Comparing Models to QCD and Accepted Values

Comparison of the Quantized Vortex Superfluid Proton Model and the 2-Bell Harmonic Analogy

This document provides a detailed comparison of two theoretical models—the Quantized Vortex Superfluid Proton Model and the 2-Bell Harmonic Analogy—focusing on their predictions for the proton radius and correlations with proton resonances (e.g., Δ and N) and higher-order resonances (e.g., Higgs boson). The summary includes equations, tables with accepted scientific values, errors, and a score column to quantify correlations.

1. Introduction

The goal of this comparison is to assess how well each model predicts key physical properties of the proton and its excited states. The Quantized Vortex Superfluid Proton Model views the proton as a quantized vortex in a superfluid medium, while the 2-Bell Harmonic Analogy uses the interference of two detuned frequencies to model particle energies and widths. Both models are evaluated against experimental data.

2. Model Descriptions

2.1 Quantized Vortex Superfluid Proton Model

This model treats the proton as a quantized vortex in a superfluid, characterized by a quantum number \( n \). The energy of a particle is given by:

\[ E_n = \frac{n}{4} \times 938.272 \, \text{MeV} \]

where \( 938.272 \, \text{MeV} \) is the proton's rest mass energy, and \( n = 4 \) corresponds to the proton itself (\( E_4 = 938.272 \, \text{MeV} \)). Higher values of \( n \) represent excited states or other particles, such as the Higgs boson at \( n = 533 \).

2.2 2-Bell Harmonic Analogy

The 2-Bell Harmonic Analogy models particle energies and widths using two slightly detuned "bells" with fundamental frequencies \( f_1 = 234.568 \, \text{Hz} \) and \( f_2 = 257.568 \, \text{Hz} \), where frequency in Hz is analogous to energy in MeV. The average energy and width for each harmonic \( n \) are:

\[ E_{\text{avg}, n} = n \times 246.068 \, \text{MeV} \] \[ \Gamma_n = n \times 23 \, \text{MeV} \]

These equations predict the central energy and width of particle resonances.

3. Proton Radius Solution

3.1 Quantized Vortex Model

The proton radius \( r_p \) is derived from the quantized vortex circulation, where the velocity at the proton radius equals the speed of light:

\[ r_p = \frac{n h}{2\pi m_p c} \]

For \( n = 4 \), \( h \) is Planck’s constant, \( m_p \) is the proton mass, and \( c \) is the speed of light, yielding \( r_p \approx 0.840 \, \text{fm} \) in natural units. A refined equation incorporating the fine-structure constant \( \alpha \), Rydberg constant \( R_\infty \), and hydrogen radius \( R_H \) is:

\[ \alpha^2 = \pi \cdot r_p \cdot R_\infty \cdot \frac{R_H}{R_\infty - R_H} \]

This also gives \( r_p \approx 0.840 \, \text{fm} \), closely aligning with the accepted value.

3.2 2-Bell Analogy

The 2-Bell Harmonic Analogy does not provide a direct method to calculate the proton radius.

3.3 Comparison

Model	Calculated \( r_p \) (fm)	Accepted Value (fm)	Error (%)
Quantized Vortex Model	0.840	0.841	0.12
2-Bell Analogy	N/A	0.841	N/A

The Quantized Vortex Model predicts a proton radius very close to the accepted value of \( 0.841 \, \text{fm} \) (from muonic hydrogen measurements), with a minimal error of 0.12%. The 2-Bell Analogy lacks a radius prediction.

4. Resonance Correlations

This section compares the models' predictions for the energies and widths of proton resonances (e.g., Δ(1232), N(1440)) and higher-order resonances (e.g., Higgs boson) against accepted values. A correlation is counted if the predicted energy is within 5% of the accepted mass.

Resonance	Accepted Mass (MeV)	Accepted Width (MeV)	Quantized Vortex \( E_n \) (MeV)	2-Bell \( E_{\text{avg}, n} \) (MeV)	2-Bell \( \Gamma_n \) (MeV)	Correlation (Quantized Vortex)	Correlation (2-Bell)
Proton (\( n = 4 \))	938.272	0	938.272	984.272	92	Yes	No
\( \Delta(1232) \) (\( n = 5 \))	1232	115	1172.84	1230.34	115	No	Yes
\( N(1440) \) (\( n = 6 \))	1440	300	1407.408	1476.408	138	Yes	Yes
\( N(1710) \) (\( n = 7 \))	1710	100–200	1641.976	1722.476	161	Yes	Yes
\( \Delta(1950) \) (\( n = 8 \))	1950	235	1876.544	1968.544	184	Yes	Yes
\( N(2220) \) (\( n = 9 \))	2220	400	2111.112	2214.612	207	Yes	Yes
Higgs Boson (\( n = 533 \))	125,000	4.07	125,000	N/A	N/A	Yes	No

Notes: The Quantized Vortex Model uses \( E_n \) to match masses, while the 2-Bell Analogy uses \( E_{\text{avg}, n} \) for mass and \( \Gamma_n \) for width. The Higgs boson correlation for the Quantized Vortex Model assumes \( n = 533 \) to match \( 125 \, \text{GeV} \), but the 2-Bell Analogy does not extend to such high \( n \).

5. Comparison and Scoring

The table below counts the number of correlations (within 5% of accepted mass) for each model based on the resonances= resonances analyzed.

Model	Number of Correlations
Quantized Vortex Model	6
2-Bell Analogy	5

The Quantized Vortex Model excels in predicting the proton radius and has more correlations, including the Higgs boson. The 2-Bell Analogy matches resonance widths well (e.g., \( \Delta(1232) \)) but lacks a radius prediction and fewer overall correlations.

6. References

• Proton radius: Pohl, R., et al. (2010). Nature, 466, 213–216.
• Resonance data: Particle Data Group