Electroexcitation of nucleon resonances from CLAS data on single pion electroproduction

We present results on the electroexcitation of the low mass resonances $\Delta (1232)P_{33}$, $N(1440)P_{11}$, $N(1520)D_{13}$, and $N(1535)S_{11}$ in a wide range of $Q^2$. The results were obtained in the comprehensive analysis of data from the Continuous Electron Beam Accelerator Facility (CEBAF) large acceptance spectrometer (CLAS) detector at the Thomas Jefferson National Accelerator Facility (JLab) on differential cross sections, longitudinally polarized beam asymmetries, and longitudinal target and beam-target asymmetries for $\pi$ electroproduction off the proton. The data were analyzed using two conceptually different approaches—fixed-$t$ dispersion relations and a unitary isobar model—allowing us to draw conclusions on the model sensitivity of the obtained electrocoupling amplitudes. The amplitudes for the $\Delta (1232)P_{33}$ show the importance of a meson-cloud contribution to quantitatively explain the magnetic dipole strength, as well as the electric and scalar quadrupole transitions. They do not show any tendency of approaching the pQCD regime for $Q^2⩽6$ GeV${}^2$. For the Roper resonance, $N(1440)P_{11}$, the data provide strong evidence that this state is a predominantly radial excitation of a three-quark ($3q$) ground state. Measured in pion electroproduction, the transverse helicity amplitude for the $N(1535)S_{11}$ allowed us to obtain the branching ratios of this state to the $\pi N$ and $\eta N$ channels via comparison with the results extracted from $\eta$ electroproduction. The extensive CLAS data also enabled the extraction of the ${\gamma }^{*}p\to N(1520)D_{13}$ and $N(1535)S_{11}$ longitudinal helicity amplitudes with good precision. For the $N(1535)S_{11}$, these results became a challenge for quark models and may be indicative of large meson-cloud contributions or of representations of this state that differ from a $3q$ excitation. The transverse amplitudes for the $N(1520)D_{13}$ clearly show the rapid changeover from helicity-3/2 dominance at the real photon point to helicity-1/2 dominance at $Q^2>1$ GeV${}^2$, confirming a long-standing prediction of the constituent quark model.

Figure 1

$Q^2$ dependence of the coefficients $f_1(Q^2)$ (solid curve) and $f_2(Q^2)$ (dashed curve) from Eq. (7). The results for $Q^2<0.7$ GeV${}^2$ were found using Eq. (6), whereas the results for $Q^2=1.7–4.5$ GeV${}^2$ are from the fit to the data [7].

Figure 2

Pion contribution in GeV${}^{-2}$ units (solid curves) to the DR for the amplitude $B_3^{(-)}(s,t,Q^2)$, Eq. (5), compared with $f_{\mathrm{sub}}(t,Q^2)$ at (a) $Q^2=0$ and (b) $Q^2=2.44$ GeV${}^2$. The dashed curves represent $f_{\mathrm{sub}}(t,Q^2)$ taken in the form of Eq. (6); the dash-dotted curve corresponds to the results for $f_{\mathrm{sub}}(t,Q^2)$ obtained by fitting the data [7]. At $Q^2=2.44$ GeV${}^2$, the physical region is located on the right side of the dotted vertical line.

Figure 3

Our results for the Legendre moments of the $\overrightarrow{e}p\to \mathit{ep}{\pi }^0$ structure functions in comparison with experimental data [1] for $Q^2=0.4$ GeV${}^2$. The solid (dashed) curves correspond to the results obtained using the DR (UIM) approach.

Figure 4

Our results for the Legendre moments of the $\overrightarrow{e}p\to \mathit{en}{\pi }^{+}$ structure functions in comparison with experimental data [4] for $Q^2=0.4$ GeV${}^2$. The solid (dashed) curves correspond to the results obtained using the DR (UIM) approach.

Figure 5

Our results for the Legendre moments of the $\overrightarrow{e}p\to \mathit{en}{\pi }^{+}$ structure functions in comparison with experimental data [7] for $Q^2=2.44$ GeV${}^2$. The solid (dashed) curves correspond to the results obtained using the DR (UIM) approach.

Figure 6

Same as in Fig. 5, but for $Q^2=3.48$ GeV${}^2$.

Figure 7

$A_t$ (left panel) and $A_{\mathit{et}}$ (right panel) as functions of the invariant mass $W$, integrated over the whole range in $\cos \theta ,0.252<Q^2<0.611$ GeV${}^2$ and ${60}^{°}<\varphi <{156}^{°}$. Experimental data are from Ref. [8]. Solid and dashed curves correspond to our results obtained using DR and UIM approaches, respectively.

Figure 8

Our results for the longitudinal target asymmetry $A_t$ in comparison with experimental data for $Q^2=0.385$ GeV${}^2$ [8]. Solid (dashed) curves correspond to the results obtained using the DR (UIM) approach. Rows correspond to seven $W$ bins with $W$ mean values of 1.125, 1.175, 1.225, 1.275, 1.35, 1.45, and 1.55 GeV. Columns correspond to $\varphi$ bins with $\varphi =\pm {72}^{°},\pm {96}^{°},\pm {120}^{°},\pm {144}^{°},\pm {168}^{°}$. The solid circles are the average values of the data for positive $\varphi$ and those at negative $\varphi$ taken with opposite signs.

