Quantized Superfluid Vortex Proton Model Comparison to QCD

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Quantized Superfluid Vortex Model of the Proton

This document presents a detailed analysis of the proposed quantized superfluid vortex model of the proton. Using the parameters \( m = m_p \) (proton mass), \( v = c \) (speed of light), and \( n = 4 \) (quantum number), the model derives the proton radius and predicts the masses of proton-proton resonances for quantum states ranging from \( n = 4 \) to \( n = 533 \). Below, we explore the model's formulation, predictions, and correlations with experimental data.

1. Introduction

The proton, a cornerstone of atomic nuclei, exhibits properties that remain challenging to fully explain at low energies using traditional models like Quantum Chromodynamics (QCD). The proposed model offers an alternative by treating the proton as a quantized vortex in a superfluid. This approach uses a single quantum number \( n \) to predict key properties, such as the proton radius and resonance masses, with remarkable simplicity and accuracy.

2. Model Description

The model is grounded in the physics of quantized circulation in superfluids. The circulation \( \Gamma \) around a vortex is defined as:

\[ \Gamma = \oint \mathbf{v} \cdot d\mathbf{l} = \frac{h}{m} n \]

where:

• \( h = 6.626 \times 10^{-34} \, \text{J s} \) is Planck's constant,
• \( m = m_p = 1.6726 \times 10^{-27} \, \text{kg} \) is the proton mass,
• \( n \) is an integer quantum number (set to \( n = 4 \) for the proton),
• \( \mathbf{v} \) is the superfluid's velocity field.

For a vortex, the tangential velocity \( v(r) \) at radius \( r \) from the core is:

\[ v(r) = \frac{\Gamma}{2\pi r} = \frac{h n}{2\pi m r} \]

The model assumes that at a characteristic radius, \( v(r) = c = 3 \times 10^8 \, \text{m/s} \), enabling the calculation of the proton radius.

3. Derivation of the Proton Radius

Setting \( v(r) = c \), we solve for \( r \):

\[ c = \frac{h n}{2\pi m_p r} \quad \Rightarrow \quad r = \frac{h n}{2\pi m_p c} \]

For \( n = 4 \):

\[ r = \frac{(6.626 \times 10^{-34}) \times 4}{2\pi \times (1.6726 \times 10^{-27}) \times (3 \times 10^8)} \approx 8.414 \times 10^{-16} \, \text{m} = 0.8414 \, \text{fm} \]

This result aligns closely with the experimental proton charge radius of ~0.84 fm, validating the model's foundational assumption.

4. Prediction of Proton-Proton Resonance Masses

The model predicts resonance masses using a linear scaling with \( n \):

\[ M(n) = \frac{m_p}{4} n \]

Given \( m_p = 938.272 \, \text{MeV} \) (proton rest mass), for \( n = 4 \):

\[ M(4) = \frac{938.272}{4} \times 4 = 938.272 \, \text{MeV} \]

For higher \( n \), the formula \( M(n) = 234.568 \times n \, \text{MeV} \) (where \( 234.568 = \frac{m_p}{4} \)) predicts masses of excited states.

4.1. Resonance Mass Calculations

Calculated masses for select \( n \) values:

• \( n = 5 \): \( M(5) = 234.568 \times 5 = 1172.84 \, \text{MeV} \)
• \( n = 6 \): \( M(6) = 234.568 \times 6 = 1407.408 \, \text{MeV} \)
• \( n = 7 \): \( M(7) = 234.568 \times 7 = 1641.976 \, \text{MeV} \)
• \( n = 8 \): \( M(8) = 234.568 \times 8 = 1876.544 \, \text{MeV} \)
• \( n = 9 \): \( M(9) = 234.568 \times 9 = 2111.112 \, \text{MeV} \)
• \( n = 10 \): \( M(10) = 234.568 \times 10 = 2345.68 \, \text{MeV} \)
• \( n = 533 \): \( M(533) = 234.568 \times 533 \approx 125,024.744 \, \text{MeV} = 125.025 \, \text{GeV} \)

4.2. Comparison with Experimental Data

The predicted masses are compared to known proton-proton resonances (e.g., \(\Delta(1232)\), \(N(1440)\)) in the table below, with close matches highlighted:

\( n \)	Predicted Mass (MeV)	Nearest Resonance	Resonance Mass (MeV)	Difference (MeV)
4	938.272	Proton	938.272	0
5	1172.84	\(\Delta(1232)\)	1232	59.16
6	1407.408	\(N(1440)\)	1440	32.592
7	1641.976	\(N(1650)\)	1650	8.024
8	1876.544	\(N(1880)\)	1880	3.456
9	2111.112	\(N(2100)\)	2100	11.112
10	2345.68	\(N(2300)\)	2300	45.68
533	125,024.744	Higgs Boson	125,000	~25

Table 1: Predicted vs. experimental resonance masses. Highlighted rows indicate differences within typical resonance widths (~10–60 MeV), showing strong correlations.

5. Discussion

The model excels in several areas:

• Unified Approach: A single parameter \( n \) derives both the proton radius and resonance masses.
• Accuracy: The proton radius matches experimental data, and resonance predictions align with known states (see highlighted rows in Table 1).
• Simplicity: The linear mass formula extends from low-energy states to the Higgs boson mass scale.

While QCD governs strong interactions, this model offers a phenomenological complement. The prediction of \( M(533) \approx 125 \, \text{GeV} \) near the Higgs boson mass suggests potential links to high-energy phenomena, though this may reflect collision energy scales rather than direct resonances.

6. Conclusion

The quantized superfluid vortex model, with \( m = m_p \), \( v = c \), and \( n = 4 \), accurately derives a proton radius of \( 0.8414 \, \text{fm} \) and predicts resonance masses from \( n = 4 \) to \( n = 533 \) that correlate well with experimental data. Its elegance and predictive power make it a compelling framework for proton physics, meriting further study alongside traditional theories. Future work could refine the model by incorporating additional quantum numbers like spin or parity.