Quark transverse charge densities in the \(\Delta(1232)\) from lattice QCD

We extend the formalism relating electromagnetic form factors to transverse quark charge densities in the light-front frame to the case of a spin-3/2 baryon and calculate these transverse densities for the \(\Delta(1232)\) isobar using lattice QCD. The transverse charge densities for a transversely polarized spin-3/2 particle are characterized by monopole, dipole, quadrupole, and octupole patterns representing the structure beyond that of a pure point-like spin-3/2 particle. We present lattice QCD results for the \(\Delta\)-isobar electromagnetic form factors for pion masses down to approximatively 350 MeV for three cases: quenched QCD, two-degenerate flavors of dynamical Wilson quarks, and three flavors of quarks using a mixed action that combines domain wall valence quarks and dynamical staggered sea quarks. We extract transverse quark charge densities from these lattice results and find that the \(\Delta\) is prolately deformed, as indicated by the fact that the quadrupole moment \(G_{E2}(0\)) is larger than the value -3 characterizing a point particle and the fact that the transverse charge density in a \(\Delta^+\) of maximal transverse spin projection is elongated along the axis of the spin.