[Paper Review] Proton form-factor dependence of the finite-size correction to the Lamb shift in muonic hydrogen

Jonathan Carroll, A. W. Thomas+2 more|arXiv (Cornell University)|Aug 12, 2011

Atomic and Molecular Physics3 citations

TL;DR

This paper investigates the dependence of the finite-size correction to the Lamb shift in muonic hydrogen on the shape of the proton's electric form factor. Using nonperturbative, relativistic Dirac wave functions and various idealized and experimentally constrained charge distributions, it finds no statistically significant variation in the extracted proton root-mean-square charge radius across different form factors, concluding that the proton's charge distribution shape does not explain the 4% discrepancy in the proton radius observed in muonic hydrogen experiments.

ABSTRACT

The measurement of the 2P^{F=2}_{3/2} to 2S^{F=1}_{1/2} transition in muonic hydrogen by Pohl et al. and subsequent analysis has led to the conclusion that the rms radius of the proton differs from the accepted (CODATA) value by approximately 4%, corresponding to a 4.9 sigma discrepancy. We investigate the finite-size effects - in particular the dependence on the shape of the proton electric form-factor - relevant to this transition using bound-state QED with nonperturbative, relativistic Dirac wave-functions for a wide range of idealised charge-distributions and a parameterization of experimental data in order to comment on the extent to which the perturbation-theory analysis which leads to the above conclusion can be confirmed. We find no statistically significant dependence of this correction on the shape of the proton form-factor.

Motivation & Objective

• To assess whether the shape of the proton's electric form factor affects the finite-size correction to the 2P3/2–2S1/2 Lamb shift in muonic hydrogen.
• To test the robustness of the proton radius extraction from muonic hydrogen spectroscopy under different assumptions about the proton charge distribution.
• To resolve conflicting claims—particularly from de Rujula (2011)—that the form factor shape could explain the proton radius puzzle.
• To provide a nonperturbative, relativistic bound-state QED analysis of the Lamb shift correction using realistic and idealized charge distributions.
• To confirm the validity of the perturbative analysis used in Pohl et al. (2010) by testing its sensitivity to the proton's internal charge structure.

Proposed method

• Solves the effective Dirac equation for a muon in the Coulomb potential of a finite-size proton using nonperturbative, relativistic wave functions.
• Models the proton charge distribution using three idealized forms: exponential (dipole), Gaussian, and Yukawa (monopole), each parameterized by the rms charge radius.
• Applies a three-dimensional Fourier transform to relate the form factor to the spatial charge distribution, ensuring normalization and consistency with the rms radius.
• Uses experimental data on the electric Sachs form factor to construct a realistic charge distribution for comparison.
• Calculates the finite-size correction to the Lamb shift by integrating the modified Coulomb potential over the proton's charge distribution.
• Compares the resulting energy shifts with experimental data from Pohl et al. (2010), using a fit to extract the proton radius under different distribution assumptions.

Experimental results

Research questions

• RQ1 Does the shape of the proton's electric form factor significantly alter the finite-size correction to the Lamb shift in muonic hydrogen?
• RQ2 How sensitive is the extracted proton root-mean-square charge radius to the choice of charge-distribution model (e.g., exponential, Gaussian, Yukawa)?
• RQ3 Can the observed 4% discrepancy in the proton radius be explained by uncertainties in the form factor or charge distribution shape?
• RQ4 To what extent does the perturbative analysis in Pohl et al. (2010) depend on the assumed proton charge distribution?
• RQ5 Does the realistic charge distribution derived from experimental form factor data yield a different proton radius than idealized models?

Key findings

• The extracted proton root-mean-square charge radius is insensitive to the choice of proton charge-distribution shape, with values ranging from 0.84155 fm (Gaussian) to 0.84194 fm (Yukawa), all within 0.0002 fm of each other.
• The discrepancy between the extracted radius and the 2006 CODATA value is 4.02% for Gaussian, 4.00% for exponential, and 3.98% for Yukawa distributions, indicating no significant dependence on form factor shape.
• The analysis confirms that the finite-size correction to the Lamb shift in muonic hydrogen does not vary significantly across a wide range of higher-order moments of the charge distribution.
• The realistic charge distribution derived from experimental form factor data yields a finite-size correction consistent with the idealized models, supporting the robustness of the result.
• The study rules out the proton form factor shape as a source of the 4% proton radius discrepancy, as claimed in de Rujula (2011), by demonstrating negligible sensitivity.
• The error in the extracted radius is dominated by the experimental uncertainty in the measured transition energy, not by the choice of charge distribution.