⚛️ phenomenology

Fermion mass hierarchy and flavor signals from a $D_4$ dihedral symmetry within a 2HDM-Tx with four-zero textures

This paper proposes an extended Two-Higgs Doublet Model with $D_4$ discrete symmetry and four-zero textures that naturally explains fermion mass hierarchies, suppresses flavor-changing neutral currents, incorporates a dark matter candidate, and simultaneously accommodates observed neutral resonance excesses while remaining consistent with LHC and lepton flavor violation constraints.

Original authors: M. A. Arroyo-Ureña, E. Díaz, J. L. Díaz-Cruz, H. Hernández-Arellano

Published 2026-09-09

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CC BY 4.0

Original authors: M. A. Arroyo-Ureña, E. Díaz, J. L. Díaz-Cruz, H. Hernández-Arellano

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 on a foundation of fundamental particles, but these particles do not all behave the same way. Some are heavy, like the top quark, while others are incredibly light, like the electron. This vast difference in weight, known as the mass hierarchy, is one of the most puzzling mysteries in modern physics. The Standard Model, our best current map of reality, can describe how these particles interact, but it cannot explain why they have the specific masses they do. It is as if the model provides the rules of a game but leaves the scorecards blank. To solve this, physicists often look for hidden patterns or symmetries—mathematical rules that dictate how particles should behave. One such pattern involves "textures," which are specific arrangements of zeros and numbers in the equations that describe particle masses. If these zeros exist for a deep reason, they could explain why the first generation of particles is so light compared to the others.

A team of researchers has proposed a new way to understand these patterns by introducing a specific type of symmetry called the dihedral group D4. Think of this symmetry like a set of strict rules for a dance floor that only allows certain partners to pair up, while forbidding others from ever touching. In this new model, the researchers combined two Higgs fields, which are responsible for giving particles their mass, with a new type of particle called a flavon. The flavon acts as a mediator that helps generate the masses of the heavier particles while naturally suppressing the masses of the lighter ones. This setup creates a "four-zero texture," meaning that in the mathematical description of the particles, four specific spots are forced to be empty. This structure naturally prevents dangerous interactions that would otherwise break the known laws of physics, while also offering a candidate for dark matter, the invisible substance that holds galaxies together.

The researchers did not just propose a theory; they tested it against real-world data to see if it could survive the scrutiny of the Large Hadron Collider. They found that their model could successfully explain several long-standing anomalies, including unexpected signals detected at energies of 95 GeV and 152 GeV. These signals have been a source of debate for years, and the new model provides a coherent story for why they appear. Furthermore, the model fits perfectly with the known properties of the Higgs boson discovered in 2012. The team used powerful computer simulations to map out the vast landscape of possible values for their model's parameters, checking every combination against strict experimental limits. They discovered that the model is highly constrained, meaning it only works within a very narrow range of settings, which makes it a precise and testable prediction rather than a vague guess.

One of the most exciting predictions of this work is a rare event where the Higgs boson decays into a tau particle and a muon, two different types of heavy electrons. In the Standard Model, this should almost never happen. However, in this new framework, it is a natural consequence of the symmetry rules. The researchers simulated how this event would look in the High-Luminosity Large Hadron Collider, the next major upgrade to the world's most powerful particle accelerator. They found that with enough data, the signal of this rare decay could be distinguished from the background noise of ordinary particle collisions. Their analysis suggests that if the model is correct, scientists could see clear evidence of this flavor-changing decay within the next decade of experiments.

The study also addresses the stability of the universe itself. By ensuring that the mathematical potential describing the particles remains stable, the researchers confirmed that their model does not lead to physical contradictions. They showed that the new particles predicted by the theory, including the dark matter candidate, are structurally viable, though a rigorous analysis of its cosmological relic abundance and direct detection constraints is deferred to a future standalone study. While the full implications for dark matter will be explored in future work, the current work establishes that the model is structurally sound. It offers a unified explanation for the mass of particles, the existence of dark matter, and the mysterious signals seen at the collider, all derived from a single, elegant set of symmetry rules.

Ultimately, this work represents a significant step toward understanding the hidden architecture of the universe. It moves beyond simply observing that particles have different masses to explaining why they must have those masses based on a fundamental symmetry. The model is not just a theoretical exercise; it makes specific, testable predictions that can be confirmed or ruled out by upcoming experiments. If the High-Luminosity Large Hadron Collider detects the predicted decay of the Higgs boson into a tau and a muon, it would provide strong evidence that the universe is indeed governed by the dihedral symmetry proposed here. This would transform our understanding of the fundamental forces, turning a collection of unexplained numbers into a coherent story written in the language of symmetry.

Technical Summary: Fermion Mass Hierarchy and Flavor Signals from a D4 Dihedral Symmetry within a 2HDM-Tx with Four-Zero Textures

Problem Statement
The Standard Model (SM) successfully describes a single SM-like Higgs boson but leaves the origin of the fermion mass hierarchy and the structure of the Yukawa sector unexplained. General Two-Higgs Doublet Models (2HDMs) offer a richer scalar spectrum but typically suffer from dangerous Flavor-Changing Neutral Currents (FCNCs) at tree level unless specific symmetries or textures are imposed. Previous phenomenological work by the authors introduced a "Texturized 2HDM" (2HDM-Tx) utilizing four-zero Yukawa textures to reproduce fermion masses and the Cabibbo-Kobayashi-Maskawa (CKM) matrix. However, a fundamental question remained: Can an underlying discrete symmetry naturally justify these specific four-zero textures with non-equivalent matrix configurations, rather than assuming them as an ad-hoc ansatz? Additionally, the model must accommodate recent experimental anomalies, specifically neutral resonance excesses at 95 GeV and 152 GeV, while remaining consistent with stringent limits on lepton flavor violation (LFV).

