[Paper Review] Sensitivity of the elastic electron-proton cross section to the proton radius

S. Pacetti, E. Tomasi‐Gustafsson|arXiv (Cornell University)|Dec 7, 2018

Atomic and Molecular Physics23 references8 citations

TL;DR

This paper investigates the sensitivity of elastic electron-proton scattering cross sections to the proton radius, demonstrating that the required precision on the cross section to resolve discrepancies in proton radius measurements is two orders of magnitude higher than current experimental precision. It shows that systematic uncertainties—particularly from extrapolation to Q²=0, fitting function choice, and data range—can be orders of magnitude larger than commonly assumed, challenging the reliability of radius determinations from scattering data.

ABSTRACT

The precise determination of the proton radius from recent elastic scattering electron-proton data is discussed. The necessary precision on the elastic cross section to discriminate among the values coming from atomic spectroscopy is scrutinized in terms of the relevant quantity, i.e., the derivative of the form factor. It is shown that such precision is two orders of magnitude higher than the precision on the cross section, that is the measured observable. Different fits on the available data and of their discrete derivative, with analytical constraints are shown. The systematic error associated to the radius is evaluated taking into account the uncertainties from different sources, as the extrapolation to the static point, the choice of the class of fitting functions and the range of the data sample. This error is shown to be even orders of magnitude larger than commonly assumed.

Motivation & Objective

• To assess the precision required in elastic electron-proton cross sections to distinguish between conflicting proton radius values from atomic spectroscopy and scattering experiments.
• To evaluate the systematic uncertainties in proton radius extraction due to extrapolation to Q² = 0, choice of fitting functions, and data sample range.
• To quantify the sensitivity of the cross section to the proton radius via the derivative of the electric form factor.
• To challenge the commonly assumed small systematic errors in proton radius determinations from scattering data.

Proposed method

• The analysis uses the Rosenbluth separation formalism to express the differential cross section in terms of electric and magnetic form factors, with the electric form factor G_E(Q²) directly related to the proton radius via its derivative at Q² = 0.
• The paper derives the sensitivity of the cross section to the proton radius by analyzing the derivative of G_E(Q²) at Q² = 0, which is proportional to the mean-square charge radius.
• It applies analytical constraints on form factors using inequalities derived from the Cauchy-Schwarz inequality and power moments of the charge radius distribution, ensuring physical consistency in the fitting procedure.
• The authors perform multiple fits to the A1 Collaboration data, including discrete derivatives of the cross section, with constraints on the form factor behavior to assess systematic errors.
• Systematic uncertainties are evaluated by varying the fitting function class, the data range, and the extrapolation method to Q² = 0, with results compared to standard assumptions.
• The study uses the binary decomposition of integers to recursively apply moment inequalities, proving that ⟨r²ⁿ⟩ ≥ ⟨r²⟩ⁿ, which constrains the form factor behavior and supports the robustness of the sensitivity analysis.

Experimental results

Research questions

• RQ1 What level of precision in the elastic electron-proton cross section is required to resolve the proton radius puzzle?
• RQ2 How do systematic errors from extrapolation to Q² = 0, fitting function choice, and data range affect the uncertainty in the proton radius?
• RQ3 What is the sensitivity of the cross section to the proton radius, and how does it depend on the derivative of the electric form factor?
• RQ4 How do analytical constraints on form factors affect the reliability of radius extraction from scattering data?
• RQ5 To what extent are current estimates of systematic uncertainty in the proton radius underestimated?

Key findings

• The required precision on the elastic cross section to resolve the proton radius discrepancy is two orders of magnitude higher than the current experimental precision.
• Systematic uncertainties in the proton radius due to extrapolation, fitting function choice, and data range are shown to be orders of magnitude larger than commonly assumed.
• The sensitivity of the cross section to the proton radius is governed by the derivative of the electric form factor at Q² = 0, which is directly proportional to the mean-square charge radius.
• The use of analytical constraints on form factors, derived from moment inequalities, provides a robust framework for evaluating systematic errors in radius extraction.
• The analysis reveals that the discrepancy between muonic and electronic hydrogen measurements cannot be resolved with current experimental precision, even with idealized fitting procedures.
• The study demonstrates that the proton radius determination from elastic scattering is more sensitive to systematic effects than previously recognized, challenging the reliability of some existing radius values.