⚛️ phenomenology

Eclectic flavour symmetries without flavons

This paper introduces a new framework for realizing eclectic flavor symmetries without flavons by employing the finite image of the integral Jacobi group acting on moduli $(\tau, z)$ to construct realistic supersymmetric models with spontaneous CP violation and to clarify the geometric distinctions between various toroidal orbifold building blocks.

Original authors: Ferruccio Feruglio, Antonio Marrone

Published 2026-09-16

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CC BY 4.0

Original authors: Ferruccio Feruglio, Antonio Marrone

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The origin of the variety of particles in our universe remains one of the deepest puzzles in physics. While the Standard Model successfully describes how these particles interact, it offers no explanation for why they have the specific masses they do, or why they mix together in the strange patterns observed in experiments. For decades, physicists have tried to solve this by proposing hidden symmetries—mathematical rules that govern how particles transform into one another. In many of these theories, the rules are broken by special fields called flavons, which settle into specific values to create the observed mass patterns. However, this approach often requires adding many new, unseen ingredients to the theory, making the explanation feel cluttered and less elegant. A more recent idea suggests that these patterns might arise naturally from the geometry of extra dimensions, where the shape of space itself dictates the properties of particles, removing the need for those extra fields.

In a new study, researchers have developed a framework that unifies these two approaches, showing how a complex symmetry can emerge purely from the geometry of two specific numbers that describe the shape of space, without ever needing to introduce the extra flavon fields. The team focused on a mathematical structure known as the Jacobi group, which acts on a pair of variables: one that describes the overall shape of a torus, or doughnut-shaped space, and another that marks a specific point on that surface. In this view, the traditional symmetry rules that usually require flavons are actually just the result of moving that point around the surface. The researchers demonstrated that this geometric setup naturally produces the exact group of symmetries needed to describe the known particles, specifically a structure called the eclectic group, which combines ordinary modular transformations with a finite Heisenberg symmetry.

The paper shows that this geometric parent structure can be reduced to a finite image that acts on the particles, effectively replacing the need for independent flavon fields. Instead of a separate field aligning itself to break the symmetry, the alignment is fixed by the position of the point on the torus. The researchers constructed a complete theory of supersymmetry based on this idea, defining how the particles and their interactions behave under these transformations. They found that the mathematical objects describing the interactions, known as Jacobi forms, can generate the necessary patterns for particle masses. Crucially, they showed that a specific combination of these forms can reproduce the complex mass matrices usually requiring a modular doublet and a flavon triplet, but here achieved with a single geometric object. This means the entire coupling structure is encoded in the geometry of the moduli, eliminating the need for a separate vacuum-alignment sector.

To test if this abstract framework could describe reality, the authors built a realistic model for the masses of charged fermions, such as quarks and electrons. They assigned all these particles to the same symmetry representation and used the geometric moduli to generate their mass matrices. The model successfully reproduced the observed masses and mixing angles of the quarks and charged leptons with high precision. A key discovery in their analysis was that the best-fit solution lies very close to a special geometric location on the torus, known as a two-torsion locus. At this specific point, a residual symmetry protects the structure of the mass matrices, naturally forcing the lightest particle to have zero mass and suppressing certain mixing angles. The observed small masses and mixings then arise from a tiny deviation from this perfect geometric point. This mechanism, which the authors call modular protection, provides a natural explanation for the hierarchy of masses without fine-tuning parameters by hand.

The study also addressed the origin of the complex phase responsible for the violation of charge-parity symmetry, a phenomenon where the laws of physics treat matter and antimatter slightly differently. The researchers showed that if the fundamental equations are set to be symmetric under a specific transformation, the complex phase observed in nature can still emerge spontaneously. This happens because the geometric variables that determine the particle masses settle into complex values that break the symmetry, generating the necessary phase without requiring any complex numbers in the initial setup. The model also extended to the neutrino sector, where the researchers introduced a slightly different mathematical multiplier to account for the unique properties of neutrino masses. They found that this approach could accommodate the observed neutrino oscillation patterns while maintaining the same geometric foundation used for the charged particles.

