Parametrizations of the ${\gamma }^{*}N\to \mathrm{\Delta }(1232)$ quadrupole form factors and Siegert’s theorem

The large $N_c$ limit provides relations that can be used to calculate the ${\gamma }^{*}N\to \mathrm{\Delta }(1232)$ quadrupole form factors at low and intermediate $Q^2$ under the assumption of the pion cloud dominance. There are two limitations in those parametrizations. First, the parametrization of the Coulomb quadrupole form factor underestimate the low $Q^2$ data. Second, when extrapolated for the timelike region, the form factors violate Siegert’s theorem by terms of the order $1/N_c^2$. We propose here corrections to the parametrization of the electric quadrupole form factor, which violate Siegert’s theorem only by terms of the order $1/N_c^4$. Combining the improved large $N_c$ pion cloud parametrizations with the valence quark contributions based on a covariant quark model for the quadrupole transition form factors, we obtain an extrapolation to the timelike region consistent with Siegert’s theorem, and accomplish also a very good description of the data.

Figure 1

${\gamma }^{*}N\to \mathrm{\Delta }$ quadrupole form factors estimated by the pion cloud parametrization of Eqs. (3)–(4). Data from Ref. [26].

Figure 2

Valence quark contributions for the ${\gamma }^{*}N\to \mathrm{\Delta }$ quadrupole form factors estimated by the covariant spectator quark model [16].

Figure 3

${\gamma }^{*}N\to \mathrm{\Delta }$ quadrupole form factors estimated as a combination of the valence quark and pion cloud contributions. Data from Ref. [26].

Figure 4

Results for the ${\gamma }^{*}N\to \mathrm{\Delta }$ quadrupole form factors in comparison with the MAID-SG2 parametrization from Ref. [19]. The bands indicate an estimate of the pion cloud error of 10%, equivalent to the order of $1/N_c^2$. Data from Ref. [26].