Tyler Williams | Searcharxiv

arXiv2025

Hofstadter-Herman Visualization as a Diagnostic Tool for Systematic Effects in Electromagnetic Form Factor Extractions

The internal charge and magnetization distributions of the proton are characterized by electromagnetic form factors GE and GM. They are experimentally extracted via Rosenbluth separation, which measures the elastic scattering of electrons and protons at multiple beam energies and angles at fixed momentum transfer Q2. Conventionally, form factor values are obtained by plotting reduced cross sections against the virtual photon polarization parameter epsilon and then extracting the slope and intercept of the best fit lines. An alternative visualization method, proposed by Hofstadter and Herman in 1960, plots GM2 vs. GE2 curves instead. The best fit values of GE2 and GM2 are immediately visible from the intersection region of the curves and their uncertainty bands. In this work, we apply both conventional and Hofstadter-Herman visualizations to classic 1994 SLAC elastic scattering data. We demonstrate that the Hofstadter-Herman method reveals previously obscured regions of form factor parameter space and highlights subtle experimental discrepancies among data sets. Our results motivate adopting this visualization method as a routine diagnostic cross-check at the Electron-Ion Collider and elsewhere to flag normalization shifts and related adjustments before they enter global fits.

nucl-ex↗

arXiv2021

Wavelet Representation of Singular Integral Operators

This article develops a novel approach to the representation of singular integral operators of Calderón-Zygmund type in terms of continuous model operators, in both the classical and the bi-parametric setting. The representation is realized as a finite sum of averages of wavelet projections of either cancellative or noncancellative type, which are themselves Calderón-Zygmund operators. Both properties are out of reach for the established dyadic-probabilistic technique. Unlike their dyadic counterparts, our representation reflects the additional kernel smoothness of the operator being analyzed. Our representation formulas lead naturally to a new family of $T(1)$ theorems on weighted Sobolev spaces whose smoothness index is naturally related to kernel smoothness. In the one parameter case, we obtain the Sobolev space analogue of the $A_2$ theorem; that is, sharp dependence of the Sobolev norm of $T$ on the weight characteristic is obtained in the full range of exponents. In the bi-parametric setting, where local average sparse domination is not generally available, we obtain quantitative $A_p$ estimates which are best known, and sharp in the range $\max\{p,p'\}\geq 3$ for the fully cancellative case.

math.CA↗