The work clarifies how this geometric interpretation connects to previous models that relied on separate symmetry factors. By taking a specific limit where the point on the torus is fixed at a central location, the new framework reproduces the traditional eclectic flavor structure, showing that the old models are a special case of this broader geometric picture. The researchers also explored how this idea applies to other types of orbifold geometries, finding that the rank-one Heisenberg-modular geometry works for several cases, while a different, more complex structure is required for others. They explicitly noted that one specific case, involving a six-fold symmetry, does not fit this pattern because the necessary non-commuting structure is absent, highlighting the specific conditions under which this elegant geometric solution holds.

Ultimately, the paper provides a concrete realization of a flavor theory where the diversity of particle masses and mixings is a direct consequence of the geometry of extra dimensions. By removing the need for flavon fields and unifying the symmetry breaking with the geometry of the moduli space, the authors offer a more economical and structurally unified description of the flavor puzzle. The success of the model in fitting experimental data, particularly the precise reproduction of quark masses and the natural emergence of CP violation, suggests that this geometric approach is a viable path forward. The findings indicate that the complex patterns of the subatomic world may not be the result of arbitrary choices or hidden fields, but rather the inevitable outcome of how particles move through the hidden shapes of our universe.

Technical Summary: Eclectic Flavour Symmetries without Flavons

Problem Statement
The origin of fermion flavour hierarchies and mixing patterns remains an open question in particle physics. While modular flavour symmetries offer an economical alternative to traditional flavour symmetries by eliminating the need for independent flavon fields, they often struggle to reproduce the full complexity of the observed spectrum without reintroducing flavon-like structures. Conversely, "eclectic" frameworks, which unify traditional discrete flavour symmetries with modular symmetries, typically rely on the vacuum expectation values (VEVs) of flavon multiplets to break the traditional symmetry sector. This reliance complicates the theoretical structure and introduces a separate vacuum-alignment sector. The central problem addressed is whether an eclectic-like flavour construction can be realized entirely through the geometry of moduli, thereby eliminating the need for flavon multiplets while preserving the successful group-theoretical structures of eclectic models.

Methodology
The authors propose a bottom-up framework where the flavour symmetry is realized as a finite image of the integral Jacobi group, $\Gamma_J = H(\mathbb{Z}) \rtimes SL(2, \mathbb{Z})$ , acting on two moduli: the standard modular parameter $\tau$ and an elliptic modulus $z$ .

Geometric Realization: The traditional flavour symmetry is identified with the finite image of the Heisenberg subgroup $H(\mathbb{Z})$ , which acts on $z$ while leaving $\tau$ invariant. The modular symmetry corresponds to the $SL(2, \mathbb{Z})$ factor acting on both moduli. The consistency condition linking traditional and modular transformations (Eq. 3 in the paper) arises naturally from the semidirect-product structure of the Jacobi group, rather than being imposed as an external constraint.
Finite Group: Focusing on level three, the infinite Jacobi group reduces to the finite group $J_3 \simeq \Delta(27) \rtimes T' \simeq \Omega(1)$ , where $\Delta(27)$ is the Heisenberg image and $T' \simeq SL(2, 3)$ is the modular image.
Supersymmetric Construction: The authors formulate a supersymmetric effective theory (both globally and in supergravity) defined on the Jacobi moduli space. They derive the Kähler potential, Jacobi-covariant kinetic tensors, and superpotential selection rules. Crucially, matter multiplets are assigned specific weights, indices, and representations of $J_3$ , transforming with inverse automorphy factors.
Flavon Replacement: Instead of introducing flavon fields, the coupling structures are constructed from holomorphic Jacobi forms of two moduli $(\tau, z)$ . Specifically, a Jacobi sextet at weight two and index twelve is shown to replace the tensor product of a modular doublet and a flavon triplet.
CP Symmetry: A consistent CP transformation is defined as an anti-holomorphic involution on the moduli space, $(\tau, z) \to (-\bar{\tau}, -\bar{z})$ . This allows CP violation to arise spontaneously from the complex VEVs of the moduli, provided the Lagrangian couplings are real.

