A dynamical coupled-channel model is presented for investigating the nucleon resonances (N * ) in the meson production reactions induced by pions and photons. Our objective is to extract the N * parameters and to investigate the meson production reaction mechanisms for mapping out the quark-gluon substructure of N * from the data. The model is based on an energy-independent Hamiltonian which is derived from a set of Lagrangians by using a unitary transformation method. The constructed model Hamiltonian consists of (a) Γ V for describing the vertex interactions N * ↔ M B, ππN with M B = γN, πN, ηN, π∆, ρN, σN , and ρ ↔ ππ and σ ↔ ππ, (b) v 22 for the non-resonant M B → M ′ B ′ and ππ → ππ interactions, (3) v M B,ππN for the non-resonant M B → ππN transitions, and (4) v ππN,ππN for the non-resonant ππN → ππN interactions. By applying the projection operator techniques, we derive a set of coupled-channel equations which satisfy the unitarity conditions within the channel space spanned by the considered two-particle M B states and the three-particle ππN state. The resulting amplitudes are written as a sum of non-resonant and resonant amplitudes such that the meson cloud effects on the N * decay can be explicitly calculated for interpreting the extracted N * parameters in terms of hadron structure calculations. We present and explain in detail a numerical method based on a spline-function expansion for solving the resulting coupled-channel equations which contain logarithmically divergent one-particle-exchange driving terms Z (E) M B,M ′ B ′ resulted from the ππN unitarity cut. This method is convenient, and perhaps more practical and accurate than the commonly employed methods of contour rotation/deformation, for calculating the two-pion production observables. For completeness in explaining our numerical procedures, we also present explicitly the formula for efficient calculations of a very large number of partial-wave matrix elements which are the input to the coupled-channel equations. Results for two pion photo-production are presented to illustrate the dynamical consequence of the one-particle-exchange driving term Z (E) M B,M ′ B ′ of the coupled-channel equations. We show that this mechanism, which contains the effects due to ππN unitarity cut, can generate rapidly varying structure in the reaction amplitudes associated with the unstable particle channels π∆, ρN , and σN , in agreement with the analysis of Aaron and Amado [Phys. Rev. D13, 2581(1976]. It also has large effects in determining the two-pion production cross sections. Our results indicate that cautions must be taken to interpret the N * parameters extracted from using models which do not include ππN cut effects. Strategies for performing a complete dynamical coupled-channel analysis of all of available data of meson photo-production and electro-production are discussed.

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As a first step to analyze the electromagnetic meson production reactions in the nucleon resonance region, the parameters of the hadronic interactions of a dynamical coupled-channel model, developed in Physics Reports 439, 193 (2007), are determined by fitting the πN scattering data. The channels included in the calculations are πN , ηN and ππN which has π∆, ρN , and σN resonant components. The non-resonant meson-baryon interactions of the model are derived from a set of Lagrangians by using a unitary transformation method. One or two bare excited nucleon states in each of S, P , D, and F partial waves are included to generate the resonant amplitudes in the fits. The parameters of the model are first determined by fitting as much as possible the empirical πN elastic scattering amplitudes of SAID up to 2 GeV. We then refine and confirm the resulting parameters by directly comparing the predicted differential cross section and target polarization asymmetry with the original data of the elastic π ± p → π ± p and charge-exchange π − p → π 0 n processes. The predicted total cross sections of πN reactions and πN → ηN reactions are also in good agreement with the data. Applications of the constructed model in analyzing the electromagnetic meson production data as well as the future developments are discussed.