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*)(N*) in the meson production reactions induced by pions and photons. Our objective is to extract the N*N* parameters and to investigate the meson production reaction mechanisms for mapping out the quark–gluon substructure of N*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) ΓVΓV for describing the vertex interactions N*↔MB,ππNN*↔MB,ππN with MB=γN,πN,ηN,πΔ,ρN,σNMB=γN,πN,ηN,πΔ,ρN,σN, and ρ↔ππρ↔ππ and σ↔ππσ↔ππ, (b) v22v22 for the non-resonant MB→M′B′MB→M′B′ and ππ→ππππ→ππ interactions, (c) vMB,ππNvMB,ππN for the non-resonant MB→ππNMB→ππN transitions, and (d) vππN,ππNvππN,ππN for the non-resonant ππN→ππNππN→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 MB   states and the three-particle ππNππN state. The resulting amplitudes are written as a sum of non-resonant and resonant amplitudes such that the meson cloud effects on the N*N* decay can be explicitly calculated for interpreting the extracted N*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 ZMB,M′B′(E) resulted from the ππNππN 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 ZMB,M′B′(E) of the coupled-channel equations. We show that this mechanism, which contains the effects due to ππNππN unitarity cut, can generate rapidly varying structure in the reaction amplitudes associated with the unstable particle channels πΔπΔ, ρNρN, and σNσN, 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*N* parameters extracted from using models which do not include ππNππN 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.