The Masses of Bosons and Usual Fermions on Supersymmetric $SU(3)_{C}\times SU(4)_{L}\times U(1)_{X}$ Model

The universe operates on a set of invisible rules that dictate how the smallest building blocks of matter interact. For decades, physicists have relied on a framework called the Standard Model to organize these particles and forces, much like a periodic table organizes the elements. This model successfully explains how particles such as electrons and quarks behave and how they are held together by forces like electromagnetism and the strong nuclear force. However, the Standard Model leaves several profound questions unanswered, including why particles have the specific masses they do and how gravity fits into the picture. To address these gaps, scientists often propose extensions to the theory, introducing new symmetries—mathematical patterns that suggest hidden relationships between particles. One such extension involves expanding the group of particles that interact with the weak nuclear force, a force responsible for processes like radioactive decay, to include a larger family of particles than currently observed.

In this context, a researcher named M. C. Rodriguez has explored a specific theoretical framework known as the supersymmetric 3-4-1 model. This model proposes that the universe contains a hidden layer of reality where every known particle has a heavier, unseen partner, a concept known as supersymmetry. The study focuses on a version of the theory where the mathematical symmetry governing the weak force is expanded from a group of three components to a group of four. By doing so, the model attempts to unify the behavior of quarks and leptons in a way that naturally explains why there are three generations of matter and how they acquire their masses. The paper does not claim to have discovered these new particles in a laboratory; rather, it performs a detailed mathematical construction to show that such a universe is theoretically consistent and to calculate exactly what the masses of the new particles would be if this model were true.

The core of this work involves building a complete map of the particle spectrum for this expanded universe. In the standard view, particles are grouped into families, and their interactions are governed by specific mathematical generators, which can be thought of as the keys that unlock different types of interactions. Rodriguez takes the known particles—quarks, which make up protons and neutrons, and leptons, such as electrons and neutrinos—and arranges them into larger groups called quartets. This arrangement requires the introduction of new, heavier versions of these particles, some of which carry electric charges that have never been observed, such as particles with a charge of plus five-thirds or minus four-thirds. The study meticulously tracks how these new particles fit together with the existing ones, ensuring that the mathematical rules of the universe, specifically the cancellation of anomalies that would otherwise make the theory impossible, are satisfied.

A significant portion of the paper is dedicated to understanding how these particles acquire mass. In the real world, particles gain mass through their interaction with a field that permeates all of space. In this model, the author introduces several new fields, represented by mathematical objects called scalars, which break the symmetry of the universe in stages. First, the symmetry breaks at a very high energy level, separating the new, heavy particles from the familiar ones. Then, at a lower energy level, the symmetry breaks again to give mass to the particles we know. The researcher calculates the specific values of these masses by determining how strongly the particles interact with these fields. The result is a detailed spectrum showing that the new, heavy particles would be significantly more massive than the known particles, while the familiar particles like the electron and the photon remain light and massless as expected.

The study also examines the forces that would exist in this expanded universe. Just as the electromagnetic force is carried by a particle called the photon, the new forces in this model are carried by new types of force-carrying particles. The author calculates the masses of these new bosons, which are the carriers of the weak and new forces. The calculations reveal a hierarchy of masses, with some of these new force carriers being very heavy, which explains why we do not see their effects in everyday life. The paper also addresses the behavior of neutrinos, the ghostly particles that rarely interact with matter. It shows how the model can generate tiny masses for these neutrinos through complex interactions involving the new fields, offering a potential explanation for why they are so much lighter than other particles.

Crucially, the paper integrates the concept of supersymmetry, which posits that every particle has a partner with a different spin. The author constructs the model so that these partners, known as superpartners, also acquire masses through the same symmetry-breaking process. The study assigns specific properties to these superpartners, ensuring that the model remains stable and mathematically sound. By doing this, the researcher provides a complete picture of both the ordinary particles and their supersymmetric counterparts within this specific 3-4-1 framework. The work confirms that such a model is viable, meaning it does not contradict the fundamental laws of physics as we understand them, and it provides a concrete set of predictions for what the masses of these new particles would be.

Ultimately, this research serves as a rigorous theoretical blueprint. It does not prove that this specific version of the universe is the one we inhabit, but it demonstrates that a universe with these specific properties is possible. By laying out the exact mass spectrum for the bosons and fermions in this supersymmetric 3-4-1 model, the paper offers a clear target for future experiments. If scientists ever build a machine powerful enough to create these heavy particles, they will know exactly what mass to look for. The work stands as a testament to the power of mathematical consistency in physics, showing how expanding our view of symmetry can lead to a richer, more complete understanding of the fundamental structure of reality.