Two-dimensional charge distributions of the $\mathrm{\Delta }$ baryon: Interpolation between the nonrelativistic and ultrarelativistic limit

We investigate how the charge distributions of both the unpolarized and transversely polarized $\mathrm{\Delta }$ baryon change as the longitudinal momentum~($P_z$) of the $\mathrm{\Delta }$ baryon increases from $P_z=0$ to $P_z=\mathrm{\infty }$ in a Wigner phase-space perspective. When the $\mathrm{\Delta }$ baryon is longitudinally polarized, its two-dimensional charge distribution is kept to be spherically symmetric with $P_z$ varied, whereas when the $\mathrm{\Delta }$ baryon is transversely polarized along the $x$-axis, the quadrupole contribution emerges at the rest frame ($P_z=0$). When $P_z$ grows, the electric dipole and octupole moments are induced. The induced dipole moment dominates over other higher multipole contributions and governs the deformation of the charge distribution of the $\mathrm{\Delta }$ baryon.