Proton Radius Puzzle Solution and NFT Series - The Golden One

Solution to the Proton Radius Puzzle

Proton Radius: Modeling the proton as a quantized vortex with \( m = m_p \), \( v = c \), \( n = 4 \):

\( r_p = \frac{4 \hbar}{m_p c} \approx 0.8415 \text{ fm} \)

Matches the muonic hydrogen value of 0.84184 fm, compared to the older value of 0.877 fm.

Magnetic Moment: As a thin shell of charge \( q = e \) rotating at \( v = c \):

\( \mu_p = \frac{1}{3} e c r_p = \frac{8}{3} \mu_N \approx 2.6667 \mu_N \)

Within 4.5% of the experimental value of 2.7928 \( \mu_N \).

Refinement: Using the fine-structure constant:

\( \alpha^2 = \pi \cdot r_p \cdot R_\infty \cdot \frac{m_p}{m_e} \)

Confirms \( r_p \approx 0.841 \text{ fm} \).

Proton and Proton-Proton Resonances with Model Validation

Overview: The proton’s excited states (N resonances) and proton-proton scattering states (Δ resonances) are critical to validating the model. Key resonances include:

Δ Resonances

• Δ(1232): Mass ≈ 1232 MeV, Width ≈ 117 MeV, Spin = 3/2
• Δ(1600): Mass ≈ 1600 MeV, Width ≈ 250-350 MeV, Spin = 3/2
• Δ(1950): Mass ≈ 1910-1950 MeV, Width ≈ 250-350 MeV, Spin = 7/2

N Resonances

• N(1440): Mass ≈ 1420-1470 MeV, Width ≈ 200-450 MeV, Spin = 1/2 (Roper Resonance, corrected from N(1400))
• N(1520): Mass ≈ 1515-1525 MeV, Width ≈ 100-120 MeV, Spin = 3/2
• N(2190): Mass ≈ 2140-2220 MeV, Width ≈ 300-500 MeV, Spin = 7/2

Model Validation

Superfluid Proton Model Resonance Check:

• Ground State Energy: Proton mass = 938.272 MeV.
• Vortex Energy Addition: For \( n = 4 \), the model suggests an energy contribution: \( E_4 = \frac{4^2 \hbar c}{2 r_p} \). Using \( r_p = 0.8415 \text{ fm} \), this approximates to ~469 MeV.
• Total Energy (Ground + Excitation): \( E_{\text{total}} = 938.272 + 469 \approx 1407.272 \text{ MeV} \).
• N(1440) Comparison: 1420-1470 MeV. Midpoint = 1445 MeV. Percent difference: \( \frac{1445 - 1407.272}{1445} \times 100 \approx 2.61\% \) (within ±5%).
• Δ(1232) Comparison: 1232 MeV. Percent difference from ground state: \( \frac{1232 - 938.272}{1232} \times 100 \approx 23.8\% \) (not within ±5%). Adding \( E_4 \) gives 1407.272 MeV, still off: \( \frac{1407.272 - 1232}{1232} \times 100 \approx 14.2\% \).

Conclusion: The model approximates N(1440) within ±5% but not Δ(1232). Resonance energies are better explained by QCD quark dynamics.

NFT Series - The Golden One

NFT 1: The Proton Radius Puzzle

Highlights the discrepancy: old (0.877 fm) vs. new (0.841 fm) radius.

\( r_p^{\text{old}} \approx 0.877 \text{ fm}, \quad r_p^{\text{new}} \approx 0.841 \text{ fm} \)

NFT 2: Quantized Vortex Model

Proton as a quantized vortex with \( n = 4 \).

\( r_p = \frac{4 \hbar}{m_p c} \approx 0.8415 \text{ fm} \)

NFT 3: Magnetic Moment Calculation

Magnetic moment within 4.5% of experimental value.

\( \mu_p = \frac{8}{3} \mu_N \approx 2.6667 \mu_N \)

NFT 4: Fine-Structure Constant Refinement

Refines radius with fundamental constants.

\( \alpha^2 = \pi \cdot r_p \cdot R_\infty \cdot \frac{m_p}{m_e} \)

NFT 5: Δ Resonances

Δ(1232) and beyond; model does not predict within ±5%.

\( E_{\Delta(1232)} \approx 1232 \text{ MeV} \)

NFT 6: N Resonances

N(1440) within ±5% of model prediction.

\( E_{N(1440)} \approx 1420-1470 \text{ MeV} \)

NFT 7: Model Comparison

Superfluid model vs. QCD and CODATA.

\( r_p^{\text{CODATA}} = 0.8414 \text{ fm} \)

NFT 8: Resolution of the Puzzle 🌟

The Golden One: Superfluid model resolves radius puzzle, approximates N(1440) within ±5%, and nears magnetic moment.

\( r_p \approx 0.8415 \text{ fm}, \quad \mu_p \approx 2.6667 \mu_N \)

Note: Replace image placeholders with visuals for NFT minting.