Reconstructing Transition GPDs for Delta(1232) from Helicity Amplitude A_1/2(Q^2) via Dipole Fits and Impact Parameter Analysis

TL;DR

This work addresses reconstructing the γ^* N→Δ transition GPD H_T(x,t) from helicity amplitude data by employing a dipole fit to A_{1/2}(Q^2) and a separable GPD model H_T(x,t)=h(x)F(t). By enforcing the GPD sum rule and performing a two-dimensional Fourier transform, it yields impact-parameter space distributions q(x,b) and analyzes how longitudinal shaping via a Beta-like profile h(x) modulates transverse localization. Uncertainty from the amplitude fit is propagated through to spatial diagnostics, and the approach demonstrates a transparent, modular framework adaptable to other transition channels. The results provide a physically interpretable link between amplitude behavior and the Delta(1232) substructure, with implications for understanding spatial deformation and transition dynamics in baryons.

Abstract

This work presents a modular reconstruction of the transition generalized parton distribution (GPD) H_T(x,t) for the Delta(1232) resonance, based on digitized helicity amplitude data and dipole fits to A_1/2(Q^2). From the fitted amplitude, we extract a Sachs-like form factor F(t) and define a separable GPD model H_T(x, t) = h(x)F(t), with h(x) modeled as a normalized Beta-like profile. This factorized ansatz satisfies the GPD sum rule and enables a direct two-dimensional Fourier transform to construct transverse spatial distributions q(x,b). We analyze how longitudinal shaping modulates transverse localization, and quantify spatial features using statistical diagnostics including mean radius, skewness, and kurtosis. The framework is reproducible, data-driven, and applicable to other transition channels, providing a physically interpretable map from amplitude behavior to spatial structure.

Figures (8)

• Figure 1: Dipole fit to $\Delta(1232)$ helicity amplitude $A_{1/2}(Q^2)$ with $\pm1\sigma$ uncertainty band derived from fit covariance. Digitized CLAS data taken from Refs. Burkert:2004Aznauryan:2013.
• Figure 2: Central longitudinal profile $h(x)$ with $(a = 0.5,\ b = 0.3)$ and shaded uncertainty envelope from the shape variation in profile parameters. Normalization is preserved.
• Figure 3: Normalized longitudinal profiles $h(x)$ for representative $(a,b)$ shapes. Larger $a$ sharpens high-$x$ falloff; smaller $b$ broadens low-$x$ contribution. All profiles normalized over $x \in [0,1]$.
• Figure 4: Transition GPD $H_T(x,t)$ for central profile $(a = 0.5,\ b = 0.3)$ at multiple $t$ values. Shaded bands show $\pm 1\sigma$ uncertainty from dipole parameter propagation.
• Figure 5: Transition GPD $H_T(x,t = -0.5~\mathrm{GeV}^2)$ for multiple longitudinal shapes. Shaded bands reflect propagated uncertainty from the dipole form factor fit.