Systematic study of longitudinal and transverse helicity amplitudes in the hypercentral constituent quark model

We report in a systematic way the predictions of the nonrelativistic hypercentral constituent quark model for the electromagnetic excitations of baryon resonances. The longitudinal and transverse helicity amplitudes are calculated with no free parameters for fourteen resonances, for both protons and neutrons. The calculations lead to an overall fair description of data, especially in the medium-$Q^2$ range, where quark degrees of freedom are expected to dominate.

Figure 1

(a) The experimental spectrum of the nonstrange three- and four-star resonances [38]. The states are reported in columns with the same parity $P$ and grouped into $\text{SU}\left(6\right)$ multiplets. (b) The spectrum given by the $\text{SU}\left(6\right)$-invariant part of the Hamiltonian Eq. (9).

Figure 2

The $P_{33}\left(1232\right)$ helicity amplitudes predicted by the hCQM (full curves) $A_{3/2}$ (a), $A_{1/2}$ (b), and $S_{1/2}$ (c), in comparison with the data of Ref. [49] and with the the maid2007 analysis [50] of the data by Refs. [51, 52]. The PDG points [38] are also shown.

Figure 3

The $P_{11}\left(1440\right)$ proton transverse (a) and longitudinal (b) helicity amplitudes predicted by the hCQM (full curves), in comparison with the data of Refs. [49, 60], and the maid2007 analysis [50] of the data by Refs. [51, 52, 61, 62]. The PDG point [38] is also shown.

Figure 4

The $D_{13}\left(1520\right)$ proton helicity amplitudes predicted by the hCQM (full curves) $A_{3/2}$ (a), $A_{1/2}$ (b), and $S_{1/2}$ (c), in comparison with the data of Refs. [49, 60], with the compilation reported in Refs. [65, 66] and the maid2007 analysis [50] of the data by Refs. [51, 52]. The PDG points [38] are also shown.

Figure 5

The $S_{11}\left(1525\right)$ proton transverse (a) and longitudinal (b) helicity amplitudes predicted by the hCQM (full curve), in comparison with the data of Ref. [67] (open diamonds), [49] (full diamonds), [68] (crosses), [69] (open squares), [70] (full squares), the maid2007 analysis [50] (full triangles) of the data by Ref. [51], and the compilation of the Bonn-Mainz-DESY data of Refs. [71, 72, 73, 74] (stars), presented in Ref. [70]. The PDG point [38] (pentagon) is also shown.

Figure 6

The proton helicity amplitudes predicted by the hCQM for the excitation of $S_{31}\left(1620\right)$ (a) and $S_{11}\left(1650\right)$ (b), respectively, in comparison with the data of Ref. [49] ($A_{1/2}$ open diamonds, $S_{1/2}$ full diamonds), [75] ($A_{1/2}$ open diamonds, $S_{1/2}$ full diamonds), the compilation reported in Ref. [65] and the maid2007 analysis [50] ($A_{1/2}$ up triangles, $S_{1/2}$ down triangles) of the data in Refs. [51, 52], and the compilation of the Bonn-Mainz-DESY data of Refs. [71, 72, 73, 74] (crosses) presented in Ref. [70]. The PDG points [38] (pentagons) are also shown.

Figure 7

The proton helicity amplitudes predicted by the hCQM for the excitation of the $D_{15}\left(1675\right)$ (a) and $D_{13}\left(1700\right)$ (b), respectively, in comparison with the data of Ref. [49]. The PDG points [38] are also shown.

Figure 8

The $D_{33}\left(1700\right)$ helicity amplitudes predicted by the hCQM (full curve) $A_{3/2}$ (a), $A_{1/2}$ (b), and $S_{1/2}$ (c) in comparison with the data of Ref. [75] and the maid2007 analysis [50] of the data by Ref. [52]. The PDG points [38] are also shown.

Figure 9

The $F_{15}\left(1680\right)$ proton helicity amplitudes predicted by the hCQM $A_{3/2}$ (a), $A_{1/2}$ (b), and $S_{1/2}$ (c), in comparison with the data of Ref. [75] and the maid2007 analysis [50] of the data by Ref. [52]. The PDG points [38] are also shown.

Figure 10

The proton helicity amplitudes predicted by the hCQM for the excitation of $P_{11}\left(1710\right)$ (a) and $P_{13}\left(1720\right)$ (b), respectively, in comparison with the data of Ref. [75] ($A_{3/2}$ open diamond, $A_{1/2}$ full diamond, $A_{3/2}$ full box) and the maid2007 analysis [50] ($A_{3/2}$ full up triangle, $A_{1/2}$ full down triangle, $A_{3/2}$ open down triangle) of the data by Refs. [51, 52]. The PDG point [38] (pentagon) is also shown.

Figure 11

The proton helicity amplitudes predicted by the hCQM for the excitation of $F_{35}\left(1905\right)$ (a) and $F_{37}\left(1950\right)$ (b), respectively. The PDG points [38] (pentagons) are also shown.

Figure 12

The neutron excitation strength predicted by the hypercentral CQM, in comparison with the PDG points [38] (part 1).

Figure 13

The neutron excitation strength predicted by the hypercentral CQM, in comparison with the PDG points [38] (part 2).

Figure 14

The neutron excitation strength predicted by the hypercentral CQM, in comparison with the PDG points [38] (part 3).