Proton Radius Puzzle 1

Proton Radius Puzzle Solution

The following equations relate the proton radius to fundamental constants, potentially addressing the proton radius puzzle:

• \(\mu = \frac{\alpha^2}{\pi r_p R_\infty}\)
• \(r_p = \frac{2h}{\pi c m_p}\)
• \(R_H = \frac{R_\infty}{1 + \frac{m_e}{m_p}}\)
• \(R_\infty = \frac{m_e e^4}{8 \epsilon_0^2 h^3 c}\)

Where:

• \(r_p\): proton radius
• \(\alpha\): fine-structure constant
• \(h\): Planck's constant
• \(c\): speed of light
• \(m_p\): proton mass
• \(m_e\): electron mass
• \(R_\infty\): Rydberg constant (infinite mass)
• \(R_H\): Rydberg constant for hydrogen

Calculating \(r_p\) using the second equation:

This matches the muonic hydrogen measurement (~0.842 fm), suggesting \(r_p = \frac{2h}{\pi c m_p}\) as the true radius. The first equation shows \(\mu = \frac{m_p}{m_e}\), linking these constants consistently.