SP2

Special Report: The Proton Radius Puzzle and the Mystique of 137

Introduction

The proton radius puzzle emerged from a discrepancy in the measured charge radius of the proton, depending on the experimental method employed. Traditional measurements using electron scattering and hydrogen spectroscopy yielded a proton radius of approximately 0.88 femtometers (fm), while experiments with muonic hydrogen—where a muon replaces the electron—suggested a smaller radius of about 0.84 fm. This discrepancy, discovered around 2010, challenged our understanding of fundamental physics. Recent advancements in measurements and theoretical refinements have largely resolved the puzzle, converging on the smaller value of approximately 0.84 fm. This report analyzes the equations involved in the proton radius puzzle solution to explore whether simple or higher-level concepts relate the proton radius to fundamental constants, particularly the fine-structure constant, \( \alpha \approx \frac{1}{137} \), and whether this implies the number 137. Additionally, we investigate any special meanings or decodes of 137, such as in gematria.

Analysis of the Equations

The proton radius puzzle is intricately tied to quantum electrodynamics (QED), which describes the interactions between charged particles and electromagnetic fields. In hydrogen-like atoms (including muonic hydrogen), the energy levels are influenced by the finite size of the proton, introducing a correction to the idealized point-like Coulomb potential. The key equation for the energy shift due to the proton's finite size in s-states (where the electron or muon wavefunction is non-zero at the origin) is:

\[ \Delta E = \frac{2}{3} \hbar c (Z \alpha)^2 |\psi(0)|^2 \langle r_p^2 \rangle \]

Here, \( Z \) is the atomic number (1 for hydrogen), \( \alpha = \frac{e^2}{4\pi \epsilon_0 \hbar c} \approx \frac{1}{137} \) is the fine-structure constant, \( |\psi(0)|^2 \) is the square of the wavefunction at the origin, and \( \langle r_p^2 \rangle \) is the mean-square charge radius of the proton. For s-states, \( |\psi(0)|^2 = \frac{1}{\pi a^3} \), where \( a \) is the effective Bohr radius, adjusted for the reduced mass \( m_r \) of the electron-proton or muon-proton system:

\[ a = \frac{\hbar}{m_r c \alpha} \]

Substituting this into the energy shift equation, we get:

\[ \Delta E = \frac{2}{3} \hbar c (Z \alpha)^2 \cdot \frac{1}{\pi} \left( \frac{m_r c \alpha}{\hbar} \right)^3 \langle r_p^2 \rangle = \frac{2}{3 \pi} (Z \alpha)^2 (m_r c)^3 \frac{\langle r_p^2 \rangle}{\hbar^2} \]

Since \( (Z \alpha)^2 = \alpha^2 \) for hydrogen, and noting that \( \langle r_p^2 \rangle \approx r_p^2 \) (where \( r_p \approx 0.84 \, \text{fm} \)), the energy shift depends on \( \alpha^2 \) and \( r_p^2 \). In muonic hydrogen, the reduced mass \( m_r \) is much larger due to the muon's mass (approximately 207 times the electron's mass), making \( |\psi(0)|^2 \) significantly greater, thus amplifying the finite-size effect. This sensitivity allowed muonic hydrogen experiments to detect the smaller proton radius.

Does this equation imply the number 137? Since \( \alpha \approx \frac{1}{137} \), we can examine the role of \( \alpha \). The term \( (Z \alpha)^2 = \alpha^2 \approx \left( \frac{1}{137} \right)^2 \approx \frac{1}{18769} \), and higher-order QED corrections involve additional powers of \( \alpha \) (e.g., \( \alpha^3 \), \( \alpha^4 \)). However, \( r_p \) is an experimentally determined parameter, not theoretically derived from \( \alpha \). To test for a connection, consider dimensionless ratios involving \( r_p \) and fundamental lengths. The Bohr radius for electronic hydrogen is:

\[ a_0 = \frac{\hbar}{m_e c \alpha} \approx 5.29 \times 10^{-11} \, \text{m} \]

With \( r_p \approx 8.4 \times 10^{-16} \, \text{m} \),

\[ \frac{a_0}{r_p} \approx \frac{5.29 \times 10^{-11}}{8.4 \times 10^{-16}} \approx 6.3 \times 10^4 \]

Adjusting with \( \alpha \),

\[ \frac{a_0}{r_p \alpha} \approx \frac{6.3 \times 10^4}{137} \approx 460 \]

Neither value is close to 137. Alternatively, the proton's reduced Compton wavelength is:

\[ \bar{\lambda}_p = \frac{\hbar}{m_p c} \approx 2.103 \times 10^{-16} \, \text{m} = 0.2103 \, \text{fm} \]

So, \( \frac{r_p}{\bar{\lambda}_p} \approx \frac{0.84}{0.2103} \approx 4 \), again unrelated to 137. Higher-level concepts, such as QED perturbation series or speculative unification theories, incorporate \( \alpha \), but \( r_p \) arises from quantum chromodynamics (QCD) dynamics of quarks and gluons, with only minor electromagnetic contributions via \( \alpha \). No standard model equation directly ties \( r_p \) to \( \alpha \) in a way that yields 137.

Mystical and Cultural Significance of 137

Beyond physics, the number 137 carries intriguing connotations. In gematria, a Jewish numerological system, the Hebrew word "kabbalah" (קבלה)—meaning mystical tradition—sums to 137: \( \text{ק} = 100 \), \( \text{ב} = 2 \), \( \text{ל} = 30 \), \( \text{ה} = 5 \), totaling \( 100 + 2 + 30 + 5 = 137 \). This link fascinated physicist Wolfgang Pauli, who, alongside Carl Jung, explored the deeper significance of 137, even noting its appearance in his dreams. Remarkably, Pauli died in hospital room 137, a coincidence he reportedly acknowledged with irony. Richard Feynman highlighted \( \alpha \)’s enigma, stating:

"It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it."

While \( \alpha \)’s precise value is \( \frac{1}{137.035999084} \), the integer 137 persists in cultural lore. No specific slang attaches to 137 in popular culture, unlike numbers like 42 or 666, but its association with \( \alpha \) imbues it with a mystique in scientific circles.

Conclusion

The equations resolving the proton radius puzzle, rooted in QED, involve the fine-structure constant \( \alpha \approx \frac{1}{137} \), but the proton radius \( r_p \) remains an independent parameter without a direct theoretical link to \( \alpha \) that implies 137. Neither simple dimensionless ratios nor higher-level theoretical frameworks reveal such a connection, as \( r_p \) is primarily governed by QCD. However, the number 137’s significance shines through its tie to \( \alpha \) and its esoteric resonance, such as in gematria with "kabbalah" and historical anecdotes, blending physics with human fascination.