Quark and Lepton Masses, Baryon Asymmetry, and Neutrino Mass from a Supersymmetric Preon Model

This paper proposes a supersymmetric preon model where Standard Model fermions are three-body composites confined by a Maxwell-Chern-Simons interaction, successfully reproducing observed quark mass ratios and neutrino masses via Pauli principle constraints and a Type I seesaw mechanism while simultaneously generating the baryon asymmetry of the universe through anomaly inflow and dynamically deriving R-parity.

The Gist

Imagine the universe as a giant, cosmic LEGO set. For decades, physicists have been trying to figure out why the pieces are shaped the way they are. Why is the "electron" piece so light and tiny, while the "top quark" piece is heavy and massive? Why is there more matter than antimatter? And why do neutrinos (ghostly particles) have almost no mass?

This paper proposes a radical new way to build that LEGO set. Instead of assuming the Standard Model particles (like electrons and quarks) are fundamental, unbreakable bricks, the author suggests they are actually composite structures—tiny clusters made of even smaller, more fundamental pieces called preons (or "chernons").

Here is the story of this model, explained through everyday analogies.

1. The Building Blocks: The "Preon" LEGO Set

Think of the universe as having a hidden, ultra-high-energy factory operating at a scale we can't see directly (about $10^{14}$ times heavier than a proton). In this factory, there are three types of fundamental "preon" bricks:

• Neutral Preons ($\psi_0$): Like white, neutral blocks.
• Charged Preons ($\psi_1, \psi_{-1}$): Like red and blue blocks with electric charge.
• The "Spectator" ($\chi$): A special, invisible block added just to keep the math of the universe from breaking (an "anomaly cancellation" fix).

These preons are glued together by a new, super-strong force called Metacolor (similar to how glue holds quarks together to make protons, but much stronger).

2. The Great Mass Mystery: Why is the Electron Lighter than the Up Quark?

In our current understanding, the electron is much lighter than the up quark (about 1/5th the weight). Standard physics treats this as a random number we just measure. This paper tries to calculate it.

The author runs four different complex computer simulations (like trying to solve a puzzle with four different strategies) to see how these preon clusters form.

• The Electron: It's made of three identical charged preons ($\psi_{-1}$). Imagine three people holding hands in a circle, all pulling on each other with a strong magnetic force. They bind together tightly, but because they are all the same, the binding energy is very specific.
• The Up Quark: It's made of two charged preons and one neutral one. Imagine two people holding hands, but the third person is holding a heavy, invisible anchor (the neutral preon). The "glue" holding them together is different.

The Breakthrough: The author found that if you adjust the "stiffness" of the glue (a value called string tension) to a specific, natural number, the math predicts the electron will be exactly 0.22 times the weight of the up quark. This matches the real world perfectly without needing to "tune" the numbers artificially.

3. The "Pauli Principle" Puzzle: Why is the Down Quark Heavier?

There's another mystery: The down quark is slightly heavier than the up quark. Why?

The paper uses a rule called the Pauli Exclusion Principle (which says identical particles can't occupy the exact same state).

• In the Up Quark, the two charged preons pair up nicely.
• In the Down Quark, the two neutral preons are forced into a "spin-triplet" state (like two dancers spinning in the same direction). Because of the rules of quantum mechanics, this specific dance makes the "glue" between them repulsive (pushing them apart) rather than attractive.

The Analogy: Imagine trying to glue two magnets together. If you align them right, they snap together (Up Quark). If you flip one, they push apart, making the whole structure harder to hold together and effectively "heavier" because it's less stable. This explains why the down quark is heavier than the up quark naturally, without guessing.

4. The Ghostly Neutrino: Why is it Massless?

Neutrinos are weird; they barely have any mass.

• The model suggests a neutrino is made of three neutral preons ($\psi_0$).
• Because they are all identical and neutral, the Pauli principle forces them into a very specific, high-energy "dance" (spin-3/2).
• In this dance, the "glue" (specifically a supersymmetric force) pushes them apart so hard that they cannot bind at all.

The Result: If they can't bind, they have no "binding energy" mass. They emerge from the factory as massless particles. This explains why neutrinos are so light. (A tiny bit of mass is added later via a "seesaw" mechanism involving the "spectator" block, bringing it down to the tiny value we observe).

5. The Universe's Imbalance: Why is there more Matter than Antimatter?

The universe is made of matter, not antimatter. This is a huge mystery.

• The paper suggests this imbalance was created at the moment the preons were glued together (the "Confinement Scale").
• As the universe cooled, the preons condensed into matter. Due to a subtle difference in how "fermions" (matter particles) and "bosons" (force particles) behave during this transition, a tiny bias was created.
• The Analogy: Imagine a coin toss where the coin is slightly bent. Over billions of tosses, you get slightly more heads than tails. This model calculates that the "bend" in the coin (a parameter called $\epsilon$) is about 2.2%. This tiny bias, multiplied by the huge energy of the early universe, creates exactly the amount of matter we see today.

