⚛️ phenomenology

Textures of dimension-six operators in the SMEFT with non-invertible selection rules

This paper classifies the flavor textures of baryon-number-conserving dimension-six SMEFT operators under non-invertible selection rules, revealing distinct Wilson coefficient structures and flavor patterns that deviate from Minimal Flavor Violation and offer unique predictions for B-meson and charged-lepton-flavor-violating observables.

Original authors: Tatsuo Kobayashi, Hajime Otsuka, Morimitsu Tanimoto, Kei Yamamoto

Published 2026-08-25

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CC BY 4.0

Original authors: Tatsuo Kobayashi, Hajime Otsuka, Morimitsu Tanimoto, Kei Yamamoto

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 universe is built from a small set of fundamental particles, such as quarks and electrons, which interact through forces that govern everything from the stability of atoms to the light of distant stars. Physicists have long been puzzled by the "flavor" of these particles: why do they come in three distinct generations, and why do they have the specific masses and mixing patterns that we observe? To answer this, scientists often look for hidden rules or symmetries that dictate how these particles can interact. In the standard view, these rules are like a rigid blueprint, where the way particles mix is strictly determined by their mass-generating properties. However, a new line of inquiry suggests that the universe might follow a different kind of logic, one that allows for more complex and unexpected patterns in how particles talk to one another.

A team of researchers has recently explored this alternative possibility by investigating the "flavor textures" of higher-dimensional operators within the Standard Model Effective Field Theory. In simple terms, this framework allows physicists to describe rare processes that happen at energy levels far beyond what current machines can directly reach. The researchers focused on a specific set of rules known as non-invertible selection rules. Unlike traditional symmetry rules that can be easily reversed or undone, these rules act like a one-way filter, permitting certain particle interactions while strictly forbidding others based on a mathematical structure called a fusion algebra. By applying these rules to a model of the universe that includes two types of Higgs fields, the team mapped out exactly which interactions are allowed and which are blocked.

The study reveals that these non-invertible rules create a unique landscape of allowed interactions that differs significantly from the patterns predicted by the most common theories. In the standard approach, known as Minimal Flavor Violation, the way particles mix in rare, high-energy processes is expected to be a direct copy of how they mix when they gain mass. The new research shows that this is not necessarily true. The non-invertible rules allow the "flavor" of these rare interactions to be completely independent of the mass-generating patterns. This means that the universe could have a hidden structure where the rules governing rare events are distinct from the rules governing the everyday mass of particles, a possibility that was previously difficult to realize without introducing extra, unobserved fields.

To test the implications of this discovery, the authors examined specific scenarios where these rules apply to the three generations of quarks and leptons. They found that for certain assignments of these particles, the number of independent parameters needed to describe the universe drops dramatically, yet the resulting patterns remain rich and complex. For instance, in one specific configuration, the rules allow for a vast number of interactions while strictly forbidding others, creating a "texture" of zeros and allowed values that is distinct from any pattern generated by conventional group symmetries. This texture is not random; it follows a precise mathematical logic derived from the fusion rules, which can be described by a reduced set of independent numbers.

The researchers then looked at how these theoretical textures would manifest in real-world observations, particularly in the decay of B-mesons, which are heavy particles containing a bottom quark. These decays are a hotbed for discovering new physics because they are sensitive to the subtle effects of higher-dimensional operators. The study suggests that the non-invertible rules would lead to specific, testable predictions for how often these decays occur and how they behave with different types of leptons. For example, the model predicts a distinct hierarchy in how new physics affects the decay of a B-meson into a tau lepton versus a muon or an electron. Unlike other theories that might predict a uniform suppression or enhancement, this model suggests a pattern where the third generation of particles is treated differently, but in a way that does not simply mirror their mass hierarchy.

Furthermore, the research sheds light on a phenomenon called charged-lepton flavor violation, where a heavy lepton like a tau or muon decays into a lighter one, such as an electron, by emitting a photon. This process is forbidden in the standard model but could occur if new physics exists. The authors found that the non-invertible rules impose a unique "chirality" on these decays, meaning they favor one specific orientation of the particle's spin over the other. In the specific scenario they analyzed, the decay of a muon into an electron is predicted to be dominated by a left-handed muon turning into a right-handed electron, while the reverse process is heavily suppressed. This stands in sharp contrast to other popular theories, which often predict the opposite dominance.

