Electromagnetic Multipoles — Theory Issues

Abstract

Some predictions of the Hypercentral Constituent Quark Model for the helicity amplitudes are discussed and compared with data and with the recent analysis of the Mainz group; the role of the pion cloud contribution in explaining the major part of the missing strength at low Q is emphasized. 1 The hypercentral Constituent Quark Model In the hypercentral Constituent Quark Model (hCQM) one introduces the hyperspherical coordinates, which are obtained from the standard Jacobi coordinates ~ ρ and ~λ substituting the absolute values ρ and λ by x = √ ~ ρ2 + ~λ2 , ξ = arctg( ρ λ ), (1) where x is the hyperradius and ξ the hyperangle. The potential for the three quark system, V , is assumed to depend on the hyperradius x only, that is to be hypercentral. It can be considered as a two-body interaction in the hypercentral approximation, which has been shown to be valid specially for the lower energy states [1]. It can also be viewed as a true three-body potential; actually the fundamental gluon interactions, predicted by QCD, lead to three-quark mechanisms. The situation is similar to the flux tube models, where two-body (∆-shaped) and three-body (Y -shaped) interactions are considered. For a hypercentral potential, in the three-quark wave function one can factor out the angular and hyperangular parts, which are given by the known hyperspherical harmonics [2] and the Schrodinger equation is reduced to a single equation for the hypercentral wave function. Such hypercentral equation can be solved analytically at least in two cases, that is for the h.o. potential and the hypercoulomb one. The two-body h.o. potential turns out to be exactly hypercentral, since ∑ i<j 1 2 k (~ ri − ~ rj) 2 = 32 k x 2 . The SU(6) states in the h.o. model are too degenerate with respect to the observed spectrum. The ’hypercoulomb’ potential [1, 3] Vhyc(x) = − τ x is not confining, however it leads to a power-law behaviour of the proton form factor and of all the transition form factors [4] and it has a perfect degeneracy between the first 0 excitated state and the first 1− states. The former can be identified with the Roper resonance 1 ar X iv :1 50 6. 09 18 8v 1 [ nu cl -t h] 3 0 Ju n 20 15 and the latter with the negative parity resonances. This degeneracy seems to be in agreement with phenomenology but such feature cannot be reproduced in models with only two-body forces, since the excited L = 0 state, having one more node, lies above the L = 1 state. In the hCQM [5] the confining hypercentral potential is assumed to be of the form V (x) = − x + αx, (2) A standard hyperfine interaction [6], treated as a perturbation, is added in order to describe the splittings within the SU(6) multiplets. The non strange spectrum is described with τ = 4.59 and α = 1.61 fm−2 and the standard strength of the hyperfine interaction needed for the N −∆ mass difference [6]. The model, keeping fixed these three parameters, has been applied in order to calculate, that is predict, various quantities of interest, namely the photocouplings [7], the transition helicity amplitudes [8], the elastic nucleon form factors [9] and the ratio between the electric and magnetic form factors [10]. In the following the results of this model for the transition helicity amplitudes will be discussed. The model has been modified in two respects in order to improve the description of the spectrum. First, isospin dependent terms have been added to the spin-spin ones [11]; the second modification is that to use the correct relativistic kinetic energy [12]. The resulting spectrum is considerably improved, in particular the correct ordering of the Roper resonance and the negative parity states is achieved. 2 The helicity amplitudes The electromagnetic transition amplitudes, A1/2 and A3/2, are defined as the matrix elements of the transverse electromagnetic interaction, H e.m., between the nucleon, N , and the resonance, B, states: A1/2 = 〈B, J ′, J ′ z = 12 |Hem|N, J = 1 2 , Jz = − 1 2 〉 A3/2 = 〈B, J ′, J ′ z = 32 |Hem|N, J = 1 2 , Jz = 1 2 〉 (3) The baryon states are obtained using the hCQM: V3q = − τ x + αx+Hhyp (4) with the parameters fixed in the previous section. The transverse transition operator is assumed to be