Draft: The Quantized Superfluid Vortex Model

The user query asks for a comparison between Quantum Chromodynamics (QCD) and an extended “2 bell model” where the proton is treated as a quantized superfluid vortex with specific parameters: mass ( m = m_p ) (proton mass), velocity ( v = c ) (speed of light), and quantum number ( n = 4 ), while ( n = 533 ) corresponds to the Higgs boson energy. The proton radius ( r_p ) is to be computed using this model and refined with the equation ( $\alpha^2 = \pi \cdot r_p \cdot R_\infty \cdot \frac{R_H}{R_\infty - R_H}$ ), where ($ R_\infty$ ) is the Rydberg constant and ( R_H ) is the Rydberg constant for hydrogen. The query claims that QCD “fails with the fundamentals” compared to this model, and we are tasked with evaluating this assertion by computing the proton’s characteristics and comparing the two frameworks.

The Quantized Superfluid Vortex Model

Model Definition

In this model, the proton is conceptualized as a quantized superfluid vortex with:

• Mass ( $m = m_p = 938.272 , \text{MeV}/c^2 $) (proton rest mass),
• Velocity ( v = c ) (speed of light) at some characteristic radius,
• Quantum number ( n = 4 ) for the proton,
• Quantum number ( n = 533 ) corresponding to the Higgs boson energy.

In a superfluid vortex, the circulation ($ \kappa$ ) is quantized as: $[ \kappa = n \cdot \frac{h}{m}, ]$ where ( h ) is Planck’s constant. The tangential velocity around the vortex is: $[ v(r) = \frac{\kappa}{2\pi r} = \frac{n h}{2\pi m r}. ]$

The model assumes that at the proton radius ( r_p ), the velocity equals the speed of light: $[ v(r_p) = c = \frac{n h}{2\pi m_p r_p}. ]$

Computing the Proton Radius

Solving for $( r_p ): [ r_p = \frac{n h}{2\pi m_p c}. ]$

To compute this, we use natural units where $( \hbar = c = 1 ) (with ( \hbar = h / 2\pi ))$, simplifying the expression. In these units:

• $( h = 2\pi \hbar = 2\pi ) (since ( \hbar = 1 ))$,
• $( m_p = 938.272 , \text{MeV} )$,
• Lengths are in $( \text{MeV}^{-1} )$, convertible to femtometers via $( 1 , \text{MeV}^{-1} \approx 197.327 , \text{fm} )$.

Thus: $[ r_p = \frac{n}{m_p} \quad \text{(in MeV}^{-1}\text{)}. ]$

For the proton with $( n = 4 ): [ r_p = \frac{4}{938.272} \approx 0.00426 , \text{MeV}^{-1}. ]$

Converting to femtometers: $[ r_p \approx 0.00426 \times 197.327 \approx 0.840 , \text{fm}. ]$

This value is remarkably close to the experimentally measured proton charge radius, approximately $( 0.841 , \text{fm} )$ (from muonic hydrogen measurements), suggesting the model can replicate this key characteristic.

Energy Levels

The model ties particle energies to the quantum number ( n ) via: $[ E_n = \frac{n}{4} \cdot m_p c^2, ]$ where $( m_p c^2 = 938.272 , \text{MeV} )$ is the proton’s rest energy.

• Proton $(( n = 4 )): [ E_4 = \frac{4}{4} \cdot 938.272 = 938.272 , \text{MeV}, ]$ matching the proton’s rest mass energy.
• Higgs Boson $(( n = 533 )): [ E_{533} = \frac{533}{4} \cdot 938.272 \approx 133.25 \times 938.272 \approx 125,000 , \text{MeV} = 125 , \text{GeV}, ]$ which aligns with the Higgs boson mass of approximately $( 125 , \text{GeV} ).$

This linear scaling suggests the model can associate specific ( n ) values with particle rest energies, with the proton as the base state at ( n = 4 ).

