Electroexcitation of the $\mathrm{\Delta }\left(1232\right){\frac{3}{2}}^{+}$ and $\mathrm{\Delta }\left(1600\right){\frac{3}{2}}^{+}$ in a light-front relativistic quark model

The magnetic-dipole form factor and the ratios $R_{EM}$ and $R_{SM}$ for the ${\gamma }^{*}N\to \mathrm{\Delta }\left(1232\right){\frac{3}{2}}^{+}$ transition are predicted within the light-front relativistic quark model up to photon virtuality $Q^2=12{\mathrm{GeV}}^2$. We also predict the helicity amplitudes of the ${\gamma }^{*}N\to \mathrm{\Delta }\left(1600\right){\frac{3}{2}}^{+}$ transition assuming that the $\mathrm{\Delta }\left(1600\right){\frac{3}{2}}^{+}$ is the first radial excitation of the ground state $\mathrm{\Delta }\left(1232\right){\frac{3}{2}}^{+}$.

Figure 1

The form factor $G_{M,Ash}\left(Q^2\right)$ for the ${\gamma }^{*}p\to \mathrm{\Delta }\left(1232\right){\frac{3}{2}}^{+}$ transition relative to $3G_D:G_D\left(Q^2\right)=1/\left(1+\frac{Q^2}{0.71{\mathrm{GeV}}^2}\right)$. The full boxes are the CLAS data extracted in the analysis of Ref. [8]; the open boxes correspond to the data from Ref. [23]. The bands show the model uncertainties of these data [8, 17]. The thin solid curve is the result of the global analysis of the Mainz group [24]. The results from other experiments are open triangles [25, 26, 27], open cross [28, 29, 30], open rhombuses [31], and open circle [32, 33]. The thick solid curve presents our results. The dashed curves demonstrate the sensitivity of these results to the form factor $G_3\left(Q^2\right)$ (25); they correspond to $G_3\left(Q^2\right)$ taken with $\pm 50\%$ deviation from the values obtained using the relation (25). The dotted curves show the uncertainty of our results (given by the solid curve) that is caused by the uncertainty of $c_{\mathrm{\Delta }}$ (31). The dashed-dotted curve is the meson-baryon contribution from Refs. [34, 35].

Figure 2

The ratios $R_{EM}$ and $R_{SM}$ for the ${\gamma }^{*}p\to \mathrm{\Delta }\left(1232\right){\frac{3}{2}}^{+}$ transition. The legend for experimental data and thick solid and dashed curves is as for Fig. 1.

Figure 3

Helicity amplitudes for the ${\gamma }^{*}p\to \mathrm{\Delta }\left(1600\right){\frac{3}{2}}^{+}$ transition. The full triangles at $Q^2=0$ are the Review of Particle Physics (RPP) estimates [41]. The thick solid curve presents our LF RQM predictions. The legend for dashed curves is as for Fig. 1. The dashed-dotted curves present the predictions from Ref. [9].