On the Microscopic Structure of $${\varvec{\pi }}$$ π NN, $${\varvec{\pi }}$$ π N $${\varvec{\Delta }}$$ Δ and $${\varvec{{\pi }\Delta \Delta }}$$ π Δ Δ Vertices

Abstract

We use a hybrid constituent-quark model for the microscopic description of π N N, π N and π vertices. In this model quarks are confined by an instantaneous potential and are allowed to emit and absorb a pion, which is also treated as dynamical degree of freedom. The point form of relativistic quantum mechanics is employed to achieve a relativistically invariant description of this system. Starting with an SU (6) spin-flavor symmetric wave function for N 0 and 0 , i.e. the eigenstates of the pure confinement problem, we calculate the strength of the π N 0 N 0 , π N 0 0 and π 0 0 couplings and the corresponding vertex form factors. Interestingly the ratios of the resulting couplings resemble strongly those needed in purely hadronic coupled-channel models, but deviate significantly from the ratios following from SU(6) spin-flavor symmetry in the non-relativistic constituent-quark model. Motivation and FormalismOur interest in π N N, π N and π couplings and vertex form factors is connected with our attempt to take pion-loop effects into account when describing the electromagnetic structure of N and within a constituentquark model. As it turns out, the calculation of the loop effects boils down to a purely hadronic problem, in which the quark substructure of the N and the is hidden in strong and electromagnetic form factors of "bare" baryons, i.e. eigenstates of the pure confinement problem. Since π N N, π N and π couplings and vertex form factors are basic building blocks of nuclear physics and every hadronic model of meson-baryon dynamics, their microscopic description is also highly desirable on more fundamental grounds.Our starting point for calculating the strong π N N, π N and π couplings and form factors is the mass-eigenvalue problem for three quarks that are confined by an instantaneous potential and can emit and reabsorb a pion. To describe this system in a relativistically invariant way, we make use of the point-form of relativistic quantum mechanics. Employing the Bakamjian-Thomas construction, the overall four-momentum operatorP μ can be separated into a free 4-velocity operatorV μ and an invariant mass operatorM that contains all the internal motion, i.e.P μ =MV μ [1]. Bakamjian-Thomas-type mass operators are most conveniently represented by means of velocity states |V ; k 1 , μ 1 ; k 2 , μ 2 ; . . . ; k n , μ n , which specify an n-body system by its overall velocity V (V μ V μ = 1), the CM momenta k i of the individual particles and their (canonical) spin projections μ i [1]. Since the physical baryons of our model contain, in addition to the 3q-component, also a