Multipole structure of the $N \to Δ$ Transition Generalized Parton Distributions

We establish the multipole structure of the $N \to \Delta$ transition at the level of the generalized parton distributions (GPDs). We decompose the four transition GPDs into one monopole, two dipole, and one quadrupole components in the transverse plane by a multipole expansion of the covariant transition matrix element in terms of the three-dimensional spin-transition tensors and the transverse momentum transfer. These multipole components are in one-to-one correspondence with the light-front helicity amplitudes. In the zero-skewness limit, the multipole GPDs define impact-parameter transition densities, which generalize the transverse transition charge densities to the $x$-dependent level. These transition densities arise from non-diagonal matrix elements between two distinct hadronic states and must therefore be distinguished from the diagonal densities of the nucleon and the $\Delta$. The multipole transition densities visualize the monopole, dipole, and quadrupole structures of the partonic $N \to \Delta$ transition in the transverse plane.