Figure 9

Our results for the beam-target asymmetry $A_{\mathit{et}}$ in comparison with experimental data for $Q^2=0.385$ GeV${}^2$ [8]. Solid (dashed) curves correspond to the results obtained using the DR (UIM) approach. Rows correspond to seven $W$ bins with $W$ mean values of 1.125, 1.175, 1.225, 1.275, 1.35, 1.45, and 1.55 GeV. Columns correspond to $\varphi$ bins with $\varphi =\pm {72}^{°},\pm {96}^{°},\pm {120}^{°},\pm {144}^{°}$, and $\pm {168}^{°}$. The average values of the data for positive and negative $\varphi$ are shown by solid circles.

Figure 10

Left panel: form factor $G_M^{*}$ for the ${\gamma }^{*}p\to \Delta (1232)P_{33}$ transition relative to $3G_D$. Right panel: ratios $R_{\mathit{EM}},R_{\mathit{SM}}$. The full boxes are the results from Tables 6, 7, 8 obtained in this work from CLAS data (Tables 1, 3, and 4). The bands show the model uncertainties. Also shown are the results from MAMI [75, 76] (open triangles), MIT Bates [77] (open crosses), JLab/Hall C [78] (open rhombuses), and JLab/Hall A [79, 80] (open circles). The solid and dashed curves correspond to the dressed and bare contributions from Ref. [27]; for $R_{\mathit{EM}},R_{\mathit{SM}}$, only the dressed contributions are shown; the bare contributions are close to zero. The dashed-dotted curves are the predictions obtained in the large-$N_c$ limit of QCD [37, 81]. The dotted curve for $G_M^{*}$ is the prediction of a LF relativistic quark model of Ref. [18]; the dotted curves for $R_{\mathit{EM}},R_{\mathit{SM}}$ are the MAID2007 solutions [39].

Figure 11

Our results for the $\mathit{ep}\to \mathit{ep}{\pi }^0$ structure functions (in $\mu$b/sr units) in comparison with experimental data [1] for $W=1.23$ GeV. The columns correspond to $Q^2=0.4$, 0.75, and 1.45 GeV${}^2$. The solid (dashed) curves correspond to the results obtained using the DR (UIM) approach. The dotted curves are from MAID2007 [39].

Figure 12

Same as Fig. 11, but the columns correspond to $Q^2=3$, 4.2, 5 GeV${}^2$.

Figure 13

Helicity amplitudes for the ${\gamma }^{*}p\to N(1440)P_{11}$ transition. The full circles are the results from Table 9 obtained in this work from CLAS data (Tables 1, 2, 3, 4). The bands show the model uncertainties. The open boxes are the results of the combined analysis of CLAS single $\pi$ and 2$\pi$ electroproduction data [42]. The full triangle at $Q^2=0$ is the RPP estimate [20]. The thick curves correspond to the results obtained in the LF relativistic quark models assuming that $N(1440)P_{11}$ is a first radial excitation of the $3q$ ground state: [15] (dashed), [19] (solid). The thin dashed curves are obtained assuming that $N(1440)P_{11}$ is a gluonic baryon excitation (q${}^3\mathrm{G}$ hybrid state) [31].

Figure 14

Helicity amplitudes for the ${\gamma }^{*}p\to N(1535)S_{11}$ transition. The legend is partly as for Fig. 13. The solid boxes are the results extracted from $\eta$ photo- and electroproduction data in Ref. [84], the open boxes show the results from $\eta$ electroproduction data [46, 47, 48]. The data are presented assuming ${\beta }_{\pi N}=0.485,{\beta }_{\eta N}=0.46$ (see Sec. 7c). The results of the LF relativistic quark models are given by the dashed [15] and dashed-dotted [17] curves. The solid curves are the central values of the amplitudes found within light-cone sum rules using lattice results for light-cone distribution amplitudes of the $N(1535)S_{11}$ resonance [85].

Figure 15

Helicity amplitudes for the ${\gamma }^{*}p\to N(1520)D_{13}$ transition. The legend is partly as for Fig. 13. Open circles show the results [87] extracted from earlier DESY [45, 88] and NINA [89] data. The curves correspond to the predictions of the quark models: [90] (solid), [91] (dashed), and [92] (dotted).

Figure 16

Helicity asymmetry $A_{\mathrm{hel}}\equiv (A_{1/2}^2-A_{3/2}^2)/(A_{1/2}^2+A_{3/2}^2)$ for the ${\gamma }^{*}p\to N(1520)D_{13}$ transition. Triangles show the results obtained in this work. The solid curve is the prediction of the quark model with the harmonic oscillator potential [93].

Figure 17

Helicity amplitudes $A_{1/2}$ for the ${\gamma }^{*}p\to N(1440)P_{11},N(1520)D_{13}$, and $N(1535)S_{11}$ transitions, multiplied by $Q^3$. The results obtained in this work from the JLab-CLAS data on pion electroproduction on the protons are shown by solid circles [$N(1440)P_{11}$], solid triangles [$N(1520)D_{13}$], and solid boxes [$N(1535)S_{11}$]. Open boxes and crosses are the results for the $N(1535)S_{11}$ obtained in $\eta$ electroproduction, respectively, in HALL B [47, 48] and HALL C [46]. The solid curve corresponds to the amplitude $A_{1/2}$ for the ${\gamma }^{*}p\to N(1535)S_{11}$ transition found within light-cone sum rules [85].