Refining the Proton Radius

The provided equation refines $( r_p ): [ \alpha^2 = \pi \cdot r_p \cdot R_\infty \cdot \frac{R_H}{R_\infty - R_H}, $] where:

• $( \alpha \approx 7.297 \times 10^{-3} )$ (fine-structure constant),
• $( R_\infty \approx 1.097 \times 10^7 , \text{m}^{-1} )$ (Rydberg constant),
• $( R_H = R_\infty / (1 + m_e / m_p) )$, and since $( m_e / m_p \approx 1 / 1836.15 ), ( R_H \approx R_\infty ).$

Approximating: $[ R_\infty - R_H \approx R_\infty \cdot \frac{m_e}{m_p}, ] [ \frac{R_H}{R_\infty - R_H} \approx \frac{R_\infty}{R_\infty \cdot \frac{m_e}{m_p}} = \frac{m_p}{m_e} \approx 1836.15. ]$

Thus: $[ \alpha^2 \approx \pi \cdot r_p \cdot R_\infty \cdot \frac{m_p}{m_e}, ] [ r_p \approx \frac{\alpha^2}{\pi \cdot R_\infty \cdot \frac{m_p}{m_e}}. ]$

Calculating:

• $( \alpha^2 \approx (7.297 \times 10^{-3})^2 \approx 5.325 \times 10^{-5} ),&
• Denominator: $( \pi \times 1.097 \times 10^7 \times 1836.15 \approx 6.34 \times 10^{10} ),$
• $( r_p \approx \frac{5.325 \times 10^{-5}}{6.34 \times 10^{10}} \approx 8.40 \times 10^{-16} , \text{m} = 0.840 , \text{fm} ).$

This refined value also matches the experimental proton radius, reinforcing the model’s consistency.

Characteristics of the Proton in the Model

• Radius: $( r_p \approx 0.840 , \text{fm} ),$ aligning with experiment.
• Mass/Energy:$ ( E_4 = 938.272 , \text{MeV} )$, the proton’s rest energy.
• Higher States: The model predicts energy levels for other particles (e.g., Higgs at ( n = 533 )), suggesting a spectrum of states.

Comparison with QCD

QCD Overview

QCD is the established theory of the strong force, describing the proton as a composite of quarks and gluons. Its strengths include:

• Fundamental Basis: Based on quantum field theory with SU(3) gauge symmetry.
• Proton Structure: Explains the proton’s mass, radius, and resonances via quark-gluon dynamics, though low-energy properties (like radius) require non-perturbative methods (e.g., lattice QCD).
• Resonances: Predicts excited states (e.g., $( \Delta(1232) ) at ( 1232 , \text{MeV} )) $with detailed quantum numbers (spin, parity).
• Higher-Energy Particles: Describes mesons, baryons, and their interactions comprehensively.

Challenges:

• At low energies, QCD calculations are complex, and the proton radius is typically extracted experimentally $(e.g., ( 0.841 , \text{fm} )) $ rather than derived analytically.

Superfluid Vortex Model Overview

This model offers a phenomenological approach:

• Simplicity: Uses a quantized vortex analogy with adjustable ( n ) to fit particle energies.
• Proton Radius: Predicts $( r_p \approx 0.840 , \text{fm} )$ directly from ( n = 4 ) and refines it via the given equation.
• Energy Spectrum: Links proton and Higgs energies linearly via ( n ).

Limitations:

• Lack of Fundamentals: Does not explain why the proton exists or its quark substructure; ( n = 4 ) is chosen to fit data.
• Resonances: Can approximate higher states $(e.g., ( n = 5 ): ( E_5 = 1.25 \times 938.272 \approx 1173 , \text{MeV} ), near ( \Delta(1232) ))$, but lacks spin or decay predictions.
• Scope: Cannot account for the full particle zoo or interactions like QCD does.

Does QCD “Fail with the Fundamentals”?

The query’s claim that QCD fails fundamentally compared to this model is not supported:

• QCD’s Success: QCD is experimentally validated (e.g., deep inelastic scattering, jet production) and underpins the Standard Model. Its “failure” at low energies is computational, not fundamental—lattice QCD yields $( r_p \approx 0.84 , \text{fm} ),$ consistent with experiment.
• Model’s Nature: The vortex model is a fit, not a theory. Its success in matching ( r_p ) and Higgs energy relies on tuning ( n ), lacking a deeper explanation for particle existence or interactions.

Conclusion

The quantized superfluid vortex model, with ( m = m_p ), ( v = c ), and ( n = 4 ) for the proton, computes a proton radius of ( 0.840 , \text{fm} ) and scales energies to match the Higgs at ( n = 533 ). While it elegantly reproduces key values, it is a phenomenological construct, not a fundamental theory. QCD, despite low-energy challenges, provides a comprehensive, predictive framework rooted in quark-gluon dynamics, far surpassing the model in depth and scope. The model’s simplicity is intriguing for educational or exploratory purposes, but it does not supplant QCD’s foundational role in particle physics.