavour Models are frameworks leveraging modular symmetries and minimal parameters to explain the hierarchy of quark masses and mixings.
• They utilize modular forms near fixed points to constrain Yukawa textures, achieving quantitative agreement with experimental data.
• Predictive and economical, these models generate realistic CKM phases and mass ratios without requiring flavon fields or fine-tuning.

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 (Varzielas et al., 2023, Petcov et al., 8 Jan 2026, Du et al., 2022, Petcov et al., 2023, Arriaga-Osante et al., 2023, Ding et al., 2024, Yao et al., 2020, Kikuchi et al., 2023, Ahn et al., 2023).

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 $k$ (Varzielas et al., 2023, Petcov et al., 8 Jan 2026, Ding et al., 2024, Du et al., 2022, Arriaga-Osante et al., 2023). 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 $\gamma\in SL(2,\mathbb{Z})$ (Varzielas et al., 2023, Petcov et al., 8 Jan 2026).

Typical representations include:

• $A_4$: $Q$ as triplet $3$, $u^c$, $d^c$ as singlets $1,1',1''$ (Petcov et al., 2022, Petcov et al., 2023)
• $S'_4$: $Q$ as $3$ or $2\oplus1$, $u^c,d^c$ as singlets or doublets (Varzielas et al., 2023, Liu et al., 2020, Petcov et al., 8 Jan 2026)
• $2O$: $Q$ as doublet and singlet, $u^c,d^c$ as doublets/singlets of $2O$ (Ding et al., 2024)
• $2D_3$: "2+1" assignments, e.g., $Q=(Q_1,Q_2)\sim 2$, $Q_3\sim 1$ (Arriaga-Osante et al., 2023)
• $\Gamma_6$: only singlet modular forms, enabling Z$_6$ residual symmetry near the cusp (Kikuchi et al., 2023)

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 (Varzielas et al., 2023, Petcov et al., 2022, Petcov et al., 2023, Petcov et al., 8 Jan 2026, Yao et al., 2020). 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 $\Gamma_6$ (Petcov et al., 8 Jan 2026, Kikuchi et al., 2023, Ding et al., 2024)
• $\tau=\omega\equiv e^{2\pi i/3}$: the left $Z_3$ cusp, important in $A_4$ and $S_4'$ models (Varzielas et al., 2023, Petcov et al., 2022)

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 (Varzielas et al., 2023, Petcov et al., 8 Jan 2026, Kikuchi et al., 2023, Petcov et al., 2022, Arriaga-Osante et al., 2023).

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 $\epsilon=|q_4|=e^{-\pi\Im\tau/2}\ll 1$ (Petcov et al., 8 Jan 2026). In $A_4$ models near $\tau=\omega$, the mass ratios are set by powers of $|\epsilon|=|\tau-\omega|$, e.g., $m_2/m_3 \sim |\epsilon|$, $m_1/m_3\sim|\epsilon|^2$ (Varzielas et al., 2023, Petcov et al., 2022).

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 (Varzielas et al., 2023, Yao et al., 2020, Petcov et al., 2022, Ahn et al., 2023). 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) (Varzielas et al., 2023, Petcov et al., 8 Jan 2026, Yao et al., 2020, Petcov et al., 2022, Kikuchi et al., 2023, Arriaga-Osante et al., 2023, Ding et al., 2024).

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)

(Petcov et al., 8 Jan 2026)

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 (Varzielas et al., 2023, Petcov et al., 8 Jan 2026, Petcov et al., 2022, Petcov et al., 2023, Kikuchi et al., 2023).

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 (Varzielas et al., 2023, Petcov et al., 8 Jan 2026, Yao et al., 2020, Petcov et al., 2022, Ding et al., 2024).

Sample fit results for $S_4'$ minimal model (Petcov et al., 8 Jan 2026): | 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 $|q|^n\sin\Re\tau$ (Varzielas et al., 2023, Petcov et al., 2022, Petcov et al., 8 Jan 2026).

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) (Petcov et al., 8 Jan 2026), or by assigning distinct moduli $\tau_u$ and $\tau_d$ for up and down sectors with mismatched phases (Varzielas et al., 2023, Petcov et al., 2022). 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 (Varzielas et al., 2023, Petcov et al., 8 Jan 2026, Yao et al., 2020, Arriaga-Osante et al., 2023, Kikuchi et al., 2023, Ding et al., 2024).

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 (Ding et al., 2024). 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 (Petcov et al., 8 Jan 2026, Varzielas et al., 2023, Du et al., 2022).

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 (Varzielas et al., 2023, Petcov et al., 2022, Petcov et al., 8 Jan 2026). 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 (Petcov et al., 2022)
• Employing additional modular symmetries or mixed representations ($2O$, $2D_3$, etc.) for detailed spectrum control (Arriaga-Osante et al., 2023, Ding et al., 2024)
• Unification with the lepton sector, leading to predictive cross-correlations with neutrino masses and mixings (Ding et al., 2024, Du et al., 2022, Yao et al., 2020)

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 (Varzielas et al., 2023, Petcov et al., 8 Jan 2026, Petcov et al., 2022, Yao et al., 2020, Arriaga-Osante et al., 2023, Ding et al., 2024, Kikuchi et al., 2023, Petcov et al., 2023).