#### Abstract

An effective Hamiltonian consisting of bare Δ⇆π*N*, γ*N* vertex interactions and energy-independent meson-exchange π*N*⇆π*N*,γ*N* transition operators is derived by applying a unitary transformation to a model Lagrangian with *N*,Δ,π, ρ, ω, and γ fields. With appropriate phenomenological form factors and coupling constants for ρ and Δ, the model can give a good description of π*N* scattering phase shifts up to the Δ excitation energy region. It is shown that the best reproduction of the recent LEGS data of the photon-asymmetry ratios in γ*p*→${\mathrm{\pi}}^{0}$*p* reactions provides rather restricted constraints on the coupling strengths ${\mathit{G}}_{\mathit{E}}$ of the electric *E*2 and ${\mathit{G}}_{\mathit{M}}$ of the magnetic *M*1 transitions of the bare Δ⇆γ*N* vertex and the less well-determined coupling constant ${\mathit{g}}_{\mathrm{\omega}\mathit{NN}}$ of ω meson. Within the ranges that ${\mathit{G}}_{\mathit{M}}$=1.9±0.05, ${\mathit{G}}_{\mathit{E}}$=0.0±0.025, and 7≤${\mathit{g}}_{\mathrm{\omega}\mathit{NN}}$≤10.5, the predicted differential cross sections and photon-asymmetry ratios are in an overall good agreement with the data of γ*p*→${\mathrm{\pi}}^{0}$*p*, γ*p*→${\mathrm{\pi}}^{+}$*n*, and γ*n*→${\mathrm{\pi}}^{\mathrm{-}}$*p* reactions from 180 MeV to the Δ excitation region. The predicted ${\mathit{M}}_{{1}^{+}}$ and ${\mathit{E}}_{{1}^{+}}$ multipole amplitudes are also in good agreement with the empirical values determined by the amplitude analyses. The constructed effective Hamiltonian is free of the nucleon renormalization problem and hence is suitable for nuclear many-body calculations. We have also shown that the assumptions made in the *K*-matrix method, commonly used in extracting empirically the γ*N*→Δ transition amplitudes from the data, are consistent with our meson-exchange dynamical model. It is found that the helicity amplitudes calculated from our bare γ*N*→Δ vertex are in good agreement with the predictions of the constituent quark model. The differences between these bare amplitudes and the dressed amplitudes, which are closer to the empirical values listed by the Particle Data Group, are shown to be due to the nonresonant meson exchange mechanisms. Within the range 7≤${\mathit{g}}_{\mathrm{\omega}\mathit{NN}}$≤10.5 of the ω meson coupling favored by the data of the photon-asymmetry ratios in γ*p*→${\mathrm{\pi}}^{0}$*p* reactions, our values of the *E*2/*M*1 ratio for the γ*N*→Δ transition are (0.0±1.3)% for the bare vertex and (-1.8±0.9)% for the dressed vertex. © *1996 The American Physical Society.*

DOI: http://dx.doi.org/10.1103/PhysRevC.54.2660

- Received 7 June 1996
- Published in the issue dated November 1996

© 1996 The American Physical Society