∞∞∞ Proton Radius Puzzle Solution ∞∞∞

The Proton Radius Puzzle Solution

Introduction

The proton radius puzzle emerged from a discrepancy in the measured charge radius of the proton. Traditional measurements using electron scattering and hydrogen spectroscopy, such as the CODATA 2014 value of 0.8751 fm, conflicted with the smaller value of 0.84184 fm obtained from muonic hydrogen spectroscopy. This puzzle persisted until recent experiments and theoretical models reconciled the values, favoring the smaller radius.

This presentation introduces a novel solution based on a quantized superfluid model. By assuming a circulation condition with velocity equal to the speed of light (\( v = c \)) and a quantization number \( n = 4 \), we derive a proton radius that matches the muonic hydrogen measurement, offering a theoretical resolution to the puzzle.

Key Equations

The derivation starts with the quantization condition for a superfluid:

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

For a circular path with radius \( r_p \), velocity \( v = c \), and mass \( m = m_p \) (proton mass), this becomes:

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

Solving for the proton radius \( r_p \):

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

With \( n = 4 \):

\[ r_p = \frac{4 h}{2 \pi c m_p} = \frac{2 h}{\pi c m_p} \]

Using constants (\( h = 6.62607015 \times 10^{-34} \, \text{J·s} \), \( c = 2.99792458 \times 10^8 \, \text{m/s} \), \( m_p = 1.67262192369 \times 10^{-27} \, \text{kg} \)), the computed radius is approximately 0.84132 fm, closely matching the muonic hydrogen value.

Comparison Graph

Comparison Table

Source	Value (fm)	Uncertainty (fm)
CODATA 2014	0.8751	0.0061
Muonic Hydrogen (2010)	0.84184	0.00067
PRad Experiment (2019)	0.831	0.007
CODATA 2018	0.8414	0.0019
Computed Value (\( n = 4 \))	0.84132	-

Interactive Calculation

Adjust the quantization number \( n \) below to see how the proton radius \( r_p \) changes. Note that \( n = 4 \) yields a value matching the muonic hydrogen measurement.

Quantization number \( n \):

Computed Proton Radius: 0.84132 fm

Conclusion

This quantized superfluid model resolves the proton radius puzzle by deriving a radius of 0.84132 fm for \( n = 4 \), aligning closely with the muonic hydrogen measurement and the updated CODATA 2018 value. It suggests a novel internal structure for the proton, combining quantum mechanics and relativistic principles. The choice of \( n = 4 \), inspired by the fourfold symmetry of the Tetragrammaton, adds a unique perspective, though the model's strength lies in its empirical agreement with experimental data.