# How Analytic Choices Can Affect the Extraction of Electromagnetic Form Factors from Elastic Electron Scattering Cross Section Data

## TL;DR

This paper demonstrates how different analytic choices in regression, such as constraints and data binning, significantly influence the extraction of proton electromagnetic form factors from electron scattering data.

## Contribution

It highlights the impact of analytic choices on regression outcomes in form factor extraction and emphasizes the need for physical or mathematical grounding.

## Key findings

- Adding a theory-motivated constraint removes oscillations in the magnetic form factor.
- Data binning can dramatically alter the regression results.
- Analytic choices can lead to substantially different physical parameter estimates.

## Abstract

Scientists often try to incorporate prior knowledge into their regression algorithms, such as a particular analytic behavior or a known value at a kinematic endpoint. Unfortunately, there is often no unique way to make use of this prior knowledge, and thus, different analytic choices can lead to very different regression results from the same set of data. To illustrate this point in the context of the proton electromagnetic form factors, we use the Mainz elastic data with its 1422 cross section points and 31 normalization parameters. Starting with a complex unbound non-linear regression, we will show how the addition of a single theory-motivated constraint removes an oscillation from the magnetic form factor and shifts the extracted proton charge radius. We then repeat both regressions using the same algorithm, but with a rebinned version of the Mainz dataset. These examples illustrate how analytic choices, such as the function that is being used or even the binning of the data, can dramatically affect the results of a complex regression. These results also demonstrate why it is critical when using regression algorithms to have either a physical model in mind or a firm mathematical basis