Investigation of $Δ(1232)$ resonance substructure in $pγ^* o Δ(1232)$ process through helicity amplitudes

This work investigates the substructure of the $Δ(1232)$ resonance in the $pγ^*\to Δ(1232)$ process through helicity transition amplitudes within the quark model framework. We consider the involved baryons composed of three quarks, and both the quark core and meson cloud contribute to the transition amplitudes. The comparison of theoretical results with experimental data reveals that, rather than the $L=0$ component of the $Δ(1232)$ resonance, it is the $L=2$ component that significantly affects its $S_{1/2}$ amplitude. These findings indicate that the $Δ(1232)$ resonance likely contains a substantial $L=2$ component, challenging the conventional view of the $Δ(1232)$ resonance as an $L=0$ baryon.

💡 Research Summary

In this work the authors investigate the internal substructure of the Δ(1232) resonance by calculating the helicity transition amplitudes for the process p γ* → Δ(1232) within a three‑quark model that incorporates both a quark‑core contribution and a meson‑cloud (pion) contribution. The spatial wave functions of the proton and the Δ are constructed using SU_f(2)⊗SU_s(2)⊗SU_c(3) symmetry and expressed in a harmonic‑oscillator basis in Jacobi coordinates. While the proton is taken to be a pure L = 0 (S‑wave) state, the Δ(1232) is allowed to contain an L = 0 component together with two L = 2 components: a symmetric D‑wave (spin 3/2) and a mixed‑symmetry D‑wave (spin 1/2). The relative weights (B, C, D) of these components are determined by fitting the calculated helicity amplitudes A₁/₂, A₃/₂ and S₁/₂ to the world data.

The transition amplitudes are derived from matrix elements ⟨Δ|T_q + T_MC|p⟩, where T_q describes the direct photon‑quark interaction (impulse approximation) and T_MC the photon‑pion‑quark interaction mediated by the meson cloud. The meson‑cloud dynamics are modeled with the chiral quark Lagrangians L_{qqπ}= (1/2F)∂μπ^i \barψ γ^μγ₅τ^iψ and L{ππγ}=−e ε^{3jk}π^j∂_νπ^k A^ν, using a monopole form factor with cutoff Λ_π = 0.732 GeV to regularize the loop integrals.

Fitting the data yields B = 0.725 (53 % L = 0), C = 0.485 (23 % symmetric L = 2) and D = 0.487 (24 % mixed‑symmetry L = 2). Thus the Δ(1232) is not a pure S‑wave state; a sizable D‑wave admixture is required. The most striking result concerns the longitudinal helicity amplitude S₁/₂: the L = 0 component contributes nothing, and the entire S₁/₂ strength originates from the L = 2 components. This explains the non‑zero S₁/₂ observed experimentally and provides a natural source for the electric quadrupole (E2) strength in the γ* N → Δ transition.

The meson‑cloud contribution is most important at low momentum transfer (Q² ≲ 0.5 GeV²), where it accounts for roughly 15–20 % of the total amplitude, while at higher Q² the quark‑core dominates. The calculated A₁/₂, A₃/₂ and S₁/₂ curves agree well with data from CLAS, MAMI and other facilities across the full Q² range.

Overall, the study demonstrates that incorporating both D‑wave components and pion‑cloud effects is essential for a quantitative description of Δ(1232) helicity amplitudes. The findings challenge the conventional picture of the Δ(1232) as a pure L = 0 baryon and suggest that higher orbital angular momentum plays a significant role in its structure. This work paves the way for refined quark‑model calculations, lattice QCD comparisons, and future experimental investigations of baryon resonances where D‑wave admixtures may be similarly important.