Robust extraction of the proton charge radius from electron-proton scattering data

Background: Extracting the proton charge radius from electron scattering data requires determining the slope of the charge form factor at ${Q}^{2}$ of zero. As experimental data cannot reach that limit, numerous methods for making the extraction have been proposed.Purpose: In this study, we seek to find functional forms that will allow a robust extraction of the proton radius from a wide variety of functional forms. The primary motivation of this study is to have confidence in the extraction of upcoming low-${Q}^{2}$ experimental data.Method: We create a general framework for studying various form-factor functions along with various fitting functions. The input form factors are used to generate pseudodata with fluctuations mimicking the binning and random uncertainty of a set of real data. All combinations of input functions and fit functions can then be tested repeatedly against regenerated pseudodata. Since the input radius is known, this allows us to find fitting functions that are robust for proton radius extractions in an objective fashion.Results: For the range and uncertainty of the PRad data, we find that a two-parameter rational function, a two-parameter continued fraction, and the second-order polynomial expansion of $z$ can extract the input radius regardless of the input charge form factor function that is used.Conclusions: We have created a framework to determine which functional forms allow a robust extraction of the radius from pseudodata generated from a wide variety of trial functions. By taking into account both bias and variance, the optimal functions for extracting the proton radius can be determined.