Methodology
The authors propose an extended 2HDM-Tx framework invariant under the discrete non-abelian dihedral group $D_4$ (order 8). The methodology involves the following steps:

Symmetry Construction: The model introduces two Higgs doublets ( $\Phi_1, \Phi_2$ ), a complex flavon doublet ( $\eta$ ), and a dark matter singlet ( $\chi$ ). Fields are assigned to specific irreducible representations of $D_4$ : first-generation fermions transform as $1^{++}$ singlets, while second and third generations transform as doublets ($2$). The Higgs doublets transform as $1^{+-}$ and $1^{-+}$ , and the flavon as a doublet ($2$).
Texture Generation: The $D_4$ multiplication rules strictly forbid renormalizable Yukawa couplings involving the first generation, automatically generating structural zeros at the $(1,1)$ , $(1,3)$ , and $(3,1)$ positions in both Yukawa matrices ( $Y_1, Y_2$ ). The remaining four parameters ( $a, b, c, d$ ) are generated via higher-dimensional dimension-5 operators involving the flavon doublet $\eta$ and a UV cutoff scale $\Lambda$ . This mechanism naturally suppresses first-generation masses.
Scalar Potential and Mass Spectrum: A full renormalizable scalar potential invariant under $D_4$ is constructed. The vacuum expectation values (VEVs) of the Higgs and flavon fields are determined via tadpole equations. The resulting $4\times4$ CP-even and CP-odd mass matrices are diagonalized to identify physical scalar states.
Phenomenological Constraints: The parameter space is scanned against:
- LHC Higgs signal strengths ( $\mu_X$ ).
- Experimental excesses at 95 GeV and 152 GeV.
- Upper limits on LFV processes ( $\tau \to \mu\gamma$ , $h \to \tau\mu$ , etc.).
- Theoretical constraints (vacuum stability and tree-level unitarity).
Collider Simulation: A Monte Carlo simulation is performed for the High-Luminosity LHC (HL-LHC) at $\sqrt{s} = 14$ TeV. The signal process $gg \to h \to \tau\mu \to \ell\mu + \text{MET}$ is simulated alongside dominant SM backgrounds (Drell-Yan, $t\bar{t}$ , dibosons). A multivariate analysis (MVA) using Boosted Decision Trees (BDT) via XGBoost is employed to optimize signal-background discrimination.

Key Contributions

Symmetry Justification: The paper provides a rigorous group-theoretical derivation for four-zero Yukawa textures using the $D_4$ symmetry, moving beyond phenomenological ansatzes. It demonstrates that the $D_4$ group naturally enforces the required zeros and allows for 15 distinct configurations of the remaining non-zero parameters.
Flavon Role: The introduction of the flavon doublet $\eta$ is shown to be essential, not merely optional. It generates the necessary off-diagonal entries to complete the mass matrices, provides a suppression scale for light fermion masses, and induces mixing between the Higgs and flavon sectors.
Simultaneous Anomaly Resolution: The model successfully accommodates the persistent neutral resonance excesses at 95 GeV and 152 GeV. The authors identify the second CP-even scalar ( $S_2$ ) and a CP-odd scalar ( $A_2$ ) with these excesses, achieved through specific mixing patterns in the scalar sector.
Dark Matter Candidate: The model naturally includes a stable scalar singlet ( $\chi$ ) as a dark matter candidate, though a detailed cosmological analysis is deferred to future work.

Results

Parameter Space: A systematic scan of approximately $5 \times 10^{10}$ points reveals a preferred region where $\cos(\alpha - \beta) \sim 0$ and $5 \lesssim \tan\beta \lesssim 40$ . This region satisfies all LHC Higgs data, theoretical stability bounds, and LFV constraints.
Mass Spectrum: The model predicts a SM-like Higgs boson ( $S_1$ ) at $\approx 125$ GeV, a scalar ( $S_2$ ) at $\approx 95$ GeV, and a pseudoscalar ( $A_2$ ) at $\approx 152$ GeV, consistent with observed excesses. Additional scalars ( $S_3, S_4, A_3$ ) are predicted in the range of 100–220 GeV, with masses dependent on the flavon VEV.
LFV Process $h \to \tau\mu$ : The study focuses on Case 15, where the Yukawa matrix $Y_2$ is strictly sparse. The branching ratio for $h \to \tau\mu$ is found to be sensitive to the parameters $\cos(\alpha-\beta)$ , $\tan\beta$ , and the Yukawa coupling $a$ .
Discovery Potential: For a benchmark cross-section of $\sigma(gg \to h \to \tau\mu) \approx 5.1$ fb, the multivariate analysis predicts a discovery significance of $5\sigma$ at the HL-LHC with an integrated luminosity of $3000 \text{ fb}^{-1}$ . For a cross-section of 2 fb, a $2\sigma$ limit can be imposed. The simulation framework was validated against CMS Run 2 data, showing excellent kinematic agreement.

Significance
The paper claims that the 2HDM-Tx with $D_4$ symmetry offers a structurally sound and highly predictive extension of the SM. Its significance lies in unifying three distinct phenomenological challenges: explaining the fermion mass hierarchy via symmetry, resolving the 95/152 GeV resonance anomalies through scalar mixing, and providing a testable signature for lepton flavor violation ( $h \to \tau\mu$ ) at future colliders. The authors assert that the model is not only consistent with current experimental bounds but also offers a clear path for discovery at the HL-LHC, distinguishing it from generic 2HDMs by its specific symmetry-motivated texture and the inclusion of a dark matter candidate.

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