⚛️ phenomenology

The Masses of Fermions in the context of the Supersymmetric $SU(3)_{C}\times SU(3)_{L}\times U(1)_{N}$ Model

This paper presents a detailed analysis of the fermion mass spectrum in the Minimal Supersymmetric $SU(3)_{C}\times SU(3)_{L}\times U(1)_{N}$ model under both $R$ -parity conserving and violating scenarios, demonstrating that numerical predictions for all masses and mixing angles align with current experimental data.

Original authors: M. C. Rodriguez

Published 2026-08-31

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: M. C. Rodriguez

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, but the rules that govern them are far from simple. For decades, physicists have relied on a standard model to explain how these particles interact, yet this model leaves several big questions unanswered. It cannot explain why some particles are incredibly heavy while others are nearly weightless, nor can it fully account for the mysterious force of gravity or the invisible matter that holds galaxies together. To solve these puzzles, scientists often look to a concept called supersymmetry. This idea suggests that every known particle has a heavier, invisible partner, creating a more balanced and complete picture of reality. When researchers combine this idea with a specific extension of the known forces, they create a new theoretical framework that attempts to explain the origin of mass for all the matter in the universe.

In a recent study, a physicist from the Federal Rural University of Rio de Janeiro explored the properties of a specific version of this extended model, known as the supersymmetric 331 model. This framework is a variation of the standard model that introduces new types of particles and forces to explain why there are exactly three families of matter and to address the hierarchy of particle masses. The researcher's goal was to calculate the masses of all the fermions—particles like electrons and quarks that make up ordinary matter—within this specific theoretical setup. By assuming that certain invisible fields in the universe take on specific values, the study mapped out how these particles would acquire their weight and how they would mix with one another.

The investigation focused on a scenario where the model includes a special type of symmetry that protects the universe from certain types of decay, a feature known as R-parity conservation. This constraint ensures that the theory remains stable and consistent with what we already know about the stability of matter. The researcher calculated the masses for the known particles, such as the top quark, which is the heaviest known elementary particle, and compared these theoretical predictions against the actual values measured by large particle colliders. The results showed that the model could successfully reproduce the observed masses of the six types of quarks and the three types of charged leptons, such as the electron and the tau particle. Furthermore, the calculations accounted for the mixing patterns between these particles, which determine how they transform into one another during interactions, matching the experimental data with high precision.

Beyond the familiar particles, the model predicts the existence of new, exotic quarks that are much heavier than anything seen so far. The study calculated that these new particles would have masses in the range of several thousand gigaelectronvolts, placing them well beyond the reach of current detectors but potentially accessible to future high-energy experiments. The research also examined the behavior of new force-carrying particles, including doubly charged bosons that could interact with matter in unique ways. The analysis confirmed that the lightest of these new particles would have masses below 310 gigaelectronvolts, a range that is consistent with existing experimental limits but suggests they could be discovered in upcoming collider runs.

The study also addressed the nature of neutrinos, the ghostly particles that pass through the Earth by the trillions every second. In this model, the mechanism that gives mass to charged particles also generates a mass for neutrinos, explaining why they are not massless as once thought. The calculations showed that the mixing between different types of neutrinos could follow specific patterns observed in nature, such as the "tribimaximal" mixing pattern, which aligns with current experimental observations. This suggests that the model provides a unified explanation for the masses of both the heavy quarks and the nearly massless neutrinos, linking them through the same underlying physical principles.

Ultimately, the work demonstrates that this supersymmetric extension of the standard model is a viable candidate for describing the fundamental structure of matter. The numerical predictions for all fermion masses and their mixing angles fall within the ranges established by current experimental data. The study does not claim to have proven the existence of these new particles, but it shows that the theory is mathematically consistent and capable of reproducing the known world while offering testable predictions for the future. By confirming that the model can accommodate the heavy top quark and the light electron simultaneously, the research strengthens the case for exploring these extended theories as a path toward a deeper understanding of the universe. The findings suggest that if nature has chosen this specific path, the next generation of particle accelerators may soon reveal the heavy partners and exotic forces that this model predicts.

Technical Summary: Mass Spectrum of Fermions in the Supersymmetric SU(3) $_C \times$ SU(3) $_L \times$ U(1) $_N$ Model

Problem Statement
The Standard Model (SM) successfully describes particle physics but leaves several fundamental questions unanswered, including the origin of fermion mass hierarchies, the number of fermion generations, and the unification of gauge couplings. Supersymmetry (SUSY) offers a framework to address these issues, particularly the hierarchy problem and gauge coupling unification. The Minimal Supersymmetric Standard Model (MSSM) is a leading candidate, yet extensions such as the Supersymmetric SU(3) $_C \times$ SU(3) $_L \times$ U(1) $_N$ (331) model provide alternative mechanisms for symmetry breaking and anomaly cancellation. This paper addresses the specific challenge of deriving the complete mass spectrum for all fermions within the context of the Minimal Supersymmetric 331 (MSUSY331) model, ensuring consistency with current experimental data and R-parity conservation.