By classifying all the possible baryon-number-conserving operators in this framework, the team provided a comprehensive map of what is possible under these new rules. They counted the number of independent parameters for each type of interaction, showing that while the total number of possibilities is reduced compared to a universe with no symmetry, it remains large enough to accommodate the complex reality we see. The study confirms that these non-invertible selection rules can reproduce the known masses and mixing angles of quarks and leptons while simultaneously generating a unique set of constraints on rare processes. This offers a fresh perspective on the flavor problem, suggesting that the universe's hidden rules might be more intricate and less rigid than previously thought.

The work does not claim to have solved the mystery of flavor, but it provides a concrete, testable framework for how it might be structured. The predictions made by this model, particularly regarding the specific patterns of B-meson decays and the spin-dependent nature of rare lepton decays, offer clear targets for future experiments. If upcoming measurements at particle colliders or flavor factories observe these specific patterns, it would provide strong evidence that the universe is governed by these non-invertible selection rules. Conversely, if the data shows a different pattern, it would help rule out this specific class of theories. The study stands as a detailed exploration of a viable alternative to the standard view, demonstrating that the laws of nature might allow for a richer, more diverse set of interactions than the conventional wisdom suggests.

Technical Summary: Textures of dimension-six operators in the SMEFT with non-invertible selection rules

Problem Statement
The flavor structure of the Standard Model Effective Field Theory (SMEFT) is a critical area for probing new physics (NP), particularly in light of anomalies in B-meson decays and the need to understand the origin of fermion masses and mixing angles. Conventional approaches, such as the Minimal Flavor Violation (MFV) hypothesis, posit that Yukawa couplings are the sole sources of flavor symmetry breaking, thereby aligning the flavor structures of higher-dimensional operators with those of the Yukawa couplings. While MFV and other group-based symmetries (e.g., $U(3)^5$ , $U(2)^5$ , modular symmetries) successfully constrain flavor parameters, they often require extended scalar sectors (flavons) and may not generate all possible texture zeros or relations observed in data. Recently, non-invertible selection rules have been proposed as an alternative mechanism to generate Yukawa textures without introducing flavon fields. However, the implications of these non-invertible rules for the flavor textures of dimension-six SMEFT operators remain largely unexplored. Specifically, it is unclear whether these rules enforce the alignment of higher-dimensional operators with Yukawa couplings or allow for distinct flavor structures.

Methodology
The authors investigate the flavor textures of all baryon-number-conserving dimension-six operators in the SMEFT under the constraint of non-invertible selection rules derived from a specific fusion algebra.

Theoretical Framework: The study utilizes a four-dimensional effective field theory where matter fields are assigned to conjugacy classes of a finite group $G$ , rather than irreducible representations. The selection rules are governed by the fusion algebra of these conjugacy classes. Specifically, the authors focus on the $\tilde{Z}_5$ symmetry, defined as the $Z_2$ gauging of a $Z_5$ symmetry (isomorphic to the dihedral group $D_5$ ), and its product $\tilde{Z}_5 \times \tilde{Z}_5$ .
Model Setup: To reproduce realistic Yukawa textures for quarks and leptons, the authors employ a Type-II Two-Higgs-Doublet Model (2HDM) where matter fields are assigned to basis elements of the $\tilde{Z}_5^{(1)} \times \tilde{Z}_5^{(2)}$ fusion algebra. They identify five representative charge assignments (labeled (a) through (e)) that successfully reproduce specific target Yukawa textures (diagonal for up-type and neutrino sectors, and specific off-diagonal patterns for down-type and charged lepton sectors).
Classification: Using these charge assignments, the authors systematically classify the flavor textures of all dimension-six operators in the Warsaw basis. This includes:
- Four-fermion operators: $(\bar{L}L)(\bar{L}L)$ , $(\bar{L}L)(\bar{R}R)$ , $(\bar{R}R)(\bar{R}R)$ , and $(\bar{L}R)(\bar{R}L)$ / $(\bar{L}R)(\bar{L}R)$ .
- Operators involving Higgs and fermions: $\psi^2XH$ , $\psi^2H^2D$ , and $\psi^2H^3$ .
Analysis: For each assignment, the authors determine the allowed non-zero Wilson coefficients and the resulting texture zeros. They analyze the symmetry properties of these textures (e.g., $Z_2$ , $Z_2 \times Z_2$ , $S_3$ ) and count the number of independent CP-even and CP-odd parameters after imposing Hermiticity and current-exchange symmetries.
Phenomenology: The paper focuses on Case (a) for phenomenological applications. The authors transform the Wilson coefficients from the flavor basis to the mass basis using the derived right-handed mixing matrices for quarks and leptons. They then evaluate the impact on:
- Semileptonic $B$ -meson decays ( $b \to c \ell \bar{\nu}_\ell$ and $b \to s \nu \bar{\nu}$ ).
- Charged-lepton flavor-violating (cLFV) radiative decays ( $\mu \to e \gamma$ , $\tau \to \ell \gamma$ ).

