Sum-Rule-Preserving Non-Factorized Transition-GPD Tomography of $N\to \Delta (1232)$ Multipole Structure

Abstract

A sum-rule-preserving transition-GPD reconstruction is developed for the $N\to \Delta (1232)$ electromagnetic transition. The analysis uses published CLAS $\Delta (1232)$ data, including the magnetic multipole amplitude and the electric and scalar/Coulomb quadrupole ratios, together with low-$Q^2$ ratio-sector constraints. Magnetic, electric, and scalar/Coulomb transition amplitudes are derived and fitted with a common library of dipole, modified-dipole, $z$-expansion, and low-$Q^2$ motivated candidate forms. The fitted transition form factors define the empirical momentum-transfer normalization for a family of transition GPDs constructed to preserve the measured form-factor sum rule. Factorized, correlated non-factorized, Regge-like, and double-distribution-inspired profiles are transformed into impact-parameter space to obtain transverse densities, localization radii, higher transverse-shape moments, and multipole-resolved radial kernels. The factorized baseline yields little genuine $x$-dependent transverse localization, while the non-factorized profiles generate distinct $x$-dependent spatial structures under the same empirical normalization. The magnetic channel provides the most stable tomography benchmark, whereas the electric and scalar/Coulomb sectors show stronger profile sensitivity. The results demonstrate that non-factorized transition-GPD tomography can extend the factorized amplitude-to-space approach while keeping the connection to measured $N\to \Delta (1232)$ transition form factors.