Methodology
The study constructs the MSUSY331 model by extending the gauge group to $SU(3)_C \times SU(3)_L \times U(1)_N$ . The methodology involves:

Field Content Definition: The model introduces chiral superfields for quarks and leptons. Quarks are arranged such that two generations transform as anti-triplets and one generation as a triplet under $SU(3)_L$ , a feature required for anomaly cancellation. Leptons are placed in triplet representations. To cancel chiral anomalies and implement SUSY, the model includes additional scalar fields: triplets ( $\eta, \rho, \chi$ ), their conjugates ( $\eta', \rho', \chi'$ ), and sextets/anti-sextets ( $S, S'$ ).
R-Parity Conservation: A discrete symmetry, R-parity ( $R = (-1)^{2S+3F}$ where $F \equiv B+L$ ), is defined to ensure the stability of the lightest supersymmetric particle and to forbid rapid proton decay. The superpotential is constructed to respect this invariance.
Vacuum Expectation Values (VEVs): The analysis assumes that all usual neutral scalar fields acquire VEVs. Specific numerical values (in GeV) are chosen for the VEVs ( $v_\eta, v_{\sigma_2}, v_{\eta'}, v_{\rho'}, v_\rho$ ) to facilitate numerical predictions.
Mass Matrix Construction:
- Quarks: Yukawa interactions involving the scalar triplets and singlets generate mass matrices for up-type, down-type, and exotic quarks. The matrices are diagonalized using rotation matrices to obtain physical masses and mixing angles.
- Charged Leptons: Masses are generated via couplings to the triplet $\eta$ and the anti-sextet $S$ .
- Neutralinos and Charginos: The mixing between gauginos and higgsinos is analyzed by constructing mass matrices for double-charged charginos, standard charginos, and neutralinos.
Numerical Analysis: The author performs numerical fits using the chosen VEVs and coupling parameters to reproduce known fermion masses and mixing matrices (CKM and PMNS).

Key Contributions

Complete Fermion Spectrum: The paper provides a comprehensive derivation of the mass spectrum for all fermions in the MSUSY331 model, including standard quarks, leptons, exotic quarks ( $J, j$ ), and supersymmetric partners (charginos, neutralinos).
Explanation of Mass Hierarchies: The model utilizes the distinct transformation properties of the third quark family compared to the first two to naturally explain the large mass difference between the third generation and the lighter generations. Specifically, the third family acquires mass through different VEVs and coupling structures than the first two.
Mixing Matrices: The work derives the CKM matrix and discusses the PMNS matrix within the model's framework, showing how specific mixing scenarios (such as "bi-large" or "tribimaximal" mixing) can be accommodated.
Exotic Particle Mass Limits: The study establishes theoretical lower bounds for the masses of exotic quarks ( $J, j$ ) and doubly charged gauge bosons ( $U^{\pm\pm}$ ), linking them to the scale of symmetry breaking ( $v_\chi, v_{\chi'}$ ).

Results

Quark Masses: The numerical analysis successfully reproduces the masses of up, down, charm, strange, bottom, and top quarks. The model predicts that the top and strange quarks are more massive than the charm and bottom quarks, respectively, based on the hierarchy of VEVs ( $v_\rho > v_\eta$ ). The CKM matrix elements are shown to be in agreement with experimental ranges.
Lepton Masses: The charged lepton masses ( $e, \mu, \tau$ ) are fitted using the VEVs of the anti-sextet and triplet fields. The model accommodates neutrino masses via the Majorana mass term generated by the VEV of the neutral component of the triplet ( $\langle \sigma^0_1 \rangle$ ), consistent with cosmological constraints on the sum of neutrino masses.
Exotic Quarks: The masses of the exotic quarks $J$ and $j$ are found to be proportional to the VEVs $v_{\chi'}$ and $v_\chi$ , respectively. With $v_{\chi'} \approx 8.0$ TeV, the model predicts $M_J > 4.0$ TeV.
Supersymmetric Partners:
- Double Charginos: The lightest double chargino ( $\tilde{\chi}^{\pm\pm}_1$ ) is predicted to have a mass $\le 245$ GeV, with the second state $\le 310$ GeV.
- Standard Charginos: The lightest chargino ( $\tilde{\chi}^\pm_1$ ) is predicted to have a mass $\le 31$ GeV, with subsequent states up to $\le 250$ GeV.
- Neutralinos: The lightest neutralino ( $\tilde{\chi}^0_1$ ) is predicted to have a mass $\le 65$ GeV, with the next state $\le 75$ GeV.
Agreement with Data: The author asserts that all calculated masses and mixing angles are in agreement with current experimental data.

Significance
The paper claims that the MSUSY331 model, with R-parity conservation and the inclusion of sextet/anti-sextet scalar fields, provides a viable extension of the Standard Model that successfully reproduces the observed fermion mass spectrum. The model offers a natural explanation for the family structure of quarks and the hierarchy of their masses without fine-tuning. Furthermore, the derived mass limits for supersymmetric particles and exotic quarks provide specific targets for experimental searches at the Large Hadron Collider (LHC). The work demonstrates that the 331 framework, when supersymmetrized, remains a consistent and phenomenologically viable candidate for physics beyond the Standard Model.

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