Key Contributions

Flavon-Free Eclectic Framework: The paper demonstrates that the eclectic group $\Omega(1)$ can be realized as the finite level-three image of the Jacobi group, unifying traditional and modular symmetries without independent flavon fields.
Jacobi-Invariant Effective Theory: The authors provide the necessary ingredients for constructing supersymmetric theories invariant under the Jacobi group, including explicit forms for the Kähler potential, kinetic terms, and superpotential selection rules.
Low-Weight Jacobi Forms Catalogue: A classification of low-weight holomorphic Jacobi forms at level three is provided, identifying specific forms (e.g., sextets at weight 2, index 12) capable of generating realistic fermion mass matrices.
Phenomenological Models:
- Charged Fermions: A realistic model for charged fermion masses and CKM mixing is constructed using a universal matter assignment and two independent Yukawa textures derived from Jacobi forms. The model incorporates non-universal kinetic terms generated by Jacobi-covariant tensors (specifically involving the triplet $G$ ). The fit reproduces experimental data with high accuracy ( $\chi^2_{ch} \approx 0.17$ ) without flavons.
- Neutrinos: The framework is extended to the neutrino sector using a type-I seesaw mechanism. By introducing a non-trivial scalar multiplier (Dedekind eta multiplier) for the Majorana sector, the authors generate a distinct coupling space for neutrino masses while keeping the Dirac sector identical to the charged fermions.
Modular Protection Mechanism: The analysis reveals that the best-fit points lie near a "two-torsion locus" in moduli space. On this locus, the Yukawa matrices exhibit a common zero mode protected by a residual $Z_2$ parity and the specific structure of Jacobi forms. This "modular protection" naturally suppresses the first-generation masses and mixing angles, with the observed values arising from small deviations from this symmetric point.

Results

Charged Fermions: The reference model, utilizing two independent sextet textures and specific kinetic deformations, successfully reproduces the quark mass ratios, CKM mixing angles, and charged lepton mass ratios. The fit requires the moduli to be close to the two-torsion locus ( $z \approx -1/2$ ), where a residual parity enforces a rank reduction in the mass matrices.
CP Violation: The model demonstrates that a CP-invariant Lagrangian (with real Wilson coefficients) can yield the observed CP-violating phase in the CKM matrix through the spontaneous breaking of CP by the complex moduli VEVs.
Neutrinos: The inclusion of a scalar multiplier allows for a viable neutrino sector with large mixing angles, distinct from the charged sector, without requiring higher-weight forms or additional flavons.
Geometric Limits: The paper clarifies the relationship between the Jacobi construction and conventional eclectic models. In the "zero section" limit ( $z \to 0$ ), the Jacobi forms reduce to modular forms times fixed flavon alignments, recovering the standard eclectic texture. The full Jacobi dependence represents a geometric deformation of this limit.

Significance and Claims
The authors claim that their work provides a "new framework" where eclectic flavour symmetries are realized without flavons, addressing the structural issue of independent vacuum alignment in traditional eclectic models. They assert that the Jacobi group offers a unified geometric parent for both traditional and modular symmetries, where the consistency condition between them is a direct consequence of the group structure rather than an ad hoc requirement.

The paper emphasizes that the successful phenomenological fits are achieved through "modular protection" arising from the geometry of the moduli space (specifically the two-torsion locus), rather than fine-tuned parameters. The authors note that while the approach is bottom-up, it is motivated by top-down string theory constructions (e.g., heterotic orbifolds and Wilson-line moduli) where Jacobi forms and enlarged modular symmetries naturally appear. They conclude that the Jacobi framework offers a consistent, economical, and geometrically motivated alternative to flavon-based flavour models, capable of reproducing the full charged-fermion spectrum and neutrino mixing patterns.

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