6. The "Invisible" Superpartners and Dark Matter

Supersymmetry (SUSY) predicts that every particle has a "superpartner" (e.g., an electron has a "selectron"). We haven't found them at the Large Hadron Collider (LHC).

• The Twist: In this model, superpartners aren't new, fundamental particles. They are just different arrangements of the same preon bricks.
• R-Parity: The model proves that you can't turn a matter-cluster into an anti-matter-cluster just by shaking it. This creates a natural "conservation law" (R-parity) that makes the lightest superpartner stable.
• Dark Matter: The lightest stable superpartner is likely a neutral, bosonic version of the neutrino. Since it's stable and invisible, it is a perfect candidate for Dark Matter.
• Why we haven't seen them: The LHC isn't powerful enough to break the preons apart to see the superpartners directly. However, the paper suggests that some of the "scalar mesons" (weird, short-lived particles we have seen) might actually be these superpartners in disguise!

Summary

This paper proposes a unified theory where:

1. Masses are determined by how three preons bind together (like LEGO structures).
2. Neutrino masslessness is a result of quantum rules preventing them from binding.
3. Matter vs. Antimatter is a result of a tiny bias in how these structures formed.
4. Dark Matter is the stable "shadow" of these structures.

It replaces a dozen random numbers in the Standard Model with a single, elegant mechanical story about how tiny preons glue together. While it requires complex math to prove, the core idea is simple: The universe isn't made of fundamental bricks; it's made of complex structures built from even smaller, hidden bricks.

Technical Summary

1. Problem Statement

The paper addresses two fundamental unsolved problems in particle physics and cosmology simultaneously:

1. The Flavor Problem: The Standard Model (SM) offers no explanation for the vast hierarchy of fermion masses (spanning 12 orders of magnitude from $\nu_e$ to $t$) or the specific mass ratios (e.g., $m_e/m_u \approx 0.22$, $m_d/m_u \approx 2.0$). These are treated as free parameters (Yukawa couplings).
2. The Baryon Asymmetry of the Universe (BAU): The observed excess of matter over antimatter ($\eta \approx 8.7 \times 10^{-10}$) requires Baryon number violation, CP violation, and departure from thermal equilibrium (Sakharov conditions), none of which are adequately satisfied by the SM alone.

The author proposes that both hierarchies originate from a single physical scale: the preon confinement scale $\Lambda_{cr} \sim 10^{14}$ GeV.

2. Theoretical Framework and Methodology

The Model

The model posits that SM fermions are three-body composites of fundamental fermions called preons (or chernons), confined by a Maxwell-Chern-Simons (MCS) interaction and a new metacolor gauge symmetry $SU(3)_{mc}$.

• Preon Spectrum:
  • $\psi_0$: Neutral, metacolor triplet $(0, \mathbf{3}_{mc})$.
  • $\psi_1$: Charged ($+1/3$), metacolor singlet.
  • $\psi_{-1}$: Charged ($-1/3$), metacolor singlet.
  • $\chi$: A spectator field $(0, \overline{\mathbf{3}}_{mc})$ introduced to cancel gauge anomalies.
• Composite Structure: SM fermions are formed as $SU(3)_{mc}$ singlets (e.g., $e^- \sim (\psi_{-1})^3$, $u \sim \psi_1^2 \psi_0$, $\nu \sim (\psi_0)^3$).
• Supersymmetry (SUSY): Implemented at the preon level. Superpartners are not new elementary particles but composite bosonic states of the same preons, differing only in spin-statistics structure.

Methodology

The author employs four systematic numerical methods, validated against the hydrogen atom, to solve the three-body bound state problem:

1. Non-relativistic Schrödinger Equation: Tested to show the two-body approximation fails.
2. Relativistic Dirac Equation: Used to analyze near-threshold binding.
3. Hyperspherical $K=0$ Method: Reduces the 6D three-body problem to a 1D radial equation.
4. Gaussian Variational Method: Applied to the full three-body problem with a trial wavefunction.

The analysis incorporates the Pauli Exclusion Principle applied to the full spin-color wavefunction and includes SUSY corrections (scalar $\sigma$ and gaugino $\lambda$ exchanges) to the potential.