Key Contributions and Results

Classification of Textures: The paper provides a complete classification of flavor textures for 59 independent dimension-six SMEFT operators under five distinct charge assignments. It demonstrates that non-invertible selection rules generate specific, non-trivial texture zeros and relations among Wilson coefficients that differ from those obtained via conventional group symmetries.
Decoupling from Yukawa Alignment: A central finding is that the flavor structures of dimension-six operators are not necessarily aligned with the Yukawa couplings. Unlike the MFV hypothesis, where Yukawa couplings govern the flavor dependence of higher-dimensional operators, the non-invertible selection rules allow the Wilson coefficients to possess distinct flavor patterns even when the Yukawa textures are reproduced.
Parameter Reduction: The selection rules significantly reduce the number of independent flavor parameters compared to the general SMEFT (2499 parameters). For the most restrictive case (Case a), the number of independent CP-even and CP-odd coefficients is reduced to 694 and 493, respectively. While this reduction is less severe than in strict MFV scenarios (e.g., $U(3)^5$ ), the resulting structures are qualitatively different.
Phenomenological Predictions (Case a):
- $b \to c \ell \bar{\nu}_\ell$ : The model predicts that new physics contributions to $b \to c \tau \bar{\nu}_\tau$ and $b \to c \mu \bar{\nu}_\mu$ are comparable in magnitude, while $b \to c e \bar{\nu}_e$ is suppressed. This contrasts with models favoring a strong third-generation coupling (like $U(2)$ symmetry) which typically enhance the third generation relative to the second.
- $b \to s \nu \bar{\nu}$ : The contribution to this neutral-current process is found to be suppressed due to CKM unitarity cancellations and the specific flavor structure of the Wilson coefficients.
- Chirality in cLFV: For the leptonic dipole operators, the model predicts a characteristic chirality dependence. Specifically, the amplitude for $\mu_R \to e_L \gamma$ is suppressed by $O(\lambda^2)$ relative to $\mu_L \to e_R \gamma$ . This is in sharp contrast to models based on $A_4$ modular symmetry or $U(2)$ symmetry, where the right-handed transition often dominates. Similarly, $\tau \to \ell \gamma$ amplitudes involving right-handed leptons are suppressed by small right-handed mixing angles.

Significance
The paper establishes that non-invertible selection rules provide a viable and distinct framework for flavor physics that does not rely on the MFV hypothesis. By demonstrating that higher-dimensional operators can have flavor structures decoupled from Yukawa couplings, the work opens new avenues for explaining flavor anomalies without the constraints of traditional symmetry alignment. The specific predictions regarding the suppression of certain chirality channels in cLFV decays and the relative strengths of $b \to c \ell \bar{\nu}$ transitions offer characteristic signatures that can be tested in current and future flavor experiments. These results suggest that the flavor structure of SMEFT operators can serve as a probe to distinguish between different UV completion mechanisms, including those based on non-invertible symmetries versus conventional group symmetries. The authors note that a systematic classification for other Yukawa textures remains a subject for future work.

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