Dynamical coupled-channel model of meson production reactions in the nucleon resonance region

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 * ↔ MB , π π N with MB = γ N , π N , η N , π Δ , ρ N , σ N , and ρ ↔ π π and σ ↔ π π , (b) v 22 for the non-resonant MB → M ′ B ′ and π π → π π interactions, (c) v MB , π π N for the non-resonant MB → π π N transitions, and (d) v π π N , π π N for the non-resonant π π N → π π N 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 MB states and the three-particle π π N 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 divergentone-particle-exchange driving terms Z MB , M ′ B ′ ( E ) resulted from the π π N 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 MB , M ′ B ′ ( E ) of the coupled-channel equations. We show that this mechanism, which contains the effects due to π π N unitarity cut, can generate rapidly varying structure in the reaction amplitudes associated with the unstable particle channels π Δ , ρ N , and σ N , in agreement with the analysis of Aaron and Amado [Phys. Rev. D13 (1976) 2581]. 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 π π N 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.