3. Key Contributions and Results

A. Fermion Mass Hierarchy ($m_e/m_u$ and $m_d/m_u$)

• Failure of Two-Body Approximation: Previous attempts using two-body potentials failed to reproduce $m_e/m_u \approx 0.22$, predicting ratios of $\sim 9$ (non-relativistic) or $\sim 1$ (relativistic).
• Three-Body Mechanism:
  • Electron ($e^-$): Composed of three identical charged preons $(\psi_{-1})^3$. All three pairs interact via the attractive MCS force, leading to strong binding.
  • Up Quark ($u$): Composed of $\psi_1^2 \psi_0$. Only the $(\psi_1, \psi_1)$ pair feels the MCS force; $\psi_0$ is a neutral spectator. The up quark is primarily bound by the metacolor Cornell potential ($V \sim -\alpha/r + \sigma r$), not the MCS force.
  • Result: By tuning the metacolor string tension $\sigma_{mc}$, the model reproduces the observed ratio $m_e/m_u = 0.22$ at $\sigma^*_{mc}/\theta^2 = 2.11$. This value is roughly half the QCD string tension ratio, indicating no fine-tuning.
• Down vs. Up Mass ($m_d > m_u$):
  • The down quark ($d \sim \psi_{-1}\psi_0^2$) contains a $\psi_0^2$ pair.
  • Due to the Pauli principle and color antisymmetry, the $\psi_0^2$ pair must be in a spin-triplet state.
  • In the SUSY potential, the spin-triplet state induces repulsive gaugino exchange, reducing the binding energy relative to the up quark.
  • Prediction: $m_d > m_u$ is a parameter-free structural prediction. A Hylleraas-corrected estimate yields $m_d/m_u \simeq 2.3$, consistent with the observed 2.0.

B. Neutrino Mass

• Tree-Level Masslessness: The neutrino is composed of three identical neutral preons $(\psi_0)^3$.
  • Metacolor singlet requirement $\to$ Totally antisymmetric color wavefunction.
  • S-wave ground state $\to$ Symmetric spatial wavefunction.
  • Pauli Principle $\to$ Requires a totally symmetric spin-3/2 state.
  • Consequence: All three preon pairs are in spin-triplet states, making the SUSY gaugino exchange repulsive for all pairs.
  • Result: The effective potential is strictly positive ($V_{eff} > 0$), preventing bound state formation. The composite emerges with zero rest mass at tree level.
• Seesaw Mechanism: The spectator field $\chi$ (required for anomaly cancellation) acquires a Majorana mass $M_\chi \sim \Lambda_{cr}$. This enables a Type I seesaw:
  $$m_\nu \sim \frac{\Lambda_{EW}^2}{\Lambda_{cr}} \sim \frac{(100 \text{ GeV})^2}{10^{14} \text{ GeV}} \sim 0.1 \text{ eV}$$
  This matches experimental bounds without new parameters.

C. Baryon Asymmetry of the Universe (BAU)

• Mechanism: The BAU is generated at $\Lambda_{cr}$ via the Callan-Harvey anomaly inflow mechanism.
• Process: Integrating out massive charged preons induces a topological Chern-Simons term in the effective action. The coefficient $\Delta k_{CS}$ depends on the asymmetry between fermionic and bosonic condensation rates ($\epsilon$).
• Non-Equilibrium: Intrinsic SUSY breaking at the confinement transition creates a mass splitting ($\Delta m \sim \epsilon \bar{m}$), driving the system out of equilibrium.
• Quantitative Result: Matching the observed $\eta \approx 8.7 \times 10^{-10}$ fixes the SUSY-breaking parameter $\epsilon \simeq 0.022$. This value is natural for a one-loop origin ($\sim \alpha/4\pi$).
• Washout Protection: Electroweak sphalerons are out of equilibrium at $\Lambda_{cr}$ ($\Gamma_{sph} \ll H$), preventing the asymmetry from being washed out.

D. R-Parity and Dark Matter

• Dynamical R-Parity: R-parity is not imposed but derived. A fermionic composite (antisymmetric spin-statistics) cannot transform into a bosonic composite (symmetric) via any interaction in the model, as no interaction flips the character of all three constituents simultaneously.
• Dark Matter Candidate: The lightest superpartner (LSP) is the bosonic superpartner of the neutrino (a spin-0 neutral composite). It is absolutely stable and constitutes a viable cold dark matter candidate.
• LHC Null Results: The model explains the lack of observed squarks/sleptons at the LHC. Superpartners are not elementary particles but hadronic states (scalar mesons/baryons) formed from preon composites. The LHC has likely already observed them as scalar mesons (e.g., $f_0(500)$), misidentified as QCD states.

4. Significance and Conclusion

This paper presents a unified framework where the flavor hierarchy, neutrino mass, baryon asymmetry, and dark matter stability are all consequences of a single underlying preon dynamics at $\Lambda_{cr} \sim 10^{14}$ GeV.

Key Advantages over Standard Models:

1. No Free Parameters for Mass Ratios: The ordering $m_d > m_u$ and the value of $m_e/m_u$ emerge from the Pauli principle and the specific three-body symmetry structures, not fitted Yukawa couplings.
2. Naturalness: The required SUSY-breaking parameter $\epsilon \approx 0.022$ is consistent with loop-level calculations, avoiding fine-tuning.
3. R-Parity Derivation: It provides a dynamical origin for R-parity, solving the proton decay problem and stabilizing the LSP without ad-hoc assumptions.
4. Reinterpretation of Data: It offers a coherent explanation for the absence of MSSM superpartners at the LHC by reinterpreting known scalar mesons as the bosonic superpartners of quarks.

The work suggests that the "flavor problem" is a manifestation of the composite nature of fermions, analogous to how QCD generates the hadron spectrum from a single confining interaction.