Bled Workshops in Physics Vol. 17, No. 1 p. 13 A Proceedings of the Mini-Workshop Quarks, Hadrons, Matter Bled, Slovenia, July 3 - 10, 2016 The microscopic structure of nNN, tcNA and nAA vertices in a hybrid constituent quark model* Ju-Hyun Jung and Wolfgang Schweiger Institute of Physics, University of Graz, A-8010 Graz, Austria Abstract. We present a microscopic description of the strong nNN, nNA and nAA vertices. Our starting point is a constituent-quark model supplemented by an additional 3qn non-valence component. In the spirit of chiral constituent-quark models, quarks are allowed to emit and reabsorb a pion. This multichannel system is treated in a relativistically invariant way within the framework of point-form quantum mechanics. Starting with a common SU(6) spin-flavor-symmetric wave function for N and A, we calculate the strength of the nNN, nNA and nAA couplings and the corresponding vertex form factors. Our results are in accordance with phenomenological fits of these quantities that have been obtained within purely hadronic multichannel models for baryon resonances. 1 Introduction One of the big deficiencies of conventional constituent-quark models is the fact that all states come out as stable bound states. In nature, however, excited states are rather resonances with a finite decay width. In order to remedy this situation, we study a constituent-quark model with explicit pionic degrees of freedom. The underlying physics is that of "chiral constituent-quark models". This means that the spontaneous chiral-symmetry breaking of QCD produces pions as the associated Goldstone bosons and constituent quarks as effective particles [1], with the pions coupling directly to the constituent quarks. The occurrence of pions affects then the masses and the structure of the hadrons and leads to resonance-like behavior of hadron excitations. If one assumes instantaneous confinement between the quarks, only "bare" hadrons, i.e. eigenstates of the pure confinement problem, can propagate. As a consequence, pionic effects on hadron masses and structure can be formulated as a purely hadronic problem with the hadron substructure entering pion-hadron vertex form factors1. In the present contribution we will present predictions for nNN, nNA and nAA couplings and vertex form factors, given the nqq coupling and an SU(6) spin-flavor symmetric model for the 3q wave function of the nucleon and the A. * Talk delivered by Ju-Hyun Jung 1 Strictly speaking these are vertex form factors of the bare hadrons. 14 Ju-Hyun Jung and Wolfgang Schweiger 2 Formalism Our starting point for calculating the strong nNN, nNA and nAA couplings and form factors is the mass-eigenvalue problem for 3 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 rela-tivistic quantum mechanics. Employing the Bakamjian-Thomas construction, the overall 4-momentum operator P^ can be separated into a free 4-velocity operator V ^ and an invariant mass operator M that contains all the internal motion, i.e. P^ = M [2]. Bakamjian-Thomas-type mass operators are most conveniently represented by means of velocity states |V; k, |i; k2, |2;...; kn, |n), which specify the system by its overall velocity V (V^V^ = 1), the CM momenta kt of the individual particles and their (canonical) spin projections | [2]. Since the physical baryons of our model contain, in addition to the 3q-component, also a 3qn-component, the mass eigenvalue problem can be formulated as a 2-channel problem of the form (M3°qnf Kn V l*3q) \ _ m (|*3q) \ (1) KMMfll^ _ mU3-«)J , (1) with |^3q) and |^3qn) denoting the two Fock-components of the physical baryon states |B). The mass operators on the diagonal contain, in addition to the relativis-tic particle energies, an instantaneous confinement potential between the quarks. The vertex operator K^ connects the two channels and describes the absorption (emission) of the n by one of the quarks. Its velocity-state representation can be directly connected to a corresponding field-theoretical interaction Lagrangean [2]. We use a pseudovector interaction Lagrangean for the nqq-coupling f £«qq(x)_-(lj>q(x)Y^5^q(x)) • 3^„(x), (2) mn where the "•"-product has to be understood as product in isospin space. After elimination of the 3qn-channel the mass-eigenvalue equation takes on the form [Mf + Kn(m - Mqf )-1Kt]|^q) _ m|^3q) , (3) v-V-' V>npt(m) where Vnpt(m) is an optical potential that describes the emission and reabsorption of the pion by the quarks. One can now solve Eq. (3) by expanding the (3q-components of the) eigenstates in terms of eigenstates of the pure confinement problem, i.e. |^3q) _ Y.b0 aB |Bo), and determining the open coefficients aBo. Since the particles which propagate within the pion loop are also bare baryons (rather than quarks), the problem of solving the mass eigenvalue equation (3) reduces then to a pure hadronic problem, in which the dressing and mixing of bare baryons by means of pion loops produces finally the physical baryons (see Fig. 1). As also indicated in Fig. 1, the quark substructure determines just the coupling strengths at the pion-baryon vertices and leads to vertex form factors. To set up The microscopic structure of nNN, nNA and nAA vertices 15 Fig. 1. Graphical representation of the kernel (B0|Vnpt(m)|Bo} needed to solve the mass-eigenvalue equation (3). the mass-eigenvalue equation on the hadronic level one needs matrix elements