The $Z$-transformation fitting technique purports to be the best choice for this type of extraction due to the true functional form of $G_E^p$ being mathematically guaranteed to exist within the parameter-space of the fit function. In this article, we examine the mathematical bias and variances introduced by choosing this technique as compared to the more traditional $Q^2$ fits to directly test if it is truly a better technique. A selection of $G_E^p$ parameterizations with known fit radius were statistically sampled and fit with in both $Q^2$ and in $Z$. The mean and variance of the extracted radii were compared to the input radius. The fits were performed both with the parameters unbound and bound to study the effect of choosing physically-motivated bounds. Additionally, the data are fit with Rational(N,M) functions in $Q^2$ as used by the PRad collaboration. As expected, the results in both $Q^2$ and $Z$ were poor for unbounded polynomials, with the fits in $Q^2$ showing a rather large bias with small variance while the fits in $Z$ had a small bias with a large variance. The application of bounds yield small improvements. In most cases, fits in $Q^2$ with Rational(N,M) type functions yielded improved results. We find that the $Z$-transformation technique is a useful tool, but not a universal improvement over fitting in $Q^2$. The "best" technique was dependent on the parameterization being fit and the $Q^2$ range of the data. The fits with a Rational(N,M) function in $Q^2$ typically provided improved results over other techniques when the data has sufficient $Q^2$ range, but provided unphysical results when the range was too small. We conclude that any planned fit should make use of psuedo data to determine the best basis and functional form for a given data set. In the case of new experiments, this selection should ideally be done prior to the collection of the data.

A nuclear physics example of statistical bootstrap is used on the MARATHON nucleon structure function ratio data in the quark momentum fraction regions xB→0 and xB→1. The extrapolated F2 ratio as quark momentum fraction xB→1 is F2nF2p→0.4±0.05 and this value is compared to theoretical predictions. The extrapolated ratio when xB→0 favors the simple model of isospin symmetry with the complete dominance of sea quarks at low momentum fraction. At high-xB, the proton quark distribution function ratio d/u is derived from the F2 ratio and found to be d/u→1/6. Our extrapolated values for both the F2nF2p ratio and the d/u parton distribution function ratio are within uncertainties of perturbative QCD values from quark counting, helicity conservation arguments, and a Dyson-Schwinger equation with a contact interaction model. In addition, it is possible to match the statistical bootstrap value to theoretical predictions by allowing two compatible models to act simultaneously in the nucleon wave function. One such example is nucleon wave functions composed of a linear combination of a quark-diquark state and a three-valence quark correlated state with coefficients that combine to give the extrapolated F2 ratio at xB=1.

Systematic differences in the the proton’s charge radius, as determined by ordinary atoms and muonic atoms, have caused a resurgence of interest in elastic lepton scattering measurements. The proton’s charge radius, defined as the slope of the charge form factor at Q2 = 0, does not depend on the probe. Any difference in the apparent size of the proton, when determined from ordinary versus muonic hydrogen, could point to new physics or need for the higher order corrections. While recent measurements seem to now be in agreement, there is to date no high precision elastic scattering data with both electrons and positrons. A high precision proton radius measurement could be performed in Hall B at Jefferson Lab with a positron beam and the calorimeter based setup of the PRad experiment. This measurement could also be extended to deuterons where a similar discrepancy has been observed between the muonic and electronic determination of deuteron charge radius. A new, high precision measurement with positrons, when viewed alongside electron scattering measurements and the forthcoming MUSE muon scattering measurement, could help provide new insights into the origins of the proton radius puzzle, and also provide new experimental constraints on radiative correction calculations.