Eur. Phys. J. A 1, 307–315 (1998) T HE EUROPEAN P HYSICAL JOURNAL A c  Springer-Verlag 1998 Nucleon form factors in a simple three-body quark model ⋆ E. Santopinto 1,2 , F. Iachello 1 , M.M. Giannini 2 1 Center for Theoretical Physics, Sloane Laboratory, Yale University, New Haven, CT 06520-8120, U.S.A 2 Dipartimento di Fisica dell’Universit`a di Genova, I.N.F.N., Sezione di Genova, via Dodecaneso 33, 16164 Genova, Italy (e-mail:giannini@genova.infn.it) Received: 15 November 1997 Communicated by V.V. Anisovich Abstract. We construct a simple 3-body quark model for the non strange nucleon resonances and we give results for the spectrum, the helicity amplitudes and the transition form factors. All the observables, in particular the transition form factors, are evaluated analytically and the results are compared with those of other models. PACS. 12.39.Jh Nonrelativistic quark model – 13.40.Gp Electromagnetic form factors – 14.20.Gk Baryon resonances with S = 0 Protons and neutrons 1 Introduction Recently there has been a renewed interest in the study of baryon properties [1–8]. Most attention has been devoted to the spectrum and the helicity amplitudes [9–11,1,4–7] but the problem of describing the transition form factors is still open [1–3,12]. On the other hand, the experimen- tal knowledge of the transition form factors is still poor. New accurate and systematic data are expected from the forthcoming experiments at TJNAF (CEBAF). The non relativistic constituent quark models (CQM) have given good results in the study of the static proper- ties of the nucleon [9,11], like the baryon spectrum and the magnetic moments, and in a qualitative reproduction of the photocouplings [13,14]. The use of harmonic oscil- lator gives rise to form factors which do not reproduce the experimental data and for this reason it can be interesting to investigate to which extent the results for the transition form factors depend on the choice of the wave functions. In this article we construct a simple three-body poten- tial model, which gives analytical results for all the ob- servables. In particular, we report on the transition form factors for the electromagnetic excitation of the nucleon. In Sect. 2 we introduce the hypercentral description of the 3-body problem. In Sect. 3 we construct the model and calculate the spectrum. In Sects. 4 and 5 we calculate the helicity amplitudes and the transitions form factors re- spectively and compare the results with the experimental data. A brief conclusion is given in Sect. 6. ⋆ Partially supported by EC-contract number ERB FMRX- CT96-0008 2 The three-body problem We consider non-strange baryons as a bound state of three constituent quarks. After removal of the center of mass coordinate, R, the configurations of three particles are described by the Jacobi coordinates, ρ and λ, ρ = 1 √ 2 (r 1 − r 2 ), λ = 1 √ 6 (r 1 + r 2 − 2r 3 ). (1) Instead of ρ and λ, one can introduce the hyperspherical coordinates, which are given by the angles Ω ρ =(θ ρ ,φ ρ ) and Ω λ =(θ λ ,φ λ ) together with the hyperradius, x, and the hyperangle, ξ , defined respectively by [15,16] x =  ρ 2 + λ 2 , ξ = arctg( ρ λ ). (2) As a model of light baryons, we consider three identical quarks of mass m with Hamiltonian H = p 2 ρ 2m + p 2 λ 2m + V (x), (3) where the potential V (x) is assumed to depend on x only, that is to be hypercentral. Since x =(ρ 2 + λ 2 ) 1 2 , the interaction in (3) is not purely a two-body interaction, but it contains three-body terms. Several authors [17–20,4,21] have suggested that three- body interactions can play an important role in hadrons, since the non-abelian nature of QCD leads to gluon-gluon couplings which, in turn, can produce three-body forces. It is also interesting to notice that two-body interactions can