[Paper Review] Solution of the proton radius puzzle? Low momentum transfer electron scattering data are not enough

M. O. Distler, Thomas Walcher+1 more|arXiv (Cornell University)|Nov 2, 2015

Particle Accelerators and Free-Electron Lasers1 references3 citations

TL;DR

This paper challenges recent claims that the proton radius puzzle—discrepancy between electron and muonic hydrogen measurements—can be resolved by truncating electron scattering data to low momentum transfer. It demonstrates that such truncation violates the Fourier theorem, leading to unphysical oscillations in charge distributions and invalid radius extractions, thus rejecting the method as fundamentally flawed despite apparent statistical fits to low-Q² data.

ABSTRACT

In two recent papers it is argued that the 'proton radius puzzle' can be explained by truncating the electron scattering data to low momentum transfer and fit the rms radius in the low momentum expansion of the form factor. It is shown that this procedure is inconsistent and violates the Fourier theorem. The puzzle cannot be explained in this way.

Motivation & Objective

• To challenge the validity of recent claims that restricting electron scattering data to low Q² resolves the proton radius puzzle.
• To demonstrate that truncating the momentum transfer range violates the Fourier transform relationship between form factors and charge distributions.
• To show that models based on truncated data produce unphysical oscillations in charge density, inconsistent with experimental observations.
• To argue that statistical fits to incomplete data cannot yield robust radius determinations without incorporating full Q²-range constraints.
• To emphasize that physical consistency, not just statistical fit quality, must guide extraction of the proton radius from scattering data.

Proposed method

• Analyzes the Fourier transform relationship between the proton's charge distribution ρ(r) and its form factor G(Q²), showing that both must be defined over the full range of r and Q².
• Applies the low-momentum expansion G(Q²) = 1 - (1/6)⟨r²⟩Q² + (1/120)⟨r⁴⟩Q⁴ - … to truncated data sets, revealing inconsistencies when higher-order terms are neglected.
• Introduces the 'sawtooth' form factor model (G(Q²) truncated at finite Q²) to illustrate unphysical oscillations in ρ(r) that contradict known proton structure.
• Evaluates the statistical validity of truncated fits by distinguishing M² (minimal deviation) from χ² (statistical significance), arguing that M² cannot be interpreted as χ² when models are not certain.
• Compares the continued fraction expansion of G(Q²) with standard parametrizations, showing it was excluded in prior analyses due to poor fit quality and lacks theoretical justification.
• Uses tabulated moments (⟨r²⟩, ⟨r⁴⟩, ⟨r⁶⟩, Zemach moments) to compare physical models (exponential, Gaussian, uniform) with the unphysical sawtooth model, highlighting inconsistencies.

Experimental results

Research questions

• RQ1 Can the proton radius puzzle be resolved by restricting electron scattering data to low momentum transfer Q²?
• RQ2 Does truncating the Q² range in form factor fits preserve the physical consistency required by the Fourier theorem?
• RQ3 What are the implications of using a truncated data set for extracting the proton's rms radius and higher-order moments?
• RQ4 Why do statistical fits to truncated data produce misleadingly small radii despite unphysical charge distributions?
• RQ5 How does the continued fraction expansion of G(Q²) compare to standard parametrizations in terms of physical consistency and statistical validity?

Key findings

• Truncating electron scattering data to low Q² violates the Fourier theorem, leading to unphysical oscillations in the inferred charge distribution ρ(r).
• The 'sawtooth' form factor model, which truncates G(Q²) at finite Q², produces unreasonably small Zemach moments and is inconsistent with experimental data.
• Even though fits to truncated data yield small radii close to the muonic hydrogen value, the method is invalid due to neglect of high-Q² information.
• The statistical measure M² from truncated fits cannot be interpreted as χ², invalidating claims of statistical significance based on such fits.
• The continued fraction expansion of G(Q²), while yielding a small radius, was excluded in prior analyses due to poor fit quality and lacks theoretical justification.
• A robust extraction of the proton radius requires using the full Q² range of the form factor, as the high-Q² region contains essential information about the charge distribution.