Investigation of the Δ ( 1232 ) resonance substructure in the p γ * → Δ ( 1232 ) process through helicity amplitudes

The substructure of the Δ ( 1232 ) resonance is studied in the p γ * → Δ ( 1232 ) transition via helicity amplitudes within a quark-model framework that treats baryons as three-quark systems and includes both quark-core and meson-cloud contributions. In the calculation of the transition amplitudes, the proton wave function is Lorentz boosted from the proton rest frame to the Δ ( 1232 ) resonance rest frame. The comparison of the calculated amplitudes A 1 / 2 , A 3 / 2 , and S 1 / 2 with experimental data indicates a non-negligible L = 2 admixture in the Δ ( 1232 ) , about 80% of the Δ ( 1232 ) wave function in the S wave and 20% in the D wave. In particular, the S 1 / 2 amplitude is found to be dominated by the L = 2 quark-core component rather than the L = 0 part. The results challenge the traditional understanding that the Δ ( 1232 ) baryon is dominantly an L = 0 state.